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**On the Recovery of Core and Crustal Components of Geomagnetic Potential Fields\
[L. Baratchart, C. Gerhards]{}\
**
In Geomagnetism it is of interest to separate the Earth’s core magnetic field from the crustal magnetic field. However, measurements by satellites can only sense the sum of the two contributions. In practice, the measured magnetic field is expanded in spherical harmonics and separation into crust and core contribution is achieved empirically, by a sharp cutoff in the spectral domain. In this paper, we derive a mathematical setup in which the two contributions are modeled by harmonic potentials $\Phi_0$ and $\Phi_1$ generated on two different spheres ${\mathbb{S}}_{R_0}$ (crust) and ${\mathbb{S}}_{R_1}$ (core) with radii $R_1<R_0$. Although it is not possible in general to recover $\Phi_0$ and $\Phi_1$ knowing their superposition $\Phi_0+\Phi_1$ on a sphere ${\mathbb{S}}_{R_2}$ with radius $R_2>R_0$, we show that it becomes possible if the magnetization $\mathbf{m}$ generating $\Phi_0$ is localized in a strict subregion of ${\mathbb{S}}_{R_0}$. Beyond unique recoverability, we show in this case how to numerically reconstruct characteristic features of $\Phi_0$ (e.g., spherical harmonic Fourier coefficients). An alternative way of phrasing the results is that knowledge of $\mathbf{m}$ on a nonempty open subset of ${\mathbb{S}}_{R_0}$ allows one to perform separation.
Harmonic Potentials, Hardy-Hodge Decomposition, Separation of Sources, Geomagnetic Field, Extremal Problems
33C55, 42B37, 45Q05, 53A45, 86A22
Introduction
============
The Earth’s magnetic field ${\mathbf{B}}$, as measured by several satellite missions, is a superposition of various contributions, e.g., of iono-/magnetospheric fields, crustal magnetic field, and of the core/main magnetic field, see [@kono09; @hulot10; @olsen15] for an overview and [@lesur10; @maus08; @sabaka15; @thebault15] for some recent geomagnetic field models. While iono-/magnetospheric contributions can to a certain extent be filtered out due to their temporal variations, the separation of the core/main field ${\mathbf{B}}_{core}$ and the crustal field ${\mathbf{B}}_{crust}$ is typically based on the empirical observation that the power spectra of Earth magnetic field models have a sharp knee at spherical harmonic degree 15 (see, e.g., [@langel82; @olsen15]). However, under this spectral separation, large-scale contributions (i.e., spherical harmonic degrees smaller than 15) are entirely neglected in crustal magnetic field models. In [@holschneider16], a Bayesian approach has been proposed that addresses the separation of geomagnetic sources based on their correlation structure. The correlation of certain components, e.g., internally and externally produced magnetic fields, can (to some extent) be obtained from the underlying geophysical equations. But this approach does not address the problem that some of the involved separation problems, e.g., the separation into crustal and core magnetic field contributions, are generally not unique for the given data situation. The goal of this paper is to derive conditions under which a rigorous separation of the contributions ${\mathbf{B}}_{crust}$ and ${\mathbf{B}}_{core}$ is possible, as well as to formulate extremal problems whose solutions lead to approximations of these contributions or certain features thereof. The main assumption that we make for our approach to work is that the magnetization generating $\mathbf{B}_{crust}$ is localized in a strict subregion of the crust. By linearity, this is equivalent to assuming that this magnetization is known on a spherical cap that may, in principle, be arbitrary small. For applications, this is interesting in as much as that the crustal magnetization may be estimated in certain places of the Earth from local measurements. Thus, given such a local estimation, its contribution can be substracted from global magnetic field measurements to yield a crustal contribution that stems from magnetizations localized in a strict subregion of the Earth (namely the complement of those places where a local estimate of the magnetization has been performed), thereby allowing us to apply the separation approach indicated in this paper. Similarly, if one can identify places on the Earth which are only weakly magnetized as compared to others, the separation process that we will describe may reasonably be applied by neglecting magnetizations in such places.
We assume throughout that the overall magnetic field is of the form ${\mathbf{B}}={\mathbf{B}}_{crust}+{\mathbf{B}}_{core}$ in ${\mathbb{R}}^3\setminus\overline{{\mathbb{B}}_{R_0}}$, where ${\mathbb{B}}_{R_0}=\{x\in{\mathbb{R}}^3:|x|<R_0\}$ denotes the ball of radius $R_0>0$ and overline indicates closure (here $R_0$ can be interpreted as the radius of the Earth). Since the sources of ${\mathbf{B}}_{crust}$ and ${\mathbf{B}}_{core}$ are located inside ${\mathbb{B}}_{R_0}$ (hence, the corresponding magnetic fields are curl-free and divergence-free in ${\mathbb{R}}^3\setminus\overline{{\mathbb{B}}_{R_0}}$), there exist potential fields $\Phi$, $\Phi_{crust}$, $\Phi_{core}$ such that ${\mathbf{B}}=\nabla \Phi$, ${\mathbf{B}}_{crust}=\nabla \Phi_{crust}$, and ${\mathbf{B}}_{core}=\nabla \Phi_{core}$ in ${\mathbb{R}}^3\setminus\overline{{\mathbb{B}}_{R_0}}$. Therefore, from a mathematical point of view, the problem reduces to finding unique $\Phi_{crust}$, $\Phi_{core}$ from the knowledge of $\Phi$ (but we should keep in mind that the actual measurements bear on the magnetic field ${\mathbf{B}}$).
It is known that ${\mathbf{B}}_{crust}$ is generated by a magnetization ${\mathbf{M}}$ confined in a thin spherical shell ${\mathbb{B}}_{R_0-d,R_0}=\{x\in{\mathbb{R}}^3:R_0-d<|x|<R_0\}$ of thickness $d>0$ (for the Earth, $d\approx 30$km is typical), therefore the corresponding magnetic potential can be expressed as (see, e.g., [@blakely95; @gubbins11]) $$\begin{aligned}
\Phi_{crust}(x)=\frac{1}{4\pi}\int_{{\mathbb{B}}_{R_0-d,R_0}}{\mathbf{M}}(y)\cdot\frac{x-y}{|x-y|^3}\,{{\mathrm{d}}}\lambda(y),\quad x\in{\mathbb{R}}^3,\label{eqn:phivol}\end{aligned}$$ where the dot indicates Euclidean scalar product in ${\mathbb R}^3$ and $\lambda$ the Lebesgue measure. Due to the thinness of the magnetized layer relative to the Earth’s radius, it is reasonable to substitute the volumetric ${\mathbf{M}}$ by a spherical magnetization ${\mathbf{m}}$ (i.e., $\mathbf{M}=\mathbf{m}\otimes\delta_{{\mathbb{S}}_{R_0}}$ in a distributional sense). Then, the magnetic potential becomes $$\begin{aligned}
\Phi_{crust}(x)=\frac{1}{4\pi}\int_{{\mathbb{S}}_{R_0}}{\mathbf{m}}(y)\cdot\frac{x-y}{|x-y|^3}\,{{\mathrm{d}}}\omega_{R_0}(y),\quad x\in{\mathbb{R}}^3\setminus{\mathbb{S}}_{R_0},\label{eqn:phisurf}\end{aligned}$$ where ${\mathbb{S}}_{R_0}=\{x\in{\mathbb{R}}^3:|x|=R_0\}$ denotes the sphere of radius $R_0>0$ and ${{\mathrm{d}}}\omega_{R_0}$ the corresponding surface element. When interested in reconstructing the actual magnetization ${\mathbf{M}}$, substituting a spherical magnetization $\mathbf{m}$ is of course a significant restriction (however, one that is fairly frequent in Geomagnetism). But since our main focus is on ${\mathbf{B}}_{crust}$ and the corresponding potential $\Phi_{crust}$ rather than the magnetization itself, this restriction actually involves no loss of information: in Section \[sec:harmpot\] we show that, under mild summability assumptions, any potential $\Phi_{crust}$ produced by a volumetric magnetization ${\mathbf{M}}$ in ${\mathbb{B}}_{R_0-d,R_0}$ can also be generated by a spherical magnetization ${\mathbf{m}}$ on ${\mathbb{S}}_{R_0}$.
The core/main contribution ${\mathbf{B}}_{core}$ is governed by the Maxwell equations (see, e.g., [@backus96]) $$\begin{aligned}
\label{Maxwell}
\nabla\times {\mathbf{B}}_{core}&=\sigma({\mathbf{E}}+\mathbf{u}\times {\mathbf{B}}_{core}),
\\\nabla\cdot {\mathbf{B}}_{core}&=0,
\\\nabla\times{\mathbf{E}}&=-\partial_t{\mathbf{B}}_{core},
\\\nabla\cdot{\mathbf{E}}&=\rho,\end{aligned}$$ where $\sigma$ denotes the conductivity, $\rho$ the charge density, and $\mathbf{u}$ the fluid velocity in the Earth’s outer core (the constant permeability $\mu_0$ and permittivity ${\varepsilon}_0$ have been set to $1$). The conductivity $\sigma$ is assumed to be zero outside a sphere ${\mathbb{S}}_{R_1}$ of radius $0<R_1<R_0$. The condition $R_1<R_0$ is crucial to the forthcoming arguments and is justified by common geophysical practice and results (see, e.g., [@ballani02; @puethe15]). In particular it implies that $\nabla\times {\mathbf{B}}_{core}=0$ in ${\mathbb{R}}^3\setminus\overline{{\mathbb{B}}_{R_1}}$, therefore, ${\mathbf{B}}_{core}=\nabla \Phi_{core}$ in ${\mathbb{R}}^3\setminus\overline{{\mathbb{B}}_{R_1}}$ for some harmonic potential $\Phi_{core}$. Although the geophysical processes in the Earth’s outer core can be extremely complex, of importance to us is only that $\Phi_{core}$ can be expressed in ${\mathbb{R}}^3\setminus\overline{{\mathbb{B}}_{R_1}}$ as a Poisson transform: $$\begin{aligned}
\label{eqn:phicore}
\Phi_{core}(x)=\frac{1}{4\pi R_1}\int_{{\mathbb{S}}_{R_1}} h(y)\frac{|x|^2-R_1^2}{|x-y|^3}{{\mathrm{d}}}\omega_{R_1}(y),\quad x\in{\mathbb{R}}^3\setminus\overline{{\mathbb{B}}_{R_1}},\end{aligned}$$ for some scalar valued auxiliary function $h$ on ${\mathbb{S}}_{R_1}$; this follows from previous considerations which imply that $ \Phi_{core}$ is harmonic in ${\mathbb{R}}^3\setminus\overline{{\mathbb{B}}_{R_1}}$ and continuous in ${\mathbb{R}}^3\setminus{\mathbb{B}}_{R_1}$. Summarizing, the problem we treat in this paper is the following (the setup is illustrated in Figure \[fig:setup\]):
\[prob:1\] Let $\Phi\in L^2({\mathbb{S}}_{R_2})$ be given on a sphere ${\mathbb{S}}_{R_2}\subset{\mathbb{R}}^3\setminus\overline{{\mathbb{B}}_{R_0}}$ of radius $R_2>R_0$. Assume $\Phi$ is decomposable into $\Phi=\Phi_0+\Phi_1$ on ${\mathbb{S}}_{R_2}$, where $\Phi_0=\Phi_0[{\mathbf{m}}]$ is of the form , with ${\mathbf{m}}\in L^2({\mathbb{S}}_{R_0},{\mathbb{R}}^3)$, and $\Phi_1=\Phi_1[h]$ is of the form , with $h\in L^2({\mathbb{S}}_{R_1})$ and $R_1<R_0$. Are $\Phi_0$ and $\Phi_1$ uniquely determined by the knowledge of $\Phi$ on ${\mathbb{S}}_{R_2}$, and if yes can they be reconstructed efficiently?
(0,0) circle (2); (0,0) circle (1.2); (0,0) circle (2); (0,0) circle (1.2); at (0.05,2.1) [[${\mathbf{m}}$]{}]{}; at (1.45,1.45) [${\mathbb{S}}_{R_0}$]{}; at (0.88,0.88) [${\mathbb{S}}_{R_1}$]{}; at (0.05,1.35) [$h$]{}; (0,0) circle (2.5); at (0.05,2.65) [[$\Phi=\Phi_0+\Phi_1$]{}]{}; at (1.81,1.81) [${\mathbb{S}}_{R_2}$]{};
The answer to the uniqueness issue in Problem \[prob:1\] is generally negative. But under the additional assumption that ${\textnormal{supp}}({\mathbf{m}})\subset\Gamma_{R_0}$ for a strict subregion $\Gamma_{R_0}\subset{\mathbb{S}}_{R_0}$ ( i.e. $\overline{\Gamma_{R_0}}\not={\mathbb{S}}_{R_0}$), uniqueness is guaranteed. This follows from results in [@baratchart13; @lima13] and their formulation on the sphere in [@gerhards16a], to be reviewed in greater detail in Section \[sec:seppot\]. In fact, we show in this case that $h$ and the curl-free contribution of ${\mathbf{m}}$ can be reconstructed uniquely from the knowledge of $\Phi$. Additionally, we provide a means of approximating $\langle\Phi_0,g\rangle_{L^2({\mathbb{S}}_{R_2})}$ knowing $\Phi$ on ${\mathbb{S}}_{R_2}$, where $g$ is some appropriate test function (e.g., a spherical harmonic). This allows one to separate the crustal and the core contributions to the Geomagnetic potential if, e.g., the crustal magnetization can be estimated over a small subregion on Earth by other means.
Throughout the paper, we call $\Phi_0$ the crustal contribution and $\Phi_1$ the core contribution. We should point out that the examples we provide at the end of the paper are not based on real Geomagnetic field data but they reflect some of the main properties of realistic scenarios (e.g., the domination of the core contribution at low spherical harmonic degrees). In Section \[sec:harmpot\], we take a closer look at harmonic potentials of the form and and show that the balayage onto ${\mathbb{S}}_{R_0}$ of a volumetric potential supported in ${\mathbb{B}}_{R_0-d,R_0}$ preserves divergence form. More precisely, if $\mathbf{M}$ is supported in ${\mathbb{B}}_{R_0-d,R_0}$ and its restriction to ${\mathbb{S}}_R$ is uniformly square-summable for $R\in (R_0-d,R_0)$, then there exists a spherical magnetization $\mathbf{m}$ supported on ${\mathbb{S}}_{R_0}$, which is square summable and generates the same potential as $\mathbf{M}$ in ${\mathbb{R}}^3\setminus\overline{{\mathbb{B}}_{R_0}}$. The latter property justifies the above-described modeling of the crustal magnetic field. Basic background and auxiliary material on geometry, spherical decomposition of vector fields as well as Sobolev and Hardy spaces is recapitulated in Section \[sec:aux\]. Some parts in the beginning are described in more detail than necessary for the core part of this paper and are only required again in Appendices \[sec:appendix2\] and \[sec:appendix1\]. So the reader familiar with the background and notation may directly proceed to Definition \[def:HH\]. Eventually, in Section \[sec:num\] we provide some initial examples of numerical approximation of $\Phi_0$ and $\langle\Phi_0,g\rangle_{L^2({\mathbb{S}}_{R_2})}$, followed by a brief conclusion in Section \[sec:conc\]. Some technical results on potentials of distributions, gradients, and divergence-free vector fields are gathered in the appendices.
Auxiliary Notations and Results {#sec:aux}
===============================
We start with some basic definitions of function spaces and differentiation on the sphere. For $R>0$, the sphere ${\mathbb{S}}_R$ is a smooth, compact oriented surface embedded in ${\mathbb{R}}^3$. That is, ${\mathbb{S}}_R$ can be described by finitely many charts $\psi_j:U_j\to V_j$ (for open subsets $U_j\subset{\mathbb{S}}_R$ and $V_j\subset {\mathbb{R}}^2$, $j=1,\ldots,N$), which allows a meaningful definition of the surface area measure $\omega_R$ on the sphere ${\mathbb{S}}_R$ via the Lebesgue measure $\lambda$ in ${\mathbb{R}}^2$. For $x\in U_j\subset{\mathbb{S}}_R$, the tangent space $T_x$ at $x$ is the image of the derivative ${\textrm{D}}\psi_j^{-1}[\psi_j(x)]:{\mathbb{R}}^2\to{\mathbb{R}}^3$. The tangent space may be described intrinsically as $T_x=\{y\in{\mathbb{R}}^3:x\cdot y=0\}$. A $k$-times differentiable or $C^{k}$-smooth function $f:{\mathbb{S}}_R\to{\mathbb{R}}$ is a function such that $f\circ\psi_j^{-1}$ is $k$-times differentiable or has continuous partial derivatives up to order $k$, respectively, for each $j=1,\ldots,N$. We simply say that $f$ is smooth if it is $C^{\infty}$-smooth. Due to the simple geometry of the sphere ${\mathbb{S}}_R$, this definition of differentiability is in fact equivalent to requiring that the radial extension $\bar{f}(x)=f(R\frac{x}{|x|})$ of $f$ has the corresponding regularity in ${\mathbb{R}}^3\setminus\{0\}$. This allows us to express the surface gradient $\nabla_{{\mathbb{S}}_R}f(x)$ of a differentiable function $f:{\mathbb{S}}_R\to{\mathbb{R}}$ at a point $x\in{\mathbb{S}}_R$ via the relation $\nabla_{{\mathbb{S}}_R}f(x)=\nabla\bar{f}(y)|_{y=x}$, where $\nabla$ denotes the Euclidean gradient. Formally, the surface gradient at $x$ is defined as the unique vector $v\in T_x$ such that the differential ${{\mathrm{d}}}f[x]:T_x\to {\mathbb{R}}$ can be identified by the scalar product with $v$, i.e., ${{\mathrm{d}}}f[x](y)=v\cdot y$ for $y\in T_x$. The differential of $f$ at $x\in U_j$ is the linear map ${{\mathrm{d}}}f(x):T_x\to{\mathbb R}$ given at $v\in T_x$ by ${{\mathrm{d}}}f[x](v)={{\mathrm{d}}}(f\circ \psi_j^{-1})[y](w)$, where $y=\psi_j(x)$ and $w\in{\mathbb R}^2$ is such that $v={{\mathrm{d}}}\psi_j^{-1}[y](w)$. Here, the Euclidean differential ${{\mathrm{d}}}(f\circ \phi_j)[y]$ is defined as usual: ${{\mathrm{d}}}(f\circ \psi_j^{-1})[y](w)=\partial_{y_1}(f\circ\psi_j^{-1})[y]w_1+
\partial_{y_1}(f\circ\psi_j^{-1})[y]w_2$, where $\partial_{y_i}$ indicates partial derivative with respect to $y_i$.
Furthermore, $L^2({\mathbb{S}}_{R})$ is denoted to be the space of square-integrable scalar valued functions $f:{\mathbb{S}}_{R}\to{\mathbb{R}}$, while $L^2({\mathbb{S}}_{R},{\mathbb{R}}^3)$ denotes the space of square integrable vector valued spherical functions $\mathbf{f}:{\mathbb{S}}_{R}\to{\mathbb{R}}^3$, equipped with the inner products $\langle f,h\rangle_{L^2({\mathbb{S}}_{R})}=\int_{{\mathbb{S}}_{R}}f(y)h(y){{\mathrm{d}}}\omega_{R}(y)$ and $\langle \mathbf{f},\mathbf{h}\rangle_{L^2({\mathbb{S}}_{R},{\mathbb{R}}^3)}=\int_{{\mathbb{S}}_{R}}\mathbf{f}(y)\cdot\mathbf{h}(y){{\mathrm{d}}}\omega_R(y)$, respectively. A vector field $\mathbf{f}:{\mathbb{S}}_R\to{\mathbb{R}}^3$ is said to be tangential if $\mathbf{f}(x)\in T_x$ for all $x\in{\mathbb{S}}_R$. The subspace of all tangential vector fields in $L^2({\mathbb{S}}_R,{\mathbb{R}}^3)$ is denoted by ${\mathcal{T}}_R$. Note that the smooth vector fields are dense in ${\mathcal{T}}_R$. Clearly, if $f$ is smooth, then $\nabla_{{\mathbb{S}}_R}f$ lies in ${\mathcal{T}}_R$. The Sobolev space $W^{1,2}({\mathbb{S}}_R)$ may be defined as the completion of smooth functions with respect to the norm [@Hebey] $$\begin{aligned}
\label{defSobinf}
\|f\|_{W^{1,2}({\mathbb{S}}_R)}=\Bigl(\|f\|^2_{L^2({\mathbb{S}}_R)}+
\|\nabla_{{\mathbb{S}}_R} f\|^2_{L^2({\mathbb{S}}_R,{\mathbb R}^3)}\Bigr)^{1/2}.\end{aligned}$$ Since, for an appropriate set of charts $\psi_j:U_j\to V_j$, $j=1,\ldots,N$, of the sphere, the $V_j$ are bounded and the corresponding determinants of the metric tensors are bounded from above and below by strictly positive constants, it holds that $f\in W^{1,2}({\mathbb{S}}_R)$ if and only if the functions $f\circ\psi_j^{-1}$ lie in the Euclidean Sobolev spaces $W^{1,2}(V_j)$ (see, e.g., [@lions68]). The gradient $\nabla_{{\mathbb{S}}_R} f(x)$ at $x\in{\mathbb{S}}_R$ of a function $f\in W^{1,2}({\mathbb{S}}_R)$ still satisfies the representation ${{\mathrm{d}}}f[x](y)=\nabla_{{\mathbb{S}}_R} f(x)\cdot y$ for $y\in T_x$, where ${{\mathrm{d}}}f$ has to be understood in the sense of distributional derivatives and $\nabla_{{\mathbb{S}}_R} f(x)$ needs not be a pointwise derivative in the strong sense (see [@stein70 Ch.VIII]). Let us put $$\begin{aligned}
{\mathcal{G}}_R=\{\nabla_{{\mathbb{S}}_R} f:f\in W^{1,2}({\mathbb{S}}_R)\}.\end{aligned}$$ We claim that ${\mathcal{G}}_R$ is closed in $L^2({\mathbb{S}}_R,{\mathbb{R}}^3)$. Indeed, if $\nabla_{{\mathbb{S}}_R} f_n$ is a Cauchy sequence in ${\mathcal{G}}_R$, where $f_n\in W^{1,2}({\mathbb{S}}_R)$ is defined up to an additive constant, we may pick $f_n$ so that $\int_{{\mathbb{S}}_{R}} f_n{{\mathrm{d}}}\omega_R=0$ and then it follows from the Hölder and the Poincaré inequalities [@Hebey Prop. 3.9] that $\|f_n-f_m\|_{L^2({\mathbb{S}}_R)}\leq C\|\nabla_{{\mathbb{S}}_R}f_n-\nabla_{{\mathbb{S}}_R}f_m\|_{L^2({\mathbb{S}}_R,{\mathbb{R}}^3)}$ for some constant $C$. Hence $f_n$ is a Cauchy sequence in $W^{1,2}({\mathbb{S}}_R)$, therefore it converges to some $f$ there and consequently $\nabla_{{\mathbb{S}}_R}f_n$ converges to $\nabla_{{\mathbb{S}}_R} f$ in $L^2({\mathbb{S}}_R,{\mathbb{R}}^3)$. Thus, ${\mathcal{G}}_R$ is complete and therefore it is closed in $L^2({\mathbb{S}}_R,{\mathbb{R}}^3)$, [which proves the claim]{}.
When $\mathbf{h}$ is a smooth tangential vector field on ${\mathbb{S}}_R$, its surface divergence $\nabla_{{\mathbb{S}}_R}\cdot\mathbf{h}$ is the smooth real valued function such that $$\label{divsmooth}
\int_{{\mathbb{S}}_R}f\,\nabla_{{\mathbb{S}}_R}\cdot\mathbf{h}\,{{\mathrm{d}}}\omega_R=-
\int_{{\mathbb{S}}_R}(\nabla_{{\mathbb{S}}_R} f)\,\cdot\,\mathbf{h}\,{{\mathrm{d}}}\omega_R,\quad\textnormal{ for all }
f \in C^\infty({\mathbb{S}}_R).$$ When $\mathbf{h}\in {\mathcal{T}}_R$ is not smooth, must be interpreted in a weak sense, namely $\nabla_{{\mathbb{S}}_R}\cdot\mathbf{h}$ is the distribution on ${\mathbb{S}}_R$ acting on smooth real-valued functions by $\langle f\,,\nabla_{{\mathbb{S}}_R}\cdot\mathbf{h}\rangle=-
\int_{{\mathbb{S}}_R}\nabla_{{\mathbb{S}}_R} f\,\cdot\, \mathbf{h}\,{{\mathrm{d}}}\omega_R$, for all $f \in C^\infty({\mathbb{S}}_R)$. This clearly extends by density to a linear form on $W^{1,2}({\mathbb{S}}_R)$, upon letting $f$ converge to a Sobolev function. Then it is apparent that $$\begin{aligned}
{\mathcal{D}}_R=\{\mathbf{h}\in{\mathcal{T}}_R:\nabla_{{\mathbb{S}}_R}\cdot\mathbf{h}=0\} \end{aligned}$$ is the orthogonal complement to ${\mathcal{G}}_R$ in ${\mathcal{T}}_R$. In particular, $$\begin{aligned}
{\mathcal{T}}_R={\mathcal{G}}_R\oplus {\mathcal{D}}_R,\label{eqn:hd}\end{aligned}$$ which is the so-called Helmholtz-Hodge decomposition. The particular geometry of ${\mathbb{S}}_R$ makes it easy to see that $\mathbf{f}\in{\mathcal{D}}_R$ if and only if its radial extension $\bar{\mathbf{f}}(x)=\mathbf{f}(R\frac{x}{|x|})$ is divergence free, as a ${\mathbb{R}}^3$-valued distribution on ${\mathbb{R}}^3\setminus\{0\}$.
We now consider the operator $J_x:T_x\to T_x$ given by $J_x(y)=\frac{x}{|x|}\times y$, for $y\in T_x$, where $\times$ indicates the vector product in ${\mathbb{R}}^3$; that is, $J_x$ is the rotation by $\pi/2$ in $T_x$. We define $J:{\mathcal{T}}_R\to{\mathcal{T}}_R$ to be the isometry acting pointwise as $J_x$ on $T_x$, namely $(J\mathbf{f})(x)=J_x(\mathbf{f}(x))$ for $\mathbf{f}\in{\mathcal{T}}_R$. It turns out that $J({\mathcal{G}}_R)={\mathcal{D}}_R$. This fact holds for more general sufficiently smooth surfaces embedded in ${\mathbb{R}}^3$. A proof seems not easy to find in the literature and is provided in Appendices \[sec:appendix2\] and \[sec:appendix1\] (for the special case of continuously differentiable tangential vector fields on the sphere, the assertion essentially corresponds to [@freedenschreiner09 Thm. 2.10]). This motivates the notion of a surface curl gradient ${\textnormal{L}}_{{\mathbb{S}}_R}=x\times\nabla_{{\mathbb{S}}_R}$, acting at a point $x\in{\mathbb{S}}_R$, and justifies the representation ${\mathcal{D}}_R=\{{\textnormal{L}}_{{\mathbb{S}}_R} f:f\in W^{1,2}({\mathbb{S}}_R)\}$. For convenience, we define the following ”normalized” operators: $\nabla_{\mathbb{S}}=R\nabla_{{\mathbb{S}}_R}$ and ${\textnormal{L}}_{\mathbb{S}}=\frac{x}{|x|}\times \nabla_{\mathbb{S}}$. The Euclidean gradient then has the expression $\nabla = \frac{x}{|x|}\partial_\nu+\frac{1}{|x|}\nabla_{\mathbb{S}}$, acting at a point $x\in{\mathbb{R}}^3$, where $\partial_\nu=\frac{x}{|x|}\cdot\nabla$ denotes the radial derivative.
Eventually, if we let ${\mathcal{N}}_R$ indicate the space of radial vector fields in $L^2({\mathbb{S}}_R,{\mathbb{R}}^3)$ (i.e., those functions whose value at $x$ is perpendicular to $T_x$ for each $x\in{\mathbb{S}}_R$), we get from the orthogonal decomposition $$\begin{aligned}
L^2({\mathbb{S}}_R,{\mathbb{R}}^3)={\mathcal{N}}_R\oplus{\mathcal{G}}_R\oplus {\mathcal{D}}_R.\label{eqn:shd}\end{aligned}$$ Related to the latter but of more relevance to our problem is the Hardy-Hodge decomposition that we now explain. For that purpose, we require the following definition.
\[def:HH\] The Hardy space ${\mathcal{H}_{+,R}^2}$ of harmonic gradients in ${\mathbb{B}}_R$ is defined by $$\begin{aligned}
{\mathcal{H}_{+,R}^2}=\left\{\mathbf{g}=\nabla g:\textnormal{ function } g:{\mathbb{B}}_R\to{\mathbb{R}}\textnormal{ with } \Delta g=0 \textnormal{ in }{\mathbb{B}}_{R}\textnormal{ and }\|\nabla{g}\|_{2,+}<\infty\right\},
\end{aligned}$$ where $\|\mathbf{g}\|_{2,+}=\big(\sup_{r\in[0,R)}\int_{{\mathbb{S}}_r}|\mathbf{g}(ry)|^2{{\mathrm{d}}}\omega_r(y)\big)^{\frac{1}{2}}$ and $\Delta$ is the Euclidean Laplacian in ${\mathbb{R}}^3$. Likewise, the Hardy space ${\mathcal{H}_{-,R}^2}$ of harmonic gradients in ${\mathbb{R}}^3\setminus\overline{{\mathbb{B}}_R}$ is defined by $$\begin{aligned}
{\mathcal{H}_{-,R}^2}=\left\{\mathbf{g}=\nabla g: \textnormal{ function } g:{\mathbb{R}}^3\setminus\overline{{\mathbb{B}}_R}\to{\mathbb{R}}\textnormal{ with } \Delta g=0 \textnormal{ in }{\mathbb{R}}^3\setminus\overline{{\mathbb{B}}_R}\textnormal{ and }\|\nabla{g}\|_{2,-}<\infty\right\},
\end{aligned}$$ where $\|\mathbf{g}\|_{2,-}=\big(\sup_{r\in(R,\infty)}\int_{{\mathbb{S}}_r}|\mathbf{g}(ry)|^2{{\mathrm{d}}}\omega_R(y)\big)^{\frac{1}{2}}$. Note that, by Weyl’s lemma [@Forster Theorem 24.9], it makes no difference whether the Euclidean gradient and Laplacian are understood in the distributional or in the strong sense.
Members of ${\mathcal{H}_{+,R}^2}$ and ${\mathcal{H}_{-,R}^2}$ have non-tangential limits a.e. on ${\mathbb{S}}_R$, and if $\mathbf{g}\in{\mathcal{H}}^2_{\pm,R}$, its nontangential limit has $L^2({\mathbb{S}}_R, {\mathbb{R}}^3)$-norm equal to $\|\mathbf{g}\|_{2,\pm}$, see [@stein70 VII.3.1] and [@steinweiss71 VI.4]. We still write $\mathbf{g}$ for this non-tangential limit and we regard it as the trace of $\mathbf{g}$ on ${\mathbb S}_R$. This way Hardy spaces can be interpreted as function spaces on ${\mathbb{S}}_R$ as well as on ${\mathbb{B}}_{R}$ or ${\mathbb R}^3\setminus\overline{{\mathbb{B}}_{R}}$, but the context will make it clear if the Euclidean or the spherical interpretation is meant because the argument belongs to ${\mathbb{R}}^3\setminus{\mathbb{S}}_R$ in the former case and to ${\mathbb{S}}_R$ in the latter. The Hardy-Hodge decomposition is the orthogonal sum $$\begin{aligned}
L^2({\mathbb{S}}_R,{\mathbb{R}}^3)={\mathcal{H}_{+,R}^2}\oplus {\mathcal{H}_{-,R}^2}\oplus {\mathcal{D}}_R.\label{eqn:hhd}\end{aligned}$$ Projecting onto the tangent space ${\mathcal{T}}_R$ and grouping the first two summands into a single gradient vector field yields back the Hodge decomposition . The Hardy-Hodge decomposition drops out at once from [@atfeh10] and . Its application to the study of inverse magnetization problems has been illustrated in [@baratchart13; @gerhards16a; @lima13]. Although not studied in mathematical detail, spherical versions of the Hardy-Hodge decomposition have previously been used to a various extent in Geomagnetic applications (see, e.g., [@backus96; @gerhards12; @gubbins11; @mayer06]).
By means of the reflection $\mathcal{R}_R(x)=\frac{R^2}{|x|^2}\,x$ across ${\mathbb{S}}_R$, we define the Kelvin transform ${K}_R[f]$ of a function $f$ defined on an open set $\Omega\subset{\mathbb R}^3$ to be the function on $\mathcal{R}_R(\Omega)$ given by $$\begin{aligned}
\label{defKelvin}
{K}_R[f](x)= \frac{R}{|x|} f(\mathcal{R}_R(x)), \quad x\in \mathcal{R}_R(\Omega).\end{aligned}$$ A function $f$ is harmonic in $\Omega$ if and only if ${K}_R[f]$ is harmonic in $\mathcal{R}_R(\Omega)$ (e.g., [@axler01 Thm. 4.7]).
Now, assume that $\mathbf{f}\in {\mathcal{H}_{+,R}^2}$ with $\mathbf{f}=\nabla f$ and $f(0)=0$. Then $\nabla{K}_R[f]\in {\mathcal{H}_{-,R}^2}$. In fact, if for $\mathbf{f}\in {\mathcal{H}_{+,R}^2}$ (resp. $\mathbf{f}\in {\mathcal{H}_{-,R}^2}$) we let $\int\mathbf{f}$ indicate the harmonic function $f$ in ${\mathbb{B}}_R$ (resp. in ${\mathbb R}^3\setminus\overline{{\mathbb{B}}}_R$) whose gradient is $\mathbf{f}$, normalized so that $f(0)=0$ (resp. $\lim_{|x|\to\infty}f(x)=0$), then $\mathbf{f}\mapsto \nabla{K}_R\circ\int\mathbf{f}$ maps $ {\mathcal{H}_{+,R}^2}$ continuously into $ {\mathcal{H}_{-,R}^2}$ and back [@atfeh10]. Moreover, in view of (\[defKelvin\]) we have that $$\begin{aligned}
\label{gradKel}
\nabla {K}_R[f](x) = \frac{R^3\nabla f(\mathcal{R}_R(x))}{|x|^3} - 2\, x \cdot \nabla f(\mathcal{R}_R(x)) \, \frac{R^3x}{|x|^{5}} - \, f(\mathcal{R}_R(x)) \, \frac{Rx}{|x|^3} \, .\end{aligned}$$ Clearly $f$ and ${K}_R[f]$ coincide on ${\mathbb{S}}_R$, therefore the tangential components of $\nabla f$ and $\nabla {K}_R[f]$ agree on ${\mathbb{S}}_R$ (these are the spherical gradients $\nabla_{{\mathbb{S}}_R}f$ and $\nabla_{{\mathbb{S}}_R}{K}_R[f]$). The normal components $\partial_\nu f$ and $\partial_\nu {K}_R[f]$, though, are different. Indeed, we get from that $$\begin{aligned}
\label{diffnK}
\partial_\nu {K}_R[f] (x) = -\partial_\nu f(x) - \frac{f(x)}{R},\quad x\in{\mathbb{S}}_R.\end{aligned}$$
We turn to some special systems of functions. First, let $\{Y_{n,k}\}_{n\in\mathbb{N}_0,\,k=1,\ldots,2n+1}$ be an $L^2({\mathbb{S}})$-orthonormal system of spherical harmonics of degrees $n$ and orders $k$. A possible choice is $$\begin{aligned}
Y_{n,k}(x)=\left\{\begin{array}{ll}
\sqrt{\frac{2n+1}{2\pi}\frac{(k-1)!}{(2n+1-k)!}}P_{n,n+1-k}(\sin(\theta))\cos((n+1-k)\varphi)&k=1,\ldots,n,
\\\sqrt{\frac{2n+1}{4\pi}}P_{n,0}(t),&k=n+1,
\\\sqrt{\frac{2n+1}{2\pi}\frac{(2n+1-k)!}{(k-1)!}}P_{n,k-(n+1)}(\sin(\theta))\sin((k-(n+1))\varphi)&k=n+2,\ldots,2n+1,
\end{array}\right.\end{aligned}$$for $x=(\cos(\theta)\cos(\varphi),\cos(\theta)\sin(\varphi),\sin(\theta))^T\in{\mathbb{S}}_1$, $\theta\in[-\frac{\pi}{2},\frac{\pi}{2}],\varphi\in[0,2\pi)$, and $P_{n,k}$ the associated Legendre polynomials of degree $n$ and order $k$ (see, e.g., [@freedenschreiner09 Ch. 3] for details; another common notation is to indicate the order of the spherical harmonics by $k=-n,\ldots,n$ rather than $k=1,\ldots,2n+1$). Then $H_{n,k}^R(x)=\big(\frac{|x|}{R}\big)^nY_{n,k}\big(\frac{x}{|x|}\big)$ is a homogeneous, harmonic polynomial of degree $n$ in ${\mathbb R}^3$ (sometimes also called inner harmonic and equipped with a normalization factor $\frac{1}{R}$). In fact, every homogeneous harmonic polynomial in ${\mathbb{R}}^3$ can be expressed as a linear combination of inner harmonics. The Kelvin transform $H_{-n-1,k}^R={K}_R[H_{n,k}^R]$ is a harmonic function in ${\mathbb R}^3\setminus\{0\}$ with $\lim_{|x|\to\infty}H_{-n-1,k}^R(x)=0$ (sometimes called outer harmonic). In [@atfeh10 Lemma 4] the following result was shown.
\[lem:yndense\] The vector space $\textnormal{span} \{\nabla H_{-n-1,k}^R\}_{n\in\mathbb{N}_0,\,k=1,\ldots,2n+1}$ is dense in ${\mathcal{H}_{-,R}^2}$ and the vector space $\textnormal{span} \{\nabla H_{n,k}^R\}_{n\in\mathbb{N}_0,\,k=1,\ldots,2n+1}$ is dense in ${\mathcal{H}_{+,R}^2}$.
For each fixed $x\in{\mathbb{R}}^3\setminus\overline{{\mathbb{B}}}_R$, the function $g_x(y)= \frac{1}{|x-y|}$ is harmonic in a neighborhood of $\overline{{\mathbb{B}}}_R$ and, therefore, its gradient $$\begin{aligned}
\mathbf{g}_x(y)=\nabla_x\,\, g_x(y)=-\frac{x-y}{|x-y|^3}
\end{aligned}$$ lies in ${\mathcal{H}_{+,R}^2}$. As a consequence of Lemma \[lem:yndense\], we shall prove the following density result.
\[dsh\] The vector space $\textnormal{span}\{\mathbf{g}_x: x\in {\mathbb R}^3\setminus\overline{{\mathbb{B}}}_R\}$ is dense in ${\mathcal{H}_{+,R}^2}$ and the vector space $\textnormal{span}\{\mathbf{g}_x: x\in {\mathbb{B}}_R\}$ is dense in ${\mathcal{H}_{-,R}^2}$.
As ${{K}_R}[g_x]= \frac{1}{|x|}g_{x/|x|^2}$ and $\nabla{{K}_R}\circ \int$ is an isomorphism from ${\mathcal{H}_{-,R}^2}$ onto ${\mathcal{H}_{+,R}^2}$ (see discussion before ), we need only prove the second assertion. Define $g(y)=\frac{1}{|y|}$ as a function of $y\in{\mathbb R}^3\setminus\{0\}$. For $\alpha=(\alpha_1,\alpha_2,\alpha_3)\in\mathbb{N}_0^3$ with $|\alpha|=\alpha_1+\alpha_2+\alpha_3=n$, the derivative $\partial_\alpha g(y)=\frac{\partial^{n}}{\partial^{\alpha_1}y_1 \partial^{\alpha_2}y_2\partial^{\alpha_3}y_3}g(y)$ is of the form $\frac{H_\alpha(y)}{|y|^{1+2n}}$, where $H_\alpha$ is a homogeneous harmonic polynomial of degree $n$, and actually every homogeneous harmonic polynomial $H_\alpha$ is a scalar multiple of $|y|^{(1+2n)}\partial_\alpha g(y)$ for some $\alpha$ [@axler01 Lemma 5.15]. The discussion before Lemma \[lem:yndense\] now implies that $\partial_\alpha g$ is an element of $\textnormal{span} \{H_{-n-1,k}^R\}_{n\in\mathbb{N}_0,\,k=1,\ldots,2n+1}$. Thus, by this lemma, we are done if we can show that whenever $\mathbf{f}\in {\mathcal{H}_{-,R}^2}$ is orthogonal in $L^2({\mathbb{S}}_R,{\mathbb R}^3)$ to all $\mathbf{g}_x$, $ x\in {\mathbb{B}}_R$, then it must be orthogonal to all $\nabla H_{-n-1,k}^R$. To this end, differentiating $\langle \mathbf{f},\mathbf{g}_x\rangle_{L^2({\mathbb{S}}_R,{\mathbb{R}}^3)}=0$ with respect to $x$ leads us to $$\begin{aligned}
\label{orthT}
0=\left\langle \mathbf{f}\,,\, \nabla \frac{H_\alpha(.-x)}{|\cdot-x|^{1+2n}}\right\rangle_{L^2({\mathbb{S}}_R,{\mathbb{R}}^3)}\end{aligned}$$ for all $\alpha\in\mathbb{N}_0^3$ and $n=|\alpha|$. Setting $x=0$ yields $$\begin{aligned}
0=\left\langle \mathbf{f}\,,\, \nabla \frac{H_\alpha}{|\cdot|^{1+2n}} \right\rangle_{L^2({\mathbb{S}}_R,{\mathbb{R}}^3)}=R^{-2n-1}\left\langle \mathbf{f}\,,\, \nabla {{K}_R}[H_\alpha] \right\rangle_{L^2({\mathbb{S}}_R,{\mathbb{R}}^3)}.\end{aligned}$$ Since every inner harmonic $H_{n,k}^R$ can be expressed as a linear combination of $H_\alpha$, this relation and the considerations before Lemma \[lem:yndense\] imply $\langle \mathbf{f}\,,\, \nabla H_{-n-1,k}^R \rangle_{L^2({\mathbb{S}}_R,{\mathbb{R}}^3)}=0$ for all $n\in\mathbb{N}_0$, $k=1,\ldots, 2n+1$, which is the desired conclusion.
Harmonic Potentials in Divergence-Form {#sec:harmpot}
======================================
The potential of a measure $\mu$ on ${\mathbb R}^3$ is defined by $$\begin{aligned}
p_\mu(x)=-\frac{1}{4\pi}\int_{{\mathbb{R}}^3}\frac{1}{|x-y|}{{\mathrm{d}}}\mu(y)\label{eqn:defpot}.\end{aligned}$$ It is the solution of $\Delta \Phi=\mu$ in ${\mathbb R}^3$ which is “smallest” at infinity. If $\mu\geq0$, the potential $p_\mu$ is a superharmonic function and therefore it is either finite quasi-everywhere or identically $-\infty$, see [@armitage01] for these properties and the definition of “quasi everywhere”. Decomposing a signed measure into its positive and negative parts (the Hahn decomposition) yields that $p_\mu$ is finite quasi-everywhere if $\mu$ is finite and compactly supported (i.e., if $\textnormal{supp}(\mu)$, which is closed by definition, is also bounded). If $\textnormal{supp}(\mu)\subset\overline{{\mathbb{B}}}_R$, the Riesz representation theorem and the maximum principle for harmonic functions imply that there exists a unique measure $\hat\mu$ with $\textnormal{supp}(\hat\mu)\subset{\mathbb{S}}_R$ such that $$\begin{aligned}
\int g(y) {{\mathrm{d}}}\mu(y)=\int g(y) {{\mathrm{d}}}\hat\mu(y)
\end{aligned}$$ for every continuous function $g$ in $\overline{{\mathbb{B}}}_R$ which is harmonic in ${\mathbb{B}}_R$. Since $y\mapsto 1/|x-y|$ is harmonic in a neighbourhood of $\overline{{\mathbb{B}}}_R$ when $x\notin \overline{{\mathbb{B}}}_R$, this entails that the potentials $p_\mu$ and $p_{\hat\mu}$ coincide in ${\mathbb{R}}^3\setminus\overline{{\mathbb{B}}}_R$, i.e., $$\begin{aligned}
p_\mu(x)=p_{\hat\mu}(x),\quad x\in{\mathbb{R}}^3\setminus\overline{{\mathbb{B}}}_R.\end{aligned}$$ The measure $\hat\mu$ is called the balayage of $\mu$ onto ${\mathbb S}_R$ (see, e.g., [@armitage01]). In fact, the potentials $p_\mu$ and $p_{\hat{\mu}}$ coincide quasi-everywhere on ${\mathbb S}_R$ as well. An expression for $\hat\mu$ easily follows from the Poisson representation of a function $f$ which is continuous in $\overline{{\mathbb{B}}}_R$ and harmonic in ${\mathbb{B}}_R$: $$\begin{aligned}
\label{Poisson}
f(x)=\frac{1}{4\pi R}\int_{{\mathbb{S}}_R}\frac{R^2-|x|^2}{|x-y|^3}f(y)\,{{\mathrm{d}}}\omega_R(y),\quad x\in{\mathbb{B}}_R.\end{aligned}$$ Clearly Equation , Fubini’s theorem and the definition of balayage imply that $$\begin{aligned}
\label{calcbal}
d\hat\mu(x)={{\mathrm{d}}}\mu_{|{\mathbb S}_R}(x)+\left(\frac{1}{4\pi R}\int_{{\mathbb{B}}_R}\frac{R^2-|y|^2}{|x-y|^3}{{\mathrm{d}}}\mu(y)\right){{\mathrm{d}}}\omega_R(x).\end{aligned}$$
\[lem:balayage\] Let the measure $\mu$ be supported in $\overline{{\mathbb{B}}_R}$. Furthermore, assume that $\mu$ is absolutely continuous in ${\mathbb{B}}_R$ with a density $h$ (i.e., ${{\mathrm{d}}}\mu(y)=h(y)dy$) that satisfies the Hardy condition $$\begin{aligned}
\operatorname*{\textnormal{ess.}\,\sup}_{0 \le r < R} \int_{{\mathbb S}_r} |h(y)|^2 \, {{\mathrm{d}}}\omega_r(y) < \infty.\label{eqn:esssup}\end{aligned}$$ Then the balayage $\hat\mu$ of $\mu$ on ${\mathbb{S}}_R$ is absolutely continuous with respect to $\omega_R$ (i.e., $d\hat\mu(y)=\hat h(y){{\mathrm{d}}}\omega_R(y)$) and it has a density $\hat h\in L^2({\mathbb S}_R)$.
Starting from and the assumption that $\mu$ is absolutely continuous, we find that the density $\hat h$ of $\hat \mu$ is $$\begin{aligned}
\hat h(x)=\frac{1}{4\pi R}\int_{{\mathbb{B}}_R}\frac{R^2-|y|^2}{|x-y|^3}h(y){{\mathrm{d}}}\lambda(y), \quad x\in {\mathbb{S}}_R.\end{aligned}$$ Using Fubini’s theorem and the identity $$\begin{aligned}
\left|\frac{x}{|x|}-|x|y\right|=\left|\frac{y}{|y|}-|y|x\right|, \qquad x,y\in{\mathbb R}^3\setminus\{0\},\end{aligned}$$ together with the changes of variable $\eta=\frac{\xi}{r}$, $y=\frac{rx}{R^2}$, we are led to $$\begin{aligned}
\|\hat h\|^2_{L^2({\mathbb S}_R)}=&\frac{1}{(4\pi R)^2}\int_{{\mathbb S}_R}\left(\int_{{\mathbb{B}}_R}\frac{R^2-|y|^2}{|x-y|^3}h(y)\,{{\mathrm{d}}}\lambda(y)\right)^2{{\mathrm{d}}}\omega_R(x)\nonumber
\\
=&\frac{1}{(4\pi R)^2}\int_{{\mathbb S}_R}\left(\int_0^R\left(\int_{{\mathbb S}_r}\frac{R^2-|\xi|^2}{|x-\xi|^3}h(\xi)\,{{\mathrm{d}}}\omega_r(\xi)\right)dr\right)^2{{\mathrm{d}}}\omega_R(x)\nonumber
\\
\leq&\frac{R}{(4\pi R)^2}\int_{{\mathbb S}_R}\left(\int_0^R\left(\int_{{\mathbb S}_r}\frac{R^2-|\xi|^2}{|x-\xi|^3}h(\xi)\,{{\mathrm{d}}}\omega_r(\xi)\right)^2dr\right){{\mathrm{d}}}\omega_R(x)\nonumber
\\
=&\frac{1}{(4\pi R)^2}\int_{{\mathbb S}_R}\left(\int_0^R\left(\int_{{\mathbb S}_r}\frac{1-(\frac{r}{R})^2}{|\frac{x}{R}-\frac{\xi}{R}|^3}h(\xi)\,{{\mathrm{d}}}\omega_r(\xi)\right)^2dr\right){{\mathrm{d}}}\omega_R(x)\nonumber
\\
=&\frac{1}{(4\pi R)^2}\int_{{\mathbb S}_R}\left(\int_0^R\left(\int_{{\mathbb S}_r}\frac{1-\left|\frac{rx}{R^2}\right|^2}{|\frac{\xi}{r}-\frac{rx}{R^2}|^3}h(\xi)\,{{\mathrm{d}}}\omega_r(\xi)\right)^2dr\right){{\mathrm{d}}}\omega_R(x)\nonumber
\\
=&\frac{1}{(4\pi R)^2}\int_0^Rr^4\left(\int_{{\mathbb S}_R}\left(\int_{{\mathbb S}_1}\frac{1-\left|\frac{rx}{R^2}\right|^2}{|\eta-\frac{rx}{R^2}|^3}h(r\eta)\,{{\mathrm{d}}}\omega_1(\eta)\right)^2 {{\mathrm{d}}}\omega_R(x)\right)dr\nonumber
\\
=&\int_0^Rr^4\left(\frac{1}{4\pi (\frac{r}{R})^2}\int_{{\mathbb S}_{\frac{r}{R}}}\left(\frac{1}{4\pi}\int_{{\mathbb S}_1}\frac{1-|y|^2}{|\eta-y|^3}h(r\eta)\,{{\mathrm{d}}}\omega_1(\eta)\right)^2 {{\mathrm{d}}}\omega_{\frac{r}{R}}(y)\right)dr.\label{eqn:hatgcomp}\end{aligned}$$ Now, the function $$\begin{aligned}
f(y)=\frac{1}{4\pi}\int_{{\mathbb S}_1}\frac{1-|y|^2}{|\eta-y|^3}h(r\eta)\,{{\mathrm{d}}}\omega_1(\eta)\end{aligned}$$ is the Poisson integral of $h(r\cdot)$ over the unit sphere ${\mathbb{S}}_1$ (and represents the middle integral on the right hand side of ). Thus, $f$ is harmonic in ${\mathbb{B}}_1$ and its square $|f|^2$ is subharmonic there. The latter implies that the mean of $|f|^2$ over the sphere ${\mathbb S}_{\frac{r}{R}}$, $r<R$, is not greater than its mean over ${\mathbb S}_1$, i.e., $$\begin{aligned}
\frac{1}{4\pi (\frac{r}{R})^2}\int_{{\mathbb S}_{\frac{r}{R}}}|f(y)|^2 {{\mathrm{d}}}\omega_{\frac{r}{R}}(y)&\leq \lim_{\frac{s}{R}\to 1-}\frac{1}{4\pi (\frac{s}{R})^2}\int_{{\mathbb S}_{\frac{s}{R}}}|f(y)|^2 {{\mathrm{d}}}\omega_{\frac{s}{R}}(y)= \frac{1}{4\pi}\int_{{\mathbb S}_1}|h(r\eta)|^2\,{{\mathrm{d}}}\omega_1(\eta)
\\&=\frac{1}{4\pi r^2}\int_{{\mathbb S}_r}|h(y)|^2\,{{\mathrm{d}}}\omega_r(y)\leq \frac{M}{4\pi r^2},\end{aligned}$$ where the constant $M>0$ comes from the Hardy condition . Together with , we find that $$\begin{aligned}
\|\hat h\|^2_{L^2({\mathbb S}_R)}\leq \frac{MR^3}{12\pi},\end{aligned}$$ eventually showing that $\hat{h}\in L^2({\mathbb S}_R)$ and that $\hat\mu$ is absolutely continuous with respect to $\omega_R$ with density $\hat h$.
More generally, an arbitrary distribution $D$ with compact support has a potential $p_D$ given outside of $\textnormal{supp}(D)$ by $$\begin{aligned}
\label{eqn:p_Ddef}
p_D(x)=D\left(-\frac{1}{4\pi}\frac{1}{|x-\cdot|}\right), \quad x\in{\mathbb{R}}^3\setminus \textnormal{supp} D.\end{aligned}$$ Compactness of $\textnormal{supp} (D)$ easily implies that $D$ indeed acts on $-1/(4\pi|x-\cdot|)$ when $x\notin \textnormal{supp} D$ so that $P_D$ is well-defined, see Appendix \[sec:appendix\] for details. If $D$ is supported in $\overline{{\mathbb{B}}}_R$ (in particular, if it is supported in some shell $\overline{{\mathbb{B}}_{R-d,R}}$), we define the balayage of $D$ onto ${\mathbb S}_R$ to be the distribution $\hat D$ on ${\mathbb{S}}_R$ that satisfies $$\begin{aligned}
p_{\hat D}(x)=p_D(x),\quad x\in{\mathbb{R}}^3\setminus\overline{{\mathbb{B}}}_R.\end{aligned}$$ Strictly speaking, $\hat{D}$ is a distribution on ${\mathbb{S}}_R$, so that $p_{\hat D}$ should rather be denoted by $p_{\hat D\otimes \delta_{{\mathbb{S}}_R}}$, where $\hat D\otimes\delta_{{\mathbb{S}}_R}$ is the distribution on ${\mathbb R}^3$ which is the tensor product of $\hat D$ with the measure $\delta_{{\mathbb{S}}_R}$, corresponding in spherical coordinates to a Dirac mass at $r=R$, see [@Schwartz]. Nevertheless, to alleviate notation, we do write $p_{\hat D}$. Thus, what is meant in when $D=\hat D$ is that $\hat D$ is applied to the restriction to ${\mathbb{S}}_R$ of $ -1/(4\pi|x-\cdot|)$.
We briefly comment on the existence and uniqueness of such a balayage in Appendix \[sec:appendix\]. If $D$ is (associated with) a measure $\mu$, then coincides with and the balayage was given in . The main difference between the case of a finite compactly supported measure $\mu$ and the case of a general compactly supported distribution $D$ is that usually $p_D(x)$ cannot be assigned a meaning when $x\in \textnormal{supp}(D)$ whereas $p_\mu$ is well-defined quasi everywhere on $\textnormal{supp}(\mu)$. We say that $D$ is in divergence form if $$\begin{aligned}
\label{eqn:divform}
D=\nabla\cdot \mathbf{M},\end{aligned}$$ where $\nabla\cdot$ is to be understood as the distributional divergence and $\mathbf{M}$ is a ${\mathbb R}^3$-valued distribution. If, e.g., $\mathbf{M}\in L^2({\mathbb{B}}_{R-d,R},{\mathbb{R}}^3)$ and $\textnormal{supp}(\mathbf{M})\subset\overline{{\mathbb{B}}_{R-d,R}}$, then the corresponding potential $p_D$ coincides with $\Phi_{crust}$ in . Now we can formulate the main result of this section, namely, that balayage preserves divergence form for those $\mathbf{M}$ satisfying a Hardy condition.
\[balayageH\] Let $D=\nabla\cdot \mathbf{M}$, where $\mathbf{M}\in L^2({\mathbb{B}}_R,{\mathbb R}^3)$ satisfies the Hardy condition $$\begin{aligned}
\operatorname*{\textnormal{ess.}\,\sup}_{0 \leq r < R} \int_{{\mathbb S}_r} |\mathbf{M}(y)|^2 \, {{\mathrm{d}}}\omega_r(y) < \infty.\end{aligned}$$ Then there exists ${\mathbf{m}}\in L^2({\mathbb S}_R,{\mathbb R}^3)$ such that $\hat D=\nabla\cdot ({\mathbf{m}}\otimes \delta_{{\mathbb S}_R})$ is the balayage of $D$ onto ${\mathbb S}_R$.
Let $\mathbf{M}=(M_1,M_2,M_3)^T$ denote the components of $\mathbf{M}$. The definition of $p_D$ yields $$\begin{aligned}
p_D(x)&=\frac{1}{4\pi}\int_{{\mathbb{B}}_{R}}{\mathbf{M}}(y)\cdot\frac{x-y}{|x-y|^3}\,{{\mathrm{d}}}\lambda(y)\nonumber
\\&=\frac{1}{4\pi}\sum_{j=1}^3\int_{{\mathbb{B}}_R} M_j(y)\frac{x_j-y_j}{|x-y|^3} \, {{\mathrm{d}}}\lambda(y),\quad x\in{\mathbb{R}}^3\setminus\overline{{\mathbb{B}}}_R.\label{eqn:pD}\end{aligned}$$ If we choose the measure $\mu_j$ such that ${{\mathrm{d}}}\mu_j(y)=M_j(y)dy$, we get from Lemma \[lem:balayage\] and the Hardy condition on $\mathbf{M}$ that there exists a $m_j\in L^2({\mathbb S}_R)$ such that balayage of $\mu_j$ onto ${\mathbb{S}}_R$ is given by the measure $\hat \mu_j$ with $d\hat\mu_j=m_j {{\mathrm{d}}}\omega_R$, $j=1,2,3$. Setting ${\mathbf{m}}=(m_1,m_2,m_3)^T$ and observing that ${g}_{x,j}(y)=\frac{x_j-y_j}{|x-y|^3}=-\partial_{x_j}\frac{1}{|x-y|}$ is harmonic in ${\mathbb{B}}_R$ and continuous in $\overline{{\mathbb{B}}_R}$, for fixed $x\in {\mathbb{R}}^3\setminus\overline{{\mathbb{B}}}_R$, then the definition of balayage yields together with that $$\begin{aligned}
p_D(x)&=\frac{1}{4\pi}\sum_{j=1}^3\int_{{\mathbb S}_R}m_j(t)\frac{x_j-y_j}{|x-y|^3}\, {{\mathrm{d}}}\omega_R(y)\nonumber
\\&=\frac{1}{4\pi}\int_{{\mathbb{S}}_{R}}{\mathbf{m}}(y)\cdot\frac{x-y}{|x-y|^3}\,{{\mathrm{d}}}\omega_R(y)=p_{\hat D}(x),\quad x\in{\mathbb{R}}^3\setminus\overline{{\mathbb{B}}}_R.\label{eqn:spherepot}\end{aligned}$$ The latter implies that $\hat D=\nabla\cdot ({\mathbf{m}}\otimes \delta_{{\mathbb S}_R})$, as announced.
Lemma \[balayageH\] eventually justifies the statement made in the introduction that, to every square summable volumetric magnetization $\mathbf{M}$ in the Earth’s crust ${\mathbb{B}}_{R-d,R}$ that satisfies the Hardy condition, there exists a spherical magnetization $\mathbf{m}$ on ${\mathbb{S}}_{R}$ that produces the same magnetic potential and therefore also the same magnetic field in the exterior of the Earth.
Separation of Potentials {#sec:seppot}
========================
We are now in a position to approach Problem \[prob:1\]. For this we study the nullspace of the potential operator $\Phi^{R_1,R_0,R_2}$ (cf. Definition \[def:ops\]), mapping a magnetization $\mathbf{m}$ on ${\mathbb{S}}_{R_0}$ and an auxiliary function $h\in L^2({\mathbb{S}}_{R_1})$ to the sum of the potentials and on ${\mathbb{S}}_{R_2}$. First, we show in Section \[sec:unique\] that uniqueness holds in Problem \[prob:1\] if $\textnormal{supp}\,\mathbf{m}\not= {\mathbb{S}}_{R_0}$. Similar results hold for the magnetic field operator $\mathbf{B}^{R_1,R_0,R_2}=\nabla\Phi^{R_1,R_0,R_2}$ (cf. Theorem \[thm:unique3\]), and also for a modified potential field operator $\Psi^{R_1,R_0,R_2}$ (cf. Definition \[def:ops2\] and Theorem \[thm:unique2\]). The operator $\Psi^{R_1,R_0,R_2}$ reflects the potential of two magnetizations $\mathbf{m}$ and $\mathbf{\mathfrak{m}}$ supported on two different spheres (i.e., at different depths), therefore it does not apply to the separation of the crustal and core contributions since the latter does not arise from a magnetization (cf.). Still it is of interest on its own, moreover we get it at no extra cost. In Section \[sec:extremal\], we discuss how the previous results can be used to approximate quantities like the Fourier coefficients $\langle \Phi_0,Y_{n,k}\rangle_{L^2({\mathbb{S}}_{R_2})}$ of $\Phi_0$. Finally, in Section \[sec:gammas\], we show that $\Phi=\Phi_0+\Phi_1$ may well vanish though $\Phi_0,\Phi_1\neq0$. This follows from Lemma \[NG\] and answers the uniqueness issue of Problem \[prob:1\] in the negative when $\textnormal{supp}\,\mathbf{m}= {\mathbb{S}}_{R_0}$.
Uniqueness Issues {#sec:unique}
-----------------
In accordance with the notation from Problem \[prob:1\], we define two operators: one mapping a spherical magnetization $\mathbf{m}$ to the potential $p_{\hat D}$ with $\hat D=\nabla\cdot (\mathbf{m}\otimes \delta_{{\mathbb{S}}_{R_0}})$, and the other mapping an auxiliary function $h\in L^2({\mathbb{S}}_{R_1})$ to its Poisson integral, both evaluated on ${\mathbb{S}}_{R_2}$.
\[def:ops\] Let $0<R_1<R_0<R_2$ be fixed radii and $\Gamma_{R_0}$ a closed subset of ${\mathbb{S}}_{R_0}$. Let $$\begin{aligned}
\Phi_0^{R_0,R_2}:L^2(\Gamma_{R_0},{\mathbb{R}}^3)\to L^2({\mathbb{S}}_{R_2}),\quad \mathbf{m}\mapsto\frac{1}{4\pi}\int_{\Gamma_{R_0}}{\mathbf{m}}(y)\cdot\frac{x-y}{|x-y|^3}\,{{\mathrm{d}}}\omega_{R_0}(y),\quad x\in{\mathbb{S}}_{R_2},
\end{aligned}$$ and $$\begin{aligned}
\Phi_1^{R_1,R_2}:L^2({\mathbb{S}}_{R_1})\to L^2({\mathbb{S}}_{R_2}),\quad h\mapsto\frac{1}{4\pi R_1}\int_{{\mathbb{S}}_{R_1}} h(y)\frac{|x|^2-R_1^2}{|x-y|^3}{{\mathrm{d}}}\omega_{R_1}(y),\quad x\in{\mathbb{S}}_{R_2}.
\end{aligned}$$ The superposition of the two operators above is denoted by $$\begin{aligned}
\Phi^{R_1,R_0,R_2}:L^2(\Gamma_{R_0},{\mathbb{R}}^3)\times L^2({\mathbb{S}}_{R_1})\to L^2({\mathbb{S}}_{R_2}),\quad(\mathbf{m},h)\mapsto \Phi_0^{R_0,R_2}[\mathbf{m}]+ \Phi_1^{R_1,R_2}[h].
\end{aligned}$$
We start by characterizing the potentials $p_{\hat D}$, with $\hat D$ in divergence-form, which are zero in ${\mathbb{R}}^3\setminus\overline{{\mathbb{B}}_R}$.
\[lem:nullpd\] Let $\mathbf{m}\in L^2({\mathbb{S}}_R,{\mathbb{R}}^3)$ and ${\hat D}=\nabla\cdot(\mathbf{m}\otimes\delta_{{\mathbb{S}}_R})$ be in divergence-form. Let further $\mathbf{m}=\mathbf{m}_{+}+\mathbf{m}_{-}+\mathbf{d}$ be the Hardy-Hodge decomposition of $\mathbf{m}$, i.e., $\mathbf{m}_{+}\in {\mathcal{H}_{+,R}^2}$, $\mathbf{m}_{-}\in {\mathcal{H}_{-,R}^2}$, and $\mathbf{d}\in {\mathcal{D}}_R$. Then $p_{\hat D}(x)=0$, for all $x\in {\mathbb{R}}^3\setminus\overline{{\mathbb{B}}_R}$, if and only if $\mathbf{m}_+\equiv0$. Analogously, $p_{\hat D}(x)=0$, for all $x\in {\mathbb{B}}_R$, if and only if $\mathbf{m}_-\equiv0$.
We already know that $\mathbf{g}_x(y)=\frac{x-y}{|x-y|^3}$ lies in ${\mathcal{H}_{+,R}^2}$ for every fixed $x\in{\mathbb{R}}^3\setminus\overline{{\mathbb{B}}_R}$. The orthogonality of the Hardy-Hodge decomposition and the representation of $p_{\hat D}$ yield that $\mathbf{m}_-$ and $\mathbf{d}$ do not change $p_{\hat D}$ in ${\mathbb{R}}^3\setminus\overline{{\mathbb{B}}_R}$. Conversely, if $p_{\hat D}(x)=0$ for all $x\in {\mathbb{R}}^3\setminus\overline{{\mathbb{B}}_R}$, then $$\begin{aligned}
p_{\hat D}(x)=\langle \mathbf{g}_x,\mathbf{m}\rangle_{L^2({\mathbb{S}}_R,{\mathbb{R}}^3)}=\langle \mathbf{g}_x,\mathbf{m}_+\rangle_{L^2({\mathbb{S}}_R,{\mathbb{R}}^3)}=0,\quad x\in{\mathbb{R}}^3\setminus\overline{{\mathbb{B}}_R}.
\end{aligned}$$ Since Lemma \[dsh\] asserts that $\textnormal{span}\{\mathbf{g}_x: x\in {\mathbb R}^3\setminus\overline{{\mathbb{B}}}_R\}$ is dense in ${\mathcal{H}_{+,R}^2}$, the above relation implies $\mathbf{m}_+\equiv0$. The assertion for the case where $p_{\hat D}(x)=0$, for all $x\in {\mathbb{B}}_R$ likewise follows by observing that $\mathbf{g}_x(y)=\frac{x-y}{|x-y|^3}$ lies in ${\mathcal{H}_{-,R}^2}$ for fixed $x\in{\mathbb{B}}_R$.
Since $\Phi_0^{R_0,R_2}[\mathbf{m}]=p_{\hat D}$, we may use Lemma \[lem:nullpd\] to characterize the nullspace of $\Phi_0^{R_0,R_2}$ (extending the magnetization $\mathbf{m}\in L^2(\Gamma_{R_0},{\mathbb{R}}^3)$ by zero on ${\mathbb{S}}_{R_0}\setminus\Gamma_{R_0}$ if the latter is nonempty). As to $\Phi^{R_1,R_2}_1$, we know its nullspace reduces to zero because the Poisson integral yields the unique harmonic extension of $h\in L^2({\mathbb{S}}_{R_1})$ to ${\mathbb{R}}^3\setminus\overline{{\mathbb{B}}_{R_1}}$ which is zero at infinity ( i.e. $h$ is the nontangential limit of its Poisson extension a.e. on ${\mathbb{S}}_{R_1}$, see [@axler01 Thm. 6.13]). This motivates the following statement on the nullspace $N(\Phi^{R_1,R_0,R_2})$ of $\Phi^{R_1,R_0,R_2}$.
\[thm:unique\] Let the setup be as in Definition \[def:ops\] and assume that $\Gamma_{R_0}\not={\mathbb{S}}_{R_0}$. Then the nullspace of $\Phi^{R_1,R_0,R_2}$ is given by $$\begin{aligned}
N(\Phi^{R_1,R_0,R_2})=\{(\mathbf{d},0):\mathbf{d}\in \mathcal{D}_{R_0},\
\textnormal{supp}(\mathbf{d})\subset\Gamma_{R_0}\}.
\end{aligned}$$
Clearly $\Phi^{R_1,R_0,R_2}[(\mathbf{m},h)]$ is harmonic in ${\mathbb{R}}^3\setminus\{\Gamma_{R_0}\cup \,{\mathbb{S}}_{R_1}\}$ and vanishes at infiity. If $\Phi^{R_1,R_0,R_2}[(\mathbf{m},h)](x)=0$ for $x\in{\mathbb{S}}_{R_2}$, then it follows from the maximum principle that $\Phi^{R_1,R_0,R_2}[(\mathbf{m},h)](x)=0$ for all $x\in {\mathbb{R}}^3\setminus{\mathbb{B}}_{R_2}$. Subsequently, by real analyticity, $\Phi^{R_1,R_0,R_2}[(\mathbf{m},h)]$ must vanish identically in ${\mathbb{R}}^3\setminus\{\Gamma_{R_0}\cup \overline{{\mathbb{B}}_{R_1}}\}$ which is connected because $\Gamma_{R_0}\not={\mathbb{S}}_{R_0}$. Thus, $\Phi^{R_1,R_0,R_2}[(\mathbf{m},h)]$ extends harmonically (by the zero function) across $\Gamma_{R_0}$: $$\begin{aligned}
\label{eqn:zeropot}
\Phi^{R_1,R_0,R_2}[(\mathbf{m},h)](x)=0,\quad x\in {\mathbb{R}}^3\setminus\overline{{\mathbb{B}}_{R_1}}. \end{aligned}$$ Since $\Phi^{R_1,R_0,R_2}[(\mathbf{m},h)]=
\Phi_0^{R_0,R_2}[\mathbf{m}]+ \Phi_1^{R_1,R_2}[h]$, where $\Phi_1^{R_1,R_2}[h]$ is harmonic on ${\mathbb{R}}^3\setminus\overline{{\mathbb{B}}_{R_1}}$, we find that $\Phi_0^{R_0,R_2}[\mathbf{m}]$ in turn extends harmonically across $\Gamma_{R_0}$, therefore it is harmonic in all of ${\mathbb{R}}^3$. Additionally $\Phi_0^{R_0,R_2}[\mathbf{m}]$ vanishes at infinity, hence $\Phi_0^{R_0,R_2}[\mathbf{m}](x)=0$ for all $x\in{\mathbb{R}}^3$ by Liouville’s theorem. Since $\Phi_0^{R_0,R_2}[\mathbf{m}]=p_{\hat D}$ for $\hat D=\nabla\cdot (\mathbf{m}\otimes \delta_{{\mathbb{S}}_{R_0}})$, Lemma \[lem:nullpd\] now implies that $\mathbf{m}=\mathbf{d}\in{\mathcal{D}}_{R_0}$ with $\textnormal{supp}\,\mathbf{d}\subset\Gamma_{R_0}$. Next, as $\Phi_0^{R_0,R_2}[\mathbf{m}]$ vanishes identically on ${\mathbb{R}}^3$, we get from that $\Phi_1^{R_1,R_2}[h](x)=0$ for all $x\in {\mathbb{R}}^3\setminus \overline{{\mathbb{B}}_{R_1}}$. Then, injectivity of the Poisson transform entails that $h\equiv0$, hence $N(\Phi^{R_1,R_0,R_2})\subset\{(\mathbf{m},0):\mathbf{m}\in \mathcal{D}_{R_0}\,\textrm{supp}({\mathbf{m}})\subset\Gamma_{R_0}\}$.
The reverse inclusion $N(\Phi^{R_1,R_0,R_2})\supset\{(\mathbf{m},0):\mathbf{m}\in \mathcal{D}_{R_0},\,\textrm{supp}({\mathbf{m}})\subset\Gamma_{R_0}\}$ is clear because Lemma \[lem:nullpd\] yields that $\Phi^{R_1,R_0,R_2}[(\mathbf{m},0)](x)=
\Phi_0^{R_0,R_2}[\mathbf{m}](x)=0$, for all $x\in{\mathbb{R}}^3\setminus\Gamma_{R_0}$ if $\mathbf{m}\in \mathcal{D}_{R_0}$.
\[cor:unique\] Notation being as in Definition \[def:ops\] with $\Gamma_{R_0}\not={\mathbb{S}}_{R_0}$, let $\Phi=\Phi^{R_1,R_0,R_2}[(\mathbf{m},h)]$ for some $\mathbf{m}\in L^2(\Gamma_{R_0},{\mathbb{R}}^3)$ and some $h\in L^2({\mathbb{S}}_{R_1})$. Then, a pair of potentials of the form $\bar{\Phi}_0=\Phi_0^{R_0,R_2}[\bar{\mathbf{m}}]$ and $\bar{\Phi}_1=\Phi_1^{R_1,R_2}[\bar{h}]$, with $\bar{\mathbf{m}}\in L^2(\Gamma_{R_0},{\mathbb{R}}^3)$ and $\bar{h}\in L^2({\mathbb{S}}_{R_1})$, is uniquely determined by the condition $\Phi(x)=\bar{\Phi}_0(x)+\bar{\Phi}_1(x)$, $x\in{\mathbb{S}}_{R_2}$.
From Theorem \[thm:unique\] we get that $h$ is uniquely determined by the values of $\Phi$ on ${\mathbb{S}}_{R_2}$, and also that the components $\mathbf{m}_+\in{\mathcal{H}_{+,R_0}^2}$ and $\mathbf{m}_-\in{\mathcal{H}_{-,R_0}^2}$ of the Hardy-Hodge decomposition of $\mathbf{m}$ are uniquely determined. The former implies $\bar{h}\equiv h$ and the latter $\bar{\mathbf{m}}\equiv\mathbf{m}+\bar{\mathbf{d}}$, for some $\bar{\mathbf{d}}\in{\mathcal{D}}_{R_0}$. By Lemma \[lem:nullpd\] we have that $\Phi_0^{R_0,R_2}[\mathbf{m}](x)=\Phi_0^{R_0,R_2}[\mathbf{m}+\bar{\mathbf{d}}](x)$ for $x\in{\mathbb{R}}^3\setminus{\mathbb{S}}_{R_0}$, so we eventually find that $\bar{\Phi}_0$ and $\bar{\Phi}_1$ are uniquely determined.
Corollary \[cor:unique\] answers the uniqueness issue of Problem \[prob:1\] in the positive provided that $\textnormal{supp}(\mathbf{m})\not={\mathbb{S}}_{R_0}$. In other words, assuming a locally supported magnetization, it is possible to separate the contribution of the Earth’s crust from the contribution of the Earth’s core if only the superposition of both magnetic potentials is known on some external orbit ${\mathbb{S}}_{R_2}$. Of course, in Geomagnetism, it is the magnetic field $\mathbf{B}=\nabla \Phi$ which is measured rather than the magnetic potential $\Phi$. However, the result carries over at once to this setting. More in fact is true: if $\textnormal{supp}(\mathbf{m})\not={\mathbb{S}}_{R_0}$, separation is possible if only the normal component of $\mathbf{B}$ is known on ${\mathbb{S}}_{R_2}$. Indeed, we have the following theorem.
\[thm:unique3\] Let the setup be as in Definition \[def:ops\] with $\Gamma_{R_0}\not={\mathbb{S}}_{R_0}$, and consider the operator $$\begin{aligned}
\mathbf{B}^{R_1,R_0,R_2}:L^2(\Gamma_{R_0},{\mathbb{R}}^3)\times L^2({\mathbb{S}}_{R_1},{\mathbb{R}}^3)\to L^2({\mathbb{S}}_{R_2},{\mathbb{R}}^3),\quad (\mathbf{m},h)\mapsto \nabla\Phi_0^{R_0,R_2}[\mathbf{m}]+ \nabla\Phi_1^{R_1,R_2}[h].
\end{aligned}$$ Define further the normal operator: $$\begin{aligned}
&\mathbf{B}_\nu^{R_1,R_0,R_2}:L^2(\Gamma_{R_0},{\mathbb{R}}^3)\times L^2({\mathbb{S}}_{R_1},{\mathbb{R}}^3)\to L^2({\mathbb{S}}_{R_2}),\quad (\mathbf{m},h)\mapsto \partial_{\nu}\left(\Phi_0^{R_0,R_2}[\mathbf{m}]+ \Phi_1^{R_1,R_2}[h]\right).
\end{aligned}$$ Then the nullspaces of $\mathbf{B}^{R_1,R_0,R_2}$ and $\mathbf{B}_\nu^{R_1,R_0,R_2}$ are all given by $$\begin{aligned}
N(\mathbf{B}^{R_1,R_0,R_2})= N(\mathbf{B}_\nu^{R_1,R_0,R_2})
=\{(\mathbf{d},0):\mathbf{d}\in \mathcal{D}_{R_0},\textnormal{ supp}(\mathbf{d})\subset\Gamma_{R_0}\}.
\end{aligned}$$
Let $\mathbf{B}_\nu^{R_1,R_0,R_2}[(\mathbf{m},h)](x)=0$ for $x\in{\mathbb{S}}_{R_2}$. Then $\Phi^{R_1,R_0,R_2}[(\mathbf{m},h)]$ has vanishing normal derivative on ${\mathbb{S}}_{R_2}$, and is otherwise harmonic in ${\mathbb{R}}^3\setminus\overline{{\mathbb{B}}_{R_2}}$. Note that $\Phi^{R_1,R_0,R_2}[(\mathbf{m},h)]$ is even harmonic across ${\mathbb{S}}_{R_2}$ onto a slightly larger open set, hence there is no issue of smoothness to define derivatives everywhere on ${\mathbb{S}}_{R_2}$. Since $\Phi^{R_1,R_0,R_2}[(\mathbf{m},h)]$ vanishes at infinity, its Kelvin transform $u=K_{R_2}[\Phi^{R_1,R_0,R_2}[(\mathbf{m},h)]]$ is harmonic in ${\mathbb{B}}_{R_2}$ with $u(0)=0$ [@axler01 Thm. 4.8], and by it holds that $\partial_\nu u(x)+ u(x)/R_2=0$ for $x\in{\mathbb{S}}_{R_2}$. Now, if $u$ is nonconstant and $x$ is a maximum place for $u$ on ${\mathbb{S}}_{R_2}$, then $\partial_\nu u(x)>0$ by the Hopf lemma [@axler01 Ch. 1, Ex. 25]. Hence $ u(x)<0$, implying that $u<0$ on ${\mathbb{B}}_{R_2}$, which contradicts the maximum principle because $u(0)=0$. Therefore $u$ vanishes identically and so does $\Phi^{R_1,R_0,R_2}[(\mathbf{m},h)]$ on ${\mathbb{S}}_{R_2}$. Appealing to Theorem \[thm:unique\] now achieves the proof.
The next corollary follows in the exact same manner as Corollary \[cor:unique\]. To state it, we indicate with a subscript $\nu$ the normal component of a field in $L^2({\mathbb{S}}_{R_2},{\mathbb{R}}^3)$ while a subscript $\tau$ denotes the tangential component.
\[cor:unique3\] Let the setup be as in Definition \[def:ops\] with $\Gamma_{R_0}\not={\mathbb{S}}_{R_0}$, and let the operator $\mathbf{B}^{R_1,R_0,R_2}$, be as in Theorem \[thm:unique3\]. Define further the operators $$\begin{aligned}
\mathbf{B}_0^{R_0,R_2}:L^2(\Gamma_{R_0},{\mathbb{R}}^3)\to L^2({\mathbb{S}}_{R_2},{\mathbb{R}}^3),\quad \mathbf{m}\mapsto\nabla\Phi_0^{R_0,R_2}[\mathbf{m}],
\end{aligned}$$ and $$\begin{aligned}
\mathbf{B}_1^{R_1,R_2}:L^2({\mathbb{S}}_{R_1})\to L^2({\mathbb{S}}_{R_2},{\mathbb{R}}^3),\quad h\mapsto\nabla\Phi_1^{R_1,R_2}[h].
\end{aligned}$$ Let further $\mathbf{B}=\mathbf{B}^{R_1,R_0,R_2}[(\mathbf{m},h)]$, with $\mathbf{m}\in L^2(\Gamma_{R_0},{\mathbb{R}}^3)$ and $h\in L^2({\mathbb{S}}_{R_1})$. A pair of fields of the form $\bar{\mathbf{B}}_0=\mathbf{B}_0^{R_0,R_2}[\bar{\mathbf{m}}]$ and $\bar{\mathbf{B}}_1=\mathbf{B}_1^{R_1,R_2}[\bar{h}]$, with $\bar{\mathbf{m}}\in L^2(\Gamma_{R_0},{\mathbb{R}}^3)$ and $\bar{h}\in L^2({\mathbb{S}}_{R_1})$, is uniquely determined by the condition $\mathbf{B}_\nu(x)=(\bar{\mathbf{B}}_0)_\nu(x)+(\bar{\mathbf{B}}_1)_\nu(x)$ and thus, a fortiori, by the condition $\mathbf{B}(x)=\bar{\mathbf{B}}_0(x)+\bar{\mathbf{B}}_1(x)$ for $x\in{\mathbb{S}}_{R_2}$.
Opposed to the normal component, it does not suffice to know the tangential component ${\mathbf{B}}_\tau$ on ${\mathbb{S}}_{R_2}$ in order to obtain uniqueness of ${\mathbf{B}}_0$ and ${\mathbf{B}}_1$. Namely, letting ${\mathbf{m}}\equiv0$ and $h$ be any nonzero constant function on ${\mathbb{S}}_{R_1}$, then ${\mathbf{B}}_\tau(x)=({\mathbf{B}}_0)_\tau(x)+({\mathbf{B}}_1)_\tau(x)=\nabla_{{\mathbb{S}}_{R_2}}\Phi_0^{R_0,R_2}[{\mathbf{m}}](x)+\nabla_{{\mathbb{S}}_{R_2}}\Phi_1^{R_1,R_2}[h](x)=0$ and ${\mathbf{B}}_0(x)=\nabla \Phi_0^{R_0,R_2}[{\mathbf{m}}](x)=0$ but ${\mathbf{B}}_1(x)=\nabla\Phi_1^{R_1,R_2}[h](x)=-\frac{hR_1}{|x|^3}x\not=0$ for $x\in{\mathbb{S}}_{R_2}$.
Analogously to the previous considerations, one can separate two potentials produced by two magnetizations located on two distinct spheres of radii $R_1<R_0$ (of which the outer magnetization again has to be supported on a strict subset of ${\mathbb{S}}_{R_0}$). We need only slightly change the setup of Definition \[def:ops\]:
\[def:ops2\] Let $0<R_1<R_0<R_2$ be fixed radii and $\Gamma_{R_0}\subset{\mathbb{S}}_{R_0}$ a closed subset. We define $$\begin{aligned}
\Psi_0^{R_0,R_2}:L^2(\Gamma_{R_0},{\mathbb{R}}^3)\to L^2({\mathbb{S}}_{R_2}),\quad \mathbf{m}\mapsto\frac{1}{4\pi}\int_{\Gamma_{R_0}}{\mathbf{m}}(y)\cdot\frac{x-y}{|x-y|^3}\,{{\mathrm{d}}}\omega_{R_0}(y),\quad x\in{\mathbb{S}}_{R_2},
\end{aligned}$$ and $$\begin{aligned}
\Psi_1^{R_1,R_2}:L^2({\mathbb{S}}_{R_1},{\mathbb{R}}^3)\to L^2({\mathbb{S}}_{R_2}),\quad \mathbf{m}\mapsto\frac{1}{4\pi}\int_{{\mathbb{S}}_{R_1}}{\mathbf{m}}(y)\cdot\frac{x-y}{|x-y|^3}\,{{\mathrm{d}}}\omega_{R_0}(y),\quad x\in{\mathbb{S}}_{R_2}.
\end{aligned}$$ The superposition of these two operators is denoted by $$\begin{aligned}
\Psi^{R_1,R_0,R_2}:L^2(\Gamma_{R_0},{\mathbb{R}}^3)\times L^2({\mathbb{S}}_{R_1},{\mathbb{R}}^3)\to L^2({\mathbb{S}}_{R_2}),\quad(\mathbf{m},\mathfrak{m})\mapsto \Psi_0^{R_0,R_2}[\mathbf{m}]+ \Psi_1^{R_1,R_2}[\mathfrak{m}].
\end{aligned}$$
\[thm:unique2\] Let the setup be as in Definition \[def:ops2\] and assume that $\Gamma_{R_0}\not={\mathbb{S}}_{R_0}$. Then the nullspace of $\Psi^{R_1,R_0,R_2}$ is given by $$\begin{aligned}
\label{decnomag}
N(\Psi^{R_1,R_0,R_2})=\{(\mathbf{d},\mathfrak{m}_-+\mathfrak{d}):\mathfrak{m}_-\in{\mathcal{H}_{-,R_1}^2},\,\mathbf{d}\in \mathcal{D}_{R_0}, \textnormal{ supp}(\mathbf{d})\subset
\Gamma_{R_0},\,\mathfrak{d}\in \mathcal{D}_{R_1}\}.
\end{aligned}$$
Let $\Psi^{R_1,R_0,R_2}[(\mathbf{m},\mathfrak{m})](x)=0$ for all $x\in{\mathbb{S}}_{R_2}$. The same argument as in the proof of Theorem \[thm:unique\] then leads us to $\Psi_0^{R_0,R_2}[\mathbf{m}](x)=0$, $x\in{\mathbb{R}}^3$, and $\Psi_1^{R_1,R_2}[\mathfrak{m}](x)=0$, $x\in {\mathbb{R}}^3\setminus \overline{{\mathbb{B}}_{R_1}}$. The former yields $\mathbf{m}=\mathbf{d}\in{\mathcal{D}}_{R_0}$, like in Theorem \[thm:unique\]. As to the latter, we observe that $\Psi_1^{R_1,R_2}[\mathfrak{m}]=p_{\hat D}$ with $\hat D=\nabla\cdot (\mathfrak{m}\otimes \delta_{{\mathbb{S}}_{R_1}})$, so Lemma \[lem:nullpd\] yields that $\mathfrak{m}=\mathfrak{m}_-+\mathfrak{d}$, where $\mathfrak{m}_-\in{\mathcal{H}_{-,R_1}^2}$ and $\mathfrak{d}\in{\mathcal{D}}_{R_1}$. Thus, the left hand side of is included in the right hand side. The reverse inclusion is a direct consequence of Lemma \[lem:nullpd\].
\[cor:unique2\] Let the setup be as in Definition \[def:ops2\] and assume that $\Gamma_{R_0}\not={\mathbb{S}}_{R_0}$. Let further $\Psi=\Psi^{R_1,R_0,R_2}[(\mathbf{m},\mathfrak{m})]$, with $\mathbf{m}\in L^2(\Gamma_{R_0},{\mathbb{R}}^3)$ and $\mathfrak{m}\in L^2({\mathbb{S}}_{R_1},{\mathbb{R}}^3)$. A pair of potentials of the form $\bar{\Psi}_0=\Psi_0^{R_0,R_2}[\bar{\mathbf{m}}]$ and $\bar{\Psi}_1=\Psi_1^{R_1,R_2}[\bar{\mathfrak{m}}]$, with $\bar{\mathbf{m}}\in L^2(\Gamma_{R_0},{\mathbb{R}}^3)$ and $\bar{\mathfrak{m}}\in L^2({\mathbb{S}}_{R_1},{\mathbb{R}}^3)$, is uniquely determined by the condition $\Psi(x)=\bar{\Psi}_0(x)+\bar{\Psi}_1(x)$, $x\in{\mathbb{S}}_{R_2}$.
From Theorem \[thm:unique\] we get that the components $\mathbf{m}_+\in{\mathcal{H}_{+,R_0}^2}$ and $\mathbf{m}_-\in{\mathcal{H}_{-,R_0}^2}$ of the Hardy-Hodge decomposition of $\mathbf{m}$ are uniquely determined by the knowledge of $\Psi$ on ${\mathbb{S}}_{R_2}$, while for $\mathfrak{m}$ only the component $\mathbf{\mathfrak{m}}_+\in{\mathcal{H}_{+,R_1}^2}$ is uniquely determined. The former implies $\bar{\mathbf{m}}\equiv\mathbf{m}+\bar{\mathbf{d}}$, for some $\bar{\mathbf{d}}\in{\mathcal{D}}_{R_0}$, and the latter yields $\bar{\mathfrak{m}}\equiv\mathfrak{m}+\bar{\mathfrak{m}}_-+\bar{\mathfrak{d}}$, for some $\bar{\mathfrak{m}}_-\in{\mathcal{H}_{-,R_1}^2}$ and $\bar{\mathfrak{d}}\in{\mathcal{D}}_{R_1}$. Since Lemma \[lem:nullpd\] yields that $\Psi_0^{R_0,R_2}[\mathbf{m}](x)=\Psi_0^{R_0,R_2}[\mathbf{m}+\mathbf{d}](x)$ and $\Psi_1^{R_1,R_2}[\mathfrak{m}](x)=\Psi_1^{R_1,R_2}[\mathfrak{m}+\bar{\mathfrak{m}}_-+\bar{\mathfrak{d}}](x)$ for $x\in{\mathbb{S}}_{R_2}$, we eventually find that $\bar{\Psi}_0$ and $\bar{\Psi}_1$ are uniquely determined.
Analogs of Theorems \[thm:unique\], \[thm:unique2\] and Corollaries \[cor:unique2\], \[cor:unique2\] are easily seen to hold for the case of finitely many magnetizations $\mathbf{m}_{1},\ldots,\mathbf{m}_n$, and $\mathbf{\mathfrak{m}}$ supported respectively on spheres ${\mathbb{S}}_{R_{0,1}},\ldots,{\mathbb{S}}_{R_{0,n}}$ and ${\mathbb{S}}_{R_1}$ of radii $R_1<R_{0,1}<\ldots<R_{0,n}<R_2$, under the localization assumptions that $\textnormal{supp}(\mathbf{m}_{i})$ is a strict subset of ${\mathbb{S}}_{R_{0,i}}$, $i=1,\ldots,n$. The corresponding separation properties may be of interest when investigating the depth profile of crustal magnetizations.
Reconstruction Issues {#sec:extremal}
---------------------
In this section, we discuss how quantities such as the Fourier coefficients $\langle \Phi_0,Y_{n,k}\rangle_{L^2({\mathbb{S}}_{R_2})}$ of $\Phi_0$ can be approximated knowing $\Phi$, without having to reconstruct $\Phi_0$ itself. Such Fourier coefficients are of interest, e.g., when looking at the power spectra of $\Phi$ and $\Phi_0$ (cf. the empirical way of separating the crustal and the core magnetic fields mentioned in the introduction). As an extra piece of notation, given $\Gamma_R\subset{\mathbb{S}}_R$ and $f:{\mathbb{S}}_R\to {\mathbb{R}}^k$, we let $f_{|\Gamma_R}:\Gamma_R\to{\mathbb{R}}^k$ designate the restriction of $f$ to $\Gamma_R$.
\[thm:errestfunc\] Let the setup be as in Definition \[def:ops\] and assume that $\Gamma_{R_0}\not={\mathbb{S}}_{R_0}$. Then, for every ${\varepsilon}>0$ and every function $\mathbf{g}\in{\mathcal{H}_{+,R_0}^2}\oplus{\mathcal{H}_{-,R_0}^2}$, there exists $f\in L^2({\mathbb{S}}_{R_2})$ (depending on ${\varepsilon}$ and $\mathbf{g}$) such that $$\begin{aligned}
\left|\langle \Phi^{R_1,R_0,R_2}[\mathbf{m},h],f\rangle_{L^2({\mathbb{S}}_{R_2})}-\langle \mathbf{m},\mathbf{g}_{|\Gamma_{R_0}}\rangle_{L^2(\Gamma_{R_0}, {\mathbb{R}}^3)}\right|\leq {\varepsilon}\|(\mathbf{m},h)\|_{L^2(\Gamma_{R_0},{\mathbb{R}}^3)\times L^2({\mathbb{S}}_{R_1})},
\end{aligned}$$ for all $\mathbf{m}\in L^2(\Gamma_{R_0},{\mathbb{R}}^3)$ and $h\in L^2({\mathbb{S}}_{R_1})$.
According to Theorem \[thm:unique\] and the orthogonality of the Hardy-Hodge decomposition, $(\mathbf{g}_{|\Gamma_{R_0}},0)$ is orthogonal to the nullspace $N(\Phi^{R_1,R_0,R_2})$ of $\Phi^{R_1,R_0,R_2}$, for if $\textnormal{supp}\,\mathbf{d}\subset\Gamma_{R_0}$ then $\langle \mathbf{g}_{|\Gamma_{R_0}},\mathbf{d}\rangle_{L^2(\Gamma_{R_0},{\mathbb{R}}^3)}=
\langle \mathbf{g},\mathbf{d}\rangle_{L^2({\mathbb{S}}_{R_0},{\mathbb{R}}^3)}=0$. Therefore, $(\mathbf{g}_{|\Gamma_{R_0}},0)$ lies in the closure of the range of the adjoint operator $\big(\Phi^{R_1,R_0,R_2}\big)^*$, i.e., to each ${\varepsilon}>0$ there is $f\in L^2({\mathbb{S}}_{R_2})$ with $$\begin{aligned}
\label{eqn:adjest}
\left\|\left(\Phi^{R_1,R_0,R_2}\right)^*[f]-(\mathbf{g}_{|\Gamma_{R_0}},0)\right\|_{L^2(\Gamma_{R_0},{\mathbb{R}}^3)\times L^2({\mathbb{S}}_{R_1})}\leq {\varepsilon}.
\end{aligned}$$ Taking the scalar product with $(\mathbf{m},h)$, we get from and the Cauchy-Schwarz inequality: $$\begin{aligned}
&\left|\langle \Phi^{R_1,R_0,R_2}[\mathbf{m},h],f\rangle_{L^2({\mathbb{S}}_{R_2})}-\langle \mathbf{m},\mathbf{g}_{|\Gamma_{R_0}}\rangle_{L^2(\Gamma_{R_0}, {\mathbb{R}}^3)}\right|
\\&=\left|\left\langle(\mathbf{m},h),\left(\Phi^{R_1,R_0,R_2}\right)^*[f]-(\mathbf{g}_{|\Gamma_{R_0}},0)\right\rangle_{L^2(\Gamma_{R_0}, {\mathbb{R}}^3)\times L^2({\mathbb{S}}_{R_2})}\right|
\\&\leq\left\|\left(\Phi^{R_1,R_0,R_2}\right)^*[f]-(\mathbf{g}_{|\Gamma_{R_0}},0)\right\|_{L^2(\Gamma_{R_0},{\mathbb{R}}^3)\times L^2({\mathbb{S}}_{R_1})}\|(\mathbf{m},h)\|_{L^2(\Gamma_{R_0},{\mathbb{R}}^3)\times L^2({\mathbb{S}}_{R_1})}
\\&\leq {\varepsilon}\|(\mathbf{m},h)\|_{L^2(\Gamma_{R_0},{\mathbb{R}}^3)\times L^2({\mathbb{S}}_{R_1})},
\end{aligned}$$ which is the desired result.
\[cor:approxcoeffs\] Let the setup be as in Definition \[def:ops\] with $\Gamma_{R_0}\not={\mathbb{S}}_{R_0}$. Then, for every ${\varepsilon}>0$ and every function $g\in L^2({\mathbb{S}}_{R_2})$, there exists $f\in L^2({\mathbb{S}}_{R_2})$ (depending on ${\varepsilon}$ and $g$) such that $$\begin{aligned}
\left|\langle \Phi^{R_1,R_0,R_2}[\mathbf{m},h],f\rangle_{L^2({\mathbb{S}}_{R_2})}-\langle \Phi_0^{R_0,R_2}[\mathbf{m}],g\rangle_{L^2({\mathbb{S}}_{R_2})}\right|\leq {\varepsilon}\|(\mathbf{m},h)\|_{L^2(\Gamma_{R_0},{\mathbb{R}}^3)\times L^2({\mathbb{S}}_{R_1})},
\end{aligned}$$ for all $\mathbf{m}\in L^2(\Gamma_{R_0},{\mathbb{R}}^3)$ and $h\in L^2({\mathbb{S}}_{R_1})$.
First observe that $$\begin{aligned}
\label{eqn:adjrel}
\left\langle \Phi_0^{R_0,R_2}[\mathbf{m}],g\right\rangle_{L^2({\mathbb{S}}_{R_2})}=\left\langle\mathbf{m},\left( \Phi_0^{R_0,R_2}\right)^*[g]\right\rangle_{L^2(\Gamma_{R_0},{\mathbb{R}}^3)},\end{aligned}$$ where the adjoint operator of $\Phi_0^{R_0,R_2}$ is given by $$\begin{aligned}
\label{adp}
\left( \Phi_0^{R_0,R_2}\right)^*:L^2({\mathbb{S}}_{R_2})\to L^2(\Gamma_{R_0},{\mathbb{R}}^3),\quad g\mapsto \mathbf{H}[g]_{|\Gamma_{R_0}},\nonumber\\
\mathbf{H}[g](x)=-\frac{1}{4\pi}\int_{{\mathbb{S}}_{R_2}}g(y)\frac{x-y}{|x-y|^3}{{\mathrm{d}}}\omega_{R_2}(y),\quad x\in
{\mathbb{S}}_{R_0}.\end{aligned}$$ Clearly $\mathbf{H}[g]\in {\mathcal{H}}^2_{+,R_0}$ whenever $g\in L^2({\mathbb{S}}_{R_2})$, therefore, together with Theorem \[thm:errestfunc\] yield the desired result.
\[rem:empsep\] The interest of Corollary \[cor:approxcoeffs\] from the Geophysical viewpoint lies with the fact that $\Phi^{R_1,R_0,R_2}[\mathbf{m},h]$ (more specifically: its gradient) corresponds to the measurements on ${\mathbb{S}}_{R_2}$ of the superposition of the core and crustal contributions, whereas $\Phi_0^{R_0,R_2}[\mathbf{m}]$ corresponds to the crustal contribution alone. Thus, if we can compute $f$ knowing $g$, we shall in principle be able to get information on the crustal contribution up to arbitrary small error. Note also that $(\mathbf{g},0)\not\in\textnormal{Ran}\,\bigl(\Phi^{R_1,R_0,R_2}\big)^*$ unless $\mathbf{g}\equiv0$, due to the injectivity of the adjoint of the Poisson transform (which is again a Poisson transform). Therefore we can only hope for an approximation of $\langle \Phi_0^{R_0,R_2}[\mathbf{m}],g\rangle_{L^2({\mathbb{S}}_{R_2})}$ in Corollary \[cor:approxcoeffs\], up to a relative error of ${\varepsilon}>0$, but not for an exact reconstruction.
Results analogous to Theorem \[thm:errestfunc\] and Corollary \[cor:approxcoeffs\] mechanically hold in the setup of Theorem \[thm:unique3\] and Corollary \[cor:unique3\] (i.e., separation of the crustal and core magnetic fields $\mathbf{B}_0$ and $\mathbf{B}_1$ instead of the potentials) and in the setup of Theorem \[thm:unique2\] and Corollary \[cor:unique2\] (i.e., separation of the potentials $\Psi_0$ and $\Psi_1$ due to magnetizations on ${\mathbb{S}}_{R_0}$ and ${\mathbb{S}}_{R_1}$). Below we state the corresponding results but we omit the proofs for they are similar to the previous ones.
\[thm:errestfunc3\] Let the setup be as in Theorem \[thm:unique3\]. Then, for every ${\varepsilon}>0$ and every field $\mathbf{g}\in{\mathcal{H}_{+,R_0}^2}\oplus{\mathcal{H}_{-,R_0}^2}$, there exists $\mathbf{f}\in L^2({\mathbb{S}}_{R_2},{\mathbb{R}}^3)$ (depending on ${\varepsilon}$ and $\mathbf{g}$) such that $$\begin{aligned}
\left|\langle \mathbf{B}^{R_1,R_0,R_2}[\mathbf{m},h],\mathbf{f}\rangle_{L^2({\mathbb{S}}_{R_2},{\mathbb{R}}^3)}-\langle \mathbf{m},\mathbf{g}_{|\Gamma_{R_0}}\rangle_{L^2(\Gamma_{R_0}, {\mathbb{R}}^3)}\right|\leq {\varepsilon}\|(\mathbf{m},h)\|_{L^2(\Gamma_{R_0},{\mathbb{R}}^3)\times L^2({\mathbb{S}}_{R_1})},
\end{aligned}$$ for all $\mathbf{m}\in L^2(\Gamma_{R_0},{\mathbb{R}}^3)$ and $h\in L^2({\mathbb{S}}_{R_1})$. The same holds if $\mathbf{B}^{R_1,R_0,R_2}[\mathbf{m},h]$ gets replaced by $\mathbf{B}_\nu^{R_1,R_0,R_2}[\mathbf{m},h]$, this time with $\mathbf{f}\in L^2({\mathbb{S}}_{R_2})$.
\[cor:approxcoeffs3\] Let the setup be as in Theorem \[thm:unique3\] and Corollary \[cor:unique3\]. Then, for every ${\varepsilon}>0$ and every field $\mathbf{g}\in L^2({\mathbb{S}}_{R_2},{\mathbb{R}}^3)$, there exists $\mathbf{f}\in L^2({\mathbb{S}}_{R_2},{\mathbb{R}}^3)$ (depending on ${\varepsilon}$ and $\mathbf{g}$) such that $$\begin{aligned}
\left|\langle \mathbf{B}^{R_1,R_0,R_2}[\mathbf{m},h],\mathbf{f}\rangle_{L^2({\mathbb{S}}_{R_2},{\mathbb{R}}^3)}-\langle \mathbf{B}_0^{R_0,R_2}[\mathbf{m}],\mathbf{g}\rangle_{L^2({\mathbb{S}}_{R_2},{\mathbb{R}}^3)}\right|\leq {\varepsilon}\|(\mathbf{m},h)\|_{L^2(\Gamma_{R_0},{\mathbb{R}}^3)\times L^2({\mathbb{S}}_{R_1})},
\end{aligned}$$ for all $\mathbf{m}\in L^2(\Gamma_{R_0},{\mathbb{R}}^3)$ and $h\in L^2({\mathbb{S}}_{R_1})$. The same holds if $\mathbf{B}^{R_1,R_0,R_2}[\mathbf{m},h]$ gets replaced by $\mathbf{B}_\nu^{R_1,R_0,R_2}[\mathbf{m},h]$, this time with $\mathbf{f}\in L^2({\mathbb{S}}_{R_2})$.
\[thm:errestfunc2\] Let the setup be as in Definition \[def:ops2\] with $\Gamma_{R_0}\not={\mathbb{S}}_{R_0}$. Then, for every ${\varepsilon}>0$ and every $\mathbf{g}\in{\mathcal{H}_{+,R_0}^2}\oplus{\mathcal{H}_{-,R_0}^2}$, there exists $f\in L^2({\mathbb{S}}_{R_2})$ (depending on ${\varepsilon}$ and $\mathbf{g}$) such that $$\begin{aligned}
\left|\langle \Psi^{R_1,R_0,R_2}[\mathbf{m},\mathfrak{m}],f\rangle_{L^2({\mathbb{S}}_{R_2})}-\langle \mathbf{m},\mathbf{g}_{|\Gamma_{R_0}}\rangle_{L^2(\Gamma_{R_0}, {\mathbb{R}}^3)}\right|\leq {\varepsilon}\|(\mathbf{m},\mathfrak{m})\|_{L^2(\Gamma_{R_0},{\mathbb{R}}^3)\times L^2({\mathbb{S}}_{R_1},{\mathbb{R}}^3)},
\end{aligned}$$ for all $\mathbf{m}\in L^2(\Gamma_{R_0},{\mathbb{R}}^3)$ and $\mathfrak{m}\in L^2({\mathbb{S}}_{R_1},{\mathbb{R}}^3)$.
\[cor:approxcoeffs2\] Let the setup be as in Definition \[def:ops2\] with $\Gamma_{R_0}\not={\mathbb{S}}_{R_0}$. Then, for every ${\varepsilon}>0$ and every function $g\in L^2({\mathbb{S}}_{R_2})$, there exists $f\in L^2({\mathbb{S}}_{R_2})$ (depending on ${\varepsilon}$ and $g$) such that $$\begin{aligned}
\left|\langle \Psi^{R_1,R_0,R_2}[\mathbf{m},\mathfrak{m}],f\rangle_{L^2({\mathbb{S}}_{R_2})}-\langle \Psi_0^{R_0,R_2}[\mathbf{m}],g\rangle_{L^2({\mathbb{S}}_{R_2})}\right|\leq {\varepsilon}\|(\mathbf{m},\mathfrak{m})\|_{L^2(\Gamma_{R_0},{\mathbb{R}}^3)\times L^2({\mathbb{S}}_{R_1},{\mathbb{R}}^3)},
\end{aligned}$$ for all $\mathbf{m}\in L^2(\Gamma_{R_0},{\mathbb{R}}^3)$ and $\mathfrak{m}\in L^2({\mathbb{S}}_{R_1},{\mathbb{R}}^3)$.
The Case $\Gamma_{R_0}={\mathbb{S}}_{R_0}$ {#sec:gammas}
------------------------------------------
We turn to the case where $\Gamma_{R_0}={\mathbb{S}}_{R_0}$. Then, uniqueness no longer holds in Problem \[prob:1\], but one can obtain the singular value decomposition of $\Phi^{R_1,R_0,R_2}$ fairly explicitly and thereby quantify non-uniqueness. Indeed basic computations using spherical harmonics yield: $$\begin{aligned}
&(\Phi_0^{R_0,R_2})^*[Y_{n,k}](x)\nonumber
\\&=\frac{1}{4\pi}\int_{{\mathbb{S}}_{R_2}}Y_{n,k}\left(\frac{y}{|y|}\right)\nabla_x\frac{1}{|x-y|}{{\mathrm{d}}}\omega_{R_2}(y)\nonumber
\\&=\frac{1}{4\pi}\sum_{m=0}^\infty\nabla_x\int_{{\mathbb{S}}_{R_2}}\frac{1}{|y|}\left(\frac{|x|}{|y|}\right)^{m}Y_{n,k}\left(\frac{y}{|y|}\right)P_{m}\left(\frac{x}{|x|}\cdot\frac{y}{|y|}\right){{\mathrm{d}}}\omega_{R_2}(y)\nonumber
\\&=\frac{1}{4\pi}\sum_{m=0}^\infty\sum_{l=1}^{2m+1}\frac{4\pi}{2m+1}\frac{1}{R_2^{m+1}}\nabla_x\left(|x|^mY_{m,l}\left(\frac{x}{|x|}\right)\right)\int_{{\mathbb{S}}_{R_2}}Y_{n,k}\left(\frac{y}{|y|}\right)Y_{m,l}\left(\frac{y}{|y|}\right){{\mathrm{d}}}\omega_{R_2}(y)\nonumber
\\&=\frac{R_2}{2n+1}\nabla H_{n,k}^{R_2}(x)=\frac{R_2}{2n+1}\left(\frac{R_0}{R_2}\right)^n\nabla H_{n,k}^{R_0}(x),\quad x\in{\mathbb{S}}_{R_0},\label{eqn:phi0coeff}
\end{aligned}$$ and $$\begin{aligned}
&(\Phi_1^{R_1,R_2})^*[Y_{n,k}](x)\nonumber
\\&=\frac{1}{4\pi R_1}\int_{{\mathbb{S}}_{R_2}}Y_{n,k}\left(\frac{y}{|y|}\right)\frac{|y|^2-R_1^2}{|x-y|^3}{{\mathrm{d}}}\omega_{R_2}(y)\nonumber
\\&=\frac{1}{4\pi R_1}\sum_{m=0}^\infty(2m+1)\int_{{\mathbb{S}}_{R_2}}\frac{1}{|y|}\left(\frac{|x|}{|y|}\right)^mP_m\left(\frac{x}{|x|}\cdot \frac{y}{|y|}\right)Y_{n,k}\left(\frac{y}{|y|}\right){{\mathrm{d}}}\omega_{R_2}(y)\nonumber
\\&=\frac{1}{R_1R_2}\sum_{m=0}^\infty\sum_{l=1}^{2m+1}\left(\frac{R_1}{R_2}\right)^mY_{m,l}\left(\frac{x}{|x|}\right)\int_{{\mathbb{S}}_{R_2}}Y_{n,k}\left(\frac{y}{|y|}\right)Y_{m,l}\left(\frac{y}{|y|}\right){{\mathrm{d}}}\omega_{R_2}(y)\nonumber
\\&=\left(\frac{R_1}{R_2}\right)^{n-1}Y_{n,k}\left(\frac{x}{|x|}\right),\quad x\in{\mathbb{S}}_{R_1},\label{eqn:phi1coeff}
\end{aligned}$$ where $H_{n,k}^{R_2}$, $H_{n,k}^{R_0}$ are the inner harmonic from Section \[sec:aux\] and $P_m$ the Legendre polynomial of degree $m$ (see, e.g., [@freeden98; @freedenschreiner09 Ch. 3] for details). So, we get for the adjoint operator $(\Phi^{R_1,R_0,R_2})^*$ that $$\begin{aligned}
\label{eqn:adjynk}
(\Phi^{R_1,R_0,R_2})^*[Y_{n,k}]=\left(\frac{R_2}{2n+1}\left(\frac{R_0}{R_2}\right)^n\nabla H_{n,k}^{R_0}\,,\left(\frac{R_1}{R_2}\right)^{n-1}Y_{n,k}\right)^T.
\end{aligned}$$ Similar calculations also yield that $$\begin{aligned}
\Phi_0^{R_0,R_2}[\nabla H_{n,k}^{R_0}](x)= \frac{n}{R_2}\left(\frac{R_0}{R_2}\right)^n Y_{n,k}\left(\frac{x}{|x|}\right),\quad x\in {\mathbb{S}}_{R_2},
\end{aligned}$$ and $$\begin{aligned}
\Phi_1^{R_1,R_2}[Y_{n,k}](x)=\left(\frac{R_1}{R_2}\right)^{n+1}Y_{n,k}\left(\frac{x}{|x|}\right),\quad x\in {\mathbb{S}}_{R_2},
\end{aligned}$$ so we obtain for $\Phi^{R_1,R_0,R_2}$ that $$\begin{aligned}
&\Phi^{R_1,R_0,R_2}[\alpha\nabla H_{n,k}^{R_0},\beta Y_{m,l}]=\alpha \frac{n}{R_2}\left(\frac{R_0}{R_2}\right)^n Y_{n,k}+\beta \left(\frac{R_1}{R_2}\right)^{m+1} Y_{m,l},\label{eqn:phiynk}
\end{aligned}$$ with $\alpha,\beta\in{\mathbb{R}}$. Based on the representations and , further computation leads us to a characterization of the nullspace of $\Phi^{R_1,R_0,R_2}$ in Lemma \[lem:gammaes\]. Note that $\Phi^{R_1,R_0,R_2}:L^2(\Gamma_{R_0},{\mathbb{R}}^3)\times L^2({\mathbb{S}}_{R_1})\to L^2({\mathbb{S}}_{R_2})$ is a compact operator, being the sum of two compact operators (for $\Phi_0^{R_0,R_2}$ and $\Phi_1^{R_1,R_2}$ have continuous kernels).
\[lem:gammaes\] Let $\Gamma_{R_0}={\mathbb{S}}_{R_0}$, then the nullspace of $\Phi^{R_1,R_0,R_2}$ is given by $$\begin{aligned}
N(\Phi^{R_1,R_0,R_2})=&\{(\mathbf{m}_-+\mathbf{d},0):\mathbf{m}_-\in{\mathcal{H}_{-,R_0}^2},\,\mathbf{d}\in \mathcal{D}_{R_0}\}
\\&\cup\,\overline{\textnormal{span}\left\{\left(\nabla H_{n,k}^{R_0}\,,-\frac{n}{R_1}\left(\frac{R_0}{R_1}\right)^nY_{n,k}\right)^T:n\in\mathbb{N},k=1,\ldots,2n+1\right\}},
\end{aligned}$$ while the orthogonal complement reads $$\begin{aligned}
N(\Phi^{R_1,R_0,R_2})^\perp=\overline{\textnormal{span}\left\{\left(\nabla H_{n,k}^{R_0}\,,\frac{2n+1}{R_1}\left(\frac{R_1}{R_0}\right)^nY_{n,k}\right)^T:n\in\mathbb{N},k=1,\ldots,2n+1\right\}}.
\end{aligned}$$ All non-zero eigenvalues values of $(\Phi^{R_1,R_0,R_2})^*\Phi^{R_1,R_0,R_2}$ are of the form $$\begin{aligned}
\sigma_{n}=\frac{n}{2n+1}\left(\frac{R_0}{R_2}\right)^{2n}+\left(\frac{R_1}{R_2}\right)^{2n},\quad n\in\mathbb{N},
\end{aligned}$$ and the corresponding eigenvectors in $L^2({\mathbb{S}}_{R_0},{\mathbb{R}}^3)\times L^2({\mathbb{S}}_{R_1})$ are $$\begin{aligned}
\left(\nabla H_{n,k}^{R_0}\,,\frac{2n+1}{R_1}\left(\frac{R_1}{R_0}\right)^nY_{n,k}\right)^T,\quad n\in\mathbb{N},\,k=1,\ldots,2n+1.
\end{aligned}$$ \[NG\]
Lemma \[NG\] entails that the nullspace of $\Phi^{R_1,R_0,R_2}$ contains elements of the form $(\mathbf{m},h)$ with $h\not=0$, hence $\Phi^{R_1,R_0,R_2}[(\mathbf{m},h)]$ may well vanish on ${\mathbb{S}}_{R_2}$ even though $\Phi_1^{R_1,R_2}[h]$ is nonzero there, by injectivity of the Poisson representation. In other words, separation of the potentials $\Phi_0^{R_0,R_2}$ and $\Phi_1^{R_1,R_2}$ knowing their sum on ${\mathbb{S}}_{R_2}$ is no longer possible in general if $\Gamma_{R_0}={\mathbb{S}}_{R_0}$.
Extremal Problems and Numerical Examples {#sec:num}
========================================
In this section, we provide some first approaches on how the results from the previous sections can be used to approximate the Fourier coefficients of $\Phi_0$ (cf. Section \[sec:fouriernum\]), as well as $\Phi_0$ itself via the reconstruction of $\mathbf{m}$ and $h$ (cf. Section \[sec:phi0num\]). For brevity, we treat only separation of the crustal and core magnetic potentials (underlying operator $\Phi^{R_1,R_0,R_2}$) and not the separation of the crustal and core magnetic fields (underlying operator $\mathbf{B}^{R_1,R_0,R_2}$) nor the separation of potentials generated by two magnetizations on different spheres (underlying operator $\Psi^{R_1,R_0,R_2}$). The procedure in such cases is of course similar.
Reconstruction of Fourier Coefficients of $\Phi_0$ {#sec:fouriernum}
--------------------------------------------------
To get a feeling of how functions $f$ in Corollary \[cor:approxcoeffs\] behave, let us derive some of their basic properties. Recall they where isentified to be those $f\in L^2({\mathbb{S}}_{R_2})$ satisfying with $\mathbf{g}=(\Phi_0^{R_0,R_2})^*[g]$.
\[lem:feps\] Let $0\not=g\in L^2({\mathbb{S}}_{R_2})$ and set $\mathbf{g}=(\Phi_0^{R_0,R_2})^*[g]$. To each ${\varepsilon}>0$, let $f_{\varepsilon}\in L^2({\mathbb{S}}_{R_2})$ satisfy $\|(\Phi^{R_1,R_0,R_2})^*[f_{\varepsilon}]-(\mathbf{g}_{|\Gamma_{R_0}},0)\|_{L^2(\Gamma_{R_0},{\mathbb{R}}^3)\times L^2({\mathbb{S}}_{R_1})}\leq {\varepsilon}$. Then:
- $\lim\limits_{{\varepsilon}\to0}\|f_{\varepsilon}\|_{L^2({\mathbb{S}}_{R_2})}=\infty$,
- $\lim\limits_{{\varepsilon}\to0}\|(\Phi_1^{R_1,R_2})^*[f_{\varepsilon}]\|_{L^2({\mathbb{S}}_{R_1})}=0$,
- $\lim\limits_{{\varepsilon}\to0}\langle f_{\varepsilon},Y_{n,k}\rangle_{L^2({\mathbb{S}}_{R_2})}=0$, for fixed $n\in\mathbb{N}_0$, $k=1,\ldots,n$.
From the considerations in Remark \[rem:empsep\] we know that $(\mathbf{g}_{|\Gamma_{R_0}},0)\in\overline{\textnormal{Ran}\big(\big(\Phi^{R_1,R_0,R_2}\big)^*\big)}$ but $(\mathbf{g}_{|\Gamma_{R_0}},0)\not\in\textnormal{Ran}\big(\big(\Phi^{R_1,R_0,R_2}\big)^*\big)$. Thus, $\|f_{\varepsilon}\|_{L^2({\mathbb{S}}_{R_2})}$ cannot remain bounded as ${\varepsilon}\to0$, otherwise a weak limit point $f_0\in L^2({\mathbb{S}}_{R_2})$ would meet $(\Phi^{R_1,R_0,R_2})^*[f_0]=(\mathbf{g}_{|\Gamma_{R_0}},0)$, a contradiction which proves (a). Next, the relation $$\begin{aligned}
&\|(\Phi^{R_1,R_0,R_2})^*[f_{\varepsilon}]-(\mathbf{g}_{|\Gamma_{R_0}},0)\|_{L^2(\Gamma_{R_0},{\mathbb{R}}^3)\times L^2({\mathbb{S}}_{R_1})}^2
\\&=\|(\Phi_0^{R_0,R_2})^*[f_{\varepsilon}]-\mathbf{g}_{|\Gamma_{R_0}}\|_{L^2(\Gamma_{R_0},{\mathbb{R}}^3)}^2+\|(\Phi_1^{R_1,R_2})^*[f_{\varepsilon}]\|_{L^2({\mathbb{S}}_{R_1})}^2\leq {\varepsilon}^2
\end{aligned}$$ immediately implies that $\lim_{{\varepsilon}\to0}\|(\Phi_1^{R_1,R_2})^*[f_{\varepsilon}]\|_{L^2({\mathbb{S}}_{R_1})}=0$ which is (b). Finally, expanding $f_{\varepsilon}$ in spherical harmonics, one readily verifies that together with (b) yields part (c).
Next, we give a quantitative appraisal of the fact that the Fourier coefficients of $\Phi_0^{R_0,R_2}$ on ${\mathbb{S}}_{R_2}$, to be estimated up to relative precision ${\varepsilon}$ by choosing $g= Y_{p,q}$ in Corollary \[cor:approxcoeffs\], can be approximated directly by those of $\Phi^{R_1,R_0,R_2}$ (i.e., neglecting entirely the core contribution) when $\frac{R_1}{R_2}$ is small enough (i.e., the core is far from the measurement orbit) and the degree $p$ is large enough. We also give a quantitative version of Lemma \[lem:feps\] point $(c)$. This provides us with bounds on the validity of the separation technique consisting merely of a sharp cutoff in the frequency domain.
\[lem:fpq\] Let ${\varepsilon}>0$ and choose $\mathbf{g}=(\Phi_0^{R_0,R_2})^*[Y_{p,q}]$ for some $p\in\mathbb{N}_0$ and $q\in\{1,\ldots 2p+1\}$. The the following assertions hold true.
- If $R_1^2\big(\frac{R_1}{R_2}\big)^{p-1}\leq{\varepsilon}$, then $f=Y_{p,q}$ satisfies $$\label{estims}
\|(\Phi^{R_1,R_0,R_2})^*[f]-(\mathbf{g}_{|\Gamma_{R_0}},0)\|_{L^2(\Gamma_{R_0},{\mathbb{R}}^3)\times L^2({\mathbb{S}}_{R_1})}\leq {\varepsilon}.$$
- If $f\in L^2({\mathbb{S}}_{R_2})$ satisfies $\|(\Phi^{R_1,R_0,R_2})^*[f]-(\mathbf{g}_{|\Gamma_{R_0}},0)\|_{L^2(\Gamma_{R_0},{\mathbb{R}}^3)\times L^2({\mathbb{S}}_{R_1})}\leq {\varepsilon}$, then, for all $n\in\mathbb{N}_0$, $k=1,\ldots, 2n+1$, $$\label{qlfepsc}
|\langle f,Y_{n,k}\rangle_{L^2({\mathbb{S}}_{R_2})}|\leq{\varepsilon}\frac{R_2^{n-1}}{R_1^{n+1}}.$$
To prove $(a)$, note that $(\Phi^{R_1,R_0,R_2})^*=
\left((\Phi_0^{R_0,R_2})^*,(\Phi_1^{R_1,R_2})^*\right)$ and by that $$\|(\Phi_1^{R_1,R_2})^*[f]\|_{L^2({\mathbb{S}}_{R_1})}=R_1^2\left(\frac{R_1}{R_2}\right)^{p-1}\leq {\varepsilon},$$ while $\|(\Phi_0^{R_0,R_2})^*[f]-\mathbf{g}_{|\Gamma_{R_0}}\|_{L^2(\Gamma_{R_0},{\mathbb{R}}^3)}=0$ if $f=Y_{p,q}$. Hence holds.
As to $(b)$, any $f\in L^2({\mathbb{S}}_{R_2})$ satisfying $\|(\Phi^{R_1,R_0,R_2})^*[f]-(\mathbf{g},0)\|_{L^2(\Gamma_{R_0},{\mathbb{R}}^3)\times L^2({\mathbb{S}}_{R_1})}\leq {\varepsilon}$ satisfies in particular, in view of : $$\begin{aligned}
\|(\Phi_1^{R_1,R_2})^*[f]\|_{L^2({\mathbb{S}}_{R_1})}^2=\sum_{n=0}^\infty\sum_{k=1}^{2n+1}R_1^4\left(\frac{R_1}{R_2}\right)^{2(n-1)}|\langle f,Y_{n,k}\rangle_{L^2({\mathbb{S}}_{R_2})}|^2\leq {\varepsilon}^2,
\end{aligned}$$ from which follows at once.
We turn to the computation of a function $f$ as in Corollary \[cor:approxcoeffs\], regardless of assumptions on $\frac{R_1}{R_2}$ or on the degree of a spherical harmonics $Y_{n,k}$ for which we want to estinate $\langle\Phi_0^{R_0,R_2},Y_{n,k}\rangle_{L^2({\mathbb{S}}_{R_2})}$. One way is to solve the following extremal problem. Note that finding $f$ requires no data on the potential $\Phi$ that we eventually want to separate into $\Phi_0+\Phi_1$.
\[prob:2\] Let the setup be as in Definition \[def:ops\] with $\Gamma_{R_0}\not={\mathbb{S}}_{R_0}$. Fix $g\in L^2({\mathbb{S}}_{R_2})$ as well as ${\varepsilon}>0$, and set $\mathbf{g}=(\Phi_0^{R_0,R_2})^*[g]$. Then, find $f\in W^{1,2}({\mathbb{S}}_{R_2})$ such that $$\begin{aligned}
\label{eqn:prob2}
\|f\|_{W^{1,2}({\mathbb{S}}_{R_2})}=\inf_{{\genfrac{}{}{0pt}{}{\bar{f}\in W^{1,2}({\mathbb{S}}_{R_2}),}{\|(\Phi^{R_1,R_0,R_2})^*[\bar{f}]-(\mathbf{g}_{|\Gamma_{R_0}},0)\|_{L^2(\Gamma_{R_0},{\mathbb{R}}^3)\times L^2({\mathbb{S}}_{R_1})}\leq {\varepsilon}}}} \|\bar{f}\|_{W^{1,2}({\mathbb{S}}_{R_2})}.\end{aligned}$$
It may look strange to seek $f\in W^{1,2}({\mathbb{S}}_{R_2})$ whereas Corollary \[cor:approxcoeffs\] merely deals with scalar products in $L^2({\mathbb{S}}_{R_2})$. This extra-smoothness requirement, though, helps regularizing the problem.
\[exun\] Let the setup be as in Problem \[prob:2\] and $g\in L^2({\mathbb{S}}_{R_2})$ with $\|\mathbf{g}_{|\Gamma_{R_0}}\|_{L^2(\Gamma_{R_0},{\mathbb{R}}^3)}>{\varepsilon}$. Then, there exists a unique solution $0\not\equiv f\in W^{1,2}({\mathbb{S}}_{R_2})$ to Problem \[prob:2\]. Moreover, the constraint in is saturated, i.e. $\|(\Phi^{R_1,R_0,R_2})^*[f]-(\mathbf{g}_{|\Gamma_{R_0}},0)\|_{L^2(\Gamma_{R_0},{\mathbb{R}}^3)\times L^2({\mathbb{S}}_{R_1})}= {\varepsilon}$.
Since $\mathbf{H}[g]$ given by lies in ${\mathcal{H}_{+,R_0}^2}$, the same argument as in the proof of Theorem \[thm:errestfunc\] and the density of $W^{1,2}({\mathbb{S}}_{R_2})$ in $L^2({\mathbb{S}}_{R_2})$ together imply the existence of $\bar{f}\in W^{1,2}({\mathbb{S}}_{R_2})$ such that $\|(\Phi^{R_1,R_0,R_2})^*[\bar{f}]-(\mathbf{g}_{|\Gamma_{R_0}},0)\|_{L^2(\Gamma_{R_0},{\mathbb{R}}^3)\times L^2({\mathbb{S}}_{R_1})}\leq{\varepsilon}$ is satisfied, which ensures that the closed convex subset of $W^{1,2}({\mathbb{S}}_{R_2})$ defined by $$\begin{aligned}
\mathcal{C}_{\varepsilon}=\left\{\bar{f}\in W^{1,2}({\mathbb{S}}_{R_2}):\|(\Phi^{R_1,R_0,R_2})^*[\bar{f}]-(\mathbf{g}_{|\Gamma_{R_0}},0)\|_{L^2(\Gamma_{R_0},{\mathbb{R}}^3)\times L^2({\mathbb{S}}_{R_1})}\leq {\varepsilon}\right\}
\end{aligned}$$ is non-empty. Existence and uniqueness of a minimizer $f$ now follows from that of a projection of minimum norm on any nonempty convex set in a Hilbert space. From the assumption that $\|\mathbf{g}\|_{L^2(\Gamma_{R_0},{\mathbb{R}}^3)}>{\varepsilon}$, we get that $f\not\equiv0$ because $0\notin\mathcal{C}_{\varepsilon}$. If the constraint is not saturated, then there is $\delta>0$ such that, for every $\bar{f}\in W^{1,2}({\mathbb{S}}_{R_2})$ with $\|\bar{f}\|_{W^{1,2}({\mathbb{S}}_{R_2})}\leq 1$, also $f+t\bar{f}$ satisfies the constraint $\|(\Phi^{R_1,R_0,R_2})^*[f+t\bar{f}]-(\mathbf{g},0)\|_{L^2(\Gamma_{R_0},{\mathbb{R}}^3)\times L^2({\mathbb{S}}_{R_1})}\leq {\varepsilon}$ for $t\in(-\delta,\delta)$. Since $f$ is a minimizer, this implies $$\begin{aligned}
0&=\partial_t \|f+t\bar{f}\|_{ W^{1,2}({\mathbb{S}}_{R_2})}^2\Big|_{t=0}=2\left\langle f,\bar{f}\right\rangle_{ W^{1,2}({\mathbb{S}}_{R_2})},
\end{aligned}$$ for every $\bar{f}\in W^{1,2}({\mathbb{S}}_{R_2})$ with $\|\bar{f}\|_{ W^{1,2}({\mathbb{S}}_{R_2})}\leq 1$. Thus $f\equiv 0$, contradicting what precedes.
\[rem:shdisc\] Lemma \[lem:feps\] and the exponential decay of the eigenvalues of $(\Phi_1^{R_1,R_2})^*$ in suggest that most of the relevant information regarding a solution $f\in W^{1,2}({\mathbb{S}}_{R_2})$ of Problem \[prob:2\] must be contained in Fourier coefficients $\langle f,Y_{n,k}\rangle_{L^2({\mathbb{S}}_{R_2})}$ of increasingly high degrees $n$ as ${\varepsilon}\to0$. Lemma \[lem:fpq\] provides a hint at the range of accuracies ${\varepsilon}$ for which numerical solutions of Problem \[prob:2\] with $\mathbf{g}=(\Phi_0^{R_0,R_2})^*[Y_{p,q}]$ behave differently for small and large $p$.
### Discretization {#discretization .unnumbered}
For the actual solution of Problem \[prob:2\], we assume that $\|\mathbf{g}_{|\Gamma_{R_0}}\|_{L^2(\Gamma_{R_0},{\mathbb{R}}^3)}>{\varepsilon}$, hence the constraint is saturated by Lemma \[exun\], and we use a Lagrangian formulation and obtain from [@chalendar03 Thm. 2.1] that $f\in W^{1,2}({\mathbb{S}}_{R_2})$ solves for $$\begin{aligned}
\Big({\textnormal{Id}}+\lambda\,\big(\Phi^{R_1,R_0,R_2}\big)^{**}\,\big(\Phi^{R_1,R_0,R_2}\big)^*\Big)[f]=\lambda\, \big(\Phi^{R_1,R_0,R_2}\big)^{**}[(\mathbf{g}_{|\Gamma_{R_0}},0)],\label{eqn:lag}
\end{aligned}$$ where $\lambda>0$ is such that $\|(\Phi^{R_1,R_0,R_2})^*[f]-(\mathbf{g}_{|\Gamma_{R_0}},0)\|_{L^2(\Gamma_{R_0},{\mathbb{R}}^3)\times L^2({\mathbb{S}}_{R_1})}= {\varepsilon}$. Here, the operator $\big(\Phi^{R_1,R_0,R_2}\big)^{**}$ stands for the adjoint of the restriction of $\big(\Phi^{R_1,R_0,R_2}\big)^{*}$ to the domain $W^{1,2}({\mathbb{S}}_{R_2})$. In order to avoid computing $\big(\Phi^{R_1,R_0,R_2}\big)^{**}$, we rewrite in variational form: to $$\begin{aligned}
&\left\langle f,\varphi\right\rangle_{W^{1,2}({\mathbb{S}}_{R_2})}+\lambda \left\langle \big(\Phi^{R_1,R_0,R_2}\big)^*[f],\big(\Phi^{R_1,R_0,R_2}\big)^*[\varphi]\right\rangle_{L^2(\Gamma_{R_0},{\mathbb{R}}^3)\times L^2({\mathbb{S}}_{R_1})}\nonumber
\\&=\lambda\left\langle (\mathbf{g}_{|\Gamma_{R_0}},0), \big(\Phi^{R_1,R_0,R_2}\big)^*[\varphi]\right\rangle_{L^2(\Gamma_{R_0},{\mathbb{R}}^3)\times L^2({\mathbb{S}}_{R_1})},\label{eqn:varform}
\end{aligned}$$ for all $\varphi\in W^{1,2}({\mathbb{S}}_{R_2})$. Remark \[rem:shdisc\] indicates that a discretization of $f$ in terms of finitely many spherical harmonics is generally not advisable. As a remedy, we use a discretization in terms of the Abel-Poisson kernels $$\begin{aligned}
K_\gamma(t)=\frac{1}{4\pi}\frac{1-\gamma^2}{(1+\gamma^2-2\gamma t)^{\frac{3}{2}}},\quad t\in[-1,1].\label{eqn:APkernel}\end{aligned}$$ More precisely, we expand $f$ as $$\begin{aligned}
f(x)&=\sum_{m=1}^M\alpha_{m} K_{\gamma,m}(x) =\sum_{m=1}^M\alpha_{m}\sum_{n=0}^\infty\sum_{k=1}^{2n+1}\gamma^nY_{n,k}\left(\frac{x}{|x|}\right)Y_{n,k}(x_m),\quad x\in{\mathbb{S}}_{R_2},\label{eqn:discf}\end{aligned}$$ where $K_{\gamma,m}(x)=K_\gamma(\frac{x}{|x|}\cdot x_m)$. The parameter $\gamma\in(0,1)$ is fixed and controls the spatial localization of $K_{\gamma,m}$ (a parameter $\gamma$ close to one means a strong localization) while $x_{m}\in{\mathbb{S}}_1$, $m=1,\ldots M,$ denote the spatial centers of the kernels $K_{\gamma,m}$. Furthermore, one can see from that $\gamma$ relates to the influence of higher spherical harmonic degrees in the discretization of $f$. Some general properties of the Abel-Poisson kernel $K_\gamma$ can be found, e.g., in [@freeden98 Ch. 5]. Computations based on the representations in Section \[sec:gammas\] yield $$\begin{aligned}
&(\Phi^{R_1,R_0,R_2})^*[K_{\gamma,m}]\nonumber
\\&=\sum_{n=0}^{\infty}\sum_{k=1}^{2n+1}Y_{n,k}(x_m)\gamma^n\left(\frac{R_2}{2p+1}\left(\frac{R_0}{R_2}\right)^n\nabla H_{n,k}^{R_0}\,,\left(\frac{R_1}{R_2}\right)^{n-1}Y_{n,k}\right)^T\nonumber
\\&=\left(\nabla\sum_{n=0}^{\infty}\sum_{k=1}^{2n+1}\gamma^n\frac{R_2}{2p+1}\left(\frac{R_0}{R_2}\right)^n \left(\frac{|\cdot|}{R_0}\right)^nY_{n,k}(x_m)Y_{n,k}\left(\frac{\cdot}{|\cdot|}\right)\,,\left(\frac{R_2}{R_1}\right)K_{\frac{\gamma R_1}{R_2},m}\right)^T\nonumber
\\&=\left(\frac{R_2}{4\pi}\nabla F_{\frac{\gamma|\cdot|}{R_2},m}\,,\left(\frac{R_2}{R_1}\right)K_{\frac{\gamma R_1}{R_2},m}\right)^T,\label{eqn:phikm}
\end{aligned}$$ where $F_{\gamma,m}(x)=F_{\gamma}(\frac{x}{|x|}\cdot x_m)$, with $F_\gamma(t)=(1+\gamma^2-2\gamma t)^{-\frac{1}{2}}$ for $t\in[-1,1]$. Inserting and into , fixing $\mathbf{g}=(\Phi_0^{R_0,R_2})^*[Y_{p,q}]$ and choosing $\varphi=K_{\gamma,n}$ for $n=1,\ldots,M,$ as test functions, we are lead to the following system of linear equations $$\begin{aligned}
\mathbf{M}\boldsymbol{\alpha}=\mathbf{d},\label{eqn:lineqfpq}
\end{aligned}$$ where $$\begin{aligned}
&\mathbf{M}=\begin{pmatrix*}[l]\displaystyle\frac{1}{\lambda}\left\langle K_{\gamma,m} ,K_{\gamma,n}\right\rangle_{W^{1,2}({\mathbb{S}}_{R_2})}+\left(\frac{R_2}{4\pi}\right)^2\left\langle\nabla F_{\frac{\gamma|\cdot|}{R_2},m},\nabla F_{\frac{\gamma|\cdot|}{R_2},n}\right\rangle_{L^2(\Gamma_{R_0},{\mathbb{R}}^3)}
\\[1.75ex]\displaystyle+\left(\frac{R_2}{R_1}\right)^2\left\langle K_{\frac{\gamma R_1}{R_2},m},K_{\frac{\gamma R_1}{R_2},n}\right\rangle_{L^2({\mathbb{S}}_{R_1})}\end{pmatrix*}_{n,m=1,\ldots,M},
\\&\boldsymbol{\alpha}=\begin{pmatrix*} \alpha_m\end{pmatrix*}_{m=1,\ldots,M},
\\&\mathbf{d}=\begin{pmatrix*} \displaystyle\frac{R_2^2}{4\pi(2p+1)}\left(\frac{R_0}{R_2}\right)^p\langle\nabla H_{p,q}^{R_0},\nabla F_n\rangle_{L^2(\Gamma_{R_0},{\mathbb{R}}^3)}\end{pmatrix*}_{n=1,\ldots,M}.
\end{aligned}$$A function $f$ of the form , determined by coefficients $\alpha_m$, $m=1,\ldots,M$, which solve will from now on be denoted as $f_{p,q}$. We use $f_{p,q}$ as an approximation of the solution to for the choice $\mathbf{g}=(\Phi_0^{R_0,R_2})^*[Y_{p,q}]$.
Input data $\Phi$Input data $\Phi$\
![ Scaled power spectrum $NR_n^{p,q}$ for $p=1$, $q=1$ (*left*) and $p=50$, $q=1$ (*right*).\
[]{data-label="fig:spect3"}](phiinput-eps-converted-to.pdf "fig:")![ Scaled power spectrum $NR_n^{p,q}$ for $p=1$, $q=1$ (*left*) and $p=50$, $q=1$ (*right*).\
[]{data-label="fig:spect3"}](phiinput4-eps-converted-to.pdf "fig:")
True $R_p$, $R_p^0$True $R_p^0$ and reconstructions $\overline{R_p^0}$\
![ Scaled power spectrum $NR_n^{p,q}$ for $p=1$, $q=1$ (*left*) and $p=50$, $q=1$ (*right*).\
[]{data-label="fig:spect3"}](int_spect_2-eps-converted-to.pdf "fig:")![ Scaled power spectrum $NR_n^{p,q}$ for $p=1$, $q=1$ (*left*) and $p=50$, $q=1$ (*right*).\
[]{data-label="fig:spect3"}](spect_2-eps-converted-to.pdf "fig:")\
True $R_p$, $R_p^0$True $R_p^0$ and reconstructions $\overline{R_p^0}$\
![ Scaled power spectrum $NR_n^{p,q}$ for $p=1$, $q=1$ (*left*) and $p=50$, $q=1$ (*right*).\
[]{data-label="fig:spect3"}](int_spect_6-eps-converted-to.pdf "fig:")![ Scaled power spectrum $NR_n^{p,q}$ for $p=1$, $q=1$ (*left*) and $p=50$, $q=1$ (*right*).\
[]{data-label="fig:spect3"}](spect_6-eps-converted-to.pdf "fig:")
Power Spectrum of $f_{1,1}$Power Spectrum of $f_{50,1}$\
![ Scaled power spectrum $NR_n^{p,q}$ for $p=1$, $q=1$ (*left*) and $p=50$, $q=1$ (*right*).\
[]{data-label="fig:spect3"}](spect_f11-eps-converted-to.pdf "fig:")![ Scaled power spectrum $NR_n^{p,q}$ for $p=1$, $q=1$ (*left*) and $p=50$, $q=1$ (*right*).\
[]{data-label="fig:spect3"}](spect_f501-eps-converted-to.pdf "fig:")
### A Numerical Example {#a-numerical-example .unnumbered}
In order to generate input data $\Phi=\Phi^{R_1,R_0,R_2}[\mathbf{m},h]=\Phi_0^{R_0,R_2}[\mathbf{m}]+\Phi_1^{R_1,R_2}[h]$ for a test example, we choose $$\begin{aligned}
\mathbf{m}(x)&=b_1\frac{x}{|x|} L_{\gamma_1}\left(\frac{x}{|x|}\cdot y_1\right)+b_2 \frac{x}{|x|} L_{\gamma_2}\left(\frac{x}{|x|}\cdot y_2\right), \quad b_1=15,\,b_2=10,\nonumber\\h(x)&=\sum_{n=0}^5\sum_{k=1}^{2n+1}a_{n,k} Y_{n,k}\left(\frac{x}{|x|}\right),\quad a_{0,1}=a_{1,1}=2^5,\,a_{2,5}=a_{3,5}=a_{4,5}=2^{4},\,a_{5,5}=2^3,\nonumber
\\& \qquad\qquad\qquad\qquad\qquad\qquad\quad a_{n,k}=0 \textnormal{ else},\label{eqn:trueh}\end{aligned}$$with $y_1=(0,0,-1)^T$ and $y_2=(0,\frac{1}{2},-\frac{\sqrt{3}}{2})^T$. The functions $L_{\gamma_i}$ are chosen as follows: $$\begin{aligned}
L_{\gamma_i}(t)=\left\{\begin{array}{ll}
0,&t\in[-1,\gamma_i),
\\\frac{(t-\gamma_i)^k}{(1-\gamma_i)^k},&t\in[\gamma_i,1],
\end{array}\right.\end{aligned}$$ for $k=3$. These functions have been studied in more detail in [@schreiner97] and are suited for our purposes since they are compactly supported and allow a recursive computation of the Fourier coefficients of $\mathbf{m}$. The parameters $\gamma_i\in (-1,1)$ reflect the localization of $L_{\gamma_i}$ (a parameter $\gamma_i$ close to one means a strong localization). In our test examples, we investigate the two setups $\gamma_1=\frac{1}{20},\gamma_2=\frac{1}{2}$ and $\gamma_1=\frac{3}{5},\gamma_2=\frac{3}{5}$, where latter reflects a slightly stronger localization of the underlying magnetization. The (unknown) crustal contribution is then denoted by $\Phi_0=\Phi_0^{R_0,R_2}[\mathbf{m}]$ and the (unknown) core contribution by $\Phi_1=\Phi_1^{R_1,R_2}[h]$. For the involved radii, we choose $R_0=1$ and $R_2=1.06$ (at scales of the Earth, the latter indicates a realistic satellite altitude of about 380km above the Earth’s surface) and $R_1=0.5$ (at scales of the Earth, this is a rough approximation of the radius of the outer core). The subregion $\Gamma_{R_0}=\{x\in{\mathbb{S}}_{R_0}:x\cdot(0,0,1)^T\leq0\}$ is set to be the Southern hemisphere and the chosen magnetizations of the form satisfy ${\textnormal{supp}}({\mathbf{m}})\subset\Gamma_{R_0}$. For our computations, we use the localization parameter $\gamma=0.95$ and choose $M=8,499$ uniformly distributed centers $x_m\in{\mathbb{S}}_1$, $m=1,\ldots,M,$ for the kernels $K_{\gamma,m}$. All numerical integrations necessary during the procedure are performed via the methods of [@driscoll94] (when the integration region comprises the entire sphere ${\mathbb{S}}_{R_0}$, ${\mathbb{S}}_{R_1}$, or ${\mathbb{S}}_{R_2}$, respectively) and [@hesse12] (when the integration is only performed over the spherical cap ${\mathbb{S}}_{R_0}\setminus\Gamma_{R_0}$). The input data for the two different setups associated with $\gamma_1,\gamma_2$ are shown in Figure \[fig:spect1\]. These setups are not based on real geomagnetic data but they reflect a typical geomagnetic situation in the sense that the core contribution clearly dominates the crustal contribution at low spherical harmonic degrees. Figure \[fig:spect2\] shows that an empirical separation by a sharp cut-off at degree $p=2$ or $p=3$ would neglect relevant information in the crustal contribution.
According to Corollary \[cor:approxcoeffs\], an approximation of the Fourier coefficient $\langle \Phi_0,Y_{p,q}\rangle_{L^2({\mathbb{S}}_{R_2})}$ of the crustal contribution $\Phi_0$ is now given by $\langle \Phi,f_{p,q}\rangle_{L^2({\mathbb{S}}_{R_2})}$, with $f_{p,q}$ of the form described in the previous subsection. We do this for various degrees $p$ and orders $q$ and we illustrate the results in terms of power spectra: The crustal power spectrum is defined as $$\begin{aligned}
R_p^0=R_p[\Phi_0]=\sum_{q=1}^{2p+1}\left|\langle \Phi_0,Y_{p,q}\rangle_{L^2({\mathbb{S}}_{R_2})}\right|^2,\quad p\in\mathbb{N}_0.\end{aligned}$$ Our approximated power spectrum is then of the form $$\begin{aligned}
\overline{R_p^{0}}=\sum_{q=1}^{2p+1}\left|\langle \Phi,f_{p,q}\rangle_{L^2({\mathbb{S}}_{R_2})}\right|^2,\quad p\in\mathbb{N}_0.\end{aligned}$$ The power spectrum of the input signal $\Phi$ (i.e., the superposition of the crustal and core contribution) is analogously defined by $R_p=R_p[\Phi]=\sum_{q=1}^{2p+1}|\langle \Phi,Y_{p,q}\rangle_{L^2({\mathbb{S}}_{R_2})}|^2$.
Figure \[fig:spect2\] shows the reconstructed power spectra and we see that they yield good results (for a well-chosen parameter $\lambda$), in both setups under investigation. Stronger deviations mainly occur at lower spherical harmonic degrees $p$. The solid red spectrum in Figure \[fig:spect2\] indicated as ’Reconstruction for best $\lambda$’ does not reflect the result for a single choice of $\lambda$ but rather for (possibly different) best $\lambda$ in each degree $p$ of the spectrum. The setup for magnetizations ${\mathbf{m}}$ with parameters $\gamma_1=\frac{3}{5},\gamma_2=\frac{3}{5}$ was chosen to investigate magnetizations with a slightly stronger localization, meaning that the corresponding potential $\Phi_0$ has slightly stronger contributions at higher spherical harmonic degrees than for the setup $\gamma_1=\frac{1}{20},\gamma_2=\frac{1}{2}$ (compare the right hand images in Figure \[fig:spect2\]). In Figure \[fig:spect3\], we illustrate the effects mentioned in Remark \[rem:shdisc\] by observing the scaled power spectrum $N_n^{p,q}=NR_n[f_{p,q}]=\frac{1}{2n+1}R_n[f_{p,q}]=\frac{1}{2n+1}\sum_{k=1}^{2n+1}|\langle f_{p,q},Y_{n,k}\rangle_{L^2({\mathbb{S}}_{R_2})}|^2$, $n\in\mathbb{N}_0$, for $p=1$, $q=1$, and $p=50$, $q=1$ (we scaled by a factor $\frac{1}{2n+1}$ solely to get a better idea of the average strength of the Fourier coefficients $|\langle f_{p,q},Y_{n,k}\rangle_{L^2({\mathbb{S}}_{R_2})}|$, $k=1,\ldots, 2n+1$, for fixed degree $n$). As expected from Remark \[rem:shdisc\], larger Lagrange parameters $\lambda$ (which correspond to smaller ${\varepsilon}$) result in a shift of the major contributions of the power spectrum towards higher spherical harmonic degrees. However, for $p=50$, $q=1$, the major spike around $n=50$ remains, somewhat motivating a different behaviour of the Fourier coefficients of $f_{p,q}$ for larger degrees $p$ compared to smaller $p$.
Approximate Reconstruction of $\Phi_0$ {#sec:phi0num}
--------------------------------------
While the previous section aimed at the reconstruction of the Fourier coefficients of $\Phi_0$, we are now concerned with the reconstruction the magnetization $\mathbf{m}$ that generates $\Phi_0$. Actually, the goal is still an approximation of $\Phi_0$, but instead of solving multiple extremal problems like Problem \[prob:2\] we rather solve a single least-squares problem to get an approximation $\bar{\mathbf{m}}$ of $\mathbf{m}$, and then we compute $\bar{\Phi}_0=\Phi_0^{R_0,R_2}[\bar{\mathbf{m}}]$ to approximate $\Phi_0$. Beyond the instrumental parametrizations from the previous section, the only input we retain from the rest of the paper is that, since we apply the technique on an example where $\Gamma_{R_0}\not={\mathbb{S}}_{R_0}$, we know that separation of the core and crustal potentials is possible by Corollary \[cor:unique\]. Still, we gather from Theorem \[thm:unique\] that $\mathbf{m}$ is not uniquely determined though $\Phi_0$ is. So, in order to regularize the problem, we use standard penalization term to compute a candidate $\bar{\mathbf{m}}$ of small norm (weighted by $\alpha$). More precisely, we consider the following extremal problem.
\[prob:3\] Let the setup be as in Definition \[def:ops\] with $\Gamma_{R_0}\not={\mathbb{S}}_{R_0}$, and let $\Phi\in L^2({\mathbb{S}}_{R_2})$ be given. Then, for fixed parameters $\alpha,\beta>0$, find $\bar{\mathbf{m}}\in W^{2,2}({\mathbb{S}}_{R_0},{\mathbb{R}}^3)$ and $\bar{h}\in W^{2,2}({\mathbb{S}}_{R_1})$ to minimize $$\begin{aligned}
\inf_{{\genfrac{}{}{0pt}{}{\bar{\mathbf{m}}\in W^{2,2}({\mathbb{S}}_{R_0},{\mathbb{R}}^3),}{\bar{h}\in W^{2,2}({\mathbb{S}}_{R_1})}}}&\left\|\Phi - \Phi^{R_1,R_0,R_2}[\bar{\mathbf{m}},\bar{h}]\right\|_{L^2({\mathbb{S}}_{R_2})}^2+\alpha\|(\bar{\mathbf{m}},\bar{h})\|_{W^{2,2}({\mathbb{S}}_{R_0},{\mathbb{R}}^3)\times W^{2,2}({\mathbb{S}}_{R_1})}^2+\beta \|\bar{\mathbf{m}}\|_{L^2({\mathbb{S}}_{R_0}\setminus\Gamma_{R_0},{\mathbb{R}}^3)}^2.\end{aligned}$$Note that in this particular setup, the integration in the definition of the operator $\Phi_0^{R_0,R_2}$ is meant over the entire sphere ${\mathbb{S}}_{R_0}$ and not just over $\Gamma_{R_0}$.
Another (more natural) choice to obtain approximations of $\mathbf{m}$ and $h$ would be to minimize $$\begin{aligned}
\inf_{{\genfrac{}{}{0pt}{}{\bar{\mathbf{m}}\in W^{2,2}(\Gamma_{R_0},{\mathbb{R}}^3),}{\bar{h}\in W^{2,2}({\mathbb{S}}_{R_1})}}}&\left\|\Phi - \Phi^{R_1,R_0,R_2}[\bar{\mathbf{m}},\bar{h}]\right\|_{L^2({\mathbb{S}}_{R_2})}^2+\alpha\|(\bar{\mathbf{m}},\bar{h})\|_{W^{2,2}(\Gamma_{R_0},{\mathbb{R}}^3)\times W^{2,2}({\mathbb{S}}_{R_1})}^2,\label{eqn:minfunc}\end{aligned}$$ where this time the integration defining $\Phi_0^{R_0,R_2}$ is only over $\Gamma_{R_0}$ (as always in this paper, with the exception of Problem \[prob:3\] and Section \[sec:gammas\]). Solving leads to magnetizations $\bar{\mathbf{m}}$ that are of class $W^{2,2}(\Gamma_{R_0},{\mathbb{R}}^3)$, while solving Problem \[prob:3\] leads to magnetizations $\bar{\mathbf{m}}$ that are of class $W^{2,2}({\mathbb{S}}_{R_0},{\mathbb{R}}^3)$ and localization in $\Gamma_{R_0}$ has to be enforced by adding a penalty term (weighted by $\beta$). However, for the upcoming example, the minimization proposed in Problem \[prob:3\] yielded slightly better results. Furthermore, it allowed an easier illustration of the effect of the localization constraint by simply dropping the penalty term (i.e., setting $\beta=0$). Existence of minimizers is guaranteed in both cases by standard arguments. The typically difficult choice of parameters $\alpha, \beta$ will not be discussed here. In the provided examples, we simply chose those parameters that seemed to yield the best results when compared to the ground truth.
### Discretization {#discretization-1 .unnumbered}
In order to discretize Problem \[prob:3\], we expand $\bar{\mathbf{m}}$ and $\bar{h}$ in terms of Abel-Poisson kernels the way indicated in Section \[sec:fouriernum\]: $$\begin{aligned}
\bar{\mathbf{m}}(x)&=\sum_{i=1}^3\sum_{n=1}^N\bar{\alpha}_{i,n} \,o^{(i)} K_{\gamma,n}\left(x\right),\quad x\in{\mathbb{S}}_{R_0},
\\\bar{h}(x)&=\sum_{n=1}^N\bar{\beta}_{n}\,K_{\gamma,n}\left(x\right), \quad x\in{\mathbb{S}}_{R_1}.\end{aligned}$$ For brevity, the vectorial operators $o^{(i)}$ have been introduced to denote $o^{(1)}=\nu\,\textnormal{Id}$, $o^{(2)}=\nabla_{\mathbb{S}}$, and $o^{(3)}=\textnormal{L}_{\mathbb{S}}$ (with $\nu$ denoting the unit normal vector). Such localized kernels are suitable here since we know/assume in advance that the sought-after magnetization ${\mathbf{m}}$ is localized in some subregion $\Gamma_{R_0}$. Using this discretization, the minimization of Problem \[prob:3\] reduces to solving the following set of linear equations for the coefficients $\bar{\alpha}_{i,n}$ and $\bar{\beta}_n$: $$\begin{aligned}
\mathbf{M}\boldsymbol{\gamma}=\mathbf{d},\label{eqn:linsolve}\end{aligned}$$ where $$\begin{aligned}
\mathbf{M}&=\left(\begin{array}{c|c}
\mathbf{A}&\mathbf{B}^T
\\\hline\mathbf{B}&\mathbf{C}
\end{array}\right)\in{\mathbb{R}}^{4N\times4N},\quad \boldsymbol{\gamma}&\!\!\!\!\!\!\!\!=(\overline{\boldsymbol{\beta}}\,\vline \,\overline{\boldsymbol{\alpha}}_j)_{j=1,2,3}^T\in\mathbb{R}^{4N},\quad\mathbf{d}&=(\mathbf{a}\,|\,\mathbf{b}_i)_{i=1,2,3}^T\in\mathbb{R}^{4N},
\end{aligned}$$ with $$\begin{aligned}
\mathbf{A}&=\begin{pmatrix*}[l]\langle\Phi^1_{n},\Phi^1_{k}\rangle_{L^2({\mathbb{S}}_{R_2})}+\alpha\langle K_{\gamma,n},K_{\gamma,k}\rangle_{W^{2,2}({\mathbb{S}}_{R_1})} \end{pmatrix*}_{n,k=1,\ldots,N},\nonumber
\\\mathbf{B}&=(\mathbf{B}_{i})_{i=1,2,3},\quad \mathbf{B}_i=\begin{pmatrix*}\langle\Phi^0_{i,n},\Phi^1_{k}\rangle_{L^2({\mathbb{S}}_{R_2})}\end{pmatrix*}_{n,k=1,\ldots,N},
\\\mathbf{C}&=(\mathbf{C}_{i,j})_{i,j=1,2,3},
\\\mathbf{C}_{i,j}&=\begin{pmatrix*}[l]\langle\Phi^0_{i,n,}\Phi^0_{j,k}\rangle_{L^2({\mathbb{S}}_{R_2})}+\alpha\left\langle o^{(i)} K_{\gamma,n},o^{(j)} K_{\gamma,k}\right\rangle_{W^{2,2}({\mathbb{S}}_{R_0},{\mathbb{R}}^3)}
\\[1.75ex]+\beta\left\langle o^{(i)} K_{\gamma,n},o^{(j)} K_{\gamma,k}\right\rangle_{L^2({\mathbb{S}}_{R_0}\setminus\Gamma_{R_0},{\mathbb{R}}^3)}\end{pmatrix*}_{n,k=1,\ldots,N},\nonumber
\\[1.75ex]
{\overline{\boldsymbol{\beta}}}&=(\overline{\beta}_k)_{k=1,\ldots,N},\quad\overline{\boldsymbol{\alpha}}_j=(\overline{\alpha}_{j,k})_{k=1,\ldots,N},
\\[1.75ex]\mathbf{a}&=\left(\langle\Phi^1_{n},\Phi\rangle_{L^2({\mathbb{S}}_{R_2})}\right)_{n=1,\ldots,N},\quad \mathbf{b}_i=\left(\langle\Phi^0_{i,n},\Phi\rangle_{L^2({\mathbb{S}}_{R_2})}\right)_{n=1,\ldots,N},
\end{aligned}$$ and $$\begin{aligned}
\Phi^0_{i,n}(x)&=\frac{1}{4\pi}\int_{{\mathbb{S}}_{R_0}} \left(o^{(i)} K_{\gamma,n}(y)\right)\cdot\frac{x-y}{|x-y|^3}{{\mathrm{d}}}\omega_{R_0}(y),
\\\Phi^1_{n}(x)&=\frac{1}{4\pi R_1}\int_{{\mathbb{S}}_{R_1}} K_{\gamma,n}(y)\frac{|x|^2-R_1^2}{|x-y|^3}{{\mathrm{d}}}\omega_{R_1}(y).\end{aligned}$$ Again, all necessary numerical integrations are performed via the methods of [@driscoll94] (when the integration region comprises the entire sphere ${\mathbb{S}}_{R_0}$, ${\mathbb{S}}_{R_1}$, or ${\mathbb{S}}_{R_2}$, respectively) and [@hesse12] (when the integration is only performed over the spherical cap ${\mathbb{S}}_{R_0}\setminus\Gamma_{R_0}$).
### A Numerical Example {#a-numerical-example-1 .unnumbered}
Input data $\Phi$\
![Results for radii $R_1=0.8$, $R_0=1$, and $R_2=1.06$: Input data $\Phi=\Phi_0+\Phi_1$ (*top*), ground truth $\Phi_0$, $\Phi_1$ (*bottom left*), and reconstructed $\bar{\Phi}_0$, $\bar{\Phi}_1$ with localization constraint (*bottom right*).[]{data-label="fig:ex1b"}](phiinput-eps-converted-to.pdf "fig:")
True $\Phi_0$= Reconstructed $\bar{\Phi}_0$ = Reconstructed $\bar{\Phi}_0$\
($\alpha=5\cdot10^{-16}$, $\beta=1$)($\alpha=5\cdot10^{-16}$, $\beta=0$)\
![Results for radii $R_1=0.8$, $R_0=1$, and $R_2=1.06$: Input data $\Phi=\Phi_0+\Phi_1$ (*top*), ground truth $\Phi_0$, $\Phi_1$ (*bottom left*), and reconstructed $\bar{\Phi}_0$, $\bar{\Phi}_1$ with localization constraint (*bottom right*).[]{data-label="fig:ex1b"}](phi0true-eps-converted-to.pdf "fig:")![Results for radii $R_1=0.8$, $R_0=1$, and $R_2=1.06$: Input data $\Phi=\Phi_0+\Phi_1$ (*top*), ground truth $\Phi_0$, $\Phi_1$ (*bottom left*), and reconstructed $\bar{\Phi}_0$, $\bar{\Phi}_1$ with localization constraint (*bottom right*).[]{data-label="fig:ex1b"}](phi0reconst-eps-converted-to.pdf "fig:")![Results for radii $R_1=0.8$, $R_0=1$, and $R_2=1.06$: Input data $\Phi=\Phi_0+\Phi_1$ (*top*), ground truth $\Phi_0$, $\Phi_1$ (*bottom left*), and reconstructed $\bar{\Phi}_0$, $\bar{\Phi}_1$ with localization constraint (*bottom right*).[]{data-label="fig:ex1b"}](phi0reconstnoloc-eps-converted-to.pdf "fig:")
True $\Phi_1$= Reconstructed $\bar{\Phi}_1$ = Reconstructed $\bar{\Phi}_1$\
($\alpha=5\cdot10^{-16}$, $\beta=1$)($\alpha=5\cdot10^{-16}$, $\beta=0$)\
![Results for radii $R_1=0.8$, $R_0=1$, and $R_2=1.06$: Input data $\Phi=\Phi_0+\Phi_1$ (*top*), ground truth $\Phi_0$, $\Phi_1$ (*bottom left*), and reconstructed $\bar{\Phi}_0$, $\bar{\Phi}_1$ with localization constraint (*bottom right*).[]{data-label="fig:ex1b"}](phi1true-eps-converted-to.pdf "fig:")![Results for radii $R_1=0.8$, $R_0=1$, and $R_2=1.06$: Input data $\Phi=\Phi_0+\Phi_1$ (*top*), ground truth $\Phi_0$, $\Phi_1$ (*bottom left*), and reconstructed $\bar{\Phi}_0$, $\bar{\Phi}_1$ with localization constraint (*bottom right*).[]{data-label="fig:ex1b"}](phi1reconst-eps-converted-to.pdf "fig:")![Results for radii $R_1=0.8$, $R_0=1$, and $R_2=1.06$: Input data $\Phi=\Phi_0+\Phi_1$ (*top*), ground truth $\Phi_0$, $\Phi_1$ (*bottom left*), and reconstructed $\bar{\Phi}_0$, $\bar{\Phi}_1$ with localization constraint (*bottom right*).[]{data-label="fig:ex1b"}](phi1reconstnoloc-eps-converted-to.pdf "fig:")
Input data $\Phi$\
![Results for radii $R_1=0.8$, $R_0=1$, and $R_2=1.06$: Input data $\Phi=\Phi_0+\Phi_1$ (*top*), ground truth $\Phi_0$, $\Phi_1$ (*bottom left*), and reconstructed $\bar{\Phi}_0$, $\bar{\Phi}_1$ with localization constraint (*bottom right*).[]{data-label="fig:ex1b"}](phiinput2-eps-converted-to.pdf "fig:")
True $\Phi_0$= Reconstructed $\bar{\Phi}_0$\
($\alpha=5\cdot10^{-15}$, $\beta=1$)\
![Results for radii $R_1=0.8$, $R_0=1$, and $R_2=1.06$: Input data $\Phi=\Phi_0+\Phi_1$ (*top*), ground truth $\Phi_0$, $\Phi_1$ (*bottom left*), and reconstructed $\bar{\Phi}_0$, $\bar{\Phi}_1$ with localization constraint (*bottom right*).[]{data-label="fig:ex1b"}](phi0true-eps-converted-to.pdf "fig:")![Results for radii $R_1=0.8$, $R_0=1$, and $R_2=1.06$: Input data $\Phi=\Phi_0+\Phi_1$ (*top*), ground truth $\Phi_0$, $\Phi_1$ (*bottom left*), and reconstructed $\bar{\Phi}_0$, $\bar{\Phi}_1$ with localization constraint (*bottom right*).[]{data-label="fig:ex1b"}](phi0reconst2-eps-converted-to.pdf "fig:")
True $\Phi_1$= Reconstructed $\bar{\Phi}_1$\
($\alpha=5\cdot10^{-15}$, $\beta=1$)\
![Results for radii $R_1=0.8$, $R_0=1$, and $R_2=1.06$: Input data $\Phi=\Phi_0+\Phi_1$ (*top*), ground truth $\Phi_0$, $\Phi_1$ (*bottom left*), and reconstructed $\bar{\Phi}_0$, $\bar{\Phi}_1$ with localization constraint (*bottom right*).[]{data-label="fig:ex1b"}](phi1true2-eps-converted-to.pdf "fig:")![Results for radii $R_1=0.8$, $R_0=1$, and $R_2=1.06$: Input data $\Phi=\Phi_0+\Phi_1$ (*top*), ground truth $\Phi_0$, $\Phi_1$ (*bottom left*), and reconstructed $\bar{\Phi}_0$, $\bar{\Phi}_1$ with localization constraint (*bottom right*).[]{data-label="fig:ex1b"}](phi1reconst2-eps-converted-to.pdf "fig:")
We use the same setup as in Section \[sec:fouriernum\] (with parameters $\gamma_1=\frac{1}{20}$, $\gamma_2=\frac{1}{2}$) to generate $\Phi=\Phi^{R_1,R_0,R_2}[\mathbf{m},h]$, $\Phi_0=\Phi_0^{R_0,R_2}[\mathbf{m}]$, and $\Phi_1=\Phi_1^{R_1,R_2}[h]$. In the discretization above, we choose $\gamma=0.9$ and take $N=10,235$ uniformly distributed centers $x_n\in {\mathbb{S}}_1$, $n=1,\ldots,N$. As in the previous example, we choose radii $R_0=1$, $R_2=1.06$, and now additionally vary $R_1$ between $0.5$ and $0.8$. The subregion $\Gamma_{R_0}$ is again the Southern hemisphere $\{x\in{\mathbb{S}}_{R_0}:x\cdot (0,0,1)^T<0\}$. Approximations of $\bar{{\mathbf{m}}}$ and $\bar{h}$ are obtained by solving .
In Figure \[fig:ex1a\], we illustrate the potentials $\bar{\Phi}_0=\Phi_0^{R_0,R_2}[\bar{\mathbf{m}}]$ and $\bar{\Phi}_1=\Phi_1^{R_1,R_2}[\bar{h}]$ corresponding to the reconstructed $\bar{\mathbf{m}}$ and $\bar{h}$ for radius $R_1=0.5$, while in Figure \[fig:ex1b\] we set $R_1=0.8$. In the first case, we see that the reconstructions yield good approximations of the ground truths $\Phi_0=\Phi_0^{R_0,R_2}[\mathbf{m}]$ and $\Phi_1=\Phi_1^{R_1,R_2}[h]$. However, Figure \[fig:ex1b\] suggests that the reconstruction of the potential $\Phi_0$ becomes numerically more critical as the spheres ${\mathbb{S}}_{R_1}$ and ${\mathbb{S}}_{R_0}$ get closer. The influence of the localization constraint on the reconstruction can be seen on the right set of images in Figure \[fig:ex1a\]: neglecting the localization constraint (i.e., choosing $\beta=0$) leads to a wrong separation of the contributions $\bar{\Phi}_0$ and $\bar{\Phi}_1$.
Conclusion {#sec:conc}
==========
In this paper, we set up a geophysically reasonable model of the core and crustal magnetic field potentials $\Phi_1$ and $\Phi_0$ respectively, for which we showed that each single potential can be recovered uniquely if only the superposition $\Phi=\Phi_0+\Phi_1$ is known on an external sphere ${\mathbb{S}}_{R_2}$. Furthermore, we supplied first approaches to the reconstruction of $\Phi_0$ and of its Fourier coefficients. The latter is particularly interesting as it would allow a comparison with the empirical approach to separation based on a sharp cut-off in the power spectrum of $\Phi$. Two main directions call for further study: (1) the geophysical post-processing of real geomagnetic data in order to back up (or deny) the assumption that ${\mathbf{m}}$ is supported in a subregion $\Gamma_{R_0}$ of the Earth’s surface; (2) improving numerical schemes allowing reconstruction of $\Phi_0$ or its Fourier coefficients when the core contribution $\Phi_1$ is clearly dominating (as is expected at lower spherical harmonic degrees in realistic geomagnetic field models) and when ${\mathbb{S}}_{R_1}$ is close to ${\mathbb{S}}_{R_0}$. The domination of the core contribution has been simulated to some extent in the presented examples but is expected to be stronger in real scenarios.\
The work of CG was partly supported by DFG GE 2781/1-1.
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Appendix: Balayage of Distributions {#sec:appendix}
===================================
Since potentials of distributions do not seem to be widely treated in the literature, let us briefly justify the statements made in Section \[sec:harmpot\]. For any distribution $D$ supported in a compact set $\Omega\subset{\mathbb{R}}^3$, the corresponding potential $p_D$ has been formally defined in via $$\begin{aligned}
\label{eqn:ap_Ddef}
p_D(x)=D\left(-\frac{1}{4\pi}\frac{1}{|x-\cdot|}\right), \quad x\in{\mathbb{R}}^3\setminus \Omega.\end{aligned}$$ Strictly speaking, this definition is not valid in that $-1/(4\pi|x-\cdot|)$ is neither smooth nor compactly supported in ${\mathbb R}^3$. However, for any compactly supported $\varphi_x\in C^{\infty}({\mathbb{R}}^3)$ with $\varphi_x\equiv1$ in a neighborhood of $\Omega$ and $\varphi_x\equiv 0$ in a neighborhood of $x$, the function $g_{\varphi_x}(y)=-\frac{1}{4\pi}\frac{1}{|x-y|}\varphi_x(y)$ is in $C^{\infty}({\mathbb{R}}^3)$ and compactly supported. Clearly $D(g_{\varphi_x})$ is independent of the choice of $\varphi_x$, for if $\psi_x$ is another function with the same properties then $g_{\psi_x}-g_{\varphi_x}$ is supported in ${\mathbb R}^3\setminus \Omega$ so that $D(g_{\psi_x}-g_{\varphi_x})=0$. Therefore makes good sense if we understand the latter to mean $p_D(x)=D(g_{\varphi_x})$. In what follows, we restrict ourselves to the case where $\Omega$ has smooth boundary $\partial \Omega$. This is no loss of generality for the matter discussed in the paper, because we only consider situations where $\Omega$ is a closed ball and we want to define balayage onto the boundary sphere. The lemma below is a simple consequence of known density results for the fundamental solution of the Laplacian in $L^2(\partial \Omega)$, $C^{0}(\partial \Omega)$, and $W^{k,2}(\partial \Omega)$, see, e.g., [@freeden80; @freedenmichel04a; @grothaus10].
\[lem:density\] Let $\Omega \subset {\mathbb{R}}^3$ be a compact, simply connected set with $C^{\infty}$-boundary $\partial \Omega$ and let $g_x(y)=\frac{1}{|x-y|}$. Then, the set of functions $\textnormal{span}\{g_x:x\in{\mathbb{R}}^3\setminus \Omega\}$ is dense in $C^{k}(\partial \Omega)$, for any $k\in \mathbb{N}_0$.
For every $f\in C^{k}(\partial \Omega)$ and ${\varepsilon}>0$, there exists $\bar{f}\in C^{\infty}(\partial \Omega)$ with $\|f-\bar{f}\|_{C^{k}(\partial \Omega)}<{\varepsilon}$. In particular, $\bar{f}$ is an element of the Sobolev space $W^{k+2,2}(\partial \Omega)$. By [@grothaus10 Thm. 8.8] we can find $N>0$, coefficients $a_i\in\mathbb{R}$, and points $x_i\in{\mathbb{R}}^3\setminus \Omega$, $i=1,\ldots,N$, such that $$\begin{aligned}
\left\|\bar{f}-\sum_{i=1}^N a_i \frac{1}{|x_i-\cdot|}\right\|_{W^{k+2,2}(\partial \Omega)}<{\varepsilon}.\end{aligned}$$ The Sobolev embedding theorem (see, e.g., [@adams03]) now yields that $W^{k+2,2}(\partial \Omega)\subset C^{k}(\partial \Omega)$ and $$\begin{aligned}
\left\|\bar{f}-\sum_{i=1}^Na_i \frac{1}{|x_i-\cdot|}\right\|_{C^ {(k)}(\partial \Omega)}\leq M\left\|\bar{f}-\sum_{i=1}^N a_i\frac{1}{|x_i-\cdot|}\right\|_{W^{k+2,2}(\partial \Omega)}<M{\varepsilon}\end{aligned}$$ for some constant $M>0$ depending only on $k$, which finishes the proof.
Let $ \Omega\subset {\mathbb{R}}^3$ be a compact, simply connected set with $C^{\infty}$-boundary $\partial \Omega$, and let $D$ be a distribution with support in $\Omega$. Then, there exists a unique distribution $\hat D$ on $\partial \Omega$ such that $$\begin{aligned}
p_D(x)=p_{\hat D}(x), \quad x\in{\mathbb{R}}^3\setminus \Omega.
\end{aligned}$$ We call $\hat D$ the balayage of $D$ onto $\partial \Omega$.
First, we deal with the existence of a balayage. Since $D$ is compactly supported, it is known that there are finitely many compactly supported continuous functions $\Phi_j$ and multiindices $\alpha_j\in\mathbb{N}_0^3$, $j=1,\ldots,m$, such that $D=\sum_{j=1}^m\partial_{\alpha_j}\Phi_j$ (see, e.g., [@rudin91]). Due to this representation, $D$ acts on compactly supported functions $g\in C^{M}({\mathbb{R}}^3)$, with $M=\max_{i=1,\ldots,N}|\alpha_i|$. Let $f$ be a function in $C^{\infty}(\partial \Omega)$ and $h$ its unique harmonic continuation to the interior of $\Omega$ with $h=f$ on $\partial \Omega$ [@lions68 Ch. 2]. A compactly supported function $g_f\in C^{M}({\mathbb{R}}^3)$ satisfying $g_f=h$ in $\Omega$ can be computed as follows. The smoothness of $\partial \Omega$ implies there is an open cover $\{{U}_i\}_{i\in\mathbb{N}}$ of $\partial \Omega$ by open sets in ${\mathbb R}^3$ and diffeomorphisms $\Psi_i\in C^{\infty}({U}_i,\mathbb{B}_1)$ that satisfy $\Psi_i({U}_i\cap \partial \Omega)\subset{\mathbb{R}}^2\times\{0\}$, $\Psi_i({U}_i\cap \Omega)\subset{\mathbb{R}}^3_-$, and $\Psi_i({U}_i\cap (\mathbb{R}^3\setminus \Omega))\subset{\mathbb{R}}^3_+$. Here, ${\mathbb{R}}^3_\pm$ refer to upper and lower half spaces. Let $\{\varphi_i\}_{i\in\mathbb{N}}\subset C^{\infty}({\mathbb{R}}^3)$ be a partition of unity subordinated to the cover $\{{U}_i\}_{i\in\mathbb{N}}$. According to the construction in [@lions68 (2.21)], there exist functions $\bar{g}_i\in C^{M}({\mathbb{R}}^3)$, $1\leq i\leq N$, compactly supported in ${\mathbb{B}}_1$, with $\bar{g}_i=(h\varphi_i)\circ\Psi_i^{-1}$ on ${\mathbb{R}}^3_-\cap{\mathbb{B}}_1$ for every $i$. The function $g_f=\sum_{i=1}^\infty\bar{g}_i\circ \Psi_i$ gives us the desired extension of $h$. We now define $\hat D$ for any $f\in C^{\infty}(\partial \Omega)$ by $$\begin{aligned}
\label{eqn:hD}
\hat D(f)=D(g_f).\end{aligned}$$ Since any two $C^M$-smooth extensions of $h$ have the same derivatives of order less than or equal to $M$ on $\Omega$, we see that $\hat D$ does not depend on the particular extension of $h$ that we use. Thus, it holds that $$\begin{aligned}
p_D(x)=p_{\hat D}(x), \quad x\in{\mathbb{R}}^3\setminus \Omega,\end{aligned}$$ because when $x\notin \Omega$, then $g_x(y)$ is a harmonic function of $y$ in a neighborhood of $\Omega$. Uniqueness of $\hat D$ is a direct consequence of the requirement $\hat{D}(g_x)=p_{\hat D}(x)=p_D(x)=D(g_x)$ for $x\in{\mathbb{R}}^3\setminus \Omega$, and of Lemma \[lem:density\] which guarantees the density of $\{g_x:x\in{\mathbb{R}}^3\setminus \Omega\}$ in $C^{k}(\partial \Omega)$ for all $k\in\mathbb{N}_0$.
Appendix: Differential forms and Hodge theory {#sec:appendix2}
=============================================
Below we gather some basic definitions and facts from Hodge theory on a smooth simply connected surface $\mathcal{M}$ embedded in ${\mathbb R}^3$, that will be used to prove the rotation lemma in Appendix \[sec:appendix1\]. A detailed and more general treatment can be found, [e.g.,]{} in [@Warner Ch. 6].
Tangent spaces, smooth functions, vector fields, metric tensor, area measure and Lebesgue spaces are defined as in Section \[sec:aux\]. Note that $\mathcal{M}$ must be a finite union of topological spheres, as follows from the classification theorem for surfaces [@Massey] and the fact that $g$-holed tori are not simply connected while projective planes cannot embed in ${\mathbb R}^3$. In particular $\mathcal{M}$ is orientable.
For $\mathcal{V}$ a real vector space of dimension 2, let $\mathcal{V}^*$ indicate its dual and $\mathcal{A}_2\mathcal{V}$ the bilinear alternating forms on $\mathcal{V}$. If $(v_1,v_2)$ is a basis of $\mathcal{V}$, the linear maps $v^*_1,v^*_2:\mathcal{V}\to{\mathbb R}$ such that $v^*_j(v_k)=\delta_{jk}$ form a basis of $\mathcal{V}^*$, dual to $(v_1,v_2)$. The bilinear alternating form $v^*_1\wedge v_2^*$ defined by $$v_1^*\wedge v_2^*(w_1,w_2)= \textnormal{det}\big((v_j^*(w_k))_{j,k=1,2}\big)=v_1^*(w_1)v_2^*(w_2)-v_1^*(w_2)v_2^*(w_1)$$ is a basis of the 1-dimensional space $\mathcal{A}_2\mathcal{V}$. Hereafter we put $$\mathcal{E}\mathcal{V}={\mathbb R}\oplus\mathcal{V}^*\oplus\mathcal{A}_2\mathcal{V}.$$ If $(w_1,w_2)$ is another basis of $\mathcal{V}$, we say that $(w_1,w_2)$ has the same orientation as $(v_1,v_2)$ if $v_1^*\wedge v_2^*(w_1,w_2)>0$, the opposite orientation if $v_1^*\wedge v_2^*(w_1,w_2)<0$. We orient $\mathcal{V}$ by choosing one of the two equivalence classes of bases with the same orientation. If $\mathcal{V}$ is equipped with a Euclidean scalar product $\langle \cdot,\cdot\rangle$, then each $L\in\mathcal{V}^*$ is of the form $L(v)=\langle w,v\rangle$ for some unique $w\in\mathcal{V}$. This way we identify $\mathcal{V}^*$ with $\mathcal{V}$ and $\mathcal{A}_2\mathcal{V}$ with the exterior product $\mathcal{V}\wedge\mathcal{V}$ (the tensor product $\mathcal{V}\otimes\mathcal{V}$ quotiented by all relations $v\otimes v=0$). Under this identification, given a positively oriented orthonormal basis $(e_1,e_2)$ of $\mathcal{V}$, we define the star operator $\mathcal{E}\mathcal{V}\to\mathcal{E}\mathcal{V}$ to be the linear map such that $*1=e_1\wedge e_2$, $*(e_1)=e_2$, $*(e_2)=-e_1$, $* (e_1\wedge e_2)=1$. The star operator does not depend on the positively oriented orthonormal basis we use to define it. Clearly, $**=\textnormal{id}$ on ${\mathbb R}\oplus\mathcal{A}_2\mathcal{V}$ and $**=-\textnormal{id}$ on $\mathcal{V}^*$.
We now introduce differential forms on $\mathcal{M}$. A 0-form is a function $\mathcal{M}\to{\mathbb R}$, a 1-form is a map associating to each $x\in\mathcal{M}$ a member of $T_x^*$, a 2-form is a map associating to $x$ a member of $\mathcal{A}_2 T_x$; here and below, $T_x$ indicates the tangent space to $\mathcal{M}$ at $x$. Given a $k$-form $\omega$ and a chart $(U,\psi)$ on $\mathcal{M}$ with $\psi(U)=V\subset{\mathbb R}^2$, one can define a $k$-form $\tilde{\omega}$ on $V$ by the rule $$\label{locform}
\tilde{\omega}[y](v_1,\cdots,v_k)=\omega[\psi^{-1}(y)]({\textrm{D}}\psi^{-1}(v_1),\cdots,
{\textrm{D}}\psi^{-1}(v_k)),\quad y\in V,\quad v_1,\cdots, v_k\in{\mathbb R}^2,$$ which represents $\omega$ in local coordinates using the isomorphism ${\textrm{D}}\psi^{-1}(y):{\mathbb R}^2\to T_{\psi^{-1}(y)}$. This way a form on $\mathcal{M}$ may be regarded as a collection of forms on images of charts which define the same form $\omega$ on overlaps [via]{} . Hence if we use a superscript prime to denote another system of local coordinates and if we set $h=\psi'\circ\psi^{-1}$ for the corresponding change of charts, we have if $k=2$ that $$\label{chvar2}
\tilde{\omega}[y](v_1,v_2)=
(\textnormal{det} ({\textrm{D}}h(y)))
\,\tilde{\omega}'[h(y)](v_1,v_2),
\quad y\in V\cap h^{-1}(V').$$ A 1-form $\omega$ can be written in local coordinates as $\tilde{\omega}[y]=a(y){{\mathrm{d}}}y_1+b(y){{\mathrm{d}}}y_2$, where $a$, $b$ are real functions of $y\in \psi(U)$ and ${{\mathrm{d}}}y_1$, ${{\mathrm{d}}}y_2$ is the basis of $({\mathbb R}^2)^*$ dual to the canonical basis of ${\mathbb R}^2$. If $\omega$ is a 2-form, then $\tilde{\omega}[y]=c(y){{\mathrm{d}}}y_1\wedge {{\mathrm{d}}}y_2$ where $c$ is real-valued on $V$. The wedge product is an associative binary operation on forms, bilinear over functions, that associates to a $k_1$-form $\omega_1$ and a $k_2$-form $\omega_2$ a $k_1+k_2$-form $\omega_1\wedge\omega_2$ such that, in local coordinates, $*({{\mathrm{d}}}y_1)\wedge *({{\mathrm{d}}}y_2)={{\mathrm{d}}}y_1\wedge {{\mathrm{d}}}y_2=-
*({{\mathrm{d}}}y_2)\wedge *({{\mathrm{d}}}y_1)$ and $*({{\mathrm{d}}}y_1)\wedge*({{\mathrm{d}}}y_1)=*({{\mathrm{d}}}y_2)\wedge*({{\mathrm{d}}}y_2)=0$. Note that $k$-forms with $k>2$ (mapping $x\in\mathcal{M}$ to a $k$-linear alternating map on $(T_x)^k$) are identically zero for $T_x$ has dimension 2. The wedge product is independent of the chart used to compute a local representative. We say that a 1-form or a 2-form is smooth if its coefficients $a,b$ or $c$ are smooth functions in every chart. We write $\Lambda^k\mathcal{M}$ for the space of smooth forms of degree $k$ on $\mathcal{M}$, and we let $\Lambda\mathcal{M}=\oplus_{k=0}^2\Lambda^k\mathcal{M}$ for the direct sum.
A smooth 2-form $\omega$ can be integrated over a Borel set $E\subset\mathcal{M}$: if $(U,\psi)$ is a chart with $\psi(U)=V$ and $\tilde{\omega}[y]=c(y){{\mathrm{d}}}y_1\wedge {{\mathrm{d}}}y_2$, and if moreover $E\subset U$, we set $\int_E\omega=\int_{\psi(E)}c(y){{\mathrm{d}}}\lambda(y)$ where $\lambda$ indicates Lebesgue measure. In the general case we cover $E$ with finitely many domains of charts and we use a partition of unity; relation and the change of variable formula ensure that the definition does not depend on which charts or partition we use. The exterior differential ${{\mathrm{d}}}:\Lambda^k\mathcal{M}\to\Lambda^{k+1}\mathcal{M}$ is defined as follows. If $g$ is a function, then ${{\mathrm{d}}}g$ is the usual differential, namely in local coordinates $\widetilde{{{\mathrm{d}}}g}=\partial \tilde{g}/\partial y_1{{\mathrm{d}}}y_1+\partial \tilde{g}/\partial y_2 {{\mathrm{d}}}y_2$. If $\tilde{\omega}=a{{\mathrm{d}}}y_1+b{{\mathrm{d}}}y_2$ is a 1-form in local coordinates, then $\widetilde{{{\mathrm{d}}}\omega}=
(\partial b/\partial y_1-\partial a/\partial y_2){{\mathrm{d}}}y_1\wedge {{\mathrm{d}}}y_2$. The differential of a 2-form is zero. Differentiation is meaningful in that it is independent of the chart used to compute its local representative. Moreover it holds that ${{\mathrm{d}}}\circ {{\mathrm{d}}}=0$. If ${{\mathrm{d}}}\omega=0$, we say that $\omega$ is closed, and if $\omega={{\mathrm{d}}}\nu$ for some $\nu$ we say that $\omega$ is exact. Exact forms are closed, and the quotient space of closed $k$-forms by exact $k$-forms is called the $k$-th (de Rham) cohomology group $H^k(\mathcal{M})$. The simple connectedness of $\mathcal{M}$ means that $H^1(\mathcal{M})=0$, i.e. every closed 1-form on $\mathcal{M}$ is exact [@Warner Ch. 5].
The Hodge-star operator maps $\Lambda^k\mathcal{M}$ to $\Lambda^{2-k}\mathcal{M}$ for $0\leq k\leq 2$, by acting pointwise as the star operator on $\mathcal{E} T_x$ for each $x\in\mathcal{M}$. If we identify a 1-form $\omega$ with the tangent vector field $\mathbf{v}_\omega$ such that $\omega[x](w)=\mathbf{v}_\omega(x)\cdot w$ for $w\in T_x$, then the Hodge star operator merely rotates $\mathbf{v}_\omega$ by $\pi/2$ in the tangent space at each point. To check that it maps smooth forms to smooth forms, we need only produce in a neighborhood of each $x_0\in\mathcal{M}$ a positively oriented orthonormal basis $(e_1(x),e_2(x))$ of $T_x$ that varies smoothly with $x$. If $(U,\psi)$ is a chart with $x_0\in U$ and $V=\psi(U)$, we may choose $e_j(\psi^{-1}(y))=D\psi^{-1}(y)
\mathbf{G}(y)^{-1/2}\kappa_j$ for $y\in V$, where $\mathbf{G}$ is the metric tensor and $\kappa_1,\kappa_2$ the canonical basis of ${\mathbb R}^2$. We denote the action of the Hodge star operator on a form $\omega$ by $*\omega$, as no confusion should arise with the star operator acting on $\mathcal{E} T_x$ for fixed $x$. Next, one defines a pairing on $\Lambda^k\mathcal{M}$ by letting $$\label{scf}
\langle\omega_1,\omega_2\rangle=\int_{\mathcal{M}}\omega_1\wedge*\omega_2.$$ Identifying $T_x^*$ and $T_x$ [via]{} the scalar product in ${\mathbb R}^3$, it follows from the definitions, with the notation of , that in local coordinates $\widetilde{e_1\wedge *e_2}=\widetilde{e_2\wedge *e_1}=0$ and, in addition, $$\widetilde{1\wedge*1}=\widetilde{e_1\wedge *e_1}=\widetilde{e_2\wedge*e_2}=
\widetilde{(e_1\wedge e_2)\wedge*
(e_1\wedge e_2)}=\sqrt{g}\,{{\mathrm{d}}}y_1\wedge {{\mathrm{d}}}y_2.$$ Hence is symmetric and positive definite, moreover we have that $$\label{L2vecform}
\langle f,f\rangle=\|f\|^2_{L^2(\mathcal{M})}\quad\textnormal{and}\quad
\langle \omega,\omega\rangle=\|\mathbf{v}_\omega\|^2_{L^2(\mathcal{M},{\mathbb R}^3)},
\qquad f\in\Lambda^0\mathcal{M},\ \omega\in\Lambda^1\mathcal{M}.$$ One extends $\langle\cdot,\cdot\rangle$ to a scalar product on $\Lambda\mathcal{M}$ by requiring that forms of different degree are orthogonal. Let $\delta:\Lambda^k\mathcal{M}\to\Lambda^{k-1}\mathcal{M}$ be the operator defined by $\delta(\omega)=(-1)^{k(2-k)}*{{\mathrm{d}}}(*\omega)$. Since $**\omega=(-1)^k\omega$ when $\omega\in\Lambda^k\mathcal{M}$, it holds if $\omega_1\in\Lambda^{k-1}\mathcal{M}$ and $\omega_2\in\Lambda^{k}\mathcal{M}$ that $${{\mathrm{d}}}(\omega_1\wedge*\omega_2)={{\mathrm{d}}}\omega_1\wedge*\omega_2+(-1)^{k-1}\omega_1\wedge
{{\mathrm{d}}}(*\omega_2)=
{{\mathrm{d}}}\omega_1\wedge*\omega_2-\omega_1\wedge*\delta(\omega_2),$$ and since the left hand side integrates to $0$ over $\mathcal{M}$ by Stoke’s theorem it implies that $\delta$ is the adjoint of ${{\mathrm{d}}}$ in $\Lambda\mathcal{M}$ equipped with . In particular, we see from that $\delta$ must coincide with the divergence operator on $\Lambda^1\mathcal{M}$ when the latter is identified with smooth tangent vector fields. The operator $\Delta={{\mathrm{d}}}\delta+\delta {{\mathrm{d}}}$ which maps $\Lambda^k\mathcal{M}$ into itself is the Laplace Beltrami operator on $\Lambda\mathcal{M}$. The kernel of $\Delta$ in $\Lambda^k\mathcal{M}$ is the space of harmonic $k$-forms, denoted by $\mathcal{H}^k$. Now, a fundamental result in Hodge theory [@Warner Thm. 6.8] is the existence of an orthogonal sum: $$\label{fundHodge}
\Lambda^k\mathcal{M}={{\mathrm{d}}}(\Lambda^{k-1}\mathcal{M})\oplus\delta (\Lambda^{k+1}\mathcal{M})\oplus\mathcal{H}^k, \qquad k=0,1,2,$$ where orthogonality holds with respect to (by convention $\Lambda^{-1}\mathcal{M}=\{0\}$). Using and elliptic regularity theory, one can further show that each equivalence class in the cohomology group $H^k(\mathcal{M})$ has a unique harmonic representative [@Warner Thm. 6.11]. Since $H^1(\mathcal{M})=\{0\}$ we deduce that $\mathcal{H}^1=0$, hence the orthogonal decomposition specializes in our case to $$\label{fundHodges}
\Lambda^1\mathcal{M}={{\mathrm{d}}}(\Lambda^{0}\mathcal{M})\oplus\delta
(\Lambda^{2}\mathcal{M}).$$ Moreover, since $*$ is obviously surjective $\Lambda^2\mathcal{M}\to\Lambda^0\mathcal{M}$ (for the inverse image of a smooth function $f$ is $f{{\mathrm{d}}}e_1\wedge {{\mathrm{d}}}e_2$), we get that $$\label{divref}
\textnormal{Im}(\delta:\Lambda^2\mathcal{M}\to\Lambda^1\mathcal{M})
=\textnormal{Im}(*{{\mathrm{d}}}:\Lambda^0\to\Lambda^1\mathcal{M}).$$
Appendix: the Rotation Lemma {#sec:appendix1}
============================
In the notation of Section \[sec:aux\], we prove below that the operator $J:\mathcal{T}_R\to \mathcal{T}_R$, which rotates a tangent vector field by $\pi/2$ at every point in the positively oriented tangent plane, isometrically maps tangential gradients to divergence free vector fields and vice-versa. This we call the rotation lemma. The result actually holds on any smooth simply connected compact surface $\mathcal{M}$ embedded in ${\mathbb R}^3$, and we deal below with this more general version but restricting ourselves to the sphere would not simplify the proof.
Gradients, Sobolev spaces, tangent and divergence-free vector fields are defined as in Section \[sec:aux\]. Thus, letting $\mathcal{T}$, $\mathcal{G}$ and $\mathcal{D}$ indicate respectively tangent, gradient, and divergence free vector fields in $L^2(\mathcal{M},{\mathbb R}^3)$, we have the orthogonal decomposition: $$\label{HHMa}
\mathcal{T}=\mathcal{G}\oplus\mathcal{D}.$$ As pointed out in Appendix \[sec:appendix2\], $\mathcal{M}$ is orientable, which makes it possible to define $J$ as rotation of a tangent vector field pointwise by $\pi/2$ in the positively oriented tangent plane.
\[RGD\] For $\mathcal{M}$ a compact simply connected surface embedded in ${\mathbb R}^3$, the map $J:{\mathcal{T}}\to{\mathcal{T}}$ isometrically maps ${\mathcal{G}}$ onto ${\mathcal{D}}$ and conversely.
That $J$ is isometric is obvious for it preserves length pointwise. Moreover, since $J^2=-I$, it suffices to establish that $J({\mathcal{G}})={\mathcal{D}}$. By this amounts to prove that ${\mathcal{T}}={\mathcal{G}}\oplus J({\mathcal{G}})$, and since smooth vector fields and smooth functions are dense in ${\mathcal{T}}$ and $W^{1,2}(\mathcal{M})$ respectively, it is enough by the isometric character of $J$ to show that $$\label{smoothH}
{\mathcal{T}}_S={\mathcal{G}}_S\oplus J({\mathcal{G}}_S),$$ where the subscript ”$S$” indicates the smooth elements of the corresponding space. Now, representing a 1-form $\omega$ as the pointwise Euclidean scalar product with a tangent vector field $\mathbf{v}_\omega$ as we did in Appendix \[sec:appendix2\], we have for any smooth function $f:\mathcal{M}\to{\mathbb R}$ that $\mathbf{v}_{{{\mathrm{d}}}f}$ is just the gradient $\nabla_{\mathcal{M}}f$ and, since we observed in the latter appendix that the Hodge star operator coincides with $J$ on $\mathbf{v}_\omega$, the decomposition follows immediately from and .
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abstract: 'Supersonic jet noise reduction is important for high speed military aircraft. Lower acoustic levels would reduce structural fatigue leading to longer lifetime of the jet aircraft. It is not solely structural aspects which are of importance, health issues of the pilot and the airfield personnel are also very important, as high acoustic levels may result in severe hearing damage. It remains a major challenge to reduce the overall noise levels of the aircraft, where the supersonic exhaust is the main noise source for near ground operation. Fluidic injection into the supersonic jet at the nozzle exhaust has been shown as a promising method for noise reduction. It has been shown to speed up the mixing process of the main jet, hence reducing the kinetic energy level of the jet and the power of the total acoustic radiation. Furthermore, the interaction mechanism between the fluidic injection and the shock structure in the jet exhaust plays a crucial role in the total noise radiation. In this study, LES is used to investigate the change in flow structures of a supersonic (M=1.56) jet from a converging-diverging nozzle. Six fluidic actuators, evenly distributed around the nozzle exit, inject air in a radial direction towards the main flow axis with a total mass flow ratio of 3%. Steady injection is compared with flapping injection. With flapping injection turned on, the injection angle of each injector is varied sinusoidally in the nozzle exit plane and the variation is the same for all injectors. This fluid dynamics video is submitted to the APS DFD Gallery of Fluid Motion 2013 at the 66 the Annual Meeting of the American Physical Society, Division of Fluid Dynamics (24-26 November, Pittsburgh, PA, USA).'
---
\
Haukur Hafsteinsson$^{1}$, Lars-Erik Eriksson$^{1}$, Niklas Andersson$^{1}$\
Daniel Cuppoletti$^{2}$, Ephraim Gutmark$^{2}$\
Erik Prisell$^{3}$\
\
\
Video Description {#video-description .unnumbered}
=================
First, a general picture is brought up to make the audience acquainted with the application. A simplified sharp throat converging-diverging nozzle in a model scale, is attached to a full size aircraft to show its actual location in real a application. Then, a slice through the full three-dimensional computational domain is showed. The domain reaches approximately 70 nozzle exit diameters downstream of the nozzle exit plane and about 4 nozzle exit diameters in the upstream direction. The flow field is obtained by solving the compressible Navier-Stokes equations using an in-house finite volume LES solver based on the G3D family of codes originally developed by Eriksson [@eriksson:95]. The computational grid used for the simulations consists of approximately 20 million cells and the simulations are done on 80 CPU’s using MPI. Three cases are shown in the video; first, the baseline supersonic case without injection is shown, second a case with a steady injection at the nozzle exit is shown and finally a case with a flapping injection. For all three cases, the nozzle is operated at a nozzle pressure ratio (NPR) of 4.0, which gives a jet-exit Mach number of $M=1.56$. For all the cases a slice through the domain colored by $\nabla^2 p$ is showed. This quantity effectively shows sudden spatial pressure variations, such as those that occur across shocks. Inside the nozzle a stationary supersonic flow field is formed. It can be noticed that a shock is formed at the sharp throat. This is a conical shock which reaches further downstream, reflects on its self towards the nozzle wall at the jet center axis. Upon reaching the nozzle wall it reflects again this time passing through the nozzle exit towards the jet center axis where it reflects radially outwards. When interacting with the shear layer it reflects back again towards the jet center axis and so on generating a set of quasi-stationary compression- and expansion waves within the jet plume. Another similar set of shocks is formed at the nozzle lip. These two shocks generate a double shock cell structure which dissipates downstream. Plotting the $M=1.0$ iso-contour reveilles the location of the boundary between the supersonic jet-core region and its subsonic surroundings. A close look at the nozzle exit shows a relatively stable shear layer which quickly unfolds into circumferential vortex cores, as the flow transitions to high turbulence levels due to steep axial velocity gradient in the radial direction.
The steady-state injection consists of 6 evenly distributed actuators around the nozzle exit. The injection angle is normal to the nozzle inner wall and is therefore directed radially inward towards the jet center axis. The total mass flow of all six injectors ($\dot{m}_\mathrm{i}$) compared to the mass flow through the nozzle throat ($\dot{m}_\mathrm{j}$) is $\dot{m}_\mathrm{i}/\dot{m}_\mathrm{j}=3\,\%$, which is considered as relatively high mass flow for practical applications. The injection has a profound effect on the jet dynamics. The $M=1$ iso-surface shows how the fluidic injectors penetrate into the shear layer and create axial vortices which are convected downstream by the main jet flow. These vortices result in increased mixing of the jet plume with its ambient air and hence the length of the potential core is reduced. Since the injectors penetrate rather deepl into the jet, the main jet flow senses a blockage and its path is forced in between the injectors. Therefore, the radial location of the $M=1$ iso-contour increases in between the injectors. Thereafter, iso-contours of $\nabla^2 p$ is showed to visualize spectacular stationary bow-shocks formations upstream of each injector.
Continuing from the animation of the stationary bow-shocks formed due to the steady injection, the injection is switched to a flapping mode which shows how the position of the bow-shocks start to follow the flapping angle. The flapping amplitude is $\pm 60^{\circ}$, the frequency of the flapping is $f=1000\mathrm{Hz}$ and as mentioned earlier, the flapping angle is in the nozzle exit plane. Shifting back to a view of the $M=1$ iso-contour, shows how the mixing-rate is dramatically increased as the flapping injectors introduce a spinning motion to the shear layer. Furthermore, looking at the slice-through the domain showing $\nabla^2 p$ along the jet axis, an interesting motion of the double shock cell structure may be noticed. The flapping injection introduces a shock motion which can be referred to as “shock pumping movement”, i.e. the two shock-cells keep more or less their original structure but the distance between them shifts back and forth. This results in a constructive and destructive shock superposition. This phenomenon is thought to be responsible for strong undesirable acoustic screech harmonics observed in the far-field as shown by Hafsteinsson et al. [@Hafsteinsson].
A final view along the jet-axis towards the nozzle exit, shows how the flapping injection creates stunning spinning-shock formations with highly complex three dimensional shock interactions.
[9]{}
L.-E. Eriksson, *“Development and Validation of Highly Modular Flow Solver Versions in G2DFLOW and G3DFLOW”*, Volvo Aero Corporation, Sweden, Internal report, 9970-1162, 1995
Haukur E. Hafsteinsson, Lars-Erik Eriksson, Niklas Andersson, Daniel R. Cuppoletti, Ephraim J. Gutmark, and Erik Prisell *“Supersonic Jet Noise Reduction Using Steady Injection and Flapping Injection”* 19th AIAA/CEAS Aeroacoustics Conference, Berlin Germany, 2013
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abstract: 'Finding the formation mechanisms for bipolar configurations of strong local magnetic field under control of the relatively weak global magnetic field of the Sun is a key problem of the physics of solar activity. This study is aimed at discriminating whether the magnetic field or fluid motion plays a primary, active role in this process. The very origin and early development stage of Active Region 12548 are investigated based on SDO/HMI observations of 2016 May 20–25. Full-vector magnetic and velocity fields are analyzed in parallel. The leading and trailing magnetic polarities are found to grow asymmetrically in terms of their amplitude, magnetic flux, and the time variation of these quantities. The leading-polarity magnetic element originates as a compact feature against the background of a distributed trailing-polarity field, with an already existing trailing-polarity magnetic element. No signs of strong horizontal magnetic fields are detected between the two magnetic poles. No predominant upflow between their future locations precedes the origin of this bipolar magnetic region (BMR); instead, upflows and downflows are mixed, with some prevalence of downflows. Any signs of a large-scale horizontal divergent flow from the area where the BMR develops are missing; in contrast, a normal supergranulation and mesogranulation pattern is preserved. This scenario of early BMR evolution is in strong contradiction with the expectations based on the model of a rising $\Omega$-shaped loop of a flux tube of strong magnetic field, and an *in situ* mechanism of magnetic-field amplification and structuring should operate in this case.'
author:
- 'A. V. Getling'
- 'A. A. Buchnev'
bibliography:
- 'Getling.bib'
title: The origin and early evolution of a bipolar magnetic region in the solar photosphere
---
Introduction
============
The origin of active regions (ARs) and bipolar sunspot groups is among the key issues to be resolved to comprehend the nature of solar activity. In essence, the central problem is finding a mechanism for the formation of bipolar configurations of strong magnetic field under control of the relatively weak global magnetic field of the Sun; such bipolar magnetic regions (BMRs), giving rise to sunspot groups, trigger the whole sequence of active processes over a wide range of heliocentric distances.
In the process of sunspot formation, a primary role may be played by either magnetic field or plasma motion. In the first case, the magnetic field, having achieved a high strength before the initiation of the sunspot-forming process, proves to be able to dictate one type of solar-plasma motion or another exerting magnetic forces on the matter. In the second case, plasma motion itself produces a strong magnetic field and imparts a bipolar configuration to it according to the laws of magnetohydrodynamics.
The first situation is assumed, in particular, by the widely known *rising-tube model* (RTM), which attributes the formation of a bipolar sunspot group to the emergence of an $\Omega$-shaped loop of a coherent flux tube of strong magnetic field (by the RTM, we mean the physical view of the process rather than the computational thin-flux-tube model). The second situation is characteristic of various possible mechanisms of *in situ* magnetic-field *amplification and structuring* due to plasma flows, e.g., convection; some of these mechanisms can be classified as local dynamos. To approach the understanding of the sunspot-formation processes, it is of great importance to discriminate between these two possibilities.
According to the RTM, the general toroidal[^1] magnetic field produces a strong flux tube deep in the convection zone, down to the tachocline, whereupon a loop of the tube is formed and then lifted by the magnetic-buoyancy force [whose role was first recognized by @parker]. The RTM agrees well with such important regularities of solar activity as Hale’s polarity law and Spörer’s law of sunspot-formation latitudes. For this reason, the properties of the rising tube have become the object of numerous studies . @Fan2009 reviewed studies of the conceivable processes of magnetic-flux-tube rising, giving primary attention to both thin-flux-tube model calculations (which fail at depths of 20–30 Mm, where the cross-sectional size of the tube becomes comparable with the local scale height) and full 2D or 3D numerical MHD simulations based on nonlinear equations for a compressible fluid. Most of these studies consider initially present tubes without discussing the process of their formation. In particular, @JouveBrun2009, dealing with a spherical geometry and using the anelastic approximation, simulated latitudinally stretched, initially axisymmetric magnetic flux tubes rising in a rotating turbulent convection zone from its base and fragmenting; interaction of the tubes with convection and large-scale flows was also considered.
Studies aimed at describing the formation of flux tubes as a result of the instability of a magnetic layer [@Fan2001] and the formation of the magnetic layer itself in a velocity-shear layer [@Vasil_Brummell_2008] are not numerous. They gave no definite indications for these possibilities under the conditions of the solar convection zone.
Since a twist stabilizes the tube, maintaining its cohesion, and in view of the observed twist of the AR magnetic fields, the rising tube is typically assumed to be twisted. Some analyses of the magnetic fields observed in ARs, with determinations of the magnetic helicity, were carried out with this idea behind [@Luoni_etal:2011; @Poisson_etal:2015].
The RTM was considered a standard paradigm in the studies of AR-formation processes for several decades. In recent years, however, abundant observational data of very high spatiotemporal resolution have progressively cast more and more doubts upon the universal adequacy of this model.
As can easily be imagined, the emergence of an $\Omega$-shaped loop of strong-magnetic-flux tube should entail three striking observable effects, viz.:
1. An upflow between the two future magnetic poles of the BMR, on a scale of no less than the distance between them.
2. Strong horizontal magnetic fields at the apex of the emerging flux-tube loop.
3. Intense spreading of matter from the loop-emergence site on the scale of the entire BMR.
As we will see, there is no convincing observational evidence for the actual presence of these effects. Nevertheless, some facts can be interpreted in terms of features 1 and 3.
@Grigor'ev_etal2007 report an enhanced plasma upflow preceding the formation of a new magnetic configuration in the developing AR 10488. In their opinion, this upflow can be attributed to the flux-tube-rising process. Let us note, however, that the Doppler-velocity and magnetic-field patterns presented by these authors do not seem to be spatially correlated in a way typical of such a process.
In their MHD simulations of flux-tube emergence, @Toriumi_Yokoyama2012 [@Toriumi_Yokoyama2013] arrived at the quite expectable conclusion that a horizontal divergent flow (HDF) should precede the appearance of the magnetic flux. @Toriumi_etal_2012 observationally detected signatures of HDFs prior to the magnetic-flux emergence. @Khlystova_Toriumi_2017, using SOHO/MDI observations of the emergence of small AR 9021 and AR 10768, found strong upflows on a mesogranular scale at the initial stage of active-region formation. They noted good agreement in the time variation of the plasma-upflow velocity and area between these observations and numerical simulations of flux-tube emergence carried out by @Toriumi_etal_2011. Strong HDFs in a number of emerging ARs were also revealed by @Khlystova2013a and @Toriumi_etal_2014 on the basis of SOHO/MDI observations. In these studies, Doppler measurements were carried out away from the disk center to determine the horizontal velocities by properly projecting the line-of-sight velocities. Although the horizontal velocity can be determined in this way more accurately at larger distances from the disk center, it should be kept in mind that the resolution of the velocity pattern on the solar surface degrades with this distance. Moreover, the discrimination between the spread velocity related to the AR development and the regular supergranulation flow is a particular, not simple task.
There are, however, observational facts definitely contradicting the above-mentioned features of the tube-rising process. In particular, @PevtsovLamb:2006 “observed no consistent plasma flows at the future location of an active region before its emergence,” and @kosov detected no “large-scale flow patterns on the surface, which would indicate emergence of a large flux-rope structure”; instead, local updrafts and downdrafts were observed. As shown by @Birch_etal_2016, the velocity fields around emerging BMRs are statistically very similar to the velocity fields in the quiet-Sun photosphere in terms of the presence of HDFs (we will return to this finding in Subsection \[velfield\]).
A further example of the AR-development pattern at an early formation stage of a new BMR within already existing AR 11313 was given by @getling_etal_hinode [@getling_etal_hinode_2] (hereinafter, Papers I and II, respectively). Neither a horizontal spreading on the scale of the whole developing subregion, nor a strong horizontal magnetic field between the growing sunspots, nor a strong upflow at that site was detected. Thus, a noticeable discrepancy was found between the observed evolutionary scenario of the magnetic and velocity fields and the RTM-based expectations.
Doubts about the adequacy of the rising-tube model are based not only on the absence of convincing observational evidence for effects 1–3 but also on the following:
1. No quite satisfactory explanation of the origin of a coherent flux tube of strong magnetic field deep in the convection zone has been suggested. The known hypotheses differ in their plausibility and the appropriateness of their starting points [see, e.g., the already mentioned works: @Fan2001; @Vasil_Brummell_2008]. It is important that an intense flux tube should affect the structure of the convective velocity field before the emergence on the photospheric surface; such an influence is not actually observed.
2. The tilt of sunspot groups is typically interpreted as an effect of the Coriolis force on the emerging $\Omega$-shaped flux-tube loop. Therefore, the tilt should be the smaller, the stronger the magnetic field counteracting the turning of the loop. However, as @kosovstenflo and @kosov note, observations do not demonstrate such a magnetic-flux dependence of the tilt angle; their “study of the variations of the tilt angle of bipolar magnetic regions (BMRs) during the flux emergence questions the current paradigm that the magnetic flux emerging on the solar surface represents large-scale magnetic flux ropes ($\Omega$-loops) rising from the bottom of the convection zone.”
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Among others, @warnetal2013 [@Jabbari_etal2016; @warneckeetal2016] critically discuss the appropriateness of the RTM. Their reasoning is based not only on the data of immediate observations but also on helioseismological inversions and direct numerical simulations. In particular, they remark that no signs of rising flux tubes have yet been found in helioseismology. These researchers treat the formation of sunspots as a shallow phenomenon and investigate the possible role of the so-called negative-effective-magnetic-pressure instability (NEMPI; the negative pressure is due to the suppression of the total turbulent pressure – the sum of hydrodynamic and magnetic contributions – by the magnetic field).
As alternatives to the RTM, various mechanisms of local *(in situ)* magnetic-field amplification and structuring have been suggested. Among them are a hydromagnetic instability related to quenching of eddy diffusivity by the enhanced magnetic field and cooling-down of the plasma [@kitmaz], the already mentioned NEMPI, and various MHD mechanisms of inductive excitation of magnetic fields strongly coupled with fluid motions [*local dynamos;* see, e.g., numerical simulations by @stein_nord2012 in which the initial presence of a uniform, untwisted, horizontal magnetic field is assumed]. In particular, based on both observations and theory, @Cheung_etal2017 note that the convective dynamo should operate in the convection zone over various spatial scales, without a clear separation between the large and small scales. We discussed some local formation mechanism for BMRs and sunspots in Paper I (and briefly in Paper II).
The vulnerability of the view of BMR origin as the emergence of a strong coherent flux tube is even reflected in the currently used terminology: the expression “flux-tube emergence” is now usually replaced with “flux emergence.” A comprehensive review of possible flux-emergence processes is given by [@Cheung_Isobe2014].
Nevertheless, many researchers still consider the RTM to be a plausible mechanism of BMR formation. Extensive analyses of numerous ARs from the standpoint of discerning various possible evolutionary scenarios are important.
We study here, on a qualitative level, the very origin and early evolutionary stage of a BMR and a sunspot group in AR 12548. In contrast to the content of Papers I and II, we now consider a “naked” emergence of an AR [i.e, after @Centeno2012 “the flux emergence that is isolated from and unrelated to pre-existing magnetic activity”]. Our approach is based on the parallel consideration of the full-vector magnetic and velocity field in the growing BMR. The time cadence of the data used is 12 min, so that we are able to keep track of the process under a temporal “magnifying glass.” We will basically discuss the observed scenario in the context of its affinity with the above-mentioned implications of the tube-rising process, 1–3. As it will be seen, the development of AR 12548 appears to be strongly dissimilar to the RTM scenario. In general, the formation mechanism must not necessarily be unique for all ARs. We consider verifying the adequacy of the RTM for a wider set of ARs to be our “tactical” aim, which could naturally be a step toward solving the “strategic” problem of understanding the mechanism (or mechanisms) of sunspot formation.\
\
Observations and Data Processing {#obs}
================================
We use here data from the Helioseismic and Magnetic Imager (HMI) of the Solar Dynamics Observatory (SDO), which are stored at and available from the Joint Science Operations Center (JSOC, <http://jsoc.stanford.edu>). A BMR that gave rise later to a sunspot group in AR 12548 originated on 2016 May 23 near the central meridian. Diffuse magnetic fields around the future BMR location were observed since their emergence at the eastern limb on May 16. The early development stage of the BMR that we will analyze here fell on May 23, and the sunspot-group formation was mainly completed by May 27.
Our analysis of the magnetic fields is based on a Spaceweather HMI Active Region Patch [SHARP; see @Bobra_etal_2014] with the data remapped to a Lambert cylindrical equal-area projection (CEA). This automatically selected patch is centered at the flux-weighted centroid of the AR. The magnetic-field vector is decomposed into a radial (vertical), latitudinal and longitudinal components. The data used to determine the velocity-field vector were taken for an area of a size specified by us, centered at the same point. The Dopplergrams are also CEA-remapped but not projected, still representing the line-of-sight, rather than radial, velocity component (we neglect the projection effects taking advantage of the fact that the BMR was not far from the disk center on May 23, and the difference between the line-of-sight and radial, or vertical, component is not important at the moment). We compute the horizontal velocities from a series of white-light images of the same CEA-remapped area using a modified local-correlation-tracking (LCT) technique [@gbuch].
The pixel size is 0.5 arcsec $\approx$ 366 km. The SHARP under study measures 547 $\times$ 372 pixels, or 200 $\times$ 136 Mm$^2$, and the size of the area used for velocity determinations is 300 $\times$ 300 pixels, or 109.8 $\times$ 109.8 Mm$^2$.
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We applied Fourier subsonic filtering [@Title_etal_1989] with a cutoff phase velocity of 4 kms$^{-1}$ to the continuum images and Dopplergrams taken with a cadence of 45 s. To eliminate the velocity fluctuations on a granular scale, we smoothed the line-of-sight velocities and reduced each smoothed Dopplergram to zero average.
The LCT procedure was applied to a sequence of images with a cadence of 135 s. For this procedure to be successful, we magnified the images doubling the number of pixels in each horizontal dimension with the use of a standard subroutine based on bilinear interpolation. To obtain final representations of the horizontal-velocity-vector field, we either averaged the measured velocities over nine time steps (20 m 15 s) or integrated the displacements of imaginary corks distributed over the area of interest, thus constructing cork trajectories for time intervals of 2 to 4 hours.\
Results
=======
Evolution in White Light
------------------------
As a reference time (RT) for the data series that we analyze, we assume the time 2016 May 23, 20:00 TAI, when the last SHARP magnetogram without signs of the growing BMR was obtained in the 12-min-cadence series. The white-light SHARP images (Figure \[white\]) show that, while the photosphere in the lower left quadrant of the patch seems completely unperturbed at the RT, two clear-cut pores are present 3 h later. During the first two days starting from the RT, the sunspot group originates and acquires an appearance typical of bipolar groups, with a well-defined umbral–penumbral structure of the leading and trailing spots. At later times, the structure of the group becomes more complex and less ordered; we will not consider here these development stages.
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Evolution of the Magnetic Field
-------------------------------
The evolution of the magnetic field in the growing BMR at the early development stage of the AR under study is illustrated in the left column of Figure \[images\_vs\_profiles\]. A map of the vertical magnetic field, $B_\mathrm v$, for the RT and four full-vector magnetic-field maps for four subsequent times are shown. We do not show the horizontal component of the field in the first map (for the RT) to clearly indicate a line segment assumed to be the BMR axis. **For each time, we draw such an axis approximately through the centroids of the main magnetic elements of the BMR. The very small differences between the line segments thus obtained are due to deformations of the magnetic-element areas.** In the right column of the same figure, profiles of $B_\mathrm v$ variation along this axis are given for the respective times **together with similar profiles of the longitudinal field, $B_\mathrm l$ – the projection of the magnetic field onto the BMR axis, and the vertical velocity-field component, $u$**.
It can be seen that, at the RT, weak diffuse magnetic fields with predominantly positive vertical component (i.e., of the trailing polarity) and a few small magnetic elements, in some cases corresponding to pores, occur over the whole SHARP. At the future location of the BMR, there are two magnetic elements of the positive (trailing) polarity, which are not yet associated with pores. They can clearly be identified in the $B_\mathrm v$ profile for the RT as two peaks with amplitudes of about 600 G.
At 20:12 TAI, these two positive magnetic elements are still present (and, at their locations, two pores are now distinguishable; they can be seen in the plate of Figure \[white\] for 23:00). However, in an enlarged $B_\mathrm v$ map (not presented here), an extremely faint shadow of the leading (negative) polarity closely adjacent to the positive element that occupies a leading position can be noticed for the first time. As the profile for 20:12 TAI in Figure \[images\_vs\_profiles\] shows, this shadow can be associated with a very shallow minimum of $B_\mathrm v$ located in the immediate neighbourhood of the positive-polarity magnetic element occupying the leading position (the local $B_\mathrm v$ extrema in the magnetic elements may not be located exactly on the line segment chosen as the BMR axis, which is why the minimum of $B_\mathrm v$ is almost imperceptible in the profile for time 20:12 TAI). This minimum becomes deeper and forms a distinct leading-polarity magnetic element by 20:48 TAI, after which the neighbouring local maximum (i.e., the positive-polarity element that was originally present and had the leading position) disappears within an hour. The growing negative (leading) element of the BMR remains in close contact with the “old” positive element as long as the latter exists. By 22:48 TAI, both the leading negative and trailing positive $B_\mathrm v$ extrema become comparable in magnitude, a well-defined BMR has formed, and its magnetic elements are related to a bipolar couple of pores (see Figure \[white\]), which subsequently develops into a bipolar sunspot group.
A consideration of the maps in the left column of Figure \[images\_vs\_profiles\] indicates that, quite expectedly, the vectors of the horizontal magnetic-field component, $\mathbf{B}_\mathrm h$, diverge from the trailing-polarity elements (where $B_\mathrm v>0$) and converge to the leading-polarity elements (where $B_\mathrm v<0$). In the growing BMR, this convergence becomes progressively more pronounced with the formation of the leading-polarity element. The magnetic field directed from the trailing to the leading element is smeared over some area, and its characteristic values can be inferred from the longitudinal field, $B_\mathrm l$.
The behavior of $B_\mathrm l$ deserves a special discussion, since it is directly related to the expectable implication of the rising-tube process noted as feature 2 in the Introduction. To this end, we present the profiles of $B_\mathrm l$ variation along the BMR axis **in the right column of Figure \[images\_vs\_profiles\], on the same panels where the profiles of $B_\mathrm v$ are shown**. It is remarkable that $|B_\mathrm l|$ in between the two magnetic poles of the BMR is typically below a level of 200 G. This field achieves considerably larger magnitudes only after the formation of the BMR in the neighbourhood of its magnetic poles (passing through zero exactly at the poles). Therefore, it demonstrates the feature noted in Paper II as the bordering effect: it reaches two extrema, opposite in sign, on both sides of either extremum of $B_\mathrm v$; in maps of $B_\mathrm h$, which we do not present here for the AR at hand but have presented in Paper II for another AR (in Figures 2–5, left), this feature appears as a segment of a bright ring bordering the dark central spot, where $|B_\mathrm v|$ is large and $B_\mathrm h$ is small. This reflects the fountainlike spatial configuration of magnetic field lines, which are mainly vertical in the center of the magnetic element and diverge around the center above, progressively inclining with the distance from the center. There are no signs of strong horizontal magnetic field between the future pole positions, which would be indicative of the emergence of the flux-tube-loop apex.
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The time variations of the amplitude magnetic-field values and magnetic flux are descriptive of the BMR-evolution process. To obtain characteristics of the sort, we selected an area, in which dramatic changes in the magnetic-field pattern occur, and chose a time interval somewhat longer than that considered up to now, starting 21 h before and ending 36 h after the RT (Figure \[area\]). As can be seen from the left panel of Figure \[ExtrFlux\], the positive and negative extrema of $B_\mathrm v$ vary in considerably different ways. The growth of the leading-polarity (negative) field described as the variation in $\min (B_\mathrm v)$ sets in quite abruptly near the RT (zero time in the graph) from values of about 100 G, indicating that the formation of the BMR begins, and is more rapid. The magnitude of this quantity changes by a factor of about 25 in less than 40 h. In contrast, $\max (B_\mathrm v)$ is initially of order 1000 G, varies more smoothly, and this trailing-polarity (positive) field proves to be amplified only by a factor of about 2.
In addition to the strong dissymmetry between the leading-polarity and trailing-polarity evolution, we have to note a remarkable feature of the variations in $\max B_\mathrm h$ (the dashed curve in the left panel of Figure \[ExtrFlux\]). In a similar way to $\max B_\mathrm v$, this quantity does not exhibit dramatic changes in the rate of its variation, also growing by a factor of about 2. Thus, no signs of the emergence of a flux tube – the process of which the variation curve should be indicative – can be noted.
The dissymmetry between the leading and trailing polarities in their evolution can also be clearly seen in the variation of the magnetic fluxes of either sign (the right panel of Figure \[ExtrFlux\]). On the whole, the pattern of variation of the magnetic flux is similar to that of the $B_\mathrm v$ extrema, although the onset of the BMR formation is not so pronounced in the flux variation. In the time interval considered, the total positive magnetic flux through the selected area, $F_\mathrm +$, changes by a factor of about 25, while the negative flux, $F_\mathrm -$, has approximately doubled.
In terms of the behavior of the magnetic polarities, this AR is in striking contrast to, e.g., the ARs described by [@Centeno2012], where the positive and negative fluxes are very well balanced during the first 15 h of the BMR development.
The Behavior of the Velocity Field {#velfield}
----------------------------------
In view of evaluating the applicability of the RTM to the origin of the AR under study, it is instructive to consider the profiles of variation of the vertical velocity, $u$, along the BMR axis at different times (see again the right column of Figure \[images\_vs\_profiles\]), and this is worth doing in comparison with the profiles of $B_\mathrm v$ variation (the right column of Figure \[images\_vs\_profiles\]). Remember that the $B_\mathrm v$ profile exhibits two well-defined extrema (magnetic elements) starting, roughly speaking, from time 21:48 TAI; they are located at $l\approx 2.5$ and 15 ($l$ being the coordinate measured along the BMR axis). At the RT (20:48 TAI), the $u$ profile indicates the presence of two pronounced vertical flows, an upflow and a downflow. Both of them are in between the future positions of the magnetic elements but the downflow almost coincides with the location where the leading polarity will appear. By 21:48 TAI, the upflow has degenerated into a fairly narrow and weak stream, making room for a wider downflow at $l\lesssim 9$, while the downflow at the location of the leading-polarity element ($l\approx 15$) still exists. At later times (e.g., 22:48 TAI), there are three downflows and two upflows at the BMR axis. It can therefore be concluded that no predominant upflow precedes the origin of the BMR. Generally, upflows and downflows are mixed, with some prevalence of downflows.
Now let us consider the entire pattern of the full-vector velocity field in the surroundings of the growing BMR. Figure \[vel\] shows this field for three selected times: shortly after the RT and about one and two hours later. It can easily be seen from the left column of panels that both the vertical and the horizontal velocity field are distributed very similarly at all these times. In particular, these maps confirm our inference that there is no upflow dominating in the area where the BMR forms; moreover, downflows even prevail in this area. Another important feature is the absence of any signs of spreading, or HDF, from the emergence area of the BMR. In contrast, as demonstrated by the maps of cork trajectories traced over an interval of 2 h 17 m (right column), the pattern of regular supergranules and mesogranules is preserved in the horizontal-velocity field; it varies little during the two-hour interval (this pattern **being virtually the same but even more** pronounced if the cork displacements are integrated over a 4-h interval; **we do not present this map here**). The accumulation of corks at the cell boundaries outlines the supergranulation and mesogranulation network and emphasizes its stability.
In the context of the observed horizontal velocities, it is worth mentioning again the study by @Birch_etal_2016. They tried to reveal HDFs deriving horizontal velocities from SDO/MHI observations of the solar surface around emerging active regions and using in parallel their numerical simulations of solar magnetoconvection in the presence of an emerging model flux tube. For 70 ARs considered, the one-$\sigma$ range of azimuthally averaged radial-outflow speeds at a distance of 15 Mm from the expected emergence location, at 3 hours before the emergence time, was found to be $-8 \pm 50\ \mathrm{m\,s}^{-1}$, while the similar range for quiet-Sun regions chosen for control purposes was $-5 \pm 40\ \mathrm{m\,s}^{-1}$. If the rising-tube mechanism is assumed, the observed flow patterns can be associated with tube-rise speeds not exceeding 150 ms$^{-1}$ at a depth of 20 Mm. This figure agrees with the estimated convection velocities at this depth but is well below the prediction of the emerging-flux-tube model. The authors conclude that the dynamics of the emerging magnetic field in the subphotospheric layers is controlled by convective flows.
Summary of Results
==================
The following remarkable traits are characteristic of the origin and early development stage of AR 12548 considered here:
1. The leading-polarity (negative) magnetic element of the BMR originates as a compact feature with a fountainlike magnetic-field structure against the background of a distributed trailing-polarity field, in which a nucleus of the trailing-polarity (positive) element is already present. The negative element is in close contact with another pre-existing positive element, which subsequently disappears.
2. There are no signs of a strong horizontal magnetic field between the nuclei of the magnetic poles of the BMR, which would indicate the emergence of the apex of an intense magnetic-flux tube. The horizontal magnetic field does not exhibit dramatic changes. Immediately before the origin of the BMR and during its early development stage, the projection of the magnetic-field vector onto the BMR axis is typically below 200 G in between the future positions of the two magnetic poles, thus being not associated with the emergence of a strong flux tube.
3. No predominant upflow between the future locations of the magnetic poles precedes the origin of the BMR. Instead, upflows and downflows are mixed, and downflows even prevail in this area. The leading-polarity magnetic element nucleates against the background of a downflow.
4. There are no signs of large-scale spreading, or HDF, from the area where the BMR develops. In contrast, a regular supergranulation and mesogranulation pattern remains intact.
5. There is a strong dissymmetry between the time variations of the negative and positive extrema of the magnetic field and between the time variations of the negative and positive magnetic fluxes through some area encompassing the BMR: the growth of the leading (negative) polarity sets in abruptly and occurs rapidly while the trailing (positive) polarity grows smoothly and more slowly. In a 57-hour interval encompassing the abrupt onset of the leading-polarity growth, the amplitude and the magnetic flux of the leading polarity increase by a factor of about 25, while those of the trailing polarity only double.
Discussion and Conclusion
=========================
Our analysis of the data on the early development stage of AR 12548 suggests a number of conclusions concerning the phenomena involved. Items 2–4 in the above list of results – the lack of a strong horizontal magnetic field, which should reflect the emergence of the apex of the flux-tube loop; the lack of an overall upflow on the scale of the growing AR, which should be indicative of the flux-tube emergence; and the lack of a spreading flow (HDF) around the area of the growing BMR – are in strong contradiction with the idea of the emergence of an $\Omega$-shaped intense-flux-tube loop. It is also worth noting some other details of the process.
The pattern of the BMR development demonstrates a great dissimilarity between the leading and trailing magnetic polarities in their behavior. The leading polarity nucleates as a compact isolated feature against the background of a distributed trailing-polarity magnetic field. The growth of the leading polarity starts from noise values, which scarcely exceed 100 G – in essence, from the complete absence of any signature of the future leading magnetic pole of the BMR. For some time, the negative (leading-polarity) magnetic element grows in close contact with a pre-existing positive element, which rapidly decays. In contrast, the starting strength of the trailing polarity is slightly below 1000 G, and the trailing magnetic pole develops as a “condensation” of the pre-existing background field. Both of these radically different scenarios appear to be hardly compatible with the notion of the emergence of an $\Omega$-shaped loop.
The persistence of the supergranulation and mesogranulation pattern during the formation of the BMR brings back memories to the observations reported many years ago by @bum63 [@bum] and @bumhow. According to these researchers, the growing magnetic fields of BMRs do not break down the pre-existing convective-velocity field but come from below “seeping” through the network of convection cells.
Thus, our principal conclusion is the inconsistency of the scenario of the origin and early evolutionary stage of AR 12548 with the idea of emergence of an $\Omega$-shaped flux-tube loop carrying a strong magnetic field. We were able to catch the origin of the BMR within several minutes and keep track of the process in its most refined (“naked”) appearance, without interference from other magnetic features complicating the pattern. The observed scenario suggests that an *in situ* mechanism should operate in this case, and plasma motion rather than the magnetic field seems to be basically responsible for the formation of the BMR.
The BMR-development pattern in AR 12548 should not necessarily be typical of most ARs. Nevertheless, both our case studies – that described in Papers I and II and especially the present one – clearly indicate that the RTM offers by far not a universal possibility of AR and sunspot-group formation.
Gathering observational data and systematizing various evolutionary scenarios of AR formation appear to be necessary to comprehend the complex of physical mechanisms responsible for the development of solar-activity processes in the convection zone and atmosphere of the Sun. The above-described study can be considered a particular contribution to the implementation of this general program. The currently available abundant and detailed observational data for AR dynamics offer possibilities for an enormous extension of the scope of studies similar to the present one. We plan further steps on this avenue, and the elaborated techniques of data processing are a favourable prerequisite for such investigations.
The observational data were used here by courtesy of NASA/SDO and the HMI science teams. The kind assistance by Arthur Amezcua, Philip Scherrer, Todd Hoeksema and Xudong Sun in dealing with HMI data available via JSOC is gratefully acknowledged.
[^1]: As is typical of the literature on stellar and planetary dynamos, we use here the terms *toroidal* and *azimuthal* as synonyms, although they are not mathematically equivalent.
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---
abstract: 'We study bubble universe collisions in the ultrarelativistic limit with the new feature of allowing for nontrivial curvature in field space. We establish a simple geometrical interpretation of such collisions in terms of a double family of field profiles whose tangent vector fields stand in mutual parallel transport. This provides a generalization of the well-known flat field space limit of the free passage approximation. We investigate the limits of this approximation and illustrate our analytical results with a numerical simulations.'
author:
- Pontus Ahlqvist
- Kate Eckerle
- Brian Greene
title: Kink Collisions in Curved Field Space
---
Introduction
============
Some time ago, a series of seminal papers [@Coleman; @ColemanCallan; @ColemanDeLuccia], showed that the universe can undergo quantum tunneling from one local minima of a potential to another. Much more recently, a classical mechanism for vacuum transitions was introduced by [@Easther;; @Giblin; @Giblin] involving coherent collisions between bubble universes. While these processes, both quantum and classical, are of significant theoretical interest in and of themselves, their relevance may be far greater due to the fact that models of inflationary cosmology typically yield numerous expanding bubble universes whose centers of nucleation are sufficiently close to allow for such collisions. Moreover, string theory offers the possibility of an enormous landscape of local minima, making it important to determine if tunneling events and bubble collisions are essential cosmological processes. These considerations have inspired numerous authors to study transitions between collections of metastable vacua, with results of possible importance to key outstanding issues, such as the cosmological constant problem and the search for experimental signatures of a multiverse [@Susskind; @BoussoPolch; @persistence-memory; @Aguirre-Johnson; @Johnson; @BrianAli; @Shiu; @Tye; @BrownSarangi; @Brown].
In this paper, we focus on aspects of bubble universe collisions. An important observation in this regard was reported in [@Easther;; @Giblin; @Giblin] which found that at sufficiently high relative velocity the physics vastly simplifies. Namely, at at high impact velocity, colliding bubble walls are so Lorentz contracted that the time it takes them to pass through one another is less than the time for interactions between the bubbles to contribute significantly. Thus, the field configurations in such situations simply superpose–the “free passage" approximation. As the relative velocity of colliding bubbles is proportional to the initial separation between their centers of nucleation, the free passage approximation becomes ever more accurate for ever larger separations.
When free passage holds, there is thus a finite window of time during which the field’s value in the widening spatial region through which both walls have passed in opposite directions – the collision region – stays nearly homogeneous. The field’s value in this region is given by the ambient background (the “parent field value") plus the sum of the field value changes across each bubble wall, i.e. the sum of the bubble field values minus the parent value. The general expectation is that post free passage, the field will be driven by the slope of the potential at this shifted field value, causing it to settle into the nearest local minimum. Thus if the free-passage-kicked field value is in the basin of attraction of a different local minimum (neither bubble, nor parent) then the collision will have spawned a new bubble universe in which the field has acquired the new value. To be sure, in [@us] we pointed out some subtleties in this picture (in which the strength of the forcing function at the free passage induced field value can be sufficient to pull the field out of the new basin of attraction, causing it to finally settle at the original field value and thus thwarting the creation of a new bubble universe), but we anticipate that for many situations, these subtleties will not arise.
To date, studies of bubble collisions have generally considered theories with a single real scalar field described by canonical kinetic terms, and for the most part ignoring gravity [@Easther;; @Giblin; @Giblin; @Aguirre-Johnson; @Johnson] [^1]. In many common theoretical scenarios, however, bubble collisions occur in theories whose scalar fields parameterize a many-dimensional, curved manifold. For instance, Calabi-Yau compactifications of string theory involve scalar fields which are local coordinates on moduli spaces that are generally complex K[ä]{}hler manifolds with nontrivial curvature–so-called “moduli fields". The string landscape is due, in part, to the various local minima of flux potentials that govern the dynamics of these moduli.
A natural issue, then, is the impact of a nontrivial metric on bubble universe collisions, which is the issue we take up in this paper. We address key subtleties in bubble collisions, even at high relative velocity, that arise from the inherent nonlinearity of nontrivial curvature, and find a satisfying geometrical interpretation of our result. Specifically, in Section II we generalize the notion of the free passage approximation from flat to curved field spaces. We derive a geometrical interpretation of the result in terms of the parallel transport of integral curves on moduli space. In Section III we argue that there always exists a regime in which our generalized free passage approximation applies, and in Section IV we numerically study some bubble collision examples (in the setting of 1+1D) and compare the results to those from our analytic expressions in Section II. Finally, we end with some conclusions and suggestions for further work.
Generalization of Free Passage
==============================
We take as our starting point the action $$S=\int_{}^{}d^2 x \left(\frac{1}{2}g_{ij}\partial_\mu\phi^i\partial^\mu\phi^j-V(\phi)\right)
\label{Action}$$ where $g_{ij}$ is the generally curved metric on the field space. We assume the potential $V(\phi)$ has three (or more) degenerate minima at the field space locations $A$, $B$, and $C$ (the minimum necessary to study collisions as a source of vacuum transitions). The Euler-Lagrange equation takes the form, $$\Box\phi^i +\Gamma^i_{j\thickspace k}\partial^\mu\phi^j\partial_\mu\phi^k =- \frac{\partial V}{\partial \phi_i}\label{curved-EOM}$$ The $A$ vacuum will play the role of the parent vacuum, and $B$ and $C$ those of the two bubble vacua we seek to collide. Static solutions to (\[curved-EOM\]) that interpolate between distinct yet degenerate local minima of the potential are solitons. We define $f^i(x)$, and $h^i(x)$ as the components of those solitons centered at $x=0$ with the following asymptotics, $$\begin{aligned}
\lim_{x\to -\infty}f^i(x)&=B^i\\
\lim_{x\to +\infty}f^i(x)&=A^i\\
\lim_{x\to -\infty}h^i(x)&=C^i\\
\lim_{x\to +\infty}h^i(x)&=A^i
\label{solitonBCs}\end{aligned}$$ Our intent is to work out the formalism to describe the collisions between these solitons, taking into account the curved moduli space metric. Of particular interest is the limiting behavior that emerges at ultrarelativistic impact velocity.
The collision of two initially widely separated solitons, say, right-moving $f^i$, and left-moving $h^i$ (that interpolate between the parent vacuum $A$ and the other local minima $B$ and $C$, respectively) is described by an observer in the center of the rest frame of the collision by the following initial value problem: $$\begin{aligned}
&\Box\phi^i +\Gamma^i_{j\thickspace k}\partial^\mu\phi^j\partial_\mu\phi^k =- \frac{\partial V}{\partial \phi_i} \label{collisionIVP}\\
\phi^i(-T,x)&=f^i(\gamma(x-u(-T)))+h^i(-\gamma(x+u(-T)))-A^i\label{collisionIC1}\\
\partial_t \phi^i(-T,x)&=-u\gamma\left({f^i}'(\gamma(x-u(-T)))+{h^i}'(-\gamma(x+u(-T)))\right)\label{collisionIC2}\end{aligned}$$ where we’ve shifted the time coordinate so that the observer’s clock is zero when the trajectories of the centers of the colliding solitons (given by $x_{R,0}=ut$ for right-moving $f^i$, and $x_{L,0}=-ut$ for left-moving $h^i$) coincide.
In order to be a legitimate representation of a soliton collision, the solitons must be widely separated at the initial time, $-T$. Thus, valid values of $T$ are those for which $uT$ is much much greater than the width of all components of both Lorentz contracted solitons (so the observer measures field value $\phi^i=A^i$ to an exceedingly good approximation initially, since outside this width the solitons approach their asymptotic values as decaying exponentials). To make this precise we’ll define a positive constant $w$ such that all components of the solitons we wish to collide, $f^i(x)$ and $h^i(x)$, differ from the relevant vacuum value by an insignificant amount outside of $x\in[-w/2,w/2]$. That is, $$\begin{aligned}
&\frac{|B^i-f^i(-w/2)|}{|A^i-B^i|}\ll 1\\
&\frac{|A^i-f^i(w/2)|}{|A^i-B^i|}\ll 1\\
&\frac{|C^i-h^i(-w/2)|}{|A^i-C^i|}\ll 1\\
&\frac{|A^i-f^i(w/2)|}{|A^i-C^i|}\ll 1.\end{aligned}$$
So, the initial time $-T$ is any time such that $uT> w/2\gamma$. We’ll view the collision as commencing at time $t_{start}=-w/2\gamma$, and lasting until $t_{end}=+w/2\gamma$. The “usefulness" then of an approximation to the actual solution to the collision initial value problem (defined by \[collisionIVP\], \[collisionIC1\], and \[collisionIC2\]), at given impact velocity $u$, is proportional to the fraction of the collision for which the approximation remains valid. Labeling the time until which an approximation accurately captures the dynamics by $t_{approx}$, the approximation’s usefulness is gleaned from $t_{approx}/t_{end}$. The greater this is the more useful the approximation is, and it will be deemed “fully realized" if $t_{approx}\ge t_{end}$.
The task of understanding the dynamics of collisions in the ultrarelativistic limit amounts to finding a one parameter family of approximations– “free passage field configurations" we’ll denote by $\{\phi^i_{FP}(t,x;u)\}_{u\in(0,1)}$– that are ever more useful approximations to the collision initial value problem’s true solution as $u$ is taken to $1$. After constructing $\{\phi^i_{FP}(t,x;u)\}_{u\in(0,1)}$ we conclude this section with a proof that there always exists an impact velocity close enough to $1$ to ensure that the free passage configuration is fully realized.
We first perform a change of variables from $t$ and $x$ to the natural dynamical variables of the problem, namely the spatial coordinates of the boosted observers riding on the soliton walls, $\sigma\equiv\gamma(x-ut)$, and $\omega\equiv\gamma(x+ut)$, which we’ll refer to as the Lorentz variables. This choice of variables enables us to isolate the effect of one soliton, say, left-moving $h$, on the field at a fixed location on the other soliton, in this case $f$. By holding $\sigma$ constant and letting $\omega$ vary from minus infinity to infinity one focuses on a fixed location on the right-moving soliton and follows how the field evolves under the influence of the collision with the left-moving soliton. Similarly, the impact of right-moving $f$ on $h$ can be ascertained by holding $\omega$ constant and varying $\sigma$. Expressing $\phi$ in terms of these, the equation of motion takes the form, $$\begin{aligned}
-4(1-\epsilon)\gamma^2\left[\frac{\partial^2 {\phi}^i}{\partial\sigma\partial\omega}+\Gamma^i_{j\thickspace k}\frac{\partial{\phi}^j}{\partial\sigma}\frac{\partial \phi^k}{\partial\omega} \right]-2\gamma^2\epsilon\left[\frac{\partial^2\phi^i}{\partial\sigma^2}+\frac{\partial^2\phi^i}{\partial\omega^2}+\Gamma^i_{j\thickspace k} \left(\frac{\partial{\phi}^j}{\partial\sigma}\frac{\partial{\phi}^k}{\partial\sigma}+ \frac{\partial{\phi}^j}{\partial\omega}\frac{\partial{\phi}^k}{\partial\omega}\right)\right]= -\frac{\partial V}{\partial {\phi}_i}\label{LorentzVarsEOM}\end{aligned}$$
where we’ve expanded in $\epsilon=1-u$, since we are interested in the limiting dynamics that emerge when $u\rightarrow 1$. Rearranging and using $1/\gamma^2=2\epsilon$ we have, $$\begin{aligned}
\frac{\partial^2 {\phi}^i}{\partial\sigma\partial\omega}+\Gamma^i_{j\thickspace k}\frac{\partial{\phi}^j}{\partial\sigma}\frac{\partial \phi^k}{\partial\omega}=\frac{\epsilon}{2}\frac{\partial V}{\partial {\phi}_i}-\frac{\epsilon}{2}\left[\Gamma^i_{j\thickspace k} \left(\frac{\partial{\phi}^j}{\partial\sigma}\frac{\partial{\phi}^k}{\partial\sigma}+ \frac{\partial{\phi}^j}{\partial\omega}\frac{\partial{\phi}^k}{\partial\omega}\right)+\frac{\partial^2\phi^i}{\partial\sigma^2}+\frac{\partial^2\phi^i}{\partial\omega^2}\right].\label{EPSparallelPDE}\end{aligned}$$ The initial conditions take the form, $$\begin{aligned}
\phi^i(\sigma,\omega)\bigg|_{\partial\Omega_\gamma}=&\left(f^i(\sigma)+h^i(-\omega)-A^i\right)\bigg|_{\partial\Omega_\gamma}\\
\gamma u\left(\frac{\partial}{\partial\omega}-\frac{\partial}{\partial\sigma}\right)\phi^i\bigg|_{\partial\Omega_\gamma}=&-\gamma u\left[{f^i}'(\sigma)+{h^i}'(-\omega)\right]\bigg|_{\partial\Omega_\gamma}\end{aligned}$$ or, $$\begin{aligned}
\phi^i(\sigma,\omega)\bigg|_{\partial\Omega_\gamma}=&\left(f^i(\sigma)+h^i(-\omega)-A^i\right)\bigg|_{\partial\Omega_\gamma}\label{initialdata1}\\
\left(\frac{\partial}{\partial\omega}-\frac{\partial}{\partial\sigma}\right)\phi^i\bigg|_{\partial\Omega_\gamma}=&-\left[{f^i}'(\sigma)+{h^i}'(-\omega)\right]\bigg|_{\partial\Omega_\gamma}\label{initialdata2}\end{aligned}$$ where $\partial\Omega_\gamma$ is the surface in the $\sigma$-$\omega$ plane of constant time $t=-T$. This boundary is simply the line, $\omega=\sigma-2\gamma uT$, which note lies only in the first, third, and fourth quadrants. Its $\omega$-intercept, $-2\gamma uT$, is less than $-w$ for any valid choice of $T$. We bisect $\partial\Omega_\gamma$ at the point $(\gamma uT,-\gamma uT)$ and name the half that lies in the third and lower fourth quadrants as $\partial\Omega_f$, and the half that lies in the first and upper fourth quadrants as $\partial\Omega_h$. These are indicated in Figure \[boundary-split\] by the highlighted yellow, and blue rays, respectively.
Since all points on $\partial\Omega_f$ have $\sigma<\gamma u T<w/2$, they satisfy,
$$\begin{aligned}
\rightarrow \omega\bigg|_{\partial\Omega_f}=(\sigma-2\gamma u T)\bigg|_{\partial\Omega_f} \leq-\gamma u T\leq -w/2\end{aligned}$$
Similarly, all points on $\partial\Omega_h$ have $\omega>-\gamma u T>-w/2$, so their corresponding $\sigma$ coordinate satisfies, $$\begin{aligned}
\sigma\bigg|_{\partial\Omega_h}&\geq2\gamma u T\geq w/2\end{aligned}$$ The boundary conditions can then be rewritten as, $$\begin{aligned}
\partial\Omega_f:&\thickspace \sigma<w/2\rightarrow \sigma-2\gamma uT<-w/2\\
&\phi^i(\sigma,\sigma-2\gamma uT)=f^i(\sigma)+h^i(-(\sigma-2\gamma uT))-A^i\thickapprox f^i(\sigma)+A^i-A^i=f^i(\sigma)\\
&\left(\frac{\partial}{\partial\omega}-\frac{\partial}{\partial\sigma}\right)\phi^i(\sigma,\omega)=-\left({f^i}'(\sigma)+{h^i}'(-\sigma+2\gamma uT)\right)\thickapprox-{f^i}'(\sigma)\\
\partial\Omega_h: &\thickspace \omega>-w/2\rightarrow \omega+2\gamma uT>w/2\\
&\phi^i(\omega+2\gamma u T,\omega)=f^i(\omega+2\gamma u T)+h^i(-\omega)-A^i\thickapprox A^i+h^i(-\omega)A^i=h^i(\omega)\\
&\left(\frac{\partial}{\partial\omega}-\frac{\partial}{\partial\sigma}\right)\phi^i(\sigma,\omega)=-\left({f^i}'(\omega+2\gamma uT)+{h^i}'(-\omega)\right)\thickapprox-{h^i}'(-\omega)\end{aligned}$$ Thus, the entire collision initial value problem stated in the Lorentz variables takes the approximate form: $$\begin{aligned}
\frac{\partial^2 {\phi}^i}{\partial\sigma\partial\omega}+\Gamma^i_{j\thickspace k}\frac{\partial{\phi}^j}{\partial\sigma}\frac{\partial \phi^k}{\partial\omega}=\frac{\epsilon}{2}&\frac{\partial V}{\partial {\phi}_i}+\frac{\epsilon}{2}\left[\Gamma^i_{j\thickspace k} \left(\frac{\partial{\phi}^j}{\partial\sigma}\frac{\partial{\phi}^k}{\partial\sigma}+ \frac{\partial{\phi}^j}{\partial\omega}\frac{\partial{\phi}^k}{\partial\omega}\right)-\frac{\partial^2\phi^i}{\partial\sigma^2}-\frac{\partial^2\phi^i}{\partial\omega^2}\right]\\
&\phi^i(\sigma,\omega) \bigg|_{\partial\Omega_f} \thickapprox f^i(\sigma)\label{approxbc1}\\
&\frac{\partial}{\partial\sigma}\phi^i(\sigma,\omega)\bigg|_{\partial\Omega_f}\thickapprox{f^i}'(\sigma)\label{approxbc2}\\
&\phi^i(\sigma,\omega)\bigg|_{\partial\Omega_h}\thickapprox h^i(\omega)\label{approxbc3}\\
&\frac{\partial}{\partial\omega}\phi^i(\sigma,\omega)\bigg|_{\partial\Omega_h}\thickapprox -{h^i}'(-\omega)\label{approxbc4}\end{aligned}$$
Now we’ll obtain the limiting form of these equations when $\gamma\rightarrow 1$. First we’ll turn our attention to the boundary conditions. As $\gamma$ is increased the boundary $\partial\Omega_\gamma$ is pushed along the diagonal with negative slope toward the fourth quadrant. This causes the $\omega$ values of points on $\partial\Omega_f$ to become increasingly negative, and the $\sigma$ values on $\partial\Omega_h$ to become increasingly positive. Consequently, the approximations made in the boundary conditions (\[approxbc1\], \[approxbc2\], \[approxbc3\], \[approxbc4\]) become ever more accurate.
This can be seen visually as well. The center of the $f$ soliton occurs, by definition, at the $\omega$-intercept of $\partial\Omega_\gamma$, and the center of the $h$ soliton occurs at the $\sigma$-intercept. As $\partial\Omega_\gamma$ is pushed along the diagonal toward the fourth quadrant the intercepts move away from each other. The distance between the center of each soliton and the place where the boundary is bisected (the endpoint of both half boundaries) increases, resulting in ever more of the $f$ soliton fitting on $\partial\Omega_f$, and the $h$ soliton fitting on $\partial\Omega_h$.[^2]
Thus, the limiting form of the boundary conditions is obtained by replacing the approximate equalities in \[approxbc1\], \[approxbc2\], \[approxbc3\], and \[approxbc4\] with equalities. Further, note that the two conditions involving the first derivatives (\[approxbc2\], \[approxbc4\]) no longer contain any additional information than what is captured by the two conditions on $\phi^i$, (\[approxbc1\], \[approxbc3\]). Clearly the limit of the differential equation, \[EPSparallelPDE\], is obtained by dropping the $\mathcal{O}(\epsilon)$ term on the righthand side.
At the risk of stating the obvious we’ll identify this limiting set of equations with the appropriate collision– that of the non-Lorentz contracted profiles $f^i$ and $h^i$ each propagating toward one another with speed $u=1$ in the *free* theory. To see why this is the case, take the equation of motion and initial conditions associated with this collision, $$\begin{aligned}
&\Box\phi^i+\Gamma^i_{j\thickspace k}\partial_\mu\phi^j\partial^\mu\phi^k=0\\
&\lim_{t_\rightarrow-\infty}\phi^i(t,x)=f^i(x-t)+h^i(-(x+t))-A^i\end{aligned}$$ and transform to the characteristics, $\xi=x-t$, and $\eta=x+t$. Doing so yields, $$\begin{aligned}
\frac{\partial^2 {\phi}^i}{\partial\xi\partial\eta}&+\Gamma^i_{j\thickspace k}\frac{\partial{\phi}^j}{\partial\xi}\frac{\partial \phi^k}{\partial\eta}=0\label{freePDE}\\
\phi^i(\xi,-\infty)&=f^i(\xi)\label{ic1}\\
\phi^i(\infty,\eta)&=h^i(-\eta)\label{ic2}\end{aligned}$$ Since the righthand side of the resulting differential equation is *identically* zero, we view *this* problem as the limit of the original one (the collision of boosted solitons in the model with nontrivial potential, $V$). So, the approximation to the true solution of the $u<1$ collision problem should be defined by obtaining the solution to the free problem (\[freePDE\], subject to \[ic1\], and \[ic2\]), and then evaluating it at the Lorentz variables as opposed to the characteristics. If we denote the solution to the free problem by $\Phi^i(\xi,\eta)$, we mean the approximation for impact velocity $u$ ought to be defined by, $$\phi^i_{FP}(t,x;u)\equiv \Phi^i(\gamma(x-ut),\gamma(x+ut))\label{FP-definition}$$ Turning our attention to $\Phi^i$, we note that it maps $\mathbb{R}^2$ to a submanifold of the field space manifold, $N\subset M$.[^3] The submanifold is a patch of field space, bounded by four curves. Two of these are simply the original soliton curves (traced out in field space) that we are colliding, since $\Phi^i(\xi,-\infty)=f^i(\xi)$ for $\xi\in\mathbb{R}$, and $\Phi^i(\infty,\eta)=h^i(-\eta)$ for $\eta\in \mathbb{R}$. Significant insight is gained by viewing $\Phi^i$ as the coordinates of two sets of integral curves– those of one set obtained by varying the first argument and fixing the second, and those of the second set obtained by fixing the first argument and varying the second. Let us name two vector fields these sets of integral curves define as follows, $$\begin{aligned}
U\equiv\frac{\partial}{\partial\xi}=\frac{\partial\Phi^i(\xi,\eta)}{\partial\xi}e_i|_{\Phi(\xi,\eta)}\\
W\equiv\frac{\partial}{\partial\eta}=\frac{\partial\Phi^i(\xi,\eta)}{\partial\eta}e_i|_{\Phi(\xi,\eta)}\\end{aligned}$$ where we’ve expanded in the coordinate basis $\{e_i\}=\{ \frac{\partial}{\partial\Phi^i}\}$. Expressed in terms of the vector fields $U$ and $W$, the boundary conditions for $\Phi^i$ simply indicate that the vector fields at the relevant two edges of the submanifold line up with the tangent vectors to the two original soliton curves. In an effort to minimize confusion with the negative signs, we explicitly point out where the vacuum values are in the submanifold, parameterized by $\xi$ and $\eta$: $\Phi(\infty,-\infty)=A$, $\Phi(-\infty,-\infty)=B$, and $\Phi(\infty,\infty)=C$.
The differential equation takes the form, $$\begin{aligned}
0=\frac{\partial^2\Phi^i}{\partial\eta\partial\xi}+\Gamma^i_{j\thickspace k}\frac{\partial\Phi^j}{\partial\xi}\frac{\partial\Phi^k}{\partial\eta}&=\frac{\partial U^i}{\partial \eta}+\Gamma^i_{j\thickspace k}U^j\frac{\partial\Phi^k}{\partial\eta}\\
&=\frac{\partial\Phi^\ell}{\partial\eta}\frac{\partial}{\partial\Phi^\ell} U^i+\frac{\partial\Phi^k}{\partial\eta}U^j\Gamma^i_{j\thickspace k}\\
&=\frac{\partial\Phi^\ell}{\partial\eta}\left(e_\ell[U^i]+U^j\Gamma^i_{j\thickspace\ell}\right)\\
&=W^\ell\left(e_\ell[U^i]+U^j\Gamma^i_{j,\ell}\right)\end{aligned}$$ Since the equality holds for each component we have, $$\begin{aligned}
0&=W^\ell\left(e_\ell[U^i]e_i+U^j\Gamma^i_{j\thickspace\ell}e_i\right)\\
&=W^\ell\left(\nabla_{e_\ell}(U^i e_i)\right)\\
&=\left(\nabla_{W^\ell e_\ell}(U^i e_i)\right)=\nabla_W U\label{parallel-trans}\end{aligned}$$ Similarly, we obtain $\nabla_U W=0$ by the analogous series of steps (when $\xi$ and $\eta$ are swapped, since \[freePDE\] is symmetric under exchange of these). We thus arrive at the geometrical description of bubble collisions in the ultrarelativistic limit: the resulting field profiles post-collision are determined by the mutual parallel transport of each soliton’s tangent vector field along that of the other soliton. That is, the tangent vector fields of the soliton profiles are parallel transported along each other everywhere in $N$.
The remaining two curves that together with $\Phi^i(\xi,-\infty)$, and $\Phi^i(\infty,\eta)$ form the boundary of $N$ are simply $\Phi^i(\xi,\infty)$ and $\Phi^i(-\infty,\eta)$. The first of these, $\Phi^i(\xi,\infty)$, goes between $C^i$ when $\xi=\infty$, and the point $\Phi^i(-\infty,\infty)\equiv D^i$. The second, $\Phi^i(-\infty,\eta)$, has endpoint $B^i$ when $\eta=\infty$, and the other at $D^i$ as well, when $\eta=-\infty$. This is shown schematically in Figure \[submani-Fig\].
![The soliton collision initial value problem, expressed in the Lorentz variables takes the form \[freePDE\], with \[ic1\], \[ic2\] when the limit that the impact velocity goes to $1$ is taken. The solution to this limiting set of equations, denoted by $\Phi^i$ maps $\mathbb{R}^2$ to a submanifold, $N$, of the field space $M$. Since the map is smooth the image of the $\xi$ coordinate lines and the $\eta$ coordinate lines are the integral curves of two vector fields. The partial differential equation \[freePDE\] indicates these two vector fields are parallel transported along one another everywhere in $N$. The initial conditions require that $\Phi(\xi,\eta)$ go to $f(\xi)$ as $\eta\rightarrow -\infty$, and go to $h(-\eta)$ as $\xi\rightarrow 1$. This is shown in the cartoon/schematic illustration above, where $\mathbb{R}^2$ is drawn as a (finite) square. The purple horizontal line in the lower half of the $\xi$-$\eta$ plane is mapped to the purple curve with endpoints at $B$ and $A$ in the $\{\phi^i\}$ coordinate plane($f$ soliton), and the green vertical line in the right half plane is mapped to the green curve with endpoints $C$ and $A$ ($h$ soliton). The remaining two curves that form the rest of the boundary of $N$ are shown in blue and pink. They are the images of the $\xi$ coordinate line at $\eta\rightarrow \infty$ and $\eta$ coordinate line at $\xi\rightarrow- \infty$, so are in a sense the curves obtained by completing the transport of $h$ along $f$ and $f$ along $h$. At sufficiently high impact velocity the field in the collision region takes on value $D$, and the outgoing walls interpolate between $D$ and the original bubble vacua, $B$ to the right and $C$ to the left. For such a collision the parametric plot of the two walls differs negligibly from the prediction via parallel transport– the blue and pink curves.[]{data-label="submani-Fig"}](PHImap_square_to_N_unrotated.png){width="\textwidth"}
This result agrees with heuristic expectations motivated by the flat field space limit. Namely, note that in the flat limit, a right moving soliton that interpolates between the parent vacuum $A$ and a local minimum $B$ leaves in its wake (to the left of the soliton’s transition wall) a field value shifted by $\Delta_L$ = $B - A$, while a left moving soliton that interpolates between the parent vacuum $A$ and a local minimum $C$ leaves in its wake (to the right of the soliton’s transition wall) a field value shifted by $\Delta_R$ = $C - A$. Thus, after free-passage collision, the collision region – which, by definition is to the left of the right-mover and to the right of the left-mover – is shifted by $\Delta_L + \Delta_R$ (which equals $B + C - 2 A$). In the case of curved field space, we divide the field shifts, both $\Delta_L$ and $\Delta_R$, into infinitesimals, which geometrically are the tangent vector fields of the soliton field profiles. Each such infinitesimal leaves in its immediate wake a field whose value is parallel transported along the infinitesimal shift vector, thus resulting in the geometrical picture we’ve described. When all tangent vectors of nontrivial magnitude have been mutually transported, they leave a widening interior of field in $\phi=D$.
This type of reasoning indicates that \[parallel-trans\] is the simplest partial differential equation that reduces to free passage in the flat field space limit. Namely, in the flat limit, the infinitesimal description of free passage is clearly the requirement that the vanishing of the directional derivatives of $U$ with respect to $W$ and $W$ with respect to $U$. The covariant version of these statements is just \[parallel-trans\]. This heuristic argument is suggestive but not sufficient since it is insensitive to any terms in the limiting form of the partial differential equation that vanish in the flat field space limit but which could nonetheless be present in the curved case. The analysis we’ve performed so far, together with that in the following section verifies that there are no such terms.
It is worth confirming that our free passage field configuration, \[FP-definition\], does indeed have the qualitative features outlined above. First note that the configuration correctly approaches the $B$ vacuum asymptotically to the left, and the $C$ vacuum to the right for any finite time $t$, since this amounts to evaluating $\Phi$ at $(\gamma(x-ut),\gamma(x+ut))\rightarrow(-\infty,-\infty)$, and $(\infty,\infty)$, respectively. At a fixed time *before* collision, any time $t\lesssim -w/2\gamma u$, the free passage field differs from $B$ vacuum by an insignificant amount at $x<ut-w/2\gamma$ since we’d effectively be evaluating $\Phi^i$ in \[FP-definition\] at $(-\infty,-\infty)$. As we march rightward the first argument increases, and reaches zero at $x=ut$ while second argument remains essentially unchanged. $\Phi(0,-\infty)$ is simply the center of the $f$ soliton. So, as one moves between the positions $-ut-w/2\gamma u$, and $-ut+w/2\gamma u$ in free passage field configuration they run through the $f$ soliton’s field configuration. If they continue moving rightward they’ll reach a stretch of $x$ values where both the arguments of $\Phi^i$ in \[FP-definition\] are effectively negative infinity, and so the $A$ vacuum is measured.
If we continue on rightward the analogous procedure leads us to realize that the free passage field configuration interpolates between the $A$ and $C$ vacua by the (reflected) $h$ soliton, centered at $-ut$. So, pre-collision the spatial profile of the free passage field configuration looks like the usual linear superposition: a nearly homogeneous interior of diminishing size in the parent vacuum, separated from the bubble vacua by the relevant solitons, whose centers lie at $ut$, and $-ut$.
The same line of reasoning can be used to deduce that post-collision, any time $t\gtrsim w/2\gamma u$, the free passage configuration again consists of three approximately homogeneous regions: a widening interior, or “collision region" with field value $\thickapprox D^i$, separated from regions of original bubble vacua on either side, by walls whose centers follow the same trajectories $x=\pm ut$. The shapes however of the spatial profiles of the field components across the walls are *not* in general the same as those of the incoming solitons. A parametric plot of the free passage configuration at a given time (with the spatial variable as the parameter) in the $\{\phi^i\}$ coordinate plane would consist of the composition of a curve that interpolates between $B$ and $D$, together with the one between $C$ and $D$. These would be nearly identical those obtained by completing parallel transport, and approaches the union of these two curves, $\Phi^i(-\infty,\infty)$, and $\Phi^i(\infty,\infty)$ asymptotically as $t\rightarrow\infty$.
We claim that there always exists an impact velocity sufficiently close to the speed of light such that the actual solution to \[collisionIVP\] is well approximated by the above free passage evolution throughout the entirety of the collision– i.e. for longer than the amount of time it would take for the incoming Lorentz contracted walls to fully pass through each other. We prove this in the following section.
Proof of Realization of Free Passage
====================================
The solution to the parallel transport problem, $\Phi^i$, and the free passage evolution function defined from it, $\phi^i_{FP}$, is, of course, only useful in predicting the outcome of a particular collision if deviations from $\phi^i_{FP}$ remain sufficiently small throughout the entirety of the collision (or longer). As we’ve mentioned previously, the amount of time it takes the solitons to fully pass through each other is $w/\gamma u$, and since we’ve chosen to set our $t=0$ at the middle of the collision we’re interested in the time period, $t\in[-w/2\gamma u, w/2\gamma u]$.
We begin by expanding the actual solution (to \[collisionIVP\]) about the free passage configuration, $$\phi^i(t,x;u)=\phi^i_{FP}(t,x;u)+\psi^i(t,x;u)
\label{expansion}$$ Simply substituting \[expansion\] into the equation of motion and expanding in powers of $\psi$ yields, $$\begin{aligned}
\Box\psi^i&=-\frac{\partial V}{\partial\phi_i}\bigg|_{\phi_{FP}}\negthickspace\negthickspace\negthickspace\negthickspace-\frac{\partial^2 V}{\partial\phi^\ell\partial\phi_i}\bigg|_{\phi_{FP}} \negthickspace\negthickspace \psi^\ell +\mathcal{O}(\psi^2)-\left[\Box\phi^i_{FP}+\Gamma^i_{j k}|_{\phi_{FP}} \partial_\mu\phi^j_{FP} \partial^\mu\phi^k_{FP} \right]\\
&-2\Gamma^i_{j k}|_{\phi_{FP}}\partial_\mu\phi_{FP}^j \partial^\mu\psi^k-\Gamma^i_{j k,\ell}|_{\phi_{FP}}\psi^\ell \partial_\mu\phi^j_{FP} \partial^\mu\phi^k_{FP}+\mathcal{O}(\psi(\partial\psi))+\mathcal{O}((\partial\psi)^2)\\
&=-\frac{\partial V}{\partial\phi_i}\bigg|_{\phi_{FP}}\negthickspace\negthickspace\negthickspace\negthickspace-\left[\Box\phi^i_{FP}+\Gamma^i_{j k}|_{\phi_{FP}} \partial_\mu\phi^j_{FP} \partial^\mu\phi^k_{FP} \right]+\mathcal{O}(\psi)
\label{GreenFN}\end{aligned}$$ We can write an implicit expression for $\psi^i(t,x)$ by integrating the right hand side of \[GreenFN\] as follows, $$\psi^i(t,x)=\int_{-w/2\gamma u}^{t}dt'\int_{x-t'}^{x+t'}dx'G^i(t',x')
\label{psiInt}$$ We now truncate at zeroth order in $\psi$, and bound the above integral. The term in the square brackets in \[GreenFN\] is,
$$\begin{aligned}
-\left[\Box\phi^i_{FP}+\Gamma^i_{j k}|_{\phi_{FP}} \partial_\mu \phi^j_{FP} \partial^\mu\phi^k_{FP} \right]&=
4(1-\epsilon)\gamma^2\left[\frac{\partial^2 \Phi^i}{\partial\sigma\partial\omega}+\Gamma^i_{j\thickspace k}\frac{\partial\Phi^j}{\partial\sigma}\frac{\partial \Phi^k}{\partial\omega} \right]\label{mismatch}\\
&-2\gamma^2\epsilon\left[\frac{\partial^2\Phi^i}{\partial\sigma^2}+\frac{\partial^2\Phi^i}{\partial\omega^2}+\Gamma^i_{j\thickspace k} \left(\frac{\partial\Phi^j}{\partial\sigma}\frac{\partial\Phi^k}{\partial\sigma}+ \frac{\partial\Phi^j}{\partial\omega}\frac{\partial\Phi^k}{\partial\omega}\right)\right]\nonumber\end{aligned}$$
where we’ve expressed the operators $\Box$ and $\partial_\mu$ in terms of the the Lorentz variables, and retained terms up to first order in $\epsilon$. This is identical to the step we took at the outset to obtain \[LorentzVarsEOM\]. Note that the first term in square brackets on the righthand side of $\ref{mismatch}$ is, by definition, zero. The second term, however, does not vanish. It results from the mismatch between the Lorentz variables and the characteristics. The nonvanishing piece can be expressed in terms of the vector fields $U$ and $W$ as follows, $$\begin{aligned}
-2\gamma^2\epsilon\left[\frac{\partial^2\Phi^i}{\partial\sigma^2}+\Gamma^i_{j\thickspace k} \frac{\partial\Phi^j}{\partial\sigma}\frac{\partial\Phi^k}{\partial\sigma}+\frac{\partial^2\Phi^i}{\partial\omega^2} +\Gamma^i_{j\thickspace k} \frac{\partial\Phi^j}{\partial\omega}\frac{\partial\Phi^k}{\partial\omega}\right]=-2\gamma^2\epsilon\left[\left(\nabla_U U\right)^i+\left(\nabla_W W\right)^i\right]\end{aligned}$$
A bound on the magnitude of $\psi^i$ can now be computed straightforwardly, $$\begin{aligned}
|\psi^i(t,x)|&\leq \int_{-w/2\gamma}^{t}dt'\int_{x-t'}^{x+t'}dx' |G^i(t',x')|\\
&\leq \int_{-w/2\gamma}^{t}dt'\int_{x-t'}^{x+t'}dx' \bigg|\frac{\partial V}{\partial \phi_i}\bigg|+2\gamma^2\epsilon\left|\left(\nabla_U U\right)^i\right|+2\gamma^2\epsilon\left|\left(\nabla_W W\right)^i\right|\end{aligned}$$ where the terms involving the vector fields are evaluated at $(\gamma(x'-ut'),\gamma(x'+ut'))$. Now, we’re only interested in the deviation at points $x$ in the collision region (outside of here the field persists very nearly equal to the bubble vacuum field values), and times $t\in[-w/2\gamma u,+w/2\gamma u]$. For this time period the collision region is always contained within $[-w/\gamma,+w/\gamma]$. This means the $x'$ interval we need to integrate over in our expression for $\psi^i(t,x)$ is always contained within $[-3w/2\gamma,+3w/2\gamma]$. So we can write, $$\begin{aligned}
|\psi^i&(t,x)|\leq \int_{-w/2\gamma}^{t}dt'\int_{-3w/2\gamma}^{3w/2\gamma}dx'|G^i(t',x')|\\
&\leq\left\{ \sup_{\phi\in N}\left(\bigg|\frac{\partial V}{\partial \phi_i}\bigg|\right)+2\gamma^2\epsilon \negthickspace\negthickspace \sup_{(\sigma,\omega)\in\mathbb{R}^2}\negthickspace \left(\left|\left(\nabla_U U(\sigma,\omega)\right)^i\right|+\left|\left(\nabla_W W(\sigma,\omega)\right)^i\right|\right)\right\} \negthickspace \int_{-w/2\gamma u}^{t}\negthickspace\negthickspace\negthickspace\negthickspace dt'\int_{-3w/2\gamma}^{3 w/2\gamma}\negthickspace\negthickspace\negthickspace\negthickspace dx'\\
&=(k_1^i+2\gamma^2\epsilon k_2^i)\frac{w(t/T+1/2)}{\gamma u}\frac{3 w}{\gamma}=\left(\frac{k_1^i}{\gamma^2}+2\epsilon k_2^i\right)3w^2=\left(2\epsilon k_1^i+2\epsilon k_2^i\right)3w^2\end{aligned}$$ So, $$|\psi^i(t,x)|\leq k^i\epsilon$$ where the positive constants, $$\begin{aligned}
k^i\equiv 6w^2\left\{ \sup_{\phi\in N}\left(\bigg|\frac{\partial V}{\partial \phi_i}\bigg|\right)+2\gamma^2\epsilon \negthickspace\negthickspace \sup_{(\sigma,\omega)\in\mathbb{R}^2}\negthickspace \left(\left|\left(\nabla_U U(\sigma,\omega)\right)^i\right|+\left|\left(\nabla_W W(\sigma,\omega)\right)^i\right|\right)\right\},\end{aligned}$$ are finite due to the smoothness of the potential and the field space manifold.
Since the difference in the coordinates of the true field configuration and free passage configuration can be made arbitrarily small, we conclude that the post collision field (for any two solitons in any curved multi-scalar field theory) successfully realizes the late-time free passage field configuration, provided the impact velocity was sufficiently relativistic. The threshold above which the impact velocity ought to be is dependent on both the model and the choice of the two colliding solitons. This threshold can be estimated by requiring that the distance in field space between the free passage field (say for the center of the collision region) at time $t$, and the free passage plus deviation location be much much smaller than the length of the path the observer at the center of the collision region has through field space from the parent vacuum until time $t$. Since the walls of bubbles nucleated via Coleman-De Luccia tunneling accelerate as they move outwards, we expect our parallel transport procedure to be a useful means of predicting the field configuration following the collision of two bubbles, provided they were nucleated sufficiently far apart (and they’re radii upon nucleation is sufficiently small compared to the separation distance such that high enough impact velocity is reached upon collision).
Numerical Simulations
=====================
We simulated soliton collisions at a variety of impact velocities in three different models with actions of the form \[Action\]. Each model featured a different two dimensional curved field space. We reiterate that the field space is curved in the sense that the matrix of $\{\phi^i\}$ dependent functions, $g^{ij}$, in the noncanonical kinetic term in the Lagrangian is the coordinate representation of the metric on a curved manifold (clearly the field components $\{\phi^i\}$ are identified as coordinates on the manifold). The particular manifolds we considered were the sphere, the ring torus, and the “teardrop"– our own creation named for obvious reasons.
For both the sphere and teardrop we used the polar angle and azimuthal angle as our two coordinates. For the torus we used the angles about the major axis, and the minor axis. To minimize the possibility for confusion we adopt standard naming conventions used for these coordinate systems, and refer to the field components ($\phi^1,\phi^2$) as ($\theta, \phi$) for the sphere and teardrop, and as ($u, v$) for the torus. For the explicit form of the metric components, as well as vacuum locations refer to Table, \[Models-Table\].
---------- ---------------------------------------------------------------------------------------- ---------------------------------------------- -------------------------------------------- ------------------------------------------- -------------
Geometry [$g^{ij}$]{} Free Passage
A B C D
Sphere $g_{\theta\theta}=1 $ $(\frac{\pi}{12}, \frac{2\pi}{15} )$ $(\frac{5\pi}{12},\frac{2\pi}{11})$ $(\frac{3\pi}{12},\frac{6\pi}{13} )$ (1.77,1.08)
$g_{\phi\phi}= \sin^2(\theta)$
Teardrop $g_{\theta\theta}= \cos^2(\theta)+\left[\frac{\sin(\theta)}{2}+(\theta-\pi) \right]^2$ $ (\frac{7 \pi}{60} ,\frac{\pi}{13})$ $(\frac{118 \pi}{327},-\frac{\pi}{13})$ $(\frac{73\pi}{327},\frac{35\pi}{109})$ (1.52,0.33)
$g_{\phi\phi}= \sin^2(\theta)$
Torus $g_{uu}= (1+.7\cos(v))^2$ $ (\frac{145 \pi}{654} ,\frac{-20\pi}{109})$ $(-\frac{145 \pi}{654},\frac{20\pi}{109})$ $(-\frac{35\pi}{654},\frac{235\pi}{654})$ (1.36,0.09)
$g_{vv}= .7^2$
---------- ---------------------------------------------------------------------------------------- ---------------------------------------------- -------------------------------------------- ------------------------------------------- -------------
: Metric Components and Vacuum Locations
\[Models-Table\]
We numerically approximated the solutions to the initial value problem \[collisionIVP\] associated with the collision of two non-identical solitons in the given theory, as well as solutions to the mutual parallel transport of two tangent vector fields problem, \[freePDE\]-\[ic2\], using Mathematica’s finite difference partial differential equation solver, NDSolve. The potential was engineered to have three degenerate vacua with generic looking wells and barriers by using the product of two trigonometric functions, and then isolating only three minima by multiplying by a superposition of hyperbolic tangents which served as smooth approximations to characteristic functions and hat functions. For the explicit form of the three potentials see Table \[potentials-Table\].
We wanted the potential to be flat outside the neighborhood immediately surrounding the three vacua so as to minimize the influence of the potential on field dynamics, both throughout the collision and after, so that the free passage behavior could feasibly be extracted. Though we absolutely assert that parallel transport is generic (there always exists a speed high enough such that it is fully realized), we wanted to design a nontrivial scenario where the boost needed was small enough, and so the grid size large enough, that we’d have a hope of resolving this in Mathematica, and on a desktop computer.
![Plot of the potential for the teardrop model. Note the cylindrical well carved out of the plateau at $D$. This addition does not change the solitons $f$, and $h$.[]{data-label="teardrop-potential"}](teardrop_V_fig.png){width="110.00000%"}
There is a final step to designing a potential that enables us to extract the free passage dynamics– the placement of a fourth degenerate vacuum at the free passage location, $D$, which of course is not known a priori. Had the potential been left as a plateau outside the three vacua the post collision field dynamics would be tainted by the pressure gradient across outgoing walls resulting from the energy density in the collision region differing from that in the surrounding region (which is still simply that of the degenerate bubble vacua). In order to prolong the amount of time after which free passage would remain a good approximation, without unduly biasing the field toward the free passage field location we carved a cylindrical well out of the plateau at the free passage field location, for each model. Clearly then, the parallel transport solution for each geometry was obtained before any collisions were simulated, so that each of their potentials could be modified in the manner described.
Note that the parallel transport solution, $\Phi$, is by definition independent of the potential provided that the soliton curves between the parent and two bubble vacua remain unchanged (since these are the boundary conditions in the parallel transport problem). Clearly the potential in the neighborhood of the three original vacua is unaffected by the addition of the narrow cylindrical well placed out on the plateau away from the original three vacua. A plot of the potential in the teardrop model is included as an example in Figure \[teardrop-potential\], and the explicit form of the potentials used for all three geometries can be found in Table \[potentials-Table\]. [^4]
Model Potential
---------- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Sphere $V(\theta,\phi)=V_0\sin(6 \theta)\sin\left(\negthickspace \frac{(3 \theta - 4\pi) \phi}{\pi}\right)\negthickspace *\negthickspace \chi_{sphere}(\theta,\phi)-\negthickspace \frac{V_0}{2}\tanh(40( (\theta-D_\theta)^2\negthickspace+\negthickspace (\phi-\negthickspace D_\phi \negthickspace -\negthickspace .3)^2))\negthickspace +2.5)$
$\qquad+\frac{V_0}{2}\tanh(40 ((\theta-D_\theta)^2+(\phi-D_\phi-.3)^2)-2.5)$
Teardrop $V(\theta,\phi)=V_0\sin(4\phi-1.2\theta+.3\pi)\sin(6\theta-1.5\pi+1.8 \phi)*\chi_{tear}(\theta,\phi)$
$\qquad-\frac{V_0}{2}\left[\tanh( 2.7- 60 (((\theta - D_\theta)^2 + (\phi - D_{\phi})^2))) +1\right]$
Torus $V(u,v)=V_0\sin(2 v - .6 u) \sin(3 u + .9v)*\chi_{torus}(u,v)$
$\qquad-\frac{V_0}{2}\left[\tanh( 3- 30((u -D_u)^2 + (v - D_v)^2))) +1\right]$
: Below is the explicit form of the potentials we used in our simulations of solitons collisions for each of the field space geometries we considered.
\[potentials-Table\]
Lastly it is necessary to discuss how the initial conditions and boundary conditions were formulated. Both are defined in terms of the components of the two solitons we are colliding. Solitons are, by definition, static solutions to the equations of motion \[curved-EOM\] that approach two distinct (obviously degenerate) minima of the potential asymptotically. In a multi-scalar field theory solitons are unique to the vacua they interpolate between, and furthermore are the minimum energy field configurations that satisfy the given pair of boundary conditions. Since the coupled ordinary differential equations that define the solitons are nonlinear, analytic solutions generally cannot be found. However, if an initial profile that satisfies the boundary conditions is evolved in time by the equations of motion plus a damping term, the profile ultimately settles down to the soliton, provided the initial guess was sufficiently close to the true soliton and the damping coefficient was not too large. We performed this relaxation procedure numerically, once again with NDSolve in Mathematica.[^5]
Note that analytic expressions for the four soliton components, $f^i$, and $h^i$, were needed in order for simulations of the collision to be feasible. At such small grid spacing time evolving initial conditions constructed out of the interpolating functions relaxation yielded was not possible. So the final step was to engineer analytic expressions that approximated each of the soliton components from relaxation (four total, $f^1$, $f^2$, $h^1$, $h^2$). All were modifications of (scaled and shifted) hyperbolic tangents, typically with the addition of small gaussians and nonlinear terms in the argument of the hyperbolic tangent.
The free passage field configuration was indeed fully realized in all three models at sufficiently large impact velocity. Snapshots of the spatial profile of each field components during such collisions can be found in Figures \[snapshots-tear\]. Note that each field component’s collision region is homogenous, with the precise value predicted by the parallel transport solution, indicated by the contrasting dashed line. Furthermore, the shapes of the outgoing soliton profiles matched the prediction as well. Figures \[sphere-results\], \[tear-results\], and \[torus-results\].
[![Here we show snapshots of each field component of the configuration during a collision simulated in the teardrop model at impact velocity $u=.995$ ($\theta$ component is on the top row, and $\phi$ on the bottom row). The prediction of each component’s value inside the collision region obtained by parallel transport, $D_i$ are indicated by the dashed teal and purple horizontal lines for $\theta$, and $\phi$, respectively. Note both the homogeneity of the field in the collision region, and its extraordinarily strong agreement with the free passage prediction.[]{data-label="snapshots-tear"}](snapshot1_tear_theta.png "fig:"){width=".35\textwidth"}![Here we show snapshots of each field component of the configuration during a collision simulated in the teardrop model at impact velocity $u=.995$ ($\theta$ component is on the top row, and $\phi$ on the bottom row). The prediction of each component’s value inside the collision region obtained by parallel transport, $D_i$ are indicated by the dashed teal and purple horizontal lines for $\theta$, and $\phi$, respectively. Note both the homogeneity of the field in the collision region, and its extraordinarily strong agreement with the free passage prediction.[]{data-label="snapshots-tear"}](snapshot3_tear_theta.png "fig:"){width=".35\textwidth"}![Here we show snapshots of each field component of the configuration during a collision simulated in the teardrop model at impact velocity $u=.995$ ($\theta$ component is on the top row, and $\phi$ on the bottom row). The prediction of each component’s value inside the collision region obtained by parallel transport, $D_i$ are indicated by the dashed teal and purple horizontal lines for $\theta$, and $\phi$, respectively. Note both the homogeneity of the field in the collision region, and its extraordinarily strong agreement with the free passage prediction.[]{data-label="snapshots-tear"}](snapshot4_tear_theta.png "fig:"){width=".35\textwidth"}]{}\
[![Here we show snapshots of each field component of the configuration during a collision simulated in the teardrop model at impact velocity $u=.995$ ($\theta$ component is on the top row, and $\phi$ on the bottom row). The prediction of each component’s value inside the collision region obtained by parallel transport, $D_i$ are indicated by the dashed teal and purple horizontal lines for $\theta$, and $\phi$, respectively. Note both the homogeneity of the field in the collision region, and its extraordinarily strong agreement with the free passage prediction.[]{data-label="snapshots-tear"}](snapshot1_tear_phi.png "fig:"){width=".35\textwidth"}![Here we show snapshots of each field component of the configuration during a collision simulated in the teardrop model at impact velocity $u=.995$ ($\theta$ component is on the top row, and $\phi$ on the bottom row). The prediction of each component’s value inside the collision region obtained by parallel transport, $D_i$ are indicated by the dashed teal and purple horizontal lines for $\theta$, and $\phi$, respectively. Note both the homogeneity of the field in the collision region, and its extraordinarily strong agreement with the free passage prediction.[]{data-label="snapshots-tear"}](snapshot3_tear_phi.png "fig:"){width=".35\textwidth"}![Here we show snapshots of each field component of the configuration during a collision simulated in the teardrop model at impact velocity $u=.995$ ($\theta$ component is on the top row, and $\phi$ on the bottom row). The prediction of each component’s value inside the collision region obtained by parallel transport, $D_i$ are indicated by the dashed teal and purple horizontal lines for $\theta$, and $\phi$, respectively. Note both the homogeneity of the field in the collision region, and its extraordinarily strong agreement with the free passage prediction.[]{data-label="snapshots-tear"}](snapshot4_tear_phi.png "fig:"){width=".35\textwidth"}]{}
[![Pictured here is a comparison the results of a collision at impact velocity $u=.995$ in the sphere model with the prediction from parallel transport. On the left we plot the field configuration at various times throughout the collision parametrically in the $\{\phi^i\}$ coordinate plane (by treating the spatial variable as the parameter) with solid purple curves. We identify the field at the origin, $x=0$, throughout the collision with the dot-dashed dark purple line. This is the path taken through field space over the course of the collision by an observer at the center of the collision’s rest frame. The solution to the parallel transport problem is shown with dashed lines. Those in pink are lines of constant $\eta$, those in orange are lines of constant $\xi$, i.e. the integral curves of the vector fields $U$ and $W$. The curves that form the boundary of the submanifold are drawn brighter and are overlaid so that they can easily be compared to the results of the collision. On the right these results are plotted on the field space manifold embedded in three space (the $\{\phi^i\}$ coordinate lines are shown in light green). The field configuration in the collision problem is again shown in solid purple for a variety of times. The post collision prediction made by parallel transport (that is, the integral curves obtained by completing the parallel transport procedure which yields the remaining two curves that form the boundary of $N$) are shown in dashed orange. The path taken through field space by an observer at the origin is shown in dashed pink. The fact that the boundary of $N$ lines up nearly perfectly with the parametric plots of the initial and final field configuration in the collision problem indicates that there is extraordinarily good agreement between the prediction, computed via parallel transport, and the actual outcome of the collision.[]{data-label="sphere-results"}](sphere2D.png "fig:"){width=".5\textwidth"}![Pictured here is a comparison the results of a collision at impact velocity $u=.995$ in the sphere model with the prediction from parallel transport. On the left we plot the field configuration at various times throughout the collision parametrically in the $\{\phi^i\}$ coordinate plane (by treating the spatial variable as the parameter) with solid purple curves. We identify the field at the origin, $x=0$, throughout the collision with the dot-dashed dark purple line. This is the path taken through field space over the course of the collision by an observer at the center of the collision’s rest frame. The solution to the parallel transport problem is shown with dashed lines. Those in pink are lines of constant $\eta$, those in orange are lines of constant $\xi$, i.e. the integral curves of the vector fields $U$ and $W$. The curves that form the boundary of the submanifold are drawn brighter and are overlaid so that they can easily be compared to the results of the collision. On the right these results are plotted on the field space manifold embedded in three space (the $\{\phi^i\}$ coordinate lines are shown in light green). The field configuration in the collision problem is again shown in solid purple for a variety of times. The post collision prediction made by parallel transport (that is, the integral curves obtained by completing the parallel transport procedure which yields the remaining two curves that form the boundary of $N$) are shown in dashed orange. The path taken through field space by an observer at the origin is shown in dashed pink. The fact that the boundary of $N$ lines up nearly perfectly with the parametric plots of the initial and final field configuration in the collision problem indicates that there is extraordinarily good agreement between the prediction, computed via parallel transport, and the actual outcome of the collision.[]{data-label="sphere-results"}](Sphere3D.png "fig:"){width=".55\textwidth"}]{}
[![Pictured here is a comparison the results of a collision at impact velocity $u=.995$ in the teardrop model with the prediction from parallel transport. The same coloring scheme is the same as that in Figure \[sphere-results\].[]{data-label="tear-results"}](tear2D.png "fig:"){width=".5\textwidth"}![Pictured here is a comparison the results of a collision at impact velocity $u=.995$ in the teardrop model with the prediction from parallel transport. The same coloring scheme is the same as that in Figure \[sphere-results\].[]{data-label="tear-results"}](tear3D.png "fig:"){width=".55\textwidth"}]{}
[![Pictured here is a comparison the results of a collision at impact velocity $u=$ in the teardrop model with the prediction from parallel transport. The same coloring scheme is the same as that in Figure \[sphere-results\].[]{data-label="torus-results"}](torus2D.png "fig:"){width=".5\textwidth"}![Pictured here is a comparison the results of a collision at impact velocity $u=$ in the teardrop model with the prediction from parallel transport. The same coloring scheme is the same as that in Figure \[sphere-results\].[]{data-label="torus-results"}](torus3D.png "fig:"){width=".55\textwidth"}]{}
Discussion
==========
In this paper we pushed the understanding of bubble universe collisions one step forward by considering the impact of working in the context of a curved field space. Far from an esoteric exercise, this situation arises in prominent contexts such as inflation on the string landscape, in which the relevant fields can be taken to be moduli on Calabi-Yau compactifications. The moduli fields generally span K[ä]{}hler manifolds with nontrivial curvature, and so the results of this paper would directly apply.
We have found a simple generalization of the free passage approximation developed in the flat space limit, which admits a satisfying geometrical interpretation. Namely, the free passage evolution is described in field space by a double family of field profiles that interpolate from the two initial solitons along a set of curves, each of whose tangent vectors is parallel transported along the tangent vector field of the other. We have analyzed the conditions under which this curved-field-space free-passage approximation holds and also illustrated its utility in a number of numerical examples.
A natural next step in this program is to include the effects of gravity on bubble collisions, an issue to which we intend to shortly return.
Acknowledgements {#acknowledgements .unnumbered}
================
We thank Andrew Brainerd, I-Sheng Yang, Eugene Lim and John T. Giblin for helpful discussions. This work was supported by the U.S. Department of Energy under grants DE-SC0011941 and DE-FG02-92ER40699.
[6]{} S. R. Coleman, Phys. Rev. D 15, 2929 (1997). C. G. Callan Jr., S.R. Coleman, Phys. Rev. D 16, 1762 (1977). S. R. Coleman, F. De Luccia, Phys. Rev. D 21, 3305 (1980). R. Easther, J. T. Giblin, L. Hui, E. Lim, Phys. Rev. D 80, 123519 (2009), arXiv:0907.3234 \[hep-th\]. J. T. Giblin, L. Hui, E. Lim, I. Yang (2010), arXiv:1005.3493 \[hep-th\]. L. Susskind, “The anthropic landscape of string theory", arXiv:hep-th/0302219. R. Bousso, J. Polchinski, “Quantization of four-form fluxes and dynamical neutralization of the cosmological constant", JHEP 0006:006 (2000), arXiv:hep-th/0004134. J. Garriga, A. H. Guth and A. Vilenkin, “Eternal inflation, bubble collisions, and the persistence of memory"� Phys. Rev. D 76, 123512 (2007), arXiv:hep-th/0612242. A. Aguirre, M. C. Johnson, and M. Tysanner, Phys. Rev. D 79, 123514 (2009). M. C. Johnson, H. V. Peiris, L. Lehner, Phys. Rev. D 85, 083516 (2012), arXiv:1112.4487 \[hep-th\]. B. Greene, D. Kagan, A. Masoumi, D. Mehta, E. J. Weinberg, and X. Xiao, Phys. Rev. D 88, 026005 (2013), arXiv:1303.4428 \[hep-th\]. S. Sarangi, G. Shiu, and B. Shlaer, Int. J. Mod. Phys. A 24, 741 (2009), arXiv:0708.4375 \[hep-th\]. S. -H. H. Tye, arXiv:0708.4374 \[hep-th\]. A. R. Brown, S. Sarangi, B. Shlaer, and A. Weltman, Phys. Rev. Lett. 99, 161601 (2007), arXiv:0706.0485 \[hep-th\]. A. R. Brown and A. Dahlen, Phys. Rev. D 82, 083519 (2010), arXiv:1004.3994 \[hep-th\]. P. Ahlqvist, K. Eckerle, B. Greene, (2013) arXiv:1310.6069 \[hep-th\].
[^1]: A notable exception is the paper [@Johnson]
[^2]: If one is uncomfortable with this argument for the splitting of the boundary where the initial data is given into two independent pieces, a conformal map can be performed before limit that $\gamma$ goes to infinity is taken. Under a conformal transformation from $\sigma$, and $\omega$ to $\alpha=\tan^{-1}(\sigma)$ and $\beta=\tan^{-1}(\omega)$ the boundary $\partial\Omega_\gamma$ becomes a hyperbola in the $\alpha-\beta$ domain (which is the square $[-\pi/2,\pi/2]\times[-\pi/2,\pi/2]$). As $\gamma$ is increased the hyperbola is pushed further and further into the lower right of the square, and ultimately becomes the union of the horizontal edge at $\beta=-\pi/2$ ($\sigma$ varying edge), and the vertical at $\sigma=\pi/2$ ($\omega$ varying edge). Though it provides a perhaps a more visually satisfying argument in favor of the split, we do not view the conformal map as necessary.
[^3]: Technically $\Phi^i$ should be viewed as a map from the square $[-s/2,s/2]^2$ in the limit that $s\rightarrow \infty$ boundary conditions given on the edges of the square defined by $(\xi,\eta)=(\xi,-s/2)$ for $\xi\in[-s/2,s/2]$, and $(\xi,\eta)=(s/2,\eta)$ for $\eta\in[-s/2,s/2]$.
[^4]: where, $$\begin{aligned}
&\chi_{sphere}(\theta,\phi)= (1 - \tanh((\theta - .25\phi - 5\pi/12 - .2) 30)))(1 - \tanh((\phi + 1.5 \theta - 3.2) 30))/4\\
& \chi_{tear}(\theta,\phi)= \frac{1}{2^6} (1 + \tanh[40 (-0.05 + (-1.56 + u)^2 + (0.06 + v)^2)]) (1 +
\tanh[20 (-0.5 + (-2.42 + 2 u)^2 + (-2.3 + 2 v)^2)]) \\
& \qquad *(1 + \tanh[20 (-0.75 + (-0.07 + 2 u)^2 + (-2 + 2 v)^2)]) (1 + \tanh[10 (-3.2 + (-3.87 + 2 u)^2 + 1/2 (-1.9 + 2 v)^2)]) \\
&\qquad*(1 + \tanh[20 (-1 + (-2.27 + 2 u)^2 + (-1 + 2 v)^2)]) (1 + \tanh[10 (2.1 - (-1.77 + 2 u)^2 - 1/2 (-0.8 + 2 v)^2)])\\
&\chi_{torus}(u,v)= \frac{1}{2^6}(\tanh(10 (2.1 - (v - .8)^2/2 - (u - .2)^2)) + 1)* (1 - \tanh(30 ((.15 v + u) - .9)))\\
&\qquad*(\tanh(10 ((v - 1.9)^2/2 + (u - 2.3)^2 - 3.2)) + 1)* (\tanh(20 ((v - 1)^2 + (u - .7)^2 - 1.2)) + 1)\\
&\qquad* (\tanh(20 ((v - 2.3)^2 + (u - .85)^2 - .5)) + 1) *(\tanh(20 ((v - 2)^2 + (u + 1.5)^2 - .75)) + 1) \end{aligned}$$
[^5]: It is important to mention how the initial guesses for the soliton components in relaxation were chosen. Since the (true) soliton is defined by both the geometry and the potential, we sought to allow both to play a role in our guesses. For a given pair of vacua we first parameterized the geodesic connecting them by writing one field component in terms of the other (for instance, in the case of the sphere the geodesics were great circles and the polar angle was parameterized in terms of the azimuthal). The potential was then evaluated along the geodesic, and the resulting function was approximated as a double well potential, which has a single free parameter after the distance between the minima is fixed. This parameter was tuned such that the approximate potential not only qualitatively resembled the true one along the geodesic, but also so that their integrals of the inverse square root of the difference between the vacuum value, $-V_0$, and the potential, between the minima were nearly identical for the. For example, for the sphere initial guess we’d compute $$\int_{\phi_A}^{\phi^B}d\phi/\sqrt{V_{sphere}(c_{geo,{AB}}(\phi),\phi)+V_0}$$ numerically and tune the double well potential’s curvature parameter until its integral matched this. The double well approximation then provides us with an initial guess, a (scaled and shifted) hyperbolic tangent, for one of the two soliton components– that which the geodesic is parameterized in terms of. To obtain a guess for the remaining component the expression for the geodesic was simply evaluated at the guess function that was just obtained for the former component– resulting in a spatial profile.
|
\#1
LIGHT QUARK MASSES AND MIXING ANGLES
JOHN F. DONOGHUE
Department of Physics and Astronomy, University of Massachusetts
Amherst MA 01002 USA
Introduction
============
The Standard Model is clearly one of the triumphs of modern science. However one of the less pleasant aspects of the theory is that it contains so many free parameters. Some of these parameters form the topic of these lectures, namely the masses $m_u , m_d$ and $m_s$ and the weak mixing elements $V_{ud}$ and $V_{us}$. Within the model, all are products of the Higgs sector. They seem to be almost arbitrary numbers, but perhaps they are clues as to the structure beyond the Standard Model. Perhaps someday we will learn to decode these clues.
There is also a second topic hidden below the surface in these lectures, i.e., how to make reliable calculations at low energy. We will see that $V_{ud}$ is known to 0.1%, $V_{us}$ to 1% and at least one mass ratio to 10%. For the physics of hadrons these accuracies are remarkably good. \[For example, $\alpha_s (M_z)$ is also only known to 10%\]. The key is the use of symmetries as a dynamical tool. In particular, we will be using chiral perturbation theory. While we do not have the time for a full pedagogical presentation of this \[1,2\], we will see what it is and how it is used.
My approach here will reserve the heavy formalism as long as possible. I will treat quark masses crudely at first in order to get a basic feel for them with a minimum of formalism. Following that is the description of $V_{ud}$. Before proceeding on to describe the extraction of $V_{us}$, I will spend some time introducing chiral perturbation theory. Finally I return to quark masses and try to be as precise as possible.
Quark Masses I
==============
Before turning to my real topic, we need to have a brief digression on ’constituent’ vs. ’Lagrangian’ or ’current’ masses. The Lagrangian of QCD
$${\cal L}_{QCD} = - {1 \over 4} F^A_{\mu \nu} F^{A \mu \nu} +
\bar{\psi} (i D - m) \psi$$
is a nonlinear field theory which contains small mass parameters $m_u, m_d , m_s$. Because these masses are small, the theory is almost chirally symmetric, as well as almost classically scale invariant. Masses also enter into the quark model of hadron structure, with
$${\cal H}_{QM} = \sum_i {p^2_i \over 2M_i} + V(r_1 - r_2)$$
Given that this is also supposed to represent the strong interactions, it is remarkable how far this is from QCD. The mass parameters are large, $M \sim m_p /3$, and there is no trace of the symmetries of QCD. The large mass of the quark model has very little relation to the mass in the Lagrangian. The former is commonly referred to as a ’constituent’ mass. Our topic here concerns only the mass parameters in the Lagrangian. In many ways these are defined by the symmetry properties and they are called ’current’ (i.e., from divergences of Noether currents) or ’Lagrangian’ masses.
Our first task is to learn to treat quark masses in the same way that we do coupling constants. Our mass parameters are not inertial masses of hadrons, and because of confinement one cannot find any poles in quark propagators. How then can we come up with a way to actually measure masses? The procedure is the same as with coupling constants. Observables depend on the masses, i.e.,
$$\begin{aligned}
M & = & M(m) \nonumber\\
& = & M_0 + am + bm^2 + \dots\end{aligned}$$
We measure the quark mass m by its effect on observables. But we have a problem; we cannot reliably calculate observables at low energy, and so it is tough to learn how masses influence the observables. It is here that symmetry comes to the rescue. There will be exact relations between observables in the symmetry limit. Quark masses break the symmetry and disturb these relations. That means that the deviations from the symmetry predictions are measures of quark mass. In the most basic of examples we will see that the pion and kaon masses start off as
$$\begin{aligned}
m^2_{\pi} & = & 0 + (m_u + m_d) B_0 + \dots \nonumber\\
m^2_{K^+} & = & 0 + (m_u + m_s) B_0 + \dots\end{aligned}$$
where $B_0$ is same constant. This lets us measure the ratio
$${m_u + m_d \over m_u + m_s} = {m^2_{\pi} \over m^2_{K^+}} + \dots$$
This is the general plan for measuring quark masses \[3\].
Once we are treating masses as coupling constants, we are led to the issue of renormalization. If the Lagrangian is written in terms of bare parameters, the interactions will induce mass shifts and we need to define renormalized masses. What then are the renormalization conditions and how are these connected to observables? I must admit that for the light quarks the answer to this question has not been completely satisfactorily found at present. In perturbation theory, of course, renormalization can be carried out. However we do not have a full connection between perturbation theory and low energy measurements. One key feature of perturbative renormalization in QCD is that the mass shift of a fermion is proportional to the mass of that fermion. In general then we will find
$$m^{(R)}_i = m^{(bare)}_i \left[ Z_0 + Z_1 m_i + Z^{\prime}_1 \sum_{j \neq i}
m_i +
\ldots \right]$$
To first order (in m) we always have
$$m^{(R)}_i \alpha \, m^{(bare)}_i$$
so that ratios of the renormalized masses are equally ratios of the bare parameters. This nice feature can be preserved in mass independent perturbative renormalization schemes.
In perturbative theory one can also choose to define running masses, $m_i
(q^2 )$. In QCD, these get smaller as $q^2$ increases. For light quarks there is not much value for using these in the measurement of mass. We have our best information on ratios of masses, and in a mass independent renormalization scheme, ratios are independent of the scale. Another point to be emphasized is that running masses for light quarks, despite getting large at low $q^2$, do not make a good model for constituent masses. This is because all of the running masses vanish at all $q^2$ in the chiral limit $(m^{(bare)}_i \rightarrow 0 \Rightarrow m_i (q^2) \rightarrow 0)$, in contrast to constituent masses which approach a constant $(\approx 300
MeV)$ in this limit.
Non-perturbative effects can also induce mass shifts. One possible new form has been suggested by instanton calculations \[4\] with a mass shift
$$\delta m_u ~\alpha ~m_d m_s$$
We will see later that this in fact is consistent with the symmetries of QCD. It raises the question of what mass we are measuring in a given observable. However let us save these issues for later and now turn to the simple lowest order estimates of mass.
Consider first a world with massless $u, d, s$ quarks. The quark helicity (L, R) is not changed by QCD interactions in this limit, and is unchanged under all Lorentz boosts. There are then two separate worlds, with left handed and right handed quarks being separately conserved. This implies an $SU(3)_L \times SU(3)_R$ symmetry. Any mass will break this symmetry because, at the very least, one can boost a massive left handed quark to a frame where it is right handed. However, if $m$ is ’small’ we are close to the symmetry limit. More precisely, in the massless limit, we have separate global invariance under
$$\begin{aligned}
\psi_L \rightarrow \psi_L^\prime = L \psi_L \nonumber\\
\psi_R \rightarrow \psi_R^\prime = R \psi_R\end{aligned}$$
with
$$\psi = \left( \begin{array}{c}
u \\ d \\ s
\end{array} \right)$$
and L in $SU(3)_L$, R in $SU(3)_R$. If there is a common mass $m_u = m_d = m_s$, this chiral symmetry is explicitly broken to $SU(3)_V$, and separate masses for $u, d, s$ breaks even this latter symmetry.
However, while we see approximate $SU(3)_V$ multiplets in the spectrum of hadrons, we do not see even approximate multiplets for $SU(3)_L \times
SU(3)_R$. This is because the symmetry is hidden by the phenomena of dynamical symmetry breaking. This is characterized by a vacuum which is not invariant under the symmetry, and the appearance of Goldstone bosons. The $\pi, K, \eta$ are the Goldstone bosons, and would be massless if the quarks were massless. This fact can be used to yield the best known measure of quark masses. For it, we need to use only first order perturbation theory, i.e., that the energy shift results from taking the matrix element of the perturbing Hamiltonian between unperturbed wavefunctions. The perturbation is
$${\cal H}_m = m_u \bar{u}u + m_d \bar{d}d + m_d \bar{s}s$$
and the results are
$$\begin{aligned}
< \pi \mid {\cal H}_m \mid \pi > & = & m^2_{\pi} = (m_u + m_d) B_0
\nonumber \\
< K^+ \mid {\cal H}_m \mid K^+ > & = & m^2_{K^+} = (m_u + m_s) B_0
\nonumber \\
< K^0 \mid {\cal H}_m \mid K^0 > & = & m^2_{K^0} = (m_d + m_s) B_0
\nonumber \\
< \eta \mid {\cal H}_m \mid \eta > & = & m^2_{\eta} = {1 \over 3} (4m_s +
m_u + m_d) B_0\end{aligned}$$
where $B_0$ is a constant (the reduced matrix element).
Defining
$$\hat{m} = {m_u + m_d \over 2}$$
we have
$$\begin{aligned}
{\hat{m} \over m_s} & = & {m^2_{\pi} \over 2m^2_K - m^2_{\pi}} \nonumber\\
{m_d - m_u \over m_s - \bar{m}} & = & {m^2_{K^0} - m^2_{K^+} \over
m^2_K - m^2_{\pi}}\end{aligned}$$
valid to first order in the quark masses. Actually the second of these needs to be corrected for electromagnetic effects, which can also influence the $K^0 - K^+$ mass difference. Here we use Dashen’s theorem \[5\], i.e., that to lowest order in chiral SU(3) \[that is, with no quark mass effects\], the kaon and pion electromagnetic splitting are the same
$$(m^2_{K^+} - m^2_{K^0})_{EM} = m^2_{\pi^+} - m^2_{\pi^0}.$$
Subtracting off this contribution leads to
$${m_d - m_u \over m_s - \hat{m}} = {m^2_{K^0} - m^2_{K^+} -
m^2_{\pi^0} + m^2_{\pi^+} \over m^2_K - m^2_{\pi}} = 1/43$$
or
$${m_d - m_u \over m_d + m_u} = {m^2_{K^0} - m^2_{K^+} -
m^2_{\pi^0} + m^2_{\pi^+} \over m^2_{\pi}} = 0.28$$
This is the estimate that most of the community is familiar with. However, the full story on quark masses is considerably more involved (or else my lectures would stop here).
Even at first order in the masses, there are other measures of quark mass ratios. Another interesting example is the decay $\eta \rightarrow 3 \pi$, which is forbidden by isospin. The electromagnetic effect vanishes at lowest order in chiral SU(2) (Sutherland-Veltman theorem \[6\]) and all estimates beyond this order indicate that electromagnetism has a negligible effect. This leaves the isospin breaking $m_d - m_u$ as the feature which induces the decay. Soft pion theorems can relate the amplitude to
$$\langle \pi^0 \mid {\cal H}_m \mid \eta \rangle = \sqrt{1 \over 3} (m_u - m_d)
B_0$$
or the result can be read off from the effective Lagrangian described later. One finds
$$\begin{aligned}
{m_d - m_u \over m_d + m_u} & = & {3 \sqrt{3} \, F^2_{\pi} A_0
(\eta \rightarrow 3 \pi^+ \pi^- \pi^0) \over m^2_{\pi}} \nonumber \\
& = & 0.56\end{aligned}$$
where $A_0$ is the amplitude in the center of the Dalitz plot and the error bars are purely experimental. This is considerably larger than the previous result, and would imply $m_u/m_d = 1/3.5$. However in this case we do know some of the higher order effects (described later) are sizeable, and will modify this result \[7\]. This result does indicate that first order measurements do not agree, and that we will need to confront the analyses at second order.
A third measurement of quark masses at first order involves $\psi^\prime
\rightarrow J/\psi \pi^0$ and $\psi^\prime \rightarrow J/\psi \eta$. The former violates isospin and the second violates SU(3). Again an electromagnetism is estimated to play a very minor role, so that these decays are driven by $m_d - m_u$ and $m_s - \hat{m}$ respectively. The analysis, using degenerate perturbative theory, yields the result
$$\begin{aligned}
{m_d - m_u \over m_s - \hat{m}} & = & \left[{16 \over 27} {\Gamma
\left(\psi^{\prime}
\rightarrow
J / \psi + \pi^0 \right) \over \Gamma \left( \psi^{\prime} \rightarrow J /
\psi + \eta \right) }
{\rho^3_{\eta} \over P^3_{\pi}} \right]^{1/3} \nonumber \\
& = & 0.033 \pm 0.004\end{aligned}$$
This calculation uses only vectorial SU(3), not chiral SU(3). The result lies almost exactly halfway between the answer given by meson masses and by $\eta \rightarrow 3 \pi$ (which yields 0.023 and 0.046 respectively). If we look at the spread around the central value, the first order values have a standard SU(3) breaking spread of $1 \pm 30 \%$.
At this stage, one might ask about the absolute values of the masses. However for the light quarks there is no measurement of the light quark masses in the sense that I am using measurement. The basic problem is that the mass enters the theory multiplied by $\bar{\psi} \psi$, i.e., ${\cal H}_m
= m \bar{\psi}\psi$. While their product is well defined, both m and $\bar{\psi} \psi$ are separately renormalization scheme dependent, and the measurements of the product do not measure m or $\bar{\psi} \psi$ separately. A very rough determination is as follows \[9\]. Since $m_{u,d}
<< m_s$, we have at first order
$$\begin{aligned}
m_{\Lambda} - m_p & = & < \Lambda \mid {\cal H}_m \mid \Lambda > - <
P \mid {\cal H}_m \mid P > \nonumber \\
& \approx & < \Lambda \mid m_s \bar{s} s \mid \Lambda > - < P \mid m_s
\bar{s} s \mid P > \nonumber \\
& \equiv & m_s Z \nonumber \\
& \approx & 180 MeV\end{aligned}$$
where
$$Z = \, < \Lambda \mid \bar{s} s \mid \Lambda > - < P \mid \bar{s} s \mid P >$$
Because $< \Lambda \mid \bar{s} \gamma_0 s \mid \Lambda > =
1$, we might expect $Z \sim O(1)$. \[However, for the vacuum state $< 0
\mid \bar{s} \gamma_0 s \mid 0 > =0$ but $< 0 \mid \bar{s} s
\mid 0 >$ is quite large.\] Explicit quark model calculation \[1\] yields $Z =
0.5
\rightarrow 0.75$, which seem reasonable, but not extremely solid. If we use these we get
$$\begin{aligned}
m_s & \sim & 150 \rightarrow 300 MeV \nonumber \\
m_\alpha & \sim & 8 \rightarrow 16 MeV \nonumber \\
m_u & \sim & 3 \rightarrow 9 MeV\end{aligned}$$
However, since these and other estimates of light quark masses are based on models, not on measurements, we will not consider absolute values further.
The CKM Elements $V_{ud}, V_{us}$
=================================
The weak mixing elements $V_{ud}$ and $V_{us}$ are best measured in semileptonic decays, as nonleptonic transitions are not under theoretical control. The focus of theoretical analysis in the semileptonic decays is the quest for precision in handling the strong interactions. With $V_{ud}$, the main issues are the electroweak radiative correction and small effects due to isospin breaking. For $V_{us}$, the primary concern is SU(3) breaking in the current matrix elements.
The reference standard, to which the hadronic decays are compared, is $\mu^- \rightarrow e^- \bar{\nu}_e \nu_{\mu}$. With the Hamiltonian
$${\cal H}_w = {G_{\mu} \over \sqrt{2}} \bar{\nu}_{\mu} \gamma_{\mu}
(1 + \gamma_5 ) \mu \bar{e} \gamma^{\mu} (1 + \gamma_5 ) \nu_e$$
and including the electroweak radiative correction, one has the rate
$$\begin{aligned}
\Gamma (\mu \rightarrow e \nu \bar{\nu}) & = & {G^2_{\mu} m^5_{\mu} \over 192
\pi^3}
\left[1 -{\alpha \over 2 \pi} \left( \pi^2 - {25 \over 4} \right) -
{8m^2_e \over m^2_{\mu}} +
{3 \over 5} {m^2_{\mu} \over m^2_W} + \ldots \right] \nonumber \\
& = & {G^2_{\mu} m^5_{\mu} \over 192 \pi^3} \left[ 1 + (4203.85 -
187.12 + 1.05) \times 10^{-6} \right]\end{aligned}$$
where the corrections, in the order written, are due to photonic radiative effects, phase space, and the W propagator. The value of $G_{\mu}$ thus extracted is
$$G_{\mu} = 1.16637(2) \times 10^{-5} GeV^{-2}$$
For $\Delta S = 0$ beta decays we have
$${\cal H}_w = {G_{\beta} \over \sqrt{2}} V_{ud} \bar{u} \gamma^u (1 +
\gamma_5) d \bar{e} \gamma_{\mu} (1 + \gamma_5) \nu_e$$
At tree level $G_{\mu} = G_{\beta}$, but at one loop this is no longer true as there is an important difference in the radiative correction. For the weak transition $1 + 3 \rightarrow 2 + 4$ some of the radiative corrections are shown in Fig. 1. Diagrams a, b are ultraviolet finite. This can be understood by noting that the calculation is the same as the vertex renormalization of a conserved current, which we know leads to no renormalization at $q^2 = 0$. Figures c, d are similar if we use the Fierz transformation
$$\bar{\psi}_2 \gamma_{\mu} (1 + \gamma_5) \psi_1 \bar{\psi}_4
\gamma^{\mu} (1 + \gamma_5) \psi_3 = \bar{\psi}_4 \gamma_{\mu} (1 +
\gamma_5) \psi_1 \bar{\psi}_2 (1 + \gamma_5) \psi_1$$
However diagrams e, f fall into a different class and are log divergent if we use the Fermi interaction with no propagator. The ultraviolet portion is then proportional to $(Q_1 Q_3 + Q_2 Q_4)$, i.e.,
$$M^{(u.v.)}_{e,f} = -M^{(0)} {3\alpha \over 2\pi} (Q_1 Q_3 + Q_2 Q_4)
ln \Lambda / \mu_L$$
where $\Lambda$ is a high energy cutoff and $\mu_l$ is a low energy scale. In muon decay $(1,2,3,4) = (\mu^-, \nu_{\mu}, \nu_e, e^-)$ so that
$$Q_{\mu} Q_{\nu_e} + Q_{\nu_\mu} Q_e = 0 .$$
However in beta decay $(1,2,3,4) = (d, u, \nu_e, e^-)$ with
$$Q_d Q_{\nu_e} + Q_{u_\mu} Q_{e^-} = - {2 \over 3} .$$
In a full treatment, including the $W$ propagator and $\gamma, Z$ loops one finds the cutoff $\Lambda = m_Z$, so that there is a universal ’model independent’ correction \[10\] which can be absorbed in the definition of $G_{\beta}$
$$G_{\beta} = G_{\mu} \left( 1 + {\alpha \over \pi} ln {M_Z \over \mu_L}
\right )$$
To this also needs to be added smaller ’model dependent’ low energy effects and coulomb corrections.
For $\Delta S = 0$ decays, the key to mastering the strong interactions is that the vector current is conserved (in the limit $m_u = m_d)$, so that the matrix element is absolutely normalized. In contrast it is not possible to predict axial current matrix elements to high accuracy. In neutron beta decay, $n \rightarrow p e \nu $, where both vector and axial currents contribute, one needs to measure the axial form factor $g_A$ in order to be able to predict the rate and measure $V_{ud}$. This works, but at present the statistical accuracy is not the best. Pion beta decay, $\pi^{\pm}
\rightarrow \pi^0 e^{\pm} \bar{\nu}$, only involves the vector current and would be the ideal channel to study, but there are not yet enough events. The most sensitive process is $0^+ \rightarrow 0^+$ nuclear beta decay between isospin partners \[11\]. This also only involves the vector current, and has very high statistics.
The superallowed $0^+ \rightarrow 0^+$ transitions have a single form factor
$$< N_2 (I_z = 0) \mid V_{\mu} \mid N_1 (I_z = 1) > = a(q^2)(p_1 + p_2) .$$
with $a(0) = \sqrt{2}$. One calculates the half life $t_{1/2}$ times a kinematical phase space factor F, and adds hard and soft radiative corrections, Coulomb corrections to the wavefunction and finite size effect
$$Ft_{1/2} = {2 \pi^3 ln 2 \over G^2_{\beta} m^5_e \mid V_{ud} \mid^2
a^2 (0)} [1 + \dots]$$
Present efforts center on the nuclear wavefunction mismatch. When one plots the Ft values for different nuclei vs. Z, the result should be a constant value if all the nuclear effects have been taken into account completely. In practice there seems to be some indication for a slope to this line \[12\], indicating that some effect linear in Z is not fully accounted for. In the recent analysis of Ref. 11 this has been corrected for phenomenologically be extrapolating the Ft values to Z = 0, with the result
$$V_{ud} = 0.9751 \pm 0.0005$$
One obtains a values for $V_{ud}$ about $2 \sigma$ lower if one simply averages the Ft measurements. Neutron and pion beta decays are consistent with Equation 35.
Effective Lagrangian Description
================================
Before going on to the measurement of $V_{us}$, I need to describe the uses of effective Lagrangian techniques in chiral perturbation theory. In these notes, I will be somewhat brief as Andy Cohen covers effective field theory in these TASI lectures \[2\] and I have elsewhere \[1\] had the opportunity to present the subject in considerably greater depth.
The main idea is that if predictions follows from symmetry alone, then any general Lagrangian with the right symmetry will yield the correct predictions \[13\]. For physics of the light mesons, we seek then the most general Lagrangian with chiral SU(3) symmetry containing only the $\pi, K, \eta$ fields. This can be accomplished with the $3 \times 3$ matrix representation
$$U = exp \left[ i {\vec{\lambda} \cdot \vec{\phi} \over F_{\pi}} \right] \; \; ,$$
with transformation
$$U \rightarrow LUR^{\dagger}$$
with L in $SU(3)_L$ and R in $SU(3)_3$. The only Lagrangian invariant under chiral SU(3) with 2 derivatives (there are none with zero derivatives) is
$${\cal L}_{SYM} = {F^2_{\pi} \over 4} Tr(\partial_{\mu} U
\partial^{\mu} U^{\dagger}) = {1 \over 2} \partial_{\mu} \phi^A \partial^{\mu}
\phi^A + \ldots$$
For QCD we also need some explicit chiral symmetry breaking, which at lowest order will be linear in the quark masses. It preserves parity and has the same chiral properties as $\bar{\psi} m \psi = \bar{\psi}_L m
\psi_R + \bar{\psi}_R m \psi_L$. At lowest order the unique choice is
$${\cal L}_{Breaking} = {F^2_{\pi} B_0 \over 2} Tr(m U + U^{\dagger}
m)$$
where $B_0$ has been chosen to be the same constant as in Section II. The full lowest order Lagrangian
$${\cal L} = {\cal L}_{sym} + {\cal L}_{Breaking}$$
when applied at tree level reproduces all of the lowest order predictions of chiral symmetry, such as the mass relations given previously.
What about effective Lagrangian with more derivatives or more powers of the quark masses? These may also have the correct chiral SU(3) properties. The key to practical applications is the energy expansion. Consider two possible chirally symmetric Lagrangians
$${\cal L}^1 = a Tr (\partial_{\mu} U \partial^{\mu} U^{\dagger} ) + b Tr
(\partial_{\mu} U \partial_{\nu} U^{\dagger} \partial^{\mu} U
\partial^{\nu} U^{\dagger})$$
The Lagrangian has dimension $(mass)^4$, which implies that $a$ has dimension $mass^2$ and $b/a \sim 1/mass^2$. When matrix elements are taken, derivatives turn into powers of momentum so that
$$M = a q^2 \left[1 + {b \over a} q^2 \right]$$
If we define $b/a \equiv 1/\Lambda^2$, then for $q^2 \ll
\Lambda^2$ there is little effect of the higher derivative terms. As $q^2$ increases, the four derivative term provides a correction to the lowest order result. In practice we most often find $\Lambda \sim m_{\rho}$, so that lowest order chiral predictions are modified as momenta approach $m_{\rho}$.
In constructing the effect of quark masses it is useful to consider an external field of the form \[14\]
$${\cal L}_{QCD} = \bar{\psi} i \rlap/{D} \psi - {1 \over 2B_0} \left(
\bar{\psi}_L \chi
\psi_R + \bar{\psi}_R \chi^{\dagger} \psi_L \right)$$
such that QCD is obtained with $\chi = 2B_0 m$. However, if we allow a transformation rule
$$\chi \rightarrow L \chi R^{\dagger}$$
the Lagrangian will be chirally invariant. The effect of masses is then found by writing chirally invariant Lagrangians containing $\chi$. We do this in Sec. 5.
Finally loop diagrams can, and must, be included. Divergences appear, but these just go into the renormalization of the parameters in the effective Lagrangian. Finite effects left over after renormalization account for the low energy propagation of the pions and kaons.
The application of effective Lagrangians to the chiral interactions of $\pi, K,
\eta$ is called Chiral Perturbation Theory. To next to leading order (i.e., $O(E^4)$) the instructions are:
1. Write the most general Lagrangians to $O(E^2)$ and $O(E^4)$; ${\cal L}_2$ contains two derivative or one power of the quark masses, and ${\cal L}_4$ has either 4 derivatives, 2 derivatives and one mass, or two powers of the mass.
2. Calculate all one loop diagram involving ${\cal L}_2$
3. Renormalize the parameters in the Lagrangian, determining the unknown parameters from experiment.
4. Find relations between different observables
These relations are the predictions of chiral symmetry.
$V_{us}$ and SU(3) Breaking
===========================
One of the applications of chiral perturbation theory is in the determination of $V_{us}$. We will need to obtain the form factors in $\Delta S = 1$ processes such as $K \rightarrow \pi e \nu$ and $\Lambda \rightarrow p e \nu$. These are related by SU(3) to the $\Delta S = 0$ form factors which we have already discussed $(\pi^+ \rightarrow \pi^0 e \nu, n \rightarrow p e
\nu)$. However typical SU(3) breaking enters into other processes at the 30% level. We want to be more accurate than this.
A crucial ingredient here is the Ademollo Gatto theorem \[15\] which says that the vector form factors are modified from their SU(3) values only by terms second order in the SU(3) breaking mass difference $m_s - \hat{m}$. This again points to the value of using vector form factors in the extraction of $V_{us}$. The two possible sources of data are hyperon decays and $K
\rightarrow \pi e \nu$.
Hyperon decays involve many modes and high statistics. The axial form factors cannot be predicted reliably from theory and must be measured. SU(3) parameterizes these form factors in terms of two reduced matrix elements, the neutron to proton axial coupling $g_A$ and a D/F ratio. The vector form factors are predicted via SU(3) plus the Ademollo Gatto theorem. The history of our ability to treat these decays has undergone fluctuations. Before 1982, SU(3) fits worked well. In 1982, the data improved enough that SU(3) breaking at the 5% level was observed and caused troubles with fits based on SU(3) symmetry, invalidating any fits using SU(3) symmetry \[16\]. A few years later the quark model was used to provide an SU(3) breaking pattern that was consistent with the data, allowing a good fit and the extraction of $V_{us}$ \[17\]. Unfortunately by 1990, the data was again better than theory, and the simple quark model pattern does not fit without modification \[18\]. Unless theory can recover once again, hyperon decays can not be analysed in any greater precision than this, because future increased statistics will only tell us more details about SU(3) breaking.
Kaon semileptonic decays involves only two modes ($K^0$ and $K^+$ decay). However the analysis is particularly strong since it can make use of a body of work on chiral perturbative theory. In addition these modes have very high statistics. For these reasons, kaon decay is the prime mode for measuring $V_{us}$.
In order to be convinced that the theory of $K \rightarrow \pi e \nu$ is under control, we have to turn to internal consistency checks. The analysis is due to Gasser and Leutwyler \[19\]. There are two form factors, $f_+$ and $f_-$
$$\begin{aligned}
< \pi^- \mid \bar{s} \gamma_{\mu} u \mid K^0 > & = & f^{K^0 \pi^-}_+ (k +
p)_{\mu} + f^{K^0 \pi^-}_- (k - p)_{\mu} \nonumber \\
< \pi^0 \mid \bar{s} \gamma_{\mu} u \mid K^+ > & = & {1 \over \sqrt{2}}
\left[ f^{K^+ \pi^0}_+ (k + p)_{\mu} + f^{K^+ \pi^0}_- (k - p)_{\mu}
\right].\end{aligned}$$
If one includes the next-to-leading order Lagrangian, as well as one loop diagrams, one obtains lengthy expressions for the form factors. Among the highlights of the results are
1. The Ademollo Gatto theorem has a correction due to isospin breaking
$$\begin{aligned}
{f^{K^+ \pi^0}_+ (0) \over f^{K^0 \pi^-}_+ (0)} & = & 1 + {3 \over 4}
{m_d - m_{\mu} \over m_s - \hat{m}} + l_{K \pi} \nonumber \\
& = & 1.029 \pm 0.010 (Data)\end{aligned}$$
where $l_{K \pi} = 0.004$ arise from loop diagrams. This value is consistent, because of the large uncertainty, with all of our previous estimates of the quark mass ratio.
2. The form factors are related to the chiral constant $L_9$ determined in the pion form factor,
$$\begin{aligned}
{f_- (0) \over f_+ (0)} & = & - \left[ 1 - {F_K \over F_{\pi}} + {2L_9 \over
F^2_{\pi}} \left(m^2_K - m^2_{\pi}\right) \right] \nonumber \\
& = & -0.13 \, theory \nonumber \\
& = & -0.20 \pm 0.08 \, data.\end{aligned}$$
3. The slopes of the form factors are predicted in agreement with the data (although the data presently have a few internal inconsistencies).
Given that the theory appears to be under control Leutwyler and Roos \[20\] have extracted
$$V_{us} = 0.2196 \pm 0.0023$$
(a 10% measurement). This value is consistent with the results of hyperon decay, and implies the check of the unitarity of the KM matrix
$$\mid V_{ud} \mid^2 + \mid V_{us} \mid^2 + \mid V_{ub} \mid^2 =
0.9990 \pm 0.0022$$
with $\mid V_{ub} \mid^2 \leq 10^{-5}$.
Quark Masses Beyond Leading Order
=================================
Now we turn to the most difficult issue in these lectures; the analysis of quark masses at second order. There are several motivations for pursuing such an analysis. First of all, we have seen how the lowest order predictions lead to some discrepancies. In addition, there is the strong CP problem \[21\], where the effect of CP violation by the $\theta$ term of QCD would vanish if $m_u \rightarrow 0$. We then must question how well we know that $m_u \neq 0$. This solution to the strong CP problem is not natural, in the technical sense, within the Standard Model, but perhaps might be possible within an extension of our present theory. Finally there are several subtle issues which arise at second order in the mass, most notably the reparameterization ambiguity described below.
The mass sector of the theory is described by
$${\cal L} = \ldots + {F^2 \over 4} Tr \left( \chi^{\dagger} U + U^{\dagger}
\chi \right)$$
at lowest order \[recall Equation 43\], and at higher order by
$$\begin{aligned}
{\cal L}_4 \ldots & + & L_6 \left[ Tr \left( \chi^{\dagger} U + U^{\dagger}
\chi \right) \right]^2 + L_7 \left[ TR \left( \chi^{\dagger} U - U^{\dagger}
\chi \right) \right]^2 \nonumber \\
& + & L_8 Tr \left( \chi U^{\dagger} \chi U^{\dagger} + \chi^{\dagger} U
\chi^{\dagger} U \right)\end{aligned}$$
where $L_{6,7,8}$ are dimensionless unknown reduced matrix elements in the basis of Gasser and Leutwyler \[14\].
The $\pi, K, \eta$ masses can be analysed to second order in the quark masses \[14\].
$$\begin{aligned}
F^2_{\pi} m^2_{\pi} & = & 2 \hat{m} F^2 B_0 \left[1 + {32 L_6 B_0 \over F^2}
\left( m_u + m_d
+ m_s \right) + {32 L_8 B_0 \hat{m} \over F^2} \right. \nonumber \\
& &\left. - 3 \mu_{\pi} - 2 \mu_K - {1 \over 3} \mu_{\eta} \right] \nonumber
\\
F^2_{K^+} m^2_{K^+} & = & \left( m_s + m_u \right) F^2 B_0 \left[ 1 + {32 L_6
B_0 \over F^2}
\left( m_u + m_d + m_s \right) \right. \nonumber \\
& &\left. + {16 L_8 B_0 \over F^2} \left( m_u + m_s \right) - {3 \over 2}
\mu_{\pi} - 3 \mu_K - {5
\over 6} \mu_{\eta} \right] \nonumber \\
F^2_{K^0} m^2_{K^0} & = & \left( m_s + m_d \right) F^2 B_0 \left[ 1 + {32
L_6 B_0 \over F^2} \left( m_u + m_d + m_s \right) \right. \nonumber \\
& &\left. + {16 L_8 B_0 \over F^2} \left( m_s + m_d \right) - {3 \over 2}
\mu_{\pi} - 3 \mu_{K} -
{5 \over 6} \mu_{\eta} \right] \nonumber \\
F^2_{\eta} m^2_{\eta} & = & {4 \over 3} F^2_K m^2_K - {1 \over3} F^2_{\pi}
m^2_{\pi} -
{64 \over 3} \left( 2 L_7 + L_8 \right) B_0 \left( m_s - \hat{m} \right)^2
\nonumber \\
& &+ \left( 2 \mu_{\pi} - {4 \over 3} \mu_K - {2 \over3} \mu_{\eta} \right)
\left( m^2_K -
m^2_{\pi} \right)\end{aligned}$$
where
$$\eta_i = {m^2_i \over 32 \pi^2 F^2_\pi} ln m^2_i / \eta^2$$
There are two main results of this analysis. One is that the deviation of the Gell Mann Okubo formula measures a useful combination of chiral coefficients
$$\begin{aligned}
\delta_{GMO} & = & {4F^2_K m^2_K - 3F^2_{\eta} m^2_{\eta} - F^2_{\pi}
m^2_{\pi} \over 4 \left(F^2_K m^2_K - F^2_{\pi} m^2_{\pi} \right)}
\nonumber \\
& = & {16 \over F^2_{\pi}} \left(2L_7 + L_8 \right) \left( m^2_{\pi} -
m^2_K \right) - {3 \over 2} \mu_{\pi} + \mu_K + {1 \over 2} \mu_{\eta}
\nonumber \\
& = & -0.06 (Data)\end{aligned}$$
which yields
$$2 L_7 + L_8 = 0.2 \times 10^{-3}$$
at the chiral scale $\mu = m^2_{\eta}$. The other prediction is more important, producing a ratio of quark masses which are free from unknown parameters
$$\begin{aligned}
{m_d - m_u \over m_s - \hat{m}} {2 \hat{m} \over m_s + \hat{m}} & = &{m^2_{\pi}
\over
m^2_K}
{\left( m^2_{K^0} - m^2_{K^+} \right)_{QM} \over m^2_K - m^2_{\pi}} \nonumber
\\
\left( m^2_{K^0} - m^2_{K^+} \right)_{QM} & \equiv & \left( m^2_{K^+} -
m^2_{K^+}
\right)_{expt}
- \left( m^2_{K^0} - m^2_{K^+} \right)_{EM}\end{aligned}$$
The only flaw in this wonderful relation is that we do not know $\left( m^2_{K^0} - m^2_{K^+} \right)_{EM}$ to the order that we are working. Recall that Dashen’s theorem was only valid to zeroth order in the quark mass. The next order results have not been fully explored in chiral perturbation theory.
Gasser and Leutwyler have also analysed $\eta \rightarrow 3 \pi$ to second order \[7\]. The result can be expressed in parameter free form as
$${m_d - m_u \over m_s - \hat{m}} ~{2 \hat{m} \over m_s + \hat{m}} = {3
\sqrt{3} F^2_{\pi} Re A_{\eta \rightarrow 3 \pi} (0) \over \left[ 1 +
\Delta_{\eta 3 \pi} \right] \left( m^2_K - m^2_{\pi} \right)} ~{m^2_{\pi}
\over m^2_K} = 2.35 \times 10^{-3}$$
where $\Delta_{\eta 3 \pi} = 0.5$. Recall that this ratio was $1.7
\times 10^{-3}$ from meson masses and $3.5 \times 10^{-3}$ from $\eta
\rightarrow 3 \pi$, both at lowest order. The effects of the $\em{O}(E^4)$ analysis has been to produce a compromise value for the ratio.
One of the advances of the past year is that it is now reasonable to expect consistency between the analysis of the kaon mass difference and that of $\eta \rightarrow 3 \pi$. The agreement of Eq. 54 and Eq. 55 would require $\left(
\Delta m^2_K \right)_{QM} = 7.0 MeV$. However Dashen’s theorem implies $\left( \Delta m^2_K \right)_{EM} = 5.3 MeV$. Are there significant violations of Dashen’s theorem? Recent analyses suggest that there are \[22\]. I am, of course, most partial to the work which I participated in. We used a series of powerful constraints on the $\gamma \pi \rightarrow
\gamma \pi$ and $\gamma K \rightarrow \gamma K$ amplitudes which serve to predict the electromagnetic mass difference when the photons are contracted into a propagator. These constraints include 1) data on $\gamma
\gamma \rightarrow \pi \pi$, 2) low energy chiral constraints, 3) the dispersion theory of $\gamma \gamma \rightarrow \pi \pi$, 4) soft pion theorems and, 5) the generalized Weinberg sum rules. These features are compatible with a vector dominance model which yields
$${\left( \Delta m^2_K \right)_{EM} \over \left( \Delta m^2_{\pi}
\right)_{EM}} = 1.8$$
whereas Dashen’s theorem says that the ratio should be unity. The difference has a rather simple origin: it is due to factors of $m^2_K$ in propagators instead of $m^2_{\pi}$. The larger electromagnetic contribution brings the kaon mass difference and $\eta \rightarrow 3 \pi$ into considerably better agreement (10 %).
The most interesting feature of the study of quark masses beyond leading order is the reparameterization invariance, first made explicit by Kaplan and Manohar \[23\]. The crude statement is that when using SU(3) symmetry one obtains the same physics using either the masses $(m_u , m_d , m_s )$ or the set
$$\begin{aligned}
m^{(\lambda)}_u & = & m_u + \bar{\lambda} m_d m_s \nonumber \\
m^{(\lambda)}_d & = & m_d + \bar{\lambda} m_u m_s \nonumber \\
m^{(\lambda)}_s & = & m_s + \bar{\lambda} m_u m_d\end{aligned}$$
for an $\bar{\lambda}$! The reason is that $m_i$ and $m^{(\lambda)}_i$ both have the same chiral SU(3) properties. This can be seen using the Cayley-Hamilton theorem for a $3 \times
3$ matrix A
$$det A = A^3 - A^2 Tr A - {A \over 2} \left[ Tr (A^2) - (Tr (A))^2 \right]$$
If we apply this to the matrix $\chi$ \[recall $\chi = 2 B_0 m$ for pure QCD\] defining
$$\chi^{(\lambda)} = \chi + \lambda [det \chi^{\dagger} ] \chi {1 \over
\chi^{\dagger} \chi}$$
we have
$$\begin{aligned}
\left[ det \chi^{\dagger} \right] \chi \, {1 \over \chi^{\dagger} \chi} & = &
\left[ det U
\chi^{\dagger} \right] \chi \,{1 \over \chi^{\dagger} \chi} \nonumber \\
& = & U \chi^{\dagger} U \chi^{\dagger} U - U \chi^{\dagger} U Tr \left( U
\chi^{\dagger} \right) \nonumber \\
& - & {U \over 2} \left[ Tr \left( U \chi^{\dagger} U \chi^{\dagger} \right) -
\left( Tr U \chi^{\dagger} \right)^2 \right]\end{aligned}$$
and
$$Tr (\chi^{\lambda} U^{\dagger}) = Tr (\chi U^{\dagger} ) - {\lambda \over 2}
\left[ Tr
(\chi^{\dagger} U \chi^{\dagger} U) - \left(Tr (\chi^{\dagger} U) \right)^2
\right]$$
In an effective Lagrangian the use of $\chi^{(\lambda)}$ instead of $\chi$ leads to a Lagrangian of the same general form since
$$\begin{aligned}
Tr (\chi^{(\lambda)} U^{\dagger} + U \chi^{(\lambda)\dagger}) & = & Tr (\chi
U^{\dagger} + U
\chi^{\dagger}) \nonumber \\
& + & {\lambda \over 2} \left[ Tr (\chi U^{\dagger} + U \chi^{\dagger})
\right]^2 \nonumber \\
& + & {\lambda \over 2} \left[ Tr (\chi U^{\dagger} - U \chi^{\dagger})
\right]^2 \nonumber \\
& - & \lambda Tr (\chi U^{\dagger} \chi U^{\dagger} + \chi^{\dagger} U
\chi^{\dagger} U)\end{aligned}$$
The last three terms lead to a modification of the chiral coefficients which we called $L_6, L_7, L_8$ previously. However the total effective Lagrangian has the same form. Use of $\chi^{(\lambda)}$ and one set of $L_6, L_7, L_8$ is equivalent to the use of $\chi$ and a different set of $L_6, L_7, L_8$. This property of $\chi$ is the same as that of the masses, when we use $\chi = 2 B_0 m$, and
$$\chi^{(\lambda)} \equiv 2 B_0 m^{\lambda} = 2 B_0 \left[ m + (2 B_0 \lambda)
m_u m_d m_s {1
\over m} \right]$$
and identify $\bar{\lambda} = 2 B_0 \lambda$. The precise statement of the reparameterization ambiguity is then that, using either SU(3) or chiral SU(3) any physics described by $(m_u, m_d, m_s)$ and $(L_6, L_7, L_8)$ can be equally well described by
$$\begin{aligned}
m^{(\lambda)}_u = m_u + \bar{\lambda} m_d m_s & L^{(\lambda)}_6 =
L_6 - \tilde{\lambda} \nonumber \\
m^{(\lambda)}_d = m_d + \bar{\lambda} m_u m_s & L^{(\lambda)}_7 =
L_7 - \tilde{\lambda} \nonumber \\
m^{(\lambda)}_s = m_s + \bar{\lambda} m_u m_d & L^{(\lambda)}_8 =
L_8 - 2 \tilde{\lambda} \end{aligned}$$
with $\bar{\lambda} = 2 B_0 \lambda ; \tilde{\lambda} = F^2_{\pi}
\lambda / 16$, for any reasonable $\lambda$.
Let us see examples of how this works. For the ratio of quark masses measured above, we have
$$\begin{aligned}
{m^{(\lambda)}_d - m^{(\lambda)}_u \over m^{(\lambda)}_s -
\hat{m}^{(\lambda)}} \,
{2 \hat{m}^{(\lambda)} \over {m^{(\lambda)}_s - \hat{m}^{(\lambda)}}}
& = & {(m_d - m_u)
(1 - \bar{\lambda} m_s) \over (m_s - \hat{m})(1 - \bar{\lambda} \hat{m})} \, {2
\hat{m} (1 +
\bar{\lambda} m_s) \over (m_s + \hat{m})(1 + \bar{\lambda} \hat{m})} \nonumber
\\
& = & {m_d - m_u \over m_s - \hat{m}}\, {2 \hat{m} \over m_s + \hat{m}} + {\cal
O} (m^2)\end{aligned}$$
i.e., the ratio is invariant. Similarly the combination
$$2 L^{(\lambda)}_7 + L^{(\lambda)}_8 = 2 (L_7 - \tilde{\lambda}) + (L_8 + 2
\tilde{\lambda}) =
2
L_7 + L_8$$
is invariant. Finally
$$\begin{aligned}
m^2_{\pi} & = & 2 B_0 \hat{m}^{(\lambda)} \left[ 1 + {32 B_0 \over F^2_{\pi}}
\left( \hat{m}
L^{(\lambda)}_8 + (2 \hat{m} + m_s) L^{(\lambda)}_6 \right) + \ldots \right]
\nonumber \\
& = & 2 B_0 \hat{m} (1 + \bar{\lambda} m_s) ]\left[ 1 - \bar{\lambda} m_s + {32
B_0 \over
F^2_{\pi}} \left( \hat{m} L_8 + (2 \hat{m} + m_s) L_6 \right) + \ldots \right]
\nonumber \\
& = & 2 B_0 \hat{m} \left[ 1 + {32 B_0 \over F^2_{\pi}} \left( \hat{m}
L_8 + (2 \hat{m} + m_s) L_6 \right) + \ldots \right] \nonumber \\
& & + {\cal O} (m^3)\end{aligned}$$
is also unchanged in form under the reparameterization.
Physical quantities are invariant under the reparameterization transformation. Quark mass ratios (or the $L_i \, 's$) are not invariant and hence can not be uniquely measured by any analysis using SU(3) or chiral SU(3). This conclusion is general and extends to other systems, such as baryons or heavy mesons, when analysed to second order (or beyond). The best that we can do is to measure a one parameter family of masses.
There is a weak restriction on the transformation in that we can’t choose $\lambda$ so large as to destroy the energy expansion. The typical sizes of the chiral coefficients are of order a few times $10^{-3}$. We should not allow any $\tilde{\lambda}$ that makes $L_6, L_7,
L_8$ unnaturally large. In practice this does not happen for the mass range that we are most interested in.
A conventional choice for masses and chiral parameters is
$$\begin{aligned}
{m_u \over m_s} = {1 \over 34} \; & ; & \; {m_d \over m_s} = {1 \over 19}
\nonumber \\
L_7 = -0.4 \times 10^{-3} \; & ; & \; L_8 = 1.1 \times 10^{-3}\end{aligned}$$
A second set which is equally consistent is one with $m_u = 0$
$$\begin{aligned}
{m_u \over m_s} = 0 \; & ; & \; {m_d \over m_s} = {1 \over 26} \nonumber \\
L_7 = 0.2 \times 10^{-3} \; & ; & \; L_8 = -0.1 \times 10^{-3}\end{aligned}$$
obtained by a reparameterization transformation. A third compatible set is
$$\begin{aligned}
{m_u \over m_s} = {1 \over 22} \; & ; & \; {m_d \over m_s} = {1 \over 16}
\nonumber \\
L_7 = -0.8 \times 10^{-3} \; & ; & \; L_8 = 1.9 \times 10^{-3}\end{aligned}$$
In all cases $L_7$ and $L_8$ are natural in size. (Nothing is known about the magnitude of $L_6$). Note that since $m_u$ is the smallest mass, it changes the most. This is to be expected since we have
$$\Delta m_u \sim m_d m_s \sim m_d {m^2_K \over \Lambda^2} \sim {1 \over 3} m_d
\sim m_u$$
so that the change in $m_u$ is of the same order as $m_u$ itself.
The reparameterization transformation is an invariance of SU(3) effective Lagrangians, not of the fundamental QCD Lagrangian. However, there may be physics in QCD which generates effects like this \[24\]. Let us consider the allowed forms of radiative corrections to the masses in various limits.
1. If $m_u = m_d = m_s = 0$, we have an exact $SU(3)_L \times SU(3)_R$ chiral symmetry. There are no modifications to masses due to radiative corrections, as the quarks are protected by the chiral symmetry from picking up a mass.
2. If $m_u = m_d = 0$ and $m_s \neq 0$, there is an exact chiral SU(2) symmetry which protects $m_u$ and $m_d$ from any quantum shifts. Likewise $m_u$ and $m_s$ would be protected in an $m_u = m_s = 0, m_d \neq 0$ world.
3. Now consider $m_u = 0$, but $m_d \neq 0$ and $m_s \neq 0$. Now there is no symmetry protection at all, because chiral SU(2) is broken and axial U(1) is not a quantum symmetry. There can be radiative corrections to $m_u$. However, since the corrections must vanish as $m_d \rightarrow 0$ or as $m_s \rightarrow 0$, it must have the form
$$m_u = c m_d m_s$$
There is in the literature an interesting example of just such a renormalization, where instantons lead to this form of radiative correction, with the overall coefficient depending on the cutoff in instanton sizes \[4\]. We don’t need to take the details of this calculation too seriously, but we must acknowledge that this form of radiative correction can occur in QCD. It is always associated with the $U(1)_A$ anomaly. By permutation symmetry, if $m_u \neq 0$ we would have $\Delta m_d = c m_u m_s, \Delta m_s = c m_u m_d$.
These radiative corrections can produce different definitions of quark masses. For example in a mass independent renormalization scheme, one has
$$\left[ m^{(r)}_i \right]_1 = Z m_i$$
with a common factor of Z. In a second renormalization scheme one might include the low energy effects (such as the instantons) which induce the radiative corrections of the preceding paragraph. The two schemes would be related by a finite renormalization
$$\left[ m^{(r)}_u \right]_2 = \left[ Z' m^{(r)}_u + \bar{\lambda} m^{(r)}_d
m^{(r)}_s \right]_1$$
for some $\bar{\lambda}$. For consistency, the various other parameters in the theory would also have to be related
$$\left[ L_7 \right]_2 = \left[L_7 - \tilde{\lambda} \right]_2$$
such that observables are unchanged. From this point of view, there is the possibility of a renormalization scheme ambiguity in QCD which mirrors the reparameterization invariance.
An caveat to the above argument involves the $U(1)_A$ dependence. In the presence of a non- zero vacuum angle $\theta$ in QCD the mass shift due to the instanton effect is actually \[4\]
$$\Delta m_u = c m_d m_s e^{i \theta}$$
The various masses of different renormalization schemes have different $\theta$ dependence, and can in principle be differentiated by their behavior under $U(1)_A$ transformations. This can also be seen in the transformation of the $\chi$ and $\chi^{(\lambda)}$ under $U(1)_A, L = e^{i \alpha}, R = e^{-i \alpha}$, in that
$$\begin{aligned}
\chi & \rightarrow & e^{2i \alpha} \chi \nonumber \\
\chi^{\lambda} & \rightarrow & e^{2i \alpha} \left[ \chi + \lambda e^{-6i
\alpha} \left[ det
\chi^+ \right] \chi {1 \over \chi^+ \chi} \right]\end{aligned}$$
so that $m$ and $m^{(\lambda)}$ are not equivalent in their $U(1)_A$ properties. Of course, $U(1)_A$ is not a symmetry, but there are a set of anomalous Ward identities \[25\] which can in principle probe the $U(1)_A$ behavior. In practice, none of the measurements discussed above involve $U(1)_A$.
There is an example which shows how the $U(1)_A$ properties can measure masses independent of the reparameterization \[24\] transformation. Briefly summarized one adds the $\theta F
\tilde{F}$ term to the QCD Lagrangian but with $\theta$ treated as an external source so that functional derivatives with respect to $\theta (x)$ yield matrix elements of $F
\tilde{F}$. The $U(1)_A$ properties determine how $\theta (x)$ enters the effective Lagrangian, and these matrix elements are calculated to $O(E^4)$. The example shown was
$$\begin{aligned}
{< 0 \mid F \tilde{F} \mid \pi^0 > \over < 0 \mid F \tilde{F} \mid \eta >} = {3
\sqrt{3} \over 4}
\left[ {m_d - m_u \over m_s - \hat{m}} \right] {F_{\eta} \over F_{\pi}}
\nonumber \\
\left[ 1 - {32 B_0 \over F^2_{\pi}} (m_s - \hat{m}) (L_7 + L_8 ) + \ldots
\right]\end{aligned}$$
This matrix element is not reparameterization invariant so that, if it could be measured, it could be used to disentangled the individual mass ratios.
What can be done in such a situation? There is at present no completely satisfactory solution. However, some possible directions have been at least partially explored. One possibility is to choose a definition of mass which is automatically reparameterization invariant. For example we can simply define invariant masses $m^*_i$ by \[24\]
$$\begin{aligned}
F^2_{\pi} m^2_{\pi} & \equiv & F^2_0 B_0 \left[ m^{\ast}_u + m^{\ast}_d \right]
\nonumber \\
F^2_{K^+} m^2_{K^+} & \equiv & F^2_0 B_0 \left[ m^{\ast}_s + m^{\ast}_u \right]
+
\delta_{GMO} F^2_K \left( m^2_K - m^2_{\pi} \right) \nonumber \\
F^2_{K^0} m^2_{K^0} & \equiv & F^2_0 B_0 \left[ m^{\ast}_s + m^{\ast}_d \right]
+
\delta_{GMO} F^2_K \left( m^2_K - m^2_{\pi} \right) \nonumber \\
F^2_{\eta} m^2_{\eta} & \equiv & F^2_0 B_0 \left[ {4 \over 3} m^{\ast}_s + {2
\over 3}
\hat{m}^{\ast} \right]\end{aligned}$$
This results in
$$\begin{aligned}
m^{\ast}_u & = & m_u \left[ 1 + {32 B_0 \over F^2_{\pi}}
\left( L_6 (m_u + m_d + m_s) + L_8 m_u \right) - 3 \mu_{\pi} - 2 \mu_K \right.
\nonumber \\
& & + \left.{1 \over 2} \left( {m_d - m_u \over m_s - \hat{m}} \right)
( \mu_{\eta} - \mu_{\pi}) \right] \nonumber \\
& & + \, 32 \, {L_7 B_0 \over F^2_{\pi}} (m_u - m_d) (m_u - m_s) \nonumber \\
\mu_i^2 & = & {m^2_i \over 32 \pi^2 \, F^2_{\pi}} \, ln \,{m^2_i / \mu^2}\end{aligned}$$
with $m^{\ast}_d$ similar with $(m_u, m_d, m_s) \rightarrow (m_d,
m_u, m_s)$, and $m^{\ast}_s$ likewise with $(m_u, m_d, m_s) \rightarrow (m_s, m_d, m_u)$, plus some rearrangement of the chiral logs \[24\]. Each of these invariant masses $m^{\ast}_i$ is also invariant under changes in the scale $\mu$ which enters when using dimensional regularization. Many ratios of the $m^{\ast}_i$ are physical and can be evaluated
$${m^{\ast}_d \over m^{\ast}_s} = {1 \over 22} \; ; \; {m^{\ast}_u \over
m^{\ast}_d} = 0.2$$
and are fine measures of the breaking of chiral SU(3) and SU(2) symmetry. They do not address the $U(1)_A$ properties and cannot answer the question of whether strong CP violation disappears due to the $m_u = 0$ option.
A second possible direction is to try to use a model to calculate one of the chiral coefficients. Leutwyler has given a sum rule for $L_7$ and saturated it with an $\eta^{\prime}$ pole \[26\]. This is reasonable, but it is a model, and it is being applied in a sector where we have no previous experience to see if resonance saturation works in the presence of the reparameterization transformation. As with many models, one often finds other contributions which upset the original conclusion – as has been suggested for the $\pi^{\prime} (1300)$ intermediate state in the sum rule \[27\].
Ultimately the most promising way would be to find a way to measure observables connected to $U(1)_A$ anomalous Ward identities. Wyler and I proposed to use $\psi^{\prime}
\rightarrow
J/\psi \pi^0$ and $\psi^{\prime} \rightarrow J/\psi \eta$ to do this \[24, 28\]. These unlikely reactions were chosen because an analysis by Voloshin and Zakharov \[29\] claimed that by using a QCD multipole \[30\] expansion, these decays were mediated by the local operator $F
\tilde{F}$, such that a ratio of the decay rates can be converted into the ratio of Eq.(80). This yielded a set of ratios with $m_u \neq 0$. However, the Voloshin Zakharov analysis has been criticized by Luty and Sundrum \[31\], and unless I am missing something it seems to me that the criticism is justified. I have some hopes of getting around this problem in the future, but it is otherwise difficult to measure masses in $U(1)_A$ processes.
Where do we stand?
==================
We have been using symmetries to measure masses and mixing angles. The results
$$\begin{aligned}
V_{ud} & = & 0.9751 \pm 0.0005 \nonumber \\
V_{us} & = & 0.220 \pm 0.002 \nonumber \\
\mid V_{ud} \mid^2 + \mid V_{us} \mid^2 & = & 0.999 \pm 0.002\end{aligned}$$
are gratifyingly precise. For the expert the interest lies in the error bars, which are dominated by nuclear uncertainties in the case of $V_{ud}$, and SU(3) breaking for $V_{us}$.
In the case of light quark masses, we have one firm ratio
$${m_d - m_u \over m_s - \hat{m}} {2 \hat{m} \over m_s + \hat{m}} = 2.3 \times
10^{-3} ,$$
accurate to about 10%. We cannot at present measure a second ratio when we work beyond leading order, due to the reparameterization transformation. We are left instead with a one parameter family of mass ratios. The up quark mass has the widest range, presently including $m_u = 0$. Somewhat better known is $m_d / m_s \sim .05 (1 \pm 0.3)$. More precise statements than this are model dependent. In order to do better in the measurement process, we need to find a way to exploit axial $U(1)$ anomalous Ward identities.
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|
[DFTT-47/98]{}\
hep-th/9808146
[All roads lead to Rome:\
Supersolvables and Supercosets]{}
0.3cm
10.mm
[**I. Pesando** ]{}[^1]
[Dipartimento di Fisica Teorica , Universiá di Torino, via P. Giuria 1, I-10125 Torino, Istituto Nazionale di Fisica Nucleare (INFN) - sezione di Torino, Italy]{}
[ **ABSTRACT**]{}
> We show explicitly that the two recently proposed actions for the type IIB superstring propagating on $ AdS_{5}\times S_{5} $ agree completely.
In the last period there has been quite a lot of activity in finding the GS action for a type IIB string propagating on $ AdS_{5}\times S_{5} $, this has resulted in two proposals which are apparently different.
It is purpose of this letter to show that they are exactly the same thus clarifying some doubts.
The first proposed action ([@IP]) reads
$$\begin{aligned}
& S=\int d^{2}\xi \, \sqrt{-g}\, g^{\alpha \beta }\frac{1}{2}\times & \nonumber \\
& \left\{ \eta _{pq}\frac{\rho ^{2}}{e^{2}}\left[ \partial _{\alpha }x^{p}+\frac{ie}{4}\left( \vartheta ^{\dagger N}\widetilde{\sigma }^{p}\partial _{\alpha }\theta _{N}-\partial _{\alpha }\vartheta ^{\dagger N}\widetilde{\sigma }^{p}\theta _{N}\right) \right] \left[ \partial _{\beta }x^{q}+\frac{ie}{4}\left( \vartheta ^{\dagger M}\widetilde{\sigma }^{q}\partial _{\beta }\theta _{M}-\partial _{\beta }\vartheta ^{\dagger M}\widetilde{\sigma }^{q}\theta _{M}\right) \right] \right. \, & \nonumber \\
& \left. -\frac{1}{e^{2}}\frac{\partial _{\alpha }\rho \partial _{\beta }\rho }{\rho ^{2}}-\frac{4}{e^{2}}\delta _{ij}\frac{\partial _{\alpha }z^{i}\partial _{\beta }z^{j}}{(1-z^{2})^{2}}\right\} & \nonumber \\
& -\frac{i}{4e}\rho \: \left[ d\theta ^{\dagger N}\sigma _{2}d\theta ^{*M}\, \overline{\eta }_{N}(z)\eta _{cM}(z)+d\theta ^{T}_{N}\sigma _{2}d\theta _{M}\, \overline{\eta }_{c}^{N}(z)\eta ^{M}(z)\right] & \label{mia} \end{aligned}$$
and it was obtained using the supersolvable algebra tecnique ([@gruppoTo]) while the other ([@Kall-act][@Kall-Tsey][@MatsaevTseytlin][@Kallosh][@Kall-fixing]) reads
$$\begin{aligned}
S=-\frac{1}{2}\int d^{2}\xi \, & \left[ \sqrt{-g}\, g^{\alpha \beta }y^{2}\left( \partial _{\alpha }x^{p}+2i\: \widehat{\overline{\vartheta }}\Gamma ^{p}\partial _{\alpha }\widehat{\theta }\right) \left( \partial _{\beta }x_{p}+2i\: \widehat{\overline{\vartheta }}\Gamma _{p}\partial _{\beta }\widehat{\theta }\right) +\frac{1}{y^{2}}\partial _{\alpha }y^{t}\partial _{\beta }y_{t}\right. & \nonumber \\
& \left. +4i\epsilon ^{\alpha \beta }\: \partial _{\alpha }y^{t}\: \widehat{\overline{\vartheta }}\Gamma _{t}\partial _{\beta }\widehat{\theta }\right] & \label{loro} \end{aligned}$$
Let us now specify the notations and spell out the differences of the two actions. The first thing to notice and the most trivial is that the first one uses a mostly minus metric while the second one uses a mostly plus metric. Second the indeces run as follows $ p,q,\ldots =0,\ldots ,3 $, $ i,j,\ldots =5,\ldots ,9 $ , $ t,u,\ldots =4,\ldots ,9 $ because the first one uses horospherical coordinates $ \left\{ x^{p},\rho \right\} $ on $ AdS_{5} $ and projective $ \left\{ z^{i}\right\} $ coordinates on $ S_{5} $ while the second one uses cartesian coordinates $ \left\{ x^{p},y^{t}\right\} $. But the most striking difference is in the fermionic sector: the fermionic coordinates in ( \[mia\]) $ \theta _{N} $ are a set of $ N=4 $ Weyl spinor in $ D=4 $ (or that is the same half a spinor in $ D=5 $ : this is the effect of fixing the $ \kappa $ symmetry on $ AdS_{5|4} $ ) while the ones used in (\[loro\]) $ \widehat{\theta } $ are a Majorana -Weyl spinor in $ D=10 $ . In addition to this in (\[mia\]) enter the c-number Killing spinors on $ S_{5} $ $ \eta ^{N} $ and their conjugate $ \eta ^{N}_{c}\equiv C_{5}\eta ^{\dagger }_{N} $ [^2]which satisfy the equation $$D_{SO(5)}\eta ^{N}\equiv \left( d-\frac{1}{4}\varpi ^{ij}\tau _{ij}\right) \eta ^{N}=-\frac{e}{2}\tau _{i}\eta ^{N}E^{i}$$
In order to compare the two actions we need the explicit form of these Killing spinors; this is not too hard to obtain with the help of the ansatz
$$\eta ^{N}=\left( a(z^{2})+b(z^{2})\: z^{i}\tau _{i}\right) \epsilon ^{N}$$ where $ \epsilon ^{N} $ are constant spinors. The result is[^3]
$$\begin{aligned}
\eta ^{N} & = & \frac{1}{\sqrt{1-z^{2}}}\left( 1-z^{i}\tau _{i}\right) \epsilon ^{N}\nonumber \\
\eta ^{N}_{c} & = & \frac{1}{\sqrt{1-z^{2}}}\left( 1+z^{i}\tau _{i}\right) \epsilon ^{N}\label{killingS5} \end{aligned}$$
with $ \epsilon ^{\dagger }_{N}\epsilon ^{M}=\delta ^{M}_{N} $ in order to satisfy the normalisation condition $ \eta ^{\dagger }_{M}\eta ^{N}=\delta ^{N}_{M} $; we can therefore choose the normalisation $ \epsilon ^{N}_{\alpha }=\delta ^{N}_{\alpha } $ ($ \alpha $ is the $ 4D $ spinor index).
When we insert this expression in (\[mia\]) we get
$$\begin{aligned}
& S=\int d^{2}\xi \, \sqrt{-g}\, g^{\alpha \beta }\frac{1}{2}\times & \nonumber \\
& \left\{ \eta _{pq}\frac{\rho ^{2}}{e^{2}}\left[ \partial _{\alpha }x^{p}+\frac{ie}{4}\left( \vartheta ^{\dagger N}\widetilde{\sigma }^{p}\partial _{\alpha }\theta _{N}-\partial _{\alpha }\vartheta ^{\dagger N}\widetilde{\sigma }^{p}\theta _{N}\right) \right] \left[ \partial _{\beta }x^{q}+\frac{ie}{4}\left( \vartheta ^{\dagger M}\widetilde{\sigma }^{q}\partial _{\beta }\theta _{M}-\partial _{\beta }\vartheta ^{\dagger M}\widetilde{\sigma }^{q}\theta _{M}\right) \right] \right. \, & \nonumber \\
& \left. -\frac{1}{e^{2}}\frac{\partial _{\alpha }\rho \partial _{\beta }\rho }{\rho ^{2}}-\frac{4}{e^{2}}\delta _{ij}\frac{\partial _{\alpha }z^{i}\partial _{\beta }z^{j}}{(1-z^{2})^{2}}\right\} & \nonumber \\
& -\frac{i}{4e}\rho \: \frac{1+z^{2}}{1-z^{2}}\left[ d\theta ^{\dagger N}\sigma _{2}d\theta ^{*M}\, \epsilon ^{\dagger }_{N}C_{5}\epsilon _{M}^{*}+d\theta ^{T}_{N}\sigma _{2}d\theta _{M}\, \epsilon ^{T\, N}C^{-1}_{5}\epsilon ^{M}\right] & \nonumber \\
& -\frac{i}{4e}\rho \: \frac{2z^{i}}{1-z^{2}}\left[ d\theta ^{\dagger N}\sigma _{2}d\theta ^{*M}\, \epsilon ^{\dagger }_{N}\tau _{i}C_{5}\epsilon _{M}^{*}-d\theta ^{T}_{N}\sigma _{2}d\theta _{M}\, \epsilon ^{T\, N}C^{-1}_{5}\tau _{i}\epsilon ^{M}\right] & \nonumber \label{mia1} \end{aligned}$$
The comparison between the two bosonic kinetic terms and the two WZ terms suggests the following change of variables $$\begin{aligned}
y^{4} & = & \rho \frac{1+z^{2}}{1-z^{2}}\nonumber \\
y^{i} & = & \rho \: \frac{2z^{i}}{1-z^{2}}\label{bos-chan} \end{aligned}$$ In this way the bosonic part of the two actions (\[mia\]) and (\[loro\]) agrees perfectly; in particular we get: $$\frac{dy^{2}}{y^{2}}=\frac{{d\rho ^{2}}}{\rho ^{2}}-4\: \frac{dz^{2}}{\left( 1-z^{2}\right) ^{2}}$$
We are left with the task of making a $ 10D $ Majorana-Weyl spinor $ \widehat{\theta } $ out of 4 $ 4D $ Weyl spinors $ \theta _{N} $ and 4 constant $ 5D $ spinors $ \epsilon ^{N} $ . To this purpose we notice that with our conventions the following $ 10D $ spinor is Majorana-Weyl for all the $ 4D $ spinors $ \alpha _{N} $ :
$$\Theta _{MW}=\left( \begin{array}{c}
\left( \begin{array}{c}
\alpha _{N}\\
\sigma _{2}\alpha ^{*M}\, C_{5NM}
\end{array}\right) \otimes \epsilon ^{N}\\
0_{16}
\end{array}\right)$$
We can now compute all the relevant two fermions currents $ \overline{\Theta }_{MW\, 1}\Gamma ^{\widehat{a}}\Theta _{MW\, 2} $ using the explicit expression for $ \epsilon _{N} $ explictly:
$$\begin{aligned}
\overline{\Theta }_{MW\, 1}\Gamma ^{p}\Theta _{MW\, 2} & = & \alpha _{1}^{\dagger N}\widetilde{\sigma }^{p}\alpha _{N\, 2}-\alpha ^{\dagger N}_{2}\widetilde{\sigma }^{p}\alpha _{N\, 1}\\
\overline{\Theta }_{MW\, 1}\Gamma ^{4}\Theta _{MW\, 2} & = & i\: \alpha _{1}^{\dagger N}\sigma _{2}\alpha _{2}^{*M}\; \epsilon ^{\dagger }_{N}C_{5}\epsilon ^{*}_{M}-i\: \alpha ^{T}_{N\, 1}\sigma _{2}\alpha _{M\, 2}\; \epsilon ^{T\, N}C^{-1}_{5}\epsilon ^{M}\\
\overline{\Theta }_{MW\, 1}\Gamma ^{i}\Theta _{MW\, 2} & = & -i\: \alpha _{1}^{\dagger N}\sigma _{2}\alpha _{2}^{*M}\; \epsilon ^{\dagger }_{N}\tau ^{i}C_{5}\epsilon ^{*}_{M}-i\: \alpha ^{T}_{N\, 1}\sigma _{2}\alpha _{M\, 2}\; \epsilon ^{T\, N}C^{-1}_{5}\tau ^{i}\epsilon ^{M}\end{aligned}$$
In order to make the WZ term to agree we have to set $ \alpha _{N}=\frac{1}{2\sqrt{2}}e^{i\pi /4}\theta _{N} $, explicitly
$$\label{chang-ferm}
\widehat{\theta }=\frac{1}{2\sqrt{2}}e^{\frac{\iota \pi }{4}}\left( \begin{array}{c}
\left( \begin{array}{c}
\theta _{N}\\
-i\: \sigma _{2}\theta ^{*M}\, C_{5NM}
\end{array}\right) \otimes \epsilon ^{N}\\
0_{16}
\end{array}\right)$$
With this substitution our action (\[mia\]) becomes
$$\begin{aligned}
S=-\frac{1}{2}\int d^{2}\xi \, & \sqrt{-g}\, g^{\alpha \beta }\left[ \frac{y^{2}}{e^{2}}\left( \partial _{\alpha }x^{p}+2ie\: \widehat{\overline{\vartheta }}\Gamma ^{p}\partial _{\alpha }\widehat{\theta }\right) \left( \partial _{\beta }x_{p}+2ie\: \widehat{\overline{\vartheta }}\Gamma _{p}\partial _{\beta }\widehat{\theta }\right) +\frac{1}{e^{2}\, y^{2}}\partial _{\alpha }y^{t}\partial _{\beta }y_{t}\right] & \nonumber \\
& -4\frac{i}{e}y^{t}\: d\widehat{\overline{\vartheta }}\Gamma _{t}d\widehat{\theta } & \nonumber \label{loro} \end{aligned}$$
which coincides with (\[loro\]) exactly after performing an integration by part of the WZ term, thus proving the exact equivalence of the two proposed actions (at least on a word sheet without holes).
**Acknowledgments**
It is a pleasure to thank the Niels Bohr Institute for the hospitality.
R.R Metsaev and A.A. Tseytlin, Type IIB Superstring Action in $ AdS_{5}\times S_{5} $ Background, hepth/9805028;\
Supersymmetric D3 Brane Action $ AdS_{5}\times S_{5} $, hep-th/9506095 R. Kallosh, J. Rahmfeld and A. Rajaraman, Near Horizon Superspace, hep-th/9805217 R. Kallosh, Superconformal Actions in Killing Gauge, hep-th/9807206 R. Kallosh and J. Rahmfeld, The GS String Action on $ AdS_{5}\times S_{5} $, hep-th/9808038 R. Kallosh and A.K. Tseytlin, Simplifying Superstring Action on $ AdS_{5}\times S_{5} $, hep-th/9808088 G. Dall’Agata, D. Fabbri, C. Fraser, P. Fre’, P. Termonia and M. Trigiante, The $ Osp(8|4) $ Singleton Action from the Supermembrane, hep-th/9807115 I. Pesando, A $ \kappa $ Gauge Fixed type IIB Superstring Action on $ AdS_{5}\times S_{5} $ ,hep-th/9802020
[^1]: e-mail: ipesando@to.infn.it, pesando@alf.nbi.dk\
Work supported by the European Commission TMR programme ERBFMRX-CT96-004
[^2]: We use the following 10D $ \Gamma $ representation in terms of the $ AdS_{5} $ $ \gamma ^{a} $ matrices and of the corresponding $ S_{5} $ $ \tau ^{i} $ matrices $$\Gamma \widehat{^{a}}=\left\{ \gamma ^{a}\otimes 1_{4}\otimes \sigma _{1}\, ,\, 1_{4}\otimes \tau ^{i}\otimes (-\sigma _{2})\right\}$$
with $ \widehat{a},\widehat{b},\ldots =0\ldots 9 $ , $ a,b,\ldots =0\ldots 4 $ and $ i,j,\ldots =5\ldots 9 $.
And we write the 10D charge conjugation as
$$\widehat{C}=C\otimes C_{5}\otimes \sigma _{2}$$
where $ C $ , $ C_{5} $ are the $ AdS_{5} $ and $ S_{5} $ charge conjugation matrices.
All $ C $ .s , $ C^{-1}\gamma ^{a} $ , $ C^{-1}_{5}\tau ^{i} $ are antisymmetric while $ \widehat{C}^{-1}\Gamma ^{\widehat{a}} $ are symmetric.
Moreover we use the following (1+4)D $ \gamma $ explicit representation
$$\gamma _{p}=\left( \begin{array}{cc}
& \sigma _{p}\\
\widetilde{\sigma _{p}} &
\end{array}\right) \; \gamma _{4}=\left( \begin{array}{cc}
i\, 1_{2} & \\
& -i\, 1_{2}
\end{array}\right) \; C=\left( \begin{array}{cc}
i\sigma _{2} & \\
& i\sigma _{2}
\end{array}\right)$$
with $ p,q,\ldots =0\ldots 3 $, $ \sigma _{p}=\left\{ 1_{2},-\sigma _{1},-\sigma _{2},-\sigma _{3}\right\} $ and $ \widetilde{\sigma }_{p}=\left\{ 1_{2},\sigma _{1},\sigma _{2},\sigma _{3}\right\} $
and the following 5D $ \tau $
$$\tau _{5}=i\gamma _{0}\; \tau _{6,\ldots ,9}=\gamma _{1,\ldots ,4}\; C_{5}=C$$
[^3]: Our conventions for the $ S_{5} $ coset manifold with the Killing induced metric, i.e. negative definite, are
$$\begin{aligned}
dE^{i}-\varpi ^{i}_{.j}E^{j} & = & 0\\
d\varpi ^{ij}-\varpi ^{ik}\varpi _{k}^{.j} & = & e^{2}E^{i}E^{j}\end{aligned}$$
where all the potentials depend on the coordinate $ z $. Explicitly we have ( $ z^{2}=\eta _{ij}z^{i}z^{j}=-\delta _{ij}z^{i}z^{j} $ )
$$\begin{aligned}
E^{i} & = & \frac{2}{e}\frac{dz^{i}}{1-z^{2}}\\
\varpi ^{ij} & = & \frac{4z^{[i}dz^{j]}}{1-z^{2}}\end{aligned}$$
|
---
abstract: 'We work in set-theory without choice $\ZF$. Given a closed subset $F$ of $[0,1]^I$ which is a bounded subset of $\ell^1(I)$ ([*resp.*]{} such that $F \subseteq \ell^0(I)$), we show that the countable axiom of choice for finite subsets of $I$, ([*resp.*]{} the countable axiom of choice $\ACD$) implies that $F$ is compact. This enhances previous results where $\ACD$ ([*resp.*]{} the axiom of Dependent Choices $\DC$) was required. Moreover, if $I$ is linearly orderable (for example $I=\IR$), the closed unit ball of $\ell^2(I)$ is weakly compact (in $\ZF$).'
address: 'ERMIT, Département de Mathématiques et Informatique, Université de La Réunion, 15 avenue René Cassin - BP 7151 - 97715 Saint-Denis Messag. Cedex 9 FRANCE'
author:
- Marianne Morillon
bibliography:
- '../../biblio.bib'
title: Uniform Eberlein spaces and the Finite Axiom of Choice
---
EQUIPE RÉUNIONNAISE DE MATHÉMATIQUES ET INFORMATIQUE THÉORIQUE (ERMIT)
Introduction {#sec:intro}
============
We work in the set-theory without the Axiom of Choice $\ZF$. It is a well known theorem of Kelley (see [@Kel]) that, in $\ZF$, the Axiom of Choice (for short $\AC$) is equivalent to the Tychonov axiom $\T$: [*“Every family $(X_i)_{i \in I}$ of compact topological spaces has a compact product.”*]{} Here, a topological space $X$ is [*compact*]{} if every family $(F_i)_{i \in I}$ of closed subsets of $X$ satisfying the finite intersection property ([*FIP*]{}) has a non-empty intersection. However, some particular cases of the Tychonov axiom are provable in $\ZF$, for example:
\[rem:finite-prod-comp\] A finite product of compact spaces is compact (in $\ZF$).
Say that the topological space $X$ is [*closely-compact*]{} if there is a mapping $\Phi$ associating to every family $(F_i)_{i \in I}$ of closed subsets of $X$ satisfying the [*FIP*]{} an element $\Phi((F_i)_{i \in I})$ of $\cap_{i \in I}F_i$: the mapping $\Phi$ is a [*witness of closed-compactness*]{} on $X$. Notice that a compact topological space $X$ is closely-compact if and only if there exists a mapping $\Psi$ associating to every non-empty closed subset $F$ of $X$ an element of $F$.
\[ex:otc-comp\] Given a linear order $(X,\le)$ which is complete (every non-empty subset of $X$ has a least upper bound), then the order topology on $X$ is closely compact. In particular, the closed bounded interval $[0,1]$ of $\IR$ is closely compact.
The space $X$ is compact (the classical proof is valid in $\ZF$). Moreover $X$ is closely compact since one can consider the choice function associating to every non-empty closed subset its first element.
The following Theorem is provable in $\ZF$:
\[theo\*:prod-clos-comp\] Let $\alpha$ be an ordinal. If $(X_i,\Phi_i)_{i \in \alpha}$ is a family of witnessed closely-compact spaces, then $\prod_{i \in \alpha} X_i$ is closely-compact, and has a witness of closed-compactness which is definable from $(X_i,\Phi_i)_{i \in \alpha}$.
For every ordinal $\alpha$, the product topological space $[0,1]^{\alpha}$ is closely compact in $\ZF$.
Given a set $I$, denote by $B_1(I)$ the set of $x=(x_i)_{i \in I} \in \IR^I$ such that $\sum_{i \in I} |x_i| \le 1$: then $B_1(I)$ is a closed subset of $[-1,1]^I$. In this paper, we shall prove that $B_1(I)$ is compact using the [*countable axiom of choice for finite subsets of $I$*]{} (see Theorem \[theo:ac-fin2-ball-hilb-base\] in Section \[subsec:acfin2-u-eber\]). This enhances Corollary 1 of [@Mo07] and partially solves Question 2 in [@Mo07]. We shall deduce (see Corollary \[cor:B1deLO\]) that, if $I$ is linearly orderable, every closed subset of $[0,1]^I$ which is contained in $B_1(I)$ is closely compact. In particular, the closed unit ball of the Hilbert space $\ell^2(\IR)$ is compact in $\ZF$, and this solves Question 3 of [@Mo07]. Notice that $\{0,1\}^{\IR}$ (and $[0,1]^{\IR}$) is not compact in $\ZF$ (see [@Ker01]). We shall also prove that [*Eberlein*]{} closed subsets of $[0,1]^I$ are compact using the [*countable axiom of choice for subsets of $I$*]{} (see Corollary \[cor:acd2comp-eberlein\]) of Section \[subsec:acd2eberl\]. This enhances Corollary 3 in [@Mo07] where the same result was proved using the axiom of [*Dependent Choices*]{} $\DC$. This also solves Questions 4 and 5 thereof.
The paper is organized as follows: in Section \[sec:various-consAC\] we review various consequences of $\AC$ (in particular the [*countable axiom of choice $\ACD$*]{} and the axiom of choice [*restricted to finite subsets $\ACDF$*]{}) and the known links between them. In Section \[sec:var-eberlein\] we present definitions of uniform Eberlein spaces, strong Eberlein spaces and Eberlein spaces. In Section \[sec:comp-ZF\] we give some tools for compactness or sequential compactness in $\ZF$. In Section \[sec:alex-comp\], we recall the one-point compactification $\hat X$ of a discrete space $X$, and we show that for every ordinal $\alpha \ge 1$, the closed-compactness of ${\hat X}^{\alpha}$ is equivalent to the axiom of choice restricted to finite subsets of $X$. Finally, in Section \[subsec:ac-fin2compB1\] ([*resp.*]{} \[sec:dc-and-comp\]) we prove that the countable axiom of choice for finite sets ([*resp.*]{} the countable axiom of choice) implies that uniform Eberlein spaces ([*resp.*]{} Eberlein spaces) are closely compact ([*resp.*]{} compact.) A basic tool for these two last Sections is a “dyadic representation” of elements of powers of $[0,1]$ (see the Theorem in Section \[subsec:dyadic\]) which we found in [@B-R-W Lemma 1.1], and for which the authors cite [@Sim76].
Some weak forms of $\AC$ {#sec:various-consAC}
========================
In this Section, we review some weak forms of the Axiom of Choice which will be used in this paper and some known links between them. For detailed references and much information on this subject, see [@Ho-Ru].
Restricted axioms of choice
---------------------------
Given a formula $\phi$ of set-theory with one free variable $x$, consider the following consequence of $\AC$, denoted by $\AC(\phi)$: [*“For every non-empty family $A=(A_i)_{i \in I}$ of non-empty sets such that $\phi[x/A]$ holds, then $\prod_{i \in I}A(i)$ is non-empty.”*]{}
In the particular case where the formula $\phi$ says that “$x$ is a mapping with domain $I$ with values in some $\ZF$-definable class $\mathcal C$”, the statement $\AC(\phi)$ is denoted by $\AC_I^{\mathcal C}$.
The statement $\forall I \AC_I^{\mathcal C}$ is denoted by $\AC^{\mathcal C}$. The statement $\AC_I^{\mathcal C}$ where $\mathcal C$ is the collection of all sets is denoted by $\AC_I$.
For every set $X$, we denote by $fin(X)$ the set of finite subsets of $X$. We denote by $fin$ the (definable) class of finite sets.
So, given a set $X$, $\AC^{fin(X)}$ is the following statement: [*“For every non-empty family $(F_i)_{i \in I}$ of non-empty finite subsets of $X$, $\prod_{i \in I} F_i$ is non-empty.”*]{}, and $\AC^{fin}$ is the following statement: [*“For every non-empty family $(F_i)_{i \in I}$ of non-empty finite sets, $\prod_{i \in I} F_i$ is non-empty.”*]{} The [*countable Axiom of Choice*]{} says that:
> $\ACD$: [*If $(A_n)_{n \in \IN}$ is a family of non-empty sets, then there exists a mapping $f : \IN \to \cup_{n \in \IN}A_n$ associating to every $n \in \IN$ an element $f(n) \in A_n$.*]{}
And the [*countable Axiom of Choice for finite sets*]{} says that:
> $\ACDF$: [*If $(A_n)_{n \in \IN}$ is a family of finite non-empty sets, then there exists a mapping $f : \IN \to \cup_{n \in \IN}A_n$ associating to every $n \in \IN$ an element $f(n) \in A_n$.*]{}
Well-orderable union of finite sets
-----------------------------------
Given an infinite ordinal $\alpha$, and a class $\mathcal C$ of sets, we consider the following consequence of $\AC^{\mathcal C}$:
> $\Uwo_{\alpha}^{\mathcal C}$: [*For every family $(F_i)_{i \in \alpha}$ of elements of $\mathcal C$, the set $\cup_{i \in \alpha} F_i$ is well-orderable.*]{}
$\AC^{fin(X)}$ implies $\Uwo_{\alpha}^{fin(X)}$.
Dependent Choices
-----------------
The axiom of [*Dependent Choices*]{} says that:
> $\DC$: [*Given a non-empty set $X$ and a binary relation $R$ on $X$ such that $\forall x \in X \exists y \in X \; xRy$, then there exists a sequence $(x_n)_{n \in \IN}$ of $X$ such that for every $n \in \IN$, $x_n R x_{n+1}$.*]{}
Of course, $\AC \Rightarrow \DC \Rightarrow \ACD \Rightarrow \ACDF$. However, the converse statements are not provable in $\ZF$, and $\ACDF$ is not provable in $\ZF$ (see references in [@Ho-Ru]).
The “Tychonov” axiom {#subsec:tycho}
--------------------
Given a class $\mathcal C$ of compact topological spaces and a set $I$, we consider the following consequence of the Tychonov axiom:
> $\T_I^{\mathcal C}$: [*Every family $(X_i)_{i \in I}$ of spaces belonging to the class $\mathcal C$ has a compact product.*]{}
For example $\T_{\IN}^{fin(X)}$ is the statement [*“Every sequence of finite discrete subsets of $X$ has a compact product.”*]{}
\[rem:acfin-alpha\]
(i) \[it:rem-acfin1\] Given a set $X$, for every ordinal $\alpha$, $$\AC^{fin(X)} \Rightarrow \Uwo_{\alpha}^{fin(X)} \Rightarrow \T_{\alpha}^{fin(X)} \Rightarrow \AC_{\alpha}^{fin(X)}$$
(ii) \[it:rem-acfin2\] For every ordinal $\alpha$, $\Uwo_{\alpha}^{fin} \Leftrightarrow \T_{\alpha}^{fin} \Leftrightarrow \AC_{\alpha}^{fin}$.
$\Uwo_{\alpha}^{fin(X)} \Rightarrow \T_{\alpha}^{fin(X)}$: Given a family $(F_i)_{i \in \alpha}$ of finite subsets of $X$, the statement $\Uwo_{\alpha}^{fin(X)}$ implies the existence of a family $(\Phi_i)_{i \in \alpha}$ such that for each $i \in \alpha$, the discrete space is closely compact with witness $\Phi_i$. Using the Theorem of Section \[sec:intro\], it follows that $\prod_{i \in \alpha}F_i$ is (closely) compact. $ \T_{\alpha}^{fin(X)} \Rightarrow \AC_{\alpha}^{fin(X)}$: one can use Kelley’s argument (see [@Kel]).\
For $\AC_{\alpha}^{fin} \Rightarrow \Uwo_{\alpha}^{fin}$: given some family $(F_i)_{i \in \alpha}$ of finite non-empty sets, then, for each $i \in \alpha$, denote by $c_i:=\{0..c_{i-1}\}$ the (finite) cardinal of $F_i$; thus set $G_i$ of one-to-one mappings from $F_i$ to $c_i$ is finitel, and, by $\AC_{\alpha}^{fin}$, the set $\prod_{i \in \alpha}G_i$ is non-empty. This implies a well-order on the set $\cup_{i \in \alpha}F_i$.
Some classes of closed subsets of $[0,1]^I$ {#sec:var-eberlein}
===========================================
Let $I$ be a set. Given some element $x=(x_i)_{i \in I} \in \IR^I$, denote by $supp(x)$ the [*support*]{} $\{i \in I : x_i \neq 0\}$. Given some subset $A$ of $\IR$ containing $0$, denote by $A^{(I)}$ the set of elements of $A^I$ with [*finite*]{} support.
We endow the space $\IR^I$ with the product topology, which we denote by $\mathcal T_I$.
Eberlein closed subsets of $[0,1]^I$
------------------------------------
Given a set $I$, we denote by $\ell^{\infty}(I)$ the Banach space of bounded mappings $f:I \to \IR$, endowed with the “sup” norm. If $I$ is infinite, we denote by $c_0(I)$ the closed subspace of $\ell^{\infty}(I)$ consisting of $f \in \ell^{\infty}(I)$ such that $f$ converges to $0$ according to the Fréchet filter on $I$ ([*i.e.*]{} the set of cofinite subsets of $I$). Thus $$\ell^0(I):=\{x=(x_i)_{i \in I} : \; \forall \varepsilon>0 \exists F_0 \in \mathcal P_f(I) \forall i \in I \backslash F_0 \;
|x_i| \le \varepsilon\}$$ If $I$ is finite, then we define $c_0(I):=\ell^{\infty}(I)=\IR^I$.
A topological space $F$ is [*$I$-Eberlein*]{} if $F$ is a closed subset of $[0,1]^I$ and if $F \subseteq c_0(I)$. A topological space $X$ is [*Eberlein*]{} if $X$ is homeomorphic with some $I$-Eberlein space.
Amir and Lindenstrauss ([@Ami-Lin]) proved in $\ZFC$ that every weakly compact subset of a normed space is an Eberlein space. This result relies on the existence of a Markhushevich basis in every weakly compactly generated Banach space, and the proof of the existence of such a basis (see [@Fab-et-al]) relies on (much) Axiom of Choice.
Consider the compact topological space $X:=[0,1]^{\IN}$. Then, the closed subset $X$ of $[0,1]^{\IN}$ is not $\IN$-Eberlein. However, the mapping $f : X \to [0,1]^{\IN} \cap c_0(\IN)$ associating to each $x=(x_n)_{n \in \IN} \in X$ the element $(\frac{x_n}{n+1})_{n \in \IN}$ is continuous and one-to-one, so $X$ is homeomorphic with the compact (hence closed) subset $f[X]$ of $[0,1]^{\IN} \cap c_0(\IN)$. It follows that $X$ is homeomorphic with some $\IN$-Eberlein space.
\[prop:Eberlein-ppties\]
(i) \[it:Eber1\] Every closed subset of a $I$-Eberlein ([*resp.*]{} Eberlein) space is $I$-Eberlein ([*resp.*]{} Eberlein).
(ii) \[it:Eber2\] Let $(I_n)_{n \in \IN}$ be a sequence of pairwise disjoint sets, and denote by $I$ the set $\sqcup_{n \in \IN}I_n$. Let $(F_n)_{n \in \IN}$ be a sequence of topological spaces such that each $F_n$ is $I_n$-Eberlein . Then the closed subset $\prod_{n \in \IN} F_n$ of $[0,1]^I$ is homeomorphic with a $I$-Eberlein space.
is trivial. We prove . For every $n \in \IN$, let $f_n : F_n \to [0,1]^{I_n}$ be the mapping associating to each $x \in F_n$ the element $\frac{1}{n+1}f_n(x)$ of $[0,1]^{I_n}$. Let $f:=\prod_{n \in \IN }f_n : \prod_{n \in \IN} F_n \to [0,1]^I$. Then $f$ is one-to-one and continuous. Moreover, the subset $F:=Im(f)$ of $[0,1]^I$ is closed since $F$ is the product $\prod_{n \in \IN} \tilde F_n$ where for each $n \in \IN$, $\tilde F_n$ is the closed subset $\frac{1}{n+1}.F_n$ of $[0,1]^{I_n}$. Finally, it can be easily checked that $F \subseteq c_0(I)$.
Uniform Eberlein closed subsets of $[0,1]^I$
--------------------------------------------
### The ball $B_p(I)$, for $1 \le p < +\infty$.
For every real number $p \ge 1$, define as usual the normed space $\ell^p(I):=\{(x_i)_{i \in I} : \sum_i |x_i|^p < +\infty\}$ endowed with the norm $N_p : x=(x_i)_{i \in I} \mapsto ({\sum_{i}|x_i|^p})^{1/p}$. We denote by $B_p(I)$ the large unit ball $\{x \in \IR^I : \sum_i |x_i|^p \le 1\}$ of $\ell^p(I)$. Notice that for $p=1$ ([*resp.*]{} $1<p<+\infty$) the topology induced by $\mathcal T_I$ on $B_p(I)$ is the topology induced by the weak\* topology $\sigma(\ell^1(I),\ell^0(I))$ ([*resp.*]{} the topology induced by the weak topology $\sigma(\ell^p(I),\ell^q(I))$ where $q=\frac{p}{p-1}$ is the conjuguate of $p$). Also notice that for $1 \le p < +\infty$, $B_p(I)$ is a closed subset of $[0,1]^I$.
\[prop:B1-to-Bp\] If $1 \le p< +\infty$, then $B_p(I)$ is homeomorphic with $B_1(I)$.
Consider the mapping $h_p: B_1(I) \to B_p(I)$ associating to every $x=(x_i)_i \in B_1(I)$ the family $(\operatorname{sgn}(x_i) |x_i|^{1/p})_{i \in I}$.
It follows that for every $p,q \in [1,+\infty[$, spaces $B_p(I)$ and $B_q(I)$ are homeomorphic [*via*]{} $h_{p,q}:=h_q \circ h_p^{-1}: B_p(I) \to B_q(I)$.
### Uniform Eberlein spaces
Given a set $I$, and some real number $p \in [1,+\infty[$, we denote by $B^+_p(I)$ the positive ball of $\ell^p(I)$: $$B^+_p(I) := \{x=(x_i)_{i \in I} \in [0,1]^I : \sum_{i \in I} x_i^p \le 1\}$$
A topological space $F$ is [*$I$-uniform Eberlein*]{} if there exists a real number $p \in [1,+\infty[$ such that $F$ is a closed subset of $B^+_p(I)$. A topological space $X$ is [*uniform Eberlein*]{} if $X$ is homeomorphic with some $I$-uniform Eberlein space.
Of course, every $I$-uniform Eberlein space is $I$-Eberlein. Moreover, using Proposition \[prop:B1-to-Bp\], every $I$-uniform Eberlein space is homeomorphic with a closed subset of $B^+_1(I)$.
\[prop:unif-eber-prod-den\]
(i) \[it:unif-Eberl1\] For every set $I$, every closed subset of a $I$-uniform Eberlein space is $I$-uniform Eberlein.
(ii) \[it:unif-Eberl2\] Let $(I_n)_{n \in \IN}$ be a sequence of pairwise disjoint sets, and denote by $I$ the set $\sqcup_{n \in \IN}I_n$. Let $(F_n)_{n \in \IN}$ be a sequence of topological spaces such that each $F_n$ is a $I_n$-uniform Eberlein space. Then the closed subset $F:=\prod_{n \in \IN} F_n$ of $[0,1]^I$ is $I$-uniform Eberlein.
is easy. The proof of is similar to the proof of Proposition \[prop:Eberlein-ppties\]-.
In particular, the compact space $[0,1]^{\IN}$ (and thus every metrisable compact space) is $\IN$-uniform Eberlein. For every set $I$, ${B^+_1(I)}^{\IN}$ is $(I \times \IN)$-uniform Eberlein.
Let $Z:=\cap_{i \in I} \{(x,y) \in B^+_1(I) \times B^+_1(I) : \; x_i .y_i=0\}$: then $Z$ is a closed subset of $B^+_1(I) \times B^+_1(I)$, and the mapping $- : Z \to B_1(I)$ is an homeomorphism; it follows that $B_1(I)$ is homeomorphic with a $(I \times \{0,1\})$-uniform Eberlein space.
### Weakly closed bounded subsets of a Hilbert space
Given a Hilbert space $H$ with a Hilbert basis $(e_i)_{i \in I}$, then its closed unit ball (and thus every bounded weakly closed subset of $H$) is (linearly) homeomorphic with the uniform Eberlein space $B_2(I)$.
Consider the following statements (the first two ones were introduced in [@De-Mo] and [@Mo04] and are consequences of the Alaoglu theorem):
- $ \AUc$: The closed unit ball (and thus every bounded subset which is closed in the convex topology) of a uniformly convex Banach space is compact in the convex topology.
- $ \AH$: (Hilbert) The closed unit ball (and thus every bounded weakly closed subset) of a Hilbert space is weakly compact.
- $ \AHb$: (Hilbert with hilbertian basis) For every set $I$, the closed unit ball of $\ell^2(I)$ is weakly compact.
- $ \AHbf$: For every sequence $(F_n)_{n \in \IN} $ of finite sets, the closed unit ball of $\ell^2(\cup_{n \in \IN} F_n)$ is weakly compact.
Of course, $ \AUc \Rightarrow \AH \Rightarrow \AHb \Rightarrow \AHbf$.
\[theo:acd2wc-uc\]
(i) \[it:ACD2A1\] $\ACD \Rightarrow \AUc$.
(ii) \[it:A1not2ACD\] $ \AUc \not \Rightarrow \ACD$.
(iii) \[it:ACD2A2\] $ \AHbf \Rightarrow \ACDF$.
In this paper, we will prove that the following statements are equivalent: $\AHb$, $\AHbf$, $\AC^{fin}_{\omega}$ (see Corollary \[cor:acdf&Ahb\]).
Does $\AH$ imply $\AUc$? Does $\AHb$ imply $\AH$?
\[rem:ah-wo\] If a Hilbert space $H$ has a well orderable dense subset, then $H$ has a well orderable hilbertian basis, thus $H$ is isometrically isomorphic with some $\ell^2(\alpha)$ where $\alpha$ is an ordinal. In this case, the closed unit ball of $H$ endowed with the weak topology is homeomorphic with a closed subset of $[-1,1]^{\alpha}$, so this ball is weakly compact.
Strongly Eberlein closed subsets of $[0,1]^I$
---------------------------------------------
A topological space $F$ is [*$I$-strong Eberlein*]{} if $F$ is a closed subset of $[0,1]^I$ which is contained in $\{0,1\}^{(I)}$. A topological space $X$ is [*strong Eberlein*]{} if $X$ is homeomorphic with some $I$-strong Eberlein space.
Of course, every $I$-strong Eberlein set is $I$-Eberlein.
For every set $I$, every closed subset of a $I$-strong Eberlein space is $I$-strong Eberlein.
Compactness (in $\ZF$) {#sec:comp-ZF}
======================
Lattices and filters {#subsec:filters}
--------------------
Given a lattice $\mathcal L$ of subsets of a set $X$, say that a non-empty proper subset $\mathcal F$ of $\mathcal L$ is a [*filter*]{} if it satisfies the two following conditions:
(i) $\forall A, B \in \mathcal F, \; A \cap B \in \mathcal F$
(ii) $\forall A \in \mathcal F, \; \forall B \in \mathcal L, (A \subseteq B \Rightarrow B \in \mathcal F)$
Say that an element $A \in \mathcal L$ is [*$\mathcal F$-stationar*]{} if for every $F \in \mathcal F$, $A \cap F \neq \varnothing$.
\[rem:stat-comp\] Let $X$ be a topological space, let $\mathcal L$ be a lattice of closed subsets of $X$, and let $\mathcal F$ be a filter of $\mathcal L$. Let $K \in \mathcal L$. If $K$ is a compact subset of $X$ and if $K$ is $\mathcal F$-stationar, then $\cap \mathcal F$ is non-empty.
Given a family $(X_i)_{i \in I}$ of topological spaces, and denoting by $X$ the topological product of this family, a closed subset $F$ of $X$ is [*elementary*]{} if $F$ is a finite union of sets of the form $\prod_{i \neq i_0}X_i \times C$ where $i_0 \in I$ and $C$ is a closed subset of $X_{i_0}$.
Given a family $(X_i)_{i \in I}$ of topological spaces with product $X$, the set of elementary closed subsets of $X$ is a lattice of subsets of $X$ that we denote by $\mathcal L_X$. Notice that given a elementary closed subset $F$ of $X$, and some subset $J$ of $I$, the projection $p_J[F]$ is a closed subset of $\prod_{j \in J}X_j$.
Continuous image of a compact space {#subsec:comp&close-comp}
-----------------------------------
The following Proposition is easy:
\[prop-im-cont\] Let $X$, $Y$ be topological spaces and let $f: X \twoheadrightarrow Y$ be a continuous onto mapping. If $X$ is compact ([*resp.*]{} closely-compact), then $Y$ is also compact ([*resp.*]{} closely compact). If $\Phi$ is a witness of closed-compactness on $X$, then $Y$ is closely-compact, and has a witness of closed-compactness which is definable from $f$ and $\Phi$.
Sequential compactness {#subsec:seq-compact}
----------------------
We denote by $[\IN]^{\omega}$ the set of infinite subsets of $\IN$.
A topological space $X$ is [*sequentially compact*]{} if every sequence $(x_n)_{n \in \IN}$ of $X$ has an infinite subsequence which converges in $X$. A [*witness of sequential compactness*]{} on $X$ is a mapping $\Phi : X^{\IN} \to [\omega]^{\omega} \times X $ associating to each sequence $(x_n)_{n \in \IN}$ of $X$ an element $(A,l) \in [\omega]^{\omega} \times X$ such that $(x_n)_{n \in A}$ converges to $l$.
If $(X,\le)$ is a complete linear order, then $X$ is sequentially compact, with a witness definable from $(X,\le)$: given a sequence $(x_n)_{n \in \IN}$, build some infinite subset $A$ of $\IN$ such that $(x_n)_{n \in A}$ is monotone; then if $(x_n)_{n \in A}$ is ascending ([*resp.*]{} descending), then $(x_n)_{n \in A}$ converges to $\sup_{n \in A} x_n$ ([*resp.*]{} $\inf_{n \in A} x_n$).
\[ex:top-kelley\] Given an infinite set $X$, and some set $\infty \notin X$, consider the topology on $\tilde X:=X \cup \{\infty\}$ generated by cofinite subsets of $\tilde X$ and $\{\infty\}$. This topology is compact and $T_1$ but it is not $T_2$. This topology is sequentially compact, and, given a point $a \in X$, there is a witness of sequential compactness which is definable from $X$ and $a$: given a sequence $(x_n)_{n \in \IN}$ of $\tilde X$, either the set of terms $\{x_n : n \in \IN\}$ is finite, and then one can define by induction an infinite subset $A$ of $\IN$ such that $\{x_n : n \in A\}$ is constant; else one can define by induction an infinite subset $A$ of $\IN$ such that $\{x_n : n \in A\}$ is one-to-one, thus it converges to $a$ (and also to every point in $X$).
Notice that the topology in Example \[ex:top-kelley\] is the one used by Kelley (see [@Kel]) to prove that “Tychonov implies $\AC$”. The following Lemma is easy:
\[lem:imag-cont-seq-comp\] Let $X,Y$ be two topological spaces and let $f: X \twoheadrightarrow Y$ be an onto continuous mapping which has a section $j$ (for example if $f$ is one-to-one). If $X$ is sequentially compact, then $Y$ is also sequentially compact. Moreover, if there is a witness $\Phi$ of sequential compactness on $X$, there also exists a witness of sequential compactness on $Y$ which is definable from $f$,$\Phi$ and $j$.
\[lem:prod-wit-seq-comp\] Let $(X_n,\phi_n)_{n \in \IN}$ be a sequence of witnessed sequentially compact spaces. The space $\prod_{n \in \IN} X_n$ is sequentially compact, and has a witness definable from $(X_n,\phi_n)_{n \in \IN}$.
Usual diagonalization.
\[ex:seq-compact\] If $D$ is a countable set, then the topological space $[0,1]^D$ is sequentially compact, a witness of sequential compactness beeing definable from every well order on $D$.
Say that a sequentially compact topological space $X$ is [*witnessable*]{} if there exists a witness of sequential compactness on $X$. It follows from Lemma \[lem:prod-wit-seq-comp\], that with $\ACD$, every sequence $(K_n)_{n \in \IN}$ of witnessable sequentially compact spaces has a product which is sequentially compact.
$\ACD$ and countable products of compact spaces
-----------------------------------------------
Denote by $T^{comp}_{\omega}$ the following statement: [*“Every sequence of compact spaces has a compact product.”*]{} Then Kelley’s argument shows that $T^{comp}_{\omega} \Rightarrow \ACD$. However, it is an open question (see [@Br85], [@HKRS]) to know whether $\ACD$ implies $T^{comp}_{\omega}$.
A topological space $X$ is [*$\omega$-compact*]{} if every descending sequence $(F_n)_{n \in \IN}$ of non-empty closed subsets of $X$ has a non-empty intersection. Say that the space $X$ is [*cluster-compact*]{} if every sequence $(x_n)_{ \in \IN}$ of $X$ has a [*cluster point*]{} [*i.e.*]{} the set $\cap_{n \in \IN} \overline{\{x_k : k \ge n\}}$ is non-empty.
\[rem:acd-seq-comp\]
(i) \[it:acd-seq-comp1\] Notice that $\text{sequentially compact} \Rightarrow \text{``cluster-compact''}$. Also notice that $\text{``$\omega$-compact''} \Rightarrow \text{``cluster-compact''}$ and that the converse holds with $\ACD$ (see [@HKRS Lemma 1]).
(ii) Given a sequence $(K_n)_{n \in \IN}$ of compact spaces, then, denoting by $K$ the product of this family, $K$ is compact [*iff*]{} $K$ is $\omega$-compact (see [@HKRS Theorem 6]).
(iii) \[it:acd-prod-seq-comp\] If the product $K$ of a sequence $(K_n)_{n \in \IN}$ of compact spaces is sequentially compact, then $\ACD$ implies that $K$ is compact.
\[prop:acd-prod-comp\] $\ACD$ is equivalent to the following statement: [*“Every sequence $(K_n)_{n \in \IN}$ of witnessable sequentially compact spaces which are also compact has a compact product.”*]{}
$\Rightarrow$: Given a sequence $(K_n)_{n \in \IN}$ of witnessable sequentially compact spaces which are also compact, then, using $\ACD$, one can choose a witness of sequential compactness on every space $K_n$. It follows by Lemma \[lem:prod-wit-seq-comp\] that $K$ is sequentially compact, whence $K$ is compact by Remark \[rem:acd-seq-comp\]-.\
$\Leftarrow$: We use Kelley’s argument (see [@Kel]). Let $(A_n)_{n \in \IN}$ be a sequence of non-empty sets. Consider some element $\infty \notin \cup_{n \in \IN} A_n$, and for every $n \in \IN$, denote by $K_n$ the set $A_n \cup \{\infty\}$ endowed with the topology generated by $\{\infty\}$ and cofinite subsets of $K_n$ (see Example \[ex:top-kelley\]). Then each $K_n$ is compact and sequentially compact; moreover, given an element $a \in A_n$, there is a witness of sequential compactness on $K_n$ which is definable from $A_n$,$\infty$ and $a$. So each $K_n$ is a witnessable sequentially compact space. It follows from the hypothesis that the product $K:=\prod_{n \in \IN}K_n$ is compact. We end as in Kelley’s proof: for every $n \in \IN$, let $F_n$ be the closed set $A_n \times \prod_{i \neq n} K_i$. By compactness of $K$, the set $\cap_{n \in \IN} F_n$ is non-empty. This yields an element of $\prod_{n \in \IN} A_n$.
One-point compactifications and related spaces {#sec:alex-comp}
==============================================
The one-point compactification of a set {#subsec:alex}
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Given a set $X$, we denote by $\hat X$ the Alexandrov compactification of the (Hausdorff locally compact) discrete space $X$: $\hat X := X \cup \{\infty\}$ where $\infty$ is some set $\notin X$ (for example $\infty:=\{x \in X : x \notin x\}$; if $X$ is finite, then $\hat X$ is discrete else open subsets of the space $\hat X$ are subsets of $X$ or cofinite subsets of $\hat X$ containing $\infty$. Notice that the space $\hat X$ is compact and Hausdorff in $\ZF$.
\[ex:hatX\_unif-eb\] Given a discrete topological space $X$, the one-point compactification ${\hat{X}}$ of $X$ is $X$-uniform Eberlein: consider the Hilbert space $\ell^2(X)$; and denote by $(e_i)_{i \in X}$ the canonical basis of the vector space $\IR^{(X)}$; then the subspace $X=\{e_i : i \in X\}$ of $\IR^{(X)}$ is discrete and the weakly closed and bounded subset $X \cup 0_{\IR^X}$ is the one-point compactification $\hat X$ of $X$.
Various notions of compactness for $\hat X^{\alpha}$, $\alpha$ ordinal
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### $\hat X^{\IN}$ is sequentially compact
\[prop:hatX-seq-comp\] Let $X$ be an infinite set.
(i) \[it:X-chap-2\] The space $\hat X$ is sequentially compact and has a witness of sequential compactness, definable from $X$.
(ii) \[polyad-den-seq\] The space $\hat X^{\IN}$ is sequentially compact with a witness definable from $X$.
We define a witness $\Phi$ of sequential compactness on $X$ as follows: given a sequence $x=(x_n)_{n \in \IN}$ of $\hat X$, if the set $T:=\{x_k : k \in \IN\}$ is infinite, we build (by induction) some infinite subset $A$ of $\IN$ such that $\{x_k : k \in A\}$ is one-to-one, and we define $\Phi(x):=(A,\infty)$; else the set $T$ is finite, so we build by induction some infinite subset $A$ of $\IN$ such that the sequence $\{x_k : k \in A\}$ is a singleton $\{l\}$, and we define $\Phi(x):=(A,l)$.\
We apply and Lemma \[lem:prod-wit-seq-comp\] in Section \[subsec:seq-compact\].
### $\AC^{fin}$ and closed-compactness
\[prop:X-hat-comp\] Let $X$ be a set.
(i) \[it:X-hat-0\] There is a mapping associating to every non-empty closed subset $F$ of $\hat X$, a finite non-empty closed subset $\tilde F$ of $F$.
(ii) \[it:X-hat-2\] $\AC^{fin(X)} \Leftrightarrow \text{ The space $\hat X$ is closely compact}$.
We may assume that $X$ is infinite.\
Given a non-empty closed subset $F$ of $\hat X$, define $\tilde F:= \{\infty\}$ if $\infty \in F$ and $\tilde F:=F$ if $F$ is finite and $\infty \notin F$.\
Use .
Spaces ${\hat X}^{\alpha}$, $\alpha$ ordinal
--------------------------------------------
For every set $X$, the space $\hat X$ is $X$-uniform Eberlein, so, given an ordinal $\alpha$, ${\hat X}^{\alpha}$ is $X \times \alpha$-uniform Eberlein (see Proposition \[prop:unif-eber-prod-den\]-).
\[theo:acdf-comp-polyad\] Let $X$ be a set. Let $\alpha$ be an ordinal $\ge 1$.
(i) \[it:hatx-puiss-alpha-1\] $\T_{\alpha}^{fin(X)} \Leftrightarrow \text{``$\hat X^{\alpha}$ is compact''}$.
(ii) \[it:hatx-puiss-alpha-2\] $\AC^{fin(X)} \Leftrightarrow \text{``$\hat X^{\alpha}$ is closely compact''}$.
$\Rightarrow$: Let $P$ be the topological product space $\hat X^{\alpha}$. Let $\mathcal F$ be a filter of the lattice $\mathcal L_X$ of elementary closed subsets of $P$. We are going to define by transfinite recursion a family $(G_n)_{n \in \alpha}$ of finite subsets of $\hat X$ such that, denoting for every $n \in \alpha$ by $Z_n$ the elementary closed subset $G_n \times {\hat X}^{\alpha \backslash \{n\}}$ of $P$, the set $\mathcal F \cup \{Z_i : i < n\}$ satisfies the finite intersection property. Given some $n \in \alpha$, we define $G_n$ in function of $(G_i)_{i < n}$ as follows: denote by $\mathcal G$ the filter generated by $\mathcal F \cup \{Z_i : i <n\}$; since $\hat X$ is compact, the closed subset $F_n:=\cap p_{\{n\}}[\mathcal G]$ is non-empty, so let $G_n:=\tilde F_n$ and let $Z_n:=G_n \times {\hat X}^{\alpha \backslash \{n\}}$. Denote by $\tilde {\mathcal F}$ the filter generated by $\mathcal F \cup \{Z_i : i <\alpha\}$. Using $\T_{\alpha}^{fin(X)}$, the product space $F:=\prod_{n \in \alpha} G_n$ is compact, and non-empty since $\T_{\alpha}^{fin(X)}$ implies $\T_{\alpha}^{fin(X)}$ (see Remark \[rem:acfin-alpha\]). Moreover, the closed subset $F$ of $P$ is $\tilde{\mathcal F}$-stationnar: it follows from Remark \[rem:stat-comp\] of Section \[subsec:filters\] that $\cap \tilde {\mathcal F}$ is non-empty, whence $\cap \mathcal F \neq \varnothing$. $\Leftarrow$: Let $(F_i)_{i < \alpha}$ be a family of finite subsets of $X$, endowed with the discrete topology. Then $\prod_{i < \alpha} F_i$ is compact because it is a closed subset of the compact Hausdorff space ${\hat X}^{\alpha}$.\
$\Rightarrow$: Use Proposition \[prop:X-hat-comp\]- and the Theorem of Section \[subsec:comp&close-comp\]. $\Leftarrow$: If $\hat X^{\alpha}$ is closely compact, then so is its continuous image $\hat X$, whence $\AC^{fin(X)}$ holds (using Proposition \[prop:X-hat-comp\]-).
Spaces $\sigma_n(X)$, $n$ integer $\ge 1$
-----------------------------------------
Given a set $X$, for every integer $n \ge 1$, let $$\sigma_n(X):=\{x \in \{0,1\}^{(X)}: |supp(x)| \le n\}$$
Thus $\sigma_n(X)$ is the set of elements of $\IR^{(X)}$ with support having at most $n$ elements. Notice that the space $\sigma_n(X)$ is strong Eberlein.
\[rem:sigma-n\]
(i) \[it:sigma1-ue\] The space $\sigma_1(X)$ is the one-point compactification of the discrete space $X$ (thus $\sigma_1(X)$ is uniform Eberlein).
(ii) \[it:sigma-n-im-sigma1\] The mapping $\mathcal U_n: (\sigma_1(X))^n \to \sigma_n(X)$ associating to each $(x_1, \dots,x_n)$ the set $\cup_{1 \le i \le n} x_i$ is continuous.
Use Example \[ex:hatX\_unif-eb\]. : easy.
### Compactness and closed compactness of $\sigma_n(X)$
\[prop:sigma-n-comp\] Let $X$ be a set, and let $n$ be some integer $ \ge 1$.
(i) \[it:sigma-n-comp1\] Both spaces $(\sigma_1(X))^n$ and $\sigma_n(X)$ are compact.
(ii) \[it:sigma-n-comp2\] With $\AC^{fin(X)}$, both spaces $(\sigma_1(X))^n$ and $\sigma_n(X)$ are closely compact (with witnesses of closed compactness definable from $X$, $n$ and some choice function on non-empty finite subsets of $X$).
The results and for $\sigma_n(X)$ follow from the result on $(\sigma_1(X))^n$ thanks to Proposition \[prop-im-cont\] and the continuous onto mapping $\mathcal U_n: (\sigma_1(X))^n \to \sigma_n(X)$ defined in Remark \[rem:sigma-n\]. The result for $(\sigma_1(X))^n$ comes from Remark \[rem:finite-prod-comp\]. We prove for $(\sigma_1(X))^n$: with $\AC^{fin(X)}$, $\sigma_1(X)$ is closely compact, so the space $(\sigma_1(X))^n$ is also closely compact because it is a finite power of a closely compact space (use the Theorem of Section \[sec:intro\]).
### Sequential compactness of $\prod_{n \in \IN}\sigma_n(X)$
\[prop:seq-comp\] Let $X$ be a set, and let $n$ be some integer $ \ge 1$.
(i) \[sigma-n-seq-comp\] The space $\sigma_n(X)$ is sequentially compact, with a witness definable from $X$ and $n$.
(ii) \[it:prod-sigma-seq-comp\] The space $\prod_{k \in \IN} \sigma_k(X)$ is sequentially compact, with a witness definable from $X$.
The proof is by induction on $n$. For $n=1$, we already know that $\sigma_1(X)=\hat X$ is sequentially compact with a witness definable from $X$ (use Proposition \[prop:hatX-seq-comp\]-). We now assume that for some integer $n\ge 1$, each space $\sigma_k(X)$ ($1 \le k \le n$) is sequentially compact with a witness $\Phi_k$ definable from $X$ and $k$. Let $(F_k)_ {k \in \IN}$ be a sequence of $\sigma_{n+1}(X)$. For every $\nu \in \IN$, let $\mathcal A_{\nu}:= \{A \in [\IN]^{\omega} : \; \forall i \neq j \in A \; |F_i \cap F_j| =\nu\}$. Let $\nu_0$ be the first element of $\IN$ such that the set $\mathcal A_{\nu_0}$ is non-empty. One can build by induction some element $A \in \mathcal A_{\nu_0}$, which is definable from $X$ and $(F_n)_{n \in \IN}$. If $\nu=0$, then the subsequence $(F_n)_{n \in A}$ converges to $\infty$. Else, there exists $a \in X$ such that the set $D_a := \{n \in A : a \in F_n\}$ is infinite. Build by induction some infinite subset $B$ of $A$ such that there exists some element $a \in X$ satisfying $\forall n \in B \; a \in F_n$. Let $R$ be the non-empty finite set $\cap_{n \in B}F_n$; let $p$ be the cardinal of $R$. The sequence $(F_n \backslash R)_{n \in B}$ lives in $\sigma_{n+1 - p}(X)$ thus, using the witness $\Phi_{n+1-p}$, it has an infinite subsequence $(F_n \backslash R)_{n \in C}$ which converges to some $L \in \sigma_{n+1 - p}(X)$. It follows that $(F_n \cup R)_{n \in C}$ converges to $L \cup R$ in $\sigma_{n+1}(X)$.\
Use Proposition \[prop:hatX-seq-comp\]- or Lemma \[lem:prod-wit-seq-comp\].
### $\AC^{fin(X)}$ and closed-compactness of the space $\prod_{n \in \IN}\sigma_n(X)$
\[theo:pros-sigma-n\] Let $X$ be a set.
(i) \[it:prod-sigma-closely-comp\] $\AC^{fin(X)} \Leftrightarrow \text{``$\prod_{n \in \IN} \sigma_n(X)$ is closely compact''}$.
(ii) \[it:prod-sigma-comp\] $\T_{\IN}^{fin(X)} \Leftrightarrow \text{``$\prod_{n \in \IN} \sigma_n(X)$ is compact''}$.
We may assume that $X$ is infinite. In both cases, we use Proposition \[prop-im-cont\] and the fact that the space $\prod_{n \in \IN } \sigma_n(X)$ is a continuous image of $\prod_{n \in \IN} \sigma_1(X)^n$, which is homeomorphic with ${\hat X}^{\IN}$.\
$\Rightarrow$: with $\AC^{fin(X)}$, ${\hat X}^{\IN}$ is closely compact (see Proposition \[theo:acdf-comp-polyad\]), and so is its continuous image $\prod_{n \in \IN } \sigma_n(X)$ . $\Leftarrow$: if $\prod_{n \in \IN} \sigma_n(X)$ is closely compact, then so is its continuous image $\sigma_1(X)=\hat X$, thus $\AC^{fin(X)}$ holds.\
$\Rightarrow$: Using Proposition \[theo:acdf-comp-polyad\]-, $\T_{\IN}^{fin(X)}$ implies that “${\hat X}^{\IN}$ is compact”. Using Remark \[rem:sigma-n\], it follows that “$\prod_{n \in \IN} \sigma_n(X)$ is compact”. $\Leftarrow$: if $\prod_{n \in \IN} \sigma_n(X)$ is compact, then its closed subset ${\sigma_1(X)}^{\IN}$ is also compact, thus $\T_{\IN}^{fin(X)}$ holds by Proposition \[theo:acdf-comp-polyad\]-.
$\AC^{fin(I)}$ and closed compactness of $B_1(I)$ {#subsec:ac-fin2compB1}
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Dyadic representations {#subsec:dyadic}
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For every $n \in \IN$, let $\varepsilon_n := \frac{1}{2^{n+1}}$. Then the mapping $\phi : \{0,1\}^{\IN} \to [0,1]$ associating to every $(x_n)_{n \in \IN}$ the real number $\sum_n \varepsilon_n x_n$ is continuous (a uniformly convergent series of continuous functions), onto, and $\phi$ has a (definable) section.
[@B-R-W Lemma 1.1] Let $I$ be a set, and for every $n \in \IN$, let $I_n:= \{n\} \times I$. Let $F$ be a closed subset of $[0,1]^I$. Consider the power mapping $g:=\phi^I : \{0,1\}^{\IN \times I} \to [0,1]^I$. For every $n \in \IN$, let $j_n : \{0,1\}^{I_n} \to \{0,1\}^{\IN \times I}$ be the canonical inclusion mapping. Let $Z:= g^{-1}[F]$ and, for every $n \in \IN$, let $Z_n := j_n^{-1}[Z]$: thus $Z$ is a closed subset of $\prod_{n \in \IN}Z_n$ and $g: Z \to F$ is continuous, onto, with a definable section
(i) \[it:dyadic2\] If $F \subseteq B^+_1(I)$, then for every $n_0 \in \IN$, $Z_{n_0} \subseteq \sigma_{M_{n_0}}(I_{n_0})$ where $M_{n_0}:= \lfloor \frac{1}{\varepsilon_n} \rfloor$ (the integral part of $\frac{1}{\varepsilon_{n_0}}$), thus $F$ is the continuous image of some closed subset of ${\hat I}^{\IN}$.
(ii) \[it:dyadic3\] If $F \subseteq \ell^0(I)$, then for every $n_0 \in \IN$, $Z_{n_0} \subseteq \{0,1\}^{(I_{n_0})}$
Let $n_0 \in \IN$ and let $(x^{n_0}_i)_{i \in I} \in Z_{n_0}$; let $x=(x^n_i)_{n \in \IN, i \in I} \in Z$ such that $j_{n_0}((x^{n_0}_i)_{i \in I})=x$. Since $F \subseteq B^+_1(I)$, $\sum_{i,n} \varepsilon_n x^n_{i} \le 1$, thus $\sum_{i \in I} \varepsilon_{n_0} x^{n_0}_{i} \le 1$; it follows that the set $\{i \in I : x^{n_0}_{i}=1 \}$ has a cardinal $\le \frac{1}{\varepsilon_{n_0}}$.\
Since $F \subseteq \ell^0(I)$, $(x^{n_0}_i)_{i \in I} \in \ell^0(I) \cap \{0,1\}^I$ thus $Z_{n_0} \subseteq \{0,1\}^{(I_{n_0})}$.
\[rem:B+&B\] The mapping $- : B^+_1(I) \times B^+_1(I) \to B_1(I)$ is continuous and onto thus $B_1(I)$ is also the continuous image of a closed subset of ${\hat I}^{\IN}$.
Aviles ([@Avi07]) proved that $B^+_1(I)$ -and thus $B_1(I)$- is a continuous image of ${\hat I}^{\IN}$ (and not only of a closed subset of ${\hat I}^{\IN}$).
Another equivalent of $\AC^{fin(I)}$ {#subsec:acfin2-u-eber}
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\[theo:ac-fin2-ball-hilb-base\] Let $I$ be a set.
(i) \[it:acf&B1\] $\AC^{fin(I)} \Leftrightarrow \text {``$B_1(I)$ is closely compact.''}$ Moreover, a witness of closed compactness on $B_1(I)$ is definable from $I$ and a choice function for non-empty finite subsets of $I$ and conversely.
(ii) \[it:tych&B1\] $\T_{\IN}^{fin(I)}$ implies that $B_1(I)$ is compact.
(iii) \[it:ZF&B1\] The space $B_1(I)$ is sequentially compact, with a witness definable from $I$.
Using the previous Theorem and Remark \[rem:B+&B\], consider some sequence $(M_n)_{n \in \IN}$, some closed subset $Z$ of $\prod_{n \in \IN} \sigma_{M_n}(I)$ and some continuous onto mapping $g : Z \to B_1(I)$, with a (definable) section.\
$\Rightarrow$: Using $\AC^{fin(I)}$ and Theorem \[theo:pros-sigma-n\]-, $Z$ is closely compact thus $B_1(I)=g[F]$ is closely compact. $\Leftarrow$: If $B_1(I)$ is closely compact, then $\hat I$ (which is a closed subset of $B_1(I)$ -see Example \[ex:hatX\_unif-eb\] in Section \[subsec:alex\]-) is also closely compact.\
Using $\T_{\IN}^{fin(I)}$ and Theorem \[theo:pros-sigma-n\]-, $Z$ is compact thus $B_1(I)=g[F]$ is also compact.\
By Proposition \[prop:seq-comp\]-, $Z$ is sequentially compact and $g$ is continuous with a section, thus Lemma \[lem:imag-cont-seq-comp\] implies that $B_1(I)$ is also sequentially compact with a witness definable from $I$.
\[cor:equ-B1-comp\] Given a set $I$, the following statements are equivalent:
(i) \[it:tych-omega-fin-1\] $\Uwo_{\IN}^{fin(I)}$
(ii) \[it:tych-omega-fin-2\] $\T_{\IN}^{fin(I)}$
(iii) \[it:tych-omega-fin-3\] The space $B_1(I)$ is compact.
(iv) \[it:tych-omega-fin-4\] For every sequence $(F_n)_{n \in \IN}$ of finite subsets of $I$, the space $B_1(\cup_{n \in \IN} F_n)$ is compact.
$\Rightarrow$ is easy and $\Rightarrow$ follows from Theorem \[theo:ac-fin2-ball-hilb-base\]. $\Rightarrow$ is easy. We show that $\Rightarrow$ . The idea of the implication is in [@Fo-Mo th. 9 p. 16]: we sketch it for sake of completeness. Let $(F_n)_{n \in \IN}$ be a disjoint sequence of non-empty finite sets of $I$. Let us show that $D:=\cup_{n \in \IN}F_n$ is countable. The Hilbert spaces $H:=\ell^2(D)$ and $\oplus_{\ell^2(\IN)} \ell^2(F_n)$ are isometrically isomorph. For every $n \in \IN$, let $\varepsilon_n : |F_n| \to ]0,1[$ be a strictly increasing mapping such that $\sum_{n \in \IN} \sum_{0 \le i < |F_n|}\varepsilon_n(i)^2=1$. For every $n \in \IN$, let $\tilde F_n :=\{x \in B_{H} : \; \forall m \neq n, x_{\restriction F_m}=0 \text{ and } x_{\restriction F_n} \text{ is one-to-one from } F_n \text{ onto } rg(\varepsilon_n)\}$. Then each $\tilde F_n$ is a weakly closed subset of $B_{\ell^2(D)}$ and the sequence $(\tilde F_n)_{n \in \IN}$ satisfies the finite intersection property. The compactness of $B_2(D)$ implies that $Z:=\cap_{n \in \IN} \tilde F_n$ is non-empty. Given an element $f=(f_n)_{n \in \IN}$ of $Z$, each $f_n$ defines a well-order on the finite set $F_n$, thus $\cup_{n \in \IN} F_n$ is countable.
For $I=\mathcal P(\IR)$, none of the equivalent statements in Corollary \[cor:equ-B1-comp\] is provable in $\ZF$. Indeed, there is a model of $\ZF$ where there exists a sequence $(P_n)_{n \in \IN}$ of pairs of subsets of $\IR$ such that $\prod_{n \in \IN} P_n$ is empty.
Thus, the statement [*“The closed unit ball of $\ell^2(\mathcal P(\IR))$ is weakly compact.”*]{} is not provable in $\ZF$.
\[cor:acdf&Ahb\] The following statements are equivalent: $\AHb$, $ \AHbf$, $\ACDF$.
Consequences
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\[cor:B1deLO\] If a set $I$ is linearly orderable, then $B_1(I)$ is compact.
If $I$ is linearly orderable, then $\AC^{fin(I)}$ holds.
For every ordinal $\alpha$, the set $\mathcal P(\alpha)$ is linearly orderable, thus $\AC^{fin(\mathcal P(\alpha))}$ holds. In particular, $\IR$ is equipotent with $\mathcal P(\IN)$ so the closed unit ball of $\ell^2(\IR)$ is closely compact. This solves Question 3 of [@Mo07].
Does $\AC^{fin}$ imply $\AH$? What is the power of the statement [*“Every Hilbert space has a hilbertian basis”*]{}? Is this statement provable in $\ZF$? Does it imply $\AC$?
$\ACD$ and Eberlein spaces {#sec:dc-and-comp}
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Given a set $I$, a closed subset $F$ of $[0,1]^I$ is [*$I$-Corson*]{} if every element $x \in F$ has a countable support.
Sequential compactness of $I$-Eberlein spaces
---------------------------------------------
Given a set $I$, denote by $count(I)$ the set of finite or countable subsets of $I$. Consider the following consequence of $\ACD$:\
$\Uwo_{\IN}^{count(I)}$: [*“Every countable union of countable subsets of $I$ is countable.”*]{}
\[prop:seq-comp-corson\] Let $I$ be a set and let $F$ be a closed subset of $[0,1]^I$.
(i) \[it:strong-eber-comp2\] $\Uwo^{fin(I)}_{\IN}$ implies that: -if $F \subseteq [0,1]^{(I)}$, then $F$ is sequentially compact. -if $F \subseteq c_0(I)$, then $F$ is $I$-Corson.
(ii) $\Uwo^{count(I)}_{\IN}$ implies that if $F$ is $I$-Corson, then $F$ is sequentially compact. Thus $\ACD$ implies that every Eberlein space is sequentially compact.
(iii) $\AC^{fin(I)}$ implies that if $F \subseteq [0,1]^{(I)}$, then $F$ is sequentially compact and has a witness of sequential compactness.
If $F \subseteq [0,1]^{(I)}$, then $F$ is sequentially compact using the fact that $[0,1]^{\IN}$ is sequentially compact (see Example \[ex:seq-compact\]).
Countable product of finitely restricted spaces
-----------------------------------------------
\[theo:acd-stron-eber-comp\] Let $F$ be a closed subset of $[0,1]^I$ which is contained in $[0,1]^{(I)}$. Then $\ACD$ implies that $F$ is compact. In particular, $\ACD$ implies that every strong Eberlein space is compact.
For every $n \in \IN$, recall that the subset $\sigma_n(I) :=\{x \in [0,1]^{(I)} : \; |supp(x)| \le n\}$ is compact (see Proposition \[prop:sigma-n-comp\]). Let $\mathcal F$ be a filter of the lattice of closed subsets of $F$. If there exists an integer $n$ such that $\sigma_n(I) $ is $\mathcal F$-stationar, then $\cap \mathcal F$ is non-empty by compactness of $\sigma_n(I)$ and using Remark \[rem:stat-comp\]. Else, using $\ACD$, consider a sequence $(F_n)_{n \in \IN}$ of closed subsets of $F$ belonging to $\mathcal F$ such that for every $n \in \IN$, $F_n \cap \sigma_n(I) =\varnothing$. Re-using $\ACD$, choose for every $n \in \IN$ an element $x_n \in F_n$. A new use of $\ACD$ and Proposition \[prop:seq-comp-corson\]- implies the existence of some infinite subset $A$ of $\IN$ such that $(x_n)_{n \in A}$ converges to some element $x \in F$. Then, for every $n \in \IN$, $x \in F_n$ (which is disjoint with $\sigma_n(I)$) so the element $x$ of $F$ has an infinite support: this is contradictory!
It does not seem provable in $\ZF$ that every closed subset of $[0,1]^I$ contained in $[0,1]^{(I)}$ is compact, or has a witness of sequential compactness.
Countable products of strong Eberlein spaces
--------------------------------------------
\[theo:prod-strong-eber\] Let $(I_n)_{n \in \IN}$ be a sequence of pairwise disjoint sets and let $I:=\cup_{n \in \IN} I_n$. For every $n \in \IN$, let $F_n$ be a closed subset of $\{0,1\}^{(I_n)}$. Let $F$ be the closed subset $\prod_{n \in \IN} F_n$ of $\{0,1\}^I$. Then $\ACD$ implies that $F$ is sequentially compact and compact.
Using $\ACD$, $F$ is sequentially compact: given a sequence $(x_n)_{n \in \IN}$ of $F$, $\ACD$ implies that for every $n \in \IN$, the support $D_n$ of $x_n$ is countable, thus re-using $\ACD$, the set $D:=\cup_{n \in \IN}D_n$ is also countable; since each $x_n$ belongs to $[0,1]^D \times \{0\}^{I \backslash D}$, and since $[0,1]^D$ is sequentially compact (see Example \[ex:seq-compact\]), it follows that $(x_n)_{n \in \IN}$ has an infinite subsequence which converges in $F$. Using $\ACD$ and Theorem \[theo:acd-stron-eber-comp\], each $F_n$ is compact. Using $\ACD$ and Remark \[rem:acd-seq-comp\]-, it follows that $F$ is compact.
$\ACD$ and Eberlein closed subsets of $[0,1]^I$ {#subsec:acd2eberl}
-----------------------------------------------
\[cor:acd2comp-eberlein\] $\ACD$ implies that every Eberlein space is both sequentially compact and compact.
Let $X$ be an Eberlein space. Then $X$ is sequentially compact by Proposition \[prop:seq-comp-corson\]-. Let $I$ be a set such that $X$ is homeomorphic with a closed subset $F$ of $[0,1]^I$, with $F \subseteq [0,1]^{(I)}$. Using the Theorem of Section \[subsec:dyadic\], there exists a family $(Z_k)_{ k \in \IN}$ such that for each $ k \in \IN$, $Z_k$ is $I$-strong Eberlein, and such that $F$ is the continuous image of a closed subset $Z$ of $\prod_{k \in \IN} Z_k$. Using $\ACD$ and Theorem \[theo:prod-strong-eber\], the space $\prod_{k \in \IN} Z_k$ is sequentially compact. With $\ACD$, each $Z_k$ is compact; with Remark \[rem:acd-seq-comp\]- and $\ACD$, it follows that $prod_{k \in \IN} Z_k$ (and thus its continuous image $F$) is also compact.
Recall that (see Theorem \[theo:ac-fin2-ball-hilb-base\]-) every $I$-uniform Eberlein space is sequentially compact, with a witness definable from $I$.
Does $\ACD$ implies that every Eberlein space is sequentially compact with a witness?
Convex-compactness and the Hahn-Banach property
-----------------------------------------------
Given a set $I$, say that a subset $F$ of $\IR^I$ is [*convex-compact*]{} if for every set $\mathcal C$ of closed convex subsets of $\IR^I$ such that $\{F \cap C : \; C \in \mathcal C\}$ satisfies the [*FIP*]{}, $C \cap \bigcap \mathcal C$ is non-empty; moreover, if there is a mapping associating to each such $\mathcal C$ an element of $C \cap \bigcap \mathcal C$, then say that $F$ is [*closely*]{} convex-compact. Given a topological vector space $E$, say that $E$ satisfies the [*continuous Hahn-Banach property*]{} if, for every continuous sublinear functional $p : E \to \IR$, for every vector subspace $F$ of $E$, and every linear functional $f : F \to R$ such that $f \le p|F$ , there exists a linear functional $g : E\to \IR$ that extends $f$ and such that $g \le p$. Moreover, if there is a mapping associating to each such $f$ some $g$ satisfying the previous conditions, then say that $E$ satisfies the [*effective*]{} continuous Hahn-Banach property.
Given a set $I$, the normed space $\ell^0(I)$ satisfies the effective continuous Hahn-Banach property. For every real number $p \in [1,+\infty[$, $\ell^p(I)$ satisfies the effective continuous Hahn-Banach property.
For every set $I$, and every real number $p \in [1,+\infty[$, $B_p(I)$ is closely convex-compact.
The continuous dual of $\ell^0(I)$ is (isometrically isomorphic with) $\ell^1(I)$ and, for every $p \in ]1,+\infty[$, the continuous dual of $\ell^p(I)$ is $\ell^q(I)$ where $\frac{1}{p} + \frac{1}{q}=1$ thus $1<q<+\infty$. We end the proof using the fact (see [@Fo-Mo]) that if a normed space $E$ satisfies the (effective) continuous Hahn-Banach property, then the closed unit ball of the continuous dual $E'$ is ([*closely*]{}) weak\* compact.
Given a set $I$, and a closed convex subset $C$ of $[0,1]^I$ which is $I$-Eberlein, is $C$ convex-compact (in $\ZF$)?
Same question if the closed convex subset $C$ of $[0,1]^I$ is $I$-Corson.
|
---
abstract: 'Diversity is a long-studied topic in information retrieval that usually refers to the requirement that retrieved results should be non-repetitive and cover different aspects. In a conversational setting, an additional dimension of diversity matters: an engaging response generation system should be able to output responses that are diverse and interesting. Sequence-to-sequence (Seq2Seq) models have been shown to be very effective for response generation. However, dialogue responses generated by Seq2Seq models tend to have low diversity. In this paper, we review known sources and existing approaches to this low-diversity problem. We also identify a source of low diversity that has been little studied so far, namely model over-confidence. We sketch several directions for tackling model over-confidence and, hence, the low-diversity problem, including confidence penalties and label smoothing.'
author:
- |
Shaojie Jiang\
University of Amsterdam\
Amsterdam, The Netherlands\
[s.jiang@uva.nl]{}\
Maarten de Rijke\
University of Amsterdam\
Amsterdam, The Netherlands\
[m.derijke@uva.nl]{}
bibliography:
- 'seq2seq.bib'
title: |
Why are Sequence-to-Sequence Models So Dull?\
Understanding the Low-Diversity Problem of Chatbots
---
=1
[[^1]]{}
Acknowledgments {#acknowledgments .unnumbered}
===============
This research was supported by the China Scholarship Council.
[^1]: [Proceedings of the 2018 EMNLP Workshop SCAI: The 2nd International Workshop on Search-Oriented Conversational AI; ISBN 978-1-948087-75-9]{}
|
**Bifurcation of limit cycles from the global center of a class of**
0.2 true cm
**integrable non-Hamilton system under perturbations**
0.2 true cm
**of piecewise smooth polynomials**
0.3 true cm
**Shiyou Sui, Liqin Zhao$^{*}$**
School of Mathematical Sciences, Beijing Normal University,
Laboratory of Mathematics and Complex Systems, Ministry of Education,
Beijing 100875, The People’s Republic of China
0.2 true cm [**Abstract**]{} In this paper, we perturb the global center of the planar polynomial vector fields $\mathcal{X}(x,y)=(-y(x^2+a^2),x(x^2+a^2))$ ($a\neq0$) inside cubic piecewise smooth polynomials with switching line $y=0$. By using average function of first order, we prove that the sharper bound of the number of limit cycles bifurcating from the period annulus is 6.
[**Keywords**]{} [piecewise smooth vector fields; averaging theory; limit cycle]{}
0.4 true cm
[**$\S 1$. Introduction and the main results**]{}
0.2 true cm
One of the main problems inside the qualitative theory in the qualitative theory of real planar differential systems is to determine the number of limit cycles which is related to the Hilbert 16th problem \[6,11\]. A limit cycle is an isolated periodic orbit defined by Poincaré \[8\]. There are many phenomena in real world related with the existence of limit cycles, some examples are the van der Pol oscillator \[20,21\], or the Belousov-Zhabotinskii reaction \[1,24\]. For more about limit cycles, one can see \[2,23\].
The notion of a center of real planar differential system is an isolated equilibrium point having a neighborhood such that all the orbits of this neighborhood are periodic with the unique exception of the equilibrium point, which is defined by Poincaré \[18\]. Late on a classic way to obtain limit cycles is perturbing the periodic orbits of a center.
In 1999, Iliev \[10\] considered the polynomial vector fields $\mathcal{X}(x,y)=(-y,x)$ by perturbing it with polynomials of degree $n$. He studied how many limit cycles can bifurcate from the periodic orbits of the linear center. Later on many people studied the perturbations of polynomial vector fields of the form $\mathcal{X}(x,y)=(-yf(x,y),xf(x,y))$, where $f(0,0)\neq0$, see \[13,19,22\] and the references they cited.
There are many studies of the limit cycles of continuous and discontinuous piecewise differential systems in $\mathbb{R}^2$ with two pieces separated by a straight line. In general these differential systems are linear, see for instance \[5,8,9,16\]. T. Carvalho, J. Llibre, and D. J. Tonon \[4\] studied the number of limit cycles which can bifurcate from the nonlinear center $\mathcal{X}(x,y)=(-y((x^2+y^2)/2)^m,x((x^2+y^2)/2)^m)$, when it is perturbed inside a class of discontinuous piecewise polynomial differential systems of degree $n$ with $k$ pieces. S. Li and Ch. Liu \[12\] studied the piecewise smooth differential system $$\mathcal{X}(x,y)=\left\{
\begin{array}{lc}
(-y(1+ax)+\varepsilon P^+(x,y),x(1+ax)+\varepsilon Q^+(x,y)),~~{\text {if}}~x>0,\\
(-y(1+bx)+\varepsilon P^-(x,y),x(1+bx)+\varepsilon Q^-(x,y),~~{\text {if}}~x<0,
\end{array}
\right.$$ where $P^\pm(x,y),~Q^\pm(x,y)$ are polynomials of degree $n$.
In this paper, we will study the number of limit cycles which can bifurcate from the center of $$\mathcal{X}(x,y)=(-y(x^2+a^2),x(x^2+a^2)),\eqno{(1.1)}$$ when it is perturbed inside discontinuous piecewise cubic polynomials, where $a\neq0$. Let $x_1=\displaystyle\frac{1}{a}x,~y_1=\displaystyle\frac{1}{a}y,~t_1=a^2t$, then system (1.1) is transformed into $$\mathcal{X}(x,y)=(-y(x^2+1),x(x^2+1)),\eqno{(1.2)}$$ here we omit the subscript 1. Hence, we consider the following perturbations of system (1.2) $$\mathcal{X}_{\varepsilon}(x,y)=\mathcal{X}(x,y)+\varepsilon\sum\limits_{k=1}^2\mathcal{X}_{S_k}(x,y)(P_k(x,y),Q_k(x,y)),\eqno{(1.3)}$$ where $P_k(x,y)=\sum\limits_{i+j=0}^3a_{ij}^kx^iy^j,~Q_k(x,y)=\sum\limits_{i+j=0}^3b_{ij}^kx^iy^j$, $a_{ij}^k,~b_{ij}^k$ are arbitrary constants, the characteristic function $\mathcal{X}_S$ of a set $S\subset\mathbb{R}^2$ is defined by $$\mathcal{X}_S(x,y)=\left\{\begin{array}{lc}
1,~~{\rm if} ~(x,y)\in S,\\
0,~~{\rm if} ~(x,y)\notin S,
\end{array}
\r.$$ and $S_1=\{(x,y)\left|y>0\right.\},~~S_2=\{(x,y)\left|y<0\right.\}$.
The system $(1.3)_{\varepsilon=0}$ has a periodic annulus surround the origin. Then, using average function of first order (see Section 2), we find the maximum number of limit cycles of system (1.3). The main results of this paper is the following.
[**Theorem 1.1.**]{} Suppose that the average function of first order associated to the discontinuous piecewise polynomial differential system (1.3) is non-zero. Then for $\left|\varepsilon\right|>0$ sufficiently small the sharper bound of the number of limit cycles of system (1.3) is 6.
[**Corollary 1.2.**]{} Under the assumption of theorem 1.1, if $a_{ij}^1=a_{ij}^2,~b_{ij}^1=b_{ij}^2$, for $\left|\varepsilon\right|>0$ sufficiently small the sharper bound of the number of limit cycles of system (1.3) is 3.
[**Remark 1.3.**]{} Using the results of \[22\], we know that system $(1.3)_{\varepsilon=0}$ under the perturbation of smoth polynomials with degree $n$ has at most $2n+2-(-1)^n$.
This paper is organized as follows. In Section 2, we introduce averaging theory for computing periodic solution and extend complete Chebyshev system for studying the number of zeros of average function. The main results is proved in Section 3.
0.2 true cm
[**$\S 2$. Preliminary results**]{}
0.2 true cm In this section we summarize the main tools that we will use to study the bifurcation of limit cycles for system (1.3). First, we introduce the averaging theory for discontinuous piecewise differential systems. The following results stated on the averaging theory are valid for discontinuous piecewise polynomial vector field defined in $\mathbb{R}^n$ and are proved in \[15\], but we shall state them for our discontinuous piecewise polynomial vector field (1.3) in polar coordinates.
Consider a non-autonomous discontinuous piecewise vector field $$\frac{{\rm d}r}{{\rm d}\theta}=\mathcal{X}(\theta,r)=\varepsilon F(\theta,r)+\varepsilon^2R(\theta,r,\varepsilon),\eqno{(2.1)}$$ where $r\in\mathbb{R}$, $\theta\in\mathbb{R}/(2\pi\mathbb{Z})$ and $$F(\theta,r)=\sum\limits_{i=1}^k\mathcal{X}_{S_i}(\theta)F_i(\theta,r),~~~R(\theta,r,\varepsilon)=\sum\limits_{i=1}^k\mathcal{X}_i(\theta)R_i(\theta,r,\varepsilon),$$ where $F_i:\mathbb{S}^1\times D\rightarrow \mathbb{R}^2$, $R_i:\mathbb{S}^1\times D\times (-\varepsilon_0,\varepsilon_0)\rightarrow \mathbb{R}^2$ for $i=1,\ldots,k$ are continuous functions, $2\pi$-periodic in the variable $\theta$, and $D$ is an open interval of $\mathbb{R}$. Here the $\mathbb{S}_i$ are the open intervals $(\theta_i,\theta_{i+1})$ for $i=1,\ldots,k$ and $0\leq\theta_1\leq\cdots\leq\theta_k\leq2\pi\leq\theta_{k+1}=\theta_1+2\pi$. We define $$D_rF(\theta,r)=\sum\limits_{i=1}^k\mathcal{X}_{S_i}(\theta,r)D_rF_i(\theta,r).$$ The [*average function*]{} $f:D\rightarrow \mathbb{R}$ is defined by $$f(r)=\int_0^TF(\theta,r){\rm d}\theta.$$ We recall that if $r(\theta,r_0)$ is the solution of the vector field $\mathcal{X}(\theta,r)$ such that $r(0,r_0)=r_0$, then we have $$r(2\pi,r_0)-r_0=\varepsilon f(r)+O(\varepsilon^2).$$ So for $\varepsilon>0$ suffieiently small the simple zeros of the averaged function $f(r)$ provides limit cycles of the vector field $\mathcal{X}(\theta,r)$.
In the next result we present a version of the averaging theory for discontinuous piecewise vector fields, that is proved in \[15\], adapted to differential equation (2.1). We note that in \[15\] the averaging theory uses that the Brouwer degree of a function $f$ in a neighborhood of a zero $\bar{r}$ of the function $f(r)$ is non-zero, while here we substitute this condition saying that the zero $\bar{r}$ is simple (i.e. $\frac{{\rm d}f}{{\rm d}r}(\bar{r})\neq0$), because this last condition implies that the mentioned Brouwer degree non-zero. See for more details \[3,17\].
[**Lemma 2.1.**]{} Assume that the following conditions hold for the discontinuous piecewise vector field $\mathcal{X}(\theta,r)$.\
(i) For $i=1,\ldots,k$ the functions $F_i(\theta,r)$ and $R_i(\theta,r)$ are locally Lipschitz with respect to $r$, and $2\pi$-periodic with respect to $\theta$.\
(ii) Let $\bar{r}\in D$ be a simple zero of the average function $f(r)$.\
Then for $\varepsilon>0$ sufficiently small, there exists a $2\pi$-periodic solution $r(\theta,\varepsilon)$ of the vector field $\mathcal{X}(\theta,r)$ such that $r(0,\varepsilon)\rightarrow \bar{r}$ as $\varepsilon\rightarrow 0$.
In order to study the number of zeros of the averaging function we will use the following results.
[**Definition 2.2.**]{}(\[14\]) Let $\mathbb{U}$ be a set and let $f_1,f_2,\ldots,f_n:\mathbb{U}\rightarrow\mathbb{R}$, we say that $f_1,\ldots,f_n$ are linearly independent functions if and only if we have that $$\sum\limits_{i=1}^nk_if_i(x)=0~~{\text{for all}}~x\in \mathbb{U}~~\Rightarrow~~~k_1=k_2=\ldots=k_n=0.$$
[**Lemma 2.3.**]{}(\[14\]) If $f_1,f_2,\ldots,f_n:\mathbb{U}\rightarrow\mathbb{R}$ are linearly independent then there exit $x_1,x_2,\ldots,x_{n-1}\in \mathbb{U}$ and $k_1,k_2,\ldots,k_n\in \mathbb{R}$ such that for every $i\in\{1,\ldots,n-1\}$ $$\sum\limits_{j=1}^nk_jf_j(x_i)=0.$$
[**Definition 2.4.**]{}(\[7\]) Let $f_0,f_1,\ldots,f_{n-1}$ be analytic functions on an open interval $L$ of $\mathbb{R}$.\
(a) $(f_0,f_1,\ldots,f_{n-1})$ is a *Chebyshev system* (in short, T-system) on $L$ if any nontrivial linear combination $$\alpha_0f_0(x)+\alpha_1f_1(x)+\ldots+\alpha_{n-1}f_{n-1}(x)$$ has at most $n-1$ isolated zeros on $L$.\
(b) $(f_0,f_1,\ldots,f_{n-1})$ is an *complete Chebyshev system* (in short, CT-system) on $L$ if $(f_0,f_1,\ldots,f_{k-1})$ is a T-system for all $k=1,2,\ldots,n$.\
(c) $(f_0,f_1,\ldots,f_{n-1})$ is an *extend complete Chebyshev system* (in short, ECT-system) on $L$ if, for all $k=1,2,\ldots,n$, any nontrivial linear combination $$\alpha_0f_0(x)+\alpha_1f_1(x)+\ldots+\alpha_{k-1}f_{k-1}(x)$$ has at most $k-1$ isolated zeros on $L$ counted with multiplicities.
It is clear that if $(f_0,f_1,\ldots,f_{n-1})$ is an EXT-system on $L$, then $(f_0,f_1,\ldots,f_{n-1})$ is a CT-system on $L$. However, the reverse implication is not true.
[**Lemma 2.5.**]{}(\[7\]) $(f_0,f_1,\ldots,f_{n-1})$ is an ECT-system on $L$ if and only if for each $k=1,2,\ldots,n$ the continuous Wronskian of $(f_0,f_1,\ldots,f_{k-1})$ at $x\in L$ is not zero, that is $$W[f_0,f_1,\dots,f_{k-1}](x)=\left|\begin{matrix}
f_0(x) & f_1(x) & \cdots & f_{k-1}(x)\\
f_0'(x)&f_1'(x)&\cdots&f_{k-1}'(x)\\
\cdots&\cdots&\cdots&\cdots\\
f_0^{(k-1)}(x)&f_1^{(k-1)}(x)&\cdots&f_{k-1}^{(k-1)}(x)
\end{matrix}\right|\neq 0~~(x\in L).$$
By Definition 2.2 and Lemma 2.5, it is easy to get the following.
[**Proposition 2.6.**]{} If $(f_0,f_1,\ldots,f_{n-1})$ is an ECT-system on $L$, then they are linearly independent.
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[**$\S 3$. Proof of main results**]{}
0.2 true cm
Using the change to polar coordinates $x=r\cos\theta,~y=r\sin\theta$, we transform the differential system (1.3) into $${\begin{split}
{\dot{r}}&=\varepsilon\sum\limits_{k=1}^2\left[\cos\theta(\mathcal{X}_{S_k}\cdot P_k)(r\cos\theta,r\sin\theta)+\sin\theta(\mathcal{X}_{S_k}\cdot Q_k)(r\cos\theta,r\sin\theta)\right],\\
{\dot{\theta}}&=r^2\cos^2\theta+\frac{\varepsilon}{r}\sum\limits_{k=1}^2\left[\cos\theta(\mathcal{X}_{S_k}\cdot Q_k)(r\cos\theta,r\sin\theta)-\sin\theta(\mathcal{X}_{S_k}\cdot P_k)(r\cos\theta,r\sin\theta)\right].
\end{split}}$$ Taking $\theta$ as the new independent variable the previous differential system becomes the differential equation $$\displaystyle\frac{{\rm d}r}{{\rm d}\theta}=\varepsilon F(\theta,r)+\varepsilon^2R(\theta,r,\varepsilon),\eqno{(3.1)}$$ where $$F(\theta,r)=\sum\limits_{k=1}^2\frac{\cos\theta (\mathcal{X}_{S_k}\cdot P_k)(r\cos\theta,r\sin\theta)+\sin\theta (\mathcal{X}_{S_k}\cdot Q_k)(r\cos\theta,r\sin\theta)}
{r^2\cos^2\theta+1},$$ $${\begin{split}
R(\theta,r,\varepsilon)&=-\sum\limits_{k=1}^2\frac{\cos\theta (\mathcal{X}_{S_k}\cdot P_k)(r\cos\theta,r\sin\theta)+\sin\theta (\mathcal{X}_{S_k}\cdot Q_k)(r\cos\theta,r\sin\theta)}
{r^2\cos^2\theta+1}\\
&\cdot\frac{\cos\theta(\mathcal{X}_{S_k}\cdot Q_k)(r\cos\theta,r\sin\theta)-\sin\theta(\mathcal{X}_{S_k}\cdot P_k)(r\cos\theta,r\sin\theta)}{r(r^2\cos^2\theta+1)+\varepsilon\left[\cos\theta(\mathcal{X}_{S_k}\cdot Q_k)(r\cos\theta,r\sin\theta)-\sin\theta(\mathcal{X}_{S_k}\cdot P_k)(r\cos\theta,r\sin\theta)\right]}.\end{split}}$$ Therefore, the average function of first order is $${\begin{split}
f(r)&=\int_0^{2\pi}F(\theta,r){\rm d}\theta\\
&=\sum\limits_{i+j=0}^3\int_0^{\pi}\frac{a_{ij}^1r^{i+j}\cos^{i+1}\theta\sin^j\theta+b_{ij}^1r^{i+j}\cos^{i}\theta\sin^{j+1}\theta}
{r^2\cos^2\theta+1}{\rm d}\theta\\
&~+\sum\limits_{i+j=0}^3\int_{\pi}^{2\pi}\frac{a_{ij}^2r^{i+j}\cos^{i+1}\theta\sin^j\theta+b_{ij}^2r^{i+j}\cos^{i}\theta\sin^{j+1}\theta}
{r^2\cos^2\theta+1}{\rm d}\theta\\
&=\sum\limits_{i=0}^3r^i\sum\limits_{j=0}^{i+1}\omega_{i+1-j,j}^1\int_0^{\pi}\frac{\cos^{i+1-j}\theta\sin^j\theta}
{r^2\cos^2\theta+1}{\rm d}\theta\\
&~+\sum\limits_{i=0}^3r^i\sum\limits_{j=0}^{i+1}\omega_{i+1-j,j}^2\int_{\pi}^{2\pi}\frac{\cos^{i+1-j}\theta\sin^j\theta}
{r^2\cos^2\theta+1}{\rm d}\theta
\end{split}}$$ where $\omega_{i,j}^k=a_{i-1,j}^k+b_{i,j-1}^k,~1\leq i+j\leq 4,~a_{-1,j}^k=b_{i,-1}^k=0,~k=1,2$. Note that $\omega_{i,j}^k$ are also arbitrary, since $a_{i,j}^k,~b_{i,j}^k$ are arbitrary.
In order to simplify the notation, we define the following functions: $$I_{i,j}(r)=\int_0^{\pi}\frac{\cos^i\theta\sin^j\theta}{r^2\cos^2\theta+1}{\rm d}\theta,$$ $$J_{i,j}(r)=\int_{\pi}^{2\pi}\frac{\cos^i\theta\sin^j\theta}{r^2\cos^2\theta+1}{\rm d}\theta.$$ Then, it is easy to check that $$J_{i,j}(r)=(-1)^{i+j}I_{i,j}(r).$$ Notice that in the interval $(0,+\infty)$, the zeros of the function $f(r)$ coincide with the zeros of the function $F(r)=rf(r)$. Therefore, in order to simplify further computation, we will study the function $F(r)$ instead of $f(r)$.
Using above notation, we can obtain that $$\begin{aligned}
F(r)&=rf(r)=\sum\limits_{i=1}^4r^i\sum\limits_{j=0}^i\omega_{i-j,j}^1I_{i-j,j}(r)+\sum\limits_{i=1}^4r^i\sum\limits_{j=0}^i\omega_{i-j,j}^2J_{i-j,j}(r)\notag\\
&=\sum\limits_{i=1}^4r^i\sum\limits_{j=0}^i\left(\omega_{i-j,j}^1+(-1)^{i}\omega_{i-j,j}^2\right)I_{i-j,j}(r)\notag\\
&=\sum\limits_{i=1}^4r^i\sum\limits_{j=0}^i\mu_{i-j,j} I_{i-j,j}(r)\tag{3.2}\end{aligned}$$ where $\mu_{i-j,j}=\omega_{i-j,j}^1+(-1)^{i}\omega_{i-j,j}^2$. Note that $\mu_{i-j,j}$ are independent, since $\omega_{i-j,j}^1,~\omega_{i-j,j}^2$ are arbitrary. By direct computation we have the following: $$\lf\{\begin{array}{lc}
I_{1,0}(r)=I_{1,1}(r)=I_{3,0}(r)=I_{1,2}(r)=I_{3,1}(r)=I_{1,3}(r)=0,\\
I_{0,0}(r)=\frac{\pi}{\sqrt{1+r^2}},~~I_{0,1}(r)=2\frac{\arctan r}{r},\\
I_{2,0}(r)=\frac{\pi}{r^2}(1-\frac{1}{\sqrt{1+r^2}}),~~I_{2,1}(r)=\frac{2}{r^3}(r-\arctan r),\\
I_{0,2}(r)=I_{0,0}(r)-I_{2,0}(r),~~I_{0,3}(r)=I_{0,1}(r)-I_{2,1}(r),~~I_{4,0}(r)=\frac{\pi}{2r^2}-\frac{1}{r^2}I_{2,0}(r),\\
I_{2,2}(r)=I_{2,0}(r)-I_{4,0}(r),~~I_{0,4}(r)=I_{4,0}(r)-2I_{2,0}(r)+I_{0,0}(r).
\end{array}
\r.\eqno{(3.3)}$$
Substituting (3.3) into (3.2), we have that $$\begin{aligned}
F(r)&=\mu_{0,1}rI_{0,1}(r)+r^2(\mu_{2,0}I_{2,0}(r)+\mu_{0,2}I_{0,2}(r))+r^3(\mu_{2,1}I_{2,1}(r)+\mu_{0,3}I_{0,3}(r))\notag\\
&~~+r^4(\mu_{4,0}I_{4,0}(r)+\mu_{2,2}I_{2,2}(r)+\mu_{0,4}I_{0,4}(r))\notag\\
&=(\mu_{0,2}r^2+\mu_{0,4}r^4)I_{0,0}(r)+(\mu_{0,1}r+\mu_{0,3}r^3)I_{0,1}(r)+((\mu_{2,0}-\mu_{0,2})r^2+(\mu_{2,2}-2\mu_{0,4})r^4)I_{2,0}(r)\notag\\
&~~+(\mu_{2,1}-\mu_{0,3})r^3I_{2,1}(r)+(\mu_{4,0}-\mu_{2,2}+\mu_{0,4})r^4I_{4,0}\notag\\
&=2\nu_1r+\pi\nu_2r^2+\pi \nu_3\frac{r^2}{\sqrt{1+r^2}}+\pi\nu_4\frac{r^4}{\sqrt{1+r^2}}+2\nu_5\arctan r \notag\\
&~~+2\nu_6r^2\arctan r +\pi\nu_7(1-\frac{1}{\sqrt{1+r^2}})\tag{3.4}\end{aligned}$$ where
$$\lf\{\begin{array}{lc}
\nu_1=\mu_{2,1}-\mu_{0,3},\\
\nu_2=\frac{1}{2}\mu_{4,0}+\frac{1}{2}\mu_{2,2}-\frac{3}{2}\mu_{0,4},\\
\nu_3=\mu_{0,2}-\mu_{2,2}+2\mu_{0,4},\\
\nu_4=\mu_{0,4},\\
\nu_5=\mu_{0,1}-\mu_{2,1}+\mu_{0,3},\\
\nu_6=\mu_{0,3},\\
\nu_7=\mu_{2,0}-\mu_{0,2}-\mu_{4,0}+\mu_{2,2}-\mu_{0,4}.
\end{array}
\r.$$
It follows from direct computation that $${\rm det}\frac{\partial(\nu_1,\nu_2,\nu_3,\nu_4,\nu_5,\nu_6,\nu_7)}{\partial(\mu_{2,1},\mu_{4,0},\mu_{0,2},\mu_{0,4},\mu_{0,1},\mu_{0,3},\mu_{2,0})}=\frac{1}{2}\neq0.$$ By the arbitrariness of $\mu_{ij}$, we have that $\nu_i$ $(i=1,\ldots,7)$ are independent. Hence, the generating functions of $F(r)$ are the following: $${\begin{split}
&f_1(r):=r,~~f_2(r):=r^2,~~f_3(r):=\frac{r^2}{\sqrt{1+r^2}},~~f_4(r):=\frac{r^4}{\sqrt{1+r^2}},\\
&f_5(r):=\arctan r,~~f_6(r):=r^2\arctan r,~~f_7(r):=1-\frac{1}{\sqrt{1+r^2}}.
\end{split}}$$
Next, we will prove that $(f_1(r),\ldots,f_7(r))$ is an ECT-system. We introduce the notation $W_k(r)=W[f_1(r),f_2(r),\ldots,f_k(r)]$. Direct computation, we have $$W_2(r)=r^2,~~W_3(r)=-\frac{3r^4}{(r^2+1)^{\frac{5}{2}}},~~W_4(r)=-\frac{6r^7(4r^2+5)}{(r^2+1)^5}.$$ It is obvious that $W_k(r)\neq0~(k=2,3,4)$ on $r\in(0,+\infty)$.
For $k=5$, using mathematical soft such as Maple, we get that $$W_5(r)=\frac{12r^3}{(r^2+1)^9}g_1(r),$$ where $${\begin{split}
g_1(r)&=12r^{10}\arctan r-27r^8\arctan r+12r^9-258r^6\arctan r+169r^7\\
&~-507r^4\arctan r+411r^5-408r^2\arctan r+373r^3-120\arctan r+120r.
\end{split}}$$ Then, $${\begin{split}
g_1'(r)&=r(120r^8\arctan r-216r^6\arctan r+120r^7-1548r^4\arctan r+1144r^5\\
&-2028r^2\arctan r+1836r^3-816\arctan r+831r).
\end{split}}$$ From the curve (Figure 1) of $\frac{g_1'(r)}{r^7}$, we know that $g_1'(r)>0$ on $(0,+\infty)$. So, we get that $g_1(r)>g_1(0)=0$. Hence, $W_5(r)\neq0$ on $(0,+\infty)$.
When $k=6$, we have $$W_6(r)=-\frac{24r}{(r^2+1)^{13}}g_2(r),$$ where $${\begin{split}
g_2(r)&=216r^{12}\arctan r+168r^{10}\arctan r+216r^{11}+843r^8\arctan r+96r^9\\
&+4986r^6\arctan r-3061r^7+8175r^4\arctan r-6655r^5\\
&+5160r^2\arctan r-4800r^3+1080\arctan r-1080r.
\end{split}}$$ By the curve (see Figure 2) of $\frac{g_2(r)}{r^{10}}$, we have that $W_6(r)\neq0$ on $(0,+\infty)$.
When $k=7$, we have $$W_7(r)=\frac{1728}{(x^2+1)^{\frac{35}{2}}}g_3(r),$$ where $${\begin{split}
g_3(r)&=3120r^{11}\arctan r-4864r^{10}\sqrt{r^2+1}+10500r^9\arctan r+3120r^{10}-14048r^8\sqrt{x^2+1}\\
&+13155r^7\arctan r+9460r^8-14224r^6\sqrt{r^2+1}+7350r^5\arctan r+10279 r^6-6020r^4\sqrt{x^2+1}\\
&+1575r^3\arctan r+4985r^4-1120r^2\sqrt{r^2+1}+1200r^2-160\sqrt{r^2+1}+160.
\end{split}}$$
Then, $${\begin{split}
g_3'(r)&=\frac{r}{\sqrt{r^2+1}}\left(34320r^9\sqrt{r^2+1}\arctan r-53504r^{10}+94500r^7\sqrt{r^2+1}\arctan r\right.\\
&+34320r^8\sqrt{r^2+1}-175072r^8+92085r^5\sqrt{x^2+1}\arctan r+83060r^6\sqrt{r^2+1}\\
&-211952r^6+36750r^3\sqrt{r^2+1}\arctan r+67449r^4\sqrt{r^2+1}\\
&-115444r^4+4725r\sqrt{r^2+1}\arctan r+21515r^2\sqrt{r^2+1}\\
&\left.-27440r^2+2400\sqrt{r^2+1}-2400\right)
\end{split}}$$ and $$\frac{{\rm d}}{{\rm d}r}\left(\frac{g_3'(r)\sqrt{r^2+1}}{r}\right)=-\frac{1}{\sqrt{r^2+1}}g_{31}(r)$$ where $${\begin{split}
g_{31}(r)&=-343200r^{10}\arctan r+535040r^9\sqrt{r^2+1}-1064880r^8\arctan r-343200r^9\\
&+1400576r^7\sqrt{r^2+1}-1214010r^6\arctan r-950480r^7+1271712r^5\sqrt{r^2+1}\\
&-607425r^4\arctan r-927690r^5+461776r^3\sqrt{r^2+1}-119700r^2\arctan r\\
&-371091r^3+54880r\sqrt{r^2+1}-4725\arctan r-50155r.
\end{split}}$$ From the curve (see Figure 3) of $\frac{g_{31}(r)}{r^8}$, we know $g_{31}(r)<0$ on $(0,+\infty)$.
Hence, $\frac{{\rm d}}{{\rm d}r}\left(\frac{g_3'(r)\sqrt{r^2+1}}{r}\right)>0$ on $(0,+\infty)$. Therefore, we have that $\frac{g_3'(r)\sqrt{r^2+1}}{r}>\lim\limits_{r\rightarrow0}\frac{g_3'(r)\sqrt{r^2+1}}{r}=0$. That is equivalent to $g_3'(r)>0$ on $(0,+\infty)$. Thus, $g_3(r)>g_3(0)=0$, which implies $W_7(r)\neq0$ on $(0,+\infty)$. By above analysis we have proved that $(f_1(r),\ldots,f_7(r))$ is an ECT-system on $(0,+\infty)$. So, by Lemma 2.5, $F(r)$ has at most 6 zeros on $(0,+\infty)$. Using proposition 2.6 and Lemma 2.3, we have that $F(r)$ can have 6 zeros on $(0,+\infty)$. Hence, by Lemma 2.1, theorem 1.1 is proved.
If $a_{ij}^1=a_{ij}^2,~b_{ij}^1=b_{ij}^2$, then $\omega_{i-j,j}^1=\omega_{i-j,j}^2$, which imply $\mu_{i-j,j}=0$ (i is odd). Then, the generating functions of $F(r)$ become $$f_2(r):=r^2,~~f_3(r):=\frac{r^2}{\sqrt{1+r^2}},~~f_4(r):=\frac{r^4}{\sqrt{1+r^2}},~~
f_7(r):=1-\frac{1}{\sqrt{1+r^2}}.$$ It is easy to check that $(f_2(r),f_3(r),f_4(r),f_7(r))$ is an ECT-system on $(0,+\infty)$. Then, we ends the proof of Corollary 1.2.
[30]{}
|
---
abstract: 'There are various situations where the classical Fourier’s law for heat conduction is not applicable, such as heat conduction in heterogeneous materials [@Botetal16; @Vanetal17] or for modeling low-temperature phenomena [@KovVan15; @KovVan16; @KovVan18]. In such cases, heat flux is not directly proportional to temperature gradient, hence, the role – and both the analytical and numerical treatment – of boundary conditions becomes nontrivial. Here, we address this question for finite difference numerics via a shifted field approach. Based on this ground, implicit schemes are presented and compared to each other for the Guyer–Krumhansl generalized heat conduction equation, which successfully describes numerous beyond-Fourier experimental findings. The results are validated by an analytical solution, and are contrasted to finite element method outcomes obtained by COMSOL.'
address: |
$^1$Department of Energy Engineering, Faculty of Mechanical Engineering, BME, Budapest, Hungary\
$^2$Department of Theoretical Physics, Institute for Particle and Nuclear Physics, Wigner Research Centre for Physics, Budapest, Hungary\
$^3$Montavid Thermodynamic Research Group, Budapest, Hungary
author:
- 'Á. Rieth$^{1}$, R. Kovács$^{123}$, T. Fülöp$^{13}$'
title: Implicit numerical schemes for generalized heat conduction equations
---
Implicit scheme ,shifted fields ,boundary conditions ,nonequilibrium thermodynamics
Introduction {#intro}
============
The need to go beyond the Fourier heat conduction equation – which reads in one spatial dimension $$\label{foueq}
\partial_t T = \alpha \partial_{xx } T$$ for temperature $T$, with thermal diffusivity $\alpha$, and which contains only first order time derivative $\partial_t$ and second order space derivative $\partial_{xx}$ – is experimentally proved under various conditions since decades [@Botetal16; @Vanetal17; @Pesh44; @JacWalMcN70; @JacWal71; @NarDyn72a; @Kam90; @Jaetal08]. These circumstances are related partly to the material structure [@Mari14; @PMMar17conf], and partly to the environment like temperature and excitation [@myphd2017]. The characteristics of the interaction between the sample and the environment are condensed into the boundary conditions, the role of which are therefore crucial during the modeling. For theories beyond the Fourier one, the common starting point is the balance equation of internal energy $e$, $$\label{baleneq}
\rho \partial_t e + \partial_x q = 0,$$ also written in one spatial dimension, with density $\rho$ and heat flux $q$. For many applications, a constant specific heat $c$ can be assumed, yielding $e=cT$.
Then, if one takes Fourier’s law, $$\label{foueq2}
q = - \qlam \partial_x T,$$ where $\qlam$ is thermal conductivity, then (\[foueq\]) can be obtained. In parallel, heat flux boundary conditions – like a heat pulse on one end and an adiabatic insulation on the other one, the case considered hereafter – can be written directly for temperature, prescribing its gradient.
However, for generalized heat conduction models, the picture is not so simple any more. For example, in the first known extension to Fourier’s law, the so-called Maxwell–Cattaneo–Vernotte (MCV) constitutive equation [@Max1867; @Cattaneo58; @Vernotte58; @Gyar77a] $$\label{mcveq}
\qtau \partial_t q + q = - \qlam \partial_x T,$$ time derivative of heat flux also appears, accompanied by a coefficient $\qtau$ called relaxation time. In this case a heat flux boundary condition cannot be translated to a Neumann-type boundary condition on temperature. The situation becomes even more involved for the Guyer–Krumhansl (GK) equation [@GuyKru66a1; @GuyKru66a2; @Van01a], $$\label{gkeq}
\qtau \partial_t q + q = - \qlam \partial_x T + \kappa^2 \partial_{xx} q,$$ where $\kappa^2$ is a parameter strongly related to the mean free path from the aspect of kinetic theory [@MulRug98]. According to room-temperature experiments [@Botetal16; @Vanetal17; @myphd2017], measured deviation from the Fourier prediction always occurs in the overdamped ($\kappa^2>\qtau$) region (as opposed to the near-to-MCV region $\kappa^2<\qtau$), thus usage of the GK equation is inescapable.
Combining (\[gkeq\]) with (\[baleneq\]) provides the temperature-only version of the GK model: $$\label{gkeq2}
\qtau \partial_{tt} T + \partial_t T = \alpha \partial_{xx} T + \kappa^2 \partial_{txx} T.$$ Solving this equation with heat flux boundary conditions, especially with time dependent ones needed for evaluating heat pulse experiments [@Botetal16; @Vanetal17], is difficult. It is not clear how to translate conditions on $q(t)$ to temperature $T(t)$, the two quantities being related to one another according to a constitutive equation (\[gkeq\]). This was the motivation to develop a simple and fast numerical scheme, a scheme of shifted fields [@KovVan15], that was specifically devised to be suitable for this type of problem. The term shifted fields refers here to the spatial discretization method. Namely, instead of solving (\[gkeq2\]) for $T$, the set of equations (\[baleneq\]) and (\[gkeq\]) are solved for $T$ and $q$ both, where spatial locations of temperature values are shifted by a half space step with respect to locations of $q$ values (see Fig. \[fig:xxx\]). This enables us to prescribe boundary conditions only for heat flux.
![Discretization method of shifted fields for a heat pulse setting. Prescribed boundary values are $q_1$ as a function of time, and $q_J = 0$ representing adiabatic insulation. All other values (illustrated by filled rectangles and circles) can be computed from neighboring and previous values. Temperature values sit at cell midpoints while heat flux values reside at cell boundaries.[]{data-label="fig:xxx"}](fig-scheme.pdf){width=".75\textwidth"}
A physical interpretation of such a scheme is the distinction between surface-related and volume-related quantities of the discrete cells. More closely, temperature represents the average value over the volume while heat flux describes energy flow at the cell boundary.
Notably, similar but different schemes like the two-step Lax-Wendroff, leapfrog or Finite-Difference-Time-Domain (FDTD) methods are known in the literature. All these apply values at half time step or half space step to update a grid point at the next time instant [@NumRec07b]. Furthermore, the present shifted field concept also differs from the multigrid method where the goal is to increase accuracy by applying finer and finer meshes [@NumRec07b; @Chapra98]. One should also mention Feynman [@Feynman1b] who presents a technique for time integration for the dynamics of a point particle where the shifting is used for time steps only. Moreover, Yee discusses the problem of electromagnetic wave propagation and applies the FDTD method to solve the Maxwell equations, and also discusses the possible boundary conditions for such wave propagation problem [@Yee97]. However, none of the mentioned techniques address the question of boundary conditions, and the advantage of the shifted strategy for boundary conditions – especially such nontrivial ones – is not realized. One should also pay attention to the work of Berezovski et al. [@KolEta17a; @BerEta10p; @Ber11a; @BerVan17p; @BerVan17b] where remarkably efficient numerical schemes are developed and tested for wave propagation problems.
Our approach uses simple finite differences to approximate the partial derivatives. An explicit version has already been developed [@KovVan15]; here we present the realization of the corresponding implicit version, which turns out to be remarkably superior in performance aspects. The outcomes are also compared to analytical and finite element solutions in respect of efficiency and speed.
Explicit scheme
===============
For the explicit scheme, all related analysis and detailed discussion are published in [@KovVan15], and are only summarized here for the sake of completeness. Hereafter, dimensionless quantities [@KovVan15] are used, which is a framework that is simple yet satisfactory for the current numerics-related considerations.
The discretized form of the balance equation of internal energy (\[baleneq\]) is $$\label{disc:inten}
T^{n+1}_j=T^n_j-\frac{\Delta t}{\tau_{\Delta} \Delta x} ( q^n_{j+1}-q^n_j),$$ where $n$ indexes time steps and $j$ the space steps, $\tau_\Delta$ denotes the dimensionless pulse duration time and $\Delta t$ the time step. The Guyer–Krumhansl constitutive equation is discretized as $$\label{GKdisc}
q^{n+1}_j=q^n_j-\frac{\Delta t}{\qtau} q^n_j - \frac{\tau_{\Delta}
\Delta t}{\qtau \Delta x} (T^n_j -T^n_{j-1}) + \frac{\kappa^2 \Delta
t}{\qtau \Delta x^2} (q^n_{j+1}-2q^n_j +q^n_{j-1}) ,$$ which is able to reproduce the solutions of the MCV model ($\kappa=0$) and of the Fourier one ($\qtau = \kappa^2$). In these formulae, forward time differencing is applied, which makes all the schemes first order in time. Let us draw attention again to the boundary conditions, thanks to which $T$ values can be updated without prescribing anything for temperature at the boundaries.
The scheme being explicit, one has to calculate the stability criteria as well. In [@KovVan15], von Neumann and Jury methods [@NumRec07b; @Jury74] are used to determine the stability conditions. In order to prove the convergence of such a scheme, the Lax–Richtmyer theorem [@Lax56stab] is exploited by proving the consistency of the schemes together with their stability. Regarding consistency, although only its weak form is proved [@myphd2017], it is enough to fulfill the Lax–Richtmyer theorem and ensure the presence of convergence [@FDA1; @Gerdt12].
Implicit schemes
================
When quantities at time instant $t^{n+1}$ are also considered, the scheme becomes implicit, leading to the following discretized form of the balance equation of internal energy: $$\begin{aligned}
\label{IMPSEPEN}
\tau_{\Delta} \frac{1}{\Delta t} (T^{n+1}_j - T^n_j) = -\frac{1}{\Delta
x} \left [ (1-\Theta) \left( q^n_{j+1} - q^n_j \right) +
\Theta \left( q^{n+1}_{j+1} -
q^{n+1}_j \right) \right ] ,
\end{aligned}$$ and of the GK-type constitutive equation: $$\begin{aligned}
\label{IMPSEPGK}
\frac{\qtau}{\Delta t} \left ( q^{n+1}_j - q^n_j \right )
+ \left [ (1-\Theta) q^n_j+ \Theta q^{n+1}_j \right ] &
\nonumber \\
+ \frac{\tau_{\Delta}}{\Delta x} \left [ (1-\Theta) ( T^n_j - T^n_{j-1}) +
\Theta (T^{n+1}_j- T^{n+1}_{j-1}) \right ] &
\\ \nonumber
- \frac{\kappa^2}{\Delta x^2} \left [ (1-\Theta) \big ( q^n_{j+1} - 2q^n_j + q^n_{j-1} \big ) + \Theta \big (q^{n+1}_{j+1} - 2q^{n+1}_j + q^{n+1}_{j-1} \big ) \right ]
& = 0,
\end{aligned}$$ where the convex combination of explicit and implicit terms is characterized by the parameter $\Theta$, with $\Theta=0$ removing the implicit terms and returning the purely explicit scheme. Analogously, for $\Theta=1$, all the explicit terms vanish, making (\[IMPSEPEN\]) and (\[IMPSEPGK\]) purely implicit. Choosing $\Theta
= 1/2$ gives the so-called Crank–Nicolson scheme, which preserves the unconditionally stable property of implicit schemes and provides one order higher accuracy. Here, accuracy is not analyzed in detail. We test the implicit scheme with settings $\Theta=1/2$ and $\Theta = 1$, for various parameter values for $\qtau$ and $\kappa^2$.
In order to prove that no stability condition is needed for these implicit schemes, we use the methods of von Neumann and Jury as before [@KovVan15], i.e., let us assume the solution of the difference equations (\[IMPSEPEN\]) and (\[IMPSEPGK\]) in the form $$\label{NSOL}
\phi^n_j=\xi^n e^{ikj \Delta x},$$ where $i$ is the imaginary unit, $k$ is the wave number parameter of the solution, $j \Delta x$ denotes the $j^{\text{th}}$ discrete spatial position, and the complex number $\xi$ is called the growth factor [@NumRec07b]. The scheme is stable if and only if $|\xi| \leq 1$ holds. Now, using (\[NSOL\]) one can express each term from (\[IMPSEPEN\])–(\[IMPSEPGK\]), for example $q^{n+1}_{j+1} =
\xi^{n+1} e^{ik(j+1)\Delta x} \cdot q_0$. Substituting back (\[NSOL\]) into (\[IMPSEPEN\]) and (\[IMPSEPGK\]) yields $$\begin{aligned}
\label{DISCSYS}
T_0 (\xi -1) + q_0 \frac{\Delta t}{\tau_{\Delta} \Delta x} \left [ ( 1-
\Theta ) \big ( e^{i k \Delta x} -1 \big ) + \Theta \xi \big ( e^{i k
\Delta x} - 1 \big ) \right ] & = 0,
\\
q_0 ( \xi -1) + q_0 \frac{\Delta t}{\qtau}
\left [ (1-\Theta) + \Theta \xi \right ] &
\nonumber \\
+ T_0 \frac{\tau_{\Delta} \Delta t}{\Delta x \qtau} \left [ (1-\Theta)
\big ( 1- e^{-i k \Delta x} \big ) + \Theta \xi \big (1- e^{-i k \Delta
x} \big ) \right ]
\nonumber \\
- q_0 \frac{\kappa^2 \Delta t}{\Delta x^2 \qtau} \left [ ( 1-\Theta)
\left ( e^{i k \Delta x} -2 + e^{- i k \Delta x} \right ) + \Theta \xi
\left (e^{i k \Delta x} - 2 + e^{-i k \Delta x} \right ) \right ] & = 0.
\end{aligned}$$ Then constructing a coefficient matrix and calculating its determinant leads to the characteristic polynomial of system (\[DISCSYS\]) in the form $F(\xi) = a_2 \xi^2 + a_1 \xi + a_0$ with the coefficients: coefficients $$\begin{aligned}
a_0 & = 1- \frac{\Delta t}{\qtau} ( 1-\Theta) +\left [ 2 \cos(k \Delta
x) -2 \right ] \left ( 1 - \Theta \right ) \frac{\Delta t}{\Delta x^2
\qtau} \left [ \kappa^2 - \Delta t ( 1-\Theta) \right ],
\nonumber \\
a_1 & = -2 + \frac{\Delta t}{\qtau} (1- 2 \Theta) +
\left [ 2 \cos(k \Delta x) -2 \right ]
\frac{\Delta t}{\Delta x^2 \qtau} \left [ \kappa^2 (2 \Theta -1)
- 2 \Delta t \left ( 1 - \Theta \right ) \Theta \right ] ,
\nonumber \\
a_2 & = 1 + \frac{\Delta t}{\qtau} \Theta - \left [
2 \cos(k \Delta x) -2 \right ] \frac{\Delta t}{\Delta x^2 \qtau} \Theta \big (\kappa^2 +\Delta t \Theta \big ).
\end{aligned}$$ The Jury criteria [@Jury74] can be used to obtain the requirements in order to ensure that the roots of characteristic polynomial remain within the unit circle in the complex plane. These criteria are, for the polynomial $F$:
1. $F(\xi=1) > 0$,
2. $F(\xi=-1) > 0$,
3. $|a_0| < a_2$.
Calculating each condition for $\Theta=1$ gives us
1. $\frac{4 \Delta t^2}{\qtau \Delta x^2}>0$,
2. $4 + 2\frac{\Delta t}{\qtau} + 4 \frac{\Delta t}{\qtau \Delta x^2} ( 2\kappa^2 + \Delta t) >0$,
3. $1 < 1 + \frac{\Delta t}{\qtau} + 4 \frac{\Delta t}{\qtau \Delta x^2} ( \kappa^2 + \Delta t)$,
hence, the scheme has met the requirements as long as all parameters are positive. In case of $\Theta=1/2$, we have
1. $\frac{4 \Delta t^2}{\qtau \Delta x^2}>0$,
2. $4 >0$,
3. $0 < 1 + \frac{4 \kappa^2}{\Delta x^2}$,
that is, the first Jury criterion gives the same result and the other two conditions are simpler and naturally fulfilled again. We remark that, for the MCV equation ($\kappa=0$), each criteria are fulfilled, too. Therefore, the schemes are stable and convergent.
Comparison with analytical solutions
====================================
Analytical solution for the GK equation is known for several cases [@Zhukov16; @Zhu16a; @Zhu16b; @ZhuSri17], even for boundary conditions related to heat pulse experiments with adiabatic condition on the rear side [@Kov18gk]. The analytical solution is available in an infinite sum form [@Kov18gk]. For the benchmark comparison between analytical and numerical solutions presented here, the following parameters have been applied:
1. Solution of Fourier equation: $\qtau = \kappa^2$,
2. Solution of MCV equation: $\qtau=0.02$, $\kappa^2=0$,
3. Over-diffusive solution of the GK equation: $\qtau=0.02$, $\kappa^2=10 \qtau$.
Moreover, heat pulse duration $\tau_\Delta=0.04$ is used in all cases, and the simulated time interval (dimensionless time $t =1$) and the number of cells ($300$) are also fixed. The various schemes are compared based on the computational run time measured by MATLAB. Although the run time itself is not representative in a single scheme and depends on many other conditions like programming language, realization of a scheme, properties of hardware, etc., for comparative reasons it is useful and representative. It is important to emphasize that the over-diffusive range ($\kappa^2>\qtau$) is distinguished by experiments, i.e., the measured non-Fourier behavior always occurs in that region of parameters.
1. Case of the Fourier equation:
1. $\Theta=1$ scheme requires $100$ time steps and the solution takes $0.119$ s (see Fig. \[fig:impfou\_anal\]).
2. $\Theta=1/2$ scheme shows no difference either in accuracy or in run time.
3. $\Theta=0$ explicit scheme requires ca. $10^6$ time steps, which takes $142.9$ s.
4. The analytical solution requires $50$ terms and takes $0.08$ s with $100$ time steps (see Fig. \[fig:impfou\_anal\]).
2. Case of the MCV equation:
1. For $\Theta=1$ scheme, $1000$ time steps are not sufficient. The solution was not accurate enough (see Fig. \[fig:impmcv1\]). With a new setting of $10^5$ time steps, solution takes $15.4$ s (Fig. \[fig:impmcv2\]).
2. $\Theta=1/2$ scheme shows significant difference especially for hyperbolic equations like the MCV one. The vicinity of the wave front is more accurate than in the previous case, $1000$ time steps are sufficient and it requires $0.2$ s.
3. $\Theta=0$ explicit scheme requires ca. $10^6$ time steps again, which takes $145.5$ s.
4. The analytical solution requires $200$ terms and takes $6.6$ s with $500$ time steps (see Fig. \[fig:impmcv\_anal\]).
3. Case of the GK equation:
1. $\Theta=1$ scheme requires $100$ time steps and the solution takes $0.261$ s (see Fig. \[fig:impgk\_anal\]).
2. $\Theta=1/2$ scheme shows no difference either in accuracy or in run time.
3. $\Theta=0$ explicit scheme requires ca. $10^7$ time steps, which takes $1640$ s.
4. The analytical solution requires $5$ terms and takes $0.07$ s with $200$ time steps (see Fig. \[fig:impgk\_anal\]).
As it is clear, the implicit schemes reproduce the analytical solution in every case. In fact, they could be faster for solutions containing jumps like in case of the MCV equation. Moreover, the capabilities of the analytical solution approach are more limited – for example, the GK equation for finite time heat pulse excitation with cooling boundary conditions is not yet solved. In such cases the numerical methods are the only way to obtain the solution. It is important to observe the significant difference between $\Theta=1$ and $1/2$ schemes for hyperbolic equations. The explicit scheme was the slowest and less efficient, not surprisingly.
Comparison with finite element method
=====================================
In this section, the finite element implementation of the same problem is presented, using the software COMSOL v5.3a. Theoretically, it is possible to implant any kind of partial differential equation within the COMSOL environment. However, to obtain a solution of a generalized heat equation is not as easy as it seems to be. Let us begin with the MCV equation. In order to achieve a smooth solution around the wave front, $100$ elements were used together with the Runge–Kutta (RK34) time stepping method, which requires $600$ time steps. Its run time was $44$ s, and for the solution see Fig. \[fig:vemcom\_mcv\]. It is to be noted that the simulated time interval was shorter, $0.6$ instead of $1$. The COMSOL solution is hardly faster than the explicit scheme presented above, and is much slower than the Crank–Nicolson-type implicit scheme.
When we turn towards the full GK equation, obtaining the solution is not straightforward at all. Although COMSOL reproduces temperature history at the rear side (Fig. \[fig:vemcom\_gk1\]) in the Fourier-type special case, Fig. \[fig:vemcom\_gk2\] presents a false one with $\qtau=0.02$ and $\kappa^2=0.2$. Instead of the breakage (like the one in Fig. \[fig:impgk\_anal\]), a false wave-shaped solution appears that does not exist either in the finite difference solution or in the analytical one. Moreover, the appearance of this numerical artifact is independent of mesh and time step sizes, and becomes bigger as $\kappa^2$ is increased. Applying again $100$ elements together with a time step of $0.001$, it takes $65$ s to run (Fig. \[fig:vemcom\_gk2\]). Hence, COMSOL seems not to be applicable to solve the GK equation in the highly over-damped domain.
Outlook for the ballistic-conductive equation
=============================================
The ballistic-conductive (BC) equation is a next-level generalization beyond the GK equation, and is strongly related to the low-temperature phenomenon called ballistic heat conduction [@KovVan15; @DreStr93a; @FriCim95; @FriCim96; @GrmLeb05]. Indeed, the BC model has been found to be necessary for explaining low-temperature experiments [@KovVan16; @KovVan18]. Such a hyperbolic equation is more challenging to solve, due to the double characteristic speed and the sharper jumps in certain parameter range. It is possible to derive the implicit schemes for this model as well, as presented below. The system of partial differential equations in question, in dimensionless form, reads $$\begin{aligned}
\qtau \frac{\partial q}{\partial t} +q +\tau_{\Delta}\frac{\partial
T}{\partial x} +\kappa \frac{\partial Q}{\partial x} & = 0,
\\ \nonumber
\tau_Q \frac{\partial Q}{\partial t} +Q+ \kappa \frac{\partial q}{\partial x}
& = 0,
\end{aligned}$$ and in the discretized form: $$\begin{aligned}
\frac{\qtau}{\Delta t} (q^{n+1}_j - q^n_j ) + \left ( (1-\Theta) q^n_j +
\Theta q^{n+1}_j \right ) &
\nonumber \\
+ \frac{\tau_{\Delta}}{\Delta x} \left ( (1-\Theta) ( T^n_j - T^n_{j-1}) + \Theta (T^{n+1}_j - T^{n+1}_{j-1}) \right )
\nonumber \\
+ \frac{\kappa}{\Delta x} \left ( (1-\Theta) ( Q^n_j - Q^n_{i-1}) + \Theta (Q^{n+1}_j - Q^{n+1}_{j-1}) \right )
& = 0,
\label{IMPSEPBC} \\ \nonumber
\frac{\tau_Q }{\Delta t} (Q^{n+1}_j - Q^n_j) +\left ( (1-\Theta) Q^n_j + \Theta Q^{n+1}_i \right ) &
\nonumber \\ \label{BCBC}
+ \frac{\kappa}{\Delta x} \left [ (1-\Theta) (q^n_{j+1} - q^n_j )+ \Theta (q^{n+1}_{j+1} - q^{n+1}_j ) \right ]
& = 0,
\end{aligned}$$ where a further variable $Q$ appears as a current density of heat flux [@KovVan15]. The related relaxation time is denoted by $\tau_Q$. The system (\[IMPSEPBC\])–(\[BCBC\]) can be solved together with the balance equation of internal energy (\[IMPSEPEN\]). Let us use the parameters $\tau_\Delta=0.0076$, $\qtau=0.0186$, $\tau_Q=0.007$, $\kappa=0.108$, which are taken from [@KovVan18] and are related to the evaluation of a ballistic heat conduction phenomenon. The same accuracy properties of implicit schemes are experienced as previously in case of the MCV equation, namely, the one with $\Theta=1/2$ is more accurate in the vicinity of wave front than the one with $\Theta=1$, see Fig. \[fig:impbc\] for details. It is sufficient for Crank–Nicolson-type scheme to use $300$ cells with $5000$ time steps which takes $1.8$ s to solve.
In contrast, the COMSOL software is much slower and less accurate, i.e., it produces the same mesh and time step dependent oscillations and jumps, see Fig. \[fig:vemcom\_bc\]. Only the last jump corresponds to a real solution, despite of the $1000$ cells used in the simulation with $9000$ time steps. The run time was $186$ s. Should one want to avoid these artificial oscillations, the simulation would require at least ten times more cells and time steps, and it would take hours for COMSOL to solve the BC model with these settings, in contrast with the $1.8$ s run time of the Crank–Nicolson-type approach.
Summary
=======
Finite difference numerical schemes based on the shifted field concept have been presented and tested in several cases. It turned out that the Crank–Nicolson-type implicit scheme is the most accurate, especially in solving hyperbolic partial differential equations. Not surprisingly, the presented implicit schemes proved much faster than the explicit one. We also focused on the validation of numerical schemes using the analytical solution of the GK equation. It is important to highlight that the analytical solutions are strongly limited as, for more natural boundary conditions like heat transfer at the boundary is not yet obtained. However, having analytical solution is not absolutely necessary in presence of such a fast and reliable numerical scheme.
The commercial software COMSOL has also been applied for comparison. We have demonstrated that solving generalized heat equations is challenging for finite element methods, and leads in some cases to false solutions so result have to be validated as extensively as possible.
Acknowledgements {#ackn}
================
This work was supported by the National Research, Development and Innovation Office of Hungary (NKFIH) via grants NKFIH K116197, K116375, K124366 and K124508.
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---
abstract: 'We study spatio-temporal pattern formation in a ring of $N$ oscillators with inhibitory unidirectional pulselike interactions. The attractors of the dynamics are limit cycles where each oscillator fires once and only once. Since some of these limit cycles lead tothe same pattern, we introduce the concept of pattern degeneracy to take it into account. Moreover, we give a qualitative estimation of the volume of the basin of attraction of each pattern by means of some probabilistic arguments and pattern degeneracy, and show how are they modified as we change the value of the coupling strength. In the limit of small coupling, our estimative formula gives a perfect agreement with numerical simulations.'
address: ' Departament de Física Fonamental, Universitat de Barcelona, Diagonal 647, E-08028 Barcelona, Spain\'
author:
- 'X. Guardiola[@xaviermail] and A. Díaz-Guilera[@albertmail]'
title: 'Pattern selection in a lattice of pulse-coupled oscillators'
---
=msbm10 \#1
Introduction
============
The study of the collective behavior of populations of interacting nonlinear oscillators has attracted the interest of physicists and mathematicians for many years since they can be used to modelize several chemical, biological and physical systems[@Kura84; @Win]. Among them, we should mention cardiac pacemakers cells[@Pes], integrate and fire neurons[@Kura91] and other systems made of excitable units[@Treves]. Most of the theoretical papers that have appeared in the scientific literature deal with oscillators interacting through continuous-time couplings, allowing them to describe the system by means of coupled differential equations and apply most of the modern nonlinear dynamics techniques. More challenging from a theoretical point of view is to consider a pulse-coupling, or in other words, oscillators coupled through instantaneous interactions that take place in a very specific moment of its period. The richness of behavior of these pulse-coupled oscillatory systems includes synchronization phenomena[@Mir], spatio-temporal pattern formation[@PhysD] (we could mention, for instance, traveling waves[@Bres2], chessboard structures[@PhysD], and periodic waves[@Rit] ), rhythm anihilation[@nature], self-organized criticality[@PRL1],...
Most of the work on pattern formation has been done in mean-field models or populations of just a few oscillators. However, such restrictions do not allow to consider the effect of certain variables whose effect can be crucial for realistic systems. The specific topology of the connections or geometry of the system are some typical examples which usually induce important changes in the collective behavior of these models. Pattern formation usually takes place when oscillatory units interact in an inhibitory way, although it has also been shown that the shape of the interacting pulse, when the spike lasts for a certain amount of time, or time delays in the interactions can lead to spatio-temporal pattern formation also in the case of excitatory couplings[@Abbot; @Geisel]. Only recently, general solutions for the general case, where the patterns existence and stability is proved, have been worked out[@Bres1; @PRE]. The aim of this paper is to study some pattern properties and get a quantitative estimation of the probability of pattern selection under arbitrary initial conditions or, in the language of dynamical systems, the volume of the basin of attraction of each pattern. Keeping this goal in mind, we will use the general results given in [@PRE] where assuming a system defined on a ring the authors developed a mathematical formalism powerful enough to get analytic information of the system. Not only about the mechanisms which are responsible for synchronization and formation of spatio-temporal structures, but also, as a complement, to proof under which conditions they are stable solutions of the dynamical equations.
Despite the apparent simplicity of the model, some ring lattices of pulse-coupled oscillators are currently used to modelize certain types of cardiac arhythmias where there is an abnormally rapid heartbeat whose period is set by the time that an excitation takes to travel the circuit [@Ito]. Moreover, there are experiments where rings of a few R15 neurons from [*Aplysia*]{} are constructed and stable patterns are reported [@Dror]. Our 1d model allows us to study analytically the most simple patterns and understand their mechanisms of selection.
The structure of this paper is as follows. In Sec II we review the model introduced in [@PRE] as well as set the notation used throughout the paper. In Section III we study some pattern properties which will be useful for, in Section IV, propose an estimation of the probability of selection of each pattern. In the last section we present our conclusions.
The model
=========
Our system consists in a ring of $(N+1)$ pulse-coupled oscillators. The phase of each oscillator $\phi_i$ evolves linearly in time
$$\frac{d\phi_i}{dt}=1 \hspace{2em}\forall i=0,\ldots ,N$$
until one of them reaches the threshold value $\phi_{th}=1$. When this happens the oscillator fires and changes the state of its rightmost neighbor according to
$$\phi_{i}\geq 1 \Rightarrow \left\{
\begin{array}{l}
\phi_{i}\rightarrow 0 \\ \phi_{i+1}\rightarrow\phi_{i+1}+
\varepsilon\phi_{i+1}\equiv\mu\phi_{i+1}
\end{array}
\right.$$ subjected to periodic boundary conditions, i.e. $N+1\equiv 0$, and where $\varepsilon $ denotes the strength of the coupling and $\mu=1+\varepsilon$. Where we have assumed that, from an effective point of view, the pulse-interaction between oscillators, as well as the state of each unit of the system, can be described in terms of changes in the phase, or in other words, in terms of the so called phase response curve (PRC), $\varepsilon \phi$ in our case. A PRC for a given oscillator represents the phase advance or delay as a result of receiving an external stimuli (the pulse) at different moments in the cycle of the oscillator. We will assume $\varepsilon<0$ througout the paper, as we are only interested in spatio-temporal pattern formation and $\varepsilon>0$ always leads to the globally synchronized state[@PRE]. This linear PRC has physical sense in some situations. For instance, it shows up when we expand the non-linear PRC for the Peskin model of pacemaker cardiac cells [@Pes] in powers of the convexity of the driving or in neuronal modelling[@Torras]. In practice, however, this condition can be relaxed since a nonlinear PRC does not change the qualitative behavior of the model provided the number of fixed points of the dynamics is not altered. Moreover, a linear PRC has the advantage of making the system tractable from an analytical point of view.
Let us describe the notation used in the paper. The population is ordered according to the following criterion: The oscillator which fires will be always labeled as unit 0 and the rest of the population will be ordered from this unit clockwise. After the firing, the system is driven until another oscillator reaches the threshold. Then, we relabel the units such that the oscillator at $\phi =1$ is now unit number 0, and so on. This firing + driving (FD) process for $N+1$ oscillators can be described through a suitable transformation
$$\vec{\phi}'=T_{k}(\vec{\phi})\equiv\vec{1}+{\Bbb{M}}_k\vec{\phi},$$
where ${\Bbb{M}}_k$ is a $N \times N$ matrix, $\vec{\phi}$ is a vector with $N$ components, $\vec{1}$ is a vector with all its components equal to one and $k$ stands for the index of the oscillator which will fire next. We call this kind of transformation a firing map, and we have to define as many firing maps as oscillators could fire, that is, index $k$ must run from $k=1$ ($\phi_1$ fires) to $N$ ($\phi_N$ fires). For example, in the $N+1=4$ oscillators case we have that the firing map corresponding to the FD process where $\phi_2$ is the next oscillator which do fire,
------------ --------------------------------------------- ------------- ---------------------------------------------- -------------------------------
$\phi_0=1$ $\stackrel{\mbox{firing}}{\longrightarrow}$ $0$ $\stackrel{\mbox{driving}}{\longrightarrow}$ $1-\phi_2=\phi'_2 $
$\phi_1$ $\longrightarrow$ $\mu\phi_1$ $\longrightarrow$ $\mu\phi_1+ 1-\phi_2=\phi'_3$
$\phi_2$ $\longrightarrow$ $\phi_2$ $\longrightarrow$ $1=\phi'_0$
$\phi_3$ $\longrightarrow$ $\phi_3$ $\longrightarrow$ $\phi_3+ 1-\phi_2=\phi'_1$
------------ --------------------------------------------- ------------- ---------------------------------------------- -------------------------------
would be $T_2(\vec{\phi})$
$$\left(
\begin{array}{c}
\phi^{\prime}_1 \\ \phi^{\prime}_2 \\ \phi^{\prime}_3
\end{array}
\right) = \left(
\begin{array}{c}
1 \\ 1 \\ 1
\end{array}
\right) + \underbrace{\left(
\begin{array}{ccc}
0 & -1 & 1 \\ 0 & -1 & 0 \\ -\mu & - 1 & 0
\end{array}
\right) }_{{\Bbb{M}}_2} \left(
\begin{array}{c}
\phi_1 \\ \phi_2 \\ \phi_3
\end{array}
\right)$$
and so on. Once we have defined all possible firing maps for a given number of oscillators we can proceed to deal with the attractors or fixed points of the system dynamics. As has been proved in [@PRE] these fixed points must be cycles of $N+1$ firings. We define a cycle as a sequence of consecutive firings where each oscillator fires once and only once. Mathematically, each cycle is described by means of a return map. The return map is the transformation that gives the evolution of $\vec{\phi}$ during a cycle and is the composition of all firing maps involved in the firing sequence of that cycle
$$\vec{\phi}'=T_{c_1} \circ T_{c_2}\ldots \circ T_{c_{N+1}}
(\vec{\phi})\equiv\
\vec{R}_{c}+{\Bbb{M}}_{c}\vec{\phi},$$
where $T_{c_i} \circ T_{c_j}(\phi)$ is the usual composition operation $T_{c_i}(T_{c_j}(\phi))$ and
$$\vec{R}_{c}=\vec{1}+\sum_{i=c_1}^{c_N}(\prod_{j=c_1}^{i}
{\Bbb{M}}_{j}) \cdot\vec{1}
\hspace{2em} \mbox{and} \hspace{2em} {\Bbb{M}}_{c}=
\prod_{j=c_1}^{c_{N+1}}{\Bbb{M}}_{j}.$$ Note that not all possible combinations of firing maps are allowed, just those ones whose indices $c_i$ sum $p(N+1)$ without any partial sum equal to $q(N+1)$, where $p>q$ are positive integers.
As all firing maps are linear transformations, return maps are also linear. There are $N!$ possible cycles in the $N+1$ oscillators case (all permutations of firing sequences with the initial firing oscillator $\phi_0$ fixed). Following our previous example, for the four oscillators case all possible firing sequences and their associated return maps are
$$\begin{aligned}
A: 0,1,2,3 \rightarrow T_1 \circ T_1 \circ T_1 \circ T_1
\\ B: 0,1,3,2 \rightarrow T_2
\circ T_{3} \circ T_{2} \circ T_{1} \\ C: 0,2,1,3 \rightarrow
T_{1} \circ T_{2} \circ
T_{3} \circ T_{2} \\ D: 0,2,3,1 \rightarrow T_{3} \circ
T_{2} \circ T_{1} \circ T_{2}
\\ E: 0,3,1,2 \rightarrow T_{2} \circ T_{1} \circ T_{2}
\circ T_{3} \\ F: 0,3,2,1
\rightarrow T_{3} \circ T_{3} \circ T_{3} \circ T_{3}\end{aligned}$$
Now, in order to find the attractors of the dynamics, we must solve the fixed point equation
$$\vec{\phi}_{c}^{*}=\vec{R}_{c}+\Bbb{M}_{c}\vec{\phi}_{c}^{*},$$
for every cycle $c$. Formally,
$$\vec{\phi}_{c}^{*}=\vec{R}_{c}\cdot({\Bbb{I}}-{\Bbb{M}}_{c})^{-1}.$$
As was shown in [@PRE], there are $N$ different stable solutions to the whole set of fixed point equations. Their stability is assured by the fact that $\varepsilon<0$, since it guarantees that all eigenvalues of $\Bbb{M}_{c}$ lie inside the unit circle for all cycles $c$. In our four oscillators example these solutions are
$$\begin{aligned}
\nonumber \vec{\phi}_{A}^{*}= & (1,\frac{3}{4+3\varepsilon},
\frac{2}{4+3\varepsilon},
\frac{1}{4+3\varepsilon}) & \hspace{1em} \\ \nonumber
\vec{\phi}_{B}^{*}=\vec{\phi}_{C}^{*}=\vec{\phi}_{D}^{*}
=\vec{\phi}_{E}^{*}= &
(1,\frac{1} {2+\varepsilon},1,\frac{1}{2+\varepsilon})
& \hspace{1em} \\
\vec{\phi}_{F}^{*}= & (1,\frac{1}{4+\varepsilon},
\frac{2+\varepsilon}{4+
\varepsilon},\frac{3+\varepsilon}{4+\varepsilon}) &
\hspace{1em}\end{aligned}$$
Which are a kind of four-oscillators traveling wave, chessboard and inverse traveling wave structures.
From now on we will label such solutions with index $m$ ($m=1...N$) since their first component always satisfy
$$\phi_{1}^{*}=\frac{m}{N+1+m\varepsilon}.$$
Therefore, in the example, we relabel patterns $\vec{\phi}_{A}^{*}$ as $m=3$, $\vec{\phi}_{B}^{*}$, $\vec{\phi}_{C}^{*}$, $\vec{\phi}_{D}^{*}$, $\vec{\phi}_{E}^{*}$ as $m=2$ and $\vec{\phi}_{F}^{*}$ as $m=1$.
Since there are $N!$ possible cycles and $N$ solutions to Eq. (7) there will be some fixed points or patterns which will appear more than once, so, we shall use $C(N+1,m)$ to characterize these degeneracies. In the example, the values of the degeneracies are $C(4,1)=C(4,3)=1$ and $C(4,2)=4$. In general, patterns which are solutions of cycle consisting in the iterative application of the same firing map (like A and F in our example) have no periodicities whereas the ones solution of mixtures of differents firing maps (B,C,D and E) have some periodic structure that are also solution of Eq. (7) for a case with less oscillators. In Fig. 1 we can visualize the solutions for $N+1=2,3$ and $4$ oscillators and realize that solution $m=2$ for the four oscillators case is a periodic composition of solution $m=1$ for the two oscillators case.
= 8.0cm
Pattern properties
==================
As we have seen, the stability of all patterns solution of Eq. (6) is guaranteed by the fact that $\varepsilon<0$, but the existence of such solutions is not ensured. In fact, for small values of the coupling strength $|\varepsilon|$ all patterns do exist, but, as we increase it, some patterns disappear. The reason is that the solution loses its physical meaning because $\phi_1^*>1$. Their first component is always the one that becomes larger than unity earlier and this happens, for each $m$ and according to Eq. (9), when
$$\varepsilon<\varepsilon_m^*=1-\frac{N+1}{m}.$$
Our coupling strength range of interest ends at $\varepsilon=-1$, since at $\varepsilon\leq -1$ we always find the same pathological dynamics which does not have any physical or biological sense. Realistic couplings never reach such higher values. Therefore, as $\varepsilon$ runs from $0$ to $-1$, all patterns whose $m$ satisfy $m>\frac{N+1}{2}$, disappear.
There is another interesting pattern property which has to do with the calculation of the pattern degeneracy $C(N+1,m)$. In principle, to calculate such degeneration, we should solve fixed point Eq. (6) for all possible cycles and count how many of them lead to the same pattern. Although for few oscillators the problem is quite straightforward, as we deal with higher and higher number of oscillators, the number of cycles increases (it grows as $N!$) and solving Eq. (6) becomes more difficult. Fortunately, there is another way of calculating $C(N+1,m)$ which reduces the problem to a combinatorial question. Lets show it through an example, in the previous four oscillators case, if we count, for each firing sequence, the number of oscillators which have received the pulse before firing, we can easily realize that this number is the same as its value of $m$
$$\begin{aligned}
A: & 0,\overline{1},\overline{2},\overline{3} &
\hspace{2em} m=3 \\ B: &
0,\overline{1},3,\overline{2} & \hspace{2em} m=2 \\ C:
& 0,2,\overline{1},\overline{3} &
\hspace{2em} m=2 \\ D: & 0,2,\overline{3},\overline{1}
& \hspace{2em} m=2 \\ E: &
0,3,\overline{1},\overline{2} & \hspace{2em} m=2 \\ F:
& 0,3,2,\overline{1} &
\hspace{2em} m=1\end{aligned}$$
Here an upper bar means that the oscillator has already received a pulse during the cycle. The point is that it turns out that every pattern $m$ corresponds to a sequences of firings involving exactly $m$ oscillators that, when they do fire, had already received a pulse from their leftmost neighbor. Therefore, this property (we have checked for several values of $N+1$) allows us to associate every cycle with the pattern it leads to, just by counting these kind of firings. Now, calculating $C(N+1,m)$ becomes a straightforward matter. In Table I we have computed $C(N+1,m)$ for several values of $N+1$.
1 2 3 4 5 6 7 8 9
---- --- ----- ------- ------- -------- ------- ------- ----- ---
2 1
3 1 1
4 1 4 1
5 1 11 11 1
6 1 26 66 26 1
7 1 57 302 302 57 1
8 1 120 1191 2416 1191 120 1
9 1 247 4293 15619 15619 4293 247 1
10 1 502 14608 88234 156190 88234 14608 502 1
: Pattern degeneracy $C(N+1,m)$. First column stands for the number $N+1$ of oscillators and first row for $m$.
Apart from brute force counting, degeneracy distribution $C(N+1,m)$ can also be determined from the following relation
$$\begin{aligned}
\nonumber C(N+1,m)& = & mC(N,m)+ \\ & &(N+1-m)C(N,m-1),\end{aligned}$$
for $2\leq m\leq N-1$. This recursion relation is closed by
$$C(N+1,1)=C(N+1,N)=1,$$
which correspond to the firing sequences
$$0,N,(N-1)...2,\overline{1} \hspace{2em} \mbox{and} \hspace{2em}
0,\overline{1},\overline{2}...\overline{(N-1)},\overline{N},$$ respectively.
From the previous relations one can deduce by induction the symmetry of the distribution with respect to its extremes at $m=1$ and $m=N$
$$C(N+1,m)=C(N+1,N+1-m),$$
and
$$\sum_{m}C(N+1,m)=N!.$$
Another interesting property is the period $\Delta_{m}^{N+1}$ of each spatio-temporal pattern $m$. Since all oscillators are in a phase-locked state, they must oscillate with the same period. Then, as the intrinsic period of each oscillator is one, and when any oscillator receives the delaying pulse from its neighbor it has a phase equal to $\phi_{1}^{*}$, one can easily realize that the effective period is
$$\Delta_{m}^{N+1}=1+\varepsilon \phi_{1}^{*}=
\frac{N+1+2m\varepsilon}{N+ 1+m\varepsilon}.$$
Therefore, the larger the value of $m$, the longer the period of its associated pattern. It is important to notice that we have not fixed the value of such periods (each pattern has its own which is different from the others), since there are some authors who fix all periods equal to some constant, and use it as a condition to find the structures[@Dror].
Pattern selection
=================
Once we have characterized all spatio-temporal patterns, we proceed to find some general formula which give us some estimation of the probability of each pattern to be selected, or in other words, an estimation of the volume of its basin of attraction. In order to achieve this objective, we should understand the mechanism which lead to the selection of a certain spatio-temporal structure and how is it modified as the parameters of the model ($\varepsilon$ in our case) change.
There is an easy and straightforward way to get the essential features of this mechanism assuming that the probability of one oscillator to fire next is, basically, proportional to its phase (that is, if it has a phase slightly below $1$ it has a higher probability to be the next firing oscillator, whereas if it has a smaller phase, it will rarely fire next). Imagine the phases of all oscillators randomly distributed over the interval $(0,1)$. Then we let the system evolve till one of the oscillators reaches a phase $\phi_i=1$ and emits a pulse that is received by its rightmost neighbor which lows its phase by an amount $\varepsilon \phi_{i+1}$. Now we assume that all phases are again randomly distributed over $(0,1)$ except the one which received the pulse whose phase is distributed over $(0,1+\varepsilon)$. So, we get rid of memory effects (we know the oscillator that has fired should, now, have a phase equal to zero) and just keep in mind if each oscillator has received a pulse or has not. Therefore, the point is that under this conditions, the probability that one oscillator which has still not received a pulse do fire is some constant and, on the other hand, for the ones which had, is this constant times the factor $(1+\varepsilon)$. Then, we can characterize the probability of having some cycle just by recalling how many oscillators do fire having previously received a pulse during that cycle. Basically, this probability is proportional to $(1+\varepsilon)^{n}$ where $n$ stands for the number of oscillators which do fire having already received a pulse (the product of all constant terms will be absorbed in a normalization factor). This approach, where we assume all firings as almost-independent events, can be viewed as a kind of mean-field approximation. Then,as has been shown before, since cycles leading to the same pattern $m$ always exactly have $m$ oscillators that do fire having received the interacting pulse, we can give an estimation of the probability for pattern $m$ selection in the $N+1$ oscillators case
$$p_m^{N+1}(\varepsilon) \simeq {\cal N}(\varepsilon)C(N+1,m)
(1+\varepsilon)^m.$$
Here ${\cal N}(\varepsilon)$ is chosen so that summation of the probabilities over m gives 1
$$\sum_m p_m^{N+1}(\varepsilon)=1.$$
In the limit of small coupling strength $\varepsilon\rightarrow 0$, which is the more interesting case for the majority of physical and biological systems, one can assume that interaction plays almost no role when pattern selection takes place. That is, the fact that one oscillator has received the pulse from its neighbor does not low its probability to fire as the pulse does not modify appreciably its phase. Then, we can consider that all cycles have approximately the same probability to be selected, $(1+\varepsilon)^m \rightarrow 1$, and only pattern degeneracy has to be considered to get a good estimation of $p_m^{N+1}$
$$p_m^{N+1} \simeq \frac{C(N+1,m)}{N!}.$$
The dominant pattern, that is, the one which has the larger probability to be selected coincides with the mean value of $m$ (due to the symmetric behavior of $C(N+1,m)$).
$$<m>_{N+1}=\sum_{m}m\frac{C(N+1,m)}{N!}=\frac{N+1}{2}.$$
For an odd number of oscillators $<m>_{N+1}$ does not exist and we have a competition between the two closest patterns $m=N/2$ and $m=(N+2)/2$. Recall that the most probable patterns turn out to be the ones with “shortest wavelengths”, a fact that was already reported in simulations of these sort of systems[@PhysD]. In Figs. 2 and 3 we check this new approximation for the $N+1=10$ and $9$ case and realize that expected results are in good agreement with simulations data.
There also is the interesting question of how does this probability distribution modifies when the number of oscillators increases. In Fig. 4 we show $p_{m}$ for different values of $N+1$. Since there are more possible values of $m$ available, as we increase $N+1$, $p^{N+1}_{m}$ diminishes. The distribution also gets narrower as we increase $N+1$ and this becomes clear when one studies the variance of $p_{m}$. It can be found that
$$\begin{aligned}
\nonumber <m^{2}>_{N+1} & = & \sum_{m}m^{2}
\frac{C(N+1,m)}{N!} \\ & =
&\frac{(N+1)^{2}}{4}+\frac{N+1}{12}.\end{aligned}$$
We could not prove this without an explicit expression for $C(N+1,m)$ but we have checked it N up to $170$. Therefore
$$\sigma^{2}_{N+1}=\frac{N+1}{12}=\frac{<m>_{N+1}}{6}.$$
It turns out that for a large number of oscillators almost all initial conditions lead to a pattern whose $m$ approximately falls in the interval $<m>_{N+1}\pm
\sqrt{<m>_{N+1}}$. In order to compare it for different number of oscillators we have to normalize $m$ dividing by $N+1$. In that case, one observes that $\sigma^{2}_{N+1}\sim
1/\sqrt{N+1}$ so that as we increase $N+1$, the spread of $p^{N+1}_{m}$ diminishes getting the distribution sharpened.
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As Eq. (14) does not take into account the disappearance of the different patterns $m$ at the different values of $\varepsilon_{m}^{*}$ predicted by Eq. (9), it can not give a good quantitative estimation of pattern selection for higher coupling values. Nevertheless we can expand Eq. (14) to the leading order in $\varepsilon$. For small $\varepsilon$, $p_m^{N+1}$ are approximated by $$p_m^{N+1}\simeq \frac{C(N+1,m)}{N!}(1+(m-\frac{N+1}{2})\varepsilon).$$ In Fig. 5 we compare this approximation with simulated data. The slopes near $\varepsilon=0$ do agree with Eq. (21).
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In our simulations we calculate the probability of each pattern to be selected just by counting how many realizations (with $\phi_0=1$ and the rest of oscillators with random initial conditions) lead to each pattern $m$ and divide over the total number of realizations.
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Although we only have a good quantitative estimation of $p^{N+1}_{m}$ for small values of $\varepsilon$, Eq. (15) catches the two basic mechanisms responsible of pattern selection. On the one hand, it is clear that for higher values of the coupling strength $|\varepsilon|$, when one oscillator receives a pulse, it lows its phase to almost zero and, consequently, its firing probability also does. Therefore pattern selection probability $p_m^{N+1}(\varepsilon)$ is strongly controlled by the number of oscillators which have to fire having already received a pulse, that is, the probabilistic factor $(1+\varepsilon)^m$. As a consequence, $p^{N+1}_{m}$ begin to decrease sooner when $|\varepsilon|$ increases, the larger $m$ is. On the other hand, for small values of the coupling strength, interaction plays almost no role and $p_m^{N+1}(\varepsilon)$ is dominated by the degeneracy factor $C(N+1,m)$. Therefore $p_m^{N+1}(\varepsilon)$ for the different values of $m$ are basically ordered as $C(N+1,m)$. In Fig. $6$, $7$ and $8$ we show results from simulations of $p_m^{N+1}(\varepsilon)$ for different number of oscillators.
Conclusions
===========
In this paper we have studied some properties of the spatio-temporal patterns that appear in a ring of pulse-coupled oscillators with inhibitory interactions. We have focused our attention in estimating the probability of selecting a certain pattern under arbitrary initial conditions and have shown the two basic mechanisms responsible of that: the degeneracy distribution $C(N+1,m)$, for small values of $\varepsilon$, and $m$, the number of oscillators that do fire having already received a pulse, for higher values of $\varepsilon$. According to this, the different probabilities of selecting pattern $m$ start being distributed following the degeneracy distribution $C(N+1,m)$, and, as $\varepsilon$ decreases, these probabilities diminish in a hierarchical way: the larger the value of $m$, the sooner its selection probability is going to decrease, so that only patterns with smaller m will survive for higher values of $\varepsilon$. Moreover, some of the structures disappear, at the different values of $\varepsilon_m^*$, during this process. We have found out an approximation formula for $p_m^{N+1}(\varepsilon)$ which takes into account all these mechanisms and gives us a quantitative estimation of the different selection probabilities for small $\varepsilon$.
The estimation of the volume of the basin of attraction of each spatio-temporal pattern $m$ also gives us an idea of the stability of the different structures with respect to additive noise fluctuations (for instance, we can add some random quantity $\eta$ to all phases after each firing event or a continuous-time $\eta(t)$ in the driving). Simulations of arrays of noisy pulse coupled oscillators showed that our most probable patterns were also the most stable[@PhysD]. The present paper only concerns spatio-temporal pattern formation in a ring of oscillators, nevertheless, all results are trivially generalized to bidirectional couplings. Although the question of what happens when dealing with higher dimension lattices remains opened, some simulations results in 2d [@PhysD] showed that almost all realizations lead to a chessboard pattern in analogy with our results in the ring. That makes us believe we have caught the basic features of the problem in our 1d model.
Acknowledgments {#acknowledgments .unnumbered}
===============
The authors are indebted to C.J.Pérez and A.Arenas for very fruitful discussions. They also acknowledge extremely constructive suggestions from an anonymous referee. This work has been supported by DGICYT of the Spanish Government through grant PB96-0168 and EU TMR Grant ERBFMRXCT980183. X.G. also acknowledges financial support from the Generalitat de Catalunya.
e-mail: [xguardi@ffn.ub.es]{} e-mail: [albert@ffn.ub.es]{}
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---
abstract: 'An interesting fundamental problem in density-functional theory of electronic structure of matter is to construct the exact Kohn-Sham (KS) potential for a given density. The exact potential can then be used to assess the accuracy of approximate functionals and the corresponding potentials. Besides its practical usefulness, such a construction by itself is a challenging inverse problem. Over the past three decades, many seemingly disjoint methods have been proposed to solve this problem. We show that these emanate from a single algorithm based on the Euler equation for the density. This provides a mathematical foundation for all different density-based methods that are used to construct the KS system from a given density and reveals their universal character.'
author:
- Ashish Kumar
- Rabeet Singh
- 'Manoj K. Harbola'
bibliography:
- 'mybib.bib'
title: 'Universal nature of different methods of obtaining the exact Kohn-Sham exchange-correlation potential for a given density'
---
Density functional theory (DFT) [@Yang; @Drei; @Gross; @Hohenberg_PR.136.B864] is the most widely used method to study electronic properties of materials [@JoneRMP_89; @SpruchRMP_1991; @JoneRMP] because of its ever increasing accuracy [@SCANACC] and computational ease of implementation. As is well known, in DFT the ground-state energy is written as a functional $E[\rho]$ of the ground-state density $\rho(\vec{r})$. The energy is the sum of the kinetic energy functional $T[\rho]$, external energy $\int v_{ext}(\vec{r})\rho(\vec{r})d\vec{r}$ where $v_{ext}(\vec{r})$ is the external potential, and the expectation value $\langle V_{ee}\rangle$, also a functional of $\rho(\vec{r})$, where ${V_{ee}}$ is the electron-electron interaction energy operator. In the Kohn-Sham [@Kohn_PR.140.A1133] approach to DFT (KSDFT) the interacting electron system is mapped to a fictitious non-interacting system and the energy $E[\rho]$ is expressed as the sum of the non-interacting kinetic energy $T_S[\rho]$ of the same density, the external energy, the Hartree energy $E_{H}[\rho]=\frac{1}{2}\iint\frac{\rho(\vec{r})\rho(\vec{r}')}{|\vec{r}-\vec{r}'|}d\vec{r}d\vec{r}'$ and the exchange-correlation energy $E_{xc}[\rho]$ with the difference $T_c= T[\rho]-T_{S}[\rho]$ absorbed in it. The equation for the density obtained by minimizing the energy $E[\rho]$ with respect to density is $$\Big[ \frac{\delta T_S[\rho]}{\delta \rho}+ v_{ext}(\vec{r})+v_H(\vec{r})+v_{xc}(\vec{r})\Big]=\mu \label{euler},$$ where $\mu$ is the Lagrange multiplier to ensure the constraint that $\int \rho(\vec{r})d\vec{r}=N=$ total number of electron and has the interpretation [@PPLB] of being the chemical potential. In Eq. \[euler\], the Hartree potential $$v_H(\vec{r})=\int\frac{\rho(\vec{r}')}{|\vec{r}-\vec{r}'|}d\vec{r}' \label{hr_pot}$$ and the exchange-correlation potential $$v_{xc}(\vec{r})= \frac{\delta E_{xc}[\rho]}{\delta \rho(\vec{r})} \label{xc_pot} .$$ We note that the exact expressions for $T_S[\rho]$ and $E_{xc}[\rho]$ in terms of the density are not known in general. Thus $\frac{\delta T_S[\rho]}{\delta \rho}$ and $v_{xc}[\vec{r}; \rho]$ are also not known exactly. In the Kohn-Sham(KS) formulation, the density is expressed in terms of single particle orbitals $\phi_i (\vec{r})$, occupying appropriately those with lowest energies so that $\rho(\vec{r})=\sum\limits_{i} f_i|\phi_i(\vec{r})|^2$ where $f_i$ is the occupancy of each orbital. These orbitals are the solutions of the KS equation $$\Big[ -\frac{1}{2}\nabla^2 +v_{ext}(\vec{r})+v_H(\vec{r})+v_{xc}(\vec{r})\Big]\phi_i(\vec{r})= \epsilon_i\phi_i(\vec{r}) \label{KS_eqn}.$$ Although writen in terms of orbitals, Eq. \[KS\_eqn\] is equivalent to Eq. \[euler\] for the density. Furthermore, the non-interacting kinetic energy $T_S[\rho]$ in the terms of KS orbitals is given exactly as $$T_S[\rho]=\sum\limits_{i}f_i\langle\phi_i(\vec{r})|-\frac{1}{2}\nabla^2|\phi_i(\vec{r})\rangle \label{ks_ke} .$$ Finally the orbital energies $\epsilon_i$ in general carry no physical meaning except that for the highest occupied [@PPLB] orbital which is equal to the chemical potential $\mu$.
In KSDFT, the energy functional $E_{xc}[\rho]$ is approximated and consequently the KS equation can be solved only approximately. Development of more and more accurate functionals for $E_{xc}[\rho]$ [@Medv_2017; @TruhF] is of central interest for application of DFT since the accuracy of the energy and density obtained depends on the quality of $E_{xc}[\rho]$ and the corresponding $v_{xc}([\rho];\vec{r})$.
While accurate exchange-correlation energy functionals are being developed [@Perdew_PRL.77.3865; @Perdew_PRL.82.2544; @Jianmin_PRL.91.146401; @Sun_PNAS; @Sun_PRL.115.036402; @Becke_JCP.98.5648; @Lee_PRB.37.785; @Vosko_CJP.58.1200; @Stephens_JPC.98.11623] and applied [@Chen_2017], it is equally important to know the exact KS solution for a many-electron density wherever the latter is available. This gives [@Stott_1988; @Gorling_1992; @Zhao_1992; @Zhao_1993; @Wang_1993; @Zhao_1994; @Vlb_1994; @WY1; @WY2; @Peirs_2003; @Stott_2004; @Viktor_2012; @Viktor_2013; @Wagner_2014; @Viktor_2015; @Wasserman_2017] the exact Kohn-Sham orbitals, the corresponding non-interacting kinetic energy and the related $T_c$. Furthermore, through construction of the KS systems, we also learn [@Buijse_1989; @Gritsenko_1996; @Teal2; @Teal3; @Teal4; @Makmal_2011; @Wagner_2012; @Kohut_2016; @Proetto_2016; @Hollins_2017] about their other interesting aspects. Thus, the exact KS systems set a benchmark to test the accuracy of approximate energy functionals. We point out that for the exact density of electrons in a given external potential, constructing the KS system boils down to finding the exact exchange-correlation potential.
The problem of finding the KS system for a given ground-state density falls in the general category of inverse problems in physics [@Newton_1970]. For a pedagogical review of such problems, we refer the reader to [@Carter_2000]. In the present context, the direct problem is to find the ground-state density of a system of electrons in an external potential by solving the Schrödinger equation for the wavefunction. The inverse problem [@Wasserman_2017], whose solution is warranted by the Hohenberg-Kohn theorem [@Hohenberg_PR.136.B864], is to find the external potential or the wavefunction for a ground-state electronic density. An interesting application [@Jayatilaka_PRL.80.798] of the inverse problem in this context has been to find the Hartree-Fock wavefunction for electrons in Beryllium crystal from its X-ray diffraction data. Finding the KS system for a ground-state density also falls in the same class of inverse problems and is of significant value in density-functional theory, as discussed above.
Given its importance, many different methods have been developed over the years to obtain the exact Kohn-Sham system for a given density. Some of these [@WY1; @WY2] are based on direct optimization of a functional while others are iterative [@Stott_1988; @Gorling_1992; @Zhao_1992; @Zhao_1993; @Zhao_1994; @Wang_1993; @Vlb_1994; @Peirs_2003; @Stott_2004; @Wagner_2014]. The latter methods converge towards the exact Kohn-Sham potential using a density based quantity to update the potential in each step of the iterative process. For example in reference [@Stott_2004], the iterative method utilizes the difference between a given density and densities obtained during iterative steps to modify the potential. Interestingly, in the same paper iterative scheme has also been linked to an optimization method. However, in general a connection between different iterative methods and their relationship with the variational principle is not known.
The purpose of this paper is to show that all the inversion schemes (except that of [@Stott_1988]) referenced above are a result of obtaining the Levy-Lieb functional [@Levy_1979; @Lieb_1983] for a given density and emanate from a single method that utilizes Eq. \[euler\] and Eq. \[KS\_eqn\] in tandem. This method has its origin in the Levy-Perdew-Sahni (LPS) equation [@LPS_1984] for the density. Hence in the next section we first derive the method for the LPS equation and then generalize it to show how apparently different methods emerge from it. The general method is demonstrated by applying it in its different forms to some spherical system in section \[result\]. Sections \[Gks\] and \[result\] thus reveal the universal character of all these methods. Using this universality, in section \[Theory\] we prove that the inversion from density to Kohn-Sham system through any of these methods is equivalent to maximization of the functional $E[v]-\int v(\vec{r})\rho_0(\vec{r})d\vec{r}$ with respect to $v(\vec{r})$ to obtain the Levy-Lieb functional [@Levy_1979; @Lieb_1983] for a given density $\rho_0(\vec{r})$; here $E[v]$ denotes the energy of the given number of electrons moving in the potential $v(\vec{r})$. In the process we also derive a criterion for the convergence of the inversion process. In section \[Equiv\] we show the equivalence of the general algorithm to different methods referenced above. Finally we conclude the paper in section \[summary\].
A general method to obtain the Kohn-Sham potential {#Gks}
==================================================
Kohn-Sham potential from the LPS equation {#GksA}
-----------------------------------------
Consider the LPS equation for the density $$\big[-\frac{1}{2}\nabla^2 + v_{eff}(\vec{r})\big] \rho^{1/2}(\vec{r})= \mu\rho^{1/2}(\vec{r})\label{lps_eqn},$$ where $v_{eff}(\vec{r})$ is given in the terms of the wavefunction. However, by writing the non-interacting kinetic energy as $$T_S[\rho]=T_W[\rho] +T_P[\rho],$$ where $$T_W[\rho]=-\frac{1}{2}\int \rho^{1/2}(\vec{r}) \nabla^2\rho^{1/2}(\vec{r}) d\vec{r} \label{ws}$$ is the Weizsäcker kinetic energy or kinetic energy of Bosons of density $\rho(\vec{r})$ in the ground state, and $T_{P}[\rho]$ is the Pauli kinetic energy, it is easy to see that $$v_{eff}(\vec{r})= v_{ext}(\vec{r})+v_P(\vec{r})+v_H(\vec{r})+v_{xc}(\vec{r}),$$ where $v_P=\frac{\delta T_P}{\delta \rho}$ is the Pauli potential [@MARCH_1985; @LEVY_1988] . Thus with $\frac{\delta T_W}{\delta \rho}= -\frac{1}{2}\frac{\nabla^2\rho^{1/2}(\vec{r})}{\rho^{1/2}(\vec{r})}$ Eq. \[euler\] for the density is $$-\frac{1}{2}\frac{\nabla^2\rho^{1/2}(\vec{r})}{\rho^{1/2}(\vec{r})}+ v_{ext}(\vec{r})+v_P(\vec{r})+v_H(\vec{r})+v_{xc}(\vec{r})=\mu \label{LPS_KS}$$ Now for a given exact density $\rho_0(\vec{r})$, if we denote the corresponding quantities with superscript $\lq 0$’, Eq. \[LPS\_KS\] can be rewritten for the exact exchange-correlation potential as $$v_{xc}^{0}(\vec{r})= \mu+\frac{1}{2}\frac{\nabla^2\rho^{1/2}_0(\vec{r})}{\rho^{1/2}_0(\vec{r})}-v_{ext}(\vec{r})-v_P^0(\vec{r})-v_H^0(\vec{r}) \label{a}$$ and for the exact Pauli potential as $$v_{P}^{0}(\vec{r})= \mu+\frac{1}{2}\frac{\nabla^2\rho^{1/2}_0(\vec{r})}{\rho^{1/2}_0(\vec{r})}-v_{ext}(\vec{r})-v_H^0(\vec{r})-v_{xc}^0(\vec{r}) \label{b}$$ Note that $\lq \mu$’ is given by density $\rho_0(\vec{r})$ from its asymptotic behavior. We use Eq. \[a\] and Eq. \[b\] to write exchange-correlation potential corresponding to density $\rho_0(\vec{r})$ for $(i+1)^{th}$ iteration if at $i^{th}$ iteration the density is $\rho_i(\vec{r})$. Accordingly $$v_{xc}^{i+1}(\vec{r})= \mu+\frac{1}{2}\frac{\nabla^2\rho^{1/2}_0(\vec{r})}{\rho^{1/2}_0(\vec{r})}-v_{ext}(\vec{r})-v_P^i(\vec{r})-v_H^i(\vec{r}), \label{c}$$ where $$v_{P}^i(\vec{r})= \mu^i+\frac{1}{2}\frac{\nabla^2\rho^{1/2}_i(\vec{r})}{\rho^{1/2}_i(\vec{r})}-v_{ext}(\vec{r})-v_H^i(\vec{r})-v_{xc}^i(\vec{r}). \label{d}$$ Substituting Eq. \[d\] in Eq. \[c\] gives (We have dropped the constant term $\mu-\mu^i$.) $$v_{xc}^{i+1}(\vec{r})=v_{xc}^i(\vec{r}) -\frac{1}{2}\frac{\nabla^2\rho^{1/2}_i(\vec{r})}{\rho^{1/2}_i(\vec{r})}+\frac{1}{2}\frac{\nabla^2\rho^{1/2}_0(\vec{r})}{\rho^{1/2}_0(\vec{r})} \label{master1}$$ The constant $(\mu-\mu^i)$ can be fixed either by adjusting the potential to get the correct $\mu$ or fixing its value at a large distance. Eq. \[master1\] is the working equation for obtaining the exchange-correlation potential $v^0_{xc}(\vec{r})$ up to a constant for the ground state density $\rho_0(\vec{r})$.
Using Eq. \[master1\], the algorithm to find the exchange-correlation potential for a density $\rho_0(\vec{r})$ is as follows:
- Start with a trial exchange-correlation potential $v_{xc}(\vec{r})$ and solve the KS equation to obtain the corresponding Kohn-Sham orbitals, the density and $\mu^i=\epsilon^{max}$. The external and Hartree potentials in the KS equations are the exact ones with the latter being calculated from the density $\rho_0({\vec{r}})$.
At the $i^{th}$ iteration this step gives the ground state density $\rho_i(\vec{r})$ corresponding to the exchange-correlation potential $v_{xc}^i(\vec{r})$. The density $\rho_i(\vec{r})$ also serves as an approximation to $\rho_0(\vec{r})$ and is expected to get closer to it with the increasing number of iterations ;
- Find the new potential using Eq. \[master1\]. At this step one can either use $\mu-\epsilon^{max}$ explicitly or fix the potential asymptotically by using the boundary condition for it;
- Use the new potential in the KS equation again until the density obtained from its solutions matches with the given density.
For completeness we point out that the expression for $v_{eff}(\vec{r})$ of Eq. \[lps\_eqn\] in terms of the many-body wavefunction was given by LPS [@LPS_1984] and has been employed [@Buijse_1989; @Gritsenko_1994; @Gritsenko_1996; @Gritsenko_1998] extensively to study properties of the Kohn-Sham potential. Secondly, if $\frac{1}{2}\frac{\nabla^2 \rho^{1/2}_0}{\rho^{1/2}_0}$ and $\frac{1}{2}\frac{\nabla^2 \rho^{1/2}_i}{\rho^{1/2}_i}$ in Eq. \[master1\] are replaced by $v_{eff} (\vec{r})$ derived from the true wavefunction and the Kohn-Sham orbitals in the $i^{th}$ iteration, respectively, an expression for $v_{xc}^{i+1}$ is obtained in terms of quantities that depend explicitly on the wavefunction and the KS orbitals. This approach has been utilized to get $v_{xc}$, or $v_{x}$ in Hartree-Fock (HF) theory, directly from wavefunctions and is discussed in the Appendix.
Use of a general functional Lg to obtain the KS potential {#GksB}
---------------------------------------------------------
To generalize Eq. \[master1\] to find the KS potential we split the kinetic energy functional $T_S[\rho]$ as $$T_S[\rho]= S[\rho]+\tilde{T}_P[{\rho}],$$ where $S[\rho]$ is a functional with the dimensions of energy and $\tilde{T}_P[{\rho}]= T_S[\rho]-S[\rho]$ is the generalized Pauli kinetic energy. An important property of $S[\rho]$ will be derived in section (\[Theory\]). In terms of $S[\rho]$, the equation for the density is $$\frac{\delta S}{\delta \rho}+v_{ext}(\vec{r})+\tilde{v}_P(\vec{r})+v_H[\rho(\vec{r})]+v_{xc}(\vec{r})= \mu, \label{gn-ks-lps}$$ where $\tilde{v}_P =\frac{\delta \tilde{T}_P}{\delta \rho}$. Analogous to the manner in which Eq. \[LPS\_KS\] leads to Eq. \[master1\] relating $v_{xc}^{i+1}(\vec{r})$ to $v_{xc}^{i}(\vec{r})$, Eq. \[gn-ks-lps\] gives $$v_{xc}^{i+1}(\vec{r}) = v_{xc}^i(\vec{r})+\frac{\delta S}{\delta \rho}\Big|_{\rho _i(\vec{r})}-\frac{\delta S}{\delta \rho}\Big|_{\rho _0(\vec{r})}, \label{master2}$$ where $\frac{\delta S}{\delta \rho}\Big|_{\rho_i(\vec{r})}$ implies that the functional derivative is evaluated at density $\rho_i(\vec{r})$. This is the general equation for obtaining the exchange-correlation potential $v_{xc}^0(\vec{r})$ corresponding to given density $\rho_0(\vec{r})$. Following the steps given at the end of section \[GksA\], it can be employed iteratively to obtain the exact exchange-correlation potential for a given density $\rho_0({\vec{r}})$ with the functional $S[\rho]$ of one’s choice.
Notice that if $S[\rho]$ is taken to be the Weizsäcker functional $T_W[\rho]$, Eq. \[master1\] is recovered. However, with Eq. \[master2\] we have the flexibility of choosing $S[\rho]$ to be a more general functional. For example, $S[\rho]$ can be chosen to be $\int f(\vec{r})\rho^n(\vec{r})d\vec{r}$ $(n>1) $, where $f(\vec{r})$ is an appropriately chosen function, or the Hartree energy $\frac{1}{2}\iint\frac{\rho(\vec{r})\rho(\vec{r}')}{|\vec{r}-\vec{r}'|}d\vec{r}d\vec{r}' $. This is somewhat along the lines of deriving the generalized density functional theory [@Siedl_1996] where the functional $\langle T+V_{ee}\rangle$ (or $ F[\rho]$) is split differently from KSDFT into the Hartree-Fock energy functional and a correlation energy functional.
 
![ \[Fig2\] Correlation potential calculated Hookium atom using using different form of $S[\rho]$ in the inset.](vc_hk.eps)
![ \[Fig3\] Exchange potential calculated for N=40 jellium sphere from the Harbola-Sahni density. Different $S[\rho]$ used are shown in the inset.](vx_n40.eps)
Hybrid type functional Lg
-------------------------
An advantage of having many functionals $S[\rho]$ that can be used in Eq. \[master2\] is that we can choose different functionals in different regions of a system. This is useful if one particular functional $S_1[\rho]$ gives better convergence in one region of the system but some other functional $S_2[\rho]$ in other regions. For example, in the asymptotic regions where density is very small, the functional $S_1[\rho]=\int\rho^n(\vec{r})d\vec{r}$ with $n$ slightly larger than $1$ (for example $1.05$) gives accurate answers because $ \rho^{0.05}$ is relatively larger there. Thus, for spherical systems we can choose $S[\rho]$ so that $$\begin{aligned}
\frac{\delta S[\rho]}{\delta \rho}= \operatorname{erf}(\alpha r)\rho^{0.05} + (1-\operatorname{erf}(\alpha r))\rho^{0.5}, \label{hyb_eq}\end{aligned}$$ where $\operatorname{erf}(\alpha r)$ is the error function with a suitably chosen parameter $\alpha$. We make use of such mixing of different $S[\rho]$ in the results section below.
Results {#result}
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We now demonstrate the ideas presented above through spherically symmetric systems. All the numerical calculation of these systems are carried out using the Herman-Skillman program [@Herman_PHP] by modifying it suitably. In Fig. \[Fig1\] we show the exchange potential for the Hartree-Fock density of Na and Ar atoms [@Bunge_1993] using the functionals: $$S[\rho] =
\begin{cases}
-\frac{1}{2} \int \rho(\vec{r})^{1/2}\nabla^2\rho(\vec{r})^{1/2}d\vec{r} & (i) \\
\\
\int\rho^n(\vec{r})d\vec{r}, \quad \quad (1<n\leq 2) & (ii)\\ \\
\frac{1}{2}\iint\frac{\rho(\vec{r})\rho(\vec{r}')}{|\vec{r}-\vec{r}'|}d\vec{r}d\vec{r}'& (iii)
\end{cases}$$ The output exchange potential matches with the corresponding optimized effective potential (OEP) [@Talman_1978] for all the functional forms mentioned above with $n$ varying over a large number of values. We have also done calculation with $\frac{\delta S[\rho]}{\delta \rho}$ given in Eq. \[hyb\_eq\] and found that due to $\rho^{0.05}$ in it, the potential in the asymptotic region is reproduced with ease. We comment on this further in the paragraph below.
Next in Fig. \[Fig2\] we display the correlation potential of Hookium atom calculated using $S[\rho] =-\frac{1}{2} \int \rho(\vec{r})^{1/2}\nabla^2\rho(\vec{r})^{1/2}d\vec{r}$, $\int \rho^{1.05}(\vec{r}) d\vec{r}$ and the hybrid functional Eq. \[hyb\_eq\]. The output potential matches perfectly with the exact correlation potential [@Kais_1993]. We wish to point out that although functional $S[\rho]= \int \rho^{1.05}(\vec{r}) d\vec{r}$ gives exact result here, for inner regions a functional with large power of $\rho$ is equally good. It is in the outer regions where the density becomes very small of because of its $e^{-r^2}$ dependence on $r$ that the functional $\int \rho^{1.05}(\vec{r}) d\vec{r}$ become really useful. This is a good example of how hybrid functionals are effective in such situations.
Finally in Fig. \[Fig3\], we employ the density for a neutral jellium sphere [@Knight_PRL.52.2141; @Matthias_RMP.65.677] to get the KS exchange potential. These densities are obtained using the Harbola-Sahni (HS) exchange potential [@MKH_89] and the general method of Eq. \[master2\] reproduces it with two different forms of $S[\rho]$ viz. $\int \rho^{1.05}(\vec{r}) d\vec{r}$ and $- \frac{1}{2} \int \rho(\vec{r})^{1/2}\nabla^2\rho(\vec{r})^{1/2}d\vec{r}$ . For other forms stated above deviation from the exact potential starts for $r>20$ as the density becomes very low.
We note that, to the best of our knowledge, the Weizsäcker functional has not been used in the past to get the Kohn-Sham potential. The functional $\int f(\vec{r}) \rho^n(\vec{r})d \vec{r}$ has been employed taking $n=2$, with $f(\vec{r}) =1$ [@Wasserman_2017] and $f(\vec{r}) = r^{\beta} (0<\beta<3)$ [@Peirs_2003]. Recently the Hartree functional has been applied [@Hollins_2017] to get the local exchange potential for Hartree-Fock density of solids.
Theory {#Theory}
======
The density $\rho_0(\vec{r})$ to potential $v(\vec{r})$ map can be established through the Levy-Lieb functional [@Levy_1979; @Lieb_1983] that is defined by Lieb [@Lieb_1983] as $$F[\rho_0]= \underset{v}{\text{Supremum}}\Big[ E[v]-\int v(\vec{r})\rho_0(\vec{r})d\vec{r}\Big] \label{lieb}$$ where the search for the Supremum is done over different potentials. For the true ground-state densities the Supremum is a maximum. We now prove that with properly chosen $S[\rho]$ the method of section \[GksB\] (also therefore all such method that it encompasses) makes $E[v]-\int v(\vec{r})\rho_0(\vec{r})d\vec{r}$ larger and larger in each iterative step converging finally to the correct $F[\rho_0]$. To this end we consider the potentials $v^i$ and $v^{i+1}$ for the $i^{th}$ and $(i+1)^{th}$ steps and calculate the difference $$\begin{aligned}
\Delta F &=& \Big( E[v^{i+1}]-\int v^{i+1}(\vec{r})\rho_0(\vec{r})d\vec{r}\Big)\\
& -& \Big( E[v^i]-\int v^i(\vec{r})\rho_0(\vec{r})d\vec{r}\Big)\\
& =&E[v^{i+1}]-E[v^{i}] -\int(v^{i+1}-v^i)\rho_0(\vec{r})d\vec{r}.\end{aligned}$$ For small $(v^{i+1}-v^{i})$ - and this can always be ensured by proper mixing [@Wagner_2013] of $v^i$ and the potential calculated from equation (\[master2\]) - we have by the first order perturbation theory $$E[v^{i+1}]-E[v^i]= \int (v^{i+1}(\vec{r})-v^i(\vec{r}))\rho_i(\vec{r})d\vec{r}$$ and therefore $$\begin{aligned}
&\Delta F& = \int(v^{i+1}(\vec{r})-v^i(\vec{r}))(\rho_i(\vec{r})-\rho_0(\vec{r}))d\vec{r} \nonumber\\
&=& \int \Big(\frac{\delta S}{\delta \rho}\Big|_{\rho _i(\vec{r})}-\frac{\delta S}{\delta \rho}\Big|_{\rho _0(\vec{r})}\Big)(\rho_i(\vec{r})-\rho_0(\vec{r}))d\vec{r} \label{cond}.\end{aligned}$$ If the iterative process is to converge towards the correct potential, from Eq. \[lieb\] we should have $\Delta F \geq 0$ at each iterative step with the equality being satisfied when $\rho_i= \rho_0$. As indicated by Eq. \[cond\], this will be the case if the functional $S[\rho]$ is such that $$\int \Big(\frac{\delta S}{\delta \rho}\Big|_{\rho _i(\vec{r})}-\frac{\delta S}{\delta \rho}\Big|_{\rho_0(\vec{r})}\Big)(\rho_i(\vec{r})-\rho_0(\vec{r}))d\vec{r} \geq 0 \label{cond1}.$$ Eq. \[cond1\] therefore is the condition on $S[\rho]$ for finding the Kohn-Sham potential using iterative methods. We note that for the KS system, $F[\rho_0]=T_S[\rho_0]$. The proof above is akin to the demonstration [@Wagner_2013] that the iterative Kohn-Sham solution always converges towards minimum energy. A strong condition on $S[\rho]$ will be that the integrand $$\Big(\frac{\delta S}{\delta \rho}\Big|_{\rho _i(\vec{r})}-\frac{\delta S}{\delta \rho}\Big|_{\rho _0(\vec{r})}\Big)(\rho_i(\vec{r})-\rho_0(\vec{r})) \geq0 \label{cond2}.$$ This condition is easy to understand physically: it means that in each iterative step the potential increases (decreases) if the density $\rho(\vec{r})$ in the previous step is larger(smaller) than the target density $\rho_0(\vec{r})$.
To sum up, we have shown that with a properly chosen $S[\rho]$, the process of obtaining the Kohn-Sham potential for a given density converges by maximizing $E[v]-\int v\rho_0(\vec{r})d\vec{r}$ iteratively. Therefore the iterative method is equivalent to the direct optimization method [@WY2; @Stott_2004] for finding the Kohn-Sham system. This is complementary to the equivalence of the minimization of energy functional for a given $v_{ext}(\vec{r})$ and solving the corresponding Kohn-Sham equations to get the ground-state density [@Wagner_2013]. It further requires the corresponding functional $S[\rho]$ to satisfy the condition given by Eq. \[cond1\] or Eq. \[cond2\].
It is easy to see that the convergence condition on $S[\rho]$ is satisfied in its strong form Eq. \[cond2\] for the functionals $S[\rho]= \int \rho^n(\vec{r})d\vec{r}$ with $n>1$. It is also satisfied for $S[\rho] = \frac{1}{2}\iint\frac{\rho(\vec{r})\rho(\vec{r}')}{|\vec{r}-\vec{r}'|}d\vec{r}d\vec{r}'$ since in that case Eq. \[cond1\] is equivalent to $ \frac{1}{2}\iint\frac{(\rho(\vec{r})-\rho_0(\vec{r}))(\rho(\vec{r}')-\rho_0(\vec{r}')}{|\vec{r}-\vec{r}'|}d\vec{r}d\vec{r}' \geq 0 $ which is always true. We now show this to be true for the Weizsäcker functional also. In that case the condition is $$\int \Big( \frac{\nabla^2 \rho_0^{1/2}(\vec{r})}{\rho_0^{1/2}(\vec{r})} - \frac{\nabla^2 \rho^{1/2}(\vec{r})}{\rho^{1/2}(\vec{r})}\Big)(\rho(\vec{r})-\rho_0(\vec{r}))d\vec{r} \geq 0.$$ The condition can easily be shown to be equivalent to $$\begin{aligned}
& & \int \Big[ \nabla \rho^{1/2}(\vec{r}) -\Big(\frac{\rho(\vec{r})}{\rho_0(\vec{r})}\Big)^{1/2}\nabla \rho_0^{1/2}(\vec{r})\Big]^2 d\vec{r} \\
&+& \int \Big[ \nabla \rho_0^{1/2}(\vec{r}) -\Big(\frac{\rho_0(\vec{r})}{\rho(\vec{r})}\Big)^{1/2}\nabla \rho^{1/2}(\vec{r})\Big]^2 d\vec{r} \geq 0\end{aligned}$$ which is always satisfied.
Finally we note that recently the method of obtaining a local potential for a given wavefunction [@Viktor_2013; @Viktor_2015] discussed in the Appendix has also been related [@Gidopoulos_2011; @Teal_2017] to finding the Levy-Lieb functional. In essence it is also equivalent to finding Levy-Lieb functional by maximizing $E[v]-\int v(\vec{r})\rho_0(\vec{r})d\vec{r}$.
Equivalence of different iterative methods {#Equiv}
==========================================
In this section we show that different density-based inversion schemes suggested in the literature are equivalent to using an appropriate functional $S[\rho]$. We consider them one by one in the following:\
van-Leeuwen Baerends (vLB) method Lg and its variants Lg
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In the vLB method, the potential $v_{Hxc}(\vec{r})= v_H(\vec{r})+v_{xc}(\vec{r})$ due to the electron-electron interaction is updated in each cycle as $$\begin{aligned}
v_{Hxc}^{i+1}(\vec{r})&=& \frac{\rho_{i}(\vec{r})}{\rho_0(\vec{r})}v_{Hxc}^i(\vec{r}) \notag \\
&=&\frac{(\rho_{i}(\vec{r})-\rho_0(\vec{r}))}{\rho_0(\vec{r})}v_{Hxc}^i(\vec{r})+v_{Hxc}^i(\vec{r}). \label{vlbeq}
$$ Thus the functional $$S[\rho] = \frac{1}{2} \int \frac{v^i_{Hxc}(\vec{r})}{\rho_0(\vec{r})}\rho^2(\vec{r})d\vec{r},$$ where $v^i_{Hxc}(\vec{r})$ is Hartree-exchange-correlation potential for the $i^{th}$ iteration, leads to Eq. \[vlbeq\] when substituted in Eq. \[master2\]. It is pointed out that $v^i_{Hxc}(\vec{r})$ is the potential in the $i^{th}$ iteration and therefore remains unchanged when $\rho$ is varied. Furthermore $$\Delta F = \int \frac{v^i_{Hxc}(\rho(\vec{r})-\rho_0(\vec{r}))^2}{\rho_0(\vec{r})}d\vec{r} \ge 0$$ so that the procedure satisfies the condition for it to converge. Other variants of the method are with different powers of density in $S[\rho]$ or that given in [@Wasserman_2017]. Among these we note that the method of Wang and Parr [@Wang_1993] will converge only if bound states have negative eigenvalues.
An alternative to the vLB method for $v_{ext}(\vec{r})<0$ is obtained with $$S[\rho] = - \frac{1}{2} \int \frac{v_{ext}(\vec{r})}{\rho_0(\vec{r})}\rho^2(\vec{r})d\vec{r},$$ so that $$v_{Hxc}^{i+1}(\vec{r})= -v_{ext}(\vec{r})\frac{(\rho_{i}(\vec{r})-\rho_0(\vec{r}))}{\rho_0(\vec{r})}+v_{Hxc}^i(\vec{r}) \label{vlbeq2}.
$$ The negative sign here is to ensure that for $v_{ext}(\vec{r})<0$ the convergence condition Eq. \[cond1\] satisfied. For $v_{ext}(\vec{r})>0$, the sign above will be positive.
Görling Lg, Gaudoin and Burke method Lg
---------------------------------------
In this method, the change in potential is calculated using $$v^{i+1}(\vec{r})-v^i(\vec{r})= \int \chi^{-1}_i(\vec{r}, \vec{r}')(\rho_0(\vec{r}')-\rho(\vec{r}'))d\vec{r}' \label{buke},$$ where $\chi^{-1}_i(\vec{r}, \vec{r}')[\rho_i]$ is the non-interacting inverse response function for the system at $i^{th}$ iteration with exchange-correlation potential $v_{xc}^i(\vec{r})$ and density $\rho_i(\vec{r})$. Thus if we take $S[\rho]= -\frac{1}{2}\iint \chi^{-1}_i(\vec{r},\vec{r}')\rho(\vec{r})\rho(\vec{r}')d\vec{r}'d\vec{r}$ we get the updated potential as given by Eq. \[buke\] . Observe that while taking the functional derivative of $S[\rho]$, the inverse response function $\chi^{-1}_i(\vec{r},\vec{r}')$ does not contribute to it because it is independent of the variable $\rho(\vec{r})$. Therefore $$\begin{aligned}
\Delta F= -\iint \chi^{-1}_i(\vec{r},\vec{r}')(\rho_0(\vec{r}')-\rho(\vec{r}'))(\rho(\vec{r})-\rho_0(\vec{r}))d\vec{r}'d\vec{r}.\end{aligned}$$ Following [@Vlb_2003], it is easy to show that $$\iint \chi^{-1}_i(\vec{r},\vec{r}') f(\vec{r})f(\vec{r}')d\vec{r}'d\vec{r} < 0$$ for a function $f(\vec{r})$. The iterative scheme therefore follows the convergence criterion to maximize $E[v]-\int v(\vec{r})\rho_0(\vec{r}) d\vec{r}$ with respect to $v(\vec{r})$.
Peirs, Van Neck and Waroquier (PNW) method Lg
---------------------------------------------
PNW use the following update algorithm to find the exchange-correlation potential for spherical systems: $$v_{xc}^{i+1}(r)=v_{xc}^i(r)+ \lambda r^{\beta} (\rho_i -\rho_0) + f(r)(\mu^i -\mu^0), \label{pnweq1}$$ where $0.5 <\lambda <3.5$ and $0<\beta<3$. The function $f(r)$ is a switching function used to tune the asymptotic behavior of potential. Leaving the last term in Eq. \[pnweq1\], the functional leading to the PNW algorithm is $$S[\rho]=\lambda\int r^\beta \rho^n(\vec{r})d\vec{r} \label{pnweq2}$$ with $n=2$ and an optimized value of $\lambda$, $\beta$. It is easy to see that functional satisfies the strong condition of Eq. \[cond2\] for the convergence of the algorithm.
The present work implies that the PVN method can be generalized by using any $n>1$ in Eq. \[pnweq2\].
Hollins, Clark, Refson and Gidopoulos (HCRG) method Lg
------------------------------------------------------
As pointed out earlier, recently the Hartree potential has been used by HCRG [@Hollins_2017] to calculate the exchange-correlation potential corresponding to the Hartree-Fock density. In this method the exchange-correlation potential is updated according to the equation $$v^{i+1}_{xc}(\vec{r}) = v^{i}_{xc}(\vec{r}) + \epsilon \int \frac{\rho_i(\vec{r}') -\rho_0(\vec{r}')}{|\vec{r}-\vec{r}'|}d\vec{r}' \label{hcrg_eqn};$$ where $\epsilon$ is small positive number. As is evident, the functional $S[\rho]= \frac{\epsilon}{2} \iint \frac{\rho(\vec{r})\rho(\vec{r}')}{|\vec{r}-\vec{r}'|}d\vec{r}'d\vec{r}$ gives the working Eq. \[hcrg\_eqn\]. We have already discussed that this $S[\rho]$ satisfies the condition for convergence given by Eq. \[cond1\].
Zhao-Maorrison-Parr (ZMP) Lg mehod
----------------------------------
In the ZMP method, the KS potential is obtained as the Hartree-potential of difference in the given density $\rho_0(\vec{r})$ and the solution density $\rho(\vec{r})$ multiplied by a large constant $\lambda$. The equation to be solved in the ZMP method is [@Zhao_1994] $$\Big[ -\frac{1}{2}\nabla^2 +v_{ext}(\vec{r})+(1-\frac{1}{N})v_H(\vec{r}) + v_{ZMP}(\vec{r}) \Big]\phi_i= \epsilon_i \phi_i \label{zmp_eqn}$$ with $$v_{ZMP}(\vec{r}) =\lambda \int \frac{\rho(\vec{r}')-\rho_0(\vec{r}')}{|\vec{r}-\vec{r}'|}d\vec{r}'.$$ Here $v_H(\vec{r})$ is Hartree potential of given density $\rho_0(\vec{r})$ and $\rho(\vec{r})= \sum_{i} |\phi_i(\vec{r})|^2$. We point out that the self-interaction component of the exchange-correlation potential has been included with Hartree potential and that make achieving self-consistency easier. In using this method, one usually starts with a small value of $\lambda$ and then increases it to obtain better and better density $\rho(\vec{r})$. Finally the exchange-correlation potential is obtained as $$v_{xc}(\vec{r})= \lim\limits_{\lambda \to \infty} v_{ZMP}(\vec{r})-\frac{v_H(\vec{r})}{N}.$$ For a given $\lambda$ Eq. \[zmp\_eqn\] is solved self-consistently. Thus one starts with some initial guess of $v_{ZMP}(\vec{r})$, say $v^1_{ZMP}(\vec{r})$, and at the $(i+1)^{th}$ cycle of the self-consistent procedure the potential is updated as $$\begin{aligned}
v^{i+1}_{ZMP}(\vec{r}) =(1-\alpha)v^i_{ZMP}(\vec{r})+ \alpha \lambda \int\frac{\rho_i(\vec{r}')-\rho_0(\vec{r}')}{|\vec{r}-\vec{r}'|}d\vec{r}' \label{zmp_eqn2},\end{aligned}$$ where $\alpha (< 1)$ is the mixing parameter and $v^i_{ZMP}(\vec{r})$ is the potential for the $i^{th}$ iteration with solution density $\rho_i(\vec{r})$. If one takes $v^1_{ZMP}(\vec{r}) =0$ then Eq. \[zmp\_eqn2\] leads to $$\begin{aligned}
v^{i+1}_{ZMP}(\vec{r}) =\alpha \lambda\Big[\sum_{m=1}^{i} (1-\alpha)^{i-m}\int\frac{\rho_m(\vec{r}')-\rho_0(\vec{r}')}{|\vec{r}-\vec{r}'|}d\vec{r}' \label{zmp_eqn3}\Big].\end{aligned}$$ In the HCRG method, the potential $v_{ZMP}(\vec{r})$ in Eq. \[zmp\_eqn\] is replaced by $v_{HCRG}(\vec{r})$ and it is updated as (by taking $\epsilon = \alpha$) $$v^{i+1}_{HCRG}(\vec{r}) = v^i_{HCRG}(\vec{r})+ \alpha\int\frac{\rho_i(\vec{r}')-\rho_0(\vec{r}')}{|\vec{r}-\vec{r}'|}d\vec{r}'\label{hcrg_eqn2}.$$ After achieving the convergence, the exchange-correlation potential $v_{xc}(\vec{r})$ is calculated as $$v_{xc}(\vec{r})= v_{HCRG}(\vec{r})- \frac{v_H(\vec{r})}{N}.$$ Again taking $v^1_{HCRG}(\vec{r}) =0$, Eq. \[hcrg\_eqn2\] becomes $$\begin{aligned}
v^{i+1}_{HCRG}(\vec{r}) =\alpha\Big[\sum_{m=1}^{i} \int\frac{\rho_m(\vec{r}')-\rho_0(\vec{r}')}{|\vec{r}-\vec{r}'|}d\vec{r}' \label{hcrg_eqn3}\Big].\end{aligned}$$ From Eq. \[zmp\_eqn3\] and Eq. \[hcrg\_eqn3\] it is evident that the methods of ZMP and HCRG are equivalent as both use the electrostatic potential of charge density $\rho_i(\vec{r})-\rho_0(\vec{r})$ for improvement of the potentials at each iterative step. However, the way these corrections are added during the process is different. In the ZMP method (Eq. \[zmp\_eqn3\]), the contribution of the potentials from previous iterations keeps on diminishing as the number of iterations increases and self-consistency is approached with the density difference becoming smaller and smaller. Thus to keep the potential finite, a large value of $\lambda$ is needed. On the other hand, in the HCRG method (Eq. \[hcrg\_eqn3\]), potential at each iteration contributes equally.
We note that to satisfy the convergence condition of Eq. \[cond1\] for the ZMP method, the value of $\alpha$ should be very small. This is because $$\begin{aligned}
v^{i+1}_{ZMP}(\vec{r})-v^{i}_{ZMP}(\vec{r}) = \lambda \Big[ \alpha \int\frac{\rho_i(\vec{r}')-\rho_0(\vec{r}')}{|\vec{r}-\vec{r}'|}d\vec{r}' \notag \\
-\alpha^2 \Big\{\sum_{m=1}^{i-1} (1-\alpha)^{i-1-m}\int\frac{\rho_m(\vec{r}')-\rho_0(\vec{r}')}{|\vec{r}-\vec{r}'|}d\vec{r}' \label{zmp_eqn4}\Big\} \Big].\end{aligned}$$ Therefore the contribution to $\Delta F$ from the term proportional to $\alpha$ (which is always positive) will be larger than that proportional to $\alpha^2$ (which could be positive or negative) if $\alpha <<1$ thereby ensuring $\Delta F \geq 0$ for each iterative step. This is seen to be the case while performing ZMP calculations.
Summary
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Exact results, whenever they can be obtained, are important to understand a theory properly. This is particularly important in density functional theory since it is the most widely used theory of electronic structure but can be applied only approximately. For example, exact conditions on the exchange-correlation energy functionals have played an important role in their development. An important part of research in density functional theory has also been to construct the exact Kohn-Sham system for known densities. Not only the problem by itself is challenging, it also sets a benchmark against which approximate exchange-correlation functionals can be tested.
Over the past thirty years or so, a variety of methods have been proposed to construct the Kohn-Sham system for a given density. These methods emerge from different ways of formulating the inverse problem. In this paper we have shown a majority of these methods (those formulated in terms of the density) to be results of the Euler equation for the density and have also provided an understanding of all these methods based on Lieb’s definition of the Hohenberg-Kohn universal functional $F[\rho]$. Our work thus connects these different methods through a fundamental principle of DFT and gives a unified theory for the construction of Kohn-Sham system for a given density. As a result it also provides flexibility in ways through which the Kohn-Sham system can be constructed for a given density as has been demonstrated in the paper.
Appendix: Derivation of the Kohn-Sham potential from Hartree-Fock wave function
===============================================================================
The LPS equation for HF density can be written as $$[-\frac{1}{2}\nabla^2 + v_{eff}^{HF}(\vec{r})]\rho^{\frac{1}{2}}_{HF}(\vec{r})= \mu^{HF}\rho^{\frac{1}{2}}_{HF} \label{lps_hf}$$ where [@Buijse_1989; @Gritsenko_1994]
$$\begin{split}
v^{HF}_{eff}&=v_{ext}+v^{HF}_S+v^{HF}_H+ \frac{1}{\rho_{HF}} \sum_j (\epsilon_{max}^{HF}-\epsilon^{HF}_j)
|\phi_j^{HF}|^2 \\
&+\frac{1}{2} \sum_j \frac{|\nabla \phi^{HF}_j|^2}{\rho_{HF}} -\frac{1}{8} \frac{|\nabla \rho_{HF}|^2}{\rho^2_{HF}}. \label{apn_veff_HF}
\end{split}$$
In the expression above $\phi^{HF}_j$ and $\epsilon^{HF}_j$ are the HF orbitals and their eigenenergies, respectively. The quantity $v^{HF}_S$ is the Slater potential [@Slater_1951] calculated from HF orbitals.\
Similarly, for the Kohn-Sham equation, the effective potential [@LEVY_1988] for the corresponding LPS equation is $$\begin{split}
v_{eff}^{KS}&=v_{ext}+v_H+ v_{x}+ \frac{1}{\rho_{KS}} \sum_j (\epsilon_{max}^{KS}-\epsilon^{KS}_j)
|\phi_j^{KS}|^2 \\
&+\frac{1}{2} \sum_j \frac{|\nabla \phi^{KS}_j|^2}{\rho_{KS}} -\frac{1}{8} \frac{|\nabla \rho_{KS}|^2}{\rho^2_{KS}}. \label{apn_veff_ks}
\end{split}$$ Now using $ \frac{1}{2}\frac{\nabla^2 \rho}{\rho}= \mu-v_{eff}$ to write Eq. \[master1\] of the main text in terms of effective potentials, we get $$v_{xc}^{i+1}(\vec{r})=v_{xc}^i(\vec{r}) + v_{eff}^{WF}(\vec{r})- v^{i,KS}_{eff}(\vec{r}),\label{lps_itr1}$$ where $v_{eff}^{WF}(\vec{r})$ is the effective potential for the interacting system. For the HF wavefunction its expression is given by Eq. \[apn\_veff\_HF\]; the general expression for it is given in [@LPS_1984; @Buijse_1989]. In addition, $v^{i,KS}_{eff}(\vec{r})$ is the effective potential corresponding to non-interacting KS system at the $i^{th}$ iteration. In particular for HF wavefunction, the equation above becomes $$\begin{split}
v^{i+1}_{x} &= \epsilon_{max}^{HF}-\epsilon_{max}^{i,KS}+ v^{HF}_S\\
&-\frac{\sum_j \epsilon^{HF}_j
|\phi_j^{HF}|^2 }{\rho_{HF}} +\frac{\sum_j \epsilon^{i,KS}_j
|\phi_j^{i,KS}|^2 }{\rho_{KS}^i} \\
&+ \frac{1}{2} \sum_j \frac{|\nabla \phi^{HF}_j|^2}{\rho_{HF}}-\frac{1}{2} \sum_j \frac{|\nabla \phi^{i,KS}_j|^2}{\rho_{KS}^i}. \label{viktor_exch}
\end{split}$$ In writing Eq. \[viktor\_exch\] all explicitly density dependent terms are canceled.
Eq. \[viktor\_exch\] was first derived and implemented by Ryabinkin, Kananenka and Staroverov (RKS) [@Viktor_2013] to generate KS exchange potential corresponding to HF wavefunction generated from finite Gaussian basis set. This method gives approximate but highly accurate local exchange potential (essentially the same as the optimized effective potential) that is free from unwanted oscillatory features that arise [@Schipper1997; @Gidopoulos_2012; @Viktor_2013] near the nucleus if Gaussian basis set is used to generate density. Staroverov and coworkers further extended it to many-body wavefunction [@Viktor_2015_JCP; @Viktor_2015]. Eq. \[viktor\_exch\] was also used by Nagy [@Nagy_1997] to derive the Krieger, Li, and Iafrate approximation [@KLI_1992] to OEP.
An alternate method to obtain the Kohn-Sham like non-interacting system for a given wave function has also been proposed in [@Gidopoulos_2011]. It has recently been applied [@Hollins_2017] to obtain the local exchange potential for the Hartree-Fock wavefunction and the corresponding band-structure of solids. However the potential is generated using the Hartree-Fock density with $S[\rho]=\frac{\epsilon}{2}\iint\frac{\rho(\vec{r})\rho(\vec{r}')}{|\vec{r}-\vec{r}'|}d\vec{r}d\vec{r}' $ and a basis consisting of a large number of plane waves.
|
---
abstract: |
We present mid-infrared imaging and far-infrared (FIR) spectroscopy of 5 IBm galaxies observed by [*ISO*]{} as part of our larger study of the interstellar medium of galaxies. Most of the irregulars in our sample are very actively forming stars, and one is a starburst system. Thus, most are not typical Im galaxies. The mid-infrared imaging was in a band centered at 6.75 that is dominated by polycyclic aromatic hydrocarbons (PAHs) and in a band centered at 15 that is dominated by small dust grains. The spectroscopy of 3 of the galaxies includes \[\]$\lambda$158 and \[\]$\lambda$63 , important coolants of photodissociation regions (PDRs), and \[\]$\lambda$88 and \[\]$\lambda$122 , which come from ionized gas. \[\]$\lambda$145 and \[\]$\lambda$52 were measured in one galaxy as well. These data are combined with PDR and region models to deduce properties of the interstellar medium of these galaxies.
We find a decrease in PAH emission in our irregulars relative to small grain, FIR, and emissions for increasing FIR color temperature which we interpret as an increase in the radiation field due to star formation resulting in a decrease in PAH emission. The ratio is constant for our irregulars, and we suggest that the 15 emission in these irregulars is being generated by the transient heating of small dust grains by single photon events, possibly Ly$\alpha$ photons trapped in regions. The low ratio, as well as the high / ratio, in our irregulars compared to spirals may be due to the lower overall dust content, resulting in fewer dust grains being, on average, near heating sources.
We find that, as in spirals, a large fraction of the \[\] emission comes from PDRs. This is partly a consequence of the high average stellar effective temperatures in these irregulars. However, our irregulars have high \[\] emission relative to FIR, PAH, and small grain emission compared to spirals. If the PAHs that produce the 6.75 emission and the PAHs that heat the PDR are the same, then the much higher / ratio in irregulars would require that the PAHs in irregulars produce several times more heat than the PAHs in spirals. Alternatively, the carrier of the 6.75 feature tracks, but contributes only a part of, the PDR heating, that is due mostly to small grains or other PAHs. In that case, our irregulars would have a higher proportion of the PAHs that heat the PDRs compared to the PAHs that produce the 6.75 feature.
The high f$_{[OIII]}$/f$_{[CII]}$ ratio may indicate a smaller solid angle of optically thick PDRs outside the regions compared to spirals. The very high L$_{[CII]}$/L$_{CO}$ ratios among our sample of irregulars could be accounted for by a very thick \[\] shell around a tiny CO core in irregulars, and PDR models for one galaxy are consistent with this.
The average densities of the PDRs and far-ultraviolet stellar radiation fields hitting the PDRs are much higher in two of our irregulars than in most normal spirals; the third irregular has properties like those in typical spirals. We deduce the presence of several molecular clouds in each galaxy with masses much larger than typical GMCs.
author:
- 'Deidre A. Hunter[^1]'
- Michael Kaufman
- 'David J. Hollenbach, Robert H. Rubin'
- Sangeeta Malhotra
- 'Daniel A. Dale, James R. Brauher, Nancy A. Silbermann, George Helou, Alessandra Contursi, and Steven D. Lord'
title: 'The Interstellar Medium of Star-Forming Irregular Galaxies: The View with [*ISO*]{}[^2] '
---
Introduction
============
The infrared wavelength region offers a variety of diagnostics of the physical state of the interstellar medium (ISM) in galaxies, and the [*Infrared Space Observatory*]{} ([*ISO*]{}, Kessler 1996) has enabled astronomers to observe many of these infrared features to unprecedented sensitivity levels. Under the US Guaranteed Time project to examine the ISM of normal galaxies, we have used [*ISO*]{} to observe a suite of atomic and ionic fine structure lines and the infrared continuum of a large sample of galaxies (Helou 1996). The 69 galaxies in our sample span the full range of Hubble types from early-type E/S0 galaxies to late-type irregulars. For a list of galaxies in the larger sample see Dale (2000). In this paper we report on the group of irregular galaxies in our sample and how they compare to our larger sample of spiral galaxies. The irregulars are interesting because they are more similar to each other and more different as a group from spirals in, for example, dust column depth and because they are the extreme end of the Hubble sequence of disk galaxies in many galactic properties. However, the reader should keep in mind that the irregulars in our sample are not all typical of the Im class since most are very actively forming stars and one is a starburst. The irregular galaxies in our sample are described in detail in the next section.
The data obtained with [*ISO*]{} for our project include broad-band images of galaxies at 6.75 and 15 $\mu$m taken with CAM (Cesarsky 1996a) and fluxes of atomic and ionic fine-structure transitions measured using the Long Wavelength Spectrometer (LWS, Clegg 1996). The mid-infrared CAM images are in the regime where light from red stars is decreasing with wavelength and emission from ISM processes has become dominant. The CAM images at 6.75 (LW2, $\Delta\lambda$ of 3.5 ) include major Polycyclic Aromatic Hydrocarbon (PAH) features at 6.2 , 7.7 , and 8.6 (Helou 2000). The CAM images at 15 (LW3, $\Delta\lambda$ of 6 ), on the other hand, are most sensitive to thermal emission due to very small grains attaining high temperatures (Dale 2000, Malhotra 2001).
The LWS was used in grating line-mode to obtain emission spectra of primary diagnostics of the ISM. These lines include \[\]$\lambda$158 , \[\]$\lambda$122 , \[\]$\lambda$88 , and \[\]$\lambda$63 . The \[\] and \[\] are cooling lines for the atomic gas and probe the conditions in photodissociation regions (PDRs), the warm neutral gas cloud surfaces, which constitute a large fraction of the neutral medium in a galaxy. The \[\] and \[\] emission lines probe conditions in the regions. A subset of galaxies, including one irregular, was also observed at \[\]$\lambda$145 and \[\]$\lambda$52 . The LWS beam of $\sim$75 diameter included most of the optical galaxy for our distant sample of 60 galaxies and was used to explore different environments within our nearby sample of 9 galaxies. (The “distant” sample is defined as galaxies that are not well resolved with respect to the size of the LWS aperture).
We supplement the [*ISO*]{} data with infrared continuum fluxes measured with the [*Infrared Astronomical Satellite*]{} ([*IRAS*]{}) at 12, 25, 60, and 100 . We also have optical images that show us what the optical counterparts are to the infrared features seen in the CAM images, provide B-band luminosities for comparison to the infrared, and show us the amount and locations of current star formation in these galaxies. Below we discuss what is known of the galaxies in our sample, the observations that we use here, the results from the mid-infrared CAM imaging—both in terms of morphology and flux ratios, and the infrared line ratios for the ionized and neutral gas and what they imply about the physical conditions in the PDRs.
The Galaxies
============
The irregular galaxies in our sample include NGC 1156, NGC 1569, NGC 2366, NGC 6822, and IC 4662. All of the galaxies that we discuss here are classed as IBm, barred Magellanic-type irregulars, by de Vaucouleurs (1991; $\equiv$RC3). Most of the irregulars in our sample are [*not typical*]{} of the Im class since the truly typical Im galaxy is too faint in the far-infrared (FIR) for us to have considered observing with [*ISO*]{}, particularly with the LWS (Hunter 1989, Melisse & Israel 1994). NGC 1156, NGC 1569, and IC 4662 are at the high end of the distribution of surface brightness and star formation activity. NGC 2366 is not, but it contains a supergiant region. NGC 6822 [*is*]{} a typical irregular and we did detect it (barely) with CAM but it is not bright enough in the FIR for LWS observations within our time allocation. The galaxies and pertinent global characteristics are given in Table \[tabgal\]. Optical images are shown in Figure \[figvha\] with V-band image contours superposed.
At an M$_B$ of $-$18.0, NGC 1156 is at the high end of the range of luminosities seen in normal, non-interacting irregular galaxies (Hunter 1997), and is about 25% brighter than the Large Magellanic Cloud in the B band. Karachentsev, Musella, & Grimaldi (1996) describe NGC 1156 as “the less disturbed galaxy in the Local Universe,” and there are no catalogued galaxy neighbors within 0.7 Mpc and $\pm$150 . NGC 1156 has a high rate of star formation and regions are crowded over the disk.
NGC 1569 has the highest star formation rate of the sample of 69 irregulars examined by Hunter (1997) as determined from its luminosity. It has just undergone a true burst of star formation (Gallagher, Hunter, & Tutukov 1984; Israel & van Driel 1990; Vallenari & Bomans 1996; Greggio 1998), meaning that its recent global star formation rate (SFR) is statistically significantly elevated compared to its average past rate. In addition it contains two luminous super star clusters that are likely young versions of globular clusters (Arp & Sandage 1985; O’Connell, Gallagher, & Hunter 1994), and ionized gas filaments that extend well beyond the main optical body of the galaxy (de Vaucouleurs, de Vaucouleurs, & Pence 1974; Hunter & Gallagher 1990, 1992, 1997; Hunter, Hawley, & Gallagher 1993). The starburst is said to have ended $\sim$4–10 Myrs ago but the presence of regions and 5 giant molecular clouds (GMCs) (Taylor 1999) suggests that we can consider star formation as on-going. Virial masses of the molecular clouds suggest that the CO–H$_2$ conversion factor should be 6.6$\pm$1.5 times the Galactic value (Taylor 1999). Stil & Israel (1998) have detected a $7\times10^6$ M cloud at 5 kpc from NGC 1569 and the hint of a bridge in connecting them. The suggestion is that interaction with this cloud is responsible for the recent starburst.
NGC 2366, located in the M81 Group of galaxies, is a lower luminosity irregular, having almost the same absolute B magnitude as the Small Magellanic Cloud. It contains the supergiant complex NGC 2363 at the southwest end of the bar and labelled in Figure \[figvha\]. NGC 2363 is nearly twice as bright in as the 30 Doradus nebula in the Large Magellanic Cloud and contains a large fraction of the total current star-formation activity of the galaxy (Aparicio 1995, Drissen 2000). However, there is also another large complex near NGC 2363 and numerous smaller regions scattered along the disk.
NGC 6822 is a low luminosity irregular in the Local Group. It contains a modest population of small regions (Killen & Dufour 1982; Hodge, Kennicutt, & Lee 1988; Massey 1995), and a modest star formation activity (Hoessel & Anderson 1986, Hodge 1991, Gallart 1996c). The structure and stellar content of the galaxy have been discussed by Hodge (1977), and the SFR over the past 1 Gyr has been found to be approximately continuous (Marconi 1995, Gallart 1996b, Cohen & Blakeslee 1998). NGC 6822 is one of the few irregulars in which a population of asymptotic giant branch stars has been measured (Gallart 1994).
IC 4662 is the lowest luminosity irregular galaxy in our sample. Nevertheless, it has a SFR per unit area that is only a factor of two lower than that of NGC 1569. The physical characteristics of this galaxy have been studied by Pastoriza & Dottori (1981) and Heydari-Malayeri, Melnick, & Martin (1990). Our optical images show a collection of stars with associated emission that appears to be detached from the main body of the galaxy. It is located 1.5, or 890 pc at IC 4662’s distance, to the southeast from the center of the galaxy. Whether this is a very tiny, but separate, companion galaxy, or a very unusually placed OB association we cannot tell. For this paper, we will refer to it as IC 4662-A, and it is labelled as such in Figure \[figvha\]. In the optical IC 4662 does not look disturbed, as could be the case if it were interacting with a companion, but its SFR is unusually high. In addition there are 3 tiny regions located well away from the main body of the galaxy and roughly halfway between IC 4662 and IC 4662-A.
In addition to these 5 irregular galaxies that form part of our distant sample of galaxies, our team has observed IC 10 with [*ISO*]{} as part of our sample of nearby, resolved (with respect to the LWS beam) galaxies. Three positions in IC 10 were observed with the LWS. The [*ISO*]{} LWS observations of IC 10 are the subject of another paper (Lord 2000), and the CAM observations have been discussed by Dale (1999). However, we will call upon those results for comparison here. IC 10 has a B-band luminosity that is comparable to that of NGC 2366 but a current SFR per unit area that is 17 times higher. Massey & Johnson (1998) found that the density of evolved massive stars in the Wolf-Rayet phase is several times higher than that of the Large Magellanic Cloud which is a considerably more luminous galaxy. For this reason, Massey & Armandroff (1995) have suggested that IC 10 too is undergoing a burst of star formation. IC 10 is also known to be unusual in having gas that extends about 7 times the optical dimensions of the galaxy (Huchtmeier 1979). It is now known that that extended gas contains a large cloud (Wilcots & Miller 1998), like that near NGC 1569, and which has been suggested is falling into the galaxy causing the current heightened star formation activity (Saito 1992). The \[\]158 emission in IC 10 has also been mapped from NASA’s Kuiper Airborne Observatory by Madden (1997), and \[CI\] at 492 GHz and CO rotational transitions have been observed by Bolatto (2000).
All of the irregular galaxies in our sample are metal-poor compared to spirals, a general characteristic of Im galaxies. The oxygen abundances are given in Table \[tabgal\]. The oxygen abundances \[12$+$log(O/H)\] range from 8.06 to 8.39 for these galaxies. This corresponds to $Z$ of 0.003 to 0.006, which are 20–40% of solar.
Observations
============
[*ISO*]{} Data
--------------
The observations of irregular galaxies obtained with [*ISO*]{} by our team are listed in Table \[tabisoobs\]. All of the irregular galaxies have been imaged in the mid-infrared through the LW2 and LW3 filters of CAM; NGC 1569 has been observed with additional filters but those observations will not be discussed here (see Lu 2000). The full galaxy sample in our program observed with CAM, the nature of the observations, and the data reduction are discussed by Dale (2000). The CAM frames of each galaxy are combined to produce a final 9 resolution. The field of view of NGC 1156, NGC 1569, and IC 4662 is 4.45, that of IC 10 is 7.25, that of NGC 2366 is 8.2, and that of NGC 6822 is 12.65. The LW2 and LW3 fluxes, and , for the irregular galaxies discussed here are given in Table \[tabcam\]. Note that foreground stars and, in the case of NGC 2366, a background galaxy have been removed from the images prior to measuring the fluxes.
To compare the CAM images with and V-band images we needed to determine the coordinate system of all the images. The CAM images were rotated according to the information in the headers so that north would be up and east to the left. For ground-based images we used stars identified on the optical image to determine an astrometric solution for the optical image. When we compared the LW2 and LW3 CAM images with the V-band images, where there are stars in common, we found that the two were offset with respect to each other. Since the pointing accuracy of the CAM images is not expected to be better than 12, we have assumed that the coordinate systems of the CAM images needed minor corrections and offset the LW2 and LW3 images to match the V-band images using the stars in common. The offsets were $\leq$10.
The irregular galaxies have also been observed in FIR atomic and ionic fine-structure lines. The lines that have been observed are listed in Table \[tabisoobs\] and the spectra are shown in Figure \[figlws\]. The full sample of galaxies observed with the LWS, details concerning the observations and data analysis, and the statistical results for the full sample are given by Malhotra (2001). The emission-line fluxes for the irregular galaxies are given in Table \[tablws\]. Upper limits are 3$\sigma$. The \[\] fluxes of NGC 1156 and NGC 1569 have been corrected for overlying foreground \[\] emission in the Milky Way using observations of a nearby off-galaxy position. The corrections are of order 14%. The IC 10 emission-fluxes have been corrected for the fact that IC 10 is extended with respect to the LWS beam, and that affects the derivation of the surface brightness. The correction factors we used were 0.59 for \[\]$\lambda$158, 0.68 for \[\]$\lambda$88, and 0.84 for \[\]$\lambda$63.
When we compare the irregulars to the rest of the galaxies in our distant galaxy sample, we will delete NGC 4418. It is a Seyfert and does not represent a normal galaxy. In plots we will label the distant galaxy sample minus the irregulars as spirals.
Ground-based and Other Data
---------------------------
Integrated infrared fluxes at 12, 25, 60, and 100 of these galaxies were measured with [*IRAS*]{}. Those fluxes and the FIR luminosity integrated from 40–120 are given in Table \[tabiras\]. The FIR flux is determined from $=1.26\times10^{-14}(2.58f_{60} + f_{100})$ W m$^{-2}$, where $f_{60}$ and $f_{100}$ are the fluxes at 60 and 100 , respectively, in Jy (Helou 1988). The total infrared flux, TIR, for 3–1100 is estimated from [*ISO*]{} and [*IRAS*]{} data using the empirical formulation of Dale (2001).
We also obtained optical ground-based images of the irregulars at the Perkins 1.8 m telescope at Lowell Observatory from 1992 to 1995 using various TI 800$\times$800 CCDs. The images were obtained through a filter centered at 6566 Å with a FWHM of 32 Å. An off-band image was taken through a filter centered at 6440 Å with a FWHM of 95 Å. The image of NGC 2366 was constructed from a mosaic of observations at multiple positions. We observed IC 4662 at the Cerro Tololo Interamerican Observatory (CTIO) on 1999 September 17. We used a SITe 2048$\times$2048 CCD on the 1.5 m telescope with a 75 Å FWHM filter centered at 6600 Å. The off-band filter was a broad-band R filter. NGC 6822 was observed in 1988 with a 0.2 m Takahashi telescope that was bolted to the side of the Hall 1.1 m telescope at Lowell Observatory. Those observations are reported by Gallagher (1991).
The off-band image was shifted, scaled, and subtracted from the image to produce an image containing only emission from regions. These images are used to trace the current star formation activity. Irregular galaxies contain significantly less dust than spirals. Therefore, we do not expect that the optical images have missed significant amounts of star formation (Hunter 1989).
We used both spectral photometric standard stars and regions in nearby galaxies to calibrate the flux. The calibrations took into account the blueshift of the passband with temperatures below 20 C and the different transmissions of the filter at the redshift of the calibrating region and the object. We note that the flux that we measure for IC 4662 is 58% of that measured by Heydari-Malayeri (1990).
In IC 4662 there are three tiny regions to the extreme southeast of the galaxy, two close together and a third more separated from them. The two close together have a combined reddening corrected luminosity of 5.2$\times10^{36}$ ergs s$^{-1}$, and the third has a luminosity of 3.3$\times10^{36}$ ergs s$^{-1}$. Thus, these regions are comparable to the Orion nebula. IC 4662-A has an luminosity of 2.6$\times10^{38}$ ergs s$^{-1}$, 1/22 that of IC 4662 itself and consistent with a large OB association, $\sim$25 O7 stars.
To trace the stars, we use broad-band V images. V-band images of NGC 1156, NGC 1569, and NGC 2366 were obtained by P. Massey with the Kitt Peak National Observatory[^3] 4 m telescope and a SITe 2048$\times$2048 CCD from 1997 to 1998. NGC 6822 was observed by C. F. Claver with the 4 m telescope and a SITe 2048$\times$2048 CCD at CTIO in 1996. We obtained V-band images of IC 4662 during our observing run at CTIO in 1999 September.
The abundances and reddening of the ionized gas were measured for individual regions in NGC 1156 and NGC 2366 using a long-slit spectrograph mounted on the Perkins 1.8 m telescope. Details of the observations and analysis are given by Hunter & Hoffman (1999).
The luminosities of the irregulars have been corrected for reddening using total gas E(B$-$V)$_g$ given in Table \[tabgal\]. These were determined from Balmer decrements in emission-line spectra. The reddening correction uses the reddening curve of Cardelli (1989). Most stars, however, will not be as extincted as those in regions. Therefore, integrated B-band luminosities, taken from RC3, are corrected for reddening using the foreground E(B$-$V)$_f$ plus 0.05 magnitude to account for internal reddening of the stars.
Star formation rates are derived from the luminosities using a Salpeter stellar initial mass function from 0.1 to 100 M(Hunter & Gallagher 1986). The star formation rates are normalized to the area of the optical galaxy using $\pi$R$_{25}^2$, where R$_{25}$ in kpc is the radius at a B-band surface brightness of 25 magnitudes arcsec$^{-2}$.
In plots we will include spiral galaxies from the larger distant galaxy sample for comparison. fluxes for these galaxies have also been measured from images obtained at Lowell Observatory, Palomar Observatory, and Kitt Peak National Observatory. The fluxes of the spirals are corrected for reddening assuming a total E(B$-$V)$_g$$=$E(B$-$V)$_f$$+$0.8 mag, where E(B$-$V)$_f$ is the Milky Way reddening determined by Burstein & Heiles (1984). The 0.8 term is added to account on average for typical reddening of regions in spiral galaxies. Their luminosities are converted to normalized star formation rates in the same way as was done for the irregulars. Integrated B-band luminosities are taken from RC3 and corrected for reddening using the foreground E(B$-$V)$_f$ plus 0.15 magnitude to account for internal reddening of the stars.
Mid-Infrared Imaging
====================
Morphology
----------
In Figure \[figcam\] we show the CAM LW2 images of the irregulars in our sample with contours superposed. One can see that to a large degree the emission traces emission: 1) [**NGC 1156**]{}: In NGC 1156 most of the emission is found in two blobs that coincide with the brightest peaks. There is also some diffuse emission and it is outlined by a lower surface brightness contour. The emission is found entirely from the two brightest peaks. 2) [**NGC 1569**]{}: In NGC 1569 the and emissions are found in two concentrations at the center of the galaxy where the emission is also brightest. In Figure \[figcam\] the positions of the 5 GMCs mapped by Taylor (1999) are marked along with the positions of the two superstar clusters. The main peak of emission in NGC 1569, the northwestern blob, is resolvable into two peaks itself. The brightest peak to the northwest coincides with the bright region there and the 5 molecular clouds are situated around the edges. The fainter peak to the southeast sits immediately to the north of a fainter peak and offset to the south of super star cluster A (the northwestern one of the two). A fainter peak to the northeast of this concentration is associated with an region that is slightly detached from this complex. Another much fainter peak well detached to the west is not obviously associated with any feature. The second brightest concentration of emission, the southeastern blob, is primarily associated with the second brightest concentration. The faint peak of emission to the extreme southeast, and just to the west of the object labeled as a star, is also not obviously associated with any feature in NGC 1569. 3) [**NGC 2366**]{}: In NGC 2366 most of the mid-infrared emission coincides with the supergiant region NGC 2363 in the southwestern part of the galaxy. There is a much fainter region of emission to the northeast of that which does not obviously correspond to anything in particular in NGC 2366 in the optical, but it is comparable in brightness to other noise elsewhere in the image beyond the optical galaxy. 4) [**NGC 6822**]{}: In NGC 6822 the mid-infrared emission is found in the direction of 5 regions which are identified in Figure \[figcam\]. There is a bit of diffuse emission towards the center of the galaxy but it is of too low signal-to-noise to measure with any degree of confidence. The fluxes given in Table \[tabcam\] are the combined fluxes of the regions. 5) [**IC 4662**]{}: In IC 4662 we find the emission primarily in two bright concentrations with some low surface brightness diffuse emission around them. These are located near the center of the galaxy. Comparison with the image suggests that these two concentrations are also the brightest complexes in the galaxy. At 15 we primarily see the brighter of these three regions. We do not detect the small detached regions to the southeast or IC 4662-A, further to the south.
This theme of the coincidence of mid-infrared emission and regions extends to a large degree to IC 10 as well. There one sees a very bright, amorphous blob in the southeast that coincides with the brightest region of . There are also several smaller blobs but without a strong correlation between intensity and intensity, and there is some lower surface brightness emission that is not obviously connected with emission at all. Then to the northwest of the brightest region, the and 6.75 emission together trace a partial shell of diameter 470 pc (for a distance to the galaxy of 1 Mpc).
Thus, in our sample of irregulars the mid-infrared emission that we measure is associated with regions of star-formation. This may be an issue of detectability. Irregulars have less dust and, hence, the emission over-all in the infrared is reduced relative to spirals. Furthermore, it is the higher surface brightness regions that will preferentially be detected and the fainter, lower surface brightness emission will be lost. In a study of the mid-infrared surface brightness of several nearby galaxies, including IC 10, Dale (1999) found that variations in dust column density are the primary drivers of infrared surface brightness differences. Therefore, it is reasonable to expect that regions and associated gas clouds, which have higher columns of gas and dust compared to the general ISM, will be the places that are easiest to detect.
Flux Ratios
-----------
Dale (1999) show that the ratio is sensitive to the intensity of heating of the ISM and, hence, is a function of the mid-infrared surface brightness. In regions of intense heating this ratio is $\leq$1, whereas in regions of less intense heating, such as interarm regions in spirals, this ratio is $>$1. Furthermore, the ratio decreases as the surface brightness rises. In the Milky Way, regions far from regions have ratios of 1–1.8 (Cesarsky 1996b) whereas in or near regions values of order 0.2 are seen (see, for example, Cesarsky 1996b,c; Contursi 1998). The decrease in the ratio with star formation activity is most likely due to a combination of PAH destruction in high SFR environments and the increasing contribution of very small grain thermal emission in the LW3 passband. The decrease in PAH features, however, does not necessarily mean physical destruction of the PAH itself. If, for example, only singly ionized PAHs produce the 6.75 feature (Hudgins 1997), a change in the ionization state of the PAH could decrease the 6.75 intensity. Similarly, coagulation of PAHs onto larger grains in dense regions could suppress the fluorescence that leads to the 6.75 emission.
In Table \[tabcam\] we give the integrated ratios for the irregulars as well as the values measured for individual regions in these galaxies. One can see that the supergiant complex NGC 2363 and, because it is dominated by this complex, its galaxy NGC 2366 have the lowest ratios, a value of 0.1. Given the large number of hot O stars that have recently formed in NGC 2363 it is not surprising that NGC 2363 has a very low value of . The rest of the galaxies, as well as the rest of the regions, have higher values of , from 0.2–0.6. This range is significantly lower than the value of unity typical for quiescent environments, and the values are typical of star-forming regions. The variation in the ratio is presumably due to variations in the intensity of the local stellar radiation field and dust content, although hardness of the radiation field is also a possibility, but it is clear that the mid-infrared emission is being dominated by the dust in or near regions in the irregulars.
In Figure \[figlwcolor\] we plot the ratio against the FIR color temperature ratio , after the figure of Dale (2000), in order to compare the irregulars to the large sample of spirals. The dashed line marks a ratio of 1. First, we see that the irregulars in our sample all lie below this line, with values of $<$1, as we have already discussed. However, it is also true that most of the galaxies in the sample of spirals also have ratios $<$1. This is due to the fact that the integrated mid-infrared fluxes are dominated by the higher surface brightness regions in the beam, which have low values of (Dale 1999); the observations are not as sensitive to the lower surface brightness emission that has a higher ratio.
Second, we see that, compared to the spirals, the high infrared-luminosity irregular galaxies in our sample do not have unusual ratios of , but they do extend the range to the lowest values; all but one of the irregulars and IC 10 have values at the low end of the range. This is consistent with the mid-infrared emission that we detect in the irregular galaxies being more dominated by star-forming regions than is the case in spirals. We speculate that irregular galaxies with lower star formation rates than those in our sample would show more moderate values of this ratio if they could be detected outside the brightest few regions. For example, NGC 6822 is a more typical irregular, but with [*ISO*]{} we have only detected the brightest regions.
Third, as Dale (2000) found, we see that the ratio drops as the FIR ratio becomes larger, implying warmer large dust grains. One interpretation of this effect is, that as the dust radiating in the FIR becomes warmer, the emission from very small grains at 15 increases more than the PAH features at 6.75 . All populations of dust grains should be sensitive to the intensity of the stellar radiation field, but the 15 emission is most sensitive because it lies on the Wien side of the blackbody spectrum. An alternate possibility is that stronger fields destroy the carriers of the 6.75 PAH features in irregulars, and we discuss support for this hypothesis below. The exception to this trend is NGC 6822 which has a low value of and a low . However, the CAM fluxes in NGC 6822 are quite weak, and the mid-infrared fluxes may be more uncertain than the formal error-bars suggest.
If the mid-infrared ratio is sensitive to the stellar radiation field, particularly in and around star-forming regions, one might expect a correlation with the SFR per unit area in a galaxy. That is, the more star formation packed into a given area on average, the more intense the hot star radiation, and the lower the ratio . We plot against the SFR in Figure \[figlwcolor\], where the SFR is a globally averaged value. However, there is no correlation. In fact, four of the irregulars themselves span a range of a factor of 50 in normalized SFR and yet have very similar ratios. Most likely, the ratio depends on [*very*]{} local conditions (Contursi 2000). The early type stars are only a few parsecs away from the absorbing dust for these ratios, whereas the SFR is averaged over kiloparsec scales. In other words, the SFR per unit area depends on how many star-forming clouds populate a square kiloparsec of galactic disk, whereas the FIR dust temperatures and the ratios depend only on how far away the exciting stars lie from the absorbing dust in an individual cloud. There is, therefore, little correlation between the two. Since the mid-infrared emission spatially traces the emission so well in our irregulars, we have considered the ratio of the mid-infrared fluxes to the flux, and , both for galaxies as a whole and for individual regions. The integrated ratios are shown for our irregular galaxies, as well as for the spirals, in Figure \[figlwhalpha\], and are listed in Table \[tabcam\]. The and ratios depend on a proper correction for extinction at . We have used an average internal E(B$-$V) for the spirals and others might make a different choice (see, for example, Kennicutt 1983). Thus, the spiral are uncertain by factors of 2–3, but that is still small compared to the trend seen in Figure \[figlwhalpha\] and small compared to the difference between the bulk of the spirals and the irregulars. The irregular with the lowest and most unusual mid-infrared to ratios is NGC 6822, but again we note the possibility of uncertainty in the mid-infrared fluxes in this galaxy.
We see that compared to the spirals, the irregulars in our sample have less mid-infrared emission relative to their emission, especially for regions of high FIR color temperature. The ratio for spirals, irregulars, and regions varies by a factor of 1000, and the irregulars have ratios at the low end or below those of spirals. There is a little less scatter among the entire sample in than in . According to Dale (1999), the surface brightness in the mid-infrared is sensitive to both heating intensity and dust column density, although the dust column density usually plays the larger role in the surface brightness differences from galaxy to galaxy. The irregulars generally have a lower dust content than spirals (Hunter 1989) due to their lower metallicity. So, dust column density is likely a factor in why the irregulars have lower ratios with respect to than the spirals. However, we note that the PAH emission appears to be more sensitive to this than the small dust grain emission since the irregulars differ from the spirals more in than in .
Among the irregulars themselves, we see that, except for NGC 6822, the irregulars in our sample with the smallest emission per emission also have the warmest dust in the FIR whereas there is no difference with emission; our irregulars have very similiar ratios. Since the luminosities of the irregulars in our sample vary by a factor of 10, the constancy of suggests that the emission from small grains is keeping pace with the emission. It seems unlikely that the dust column density is varying among our irregulars in just the right way to compensate for variations in emission. (The L$_{FIR}$/L$_B$ ratio varies by only a factor of 3 among our sample of irregulars). Thus, the small grains appear to be reacting to the hot stars that are also ionizing the gas that emits at . In other words, more small grains are heated as more hot stars are formed (but see Bolatto \[2000\] who argue for the destruction of small dust grains in low metallicity, high ultraviolet radiation environments). In addition, they are heated in such a way that their 15 flux rises linearly with , which tracks the ionizing luminosity of the exciting stars if the regions are significantly ionization-bounded. In the ionization-bounded regime, essentially “counts ionizing photons” from the stars, and therefore it appears that the 15 flux is similarly counting exciting photons from the stars. This could occur if the 15 flux is dominated by the transient heating of small grains by single photons, for example. If those photons were trapped Ly$\alpha$ photons in the regions, then the dust column density is not a factor since the photons scatter until absorbed by dust.
However, if this is the case, then, since the 6.75 flux from PAHs is excited by single photons as well, the decline in the ratio with increasing seen in Figure \[figlwcolor\] must be explained with a destruction of the 6.75 PAH carrier with increasing in irregular galaxies. This appears consistent with the decline in with increasing . While the ratio is constant among the irregulars, the ratio varies by a factor of 6 if we exclude NGC 6822. In other words, as the emission increases, the emission by PAHs in the LW2 passband does not keep pace (see also the UV/[*ISO*]{} comparison of Boselli 1997). For NGC 6822 we are not sufficiently confident of the mid-infrared fluxes to argue that NGC 6822 is truly abnormal. From studies of star-forming regions in the Milky Way, we expect that PAHs are more easily destroyed than small grains in the presence of intense stellar radiation fields (Cesarsky 1996b,c; Contursi 1998) or they are charged in such a way that they do not produce the 6.75 feature. Thus, we are most likely seeing an increase in star formation intensity causing a decrease in PAH 6.75 emission. This is given additional credence by Figure \[figlwfir\] which plots the ratios of / and / against . We see that as the FIR color temperature goes up, the ratio of PAH to FIR emission drops, while the ratio of emission from small dust grains to FIR remains more nearly constant, dropping only slightly. (See Dale 2001 for a more thorough discussion of this relationship).
There is a different trend among the spirals. First, there is a scatter of a factor of 40 in in the spirals that is much greater than in the irregulars. Second, the ratio in the spirals increases with increasing FIR color temperature. The higher metallicity and, therefore, higher dust column densities in spirals may mean that the small dust particles responsible for the 15 flux are on average closer to the OB stars and warmer than in irregulars. The 15 flux in spirals may not be dominated by single photon transient heating, but by the steady emission of warm grains radiating on the Wien side of the blackbody spectrum (Dale 2001, Helou 2001). The scatter is then caused by different density regions that give somewhat different temperature small grains (even for the same which indicates the temperature of larger grains further away). The rise in with is caused by the increased temperature of the small grains as the large grains get warmer.
In the right panels of Figure \[figlwhalpha\] we plot the mid-infrared to flux ratios against the FIR to B-band luminosity ratio. The more FIR emission there is relative to optical emission (that is, the more cool, large-grain dust emission there is) the higher the mid-infrared emission is relative to emission for both the PAHs and the small dust grains. The irregulars in our sample follow both of these trends, but sit at the low /end of the distribution although, as previously noted, their values are more constant in than in for their /. Thus, in these irregulars, the amount of cold dust emission is related to the emission from PAHs, which is consistent with the premise that dust column densities are important in determining the PAH emission and, possibly, that PAH emission originates in the cooler neutral gas (PDR) and dust that dominates the FIR dust emission. However, the amount of emission from small dust grains is also related to the amount of total dust emission from larger grains as well as would be expected for a continuous dust population (Giard 1994).
Motivated by Helou ’s (2001) observation that the ratio of \[\]158 emission to the emission of PAHs in the LW2 passband is constant in galaxies, we have plotted the ratio of the \[\] emission to the mid-infrared emission against the FIR ratio and the ratio L$_{FIR}$/L$_B$ in Figure \[figlwcii\]. As Helou found, we see that the ratio varies by only a factor of 5 for the spiral galaxies. We see that in our irregulars compared to spirals the \[\] emission is elevated relative to that of the PAH emission by a factor of nearly 3, but that it also stays quite constant as changes. The modest scatter in for a large variation in among spirals is interpreted by Helou (2001) as due to the fact that either the same PAHs which heat the PDR gas also produce the emission, or that the ratio of the 6.75 carrier to the grain or PAH population that dominates the PDR gas heating stays constant. In the irregulars in our sample the former interpretation is hard to understand because the ratio of PDR gas heating to 6.75 emission is up by a factor of 3. This implies that in irregulars the PAHs which produce the 6.75 feature and do the heating are quite different from their counterpart PAHs in spirals. This could result from the different processing history and make-up of the ISM in irregulars compared to spirals. In the latter interpretation, the increase in the /f$_{6.75}$ ratio in irregulars could be caused by an increase in the PDR heating population relative to the 6.75 carrier population (although some \[\] emission comes from regions where PAHs may be destroyed). However, this seems inconsistent with observations that show that the relative PAH and dust contents remain the same throughout the Milky Way (Giard 1994) although other studies conclude that PAH fractions vary on small scales (see discussion by Helou 2001).
In Figure \[figlwcii\] we see that for a given the ratio for our irregulars is higher than in spirals and that there is a noisy trend such that the warmer the large dust grains in the FIR, the less \[\] emission per small grain emission. Furthermore, there is no trend of the ratio with L$_{FIR}$/L$_B$ but a very noisy trend of with L$_{FIR}$/L$_B$ (see also Malhotra 2001). The higher at a given may be caused by the lower dust content in irregulars due to their lower metallicity, and that results in the average radiating grain being further away from the OB stars and, hence, cooler on average. Perhaps as well, small dust grains subjected to transient heating by single photons, as we have suggested from the ratio, are more efficient at radiating at 15 . The trend with suggests that \[\] increases more slowly with increasing radiation fields than does the 15 emission. This is consistent with PDR models that show a weak dependence of \[\] with increasing UV fields (Hollenbach, Takahashi, & Tielens 1991). Variations in the incident spectral energy distribution may play a role in variations in the observed ratios as well (Dale 2001).
Infrared Line Data
==================
Far-infrared emission lines observed with [*ISO*]{} are useful diagnostics of the ISM of galaxies. \[\]$\lambda$158 and \[\]$\lambda$63 are important cooling lines of the neutral atomic medium. \[\]$\lambda$122 , \[\]$\lambda$88 , \[\]$\lambda$57 , and \[\]$\lambda$52 emission come from regions. The regions contribute to the \[\]$\lambda$158 emission as well. \[\]$\lambda$88 and \[\]$\lambda$52 together provide a measure of the electron density $n_e$ of the region, or more strictly, of the O$^{++}$ region. Here we examine the [*ISO*]{} emission line ratios observed for the irregulars NGC 1156, NGC 1569, and IC 4662. The other irregulars included in this study, NGC 2366 and NGC 6822, were not observed with the LWS. The observations refer to integrated emission from a large fraction of the optical galaxy, and thus include the full range of galactic environments. Furthermore, the contributions of the different galactic environments are luminosity-weighted.
The Ionized Gas
---------------
### \[\]$\lambda$52,88 and $n_e$
The f$_{[OIII]52}$/f$_{[OIII]88}$ ratio is sensitive to the electron density of the O$^{++}$ regions of regions. This ratio is given in Table \[tablws\] for NGC 1569 (see also Malhotra 2001). It has also been measured from [*ISO*]{} observations for IC 10. From the methodology of Rubin (1994) we find that the O$^{++}$ regions covered by the LWS beam in NGC 1569 have an average $n_e$ of $\leq$300 cm$^{-3}$. The emission ratio places $n_e$ in the low density limit and we can only place an upper limit on $n_e$. This value of $n_e$ is comparable to the values measured for two spirals in our sample (Malhotra 2001). The low values of region densities are also consistent with the low ($\leq$200 cm$^{-3}$) values observed in giant regions in irregulars from the ratios of optical \[\]$\lambda$3726,3729 lines (see, for example, Hunter & Hoffman 1999). However, the \[\] lines measure the density of the O$^+$ regions and are more appropriate for the C$^+$ portions of regions that contribute to the observed \[\] emission.
### \[\]$\lambda$88 and T$_{eff}$
The /L$_{[OIII]88}$ ratio can be used to estimate the effective temperature of the ionizing stars when the oxygen abundance of the gas is taken into account. We have measured this ratio for NGC 1569 and IC 4662, and the values are given in Table \[tablws\]. We have combined the observed ratios with the models of Rubin (1985) with a density of 100 cm$^{-3}$ and scaled for the oxygen abundances given in Table \[tabgal\]. We use a density of 100 cm$^{-3}$ since in the previous section we found $n_e$ to be in the low density limit ($n_e\leq$300 cm$^{-3}$). Furthermore, in this regime and \[\] emissions have the same functional dependence on $n_e$, so the ratio L$_{[OIII]}$ is not dependent on the density. We examine the resulting T$_{eff}$ for three model choices of the total nebular Lyman continuum photons per second. These models are appropriate for regions that are smaller than the supergiant regions. Because the LWS beam encompasses a large fraction of the galaxy, these are rough estimates of the average stellar effective temperature.
We find that T$_{eff}$ in NGC 1569 is about 40,000 K (log g$=$4.0 model) for all three models with the uncertainty in the measured ratio producing an uncertainty of order 1000 K. This T$_{eff}$ corresponds to an O6.5V star (Conti & Underhill 1988). The total luminosity of NGC 1569 would correspond to the equivalent of 4500 O6.5V stars (Panagia 1973). This large number of O stars is consistent with the recent starburst in that galaxy. It has been suggested that the starburst ended 5–10 Myrs ago (Greggio 1998), 4 Myrs ago (Vallenari & Bomans 1996), and 10 Myrs ago (Israel & van Driel 1990). The hydrogen-burning lifetime of an O6.5 star is about 6.4 Myrs at a metallicity of 0.001 (Schaller 1992). This lifetime is more consistent with the most recent estimates for the end of the starburst; 10 Myrs would be too long ago.
The models yield an average T$_{eff}$ in IC 4662 of somewhat hotter than about 45,000 K (log g$=$4.5 model) unless the regions are all Orion-like in size, which is unlikely. The /L$_{[OIII]}$ ratio yields a T$_{eff}$ that is hotter than 45,000 K even if the ratio is larger by 1$\sigma$. A T$_{eff}$ of 45,000 K corresponds to an O4V star (Conti & Underhill 1988). The total luminosity of IC 4662, however, would correspond to only 50 equivalent O4V stars (or $\sim$400 O6.5V stars). Thus, the total number of O stars in IC 4662 is significantly lower than in NGC 1569, but we are catching those regions at a much younger age unless the stellar initial mass function is highly top heavy which seems unlikely (Massey & Hunter 1998). The hydrogen-burning lifetime of an O4 star is about 3 Myrs (Schaller 1992).
In both galaxies the average T$_{eff}$ are hotter than what are usually determined for other galaxies. For example, T$_{eff}$ determined from an \[\]88 measurement in the starburst nucleus of NGC 253 gives $\geq$34,500 K, which is equivalent to an O8.5V star (Carral 1994). For a zero-age main sequence stellar population of the same stellar initial mass function, a higher T$_{eff}$ could result if individual star-forming regions are richer in stars, rich enough to allow the rarer highest mass stars to form (Massey & Hunter 1998). In that case, then, we would expect to see supergiant regions associated with these large concentrations of massive stars or perhaps super star clusters. The luminosity function of regions in NGC 1569 does show that the region population in NGC 1569 extends to luminous regions ($\sim10^{40}$ ) (Youngblood & Hunter 1999; we do not have an region luminosity function for IC 4662), and NGC 1569 does contain two super star clusters. In addition, since NGC 1569 is a starburst system and, hence, the stellar population is coeval to a greater extent than in most galaxies, the average age of the O-star population is expected to be low compared to spirals with more continuous star formation.
### \[\]$\lambda$88 and \[\]
The / and / ratios are much higher in the irregulars than they are in spirals. This is shown for / in Figure \[figoiiicii\]. Plots of / look similar to this since \[\] and \[\] both come mostly from PDRs. We see that the irregulars have / ratios that are as much as 16 times higher than those in most spirals. We use the \[\] observations in §5.3.1 to show that most of the \[\] originates in PDRs as opposed to gas whereas \[\] comes from regions.
Even if there were no PDR contribution to the \[\] in irregulars, we would not expect differences of order 16 from spirals if the effective temperatures of the stars were not considerably higher in irregulars compared with spirals. In spirals, the regions contribute of order 0.2 to 0.5 of the \[\] emission (Malhotra et al 2001). Therefore, eliminating the PDR contribution would only increase the / ratio by factors of 2-5 in spirals. The irregulars can only gain the large factor of 16 by also having less \[\] compared with \[\] in the region. This is accomplished by having higher effective temperature stars, which doubly ionize most species and leave only small quantities of C$^+$ in the regions.
However, if regions in the irregulars in our sample were surrounded by optically thick PDRs (A$_V > 1$), the / ratio would be much smaller than observed, because such PDRs produce copious emission in \[\], quite independent of metallicity (Kaufman 1999). Therefore, the extremely high ratios of / observed in the irregulars in our sample can only be achieved by having both high effective temperature stars and small covering factors of optically thick PDRs outside the regions compared with normal spirals.
The PDR and Neutral Gas
-----------------------
### The \[\]-to-CO Ratio
Integrated /(1–0) ratios have been shown to be related to the global star formation activity of a galaxy (see, for example, Pierini 1999). Stacey (1991) have shown that galaxies with starburst nuclei have ratios comparable to those seen in Galactic star-forming regions, while more quiescent galactic nuclei have ratios that are about 3 times smaller and comparable to values averaged over entire Milky Way GMCs. Stacey measured / values of 900–6100 for normal spirals and values of 1000–1800 for GMCs. By contrast Stacey measured a ratio of 6100 for the starburst galaxy M82 and values of 5900–14000 for Milky Way regions. Poglitsch (1995) measure a ratio of 60000 in the supergiant complex 30 Doradus, and Madden (1997) find values of 1500–10000 throughout IC 10. Smith & Madden (1997) measured ratios of 14000 and 16000 for two Virgo spirals. There are, however, also variations among similar clouds. Israel (1996) report variations of up to a factor of 100 among individual clouds in the LMC; they measured ratios of 400–34000 in 4 clouds, and attribute the variations to evolutionary effects. At the other extreme, Malhotra (2000) extend the /range to low values with [*ISO*]{} observations of early-type galaxies where presumably the environment is quite quiescent. Generally, regions of intense star formation, including kpc-sized regions in galaxies, have much higher / compared to more quiescent regions.
To place our irregular galaxies in this context, we have examined the / ratios for NGC 1156, NGC 1569, and IC 4662, and they are given in Table \[tablws\]. The $^{12}$CO(1–0) observations are taken from Hunter & Sage (1993), Young (1995), and Heydari-Malayeri (1990), and $=0.119 D^2 S_{CO}$ from Kenney & Young (1989) where $S_{CO}$ is the CO flux in Jy and D is the distance to the galaxy in Mpc. The lower limits on / for NGC 1156 and IC 4662 result from upper limits on . The observed ratios in the irregulars are very high: at least several times $10^4$. Galactic values of this level are also seen in the irregulars IC 10 (Madden 1997) and the LMC (Mochizuki 1994).
These values are much higher than values Stacey (1991) and others measured for spiral nuclei and giant molecular clouds. But, they are also higher than values measured for all but one of the Milky Way regions and higher than that measured for the starburst galaxy M82. This can also be seen graphically in Figure 2(b) of Pierini (1999) where they plot / against /. All of the galaxies in their sample and our irregulars have similar /, but our irregulars sit at higher / than all of the extragalactic and Galactic sources. The ratios for our three irregulars are not corrected for the smaller beam size of the CO observations, although the ratios of the beam sizes are given in Table \[tablws\], and in two cases the ratios are lower limits because of upper limits on the CO flux. Thus, the / ratios of these irregulars could be even higher.
Pierini (1999) show that there is a correlation between / and equivalent width in the sense that galaxies with high equivalent widths, indicating a higher relative star formation rate, have higher /. In our sample NGC 1569 has just undergone a starburst, and NGC 1156 and IC 4662 relatively high SFRs. In NGC 1569 the starburst has blown a hole in the (Israel & van Driel 1990) and severely disrupted the ISM. Yet in spite of this, NGC 1569’s / ratio is of the same order of magnitude as that of the other two irregulars. Perhaps the five GMCs detected in NGC 1569 (Taylor 1999) are dominating both the \[\] and CO emission.
The high / ratio in the irregulars could be an indication of a different geometry of clouds in irregulars compared to spirals (e.g., Pak 1998, Kaufman 1999, and references therein). If the PDR is a thin skin around a CO core, both the \[\] and CO luminosity from a cloud go as R$^2$ and so the ratio is approximately constant for a dense warm PDR layer. However, if the CO core is small and the \[\] layer around the outside is very thick, the CO luminosity goes as R$^2$ but the \[\] goes as (R$+\Delta$R)$^2$ and the / ratio will be higher. Thus, Im galaxies could have very thick PDR regions around tiny molecular cores. This would also be consistent with the picture in which the molecular clouds are the dense cores of large, massive atomic gas complexes (Rubio 1991). In fact, below, we derive physical conditions of the PDRs from models, and find that for NGC 1569 the PDR is 6 pc thick, thicker than is usual in spirals.
It should also be kept in mind that in irregulars is probably an underestimate of the amount of molecular hydrogen that is present compared to the case in higher metallicity spirals. There is strong evidence that the conversion of I$_{CO}$ to N(H$_2$) depends on the metallicity and that the CO component of a cloud in metal poor irregulars is a much smaller fraction of the H$_2$ cloud (see, for example, Maloney & Black 1988; Dettmar & Heithausen 1989; Stacey 1993; Mochizuki 1994; Poglitsch 1995; Verter & Hodge 1995; Wilson 1995; Madden 1997; Smith & Madden 1997; but see also Wilson & Reid 1991). In the LMC where the metallicity is about half solar, virial masses of molecular clouds yield a correction to the standard ratio of a factor of 6 (Cohen 1988). A factor of 6.6 was also found for giant molecular clouds in NGC 1569 (Taylor 1999), but the correction factor to L$_{CO}$ remains somewhat controversial.
### \[\]$\lambda$158 and FIR Emission
Malhotra (1997, 2001) report a correlation between integrated and and for the distant galaxy sample. The bulk of the galaxies have similiar ratios, but then as the FIR dust temperature increases, the ratio decreases and increases. For most galaxies L$_{[CII]}$/L$_{FIR}$ is about the same because $G_0\propto n^{1.4}$. Here $G_0$ is the far-ultraviolet (FUV) stellar radiation flux, in units of the value measured for the solar neighborhood, and $n$ is the neutral gas density. Therefore, $G_0$/$n$ rises slowly with $G_0$, which keeps the grain charge fixed and the gas heating efficiency via the grain and PAH photoelectric heating mechanism fixed. This implies similarities in star-forming clouds in different galaxies. Malhotra (2001) interpret the decrease in at high values of (FIR dust temperatures) in some galaxies as due to less efficient gas heating resulting from charged dust grains in high $G_0$/$n$ regimes (Bakes & Tielens 1994). However, one should also question the normalization of given the role of PAHs (Helou 2001).
Figure \[figciifir\] shows the irregular galaxies in our sample relative to the spirals in L$_{[CII]}$/L$_{FIR}$. We see that NGC 1156 and a position in IC 10 have among the highest ratios of the entire sample of galaxies. The other two irregulars and other positions in IC 10 have ratios that are comparable to that of the bulk of the spirals. Furthermore, for a given FIR dust temperature, the irregular galaxies are seen to have a higher value of than irregulars.
For comparison, Israel (1996) report a ratio of 0.01 for the supergiant complex 30 Doradus in the LMC and, presumably because it is dominated by 30 Doradus, for the LMC as a whole. The ratio observed in 30 Doradus is, therefore, 2–4 times higher than the integrated values for the irregulars in this sample. Even though NGC 1569 is a starburst galaxy, it is significantly lower in compared to a supergiant star-forming region, implying that average conditions in NGC 1569 today are not too extreme. This could mean that the FIR emission in NGC 1569 is being dominated by the remaining GMCs and is not reflecting the disrupted nature of the rest of the ISM of the galaxy. The LMC, on the other hand, has a much higher integrated value of than NGC 1569 even though NGC 1569 has a recent SFR per unit area that is 30 times higher. This suggests that a supergiant region, when it is present, can dominate the integrated properties of a galaxy. We would predict, therefore, that NGC 2366 should have a ratio that is like that of the LMC since it is dominated by the supergiant complex NGC 2363. Unfortunately, we do not have LWS observations of NGC 2366.
The lower ratio of in spirals compared to our irregulars for the same , might be an indication that cooler sources of radiation (than hot OB stars) may be contributing to the FIR luminosity in those sources with low . Another possibility is that the ratio of $G_0/n$ is somewhat higher in those sources with low , resulting in more highly charged grains and less efficient gas heating.
In Figure \[figciifir\] we also plot against the global SFR per unit area for the irregulars and other galaxies in the distant sample. We see that the bulk of the sample has a similar ratio with some decrease or more scatter of as the SFR per unit area decreases for the spirals (the irregulars appear to go counter to this trend, but the statistics are poor). This trend is similar to what is seen by Pierini (1999) for a sample of Virgo cluster spirals, where they see a plateau value of 0.004 in and a drop to lower values for lower equivalent widths. They suggest that this plateau is an average of compact and diffuse regions in the galaxy and that diffuse regions begin to dominate as the SFR declines. However, PDR models (Kaufman 1999) and observations of diffuse regions in the Galaxy show that diffuse regions have lower $G_0$/n and, hence, higher . The near constancy with SFR means that a higher level of star formation in a given area does not imply a higher local $G_0$/$n$ ratio. Although $G_0$ goes up in regions of star formation, Malhotra (2001) argue that $n$ goes up too, so that ordinarily the average $G_0$/$n$, and hence , rises slowly. However, individual star-forming regions may have enhanced $G_0$/$n$ ratios, which lead to inefficient heating by grains and lower . The large scatter for low values of SFR may be caused by a small number of star-forming regions dominating this regime, so that the local conditions in a particular star-forming region become more evident.
### \[\]$\lambda$145,63
The ratio is a measure of the temperature of the gas or an indicator of optical depth in the 63 line. Malhotra (2001) interpret the increase of with increasing in the large sample of galaxies as an indication of optical depth effects in \[\]63. We have for NGC 1569 and that measurement is shown in Figure \[figoioi\]. We see that NGC 1569 falls at the tail end of the relationship traced by the spirals. Thus, \[\]63 in NGC 1569 appears to be affected by optical depth as in spiral galaxies, but NGC 1569 lies at the low optical depth end of the distribution.
Physical Conditions in the PDRs in Irregulars
---------------------------------------------
The FIR emission-line ratios can be combined with PDR models in order to derive fundamental parameters that describe the state of the ISM in the irregular galaxies in our sample (Kaufman 1999). In particular, the FIR line ratios yield information on the far-ultraviolet stellar radiation field, $G_0$, and the neutral gas density, $n$, of H nuclei in PDRs.
### The PDR and Region Contributions to \[\]
A primary diagnostic is the \[\] line, but both PDRs and dense and diffuse regions can contribute to this line flux. In Figure \[figciiha\] we plot the / ratio against global L$_{FIR}$/L$_B$. The correlation of / with /Ł$_B$ resembles that of and with /L$_B$ shown in Figure \[figlwhalpha\] although there is more scatter in the / plot. As cold dust emission increases relative to the optical, the \[\], PAH, and small grain emissions all increase to some extent relative to . Thus, the amount of \[\] emisison is related to the amount of cold dust emission, as expected. We see that the three irregulars in our sample have the lowest / ratio of the distant galaxy sample. However, regions could be a significant source of \[\] emission, and that contribution needs to be removed in order to examine the emission from PDRs.
We have, therefore, determined corrections to for the contribution following the procedure of Malhotra (2001). They used the fact that \[\]$\lambda$122 emission comes only from the dense and diffuse ionized gas (not the PDR gas) and derive the ionized gas contribution to \[\] as 4.5 times the flux of \[\]. This correction factor assumes a C/N ratio of 1.9 and that may not be accurate for metal-poor irregulars. If irregular galaxies such as those for which we have LWS spectra have higher C/N ratios, as indicated by Garnett (1995), the correction to \[\] would be correspondingly higher. On the other hand, for the three irregulars we only have upper limits to L$_{[NII]}$ and hence lower limits to the PDR contribution to the \[\] flux, L$_{[CII],PDR}$. At least 61% of the \[\] emission in the irregulars is from PDRs. The L$_{[OI]}$/L$_{[CII],PDR}$ ratios are given in Table \[tablws\] as a range of values with the lower end of the range determined assuming that all of \[\] comes from the PDRs and the upper end of the range determined assuming a correction to \[\] for regions that assumes the upper limit of the \[\] flux.
For IC 10 the contribution of ionized gas to \[\] was found to be $\sim$20% by Madden (1997). Stacey (1993) also found that 25%–30% of \[\] emission in the spirals NGC 6946 and NGC 891 comes from diffuse ionized gas, and Malhotra (2001) find that the correction is typically 30%. Thus, unless we are underestimating the amount of \[\] coming from regions because of a different C/N ratio, at most the irregulars in our sample are similar to other galaxies in proportions of \[\] coming from PDR and ionized gas, and potentially our irregulars have a somewhat lower proportion coming from ionized gas. This is a surprise because other parameters have led us to conclude that the integrated properties of irregular galaxies are dominated by regions more so than the spirals. However, because of the high effective temperature of the exciting stars, the regions in these irregulars will be in high ionization states, resulting in less C$^+$ in the gas than would otherwise be expected.
The ratios of \[\] to \[\] are shown in Figure \[figoicii\] for the corrected and uncorrected \[\] flux. The points for the 3 irregulars are upper limits for , but those for two regions in IC 10 are measurements since the \[\] line was detected there. We see that our irregulars have ratios that are lower for a given ratio compared to spirals.
### Results from PDR Models
We have used the LWS line strengths in conjunction with models that are slightly modified from those presented by Kaufman (1999). The analysis by Kaufman is most appropriate for individual nearby molecular clouds where FUV illumination occurs primarily from one side of a molecular cloud. The models have been modified to consider integrated emission from a large fraction of a galaxy by assuming that the interstellar clouds can be illuminated from any side and that optically thin \[\] and FIR continuum reach the observer from any side. Because it is, however, optically thick, \[\] emission is assumed to come only from the side closest to the observer. The ratio ($+$L$_{[OI]}$)/ measures the efficiency with which FUV photons are converted to emission lines. This is combined with the ratio to derive $G_0$, $n$, and the average gas surface temperature T$_{gas}$ and pressure $P$ of the interstellar gas producing the line and the FIR continuum in PDRs.
In this manner, we derive $G_0$, $n$, T$_{gas}$, and $P$ given in Table \[tabmodel\] for PDRs in NGC 1156, NGC 1569, and IC 4662. We see that $G_0$ is $1.1\times10^3$–$2.4\times10^4$, $n$ is $1.3\times10^3$–$2.8\times10^4$ cm$^{-3}$, and T$_{gas}$ is 310–725 K. The conditions in IC 4662 seem to be the most severe of the three irregulars.
For the larger group of spiral galaxies, Malhotra (2001) derived $G_0$ of 10$^2$–10$^{4.5}$ and $n$ of 10$^2$–10$^{4.5}$ cm$^{-3}$. A comparison of the irregulars with Figure 9 of Malhotra (2001) shows how $G_0$ in the irregulars compares with typical values in spirals: NGC 1156 has a $G_0$ that is comparable to the bulk of spirals, IC 4662’s $G_0$ is higher than those of all but a few spirals, and NGC 1569 has a $G_0$ that is higher than those of most of the spirals. The values of $G_0$ and $n$ derived for the irregulars, especially NGC 1569 and IC 4662, are similar to the values derived for the peculiar galaxy M82. For M82 Colbert (1999) analyze LWS spectra to find that $G_0$ is 630 and $n$ is 2000 cm$^{-3}$, but Kaufman (1999) find a higher $G_0$ of 10$^{3.5}$ and a higher $n$ of 10$^4$ cm$^{-3}$ for the center of M82 from analysis of Kuiper Airborne Observatory data (Lord 1996). Also, the central regions of IC 10, a galaxy that has many similarities to NGC 1569, have $G_0$ that are a factor of 6 lower, 300–500, with densities that are similar, 10$^3$–10$^4$ cm$^{-3}$ (Madden 1997). However, in the starburst nuclei of two spirals, Carral (1994) measure $G_0$, 10$^3$–10$^4$ that are similar to the three irregulars. Thus, the illuminated neutral clouds in irregulars are within the range of values observed for spiral galaxies but are at the high end, having a more intense stellar radiation field reaching the clouds and a higher density than most. Malhotra interpret the high T$_{gas}$ and $P$ they derive for the spiral sample as indicating that most of the grain and gas heating is occuring very close to the concentrations of young, hot stars in these GMCs, and this must be the case in the irregulars as well.
We derive pressures $P$ in the PDRs of 4–200$\times10^5$ K cm$^{-3}$. Elmegreen & Hunter (2000) have estimated pressures of regions in a sample of 6 irregular galaxies (unfortunately, none of the galaxies are in common with our LWS observations). They find region pressures of 10$^4$–10$^5$ K cm$^{-3}$. The pressures estimated from regions are therefore lower than those in the PDRs of our galaxies, although both pressure determinations are uncertain. The region pressures measured by Elmegreen and Hunter are lower probably because these regions are evolved enough that they no longer have PDRs around them.
The differences in physical PDR parameters between the irregulars NGC 1569 and IC 4662 and normal spiral galaxies imply that the star-forming clouds in NGC 1569 and IC 4662 are different. This is perhaps not a surprise since at least one of the galaxies is a starburst. NGC 1569 is a starburst galaxy with a high population of massive stars providing an intense FUV field, and the ISM has been disrupted in many ways. The very high $G_0$ and $n$ of IC 4662 suggest that maybe this galaxy has experienced some recent peculiar event like that of NGC 1569, these properties are also consistent with the higher average stellar T$_{eff}$ found in §5.1.2. NGC 1156, on the other hand, is a normal, albeit active, irregular and its cloud properties are correspondingly more normal.
In order to further investigate the properties of the ISM in these three irregular galaxies, we use the technique of Wolfire, Tielens & Hollenbach (1990) along with the model results of Kaufman (1999) modified for application to extragalactic sources. Given the PDR parameters derived from \[\], \[\] and FIR observations, the PDR models make specific predictions for the \[\] cooling rate per atom and the \[\] intensity. By comparing the cooling rate per atom with the observed \[\] flux from PDRs, L$_{[CII],PDR}$, an estimate of the atomic gas mass M$_a$ in PDRs can be found. Following the measurements of Taylor et al. (1999) for NGC1569, we assume a carbon abundance of 1/6 solar, so $x_C=2.3\times 10^{-5}$ in all three galaxies. From this, we find the mass of atomic gas on the outsides of molecular clouds in these three galaxies varies from $\sim 1.3\times 10^5 - 4\times 10^7\,M_{\sun}$; values are given in Table \[tabmodel\]. An estimate of the area filling factor of PDR gas in a galaxy’s central region producing \[\] emission can be made by comparing the observed \[\] intensity for this region with the predicted \[\] intensity from the PDR models. In order to make this comparison, we assume that the \[\] emitting region lies within the contours shown on the 6.75 images in Figure \[figcam\] (see figure caption for surface brightness levels). Estimates of the PDR area filling factor range from $\sim 6-10\%$ in the three galaxies.
To further extend the analysis, we need to find the mass of molecular gas in the clouds whose surfaces produce the PDR emission. NGC 1569 is the only galaxy of the three for which we have a measured value of the CO luminosity, $L_{CO}=35\,L_{\sun}$, from Young (1995). The standard conversion factor, $L_{CO}/M_{H_2}\sim 8\times 10^{-6}L_{\sun}/M_{\sun}$ (e.g. Young 1986) has been shown by Taylor (1999) to be too large by a factor of $\sim 6$ in this galaxy, reflecting a metallicity about 1/6 solar. By applying this correction to the standard conversion factor, we find a molecular gas mass M(H$_2$) in NGC 1569 of $1.9\times 10^7\,M_{\sun}$, about 1.5 times the atomic gas mass. Values of M(H$_2$) are given in Table \[tabmodel\].
A comparison of the surface area of the clouds (the \[\] measurement) with the volume of the clouds (the CO measurement) allows us to characterize the molecular clouds in NGC 1569, and the derived parameters are given in Table \[tabcloud\]. If we assume that the \[\] and CO regions have equal density (that is, $n$(H$_2$)/$n_a$$\sim$1), then we find the emission is fit by an ensemble of 4 giant molecular clouds with typical radius $\sim 25$ pc and cloud mass $\sim 8\times 10^6 M_{\sun}$. If, instead, we assume that the CO region has a density 3 times that of the atomic gas ($n$(H$_2$)/$n_a$$\sim$3), the number of clouds increases to 8, the mass per cloud decreases by a factor 2, and the cloud radius falls to 18 pc. These results for number of clouds and cloud size are similar to those of Taylor (1999), who found from CO observations that the GMCs in NGC 1569 could be resolved into $\sim 4-5$ clouds with radii $\sim 20$ pc, though we note that they report cloud masses which are significantly lower than ours. Our clouds are $\sim$20 times more massive than a typical Milky Way GMC (Scoville & Sanders 1987).
The PDR surface layer on our clouds would extend $\sim$6 pc in from the cloud surface, a distance larger than would be found if the clouds had solar metallicities. Given the derived density of the PDR gas, $n=3.5\times
10^3\,\rm cm^{-3}$, these GMCs have a gas column density from surface to center of $\sim 3\times 10^{23}\,
\rm cm^{-2}$, corresponding to a visual extinction of A$_V$$\sim 22$ assuming that dust is depleted by the same factor as the carbon. As a consistency check, we compare these results with those of Kaufman (1999) who find that clouds with $A_V\sim 10$, $n=10^3\,\rm cm^{-3}$, and metallicities of a few tenths solar have L(CO)/M(H$_2$) conversion factors of $\sim 10^{-6}L_{\odot}/M_{\odot}$, very close to the corrected value of Taylor (1999). The results for the NGC 1569 clouds give A$_V$ higher by a factor of $\sim 3-4$ than the A$_V$ predicted by the photoionization-regulated star formation model of McKee (1989). This may imply that the clouds in NGC 1569 have not yet reached equilibrium, which may be consistent with the disruption of the ISM from the recent starburst and the proximity of the GMCs to the super star clusters (Taylor 1999). (Note that these high extinction values only apply to the GMCs which are not necessarily visible in the optical).
One thing that is clear from these observations is that the PDR observations in irregulars are sensitive, not to the average density of GMCs, but to the conditions in the denser clumps within the GMCs. In the Milky Way, the average GMC density is of order 100 cm$^{-3}$, but observations of FUV illuminated gas give PDR densities of order $10^3\,\rm cm^{-3}$ and higher. Such is the case, for example, in Orion where the average density of the entire molecular cloud complex is far lower than that in the FUV illuminated clumps region at the edge of the Orion Nebula.
A similar analysis to the one carried out above may be done for the other two galaxies, NGC 1156 and IC 4662. However, in both cases we only have upper limits to the molecular mass. As a result, we can only derive ranges for the numbers and masses of GMCs in these galaxies with large uncertainties. In NCG 1156, we find between 1 and 6 GMCs with masses ranging from $\sim 7\times 10^7$ M to $3\times 10^8$ M; for IC 4662, we find between 1 and 2 GMCs with masses from $\sim 3\times 10^6$ M to $8\times 10^6$ M. These GMCs are also large compared to those in the Milky Way (Scoville & Sanders 1987): 175–750 times more massive in NGC 1156 and 7–20 times in IC 4662.
### Uncertainties in PDR Modelling
To quantify the uncertainties in deriving $G_0$ and $n$, we have used several different methods for each galaxy. In the PDR models, we typically compare one line ratio (usually, L$_{[CII]}$/L$_{[OI]}$) with the infrared line-to-continuum ratio, (L$_{[OI]+[CII]}$/). If we do this using the raw \[\], \[\] and FIR values, we get certain values for $G_0$ and $n$, but we suspect that much of the \[\] emission comes from ionized gas, not PDRs. Thus, we correct \[\] to get \[\]$_{PDR}$. The correction is based on the observed \[\] flux with the assumption that \[\] only comes from ionized gas; given an assumed \[\]-to-\[\] abundance ratio, we can calculate the \[\] from ionized gas and subtract that portion from the total \[\], leaving only the PDR contribution. In all three galaxies, we assumed that the C/N ratio was 4.6. However, for all three galaxies we only have upper limits on the \[\] flux, so we have a range of possible \[\] from PDRs. In one variation on the standard method, if one assumes that there is no \[\], then all of the \[\] comes from PDRs, and the models give significantly different answers for $G_0$ and $n$. A second variation is to solve for $G_0$ and $n$ using other ratios. In this case, we compared $L_{[OI]}$/ with $L_{60}/L_{100}$. This has the advantage that we no longer rely on the \[\] measurements, though we do have to assume that the 60 and 100 micron fluxes are produced in PDRs. Again, we get somewhat different results for $G_0$ and $n$.
In Figure \[figuncertain\], we summarize the results of the three methods. These three methods together give an estimate of the uncertainty in the model values of $G_0$ and $n$ and hence clouds masses and sizes. One can see that the three methods give answers that can be uncertain by up to an order of magnitude in $G_0$ and $n$. These uncertainties almost surely are larger than the uncertainties in the measurements. What is heartening is that the preferred method gives results within a factor $\sim$3 of the other two methods.
Summary and Comments
====================
We have discussed mid-infrared imaging and FIR spectroscopy of a sample of 5 IBm galaxies observed by [*ISO*]{} as part of a larger study of galaxies of all Hubble types. The galaxies include NGC 1156, NGC 1569, NGC 2366, NGC 6822, and IC 4662. The galaxies NGC 1156, NGC 1569, and IC 4662 are high luminosity, high surface brightness, and high star formation rate systems relative to median properties of larger samples of irregulars, and so are not typical irregulars. NGC 2366 is dominated by a supergiant complex, and NGC 6822, although more of a typical irregular, yielded useful data with CAM.
The mid-infrared imaging of all 5 galaxies is in two bands: one at 6.75 that is dominated by PAHs and one at 15 that is dominated by small dust grains. The spectroscopy of 3 of the galaxies (NGC 1156, NGC 1569, and IC 4662) includes \[\]$\lambda$158 and \[\]$\lambda$63 , important coolants of PDRs, and \[\]$\lambda$88 and \[\]$\lambda$122 , that come from ionized gas regions. \[\]$\lambda$145 and \[\]$\lambda$52 were measured in NGC 1569 as well. We compare the observations of the irregulars with the larger sample of spiral galaxies observed as part of our [*ISO*]{} program.
We observe the following:
- [In the mid-infrared images most of the emission we detect is associated with the brightest regions in the galaxies, and the integrated ratios are comparable to what is observed in regions of star formation.]{}
- [The ratio of PAH-to-small grain emission drops as the FIR color temperature becomes hotter in all galaxies, and the irregulars in our sample lie mostly at the warmer and lower end of the distribution.]{}
- [The ratio of PAH-to-FIR emission drops as the FIR color temperature becomes hotter while the small grain emission-to-FIR ratio remains more nearly constant in all galaxy types.]{}
- [The irregular galaxies in our sample have low PAH emission at 6.75 and low small grain emission at 15 relative to emission, compared to spirals. The ratio drops with increasing FIR color temperature, while the ratio is constant among our irregulars.]{}
- [The PAH and small grain emissions relative to both increase as / increases for all galaxies, possibly with a plateau in the value of /for low values of /. Our irregular galaxies lie at the low / end of the distributions.]{}
- [The \[\] emission is higher relative to FIR emission in these irregulars than in spirals of the same FIR color temperature. The \[\] line represents 0.3% to 1% of the FIR emission of the irregulars.]{}
- [The \[\] emission in our irregulars is high relative to the PAH 6.75 emission compared to spirals, and / is constant among the irregulars. On the other hand, for small grains, the /ratio drops as the FIR color temperature increases for all galaxies, but the / ratio is high in the irregulars compared to spirals with the same FIR color temperature.]{}
- [For our irregulars the ratio of \[\] to \[\] is very high compared to spirals. In addition, the high \[\] to 6.75 and FIR emissions suggest that \[\] is high relative to the infrared continuum and other lines as well.]{}
- [For all galaxies the / ratio increases as / increases, and our irregulars are at the low /, low / end of the distribution.]{}
- [In the irregulars in our sample the \[\] emission is high with respect to \[\].]{}
- [ The L$_{[CII]}$/L$_{CO}$ ratio is very high among our sample of irregulars.]{}
- [From PDR models we derive physical conditions in the PDRs of NGC 1156, NGC 1569, and IC 4662: Radiation fields relative to the solar neighborhood $G_0$ are $10^{3.0}$–$10^{4.4}$, gas densities $n$ are $10^{3.1}$–$10^{4.4}$ cm$^{-3}$, and pressures $P$ are $10^{5.6}$–$10^{7.3}$ cm$^{-3}$ K. In NGC 1569, where L$_{CO}$ has been measured, we deduce the presence of 4 GMCs with masses of that are about 20 times higher than that of a typical GMC in the Milky Way, and the PDR is 6 pc thick. ]{}
The upper limits on the \[\]$\lambda$122 emission in our irregulars indicate that a large fraction of the \[\] emission detected in these irregulars comes from PDR gas and the component from regions is small although these conclusions do depend on physical conditions being similar to spirals. Less than 39% of the \[\] originates from regions. In spite of the high \[\] and emission relative to \[\], the contribution of ionized gas regions to \[\] emission is similar to or less than the fractions observed for spirals.
The average stellar effective temperatures in NGC 1569 and IC 4662 are high compared to measurements in other galaxies. In NGC 1569 this is consistent with the presence of two young super star clusters and luminous regions, and suggests an end to the recent starburst of $\leq$6 Myrs ago. This result is also consistent with the measurement of exceptionally high / ratios in our irregulars. The high excitation stars ionize most of the regions to higher ionization states, reducing the amount of \[\] in regions. Our irregulars also have higher / ratios for their FIR color temperatures relative to spirals. This can be understood if the spirals have a larger contribution to heating of the dust from cooler stars, presumbably from the general interstellar radiation field, than do the irregulars.
The increase in and with increasing /tells us that the PAH and small grain emissions are related to the total dust emission. However, the decrease in PAH emission in the irregulars in our sample relative to small grain, FIR, and emissions for increasing FIR color temperature is interpreted as a decrease in PAH emission resulting from an increase in the radiation field due to star formation, either through destruction of the PAH itself or of the 6.75 carrier on the PAH.
That the ratio is constant among the irregulars that we observed means that the more hot stars that are formed, the more small grains are heated. The 15 emission may come primarily from small grains in regions. We interpret the linear dependence of the 15 flux with H$\alpha$ to mean that the 15 emission is being generated by the transient heating of small dust grains by single photon events, possibly Ly$\alpha$ photons trapped in regions. The low ratio, as well as the high / ratio, in our irregulars compared to spirals may be due to the lower dust content overall resulting in dust grains being, on average, further away from the heating source. By contrast, it has been suggested that the 15 emission in spirals may be warm dust grains radiating on the Wien side of the blackbody spectrum rather than transient single photon events. This interpretation is consistent with the increase in ratio with hotter FIR color temperature seen in spirals, but not irregulars.
The high \[\] emission relative to FIR, PAH emission at 6.75 , and small grain emission at 15 in our irregulars is harder to interpret. The small scatter in the / ratio among spirals has been interpreted as meaning that the PAHs that produce the 6.75 emission and the PAHs that heat the PDR and hence produce the \[\] emission are the same entities. But, then the much higher / ratio in our irregulars compared to spirals would require that the PAHs in irregulars produce several times more heat than the PAHs in spirals. Alternatively, the small scatter in / among spirals could be explained if the carrier of the 6.75 feature tracks the heating but contributes only a part of the heating, which is due mostly to small grains or other PAHs. In this scenario these irregulars are understood if they have a higher proportion of the PAHs that heat the PDRs to the PAHs that produce the 6.75 feature. A different incident spectral energy distribution could also play a role.
Our data give some clues to the geometry of the clouds in the irregulars in our sample. The high \[\]/\[\] ratio requires a small solid angle of optically thick ($A_V>1$) PDRs outside the regions. That is, if the region is surrounded by an optically thick PDR, the copious production of \[\] there would lower the / ratio. So, the optically thick component of the PDR must present a smaller covering factor in our irregulars compared to spirals. This picture is also consistent with the fact that PAH 6.75 and \[\] emissions, both of which come primarily from PDR gas, are lower compared to . On the other hand, the L$_{[CII]}$/L$_{CO}$ ratio is very high among our sample of irregulars, higher than values measured in spirals or regions. A picture in which the clouds in our irregulars have very thick \[\] shells around tiny CO cores compared to clouds in spirals is consistent with physical parameters we derive for NGC 1569’s PDRs.
From models we deduce the physical conditions in the PDRs. The FUV stellar radiation field $G_0$ is like that in typical spirals in NGC 1156 but much higher in the starburst galaxy NGC 1569 and in IC 4662. That NGC 1569 stands out in $G_0$ makes sense since it is a starburst system. We emphasize, however, that the conditions measured by the data presented here apply to very local conditions in star-forming clouds, particularly the denser clumps within GMCs. That is why NGC 1569, in fact, does not appear far more extreme than it does.
Finally, we estimated the number of molecular clouds, their masses, and their sizes within the LWS beam in NGC 1156, NGC 1569, and IC 4662. We deduce the presence of a few clouds in each galaxy with masses much larger than typical Milky Way GMCs. These extraordinarily large clouds may also be necessary to form super star clusters, as NGC 1569 has recently done.
The irregular galaxies in our sample have star formation rates normalized to their size that are comparable to those in spirals. In fact, the star formation rates span the entire range observed in our spiral sample, including the high end of the range. However, the infrared properties of the irregulars are dominated by star-formation, perhaps more than is the case in spirals. This implies that the ISM beyond the regions and associated PDRs in irregulars is too faint to contribute measureably to the [*ISO*]{} observations. We know that the ISM in irregulars can be quite clumpy and that ionizing photons can travel large distances. But, the low contribution of the ISM beyond regions to the infrared may be a consequence of a lower stellar radiation field outside star-forming regions and reflect the general low surface brightness disk of irregulars. In galaxies with low dust column densities, without concentrations of hot stars to heat the dust and gas, the ISM becomes very hard to detect in PAH, small grain, and PDR emission features.
What does this imply about more normal irregulars? Most of the irregular galaxies in our sample are not representative of typical irregular galaxies. Several in our sample are very high surface brightness and one is a starburst. Most irregulars are lower in surface brightness and star formation activity. Unfortunately, we were not able to observe these galaxies with [*ISO*]{}. However, it is likely that more typical irregulars, if they could be observed, would also turn out to be dominated by star-forming regions since these would be the primary contributors to the FIR and mid-infrared emission. The irregulars can be seen as containing glowing islands of star-forming regions in a sea of otherwise relatively dim gas and stars.
We are grateful to P. Massey and C. Claver for the use of their optical images. This work was supported in part by [*ISO*]{} data analysis funding from the US National Aeronautics and Space Administration through the Jet Propulsion Laboratory (JPL) of the California Institute of Technology and in part by support to D.A.H. from the Lowell Research Fund and grant AST-9802193 from the National Science Foundation. The imaging would not have been possible without filters purchased through funds provided by a Small Research Grant from the American Astronomical Society, National Science Foundation grant AST-9022046, and grant 960355 from JPL. D.A.H. wishes to thank Lowell Observatory for the observing time on the Perkins 1.8 m telescope that produced the northern images and optical long-slit spectroscopy.
\[tabgal\]
\[tabisoobs\]
\[tabcam\]
\[tablws\]
\[tabiras\]
\[tabmodel\]
\[tabcloud\]
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[^1]: Visiting Astronomer, Cerro Tololo Interamerican Observatory, National Optical Astronomy Observatory, which is operated by the Association of Universities in Astronomy, Inc., under cooperative agreement with the National Science Foundation.
[^2]: Based on observations made with [*ISO*]{}, an ESA project with instruments funded by ESA Member States (especially the PI countries: France, Germany, the Netherlands, and the United Kingdom), and with the participation of ISAS and NASA.
[^3]: A division of the National Optical Astronomy Observatory, which is operated by the Association of Universities in Astronomy, Inc., under cooperative agreement with the National Science Foundation.
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abstract: 'We construct an X-ray spectral model for the clumpy torus in an active galactic nucleus (AGN) using Geant4, which includes the physical processes of the photoelectric effect, Compton scattering, Rayleigh scattering, $\gamma$ conversion, fluorescence line, and Auger process. Since the electrons in the torus are expected to be bounded instead of free, the deviation of the scattering cross section from the Klein-Nishina cross section has also been included, which changes the X-ray spectra by up to 25% below $10$ keV. We have investigated the effect of the clumpiness parameters on the reflection spectra and the strength of the fluorescent line Fe K$\alpha$. The volume filling factor of the clouds in the clumpy torus only slightly influences the reflection spectra, however, the total column density and the number of clouds along the line of sight significantly change the shapes and amplitudes of the reflection spectra. The effect of column density is similar to the case of a smooth torus, while a small number of clouds along the line of sight will smooth out the anisotropy of the reflection spectra and the fluorescent line Fe K$\alpha$. The smoothing effect is mild in the low column density case (${N_\mathrm{H}}=10^{23}$ cm$^{-2}$), whereas it is much more evident in the high column density case (${N_\mathrm{H}}=10^{25}$ cm$^{-2}$). Our model provides a quantitative tool for the spectral analysis of the clumpy torus. We suggest that the joint fits of the broad band spectral energy distributions of AGNs (from X-ray to infrared) should better constrain the structure of the torus.'
author:
- 'Yuan Liu (ÁõÔª) and Xiaobo Li (ÀîС²¨)'
title: 'An X-ray spectral model for clumpy tori in active galactic nuclei'
---
[GB]{}[gbsn]{}
Introduction
============
In the unified model of active galactic nuclei (AGNs), a dusty torus is proposed to account for the apparent difference between type 1 and 2 AGNs, i.e., the central source is obscured by the torus in type 2 AGNs but not in type 1 AGNs (Antonucci 1993; Urry & Padovani 1995). Despite the key ingredient of the unification model, the origin and structure of the torus are still not clear. An important debate about the torus structure is whether the material in a torus is smooth or clumpy (Feltre et al. 2012; Hönig 2013). Both models have their advantages and drawbacks. The torus absorbs high-energy photons from an accretion disk/corona and converts them into the infrared band. Therefore, it is possible to constrain the structure of a torus by its infrared spectrum. The smooth model predicts strong silicate features in the infrared band; however, the observed feature is much weaker than the model’s prediction (Laor & Draine 1993; Nenkova et al. 2002; Nikutta et al. 2009). The clumpy model can naturally explain the weakness of the silicate feature but cannot produce sufficient near-infrared emission from hot dust (Mor et al. 2009; Vignali et al. 2011). For a few nearby sources, the near-infrared interferometers can directly constrain the structure of the tori and indeed favor the clumpy model (Tristram et al. 2007). However, this method is limited by the low sensitivity of the infrared interferometer and cannot be applied to a large sample. By fitting the infrared energy spectral distribution with a specific model (smooth or clumpy), we can also obtain the parameters of the particular model. It has recently been claimed that the covering factors of type 2 AGNs are systematically larger than those of type 1 AGNs (Elitzur 2012).
Besides the infrared emission, the torus will also absorb and reflect the X-ray photons. Therefore, the X-ray spectrum can provide independent constraints on the structure of a torus. There are many works on the reflection spectra of different geometries. The commonly used model `pexrav` in *XSPEC* accounts for the reflection from a flat disk with infinite optical depth (Magdziarz & Zdziarski 1995). However, fluorescence lines are not self-consistently included in this model. The Gaussian lines should be manually added in the fit model of real X-ray spectra. This work is then improved by including the fluorescence line (e.g., `pexmon`), the ionization state of the reflection disk, and relativistic effects around a black hole (Nandra et al. 2007; Ross & Fabian 2005; Brenneman & Reynolds 2006). For the smooth torus, detailed X-ray models are already available (Ikeda et al. 2009; Murphy & Yaqoob 2009; Brightman & Nandra 2011), though the results are slightly different due to the discrepancies in geometry, cross section, and simulation methods used. Some works have also discussed the X-ray spectrum for clumpy media but do not dedicate to tori (Nandra & George 1994; Tatum et al. 2013). Yaqoob (2012) discussed the qualitative effects on the reflection spectrum and Fe K$\alpha$ line due to the clumpy structure.
It is necessary to utilize Monte Carlo simulations to deal with clumpy geometry. In this paper, we present detailed results of simulations on a clumpy torus. In Section 2, we will explain the simulation process and other assumptions about the clumpy torus. Then, the results of the reflection continuum and Fe K$\alpha$ are presented in Sections 3 and 4, respectively. In Section 5, we summarize our results and discuss the implications for observations. As the first part of a series of papers, we will only discuss the spectrum in this paper. The results of temporal response and polarization will be given in subsequent papers.
Simulation Method and assumptions
=================================
We utilized an object-oriented toolkit, Geant4 (version 4.9.4), to perform the simulations. Three classes are necessary to construct the simulation model in Geant4.
1 The geometry and material of the interaction region are described by the class derived from G4VUserDetectorConstruction.
2 The class derived from G4VUserPhysicsList is required to construct the particles and physical processes to be activated in the simulation.
3 The class derived from G4VUserPrimaryGeneratorAction will generate the primary events, including the type, energy, direction, and position of the initial particles.
After the three classes have been defined, Geant4 will treat the particles one by one and track the trajectories of primary particles and secondary particles step by step. These particles will participate in the physical processes activated in the simulation. To determine the position of the interaction point of a given physical process, Geant4 first calculates the mean free path $\lambda$ as the function of energy $$\label{cs}
\lambda(E)=(\sum_i[n_i\cdot\sigma(Z_i,E)])^{-1},$$ where $n_i$ is the number density of the $i$th element, $\sigma(Z_i,E)$ is the cross section per atom of the process, and $\sum_i[]$ means the sum of all elements of the torus. Then the number of the mean free paths the particle travels before the interaction point is randomly determined by $-\log(\eta)$, where $\eta$ is uniformly distributed in the range (0, 1). If the particle is absorbed or escapes from the world boundary, the tracking of it will be ended. The result of the relaxation of an exited atom is randomly determined by the atomic data adopted, e.g., fluorescent yields. The secondary particles (e.g., the Fe K$\alpha$ photons) are also tracked according to the method above (Agostinelli et al. 2003).
At each step, Geant4 will record the information of particles, e.g., kinetic energy, momentum, position, time, and physical process involved. Then the recorded information of every step can be used to select the particles of interest. In the simulations in this paper, photons that escape from the world boundary are selected to construct the X-ray spectra in different situations. In the following sections, we will discuss the assumptions in the three classes of our simulations.
Geometry and Constituents
-------------------------
For comparison, we also performed simulations of a smooth torus. The geometry of the smooth torus is shown in Figure \[fig1\] (left). The boundaries are defined by the inner radius $R_\mathrm{in}=0.1$ pc, the outer radius $R_\mathrm{out}=2.0$ pc, and the half-opening angle $\sigma=60^\circ$. We assume that the gas is uniformly distributed in the torus. For the clumpy torus, we use the same parameters ($R_\mathrm{in}$, $R_\mathrm{out}$, and $\sigma$) to define the envelope of the torus, within which numerous spherical clouds are uniformly distributed. In addition, we need parameters to describe the clumpiness of the torus. There can be different choices of the free parameters. In our simulation, we use the volume filling factor $\phi$, the number of clouds along the line of sight $N$, and total column density ${N_\mathrm{H}}$ as the input parameters. Other quantities, e.g., the radius of the cloud and the density of the gas in the cloud, can be derived from these input parameters. $\phi$ and $N$ determine the total number and the size of clouds and ${N_\mathrm{H}}$ further determines the density of the gas in clouds. Figure \[fig1\] (right) is the configuration of the clumpy torus. In the following simulations, we fix the parameters of the envelope, since we focus on the comparison between the smooth and clumpy cases. It is easy to change the parameters of the envelope in the subsequent simulations if necessary. We further assume that the sizes of the clouds are the same and the density of the gas in the cloud is uniform. These assumptions are surely simplified compared with the realistic tori of AGNs. However, the assumption “uniform distribution” is widely used in the previous simulations about the smooth torus. Hence, we follow this assumption and mainly focus on how the clumpiness influences the spectra. More complex and realistic distributions will be investigated in future simulations. We have included elements with the abundances from Anders & Grevesse (1989). The gas in the cloud is assumed to be cold, i.e., all atoms are in their ground states.
Physical Processes
------------------
We invoked the low-energy electromagnetic process in Geant4, which is valid in the energy range from 0.25 keV to 100 GeV. For photons, we considered the photoelectric effect, Compton scattering, Rayleigh scattering, and $\gamma$ conversion. Fluorescence and Auger processes were also loaded in the photoelectric effect. For electrons, ionization, bremsstrahlung, and multiple scattering were added in the process. The relevant cross sections and atomic data are adopted from EPDL97. In spite of slightly different cross sections of the photoelectric effect used in previous simulations of torus, the most important difference is the cross section of scatterings in our simulations. Since the electrons in a torus are bounded, the Klein-Nishina cross section is not appropriate (Hubbell et al. 1975). For the scattering by bounded electrons, the cross section is divided into two parts, Rayleigh scattering (coherent scattering, i.e., the atom is still in the ground state after the scattering) and Compton scattering (incoherent scattering, i.e., the atom is excited after the scattering). The incoherent scattering dominates high-energy band and tends to the Klein-Nishina cross section as the energy increasing; while the coherent scattering is more important at low-energy band and its cross section is proportional to the square of the total number of the electrons in one atom. As a result, both the total cross section and the angular distribution of scattered photons will deviate from the Klein-Nishina cross section. The importance of this correction further depends on the abundance of the gas. To evaluate the amplitude of the deviation in the spectra, we replaced the cross section of scattering in Geant4 with the Klein-Nishina cross section and then compared the spectra with that using the default cross section in Geant4. An illustration of this comparison is shown in Figure \[figcom\]. This correction is indeed more important for low-energy photons. The deviation is negligible above 10 keV and increases below 10 keV to 25% at 1 keV. Under the current abundance, helium is responsible for most of the deviation in the spectra. Therefore, this effect should be included in future simulations on the X-ray spectra of tori.
Incident Spectrum
-----------------
We adopted a single power law as the incident spectrum, i.e., $\mathrm{flux} \propto E^{-\Gamma}$ (1 keV $\leq E\leq$ 500 keV), where $\Gamma$ is the photon index and fixed at 1.8 throughout the simulations in this paper. The photons are isotropically emitted from the center $O$ (see Figure \[fig1\]), which is the location of the accretion disk/corona. The realistic X-ray spectrum of an AGN is usually more complex than a single power law and more components will induce curvature in the spectra. However, we intend to show the curvature produced by the torus itself and hence adopt this simple incident spectrum. It is convenient to include more complicated incident spectra in our simulation when we fit the observed spectrum.
Reflection spectra
==================
The ${N_\mathrm{H}}$ of the clumpy torus is actually the total column density if $N$ clouds are exactly aligned along our line of sight. However, for a set of clouds, the number of clouds in different directions is nearly a Poisson distribution (Nenkova et al. 2008) with a mean of $N$ and the portion of one particular cloud along the line of sight can be smaller than its diameter. As a result, the average column density is smaller than ${N_\mathrm{H}}$. Figure \[disn\] (left) shows the distribution of the number of clouds in some randomly selected directions for ${N_\mathrm{H}}=10^{24}$ cm$^{-2}$, $N=10$, and $\phi=0.01$ (the total number of clouds in the torus is about $1.4\times10^7$ under these parameters). The distribution of the cumulative column density (the sum of the column density of the intersected clouds in a particular direction) in different directions is shown in Figure \[disn\] (right). The mean column density (or the equivalent ${N_\mathrm{H}}$) is smaller than ${N_\mathrm{H}}$ by a factor of 0.66. If we reshape the spherical cloud into a cylinder with the same number density and with the axis pointing to the center, the height of the cylinder will be two-thirds of the diameter of the spherical cloud. This “geometry-average factor” is very close to the mean value in Figure \[disn\] (right). Therefore, we compare the results of the clumpy torus with those of the smooth torus with 0.66${N_\mathrm{H}}$. We should stress that this is not a unique method to calculate the “equivalent ${N_\mathrm{H}}$” and there is actually no exact equivalence between a smooth torus and a clumpy torus. We intend to present a more meaningful comparison, but the clumpy torus is intrinsically different from the smooth case, which is actually the motivation of this paper.
{width="0.49\linewidth"} {width="0.49\linewidth"}
For the smooth torus, the direct component (photons that escape from the torus without any interaction) of the transmitted spectrum is simply the incident spectrum weakened by the optical depth (determined by a single ${N_\mathrm{H}}$) due to photoelectric absorption and scattering along the line of sight. For the clumpy torus, if the size of the cloud is much larger than the compact corona of AGNs ($\lesssim$10 Schwarzschild radii), a single ${N_\mathrm{H}}$ for the direct component is still appropriate, which depends on the position of the cloud relative to the X-ray source (the corona). However, if our model is applied to a more extended X-ray source (i.e., the partial covering case), a geometry-averaged ${N_\mathrm{H}}$ should be applied to the direct component, which depends on the density distribution within the cloud and the brightness profile of the X-ray source. We will explore various possibilities in the comparison with the observed spectrum in future works. Then we will only discuss the reflection component, i.e., the scattered component, and the strength of the fluorescence line is presented in the next section.
In the following discussion, we divide the direction of the photons in the reflection spectra into 10 uniform bins according to ${\cos\theta}$ ($\theta$ is the angle between the direction of the photon and the $z$-axis in Figure \[fig1\]). The bin ${\cos\theta}=0-0.1$ and ${\cos\theta}=0.9-1.0$ are defined as the edge-on and face-on directions, respectively. In Figure \[refl\], we show the reflection spectra for different ${N_\mathrm{H}}$, $N$, $\phi$, and ${\cos\theta}$. For clarity, only the spectra of edge-on and face-on cases are shown. The results of the smooth torus are also plotted for comparison. Next, we discuss the effect of the three clumpiness parameters (${N_\mathrm{H}}$, $\phi$, and $N$). The general effect of ${N_\mathrm{H}}$ is similar to the smooth case, e.g., the Compton hump and the anisotropy of the reflection spectra become more evident with increasing ${N_\mathrm{H}}$. The curvature above 200 keV is due to the decrease of the cross section of Compton scattering. Since the column density of one cloud is solely determined by $N$ and ${N_\mathrm{H}}$, the filling factor $\phi$ only slightly impacts the spectra. However, the number of clouds along the line of sight significantly changes the column density of one cloud and further the shape of the spectra. As more low-energy photons can escape from the torus with a smaller $N$ (the photons scattered by the clouds in the far side of the torus can leak from “holes" in the near side of the torus), the reflection spectra become more isotropic. We show the distribution of photons in the 5-6 keV band as a function of $\cos\theta$ in Figure \[dis\_con\], where the curves are normalized at the minima. With increasing ${N_\mathrm{H}}$, more photons in the edge-on direction are absorbed in the smooth case; however, the anisotropy of the reflection spectra is significantly weakened in the case of ${N_\mathrm{H}}=10^{25}$ cm$^{-2}$ and $N=2$.
{width="0.33\linewidth"} {width="0.33\linewidth"} {width="0.33\linewidth"}
To better understand the effect of ${N_\mathrm{H}}$ and $N$ on the reflection spectra, we show the single and multi-scattering spectra separately in Figure \[sing\]. A larger ${N_\mathrm{H}}$ is helpful to suppress the fraction of single scattering, i.e., the photons have a higher probability of scattering with the clouds before escaping from the torus. However, the spectrum of single scattering still dominates at the lower energies; the multi-scattering spectrum is more important at higher energies since it will experience more absorption during the scatterings. The multi-scattering spectrum is more evident for larger $N$, as there are more interfaces to produce scatterings. We show the histogram of the number of scatterings in Figure \[sinn\] for ${N_\mathrm{H}}=10^{23}$ cm$^{-2}$ and $10^{25}$ cm$^{-2}$. Both the maximum number of scatterings and the fraction of multi-scatterings increase with increasing ${N_\mathrm{H}}$. In addition, since the geometry covering factor of the clumpy torus is $(1 - {e^{ - N}})\cos \sigma
$ (Nenkova et al. 2008), the covering factor of $N=2$ is smaller than that of $N=10$ by a factor of 0.86.
The strength of Fe K$\alpha$ line
=================================
The fluorescent line Fe K$\alpha$ is one of the most important lines in the X-ray band, which can reflect the structure and density of the torus. If the torus is optically thin to the photons of Fe K$\alpha$, the strength of Fe K$\alpha$ can be simply calculated by a linear relation of the properties of the torus, e.g., column density, covering factor, and the abundance of iron (Krolik & Kallman 1987). Moreover, the Fe K$\alpha$ photons are isotropically distributed. However, the optically thin approximation is not valid for Fe K$\alpha$ photons if the column density of the torus is larger than $2\times10^{22}$ cm$^{-2}$ (Yaqoob et al. 2010). In this case, the geometry will impact the distribution of Fe K$\alpha$ photons and numerical simulation is required to determine the luminosity of Fe K$\alpha$. We will investigate the relation between the anisotropy of Fe K$\alpha$ and the parameters of clumpiness. In the following discussion, we have included the scattered Fe K$\alpha$ photons (the so-called “Compton shoulder”) into the total flux of Fe K$\alpha$. In Figure \[fedis\], we plot the distribution of Fe K$\alpha$ photons as a function of ${\cos\theta}$, where the curves are normalized at the minima. Since the effect of $\phi$ is minor as shown in Figure \[refl\], we will only discuss the results with different ${N_\mathrm{H}}$ and $N$ but fix $\phi=0.01$. The anisotropy increases with increasing ${N_\mathrm{H}}$, which is the simple result of more absorption in the edge-on direction in the high column density case. For the same equivalent ${N_\mathrm{H}}$, the anisotropy of Fe K$\alpha$ in clumpy cases are weaker than the smooth case. More specifically, a smaller $N$ will further suppress the anisotropy of Fe K$\alpha$. This effect is much more significant in the ${N_\mathrm{H}}=25$ cm$^{-2}$ case, which is similar to the situation of the reflection spectrum. Due to the limited statistics of the current simulations, we can only investigate the strength of Fe K$\alpha$. However, it is possible to extend our simulation and discuss other fluorescence lines.
{width="0.33\linewidth"} {width="0.33\linewidth"} {width="0.33\linewidth"}
Summary and discussion
======================
To construct an X-ray spectral model for the clumpy torus in AGNs, we have performed simulations using an object-oriented toolkit Geant4, by which it is convenient to deal with complex geometry. Besides the necessary physical processes, e.g., photoelectric absorption, Compton scattering, and fluorescence lines, considered in previous simulations of tori, we have included corrections to the treatment of scattering by explicitly considering Rayleigh (coherent) and Compton (incoherent) scattering from bound, rather than free, electrons. This correction can induce a deviation on the X-ray spectra up to 25% at 1 keV, therefore we cannot ignore it in the simulation of neutral tori. There are indications from the widths of the Fe K$\alpha$ lines (Shu et al. 2010, 2011) that the location of the line-emitting material may be closer to the central engine than the traditional torus in some AGNs. Hence, the observed effects of the corrections may vary from AGN to AGN.
Different combinations of ${N_\mathrm{H}}$, $\phi$, and $N$ have been investigated. The filling factor only slightly changes the reflection spectra, while the number of clouds along the line of sight significantly influences the spectra. If there are more clouds, i.e., $N=10$, the result is similar to the smooth case; while for the extreme case (${N_\mathrm{H}}=10^{25}$ cm$^{-2}$ and $N=2$), the shapes of the reflection spectra in different directions are quite similar expect for the somewhat lower amplitude in the edge-on direction, i.e., the reflection spectra become more isotropic. Therefore, if strong reflection components are found in the observed spectra of type 2 AGNs, the “clumpy" scenario could be invoked to explain the spectra. In this case, the quantitative spectral model presented in this paper is necessary for the measurement of the structure of the clumpy torus.
Besides the reflection continuum, the anisotropy of the Fe K$\alpha$ line will also be impacted by clumpiness. The situation is quite similar to the reflection continuum. In the low column density case (${N_\mathrm{H}}=10^{23}$ cm$^{-2}$), the distribution of Fe K$\alpha$ photons is nearly isotropic and only slightly changed by the parameters of clumpiness, while for the high column density case (${N_\mathrm{H}}=10^{25}$ cm$^{-2}$), a smaller $N$ significantly degrades the anisotropy of Fe K$\alpha$ photons. The strength of Fe K$\alpha$ has already been investigated in previous simulations of smooth tori and found to be anisotropic (though the result is expressed in the equivalent width of Fe K$\alpha$). The clumpy torus can further smooth out the anisotropy of the Fe K$\alpha$, which depends on the column density of the clouds. This result is similar to the explanation of the weakness of the silicate feature in the infrared spectra of AGNs (Nenkova et al. 2002).
In the observational aspect, since the X-ray continua of AGNs can be contaminated by other components not related to the torus, the luminosity of the Fe K$\alpha$ line will provide independent evidence of the structure of the torus. For example, if there is no significant difference between the luminosities of Fe K$\alpha$ lines in type 1 and 2 (${N_\mathrm{H}}>10^{23}$ cm$^{-2}$) AGNs, the “clumpy" torus is required according to the curves in Figure \[fedis\] or we should modify the unified model of AGNs as claimed by Elitzur (2012). The current observations have already provided some clues but are not conclusive. Liu & Wang (2010) found that the luminosities of the narrow Fe K$\alpha$ lines in Compton-thin and Compton-thick type 2 AGNs are weaker than those in type 1 AGNs by a factor of 2.9 and 5.6, respectively. This difference is broadly consistent with the results for a smooth torus. However, from a smaller sample from *Chandra HETG*, Shu et al. (2011) found the Fe K$\alpha$ line flux of type 2 AGNs is only marginally lower than that of type 2 AGNs, which will require the smoothing effect of a clumpy torus since the observed column density of Compton-thick AGN is already larger than $10^{24}$ cm$^{-2}$. A more complete sample of the luminosity of Fe K$\alpha$ should be helpful in determining the geometry of a torus and the curves presented in Figure \[fedis\] will further constrain the parameters.
In principle, it is possible to combine the infrared spectral energy distribution and X-ray spectra to better constrain the structure of tori. However, it should be cautioned that the gas in a torus is only sensitive to X-ray photons but the dust can also absorb optical photons. Therefore, if the dust-to-gas ratio is not uniform in a torus (which is likely to be the case due to the temperature gradient in the torus), the structure of the torus probed by the X-ray photons can be different from that obtained from the infrared spectral energy distribution. We only present the X-ray spectral model for clumpy tori here, and the details of the comparison between the structure from X-ray and infrared bands will be presented in future works. The results from temporal and polarization observations should be further helpful to break the degeneracy of the clumpiness parameters of the torus (Hönig & Kishimoto 2010; Ramos Almeida et al. 2011).
The authors thank the referee for useful comments which clarified the paper. This work is supported by 973 Program of China under grant 2009CB824800, and by the National Natural Science Foundation of China under grant Nos. 11103019, 11133002, and 11103022.
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abstract: 'In the past years, large particle-physics experiments have shown that muon rate variations detected in underground laboratories are sensitive to regional, middle-atmosphere temperature variations. Potential applications include tracking short-term atmosphere dynamics, such as Sudden Stratospheric Warmings. We report here that such sensitivity is not only limited to large surface detectors under high-opacity conditions. We use a portable muon detector conceived for muon tomography for geophysical applications and we study muon rate variations observed over one year of measurements at the Mont Terri Underground Rock Laboratory, Switzerland (opacity of $\sim 700$ meter water equivalent). We observe a direct correlation between middle-atmosphere seasonal temperature variations and muon rate. Muon rate variations are also sensitive to the abnormal atmosphere heating in January-February 2017, associated to a Sudden Stratospheric Warming. Estimates of the effective temperature coefficient for our particular case agree with theoretical models and with those calculated from large neutrino experiments under comparable conditions. Thus, portable muon detectors may be useful to 1) study seasonal and short-term middle atmosphere dynamics, especially in locations where data is lacking such as mid-latitudes; and 2) improve the calibration of the effective temperature coefficient for different opacity conditions. Furthermore, we highlight the importance of assessing the impact of temperature on muon rate variations when considering geophysical applications. Depending on latitude and opacity conditions, this effect may be large enough to hide subsurface density variations due to changes in groundwater content, and should therefore be removed from the time-series.'
title: 'Middle-atmosphere dynamics observed with a portable muon detector'
---
We report muon rate variations associated to temperature changes in the middle atmosphere observed with a portable muon detector
The effect is significant both for seasonal and short-term temperature variations, even under low-opacity conditions at mid-latitudes
We highlight potential applications on atmosphere dynamics and the need to account for these phenomena in geophysical applications\
Introduction
============
First observed in 1952 using radiosonde measurements [@scherhag52], Sudden Stratospheric Warmings (SSWs) are extreme wintertime circulation anomalies that produce a rapid rise in temperature in the mid to upper polar stratosphere (30-50 km). SSW effects on middle-atmosphere dynamics have lifetimes of approximately 80 days [@limpasuvan2004life]. They are the clearest and strongest manifestation of dynamic coupling throughout the whole atmosphere-ocean system [@o2014effects; @goncharenko2010unexpected; @liu2002study]. Following a major SSW, the high altitude winds reverse to flow westward instead of their usual eastward direction. This reversal often results in dramatic surface temperature reductions in mid-latitudes, particularly in Europe, which suggests the possibility of monitoring the stratosphere for predicting extreme tropospheric weather [@thompson2002stratospheric]. The frequency of SSWs may increase due to global warming [@schimanke2013variability; @kang2017more]. While many studies have focused on the characterization of SSWs through observation and modeling dynamics at high latitude regions, observation studies at mid-latitudes are rare and could be crucial to better understand the phenomena [@yuan2012wind; @sox2016connection].
Cosmic muons represent the largest proportion of charged particles reaching the surface of the Earth, yielding a flux of $\sim$ 70 m${}^{-2}$s${}^{-1}$sr${}^{-1}$ for particles above 1 GeV [@PDG18]. They are a product of the primary cosmic rays interaction with the atmosphere, which produces short-lived mesons, in particular, charged pions and kaons. These particles decay into muons that easily penetrate the atmosphere and may reach the surface of the Earth. The flux of muons decreases as muons travel through an increasing amount of matter. Thus, only the most energetic muons can reach underground detectors [@gaisser2016cosmic]. The muon production process requires that the parent mesons did not undergo destructive interactions with the propagating medium before they decay [@grashorn2010atmospheric]. Thus, changes in the atmospheric properties, in particular in its density, may have large impacts on the muon flux measured at ground level, either by affecting the parent mesons survival probabilities before decay or by affecting the rate of absorption of the muons themselves along their path down from their production level.
An increase in the atmospheric temperature lowers the atmospheric density. Temperature changes in the atmosphere may therefore affect the production of muons [@gaisser2016cosmic]. The decrease in atmospheric density increases the mean free path of the mesons and therefore their decay probability, thus increasing the muon flux. The effect is more important for high-energy muons, which result from high-energy mesons with larger lifetime due to time dilation and therefore with longer paths in the atmosphere. This increases their interaction probability before decay [@grashorn2010atmospheric], thus one expects high-energy muons to be more sensitive to temperature changes. The opacity is the integrated density along a travel path. It is used to quantify the amount of matter encountered by the muons and is generally expressed in meter water equivalent (mwe). Detectors in high-opacity conditions are more likely to register the effects of temperature variations in the atmosphere. Notice that the low-energy muons may also be affected by temperature changes because their own interaction probability with the atmosphere along their path down to the Earth depends on the atmospheric density. Indeed, this effect has been observed in low opacity conditions [e.g. @jourde2016monitoring], but is not relevant for detectors deeper than 50 mwe [@ambrosio97]. The variations in the cosmic muon flux caused by atmospheric temperature changes can be treated in terms of an effective temperature [@barrett52; @ambrosio97]. This effective temperature is a weighted average of the atmosphere’s temperature profile, with weights related to the altitudes where muons are produced [@grashorn2010atmospheric].
Modulation of the cosmic muon flux produced by seasonal variations in the atmospheric temperature have been reported for large detectors (AMANDA: @bouchta1999seasonal, Borexino: @agostini2019modulations, Daya Bay: @an2018seasonal, Double Chooz: @abrahao2017cosmic, GERDA: @GERDA16, IceCube: @desiati2011seasonal, LVD: @vigorito2017underground, MACRO: @ambrosio97, MINOS: @adamson2014observation [@adamson2010observation], OPERA: @agafonova2018measurement). @osprey09 and @agostini2019modulations also report that measured muon rates are sensitive to short-term variations (day scale) in the thermal state of the atmosphere, such as the occurrence of SSWs. [@agafonova2018measurement] observed short-term, non-seasonal variations in latitudes as low as 42$^\circ$ N, in Italy.
The previously mentioned studies highlight the potential of muon measurements to characterize and monitor middle atmosphere dynamics. However, all these studies were conducted by large-scale, general-purpose particle detectors, specifically built for neutrino and high-energy particle experiments. Most of them were placed hundreds of meters underground, which improves data sensitivity to atmospheric effects by filtering out low-energy muons. The detection surface of these systems are huge compared to portable ones, which are used for geoscience applications such as characterizing the density structure of volcanoes [e.g. @rosas2017three]. Recently, muon rate variations following the passage of a thundercloud were reported by @hariharan2019measurement using a relatively large detector (6$\times 6$$\times 2$ m${}^3$). To the best of our knowledge, no experiment has reported the sensitivity of portable muon detectors to middle atmosphere dynamics, especially under relatively low opacity conditions.
In this paper, we study seasonal and short-term variations in the muon rate observed with a portable muon detector installed at the Mont Terri Underground Rock Laboratory (Switzerland, 47.4$^\circ$ N). We first present our detector and the general conditions under which the measurements were taken. We then analyze the variations observed and compare them to atmospheric temperature and middle-atmosphere dynamics data. Finally, we discuss the implications of our observations both for the atmospheric science and geophysics communities, the latter aiming to characterize density variations in the subsurface with muon data.
The muon detector
=================
Our portable muon detector was conceived for geoscience applications by the DIAPHANE project [e.g., @marteau2012muons; @marteau2017diaphane]. It is equipped with 3 plastic scintillator matrices of 80 cm width composed by $N_x=N_y=16$ scintillators bars, in the horizontal and vertical directions, whose interceptions define $16 \times 16$ pixels of $5 \times 5$ cm${}^2$. When a muon passes through the 3 matrices (i.e., an “event” is registered), 3 hits are recorded in time coincidence, with a resolution better than 1 ns [@marteau2014telescope], enabling us to reconstruct its trajectory from the sets of pixels fired in each matrix. We apply a selection based on the goodness of the reconstructed trajectory in order to filter out random coincidences, i.e, three coincident fired pixels that do not align. If the reconstructed trajectories using two consecutive matrices differ by more than one pixel, in either the horizontal or the vertical direction, the event is discarded. More details on the hit selection and the technique applied to determine the propagation directions of muons through the detector matrices can be found in @jourde2015thesis and in @marteau2014telescope. The distance between the front and rear matrices is set to 100 cm for this study (Fig. \[fig:telescope\]a). Because of the large volume of rock studied compared to the detector size, we admit a point-like approximation of the detector [@lesparre2010geophysical]. With this approximation, given that two points are sufficient to uniquely determine a direction, events whose pair of pixels in the front and the rear matrices share the same relative direction are considered to correspond to the same trajectory. This yields a total of $(2N_x-1) \times (2N_y-1) = 961$ axes of observation studied (represented in Fig. \[fig:telescope\]b).
The passage of muons is detected with wave-length shifting optical fibers that transport the photons generated by the scintillators to the photomultiplier, where they are detected based on a time coincidence logic. The optoelectronic chain has been developed from high-energy particle experiments on the concept of the autonomous, Ethernet-capable, low power, smart sensors [@marteau2014telescope]. In order to support strenuous field conditions, besides being sensitive the detector is also robust, modular and transportable [@lesparre2012design]. In this experiment, the muon detector was deployed in the Mont Terri Underground Rock Laboratory (URL) and acquired data for 382 days between October 2016 and February 2018. The minimum and the maximum amount of rock traversed by muons registered by the detector are of approximately 200 and 500 m, respectively. Prior to the underground measurements, a calibration experiment was performed by measuring the open-sky muon flux at the zenith, from which we register a total acceptance of 1385 cm${}^2$ sr for our data set [@lesparre2010geophysical].
![a) The muon telescope deployed in the Mont Terri URL. b) Telescope’s position (blue) and axes of observation (red), along with the topography.[]{data-label="fig:telescope"}](detectorAandB_axesOff.png){width="1\linewidth"}
Methodology
===========
Our data set consists of a list of muon detections called “events”. Each event is characterized by the arrival time and the direction of the particle (possible directions shown in Fig. \[fig:telescope\]b). From these data, we compute the average cosmic muon rate, $R$, using a 30-day width Hamming moving average window [@hamming1998digital]. In order to increase the signal to noise ratio and, therefore, to improve the statistics in our analysis, we merge the signals from all the directions together [e.g. @jourde2016muon]. Such a merging is done exclusively to compute $R$.
Seasonal variations in $R$, caused by the temperature changes in the atmosphere, can be treated in terms of an effective temperature [@barrett52], $T_{\text{eff}}$: $$\dfrac{\Delta R}{\left\langle R \right\rangle } = \alpha_\text{T} \dfrac{\Delta T_{\text{eff}}}{\left\langle T_{\text{eff}} \right\rangle} \ ,
\label{eq:relationship}$$ where $\alpha_\text{T}$ is the effective temperature coefficient, $\left\langle R \right\rangle$ is the mean muon rate and $\left\langle T_{\text{eff}} \right\rangle$ is the mean effective temperature. $T_{\text{eff}}$ is defined as the temperature of an isothermal atmosphere that produces the same meson intensities as the actual atmosphere. Thus, it is related to the atmosphere’s temperature profile, and it is associated to the altitudes where observed muons are produced. We use the parametrization given by @grashorn2010atmospheric:
$$\begin{aligned}
T_{\text{eff}} = \dfrac{\int_{0}^{\infty} W(X) T(X) dX}{\int_{0}^{\infty} W(X) dX} \ ,
\label{eq:Teff}\end{aligned}$$
where the temperature, $T(X)$, is measured as a function of atmospheric depth, $X$. The weights, $W(X)$, contain the contribution of each atmospheric depth to the overall muon production. These weights depend on the threshold energy $E_{\text{th}}$, that is, the minimum energy required for a muon to survive a particular opacity in order to reach the underground detector. Since $T(X)$ is measured at discrete levels of $X$, we perform a numerical integration based on a quadratic interpolation between temperature measurements to obtain $T_{\text{eff}}$.
The effective temperature will be different for different zenith angles. To compare $T_\text{eff}$ variations to our measured muon rates, we need to account for this dependence. Following @adamson2014observation, we bin the zenith angle distribution and calculate a weighted effective temperature, $T_\text{eff}^\text{weight}$, as:
$$T_\text{eff}^\text{weight} = \sum_{i=1}^{M} F_i \cdot T_\text{eff}(\theta_i) \ ,$$
where $M$ is the number of zenith-angle bins, $T_\text{eff}(\theta_i)$ is the effective temperature in bin $i$ and $F_i$ is the fraction of muons observed in that bin. The formula for $T_\text{eff}(\theta_i)$ is similar to Eq. (\[eq:Teff\]), but the atmospheric depth is replaced by $X/\cos\theta$ and $E_{\text{th}}$ is calculated for each zenith angle as well. From now on, we will refer to $T_\text{eff}^\text{weight}$ as $T_\text{eff}$. These values are calculated four times a day and then day-averaged, and the resulting standard deviation is used as an uncertainty estimate of the effective temperature daily mean value. Thus, a representative value of effective temperature is calculated for each day, which fully accounts for the particular setup of our experiment.
The goodness of fit of the linear relationship in Eq. (\[eq:relationship\]) can be quantified by the Pearson correlation coefficient $r$. This parameter is equal to $\pm 1$ for a full positive/negative linear correlation, respectively, and 0 for no correlation. We perform a linear regression between the relative muon rate and effective temperature variations using Monte Carlo simulations. In this way, we can account for error bars in both variables and compute the uncertainty of the fitted parameters. Following @adamson2010observation, the intercept is fixed at zero and the slope of the linear fit is the effective temperature coefficient, $\alpha_\text{T}$. To evaluate the effects of systematic uncertainties we modify $\left\langle T_{\text{eff}} \right\rangle$ and the parameters involved in the computation of $T_{\text{eff}}$ [i.e. the twelve input parameters in $W(X)$ , c.f. @adamson2010observation] and recalculate the effective temperature coefficient, $\alpha_{\text{T}}$. These systematic errors are added in quadrature to the statistical error obtained from the linear fit in orden to obtain the experimental value of $\alpha_{\text{T}}$.
We also use Monte Carlo simulations to determine the theoretical expected value of the effective temperature coefficient, $\alpha_\text{T}^{\text{theory}}$, in order to compare it with the experimental one. Muon energy, $E_\mu$, and zenithal angle, $\theta$, are randomly sampled from the differential muon spectrum given by @gaisser2016cosmic and corrected for altitude according to @hebbeker2002compilation. Then, the muon is randomly assigned an azimuthal angle, $\phi$, according to a uniform probability distribution. The overburden opacity in the Mont Terri URL is determined for each combination of ($\phi$, $\theta$) from our muon data set, together with the corresponding $E_{\text{th}}$ [@PDG18]. We continue the Monte Carlo sampling until we obtain 10,000 successful events that satisfy $E_\mu >
E_{\text{th}}$, for which we compute the $\alpha_\text{T}^{\text{theory}}$ distribution using the expression derived by @grashorn2010atmospheric. Next, we determine the value of $\alpha_\text{T}^{\text{theory}}$ and its uncertainty as the mean and standard deviation of the distribution, respectively. The systematic uncertainty is the one reported by @adamson2014observation.
We look for the ocurrence of SSWs during the acquisition period using the definition of a major SSW given by @charlton2007. A major mid-winter warming is considered to occur when the zonal mean zonal wind at 60$^\circ$N and 10 hPa become easterly during winter. The first day on which this condition is met is defined as the central date of the warming. The zonal mean zonal wind is the average east-west (zonal) wind speed along a latitude circle. To ensure that only major mid-winter warmings are identified, cases where the zonal mean zonal wind does not reverse back to westerly for at least 2 weeks prior to their seasonal reversal to easterly in spring are assumed to be final warmings, and as such are discarded. SSWs typically manifest as a displacement or a splitting of the polar vortex [@charlton2007], a cyclone residing on both of the Earth’s poles that goes from the mid-troposphere into the stratosphere.
Results {#sec:Results}
=======
Based on 382 days of data, the average daily rate of cosmic muons in the Mont Terri URL is of $(800 \pm 10)$ d$^{-1}$, calculated by counting all the muons detected each day no matter their direction or the altitude at which they were produced. We also compute an average muon rate for each axis of observation, which we use to estimate the corresponding opacity values. Minimum and maximum opacities are of approximately 500 and 1500 mwe, respectively, while the average opacity considering all possible directions is of $(700 \pm 160)$ mwe. The cosmic muon rate presents significant variations in time (Fig. \[fig:absoluteFlux\]). Maximum rate values occur close to the summer periods while minimum rate values occur during winter times.
We use the ERA5 data set offered by the European Centre for Medium-range Weather Forecast (ECMWF), which is a climate reanalysis data set produced using 4D-Var data assimilation [@era5]. Temperature data consist of interpolated (${0.25}^{\circ}$ by ${0.25}^{\circ}$) globally gridded data on 37 atmospheric pressure levels from 0 to 1000 hPa, listed four times a day (00:00 h, 06:00 h, 12:00 h and 18:00 h). From this data set, we interpolate the temperature profiles at Mont Terri URL location. In Fig. \[fig:TandW\] we present the typical atmospheric temperature profiles at Mont Terri for summer, winter and a year average over the analysis period. We also display in the same plot the corresponding normalized weighting coefficients $W$ as a function of pressure levels, used to compute $T_\text{eff}$. The largest temperature changes occur above $\sim$16 km, where the weighting coefficients are more significant. The effective temperatures corresponding to the average curves and $\theta = 0^{\circ}$ are given by $T_\text{eff}^{\text{year}}=(217\pm1)$ K, $T_\text{eff}^{\text{summer}}=(225\pm1)$ K and $T_\text{eff}^{\text{winter}}=(214\pm1)$ K. There is thus a difference of $\sim$10 K between typical summer and winter conditions.
![Average cosmic muon rate as a function of time, computed using a 30-day width Hamming moving average window. The colored surface delimits the 95% confidence interval. Gray bars indicate periods where the acquisition was interrupted for work in the Mont Terri URL.[]{data-label="fig:absoluteFlux"}](absoluteFlux_V3.png){width="1\linewidth"}
![Atmospheric temperature profiles (solid lines) above the Mont Terri site, and weighting coefficients (dashed line) used to calculate $T_\text{eff}$, as a function of pressure level and altitude. The dots represent the 37 pressure levels for which the temperature data sets are provided by the ECMWF. The right vertical axis represents approximate altitudes corresponding to the pressure levels on the left vertical axis. The summer average temperature (solid red line) and the winter average temperature (solid blue line) are computed considering a period of 1.5 months in each season during 2017. The colored surfaces represent the $\pm 1$ standard deviation in each curve. The effective temperatures of each profile are: , $T_\text{eff}^{\text{summer}}=(225\pm1)$ K and $T_\text{eff}^{\text{winter}}=(214\pm1)$ K. []{data-label="fig:TandW"}](T_and_W.png){width="1\linewidth"}
We compare the variations in the muon rate to the variations in the effective temperature in Fig. \[fig:dfm\] in terms of relative variations (see Eq. \[eq:relationship\]). For consistency, we also apply a Hamming moving average window of 30 days to the $T_\text{eff}$ time series. The two average curves evolve similarly in time. Indeed, the Pearson correlation coefficient between the deviation from mean of the average muon rate and that of average effective temperature yield a value of 0.81. We compute a linear fit between the two data sets (see Methodology), which yields an effective temperature coefficient of $\alpha_{\text{T}} = 0.68 \pm
0.03_{stat} \pm 0.01_{syst}$, with $\chi^2/\text{NDF} = 414/381$ being the reduced $\chi^2$ of the fit (Fig. \[fig:alpha\]). The largest contribution to the systematic error in $\alpha_{\text{T}}$ comes from the $\pm 0.06$ uncertainty in the meson production ratio [@barr2006uncertainties], the $\pm 0.31$ K uncertainty in the mean effective temperature [@adamson2010observation] and the $\pm
0.026$ TeV uncertainty in $E_\text{th}$, which results from the distribution of opacities along the axes of observation. To discard possible systematic biases, we also performed a linear fit allowing for a non-zero y intercept. The fit resulted in an estimated value of zero within one standard deviation uncertainty for this intercept, and a slightly lower value of $\alpha_{\text{T}} = 0.67 \pm
0.03_{stat} \pm 0.01_{syst}$ for the effective temperature coefficient.
![Daily percent deviations from the mean of the average cosmic muon rate, the daily effective temperature, and the average effective temperature computed using a 30 days width Hamming moving average window. The colored surfaces delimit the 95% confidence interval associated to each curve. The inset displays a zoom around the period of time in which a major SSW is detected.[]{data-label="fig:dfm"}](DeviationFromMean_percentage_alphaWeighted_wInterpolation_inset_V6.png){width="1\linewidth"}
![Average cosmic muon rate relative variation versus average effective temperature relative variation, fitted with a line with the *y*-intercept fixed at 0. The resulting slope is $\alpha_T = 0.68 \pm 0.03_{\text{stat}} \pm 0.01_{\text{syst}}$ and is represented with a red line. The blue line represents the theoretical expected value of $\alpha_{\text{T}}^{theory} = 0.65 \pm 0.02_{stat} \pm 0.03_{syst}$. The dotted lines represent the uncertainty of each one of the values.[]{data-label="fig:alpha"}](alpha_weighted_M10_std_wInterpolation_wTheoretical.png){width="1\linewidth"}
The theoretical expected value was found to be $\alpha_{\text{T}}^{theory} = 0.65 \pm 0.02_{stat} \pm 0.03_{syst}$. Thus, the experimentally estimated value is consistent with the theoretical one within one standard deviation. In Fig. \[fig:expValues\] we present our estimated value of $\alpha_{\text{T}}$ along with a theoretical model accounting for pions and kaons [@agafonova2018measurement], and estimates from other experiments. Our estimate is consistent with the one obtained by @an2018seasonal in similar opacity conditions, and with the theoretical model.
![Experimental values of the effective temperature coefficient as a function of . The red dot represents the present study. The continuous black line represents a theoretical model. The insert plot show the experiments performed at the underground Gran Sasso Laboratory. Figure adapted from @agafonova2018measurement[]{data-label="fig:expValues"}](experimentalValues_V4.png){width=".8\linewidth"}
Taking a closer look at Fig. \[fig:dfm\], we can see that an anomalous increase in the effective temperature occurs between January and February 2017. The same anomalous behavior can be observed in the muon rate (see inset in Fig. \[fig:dfm\]). We used the [@charlton2007] definition and the Modern-Era Retrospective analysis for Research and Applications, Version 2 (MERRA-2), produced by the Goddard Earth Observing System Data Assimilation System (GEOS DAS) [@gelaro2017modern] to determine if a major SSW occurred during this time period. We found that a major SSW took place during winter 2016-2017, with February 1 as the central date of the warming. In a few days, it increased the zonal mean temperature in the polar region by more than 20 K (Fig. \[fig:algorithm\] a).
Finally, we analyzed changes produced by the SSW using Ertel’s potential vorticity [@matthewman2009new]. This parameter quantifies the location, size, and shape of the winter polar vortex. Figure \[fig:ssw\] shows the spatial distribution of Ertel’s potential vorticity at the 850 K potential temperature surface ($\sim 10$ hPa, $\sim 32$ km) for 3 different days, which are representative of the changes provoked. The figure also shows the effective temperature spatial distribution during these 3 days. On January 1 (Fig. \[fig:ssw\] a) the vorticity and temperature exhibit “typical” winter conditions: the polar vortex is centered on the Pole, together with the minimum effective temperature. On January 17, a reshaping on the polar vortex can be already observed. It is at this moment also that the largest effective temperature anomaly occurs in the Mont Terri region (Fig. \[fig:ssw\] b). On February 2, that is, one day after the event can be properly classified as a major SSW due to the reversal of the zonal mean zonal wind (see Fig. \[fig:algorithm\] b), the polar vortex shape is still anomalous with the “comma” shaped maximum of potential vorticity now closer to the Mont Terri URL (Fig. \[fig:ssw\] c). At the same time, the effective temperature in the Mont Terri region has decreased to values similar to those in January 1.
![GEOS DAS MERRA-2 data used to define SSW events. a) zonal mean temperatures averaged over $60^{\circ}$N-$90^{\circ}$N. b) zonal mean zonal wind at $60^{\circ}$N. The red curve denotes values for the 2016-2017 period and the thick black curve corresponds to climatological values averaged from 1978 to 2018. The vertical blue lines reference a major SSW for that winter.[]{data-label="fig:algorithm"}](temperature_and_wind_2016-2017.png){width="100.00000%"}
![Potential vorticity at the 850 K potential temperature surface (top) and effective temperature (bottom) for January 1, January 17 and February 2, 2017, derived from the ECMWF data set. The maps are centered on the North Pole and the location of the Mont Terri Underground Laboratory ($47.38^{\circ}$N, $7.17^{\circ}$E), close to the town of Saint-Ursanne, Switzerland, is represented with a star. 1 PVU = $10^{-6} \ \text{K} \
\text{m}^2 \ \text{Kg}^{-1} \ \text{s}^{-1}$.[]{data-label="fig:ssw"}](SSW_sequence_landscape_V2.png){width="1\linewidth"}
Discussion
==========
After a year of continous muon measurements with a portable muon detector under relatively low-opacity conditions, we found that changes in the thermal state of the atmosphere represent the largest cause of muon rate variations. The correlation between these variables was first suggested by a simple comparison of the relative variation time-series. Then, it was confirmed by the large correlation coefficient (0.81), and by the fitted effective temperature coefficient, which is in agreement with the theoretical value predicted for our particular opacity and zenith angle conditions. Furthermore, our experiment was by chance performed under similar opacity conditions to the Daya Bay detector, an established underground muon detector especially built for neutrino experiments [@an2018seasonal]. Its corresponding estimate of the effective temperature coefficient is also in agreement with ours (Fig. \[fig:expValues\]).
Our muon detector is sensitive to both seasonal and short-term temperature variations. The regional thermal anomaly reaching its maximum around January 17, 2017 (Fig. \[fig:dfm\]), is coincident with the polar vortex changing its shape from a normal pole-centered circle to a displaced “comma shaped" one (Fig. \[fig:ssw\]). This is a typical feature of a SSW [@ONeill2003encyclopedia]. Furthermore, the criteria by [@charlton2007] for declaring a major SSW is accomplished 15 days later. The time difference can be potentially explained by the zonally-averaged wind criteria used to define major SSWs, against the local character of the temperature variations affecting the production of high-energy muons.
Under much higher opacity conditions (3,800 in mwe, i.e., more than 5 times the Mont Terri URL opacity), the large muon detector of the Borexino experiment, Gran Sasso, Italy, also reported muon rate variations related to this SSW in 2017 [@agostini2019modulations]. Given the large opacity, most of the muons completely loose their energy before reaching the detector. Thus, only high-energy muons resulting from the decay of high-energy parent mesons are detected. As explained by [@grashorn2010atmospheric], high-energy mesons are most sensitive to middle-atmosphere temperature variations due to their relatively longer lifetime, and thus a higher probability of interacting with the atmosphere before decaying. This results in a higher sensitivity to temperature variations, which translates into a larger effective temperature coefficient (see Fig. \[fig:expValues\]). Despite being in less advantageous conditions in terms of detector acceptance and tunnel depth, our portable muon detector was also able to detect these short-term effect (15-days) directly linked to middle-atmosphere dynamics (Fig. \[fig:dfm\]).
Compared to lidar measurements, which can obtain temperature profiles over tens of kilometers in altitude but have very narrow global coverage (only as wide as the laser beam), muon detectors naturally provide integrated measurements in altitude, and a larger horizontal coverage. Our results therefore imply that small and affordable muon detectors could be used to study middle-atmosphere temperature variations without resorting to, for example, expensive lidar systems. Besides being transportable, the advantage is that no high-opacity conditions are needed. A minimum opacity of 50 mwe would be required to filter out the temperature-dependent lowest-energy muons [@grashorn2010atmospheric]. Besides being temperature dependent, low-energy muons can also be influenced by other phenomena such as atmospheric pressure variations [@jourde2016monitoring], which is why we consider optimal to remove them. However, open-sky conditions may also reveal new insights into atmospheric phenomena (e.g., [@hariharan2019measurement]) and more experimental studies are needed to better understand the limits of the methodology. Thus, detectors could be installed in any buried facility with access to electrical power and real-time data transmission, for example with a wi-fi network., such as road tunnels. In Europe, many underground research facilities exist in this condition (e.g. Mont Terri UL in Switzerland, 47.4$^{\circ}$N; the LSBB UL in France, 43.9$^{\circ}$N; Canfranc UL in Spain, 42.7$^{\circ}$N). These experiments could be crucial to fill the current data gap related to middle-atmospheric dynamics, in particular the study of temperature anomalies associated to SSW in mid-latitudes [@sox2016connection]. Furthermore, the technique may be used to study similar phenomena in the Southern Hemisphere.
The effective atmospheric temperature to which the muon rate is sensitive is a weighted average of a temperature profile from 0 to 50 km, with increasingly significant weights at higher altitudes [@grashorn2010atmospheric]. Indeed, 70 $\%$ of the total weights are given between 50 and 26 km, 90 $\%$ between 50 and 18 km and 95 $\%$ between 50 and 15 km (see Fig. \[fig:TandW\]). Thus, muon rate variations are mostly sensitive to temperature variations in the high stratosphere. Muon measurements can therefore complement lidar mesospheric studies (e.g., @sox2016connection [@yuan2012wind]). In terms of the spatial support, in the configuration used for this experiment (see Section 2), the total angular aperture of the detector is of approximately $\pm 40^{\circ}$, but more than 95% of the muons are registered within an aperture of $\pm 30^{\circ}$. At 50 km, this represents a surface of 50$\times 50$ km${}^2$. Therefore, muon measurements may be used to sample more regional atmospheric behavior.
Besides the potential applications to atmospheric studies, portable muon detectors may be used to precisely calibrate the effective temperature curve (Fig. \[fig:expValues\]). The experimental setups used to estimate these values, so far, are concentrated in either high or low-opacity conditions, whereas with our approach we could sample the curve rather uniformly, even in the same tunnel by varying the orientation of our detector and thus the opacity and zenith angle conditions.
Our findings have direct implications for applications aiming to characterize density variations in the subsurface (e.g. @jourde2016muon). Indeed, synchronous tracking of the open-sky muon rate while performing a continuous imaging of a geological body (e.g. density monitoring) may not be sufficient to characterize the influence of high-atmosphere temperature variations since the relative effect on the total amount of muons registered increases with opacity. In turn, the mentioned possibility to improve the calibration of the muon-rate dependence with middle-atmosphere dynamics will be crucial to safely remove this effect. The effect will be increasingly important at higher latitudes due to the increase of seasonal temperature variations, and for increasing rock opacities. At Mont Terri ($47.38^{\circ}$N), relative effective temperature variations can be as high as 4$\%$, which given the effective temperature coefficient estimated, imply changes in muon rate as high as $3\%$ (c.f. Fig. \[fig:dfm\]). However, muon rate changes would be at maximum $1\%$ if the opacity would be reduced by one order of magnitude to 70 mwe, or equivalently 26 m of standard rock, and for vertical observations.
Finally, relative temperature and muon rate variations are not always coincident in Fig. \[fig:dfm\], despite using the same time-averaging window. Equivalently, deviations from the linear relationship up to 2% and mostly around 1% can be observed in Fig. \[fig:alpha\]. The deviations from a perfect correspondence are presumably due to physical phenomena influencing the muon rate other than the effective atmospheric temperature. Variations arising from changes in the primary cosmic rays, or changes in the geomagnetic field induced by solar wind typically have temporal scales that are much smaller (e.g. seconds to hours) or much larger (e.g. a solar cycle of $\sim$11 years). Changes reported recently as induced by lower altitude atmospheric phenomena such as thunderclouds only lasted 10 minutes [@hariharan2019measurement], and the low-energy muons affected by atmospheric pressure variations [@jourde2016monitoring] get filtered in the first meters of rock in our experiment. A much more likely explanation may be given by changes in the groundwater content of the rock overlying the Mont Terri URL and will be the subject of forthcoming publications.
Conclusion
==========
We report for the first time sensitivity to middle-atmosphere temperature variations using a portable muon detector. Changes detected are associated not only to seasonal variations but also short-term (15-days) variations caused by a Sudden Stratospheric Warming. The occurrence of this event was verified by applying a standard definition of SSWs, and also observed by regional temperature and polar vortex variations obtained from ECMWF and MERRA-2 reanalysis data. Previous reports on the sensitivity of muon rate to these phenomena exist only for large, expensive and immobile muon detectors often times associated to neutrino experiments and high-opacity conditions. Our findings imply that portable muon detectors may be used to further study short-term temperature variations, and to improve the calibration curve of muon rate dependence with an effective temperature value. This, in turn, is crucial for geoscience applications aiming at studying subsurface processes by characterizing density changes with muons.
This study is part of the DIAPHANE project and was financially supported by the ANR-14-CE 04-0001 and the MD experiment of the Mont Terri project ([www.mont-terri.ch](www.mont-terri.ch)) funded by Swisstopo. MRC thanks the AXA Research Fund for their financial support. We are grateful to Thierry Theurillat and Senecio Schefer for their technical and logistical assistance at Mont Terri URL. The MERRA data are available from <https://acd-ext.gsfc.nasa.gov/Data_services/met/ann_data.html>, and the ECMWF data from <https://www.ecmwf.int/>. Muon data used for all calculations are displayed in figures and are available in the Supplementary Table S1. This is IPGP contribution number 4049. We thank the editor and two anonymous reviewers for their constructive comments and suggestions, which helped to improve our work.
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|
---
author:
- 'G. Dagvadorj'
- 'J. M. Fellows'
- 'S. Matyjaśkiewicz'
- 'F. M. Marchetti'
- 'I. Carusotto'
- 'M. H. Szymańska'
title: 'Non-equilibrium Berezinskii-Kosterlitz-Thouless Transition in a Driven Open Quantum System'
---
**The Berezinskii-Kosterlitz-Thouless mechanism, in which a phase transition is mediated by the proliferation of topological defects, governs the critical behaviour of a wide range of equilibrium two-dimensional systems with a continuous symmetry, ranging from superconducting thin films to two-dimensional Bose fluids, such as liquid helium and ultracold atoms. We show here that this phenomenon is not restricted to thermal equilibrium, rather it survives more generally in a dissipative highly non-equilibrium system driven into a steady-state. By considering a light-matter superfluid of polaritons, in the so-called optical parametric oscillator regime, we demonstrate that it indeed undergoes a vortex binding-unbinding phase transition. Yet, the exponent of the power-law decay of the first order correlation function in the (algebraically) ordered phase can exceed the equilibrium upper limit – a surprising occurrence, which has also been observed in a recent experiment [@Roumpos2012a]. Thus we demonstrate that the ordered phase is somehow more robust against the quantum fluctuations of driven systems than thermal ones in equilibrium.**
The Hohenberg-Mermin-Wagner theorem prohibits spontaneous symmetry breaking of continuous symmetries and associated off-diagonal long-range order for systems with short-range interactions at thermal equilibrium in two (or fewer) dimensions [@Mermin1966]. This is because long-range fluctuations due to the soft Goldstone mode are so strong as to be able to “shake apart” any possible long-ranged order. The Berezinskii-Kosterlitz-Thouless (BKT) mechanism (for an overview see Refs. [@Chaikin2000; @Minnhagen1987]) provides a loophole to the Hohenberg-Mermin-Wagner theorem: Two-dimensional (2D) systems can still exhibit a phase transition between a quasi-long-range ordered phase below a critical temperature, where correlations decay algebraically and topological defects are bound together, and a disordered phase above such a temperature, where defects unbind and proliferate, causing exponential decay of correlations. Further, it can be shown [@Nelson1977] that the algebraic decay exponent in the ordered phase cannot exceed the upper bound value of $1/4$.
The BKT transition is relevant for a wide class of systems, perhaps the most celebrated examples are those in the context of 2D superfluids, as in ${}^4$He and ultracold atoms: Here, despite the absence of true long-range order, as well as a true condensate fraction, clear evidence of superfluid behaviour has been observed in the ordered phase [@Bishop1978]. Particularly interesting, and far from obvious, is the case of harmonically trapped ultracold atomic gases [@Hadzibabic2006]. While, in an ideal gas, trapping modifies the density of states to allow Bose-Einstein condensation and a true condensate [@Shlyapnikov2004; @Pitaevskii2003], weak interactions change the phase transition from normal-to-BEC to normal-to-superfluid and recover the BKT physics despite the system’s harmonic confinement.
These considerations are applicable to equilibrium, where the BKT transition can be understood in terms of the free energy being minimised in either the phase with free vortices or in the one with bound vortex-anti-vortex pairs. However, in recent years a new class of 2D quantum systems has emerged: strongly driven and highly dissipative interacting many-body light-matter systems such as, for example, polaritons in semiconductor microcavities [@Carusotto2012], cold atoms in optical cavities [@Ritsch2013] or cavity arrays [@Carusotto2009; @Hartmann2008]. Due to the dissipative nature of the photonic part, a strong drive is necessary to sustain a non-equilibrium steady-state. In spite of this, a transition from a normal to a superfluid phase in driven microcavity polaritons has been observed [@Kasprzak2006; @Deng2010], and the superfluid properties of the ordered phase have started being explored [@Amo2009; @Amo2009a; @Sanvitto2010; @Marchetti2010; @Tosi2011]. Being strongly driven, the system does not obey the principle of free energy minimisation, and so it is not obvious whether the transition between the normal and superfluid phases, as the density of particles is increased, is of the BKT type i.e. due to vortex-antivortex pairs unbinding.
Current experiments are not yet able to resolve single shot measurements, and so are not sufficiently sensitive to detect randomly moving vortices. Algebraic decay of correlations was reported from averaged data [@Roumpos2012a; @Roumpos2012], however, the power-law decay displays a larger exponent than is possible in equilibrium, which posed questions as to the actual mechanism of the transition. On the theory side, by mapping the complex Ginzburg-Landau equation describing long-wavelength condensate dynamics onto the anisotropic Kardar-Parisi-Zhang (KPZ) equation, Altman et al. [@Altman2013] concluded that although no algebraic order is possible in a truly infinite system, the KPZ length scale is certainly much larger than any reasonable system size in the case of microcavity polaritons.
In this work, we consider the case of microcavity polaritons coherently driven into the optical parametric oscillator regime [@Stevenson2000; @Baumberg2000] as the archetype of a 2D driven-dissipative system. Another popular pumping scheme is incoherent injection of hot carriers, which relax down to the polariton ground state by exciton formation and interactions with the lattice phonons [@Kasprzak2006; @Deng2010]. However, the incoherently pumped polariton system is challenging to model due to the complicated and not yet fully understood processes of pumping and relaxation. As a result, one is typically forced to use phenomenological models [@Chiocchetta2014], which often suffer from spurious divergences. From this point of view, the parametric pumping scheme is particularly appealing, as an ab initio theoretical description can be developed in terms of a system Hamiltonian [@Vogel1989], and its predictions can be directly compared to experiment.
Analysing the non-equilibrium steady-state, we show that despite the presence of a strong drive and dissipation the transition from the normal to the superfluid phase in this light-matter interacting system is of the BKT type i.e governed by binding and dissociation of vortex-antivortex pairs as a function of particle density, and bares a lot of similarities to the equilibrium counterpart. However, as recent experiments suggested [@Roumpos2012a], we find that larger exponents of the power-law decay are possible before vortices unbind and destroy the quasi-long-range order leading to exponential decay. This suggests that the external drive, decay and associated noise favours excitations of collective excitations, the Goldstone phase modes, which lead to faster spatial decay, over unpaired vortices which would destroy the quasi-order all together. This externally over-shaken but not stirred quantum fluid constitutes an interesting new laboratory to explore non-equilibrium phases of matter.
![**Polariton system in the OPO regime.** Upper panel: 2D map of the photonic OPO spectrum $|\psi_{C, k_x, k_y=0 }^{}
(\omega)|^2$ (logarithmic scale) of energy $\omega$ versus the $k_x$ momentum component (cut at $k_y=0$) for a single noise realisation and at a pump power $f_p=1.02436 f_p^{\text{th}}$, where $f_p^{^\textrm{th}}$ is the mean-field OPO threshold. The arrows show schematically the parametric process scattering polaritons from the pump state into the signal and idler modes. Dashed (green) lines show the bare upper (UP) and lower polariton (LP) dispersions, while dotted (black) lines are the cavity photon ($C$) and exciton ($X$) dispersions. The solid (black) line underneath the spectrum is the $k_y=0$ cut of the single-shot time-averaged in the steady-state momentum distribution $\int dt |\psi_{C, k_x, k_y=0}^{} (t)|^2$, clearly showing the macroscopic occupation of the three OPO pump, signal, and idler states. Lower panels: 2D maps of the filtered space profiles $|\psi_{s,p,i} ({{\mathbf r}},t)|^2$ at a fixed time $t$ for which a steady-state regime is reached — the pump emission intensity is rescaled to $1$. Blue (red) dots indicate the vortex (antivortex) core positions.[]{data-label="fig:figu1"}](fig_Spectrum.png){width="1\linewidth"}
Simulating driven-dissipative open systems {#simulating-driven-dissipative-open-systems .unnumbered}
==========================================
We describe the dynamics of polaritons in the OPO regime, including the effects of fluctuations, by starting from the system Hamiltonian for the coupled exciton and cavity photon field operators $\hat{\psi}_{X,C}^{} ({{\mathbf r}},t)$, depending on time $t$ and 2D spatial coordinates ${{\mathbf r}}=(x,y)$ ($\hbar=1$): $$\hat{H}_S = \int d{{\mathbf r}} \begin{pmatrix} \hat{\psi}_{X}^{\dag} &
\hat{\psi}_{C}^{\dag} \end{pmatrix} \begin{pmatrix}
\frac{-\nabla^2}{2m_X} + \frac{g_X}{2} |\hat{\psi}_{X}|^2 &
\frac{\Omega_R}{2} \\ \frac{\Omega_R}{2} &
\frac{-\nabla^2}{2m_C} \end{pmatrix} \begin{pmatrix}
\hat{\psi}_{X}^{} \\ \hat{\psi}_{C}^{} \end{pmatrix}\; .$$ Here, $m_{X,C}$ are the exciton and photon masses, $g_X$ the exciton-exciton interaction strength, and $\Omega_R$ the Rabi splitting [@Carusotto2012]. In order to introduce the effects of both an external drive (pump) and the incoherent decay, we add to $\hat{H}_S$ a system-bath Hamiltonian $\hat{H}_{SB}$ $$\begin{gathered}
\hat{H}_{SB} = \int d{{\mathbf r}} \left[F({{\mathbf r}},t)
\hat{\psi}_{C}^{\dag} ({{\mathbf r}},t) +
\text{H.c.}\right] \\
+ \sum_{{{\mathbf k}}} \sum_{l=X,C} \left\{\zeta^{l}_{{{\mathbf k}}}
\left[\hat{\psi}_{l, {{\mathbf k}}}^{\dag} (t) \hat{B}_{l, {{\mathbf k}}}^{}
+ \text{H.c.}\right] + \omega_{l, {{\mathbf k}}} \hat{B}_{l,
{{\mathbf k}}}^{\dag} \hat{B}_{l, {{\mathbf k}}}^{} \right\}\; ,\end{gathered}$$ where $\hat{\psi}_{l, {{\mathbf k}}}^{} (t)$ are obtained Fourier transforming to momentum space ${{\mathbf k}}=(k_x, k_y)$ the corresponding field operators in real space $\hat{\psi}_{l}^{}
({{\mathbf r}},t)$. $\hat{B}_{l, {{\mathbf k}}}^{}$ and $\hat{B}_{l,
{{\mathbf k}}}^{\dag}$ are the bath’s bosonic annihilation and creation operators with momentum ${{\mathbf k}}$ and energy $\omega_{l, {{\mathbf k}}}$, describing the decay processes for both excitons and cavity photons. To compensate the decay, the system is driven by an external homogeneous coherent pump $F({{\mathbf r}},t) = f_p e^{i ({{\mathbf k}}_p \cdot
{{\mathbf r}} - \omega_p t)}$, which continuously injects polaritons into a pump state, with momentum ${{\mathbf k}}_p$ and energy $\omega_p$.
Within the Markovian bath regime, standard quantum optical methods [@Szymanska2007; @Walls2007] can be used to eliminate the environment and obtain a description of the system dynamics in terms of a master equation. As the full quantum problem is, in practice, intractable, a simple yet useful description of the parametric oscillation process properties is provided by the mean-field approximation, where the quantum fields $\hat{\psi}_{l}^{}
({{\mathbf r}},t)$ are replaced by the classical fields $\psi_{l}^{}
({{\mathbf r}},t)$, whose dynamics is governed by the following generalised Gross-Pitaevskii equation [@Carusotto2012] $$\begin{aligned}
\label{eq:mfdyn}
i\partial_t \begin{pmatrix} \psi_{X}^{} \\ \psi_{C}^{} \end{pmatrix}
&= H_{MF} \begin{pmatrix} \psi_{X}^{} \\ \psi_{C}^{} \end{pmatrix}
+ \begin{pmatrix} 0 \\ F({{\mathbf r}},t) \end{pmatrix}\\ H_{MF}
&= \begin{pmatrix} \frac{-\nabla^2}{2m_X} + g_X |\psi_{X}|^2 -
i\kappa_X & \frac{\Omega_R}{2} \\ \frac{\Omega_R}{2} &
\frac{-\nabla^2}{2m_C} -i \kappa_C\end{pmatrix} \; ,
\nonumber\end{aligned}$$ where $\kappa_{X,C}$ are the exciton and photon decay rates. By solving both analytically and numerically, much work has been carried out on the mean-field dynamics for polaritons in the OPO regime and its properties analysed in detail [@whittaker05; @wouters07:prb; @marchetti_review]. Here, polaritons resonantly injected into the pump state, with momentum ${{\mathbf k}}_p$ and energy $\omega_p$, undergo parametric scattering into the signal $({{\mathbf k}}_s, \omega_s)$ and idler $({{\mathbf k}}_i, \omega_i)$ states — see Fig. \[fig:figu1\]. As explained in detail in Ref. [@SM], as well as in other works [@marchetti_review], the full steady-state OPO photon emission $\psi_{C}^{} ({{\mathbf r}},t)$ is filtered in momentum around the values of the signal, pump and idler momenta ${{\mathbf k}}_{s,p,i}$ in order to get their corresponding steady-state profiles, i.e., $\psi_{s,p,i} ({{\mathbf r}},t) = \sum_{|{{\mathbf k}} - {{\mathbf k}}_{s,p,i}| <
\tilde{k}_{s,p,i}} \psi_{C, {{\mathbf k}}}^{} (t) e^{i {{\mathbf k}} \cdot
{{\mathbf r}}}$. The choice of each state filtering radius, $\tilde{k}_{s,p,i}$, is such that the filtered profiles $\psi_{s,p,i}
({{\mathbf r}},t)$ are not affected by them — for details, see [@SM]. The mean-field onset of OPO is shown in the inset of Fig. \[fig:figu3\], where the mean-field densities of both pump and signal are plotted as a function of the increasing pump power $f_p$. At mean-field level, parametric processes lock the sum of the phases of signal and idler fields $\psi_{s,i}$ to that of the external pump, while allowing a global $U(1)$ gauge symmetry for their phase difference to be spontaneously broken into the OPO phase — a feature which implies the appearance of a Goldstone mode [@Wouters2007]. As shown below, fluctuations above mean-field can lift, close to the OPO threshold, this perfect phase locking.
Fluctuations beyond the Gross-Pitaevskii mean-field description can be included by making use of phase-space techniques — for a general introduction, see Ref. [@Gardiner2004] , while for recent developments in quantum fluids of atoms and photons, see, e.g., Refs. [@Carusotto2005; @Giorgetti2007; @Foster2010]. Here, the quantum fields $\hat{\psi}_{l}^{} ({{\mathbf r}},t)$ are represented as quasiprobability distribution functions in the functional space of C-number fields $\psi_{l}^{} ({{\mathbf r}},t)$. Under suitable conditions, the Fokker-Planck partial differential equation, which governs the time evolution of the quasiprobability distribution, can be mapped on a stochastic partial differential equation, which in turn can be numerically simulated on a finite $N
\times N$ grid with spacing $a$ (along both $x$ and $y$ directions) and a total size $L_{x,y}=Na$ comparable to the polariton pump spot size in state-of-the-art experiments. For the system under consideration here, the Wigner representation – one of the many possible quasiprobability distributions – is the most suitable to numerical implementation: in the limit $g_X/(\kappa_{X,C} dV) \ll
1$, where $dV=a^2$ is the cell area, it appears in fact legitimate [@Drummond1980; @Vogel1989] to truncate the Fokker-Planck equation, retaining the second-order derivative term only, thus obtaining the following stochastic differential equation: $$i d \begin{pmatrix} \psi_X \\ \psi_C \end{pmatrix} =
\left[H_{MF}' \begin{pmatrix} \psi_X \\ \psi_C \end{pmatrix}
+ \begin{pmatrix} 0 \\ F \end{pmatrix}\right] dt +
i \begin{pmatrix} \sqrt{\kappa_X} dW_X \\ \sqrt{\kappa_C}
dW_C \end{pmatrix}\; .
\label{eq:wigne}$$ Here, $dW_{l=X,C}$ are complex valued, zero-mean, independent Weiner noise terms with $\langle
dW^{*}_l ({{\mathbf r}},t) dW_m ({{\mathbf r}}',t) \rangle =
\delta_{{{\mathbf r}},{{\mathbf r}}'} \delta_{l,m} \frac{dt}{dV}$, and the operator $H_{MF}'$ coincides with $H_{MF}$ in Eq. with the replacement $|\psi_X|^2 \mapsto |\psi_X|^2-\frac{1}{dV}$. The same stochastic equation can be alternatively derived applying a Keldysh path integral formalism to the Hamiltonian $\hat{H}_S + \hat{H}_{SB}$, integrating out the bath fields, and keeping only the renormalisation group relevant terms [@Sieberer2013]. Note that, remarkably, some of the difficulties of the truncated Wigner method met in the context of equilibrium systems, such as for cold atoms, are suppressed here by the presence of loss and pump terms, i.e., the existence of a small parameter $g_X/(\kappa_l dV)$ which controls the truncation [@Carusotto2012]. Note, however, that the bound on this truncation parameter involves the cell area $dV$ of the numerical grid, that is the UV cut-off of the stochastic truncated Wigner equation. For typical OPO parameters, it is possible to choose $dV$ small enough to capture the physics, but at the same time large enough to keep the UV issues under control.
We reconstruct the steady-state Wigner distributions $\psi_{l}^{}
({{\mathbf r}},t)$ by considering a monochromatic homogeneous continuous-wave pump $F({{\mathbf r}},t) = f_p e^{i ({{\mathbf k}}_p \cdot
{{\mathbf r}} - \omega_p t)}$ as before and letting the system evolve to its steady-state. In order to rule out any dependence on the chosen initial conditions, we have considered four extremely different cases: empty cavity with random noise initial conditions and adiabatic increase of the external pump power strength; mean-field condensate initial conditions; either random or mean-field initial conditions in the presence of an unpumped region at the edges of the numerical box, so to model a sort of “vortex-antivortex reservoir. The different initial stage dynamics, and their physical interpretation for each of these four different initial conditions, are carefully described in [@SM]; in all four cases we always reach the very same steady-state regime, i.e., all noise averaged observable quantities discussed in the following lead to the same result — this could not be a priori assumed for a non-linear system.
Below we first analyse results from single noise realisations (concretely, here, for the case of mean-field initial conditions and no “V-AV reservoir” present), by filtering the photon emission at the signal, pump and idler momenta as also previously done at mean-field level. The filtered profiles are again indicated as $\psi_{s,p,i} ({{\mathbf r}},t)$ — for details on filtering see [@SM]. Second, we consider a large number of independent noise realisations and perform stochastic averages of appropriate field functions in order to determine the expectation values of the corresponding symmetrically ordered quantum operators. In particular, we evaluate the signal first-order correlation function as $$g^{(1)} ({{\mathbf r}}) = {\displaystyle\frac{\langle \psi_{s}^* ({{\mathbf r}} + {{\mathbf R}},t)
\psi_{s}^{} ({{\mathbf R}},t) \rangle}{\sqrt{ \langle \psi_{s}^*
({{\mathbf R}},t) \psi_{s}^{} ({{\mathbf R}},t) \rangle \langle \psi_{s}^*
({{\mathbf r}} + {{\mathbf R}},t) \psi_{s}^{} ({{\mathbf r}} + {{\mathbf R}},t)
\rangle}}}\; ,
\label{eq:corre}$$ where the averaging $\langle \dots \rangle$ is taken over both noise realisations as well as the auxiliary position ${{\mathbf R}}$, and where $t$ is either a fixed time after a steady-state is reached, or we take additional time average in the steady-state [@SM].
Vortices and densities across the transition
--------------------------------------------
It is particularly revealing to explore the steady-state profiles, i.e. $\psi_{s,p,i} ({{\mathbf r}},t)$, of the signal, pump and idler states. Fig. \[fig:figu1\] shows a cut at $k_y=0$ of the OPO spectrum, $|\psi_{C, k_x, k_y=0 }^{} (\omega)|^2$, determined by solving Eq. for $\psi_{X,C}^{} ({{\mathbf r}},t)$ to a steady-state and evaluating the Fourier transforms in both space and time. Note, that the logarithmic scale of this 2D map plot (which we employ to clearly characterise all three OPO states) makes the emission artificially broad in energy, while in reality this is sharp (as required by a steady-state regime), as well as it is very narrow in momentum. The filtered space profiles $\psi_{s,p,i} ({{\mathbf r}},t)$ shown in the bottom panels of Fig. \[fig:figu1\] reveal that while the pump state is homogeneous and free from defects, vortex-antivortex (V-AV) pairs are present for both signal and idler states. Note, that while at the mean-field level the sum of the signal and idler phases is locked to the one of the pump (and thus a V in the signal implies the presence of an AV at the same position in the idler), the large fluctuations occurring in the vicinity of the OPO threshold make this coherent phase-locking mechanism only weakly enforced, resulting in a different number (and different core locations) of V-AV pairs in the signal and idler states. Because the density of photons in the idler state is much lower than the one at the signal (see, e.g., the photonic momentum distribution plotted as a solid black line inside the upper panel of Fig. \[fig:figu1\]), while both states experience the same noise strength, the number of V-AV pairs in the filtered photonic signal profile is much lower than the number of pairs in the filtered photonic idler profile. Phase locking between signal and idler is recovered instead for pump powers well above the OPO threshold, where long-range coherence over the entire pumping region is re-established.
![**Binding-unbinding transition and vortex-antivortex proliferation across the OPO threshold.** Phase (colour map) of the filtered OPO signal $\psi_{s} ({{\mathbf r}},t)$ and position of vortices (black dots) and antivortices (red dots) for increasing values of the pump power, in a narrow region close to the mean-field OPO threshold $f_p^{\text{th}}$: (a) $f_p=1.00287f_p^{\textrm{th}}$, (b) $f_p=1.01648f_p^{\textrm{th}}$, (c) $f_p=1.01719f_p^{\textrm{th}}$, and (d) $f_p=1.02436f_p^{\textrm{th}}$. We observe a dramatic decrease of both the number of Vs and AVs, as well as the typical distance between pairs, as a function of the increasing pump power. The filtered profiles are plotted at a late stage of the dynamics, at which a steady-state is reached.[]{data-label="fig:figu2"}](fig_PhaseVortices.png){width="1\linewidth"}
The proliferation of vortices below the OPO transition, followed by a sharp decrease in their density and their binding into close vortex-antivortex pairs is illustrated in Fig. \[fig:figu2\]. Here, we plot the 2D maps of the phase for the single noise realisation of the filtered OPO signal $\psi_{s}
({{\mathbf r}},t)$ (photonic component) for increasing values of the pump power $f_p$ in a narrow region close to the mean-field OPO threshold $f_p^{\text{th}}$; the position of the generated vortices (antivortices) are marked with blue (black) dots. While at lower pump powers there is a dense “plasma” of Vs and AVs, the number of V-AV pairs decrease with increasing pump powers till eventually disappearing altogether (not shown). We do also record a net decrease in the distance between nearest neighbouring vortices with opposite winding number with respect to that between vortices with the same winding number. In order to quantify the vortex binding across the OPO transition, we measure, for each detected vortex, the distance to its nearest vortex, $r_{\text{V-V}}$ and to its nearest antivortex $r_{\text{V-AV}}$; and similarly, for each detected antivortex, we measure $r_{\text{AV-AV}}$ and $r_{\text{AV-V}}$. We then consider the symmetrised ratio $b =
\frac{r_{\text{V-V}}+r_{\text{AV-AV}}}{r_{\text{V-AV}}+r_{\text{AV-V}}}$. In order to extract a noise realisation independent quantity, an average over many different realisations, as well as over individual vortex positions, is performed to obtain $\langle b
\rangle$; this quantity $\langle b \rangle \to 1$ for an unbound vortex plasma, while $\langle b \rangle \to 0$ when vortices form tightly bound pairs. We observe a dramatic drop in $\langle b \rangle$ (green squares in Fig. \[fig:figu3\]) when increasing the pump power across the OPO threshold, indicating that vortices and antivortices are indeed binding, as it is expected for a BKT transition.
![**The phase diagram and the BKT transition.** Inset: Mean-field photonic OPO densities for pump (black) $n_{p}$ and signal (orange) $n_{s}$ states as a function of increasing pump power $f_p$ rescaled by the threshold value $f_p^{\text{th}}$ (vertical black dashed line). The black square at $f_p \simeq
f_p^{\text{th}}$ indicates the tiny pump strength interval close to mean-field threshold analysed in the main panel. Main panel: We plot with (orange) squares the same mean-field signal density $n_{s}$ as in the inset. All other data are noise averaged properties from stochastic simulations as a function of the pump strength. The noise averaged signal density $n_s$ is plotted with (blue) dots; the average vortex number in the signal rescaled by its average maximum value, $N_{\textrm{max}}=222.8$ with (red) diamonds; the noise averaged and symmetrised distance ratio $\langle b \rangle$ between nearest neighbouring V-V and AV-AV over V-AV pairs with (green) empty squares. The shaded region indicates the pump region for the BKT transition.[]{data-label="fig:figu3"}](fig_PhaseDiagram.png){width="1\linewidth"}
By evaluating other relevant noise averaged observable quantities, we are able to construct a phase diagram for the OPO transition in Fig. \[fig:figu3\] and link it with the properties of the BKT transition. We evaluate the averaged signal photonic density at some time $t$ in the steady-state, $n_{s} = \int d{{\mathbf r}} \langle |\psi_{s}
({{\mathbf r}},t)|^2 \rangle /V$, where $V=(Na)^2$ is the system area and $\langle \dots \rangle$ indicates the noise average for the stochastic dynamics (blue dots). We also show the steady state signal density in the mean field (orange squares). The corresponding mean-field densities for both signal (orange line) and pump (black line) are presented for comparison in the inset of Fig. \[fig:figu3\]. At mean-field level, both signal and idler (not shown) suddenly switch on at the OPO threshold pump power, $f_p = f_p^{\text{th}}$ and both states are macroscopically occupied above threshold. The effect of fluctuations is to smoothen the sharp mean-field transition, as clearly shown by the (blue) dots in the main panel of Fig. \[fig:figu3\], where we plot the noise average signal density $n_{s}$. This is because, even below the mean-field threshold, incoherent fluctuations weakly populate the signal. Note also that, even though somewhat smoothened, we can still appreciate a kink in the $n_{s}$ density, but at higher values of the pump power compared to the mean-field threshold $f_p^{\text{th}}$. We identify this as the novel BKT transition for our out-of-equilibrium system, as discussed more in detail below. This is further confirmed by a sudden decrease of the averaged number of vortices in the signal (red diamonds), and of the averaged distance between nearest neighbouring vortices of opposite winding number, $\langle b \rangle$ (green squares), as a function of the pump power concomitant with the observed kink for $n_{s}$. These results suggest that the system undergoes an OPO transition which, by including fluctuations above mean-field, is indeed analogous to the equilibrium BKT transition. Both vortices and antivortices proliferate below some threshold and, above, they bind to eventually disappear altogether. As indicated by the black square in the inset of Fig. \[fig:figu3\], the region for such a crossover is indeed narrow in the pump strength.
![**Algebraic and exponential decay of the first order correlation function across the BKT transition.** Main panel: Long-range spatial dependence of $g^{(1)} ({{\mathbf r}})$ for different pump powers $f_p/f_p^{\text{th}}$ close to the mean-field pump threshold (the symbols are the same ones as in the inset and correspond to the same values of $f_p/f_p^{\text{th}}$). Thick solid (thick dashed) lines are power-law (exponential) fitting, from which values of the exponent $\alpha$ are derived. The $f_p/f^\textrm{th}_p=1.0129$ case (orange squares) is a marginal case where both algebraic and exponential fits apply almost equally well, signalling the BKT transition region. Inset: Power-law algebraic decay exponent $\alpha$ for different pump powers $f_p/f_p^{\text{th}}$; error bars are standard deviations of the time-average.[]{data-label="fig:figu4"}](g1_alpha.png){width="1\linewidth"}
First-order spatial correlations {#first-order-spatial-correlations .unnumbered}
================================
For systems in thermal equilibrium, the BKT transition is associated with the onset of quasi-off-diagonal long-range order, i.e., with the algebraic decay of the first-order correlation function in the ordered phase, where vortices are bound, and exponential decay in the disordered phase, where free vortices do proliferate. In order to investigate whether the same physics applies to our out-of-equilibrium open-dissipative polariton system, we evaluate the signal first-order correlation function $g^{(1)} ({{\mathbf r}})$ according to the prescription of Eq. and characterise its long-range behaviour in Fig. \[fig:figu4\]. We observe the ordering transition as a crossover in the long-distance behaviour between an exponential decay in the disordered phase, $g^{(1)}({{\mathbf r}}) \sim e^{-r/\xi}$, and an algebraic decay in the quasi-ordered phase, $g^{(1)} ({{\mathbf r}}) \sim (r/r_0)^{-\alpha}$. We therefore fit the tail of the calculated correlation function to both of these functional forms and observe that, at the onset of vortex binding-unbinding and proliferation, the signal’s spatial correlation function changes its long-range nature, from exponential at lower pump powers to algebraic at higher (see Fig. \[fig:figu4\]).
However, in contrast with the thermal equilibrium case, we do observe that the exponent $\alpha$ of the power-law decay (inset of Fig. \[fig:figu4\]) can exceed the equilibrium upper bound of $1/4$ [@Nelson1977], and can reach values as high as $\alpha \simeq
1.2$ for $f_p/f_p^{\text{th}} = 1.0136$, just within the ordered phase. Further, as thoroughly discussed in Ref. [@SM], it is interesting to note that, close to the transition, we do observe a critical slowing down of the dynamics: Here, the convergence to a steady-state is dramatically slowed down compared to cases above or below the OPO transition, a common feature of other phase transitions. At the same time, close to threshold, the convergence of noise averaged number of vortices is much faster than the convergence of the power-law exponent $\alpha$. This indicates that the fluctuations induced by both the external drive and the decay preferentially excite collective excitations, rather than topological excitations, resulting in vortices being much more dynamically stable. Finally, note that for sufficiently strong pump powers, the power-law exponent becomes extremely small and thus quasi-long-range order is difficult to distinguish from the true long-range order over the entire system size.
Our findings explain why recent experimental studies, both in the OPO regime [@Spano2013], as well as for non-resonant pumping [@Roumpos2012a], experienced noticeable difficulties in investigating the power-law decay of the first order correlation function across the transition. We do indeed find that the pump strength interval over which power-law decay can be clearly observed is extremely small, and the system quickly enters a regime where coherence extends over the entire system size, as measured in [@Spano2013]. This was also observed in non-resonantly pumped experiments when using a single-mode laser [@Kasprzak2006]. However, by intentionally adding extra fluctuations with a multimode laser pump, as in [@Roumpos2012a], power-law decay was finally observed in the correlated regime, with an exponent in the range $\alpha \simeq
0.9-1.2$, in agreement with our results.
Discussion {#discussion .unnumbered}
==========
Using microcavity polaritons in the optical parametric oscillator regime as the prototype of a driven-dissipative system, we have numerically shown that a mechanism analogous to the BKT transition, which governs the equilibrium continuous-symmetry-breaking phase transitions in two dimensions, occurs out of equilibrium for a driven-dissipative system of experimentally realistic size. Notwithstanding the novelty and significance of this result, there are a number of novel features which warrant discussion as they are peculiar to non-equilibrium phase transitions. We have observed that the exponent of algebraic decay in the quasi-long-range ordered phase exceeds what would be attainable in equilibrium. This recovers a recent observation [@Roumpos2012a], and strongly suggests that indeed a non-equilibrium BKT may have been seen there. Moreover, our findings imply that the ordered phase is more robust to fluctuations induced by the external drive and decay than an analogous equilibrium ordered phase would be to thermal fluctuations. Although for realistic experimental conditions, we have found that the region for BKT physics, before the pump power is strong enough to induce perfect spatial coherence over the entire system size, is indeed narrow, we believe our work will encourage further experimental investigations in the direction of studying the non-equilibrium BKT phenomena. Even though the small size of the critical region has so far hindered its direct experimental study, our calculations indicate that the macroscopic coherence observed in past polariton experiments [@Keeling2007; @Kasprzak2006; @Stevenson2000; @Baumberg2000; @Carusotto2012; @Deng2010; @Spano2013] results from a non-equilibrium phase transition of the BKT rather than the BEC kind.
Methods {#methods .unnumbered}
=======
We simulate the dynamics of the stochastic equations with the XMDS2 software framework [@Dennis2012] using a fixed-step (where the fixed step-size ensures stochastic noise consistency) 4th order Runge-Kutta (RK) algorithm, which we have tested against fixed-step 9th order RK, and a semi-implicit fixed-step algorithm with 3 and 5 iterations. We choose the system parameters to be close to current experiments [@Sanvitto2010]: The Rabi frequency is chosen as $\Omega_R=4.4$ meV, the mass of the microcavity photons is taken to be $m_C=2.3\times10^{-5}m_e$, where $m_e$ is the electron mass, the mass of the excitons is much greater than this so we may take $m_X^{-1}\to0$, the exciton and photon decay rates as $\kappa_X =
\kappa_C = 0.1$ meV, and the exciton-exciton interaction strength $g_X=0.002$ meV$\mu$m$^2$ [@Ferrier2011]. The pump momentum ${{\mathbf k}}_p = (k_p,0)$, with $k_p =
1.6$ $\mu$m$^{-1}$, is fixed just above the inflection point of the LP dispersion, and its frequency, $\omega_p - \omega_X (0)=1.0$ meV, just below the bare LP dispersion. In order to satisfy the condition necessary to derive the truncated Wigner equation , $g_X/(\kappa_{X,C} d V) \ll 1$, whilst maintaining a sufficient spatial resolution and, at the same time, a large enough momentum range so that to resolve the idler state, simulations are performed on a 2D finite grid of $N \times N =
280 \times 280$ points and lattice spacing $a = 0.866$ $\mu$m. Thus, the only system parameter left free to be varied is the pump strength $f_p$: We first solve the mean-field dynamics in order to determine the pump threshold $f_p^{\text{th}}$ for the onset of OPO. We then vary the value of $f_p$ around $f_p^{\text{th}}$ in presence of the noise in order to investigate the nature of the OPO transition. We analyse the results from single noise realisations by filtering the full photonic emission for signal pump and idler, as described in the main text, as well as in [@SM]. Further, we average all of our results over many independent realisations, which are either taken from $96$ independent stochastic paths or from multiple independent snapshots in time after the steady-state is reached: As thoroughly discussed in [@SM], 96 stochastic paths is shown to be sufficient to ensure the convergence of noise averaged observable quantities. For each noise realisation, vortices are counted by summing the phase difference (modulo $2\pi$) along each link around every elementary plaquette on the filtered grid. In the absence of a topological defect this sum is zero, while if the sum is $2\pi$ ($-2\pi$) we determine there to be a vortex (antivortex) at the center of the plaquette. The number of vortices is then averaged over the different stochastic paths or over time in the steady-state (see Ref. [@SM]): We consider the average number of vortices to be converged in time when its variation is less then $5\%$. Finally, the first order correlation function $g^{(1)} ({{\mathbf r}})$ is evaluated according to Eq. , by averaging over both the noise and the auxiliary position ${{\mathbf R}}$; as discussed in [@SM], this can be computed efficiently in momentum space.
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**Acknowledgments**\
We thank J. Keeling for stimulating discussions. MHS acknowledges support from EPSRC (grants EP/I028900/2 and EP/K003623/2), FMM from the programs MINECO (MAT2011-22997) and CAM (S-2009/ESP-1503), and IC from ERC through the QGBE grant and from the Autonomous Province of Trento, partly through the project “On silicon chip quantum optics for quantum computing and secure communications” (“SiQuro”).
**Supplemental Material for “Non-equilibrium Berezinskii-Kosterlitz-Thouless Transition in a Driven Open Quantum System”**
In this supplementary information we provide a more detailed account of some of the technical issues arising from our numerical methods, which are not of sufficiently general interest to warrant a discussion in the main text, but which are germane to the validity of our conclusions. In particular, we discuss the filtering we perform in order to isolate the signal state, and the loss in resolution this incurs, and we describe various tests we have undertaken in order to ensure that the results we present are properly converged.
Momentum filtering
==================
In order to study the condensate at the signal (or idler), we have to filter the full emission $\psi_{C}^{} ({{\mathbf r}},t)$ in such a way as to omit contributions with momentum outside a set radius about the signal (idler) states in $\mathbf{k}-$space. This applies both to the theory and the experimental data, and this procedure could equivalently be performed in either momentum or energy space. Here we filter in momentum space, and define: $$\psi_{s,p,i} ({{\mathbf r}},t) = \sum_{|{{\mathbf k}} - {{\mathbf k}}_{s,p,i}| <
\tilde{k}_{s,p,i}} \psi_{C, {{\mathbf k}}}^{} (t) e^{i {{\mathbf k}} \cdot
{{\mathbf r}}},$$ where ${{\mathbf k}}_{s,p,i}$ is the momentum of the signal, pump and idler states respectively and $\tilde{k}_{s,p,i}$ is the filtering radius. We fix the filtering radius for the signal and idler to be $\tilde{k}_{s,i}=\tfrac{1}{2}|\mathbf{k}_{s,i}-\mathbf{k}_p|$ and choose to operate in the frame of reference co-moving with the signal (idler) so that the (physically irrelevant) background current does not show in the data. The filtering reveals the phase-freedom of the signal (idler) state at the expense of spatial resolution. Note, that this limits our ability to distinguish vortex-antivortex (V-AV) pairs with a separation less than a distance $\pi/\tilde{k}_{l}$, which for our chosen parameters is $\approx2.708\mu\textrm{m}$. However, such extremely close vortex-antivortex pairs do not affect the spatial correlations of the field at large distances.
Numerical Convergence
=====================
Number of stochastic realisations
---------------------------------
The stochastic averages over the configurations of different realisations of the fields provide the expectation value of the corresponding symmetrically ordered operators. The realisation-averaged results presented in the main paper have been averaged over 96 realisations taken from independent stochastic paths as we assessed this to be sufficiently many to give a reliably converged result. In Fig. \[fig:g1\_vs\_np\] we show the first order correlation function $g^{(1)}(x)$ for $f_p=1.017 f_p^{\textrm{th}}$, averaged over different numbers of realisations (96, 192, 288, 384 and 480). We conclude that 96 is sufficient to determine the nature of the correlations.
The average number of vortices also does not change significantly (no more than $\pm0.5$) beyond 96 realisations, and so we conclude that this number is sufficient to ensure consistent results in both the smooth and topological sectors of the model.
![ [**Correlation function averaged over different numbers of realisations.**]{} The strength of the pump is fixed at $f_p=1.017 f_p^{\textrm{th}}$, and each simulation is run well into the steady state. We calculate the correlation function averaged over 96, 192, 288, 384 and 480 realisations and observe no significant improvement in the convergence past 96 realisations. []{data-label="fig:g1_vs_np"}](fig_g1_vs_np.png){width="1\linewidth"}
Convergence in time to a steady state
-------------------------------------
In order to assess when a time evolution has reached its steady state we consider the average (over realisations) of both the signal density, $|\psi_\textrm{s}|^2$, and the number of vortices. In Figs. \[fig:ns\_vs\_t\] and \[fig:nv\_vs\_t\] we show the evolution of these quantities toward a steady state, assuming the initial condition wherein the system is allowed to reach its mean field steady state (at time $t=0$) and stochastic processes are adiabatically switched on. In practice this means that the noise terms are multiplied by a ramp function $$r(t) = \frac{1}{2}\left[ \tanh\left(\frac{t-t_0}{t_r}\right)+1 \right]$$ such that the noise is slowly increases from $0$, achieving half its maximum at $t_0$, at a rate determined by $t_r$. For the empty cavity with noise initial condition, in which the pump is adiabatically increased, it is the pump term that is multiplied by this ramp function. We choose $t_0 = 450\textrm{ps}$ and $t_r =
150\textrm{ps}$ as these prove to give a fairly rapid convergence to the steady state. It is, however, important to note that the steady state is unique and does not depend upon the specific values of $t_0$ and $t_r$ we choose nor on the initial conditions, which we start the dynamics from.
![ [**Convergence in time of the average signal density.**]{} Here we show $|\psi_\textrm{s}|^2$ evolving in time for a range of pump powers. The initial jump in the signal occupation is induced by the introduction of stochastic processes, and dies away as the system resolves toward its non-equilibrium steady state. []{data-label="fig:ns_vs_t"}](fig_ns_vs_t.png){width="1\linewidth"}
![ [**Convergence in time of the average number of vortices.**]{} Here we show the average number of vortices in the system evolving in time for a range of pump powers. The introduction of stochastic processes increases the number of vortices (which is always zero in the spatially homogeneous mean field steady-state), which then evolves toward its steady state value through pair creation and annihilation events. []{data-label="fig:nv_vs_t"}](fig_nv_vs_t.png){width="1\linewidth"}
In all cases the steady state is reached within around twenty nanoseconds. The slowest convergence occurs in the vicinity of the BKT transition. This critical slowing of the dynamics is to be anticipated given the divergence of the correlation length as the transition is approached.
Realisations from independent stochastic paths vs. independent time snapshots in a single path
----------------------------------------------------------------------------------------------
In Fig. \[fig:nv\_vs\_fp\] we show the number of vortices for a broad range of pump powers across the transition averaged over realisations taken from independent stochastic paths (red dots) and from multiple snapshots over time (blue dots) once the steady-state is reached. Averaging over time within the steady state is a less numerically intensive approach, and shows an excellent agreement with the average over realisations even in the critical region. In practice, averaging over time is computationally efficient away from the critical region, where the steady state is reached quickly and memory is quickly lost during the time evolution. Around the critical region it takes a significant time to reach a steady state and then to decorrelate the snapshots, so averaging over stochastic paths becomes more computationally effective.
![ [**Time versus stochastic realisations averaging.**]{} The red dots show number of vortices averaged over many stochastic paths. The blue dots show the number of vortices averaged over the final 500 frames, running from $15\textrm{ns}$ to $30\textrm{ns}$, of a single stochastic path, once the system has converged to a steady state. We observe an excellent agreement between these two approaches and so away from the critical region we can consider the (less numerically intensive) average over time. []{data-label="fig:nv_vs_fp"}](fig_nv_vs_fp.png){width="1\linewidth"}
Dependence on different initial conditions
------------------------------------------
We have investigated a number of physically diverse initial conditions in order to rule out dependence of the final steady state on the starting configuration. To initiate our simulations we either adiabatically increase the pump strength atop a white noise background or adiabatically increase the strength of the stochastic terms starting from the mean field steady state. We also wish to eliminate the possible effects of trapped vortices, and so in addition to simulations with a uniform pump region we have performed simulations, where a small strip around the numerical integration box is left un-pumped so as to act as a source/sink for vortices. The four distinct initial conditions we consider are therefore those classified in the following table:
Increasing Pump Increasing Noise
-------------- ------------------- --------------------
No Reservoir Scheme A Scheme B
Reservoir Scheme D Scheme C
In schemes A and D, there are initially very many vortices, which then proceed to annihilate with one another so that the overall vorticity tends to decrease toward the steady state. In schemes B and C there are no vortices at the outset but as stochastic processes shake the system the vorticity increases toward the steady state. Therefore even before the true steady state is reached, we can treat schemes A and D as upper bounds for the vorticity and schemes B and C as lower bounds. In schemes C and D we incorporate a reservoir (region with no drive and thus of very low density) of width $34.72\textrm{nm}$ along the two sides of the numerical integration box parallel to $k_p$, which we then exclude from calculations of the vorticity, signal density, and correlations. The width of the reservoir does not change the result except in that it needs to be wide enough that the condensate has room to decay away to zero.
In Supplementary Movie 1 we show the evolution towards the steady state for all four of these schemes with a pump power $f_p=1.017
f_p^{\textrm{th}}$. We plot the vortex density (top panel) as well as dynamics of vortices (bottom panel, where blue and red dots show positions of V and AV cores) for each scheme (A, B, C and D starting from the left). We see that every scheme converges fairly rapidly towards the same steady state. From the animations it is evident that the vortices always appear in pairs but that these pairs are not always tightly bound. It was not [a priori]{} obvious that all four schemes should lead to the same steady state but we take this as evidence that the physical process leading to the steady state is universal for this system. We observe that the schemes incorporating an empty reservoir converge more quickly to the steady state than their counterparts without a reservoir. Nevertheless we present data for scheme B because the reservoirs have no counterpart in the traditional BKT transition to which we wish to compare our results.
Fitting to the spatial correlation function
===========================================
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The behaviour of the first order correlation function $g^{(1)}(x)$, whether it decays as an exponential or as a power law, is determined by fitting the tail of the data to each functional form and taking the better fit of the two. The fitting is illustrated in Fig. \[fig:g1\_linlin\]. For pump powers close but above the critical region it is clear that the power-law fits the data better (which is also confirmed by a larger correlation coefficient), while below the critical region the exponential fit is closer to the data. For stronger pump powers, away from the transition, $g^{(1)}(x)$ is practically constant on these length-scales (as reported in experiments and discussed in the main text).
|
---
abstract: 'In this paper, the effect of the space-time dimension on effective thermodynamic quantities in (n+2)-dimensional Reissoner-Nordstrom-de Sitter space has been studied. Based on derived effective thermodynamic quantities, conditions for the phase transition are obtained. The result shows that the accelerating cosmic expansion can be attained by the entropy force arisen from the interaction between horizons of black holes and our universe, which provides a possible way to explain the physical mechanism for the accelerating cosmic expansion.'
author:
- 'Yang Zhang$^{a,b}$, Wen-qi Wang$^{a}$, Yu-bo Ma$^{a,b}$, and Jun Wang$^{c}$[^1]'
title: 'Phase transition and entropy force between two horizons in (n+2)-dimensional de Sitter space'
---
Introduction
============
It is well known that the cosmic accelerated expansion indicates that our universe is a asymptotical de Sitter one. Moreover, due to the success of AdS / CFT, it prompts us to search for the similar dual relationships in de Sitter space. Therefore, the research of de Sitter space is not only of interest to the theory itself, but also the need of the reality.
In de Sitter space, the radiation temperature on the horizon of black holes and the universe is generally not the same. Therefore, the stability of the thermodynamic equilibrium can not be protected in it, which makes troubles to corresponding researches. In recent years, study on thermodynamic properties of de Sitter space is getting more and more attention [1,2,3,4,5,6,7,8,9,10,11,12]{}. In the inflationary period, our universe seem to be a quasi de Sitter space, in which the cosmological constant is introduced as the vacuum energy, which is a candidate for dark energy. If the cosmological constant corresponds to dark energy, our universe will goes into a new phase in de Sitter space. In order to construct the entire evolutionary history of our universe, and understand the intrinsic reason for the cosmic accelerated expansion, both the classic and quantum nature of de Sitter space should be studied.
For a multi-horizon de Sitter space, although different horizons have different temperatures, thermodynamic quantities on horizons of black holes and the universe are functions depended on variables of mass, electric charge, cosmological constant and so on. Form this point of view, thermodynamic quantities on horizons are not individual. Based on this fact, effective thermodynamic quantities can be introduced. Considering the correlation between horizons of black holes and the universe, we have studied the phase transition and the critical phenomenon in RN-dS black holes with four-dimension and high-dimension by using effective thermodynamic quantities, respectively. Moreover, the entropy for the interaction between horizons of black holes and the universe is also obtained [@13; @14; @15; @16; @17]. When we consider the cosmological constant as a thermodynamic state parameter with the thermodynamic pressure, the result shows that de Sitter space not only has a critical behavior similar to the van der Waals system [@17; @18], but also take second-order phase transition similar to AdS black hole [@19; @20; @21; @22; @23; @24; @+1; @+2; @+3; @+4; @+5]. However, first-order phase transition similar to AdS black hole is not existed. In this work, we investigate the issue of the phase transition in a high-dimensional de Sitter space, and analyze the effect of the dimension on the phase transition and the entropy produced by two interactive horizons.
Nine years ago, Verlinde [@25] proposed to link gravity with an entropic force. The ensuing conjecture was proved recently [@26; @27], in a purely classical environment and then extended to a quantal bosonic system in Ref. [@26]. In 1998, the result of the observational data from the type Ia supernovae (SNe Ia) [@40; @41] indicates that our universe presently experiences an accelerating expansion, which contrasts to the one given in general relativity (GR) by Albert Einstein. In order to explain this observational phenomenon, a variety of proposal have been proposed. The theory of “early dark energy” proposed by Adam Riess [@29; @30] is one of them, where dark energy [@42; @43] as an exotic component with large negative pressure seems to be the cause of this observational phenomenon. According to the observations, dark energy occupies about $73\%$ in cosmic components. Therefore, one believe that the present accelerating expansion of our universe should be caused by dark energy. Then a lot of dark energy models have been proposed. However, up to now, the nature of dark energy is not clear.
Based on the entropy caused by the interaction between horizons of black holes and the universe, the relationship between the entropy force and the position ratio of the two horizons is obtained. When the position ratio of the black hole horizon to the universe horizon is greater (less) than a certain value, the entropy force between the two horizons is repulsive (attractive), which indicates that the expansion of the universe horizon is accelerating (decelerating). While when it equal to the certain value, the entropy force is absent, and then the expansion of the universe horizon is uniform. According to this, we suppose that the entropy force between the two horizons can be seen as a candidate to cause the cosmic accelerated expansion.
This paper is organized as follows. According to Refs. [@16; @17; @18], a briefly review for the effective thermodynamic quantities, the conditions for the phase transition and the effect of the dimension on the phase transition in $(n+2)-$dimensional Reissoner-Nordstrom-de Sitter (DRNdS) space is given in the next section. In section 3, the entropy force of the interaction between horizons of black holes and the universe is derived, and then the effect of the dimension on it is explored. Moreover, the relationship between the entropy force and the position ratio of the two horizons is obtained. Conclusions and discussions are given in the last section. The units$G=\hbar =k_{B}=c=1$ are used throughout this work.
Effective thermodynamic quantities
==================================
The metric of $(n+2)-$dimensional DRNdS space is [@35]: $$d{s^2} = - f(r)d{t^2} + {f^{ - 1}}(r)d{r^2} + {r^2}d\Omega _n^2 \label{2.1}$$where the metric function is $$f(r) = 1 - \frac{{{\omega _n}M}}{{{r^{n - 1}}}} + \frac{{n\omega _n^2{Q^2}}}{{8(n - 1){r^{2n - 2}}}} - \frac{{r^2}}{{l^2}}, {\omega _n} = \frac{{16\pi G}}{{nVol({S^n})}}.$$Here$G$is the gravitational constant in $n+2-$dimensional space,$l$ is the curvature radius of dS space,$Vol({S^n})$ denotes the volume of a unit $n-$sphere $d\Omega _n^2$,$M$ is an integration constant and $Q$ is the electric/magnetic charge of Maxwell field.
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In $n+2-$dimensional DRNdS space, positions of the black hole horizon ${r_{+}}$ and the universe horizon ${r_{c}}$ can be determined when $f({r_{+,c}})=0$. Moreover, thermodynamic quantities on these two horizons satisfy the first law of thermodynamics, respectively [@3; @5; @35]. However, thermodynamic systems denoted by the two horizons are not independent, since thermodynamic quantities on them are functions depended on variables of mass $M$, electric charge $Q$ and cosmological constant ${l^{2}}$ satisfy the first law of thermodynamics. When parameters of state of $n+2-$dimensional DRNdS space satisfy the first law of thermodynamics, the entropy is [@16; @17; @18] $$S=\frac{{Vol({S^{n}})}}{{4G}}r_{c}^{n}(1+{x^{n}}+{f_{n}}(x))={S_{c,+}}+{S_{AB}},$$where $x={r_{+}}/{r_{c}}$,${S_{c,+}}=\frac{{Vol({S^{n}})}}{{4G}}r_{c}^{n}(1+{x^{n}})$ and ${S_{AB}}=\frac{{Vol({S^{n}})}}{{4G}}r_{c}^{n}{f_{n}}(x)$ are entropies with and without the interaction between the two horizons, respectively, and $${f_{n}}(x)=\frac{{3n+2}}{{2n+1}}{(1-{x^{n+1}})^{n/(n+1)}}-\frac{{(n+1)(1+{x^{2n+1}})+(2n+1)(1-2{x^{n+1}}-{x^{2n+1}})}}{{(2n+1)(1-{x^{n+1}})}}.$$The volume of $n+2-$dimensionalDRNdS space is[@3; @7; @13] $$V={V_{c}}-{V_{+}}=\frac{{Vol({S^{n}})}}{{(n+1)}}r_{c}^{n+1}(1-{x^{n+1}}).$$When parameters of state of $n+2-$ dimensional DRNdS space satisfy the first law of thermodynamics, the effective temperature is [@16; @17; @18] $$\begin{aligned}
{T_{eff}} &=&(1-{x^{n+1}})\frac{{{{(\partial M/\partial x)}_{{r_{c}}}}(1-{x^{n+1}})+{r_{c}}{x^{n}}{{(\partial M/\partial {r_{c}})}_{x}}}}{{Vol({S^{n}})r_{c}^{n}{x^{n-1}}(1+{x^{n+2}})}} \notag \\
&=&\frac{{B(x)}}{{Vol({S^{n}}){r_{c}}{x^{2n-1}}{\omega _{n}}(1+{x^{n+2}})}},\end{aligned}$$where $$\begin{aligned}
B(x) &=&{x^{n}}[(n-1){x^{n-2}}-(n+1){x^{n}}+2{x^{2n-1}}+(n-1){x^{2n-1}}(1-{x^{2}})] \notag \\
&&-\frac{{n\omega _{n}^{2}{Q^{2}}[(n-1){x^{n+1}}(1-{x^{2n}})-2n{x^{n+1}}+(n-1)+(n+1){x^{2n}}]}}{{8(n-1)r_{c}^{2n-2}}} \notag \\
&=&{x^{n}}[(n-1){x^{n-2}}-(n+1){x^{n}}+2{x^{2n-1}}+(n-1){x^{2n-1}}(1-{x^{2}})] \notag \\
&&-\frac{{2\phi _{c}^{2}(n-1)[(n-1){x^{n+1}}(1-{x^{2n}})-2n{x^{n+1}}+(n-1)+(n+1){x^{2n}}]}}{n},\end{aligned}$$where ${\phi _{c}}=\frac{n}{{4(n-1)}}\frac{{{\omega _{n}}Q}}{{r_{c}^{n-1}}}$ is electric potential on the universe horizon. The effective pressure $P_{eff}$, isochoric heat capacity $C_{veff}$ and isobaric heat capacity $C_{P_{veff}}$ in $n+2-$dimensional DRNdS spaceare $${P_{eff}}=\frac{{D(x)}}{{{\omega _{n}}Vol({S^{n}})(1-{x^{n+1}})r_{c}^{2}{x^{n-1}}(1+{x^{n+2}})}},$$where $$\begin{aligned}
D(x) &=&\left[ {(n-1){x^{n-2}}-(n+1){x^{n}}+2{x^{2n-1}}-\frac{{n\omega
_{n}^{2}{Q^{2}}(2n{x^{n+1}}-(n-1)-(n+1){x^{2n}})}}{{8(n-1)r_{c}^{2n-2}{x^{n}}}}}\right] \times \notag \\
&&(1+{x^{n}}+f(x)) \\
&&-\left[ {(n-1){x^{n-1}}(1-{x^{2}})-\frac{{n\omega _{n}^{2}{Q^{2}}(1-{x^{2n}})}}{{8r_{c}^{2n-2}{x^{n-1}}}}}\right] \left( {{x^{n-1}}+\frac{{f^{\prime
}(x)}}{n}}\right) (1-{x^{n+1}}), \notag\end{aligned}$$$$\begin{aligned}
{C_{V}} &=&{T_{eff}}{\left( {\frac{{\partial S}}{{\partial {T_{eff}}}}}\right) _{V}}={T_{eff}}\frac{{{{\left( {\frac{{\partial S}}{{\partial {r_{c}}}}}\right) }_{x}}{{\left( {\frac{{\partial V}}{{\partial x}}}\right) }_{{r_{c}}}}-{{\left( {\frac{{\partial S}}{{\partial x}}}\right) }_{{r_{c}}}}{{\left( {\frac{{\partial V}}{{\partial {r_{c}}}}}\right) }_{x}}}}{{{{\left( {\frac{{\partial V}}{{\partial x}}}\right) }_{{r_{c}}}}{{\left( {\frac{{\partial {T_{eff}}}}{{\partial {r_{c}}}}}\right) }_{x}}-{{\left( {\frac{{\partial V}}{{\partial {r_{c}}}}}\right) }_{x}}{{\left( {\frac{{\partial {T_{eff}}}}{{\partial x}}}\right) }_{{r_{c}}}}}} \\
&=&\frac{1}{{4G(1-{x^{n+1}})}}\times \notag \\
&&\frac{{-Vol({S^{n}})r_{c}^{n}B(x)n{x^{n}}{{(1+{x^{n+2}})}^{2}}}}{{{\bar{B}(x){x^{n+1}}(1+{x^{n+2}})-(1-{x^{n+1}}){x(1+{x^{n+2}})B^{\prime
}(x)-B(x)[2n-1+(3n+1){x^{2n+2}}]}}}} \notag\end{aligned}$$where $$\begin{aligned}
\bar{B}(x) &=&{x^{n}}[(n-1){x^{n-2}}-(n+1){x^{n}}+2{x^{2n-1}}+(n-1){x^{2n-1}}(1-{x^{2}})] \notag \\
&&-\frac{{n\omega _{n}^{2}{Q^{2}}(2n-1)[(n-1){x^{n+1}}(1-{x^{2n}})-2n{x^{n+1}}+(n-1)+(n+1){x^{2n}}]}}{{8(n-1)r_{c}^{2n-2}}}, \notag \\
B^{\prime }(x) &=&\frac{{dB(x)}}{{dx}},D^{\prime }(x)=\frac{{dD(x)}}{{dx}},\end{aligned}$$$$\begin{aligned}
{C_{{P_{eff}}}} &=&{T_{eff}}{\left( {\frac{{\partial S}}{{\partial {T_{eff}}}}}\right) _{{P_{eff}}}}={T_{eff}}\frac{{{{\left( {\frac{{\partial S}}{{\partial {r_{c}}}}}\right) }_{x}}{{\left( {\frac{{\partial {P_{eff}}}}{{\partial x}}}\right) }_{{r_{c}}}}-{{\left( {\frac{{\partial S}}{{\partial x}}}\right) }_{{r_{c}}}}{{\left( {\frac{{\partial {P_{eff}}}}{{\partial {r_{c}}}}}\right) }_{x}}}}{{{{\left( {\frac{{\partial {P_{eff}}}}{{\partial x}}}\right) }_{{r_{c}}}}{{\left( {\frac{{\partial {T_{eff}}}}{{\partial {r_{c}}}}}\right) }_{x}}-{{\left( {\frac{{\partial {P_{eff}}}}{{\partial {r_{c}}}}}\right) }_{x}}{{\left( {\frac{{\partial {T_{eff}}}}{{\partial x}}}\right) }_{{r_{c}}}}}} \notag \\
&=&r_{c}^{n}\frac{{Vol({S^{n}})B(x)E(x)}}{{4GH(x)}},\end{aligned}$$where $$\begin{aligned}
E(x) &=&[n{x^{n-1}}+f^{\prime }(x)][\bar{D}(x)-2D(x)](1-{x^{n+1}})x(1+{x^{n+2}}) \notag \\
&&-n[1+{x^{n}}+f(x)]\{D^{\prime }(x)x(1-{x^{n+1}})(1+{x^{n+2}}) \notag \\
&&-D(x)[(n-1)-2n{x^{n+1}}+(2n+1){x^{n+2}}-(3n+2){x^{2n+3}}]\}, \notag \\
H(x) &=&\bar{B}(x)\{D^{\prime }(x)x(1-{x^{n+1}})(1+{x^{n+2}})-D(x)[(n-1)
\notag \\
&&-2n{x^{n+1}}+(2n+1){x^{n+2}}-(3n+2){x^{2n+3}}]\} \\
&&+(1-{x^{n+1}})[\bar{D}(x)-2D(x)]\left[ {x(1+{x^{n+2}})B^{\prime
}(x)-B(x)[2n-1+(3n+1){x^{2n+2}}]}\right] . \notag \\
\bar{D}(x) &=&\frac{{n\omega _{n}^{2}{Q^{2}}(2n{x^{n+1}}-(n-1)-(n+1){x^{2n}})}}{{4r_{c}^{2n-2}{x^{n}}}}(1+{x^{n}}+f(x)) \notag \\
&&-\frac{{n(n-1)\omega _{n}^{2}{Q^{2}}(1-{x^{2n}})}}{{4r_{c}^{2n-2}{x^{n-1}}}}\left( {{x^{n-1}}+\frac{{f^{\prime }(x)}}{n}}\right) (1-{x^{n+1}}). \notag\end{aligned}$$The coefficient of isobaric volume expansion and isothermal compressibility in $n+2-$ dimensional DRNdS spaceis given by $$\begin{aligned}
\alpha &=&\frac{1}{V}{\left( {\frac{{\partial V}}{{\partial {T_{eff}}}}}\right) _{{P_{eff}}}}=\frac{1}{V}\frac{{{{\left( {\frac{{\partial V}}{{\partial {r_{c}}}}}\right) }_{x}}{{\left( {\frac{{\partial {P_{eff}}}}{{\partial x}}}\right) }_{{r_{c}}}}-{{\left( {\frac{{\partial V}}{{\partial x}}}\right) }_{{r_{c}}}}{{\left( {\frac{{\partial {P_{eff}}}}{{\partial {r_{c}}}}}\right) }_{x}}}}{{{{\left( {\frac{{\partial {P_{eff}}}}{{\partial x}}}\right) }_{{r_{c}}}}{{\left( {\frac{{\partial {T_{eff}}}}{{\partial {r_{c}}}}}\right) }_{x}}-{{\left( {\frac{{\partial {P_{eff}}}}{{\partial {r_{c}}}}}\right) }_{x}}{{\left( {\frac{{\partial {T_{eff}}}}{{\partial x}}}\right) }_{{r_{c}}}}}} \notag \\
&=&-\frac{{{\omega _{n}}(n+1)Vol({S^{n}}){x^{2n-1}}(1+{x^{n+2}})}}{{H(x)}}{r_{c}}\{{x^{n+1}}[\bar{D}(x)-2D(x)](1+{x^{n+2}}) \\
&&+D^{\prime }(x)x(1-{x^{n+1}})(1+{x^{n+2}})-D(x)[(n-1)-2n{x^{n+1}}+(2n+1){x^{n+2}}-(3n+2){x^{2n+3}}]\}. \notag\end{aligned}$$$$\begin{aligned}
{\kappa _{{T_{eff}}}} &=&-\frac{1}{V}{\left( {\frac{{\partial V}}{{\partial {P_{eff}}}}}\right) _{{T_{eff}}}}=\frac{1}{V}\frac{{{{\left( {\frac{{\partial
V}}{{\partial {r_{c}}}}}\right) }_{x}}{{\left( {\frac{{\partial {T_{eff}}}}{{\partial x}}}\right) }_{{r_{c}}}}-{{\left( {\frac{{\partial V}}{{\partial x}}}\right) }_{{r_{c}}}}{{\left( {\frac{{\partial {T_{eff}}}}{{\partial {r_{c}}}}}\right) }_{x}}}}{{{{\left( {\frac{{\partial {P_{eff}}}}{{\partial x}}}\right) }_{{r_{c}}}}{{\left( {\frac{{\partial {T_{eff}}}}{{\partial {r_{c}}}}}\right) }_{x}}-{{\left( {\frac{{\partial {P_{eff}}}}{{\partial {r_{c}}}}}\right) }_{x}}{{\left( {\frac{{\partial {T_{eff}}}}{{\partial x}}}\right) }_{{r_{c}}}}}} \notag \\
&=&\frac{{r_{c}^{2}{\omega _{n}}(n+1)Vol({S^{n}})(1-{x^{n+1}}){x^{n-1}}(1+{x^{n+2}})}}{{H(x)}}\times \\
&&\left\{ {(1-{x^{n+1}})\left[ {x(1+{x^{n+2}})B^{\prime }(x)-B(x)[2n-1+(3n+1){x^{2n+2}}]}\right] -{x^{n+1}}(1+{x^{n+2}})\bar{B}(x)}\right\} \notag\end{aligned}$$
Numerical solutions for the isobaric heat capacity $C_{p_{eff}}$ and coefficients of isobaric volume expansion $\alpha$ and isothermal compressibility $\kappa_{T_{eff}}$ with the position ratio of the black hole horizon to the universe horizon $x$ have been given in Fig. 1, Fig. 2 and Fig. 3, respectively. Form the figures, it is clear that values of $C_{p_{eff}}$, $\alpha$ and $\kappa_{T_{eff}}$ have sudden change with the charge of the spacetime is a constant, which is similar to the Van der Waals system. Moreover, as the dimension of the space increases, the value of $x$ to denote the sudden change also increases. This indicates that the point of the phase transition is closely related to the dimensions of the space time.
$ n=2 $ $n=4 $ $n=6$
--------------- --------- -------- -------- -- -- --
$ x_c $ 0.5894 0.7053 0.7674
$ T_{eff}^c $ 0.0301 0.1127 0.2095
$ P_{eff}^c$ 0.0238 0.0952 0.1825
\[tab:1\]
From Table 1, it is clear that the phase transition point is different with different dimensions. Moreover, as the dimension increases, the critical value of the phase transition point and the effective pressure and temperature are all increased.
Entropy force
=============
The entropy force of a thermodynamic system can be expressed as [25,26,27,36,37,38,39]{} $$F = - T\frac{{\partial S}}{{\partial r}},$$ where $T$ is the temperature and $\gamma$ is the radius.
From Eq.(2.3), the entropy caused by the interaction between horizons of black holes and the universe is $${S_{AB}} = \frac{{Vol({S^n})}}{{4G}}r_c^n{f_n}(x).$$
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From Fig.4, it shows that as the dimension increases, the intersectional point of the curve and the $x-$axis is moving to the right.In other words, the value of ${x_0}$ increases with the dimension,which denotes the point where the entropy caused by the interaction between horizons of black holes and the universe changes between positive and negative values. The entropy given in Eq. (2.4) does not contain explicit electric charge Q dependent $Q$ terms.
From Eq. (3.1), the entropy force of the two interactive horizons can be given as $$F=-{T_{eff}}{\left( {\frac{{\partial {S_{AB}}}}{{\partial r}}}\right) _{{T_{eff}}}},$$where ${T_{eff}}$ is the effective temperature of the considering case and $r={r_{c}}-{r_{+}}={r_{c}}(1-x)$. Then it gives $$\begin{aligned}
F(x) &=&-{T_{eff}}\frac{{{{\left( {\frac{{\partial {S_{f}}}}{{\partial {r_{c}}}}}\right) }_{x}}{{\left( {\frac{{\partial {T_{eff}}}}{{\partial x}}}\right) }_{{r_{c}}}}-{{\left( {\frac{{\partial {S_{f}}}}{{\partial x}}}\right) }_{{r_{c}}}}{{\left( {\frac{{\partial {T_{eff}}}}{{\partial {r_{c}}}}}\right) }_{x}}}}{{(1-x){{\left( {\frac{{\partial {T_{eff}}}}{{\partial x}}}\right) }_{{r_{c}}}}+{r_{c}}{{\left( {\frac{{\partial {T_{eff}}}}{{\partial {r_{c}}}}}\right) }_{x}}}} \\
&=&\frac{{-B(x)r_{c}^{n-2}}}{{4G{x^{2n-1}}{\omega _{n}}(1+{x^{n+2}})}}\times
\notag \\
&&\frac{{n{f_{n}}(x)\left[ {x(1+{x^{n+2}})B^{\prime }(x)-B(x)[2n-1+(3n+1){x^{2n+2}}]}\right] +x(1+{x^{n+2}})\bar{B}(x)f{_{n}^{\prime }}(x)}}{{(1-x)\left[ {x(1+{x^{n+2}})B^{\prime }(x)-B(x)[2n-1+(3n+1){x^{2n+2}}]}\right] +{x^{2}}\bar{B}(x)(1+{x^{n+2}})}}. \notag\end{aligned}$$
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Fig.5 shows that the entropy force increases with the dimension. Moreover, when $n=2$ and $x={x_{0}}=0.9009$, $n=4$ and $x={x_{0}}=0.9035$ , and $n=6$ and $x={x_{0}}=0.9224$ , $F({x_{0}})=0$, respectively. It indicates that the value of ${x_{0}}$ increases with the dimension, which denotes the point where the direction of the entropy force changes.
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Fig. 6 shows that when $Q=0.001$ and $x={x_{0}}=0.9014,$ $Q=0.01$ and $x={x_{0}}=0.9009$, and $Q=0.1$ and $x={x_{0}}=0.8120,F({x_{0}})=0$ respectively. It implies that as the electric charge increases, the value of ${x_{0}}$ decreases, which denotes the point where the entropy force changes between positive and negative values.
From Fig. 5, we can obtain that when $x\rightarrow 1,$ $F(x\rightarrow
1)\rightarrow \infty $, and then according to Eq. (2.6), ${T_{eff}}\rightarrow 0$. This result indicates that the interaction between horizons of black holes and the universe tends to infinity, which contrast to the third law of thermodynamics. In order to protect the laws of thermodynamics, the black hole horizon and the cosmological horizon can not coincide with each other. Based on this fact, we take $1-\Delta x$ as the maximum value of $x$, where $\Delta x$ is a minor dimensionless quantity. The value of $\Delta x$ can be determined by the speed of the cosmic accelerated expansion at the position $x$.
According to the expression of the entropy force, when ${x_{0}}<x<1-\Delta
x, $ ${F}(x)>0$ which indicates that the interaction between horizons of black holes and the universe is repulsive. Consequently,the expansion of the cosmological horizon can be accelerated by the entropy force in the absence of other forces. In Fig. 5, it is known that the entropy force is different at different positions. Thus the expansion of the universe is variable acceleration in the interval of ${x_{0}}<x<1-\Delta x$. While when $0<x<{x_{0}}$, ${F}(x)<0$, which indicates that the interaction between horizons of black holes and the universe is attractive, and then the expansion of the universe is variable deceleration in this interval.
From Fig. 5, we find that when the area enclosed by the curve $F(x)-x$ and the $x-$axis with the interval of ${x_{0}}<x<1-\Delta x$ is larger than the area enclosed by the same curve and the $x-$axis with the interval of $0<x<{x_{0}}$,the cosmic expansion is from acceleration to deceleration. It gives an expanding universe. While when the former area is less than or equal to the latter one, the cosmic expansion is from acceleration to deceleration. Moreover, when these two areas are equal at the position ratio $x$ , which belongs to the interval of $\bar{x}<x<{x_{0}},$ the universe is accelerated shrinkage from the position ratio $\bar{x}$ to the position ratio ${x_{0}}$, where $\bar{x}$ is determined when the area between the curve and the x-axis with the interval of $[\bar{x},1-\Delta x]$ is zero. After the universeshrink to the position ratio $x=1-\Delta x,$ the evolution of the universe begins the next cycle. It gives a oscillating universe.
Conclusions
===========
When horizons of black holes and the universe are irrelevant, thermodynamic systems of them are independent. Since the radiational temperature on them is different, the requirement of thermodynamic equilibrium stability can not be meet. Therefore, the space is unstable. While when they are related, the effective temperature ${T_{eff}}$ and pressure ${P_{eff}}$ for DRNdS space can be obtained from Eqs.(2.6) and (2.8). According to curves ${C_{{P_{eff}}}}-x$, $\alpha -x$, and ${\kappa _{{T_{eff}}}}-x$, when $x={x_{c}}$, the phase transition of DRNdS space time occurs. Since its entropy and volume are continuous, the phase transition is the second-order one according to Ehrenfest’s classification. It is similar to the case occured in AdS black holes [@19; @20; @21; @22; @23; @24; @44; @45]. From Eq. (2.10), we find that the isochoric heat capacity ${C_{v}}$ of DRNdS space is non-trivial, which is similar to the system of Van der Waals, but different from AdS black holes. In second 2, the effect of the dimension on the phase transition point is analyzed, which lays the foundation for the further study of the thermodynamic characteristics of the high-dimensional complex dS space.
From Fig. 5, we find that when the area enclosed by the curve $F(x)-x$ and the $x-$axis with the interval of ${x_{0}}<x<1-\Delta x$ is larger than the area enclosed by the same curve and the $x-$axis with the interval of $0<x<{x_{0}}$, the cosmic expansion is from acceleration to deceleration. It gives an expanding universe. While when the former area is less than or equal to the latter one, the cosmic expansion is from acceleration to deceleration. Moreover, when these two areas are equal at the position ratio $x$, which belongs to the interval of $\bar{x}<x<{x_{0}}$, the universe is accelerated shrinkage from the position ratio $\bar{x}$ to the position ratio ${x_{0}}$, where $\bar{x}$ is determined when the area between the curve and the $x-$axis with the interval of $[\bar{x},1-\Delta x]$ is zero. After the universe shrink to the position ratio $x=1-\Delta x$, the evolution of the universe begins the next cycle. It gives a oscillating universe.
Whether the universe is an expanding one or a oscillating one is determined by the value of the minor dimensionless quantity. From Fig. 5 and Fig.6, we find that the position, where the entropy force changes between positive and negative values, is greatly affected by the dimension, but commonly by the electric charge. Therefore, the effect of the dimension on the cosmic expansion is greater than the electric charge. Moreover, since the curve $F(x) - x$ is continuous at the phase transition point ${x_c}$, the entropy force can not be affected by the phase transition in the space with a given dimension and electric charge. The amplitude and the value of the entropy force is only determined by the position ratio $x$. According to our research result, the entropy force between horizons of black holes and the universe can be taken as one of the reasons for the cosmic expansion, which provides a new approach for people to explore the physical mechanism of the cosmic expansion.
Acknowledgements {#acknowledgements .unnumbered}
================
We thank Prof. Z. H. Zhu for useful discussions. This work was supported by the National Natural Science Foundation of China (Grant No. 11847123, 11847128, 11475108, 11705106, 11705107, 11605107), Science $\&$ Technology Department of Yunnan Province - Yunnan University Joint Funding (Grant No. 2019FY003005) and Donglu Youth Teacher Plan of Yunnan University.
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[^1]: [E-mail: wjun@ynu.edu.cn](mailto:wjun@ynu.edu.cn)
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Introduction
============
Colloids are all around us. From milk to microreactors, from pie filling to paint, and from suspensions to sieves, colloidal materials are important in a variety of products and technologies and are the basis for a new class of functional materials. The advent of applications based on engineered crystalline materials, such as photonic bandgap materials, [@Yablonovitch87; @John87] has put even greater focus on colloids. Both the size of colloidal particles and the interparticle interaction are tunable, which provides the basis for the manufacture of ordered structures with desired lattice spacings and space groups as well as mechanical, thermal, and electrical properties.
Over the years, considerable amounts of experimental crystallographic data have been collected for various colloidal systems. [@Russel89] While the tabulation of the relationship between the interparticle interaction and the symmetry of the crystal lattice is undoubtedly valuable, it is desirable, if not necessary, that the synthesis of colloidal crystals with specific space group be based on analytic rather than empirical insight into the mechanisms of self-assembly. In principle this should be possible because the interaction between the particles is much simpler than in atomic or molecular crystals: colloidal particles are made of hundreds or thousands of atoms, and the effective interaction between the particles is less specific and thus much simpler than interatomic interactions. Thus the interaction between colloidal particles is chiefly determined by their mechanical as opposed to chemical structure. Here we will address those colloidal particles that are characterized by a relatively dense core and a fluffy corona. In this case, the interparticle potential may be approximated fairly well by the simple hard-core repulsion dressed with a repulsive short-range interaction of finite strength.
Present theoretical understanding of the phase behavior and stability of the various colloidal systems is quite impressive, especially in the case of charged colloidal suspensions interacting via screened Coulomb potentials. [@Hone83; @Kremer86; @Rosenberg87; @Rascon97] It is now well established both by analytical approaches [@Hone83; @Rascon97] and by simulations [@Kremer86] that the solid part of the phase diagram of charged colloids includes face-centered cubic (FCC) and body-centered cubic (BCC) lattices. In the case of a convex interparticle potential – its simplest variant being the square shoulder interaction [@Bolhuis97] – the phase diagram is even more complex and includes the dense and the loose FCC and BCC lattices. [@Stell72] More elaborate soft potentials such as the interaction between star polymers lead to body-centered orthorhombic (BCO) and diamond lattices in addition to FCC and BCC lattices. [@Watzlawek99]
While extremely important, these theories do not provide a robust explanation of the stability of colloidal crystals, and the aim of this study is to look at the problem from a more geometrical point of view and to capture the statistical mechanics of colloids in a new self-organization principle. One such principle is the maximum packing fraction rule, which states that pure excluded-volume interactions favor an expanded close-packed structure, thereby maximizing the configurational entropy. In the case of monodisperse hard spheres, such an arrangement corresponds to the FCC lattice. [@Mau99; @Hales00] However, many soft-sphere systems form non-close-packed lattices, including the BCC, BCO, and diamond lattices as well as the A15 lattice [@Rivier94] observed in crystals of self-assembled micelles of some dendritic polymers. [@Balagurusamy97] Can we understand the existence of this rather loosely packed structure? Is there another simple geometrical principle that describes the self-organization of soft spheres and is analogous but opposing to the maximum packing fraction rule? We pursue this question using an idealized model and find an analogy between the soft-sphere crystals and dry soap froths, the latter being described by Kelvin’s problem of finding the minimal-area regular partition of space into cells of equal volume. Within this framework, we propose a new principle of area-minimization that can favor these loosely packed lattices of soft spheres. [@Ziherl00]
Having proposed our semiquantitative and intuitive explanation for the behavior of soft-sphere colloids, we have checked our predictions within a more rigorous statistical-mechanical model to see whether the theoretically calculated phase diagram includes some of these loose-packed structures – the A15 lattice in particular. In previous theoretical studies this structure was not considered as a trial state of a repulsive colloidal system, and the solid part of the phase diagram was shared virtually exclusively by FCC and BCC phases. [@Hone83; @Kremer86; @Rascon97] We scan the phase diagram for the case of square-shoulder soft potential, the focus of other studies, and determine the range of widths of the soft potential where the A15 lattice is the colloidal ground state.
The paper is organized as follows: in Section II, we describe our model of colloidal self-organization and establish the analogy between these systems and soap froth. We digress to describe the Kelvin problem and its conjectured solution. We show that this analogy leads to a second global principle of self-assembly – the primary one being the principle of maximum packing entropy – and that the two mechanisms give rise to frustration. We apply these ideas to a dendrimer compound (which crystallizes into the A15 lattice) as well as to other systems. In Section III we reexamine the phase diagram of the hard-core–square-shoulder interaction, which captures many features of real colloidal systems. We analyze it within the cellular free volume approximation using a numerical model whose main advantage is that the colloidal interaction is treated nonperturbatively. Section IV concludes the paper.
Colloidal systems as area-minimizing structures
===============================================
Recently, dendrimers composed of a poly(benzyl ether) core segments decorated with dodecyl chains [@dendrimer] were synthesized with the intention of producing a molecule with a conical, fan-shaped architecture. [@Balagurusamy97] Since it is known that many dendrimers spontaneously self-assemble into supramolecular clusters, the underlying rationale was motivated by the possibility of creating spherical micellar-like objects a few nanometers in diameter. These dendrimers do indeed form spheres which, in turn, form a crystal lattice with the Pm$\overline{3}$n space group, also known as the A15, Q$^{223}$, and $\beta$-Tungsten lattice. This lattice belongs to the cubic system, and its unit cell includes 8 sites which can be divided into 3 pairs of columnar sites and 2 interstitial sites. The columnar sites lie evenly spaced along the bisectors of the faces of the unit cell and can be thought of as forming three mutually perpendicular and interlocking columns. The interstitial sites fill out the space between the columns: one is at the center of the cell and the other one is at the vertex (Fig. \[lattices\]).
Though the structure of dendrimer crystals has been studied in great detail, [@Percec98; @Hudson99] it remains unclear why the dendrimer assemblies should make the A15 lattice. The micelles are nearly spherically symmetric and very monodisperse: [@Balagurusamy97] given their chemical composition, the interaction between the micelles must be predominantly steric. If they were of uniform density, the interaction arising from the impenetrability of the micelles would be well-described by a spherically symmetric hard-core potential. In this case, one might expect the spheres to assemble into an FCC lattice to maximize their positional entropy – because it has the largest packing fraction, it maximizes the volume available to each sphere. [@Mau99; @Hales00] On the other hand, the A15 lattice is rather loosely packed, with the same packing fraction as that of the simple cubic (SC) lattice. Thus the A15 lattice is very inefficient from the point of view of the center-of-mass entropy of each sphere.
If the micelles were structureless hard spheres, their free energy would depend only on their position. However, the dendrimers have a well-defined structure consisting of a more or less compact core of benzyl ether rings and a floppy, squishy corona of alkyl chains. In this case, the stability of a certain arrangement of micelles does not depend only on the hard-core repulsion between the cores but also on the interaction between the brush-like coronas: the larger the overlap between the neighboring micelles, the more constrained the conformations of the chains within the coronas and the smaller their orientational entropy (Fig. \[inter\]). This effective interaction is, of course, repulsive but short-ranged: at distances larger than the diameter of the micelles, their coronas do not overlap and the interaction vanishes.
Though the hard and the soft part of the repulsion both arise from the steric interaction between the dendrimers, their dependence on the density is very different. The hard cores lead to an inaccessible volume, while the matrix of interpenetrating soft coronas leads to a softer entropic repulsion. We can regard the matrix of coronas as a bilayer of dodecyl chains wrapped around each hard core. The free energy of these bilayers decreases monotonically with thickness. However, the volume of this soft matrix is fixed by the difference between the total volume and that of the hard cores and can be written as the product of the total area ($A$) of these bilayers and their average thickness ($d$), so that at a given density $$Ad={\rm constant.}$$ Since the soft repulsion of the tails favors larger bilayer thicknesses $d$, the coronal entropy is maximized when the area $A$ is minimized, or, in other words, when the interfacial area between the neighboring micelles is a minimal surface (Fig. \[minimalarea\]). We should emphasize that the interface between the micelles is a mathematical concept that embodies the membrane-like, two-dimensional character of the interdigitated coronas, and it does not correspond to, say, the position of a particular segment of the dendrimer molecule.
It is essential to note that this minimal-area principle is incompatible with close packing and can favor lattices other than FCC, thereby giving rise to frustration in these systems and a rich phase diagram. If the free energy of the lattice of micelles depended only on the interfacial energy, the system would behave as periodic membrane enclosing bubbles of equal volume – as an ideal, dry soap froth. At this juncture the solution to the problem of minimizing the interfacial area of a set of equal volume, space-filling cells is not known. However, it is known that the area of the BCC “foam” is smaller than that of the FCC foam and that the A15 foam has a smaller area still.
Kelvin’s problem
----------------
The problem of finding the ideal configuration of a soap froth was introduced by Kelvin in 1887 while studying how light propagated in a crystal, the relation being based on the now abandoned notion of aether. He realized that the dry soap froth problem could be cast in the mathematically precise form: What regular partition of space into cells of equal volume has the smallest area of cells? As is almost always the case, the plain and simple formulation of the problem is a harbinger of its complexity.
Kelvin built on Plateau’s investigations of the stability of soap films, summarized by two rules which represent mechanical equilibrium: In an equilibrium froth, (i) adjacent faces meet at an angle of $120^\circ$ and (ii) the edges (the so-called Plateau borders) must form a tetrahedral angle of $109^\circ28'$. It follows that arrangements with more than 3 faces meeting at a common edge are unstable as are junctions with more than 4 edges. At that time, these rules did not have a theoretical background and neither did Kelvin’s work. Kelvin’s approach relied on experiments based on soldered wire frames dipped into soap solution and the observation of the evolution of the soap film spanned by the frame. [@Thomson87] Kelvin was led to the conclusion that his problem was solved by a lattice of polygons with the topology of an orthic tetrakaidecahedron consisting of 6 quadrilateral and 8 hexagonal faces. In more modern terms these shapes are known as the Wigner-Seitz cell of the BCC lattice (Fig. \[bubbles\]). Kelvin subsequently worked out the exact shape of the faces and showed that to satisfy the Plateau rules, the edges of the tetrakaidecahedra must be slightly curved and the hexagonal faces must be somewhat nonplanar.
Though a conjecture, Kelvin’s partition of space was thought to be the solution of the problem by the mathematics community, despite the realization the proof might be highly elusive. [@Klarreich00] Indeed, these types of problems are notoriously difficult. For example, even the Plateau rules themselves were nothing but experimental facts until 1976 when they were put on a firm theoretical footing by Taylor [@Taylor76] and it was only in 1999 that Hales proved that the regular hexagon is the solution of the two-dimensional variant of the Kelvin problem. [@Hales01] We also note that the related Kepler problem of packing hard spheres as densely as possible turned out to be equally challenging: the well-known FCC or HCP packing was demonstrated to be the most efficient in 1998, the proof being furnished again by Hales. [@Hales00]
Kelvin’s conjecture stood unchallenged for more than a century until 1994 when Weaire and Phelan discovered that a froth with the symmetry of the A15 lattice has an area smaller than BCC lattice by 0.3%. [@Weaire94a] While this may seem to be a small difference, it is, in fact, significant: the relative difference in the areas of Kelvin’s BCC structure and the FCC-type partition of rhombic dodecahedra (unstable as a froth because it contains vertices joining 8 edges) is 0.7%. Thus the A15 foam is a 50% improvement on this scale.
The A15 foam consists of 6 Goldberg tetrakaidecahedra, each with 2 hexagonal and 12 pentagonal faces, which form three sets of interlocking columns, and 2 irregular pentagonal dodecahedra at the interstices. Since it is composed of two types of cells, the Weaire-Phelan soap froth is obviously less symmetric than Kelvin’s froth. However, it was not surprising to some that it has a rather small surface area. The A15 lattice is derived from the polytope $\{3,3,5\}$, a partition of positively curved space consisting of 120 regular dodecahedral bubbles. As suggested by Kl' eman and Sadoc, positive spatial curvature relieves the frustration brought about by the incompatibility of the optimal local arrangement of bubbles and the structure of flat three-dimensional space. The polytope $\{3,3,5\}$ should be regarded as the ideal template whence the various tetrahedrally close-packed (TCP) lattices (the layered TPC lattices are also known as Frank-Kasper phases) are derived via decurving, [*i.e.*]{}, by substituting some of the dodecahedral cells by bubbles with 14 ($=12$ pentagonal $+$ 2 hexagonal), 15 ($=12$ pentagonal $+$ 3 hexagonal), or 16 ($=12$ pentagonal $+$ 4 hexagonal) faces. [@Rivier94; @Kleman79; @Sadoc99] Cells with 13 faces are forbidden for topological reasons, and those with more than 16 faces are dynamically unstable. [@Rivier94]
There are 24 known ways of decurving the polytope $\{3,3,5\}$ and thus 24 TCP crystal lattices. [@Rivier94] With 2 types of bubbles and 8 bubbles per unit cell, the A15 lattice is among the simplest: the unit cell of the most complicated structure, the so-called I lattice, consists of 228 bubbles that include all 4 types of bubbles. [@Rivier94] Given the rationale for the success of the A15 lattice, it is not unreasonable to expect that other TPC lattices may have an even smaller surface area. Indeed, the discovery of Weaire and Phelan renewed the interest in the field. Subsequently some of the remaining TCP-type soap froths have been studied, [@Weaire97] facilitated by Surface Evolver, a remarkable software package developed by Brakke. [@Brakke92] In addition, other classes of periodic and quasiperiodic partitions based on Kelvin’s BCC and Williams’ body-centered-tetragonal (BCT) bubbles have been suggested. [@Glazier94] At this time, however, the A15 foam has not been bested – it stands as the tentative solution of Kelvin’s problem.
The connection between the observed crystal structure and the soft part of the intermicellar potential should now be clear: if it were absent, the hard cores would favor an FCC lattice, but instead the equilibrium structure is a different lattice with a larger bulk free energy but a smaller surface free energy. The only variables in this model are the parameters of the surface interaction, which we will estimate roughly. In view of the experimental data on soft spheres, we will compare the FCC, BCC, and A15 lattices and show that in the case of the dendrimer micelles, it is reasonable to expect that the area-minimizing A15 lattice is favored over the close-packed FCC lattice.
Bulk free energy
----------------
Within the framework of the two competing ordering principles, we propose a simple and approximate theory of colloidal crystals where the bulk and the surface terms are coupled only through the constraint of fixed volume. As far as the bulk free energy is concerned, the micelles will be treated as hard spheres of diameter $\sigma$, and the surface interaction will be calculated as if the matrix of coronas were a thin structureless layer so that we may neglect the effects of curvature.
We start our analysis with the bulk term. The configuration integral of a one-component classical system of $N$ particles confined to a volume $V$ is $$Z=\frac{1}{\lambda^{3N}N!}\int_V\prod_{i=1}^N{\rm d\bf r}_i
\,\exp\biglb(-U({\bf r}_1,{\bf r}_2,\ldots{\bf r}_N)/k_{\scriptscriptstyle
B}T\bigrb),$$ where $\lambda=\sqrt{h^2/2\pi mk_{\scriptscriptstyle B}T}$ is the thermal de Broglie wavelength and $U({\bf r}_1,{\bf r}_2,\ldots{\bf r}_N)=\frac{1}{2}
\sum_{i,j=1}^Nu({\bf r}_i,{\bf r}_j)$ is the total interaction energy consisting of pairwise interactions $u({\bf r}_i,{\bf r}_j)=u(|{\bf r}_i-{\bf r}_j|)$. Despite the simplicity of the hard-core potential, $Z$ cannot be calculated analytically, and one must resort either to numerical approaches, such as Monte Carlo simulations, or to approximate analytical methods. The latter are more appropriate for our purposes since we are seeking a simple, heuristic explanation of crystal structure. When considering crystal phases we can assume that the particles are localized within cells formed by their neighbors and thus the configuration integral breaks up into $N$ single-particle integrals so that $$Z\approx\left(\lambda^{-3}\int_{V_0}{\rm d}{\bf r}\,\exp(-u_{\rm eff}({\bf r})/
k_{\scriptscriptstyle B}T)\right)^N,
\label{cellmodel}$$ where $V_0$ is the volume of the cell (usually the Wigner-Seitz cell), $u_{\rm eff}(\bf r)$ is the effective potential felt by the particle, and $N!$ has been absorbed by factorization of the partition function. This approximation is valid only for lattices where all sites are equivalent; if the unit cell of the crystal consists of inequivalent sites, Eq. (\[cellmodel\]) can be embellished accordingly.
As natural as this approximation may seem for a periodic arrangement of particles in a crystal, it neglects any correlated motion of neighbors and the associated communal entropy. [@Hill56; @Barker63] Nevertheless, the cellular model often gives quantitatively accurate results that are in good agreement with more complete numerical simulations. For hard spheres, where the cellular theory becomes exact in the high-density limit, the agreement with Monte Carlo results is in fact excellent. [@Barker63; @Curtin87] In this case, each particle is assumed to be uniformly smeared over its reduced Wigner-Seitz so that it cannot overlap with its neighbors. As a result, the effective potential is quite simple: 0 within the volume the center of mass is allowed to trace out (the so-called free volume) and infinite otherwise. Thus $$Z\approx\left(\lambda^{-3}\int_{V_{F}}{\rm d}{\bf r}\right)^N=\left(V_{F}/
\lambda^3\right)^N,
\label{cellmodel3}$$ where $V_{F}$ is the free volume whose shape reflects the symmetry of the lattice.[@Barker63] We note that temperature drops out of this problem and that the interaction is entirely entropic.
Within the cellular free-volume theory, the free energy of hard spheres becomes a purely geometrical issue. All one has to do is to calculate the volume of the Wigner-Seitz cell after a layer of thickness $\sigma/2$ has been peeled off of its faces, which is the volume accessible to the particle’s center of mass. In the FCC lattice, the free volume has the shape of a rhombic dodecahedron just as the Wigner-Seitz cell. [@Kittel53] On the other hand, in the BCC lattice it remains an orthic tetrakaidecahedron only at rather low densities far below the freezing point. At higher densities, where the hard spheres form a solid phase, the square faces become absent rendering the free volume a regular octahedron. For the FCC and BCC lattices, the bulk free energy per dendrimer micelle reads $$F^X_{\rm bulk}=-k_{\scriptscriptstyle B}T\ln\Bigglb(\alpha^X
\left(\frac{\beta^X}{n^{1/3}}-1\right)^3\Biggrb),
\label{fccbcc}$$ where $X$ stands for FCC or BCC, $n=\rho\sigma^3$ is the reduced number density ($\sigma$ being the hard-core diameter), and the term $-3k_{\scriptscriptstyle B}T\ln(\sigma/\lambda)$ which is independent of the lattice structure has been dropped. The coefficients $\alpha^{\rm FCC}=2^{5/2}$ and $\alpha^{\rm BCC}=2^23^{1/2}$ reflect the shape of the free volume, and $\beta^{\rm FCC}=2^{1/6}$ and $\beta^{\rm BCC}=2^{-2/3}3^{1/2}$ specify their size.
The free volume of the A15 lattice, which includes two types of sites, is a bit more complicated. As determined by the Voronoi construction subject to the constraint that all cells have equal volume, the shapes of the free volumes are irregular pentagonal dodecahedra and tetrakaidecahedra with two hexagonal and twelve pentagonal faces rather than regular polyhedra. Their volumes cannot be expressed in an amenable analytical form, and thus we have calculated them numerically for a range of densities. However, we note that the exact free volumes can be approximated very well by replacing the dodecahedra and tetrakaidecahedra by spheres and cylinders, respectively (Fig. \[spheres\]).
This [*ansatz*]{} takes into account that the free volume of a columnar site changes anisotropically with density and approaches a flattened shape in the close-packing limit, whereas the shape of the free volume of interstitial sites does not depend on density. We also introduce two adjustable parameters that quantify the fact that the actual volumes of the Wigner-Seitz cells are larger than the volumes of spheres and cylinders, which leave empty voids between them. Given that the ratio of columnar and interstitial sites is 3:1, this leads to the average bulk free energy per micelle of $$\begin{aligned}
F_{\rm bulk}^{\rm A15}&=&-k_{\scriptscriptstyle B}T\left[\frac{1}{4}\ln
\Bigglb(\frac{4\pi S}{3}\left(\frac{\sqrt{5}}{2n^{1/3}}-1\right)^3\Biggrb)
\right.\\ \nonumber
&&+\left.\frac{3}{4}\ln\Bigglb(2\pi C\left(\frac{\sqrt{5}}{2n^{1/3}}-1
\right)^2\left(\frac{1}{n^{1/3}}-1\right)\Biggrb)\right].
\label{a15}\end{aligned}$$ This formula best agrees with the numerical results for $S=1.638$ and $C=1.381$, where the relative deviation from the true bulk free energy is below 0.1% at densities higher than $n\approx0.8$ and not significantly larger at lower densities. $F_{\rm bulk}^{\rm FCC}$, $F_{\rm bulk}^{\rm BCC}$, and $F_{\rm bulk}^{\rm A15}$ are plotted in Fig. \[fbulk\]. [@Ziherl00]
Surface free energy
-------------------
Having calculated the bulk free energy for the three lattices, we now turn to the surface free energy, which requires a specific model of the soft interaction between the particles. To determine which model is the most appropriate, we should first find out whether the overlap of the neighboring particles is large or small. At this point, the analysis becomes somewhat less general, because the overlap differs from one system to another. Being interested primarily by the stability of the loosely-packed crystal lattices, we now focus on the dendrimers that form the A15 crystal. [@Balagurusamy97] The relevant quantitative data include measurements of the radii of dendrimer molecules and the lattice constants of the micellar crystals. Since the dendrimers consist of 2, 3, or 4 generations of branching benzyloxy segments crowned by dodecyl chains with bare radii of $\sigma_{\rm bare}/2=2.6,$ 3.2, and 3.8 nm, respectively, we can deduce that the lengths of the benzyloxy core segment and the dodecyl chain are $l_{\rm core}=
0.6$ nm and $l_{\rm corona}=1.4$ nm (Fig. \[corechain\]). The effective diameter of the micelles $\sigma_{\rm eff}$ can be calculated from the lattice constant $a$. According to Fig. \[lattices\], $\sigma_{\rm eff}=a/2$. This gives $\sigma_{\rm
eff}=3.4,$ 4.0, and 4.2 nm. Obviously, $\sigma_{\rm eff}$ is considerably smaller than $\sigma_{\rm bare}$ for all generations. Actually, the most conclusive information can be extracted from 4$^{\rm th}$ generation data: 2$^{\rm nd}$ and 3$^{\rm rd}$ generation micelles probably have an empty center and their true diameter is most likely larger than $\sigma_{\rm bare}$. These data show that the effective diameter of the hemispheric 4$^{\rm th}$ generation micelles, 4.2 nm, does not exceed the diameter of the their benzyloxy core, $8l_{\rm core}=4.8$ nm. Thus we can conclude that the hard-core diameter of the monodendrons must be smaller than the diameter of the benzyloxy core. Thus not only is there a considerable overlap between the coronas, but also the dodecyl chains penetrate into the core itself. This indicates that the interfacial effects, related to the limited orientational entropy of the chains, are important in the dendrimer system.
In the absence of a quantitative insight into the intermicellar potential – such as a direct measurement of the interaction via optical trapping [@Crocker94] – we model the interaction of the interpenetrating dodecyl chains as the interaction between grafted polymer brushes. [@Milner88] In the high-interdigitation limit, which is certainly applicable in our case, the free energy of a compressed brush reduces to the excluded-volume repulsion of the chains. An argument in the spirit of Flory theory gives $$F_{\rm surf}=\frac{\ell N_0k_{\scriptscriptstyle B}T}{h}=\frac{2\ell
N_0k_{\scriptscriptstyle B}T}{d},$$ where $\ell$ is a parameter with the dimension of length, which determines the strength of repulsion, $N_0$ is the number of chains per micelle, and $h$, the thickness of the single corona, is half the average thickness of the interdigitated matrix of the chains $d$. This approximation neglects some details of the actual interaction, most notably the curvature of the dodecyl brushes in the coronas. However, these effects are subdominant when the dendrimers are packed very closely. In fact, it is known that the density is nearly constant in the volume occupied by the chains, [@Balagurusamy97] and we thus expect that our model should provide a robust description of the system.
The thickness $d$ of the coronal matrix depends on the density since the dodecyl bilayer must fill the space between the hard cores. [@Ziherl00] Thus $$d=\frac{2(n^{-1}-\pi/6)\sigma^3}{A_M},$$ where $A_M$ is the interfacial area per micelle. Since the area is proportional to the square of the lattice constant, the surface free energy per micelle reads $$F^X_{\rm surf}=\frac{\ell N_0k_{\scriptscriptstyle
B}T}{\sigma}\frac{\gamma^Xn^{-2/3}}{n^{-1}-\pi/6},
\label{surf}$$ where $\gamma^X$ is the coefficient determined by the symmetry of the lattice and defined by $A_M=\gamma^X\sigma^2n^{-2/3}$. Lower values of $\gamma^X$ correspond to more area-efficient partitions. The typical magnitude of this coefficient is set by the simple cubic (SC) lattice, which gives $\gamma^{\rm SC}=6$. The more efficient area-minimizing lattices have smaller values of $\gamma^X$: $\gamma^{\rm FCC}=2^{5/6}3=5.345$, $\gamma^{\rm BCC}=5.306$, and $\gamma^{\rm A15}=5.288$. [@Weaire94a] $\gamma^{\rm BCC}$ and $\gamma^{\rm A15}$ can only be computed numerically, e.g., using Surface Evolver. [@Brakke92] As a comparison, the ultimate lower bound for $\gamma$ corresponds to a sphere (which, of course, is not a space-filling body): $\gamma^{\rm sphere}=2^{2/3}3^{2/3}\pi^{1/3}=4.836.$
Combining Eqs. (\[fccbcc\]), (\[a15\]), and (\[surf\]), we arrive at a single-parameter ($\ell$) free energy of micellar crystals: $$F^X=F_{\rm bulk}^X+F_{\rm surf}^X.$$ Before we calculate the minimal strength of the soft repulsion necessary to stabilize the area-minimizing structures, we need to determine the density of the crystal. Not knowing the hard-core radius of the micelles, we can only provide an order-of-magnitude estimate of $\ell$, and for this purpose, it is sufficient to note that unless the observed A15 lattice approaches the packing limit of $n=1.0$, its bulk free energy per micelle is roughly $1k_{\scriptscriptstyle B}T$ larger than $F_{\rm bulk}^{\rm BCC}$ and $2k_{\scriptscriptstyle B}T$ larger than $F_{\rm bulk}^{\rm FCC}$. To be concrete, we compare the total free energies at $n=0.95$, which is well within the high-density regime but not quite at the close-packing limit and thus consistent with the structural data. At this density, the BCC lattice becomes more favorable than FCC at $\ell\gtrsim0.05 \sigma$ whereas the BCC-A15 transition occurs at $\ell\approx0.15\sigma$. Given that there are $N_0=162$ dodecyl chains per 3rd and 4th generation micelle, these values of $\ell$ correspond to an entropy of about $0.5k_{\scriptscriptstyle B}$ and $1.5k_{\scriptscriptstyle B}$ per chain, respectively. In other words, this means that if the overlap of the micelles is so large that the decrease of orientational entropy of a chain due to interdigitation reaches $0.5k_{\scriptscriptstyle B}$ and $1.5k_{\scriptscriptstyle B}$, the differences between the surface entropies of the BCC and FCC lattices and the A15 and BCC lattices overcome the corresponding differences of the bulk entropies and thus favor the area-minimizing structures.
These values are physically reasonable and of the correct order of magnitude. When unrestricted by other chains, each chain has a few orientational and conformational degrees of freedom. Thus its entropy is a few $k_{\scriptscriptstyle B}$, and so it can easily loose $0.5k_{\scriptscriptstyle B}$, $1.5k_{\scriptscriptstyle B}$, or more entropy upon interdigitation. If we plug these numbers back into the free energy, we learn one more thing about these systems: their energetics is controlled primarily by the surface term, which is a direct consequence of the three center-of-mass degrees of freedom associated with the hard cores being greatly outnumbered by the several hundreds of internal degrees of freedom associated with the dodecyl chains. In other words, in the dendrimer system the “squishiness” of the particles wins over their hard cores.
A new paradigm
--------------
Our proposed model provides a novel way of looking at the self-organization of colloidal crystals. By complementing the well-known close-packing rule, which controls the stability of hard spheres, with the minimal-area rule, which stems from the additional short-range repulsion between the particles, we have shown that the equilibrium ordered structures are a result of frustration between two incompatible requirements. This picture implies that if the corona is thin compared to the core, the colloids will behave as hard spheres and form a close-packed lattice such as FCC, but as it grows thicker, an area-minimizing structure – the A15 lattice – should be observed. In between the two extremes, there may be a spectrum of lattices that neither maximize the packing fraction nor minimize the interfacial area but represent a reasonable compromise for a given intermicellar potential. However, in some systems coexistence of the FCC and A15 lattices could be energetically preferable to an intermediate structure such as BCC, provided that the density is not too high to destabilize the A15 lattice.
This observation is consistent with the experimentally determined structures found in other colloidal materials, such as crystals of aurothiol particles consisting of a gold crystallite core and covered by about 50 $n$-alkylthiols where $n=4$, $6$, or $12$. This system is remarkably close to the dendrimer system in both size and structure. The diameter of the metallic cores can be varied from 1.6 to 3.1 nm, and the length of the (fully extended) coronal chains is between 0.6 to 1.56 nm. The alkylthiol chains were adsorbed to the gold core with the sulfur atom, leaving the outer part of the corona chemically identical to the dendrimer coronas. Depending on the relative size of the corona with respect to the diameter of the core, either the FCC or BCC lattice was observed. In addition, in some samples the BCT lattice was found as a moderately anisotropic variation of the BCC lattice with $c/a\sim1.15$. (We will see later on that this lattice is not really unexpected although it departs somewhat from our model: in a BCT crystal, both the bulk and surface entropy are larger than their BCC counterparts.)
A particularly interesting feature of this system is that the size of the core can be varied continuously, so that the FCC-BCC transition can be located very precisely. For the range of core radii explored in the study, [@Whetten99] an FCC-BCC transition was found in particles covered with hexylthiol chains: the FCC lattice is stable if the ratio of the thickness of corona and the core radius is smaller than about 0.73, and BCC or BCT lattices are observed otherwise. The structural parameters of the aurothiol particles were measured in some detail and if we identify the hard core with the gold nanocrystal, we find that the reduced density at the transition is about $n_{\rm FCC-BCC}=0.4$. Proceeding along the same lines as before, this gives an entropy decrease of about $0.05k_{\scriptscriptstyle B}$ for each of the approximately 150 chains in the corona. Although still reasonable, this estimate is by an order of magnitude smaller than in the dendrimer case. However, we note that because of the high coverage of the gold cores with the adsorbed alkylthiols, the effective hard-core diameter of the particles is most likely larger than the gold core diameter. In this case, the reduced density at the transition would be larger, and this would lead to a larger entropy decrease per alkylthiol chain. Nonetheless we can still combine this figure with its dendrimer counterpart to bound the value of the entropy per coronal chain at the FCC-BCC transition between 0.05 and $0.5k_{\scriptscriptstyle B}$, which can serve as an estimate for future studies.
Similar behavior was discovered in polystyrene-polyisoprene diblock copolymers dispersed in decane, a solvent preferential for the polyisoprene. [@McConnell93; @McConnell96] These diblocks spontaneously form micelles with a polystyrene core and polyisoprene corona. Indeed, micelles based on diblocks with a core segment containing from 1.5 to 2 times as many monomers as the coronal segment crystallize into an FCC lattice. [@McConnell96] On the other hand, in copolymers made of blocks with the same number of monomers the BCC lattice was observed. Although the A15 lattice was not seen in this system, we conjecture that it could be found in copolymers with the polyisoprene block sufficiently longer than the polystyrene block.
While rare in crystalline systems, we mention that the A15 structure is not uncommon in lyotropic systems, typically containing lipids dispersed in water matrix. These systems self-arrange in either direct or inverted micelles, which are known to form a variety of cubic structures. [@Charvolin88; @Luzzati96; @Mariani94; @Clerc96] However, the existence of the A15 lattice in lyotropic systems should not be as surprising as in colloidal systems. The cohesive force of these micelles is the hydrophobic interaction which means that their size is not as well-defined as in colloids. If there can be several types of micelles the A15 lattice is not hard to assemble. The other difference with respect to colloids is that the water matrix that encloses the micelles is truly fluid and thus much more similar to actual wet soap froths with relatively large liquid content. At the same time, in the lyotropic systems the effective intermicellar potential may not be necessarily dominated by steric effects, implying that the stability of a particular ordered micellar structure can depend on the chemical composition of lipids.
Specific effects like these can not easily be incorporated into our framework: our goal was not to develop an elaborate description that would cover the many details that are captured in more or less involved theories, such as self-consistent field theory [@Matsen97] and various Monte Carlo schemes. [@Watzlawek99; @Dotera99] Instead, we have proposed a heuristic model that describes the essential physics of many colloids and clearly exposes the frustration that is introduced by dressing hard spheres with a soft repulsive interaction. This model provides an explanation of the results of the more rigorous theories with the added advantage that it is very simple, yet can give reasonable semiquantitative predictions about the stability of the different lattices. Obviously, it can be improved by introducing a more refined model of the interfacial interaction which would account for the curvature of the coronal matrix, the strain of the coronas into the interstitial regions, solvent effects, and related phenomena.
As useful as it might be, like all coarse-grained approaches, the proposed theory [@Ziherl00] has some limitations. Our model is tailored for a broad yet specific class of colloidal particles that interact by short-range potentials, and it may not be suitable for systems characterized by long-range forces, such as unscreened or partly screened electrostatic forces. The range of potential is determined by the requirement that the nearest-neighbor interaction be much stronger than the interaction with the rest of the particles; otherwise we could not define the concept of the interface. Additionally, we have decoupled the bulk and interfacial free energies, or the hard and soft parts of the potential. We suspect that a limit of the full problem exists in which the minimum-area principle is a mean-field approximation.
In order to gain insight into colloidal crystal structure and to strengthen the case for our new principle, we will, in the next Section, calculate the phase diagram for a certain interparticle potential, and identify the range of stability of the A15 lattice using a “single-particle” Monte Carlo algorithm. The significance of these results is twofold: at the qualitative level, they suggest that the phase diagram of soft spheres can include a variety of solid phases, not just FCC and BCC lattices, and at the quantitative level they will provide guidance for a more complete numerical analysis of the system.
Short-range potentials and solid-solid phase transitions
========================================================
Crystal architecture
--------------------
The FCC lattice, which, as we know now for certain, [@Hales00] is the closest regular packing of hard spheres with cubic symmetry, with each site enclosed completely by the 12 equidistant neighbors. In the case of a purely hard core interaction, the free energy consists solely of the entropic contribution and is a convex function of volume. To model the effect of the soft coronas in the systems we are discussing, we add to the hard core a soft potential of range $\delta$. Note that we are replacing an entropic interaction with an energetic term which should make no difference to our argument but is more convenient in our calculational scheme. At densities low enough so that the average distance between the particles is much larger than $\delta$, the overlap of the coronas is small and so the free energy is dominated by the entropy, as if the extra potential were not there. On the other hand, in the high-density limit where the separation between the hard cores is smaller than $\delta$, each particle feels the soft potential of all neighbors. If the potential is flat enough, the average field may in fact not change dramatically with position and then the particle will behave essentially as if it were a hard sphere moving in a constant potential, its free energy being given by the entropic contribution of the hard-core interaction shifted by the energy of the overlapping coronas. For certain forms of the soft potential, the transition from the low-density to high-density regime can be rather abrupt (Fig. \[feschematic\]). Upon compression, the free energy of the colloid will then change quite sharply and may no longer be a convex function of volume. In this case, the low-density and high-density behavior will be separated by a region of coexistence between expanded and condensed FCC lattices, implying an isostructural transition between them. [@Bolhuis97]
Coexistence between the expanded and condensed FCC phases is not the only possibility: other crystal lattices may exist in this density regime where it may be favorable for the spheres to be configured in a structure with a somewhat higher entropy but lower energy. For example, the BCC lattice has a more open architecture compared to FCC where the “cage” of each particle is defined by the 12 nearest neighbors. In the BCC lattice, the 8 nearest neighbors do not enclose the particle completely: as the Wigner-Seitz cell shows us, even in the hard-core limit, each particle also interacts with the 6 next-to-nearest neighbors. At any fixed density, the 8 neighbors from the first coordination shell are closer to the particle than in the FCC crystal, but the 6 second-shell neighbors are further away. As a result, BCC hard spheres have a lower entropy than the FCC hard spheres, but the additional soft potential can stabilize the BCC lattice. The more structured arrangement of sites implies that upon compression, the energy of the BCC crystal increases more gradually than its FCC counterpart: the central particle first overlaps with the first-shell neighbors alone and in this density regime the BCC free energy can be lower than the free energy of the coexisting expanded and condensed FCC phases, provided that the soft potential overlaps with all 12 neighbors of the FCC lattice. At somewhat higher densities the particles in the BCC lattice interact with all of the 14 neighbors and then both the entropic and energetic terms disfavor the BCC lattice. Similarly, the stability of the A15 lattice and other more structured phases is facilitated by their even smoother, less step-like free energy as a function of volume.
This view of the energetics is closely related to the structure of the pair distribution function of a particular lattice. As long as the interaction between the particles is short-range, one only has to consider the pair distribution function at short separations, which directly reflects the shape of the corresponding Wigner-Seitz cell. If the distribution function has a single peak as in the highly symmetric FCC crystal, a soft short-range repulsion between the particles can destabilize the lattice with respect to other lattices with more complex pair distribution functions. As noted above, these include the BCC lattice with two peaks at short distances (Fig. \[rdf\]) but also the anisotropic variants such as the BCT lattice [@Whetten99] where the 6 BCC second-shell sites are split into a subshell of 4 sites and a subshell of 2 sites, and the BCO lattice [@Watzlawek99] where the fourfold BCT next-to-nearest-neighbor subshell is further split in two subshells. (Curiously, the anisotropic version of the FCC lattice – the face-centered orthorombic (FCO) structure has not yet been observed in a colloidal or related system.)
This argument can be extended to include other, less symmetric lattices. An example is the A15 lattice. By considering the pair distribution function, we see that, on average, each site interacts with 13.5 neighbors: through the faces of their Wigner-Seitz cells, the columnar and interstitial sites interact with 14 and 12 neighbors, respectively, and there are 3 times as many columnar sites as the interstitial sites. Of these 13.5 neighbors, 1.5 particles are closest to the central particle, followed by two shells of 6 particles at larger distances (Fig. \[rdf\]). While it is obvious that this lattice must have a smaller entropy than BCC, for a suitably chosen soft potential it could have a lower net free energy. An extreme case of the tradeoff between entropy and energy is the diamond phase observed in a numerical study of packing of star polymers which interact with an extremely soft potential. [@Watzlawek99] The diamond lattice with 4 nearest and 12 next-to-nearest neighbors is clearly unfavorable from the entropic point of view, but apparently the pair distribution function is so strongly peaked at next-to-nearest neighbors that for certain types of short-range interparticle potentials the energy of the diamond lattice is very low.
While these ideas provide additional perspective and complement our foam model described above, it is not completely clear whether the structural characteristics captured by the pair distribution function can be easily related to the minimal-area principle and Kelvin problem. We will not pursue this interesting question here; instead, we now turn to statistical mechanics of soft spheres.
Phase diagrams
--------------
The phase behavior of soft-sphere systems has had ongoing study over the years. A variety of purely repulsive forces have been considered, including power law potentials as the high-temperature limit of the van der Waals interaction, [@Hoover70] screened Coulomb potentials with [@Meijer97] or without a hard core, [@Kremer86; @Hone83] and square shoulder potentials [@Bolhuis97; @Rascon97; @Lang99] as well as their variants. [@Velasco00] With its flat plateau and as sharp a cutoff as possible, the square-shoulder potential is very sensitive to the structure of the pair distribution function and thus the quintessential short-ranged interaction. This is the reason why it is used so often to analyze solid-solid transitions. We shall use the square-shoulder interaction in the following, in order to provide a comparison with earlier studies.
The hard-core square-shoulder (HCSS) system is characterized by a pairwise potential of the form $$u(r) = \left\{
\begin{array}{lcl}
\infty & &r<\sigma\\
\epsilon & & \sigma\leq r<\sigma+\delta\\
0 &&r>\sigma+\delta\\
\end{array}
\right.$$ where $\delta$ and $\epsilon>0$ are the thickness and the height of the shoulder, respectively (Fig. \[hcss\]). This system has already been studied [@Rascon97; @Velasco98] by using density-functional perturbation theory, which treats the behavior of the system primarily by the hard-core interaction and treats the additional soft potential as a perturbation. If the free energy and the structure of the reference state ($F_{\rm ref}$) are known, the total free energy may be expanded to linear order in the perturbative term [@Hansen86]: $$F[\rho({\bf r})]=F_{\rm ref}[\rho({\bf r})]+\frac{1}{2}\int{\rm d}{\bf r}_1\,
{\rm d}{\bf r}_2\,\rho_{\rm ref}^{(2)}({\bf r}_1,{\bf r}_2)\,\phi(r_{12}),$$ where ${\bf r}={\bf r}_2-{\bf r}_1$ describes the relative position of the two particles, $\rho_{\rm ref}^{(2)}({\bf r}_1,{\bf r}_2)$ is the pair distribution function of the reference state, and $\phi_{\rm pert}$ is the perturbative part of the potential. The inputs of this scheme are the pair distribution function and the free energy of the reference system. Typically, the free energy of the solid HC phases are described within the so-called cellular free volume theory [@Barker63; @Hill56] and the corresponding pair distribution is calculated from the Gaussian one-particle density, whereas the fluid free energy and pair distribution are described by the Carnahan-Starling formula and the Verlet-Weis formula, respectively. [@Hansen86]
Within perturbation theory, it has been found that the solid part of the phase diagram can indeed be very complex, and that its topology depends very delicately on
the parameters of the potential. For relatively narrow shoulders, the phase diagram differs from its hard-core counterpart only in the expanded FCC–condensed FCC transition that occurs at densities near the close-packing density and terminates at a critical point. As the shoulder becomes broader, the region of coexistence between the expanded and condensed phases shifts towards lower densities. Eventually, the expanded FCC structure is replaced in part by the BCC at intermediate temperatures, [*i.e.*]{}, at temperatures high enough so that the shoulder does not appear very high, yet low enough so that it is not irrelevant. According to this analysis, the transition between the fluid and condensed FCC phase can be either direct or indirect, the intervening phases being BCC, expanded FCC, or both. Similar results were obtained for a sloped shoulder and a square shoulder followed by a linear ramp, except that these potentials give rise to several new features of the phase diagram, such as the expanded BCC–condensed BCC transition and a triple point. These predictions are consistent with the limited Monte Carlo results available, [@Bolhuis97] but most of the features remain to be verified by rigorous numerical studies.
The perturbation analysis has provided valuable information on the behavior of HCSS spheres, but its predictive power is limited to high temperatures where the expansion is valid. Although some additional information can be obtained by interpolating between the high-temperature and zero-temperature data, this leaves out the most interesting part of the phase diagram. A natural extension of this approach is to calculate the configuration integral nonperturbatively. This can be done, for example, within a mean-field approximation in the cellular model where each particle is assumed to move independently in the average field of its neighbors. Such an approximation can be used within the cellular model by assuming an appropriate [*ansatz*]{} for the probability density – usually a sum of Gaussians – which then leads to a self-consistency relation determined by the requirement that probability densities of all particles be the same, thereby fixing the parameters of the [*ansatz*]{}. [@Hone83] Here we follow the spirit of this approach but we note that in the case of a system interacting with flat potential such as HCSS, the Gaussian ansatz may not be adequate. Moreover, in the A15 lattice the probability distribution of particles in columnar positions should be very anisotropic and in fact nonspherical. Given that our main task here is to examine the possibility that the phase diagram of the HCSS system includes the A15 lattice, it seems appropriate to capture the geometrical details of the different structures by calculating the configuration integral numerically.
Our procedure places the central particle within a cage of its neighbors that are themselves allowed to sample a distribution of positions such that the probability densities of all particles at equivalent sites are the same. This is achieved by sampling the actual probability density of the central particle continuously, and recreating identical distributions at the equivalent neighbor positions. [@MCremark] After equilibrating the system for a certain number of steps and checking for the consistency of the distributions of the central particle and the neighbors, we calculate the configuration integral and the free energy. This approach could be dubbed a “single-particle Monte Carlo method”; we note, however, that it is in fact a numerical variant of the self-consistent mean-field approximation and a relative of the variational method proposed by Kirkwood [@Kirkwood50] rather than a simplified variant of a true multiparticle Monte Carlo integration. Our procedure is single particle because we approximate the multiparticle probability density by a product of single-particle probability densities and thus ignore long-wavelength excitations.
To ensure that the model reduces to the quantitatively successful cellular free-volume theory used in Section II [@Curtin87], we modify the HCSS interparticle potential by not allowing the hard core of the particle to leave the cell. This gives the correct free energy for small shoulders, ($\epsilon<k_{\scriptscriptstyle B}T$) in good agreement with Monte Carlo analyses, [@Rascon97; @Velasco98] as well as for large shoulders, where the particles behave essentially as hard spheres moving in a constant potential. We expect that between these two limiting cases, the correlations between nearest neighbors (and thus the free energy) should also be well approximated. While this method can not include the fluid phase, we are really interested in the stability of the A15 lattice so that the only relevant aspect of the fluid-solid transition is melting. Instead of going into the details of the fluid HCSS phase and analyzing the transition with, for example, Ramakrishnan-Yussouff theory, [@Ramakrishnan79] we can estimate the melting curve by extrapolating the zero-temperature FCC melting point, which is, as we will see, far enough from the other phase boundaries to suggest that the latter persist within the temperature range that we study.
Using this approach, we have analyzed the solid part of the phase diagram at low temperatures, focusing on the FCC, BCC, and A15 lattices. The coexistence between the phases was determined using the Maxwell double-tangent construction, and the results are shown in Figs. \[phasediagrama\] and \[phasediagramb\] for shoulder widths $\delta/\sigma=0.2,0.25,0.3,$ and 0.35.
The main features of the phase diagram can be summarized as follows:
\(i) We find that the A15 lattice can be a stable state of the HCSS system between the expanded and condensed FCC phase. As expected from the structure of the pair distribution function, this lattice is stable for shoulders neither too narrow (which would make the particles too similar to ordinary hard spheres) nor too broad (which would destroy the comparative advantage of the A15 lattice over the FCC and BCC lattices). The minimal and the maximal shoulder widths roughly correspond to $\delta=0.2$ and 0.35, respectively. Note that these widths are consistent with the structural parameters of the dendrimer compound where this lattice was observed experimentally, although the effective hard core of the micelles may not necessarily coincide with the benzyl inner segment of the dendrimers.
Temperature is also a crucial parameter of the stability of the A15 lattice and should not be too low nor too high. The A15 island in the phase diagram appears to be centered around $k_{\scriptscriptstyle B}T/\epsilon\approx1.4$ and its temperature range spans about $2k_{\scriptscriptstyle B}T/\epsilon$ at most, the maximum being at $\delta\approx0.27$. The A15 lattice is a delicate structure: for all $\delta$ and $T$, the pure A15 structure occurs only within a rather narrow range of density centered at $n\approx0.85$ with width $\Delta n\approx0.05$.
\(ii) In the density-temperature plane, islands of stability of the intermediate-density solid phases ([*i.e.*]{}, expanded FCC, expanded BCC, A15, and condensed BCC lattice) are elongated along the temperature axis, implying that over broad ranges of temperature, the phase sequence does not change dramatically with density. This departs from the low-temperature extrapolations of the results of perturbative theory: [@Velasco98] the stripe-like topology of the phase diagrams presented in Fig. \[phasediagrama\] and Fig. \[phasediagramb\] is free of critical and/or triple points as well as the corresponding isostructural transitions. This is most likely a consequence of the rather broad shoulder.
\(iii) For shoulders as broad as necessary to stabilize the A15 lattice, the phase diagram is characterized by relatively small islands of stability of the intermediate phases: two-phase coexistence dominates the phase diagram. This feature appears to be specific to convex interparticle potentials, whereas in case of concave potentials the regions of coexistence are typically much narrower. [@Lang00]
While the stability of the A15 lattice is certainly not limited to HCSS potentials, our findings indicate that at least in some systems it could coexist with FCC, BCC, or perhaps another lattice over a rather wide density range. As far as the structural identification of the samples is concerned, this simple fact may have important consequences. In the case of phase coexistence, the interpretation of X-ray measurements is difficult. In practice, the space groups of the coexisting phases can only be determined unambiguously by varying the external parameters and moving from one island of stability across the coexistence region to the other island of stability. Given the stripe-like topology of the phase diagram, the parameter to be varied should be the density and not temperature. Even with this proviso, the identification can be difficult because of a considerable overlap between the diffraction peaks of the structures involved. For example, the difference between the patterns of A15 and BCC lattice is not very striking – the reflections being at $\sqrt{2},\sqrt{4},\sqrt{5},\sqrt{6},\ldots$ and at $\sqrt{2},\sqrt{4},\sqrt{6},\ldots$ for the A15 and BCC lattices, respectively [@Hahn83] – and could be masked by the form factor of the particles, which usually falls off quite rapidly with the wavevector. [@McConnell93] These shortcomings could be overcome by complementing the X-ray studies by calorimetric measurements.
Last but not least, we note that the relative rarity of the A15 lattice in real colloidal crystals may be caused by other, non-equilibrium mechanisms that can slow down the formation of a pure A15 colloidal crystal. For example, it is conceivable that the relatively simpler BCC lattice is kinetically favored over the A15 lattice in experiments which evaporate solvent to form crystallites. Given that the free energies of the different lattices are typically rather small, the equilibration can take very long and the actual ground state may be hard to observe.
Nevertheless, along with recent experimental [@Balagurusamy97] as well as theoretical studies [@Watzlawek99] our preliminary results point to the necessity of extending the phase diagram of soft spherical particles by the A15 lattice and possibly other non-close-packed lattices. At the same time, the more intricate phase diagram may be more difficult to determine and understand.
Conclusions
===========
In this study, we have extended the geometrical interpretation of the free energy of weakly interacting classical particles, and we have complemented the well-known close-packing rule with a minimal-area rule. The incompatibility of the two principles leads to frustration which gives rise to range of possible equilibrium structures, depending on the relative weight of the two terms. Our proposed scheme is a computationally simple, zeroth-order description of soft-sphere crystals which is more transparent than detailed numerical models, most notably molecular modeling.
As such, our theory provides a robust insight into the self-organization of such objects, which should be useful for the engineering of colloidal crystals. The relevance of these universal guidelines is as broad as the use of colloids themselves, ranging from photonic bandgap crystals [@Tarhan96; @Busch98] to micro- or mesoporous materials used for chemical microreactors and molecular sieves. [@Jenekhe99; @Sakamoto00] To meet the demands of a particular application, these designer materials must be characterized by a given lattice constant, symmetry, and mechanical properties, and, in the case of porous structures, void size and connectivity. All these parameters can be controlled by tuning the structure and size of the (template) colloidal particles and the interaction between them, and our model establishes a semi-quantitative relationship between particle geometry and bulk material properties.
We envision this work to be extended in several directions. One problem to be addressed is to locate the A15 lattice within the phase diagram using Monte Carlo analysis, starting with a square-shoulder potential but also employing less generic short-range potentials. In addition, we will further explore the analogy between colloidal crystals and soap froths in view of the geometrical approach that we have adopted here. Another interesting aspect of future work could be to use the model to derive some of the mechanical properties of colloidal crystals, such as the shear and Young moduli. One could also study the stability of non-cubic lattices, which have been mostly neglected so far, in an attempt to understand non-spherical colloidal particles. Work along these lines should lead to easily verifiable predictions and a deeper insight into the physics of colloids.
We gratefully acknowledge stimulating conversations with M. Clerc-Impéror, G.H. Fredrickson, W.M. Gelbart, P.A. Heiney, C.N. Likos, T.C. Lubensky, V. Percec, J.-F. Sadoc, and A.G. Yodh.
This work was supported in part by NSF Grants DMR97-32963, DMR00-79909 and INT99-10017, the Donors of the Petroleum Research Fund, administered by the American Chemical Society, and a gift from L.J. Bernstein. R.D.K. was also supported by the Alfred P. Sloan Foundation.
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---
abstract: |
A bipartite graph $G=(L,R;E)$ with at least one edge is said to be *identifiable* if for every vertex $v\in L$, the subgraph induced by its non-neighbors has a matching of cardinality ${\vert{}L\vert{}}-1$. An of $G$ is an induced subgraph of $G$ obtained by deleting from it some vertices in $L$ together with all their neighbors. The [Identifiable Subgraph]{} problem is the problem of determining whether a given bipartite graph contains an identifiable $\ell$-subgraph.
We show that the [Identifiable Subgraph]{} problem is polynomially solvable, along with the version of the problem in which the task is to delete as few vertices from $L$ as possible together with all their neighbors so that the resulting $\ell$-subgraph is identifiable. We also complement a known [$\mathsf{APX}$]{}-hardness result for the complementary problem in which the task is to minimize the number of remaining vertices in $L$, by showing that two parameterized variants of the problem are [$\mathsf{W[1]}$]{}-hard.
author:
- |
Stefan Kratsch\
University of Bonn, Institute of Computer Science, Friedrich-Ebert-Allee 144, D-53113 Bonn, Germany\
`kratsch@cs.uni-bonn.de`
- |
Martin Milanič\
University of Primorska, UP IAM, Muzejski trg 2, SI-6000 Koper, Slovenia\
University of Primorska, UP FAMNIT, Glagoljaška 8, SI-6000 Koper, Slovenia\
`martin.milanic@upr.si`
title: 'On the complexity of the identifiable subgraph problem, revisited'
---
Introduction {#sec:Introduction}
============
A *matching* in a graph is a subset of pairwise disjoint edges. A bipartite graph $G=(L,R;E)$ with at least one edge is said to be *identifiable* if for every vertex in $L$, the subgraph of $G$ induced by its non-neighborhood has a matching of cardinality ${\vert{}L\vert{}}-1$. Identifiable bipartite graphs were studied in several papers [@DAM1; @ALGO; @COC; @DAM2]; the property arises in the context of low-rank matrix factorization and has applications in data mining, signal processing, and computational biology. For further details on applications of notions and problems discussed in this paper, we refer to [@DAM1; @ALGO].
While the recognition problem for identifiable bipartite graphs is clearly polynomial using bipartite matching algorithms, several natural algorithmic problems concerning identifiable graphs turn out to be [NP]{}-complete (see [@DAM1; @ALGO; @DAM2]). In [@DAM1], three problems related to finding specific identifiable subgraphs were introduced. To state these problems, we need to recall the notion of an $\ell$-subgraph of a bipartite graph (which appeared first in [@DAM1] and, in a slightly modified form, which we will adopt, in [@DAM2]). For a bipartite graph $G=(L,R;E)$ and vertex sets $X \subseteq L$, $Y \subseteq R$, we denote by $G[X,Y]$ the subgraph of $G$ induced by $X \cup Y$.
Let $G=(L,R;E)$ be a bipartite graph. For a subset $J\subseteq L$, the *$\ell$-subgraph of $G$ induced by $J$* is the subgraph $G(J) = G[J, R \setminus N(L\setminus J)]$, where $N(L\setminus J)$ denotes the set of all vertices in $R$ with a neighbor in $L\setminus J$. We say that a graph $G'$ is an *$\ell$-subgraph of $G$* if there exists a subset $J\subseteq L$ such that $G'=G(J)$.
The following three problems are all related to finding identifiable $\ell$-subgraphs of a given graph:
In [@DAM1], the optimization version of the [<span style="font-variant:small-caps;">Min-Identifiable Subgraph</span>]{}problem was shown to be [$\mathsf{APX}$]{}-hard. In the same paper it was shown that all three problems are polynomially solvable for trees, as well as for bipartite graphs $G = (L,R;E)$ such that the maximum degree of vertices in $L$ is at most $2$. In [@DAM2], restricted versions of the [<span style="font-variant:small-caps;">Identifiable Subgraph</span>]{}problem were studied, parameterizing the instances according to the maximum degree $\Delta(R)$ of vertices in $R$. Formally:
It was shown in [@DAM2] that the [$k$-bounded Identifiable Subgraph]{} problem for $k\ge 3$ is as hard as [<span style="font-variant:small-caps;">Identifiable Subgraph</span>]{}problem in general and that the [$2$-bounded Identifiable Subgraph]{} problem is solvable in linear time. The complexity of the [<span style="font-variant:small-caps;">Identifiable Subgraph</span>]{}and [<span style="font-variant:small-caps;">Max-Identifiable Subgraph</span>]{}problems in general bipartite graphs was left open by previous works.
In this paper, we establish the computational complexity of the [<span style="font-variant:small-caps;">Identifiable Subgraph</span>]{}and [<span style="font-variant:small-caps;">Max-Identifiable Subgraph</span>]{}problems, showing that both problems are solvable in polynomial time. The key idea to our approach is the observation that if the input graph $G = (L,R;E)$ is not identifiable, then one can compute in polynomial time a maximal subset $K\subseteq L$ no vertex of which is contained in any identifiable $\ell$-subgraph of $G$. Such a set $K$ is non-empty and can be safely deleted from the graph together with all its neighbors, thus reducing the problem to a smaller graph. If the algorithm finds an identifiable $\ell$-subgraph of $G$, then it in fact finds an identifiable $\ell$-subgraph of $G$ induced by a largest possible subset of $L$, thereby also solving the [<span style="font-variant:small-caps;">Max-Identifiable Subgraph</span>]{}problem. The proof also shows that such a subgraph is unique.
In the second part of the paper, we complement the [$\mathsf{APX}$]{}-hardness result for the optimization version of the [<span style="font-variant:small-caps;">Min-Identifiable Subgraph</span>]{}problem from [@DAM1] by studying the problem from the parameterized complexity point of view. We introduce two natural parameterized variants of the [<span style="font-variant:small-caps;">Min-Identifiable Subgraph</span>]{}problem and prove that both are [$\mathsf{W[1]}$]{}-hard, by giving parameterized reductions from the well-known [$\mathsf{W[1]}$]{}-hard [<span style="font-variant:small-caps;">Multicolored Clique</span>($k$)]{}problem.
The paper is structured as follows. In Section \[sec:prelim\], we give the necessary definitions. In Section \[sec:algorithm\], we give a polynomial time algorithm that simultaneously solves the [<span style="font-variant:small-caps;">Identifiable Subgraph</span>]{}and the [<span style="font-variant:small-caps;">Max-Identifiable Subgraph</span>]{}problems. In Section \[sec:parameterized\], we study the [$\mathsf{NP}$]{}-hard [<span style="font-variant:small-caps;">Min-Identifiable Subgraph</span>]{}problem from the parameterized complexity point of view. Section \[sec:conclusion\] concludes the paper with some open questions.
Preliminaries {#sec:prelim}
=============
All graphs considered in this paper are finite, simple, and undirected. For a graph $G$, we denote by $V(G)$ the vertex set of $G$ and by $E(G)$ its edge set. A *bipartite graph* is a graph $G = (V,E)$ such that there exists a partition of $V$ into two sets $L$ and $R$ such that $L\cap R=\emptyset$ and $E\subseteq \{\{\ell, r\}~;\ell\in L \textrm{~and~} r\in R\}$. In this paper, we will regard bipartite graphs as already *bipartitioned*, that is, given together with a fixed bipartition $(L,R)$ of their vertex set, and hence use the notation $G = (L,R;E)$. For a graph $G=(V,E)$ and a subset of vertices $X\subseteq V$, $N_G(X)$ denotes the neighborhood of $X$, i.e., the set of all vertices in $V \setminus X$ that have a neighbor in $X$. For a vertex $x\in V$, we write $N_G(x)$ for $N_G(\{x\})$, and denote the *degree* of $x$ with $d_G(x)=|N_G(x)|$. In $N_G(X)$, $N_G(x)$, $d_G(x)$, we shall omit the subscript $G$ if the graph is clear from the context. A [*clique*]{} in a graph is a set of pairwise adjacent vertices.
A *parameterized problem* is a language $Q\subseteq\Sigma^*\times{\mathbb{N}}$; the second component, $k$, of instances $(x,k)\in\Sigma^*\times{\mathbb{N}}$ is called the *parameter*. A parameterized problem $Q$ is *fixed-parameter tractable* (FPT) if there is a function $f\colon{\mathbb{N}}\to{\mathbb{N}}$, a constant $c$, and an algorithm $A$ that decides $(x,k)\in Q$ in time $f(k)|x|^c$ for all $(x,k)\in\Sigma^*\times{\mathbb{N}}$. Let ${\ensuremath{\mathsf{FPT}}\xspace}$ denote the class of all fixed-parameter tractable parameterized problems. A *parameterized reduction* from $Q\subseteq\Sigma^*\times{\mathbb{N}}$ to $Q'\subseteq\Sigma'^*\times{\mathbb{N}}$ is a mapping $\pi\colon\Sigma^*\times{\mathbb{N}}\to\Sigma'^*\times{\mathbb{N}}$ such that there are functions $g,h\colon{\mathbb{N}}\to{\mathbb{N}}$ and a constant $c$ with: $(x,k)\in Q$ if and only if $\pi((x,k))\in Q'$, the parameter value $k'$ of $(x',k')=\pi((x,k))$ is at most $g(k)$, and $\pi((x,k))$ can be computed in time $h(k)|x|^c$. It is well known that the existence of a parameterized reduction from $Q$ to $Q'$ and $Q'\in{\ensuremath{\mathsf{FPT}}\xspace}$ imply that $Q\in{\ensuremath{\mathsf{FPT}}\xspace}$ as well, and that parameterized reducibility is transitive. Accordingly, similarly to $\mathsf{P}$ vs. ${\ensuremath{\mathsf{NP}}\xspace}$, there are hardness classes of problems that are suspected not to be FPT. In particular, it is believed that ${\ensuremath{\mathsf{FPT}}\xspace}\neq{\ensuremath{\mathsf{W[1]}}\xspace}$ and, under this assumption, a parameterized reduction from any [$\mathsf{W[1]}$]{}-hard problem rules out fixed-parameter tractability. (Here [$\mathsf{W[1]}$]{}-hardness is with respect to parameterized reductions.)
For graph-theoretic definitions not given in the paper we refer to [@MR2744811; @MR1367739], for further background in matching theory to [@MR859549], and for background in parameterized complexity to [@MR3380745; @DowneyF13].
A polynomial time algorithm for the [<span style="font-variant:small-caps;">Identifiable Subgraph</span>]{}and the [<span style="font-variant:small-caps;">Max-Identifiable Subgraph</span>]{}problems {#sec:algorithm}
======================================================================================================================================================================================================
In this section we give a polynomial time algorithm for the [<span style="font-variant:small-caps;">Identifiable Subgraph</span>]{}problem, the problem of determining whether a given graph $G=(L,R;E)$ has an identifiable $\ell$-subgraph. As a corollary of our approach we will also obtain a polynomial time algorithm for the [<span style="font-variant:small-caps;">Max-Identifiable Subgraph</span>]{}problem.
The key ingredient for the algorithm is the following lemma.
\[lemma:ptime:key\] Let $G=(L,R;E)$ be a non-identifiable bipartite graph with at least one edge and let $v\in L$ such that there is no matching of $L\setminus \{v\}$ into $R\setminus N(v)$. Let $K$ be an inclusion-wise minimal subset of $L\setminus \{v\}$ that has no matching into $R\setminus N(v)$. Such a set $K$ is nonempty and always exists. Moreover, no identifiable $\ell$-subgraph of $G$ contains a vertex of $K$.
Let $K$ be a minimal subset of $L\setminus \{v\}$ that has no matching into $R \setminus N(v)$. By Hall’s Theorem there must be a subset $K'\subseteq K$ with $|N(K')\cap (R\setminus N(v))|<|K'|$. Any such set $K'$ has no matching into $R\setminus N(v)$. Because $K$ is a minimal set without a matching into $R\setminus N(v)$ it follows that $|N(K)\cap (R\setminus N(v))|<|K|$. Furthermore, every proper subset of $K$ does have a matching into $R\setminus N(v)$.
Now, fix an arbitrary set $J\subseteq L$ such that the induced $\ell$-subgraph $G'=G[J,R\setminus N(L\setminus J)]$ is identifiable. We need to show that $J\cap K=\emptyset$.
Assume for contradiction that $K\cap J\neq \emptyset$. Let $K_{\it in}=K\cap J$ and $K_{\it out}=K\setminus J$. Because $K_{\it in}\neq\emptyset$ we have that $K_{\it out}$ is a proper, possibly empty, subset of $K$. Hence, by the first paragraph, we have that $|K_{\it out}|\leq |N(K_{\it out})\cap (R\setminus N(v))|$. In the $\ell$-subgraph $G_J$ induced by $J$, by definition, none of the neighbors of $K_{\it out}$ are present. Thus, the vertices in $K_{\it in}$ have at most those vertices as neighbors that are adjacent to $K_{\it in}$ but not to $K_{\it out}$. (Further vertices in $L\setminus (J\cup K)$ may imply that further neighbors of $K_{\it in}$ are not present, but this will not be important.) Thus, the number of neighbors that $K_{\it in}$ has in the vertices of $R\setminus N(v)$ that are present in $G_J$ is at most $$|N_{G_J}(K)|-|N_{G_J}(K_{\it out})|< |K| - |K_{\it out}|=|K_{\it in}|.$$ It follows immediately that $K_{\it in}$ has no matching into $R\setminus N(v)$ in $G_J$. If $v\in J$ then testing the identifiability condition for $v$ would require such a matching. If $v\notin J$ then using that $G_J$ must have a matching of $J$ into $N_{G_J}(J)$ means that we would need a matching of $K_{\it in}$ into $N_{G_J}(K_{\it in})\subseteq R\setminus N(v)$. Thus, either way we get a contraction. This implies that $J\cap K=\emptyset$, as claimed.
Given a graph $G$ and vertex $v$ as in Lemma \[lemma:ptime:key\] the set $K$ can be found in a straightforward way by folklore knowledge about bipartite matchings. We sketch a very simple algorithm by self-reduction for completeness.
\[lemma:ptime:minimalset\] Given a non-identifiable graph $G=(L,R;E)$ and vertex $v\in L$ such that there is no matching of $L\setminus \{v\}$ into $R\setminus N(v)$, a minimal set $K$ as in Lemma \[lemma:ptime:key\] can be found in polynomial time.
Set $K:=L\setminus \{v\}$ and repeat the following routine: Try each vertex $w\in K$ and test whether there is a matching of $K\setminus \{w\}$ into $R\setminus N(v)$. If there is then try the next vertex. If not then update $K:=K\setminus \{w\}$ and repeat. Output the current set $K$ if each $K\setminus \{w\}$ has a matching of $K\setminus \{w\}$ into $R\setminus N(v)$.
As an invariant, the set $K$ never has a matching into $R\setminus N(v)$. In particular, we can never reach an empty set (and we can only reach a singleton vertex if it is isolated). Thus, the algorithm must terminate with a nonempty set $K$ such that each set $K\setminus \{w\}$ has a matching into $R\setminus N(v)$. This also means that all smaller subsets of $K$ have matchings into $R\setminus N(v)$. Thus, $K$ is a minimal set with no matching into $R\setminus N(v)$.
We know now that if $G=(L,R;E)$ is not identifiable then we can efficiently find a subset $K\subseteq L$ such that no vertex of $K$ is contained in any identifiable $\ell$-subgraph of $G$. We now prove formally that we may safely delete $K$ and $N(K)$ from $G$ while still retaining the same set of identifiable $\ell$-subgraphs.
\[lemma:lsubgraph:deletion\] Let $G=(L,R;E)$ a bipartite graph and let $K\subseteq L$. Every $\ell$-subgraph of $G$ that contains no vertex of $K$ is also an $\ell$-subgraph of the $\ell$-subgraph of $G$ induced by $L\setminus K$, and vice versa. Moreover, these $\ell$-subgraphs are induced by the same sets $J\subseteq L\setminus K$.
Every $\ell$-subgraph of a graph is defined by the left part of its bipartition. We show that taking the induced $\ell$-subgraph for any $J\subseteq L\setminus K$ gives the same graph from both $G$ and $G-N[K]=G[L\setminus K,R\setminus N(K)]$. Fix an arbitrary set $J\subseteq L\setminus K$.
Clearly, since $J$ is a subset of the left part of the bipartition in both graphs, we get $\ell$-subgraphs of the form $H_1=G[J,R_1]$ and $H_2=(G-N[K])[J,R_2]$. The latter is also an induced subgraph of $G$ so it simplifies to $H_2=G[J,R_2]$. It suffices to prove that $R_1=R_2$.
By definition of $\ell$-subgraph we have $R_1=R\setminus N_G(L\setminus J)$. Similarly, for the $\ell$-subgraph of $J$ in $G'=G-N_G[K]=G[L\setminus K,R\setminus N_G(K)]$ we get $$R_2=(R\setminus N_G(K))\setminus N_{G'}((L\setminus K)\setminus J)\,.$$ We can safely replace $N_{G'}((L\setminus K)\setminus J)$ by $N_G((L\setminus K)\setminus J)$ because $G'$ is an induced subgraph of $G$ so the neighborhood is only affected by restriction to $R\setminus N_G(K)$, the right part of the bipartition of $G'$. Thus, in $R_2$ we have the vertices of $R$ that do not have a neighbor in $K$ and that do not have a neighbor in $(L\setminus K)\setminus J$. Because $K$ and $J$ are disjoint subsets of $L$ this is the same as taking out the neighbors of $L\setminus J$ from $R$, i.e., taking $R\setminus N(L\setminus J)=R_1$. Thus, both graphs are induced subgraphs of $G$ with left part $J$ and right part $R\setminus N(L\setminus J)$, so they are identical as claimed.
In particular, the lemma implies that if no identifiable $\ell$-subgraph contains a vertex of a nonempty set $K\subseteq L$ then $G$ and $G-N[K]$ contain the same identifiable $\ell$-subgraphs. Thus, when seeking identifiable $\ell$-subgraphs it is safe to eliminate $N[K]$ for sets $K$ obtained via Lemma \[lemma:ptime:key\].
Now we can put together the claimed polynomial time algorithm.
\[theorem:ids:ptime\] The [<span style="font-variant:small-caps;">Identifiable Subgraph</span>]{}problem can be solved in polynomial time.
The algorithm works as follows. Given an input graph $G=(L,R;E)$ it first tests if $E = \emptyset$. If $E = \emptyset$, then $G$ is not identifiable and has no identifiable $\ell$-subgraph; the algorithm reports this fact and halts. If $E\neq\emptyset$, the algorithm proceeds iteratively. Identifiability can be efficiently tested by $|L|$ bipartite matching computations. If $G$ is identifiable then graph $G$ is output as an identifiable $\ell$-subgraph. If $G$ is not identifiable then the algorithm picks an arbitrary $v$ such that there is no matching of $L\setminus \{v\}$ into $R\setminus N(v)$. By Lemma \[lemma:ptime:key\] there is a nonempty set $K\subseteq L$ such that no identifiable $\ell$-subgraph of $G$ contains a vertex of $K$; such a set can be found efficiently by Lemma \[lemma:ptime:minimalset\]. Thus, if $G$ has any identifiable $\ell$-subgraph then every such subgraph must avoid $K$ and, hence, it is also an $\ell$-subgraph of $G-N[K]$ by Lemma \[lemma:lsubgraph:deletion\]. Conversely, $G-N[K]$ contains no further identifiable $\ell$-subgraphs. The algorithm thus replaces $G$ by $G-N[K]$ and starts over. In case a graph is output, Lemma \[lemma:lsubgraph:deletion\] implies that the output graph is also an $\ell$-subgraph of the initial graph $G$.
In fact, it can be easily seen that the algorithm always returns a maximum identifiable $\ell$-subgraph and thus also solves the maximization variant of the problem, [<span style="font-variant:small-caps;">Max-Identifiable Subgraph</span>]{}, in polynomial time. (The proof also shows that this graph is unique.)
\[corollary:max-ids:ptime\] The [<span style="font-variant:small-caps;">Max-Identifiable Subgraph</span>]{}problem can be solved in polynomial time.
Clearly, if the input graph is identifiable then returning it is optimal. If not then either $E = \emptyset$ (in which case $G$ has no identifiable $\ell$-subgraph) or there is a vertex $v$ such that $L\setminus \{v\}$ cannot be matched into $R\setminus N(v)$ and, by Lemma \[lemma:ptime:key\] the algorithm finds a nonempty set $K$ that is avoided by all identifiable $\ell$-subgraphs. Since $G$ and $G-N[K]$ have the same $\ell$-subgraphs induced by $J\subseteq L\setminus K$, in particular, any maximum identifiable $\ell$-subgraph of $G$ is also an identifiable $\ell$-subgraph of $G-N[K]$. Thus, continuing the iterative approach on $G-N[K]$ will find a maximum solution, if one exists.
Parameterized complexity of Min-Identifiable Subgraph {#sec:parameterized}
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In this section we study the parameterized complexity of the [<span style="font-variant:small-caps;">Min-Identifiable Subgraph</span>]{}problem, which was proved [$\mathsf{NP}$]{}-hard in a previous work [@DAM1]. We consider the following parameterized variants [<span style="font-variant:small-caps;">Min-Identifiable Subgraph</span>($k$)]{}and [<span style="font-variant:small-caps;">Min-Identifiable Subgraph</span>($|L|-k$)]{}.
The two problems differ only in the choice of parameter; the [<span style="font-variant:small-caps;">Min-Identifiable Subgraph</span>($|L|-k$)]{}problem can be reformulated as the problem of finding a set $L_0\subseteq L$ of size at least $k$ such that $G-N[L_0]$ is identifiable. We show that both parameterizations are [$\mathsf{W[1]}$]{}-hard, i.e., they are not fixed-parameter tractable unless ${\ensuremath{\mathsf{FPT}}\xspace}={\ensuremath{\mathsf{W[1]}}\xspace}$, which is deemed unlikely. For both problems we give parameterized reductions from the well-known [$\mathsf{W[1]}$]{}-hard [<span style="font-variant:small-caps;">Multicolored Clique</span>($k$)]{}problem, defined as follows.
[<span style="font-variant:small-caps;">Min-Identifiable Subgraph</span>($k$)]{}is [$\mathsf{W[1]}$]{}-hard.
We give a parameterized reduction from [<span style="font-variant:small-caps;">Multicolored Clique</span>($k$)]{}to [<span style="font-variant:small-caps;">Min-Identifiable Subgraph</span>($k$)]{}. Let $(G=(V,E),\phi,k)$ be an instance of [<span style="font-variant:small-caps;">Multicolored Clique</span>($k$)]{}. Without loss of generality assume that $k\geq 3$ or else solve the instance in polynomial time (finding a clique of size $k\in\{1,2\}$). Let $V_i:=\phi^{-1}(i)$ for $i\in\{1,\ldots,k\}$. We will construct a bipartite graph $G'=(L,R;E')$ such that $(G,k,\phi)$ is yes for [<span style="font-variant:small-caps;">Multicolored Clique</span>($k$)]{}if and only if $(G',k')$ is yes for [<span style="font-variant:small-caps;">Min-Identifiable Subgraph</span>($k$)]{}.
*Construction.* We create an instance of [<span style="font-variant:small-caps;">Min-Identifiable Subgraph</span>($k$)]{}with bipartite graph $G'=(L,R;E')$ and parameter $k'=2k$. The set $L$ consists of the vertices in $V$ along with $k$ special vertices $t_1,\ldots,t_k$. We now describe the set $R$ along with the adjacencies between $L$ and $R$:
- For each choice of $1\leq i<j\leq k$ we create a set $E_{ij}$ of vertices, which is then added to $R$.
- For each edge $\{u,v\}$ in $G$ with $u\in V_i$ and $v\in V_j$ we add a vertex to $E_{ij}$ and make it adjacent to $u$ and $v$ in $G'$. (We could also achieve this by starting with $G$, dropping the (irrelevant) edges between vertices of the same set $V_i$, and then subdividing every edge.)
- Make the special vertex $t_i$ adjacent to all vertices of $E_{ab}$ with $1\leq a<b\leq k$ and $i\notin \{a,b\}$.
- For each $1\leq i\leq k$ create a set $F_i$ of $k\cdot |V_i|$ vertices and add it to $R$.
- Make each vertex $v\in V_i$ adjacent to $k$ private vertices in $F_i$. No other vertices of $V$ will be adjacent to these vertices.
- Make each special vertex $t_j$ adjacent to all vertices of $F_i$ with $i\neq j$.
This completes the construction of $G'=(L,R;E')$. It can be helpful to keep in mind that vertices in $V_i$ are only adjacent to (some) vertices in $F_i$ or in $E_{ab}$ with $i\in\{a,b\}$, whereas each special vertex $t_i$ is adjacent to all vertices in $F_j$ with $i\neq j$ and all vertices in $E_{ab}$ with $i\notin\{a,b\}$. An example construction is shown in Fig. \[fig:reduction\].
![An example construction of the bipartite graph $G' = (L,R;E')$ from an input $(G,k,\phi)$ to [<span style="font-variant:small-caps;">Multicolored Clique</span>($k$)]{}. In the example $k = 3$, vertices in $L$ and $R$ are colored black and white, respectively, and a thick edge between vertex $t_i$ and a set $F_j$ means that $t_i$ is adjacent to all vertices in $F_j$.[]{data-label="fig:reduction"}](reduction.pdf){width="160mm"}
Clearly, the construction can be performed in polynomial time. It remains to prove correctness, that is, that $G$ has a $k$-clique containing exactly one vertex of each set $V_i$ if and only if $(G',k')$ is a yes instance of [<span style="font-variant:small-caps;">Min-Identifiable Subgraph</span>($k$)]{}for $k'=2k$.
*Correctness.* Assume first that $G$ contains a $k$-clique $C$ with exactly one vertex from each set $V_i$ and let $\{v_i\}=C\cap V_i$. We claim that $L'=\{v_1,\ldots,v_k,t_1,\ldots,t_k\}$ induces an identifiable $\ell$-subgraph. Let $R'\subseteq R$ denote the vertices in $R$ of the induced $\ell$-subgraph, i.e., the vertices that have no neighbor among $L\setminus L'$.
Let us first check that there is a matching of $L'\setminus\{t_i\}$ into $R'\setminus N(t_i)$: The vertex $t_i$ is not adjacent to $F_i$ nor to sets $E_{ab}$ with $i\notin\{a,b\}$. There are $k$ vertices in $F_i$ that are adjacent to $t_1,\ldots,t_{i-1},t_{i+1},\ldots,t_k$ as well as the vertex $v_i$. These are contained in $R'$ since all their neighbors are in $L'$. We can match the mentioned vertices of $L'$ to them. Because $C$ is a clique there are edges from $v_i$ to each other vertex of the clique; these give rise to vertices in $E_{1i},\ldots,E_{i-1,i}, E_{i,i+1}, \ldots,E_{i,k}$ corresponding to these edges that have no other neighbors in $V\subseteq L$ (in $G'$). Because $t_i$ is not adjacent to such sets $E_{ab}$ all these vertices are present and each $v_1,\ldots,v_{i-1},v_{i+1},\ldots,v_k$ can be matched to the vertex representing its edge to $v_i$.
Let us now check that there is a matching of $L'\setminus\{v_i\}$ into $R'\setminus N(v_i)$: For each $v_j$ with $j\neq i$ all its neighbors in $F_j$ are present since they have no other neighbor in $V$ and all $t_1,\ldots,t_k$ are in the $\ell$-subgraph. Thus, each $v_j$ can be matched to such a neighbor. Because $k\geq 3$ there are at least two sets $F_j$ and $F_{j'}$ with $i\notin \{j,j'\}$ and a total of $2k-2$ vertices therein are not yet matched to. Thus, all vertices $t_1,\ldots,t_k$ can be matched to these vertices. (E.g., all but $t_j$ to vertices of $F_j$ and $t_j$ to a vertex of $F_{j'}$.)
Thus, the $\ell$-subgraph induced by $L'$ is indeed identifiable. Since $|L'|=2k$ this implies that $(G',2k)$ is a yes instance of [<span style="font-variant:small-caps;">Min-Identifiable Subgraph</span>($k$)]{}.
Now assume that $(G',2k)$ is yes for [<span style="font-variant:small-caps;">Min-Identifiable Subgraph</span>($k$)]{}. Let $L'\subseteq L$ be a set of size at most $2k$ such that the $\ell$-subgraph of $G'$ induced by $L'$, namely $G'[L',R']$ with $R'=R\setminus N(L\setminus L')$, is identifiable. Our goal is to show that it includes all special vertices $t_1,\ldots,t_k$ along with one vertex per set $V_i$ and that the latter vertices form a $k$-clique in $G$.
We first observe that $L'\cap V\neq\emptyset$ and $L'\cap \{t_1,\ldots,t_k\}\neq \emptyset$ is required: Excluding either type of vertex implies $R'=\emptyset$ since each vertex in $R$ is adjacent to at least one vertex of $V$ and at least one vertex $t_i$.
Assume for contradiction that at least two special vertices, say $t_i$ and $t_j$ with $i\neq j$, are not in $L'$. Thus, picking a third vertex $t_\ell\notin \{t_i,t_j\}$ we need a matching of $L'\setminus \{t_i,t_k,t_\ell\}$ into the vertices of $R$ that are not neighbors of (at least) $t_i$, $t_j$, and $t_\ell$, but no such vertices exist: There is no $F_a$ with $a= i$ and $a=j$ and there is no $E_{ab}$ with $i\in\{a,b\}$, $j\in\{a,b\}$, $\ell\in\{a,b\}$. Since we have at least one vertex $v\in L'\cap V$ we can observe that this vertex cannot be matched; a contradiction.
Now assume for contradiction that exactly one special vertex, say $t_1$, is not contained in $L'$. This requires a matching of $L'\setminus\{t_2\}$ into $R'\setminus N(t_2)$. Since both $t_1,t_2\notin L'\setminus \{t_2\}$, no vertex of a set $E_{ab}$ may exist in $R'\setminus N(t_2)$ except possibly for vertices of $E_{12}$. Thus, no vertex of $V_3,\ldots,V_k$ may be in $L'$ since they would have no neighbors to match to. This in turn implies that no other set $E_{ab}$ except for $E_{12}$ has any vertices in $R'$. We complete the contradiction by considering the requirement of a matching of $L'\setminus \{t_3\}$ into $R'\setminus N(t_3)$: Now, the vertices of $E_{12}$ are not available since they are adjacent to $t_3$. Thus, there are no $E_{ab}$ vertices to match to. Similarly, absence of $t_1$ and $t_3$ eliminates all vertices of sets $F_a$. Since $L'\setminus\{t_3\}$ must contain at least one vertex of $V$, we find that such a vertex cannot be matched into $R'\setminus N(t_3)$; a contradiction.
We now have the remaining case that $\{t_1,\ldots,t_k\}\subseteq L'$. We also know already that at least one vertex of $V$ must be contained in $L'$, say $V_1\cap L'\neq \emptyset$ and pick $v_1\in V_1\cap L'$. Assume for contradiction that some set $V_i$ with $i\neq 1$ has an empty intersection with $L'$. It follows that in $R'$ there are no vertices of sets $E_{ab}$ with $i\in\{a,b\}$ since each such vertex is adjacent to some vertex in $V_i$. We now consider the requirement of a matching of $L'\setminus\{t_i\}$ into $R'\setminus N(t_i)$ to complete the contradiction: This additionally ensures that there are no vertices of $F_1$ left in $R'\setminus N(t_i)$ as well as no vertices of $E_{ab}$ with $i\not\in \{a,b\}$, implying that there are no neighbors for $v_1$ to match to; a contradiction.
Thus, we have $\{t_1,\ldots,t_k\}\subseteq L'$ and $L'$ has a nonempty intersection with each set $V_1,\ldots,V_k$. Because $L'$ has size at most $2k$ this directly implies that its size is exactly $2k$ and that it contains exactly one vertex of each set $V_i$, say $\{v_i\}=L'\cap V_i$. It remains to show that $\{v_1,\ldots,v_k\}$ is a clique in $G$. Assume for contradiction that this is not the case, say that $v_i$ and $v_j$ for $i\neq j$ are not adjacent in $G$. Consider the requirement of a matching of $L'\setminus \{t_i\}$ into $R'\setminus N(t_i)$: Absence of $t_i$ ensures that no vertex of $F_j$ is present. Moreover, no vertices of $E_{ab}$ for $i\notin \{a,b\}$ are in $R'\setminus N(t_i)$. In particular, for vertex $v_j$ this only leaves vertices in $E_{ij}$ or $E_{ji}$ (depending on whether $i<j$ or $i>j$). Because $v_i$ and $v_j$ are not adjacent, however, and no other vertex of $V_i$ is in $L'$, no such vertices exist in $R'\supseteq R'\setminus N(t_i)$. Thus, $v_j$ cannot be matched; a contradiction.
It follows that the vertices $v_1,\ldots,v_k$ must indeed form a clique in $G$. This completes the proof.
The above proof also shows that the problem of testing whether a given bipartite graph has an identifiable $\ell$-subgraph induced by a set $J\subseteq L$ with $|J|= k$ is [$\mathsf{W[1]}$]{}-hard (with respect to parameter $k$).
\[theorem:minidsnk\] [<span style="font-variant:small-caps;">Min-Identifiable Subgraph</span>($|L|-k$)]{}is [$\mathsf{W[1]}$]{}-hard.
We give a parameterized reduction from [<span style="font-variant:small-caps;">Multicolored Clique</span>($k$)]{}to [<span style="font-variant:small-caps;">Min-Identifiable Subgraph</span>($|L|-k$)]{}. Let $(G=(V,E),\phi,k)$ be an instance of [<span style="font-variant:small-caps;">Multicolored Clique</span>($k$)]{}. W.l.o.g., assume $k\geq 3$ or else solve the instance in polynomial time. Let $n:=|V|$ and let $V_i:=\phi^{-1}(i)$ for $i\in\{1,\ldots,k\}$. Assume w.l.o.g. that each set $V_i$ contains at least two vertices (else we can restrict the graph to the subgraph induced by the neighborhood of $v_i\in V_i$ and drop color $i$ to get an equivalent instance).
We will construct a bipartite graph $G'=(L,R;E')$ such that $(G,k,\phi)$ is yes for [<span style="font-variant:small-caps;">Multicolored Clique</span>($k$)]{}if and only if $(G',k')$ is yes for [<span style="font-variant:small-caps;">Min-Identifiable Subgraph</span>($|L|-k$)]{}for $k'=|L|-k$. That is, $(G,k,\phi)$ should be yes if and only if the graph $G'$ contains an $\ell$-identifiable subgraph that is induced by a set $L'$ of size at most $|L|-k$. Note that $(G',k')$ has a parameter value of $|L|-k'=|L|-(|L|-k)=k$. Recall that we can equivalently ask for the existence of a set $L_0\subseteq L$ of size at least $k$ such that $G'-N[L_0]$ is identifiable since then $L\setminus L_0$ is of size at most $|L|-k$ and can play the role of the requested set $L'$ (and conversely $L\setminus L'$ is a feasible choice for $L_0$). Define $r:=n+k$; this value will be used in the construction.
*Construction.* The graph $G'=(L,R;E')$ is defined as follows:
- The vertex set $L$ consists of $V$ as well as a set $T$ of special vertices $t_1,\ldots,t_k$.
- The set $R$ contains for each vertex $v\in V$ a set of $k+1$ vertices $p_{v,1},\ldots,p_{v,k+1}$ whose only neighbor in $V$ will be $v$ (so they are in a limited sense a private neighbors of $v$). Let $F_i$ denote the set of vertices $p_{v,\ell}$ with $v\in V_i$ for each $i\in\{1,\ldots,k\}$. (The exact number of these vertices per vertex $v$ will be immaterial so long as they are at least $k+1$.)
- The set $R$ furthermore contains vertices derived from the edges of $G$. Let $e=\{v_i,v_j\}$ be any edge of $G$ with $v_i\in V_i$ and $v_j\in V_j$. Create $r$ vertices $q_{e,1},\ldots,q_{e,r}$ and add them to $R$. Make each of them adjacent to all vertices of $V_i\setminus \{v_i\}$ and all vertices of $V_j\setminus\{v_j\}$. Do this for all edges for any $1\leq i<j\leq k$ and let $E_{ij}$ contain the vertices $q_{e,1},\ldots,q_{e,r}$ for edges $e$ between $V_i$ and $V_j$ in $G$. The set $R$ is thus the union of sets $F_a$ for $1\leq a\leq k$ and sets $E_{ab}$ for $1\leq a<b\leq k$. (Again, the exact value of $r$ is not important so long as $r\geq n+k$.)
- Make each vertex $t_i\in T\subseteq L$ adjacent to all vertices of each set $F_a$ with $1\leq a\leq k$. Furthermore, make each $t_i$ adjacent to all vertices of each set $E_{ab}$ with $i\notin\{a,b\}$.
Define $k'=|L|-k$ and return the instance $(G',k')$. An example construction is shown in Fig. \[fig:reduction-2\].
![An example construction of the bipartite graph $G' = (L,R;E')$ from an input $(G,k,\phi)$ to [<span style="font-variant:small-caps;">Multicolored Clique</span>($k$)]{}. In the example $k = 3$, vertices in $L$ and $R$ are colored black and white, respectively, and an edge between vertex $u\in L$ and a set $S$ of vertices in $R$ means that $u$ is adjacent to all vertices in $S$.[]{data-label="fig:reduction-2"}](reduction-2.pdf){width="170mm"}
Clearly this construction can be performed in polynomial time and we already pointed out that the parameter value of $(G',k')$ is equal to $k$. (Parameter value bounded by any function of $k$ would be enough for a parameterized reduction.) It remains to prove correctness, that is, that $G$ has a $k$-clique containing exactly one vertex of each set $V_i$ if and only if $(G',k)$ is a yes instance of [<span style="font-variant:small-caps;">Min-Identifiable Subgraph</span>($|L|-k$)]{}.
*Correctness.* Assume first that $(G,k,\phi)$ is yes for [<span style="font-variant:small-caps;">Multicolored Clique</span>($k$)]{}. Thus, $G$ contains a clique $C$ containing exactly one vertex of each set $V_i$. We claim that $L':=L\setminus C$ induces an identifiable $\ell$-subgraph in $G'$. The $\ell$-subgraph induced by $L'$ is exactly $G''=G'[L',R']$ where $R'=R\setminus N(L\setminus L')=R\setminus N(C)$. We need to show that $G'$ is identifiable.
Let us first see that $G''$ has a matching of $L'\setminus \{w\}$ into $R'\setminus N(w)$ for any $w\in V\cap L'$. Fix any such vertex $w$: Recall that in the construction we made for each vertex $v\in V$ vertices $p_{v,1},\ldots,p_{v,k+1}$ such that $v$ is their only neighbor in $V$. For each vertex $v\in V\setminus (L'\cup\{w\})$ all these vertices $p_{v,\ell}$ exist in $R'\setminus N(w)=R\setminus N(C\cup\{w\})$ since $C\cup\{w\}\subseteq V$ so we can match each $v$ to $p_{v,1}$. Moreover, for any $v\in L'\setminus \{w\}$ we can match all $k$ vertices of $T$ to $p_{v,2},\ldots,p_{v,k+1}$. (Here we tacitly assume that $G$ has more than $k+1$ vertices, which is w.l.o.g.) This completes the required matching.
Let us now exhibit a matching in $G''$ of $L'\setminus \{t_i\}$ into $R''=R'\setminus N(t_i)$ for any $t_i\in T\subseteq L'$: Note that $R''$ in particular does not contain vertices of sets $F_a$ for $1\leq a\leq k$ nor vertices of $E_{ab}$ for $i\notin \{a,b\}$ since all those are adjacent to $t_i$ in $G'$ (so they are also adjacent to $t_i$ in $G''$). It remains to use vertices of $E_{ab}$ with $i\in\{a,b\}$, recalling that many of them are not present already in $R'=R\setminus N(C)$. Fix $j\neq i$. Let $\{v_i\}:= C\cap V_i$ and $\{v_j\}:= C\cap V_j$. Since $C$ is a clique, vertices $v_i$ and $v_j$ are adjacent in $G$. For the corresponding edge $e=\{v_i,v_j\}$ we created vertices $q_{e,1},\ldots,q_{e,r}$ in $E_{ij}\subseteq R$ (or $E_{ji}$ if $j<i$). Each $q_{e,l}$ is adjacent to all of $V_i\setminus \{v_i\}$ and $V_j\setminus\{v_j\}$ but not to $v_i$ or $v_j$; they are not adjacent to any vertex of $V\setminus(V_i\cup V_j)$. Thus, all of these vertices are present in $R'$ and hence also in $R''=R'\setminus N(t_i)$. We can therefore match all vertices of $V_i\setminus \{v_i\}$, $V_j\setminus\{v_j\}$, and $T\setminus\{t_i,t_j\}$ to them since these are in total less than $r=n+k$ vertices. By repeating the argument for all $j'\in\{1,\ldots,k\}\setminus\{i,j\}$ we can also match vertices in $\bigcup_{j'}\left(V_{j'}\setminus C\right)\cup\{t_j\}$, obtaining a matching for all of $L'\setminus\{t_i\}=L\setminus (C\cup \{t_i\})$. (Here we need that $k\ge 3$ so that we can match $t_j$.)
It follows that the $\ell$-subgraph induced by $L'$ is indeed identifiable. Since $|L'|\leq |L|-k$ it follows that $(G',k')$ is yes for [<span style="font-variant:small-caps;">Min-Identifiable Subgraph</span>($|L|-k$)]{}.
Assume now that $(G',k')$ is yes for [<span style="font-variant:small-caps;">Min-Identifiable Subgraph</span>($|L|-k$)]{}and let $L'$ be a subset of $L$ of size at most $k'=|L|-k$ such that the $\ell$-subgraph induced by $L'$, namely $G'':=G'-N[L']=G'[L',R']$ where $R'=R\setminus N(L\setminus L')$, is identifiable. Let $C:=L\setminus L'$. This is a set of size at least $k$. We will show that $C$ is a subset of $k$ vertices of $V$ that form a clique in $G$ with $|C\cap V_i|=1$ for $1\leq i\leq k$. Note that $R'=R\setminus N(C)$.
We begin with some observations: If $V\subseteq C$ then $R'=\emptyset$ because each vertex of $R$ is adjacent to at least one vertex of $V$. In this case, $G''$ could not be identifiable since that requires having at least one edge. Thus, $V\setminus C\neq \emptyset$ and we pick an arbitrary vertex $v_0\in V\setminus C\subseteq L'$ to be used later. Similarly, if $T\subseteq C$ then again $R'=\emptyset$ and $G'[L',R']$ cannot be identifiable. We pick $t_0\in T\setminus C$ to be used later. (Note that $t_0=t_i$ for some $1\leq i\leq k$.)
We will now prove several restrictions on $C$ by contradiction-based arguments. The first two aim at proving that $T\cap C=\emptyset$.
Assume for contradiction that $|T\cap C|\geq 2$. Thus, $G''$ being identifiable implies that there must be a matching of $L'\setminus \{t_0\}$ to $R'\setminus N(t_0)$. Say $t_i,t_j\in T\cap C$ with $i\neq j$, then $R'\setminus N(t_0)$ contains no vertices of $R$ that are adjacent to any of $t_0$, $t_i$, or $t_j$. This is a contradiction since every vertex of $R$ is adjacent to at least one of them: Vertices in any $F_a$ are adjacent to each vertex of $T$ and vertices in any $E_{ab}$ are only not adjacent to two vertices of $T$, namely $t_a$ and $t_b$. Thus, $|T\cap C|\leq 1$.
Assume for contradiction that $|T\cap C|=1$ and assume w.l.o.g. that $\{t_1\}=T\cap C$. Because $G''$ is identifiable there must be a matching of $L'\setminus\{t_2\}$ into $R'\setminus N(t_2)$ in $G''$. Note that in $R'\setminus N(t_2)$ there is no vertex of any set $F_a$. Similarly, vertices of $E_{ab}$ are not in $R'\setminus N(t_2)$ unless $1\in\{a,b\}$ and $2\in\{a,b\}$ since otherwise they are adjacent to at least one of $t_1$ or $t_2$. This in turn implies that $L'$ contains no vertices from $V_a$ for $a\notin\{1,2\}$ since they have no neighbors in $R'\setminus N(t_2)$. Consequently, even $R'\supseteq R'\setminus N(t_2)$ contains no vertex of any set $E_{ab}$ with $(a,b)\neq(1,2)$ because the vertices of other sets $E_{ab}$ are all adjacent to some vertex of $V\setminus(V_1\cup V_2)\subseteq L\setminus L'=C$. Now, consider the requirement of a matching of $L'\setminus\{t_3\}$ into $R'\setminus N(t_3)$ in $G''$. We now get that there are no vertices of sets $F_a$ nor of sets $E_{ab}$. The latter holds because in $R'$ only vertices of $E_{12}$ can exist but all those are adjacent to $t_3$ and hence not in $R'\setminus N(t_3)$. Thus, $R'\setminus N(t_3)$ is empty and we cannot match the vertex $v_0$ anywhere; a contradiction.
We now know that $C\cap T=\emptyset$. Assume that $C$ contains at least two vertices of the same set $V_i$, i.e., that $|C\cap V_i|\geq 2$. Let $v_i,v'_i\in C\cap V_i$ with $v_i\neq v'_i$. Crucially, for any $1\leq a<b\leq k$ with $i\in\{a,b\}$, all vertices in $E_{ab}$ are adjacent to at least one of $v_i$ and $v'_i$, by construction: A vertex $q_{e,l}$ for $e=\{p,q\}$ with $p\in V_i$ and $q\in V_j$ is adjacent to all vertices of $V_i\setminus \{p\}$. Because only one of $v_i$ and $v'_i$ can be equal to $p$ it follows that $q_{e,l}$ is adjacent to at least one of the two. Thus, no vertex of $E_{a,b}$ is present in $R'\subseteq R\setminus N(\{v_i,v'_i\})$ for $a$ and $b$ with $i\in\{a,b\}$.
Now, because $G''$ must be identifiable there must be a matching of $L'\setminus\{t_i\}$ into $R''=R'\setminus N(t_i)$. There are no vertices of any set $F_a$ in $R''$ because all of them are adjacent to $t_i$. For the same reason there are no vertices of sets $E_{ab}$ when $i\notin \{a,b\}$. From above we know that for $E_{ab}$ with $i\in\{a,b\}$ no such vertices are present in $R'\supseteq R''$. Thus, vertex $v_0\in V\setminus C\subseteq L'$ cannot be matched; a contradiction.
At this point we know that $C$ contains no vertices of $T$ and at most one vertex of each set $V_i$. Because the size of $C$ is at least $k$ this implies that $C$ contains exactly one vertex of each set $V_i$ and no further vertices (it is of size exactly $k$). Let $\{v_i\}=C\cap V_i$ for $1\leq i\leq k$. It remains to show that the vertices $v_1,\ldots,v_k$ form a clique in $G$.
Assume for contradiction that $v_i\in V_i$ and $v_j\in V_j$ are not adjacent in $G$ for some $i\neq j$; w.l.o.g. $i<j$. Again the construction of the edge-related vertices $q_{e,l}$ is important here: Consider any edge $e=\{v'_i,v'_j\}$ with $v'_i\in V_i$ and $v'_j\in V_j$. s(Here we also tacitly assume that there is such an edge, which is w.l.o.g. as otherwise $(G,k,\phi)$ is a no instance to [<span style="font-variant:small-caps;">Multicolored Clique</span>($k$)]{}.) Because there is no edge between $v_i$ and $v_j$, we must have $v'_i\neq v_i$ or $v'_j\neq v_j$ (or both). This directly implies that each vertex $q_{e,l}\in E_{ij}$ is adjacent to $v_i$ or $v_j$ (or both) because it is adjacent to all of $V_i\setminus\{v'_i\}$ and all of $V_j\setminus\{v'_j\}$. It follows that there are no vertices of $E_{ij}$ in $R'\subseteq R\setminus N(\{v_i,v_j\})$.
Because $G''$ is identifiable there must be a matching of $L'\setminus\{t_i\}$ into $R'\setminus N(t_i)$. In the latter set there are no vertices of any set $F_a$ because they are subsets of $N(t_i)$ and similarly no vertices of $E_{ab}$ if $i\notin\{a,b\}$. Consider any vertex $v''_j\in V_j\setminus \{v_j\}\subseteq L'$ (using that $C\cap V_j=\{v_j\}$ and $|V_j|\geq 2$): Its neighbors in $G'$ are in $F_j$ and in sets $E_{ab}$ with $j\in\{a,b\}$. This leaves only the set $E_{ij}$ since we need $i\in\{a,b\}$ and $j\in\{a,b\}$ or else none of the vertices are in $R'$, but we already know that no vertices of $E_{ij}$ are in $R'\supseteq R'\setminus N(t_j)$; a contradiction.
It follows that the vertices of $C$ form a clique in $G$, with exactly one vertex from each set $V_i$; this proves that $(G,k,\phi)$ is yes for [<span style="font-variant:small-caps;">Multicolored Clique</span>($k$)]{}and completes the proof.
Concluding remarks {#sec:conclusion}
==================
In this paper, we showed that the [<span style="font-variant:small-caps;">Identifiable Subgraph</span>]{}and the [<span style="font-variant:small-caps;">Max-Identifiable Subgraph</span>]{}problems are polynomially solvable and that two natural parameterized variants of the [<span style="font-variant:small-caps;">Min-Identifiable Subgraph</span>]{}problem are [$\mathsf{W[1]}$]{}-hard. Regarding approximation issues, the [<span style="font-variant:small-caps;">Min-Identifiable Subgraph</span>]{}problem was shown to be [$\mathsf{APX}$]{}-hard [@DAM1], however, its exact (in)approximability status remains an open question.
In [@DAM1], two other [$\mathsf{NP}$]{}-hard problems related to identifiability were studied: finding the minimum number of edges that one must delete from a given identifiable graph to destroy identifiability and finding the smallest size of a set $R'\subseteq R$ such that the graph $G[L,R']$ is identifiable. The hardness proof for the former problem shows that the problem is also [$\mathsf{W[1]}$]{}-hard with respect to its natural parameterization. More precisely, one can combine the [$\mathsf{NP}$]{}-hardness proof from [@DAM1] and the proof of the [$\mathsf{NP}$]{}-hardness of the problem from which that reduction was made (finding the minimum number of edges that one must delete from a given bipartite graph in order to decrease its matching number) [@MR2519166 Theorems 3.2 and 3.3 and their proofs] to obtain a parameter-preserving reduction from the parameterized clique problem, with respect to its natural parameterization, which is [$\mathsf{W[1]}$]{}-hard. We leave for future research the determination of the parameterized complexity status of the latter problem, as well as the (in)approximability status of both problems.
Acknowledgements {#acknowledgements .unnumbered}
================
This work was supported in part by the Slovenian Research Agency (I$0$-$0035$, research program P$1$-$0285$ and research projects N$1$-$0032$, J$1$-$5433$, J$1$-$6720$, J$1$-$6743$, and J$1$-$7051$). Part of this research was carried out during the visit of M.M. to S.K. at University of Bonn; their hospitality and support is gratefully acknowledged.
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E. Fritzilas, Y. A. Rios-Solis, and S. Rahmann. Structural identifiability in low-rank matrix factorization. In [*Computing and combinatorics*]{}, volume 5092 of [*Lecture Notes in Comput. Sci.*]{}, pages 140–148. Springer, Berlin, 2008.
M. Kami[ń]{}ski and M. Milani[č]{}. On the complexity of the identifiable subgraph problem. , 182:25–33, 2015.
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R. Zenklusen, B. Ries, C. Picouleau, D. de Werra, M.-C. Costa, and C. Bentz. Blockers and transversals. , 309(13):4306–4314, 2009.
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abstract: 'We present evidence that nonlinear resonances govern the tunneling process between symmetry-related islands of regular motion in mixed regular-chaotic systems. In a similar way as for near-integrable tunneling, such resonances induce couplings between regular states within the islands and states that are supported by the chaotic sea. On the basis of this mechanism, we derive a semiclassical expression for the average tunneling rate, which yields good agreement in comparison with the exact quantum tunneling rates calculated for the kicked rotor and the kicked Harper.'
author:
- Christopher Eltschka
- Peter Schlagheck
title: 'Resonance- and chaos-assisted tunneling in mixed regular-chaotic systems'
---
Despite its genuine quantal character, dynamical tunneling [@DavHel81JCP] is strongly sensitive to details of the underlying classical phase space [@Cre98]. A particularly prominent scenario in this context is “chaos-assisted” tunneling [@LinBal90PRL; @BohTomUll93PR; @TomUll94PRE; @DorFri95PRL] which takes place between quantum states that are localized on two symmetry-related regular islands in a mixed regular-chaotic phase space. The presence of an appreciable chaotic layer between the islands dramatically enhances the associated tunneling rate as compared to the integrable case, and induces strong fluctuations of the rate at variations of external parameters [@LinBal90PRL; @BohTomUll93PR]. This phenomenon is attributed to the influence of “chaotic states” that are distributed over the stochastic sea. Since such chaotic states typically exhibit an appreciable overlap with the boundary regions of both islands, they may provide efficient “shortcuts” between the two regular quasimodes in the islands [@BohTomUll93PR; @TomUll94PRE; @DorFri95PRL]. Indeed, chaos-assisted tunneling processes arise in a number of physical systems, e.g. in the ionization of resonantly driven hydrogen [@ZakDelBuc98PRE], in microwave or optical cavities [@NoeSto97N; @DemO00PRL], as well as in the effective pendulum dynamics describing tunneling experiments of cold atoms in optical lattices [@MouO01PRE; @HenO01N].
While the statistical properties of the chaos-assisted tunneling rates are well reproduced by a random matrix description of the chaotic part of the Hamiltonian [@LeyUll96JPA], the formulation of a tractable and reliable semiclassical theory for the average tunneling rate is still an open problem. Promising progress in this direction was reported by Shudo and coworkers [@ShuIke95PRL] who obtain a good quantitative reproduction of classically forbidden propagation processes in mixed systems by incorporating complex trajectories into the semiclassical propagator Their approach requires, however, the study of highly nontrivial structures in complex phase space, and cannot be straightforwardly connected to single coupling matrix elements between regular and chaotic states. A complementary ansatz, based on a Bardeen-type expression for the coupling to the chaos, was presented by Podolsky and Narimanov [@PodNar03PRL]. In comparison with tunneling rates from the driven pendulum, good agreement was obtained for large and moderate $\hbar$, whereas significant deviations seem to occur deep in the semiclassical regime [@PodNar03PRL].
In the present Letter, we shall point out that [*nonlinear resonances*]{} between different classical degrees of freedom play a crucial role in chaos-assisted tunneling processes. Such nonlinear resonances are known to govern tunneling between symmetry-related wells in near-integrable systems [@Ozo84JPC; @BonO98PRE; @BroSchUll01PRL; @Kes03JCP], where they induce transitions to highly excited states inside the well and thereby strongly enhance the tunneling rate [@BroSchUll01PRL]. We shall argue that the same mechanism is also responsible for the semiclassical coupling between regular and chaotic states in mixed systems, and determines the average tunneling rate in chaos-assisted tunneling. A simple semiclassical expression derived from this principle shows indeed reasonably good agreement with the exact quantum splittings.
We restrict our study to systems with one degree of freedom that evolve under a periodically time-dependent Hamiltonian $H(p,q,t) = H(p,q,t + \tau)$ and are visualized by a stroboscopic Poincar[é]{} section evaluated at $t = n \tau$ ($n \in \mathbb{Z}$). We suppose that $H$ possesses a discrete symmetry which, for a suitable choice of a parameter of $H$, leads to a mixed phase space with two symmetric regular islands that are separated by a chaotic sea. We furthermore assume that each of the symmetric islands exhibits a prominent $r$:$s$ resonance—i.e., where $s$ internal oscillations around the island’s center take place within $r$ periods of the driving—which manifests itself in the stroboscopic section as a chain of $r$ sub-islands that are embedded in the torus structure of the regular island.
The motion in the vicinity of the $r$:$s$ resonance is approximately integrated by secular perturbation theory [@LicLie]. For this purpose, we formally introduce a time-independent Hamiltonian $H_0(p,q)$ that approximately reproduces the regular motion in the islands, and denote by $(I,\theta)$ the action-angle variables describing the dynamics within each of the islands. After the canonical transformation $\theta \mapsto \vartheta = \theta - s/r \cdot 2\pi t/\tau$ to the frame that corotates with the resonance, and after averaging the resulting Hamiltonian over $r$ periods of the external driving, which is justified since $\vartheta$ varies slowly with time near resonance, we obtain in lowest nonvanishing order $$H_{\rm eff}(I,\vartheta) = \frac{(I - I_{r:s})^2}{2 m_{r:s}} + 2 V_{r:s} \cos r \vartheta
\label{eq:heff}$$ as effective integrable Hamiltonian for the dynamics near the resonance. Here, $I_{r:s}$ denotes the action variable at resonance, $1/m_{r:s}$ parametrizes the variation of the internal oscillation frequency with $I$ at resonance, and $V_{r:s}$ characterizes the strength of the perturbation.
Comparing the pendulum-like dynamics of this effective Hamiltonian with the actual classical dynamics generated by $H$ provides an access to the parameters of $H_{\rm eff}$ without explicitly using the functional form of $H_0(p,q)$. To this end, we numerically calculate the monodromy matrix $M_{r:s}$ of a stable periodic point of the resonance (which involves $r$ iterations of the stroboscopic map) as well as the phase space areas $S_{r:s}^+$ and $S_{r:s}^-$ that are enclosed by the outer and inner separatrices of the resonance, respectively. Using the fact that the trace of $M_{r:s}$ as well as the phase space areas $S_{r:s}^\pm$ remain invariant under the canonical transformation to $(I,\vartheta)$, we infer $$\begin{aligned}
I_{r:s} & = & \frac{1}{4 \pi} ( S_{r:s}^+ + S_{r:s}^- ) \, , \label{eq:area} \\
\sqrt{2 m_{r:s} V_{r:s}} & = & \frac{1}{16} ( S_{r:s}^+ - S_{r:s}^- ) \, ,
\label{eq:sep} \\
\sqrt{\frac{2 V_{r:s}}{m_{r:s}}} & = & \frac{1}{r^2 \tau} \arccos({\rm tr} \,
M_{r:s}/2) \label{eq:trm}\end{aligned}$$ from the integration of the dynamics generated by $H_{\rm eff}$, which allows us to determine $I_{r:s}$, $m_{r:s}$, and $V_{r:s}$.
The implications of the nonlinear resonance for the corresponding quantum system can be directly seen from the representation of the quantized version of $H_{\rm eff}$ in the eigenbasis of $H_0$, which consists of “even” and “odd” functions with respect to the discrete symmetry of $H$. In the action-angle variable representation, the eigenfunctions of $H_0$ are, for a fixed parity, essentially given by plane waves $\psi_n(\vartheta) \sim \exp(i n \vartheta)$ as a function of the angle variable, where the integer index $n$ denotes the excitation as counted from the center of the island. The first, “kinetic” term of $H_{\rm eff}$ is therefore diagonal in this basis with the matrix elements $$E_n = [\hbar(n + 1/2) - I_{r:s}]^2/(2m_{r:s}) \, , \label{eq:en}$$ while the “potential” term $2 V_{r:s} \cos r \vartheta$ induces couplings between $\psi_n$ and $\psi_{n\pm r}$ with the matrix element $V_{r:s}$. In this way, a perturbative chain is created that connects the “ground state” $\psi_0$ of the island to the excited states $\psi_{lr}$ with integer $l$. As was shown in Ref. [@BroSchUll01PRL], this coupling mechanism generally leads to a strong enhancement of the level splitting between the even and the odd ground state in the near-integrable regime, since the unperturbed tunneling rate of a highly excited state $\psi_{lr}$ is much larger than that of $\psi_0$.
In the mixed regular-chaotic case, the above tridiagonal structure of the effective Hamiltonian becomes invalid beyond a maximum excitation index $n_c$ that marks the chaos border, i.e. for which $2\pi \hbar (n_c + 1/2)$ roughly equals the size of the island. Basis states $\psi_n$ with $n > n_c$ are defined on tori of $H_0$ that are destroyed by the presence of other strong resonances, and therefore exhibit on average a more or less equally strong coupling to each other. In the simplest possible approximation, which neglects the presence of partial barriers in the chaos [@BohTomUll93PR], the “chaotic block” $(H_{n,n'})_{n,n' > n_c}$ of the effective Hamiltonian is therefore represented by a random matrix from the Gaussian orthogonal ensemble [@TomUll94PRE; @LeyUll96JPA].
The probability density $P(\Delta E)$ for obtaining the level splitting $\Delta E$ between the ground state energies of the two symmetry classes can now be calculated by performing the random matrix average over the chaotic part of the Hamiltonian. As was worked out by Leyvraz and Ullmo [@LeyUll96JPA], this leads to a Cauchy distribution $$P(\Delta E) = \frac{4 N_c \Delta_c V_{\rm eff}^2}{(N_c \Delta_c \, \Delta E)^2 + 4 \pi^2 V_{\rm eff}^4}
\label{eq:peff}$$ with a cutoff at $\Delta E \sim 2 V_{\rm eff}$, where $N_c$ and $\Delta_c$ denote the number of chaotic states and their mean level spacing at energy $E_0$, respectively, and $V_{\rm eff}$ represents the effective coupling matrix element between the ground state and the chaotic block. In the presence of the nonlinear resonance inside the island, the latter is evaluated by means of the tridiagonal structure within the regular part of the Hamiltonian: assuming $V_{r:s}$ to be much smaller than the intermediate energy differences, we obtain $$V_{\rm eff} = V_{r:s} \prod_{l=1}^{k - 1}\frac{V_{r:s}}{E_0 - E_{lr}} \label{eq:veff}$$ where the energies $E_{lr}$ are computed from Eq. (\[eq:en\]). The elimination of intermediate regular states is performed up to the first state $\psi_{kr}$ that is already located beyond the chaos border \[i.e., $(k-1)r < n_c < kr$\].
Since tunneling rates and their parametric variations are typically studied in a logarithmic representation, the relevant quantity to be calculated from Eq. (\[eq:peff\]) and compared to quantum data is not the mean value of $\Delta E$ (which would diverge if the cutoff is not taken into account), but rather the average of the [*logarithm*]{} of $\Delta E$. We therefore obtain the “mean” level splitting $\overline{\Delta E}$ as $$\overline{\Delta E} \equiv \frac{V_{\rm eff}^2}{N_c \Delta_c} \exp \left( \left\langle \log
\frac{N_c \Delta_c \, \Delta E}{V_{\rm eff}^2} \right\rangle \right)
= \frac{2 \pi V_{\rm eff}^2}{N_c \Delta_c}$$ where $\langle\ldots\rangle$ denotes the average with respect to the probability distribution (\[eq:peff\]). The expression for the mean splitting further simplifies for our case of periodically driven systems, where the eigenphases of the time evolution operator are calculated. Using the fact that the chaotic eigenphases are more or less uniformly distributed in the interval $0 \leq \varphi < 2 \pi$, we obtain $N_c \Delta_c = \hbar \omega = 2 \pi \hbar / \tau$. This results in the mean eigenphase splitting $$\overline{\Delta \varphi} = \frac{\tau \overline{\Delta E}}{\hbar} =
\left( \frac{\tau V_{\rm eff}}{\hbar} \right)^2 \, .$$
To illustrate our theory, we apply it to one-dimensional systems that are subject to time-periodic kicks. Their classical Hamiltonian is given by $$H(p,q,t) = T(p) + \sum_{n=-\infty}^\infty \tau \delta(t - n \tau) V(q)$$ where $T(p)$ and $V(q)$ denote the kinetic energy and the potential associated with the kick, respectively. The time evolution can be represented by the map $(p,q) \mapsto
(\tilde{p},\tilde{q})$ with $\tilde{p} = p - \tau V'(q)$ and $\tilde{q} = q + \tau T'(p)$, which describes the stroboscopic Poincar[é]{} section at times immediately before the kick. The corresponding quantum dynamics is generated by the unitary operator $$U = \exp\left( - \frac{i \tau}{\hbar} T(\hat{p}) \right)
\exp\left( - \frac{i \tau}{\hbar} V(\hat{q}) \right)$$ where $\hat{p}$ and $\hat{q}$ denote the momentum and position operator, respectively.
Specifically, we consider the kicked rotor given by $T(p) = \frac{1}{2} p^2$ and $V(q) = K \cos q$ with $\tau \equiv 1$, and the kicked Harper given by $T(p) = \cos p$ and $V(q) = \cos q$ [@LebO90PRL]. Using Bloch’s theorem, we restrict our study to eigenfunctions of $U$ that are periodic in position. We furthermore choose $\hbar = 2 \pi / N$ where $N$ is an even integer. This allows us, for both the kicked rotor and the kicked Harper, to write the periodic eigenfunctions as Bloch functions in momentum—i.e., with $\tilde{\psi}(p + 2 \pi) = \tilde{\psi}(p) \exp(i \xi)$ where $\tilde{\psi}$ is the Fourier transform of $\psi$. Since the subspace of such functions is $N$-dimensional for fixed $\xi$ [@LebO90PRL], the eigenphases and eigenvectors of $U$ can be calculated by diagonalizing finite $N \times N$ matrices.
Quantum tunneling can take place between a regular island in the fundamental phase space cell and its periodically shifted counterparts. As a consequence, different Bloch phases $\xi$ lead to slightly different eigenphases for states that are localized on a given torus in the island. The spectral quantity that we discuss in the following is the difference $\Delta\varphi = |\varphi^{(\xi=0)} - \varphi^{(\xi=\pi)}|$ between the eigenphases of the island’s ground state for $\xi=0$ and $\xi=\pi$. $\varphi^{(\xi=0)}$ and $\varphi^{(\xi=\pi)}$ are calculated by diagonalizing $U$ in a suitable basis [@BroSchUll01PRL], and by identifying the ground state from the localization properties of the eigenstates near the center of the island. Multiple precision arithmetics is used in order to calculate splittings below $\Delta\varphi \sim 10^{-15}$.
Figs. \[fg:spkr\](a), \[fg:spkr\](b), and \[fg:spkh\](a) show the eigenphase splittings for the kicked rotor and the kicked Harper, respectively, as a function of $N = 2 \pi / \hbar$, calculated for $K = 2$ and $3$ in Fig. \[fg:spkr\] as well as for $\tau = 2$ in Fig. \[fg:spkh\](a). The step-like curves show our semiclassical predictions of the eigenphase splittings, which are based on prominent resonance chains boldly marked in the corresponding phase space. The relevant parameters $m_{r:s}$, $V_{r:s}$ and $I_{r:s}$ are computed from phase space areas and periodic points via Eqs. (\[eq:area\]–\[eq:trm\]). From the numerically calculated phase space area $S$ covered by the island, we infer the number $k$ of intermediate steps that are necessary to couple the ground state to the chaos. An artificially sharp decrease of the semiclassical splitting $\overline{\Delta \varphi}$ therefore occurs whenever $\hbar$ passes through a value where $S = 2 \pi \hbar (kr + 0.5)$ with integer $k$.
In spite of the number of simplifications and approximations that are involved in the derivation of the semiclassical expression for the mean eigenphase splittings, we obtain a relatively good agreement between $\Delta\varphi$ and $\overline{\Delta \varphi}$. In particular, the first major plateau in the quantum splittings is remarkably well matched by the semiclassical curve, which clearly indicates that the coupling to the chaotic sea is mediated by the nonlinear resonance there. Our method fails to reproduce the quantum splittings in the “anticlassical” limit of large $\hbar$, e.g. for $N < 50$ in Fig. \[fg:spkr\](b). Preliminary calculations show, however, that a better agreement in this regime might be obtained by properly taking into account the action dependence of $V_{r:s}$ in the effective Hamiltonian (\[eq:heff\]), which is completely neglected in the present treatment. More details will be presented in a subsequent publication.
Apart from Fig. \[fg:spkr\](b), where also plateaus of higher order are well reproduced by the semiclassical theory, we observe a systematic tendency to overestimate the exact quantum splittings for low and moderate values of $1/\hbar$. We tentatively attribute this fact to the existence of partial barriers in the chaotic part of the phase space, which may enhance the effective size of the island for the quantum tunneling process. In particular, it is known that “Cantori” (i.e., broken tori) in the chaos inhibit the quantum transport in a similar way as invariant tori in the island, as long as the phase space area associated with the classical flux through the Cantorus is smaller than $\pi \hbar$ [@GeiRadRub86PRL]. A better agreement with the quantum splittings might therefore be obtained by properly incorporating [*hierarchical*]{} states [@KetO00PRL], which are localized in the immediate vicinity of the island, into the semiclassical description (see in this context also [@DorFri95PRL]).
Finally, Fig. \[fg:spkh\](b) shows the case of tunneling between two symmetric regular islands in the kicked Harper at $\tau = 3$, which arise from a bifurcation of the central island taking place at $\tau = 2$. The quantum splittings are now given by the eigenphase difference between the symmetric and the antisymmetric state associated with the pair of islands, calculated here at fixed $\xi = 0$. We see that the splittings display a prominent plateau at $N \simeq 300 \ldots 500$, which is well reproduced by the semiclassical prediction based on a $9$:$1$ resonance inside the islands.
In conclusion, we have presented a straightforward semiclassical scheme to reproduce tunneling rates between regular islands in mixed systems. Our approach is based on the existence of a prominent nonlinear resonance inside the island, and uses elementary classical parameters associated with this resonance to estimate the coupling rate from the island to the chaos. In combination with a random matrix description of the chaotic part of the Hamiltonian, we obtain a simple expression for the average level splittings between symmetry-related islands, which agrees reasonably well with the exact quantum splittings calculated for the kicked rotor and the kicked Harper. Our study underlines that nonlinear resonances govern the coupling between regular islands and the surrounding chaotic sea in the semiclassical limit. We expect that they play an equally prominent role also in complex systems with more degrees of freedom, for which our approach could develop into a useful method to quantitatively estimate tunneling rates in presence of chaos.
We thank O. Brodier, S. Keshavamurthy, S. Tomsovic, and D. Ullmo for fruitful and inspiring discussions. Support from the Deutsche Forschungsgemeinschaft is gratefully acknowledged.
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author:
- |
\
European Southern Observatory, Karl-Schwarzschild-Str. 2, D-85748 Garching bei München, Germany\
E-mail:
title: 'Concluding Remarks at the Multifrequency Behaviour of High Energy Cosmic Sources - XIII Workshop – II'
---
The end of an era {#sec:Intro}
=================
If I had to summarise in one sentence this workshop, I would say this: [ *It marked the end of an era and the beginning of a new one.*]{} People have a tendency to think that their epoch is somewhat special compared to previous ones but in our case, it really is! In the last couple of years we have witnessed the birth of (non-stellar) multi-messenger astrophysics[^1]. First, there was the detection of the first gravitational wave (GW) event by LIGO in 2016 [@Abbott_2016] \[Gondek-Rosinka, Poggiani[^2]\], followed by the association in 2017 between a LIGO/Virgo event (GW170817) and electromagnetic emission from a binary neutron star merger (GRB 170817A) in the galaxy NGC 4993 at $z=0.01$ [@Abbott_2017_b] \[D’Avanzo\]. Then there was the first association of high-energy IceCube neutrinos with a blazar at $z=0.3365$, TXS 0506+056, in July 2018 [@2018Sci...361.1378I; @2018Sci...361..147I; @Padovani_2018] \[Righi, Paredes, Padovani\]. And, as a bonus, this year we also had the first image of the “shadow” of a black hole (BH) [@EHT_2019] \[De Laurentis\]. What an amazing time to be an astronomer!
While the astrophysical relevance of the GW170817/GRB 170817A event has been discussed at length in the literature, I feel this is not the case for the neutrino result. This has a number of astrophysical implications [@Padovani_2018]: 1. with neutrinos we are now exploring an energy range ($\sim$ PeV $= 10^{15}$ eV) which is, and will always be, inaccessible with photons at this (or any!) redshift. That is because these very high-energy photons collide with IR/mm photons (the so-called extragalactic background light \[Costamante, Paredes\]) and are annihilated with the resulting production of electron – positron pairs. Neutrinos provide us then with a new and unique window on blazar physics; 2. the spectral energy distribution of at least one blazar has to be modelled using protons (the so-called lepto-hadronic scenario), laying to rest a debate (leptons or hadrons?), which has been around for decades; 3. the number of known neutrino sources has jumped by 50% from two (the Sun and SN 1987A) to three. And we have identified the first non-stellar neutrino source; 4. the [*first*]{} cosmic ray (CR) source has been identified. The IceCube results, in fact, imply the existence of protons with energies $\gtrsim 1$ PeV, around the so-called “knee” of the CR flux distribution \[Caccianiga\], in the blazar TXS 0506+056. And as it often happens when new, ground-breaking observations are available, theorists have some difficulty in explaining in a coherent way the electromagnetic and neutrino emission in TXS 0506+056 (e.g. [@Keivani_2018; @Winter_2019]) \[Righi\].
Selected topics at this workshop {#sec:topics}
================================
![The distribution of talks at the workshop grouped by topics: 21% of the talks dealt with multi-messenger astronomy.[]{data-label="fig:chart"}](Pie_Chart_Mondello.png){width="100.00000%"}
The advent of this “new era” is also apparent in the topics discussed at the workshop, which are shown in Fig. \[fig:chart\]: a conservative estimate shows that 21% of the talks dealt with multi-messenger (i.e., non “photon-based”) astronomy. This is a considerable fraction, which reflects also the fact that this was the first Frascati Workshop after the LIGO/Virgo and IceCube results. I now touch upon some selected (and biased) topics, which were discussed at the workshop.
Peak luminosities of Gravitational Wave events
----------------------------------------------
The GW events reach extraordinary values of peak luminosity, so far all but one above $10^{56}$ erg s$^{-1}$ (Tab. III of [@Abbott_2019]), albeit over a very short time ($\approx$ a few ms) \[Gondek-Rosinka, Poggiani\]. These can be compared to the powers of Active Galactic Nuclei (AGN), with $L \lesssim 3 \times
10^{48}$ erg s$^{-1}$ over more than tens of millions of years, and $\gamma$-ray bursts, with $L \lesssim 5 \times 10^{54}$ erg s$^{-1}$ over a few seconds. Amazingly enough, the GW luminosities are even larger than the power emitted by all the stars in the Universe! An upper limit to this value can be roughly calculated by multiplying the luminosity of the Milky Way ($ \sim 8 \times 10^{43}$ erg s$^{-1}$) by the number of galaxies in the Universe ($\approx 10^{12}$: [@Conselice_2016]), which gives $L_{\rm all~stars} < 8 \times 10^{55}$ erg s$^{-1}$ (as most galaxies are less luminous than the Milky Way).
These powers are also quite close to the [*maximum*]{} luminosity of any physical system, which is sometimes called the Planck luminosity $L_{\rm
Planck}$ [@Thorne_1983; @Abbott_2017_a]. This can be derived by dividing the rest mass energy of a body ($M c^2$) by the crossing time of its event horizon ($\sim [2 G M/c^2]/c$), which yields $L_{\rm Planck}/2$, where $L_{\rm Planck} = c^5/G = 3.63 \times 10^{59}$ erg s$^{-1}$.
On a separate note, although the LIGO/Virgo data are not sensitive to the signal of binary supermassive black hole (SMBH), GW data have already been used in this context: [@Jenet_2004] have excluded the presence of a binary SMBH in 3C 66B, a radio galaxy at $z=0.02$, using pulsar timing \[Possenti\]. Were this binary BH there, in fact, as had been suggested by radio observations, it would have been apparent in the 7 yr of timing data from the radio pulsar PSR B1855+09.
In general, accreting systems, ranging from white dwarf binaries to cataclysmic variables to X-ray binaries to AGN, are potential gravitational wave emitters at different scales \[Poggiani\]. The spectrum of GWs from these systems is very broad and their detection requires ground based interferometry, space based interferometry, and pulsar timing.
Getting really close to black holes {#sec:BH_close}
-----------------------------------
The Event Horizon Telescope (EHT) has given us the first ever images of the “shadow” of the BH at the centre of M 87 [@EHT_2019] \[De Laurentis\], which has allowed the EHT collaboration to determine its mass ($6.5\pm0.7
\times 10^9$ M$_{\odot}$). The diameter of the shadow is $\sim 5.5$ times the Schwarzschild radius of the SMBH. It is estimated that these images have been seen by about three billion people (H. Falcke, p.c.), which would make these the most popular images of all times. With a jet inclination angle $\sim 17^{\circ}$ [@Walker_2018] M 87 is [*almost*]{} a blazar of the BL Lac type [@Urry_1995] (see also Sect. \[sec:blazars\]). GRAVITY at the ESO/VLT is also getting very close to the BH at the centre of the Milky Way by studying flares at distances $\sim 3 - 5$ times the Schwarzschild radius, with significant and continuous positional changes of the emission centroid corresponding to $\sim 30\%$ the speed of light [@Gravity_2018] \[Borkar\]. The Extremely Large Telescope[^3], which in 2025 will be the largest optical-near-IR telescope in the world with a diameter of 39m \[Padovani\], will be able to detect a $10^6$ M$_{\odot}$ BH up to $\sim 30$ Mpc and a $10^9$ M$_{\odot}$ one up to $\sim 1$ Gpc. This will provide an increase in the distances reachable with current 8-10m telescopes of a factor $\approx 5$. The ELT, thanks to its much better resolution, will also be able to get even closer to BHs than currently possible.
Black hole spins
----------------
We have also heard about BH spins \[Aschenbach, Bambi\]. The mantra in the AGN community has been for years that jetted AGN [@Padovani_2017] (also called, I think misleadingly, “radio-loud”) are powered by a rotating BH while non-jetted AGN are not [@Padovani_2017_b]. X-ray reflection spectroscopy is currently the only available method to measure the spin of SMBHs and $70\%$ of AGN have very high $a_*$ ($> 0.9$, where $a_* = c J/G M^2$ is the dimensionless BH spin parameter and $J$ is the angular momentum) \[Bambi\]. But most of these are non-jetted AGN, which should be non-rotating. However, these spin measurements have to be taken carefully, as they may be affected by systematics related to the model employed to infer them [@Bambi_2018].
M 87 is a classical jetted AGN. I would have then expected the EHT collaboration (Sect. \[sec:BH\_close\]) to find a relatively high value of $a_*$. However, the M 87 papers just say that “compact 1.3 mm emission in M87 arises within a few $r_g$ of a Kerr BH" [@EHT_2019_1]. So we know that the BH at the centre of M 87 is rotating but more data are needed before a specific value of $a_*$ can be provided. This reflects the fact that the size of the shadow of a Kerr BH depends only weakly on spin (and inclination) [@Psaltis_2019].
The LIGO/Virgo GW data can also constrain spins \[Gondek-Rosinka, Poggiani\], in this case of stellar BHs. Table III of [@Abbott_2019] provides values of $\chi_{\rm eff}$, which is a mass-weighted linear combination of the spins of the two merging BHs projected onto the Newtonian angular momentum. Only in 2/11 cases can $\chi_{\rm eff} = 0$ be excluded at $>
90\%$ confidence, which means that the data disfavour scenarios in which most BH merge with large spins aligned with the binary’s orbital angular momentum.
Magnetars
---------
I really enjoyed learning about magnetars \[Mereghetti, Nakagawa, Dainotti, D’Avanzo, Ferrazzoli\]; see also [@Mereghetti_2015]. A magnetar is a type of neutron star having a huge external magnetic field $\sim 10^{13} -
10^{15}$ G, i.e., up to $\sim 1,000$ above the average. This is the main source of energy, instead of the rotation, accretion, nuclear reactions, or cooling, which power the more normal neutron stars. What are now called magnetars were initially split into two separate classes of sources, soft $\gamma$-ray repeaters and anomalous X-ray pulsars. The main properties of the two dozen magnetars known in the galaxy and the Magellanic Clouds are their slow rotation periods (P $\sim 2 - 12$ s), persistent X-ray powers ($L_{\rm X} \approx 10^{34} - 10^{36}$ erg s$^{-1}$), faint optical/NIR counterparts ($K \sim 20$), and strong variability, with powerful short bursts in the X-rays and soft $\gamma$-rays, often reaching super-Eddington luminosities. Three [*giant*]{} flares have also been observed with huge peak powers. The most powerful one ($\approx 10^{47}$ erg s$^{-1}$) came from SGR 1806–20 on December 27, 2004 and was mind-boggling: more than 20 satellites recorded this exceptional event, which started with a hard pulse so intense that it saturated most detectors and significantly ionised the Earth’s upper atmosphere [@Mereghetti_2005]. The flare was brighter than anything ever detected from beyond our Solar System with a fluence $\sim 2$ erg cm$^{-2}$ ($E > 80$ keV) and lasted over a tenth of a second. Magnetars are also possible GW sources both through their fast rotation, which leads to deformations, and their impulsive activity.
Blazars {#sec:blazars}
-------
Blazars are AGN hosting a relativistic jet oriented at a small angle ($\lesssim 15 - 20^{\circ}$) w.r.t. the line of sight [@Urry_1995; @Padovani_2017]. This translates into very interesting and somewhat extreme properties, including relativistic beaming, which makes blazars appear orders of magnitude more powerful than they really are, superluminal motion, and strong, non-thermal emission over the entire electromagnetic spectrum and beyond, i.e., into neutrino territory. At the meeting we heard the latest news about blazars \[Costamante, Böttcher, Pittori\]. These include the image of the “shadow” of the BH at the centre of M 87, which is “almost” a blazar (Sect. \[sec:BH\_close\]) and the first association of high-energy IceCube neutrinos with a blazar of the BL Lac type at $z=0.3365$, TXS 0506+056 (Sect. \[sec:Intro\]). Apparently Nature loves disks and jets, as they seem to be present in a variety of astronomical objects ranging from stars and planetary systems, X-ray binaries, and AGN, including blazars of the “flat spectrum radio quasar” (FSRQ) type. In the latter, however, it looks like the power of relativistic jets is larger than the luminosity of their accretion disks [@Ghisellini_2014]. $\gamma$-ray emission in FSRQs is generally explained as inverse Compton radiation of relativistic electrons in the jet scattering optical-UV photons from the broad-line region (BLR), the so-called BLR external Compton (EC) scenario. However, [@Costamante_2018] have found no evidence for the expected BLR absorption, with only 1 object out of 10 being compatible with substantial attenuation, which essentially rules out the EC mechanism and implies that $\gamma$-ray emission originates predominantly outside the BLR. This has important implications for the theoretical interpretation of the spectral energy distributions of blazars and it also means that CTA should see many more FSRQs than previously thought. Finally, the Astro-rivelatore Gamma a Immagini Leggero (AGILE) satellite, has found three transient $\gamma$-ray sources ($E > 100$ MeV) temporally and spatially coincident with recent high-energy neutrino IceCube events [@Lucarelli_2019]. The post-trial chance probability for this to happen is $\sim 4.7 \sigma$. One of the objects is the already known neutrino source TXS 0506+056 (Sect. \[sec:Intro\]). For the other two there are no obvious counterparts, although one of the most interesting sources is (again) a blazar of the BL Lac type (3FGL J0627.9–1517).
Supernovae (in the optical band)
--------------------------------
Last but not least, I wanted to mention two examples of synergy between supernovae in the optical band and cosmology. We learnt that extinction in starburst clusters is temporarily altered by type II SNe for $\sim 50 -
100$ Myr after the star formation episode, which has important implications for extinction corrections in the early Universe \[De Marchi\]. And also that SNe Ia are dimmer in ellipticals and brighter in spirals, which implies that the supernova properties depend on their environment. Said differently, one needs to include a host galaxy term into the Hubble diagram fit, which is used to constrain the shape of the Universe \[Pruzhinskaya\].
New facilities and the era of “even bigger data” {#sec:facilities}
================================================
We have also heard about some of the new facilities, which will come online in the next few years (or have been recently starting taking data) and will be very relevant for the topics discussed at the workshop. These include, without any claim to completeness (and in rough chronological order within a band):
- Radio: ASKAP, MeerKAT, e-MERLIN, APERTIF, SKA
- IR: JWST, Tokyo Atacama Observatory, Euclid, WFIRST, SPICA
- Optical/near-IR: Zwicky Transient Facility, LSST, ELT, GMT, TMT
- X-ray: eROSITA, IXPE, SVOM, eXTP, XIPE, Athena, Theseus, FORCE, XRISM, Colibrì
- $\gamma$-ray: Large High Altitude Air Shower Observatory, CTA
... and certainly more, including CubeSats \[Bernardini, Caiazzo, Ferrazzoli, Hudec, Ishida, Mori, Padovani\]. These are going to move us from the “big data” era into the “even bigger data” era. For example, while the volume of the Sloan Digital Sky Survey was around 40 Terabytes, the LSST will reach 200 Petabytes, while the SKA will get into Exabyte territory [@Zhang_2015].
Quantity vs. quality
--------------------
![The number of astronomical papers (red line) and publishing astronomers (blue line) per year vs. year. Note the logarithmic scale on the y-axis. Courtesy of Robert Simpson: see <https://orbitingfrog.com/2012/08/04/authorship-in-astronomy/>.[]{data-label="fig:papers"}](Papers_authors_per_year.pdf){width=".7\textwidth"}
More data might mean more papers. Can we see signs of increasing astronomical output also in terms of published papers? Yes, as shown in Fig. \[fig:papers\] (red line). After a slow rise the numbers have started to pick up after around 1960. Is this a good sign? Well, about 50% of all science papers have $\leq 1$ citations, based on a study of about 58 million papers published since 1900 [@VanNoorden_2014]. And the percentage of papers published in 2009 with no citations other than self-citations after 5 years, based on 120 million academic papers, is 72.1%, down from even higher values in previous years [@Fire_2019].
The case for astronomy looks much better, although the studies I found were old and/or limited in numbers: 3.3% of papers (283/7724) published in 2001 and 2002 in 20 journals are never cited during the three calendar years after the one in which they were published [@Trimble_2007]; and 6.1% of papers (20/326) published in 1961 in American journals are never cited during the 18 years after publication [@Abt_1981]. To get some better statistics I picked two random contiguous years (2000 and 2001) and used the Astrophysics Data System. Out of the 39,972 astronomical refereed papers published in that time period 15.8% (6327) gathered zero citations to date. Most of these are in (many) minor journals. If I consider only AAS journals, A&A and A&AS, MNRAS, PASP, and PASJ the fraction of never cited papers gets much smaller, i.e., 1.3% (159/12568).
But is quantity correlated with quality? Or is there actually an inverse correlation? A “destructive feedback between the production of poor-quality science, the responsibility to cite previous work and the compulsion to publish” has been pointed out [@Sarewitz_2016] (see also Franco Giovannelli’s introductory remarks at this workshop). The same paper talks also about the fact that “Current trajectories threaten science with drowning in the noise of its own rising productivity ... Avoiding this destiny will, in part, require much more selective publication. Rising quality can thus emerge from declining scientific efficiency and productivity. We can start by publishing less, and less often”. Granted, this paper deals with biology but some of these points apply to astronomy as well. Whose fault is it? At least partly ours. We as referees, together with the journal editors, allow way too many papers to get published[^4]. Moreover, I have the feeling that the system, to which we all belong, tends to give way too much importance to quantity, which is easier to evaluate, and less to quality.
![The number of astronomical papers per capita per year vs. year. This is the ratio of the red and blue lines in Fig. \[fig:papers\]. Note the logarithmic scale on the y-axis. Courtesy of Robert Simpson: see <https://orbitingfrog.com/2012/08/04/authorship-in-astronomy/>.[]{data-label="fig:papers_per_author"}](Papers_per_author_Astronomy.pdf){width=".7\textwidth"}
But are we really publishing more? No: per capita we are publishing less! Fig. \[fig:papers\_per\_author\] shows that while in 1960 astronomers were publishing about 1 paper each per year, we are now getting close to 1 paper every three years. Every colleague I show this figure to is shocked, as we are all under the impression that we are publishing a lot. But the number of papers has not caught up with the increasing number of astronomers, which means we are getting less efficient. My interpretation of this result has to do with the increasing size of astronomical collaborations: while a group of $2-4$ astronomers might easily publish $2-4$ papers per year, a large collaboration of, say, 100 people, is [ *not*]{} going to publish 100 papers per year. One might however argue that papers in 1960 were shorter [@Abt_2000] and therefore easier to write, although I am not sure this effect by itself can explain the trend shown in Fig. \[fig:papers\_per\_author\].
Getting ready {#sec:ready}
=============
Let us look at the bright side: we will soon be (even more) flooded with data relevant to the topics discussed at this workshop. And we need to be ready for that. What follows is some advice (mostly for the younger members of the astronomical community):
- Think out of the box: even now there are lots of data but very few new ideas. Spend less time running around writing papers and more time thinking about the important open issues.
- Ask the right questions: doing that is the toughest part of solving problems.
- Change topic every once in a while. I love this quote from Pablo Picasso: “Success is dangerous. One begins to copy oneself, and to copy oneself is more dangerous than to copy others. It leads to sterility.” Astronomy is great also because one can change band and topic relatively easily. It takes some courage and humility, as you need to start from scratch in a new field, but it is very rewarding.
- Learn the right tools. Be they “data mining”, “virtual observatory”, “neural networks”, “artificial intelligence”, whatever. It is clear that we cannot handle the huge amount of data we will soon get without changing the way we deal with them. As a first step, have a look at the presentations given at the Workshop on “Artificial Intelligence in Astronomy” held at ESO in June 2019 ([<https://www.eso.org/sci/meetings/2019/AIA2019.html>]{}).
- Have fun! This is the most important advice. If you do not enjoy what you are doing you will be much less productive and also less happy.
I want to conclude with a plea to Franco Giovannelli to change the workshop name to [*Multimessenger Behaviour of High Energy Cosmic Sources*]{}!
I thank the organisers of the workshop for their kind invitation, Evanthia Hatziminaoglou, Elisa Resconi, and Eva Villaver for their careful reading of the paper, and Mariafelicia De Laurentis, Sandro Mereghetti, and Rosa Poggiani for useful discussions.
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[^1]: This birth happened actually in 1968 with the discovery of solar neutrinos and then (again) with the detection of neutrinos from SN 1987A. Hence the addition of “non-stellar”.
[^2]: I include the names of the speakers who dealt with the topic at hand during the Workshop.
[^3]: <http://www.eso.org/sci/facilities/eelt/>
[^4]: I am doing my best to go against the tide: in the past four years I have rejected 73% of the papers I have refereed. I simply require that papers add something [*substantially new*]{} to our understanding of the topic at hand.
|
[**Betrachtungen jenseits des Standardmodells der Teilchenphysik**]{}\
\
Diplomarbeit\
von\
Martin Kober\
Institut für Theoretische Physik\
Johann Wolfgang Goethe Universität\
Frankfurt am Main\
Einleitung {#einleitung .unnumbered}
==========
Die vorliegende Diplomarbeit besteht aus zwei Teilen. Im ersten Teil der Arbeit geht es um die Erweiterung der Allgemeinen Relativitätstheorie durch die Annahme zusätzlicher Dimensionen, während im zweiten Teil die Konsequenzen einer veränderten Raumzeitgeometrie unter der Annahme einer Vertauschungsrelation zwischen den Ortskoordinaten beschrieben werden. Beide Teile umfassen jeweils vier Kapitel, wobei die ersten drei Kapitel der Beschreibung des grundsätzlichen Rahmens dienen, während die jeweils letzten Kapitel die neuen Ergebnisse dieser Diplomarbeit enthalten.
Der erste Teil “Gravitation und zusätzliche Dimensionen” beginnt mit einer Beschreibung der Allgemeinen Relativitätstheorie. Am Anfang steht eine historische Einleitung, da ein wahres Verständnis der Theorie wohl am besten in Bezug auf den historischen Zusammenhang erreicht werden kann. Die Darstellung muss sich zwangsläufig auf die grundlegenden Aussagen der Theorie beschränken, wobei zunächst eine kurze Einleitung in die bei der Formulierung der Theorie verwendete Mathematik gegeben wird. Im darauffolgenden Kapitel wird die Idee der Annahme zusätzlicher kompaktifizierter Dimensionen erläutert, welche auf Kaluza und Klein zurückgeht. Damit sind die Voraussetzungen geschaffen, um im vierten Kapitel eine effektive Quantenfeldtheorie der Gravitation unter Einbeziehung der zusätzlichen kompaktifizierten Dimensionen zu entwickeln, welche sich der Methode der Pfadintegralquantisierung bedient. Im letzten Kapitel des ersten Teiles soll dann schließlich die Konsequenz für einen Wechselwirkungsprozess der Teilchenphysik untersucht werden, nämlich die Produktion eines ZZ-Paares durch Vernichtung zweier Protonen. Ein solcher Vorgang vermittelt durch Gravitonen könnte am LHC relevant werden. Abhängig von der Zahl der zusätzlichen Dimensionen und dem Kompaktifizierungsradius und damit der modifizierten Planckmasse ergibt sich ein Beitrag zu dem Wirkungsquerschnitt für den genannten Prozess. Dieser wird mit dem Wert verglichen, der bei alleiniger Zugrundelegung des Standardmodells erwartet würde, was Rückschlüsse darauf zulässt, ob die Existenz zusätzlicher Dimensionen sich in einer signifikant erhöhten ZZ-Produktionsrate äußern könnte. Das Ergebnis der Untersuchung der ZZ-Produktionsrate im ADD Modell stellt das eigentlich neue wissenschaftliche Ergebnis des ersten Teiles dar und ist in [@Kober:2007bc] veröffentlicht. Natürlich handelt es sich bei der hier verwendeten effektive Beschreibungsweise der Gravitation um einen Versuch, die Gravitation in den Rahmen der bisherigen empirisch bestätigten Teilchenphysik zu integrieren, welche durch relativistische Quantenfeldtheorien beschrieben wird. Dies rechtfertigt den Titel der Arbeit “Betrachtungen jenseits des Standardmodells der Teilchenphysik” auch in Bezug auf den ersten Teil. Es kann sich bei der hier verwendeten linearen Entwicklung des metrischen Feldes um die Metrik der flachen Minkowskiraumzeit a priori nur um eine Näherung handeln. Wichtiger aber ist, dass man es mit einer nichthintergrundunabhängigen Theorie zu tun hat. Damit wird sie der eigentlich konzeptionell wichtigsten Aussage der Allgemeinen Relativitätstheorie nicht ganz gerecht, die darin besteht, dass die Raumzeit und ihre Struktur an sich eine dynamische Entität darstellt, das Gravitationsfeld also identisch mit der Raumzeit selbst ist. Dennoch ist die Theorie durchaus adäquat in Bezug auf die Untersuchung der Konsequenzen der in der hier beschriebenen Weise zusätzlich eingeführten Dimensionen für die Teilchenphysik.
Das erste Kapitel des zweiten Teiles “Nichtkommutative Geometrie” beinhaltet zunächst eine Darstellung des theoretischen Gebäudes von Eichtheorien. Diese führen die Bedeutung von Symmetrieprinzipien in der Teilchenphysik vor Augen, denen wohl grundsätzlich eine konstitutive Rolle bei der Beschreibung der Wirklichkeit zukommt. In Anschluss daran wird die elektroschwache Theorie und der damit verknüpfte Higgsmechanismus vorgestellt. Das folgende Kapitel gibt eine Einführung in die Grundidee der nichtkommutativen Geometrie, welche unter anderem die Definition eines Sternproduktes für die Multiplikation von Feldern zur Folge hat. Schließlich wird sich speziell der Formulierung von Eichfeldtheorien auf einer nichtkommutativen Raumzeit zugewendet, die das Konzept von sogenannten Seiberg-Witten-Abbildungen beinhaltet, welche die Theorie der nichtkommutativen Raumzeit auf eine Theorie mit kommutativer Raumzeit abbilden. Der letzte Teil ist einer neuen Untersuchung gewidmet, welche sich auf den Higgsmechanismus bezieht und rein theoretischer Natur ist. Es wird gezeigt, dass die spontane Symmetriebrechung auf der nichtkommutativen Raumzeit nach Abbildung auf die gewöhnlichen Felder zum gleichen Ergebnis führt wie die spontane Symmetriebrechung nach der Abbildung, wie sie gewöhnlich in Betracht gezogen wird.
Die Allgemeine Relativitätsheorie
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Historische Einleitung
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> [“General Relativity is the discovery that spacetime and the gravitational field are the same entity. What we call >spacetime< is itself a physical object, in many respects similar to the electromagnetic field.” (Carlo Rovelli)]{}
Gegen Ende des neunzehnten Jahrhunderts gab es zwei Bereiche in der theoretischen Physik, die jeweils in sich widerspruchsfrei waren und den ihnen entsprechenden Erfahrungsbereich angemessen beschreiben konnten. Es handelte sich um die klassische Mechanik und die Elektrodynamik. Die klassische Mechanik stellt den ersten Versuch einer allgemeinen und exakten Naturtheorie dar. Newton gelang es bekanntlich, alle mechanischen Erscheinungen, von der Planetenbewegung bis hin zum harmonischen Oszillator in eine allgemeine begriffliche Form zu bringen, die durch die bekannten Newtonschen Axiome ausgedrückt wird. Später wurde sie durch Lagrange und Hamilton in einer noch eleganteren Weise mathematisch formuliert. Die klassische Mechanik kennt, um es in der Sprache der Ontologie auszudrücken, vier basale Entitäten. Es handelt sich um Raum, Zeit, Körper und Kräfte. Raum und Zeit stellen so etwas wie eine starre Bühne dar, auf der sich das physikalische Geschehen abspielt, welche selbst jedoch nicht in das dynamische Geschehen miteinbezogen ist. Es sind die Körper, welche den Gegenstand einer dynamischen Beschreibung darstellen. Gemäß dem zweiten Newtonschen Axiom ist die Beschleunigung eines Körpers proportional zu der auf ihn einwirkenden Kraft. Damit ein physikalischer Vorgang eindeutig bestimmt ist, bedarf es neben den Anfangsbedingungen für die Geschwindigkeit und die Position des entsprechenden Körpers eines speziellen Kraftgesetzes, dass eine Abhängigkeit der Kraft vom Raumzeitpunkt festlegt. Im Falle der Astronomie liefert dies das Newtonsche Gravitationsgesetz demgemäß zwischen zwei Körpern eine Kraft proportional zum reziproken des Abstandsquadrates und dem Produkt der beiden Massen wirkt.
Es stellte sich jedoch heraus, dass die klassische Mechanik nicht in der Lage war, auch die im achtzehnten und neunzehnten Jahrhundert untersuchten elektromagnetischen Phänomene zu erklären. Den Schlüssel zu einer allgemeinen Theorie, die genauso wie die Newtonsche Theorie im Bereich der Mechanik alle Phänomene im Bereich der Elektrodynamik umfasste, lieferte die Einführung des völlig neuartigen Begriffs des Kraftfeldes durch Faraday. Faraday hatte die Idee, die in der klassischen Mechanik als starr vorausgesetzten Kräfte selbst einer inneren Dynamik zu unterwerfen. Schließlich gelang Maxwell aufbauend auf der konzeptionellen Revolution durch Faraday die exakte mathematische Formulierung der Elektrodynamik. Er entwickelte ein System von vier Gleichungen, welche die Dynamik der elektromagnetischen Kraftfelder vollständig beschreiben.
Nun entdeckte jedoch Lorentz, dass die Maxwellschen Gleichungen nicht invariant unter der Transformationsgruppe der klassischen Mechanik, den Galileitransformationen, sondern invariant unter einer neuen Transformationsgruppe sind, den nach ihm benannten Lorentztransformationen. Diese haben die Eigenschaft, dass sie die Lichtgeschwindigkeit invariant lassen. Klassische Mechanik und Elektrodynamik sind also invariant unter unterschiedlichen Transformationsgruppen und damit unvereinbar. Daneben stand das Experiment von Michelson und Morley, das ebenfalls die Unabhängigkeit der Lichtgeschwindigkeit vom Bezugssystem des Beobachters nahelegte. Es war nun eine der großen Leistungen Einsteins, dass er erkannte, dass sich der Widerspruch zwischen klassischer Mechanik und Elektrodynamik durch Aufgabe der bisherigen Bedeutung des Begriffs der Gleichzeitigkeit aufheben ließ. Einstein relativierte die Begriffe des absoluten Raumes und der absoluten Zeit, was zur speziellen Relativitätstheorie und damit zur Vereinheitlichung von klassischer Mechanik und Elektrodynamik führte [@Einstein:1905ve].
Die Revolution der speziellen Relativitätstheorie, welche erstmals auf konkret wissenschaftlichem Wege vor Augen führte, was in der Erkenntnistheorie schon lange vorher behauptet wurde, dass nämlich unsere grundlegenden Denkstrukturen und damit die Art wie wir die Welt wahrnehmen und denken zu unterscheiden sind von ihrer realen Beschaffenheit, hätte bereits ausgereicht, um Einstein eine entscheidende Rolle in der Geistegeschichte der Menschheit zuzuordnen. Dieses Urteil behält seine Gültigkeit auch dann, wenn man einräumt, dass es andere Wissenschaftler wie Poincaré gab, die ähnliche Gedanken hatten. Tatsächlich war Einstein mit der neu erreichten begrifflichen Einheit jedoch noch nicht zufrieden. Er sah grundsätzlich zwei Probleme. Erstens trug die spezielle Relativitätstheorie noch nicht die Gleichberechtigung beliebiger Bezugssysteme sondern nur die aller Inertialsysteme in sich. Zweitens war die Theorie der Gravitation in ihrer alten Form nicht mit der speziellen Relativitätstheorie vereinbar, da sie eine instantane Wirkung zwischen Massen voraussetzte, was natürlich dem Postulat widerspricht, dass die Lichtgeschwindigkeit eine obere Grenze für den Austausch von Information darstellt. Einstein versuchte also, die Gravitation in Analogie zur Elektrodynamik als Feldtheorie zu formulieren. Es war allerdings höchst bemerkenswert, dass im Falle der Gravitation die Größe, von welcher die Wechselwirkung des Körpers mit dem Feld abhängt, nämlich die schwere Masse, äquivalent ist zur trägen Masse, also einer Größe, welche allgemeine mechanische Eigenschaften eines Körpers bestimmt. Dies verhält sich also anders als im Fall der Elektrodynamik, wo der Kopplungsparameter, nämlich die Ladung, völlig unabhängig ist von der trägen Masse eines Körpers. Die Gleichheit des Kopplungsparameters der Gravitation mit der Größe, welche bestimmt, wie sehr ein Körper auf Kräfte reagiert, brachte Einstein schließlich auf die geniale Idee, dass bei der Gravitation die Raumzeit selbst das Wechselwirkungsfeld sein könnte. Diese erhielte dann selbst eine innere Dynamik und träte mit den anderen Gegenständen physikalischer Beschreibung in Wechselwirkung. Durch die Riemannsche Geometrie wurde die mathematische Sprache geliefert, um eine solche dynamische Raumzeit zu beschreiben. Der Gedanke, dass der physikalische Raum eine von der euklidischen Geometrie abweichende Struktur haben könnte, findet sich bereits bei Gauss, welcher Experimente anstellte, um genau dies herauszufinden, aber keine Abweichung von der euklidischen Geometrie feststellen konnte. Freilich geht die Einsteinsche Beschreibungsweise der Raumzeit noch viel weiter [@Einstein:1916vd]. Die Allgemeine Relativitätstheorie stellt den Abschluss der klassischen Physik dar und ist wahrscheinlich der Gipfel begrifflich-konzeptionellen Denkens in der Geschichte der Naturwissenschaft überhaupt. In der folgenden Darstellung soll zunächst auf einige für die mathematische Formulierung der Allgemeinen Relativitätstheorie bedeutsame differentialgeometrische Aspekte eingegangen werden, ehe mit ihrer Hilfe eine kurze Beschreibung der Theorie selbst folgt. Natürlich kann im Rahmen dieser Arbeit nur auf die basalen Grundprinzipien eingegangen werden. Ausführlichere Darstellungen können beispielsweise in [@WeinbergGC],[@Wheeler] und [@DeFeliceClarke] gefunden werden.
Mathematischer Formalismus
--------------------------
Für die folgende Betrachtung soll zunächst einmal in Erinnerung gerufen werden, dass man es bei der mathematischen Beschreibung physikalischer Vorgänge grundsätzlich mit Räumen zu tun hat. Diese müssen nicht mit dem physikalischen Anschauungsraum zu tun haben. Es kann sich auch um rein abstrakte mathematische Räume handeln, die bestimmte Zusammenhänge in der Natur widerspiegeln sollen, wie etwa die Zustandsräume in der Quantenmechanik, welche eine Hilbertraumstruktur aufweisen. Hierbei ist es wichtig, dass die Struktur solcher Räume nicht a priori gegeben ist, sondern dass man diese Struktur durch Einführung bestimmter Konzepte zunächst definieren muss.
### Differenzierbare Mannigfaltigkeiten
Am Anfang sei der Begriff der differenzierbaren Mannigfaltigkeit eingeführt, da er das mathematische Grundgerüst darstellt, auf dem alle folgenden Konstruktionen der Allgemeinen Relativitätstheorie aufbauen.
Es sei zunächst eine abstrakte Punktmenge M gegeben, welche die Struktur eines topologischen Raumes aufweise, was bedeutet, dass ein System offener Mengen definiert ist, welches den Axiomen einer Topologie genügt. Als Karte bezeichnet man eine bijektive Abbildung $\phi$ einer offenen Teilmenge U aus M in eine offene Menge des $\mathcal{R}^d$
$$\phi : U \rightarrow \mathcal{R}^d.$$
Durch eine solche Karte werden die einzelnen Punkte p aus U mit Koordinaten $x^\alpha$ bezeichnet, die den Punkten im $\mathcal{R}^d$ entsprechen. Die Karten fungieren also als Koordinatensysteme für die Mengen, auf denen sie definiert sind. Wenn man nun zwei verschiedene Karten $\phi$ und $\phi^{'}$ betrachtet, die auf zwei verschiedenen Mengen $U$ und $U^{'}$ definiert sind, die einander schneiden, so ist die Schnittmenge $U \bigcap U^{'}$ mit zwei Koordinatensystemen $x^\alpha$ und $x^{\alpha '}$ überdeckt. Diese müssen also zwangsläufig direkt miteinander verknüpft sein, was bedeutet, dass man die einen Koordinaten als Funktionen der anderen darstellen kann
$$x^{\alpha '}=y^{\alpha '}(x^\beta) \quad,\quad
x^{\alpha}=y^{\alpha}(x^{\beta '}).$$
Als Atlas bezeichnet man eine Menge von Karten, welche den gesamten Raum M überdecken. Eine differenzierbare $C^k$-Mannigfaltigkeit ist nun eine Menge, auf der ein solcher Atlas definiert ist, wobei für die Karten, die der Atlas enthält, gelten muss, dass die Funktionen, welche auf der Schnittmenge zweier Karten die Koordinaten der einen Karte in die der anderen abbilden, k mal stetig differenzierbar sein müssen. Die Dimension der Mannigfaltigkeit ist durch die Anzahl der Koordinaten bestimmt, welche notwendig ist, um einen Punkt zu Kennzeichnen, also die Dimension d des $\mathcal{R}^d$, in welchen die Karten die Punkte der Mannigfaltigkeit abbilden.
### Tangentialräume und Tensorfelder
Um den Begriff des Tangentialvektors bzw. des Tangentialraumes einzuführen, kann man zunächst davon ausgehen, dass man es mit einer differenzierbaren d-dimensionalen Mannigfaltigkeit zu tun hat, auf der eine Funktion f definiert sei. In einer Umgebung eines Punktes p ist nun eine Karte definiert, welche es gestattet, den entsprechenden Punkt durch die d Koordinaten $(x^1(p),...,x^d(p))$ zu kennzeichnen. Desweiteren bezeichne k eine Kurve auf der Mannigfaltigkeit, welche durch s parametrisiert sei. In der Umgebung des Punktes p können die Koordinaten der Punkte, welche die Kurve durchläuft, also in Abhängigkeit von s ausgedrückt werden. Für die Werte der Funktion f entlang der Kurve gilt also
$$f(s)=f(x^1(s),...,x^d(s)).$$
Es gelte $k(s_p)=p$. Man kann nun die Änderung der Funktion entlang der Kurve k im Punkt p betrachten, indem man sie dort nach dem Parameter s ableitet
$$\frac{df}{ds}\mid_{s=s_p}=\frac{\partial f(x^1(s),...,x^d(s))}{\partial x^1}\frac{\partial x^1(s)}{\partial s}+...
+\frac{\partial f(x^1(s),...x^d(s))}{\partial x^d}\frac{\partial x^d(s)}{\partial s}\mid_{s=s_p}.$$
Die Abbildung, welche der Funktion ihre Ableitung entlang einer Kurve zuordnet, bezeichnet man als Derivation. Man kann sie gewissermaßen durch den Operator
$$\frac{d}{ds}=\frac{\partial}{\partial x^1}\frac{\partial x^1(s)}{\partial s}+...+\frac{\partial}{\partial x^d}\frac{\partial x^d(s)}{\partial s}$$
darstellen. Da man solche Derivationen nun linear kombinieren kann und das Ergebnis wieder eine Derivation darstellt, bildet die Menge der Derivationen in einem Punkt p einen Vektorraum, den Tangentialvektorraum $T_p$ an p, wobei jede einzelne Derivation einem Tangentialvektor repräsentiert, dessen Komponenten den Ableitungen der Koordinaten nach dem Parameter entsprechen $(\frac{\partial x^1}{\partial s},...,\frac{\partial x^d}{\partial s})$. Die Menge der Linearformen $\lambda$ auf einem Vektorraum V, also Abbildungen der Form
$$\lambda: V \rightarrow \mathcal{R}\ \ \ \ \ mit\ \ \lambda(av_1+bv_2)=a\lambda(v_1)+b\lambda(v_2)\ \ v_1,v_2\ aus\ V,$$
bilden den Dualraum. Im Falle eines Tangentialvektorraumes bezeichnet man den Dualraum als Kotangentialraum. Die Menge aller Tangentialvektorräume bzw. Kotangentialvektorräume bezeichnet man als das Tangentialbündel bzw. Kotangentialbündel. Ein Tensor W ist eine Multilinearform, also eine Abbildung der folgenden Form
$$W:\underbrace{V^* \times ... \times V^*}_{n-mal} \times \underbrace{V \times ... \times V}_{m-mal} \rightarrow \mathcal{R},$$
wobei $V^*$ den Dualraum eines Vektorraums V von Verschiebungsvektoren, also im Falle einer Mannigfaltigkeit den Kotangentialvektorraum zum Vektorraum V darstellt. Ein solcher Tensor wird als n-fach kontravariant und m-fach kovariant bezeichnet. Die Begriffe ko- und kontravariant rühren daher, dass sich die Komponenten bei einer Basistransformation gemäß oder umgekehrt zu den Basisvektoren transformieren. Im folgenden wird ein Tensor durch die Indizes beschrieben werden, welche die einzelnen Komponenten bezüglich einer Basis kennzeichnen, wobei n Indizes oben und m Indizes unten stehen. Eine Abbildung, welche jedem Punkt einer Mannigfaltigkeit einen Tensor zuordnet, bezeichnet man als Tensorfeld.
### Affine Zusammenhänge
Als Parallelverschiebung von Vektoren wird eine Verschiebung bezeichnet, welche einen Vektor konstant lässt. In einem affinen Raum können Vektoren einfach von einem Punkt zu einem anderen verschoben werden, ohne dass eine besondere Vorschrift angegeben werden müsste, wie zwei Vektoren an unterschiedlichen Punkten miteinander zu vergleichen sind. Daher ist dort die Verschiebung von Vektoren auch grundsätzlich wegunabhängig. Dies ist jedoch anders im Falle gekrümmter Räume. Hier hat man es an jedem Raumpunkt mit einem lokalen Tangentialvektorraum zu tun und es muss eine Struktur definiert werden, die bestimmt, wie Vektoren an unterschiedlichen Raumpunkten miteinander verglichen werden müssen. Hierzu führt man eine kovariante Ableitung ein. Diese ordnet einem kontravarianten Vektor einen einfach ko- und kontravarianten Tensor zu und ist durch ihre Wirkung auf einen Satz von Basisvektoren $e_\mu$ wie folgt bestimmt
$$\nabla_\mu e_\nu =\Gamma^\rho_{\mu\nu} e_\rho.$$
Hierbei bezeichnet man die $\Gamma^\rho_{\mu\nu}$ als Zusammenhangkoeffizienten. Mit diesen kann man nun die Wirkung der kovarianten Ableitung auf einen kontravarianten Vektor mit den Komponenten $X^\nu$ bezüglich der Basis $e_\nu$ wie folgt ausdrücken
$$\nabla_\mu X^\nu=\partial_\mu X^\nu+\Gamma^\nu_{\mu\rho} X^\rho.
\label{kovAbleitung1}$$
Für kovariante Vektoren mit den Komponenten $X_\nu$ ergibt sich
$$\nabla_\mu X_\nu=\partial_\mu X_\nu-\Gamma^\rho_{\mu\nu} X^\nu.
\label{kovAbleitung2}$$
Der Riemannsche Krümmungsstensor ist durch den Kommutator zweier kovarianter Ableitungen angewandt auf einen Verschiebungsvektor wie folgt definiert
$$(\nabla_\mu \nabla_\nu-\nabla_\nu \nabla_\mu)X^\sigma=R_{\mu\nu\rho}^{\ \ \ \sigma} X^\rho.
\label{Definition_Riemann-Tensor}$$
Da die Nichtkommutativität der Parallelverschiebung von Vektoren die Krümmung eines Raumes beschreibt, ist der Riemanntensor ein Maß für jene Krümmung. Man kann natürlich unmittelbar erkennen, dass er eine Antisymmetrie bezüglich $\mu$ und $\nu$ aufweist
$$R_{\mu\nu\rho}^{\ \ \ \ \sigma}=-R_{\nu\mu\rho}^{\ \ \ \ \sigma}.
\label{Riemann_Anti-Symmetrie}$$
Desweiteren erfüllt er die zyklische Identität
$$R_{[\mu\nu\rho]}^{\ \ \ \ \sigma}=0,
\label{Riemann_zyklische_Identitaet}$$
und die sogenannte Bianchiidentität
$$\nabla_{[\epsilon} R_{\mu\nu]\rho}^{\ \ \ \ \sigma}=0.
\label{Bianchi-Identitaet}$$
Diese spielt in Bezug auf die Einsteinschen Feldgleichungen eine entscheidende Rolle. Wenn man ($\ref{kovAbleitung1}$) und ($\ref{kovAbleitung2}$) in ($\ref{Definition_Riemann-Tensor}$) verwendet, kann man den Riemanntensor in Abhängigkeit der Zusammenhangkoeffizienten ausdrücken
$$R_{\mu\nu\rho}^{\ \ \ \ \sigma}=\partial_\mu \Gamma_{\nu\rho}^\sigma-\partial_\nu \Gamma^\sigma_{\mu\rho}
+\Gamma^\sigma_{\mu\epsilon} \Gamma^\epsilon_{\nu\rho} -\Gamma^\sigma_{\nu\epsilon} \Gamma^\epsilon_{\mu\rho}.
\label{RiemannZusammenhangkoeffizienten}$$
Wichtig ist schließlich noch die Definition der Torsion. Die Torsion entspricht der Nichtkommutativität der kovarianten Ableitungen bezüglich ihrer Anwendung auf skalarwertige Funktionen f. In der Allgemeinen Relativitätstheorie werden ausschließlich torsionsfreie Zusammenhänge betrachtet, für die demnach gilt
$$\nabla_\mu \nabla_\nu f=\nabla_\nu \nabla_\mu f.
\label{Torsionsfreiheit}$$
Die Zusammenhangkoeffizienten $\Gamma^\rho_{\mu\nu}$ eines torsionsfreien Zusammenhangs sind symmetrisch bezüglich der unteren beiden Indizes. Durch Verwendung von ($\ref{kovAbleitung1}$) und ($\ref{kovAbleitung2}$) in ($\ref{Torsionsfreiheit}$) ergibt sich nämlich
$$\partial_\mu \partial_\nu f+\Gamma^\rho_{\mu\nu} \partial_\rho f=\partial_\nu \partial_\mu f+\Gamma^\rho_{\nu\mu}\partial_\rho f.$$
Die gewöhnlichen Ableitungen sind vertauschbar, womit man direkt
$$\Gamma^\rho_{\mu\nu}=\Gamma^\rho_{\nu\mu}
\label{Koeffizientensymmetrie}$$
ablesen kann.
### Riemannsche Mannigfaltigkeiten
Bisher wurden nur Mannigfaltigkeiten betrachtet, die mit einem affinen Zusammenhang ausgestattet waren. Auf solchen Mannigfaltigkeiten ist allerdings noch nicht zwangsläufig eine metrische Struktur definiert. Um Begriffen wie dem Abstand zweier Punkte oder der Orthogonalität zweier Tangentialvektoren einen Sinn zu verleihen, bedarf es der Einführung einer Metrik. Eine Metrik g in einem Vektorraum ist ein symmetrischer zweifach kovarianter Tensor. Er ordnet also einem Paar zweier kontravarianter Vektoren eine reelle Zahl zu. Zwei Vektoren mit Komponenten $X^\mu$ und $X^\nu$ werden orthogonal genannt, wenn gilt
$$g_{\mu\nu}X^\mu X^\nu=0.$$
Ein metrisches Feld ist damit ein symmetrisches zweifach kovariantes Tensorfeld, das in jedem Punkt der Mannigfaltigkeit den entsprechenden Tangentialvektorraum mit einem inneren Produkt ausstattet. Mit der Auszeichnung eines solchen metrischen Feldes ist also die geometrische Struktur einer Mannigfaltigkeit vollkommen bestimmt. Die Metrik ordnet damit jedem Vektor $X^\mu$ in direkter Weise einen dualen Vektor $X_\nu$ zu
$$g_{\mu\nu}X^\mu X^\nu=X_\nu X^\nu.$$
Man bezeichnet eine solche Mannigfaltigkeit auch als Riemannsche Mannigfaltigkeit.
Es ist nun möglich, den affinen Zusammenhang, welcher verschiedene Tangentialvektorräume miteinander verbindet, mit Hilfe der Metrik zu definieren. Im Rahmen der Allgemeinen Relativitätstheorie ist insbesondere der torsionsfreie Zusammenhang von Bedeutung, welcher durch die Bedingung definiert ist, dass innere Produkte zwischen Vektoren bei Verschiebung konstant gehalten werden sollen, was einem Verschwinden der kovarianten Ableitung bei Anwendung auf den metrischen Tensor entspricht. Dieser Zusammenhang ist eindeutig bestimmt und heißt Levy-Civita-Zusammenhang. Wenn man nun die Zusammenhangkoeffizienten des Levy-Civita-Zusammenhangs bestimmen möchte, muss die Bedingung
$$\nabla_\mu g_{\rho\sigma}=0$$
entsprechend umformuliert werden. Durch Anwenden der kovarianten Ableitung auf jeden Index ergibt sich
$$\partial_\mu g_{\rho\sigma}-\Gamma_{\mu\rho}^{\nu} g_{\nu\sigma}-\Gamma_{\mu\sigma}^{\nu} g_{\rho\nu}=0.$$
Zyklisches Vertauschen der Indizes führt auf zwei weitere Gleichungen
$$\begin{aligned}
\partial_\mu g_{\rho\sigma}-\Gamma_{\mu\rho}^{\nu} g_{\nu\sigma}-\Gamma_{\mu\sigma}^{\nu} g_{\rho\nu}&=&0\nonumber\\
\partial_\rho g_{\sigma\mu}-\Gamma_{\rho\sigma}^{\nu} g_{\nu\mu}-\Gamma_{\rho\mu}^{\nu} g_{\sigma\nu}&=&0\nonumber\\
\partial_\sigma g_{\mu\rho}-\Gamma_{\sigma\mu}^{\nu} g_{\nu\rho}-\Gamma_{\sigma\rho}^{\nu} g_{\mu\nu}&=&0.\end{aligned}$$
Wenn man nun die zweite Gleichung zur ersten addiert und die zweite subtrahiert, so ergibt sich unter Verwendung der Symmetrieeigenschaft des metrischen Tensors sowie ($\ref{Koeffizientensymmetrie}$) folgende Gleichung
$$-2\Gamma_{\mu\rho}^{\nu} g_{\nu\sigma}+\partial_\mu g_{\rho\sigma}+\partial_\rho g_{\sigma\mu}-\partial_\sigma g_{\mu\rho}=0.$$
Durch umstellen erhält man
$$\Gamma_{\mu\rho}^{\nu}=\frac{1}{2} g^{\nu\sigma}(\partial_\mu g_{\rho\sigma}+\partial_\rho g_{\sigma\mu}-\partial_\sigma g_{\mu\rho}).
\label{Christoffelsymbole}$$
In diesem speziellen Falle bezeichnet man die Zusammenhangkoeffizienten als Christoffelsymbole. Wenn man den Kommutator der kovarianten Ableitung des Levy-Civita-Zusammenhangs auf die Metrik anwendet, so ergibt sich
$$(\nabla_\mu \nabla_\nu-\nabla_\nu \nabla_\mu)g^{\rho\sigma }=
R_{\mu\nu\lambda}^{\ \ \ \ \rho} g^{\lambda\sigma}+R_{\mu\nu\lambda}^{\ \ \ \ \sigma} g^{\rho\lambda}=R_{\mu\nu}^{\ \ \ \sigma\rho}+R_{\mu\nu}^{\ \ \ \rho\sigma}=0.$$
Das bedeutet, dass der Riemanntensor in diesem Fall antisymmetrisch bezüglich der hinteren beiden Indizes ist
$$R_{\mu\nu}^{\ \ \ \sigma\rho}=-R_{\mu\nu}^{\ \ \ \rho\sigma}.
\label{Riemann_LCAntiSymmetrie}$$
Physikalische Prinzipien
------------------------
Das gesamte theoretische Gebäude der Allgemeinen Relätivitätstheorie geht aus dem Äquivalenzprinzip hervor, dass ausgehend von der bereits in der Einleitung erwähnten Gleichheit von schwerer und träger Masse, die Unmöglichkeit der Unterscheidung eines beschleunigten Bezugssystems von der Wirkung eines Gravitationsfeldes beinhaltet. Dies bedeutet, dass man die Wirkung eines Gravitationsfeldes als Eigenschaft der Raumzeit selbst deuten kann, die eine in sich gekrümmte Struktur aufweist, welche sich im Rahmen der Riemannschen Geometrie dann in einer von der euklidischen abweichenden Metrik äußert. Der oben entwickelte Formalismus wird also gewissermaßen auf die reale Raumzeit übertragen.
### Die Postulate der Allgemeinen Relativitätstheorie
Die Allgemeine Relativitätstheorie basiert auf zwei Grundpostulaten [@Einstein:1916vd].\
1) Die Raumzeit wird durch eine vierdimensionale Riemannsche Mannigfaltigkeit beschrieben, auf der demnach eine Metrik ausgezeichnet ist. Es handelt sich um eine Lorentzmetrik, die grundsätzlich durch eine geeignete Koordinatentransformation lokal in die Gestalt $\eta_{\mu\nu}=diag(1,-1,-1,-1)$ gebracht werden kann. Die genaue Geometrie der Raumzeit wird durch die Materieverteilung bestimmt, wobei dieser Zusammenhang in mathematisch exakter Weise durch die Einsteinschen Feldgleichungen ausgedrückt wird
$$G_{\mu\nu}=-8\pi G T_{\mu\nu}.$$
2\) Die Bewegung von kräftefreien Körpern erfolgt auf der kürzesten Linie durch die Raumzeit. Eine solche wird im verallgemeinerten Fall gekrümmter Räume als Geodäte bezeichnet. Weltlinien von Körpern gehorchen also der Geodätengleichung
$$\ddot X^\mu+\Gamma^\mu_{\rho\sigma}\dot X^\rho \dot X^\sigma=0,$$
wobei die $\Gamma^\mu_{\rho\sigma}$ die Christoffelsymbole bezeichnen.
### Die Einsteinschen Feldgleichungen
Bei der Suche nach den Feldgleichungen, welche die Raumzeitstruktur bestimmen, wurde Einstein durch das Analogon aus der Elektrodynamik geleitet. Hier koppelt das Elektromagnetische Feld an den elektrischen Viererstrom $J^\nu$, welcher die Ladungs- und Ladungsstromdichte enthält. Dieser Zusammenhang wird durch die inhomogenen Maxwellschen Gleichungen ausgedrückt, die in kovarianter Formulierung in folgender Gleichung enthalten sind
$$\partial_\mu F^{\mu\nu}=-J^\nu.$$
Das Gravitationsfeld, also die Struktur der Raumzeit, kann aufgrund des klassischen Grenzfalles, der sich im Newtonschon Gravitationsgesetz ausdrückt, nur durch die Materieverteilung definiert sein, genauer gesagt die Energieverteilung. Diese aber ist im Rahmen der Relativitätstheorie mit dem Impuls zum Viererimpuls verbunden. Es ist nun der Energie-Impuls-Tensor, bezeichnet als $T_{\mu\nu}$, welcher analog der Viererstromdichte, welche die Ladungsverteilung bestimmt, die Energie- und Impulsverteilung der Materie beschreibt. Da die Energie-Impuls-Verteilung die Struktur der Raumzeit festlegen soll, muss es einen direkten Zusammenhang zu den Krümmungsgrößen der Raumzeit geben. Hierbei ist nun folgendes wichtig. Aufgrund des Energie- und Impulserhaltungssatzes muss der Energie-Impuls-Tensor divergenzfrei sein. Dies bedeutet nichts anderes als
$$\nabla^\mu T_{\mu\nu}=0.$$
Daneben gibt es nur einen divergenzfreien Tensor zweiter Stufe, welcher sich aus den Krümmungsgrößen konstruieren lässt. Dieser kann unter Verwendung der Bianchiidentität ($\ref{Bianchi-Identitaet}$) gefunden werden. Aus dieser ergibt sich mit ($\ref{Riemann_Anti-Symmetrie}$)
$$\nabla_\epsilon R_{\mu\nu\rho}^{\ \ \ \sigma}+\nabla_\mu R_{\nu\epsilon\rho}^{\ \ \ \sigma}+\nabla_\nu R_{\epsilon\mu\rho}^{\ \ \ \sigma}=0.$$
Durch erneute Anwendung von ($\ref{Riemann_Anti-Symmetrie}$) im zweiten Ausdruck und Kontraktion der Indizes $\nu$ und $\sigma$ erhält man
$$\nabla_\epsilon R_{\mu\rho}-\nabla_\mu R_{\epsilon\rho}+\nabla_\nu R_{\epsilon\mu\rho}^{\ \ \ \nu}=0.$$
Wenn man nun im letzten Term ein weiteres Mal ($\ref{Riemann_Anti-Symmetrie}$) sowie ($\ref{Riemann_LCAntiSymmetrie}$) benutzt und anschließend $\rho$ mit $\epsilon$ kontrahiert, so ergibt sich
$$\nabla^\rho R_{\mu\rho}-\nabla_\mu R+\nabla^\nu R_{\mu\nu}.$$
Weiteres Umformen führt schließlich auf
$$\nabla^\nu(R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu})=0.
\label{Divergenzfreiheit}$$
Der Tensor
$$G_{\mu\nu}=R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}$$
ist also divergenzfrei und wird als Einsteintensor bezeichnet. Der darin auftauchende Tensor $R_{\mu\nu}=R_{\mu\lambda\nu}^{\ \ \ \lambda}$ heißt Riccitensor und die Größe $R=g^{\mu\nu}R_{\mu\nu}$ wird als Ricciskalar bezeichnet. Aufgrund der Eigenschaft ($\ref{Divergenzfreiheit}$) vermutete Einstein nun den Zusammenhang
$$G_{\mu\nu}=\kappa T_{\mu\nu}.$$
Dies ist die Einsteinsche Feldgleichung, wobei der Proportionalitätsfaktor $\kappa$ durch Analogieschlüsse zum klassischen Grenzfall, den die Einsteinschen Feldgleichungen natürlich imlizit enthalten müssen, bestimmt werden kann. Der Proportionalitätsfaktor hat jedoch keine prinzipielle Bedeutung und deshalb soll er hier einfach nur angegeben werden, womit man schließlich
$$G_{\mu\nu}=-8\pi G T_{\mu\nu}$$
erhält.
### Bewegung von Körpern auf der Raumzeit
Die kürzeste Linie durch die Raumzeit zwischen zwei Punkten ist dadurch definiert, dass sich ein Weltlinientangentialvektor entlang der gesamten Kurve nicht ändert, dass er also Parallelverschoben wird. Dies bedeutet, dass die kovariante Ableitung in Richtung der Kurve also des Tangentialvektors selbst angewandt auf diesen Vektor an jeder Stelle der Kurve gleich null ist. Ein Tangentialvektor ist durch die Ableitung der Koordinaten $X^\mu$, die zu einer Karte an einem bestimmten Punkt gehören, nach dem Kurvenparameter s charakterisiert, welche hier wie folgt notiert werden soll $\dot X^\mu=\frac{\partial X^\mu}{\partial s}$. Da die Ableitung in Richtung des Tangentialvektors von Interesse ist, werden die Ableitungen in Richtung der die Karte an einer bestimmten Stelle charakterisierenden Koordinaten mit den Komponenten des Tangentialvektors in eben diesen Koordinaten gewichtet. Dies bedeutet, dass sich die Forderung des Verschwindens der kovarianten Ableitung des Tangentialvektors entlang der Kurve
$$\nabla_s \dot X^\nu=0$$
wie folgt ausdrücken lässt
$$\dot X^\nu \nabla_\nu \dot X^\mu=\dot X^\nu \partial_\nu \dot X^\mu+\dot X^\nu \Gamma^\mu_{\nu\rho} \dot X^\rho=\ddot X^\mu+\Gamma^\mu_{\nu\rho}\dot X^\nu \dot X^\rho=0.$$
Dies ist aber die Geodätengleichung, welche festlegt, wie sich der Tangentialvektor der Weltlinie eines Körpers und damit die Bewegung des Körpers bei vorgegebener Raumzeitstruktur verhält. Es handelt sich hierbei um eine Verallgemeinerung des Galileischen Trägheitsprinzips für Räume mit beliebiger Geometrie.
Die Einführung zusätzlicher Dimensionen
=======================================
Die Allgemeine Relativitätstheorie in der beschriebenen Form hatte zwar die Gravitation mit der speziellen Relativitätstheorie in einer einheitlichen Theorie formuliert. Dennoch stellte auch sie, selbst bei Beschränkung auf den Rahmen der klassischen Physik, noch keine völlig einheitliche Naturbeschreibung dar, denn die Elektromagnetischen Felder blieben zur geometrischen Beschreibungsweise des Gravitationsfeldes wesensfremd. Darüber hinaus enthält die Einsteinsche Feldgleichung auf der rechten Seite den Energie-Impuls-Tensor, dessen spezielle Gestalt aus einer anderen Theorie übernommen werden muss, während die das Gravitationsfeld beschreibenden Größen auf der linken Seite rein geometrischer Natur sind. Einstein strebte nun eine Beschreibungsweise an, bei der auch die Materie letztlich auf Geometrie zurückgeführt wird.
Kaluza-Klein-Theorie
--------------------
Eine Erweiterung der Allgemeinen Relativitätstheorie in dieser Richtung entwickelte der Mathematiker Theodor Kaluza [@Kaluza:1921tu]. Seine Theorie beinhaltete unter Einbeziehung einer fünften Dimension auch den Elektromagnetismus, der damit also auch einer geometrischen Beschreibung zugänglich gemacht wurde. Die Tatsache, dass die fünfte räumliche Dimension nicht direkt in Erscheinung tritt, kann durch den Ansatz Oskar Kleins auf elegante Art und Weise erklärt werden, welcher die Kaluzasche Theorie dahingehend modifizierte, dass er die zusätzliche Dimension kompaktifizierte [@Klein:1926tv].
### Die Struktur der Raumzeit nach Kaluza und Klein
Es wird also davon ausgegangen, dass der Raum neben den vier Dimensionen der gewöhnlichen Raumzeit, die mit griechischen Indizes bezeichnet seien, welche von 0 bis 3 laufen, noch eine weitere Dimension $x^4$ enthält, die in der Weise kompaktifiziert ist, dass sie periodisch unter folgender Transformation ist
$$x^4 \rightarrow x^4+2\pi R,$$
was also der Kompaktifizierung zu einem Kreis entspricht. Die Metrik der vollständigen Raumzeit $g_{MN}$ unter Einbeziehung der kompaktifizierten Dimension besteht also aus den Komponenten, die sich auf die gewöhnliche Raumzeit beziehen, den Komponenten, deren einer Index sich auf die fünfte Raumzeitdimension bezieht und der Komponente, deren beide Indizes sich auf die kompaktifizierte Komponente beziehen. Es wird davon ausgegangen, dass die Metrik nur von den nichtkompaktifizierten Koordinaten abhängt. Der folgende Ansatz für den metrischen Tensor
$$g_{MN}=\left(\begin{array}{cc}g_{\mu\nu}+g_{44}A_\mu A_\nu & 2g_{44}A_\mu\\ 2g_{44}A_\mu & g_{44}\end{array}\right)$$
und damit für ein infinitesimales Linienelement
$$ds^2=g_{MN} dx^M dx^N=g_{\mu\nu}dx^\mu dx^\nu+g_{44}(dx^4+A_\mu dx^\mu)^2$$
ist invariant unter Transformationen der Form
$$x^d \rightarrow x^d+\lambda(x^\mu)\quad,\quad A_\mu \rightarrow A_\mu - \partial_\mu \lambda.$$
Dies entspricht also der Form nach einer Eichtransformation des Elektromagnetischen Feldes. Damit lässt sich der Vierervektor $A_\mu$ mit dem Elektromagnetischen Potential identifizieren. Tatsächlich ergeben sich aus diesem Ansatz die Maxwellschen Gleichungen. Die Herleitung, aus der dies hervorgeht, sowie eine Darstellung des Kaluza-Klein-Formalismus im Allgemeinen findet sich in [@Polchinsky].
### Generierung von Massen
Es soll nun ein skalares Feld auf einer solchen fünfdimensionalen Raumzeit aufgespalten werden in einen Anteil, welcher die Abhängigkeit von der vierdimensionalen Untermannigfaltigkeit der normalen Raumzeit ausdrückt, und eine Fourierentwicklung nach der kompaktifizierten Koordinate, deren Anregungen natürlich aufgrund der Periodizität der Koordinate gequantelt sind
$$\Phi_n(x^M)=\sum_{n=-\infty}^\infty \Phi (x^\mu) exp\left(\frac{inx^4}{R}\right).
\label{SkalarfeldKK}$$
Bei einem masselosen Skalarfeld, dass der Gleichung $\partial_M \partial^M \phi_n(x^M)=0$ genügt, ergibt sich durch Aufspaltung des Operators $\partial_M \partial^M$ in den d’Alembertoperator auf der vierdimensionalen Raumzeit und die zweite Ableitung nach der kompaktifizierten Koordinate
$$\partial_M \partial^M=\partial_\mu \partial^\mu+\partial_4 \partial^4$$
und anschließender Anwendung von $\partial_4 \partial^4$ die folgende Gleichung
$$\partial_\mu \partial^\mu \Phi_n (x^\mu)=\frac{n^2}{R^2} \Phi_n (x^\mu),$$
also der Form nach eine Klein-Gordon-Gleichung mit einem Massenterm $\frac{n^2}{R^2}$. Das Feld $\Phi_n$ erhält also eine Masse, die dem Quadrat der Schwingungszahl der Anregung in der kompaktifizierten Dimension proportional ist. Auf die gleiche Weise erhalten im nächsten Kapitel Gravitonen eine Masse.
Das ADD-Modell
--------------
### Die Grundidee
In einer anderen Weise werden im Rahmen des in jüngerer Zeit entwickelten Randall-Sundrum-Modells [@Randall:1999ee],[@Randall:1999vf] und dem dazu verwandten ADD-Modell [@Antoniadis:1997zg],[@Arkani-Hamed:1998rs],[@Antoniadis:1998ig] zusätzliche Dimensionen eingeführt. Hier geht man davon aus, dass die üblichen Materiefelder auf einer 3+1-dimensionalen Untermannigfaltigkeit eines höherdimensionalen Raumes leben, der dem üblichen Raum entspricht und im Falle des Randall-Sundrum-Modells eine zusätzliche und im Rahmen des ADD-Modells eine zunächst nicht festgelegte Anzahl an zusätzlichen Dimensionen enthält, die wie in der Theorie von Kaluza und Klein auf eine bestimmte Art und Weise kompaktifiziert sind. Im Gegensatz zu allen anderen Feldern ist das Gravitationsfeld jedoch nicht auf die Untermannigfaltigkeit des üblichen Raumes beschränkt. Dies könnte eine mögliche Lösung des Hierarchieproblems liefern, welches in der Nichterklärbarkeit der unglaublichen Schwäche der Gravitation im Vergleich zu allen anderen Wechselwirkungen besteht. Aus dem Newtonschen Gravitationsgesetz ergibt sich eine neue Größe, eine D-dimensionale Planckmasse $M_D$, die in folgender Relation zur gewöhnlichen Planckmasse $M_P$ steht
$$M_P^2=c M_D^{\delta+2} R^\delta,\quad c=const,$$
wobei $\delta$ die Zahl der zusätzlichen Dimensionen und R den Kompaktifizierungsradius beschreibt. Bei der Voraussetzung, dass die Gravitation im Falle der Existenz zusätzlicher Dimensionen nicht auf die übliche 3+1-dimensionale Raumzeit beschränkt ist, handelt es sich um eine vollkommen natürliche Annahme. Wenn man die Gravitation nämlich, wie das in der Allgemeinen Relativitätstheorie getan wird, als geometrische Eigenschaft der Raumzeit selbst deutet, so bedeutet die Annahme, dass die Gravitation auch auf die zusätzlichen Dimensionen ausgedehnt ist, nichts anderes, als dass deren Geometrie ebenfalls einer gewissen Dynamik unterworfen ist.
Es wird also von einer D-dimensionalen Raumzeit ausgegangen. Ein beliebiger Punkt einer solchen Raumzeit wird durch ein D-Tupel an Koordinaten $z=(z_1,...,z_D)$ beschrieben. Alle zu den 3+1 Dimensionen der gewöhnlichen Raumzeit zusätzlichen Dimensionen sollen nun torusförmig kompaktifiziert werden. Das D-Tupel an Koordinaten wird also aufgespalten in die Koordinaten, welche die Untermannigfaltigkeit der gewöhnlichen Raumzeit beschreibt, und jene der zusätzlichen Dimensionen
$$z=(x_0,{{\bf x}},y_1,...,y_\delta),\quad\delta=D-4,$$
wobei für die Koordinaten der zusätzlichen Dimensionen die gleiche Periodizität wie im Falle der Theorie von Kaluza und Klein gilt
$$y_j\rightarrow y_j+2\pi R\quad,\quad j=1,...,\delta.$$
Dies entspricht der Kompaktifizierung zu einem $\delta$-dimensionalen Torus, dessen Volumen durch $V_\delta=(2\pi R)^\delta$ gegeben ist. Damit gilt für die Relation zwischen $M_P$ und $M_D$
$$M_P^2=8\pi R^\delta M_D^{2+\delta}.$$
### Metrische Struktur und Materieverteilung
Die metrische Struktur des Raumes kann näherungsweise durch die im letzten Kapitel beschriebenen Einsteinschen Feldgleichungen erweitert auf D Dimensionen beschrieben werden
$$G_{MN}=-\frac{T_{MN}}{\bar M_D^{2+\delta}},$$
wobei $\bar M_D=(2\pi)^{-\frac{\delta}{2+\delta}} M_D$. Damit ist der metrische Tensor $g_{MN}$ mit $M,N=0,...,D$ bestimmt. Interessant ist jedoch nun der Zusammenhang der Metrik des höherdimensionalen Raumes zur Metrik der 3+1-dimensionalen Untermannigfaltigkeit. Für ein infinitesimales Linienelement gilt
$$\begin{aligned}
ds^2&=&G_{MN}(Y(x)) dY^M dY^N\nonumber\\
&=&G_{MN}(Y(x))\frac{\partial Y^M}{\partial x^\mu}dx^\mu \frac{\partial Y^N}{\partial x^\nu}dx^\nu.\end{aligned}$$
Wenn man aber nun berücksichtigt, dass das Linienelement gleichzeitig durch $ds^2=g_{\mu\nu}dx^\mu dx^\nu$ beschrieben werden kann, bedeutet das für die Metrik $g_{\mu\nu}(x)$ der in den höherdimensionalen Raum eingebetteten gewöhnlichen Raumzeit
$$g_{\mu\nu}=G_{MN}(Y(x))\partial_\mu Y^M \partial_\nu Y^N.$$
Wie ist aber nun der die Materieverteilung bestimmende Energie-Impuls-Tensor zu beschreiben ? Wenn davon ausgegangen wird, dass die Materie- und Wechselwirkungsfelder auf der 3+1-dimensionalen Untermannigfaltigkeit leben und man das Gravitationsfeld und damit seine Selbstenergie als so schwach annimmt, dass sie im Vergleich zu der Energie der übrigen Felder vernachlässigt werden kann, so muss auch der Energie-Impuls-Tensor auf die gewöhnliche Raumzeit beschränkt sein. Dies drückt sich mathematisch in einer Deltafunktion bezüglich der zusätzlichen Dimensionen aus. Der Energie-Impuls-Tensor der höherdimensionalen Raumzeit besitzt also folgende Form
$$T_{MN}(z)=\eta^\mu_M \eta^\nu_N T_{\mu\nu} (x) \delta(y).$$
Die Formulierung von Feldtheorien auf einer Raumzeit der beschriebenen Struktur wird in [@Sundrum:1998sj] gegeben.
Eine Quantenfeldtheorie der Gravitation
=======================================
In diesem Kapitel soll es nun um die Formulierung der Gravitation als Quantenfeldtheorie unter Berücksichtigung der zusätzlichen kompaktifizierten Dimensionen gehen. Grundsätzlich besteht natürlich wie im klassischen Falle auch hier die Formulierung aus zwei Teilbereichen. Einerseits muss die Art und Weise festgelegt werden, wie die quantentheoretisch beschriebenen Materiefelder an das Gravitationsfeld koppeln und andererseits muss das Gravitationsfeld selbst quantentheoretisch beschrieben werden. Bezüglich des ersten Teilproblems gibt es eine unumstrittene Beschreibungsmethode, welche wie im Falle der anderen Wechselwirkungen auf ein Eichprinzip zurückgeführt werden kann. Im Gegensatz zu den anderen Wechselwirkungen, bei denen Invarianz unter lokalen Transformationen bezüglich innerer Symmetriegruppen gefordert wird, handelt es sich hierbei um die Forderung lokaler Invarianz unter Lorentztransformationen. Im Hinblick auf eine Vereinheitlichung mit den anderen Wechselwirkungen ist es sehr bemerkenswert, dass eine solche eichtheoretische Formulierung auch im Falle der Gravitation möglich ist. Dies ist ein wichtiges Bindeglied zu den übrigen Wechselwirkungen.
Die quantentheoretische Beschreibungsweise des Gravitationsfeldes selbst ist jedoch das eigentlich entscheidende Problem. Die hier verwendete Theorie geht von einer Entwicklung des metrischen Feldes um die Minkowskimetrik in linearer Näherung aus, welche dann in die den Einsteinschen Feldgleichungen korrespondierende Lagrangedichte eingesetzt wird. Ebenso wie bei der Quantisierung von üblichen Eichtheorien wird die Methode der Pfadintegralquantisierung verwendet. Diese von Feynman eingeführte quantenfeldtheoretische Beschreibungsweise führt direkt von der Lagrangedichte auf die Propagatoren. Es muss natürlich nicht erwähnt werden, dass es sich hierbei nur um eine effektive Theorie handelt, die in keiner Weise den Anspruch hat, die Gravitation auf fundamentale Weise quantentheoretisch zu beschreiben. Dies ist wie bereits in der Einleitung erwähnt schon aufgrund der fehlenden Hintergrundunabhängigkeit ausgeschlossen, welche durch die Entwicklung um die flache Minkowskimetrik von vorneherein nicht gegeben ist.
Eichtheorie der Gravitation
---------------------------
Gemäß den anderen fundamentalen Wechselwirkungen kann wie bereits erwähnt auch die Gravitation als Eichtheorie beschrieben werden. Man geht also ebenfalls zunächst von einer freien Materiefeldgleichung aus, deren Kopplung an das Gravitationsfeld dann durch eine Symmetrieforderung bestimmt wird. Im Gegensatz zu der bei anderen Wechselwirkungen geforderten Invarianz unter lokalen Transformationen bezüglich innerer Symmetrien fordert man im Falle der Gravitation Invarianz unter lokalen Lorentztransformationen.
### Lokale Lorentzinvarianz und Vierbein
Um das Verhalten von Fermionen in einem Gravitationsfeld eichtheoretisch zu beschreiben, muss zunächst ein neuer mathematischer Begriff eingeführt werden. Es handelt sich um das Vierbein bzw. im Falle zusätzlicher Dimensionen um das Vielbein. Hierzu muss man sich das (schwache) Äquivalenzprinzip in Erinnerung rufen, demgemäß ein beliebiges Gravitationsfeld durch Wahl geeigneter Koordinaten zum Verschwinden gebracht werden kann. In dem ensprechenden Koordinatensystem gilt also
$$g_{mn}=\eta_{mn}.$$
Das Vierbein transformiert nun beliebige globale Koordinaten $x^\mu$ in diejenigen lokalen Koordinaten $y^m$, in welchen die Krümmung verschwindet. Das bedeutet
$$x^m=e^m_\mu y^\mu \quad,\quad \eta_{mn}=e^\mu_m e^\nu_n \eta_{\mu\nu}.$$
Damit stellt das Vierbeinfeld eine vollständige zum metrischen Feld äquivalente Beschreibungsweise des Gravitationsfeldes dar. Man geht also von der freien Lagrangedichte des Diracfeldes aus
$$\mathcal{L}=\bar \Psi(i\gamma^\mu\partial_\mu-m)\Psi.$$
Nun wird Invarianz unter lokalen Lorentztransformationen
$$\psi\ \rightarrow\ U(x)\Psi\quad,\quad\partial_\mu\ \rightarrow\ \Lambda_\mu^\nu(x)\partial_\nu,$$
gefordert. Da es sich bei $\Psi$ um ein Spinorfeld handelt, wird die Transformation durch die Generatoren der Lorentzgruppe in Diracspinordarstellung $\Sigma_{mn}=\frac{i}{4}[\gamma_m,\gamma_n]$ vermittelt. Durch die Einführung einer kovarianten Ableitung der folgenden Form
$$D_m=e_m^\mu(\partial_\mu+i\omega_\mu^{mn}\Sigma_{mn}),$$
wobei die Zusammenhangkoeffizienten mit dem Vierbein in folgender Beziehung stehen
$$\omega_\mu^{mn}=2e^{\nu [m}\partial_{[\mu}e_{\nu]}^{n]}+e_{\mu p}e^{\nu m}e^{\sigma n}\partial_{[\sigma}e_{\nu]}^p,$$
kann man nun eine unter lokalen Lorentztransformationen invariante Lagrangedichte erhalten
$$\mathcal{L}=\bar \Psi(i\gamma^\mu D_\mu-m)\Psi.$$
Eine ausführlichere Beschreibung der Idee von Eichtheorien im Allgemeinen ist in Kapitel 5 zu finden.
### Wirkungen von Feldern im Gravitationsfeld
Wenn man nun die zur Lagrangedichte gehörige Wirkung formulieren möchte, so muss man beachten, dass für ein infinitesimales Volumenelement $dV$ in einem Raum mit der Metrik $g_{\mu\nu}$ gilt
$$dV=\epsilon_{\mu\nu\rho\sigma} dx^\mu dx^\nu dx^\rho dx^\sigma,$$
was zu folgendem Volumen eines Raumzeitbereiches $\omega$ führt
$$V=\int_\omega \sqrt{-det(g_{\mu\nu})} dx^1 dx^2 dx^3 dx^4=\int_\omega \sqrt{-det(g_{\mu\nu})} d^4 x.$$
Dies bedeutet, dass das Volumenelement in den lokalen Koordinaten, die zu einer flachen Metrik führen, mit den globalen Koordinaten in folgender Beziehung steht
$$d^4 y=\sqrt{-g} d^4 x,$$
wodurch sich die folgende Wirkung ergibt
$$S_{Fermion}=\int d^4 x \sqrt{-g}\{\bar \Psi(\gamma^m e_m^\mu(\partial_\mu+i\omega_\mu^{mn}\Sigma_{mn})-m)\Psi\}.
\label{FermionGravWirkung}$$
Hierbei steht $g$ für $det(g_{\mu\nu})$. Für weitere Aspekte bezüglich der eichtheoretischen Formulierung der Gravitation und des Vierbeinformalismus sei auf [@Ramond],[@Nakahara] und [@Rovelli] verwiesen. Die Formulierung der Wirkung von Bosonen und üblichen Eichfeldern ist bei weitem unproblematischer, da hier der metrische Tensor der Minkowskiraumzeit explizit auftaucht und durch den allgemeinen metrischen Tensor einer gekrümmten Raumzeit ersetzt wird, was zu den folgenden Wirkungen führt. Für ein Boson ergibt sich
$$S_{Boson}=\int d^4 x \sqrt{-g} \{g^{\mu\nu} D_\mu \Phi D_\nu \Phi-V(\Phi)\},
\label{BosonGravWirkung}$$
wobei die kovarianten Ableitungen sich hier auf die üblichen Eichfelder beziehen. Die kovariante Ableitung in Bezug auf das Gravitationsfeld, also die Raumzeit, entspricht bei einem Skalarfeld natürlich der einfachen Ableitung. Die Wirkung der Eichfelder sieht wie folgt aus
$$S_{Eichfeld}=\int d^4 x \{-\frac{g^{\mu\rho}g^{\nu\sigma}}{4}F_{\rho\sigma}F_{\mu\nu}\}.
\label{EichfeldGravWirkung}$$
Die Methode der Pfadintegralquantisierung
-----------------------------------------
Um das Gravitationsfeld selbst im Rahmen einer Quantenfeldtheorie zu beschreiben und einen Propagator herzuleiten, soll zunächst eine kurze Beschreibung der Methode der Pfadintegralquantisierung gegeben werden, wie sie in [@WeinbergQTF1] gefunden werden kann. Feynman wurde bei der Suche nach einer quantenmechanischen Beschreibungsweise, welche vom Prinzip der kleinsten Wirkung ausgeht auf die Möglichkeit geführt, Propagatoren direkt aus der Lagrangedichte herzuleiten [@Feynman:1948ur]. Ein Vorteil dieser Beschreibungsweise ist die Tatsache, dass sie explizit kovariant ist.
### Allgemeine Formulierung
Gegeben sei ein quantenmechanisches System, dass durch einen Satz kommutierender hermitescher Operatoren $Q_a$ vollständig beschrieben sei, wobei die kanonisch konjugierten Operatoren mit $P_a$ bezeichnet seien und durch die Vertauschungsrelationen der Heisenbergalgebra definiert sind
$$[Q_a,P_b]=i\delta_{ab} \quad,\quad [Q_a,Q_b]=[P_a,P_b]=0.$$
$| q \rangle$ sei Eigenzustand zu allen Operatoren $Q_a$ und die zeitliche Entwicklung werde durch den Hamiltonoperator $H$ beschrieben. Die Wahrscheinlichkeit ein Teilchen zur Zeit t’ im Eigenzustand $| q' \rangle$ zu finden, wenn es sich zum Zeitpunkt $t$ im Eigenzustand $| q \rangle$ befunden hat, kann im Rahmen des Feynmanschen Pfadintegralformalismus wie folgt ausgedrückt werden
$$\begin{aligned}
\langle q',t'|q,t \rangle=\lim_{d\tau \to 0} \int \left[ \prod_{k=1}^{N-1} \prod_a dq_{k,a}\right] \left[\prod_b \prod_{k=0}^{N-1} \frac{dp_{k,b}}{2\pi}\right]\nonumber\\
exp\left[i\sum_{k=1}^{N}\left(\sum_a(q_{k,a}-q_{k-1,a})p_{k-1,a}-H(q_k,p_{k-1})d \tau\right)\right],\end{aligned}$$
wobei das Zeitintervall $t'-t$ in N Zeitintervalle $d\tau=\frac{t'-t}{N}$ unterteilt sei. Im Grenzfall $d\tau\rightarrow 0$ ergibt sich damit
$$\begin{aligned}
\langle q',t'|q,t \rangle=\int_{q_a(t)}^{q_a'} \prod_{\tau,a} dq_a(\tau) \prod_{\tau,b} \frac{dp_b(\tau)}{2\pi}\nonumber\\
exp\left[i\int_t^{t'} d\tau \left(\sum_a(\dot q_a (\tau) p_a (\tau)-H(q(\tau),p(\tau))d \tau\right)\right]. \end{aligned}$$
Dieser Zusammenhang kann so interpretiert werden, dass ein quantenmechanisches System in gewissem Sinne alle möglichen dynamischen Entwicklungen gleichzeitig durchläuft. Die Wahrscheinlichkeit es zur Zeit t’ in einem Zustand $| q' \rangle$ zu finden, wenn es sich zur Zeit t im Zustand $|q \rangle$ befunden hat entspricht der Überlagerung aller dynamischen Entwicklungen die zwischen diesen beiden Zeitpunkten formal möglich sind. Im Exponenten steht ein Ausdruck, der in den physikalisch relevanten Fällen der Wirkung des Systems entspricht. Nun werden sich die Amplituden von möglichen Entwicklungswegen, deren Wirkung sich stark unterscheidet, im Mittel gegenseitig aufheben, da sie aufgrund des Auftauchens der Wirkung in der Exponentialfunktion nicht kohärent sind. Es tragen also im Wesentlichen nur die Entwicklungswege zum letztlich messbaren Betragsquadrat des inneren Produktes zwischen Anfangs- und Endzustand bei, denen ein Wert der Wirkung in der Nähe des Maximums entspricht, da die Amplituden hier nahezu kohärent sind und sich aufsummieren. Dies bedeutet im klassischen Grenzfall die Implikation des Hamiltonschen Prinzips der kleinsten Wirkung. Im Rahmen einer quantenfeldtheoretischen Beschreibungsweise werden die Operatoren $Q_a$ und $P_a$ durch Feldoperatoren $\Phi_m (x)$ und ihre kanonisch konjugierten Feldoperatoren $\Pi_m (x)$ ersetzt, welche von den Raumzeitkoordinaten abhängen und einen Index m für den Spinfreiheitsgrad besitzen. Als Vakuum-Vakuum-Amplitude für ein zeitgeordnetes Produkt von Operatoren ergibt sich hier
$$\begin{aligned}
\langle Vac | T\{\mathcal{O}_A [\Phi_A(t_A), \Pi(t_A)], \mathcal{O}_B [\Phi_B(t_B), \Pi(t_B)], ...| Vac \rangle \nonumber\\
=|\mathcal{N}|^2 \int \left[\prod_{\tau,x,m} d\phi_m(x,\tau)\right]
\left[\prod_{\tau,x,m} \frac{d\pi_m(x,\tau,m)}{2\pi}\right]\nonumber\\
\mathcal{O}_A [\pi(t_A),\phi(t_A)] \mathcal{O}_B [\pi(t_B),\phi(t_B)]\nonumber\\
\times exp\left[i\int_{-\infty}^{\infty}d\tau \left(\int d^3 x (\sum_m \dot \phi_m(x,\tau)\pi_m(x,\tau)-H[\phi(\tau),\pi(\tau)])+i\epsilon \right)\right].\nonumber\\\end{aligned}$$
Der Integrand in der Exponentialfunktion hat die Gestalt einer Lagrangedichte. Da die Impulsoperatoren jedoch zunächst unabhängige Größen sind, darf dieser nicht einfach der Lagrangedichte gleichgesetzt werden. Dies ist nur dann der Fall, wenn der Hamiltonoperator eine quadratische Abhängigkeit von den kanonisch konjugierten Impulsoperatoren aufweist, denn man kann zeigen, dass dann gilt
$$\dot \Psi(x,\tau)=\frac{\delta H[\Psi(\tau),\Pi(\tau)]}{\delta \pi(x,\tau)},$$
was bedeutet
$$L[\Psi(\tau),\dot \Psi(\tau)]=\int d^3 x (\sum_n \dot \Psi(x,\tau) \Pi(x,\tau)-H[\Psi(\tau),\Pi(\tau)]),$$
und damit
$$\begin{aligned}
\langle Vac | T\{\mathcal{O}_A [\Phi_A(t_A), \Pi(t_A)], \mathcal{O}_B [\Phi_B(t_B), \Pi(t_B)], ...| Vac \rangle\nonumber\\
=|\mathcal{N}|^2 \int \left[\prod_{\tau,x,m} d\phi_m(x,\tau)\right] \left[\prod_{\tau,x,m} \frac{d\pi_m(x,\tau,m)}{2\pi}\right]\nonumber\\
\mathcal{O}_A [\pi(t_A),\phi(t_A)] \mathcal{O}_B [\pi(t_B),\phi(t_B)] exp\left[i\int_{-\infty}^{\infty}d\tau L\left[\Psi(\tau),\dot \Psi(\tau)\right]+i\epsilon \right].\end{aligned}$$
### Herleitung eines Propagators
Um nun den Propagator des freien Feldes zu erhalten, zerlegt man die Lagrangedichte $\mathcal{L}$ in einen Term $\mathcal{L}_0$, welcher das freie Feld beschreibt, und einen Wechselwirkungsterm $\mathcal{L}_1$
$$L[\Psi(\tau),\dot \Psi(\tau)]=\int d^3 x [\mathcal{L}_0(\Psi({{\bf x}},\tau),\partial_\mu \Psi({{\bf x}},\tau))+
\mathcal{L}_1(\Psi({{\bf x}},\tau),\partial_\mu \Psi({{\bf x}},\tau))].$$
Der Term $\mathcal{L}_0$, welcher das freie Feld beschreibt, ist eine quadratische Funktion des Feldes und daher kann der freie Teil der Wirkung wie folgt ausgedrückt werden
$$S_0=\int d^4 x \Psi_m (x) \mathcal{D}_{mm'} \Psi_{m'}.$$
Wenn man $\mathcal{D}$ nun im Impulsraum ausdrückt, so kann man durch eine recht lange Herleitung zeigen [@WeinbergQTF1], dass für den Propagator im Ortsraum gilt
$$\Delta(y,x)=\frac{1}{(2\pi)^4}\int d^4 p e^{ip(y-x)}\mathcal{D}^{-1}(p).
\label{Propagator-Lagrangian}$$
Im Falle einer Quantisierung von Eichtheorien ist noch die Einführung eines Eichfixierungstermes von Nöten, der die Gestalt einer Deltafunktion hat, sodass der Gesamtausdruck nur einen Beitrag liefert, wenn die Eichbedingung erfüllt ist. Es ergibt sich in diesem Falle ein Ausdruck der folgenden Form für das Pfadintegral eines Feldes $\Psi$
$$\langle Vac | Vac \rangle = \int \left[\prod d\Psi(x)\right]\ \delta[(G(\Psi)]\ det|\frac{\delta G}{\delta \alpha}|\ exp(iS[\Psi(x)]),$$
wobei $G$ die Eichfixierungsfunktion und $\alpha$ der Eichparameter ist. Wenn man die Eichfixierung und die Determinante in eine Exponentialfunktion umschreibt erhält man einen zusätzliche Eichfixierungsterm im quadratischen Teil der Lagrangedichte sowie sogenannte Geisterfelder (siehe beispielsweise [@WeinbergQTF2] oder [@Rivers]). Aber diese Thematik spielt hier keine weitere Rolle, da es hier um Gravitonen gehen wird, deren Schwingungszustände in den zusätzlichen Dimensionen ihnen eine Masse verleihen, wodurch die Eichinvarianz aufgehoben wird.
Quantisierung des Gravitationsfeldes
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### Die Lagrangedichte des Gravitationsfeldes
Bezüglich der Herleitung eines Propagators für das Graviton, ist es zunächst einmal wichtig, dass man die Einsteinsche Feldgleichung, wie alle anderen dynamischen Gleichungen durch eine Wirkung beschreiben kann, die durch das bekannte Hamiltonsche Variationsprinzip dann wieder auf die Einsteinschen Gleichungen führt. Es handelt sich um die Einstein-Hilbert-Wirkung, welche folgende Gestalt hat
$$S_{EH}=\frac{1}{16\pi G}\int d^4 x \sqrt{-g} R.$$
Im allgemeineren Fall einer beliebigen Anzahl D an Raumzeitdimensionen hat man es mit der folgenden Wirkung zu tun
$$S=-\frac{1}{2}M^{D-2} \int d^D x \sqrt{g} R.$$
Diese muss zunächst in eine geeignete Gestalt gebracht werden, um anschließ end den Propagator für das Graviton aus ihr ableiten zu können. Man verwendet nun eine lineare Entwicklung der Metrik $g_{MN}$ um die Minkowskimetrik der flachen Raumzeit $$g_{MN}=\eta_{MN}+2M^\frac{(D-2)}{2} h_{MN}.
\label{Entwicklung-Metrik}$$ Die hier gegeben Darstellung der Quantisierung des Gravitationsfeldes folgt im Wesentlichen der in [@Callin:2004zm] gelieferten Herleitung. Eine Beschreibung der entsprechenden effektiven Quantenfeldtheorie im Falle einer gewöhnlichen Raumzeit wird hier übergangen. Hierfür sei auf [@Donoghue:1995cz] verwiesen. In der Einstein-Hilbert-Wirkung taucht der Ricci-Skalar R auf, welchen man aus dem Riemanntensor durch Kontraktion erhält. Da der Riemanntensor gemäß des Zusammenhangs aus dem ersten Kapitel ($\ref{RiemannZusammenhangkoeffizienten}$) über die Zusammenhangkoeffizienten bestimmt ist, welche wiederum im Falle der Allgemeinen Relativitätstheorie als Christoffelsymbole über die Metrik definiert sind ($\ref{Christoffelsymbole}$), führt ($\ref{Entwicklung-Metrik}$) auf einen Ausdruck für R in Abhängigkeit von h, womit sich folgender Ausdruck für die Lagrangedichte $\mathcal{L}_h=\sqrt{-g} R$ ergibt
$$\mathcal{L}_h=-\frac{1}{2}\partial_M \partial^M h+\frac{1}{2}\partial_R h_{MN} \partial^R h^{MN}
+\partial_M h^{MN}\partial_N h-\partial_M h^{MN} \partial_R h^R_N.$$
Diese Lagrangedichte ist zunächst invariant unter Eichtransformationen der folgenden Form
$$x^M \rightarrow x^M+2M^{2-D}{2} \alpha^M (x)\quad,\quad h_{MN} \rightarrow h_{MN}-(\partial_M \alpha_N+\partial_N \alpha_M).$$
Dies würde das Hinzufügen eines Eichfixierungstermes der Gestalt $\mathcal{L}_{hg}=\frac{1}{\alpha}C_M C^M$ notwendig machen. Im Rahmen dieser Arbeit sollen aber wie bereits erwähnt massebehaftete Gravitonen betrachtet werden. Durch Einführung eines sogenannten Fierz-Pauli-Termes $-\frac{1}{2}m^2(h^{MN} h_{MN}-h^2)$ erhält man folgende Lagrangedichte für das Graviton, welches nun massebehaftet ist
$$\begin{aligned}
\mathcal{L}_h=-\frac{1}{2}\partial_M h \partial^M h+\frac{1}{2}\partial_R h_{MN} \partial^R h^{MN}
+\partial_\mu h^{MN}\partial_N h-\partial_M h^{MN} \partial_R h^R_N\nonumber\\
-\frac{1}{2}m^2 (h^{MN}h_{MN}-h^2).
\label{freieLagrangedichteMasse}\end{aligned}$$
Diese besitzt nun keine Eichfreiheit mehr, womit keine zusätzlichen Terme in das entsprechende Pfadintegral eingeführt werden müssen.
### Der Gravitonpropagator
Als Gravitonpropagator ergibt sich schließlich unter Verwendung der allgemeinen Relation zwischen dem Anteil des freien Feldes in der Lagrangedichte und dem entsprechenden Propagator ($\ref{Propagator-Lagrangian}$) der folgende Ausdruck für den Gravitonpropagator
$$\Delta_{MNRS}(x,y)=\int \frac{d^D k}{(2\pi)^D}\frac{P_{MNRS}(k)}{k^2-m^2}e^{-ik(x-y)},
\label{GravitonpropagatorMasseD}$$
wobei der Polarisationstensor $P_{\mu\nu\rho\sigma}$ wie folgt aussieht
$$\begin{aligned}
P_{MNRS}(k)=\frac{1}{2}(\eta_{MR}\eta_{NS}+\eta_{MS}\eta_{NR})
-\frac{1}{D-2}\eta_{MN}\eta_{RS}\\\nonumber
-\frac{1}{2m^2}(\eta_{MR}k_N k_\sigma+\eta_{NS}k_M k_R+\eta_{MR}k_N k_S+\eta_{NS}k_M k_R)\nonumber\\
+\frac{1}{(D-1)(D-2)}(\eta_{MN}+\frac{D-2}{m^2} k_M k_N)(\eta_{RS}+\frac{D-2}{m^2}k_R k_S).\end{aligned}$$
Es wurde also ein Propagator für massebehaftete Gravitonen in einer D-dimensionalen Raumzeit hergeleitet. Innerhalb des ADD-Modells sind die zusätzlichen Dimensionen aber kompaktifiziert. Deshalb wählt man für das Gravitationsfeld, das im Rahmen der obigen Entwicklung ($\ref{Entwicklung-Metrik}$) durch $h_{\mu\nu}$ beschrieben wird einen Ansatz, der dem bereits im letzten Kapitel beschriebenen für das Skalarfeld entspricht ($\ref{SkalarfeldKK}$), mit dem Unterschied, dass von einer beliebigen Anzahl $\delta$ an zusätzlichen Dimensionen ausgegangen wird. Das führt auf folgenden Ausdruck für das Gravitationsfeld
$$h_{MN}(z)=\sum_{j=1}^\delta \sum_{n_\delta=-\infty}^{\infty} \frac{h_{MN}(x)}{\sqrt{V_\delta}}e^{i\frac{n^j y_j}{R}}.$$
Gemäß der betrachteten Situation eines Skalarfeldes bei einer zusätzlichen Dimension verleihen die Anregungen in den kompaktifizierten Dimensionen dem Gravitationsfeld eine Masse, wobei die Masse nun der Summe der Quadrate des Verhältnisses der Schwingungszahlen zum Radius entspricht
$$\sum_{j=1}^{\delta}|\hat n_j|^2 \quad,\quad |\hat n_j|^2=\frac{|n_j|^2}{R^2}.$$
Der in der obigen Betrachtung eingeführte Massenterm ergibt sich also unter der Annahme, dass das Gravitationsfeld in den zusätzlichen kompaktifizierten Dimensionen Anregungszustände besitzt, die aufgrund der Periodizität gequantelt sein müssen [@Giudice:1998ck], [@Han:1998sg]. Die Masse ist hier jedoch nicht wie gewöhnlich eine Konstante, sondern eben auch eine Eigenschaft eines Zustandes. In dieser Beschreibungsweise wird die Störung der Metrik, welche dem Gravitationsfeld entspricht, also aufgespalten in einen Anteil der die Geometrie der Untermannigfaltigkeit beschreibt, auf der die bekannten Materiefelder leben, und die zusätzlichen Freiheitsgrade der weiteren Dimensionen. Bezüglich der Wechselwirkung mit den Materiefeldern ist also nur der erste Anteil von Bedeutung, wobei die Schwingungen in den zusätzlichen Dimensionen natürlich das Verhalten der Gravitonen insofern beeinflussen als sie ihnen eben eine Masse verleihen. Dies bedeutet, dass sich letztlich für den Gravitonpropagator in dem hier vorausgesetzten Modell der oben hergeleitete Gravitonpropagator ($\ref{GravitonpropagatorMasseD}$) für den Fall von 4 Dimensionen ergibt
$$\Delta_{\mu\nu\rho\sigma}(x,y)=\int \frac{d^D k}{(2\pi)^D}\frac{P_{\mu\nu\rho\sigma}(k)}{k^2-m^2}e^{-ik(x-y)},
\label{GravitonpropagatorMasse}$$
mit Polarisationstensor
$$\begin{aligned}
P_{\mu\nu\rho\sigma}(k)=\frac{1}{2}(\eta_{\mu\rho}\eta_{\nu\sigma}+\eta_{\mu\sigma}\eta_{\nu\rho})
-\frac{1}{2}\eta_{\mu\nu}\eta_{\rho\sigma}\\\nonumber
-\frac{1}{2m^2}(\eta_{\mu\rho}k_\nu k_\sigma+\eta_{\mu\sigma}k_\nu k_\rho+\eta_{\nu\rho}k_\mu k_\sigma+\eta_{\nu\sigma}k_\mu k_\rho)\\\nonumber
+\frac{1}{6}(\eta_{\mu\nu}+\frac{2}{m^2} k_\mu k_\nu)(\eta_{\rho\sigma}+\frac{2}{m^2}k_\rho k_\sigma),
\label{PolarisationstensorGraviton}\end{aligned}$$
wobei für die Masse m gilt
$$m^2=\sum_{j=1}^{\delta}\left(\frac{n_j^2}{R^2}\right).$$
Durch Fouriertransformation erhält man wie üblich den Propagator im Impulsraum
$$\delta_{\mu\nu\rho\sigma}=\frac{P_{\mu\nu\rho\sigma}(k)}{k^2-m^2}.$$
Der Polarisationstensor erfüllt die folgenden Relationen
$$\eta^{\mu\nu}P_{\mu\nu\rho\sigma}=0 \quad,\quad k^\mu P_{\mu\nu\rho\sigma}=0.$$
Der Gravitonpropagator ($\ref{GravitonpropagatorMasse}$) wird in [@Giudice:1998ck] auf eine etwas andere Art und Weise hergeleitet.
### Kopplungen und Vertizes
Wenn man nun die Wechselwirkung mit Materiefeldern konkret beschreiben will, so muss man den Energie-Impuls-Tensor in die Lagrangedichte integrieren. Die oben angegebene Einstein-Hilbert-Wirkung bezieht sich nur auf die Einsteingleichung ohne Materie, bei der also $T_{\mu\nu}=0$ angenommen wird. Es ist also notwendig, zur Lagrangedichte für das freie massebehaftete Graviton einen zusätzlichen Term zu addieren, welcher auf den Energie-Impuls-Tensor in den Einsteinschen Feldgleichungen führt. Insgesamt lautet damit die Lagrangedichte
$$\begin{aligned}
\mathcal{L}_h=-\frac{1}{2}\partial_\mu h \partial^\mu h+\frac{1}{2}\partial_R h_{\mu\nu} \partial^R h^{\mu\nu}
+\partial_\mu h^{\mu\nu}\partial_\nu h-\partial_\mu h^{\mu\nu} \partial_R h^R_\nu\nonumber\\
-\frac{1}{2}m^2 (h^{\mu\nu}h_{\mu\nu}-h^2)-\frac{1}{\bar M_P}h^{\mu\nu}T_{\mu\nu}.\end{aligned}$$
Die spezifische Form des Energie-Impuls-Tensors ist natürlich durch die Wirkungen ($\ref{FermionGravWirkung}$),($\ref{BosonGravWirkung}$),($\ref{EichfeldGravWirkung}$) definiert, welche das Verhalten von Materiefeldern in gekrümmten Raumzeiten beschreibt und um welche die Einstein-Hilbert-Wirkung des freien Gravitationsfeldes erweitert werden muss. Aus den in diesen enthaltenen Kopplungstermen bestimmt man die Vertizes. Für die Wechselwirkung eines Fermion bzw. Vektorbosons ergeben sich die folgenden Vertizes [@Giudice:1998ck].\
[**Fermion-Fermion-Graviton**]{}
$$\begin{aligned}
-\frac{i}{4 \bar M_P}[W_{\mu\nu}+W_{\nu\mu}]\\\nonumber
\\\nonumber
W_{\mu\nu}=(p_1+p_2)_\mu \gamma_\nu
\label{VertexFFG}\end{aligned}$$
\
[**Vektorboson-Vektorboson-Graviton**]{}
$$\begin{aligned}
&&-\frac{i}{4 \bar M_P}\delta^{ab}[W_{\mu\nu\rho\sigma}+W_{\nu\mu\rho\sigma}]\\\nonumber
\\\nonumber
W_{\mu\nu\rho\sigma}&=&\frac{1}{2} \eta_{\mu\nu}(k_{1\sigma}k_{2\rho}-k_1\cdot k_2 \eta_{\rho\sigma})+\eta_{\rho\sigma}k_{1\mu}k_{2\nu}\\\nonumber
&&+\eta_{\mu\rho}(k_1 \cdot k_2 \eta_{\nu\sigma}-k_{1\sigma}k_{2\nu})-\eta_{\mu\sigma}k_{1\nu}k_{2\rho}
\label{VertexVbVbG}\end{aligned}$$
\
[**Fermion-Fermion-Vektorboson-Graviton**]{}
$$\begin{aligned}
-\frac{i}{2\bar M_P} gT^a (X_{\mu\nu\alpha}+X_{\nu\mu\alpha})\quad\quad\quad X_{\mu\nu\alpha}=\gamma_\mu \eta_{\nu\alpha}\end{aligned}$$
\
[**Vektorboson-Vektorboson-Vektorboson-Graviton**]{}
$$\begin{aligned}
&\frac{g}{\bar M_P}f^{abc}&[Y(k_1)_{\mu\nu\alpha\beta\gamma}+Y(k_2)_{\mu\nu\beta\gamma\alpha}+Y(k_3)_{\mu\nu\gamma\alpha\beta}\nonumber\\
&&+Y(k_1)_{\nu\mu\alpha\beta\gamma}+Y(k_2)_{\nu\mu\beta\gamma\alpha}+Y(k_3)_{\nu\mu\gamma\alpha\beta}]\nonumber\\
\nonumber\\
Y(k)&=&k_\mu(\eta_{\nu\beta}\eta_{\alpha\gamma}-\eta_{\nu\gamma}\eta_{\alpha\beta})\nonumber\\
&&+k_{\beta}\left(\eta_{\mu\alpha}\eta_{\nu\gamma}-\frac{1}{2}\eta_{\mu\nu}\eta_{\alpha\gamma}\right)-k_{\gamma}\left(\eta_{\mu\alpha}\eta_{\nu\beta}-\frac{1}{2}\eta_{\mu\nu}\eta_{\alpha\beta} \right)\nonumber\\\end{aligned}$$
ZZ-Produktion durch Gravitonenvermittlung
=========================================
In diesem Kapitel soll die in den vorigen Kapiteln dargestellte Theorie nun auf einen konkreten physikalischen Vorgang angewandt werden. Es soll der Vorgang betrachtet werden, bei dem ein Teilchen und ein Antiteilchen sich gegenseitig vernichten und dabei ein virtuelles Graviton erzeugen, das sich anschließend in zwei Z-Teilchen umwandelt. Diese stellen neben den $W^+$ und den $W^-$-Teilchen die Austauschteilchen der schwachen Wechselwirkung dar (siehe Kapitel 5). Hierbei handelt es sich um einen Prozess, der im Standardmodell ziemlich unwahrscheinlich ist. Daher könnte der Gravitonenaustausch, welcher unter der Annahme zusätzlicher Dimensionen und einer entsprechend modifizierten Planckmasse $M_D$ sehr viel wahrscheinlicher wird, zu einer deutlichen Abweichung gegenüber dem gemäß dem Standardmodell erwarteten Wert für die Produktionsrate führen. Im Speziellen soll hier das Aufeinandertreffen zweier Protonen untersucht werden, da dies der Situation am LHC entspricht. Ein Proton setzt sich aus Quarks und Gluonen zusammen. Durch letztere wechselwirken die Quarks gemäß der Quantenchromodynamik miteinander und werden dadurch zusammengehalten. Zunächst müssen die Wirkungsquerschnitte für die Einzelprozesse berechnet werden, ehe diese mit den Verteilungsfunktionen für Partonen innerhalb eines Protons gefaltet werden. Diese Verteilungsfunktionen sind aufgrund der unglaublichen Komplexität der physikalischen Verhältnisse innerhalb eines Protons nur durch Messungen bekannt, können also nicht selbst auf theoretischem Wege ermittelt werden.
S-Matrix und Feynmanamplitude für die ZZ-Produktion durch Gluonen
-----------------------------------------------------------------
Der erste Schritt zur Berechnung der ZZ-Produktionsrate durch Vernichtung zweier Protonen besteht in der Berechnung der S-Matrix für den ZZ-Produktionsprozess durch Vernichtung der im Proton enthaltenen Partonen. Hierbei wird die übliche Störungstheorie zu Grunde gelegt, die eigentlich in jedem Buch über Quantenfeldtheorie wie beispielsweise [@BjorkenDrellRQFT], [@WeinbergQTF1] und [@PeskinSchroeder] zu finden ist und hier nicht weiter thematisiert werden soll. Es wird eine Rechnung in erster Ordnung durchgeführt. Zunächst ist zu erwähnen, dass alle hier auftauchenden Feynmangraphen, die einen Vierervertex der im letzten Kapitel angegebenen Art enthalten, entweder überhaupt keinen Beitrag liefern oder im Rahmen einer Störungsentwicklung in erster Ordnung nicht berücksichtigt werden müssen. Bei zwei einlaufenden Gluonen und zwei auslaufenden Z-Bosonen kann ohnehin nur ein Graviton ausgetauscht werden, da diese im Standardmodell nicht aneinander koppeln. Im Falle eines einlaufenden Quark-Antiquark-Paares gibt es zwei mögliche Graphen mit Vierervertizes. Einerseits kann das Quark-Antiquark-Paar direkt in einen Vierervertex laufen und ein Graviton und ein Teilchen der schwachen Wechselwirkung erzeugen. In diesem Falle verschwindet der zweite Vertex mit den auslaufenden Z-Bosonen aufgrund der im Vertex auftauchenden Strukturkonstante $f^{abc}$, die bei zwei gleichen Indizes, die den gleichen zu den beiden Z-Teilchen gehörigen Generatoren entsprechen, gleich null ist. Andererseits kann aber auch das eine Quark ein Z-Teilchen emittieren und gemeinsam mit einem im Vertex erzeugten Graviton weiterlaufen, um sich mit diesem und dem Antiquark dann im zweiten Vertex zu vernichten und dabei das andere Z-Boson zu erzeugen. In den beiden Vertizes steht jedoch neben der inversen Planckmasse auch noch die Kopplungskonstante der schwachen Wechselwirkung. Das bedeutet jedoch, dass dieser Beitrag von höherer Ordnung ist und damit nicht berücksichtigt werden muss. Dies gilt natürlich nur dann, wenn die Gravitonenkopplung in der Störungsentwicklung wie die Kopplung im Standardmodell, also in diesem Falle die der elektroschwachen Wechselwirkung, behandelt wird. Dies ist aber unumgänglich, wenn eine Rechnung in erster Ordnung überhaupt einen Sinn haben soll. Damit bleiben die Prozesse übrig, bei denen sich ein Quark-Antiquark- bzw. ein Gluon-Paar direkt vernichten und dabei ein einzelnes Graviton erzeugen, welches anschließend in zwei Z-Bosonen übergeht (Graphen \[Graph1\] und \[Graph2\]).
Die S-Matrix des Feynmangraphen für den Quark-Antiquark-Prozess ergibt exakt 0 (siehe Anhang). Da im Rahmen des Gravitonaustausches also einerseits nur die Gluonen entscheidend beitragen und im Standardmodell andererseits nur die Quarks überhaupt einen Beitrag liefern, weil die Gluonen nur mit sich selbst und nicht elektroschwach wechselwirken (siehe Abschnitt 5.3), muss bezüglich des zusätzlichen Beitrages nur ein Feynmandiagramm ausgewertet werden, um die entsprechende S-Matrix zu erhalten. Aufgrund der Unabhängigkeit der S-Matrix des Quarkprozesses von der des Gluonenprozesses, kann der zum Standardmodell zusätzliche Wirkungsquerschnitt separat berechnet werden. Zunächst werden die Polarisationszustände für das einlaufende Gluon-Antigluon-Paar und das auslaufende ZZ-Paar, sowie die Ausdrücke für die Vertizes und den Gravitonpropagator zusammengefügt, die aus dem letzten Kapitel bekannt sind. Hierbei sollen die entsprechenden Ausdrücke im Impulsraum verwendet werden. Entscheidend ist, dass im Gravitonpropagator über alle Anregungszustände in den zusätzlichen kompaktifizierten Dimensionen summiert werden muss, die verschiedenen Massen des Gravitons entsprechen. Die Vertizes erhalten aufgrund der Impulserhaltung wie üblich eine $\delta$-Funktion. Über den Impuls des virtuellen Gravitons muss integriert werden, da er keine direkte Messgröße darstellt. Damit erhält man folgenden Ausdruck für die S-Matrix
$$\begin{aligned}
S(g(k_1,g_1)+g(k_2,g_2) \rightarrow Z(l_1,Z_1)+Z(l_2,Z_2))\nonumber\\
=\frac{1}{(2\pi)^4}\int d^{4}k \frac{g^1_{\alpha a}}{(2\pi)^\frac{3}{2}\sqrt{k_{10}}} \frac{g_{2\beta b}}{(2\pi)^\frac{3}{2}\sqrt{k_{20}}}
\left(-\frac{i}{\bar M_P}\delta^{ab}\left[W^{\mu\nu\alpha\beta}+W^{\nu\mu\alpha\beta}\right] \right)\nonumber\\
\cdot (2\pi)^4\delta^4(k_1+k_2-k)
\sum_n \frac{iP_{\mu\nu\rho\sigma}}{k^2-m^2}
(2\pi)^4\delta^4(k-l_1-l_2)\nonumber\\
\cdot \left(-\frac{i}{\bar M_P}\delta^{cd}\left[W^{\rho\sigma\gamma\delta}+W^{\sigma\rho\gamma\delta}\right] \right)
\frac{Z_{1\gamma c}}{(2\pi)^\frac{3}{2}\sqrt{l_{10}}} \frac{Z_{2\delta d}}{(2\pi)^\frac{3}{2}\sqrt{l_{20}}}.
\label{S-Matrix}\end{aligned}$$
Hierbei sind $W_{\mu\nu\alpha\beta}$ und $P_{\mu\nu\rho\sigma}$ gemäß ($\ref{VertexVbVbG}$) bzw. ($\ref{PolarisationstensorGraviton}$) wie folgt definiert
$$\begin{aligned}
W_{\mu\nu\alpha\beta}&=&\frac{1}{2} \eta_{\mu\nu}(k_{1\beta}k_{2\alpha}-k_1\cdot k_2 \eta_{\alpha\beta})+\eta_{\alpha\beta} k_{1\mu} k_{2\nu}\nonumber\\
&&+\eta_{\mu\alpha}(k_1 \cdot k_2 \eta_{\nu\beta}-k_{1 \beta} k_{2 \nu})-\eta_{\mu\beta} k_{1 \nu} k_{2 \alpha},
\label{W-Ausdruck}\end{aligned}$$
$$\begin{aligned}
P_{\mu\nu\rho\sigma}&=&\frac{1}{2}(\eta_{\mu\alpha}\eta_{\nu\beta}+\eta_{\mu\beta}\eta_{\nu\alpha}-\eta_{\mu\nu}\eta_{\alpha\beta})\nonumber\\
&&-\frac{1}{2m^2}(\eta_{\mu\alpha}k_\nu k_\beta+\eta_{\nu\beta}k_\mu k_\alpha+\eta_{\mu\beta}k_\nu k_\alpha+\eta_{\nu\alpha}k_\mu k_\beta)\nonumber\\
&&+\frac{1}{6}\left(\eta_{\mu\nu}+\frac{2}{m^2}k_\mu k_\nu\right)\left(\eta_{\alpha\beta}+\frac{2}{m^2}k_\alpha k_\beta \right).
\label{P-Ausdruck}\end{aligned}$$
Wichtig ist, dass der Tensor $P_{\mu\nu\rho\sigma}$ “on mass-shell” ist und damit $p^2=m^2$ gilt. Der Ausdruck für die S-Matrix ($\ref{S-Matrix}$) kann umformuliert werden zu
$$\begin{aligned}
S&=&\frac{-i}{(2\pi)^2 \sqrt{2k_{10}} \sqrt{2k_{20}} \sqrt{2l_{10}} \sqrt{2l_{20}}} \frac{1}{\bar M_P^2} \sum_n \frac{1}{p^2-m^2} \nonumber\\
&&\cdot g_{1\alpha} g_{2\beta} \left[W^{\mu\nu\alpha\beta}+W^{\nu\mu\alpha\beta}\right] P_{\mu\nu\rho\sigma} \left[W^{\rho\sigma\gamma\delta}+W^{\sigma\rho\gamma\delta}\right] Z_{1\gamma} Z_{2\delta} \nonumber\\
&&\cdot \delta^4(p-l_1-l_2).\end{aligned}$$
Es wurde unter anderem über die erste Deltafunktion integriert und p als $p=k^1+k^2$ definiert. Desweiteren soll nun der folgende Zusammenhang zwischen der S-Matrix und der Feynmanamplitude M verwendet werden
$$S=-2\pi iM\delta^4(p-l_1-l_2).
\label{SMatrixFeynman}$$
Damit erhält man für die Feynmanamplitude M folgenden Ausdruck
$$\begin{aligned}
M&=&\frac{1}{(2\pi)^3 \sqrt{2k_{10}} \sqrt{2k_{20}} \sqrt{2l_{10}} \sqrt{2l_{20}}} \frac{1}{\bar M_P^2} \sum_n \frac{1}{p^2-m^2}\nonumber\\
&&\cdot g_{1\alpha} g_{2\beta} [W^{\mu\nu\alpha\beta}+W^{\nu\mu\alpha\beta}]
P_{\mu\nu\rho\sigma}
\left[W^{\rho\sigma\gamma\delta}+W^{\sigma\rho\gamma\delta}\right] Z_{1\gamma} Z_{2\delta}.\end{aligned}$$
Wenn man A wie folgt definiert $A=\frac{1}{(2\pi)^3 \sqrt{2k_{10}} \sqrt{2k_{20}} \sqrt{2l_{10}} \sqrt{2l_{20}}}\sum_n \frac{1}{\bar M_P^2} \frac{1}{p^2-m^2}$, erhält man
$$M=A \cdot g_{1\alpha} g_{2\beta} \left[W^{\mu\nu\alpha\beta}+W^{\nu\mu\alpha\beta}\right]
P_{\mu\nu\rho\sigma}
\left[W^{\rho\sigma\gamma\delta}+W^{\sigma\rho\gamma\delta}\right] Z_{1\gamma} Z_{2\delta}.$$
Da der Gravitonpropagator einerseits symmetrisch bezüglich der Indizes $\mu$ and $\nu$ und andererseits symmetrisch bezüglich der Indizes $\rho$ and $\sigma$ ist, kann man obigen Ausdruck wie folgt schreiben
$$M=4A \cdot g_{1\alpha} g_{2\beta} \cdot W^{\mu\nu\alpha\beta} \cdot P_{\mu\nu\rho\sigma} \cdot W^{\rho\sigma\gamma\delta} \cdot Z_{1\gamma} Z_{2\delta}.
\label{Feynman-Amplitude1}$$
Indem man die Ausdrücke für $W_{\mu\nu\alpha\beta}$ und $P_{\mu\nu\rho\sigma}$ aus ($\ref{W-Ausdruck}$) und ($\ref{P-Ausdruck}$) in ($\ref{Feynman-Amplitude1}$) einsetzt, erhält man
$$\begin{aligned}
M&=&4A \cdot g_{1\alpha} g_{2\beta}
\cdot[\frac{1}{2}\eta^{\mu\nu}(k_1^\beta k_2\alpha-k_1 k_2\eta^{\alpha\beta})+\eta^{\alpha\beta}k_1^\mu k_2^\nu\nonumber\\
&&+\eta^{\mu\alpha}(k_1 k_2\eta^{\nu\beta}-k_1^\beta k_2^\nu-\eta^{\mu\beta}k_1^\nu k_2^\alpha] \nonumber\\
&&\cdot[\frac{1}{2}(\eta_{\mu\rho}\eta_{\nu\sigma}+\eta_{\mu\sigma}\eta_{\nu\rho}+\eta_{\mu\nu}\eta_{\rho\sigma})\nonumber\\
&&-\frac{1}{2m^2}(\eta_{\mu\rho}p_{\nu}p_{\sigma}+\eta_{\nu\sigma}p_{\mu}p_{\rho}+\eta_{\mu\sigma}p_{\nu}p_{\rho}+\eta_{\nu\rho}p_{\mu}p_{\sigma})\nonumber\\
&&+\frac{1}{6}\left(\eta_{\mu\nu}+\frac{2}{m^2}p_{\mu}p_{\nu}\right)\left(\eta_{\rho\sigma}+\frac{2}{m^2}p_{\rho}p_{\sigma}\right)]\nonumber\\
&&\cdot[\frac{1}{2}\eta^{\rho\sigma}(k_1^\delta k_2\gamma-k_{1}k_{2}\eta^{\gamma\delta})+\eta^{\gamma\delta}k_1^\rho k_2^\sigma\nonumber\\
&&+\eta^{\rho\gamma}(k_1 k_2 \eta^{\sigma\delta}-k_1^\delta k_2^\sigma)-\eta^{\rho\delta}k_1^\sigma k_2^\gamma]\cdot Z_{1\gamma} Z_{2\delta}.\end{aligned}$$
Die Polarisationsvektoren für die Gluonen und Z-Teilchen stehen immer senkrecht auf dem jeweiligen Impuls und erfüllen damit folgende Relationen
$$k_1^\mu g_1^\mu=0\quad,\quad k_2^\mu g_2^\mu=0\quad,\quad l_1^\mu Z_{1\mu}=0\quad,\quad l_2^\mu Z_{2\mu}=0.$$
Außerdem soll die Rechnung von nun an im Schwerpunktsystem betrachtet werden, da sich hierdurch vieles vereinfacht. Dies bedeutet, dass gilt
$${{\bf k}}_1=-{{\bf k}}_2\quad,\quad p=0\quad,\quad {{\bf l}}_1=-{{\bf l}}_2.$$
(Die fettgedruckten Größen sollen hier den räumlichen Anteil der Vierervektoren bezeichnen.) Damit erhält man für die Feynmanamplitude folgenden Ausdruck
$$\begin{aligned}
M&=&4A \cdot g_{1\alpha} g_{2\beta} \cdot
\left[-\frac{1}{2}\eta^{\mu\nu}(k_1 \cdot k_2)(g_1 \cdot g_2)+(g_1 \cdot g_2)(k_1\mu k_2\nu)+(k_1 \cdot k_2)g_1\mu g_2^\nu\right]\nonumber\\
&&\cdot[\frac{1}{2}(\eta_{\mu\rho}\eta_{\nu\sigma}+\eta_{\mu\sigma}\eta_{\nu\rho})-\frac{1}{3}\eta_{\mu\nu}\eta_{\rho\sigma}\nonumber\\
&&-\frac{1}{2m^2}(\eta_{\mu\rho}p_{\nu}p_{\sigma}+\eta_{\nu\sigma}p_{\mu}p_{\rho}+\eta_{\mu\sigma}p_{\nu}p_{\rho}+\eta_{\nu\rho}p_{\mu}p_{\sigma})\nonumber\\
&&+\frac{1}{3m^2}\eta_{\rho\sigma}p_\mu p_\nu+\frac{1}{3m^2}\eta_{\mu\nu}p_\rho p_\sigma+\frac{2}{3m^4} p_\mu p_\nu p_\rho p_\sigma]\nonumber\\
&&\cdot\left[-\frac{1}{2}\eta^{\rho\sigma}(l_1 \cdot l_2)(Z_1 \cdot Z_2)+(Z_1 \cdot Z_2)(l_1^\rho l_2^\sigma)+(l_1 \cdot l_2) Z_1^\rho Z_2^\sigma\right].
\label{Feynman-Amplitude2}\end{aligned}$$
Wirkungsquerschnitt für die ZZ-Produktion durch Gluonen
-------------------------------------------------------
### Von der Feynmanamplitude zum Wirkungsquerschnitt
Der differentielle Wirkungsquerschnitt für den Übergang von einem Zustand $\alpha$ in einen Zustand $\beta$ hat allgemein folgende Gestalt
$$d\sigma(\alpha \rightarrow \beta)=(2 \pi)^4 u_\alpha^{-1} \sum_\sigma |M|^2 \delta^4(p_\beta-p_\alpha) d\beta,$$
wobei gilt $$u_\alpha=\frac{\sqrt{(p_1 \cdot p_2)^2-m_1^2 m_2^2}}{E_1 E_2}.$$ (siehe [@WeinbergQTF1])\
In dem Fall, der hier betrachtet wird, bedeutet dies
$$d \sigma=\frac{(2 \pi)^4}{2} \sum_\sigma |M|^2 \delta^4 (l_1+l_2-k_1-k_2)d^4 l_1 d^4 l_2.$$
Durch Aufspaltung der $\delta$-Funktion und Ausnutzung der relativistischen Energie-Impuls-Beziehung erhält man
$$\begin{aligned}
d \sigma&=&\frac{(2 \pi)^4}{2} \sum_\sigma |M|^2 \delta^3 ({{\bf l}}_1+{{\bf l}}_2-({{\bf k}}_1+{{\bf k}}_2))\nonumber\\
&&\cdot\delta(\sqrt{|{{\bf l}}_1|^2+m_Z^2}+\sqrt{|{{\bf l}}_2|^2 +m_Z^2}-2E) d^3 l_1 d^3 l_2.\end{aligned}$$
Im Schwerpunktsystem gilt
$${{\bf k}}_1+{{\bf k}}_2=0.$$
Integrieren über $d^3 l_1$ liefert damit
$$d \sigma=\frac{(2 \pi)^4}{2} \sum_\sigma |M|^2 \delta(\sqrt{|{{\bf l}}_2|^2+m_Z^2}-E)d^3 l_2.$$
Das ist gleichbedeutend mit
$$d \sigma=\frac{(2 \pi)^4}{2} \sum_\sigma |M|^2 \delta(\sqrt{|{{\bf l}}_2|^2+m_Z^2}-E)|{{\bf l}}_2|^2 sin(\theta) d|{{\bf l}}_2|d \Omega.$$
Integrieren über $d|{{\bf l}}_2|$ liefert schließlich
$$d \sigma=\frac{(2 \pi)^4}{2} \sum_\sigma |M|^2 E \sqrt{E^2-m_Z^2} sin(\theta) d \Omega.$$
Wenn man nun noch über den Azimutalwinkel integriert, bekommt man folgenden Ausdruck
$$d \sigma=\frac{(2 \pi)^5}{2} \sum_\sigma |M|^2 E \sqrt{E^2-m_Z^2} sin(\theta) d \theta.$$
Der Wirkungsquerschnitt enthält nicht die Feynmanamplitude selbst, sondern das Betragsquadrat. Nach Ausmultiplizieren des Ausdruckes für die Feynmanamplitude ($\ref{Feynman-Amplitude2}$) wird das im Wirkungsquerschnitt auftauchende Betragsquadrat der Feynmanamplitude gebildet. Da keine spezielle Polarisation der Gluonen angenommen wird und die Polarisation der Z-Teilchen keine Rolle spielt, muss über alle möglichen Polarisationen summiert werden. Eine konkrete Berechnung von $\sum_\sigma |M|^2$ unter Ausnutzung der Relationen
$$\sum_\sigma g_\mu g_\nu^*=-\eta_{\mu\nu},$$
und
$$\sum_\sigma Z_\mu Z_\nu^*=\left(-\eta_{\mu\nu}+\frac{l_\mu l_\nu}{m_Z^2}\right)$$
für die Polarisationsvektoren der masselosen Gluonen, sowie der massebehafteten Z-Teilchen, führt nach Integration über den Streuwinkel $\theta$ schließlich auf folgende Ausdruck für den absoluten Wirkungsquerschnitt $\sigma$
$$\sigma=\frac{D^2}{\bar M_p^4}\frac{E \sqrt{E^2-m_Z^2}(3552 E^8-7400 E^6 m_Z^2+4977 E^4 m_Z^4-1257 E^2 m_Z^6+98 m_Z^8)}{30 \pi m_Z^4}.$$
Hierbei ist D wie folgt definiert
$$D=\sum_n \frac{1}{p^2-m^2},
\label{Kaluza-Klein-Summierung}$$
wobei die Masse den Anregungen in den zusätzlichen Dimensionen entspricht. Die Energie bezieht sich natürlich auf das Schwerpunktsystem. Es ist daher sinnvoll, den Wirkungsquerschnitt mit Hilfe der lorentzinvarianten Mandelstamvariable s audszudrücken. In Abhängigkeit von s ergibt sich folgender Ausdruck
$$\begin{aligned}
\sigma=\frac{D^2}{\bar M_p^4}\frac{s}{2}\sqrt{\frac{s}{4}-m_Z^2}
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \nonumber\\
\cdot\frac{(13.875 s^4-115.625 s^3 m_Z^2+311.0625 s^2 m_Z^4-314.25 s m_Z^6+98 m_Z^8)}
{30 \pi m_Z^4}.
\label{WirkungsquerschnittsD}\end{aligned}$$
### Summierung über die Kaluza-Klein-Anregungen
Die Summation über die Kaluza-Klein-Zustände in (\[Kaluza-Klein-Summierung\]) kann umformuliert werden, indem man die Tatsache berücksichtigt, dass die Masse $m$ des Gravitons durch die Kaluza-Klein-Anregungen über den Zusammenhang
$$m^2=\hat n^2=\sum_{j=1}^{\delta}|\hat n_j|^2
\label{Gravitonmasse_Anregungen}$$
bestimmt ist. In [@Giudice:1998ck] ist die Summierung über die Kaluza-Klein Anregungen ausgeführt. Dieser Darstellung soll hier gefolgt werden. Die Summe über die einzelnen Anregungen kann näherungsweise durch ein Integral über das Massenquadrat ersetzt werden, wobei eine Dichteverteilung bezüglich des Massenquadrats auftaucht, da unterschiedliche Massen unterschiedliche Gewichtungen haben. Wenn man Gleichung ($\ref{Gravitonmasse_Anregungen}$) betrachtet, sieht man, dass die Gewichtung des Massenquadrates der Zahl der Anregungskombinationen entspricht, die auf ein bestimmtes $\hat n^2$ führen. Diese entspricht im Limes eines großen $\hat n^2$ der Oberfläche einer n-dimensionalen Sphäre. Man erhält den folgenden Ausdruck
$$D=\sum_n \frac{1}{p^2-m^2}=\int_0^{\infty} dm^2 \frac{\rho(m)}{s-m^2}$$
mit
$$\rho(m)=\frac{R^n m^{(\delta-2)}}{(4\pi)^{\frac{n}{2}}\Gamma (\frac{\delta}{2})},$$
wobei $\Gamma$ die übliche Gammafunktion beschreibt. Es wird nun das Verfahren dimensionaler Regularisierung angewandt. Unter Verwendung des niedrigst dimensionalen Beitrages ($c_1=1$ and $c_i=0$ for $i \neq 1$), bei Wahl einer Regularisierungsskala in der Größenordnung der neuen Massenskala $\Lambda=M_D$ und der Relation zwischen der $(4+\delta)$-dimensionalen Planckmasse und der gewöhnlichen Planckmasse $M_P^2=8\pi M_D^{(2+\delta)}R^\delta$ wird man auf folgenden Ausdruck geführt $$\sum_n \frac{1}{s-m_n^2}\approx\frac{\bar M_P^2\pi^{\frac{\delta}{2}}}{\Gamma (\frac{\delta}{2})M_D^4} \quad.
\label{KKSummierung}$$ ein andere Näherungsverfahren für die Kaluza-Klein Summierung findet man in [@Han:1998sg]. Wir beschränken uns hier jedoch auf (\[KKSummierung\]).\
Das führt schließlich auf den folgenden totalen Wirkungsquerschnitt
$$\sigma(gg \rightarrow ZZ)=\frac{\pi^\delta \sqrt{\frac{\hat s}{4}}\sqrt{\frac{\hat s}{4}-m_Z^2}Z}{\Gamma^2 (\frac{\delta}{2}) M_D^8 30 \pi m_Z^4}
\label{ADDWirkungsquerschnitt}$$
mit
$$\begin{aligned}
Z&=&13.875 \hat s^4-115.625 \hat s^3 m_Z^2+311.0625 \hat s^2 m_Z^4\nonumber\\
&&-314.250 \hat s m_Z^6+98 m_Z^8\quad.\end{aligned}$$
Das ist der totale Wirkungsquerschnitt als Funktion der Zahl der Extra Dimensionen $\delta$ und der Planckmasse $M_D$ in $4+\delta$ Dimensionen.
Wirkungsquerschnitt nach Faltung über die Verteilungsfunktionen im Proton und Vergleich mit dem Standardmodell
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### Faltung über die Partonenverteilungsfunktionen
Um nun den Wirkungsquerschnitt für den Proton-Proton-Prozess zu erhalten, müssen die Wirkungsquerschnitte für die einzelnen Partonenprozesse mit der entsprechenden Verteilungsfunktion für die Gluonen bzw. Quarks innerhalb des Protons gefaltet werden. Die Formel für den Proton-Proton-Wirkungsquerschnitt bei gegebenen Partonenwirkungsquerschnitten lautet wie folgt
$$\begin{aligned}
\sigma(p^+(k_1) p^-(k_2)\rightarrow Z(l_1)Z(l_2))\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ &&\nonumber\\
=\int_0^1 dx_2 \int_0^1 dx_1 \sum_f f_f(x_1) f_f(x_2)\sigma(q(x_1 k_1)\bar q(x_2 k_2)\rightarrow Z(l_1)Z(l_2)),&&\end{aligned}$$
(siehe [@PeskinSchroeder]) wobei die $f_f$ die Partonenverteilungsfunktionen darstellen. Die Summierung bezieht sich auf die verschiedenen Sorten von Partonen, also die verschiedenen Quarksorten sowie die Gluonen. Da im obigen Ausdruck über die Beiträge der verschiedenen Partonen summiert wird, ist der durch die Gluonen bedingte Wirkungsquerschnitt und damit der Wirkungsquerschnitt der durch den Gravitonenaustausch hinzukommt weiterhin unbhängig von dem der Quarks. Für die Differenz zum Standardmodellwirkungsquerschnitt gilt damit
$$\begin{aligned}
\Delta\sigma(p^+(k_1) p^-(k_2)\rightarrow Z(l_1)Z(l_2))\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ &&\nonumber\\
=\int_0^1 dx_2 \int_0^1 dx_1 f_g(x_1) f_g(x_2)\sigma(g(x_1 k_1) g(x_2 k_2)\rightarrow Z(l_1)Z(l_2)).&&\end{aligned}$$
### ZZ-Produktion im Standardmodell
Um nun einen Vergleich des zusätzlichen Wirkungsquerschnitts im ADD-Modell zu dem des Standardmodells herzustellen, muss analog der obigen Faltung über die Wirkungsquerschnitte für die Einzelprozesse innerhalb des Standardmodells gefaltet werden. Der Übergang von zwei Gluonen in zwei Z-Teilchen ist unmöglich, da Gluonen als Austauschteilchen der starken Wechselwirkung nur mit sich selbst also nicht schwach wechselwirken, worauf bereits hingewiesen wurde. Daher ergeben sich nur Wirkungsquerschnitte für den Übergang eines Quark-Antiquark-Paares in ein ZZ-Paar. Die folgenden Prozesse (Graphen \[Graph3\], \[Graph4\] und \[Graph5\]) liefern im Standardmodell in erster Ordnung einen Beitrag zur ZZ-Produktion
Die Graphen (\[Graph3\]),(\[Graph4\]) und (\[Graph5\]) beziehen sich jeweils auf ein Quark-Antiquark-Paar jeder Sorte. Der Prozess, welcher dem Graphen (\[Graph3\]) entspricht, ist aufgrund der großen Masse des Higgsteilchens zu vernachlässigen. Durch Auswertung dieser Feynmangraphen gelangt man zu den Wirkungsquerschnitten für das up-quark/charme-quark und das down-quark/strange-quark Da aufgrund des vorausgesetzten hohen Impulses die Massen der Quarks in der relativistischen Energie-Impuls-Beziehung vernachlässigbar sind, kann von folgender Relation $E^2 \approx p^2$ ausgegangen werden. Dies bedeutet, dass der Wirkungsquerschnitt des charme-Quarks dem des up-quarks und der des strange-quarks dem des down-quarks entspricht. Die Wirkungsquerschnitte für das bottom- und das top-Quark können aufgrund ihrer riesigen Masse und damit viel geringeren Produktionswahrscheinlichkeit innerhalb des Protons vernachlässigt werden. Da eine Raumzeit mit zwei oder weniger zusätzlichen Dimensionen im Rahmen des ADD Modells empirisch ausgeschlossen ist, wird hier von drei oder mehr Dimensionen ausgegangen. Außerdem wird im Allgemeinen angenommen, dass die modifizierte Planckmasse $M_D$ mindestens einen Wert von 1 TeV annimmt.
### Ergebnisse und Diskussion
Um den Wirkungsquerschnitt für den Proton-Proton Prozess zu erlangen, integriert man über die Partonenverteilungsfunktionen gemäßdem vorletzten Unterkapitel. Die Partonenverteilungsfunktionen (gegeben in [@CTEQ6]) werden für den Prozess des Graphen (\[Graph1\]) bei einer Skala von $Q=\sqrt{\hat s}$ und für die Prozesse der Graphen (\[Graph3\]),(\[Graph4\]) und (\[Graph5\]) bei einer Skala von $Q=m_Z$ ausgewertet.
In Graph (\[sig2\]) ist der totale Wirkungsquerchnitt bei der Fermilabenergie von $2000$ GeV als eine Funktion der fundamentalen Massenskala $M_D$ aufgetragen [@Giudice:1998ck]). Man sieht hier, dass die Gravitonenvermittlung gemäßder in Kapitel 3 beschriebenen Theorie bei $M_D > 2500$ GeV keinen beobachtbaren Einfluss auf die ZZ-Produktionsrate [@Acosta:2005pq] am Fermilab hat.
Die gleiche Analyse wird in Graph (\[sig14\]) für eine Proton-Proton-Reaktion bei einer Energie von $14000$ GeV gezeigt, wie sie am LHC verfügbar ist. Hier könnte die drastische Differez zwischen der ZZ-Bosonen Rate des Standardmodells und der gravitonenvermittelten ZZ-Bosonen Rate eine Beobachtung von Effekten großer zusätzlicher Dimensionen sogar für eine fundamentale Skala von $M_D\sim 18000$ GeV erlauben. Graph (\[sig14\]) zeigt, dass das ADD-Resultat bei $\sqrt{s}$=14000 GeV die Standardmodellvorhersage für kleines $M_D\ll \sqrt{s}$ übertrifft. Dies könnte Ausdruck der Tatsache sein, dass die Regularisierungsmethode und der Zugang der störungstheoretischen Quantenfeldtheorie in diesem Bereich ihre Gültigkeit verlieren. Deshalb ist es sinnvoll, die in den Graphen (\[sig2\]) und (\[sig14\]) aufgezeichneten Resultate nur nahe im Bereich ihrer Gültigkeit zu verwenden $\sqrt{s}\sim M_D$. Dies erlaubt eine Aussage darüber, ab welchem Wert für die modifizierte Planckmasse $M_D$ experimentelle Abweichungen von der ZZ-Produktionsrate des Standardmodells am LHC [@Ohnemus:1995gb] erwartet werden sollten. In Graph (\[Mdschnitt\]) ist der überprüfbare Parameterraum sowohl für den LHC alsauch das Tevatron aufgeführt.
Für die experimentelle Beobachtung müsste man nach zwei hochenergetischen und korrelierten Leptonenpaaren im Endzustand $Z\rightarrow l^+ l^-$ suchen. Indem man den totalen Wirkungsquerschnitt mit dem Verhältnis $\eta$ multipliziert, kann der Wirkungsquerschnitt abgeschätzt werden. Dieses wiederum kann erhalten werden, indem man das Verhältnis der Kopplungen in den Leptonen-Kanälen zu den Kopplungen in allen Fermionen-Kanälen betrachtet (es erscheint das Quadrat, weil beide Z-Bosonen zu einem di-leptonen Paar umgewandelt werden). $$\eta=\left(\frac{(\frac{1}{2}-\sin^2(\theta_W))^2+\sin^4(\theta_W)}{2-4\sin^2(\theta_W)+\frac{16}{3}\sin^4(\theta_W)}\right)^2
\approx 0.01 \quad,$$ wobei $\theta_W$ den Weinbergwinkel im Bereich der Z-Skala bezeichnet und desweiteren $\sin^2(\theta_W)\approx0.23$ gilt.
Es wurde also der zusätzliche Beitrag zur ZZ-Produktion durch Gravitonenvermittlung innerhalb des ADD Modells in Proton-Proton Reaktionen bei hohen Energien berechnet. Die Rechnung wurde in niedrigster Ordnung (in $\sqrt{\alpha_{ew,strong}}$ und dem Verhältnis $m_{X}/M_D$) Störungstheorie durchgeführt. Störungstheorie höherer Ordnung wäre aufgrund der Nichtrenormalisierbarkeit der effektiven Quantenfeldtheorie der Gravitation nicht durchführbar gewesen. Aber auch der hier erreichte Genauigkeitsgrad lässt sehr signifikante Aussagen zu. Es wurde gezeigt, dass der ZZ-Produktions-Wirkungsquerschnitt des Standardmodells im Vergleich zu diesem entscheidend erhöht würde, wenn die fundamentale Massenskala des ADD-Modells kleiner als $15000$ GeV im Falle des LHC, beziehungsweise $1700$ GeV im Falle des Tevatron wäre. Für den Fall von sieben Extra Dimensionen könnte sogar $M_D=18000$ GeV am LHC getested werden.
In Betracht dieser Ergebnisse ist es wichtig, den Leser daran zu erinnern, dass die Größe des Gravitonenbeitrages des ADD Modells (\[ADDWirkungsquerschnitt\]) direkt von der gewählten Regulasrisierungsskala $\Lambda$ in Gleichung (\[KKSummierung\]) abhängt. Aber eine solche Wahl $\Lambda=M_D$ erscheint natürlich, da $M_D$ ja das analogon zur Planckmasse darstellt, welche wiederum allgemein als eine absolute Gültigkeitsgrenze effektiver Quantenfeldtheorien angesehen wird. Wenn also eine Vergrößerung des Wirkungsquerschnittes der ZZ-Rate im Standardmodell bei LHC-Energien beobachtet würde, könnte dies wichtige Einsichten in die möglicherweise höherdimensionale Struktur der Raumzeit liefern.
Eichtheorien und Higgsmechanismus
=================================
Die Idee von Symmetrieprinzipien
--------------------------------
> [“>Am Anfang war die Symmetrie<, das ist sicher richtiger als die Demokritsche Behauptung >Am Anfang war das Teilchen<. Die Elementarteilchen verkörpern die Symmetrien, sie sind ihre einfachsten Darstellungen, aber sie sind erst eine Folge der Symmetrien.” (Werner Heisenberg)]{}
Grundsätzlich kann man die fundamentalen Teilchen des Standardmodells in zwei Klassen einteilen, in die Klasse der eigentlichen Materieteilchen und die der Austauschteilchen der Wechselwirkungsfelder. Alle Teilchen der ersten Klasse sind Fermionen und alle der zweiten Klasse Bosonen. Fermionen unterscheiden sich dadurch von Bosonen, dass sie einen halbzahligen Spin besitzen, während der Spin letzterer ganzzahlig ist. Dies bedeutet, dass Fermionen im Gegensatz zu Bosonen dem Paulischen Ausschließungsprinzip gehorchen, das besagt, dass sich zwei Elementarteilchen mit halbzahligem Spin niemals im gleichen Zustand befinden dürfen. Gemäß dem von Pauli bewiesenen Spin-Statistik-Theorem gehorchen Fermionen daher einer anderen Statistik als Bosonen.
Die unterschiedlichen Spins der einzelnen Elementarteilchen spiegeln sich mathematisch in einer verschiedenen Struktur der sie beschreibenden Felder wider. Nun müssen aber alle diese Felder, da sie nun einmal auf der Raumzeit leben, invariant unter der Symmetriegruppe der speziellen Relativitätstheorie sein, also der Poincarégruppe. Die Transformationen der Poincarégruppe müssen also auf dem entsprechenden mathematischen Raum dargestellt werden, der die Spinstruktur des Teilchens beschreibt.
In Rahmen relativistischer Quantenfeldtheorien beschreibt man Elementarteilchen bzw. die ihnen entsprechenden Felder daher als irreduzible Darstellungen der Poincarégruppe. Neben der Spinstruktur spielt hier natürlich auch noch die Frage eine Rolle, ob es sich um massebehaftete oder Teilchen ohne Ruhemasse handelt. Das Quadrat der Ruhemasse des Teilchens ist neben dem Spin Casimiroperator der Poincarégruppe und stellt damit wie der Spin eine lorentzinvariante Eigenschaft des Teilchens selbst dar. Die Struktur der Poincarégruppe wird durch die folgende Liealgebra beschrieben
$$\begin{aligned}
i[J^{\mu\nu},J^{\rho\sigma}]&=&\eta^{\nu\rho}J^{\mu\sigma}-\eta^{\mu\rho}J^{\nu\sigma}
-\eta^{\sigma\mu}J^{\rho\nu}+\eta^{\sigma\nu}J^{\rho\mu},\nonumber \\
i[P^{\mu}, J^{\rho\sigma}]&=&\eta^{\mu\rho} P^{\sigma}-\eta^{\rho\sigma} P^{\rho},\nonumber\\
i[P^{\mu},P^{\sigma}]&=& 0.\end{aligned}$$
Die $J^{\mu\nu}$-Generatoren beschreiben die homogene Lorentzgruppe. Diejenigen Generatoren, deren Indizes Werte zwischen 1 und 3 annehmen, entsprechen den Drehimpulsoperatoren, also den Generatoren der Drehungen in den drei Raumrichtungen, wohingegen die Generatoren mit einer 0 als Index die Lorentzboosts in die drei Raumrichtungen darstellen. Aufgrund der Antisymmetrie verschwinden die Elemente mit zwei gleichen Indizes. Insgesamt gibt es also sechs unabhängige Symmetrietransformationen innerhalb der homogenen Lorentzgruppe. Hinzu kommen die vier Raumzeittranslationen, welche durch die Impulsoperatoren $P^{k}$ und den Hamiltonoperator $P^0$ beschrieben werden und untereinander kommutieren. Insgesamt ergibt sich also eine zehndimensionale Gruppe.
Jede Sorte Elementarteilchen gehorcht einer bestimmten Feldgleichung, die über das Hamiltonsche Wirkungsprinzip mit einer Lagrangedichte verbunden werden kann. Die Dynamik jedes freien Feldes, das einem bestimmten Teilchen entspricht, kann also mit Hilfe der entsprechenden Feldgleichung beschrieben werden. Nun beruht aber das reale physikalische Geschehen im Wesentlichen auf Wechselwirkung zwischen verschiedenen Objekten. Es drängt sich also die Frage auf, wie die Kopplung zwischen den verschiedenen Materieteilchen bzw. Feldern und den Wechselwirkungsfeldern beschrieben werden kann. Eine der faszinierendsten Tatsachen der modernen Physik ist, dass man die Wechselwirkung der Felder untereinander auch auf Symmetrieprinzipien zurückführen kann. Hierzu muss zunächst einmal erwähnt werden, dass Felder neben der äußeren raumzeitlichen Struktur und dem Spin als irreduzible Darstellungen der Poincarégruppe auch noch zusätzliche innere Strukturen aufweisen. Die Idee der zusätzlichen inneren Symmetrien geht auf Heisenberg zurück, der im Jahr 1932 den Isospin einführte. Der Unterschied zwischen dem Proton und dem Neutron stellt sich hier in den beiden unterschiedlichen Einstellungen des Isospins dar. In Bezug auf die starke Wechselwirkung verhalten sich die beiden Teilchen vollkommen gleich. Die Symmetrie der Naturgesetze bezüglich des Isospins wird also durch die unterschiedliche Ladung und die leicht unterschiedlichen Massen gebrochen.
Eine ähnliche Idee liegt der Beschreibung der Wechselwirkung von Teilchen im Rahmen von Eichtheorien zu Grunde. Man betrachtet zunächst ein Teilchen, dass bestimmte zusätzliche Eigenschaften aufweist. Diese Klasse von Eigenschaften beschreibt man dann formal durch Zuordnung einer neuen Quantenzahl, welche die Beschreibung durch einen neuen inneren Raum zur Folge hat, der natürlich gemäß den Postulaten der Quantentheorie eine Hilbertraumstruktur aufweisen muss. Da aber nur innere Produkte zwischen Zuständen und nicht Zustände an sich physikalisch relevant sind, muss die Theorie invariant unter der bezüglich dieses Raumes fundamentalen unitären Transformationsgruppe sein. Mit einer Symmetrietransformation bezüglich des inneren Raumes, die einer solchen Transformationsgruppe angehört, meint man zunächst eine Transformation, die keine Raumzeitabhängigkeit aufweist und die man daher als global bezeichnet. Ein im Hinblick auf Eichtheorien relevantes Beispiel eines zusätzlichen inneren Freiheitsgrades liefert die Farbladung der Quarks.
In Rahmen von Eichfeldtheorien fordert man nun die Invarianz nicht nur unter globalen Symmetrien, sondern unter inneren Symmetrietransformationen, die vom speziellen Raumzeitpunkt des Feldes abhängen. Die freien Materiefeldgleichungen erfüllen diese Forderung zunächst nicht. Durch Einführung einer kovarianten Ableitung, welche einen Kopplungsterm in der Lagrangedichte zur Folge hat, kann man jedoch Invarianz auch unter lokalen Symmetrien gewährleisten. Die Zustände der Felder leben mathematisch in einem Raum, der dem Tensorprodukt des Hilbertraumes der quadratintegrablen Funktionen mit dem Spinraum und dem entsprechenden Raum der zusätzlichen inneren Symmetrien entspricht. Wenn man also eine Symmetrietransformation betrachtet, die an jedem Raumzeitpunkt anders wirkt, betrachtet man in gewisser Weise an jedem Raumzeitpunkt einen separaten inneren Raum gleicher Struktur. Man hat es also nicht mehr mit einem globalen Tensorprodukt zu tun, sondern es muss eine mathematische Struktur geben, welche diese einzelnen Räume miteinander verbindet. Es handelt sich hierbei um die gleiche formale Beschreibungsweise wie im Falle der Allgemeinen Relativitätstheorie. Die in der kovarianten Ableitung auftauchenden Zusammenhangkoeffizienten bestimmen, wie zwei Elemente der entsprechenden Räume verglichen werden müssen. Der Unterschied zur Allgemeinen Relativitätstheorie besteht darin, dass es dort die Tangentialräume an die einzelnen Raumzeitpunkte, also Verschiebungsvektoren auf der Raumzeit sind, die miteinander verglichen werden. Im Falle von Eichtheorien sind es die Elemente der inneren Räume. Im folgenden soll es nun um die konkrete Beschreibungsweise von Eichtheorien gehen, welche von Yang und Mills in die Teilchenphysik eingeführt wurden und beispielsweise in [@Ramond], [@WeinbergQTF2] und [@Pokorski] dargestellt ist.
Das Prinzip lokaler Eichinvarianz
---------------------------------
### Innere Symmetrien einer Lagrangedichte
$\Psi$ sei ein Materiefeld, dass einer freien Materiefeldgleichung gehorchen möge. Da alle fundamentalen Materiefelder außer dem Higgsfeld, auf das später noch zu sprechen zu kommen sein wird, Fermionen sind, ist dies die Diracgleichung. Desweiteren enthalte $\Psi$ einen zusätzlichen inneren Freiheitsgrad und soll damit also n Komponenten beschreiben, die ihrerseits jeweils wieder einen Diracspinor darstellen. Die entsprechende Lagrangedichte lautet damit
$$\mathcal{L}=\bar \Psi ^n (i\gamma^\mu \partial_\mu-m) \Psi _n,$$
wobei n die Komponenten des inneren Freiheitsgrades beschreiben soll. Wenn man $\Psi$ nun einer Transformation der unitären Symmetriegruppe des inneren Raumes aussetzt, also der SU(N)
$$\Psi(x)\quad\rightarrow\quad U \Psi(x)\quad,\quad \Psi^{\dagger}(x) \quad
\rightarrow \quad \Psi^{\dagger}(x) U^{\dagger},$$
so heben sich die beiden Operatoren U und $U^{\dagger}$ gegenseitig auf, denn der Operator U wirkt auf einen anderen Raum als die Gammamatrizen und der Ableitungsoperator und kommutiert daher mit diesen. Damit ist die Lagrangedichte invariant unter dieser Transformation. Der tiefere Grund hierfür ist, wie bereits im letzten Abschnitt erwähnt, die Tatsache, dass nur innere Produkte zwischen Zuständen physikalisch von Bedeutung sind, und diese sind unter unitären Transformationen invariant. Die unitären Operatoren haben folgende Gestalt
$$U=e^{i\omega_a T^{a}},$$
wobei die $T^{a}$ die Generatoren der Gruppe darstellen, welche hermitesch sind und eine Liealgebra bilden. Diese wiederum ist durch die Vertauschungsrelationen gegeben
$$[T^a,T^b]=if_{c}^{ab} T^c.$$
Hierbei beschreiben die $f_c^{ab}$ die sogenannten Strukturkonstanten der Gruppe.
### Übergang zu lokalen Symmetrien und Wechselwirkung
Wenn der Transformationsparameter $\omega$ aber jetzt nicht mehr konstant ist, sondern vom Raumzeitpunkt abhängt, was zu einer Transformation der folgenden Gestalt führt
$$\Psi \quad \rightarrow \quad U(x) \Psi,$$
so gilt für die Transformation der in der Lagrangedichte auftauchenden Ableitung des Feldes $\partial_\mu \Psi$
$$\partial_\mu \Psi(x) \rightarrow \partial_\mu U(x) \Psi(x)=[\partial_\mu U(x)] \Psi(x)+U(x)\partial_\mu \Psi(x) \neq
U(x) \partial_\mu \Psi(x).$$
Dies hat zur Folge, dass die Lagrangedichte nicht mehr invariant unter der nun lokalen unitären Transformation ist. Um die Invarianz auch unter lokalen Transformationen $U(x)$ zu gewährleisten, muss die Ableitung $\partial_\mu$ durch eine kovariante Ableitung $D_\mu$ ersetzt werden, die folgende Transformationseigenschaft erfüllt
$$D_\mu \quad \rightarrow \quad U(x) D_\mu U^{\dagger}(x).
\label{TranskovAbleitung}$$
Wenn diese Transformationseigenschaft erfüllt ist, gilt nämlich
$$D_\mu \Psi(x) \quad \rightarrow \quad U(x) D_\mu U^{\dagger}(x) U(x) \Psi(x)=U(x) D_\mu \Psi(x).$$
Damit ist die Lagrangedichte
$$\mathcal{L}=\bar \Psi^n(i\gamma^\mu D_\mu-m)\Psi_n
\label{LagrangekovAbl}$$
invariant unter den entsprechenden lokalen Symmetrietransformationen. Es muss also eine entsprechende kovariante Ableitung gebildet werden, welche die Bedingung ($\ref{TranskovAbleitung}$) erfüllt. Hierzu sei folgender Ansatz gewält
$$D_\mu=\partial_\mu+iA_\mu(x),
\label{kovAbleitung}$$
wobei $A_\mu(x)$ ein Vektorfeld ist, dass später mit dem Wechselwirkungsfeld identifiziert wird und Werte in der Liealgebra der Symmetriegruppe annimmt. Es kann daher als Linearkombination der Generatoren der Gruppe ausgedrückt werden
$$A_\mu(x)=A_\mu^a(x) T^a.$$
Einsetzen des Ausdruckes für die kovariante Ableitung ($\ref{kovAbleitung}$) in (\[TranskovAbleitung\]) ergibt
$$\partial_\mu+iA_\mu \quad\rightarrow\quad U(x)(\partial_\mu+iA_\mu)U^{\dagger}(x).$$
Dies kann umgeformt werden zu
$$A_\mu \quad \rightarrow \quad -iU(x)[\partial_\mu U(x)^{\dagger}]
+U(x) A_\mu U(x)^{\dagger}.
\label{TransVektorfeld}$$
Es ist also eine Transformationsbedingung für $A_\mu$ gefunden. Um dies weiter umzuformen betrachten wir nun infinitesimale Transformationen. Unter dieser Voraussetzung können wir die Entwicklung des Transformationsoperators $U(x)$ in erster Ordnung verwenden
$$U(x)=1+i\omega^a T^a+\mathcal{O}(\omega^2).$$
Einsetzen in die Transformationsbedingung für das Vektorfeld $A_\mu$ ergibt
$$A_\mu \quad \rightarrow \quad A_\mu-\partial_\mu \omega^a T^a+i[\omega^a T^a, A_\mu]+\mathcal{O}(\omega^2).$$
Die Lagrangedichte ($\ref{LagrangekovAbl}$) mit der kovarianten Ableitung ($\ref{kovAbleitung}$) wird also unter einer infinitesimalen Transformation der Form
$$\begin{aligned}
\delta \Psi &=& i \omega^a T^a \Psi\nonumber,\\
\delta A_\mu &=& -\partial_\mu \omega^a T^a+i[\omega^a T^a, A_\mu]\end{aligned}$$
invariant gelassen. Der erste Teil der Transformation des Vektorfeldes entspricht also der Form nach einer Eichtransformation des elektromagnetischen Feldes. Hinzu kommt jedoch, da das Vektorfeld liealgebrawertig ist und die Generatoren nicht kommutieren, der Kommutator des Eichparameters mit dem Vektorfeld. Es ist nun aufgrund der Transformationseigenschaften sinnvoll, das Vektorfeld in der kovarianten Ableitung mit einem neuen Wechselwirkungsfeld zu identifizieren, dessen Existenz so gewissermaßen aus der Forderung nach lokaler Eichinvarianz unter einer gewissen Symmetriegruppe hergeleitet wurde. Die Nichtkommutativität der Generatoren der Eichgruppe und damit der entsprechenden Felder entspricht hierbei ihrer Selbstwechselwirkung. Natürlich ist dies eigentlich so nicht ganz richtig, denn bisher wurde eigentlich nichts anderes getan, als an jedem Raumzeitpunkt ein anderes Koordinatensystem bezüglich des inneren Raumes eingeführt. Die kovariante Ableitung enthält die Information, wie die Komponenten des inneren Raumes an verschiedenen Raumzeitpunkten miteinander verglichen werden müssen. Dies hat aber zunächst eine rein mathematische Bedeutung. Um der kovarianten Ableitung $D_\mu$ und dem mit ihr eingeführten Feld $A_\mu$ physikalische Signifikanz zu verleihen, bedarf es einer zusätzlichen Annahme, die analog dem Äquivalenzprinzip in der Allgemeinen Relativitätstheorie ist. Auf die mit dieser Frage verbundenen tiefgründigen philosophischen Probleme, denen eine herausragende Bedeutung in der zeitgenössischen Naturphilosophie zukommt, kann an dieser Stelle nicht näher eingegangen werden. Hierzu sei auf [@Lyre] verwiesen.
### Die Dynamik des Wechselwirkungsfeldes
Was bisher beschrieben wurde, ist die Art und Weise wie ein zunächst freies Materiefeld an das jeweilige Wechselwirkungsfeld koppelt. In der neuen Lagrangedichte erscheint aber bisher noch kein Term, der die innere Dynamik des Wechselwirkungsfeldes selbst beschreibt. Ein solcher Term muss natürlich auch die Forderung nach lokaler Eichinvarianz erfüllen. Zunächst soll ein Feldstärketensor $F_{\mu\nu}$ wie folgt definiert werden
$$F_{\mu\nu}=-i[D_\mu, D_\nu].
\label{Feldstaerketensor}$$
Ein solcher Ausdruck muss eichinvariant sein, da die in ihm enthaltenen kovarianten Ableitungen eichinvariant sind. Durch Einsetzen des Ausdruckes ($\ref{kovAbleitung}$) in ($\ref{Feldstaerketensor}$) erhält man
$$F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu+i[A_\mu, A_\nu].$$
Es handelt sich also um eine Verallgemeinerung des elektromagnetischen Feldstärketensors. Im Spezialfall kommutierender Felder $A_\mu$ geht er in diesen über. Da der Feldstärketensor ebenso wie das Vektorfeld $A_\mu$ liealgebrawertig ist, kann er ebenso nach den Generatoren der Eichgruppe entwickelt werden
$$F_{\mu\nu}=F^a_{\mu\nu}T^a,$$
wobei für die einzelnen Komponenten $F^a_{\mu\nu}$ gilt
$$F^a_{\mu\nu}=\partial_\mu A^a_\nu-\partial_\nu A^a_\mu-f^{abc} A^b_\mu A^c_\nu.$$
Um nun den entsprechenden Term in der Lagrangedichte zu erhalten, muss ein skalarer Ausdruck aus dem Feldstärketensor konstruiert werden. Nur eine Lagrangedichte der folgenden Form ist mit Lorentzinvarianz und Paritätserhaltung verträglich
$$\mathcal{L}=-\frac{1}{4} g_{ab} F^a_{\mu\nu}F^{b \mu\nu},$$
wobei durch Definition einer geeigneten Skala grundsätzlich erreicht werden kann, dass $g_{ab}=\delta_{ab}$. Damit nimmt die Gesamtlagrangedichte die folgende Gestalt an
$$\mathcal{L}=\bar \Psi^n(i\gamma^\mu D_\mu-m)\Psi_n-\frac{1}{4} F^a_{\mu\nu} F^{a \mu\nu}.$$
Es ist nun wichtig, dass Eichinvarianz nur dann gewährleistet werden kann, wenn die entsprechenden Vektorbosonen des Wechselwirkungsfeldes $A_\mu$ keine Masse besitzen. Ein zusätzlicher Term etwa der Form $m^2 A_\mu A^\mu$ in der Lagrangedichte würde die Eichinvarianz aufheben.
Elektroschwache Vereinheitlichung und Higgsmechanismus
------------------------------------------------------
Alle bekannten in der Natur vorkommenden Wechselwirkungen können durch Eichtheorien beschrieben werden. Im Falle des Elektromagnetismus ist es die $U(1)$-Gruppe, bei der schwachen Wechselwirkung die Symmetriegruppe des schwachen Isospins $SU(2)$, bei der starken Wechselwirkung die $SU(3)$ colour und bei der Gravitation schließlich die Lorentzgruppe (siehe Kapitel 3), welche als Eichgruppen zu Grunde gelegt werden. Nun ist es aber eine empirische Tatsache, dass die Austauschteilchen der schwachen Wechselwirkung eine Masse aufweisen. Dies steht jedoch im Widerspruch zu der erwähnten Eigenschaft, dass Massenterme die Eichinvarianz brechen. Eine Lösung dieses Problems liefert die Generierung der Massen durch eine sogenannte spontane Symmetriebrechung, die auch als Higgsmechanismus bezeichnet wird, benannt nach dem schottischen Physiker Peter Higgs [@Higgs:1964pj].
### Die Elektroschwache Theorie
Im folgenden soll nun der Higgsmechanismus im Rahmen der Eichtheorie, welche die elektromagnetische und die schwache Wechselwirkung in einem einheitlichen Schema beschreibt, betrachtet werden. Sie wurde von Glashow [@Glashow:1961tr], Weinberg [@Weinberg:1967tq] und Salam [@Salam:1968rm] entwickelt und ist beispielsweise in [@WeinbergQTF2],[@Greiner] und [@Quigg] zu finden. Als Eichgruppe wird hier die Gruppe $SU(2)$ $\times$ $U(1)$ zu Grunde gelegt, wobei sich die $SU(2)$-Gruppe auf den schwachen Isospin, also beispielsweise das Dublett bestehend aus Elektron und Elektronneutrino bezieht. Da Neutrinos nur linkshändig vorkommen und der rechtshändige Teil des entsprechenden Diracspinors demnach bezüglich des schwachen Isospins ein Singlett darstellt, muss in diesem Falle die $SU(2)$ genau genommen auf die linkshändige Komponente des Spinors eingeschränkt werden. Die gesamte Gruppe besitzt insgesamt vier Transformationsfreiheitsgrade, die den drei Generatoren der $SU(2)_L$ und der Phasentransformation der $U(1)$ entsprechen. Eine allgemeine Symmetrietransformation angewandt auf einen Zustand, welcher ein Dublett aus Elektron und Elektronneutrino beschreibt, hat also folgende Gestalt
$$\Psi\quad\rightarrow\quad exp\left(i\left[\omega^a T^a+\omega^y Y \right]\right) \Psi,$$
wobei die drei Generatoren $T^a$ der $SU(2)_L$ und der Generator $Y$ der $U(1)$, auch als schwache Hyperladung bezeichnet, wie folgt definiert sind
$$\begin{aligned}
T^a&=&g\left[\frac{1}{2}\left(\frac{1+\gamma_5}{2}\right)
\sigma^a\right] \\
Y&=&g^{'}\left[\frac{1}{2}\left(\frac{1+\gamma_5}{2}\right)
{\bf 1}+\left(\frac{1-\gamma_5}{2}\right)\right].\end{aligned}$$
Hierbei ist ${\bf 1}$ die Einheitsmatrix in zwei Dimensionen und die $\sigma^a$ bezeichnen die Paulischen Spinmatrizen
$${\bf 1}=\left(\begin{array}{cc} 1&0\\0&1\end{array}\right),
\sigma^1=\left(\begin{array}{cc} 0&1\\1&0\end{array}\right),
\sigma^2=\left(\begin{array}{cc} 0&-i\\i&0\end{array}\right),
\sigma^3=\left(\begin{array}{cc} 1&0\\0&-1\end{array}\right).$$
Die Operatoren $\frac{1+\gamma_5}{2}$ und $\frac{1-\gamma_5}{2}$ stellen die Projektion auf die links- bzw. rechtshändige Komponente des Diracspinors dar. Der elektrische Ladungsoperator ist eine Linearkombination des Transformationsoperators der $U(1)$, nämlich Y, und der dritten Komponente der Transformation im schwachen Isospinraum $T^3$
$$Q=\frac{e}{g}T^3-\frac{e}{g^{'}}Y.
\label{Ladung-Hyperladung}$$
Damit ist ein reiner Elektronzustand im schwachen Isospinraum Eigenzustand zur elektrischen Ladung, worin sich widerspiegelt, dass nur die Elektronen, nicht aber die Neutrinos eine Ladung tragen. Nun werden die neuen Felder $W^\mu, W^{\mu\dagger}, Z^\mu$ und $A^\mu$ eingeführt, die wie folgt definiert sind
$$\begin{aligned}
W^\mu&=&\frac{1}{\sqrt{2}}(\mathcal{A}_1^\mu+i\mathcal{A}_2^\mu)
\label{DefinitionW+Feld}\\
W^{\mu\dagger}&=&\frac{1}{\sqrt{2}}(\mathcal{A}_1^\mu-i\mathcal{A}_2^\mu),
\label{DefinitionW-Feld}\end{aligned}$$
die dem $W^+$-und dem $W^-$-Teilchen der schwachen Wechselwirkung entsprechen, sowie
$$\begin{aligned}
Z^\mu&=&cos(\theta_W)\mathcal{A}_3^\mu+sin(\theta_W)\mathcal{B}^\mu
\label{DefinitionZ-Feld}\\
A^\mu&=&-sin(\theta_W)\mathcal{A}_3^\mu+cos(\theta_W)\mathcal{B}^\mu.\end{aligned}$$
Hier beschreibt $\theta$ den sogenannten Weinbergwinkel, den Mischungswinkel zwischen dem elektromagnetischen Feld $A^\mu$, das dem Photon und dem Feld, das dem Z-Teilchen der schwachen Wechselwirkung entspricht. Der Weinbergwinkel ist auf folgende Weise mit den Parametern $g$ und $g^{'}$ verknüpft, welche das Verhältnis der $SU(2)_L$- zu den $U(1)$ Transformationen bestimmen
$$g=-\frac{e}{sin{\theta_W}}\quad,\quad g^{'}=-\frac{e}{cos{\theta_W}}.$$
### Generierung der Massen der Eichbosonen
In der bisherigen Beschreibung sind die verschiedenen Wechselwirkungsfelder zwangsläufig noch masselos geblieben, da wie bereits erwähnt die Forderung der Eichinvarianz keine Massenterme zulässt. Die Austauschteilchen der schwachen Wechselwirkung müssen ihre Massen also gewissermaßen auf indirektem Wege erhalten. Aus diesem Grunde postuliert man ein skalares Hintergrundfeld, das Higgsfeld, dessen Selbstwechselwirkung durch einen zusätzlichen Potentialterm beschrieben wird. Das Higgsfeld stellt ebenfalls ein Dublett bezüglich des schwachen Isospins dar, wobei vorausgesetzt wird, dass der obere Zustand Eigenzustand zum Ladungsoperator mit positiver Ladung ist $$\Phi=\left(\begin{array}{c}\Phi^{+}\\ \Phi^0 \end{array}\right).$$ Der Grund hat damit zu tun, dass auch das Elektron seine Masse durch das Higgsfeld durch eine Yukawakopplung erhalten soll. Da im Falle des Higgsfeldes beide Komponenten sowohl rechts- alsauch linkshändig vorkommen sollen, tauchen in den Generatoren der Transformationen keine Projektionsoperatoren auf die rechts- bzw. linkshändige Komponente auf. Unter Berücksichtigung des Zusammenhangs zwischen Ladung und Hyperladung ($\ref{Ladung-Hyperladung}$) und der Gestalt des Ladungsoperators des Higgsfeldes $$q=-\frac{g^{'}}{2}\left(\begin{array}{cc}1&0\\0&1\end{array}\right)$$ ergeben sich damit folgende Generatoren für die Transformation des Higgsfeldes
$$\begin{aligned}
\bar T^a=\frac{g}{2}\sigma^a,\\
\bar Y=-\frac{g^{'}}{2}{\bf 1}.
\label{GeneratorenHiggsFeld}\end{aligned}$$
Die entsprechende eichinvariante Lagrangedichte lautet
$$\mathcal{L}_{Higgs}=\frac{1}{2}(D_\mu \Phi)(D^\mu \Phi)^\dagger+
\frac{\mu^2}{2}\Phi^\dagger\Phi-\frac{\lambda}{4}(\Phi^\dagger \Phi)^2,$$
wobei für die kovariante Ableitung $D_\mu$ gilt
$$D_\mu=\partial_{\mu}+i\mathcal{A}_\mu^a T^a+i\mathcal{B}_\mu Y.$$
Nun besitzt das skalare Feld bei dieser Lagrangedichte allerdings einen von 0 verschiedenen Vakuumerwartungswert $\nu$, der dem Minimum des Potentialterms in der Higgslagrangedichte $\frac{\mu^2}{2}\Phi^\dagger\Phi-\frac{\lambda}{4}(\Phi^\dagger \Phi)^2$ entspricht. Für das Betragsquadrat gilt
$$\nu^2= \langle \Phi \rangle ^{\dagger} \langle \Phi \rangle = \frac{\mu^2}{\lambda}.$$
Es ist grundsätzlich möglich, eine entsprechende $SU(2) \times U(1)$ Eichtransformation durchzuführen, welche die obere Komponente des Higgsfeldes zum verschwinden bringt und die untere reell macht. Eine solche Eichung wird als unitäre Eichung bezeichnet. Damit gilt für die Vakuumerwartungswerte
$$\langle \Phi^+ \rangle=0\quad,\quad \langle \Phi^0 \rangle=\nu.$$
Wenn man nun das Feld $\Phi$ um den Vakuumerwartungswert entwickelt
$$\Phi=\left(\begin{array}{c}0\\ \nu+\phi \end{array}\right),$$
so taucht im Kopplungsterm der Wechselwirkungsfelder an das Higgsfeld der Ausdruck
$$\left(i\left[ (\mathcal{A}_\mu^a \bar T^a+\mathcal{B}_\mu \bar Y)\left(\begin{array}{c}0\\v \end{array}\right)\right]\right)
\left(i\left[ (\mathcal{A}_\mu^a \bar T^a+\mathcal{B}_\mu \bar Y)\left(\begin{array}{c}0\\v \end{array}\right)\right]\right)^{\dagger}$$
auf. Dieser kann unter Verwendung von ($\ref{GeneratorenHiggsFeld}$) wie folgt umgeschrieben werden
$$\left|\frac{g}{2}A_\mu^1\left(\begin{array}{c}\nu\\0\end{array}\right)
-i\frac{g}{2}A_\mu^2\left(\begin{array}{c}\nu\\0\end{array}\right)
-\frac{g}{2}A_\mu^3\left(\begin{array}{c}0\\\nu\end{array}\right)
-\frac{g^{'}}{2}YB_\mu^1\left(\begin{array}{c}0\\\nu\end{array}\right)\right|^2.
\label{KopplungVakuum}$$
Wenn man nun ($\ref{DefinitionW+Feld}$) und ($\ref{DefinitionW-Feld}$) in Erinnerung ruft, sowie ($\ref{DefinitionZ-Feld}$) nach $\mathcal{B}$ umstellt und in obige Gleichung ($\ref{KopplungVakuum}$) einsetzt, so heben sich die $\mathcal{A}_\mu^3$-Terme auf und man erhält
$$\left|\frac{g}{\sqrt{2}} W_\mu^\dagger
\left(\begin{array}{c}\nu\\0\end{array}\right)
+\frac{g g^{'}}{e} Z_\mu
\left(\begin{array}{c}0\\\nu\end{array}\right)\right|^2
=\frac{\nu^2 g^2}{4}W_\mu^\dagger W^\mu
+\frac{\nu^2 (g^2+g^{'2})}{8} Z_\mu Z^\mu.
\label{KopplungVakuumEndausdruck}$$
Diese haben die Gestalt von Massentermen für die W-Felder und das Z-Feld, welche so also eine Masse erhalten haben, wobei man unter Berücksichtigung der Tatsache, dass die Massenterme von Skalar- und Vektorfeldern immer das Massenquadrat enthalten, für die Massen der W- und Z-Teilchen aus obigem Ausdruck folgende Werte herausliest
$$m_W=\frac{\nu g}{2}\quad,\quad m_Z=\frac{\nu\sqrt{g^2+g{'}^2}}{2}.$$
Da kein Kopplungsterm für das Photonenfeld $A_\mu$ an den Vakuumerwartungswert des Higgsfeldes auftaucht, bleiben diese masselos. Die Eigenschaft der Masse stellt sich also als Wechselwirkung mit einem skalaren Hintergrundfeld dar. Dies bedeutet in gewisser Weise die Rückführung des Begriffes der Masse auf den der Wechselwirkung.
Das Grundprinzip der nichtkommutativen Geometrie
================================================
Die Einführung einer neuen Algebra für Ort und Impuls
-----------------------------------------------------
Beim Übergang von einer klassischen Theorie zu der entsprechenden Quantentheorie ersetzt man die klassischen Größen durch hermitesche Operatoren, die man bestimmten Vertauschungsrelationen unterwirft und deren Eigenwerte die möglichen Messwerte der entsprechenden Größen darstellen. Im speziellen Fall des Übergangs von der klassischen Punktteilchenmechanik zur Quantenmechanik, dem historisch ersten Beispiel einer Quantisierung, werden an die Operatoren, die den Ort x und den Impuls p des Teilchens beschreiben sollen, die folgenden Vertauschungsrelationen gefordert
$$[\hat x^{i},\hat x_{j}]_-=0\quad,\quad [\hat p^{i},\hat p_{j}]_-=0\quad,\quad [\hat x^{i},\hat p_{j}]_-=\delta^{i}_{j}.
\label{Vertauschungsrelationen}$$
### Vertauschungsrelationen zwischen den Koordinaten
Die Idee der nichtkommutativen Geometrie geht nun davon aus, die Vertauschungsrelation ($\ref{Vertauschungsrelationen}$) dahingehend abzuändern, dass der Kommutator der Raumzeitkoordinaten nicht mehr verschwindet. Man fordert also eine neue Vertauschungsrelation zwischen den Ortskoordianten
$$[\hat x^{i},\hat x^{j}]_-\neq 0.$$
Eine solche Vertauschungsrelationen lässt jedoch die Vertauschungsrelationen zwischen den Ortskoordinaten und den Ableitungen nach den Ortskoordinaten und damit auch den Impulsoperatoren unberührt. Die Auswirkung auf die Vertauschungsrelation der Ableitungsoperatoren untereinander ist zunächst variabel. Die Idee, einen nichtverschwindenden Kommutator der Ortskoordinaten einzuführen, geht auf [@Snyder:1946qz] zurück. In neuerer Zeit wurde durch Seiberg und Witten gezeigt, dass sich eine Nichtkommutativität als Konsequenz aus Stringtheorien ergibt [@Seiberg:1999vs]. Die Anwesenheit eines magnetischen Hintergrundfeldes kann dafür sorgen, dass sich normale Feldtheorien effektiv so verhalten, als läge eine nichtkommutative Geometrie vor [@Gorbar:2004ck]. Es gibt verschiedene Formen der Nichtkommutativität der Ortskoordinaten. Eine Möglichkeit ist, dass sie eine Liealgebra bilden und die Vertauschungsrelation folgende Gestalt hat
$$[\hat x^{i},\hat x^{j}]_-=i\lambda^{ij}_k \hat x^k,$$
wobei die $\lambda^{ij}_k$ die Strukturkonstanten der Gruppe darstellen. Dem Kommutator zweier Ortskoordinaten wird also selbst wieder eine Ortskoordinate zugeordnet. Eine solche Struktur ist analog den Liealgebren der Generatoren der Eichgruppen. Eine weitere Möglichkeit ist die Zuordnung eines bezüglich der Koordinaten quadratischen Ausdrucks der folgenden Form
$$[\hat x^{i},\hat x^{j}]_-=\left(\frac{1}{q}\hat R^{ij}_{kl}
-\delta^i_l \delta^j_k\right)\hat x^k \hat x^l.$$
Im kanonischen Fall wird der Kommutator gleich einem antisymmetrischen Tensor $i\theta^{ij}$ gesetzt, sodass sich die folgende Vertauschungsrelation ergibt
$$[\hat x^{i},\hat x^{j}]_-=i\theta^{ij}.
\label{Koordinatenvertauschungsrelation}$$
In den folgenden Betrachtungen soll grundsätzlich eine kanonische Vertauschungsrelation zu Grunde gelegt werden. Desweiteren soll davon ausgegangen werden, dass das in diesem Falle auftretende $\theta^{ij}$ konstant ist, also nicht von den Koordinaten abhängt.
### Implikation für die Ableitungsoperatoren
Es stellt sich nun noch die Frage nach den Vertauschungsrelationen der Ableitungsoperatoren. Die Größe $\hat x^i-i\theta^{ij}\hat \partial_j$ kommutiert mit den Ortskoordinaten, denn
$$\begin{aligned}
[\hat x^i-i\theta^{ij}\hat \partial_j,\hat x^{k}]_-
=[\hat x^i,\hat x^{k}]_--[i\theta^{ij}\hat \partial_j,\hat x^{k}]_-\nonumber\\
=i\theta^{ik}-i\theta^{ij}\delta^k_j
=i\theta^{ik}-i\theta^{ik}=0.\end{aligned}$$
Es ist also sinnvoll, diese Größe gleich einer Konstanten zu setzen. Wenn man nun 0 als Konstante wählt, ergibt sich
$$\hat \partial_j=-i\theta_{ij}^{-1} \hat x^i.
\label{DefnkAbleitung}$$
Dies hat folgende Vertauschungsrelation für die Ableitungsoperatoren zur Folge
$$\begin{aligned}
[\hat \partial_i,\hat \partial_j]_-&
=&[-i\theta_{ik}^{-1} \hat x^k,-i\theta_{jl}^{-1} \hat x^l]_-
=-\theta_{ik}^{-1}\theta_{jl}^{-1}[x^k,x^l]_-\nonumber\\
&=&-\theta_{ik}^{-1}\theta_{jl}^{-1}i\theta^{kl}
=-i\theta_{ik}^{-1}\delta_j^k
=-i\theta_{ij}^{-1}.\end{aligned}$$
Bei der folgenden Definition
$$\hat \partial_i \hat f=-i\theta_{ij}^{-1}[\hat x^j,\hat f]_-,$$
erhielte man kommutierende Ableitungsoperatoren
$$[\hat \partial_i,\hat \partial_j]_-=0.$$
Wie dem auch sei. Falls eine kanonische Vertauschungsrelation ($\ref{Koordinatenvertauschungsrelation}$) sowie Zusammenhang ($\ref{DefnkAbleitung}$) als Definition des Ableitungsoperators zu Grunde gelegt werden, erhält man insgesamt folgenden Satz von Vertauschungsrelationen zwischen Orts- und Impulsoperatoren
$$[\hat x^i,\hat x_j]_-=i\theta^i_j\quad,\quad [x^i,p_j]_-=i\delta^i_j \quad,\quad [p^i,p_j]=i(\theta^i_j)^{-1}.$$
Weitere allgemeine Aspekte der nichtkommutativen Geometrie werden in [@Wohlgenannt:2003de] ausgeführt.
Das Sternprodukt
----------------
Wenn man nun eine Quantenfeldtheorie auf einer Raumzeit, welche die Struktur einer nichtkommutativen Geometrie trägt, formulieren möchte, so stellt sich die Frage, wie Produkte von Feldern zu formulieren sind, die von nicht-kommutativen Koordinaten abhängen und in den Lagrangedichten auftauchen. Die Antwort liefert die Einführung des Sternproduktes, wie es beispielsweise in [@Wohlgenannt:2003de] beschrieben wird. Hierzu wird die Methode der sogenannten Weylquantisierung verwendet. Mit Weylquantisierung bezeichnet man die Beschreibung der Felder durch eine Überlagerung ebener Wellen, die nun von Operatoren abhängen
$$\psi(\hat x)=\frac{1}{2\pi}\int d^n k e^{ik_{j} \hat x^{j}} \bar \psi(k),
\label{Weyl}$$
wobei $\bar \psi(k)$ die Gewichtungsfunktion der ebenen Wellen darstellt. Man erhält sie durch Fouriertransformation des von den üblichen Koordinaten abhängigen Feldes $\psi(x)$
$$\psi(k)=\frac{1}{2\pi}\int d^n x e^{-ik_j x^j} \psi(x).
\label{Fouriertransformation}$$
Multipliziert man nun zwei Felder auf der nichtkommutativen Raumzeit, so ergibt sich unter Berücksichtigung von ($\ref{Weyl}$)
$$\psi(\hat x)\cdot \phi(\hat x)=\frac{1}{(2\pi)^4}\int d^n k d^n p e^{ik_{j}\hat x^{j}} e^{ip_{j}\hat x^{j}} \bar \psi(k) \bar \phi(p).
\label{ProduktnkF}$$
Unter Verwendung der Campbell-Baker-Hausdorff-Formel
$$e^{A}\cdot e^{B}=e^{A+B+\frac{1}{2}[A,B]_+\frac{1}{12}[[A,B],B]-\frac{1}{12}[[A,B],A]+...},
\label{C-B-H_F}$$
welche das Produkt zweier Exponentialfunktionen bestimmt, in deren Exponenten Matrizen stehen, erhält man
$$exp(ik_j \hat x^{j})exp(ip_j \hat x^{j})=exp ( i(k_j+p_j)\hat x^{j}-\frac{i}{2} k_{i} \theta^{ij} p_{j} ).$$
Hierbei wurde die kanonische Vertauschungsrelation zwischen den Koordinaten ($\ref{Koordinatenvertauschungsrelation}$) zu Grunde gelegt und die Tatsache ausgenutzt, dass alle Kommutatoren in der Baker-Campbell-Hausdorff-Formel, die selbst einen Kommutator enthalten, in diesem Falle verschwinden. Dies liegt daran, dass gemäß der oben getroffenen Annahme der Kommutator zweier Koordinaten ($\ref{Koordinatenvertauschungsrelation}$) konstant sein soll, also selbst nicht von den Koordinaten abhängen soll. Wenn man diesen Ausdruck in ($\ref{ProduktnkF}$) verwendet, so ergibt sich
$$\psi(\hat x)\cdot \phi(\hat x)=\frac{1}{(2\pi)^4}\int d^n k d^n p e^{i(k_j+p_j)\hat x^{j}-\frac{i}{2}k_{i}\theta^{ij}p_{j}} \bar \psi(k) \bar \phi(p).$$
Unter Einsetzung von ($\ref{Fouriertransformation}$) und Verwendung der Ortsdarstellung für die Impulsoperatoren ergibt sich
$$\psi(\hat x)\cdot \phi(\hat x)=exp\left(\frac{i}{2}\frac{\partial}{\partial x^{i}}\theta^{ij}\frac{\partial}{\partial y^{j}} \right)
\psi(x)\phi(y)\mid_{y \rightarrow x}.$$
Das Produkt von Feldern auf einer Raumzeit mit nichtkommutativer Geometrie ist also auf einen Ausdruck zurückgeführt worden, der die von kommutativen Koordinaten abhängigen Felder enthält. Es ist deshalb sinnvoll, das sogenannte Sternprodukt zwischen Feldern zu definieren
$$\psi \star \phi (x)=exp\left(\frac{i}{2}\frac{\partial}{\partial x^{i}}\theta^{ij}\frac{\partial}{\partial y^{j}}\right)
\psi(x)\phi(y)\mid_{y \rightarrow x}.$$
Multipliziert man also zwei normale Felder auf einer kommutativen Raumzeit mit dem Sternprodukt entspricht dies der Multiplikation der entsprechenden Felder, die von nichtkommutativen Koordinaten abhängen.
Nichtkommutative Eichtheorien
=============================
Nachdem das vorige Kapitel die Grundidee der nichtkommutativen Geometrie im Allgemeinen zum Thema hatte, sollen hier nun im Speziellen die Konsequenzen für die Formulierung von Eichtheorien erörtert werden. Hierbei werden die Darstellungen in [@Wohlgenannt:2003de], [@Calmet:2003jv], [@Calmet:2004yj],[@Madore:2000en],[@Jurco:2000ja] und [@Jurco:2001rq] zu Grunde gelegt. Dies macht schließlich die Einführung von sogenannten Seiberg-Witten-Abbildungen notwendig. Im ersten Kapitel wurde das Prinzip lokaler Eichtheorien beschrieben. Dort werden Symmetrietransformationen beschrieben, deren (liealgebrawertiger) Parameter von Raumzeitpunkt zu Raumzeitpunkt verschieden ist und damit eine Funktion auf der Raumzeit darstellt. Wenn man nun eine infinitesimale Eichtransformation eines Materiefeldes $\Psi$ zunächst im kommutativen Fall betrachtet
$$\delta \Psi(x)=i\omega^a(x) T^a \Psi(x)=i\alpha(x)\Psi(x)\quad,\quad \alpha(x)=\omega^a (x) T^a,$$
so werden beim Übergang zu einer nichtkommutativen Raumzeit sowohl der Eichparameter $\alpha$ alsauch das Materiefeld $\Psi$ zu Feldern, welche von nichtkommutativen Koordinaten abhängen und daher muss bei einer Multiplikation das Sternprodukt zu Grunde gelegt werden. Als infinitesimale Transformation für das Materiefeld ergibt sich damit
$$\hat \delta \Psi=i\alpha(\hat x)\Psi(\hat x)=i \alpha\star\Psi,$$
wobei $\hat \delta$ eine Eichtransformation auf einer nichtkommutativen Raumzeit bezeichnet. Damit hat die Nichtkommutativität der Raumzeit also auch eine Auswirkung auf eine lokale Eichtransformation. Im folgenden soll nun den sich daraus ergebenden Konsequenzen Rechnung getragen werden.
Kovariante Koordinaten
----------------------
Zunächst muss darauf hingewiesen werden, dass sich die Koordinaten der nichtkommutativen Raumzeit $\hat x$ unter einer Eichtransformation nicht transformieren, also $\hat \delta \hat x=0$. Wenn man sich nun die Wirkung einer infinitesimalen Eichtransformation auf das Sternprodukt einer Koordinate mit einem Materiefeld ansieht, so erkennt man, dass dieses sich nicht wie das Materiefeld selbst transformiert, denn
$$\hat \delta(x\star\Psi)=\hat\delta x\star\Psi+x\star\hat\delta \Psi=ix\star\alpha(x)\star\Psi \neq i\alpha(x)\star x\star\Psi.
\label{Transformationprodukt}$$
Dies verhält sich in Analogie zur Transformation der Ableitung des Materiefeldes innerhalb gewöhnlicher Eichtheorien, die im vorletzten Kapitel thematisiert wurden und wo die einfache Ableitung $\partial_\mu$ durch eine kovariante Ableitung $D_\mu=\partial_\mu+iA_\mu$ ersetzt werden musste. Bei nichtkommutativen Eichtheorien müssen daher die Koordinaten $\hat x^\mu$ durch kovariante Koordinaten $\hat X^\mu$ der folgenden Form ersetzt werden
$$\hat x^\mu \rightarrow \hat X^\mu=\hat x^\mu+B^\mu,$$
wobei an $B_\mu$ eine bestimmte Transformationsforderungen zu stellen ist. Wenn man diese Ersetzung in ($\ref{Transformationprodukt}$) vornimmt, so ergibt sich
$$\begin{aligned}
\hat \delta(X^\mu \star \Psi)=\hat \delta((x^\mu+B^\mu) \star \Psi)=(\hat \delta B^\mu)\star\Psi+(x^\mu+B^\mu)\star\delta \Psi\nonumber\\
=\hat \delta B^\mu \star\Psi+i(x^\mu+B^\mu)\star\alpha\star\Psi.\end{aligned}$$
Um zu gewährleisten, dass sich das Produkt $X^\mu*\Psi$ gemäß
$$\hat \delta (X^\mu \star \Psi)=i\alpha\star X^\mu \star \Psi$$
transformiert, muss sich $B^\mu$ also wie folgt transformieren
$$\delta B^\mu=i[\alpha\ _{,}^{*}\ x^\mu]+i[\alpha\ _{,}^{*}\ B^\mu]=i[\alpha\ _{,}^{*}\ X^\mu].$$
Das Symbol $[\ _{,}^{*}\ ]$ bezeichnet hierbei den Kommutator bezüglich des Sternproduktes. Es ist insofern nicht überraschend, dass die Koordinaten bei nichtkommutativen Eichtheorien durch kovariante Koordinaten ersetzt werden müssen, als sie ja wie im letzten Kapitel gezeigt als proportional zu den entsprechenden Ableitungen definiert werden können, um die entsprechenden Vertauschungsrelationen zu gewährleisten. Unter Verwendung dieser Definition und Betrachtung der kovarianten Ableitung $\partial_\mu+iA_\mu$ ergibt sich folgender Zusammenhang zwischen $B^\mu$ und dem Eichpotential $A^\mu$
$$B^\mu=-\theta^{\mu\nu} A_\nu.$$
Bei der Transformation des Vektorpotentials und der Definition des Feldstärketensors tauchen naturgemäß ebenfalls die Kommutatoren bezüglich des Sternprodukts auf. Für die Transformation des Vektorpotentials $A_\mu$ gilt damit
$$\hat \delta A_\mu=\partial_\mu \alpha+i[\alpha\ _{,}^{*}\ A_\mu].$$
Der Feldstärketensor ist natürlich wie üblich über den Kommutator der kovarianten Ableitungen definiert. Auf einer nichtkommutativen Raumzeit enthält er aber nun das Sternprodukt
$$F_{\mu\nu}=i[D_\mu\ _{,}^{*}\ D_\nu],$$
was auf folgenden Ausdruck für den Feldstärketensor führt
$$F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu+i[A_\mu\ _{,}^{*}\ A_\nu].$$
Dies bedeutet insbesondere, dass auch im Falle einer Abelschen Eichtheorie aufgrund der Nichtkommutativität des Sternproduktes Kommutatorterme auftauchen.
Seiberg-Witten-Abbildungen
--------------------------
### Die Bedingung der Geschlossenheit
Natürlich ist die Voraussetzung einer jeden Symmetriegruppe die Geschlossenheit, was bedeutet, dass die Anwendung zweier Symmetrietransformationen wieder zu einer Symmetrietransformation führt, die Element der Gruppe ist. Im Falle kommutativer Eichtheorien gilt für den Kommutator zweier Eichtransformationen bezüglich der Eichparameter $\alpha$ und $\beta$
$$(\delta_\alpha \delta_\beta-\delta_\beta \delta_\alpha)\Psi=\delta_{i[\alpha,\beta]}\Psi.
\label{TransformationKommutator}$$
Der Kommutator zweier zu den Parametern $\alpha$ und $\beta$ gehöriger Eichtransformationen ist also gleich der Eichtransformation des Kommutators von $\alpha$ und $\beta$ multipliziert mit i. Da die Generatoren $T^a$ der Eichgruppe aber eine Liealgebra bilden $[T^a,T^b]=if^{abc} T^c$, sieht der Kommutator zweier Eichtransformationen wie folgt aus
$$i[\alpha,\beta]=i\omega^a \omega^b [T^a,T^b]=\omega^a \omega^b f^{bac} T^c.$$
Es ergibt sich also erneut eine Eichtransformation innerhalb der Gruppe. Nun stellt sich die Frage, wie dies bei nichtkommutativen Eichtheorien aussieht. Prinzipiell ist die Situation hier natürlich analog. Allerdings enthält der Kommutator jetzt natürlich das Sternprodukt. Für nichtkommutative Eichtransformationen ergibt sich damit als Kommutator der Eichparameter
$$[\alpha\ _{,}^{*}\ \beta]=[\omega^a T^a\ _{,}^{*}\ \omega^b T^b]=\frac{1}{2}\{\alpha_a\ _,^{*}\ \beta_b\}[T^a\ ,\ T^b]+\frac{1}{2}[\alpha_a\ _,^{*}\ \beta_b]\{T^a\ ,\ T^b\}.$$
Die Kommutatoren und Antikommutatoren der Vorfaktoren sind bezüglich der Frage, ob sich die Gruppe geschlossen ist, nicht von Bedeutung. Der Kommutator der Generatoren im ersten Term ist natürlich wie bei gewöhnlichen Eichtheorien auch hier umproblematisch. Probleme bereitet jedoch der Antikommutator im zweiten Term. Dieser ist nur liealgebrawertig im Falle der Generatoren der U(N), nicht jedoch bei der SU(N).
### Erweiterung zur einhüllenden Algebra
Es gibt allerdings die Möglichkeit, eine geschlossene Algebra zu erreichen, indem man die Gruppe der Eichparameter zu der Einhüllenden der Liealgebra erweitert. Diese beinhaltet im Gegensatz zu einer einfachen Liealgebra nicht nur beliebige Linearkombinationen eines Satzes von Generatoren, sondern enthält auch beliebige Produkte der Generatoren. Ein Eichparameter $\hat \alpha$ in der Einhüllenden der Liealgebra mit Generatoren $T^a$ hat dann folgende allgemeine Gestalt
$$\hat \alpha=\hat \alpha_0^a T^a+\hat \alpha_1^{ab} \{T^a,T^b\}+\hat \alpha_2^{abc} \{T^a,T^b,T^c\}+...,$$
wobei die geschwungenen Klammern eine Summierung über alle Reihenfolgen der Generatoren darstellt und insofern eine Verallgemeinerung des Antikommutators für mehr als zwei Elemente beschreibt. Gemäß den Eichparametern sind natürlich auch die nichtkommutativen Eichpotentiale und damit die Feldstärketensoren in der einhüllenden Algebra definiert. Ebenso wie das im letzten Kapitel behandelte Produkt zwischen Feldern auf einer nichtkommutativen Raumzeit, können auch die Eichparameter und die nichtkommutativen Felder selbst nach dem Parameter $\theta^{\mu\nu}$ entwickelt werden, welcher die Nichtkommutativität der Koordinaten angibt. Damit werden die kommutativen Größen auf nichtkommutative Größen abgebildet. Diese Abbildungen wurden erstmals von Seiberg und Witten gefunden [@Seiberg:1999vs] und werden daher nach ihnen als Seiberg-Witten-Abbildungen bezeichnet. Es soll nun beschrieben werden, wie die Seiberg-Witten-Abbildungen zu bestimmen sind. Aus der Relation ($\ref{TransformationKommutator}$) für die Eichparameter wird auf einer nichtkommutativen Raumzeit die folgende Bedingung, welche nun das Sternprodukt enthält
$$\begin{aligned}
(i\hat \delta_\alpha \hat \delta_\beta-\hat \delta_\beta \hat \delta_\alpha)\star\Psi=\hat \delta_{i[\alpha,\beta]}\hat \Psi \nonumber\\
\Leftrightarrow i\hat \delta_\alpha \hat \beta-i\hat \delta_\beta \hat \alpha+[\hat \alpha\ _{,}^*\ \hat \beta]*\Psi=\hat {[\alpha,\beta]}\star\hat \Psi.\end{aligned}$$
Wenn man nun den Eichparameter $\hat \alpha$ in der einhüllenden Algebra in einer Reihe nach der Größe $\theta^{\mu\nu}$ entwickelt, welche die Nichtkommutativität der Raumzeit angibt und damit im Sternprodukt auftaucht
$$\hat \alpha=\alpha_0+\alpha_1(\theta)+...,$$
so ergibt sich bei einer Entwicklung in erster Ordnung in $\theta^{\mu\nu}$ für den Term $\alpha_1$
$$\alpha_1=\frac{1}{4}\theta^{\mu\nu}\{\partial_\mu,A_\nu \}.$$
Die Entwicklung von $\hat \alpha$ in erster Ordnung hat also folgende Form
$$\hat \alpha=\alpha+\frac{1}{4}\theta^{\mu\nu}\{\partial_\mu,A_\nu \}+\mathcal{O}(\theta^2).$$
### Abbildungen der Felder
Die Seiberg-Witten-Abbildungen für die Felder sind nun dadurch bestimmt, dass eine gewöhnliche Eichtransformation der nichtkommutativen Felder, welche also auf die kommutativen Felder wirkt, von denen sie abhängen, gleich einer nichtkommutativen Eichtransformation der nichtkommutativen Felder ist. Dies bedeutet für eine infinitesimale Eichtransformation eines Feldes A
$$\hat A[A+\delta_\alpha A]=\hat A[A]+\hat \delta_\alpha \hat A[A].$$
Für ein Materiefeld $\hat \Psi$ ergibt sich damit konkret die Bedingung
$$\delta_\alpha \hat \Psi=\hat \delta_\alpha \hat \Psi=i\hat \alpha * \hat \Psi.$$
Eine Entwicklung in erster Ordnung
$$\hat \Psi=\Psi^0+\Psi^1+\mathcal{O}(\theta^2)$$
führt hierbei auf folgende Bedingung
$$\delta_\alpha \Psi_0+\delta_\alpha \Psi_1=i\alpha \Psi_0-\frac{1}{2}\theta^{\mu\nu}\partial_\mu \alpha \partial_\nu \Psi_0+i\alpha_1 \Psi_0+i\alpha \Psi_1+\mathcal{O}(\theta^2).$$
Da im kommutativen Grenzfall, welcher $\theta=0$ impliziert, sich natürlich wieder $\Psi$ ergeben muss, gilt $\Psi_0$=$\Psi$. Insgesamt findet man für die Seiberg-Witten-Abbildung in erster Ordnung
$$\hat \Psi=\Psi-\frac{1}{2}\theta^{\mu\nu}A_\nu \partial_\mu \Psi+\frac{i}{8}\theta^{\mu\nu}[A_\mu,A_\nu] \Psi+\mathcal{O}(\theta^2).$$
Hierbei ist zu beachten, dass bei einer kommutativen Eichtransformation des nichtkommutativen Materiefeldes $\hat \Psi$, welches nicht nur eine Funktion des kommutativen Materiefeldes $\Psi$, sondern auch des Eichpotentiales $A_\mu$ ist, letzteres bei einer Eichtransformation mittransformiert werden muss. Wenn man die analogen Bedingungen für das Vektorpotential und den Feldstärketensor aufstellt
$$\delta_\alpha \hat A_\mu=\partial_\mu \hat \alpha+i[\hat \alpha\ _,^*\ \hat A_\mu]$$
und
$$\hat \delta \hat F_{\mu\nu}=i[\hat \alpha\ _{,}^*\ \hat F_{\mu\nu}],$$
so erhält man die folgenden Seiberg-Witten-Abbildungen in erster Ordnung in $\theta^{\mu\nu}$
$$\begin{aligned}
\hat A_\mu=A_\mu-\frac{1}{4}\theta^{\rho\nu}\{A_\rho,\partial_\nu A_\mu+F_{\nu\mu}\}+\mathcal{O}(\theta^2)\nonumber\\
\hat F_{\mu\nu}=F_{\mu\nu}+\frac{1}{2}\theta^{\rho\sigma}\{F_{\mu\rho},F_{\nu\sigma}-\frac{1}{4}\theta^{\rho\sigma}\{A_\rho,(\partial_\sigma+D_\sigma)F_{\mu\nu}\}+\mathcal{O}(\theta^2).\end{aligned}$$
Diese machen nun die Formulierung des Standardmodells auf einer nichtkommutativen Raumzeit möglich, wie sie in [@Calmet:2001na], [@Melic:2005fm] und [@Melic:2005am] gegeben wird.
Higgsmechanismus und nichtkommutative Geometrie
===============================================
Bei der Formulierung nichtkommutativer Eichtheorien spielen zwangsläufig die neu eingeführten nichtkommutativen Felder eine Rolle, welche die Eichinvarianz auch auf einer nichtkommutativen Raumzeit gewährleisten. Da nun aber dem Higgsmechanismus eine prinzipielle Bedeutung zukommt, weil die Austauschteilchen der schwachen Wechselwirkung nur so eine Masse bekommen können, ohne dass die Eichinvarianz unter der Symmetriegruppe des schwachen Isospins verletzt würde, muss dieser natürlich ebenfalls in die nichtkommutative Eichtheorie integriert werden. In [@Calmet:2001na],[@Melic:2005fm] und [@Petriello:2001mp] wurde auf den Higgsmechanismus im Rahmen nichtkommutativer Eichtheorien eingegangen. Wenn man zunächst einmal die Lagrangedichte des Higgsfeldes auf einer nichtkommutativen Raumzeit formuliert, indem man das Produkt zwischen Feldern durch das Sternprodukt und die Felder durch die nichtkommutativen Felder ersetzt, erhält man
$$\mathcal{L}_{Higgs}=(D_\mu \hat \Phi)^{\dagger}\star (D^\mu \hat \Phi)-\mu^2 (\hat \Phi)^\dagger \star \hat \Phi-\lambda(\hat \Phi ^\dagger \star \hat \Phi)\star (\hat \Phi ^\dagger \star \hat \Phi).$$
Es sind nun zwei Möglichkeiten denkbar, wie man die entsprechende Lagrangedichte mit den entsprechenden Massentermen in Abhängigkeit der ublichen Felder erhält. Man kann entweder zunächst die Seiberg-Witten-Abbildungen ausführen und dann die spontane Symmetriebrechung betrachten. Dies ist die übliche Reihenfolge. Es ist aber auch denkbar, die spontane Symmetriebrechung schon bei den nichtkommutativen Feldern auszuführen und anschließend die Seiberg-Witten-Abbildungen zu verwenden. Allerdings stellt sich die Frage, ob man in beiden Fällen das gleiche Resultat erhält. Es wird hier gezeigt werden, dass diese beiden Prozeduren tatsächlich äquivalent zueinander sind. Zunächst soll der Abelsche Fall betrachtet werden. Hier fällt der konstante Vakuumerwartungswert bei allen Termen mit $\theta$ heraus, sodass sich sie Situation als völlig unproblematisch erweist. Bei nicht-Abelschen Eichtheorien muss die Bedingung an die nichtkommutativen Felder
$$\delta \hat \Phi=\hat \delta \hat \Phi=i\hat \alpha \star \hat \Phi
\label{SWcondition}$$
betrachtet werden, welche auf die Seiberg-Witten-Abbildung
$$\hat \Phi=\Phi-\frac{1}{2}\theta^{\mu\nu}A_\mu \partial_\nu \Phi
+\frac{i}{8}\theta^{\mu\nu}[A_\mu,A_\nu]_-\Psi+\mathcal{O}(\theta^2)
\label{SeibergWittenPhi}$$
führt. Die Seiberg-Witten-Abbildung des Eichparameters ist durch
$$\hat \alpha=\alpha+\frac{1}{4}\theta^{\mu\nu}\{\partial_\mu \alpha,A_\nu\}+\mathcal{O}(\theta^2)$$
gegeben.
Abelscher Fall
--------------
Im Abelschen Fall reduziert sich ($\ref{SeibergWittenPhi}$) auf folgende Seiberg-Witten-Abbildung für das Materiefeld $\Phi$
$$\hat \Phi=\Phi-\frac{1}{2}\theta^{\mu\nu}A_\mu \partial_\nu \Phi
+\mathcal{O}(\theta^2).
\label{AbelianSeiberg-Witten}$$
Die Abbildung für den Eichparameter wird zu
$$\hat \alpha=\alpha-\frac{1}{2}\theta^{\mu\nu} A_\mu \partial_\nu \alpha+\mathcal{O}(\theta^2).$$
Die Seiberg-Witten-Abbildungen sowohl für das Eichpotential $A_\mu$ alsauch für den Feldstärketensor $F_{\mu\nu}$ müssen nicht betrachtet werden, da die spontane Symmetriebrechung keine direkte Auswirkung auf sie hat.
### Zunächst Seiberg-Witten-Abbildung und dann Symmetriebrechung
Zunächst soll die spontane Symmetriebrechung auf Seite der kommutativen Felder erfolgen. Dies führt auf folgende Relation
$$\Phi=\nu+h+i\sigma,
\label{symmetry-breaking}$$
wobei $\nu$ der konstante Vakuumerwartungswert des Feldes $\Phi$ ist. Die spontane Symmetriebrechung hat das Erscheinen von Goldstonebosonen zur Folge. Dies spielt hier jedoch keine Rolle. Indem man definiert
$$\phi=h+i\sigma,$$
wird aus ($\ref{symmetry-breaking}$)
$$\Phi=\nu+\phi.
\label{symmetry-breaking-ugauge}$$
Einsetzen von ($\ref{symmetry-breaking-ugauge}$) in ($\ref{AbelianSeiberg-Witten}$) führt auf
$$\hat \Phi=\nu+\phi-\frac{1}{2}\theta^{\mu\nu}A_\mu \partial_\nu (\nu+\phi)+\mathcal{O}(\theta^2).
\label{Abelianresult1}$$
Da $\nu$ eine Konstante ist, bleibt dieser Ausdruck übrig
$$\hat \Phi=\nu+\phi-\frac{1}{2}\theta^{\mu\nu}A_\mu \partial_\nu h+\mathcal{O}(\theta^2).
\label{result1}$$
### Zunächst Symmetriebrechung and dann Seiberg-Witten-Abbildung
Die andere Möglichkeit besteht darin, eine Symmetriebrechung der nichtkommutativen Felder zu betrachten.
$$\hat \Phi=\nu+\hat \phi.
\label{nonc-symmetry-breaking-ug}$$
Nun muss man jedoch die Seiberg-Witten-Abbildung für $\hat \phi$ festlegen. Deshalb ist es notwendig, die Bedingung für die Seiberg-Witten-Abbildung von $\Phi$ zu betrachten. Einsetzen von ($\ref{nonc-symmetry-breaking-ug}$) in ($\ref{SWcondition}$) liefert
$$\delta (\nu +\hat \phi)=\hat \delta (\nu+\hat \phi).$$
Da $\nu$ eine Konstante ist, transformiert es sich nicht. Aus diesem Grund bleibt folgende Relation
$$\delta \hat \phi=\hat \delta \hat \phi.
\label{SWcondition-h}$$
Diese Bedingung ist von der gleichen Gestalt wie die Bedingung an $\Phi$, aber hierbei ist zu berücksichtigen, dass $\phi$ sich unter einer Eichtransformation anders als $\Phi$ transformiert. Dies kann man wie folgt sehen
$$\delta \Phi=i \alpha \Phi=i \alpha (\nu+\phi)\quad,\quad\delta \Phi=\delta(\nu+\phi)=\delta \phi.$$
Gleichsetzen der beiden Ausdrücke zeigt, dass sich $\phi$ wie folgt transformiert
$$\delta \phi=i\alpha(\nu+\phi).
\label{gauge-transformation-h}$$
Indem man die analoge Argumentation im Falle des nichtkommutativen Feldes $\hat \phi$ unter einer nichtkommutative Eichtransformation anwendet, sieht man, dass sich $\hat \phi$ analog transformiert
$$\hat \delta \hat \phi=i\hat \alpha \star(\nu+\hat \phi).
\label{gauge-transformation-hath}$$
Wie dem auch sei, im Abelschen Falle sieht die Seiberg-Witten-Abbildung für $h$ wie die Seiberg-Witten-Abbildung für $\Phi$ aus
$$\hat \phi=\phi-\frac{1}{2}\theta^{\mu\nu}A_\mu \partial_\nu \phi
+\mathcal{O}(\theta^2).$$
\[SWmaph\] Dies kann man sehen, indem man überprüft, ob sie die Bedingung an die Seiberg-Witten-Abbildung ($\ref{SWcondition-h}$) erfüllt. Einsetzen von ($\ref{SWmaph}$) in ($\ref{SWcondition-h}$) führt auf die folgende Gleichung in erster Ordnung in $\theta$
$$\delta(\phi-\frac{1}{2}\theta^{\mu\nu}A_\mu \partial_\nu \phi)
=i \hat \alpha \star (\phi-\frac{1}{2}\theta^{\mu\nu}A_\mu \partial_\nu \phi).$$
Benutzung von ($\ref{gauge-transformation-h}$), ($\ref{gauge-transformation-hath}$) und der Transformationseigenschaft des Vektorpotentials $A_\mu$
$$\delta A_\mu=\partial_\mu \alpha$$
liefert
$$\begin{aligned}
i\alpha(\nu+\phi)-\frac{1}{2}\theta^{\mu\nu}A_\mu \partial_\nu i\alpha(\nu+\phi)-\frac{1}{2}\theta^{\mu\nu}\partial_\mu \alpha \partial_\nu \phi =\nonumber\\
i\hat \alpha \star (\nu+\phi-\frac{1}{2}\theta^{\mu\nu}A_\mu \partial_\nu \phi).\end{aligned}$$
Durch Ausführen der Ableitung auf der linken und des Sternproduktes auf der rechten Seite erhält man
$$\begin{aligned}
i\alpha(\nu+\phi)-i\frac{1}{2}\theta^{\mu\nu}A_\mu [(\partial_\nu \alpha)(\nu+\phi)+\alpha (\partial_\nu \phi)]-\frac{1}{2}\theta^{\mu\nu}\partial_\mu \alpha \partial_\nu \phi \nonumber\\
=i\alpha(\nu+\phi)-\frac{1}{2}\theta^{\mu\nu}\partial_\mu \alpha \partial_\nu \phi-i\alpha \frac{1}{2}\theta^{\mu\nu}A_\mu \partial_\nu \phi-i\frac{1}{2}\theta^{\mu\nu}A_\mu \partial_\nu \alpha(\nu+\phi).\end{aligned}$$
Nun kann man sehen, dass die beiden Seiten gleich sind und ($\ref{SWmaph}$) in der Tat die richtige Seiberg-Witten-Abbildung für $\hat \phi$ ist. Indem wir ($\ref{SWmaph}$) in ($\ref{nonc-symmetry-breaking-ug}$) einsetzen erhalten wir ($\ref{result1}$)
$$\hat \Phi=\nu+\phi-\frac{1}{2}\theta^{\mu\nu}A_\mu \partial_\nu \phi+\mathcal{O}(\theta^2).$$
Somit ist gezeigt, dass spontane Symmetriebrechung und Abbilden auf kommutative Felder im Abelschen Fall umkehrbar ist.
Nicht-Abelscher Fall
--------------------
Im Falle Nicht-Abelscher Eichtheorien sieht die Situation etwas anders aus.
### S.-W.-Abbildung und dann Symmetriebrechung
Die folgende Betracht gilt für beliebige nichtabelsche Eichgruppen. Dazu gehört also auch die Eichgruppe, bei welcher der Higgsmechanismus im Standardmodell relevant ist. Es handelt sich also um die in Kapitel 5 thematisierte $SU(2) \times U(1)$ Eichgruppe. In einer allgemeinen Darstellung ohne Wahl einer speziellen Eichung führt die Symmetriebrechung auf folgende Relation
$$\Phi=\nu_n+\phi_n,
\label{NAsymmetrybreaking}$$
wobei der Index n andeutet, dass es sich bei dem Vakuumerwartungswert $\nu$ und dem Feld $\phi$ um Multipletts handelt. Durch Einsetzen von ($\ref{NAsymmetrybreaking}$) in ($\ref{SeibergWittenPhi}$), erhält man
$$\hat \Phi=\nu_n+\phi_n
-\frac{1}{2}\theta^{\mu\nu}A_\mu \partial_\nu (\nu_n+\phi_n)
+\frac{i}{8}\theta^{\mu\nu}[A_\mu,A_\nu]
(\nu_n+\phi_n)
+\mathcal{O}(\theta^2).$$
Der $\nu$-Term im zweiten Term verschwindet und so ergibt sich folgender Ausdruck
$$\hat \Phi=\nu_n+\phi_n
-\frac{1}{2}\theta^{\mu\nu}A_\mu \partial_\nu (\nu_n+\phi_n)
+\frac{i}{8}\theta^{\mu\nu}[A_\mu,A_\nu]
(\nu_n+\phi_n)
+\mathcal{O}(\theta^2).
\label{NAresult1}$$
### Symmetriebrechung und dann S.-W.-Abbildung
Wenn man Symmetriebrechung auf der nichtkommutativen Seite betrachtet
$$\hat \Phi=\nu_n+\hat \phi_n,
\label{NAsymmetrybreakingnc}$$
sieht die Bedingung an die Seiberg-Witten-Abbildung des Materiefeldes ($\ref{SWcondition}$) wie folgt aus
$$\delta (\nu_n+\hat \phi_n)
=i \hat \alpha \star (\nu_n+\hat \phi_n).
\label{SWconditionnA}$$
Die Bedingung ($\ref{SWconditionnA}$) kann in folgender Weise umformuliert werden
$$\delta \hat \phi_n=\hat \delta \hat \phi_n.
\label{SWconditionh-d}$$
$\phi_n$ und $\hat \phi_n$ transformieren sich in Analogie zum Abelschen Fall gemäß
$$\delta \phi_n=i\alpha(\nu_n+\phi_n)\quad,\quad \hat \delta \hat \phi_n=i\hat \alpha \star(\nu_n+\hat \phi_n).
\label{gauge-transformation-h-d}$$
Es wird die Annahme gemacht, dass $\phi_n$ auf die folgende Art und Weise abgebildet wird
$$\begin{aligned}
\hat \phi_n&=&\phi_n-\frac{1}{2}\theta^{\mu\nu}A_\mu \partial_\nu \phi_n
+\frac{i}{8}\theta^{\mu\nu}[A_\mu,A_\nu]\phi_n\nonumber\\
&&+\frac{i}{8}\theta^{\mu\nu}[A_\mu,A_\nu]\nu_n
+\mathcal{O}(\theta^2).
\label{SeibergWittenhd}\end{aligned}$$
Einsetzen von ($\ref{SeibergWittenhd}$) in ($\ref{SWconditionh-d}$) führt zu der folgenden Gleichung in erster Ordnung in $\theta^{\mu\nu}$
$$\begin{aligned}
\delta
(\phi_n-\frac{1}{2}\theta^{\mu\nu}A_\mu \partial_\nu \phi_n
+\frac{i}{8}\theta^{\mu\nu}[A_\mu,A_\nu]\phi_n
+\frac{i}{8}\theta^{\mu\nu}[A_\mu,A_\nu]\phi_n)\nonumber\\
=i \hat \alpha \star
(\phi_n-\frac{1}{2}\theta^{\mu\nu}A_\mu \partial_\nu \phi_n
+\frac{i}{8}\theta^{\mu\nu}[A_\mu,A_\nu]\phi_n
+\frac{i}{8}\theta^{\mu\nu}[A_\mu,A_\nu]\nu_n).\end{aligned}$$
Durch Benutzung von ($\ref{gauge-transformation-h-d}$) und der Eichtransformation eines liealgebrawertigen Vektorfeldes
$$\delta A_\mu=\partial_\mu \alpha+i[\alpha,A_\mu]$$
kann dies geschrieben werden als
$$\begin{aligned}
i\alpha(\nu_n+\phi_n)-\frac{1}{2}\theta^{\mu\nu} A_\mu \partial_\nu (i\alpha(\nu_n+\phi_n))-\frac{i}{8}\theta^{\mu\nu}[A_\mu,A_\nu]i\alpha(\nu_n+\phi_n)&&\nonumber\\
-\frac{1}{2}\theta^{\mu\nu}\partial_\mu \alpha \partial_\nu \phi_n-\frac{1}{2}\theta^{\mu\nu}i[\alpha,A_\mu]\partial_\nu \phi_n&&\nonumber\\
+\frac{i}{8}\theta^{\mu\nu}[\partial_\mu \alpha, A_\nu](\nu_n+\phi_n)+\frac{i}{8}\theta^{\mu\nu}[A_\mu,\partial_\nu \alpha](\nu_n+\phi_n)&&\nonumber\\
+\frac{i}{8}\theta^{\mu\nu}[i[\alpha,A_\mu],\partial_\nu \alpha](\nu_n+\phi_n)+\frac{i}{8}\theta^{\mu\nu}[A_\mu,i[\alpha,A_\nu]](\nu_n+\phi_n)&&\nonumber\\
=i\alpha(\nu_n+\phi_n)-\frac{1}{2}\theta^{\mu\nu}\partial_\mu \alpha \partial_\nu \phi_n-i\alpha\frac{1}{2}\theta^{\mu\nu}A_\mu \partial_\nu \phi_n\nonumber\\
-i\alpha\frac{i}{8}\theta^{\mu\nu}[A_\mu,A_\nu](\nu_n+\phi_n)+\frac{i}{4}\theta^{\mu\nu}\{\partial_\mu \alpha,A_\nu\}(\nu_n+\phi_n).\end{aligned}$$
Die Ausdrücke $\ i\alpha(\nu_n+\phi_n)\ $ und $\ \frac{1}{2}\theta^{\mu\nu}\partial_\mu \alpha \partial_\nu \phi_n\ $ erscheinen auf beiden Seiten und heben sich gegenseitig auf. Deshalb kann man schreiben
$$\begin{aligned}
-\frac{1}{2}\theta^{\mu\nu}A_\mu i(\partial_\nu \alpha)(\nu_n+\phi_n)-\frac{1}{2}\theta^{\mu\nu}A_\mu i\alpha \partial_\nu \phi_n
+\frac{i}{8}\theta^{\mu\nu}[A_\mu,A_\nu]i\alpha(\nu_n+\phi_n)&&\nonumber\\
-\frac{1}{2}\theta^{\mu\nu}i[\alpha,A_\mu]\partial_\nu \phi_n
+\frac{i}{8}\theta^{\mu\nu}[\partial_\mu \alpha,A_\nu](\nu_n+\phi_n)+\frac{i}{8}\theta^{\mu\nu}[A_\mu,\partial_\nu \alpha](\nu_n+\phi_n)&&\nonumber\\
+\frac{i}{8}\theta^{\mu\nu}[i[\alpha,A_\mu],A_\nu](\nu_n+\phi_n)+\frac{i}{8}\theta^{\mu\nu}[A_\mu,i[\alpha,A_\nu]](\nu_n+\phi_n)&&\nonumber\\
=-i\alpha\frac{1}{2}\theta^{\mu\nu}A_\mu\partial_\nu \phi_n+i\alpha\frac{i}{8}[A_\mu,A_\nu](\nu_n+\phi_n)+\frac{i}{4}\theta^{\mu\nu}\{\partial_\mu \alpha, A_\nu\}(\nu_n+\phi_n).&&\nonumber\\\end{aligned}$$
Wenn man beachtet, dass die folgende Relation gültig ist
$$-\frac{1}{2}\theta^{\mu\nu}A_\mu i\alpha \partial_\nu \phi_n-\frac{1}{2}\theta^{\mu\nu}i[\alpha,A_\mu]\partial_\nu \phi_n=-i\alpha\frac{1}{2}\theta^{\mu\nu}A_\mu\partial_\nu \phi_n,$$
heben sich die entsprechenden Terme auf und man wird auf folgende Gleichung geführt
$$\begin{aligned}
-\frac{1}{2}\theta^{\mu\nu}A_\mu i(\partial_\nu \alpha)(\nu_n+\phi_n)
+\frac{i}{8}\theta^{\mu\nu}[A_\mu,A_\nu] i\alpha(\nu_n+\phi_n)\nonumber\\
+\frac{i}{8}\theta^{\mu\nu}[\partial_\mu \alpha,A_\nu](\nu_n+\phi_n)
+\frac{i}{8}\theta^{\mu\nu}[A_\mu,\partial_\nu \alpha](\nu_n+\phi_n)\nonumber\\
+\frac{i}{8}\theta^{\mu\nu}[i[\alpha,A_\mu],A_\nu](\nu_n+\phi_n)
+\frac{i}{8}\theta^{\mu\nu}[A_\mu,i[\alpha,A_\nu]](\nu_n+\phi_n)\nonumber\\
=i\alpha\frac{i}{8}[A_\mu,A_\nu](\nu_n+\phi_n)+\frac{i}{4}\theta^{\mu\nu}\{\partial_\mu \alpha, A_\nu\}(\nu_n+\phi_n).\end{aligned}$$
Aufgrund der Antisymmetrie von $\theta^{\mu\nu}$ gilt
$$\frac{i}{8}\theta^{\mu\nu}[\partial_\mu \alpha, A_\nu]_-=\frac{i}{8}\theta^{\mu\nu}[A_\mu,\partial_\nu \alpha]$$
und somit
$$\begin{aligned}
-\frac{1}{2}\theta^{\mu\nu}A_\mu i(\partial_\nu \alpha)(\nu_n+\phi_n)+\frac{i}{8}\theta^{\mu\nu}[\partial_\mu \alpha,A_\nu](\nu_n+\phi_n)\nonumber\\
+\frac{i}{8}\theta^{\mu\nu}[A_\mu,\partial_\nu \alpha](\nu_n+\phi_n)\nonumber\\
=-\frac{1}{2}\theta^{\mu\nu}A_\mu i(\partial_\nu \alpha)(\nu_n+\phi_n)+\frac{i}{4}\theta^{\mu\nu}[\partial_\mu \alpha, A_\nu](\nu_n+\phi_n)\nonumber\\
=-\frac{1}{2}\theta^{\mu\nu}A_\mu i(\partial_\nu \alpha)(\nu_n+\phi_n)+\frac{i}{4}\theta^{\mu\nu}\partial_\mu \alpha A_\nu(\nu_n+\phi_n)\nonumber\\
-\frac{i}{4}\theta^{\mu\nu}A_\nu \partial_\mu \alpha(\nu_n+\phi_n)
=\frac{i}{4}\theta^{\mu\nu}\{\partial_\mu \alpha,A_\nu\}.\end{aligned}$$
Es verbleibt die folgende Gleichung
$$\begin{aligned}
\frac{i}{8}\theta^{\mu\nu}[A_\mu,A_\nu](\nu_n+\phi_n)+\frac{i}{8}\theta^{\mu\nu}[i[\alpha,A_\mu],A_\nu](\nu_n+\phi_n)\nonumber\\
+\frac{i}{8}\theta^{\mu\nu}[A_\mu,i[\alpha,A_\nu]](\nu_n+\phi_n)
=i\alpha\frac{i}{8}[A_\mu,A_\nu]_-(\nu_n+\phi_n).\end{aligned}$$
Indem man nun die verschachtelten Kommutatoren ausrechnet, ergibt sich die Gültigkeit auch dieser Gleichung. Dies bedeutet, dass ($\ref{SeibergWittenhd}$) in der Tat die richtige Seiberg-Witten-Abbildung liefert. Indem man nun ($\ref{SeibergWittenhd}$) in ($\ref{NAsymmetrybreakingnc}$) einsetzt, erhält man
$$\hat \Phi=(\nu_n+\phi_n)
+\frac{1}{2}\theta^{\mu\nu}A_\mu \partial_\nu (\nu_n+\phi_n)
+\frac{i}{8}\theta^{\mu\nu}[A_\mu,A_\nu]
(\nu_n+\phi_n)
+\mathcal{O}(\theta^2).$$
Die entspricht exakt dem Resultat aus (\[NAresult1\]). Damit wurde gezeigt, dass es keine Rolle spielt, ob man zunächst die Seiberg-Witten-Abbildungen ausführt und dann die Symmetrie bricht oder ob die Symmetriebrechung den Seiberg-Witten-Abbildungen voran geht.
Schlussbemerkungen {#schlussbemerkungen .unnumbered}
==================
Am Ende dieser Arbeit soll nun das hier Beschriebene in den größeren Rahmen der zeitgenössischen Physik eingeordnet werden. Die in dieser Diplomarbeit vorgestellten Erweiterungen der bisherigen Physik gehen letztlich aus dem Versuch hervor, das Gebäude der theoretischen Physik in einer Weise zu erweitern, die einer einheitlicheren Beschreibung der Natur zustrebt. Dieses Bestreben besteht im Grunde aus zwei Teilen, die natürlich eng miteinander verwoben, aber nicht identisch sind. Die Quantentheorie stellt der heutigen Auffassung nach ein allgemeines Beschreibungsschema für beliebige dynamische Objekte in der Natur dar. In der Teilchenphysik ist es nun das Ziel, eine ebenso allgemeine Theorie zu finden, aus der hervorgeht, welche Arten von Objekten es gibt und welche Eigenschaften diese besitzen. Dies würde insbesondere die Rückführung der bekannten Wechselwirkungen auf ein einheitliches Prinzip bedeuten. Durch diese Fragestellung wird man jedoch direkt auf das zweite Problem geführt, dessen Lösung in gewisser Hinsicht die Voraussetzung für die Behandlung des ersten ist. Dieses besteht nämlich in der Wesensfremdheit der Gravitation zu den anderen Wechselwirkungen. Denn einerseits wird die Gravitation im Rahmen der Allgemeinen Relativitätstheorie als Eigenschaft der Raumzeit beschrieben, während die anderen fundamentalen Wechselwirkungen durch Felder auf der Raumzeit beschrieben werden. Andererseits ist die Allgemeine Relativitätstheorie in ihrer bisher erreichten Formulierung eine klassische Theorie, während die anderen Wechselwirkungen in den Rahmen relativistischer Quantenfeldtheorien eingebettet sind. Es muss also eine Quantentheorie der Gravitation, damit aber letztlich eine quantentheoretische Beschreibung der Raumzeitstruktur selbst gefunden werden. Die nichtkommutative Geometrie ist ein Versuch ein quantentheoretisches Element in die Beschreibung der Raumzeit hineinzubringen, indem man Vertauschungsrelationen zwischen den Koordinaten fordert, was in gewisser Hinsicht einem Prozess der Quantisierung ähnelt. Hierbei spielt die Gravitation aber im Grunde noch keine Rolle. Wie bereits erwähnt ergibt sich im Rahmen von Stringtheorien eine nichtkommutative Geometrie [@Seiberg:1999vs]. In diesen wird die Gravitation grundsätzlich garnicht als Struktur der Raumzeit, sondern als Konsequenz bestimmter Anregungszustände von Strings beschrieben. Aus ihnen geht natürlich auch die Motivation hervor, eine Raumzeit mit zusätzlichen kompaktifizierten Dimensionen zu postulieren, auch wenn diese im Gegensatz zu der im ersten Teil dieser Arbeit untersuchten Annahme von einer Kompaktifizierung ausgehen, die nur Voraussagen jenseits des in absehbarer Zeit empirisch zugänglichen zulassen [@Polchinsky]. Daneben steht das Hierarchieproblem, das durch die Annahme zusätzlicher Dimensionen gelöst werden könnte. Ihm scheint aber keine so prinzipielle Bedeutung zuzukommen wie den beiden ersten grundsätzlichen Schwierigkeiten. Bei beiden Annahmen, sowohl derer der nichtkommutativen Geometrie alsauch jener der zusätzlichen kompaktifizierten Dimensionen, handelt es sich in jedem Falle um wichtige Ansätze zu einer Erweiterung der Physik, denen weiterhin eine große Bedeutung in der theoretischen Physik zukommen wird und die auch rein mathematisch von großem Interesse sind. Es muss aber auch kritisch erwähnt werden, dass sie in Bezug zur Lösung der oben angesprochenen konzeptionellen Probleme grundsätzlicher Natur nur teilweise einen Fortschritt darstellen. Diese bestehen eben gerade darin, die Dualität zwischen einer Beschreibung von Objekten auf einer vorgegebenen Raumzeit und der Beschreibung der Gravitation als Eigenschaft einer dynamischen Raumzeit aufzuheben und zu einer einheitlichen quantentheoretischen Beschreibung zu gelangen. Der Ansatz der kanonischen Quantisierung der Allgemeinen Relativitätstheorie, wie er im Rahmen der Loop-Quantengravitation untersucht wird, scheint diesbezüglich noch überzeugender. Dies liegt einerseits daran, dass er den wichtigsten Grundaussagen der Quantentheorie und der Allgemeinen Relativitätstheorie gerecht wird, zu denen bezüglich letzterer Theorie vor allem die Hintergrundunabhängigkeit gehört. Andererseits kommt er ohne über die beiden Theorien hinausgehende Ad-hoc-Annahmen aus [@Rovelli], was dem heuristischen Gebot Rechnung trägt, eine Theorie auf so wenige zusätzliche Annahmen wie möglich zu gründen. Dies gilt auch dann, wenn man hinzufügt, dass er zunächst nichts über eine Vereinheitlichung mit den anderen Wechselwirkungen aussagt.
Anhang {#anhang .unnumbered}
======
In diesem Anhang soll gezeigt werden, dass der dem Feynmangraphen ($\ref{Graph2}$) entsprechende Beitrag zur S-Matrix bzw. Feynmanamplitude für den Quarkprozess verschwindet. Unter Verwendung von ($\ref{GravitonpropagatorMasse}$),($\ref{VertexFFG}$) und ($\ref{VertexVbVbG}$) ergibt sich folgender Ausdruck für den Beitrag zur S-Matrix
$$\begin{aligned}
S_{Graviton}(q(p_1,u_1)+\bar q(p_2,v_2) \rightarrow Z(k_1,Z_1)+Z(k_2,Z_2))\nonumber\\
=\frac{1}{(2\pi)^4}\int d^4 k \frac{u_1}{(2\pi)^{\frac{3}{2}}}\frac{\bar v_2}{(2\pi)^{\frac{3}{2}}}
\left(-\frac{i}{4\bar M_P}[(p_1+p_2)^\mu \gamma^\nu+(p_1+p_2)^\nu \gamma^\mu]\right)\nonumber\\
\cdot (2\pi)^4\delta^4(p_1+p_2-k)
\sum_n \frac{iP_{\mu\nu\rho\sigma}}{k^2-m^2}
(2\pi)^4\delta^4(k-k_1-k_2)\nonumber\\
\cdot \left(-\frac{i}{\bar M_P}\delta^{cd}\left[W^{\rho\sigma\gamma\delta}+W^{\sigma\rho\gamma\delta}\right] \right)
\frac{Z_{1\gamma c}}{(2\pi)^\frac{3}{2}\sqrt{k_{10}}} \frac{Z_{2\delta d}}{(2\pi)^\frac{3}{2}\sqrt{k_{20}}},\nonumber\end{aligned}$$
wobei $W^{\rho\sigma\gamma\delta}$ und $P_{\mu\nu\rho\sigma}$ wieder gemäß ($\ref{VertexVbVbG}$) und ($\ref{PolarisationstensorGraviton}$) definiert sind. Dies kann vereinfacht werden zu
$$\begin{aligned}
S_G&=&\frac{-i}{(2\pi)^2 \sqrt{2k_{10}} \sqrt{2k_{20}}} \frac{1}{\bar M_P^2} \sum_n \frac{1}{p^2-m^2} \nonumber\\
&&\cdot u_1 \bar v_2 [(p_1+p_2)^\mu \gamma^\nu+(p_1+p_2)^\nu \gamma^\mu] P_{\mu\nu\rho\sigma} \left[W^{\rho\sigma\gamma\delta}+W^{\sigma\rho\gamma\delta}\right] Z_{1\gamma} Z_{2\delta} \nonumber\\
&&\cdot \delta^4(p-k_1-k_2).\nonumber\end{aligned}$$
Mit $p=p_1+p_2$, $A=\frac{1}{(2\pi)^3 \sqrt{2k_{10}}\sqrt{2k_{20}}}\frac{1}{\bar M_P^2} \sum_n \frac{1}{p^2-m^2}$ und ($\ref{SMatrixFeynman}$) erhält man folgenden Ausdruck für den Beitrag zur Feynmanamplitude
$$M_G=A u_1 \bar v_2 [p^\mu \gamma^\nu+p^\nu \gamma^\mu]P_{\mu\nu\rho\sigma}
\left[W^{\rho\sigma\gamma\delta}+W^{\sigma\rho\gamma\delta}\right]Z_{1\gamma} Z_{2\delta}\nonumber$$
und durch Ausnutzung der Symmetrieeigenschaften des Polarisationstensors des Gravitons $P_{\mu\nu\rho\sigma}$
$$M_G=4A\cdot u^1\bar v^2 \cdot p^\mu \gamma^\nu \cdot P_{\mu\nu\rho\sigma} \cdot W^{\rho\sigma\gamma\delta} \cdot Z_\gamma Z_\delta.\nonumber$$
Im Schwerpunktsystem gilt
$$k_1^\mu Z_1^\mu=0\quad,\quad k_2^\mu Z_2^\mu=0\nonumber$$
und
$${{\bf p}}_1=-{{\bf p}}_2\quad,\quad {{\bf p}}=0\quad,\quad {{\bf k}}_1=-{{\bf k}}_2.\nonumber$$
(Die fettgedruckten Größen bezeichnen Dreiervektoren.) Durch Einsetzen der Ausdrücke für $W^{\rho\sigma\gamma\delta}$ und $P_{\mu\nu\rho\sigma}$ und Ausnutzen der Relationen für die Polarisation des Z-Teilchens ergibt sich
$$\begin{aligned}
M_G&=&4A u_1 \bar v_2 p^\mu \gamma^\nu
\cdot[\frac{1}{2}(\eta_{\mu\rho}\eta_{\nu\sigma}+\eta_{\mu\sigma}\eta_{\nu\rho})-\frac{1}{3}\eta_{\mu\nu}\eta_{\rho\sigma}\nonumber\\
&&-\frac{1}{2m^2}(\eta_{\mu\rho}p_{\nu}p_{\sigma}+\eta_{\nu\sigma}p_{\mu}p_{\rho}+\eta_{\mu\sigma}p_{\nu}p_{\rho}+\eta_{\nu\rho}p_{\mu}p_{\sigma})\nonumber\\
&&+\frac{1}{3m^2}\eta_{\rho\sigma}p_\mu p_\nu+\frac{1}{3m^2}\eta_{\mu\nu}p_\rho p_\sigma+\frac{2}{3m^4} p_\mu p_\nu p_\rho p_\sigma]\nonumber\\
&&\cdot\left[-\frac{1}{2}\eta^{\rho\sigma}(k_1 \cdot k_2)(Z_1 \cdot Z_2)+(Z_1 \cdot Z_2)(k_1^\rho k_2^\sigma)+(k_1 \cdot k_2) Z_1^\rho Z_2^\sigma \right].\nonumber\end{aligned}$$
Da ${{\bf p}}=0$ gilt, liefern bei der Multiplikation des letzten Termes der unteren eckigen Klammer mit der oberen eckigen Klammer nur die Terme, die $\eta_{\rho\sigma}$ enthalten (wobei $\rho$ und $\sigma$ hier die Summationsindizes bezüglich $Z^\rho$ und $Z^\sigma$ bezeichnen), einen Beitrag. Damit tauchen aber keine Terme auf, welche den Faktor $\gamma^\mu Z_\mu$ enthalten. Das bedeutet, dass mn der Form nach einen Ausdruck der folgenden Gestalt erhält
$$M_G=4A\cdot u_1 \bar v_2 \cdot [a \gamma^\mu p_\mu+b \gamma^\mu k_{1\mu}+c \gamma^\mu k_{2\mu}](Z_1 \cdot Z_2),\nonumber$$
wobei a, b und c skalare Größen sind, deren genaue Gestalt für die weitere Argumentation keine Rolle spielt. Da $M_G$ aber symmetrisch in $k_1$ und $k_2$ ist, muss $b=c$ gelten. Es gilt aber auch $k_1+k_2=p$ und damit $\gamma^\mu k^{1}_\mu+\gamma^\mu k^2_\mu=\gamma^\mu p_\mu$. Das führt auf folgenden Ausdruck
$$M_G=4A\cdot u_1 \bar v_2 \cdot d \gamma^\mu p_\mu (Z_1 \cdot Z_2),\nonumber$$
mit $d=a+2b$. Wenn man nun das Quadrat der Feynmanamplitude bildet, wobei über die Spineinstellungen summiert wird, ergibt sich
$$\sum_\sigma |M|_G^2=16 A^2 d^2 \sum_\sigma (\bar v_2 \gamma^\mu p_\mu u_1)(\bar u_1 \gamma^\mu p_\mu v_2)(Z_1 \cdot Z_2).\nonumber$$
Wenn man sich nun den Term $\sum_\sigma (\bar v_2 \gamma^\mu p_\mu u_1)(\bar u_1 \gamma^\nu p_\nu v_2)$ alleine betrachtet und
$$\sum_\sigma(\bar v_2 \gamma^\mu u_1)(\bar u_1 \gamma^\nu v_2)=Tr\{\gamma^\mu(\gamma^\rho p_{2\rho}-m_f) \gamma^\nu(\gamma^\sigma p^1_\sigma+m_f)\}\nonumber$$
ausnutzt (wobei $m_f$ die Fermionenmasse bezeichnet), erhält man
$$\begin{aligned}
\sum_\sigma (\bar v_2 \gamma^\mu p_\mu u_1)(\bar u_1 \gamma^\nu p_\nu v_2)\nonumber\\
=p_\mu p_\nu Tr\{\gamma^\mu(\gamma^\rho p_{2\rho}-m_f) \gamma^\nu(\gamma^\sigma p_{1\sigma}+m_f)\}.\nonumber\end{aligned}$$
Die Spur einer ungeraden Anzahl von Gammamatrizen verschwindet. Desweiteren gelten die beiden Relationen
$$Tr\{\gamma^\mu \gamma^\nu\}=\eta^{\mu\nu}\quad,\quad Tr\{\gamma^\mu \gamma^\rho \gamma^\nu \gamma^\sigma\}=\eta^{\mu\rho}\eta^{\nu\sigma}-\eta^{\nu\mu}\eta^{\rho\sigma}+\eta^{\mu\sigma}\eta^{\rho\nu}.\nonumber$$
Damit erhält man
$$\begin{aligned}
\sum_\sigma (\bar v_2 \gamma^\mu p_\mu u_1)(\bar u_1 \gamma^\nu p_\nu v_2)\nonumber\\
=p_\mu p_\nu [p_{2\rho} p_{1\sigma} (\eta^{\mu\rho}\eta^{\nu\sigma}-\eta^{\nu\mu}\eta^{\rho\sigma}+\eta^{\mu\sigma}\eta^{\rho\nu})-m_f^2\eta^{\mu\nu}]\nonumber\\
=p_\mu p_\nu[p_2^\mu p_1^\nu-(p_1 \cdot p_2)\eta^{\mu\nu}+p_2^\mu p_1^\nu)]-m_f^2 p^2\nonumber\\
=(p\cdot p_1)(p\cdot p_2)-p^2(p_1\cdot p_2)+(p_1\cdot p)(p_2\cdot p)-m_f^2 p^2\nonumber\\
=4E^4-4 E^2(E^2-{{\bf p}}_1 {{\bf p}}_2)+4E^4-4E^2 m_f^2\nonumber\\
=8E^4-4 E^2(2 E^2-m_f^2)-4E^2 m_f^2=0.\nonumber\end{aligned}$$
Im vorletzten Schritt wurde ${{\bf p}}_2=-{{\bf p}}_1$ und $E^2=(p_1)^2+m_f^2=(p_2)^2+m_f^2$ ausgenutzt. Damit verschwindet aber $\sum_\sigma |M|_G^2$ und damit auch $M_G$ selbst.
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Danksagung {#danksagung .unnumbered}
==========
Natürlich ist es unmöglich, all jenen zu danken, die letztlich indirekt zur Entstehung dieser Diplomarbeit beigetragen haben, indem sie mir beim Erwerb des Wissens halfen, das die unabdingbare Voraussetzung für die Behandlung der hier vorgestellten Themen darstellt. Hier wären zahllose Tutoren, Professoren und Buchautoren und wenn man die Entwicklung nur weit genug zurückverfolgt, auch viele ehemalige Lehrer aufzuführen, denen mein innigster Dank ausgesprochen werden müsste. Vier Namen sind jedoch zu nennen, denen in besonderer Weise Dank gebührt. Es handelt sich um Benjamin Koch, Xavier Calmet, Marcus Bleicher und Horst Stöcker.
Auf Benjamin Koch geht die Idee zurück, die ZZ-Produktionsrate in der beschriebenen effektiven Quantenfeldtheorie der Gravitation unter Einbeziehung zusätzlicher Dimensionen zu untersuchen. Ihm verdanke ich unzählige Ratschläge und hilfreiche Diskussionen. Xavier Calmet hat mir den Vorschlag gemacht, die oben durchgeführte Untersuchung bezüglich des Higgsmechanismus auf einer nichtkommutativen Raumzeit durchzuführen. Während zweier sehr angenehmer einwöchiger Aufenthalte bei ihm in Brüssel und über das Internet kam es zu einem fruchtbaren wissenschaftlichen Dialog. Marcus Bleicher danke ich dafür, dass er mich zu Xavier Calmet vermittelte und mir die beiden Reisen nach Brüssel ermöglichte. Darüber hinaus hat er mich in den letzten Monaten in vielerlei Hinsicht unterstützt. Hierbei ragen vor allem seine vielen wertvollen Hinweise in Zusamenhang mit der Veröffentlichung eines gemeinsamen Artikels hervor. Schließlich danke ich meinem offiziellen Betreuer Horst Stöcker. Er hat mir durch die Aufnahme ans Institut für Theoretische Physik überhaupt erst die Möglichkeit eröffnet, in diesem Gebiet zu arbeiten. Der erwähnte Artikel geht auf seine Idee zu einem interessanten Sommerprojekt zurück. Vor allem aber hat er mit einer hervorragenden sechs Semester währenden Einführungsvorlesung in die Theoretische Physik die Grundlage für alle weitere Beschäftigung mit dieser faszinierenden Wissenschaft gelegt.
Am meisten aber schulde ich meinen Eltern Dank. Sie sorgten dafür, dass ich mich während meines Studiums ganz auf die Physik konzentrieren konnte. Noch bedeutender aber ist, dass ich es im Grunde ihnen zu verdanken habe, mit jener Literatur in Berührung gekommen zu sein, die mir die Dimension so vieler Fragen in Naturwissenschaft und Philosophie eröffnete und mich die für unser Weltbild konstitutive Rolle der Physik als basalster aller Naturwissenschaften erkennen ließ.
Ich versichere hiermit, dass ich die vorliegende Arbeit selbständig verfasst, keine anderen als die angegebenen Hilfsmittel verwendet und sämtliche Stellen, die benutzten Werken im Wortlaut oder dem Sinne nach entnommen sind, mit Quellen- bzw. Herkunftsangaben kenntlich gemacht habe.\
\
Martin Kober
|
---
author:
- 'Yu Chen[^1],Jin Cheng[^2],Yu Jiang[^3]and Keji Liu[^4]'
title: '**A Time Delay Dynamic System with External Source for the Local Outbreak of 2019-nCoV**'
---
**Abstract.** How to model the 2019 CoronaVirus (2019-nCov) spread in China is one of the most urgent and interesting problems in applied mathematics. In this paper, we propose a novel time delay dynamic system with external source to describe the trend of local outbreak for the 2019-nCoV. The external source is introduced in the newly proposed dynamic system, which can be considered as the suspected people travel to different areas. The numerical simulations exhibit the dynamic system with the external source is more reliable than the one without it, and the rate of isolation is extremely important for controlling the increase of cumulative confirmed people of 2019-nCoV. Based on our numerical simulation results with the public data, we suggest that the local government should have some more strict measures to maintain the rate of isolation. Otherwise the local cumulative confirmed people of 2019-nCoV might be out of control.
ß**Key words.** Dynamic system, external source, prediction, 2019-nCoV.
ß[**MSC classifications**]{}. 35R30, 65N21.
Introduction {#sec:intro}
=============
In late December 2019, a cluster of serious pneumonia cases in Wuhan was caused by a novel coronavirus, and the outbreak of pneumonia began to attract considerable attention in the world. Coronaviruses are enveloped nonsegmented positive-sense RNA viruses belonging to the family Coronaviridae and the order Nidovirales which are discovered and characterized in 1965 and are broadly distributed in humans and other mammals. In humans, most of the coronaviruses cause mild respiratory infections, but rarer forms such as the “Severe Acute Respiratory Syndrome” (SARS) outbreak in 2003 in China and the “Middle East Respiratory Syndrome” (MERS) outbreak in 2012 in Saudi Arabia and outbreak in 2015 in South Korea had cased more than 10000 cumulative cases. In more details, there are more than 8000 confirmed SARS cases and 2200 confirmed MERS 2000 cases separately. Although a lot of coronaviruses had been identified and characterized, they might be a tip of the iceberg and lots of potential severe and novel zoonotic coronaviruses needed to be revealed.
The World Heath Organization (WHO) designated the causative agent as the 2019 novel coronavirus (2019-nCoV), which was identified by the Chinese authorities. Because Wuhan is the capital of Hubei province and the 7th largest city China and the largest transport hub in the central part of China, it transports millions of people to lots of cities in China and many countries in the world everyday. Based on the special location and transport hub of Wuhan, the Chinese government has revised the law provisions of infectious diseases to add the novel 2019-nCoV as class [**A**]{} agent on January 20th 2020. Moreover, a series of non-pharmaceutical interventions were implemented, say, rigorous isolation of symptomatic, suspected person, 14 days of isolation for the people who traveled from one city to another, strictly prohibit the travel in the many provinces (especially the Hubei province), the public transport is partially shut down in lots of cities, etc. However, the effectiveness and efficiency of these interventions during the early stage is questionable. So far, there are more than 8000 confirmed cases in Wuhan and more than 24000 confirmed cases in China, and the cumulative confirmed cases of 2019-nCoV from January 23rd to February 4th in Wuhan and mainland China are shown in Figure \[fig:demo\](a) and Figure \[fig:demo\](b) respectively. In addition, several exported cases have been confirmed in many other countries including Japan, South Korea, Singapore, USA, Canada, Germany, France, UK, Spain, etc.
{width="47.00000%"} {width="47.00000%"}
\(a) (b)
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The WHO has recognized that the mathematical models of epidemic play a significant role in informing evidence-based decisions by health decision and policy makers. In order to determine the impact of prevention and control of infection in different positions (i.e. provinces and cities), the strength and duration of isolation, the value for the rate of recovery, we propose a novel dynamic system with time delay and external source in this paper. With the help of this novel system, we are able to not only predict the trend for the outbreak of 2019-nCoV in different districts of China but also provide some helpful suggestions to achieve the maximal protection of population with the minimal interruption of social-economic activities.
Recently we propose a novel time delay dynamical system to describe the 2019-nCoV spread in China [@ChenArxiv2020; @Yan2020], but it is not suitable to describe the trend of local outbreak for the 2019-nCoV. In this paper, we propose a novel time delay dynamic system with external source. In the newly proposed system, the external source term is added which can be considered as the suspected people of Wuhan (or other cities) travel to other districts of China. Moreover, we apply different kernel function in the novel system which can describe the local outbreak of 2019-nCoV more accurately.
The rest of paper is organized as follows: in section 2, we shall propose the notations, the assumptions and the corresponding novel time delay dynamic system with external source. The effective approach for estimating the parameters of novel dynamic system and the prospective cumulative confirmed people are provided in Section 3. Based on the public data, several numerical examples are exhibited in Section 4 to verify the accuracy and effectiveness of our estimation scheme and dynamic system. Finally, we present some concluding remarks and suggestions in section 5.
The Time Delay Dynamic System with External Source
==================================================
In this section, we shall state a novel dynamic system with time delay and external source to describe the local outbreak of 2019-nCoV in China. The people in our novel dynamic system are separated into 5 kinds: external suspected people (external source), infected people, confirmed people, isolated people and cured people. Furthermore, we apply following notations to describe them,
- $I(t)$: cumulative infected people at time $t$;
- $J(t)$: cumulative confirmed people at time $t$;
- $G(t)$: currently isolated people who are infected but still in latent period at time $t$;
- $R(t)$: cumulative cured people at time $t$.
The assumptions of external source, spread rate $\beta$, latent period $\tau_1$, delay period $t-\tau_1$, exposed people and cured rate $\kappa$ in our novel system are presented as follows:
1. The transfer of infected people is assumed to be 1-to-1. And the area with external source and the area of destination would be specified in the system.
2. Suppose the infected person can transfer the coronavirus to others at a spread rate $\beta$, which is defined by the average amount of people becoming infected by this person in unit time.
3. In average, the infected people experience a latent period of $\tau_1$ days before they display obvious symptoms. Moreover, we assume the infected person with palpable symptoms would seek for treatment and therefore become confirmed people.
4. Some of the infected people would be exposed in the latent period $\tau_1$ until they are confirmed. The average exposed period of these people are $\tau_1-\tau_1'$ days, which means they would be confirmed in the next $\tau_1'$ days. Some other part of the infected people are isolated during latent period according to investigation of diagnosed cases.
5. No matter the cumulative confirmed people $J(t)$ are isolated before diagnosed or not, they are consist of the population infected at time $t-\tau_1$ averagely.
6. Suppose the individual would no longer transmit the coronavirus to others when he/she is isolated or in the treatment. Consequently, the exposed people at time $t$ are $I(t)-J(t)-G(t)$.
7. It is $\tau_2$ days in average for the confirmed people become cured with rate $\kappa$ or dead with rate $1-\kappa$.
With the help of the notations and assumptions stated previously, the novel time delay dynamic system with external source to describe the local outbreak of 2019-nCoV is shown in Figure \[fig:demo2\].
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(0 , 5.5) node\[draw\] (aI) (2 , 3.5) node\[draw\] (aG) (4.2, 5.5) node (aJ1) (4.2, 4.5) node (aJ) (4.2, 3.5) node (aJ2) (7.5, 5.5) node (aR) (7.4, 4.5) node (aRD) (7.5, 3.5) node (aD) ;(aI) node [$\Delta I_{a}$]{};(aG) node [$\Delta G_{a}$]{};(aJ) node [$\Delta J_{a}$]{};(3.75,5.95) rectangle (4.65,2.95);(aR.south) node [$\substack{\displaystyle\Delta R_{a}\\[0.2ex]( \kappa)}$]{};(aD.north) node [$\substack{\displaystyle\Delta D_{a}\\[0.2ex](1-\kappa)}$]{};(6.95,5.95) rectangle (8.05,2.95);(aI.east) – node\[pos=0.50,above\] [$\tau_{1}$ Delay]{} (aJ1.west);(aI.south) |- node\[pos=0.75,above\] [$\tau_{1}-\tau_{1}^{\,\prime}$]{} (aG.west);(aG.east) – node\[pos=0.50,above\] [$\tau_{1}^{\,\prime}$]{} (aJ2.west);(aJ.east) – node\[pos=0.50,above\] [$\tau_{2}$ Delay]{} (aRD.west);(aI.north) – (0,6.7) – node\[pos=0.50,above\] [$\tau_{1}+\tau_{2}$ Delay]{} (7.5,6.7) – (aR.north);(-0.7,2.7) rectangle (8.3,7.4);(0 , 0) node\[draw\] (bI) (2 ,-2) node\[draw\] (bG) (4.2, 0) node (bJ1) (4.2,-1) node (bJ) (4.2,-2) node (bJ2) (7.5, 0) node (bR) (7.4,-1) node (bRD) (7.5,-2) node (bD) ;(bI) node [$\Delta I_{b}$]{};(bG) node [$\Delta G_{b}$]{};(bJ) node [$\Delta J_{b}$]{};(3.75,0.45) rectangle (4.65,-2.45);(bR.south) node [$\substack{\displaystyle\Delta R_{b}\\[0.2ex]( \kappa)}$]{};(bD.north) node [$\substack{\displaystyle\Delta D_{b}\\[0.2ex](1-\kappa)}$]{};(6.95,0.45) rectangle (8.05,-2.45);(bI.east) – node\[pos=0.50,above\] [$\tau_{1}$ Delay]{} (bJ1.west);(bI.south) |- node\[pos=0.75,above\] [$\tau_{1}-\tau_{1}^{\,\prime}$]{} (bG.west);(bG.east) – node\[pos=0.50,above\] [$\tau_{1}^{\,\prime}$]{} (bJ2.west);(bJ.east) – node\[pos=0.50,above\] [$\tau_{2}$ Delay]{} (bRD.west);(bI.north) – (0,1.2) – node\[pos=0.50,above\] [$\tau_{1}+\tau_{2}$ Delay]{} (7.5,1.2) – (bR.north);(-0.7,-2.7) rectangle (8.3,1.9);(3.8,2.7) – (3.8,1.9);
\[fig:demo2\]
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Since the novel dynamic model would be more complicated compared with the one proposed in [@ChenArxiv2020], we first provide the following general form of time delay model that is valid for various cases including those with source or sink, $$\left\{
\begin{aligned}
\frac{\mathrm{d}I}{\mathrm{d}t}&=\tilde{\mathcal{I}}(t)\\[2mm]
\frac{\mathrm{d}J}{\mathrm{d}t}&=\gamma \int_0^t h_1(t-\tau_1,t')\tilde{\mathcal{I}} (t')\mathrm{d}t',\\[2mm]
\frac{\mathrm{d}G}{\mathrm{d}t}&=\tilde{\mathcal{G}}(t)-\int_0^t h_2(t-\tau_2,t')\tilde{\mathcal{G}}(t')\mathrm{d}t' \\[2mm]
\frac{\mathrm{d}R}{\mathrm{d}t}&=\kappa\int_0^t h_3(t-\tau_1-\tau_2,t')\tilde{\mathcal{I}}(t') \mathrm{d}t'.
\end{aligned}
\right.
\label{eq:newmodel-basic}$$
In order to provide a better understanding of the dynamic system , we present some detailed explanations as follows:
1. $\tilde{\mathcal{I}}$ is generally the increase rate of cumulative infected people $I(t)$ at time $t$. The specific form of $\tilde{\mathcal{I}}$ depends on the system is closed or has outflow/inflow, and it would be illustrate in detail later.
2. Because all the cumulative confirmed people $J(t)$ come from the previously infected population, the increment of $I(t)$ at time $t'$ ($t'<t$), i.e., $\tilde{\mathcal{I}}(t')$, which means the increment of $J(t)$ depends on the history of $\tilde{\mathcal{I}}(t)$. If the average delay between infected time and confirmed time is $\tau_1$, the increase rate of $J$ can be represented as $$\gamma\int_0^t h_1(t-\tau_1,t')\tilde{\mathcal{I}}(t')\mathrm{d}t',\label{1delay}$$ where $\gamma$ is the morbidity, and $h_1(\hat{t},t')$ ($\hat{t}=t-\tau_1$) is a distribution which should be normalized as $$\int_0^t h_1(\hat{t}, t')\mathrm{d}t'=1,\quad \hat{t}\in (0,t).$$ We are able to observe that $h_1(\hat{t},t)$ can be regarded as the probability distribution of infection time $t'$, and here we take the normal distribution $$h_1(\hat{t},t')=c_1 e^{-c_2 (\hat{t}-t')^2}$$ with $c_1$ and $c_2$ be constants.
3. The function $\tilde{\mathcal{G}}(t)$ is the newly isolated and infected people. The integral term in the equation for $G(t)$ means the people isolated $\tau_2$ days ago (averagely) and would be confirmed and sent for treatment, who would no longer be counted into to the instant isolated population. In addition, the kernel function in the integral has the following expression $$h_2(\hat{t},t')=c_3 e^{-c_4 (\hat{t}-t')^2}$$ with $c_3$ and $c_4$ be constants.
4. The parameter $\kappa$ is the cured rate. The time delay term is obtained similarly as that of $J(t)$’s, and $h_3(\hat{t},t')=c_5 e^{-c_6 (\hat{t}-t')^2}$ with $c_5$ and $c_6$ be constants.
We next derive the specific form of cases with inflow or outflow sources separately, i.e. the specific form of $\tilde{\mathcal{I}}$. For simplicity we consider the half open cases, i.e. with single outflow or single inflow. The situation with both inflow and outflow can be treated based on similarly. It is assumed that some of the infected people of Area [**a**]{} would transfer to Area [**b**]{}. We employ [**a**]{} and [**b**]{} to distinguish the source area and destination respectively. The dynamic system for destination [**b**]{} applies the output of source area [**a**]{} as input. Although the cured rate and dead rate are the same for the two dynamic systems, the isolation ratio $\ell$ and infection rate $\beta$ are different for these two dynamic systems.
#### Area a with single outflow to Area b
For the case with merely outward transfer, the exposed people at time $t$ is $I_a(t)-J_a(t)-G_a(t)$, which further make $\beta_a(I_a(t)-J_a(t)-G_a(t))$ people infected with $\beta_a$ as the infection rate for this area. Meanwhile, $\nu(t)\theta(I_a(t)-J_a(t)-G_a(t))$ of them transfers to other regions, where $\theta$ is the coefficient of transport activity and $\nu(t)$ is the time-dependent distribution of exposed people that are likely to move out. Consequently, at time $t$ the net increment of the infected number in this region is $$\tilde{\mathcal{I}}_a(t)=\Big(\beta_a-\nu(t)\theta\Big) \Big(I_a(t)-J_a(t)-G_a(t)\Big).
\label{eq-I1}$$ We further assume that $$\tilde{\mathcal{G}}_a(t)=\ell_a \Big(I_a(t)-J_a(t)-G_a(t)\Big)
\label{eq-G1}$$ which means the currently exposed people are isolated at rate $\ell_a$. By substituting and into , we arrive at the following expressions for the single outflow system $$\left\{
\begin{aligned}
\frac{\mathrm{d}I_a}{\mathrm{d}t}&=\tilde{\mathcal{I}}_a(t),\\[2mm]
\frac{\mathrm{d}J_a}{\mathrm{d}t}&=\gamma \int_0^t h_1(t-\tau_1,t')\,\tilde{\mathcal{I}}_a(t')\mathrm{d}t',\\[2mm]
\frac{\mathrm{d}G_a}{\mathrm{d}t}&=\tilde{\mathcal{G}}_a(t)
-\int_0^t h_2(t-\tau'_1,t') \,\tilde{\mathcal{G}}_a(t') \mathrm{d}t',\\[2mm]
\frac{\mathrm{d}R_a}{\mathrm{d}t}&=\kappa\int_0^t h_3(t-\tau_1-\tau_2,t')\, \tilde{\mathcal{I}}_a(t')\mathrm{d}t'.
\end{aligned}
\right.
\label{eq:newmodel-output}$$
#### Area b with single inflow from Area a
We next concern about another system with single external source from Area [**a**]{}. The instant increment of infected people caused by existed exposed people at time $t$ is also assumed as $\beta_b \Big(I_b(t)-J_b(t)-G_b(t)\Big)$, while the external transport contribute to the increment is $\tilde{\mathcal{I}}_{In}(t)$, thus the total increment at that time is $$\tilde{\mathcal{I}}_b(t): =\beta_b \Big(I_b(t)-J_b(t)-G_b(t)\Big)+\tilde{\mathcal{I}}_{In}(t),$$ where $\tilde{\mathcal{I}}_{In}(t)$ is the output of source region, namely, $$\tilde{\mathcal{I}}_{In}=\nu(t)\theta\Big(I_a(t)-J_a(t)-G_a(t)\Big).\label{eq:externalsource}$$ The rate of isolation is still depends on the existed exposed amount of people, so we possess the following form similarly as $$\tilde{\mathcal{G}}_b(t)=\ell_b \Big(I_b(t)-J_b(t)-G_b(t)\Big),
\label{eq-G1}$$ where the currently exposed people are isolated at rate $\ell_b$. According to , we are able to state the novel time delay dynamic system for Area [**b**]{} with the external source from Area [**a**]{} as follows: $$\left\{
\begin{aligned}
\frac{\mathrm{d}I_b}{\mathrm{d}t}&=\tilde{\mathcal{I}}_b(t),\\[2mm]
\frac{\mathrm{d}J_b}{\mathrm{d}t}&=\gamma \int_0^t h_1(t-\tau_1,t')\,\tilde{\mathcal{I}}_b(t')\mathrm{d}t',\\[2mm]
\frac{\mathrm{d}G_b}{\mathrm{d}t}&=\tilde{\mathcal{G}}_b(t)
-\int_0^t h_2(t-\tau'_1,t') \,\tilde{\mathcal{G}}_b(t') \mathrm{d}t',\\[2mm]
\frac{\mathrm{d}R_b}{\mathrm{d}t}&=\kappa\int_0^t h_3(t-\tau_1-\tau_2,t')\, \tilde{\mathcal{I}}_b(t') \mathrm{d}t'.
\end{aligned}
\right.
\label{eq:newmodel-input}
$$
For the purpose of understanding our novel system expressly, we now illustrate the system in detail in the following paragraphs.
We would remark that the rate of spread $\beta$ is a function of $t$ in general since coronaviruses may mutate into new forms and the environment (i.e. temperature, humidity, etc.) may change. However, we only concern about a short period (e.g. 30 days), so the spread rate $\beta$ is assumed to be a constant in each considered province (or city).
(This part should be revised to consist with the new model) Because all the cumulative confirmed people $J(t)$ come from the previously infected population, the increment of $I(t)$ at time $t'$ ($t'<t$), i.e., $\tilde{\mathcal{I}}(t')$, would contribute to $J_1(t)$, which means $J_1(t)$ depends on the history of $\tilde{\mathcal{I}}(t)$. If the average delay between infected time and confirmed time is $\tau_1$, the form of $J_1$ can be represented as $$J_1(t)=\gamma\int_0^t h_1(t-\tau_1,t')\Big[\beta \tilde{\mathcal{I}}(t')+I_{1S}(t')\Big]\mathrm{d}t',\label{1delay}$$ where $\gamma$ is the morbidity, and $h_1(\hat{t},t')$ ($\hat{t}=t-\tau_1$) is a distribution which should be normalized as $$\int_0^t h_1(\hat{t}, t')\mathrm{d}t'=1,\quad \hat{t}\in (0,t).$$ We are able to observe that $h_1(\hat{t},t)$ can be regarded as the probability distribution of infection time $t'$, and we usually take the normal distribution $h_1(\hat{t},t')=c_1 e^{-c_2 (\hat{t}-t')^2}$ with $c_1$ and $c_2$ be constants. In the simplest case, $h_1$ can also be the $\delta$-function $h_1(\hat{t},t')=\delta(\hat{t}-t')$, which means that every infected individual experienced the same latent period and treatment period.
The instant change in $G(t)$ is defined in the following form $$\ell\beta \tilde{\mathcal{I}}(t)+\ell I_{1S}(t)-\int_0^t h_2(t-\tau'_1,t')\ell\beta\Big[\tilde{\mathcal{I}}(t')+I_{1S}(t')\Big] \mathrm{d}t',\label{2delay}$$ where $\ell$ is the rate of isolation for the currently exposed people, and $h_2(\hat{t},t')=c_3 e^{-c_4 (\hat{t}-t')^2}$ with $c_3$ and $c_4$ be constants. This means some of the exposed infectors are newly isolated, and some existent isolated infectors are diagnosed and sent to hospital for treatment. The time delay term $\int_0^t h_2(t-\tau'_1,t')G(t') \mathrm{d}t'$ stands for the newly diagnosed people among $G(t)$ depending on the history of $G(t)$.
As illustrated above, the accumulated cured people at time $t$ comes from the ones infected at $t-\tau_1-\tau_2$ (in average). We apply the time delay term $$\kappa\int_0^t h_3(t-\tau_1-\tau_2,t')\Big[\beta \tilde{\mathcal{I}}(t')+I_{1S}(t')\Big]\mathrm{d}t'\label{3delay}$$ to describe $\frac{dR}{dt}$, where $h_3(\hat{t},t')=c_5 e^{-c_6 (\hat{t}-t')^2}$ with $c_5$ and $c_6$ be constants.
Remark: for a general case, we can assume
$$\tilde{\mathcal{I}}(t)=\beta\Big(I(t)-G(t)-J(t)\Big)+\tilde{\mathcal{I}}_{In}-\tilde{\mathcal{I}}_{Out} .
\label{eq-I1}$$
In sum, the above 1-to-1 single direction model is consistent of two sets of equations corresponding to the source system and destination system. It can be observed that the source system of Area [**a**]{} is independent, which means it can be solved without any information of system of Area [**b**]{}. Conversely, the system of Area [**b**]{} relies on the source system of Area [**a**]{}, so the output of Area [**a**]{} is applied as its input in the computation.
The Estimation and Prediction Scheme
====================================
In this section, we shall first state the optimization method to estimate some parameters of the dynamic system from the official data, and a prediction scheme would be presented to forecast the tendency of outbreak for the 2019-nCoV.
Based on the information of parameters $\{\beta_a,\kappa,\ell_a,\gamma,\tau_1,\tau'_1,\tau_2\}$ and initial conditions $\{I_a(t_0), $ $G_a(t_0), J_a(t_0),R_a(t_0)\}$ in the novel dynamic system , the cumulative cured people $R_a(T)$ and the cumulative confirmed people $J_a(T)$ at any given time $T$ are readily to attain by solving the novel dynamic system numerically. Likewise, the other novel dynamic system can be solved with the extra information of parameters $\{\beta_b,\ell_b\}$, the external source $\tilde{\mathcal{I}}_{In}$ and the initial conditions $\{G_b(t_0), J_b(t_0),R_b(t_0)\}$. In the practical applications, we suggest to apply the Matlab$\circledR$ inner-embedded program [**dde23**]{} to solve the novel dynamic systems and .
In addition, the following conditions for the initial time and some parameters are assumed in the practical applications:
1. [**initial conditions:**]{} On the initial day $t_0$, we suppose $5$ people in the Area [**a**]{} are infected the 2019-nCoV from unknown sources. Moreover, the confirmed, isolated and recovered people are all 0 on the initial day. All these assumptions represent $I_a(t_0)=1$, $G_a(t_0)=J_a(t_0)=R_a(t_0)=0$ and $G_b(t_0)=J_b(t_0)=R_b(t_0)=0$. In the numerical simulation, we further assume that there are no isolation measures implemented before $T=t_0+15$.
2. [**parameters:**]{} According to the present data, we suppose a relatively high cure rate as $\kappa=0.97$, the morbidity is relatively high with $\gamma=0.99$. The average latent period $\tau_1$ and treatment period $\tau_2$ are also regarded as known according to the official data. The average period between getting isolated and diagnosed $\tau_1'$ satisfies $0<\tau_1'<\tau_1$. The known parameter set is summarized in Table \[par-known\].
$\kappa $ $\gamma$ $\tau_1$ $\tau'_1$ $\tau_2$
----------- ---------- ---------- ----------- ----------
0.97 $0.99$ $7$ $5$ $12$
Accordingly, the rest parts of parameters that need to be estimated are as follows, $$\Theta_a:=[\beta_a,\ell_a]\quad\text{and}\quad \Theta_b:=[\beta_b,\ell_b],$$ and the identifications of parameters $\Theta_a$ and $\Theta_b$ come to the following two optimization problems, $$\label{eq:op1}
\min_{\Theta_a}\|J_a(\Theta_a;t)-J^a_{Obs}\|_2,$$ and $$\label{eq:op2}
\min_{\Theta_b}\|J_b(\Theta_b;t)-J^b_{Obs}\|_2,$$ where $J^a_{Obs}$ and $J^b_{Obs}$ are separately the daily official data in Area [**a**]{} and Area [**b**]{} reported by the National Health Commission of China.
Consequently, we solve the optimization problem to derive the parameter $\Theta_a$ initially. In addition, with the knowledge of parameters $\Theta_a$, we are able to predict the trend of local outbreak in Area [**a**]{} and the external source for Area [**b**]{}. Furthermore, solve the optimization problem , we obtain the parameter $\Theta_b$ and are able to predict the tendency of local outbreak in Area [**b**]{}. The whole procedure is concluded as follows:\
[**The Estimation and Prediction Scheme**]{}:
1. Based on the official data $J^a_{Obs}$, we apply the Levenberg-Marquad (LM) method or the Markov chain Monte Carlo (MCMC) method [@KNO; @KS] to solve the optimization problem , and the estimated parameter $\Theta_a^*$ is obtained.
2. With the reconstructed $\Theta_a^*$ , one could acquire the predictions of $\{I_a(t),J_a(t), G_a(t), R_a(t)\}$ and $\tilde{\mathcal{I}}_{In}$.
3. Based on the official data $J^b_{Obs}$, we solve the optimization problem and attain the estimated parameter $\Theta_b^*$ in account of $\tilde{\mathcal{I}}_{In}$.
4. The values of $\{I_b(t),J_b(t), G_b(t), R_b(t)\}$ are obtained by solving the novel dynamic system numerically.
Numerical Simulations
=====================
In this section, we shall present some numerical experiments to verify the accuracy and efficient of the estimation and prediction scheme. It is worth mentioning that the data employed in our novel dynamic system are acquired from the Health Commission of each province and city of China and the National Health Commission of China. Moreover, the data includes the cumulative confirmed people, and the cumulative cured people and the cumulative dead people from Jan. 23rd 2020 to Feb. 4th 2020.
In order to predict the external source, we design the time distribution of exposed people moving out as $$\nu(t)=e^{-0.1(t-t_1)^2}+e^{-0.1(t-t_2)^2},$$ which illuminates there would be two peaks around February 8th and February 20th separately. The reason for these two peaks is that the people would go back to work and school after the Lunar New Year on the specified two days. By implementing the estimation and prediction scheme of section 3, the estimations of parameters are $\beta_a=0.27$ and $\ell_a=0.482$, and $\tilde{\mathcal{I}}_{In}(t)$ with diffident $\theta$ are exhibited in Figure \[fig:ex-source\].
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For the purpose of comparing the tendency of Area [**b**]{} with the external source and without it, we first show the prediction of area [**b**]{} without external source, which means the case with the parameter $\theta=0$, in Figure \[Ideal60\].
{width="50.00000%"}
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It is obviously that the situation of pneumonia would tend to end at the beginning of March and the final cumulative confirmed people would be about 500. However, by observing the prediction of Area [**b**]{} with the external source in Figure \[Ideal5\], the increase of exposed people would lead to the increase of final cumulative confirmed people.
\
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In addition, the ending time of this pneumonia would be absolutely postpone, we show Table \[tab::source\] for reference. We shall remark that the same $\beta_b=0.2413$ and $\ell_b=0.5384$ are employed for the prediction.
$\theta$ cumulative confirmed people
---------- ----------------------------- --
$0\%$ $\approx$500
$1\%$ $\approx$1000
$5\%$ $\approx$4000
$10\%$ $\approx$7500
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Now the question comes to how to decrease the final cumulative confirmed people since the external source is inevitable. One feasible way is the increase of isolated ratio $\ell_b$, and the numerical simulations are shown in Figure \[Ideal7\]. We can note that the cumulative confirmed people are sharply reduce with the high rate of isolation.
\
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In order to provide a better observation of the cumulative confirmed people with different impacts of isolated ratio, we exhibit the numerical results in Table \[tab::isolatione\].
$\ell_b$ cumulative confirmed people
--------------- ----------------------------- --
$\ell_b$ $\approx$4500
50% $\ell_b$ $\approx$7000
90% $\ell_b$ $\approx$4500
180% $\ell_b$ $\approx$3500
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From the numerical simulations, we possess following conclusions
1. The external source plays a significant role in the dynamic system, and the prediction of outbreak with the external source is more reliable than the one without it.
2. The inflow of exposed people would increase the final cumulative confirmed people, and the ending time of this pneumonia would postpone.
3. To avoid of rapid increasing of final cumulative confirmed people, the local government need to implement some more efficient restrictive policies to maintain the rate of isolation.
Conclusions {#sec:con}
===========
In this paper, we have proposed a novel time delay dynamic system with external source. In this system, the suspected people of Area [**a**]{} transfer to Area [**b**]{} is concerned, and it is more reasonable and appropriate than the one in [@ChenArxiv2020; @Yan2020] to describe the trend of local outbreak for the 2019-nCoV. The numerical simulations are carried out to verify the effectiveness and accuracy of the novel time delay dynamic system with external source. Moreover, the newly proposed dynamic system can approximate the true data quite well in this event, and it could further forecast the trend of local event. From the numerical simulations, we would like to advice that the local government apply some more efficient and strict measures to maintain the rate of isolation. Otherwise the local cumulative confirmed people of 2019-nCoV might be out of control.
At present, the parameters involved in the novel dynamic system are time-independent. In the future work, the change in the impact of isolation and spread rate may be assumed to depend on time so as to improve the agreement between real data and estimated solution. Besides, the complex network and stochastic process would be concerned in our dynamic system and the machine learning techniques would be applied to provide a better prediction of tendency for the outbreak of 2019-nCoV.
Acknowledgements
================
This work of Jin Cheng was supported in part by the National Science Foundation of China (NSFC: No. 11971121), and the work of Keji Liu was substantially supported by the Science and Technology Commission of Shanghai Municipality under the “Shanghai Rising-Star Program” No. 19QA1403400. The authors thank for the helpful discussions with Prof. Guanghong Ding, Prof. Wenbin Chen and Prof. Shuai Lu in Fudan University, Prof. Xiang Xu in Zhejiang University, Dr. Yue Yan and Dr. Boxi Xu in Shanghai University of Finance and Economics, and the collection of official data by Mrs. Jingyun Bian. Cheer up Wuhan!
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<span style="font-variant:small-caps;">B. Kaltenbacher, A. Neubauer and O. Scherzer</span>, [*Iterative regularization methods for nonlinear ill-posed problems*]{}, Radon Series on Computational and Applied Mathematics 6. Walter de Gruyter GmbH & Co. KG, Berlin, 2008.
<span style="font-variant:small-caps;">J. Kaipio and E. Somersalo</span>, [*Statistical and Computational Inverse Problems*]{}, Springer, New York, USA, 2005.
<span style="font-variant:small-caps;">Y. Chen, J. Cheng, Y. Jiang and K. Liu</span>, [*A Time Delay Dynamical Model for Outbreak of 2019-nCoV and the Parameter Identification*]{}, arXiv:2002.00418, 2020.
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[^1]: School of Mathematics, Shanghai University of Finance and Economics, 777 Guoding Road, Shanghai 200433, P.R. China. ([yuchen@sufe.edu.cn]{}).
[^2]: School of Mathematical Sciences, Fudan University, Shanghai, 200433, China, P. R. China. ([jcheng@fudan.edu.cn]{}).
[^3]: School of Mathematics, Shanghai University of Finance and Economics, 777 Guoding Road, Shanghai 200433, P.R. China. ([jiang.yu@mail.shufe.edu.cn]{}).
[^4]: School of Mathematics, Shanghai Key Laboratory of Financial Information Technology, Shanghai University of Finance and Economics, 777 Guoding Road, Shanghai 200433, P.R. China. ([liu.keji@sufe.edu.cn; kjliu.ip@gmail.com]{}).
|
---
abstract: |
Recently, a combined approach of CFIE–BAE has been proposed by authors for solving external scattering problems in acoustics. CFIE stands for combined-field integral equations, and BAE is the method of boundary algebraical equation. The combined method is, essentially, a discrete analogue of the boundary element method (BEM), having none of its disadvantages. Namely, due to the discrete nature of BAE one should not compute quadratures of oversingular integrals. Moreover, due to CFIE formulation, the method does not possess spurious resonances.
However, the CFIE–BAE method has an important drawback. Since the modelling is performed in a regular discrete space, the shape of the obstacle should be assembled of elementary “bricks”, so smooth scatterers (like spheres, cylinders, etc) are approximated with a poor accuracy. This loss of accuracy becomes the bottleneck of the method. Here this disadvantage is overcome. The CFIE–BAE method developed for regular meshing of the outer space is coupled in a standard way with a relatively small irregular mesh enabling one to describe the shape of the obstacle accurately enough.
author:
- 'J. Poblet-Puig[^1]'
- 'A. V. Shanin[^2]'
title: Coupling of finite element method with boundary algebraic equations
---
[***Keywords: *** boundary integral, Helmholtz, FEM, wave, scattering]{}
List of symbols and acronyms {#list-of-symbols-and-acronyms .unnumbered}
============================
------------------------------------------ -------------------------------------------------------------------------------
$\alpha,\beta$ coefficients of the numerical technique considered in each domain
${\mathbf{A}},{\mathbf{B}},{\mathbf{C}}$ matrices of the method
${\mathbf{\Pi}}$ projector matrix
$BAE$ Boundary Algebraic Equations
$BEM$ Boundary Element Method
$CFIE$ Combined-Field Integral Equations
$DtN$ Dirichlet-to-Neumann
$\delta_{j,m}$ Dirac delta
$FEM$ Finite Element Method
$f$ force term, sources of the field
$G_{j,m}$ discrete Green’s function
$\Gamma_{{\rm int}}$ scaterer surface (or curve)
$\Gamma_{{\rm ext}}$ boundary between domains $\Omega_{{\rm int}}$ (solved with FEM)
and $\Omega_{{\rm ext}}$ (solved with BAE)
$\gamma_{{\rm ext}}$ set of nodes on $\Gamma_{{\rm ext}}$
$\gamma_{\rm o}$ set of nodes surrounding $\gamma_{{\rm ext}}$ and $\gamma_{{\rm ext}}$ itself
$h$ grid or finite element size
$h^{{\rm int}}, h^{{\rm ext}}$ fluxes across $\Gamma_{{\rm ext}}$
${\mathcal{K}}$ wavenumber
$R$ radius of the circular scatterer
$u$ main variable (scattered field)
$\Omega_{{\rm int}}$ domain inside $\Gamma_{{\rm ext}}$ and around the scatterer
$\omega_{{\rm int}}$ set of nodes in $\Omega_{{\rm int}}$
$\omega'_{{\rm int}}$ set of elements in $\Omega_{{\rm int}}$
$\Omega_{{\rm ext}}$ infinite domain outside $\Gamma_{{\rm ext}}$
$\omega_{{\rm int}}$ set of nodes in $\Omega_{{\rm ext}}$
$\omega'_{{\rm int}}$ set of elements in $\Omega_{{\rm ext}}$
$\Omega$ entire space covered with uniform (periodic) mesh
$\omega$ set of nodes in $\Omega$
$\omega'$ set of elements in $\Omega$
------------------------------------------ -------------------------------------------------------------------------------
Introduction
============
The problem of external acoustic scattering has recently been solved [@poblet-PVS:2015] by means of the boundary algebraic equations method (BAE [@Martinsson-Rodin:2009; @Gillman-Martinsson:2010; @Tsukerman:2011; @Bhat-Osting:2009]) and considering a combined-field integral formulation (CFIE, [@Burton-Miller:1971; @kirkup:1998]). The resulting method is, essentially, a discrete analogue of the boundary element method (BEM) that inherits the good properties of BAE and the advantages of CFIE, avoiding most of the BEM drawbacks. On the one hand, no quadratures of oversingular integrals have to be computed due to the discrete nature of BAE. On the other hand, the resulting integral equations are free of spurious resonances due to the CFIE formulation [@poblet-PVS:2014].
However, the main drawback of the CFIE–BAE method is the reduction of accuracy when smooth scatterers with curved surfaces such as spheres or cylinders are considered. This is because the method is based on a regular discretisation of the space (grid) and the obstacles must be approximated by means of the closest brick-description.
Our goal here is to present a complementary formulation where the CFIE–BAE method is coupled with some more versatile numerical technique in order to deal with arbitrary shaped scatterers. This will typically be a thin layer of finite elements (FEM) between the obstacle surface and a close grid-shaped boundary that surrounds the obstacle. The FEM domain has on the one side the boundary conditions corresponding to the scatterer and on the other side the coupling with the CFIE–BAE. This acts as a method for domain truncation and exactly imposes the radiation boundary conditions.
The coupling of numerical methods in order to maximize the benefits and reduce the disadvantages of each one has been often used. See for example [@Zienkiewicz-ZKB:1977] where the FEM was complemented with a boundary integral method to deal with radiation conditions or [@JohnsonNedelec:1980] where the stability conditions of FEM–BEM couplings were studied.
Some more recent works on the FEM–BEM coupling applied to the scattering of waves can be found, see for example [@hsiao:1991; @Chiang:2000; @Gatica:2009]. However, to the best of the authors knowledge, the coupling BAE–FEM has not been considered.
The method presented here can also be understood as an alternative to impose the radiation boundary condition and truncate the computation of domains. It has the added value that the obtained solutions are ‘exact’ in the sense that no numerical artefact is required. In some popular alternatives such as the perfectly matched layers (PML) the reflected waves are attenuated by means of a virtual damping medium placed in the surrounding of the problem domain. It certainly diminishes the reflected waves but it is well know that their parameters (i.e. complex wave number of the medium) must be calibrated properly, see for example [@lin:2009]. Moreover, evanescent waves can remain undamped (see [@zampolli:2008; @basu:2003]) and the quality of solution can be diminished in some zones close to the layer such as the corners. The shape of the PML, the thickness of the layer and the distance from the scatterer are important aspects also for the quality of the solution and in order to derive the PML equations. On the contrary, the approach presented here is more flexible in the sense that it is independent of the shape and the outer boundary can be placed very close to the scatterer without affecting the quality of the solution. This will be illustrated later in [Section \[sec:NumericalExamples\]]{}.
In the remainder of the document, the formulation of the problem is presented in [Section \[sec:formulation\]]{} and the method is detailed in [Section \[sec:method\]]{}. Its properties are shown with the numerical examples in [Section \[sec:NumericalExamples\]]{} before the conclusions. The parts of the development that are not essential have been grouped in the appendices: some details of the derivation of BAE equations in [Appendix \[sec:AppA\]]{} and a proof of solvability in [Appendix \[sec:AppB\]]{}.
Formulation of continuous and discrete problems {#sec:formulation}
===============================================
We consider a 2D of 3D external acoustic stationary problem. The scatterer is approximated by a surface (or a curve) $\Gamma_{{\rm int}}$. The inhomogeneous Helmholtz equation $$\Delta u + {\mathcal{K}}^2 u = f
\label{eq0201}$$ is assumed to be fulfilled in the medium. Variable $u$ may correspond to acoustical pressure or acoustical potential. We assume that the boundary is acoustically hard (Neumann).
Function $f$ represents the sources of the field, i. e. a radiation problem is studied. If the sources are put on the surface $\Gamma_{{\rm int}}$ then one can study radiation of wave by a vibrating boundary. Typically it is necessary to find directivity of the field as the result.
We assume that the exponential factor of an outgoing wave has form of $\exp \{ i {\mathcal{K}}r\}$ for big $r$, where $r$ is the distance from the origin. We assume that ${\mathcal{K}}$ has a vanishing positive imaginary part. Thus, an outgoing wave should decay exponentially at infinity. The same property (it is the [*radiation condition*]{}) should be obeyed by any numerical approximation of $u$.
Split the domain external with respect to $\Gamma_{{\rm int}}$ into two subdomains $\Omega_{{\rm int}}$ and $\Omega_{{\rm ext}}$ (one inside another, see [Fig. \[fig01\]]{}). The boundary between these domains, $\Gamma_{{\rm ext}}$, should have a simple shape. For example, the interior of $\Gamma_{{\rm ext}}$ should be a union of equal cubes/squares. This property will enable us to apply the BAE method to $\Gamma_{{\rm ext}}$.
The boundary $\Gamma_{{\rm ext}}$ does not correspond to any physical interface, but it divides the space into two parts, which be treated numerically in a different manner. The wave process in domain $\Omega_{{\rm ext}}$ will be modelled by the BAE method, thus giving a boundary condition (an approximation of a DtN operator) on $\Gamma_{{\rm ext}}$. This boundary condition should establish the absence of waves coming from infinity on $\Gamma_{{\rm ext}}$. The internal domain $\Omega_{{\rm int}}$ will be treated by a usual finite element method.
We assume that all sources lie inside $\Gamma_{{\rm ext}}$, i. e. belong to $\Omega_{{\rm int}}$.
Consider the entire space $\Omega$ covered with uniform (periodic) mesh (see [Fig. \[fig01\]]{}, right). The mesh in our understanding consists of nodes and finite elements (polygons or polyhedra). The set of all nodes belonging to the uniform mesh will be denoted by $\omega$, and the set of all finite elements by $\omega'$. Let $\bar \Omega_{{\rm int}}$ be a domain composed of some finite elements of the uniform mesh $\bar \omega_{{\rm int}}'$. Denote the set of nodes adjacent to these selected elements by $\bar \omega_{{\rm int}}$. Denote the set of nodes adjacent to the finite elements $\omega_{{\rm ext}}' = \omega' \setminus \bar \omega'_{{\rm int}}$ of the uniform mesh by $\omega_{{\rm ext}}$. The boundary nodes form the set $\gamma_{{\rm ext}}= \bar \omega_{{\rm int}}\cap \omega_{{\rm ext}}$. Obviously, these nodes belong to $\Gamma_{{\rm ext}}$.
Now consider a non-uniform mesh defined in domain $\Omega_{{\rm int}}$ ([Fig. \[fig01\]]{}, left). Denote the set of nodes of this mesh by $\omega_{{\rm int}}$ and the elements of this mesh by $\omega'_{{\rm int}}$. The nodes of $\omega_{{\rm int}}$ lying on the boundary $\Gamma_{{\rm ext}}$ should coincide with $\gamma_{{\rm ext}}$, i. e. the uniform mesh on $\Omega_{{\rm ext}}$ and the arbitrary mesh on $\Omega_{{\rm int}}$ should form together a valid mesh on $\Omega_{{\rm int}}\cup \Omega_{{\rm ext}}$. Also, $\omega_{{\rm int}}\cap \omega_{{\rm ext}}= \gamma_{{\rm ext}}$.
Let equation $$\Delta u + {\mathcal{K}}^2 u = g,
\label{eq0201a}$$ valid in the entire space $\Omega$, be approximated on a uniform mesh $\omega, \omega'$ using the finite element method. Let the nodal values of $u$ and $g$ be denoted by $u_j$, $g_j$. Write the approximation in the form $$\sum_{k \in \omega} \beta_{j,k} u_k = g_j ,
\qquad
j \in \omega .
\label{eq0202}$$ Assume that the coefficients $\beta_{j,k}$ possess the following properties:
- $\beta_{j,k} \ne 0$ only for nodes $j$ and $k$ adjacent to the same finite element;
- the matrix is symmetrical $\beta_{j,k} = \beta_{k,j}$ ;
- since the mesh is periodical, the coefficients do not change when the pair of nodes is translated along the mesh.
Now consider the approximation of equation (\[eq0201\]) in the domain $\Omega_{{\rm int}}\cup \Omega_{{\rm ext}}$. Let this approximation be written in the form $$\sum_{k \in (\omega_{{\rm int}}\cup \omega_{{\rm ext}}) }\alpha_{j,k} u_k
= f_j, \qquad j \in \omega_{{\rm int}}\cup \omega_{{\rm ext}},
\label{eq0203}$$ Let the coefficients $\alpha_{j,k}$ have the following properties:
- $\alpha_{j,k} \ne 0$ only for nodes $j$ and $k$ both belonging to the same finite element;
- the matrix is symmetrical $\alpha_{j,k} = \alpha_{k,j}$ ;
- $\alpha_{j,k} = \beta_{j,k}$ if $j, k \in \omega_{{\rm ext}}$ and at least one of the nodes $j$, $k$ belongs to $\omega_{{\rm ext}}\setminus \gamma_{{\rm ext}}$.
The last point means that the discretisation (\[eq0203\]) is uniform in $\Omega_{{\rm ext}}$.
Since the Neumann boundary condition is imposed on $\Gamma_{{\rm int}}$, equation (\[eq0203\]) naturally incorporates the boundary condition. The method, though, can be easily modified to the case of arbitrary boundary conditions.
Our aim is to present a method for solving (\[eq0203\]). Equation (\[eq0202\]) is auxiliary for the method.
FEM–BAE method {#sec:method}
==============
Split equation (\[eq0203\]) into two equations: $$\sum_{k \in \omega_{{\rm int}}} \alpha_{j,k}^{{\rm int}}u_{k}^{{\rm int}}= f_j + h_j^{{\rm int}},
\qquad
j \in \omega_{{\rm int}}\label{eq0301}$$ $$\sum_{k \in \omega_{{\rm ext}}} \alpha_{j,k}^{{\rm ext}}u_{k}^{{\rm ext}}= h_j^{{\rm ext}},
\qquad
j \in \omega_{{\rm ext}}.
\label{eq0302}$$ The matrices $\alpha^{{\rm int}}_{j,k}$, $\alpha^{{\rm ext}}_{j,k}$ and the flows $h^{{\rm ext}}_j$, $h^{{\rm int}}_j$ should posses the following properties:
- $\alpha_{j,k}^{{\rm int}}= \alpha_{j,k}$ if $j, k \in \omega_{{\rm int}}$, and at least one of the nodes $j, k$ belongs to $\omega_{{\rm int}}\setminus \gamma_{{\rm ext}}$;
- $\alpha_{j,k}^{{\rm ext}}= \alpha_{j,k} = \beta_{j,k}$ if $j, k \in \omega_{{\rm ext}}$, and at least one of the nodes $j, k$ belongs to $\omega_{{\rm ext}}\setminus \gamma_{{\rm ext}}$;
- $\alpha_{j,k}^{{\rm ext}}+ \alpha_{j,k}^{{\rm int}}= \alpha_{j,k}$ if $j, k \in \gamma_{{\rm ext}}$ ;
- matrices are symmetrical: $\alpha^{{\rm ext}}_{j,k} = \alpha^{{\rm ext}}_{k,j}$, $\alpha^{{\rm int}}_{j,k} = \alpha^{{\rm int}}_{k,j}$;
- $h_j^{{\rm ext}}\ne 0$ or $h_j^{{\rm int}}\ne 0$ only if $j \in \gamma_{{\rm ext}}$
- $h^{{\rm ext}}_j = - h^{{\rm int}}_j$ if $j \in \gamma_{{\rm ext}}$.
Matrices $\alpha_{j,k}^{{\rm ext}}$ and $\alpha_{j,k}^{{\rm int}}$ possessing the listed properties can be obtained by assembling the standard FEM matrices performing summation only over the elements belonging to $\omega_{{\rm ext}}'$ or over $\omega_{{\rm int}}'$, respectively. The flows $h^{{\rm ext}}_j$, $h^{{\rm int}}_j$ remain unknown at this stage.
Let also be $u^{{\rm ext}}_j = u^{{\rm int}}_j$ for $j \in \gamma_{{\rm ext}}$.
By summing (\[eq0301\]) and (\[eq0302\]) it is easy to check that the function $$u_j = \left\{ \begin{array}{ll}
u_j^{{\rm ext}}& j \in \omega_{{\rm ext}}\\
u_j^{{\rm int}}& j \in \omega_{{\rm int}}\end{array}\right.
\label{eq0303}$$ is a solution of (\[eq0203\]). Our plan is to substitute (\[eq0302\]) by a relation of the form $$h^{{\rm ext}}_j = \sum_{k \in \gamma_{{\rm ext}}} B_{j,k} u^{{\rm ext}}_k ,
\qquad
j \in \gamma_{{\rm ext}}\label{eq0304}$$ for some matrix $B$, and then represent (\[eq0301\]) in the form $$\left( \alpha^{{\rm int}}_{j,k} + \sum_{m,n \in \gamma_{{\rm ext}}}\Pi^T_{j,m}B_{m,n}\Pi_{n,k} \right) u_k = f_j,
\label{eq0305}$$ where $\Pi_{m,n}$, $m \in \gamma_{{\rm ext}}$, $n \in \omega_{{\rm int}}$ is a projector matrix $$\Pi_{m,n} = \left\{ \begin{array}{ll}
1, & m=n,\quad n\in \gamma_{{\rm ext}}\\
0, & \mbox{otherwise}
\end{array} \right.
\label{eq0306}$$ and $\Pi^T_{m,n} = \Pi_{n,m}$. Then (\[eq0305\]) can be solved as a linear system.
Expression (\[eq0304\]) can be obtained from the BAE–CFIE method [@poblet-PVS:2015]. Here we follow the consideration of [@poblet-PVS:2015]. Let $G_{m,n}$ be an approximation of the Green’s function of equation (\[eq0201a\]), i. e. let $G_{m,n}$ obey equation $$\sum_{k \in \omega}\beta_{j,k} G_{k,m} = \delta_{j,m},
\qquad
j,m \in \omega,
\label{eq0307}$$ and the radiation condition. Here $\delta_{j, m}$ is the Kronecker’s delta. Since (\[eq0201a\]) is an equation on a uniform (periodic) mesh covering the whole space, function $G$ can be computed analytically by the Fourier transformation method. Matrix $G_{m,n}$ is symmetrical: $G_{m,n} = G_{n,m}$ (see [@poblet-PVS:2015]). Introduce a notation $$b_{j,m} =
\sum_{n \in \omega_{{\rm ext}}}
\alpha^{{\rm ext}}_{j,n} G_{n,m} - \delta_{j,m} ,
\qquad
j,m \in \omega_{{\rm ext}}.
\label{eq0308a}$$ where $b_{j,m} \ne 0$ only if $j \in \gamma_{{\rm ext}}$ (note that for $j \in (\omega_{{\rm ext}}\setminus \gamma_{{\rm ext}})$ $\alpha^{{\rm ext}}_{j,n} = \beta_{j,n}$, and (\[eq0307\]) can be applied).
According to [@poblet-PVS:2015], the BAE–CFIE equation connecting $h^{{\rm ext}}_j$ and $u^{{\rm ext}}_j$, $j \in \gamma_{{\rm ext}}$ is as follows: $$\sum_{j \in \gamma_{{\rm ext}}} u^{{\rm ext}}_j A_{j,m} =
\sum_{j \in \gamma_{{\rm ext}}} h^{{\rm ext}}_j
C_{j,m},
\qquad
j,m \in \gamma_{{\rm ext}},
\label{eq0308b}$$ $$A_{j,m} = \delta_{j,m} + b_{j,m} + \nu \sum_{n \in \omega_{{\rm ext}}} b_{j,n} \alpha_{n,m}^{{\rm ext}},
\label{eq0309}$$ $$C_{j,m} =
- \nu \delta_{j,m} + G_{j,m} + \nu \sum_{n \in \omega_{{\rm ext}}}G_{j,n} \alpha_{n,m}^{{\rm ext}}.
\label{eq0310}$$ $\nu$ is an arbitrary complex number with a non-zero imaginary part.
It follows from (\[eq0308b\]) that matrix $B$ from (\[eq0304\]) can be written as $${\mathbf{B}}= ({\mathbf{A}}{\mathbf{C}}^{-1})^T.
\label{eq0311}$$
A known problem associated with the boundary integral equation is linked with formula (\[eq0311\]) or a similar one. Although ${\mathbf{B}}$ should exist for all temporal frequencies, if no special measures are undertaken matrices ${\mathbf{A}}$ and ${\mathbf{C}}$ may be singular. This feature is named spurious resonances. For example, if $\nu =0$ (\[eq0308a\]) corresponds to Kirchhoff formulation of boundary integral equations. The Kirchhoff boundary integral equations are known to be prone to spurious resonances [@Schenck:41; @Benthien-Schenck:1997; @chien:1990]. The CFIE approach is necessary to suppress the spurious resonances. The case ${\rm Im}[\nu] \ne 0$ corresponds to a CFIE formulation.
A sketch of derivation of (\[eq0308b\]) and a proof of invertibility of ${\mathbf{C}}$ under some general condition can be found in the Appendix.
Introduce the set of nodes $\gamma_{\rm o}$ belonging to $\omega_{{\rm ext}}$ and neighbouring $\gamma_{{\rm ext}}$ (i. e. they are the nodes adjacent to the finite elements adjacent to nodes from $\gamma_{{\rm ext}}$). The set $\gamma_{\rm o}$ is finite. By construction, $\gamma_{{\rm ext}}\subset \gamma_{\rm o}$. The summation in (\[eq0309\]) and (\[eq0310\]) can be held along $\gamma_{\rm o}$ instead of $\omega_{{\rm ext}}$.
Let us summarize the procedure of solving (\[eq0203\]).
- The Green’s function $G_{m,n}$ and values $b_{m,n}$ should be tabulated for $m\in \gamma_{{\rm ext}}$, $n \in \gamma_{\rm o}$.
- Matrices ${\mathbf{A}}$, ${\mathbf{C}}$ should be calculated from (\[eq0309\]), (\[eq0310\]) for $j,m \in \gamma_{{\rm ext}}$.
- Matrix ${\mathbf{B}}$ should be found from (\[eq0311\]).
- Equation (\[eq0305\]) should be solved.
As the result of this procedure, one obtains the nodal values of field $u_j^{{\rm int}}$. Thus, the near field becomes known. To get the far field, one needs to perform an additional step of post-processing. Namely, for any $m \in \omega_{{\rm int}}$ $$u_m^{{\rm ext}}= \sum_{j \in \gamma_{{\rm ext}}} (h_j^{{\rm ext}}G_{j,m} - u_j^{{\rm int}}b_{j,m}).
\label{eq0312a}$$ Substituting (\[eq0304\]), obtain $$u_m^{{\rm ext}}= \sum_{j \in \gamma_{{\rm ext}}} u_j^{{\rm int}}\left(
\sum_{k\in \gamma_{{\rm ext}}} B_{k,j} G_{k,m} - b_{j,m}
\right).
\label{eq0312b}$$ If node $m$ is located far enough, asymptotic expressions for $G_{j,m}$ and $b_{m,j}$ can be found. Formula (\[eq0312b\]) provides the solution in the far field (a directivity can be taken from it).
It can be convenient to solve the whole problem at the same time and avoid the explicit inversion of matrix ${\mathbf{C}}$. One should consider a linear system of equations where the unknowns are $\mathbf{u}^{{\rm ext}}$ and $\mathbf{u}^{{\rm int}}$ that contain the nodal values in $\gamma_{{\rm ext}}$ and $\omega_{{\rm int}}$ respectively, and $\mathbf{h}^{{\rm ext}}$ that contain the fluxes $h^{{\rm ext}}$ defined in (\[eq0302\]). The coupled linear system of equations is $$\label{eq:CoupledSystem}
\begin{bmatrix}
{\mathbf{A}}& {\mathbf{0}}& -{\mathbf{C}}\\[2.5ex]
{\mathbf{0}}& {\mathbf{A}}^{{\rm int}}& {\mathbf{\Pi}}^T\\[2.5ex]
{\mathbf{I}}& -{\mathbf{\Pi}}& {\mathbf{0}}\end{bmatrix}
\begin{bmatrix}
\mathbf{u}^{{\rm ext}}\\[2.5ex]
\mathbf{u}^{{\rm int}}\\[2.5ex]
\mathbf{h}^{{\rm ext}}\end{bmatrix} =
\begin{bmatrix}
{\mathbf{0}}\\[2.5ex]
\mathbf{f} \\[2.5ex]
{\mathbf{0}}\end{bmatrix}$$ where ${\mathbf{A}}$ and ${\mathbf{C}}$ are the matrices defined in (\[eq0309\]) and (\[eq0310\]), ${\mathbf{A}}^{{\rm int}}$ is the matrix obtained from (\[eq0301\]) which is typically the usual FEM matrix, ${\mathbf{0}}$ is a null matrix, ${\mathbf{I}}$ the identity and ${\mathbf{\Pi}}$ the projector matrix defined in (\[eq0306\]) (rows for the nodes in $\gamma_{{\rm ext}}$ and columns for the nodes in $\omega_{{\rm int}}$). The force vector includes $\mathbf{f}$ from (\[eq0301\]).
In the linear system (\[eq:CoupledSystem\]) the first block of equations represent (\[eq0308b\]), the second block of equations accounts for (\[eq0301\]) and the continuity of fluxes $h^{{\rm ext}}_j = - h^{{\rm int}}_j$ if $j \in \gamma_{{\rm ext}}$. And finally the third block imposes continuity of variable $u$: $u^{{\rm ext}}_j = u^{{\rm int}}_j$ for $j \in \gamma_{{\rm ext}}$.
Numerical results {#sec:NumericalExamples}
=================
The efficiency of the numerical method is illustrated in a two-dimensional problem with circle-shaped scatterer (see [Fig. \[fig:TheMesh\]]{}(a)). It has analytical solution that is used as reference. The scatterer has a curved surface. This is important in order to demonstrate the improvement caused by the better geometry description of the FEM layer (coupled model) with respect to a staircase approximation based on the regular grid (use of only BAE [@poblet-PVS:2015]). The force, which represents the imposed normal derivative of the variable $u$ at the contour, is chosen in order to generate a scattered wave described by means of only one cylindrical harmonic. The nodal values of the force vector are $$f_{i} = \cos\left( N \varphi_{i} \right), \qquad i \in \gamma_{{\rm int}}$$ The angle $\varphi$ and the radius $R$ of the circle are shown in the sketch of [Fig. \[fig:TheMesh\]]{}(a). $N$ is related with the spatial frequency of the imposed force, $N$ waves exist over the circle. The expression of the scattered field on the circle surface is $$\label{eq:AnalyticalSolution}
u(R,\varphi) = \frac{2 H_{N}^{(1)}\left( {\mathcal{K}}R\right)}{H_{N-1}^{(1)}\left( {\mathcal{K}}R\right) - H_{N+1}^{(1)}\left( {\mathcal{K}}R\right)} \cos\left( N \varphi \right)$$ where $H_{N}^{(1)}$ is the Hankel function of the first kind and order $N$ and ${\mathcal{K}}$ is the wavenumber of the problem.
Different error types play an important role in the numerical solution of this problem: *i)*interpolation and dispersion error of the scattered field; *ii)*error in the description of the oscillatory force imposed on the scatterer surface; and *iii)*geometry error in the approximation of the scatterer shape. Error types *i)* and *ii)* are related with the number of nodes per wave length of the scattered field or the imposed force, respectively. Error type *iii)* is related with the curvature of the scatterer. Each error type can be the dominant error source depending on the frequency range and the geometrical or material parameters of the model.
The mesh in [Fig. \[fig:TheMesh\]]{}(b) is designed in order to have a transition zone between the circle (boundary $\Gamma_{{\rm int}}$) and a closed grid shape. It is forced to be thin in order to use the minimum number of finite elements. This mesh has nodes $\omega_{{\rm int}}$ and elements $\omega'_{{\rm int}}$. The nodes over the internal boundary $\gamma_{{\rm int}}$ are placed exactly on the circle (equally distributed). The force vector is null for nodes not belonging to $\gamma_{{\rm int}}$. The nodes on the external boundary $\gamma_{{\rm ext}}$ are considered in the BAE part of the problem. The mesh is built with the GMSH software [@Geuzaine-Remacle:2009].
The error is measured as $$e = \frac{\left|\left| \mathbf{u}^{{\rm ext}}_{\mathrm{num}} - \mathbf{u}^{{\rm ext}}_{\mathrm{exact}}\right|\right|}{\left|\left| \mathbf{u}^{{\rm ext}}_{\mathrm{exact}}\right|\right|} \simeq \sqrt{\frac{\sum_{i \in \gamma_{{\rm ext}}}^{n} \left| u^{{\rm ext}}_{\mathrm{num},i} - u^{{\rm ext}}_{\mathrm{exact},i}\right|}{\sum_{i \in \gamma_{{\rm ext}}}^{n} \left|u^{{\rm ext}}_{\mathrm{exact},i}\right|}}$$ where ‘num’ is the numerical solution and ‘exact’ the solution obtained with (\[eq:AnalyticalSolution\]).
In all the examples the grid spacing is $h = 1$. If nothing else is specified, the mean finite element size is also $h = 1$ and the layer of finite elements that surrounds the circle has an approximate external radius of $R_{ext} = R + h$.
[Fig. \[fig:InfluenceOfRadiusAndHarmonic\]]{}(a) shows the error evolution with respect to the dimensionless wavenumber ${\mathcal{K}}h$ for several cylindrical scatterers of different size and the harmonic $N=0$. In all the cases the slope of the error curve is close to $2$. This is the expected result for the interpolation error of linear finite elements where $e = \theta h^{2}$, with $\theta$ a constant value [@Bouillard-Ihlenburg:99].
It is observed that the numerical error has a different lower bound for each curve. This value is larger for smaller scatterers (with a more pronounced curvature compared to the element size) due to the geometry error of the linear finite element approximation of the circular shape. This error is invariant with respect to the wavenumber of the problem because it only depends on the relationship between the element size and the curvature of the circle.
The geometry error is comparatively not important for large values of dimensionless wave number (${\mathcal{K}}h \approx 0.3-1.0$) where the interpolation and dispersion error of the scattered field is dominant. On the contrary, geometry error becomes dominant at low frequencies when the scattered field is oscillating with a larger spatial wave length. As an example, consider the circle of radius $R = 3h$ where the exact curved piece of surface that contributes to each node is $ds \simeq {2\pi R / n} = 0.94247781$ ($n=20$ elements around the circle $\Gamma_{{\rm int}}$). Its equivalent finite element length is $0.93860679$ which is slightly different.
For all this, it can be seen in [Fig. \[fig:InfluenceOfRadiusAndHarmonic\]]{}(a) how the theoretical convergence slope is lost for ${\mathcal{K}}h < 0.4$ in the circle of radius $R = 3h$ and for ${\mathcal{K}}h < 0.15$ in the circle of radius $R = 10h$. The circle of radius $R = 30h$ is not sensitive to the geometrical error in the studied frequency range.
The influence of the spatial wavenumber of the imposed force for a scatterer of radius $R = 10 h$ is shown in [Fig. \[fig:InfluenceOfRadiusAndHarmonic\]]{}(b). There are $64$ nodes on the circle. The imposed force describes $N$ complete waves around the circle. Consequently, there are: $64$, $32$ and $16$ nodes per excitation wave length in the harmonics $N=1,2,3$ respectively. This amount of nodes is related with the precision in the computation of the force vector.
In the results of [Fig. \[fig:InfluenceOfRadiusAndHarmonic\]]{}(b) two different zones can be clearly distinguished: large wavenumbers where the interpolation and dispersion error in $u^{{{\rm ext}}}$ is dominant and low frequencies where the error due to the force description is more important. Each curve has a limit wavenumber ${\mathcal{K}}$ for which the error in the solution becomes more or less constant and cannot be reduced with a decrease of ${\mathcal{K}}h$. This limit value of the wavenumber ${\mathcal{K}}$ is related with the number of the harmonic $N$: ${\mathcal{K}}h \simeq 0.4$ for $N=3$, ${\mathcal{K}}h \simeq 0.3$ for $N=2$, and ${\mathcal{K}}h \simeq 0.18$ for $N=1$. The curve corresponding to $N = 0$ is not affected by the error in the description of the force because it is constant all around the scatterer.
[Fig. \[fig:InfluenceOfFemSize\]]{} illustrates which is the effect of reducing the finite element size only on the circle (increase the number of nodes in $\gamma_{{\rm int}}$) and not on the BAE contour (the number of nodes on $\gamma_{{\rm ext}}$ remains constant). The element size on $\Gamma_{{\rm int}}$ is $\sigma h$, with $\sigma = 0.25, 0.5$ and $1$. The results are shown for two circles with radius $R = 3h$ and $R = 10h$. The improvement is more important for the case $R = 3h$ which is more sensitive to the geometry error at small wavenumbers. The reduction of the finite element size around the scatterer reduces the error in the whole frequency range. However, a lower bound (frequency invariant) is found for each $\sigma$ which shows again that it is due to approximation of the scatterer geometry and not due to the proper interpolation of the scattered field.
[Fig. \[fig:InfluenceOfBoxSize\]]{} shows the effect of the finite element mesh truncation. First, the boundary $\Gamma^{{{\rm ext}}}$ is placed at several distances: $R+h$, $R+5h$ and $R+10h$ with a circular scatterer of radius $R = 10h$. It can be seen how the results are almost insensitive (or without clear meaningful trend) to the truncation distance. This is important because it allows the use of the thinnest finite element mesh around the scatterer, only conditioned by scatterer shape and meshing procedures. The use of a small mesh contributes to the reduction of computational costs. On the one hand, there are less unknowns. On the other hand, the range of required values of the discrete Green’s function is smaller.
[Fig. \[fig:InfluenceOfBoxSize\]]{}(b) shows a comparison between the case when the finite element layer is used (‘FEM+BAE’) and the case when it is not considered (‘Only BAE’). In this second case the circular shape of the scatterer is approximated by means of a staircased geometry, defined by the closest grid (as it was done in [@poblet-PVS:2015]). One can observe the improvement caused by the description of the scatterer geometry by means of triangular finite elements comparatively to a grid approximation of the circle. The difference is larger for higher wavenumbers. But the slope or general trend is similar.
Conclusions {#sec:conclusions}
===========
A numerical technique to deal with scattering problems has been presented. On the one hand, it can be understood as a complement to the CFIE–BAE method where a FEM layer is placed around the scatterer in order to better approximate its shape and reduce the geometry error. On the other hand, it can be understood as the use of BAE in order to exactly impose the radiation boundary condition in a FEM model.
It is shown how the resulting method keeps the properties of finite elements. Since linear triangles are considered for the FEM layer, order two convergence is observed. This behaviour is only truncated at very low values of dimensionless wavenumber ${\mathcal{K}}h$ by the geometrical error in the discretisation of the scatterer shape or the approximation of the force vector.
The coupling with FEM largely reduces the numerical error of BAE solutions and helps to overcome its main drawback in problems involving curve-shaped scatterers. This was caused by the staircase approximation of that shapes. That shapes are now approximated by means of standard finite elements without loosing any of the good properties of BAE for scattering problems: no need to compute boundary integrals (which are usually singular in other methods such as BEM), non-singularity of the problem even for the spurious eigenfrequencies of the scatterer and exact representation of the domain truncation.
Appendix A. Derivation of equations (\[eq0312a\]) and (\[eq0308b\]) {#sec:AppA}
===================================================================
First, derive (\[eq0312a\]). Formally the proof can be written as follows. Consider the expression $$\sum_{j,k \in \omega_{{\rm ext}}} u_j^{{\rm ext}}\alpha_{j,k}^{{\rm ext}}G_{k,m}$$ On the one hand, due to (\[eq0302\]) $$\sum_{j,k \in \omega_{{\rm ext}}} u_j^{{\rm ext}}\alpha_{j,k}^{{\rm ext}}G_{k,m} = \sum_{k \in \omega_{{\rm ext}}} h_k G_{k, m}.
\label{eqA01}$$ On the other hand, due to (\[eq0308a\]), $$\sum_{j,k \in \omega_{{\rm ext}}} u_j^{{\rm ext}}\alpha_{j,k}^{{\rm ext}}G_{k,m} =
\sum_{j \in \omega_{{\rm ext}}} u_j^{{{\rm ext}}} (\delta_{j, m} + b_{j, m})
\label{eqA02}$$ If $m \in \omega_{{\rm ext}}$, combining the expressions \[eqA01\] and \[eqA02\], obtain $$u_m^{{\rm ext}}= \sum_{j \in \gamma_{{\rm ext}}} (h_j^{{\rm ext}}G_{j,m} - u_j^{{\rm ext}}b_{j,m}).
\label{eqA03}$$ After substitution $u^{{\rm ext}}_j = u^{{\rm int}}_j$ for $j \in \omega_{{\rm ext}}$ get (\[eq0312a\]). Note that (\[eqA03\]) is valid only for the solution $u^{{\rm ext}}_j$ obeying the radiation condition.
However, this method cannot be applied directly, since the summation is held over an infinite set of nodes $\omega_{{\rm ext}}$. In [@poblet-PVS:2014] one can find a refined procedure. One should truncate the area $\Omega_{{\rm ext}}$, say, by a large square/cube, and apply (\[eqA01\]), (\[eqA02\]) to the truncated mesh. Then one should consider the limit of the size of the square/cube growing to infinity. The radiation condition obeyed by $u_j^{{\rm ext}}$ and $G_{j,m}$ guarantee that the integral over the outer boundary of the sphere/cube vanishes.
Now apply matrix $\alpha^{{\rm ext}}_{m,n}$ to (\[eqA03\]): $$\sum_{m \in \omega_{{\rm ext}}} u_m^{{\rm ext}}\alpha^{{\rm ext}}_{m,n} =
\sum_{m \in \omega_{{\rm ext}}}\sum_{j \in \gamma_{{\rm ext}}} (h_j^{{\rm ext}}G_{j,m} - u_j^{{\rm ext}}b_{j,m}) \alpha^{{\rm ext}}_{m,n}.
\label{eqA04}$$ Here the summation over $m$ causes no problem, since for each $n$ it is held only over the neighbors of $n$, where the coefficients $\alpha^{{\rm ext}}_{m,n}$ are non-zero. Changing the order of summation in (\[eqA04\]) and taking into account (\[eq0302\]), get $$h^{{\rm ext}}_n =
\sum_{j \in \gamma_{{\rm ext}}} \left( h_j^{{\rm ext}}\sum_{m \in \omega_{{\rm ext}}}G_{j,m} \alpha^{{\rm ext}}_{m,n} -
u_j^{{\rm ext}}\sum_{m \in \omega_{{\rm ext}}} b_{j,m} \alpha^{{\rm ext}}_{m,n} \right) .
\label{eqA05}$$ Now multiply (\[eqA05\]) by an arbitrary complex number $\nu$ with a non-zero imaginary part and add to (\[eqA03\]). The result is (\[eq0308b\]).
Appendix B. On invertibility of ${\mathbf{C}}$ {#sec:AppB}
==============================================
The invertibility of ${\mathbf{C}}$ depends on details of realization of the finite element method, so here we can prove a general but relatively weak theorem:
[*If a homogeneous Dirichlet problem on $\Omega_{{\rm ext}}$ has no non-trivial solutions, then matrix ${\mathbf{C}}$ is invertible.*]{}
A homogeneous Dirichlet problem on $\Omega_{{\rm ext}}$ is as follows: Find a function $w_j$ obeying equation $$\sum_{j \in \omega_{{\rm ext}}} \alpha^{{\rm ext}}_{m,j} w_{j} = 0,
\qquad
m \in (\omega_{{\rm ext}}\setminus \gamma_{{\rm ext}}),
\label{eqB01}$$ boundary condition $$w_j = 0 ,
\qquad
j \in \gamma_{{\rm ext}},
\label{eqB02}$$ and the radiation condition.
The uniqueness of solution of a homogeneous Dirichlet problem can be proven in many particular cases.
The proof of the theorem is analogous to that of [@poblet-PVS:2014]. Assume that all coefficients $\beta_{m,n}$ and $\alpha_{j,m}^{{\rm ext}}$ are real. Let matrix ${\mathbf{C}}$ be not invertible. This means that there exists a non-zero vector $v_j$, $j \in \gamma_{{\rm ext}}$ such that $v {\mathbf{C}}$ is a zero vector, i. e. $$\sum_{j \in \gamma_{{\rm ext}}} v_j G_{j,m} = \nu \sum_{j \in \gamma_{{\rm ext}}} v_j
\left(
\delta_{j,m} - G_{j,m} \alpha_{j,m}^{{\rm ext}}\right)
,
\qquad
m \in \gamma_{{\rm ext}}.
\label{eqB03}$$ Consider function $v_j$ on $\gamma_{{\rm ext}}$. Introduce a “single-layer potential” on the uniform mesh $\omega$: $$w_m = \sum_{j \in \gamma_{{\rm ext}}} G_{m,j} v_j,
\qquad
m \in \omega.
\label{eqB04}$$ This function obeys equation (\[eqB01\]) and the radiation condition by construction. Note that $$v_m = \sum_{j \in \bar \omega} G_{m,j} w_j.
\label{eqB05}$$ Thus, (\[eqB03\]) can be written in the form $$w_m = \nu \sum_{j \in \bar \omega_{{\rm int}}} \beta^{{\rm int}}_{m,j} w_j,
\qquad
m \in \gamma_{{\rm ext}},
\label{eqB06}$$ where $$\beta_{m,n}^{{\rm int}}=
\left\{ \begin{array}{ll}
\beta_{m,n} & \mbox{if } m \mbox{ or } n \mbox{ belongs to }\bar \omega_{{\rm int}}\setminus \gamma_{{\rm ext}}\\
\beta_{m,n} - \alpha_{m,n}^{{\rm ext}}& \mbox{otherwise}
\end{array} \right.
\label{eqB07}$$ Note that $\beta^{{\rm int}}_{m,n} \ne 0$ only if $m,n \in \bar \omega_{{\rm int}}$. Note also that $$\sum_{j \in \bar \omega_in} \beta^{{\rm int}}_{m,j} w_j =
\sum_{j \in \bar \omega_in} \beta_{m,j} w_j = 0
\qquad
m \in (\bar \omega_{{\rm int}}\setminus \gamma_{{\rm ext}}).
\label{eqB08}$$
Consider a combination $$\sum_{m,n \in \bar \omega_{{\rm int}}}
w^*_m \beta^{{\rm int}}_{m,n} w_n$$ where $\cdot^*$ denotes complex conjugation. Using (\[eqB06\]) and (\[eqB08\]) one can obtain two representations for this combinations: $$\sum_{m,n \in \bar \omega_{{\rm int}}}
w^*_m \beta^{{\rm int}}_{m,n} w_n =
\nu^{-1} \sum_{m \in \gamma_{{\rm ext}}} w^*_m w_m =
(\nu^*)^{-1} \sum_{m \in \gamma_{{\rm ext}}} w^*_m w_m.
\label{eq09}$$ Thus, we can conclude that $$w_j = 0 ,
\qquad
j \in \gamma_{{\rm ext}},
\label{eqB10}$$ and $w_j$ is a solution of the homogeneous Dirichlet problem. It is non-trivial on $\omega_{{\rm ext}}$, since equations (\[eqB05\]) and $$\sum_{j \in \bar \omega_{{\rm int}}} \beta^{{\rm int}}_{m,j} w_j =0 ,
\qquad
m \in \gamma_{{\rm ext}},
\label{eqB11}$$ (following from (\[eqB06\])), are valid.
Acknowledgements {#acknowledgements .unnumbered}
================
The authors acknowledge the Euro-Russian Academic Network-Plus program (grant number 2012-2734/001-001-EMA2). J. Poblet-Puig from the LaCàN research group is grateful for the sponsorship/funding received from Generalitat de Catalunya (Grant number 2014-SGR-1471). A.V.Shanin has been also supported by Russian Scientific school grant 7062.2016.2 and the Russian Foundation for Basic Research grant 14-02-00573.
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[^1]: correspondence: UPC, Campus Nord B1, Jordi Girona 1, E-08034 Barcelona, Spain, e-mail: jordi.poblet@upc.edu
[^2]: e-mail: a.v.shanin@gmail.com
|
---
abstract: 'We probe a non-supersymmetric D6 + D0 state with D6-branes and find agreement at subleading order between the supergravity and super Yang-Mills description of the long-distance, low-velocity interaction.'
---
Imperial/TP/97-98/58\
hep-th/9806186\
[ Probing a D6 + D0 state with D6-branes:\
SYM - Supergravity correspondence\
at subleading level [^1] ]{}\
\
3 cm
Introduction
============
During the last year and a half there was an intensive study of the correspondence between the Supergravity and the Super Yang-Mills (SYM) descriptions of the long distance interactions between D$p$-branes and their bound states [@bac9511; @lif9610; @bfss9610; @che-tse9705; @mal9709; @kab-tay9712] [^2]. Most of this work considered only the leading $F^4$ terms in the 1-loop SYM effective action, for which there is a known expression for a general gauge field background. For the specific case of D0-brane - D0-brane interaction, the subleading terms were directly computed in SYM theory, and once again agreement was found with the supergravity calculations [@bec-bec9705; @bbpt9706].
This led to the conjecture [@che-tse9709; @mal9709] (see also [@bgl9712; @kes-kra9709]) that the leading part of the $L$-loop term in the SYM effective action in $D=p+1$ dimensions has a universal $F^{2L+2}/M^{(7-p)L}$ structure, and that this term, when computed for a SYM background representing a configuration of interacting branes and in the large $N$ limit, should reproduce the $1/r^{(7-p)L}$ term in the corresponding long-distance supergravity potential. The authors of [@che-tse9709] then proposed a specific form for the $F^6$ terms in the 2-loop effective action, and proceeded to show that it reproduces the subleading supergravity potentials for a variety of configurations. However, only configurations that still preserved some ($1/2$, $1/4$ or $1/8$) supersymmetry were considered. The proposed $F^6$ term has not yet been tested for a [*non-supersymmetric*]{} configuration.
The existence of extreme but non-supersymmetric black holes was shown in [@khu-ort9512], where a supergravity background that after compactification to 10 dimensions describes a dyonic non-supersymmetric black hole was found. A different compactification scheme was used in [@she9705] to obtain an extremal non-supersymmetric solution to low energy type IIA string theory carrying D0- and D6-brane charges. This solution, for non-zero values of both charges, breaks all supersymmetries. The scattering of D0- and D6-branes from these states was studied in [@kes-kra9706; @pie9707], and considering a more general supergravity background in [@bisy9711] (applying the SYM results of [@mal9705; @mal9709]). Once again agreement was found between the leading terms in supergravity and the SYM expressions at 1-loop[^3].
Our aim in the present paper is to extend the supergravity calculation [@bisy9711] of the source-probe potential to subleading terms in $1/r$, and compare these with those obtained using the ansatz of [@che-tse9709] for the 2-loop SYM effective action.
Supergravity description
========================
In this section, we use the same normalisation as in [@bisy9711], $\alpha^{\prime}=1$.
In the supergravity calculation, one considers a probe moving in the background created by a source. Both the probe and source can be clusters of branes or of bound states of branes.
The relevant part of the action of a D$p$-brane (containing no lower dimensional branes) in a supergravity background is
$$S = T_{p}\int d^{p+1} x \; \left[e^{-\phi}
\sqrt{-\det (G_{\mu\nu}+B_{\mu\nu})} +
\frac{1}{(p+1)!}\epsilon^{\mu_1\mu_2\ldots\mu_{p+1}}
C_{\mu_1\mu_2\ldots\mu_{p+1}}\right],
\label{pbaction}$$
where $G_{\mu\nu}$, $B_{\mu\nu}$ and $C_{p+1}$ are the pullbacks to the world-volume of the background 10D metric, 2-form and R-R $(p+1)$-form (for ‘electric’ branes) or the Hodge dual of the R-R $(7-p)$-form (for ‘magnetic’ branes), respectively, the brane tension is [@pol9611]
$$T_{p}=n_p g_s^{-1} (2\pi)^{(1-p)/2}T^{(p+1)/2}\;,
\label{tension}$$
with $g_s$ the string coupling constant, and the string tension is $T=(2\pi \alpha ')^{-1}$.
The supergravity background
---------------------------
In [@bisy9711], to which we refer the reader for further details, a solution to the low energy effective action of type IIA string theory,
$$S = \frac{1}{(2\pi)^7 g_s^2}\int d^{10} x \: \sqrt{-g_{10}}
\left[e^{-2\phi}\left(R_{10}+4(\nabla\phi)^2\right)
-\frac{1}{4}F_{\mu\nu}^2\right],$$
was found describing a system with D0- and D6-branes.
The spherically symmetric, time-independent solutions are parametrised by the mass ($M$), electric charge ($Q$) and magnetic charge ($P$). The dilaton charge ($\Sigma$) is related to these by
$$\frac{8}{3}\Sigma = \frac{Q^2}{\Sigma + \sqrt{3}M} +
\frac{P^2}{\Sigma - \sqrt{3}M}\;.
\label{dilcharge}$$
The explicit form of the fields is
$$\begin{aligned}
e^{4\phi/3} & = & \frac{B}{A}\;, \label{dil} \\
A_{\mu} dx^{\mu} & = & \frac{Q}{B}(r-\Sigma)dt + P \cos \theta d\phi\;,
\label{1form} \\
g_{\mu\nu}dx^{\mu}dx^{\nu} & = & -\frac{F}{\sqrt{AB}}dt^2 +
\sqrt{\frac{B}{A}}(dx_1^2 + \cdots + dx_6^2) +
\frac{\sqrt{AB}}{F}dr^2 \nonumber \\
& & + \sqrt{AB}(d\theta^2 + \sin^2 \theta d\phi^2)\;,\end{aligned}$$
where $$\begin{aligned}
F & = & (r-r_{+})(r-r_{-})\;, \nonumber \\
A & = & (r-r_{A+})(r-r_{A-})\;, \\
B & = & (r-r_{B+})(r-r_{B-})\;, \nonumber\end{aligned}$$
and $$\begin{aligned}
r_{\pm} & = & M \pm \sqrt{M^2 + \Sigma^2 -
\frac{P^2}{4}-\frac{Q^2}{4}}\;, \nonumber \\
r_{A\pm} & = & \frac{\Sigma}{\sqrt{3}}\pm
\sqrt{\frac{P^2\Sigma/2}{\Sigma-\sqrt{3}M}}\;, \\
r_{B\pm} & = & -\frac{\Sigma}{\sqrt{3}}\pm
\sqrt{\frac{Q^2\Sigma/2}{\Sigma+\sqrt{3}M}}\;. \nonumber\end{aligned}$$
The extremality condition, $r_+ = r_-$, is equivalent to
$$M^2 + \Sigma^2 - \frac{P^2}{4}-\frac{Q^2}{4} = 0\;.
\label{extremal}$$
When imposing this condition it turns out to be convenient to perform the coordinate change $r \rightarrow r^{\prime} = r - M$. The extremal solution is then written as (we drop the prime on $r$)
$$ds^2= -f_1(r)dt^2 + f_2(r)dx_i dx_i + f_1^{-1}(r)(dr^2+r^2d\Omega^2)\;,
\label{06back}$$
where
$$f_1(r) = \frac{r^2}{\sqrt{AB}}\;,\;\;\;\;\;\;\;
f_2(r) = \sqrt{\frac{B}{A}}\;.$$
It is easy to see that the pure magnetic solution
$$P=4M\;, \;\;\;\; Q=0\;,\; \;\;\;\Sigma=-\sqrt{3}M\;,$$
solves the extremality condition [(\[extremal\])]{} and describes a D6-brane.
At the same time, the pure electric solution
$$P=0\;, \;\;\;\;Q=4M\;, \;\;\;\;\Sigma=\sqrt{3}M\;,$$
also solves [(\[extremal\])]{} and describes D0-branes smeared over a 6-torus.
We expect that solutions with both electric and magnetic charge interpolate between these and so describe a system with both D0- and D6-branes.
It turns out ([@bisy9711] and references therein) that the mass, electric and magnetic charge are related to the number of branes as
$$\begin{aligned}
M & = & \frac{g_s N_6}{8}\;, \nonumber \\
P & = & \frac{g_s N_6}{2}\;, \label{MPQ} \\
Q & = & \frac{g_s N_0(2\pi)^2}{2V_6}\;. \nonumber\end{aligned}$$
We can now describe a system with a large number of D6-branes and a relatively small number of D0-branes, i.e., $P \gg Q $. To do so we consider some fixed $M$, and move away from the pure magnetic solution, by taking
$$\Sigma = M(-\sqrt{3}+\epsilon)\;,\;\;\;\;\;\;\;\; \epsilon \ll 1\;.$$
Note that by doing this we are moving away from a $1/2$ supersymmetric solution, a D6-brane, towards one that breaks all supersymmetries.
The dilaton charge equation [(\[dilcharge\])]{} and the extremality condition [(\[extremal\])]{} allow us to determine the electric and magnetic charge of this system as
$$\begin{aligned}
P & = & \frac{1}{3}\sqrt{2}\sqrt{72M^2-36\sqrt{3}\epsilon M^2
+18\epsilon^2 M^2-\sqrt{3}\epsilon^3 M^2}\;, \\
Q & = & \frac{\sqrt{2}\epsilon^{3/2}M}{3^{3/4}}\;,\end{aligned}$$
or, to leading order in $\epsilon$,
$$\begin{aligned}
P & = & 4M-\sqrt{3}M\epsilon+\frac{M\epsilon^2}{8} + O(\epsilon^3)\;,
\label{P} \\
Q & = & \frac{\sqrt{2}\epsilon^{3/2}M}{3^{3/4}}\;. \label{Q}\end{aligned}$$
For these values of the parameters our solution is an extremal (in the sense of [(\[extremal\])]{}), near-supersymmetric one, with the parameter $\epsilon$ measuring deviation from supersymmetry.
Supergravity calculation
------------------------
We now consider a D6-brane probe moving in this background. We take the D6 probe to be parallel to the D6 source, and assume the static gauge, i.e., that the worldvolume coordinates of the D6 probe are the same as those of the source, and that the transverse coordinates do not depend on the spatial worldvolume coordinates.
With these assumptions, the long distance, low velocity action of the D6 probe can be obtained from [(\[pbaction\])]{} by plugging in the background [(\[dil\])]{}, [(\[1form\])]{}, [(\[06back\])]{} for the values of the parameters [(\[P\])]{}, [(\[Q\])]{} and expanding in $1/r$ and $v$, with $r$ the transverse distance between branes and $v$ the transverse velocity.
We obtain [^4]
$$\begin{aligned}
S & = & \frac{n_6}{g_s (2\pi)^6} V_6 \times \nonumber \\
&& \int dt
\left[ 1+\frac{\epsilon^2 M}{8r} -
\left(\frac{1}{2}+\frac{3\epsilon^2 M^2}{4 r^2}+
\frac{\sqrt{3}\epsilon M}{2r} \right)v^2 \right. \nonumber \\
&& \;\;\;\;\;\;\;\;\;\;
-\left(\frac{1}{8}+\frac{\sqrt{3}\epsilon M^2}{r^2}-
\frac{\epsilon^2 M^2}{16 r^2}+\frac{M}{2r}+
\frac{\sqrt{3}\epsilon M}{8r}\right)v^4 \nonumber \\
&& \;\;\;\;\;\;\;\;\;\;
\left. -\left(\frac{1}{16}+\frac{M^2}{r^2}+
\frac{\sqrt{3}\epsilon M^2}{r^2} -
\frac{5\epsilon^2 M^2}{32r^2}+\frac{M}{2r}+
\frac{\sqrt{3}\epsilon M}{16r}\right) v^6\right],
\label{Ssugra06}\end{aligned}$$
where $n_6$ is the number of D6 branes in the probe.
Note that in the limit $\epsilon \rightarrow 0$ the static term of the potential vanishes, and the corrections to the energy start only at $v^4$. This is what we should expect [@tse9609], since in this limit our background reduces just to a collection of overlapping D6-branes, and so the full system is just D6-branes parallel to D6-branes, which is a BPS configuration.
SYM description
===============
In this section, we use the normalisation of [@che-tse9709], $T=1$.
The low energy dynamics of $N$ D-branes is described by the dimensional reduction to $p+1$ dimensions of $\mathcal{N}=1$ SYM in 10 dimensions with $U(N)$ gauge symmetry (for a recent review and references see [@tay9801]).
In [@che-tse9709; @che-tse9801] it was argued that the sum of leading large $N$ IR contributions to the effective action can be written as
$$\Gamma = \sum_{L=1}^{\infty} \Gamma^{(L)} =
\frac{1}{2}\sum_{L=1}^{\infty} \int d^{p+1}x \left(\frac{a_{p}}
{M^{7-p}}\right)^L ({\ensuremath{g^{2}_{\mathrm{YM}\ }}})^{L-1}\hat{C}_{2L+2}(F)\;,
\label{effsym}$$
where $F$ is a background field, the coefficients $a_p$ are given by ($T=1$)
$$a_{p} = 2^{2-p}\pi^{-(p+1)/2}\Gamma\left(\frac{7-p}{2}\right),$$
and the coefficients $\hat{C}_{2L+2}$ are polynomials of $F$. The only term which was explicitly computed is the 1-loop one, with the result
$$\hat{C}_4 = {\mbox{STr}}\, C_4\;,
\label{hc4}$$
where STr is the symmetrised trace in the [*adjoint*]{} representation, and $C_4$ is as given below.
For the corresponding 2-loop term the authors of [@che-tse9709] have proposed the ansatz
$$\hat{C}_6 = \widehat{{\mbox{STr}}}\, C_6\;,
\label{hc6}$$
where $\widehat{{\mbox{STr}}}$ is a modified symmetrised trace,
$$\begin{aligned}
\widehat{{\mbox{STr}}}\,(X_{i_1}\ldots X_{i_6})& = &
2N{\mbox{tr}}(X_{(i_1}\ldots X_{i_6)})+
60\,{\mbox{tr}}(X_{(i_1}\ldots X_{i_4}){\mbox{tr}}(X_{i_5}X_{i_6)}) \nonumber \\
&& -50\,{\mbox{tr}}(X_{(i_1}\ldots X_{i_3}){\mbox{tr}}(X_{i_4}\ldots X_{i_6)})
\nonumber \\
&& -30 N^{-1} {\mbox{tr}}(X_{(i_1}X_{i_2}){\mbox{tr}}(X_{i_3}X_{i_4})
{\mbox{tr}}(X_{i_5}X_{i_6)})\;.\end{aligned}$$
The coefficients $C_4$ and $C_6$ are the same as the polynomials appearing in the expansion of the abelian Born-Infeld action ($T=1$),
$$\sqrt{-\det(\eta_{\mu\nu}+F_{\mu\nu})}= \sum_{n=0}^{\infty}
C_{2n}(F)\;,$$
with $$\begin{aligned}
& C_{0} = 1\;,\;\;\;\; C_{2}=-\frac{1}{4}F^{2}\;,\;\;\;\;
C_{4} = -\frac{1}{8}
\left[F^{4}-\frac{1}{4}(F^{2})^{2}\right]\;, & \nonumber \\
& C_{6} = -\frac{1}{12}
\left[F^{6}-\frac{3}{8}F^{4}F^{2}+\frac{1}{32}(F^2)^3\right]\: , \:
\ldots & \label{BIc}\end{aligned}$$
where $F^{k}$ is the trace of the matrix product over Lorentz indices,
$$F^{2}=F_{\mu\nu}F^{\nu\mu}\;, \;\;\;\ldots \; ,\;\;\;
F^{2k}=F_{\mu_1\mu_2}F^{\mu_2\mu_3} \ldots
F_{\mu_{2k-1}\mu_{2k}}F^{\mu_{2k}\mu_1}\;.$$
The SYM background
------------------
A SYM background describing a system with D0- and D6-branes but no D2- nor D4-branes was found in [@tay9705].
Let [^5] $$F^{S}_{12} = F_0 J^{S}_1\;,\;\;\;\; F^{S}_{34} = F_0 J^{S}_2\;,\;\;\;\;
F^{S}_{56}=F_0 J^{S}_3\;,$$
where $F_0$ is an arbitrary constant, the $J_i^{S}$’s are $N_6 \times N_6$ block-diagonal matrices built out of $\frac{1}{4} N_6$ copies of $\mu_i$, $N_6$ a multiple of four,
$$J^{S}_i = {\mbox{diag}\,}(\mu_i, \ldots, \mu_i)\;,
\;\;\;\; i=1,2,3,
\label{Jsi}$$
and $\mu_i$ are the $su(4)$ Cartan subalgebra matrices $$\begin{aligned}
\mu_1 & = & {\mbox{diag}\,}(1,1,-1,-1)\;, \nonumber \\
\mu_2 & = & {\mbox{diag}\,}(1,-1,-1,1)\;, \\
\mu_3 & = & {\mbox{diag}\,}(1,-1,1,-1)\;. \nonumber \end{aligned}$$
They have the properties [@tay9705] $$\begin{aligned}
& & \;\;{\mbox{tr}}\, (\mu_i^2) = 4\;, \nonumber \\
& & \left. \begin{array}{l}
{\mbox{tr}}\,({\mu_i}) = 0\;,\\
\mu_i \mu_j = |\epsilon_{ijk}| \mu_k\,, \;\; i \neq j\, ,\\
\end{array} \right\}
\Rightarrow {\mbox{tr}}\,(\mu_i \mu_j) = 0\, ,\;\; i\neq j\;.\end{aligned}$$
Using these we easily evaluate the D0-, D2- and D4-brane charges induced by the flux of the background in the worldvolume,
$$\begin{aligned}
N_4 & \propto & \int d^2 x \;{\mbox{tr}}\,(F^{S}) = 0\;, \nonumber \\
N_2 & \propto & \int d^4 x \;{\mbox{tr}}\,(F^{S}\wedge F^{S}) = 0\;, \nonumber \\
N_0 & = & \frac{T^3}{6(2\pi)^3}
\int d^6x\;{\mbox{tr}}\,(F^{S}\wedge F^{S}\wedge F^{S}) \nonumber \\
& = & \frac{F_0^3 N_6 V_6 T^3}{(2\pi)^3}\;, \label{N0}\end{aligned}$$
so that, as required, there are no D2- or D4-brane charges, but only D0- and D6-brane ones. It can easily be seen [@tay9705] that, for $F_0 \neq 0$, this is a non-supersymmetric state. This state is also not a bound state, since its energy is larger than that of the separated constituents. It is expected nevertheless that it should represent a meta-stable state.
We will probe this system with D6-branes. We consider a D6-brane probe parallel to the D6 source, and for simplicity consider them aligned with the coordinate axis $(0)123456$. The probe will have transverse velocity $v$, and we will take its direction to be along the $9$ axis.
The SYM background describing the full system (source + probe) is (see e.g. [@che-tse9709]),
$$F_{12} = F_0 J_1\;,\;\;\; F_{34} = F_0 J_2\;,\;\;\; F_{56}=F_0 J_3\;,
\;\;\;F_{09} = v J_0\;,
\label{F}$$
where the $J$’s are block-diagonal matrices,
$$\begin{aligned}
J_i & = & {\mbox{diag}\,}\!\left(0_{n_6 \times n_6},(J^{S}_i)_{N_6\times N_6}\right)
,\;\;\;\;i=1,2,3,
\label{Ji} \\
J_{0} & = & \frac{1}{N} {\mbox{diag}\,}\!\left(N_6 I_{n_6 \times n_6} ,
-n_6 I_{N_6 \times N_6}\right)\;, \;\;\;\;N \equiv n_6 + N_6\;.
\label{J0}\end{aligned}$$
SYM calculation
---------------
Note that the $F$ matrices in [(\[F\])]{} are in the [*fundamental*]{} representation of $su(N)$, and the trace in [(\[hc4\])]{}, [(\[hc6\])]{} is in the adjoint representation. We need therefore to know how to relate traces in the two representations. To that effect, we collect some useful formulas in Appendix A.
We start by evaluating the $F^4$. Plugging the SYM background [(\[Ji\])]{},[(\[J0\])]{} into [(\[BIc\])]{} we get
$$\begin{aligned}
C_4 & = & -\frac{1}{8}\left[F_0^4(J_1^4 + J_2^4 + J_3^4)
- 2 F_0^4(J_1^2 J_2^2 + J_1^2 J_3^2 + J_2^2J_3^2)
\right. \nonumber \\
& & \left.\;\;\;\;\;\;\;
+\,2 F_0^2 (J_1^2 + J_2^2 + J_3^2)J_0^2 v^2
+ J_0^4 v^4\right],\end{aligned}$$
from which, using the trace results of Appendix A,
$$\hat{C}_4 = \frac{1}{4}n_6 N_6 (3 F_0^4 - 6 F_0^2 v^2 - v^4)\;.$$
From [(\[effsym\])]{} we read the expression for the 1-loop effective action,
$$\begin{aligned}
\Gamma^{(1)} & = & \frac{a_6}{2M}\int d^7x\;\hat{C}_4(F)
\nonumber \\
& = & \frac{n_6 N_6}{16 (2\pi)^3 r}V_6
\int dt (3 F_0^4 - 6 F_0^2 v^2 - v^4)\;. \label{Gamma1}\end{aligned}$$
Repeating the procedure for the $F^6$ term, we obtain
$$\begin{aligned}
C_6 & = & \frac{F_0^6}{16}\left(J_1^6 + J_2^6 + J_3^6 - J_1^4J_2^2
- J_1^4J_3^2 - J_2^4J_3^2 - J_2^4J_1^2 - J_3^4J_1^2 - J_3^4J_2^2
\right. \nonumber \\
& & \hspace{0.9cm} \left. + \, 2 J_1^2J_2^2J_3^2\right) \nonumber \\
& & + \frac{F_0^4}{16}\left[J_0^2(J_1^4 + J_2^4 + J_3^4)
-2J_0^2(J_1^2J_2^2 + J_1^2J_3^2 + J_2^2J_3^2)\right]v^2 \\
& & -\frac{F_0^2}{16}\left[J_0^4(J_1^2 + J_2^2 + J_3^2)\right]v^4 -
\frac{1}{16}J_0^6 v^6\;. \nonumber \end{aligned}$$
A straightforward but lengthy calculation of the several traces in [(\[hc6\])]{} results in
$$\begin{aligned}
\hat{C}_6 & = & -F_0^6
\frac{N_6(n_6^2 + 4n_6 N_6 + 32 N_6^2)}{8(n_6 + N_6)}
- F_0^4\frac{3n_6 N_6(n_6 + 3N_6)}{8(n_6 + N_6)} v^2 \nonumber \\
& & - F_0^2 \frac{3n_6 N_6(n_6 + 2N_6)}{8 (n_6 + N_6)}v^4 -
\frac{n_6 N_6}{8}v^6\;. \end{aligned}$$
Reading from [(\[effsym\])]{} the 2-loop effective action, we have
$$\begin{aligned}
\Gamma^{(2)} & = & \frac{1}{2} \left(\frac{a_6}{M}\right)^2
N {\ensuremath{g^{2}_{\mathrm{YM}\ }}}\int d^7x\;\hat{C}_6(F) \nonumber \\
& = & \frac{g_s }{2^6(2\pi)^{7/2} r^2} V_6 \times \nonumber \\
& & \int dt \left[-F_0^6
N_6(n_6^2 + 4n_6 N_6 + 32 N_6^2)
- 3 F_0^4 n_6 N_6(n_6 + 3N_6)v^2
\right. \nonumber \\
& & \;\;\;\;\;\;\;\;\;\left. - 3 F_0^2 n_6 N_6(n_6 + 2N_6)v^4 -
N n_6 N_6 v^6 \right].
\label{Gamma2}\end{aligned}$$
Comparison
==========
We proceed to compare the supergravity and SYM results. In the two pictures there is a parameter that measures deviation from a supersymmetric state, $\epsilon$ in supergravity and $F_0$ in SYM. We expect them to be related.
In fact, from [(\[N0\])]{} we have
$$N_0 = \frac{F_0^3 N_6 V_6 }{(2\pi)^6}\;.$$
Then [(\[MPQ\])]{} imply
$$F_0^3 = \frac{Q}{P}\;,$$
or, after replacing [(\[P\])]{}, [(\[Q\])]{}, to leading order in $\epsilon$,
$$F_0^2 = \frac{\epsilon }{2\sqrt{3}}+ \frac{\epsilon^2}{12} +
O(\epsilon^{3})\;.
\label{F02k}$$
So far we have used different normalisations in the supergravity and SYM calculations. In order to compare those results, we will from now on express the SYM results in $\alpha^{\prime} = 1$ normalisation.
At 1-loop, restoring $T$ in [(\[Gamma1\])]{} and replacing the results [(\[MPQ\])]{}, [(\[F02k\])]{}, we have, to leading order in $\epsilon$ for each order in $v$ and $1/r$,
$$\Gamma^{(1)} = \frac{n_6}{g_s (2\pi)^6}V_6
\int dt \left(\frac{\epsilon^2 M}{8r}
-\frac{\epsilon \sqrt{3}M}{2r}v^2
-\frac{M}{2r}v^4\right).$$
Comparing with the supergravity result [(\[Ssugra06\])]{} we see that the SYM result reproduces exactly the $1/r$ and $v^2/r$ terms, a result already obtained in [@bisy9711].
At 2-loop level, repeating this procedure and considering the ‘source much heavier than probe’ limit, $N_6 >> n_6$, we obtain, to leading order in $\epsilon$ in each order in $v$ and $1/r$,
$$\Gamma^{(2)} = \frac{n_6}{g_s (2\pi)^6}V_6
\int dt \left(-\frac{3 \epsilon^2 M^2}{4r^2}v^2
-\frac{\sqrt{3}\epsilon M^2}{r^2}v^4
-\frac{M^2}{r^2}v^6\right).$$
Comparing with [(\[Ssugra06\])]{}, we see that the SYM result reproduces the $v^2/r^2$, $v^4/r^2$ and $v^6/r^2$ terms to leading order in $\epsilon$.
Conclusion
==========
We studied the long distance, low velocity interaction potential between a D6-brane probe and a non-supersymmetric source containing D0- and D6-branes. We extended the supergravity calculation of [@bisy9711] to subleading order in $1/r$, and compared the resulting potential with the one obtained using the ansatz of [@che-tse9709] for the 2-loop SYM effective action. We found agreement at subleading order, thus providing a further non-trivial check for this ansatz.
It would be interesting to have a direct SYM calculation of the 2-loop terms, and to see how they compare to the ones obtained here.
0.2cm
We would like to thank Arkady Tseytlin for proposing the problem and for many useful discussions.
Tr vs. tr
=========
This section is based in [@che-tse9705; @che-tse9709] and references therein.
Let Tr denote the trace in the adjoint representation, and tr denote trace in the fundamental representation. For $su(N)$ generators, $T_a$, and an element of the algebra, $X=X^a T_a$, the traces are related as
$$\begin{aligned}
{\mbox{Tr}}(T_a T_b) & = & N \delta _{ab}\;, \\
{\mbox{tr}}(T_a T_b) & = & \frac{1}{2} \delta_{ab}\;,\\
{\mbox{Tr}}(X^2) & = & 2N{\mbox{tr}}(X^2)\;,\\
{\mbox{Tr}}(X^4)& = & 2N{\mbox{tr}}(X^4)+6[{\mbox{tr}}(X^2)]^2\;,\\
{\mbox{Tr}}(X^6) & = & 2N{\mbox{tr}}(X^6)+30\,{\mbox{tr}}(X^4){\mbox{tr}}(X^2)-20[{\mbox{tr}}(X^3)]^2\;.\end{aligned}$$
Similar relations apply to symmetrised products of generators,
$$\begin{aligned}
{\mbox{STr}}\left(X_{i_1}X_{i_2}X_{i_3}X_{i_4}\right) & = &
{\mbox{Tr}}\!\left(X_{(i_1}X_{i_2}X_{i_3}X_{i_4)}\right)= \\
& = & 2N{\mbox{tr}}\!\left(X_{(i_1}X_{i_2}X_{i_3}X_{i_4)}\right)+
6\,{\mbox{tr}}\!\left(X_{(i_1}X_{i_2}\right)
{\mbox{tr}}\!\left(X_{i_3}X_{i_4)}\right), \\
{\mbox{STr}}\left(X_{i_1}X_{i_2}\ldots X_{i_6}\right) & = &
{\mbox{Tr}}\!\left(X_{(i_1}X_{i_2}\ldots X_{i_6)}\right) \\
& = & 2N{\mbox{tr}}\!\left(X_{(i_1}X_{i_2}\ldots X_{i_6)}\right)+
30\,{\mbox{tr}}\!\left(X_{(i_1}\ldots X_{i_4}\right)
{\mbox{tr}}\!\left(X_{i_5}X_{i_6)}\right) \\
&& -20\,{\mbox{tr}}\!\left(X_{(i_1}X_{i_2}X_{i_3}\right)
{\mbox{tr}}\!\left(X_{i_4}X_{i_5}X_{i_6)}\right).\end{aligned}$$
Note that even for commuting backgrounds, where obviously STr = Tr, it is often convenient to keep STr in order to change from Tr to tr.
We list some results that will be used. For arbitrary [*commuting*]{} $X_0$, $X_1$
$${\mbox{STr}}\left(X_0^2 X_1^2\right) = 2N{\mbox{tr}}\!\left(X_0^2 X_1^2\right)
+ 2\,{\mbox{tr}}\!\left(X_0^2\right){\mbox{tr}}\!\left(X_1^2\right) +
4\,\left[{\mbox{tr}}\left(X_0 X_1\right)\right]^2.$$
Another common term is of the form $X_0^4 X_1^2$. Let $X$ represent either $X_0$ or $X_1$, such that $X^6 = X_0^4 X_1^2$. Then
$$\begin{aligned}
sym\;{\mbox{tr}}\!\left(X^4\right){\mbox{tr}}\!\left(X^2\right)
& = & \frac{1}{15}\left[{\mbox{tr}}\!\left(X_0^4\right){\mbox{tr}}\!\left(X_1^2\right)+
8\,{\mbox{tr}}\!\left(X_0^3 X_1\right){\mbox{tr}}\!\left(X_0 X_1\right)\right. \\
& & \;\;\;\;\;\;\; +
\left. 6\,{\mbox{tr}}\!\left(X_0^2 X_1^2\right)
{\mbox{tr}}\!\left(X_0^2\right)\right], \\
sym\;{\mbox{tr}}\!\left(X^3\right){\mbox{tr}}\!\left(X^3\right)
& = & \frac{1}{5}\left\{2\,{\mbox{tr}}\!\left(X_0^3\right)
{\mbox{tr}}\!\left(X_0 X_1^2\right)+
3\left[{\mbox{tr}}\!\left(X_0^2 X_1\right)\right]^2\right\}, \\
sym\;{\mbox{tr}}\left(X^2\right){\mbox{tr}}\left(X^2\right){\mbox{tr}}\left(X^2\right)
& = & \frac{1}{5}
\left\{\left[{\mbox{tr}}\!\left(X_0^2\right)\right]^2{\mbox{tr}}\!\left(X_1^2\right)+
4\,{\mbox{tr}}\!\left(X_0^2\right)
\left[{\mbox{tr}}\!\left(X_0 X_1\right)\right]^2\right\}, \end{aligned}$$
where $sym$ denotes the symmetrisation operator.
Yet another common term is of the form $X_1^2 X_2^2 X_3^2$, all $X_i$’s commuting. Let $X$ represent one of the $X_i$, $i=1,2,3$, such that $X^6 = X_1^2 X_2^2 X_3^2$. Then
$$\begin{aligned}
sym\;{\mbox{tr}}\left(X^4\right){\mbox{tr}}\left(X^2\right) & = & \frac{1}{15}
\left\{\left[{\mbox{tr}}\!\left(X_1^2\right)
{\mbox{tr}}\!\left(X_2^2 X_3^2\right)+ \; 2 \; terms\right]\right. \\
& & \;\;\;\;\;\;\;
\left. + \; 4 \left[{\mbox{tr}}\!\left(X_1 X_2\right)
{\mbox{tr}}\!\left(X_1 X_2 X_3^2\right)
+ 2 \; terms\right]\right\}, \\
sym\;{\mbox{tr}}\left(X^3\right){\mbox{tr}}\left(X^3\right) & = & \frac{1}{5}
\left\{2\left[{\mbox{tr}}\!\left(X_1 X_2 X_3\right)\right]^2 +
\left[{\mbox{tr}}\!\left(X_1^2 X_2\right)
{\mbox{tr}}\!\left(X_2 X_3^2 \right) + 2 \; terms\right]\right\}, \\
sym\;{\mbox{tr}}\left(X^2\right){\mbox{tr}}\left(X^2\right){\mbox{tr}}\left(X^2\right)
& = & \frac{1}{15}
\left\{{\mbox{tr}}\!\left(X_1^2\right){\mbox{tr}}\!\left(X_2^2\right)
{\mbox{tr}}\!\left(X_3^2\right) \right. \\
& & \;\;\;\;\;\;\;
+ \; 2 \left[{\mbox{tr}}\!\left(X_1^2 \right){\mbox{tr}}\!\left(X_2 X_3\right)
{\mbox{tr}}\!\left(X_2 X_3\right) + 2 \; terms\right] \\
& & \;\;\;\;\;\;\;
\left. + \; 8\,{\mbox{tr}}\!\left(X_1 X_2\right){\mbox{tr}}\!\left(X_2 X_3\right)
{\mbox{tr}}\!\left(X_3 X_1\right)\right\}. \end{aligned}$$
A common matrix appearing in these calculations is
$$J_0 = \frac{1}{N}\left(
\begin{array}{cc}
N_6 I_{n_6 \times n_6} & 0 \\
0 & -n_6 I_{N_6 \times N_6}
\end{array} \right)$$
where $N = n_6 + N_6\;$.
The adjoint trace of its even powers is given by
$${\mbox{Tr}}(J_0^{2k}) = 2 n_6 N_6\;.$$
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[^1]: Work supported by FCT - Programa PRAXIS XXI, under contract BD/9347/96
[^2]: This is by no means an exhaustive list. For more complete references see e.g. [@kab-tay9712].
[^3]: A very similar supergravity background was found in [@dha-man9803], describing a 4-dimensional black hole carrying D0- and D6-brane charges. The black hole was probed with D0-branes both in the supergravity and M(atrix) theory formalism, with the by now expected agreement at 1-loop.
[^4]: We have dropped here a term linear in $v$, since we will not compare it with the SYM results. A term of this type was recently discussed in [@bvflrs9805] for the potential between a D0- and a D6-brane in relative motion.
[^5]: We use the superscript ‘$S$’ for ‘source’, i.e., the background describing just the D0+D6 system, in order to avoid confusion with the full background [(\[F\])]{}, [(\[Ji\])]{}.
|
---
abstract: 'Linear Logic and Defeasible Logic have been adopted to formalise different features of knowledge representation: consumption of resources, and non monotonic reasoning in particular to represent exceptions. Recently, a framework to combine sub-structural features, corresponding to the consumption of resources, with defeasibility aspects to handle potentially conflicting information, has been discussed in literature, by some of the authors. Two applications emerged that are very relevant: energy management and business process management. We illustrate a set of guide lines to determine how to apply linear defeasible logic to those contexts.'
author:
- 'Francesco Olivieri, Guido Governatori'
- 'Claudio Tomazzoli, Matteo Cristani'
bibliography:
- 'thisbiblio.bib'
title: 'Applications of Linear Defeasible Logic: combining resource consumption and exceptions to energy management and business processes'
---
|
---
abstract: 'We consider a superfluid state in a two-component gas of fermionic atoms with equal densities and unequal masses in the BCS limit. We develop a perturbation theory along the lines proposed by Gorkov and Melik-Barkhudarov and find that for a large difference in the masses of heavy ($M$) and light ($m$) atoms one has to take into account both the second-order and third-order contributions. The result for the critical temperature and order parameter is then quite different from the prediction of the simple BCS approach. Moreover, the small parameter of the theory turns out to be $(p_{F}|a|/\hbar) \sqrt{M/m}\ll1$, where $p_{F}$ is the Fermi momentum, and $a$ the scattering length. Thus, for a large mass ratio $M/m$ the conventional perturbation theory requires significantly smaller Fermi momenta (densities) or scattering lengths than in the case of $M\sim m$, where the small parameter is $p_{F}|a|/\hbar\ll1$. We show that $3$-body scattering resonances appearing at a large mass ratio due to the presence of $3$-body bound Efimov states do not influence the result, which in this sense becomes universal.'
author:
- 'M.A. Baranov'
- 'C. Lobo'
- 'G. V. Shlyapnikov'
title: Superfluid pairing between fermions with unequal masses
---
Introduction
============
Superfluid pairing in a two-component gas of fermions is a well-known problem [@LL9] lying in the background of extensive studies in condensed matter and nuclear physics [@SC; @He3; @Migdal]. Recently, this problem was actively investigated in cold gases of fermionic atoms (see [@Trento] for a review). Experimental efforts were focused on $^{6}$Li or $^{40}$K atoms in two different internal (hyperfine) states, where one can use Feshbach resonances for switching the sign and tuning the absolute value of the interspecies interaction (scattering length $a$), which at resonance changes from $-\infty$ to $+\infty$. In this respect, one encounters the problem of BCS-BEC crossover discussed earlier in the context of superconductivity [@Eag; @Leg; @Noz; @Melo; @Rand] and for superfluidity of two-dimensional $^{3}$He films [@M; @MYu]. On the negative side of the resonance ($a<0$), one should have the Bardeen-Cooper-Schrieffer (BCS) superfluid pairing at sufficiently low temperatures, and on the positive side ($a>0$) one expects Bose-Einstein condensation of diatomic molecules formed by atoms of different components. Remarkable achievements of cold-atom physics in the last years include the observation of superfluid behavior through vortex formation in the strongly interacting regime ($n\left\vert a\right\vert ^{3}\gtrsim1$, where $n$ is the gas density) [@zw1], and the formation and Bose-Einstein condensation of long-lived weakly bound diatomic molecules at $a>0$ [@bec]. Ongoing experiments with atomic Fermi gases have reached temperatures in the nanokelvin regime, where at achieved densities one has $T\sim0.1E_{F}$, with $E_{F}$ being the Fermi energy. For $a<0$ the experiments are now approaching superfluidity in the BCS limit where $n|a|^{3}\ll1$.
Currently, a new generation of experiments is being set up. In particular, it is dealing with mixtures of different fermionic atoms or mixtures of fermions and bosons. The main goal is to reveal the influence of the mass difference on superfluid properties and to search for novel types of superfluid pairing. The first experiments demonstrating a possibility of using Feshbach resonances and creating collisionally stable mixtures of $^{40}$K with $^{6}$Li and/or with $^{87}$Rb, and $^{6}$Li with $^{23}$Na have already been performed [@Jin; @Ket; @Ospelkaus; @Ing; @kai; @Jook]. Recent theoretical literature on mixtures of different fermionic atoms contains a discussion of the BCS limit [@Liu; @Caldas; @He], the limit of molecular BEC [@PSSJ], BCS-BEC crossover [@Iskin1; @Iskin2; @Parish], and the strongly interacting regime [@Pao].
In this paper we consider a two-component mixture of fermionic atoms with different masses and attractive intercomponent interaction in the BCS limit. It is assumed that the densities of the two species are equal which means that there is no mismatch between their Fermi surfaces, leading to the usual BCS type of superfluid pairing. Other kinds of pairing that can occur and compete with BCS, especially for unequal densities, will be discussed elsewhere [@FFLO]. Here, we generalize the perturbation treatment of the gap equation, introduced by Gorkov and Melik-Barkhudarov [@GMB] for equal masses of fermions belonging to different components. This approach takes into account the interaction between the atoms in a Cooper pair due to the polarization of the medium and allows one to correctly determine the dependence of the zero-temperature gap $\Delta_{0}$ and superfluid transition temperature $T_{c}$ on the masses of heavy ($M$) and light ($m$) fermionic atoms. As we shall see below, already the second order of the perturbation, the so-called Gorkov-Melik-Barkhudarov contribution, leads to a very different dependence of the preexponential factor in the expressions for $\Delta_{0}$ and $T_{c}$ on the mass ratio $M/m$, compared to the prediction of the simple BCS theory.
For a large mass ratio $M/m\gg1$, we include higher order contributions and show that the actual small parameter of the theory is $(p_{F}|a|/\hbar
)\sqrt{M/m}\ll1$ ($p_{F}$ is the Fermi momentum), not simply $p_{F}%
|a|/\hbar\ll1$ as in the case of $M\sim m$. We give a physical interpretation of this fact and calculate effective masses of heavy and light fermions.
Large mass ratios $M/m$ are realized in electron-ion plasmas, where the heavy ion component is usually considered as non-degenerate [@LL5]. The electron-proton pairing in the hydrogen plasma, assuming quantum degeneracy for both electrons and protons, was discussed by Moulopoulos and Ashcroft [@MA]. They found that at low temperatures the Coulomb electron-proton attraction leads to the appearance of a (momentum-dependent) gap which for sufficiently high densities is comparable with the Coulomb interaction at the mean interparticle separation. Note that this problem is quite different from ours where the attractive interaction between heavy and light fermions is short-ranged.
Before proceeding with our analysis we make two important remarks. First of all, if the masses of heavy and light fermionic atoms are very different from each other and the mass ratio exceeds a critical value, $M/m>13.6$, then two heavy and one light fermion can form $3$-body weakly bound states. The appearance of these states, predicted by Efimov [@Efimov], can be easily understood in the Born-Oppenheimer picture [@Fonseca]. If we fix the two heavy atoms at a relative distance $R<\left\vert a\right\vert $, a localized state for the light atom appears due to the presence of the heavy pair, which in turn mediates an attractive interaction $\sim-\hbar^{2}/mR^{2}$ between the heavy atoms (see, e.g. [@PSSJ] and references therein). For a large mass ratio, $M/m>13.6$, this mediated attraction overcomes the kinetic energy of the relative motion of the heavy atoms and one has the well-known phenomenon of “fall into center" [@LL3]. The energy of this state is bounded from below only due to short-range repulsion. The corresponding wave function of the relative motion of heavy atoms acquires a large number of nodes thus showing the presence of many bound states. This makes the $3$-body problem non-universal in the sense that aside from the $2$-body scattering length $a$, the description of this problem requires one more parameter - the so-called $3$-body parameter coming from short-range physics. Also, the presence of weakly bound Efimov states introduces a resonant character to the $3$-body scattering problem. This is especially important for the Gorkov-Melikh-Barkhudarov contribution as it is actually dealing with processes involving $3$ particles. We, however, have found that the $3$-body resonances are rather narrow and their contribution is not important. This makes the Gorkov-Melikh-Barkhudarov approach universal at any mass ratio $M/m$.
Our second remark is related to analogies between BCS pairing of unequal-mass particles in cold-atom and high energy physics. We wish to emphasize that there are strong physical differences between the pairing of particles of different masses in relativistic and in nonrelativistic systems such as cold atoms. The problem arises in relativistic systems in the study of hadronic matter [@hadron]. It is thought that, at the high densities achieved in neutron stars, quarks become deconfined and the different types of them (e.g. up, down and strange quarks) will tend to form Cooper pairs with each other. These different types of quarks have different masses. This relativistic limit has been investigated by Kundu and Rajagopal [@Kundu] and is characterized by the Fermi momentum being larger than the bare mass: $p_{F}\gg mc$. Linearizing the momentum near the Fermi surface $p=p_{F}+\delta p$, ($\delta
p/p_{F}\ll1$), we can expand the free particle energy to the lowest nonvanishing order in $\delta p/p_{F}$ and $mc/p_{F}$: $$E=\sqrt{p^{2}c^{2}+m^{2}c^{4}}\simeq p_{F}c\left( 1+\frac{\delta p}{p_{F}%
}+\frac{m^{2}c^{2}}{2p_{F}^{2}}\right) .$$ We see that, as far as the kinetic energy is concerned, a change in mass will amount to a shift in the chemical potential of the species which is proportional to $m^{2}$ and depends inversely on $p_{F}$. Therefore, pairing between particles with different masses in the relativistic limit is equivalent to studying the problem of pairing of equal mass particles in the presence of a difference between the chemical potentials of the two species. In the nonrelativistic limit ($p_{F}\ll mc$) the situation is different: $$E\simeq mc^{2}+\frac{p^{2}}{2m}%$$ and so the mass change cannot be incorporated into the chemical potential, requiring a very different analysis which we carry out here.
The paper is organized as follows. In Section \[sec:bcs\] we present general equations, and in Sections III and IV we calculate the critical temperature $T_{c}$. Section V is dedicated to the discussion of the small parameter of the theory, and in Section VI we discuss the order parameter and excitation spectrum. In Sec. VII we analyze the three-body resonances at a large mass ratio $M/m$ and show that they do not change the result of the Gorkov-Melik-Barkhudarov approach. In Sec. VIII we conclude.
General equations {#sec:bcs}
=================
We consider a uniform gas composed of heavy and light fermionic atoms with masses $M$ and $m$, respectively. Both heavy and light atoms are in a single hyperfine state, and considering low temperatures we omit heavy-heavy and light-light interactions. The interaction of heavy atoms with light ones is assumed to be attractive and characterized by a negative s-wave scattering length $a<0$. The Hamiltonian of the system has the form:$$\label{H}H=\int d\mathbf{r}\left[ \sum_{i=1,2}\widehat{\psi}_{i}%
^{+}(\mathbf{r})\left( -\frac{\hbar^{2}}{2m_{i}}\nabla^{2}-\mu_{i}\right)
\widehat{\psi}_{i}(\mathbf{r})+g\widehat{\psi}_{1}^{+}(\mathbf{r}%
)\widehat{\psi}_{1}(\mathbf{r})\widehat{\psi}_{2}^{+}(\mathbf{r})\widehat
{\psi}_{2}(\mathbf{r})\right] ,$$ where $\widehat{\psi}_{i}(\mathbf{r})$ and $\widehat{\psi}_{i}^{+}$ are the field operators of fermionic atoms labeled by indices $i=1$ (heavy) and $i=2$ (light), $\mu_{i}$ is the corresponding chemical potential, and $g=2\pi
\hbar^{2}a/m_{r}$ is the coupling constant, with $m_{r}=Mm/(M+m)$ being the reduced mass. Since the densities of the two species are equal, $n_{1}%
=n_{2}=n$, they have the same Fermi momentum $p_{F}=\hbar(6\pi^{2}n)^{1/3}$, and, hence, $\mu_{1}=p_{F}^{2}/2M$ and $\mu_{2}=p_{F}^{2}/2m$. Finally, we require that the system be in the weakly interacting regime, which requires the inequality $p_{F}|a|/\hbar\ll1 $.
We now consider the usual BCS scheme where a heavy atom with momentum $\mathbf{p}$ is paired to a light one having momentum $-\mathbf{p}$. This leads to the gap equation $$\Delta(\mathbf{p})=-\int\frac{d\mathbf{p}^{\prime}}{(2\pi\hbar)^{3}%
}V_{\mathrm{eff}}(\mathbf{p},\mathbf{p}^{\prime})\frac{1-f(E_{+}%
(\mathbf{p}^{\prime}))-f(E_{-}(\mathbf{p}^{\prime}))}{E_{+}(\mathbf{p}%
^{\prime})+E_{-}(\mathbf{p}^{\prime})}\Delta(\mathbf{p}^{\prime})
\label{eq:bcsgap}%$$ where $f(E)=[\exp(E/T)+1]^{-1}$ is the Fermi-Dirac distribution function, and we assume that the order parameter is real. The dispersion relations for the two branches of single-particle excitations are written as $$E_{\pm}(\mathbf{p})=\pm\left( \frac{\xi_{1}(\mathbf{p})-\xi_{2}(\mathbf{p}%
)}{2}\right) +\sqrt{\left( \frac{\xi_{1}(\mathbf{p})+\xi_{2}(\mathbf{p})}%
{2}\right) ^{2}+\Delta^{2}}, \label{eq:dispersion}%$$ and the quantities $\xi_{1,2}$ are given by $\xi_{1}(\mathbf{p})=(p^{2}%
-p_{F}^{2})/2M$ and $\xi_{2}(\mathbf{p})=(p^{2}-p_{F}^{2})/2m$. The function $V_{\mathrm{eff}}(\mathbf{p},\mathbf{p}^{\prime})=g+\delta V(\mathbf{p}%
,\mathbf{p}^{\prime})$ is an effective interaction between particles in the medium, where the quantity $\delta V(\mathbf{p},\mathbf{p}^{\prime})$ originates from many-body effects and is a correction to the bare interparticle interaction $g$. The leading correction is second order in $g$ and the corresponding diagram is shown in Fig. \[Fig1\].
![The leading many-body contribution (second order, or Gorkov-Melik-Barkhudarov correction) to the effective interaction between heavy (thick line) and light (thin line) fermions. The dashed line corresponds to the coupling constant $g$.[]{data-label="Fig1"}](Fig1.eps){width="10cm"}
The integral in Eq. (\[eq:bcsgap\]) diverges at large momenta due to the first term in $V_{\mathrm{eff}}$. This divergency can be eliminated by expressing the bare interaction $g$ in terms of the scattering length $a$ [@GMB; @LL9; @AGD]. If we confine ourselves to the second order in perturbation theory with respect to $g$, then the renormalized gap equation reads $$\begin{aligned}
\Delta(\mathbf{p}) & =-\frac{2\pi\hbar^{2}a}{m_{r}}\int\frac{d\mathbf{p}%
^{\prime}}{(2\pi\hbar)^{3}}\left[ \frac{1-f(E_{+}(\mathbf{p}^{\prime
}))-f(E_{-}(\mathbf{p}^{\prime}))}{E_{+}(\mathbf{p}^{\prime})+E_{-}%
(\mathbf{p}^{\prime})}-\frac{2m_{r}}{p^{\prime2}}\right] \Delta
(\mathbf{p}^{\prime})\nonumber\\
& -\int\frac{d\mathbf{p}^{\prime}}{(2\pi\hbar)^{3}}\delta V(\mathbf{p}%
,\mathbf{p}^{\prime})\frac{1-f(E_{+}(\mathbf{p}^{\prime}))-f(E_{-}%
(\mathbf{p}^{\prime}))}{E_{+}(\mathbf{p}^{\prime})+E_{-}(\mathbf{p}^{\prime}%
)}\Delta(\mathbf{p}^{\prime}). \label{eq:renorm_gap}%\end{aligned}$$ The convergence of the integral over $p^{\prime}$ in the first term of the right-hand side of Eq. (\[eq:renorm\_gap\]) is now obvious, while the convergence of the second term is due to the decay of $\delta V$ at large momenta (see Eq. (\[eq:V2\]) below). The gap equation accounting for higher orders in $g$ will be derived and discussed in Section IV.
In the limit of $\Delta\rightarrow0$ one can reduce Eq. (\[eq:renorm\_gap\]) to a linearized gap equation: $$\begin{aligned}
\Delta(\mathbf{p}) & =-\frac{2\pi\hbar^{2}a}{m_{r}}\int\frac{d\mathbf{p}%
^{\prime}}{(2\pi\hbar)^{3}}\left[ \frac{\tanh[\xi_{1}(\mathbf{p}^{\prime
})/2T]+\tanh[\xi_{2}(\mathbf{p}^{\prime})/2T]}{2[\xi_{1}(\mathbf{p}^{\prime
})+\xi_{2}(\mathbf{p}^{\prime})]}-\frac{2m_{r}}{p^{\prime2}}\right]
\Delta(\mathbf{p}^{\prime})\nonumber\\
& -\int\frac{d\mathbf{p}^{\prime}}{(2\pi\hbar)^{3}}\delta V(\mathbf{p}%
,\mathbf{p}^{\prime})\frac{\tanh[\xi_{1}(\mathbf{p}^{\prime})/2T]+\tanh
[\xi_{2}(\mathbf{p}^{\prime})/2T]}{2[\xi_{1}(\mathbf{p}^{\prime})+\xi
_{2}(\mathbf{p}^{\prime})]}\Delta(\mathbf{p}^{\prime})
\label{eq:linear_ren_gap}%\end{aligned}$$ The critical temperature $T_{c}$ is determined from Eq. (\[eq:linear\_ren\_gap\]) as the highest temperature at which this equation has a non-trivial solution for $\Delta$.
Critical temperature. BCS and GM approaches
===========================================
The first line of Eq. (\[eq:linear\_ren\_gap\]) corresponds to the linearized gap equation in the traditional BCS approach: $$\Delta(\mathbf{p})=-\frac{2\pi\hbar^{2}a}{m_{r}}\int\frac{d\mathbf{p}^{\prime
}}{(2\pi\hbar)^{3}}\left[ \frac{\tanh[\xi_{1}(\mathbf{p}^{\prime}%
)/2T]+\tanh[\xi_{2}(\mathbf{p}^{\prime})/2T]}{2[\xi_{1}(\mathbf{p}^{\prime
})+\xi_{2}(\mathbf{p}^{\prime})]}-\frac{2m_{r}}{p^{\prime2}}\right]
\Delta(\mathbf{p}^{\prime}). \label{eq:linearBCS_gap_eq}%$$ In this case the order parameter is momentum independent, $\Delta
(\mathbf{p})=\Delta$, and Eq. (\[eq:linearBCS\_gap\_eq\]) reduces to the equation for the critical temperature: $$1=\frac{\lambda}{2}\left[ \ln\frac{8\mu_{1}\exp(\gamma-2)}{\pi T_{BCS}}%
+\ln\frac{8\mu_{2}\exp(\gamma-2)}{\pi T_{BCS}}\right] ,
\label{BCSlinear_gap_eq}%$$ where $\gamma=0.5772$ is the Euler constant and we introduced a small parameter $$\label{lambda}\lambda=2p_{F}\left\vert a\right\vert /\pi\hbar\ll1.$$
Equation (\[BCSlinear\_gap\_eq\]) is obtained straightforwardly. First, integrating Eq. (\[eq:linearBCS\_gap\_eq\]) over the angles one has $$\begin{aligned}
1 & =\frac{\lambda}{2}\int_{0}^{\infty}dx\left[ \tanh[(x^{2}-1)\mu
_{1}/2T_{c}]+\tanh[(x^{2}-1)\mu_{2}/2T_{c}]-2\right] \\
& +\frac{\lambda}{2}\int_{0}^{\infty}dx\frac{\tanh[(x^{2}-1)\mu_{1}%
/2T_{c}]+\tanh[(x^{2}-1)\mu_{2}/2T_{c}]}{x^{2}-1},\end{aligned}$$ where $x=p/p_{F}$. For $\mu_{i}/T_{c}>>1$ the integrand of the first integral is equal to $-4$ for $x<1$, and for $x>1$ it rapidly drops to $0$ in a narrow interval of $x$, where $\left\vert x-1\right\vert \lesssim T_{c}/\mu<<1$. The contribution of this interval can be neglected and, therefore, the first term equals $-2 \lambda$. Then, after integrating the second term by parts, we obtain $$1=\lambda\left\{ -2-\frac{1}{2}\int_{0}^{\infty}dx\left[ \frac{x\mu
_{1}/T_{c}}{\cosh^{2}[(x^{2}-1)\mu_{1}/2T_{c}]}+\frac{x\mu_{2}/T_{c}}%
{\cosh^{2}[(x^{2}-1)\mu_{2}/2T_{c}]}\right] \ln\left\vert \frac{x-1}%
{x+1}\right\vert \right\} .$$ The final integration can easily be performed by using the fact that the integrand is non-zero only in a narrow range of $x$, where $\left\vert
x-1\right\vert \lesssim T_{c}/\mu<<1$. We can therefore introduce a new variable $y=x-1$ and extend the limits of integration over $y $ from $-\infty$ to $+\infty$. The equation then reads: $$1=\lambda\left\{ -2-\frac{1}{2}\int_{-\infty}^{\infty}dy\left[ \frac{\mu
_{1}/T_{c}}{\cosh^{2}(y\mu_{1}/T_{c})}+\frac{\mu_{2}/T_{c}}{\cosh^{2}(y\mu
_{2}/T_{c})}\right] \ln\frac{\left\vert y\right\vert }{2}\right\} ,$$ and performing the integration one arrives at Eq. (\[BCSlinear\_gap\_eq\]). This equation gives the critical BCS temperature (cf. [@Caldas]): $$T_{BCS}=\frac{8}{\pi}\exp(\gamma-2)\sqrt{\mu_{1}\mu_{2}}\exp\left( -\frac
{1}{\lambda}\right) . \label{eq:TcBCS}%$$
However, the linearized BCS gap equation (\[eq:linearBCS\_gap\_eq\]) can only be used for the calculation of the leading contribution to the critical temperature, corresponding to the term $\sim\lambda^{-1}$ in the exponent of Eq. (\[eq:TcBCS\]). Therefore, only the exponent in this equation is correct. As was shown by Gorkov and Melik-Barkhudarov [@GMB], the preexponential factor in Eq. (\[eq:TcBCS\]) is determined by next-to-leading order terms, which depend on many-body effects in the interparticle interaction. These are the interactions between particles in a many-body system through the polarization of the medium - virtual creation of particle-hole pairs.
The importance of the many-body effects for the preexponential factor can be understood as follows. After performing the integration over momenta, the gap equation (\[eq:linear\_ren\_gap\]) can be qualitatively written as $$1=\nu_{F}V_{\mathrm{eff}}\left[ \ln\frac{\mu}{T_{c}}+C\right] ,
\label{eq:approx_eqTc}%$$ where $\nu_{F}=m_{r}p_{F}/\pi^{2}\hbar^{3}$. In this formula, the large logarithm $\ln\mu/T_{c}$ comes from the integration over momenta near the Fermi surface, whereas the momenta far from the Fermi surface contribute to the constant $C$ which is of the order of unity. We then write $\nu
_{F}V_{\mathrm{eff}}=-\lambda+a\lambda^{2}$, where the first term is the direct interparticle interaction and we keep only the second order term in the many-body part $\delta V$ of the effective interaction. It is now easy to see that $\lambda\ln\mu/T_{c}\sim1$ and, therefore, the terms $a\lambda^{2}\ln
\mu/T_{c}$ and $\lambda C$ are of the same order of magnitude. As a result, both terms have to be taken into account for the calculation of the preexponential factor. Also, note that the contribution of the many-body part of the interparticle interaction comes from momenta near the Fermi surface, which are responsible for the large logarithm $\ln\mu/T_{c}\sim\lambda^{-1}$. Hence, only the values of $\delta V$ at the Fermi surface are important.
We now calculate the contribution of the many-body effects to the preexponential factor for the critical temperature. They are usually called Gorkov-Melik-Barkhudarov (GM) corrections. As it was argued above, in the weak coupling limit the most important contributions to the effective interaction are second order in $g$ (the role of high order terms will be discussed later). In the considered case of a two-component Fermi gas with an $s$-wave interaction, there is only the contribution shown in Fig. \[Fig1\], and the corresponding analytical expression reads: $$\delta V(\mathbf{p},\mathbf{p}^{\prime})=-g^{2}\int\frac{d\mathbf{k}}%
{(2\pi)^{3}}\frac{f[\xi_{1}(\mathbf{k}+\mathbf{q}/2)]-f[\xi_{2}(\mathbf{k}%
-\mathbf{q}/2)]}{\xi_{1}(\mathbf{k}+\mathbf{q}/2)-\xi_{2}(\mathbf{k}%
-\mathbf{q}/2)}, \label{eq:V2}%$$ where $\mathbf{q}=\mathbf{p}+\mathbf{p}^{\prime}$. In obtaining this expression we used the zero-temperature distribution function $f[\xi
_{1,2}(p)]=\theta(-\xi_{1,2}(p))$, with $\theta(x)$ being the step function. This is legitimate because the finite temperature corrections are proportional to the ratio of the critical temperature to the chemical potential and, therefore, are exponentially small. As can be seen from Eq. (\[eq:V2\]), the effective interaction $\delta V(\mathbf{p},\mathbf{p}^{\prime})$ changes on the momentum scale $p\sim p^{\prime}\sim p_{F}$.
We now solve Eq. (\[eq:linear\_ren\_gap\]). In this equation, the momentum dependence of the order parameter originates only from the momentum dependence of the many-body contribution to the interparticle interaction and, therefore, contains an extra power of the small parameter $\lambda$. As a result, this dependence can be ignored in the first integral on the right-hand side of Eq. (\[eq:linear\_ren\_gap\]), and we can simply replace there the order parameter $\Delta(p^{\prime})$ by its value on the Fermi surface $\Delta(p_{F})$. This does not affect the convergence of the integral at large momenta and, hence, changes only the constant $C$ in Eq. (\[eq:approx\_eqTc\]). The corresponding modification, however, is proportional to the small parameter $\lambda$ and can be neglected. In the second integral on the right-hand side of Eq. (\[eq:linear\_ren\_gap\]), as we have discussed earlier, only momenta $p^{\prime}$ near the Fermi surface ($p^{\prime}\approx p_{F}$) are important, and we can also put $p^{\prime}=p_{F}$ in $\Delta(p^{\prime})$ and $\delta
V(\mathbf{p},\mathbf{p}^{\prime})$. The gap equation then reads: $$\begin{aligned}
\Delta(\mathbf{p}) & =-\frac{2\pi\hbar^{2}a}{m_{r}}\int\frac{d\mathbf{p}%
^{\prime}}{(2\pi\hbar)^{3}}\left[ \frac{\tanh[\xi_{1}(\mathbf{p}^{\prime
})/2T_{c}]+\tanh[\xi_{2}(\mathbf{p}^{\prime})/2T_{c}]}{2[\xi_{1}%
(\mathbf{p}^{\prime})+\xi_{2}(\mathbf{p}^{\prime})]}-\frac{2m_{r}}{p^{\prime
2}}\right] \Delta(\mathbf{n}^{\prime}p_{F})\nonumber\\
& -\int_{p<\Lambda p_{F}}\frac{d\mathbf{p}^{\prime}}{(2\pi\hbar)^{3}}\delta
V(\mathbf{p},\mathbf{n}^{\prime}p_{F})\frac{\tanh[\xi_{1}(\mathbf{p}^{\prime
})/2T_{c}]+\tanh[\xi_{2}(\mathbf{p}^{\prime})/2T_{c}]}{2[\xi_{1}%
(\mathbf{p}^{\prime})+\xi_{2}(\mathbf{p}^{\prime})]}\Delta(\mathbf{n}^{\prime
}p_{F}), \label{eq:linear_ren_gap1}%\end{aligned}$$ where $\mathbf{n}^{\prime}$ is the unit vector in the direction of $\mathbf{p}^{\prime}$ and we introduced an upper cut-off $\Lambda p_{F}$ with $\Lambda\sim1$, for the purpose of convergence at large momenta. The exact value of $\Lambda$ is not important because, as we mentioned above, the contribution of large momenta to this integral has to be neglected. To derive an equation for the critical temperature, we consider Eq. (\[eq:linear\_ren\_gap1\]) for $\mathbf{p}=\mathbf{n}p_{F}$ and average it over the directions of $\mathbf{n}$. Taking into account that the order parameter for the $s$-wave pairing can only depend on the absolute value of the momentum, we obtain: $$\begin{aligned}
1 & =-\frac{2\pi\hbar^{2}a}{m_{r}}\int\frac{d\mathbf{p}^{\prime}}{(2\pi
\hbar)^{3}}\left[ \frac{\tanh[\xi_{1}(\mathbf{p}^{\prime})/2T_{c}]+\tanh
[\xi_{2}(\mathbf{p}^{\prime})/2T_{c}]}{2[\xi_{1}(\mathbf{p}^{\prime})+\xi
_{2}(\mathbf{p}^{\prime})]}-\frac{2m_{r}}{p^{\prime2}}\right] \nonumber\\
& -\overline{\delta V}\int_{p<\Lambda p_{F}}\frac{d\mathbf{p}^{\prime}}%
{(2\pi\hbar)^{3}}\frac{\tanh[\xi_{1}(\mathbf{p}^{\prime})/2T_{c}]+\tanh
[\xi_{2}(\mathbf{p}^{\prime})/2T_{c}]}{2[\xi_{1}(\mathbf{p}^{\prime})+\xi
_{2}(\mathbf{p}^{\prime})]}\nonumber\\
& =\lambda\ln\frac{8\sqrt{\mu_{1}\mu_{2}}\exp(\gamma-2)}{\pi T_{c}}-\nu
_{F}\overline{\delta V}\ln\frac{8\sqrt{\mu_{1}\mu_{2}}\exp(\gamma+\Lambda
-2)}{\pi T_{c}}\approx(\lambda-\nu_{F}\overline{\delta V})\ln\frac{8\sqrt
{\mu_{1}\mu_{2}}\exp(\gamma-2)}{\pi T_{c}}, \label{eq:linear_ren_gap3}%\end{aligned}$$ where $$\overline{\delta V}=\int\frac{d\mathbf{n}}{4\pi}\int\frac{d\mathbf{n}^{\prime
}}{4\pi}\delta V(\mathbf{n}p_{F},\mathbf{n}^{\prime}p_{F}) \label{deltaVint}%$$ is the $s$-wave component of the many-body interaction. Using Eq. (\[eq:V2\]) and integrating over the angles in Eq. (\[deltaVint\]), we obtain$$\overline{\delta V}=\nu_{F}g^{2}\frac{1+\ln4}{3}[f(\kappa)+f(\kappa^{-1})],
\label{deltaVf}%$$ where $\kappa=M/m$ and the function $f(\kappa)$ is given by $$f(\kappa)=-\frac{3(1+\kappa)}{4(1+\ln4)}\int_{0}^{1}dq\int_{0}^{1}%
pdp\ln\left\vert \frac{(p^{2}-1)(\kappa-1)+4q(p-q)}{(p^{2}-1)(\kappa
-1)-4q(p+q)}\right\vert . \label{eq:f}%$$ A straightforward lengthy integration of Eq. (\[eq:f\]) yields $$\begin{aligned}
f(\kappa) & =-\frac{3}{4(1+\ln4)}(1+\kappa)\left[ -\frac{\kappa+1}{3\kappa
}\ln(2)+\frac{\kappa-1}{3\kappa}\ln\left\vert \kappa-1\right\vert \right.
\nonumber\\
& \left. +\frac{4}{3(\kappa-1)}-\left( \frac{(\kappa+3)^{2}}{6(\kappa
-1)^{2}}+\frac{\kappa+2}{6\kappa}\right) \ln\frac{\kappa+1}{2}\right] .
\label{f_function}%\end{aligned}$$ From Eqs. (\[eq:linear\_ren\_gap3\]) and (\[deltaVf\]) we obtain the following expression for the critical temperature:$$T_{GM}=\frac{8}{\pi}e^{\gamma-2}\sqrt{\mu_{1}\mu_{2}}\exp[-(\lambda-\nu
_{F}\overline{\delta V})^{-1}]\approx T_{BCS}\exp(-\nu_{F}\overline{\delta
V}/\lambda^{2})$$$$=\frac{e^{\gamma}}{\pi}\left( \frac{2}{e}\right) ^{7/3}\exp\left\{
-\frac{1+\ln4}{3}[f(\kappa)+f(\kappa^{-1})-1]\right\} \sqrt{\mu_{1}\mu_{2}%
}\exp\left( -\frac{1}{\lambda}\right) . \label{eq:TcGMB}%$$ It can be rewritten in the form$$T_{GM}=\frac{e^{\gamma}}{\pi}\left( \frac{2}{e}\right) ^{7/3}\mu_{1}%
F(\kappa)\exp\left( -\frac{1}{\lambda}\right) \equiv0.277\,\frac{p_{F}^{2}%
}{2M}F(\kappa)\exp(-\pi\hbar/2p_{F}|a|), \label{TcF}%$$ where we expressed the Fermi energy of light atoms $\mu_{2}$ through the heavy-atom Fermi energy $\mu_{1}=p_{F}^{2}/2M$, and the function $F(\kappa)$ is given by $$F(\kappa)=\sqrt{\kappa}\exp\left\{ -\frac{1+\ln4}{3}[f(\kappa)+f(\kappa
^{-1})-1]\right\} . \label{F_function}%$$
For equal masses one has $\kappa=M/m=1$, and Eqs. (\[f\_function\]) and (\[F\_function\]) give $f(1)=1/2$, $F(1)=1$. Then Eq. (\[eq:TcGMB\]) reproduces the original result of Ref. [@GMB]. The function $F(\kappa)$ is shown in Fig. \[Fig2\].
![The function $F(\kappa)$ in the preexponential factor of Eq. (\[TcF\]).[]{data-label="Fig2"}](Fig2.eps){width="8cm"}
For a large mass ratio $\kappa=M/m\gg1$, this function tends to a constant value, $F(\kappa\rightarrow\infty)=2^{4/3}e^{1/6}$. The critical temperature is then given by$$T_{GM}=\frac{8e^{\gamma-2}}{\pi}2^{2/3}e^{-1/6}\mu_{1}\exp(-1/\lambda
)\equiv0.825\,\frac{p_{F}^{2}}{2M}\exp(-\pi\hbar/2p_{F}|a|),\quad\kappa\gg1.
\label{TcLargeratio}%$$
Note that this result is quite different from the BCS critical temperature of Eq. (\[eq:TcBCS\]). Aside from a constant of the order of unity, it contains an extra small factor $1/\sqrt{\kappa}=\sqrt{m/M}\ll1$. Thus, for a fermionic mixture with a large mass ratio of the components the second order contribution significantly reduces the critical temperature compared to the prediction of the simple BCS approach.
In Table 1 we show the critical temperature $T_{GM}$ following from Eq. (\[TcF\]) with $F(\kappa)$ given by Eq. (\[F\_function\]), for various mixtures of fermionic atoms. The critical temperature is given in units of $T_{GM}$ for a $^{6}$Li-$^{6}$Li mixture, and it is assumed that the quantity $p_{F}^{2}\exp(-\pi\hbar/2p_{F}|a|)$ is the same for all mixtures. One clearly sees that replacing one species in a mixture by a lighter one increases the critical temperature, whereas the replacement with a heavier one decreases $T_{GM}$.
\[c\][|c|c|c|c|c|]{}& $^{6}$Li & $^{40}$K & $^{87}$Sr & $^{171}$Yb\
$^{6}$Li & 1.000 & 0300 & 0.161 & 0.090\
$^{40}$K & & 0.150 & 0.097 & 0.062\
$^{87}$Sr & & & 0.069 & 0.048\
$^{171}$Yb & & & & 0.035\
Critical temperature. Higher order contributions
================================================
Let us now consider the contribution of higher order ($\sim\lambda^{3}$) many-body corrections. As can be seen from Eq. (\[eq:TcGMB\]), these corrections enter the exponent for the critical temperature being divided by $\lambda^{2}$ and, therefore, the corresponding term is $\lambda Q(M/m)$, where $Q$ is a function of the mass ratio. For moderate values of $M/m$, the function $Q$ is of the order of unity, and, therefore, the corresponding corrections can be neglected. However, as we will see later, for a large mass ratio, the function $Q$ becomes proportional to $M/m$, and the related corrections in the exponent are $\sim k_{F}aM/m$. The applicability of the perturbation theory requires $(k_{F}a)^{2}M/m\ll1$ (see Eq. (\[small\_parameter\]) below), but the quantity $k_{F}aM/m$ should not necessarily be small for $M/m\gg1$. As a result, the third-order many-body corrections proportional to $M/m$ have to be taken into account.
For calculating the terms of the order of $\lambda^{3}$, we have to modify the gap equation (\[eq:bcsgap\]) in order to include the difference between particles and quasiparticles, or single-particle excitations [@foot]. In the considered case of a two-component Fermi gas, the quasiparticles are characterized by the effective masses $M^{\ast}$ and $m^{\ast}$ and by the $Z$-factors $Z_{M}$ and $Z_{m}$. The $Z$-factors are related to the amplitude of creating a quasiparticle by adding an extra particle to the system (see [@LL9] for rigorous definitions and details). For a given (effective) interaction $V_{\mathrm{eff}}$ between heavy and light fermions, the interaction between the corresponding quasiparticles is simply equal to $Z_{M}Z_{m}V_{\mathrm{eff}}$. It is important for our approach that the ratios $M^{\ast}/M$, $m^{\ast}/m$, and the constants $Z_{M}$, $Z_{m}$ differ from unity only by a small amount proportional $\lambda$.
We can now extend the analysis of the gap equation to higher order terms. As follows from the previous discussions, the qualitative form of the gap equation can be written as$$1=\nu_{F}^{\ast}Z_{M}Z_{m}V_{\mathrm{eff}}\left[ \ln\frac{\mu}{T_{c}%
}+C\right] , \label{gap3}%$$ where $\nu_{F}^{\ast}=m_{r}^{\ast}k_{F}/\pi^{2}\hbar^{2}$ and the reduced mass $m_{r}^{\ast}$ is determined by the effective masses $M^{*}$ and $m^{*}$. The effective interaction $V_{\mathrm{eff}}=g+\delta V+\delta V^{(3)}$ now includes also the third-order many-body contribution $\delta V$ $^{(3)}$. We are interested in the terms of the order of $\lambda^{2}$ and $\lambda^{3}%
\ln\mu/T_{c}$ on the right-hand side of Eq. (\[gap3\]), where the $\lambda^{3}\ln\mu/T_{c}$ contributions come from the integration over momenta near the Fermi surface, whereas the momenta far from the Fermi surface result in $\lambda^{2}$ contributions. The term of the order $\lambda^{2}$ in Eq. (\[gap3\]) comes only from the second-order term $\delta V$ in the effective interaction. For a large mass ratio this term contains only $\ln(M/m)$ for large mass ratio (see. Eqs. (\[deltaVf\]) and \[f\_function\]) and, therefore, can be neglected. The term $\lambda^{3}\ln\mu/T_{c}$ results from the third-order term $\delta V^{(3)}$ in the effective interaction (with $\nu_{F}^{\ast}Z_{M}Z_{m}\rightarrow\nu_{F}$) and from the difference between particles and quasiparticles, $(\nu_{F}^{\ast}Z_{M}Z_{m}-\nu_{F})\sim
\lambda^{2}$, multiplied by the first order term $g$ in the effective interaction. As a result, up to terms of the order of $\lambda^{2}$, we can write the linearized renormalized gap equation as$$\begin{aligned}
\Delta(\mathbf{p}) & =-\frac{2\pi\hbar^{2}a}{m_{r}}\frac{m_{r}^{\ast}}%
{m_{r}}Z_{M}Z_{m}\int\frac{d\mathbf{p}^{\prime}}{(2\pi\hbar)^{3}}\left[
\frac{\tanh[\xi_{1}(\mathbf{p}^{\prime})/2T_{c}]+\tanh[\xi_{2}(\mathbf{p}%
^{\prime})/2T_{c}]}{2[\xi_{1}(\mathbf{p}^{\prime})+\xi_{2}(\mathbf{p}^{\prime
})]}-\frac{2m_{r}}{p^{\prime2}}\right] \Delta(\mathbf{n}^{\prime}%
p_{F})\nonumber\\
& -\int_{p<\Lambda p_{F}}\frac{d\mathbf{p}^{\prime}}{(2\pi\hbar)^{3}}\left[
\delta V(\mathbf{p},\mathbf{n}^{\prime}p_{F})+\delta V^{(3)}(\mathbf{p}%
,\mathbf{n}^{\prime}p_{F})\right] \frac{\tanh[\xi_{1}(\mathbf{p}^{\prime
})/2T_{c}]+\tanh[\xi_{2}(\mathbf{p}^{\prime})/2T_{c}]}{2[\xi_{1}%
(\mathbf{p}^{\prime})+\xi_{2}(\mathbf{p}^{\prime})]}\Delta(\mathbf{n}^{\prime
}p_{F}). \label{lingap3}%\end{aligned}$$ As in Eq. (\[eq:linear\_ren\_gap1\]), we introduce an upper cut-off $\Lambda
p_{F}$ for the purpose of convergence of integrals at large momenta. Eq. (\[lingap3\]) can be solved in the same way as Eq. (\[eq:linear\_ren\_gap1\]) and we obtain$$\begin{aligned}
T_{c} & =\frac{8}{\pi}e^{\gamma-2}\sqrt{\mu_{1}\mu_{2}}\exp\left\{ -\left[
\frac{m_{r}^{\ast}}{m_{r}}Z_{M}Z_{m}\lambda-\nu_{F}\left( \overline{\delta
V}+\overline{\delta V^{(3)}}\right) \right] ^{-1}\right\} \nonumber\\
& \approx T_{BCS}\exp\left\{ -\frac{1}{\lambda^{2}}\left[ \left(
\frac{m_{r}^{\ast}}{m_{r}}Z_{M}Z_{m}-1\right) \lambda+\nu_{F}\left(
\overline{\delta V}+\overline{\delta V^{(3)}}\right) \right] \right\}
\nonumber\\
& =T_{cGM}\exp\left\{ -\frac{1}{\lambda^{2}}\left[ \left( \frac
{m_{r}^{\ast}}{m_{r}}Z_{M}Z_{m}-1\right) \lambda+\nu_{F}\overline{\delta
V^{(3)}}\right] \right\} . \label{Tc3}%\end{aligned}$$ Note that only the contributions that are linear in $M/m$ for $M/m\gg1$ should be kept in $m_{r}^{\ast}/m_{r}$, $Z_{M}$, $Z_{m}$, and $\overline{\delta
V^{(3)}}$.
The effective masses $M^{\ast}$, $m^{\ast}$ and the constants $Z_{M}$, $Z_{m}$ can be obtained from the derivatives of the corresponding self-energies $\Sigma_{M}(\omega,p)$ and $\Sigma_{m}(\omega,p)$ with respect to the frequency $\omega$ and momentum $p$, evaluated at $\omega=0$ and $p=p_{F}$:$$\begin{aligned}
Z_{M(m)} & =\left( 1-\left. \frac{\partial\Sigma_{M(m)}(\omega
,p)}{\partial\omega}\right\vert _{\omega=0,p=p_{F}}\right) ^{-1},\label{Z}\\
M^{\ast}/M & =Z_{M}^{-1}\left( 1+\frac{M}{p_{F}}\left. \frac
{\partial\Sigma_{M}(\omega,p)}{\partial p}\right\vert _{\omega=0,p=p_{F}%
}\right) ^{-1},\label{Meff}\\
m^{\ast}/m & =Z_{m}^{-1}\left( 1+\frac{m}{p_{F}}\left. \frac
{\partial\Sigma_{m}(\omega,p)}{\partial p}\right\vert _{\omega=0,p=p_{F}%
}\right) ^{-1}. \label{meff}%\end{aligned}$$ The diagrams for the self energies $\Sigma_{M}$ and $\Sigma_{m}$ up to the second order in $g$ are shown in Fig. \[FigX\].
![The first and second order diagrams to the self-energy $\Sigma
_{\alpha}(\omega,p)$. Note that we present here only irreducible diagrams, and, therefore, the diagram of the second order containing the first order self-energy insertion to the Green function of the $\beta$-fermion in the first diagram is omitted.[]{data-label="FigX"}](Fig3.eps){width="15cm"}
The corresponding analytical expressions read$$\begin{aligned}
\Sigma_{\alpha}(\omega,p) & =\frac{2\pi\hbar^{2}a}{m_{r}}n_{\beta}+\left(
\frac{2\pi\hbar^{2}a}{m_{r}}\right) ^{2}\int\frac{d\omega_{2}}{2\pi}%
\frac{d\mathbf{p}_{2}}{(2\pi\hbar)^{3}}G_{\beta}(\omega_{2},\mathbf{p}%
_{2})\nonumber\\
& \times\int\frac{d\omega_{1}}{2\pi}\frac{d\mathbf{p}_{1}}{(2\pi\hbar)^{3}%
}\left[ G_{\beta}(\omega_{1}+\omega_{2},\mathbf{p}_{2}+\mathbf{p}%
_{1})G_{\alpha}(-\omega_{1}+\omega,-\mathbf{p}_{1}+\mathbf{p})-G_{\beta}%
^{(0)}(\omega_{1},\mathbf{p}_{1})G_{\alpha}^{(0)}(-\omega_{1},-\mathbf{p}%
_{1})\right] , \label{sigma}%\end{aligned}$$ where $\alpha=M,\,\beta=m$ or $\alpha=m,\,\beta=M$, and the Green functions are given by $$G_{\alpha(\beta)}(\omega,\mathbf{p})=\frac{1}{\omega-\xi_{\alpha(\beta
)}(\mathbf{p})+i\delta\mathrm{sign}[\xi_{\alpha(\beta)}(\mathbf{p})]},$$$$G_{\alpha(\beta)}^{(0)}(\omega,\mathbf{p})=\frac{1}{\omega-p^{2}%
/2m_{\alpha(\beta)}+i\delta}%$$ with $\delta=+0$. The divergent integral in the second-order contribution is renormalized in a standard way by replacing the coupling constant $g$ with the scattering amplitude $a$ and subtracting the product $G_{\beta}^{(0)}%
(\omega_{1},\mathbf{p}_{1})G_{\alpha}^{(0)}(-\omega_{1},-\mathbf{p}_{1})$ of the two Green functions in vacuum ($\mu_{1,2}=0$), which corresponds to the second order Born contribution to the scattering amplitude, from the integrand. After integrating over the frequencies $\omega_{1}$ and $\omega
_{2}$ in Eq. (\[sigma\]), we obtain$$\begin{aligned}
\Sigma_{\alpha}(\omega,p) & =\frac{2\pi\hbar^{2}a}{m_{r}}n+\left(
\frac{2\pi\hbar^{2}a}{m_{r}}\right) ^{2}\int\frac{d\mathbf{p}_{2}}{(2\pi
\hbar)^{3}}\frac{d\mathbf{p}_{1}}{(2\pi\hbar)^{3}}\left\{ \frac
{(1-f[\xi_{\beta}(\mathbf{p}_{2})])f[\xi_{\alpha}(-\mathbf{p}_{1}%
+\mathbf{p})]f[\xi_{\beta}(\mathbf{p}_{2}+\mathbf{p}_{1})]}{\omega-\xi
_{\alpha}(-\mathbf{p}_{1}+\mathbf{p})-\xi_{\beta}(\mathbf{p}_{2}%
+\mathbf{p}_{1})+\xi_{\beta}(\mathbf{p}_{2})-i\delta}\right. \nonumber\\
& +\left. \frac{f[\xi_{\beta}(\mathbf{p}_{2})](1-f[\xi_{\alpha}%
(-\mathbf{p}_{1}+\mathbf{p})])(1-f[\xi_{\beta}(\mathbf{p}_{2}+\mathbf{p}%
_{1})])}{\omega-\xi_{\alpha}(-\mathbf{p}_{1}+\mathbf{p})-\xi_{\beta
}(\mathbf{p}_{2}+\mathbf{p}_{1})+\xi_{\beta}(\mathbf{p}_{2})+i\delta}%
+\frac{f[\xi_{\beta}(\mathbf{p}_{2})]}{p_{1}^{2}/2m_{r}-i\delta}\right\} .
\label{sigmaP}%\end{aligned}$$ As we discussed above, for a large $M/m$ only the leading contributions that are linear in $M/m$ should be kept, and lengthy calculations with the use of Eqs. (\[Z\])-(\[sigmaP\]) give$$\begin{aligned}
Z_{M} & =1,\label{ZM}\\
\frac{M^{\ast}}{M} & =1+\frac{1}{5}\left( 2\ln2-1\right) \left(
\frac{ap_{F}}{\pi\hbar}\right) ^{2}\frac{M}{m},\label{Meffres}\\
Z_{m} & =1-\frac{1}{3}(1+2\ln2)\left( \frac{ap_{F}}{\pi\hbar}\right)
^{2}\frac{M}{m},\label{Zm}\\
\frac{m^{\ast}}{m} & =1+\frac{1}{3}(1+2\ln2)\left( \frac{ap_{F}}{\pi\hbar
}\right) ^{2}\frac{M}{m}. \label{meffres}%\end{aligned}$$
The diagrams for third-order contributions to the effective interaction $V_{\mathrm{eff}}(\mathbf{p},\mathbf{p}^{\prime})$ are shown in Fig. \[FigY\],
![The third-order contributions to the effective interaction between heavy (thick line) and light (thin line) fermions. The dashed line corresponds to the coupling constant $g$.[]{data-label="FigY"}](Fig4.eps){width="14cm"}
where we omit the diagrams that can be obtained by inserting the first order self-energy blocks (the first diagram in Fig. \[FigX\]) into the internal lines of the second-order diagram from Fig. \[Fig1\] . These self-energy contributions (the first term in Eq. (\[sigma\])) simply shift the chemical potentials. It turns out that only diagrams a, b, and c could contain terms linear in $M/m$, whereas the rest of the diagrams are proportional to $\ln(M/m)$. The divergencies at large momenta in diagrams d and f can be removed by renormalizing the coupling constant $g$ in the second-order diagram in Fig. \[Fig1\]. Analytical expressions for the diagrams a,b, and c are the following: $$\delta V_{a}^{(3)}(\mathbf{p},\mathbf{p}^{\prime})=g^{3}\int\frac{d\omega
_{1}d\mathbf{p}_{1}}{(2\pi)^{4}\hbar^{3}}G_{M}(\omega_{1},\mathbf{q}%
+\mathbf{p}_{1})G_{M}(\omega_{1},\mathbf{p}_{1})\int\frac{d\omega
_{2}d\mathbf{p}_{2}}{(2\pi)^{4}\hbar^{3}}G_{m}(\omega_{2},\mathbf{q}%
+\mathbf{p}_{2})G_{m}(\omega_{2},\mathbf{p}_{2}),$$ $$\delta V_{b}^{(3)}(\mathbf{p},\mathbf{p}^{\prime})=g^{3}\int\frac{d\omega
_{1}d\mathbf{p}_{1}}{(2\pi)^{4}\hbar^{3}}G_{M}(\omega_{1},\mathbf{p}%
+\mathbf{p}_{1})G_{M}(\omega_{1},\mathbf{p}^{\prime}+\mathbf{p}_{1})\int
\frac{d\omega_{2}d\mathbf{p}_{2}}{(2\pi)^{4}\hbar^{3}}G_{m}(\omega
_{2},\mathbf{p}_{2})G_{m}(\omega_{1}+\omega_{2},\mathbf{p}_{1}+\mathbf{p}%
_{2}),$$ $$\delta V_{c}^{(3)}(\mathbf{p},\mathbf{p}^{\prime})=g^{3}\int\frac{d\omega
_{1}d\mathbf{p}_{1}}{(2\pi)^{4}\hbar^{3}}G_{m}(\omega_{1},\mathbf{p}%
+\mathbf{p}_{1})G_{m}(\omega_{1},\mathbf{p}^{\prime}+\mathbf{p}_{1})\int
\frac{d\omega_{2}d\mathbf{p}_{2}}{(2\pi)^{4}\hbar^{3}}G_{M}(\omega
_{2},\mathbf{p}_{2})G_{M}(\omega_{1}+\omega_{2},\mathbf{p}_{1}+\mathbf{p}%
_{2}),$$ where $\mathbf{q}=\mathbf{p}-\mathbf{p}^{\prime}$. The integration over frequencies $\omega_{1}$ and $\omega_{2}$ in the above expressions is straightforward and gives$$\begin{aligned}
\delta V_{a}^{(3)}(\mathbf{p},\mathbf{p}^{\prime}) & =g^{3}\nu_{M}\nu
_{m}\int\frac{d\mathbf{p}_{1}}{(2\pi\hbar)^{3}}\frac{f[\xi_{1}(\mathbf{p}%
_{1}+\mathbf{q})]-f[\xi_{1}(\mathbf{p}_{1})]}{\xi_{1}(\mathbf{p}%
_{1}+\mathbf{q})-\xi_{1}(\mathbf{p}_{1})}\int\frac{d\mathbf{p}_{2}}{(2\pi
\hbar)^{3}}\frac{f[\xi_{2}(\mathbf{p}_{2}+\mathbf{q})]-f[\xi_{2}%
(\mathbf{p}_{2})]}{\xi_{2}(\mathbf{p}_{2}+\mathbf{q})-\xi_{2}(\mathbf{p}_{2}%
)}\\
& =g^{3}\nu_{M}\nu_{m}\frac{1}{4}\left[ 1+\frac{p_{F}}{q}\left(
1-\frac{q^{2}}{4p_{F}^{2}}\right) \ln\frac{2p_{F}+q}{\left\vert
2p_{F}-q\right\vert }\right] ^{2},\end{aligned}$$ where $\nu_{M}=Mp_{F}/2\pi^{2}\hbar^{3}$ and $\nu_{m}=mp_{F}/2\pi^{2}\hbar
^{3}$ are the densities of states at the Fermi level for heavy and light fermions, respectively. For the other two contributions we obtain: $$\begin{aligned}
\delta V_{b}^{(3)}(\mathbf{p},\mathbf{p}^{\prime}) & =g^{3}\int
\frac{d\mathbf{p}_{1}}{(2\pi\hbar)^{3}}\int_{0}^{\infty}dsA_{M}(s,\mathbf{p}%
_{1})\left[ \frac{f[\xi_{2}(\mathbf{p}_{1}+\mathbf{p})]}{\xi_{2}%
(\mathbf{p}_{1}+\mathbf{p})-\xi_{2}(\mathbf{p}_{1}+\mathbf{p}^{\prime
})+i\delta}\,\,\,\,\frac{1}{\xi_{2}(\mathbf{p}_{1}+\mathbf{p})-s}\right. \\
& \left. +\frac{1-f[\xi_{2}(\mathbf{p}_{1}+\mathbf{p})]}{\xi_{2}%
(\mathbf{p}_{1}+\mathbf{p})-\xi_{2}(\mathbf{p}_{1}+\mathbf{p}^{\prime
})-i\delta}\,\,\,\,\frac{1}{\xi_{2}(\mathbf{p}_{1}+\mathbf{p})+s}%
+(\mathbf{p}\leftrightarrow\mathbf{p}^{\prime})\right]\end{aligned}$$ with$$A_{M}(s,\mathbf{p})=\frac{Mp_{F}^{2}}{8\pi^{2}\hbar^{3}p}\left\{
\begin{array}
[c]{l}%
\displaystyle{1-\left( \frac{Ms}{pp_{F}}-\frac{p}{2p_{F}}\right) ^{2}%
,\quad\frac{p}{2M}\left\vert p-2p_{F}\right\vert \leq s\leq\frac{p}%
{2M}(p+2p_{F})}\\
\displaystyle{\frac{2Ms}{p_{F}^{2}},\quad0\leq s\leq\frac{p}{2M}(2p_{F}%
-p)};\,\,\,\,\,p\leq2p_{F},
\end{array}
\right. , \label{A1}%$$ and zero otherwise. A similar expression is obtained for $\delta V_{c}%
^{(3)}(\mathbf{p},\mathbf{p}^{\prime})$, with the replacements $\xi
_{2}\rightarrow\xi_{1}$ and $A_{M}(s,\mathbf{p})\rightarrow A_{m}%
(s,\mathbf{p})$.
The corresponding contributions to the $s$-wave scattering channel can be obtained by averaging over the directions of the momenta $\mathbf{p}$ and $\mathbf{p}^{\prime}$:$$\overline{\delta V_{j}^{(3)}}=\int\frac{d\widehat{\mathbf{p}}}{4\pi}\int
\frac{d\widehat{\mathbf{p}}^{\prime}}{4\pi}\delta V_{j}^{(3)}(\mathbf{p}%
,\mathbf{p}^{\prime}),\quad j=a,b,c.$$ In the limit of $M/m\gg1$, the leading terms in these contributions are$$\begin{aligned}
\overline{\delta V_{a}^{(3)}} & =g\frac{2+7\zeta(3)}{16}\left( \frac
{ap_{F}}{\pi\hbar}\right) ^{2}\frac{M}{m},\label{Va}\\
\overline{\delta V_{b}^{(3)}} & =0,\label{Vb}\\
\overline{\delta V_{c}^{(3)}} & =-g\frac{1+4(2+3\ln2)\ln2}{18}\left(
\frac{ap_{F}}{\pi\hbar}\right) ^{2}\frac{M}{m}, \label{Vc}%\end{aligned}$$ where $\zeta(x)$ is the Riemann zeta-functions ($\zeta(3)=1.202$). As a result, the quantity $\overline{\delta V^{(3)}}$ in Eq. (\[Tc3\]) is $$\overline{\delta V^{(3)}}=\overline{\delta V_{a}^{(3)}}+\overline{\delta
V_{b}^{(3)}}+\overline{\delta V_{c}^{(3)}}. \label{V3}%$$
Note that the validity of the perturbation theory requires the quantity $\overline{\delta V^{(3)}}$ be smaller than the coupling constant $g$. This leads to the condition $$(ap_{F}/\hbar)^{2}M/m\ll1. \label{small_parameter}%$$ Thus, the actual small parameter of the theory in the limit of a large mass ratio $M/m$ is $(p_{F}|a|/\hbar)\sqrt{M/m}$.
After substituting Eqs. (\[ZM\])-(\[meffres\]) and (\[V3\]) into Eq. (\[Tc3\]) we find the critical temperature in the limit of $M/m\gg1$: $$T_{c}=\frac{8e^{\gamma-2}}{\pi}2^{2/3}e^{-1/6}\frac{p_{F}^{2}}{2M}\exp\left(
-\frac{\pi\hbar}{2\left\vert a\right\vert p_{F}}-0.034\frac{\left\vert
a\right\vert p_{F}}{\pi\hbar}\frac{M}{m}\right) =T_{GM}\exp\{-0.011(p_{F}%
|a|/\hbar)M/m\}. \label{Treal}%$$ Compared to the transition temperature in the GM approach, Eq.(\[Treal\]) contains an extra exponential factor which, in principle, can be large. However, this requires a very high mass ratio $M/m$. The extra term in the exponent of Eq.(\[Treal\]) can be written as $0.01\sqrt{M/m}\times
(p_{F}|a|/\hbar)\sqrt{M/m}$ and, since the second multiple in this expression is small, one should have the mass ratio at least of the order of thousands in order to get a noticeable change of $T_{c}$ compared to the GM result. In this case Eq.(\[small\_parameter\]) shows that the parameter $\lambda
=2p_{F}|a|/\pi\hbar$ should be very small and, hence, the transition temperature itself is vanishingly low.
We thus see that for reasonable values of $p_{F}|a|/\hbar$ satisfying Eq.(\[small\_parameter\]), let say $p_{F}|a|/\hbar\sim0.1$ and $M/m<100$, the higher order contributions do not really change the GM result for the transition temperature.
At the same time, our analysis shows that for $M/m\gg1$ the conventional weakly interacting regime requires much lower values of $\lambda=2p_{F}%
|a|/\pi\hbar$ than in the case of equal masses and the small parameter of the perturbation theory is given by Eq.(\[small\_parameter\]) In the next section we discuss the physical origin of this parameter.
Small parameter of the theory
=============================
There are several conditions that allow one to develop a perturbation theory for a many-body fermionic system on the basis of Hamiltonian (\[H\]). First of all, this is the condition of the weakly interacting regime, which assumes that the amplitude $a$ of the interspecies interaction is much smaller than the mean separation between particles. The latter is of the order of $\hbar/p_{F}$, and we immediately have the inequality $$\label{ka}p_{F}|a|/\hbar\ll1.$$ At the same time, inequality (\[ka\]) allows one to use the binary approach for the interparticle interaction. Then, assuming a short-range character of the interatomic potential, one can consider the interaction between particles as contact and write the interaction part of the Hamiltonian as $g\int
d\mathbf{r} \widehat{\psi}_{1}^{+}(\mathbf{r})\widehat{\psi}_{1}%
(\mathbf{r})\widehat{\psi}_{2}^{+} (\mathbf{r})\widehat{\psi}_{2}(\mathbf{r})$.
In the weakly interacting regime only fermions near the Fermi surface participate in the response of the system to external perturbations. Therefore, there is another condition that is needed for constructing the perturbation theory. Namely, we have to assume that for both light and heavy fermions the density of states near the Fermi surface is not strongly distorted by the interactions. This is certainly the case if both Fermi energies, $p_{F}^{2}/2m$ and $p_{F}^{2}/2M$, greatly exceed the mean-field interaction $ng$. For $M\sim m$ this condition is equivalent to inequality (\[ka\]). In contrast, for $M\gg m$ Eq. (\[ka\]) only guarantees that the Fermi energy of light fermions is $p_{F}^{2}/2m\gg ng$, whereas the condition $p_{F}^{2}/2M\gg ng$ leads to the inequality $(p_{F}|a|/\hbar)M/m\ll1$. This mean-field condition, however, is far too strong because at the mean-field level, the interaction shifts uniformly all energy states and, hence, results only in the change of the chemical potential. Actually, the interaction-induced modification of the density of states is determined by the momentum and frequency dependence of the fermionic self-energy (see Eqs. (\[Z\])-(\[meff\])) and appears in the second order of the perturbative expansion in $g$ (the second diagram in Fig. \[FigX\]). The corresponding contribution describes the process in which a heavy fermion pushes a light one out of the Fermi sphere and then, interacting once more with this light fermion, puts it back to the initial state. Due to the Pauli principle, the momenta of both light and heavy fermions in the intermediate state should be larger than the Fermi momentum. As a result, for the initial heavy-fermion state close to the Fermi surface, the most important intermediate states will be those with momenta close to the Fermi momentum. Therefore, the resulting contribution should be proportional to the product of the densities of states of heavy and light fermions at the Fermi surface, and the relative change of both densities of states is controlled by the parameter $g^{2}\nu_{M}\nu_{m}\sim(p_{F}a/\hbar)^{2}M/m$. Thus, this parameter should be small, i.e. we arrive at Eq. (\[small\_parameter\]): $$(p_{F}a/\hbar)^{2}M/m\ll1.$$
A complementary physical argument on support of this small parameter comes from the consideration of the effective interaction between a light and a heavy fermion in the medium. For example, the process described by the diagram in Fig. 4a can be viewed in the following way. Incoming heavy and light fermions interact with fermions inside the filled Fermi spheres and transfer them to the states above the Fermi surfaces. Then the transferred heavy and light fermions interact with each other and return to their initial states. The important point is that the intermediate state of this process contains excitations (particle-hole pairs) near the Fermi surface of the filled Fermi sphere of heavy fermions. Therefore, the corresponding contribution to the effective interaction is $g_{\mathrm{eff}}\sim g^{3}\nu_{m}\nu_{M}$, where the densities of states of heavy and light fermions near the Fermi surface are $\nu_{M}=Mp_{F}/2\pi^{2}\hbar^{3}$ and $\nu_{m}=mp_{F}/2\pi^{2}\hbar^{3}$, respectively. This leads to $g_{\mathrm{eff}}\sim g(p_{F}a/\hbar)^{2}M/m$. Comparing it with the direct interaction $g$ and requiring the inequality $|g_{\mathrm{eff}}|\ll|g|$ which allows one to use a perturbation theory, we again obtain a small parameter of the theory $(p_{F}a/\hbar)^{2}M/m\ll1$.
Let us now understand in which physical quantities the parameter (\[small\_parameter\]) enters directly. In the limit of $M/m\gg1$ heavy fermions occupy the energy interval $p_{F}^{2}/2M$ which is much narrower than the energy interval $p_{F}^{2}/2m$ occupied by light fermions. However, the heavy-fermion density of states is much larger: $\nu_{M}=Mp_{F}/(2\pi^{2}%
\hbar^{3})\gg\nu_{m}=mp_{F}/(2\pi^{2}\hbar^{3})$. The high density of states of heavy fermions manifests itself in any quantity characterized by processes where, for the heavy fermions, only the states near the Fermi surface are important. This is the case for the effective masses of atoms, critical temperature, and (see the next section) for the zero-temperature order parameter $\Delta_{0}$. If, however, all states of the heavy fermions are important, then the peak at $E\sim p_{F}^{2}/2M$ in the energy distribution of heavy fermions is integrated out, and the result does not contain the parameter (\[small\_parameter\]). This is exactly what is happening in the calculation of the second order correction to the energy of the system, which involves the sum over all energy states.
![The second order contribution to the energy of the system.[]{data-label="FigE"}](Fig5.eps){width="8cm"}
The second order contribution to the energy is shown diagrammatically in Fig. \[FigE\], and the corresponding analytical expression reads$$E^{(2)}=g^{2}\int\frac{d\omega}{2\pi}\frac{d\mathbf{q}}{(2\pi\hbar)^{3}}%
\Pi_{m}(\omega,q)\Pi_{M}(-\omega,q), \label{energy2}%$$ where$$\begin{aligned}
\Pi_{\alpha}(\omega,q) & =\int\frac{d\omega_{1}}{2\pi}\frac{d\mathbf{p}_{1}%
}{(2\pi\hbar)^{3}}G_{\alpha}(\omega+\omega_{1},\mathbf{q}+\mathbf{p}%
_{1})G_{\alpha}(\omega_{1},\mathbf{p}_{1})\\
& =-\int\frac{d\mathbf{p}_{1}}{(2\pi\hbar)^{3}}\frac{f[\xi_{\alpha
}(\mathbf{p}_{1}+\mathbf{q})]-f[\xi_{\alpha}(\mathbf{p}_{1})]}{\omega
-(\xi_{\alpha}(\mathbf{p}_{1}+\mathbf{q})-\xi_{\alpha}(\mathbf{p}%
_{1}))+i\delta(\mathrm{sign}[\xi_{\alpha}(\mathbf{p}_{1}+\mathbf{q}%
)]-\mathrm{sign}[\xi_{\alpha}(\mathbf{p}_{1})])}%\end{aligned}$$ is the polarization operator (the bubble in the diagrammatic language) for $\alpha$-fermions ($\alpha=M$ or $m$). As it can be seen, the integration over $\omega$ in Eq. (\[energy2\]) results in an integral that diverges at large $q$. This divergence, however, can be eliminated by subtracting the second-order Born contribution to the interparticle scattering amplitude multiplied by the densities of fermions. This corresponds to the renormalization of the coupling constant $g$ in the first order (mean-field) contribution to the energy $E^{(1)}=gn^{2}$ (see [@LL9] for more details). The resulting expression then coincides with equation (6.12) in [@LL9].
After using the spectral representation for the polarization operator $\Pi_{\alpha}(\omega,p)$: $$\Pi_{\alpha}(\omega,q)=\int_{0}^{\infty}dsA_{\alpha}(s,q)\left[ \frac
{1}{\omega-s+i\delta}-\frac{1}{\omega+s-i\delta}\right] ,$$ with the function $A_{\alpha}(s,p)$ from Eq. (\[A1\]), equation (\[energy2\]) can be rewritten in the form$$E^{(2)}=\frac{1}{2}g^{2}\int\frac{d\mathbf{q}}{(2\pi\hbar)^{3}}\left[
\int_{0}^{\infty}ds_{1}\int_{0}^{\infty}ds_{2}\frac{A_{M}(s_{1},q)A_{m}%
(s_{2},q)}{s_{1}+s_{2}}-\frac{2m_{r}}{q^{2}}n^{2}\right] , \label{E2A}%$$ where the second term in the brackets corresponds to the renormalization. In the limit of $M/m\gg1$, as follows from Eq. (\[A1\]), typical values of $s_{1}$ are much smaller than typical values of $s_{2}$. We therefore can neglect $s_{1}$ in the denominator of the first term in Eq. (\[E2A\]) and replace $m_{r}$ by $m$. This gives$$\begin{aligned}
E^{(2)} & =\frac{1}{2}g^{2}\int\frac{d\mathbf{q}}{(2\pi\hbar)^{3}}\left[
\int_{0}^{\infty}ds_{1}A_{M}(s_{1},q)\int_{0}^{\infty}ds_{2}\frac{A_{m}%
(s_{2},q)}{s_{2}}-\frac{2m}{q^{2}}n^{2}\right] \\
& =\frac{1}{2}g^{2}\int\frac{d\mathbf{q}}{(2\pi\hbar)^{3}}\left[ \int
_{0}^{\infty}ds_{1}A_{M}(s_{1},q)\left( -\frac{1}{2}\right) \Pi
_{m}(0,q)-\frac{2m}{q^{2}}n^{2}\right] \\
& =\frac{1}{2}g^{2}\int\frac{d\mathbf{q}}{(2\pi\hbar)^{3}}\left[ n\left[
\theta(q-2p_{F})+\frac{3}{4}\frac{q}{p_{F}}\left( 1-\frac{q^{2}}{12p_{F}^{2}%
}\right) \theta(2p_{F}-q)\right] \left( -\frac{1}{2}\right) \Pi
_{m}(0,q)-\frac{2m}{q^{2}}n^{2}\right] \\
& =\frac{1}{2}g^{2}\int\frac{d\mathbf{q}}{(2\pi\hbar)^{3}}\left\{ n\left[
\theta(q-2p_{F})+\frac{3}{4}\frac{q}{p_{F}}\left( 1-\frac{q^{2}}{12p_{F}^{2}%
}\right) \theta(2p_{F}-q)\right] \frac{\nu_{m}}{4}\left[ 1+\frac{p_{F}}%
{q}\left( 1-\frac{q^{2}}{4p_{F}^{2}}\right) \ln\frac{2p_{F}+q}{\left\vert
2p_{F}-q\right\vert }\right] -\frac{2m}{q^{2}}n^{2}\right\} \\
& =gn^{2}\frac{9(8\ln2-9)}{140}\frac{ap_{F}}{\pi\hbar}.\end{aligned}$$ As we see, the final result does not depend on $M$, as it was anticipated above.
Order parameter and single-particle excitations
===============================================
We now calculate the order parameter and its temperature dependence. At zero temperature no quasiparticles are present since both quasiparticle energies are positive ($E_{\pm}>0$). Then, confining ourselves to second order terms in $g$, the renormalized gap equation reads: $$\begin{aligned}
\Delta_{0}(\mathbf{p}) & =-\frac{2\pi\hbar^{2}a}{m_{r}}\int\frac
{d\mathbf{p}^{\prime}}{(2\pi\hbar)^{3}}\left[ \frac{1}{E_{+}(\mathbf{p}%
^{\prime})+E_{-}(\mathbf{p}^{\prime})}-\frac{2m_{r}}{p^{\prime2}}\right]
\Delta_{0}(\mathbf{p}^{\prime})\nonumber\\
& -\int\frac{d\mathbf{p}^{\prime}}{(2\pi\hbar)^{3}}\delta V(\mathbf{p}%
,\mathbf{p}^{\prime})\frac{1}{E_{+}(\mathbf{p}^{\prime})+E_{-}(\mathbf{p}%
^{\prime})}\Delta_{0}(\mathbf{p}^{\prime}). \label{eq:rengap_zeroT}%\end{aligned}$$ Strictly speaking, the many-body contribution to the interparticle interaction $\delta V$ is affected by the superfluid pairing and, therefore, does not coincide with that of Eq. (\[eq:V2\]). However, at zero temperature the difference is proportional to $\Delta_{0}/\mu_{i}$ and, hence, is exponentially small. Therefore, we can use Eq. (\[eq:V2\]) for $\delta V$ in Eq. (\[eq:rengap\_zeroT\]).
The arguments used above for obtaining Eq. (\[TcF\]) from Eq. (\[eq:linear\_ren\_gap\]), can also be applied here, but the large logarithm $\ln(\mu/T_{c})$ should be replaced by $\ln(\mu/\Delta_{0}%
(p_{F}))$.
For the value of the order parameter at the Fermi surface, $\Delta_{0}(p_{F}%
)$, we obtain $$\Delta_{0}(p_{F})=\left( \frac{2}{e}\right) ^{7/3}\exp\left\{ -\frac
{1+\ln4}{3}[f(\kappa)+f(\kappa^{-1})-1]\right\} \frac{p_{F}^{2}}{4m_{r}}%
\exp\left( -\frac{1}{\lambda}\right) . \label{eq:delta0}%$$ Comparing Eq. (\[eq:TcGMB\]) with Eq. (\[eq:delta0\]) we obtain a relation between $\Delta_{0}(p_{F})$ and $T_{c}$: $$T_{c}=\frac{e^{\gamma}}{\pi}\frac{2}{\kappa^{1/2}+\kappa^{-1/2}}\Delta
_{0}(p_{F}). \label{TcDelta}%$$ We should emphasize that relation (\[eq:delta0\]) between the order parameter and the critical temperature remains valid after taking into account higher order terms in the gap equation, which is necessary for a large mass ratio (see the previous section for the discussion of the critical temperature). The generalization of Eq. (\[eq:rengap\_zeroT\]) in order to include the higher order terms repeats the derivation of Eq. (\[lingap3\]) and the resulting equation reads: $$\begin{aligned}
\Delta(\mathbf{p}) & =-\frac{2\pi\hbar^{2}a}{m_{r}}\frac{m_{r}^{\ast}}%
{m_{r}}Z_{M}Z_{m}\int\frac{d\mathbf{p}^{\prime}}{(2\pi\hbar)^{3}}\left[
\frac{1}{E_{+}(\mathbf{p}^{\prime})+E_{-}(\mathbf{p}^{\prime})}-\frac{2m_{r}%
}{p^{\prime2}}\right] \Delta(\mathbf{n}^{\prime}p_{F})\nonumber\\
& -\int_{p<\Lambda p_{F}}\frac{d\mathbf{p}^{\prime}}{(2\pi\hbar)^{3}}\left[
\delta V(\mathbf{p},\mathbf{n}^{\prime}p_{F})+\delta V^{(3)}(\mathbf{p}%
,\mathbf{n}^{\prime}p_{F})\right] \frac{1}{E_{+}(\mathbf{p}^{\prime}%
)+E_{-}(\mathbf{p}^{\prime})}\Delta(\mathbf{n}^{\prime}p_{F}).
\label{gapzero3}%\end{aligned}$$ This equation can be solved in a way similar to that of solving Eq. (\[lingap3\]), and the solution is$$\Delta_{0}(p_{F})=\pi e^{-\gamma}\frac{p_{F}^{2}}{4m_{r}}\exp\left\{ -\left[
\frac{m_{r}^{\ast}}{m_{r}}Z_{M}Z_{m}\lambda-\nu_{F}\left( \overline{\delta
V}+\overline{\delta V^{(3)}}\right) \right] ^{-1}\right\} . \label{Delta3}%$$ Comparing Eq. (\[Tc3\]) with Eq. (\[Delta3\]), we immediately obtain Eq. (\[TcDelta\]).
We now analyze the order parameter in the two limiting cases: $M=m$ and $M\gg
m$. In the case of equal masses we have $m_{r}=m/2$ and recover the usual expression [@GMB] for the order parameter from Eq. (\[eq:delta0\]) : $$\Delta_{0}(p_{F})=\left( \frac{2}{e}\right) ^{7/3}\frac{p_{F}^{2}}{2M}%
\exp\left( -\frac{\pi\hbar}{2p_{F}|a|}\right) =0.489\,T_{c};\quad M=m.$$
In the case of $M\gg m$, i.e. $\kappa\gg1$, we have $m_{r}\approx m$ and, including higher-order contributions, from Eqs. (\[TcDelta\]) and (\[TcLargeratio\]) we obtain $$\Delta_{0}(p_{F})=2^{8/3}e^{-13/6}\frac{p_{F}^{2}}{2\sqrt{Mm}}\exp\left(
-\frac{\pi\hbar}{2p_{F}|a|}-0.011\frac{\left\vert a\right\vert p_{F}}{\hbar
}\frac{M}{m}\right) =0.882\,\sqrt{\frac{M}{m}}T_{c}\gg T_{c};\quad M\gg m.
\label{Delta0}%$$ Note that in this limit the order parameter at the Fermi surface is much larger than the critical temperature.
In order to analyze the behavior of the order parameter $\Delta$ for temperatures close to the critical temperature, $(T_{c}-T)\ll T_{c}$, we have to expand the gap equation (\[eq:renorm\_gap\]) in powers of $\Delta/T_{c}%
\ll1$ and keep the cubic term. The result can be written as$$\Delta(p_{F})\left[ \ln\frac{T_{c}}{T}-\frac{4\kappa}{(1+\kappa)^{2}}%
\frac{7\zeta(3)}{8\pi^{2}}\left( \frac{\Delta(p_{F})}{T_{c}}\right)
^{2}\right] =0.$$ From this equation we obtain$$\Delta(p_{F})=\sqrt{\frac{8\pi^{2}}{7\zeta(3)}}\frac{\kappa^{1/2}%
+\kappa^{-1/2}}{2}T_{c}\sqrt{1-\frac{T}{T_{c}}}. \label{DeltaTc}%$$ In the limit of equal masses, $\kappa=1$, we reproduce the well-known result for the temperature dependence of the order parameter. In the opposite limit of a large mass ratio, $\kappa\gg1$, we find$$\Delta(p_{F})=\sqrt{\frac{M}{m}}\sqrt{\frac{8\pi^{2}}{7\zeta(3)}}T_{c}%
\sqrt{1-\frac{T}{T_{c}}}. \label{DeltaTcLargeratio}%$$ Note that as well as Eq. (\[Delta0\]), the obtained equation (\[DeltaTc\]) contains a large factor $\sqrt{M/m}$.
At any temperature the energies of single-particle excitations are given by (see Eq. (\[eq:dispersion\]) and Fig. \[Fig3\]) $$E_{\pm}(p)=\pm\left( \frac{p^{2}-p_{F}^{2}}{4m_{-}}\right) +\sqrt{\left(
\frac{p^{2}-p_{F}^{2}}{4m_{r}}\right) ^{2}+\Delta^{2}(p_{F})},
\label{eq:dispersion2}%$$ where $m_{-}=Mm/(M-m)$.
![Two branches $E_{\pm}(p)$ of single-particle excitations for $M/m=10$.[]{data-label="Fig3"}](Fig6.eps){width="8cm"}
Equation (\[eq:dispersion2\]) reveals a peculiar feature of fermionic mixtures with unequal masses of components. For equal masses, the minimum of $E_{\pm}(\mathbf{p})$ (i.e. the gap) is at the Fermi surface and equals $\Delta(p_{F})$. The situation for unequal masses is different: the single-particle excitation energies $E_{\pm}(p)$ reach their minimum values $E_{\pm\mathrm{min}}=\Delta(p_{F})\sqrt{1-s^{2}}=2\Delta(p_{F})/(\kappa
^{1/2}+\kappa^{-1/2})$ at momenta $p_{\pm}^{\ast2}=p_{F}^{2}\mp2m\kappa
^{1/2}(\kappa-1) \Delta(p_{F})/(\kappa+1)$. For $M/m\gg1$ the corresponding gap is much smaller than $\Delta(p_{F})$:$$E_{\pm\mathrm{min}}(M\gg m)\approx2\Delta(p_{F})\kappa^{-1/2}=2\sqrt{\frac
{m}{M}}\Delta(p_{F})\ll\Delta(p_{F}). \label{Emin}%$$ It is interesting to note that the presence of a small factor $\sqrt{m/M}$ in Eq. (\[Emin\]) restores the intuitive picture that the gap in the single-particle spectrum and the critical temperature are of the same order of magnitude even in the limit of a large mass ratio. We point out, however, that in this limit the order parameter on the Fermi surface, being much larger than the critical temperature, is not equal to the gap in the single-particle spectrum. This gap is of the order of the critical temperature, and the low-energy single-particle excitations correspond to momenta different from the Fermi momentum $p_{F}$. Owing to the former circumstance, one does not expect any dramatic changes in thermodynamic properties of the system with increasing the mass ratio $M/m$ to a large value.
Three-body resonances
=====================
Let us now discuss the influence of the three-body physics on the results of the previous sections. The diagram for the Gorkov-Melikh-Barkhudarov corrections in Fig. \[Fig1\] corresponds to collisions between three particles: two from a Cooper pair and one from the filled Fermi sea. Thus, it is a three-body process. During this process, however, the three particles undergo two successive two-body collisions and never appear simultaneously within the range of the interatomic interaction. The corresponding three-body wave function vanishes when the hyperspherical radius (see the definition before Eq. (\[wave\_function\])) is tending to zero.
In a dilute two-component Fermi gas, real three-body collisions during which the three colliding particles simultaneously approach each other, are rare. An additional smallness compared to a Bose gas is provided by the Pauli principle. Two of the three colliding particles are identical fermions and, therefore, the wave function of their relative motion should strongly decrease at small separations. As a result, the contribution of such collisions to the effective pairing interaction is small and can be neglected. However, for the case of a large mass ratio $M/m$ the situation is more subtle. If $M/m>13.6$, two heavy and one light fermions can form three-body bound states [@Efimov; @Fonseca; @PSSJ]. The most interesting case corresponds to the presence of a weakly bound trimer state because this results in a resonance $3$-body scattering at low energy. It is not clear that these resonances should be taken into account when calculating the pairing energy since there may be nontrivial issues of wave function statistics involved; nevertheless we shall estimate their possible contribution and leave such issues for further study [@CL].
To analyze the effect of three-body bound states we note that the contribution to the Gorkov-Melikh-Barkhudarov corrections in Fig. \[Fig1\] is part of a more general contribution involving the connected three-body vertex function $\Gamma_{c}^{(3)}$(see Fig. \[Fig4\]).
![The contribution of three-body processes between one light and two heavy fermions described by the connected three-body vertex $\Gamma_{c}^{(3)}%
$, to the Gorkov-Melikh-Barkhudarov corrections.[]{data-label="Fig4"}](Fig7.eps){width="8cm"}
This is a consequence of a general relation between two- and three-particle Green functions. The quantity $\Gamma_{c}^{(3)}(\left\{
p_{i}\right\} _{\mathrm{in}},\left\{ p_{i}^{\prime}\right\} _{\mathrm{out}%
})$ with $p_{i}=(\omega_{i},\mathbf{p}_{i})$, $i=1,2,3$, describes the scattering of two heavy and one light particle from the initial state with incoming energy-momenta $p_{i}$ into the final state with outgoing energy-momenta $p_{i}^{\prime}$. For $\omega_{i}=p_{i}^{2}/2m$ (the mass-shell condition) the vertex function coincides with the $T$-matrix. By definition, the connected vertex function $\Gamma_{c}^{(3)}$ does not include three-body processes in which only two out of the three particles collide (in our case, a light fermion collides with only one heavy fermion) and, therefore, $\Gamma_{c}^{(3)}$ is represented only by connected diagrams. The general three-body vertex function $\Gamma^{(3)}$(see Eq. \[Gamma3\]) contains all diagrams including disconnected ones. Those describe processes in which only two out of the three particles interact with each other. Fig. \[Fig5\]
![The lowest order contribution to the connected three-body vertex.[]{data-label="Fig5"}](Fig8.eps){width="8cm"}
shows the simplest contribution to $\Gamma_{c}^{(3)}$ that is second order and results in the Gorkov-Melikh-Barkhudarov corrections shown in Fig. \[Fig1\].
We consider the case where the size of a three-body bound state is much larger than $\left\vert a\right\vert $, but much smaller than the average distance between particles in the gas. Accordingly, the binding energy is much larger than typical kinetic energies of particles, the Fermi energies $\mu_{i}$. In this case, the influence of other particles of the gas can be neglected and the properties of the bound state can be found by solving the three-body Schrödinger equation. Introducing the hyperspherical radius $\rho
=\sqrt{\mathbf{x}^{2}+\mathbf{y}^{2}}$ in the $6$-dimensional space $(\mathbf{x},\mathbf{y})$, where $\mathbf{x}\sqrt{(2M+m)/4m}=\mathbf{r}%
_{1}-(\mathbf{r}_{2}+\mathbf{r}_{3})/2$ is the distance between the light fermion and the center of mass of two heavy ones separated from each other by a distance $\mathbf{y}=\mathbf{r}_{2}-\mathbf{r}_{3}$, the normalized wave function of a shallow bound state with the binding energy $E_{b}=-\hbar
^{2}/2Mb^{2}$ and the size $b\gg\left\vert a\right\vert $, has the form [@Petrov]: $$\varphi(\mathbf{\rho})\sim\left( \frac{m}{M}\right) ^{3/4}\frac
{1}{\left\vert a\right\vert }\times\left\{
\begin{array}
[c]{l}%
\displaystyle{\frac{\Phi_{1}(\Omega)}{\rho^{2}},\quad\rho\ll\left\vert
a\right\vert }\\
\displaystyle{\frac{\left\vert a\right\vert ^{3}\Phi_{2}(\Omega)}{\rho^{5}%
},\quad a\ll\rho\ll b}%
\end{array}
\right. . \label{wave_function}%$$ Here $\Phi_{1}$ and $\Phi_{2}$ are the functions of hyperangles $\Omega$, and we do not give explicit expressions for these functions because of their complexity. For $\rho>b$, the wave function decays exponentially. Note that the normalization of the wave function (\[wave\_function\]) is determined by distances $\rho\sim\left\vert a\right\vert $.
Most conveniently the contribution of the bound state to the vertex function can be found using the three-body Green function $G(\left\{ \mathbf{p}%
_{i}\right\} ,\left\{ \mathbf{p}_{i}^{\prime}\right\} ,\omega)$:$$G^{(3)}\left( \left\{ \mathbf{p}_{i}\right\} ,\left\{ \mathbf{p}%
_{i}^{\prime}\right\} ,\omega\right) =\left\langle \left\{ \mathbf{p}%
_{i}^{\prime}\right\} \left\vert \frac{1}{\omega-H+i0}\right\vert \left\{
\mathbf{p}_{i}\right\} \right\rangle ,$$ where the Hamiltonian $H$ has the form $H=H_{0}+\widehat{V}$ with$$H_{0}=\frac{p_{1}^{2}}{2m}+\frac{1}{2M}\left( p_{2}^{2}+p_{3}^{2}\right)$$ and$$\widehat{V}=g\delta(\mathbf{r}_{1}-\mathbf{r}_{2})+g\delta(\mathbf{r}%
_{1}-\mathbf{r}_{3})$$ in the coordinate representation (index $1$ corresponds to the light fermion and indices $2$ and $3$ to the heavy ones). The Green function satisfies the equation$$HG^{(3)}\left( \left\{ \mathbf{p}_{i}\right\} ,\left\{ \mathbf{p}%
_{i}^{\prime}\right\} ,\omega\right) =\prod\limits_{i=1,2,3}\delta
(\mathbf{p}_{i}-\mathbf{p}_{i}^{\prime}),$$ which is equivalent to the integral equation$$G^{(3)}=G_{0}^{(3)}+G_{0}^{(3)}\widehat{V}G^{(3)}, \label{G3}%$$ with $G_{0}^{(3)}=[\omega-p_{1}^{2}/2m-\left( p_{2}^{2}+p_{3}^{2}\right)
/2M+i0]^{-1}\prod\nolimits_{i}\delta(\mathbf{p}_{i}-\mathbf{p}_{i}^{\prime})$ being the Green function for free particles. This equation can be rewritten in the form$$G^{(3)}=G_{0}^{(3)}+G_{0}^{(3)}\Gamma^{(3)}G_{0}^{(3)}, \label{Gamma3}%$$ where we introduce the vertex function $\Gamma^{(3)}$. This function describes all scattering processes involving three particles, both connected (described by $\Gamma_{c}^{(3)}$) and disconnected ones (not included in $\Gamma
_{c}^{(3)}$). The vertex function $\Gamma^{(3)}$ obeys the Lipmann-Schwinger equation$$\Gamma^{(3)}=\widehat{V}+\widehat{V}G_{0}^{(3)}\Gamma^{(3)}
\label{Lippmann-Schwinger}%$$ and, as it can be seen from Eqs. (\[G3\])-(\[Lippmann-Schwinger\]), is related to the Green function $G^{(3)}$ as$$\Gamma^{(3)}=\widehat{V}+\widehat{V}G^{(3)}\widehat{V}. \label{Gamma3-G3}%$$ It is convenient to use the spectral decomposition of the Green function. In the center-of-mass reference frame, where $\sum_{i}\mathbf{p}_{i}=\sum
_{i}\mathbf{p}_{i}^{\prime}=0$, this decomposition reads:$$G^{(3)}\left( \left\{ \mathbf{p}_{i}\right\} ,\left\{ \mathbf{p}%
_{i}^{\prime}\right\} ,\omega\right) =\sum_{n}\Psi_{n}^{\ast}(\left\{
\mathbf{p}_{i}\right\} )\frac{1}{\omega-E_{n}+i0}\Psi_{n}(\left\{
\mathbf{p}_{i}^{\prime}\right\} )+\int d\lambda\Psi_{\lambda}^{(+)\ast
}(\left\{ \mathbf{p}_{i}\right\} )\frac{1}{\omega-E_{\lambda}+i0}%
\Psi_{\lambda}^{(+)}(\left\{ \mathbf{p}_{i}^{\prime}\right\} ),
\label{SpectralG3}%$$ where the summation is performed over a complete set $\left\{ \Psi_{n}%
,\Psi_{\lambda}^{(+)}\right\} $ of eigenfunctions of the three-body Hamiltonian $H$ with eigenenergies $E_{n\text{,}}$ $E_{\lambda}$, respectively. The eigenfunctions $\Psi_{n}$ correspond to bound states with energies $E_{n}<0$, and the eigenfunctions $\Psi_{\lambda}^{(+)}$ to scattering states of three particles with energies $E_{\lambda}>0$. Their asymptotic behavior at large interparticle distances contains incoming plane waves with momenta specified by the index $\lambda$, and outgoing (therefore, index $+$) scattered waves. These eigenfunctions vanish for small hyperspherical radius $\rho$ and, in particular, they describe the Gorkov-Melik-Barkhudarov corrections discussed above. On the contrary, the bound state eigenfunctions are nonzero for small $\rho$ and decay exponentially for $\rho\rightarrow\infty$.
Eqs. (\[Gamma3-G3\]) and (\[SpectralG3\]) together give the decomposition of the vertex function $\Gamma^{(3)}$ in terms of the solutions of the three-body Schrödinger equation. Obviously, the bound states contribute only to the connected vertex function $\Gamma_{c}^{(3)}$. In particular, the contribution of the bound state $\Psi_{n}$ is:$$\delta_{n}\Gamma^{(3)}\left( \left\{ \mathbf{p}_{i}\right\} ,\left\{
\mathbf{p}_{i}^{\prime}\right\} ,\omega\right) =\delta_{n}\Gamma_{c}%
^{(3)}\left( \left\{ \mathbf{p}_{i}\right\} ,\left\{ \mathbf{p}%
_{i}^{\prime}\right\} ,\omega\right) =\left[ \widehat{V}\Psi_{n}(\left\{
\mathbf{p}_{i}\right\} )\right] ^{\ast}\frac{1}{\omega-E_{n}+i0}\widehat
{V}\Psi_{n}(\left\{ \mathbf{p}_{i}^{\prime}\right\} ). \label{Gamma3bound}%$$ By using the Schrödinger equation$$(H_{0}+\widehat{V})\Psi_{n}=E_{n}\Psi_{n},$$ Eq. (\[Gamma3bound\]) can be rewritten in the form$$\delta_{n}\Gamma_{c}^{(3)}\left( \left\{ \mathbf{p}_{i}\right\} ,\left\{
\mathbf{p}_{i}^{\prime}\right\} ,\omega\right) =\Psi_{n}^{\ast}(\left\{
\mathbf{p}_{i}\right\} )\frac{[E_{n}-E_{0}(\left\{ \mathbf{p}_{i}\right\}
)][E_{n}-E_{0}(\left\{ \mathbf{p}_{i}^{\prime}\right\} )]}{\omega-E_{n}%
+i0}\Psi_{n}(\left\{ \mathbf{p}_{i}^{\prime}\right\} ), \label{Gamma3bound1}%$$ where $E_{0}(\left\{ \mathbf{p}_{i}\right\} )=p_{1}^{2}/2m+\left( p_{2}%
^{2}+p_{3}^{2}\right) /2M$.
Now we can estimate the contribution of the weakly bound state of one light and two heavy fermions to the effective interparticle interaction $V_{\mathrm{eff}}$ between light and heavy fermions with opposite momenta on the Fermi surface. The analytical expression corresponding to the diagram in Fig. \[Fig4\] is$$\delta V_{\mathrm{eff}}=\int\frac{d\omega}{2\pi}\int\frac{d\mathbf{q}}%
{(2\pi\hbar)^{3}}\Gamma_{c}^{(3)}\left( \mathbf{p},\mathbf{q},-\mathbf{p}%
;\mathbf{p}^{\prime},-\mathbf{p}^{\prime},\mathbf{q};\omega\right) \frac
{1}{\omega-(q^{2}-p_{F}^{2})/2M+i0\mathrm{sign}(q-p_{F})}. \label{effective3}%$$ The contribution of the bound state is then obtained by substituting Eq. (\[Gamma3bound1\]) into Eq. (\[effective3\]) and integrating out the frequency $\omega$:$$\delta_{n}V_{\mathrm{eff}}=\int\frac{d\mathbf{q}}{(2\pi\hbar)^{3}}\theta
(p_{F}-q)\Psi_{n}^{\ast}(\mathbf{p},\mathbf{q},-\mathbf{p})\frac{[E_{n}%
-\mu_{1}-\mu_{2}-q^{2}/2M]^{2}}{(q^{2}-p_{F}^{2})/2M+\mu_{1}+2\mu_{2}%
-E_{n}-q^{2}/[2(2M+m)]}\Psi_{n}(\mathbf{p}^{\prime},-\mathbf{p}^{\prime
},\mathbf{q}), \label{effective3bound}%$$ where the last term in the denominator corresponds to the motion of the center of mass. In the considered case, the binding energy $E_{n}$ is the largest energy scale ($E_{n}\gg\mu_{1},\mu_{2}$), and $\delta_{n}V_{\mathrm{eff}}$ in Eq. (\[effective3bound\]) can be estimated as$$\delta_{n}V_{\mathrm{eff}}\sim(p_{F}/\hbar)^{3}E_{b}\left\vert \Psi
_{n}(0,0,0)\right\vert ^{2}, \label{effective_estimate}%$$ where the factor $(p_{F}/\hbar)^{3}$ results from the integration over $d\mathbf{q}$ and we used the condition $\left\vert a\right\vert p_{F}%
/\hbar\ll1$ to set all momenta in the wave function of the bound state to zero. After using the wave function from Eq. (\[wave\_function\]), we obtain: $$\Psi_{n}(0,0,0)\sim(p_{F}/\hbar)\left( \frac{M}{m}\right) ^{3/4}(p_{F}%
/\hbar)a^{2}b^{2},$$ and therefore$$\nu_{F}\delta_{n}V_{\mathrm{eff}}\sim\nu_{F}(p_{F}/\hbar)^{5}E_{b}a^{4}%
b^{4}\left( \frac{M}{m}\right) ^{3/2}\sim(p_{F}\left\vert a\right\vert
/\hbar)^{4}(p_{F}b/\hbar)^{2}\sqrt{\frac{M}{m}}.$$ This result has to be compared with the GM contribution $\nu_{F}%
\overline{\delta V}\sim(p_{F}\left\vert a\right\vert /\hbar)^{2}\ln(M/m)$, and with the contribution of third-ordrer terms $\nu_{F}\delta V^{(3)}\sim
(p_{F}|a|/\hbar)^{3}M/m$. Under the condition $(p_{F}b/\hbar)<1$ corresponding to a not too shallow bound state, we find that the contribution of three-body resonances is small compared to both of them:$$\frac{\nu_{F}\delta_{n}V_{\mathrm{eff}}}{\nu_{F}\overline{\delta V}}\sim
(p_{F}\left\vert a\right\vert /\hbar)^{2}(p_{F}b/\hbar)^{2}\frac{\sqrt{M/m}%
}{\ln(M/m)}\ll1,$$ and $$\frac{\nu_{F}\delta_{n}V_{\mathrm{eff}}}{\nu_{F}\delta V^{(3)}}\sim
(p_{F}\left\vert a\right\vert /\hbar)(p_{F}b/\hbar)^{2}\sqrt{m/M}\ll1.$$
We thus see that three-body resonances are rather narrow, and their contribution to the effective interaction can be omitted. So, the results obtained in the previous sections for the critical temperature, effective masses, order parameter, and elementary excitations remain unchanged.
Concluding remarks
==================
We now give an outlook on the physics of attractively interacting mixtures of heavy and light fermionic atoms in view of the results obtained in this paper. We have developed a perturbation theory in the BCS limit for the heavy-light superfluid pairing along the lines proposed by Gorkov and Melik-Barkhudarov [@GMB] and found that for $M/m\gg1$ one has to take into account both the second-order and third-order contributions. The result for the critical temperature and order parameter is then quite different from the outcome of the simple BCS approach. Moreover, the small parameter of the theory is given by Eq. (\[small\_parameter\]) and reads: $(p_{F}|a|/\hbar)\ll1$. As we explained in Section V, this can be seen from the second-order correction to the fermionic self-energy, which is controlled by the parameter $g^{2}\nu
_{M}\nu_{m}\sim(p_{F}a/\hbar)^{2}M/m$. Therefore, in a mixture of heavy and light fermions the conventional perturbation theory for the weakly interacting regime requires much smaller $p_{F}$ (densities) and/or $|a|$ than in the case of $M\sim m$, where the small parameter is $(p_{F}|a|/\hbar)\ll1$.
![Regimes of superfluid pairing for $M/m\gg1$. In the intermediate regime the conventional perturbation theory is not applicable (see text).[]{data-label="C"}](Fig9.eps){width="12cm"}
Let us now discuss the cases of $M=m$ and $M\gg m$ regarding the regimes of superfluid pairing. For $M=m$ we have the strongly interacting regime for $(p_{F}|a|/\hbar\gtrsim1$, and the BCS limit for $(p_{F}|a|/\hbar\ll1$. In the former case the perturbation theory is not applicable and the results are obtained either by Monte Carlo methods or by adjusting the mean-field theory to this regime (see [@Trento] for review). For $M\gg m$ the situation is different. As we found, the conventional perturbation theory works well under the condition $(p_{F}|a|/\hbar)\ll\sqrt{m/M}$ (see Fig. \[C\]). For $(p_{F}|a|/\hbar\gtrsim1$ we have the strongly interacting regime where the perturbation theory does not work at all. However, we now have a range of densities and scattering lengths, where $\sqrt{m/M}\ll(p_{F}|a|/\hbar)\ll1$. In this intermediate regime one can still use Hamiltonian (\[H\]) and try to develop a perturbative approach, since the scattering amplitude is much smaller than the mean interparticle separation. On the other hand, the conventional perturbation theory does not work for the reasons explained in Section V. In order to construct a reliable theory one should at least renormalize the interaction between heavy and light fermions by making an exact resummation of diagrams containing loops of heavy and light fermions. We then expect a substantial renormalization of the properties of the superfluid phase.
We thus see that our findings pave a way to revealing novel types of superfluid pairing in mixtures of attractively interacting ultracold fermionic atoms with very different masses. An appropriate candidate is a gaseous mixture of $^{171}$Yb with $^{6}$Li, and one should work out possibilities for tuning the Li-Yb interaction in this system. Another candidate is a two-species system of fermionic atoms in an optical lattice with a small filling factor. The difference in the hopping amplitudes of the species can be made rather large, which corresponds to a large ratio of the heavy to light effective mass. For example, in the case of $^{6}$Li-$^{40}$K mixture one can increase the mass ratio by a factor of $20$ in a lattice with period of $250$ nm and the tunneling rates $\sim10^{3}$ s$^{-1}$ and $\sim10^{5}$ s$^{-1}$ for K and Li, respectively.
Acknowledgements {#acknowledgements .unnumbered}
================
We acknowledge fruitful discussions with B.L. Altshuler and D.S. Petrov. Gratefully acknowledged is the hospitality and support of Institute Henri Poincaré during the workshop “Quantum Gases” where part of this work has been done. The work was also supported by the Dutch Foundation FOM, by the IFRAF Institute, by ANR (grants 05-BLAN-0205 and 06-NANO-014-01), by the QUDEDIS program of ESF, by the Austrian Science Foundation (FWF), and by the Russian Foundation for Fundamental Research. C.L. acknowledges support from the EPSRC through the Advanced Fellowship EP/E053033/1. LPTMS is a mixed research unit No. 8626 of CNRS and Université Paris Sud.
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---
abstract: 'For a quarternionic projective space, the homotopy inertia group and the concordance inertia group are isomorphic, but the inertia group might be different. We show that the concordance inertia group is trivial in dimension 20, but there are many examples in high dimensions where the concordance inertia group is non-trivial. We extend these to computations of concordance classes of smooth structures. These have applications to $3$-sphere actions on homotopy spheres and tangential homotopy structures.'
address:
- |
Department of Mathematical and Computational Science\
Indian Association for the Cultivation of Science\
Kolkata - 700032\
India.
- 'Statistics and Mathematics Unit, Indian Statistical Institute, Bangalore Centre, Bangalore - 560059, Karnataka, India.'
author:
- Samik Basu
- Ramesh Kasilingam
title: Inertia Groups and Smooth Structures on Quaternionic Projective Spaces
---
Introduction
============
The study of exotic structures on manifolds is a topic of fundamental interest in differential topology. The first such example comes up in the celebrated paper of Milnor [@Mil56], in examples of manifolds that are homeomorphic to $S^7$ but not diffeomorphic. This inspires the definition of the set $\Theta_n$ of differentiable manifolds homeomorphic to $S^n$. These form a group under connected sum and are related to stable homotopy groups in [@KM63].
One defines a family of structure sets for any differentiable manifold $M$, which are denoted as $\mathcal{S}^{\mathcal{C}}(M)$ for various equivalences ${\mathcal{C}}$. The smooth structure set is the set of manifolds homotopy equivalent to $M$ modulo the relation given by diffeomorphism. One may also consider other structure sets, such as the set of manifolds homotopy equivalent to $M$ with the relation given by homeomorphism. Along with homotopy equivalence, homeomorphism and diffeomorphism, one also considers equivalences given by piecewise-linear isomorphisms.
A possible way to change the smooth structure on a smooth manifold $M^n$ without changing the homeomorphism type is by taking connected sum with an exotic sphere. This induces an action of $\Theta_n$ on the set of smooth structures on $M$. The stabilizer of $M$ under this action is the inertia group $I(M)$.
This paper deals with computations in the inertia groups of quarternionic projective spaces, and associated computations for smooth structures. Computations in the inertia group are known for certain products of spheres [@Sch71], $3$-sphere bundles over $S^4$ [@Tam62], low dimensional complex projective spaces [@Kaw68; @BK]. However, there is no systematic approach for computing inertia groups in general, and many problems are still open.
One also makes certain analogous definitions for a differentiable manifold $M$, such as the homotopy inertia group and the concordance inertia group. These groups are the same for spheres and complex projective spaces, but for ${\mathbb{H}}P^n$, the homotopy inertia group and the concordance inertia group are the same, while the inertia group may be different from these. It follows from [@AF04] that the concordance inertia group is trivial for $n\leq 4$, while [@Kas16] demonstrates that the inertia group of ${\mathbb{H}}P^2$ is non-trivial. In this paper we prove that the concordance inertia group of ${\mathbb{H}}P^5$ is trivial. On the other hand, we observe that in high dimensions there are many examples where the concordance inertia group is non-trivial.
We follow up computations in the inertia group by computing the set of concordance classes of smooth structures on ${\mathbb{H}}P^n$ for $n\leq 5$. For $n=2$, this was computed in [@Kas16], where it was shown that the concordance classes are given by taking connected sum with an exotic $8$-sphere. For ${\mathbb{H}}P^4$, the same result is proved, but this does not hold for ${\mathbb{H}}P^3$. For ${\mathbb{H}}P^5$, there are $48$ different concordance classes. We also relate the concordance classes for ${\mathbb{H}}P^n$ to tangential homotopy structures, so that these computations also imply results for tangential homotopy structures for $n\leq 5$.
Computations in the concordance group allows us to discuss free smooth actions of the unit quarternionic sphere on homotopy spheres. The study of such actions has been of considerable interest. The orbit spaces of such actions are always homotopy equivalent to ${\mathbb{H}}P^n$. We explore such actions such that the orbit spaces are $PL$-homeomorphic to ${\mathbb{H}}P^n$, and prove that for $n=2,4$, no such action exists on a non-trivial exotic sphere, while for $n=3$, there is an exotic $15$-sphere which supports such an action.
Organisation
------------
In section \[iner-quart\], we introduce some preliminaries on the inertia group and make computations for ${\mathbb{H}}P^n$. In section \[smstr-quart\], we compute the set of concordance classes of ${\mathbb{H}}P^n$. Section \[s3act\] deals with applications to $3$-sphere actions on homotopy spheres and Section \[sec5\] deals with applications to tangential homotopy structures.
Notation
--------
Denote by $O={\mathit{colim}}_n O(n)$, $Top= {\mathit{colim}}_n Top(n)$, $F={\mathit{colim}}_n F(n)$, $PL={\mathit{colim}}_n PL(n)$ the direct limit of the groups of orthogonal transformations, homeomorphisms, homotopy equivalences, and piece-wise linear isomorphisms respectively. In this paper all manifolds will be closed smooth, oriented and connected, and all homeomorphisms and diffeomorphisms are assumed to preserve orientation, unless otherwise stated. We use the notation $\mathbb{S}^m$ for the standard unit $m$-sphere in ${\mathbb{R}}^{m+1}$, and while making computations in homotopy classes, the notation $S^m$ for a space homotopy equivalent to $\mathbb{S}^m$.
Acknowledgements
----------------
The research of the first author was partially supported by NBHM project ref. no. 2/48(11)/2015/NBHM(R.P.)/R&D II/3743, and that of the second author was supported by DST-Inspire Faculty Scheme (MA-64/2015).
Inertia groups of quaternionic projective spaces {#iner-quart}
================================================
In this section, we make some computations in the inertia groups of ${\mathbb{H}}P^n$. We show that the concordance inertia group is trivial when $n=5$, but non-trivial in many cases in high dimensions. We begin by recalling some preliminaries on exotic spheres and inertia groups.
- A homotopy $m$-sphere $\Sigma^m$ is an oriented smooth closed manifold homeomorphic to $\mathbb{S}^m$.
- A homotopy $m$-sphere $\Sigma^m$ is said to be exotic if it is not diffeomorphic to $\mathbb{S}^m$.
- Two homotopy $m$-spheres $\Sigma^{m}_{1}$ and $\Sigma^{m}_{2}$ are said to be equivalent if there exists an orientation preserving diffeomorphism $f:\Sigma^{m}_{1}\to \Sigma^{m}_{2}$.
The set of diffeomorphism classes of homotopy $m$-spheres is denoted by $\Theta_m$. The class of $\Sigma^m$ is denoted by \[$\Sigma^m$\]. When $m\geq 5$, $\Theta_m$ forms an abelian group with group operation given by connected sum $\#$, and the zero element represented by the equivalence class of $\mathbb{S}^m$. M. Kervaire and J. Milnor [@KM63] showed that each $\Theta_m$ is a finite group; in particular, $\Theta_{m}\cong \mathbb{Z}_2$, where $m=8, 16$, and $\Theta_{20}\cong \mathbb{Z}_{24}$.
Let $M^m$ be a closed smooth $m$-dimensional manifold. The inertia group $I(M)\subset \Theta_{m}$ is defined as the set of $\Sigma \in \Theta_{m}$ for which there exists a diffeomorphism $\phi :M\to M\#\Sigma$.
The homotopy inertia group $I_h(M)$ is the set of all $\Sigma\in I(M)$ such that there exists a diffeomorphism $M\to M\#\Sigma$ which is homotopic to $id:M \to M\#\Sigma$. The concordance inertia group $I_c(M)$ is the set of all $\Sigma\in I_h(M)$ such that $M\#\Sigma$ is concordant to $M$.
Note that $I_c(M)\subset I_h(M) \subset I(M)$, and these might not be equal [@Sch87 Theorem 2.1]. For ${\mathbb{S}}^n$ or an exotic sphere $\Sigma^n$, all these groups are trivial. For the space ${\mathbb{C}}P^n$, these groups are all equal [@Kas].
The groups $I(M)$ and $I_h(M)$ are not homotopy invariant. One has $I({\mathbb{S}}^3 \times \Sigma^{10}) \neq I({\mathbb{S}}^3 \times {\mathbb{S}}^{10})$ from [@Kaw69 Corollary 2, 3], while from [@Bru71] one notes that $I_h({\mathbb{S}}^1\times {\mathbb{C}}P^3) \cong {\mathbb{Z}}/7$, but for certian $6$-manifolds $P_j^6\simeq {\mathbb{C}}P^3$ for $j\equiv 1 \pmod 7$, $I_h({\mathbb{S}}^1 \times {\mathbb{C}}P^3)= 0$. On the contrary, the group $I_c(M)$ is indeed homotopy invariant, and we recall the formulation below.
The set of concordance classes of smooth structures on a topological manifold $M$ is defined as $$\mathcal{C}(M) =\{(N,f)\mid f:N\to M~\mathit{homeomorphism} \}/\sim$$ Two homeomorphisms $f:N\to M$ where $f_0:N_0\to M$ and $f_1:N_1\to M$ are related by $\sim$ if there is a diffeomorphism $g:N_{0}\to N_{1}$ such that the composition $f_{1}\circ g$ is topologically concordant to $f_{0}$, i.e., there exists a homeomorphism $F: N_{0}\times [0,1]\to M\times [0,1]$ such that $F_{|N_{0}\times 0}=f_{10}$ and $F_{|N_{0}\times 1}=f_{1}\circ g$.
We note from [@KS77 p. 25 and 194] that ${\mathcal{C}}({\mathbb{S}}^n) \cong \Theta_n \cong [S^n, Top/O]$, and ${\mathcal{C}}(M) \cong [M, Top/O]$. There is a homeomorphism $h: M^m\#\Sigma^m \to M^m$ $(m\geq5)$, which induces a class in $\mathcal{C}(M)$ denoted by $[M^m\#\Sigma^m]$. Note that $[M^m\#\mathbb{S}^m]$ is the class of $(M^m, Id)$.
Let $f_{M}:M^m\to S^m $ be a degree one map (which is well-defined up to homotopy). Composition with $f_{M}$ defines a homomorphism $$f_{M}^*:[S^m, Top/O]\to [M^m ,Top/O],$$ and in terms of the identifications above , $f_{M}^*$ becomes $[\Sigma^m]\mapsto [M^m\#\Sigma^m]$. Therefore, the concordance inertia group $I_c(M)$ can be identified with ${\mathit{Ker}}(f_M^\ast)$.
In this paper, we are interested in the inertia groups of ${\mathbb{H}}P^n$. One notes from [@Kas17 Corollary 3.2] that $I_h({\mathbb{H}}P^n) = I_c({\mathbb{H}}P^n)$. On the other hand, the inertia group of ${\mathbb{H}}P^n$ may be different from this. For the concordance inertia group, we use the homotopy-theoretic description above for $M={\mathbb{H}}P^{n}$. In [@AF04 Corollary 3.4.], it was proved that $f_{{\mathbb{H}}P^{n}}^*:[\mathbb{S}^{4n}, Top/O]\to [{\mathbb{H}}P^{n}, Top/O]$ is monic for $n\leq 4$, and hence, the concordance inertia group of ${\mathbb{H}}P^{n}$ is trivial for $n\leq 4$ ; for $n=5$, the concordance inertia group is shown to have $2$-primary component of order at most $2$. In contrary to the concordance and homotopy inertia groups, the inertia group of ${\mathbb{H}}P^{n}$ has a non-trivial element even for $n=2$. In [@Kas16 Theorem 1.1], it was proved that the inertia group of ${\mathbb{H}}P^{2}$ is isomorphic to the group of homotopy $8$-spheres $\Theta_8\cong {\mathbb{Z}}_2$.
In this paper, we show that the inertia group of ${\mathbb{H}}P^{n}$ is non-trivial in many cases. For this, we prove the non triviality of the concordance inertia group of ${\mathbb{H}}P^{n}$ by using the computations of the stable homotopy group of spheres. The first unresolved case is $n=5$ where we prove that the concordance inertia group is trivial.
\[iner\] The homomorphism $f_{{\mathbb{H}}P^{5}}^*:[{\mathbb{S}}^{20}, Top/O]\to [{\mathbb{H}}P^{5}, Top/O]$ is monic.
We proceed as in [@AF04] by considering the map $Top/O \to F/O$ and using the fact $F/O_{(p)} \simeq BSO_{(p)} \times cok J_{(p)}$ from [@MM79 Theorem 5.18]. The splitting is as H-spaces if $p$ is an odd prime. So, we have a commutative diagram $$\xymatrix{ [{\mathbb{H}}P^5, Top/O]_{(p)} \ar[r]^{\alpha'} & [{\mathbb{H}}P^5, F/O]_{(p)} & [{\mathbb{H}}P^5, Cok J_{(p)}] \ar[l]^{\beta'} \\
[S^{20}, Top/O]_{(p)} \ar[r]^\alpha \ar[u]^{f^\ast_{{\mathbb{H}}P^5}} & [S^{20}, F/O]_{(p)} \ar[u] & [S^{20}, Cok J_{(p)}] \ar[l]^\beta \ar[u]^{f^\ast_{{\mathbb{H}}P^5}} \\}$$ We know from the results of Kervaire and Milnor ([@KM63]) that $\alpha$ is injective and has the same image as $\beta$. Therefore, in order to prove that $[S^{20}, Top/O]\to [{\mathbb{H}}P^{5}, Top/O]$ is monic, it suffices to prove that $[S^{20}, Cok J_{(p)}]\to [{\mathbb{H}}P^{5}, Cok J_{(p)}]$ is monic for every prime $p$. As $\pi_{20}^s \cong {\mathbb{Z}}_{24}$, the primes we need to consider are $2$ and $3$.
We start with the prime $2$. Consider the diagram $$\xymatrix{ \{S^{17}, Cok J_{(2)}\} \ar[d]^{q^\ast} \\
\{\Sigma {\mathbb{H}}P^4, Cok J_{(2)}\} \ar[r] & \{S^{20}, Cok J_{(2)}\} \ar[r] & \{{\mathbb{H}}P^5, Cok J_{(2)}\} }$$ The bottom row is part of the long exact sequence for the cofibre $S^{19} \to {\mathbb{H}}P^4 \to {\mathbb{H}}P^5$, and the vertical map is induced by $q: {\mathbb{H}}P^4 \to {\mathbb{H}}P^4/ {\mathbb{H}}P^3 \simeq S^{16}$. In [@AF04 Lemma 3.2], it was checked that $q^\ast$ is surjective. The composite $$\{S^{17}, Cok J_{(2)}\} \stackrel{q^\ast}{\to} \{\Sigma {\mathbb{H}}P^4, Cok J_{(2)}\} \to \{S^{20}, Cok J_{(2)}\}$$ is induced by the map $\alpha : S^{20} \to S^{17}$ whose cofibre is the space $\Sigma ({\mathbb{H}}P^5/{\mathbb{H}}P^3)$. We note that the mod $2$ cohomology of ${\mathbb{H}}P^n$ is given by the ring ${\mathbb{Z}}/2[y]/(y^{n+1})$ with $|y|=4$, and it is easily computed that $Sq^4(y^4)=0$. It follows that the Hopf invariant of the attaching map $\alpha$ is $0$, and hence $\alpha = 2\alpha'$ in the group $\pi_{20}(S^{17}_{(2)}) \cong {\mathbb{Z}}/8$. As $ 2\{S^{17}, Cok J_{(2)}\} =0 $, we deduce that $\alpha^\ast : \{S^{17}, Cok J_{(2)}\} \to \{S^{20}, Cok J_{(2)}\}$ is $0$. Therefore, the kernel of $ \{S^{20}, Cok J_{(2)}\} \to \{{\mathbb{H}}P^5, Cok J_{(2)}\} $ is trivial.
Next we consider the prime $3$. We have the map $F_{(3)} \to Cok J_{(3)}$, and $Cok J_{(3)}$ is a summand of $F_{(3)}$. Therefore, it suffices to show $f^\ast_{{\mathbb{H}}P^5}: [S^{20}, F_{(3)}] \to [{\mathbb{H}}P^5, F_{(3)}]$ is injective on the classes which come from cokernel of $J$. Finally, these homotopy classes may be computed using a single path component of $F$ and thus these may be computed using $\Omega^\infty S^0$, the infinite loop space associated to the sphere spectrum. Thus it suffices to compute $f^\ast_{{\mathbb{H}}P^5}: {\pi_{20}^s}\otimes {\mathbb{Z}}_{(3)}= \{ S^{20}, S^0_{(3)} \} \to \{ {\mathbb{H}}P^5, S^0_{(3)} \}$ as all the classes in ${\pi_{20}^s} \otimes {\mathbb{Z}}_{(3)}$ lie in the cokernel of the $J$-homomorphism. As in the case above, we compute $\{\Sigma {\mathbb{H}}P^4, S^0_{(3)}\}$ and prove that it is $0$ using computations in [@Rav04 Table A.3.4].
We have $\{\Sigma {\mathbb{H}}P^1, S^0_{(3)}\} = {\pi_5^s} \otimes {\mathbb{Z}}_{(3)} = 0$, and therefore, the exact sequence $$\{S^9, S^0_{(3)}\} \to \{\Sigma {\mathbb{H}}P^2, S^0_{(3)}\} \to \{\Sigma {\mathbb{H}}P^1, S^0_{(3)}\}$$ together with the fact ${\pi_9^s} \otimes {\mathbb{Z}}_{(3)} = 0$ implies that $\{\Sigma {\mathbb{H}}P^2, S^0_{(3)}\} =0$. Next we have the exact sequence $$\{\Sigma^2 {\mathbb{H}}P^2, S^0_{(3)}\}\to \{S^{13}, S^0_{(3)}\} \to \{\Sigma {\mathbb{H}}P^3, S^0_{(3)}\} \to \{\Sigma {\mathbb{H}}P^2, S^0_{(3)}\}$$ in which the right hand term is $0$. The group $\pi_{13}^s \otimes {\mathbb{Z}}_{(3)} \cong {\mathbb{Z}}/3\{\alpha_1\beta_1\}$ from [@Rav04 Table A.3.4]. For computing the term $ \{\Sigma^2 {\mathbb{H}}P^2, S^0_{(3)}\}$ we note $ \{\Sigma^2 {\mathbb{H}}P^1, S^0_{(3)}\} = \pi_6^s \otimes {\mathbb{Z}}_{(3)} =0$ and thus $ \{\Sigma^2 {\mathbb{H}}P^2, S^0_{(3)}\} \cong \pi_{10}^s \otimes {\mathbb{Z}}_{(3)} \cong {\mathbb{Z}}/3\{\beta_1\}$. Thus we have the diagram $$\xymatrix{ \{S^{10}, S^0_{(3)}\} \ar[d]^{q^\ast}_{\cong} \\
\{\Sigma^2 {\mathbb{H}}P^2, S^0_{(p)}\} \ar[r] & \{S^{13}, S^0_{(3)}\} }$$ such that the composite computes the map in the exact sequence above. This is induced by the map $S^{13} \to S^{10}$ whose cofibre is $\Sigma^2({\mathbb{H}}P^3 /{\mathbb{H}}P^1)$. We compute the mod $3$ cohomology of ${\mathbb{H}}P^n$ as ${\mathbb{Z}}/3[y]/(y^{n+1})$ with $|y|=4$, and so for the Steenrod power operation ${\mathcal{P}}^1$, ${\mathcal{P}}^1(y^2)=y^3$. As the operation ${\mathcal{P}}^1$ detects $\alpha_1$ in the stable homotopy groups of spheres, it follows that the map $S^{13} \to S^{10}$ is a non-trivial multiple of $\alpha_1$. Therefore $\{\Sigma^2 {\mathbb{H}}P^2, S^0_{(3)}\}\to \{S^{13}, S^0_{(3)}\} $ takes $\beta_1$ to $\alpha_1\beta_1$ and is thus an isomorphism. Hence $ \{\Sigma {\mathbb{H}}P^3, S^0_{(3)}\}=0$. Finally from the exact sequence $$\{S^{17}, S^0_{(3)}\} \to \{\Sigma {\mathbb{H}}P^4, S^0_{(3)}\} \to \{\Sigma {\mathbb{H}}P^3, S^0_{(3)}\}$$ and the fact that $\pi_{17}^s\otimes {\mathbb{Z}}_{(3)}=0$, we deduce $\{\Sigma {\mathbb{H}}P^4, S^0_{(3)}\}=0$. This completes the proof of the theorem.
From the above result, we have the following:
\[exotic\] For any two elements $\Sigma_1, \Sigma_2\in \Theta_{20}$, ${\mathbb{H}}P^{5}\#\Sigma_1$ is concordant to ${\mathbb{H}}P^{5}\#\Sigma_2$ if and only if $\Sigma_1=\Sigma_2$. In particular, the concordance inertia group $I_c({\mathbb{H}}P^{5})=0$.
Computations such as Theorem \[iner\] have geometric applications along the lines of [@AF04]. We may start with a quarternionic hyperbolic manifold $N$ of dimension $20$ which has a finite-sheeted cover $M$ that have a non-zero tangential map to ${\mathbb{H}}P^5$. Now from [@Oku02 Lemma 3.5, Theorem 3.6] we have a relation between the action of $\Theta_{20}$ on concordance classes of smooth structures on ${\mathbb{H}}P^5$ and on the set of smooth structures on $M$.
\[exohyp\] There exist twelve exotic spheres $\{\Sigma_i : ~~i=1~{\rm to}~ 12\}\subset \Theta_{20}$ and a closed quarternionic hyperbolic manifold $M^{20}$ of quaternion dimension $5$ such that the following is true.
- The manifolds $M^{20}$, $\{M\#\Sigma_i : ~~i=1~{\rm to}~ 12\}$ are pairwise non-diffeomorphic.
- Each of the manifolds $M\#\Sigma_i $ supports a Riemannian metric whose sectional curvatures are all negative.
For every closed quarternionic hyperbolic manifold $N$, there is a finite sheeted cover which satisfies the conclusions for $M$ above.
We have thus observed that for $n\leq 5$, $I_c({\mathbb{H}}P^5)=0$. However, this is a phenomenon only in low dimensions, as it is possible to construct a fairly large number of non-trivial elements in the inertia groups of high dimensional quarternionic projective spaces. The technique for constructing these has been used in [@BK Theorem 3.9] for complex projective spaces in dimensions $4n+2$.
We use the result from [@CNL96] : For $p\geq 7$ the classes $\alpha_1\beta_1^r\gamma_t$ are non trivial in the stable homotopy groups of $S^0$ for $2\leq t \leq p-1$ and $r\leq p-2$ (in dimension $n(t,p,r)= [2(tp^3 - t - p^2) +2r(p^2 - 1 - p) -2]$). With these assumptions $\beta_1^r\gamma_t$ is also non-trivial in dimension $n(t,p,r)-(2p -3)$. Note that whenever $r$ is even, $4 \mid n(t,p,r)$, and if $p\nmid t+r$, $p\nmid n(t,p,r)-2(p-1)$.
\[inerhigh\] Suppose that $p$ is a prime $\geq 7$, $2\leq t \leq p-1$ and $r\leq p-2$. Assume that $r$ is even and $p$ does not divide $t+r$. Under these assumptions, the map $$[S^{n(t,p,r)}, Top/O] \to [{\mathbb{H}}P^{\frac{n(t,p,r)}{4}}, Top/O]$$ induced by the degree one map ${\mathbb{H}}P^{\frac{n(t,p,r)}{4}} \to S^{n(t,p,r)}$ has non-trivial $p$-torsion in the kernel.
We note that in $H^\ast ({\mathbb{H}}P^\infty;{\mathbb{Z}}/p)\cong {\mathbb{Z}}/p[y]$, the Steenrod operation ${\mathcal{P}}^1(y^k)\neq 0$ whenever $p\nmid k$. Once we assume the stated hypothesis, it follows that ${\mathcal{P}}^1(y^{\frac{n(t,p,r)}{4} - \frac{p-1}{2}}) \neq 0$.
Let $N=\frac{n(t,p,r)}{4}$ and $M=\frac{n(t,p,r)}{4} - \frac{p-1}{2}$. We consider the map $q:{\mathbb{H}}P^N \to S^N$ which quotients out the $(N-1)$-skeleton. This is the usual degree one map from ${\mathbb{H}}P^N$ to $S^N$ up to homotopy. We deduce that for the map $q^\ast : [S^N, Cok J_{(p)}] \to [{\mathbb{H}}P^N, Cok J_{(p)}]$, ${\mathit{Ker}}(q^\ast)$ has non-trivial $p$-torsion. From the proof of Theorem \[iner\], observe that it suffices to prove this. We start by noting that $q$ splits into a composite $${\mathbb{H}}P^N \stackrel{q_1}{\to} {\mathbb{H}}P^N/{\mathbb{H}}P^{M-1} \stackrel{q_2}{\to} S^N$$ so that it suffices to verify that ${\mathit{Ker}}(q_2^\ast)$ has non-trivial $p$-torsion. The space ${\mathbb{H}}P^N/ {\mathbb{H}}P^{M-1}$ has a CW-complex structure described as $S^{4M}\cup e^{4M+4} \cup \cdots \cup e^{4N}$. Working $p$-locally and in the stable homotopy category we work out the attaching maps which are of the form $S^{4M + 4k -1} \to S^{4M}\cup e^{4M+4} \cup \cdots \cup e^{4M+4(k-1)}$ for $1\leq k \leq \frac{p-1}{2}$. We note that on stable homotopy classes one has long exact sequences for any $X$ and a cell $e^m$ attached to $X$, $$\cdots \{S^r, X \} \to \{ S^r, X\cup e^m\} \to \{ S^r, S^m \} \to \{ S^r, \Sigma X \} \cdots$$ We note that for $1\leq k \leq \frac{p-1}{2}$, $\{ S^{4M+4k -1}, S^{4M + 4s}\}_{(p)}$ equals $0$, unless $k=\frac{p-1}{2}$ and $s=0$. Therefore the map $S^{4M + 4k -1} \to S^{4M}\cup e^{4M+4} \cup \cdots \cup e^{4M+4(k-1)}$ is homotopically trivial unless $k=\frac{p-1}{2}$. For the attaching map $S^{4N-1}$ the map goes down to the sphere $S^{4M}$, that is it is homotopic to a composite $S^{4N-1} \to S^{4M} \to S^{4M}\cup e^{4M+4} \cup \cdots \cup e^{4N-4}$. Therefore, it follows that $${\mathbb{H}}P^N/ {\mathbb{H}}P^{M-1}_{(p)} \simeq (S^{4M} \cup e^{4N})_{(p)} \vee S^{4M+4}_{(p)} \vee \cdots \vee S^{4N-4}_{(p)}$$ We denote $X= (S^{4M}\cup e^{4N})_{(p)}$ in the above, so that it suffices to prove that the kernel of $q_3^\ast$ has non-trivial $p$-torsion where $q_3:X \to S^{4N}$. The attaching map $S^{4N-1} \to S^{4M}$ must be a non-trivial multiple of $\alpha_1$ as the operation ${\mathcal{P}}^1$ carries the cohomology generator in degree $4M$ to the generator in degree $4N$, as $p \nmid M$. Thus $X\simeq S^{4M} \cup_{\alpha_1} S^{4N}$. Now we have the long exact sequence $$\cdots \to [S^{4M+1}, Cok J_{(p)}] \to [S^{4N}, Cok J_{(p)}] \to [X, Cok J_{(p)}] \cdots$$ where the left map is induced by multiplication by $\alpha_1$. We proceed as in the proof of the $p=3$ case of Theorem \[iner\], and use that $Im J_{(p)}$ is trivial in degrees which are $0, 1 \pmod 4$. Thus it suffices to check that the left arrow hits a non-trivial element when $Cok J_{(p)}$ is replaced by the sphere spectrum $ S^0_{(p)}$. Here, we know that $\beta_1^r \gamma_t$ is carried to $\alpha_1 \beta_1^r \gamma_t$ by the discussion preceeding the Theorem, and thus we are done.
[ Theorem \[inerhigh\] shows that under the given hypothesis, the concordance inertia group $I_c({\mathbb{H}}P^\frac{n(t,p,r)}{4})$ has non-trivial $p$-torsion. Observe that the hypothesis may be easily satisfied. The first example arises when $t=2,r=2$ which says that $I_c({\mathbb{H}}P^{p^3 -p + \frac{p^2-5}{2}})$ has non-trivial $p$-torsion. Specializing further to $p=7$ we get that $I_c({\mathbb{H}}P^{310})$ has non-trivial $7$-torsion. It follows that in these cases the homotopy inertia group and the inertia groups are also non-trivial. ]{}
Smooth Structures on Quaternionic Projective Spaces {#smstr-quart}
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Computations of the group ${\mathcal{C}}(M)$ are very important in the study of classification of manifolds and smooth structures, and from the identification ${\mathcal{C}}(M) \cong [M,Top/O]$, this may be computed using homotopy theory. For $M={\mathbb{C}}P^n$, the group ${\mathcal{C}}({\mathbb{C}}P^n)$ was determined in [@Kas17a] for $n\leq 8$ by using of the non-triviality of certain elements and relations in the stable homotopy groups of spheres proved in [@Tod62]. In this section, we study ${\mathcal{C}}({\mathbb{H}}P^n)$ for low values of $n$.
In the case $n=2$, it was proved in [@Kas16 Theorem 2.7] that $${\mathcal{C}}({\mathbb{H}}P^2)\cong \{{\mathbb{H}}P^2\#\Sigma^8, {\mathbb{H}}P^2 \} \cong{\mathbb{Z}}_2,$$ where $\Sigma^8\in \Theta_8$ is the exotic $8$-sphere. In the following, we are interested in the group ${\mathcal{C}}({\mathbb{H}}P^n)$ for $n\geq 3$.
For the computations below, we use the cofiber sequence $$S^{4n-1} \stackrel{p}{\to} {\mathbb{H}}{P}^{n-1}\stackrel{i} {\to} {\mathbb{H}}P^{n} \stackrel{f_{{\mathbb{H}}P^{n}}} {\to} S^{4n}$$ which induces the long exact sequence $$\label{longG}
\cdots \to [\Sigma {\mathbb{H}}P^{n-1}, Top/O] \to [S^{4n}, Top/O] \stackrel{f^{*}_{{\mathbb{H}}P^{n}}}{\to}[{\mathbb{H}}P^{n}, Top/O] \stackrel{i^{*}}{\to}[{\mathbb{H}}P^{n-1}, Top/O] \cdots .$$ Using these techniques, we compute
\[main\] (i) ${\mathcal{C}}({\mathbb{H}}P^3)$ has two concordance classes and therefore is $\cong {\mathbb{Z}}_2$. These classes induce the two different concordance classes on ${\mathbb{H}}P^2$ via $i^\ast$ in .\
(ii) ${\mathcal{C}}({\mathbb{H}}P^4)$ has exactly two concordance classes. The two classes are obtained as $\left \{ [({\mathbb{H}}P^4\#\Sigma]~ |~\Sigma\in \Theta_{16} \right \}$. (iii) ${\mathcal{C}}({\mathbb{H}}P^5)$ has $48$ concordance classes and as a group it is isomorphic to $ {\mathbb{Z}}_{24}\oplus {\mathbb{Z}}_2$ or ${\mathbb{Z}}_{48}$.
We start by proving (i). In the exact sequence , we use the fact $[S^{12}, Top/O]\cong \Theta_{12}=0$, to deduce that $$i^{*}:[{\mathbb{H}}P^{3}, Top/O] {\to} [{\mathbb{H}}P^{2}, Top/O]$$ is a monomorphism. Since $f^{*}_{{\mathbb{H}}P^{2}} : [S^{8}, Top/O] \to [{\mathbb{H}}P^{2}, Top/O]$ is an isomorphism and $[S^{8}, Top/O]\cong \Theta_8\cong {\mathbb{Z}}_2$, the non-trivial element in $[{\mathbb{H}}P^{2}, Top/O]$ is represented by a map $$g:{\mathbb{H}}P^{2} \stackrel{f_{{\mathbb{H}}P^{2}}}{\to} S^{8} \stackrel{\Sigma}{\to} Top/O,$$ where $\Sigma: S^{8}\to Top/O$ represents the exotic $8$-sphere in $\Theta_8$. Therefore, the effect of $p^{*}: [{\mathbb{H}}P^{2}, Top/O]{\to} [S^{11},Top/O]\cong {\mathbb{Z}}_{992}$ on the homotopy class $[g]$ is represented by the map $$S^{11}\stackrel{p}{\to} {\mathbb{H}}P^{2}\stackrel{f_{{\mathbb{H}}P^{2}}}{\to} S^{8} \stackrel{\Sigma}{\to} Top/O.$$ Now we will use the fact that the composition $f_{{\mathbb{H}}P^{2}}\circ p:S^{11}\to S^{8}$ is multiplication by $2\nu_2$ [@Jam76 page 38], where $\nu_2 = \Sigma^4 \nu \in \pi_{11}(S^8)$ is the $4$-fold suspension of the Hopf map $\nu$. As $2 [\Sigma]=0$, we have $$p^\ast([g]) = (2\nu_2) ([\Sigma]) = \nu_2(2[\Sigma]) = 0$$ where the second equality is derived from the fact that $2$ commutes with $\nu$ in the stable range. It follows that $p^{*}: [{\mathbb{H}}P^{2}, Top/O]{\to} [S^{11},Top/O]$ is the zero map. Therefore the map $$i^{*}:[{\mathbb{H}}P^{3}, Top/O]{\to}[{\mathbb{H}}P^{2}, Top/O]$$ is an isomorphism and hence ${\mathcal{C}}({\mathbb{H}}P^3)\cong {\mathbb{Z}}_2$.
Now consider the case (ii), that is, $n=4$. We prove that the map $[{\mathbb{H}}P^3, Top/O] \to [S^{15}, Top/O]$ induced by the attaching map of ${\mathbb{H}}P^4$ is injective. Since $[{\mathbb{H}}P^3, Top/O]\cong {\mathbb{Z}}/2$ we work $2$-locally. We use the diagram $$\xymatrix{ [{\mathbb{H}}P^3, Top/O]_{(2)} \ar[r]^{\alpha} \ar[d] & [{\mathbb{H}}P^3, F/O]_{(2)} \ar[d] & [{\mathbb{H}}P^3, Cok J_{(2)}] \ar[l]^{\beta} \ar[d] \\
[S^{15}, Top/O]_{(2)} \ar[r]^\alpha & [S^{15}, F/O]_{(2)} & [S^{15}, Cok J_{(2)}] \ar[l]^\beta }$$ On both the top row and the bottom row the maps $\alpha$ are injective and has the same image as $\beta$, so that it suffices to prove injectivity of the right vertical arrow. Firstly we note that $\pi_4 (Cok J_{(2)})=0$. Therefore, it suffices to compute $$q^\ast : [{\mathbb{H}}P^3/{\mathbb{H}}P^1, Cok(J)_{(2)}] \to [S^{15}, Cok(J)_{(2)}]$$ with $q$ induced by $S^{15} \to {\mathbb{H}}P^3 \to {\mathbb{H}}P^3/{\mathbb{H}}P^1$. We use stable homotopy theory to complete the computation – we assume that the spaces used below are actually suspension spectra and we use the notation of stable stems from [@Rav04]. We are allowed to do this because the space $Cok(J)_{(2)}$ is an infinite loop space.
The space ${\mathbb{H}}P^3/ {\mathbb{H}}P^1$ is a cell complex with $2$-cells of dimensions $8$ and $12$. As $\pi_{12} Cok(J)_{(2)} = 0$ we have that the group $[{\mathbb{H}}P^3/{\mathbb{H}}P^1, Cok(J)_{(2)}]$ is ${\mathbb{Z}}/2$ generated by the class $c_0$. From [@Jam76 Page 38], we note that the attaching map of the $12$-cell onto the $8$-cell is $2h_2$ and, we also note that the composite $$S^{15} \to {\mathbb{H}}P^3/{\mathbb{H}}P^1 \to {\mathbb{H}}P^3/{\mathbb{H}}P^2 \simeq S^{12}$$ is $3h_2$.
Therefore, the map $S^{15} \to {\mathbb{H}}P^3/{\mathbb{H}}P^1$ (in the category of spectra) is described by a map $\phi: S^{15} \to S^{12}$ which is $\simeq 3h_2$, together with a specific choice of null-homotopy of $2h_2 \circ \phi$. This means that the pullback $q^\ast c_0$ is an element of the Toda bracket $\langle 3h_2,2h_2, c_0\rangle$. Note that the indeterminacy of the bracket is $\pi_7. c_0 + 3h_2 . \pi_{12}$ which is $0$ as $\pi_{12} =0$, and the product of $c_0$ with $h_3$ is $0$.
Thus it remains to compute the Toda bracket $\langle 3h_2, 2h_2, c_0 \rangle$ which is an odd multiple of $\langle h_2, h_0h_2, c_0 \rangle$. Now we can compute using relations described in [@Ko96]. We note the generator of $\pi_{15}(Cok(J)_{(2)})$ is $h_1 d_0$. Now we have the following two Toda brackets : $d_0= \langle h_0,h_1,h_2,c_0 \rangle$ [@Ko96 Page 250] and $h_0h_2 = \langle h_1,h_0,h_1 \rangle$ [@MT68 Page 179]. We can now make some manoevres with Toda brackets $$h_1d_0= h_1\langle h_0,h_1,h_2,c_0 \rangle \subset \langle \langle h_1,h_0,h_1\rangle,h_2,c_0 \rangle$$ by [@Ko96 Proposition 5.7.4 c]. Thus the above computations imply $$\langle \langle h_1,h_0,h_1\rangle,h_2,c_0\rangle = \langle h_0h_2,h_2,c_0\rangle \subset \langle h_2, h_0h_2, c_0 \rangle$$ where the last equation is implied by [@Ko96 Proposition 5.7.4 b]. Therefore, $h_1d_0$ is an element of $\langle h_2, h_0h_2, c_0 \rangle$. Now, as the indeterminacy is trivial, this must be the same element as the image of $c_0$ in $[HP^3, Cok(J)_{(2)}]$ which implies the required injectivity.
Since the maps $f^{*}_{{\mathbb{H}}P^{4}}:[S^{16}, Top/O]\to [{\mathbb{H}}P^{4}, Top/O]$ and $p^{*}: [{\mathbb{H}}P^{3}, Top/O]{\to} [S^{15},Top/O]$ are injective, it follows from the exact sequence , that the map $f^{*}_{{\mathbb{H}}P^{4}}:[S^{16}, Top/O]\to [{\mathbb{H}}P^{4}, Top/O]$ is an isomorphism. Therefore $${\mathcal{C}}({\mathbb{H}}P^4)=\left \{ [({\mathbb{H}}P^4\#\Sigma]~|~\Sigma\in \Theta_{16} \right \}\cong {\mathbb{Z}}_2.$$
Finally we consider $n=5$. Observe from Theorem \[iner\] and (ii) that the maps $$f^{*}_{{\mathbb{H}}P^{5}}:[S^{20}, Top/O]\to [{\mathbb{H}}P^{5}, Top/O]$$ is injective and $$f^{*}_{{\mathbb{H}}P^{4}}:[S^{16}, Top/O]\to [{\mathbb{H}}P^{4}, Top/O]$$ is an isomorphism. Now we use the fact that the composition $f_{{\mathbb{H}}P^{4}}\circ p: S^{19}\to S^{16}$ is multiplication by $4\nu_2$ [@Jam76 Page 38], to deduce that the homomorphism $$p^{*}: [{\mathbb{H}}P^{4}, Top/O]{\to} [S^{19},Top/O]$$ is trivial as in (i). Therefore, from the exact sequence , it follows that there is an exact sequence $$0\to {\mathbb{Z}}_{24} \to {\mathcal{C}}({\mathbb{H}}P^5) \to {\mathbb{Z}}_2 \to 0 .$$ Hence we have, ${\mathcal{C}}({\mathbb{H}}P^5) \cong {\mathbb{Z}}_{24}\oplus {\mathbb{Z}}_2$ or ${\mathbb{Z}}_{48}$.
Smooth Free actions of ${\mathbb{S}}^3$ on Homotopy Spheres {#s3act}
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The topic of the existence and classification of free actions of ${\mathbb{S}}^1$ and ${\mathbb{S}}^3$ on exotic spheres is of considerable interest. It is well known that these are the only compact connected Lie groups which have free differentiable actions on homotopy spheres [@Bor53]. In fact one may prove that ${\mathbb{Z}}_p\times {\mathbb{Z}}_p$ does not act freely on a sphere, so that any Lie group acting freely on a sphere must have rank $1$.
For spheres acting on spheres, it follows from Gleason’s lemma [@Gle50] that such an action is always a principal fibration. These actions are homotopy equivalent to the classical Hopf fibrations [@Ste51] ($S^{2n+1} \to {\mathbb{C}}P^n$ for ${\mathbb{S}}^1$-actions and $S^{4n+3}\to {\mathbb{H}}P^n$ for ${\mathbb{S}}^3$-actions). However as smooth actions, there might be many inequivalent ones. Among many results in this regard, Hsiang [@Hsi66] has shown that in fact, there are always infinitely many differentiably inequivalent free actions of ${\mathbb{S}}^3$ (respectively, of ${\mathbb{S}}^1$) on $\Sigma^{4n+3}$ (respectively on $\Sigma^{2n+1}$) for $n\geq 2$ (respectively, for $n\geq 4$). Our interest lies in the case of ${\mathbb{S}}^3$-actions.
It is also known that equivariant diffeomorphism classes of differentiable, fixed point free ${\mathbb{S}}^3$ actions on homotopy $(4n+3)$-spheres, $n\geq 2$, correspond bijectively with equivalence classes of homotopy smoothings of ${\mathbb{H}}P^n$. The correspondence is defined as follows. If ${\mathbb{S}}^3$ acts on $\Sigma^{4n+3}$, then, $\Sigma^{4n+3}/{\mathbb{S}}^3 = P \simeq {\mathbb{H}}P^n$. On the other hand if $P^{4n}$ is smooth and homotopy equivalent to ${\mathbb{H}}P^n$, then there is a pullback $$\xymatrix{M^{4n+3} \ar[d] \ar[r]^-{\tilde{h}} & {\mathbb{S}}^{4n+3} \ar[d]\\
P^{4n} \ar[r]^-{h} & {\mathbb{H}}P^n.}$$ It follows easily that the map $\tilde{h}$ is a homotopy equivalence so that $M$ is an exotic sphere $\Sigma^{4n+3}$.
G. Brumfiel [@Bru68; @Bru71] has found all possible homotopy spheres in dimensions $9$, $11$ and $13$ which admit free differentiable actions of $\mathbb{S}^1$. He also studied the free differentiable actions of $\mathbb{S}^1$ on homotopy spheres which do not bound $\pi$-manifolds. Following Brumfiel, we study here all possible homotopy spheres which admit free differentiable actions of $\mathbb{S}^3$.
We start by recalling some facts from smoothing theory [@Bru68; @Bru71]. Recall that the smooth structure set on a manifold $N$ is defined as $$\mathcal{S}^{\mathit Diff}(N) =\{(P,f)\mid f:P\stackrel{\simeq}{\to} N \}/\sim,$$ where $f_0:P_0\to N$ and $f_1:P_1\to N$ are related by $\sim$ if there is a diffeomorphism $g:P_{0}\to P_{1}$ such that the composition $f_{1}\circ g$ is homotopic to $f_{0}$.
One may slightly extend this definition of a structure set $\mathcal{S}^{\mathit Diff}$. Let $M^k$, $k\geq 6$, be a simply connected, oriented, closed combinatorial manifold with a differentiable structure in the complement of a point. Let $M^k_0=M^k\setminus {\rm int}(\mathbb{D}^k)$, where $\mathbb{D}^k$ is a combinatorially embedded disc. $M^k_0$ inherits a differentiable structure from $M^k\setminus \{p\}$, hence $\partial M^k_0$ belongs to $\Theta_{k-1}$. Following Sullivan, two homotopy smoothings, $h:(M^{'}_0, \partial M^{'}_0)\to (M_0, \partial M_0)$ and $g:(M^{''}_0, \partial M^{''}_0)\to (M_0, \partial M_0)$ are called equivalent if there is a diffeomorphism $f:M^{'}_0\to M^{''}_0$ such that $h$ is homotopic to $g\circ f$. The set of equivalence classes is denoted by $\mathcal{S}^{Diff}(M_0)$. In [@Sul67], Sullivan constructs a bijection $\theta:\mathcal{S}^{Diff}(M_0)\stackrel{\cong} {\to}[M_0, F/O]$. Thus, if $h:M_{0}^{'}\to M_0$ represents an element of $\mathcal{S}^{Diff}(M_0)$, the formula $d\theta(M_{0}^{'}, h)=\partial M_{0}^{'}-\partial M_{0}$ defines a map $d: [M_0, F/O]\to \Theta_{k-1}$. Further, if $v\in [M^k_0,PL/O]$ then $$dv=\partial^*(v)\in \pi_{k-1}(PL/O)=\Theta_{k-1},$$ where $\partial:S^{k-1}\to M^k_0$ represents the homotopy class of the inclusion of the boundary $\partial M_{0}\to M_0$. In particular, $d: [M_0, PL/O]\to \Theta_{k-1}$ is a group homomorphism. Let $M={\mathbb{H}}P^n$ and ${\mathbb{H}}P^{n+1}_0$ is regarded as the total space of the $\mathbb{D}^4$ bundle $H$ over ${\mathbb{H}}P^n$. Thus there are maps $$S^{4n+3}=\partial {\mathbb{H}}P^{n+1}_0 \stackrel{i} {\to}{\mathbb{H}}P^{n+1}_0\stackrel{H} {\to} {\mathbb{H}}P^n$$ which induces $$[{\mathbb{H}}P^n, F/O]\stackrel{H^*}{\to}[{\mathbb{H}}P^{n+1}_0, F/O]\stackrel{\theta} {\to}\mathcal{S}^{Diff}({\mathbb{H}}P^{n+1}_0) \stackrel{i^*} {\to}\Theta_{4n+3},$$ where $i^*$ is the map which assigns to a homotopy smoothing of ${\mathbb{H}}P^{n+1}_0$ its boundary, which is a homotopy sphere. Note that $i^*\circ \theta=d$. We will denote the composition $i^*\circ \theta\circ H^*:[{\mathbb{H}}P^n, F/O]\to \Theta_{4n+3}$ itself by $d$.\
The proof of [@Bru68 Preposition 1.1] and [@Bru68 Lemma 6.1] work verbatim for ${\mathbb{H}}P^n$. So, we have the following results:
\[tec1\] Let $f: P^{4n}\to {\mathbb{H}}P^n$ in $\mathcal{S}^{Diff}({\mathbb{H}}P^{n})$ correspond to the $\mathbb{S}^3$ action on $\Sigma^{4n+3}$. Then $\Sigma^{4n+3}=d(P^{4n},f)$.$$\mathcal{C}(M) =\{(N,f)\mid f:N\to M~\mathit{homeomorphism} \}/\sim$$ Two homeomorphisms $f:N\to M$ where $f_0:N_0\to M$ and $f_1:N_1\to M$ are related by $\sim$ if there is a diffeomorphism $g:N_{0}\to N_{1}$ such that the composition $f_{1}\circ g$ is topologically concordant to $f_{0}$, i.e., there exists a homeomorphism $F: N_{0}\times [0,1]\to M\times [0,1]$ such that $F_{|N_{0}\times 0}=f_{10}$ and $F_{|N_{0}\times 1}=f_{1}\circ g$.
\[tec2\] Let $g\in [{\mathbb{H}}P^n, PL/O]$ correspond to the smoothing $P^{4n}$ of ${\mathbb{H}}P^n$. Then the composition $${\mathbb{H}}P^{n+1}_0 \stackrel{H} {\to} {\mathbb{H}}P^n\stackrel{g} {\to} PL/O$$ corresponds to a smoothing of ${\mathbb{H}}P^{n+1}_0$ which coincides with the natural smooth structure, say $P^{4n+4}_0$, on the Hopf disc bundle over $P^{4n}$.
Proposition \[tec1\] and Lemma \[tec2\] imply that the homomorphism $d: [{\mathbb{H}}P^n, PL/O]\to \Theta_{4n+3}=\pi_{4n+3}(PL/O)$ ($n\geq 1$) coincides with the map $[{\mathbb{H}}P^n, PL/O]\stackrel{p^*}{\to}\pi_{4n+3}(PL/O)$ induced by the Hopf map $p:S^{4n+3}\to {\mathbb{H}}P^n$. Therefore we have :
\[actions\] A homotopy sphere $\Sigma^{4n+3}$ admits a differentiable free $\mathbb{S}^3$-action such that the orbit space is PL-homeomorphic to ${\mathbb{H}}P^n$ if and only if $\Sigma^{4n+3}$ corresponds to a composition $S^{4n+3} \stackrel{p}{\to} {\mathbb{H}}P^n \stackrel{g}{\to} PL/O$ for some map $g$, by the natural isomorphism $\Theta_{4n+3}\cong \pi_{4n+3}(PL/O)$.
Note from Proposition \[tec1\] that if $f:P^{4n}\to {\mathbb{H}}P^n$ in $\mathcal{S}^{Diff}({\mathbb{H}}P^n)$ corresponds to the $\mathbb{S}^3$ action on $\Sigma^{4n+3}$, then $d(P^{4n},h)=\Sigma^{4n+3}$. This shows that the image of the composition $$\mathcal{S}^{Diff}({\mathbb{H}}P^n)\hookrightarrow [{\mathbb{H}}P^n, F/O]\stackrel{d} {\to}\Theta_{4n+3}$$ coincides with the set of homotopy $(4n+3)$-spheres which admit free $\mathbb{S}^3$ actions.
From the proof of Theorem \[main\]((i),(ii),(iii)), we get :
\[sphere\]
- $p^{*}: [\mathbb{H} P^{2}, Top/O]{\longrightarrow} [\mathbb{S}^{11},Top/O]$ is the zero map.
- $p^{*}: [\mathbb{H} P^{3}, Top/O]{\longrightarrow} [\mathbb{S}^{15},Top/O]$ is injective,
- $p^{*}: [\mathbb{H} P^{4}, Top/O]{\longrightarrow} [\mathbb{S}^{19},Top/O]$ is the zero map.
By Lemma \[actions\] and Theorem \[sphere\], we have the following:
There exists no differentiable free action of $\mathbb{S}^3$ on an exotic sphere $\Sigma^{4n+3}$ such that the orbit space is PL-homeomorphic to the quaternionic projective space $\mathbb{H} P^n$ when $n$ is any of $2$, $4$.
There exists an exotic $15$-sphere $\Sigma^{15}$ which does not bound a manifold such that $\Sigma^{15}$ admits a free differentiable action of $\mathbb{S}^3$ such that the orbit space is PL-homeomorphic to the quaternionic projective space $\mathbb{H} P^3$.
The Smooth Tangential Structures {#sec5}
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Following [@KM63; @Mil61; @Nov64], the study of manifolds of the same tangential homotopy type as ${\mathbb{H}}P^{n}$ was given in [@Her69]. Recall that two manifolds $N$ and $M$ are called tangentially homotopy equivalent if there is a homotopy equivalence $f: N\to M$ such that for some integers $k$, $l$, $f^{*}T(M)\oplus \displaystyle{\epsilon}_{N}^{k}\cong T(N)\oplus \displaystyle{\epsilon}_{N}^{l}$. The tangential smooth structure set of $M$, $\theta(M)$ is defined as the set $$\{(N,f) ~|~f:N\to M~\mbox{tangential homotopy equivalence} \}/\sim$$ where $f_0:N_0\to M$ and $f_1:N_1\to M$ are related by $\sim$ if there is a diffeomorphism $g:N_{0}\to N_{1}$ such that the composition $f_{1}\circ g$ is homotopic to $f_{0}$.
For $M={\mathbb{H}}P^{n}$, N. Hertz [@Her69] gave an inductive geometric procedure where by representatives for all elements of $\theta({\mathbb{H}}P^{n})$ may be constructed from elements of $\theta({\mathbb{H}}P^{n-1})$ and showed that $\theta({\mathbb{H}}P^{2})$ contains atmost two elements. In [@Kas16 Corollary 3.4], it was showed that $\theta({\mathbb{H}}P^2)$ contains exactly two elements, namely $\theta({\mathbb{H}}P^2)\cong \{{\mathbb{H}}P^2\#\Sigma^8, {\mathbb{H}}P^2 \} \cong {\mathbb{Z}}_2$. In this section, we are interested to compute the structure set $\theta({\mathbb{H}}P^n)$ for $n=3$, $4$ and $5$.
Let $M$ be a closed smooth manifold homotopy equivalent to ${\mathbb{H}}P^{n}$, $n\geq 2$. By the surgery exact sequence([@Bro72]), we have $$0\to\mathcal{S}^{Diff}(M)\to [M, F/O]\stackrel{\sigma}{\longrightarrow}L_{4n}(e).$$ To study $[M, F/O]$ we use the exact sequence $$\widetilde{KO}^{-1}(M)\to [M, SF]\stackrel{\phi_* }{\longrightarrow} [M, F/O]\to \widetilde{KO}^0(M)\to \bar{J}(M).$$ induced from fibrations $$SO\to SF\stackrel{\phi }{\longrightarrow}F/O\to BSO\to BSF.$$ Since $\widetilde{KO}^0(M)$ is free abelian group ([@SS73]), it follows that the image $${\rm Im~}(\phi_*:[M, SF]\to [M, F/O])$$ is the torsion subgroup of $[M, F/O]$. If the homotopy equivalence $f : N\to M$ represents an element of $\mathcal{S}^{Diff}(M)$, its image in $\widetilde{KO}^0(M)$ is given by $$(f^{-1})^{*}T^0N-T^0M,$$ where $T^0(N)$ is the stable tangent bundle of $N$. Thus, a normal map $\varphi\in [M, F/O]$ represents a tangential smoothing structure for $M$ if and only if $\varphi\in {\rm Im~}(\phi_*:[M, SF]\to [M, F/O])$ and $\sigma(\varphi)=0$. Therefore we have the following result:
\[sur\] Let $M$ be a closed smooth manifold homotopy equivalent to $\mathbb{H} P^{n}$, $n\geq 2$. Then $\theta(M)={\mathit{Ker}}\{\sigma:{\rm Im~}(\phi_*:[M, SF]\to [M, F/O])\to L_{4n}(e)\}.$
Now we prove the following.
\[imge\] Let $M$ be a closed smooth manifold homotopy equivalent to $\mathbb{H} P^{n}$, $n\geq 2$. Then
${\mathit Im}(\phi_*:[M, SF]\to[M, F/O])={\mathit Im}(\psi_*:[M, Top/O]\to[M, F/O]).$
The proof follows from the argument given in the proof of Theorem 5.2 in [@Kas17] by noting that $\widetilde{KO}^0(M)$ is free abelian group and $H_k(M;\mathbb{Z})=0$ for all $k\not\equiv{\rm 0~ mod~ 4}$.
\[imge1\] Let $M$ be a closed smooth manifold homotopy equivalent to $\mathbb{H}P^{n}$, $n\geq 2$. If a smooth manifold $N$ is tangential homotopy equivalent to $M$, then $N$ is homeomorphic to $M$.
This follows by the same argument given in the proof of Theorem 5.4 in [@Kas17] by using Theorem \[imge\].
\[tanset\] Let $M$ be a closed smooth manifold homotopy equivalent to $\mathbb{H}P^{n}$, $n\geq 2$. Then $\theta(M)\cong \mathcal{C}(M)$.
The smooth structure set $\mathcal{S}^{Diff}(M)$ fits into the following surgery exact sequence $$\mathcal{S}^{Diff}(M) \xhookrightarrow{inj}[M, F/O]\stackrel{\sigma}{\longrightarrow} L_{4n}(\mathbb{Z}).$$ Thus, a normal map $\varphi\in [M, F/O]$ represents a homotopy smoothing structure for $M$ if and only if $\sigma(\varphi)=0$. Observe that if $\varphi$ lies in the image $$[M,Top/O]\xhookrightarrow{\psi_*}\mathcal{S}^{Diff}(M)\xhookrightarrow{inj} [M,F/O],$$ then $\sigma(\varphi)=0$. Therefore the surgery obstruction $$\sigma:{\mathit Im}(\psi_*:[M, Top/O]\to[M, F/O])\to L_{4n}(\mathbb{Z})$$ is the zero map. Now by Theorem \[imge\] and Corollary \[sur\], we get that $$\theta(M)\cong {\mathit Im}(\psi_*:[M, Top/O]\to[M, F/O]).$$ Thus, by noticing that $\psi_*:[M, Top/O]\to[M, F/O]$ is monic and $[M, Top/O]\cong\mathcal{C}(M)$, we have $\theta(M)\cong \mathcal{C}(M)$.The proof of the claim is completed.
As a consequence of Theorem \[tanset\] and Theorem \[main\], we get the following results.
\[struset\]
- $\theta(\mathbb{H}P^3)\cong \mathbb{Z}_2$.
- $\theta(\mathbb{H} P^4)=\left \{ [(\mathbb{H} P^4\#\Sigma]~~ |~~\Sigma\in \Theta_{16} \right \}\cong \mathbb{Z}_2$.
- $\theta(\mathbb{H} P^5)$ is isomorphic to either $\mathbb{Z}_{24}\oplus \mathbb{Z}_2$ or $\mathbb{Z}_{48}$.
Theorem \[struset\] immediately implies the following result.
Let $M$ be a smooth manifold tangential homotopy equivalent to $\mathbb{H} P^4$. Then there is a homotopy $16$-sphere $\Sigma$ such that $M$ is diffeomorphic to $\mathbb{H} P^4\#\Sigma$.
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---
abstract: |
A complete classification of the renormalization-group flow is given for impurity-like marginal operators of membranes whose elastic stress scales like $ (\Delta
r)^2 $ around the external critical dimension $ d_c=2. $ These operators are classified by characteristic functions on $ {\mbox{\rm I}\!\mbox{\rm R}}^2 \times {\mbox{\rm I}\!\mbox{\rm R}}^2. $
---
-6.0cm 2.5cm
[**Classification of Perturbations for Membranes**]{}
**with Bending Rigidity**
2.5cm
**Kay Jörg Wiese[^1]**
CEA, Service de Physique Théorique, CE-Saclay
F-91191 Gif-sur-Yvette Cedex, FRANCE
1.5cm
Introduction
============
Fluctuating tethered membranes have attracted much interest during the last years. Considerable theoretical advance has been made through the work of F. David, B. Duplantier and E. Guitter [@DDG1] who proved that the theory described by $$\label{e:model1}
{\cal H} = \int \mbox{d}^D\! x \,{\frac{1}{2}}r(x) (-\Delta)^{k/2} r(x) + \lambda \,\delta^d(r(x))$$ with $k\ge 2$ is a renormalizable field theory, if $D$ and $d$ are properly chosen. The case $k=2$ corresponds to the case of a $D$-dimensional Gaussian manifold imbedded in $d$ dimensions. The field $$r: x\in {\mbox{\rm I}\!\mbox{\rm R}}^D \longrightarrow r(x) \in {\mbox{\rm I}\!\mbox{\rm R}}^d$$ is the coordinate of the membrane.
For $k=4$ (\[e:model1\]) represents a manifold with vanishing tension but with bending rigidity. In this case $r(x)$ is the amplitude of the orthogonal modes, the membrane thus imbedded in $D+d$ dimensions. It is this latter object which shall be studied in the following.
The $\delta$-potential describes the interaction of the manifold with a fixed point. The case of a membrane ($D=2$) is remarkable as $r$ has dimension $-1$ in internal momentum-units such that $\nabla r$ is dimensionless. Possible marginal perturbations are thus $${\cal H}_{\mbox{\scriptsize int}} = \int \mbox{d}^D\!x \, \delta^d(r(x))\, f(\nabla r(x))$$ with an arbitrary function $f$ instead of a simple $\delta$-distribution. The model has infinite many marginal perturbations for $D=d=2$ and one expects a rich mathematical structure. The goal is to find the eigen-operators of the renormalization-group flow.
The paper is organized as follows: First a brief description of the model without interaction is given. It is shown that it is not conformal invariant. Therefore the methods of conformal field theory do not apply and the model can only be studied in the framework of perturbation-theory. After reviewing the relation between the 1-loop $\beta$-function and the leading coefficient in the operator product expansion the eigen-operators are constructed. The renormalization-group flow and the physical relevance are discussed.
Description of the free model {#s:Description of the free model}
=============================
We are interested in membranes ($ D=2$), whose motion is governed by bending rigidity. Let us therefore introduce the free Hamiltonian $ (r: {\mbox{\rm I}\!\mbox{\rm R}}^D \to
{\mbox{\rm I}\!\mbox{\rm R}}^d) $ $${\cal H}_0 = - {1 \over 2(4-D)(2-D)} \int^{ }_ x{1 \over 2} (\Delta r(x))^2
\label{1}$$ where we abbreviated the integration measure $ (S_D $ is the volume of the $ D $-dimensional unit-sphere) $$\int^{ }_ x= {1 \over S_D} \int^{ }_{ } \mbox{d}^Dx\ ,\ \ \ \ S_D = 2 {\pi^{
D/2} \over \Gamma( D/2)} \label{2}$$ At the end of the calculations we intend to take the limit $ D \to
2. $ The factor $ {1 \over 2-D} $ therefore seems to be rather strange. It is however necessary to define an analytic continuation of the model for $ D\leq 2. $ With this choice of the Hamiltonian we have $${1 \over 2} \left\langle( r_i(x)-r_j(y))^2 \right\rangle_ 0 = \delta_{ij} \, \vert x-y\vert^{
4-D} \ , \label{3}$$ thus especially $$\left\langle( r(x)-r(y))^2 \right\rangle_ 0 \geq 0 \label{4}$$ as demanded from physical arguments even for $D<2$. The factor $1/(2(4-D)S_D) $ in front of the action (\[1\]) is introduced for pure convenience, i.e. to have normalization 1 in (\[3\]).
For $D>2$ the model is positive definite, for $D<2$ negative definite. In the latter regime we understand it as analytical continuation from $D>2$. This phenomenon reflects the fact that the expression for the free 2-point correlation function $${\frac{1}{2}}\left< \left( r(x) -r(y) \right)^2 \right>_0 = 2 (4-D) (2-D) S_D D \int
\frac{\mbox d^Dp}{(2 \pi)^D} \frac{\mbox{e}^{ipx}-1}{p^4}$$ becomes IR-divergent in the limit $D\to 2$ from above.
Remark about conformal invariance {#s:Remark about Conformal Invariance}
=================================
An interesting question arising in this context is, whether the 2-dimensional biharmonic model is conformal invariant. Its free Hamiltonian is (with a change in normalization and for a scalar field for simplicity) $$S_0 ={\frac{1}{2}}\int d^2\! x \, (\Delta \varphi)^2$$ To answer the question, the stress tensor has to be calculated. It is well known that it is not uniquely defined. We only give the result for one of the symmetric versions of the stress tensor: $$\begin{aligned}
\label{e:T}
T_{\mu \nu} &= & - \delta_{\mu \nu} \left( {\frac{1}{2}}(\Delta \varphi)^2 +
\partial_\rho( \varphi \partial_\rho \Delta \varphi )\right)
+\partial_\mu \partial_\nu \varphi \Delta \varphi
+\varphi \partial_\mu \partial_\nu \Delta \varphi \end{aligned}$$ We have proven that it is impossible to render the stress tensor both symmetric and traceless. The trace of (\[e:T\]) is: $$\Theta = -2 \partial_\mu \left( \varphi
\partial_\mu \Delta \varphi \right) + \varphi \Delta^2 \varphi$$ The last term on the r.h.s. is a redundant operator which can be neglected because of the classical equation of motion: $$\Delta^2 \varphi = 0$$ So the trace of the stress tensor has the form $$\Theta = - \partial_\mu K^\mu$$ where $K^\mu$ cannot be written as a total divergence (up to redundant operators). According to [@Pol88] this implies that the free theory is scale invariant but not conformal invariant. The standard methods of 2-dimensional conformal field theories thus can not be applied.
It is interesting to note that it is possible to construct a biharmonic conformal field theory by introducing an additional gauge field, which cancels the unwanted terms in the stress tensor [@r:Ferrari95].
Renormalization and operator product expansion {#s:3}
==============================================
Before actually analyzing possible marginal perturbations, let us discuss how these perturbations generate divergencies and how these divergencies have to be treated in the framework of renormalization [@r:ZINN].
The goal of renormalization is to eliminate UV-divergences, occurring in the perturbation expansion of IR-finite physical quantities $$\langle{\cal O}\rangle_{ \lambda_ 0} = { \int^{ }_{ } D[r]{\cal O}\ \mbox{e}^{-{\cal H}_0-\lambda_ 0{\cal H}_{ {\mbox{\tiny int}}}} \over \int^{ }_{ } D[r] \mbox{e}^{-{\cal H}_0-\lambda_ 0{\cal H}_{ {\mbox{\tiny int}}}}}\ . \label{5}$$ $ {\cal O} $ e.g. may be a neutral product of vertex-operators $${\cal O }= \prod^{ }_ n {\mbox{e}}^{ik_nr \left(x_n \right)}\ \ \mbox{with} \ \
\sum^{ }_ nk_n=0 \ . \label{6}$$ Denoting the perturbations by $${\cal H}_{ {\mbox{\scriptsize int}}} = \int^{ }_ xE(x)\ , \label{7}$$ where $ E(x) $ is some local functional of $ r(x), $ the $ n $-th order term in the perturbative expansion of $\left\langle{\cal O}\right\rangle_{\lambda_0}$ becomes: $${(-\lambda_0)^n \over n!} \int^{ }_{ x_1}... \int^{ }_{ x_n}
\left\langle{\cal O}\ E \left(x_1 \right) \ldots E \left(x_n \right)
\right\rangle^{ {\mbox{\scriptsize conn}}} \label{8}$$ Use was made of the standard abbreviations $$\begin{aligned}
\langle{\cal AB}\rangle^{ {\mbox{\scriptsize conn}}} & = & \langle{\cal
AB}\rangle -\langle{\cal A}\rangle\langle{\cal B}\rangle \\
\langle{\cal ABC}\rangle^{ {\mbox{\scriptsize conn}}} & = &
\langle{\cal A B C}\rangle
-
\langle{\cal
A}\rangle\langle{\cal BC}\rangle -\langle{\cal B}\rangle\langle{\cal
AC}\rangle -\langle{\cal C}\rangle\langle{\cal AB}\rangle +2\langle{\cal
A}\rangle\langle{\cal B}\rangle\langle{\cal C}\rangle \ .\end{aligned}$$ Let us suppose that UV-divergencies occur according to the operator product expansion for $|x-y|\to 0$, $ z = {x+y \over 2} $: $$E(x)E(y) = {1 \over\vert x-y\vert^{ D-\varepsilon }} E(z) +
\mbox{less singular terms} \ .
\label{11}$$ $\varepsilon$ is a small dimensional regularization parameter, which will be defined later. We will prove that the divergences which appear for small $ \vert
x-y\vert $ are of this type. According to [@DDG1; @DDG3] these are the only divergencies which may occur. In the perturbation expansion, the first divergent term is $${\lambda^ 2_0 \over 2} \int^{ }_ x \int^{ }_ y\langle{\cal O}\ E(x)
E(y)\rangle^{ {\mbox{\scriptsize conn}}} = {\lambda^ 2_0 \over 2} \int^{ }_ z\langle{\cal O}\
E(z)\rangle \int^{ }_{x-y}{1 \over\vert x-y\vert^{ D-\varepsilon }}
+\mbox{less singular terms} \ .
\label{12}$$ In the last integral the small positive parameter $ \varepsilon $ plays the role of an regulator. An IR-cutoff $L$ is also needed. (For the regularization procedure cf. [@Wiese; @David; @1].) We get: $$\int^{ }_{\vert x-y\vert <L}{1 \over\vert x-y\vert^{ D-\varepsilon }} = \int^
L_0{ \mbox{d} s \over s} s^\varepsilon = {L^\varepsilon \over \varepsilon } \label{13}$$ At 1-loop order the theory is thus renormalized by introducing a renormalized coupling constant $$\lambda = Z^{-1} \mu^{ -\varepsilon } \lambda_ 0 \label{14}$$ where $ Z $ takes the form $$Z = 1 + {\lambda \over 2\varepsilon } \label{15}$$ This is the only necessary renormalization. Especially the field $r(x)$ has not to be renormalized as is known from [@DDG1]. Intuitively this is understood from the observation that no renormalization is needed if the membrane is far away from the origin as in this case the membrane is non-interacting. Thus divergencies are always proportional to operators localized at $r=0$.
The renormalization-group $ \beta $-function describes as usual the variation of the coupling constant $ \lambda $ with respect to a variation of the renormalization-scale $ \mu : $ $$\begin{aligned}
\beta( \lambda) & = & \mu \left.{\partial \over \partial \mu}
\right\vert_{ \lambda_ 0} \lambda \nonumber \\
& = & - \varepsilon \lambda +{1 \over 2}
\lambda^ 2+{\cal O} \left(\lambda^ 3 \right) \label{16}\end{aligned}$$ For $ \varepsilon >0 $ this equation has a non-trivial IR-stable fixed point $$\lambda^* = 2\varepsilon \ . \label{17}$$
Perturbations {#s:Perturbations}
=============
Let us analyze the canonical scaling dimensions of the free model in order to determine [*all*]{} marginal perturbations. In internal units such that $\left[ x\right] =-1$ we have:
$$[r] = {D-4 \over 2} \label{18}$$
Therefore $$[\nabla r] = {D-2 \over 2} \label{19}$$ and is dimensionless in $ D=2. $
Regarding polynomial operators, the following marginal perturbations are possible: $$H_{{\mbox{\scriptsize pol}}} = \int_x (\nabla \nabla r)^2 f(\nabla r)$$ where we did not specify the index structure for $\nabla$ and $f$ is an arbitrary function. This is a class of perturbations, we do not want to consider here. This is consistent as they are not generated in perturbation theory. We will see that below. On the other hand we may have impurity-like interactions: $$H_{ {\mbox{\scriptsize int}}} = \int^{ }_ x\tilde \delta ^ d(r(x)) \label{20}$$ which are dimensionless, if $$d = {2D \over 4-D} \label{21}$$ i.e. for $ D=2, $ if $$d=2\ . \label{22}$$ We again use convenient normalizations $$\tilde \delta ^ d(r(x)) = (4\pi)^{ d/2}\delta ^ d(r(x)) = \int^{ }_ p \mbox{e}^{ipr(x)} \label{24}$$ with $$\int_p= \pi^{ -d/2} \int^{ }_{ } \mbox{d}^dp \label{25}$$ to have $$\int^{ }_ p {\mbox{e}}^{-p^2a} = a^{-d/2} \ . \label{26}$$ The marginal perturbations for $ D=2 $ and $ d=2 $ are: $$\int^{ }_ x :{\mbox{e}}^{i\alpha^ \mu_ i\nabla_ \mu r^i(x)} \tilde \delta ^
d(r(x)): \label{23}$$ Normal-ordering has been used to eliminate contributions due to self-contractions. Let us further introduce the notation of vertex-operators $$V_{\alpha k}(x) = : {\mbox{e}}^{i\alpha \nabla r(x)} {\mbox{e}}^{ikr(x)} :
\label{27}$$ where the indices for $ \alpha $ from (\[23\]) have been suppressed. The marginal perturbations now read: $$V_\alpha( x) = \int^{ }_ kV_{\alpha k}(x) = : {\mbox{e}}^{i\alpha \nabla
r(x)}\tilde\delta^d ( r(x)) : \label{28}$$ In the spirit of [@DDG3] all possible contractions of perturbations have to be analyzed. At 1-loop order there is only one possibility: $$V_a(x) V_\beta( y) \label{29}$$ Following [@DDG3], these operators are contracted according to $ (x-y \to 0 $ and $ z = {x+y \over 2}): $ $$\begin{aligned}
\int_k\int_l V_{\alpha k}(x)V_{\beta l}(y) & = & \int_k\int_l:V_{\alpha k}(x)V_{\beta
l}(y): {\mbox{e}}^{-\langle( \alpha \nabla +k)r(x)(\beta \nabla +l)r(y)\rangle_
0} \nonumber \\
&=& \int_k\int_l: {\mbox{e}}^{(x-z)\partial_ z}V_{\alpha k}(z)\ \mbox{e}^{(y-z)\partial_ z}V_{\beta l}(z): {\mbox{e}}^{-\langle( \alpha \nabla
+k)r(x)(\beta \nabla +l)r(y)\rangle_ 0}
\label{30} \end{aligned}$$ In order to retain only the most relevant contribution in (\[30\]) three simplifications can be made. First of all, terms proportional to $ \partial_ zV_{\alpha k}(z)
$ are irrelevant and thus can be neglected. (\[30\]) becomes after the change of variables $ l
\to l-k $ $$V_\alpha( x)V_\beta( y)= \int^{ }_ k \int^{ }_ l: {\mbox{e}}^{i(\alpha +\beta)
\nabla r(z)+ilr(z)}: {\mbox{e}}^{-\langle( \alpha \nabla +k)r(x)(\beta \nabla
+l-k)r(y)\rangle_ 0} + {\mbox{less singular terms}}\label{31}$$ The integration over $ l $ yields $ \tilde \delta^d ( r(z)) $ plus its higher derivatives, which are irrelevant and thus neglected: $$\begin{aligned}
V_{\alpha +\beta}( z) \int^{ }_ k {\mbox{e}}^{-\langle( \alpha
\nabla +k)r(x)(\beta \nabla -k)r(y)\rangle_ 0} \nonumber\label{32}
&=& V_{\alpha +\beta}(
z) \int^{ }_ k {\mbox{e}}^{-k^2\vert x-y\vert^{ 4-D}-[\alpha(
x-y)][\beta( x-y)](4-D)(2-D)\vert x-y\vert^{ -D}}
\nonumber \\
& & \qquad \, {\mbox{e}}^{-(4-D) \alpha \beta \vert x-y \vert^{2-D}
+\left[{\alpha +\beta \over
2}(x-y) \right]^2(4-D)^2 \vert x-y\vert^{ -D}} \end{aligned}$$ where the integral over $ k $ was shifted to isolate the term quadratic in $ k. $ After integration over $ k $ equation (\[32\]) becomes: $$V_{\alpha + \beta}( z) \left({1 \over\vert x-y\vert^{ 4-D}}
\right)^{d/2} {\mbox{e}}^{\, \left[{\alpha +\beta \over 2} (y-x)
\right]^2(4-D )^ 2\vert x-y\vert^{ -D}-[\alpha( x-y)][\beta( x-y)](4-D ) (2-D) \vert x-y\vert^{
-D}-(4-D) \alpha \beta \vert x-y \vert^{2-D}
} \label{34}$$ As explained in section \[s:3\], the integration over the relative distance determines the renormalization of an operator. So we have to analyze the singularity for $ x \to y. $ Introducing the dimensional regularization parameter $ \varepsilon , $ $$\varepsilon = D-2d + {Dd \over 2} \label{39}$$ we get $$\left({1 \over\vert x-y\vert^{ 4-D }} \right)^{d/2} = \vert
x-y\vert^{ \varepsilon -D} \label{40}$$
\[0mm\]\[0mm\][relevant ${\varepsilon}>0$]{} \[0mm\]\[0mm\][irrelevant ${\varepsilon}<0$]{}
Integration over $ \vert x-y\vert $ thus yields pole terms in $ \varepsilon $. In addition, the only dependence of the pole term on the exponential factors in (\[34\]) comes from $ \vert
x-y\vert =0 $. In the spirit of analytic continuation we choose $D<2$ in order to have a regular expression for the exponential factors in (\[34\]). As by this way they equal 1 at $x=y$, the analytical continuation to $D\ge2$ is unique, delivering 1 for the whole range. This would not be the case, if the limit $ D \to 2$ had been performed before.
Finally we arrive at: $$\begin{aligned}
\int^{ }_{\vert x-y\vert <L}V_\alpha( x)V_\beta( y) & = &
V_{\alpha +\beta}( z) \int^ L_0{ \mbox{d} s \over s} s^\varepsilon +{\mbox{less singular terms}}\nonumber \\
& = & V_{\alpha +\beta}( z) {L^\varepsilon \over \varepsilon } +{\mbox{less singular terms}}\label{41}\end{aligned}$$ We construct now eigen-operators $E(x)$ of the contraction. Define $$E(x) = \int^{ }_ \alpha e(\alpha) V_\alpha( x) \label{42}$$ which have to satisfy $$E(x)E(y) = \vert x-y\vert^{ \varepsilon -D}E(z)+
{\mbox{less singular terms}}\ .
\label{43}$$ This fixes the normalization of $E(x)$. Plugging in the definition of $ E(x) $ results in $$\int^{ }_ \alpha \int^{ }_ \beta e(\alpha) e(\beta) \
\delta ( \gamma
-\alpha -\beta) = e(\gamma) \label{44}$$ This equation can be solved by introducing the Fourier transform of $
e(\alpha)$: $$\tilde e(p) = \int^{ }_ \alpha {\mbox{e}}^{ip\alpha} e(\alpha) \label{45}$$ (\[44\]) becomes: $$\tilde e(p)^2= \tilde e(p) \label{46}$$ Let us recall that $ \alpha $ was in $ {\mbox{\rm I}\!\mbox{\rm R}}^2 \times {\mbox{\rm I}\!\mbox{\rm R}}^2, $ hence $
p $.
Solutions of (\[46\]) are characteristic functions of (measurable) subsets $ M $ of $ {\mbox{\rm I}\!\mbox{\rm R}}^2 \times {\mbox{\rm I}\!\mbox{\rm R}}^2: $ $$\tilde e(p) = \chi_ M(p) \label{47}$$ Eigen-operators of the contraction (\[41\]) and therefore of the renormalization-group flow are: $$\begin{aligned}
E_M(x) & = & \int^{ }_ \alpha \int^{ }_ p {\mbox{e}}^{-ip\alpha
+i\alpha \nabla r(x)}\chi_ M(p) \tilde\delta ^ d(r(x)) \nonumber \\
&=& \chi_ M(\nabla r(x))\, \tilde \delta ^ d(r(x)) \label{48} \end{aligned}$$ Another interesting conclusion can be drawn: Rewriting (\[43\]) for two different perturbations $ E_{M_1}(x) $ and $ E_{M_2}(y) $ gives in the limit $ \vert
x-y\vert \to 0 $ $$E_{M_1}(x)E_{M_2}(y) = \vert x-y\vert^{ \varepsilon -D}E_{M_1\cap M_2}
\left({x+y \over 2} \right)+ \mbox{less singular terms} \ . \label{49}$$ This is an orthogonality relation for contractions.
At this point we should study what happens if in the free Hamiltonian (\[1\]) we do not introduce the factor $ {1 \over 2-D}, $ i.e. if we use $$\tilde{\cal H}_0 = - {1 \over 2(4-D)} \int^{ }_ x{1 \over 2} (\Delta
r(x))^2 \label{35}$$ instead of $ {\cal H}_0. $ Equation (\[34\]) then becomes $$V_{\alpha+\beta}( z) \left({2-D \over\vert x-y\vert^{
4-D}} \right)^{d/2} {\mbox{e}}^{\, \left[{\alpha +\beta \over 2} (y-x)
\right]^2{(4-D )^ 2 \over 2-D }\vert x-y\vert^{ -D}
-[\alpha( x-y)][\beta( x-y)](4-D )\vert x-y\vert^{ -D}-\frac{4-D}{2-D} \alpha \beta \vert x-y \vert^{2-D}
}
\label{36}$$ This equation looks rather ugly, so let us put $ \alpha =\beta =0 $ for the moment. If $ d\not= 2 $ (\[36\]) is even non-analytic in the regularization parameter $ D$ for $D\to 2$. But also the case $ d=2 $ is peculiar: $$(2-D) \ V_0(z) \cdot {1 \over\vert x-y\vert^{ 4-D }} \label{37}$$ Although the integration over $ x-y $ yields a pole term in $ 1/(2-D)
$ $$\int^{ }_ x{1 \over\vert x\vert^{ 4-D }} = \int^{ }_ \Lambda{
\mbox{d} x \over x} x^{-2(2-D) } = {1 \over 2(2-D) } \Lambda^{
2(2-D) } \label{38}$$ it will be cancelled by the factor $ (2-D) $ in (\[37\]). The system has no UV-divergence at all! For $ \alpha \not= \beta $ the situation is even worse: Strong IR-singularities appear. We conclude that the Hamiltonian (\[35\]) is too weak and thus there is no way to define a sensible model in the limit $ D \to 2. $
Interpretation of the result {#s:Interpretation of the result}
============================
The operators $E_M(x)$ defined in (\[48\]) were constructed as eigen-oprators of the contraction, equation (\[43\]) or equivalently (\[11\]). Their renormalization has been analyzed in section 4. There we showed that in the regime ${\varepsilon}>0$ the renormalization-group flow of $\lambda_M$ in $${\cal H}_M = - {1 \over 2(4-D)(2-D)} \int^{ }_ x{1 \over 2}(\Delta r(x))^2
+ \lambda_M \int^{ }_x E_M(x)$$ has an IR-stable fixed point $\lambda_M^*=2{\varepsilon}$. This result is independant of $M$ i.e. the fixed-point Hamiltonian is: $${\cal H}^* = - {1 \over 2(4-D)(2-D)} \int^{ }_ x{1 \over 2}(\Delta r(x))^2
+ 2\varepsilon \int^{ }_ x\tilde \delta^ d(r(x)) \label{51}$$ So the interaction part of $ {\cal H}^* $ does [*not*]{} depend on $ \nabla r(x). $
This result however may be false in practical cases. Suppose $${\cal H}(t) = {\cal H}_0 + \int^{ }_p f(t,p) \int^{ }_
xE_{ \{ p \} }( x) \label{52}$$ where $ f $ is normal distributed $$f(1,p) = 2\varepsilon \ {\mbox{e}}^{-p^ 2/\sigma^ 2} \label{53}$$ and $ \mu = t^{-1}\mu_ 0. $
The typical time $ \tilde t$ which is necessary until $ f(t,p) $ has reached the fixed point $ 2\varepsilon $ is approximately $$\tilde t (p) \approx {1 \over \varepsilon } {p^ 2 \over \sigma^ 2}\ , \label{54}$$ thus increases rapidly with $ p $. If the microscopical Hamiltonian is given by (\[52\]) and (\[53\]) and if the microscopical scale and the scale of experiment are related by a renormalization-group transformation with say $t=10^6$, then the modes with $t^{\ast}>10^6$ will stay nearly 0 after the renormalization-group transformation. Stated otherwise, the critical regime for these modes is not reached. Whether this line of arguments is relevant depends on the initial values of $ f(1,p)$.
Conclusions {#s:Conclusions}
===========
We discussed a 2-dimensional field theory which is not conformal invariant but which can be treated in the framework of perturbation theory. Although the question of the physical interpretation of the model, especially the normalization involved in (\[1\]), had to stay open, a complete classification of all marginal impurity like perturbations was given at 1-loop order. Those are characteristic functions on $ {\mbox{\rm I}\!\mbox{\rm R}}^2 \times {\mbox{\rm I}\!\mbox{\rm R}}^2 $. The renormalization-flow shows a rich structure which is special for the considered model.
Acknowledgements {#acknowledgements .unnumbered}
================
It is a pleasure to thank François David, Stefan Kehrein, Sergey Shabanov, Jean Zinn-Justin and Jean-Bernard Zuber for useful discussions and François David and Jean Zinn-Justin for a careful reading of the manuscript.
[9]{} F. David, B. Duplantier and E. Guitter, [*Nucl. Phys.*]{} [**B394**]{} (1993) 555-664 F. David, B. Duplantier and E. Guitter, [*Phys. Rev. Lett.*]{} [**72**]{} (1994) 311 J. Polchinski, [*Nucl. Phys.*]{} [**B303**]{} (1988) 226-236 F. Ferrari, [*hep-th*]{}/9507142 K. J. Wiese and F. David, [*Nucl. Phys.*]{} [**B450**]{} (1995) 495-557 J. Zinn-Justin, [*Quantum Field Theory and Critical Phenomena*]{}, Oxford 1989
[^1]: Email: wiese@amoco.saclay.cea.fr
|
---
abstract: |
We study the directed polymer model in dimension ${1+1}$ when the environment is heavy-tailed, with a decay exponent $\alpha\in(0,2)$. We give all possible scaling limits of the model in the *weak-coupling* regime, *i.e.* when the inverse temperature temperature $\beta=\beta_n$ vanishes as the size of the system $n$ goes to infinity. When $\alpha\in(1/2,2)$, we show that all possible transversal fluctuations $\sqrt{n} \leq h_n \leq n$ can be achieved by tuning properly $\beta_n$, allowing to interpolate between all super-diffusive scales. Moreover, we determine the scaling limit of the model, answering a conjecture by Dey and Zygouras [@cf:DZ]—we actually identify five different regimes. On the other hand, when $\alpha<1/2$, we show that there are only two regimes: the transversal fluctuations are either $\sqrt{n}$ or $n$. As a key ingredient, we use the *Entropy-controlled Last Passage Percolation* (E-LPP), introduced in a companion paper [@cf:BT_ELPP].\
*2010 Mathematics Subject Classification*: Primary: 60F05, 82D60; Secondary:60K37, 60G70.\
*Keywords*: Directed polymer, Heavy-tail distributions, Weak-coupling limit, Last Passage Percolation, Super-diffusivity.
address:
- 'Sorbonne Université, LPSM, Campus Pierre et Marie Curie, case 188, 4 place Jussieu, 75252 Paris Cedex 5, France'
- 'Sorbonne Université, LPSM, Campus Pierre et Marie Curie, case 188, 4 place Jussieu, 75252 Paris Cedex 5, France'
author:
- Quentin Berger
- Niccolò Torri
bibliography:
- 'biblioHTBN.bib'
title: 'Directed polymers in heavy-tail random environment'
---
Introduction: Directed Polymers in Random Environment
=====================================================
General setting
---------------
We consider the directed polymer model: it has been introduced by Huse and Henley [@HH85] as an effective model for an interface in the Ising model with random interactions, and is now used to describe a stretched polymer interacting with an inhomogeneous solvent.
Let $S$ be a nearest-neighbor simple symmetric random walk on $\mathbb{Z}^d$, $d\geq 1$, whose law is denoted by $\mathbf{P}$, and let $(\omega_{i,x})_{i\in \mathbb{N},\, x\in \mathbb{Z}^d}$ be a field of i.i.d. random variables (the *environment*) with law ${\mathbb{P}}$ (${\omega}$ will denote a random variable which has the common distribution of the ${\omega}_{i,x}$). The *directed* random walk $(i,S_i)_{i\in\mathbb N_0}$ represents a polymer trajectory and interacts with its environment via a coupling parameter ${\beta}>0$ (the inverse temperature). The model is defined through a Gibbs measure, $$\label{eq:DPRE}
\frac{{\textrm{d}}{\mathbf{P}}^\omega_{n,\beta}}{{\textrm{d}}{\mathbf{P}}}(s)\, :=\,
\frac{1}{{\mathbf{Z}}^\omega_{n\, \beta}} \exp\Big( \beta\, \sum_{i=1}^n\omega_{i,s_i} \Big)\, ,$$ where ${\mathbf{Z}}^\omega_{n\, \beta}$ is the *partition function* of the model.
One of the main question about this model is that of the localization and super-diffusivity of paths trajectories drawn from the measure ${\mathbf{P}}^\omega_{n,\beta}$. The transversal exponent $\xi$ describes the fluctuation of the end-point, that is ${\mathbb{E}}{\mathbf{E}}^\omega_{n,\beta} |S_n| \approx n^{\xi} $ as $n\to\infty$. Another quantity of interest is the fluctuation exponent $\chi$, that describes the fluctuations of $\log {\mathbf{Z}}_{n,{\beta}}^{{\omega}}$, *i.e.* $|\log {\mathbf{Z}}_{n,{\beta}}^{{\omega}} - {\mathbb{E}}\log {\mathbf{Z}}_{n,{\beta}}^{{\omega}}| \approx n^{\chi}$ as $n\to\infty$.
This model has been widely studied in the physical and mathematical literature (we refer to [@C17; @CSY04] for a general overview), in particular when ${\omega}_{n,x}$ have an exponential moment. The case of the dimension $d=1$ as attracted much attention in recent years, in particular because the model is in the *KPZ universality class* ($\log {\mathbf{Z}}_{n,{\beta}}^{{\omega}}$ is seen as a discretization of the Hopf-Cole solution of the KPZ equation). It is conjectured that the transversal and fluctuation exponents are $\xi=2/3$ and $\chi=1/3$ respectively. Moreover, it is expected that the point-to-point partition function, when properly centered and renormalized, converges in distribution to the GUE distribution. Such scalings has been proved so far only for some special models, cf. [@BQS11; @Se09].
A recent and fruitful approach to proving universality results for this model has been to consider is *weak-coupling limit*, that is when the coupling parameter ${\beta}$ is close to criticality. This means that we allow ${\beta}={\beta}_n$ to depend on $n$, with ${\beta}_n \to 0$ as $n\to\infty$. In [@AKQ14b; @AKQ14a] and [@CSZ13], the authors let $\beta_n = {\widehat}\beta n^{-\gamma},\, \gamma=1/4$ for some fixed ${\widehat}\beta>0$, and they prove that the model (one may focus on its partition function ${\mathbf{Z}}_{n,{\beta}_n}^{{\omega}}$) converges to a *non-trivial* (*i.e.* *disordered*) continuous version of the model. This is called the *intermediate disorder regime*, since it somehow interpolates between weak disorder and strong disorder behaviors. More precisely, they showed that $$\log {\mathbf{Z}}_{n,{\beta}_n}^{{\omega}} - n\lambda(\beta_n) \overset{({\textrm{d}})}{\longrightarrow} \log {{\ensuremath{\mathcal Z}} }_{\sqrt 2 {\widehat}\beta}, \quad \text{as}\quad n\to\infty,$$ where $\lambda(s):=\log {\mathbb{E}}[e^{s\omega}]$. The process ${\widehat}\beta \mapsto \log{{\ensuremath{\mathcal Z}} }_{\sqrt 2 {\widehat}\beta}$ is the so called *cross-over process*, and interpolates between Gaussian and GUE scalings as ${\widehat}\beta$ goes from $0$ to $\infty$ (see [@ACQ11]). These results were obtained under the assumption that ${\omega}$ has exponential moments, but the *universality of the limit* was conjectured to hold under the assumption of six moments [@AKQ14a]. In [@cf:DZ] Dey and Zygouras proved this conjecture, and they suggest that this result is a part of a bigger picture (when $\lambda(s)$ is not defined a different centering is necessary).
The case of a heavy-tail environment
------------------------------------
In the rest of the paper we will focus on the dimension $d=1$ for simplicity. We consider the case where the environment distribution ${\omega}$ is non-negative (for simplicity, nothing deep is hidden in that assumption) and has some heavy tail distribution: there is some ${\alpha}>0$ and some slowly varying function $L(\cdot)$ such that $$\label{eq:DisTail}
\mathbb P\left(\omega> x\right)=L(x) x^{-\alpha} \, .$$
In the case where ${\beta}>0$ does not depend on $n$, the $\xi=2/3,\chi=1/3$ picture is expected to be modified, depending on the value of ${\alpha}$. According to the heuristics (and terminology) of [@BBP07; @GLDBR], three regimes should occur, with different paths behaviors:
\(a) if ${\alpha}>5$, there should be a *collective* optimization and we should have $\xi=2/3$, KPZ universality class, as in the finite exponential moment case;
\(b) if ${\alpha}\in (2,5)$, the optmization strategy should be *elitist*: most of the total energy collected should be via a small fraction of the points visited by the path, and we should have $\xi=\frac{{\alpha}+1}{2{\alpha}-1}$;
\(c) if ${\alpha}\in(0,2)$, the strategy is *individual*: the polymer targets few exceptional points, and we have $\xi=1$. This case is treated in [@AL11; @HM07].
As suggested by [@cf:DZ], this is part of a larger picture, when the inverse temperature ${\beta}$ is allowed to depend on $n$. Setting ${\beta}_n= {\widehat}{\beta}n^{-\gamma}$ for some ${\widehat}{\beta}> 0$ and some $\gamma \in {\mathbb{R}}$ then we have three different classes of coupling. When $\gamma=0$ we recover the standard directed polymer model, when $\gamma>0$ we have a weak-coupling limit, while in the case $\gamma<0$ we have a strong-coupling limit. Let us stress that this last case has not been studied in the literature (for no apparent reason) and should also be of interest. In [@AKQ10] and in [@cf:DZ], the authors suggest that the fluctuation exponent depends on ${\alpha},\gamma$ in the following manner $$\label{def:xi}
\xi = \left\{
\begin{array}{ll}
\frac{2 (1-\gamma)}{3} &\qquad \text{for } {\alpha}\ge \tfrac{5-2\gamma}{1-\gamma} ,\ - \tfrac 12\le \gamma \le \tfrac14 \, ,\\
\frac{ 1+ {\alpha}(1-\gamma)}{2{\alpha}-1} &\qquad \text{for } {\alpha}\le \frac{5-2\gamma}{1-\gamma},\ \tfrac{2}{{\alpha}}-1 \le \gamma \le \tfrac{3}{2{\alpha}} \, .
\end{array}
\right.$$ The first part is derived in [@AKQ10], based on Airy process considerations, and the second part is derived in [@cf:DZ], based on a Flory argument inspired by [@BBP07]. Moreover, in the two regions of the $({\alpha},\gamma)$ plane defined by , the KPZ scaling relation $\chi=2\xi-1$ should hold (this has been proved in the case $\gamma=0,{\alpha}>2$ in [@AD13]). Outside of these regions, one should have $\xi=1/2$ ($\gamma$ large) or $\xi=1$ ($\gamma$ small). This is summarized in Figure \[fig1\] below, which is the analogous of [@cf:DZ Fig. 1].
![We identify four regions in the $(\alpha,\gamma)$ plane. Region [**A**]{} with ${\alpha}<2$ is treated in [@AL11] and Region [**B**]{} with ${\alpha}>1/2$ in [@cf:DZ]. Regions [**C**]{} and [**D**]{} are still open, and the KPZ scaling relation $\chi=2\xi-1$ should hold in these two regions. Our main result is to settle the picture in the case ${\alpha}\in(0,2)$. []{data-label="fig1"}](figPhases)
This picture is far from being settled, and so far only the border cases where $\xi=1$ or $\xi=1/2$ have been studied: Dey and Zygouras [@cf:DZ] proved that $\xi=1/2$ in the cases ${\alpha}>6, \gamma =1/4$ and ${\alpha}\in(1/2,6) , \gamma = 3/2{\alpha}$; Auffinger and Louidor [@AL11] proved that $\xi=1$ for ${\alpha}\in(0,2)$ and $\gamma = \frac{2}{{\alpha}} -1$. Here, we complete the picture in the case ${\alpha}\in(0,2)$. For ${\alpha}\in(1/2,2)$ we go beyond the cases $\xi=1/2$ or $\xi=1$: we identify the correct order for the transversal fluctuations (they interpolate between $\xi=1/2$ and $\xi=1$), and we prove the convergence of $\log {\mathbf{Z}}_{n,{\beta}_n}^{{\omega}}$ in all possible intermediate disorder regimes—this proves Conjecture 1.7 in [@cf:DZ]. For ${\alpha}<1/2$ we show that a sharp transition occurs on the line $\gamma = \frac{2}{{\alpha}} -1$, between a regime where $\xi=1$ and a regime where $\xi=1/2$.
Main results: weak-coupling limits in the case ${\alpha}\in(0,2)$
=================================================================
From now on, we consider the case of an environment ${\omega}$ verifying with ${\alpha}\in(0,2)$. For the inverse temperature, we will consider arbitrary sequences $({\beta}_n)_{n\ge 1}$, but a reference example is ${\beta}_n = n^{-\gamma}$ for some $\gamma \in {\mathbb{R}}$.
For two sequences $(a_n)_{n\ge 1}, (b_n)_{n\ge 1}$, we use the notations $a_n \sim b_n$ if $\lim_{n\to\infty} a_n/b_n =1$, $a_n \ll b_n$ if $\lim_{n\to\infty} a_n/b_n =0$, and $a_n \asymp b_n$ if $0<\liminf a_n/b_n \le \limsup a_n/b_n <\infty$.
First definitions and heuristics
--------------------------------
First of all, let us present a brief energy/entropy argument to justify what the correct transversal fluctuations of the polymer should be. Let $F(x) = {\mathbb{P}}({\omega}\le x)$ be the disorder distribution, and define the function $m(x)$ by $$\label{def:m}
m(x) := F^{-1} \big(1-\tfrac1x \big), \qquad \text{so we have }\ {\mathbb{P}}\big( {\omega}>m (x) \big) = \frac1x .$$ Note that the second identity characterizes $m(x)$ up to asymptotic equivalence: we have that $m(\cdot)$ is a regularly varying function with exponent $1/{\alpha}$.
Assuming that the transversal fluctuations are of order $h_n$ (we necessarily have $\sqrt{n}\le h_n \le n$), then the amount of weight collected by a path should be of order $m(n h_n)$ (it should be dominated by the maximal value of ${\omega}$ in $[0,n] \times [-h_n, h_n]$). On the other hand, thanks to moderate deviations estimates for the simple random walk, the entropic cost of having fluctuations of order $h_n$ is roughly $h_n^2/n$ at the exponential level – at least when $h_n \gg \sqrt{n \log n}$, see below. It therefore leads us to define $h_n$ (seen as a function of ${\beta}_n$) up to asymptotic equivalence by the relation $$\label{def:hn}
{\beta}_n m(n h_n) \sim h_n^2 /n \, .$$
In the case ${\beta}_n = n^{-\gamma}$ and ${\alpha}\in(1/2,2)$ we recover , that is we get that $h_n=n^{\xi+o(1)}$ with $\xi = \frac{1+{\alpha}(1-\gamma)}{2{\alpha}-1}$, which is in $(1/2,1)$ for $\gamma \in (\tfrac{2}{{\alpha}}-1, \tfrac{3}{2{\alpha}})$. When ${\alpha}\in(0, 1/2)$, there is no $h_n$ verifying with $\sqrt n \ll h_n \ll n$, leading to believe that intermediate transversal fluctuations (*i.e.* $\xi\in(1/2,1)$) cannot occur. In the following, we separate the cases ${\alpha}\in (1/2,2)$ and ${\alpha}\in (0,1/2)$.
A natural candidate for the scaling limit {#sec:casealpha02}
-----------------------------------------
Once we have identified in the scale $h_n$ for the transversal fluctuations, we are able to rescale both path trajectories and the field $({\omega}_{i,x})$, so that we can define the rescaled “entropy” and “energy” of a path, and the corresponding continous quantities. The rescaled paths will be in the following set $$\label{def:D}
{\mathscr{D}}:= \big\{ s: [0,1]\to {\mathbb{R}}\ ;\ s \text{ continuous and a.e.\ differentiable} \big\}\, ,$$ and the (continuum) entropy of a path $s\in {\mathscr{D}}$ will derive from the rate function of the moderate deviation of the simple random walk (see [@S67] or below), *i.e.* $$\label{def:ContinuumEntropy}
{\mathrm{Ent}}(s) = \frac12 \int_0^1 \big( s'(t) \big)^2 dt \qquad \text{for } s\in {\mathscr{D}}.$$
As far as the disorder field is concerned, we let ${{\ensuremath{\mathcal P}} }:=\{(w_i,t_i,x_i)\}_{ i\geq 1}$ be a Poisson Point Process on $[0,\infty)\times[0,1]\times\mathbb R $ of intensity $\mu(d w d t d x)=\frac{\alpha}{2} w^{-\alpha-1}{{\sf 1}}_{\{w>0\}}d w d t d x$. For a quenched realization of ${{\ensuremath{\mathcal P}} }$, the energy of a continuous path $s\in{\mathscr{D}}$ is then defined by $$\label{def:discrCont}
\pi(s) =\pi_{{{\ensuremath{\mathcal P}} }}(s):=\sum_{(w,t,x)\in {{\ensuremath{\mathcal P}} }} \, w \,{{\sf 1}}_{\{(t,x)\in s\}},$$ where the notation $(t,x)\in s$ means that $s_t=x$.
Then, a natural guess for the continuous scaling limit of the partition function is to consider an energy–entropy competition variational problem. For any ${\beta}\geq 0$ we let $$\label{def:T}
\mathcal T_{{\beta}} := \sup_{s\in {\mathscr{D}}, {\mathrm{Ent}}(s) <+\infty} \Big\{ {\beta}\pi(s) -{\mathrm{Ent}}(s) \Big\} .$$ This variational problem was originally introduced by Dey and Zygouras [@cf:DZ Conjecture 1.7], conjecturing that it was well defined as long as $\alpha\in (1/2,2)$ and that it was the good candidate for the scaling limit. In [@cf:BT_ELPP Theorem 2.7] we show that the variational problem is indeed well defined as long as ${\alpha}\in(1/2,2)$. In Theorem \[thm:alpha>12\] below, we prove the second part of [@cf:DZ Conjecture 1.7].
\[thm:TbhatTb\] For ${\alpha}\in (1/2,2)$ we have that ${{\ensuremath{\mathcal T}} }_{{\beta}}\in (0,+\infty)$ for all ${\beta}>0$ a.s. On the other hand, for ${\alpha}\in(0,1/2]$ we have ${{\ensuremath{\mathcal T}} }_{{\beta}} =+\infty$ for all ${\beta}>0$ a.s.
Let us mention here that in [@AL11], the authors consider the case of transversal fluctuations of order $n$. The natural candidate for the limit is ${\widehat}{{\ensuremath{\mathcal T}} }_{{\beta}}$, defined analogously to by ${\widehat}{{\ensuremath{\mathcal T}} }_{{\beta}} = 0$ for ${\beta}=0$, and for ${\beta}>0$ $$\label{def:hatT}
{\widehat}{{\ensuremath{\mathcal T}} }_{{\beta}} =\sup_{s\in \mathrm{Lip}_1} \Big \{ \pi(s) - \frac1{\beta}{{\widehat{\rm E} \rm nt }}(s) \Big\}.$$ Here the supremum is taken over the set $\mathrm{Lip}_1$ of $1$-Lipschitz functions, and the entropy $ {{\widehat{\rm E} \rm nt }}(s)$ derives from the rate function of the large deviations for the simple random walk, *i.e.* $${{\widehat{\rm E} \rm nt }}(s) = \int_0^1 e\big( s'(t) \big) dt \quad \text{with } e(x) = \tfrac12 (1+x) \log (1+x) + \tfrac12 (1-x)\log (1- x)\, .$$
Main results I : the case ${\alpha}\in(1/2,2)$ {#sec:resultsI}
----------------------------------------------
Our first result deals with the transversal fluctuations of the polymer: we prove that $h_n$ defined in indeed gives the correct order for the transversal fluctuations.
\[thm:fluctu\] Assume that ${\alpha}\in(1/2,2)$, that ${\beta}_n m(n^{2}) \to 0$ and that ${\beta}_n m(n^{3/2}) \to +\infty$, and define $h_n$ as in : then $\sqrt{n} \ll h_n \ll n$. Then, there are constants $c_1,c_2$ and $\nu>0$ such that for any sequences $A_n\ge 1$ we have for all $n\ge 1$ $$\label{eq:hscaling}
{\mathbb{P}}\left( {\mathbf{P}}_{n,{\beta}_n}^{{\omega}} \big( \max_{i\leq n} |S_i| \geq A_n \, h_n \big) \geq n\, e^{- c_1 A_n^2 h_n^2/n} \right) \leq c_2\, A_n^{-\nu} \, .$$
In particular, this proves that if $h_n$ defined in is larger than a constant times $\sqrt{n\log n}$, then $n e^{- c_1 A h_n^2/n}$ goes to $0$ as $n\to\infty$ provided that $A$ is large enough: the transversal fluctuations are at most $A h_n$, with high ${\mathbb{P}}$-probability. On the other hand, if $h_n$ is much smaller than $\sqrt{n\log n}$, then this theorem does not give sharp information: we still find that the transversal fluctuations must be smaller than $A\sqrt{n\log n}$, with high ${\mathbb{P}}$-probability. Anyway, in the course of the demonstration of our results, it will be clear that the main contribution to the partition function comes from trajectories with transversal fluctuations of order exactly $h_n$.
We stress that the cases ${\beta}_n m(n^2) \to {\beta}\in(0,+\infty]$ and ${\beta}_n m(n^{3/2}) \to {\beta}\in[0,\infty)$ have already been considered by Auffinger and Louidor [@AL11] and Dey and Zygouras [@cf:DZ] respectively: they find that the transversal fluctuations are of order $n$, resp. $\sqrt{n}$. We state their results below, see Theorem \[thm:AL\] and Theorem \[thm:DZ\] respectively. Our first series of results consist in identifying three new regimes for the transversal fluctuations ($\sqrt{n\log n} \ll h_n \ll n$, $h_n \asymp \sqrt{n\log n}$, and $\sqrt{n} \ll h_n \ll \sqrt{n\log n}$), that interpolate between the Auffinger Louidor regime ($h_n \asymp n$) and the Dey Zygouras regime ($h_n \asymp \sqrt{n}$). We now describe more precisely these five different regimes.
### : transversal fluctuations of order $n$ {#transversal-fluctuations-of-order-n .unnumbered}
Consider the case where
$$\label{reg1}
\tag{R1}
{\beta}_n n^{-1}m(n^2) \to {\beta}\in (0,\infty]\, ,$$
which corresponds to having transversal fluctuations of order $n$. Auffinger and Louidor showed that, properly rescaled, $\log {\mathbf{Z}}_{n,{\beta}_n}^{{\omega}}$ converges to ${\widehat}{{\ensuremath{\mathcal T}} }_{\beta}$ defined in .
\[thm:AL\] Assume ${\alpha}\in(0,2)$, and consider ${\beta}_n$ such that holds. Then we have the following convergence $$\frac{1}{\beta_n m(n^2)} \log {\mathbf{Z}}_{n,{\beta}_n}^{{\omega}} \stackrel{\rm (d)}{\longrightarrow} {\widehat}{{\ensuremath{\mathcal T}} }_{{\beta}} \quad \text{as } n\to\infty ,$$ with ${\widehat}{{\ensuremath{\mathcal T}} }_{{\beta}}$ defined in . For ${\alpha}\in [1/2,2)$, we have ${\widehat}{{\ensuremath{\mathcal T}} }_{{\beta}}>0$ a.s. for all ${\beta}>0$.
### : $ \sqrt{n \log n} \ll h_n \ll n$ {#sqrtn-log-n-ll-h_n-ll-n .unnumbered}
Consider the case when
$$\label{reg2}
\tag{R2}
{\beta}_n n^{-1}m(n^2) \to 0 \quad \text{ and } \quad {\beta}_n \log n^{-1} m(n^{3/2} \sqrt{\log n}) \to \infty\, ,$$
which corresponds to having transversal fluctuations $\sqrt{n \log n} \ll h_n \ll n$, see . We find that, properly rescaled, $\log {\mathbf{Z}}_{n,{\beta}_n}^{{\omega}}$ converges to ${{\ensuremath{\mathcal T}} }_1$ defined in —this proves Conjecture 1.7 in [@cf:DZ].
\[thm:alpha>12\] Assume that ${\alpha}\in(1/2,2)$, and consider ${\beta}_n$ such that holds. Defining $h_n$ as in , then $\sqrt{n\log n}\ll h_n \ll n$, and we have $$\label{eq:hscaling3}
\frac{1}{\beta_n m(n h_n)} \Big( \log{\mathbf{Z}}^\omega_{n,\beta_n} - n {\beta}_n {\mathbb{E}}[{\omega}] {{\sf 1}}_{\{{\alpha}\ge 3/2\}} \Big) \ \stackrel{\rm (d)}{\longrightarrow} \ {{\ensuremath{\mathcal T}} }_1 \quad \text{as } n\to\infty,$$ with ${{\ensuremath{\mathcal T}} }_1$ defined in .
We stress here that we need to recenter $\log{\mathbf{Z}}^\omega_{n,\beta_n}$ by $n {\beta}_n {\mathbb{E}}[{\omega}] $ only when necessary, that is when $n/m(nh_n)$ does not go to $0$: in terms of the picture described in Figure \[fig1\], this can happen only when $\gamma \ge 4-2{\alpha}$, and in particular when ${\alpha}\ge 3/2$ (this is stressed in the statement of the theorem).
### : $h_n \asymp \sqrt{n\log n}$ {#h_n-asymp-sqrtnlog-n .unnumbered}
Consider the case $$\label{reg3}
\tag{R3}
{\beta}_n \log n^{-1} m(n^{3/2} \sqrt{\log n}) \to {\beta}\in(0,\infty)\, ,$$ which from corresponds to transversal fluctuations $h_n\sim {\beta}^{1/2}\sqrt{n\log n}$, see . We find the correct scaling of $\log {\mathbf{Z}}_{n,{\beta}_n}^{{\omega}}$, which can be of two different natures (and go to $+\infty$ or $0$), see Theorems \[thm:cas3\]-\[thm:cas3bis\] below.
We first need to introduce a few more notations. For a quenched continuum energy field ${{\ensuremath{\mathcal P}} }$ (as defined in Section \[sec:casealpha02\]), we define for a path $s$ the number of weights $w$ it collects: $$\label{def:N}
N(s) : = \sum_{(w,t,x)\in {{\ensuremath{\mathcal P}} }} {{\sf 1}}_{\{ (t,x) \in s\}} \, .$$ Then, we define a new energy-entropy variational problem: for a fixed realization of ${{\ensuremath{\mathcal P}} }$, define for any $k\ge 1$ $$\label{def:tildeT}
\begin{split}
{\widetilde}{{\ensuremath{\mathcal T}} }_{{\beta}}^{(k)} = {\widetilde}{{\ensuremath{\mathcal T}} }_{{\beta}}^{(k)} ({{\ensuremath{\mathcal P}} })&:= \sup_{s\in {\mathscr{D}}, N(s) =k} \Big\{ \pi(s) -
{\mathrm{Ent}}(s) - \frac{k}{2{\beta}} \Big\}, \\
\text{and }\quad {\widetilde}{{\ensuremath{\mathcal T}} }_{{\beta}}^{(\ge r)}&:=\sup_{k\ge r} {\widetilde}{{\ensuremath{\mathcal T}} }_{{\beta}}^{(k)} .
\end{split}$$ When $r=0$ we denote by ${\widetilde}{{\ensuremath{\mathcal T}} }_{{\beta}}$ the quantity ${\widetilde}{{\ensuremath{\mathcal T}} }_{{\beta}}^{(\ge 0)}$. In Proposition \[propTtilde\] below, we prove that these quantities are well defined, and that there exists ${\beta}_c = {\beta}_c(\mathcal P)\in (0,\infty)$ such that $ {\widetilde}{{\ensuremath{\mathcal T}} }_{{\beta}} \in (0,\infty)$ if ${\beta}>{\beta}_c$ and ${\widetilde}{{\ensuremath{\mathcal T}} }_{{\beta}} =0$ if ${\beta}<{\beta}_c$.
\[thm:cas3\] Assume that ${\alpha}\in (1/2,2)$, and consider ${\beta}_n$ such that holds. Then from we have $h_n \asymp \sqrt{n\log n}$, and $$\frac{1}{{\beta}_n m(nh_n)} \Big( \log{\mathbf{Z}}^\omega_{n,\beta_n} - n {\beta}_n {\mathbb{E}}[{\omega}] {{\sf 1}}_{\{{\alpha}\ge 3/2\}} \Big) \stackrel{\rm (d)}{\longrightarrow} {\widetilde}{{\ensuremath{\mathcal T}} }_{{\beta}} \quad \text{as } n\to\infty \, .$$ (Recall that ${\beta}_n m(n h_n)\sim h_n^2/n \sim {\beta}\log n$.)
If ${\widetilde}{{\ensuremath{\mathcal T}} }_{{\beta}} >0$ (${\beta}>{\beta}_c$) the scaling limit is therefore well identified, and $\log {\mathbf{Z}}_{n,{\beta}_n}^{{\omega}}$ (when recentered) grows like ${\beta}{\widetilde}{{\ensuremath{\mathcal T}} }_{{\beta}}\log n$ with $ {\beta}{\widetilde}{{\ensuremath{\mathcal T}} }_{{\beta}}>0$. On the other hand, if ${\widetilde}{{\ensuremath{\mathcal T}} }_{{\beta}} =0$, then the above theorem gives only a trivial limit. By an extended version of Skorokhod representation theorem [@K97 Corollary 5.12], one can couple the discrete environment and the continuum field ${{\ensuremath{\mathcal P}} }$ in order to obtain an almost sure convergence in Theorem \[thm:cas3\] above. Hence, it makes sense to work conditionally on ${\widetilde}{{\ensuremath{\mathcal T}} }_{\beta}^{\ge 1}<0$ (${\beta}<{\beta}_c$), even at the discrete level. Our next theorem says that for ${\beta}<{\beta}_c$, $\log {\mathbf{Z}}_{n,{\beta}_n}$ decays polynomially, with a random exponent ${\beta}{\widetilde}{{\ensuremath{\mathcal T}} }_{{\beta}}^{(\ge 1)} \in (-1/2,0)$.
\[thm:cas3bis\] Assume that ${\alpha}\in(1/2,2)$ and that holds. Then, conditionally on $\{{\widetilde}{{\ensuremath{\mathcal T}} }_{{\beta}}^{(\ge 1)} <0 \}$ (i.e. ${\beta}<{\beta}_c$), $$\frac{1}{{\beta}_n m(n h_n)}\log \Big( \log{\mathbf{Z}}^\omega_{n,\beta_n} - n {\beta}_n {\mathbb{E}}[{\omega}{{\sf 1}}_{\{{\omega}\le 1/{\beta}_n\}}] {{\sf 1}}_{\{{\alpha}\ge 1\}} \Big) \stackrel{\rm (d)}{\longrightarrow} {\widetilde}{{\ensuremath{\mathcal T}} }_{{\beta}}^{(\ge 1)} \quad \text{as } n\to\infty \, .$$
Recalling that ${\beta}_n m(n h_n)\sim h_n^2/n \sim {\beta}\log n$, we note that $\exp({\beta}{\widetilde}{{\ensuremath{\mathcal T}} }_{{\beta}}^{(\ge 1)} \log n)$ goes to $0$ as a (random) power ${\beta}{\widetilde}{{\ensuremath{\mathcal T}} }_{{\beta}}^{(\ge 1)}$ of $n$, with ${\beta}{\widetilde}{{\ensuremath{\mathcal T}} }_{{\beta}}^{(\ge 1)} \in (-1/2,0)$.
### : $\sqrt{n} \ll h_n \ll \sqrt{n\log n}$ {#sqrtn-ll-h_n-ll-sqrtnlog-n .unnumbered}
Consider the case $$\label{reg4}
\tag{R4}
{\beta}_n m(n^{3/2}) \to \infty \quad \text{ and } \quad {\beta}_n \log n^{-1} m(n^{3/2} \sqrt{\log n} ) \to 0 \, ;$$ which corresponds to having transversal fluctuations $\sqrt{n} \ll h_n \ll \sqrt{n\log n}$, see . Let us define $$\label{def:W}
W_{{\beta}} := {\widetilde}{{\ensuremath{\mathcal T}} }_{{\beta}}^{(1)} + \frac{1}{2{\beta}}:= \sup_{(w,x,t) \in {{\ensuremath{\mathcal P}} }} \Big\{ w - \frac{x^2}{2 {\beta}t} \Big\} \, ,$$ which is a.s. positive and finite if ${\alpha}\in(1/2,2)$, see Proposition \[prop:W\] below.
\[thm:cas4\] Assume that ${\alpha}\in (1/2,2)$, and consider ${\beta}_n$ such that holds. Defining $h_n$ as in , then $\sqrt{n}\ll h_n \ll \sqrt{n\log n}$, and we have $$\frac{1}{{\beta}_n m(nh_n)}\log \bigg( \sqrt{n} \Big( \log{\mathbf{Z}}^\omega_{n,\beta_n} - n {\beta}_n {\mathbb{E}}[{\omega}{{\sf 1}}_{\{{\omega}\le 1/{\beta}_n \}} ] {{\sf 1}}_{\{{\alpha}\ge 1\}} \Big) \bigg) \stackrel{\rm (d)}{\longrightarrow} W_1$$ as $n\to\infty$.
Recalling that ${\beta}_n m(nh_n) \sim h_n^2/n \ll \log n$, we note that $\exp\big( W_1 h_n^2/n \big) $ goes to infinity (at some random rate), but slower than any power of $n$.
### : transversal fluctuations of order $\sqrt{n}$ {#transversal-fluctuations-of-order-sqrtn .unnumbered}
Consider the case $$\label{reg5}
\tag{R5}
{\beta}_n m(n^{3/2}) \to {\beta}\in [0,\infty) \, ;$$ this corresponds to having transversal fluctuations $h_n$ of order $\sqrt n$. Here, we state one of the results obtained by Dey and Zygouras, [@cf:DZ Theorem 1.4].
\[thm:DZ\] Assume that ${\alpha}\in(1/2,2)$, and consider ${\beta}_n$ such that holds, that is ${\beta}_n m(n^{3/2}) \to {\beta}\in [0,\infty)$. Then $$\frac{\sqrt{n}}{ {\beta}_n m(n^{3/2})} \Big( \log {\mathbf{Z}}_{n,{\beta}_n} - n {\beta}_n {\mathbb{E}}\big[ {\omega}{{\sf 1}}_{\{{\omega}\le m(n^{3/2})\}} \big] {{\sf 1}}_{{\alpha}\ge 1}\Big) \stackrel{\rm (d)}{\longrightarrow} 2 {{\ensuremath{\mathcal W}} }_{\beta}^{({\alpha})} \quad \text{as } n\to\infty \, .$$ Here, ${{\ensuremath{\mathcal W}} }_{{\beta}}^{({\alpha})}$ is some specific ${\alpha}$-stable random variable (defined in [@cf:DZ p. 4011]).
### Some comments about the different regimes {#some-comments-about-the-different-regimes .unnumbered}
The regimes 2-3-4 have different behavior due to the different behaviors for the *local moderate deviation*, see [@S67 Theorem 3]. We indeed have that for ${\sqrt n \ll h_n \ll n}$ $$p_n(h_n) :={\mathbf{P}}(S_n = h_n) = \frac{c}{\sqrt{n}}\exp\Big( - (1+o(1)) \, \frac{h_n^2}{2 n} \Big) \, ,
\label{LLT}$$ so that we identify three main possibilities: if $h_n \ll \sqrt{n\log n}$, then $p_n(h_n) = n^{-1/2 +o(1)}$; if $h_n \sim c \sqrt{n\log n} $ then $p_n(h_n) =n^{-(c^2+1)/2 +o(1)}$; if $h_n \gg \sqrt{n\log n}$ then $p_n(h_n) = e^{-(1+o(1)) h_n^2/n}$ which decays faster than any power of $n$.
This is actually reflected in the behavior of the partition function. Let us denote $\bar {\mathbf{Z}}_{n,{\beta}_n}^{{\omega}} = e^{- n {\beta}_n C_{\alpha}}\times {\mathbf{Z}}_{n,{\beta}_n}^{{\omega}}$ be the renormalized (when necessary) partition function. We recall that $C_{\alpha}$ is equal either to ${\mathbb{E}}[{\omega}] {{\sf 1}}_{\{{\alpha}\ge 3/2\}}$ (Regime 2 and 3-a) or to ${\mathbb{E}}[{\omega}{{\sf 1}}_{\{{\omega}\le 1/{\beta}_n\}}] {{\sf 1}}_{{\alpha}\ge 1}$ (Regime 3-b and 4). Then we have
In Regimes 1 and 2, transversal fluctuations are $h_n \gg \sqrt{n\log n}$, and $\bar {\mathbf{Z}}_{n,{\beta}_n}$ grows faster than any power of $n$: roughly, it is of order $e^{{\beta}{\widehat}{{\ensuremath{\mathcal T}} }_{{\beta}} n}$ in Regime 1 (for ${\beta}<\infty$), and of order $e^{{{\ensuremath{\mathcal T}} }_1 h_n^2/n}$ in Regime 2.
In Regime 3, transversal fluctuations are $h_n\asymp \sqrt{n\log n}$, and $\bar{\mathbf{Z}}_{n,{\beta}_n}$ goes to infinity polynomially in Regime 3-a, and it goes to $1$ with a polynomial correction in Regime 3-b. This could be summarized as $\bar {\mathbf{Z}}_{n,{\beta}_n}\approx 1+ n^{ {\beta}{\widetilde}{{\ensuremath{\mathcal T}} }_{{\beta}}^{(\ge 1)} }$, with ${\beta}{\widetilde}{{\ensuremath{\mathcal T}} }_{{\beta}}^{(\ge 1)} >-1/2$: the transition between regime 3-a and 3-b occurs as ${\beta}{\widetilde}{{\ensuremath{\mathcal T}} }_{{\beta}}^{(\ge 1)}$ changes sign, at ${\beta}={\beta}_c$ (note that ${\beta}{\widetilde}{{\ensuremath{\mathcal T}} }_{{\beta}}^{(\ge 1)}$ keeps a mark of the local limit theorem, see and ).
In Regime 4, $\bar {\mathbf{Z}}_{n,{\beta}_n}$ goes to $1$ with a correction of order $n^{-1/2} \times e^{W_{1} h_n^2/n}$, with $e^{W_{1} h_n^2/n}$ going to infinity slower than any power of $n$: this corresponds to the cost for a trajectory to visit a single site, at which the supremum in $W_1$ is attained. In Regime 5, $\bar {\mathbf{Z}}_{n,{\beta}_n}^{{\omega}}$ goes to $1$ with a correction of order $ n^{-1/2}$.
Main results II : the case ${\alpha}\in(0, 1/2)$ {#sec:resultsII}
------------------------------------------------
In this case, since we have $n^{-1} m(n^2) /m(n^{3/2}) \to \infty$, there is no sequence ${\beta}_n$ such that ${\beta}_n n^{-1} m(n^2) \to 0$ and ${\beta}_n m(n^{3/2}) \to +\infty$. First of all, Theorem \[thm:AL\] already gives a result, but a phase transition has been identified in [@AL11; @T14] when ${\alpha}\in(0,1/2)$.
\[thm:T14\] When ${\alpha}\in (0,1/2)$, ${\widehat}{{\ensuremath{\mathcal T}} }_{{\beta}}$ defined in undergoes a phase transition: there exists some ${\widehat}{\beta}_c = {\widehat}{\beta}_c({{\ensuremath{\mathcal P}} })$ with ${\widehat}{\beta}_c \in(0,\infty)$ ${\mathbb{P}}$-a.s., such that $ {\widehat}{{\ensuremath{\mathcal T}} }_{{\beta}} =0$ if ${\beta}\le {\widehat}{\beta}_c$ and $ {\widehat}{{\ensuremath{\mathcal T}} }_{{\beta}}>0$ if ${\beta}> {\widehat}{\beta}_c$.
The fact that ${\widehat}{{\ensuremath{\mathcal T}} }_{{\widehat}{\beta}_c} =0$ was not noted in [@AL11; @T14], but simply comes from the (left) continuity of ${\beta}\mapsto {\widehat}{{\ensuremath{\mathcal T}} }_{{\beta}}$ (the proof is identical to that for ${\beta}\mapsto {{\ensuremath{\mathcal T}} }_{{\beta}}$, see [@cf:BT_ELPP Section 4.5]).
In view of Theorem \[thm:AL\], the scaling limit of $\log {\mathbf{Z}}_{n,{\beta}_n}^{{\omega}}$ is identified when ${\widehat}{{\ensuremath{\mathcal T}} }_{{\beta}}>0$, and it is trivial when $ {\widehat}{{\ensuremath{\mathcal T}} }_{{\beta}} =0$. Again, by an extended version of Skorokhod representation theorem [@K97 Corollary 5.12], we can obtain an almost sure convergence in Theorem \[thm:AL\]. Hence, it makes sense to work conditionally on ${\widehat}{{\ensuremath{\mathcal T}} }_{{\beta}}>0$ or ${\widehat}{{\ensuremath{\mathcal T}} }_{\beta}=0$, even at the discrete level. We show here that only two regimes can hold: if ${\widehat}{{\ensuremath{\mathcal T}} }_{{\beta}} >0$, then fluctuations are of order $n$, and properly rescaled, $\log {\mathbf{Z}}_{n,{\beta}_n}^{{\omega}}$ converges to ${\widehat}{{\ensuremath{\mathcal T}} }_{{\beta}}$ (this is Theorem \[thm:AL\]); if ${\widehat}{{\ensuremath{\mathcal T}} }_{{\beta}} =0$, then fluctuations are of order $\sqrt{n}$, and properly rescaled, $\log {\mathbf{Z}}_{n,{\beta}_n}^{{\omega}}$ converges in distribution (conditionally on ${\widehat}{{\ensuremath{\mathcal T}} }_{{\beta}} =0$).
\[thm:alpha<12\] Assume ${\alpha}\in(0,1/2)$, and consider ${\beta}_n$ with ${\beta}_n n^{-1} m(n^{2}) \to {\beta}\in[0,+\infty)$. Then, on the event $\{ {\widehat}{{\ensuremath{\mathcal T}} }_{{\beta}}=0\}$ (${\beta}\le {\widehat}{\beta}_c<\infty$), transversal fluctuations are of order $\sqrt{n}$. More precisely, for any ${\varepsilon}>0$, there exists some $c_0,\nu>0$ such that, for any sequence $C_n>1$ we have $$\label{eq:alpha<12_one}
\mathbb P\bigg({\mathbf{P}}_{n,{\beta}_n}^{{\omega}} \Big( \max_{i\leq n} |S_i| \ge C_n \sqrt{n} \Big) \ge e^{-c_0 C_n^2\wedge n^{1/2}}\ \Big| \ {\widehat}{{\ensuremath{\mathcal T}} }_{{\beta}} =0\bigg)\leq {\varepsilon}.$$ Moreover, conditionally on $\{{\widehat}{{\ensuremath{\mathcal T}} }_{{\beta}}=0\}$, we have that $$\label{eq:alpha<12_two}
\frac{\sqrt{n}}{{\beta}_n m(n^{3/2})} \log {\mathbf{Z}}^\omega_{n,\beta} \stackrel{\rm (d)}{\longrightarrow} 2 {{\ensuremath{\mathcal W}} }_0^{({\alpha})} \, , \qquad \text{as } n\to+\infty \, .$$ where ${{\ensuremath{\mathcal W}} }_0^{({\alpha})}:= \int_{{\mathbb{R}}_+\times {\mathbb{R}}\times [0,1]} w \rho(t,x) {{\ensuremath{\mathcal P}} }({\textrm{d}}w ,{\textrm{d}}x,{\textrm{d}}t)$ with ${{\ensuremath{\mathcal P}} }$ a realization of the Poisson Point Process defined in Section \[sec:casealpha02\], and $\rho(t,x) = ({2\pi t})^{-1/2} e^{-x^2/2t}$ is the Gaussian Heat kernel.
Note that ${{\ensuremath{\mathcal W}} }_0^{({\alpha})}$ is well defined and has an ${\alpha}$-stable distribution, with explicit characteristic function, see Lemma 1.3 in [@cf:DZ]. Theorem \[thm:alpha<12\] therefore shows that, when ${\alpha}<1/2$, a very sharp phase transition occurs on the line ${\beta}_n \sim {\beta}n/m(n^2)$: for ${\beta}\le {\widehat}{\beta}_c$, transversal fluctuations are of order $\sqrt{n}$ whereas for ${\beta}> {\widehat}{\beta}_c$ they are of order $n$.
Some comments and perspectives {#sec:comments}
------------------------------
We now present some possible generalizations, and we discuss some open questions.
### About the case ${\alpha}=1/2$
We excluded above the case ${\alpha}=1/2$. In that case, both $n^{-1} m(n^{2})$ and $m(n^{3/2})$ are regularly varying with index $3$, and there are mostly two possibilities.
\(1) If $\frac{n^{-1}m(n^2)}{m(n^{3/2})} \to 0$ (for instance if $L(x) = e^{- (\log x)^{b}}$ for some $b\in (0,1)$), there are sequences $({\beta}_n)_{n\ge 1}$ with ${\beta}_n n^{-1} m(n^2)\to 0$ and ${\beta}_n m(n^{3/2}) \to +\infty$. The situation should be similar to that of Section \[sec:resultsI\]: there should be five regimes, with transversal fluctuations $h_n$ interpolating between $\sqrt{n}$ and $n$.
\(2) If $\frac{n^{-1}m(n^2)}{m(n^{3/2})} \to c \in (0,\infty]$ (for instance if $L(n) = (\log x)^b$ for some $b$), there is no sequence $({\beta}_n)_{n\ge 1}$ with ${\beta}_n n^{-1} m(n^2)\to 0$ and ${\beta}_n m(n^{3/2}) \to +\infty$. Then, the situation should be similar to that of Section \[sec:resultsII\]: there should be only two regimes, with transversal fluctuations either $\sqrt{n}$ or $n$.
### Toward the case ${\alpha}\in (2,5)$
When ${\alpha}\in(2,5)$ (more generally in region $\mathbf{C}$ in Figure \[fig1\]), an important difficulty is to find the correct centering term for $\log {\mathbf{Z}}_{n,{\beta}_n}^{{\omega}}$. Another problem is that the variational problem ${{\ensuremath{\mathcal T}} }_{{\beta}}$ defined in is ${{\ensuremath{\mathcal T}} }_{{\beta}}=+\infty$ a.s., since paths that collect many small weights bring an important contribution to ${{\ensuremath{\mathcal T}} }_{{\beta}}$. The main objective is therefore to prove a result of the type: there exists a function $f(\cdot)$ such that, for ${\alpha}\in (2,6)$ and any ${\beta}_n$ in region $\mathbf{C}$ of Figure \[fig1\] $$\frac{1}{{\beta}_n m(nh_n)} \Big( \log {\mathbf{Z}}_{n,{\beta}_n}^{{\omega}} - f({\beta}_n) \Big) \stackrel{({\rm d})}{\longrightarrow} \check {{\ensuremath{\mathcal T}} }_{1} \, ,$$ with $h_n$ defined as in and where $\check {{\ensuremath{\mathcal T}} }_{1}$ is somehow a “recentered” version of the variational problem (that is in which the contribution of the small weights has been canceled out). The difficulties are however serious: one needs (i) to identify the centering term $f({\beta}_n)$, (ii) to make sense of the variational problem $\check {{\ensuremath{\mathcal T}} }_1$.
### Path localization
We mention that in [@AL11], Auffinger and Louidor show some path localization: they prove that, under ${\mathbf{P}}_{n,{\beta}_n}^{{\omega}}$, path trajectories concentrate around the (unique) maximizer $\gamma^*_{n,{\beta}_n}$ of the discrete analogue of the variational problem , see Theorem 2.1 in [@AL11]; moreover this maximizer $\gamma^*_{n,{\beta}_n}$ converges in distribution to the (unique) maximizer ${\widehat}\gamma_{{\beta}}^*$ of the variational problem . This could theoretically be done in our setting: in [@cf:BT_ELPP Section 4.6] we prove the existence and uniqueness of the maximizer of the continuous variational problem . Then similar techniques to those of [@AL11] could potentially be used, and one would obtain a result analogous to [@AL11 Thm. 2.1]
### Higher dimensions
Similarly to [@AL11], our methods should work in any dimension $1+d$ (one temporal dimension, $d$ transversal dimensions). The relation is replaced by ${\beta}_n m(n h_n^d) \sim h_n^2/n$: for paths with transversal scale $h_n$, the energy collected should be of order ${\beta}_n m ( n h_n^d)$ while the entropy cost should remain of order $h_n^2/n$, at the exponential level. For ${\alpha}\in (0,1+d)$, and choosing ${\beta}_n = n^{-\gamma}$, we should therefore find that in dimension $d$ a similar picture to Figure \[fig1\] hold:
Case ${\alpha}\in (0,d/2)$\
$\gamma < \frac{1+d}{{\alpha}} -1$ $\gamma > \frac{1+d}{{\alpha}} -1$
------------------------------------ ------------------------------------
$\xi=1$ $\xi =1/2$
Case ${\alpha}\in (d/2 ,1+d)$\
$\gamma \le \frac{1+d}{{\alpha}} -1$ $\frac{1+d}{{\alpha}} -1 <\gamma < \frac{2+d}{2{\alpha}}$ $\gamma\ge \frac{2+d}{2{\alpha}}$
-------------------------------------- ----------------------------------------------------------------- -----------------------------------
$\xi=1$ $\xi = \frac{1+(1-\gamma){\alpha}}{2{\alpha}-d}\in (\frac12,1)$ $\xi =\frac12$
Organization of the rest of the paper
-------------------------------------
We present an overview of the main ideas used in the paper, and describe how the proofs are organized.
$\ast$ In Section \[sec:3\], we recall some of the notations and results of the Entropy-controlled Last-Passage Percolation (E-LPP) developed in [@cf:BT_ELPP], which will be a central tool for the rest of the paper. In particular, we introduce a discrete energy/entropy variational problem (which is the discrete counterpart of ), and state its convergence toward in Proposition \[prop:ConvVP\].
$\ast$ In Section \[sec:fluctu\], we prove Theorem \[thm:fluctu\], identifying the correct transversal fluctuations. In order to make our ideas appear clearer, we first treat the case when no centering is needed (*i.e.* ${\alpha}<3/2$) in Section \[sec:nocentering\]. In Section \[sec:centeringneeded\] we adapt the proof to the case where it is needed. In the first case, we use a rough bound ${\mathbf{P}}_{n,{\beta}_n}^{{\omega}} \big( \max_{i\le n}|S_i| \ge A_n h_n \big) \le {\mathbf{Z}}_{n,{\beta}_n}^{{\omega}} \big( \max_{i\le n}|S_i| \ge A_n h_n \big) $, the second term being the partition function restricted to trajectories with $ \max_{i\le n}|S_i| \ge A_n h_n $. The key idea is to decompose this quantity into sub-parts where trajectories have a “fixed” transversal fluctuation $${\mathbf{Z}}_{n,{\beta}_n}^{{\omega}} \big( \max_{i\le n}|S_i| \ge A_n h_n \big) = \sum_{k=\log_2 A_n +1}^{\log_2(n/h_n)} {\mathbf{Z}}_{n,{\beta}_n}^{{\omega}} \Big( \max_{i\le n}|S_i| \in [2^{k-1} h_n, 2^k h_n) \Big) \, .$$ Then, we control each term separately. Forcing the random walk to reach the scale $2^{k-1} h_n$ has an entropy cost $\exp ( - c 2^{2k} h_n^2/n )$ so we need to understand if the partition function, when restricted to trajectories with $\max_{i\le n} |S_i| \le 2^k h_n$, compensates this cost (cf. ): we need to estimate the probability of having $ {\mathbf{Z}}_{n,{\beta}_n}^{{\omega}}(\max_{i\le n} |S_i| \le 2^k h_n) \ge e^{c 2^{2k} h_n^2/n}$. This is the purpose of Lemma \[lem:Zmax\], which is the central estimate of this section, and which tediously uses estimates derived in [@cf:BT_ELPP] (in particular Proposition 2.6).
$\ast$ In Section \[secProofeq:hscaling\], we consider Regimes 2 and 3-a, and we prove Theorems \[thm:alpha>12\]-\[thm:cas3\]. The proof is decomposed into three steps. In the first step (Section \[sec:reduction2\]), we use Theorem \[thm:fluctu\] in order to restrict the partition function to path trajectories that have transversal fluctuations smaller than $A h_n$ (for some large $A$ fixed). In a second step (Section \[largeweights2\]), we show that we can keep only the largest weights in the box of height $A h_n$ (more precisely a finite number of them), the small-weights contribution being negligible. Finally, the third step (Section \[sec:2-Step3\]) consists in proving the convergence of the large-weights partition function, and relies on the convergence of the discrete variational problem of Section \[sec:3\].
$\ast$ In Section \[sec:3b4\], we treat Regime 3-b and Regime 4, and we prove Theorems \[thm:cas3bis\]-\[thm:cas4\]. We proceed in four steps. In the first step (Section \[sec:4-step1\]), we again use Theorem \[thm:fluctu\] to restrict the partition function to trajectories with transversal fluctuations smaller than $A \sqrt{n\log n}$ (for some large $A$ fixed). The second step (Section \[largeweights3\]) consists in showing that one can restrict to large weights. In the third step (Section \[reductionlogZ\]), we observe that since we consider a regime $\log {\mathbf{Z}}_{n,{\beta}_n}^{{\omega}} \to 0$, it is equivalent to studying the convergence of ${\mathbf{Z}}_{n,{\beta}_n}^{{\omega}}-1$: we reduce to showing the convergence of a finite number of terms of the polynomial chaos expansion of ${\mathbf{Z}}_{n,{\beta}_n}^{{\omega}}-1$, see Lemmas \[lem:cas3-mainterm\]-\[lem:cas4-mainterm\]. We prove this convergence in a last step: in Section \[sec:prooflem3b\], we show the convergence in Regime 3-b (Lemma \[lem:cas3-mainterm\]), relying on the convergence of a discrete variational problem. In Section \[sec:regime4\], we show the convergence in Regime 4 (Lemma \[lem:cas4-mainterm\]), which is slightly more technical since we first need to reduce to trajectories with transversal fluctuations of order $h_n \ll \sqrt{n\log n}$.
$\ast$ In Section \[sec:alpha12\], we consider the case ${\alpha}\in(0,1/2)$, and we prove Theorem \[thm:alpha<12\]. First, in Section \[sec:fluctualpha12\], we prove *i.e.* that there cannot be intermediate transversal fluctuations between $\sqrt{n}$ and $n$. We use mostly the same ideas as in Section \[sec:fluctu\], decomposing the contribution to the partition function according to the scale of the path, and controlling the entropic cost vs. energy reward for each term. Here, some simplifications occur: one can bound the maximal energy collected by a path at a given scale by the sum of all weights in a box containing the path, this sum being roughly dominated by the maximal weight in the box (this is true for ${\alpha}<1$). We then turn to the convergence of the partition function in Section \[sec:convalpha12\]. The idea is similar to that of [@cf:DZ Section 5], and consists in several steps: first we reduce the partition function to trajectories that stay at scale $\sqrt{n \log n}$; then we perform a polynomial chaos expansion of ${\mathbf{Z}}_{n,{\beta}_n}^{{\omega}}-1$ and we show that only the first term contributes; finally, we prove the convergence of the main term, see Lemma \[lem:convergence\], showing in particular that the main contribution comes from trajectories that stay at scale $\sqrt{n}$.
Discrete energy-entropy variational problem {#sec:3}
===========================================
We introduce here a few necessary notations, and state some useful results from [@cf:BT_ELPP]. Let us consider a box $\Lambda_{n,h} = \llbracket 1,n \rrbracket \times \llbracket -h,h \rrbracket$. For any set $\Delta \subset \Lambda_{n,h}$, we define the (discrete) energy collected by $\Delta$ by $$\label{energydiscrete}
\Omega_{n,h} (\Delta) := \sum_{(i,x) \in \Delta} {\omega}_{i,x} \, .$$ We can also define the (discrete) entropy of a finite set $\Delta=\big\{ (t_i,x_i) ; 1\le i\le j \big\} \subset {\mathbb{R}}^2$ with $|\Delta|=j\in \mathbb N$ and with $0\le t_1\le t_2\le \cdots \le t_j$ (with $t_0=0,x_0=0$) $$\label{def:entdelta}
{\mathrm{Ent}}(\Delta) := \frac12 \sum_{i=1}^j \frac{(x_i-x_{i-1})^2}{t_i-t_{i-1}} \, ,$$ By convention, if $t_i=t_{i-1}$ for some $i$, then ${\mathrm{Ent}}(\Delta)=+\infty$. The set $\Delta$ is seen as a set of points a (continuous or discrete) path has to go through: if $\Delta \subset {\mathbb{N}}\times {\mathbb{Z}}$ a standard calculation gives that ${\mathbf{P}}(\Delta \subset S) \le e^{-{\mathrm{Ent}}(\Delta)}$ ($\Delta\subset S$ means that $S_{t_i}=x_i$ for all $i\le |\Delta|$), where we use that ${\mathbf{P}}(S_i=x) \le e^{- x^2/2i}$ by a standard Chernoff bound argument.
We are interested in the (discrete) variational problem, analogous to $$\label{def:discreteELPP}
T_{n,h}^{{\beta}_{n,h}} := \max_{ \Delta \subset \Lambda_{n,h}} \big\{ {\beta}_{n,h} \Omega_{n,h} (\Delta) - {\mathrm{Ent}}(\Delta) \big\} \, ,$$ with ${\beta}_{n,h}$ some function of $n,h$ (soon to be specified).
We may rewrite the disorder in the region $\Lambda_{n,h}$, using the *ordered statistic*: we let $M_r^{(n,h)}$ be the $r$-th largest value of $(\omega_{i,x})_{(i,x)\in \Lambda_{n,h}}$ and $Y_r^{(n,h)}\in \Lambda_{n,h}$ its position. In such a way $$(\omega_{i,j})_{(i,j)\in \Lambda_n}{=}(M_r^{(n,h)},Y_r^{(n,h)})_{r=1}^{|\Lambda_{n,h}|} \, .$$ In the following we refer to $(M_r^{(n,h)})_{r=1}^{|\Lambda_{n,h}|}$ as the *weight* sequence. Note also that $(Y_r^{(n,h)})_{r=1}^{|\Lambda_{n,h}|}$ is simply a random permutation of the points of $\Lambda_{n,h}$. The ordered statistics allows us to redefine the energy collected by a set $\Delta \subset \Lambda_{n,h}$, and its contribution by the first $\ell$ weights (with $1\le \ell \le |\Lambda_{n,h}|$) by $$\label{def:Omega}
\Omega_{n,h}^{(\ell)} (\Delta) := \sum_{r=1}^{\ell} M_r^{(n,h)} {{\sf 1}}_{\{ Y_r^{(n,h)} \in \Delta\}}\, , \qquad \Omega_{n,h} (\Delta) := \Omega_{n,h}^{(|\Lambda_{n,h}|)} (\Delta) \, .$$ We also set $ \Omega_{n,h}^{(>\ell)} (\Delta) := \Omega_{n,h} (\Delta)- \Omega_{n,h}^{(\ell)}(\Delta)$. We then define analogues of with a restriction to the $\ell$ largest weights, or beyond the $\ell$-th weight $$\label{def:discrELPPell}
\begin{split}
T_{n,h}^{{\beta}_{n,h},(\ell)} &:= \max_{ \Delta \subset \Lambda_{n,h}} \big\{ {\beta}_{n,h} \Omega_{n,h}^{(\ell)} (\Delta) - {\mathrm{Ent}}(\Delta) \big\} \, ,\\
T_{n,h}^{{\beta}_{n,h},(>\ell)} &:= \max_{ \Delta \subset \Lambda_{n,h}} \big\{ {\beta}_{n,h} \Omega_{n,h}^{(>\ell)} (\Delta) - {\mathrm{Ent}}(\Delta) \big\} \, .
\end{split}$$ Estimates on these quantities are given in [@cf:BT_ELPP Prop. 2.6] (most useful in Section \[sec:fluctu\]). The following convergence in distribution is given in [@cf:BT_ELPP Thm. 2.7], and plays a crucial role for the convergence in Theorems \[thm:alpha>12\]—\[thm:cas4\] .
\[prop:ConvVP\] Suppose that $\frac{n}{h^2}{\beta}_{n,h} m(nh) \to \nu \in[0,\infty)$ as $n,h\to\infty$. For every ${\alpha}\in (1/2,2)$ and for any $q>0$ we have $$\label{def:TA}
\frac{n}{h^2}\, T_{n,qh}^{\beta_{n,h}} \stackrel{({\textrm{d}})}\longrightarrow {{\ensuremath{\mathcal T}} }_{\nu,q} :=\sup_{s\in {\mathscr{M}}_q}\big\{\nu \pi(s)-{\mathrm{Ent}}(s) \big\} \quad \text{as } n\to\infty,$$ with ${\mathscr{M}}_q:=\{s\in {\mathscr{D}}, {\mathrm{Ent}}(s)<\infty, \max_{t\in [0,1]} |s(t)|\le q\}.$ We also have $$\label{conv:largeweights}
\frac{n}{h^2}\, T_{n,qh}^{\beta_{n,h}, (\ell)} \stackrel{({\textrm{d}})}\longrightarrow {{\ensuremath{\mathcal T}} }_{\nu,q}^{(\ell)} :=\sup_{s\in {\mathscr{M}}_q}\big\{\nu \pi^{(\ell)}(s)-{\mathrm{Ent}}(s) \big\} \quad \text{as } n\to\infty,$$ where $\pi^{(\ell)}:= \sum_{r=1}^{\ell} M_r {{\sf 1}}_{\{Y_r\in s\}}$ with $\{(M_r,Y_r)\}_{r\ge 1}$ the ordered statistics of ${{\ensuremath{\mathcal P}} }$ restricted to $[0,1]\times [-q,q]$, see [@cf:BT_ELPP Section 5.1] for details.
Finally, we have ${{\ensuremath{\mathcal T}} }_{\nu,q}^{(\ell)} \to {{\ensuremath{\mathcal T}} }_{\nu,q}$ as $\ell\to\infty$, and ${{\ensuremath{\mathcal T}} }_{\nu,q} \to {{\ensuremath{\mathcal T}} }_{\nu}$ as $q\to\infty$, a.s.
Transversal fluctuations: proof of Theorem \[thm:fluctu\] {#sec:fluctu}
=========================================================
In this section, we have ${\alpha}\in (1/2,2)$.
First, we partition the interval $[A_n h_n,n]$ into blocks $$B_{k,n}:=[2^{k-1}h_n,2^{k} h_n),\quad k=\log_2 A_n+1,\dots, \log_2 (n/h_n)+1.$$ In such a way, $$\label{alph12EQ1}
{\mathbf{P}}^\omega_{n,\beta_n}\big( \max\limits_{i\leq n} | S_i | \ge A_n h_n\big)=
\sum_{k=\log_2 A_n+1}^{\log_2 (n/h_n)}{\mathbf{P}}^\omega_{n,\beta_n}\big( \max\limits_{i\leq n} | S_i |\in B_{k,n}\big).$$
We first deal with the case where $n/m(nh_n) \stackrel{n\to\infty}{\to} 0$ for the sake of clarity of the exposition: in that case, $\log {\mathbf{Z}}_{n,{\beta}_n}^{{\omega}}$ does not need to be recentered. We treat the remaining case (in particular we have ${\alpha}\ge 3/2$) in a second step.
Case $n/m(nh_n) \stackrel{n\to\infty}{\to} 0$ {#sec:nocentering}
---------------------------------------------
We observe that the assumption $\omega\geq 0$ implies that the partition function ${\mathbf{Z}}^\omega_{n,\beta_n}$ is larger than one. Therefore, $${\mathbf{P}}^\omega_{n,\beta_n}\big( \max\limits_{i\leq n} | S_i |\in B_{k,n}\big)\leq
{\mathbf{Z}}^\omega_{n,\beta_n}\big( \max\limits_{i\leq n} | S_i |\in B_{k,n}\big).$$ By using Cauchy-Schwarz inequality, we get that $$\label{eq:CauchySchwarz}
{\mathbf{Z}}^\omega_{n,\beta_n}\big( \max\limits_{i\leq n} | S_i |\in B_{k,n}\big)^2 \le {\mathbf{P}}\big( \max\limits_{i\leq n} \big| S_i\big| \ge 2^{k-1} h_n \big) \times
{\mathbf{Z}}^\omega_{n,2\beta_n}\big( \max\limits_{i\leq n} | S_i | \le 2^k h_n \big) \, .$$
The first probability is bounded by $ 2{\mathbf{P}}( |S_n|\ge h_n )\le 4 \exp(- 2^{2k} h_n^2/2n)$ (by Levy’s inequality and a standard Chernov’s bound). We are going to show the following lemma, which is the central estimate of the proof.
\[lem:Zmax\] There exist some constant $q_0>0$ and some $\nu>0$, such that for all $q \ge q_0$ we have $${\mathbb{P}}\Big( {\mathbf{Z}}^\omega_{n,2\beta_n}\big( \max\limits_{i\leq n} | S_i | \le qh_n \big) \ge e^{ \frac{1}{4} q^2 \frac{h_n^2 }{n} } \Big) \le q^{-\nu}\Big( 1 + 1\wedge \frac{n}{m(nh_n)} \Big) \, .$$
Therefore, if $n/m(nh_n) \stackrel{n\to\infty}{\to} 0$, this lemma gives that for $c_0 =1/8$ and for $k$ large enough (*i.e.* $A_n$ large enough), using , $${\mathbb{P}}\Big( {\mathbf{Z}}^\omega_{n,\beta_n}\big( \max\limits_{i\leq n} | S_i |\in B_{k,n}\big) \ge 4e^{- c_0 2^{2k} h_n^2/n } \Big) \le (2^{k})^{-\nu}\, .$$ Then, using that $\sum_{k> \log_2 A_n} 4e^{- c_0 2^{2k} h_n^2 /n } \le e^{- c_1 A_n^2 h_n^2/n}$, we get that by a union bound $$\begin{aligned}
{\mathbb{P}}\Big( {\mathbf{P}}^\omega_{n,\beta}\big( & \max\limits_{i\leq n} \big| S_i\big| \ge A_n h_n \big) \ge e^{- c_1 A_n^2 h_n^2/n} \Big) \notag \\
&\le \sum_{k=\log_2 A_n +1}^{\log_2(n/h_n)} {\mathbb{P}}\Big( {\mathbf{Z}}^\omega_{n,\beta}\big( \max\limits_{i\leq n} \big| S_i\big|\in B_{k,n}\big) \ge 4e^{- c_0 2^{2k} h_n^2/n} \Big) \notag \\
&\le \sum_{k > \log_2 A_n } 2^{-\nu k} \le c A_n^{-\nu} \, .
\label{unionboundBkn}\end{aligned}$$ We stress that in the case when $n/m(nh_n) \stackrel{n\to\infty}{\to} 0$, we do not need the additional $n$ in front of $e^{- c_1 A_n^2 h_n^2/n}$ in .
For simplicity, we assume in the following that $qh_n$ is an integer. We fix ${\delta}>0$ such that $(1+{\delta})/{\alpha}<2$ and $(1-{\delta})/{\alpha}>1/2$, and let $$\label{eq:defT}
\mathtt T=\mathtt T_n(qh_n) = \frac{h_n^2}{n} q^{1/{\alpha}} ( q^2 h_n^2/n)^{-(1-{\delta})^{3/2}/{\alpha}} \vee 1$$ be a truncation level. Note that if ${\alpha}\le (1-{\delta})^{3/2}$ then we have $\mathtt T =1$. We decompose the partition function into three parts: thanks to Hölder’s inequality, we can write that $$\begin{aligned}
\log {\mathbf{Z}}^\omega_{n,2\beta_n}\big( \max\limits_{i\leq n} | S_i | \le qh_n \big)
\le \frac13 \log {\mathbf{Z}}_{n,6{\beta}_n}^{(>\mathtt T)}
+ \frac13 \log {\mathbf{Z}}_{n,6{\beta}_n}^{((1,\mathtt T])}
+ \frac13 \log {\mathbf{Z}}_{n,6{\beta}_n}^{(\le1)}\, ,
\label{eq:threeparts}\end{aligned}$$ where the three partition functions correspond to three ranges for the weights ${\beta}_n {\omega}_{i,S_i}$: $$\begin{aligned}
\label{def:ZbigT}
{\mathbf{Z}}_{n,6{\beta}_n}^{(>\mathtt T)} &:= {\mathbf{E}}\Big[ \exp\Big(\sum_{i=1}^n 6 {\beta}_n {\omega}_{i,S_i} {{\sf 1}}_{\{ {\beta}_n {\omega}_{i,S_i} >\mathtt T\}} \Big) {{\sf 1}}_{\{ \max\limits_{i\leq n} | S_i | \le qh_n\}}\Big] \\
{\mathbf{Z}}_{n,6{\beta}_n}^{((1,\mathtt T])} &:= {\mathbf{E}}\Big[ \exp\Big(\sum_{i=1}^n 6 {\beta}_n {\omega}_{i,S_i} {{\sf 1}}_{\{ {\beta}_n {\omega}_{i,S_i} \in (1, \mathtt T]\}} \Big) {{\sf 1}}_{\{ \max\limits_{i\leq n} | S_i | \le qh_n\}}\Big]
\label{def:Zin1T}\\
{\mathbf{Z}}_{n,6{\beta}_n}^{(\le1)} & := {\mathbf{E}}\Big[ \exp\Big(\sum_{i=1}^n 6 {\beta}_n {\omega}_{i,S_i} {{\sf 1}}_{\{ {\beta}_n {\omega}_{i,S_i} \le 1\}} \Big)
{{\sf 1}}_{\{ \max\limits_{i\leq n} | S_i | \le qh_n\}}
\Big] \, .
\label{def:Zsmall1}\end{aligned}$$ We now show that with high probability, these three partition functions cannot be large. Note that when $\mathtt T=1$, the second term is equal to $1$ and we do not have to deal with it.
For , we prove that for any $\nu<2{\alpha}-1$, for $q$ sufficiently large, for all $n$ large enough we have $$\label{aim:part1}
{\mathbb{P}}\Big( \log {\mathbf{Z}}_{n,6{\beta}_n}^{(>\mathtt T)} \ge c_0 q^2 \frac{h_n^2}{n} \Big) \le q^{-\nu} \, .$$ We compare this truncated partition function with the partition function where we keep the first $\ell$ weights in the ordered statistics $(M_i^{(n,qh_n)})_{1\le i\le nqh_n}$. Define $$\label{def:ell0}
\ell = \ell_n(qh_n):= (q^2 h_n^2/n)^{1-{\delta}} \, , \quad \text{so }\ \mathtt T = \frac{h_n^2}{n} q^{1/{\alpha}} \times \ell^{-(1-{\delta})^{1/2}/{\alpha}} \, ,$$ and set $$\label{Zell}
{\mathbf{Z}}_{n,6{\beta}_n}^{(\ell)} := {\mathbf{E}}\Big[ \exp\Big(\sum_{i=1}^{\ell} 6{\beta}_n M_i^{(n,qh_n)} {{\sf 1}}_{\{Y_i^{(n,q h_n)} \in S\}}\Big) \Big] \, .$$ Remark that, with the definition of $\mathtt T$ and thanks to the relation verified by ${\beta}_n$, we have that for $n$ large enough $$\begin{aligned}
{\mathbb{P}}\Big( {\beta}_n M_{\ell}^{(n,qh_n)} >\mathtt T \Big) &\le {\mathbb{P}}\Big( M_{\ell}^{(n,qh_n)} \ge \frac12 q^{1/{\alpha}} \ell^{-(1-{\delta})^{1/2}/{\alpha}} m(nh_n) \Big)\end{aligned}$$ Then, since we have $q/\ell\le 1$ (see ), we can use Potter’s bound to get that for $n$ sufficiently large $$m\big( n q h_n /\ell \big) \le (q/\ell)^{(1-{\delta}^2)/{\alpha}} m(nh_n) \, ,$$ and we obtain that provided that ${\delta}$ is small enough $$\begin{aligned}
{\mathbb{P}}\Big( {\beta}_n M_{\ell}^{(n,qh_n)} >\mathtt T \Big) &\le {\mathbb{P}}\Big( M_{\ell}^{(n,qh_n)} \ge c_0 q^{{\delta}^2/{\alpha}}\ell^{{\delta}^2/{\alpha}} m\big( n q h_n /\ell \big) \Big) \le (c q \ell)^{- {\delta}^2 \ell/2} \, ,\end{aligned}$$ where we used [@cf:BT_ELPP Lemma 5.1] for the last inequality. We therefore get that, with probability larger than $1- (c \ell )^{- {\delta}\ell/2}$ (note that $\ell^{- {\delta}\ell/2} \le q^{-{\delta}\ell/2} \le q^{-4}$ for $n$ large enough), we have that $$\label{eq:inclusion}
\Big\{ (i,x) \in \llbracket 1,n \rrbracket \times \llbracket -qh_n ,qh_n \rrbracket ; {\beta}_n {\omega}_{i,x} > \mathtt T \Big\} \subset {\Upsilon}_\ell := \big\{ Y_1^{(n,qh_n)}, \ldots, Y_{\ell}^{(n,qh_n)}\big\} \, ,$$ and hence ${\mathbf{Z}}_{n,6{\beta}_n}^{(>T)} \le {\mathbf{Z}}_{n,6{\beta}_n}^{(\ell)}$.
We are therefore left to focus on the term ${\mathbf{Z}}_{n,6{\beta}_n}^{(\ell)}$: recalling the definitions and , we get that $$\label{ZTleZell}
\begin{split}
{\mathbf{Z}}_{n,6{\beta}_n}^{(\ell)} &= \sum_{{\Delta}\subset {\Upsilon}_\ell} e^{ 6 {\beta}_n \Omega_{n,qh_n}^{(\ell)}(\Delta) } {\mathbf{P}}\big( S \cap {\Upsilon}_m =\Delta \big) \\
& \le \sum_{{\Delta}\subset {\Upsilon}_\ell} \exp\big( 6{\beta}_n \Omega_{n,qh_n}(\Delta) - {\mathrm{Ent}}(\Delta) \big) \le 2^{\ell} \exp \Big( T_{n,qh_n}^{6{\beta}_n,(\ell)}\Big) \, ,
\end{split}$$ where we used that ${\mathbf{P}}(\Delta \subset S)\le \exp(-{\mathrm{Ent}}(\Delta))$ as noted below .
Note that we have $\ell \le \frac12 c_0 q^2 h_n^2/n$ for $n$ large enough (and $q\ge 1$), so we get that $${\mathbb{P}}\Big( \log {\mathbf{Z}}_{n,6{\beta}_n}^{(\ell)} \ge c_0 q^2 \frac{h_n^2}{n} \Big) \le {\mathbb{P}}\Big( T_{n,qh_n}^{6{\beta}_n, (\ell)} \ge \frac12 c_0 q^2 \frac{h_n^2}{n}\Big) \, .$$ Then, by the definition and thanks to Potter’s bound, for any $\eta>0$ there exists a constant $c_{\eta}$ such that for any $q\ge 1$ $$\frac{\big(6{\beta}_n m(nqh_n) \big)^{4/3}}{ ( q^2 h_n^2/n)^{1/3}} \le c_{\eta} q^{(1+\eta) \frac{4}{3{\alpha}} -\frac{2}{3} }\, \frac{h_n^2}{n} = c_{\eta} (q^{4/3})^{(1+\eta)/{\alpha}-2} \times q^2 \frac{h_n^2}{n} \, ,$$ where we used that for any $\eta>0$, $m(nqh_n) \le c'_{\eta} q^{(1+\eta)/{\alpha}} m(nh_n)$ provided that $n$ is large enough (Potter’s bound). Therefore, provided that $\eta$ is small enough so that $(1+\eta)/{\alpha}< 2$, an application of [@cf:BT_ELPP Prop. 2.6] gives that for $q$ large enough (so that $b_q:= \tfrac{c_0}{2 c_{\eta}} (q^{4/3})^{2- (1+\eta)/{\alpha}}$ is large), $${\mathbb{P}}\Big( T_{n,qh_n}^{6{\beta}_n, (\ell)} \ge \frac12 c_0 q^2 \frac{h_n^2}{n}\Big) \le {\mathbb{P}}\Big( T_{n,qh_n}^{6{\beta}_n, (\ell)} \ge b_q \times\frac{\big(6{\beta}_n m(nqh_n) \big)^{4/3}}{ ( q^2 h_n^2/n)^{1/3}} \Big) \le c q^{- \nu} \, ,$$ with $\nu= 2{\alpha}-1-2\eta$. This gives , since $\eta$ is arbitrary.
We now turn to We consider only the case $\mathtt{T} >1$ (and in particular we have ${\alpha}>(1-{\delta})^{3/2}$). We show that for any $\eta>0$, there is a constant $c_{\eta}>0$ such that for $q$ large enough and $n$ large enough, $$\label{aim:part2}
{\mathbb{P}}\Big( \log {\mathbf{Z}}_{n,6{\beta}_n}^{((1,\mathtt T])} \ge c_0 \big( q^2 h_n^2 /n \big)^{1-\eta} \Big) \le \exp\big( - c_{\eta} (q^2 h_n^2/n)^{1/3} \big)\, .$$ Again, we need to decompose ${\mathbf{Z}}_{n,6{\beta}_n}^{((1,\mathtt T])}$ according to the values of the weights. We set $\theta := (1-{\delta})2/{\alpha}>1$, and let $$\begin{aligned}
\ell_j &:= ( q^2 h_n^2/n)^{ \theta^j (1-{\delta})} = (\ell_0)^{\theta^j}\, , \quad \text{ with } \ell_0=\ell =(q^2 h_n^2/n)^{1-{\delta}} \text{ as in \eqref{def:ell0} } \label{def:ellj}\\
\mathtt{T}^{(j)} &:= \frac{h_n^2}{n} q^{1/{\alpha}} \times (q^2 h_n^2/n)^{- \theta^j (1-{\delta})^{3/2}/{\alpha}} = \frac{h_n^2}{n} q^{1/{\alpha}} \big( \ell_j \big)^{-(1-{\delta})^{1/2} /{\alpha}}
\label{def:Tj}\end{aligned}$$ for $j \in \{ 0,\ldots, \kappa \}$ with $\kappa$ the first integer such that $ \theta^{\kappa} >{\alpha}/(1-{\delta})^{3/2}$. We get that $\mathtt{T}^{(0)}=\mathtt T$, and $\mathtt T^{(\kappa)} <1$. Then, thanks to Hölder inequality we may write $$\begin{aligned}
\log {\mathbf{Z}}_{n,6{\beta}_n}^{((1,T])} &\le \frac{1}{\kappa} \sum_{j=1}^{\kappa} \log {\mathbf{Z}}_{n, 6 \kappa {\beta}_n}^{((\mathtt{T}^{(j)}, \mathtt{T}^{(j-1)}])} \, , \quad \text{with}\\
{\mathbf{Z}}_{n, 6 \kappa {\beta}_n}^{((\mathtt{T}^{(j)}, \mathtt{T}^{(j-1)}])} &:= {\mathbf{E}}\Big[ \exp\Big(\sum_{i=1}^n 6 \kappa {\beta}_n {\omega}_{i,S_i} {{\sf 1}}_{\{ {\beta}_n {\omega}_{i,S_i} \in (\mathtt{T}^{(j)}, \mathtt{T}^{(j-1)}]\}} \Big) {{\sf 1}}_{\{ \max\limits_{i\leq n} | S_i | \le qh_n\}}\Big]\, .\end{aligned}$$ To prove , it is therefore enough to prove that for any $1\le j\le \kappa$, since $\ell_j \ge (q^2 h_n^2/n)^{1-{\delta}}$, $$\label{aim:part2-intermed}
{\mathbb{P}}\Big( \log {\mathbf{Z}}_{n, 6 \kappa {\beta}_n}^{((\mathtt{T}^{(j)}, \mathtt{T}^{(j-1)}])} \ge 8 \kappa \big( q^2 h_n^2/n \big) \ell_j^{-{\delta}/10} \Big) \le \exp\big( - c ( q^2 h_n^2/n)^{1/3} \big) \, .$$ First of all, we notice that in view of -, with the same computation leading to , we have that with probability larger than $1- (c\ell_j )^{-{\delta}\ell_j/4}$ $$\begin{aligned}
\label{eq:inclusionj}
\Big\{ (i,x) \in \llbracket 1,n \rrbracket \times \llbracket -qh_n ,qh_n \rrbracket ; &{\beta}_n {\omega}_{i,x} > \mathtt{T}^{(j-1)} \Big\} \notag\\
& \subset {\Upsilon}_{\ell_j} := \big\{ Y_1^{(n,qh_n)}, \ldots, Y_{\ell_j}^{(n,qh_n)}\big\} \, .\end{aligned}$$ On this event, and using that $\ell_j = (\ell_{j-1})^{(1-{\delta}) 2/{\alpha}}$ and $$\mathtt{T}^{(j-1)} = \frac{h_n^2}{n} q^{1/{\alpha}} \ell_j^{- (1-{\delta})^{-1/2}/2} \le\frac{h_n^2}{n} q^{1/{\alpha}} \ell_j^{- 1/2 -{\delta}/5}$$ (if ${\delta}$ is small), we have $$\begin{aligned}
\label{eq:estimatebZ6kappa}
{\mathbf{Z}}_{n, 6 \kappa {\beta}_n}^{((\mathtt{T}^{(j)}, \mathtt{T}^{(j-1)}])} &\le {\mathbf{E}}\Big[ \exp\Big( 6\kappa \mathtt{T}^{(j-1)} \sum_{i=1}^{\ell_j} {{\sf 1}}_{\{ Y_i^{(n,qh_n)} \in S\} } \Big) \Big] \\
\nonumber
& \leq e^{ 6 \kappa q^2 \frac{h_n^2}{n} \ell_j^{-{\delta}/10} } + \mathcal{H}_j\end{aligned}$$ with $$\begin{aligned}
\mathcal{H}_j& := \sum_{k= q^{2-\frac{1}{{\alpha}} } \ell_{j}^{1/2 + {\delta}/10} }^{\ell_j} \sum_{\Delta \subset {\Upsilon}_{\ell_j} ; |\Delta|=k} e^{ 6\kappa \frac{h_n^2}{n} q^{1/{\alpha}} \ell_j^{-1/2- {\delta}/5} k } \ {\mathbf{P}}\big( S\cap {\Upsilon}_{\ell_j} =\Delta \big)\\
& \le \sum_{k= q^{2-\frac{1}{{\alpha}} }\ell_{j}^{1/2 + {\delta}/10} }^{\ell_j} \binom{\ell_j}{k} \exp\Big( 6\kappa \frac{h_n^2}{n} q^{1/{\alpha}} \ell_j^{-1/2 - {\delta}/5 } k - \inf_{\Delta\subset {\Upsilon}_{\ell_j} , |\Delta| =k} {\mathrm{Ent}}(\Delta) \Big) \, .
\nonumber\end{aligned}$$ Then, we may bound $\binom{\ell_j}{k}\le e^{ k \log \ell_j }$. We notice from the definition of $\kappa$ (and since $\theta \in (1,2)$) that there exists some $\eta>0$ such that $\ell_{j} \le \ell_{\kappa} \le (q^2h_n^2/n)^{2-\eta}$ for any $1\le j\le \kappa$: it shows in particular that $\log \ell_j \le \ell_j^{{\delta}^2} \le q^2 \frac{h_n^{2}}{n} \ell_j^{-1/2- {\delta}/5}$, provided that $n$ is sufficiently large and ${\delta}$ has been fixed sufficiently small. We end up with the following bound $$\label{eq:estimatebZ6kappa2}
\mathcal{H}_j \le \sum_{k= q^{2-\frac{1}{{\alpha}} } \ell_{j}^{1/2+{\delta}/10} }^{\ell_j} \exp\Big( c q^{2} \frac{h_n^2}{n} \ell_j^{-1/2- {\delta}/5 } k - \inf_{\Delta\subset {\Upsilon}_{\ell_j} , |\Delta| =k} {\mathrm{Ent}}(\Delta) \Big) .$$ Then, we may use relation (2.5) of [@cf:BT_ELPP] (with $m=\ell_j$, $h=qh_n$) to get that, for any $ k\ge q^{2-\frac{1}{{\alpha}} } \ell_{j}^{1/2+{\delta}/10} $ $$\begin{aligned}
\label{eq:entDelta>q}
{\mathbb{P}}\Big( \inf_{\Delta\subset {\Upsilon}_{\ell_j} , |\Delta| =k} {\mathrm{Ent}}(\Delta) \le 2 c q^{2} \frac{h_n^2}{n} \ell_j^{-1/2- {\delta}/5 } k \Big) & \le \bigg( \frac{C_0 (2c\ell_j^{-1/2- {\delta}/5} k)^{1/2} \ell_j}{k^2} \bigg)^k \nonumber\\
& \le \big( c q^{\frac{3}{2{\alpha}} - 3} \ell_j^{ -{\delta}/4} \big)^k \le \big( c \ell_j \big)^{- {\delta}k /4}\, .\end{aligned}$$ For the last inequality, we used that $q^{\frac{3}{2{\alpha}} - 3}\le 1$, since ${\alpha}>1/2$ and $q\ge 1$. Since we have that $q^2 \frac{h_n^2}{n} \ell_j^{-1/2 - {\delta}/5} \ge 1$, we get that there is a constant $c'>0$ such that $$\sum_{k\ge q^{2-\frac1{\alpha}} \ell_j^{1/2 +{\delta}/10} } e^{- c q^2 \frac{h_n^2}{n} \ell_j^{-1/2 - {\delta}/5} k} \le c' e^{ - c q^2 \frac{h_n^2}{n} \ell_j^{-{\delta}/10}} \le c'.$$ Using , we therefore obtain, via a union bound (also recalling ), that provided that $n$ is large enough $$\begin{aligned}
{\mathbb{P}}\Big( {\mathbf{Z}}_{n, 6 \kappa {\beta}_n}^{((\mathtt{T}^{(j)}, \mathtt{T}^{(j-1)}])} \ge e^{8 \kappa q^2 \frac{h_n^2}{n} \ell_j^{-{\delta}/10}} \Big)& \le (c \ell_j)^{-{\delta}\ell_j/4} + \sum_{k\ge q^{2-\frac1{\alpha}} \ell_j^{1/2 +{\delta}/10} } \big( c \ell_j \big)^{- {\delta}k /4}\\
& \le \big( c \ell_j \big)^{- c_{{\delta}} \ell_j^{1/2}} \, .\end{aligned}$$ This proves since $\ell_j \ge \ell_0 = (q^2 h_n^2/n)^{1-{\delta}}$.
[**Term 3.**]{} For the last part , we prove that for arbitrary $\eta>0$, $$\label{aim:part3}
{\mathbb{P}}\Big( \log {\mathbf{Z}}_{n,6{\beta}_n}^{(\le1)} \ge c_0 q^2 \frac{h_n^2}{n} \Big) \le c q^{-2} \times
\begin{cases}
\frac{n}{m(nh_n)} & \quad \text{ if } {\alpha}>1 \, ,\\
\frac{n}{ m(nh_n)^{(1-\eta){\alpha}}} & \quad \text{ if } {\alpha}\le 1 \, .
\end{cases}$$ Let us stress that in the case ${\alpha}\le 1$ we get that for $n$ large $m(nh_n)^{(1-\eta){\alpha}} \ge (nh_n)^{1-2\eta}$, therefore $n/(nh_n)^{(1-\eta)\alpha}$ goes to $0$ provided that $\eta$ is small enough, since we are considering the case when $h_n\ge \sqrt{n}$. Hence, we can replace the upper bound in by $1\wedge (n/m(nh_n))$.
To prove , we use that $e^{6 x {{\sf 1}}_{\{x\le 1\}}} \le 1+ e^6 x {{\sf 1}}_{\{x\le 1\}}$ for any $x$, and we get that $$\begin{aligned}
\label{EZnsmall}
{\mathbf{Z}}_{n,6{\beta}_n}^{(\le1)} & \le {\mathbf{E}}\Big[ \prod_{i=1}^{n} \big (1+ 6e^6 {\beta}_n {\omega}_{i,s_i} {{\sf 1}}_{\{ {\beta}_n {\omega}_{i,s_i} \le 1\}} \big) \Big]\, , \\
\text{and } \ {\mathbb{E}}{\mathbf{Z}}_{n,6{\beta}_n}^{(\le1)} & \le {\mathbf{E}}\Big[ \prod_{i=1}^{n} \big( 1+ 6e^6 {\beta}_n {\mathbb{E}}\big[ {\omega}{{\sf 1}}_{\{ {\omega}\le 1/{\beta}_n\}} \big] \big) \Big] \le e^{ 6e^6 n {\beta}_n {\mathbb{E}}[ {\omega}{{\sf 1}}_{\{ {\omega}\le 1/{\beta}_n\}}] } \, .\notag\end{aligned}$$ Therefore, by Markov inequality and Jensen inequality, $$\begin{aligned}
{\mathbb{P}}\Big( \log {\mathbf{Z}}_{n,6{\beta}_n}^{(\le1)} \ge c_0 q^2 \frac{h_n^2}{n} \Big)& \le \frac{1}{c_0 q^2} \frac{n}{h_n^2} \log {\mathbb{E}}{\mathbf{Z}}_{n,6{\beta}_n}^{(\le1)} \le C q^{-2} \frac{n^2 {\beta}_n }{h_n^2} {\mathbb{E}}\big[ {\omega}{{\sf 1}}_{\{ {\omega}\le 1/{\beta}_n\}} \big] \, .
\label{MarkovJensen}\end{aligned}$$ It remains to estimate ${\mathbb{E}}\big[ {\omega}{{\sf 1}}_{\{ {\omega}\le 1/{\beta}_n\}} \big]$. If ${\alpha}>1$ then it is bounded by ${\mathbb{E}}[{\omega}]<+\infty$: this gives the first part of , using also . If ${\alpha}\le1$ then for any ${\delta}>0$, for $n$ large enough we have ${\beta}_n {\mathbb{E}}\big[ {\omega}{{\sf 1}}_{\{ {\omega}\le 1/{\beta}_n\}} \big]\le {\beta}_n^{(1-\eta){\alpha}} $ for $n$ large: by using together with $h_n^2/n \geq 1$, this gives the second part of .
The conclusion of Lemma \[lem:Zmax\] follows by collecting the estimates -- of the three terms in .
Remaining case (${\alpha}\ge 3/2$) {#sec:centeringneeded}
----------------------------------
We now consider the remaining case, *i.e.* when we do not have that $n/m(nh_n) \stackrel{n\to\infty}{\to} 0$. In particular, we need to have that ${\alpha}\ge 3/2$, and hence ${\mathbb{E}}[{\omega}]=:\mu <+\infty$. Then, we do not simply use that ${\mathbf{Z}}_{n,{\beta}_n}^{{\omega}} \ge 1$ to bound ${\mathbf{P}}_{n,{\beta}_n}^{{\omega}} \big( \max_{i\le n} |S_i| \in B_{k,n} \big)$, but instead we use a re-centered partition function $\bar {\mathbf{Z}}_{n,{\beta}_n}^{{\omega}} = e^{- n{\beta}_n \mu}{\mathbf{Z}}_{n,{\beta}_n}^{{\omega}}$, so that we can write $$\begin{aligned}
{\mathbf{P}}_{n,{\beta}_n}^{{\omega}} \big( \max_{i\le n} |S_i| \in B_{k,n} \big)& = \frac{1}{ \bar {\mathbf{Z}}_{n,{\beta}_n}^{{\omega}}} {\mathbf{E}}\Big[ \exp\Big( \sum_{i=1}^n {\beta}_n ({\omega}_{i,s_i} -\mu) \Big) {{\sf 1}}_{\{ \max_{i\le n} |S_i| \in B_{k,n}\}}\Big] \nonumber \\
& =: \frac{1}{ \bar {\mathbf{Z}}_{n,{\beta}_n}^{{\omega}}}\, \bar {\mathbf{Z}}_{n,{\beta}_n}^{{\omega}} \big( \max_{i\le n} |S_i| \in B_{k,n} \big)\, .
\label{P=Zbar}\end{aligned}$$ First, we need to get a lower bound on $\bar {\mathbf{Z}}_{n,{\beta}_n}^{{\omega}}$.
\[lem:barZlowerbound\] For any ${\delta}>0$, there is a constant $c>0$ such that for any positive sequence ${\varepsilon}_n \le 1$ with ${\varepsilon}_n \ge n^{-1/2} (h_n^2/n)^{{\alpha}-3/2+{\delta}}$ (this goes to $0$ for ${\delta}$ small enough), and any $n\ge 1$ $${\mathbb{P}}\Big( \bar {\mathbf{Z}}_{n,{\beta}_n}^{{\omega}} \ge n^{-1} \, e^{ {\varepsilon}_n \frac{ h_n^2}{n} } \Big) \ge 1- e^{- c /{\varepsilon}_n^{{\alpha}-1/2-{\delta}} } - e^{- c {\varepsilon}_n h_n^2/n}.$$
We postpone the proof of this lemma to the end of this subsection, and we now complete the proof of Theorem \[thm:fluctu\]-. Lemma \[lem:barZlowerbound\] gives that $\bar {\mathbf{Z}}_{n,{\beta}_n}^{{\omega}} \ge n^{-1}$ with overwhelming probability: using combined with , we get, analogously to , $$\begin{aligned}
\label{unionboundBkn-bis}
{\mathbb{P}}&\Big( {\mathbf{P}}^\omega_{n,\beta}\big( \max\limits_{i\leq n} \big| S_i\big| \ge A_n h_n \big) \ge n e^{- c_1 A_n^2 h_n^2/n} \Big) \\
&\le {\mathbb{P}}\big( \bar {\mathbf{Z}}_{n,{\beta}_n}^{{\omega}} \le n^{-1} \big) + \sum_{k=\log_2 A_n +1}^{\log_2(n/h_n)+1} {\mathbb{P}}\Big( \bar {\mathbf{Z}}^\omega_{n,\beta}\big( \max\limits_{i\leq n} \big| S_i\big|\in B_{k,n}\big) \ge 4e^{- c_0 2^{2k} h_n^2/n} \Big)\, .\notag\end{aligned}$$ Then, we have a lemma which is the analogous of Lemma \[lem:Zmax\] for $\bar {\mathbf{Z}}_{n,{\beta}_n}^{{\omega}}$.
\[lem:barZmax\] There exist some constant $q_0>0$ and some $\nu>0$, such that for all $q \ge q_0$ we have $${\mathbb{P}}\Big( \bar {\mathbf{Z}}^\omega_{n,2\beta_n}\big( \max\limits_{i\leq n} | S_i | \le qh_n \big) \ge e^{ \frac{1}{4} q^2 \frac{h_n^2 }{n} } \Big) \le q^{-\nu} \, .$$
The proof follows the same lines as for Lemma \[lem:Zmax\]: still holds, with ${\beta}_n {\omega}_{i,S_i}$ replaced by ${\beta}_n ({\omega}_{i,S_i}-\mu)$ (outside of the indicator function). The bounds - for terms 1 and 2 still hold, since one fall back to the same estimates by using that $({\omega}_{i,S_i}-\mu)\le {\omega}_{i,S_i}$. It remains only to control only the third term: we prove that when $\mu:={\mathbb{E}}[{\omega}]<\infty$, then for any ${\delta}>0$, provided that $n$ is large enough, $$\label{aim:part3-bar}
{\mathbb{P}}\Big( \log \bar {\mathbf{Z}}_{n,6{\beta}_n}^{(\le 1)} \ge c_0 q^2 \frac{h_n^2}{n} \Big) \le c q^{-2} \times n^{-1/2} \Big( \frac{h_n^2}{n}\Big)^{{\alpha}-\frac32 +{\delta}} ,$$ where we set analogously to $$\label{def:barZsmall1}
\bar {\mathbf{Z}}_{n,6{\beta}_n}^{(\le 1)} := {\mathbf{E}}\Big[ \exp\Big(\sum_{i=1}^n 6 {\beta}_n ({\omega}_{i,S_i}-\mu) {{\sf 1}}_{\{ {\beta}_n {\omega}_{i,S_i} \le 1\}} \Big)
\Big] \, .$$ Then, using $h_n^2/n\le n$ (if ${\alpha}\ge 3/2$, the upper bound in is bounded by $c q^{-2} n^{{\alpha}-2 +{\delta}}$ which is smaller than $q^{-2}$ provided that ${\delta}$ had been fixed small enough.
To prove , we use that there is a constant $c$ such that $e^{x} \le 1+x+c x^2$ as soon as $|x|\le 6$, so that we get similarly to that $$\begin{aligned}
{\mathbb{E}}{\mathbf{Z}}_{n,6{\beta}_n}^{(\le1)} & \le \Big( 1+ {\beta}_n {\mathbb{E}}\big[ ({\omega}-\mu) {{\sf 1}}_{\{ {\omega}\le 1/{\beta}_n\}} \big] + c{\beta}_n^2 {\mathbb{E}}\big[ ({\omega}-\mu)^2 {{\sf 1}}_{\{ {\omega}\le 1/{\beta}_n\}} \big] \Big)^n \notag \\
&\le \exp\Big( c n L(1/{\beta}_n) {\beta}_n^{{\alpha}} \Big) \le \exp\Big( \frac{c}{h_n} (h_n^2/n)^{{\alpha}+{\delta}} \Big) \, .
\label{EZnsmall-bis}\end{aligned}$$ For the second inequality, we used that ${\mathbb{E}}\big[ ({\omega}-\mu) {{\sf 1}}_{\{ {\omega}\le 1/{\beta}_n\}} \big] \le 0$ (as soon a $1/{\beta}_n \ge \mu$), and also that ${\mathbb{E}}\big[ ({\omega}-\mu)^2 {{\sf 1}}_{\{ {\omega}\le 1/{\beta}_n\}} \big] \le c L(1/{\beta}_n) {\beta}_n^{{\alpha}-2}$, thanks to . The last inequality holds for any fixed ${\delta}$, provided that $n$ is large enough, and comes from using Potter’s bound and the relation to get that $L(1/{\beta}_n) {\beta}_n^{{\alpha}}\le c' {\mathbb{P}}({\omega}>1/{\beta}_n) \le (nh_n)^{-1} (h_n^2/n)^{{\alpha}+{\delta}}$. Then, applying Markov and Jensen inequalities as in , we get that $$\begin{aligned}
{\mathbb{P}}\Big( \log \bar {\mathbf{Z}}_{n,6{\beta}_n}^{(\le 1)} \ge c_0 q^2 \frac{h_n^2}{n} \Big) & \le c q^{-2} \frac{n}{h_n^3} \Big( \frac{h_n^2}{n}\Big)^{{\alpha}+{\delta}} \, ,\end{aligned}$$ which proves .
With Lemma \[lem:barZmax\] in hand, and using Cauchy-Schwarz inequality as in , we get that $${\mathbb{P}}\Big( \bar {\mathbf{Z}}^\omega_{n,\beta}\big( \max\limits_{i\leq n} \big| S_i\big|\in B_{k,n}\big) \ge 2e^{- c_0 2^{2k} h_n^2/n} \Big) \le (2^{k})^{-\nu} \, .$$ Plugged into , this concludes the proof of Theorem \[thm:fluctu\]-. It therefore only remains to prove Lemma \[lem:barZlowerbound\].
We need to obtain a lower bound on $\bar {\mathbf{Z}}_{n,{\beta}_n}$, so we use Cauchy-Schwarz inequality *backwards*: we apply Cauchy Schwarz inequality to $$\begin{aligned}
\bar {\mathbf{Z}}_{n,{\beta}_n/2}^{(>1)}&:={\mathbf{E}}\Big[ \exp\Big( \sum_{i=1}^n \frac{{\beta}_n}{2} ({\omega}_{i,s_i} -\mu) {{\sf 1}}_{\{ {\beta}_n {\omega}_{i,s_i} >1 \}} \Big) \Big] \\
&\le (\bar {\mathbf{Z}}_{n,{\beta}_n} \big)^{1/2} {\mathbf{E}}\Big[ \exp\Big( \sum_{i=1}^n -{\beta}_n ({\omega}_{i,s_i} -\mu) {{\sf 1}}_{\{ {\beta}_n {\omega}_{i,s_i} >1 \}} \Big)\Big]^{1/2}\\
&\hspace{6cm} =:
(\bar {\mathbf{Z}}_{n,{\beta}_n} \big)^{1/2} (\bar {\mathbf{Z}}_{n,-{\beta}_n}^{(\le 1)} \big)^{1/2} ,\end{aligned}$$ so that $$\bar {\mathbf{Z}}_{n,{\beta}_n} \ge \big( \bar {\mathbf{Z}}_{n,{\beta}_n/2}^{(>1)}\big)^2 \Big/ \bar {\mathbf{Z}}_{n,-{\beta}_n}^{(\le 1)}\, .$$ Hence, we get that $$\label{twotermsbarZmax}
{\mathbb{P}}\Big( \bar {\mathbf{Z}}_{n,{\beta}_n}^{{\omega}} \le n^{-1} \, e^{ {\varepsilon}_n \frac{ h_n^2}{n} } \Big) \le {\mathbb{P}}\Big( \bar {\mathbf{Z}}_{n,-{\beta}_n}^{(\le 1)} \ge e^{ {\varepsilon}_n \frac{ h_n^2}{2n} } \Big) + {\mathbb{P}}\Big( \bar {\mathbf{Z}}_{n,{\beta}_n/2}^{(>1)} \le n^{-1/2} \, e^{ {\varepsilon}_n \frac{ h_n^2}{4n} } \Big)\, ,$$ and we deal with both terms separately.
For the first term, we use that analogously to we have $$\label{EZnegativebeta}
\begin{split}
{\mathbb{E}}\bar {\mathbf{Z}}_{n,-{\beta}_n}^{(\le 1)} &\le \Big( 1- {\beta}_n {\mathbb{E}}\big[ ({\omega}-\mu) {{\sf 1}}_{\{ {\omega}\le 1/{\beta}_n\}} \big] + c{\beta}_n^2 {\mathbb{E}}\big[ ({\omega}-\mu)^2 {{\sf 1}}_{\{ {\omega}\le 1/{\beta}_n\}} \big] \Big)^n \\
&\le \Big( 1+ c L(1/{\beta}_n) {\beta}_n^{{\alpha}}\Big)^n \le \exp\Big( \frac{c}{h_n} \big( h_n^2/n \big)^{{\alpha}+{\delta}/2} \Big) \, ,
\end{split}$$ Here, the difference with is that we use for the second inequality that $- {\mathbb{E}}\big[ ({\omega}-\mu) {{\sf 1}}_{\{ {\omega}\le 1/{\beta}_n\}} \big] = {\mathbb{E}}\big[ ({\omega}-\mu) {{\sf 1}}_{\{ {\omega}> 1/{\beta}_n\}}\big] \le c L(1/{\beta}_n) {\beta}_n^{{\alpha}-1}$, thanks to . Again, the second inequality holds for any fixed ${\delta}$, provided that $n$ is large enough. Using Markov’s inequality, one therefore obtains that the first term in is bounded by $$\label{EZnegativebeta2} {\mathbb{P}}\Big( \bar {\mathbf{Z}}_{n,-{\beta}_n}^{(\le 1)} \ge e^{ {\varepsilon}_n \frac{{\varepsilon}_n h_n^2}{2n} } \Big) \le \exp\Big( \frac{c}{h_n} \big( h_n^2/n \big)^{{\alpha}+{\delta}} - {\varepsilon}_n \frac{ h_n^2}{2n} \Big) \le \exp\Big( - {\varepsilon}_n \frac{ h_n^2}{4n} \Big) \, ,$$ the second inequality holding provided that ${\varepsilon}_n$ is larger than $n^{-1/2} \Big( \frac{h_n^2}{n}\Big)^{{\alpha}-\frac32 +{\delta}}$.
As far as the second term in is concerned, we find a lower bound on ${\mathbf{Z}}_{n,{\beta}_n}^{(\ge 1)}$ by restricting to a particular set of trajectories. Consider the set $$\mathcal{O}_n :=\Big\{ (i,x) \in \llbracket n/2,n \rrbracket\times \llbracket {\varepsilon}_n^{1/2} h_n, 2 {\varepsilon}_n^{1/2} h_n \rrbracket ; {\beta}_n {\omega}_{i,x} \ge 2 x^2 /i \Big\} \, .$$ If the set $\mathcal{O}_n$ is non-empty, then pick some $(i_0,x_0)\in \mathcal{O}_n$, and consider trajectories which visit this specific site: since all other weights are non-negative ($({\omega}-\mu){{\sf 1}}_{\{{\beta}_n {\omega}>1\}} \ge 0$ provided $\mu<1/{\beta}_n$), we get that $$\begin{aligned}
\bar {\mathbf{Z}}_{n,{\beta}_n}^{(\ge 1)} & \ge e^{{\beta}_n ({\omega}_{i_0,x_0} -\mu)} {\mathbf{P}}\big( S_{i_0} = x_0 \big) \notag \\
& \ge \frac{c}{\sqrt{n}} \exp\Big( {\beta}_n {\omega}_{i_0,x_0} - \frac{x_0^2}{i_0}\Big) \ge \frac{c}{\sqrt{n}} e^{ {\varepsilon}_n \frac{h_n^2}{n} }\, .
\label{target}\end{aligned}$$ We used Stone’s local limit theorem [@S67] for the second inequality (valid provided that $n$ is large, using also that $i_0\ge n/2$). For the last inequality, we used the definition of $\mathcal{O}_n$ to bound the argument of the exponential by $x_0^2/i_0 \ge {\varepsilon}_n h_n^2/n$. Therefore, we get that $$\begin{aligned}
{\mathbb{P}}\Big( \bar {\mathbf{Z}}_{n,{\beta}_n}^{(\ge 1)} \le \frac{c}{\sqrt{n}} e^{ {\varepsilon}_n \frac{h_n^2}{n} } \Big) & \le {\mathbb{P}}\big( \mathcal{O}_n = \emptyset \big) = \prod_{i=n/2}^n \prod_{x = {\varepsilon}_n^{1/2} h_n}^{2{\varepsilon}_n^{1/2} h_n} \Big( 1- {\mathbb{P}}\big( {\beta}_n{\omega}>2 x^2/i \big) \Big)\\
&\le \Big( 1- {\mathbb{P}}\big( {\omega}> 4 {\varepsilon}_n m(n h_n)\big) \Big)^{{\varepsilon}_n^{1/2} n h_n} \, .\end{aligned}$$ For the second inequality we used that $x^2/i \ge {\varepsilon}_n h_n^2/n$ for the range considered, together with the relation characterizing ${\beta}_n$. Then, we use the definition of $m(nh_n)$ together with Potter’s bound to get that for any fixed ${\delta}>0$, we have ${\mathbb{P}}\big( {\omega}> 4 {\varepsilon}_n m(n h_n)\big) \ge c {\varepsilon}_n^{-{\alpha}+{\delta}} (nh_n)^{-1}$, provided that $n$ is large enough. Therefore, we obtain that $$\label{631}
{\mathbb{P}}\Big( \bar {\mathbf{Z}}_{n,{\beta}_n}^{(\ge 1)} \le \frac{c}{\sqrt{n}} e^{ {\varepsilon}_n \frac{h_n^2}{n} } \Big) \le \exp\Big( - c\, {\varepsilon}_n^{\frac12 -{\alpha}+{\delta}}\Big) \, ,$$ which bounds the second term in .
Regime 2 and regime 3-a {#secProofeq:hscaling}
=======================
In this section we prove Theorem \[thm:alpha>12\] and Theorem \[thm:cas3\]. We decompose the proof in three steps, Step $1$ and Step $2$ being the same for both theorems. For the third step, we give the details in regime 2, and adapt the reasoning to regime 3-a.
Step 1: Reduction of the set of trajectories {#sec:reduction2}
--------------------------------------------
Recalling $\mu={\mathbb{E}}[{\omega}]$ (which is finite for $\alpha>1$), we define $$\label{def:Zbar}
\bar{\mathbf{Z}}_{n,\beta_n}^{\omega}:= {\mathbf{E}}\Big[\exp\Big(\sum_{i=1}^n \beta_n \big({\omega}_{i,S_i}-\mu {{\sf 1}}_{\{\alpha\ge 3/2\}}\big)\Big)\Big]$$ We show that to prove Theorem \[thm:alpha>12\] and Theorem \[thm:cas3\] we can reduce the problem to the random walk trajectories belonging to $\Lambda_{n, A h_n}$ for some $A>0$ (large). For any $A>0$, we define $$\label{defBAset}
{{\ensuremath{\mathcal B}} }_n(A):=\Big \{(i,S_i)_{i=1}^n \colon \max_{i\leq n} |S_i|\le A h_n \Big \}$$ and we let $$\bar {\mathbf{Z}}_{n,\beta_n}^\omega({{\ensuremath{\mathcal B}} }_n(A)):= {\mathbf{E}}\Big[\exp\Big(\sum_{i=1}^n \beta_n \big({\omega}_{i,s_i}-\mu {{\sf 1}}_{\{\alpha\ge 3/2\}}\big)\Big)
{{\sf 1}}_{{{\ensuremath{\mathcal B}} }_n(A)}\Big].$$ Relation gives that $\mathbb P \Big({\mathbf{P}}_{n,\beta_n}^\omega\big({{\ensuremath{\mathcal B}} }_n(A)\big) \ge n e^{-c_1 A^2 h_n^2/n} \Big)\le c_2 A^{-\nu_1}$, uniformly on $n\in \mathbb N$. This implies that $$\label{eq:step1}
{\mathbb{P}}\bigg(\Big|\log \bar {\mathbf{Z}}_{n,\beta_n}^\omega-\log \bar {\mathbf{Z}}_{n,\beta_n}^\omega({{\ensuremath{\mathcal B}} }_n(A))\Big | \ge ne^{-c_1' A^2 h_n^2/n} \bigg) \leq c_2 A^{-\nu_1},$$ uniformly on $n\in\mathbb N$. Let us observe that in [Regime 2 and regime 3-a]{} we have that $h_n^2/n \ge c_{{\beta}}\log n$, therefore $ne^{-c_1' A^2 h_n^2/n}$ goes to $0$ as $n$ gets large, provided $A$ is sufficiently large.
In such a way relation implies $$\label{rel1thm23a}
\lim_{n\to\infty} \frac{n}{h_n^2}\log \bar {\mathbf{Z}}_{n,\beta_n}^\omega=\lim_{A\to\infty}\lim_{n\to\infty}\frac{n}{h_n^2}\log \bar {\mathbf{Z}}_{n,\beta_n}^\omega({{\ensuremath{\mathcal B}} }_n(A)).$$
Step 2: Restriction to large weights {#largeweights2}
------------------------------------
In the second step of the proof we show that we can only consider the partition function ${\mathbf{Z}}_{n,\beta_n}^{{\omega}, (\texttt L)}$ truncated to a finite number $\mathtt{L}$ of large weights, iwth$\texttt L$ independent of $n$. We need some intermediate truncation steps.
We start by removing the small weights. Using the notations introduced in (\[def:ZbigT\] – \[def:Zsmall1\]) and , Hölder’s inequality gives that for any $\eta \in(0,1)$ $$\begin{aligned}
\label{eq:step2-reg23a}
\Big( \bar {\mathbf{Z}}_{n,(1-\eta)\beta_n}^{(>1)} &\Big)^{\frac{1}{1-\eta}} \Big( \bar {\mathbf{Z}}_{n,-(\eta^{-1}-1)\beta_n}^{(\le 1)} \Big)^{ - \frac{\eta}{1-\eta} }
\\
& \le \bar {\mathbf{Z}}_{n,\beta_n}^\omega({{\ensuremath{\mathcal B}} }_n(A)) \le
\Big( \bar {\mathbf{Z}}_{n,(1+\eta)\beta_n}^{(>1)} \Big)^{\frac{1}{1+\eta}} \Big( \bar {\mathbf{Z}}_{n,(1+\eta^{-1})\beta_n}^{(\le 1)} \Big)^{ \frac{\eta}{1+\eta} } \, ,
\notag\end{aligned}$$ We observe that the condition $\beta_n\omega>1$ implies (if $\mu<\infty$) $$\label{barreplacement}
(1-2\eta)\beta_n\omega \le (1-\eta)\beta_n(\omega-\mu)
\ \text{ and }\
(1+\eta)\beta_n(\omega-\mu)\le (1+\eta) \beta_n \omega,$$ provided $n$ is large enough. In such a way, we can safely replace $\bar{\mathbf{Z}}_{n,(1-\eta)\beta_n}^{(>1)}$ by ${\mathbf{Z}}_{n,(1-2\eta)\beta_n}^{(>1)}$ and $\bar{\mathbf{Z}}_{n,(1+\eta)\beta_n}^{(>1)}$ by ${\mathbf{Z}}_{n,(1+\eta)\beta_n}^{(>1)}$ in . The next lemma shows that the contribution given by $\log\bar {\mathbf{Z}}_{n,\rho\beta_n}^{(\le 1)}$ is negligible.
\[lem:ZBless1neg\] Let $\rho\in \mathbb R$. Then, $$\frac{n}{h_n^2}\log\bar {\mathbf{Z}}_{n,\rho\beta_n}^{(\le 1)}\overset{{\mathbb{P}}}{\to} 0, \quad \text{as}\, n\to\infty.$$
The case $\rho>0$ is a consequence of the estimate in and , while the case $\rho<0$ is a consequence of the estimate in and
We can further reduce the partition function ${\mathbf{Z}}_{n,\nu\beta_n}^{(>1)} $ to even (intermediate) larger weights (with $\nu>0$).
We fix some ${\delta}>0$ small, and define $\ell := ( A^2 h_n^2/ n )^{1-{\delta}}$ and also $\mathtt T = A^{1/{\alpha}} \frac{h_n^2}{n} \ell^{-(1-{\delta})^{1/2}/{\alpha}}$ as in : then, Hölder’s inequality gives that for any $\eta \in(0,1)$ $$\log {\mathbf{Z}}_{n,\nu\beta_n}^{(>\mathtt T)} \le \log {\mathbf{Z}}_{n,\nu\beta_n}^{(>1)} \le \frac{1}{1+\eta} \log {\mathbf{Z}}_{n,(1+\eta)\nu\beta_n}^{(>\mathtt T)} + \frac{\eta}{1+\eta} \log {\mathbf{Z}}_{n,(1+\eta^{-1})\nu\beta_n}^{((1,\mathtt T])} \, .$$ Then, gives that for any fixed $A\ge 1$, and since $h_n^2/n \to\infty$, we have that for any $\rho>0$, $$\frac{n}{h_n^2} \log {\mathbf{Z}}_{n,\rho \beta_n}^{((1,\mathtt T])} \overset{{\mathbb{P}}}{\to} 0, \quad \text{as}\, n\to\infty.$$ Finally we show that we can only consider a finite number of large weights. We consider ${\Upsilon}_\ell = \big\{ Y_1^{(n,Ah_n)}, \ldots, Y_{\ell}^{(n,Ah_n)}\big\}$ with $\ell$ chosen above. Using , with probability larger $1- (c\ell)^{-\delta \ell/2}$ (with $\ell\to\infty$ as $n\to\infty$) we have that $$\Xi_\texttt{T}:=
\Big\{ (i,x) \in \llbracket 1,n \rrbracket \times \llbracket -Ah_n ,Ah_n \rrbracket ; {\beta}_n {\omega}_{i,x} > \mathtt T \Big\} \subset {\Upsilon}_\ell$$ and thus $ {\mathbf{Z}}_{n,\nu\beta_n}^{(>\mathtt T)}\le {\mathbf{Z}}_{n,\nu \beta_n}^{(\ell)}$ with high probability. We let $\texttt L\in \mathbb N$ be a fixed (large) constant. Since $|\Xi_\texttt{T} |\to\infty$ as $n\to\infty$ in probability, we have that ${\Upsilon}_{\texttt L}\subset \Xi_\texttt{T}$ so that, ${\mathbf{Z}}_{n,\nu\beta_n}^{( \texttt L)}\le {\mathbf{Z}}_{n,\nu\beta_n}^{(>\mathtt T)}$ for large $n$, with high probability. By using Hölder’s inequality we get, $${\mathbf{Z}}_{n,\nu\beta_n}^{(\texttt L)}\le {\mathbf{Z}}_{n,\nu \beta_n}^{(>\texttt T)}\le \Big({\mathbf{Z}}_{n,\nu(1+\eta) \beta_n}^{(\texttt L)}\Big)^{\frac{1}{1+\eta}} \Big({\mathbf{Z}}_{n,\nu(1+\eta^{-1}) \beta_n}^{(\texttt L, \ell)}\Big)^{\frac{\eta}{1+\eta}},$$ where $${\mathbf{Z}}_{n, \beta_n}^{(\texttt L, \ell)}:={\mathbf{E}}\Big[ \exp\Big(\sum_{i=\texttt L+1}^{\ell} {\beta}_n M_i^{(n,qh_n)} {{\sf 1}}_{\{Y_i^{(n,q h_n)} \in S\}}\Big) \Big] .$$
We now show that the contribution of ${\mathbf{Z}}_{n,\nu(1+\eta^{-1}) \beta_n}^{(\texttt L, \ell)}$ is negligible.
For any ${\varepsilon}\in (0,1)$ and for any $\mathtt L\in \mathbb N$ and $\rho>0$ there exists $\delta_{\mathtt L}$ such that for all $n$ $${\mathbb{P}}\Big(\frac{n}{h_n^2}\log {\mathbf{Z}}_{n,\rho \beta_n}^{(\mathtt L, \ell)}>{\varepsilon}\Big)\leq \delta_{\mathtt L} ,$$ with ${\delta}_{\mathtt L} \to 0$ as $\mathtt L \to\infty$.
We let $\rho>0$. Recalling the definition , and using that ${\mathbf{P}}(\Delta \subset S)\le e^{{\mathrm{Ent}}(\Delta)}$, we have that $$\begin{aligned}
{\mathbf{Z}}_{n,\rho \beta_n}^{(\texttt L, \ell)} &\le \sum_{{\Delta}\subset {\Upsilon}_\ell} e^{ \rho {\beta}_n \Omega_{n,qh_n}^{(>\texttt L)}(\Delta) } {\mathbf{P}}\big( S \cap {\Upsilon}_\ell =\Delta \big) \\
& \le \sum_{{\Delta}\subset {\Upsilon}_\ell} \exp\Big( \rho{\beta}_n \Omega_{n,qh_n}^{(>\texttt L)}(\Delta) - {\mathrm{Ent}}(\Delta) \Big) \le 2^{\ell} \exp \Big( T_{n,Ah_n}^{\rho {\beta}_n,(>\texttt L)}\Big) \, .\end{aligned}$$ Using that $\ell=o(h^2/n)$ and relation (5.5) of [@cf:BT_ELPP], we conclude the proof.
Collecting the above estimates, we can conclude that $$\label{finallimit3}
\lim_{n\to\infty}\frac{n}{h_n^2} \log \bar {\mathbf{Z}}_{n,\beta_n}^\omega({{\ensuremath{\mathcal B}} }_n(A))
=\lim_{\nu\to 1}\lim_{\texttt L\to\infty}\lim_{n\to\infty}\frac{n}{h_n^2}\log {\mathbf{Z}}_{n,\nu\beta_n}^{(\texttt L)} \, .$$
Step 3: Regime 2. Convergence of the main term {#sec:2-Step3}
----------------------------------------------
It remains to show the convergence of the partition function restricted to the large weights.
\[prop:convBZbig1\] For any $\nu>0$, and $\mathtt L>0$ $$\label{convBZbig1}
\frac{n}{h_n^2}\log {\mathbf{Z}}_{n,\nu \beta_n}^{(\mathtt{L})} \stackrel{({\textrm{d}})}{\to}
\begin{cases}
{{\ensuremath{\mathcal T}} }_{\nu,A}^{(\mathtt L)} &\quad \text{in Regime 2}, \\
{\widetilde}{{\ensuremath{\mathcal T}} }_{{\beta},\nu,A}^{(\mathtt L)}&\quad \text{in Regime 3-a},
\end{cases}$$ where ${{\ensuremath{\mathcal T}} }_{{\beta},A}^{(\mathtt L)}$ was introduced in and ${\widetilde}{{\ensuremath{\mathcal T}} }_{\beta,\nu,A}^{(\mathtt L)}$ is defined in below.
One readily verifies that
$\ast$ $\nu\mapsto {{\ensuremath{\mathcal T}} }_{\nu,A}^{(\mathtt L)}$ (resp. $\nu\mapsto {\widetilde}{{\ensuremath{\mathcal T}} }_{{\beta},\nu,A}^{(\mathtt L)}$) is a continuous function;
$\ast$ ${{\ensuremath{\mathcal T}} }_{1,A}^{(\mathtt L)}\to {{\ensuremath{\mathcal T}} }_{1,A}$ (resp. ${\widetilde}{{\ensuremath{\mathcal T}} }_{{\beta},1,A}^{(\mathtt L)}\to {\widetilde}{{\ensuremath{\mathcal T}} }_{{\beta}, 1,A}$) as $\texttt L\to\infty$ (see Proposition \[prop:ConvVP\], resp. Proposition \[prop:ConvVPtilde\]);
$\ast$ ${{\ensuremath{\mathcal T}} }_{1,A}\to {{\ensuremath{\mathcal T}} }_{1}$ (resp. ${\widetilde}{{\ensuremath{\mathcal T}} }_{{\beta},1,A}\to {\widetilde}{{\ensuremath{\mathcal T}} }_{{\beta}}$) as $A\to\infty$ (see Proposition \[prop:ConvVP\], resp. Proposition \[prop:ConvVPtilde\]).
Therefore, the proof of Theorem \[thm:alpha>12\] and Theorem \[thm:cas3\] is a consequence of relations , and .
We detail the proof for the Regime 2. The Regime 3-a follows similarly using the results in Section \[complement3a\] below. To keep the notation lighter we let $\nu=1$.
*Lower bound.* For any ${\texttt L}\in \mathbb N$ we consider a set $\Delta_{\texttt L}\subset {\Upsilon}_{\texttt L}$ which achieves the maximum of $T_{n, Ah_n}^{\beta_n, ({\texttt L})} $, resp. of ${\widetilde}T_{n,Ah_n}^{{\beta}_n,({\texttt L})}$ defined below in for Regime 3-a. We have $${\mathbf{Z}}_{n,\beta_n}^{({\texttt L})}\ge \exp\Big(\beta_n\Omega_{n, A h_n}(\Delta_{\mathtt L})\Big)\, {\mathbf{P}}\big( S \cap {\Upsilon}_{\texttt L}=\Delta_{\mathtt L} \big) \, .$$ Since ${\texttt L}$ is fixed, we realize that any pair of points $(i,x),(j,y)\in {\Upsilon}_{\texttt L}$ satisfies the condition $|i-j|\ge {\varepsilon}n$ and $|x-y|\ge{\varepsilon}h_n$ with probability at least $1-c_{\varepsilon}$ with $c_{\varepsilon}\to 0$ as ${\varepsilon}\to 0$. In such a way, we can use the Stone local limit theorem [@S67] to get that ${\mathbf{P}}(S \cap {\Upsilon}_{\texttt L}=\Delta_{\mathtt L} ) = n^{-\frac{|\Delta_{\mathtt L}|}{2} +o(1)}e^{-{\mathrm{Ent}}(\Delta_{\mathtt L})}$. In the Regime 2, in which ${\mathrm{Ent}}(\Delta_{\mathtt L}) \asymp h_n^2/n \gg \log n$, this implies that $$\label{end-lowerbound}
{\mathbf{Z}}_{n,\beta_n}^{({\texttt L})}
\ge
\exp\Big((1+o(1))T_{n,Ah_n}^{{\beta}_n,({\texttt L})}\Big).$$ To conclude, we use Proposition \[prop:ConvVP\]- to obtain that $T_{n,Ah_n}^{{\beta}_n,({\texttt L})}$ converges in distribution to $ {{\ensuremath{\mathcal T}} }_{1 ,A}^{(\texttt L)}$, concluding the lower bound.
In Regime 3-a, is replaced by $${\mathbf{Z}}_{n,\beta_n}^{({\texttt L})}
\ge
\exp\Big( (1+o(1)) \Big\{ \beta_n\Omega_{n, A h_n}(\Delta_{\mathtt L}) - {\mathrm{Ent}}(\Delta_{\mathtt L}) - \frac{|\Delta_{\mathtt L}|}{2} \log n \Big\} \Big),$$ so that $T_{n,Ah_n}^{{\beta}_n,({\texttt L})}$ is replaced by ${\widetilde}T_{n,Ah_n}^{{\beta}_n,({\texttt L})}$ defined in . Then the conclusion follows by Proposition \[prop:ConvVPtilde\]- below.
*Upper bound.* We have $$\begin{aligned}
{\mathbf{Z}}_{n,\beta_n}^{(\texttt L)} &= \sum_{{\Delta}\subset {\Upsilon}_{\texttt L}} e^{ {\beta}_n \Omega_{n,qh_n}^{(\texttt L)}(\Delta) } {\mathbf{P}}\big( S \cap {\Upsilon}_{\texttt L} =\Delta \big)
$$ Using the Stone local limit theorem [@S67] we have that ${\mathbf{P}}(S \cap {\Upsilon}_{\texttt L}=\Delta ) = n^{-\frac{|\Delta|}{2} +o(1)} e^{-{\mathrm{Ent}}(\Delta)}$ uniformly for all $\Delta \subset {\Upsilon}_{\mathtt L}$. Since we have only a finite number of sets, we obtain that $$\label{end-upperbound}
{\mathbf{Z}}_{n,\beta_n}^{(\texttt L)}\le 2^{\texttt L} \exp \Big( (1+o(1)) T_{n,Ah_n}^{{\beta}_n,(\texttt L)}\Big),$$ which concludes the proof of the upper bound, again thanks to the convergence proven in Proposition \[prop:ConvVP\]-. In Regime 3-a, using the Stone local limit theorem, we can safely replace $T_{n,Ah_n}^{{\beta}_n,(\texttt L)}$ by ${\widetilde}T_{n,Ah_n}^{{\beta}_n,(\texttt L)}$ defined below in , and also conclude thanks to Proposition \[prop:ConvVPtilde\]-.
Step 3: Regime 3.a. Complements for the convergence of the main term {#complement3a}
--------------------------------------------------------------------
We end here the proof of Theorem \[thm:cas3\] by stating the results needed to complete Step 3 above in the case of regime 3.a. In analogy with , and in view of the local limit theorem , we define $$\label{def:discreteELPPtilde}
\begin{split}
&{\widetilde}T_{n,h}^{{\beta}_{n,h}} := \max_{ \Delta \subset \Lambda_{n,h}} \big\{ {\beta}_{n,h} \Omega_{n,h} (\Delta) - {\mathrm{Ent}}(\Delta) - \frac{|\Delta|}{2}\log n \big\} \, ,\\
& {\widetilde}T_{n,h}^{{\beta}_{n,h}, (\ell)} := \max_{ \Delta \subset \Lambda_{n,h}} \big\{ {\beta}_{n,h} \Omega_{n,h}^{(\ell)} (\Delta) - {\mathrm{Ent}}(\Delta) - \frac{|\Delta|}{2}\log n \big\} \,
\end{split}$$ In the next result we state the convergence of $\frac{n}{h^2}{\widetilde}T_{n,h}^{{\beta}_{n,h}}$ and $\frac{n}{h^2}{\widetilde}T_{n,h}^{{\beta}_{n,h},(\ell)}$, analogously to Proposition \[prop:ConvVP\].
\[prop:ConvVPtilde\] Suppose that $ \frac{n}{h^2}{\beta}_{n,h} m(nh) \to \nu \in (0,\infty)$ as $n,h\to\infty$ and $h\sim \beta^{1/2}\sqrt{\log n}$, with $\beta>0$. Then, for every ${\alpha}\in (1/2,2)$ and for any $q>0,\, \ell\in \mathbb N$ we have the following convergence in distribution, as $n\to\infty$ $$\label{def:tildeTA}
\frac{n}{h^2}\, {\widetilde}T_{n,qh}^{\beta_{n,h}} \stackrel{\rm (d)}\longrightarrow {\widetilde}{{\ensuremath{\mathcal T}} }_{{\beta},\nu,q}:= \sup_{s\in{\mathscr{M}}_q}\Big\{\nu \pi(s)-{\mathrm{Ent}}(s)-\frac{N(s)}{2\beta} \Big\} \, ,$$ with ${\mathscr{M}}_q$ as defined in Proposition \[prop:ConvVP\]. We also have, as $n\to\infty$ $$\label{def:tildeTAL}
\frac{n}{h^2}\, {\widetilde}T_{n,qh}^{\beta_{n,h}, (\ell)} \stackrel{\rm (d)}\longrightarrow {\widetilde}{{\ensuremath{\mathcal T}} }_{{\beta},\nu,q}^{(\ell)} :=\sup_{s\in{\mathscr{M}}_q}\Big\{\nu \pi^{(\ell)}(s)-{\mathrm{Ent}}(s)-\frac{N(s)}{2\beta} \Big\}\, .$$ Moreover, we have ${\widetilde}{{\ensuremath{\mathcal T}} }_{{\beta},\nu,q}^{(\ell)}\overset{({\textrm{d}})}{\to} {\widetilde}{{\ensuremath{\mathcal T}} }_{{\beta},\nu,q}$ as $\ell\to\infty$, and ${\widetilde}{{\ensuremath{\mathcal T}} }_{{\beta},\nu,q}\overset{({\textrm{d}})}{\to} {\widetilde}{{\ensuremath{\mathcal T}} }_{{\beta},\nu}$ as $q\to\infty$.
The proof is identical to the proof of Proposition \[prop:ConvVP\] (cf. proof of [@cf:BT_ELPP Theorem 2.7], using also that $\frac{n}{h_n^2}\log n \to \frac{1}{{\beta}}$ in regime 3), for this reason it is omitted. To conclude, let us show that ${\widetilde}{{\ensuremath{\mathcal T}} }_\beta^{(\ge r)}$ defined in is well defined.
\[propTtilde\] For any $r\ge 0$ the quantities ${\widetilde}{{\ensuremath{\mathcal T}} }_\beta^{(\ge r)}$ are well defined and for any ${\beta}> 0$ $$\label{sandwichTgb}
-\frac{1}{2{\beta}} < {\widetilde}{{\ensuremath{\mathcal T}} }_\beta^{(\ge 1)}\le {\widetilde}{{\ensuremath{\mathcal T}} }_\beta <\infty.$$ Moreover ${\widetilde}{{\ensuremath{\mathcal T}} }_{{\beta}} \ge 0$, and we have ${\widetilde}{{\ensuremath{\mathcal T}} }_{\beta}>0$ if and only if ${\widetilde}{{\ensuremath{\mathcal T}} }_{\beta}^{(\ge 1)} >0$. Finally, there is a critical value ${\beta}_c = \inf\{ {\beta}\colon {\widetilde}{{\ensuremath{\mathcal T}} }_{\beta}>0 \} \in (0,\infty).$
Since ${\widetilde}{{\ensuremath{\mathcal T}} }_{{\beta}}^{(0)} =0$, we obtain that ${\widetilde}{{\ensuremath{\mathcal T}} }_{{\beta}} \in [0,\infty)$. As a by-product we also have that ${\widetilde}{{\ensuremath{\mathcal T}} }_{{\beta}}>0 $ if and only if $ {\widetilde}{{\ensuremath{\mathcal T}} }_{{\beta}}^{(\ge 1)}>0$; and in that case ${\widetilde}{{\ensuremath{\mathcal T}} }_{{\beta}} ={\widetilde}{{\ensuremath{\mathcal T}} }_{{\beta}}^{(\ge 1)}$. Additionally, we have $$W_\beta-\frac{1}{2\beta}\le {\widetilde}{{\ensuremath{\mathcal T}} }_\beta^{(\ge 1)}\le {\widetilde}{{\ensuremath{\mathcal T}} }_\beta\le \Big( {{\ensuremath{\mathcal T}} }_1 -\frac{1}{2\beta} \Big)\vee 0 ,$$ with $W_\beta$ and ${{\ensuremath{\mathcal T}} }_1$ defined in and respectively. Proposition \[prop:W\] and Theorem \[thm:TbhatTb\] ensure that for ${\beta}>0$, $W_\beta \in (0,\infty)$ and ${\widetilde}{{\ensuremath{\mathcal T}} }_{1} <\infty$, showing .
It remains to show that ${\beta}_c \in (0,\infty)$, by observing that $\beta\mapsto \beta W_\beta$ and ${\beta}\mapsto ({\beta}{{\ensuremath{\mathcal T}} }_1 -1/2 )\vee 0 $ are monotone functions which converge to $0$ as $\beta\to 0$.
Regime 3-b and regime 4 {#sec:3b4}
=======================
In this section we prove Theorem \[thm:cas3bis\] and Theorem \[thm:cas4\]. We decompose the proof in three steps (analogously to what is done in Section \[secProofeq:hscaling\]), Step $1$ and Step $2$ being the same for both regimes 3-b and 2. For the third step, we separate regime 3-b and regime 4, which have different behaviors. Note that in both regimes there is a constant $c_{{\beta}}>0$ such that $h_n \le c \sqrt{ n\log n}$ (in regime 4, we have $h_n \ll \sqrt{n\log n}$).
Let us define here, analogously to , the recntered partition function $$\label{def:Zbar2}
\bar {\mathbf{Z}}_{n,{\beta}_n}^{{\omega}} := {\mathbf{E}}\Big[ \exp\Big( \sum_{i=1}^n {\beta}_n \big( {\omega}_{i,s_i} - {\mathbb{E}}[{\omega}{{\sf 1}}_{{\omega}\le 1/{\beta}_n}] {{\sf 1}}_{\{{\alpha}\ge 1\}} \big) \Big] \, .$$ Then, roughly speaking, we show that $\log \bar {\mathbf{Z}}_{n,{\beta}_n}^{{\omega}}$ is of order $n^{-1/2}\exp( X h_n^2/n) $, with $X = {\widetilde}{{\ensuremath{\mathcal T}} }_{{\beta}}^{(\ge 1)} +\frac{1}{2{\beta}}$ in the regime 3-b (where $h_n^2/n \sim {\beta}\log n$), and with $X=W_1$ in regime 4. In all cases, we will have $\log \bar {\mathbf{Z}}^{{\omega}}_{n,{\beta}_n} =o(1)$ (recall that in regime 3-b, ${\widetilde}{{\ensuremath{\mathcal T}} }_{{\beta}}^{(\ge 1)}<0$).
Step 1. Reduction of the set of trajectories {#sec:4-step1}
--------------------------------------------
We proceed as for Step 1 in Section \[secProofeq:hscaling\]: for any $A>0$ (fixed large in a moment), we define $$\label{def:An}
{{\ensuremath{\mathcal A}} }_n := \Big\{ (i,S_i) \, : \, \max_{i\le n} |S_i| \le A \sqrt{ n \log n}\Big\} \, .$$ Then, we let $\bar{\mathbf{Z}}_{n,{\beta}_n}^{{\omega}}({{\ensuremath{\mathcal A}} }_n) $ be the (normalized) partition function restricted to trajectories in ${{\ensuremath{\mathcal A}} }_n$. Relation gives that, analogously to $${\mathbb{P}}\bigg(\Big|\log \bar {\mathbf{Z}}_{n,\beta_n}^\omega-\log \bar {\mathbf{Z}}_{n,\beta_n}^\omega({{\ensuremath{\mathcal A}} }_n)\Big | \ge n e^{-c_1 A^2 \log n } \bigg) \leq c_2 A^{-\nu_1}\, .$$ Hence, we fix $A$ large enough so that $e^{- c_0 A^2 \log n} \le n^{-3}$. This shows that with high probability $\log \bar {\mathbf{Z}}_{n,\beta_n}^\omega = \log \bar {\mathbf{Z}}_{n,\beta_n}^\omega({{\ensuremath{\mathcal A}} }_n) + O(n^{-2})$. In such a way, in the following we can safely focus only on the partition function with trajectories restricted to ${{\ensuremath{\mathcal A}} }_n$.
Step 2. Restriction to large weights {#largeweights3}
------------------------------------
We now fix $\eta \in(0,1)$, small. The same Hölder inequalities as in hold for ${\mathbf{Z}}_{n,\beta_n}^\omega({{\ensuremath{\mathcal A}} }_n) $, so that we can write, with similar notations as in - (the restriction to trajectories in ${{\ensuremath{\mathcal A}} }_n$ does not appear in the notations) $$\begin{aligned}
\label{Holder:reg34}
\log \bar {\mathbf{Z}}_{n,\beta_n}^\omega({{\ensuremath{\mathcal A}} }_n)\, \left\{
\begin{aligned}
&\le \frac{1}{1+\eta} \log {\mathbf{Z}}_{n,(1+\eta)\beta_n}^{(>1)} + \frac{\eta}{1+\eta} \log \bar {\mathbf{Z}}_{n,(1+\eta^{-1})\beta_n}^{(\le1)} \,
, \\
&\ge \frac{1}{1-\eta} \log {\mathbf{Z}}_{n,(1-2\eta)\beta_n}^{(>1)} - \frac{\eta}{1-\eta} \log \bar {\mathbf{Z}}_{n,-(\eta^{-1}-1)\beta_n}^{(\le1)} \, .
\end{aligned}
\right. \end{aligned}$$ We used also to be able to bound below $ \bar {\mathbf{Z}}_{n,(1-\eta)\beta_n}^{(>1)}$ by $ {\mathbf{Z}}_{n,(1-2\eta)\beta_n}^{(>1)}$ (using that ${\beta}_n {\mathbb{E}}[{\omega}{{\sf 1}}_{\{{\omega}\le 1/{\beta}_n\}}] \ll 1$ when ${\alpha}\ge 1$). Then, we need to get a more precise statement than Lemma \[lem:ZBless1neg\] to deal with $\bar {\mathbf{Z}}_{n,\rho {\beta}_n}^{(\le 1)}$.
\[lem:logbarZ\] For any $\rho \in {\mathbb{R}}$, $$\Big( \frac{h_n^2}{n} \Big)^{-3{\alpha}} \sqrt{n}\log \bar {\mathbf{Z}}_{n,\rho {\beta}_n}^{(\le 1)} \stackrel{{\mathbb{P}}}{ \to} 0 \, , \qquad \text{as } n\to\infty \, .$$
We will simply control the first moment of $\bar {\mathbf{Z}}_{n,\rho {\beta}_n}^{(\le 1)} -1$. The idea is similar to that used to obtain and . We divide the proof into two cases: when ${\alpha}<1$ so that there is no renormalization necessary in , and when ${\alpha}\in [1,2)$.
Let us start with the case ${\alpha}<1$: using that $ |\rho| {\beta}_n {\omega}_{i,S_i} \le |\rho|$ on the event $\{{\beta}_n {\omega}_{i,S_i} \le 1\}$, we get that there exists a constant $c_{\rho}$ such that $$\label{develop:exp}
e^{ \sum_{i=1}^n \rho {\beta}_n {\omega}_{i,S_i} {{\sf 1}}_{\{{\beta}_n {\omega}_{i,S_i} \le 1\}} } \le \prod_{i=1}^n \big(1+ c_{\rho} {\beta}_n {\omega}_{i,S_i} {{\sf 1}}_{\{{\beta}_n {\omega}_{i,S_i} \le 1\}} \big) \, .$$ By independence, and since ${\mathbb{P}}({\omega}>t )$ is regularly varying, we get that for $n$ sufficiently large $$\begin{aligned}
{\mathbb{E}}[ {\beta}_n {\omega}_{i,x} {{\sf 1}}_{\{{\beta}_n {\omega}_{i,x} \le 1\}} ] &\le \int_{0}^{1/{\beta}_n} {\beta}_n {\mathbb{P}}({\omega}>t) dt \le c \, L(1 /{\beta}_n ) {\beta}_n^{{\alpha}} \nonumber \\
& \le c {\mathbb{P}}\big( {\omega}>1/{\beta}_n \big) \le \frac{c'}{ n h_n } \Big( \frac{h_n^2}{n}\Big)^{2{\alpha}} \, .
\label{truncmean}\end{aligned}$$ For the last inequality we used Potter’s bound, and the definition of ${\beta}_n$, *i.e.* the fact that ${\beta}_n \sim \frac{h_n^2}{n} m(n h_n)$. Therefore, in view of and using that $h_n\ge \sqrt{n}$, we get that for $n$ sufficiently large (how large depends on $\rho$) $$\label{eq:EZ-1}
{\mathbb{E}}\big[\bar {\mathbf{Z}}_{n,\rho {\beta}_n}^{(\le 1)} -1 \big] \le \Big(1+ c'_{\rho} \frac{\big(h_n^2/n\big)^{2{\alpha}} }{ n^{3/2} } \Big)^n -1 \le 2 c'_{\rho} n^{-1/2} \Big( \frac{h_n^2}{n} \Big)^{2{\alpha}}\, .$$ This concludes the proof in the case ${\alpha}<1$ by using Markov’s inequality, since $h_n^2/n \to +\infty$.
In the case ${\alpha}\in[1,2)$, we use the expansion $e^x \le 1+x+ c_{\rho} x^2$ for all $|x|\le 2 |\rho|$, to get analogously to , and setting $\mu_n:= {\mathbb{E}}[{\omega}{{\sf 1}}_{\{{\omega}\le 1/{\beta}_n\}}] \ll 1/{\beta}_n$, $$\begin{aligned}
{\mathbb{E}}\Big[\bar {\mathbf{Z}}_{n,\rho{\beta}_n}^{(\le1)}\Big] & \le \Big( 1+ \rho {\beta}_n {\mathbb{E}}\big[ ({\omega}-\mu_n) {{\sf 1}}_{\{ {\omega}\le 1/{\beta}_n\}} \big] + c_{\rho}{\beta}_n^2 {\mathbb{E}}\big[ ({\omega}-\mu_n)^2 {{\sf 1}}_{\{ {\omega}\le 1/{\beta}_n\}} \big] \Big)^n \\
&\le \exp\Big( c\, n {\mathbb{P}}({\omega}>1/{\beta}_n) \Big) \le 1+ c n^{-1/2} \Big( \frac{h_n^2}{n} \Big)^{2{\alpha}}\, ,\end{aligned}$$ obtaining the same upper bound as in . To obtain the above inequality, we used that $$\begin{aligned}
&{\mathbb{E}}[ ({\omega}-\mu_n) {{\sf 1}}_{\{ {\omega}\le 1/{\beta}_n\}} ] = \mu_n {\mathbb{P}}({\omega}>1/{\beta}_n) \le {\beta}_n^{-1} {\mathbb{P}}({\omega}>1/{\beta}_n) \, ,\\
& {\mathbb{E}}[ ({\omega}-\mu_n)^2 {{\sf 1}}_{\{ {\omega}\le 1/{\beta}_n\}} ] \le {\mathbb{E}}[ {\omega}^2 {{\sf 1}}_{\{ {\omega}\le 1/{\beta}_n\}} ] \le c L(1/{\beta}_n) {\beta}_n^{{\alpha}-2} \, ,\end{aligned}$$ where the last inequality follows similarly to . One concludes that also holds when ${\alpha}\ge 1$, and the lemma follows by Markov’s inequality.
Therefore, in view of and Lemma \[lem:logbarZ\], we have that for both regimes 3-b and 4 $$\label{eq:finallimit3b}
\lim_{n\to\infty} \frac{n}{h_n^2}\log \Big( \sqrt{n} \log \bar {\mathbf{Z}}_{n,\beta_n}^\omega({{\ensuremath{\mathcal A}} }_n) \Big) = \lim_{\nu\to 1} \lim_{n\to\infty} \frac{n}{h_n^2}\log \Big(\sqrt{n} \log {\mathbf{Z}}_{n,\nu\beta_n}^{(>1)} \Big)\, .$$ Note that in the case of regime 3-b, $h_n^2/n \sim {\beta}\log n$, so the limit is that of $$\frac{1}{{\beta}\log n} \log \Big( \log {\mathbf{Z}}_{n,\nu\beta_n}^{(>1)} \Big) + \frac{1}{2{\beta}} \, .$$ For simplicity of notations, we will consider only the case $\nu=1$ in the following.
Step 3. Reduction of the main term {#reductionlogZ}
----------------------------------
In both regimes 3-b and 4, we show that $\log {\mathbf{Z}}_{n,{\beta}_n}^{(>1)}$ goes to $0$, and we identify at which rate: to do so, it is equivalent to identify the rate at which ${\mathbf{Z}}_{n,{\beta}_n}^{(>1)}-1$ goes to $0$. The behavior for regimes 3-b and 4 are different, since the main contribution to ${\mathbf{Z}}_{n,{\beta}_n}^{(>1)}-1$ may come from several large weights in whereas it comes from a single large weight in regime 4, as it will be reflected in the proof.
Let us define $\ell =\ell({\omega})$ the number of $(i,x) \in \Lambda_{n,A_n}=\llbracket 1,n \rrbracket \times \llbracket -A_n , A_n\rrbracket$ (with the notation $A_n =A\sqrt{n\log n}$ for simplicity) such that ${\beta}_n {\omega}_{i,x} \ge 1 $, and let us denote $$\begin{aligned}
\label{def:gUell}
\Big\{ (i,x) \in \Lambda_{n,A_n} & ; {\beta}_n {\omega}_{i,x} \ge 1 \Big\} = {\Upsilon}_\ell := \big\{ Y_1^{(n,A_n)}, \ldots, Y_{\ell}^{(n,A_n)}\big\} \, ,\end{aligned}$$ with $Y_i^{(n,A_n)}$ the ordered statistic, as in Section \[sec:3\]. We have that $$\label{Eell}
{\mathbb{E}}[\ell] = \sum_{(i,x)\in \Lambda_{n,A_n} } {\mathbb{P}}({\beta}_n {\omega}_{i,x} \ge 1) \le 2A n^{3/2} \sqrt{\log n} \Big(\frac{h_n^2}{n} \Big)^{2{\alpha}} \frac{1}{n h_n} \,,$$ where we used that $ {\mathbb{P}}({\omega}\ge 1/{\beta}_n) \le (h_n^2/n)^{2{\alpha}} (nh_n)^{-1}$ for $n$ large enough, thanks to and Potter’s bound. Since $h_n^2/n \le c \log n$, $h_n \gg \sqrt{n}$, implies that $\ell\le (\log n)^{3{\alpha}}$ with probability going to $1$ (we also used that $\frac12+2{\alpha}<3{\alpha}$).
Hence, decomposing ${\mathbf{Z}}_{n,{\beta}_n}^{(>1)}$ according to the number of sites in ${\Upsilon}_{\ell}$ visited, we can write for any fixed $k_0>0$, $$\begin{aligned}
\label{eq:lubZ3a}
& \sum_{k=1}^{k_0} {{\ensuremath{\mathbf U}} }_k \le {\mathbf{Z}}_{n,{\beta}_n}^{(>1)}-1 =\sum_{k=1}^{\ell} {{\ensuremath{\mathbf U}} }_k\, , \\
\text{with }\ & {{\ensuremath{\mathbf U}} }_k := \sum_{\Delta \subset {\Upsilon}_\ell , |\Delta|=k} e^{{\beta}_n\Omega_{n, A_n} (\Delta)} {\mathbf{P}}\big( S\cap {\Upsilon}_\ell = \Delta \big) \, .\notag\end{aligned}$$ In regime 3-b, the main contribution comes from one of the ${{\ensuremath{\mathbf U}} }_k$’s for some $k\ge 1$, whereas in regime 4 only the term ${{\ensuremath{\mathbf U}} }_1$ will contribute.
Let us now show that, with high probability, we can replace the upper bound in by considering only a finite number of terms. For this purpose, notice that $\ell\le (\log n)^{3{\alpha}}$ and $\min\{ |i-j| , (i,x) \neq (j,y) \in {\Upsilon}_\ell \} \ge n/ (\log n)^{10{\alpha}}$ with probability going to $1$. Then, we can use the Stone local limit theorem [@S67] to have that for any $\Delta\subset {\Upsilon}_\ell$ $${\mathbf{P}}\big( S\cap {\Upsilon}_\ell = \Delta \big)\le c n^{-(\frac12-\eta) |\Delta|}e^{-{\mathrm{Ent}}(\Delta)}\, ,$$ where $\eta>0$ is independent of $\Delta$ and can be chosen arbitrary small (by changing the value of the constant $c$).
As a consequence, using that $\binom{\ell}{k}\le \ell^k$ and $\ell\le (\log n)^{3{\alpha}}$, we have for any $1\le k_1\le \ell$ $$\begin{aligned}
\label{eq:morethenk1-3a}
\sum_{k=k_1}^{\ell} {{\ensuremath{\mathbf U}} }_k &=\sum_{k=k_1}^\ell\sum_{\Delta \subset {\Upsilon}_\ell , |\Delta|=k} e^{{\beta}_n\Omega_{n, A_n} (\Delta)} {\mathbf{P}}\big( S\cap {\Upsilon}_\ell = \Delta \big) \\\nonumber
&\le e^{T_{n,A_n}^{\beta_n}} \sum_{k=k_1}^\ell \ell^k \, n^{-k(\frac12-\eta)} \le c\, e^{T_{n,A_n}^{\beta_n}} \, n^{-k_1(\frac12-\eta')}.\end{aligned}$$ Recalling Proposition \[prop:ConvVP\] (and the fact that $h_n^2/n \le c \log n$) we have that $T_{n,A_n}^{\beta_n}\le C\log n$ with probability going to $1$ as $C\to\infty$. Therefore, we obtain that is $O(n^{-2})$ with probability close to $1$, provided that $k_1$ is sufficiently large – this will turn out to be negligible, see Lemma \[lem:cas3-mainterm\]. Hence, we have shown that with probability close to $1$, we can keep a finite number of terms in .
This can actually be improved in regime 4, where we can keep only one term: indeed, since in that case $h_n^2/n = o(\log n) $, we get that for any fixed $\gamma>0$, $T_{n,A_n}^{\beta_n}\le \gamma \log n$ with probability going to one. Hence, we get that in regime 4, we can take $k_1=2$ in and obtain that $\sum_{k=2}^\ell {{\ensuremath{\mathbf U}} }_k = O(n^{-3/4})$ with probability close to $1$, which will turn out to be negligible, see Lemma \[lem:cas4-mainterm\].
It remains to show the following lemmas, proving the convergence of the main term in regimes 3-b and 4.
\[lem:cas3-mainterm\] In regime 3 (recall $h_n^2/n \sim {\beta}\log n$), for any $K>0$ we have that $$\label{eq:convcas3}
\frac{n}{h_n^2} \log \Big(
\sum_{k = 1}^{K}
{{\ensuremath{\mathbf U}} }_k \Big) \stackrel{({\rm d})}{\longrightarrow} \sup_{1\le k\le K} {\widetilde}{{\ensuremath{\mathcal T}} }_{{\beta},A}^{(k)}\, ,$$ where ${\widetilde}{{\ensuremath{\mathcal T}} }_{{\beta},A}^{(k)} := \sup_{s \in {\mathscr{M}}_A , N(s)=k} \big\{ \pi(s) - {\mathrm{Ent}}(s) - \frac{k}{2{\beta}}\big\}\, , $ with ${\mathscr{M}}_A$ defined below .
Note that we have $ \sup_{k\ge 1} {\widetilde}{{\ensuremath{\mathcal T}} }_{{\beta},A}^{(k)} <0$ in regime 3-b: this lemma proves that $\sum_{k = 1}^{K}
{{\ensuremath{\mathbf U}} }_k$ goes to $0$ in probability, and hence ${\mathbf{Z}}_{n,{\beta}_n}^{(>1)}-1$ also goes to $0$ in probability. This is needed to replace the study of $\log {\mathbf{Z}}_{n,{\beta}_n}^{(>1)}$ by that of ${\mathbf{Z}}_{n,{\beta}_n}^{(>1)}-1$, and it is actually the only place where the definition of regime 3-b is used.
\[lem:cas4-mainterm\] In regime 4 , we have that $$\frac{n}{h_n^2} \log \Big( \sqrt{n}\, {{\ensuremath{\mathbf U}} }_1 \Big) \stackrel{({\rm d})}{\longrightarrow} W_1 \, ,$$ with $W_1$ defined in .
Here also, this proves that ${{\ensuremath{\mathbf U}} }_1 \to 0$ in probability, and hence so does ${\mathbf{Z}}_{n,{\beta}_n}^{(>1)}-1$.
Regime 3-b: convergence of the main term {#sec:prooflem3b}
----------------------------------------
In this section, we prove Lemma \[lem:cas3-mainterm\].
### Reduction to finitely many weights
First of all, we fix some $\mathtt L$ large and show that the main contribution comes from the $\texttt L$ largest weights. We define $${{\ensuremath{\mathbf U}} }_k^{(\mathtt L)} := \sum_{\Delta \subset {\Upsilon}_{\mathtt L}, |\Delta| =k} e^{{\beta}_n \Omega_{n,A_n}(\Delta)} {\mathbf{P}}\big( S\cap {\Upsilon}_{\ell} =\Delta \big) \, ,$$ where ${\Upsilon}_{\mathtt L} = \{ Y_1^{n,A_n}, \ldots, Y_{\mathtt L}^{n,A_n}\}$ is the set of $\mathtt L$ largest weights in $\Lambda_{n,A_n}$ (note that ${\Upsilon}_{\mathtt L} \subset {\Upsilon}_{\ell}$ for $n$ large enough). Then we have that ${{\ensuremath{\mathbf U}} }_k \ge {{\ensuremath{\mathbf U}} }_k^{(\mathtt L)}$, and $ \sum_{k=1}^K {{\ensuremath{\mathbf U}} }_k $ is bounded by $$\begin{aligned}
&\sum_{k=1}^{K} \sum_{\Delta \subset {\Upsilon}_{\mathtt{L}} , |\Delta| =k} \sum_{ \Delta' \subset {\Upsilon}_{\ell}\setminus {\Upsilon}_{\mathtt L}, | \Delta'| \le K}
e^{{\beta}_n \Omega_{n,A_n}(\Delta) +{\beta}_n \Omega_{n,A_n}(\Delta')} {\mathbf{P}}\big( S\cap {\Upsilon}_{\ell} =\Delta \cup \Delta' \big) \\
& \le \sum_{k=1}^{K} \sum_{\Delta \subset {\Upsilon}_{\mathtt{L}} , |\Delta| =k} e^{{\beta}_n \Omega_{n,A_n}(\Delta)} {\mathbf{P}}\big( S\cap {\Upsilon}_{\mathtt L} =\Delta \big) \times \exp\Big( K {\beta}_n M_{\mathtt L}^{(n,A_n)}\Big)\, \\
&=\exp\Big( K {\beta}_n M_{\mathtt L}^{(n,A_n)}\Big) \sum_{k=1}^K {{\ensuremath{\mathbf U}} }_k^{(\mathtt L)}.\end{aligned}$$ In the second inequality, we simply bounded $\Omega_{n,A_n}(\Delta')$ by $K M_{\mathtt L}^{(n,A_n)}$ uniformly for $\Delta' \subset {\Upsilon}_{\ell}\setminus {\Upsilon}_{\mathtt L}$, with $|\Delta'|\le K$. Then, since ${\beta}_n\sim c_{{\beta}} (\log n) / m(n h_n) \sim c_{{\beta},A} (\log n) / m(nA_n)$ as $n\to\infty$, we get that $K {\beta}_n M_{\mathtt L}^{(n,A_n)}$ is bounded above by $ 2c_{{\beta},A} K M_{\mathtt L}^{(n,A_n)} /m(nA_n) \times \log n $. For any fixed ${\varepsilon}> 0$, we can fix $\mathtt L$ large enough so that for large $n$ we have $M_{\mathtt L}^{(n,A_n)} /m(nA_n) \le {\varepsilon}/(2Kc_{{\beta},A})$ with probability larger than $1-{\varepsilon}$. We conclude that there exists some ${\varepsilon}_{\mathtt L}$ with ${\varepsilon}_{\mathtt L} \to 0$ as $\mathtt L\to\infty$ such that $$\begin{aligned}
0\le \sum_{k=1}^K ({{\ensuremath{\mathbf U}} }_k -{{\ensuremath{\mathbf U}} }_k^{(\mathtt L)} ) \le n ^{{\varepsilon}_{\mathtt L}} \sum_{k=1}^{K} {{\ensuremath{\mathbf U}} }_k^{(\mathtt L)} \, .\end{aligned}$$ Since $h_n^2/n \sim {\beta}\log n$, this proves that $$\lim_{n\to\infty} \frac{n}{h_n^2} \log \Big( \sum_{k=1}^K {{\ensuremath{\mathbf U}} }_k \Big) = \lim_{\mathtt L \to\infty} \lim_{n\to\infty} \frac{n}{h_n^2} \log \Big( \sum_{k=1}^K {{\ensuremath{\mathbf U}} }_k^{(\mathtt L)} \Big) \, .$$
### Convergence of the remaining term
We finally prove that $$\label{cas3-bfinal}
\frac{n}{h_n^2}\log \Big( \sum_{k=1}^K {{\ensuremath{\mathbf U}} }_k^{(\mathtt L)} \Big) \stackrel{({\rm d})}{\longrightarrow} \max_{1\le k\le K}{\widetilde}{{\ensuremath{\mathcal T}} }_{{\beta},A}^{(k,\mathtt L)}$$ where ${\widetilde}{{\ensuremath{\mathcal T}} }_{{\beta},A}^{(k,\mathtt L)}$ is the restriction of ${\widetilde}{{\ensuremath{\mathcal T}} }_{{\beta},A}^{(k)}$ to the $\mathtt L$ largest weights in $[0,1]\times [-A,A]$, that is $${\widetilde}{{\ensuremath{\mathcal T}} }_{{\beta},A}^{(k,\mathtt L)} := \sup_{s \in {\mathscr{M}}_A , N(s)=k} \Big\{ \pi^{(\mathtt L)}(s) - {\mathrm{Ent}}(s) - \frac{k}{2{\beta}}\Big\}$$ In analogy with Proposition \[prop:ConvVPtilde\], one shows that ${\widetilde}{{\ensuremath{\mathcal T}} }_{{\beta},A}^{(k,\mathtt L)} \to {\widetilde}{{\ensuremath{\mathcal T}} }_{{\beta},A}^{(k)}$ as $\mathtt L \to\infty$, which completes the proof.
The proof of comes from the rewriting $$\begin{aligned}
\sum_{k=1}^K {{\ensuremath{\mathbf U}} }_k^{(\mathtt L)} &= \sum_{\Delta \subset {\Upsilon}_{\mathtt L}, |\Delta| \le K} e^{{\beta}_n \Omega_{n,A_n}(\Delta)} {\mathbf{P}}\big( S\cap {\Upsilon}_{\mathtt L} = \Delta\big) \\
&= \sum_{\Delta \subset {\Upsilon}_{\mathtt L}, |\Delta| \le K} \exp\Big( {\beta}_n \Omega_{n,A_n}(\Delta) - {\mathrm{Ent}}(\Delta) - \frac{|\Delta|}{2} \log n + o (K) \Big)\, ,\end{aligned}$$ where for the last inequality we used Stone local limit theorem [@S67] (using that any two points in ${\Upsilon}_{\mathtt L}$ have abscissa differing by at least ${\varepsilon}n$ with probability going to $1$ as ${\varepsilon}\to 0$) to get that ${\mathbf{P}}\big( S\cap {\Upsilon}_{\mathtt L} = \Delta\big) = n^{-\frac{|\Delta|}{2} +o(1)} e^{-{\mathrm{Ent}}(\Delta)}$ uniformly for $\Delta \subset {\Upsilon}_{\mathtt L}$. Since there are finitely many terms in the sum, we get that analogously to -, $$\sum_{k=1}^K {{\ensuremath{\mathbf U}} }_k^{(\mathtt L)} = e^{o(\log n)} \times \exp\Big( \max_{\Delta \subset {\Upsilon}_{\mathtt L}, |\Delta |\le K }\Big\{ {\beta}_n \Omega_{n,A_n}(\Delta) - {\mathrm{Ent}}(\Delta) - \frac{|\Delta|}{2} \log n \Big\}\Big) .$$ At this stage we write $$\begin{split}
&\max_{\Delta \subset {\Upsilon}_{\mathtt L}, |\Delta |\le K }\Big\{ {\beta}_n \Omega_{n,A_n}(\Delta) - {\mathrm{Ent}}(\Delta) - \frac{|\Delta|}{2} \log n \Big\}=
\max_{1\le k\le K} {\widetilde}T_{n,h}^{{\beta}_{n,h}, (k,\mathtt L)},\\
&\text{where}\qquad
{\widetilde}T_{n,h}^{{\beta}_{n,h}, (k,\mathtt L)}:=
\max_{\Delta \subset {\Upsilon}_{\mathtt L}, |\Delta |=k }\Big\{ {\beta}_n \Omega_{n,A_n}(\Delta) - {\mathrm{Ent}}(\Delta) - \frac{k}{2} \log n \Big\}
\end{split}$$ To complete the proof of we only have to show that $$\label{convtildeTkl}
\frac{n}{h_n^2}\log \Big( \sum_{k=1}^K {{\ensuremath{\mathbf U}} }_k^{(\mathtt L)} \Big) = o(1)+
\frac{n}{h_n^2}\max_{1\le k\le K} {\widetilde}T_{n,h}^{{\beta}_{n,h}, (k,\mathtt L)}
\xrightarrow[]{({\textrm{d}})} \max_{1\le k\le K}{\widetilde}{{\ensuremath{\mathcal T}} }_{{\beta},A}^{(k,\mathtt L)}.$$ In analogy with and Proposition \[prop:ConvVPtilde\], we have that for any fixed $k$, $$\frac{n}{h_n^2} {\widetilde}T_{n,h}^{{\beta}_{n,h}, (k,\mathtt L)}
\xrightarrow[]{({\textrm{d}})} {\widetilde}{{\ensuremath{\mathcal T}} }_{{\beta},A}^{(k,\mathtt L)} \, .$$ As for the convergence of , since we have only a finite number of points, the proof is a consequence of (5.1) and (5.2) of [@cf:BT_ELPP] and the Skorokhod representation theorem—we use also that $\frac{n}{h_n^2} \log n \to \frac{1}{{\beta}}$. Since the maximum is taken over a finite number of terms, this shows and concludes the proof.
Regime 4: convergence of the main term {#sec:regime4}
--------------------------------------
First of all, we show briefly that $W_{{\beta}}$ is well defined, before we turn to the proof of Lemma \[lem:cas4-mainterm\]. One of the difficulties here is that the reduction to trajectories operated in Section \[sec:4-step1\] (to trajectories with $\max_{i\le n} |S_i| \le A\sqrt{n\log n}$) is not adapted here, since the transversal fluctuations are of order $h_n \ll \sqrt{n\log n}$. Therefore, we have to further reduce the set of trajectories in ${{\ensuremath{\mathbf U}} }_1$.
### Well-posedness and properties of $W_{{\beta}}$
We prove the following proposition.
\[prop:W\] Assume that ${\alpha}\in(1/2,1)$. Then for every ${\beta}>0$, $W_{\beta}\in (0,\infty)$ almost surely.
Recalling the definition of $W_{{\beta}}$. We fix a region $\mathcal{D}_{\varepsilon}:= [\frac{1}{2},1]\times [-{\varepsilon},{\varepsilon}]$, for ${\varepsilon}>0$. In such a way we have that $$\label{lbWb}
W_{\beta}\ge \sup_{(w,t.x)\in {{\ensuremath{\mathcal P}} }; (t,x) \in \mathcal{D}_{\varepsilon}}\big\{\, w\, \big\}- \frac{{\varepsilon}^2}{{\beta}}.$$ We observe that $$\max_{(w,t,x)\in {{\ensuremath{\mathcal P}} }; (t,x )\in\mathcal{D}_{{\varepsilon}}}\big\{\, w\, \big\}\overset{({\textrm{d}})}{=}
(2 {\varepsilon})^{1/{\alpha}} \mathrm{Exp}(1)^{-1/\alpha}.$$ Therefore, since $\frac{1}{{\alpha}}<2$, the r.h.s. of is a.s. positive provided ${\varepsilon}$ is sufficiently small.
For an upper bound, we simply observe that $W_{{\beta}} \le {{\ensuremath{\mathcal T}} }_{{\beta}} <\infty$ a.s.
### Proof of Lemma \[lem:cas4-mainterm\]
We denote $p(i,x) := {\mathbf{P}}(S_i=x)$ for the random walk kernel. For $A>0$ fixed and ${\delta}>0$, we split $\sqrt{n}\, {{\ensuremath{\mathbf U}} }_1$ into three parts: $$\begin{aligned}
\label{eq:str27-1}
\sqrt{n}\, {{\ensuremath{\mathbf U}} }_1& := \sum_{(i,x)\in {\Upsilon}_{\ell} } e^{\beta_n \omega_{i,x}}\sqrt n p(i,x) \\
& =
\bigg( {\sum_{\substack{(i,x)\in {\Upsilon}_{\ell} \\ |x|> A h_n }}}
+ {\sum_{\substack{(i,x)\in {\Upsilon}_{\ell} \\ i < {\delta}n, |x|\le A h_n }}}
+ {\sum_{\substack{(i,x)\in {\Upsilon}_{\ell} \\ i\ge {\delta}n, |x|\le A h_n }}} \bigg) e^{\beta_n \omega_{i,x}}\sqrt n p(i,x)\, . \notag\end{aligned}$$ The main term is the last one, and we now give three lemmas to control the three terms.
\[lem:term1\] There exist constants $c$ and $\nu>0$ such that for all $n$ sufficiently large, for any $A>1$ $${\mathbb{P}}\Big(\sum_{(i,x)\in {\Upsilon}_{\ell}, |x|> A h_n } e^{\beta_n \omega_{i,x}}\sqrt n p(i,x) > A \Big(\frac{h_n^2}{n}\Big)^{3\alpha} \Big) \le c A^{-\nu}\, .$$
\[lem:term2\] There exist some $c,\nu>0$ such that, for any $A>1$ and $0<{\delta}<A^{-1}$, we get that for $n$ sufficiently large, $${\mathbb{P}}\bigg( \frac{n}{h_n^2} \log \Big( \sum_{(i,x)\in {\Upsilon}_{\ell}, i < {\delta}n, |x|\le A h_n } e^{\beta_n \omega_{i,x}}\sqrt n p(i,x) \Big) \ge ({\delta}A)^{\frac{1}{4{\alpha}}} \bigg) \le c ({\delta}A)^{1/2} \, .$$
And finally, for last term, we have the convergence.
\[lem:term3\] We have that $$\frac{n}{h_n^2} \log \Big ( \sum_{(i,x)\in {\Upsilon}_{\ell}, i\ge {\delta}n, |x|\le A h_n } e^{\beta_n \omega_{i,x}}\sqrt n p(i,x) \Big ) \stackrel{({\rm d})}{\longrightarrow} W_1({\delta},A) \, ,$$ with $W_1({\delta},A):= \max\limits_{(w,t,x)\in {{\ensuremath{\mathcal P}} }, t>{\delta}, |x|\le A}\big\{w-\frac{x^2}{2 t}\big\} \, .$
Now, let us observe that taking the limit ${\delta}\downarrow 0$, and $A\uparrow \infty$, we readily obtain that $W_1( {\delta}, A)\to W_1$ (by monotonicity). Therefore, combining Lemmas \[lem:term1\]-\[lem:term2\]-\[lem:term3\], we conclude the proof of Lemma \[lem:cas4-mainterm\].
Let us consider the event $${{\ensuremath{\mathcal G}} }(n,A):= \Big \{ \, \beta_n \omega_{i,x}\le \frac{x^2}{8 i} \, \text{for any}\, |x|> Ah_n,\, 1\le i \le n \Big \}.$$ Using this event to split the probability (and Markov’s inequality), we have that, recalling the definition of ${\Upsilon}_\ell$ $$\begin{aligned}
\label{splitG}
&{\mathbb{P}}\Big(\sum_{(i,x)\in {\Upsilon}_{\ell}, |x| >A h_n } e^{\beta_n \omega_{i,x}}\sqrt n p(i,x) > A \Big(\frac{h_n^2}{n}\Big)^{3\alpha} \Big)
\\
& \le
\frac{1}{A}\Big(\frac{h_n^2}{n}\Big)^{-3\alpha}{\mathbb{E}}\Big[\sum_{i=1}^n\sum_{|x|> Ah_n} e^{x^2/8i}\sqrt{n}p(i,x) {{\sf 1}}_{\{{\beta}_n {\omega}_{i,x} \ge 1\}}\Big]
+ {\mathbb{P}}\Big({{\ensuremath{\mathcal G}} }(n,A)^c\Big) \, .\notag\end{aligned}$$ Using again that $ {\mathbb{P}}({\omega}\ge 1/{\beta}_n) \le (h_n^2/n)^{2{\alpha}} (nh_n)^{-1}$ and that $p(i,x) \le e^{-x^2/4i}$ uniformly in the range considered (provided that $n$ is large enough), we get that the first term is bounded by $$\begin{aligned}
\frac{1}{A}\Big(\frac{h_n^2}{n}\Big)^{-\alpha} \frac{\sqrt{n}}{n h_n} & \sum_{i=1}^n\sum_{|x|> Ah_n} e^{-x^2/8i} \le \Big(\frac{h_n^2}{n}\Big)^{-\alpha} \, .\end{aligned}$$ In the last inequality, we used that the sum over $x$ is bounded by a constant independent of $i$, and also that $\sqrt{n}/h_n \to 0$. The first term in therefore goes to $0$ as $n\to\infty$, and we are left to control ${\mathbb{P}}({{\ensuremath{\mathcal G}} }(n,A)^c)$. A union bound gives $$\begin{aligned}
{\mathbb{P}}\big( {{\ensuremath{\mathcal G}} }(n,A)^c \big)& \le \sum_{i=1}^n \sum_{x =A h_n}^{+\infty} {\mathbb{P}}\Big( {\beta}_n {\omega}_{i,x} \ge \frac{x^2}{8i} \Big)
\le n \sum_{k = 0}^{+\infty} \sum_{x = 2^{k} A h_n}^{2^{k+1} A h_n} {\mathbb{P}}\Big( {\beta}_n {\omega}\ge 2^{2k} A^2 \frac{h_n^2}{8n} \Big) \notag\\
& \le 2 A n h_n \sum_{k=0}^{\infty} 2^k {\mathbb{P}}\Big( {\omega}\ge \frac{1}{10} 2^{2k} A^2 m(nh_n) \Big) \, ,\end{aligned}$$ where we used the definition of $h_n$ for the last inequality, with $n$ large enough. Then, using the definition of $m(nh_n)$ and Potter’s bound, we obtain that for any $\eta>0$ (chosen such that $1-2{\alpha}+2\eta<0$) there is a constant $c>0$ such that for $n$ large enough $${\mathbb{P}}\big( {{\ensuremath{\mathcal G}} }(n,A)^c \big) \le c A n h_n \sum_{k\ge 1} 2^{k} (2^{2k} A^{2})^{- {\alpha}+\eta} \frac{1}{n h_n}\le c' A^{1-2{\alpha}+2\eta }\, ,$$ where the sum over $k$ is finite because $1-2{\alpha}+2\eta<0$. This concludes the proof of Lemma \[lem:term1\].
Decomposing over the event $${{\ensuremath{\mathcal M}} }_{n}({\delta},A) = \Big\{ \max_{ i < {\delta}n , |x| \le A h_n } {\beta}_n {\omega}_{i,x} \le \frac12 ({\delta}A)^{\frac{1}{4{\alpha}}} \frac{h_n^2}{n} \Big\} \, ,$$ and using Markov’s inequality, we get that (similarly to ) $$\begin{aligned}
&{\mathbb{P}}\bigg( \sum_{ (i,x)\in {\Upsilon}_{\ell}, i < {\delta}n, |x|\le A h_n} e^{\beta_n \omega_{i,x}}\sqrt n p(i,x) \ge \exp\Big( ({\delta}A)^{\frac{1}{4{\alpha}}} \frac{h_n^2}{n} \Big)\bigg)\\
&\le e^{- \frac12 ({\delta}A)^{\frac{1}{4{\alpha}}} \frac{h_n^2}{n} } {\mathbb{E}}\Big[ \sum_{i=1}^{{\delta}n}\sum_{|x|\le Ah_n} \sqrt n p(i,x){{\sf 1}}_{\{ \beta_n \omega_{i,x} \ge 1 \}} \Big]
+ {\mathbb{P}}\big( {{\ensuremath{\mathcal M}} }_{n}({\delta},A)^{c} \big)\, . \notag\end{aligned}$$ We use again that ${\mathbb{P}}( \omega\ge 1/ {\beta}_n ) \le (h_n^2/n)^{2{\alpha}} (nh_n)^{-1}$, and the fact that $\sum_x p(i,x) =1$ for any $i\in {\mathbb{N}}$, to get that the first term is bounded by $$e^{- \frac12 ({\delta}A)^{\frac{1}{4{\alpha}}} \frac{ h_n^2}{n} } \Big( \frac{h_n^2}{n}\Big)^{2{\alpha}} \frac{n\sqrt{n}}{nh_n} \to 0 \quad \text{as } n\to\infty \, .$$ For the remaining term, using that ${\beta}_n^{-1} h_n^2/n \sim m(n h_n)$, we have by a union bound that for $n$ large enough $$\begin{aligned}
{\mathbb{P}}\big( {{\ensuremath{\mathcal M}} }_{n}({\delta},A)^{c} \big)& \le {\delta}A n h_n {\mathbb{P}}\Big( {\omega}> \frac14 ({\delta}A)^{\frac{1}{4{\alpha}}} m(n h_n)\Big) \\
&\le c {\delta}A n h_n \times \big( ({\delta}A)^{\frac{1}{4{\alpha}}} \big)^{-2{\alpha}} \frac{1}{nh_n} \, ,\end{aligned}$$ where we used Potter’s bound (with $({\delta}A)^{\frac{1}{4{\alpha}}}$ small) and the definition of $m(nh_n)$ for the last inequality (for $n$ large). This concludes the proof of Lemma \[lem:term2\].
The Stone local limit theorem [@S67] (see ) gives that, for fixed $A>0, {\delta}>0$, there exists $c>0$ such that uniformly for ${\delta}n\le i\le n$, $|x|\le A h_n$, $$\label{LocalLT}
\frac1c\, e^{- x^2/2i} \le \sqrt{i}\, p(i,x)
\le c\, e^{- x^2/2i} \, .$$
Since $\sqrt{n/i} \ge 1$ for all $i\le n$, we get the lower bound $$\begin{aligned}
\sum_{i={\delta}n }^{n} \sum_{|x| \le A h_n} e^{{\beta}_n {\omega}_{i,x}} \sqrt{n} p(i,x) {{\sf 1}}_{\{\beta_n \omega_{i,x} \ge 1 \}} \ge c \exp\Big( {\beta}_n W_{n}( {\delta},A)\Big)\, ,
\label{lowbound}\end{aligned}$$ where $W_{n}( {\delta},A)$ is a discrete analogue of $W_1( {\delta},A)$, that is $$\label{def:Wn}
W_{n}({\delta},A) := \max\limits_{ \substack {|x|\le A h_n , \, i ={\delta}n ,\ldots, n \\
{\beta}_n {\omega}_{i,x} \ge 1 } } \Big\{ {\omega}_{i,x}-\frac{x^2}{2 {\beta}_n i} \Big\} \, .$$ On the other hand, we get that $\sqrt{n/i} \le {\delta}^{-1/2}$ for $i\ge {\delta}n$, so that from we get $$\label{eq024}
\sum_{i={\delta}n }^{n} \sum_{|x| \le A h_n} e^{{\beta}_n {\omega}_{i,x}} \sqrt{n} p(i,x) {{\sf 1}}_{\{\beta_n \omega_{i,x} \ge 1 \}}\le
\frac{c}{\sqrt{{\delta}}}\, e^{{\beta}_n W_{n}( {\delta},A)} \sum_{i=1}^n\sum_{|x|\le Ah_n} {{\sf 1}}_{\{\beta_n \omega_{i,x} \ge 1 \}} \, .$$ Now, we have that ${\mathbb{P}}( {\omega}> 1/ {\beta}_n) \le (h_n^2/n)^{2{\alpha}} (nh_n)^{-1}$ as already noticed, so that $$\label{eq024after}
{\mathbb{E}}\Big[\sum_{i=1}^n\sum_{|x|\le Ah_n} {{\sf 1}}_{\{\beta_n \omega_{i,x} \ge 1 \}} \Big]\le A \left(\frac{h_n^2}{n}\right)^{2{\alpha}} .$$ Overall, combining with -, we get that with probability going to $1$ as $n\to\infty$, $$\bigg| \log \Big( \sum_{(i,x)\in {\Upsilon}_{\ell}, i\ge {\delta}n, |x|\le A h_n } e^{\beta_n \omega_{i,x}}\sqrt n p(i,x) \Big) - {\beta}_n W_{n}({\delta},A)
\bigg| \le (2{\alpha}+1) \log \frac{h_n^2}{n} \, .$$
To conclude the proof of Lemma \[lem:term2\], it therefore remains to show that $$\label{W1deltAconv}
\frac{n}{h_n^2} \times \beta_n W_{n}( {\delta}, A) \xrightarrow[n\to\infty]{({\textrm{d}})} W_1( {\delta}, A),$$ where $W_1( {\delta},A)$ is defined in Lemma \[lem:term2\].
We fix ${\varepsilon}>0$ and we consider $\widetilde W_{n}({\varepsilon}, {\delta},A) $ the truncated version of $W_{n}( {\delta},A)$ in which we replace the condition $\{{\beta}_n {\omega}_{i,x} \ge 1\}$ by $\{{\beta}_n {\omega}_{i,x} >{\varepsilon}\frac{h_n^2}{n}\}$, that is $$\label{def:Wntilde}
\widetilde W_{n}({\varepsilon}, {\delta},A) := \max\limits_{ \substack {|x|\le A h_n , \, i ={\delta}n ,\ldots, n \\
{\beta}_n {\omega}_{i,x} >{\varepsilon}\frac{h_n^2}{n} } } \Big\{ {\omega}_{i,x}-\frac{x^2}{2 {\beta}_n i} \Big\} \, .$$ In such a way, and since ${\varepsilon}h_n^2/n \ge 1$ for large $n$, we have $$\frac{n}{h_n^2} \beta_n \widetilde W_{n}({\varepsilon},{\delta}, A)\le \frac{n}{h_n^2} \beta_n W_{n}({\delta}, A) \le \frac{n}{h_n^2} \beta_n\widetilde W_{n}({\varepsilon},{\delta}, A) +{\varepsilon}.$$ To prove we need to show that $$\label{convWtilde}
\frac{n}{h_n^2} \times \beta_n {\widetilde}W_{n}({\varepsilon}, {\delta}, A) \xrightarrow[n\to\infty]{({\textrm{d}})} {\widetilde}W_1( {\varepsilon}, {\delta}, A) :=\max\limits_{\substack{(w,t,x)\in {{\ensuremath{\mathcal P}} }\\ t>{\delta}, |x|\le A, w>{\varepsilon}}}\Big\{w-\frac{x^2}{2 t} \Big\} ,$$ and then let ${\varepsilon}\downarrow 0$ – notice that we have ${\widetilde}W_1({\varepsilon}, {\delta},A)\le W_1({\delta},A)\le {\widetilde}W_1({\varepsilon}, {\delta},A) +{\varepsilon}$ so that ${\widetilde}W_1( {\varepsilon}, {\delta}, A) \to W_1({\delta},A)$ as ${\varepsilon}\downarrow 0$.
We observe that a.s. there are only finitely many ${\omega}_{i,x}$ in $\llbracket 1,n \rrbracket \times \llbracket -A h_n, A h_n\rrbracket$ that are larger than ${\varepsilon}m(nh_n) \sim {\beta}_n^{-1} {\varepsilon}h_n^2/n$. This is a consequence of Markov’s inequality and Borel-Cantelli Lemma. Indeed, for any $K\in \mathbb N$ we have $$\begin{split}
{\mathbb{P}}\Big( \, \Big|\big\{(i,x)\in \llbracket 1,n \rrbracket \times \llbracket -A h_n, &A h_n\rrbracket \colon \omega_{i,x} \ge {\varepsilon}m(nh_n)
\big\}\Big| > 2^K\Big)\\
&\le 2^{-K} (2Anh_n) {\mathbb{P}}\Big(\omega\ge {\varepsilon}m(nh_n)\Big)\le C_{\varepsilon}2^{-K}\, .
\end{split}$$ Therefore, the convergence is a straightforward consequence of the Skorokhod representational theorem.
Case ${\alpha}\in(0,1/2)$ {#sec:alpha12}
=========================
In the first part of this section we prove . In the second part, we prove the convergence .
Transversal fluctuations: proof of {#sec:fluctualpha12}
-----------------------------------
### Paths cannot be at an intermediate scale
We start by showing that there exists $c_0,c,\nu>0$ such that for any sequences $C_n>1$ and $\delta_n\in (0,1)$ (which may go to $\infty$, resp. $0$, as $n\to\infty$) and for any $n\ge 1$ $$\label{goalalphale12}
\mathbb P \Big({\mathbf{P}}^\omega_{n,\beta_n}\big( \max\limits_{i\leq n} \big| S_i\big| \in [C_n \sqrt{n}, \delta_n n)\big) \leq e^{-c_0 C_n^2}
+e^{-c_0 n^{1/2}}\Big) \geq 1-c\delta_n^{\nu} + n^{- \frac{1-2{\alpha}}{4} +{\varepsilon}}.$$ To prove it, we use a decomposition into blocks, as we did in Section \[sec:fluctu\]. Here, we have to partition the interval $[C_n \sqrt{n}, \delta_n n)$ into $[C_n \sqrt{n}, n^{3/4})\cup [n^{3/4}, \delta_n n)$ (one of these intervals might be empty), obtaining $$\begin{aligned}
\nonumber
&{\mathbf{P}}^\omega_{n,\beta_n}\Big( \max\limits_{i\leq n} \big| S_i\big| \in [C_n \sqrt{n}, \delta n)\Big) \\
& \qquad = {\mathbf{P}}^\omega_{n,\beta_n}\Big( \max\limits_{i\leq n} \big| S_i\big| \in [C_n \sqrt{n}, n^{3/4})\Big)+
{\mathbf{P}}^\omega_{n,\beta_n}\Big( \max\limits_{i\leq n} \big| S_i\big| \in (n^{3/4}, \delta_n n)\Big).
\label{eqalphale12_1}\end{aligned}$$
For the first term, we partition the interval $[C_n\sqrt{n}, n^{3/4})$ into smaller blocks $D_{k,n}:=[2^{k}\sqrt{n}, 2^{k+1}\sqrt{n})$, with $k= \log_2 C_n, \dots,\log_2 n^{1/4}-1$. Let us define $$\Sigma(n,h) = \sum_{i=1}^n \sum_{x\in \llbracket -h ,h\rrbracket} {\omega}_{i,x}$$ the sum of all weights in $\llbracket 1,n \rrbracket \times \llbracket -h ,h\rrbracket$. Then, we write similarly to (we also use that ${\mathbf{Z}}_{n,{\beta}_n}^{{\omega}}\ge 1$, which is harmless here since no recentering term is needed) $$\begin{aligned}
{\mathbf{P}}^\omega_{n,\beta_n}\Big( \max\limits_{i\leq n} \big| S_i\big| \in & [C_n \sqrt{n}, n^{3/4})\Big) \le \sum_{k=\log_2 C_n}^{\log_2 n^{1/4} } {\mathbf{Z}}^{{\omega}}_{n,{\beta}_n}\big( \max_{i\le n} |S_i| \in D_{k,n} \big) \\
& \le \sum_{k=\log_2 C_n}^{\log_2 n^{1/4} -1} e^{\beta_n \Sigma(n,2^{k+1}\sqrt n) }{\mathbf{P}}\big( \max_{i\le n} |S_i| \in D_{k,n} \big)\\
& \le \sum_{k=\log_2 C_n}^{ \log_2 n^{1/4}} \exp\Big( \beta_n \Sigma(n,2^{k+1}\sqrt n) - c 2^{2k} \Big)\end{aligned}$$ where for the last inequality we used a standard estimate for the deviation probability of a random walk ${\mathbf{P}}\big( \max_{i\le n} |S_i| \ge 2^k \sqrt{n} \big) \le e^{ - c 2^{2k}}$, see for example [@LL10 Prop. 2.1.2-(b)]. Therefore, on the event $$\label{evA34}
\Big\{ \forall\, k= \log_2 C_n, \dots,\log_2 n^{1/4},\, \beta_n \Sigma(n,2^{k+1}\sqrt n) \leq \frac{c }{2}2^{2k}
\, \Big\}$$ we have that $${\mathbf{P}}^\omega_{n,\beta}\Big( \max\limits_{i\leq n} \big| S_i\big| \in [C_n\sqrt{n}, n^{3/4})\Big)
\leq \sum_{k=\log_2 C_n}^{\log_2 n^{1/4}} e^{-\frac{c }{2}2^{2k}}\leq c' e^{-\frac{c }{2} C_n^2}.$$
For the second term in , we partition the interval $(n^{3/4}, \delta_n n)$ into blocks $E_{n,k}:=[2^{-k-1}n, 2^{-k}n)$, $k= \log_2(1/\delta_n), \dots,\log_2 n^{1/4}-1$. Exactly as above we use the large deviation estimate ${\mathbf{P}}\big( \max_{i\le n} |S_i| \ge 2^{-k+1} n \big) \le e^{ - c 2^{-2k} n}$ (see e.g [@LL10 Prop. 2.1.2-(b)]), and we obtain that on the event $$\label{ev34delta}
\Big\{ \forall\, k=\log_2(1/\delta_n), \dots,\log_2 n^{1/4},\, \beta_n \Sigma(n,2^{-k n}) \leq \frac{c}{2}2^{-2k}n \,
\Big\}$$ we have $${\mathbf{P}}^\omega_{n,\beta}\Big( \max\limits_{i\leq n} \big| S_i\big| \in (n^{3/4}, \delta_n n)\Big)
\leq \sum_{k=\log_2(1/\delta_n)}^{\log_2 n^{1/4}} e^{ - \frac{c}{2} 2^{-2k} n } \leq c' e^{-\frac{c}{2} n^{1/2}}.$$
It now only remains to show that the complementary events of and have small probability. We start with . Using that ${\beta}_n \le 2{\beta}n /m(n^2)$ for $n$ large, we get by a union bound that $$\begin{aligned}
\label{ev34delta2}
{\mathbb{P}}\Big( \exists\, k \ge \log_2 1/{\delta}_n &\, , \, \beta_n \Sigma(n,2^{-k }n) > \frac{c}{2} 2^{-2k} n \Big) \\
& \le \sum_{k\ge \log_2 1/{\delta}_n} {\mathbb{P}}\Big( \Sigma(n,2^{-k }n) > c_{{\beta}} 2^{-2k} m(n^2) \Big) \, .
\notag\end{aligned}$$ Then, by Potter’s bound we have that $m(2^{-k+1}n^2) \le 2^{-2k} m(n^2)$ since ${\alpha}<1/2$ (recall $m(\cdot)$ is regularly varying with exponent $1/{\alpha}$). As a consequence, the last probability in is in the so-called one-jump large deviation domain (see [@Nag79 Thm. 1.1], we are using ${\alpha}<1$ here), that is $${\mathbb{P}}\Big( \Sigma(n,2^{-k }n) > c_{{\beta}} 2^{-2k} m(n^2) \Big) \sim 2^{-k+1} n^2 {\mathbb{P}}\big( {\omega}> c_{{\beta}} 2^{-2k} m(n^2)\big)\, .$$ Therefore, using again Potter’s bound, we get that for arbitrary $\eta$ there is some constant $c$ such that $${\mathbb{P}}\Big( \Sigma(n,2^{-k }n) > c_{{\beta}} 2^{-2k} m(n^2) \Big) \le c (2^{2k})^{{\alpha}+\eta} n^{-2}$$ where we also used that ${\mathbb{P}}({\omega}>m(n^2)) =n^{-2}$. Therefore, taking $\eta$ small enough so that $2{\alpha}-1 +2\eta<0$, we obtain that is bounded by a constant times $$\sum_{k\ge \log_2 1/{\delta}_n} 2^{k (2{\alpha}-1 +2\eta)} \le c {\delta}_n^{1-2{\alpha}+2\eta}\, .$$
Similarly, for , we have by a union bound that $$\begin{aligned}
\label{evA342}
{\mathbb{P}}\Big( \exists\, k \in \{ \log_2 C_n, &\dots,\log_2 n^{1/4}\}, \, \beta_n \Sigma(n,2^{k+1} \sqrt{n} ) > \frac{c}{2}2^{2k} \Big) \notag\\
&\le \sum_{k=\log_2 C_n}^{\log_2 n^{1/4}} {\mathbb{P}}\Big( \Sigma(n,2^{k+1} \sqrt{n} ) > c_{{\beta}} 2^{2k} n^{-1} m(n^2) \Big)\, .\end{aligned}$$ Then again, we notice that $m(2^{k+2}n^{3/2}) \le 2^{2k} n^{-1} m(n^2)$ (using Potter’s bound, as ${\alpha}< 1/2$). Hence, the last probability in is in the one-jump large deviation domain (see [@Nag79 Thm. 1.1]), that is $${\mathbb{P}}\Big( \Sigma(n,2^{k+1} \sqrt{n} ) > c 2^{2k} n^{-1} m(n^2) \Big) \le c 2^{k}n^{3/2} {\mathbb{P}}\big({\omega}> c_{{\beta}} 2^{2k} n^{-1} m(n^2) \big)$$ Then, we also get that for any $\eta>0$ we have that there is a constant $c>0$ such that $${\mathbb{P}}\big({\omega}> c_{{\beta}} 2^{2k} n^{-1} m(n^2) \big) \le c (2^{2k} n^{-1})^{-{\alpha}-\eta} \, ,$$ so that provided that $1-2{\alpha}-2\eta >0$, is bounded by a constant times $$\sum_{k=\log_2 C_n}^{\log_2 n^{1/4}} 2^{k(1-2\alpha-2\eta)}n^{{\alpha}-\frac12+\eta} \le c n^{- \frac14 (1-2{\alpha}-2\eta)}\, .$$
### Paths cannot be at scale $n$ conditionnaly on ${\widehat}{{\ensuremath{\mathcal T}} }_{{\beta}}=0$
We have shown in that paths cannot be on an intermediate scale: it remains to prove that on the event ${\widehat}{{\ensuremath{\mathcal T}} }_{{\beta}} =0$, paths cannot be at scale $n$. For this purpose we use [@AL11 Theorem 2.1] and [@T14 Theorem 1.8], which ensure that for any $\delta$ and ${\varepsilon}>0$ there exists $\nu>0$ such that $$\label{eq:ALThm2.1}
\mathbb P\Big({\mathbf{P}}_{n,{\beta}_n}^{{\omega}} \big( \max_{i\leq n} |S_i| \in (\delta n, n] \big) \leq e^{-n\nu}\ \Big| \ {\widehat}{{\ensuremath{\mathcal T}} }_{{\beta}} =0\Big)\ge 1-{\varepsilon}.$$
Therefore, we get that for any ${\varepsilon}>0$ and ${\delta}>0$, combining with , for any sequence $C_n >1$, provided that $n$ is large enough we have $$\mathbb P \Big({\mathbf{P}}_{n,{\beta}_n}^{{\omega}} \big( \max_{i\leq n} |S_i| \ge C_n \sqrt{n} \big) \ge e^{-c_0 C_n^2} + e^{-c_0 n^{1/2}} + e^{-n\nu} \ \Big| \ {\widehat}{{\ensuremath{\mathcal T}} }_{{\beta}} =0 \Big) \le c {\delta}^{\nu} + 2{\varepsilon}\, ,$$ which concludes the proof of .
Convergence in distribution conditionally on ${\widehat}{{\ensuremath{\mathcal T}} }_{{\beta}} =0$, proof of {#sec:convalpha12}
-------------------------------------------------------------------------------------------------------------
In the following, we consider the case where ${\beta}_n n^{-1} m(n^2) \to \beta$ with $\beta<\infty$. In the case ${\beta}=+\infty$, we would indeed have that ${\widehat}{{\ensuremath{\mathcal T}} }_{{\beta}} >0$. The proof follows the same idea as that of [@cf:DZ Thm. 1.4] (and similar steps as above), but with many adaptations (and simplifications) in our case. We focus on the case ${\beta}>0$, in which $\frac{\sqrt n}{ {\beta}_n m(n^{3/2})} $ goes to infinity as a regularly varying function with exponent $\frac{2}{{\alpha}}-\frac12 -\frac{3}{2{\alpha}} = \frac{1-{\alpha}}{2{\alpha}} >0$ (If ${\beta}=0$, it goes to infinity faster).
### Step 1. Reduction of the set of trajectories.
Equation (with $C_n=A\sqrt{\log n}$) gives that, with ${\mathbb{P}}$ probability larger than $1-{\varepsilon}$ (conditionally on ${\widehat}{{\ensuremath{\mathcal T}} }_{{\beta}}=0$), we have ${\mathbf{P}}_{n,\beta_n}^{\omega}\big(\max_{i\le n}|S_i|\leq A\sqrt{n \log n }\big) \ge 1- e^{ - c_0 A \log n }$ provided that $n$ is large enough. We therefore get $${\mathbb{P}}\Big( \big| \log {\mathbf{Z}}_{n,\beta_n}^{\omega}-\log {\mathbf{Z}}_{n,\beta_n}^{\omega}\big({{\ensuremath{\mathcal A}} }_n \big)\big| \le n^{-c_0 A} \, \Big |\, {\widehat}{{\ensuremath{\mathcal T}} }_{{\beta}} =0\Big) \ge 1-{\varepsilon}\,,$$ where ${{\ensuremath{\mathcal A}} }_n$ is defined in . Note that, provided $A$ has been fixed large enough, we have that $\frac{\sqrt n}{\beta_n m(n^{3/2})} n^{-c_0 A} \to 0$ as $n\to\infty$: we conclude that, for any ${\varepsilon}>0$ $${\mathbb{P}}\Bigg(\frac{\sqrt n}{\beta_n m(n^{3/2})} \big| \log {\mathbf{Z}}_{n,\beta_n}^{\omega}-\log {\mathbf{Z}}_{n,\beta_n}^{\omega}\big({{\ensuremath{\mathcal A}} }_n\big)\big| > {\varepsilon}\, \Big |\, {\widehat}{{\ensuremath{\mathcal T}} }_{{\beta}} =0\Bigg) \le {\varepsilon}\, ,$$ provided that $n$ is large enough. We will therefore focus on $\log {\mathbf{Z}}_{n,\beta_n}^{\omega}\big({{\ensuremath{\mathcal A}} }_n\big)$.
As in Section \[sec:3b4\], we use the notation $A_n = A\sqrt{n\log n}=C_n \sqrt n$ and $\Lambda_{n,A_n} = \llbracket 1,n \rrbracket \times \llbracket -A_n,A_n \rrbracket$.
### Step 2. Truncation of the weights.
We let $k_n:=m(n^{3/2} \log n)$ be a sequence of truncation levels, and ${\widetilde}{\omega}_x := {\omega}_{x} {{\sf 1}}_{\{{\omega}_x \le k_n\}}$ be the truncated environment. Then, we have that $$\begin{aligned}
{\mathbb{P}}\Big( {\mathbf{Z}}_{n,\beta_n}^\omega ({{\ensuremath{\mathcal A}} }_n) \neq {\mathbf{Z}}_{n,\beta_n}^{{\widetilde}\omega} ({{\ensuremath{\mathcal A}} }_n) \Big)& = {\mathbb{P}}\big( \max_{(i,x)\in \Lambda_{n,A_n}} {\omega}_{i,x} > m(n^{3/2} \log n) \big) \\
& \le \frac{2A}{\sqrt{\log n}} \stackrel{n\to\infty}{\to} 0\, ,\end{aligned}$$ where we used a union bound for the last inequality, together with the definition of $m(\cdot)$ . Henceforth we can safely replace ${\mathbf{Z}}_{n,\beta_n}^\omega ({{\ensuremath{\mathcal A}} }_n)$ with the truncated partition function ${\mathbf{Z}}_{n,\beta_n}^{{\widetilde}\omega} ({{\ensuremath{\mathcal A}} }_n)$.
### Step 3. Expansion of the partition function.
We write again $p(i,x)={\mathbf{P}}(S_i=x)$ for the random walk kernel, and let $\lambda_n(t) = \log {\mathbb{E}}[e^{t {\widetilde}{\omega}_x}]$. Then, expanding $$\exp\Big( \sum_{i=1}^n \big( {\beta}_n {\omega}_{i,S_i} -\lambda_n({\beta}_n) \big) \Big) = \prod_{(i,x)\in \Lambda_{n,A_n}} \big(1+e^{{\beta}_n {\widetilde}{\omega}_{i,x} -\lambda_n({\beta}_n)}-1 \big)^{{{\sf 1}}_{\{S_{i}=x\}}},$$ we obtain $$\begin{aligned}
\label{CEZomega}
e^{-n \lambda_n({\beta}_n)} {\mathbf{Z}}_{n,\beta_n}^{{\widetilde}\omega}({{\ensuremath{\mathcal A}} }_n)= 1 + \!\!\!\!
\sum_{(i,x)\in \Lambda_{n,A_n}} \!\!\!\! \big(e^{\beta_n {\widetilde}\omega_{i,x} - \lambda_n(\beta_n)}-1\big)p(i,x)+ {{\ensuremath{\mathbf R}} }_n,\end{aligned}$$ with $${{\ensuremath{\mathbf R}} }_n:=\sum_{k=2}^\infty \sum_{\substack{{1\leq i_1<\dots<i_k\leq n} \\ { |x_i|\leq A_n,\, i=1,\dots, k}}}
\prod_{j=1}^k\big(e^{\beta_n {\widetilde}\omega_{j,x_j} - \lambda_n(\beta_n)}-1\big)p_n(i_j-i_{j-1},x_j-x_{j-1})\, .$$
\[lem:R\] We have that for $n$ large $${\mathbb{P}}\Bigg( \frac{\sqrt{n}}{{\beta}_n m(n^{3/2}) } {{\ensuremath{\mathbf R}} }_n \ge n^{-1/4} \Bigg) \le \frac{(\log n)^{4/{\alpha}}}{\sqrt{n}} \to 0\, .$$ In particular, ${{\ensuremath{\mathbf R}} }_n \to 0$ in probability.
Note that ${\mathbb{E}}[{{\ensuremath{\mathbf R}} }_n]=0$, so it will be enough to control the second moment of ${{\ensuremath{\mathbf R}} }_n$. Since the ${\widetilde}{\omega}_{i,x}$ are independent and ${\mathbb{E}}[e^{{\beta}_n {\widetilde}{\omega}_{i,x} -\lambda_n({\beta}_n)} - 1] =0$, $$\begin{aligned}
\mathbb E[{{\ensuremath{\mathbf R}} }_n^2] & = \sum_{k = 2}^{\infty} \sum_{\substack{{1\leq i_1<\dots<i_k\leq n} \\ { |x_i|\leq A_n,\, i=1,\dots, k}}} \big(e^{\lambda_n(2\beta_n)-\lambda_n(\beta_n)}-1\big)^k\prod_{j=1}^k p_n(i_j-i_{j-1},x_j-x_{j-1})^2 \\
& \le \sum_{k =2}^{\infty} \big( e^{\lambda_n(2{\beta}_n)} -1\big)^k \Big(\sum_{i=1}^n \sum_{x\in\mathbb Z} p(i,x)^2 \Big)^k .\notag\end{aligned}$$ First, we have that $$\sum_{i=1}^n \sum_{x\in\mathbb Z} p(i,x)^2 ={\mathbf{E}}^{\otimes 2}\Big[ \sum_{i=1}^n {{\sf 1}}_{\{S_n = S'_n\}} \Big] \le c\sqrt{n} \, ,$$ where $S$ and $S'$ are two independent simple random walks. Then, since ${\beta}_n {\widetilde}{\omega}\le {\beta}_n k_n \to 0$, we can write $e^{2{\beta}_n {\widetilde}{\omega}} \le 1+ 3{\beta}_n {\widetilde}{\omega}$ for $n$ large, so that $$\begin{aligned}
e^{\lambda_n( 2 {\beta}_n) } -1 &\le 3 {\beta}_n {\mathbb{E}}[{\widetilde}{\omega}] = 3{\beta}_n \int_0^{k_n} {\mathbb{P}}({\omega}>u) {\rm d}u \notag\\
&\le c {\beta}_n L(k_n) k_n^{1-{\alpha}} \le \frac{c {\beta}_n k_n }{n^{3/2} \log n}\, .
\label{elambdan}\end{aligned}$$ To estimate the integral we used the tail behavior of ${\mathbb{P}}({\omega}>u)$ (see [@BGT89 Theorem 1.5.8]), while for the last inequality, we used that $k_n = m(n^{3/2} \log n)$ and the definition of $m(\cdot)$, so that $L(k_n) k_n^{-{\alpha}} \sim n^{-3/2} (\log n)^{-1}$. We therefore get that for $n$ large enough $${\mathbb{E}}[{{\ensuremath{\mathbf R}} }_n^2] \le \sum_{k\ge 2} \Big( \frac{ {\beta}_n k_n }{n} \Big)^k
\le 2 \Big( \frac{{\beta}_n k_n}{ n } \Big)^2\, .$$ To conclude, by Potter’s bounds we get that $k_n \le m(n^{3/2}) (\log n)^{2/{\alpha}}$ for $n$ large, so that $${\mathbb{E}}[{{\ensuremath{\mathbf R}} }_n^2] \le \Big( \frac{{\beta}_n m(n^{3/2})}{\sqrt{n}} \Big)^2 \times \frac{(\log n)^{\frac{4}{{\alpha}} }}{n}\, ,$$ and the conclusion of the lemma follows by using Markov’s inequality.
Going back to , we get that $$\begin{aligned}
&{\mathbf{Z}}_{n,\beta_n}^{{\widetilde}\omega} ({{\ensuremath{\mathcal A}} }_n) \\
& = e^{(n-1) \lambda_n({\beta}_n)} \Big( e^{\lambda_n({\beta}_n)} +
\sum_{(i,x)\in \Lambda_{n,A_n} } \big(e^{\beta_n {\widetilde}\omega_{i,x}} -e^{\lambda_n({\beta}_n)}\big)p(i,x)+ e^{\lambda_n({\beta}_n)} {{\ensuremath{\mathbf R}} }_n \Big) \notag\\
&=e^{(n-1) \lambda_n({\beta}_n)} \Big( 1 +
\mathbf{V}_n+ \mathbf{W}_n+ e^{\lambda_n({\beta}_n)} {{\ensuremath{\mathbf R}} }_n \Big)\, ,
\notag\end{aligned}$$ with $$\mathbf{V}_n:= \!\! \sum_{(i,x)\in \Lambda_{n,A_n}}\!\! \big(e^{\beta_n {\widetilde}\omega_{i,x}} -1\big)p(i,x) \ \text{ and }\ \mathbf{W}_n := (e^{\lambda_n({\beta}_n)}-1)\big( 1 - \!\!\!\! \sum_{(i,x)\in \Lambda_{n,A_n}} \!\! p(i,x) \big)\, .$$ We show below that $\lim_{n\to\infty} \mathbf{W}_n = 0$ and that $ \mathbf{V}_n$ converges in probability to $0$, so that using also Lemma \[lem:R\], we get $$\begin{aligned}
\label{almostthere}
&\frac{\sqrt{n}}{{\beta}_n m(n^{3/2})} \log {\mathbf{Z}}_{n,\beta_n}^{{\widetilde}\omega} ({{\ensuremath{\mathcal A}} }_n) \\
&= \frac{\sqrt{n}}{{\beta}_n m(n^{3/2})} \mathbf{V}_n + \frac{\sqrt{n}}{{\beta}_n m(n^{3/2})} \Big( (n-1) \lambda_n({\beta}_n) + \mathbf{W_n} \Big) + o(1)\, .
\notag\end{aligned}$$
Before we prove the convergence of the first term (see Lemma \[lem:convergence\]), we show that the second term goes to $0$—note that this implies that $\mathbf{W}_n \to 0$ since ${\beta}_n n^{-1/2} m(n^{3/2}) \to 0$. We write that $$\begin{aligned}
\label{rewriteW}
\big| (n-1) \lambda_n({\beta}_n) + \mathbf{W_n} \big| \le (n-1) \big| e^{\lambda_n({\beta}_n)} -& 1- \lambda_n({\beta}_n) \big|\\
& + \Big| n- \!\!\!\! \sum_{(i,x) \in \Lambda_{n,A_n}} \!\! p(i,x) \Big| \, .\notag\end{aligned}$$ For the second term, using standard large deviation for the simple random walk (e.g. [@LL10 Prop. 2.1.2-(b)]), there is a constant $c>0$ such that $$\label{difference}
n-\sum_{(i,x)\in \Lambda_{n,A_n}} p(i,x) =\sum_{i=1}^n {\mathbf{P}}(S_i > A \sqrt{n\log n}) \le n e^{- c A^2 \log n}\, .$$ For the first term, since we have $\lambda_n({\beta}_n) \to 0$, we get that for $n$ large enough $$\label{expolambda}
\big| e^{\lambda_n({\beta}_n)} - 1- \lambda_n({\beta}_n) \big| \le \lambda_n({\beta}_n)^2 \le \Big( \frac{{\beta}_n m(n^{3/2})}{n^{3/2}} (\log n)^{2/{\alpha}} \Big)^2 \, ,$$ where for the second inequality we used (note that $\lambda_n({\beta}_n) \le e^{\lambda_n({\beta}_n)}-1$), together with the fact that $k_n \le m(n^{3/2}) (\log n)^{2/{\alpha}}$.
Hence plugging and into , we get that provided that $A$ is large enough, $$\frac{\sqrt{n}}{{\beta}_n m(n^{3/2})} \Big| (n-1) \lambda_n({\beta}_n) + \mathbf{W_n} \Big| \le \frac{{\beta}_n m(n^{3/2})}{ n^{3/2} } (\log n)^{4/{\alpha}} + o(1) \xrightarrow{n\to\infty} 0 \, .$$ so that the second term in goes to $0$ as $n\to\infty$, proving also that $\mathbf{W}_n\to 0$ (recall also ${\beta}_n n^{-1/2} m(n^{3/2}) \to 0$).
### Step 4. Convergence of the main term.
We conclude the proof by showing the convergence in distribution of the first term in – which proves also that $\mathbf{V}_n$ goes to $0$ in probability, since ${\beta}_n n^{-1/2} m(n^{3/2}) \to 0$.
\[lem:convergence\] We have the following convergence in distribution, $$\frac{\sqrt{n}}{{\beta}_n m(n^{3/2})}\mathbf{V}_n := \frac{\sqrt{n}}{{\beta}_n m(n^{3/2})} \sum_{(i,x)\in \Lambda_{n,A_n}} \big(e^{\beta_n {\widetilde}\omega_{i,x}} -1\big)p(i,x) \ \xrightarrow[n\to\infty]{({\textrm{d}})}\ \mathcal{W}_0^{({\alpha})} \, ,$$ with $\mathcal{W}_0^{{\alpha}}$ defined in Theorem \[thm:alpha<12\].
First of all, since $\beta_n {\widetilde}{\omega}_{i,x} \le {\beta}_n k_n \to 0$ as $n\to\infty$ (and using that $0\le e^{x}-1-x \leq x^2$ for $x$ small), we have that for $n$ large $$0\le \mathbf{V}_n - {\beta}_n \sum_{(i,x) \in \Lambda_{n,A_n} } {\widetilde}{\omega}_{i,x} p(i,x) \le \sum_{(i,x) \in \Lambda_{n,A_n}} \big({\beta}_n {\widetilde}{\omega}_{i,x} \big)^2 p(i,x) \, .$$ Then, we can estimate the expectation of the upper bound, using that similarly to we have ${\mathbb{E}}[({\widetilde}{\omega})^2] \le c L(k_n) k_n^{2-{\alpha}} \sim c k_n^2 /(n^{3/2} \log n) $. Using also that $k_n\le m(n^{3/2}) (\log n)^{2/{\alpha}}$ for $n$ large, we obtain that $$\begin{aligned}
\frac{\sqrt{n}}{ {\beta}_n m(n^{3/2})}{\mathbb{E}}\Big[ \sum_{(i,x) \in \bar \Lambda_n} \big({\beta}_n {\widetilde}{\omega}_{i,x} \big)^2 p(i,x) \Big] & \le
c \, \frac{k_n}{m(n^{3/2})}\, {\beta}_n k_n \, n^{-1} \sum_{i=1} ^n \sum_{x\in \mathbb{Z}} p(i,x) \\
& \le c (\log n)^{2/{\alpha}} {\beta}_n k_n \xrightarrow{n\to\infty} 0\, .\end{aligned}$$
The proof of the lemma is therefore reduced to showing the convergence in distribution of the following term $$\begin{aligned}
\label{splitlastterm}
&\frac{\sqrt{n}}{m(n^{3/2})} \sum_{(i,x) \in \bar \Lambda_n} {\widetilde}{\omega}_{i,x} p(i,x) \\
&= \sum_{i=1}^n \sum_{|x|\le K\sqrt{n}} \frac{{\widetilde}{\omega}_{i,x}}{m(n^{3/2})} \sqrt{n} p(i,x) + \sum_{i=1}^n \sum_{ K \sqrt n < |x| \le A_n} \frac{{\widetilde}{\omega}_{i,x}}{m(n^{3/2})} \sqrt{n} p(i,x),
\notag\end{aligned}$$ where we fixed some level $K>0$ (we take the limit $K\to\infty$ in the end).
*Second term in .* To conclude the proof, it remains to show that the second term in goes to $0$ in probability as $K\to\infty$, uniformly in $n$: for any $K$ (large), we have for $n$ sufficiently large $$\label{controlK}
{\mathbb{P}}\Big( \sum_{i=1}^n \sum_{ K \sqrt n < |x| \le A_n} \frac{{\widetilde}{\omega}_{i,x}}{m(n^{3/2})} \sqrt{n} p(i,x) \ge K^{-1} \Big) \le c e^{- {\alpha}K}\, .$$ To prove , we split the sum in parts with $|x|\in (2^{k-1} K\sqrt{n} , 2^{k} K\sqrt{n}] $ for $k =1,2\ldots$. By a union bound, we have $$\begin{aligned}
{\mathbb{P}}\Big(\sum_{i=1}^n &\sum_{|x|> K\sqrt{n}} \frac{{\widetilde}{\omega}_{i,x}}{m(n^{3/2})} \sqrt{n} p(i,x) \ge K^{-1} \Big) \nonumber\\
& \le \sum_{k = 1}^{\infty} {\mathbb{P}}\Big( \sum_{i=1}^n \sum_{|x|= 2^{k-1} K\sqrt{n}}^{2^{k} K \sqrt{n}} \frac{{\omega}_{i,x}}{m(n^{3/2})} \sqrt{n} p(i,x) \ge K^{-1} 2^{-k} \Big) \nonumber\\
& \le \sum_{k\ge 1} {\mathbb{P}}\Big( \sum_{i=1}^n \sum_{|x|\le 2^{k} K \sqrt{n}} {\omega}_{i,x} \ge e^{ c' (2^k K)^2 } m\big( n^{3/2}\big) \Big)
\label{sumlargerK}\end{aligned}$$ In the last inequality, we used that there is a constant $c$ such that for any $k$, uniformly in $i\in\{1,\ldots, n\}$ and $|x| \ge 2^{k-1} K\sqrt{n}$, we have $\sqrt{n} p(i,x) \le e^{ - c (2^k K)^2 } \le 2^{-k}K^{-1} e^{ - c' (2^k K)^2 } $ (since $K2^k \ge 1$).
Now, we use that $m(2^{k+1} K n^{3/2} ) \ge (2^k K )^{-2/{\alpha}} m(n^{3/2}) $ by Potter’s bound, and also that for all $k$, $e^{ c' (2^k K)^2 }(2^k K )^{-2/{\alpha}} \ge e^{2^k K}$ if $K$ is large: the last probability in is in the one-jump large deviation domain (see [@Nag79 Thm. 1.1], we use here that ${\alpha}<1$): there is a $c>0$ such that for all $k\ge 1$ $$\begin{aligned}
{\mathbb{P}}\Big( \sum_{i=1}^n \sum_{|x|\le 2^{k} K \sqrt{n}} & {\omega}_{i,x} \ge e^{ 2^k K } m\big( 2^{k+1} Kn^{3/2}\big) \Big)\\
& \le c 2^k K n^{3/2} {\mathbb{P}}\Big( {\omega}\ge e^{ 2^k K } m\big( 2^k Kn^{3/2}\big) \Big) \le c e^{ - \frac{{\alpha}}{2} 2^k K }\, .\end{aligned}$$ The second inequality comes from Potter’s bound, provided that $ e^{2^k K}$ is large enough, and also the definition of $m(\cdot)$. Plugged in , we get $${\mathbb{P}}\Big(\sum_{i=1}^n \sum_{|x|> K\sqrt{n}} \frac{{\widetilde}{\omega}_{i,x}}{m(n^{3/2})} \sqrt{n} p(i,x) \ge {\varepsilon}\Big) \le c \sum_{k\ge 1} e^{- \frac{\alpha}2 2^k K} \le c e^{- {\alpha}K} \, ,$$ which is .
[**Acknowledgements**]{} We are most grateful to N. Zygouras for many enlightening discussions.
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---
abstract: 'Quantum protocols will be more efficient with high-dimensional entangled states. Photons carrying orbital angular momenta can be used to create a high-dimensional entangled state. In this paper we experimentally demonstrate the entanglement of the orbital angular momentum between the Stokes and anti-Stokes photons generated in a hot atomic ensemble using spontaneous four-wave-mixing. This experiment also suggests the existence of the entanglement concerned with spatial degrees of freedom between the hot atomic ensemble and the Stokes photon.'
author:
- 'Qun-Feng Chen'
- 'Bao-Sen Shi'
- 'Yong-Sheng Zhang'
- 'Guang-Can Guo'
title: Entanglement of the orbital angular momentum states of the photons generated in a hot atomic ensemble
---
Entanglement is one of the most fantastic phenomenon of quantum mechanics, and is used as a resource in quantum information field[@RevModPhys.74.347]. High-dimensional two-particle entangled states can be used to realize some quantum information protocols more efficiently [@PhysRevA.64.012306; @PhysRevLett.85.3313]. Photons carrying orbital angular momenta (OAM) are used to create high-dimensional entangled states, since OAM can be used to define an infinite-dimensional Hilbert space [@calvo:013805]. The first experiment of the entanglement of the OAM states generated via spontaneous parametric down-conversion in a nonlinear crystal was demonstrated in 2001[@Mair:N:2001:313], since then several protocols based on OAM states of photons have been realized experimentally[@Vaziri:PRL:2002:240401; @Vaziri:PRL:2003:227902; @Langford:PRL:2004:053601]. The transferring of OAM between classical light and cold atoms[@PhysRevLett.83.4967; @PhysRevLett.90.133001; @Barreiro:OL:2004:1515] and hot atoms[@Jiang:PRA:2006:043811] has also been reported in the past years. Recently, the entanglement of OAM states of the photons generated in a cold atomic system using Duan-Lukin-Cirac-Zoller (DLCZ) scheme[@Duan:N:2001:413] has been clarified by Inoue *et al.*[@inoue:053809]. So far, there is no experimental discussion about the entanglement of the OAM states of the photons generated in a hot atomic system. In this paper we demonstrate the entanglement of OAM states of the photons generated in a hot atomic ensemble using the spontaneous four-wave-mixing (SFWM)[@balic:183601; @Chen:unpublished]. Our experiment is different from the experiment done by Inoue *et al.*: In our experiment, SFWM is used to generate a photon pair, in contrast with the experiment of Ref. [@inoue:053809], in which the method based on DLCZ scheme is used. Furthermore, our experiment is based on a hot atomic ensemble, which is more easy to be realized compared with the scheme based on a cold atomci system. In our experiment, we clearly demonstrate the entanglement of the OAM between the Stokes and anti-Stokes photons generated via SFWM in a hot atomic ensemble, the concurrence got in this experiment is about 0.81. This experiment also suggests the existence of the entanglement concerned with spatial degrees of freedom between the hot atomic ensemble and the Stokes photon.
The schematic setup used in this experiment is shown in Fig. \[fig:1\]. The energy levels and the frequencies of the lasers used are shown in Fig. \[fig:1\](a). A strong coupling laser, which is resonant with the $|b\rangle\to|c\rangle$ transition, drives the populations of the atoms into level $|a\rangle$. A weak pump laser, resonant with the $|a\rangle\to|d\rangle$ transition, is applied to the system. The $|d\rangle\to|b\rangle$ transition will be induced by the pump laser and the Stokes (S) photons will be generated. When a Stokes photon is emitted, the atomic ensemble collapses into the state $\frac{1}{\sqrt{N}}
\sum_j|a_1,a_2,\ldots,b_j,\ldots,a_N\rangle$. The strong coupling laser repumps the atomic ensemble back to the state $|a_1,a_2,\ldots a_N\rangle$, and an anti-Stokes (AS) photon is generated. In this process, the energy, momentum and OAM of the photons will be conserved[@Scully:PRL:2006:010501; @Jiang:PRA:2006:043811],i. e., $$\begin{aligned}
\omega_{\rm S}+\omega_{\rm AS}&=&\omega_{\rm P}+\omega_{\rm
C},\nonumber \\
\vec k_{\rm S}+\vec k_{\rm AS}&=&\vec k_{\rm P}+\vec k_{\rm C},\nonumber\\
L_{\rm S}+L_{\rm AS}&=&L_{\rm P}+L_{\rm C}\,,
\label{cons}\end{aligned}$$ where the $\omega_i$, $\vec k_i$ and $L_i$ represent the frequency, wave vector and OAM of the corresponding photons respectively. According to Eq. (\[cons\]), when the pump and coupling lasers carry zero OAM, the Stokes and anti-Stokes photons will be in the entangled state of $$|\Psi\rangle= C\sum_{i=-\infty}^{+\infty}\alpha_i|i\rangle_{\rm
S}|-i\rangle_{\rm AS}\,,
\label{state}$$ where $C$ is the normalization coefficient, $\alpha_i$ are the relative amplitudes of the OAM states. In this work we only investigate the entanglement concerned with $i=0$ and $1$, thus the experimental expected entangled state can be written as: $$|\Psi\rangle = C(|0\rangle_{\rm S}|0\rangle_{\rm
AS} + \alpha_1 |1\rangle_{\rm S}|-1\rangle_{\rm AS})\,.
\label{}$$ Although we only discuss the two dimensional case, it is natural to presume that our discussion can be extended into high-dimensional cases over a wide range of OAM[@inoue:053809].
A Gaussian mode beam carrying the well-defined OAM is in Laguerre-Gaussian (LG) mode[@PhysRevA.45.8185], it can be described by LG$_{pl}$ mode, where $p+1$ is the number of the radial nodes, and $l$ is the number of the $2\pi$-phase variations along a closed path around the beam center. Here we only consider the cases of $p=0$. The LG$_{0l}$ mode carries the corresponding OAM of $l\hbar$ per photon and has a doughnut-shape intensity distribution: $$E_{0l}(r,\varphi) = E_{00}(r) \frac{1}{\sqrt{|l|!}}
\left(\frac{r\sqrt{2}}{w}\right)^{|l|} e^{-il\varphi},
\label{}$$ where $$E_{00}(r)= \sqrt{\frac{2}{\pi}}\frac{1}{w}\exp\left(-\frac{r^2}{w^2}\right)$$ is the intensity distribution of a Gaussian mode beam which carries zero OAM (LG$_{00}$) and $w$ is the beam waist. In most cases, computer-generated holograms (CGH) are used to create the LG modes of various orders[@Arlt:JOM:1998:1231]. The superposition of the LG$_{00}$ mode and the LG$_{01}$ mode can be achieved by shifting the dislocation of the hologram out of the beam center a certain amount[@Mair:N:2001:313; @Vaziri:JOO:2002:S47].
In this paper, a CGH combined with a single-mode fiber are used for mode discrimination. The $\pm 1$ order diffraction of the CGH increases the OAM of the input beam by $\pm 1\hbar$ per photon when the dislocation of the hologram is overlapped with the beam center. The first order diffraction of the CGH is coupled into the single-mode fiber. The single-mode fiber collects only the Gaussian mode beam, therefore the combination of the CGH and the single-mode fiber can be used to select the LG$_{0\mp 1}$ or LG$_{00}$ mode or the superposition of the them, according to which of the $\pm 1$ order diffraction of the hologram is coupled and the displacement of the hologram. It should be noted that there are also higher order LG modes in the first order diffraction, but they are very small compared with the LG$_{0\pm1}$ mode[@Vaziri:JOO:2002:S47] and the influence of them is ignored in this paper.
![(Color online) (a) Energy levels and frequencies of the lasers used in this experiment. (b) Schematic setup of our experiment. A strong coupling and a weak pump laser, which are resonant with $|5S_{1/2},F=2\rangle \to
|5P_{1/2},F=2\rangle$ and $|5S_{1/2},F=1\rangle \to
|5P_{3/2},F=2\rangle$ transitions of $^{87}$Rb respectively, are in counter propagating. Pairs of correlated Stokes and anti-Stokes photons are generated in phase-matched directions. H1 and H2 are computer-generated holograms; SMF1 and SMF2 are single-mode fibers, which are connected to single photon counting modules(SPCM); F1 and F2 are filters.[]{data-label="fig:1"}](fig1){width="8.3cm"}
The schematic experimental setup is shown in Fig. \[fig:1\](b). A natural rubidium cell with a length of 5 cm is used as the working medium. The temperature of the cell is kept at about 50$^\circ$C, corresponding to an atomic intensity of about $1\times10^{11}/{\rm cm}^3$. The coupling laser, which is vertically linear polarized, is resonant with the $|5S_{1/2},F=2\rangle \to |5P_{1/2},F=2\rangle$ transition of $^{87}$Rb. The intensity of the coupling laser is about 7 mW. The pump laser, which is counter-propagating with the coupling laser and horizontally polarized, is resonant with the $|5S_{1/2},F=1\rangle \to |5P_{3/2},F=2\rangle$ transition of ${}^{87}$Rb. The power of the pump is about $60\mu$W. The $1/e^2$ diameters of these two lasers are about 2 mm. The vertically polarized Stokes photons emitted at an angle of about 4$^\circ$ to the lasers are diffracted by a CGH (H1), and the $-1$ order diffraction of the H1 is coupled into a single-mode fiber (SMF1) after being filtered by the F1. The diffraction of the H1 decreases the OAM of the input photons by $1\hbar$ when the displacement of H1 is 0. The displacement of the CGH is defined as the distance between the dislocation of the CGH and the beam center. The horizontally polarized anti-Stokes photons in the phase matched direction are diffracted by the other CGH (H2). The $+1$ order diffraction is coupled into SMF2 after being filtered by F2, which increases the OAM of the collected anti-Stokes photons by $1\hbar$ at 0 displacement. The diffraction efficiency of the CGHs used in this experiment are about $40\%$. Each of the filters F1 and F2 consists of an optical pumped paraffin-coated $^{87}$Rb cell and a ruled diffraction grating. The optical pumped rubidium cell is used to filter out the scattering of the co-propagating laser, and the ruled diffraction grating is used to separate the photons at the D1 and D2 transitions. The collected photons are detected by photon-counting modules (Perkin-Elmer SPCM-AQR-15). The time resolved coincident statistics of the Stokes and anti-Stokes photons are accumulated by a time digitizer (FAST ComTec P7888-1E) with 2 n$s$ bin width and totally 160 bins. In this experiment the Stokes photons are used as the START of the P7888-1E and the anti-Stokes photons after certain delay are used as the STOP of the P7888-1E.
The time resolved coincident counts of the Stokes and anti-Stokes photons when the displacement of the both CGHs are far larger than the waists of the beam are shown in Fig. \[fig:2\]. When the displacement of a CGH is far larger than the waist of a beam, the CGH almost does not affect the mode of the photons, therefore Fig. \[fig:2\] shows the coincidence between the Stokes and anti-Stokes photons in LG$_{00}$ mode. The maximum coincident counts are obtained at the relative delay of 12 ns between the Stokes and anti-Stokes photons, which gives a correlation function of $g_{\rm S,AS}(12\textrm{
ns})=1.57\pm0.04$. The counting rates of the Stokes and anti-Stokes photons are $1.4\times10^4/$s and $4.0\times10^4/$s respectively. The larger counting rates of the anti-Stokes photons is caused by the atoms moving out and in the coupling beam quickly, which makes a large effective decay rate between the ground states. The atoms in the state $\left|b\right>$ moving into the coupling laser contribute to uncorrelated anti-Stokes photons. Even when the pump beam is absent the counting rate of the anti-Stokes is larger than 20000/s. These uncorrelated counts causes the large background in the coincidence between the Stokes and anti-Stokes photons, as shown in Fig. \[fig:2\]. From Fig. \[fig:2\] we found that the correlated time between the Stokes and anti-Stokes photons is less than 30 ns.
![(Color online) Time resolved coincidence counting between the Stokes and anti-Stokes photons. The data is accumulated about 1000 seconds and then normalized in time. $\tau$ is the relative delay between the Stokes and anti-Stokes photons. The delay between the Stokes photons and anti-Stokes photons is cause by time used to generate anti-Stokes photons, which is mainly determined by the Rabi frequency of coupling field[@balic:183601]. []{data-label="fig:2"}](fig2){width="8.3cm"}
In order to evaluate the quantum correlation of the OAM states, we measure the coincident counts with various displacements of the holograms. Figure \[fig:3\] shows the results when the H1 is fixed at various displacement while the displacement of H2 is swept. Every point is got by $N=\sum_{\tau =2 \rm ns}^{32 \rm ns}(N(\tau)-bg)/bg$, where $N(\tau)$ is the counting rate of each bin and $bg$ is background counting rate which is got by averaging the coincidences between the Stokes and anti-Stokes photons when $\tau>50$ ns. This guarantees that most of the correlated anti-Stokes photons are taken into account. Every point is accumulated over 500 seconds. The data are fitted with the square of the projection function[@Arlt:JOM:1998:1231]: $$\begin{aligned}
a(x0)&=&\int\!\!\!\int e^{-i
\arg(r\cos\varphi-x0, r\sin\varphi)}\nonumber\\
&&\times u_{AS}(r)u_{S}(r, \varphi)^*r\,\mathrm{d}
r\,\mathrm{d}\varphi\,,
\label{cocount}\end{aligned}$$ where $\arg(x,y)$ is the argument of the complex number $x+i\,y$, $e^{-i
\arg(r\cos\varphi-x0, r\sin\varphi)}$ represents the transmitting function of H2 with displacement of $x0$, $ u_{AS}(r)= E_{00}(r)$ is the field amplitude of the anti-Stokes photons collected by the single-mode fiber after being diffracted by the hologram, and $u_{S}(r,\varphi) = \cos\theta
E_{00}(r)+\sin\theta E_{01}(r,\varphi)$ is the field amplitude of the Stokes photons collected by the single-mode fiber. The superposition of the LG$_{00}$ and LG$_{01}$ modes can be controlled by the displacement of H1. Equation (\[cocount\]) gives the projection between the different OAM modes. In this paper the $u_{i}$s are the amplitudes of the Stokes and anti-Stokes photons respectively. This equation is tenable only when the collapse of the Stokes photons lead the anti-Stokes photons collapse into the corresponding states. Therefore if Eq. (\[cocount\]) always holds no matter the Stokes photons are collapsed to stationary states or superposition states, the Stokes photon and anti-Stokes photon should be in a quantum correlated state. In Fig. \[fig:3\] (a), the red squares show the results of the coincident counts versus the displacement of H2 when the displacement of H1 is far larger than the waist of the Stokes photons, and the green dots show the results when the displacement of H1 is 0. The red line in Fig. \[fig:3\] (a) is fitted with $\theta=0$ and the green dashed line is fitted with $\theta=\pi/2$, which means the Stokes photons are in LG$_{00}$ and LG$_{01}$ modes respectively. This figure demonstrates the collapse of the Stokes photon state into the stationary states lead the anti-Stokes photon state collapse into the corresponding stationary states. Therefore this figure indicates clearly the correlation of OAM between the Stokes and anti-Stokes photons. However, such a correlation can be obtained even in the mixture $|0\rangle_{S}|0\rangle_{AS}$ and $|1\rangle_{S}|1\rangle_{AS}$ states. To further demonstrate that the Stokes and anti-Stokes photons are in a quantum correlated state, we displace the H1 with a certain amount, which make the collected Stokes photons be in the superposition states ${1}/{\sqrt{2}}(|0\rangle\pm |1\rangle)$, and then sweep H2. The results are shown in Fig. \[fig:3\] (b). The data fit well with the theoretical prediction, which demonstrates that the anti-Stokes photon state collapses into the corresponding superposition states when the Stokes photon state collapses into the superposition states. Therefore the results shown in Fig. \[fig:3\] demonstrate that the Stokes and anti-Stokes photons are in strongly quantum correlated OAM states.
![(Color online) Coincident counts versus the displacement of H2 with different displacement of H1. (a) shows the results that the Stokes photons are in stationary states $|0\rangle$ (red squares) and $|1\rangle$ (green dots); (b) shows the results that the Stokes photons are in the superposition states $(|0\rangle\pm|1\rangle)/\sqrt{2}$. The data are fitted using the square of Eq. (\[cocount\]) with $w=0.8$ mm. []{data-label="fig:3"}](fig3){width="8.3cm"}
![(Color online) Graphical representation of the reconstructed density matrix. (a) is the real part and (b) is the imaginary part.[]{data-label="fig:4"}](fig4){width="6cm"}
To further demonstrate the entanglement of the Stokes and anti-Stokes photons, we perform a two-qubit state tomography[@PhysRevA.64.052312], and get the full state of the Stokes and anti-Stokes photons. The density matrix is reconstructed from the experimentally obtained coincidences with various combinations of the measurement basis. A graphical representation of the reconstructed density matrix is shown in Fig. \[fig:4\]. From the density matrix, the fidelity[@Nielsen:2000] to the maximally entangled state $|\Psi\rangle =
(|0\rangle_{S}|0\rangle_{AS} +
|1\rangle_{S}|-1\rangle_{AS})/\sqrt{2}$ is estimated to about $\langle \Psi|\rho|\Psi\rangle=0.89$. The concurrence[@PhysRevLett.80.2245] estimated from the density matrix is about $0.81>0$, which demonstrated the Stokes and anti-Stokes photons are in an entangled state clearly[@PhysRevLett.80.2245]. The entanglement of formation[@Nielsen:2000] is also estimated to be 0.74.
The Stokes photons and the anti-Stokes photons are not generated simultaneously in the SFWM. The atomic ensemble collapses into the state $\frac{1}{\sqrt{N}} \sum_j|a_1,a_2,\ldots,b_j,\ldots,a_N\rangle$ after emitting an Stokes photon, the information of the Stokes photons will be stored in the atomic system firstly. Lately the information of the atomic ensemble is retrieved by the coupling laser, and an anti-Stokes photon is generated[@Scully:PRL:2006:010501; @inoue:053809], the anti-Stokes photon carries the information of the atomic ensemble. The speed of the anti-Stokes photon generated is mainly determined by the Rabi frequency of the coupling laser[@balic:183601]. Therefore the entanglement of OAM between Stokes photons and the anti-Stokes photons might suggest the existence of the entanglement of OAM between the Stokes photon and the atomic ensemble. Our work is different from the work of V. Boyer *el al.*[@boyer:143601]. In their work they used a four-wave-mixing process[@boyer:143601:1] to generate the spatially multimode quantum-correlated twin beams with finite OAM in a hot atomic vapor. Their experiment is not a spontaneous process, and is not in the photon level. They also have not demonstrated the entanglement between the beams.
We estimate that the main sources of the errors in this experiment are from follows: the decay rate of the atoms is very large, which causes the large background counting; the instability of the frequency of the lasers; there are also other LG modes in the diffraction except for the LG$_{00}$, LG$_{01}$ modes and their superposition[@Arlt:JOM:1998:1231; @Vaziri:JOO:2002:S47]; the superposition of the state LG$_{00}$ and LG$_{01}$ is got by shifting the hologram, which is dependent on the beam waist, therefore the small fluctuation of the beam position also causes the error.
In summary, we have demonstrated the entanglement of OAM states between the Stokes and anti-Stokes photons generated via SFWM in a hot rubidium cell. The entanglement of the Stokes and anti-Stokes photons also suggests that the Stokes photon might entangle with the hot atomic ensemble in spatial degrees of freedom (OAM in this paper).
We thank Pei Zhang for supplying computer-generated holograms and some useful discussion. We also thank Xi-Feng Ren for some useful discussion. This work is supported by National Fundamental Research Program(2006CB921907), National Natural Science Foundation of China(60621064, 10674126, 10674127), the Innovation funds from Chinese Academy of Sciences, and the Program for NCET.
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---
abstract: 'Many of the baryons associated with a galaxy reside in its circumgalactic medium (CGM), in a diffuse volume-filling phase at roughly the virial temperature. Much of the oxygen produced over cosmic time by the galaxy’s stars also ends up there. The resulting absorption lines in the spectra of UV and X-ray background sources are powerful diagnostics of the feedback processes that prevent more of those baryons from forming stars. This paper presents predictions for CGM absorption lines (O VI, O VII, O VIII, Ne VIII, N V) that are based on precipitation-regulated feedback models, which posit that the radiative cooling time of the ambient medium cannot drop much below 10 times the freefall time without triggering a strong feedback event. The resulting predictions align with many different observational constraints on the Milky Way’s ambient CGM and explain why $N_{\rm OVI} \approx 10^{14} \, {\rm cm^{-2}}$ over large ranges in halo mass and projected radius. Within the precipitation framework, the strongest O VI absorption lines result from vertical mixing of the CGM that raises low-entropy ambient gas to greater altitudes, because adiabatic cooling of the uplifted gas then lowers its temperature and raises the fractional abundance of O$^{5+}$. Condensation stimulated by uplift may also produce associated low-ionization components. The observed velocity structure of the O VI absorption suggests that galactic outflows do not expel circumgalactic gas at the halo’s escape velocity but rather drive circulation that dissipates much of the galaxy’s supernova energy within the ambient medium, causing some of it to expand beyond the virial radius.'
author:
- 'G. Mark Voit'
title: '**Ambient Column Densities of Highly Ionized Oxygen in Precipitation-Limited Circumgalactic Media**'
---
Introduction
============
X-ray observations of galaxy clusters and groups have recently revealed a pervasive upper limit on the electron density of the ambient circumgalactic medium (CGM) surrounding a massive galaxy. Apparently, non-gravitational feedback triggered by radiative cooling and powered by either an active galactic nucleus or supernovae, or maybe a combination of the two, prevents $t_{\rm cool} / t_{\rm ff}$, the ratio of cooling time to freefall time in the ambient medium, from dropping much below $\approx 10$ [e.g., @McCourt+2012MNRAS.419.3319M; @Voit_2015Natur.519..203V; @Voit+2015ApJ...803L..21V; @Hogan_2017_tctff]. The conventional definition of the cooling time in this critical ratio is $t_{\rm cool} = 3 P / 2 n_e n_i \Lambda$, where $P$ is the gas pressure, $n_e$ and $n_i$ are the electron and ion densities, respectively, and $\Lambda$ is the usual radiative cooling function. The conventional definition of the freefall time is $t_{\rm ff} = (r / 2g)^{1/2}$, where $g$ is the gravitational acceleration and $r$ is the distance to the bottom of the potential well. Virtually all galactic systems ranging in mass from $10^{15} \, M_\odot$ down through $10^{13} \, M_\odot$ adhere to this limit [@Voit2018_LX-T-R].
Numerical simulations have shown that the limiting value of $t_{\rm cool} / t_{\rm ff}$ reflects the susceptibility of circumgalactic gas to condensation [e.g., @Sharma_2012MNRAS.420.3174S; @Gaspari+2012ApJ...746...94G; @Gaspari+2013MNRAS.432.3401G; @Li_2015ApJ...811...73L; @Prasad_2015ApJ...811..108P]. In gravitationally stratified media with $t_{\rm cool} / t_{\rm ff} \gg 1$ and a significant entropy gradient, buoyancy suppresses development of a multiphase state [@Cowie_1980MNRAS.191..399C]. Thermal instability does cause small perturbations in specific entropy to grow but results in buoyant oscillations that saturate a fractional amplitude $\sim (t_{\rm cool} / t_{\rm ff})^{-1}$ without progressing to condensation [@McCourt+2012MNRAS.419.3319M]. However, bulk uplift of lower-entropy ambient gas to greater altitudes can induce condensation if it lengthens $t_{\rm ff}$ so that $t_{\rm cool} / t_{\rm ff} \lesssim 1$ [*within the uplifted gas*]{}. That condition is relatively easy to satisfy if the global mean ratio is $t_{\rm ff} / t_{\rm cool} \lesssim 10$ but difficult if $t_{\rm ff} / t_{\rm cool} \gtrsim 20$ [@Voit_2017_BigPaper]. Drag can assist condensation by suppressing the damping effects of buoyancy [e.g., @Nulsen_1986MNRAS.221..377N; @ps05; @McNamara_2016arXiv160404629M], as can turbulence [@Gaspari+2013MNRAS.432.3401G; @Voit_2018arXiv180306036V] and magnetic fields [@Ji_2018MNRAS.476..852J].
The implications for massive galaxies are profound. Feedback from an active galactic nucleus can limit CGM condensation in those systems but requires tight coupling between radiative cooling of the CGM and energy output from the central engine [@mn07; @McNamaraNulsen2012NJPh...14e5023M]. A sharp transition to a multiphase state is essential, because it sensitively links the thermal state of the ambient medium on $\sim 10$ kpc scales with feeding of the central black hole on much smaller scales [see @Gaspari_2017MNRAS.466..677G; @Voit_2017_BigPaper and references therein]. The feedback loop works like this: If $t_{\rm cool} / t_{\rm ff}$ in the ambient medium is too large, then the black-hole accretion rate is too low for feedback energy to balance radiative cooling. The specific entropy and cooling time of the ambient medium therefore decline until $t_{\rm cool} / t_{\rm ff}$ becomes small enough for cold clouds to precipitate out of the hot medium. Those cold clouds then rain down onto the central black hole and fuel a much stronger feedback response that raises $t_{\rm cool}$ in the ambient medium. Such a system naturally tunes itself to a value of $t_{\rm cool} / t_{\rm ff}$ at which the ambient medium is marginally unstable to precipitation.
This paper proposes some observational tests that can probe whether the precipitation framework for self-regulating feedback also applies to galactic systems in the $10^{11} \, M_\odot$–$10^{13} \, M_\odot$ mass range, in which most of the feedback energy is thought to come from supernovae. X-ray observations of those systems remain extremely difficult, but the ambient CGM may also leave an imprint on UV absorption-line spectra. The ions responsible for the O VI and Ne VIII absorption lines observable with Hubble’s Cosmic Origins Spectrograph (COS) are not the dominant ones in circumgalactic gas at $\gtrsim 10^6$ K but may still produce detectable signatures. Consider, for example, the O VII absorption-line detections of the Milky Way’s CGM [e.g. @Fang_2006ApJ...644..174F; @BregmanLloydDavies_2007ApJ...669..990B; @Gupta_2012ApJ...756L...8G; @MillerBregman_2013ApJ...770..118M; @Fang_2015ApJS..217...21F], which indicate $N_{\rm OVII} \approx 10^{16} \, {\rm cm^{-2}}$ along lines of sight to extragalactic continuum sources. Collisional ionization equilibrium at $\sim 10^6 \, {\rm K}$ predicts that $N_{\rm O VI} / N_{\rm O VII} \sim 10^{-2}$ [@sd93]. O VII absorption-line gas at that temperature would therefore have $N_{\rm O VI} \sim 10^{14} \, {\rm cm^{-2}}$, which is observable with COS. There may be additional O VI absorption arising from cooler multiphase gas along those lines of sight, but the ambient gas alone should produce a detectable minimum O VI signal that depends predictably on the mass of the confining gravitational potential.
It is quite likely that such O VI absorption lines from the ambient CGM have already been detected. The most convincing candidates are moderate O VI lines ($N_{\rm OVI} \sim 10^{14} \, {\rm cm^{-2}}$) associated with broad, shallow Ly$\alpha$ absorption ($N_{\rm HI} \sim 10^{13-14} \, {\rm cm^{-2}}$, $b \sim 100 \, {\rm km \, s^{-1}}$) and comparable Ne VIII absorption ($N_{\rm NeVIII} \sim 10^{14} \, {\rm cm^{-2}}$). Such systems sometimes have no associated low-ionization gas [e.g., @Stocke_2013ApJ...763..148S; @Werk2016_ApJ...833...54W]. Both the broad Ly$\alpha$ line widths and a collisional-ionization interpretation of the NeVIII/OVI ratios imply gas temperatures $\sim 10^6$ K [@Savage2011_OVI_NeVIII_ApJ...743..180S]. At that temperature, the column densities of the broad Ly$\alpha$ lines imply a total hydrogen column density $N_{\rm H} \sim 10^{20} \, {\rm cm^{-2}}$ [@Savage_BroadOVI_2011ApJ...731...14S].
Section \[sec-CGM\_Models\] of this paper shows that the precipitation framework, when applied to the Milky Way, predicts that its CGM should indeed have a temperature $\sim 10^6$ K and $N_{\rm H} \sim 10^{20} \, {\rm cm^{-2}}$, nearly independent of projected radius. The resulting CGM models depend only on the maximum circular velocity of the galaxy’s halo, the minimum value of $t_{\rm cool} / t_{\rm ff}$, and surprisingly weakly on heavy-element abundances. Section \[sec-MilkyWay\] presents a detailed comparison of those models with a large variety of Milky-Way data and shows that the models agree with current constraints on the density, temperature, and abundance profiles of the Milky Way’s CGM, without any parameter fitting. In other words, a physically motivated model originally developed to describe feedback regulation of galaxy-cluster cores also aligns with what is currently known about the Milky Way’s ambient CGM. Section \[sec-Columns\] then extends that model to predict precipitation-limited O VI column densities of the ambient CGM in halos ranging in mass from $10^{11} \, M_\odot$ to $10^{13} \, M_\odot$. For a static CGM, the model gives $N_{\rm OVI} \approx 10^{14} \, {\rm cm^{-2}}$ out to nearly the virial radius across most of the mass range. However, radial mixing in a dynamic CGM can boost the O VI column densities to $N_{\rm OVI} \approx 10^{15} \, {\rm cm^{-2}}$ by producing large fluctuations in entropy and temperature that alter the ionization balance. Section \[sec-SpeculationCirculation\] considers the implications of that finding for CGM circulation, supernova feedback, and the dependence of the stellar baryon fraction on halo mass. Section \[sec-Summary\] summarizes the paper.
Precipitation-Limited CGM Models {#sec-CGM_Models}
================================
This section presents two simple models for a precipitation-limited CGM. Both invoke the $t_{\rm cool}/t_{\rm ff} \gtrsim 10$ criterion but make different assumptions about the potential wells and CGM entropy profiles resulting from cosmological structure formation. The first model was introduced by @Voit2018_LX-T-R, who used it to calculate $L_X$–$T$ relations for galaxy clusters and groups. It is extremely simple and serves here to illustrate the basic principles of precipitation-limited models. The second builds upon the first and is more suitable for predicting absorption-line column densities along lines of sight through the ambient CGM around lower-mass galaxies.
The pSIS Model
--------------
The simplest approximation to the structure of a precipitation-limited CGM assumes that the confining potential is a singular isothermal sphere (SIS) characterized by a circular velocity $v_c$ that is constant with radius. In that case, the corresponding cosmological baryon density profile without radiative cooling or galaxy formation would be $$\rho_{\rm cos} (r) = \frac {f_{\rm b} v_c^2} {4 \pi G r^2}
\; \; ,$$ where $f_{\rm b}$ is the cosmic baryon mass fraction. Gas with this density profile can remain in hydrostatic equilibrium in the SIS potential if it is at the gravitational temperature $kT_\phi \equiv \mu m_p v_c^2 / 2$, with an entropy profile $$K_{\rm SIS} (r) = \frac {\mu m_p} {2}
\left[ \frac {4 \pi G \mu_e m_p v_c} {f_b} \right]^{2/3}
r^{4/3}
\; \; .$$ The slight difference between the $K \propto r^{4/3}$ power-law slope of this approximate cosmological profile and the $K \propto r^{1.1}$ slope found in non-radiative numerical simulations of cosmological structure formation will be addressed in §\[sec-pNFW\].
As mentioned in the introduction, radiative cooling and the precipitation-regulated feedback that it fuels jointly prevent the ambient cooling time from dropping much below $10 t_{\rm ff}$. Together, these processes limit the ambient electron density to be no more than about $$n_{e,{\rm pre}} (r) = \frac {3 kT} {10 \, \Lambda(T)} \left( \frac {2 n_i} {n} \right)
\frac {v_c} {2^{1/2} r}
\; \; .
\label{eq-ne_pre}$$ A gas temperature $T = 2 T_\phi$ is required to maintain a gas density profile with $n \propto r^{-1}$ in hydrostatic equilibrium. Combining these expressions for density and temperature therefore gives a precipitation-limited entropy profile $$K_{\rm pre} (r) = ( 2 \mu m_p)^{1/3}
\left[ \frac {10} {3}
\left( \frac {2 n_i} {n} \right)
\, \Lambda (2 T_\phi) \right]^{2/3}
r^{2/3}
\label{eq-K_pre}$$ that expresses how the minimum specific entropy of the ambient CGM depends on radius. Notice that equations (\[eq-ne\_pre\]) and (\[eq-K\_pre\]) both assume $\min (t_{\rm cool}/t_{\rm ff}) = 10$, but the limiting $t_{\rm cool}/ t_{\rm ff}$ ratio may also be considered an adjustable parameter of the model. Observations of galaxy clusters with multiphase gas at their centers show that a large majority of them have $10 \lesssim \min ( t_{\rm cool} / t_{\rm ff} ) \lesssim 20$ [@Voit_2015Natur.519..203V; @Hogan_2017_tctff]. Sections \[sec-MilkyWay\] and \[sec-Columns\] therefore consider how the predictions of precipitation-limited CGM models change as $\min (t_{\rm cool} / t_{\rm ff})$ shifts through this range.
In the precipitation-limited CGM model originally introduced by @Voit2018_LX-T-R, which this paper will call the pSIS model, the ambient entropy profile is taken to be the sum of the SIS and precipitation-limited profiles: $$K_{\rm pSIS} (r) = K_{\rm SIS} (r) + K_{\rm pre} (r)
\; \; .$$ The assumed temperature profile, $$kT_{\rm pSIS} (r) = \frac {\mu m_p v_c^2 \cdot K_{\rm pSIS} (r)}
{2 K_{\rm SIS}(r) + K_{\rm pre}(r)}
\; \; ,$$ is designed to approach the appropriate limiting values at both small and large radii. Given these expressions for entropy and temperature, the precipitation-limited electron density profile in the pSIS model is $$n_{e,{\rm pSIS}} (r) = \left[ \frac {2 K_{\rm SIS}(r) + K_{\rm pre}(r)}
{\mu m_p v_c^2} \right]^{-3/2}
\; \; .$$ Multiplying $n_e$ by $2 r_{\rm proj}$ gives the characteristic electron column density along a line of sight through a spherical CGM at a projected radius $r_{\rm proj}$. This characteristic column density is nearly independent of $r_{\rm proj}$ within the precipitation-limited regions of the pSIS model.
The pNFW Model {#sec-pNFW}
--------------
Despite its extreme simplicity, the pSIS model makes accurate predictions for the X-ray luminosity-temperature relations among halos in the mass range $10^{12} \, M_\odot$–$10^{15} \, M_\odot$ [@Voit2018_LX-T-R]. However, if one would like to estimate circumgalactic column densities of O VI and Ne VIII, the pSIS model has some weaknesses. Primary among those weakness is its lack of a gas-temperature decline below $\max(T_\phi)$ at large radii. X-ray observations of galaxy clusters systematically show such a decline [e.g., @Ghirardini_2018arXiv180500042G], which stems in part from a drop-off in $v_c$ at larger radii and additionally from incomplete thermalization of the kinetic energy being supplied by the incoming accretion flow [e.g., @lkn09]. There is little direct evidence for a similar outer temperature decline in the ambient gas belonging to halos in the $10^{11} \, M_\odot$–$10^{13} \, M_\odot$ mass range, but if such a decline exists, it can significantly increase the predicted O VI and Ne VIII columns, relative to the pSIS model, along any given line of sight through the CGM of a Milky-Way-like galaxy.
Here we construct a slightly less simple alternative, the pNFW model, that addresses those weaknesses. It assumes a confining gravitational potential with a constant circular velocity at small radii, in order to represent the inner regions of a typical galactic potential well. At larger radii, the circular-velocity profile declines like that of an NFW halo [e.g., @nfw97] with scale radius $r_s$. These two circular-velocity profiles are continuously joined at the radius $2.163 r_s$, where the circular velocity of an NFW halo reaches its peak value. The overall circular-velocity profile is consequently flat at small radii, with $v_c (r) = v_{c,{\rm max}}$ for $r \leq 2.163 r_s$, and declines toward larger radii following $$v_c^2(r) = v_{c,{\rm max}}^2 \cdot 4.625 \left[ \frac {\ln (1 + r/r_s)} {r/r_s} -
\frac {1} {1 + r/r_s} \right]
\; \; .$$ A halo concentration $r_{200} / r_s = 10$ is assumed, implying that $v_c(r_{200}) = 0.83 \, v_{c,{\rm max}}$, with $r_{200}$ representing the radius encompassing a mean matter density 200 times the cosmological critical density $\rho_{\rm cr}$. This model gives $r_{200} = (237 \, {\rm kpc}) v_{200}$ and $M_{200} = (1.5 \times 10^{12} \, M_\odot) v_{200}^3$, for $v_{200} \equiv v_{c,{\rm max}} / 200 \, {\rm km \, s^{-1}}$ and $H = 70 \, {\rm km \, s^{-1} \, Mpc^{-1}}$, whereas the gravitational potential in the pSIS model gives $r_{200} = (286 \, {\rm kpc}) v_{200}$ and $M_{200} = (2.6 \times 10^{12} \, M_\odot) v_{200}^3$. The pSIS and pNFW models are therefore more appropriately compared at similar values of $v_{c,{\rm max}}$ than at similar values of $M_{200}$.
Within this potential well, the baseline entropy profile produced by non-radiative structure formation is taken to be $$K_{\rm base} (r) = 1.32 \, \frac {k T_\phi (r_{200})} {\bar{n}_{e,200}^{2/3}}
\left( \frac {r} {r_{200}} \right)^{1.1}
\; \; ,$$ where $\bar{n}_{e,200} \equiv 200 f_{\rm b} \rho_{\rm cr} / \mu_e m_p$ is the mean electron density expected within $r_{200}$ [@vkb05]. This expression simplifies to $$K_{\rm base} (r) = (39 \, {\rm keV \, cm^2}) \, v_{200}^2
\left( \frac {r} {r_{200}} \right)^{1.1}
\; \; ,$$ for $v_c(r_{200}) = 0.83 \, v_{c,{\rm max}}$, $H = 70 \, {\rm km \, s^{-1} \, Mpc^{-1}}$, and $f_{\rm b} = 0.16$. The modified entropy profile that results from applying the precipitation limit is then $$K_{\rm pNFW} (r) = K_{\rm base} (r) + K_{\rm pre} (r)
\; \; ,$$ with $kT_\phi = \mu m_p v_c^2(r) / 2$ used to determine $\Lambda (2 T_\phi)$ in the calculation of $K_{\rm pre}$ via equation (\[eq-K\_pre\]).
Gas temperature and density in the pNFW model are determined from $K_{\rm pNFW} (r)$ assuming hydrostatic equilibrium. The integration of $dP/dr$ to find $T(r)$ and $n_e(r)$ depends on a boundary condition that determines the pressure profile. Choosing $kT(r_{200}) = 0.25 \mu m_p v_{c,{\rm max}}^2$ ensures that the CGM gas temperature drops to roughly half the virial temperature near $r_{200}$, in agreement with observations of the outer temperature profiles of galaxy clusters [@Ghirardini_2018arXiv180500042G].
\
Figure \[fig-1\] compares the radial profiles of $K$, $T$, and $n_e$ predicted by the pSIS and pNFW models for $v_{c,{\rm max}} = 220 \, {\rm km \, s^{-1}}$. The entropy profiles predicted by the two models are nearly identical, but the pNFW model has a greater temperature gradient, primarily because of the smaller pressure boundary condition applied at $r_{200}$, but also because of the smaller circular velocity at that radius. Likewise, the density profile of the pNFW model diverges from that of the pSIS model as it approaches $r_{200}$, resulting in a steepening decline of the characteristic column density with radius. At radii larger than $r_{200}$, the precipitation limit is no longer physically well motivated, because the associated cooling times exceed the age of the universe, as indicated by the thin grey lines in the entropy panel.
Assumptions about Abundances
----------------------------
Inferences of observable CGM properties from the pSIS and pNFW models require supplementary assumptions about the total heavy-element content of the CGM and how it is distributed with radius. The precipitation framework does not constrain that radial distribution but does make predictions about how the total heavy-element content of the CGM should scale with halo circular velocity. @Voit_PrecipReg_2015ApJ...808L..30V developed models for precipitation-regulated galaxies that link their star-formation rates with enrichment of the CGM. In those simplistic models, all of the gas associated with a galaxy, including the CGM, is assumed to have a uniform metallicity. That assumption is what connects the condensation rate of the CGM, and therefore the galactic star-formation rate, to the enrichment of CGM gas. The resulting stellar mass-metallicity relationship broadly agrees with observations, and so we will adopt that relationship here. Our fiducial model therefore assumes that a galaxy like the Milky Way has a solar metallicity CGM.
However, the predicted absorption-line column densities of highly-ionized elements that emerge from precipitation-limited models are not particularly sensitive to assumptions about the metallicity. According to equation (\[eq-K\_pre\]), lowering the CGM abundances raises the limiting electron density, and therefore the total CGM column density, by lowering $\Lambda (T)$. As a result, the predicted column densities of highly-ionized elements have a dependence on abundance that is shallower than linear, as illustrated in Figure \[fig-2\]. The lines in that figure show how the column densities of O VII and O VIII predicted by pNFW models rise along lines of sight extending radially outward from a location 8.5 kpc from the center. Purple lines represent $N_{\rm OVIII}(r)$ and rise more rapidly at smaller radii because of the greater O VIII fraction there. Red lines represent $N_{\rm OVII}(r)$ and rise toward $\sim 10^{16} \, {\rm cm^{-2}}$ at larger radii, into the grey shading showing the range of Milky-Way $N_{\rm OVII}$ observations compiled by @MillerBregman_2013ApJ...770..118M. Notice that the oxygen column-density predictions of the pNFW models differ by less than a factor of 4, even though the oxygen abundance spans a factor of 10. Green symbols show the predictions at $r_{200}$ of a solar-abundance pSIS model, in which the CGM temperature exceeds the ambient temperature inferred from X-ray observations and leads to overpredictions of $N_{\rm OVIII}$ and underpredictions of $N_{\rm OVII}$.
\
Most of the following calculations assume that CGM abundances are independent of radius, but galaxy clusters and groups tend to have declining metallicity gradients, suggesting that CGM metallicity may also depend on radius in less massive galactic systems. In order to model the $L_X$–$T$ relations of galaxy clusters and groups, @Voit2018_LX-T-R assumed a metallicity gradient inspired by observations, with $Z(r)/Z_\odot = \min [ 1.0 , 0.3 (r/r_{500})^{-0.5} ]$, where $r_{500}$ is the radius encompassing a mean matter density $500 \rho_{\rm cr}$ and $Z_\odot$ represents solar abundances. This paper will call a model with that abundance gradient a “Zgrad" model.
One must also choose a standard “solar" oxygen abundance. Values that have been used as standards in recent years range from ${\rm O/H} = 4.6 \times 10^{-4}$ through ${\rm O/H} = 8.5 \times 10^{-4}$ [@AndersGrevesse_1989GeCoA..53..197A]. The lower values are in tension with helioseismology, while the higher ones are in tension with 3D solar-atmosphere models [e.g, @BasuAntia_2008PhR...457..217B]. This paper therefore adopts an intermediate value of ${\rm O/H} = 5.4 \times 10^{-4}$ as a standard.
A Milky-Way Comparison {#sec-MilkyWay}
======================
Comparing the precipitation-limited CGM models of §\[sec-CGM\_Models\] with available data on the Milky Way’s ambient CGM reveals a remarkable level of consistency, considering that the precipitation framework was originally developed to describe galaxy clusters and has simply been scaled down to a Milky-Way sized halo. Figure \[fig-3\] shows comparisons of $n_e(r)$ derived from pNFW models based on four different assumptions about the Milky Way’s CGM metallicity with a broad set of observational constraints. The observations generally imply electron density gradients that are similar to the pNFW models, which have $n_e \propto r^{-1.2}$ at small radii and $n_e \propto r^{-2.3}$ at large radii (see Figure \[fig-1\]). Differences in assumed abundances affect both the model predictions and most of the observational constraints on $n_e(r)$, but the models are generally most consistent with observations for abundances in the range $0.3 Z_\odot \lesssim Z \lesssim Z_\odot$. The rest of this section discusses in more detail those observational constraints and how they depend on assumptions about abundances.
\
Interstellar Medium Pressures
-----------------------------
Interstellar thermal gas pressures within a few kpc of the Sun can be robustly measured using ultraviolet observations of absorption lines arising from the three fine-structure levels of the carbon atom’s ground state [@JenkinsShaya_1979ApJ...231...55J]. In a comprehensive analysis of the available observational data, @JenkinsTripp_2011ApJ...734...65J found the mean thermal pressure of the local interstellar medium (ISM) to be $P_{\rm ISM} / k = 3800 \, {\rm K \, cm^{-3}}$, with a dispersion of 0.175 dex and a distribution having wings broader than those expected from a log-normal distribution. A green rectangle spanning a radial range of 6–10 kpc shows a corresponding range of electron densities derived assuming $n_e = 0.5 \, P_{\rm ISM} / k (2 \times 10^6 \, {\rm K})$.
The ISM thermal pressure can be considered an upper bound on the CGM thermal pressure at equivalent galactic radii. While additional forms of ISM support, such as turbulence, magnetic fields, and cosmic-ray pressure, may be comparable to the thermal pressure indicated by the C I lines, those same sources of additional pressure support are probably at least as important in the CGM. Given those uncertainties, the pNFW models with $10 \leq \min (t_{\rm cool}/t_{\rm ff}) \leq 20$ agree reasonably well with the ISM pressure constraint, with greater tension arising as the CGM abundances decrease. However, the pNFW models with $\min (t_{\rm cool}/t_{\rm ff}) \lesssim 20$ and abundances below $0.3 Z_\odot$ imply mean CGM pressures at $\sim 8$ kpc that are significantly greater than the observed ISM pressure.
X-ray Emission
--------------
Observations of soft X-ray emission over large portions of the sky consistently indicate that the emissivity-weighted temperature of the Milky Way’s hot ambient CGM is approximately $2 \times 10^6$ K [e.g., @KuntzSnowden_2000ApJ...543..195K; @McCammon_2002ApJ...576..188M; @Gupta_2009ApJ...707..644G; @Yoshino_2009PASJ...61..805Y]. For example, @HenleyShelton_2013ApJ...773...92H analyzed [*XMM-Newton*]{} spectra along 110 lines of sight through the Milky Way and found a fairly uniform median temperature of $2.2 \times 10^6 \, {\rm K}$ with an interquartile range of $0.63 \times 10^6 \, {\rm K}$. That range is shown with grey shading in the lower-left panel of Figure \[fig-1\]. It is consistent with the Milky Way CGM temperature predicted by the pNFW model for radii from 3 kpc to 70 kpc but is inconsistent with the pSIS model, which predicts hotter temperatures.
@HenleyShelton_2013ApJ...773...92H also found a spread in emission measure ranging over $\sim (0.4$–$7) \times 10^{-3} \, {\rm cm^{-6} \, pc}$, with a median of $1.9 \times 10^{-3} \, {\rm cm^{-6} \, pc}$, assuming solar abundances. Emission measure generally increases toward the center of the galaxy but is not strongly dependent on galactic latitude, indicating that the gas distribution is more spherical than disk-like.
@MillerBregman_2015ApJ...800...14M used an even larger sample of O VII and O VIII emission-line observations compiled by @HenleyShelton_2012ApJS..202...14H to constrain the radial density distribution of the line-emitting gas. They selected a subset of 649 [*XMM-Newton*]{} spectra sampling the entire sky and fit them with a model assuming constant-temperature gas at $T = 10^{6.3} \, {\rm K}$ and a power-law density profile $n_e \propto r^{-3 \beta}$. This isothermal power-law model yielded an excellent fit to the O VIII emission for $\beta = 0.50 \pm 0.03$, assuming optically-thin emission, and $\beta = 0.54 \pm 0.03$ after accounting for potential optical-depth effects.
Dashed blue lines in Figure \[fig-3\] show the best fit from @MillerBregman_2015ApJ...800...14M to optically-thin O VIII emission from a solar-abundance plasma. In the panels corresponding to $0.5 Z_\odot$ and $0.3 Z_\odot$, the density normalization of that fit has been multiplied by $(Z/Z_\odot)^{-1/2}$, because the line intensity scales $\propto n_e^2 (Z/Z_\odot)$. In the panel showing the Zgrad model, the abundance correction corresponds to a uniform abundance of $0.5 Z_\odot$. Each of the lines representing emission constraints extends from 9 kpc to 40 kpc because integrating over larger radii increases the emission measure by $< 10$%. Notice that the best-fitting power law from @MillerBregman_2015ApJ...800...14M has a slope ($n_e \propto r^{-1.5}$) that is similar to the pNFW models within that radial range but has a slightly lower normalization.
Solid blue lines in Figure \[fig-3\] show abundance-corrected versions of the best fit by @MillerBregman_2015ApJ...800...14M to optically-thin O VII emission, which has $\beta = 0.43 \pm 0.01$. Each of those lines therefore illustrates a density profile with $n_e \propto r^{-1.29}$, which is also quite similar to the electron-density profile shape predicted by the pNFW models in the 9 kpc to 40 kpc region. @MillerBregman_2015ApJ...800...14M report that their best fit to the O VII emission data is poorer than their best fit to the O VIII emission. In order to obtain acceptable $\chi^2$ values, they had to add systematic scatter of roughly a factor of 2 to their error budget, suggesting that the gas responsible for much of the O VII emission is inhomogeneous. Relatively modest variations in gas temperature within the observed temperature range can potentially produce that inhomogeneity, because the O VII ionization fraction rises by more than a factor of 6 as the CGM temperature declines through the @HenleyShelton_2013ApJ...773...92H range from $2.83 \times 10^6$ K to $1.57 \times 10^6$ K. Over the same temperature range, the O VIII ionization fraction changes by less than a factor of 2.
@MillerBregman_2015ApJ...800...14M hypothesized that the difference in power-law slope between the O VIII and O VII best fits may arise from a temperature gradient, because of how the ionization fractions change with temperature. Their Figure 13 shows that the necessary temperature gradient is approximately $T \propto r^{-0.08}$ if the temperature at 8.5 kpc is held fixed at $2 \times 10^6 \, {\rm K}$. In that same vicinity, the pNFW models have a temperature slope similar to $T \propto r^{-0.13}$, with a temperature $\approx 2.6 \times 10^6 \, {\rm K}$ at 8.5 kpc.
The broad cyan strips in Figure \[fig-3\] are based on the range of emission-measure observations found by @HenleyShelton_2013ApJ...773...92H and have a power-law slope $n_e \propto r^{-1.4}$, in between the slopes derived from Miller & Bregman’s O VII and O VIII best fits. Each cyan strip shows a range of electron density profiles corresponding to an emission-measure range $(1$–$4) \times 10^{-3} \, {\rm cm^{-6} \, pc}$, multiplied by a metallicity correction factor $[\Lambda(T,Z_\odot))/\Lambda(T, Z)]^{1/2}$ with $T = 2.2 \times 10^6 \, {\rm K}$. For all of the assumed metallicities, the high end of this emission-measure range is generally more consistent with the pNFW models than the low end. In that context, it is worth noting that the median emission measure from @HenleyShelton_2013ApJ...773...92H falls slightly below the emission measures found by some other studies [e.g., @Yoshino_2009PASJ...61..805Y; @Gupta_2009ApJ...707..644G].
X-ray Absorption
----------------
X-ray observations of O VII and O VIII absorption lines provide complementary constraints on the electron-density profile that help to break model degeneracies. @MillerBregman_2013ApJ...770..118M undertook a comprehensive analysis of the available O VII absorption data, which they extended in @MillerBregman_2015ApJ...800...14M. Assuming optically-thin absorption and a power-law density profile with a constant $n_{\rm OVII}/n_e$ ratio, they found a best-fit density profile with $\beta = 0.56_{-0.12}^{+0.10}$. When attempting to correct for saturation assuming an absorption-profile velocity width $b = 150 \, {\rm km \, s^{-1}}$, they found $\beta = 0.71_{-0.14}^{+0.13}$ for the whole data set and $\beta = 0.60_{-0.13}^{+0.12}$ using only the observations with signal-to-noise $> 1.1$. The lines of sight along directions that pass within $< 8.5$ kpc of the galactic center tend to receive the greatest saturation corrections, while the saturation corrections along lines of sight pointing away from the center tend be small. The $n_e \propto r^{-1.68}$ power-law density profile found without saturation correction may therefore be more representative of radii $> 8.5$ kpc, and this paper will adopt it for comparisons with the pNFW models.
Dot-dashed (orange-red) lines in Figure \[fig-3\] show the @MillerBregman_2013ApJ...770..118M electron density profiles derived from O VII absorption assuming no saturation. The abundance corrections are $\propto Z^{-1}$, with a correction for a uniform abundance of $0.5 Z_\odot$ applied in the Zgrad panel. Those power-law profiles have slopes similar to the pNFW models in the 8.5–200 kpc range and lie significantly below the pNFW predictions. However, the constant O VII ionization fraction of 0.5 assumed by @MillerBregman_2013ApJ...770..118M is inconsistent with the pNFW models, in which collisional ionization equilibrium gives an ionization fraction $n_{\rm OVII}/n_{\rm O} \lesssim 0.2$ at $< 10$ kpc and $n_{\rm OVII}/n_{\rm O} \gtrsim 0.4$ at $\sim 40$ kpc. The tendency for the O VII ionization fraction in the pNFW models to be $< 0.5$ at radii $\lesssim 30$ kpc can also be seen in Figure \[fig-2\], which shows that the cumulative O VIII column density along directions away from the galactic center rises more sharply with radius than the cumulative O VII column density in the 8.5 kpc to $\gtrsim 30$ kpc interval.
A proper comparison of the @MillerBregman_2013ApJ...770..118M data set with the pNFW models therefore requires an upward renormalization of the electron-density profiles derived from them. Brown strips in Figure \[fig-3\] show electron-density profiles with normalizations determined assuming collisional ionization equilibrium at the temperatures given by the pNFW models. All of the brown strips share the same power-law slope ($n_e \propto r^{-1.68}$) as the best fitting optically-thin model from @MillerBregman_2013ApJ...770..118M. The lower edge of each brown strip is normalized so that $N_{\rm OVII} = 5 \times 10^{15} \, {\rm cm^{-2}}$, corresponding to the low end of the @MillerBregman_2013ApJ...770..118M data set. The upper edge of each strip is normalized so that $N_{\rm OVII} = 1.5 \times 10^{16} \, {\rm cm^{-2}}$, which is the weighted mean column density found by @Gupta_2012ApJ...756L...8G, after they corrected for saturation. These brown strips generally agree well with the pNFW model in both normalization and slope, with slightly better agreement for sub-solar CGM metallicities.
@Gupta_2012ApJ...756L...8G also presented O VIII equivalent-width measurements, showing that they are comparable to the O VII equivalent widths. This finding is consistent with the pNFW models shown in Figure \[fig-2\], even though the ratio of O VII to O VIII is not constant with radius. More recently, Nevalainen et al. (2017) have published [*XMM-Newton*]{} absorption-line observations of O IV, O V, O VII, and O VIII along a particularly well-observed line of sight toward PKS 2155-304. They derived independent O VII and O VIII column-density measurements from the four different [*XMM-Newton*]{} detectors, finding column densities within a factor of two of $1 \times 10^{16} \, {\rm cm^{-2}}$ for both lines, again consistent with the pNFW models in Figure \[fig-2\] for CGM abundances in the $(0.3$–$1)~Z_\odot$ range.
LMC Dispersion Measure
----------------------
Observations of the dispersion measure toward pulsars in the Large Magellanic Cloud place an upper limit on the normalization of the $n_e$ profile within 50 kpc of the galactic center. @AndersonBregman_2010ApJ...714..320A found the dispersion measure attributable to the CGM in that radial interval to be no greater than $2.3 \times 10^{-2} \, {\rm cm^{-3} \, kpc}$. Grey lines with inverted triangles illustrate this upper limit in the four panels of Figure \[fig-3\], assuming a density profile with $n_e \propto r^{-1.2}$, as in the inner parts of the pNFW models.
@MillerBregman_2013ApJ...770..118M [@MillerBregman_2015ApJ...800...14M] showed that this upper limit, which does not depend on metallicity, places interesting constraints on the CGM metallicity when combined with inferences of $n_e(r)$ from their O VII and O VIII data sets. They found that a CGM metallicity $\gtrsim 0.3 Z_\odot$ was necessary to satisfy all of their constraints. Likewise, the LMC dispersion-measure limits place interesting constraints on the allowed metallicities of pNFW models for the Milky Way’s CGM. The entire $10 \leq \min ( t_{\rm cool} / t_{\rm ff} ) \leq 20$ range of solar-metallicity models can satisfy the dispersion-measure constraint, but the models come into increasing tension with the constraint as metallicity decreases. In the $0.3 Z_\odot$ case, only pNFW models with $\min ( t_{\rm cool} / t_{\rm ff} ) \gtrsim 20$ are permitted.
Ram-Pressure Stripping
----------------------
Additional metallicity-independent constraints come from ram-pressure stripping models of dwarf galaxies that orbit the Milky Way. Figure \[fig-3\] uses diamond-like polygons to illustrate those constraints. The horizontal span of each polygon shows the uncertainty in radius of the orbital pericenter; the vertical span shows the uncertainty in inferred CGM density at the pericenter. A purple polygon shows constraints derived from the LMC by @Salem_2015ApJ...815...77S. Red and blue polygons show constraints derived by @Gatto_2013MNRAS.433.2749G from the Carina and Sextans dwarf galaxies, respectively. Orange and blue polygons show constraints derived by @Grcevich_2009ApJ...696..385G from the Fornax and Sculptor dwarf galaxies, respectively. Constraints based on the other two dwarf galaxies modeled by @Grcevich_2009ApJ...696..385G are not shown because they are too weak to be interesting in this context. As a group, these ram-pressure constraints tend to be in tension with the uncorrected density profiles inferred from the O VII and O VIII data by @MillerBregman_2013ApJ...770..118M [@MillerBregman_2015ApJ...800...14M]. They are in better agreement with the pNFW models, particularly at the lower end of the CGM metallicity range allowed by the dispersion-measure constraints. The model with a metallicity gradient (pNFW-Zgrad) is the most successful at satisfying both the dispersion-measure and ram-pressure constraints.
High-Velocity Clouds
--------------------
Circumgalactic pressures can be derived from 21 cm observations of H I in high-velocity clouds with the help of assumptions about their distance and shape. If the clouds are roughly spherical, their extent along the line of sight can be estimated from their transverse size, given a distance estimate. A column-density measurement can then be converted into a density measurement, which becomes a pressure measurement when combined with information about the cloud’s temperature. The pressures inferred by from such observations of high-velocity clouds at distances $\sim 10$–15 kpc from the galactic center are $10^{2.7} \, {\rm K \, cm^{-3}} \lesssim P/k \lesssim 10^{3.1} \, {\rm K \, cm^{-3}}$. In Figure \[fig-3\], teal line segments bounded by triangles show the CGM density constraints that result from assuming that those clouds are in pressure equilibrium with an ambient medium at $2.2 \times 10^6 \, {\rm K}$. They tend to indicate ambient densities lower than those derived from other constraints, with increasing tension as the assumed CGM metallicity declines.
Magellanic Stream
-----------------
Similar constraints on ambient pressure can be derived from 21 cm observations of clouds in the Magellanic Stream. Inverted brown triangles in Figure \[fig-3\] show ambient density constraints that follow from pressure estimates by @Stanimirovich_2002ApJ...576..773S, who considered them upper limits on the actual thermal pressure because other forms of pressure, such as ram pressure, could also be contributing to cloud compression. Their electron-density constraint at an assumed distance of 45 kpc, which has been adjusted here for consistency with the $1.8 \times 10^6$ K ambient temperature predicted at that distance by pNFW models, is similar to the ambient densities inferred from ram-pressure stripping of dwarf galaxies.
CMB/X-ray Stacking
------------------
The final set of constraints shown in Figure \[fig-3\] is derived from galaxies more massive than the Milky Way. @Singh_stacks_2018MNRAS.478.2909S combined stacked X-ray observations of galaxies with halo masses in the $10^{12.6} M_\odot$–$10^{13.0} M_\odot$ range from @Anderson_2015MNRAS.449.3806A and stacked CMB observations from [*Planck*]{} in that same mass range . By jointly fitting those data sets with simple CGM scaling laws, @Singh_stacks_2018MNRAS.478.2909S found a best-fit density slope $n_e \propto r^ {-1.2}$ and a best-fitting CGM temperature scaling law that extrapolates to $\approx 2.2 \times 10^6 \, {\rm K}$ at the mass scale of the Milky Way.
Grey strips in Figure \[fig-3\] show where the best power-law fits of Singh et al. (2018) to CGM density profiles fall when extrapolated to a pNFW model with $M_{200} = 2 \times 10^{12} \, M_\odot$ and $v_{c,{\rm max}} = 220 \, {\rm km \, s^{-1}}$. Metallicity corrections have been made because the original power-law fits assumed a metallicity of $0.2 Z_\odot$. They have therefore been multiplied by $[\Lambda(T,0.2 Z_\odot))/\Lambda(T, Z)]^{1/2}$, with $T = 2.2 \times 10^6 \, {\rm K}$, to account for the effects of metallicity on X-ray emission. The strips span the radial range $(0.15$–$1) r_{500}$ because they are derived from projected data excluding the core region at $< 0.15 r_{500}$. The vertical span of each strip reflects an uncertainty range extending a factor of 1.6 in each direction, corresponding to the uncertainty range of the CGM baryonic gas fraction in the fits of @Singh_stacks_2018MNRAS.478.2909S.
Comparison Summary
------------------
\
Taken as a whole, these comparisons of observations with the pNFW models show that the CGM of the Milky Way is plausibly precipitation-limited, in a manner similar to the multiphase cores of galaxy clusters and central group galaxies, which also tend to have $10 \lesssim \min ( t_{\rm cool} / t_{\rm ff} ) \lesssim 20$. There are some points of tension with the data that need to be better understood through attempts to fit those data sets with parametric pNFW models. However, we will leave that task for the future. The main objective of this section has been to validate the pNFW models through comparisons with Milky Way before relying on them to make predictions for UV absorption lines from the CGM of galaxies with halo masses $10^{11} \, M_\odot$–$10^{13} \, M_\odot$.
For reference, Figure 4 shows how the total baryonic mass enclosed within a given radius rises toward radii $> r_{200}$. The stellar mass of this Milky-Way-like galaxy is assumed to be $7 \times 10^{10} \, M_\odot$, with a mass distribution giving $v_c = 220 \, {\rm km \, s^{-1}}$ at small radii. Gas-mass profiles ($M_{\rm CGM}$) in the figure are derived from pNFW models assuming $\min ( t_{\rm cool} / t_{\rm ff} ) = 10$. The total baryonic mass within $r_{200}$ predicted by the solar-abundance pNFW model corresponds to $\sim 30$% of the cosmic baryon fraction and rises to $\sim 40$% in the pNFW-Zgrad model. Potential contributions from the galactic ISM and lower-ionization phases of the CGM are not included in these estimates but are unlikely to close the baryon budget. Therefore, a galaxy like the Milky Way must push at least 50% of its baryons beyond $r_{200}$ in order to satisfy the precipitation limit.
Relationships to Similar Models
-------------------------------
Other models for the Milky Way’s CGM based on different assumptions have made comparable predictions. For example, the model of @Faerman_2017ApJ...835...52F assumes that the CGM is isothermal at $1.5 \times 10^6$ K with 60–$80 \, {\rm km \, s^{-1}}$ of turbulence and log-normal temperature fluctuations with a dispersion $\sigma_{\ln T} = 0.3$. Figure \[fig-4A\] shows that this isothermal model is similar to the pNFW model at 20–60 kpc but has a flatter electron-density profile and a larger CGM mass inside of $r_{200}$. Likewise, the isentropic CGM model of @MallerBullock_2004MNRAS.355..694M also has a flatter profile than the pNFW model and a greater CGM mass within $r_{200}$. Both of those other models are in considerable tension with the electron-density profiles inferred from X-ray spectroscopy by @MillerBregman_2013ApJ...770..118M [@MillerBregman_2015ApJ...800...14M]. In contrast, the idealized Milky-Way galaxy simulated by @Fielding_2017MNRAS.466.3810F, in which supernova-driven winds regulate the structure of the CGM, has an ambient density profile ($n_e \appropto r^{-1.5}$) consistent with the profile slopes derived from both X-ray spectroscopy and precipitation-limited models.
\
Ambient O VI Column Densities {#sec-Columns}
=============================
The preceding section demonstrated that precipitation-limited models for the Milky Way’s ambient CGM are compatible with the available observational constraints. This section uses those models to make predictions for the column densities of O VI, Ne VIII, and N V in the ambient CGM around galaxies in halos ranging from $10^{11} \, M_\odot$–$10^{13} \, M_\odot$, so that the precipitation framework can be tested with UV absorption-line observations. It first considers a static CGM with gas temperatures and ionization states that are uniform at each radius. Under those conditions, the models predict that the ambient CGM has $N_{\rm OVI} \approx 10^{14} \, {\rm cm^{-2}}$ over wide ranges in projected radius, halo mass, and CGM metallicity. However, the observed velocity structure of the O VI lines clearly shows that the CGM is not static. Gas motions in the CGM can produce temperature fluctuations that broaden the range of ionization states expected at each radius. This section shows that accounting for temperature fluctuations leads to O VI predictions that can rise as high as $N_{\rm OVI} \approx 10^{15} \, {\rm cm^{-2}}$ in $10^{12} \, M_\odot$ halos and may offer opportunities to probe how disturbances propagating through the CGM stimulate condensation and production of lower-ionization gas.
Static CGM {#sec-Static}
----------
\
The CGM models presented in §\[sec-CGM\_Models\] are completely hydrostatic, and so have a unique temperature at each radius. In collisional ionization equilibrium, that temperature determines the ion fractions at each radius. Integration along a CGM line of sight at a particular projected radius $r_{\rm proj}$ to find the column density of each ion is then straightforward but requires some assumptions about the limits of integration. The column density predictions presented here apply two limits. First, the spherical CGM models to be integrated are truncated at $2 r_{100}$, where $r_{100}$ contains a mean mass density $100 \rho_{\rm cr}$. This choice ensures that the line-of-sight integration does not extend far beyond the virialized region around the galaxy, outside of which the pNFW models are unlikely to be valid. Second, the integration is limited to within a physical radius of 500 kpc, since gas beyond that point is unlikely to be influenced by the central galaxy. This latter limit affects O VI column-density predictions for halos of mass $\gtrsim 10^{13} \, M_\odot$ but has negligible effects on smaller systems.
### Radial Profiles
Figure \[fig-5\] shows the radial profiles of $N_{\rm OVI} (r_{\rm proj})$ for pNFW models spanning the circular-velocity range $80 \, {\rm km \, s^{-1}} \leq v_{\rm c,max} \leq 350 \, {\rm km \, s^{-1}}$. The potential wells of all models have an identical shape, with $r_{200} / r_{\rm s} = 10$, and therefore all have $M_{200} = (1.5 \times 10^{12} \, M_\odot) v_{200}^3$, with a mass range $9.6 \times 10^{10} \, M_\odot \leq M_{200} \leq 8.0 \times 10^{12} \, M_\odot$.
Two features stand out: (1) the column-density profiles are generally flat to beyond 100 kpc, and (2) the characteristic column density is $N_{\rm O VI} \approx 10^{14} \, {\rm cm^{-2}}$ over the entire mass range. The flatness of the column-density profiles reflects two separate features of the pNFW models. First, the characteristic electron density profile at small radii is $n_e \propto r^{-1.2}$, as shown in Figure \[fig-1\]. Integrating density along lines of sight at a given projected radius therefore tends to give $N_{\rm CGM} \propto r_{\rm proj}^{-0.2}$. This result is close to the column-density profile slope in the middle column of Figure \[fig-5\]. Second, the primary contribution to the total O VI column density in some cases comes from radii $\gtrsim 100$ kpc, as shown by the black dotted lines in Figure \[fig-5\]. This circumstance arises when the temperature-dependent ionization correction for O VI is more favorable at large radii than at small radii. In those cases, $N_{\rm O VI}$ is nearly independent of $r_{\rm proj}$ to beyond 100 kpc because it is coming primarily from a thick shell at $\sim 100$ kpc.
### Scaling with Halo Mass {#sec-NOVI_scaling}
A simple scaling argument captures the essence of the insensitivity of $N_{\rm OVI}$ to halo mass. The total hydrogen column density along a line of sight through a precipitation-limited CGM is $$\begin{aligned}
N_{\rm H} & \, \approx \, & 2 n_e(r_{\rm proj}) \, r_{\rm proj}
\\
& \, \approx \, & \frac {2 r_{\rm proj}} {t_{\rm ff}(r_{\rm proj}) }
\left[ \frac {3kT} {10 \Lambda(T)} \right]
\label{eq-NH_step2}
\\
& \, \approx \, & \frac {3} {2^{1/2} 5}
\left[ \frac {\mu m_p v_c^3}
{\Lambda(2 T_\phi)} \right]
\label{eq-NH_step3}
\\
& \, \approx \, & 7 \times 10^{19} \, {\rm cm^{-2}}
\left( \frac {Z} {Z_\odot} \right)^{-0.7} v_{200}^{4.7}
\label{eq-NH_step4}
\; \; \end{aligned}$$ Equation (\[eq-NH\_step2\]) assumes $t_{\rm cool} / t_{\rm ff} = 10$. Equation (\[eq-NH\_step3\]) sets $T = 2 T_\phi$ in the cooling function, because the CGM temperature at small radii determines the radial structure of the ambient medium. Equation (\[eq-NH\_step4\]) assumes $\Lambda = 1.2 \times 10^{-22} \, {\rm erg \, cm^3 s^{-1}} (T/10^6 \, {\rm K})^{-0.85}(Z/Z_\odot)^{0.7} $, which approximates the cooling functions of @sd93 in the temperature range $10^{5.5} \, {\rm K} \leq T \leq 10^{6.5} \, {\rm K}$ and the abundance range $0.1 \leq Z/Z_\odot \leq 1.0$. Converting to an oxygen column density requires an expression for the oxygen abundance. This calculation assumes O/H = $5.4 \times 10^{-4} (Z/Z_\odot)$ at $v_{\rm c,max} = 200 \, {\rm km \, s^{-1}}$, so that $$N_{\rm O} \, \approx \, 4 \times 10^{16} \, {\rm cm^{-2}}
\left( \frac {Z} {Z_\odot} \right)^{0.3} v_{200}^{4.7}
\; \;
\label{eq-N_O}$$ The remaining step applies an O VI ionization correction. Fitting a power law to the O VI ionization fractions of @sd93 gives $f_{\rm OVI} = 0.006 (T/10^6 \, {\rm K})^{-2.3}$ in the temperature range $10^{5.5} \, {\rm K} \leq T \leq 10^{6.5} \, {\rm K}$. Gas at $r \sim 0.5 r_{200}$ and $T \approx T_\phi$ generally contributes the bulk of the O VI column density, and using a temperature $T = T_\phi$ to determine the O IV ionization fraction gives $$N_{\rm OVI} \, \approx \, 1 \times 10^{14} \, {\rm cm^{-2}}
\left( \frac {Z} {Z_\odot} \right)^{0.3} v_{200}^{0.1}
\; \; .$$ This value is indeed close to the characteristic column density of the profiles in Figure \[fig-5\] and has a negligible dependence on halo mass within the range corresponding to ambient temperatures between $10^{5.5}$ K and $10^{6.5}$ K. In other words, the halo-mass dependence of total column density in a precipitation-limited CGM ($N_{\rm H} \appropto M_{200}^{1.56}$) almost exactly offsets the steep decline in O VI ionization fraction ($f_{\rm OVI} \appropto M_{200}^{-1.53}$) within this mass range, while the precipitation condition mitigates the sensitivity of $N_{\rm OVI}$ to metallicity.
At the endpoints of this mass range, the pNFW model predictions assuming pure collisional ionization drop off. On the high-mass end, the increasing ambient temperature strongly suppresses the O VI ionization fraction [@Oppenheimer_2016MNRAS.460.2157O]. On the low-mass end, the ambient temperature becomes insufficient to produce observable O VI lines through collisional ionization. However, the thermal pressure in the precipitation-limited CGM of a halo with $M_{200} \lesssim 10^{11.5} M_\odot$ is $n_{\rm H} T \lesssim 5 \, {\rm K \, cm^{-3}}$ at $\gtrsim 50$ kpc, which is small enough for the metagalactic ionizing radiation at $z \sim 0$ to boost the O VI column density above the collisional-ionization prediction [e.g., @Stern_2018arXiv180305446S]. In that case, equation (\[eq-N\_O\]) gives an upper limit $N_{\rm OVI} \lesssim 10^{14} \, {\rm cm^{-2}} (Z/Z_\odot)^{0.3} (M_{200} / 10^{11} \, M_\odot)^{1.6}$, assuming $f_{\rm OVI} \lesssim 0.2$.
### Ne VIII and N V
\
Observations of Ne VIII and N V absorption lines can be used to test these models. The solid lines in Figure \[fig-6\] show how the ambient $N_{\rm NeVIII}/N_{\rm OVI}$ and $N_{\rm NV}/N_{\rm OVI}$ ratios in a static precipitation-limited CGM depend on halo mass. At $M_{200} \sim 10^{12} \, M_\odot$, Ne VIII absorption is predicted to be comparable to O VI, with $N_{\rm NeVIII} \sim 10^{14} \, {\rm cm^{-2}}$. However, the predictions for lower halo masses drop sharply because their ambient CGM temperatures are too low for significant Ne VIII absorption. In contrast, static pNFW models for $M_{200} \gtrsim 10^{11.5} \, M_\odot$ predict $N_{\rm NV}/N_{\rm OVI} \lesssim 0.1$ and $N_{\rm NeV} \sim 10^{13} \, {\rm cm^{-2}}$. The other lines in Figure \[fig-6\] illustrate the dynamic CGM models presented in §\[sec-Dynamic\].
These static-model predictions for $N_{\rm NeVIII}$ and $N_{\rm NV}$ generally agree with the available absorption-line data for the CGM in $10^{12} \, M_\odot$ halos. Observations of the COS-HALOS galaxies typically fail to detect N V [e.g., @Werk2016_ApJ...833...54W], giving mostly upper limits ($N_{\rm NV} \lesssim 10^{13.4-13.8} \, {\rm cm^{-2}}$) and just three detections with $N_{\rm NV}/N_{\rm OVI} \sim 0.1$. Fewer targets permit observations of Ne VIII absorption, but the existing detections cluster around $N_{\rm NeVIII} \sim 10^{14} \, {\rm cm^{-2}}$ [e.g., @Pachat_2017MNRAS.471..792P; @Frank_2018MNRAS.476.1356F; @Burchett_2018arXiv181006560B].
Dynamic CGM {#sec-Dynamic}
-----------
Dynamic disturbances in the CGM can alter the absorption-line predictions of precipitation-limited models by perturbing the ionization fractions at each radius. The typical velocity widths and centroid offsets of O VI lines from the central galaxy are indeed suggestive of sub-Keplerian disturbances and show that $N_{\rm OVI}$ is positively correlated with line width, as quantified by the Doppler $b$ parameter [@Werk2016_ApJ...833...54W]. Those findings motivate an extension of the pNFW model that allows for temperature fluctuations at each radius in the CGM.
### Temperature Fluctuations
\
The simplest extension assumes a distribution of gas temperatures having the same log-normal dispersion, $\sigma_{\ln T}$, at all radii [e.g., @Faerman_2017ApJ...835...52F; @McQuinnWerk_2018ApJ...852...33M]. Figure \[fig-7\] illustrates how such a dispersion affects the ion fractions when they are convolved with a log-normal temperature distribution, assuming collisional ionization equilibrium remains valid, a critical assumption that will be discussed in §\[sec-CIE\]. If it holds, the distribution of ion fractions at each radius broadens as $\sigma_{\ln T}$ increases, with greater effects on the minority ionization species. In particular, the O VI ionization fraction associated with gas at a mean temperature $\approx 10^6 \, {\rm K}$ rises by nearly an order of magnitude as the temperature dispersion approaches $\sigma_{\ln T} \approx 0.9$, causing a substantial increase in $N_{\rm OVI}$ if such a temperature dispersion is present in the CGM around real galaxies.
\
Figure \[fig-8\] shows how this extension alters the pNFW model predictions for CGM absorption lines at a projected radius of 50 kpc. The top panel presents $N_{\rm OVI}$ predictions, along with a set of predictions from the numerical simulations of @Oppenheimer_2018MNRAS.474.4740O. Both the pNFW predictions and the simulations feature a broad plateau at $N_{\rm OVI} \approx 10^{14} \, {\rm cm^{-2}}$ in the halo mass range $10^{11} \, M_\odot \lesssim M_{200} \lesssim 10^{13} \, M_\odot$, in accordance with the scaling argument in §\[sec-NOVI\_scaling\]. At $M_{200} \lesssim 10^{11} \, M_\odot$, the O VI predictions rapidly drop, because the ambient CGM temperature is not great enough to produce appreciable quantities of O$^{5+}$. However, these pNFW models do not account for production of O$^{5+}$ by photoionization, nor do they account for hot galactic outflows that may extend into the CGM at temperatures exceeding the virial temperature.
\
At $M_{200} \gtrsim 10^{11.7} \, M_\odot$, temperature fluctuations substantially enhance the ambient O VI column density of a precipitation-limited CGM. Figure \[fig-9\] compares those model predictions to a subset of COS-HALOS observations that were analyzed in detail by @Werk2016_ApJ...833...54W. They divided those observations into three categories. Two categories have low-ionization absorption lines coinciding in velocity with the O VI lines and were divided according to whether the O VI line was “broad" ($b > 40 \, {\rm km \, s^{-1}}$) or “narrow" ($b < 40 \, {\rm km \, s^{-1}}$). The third category, called “no-lows," consists solely of O VI absorption lines without associated low-ionization absorption. The “broad" category tends to have the strongest absorbers, with $10^{14.5} \, {\rm cm^{-2}} \lesssim N_{\rm OVI} \lesssim 10^{15} \, {\rm cm^{-2}}$, a level that has been difficult for simulations of the CGM to achieve [e.g., @Hummels_2013MNRAS.430.1548H]. However, the “broad" O VI absorbers agree well with pNFW models having $\sigma_{\ln T} \approx 0.7$, while the “narrow" absorbers are more consistent with nearly static pNFW models.
### Adiabatic Uplift
According to this model, the strongest CGM O VI absorption lines originate in ambient media with large temperature fluctuations. Outflows from the central galaxy can produce such fluctuations by lifting low-entropy gas to greater altitudes. It is not necessary for the uplifted gas to originate within the galactic disk. As in the cores of galaxy clusters, high-entropy bubbles that buoyantly rise through the ambient medium can lift lower-entropy CGM gas nearly adiabatically, either on their leading edges or within their wakes.
Uplifted gas that remains in pressure balance with its surroundings adiabatically cools as it rises, leading to temperature fluctuations with $$\sigma_{\rm \ln T} \approx \frac {3} {5} \sigma_{\ln K}
\; \; ,$$ where $\sigma_{\ln K}$ is the dispersion of entropy fluctuations resulting from uplift. Persistent temperature fluctuations with $\sigma_{\ln T} \gtrsim 0.6$ therefore imply a distribution of entropy fluctuations with $\sigma_{\ln K} \gtrsim 1$. In an adiabatic medium with a background profile $K \propto r^{2/3}$, entropy fluctuations of this amplitude can be achieved by lifting CGM gas a factor $\approx e^{3/2} \approx 5$ in radius.
### Internal Gravity Waves
One way to characterize the effects of CGM uplift is in terms of internal gravity waves, which oscillate at a frequency $\sim t_{\rm ff}^{-1}$. Internal gravity waves are thermally unstable[^1] in a thermally balanced medium with an entropy gradient $\alpha_K \equiv d \ln K / d \ln r \gg ( t_{\rm ff} / t_{\rm cool})^2$. Their oscillation amplitudes grow on a timescale $\sim t_{\rm cool}$ until they saturate with $\sigma_{\ln K} \sim \alpha_K^{1/2} ( t_{\rm ff} / t_{\rm cool})$ [@McCourt+2012MNRAS.419.3319M; @ChoudhurySharma_2016MNRAS.457.2554C; @Voit_2017_BigPaper]. Producing precipitation and multiphase gas in such a medium requires a mechanism that drives those oscillations nonlinear and then into overdamping, which leads to condensation.
@Voit_2018arXiv180306036V recently presented an analysis of circumgalactic precipitation showing that a gravitationally stratified medium with $K \propto r^{2/3}$ and $t_{\rm cool} / t_{\rm ff} \approx 10$ begins to produce condensates when forcing of gravity-wave oscillations causes the velocity dispersion to reach $\sigma_{\rm t} \approx 0.5 \sigma_v$, where $\sigma_v \approx v_c / \sqrt{2}$ is the one-dimensional stellar velocity dispersion corresponding to $v_c$. When expressed in terms of circular velocity, that critical velocity dispersion is $\sigma_{\rm t} \approx ( 70 \, {\rm km \, s^{-1}} ) v_{200}$, which is equivalent to $b \approx ( 100 \, {\rm km \, s^{-1}} ) v_{200}$ if thermal broadening is negligible. Gravity waves with that velocity amplitude in a CGM with $t_{\rm cool} / t_{\rm ff} \approx 10$ can no longer be considered adiabatic, because the gas in the low-entropy tail of the resulting entropy distribution has a cooling time comparable to $t_{\rm ff}$.
### Stimulation and Regulation of Condensation {#sec-Condensation}
Another way to view the significance of $\sigma_{\ln T} \gtrsim 0.6$ is in terms of isobaric cooling-time fluctuations, which have $$\sigma_{\ln t_{\rm cool}} \approx (2 - \lambda) \sigma_{\ln T}$$ in a medium with $\lambda \equiv d \ln \Lambda / d \ln T$. In the vicinity of $10^6 \, {\rm K}$, the cooling functions of @sd93 have $\lambda \approx -0.85$, implying $\sigma_{\ln t_{\rm cool}} \gtrsim 1.7$ in a medium with $\sigma_{\ln T} \gtrsim 0.6$. The low-entropy tail of such a distribution (more than $1 \sigma$ below the mean) has $t_{\rm cool} \lesssim 2 t_{\rm ff}$ if the mean ratio is $t_{\rm cool} / t_{\rm ff} \approx 10$. The lowest-entropy (shortest cooling-time) gas is therefore susceptible to condensation during a single gravity-wave oscillation. Larger temperature fluctuations, with $\sigma_{\ln T} \approx 0.9$ and $\sigma_{\ln t_{\rm cool}} \gtrsim 2.6$, imply that gas more than $1 \sigma$ below the mean cooling time has $t_{\rm cool} \lesssim 0.7 t_{\rm ff}$. In that case, a large fraction of the CGM would cool on a gravitational timescale.
Intriguingly, the ridge line of green squares representing “broad" O VI systems in Figure \[fig-9\] resides in the region corresponding to pNFW models with $0.6 \lesssim \sigma_{\ln T} \lesssim 0.9$. According to the preceding argument, this is exactly where forcing of gravity waves in a medium with a mean ratio $t_{\rm cool} / t_{\rm ff} \approx 10$ should drive it into precipitation. In the framework of precipitation-regulated feedback, the response of the galaxy should be a release of energy that raises the ambient $t_{\rm cool} / t_{\rm ff}$ ratio until it suppresses further precipitation. Low-ionization condensates might outlive the feedback event, while the CGM settles and the gravity waves damp. The “narrow" O VI systems of @Werk2016_ApJ...833...54W may be resulting from that damping process.
In the context of those interpretations of “broad" and “narrow" O VI systems, the “no-lows" would appear to arise from temperature fluctuations associated with gravity waves that are below the threshold for condensation. As a population, the “no-lows" have smaller line widths than the “broad" systems, with a mean $\langle b \rangle \approx 50 \, {\rm km \, s^{-1}}$ and $\max(b) \approx 70 \, {\rm km \, s^{-1}}$. The “broad" systems, in contrast, have $\langle b \rangle \approx 90 \, {\rm km \, s^{-1}}$ and $\max(b) \approx 160 \, {\rm km \, s^{-1}}$. Those characteristics are consistent with the notion that CGM gas within a $\sim 10^{12} \, M_\odot$ halo is driven into condensation when its velocity dispersion approaches $\sigma_{\rm t} \sim 70 \, {\rm km \, s^{-1}}$.
Collisional Ionization Equilibrium {#sec-CIE}
----------------------------------
Interpretations of the strong COS-HALOS O VI absorbers that rely on temperature fluctuations hinge on the assumption that ionization fractions remain near collisional ionization equilibrium as the CGM temperature fluctuates. If the fluctuations are produced on a dynamical timescale $\sim t_{\rm ff}$, then this assumption can be checked by comparing $t_{\rm ff}$ with the O VI recombination time of gas at the CGM’s $n_e$ and $T$. Figure \[fig-10\] shows such a comparison as a function of radius for pNFW models with $\min (t_{\rm cool} / t_{\rm ff}) = 10$ and an O VI recombination coefficient from the fits of Shull & van Steenberg (1982).
\
In a Milky-Way-like halo with $v_{\rm c,max} = 220 \, {\rm km \, s^{-1}}$, the O VI recombination time is short compared to the dynamical time at $\lesssim 100$ kpc, out to a radius depending on the CGM abundances. This dependence on abundance arises because a CGM with lower abundances can persist at greater density without violating the precipitation limit. Within such a halo, the assumption of collisional ionization equilibrium is valid for large-scale motions of CGM gas on a gravitational timescale, including internal gravity waves and slow outflows. However, it is not valid for temperature fluctuations associated with short-wavelength sound waves or small-scale turbulence.
The bottom two panels show that the assumption of collisional ionization equilibrium becomes more questionable in lower mass halos, because the precipitation-limited gas density at a given radius is substantially smaller. Consequently, the O VI recombination time in a halo with $v_c \lesssim 150 \, {\rm km \, s^{-1}}$ is long compared with the dynamical time, implying that the CGM in such a halo might not remain in collisional ionization equilibrium as adiabatic processes change its temperature. In that case, the O VI ion fractions would simply reflect the mean temperature of the ambient medium, unless the CGM pressure is low enough for photoionization to determine the O$^{5+}$ fraction. @Stern_2018arXiv180305446S have shown that photoionization dominates collisional ionization at $z \sim 0$ in a CGM with thermal pressure $n_{\rm H} T \lesssim 5 \, {\rm K \, cm^{-3}}$. Precipitation-limited pressures at $\sim 100$ kpc in halos with $M_{200} \lesssim 10^{11.5} \, M_\odot$ are lower than this threshold (see § \[sec-NOVI\_scaling\]), implying that the collisional-ionization assumption is not valid in the outer regions of those lower-mass halos.
Ambient temperature fluctuations therefore have the most consequential effects on $N_{\rm OVI}$ in systems with $v_{\rm c,max} \gtrsim 180 \, {\rm km \, s^{-1}}$, corresponding to $M_{200} \gtrsim 10^{12} \, M_\odot$. In that mass range, the response of O VI ionization to adiabatic cooling on a gravitational timescale is likely to be interesting and relevant. Coherent uplift of gas with a transverse extent comparable to the radius will then produce large, low-temperature structures in which O$^{5+}$ is enhanced. If the adiabatic temperature decrease is large enough, then the highest-density regions in those uplifted structures should have cooling times that lead to spatially correlated condensation, as discussed in §\[sec-ColdGas\].
Speculation about Circulation {#sec-SpeculationCirculation}
=============================
The observations analyzed in this paper are consistent with models in which energetic feedback heats the CGM, causing the medium to expand without necessarily unbinding it from the galaxy’s halo [e.g., @Voit_PrecipReg_2015ApJ...808L..30V]. According to those models, expansion must drive down the ambient CGM density so that it does not exceed the observed precipitation limit at $\min (t_{\rm cool} / t_{\rm ff}) \approx 10$. Otherwise, excessive condensation would lead to overproduction of stars. In such a scenario, the energy supply from the galaxy at the bottom of the potential well drives CGM circulation instead of strong radial outflows that escape the potential well. This section considers some of the potential implications of O VI absorption-line phenomenology within that context, showing that the implied supernova energy input can push much of the CGM beyond $r_{200}$, thereby regulating the fraction of baryons that form stars.
$N_{\rm OVI}$ and Active Star Formation
---------------------------------------
Actively star-forming galaxies are well-known to have O VI column densities roughly an order of magnitude greater than those around passive galaxies [@ChenMulchaey_2009ApJ...701.1219C; @Tumlinson_2011Sci...334..948T; @Johnson_OVI_2015MNRAS.449.3263J]. The models presented in this paper, particularly in Figure \[fig-9\], suggest that star formation enhances O VI absorption because the energetic outflows that star formation propels into the CGM produce temperature fluctuations with $\sigma_{\ln T} \sim 0.7$. Without a source of energy to cause fluctuations of that magnitude, the ambient CGM within a precipitation-limited halo of mass $10^{12} \, M_\odot \lesssim M_{200} \lesssim 10^{13} \, M_\odot$ should have $N_{\rm OVI} \approx 10^{13.5-14} \, {\rm cm^{-2}}$. This model prediction is consistent with the detections and upper limits observed around passive galaxies and implies that the greater O VI columns observed around star-forming galaxies signify circulation.
Circulation and Dissipation
---------------------------
Galactic outflows that lift low-entropy gas without ejecting it from the galaxy’s potential well inevitably drive circulation, because the low-entropy gas ultimately sinks back toward the bottom of the potential well. The rate of energy input required to sustain the level of circulation suggested by the O VI observations is substantial. For example, consider the CGM of a galaxy like the Milky Way, which has a mass $M_{\rm CGM} \sim 5 \times 10^{10} \, M_\odot$ within $r_{200}$ (see §\[sec-MilkyWay\]). Sustaining CGM circulation with a one-dimensional velocity dispersion $\sigma_{\rm t} \sim 70 \, {\rm km \, s^{-1}}$ and a characteristic circulation length $l_{\rm circ}$ requires a power input $$\begin{aligned}
\dot{E}_{\rm circ} & \; \approx \; & 2 \times 10^{41} \, {\rm erg \, s^{-1}}
\left( \frac {M_{\rm CGM}} {5 \times 10^{10} \, M_\odot} \right)
\nonumber \\
& ~ & \times
\left( \frac {\sigma_{\rm t}} {70 \, {\rm km \, s^{-1}}} \right)^3
\left( \frac {l_{\rm circ}} {100 \, {\rm kpc}} \right)^{-1}
\Gamma \end{aligned}$$ in order to offset turbulent dissipation of kinetic energy. In this expression, the quantity $\Gamma$ represents the dimensionless dissipation rate in units of $\sigma_{\rm t} / l_{\rm circ}$ and is of order unity.
This power input is similar in magnitude to the total supernova power of the galaxy ($\approx 3 \times 10^{41} \, {\rm erg \, s^{-1}}$ at a rate of $10^{51} \, {\rm erg}$ per century). If supernova-driven outflows are indeed responsible for stirring the CGM so that its circulation velocity remains $\sigma_{\rm t} \sim 70 \, {\rm km \, s^{-1}}$, then much of the supernova power generated within the galaxy must dissipate into heat in its CGM. Clustered supernovae that produce buoyant superbubbles may be required to transport that supernova energy out of the galaxy with the required efficiency [e.g., @Keller_2014MNRAS.442.3013K; @Fielding_2018arXiv180708758F]. Also, the inferred dissipation rate of CGM circulation exceeds the radiative luminosity of the CGM by more than order of magnitude. For example, integrating over the electron density profiles inferred by @MillerBregman_2013ApJ...770..118M [@MillerBregman_2015ApJ...800...14M] gives bolometric luminosity estimates $\lesssim 10^{40} \, {\rm erg \, s^{-1}}$ for the Milky Way’s CGM. These estimates imply that dissipation of CGM circulation in galaxies like the Milky Way adds heat energy to the CGM faster than it can be radiated away. The denser, low-entropy fluctuations may still be able to radiate energy fast enough to condense, but higher-entropy regions are likely to be gaining heat as the kinetic energy of CGM circulation dissipates. If so, then the ambient CGM responds to this entropy input by expanding at approximately constant temperature, and its expansion gently pushes the outer layers of the CGM beyond $r_{200}$.
Supernova Feedback and the Precipitation Limit
----------------------------------------------
Linking the heat input required to gently lift a galaxy’s CGM with the galaxy’s total output of supernova energy reproduces a scaling relation more commonly associated with galactic winds moving at escape speed. According to §\[sec-MilkyWay\], a galaxy like the Milky Way must push at least half of the baryons belonging to its halo outside of $r_{200}$ in order to satisfy the precipitation limit. The amount of energy necessary to lift those “missing" baryons to such an altitude is $\sim f_{\rm b} M_{200} v_c^2 \sim (2 \times 10^{59} \, {\rm erg}) v_{200}^5$, which is a significant fraction of all the supernova energy that a stellar population with $M_* \approx 7 \times 10^{10} \, M_\odot$ can produce. More generally, one can define $f_* \equiv M_* / f_{\rm b} M_{200}$ to be a galaxy’s stellar baryon fraction and $f_{\rm heat}$ to be the fraction of its supernova energy that is thermalized in the CGM. Requiring that heat input to lift a majority of the baryonic mass $f_{\rm b} M_{200}$ beyond $r_{200}$ then gives $$\begin{aligned}
f_* & \, \approx \, & \frac {v_c^2} {f_{\rm heat} \epsilon_{\rm SN} c^2} \\
& \, \approx \, & 0.2 \left( \frac {f_{\rm heat}} {0.5} \right)
\left( \frac {\epsilon_{\rm SN}} {5 \times 10^{-6}} \right)
v_{200}^2
\label{eq-fstar}
\; \; ,\end{aligned}$$ where $\epsilon_{\rm SN} \approx 5 \times 10^{-6}$ is the fraction of $M_* c^2$ that ultimately becomes supernova energy.[^2] Equation (\[eq-fstar\]) agrees with the Milky Way’s stellar mass fraction, given $v_c = 220 \, {\rm km \, s^{-1}}$. It also yields a dependence of stellar mass on halo mass ($M_* \propto M_{200}^{5/3}$) that aligns with the results of abundance matching in the mass range $10^{11} \, M_\odot \lesssim M_{200} \lesssim 10^{12} \, M_\odot$ [e.g., @Moster_2010ApJ...710..903M].
A similar result can be obtained by assuming that all of the accreting baryons ($f_{\rm b} M_{200}$) enter the central galaxy’s interstellar medium and fuel star formation that ejects a fraction $\eta/(\eta + 1)$ of the accreted gas, leaving behind a fraction $1/(\eta + 1)$ in the form of stars . If the scaling of the mass-loading factor $\eta$ is determined by requiring SN energy to eject the gas, then $\eta \propto v_c^{-2}$ and $f_* \propto v_c^2$ [@Larson_1974MNRAS.169..229L; @DekelSilk1986ApJ...303...39D].
However, a literal interpretation of the mass-loading scaling argument does not allow for recycling of gas through the CGM. Instead, it requires galactic winds to unbind a large fraction of a galaxy’s baryons from the parent halo, so that they do not return to the central galaxy. In contrast, the precipitation interpretation simply requires the supernova energy to regulate the recycling rate through subsonic pressure-driven lifting of the CGM. The precipitation interpretation therefore appears to be in better alignment with observations showing that the speeds of CGM clouds are usually sub-Keplerian [e.g., @Tumlinson_2011Sci...334..948T; @Zhu_2014MNRAS.439.3139Z; @Huang_2016MNRAS.455.1713H; @Borthakur_2016ApJ...833..259B] and simulations showing that a large proportion of the baryons that end up in stars have cycled at least once through the CGM [@Oppenheimer_2010MNRAS.406.2325O; @Angles-Alcazar_2017MNRAS.470.4698A].
Associated Low-Ionization Gas {#sec-ColdGas}
-----------------------------
Many of the intervening O VI absorption lines in quasar spectra are well-correlated in velocity with H I lines that have widths indicating a temperature $< 10^5$ K, far below the temperatures at which collisional ionization produces appreciable O$^{5+}$ [e.g., @Tripp_2008ApJS..177...39T; @ThomChen_2008ApJS..179...37T]. If the O VI absorbing gas is indeed cospatial with such cool H I gas, then it would have to be photoionized, and therefore at a pressure lower than the pNFW models presented here predict for the CGM in halos of mass $\gtrsim 10^{11.5} \, M_\odot$. However, most of the O VI absorbers in the COS-HALOS sample have low-ionization counterparts (e.g., C II, N II, Si II) indicating that the O VI gas might not be cospatial with the majority of the H I gas [@Werk2016_ApJ...833...54W].
Circulation that induces CGM precipitation is a potential origin for correlations in both velocity space and physical space among gas components that are not strictly cospatial. For example, consider an outflow that lifts ambient CGM gas by a factor of a few in radius over a large solid angle. The column density of uplifted gas would be comparable to the column density of the CGM itself. In a halo of mass $\sim 10^{12} \, M_\odot$, the adiabatic temperature drop in the uplifted gas would strongly enhance its O$^{5+}$ content, giving $N_{\rm OVI} \gtrsim 10^{14.5} \, {\rm cm^{-2}}$ (§\[sec-Dynamic\]). If the uplift were sufficient to make $t_{\rm cool} \sim t_{\rm ff}$ [*in the uplifted gas*]{} (see §\[sec-Condensation\]), then some of it would condense and enter a state of photoionization equilibrium before the uplifted gas could descend.
One likely result is “shattering" of the condensates into fragments of column density $N_{\rm H} \sim 10^{17} \, {\rm cm^{-2}}$. That is the maximum column density at which the sound crossing time remains less than the radiative cooling time as the gas temperature drops through $\sim 10^5$ K [e.g., @McCourt_2018MNRAS.473.5407M; @LiangRemming_2018arXiv180610688L]. Those fragments would collectively form a “mist" of low-ionization cloudlets embedded within the O VI absorber and would co-move with it. A cloudlet exposed to the metagalactic ionizing radiation at $z \sim 0$ would have a neutral hydrogen fraction $f_{{\rm H}^0} \approx 10^{-5.5} / U$ and column density $N_{\rm HI} \sim 10^{14.5} \, {\rm cm^{-2}} (U/10^{-3})^{-1}$, where the usual ionization parameter $U$ has been scaled to correspond with observations showing $-4 \lesssim \log U \lesssim -2$ in the low-ionization CGM clouds [@Stocke_2013ApJ...763..148S; @Werk_2014ApJ...792....8W; @Keeney_2017ApJS..230....6K].
The narrow H I absorption components associated in velocity with O VI absorption often have $10^{13.5} \, {\rm cm^{-2}} \lesssim N_{\rm HI} \lesssim 10^{15.5} \, {\rm cm^{-2}}$ [e.g., @Tripp_2008ApJS..177...39T], and are therefore are consistent with the presence of at least one and perhaps several such low-ionization cloudlets along a line of sight through a larger-scale O VI absorber. Many more cloudlets along a given line of sight would produce stronger H I absorption, but the precipitation model is not yet well-enough developed to predict either the total amount or the longevity of photoionized gas that would result from this condensation process. Certainly, the total column of low-ionization gas would not be greater than that of the ambient medium from which it originated. According to equation (\[eq-NH\_step4\]), the upper bound on the column density of low-ionization gas would be $N_{\rm H} \lesssim 10^{20} \, {\rm cm^{-2}}$, independent of projected radius, which accords with the upper bounds on $N_{\rm H}$ inferred from photoionization modeling [@Stocke_2013ApJ...763..148S; @Werk_2014ApJ...792....8W; @Keeney_2017ApJS..230....6K].
Photoionized clouds in pressure equilibrium with a hotter ambient medium have ionization levels determined by the ambient pressure. However, the pressure and density of low-ionization CGM clouds are currently somewhat uncertain because of uncertainties in the metagalactic photoionizing radiation [@Shull_UVB_2015ApJ...811....3S; @Chen_2017ApJ...842L..19C; @Keeney_2017ApJS..230....6K]. Some recent analyses favor an ionizing background at the high end of the uncertainty range [e.g. @Kollmeier_2014ApJ...789L..32K; @Viel_2017MNRAS.467L..86V], resulting in pressures and densities consistent with the ambient pressures predicted by precipitation-limited models. According to Figure 9 from @Zahedy_2018arXiv180905115Z, the relationship between gas density and ionization parameter for such a background is $n_{\rm H} \approx 10^{-5.4} \, {\rm cm^{-3}} / U$, giving $n_{\rm H} T \approx 40 \, {\rm K \, cm^{-3}}$ for $\log U \approx -3$ and $T \approx 10^4 \, {\rm K}$. For comparison, the ambient pressure at 100 kpc in the solar-metallicity pNFW model illustrated in Figure \[fig-1\] is $n_{\rm H} T \approx 40 \, {\rm K \, cm^{-3}}$; it rises to $400 \, {\rm K \, cm^{-3}}$ at $\approx 35$ kpc and drops to $4 \, {\rm K \, cm^{-3}}$ at $\approx 250$ kpc. Photoionization models of low-ionization CGM clouds with $-4 \lesssim \log U \lesssim -2$ are therefore completely consistent with pressure confinement by a precipitation-limited ambient medium, given current uncertainties in the metagalactic UV background [see also @Zahedy_2018arXiv180905115Z].
Summary {#sec-Summary}
=======
This paper has derived predictions for absorption-line column densities of O VI, O VII, and O VIII, plus N V and Ne VIII, from models in which susceptibility to precipitation limits the ambient density of CGM gas. Those models were inspired by observations showing that the $t_{\rm cool}/t_{\rm ff}$ ratio in the CGM around very massive galaxies rarely drops much below 10. Presumably, that lower limit on $t_{\rm cool}/t_{\rm ff}$ arises because ambient gas with a lower ratio is overly prone to condensation and production of cold clouds that accrete onto the galaxy and fuel energetic feedback that raises $t_{\rm cool}$.
Section \[sec-pNFW\] presented a prescription for constructing precipitation-limited models of the ambient CGM (i.e. “pNFW" models) that have declining outer temperature profiles similar to those observed in galaxy clusters and groups. Those new models are superior to the precipitation-limited models introduced by , which predict gas temperatures too hot to be consistent with X-ray observations of both emission and absorption by the Milky Way’s CGM. For the Milky Way, the pNFW models predict a CGM temperature $\gtrsim 2 \times 10^6 \, {\rm K}$ at $\lesssim 40 \, {\rm kpc}$ that declines to $\lesssim 1 \times 10^6 \, {\rm K}$ at $\gtrsim 200 \, {\rm kpc}$, as well as $N_{\rm OVII} \sim N_{\rm OVIII} \sim 10^{16} \, {\rm cm^{-2}}$ for $0.3 \lesssim Z/Z_\odot \lesssim 1.0$. Both findings are consistent with Milky Way observations. Given these temperatures and O VII column densities, the expected O VI column density of the Milky Way’s [*ambient*]{} CGM is $N_{\rm OVI} \sim 10^{14} \, {\rm cm}^{-2}$.
Section \[sec-MilkyWay\] provided further validation of the pNFW models by comparing them with a broad array of multi-wavelength Milky Way data. Collectively, the data indicate that the Milky Way’s CGM has an electron density profile between $n_e \propto r^{-1.2}$ and $n_e \propto r^{-1.5}$ from 10 kpc to 100 kpc, in agreement with the pNFW model predictions. As shown previously by @MillerBregman_2013ApJ...770..118M [@MillerBregman_2015ApJ...800...14M], combining the X-ray observations with upper limits on the dispersion measure of LMC pulsars places a lower limit of $Z \gtrsim 0.3 Z_\odot$ on the metallicity of the ambient CGM. The data are most consistent with a CGM having $10 \lesssim \min(t_{\rm cool}/t_{\rm ff}) \lesssim 20$ and a metallicity gradient going from $Z_\odot$ at $\sim 10$ kpc to $0.3 Z_\odot$ at $\sim 200$ kpc, with a total mass $\sim 5 \times 10^{10} \, M_\odot$ inside of $r_{200}$.
Section \[sec-Static\] then applied the pNFW model prescription to predict precipitation-limited O VI column densities for the ambient CGM in halos from $10^{11} \, M_\odot$ to $10^{13} \, M_\odot$, while assuming that the medium is static. Perhaps surprisingly, those models give $N_{\rm OVI} \approx 10^{14} \, {\rm cm^{-2}}$ across almost the entire mass range, with low sensitivity to metallicity. The lack of sensitivity to halo mass arises because the rise in total CGM column density with halo mass nearly offsets the decline in the O$^{5+}$ ionization fraction with increasing CGM temperature. The lack of sensitivity to metallicity arises because the total CGM column density in a precipitation-limited model is greater for lower metallicities. These static models also predict $N_{\rm NV} \sim 10^{13} \, {\rm cm^{-2}}$ and $N_{\rm NeVIII} \sim 10^{14} \, {\rm cm^{-2}}$ for the CGM in a $\sim 10^{12} \, M_\odot$ halo, in broad agreement with existing observational constraints.
Section \[sec-Dynamic\] relaxed the assumption of a static medium and considered the consequences of CGM circulation for O VI column densities. Circulation that lifts low-entropy CGM gas to greater altitudes causes adiabatic cooling that can raise the O$^{5+}$ fraction in an ambient medium with a mean temperature $> 10^{5.5}$ K. Around a galaxy like the Milky Way, circulation that produces isobaric entropy fluctuations with $\sigma_{\ln K} \gtrsim 1$ gives rise to temperature fluctuations with $\sigma_{\ln T} \gtrsim 0.6$ and boosts the O VI column density to $N_{\rm OVI} \gtrsim 10^{14.5} \, {\rm cm^{-2}}$, as long as the uplifted gas remains close to collisional ionization equilibrium. The corresponding fluctuations in cooling time have $\sigma_{\ln t_{\rm cool}} \gtrsim 1.7$, implying that the low-entropy tail of the distribution has $t_{\rm cool}/t_{\rm ff} \lesssim 1$, if the mean ratio is $t_{\rm cool}/t_{\rm ff} \approx 10$. The strongest O VI absorbers among the COS-HALOS galaxies are therefore plausible examples of CGM systems that circulation has driven into precipitation.
Section \[sec-SpeculationCirculation\] explored what the O VI absorption-line phenomenology may be telling us, if that interpretation is correct. Sustaining CGM circulation with $\sigma_{\rm t} \approx 70 \, {\rm km \, s^{-1}}$ on a length scale $\sim 100$ kpc requires a power input comparable to the total supernova power of a galaxy like the Milky Way. That may be why the CGM around a massive star-forming galaxy ($M_{200} \gtrsim 10^{12} \, M_\odot$) tends to have an O VI column density exceeding the $10^{13.5-14} \, {\rm cm^{-2}}$ value expected from a static precipitation-limited ambient medium and typically observed around comparably massive galaxies without star formation. A large cooling flow is not necessarily implied, because much of the O VI absorption can be coming from gas that uplift has caused to cool adiabatically rather than radiatively. If radiative cooling then causes a subset of that uplifted gas to condense, it will form small photoionized condensates embedded within a larger collisionally-ionized structure, accounting for the low-ionization absorption lines frequently observed to be associated in velocity with the strongest O VI lines.
More generally, requiring supernova energy input to expand the ambient CGM in the potential well of a lower-mass galaxy ($M_{200} \lesssim 10^{12} \, M_\odot$), so as to satisfy the precipitation limit, leads to the relation $f_* \approx 0.2 v_{200}^2$. The same scaling of stellar baryon fraction with circular velocity emerges from feedback models invoking mass-loaded winds driven by supernova energy, but in the precipitation framework those energy-driven outflows do not need to move at escape velocity and unbind gas from the halo. Instead, they drive dissipative circulation that causes the ambient CGM to expand subsonically, without necessarily becoming unbound.
Several observational tests of the precipitation framework emerge from these models:
- The most robust prediction is that the cooling time of the ambient CGM at radius $r$ in a precipitation-limited system should rarely, if ever, be smaller than 10 times the freefall time at that radius. As a consequence, a lower limit on the entropy profile $K(r)$ and an upper limit on the electron-density profile $n_e(r)$ can be calculated from the shape of the potential well within which the CGM resides. The Appendix provides fitting formulae for those limiting profiles in halos of mass $10^{11} M_\odot \lesssim M_{200} \lesssim 10^{13} M_\odot$ with CGM abundances ranging from $0.1 Z_\odot$ to $Z_\odot$. Table \[table-Fitting\] lists best-fit coefficients corresponding to $\min(t_{\rm cool}/t_{\rm ff}) = 10$, and also $\min(t_{\rm cool}/t_{\rm ff}) = 20$ for a sparser set of halo masses, because $\min(t_{\rm cool}/t_{\rm ff})$ is observed to range from 10 through 20 in higher-mass systems. (Greater lower limits on $t_{\rm cool}/t_{\rm ff}$ may apply in precipitation-limited systems that are rotating, because rotation at nearly Keplerian speeds significantly reduces the frequency of buoyant oscillations, thereby lengthening the effective dynamical time in the rotating frame.)
- Ambient temperatures in the central regions of precipitation-limited systems should be $T \approx \mu m_p v_c^2 / 1.2 k \approx (2.4 \times 10^6 \, {\rm K}) v_{200}^2$, because hydrostatic gas at the precipitation limit has $d \ln P / d \ln r \approx d \ln n_e / d \ln r \approx -1.2$. At larger radii, the gas temperature depends on the outer pressure boundary condition. Radial profiles of ambient gas temperature and pressure predicted by pNFW models can be calculated from the $K(r)$ and $n_e(r)$ fitting formulae in the Appendix. X-ray surface brightness predictions for imaging missions currently under development, such as Lynx and AXIS, can be derived from the $n_e(r)$ and $T(r)$ profiles for a given CGM metallicity.
- Out to radii $\sim 100$ kpc, the total hydrogen column density of a precipitation-limited CGM should be nearly independent of projected radius. Equation (\[eq-NH\_step4\]) predicts $N_{\rm H} \approx 7 \times 10^{19} \, {\rm cm^{-2}} (Z/Z_\odot)^{-0.7} v_{200}^{4.7}$ for a region in which $t_{\rm cool}/t_{\rm ff} \approx 10$ and $kT \approx \mu m_p v_c^2$. To obtain more precise $N_{\rm H}(r_{\rm proj})$ predictions, one can integrate over the $n_e(r)$ fits in the Appendix at a projected radius $r_{\rm proj}$.
- Multiplying $N_{\rm H}(r_{\rm proj})$ by the oxygen abundance gives a prediction for the total oxygen column density. For a region in which $t_{\rm cool}/t_{\rm ff} \approx 10$ and $kT \approx \mu m_p v_c^2$, equation (\[eq-N\_O\]) gives $N_{\rm O} \approx 4 \times 10^{16} \, {\rm cm^{-2}} (Z/Z_\odot)^{0.3} v_{200}^{4.7}$.
- Assuming collisional ionization equilibrium, one can derive $N_{\rm OVII}(r_{\rm proj})$ and $N_{\rm OVIII}(r_{\rm proj})$ from $N_{\rm O}(r_{\rm proj})$ by applying ionization corrections determined from $T(r)$. For galaxies like the Milky Way, pNFW models typically predict $N_{\rm OVI} \sim 2 \times 10^{16} \, {\rm cm^{-2}}$ for $\min(t_{\rm cool}/t_{\rm ff}) = 10$ and smaller values for larger $\min(t_{\rm cool}/t_{\rm ff})$. A spectroscopic X-ray observatory such as ARCUS would be capable of testing this prediction in the relatively near future [@Bregman_ARCUS_2018AAS...23123717B].
- The O VI absorption lines expected from ambient CGM gas in halos of mass $10^{11} M_\odot \lesssim M_{200} \lesssim 10^{13} M_\odot$ are currently observable, because the pNFW models predict $N_{\rm OVI} \gtrsim 10^{13.5} \, {\rm cm^{-2}}$ at nearly all projected radii (see Figure \[fig-5\]). The corresponding H I column density of the ambient medium is an order of magnitude smaller for a CGM metallicity $\sim Z_\odot$ (see Figure \[fig-7\]). If the medium is essentially static, the widths of those lines will be consistent with thermal broadening at the ambient temperature.
- Collisionally ionized gas in the ambient CGM should have $N_{\rm NV} \lesssim 0.1 N_{\rm OVI}$ in halos with $M_{200} \gtrsim 10^{11.5} \, M_\odot$ and $N_{\rm NeVIII} \approx N_{\rm OVI}$ in halos with $10^{11.7} \, M_\odot \lesssim M_{200} \lesssim 10^{12.5} \, {\rm cm^{-2}} \, M_\odot$ (see Figure \[fig-6\]).
- Circulation of CGM gas in halos of mass $\gtrsim 10^{11.7} M_\odot$ should cause $N_{\rm OVI}$ to correlate positively with the line width and/or its offset from the galaxy’s systemic velocity, because greater circulation speeds lead to greater fluctuations in specific entropy, temperature, and ionization state (see Figure \[fig-8\]). However, specific predictions for the relationship between line width and $N_{\rm OVI}$ require a more definite model for CGM circulation.
- Circulation that produces entropy fluctuations large enough for the low-entropy tail of the distribution to have $t_{\rm cool} \lesssim t_{\rm ff}$ will cause condensates to precipitate out of the ambient gas. @Voit_2018arXiv180306036V has shown that the threshold for condensation corresponds to a one-dimensional velocity dispersion $\sigma_{\rm t} \approx 0.35 v_c$ in a background medium with $10 \lesssim t_{\rm cool}/t_{\rm ff} \lesssim 20$. Low-ionization gas resulting from precipitation is therefore expected to have a dispersion of velocity offsets $\sim 70 \, {\rm km \, s^{-1}}$ at $M_{200} \approx 10^{12} \, M_\odot$ and $\sim 120 \, {\rm km \, s^{-1}}$ at $M_{200} \approx 10^{13} \, M_\odot$.
- The resulting mist of cloudlets will be photoionized by the metagalactic UV background, with an ionization level determined by the ambient CGM pressure, which can be calculated for pNFW models using the fitting formulae in the Appendix. Around a galaxy like the Milky Way, those models predict $n_{\rm H} T \approx 400 \, {\rm K \, cm^{-2}}$ at 35 kpc, $n_{\rm H} T \approx 40 \, {\rm K \, cm^{-2}}$ at 100 kpc, and $n_{\rm H} T \approx 4 \, {\rm K \, cm^{-2}}$ at 250 kpc, assuming $\min(t_{\rm cool}/t_{\rm ff}) = 10$. Those pressure predictions drop by a factor of two for $\min(t_{\rm cool}/t_{\rm ff}) = 20$.
- In lower-mass halos, the pNFW models predict smaller CGM pressures that may allow photoionization to produce the observed O VI column densities. At radii $\sim 100$ kpc in a halo with $M_{200} \lesssim 10^{11.5}$, the predicted CGM pressure is $n_{\rm H} T \lesssim 5 \, {\rm K \, cm^{-2}}$, and O$^{5+}$ is produced mainly by photoionization. In that limit, the pNFW models predict $N_{\rm OVI} \lesssim 10^{14} \, {\rm cm^{-2}} (Z/Z_\odot)^{0.3} (M_{200}/10^{11} \, M_\odot)^{1.6}$, based on multiplying $N_{\rm O}$ by $f_{\rm OVI} \lesssim 0.2$.
- The total column density of photoionized condensed gas cannot exceed that of the ambient medium. Equation (\[eq-NH\_step4\]) therefore places an upper limit of $N_{\rm H} \lesssim 7 \times 10^{19} \, {\rm cm}^{-2} (Z / Z_\odot)^{-0.7} v_{200}^{4.7}$ on the condensed phase, implying a joint dependence on halo mass and metallicity $\appropto Z^{-0.7} M_{200}^{1.6}$.
The author would like to thank J. Bregman, G. Bryan, J. Burchett, H.-W. Chen, M. Donahue, M. Gaspari, S. Johnson, N. Murray, B. Nath, B. Oppenheimer, B. O’Shea, M. Peeples, M. Shull, P. Singh, J. Stern, A. Sternberg, J. Stocke, T. Tripp, J. Tumlinson, J. Werk, and F. Zahedy for stimulating and helpful conversations. Jess Werk and Hsiao-Wen Chen receive extra credit for helpful comments on earlier drafts of the paper. Partial support for this work was provided by the Chandra Science Center through grant TM8-19006X.
Fitting Formulae for pNFW Profiles
==================================
A single power law provides a good fit to pNFW profiles for the CGM in halos with $10^{11} M_\odot \lesssim M_{200} \lesssim 10^{13} M_\odot$: $$K(r) = K_1 \left( \frac {r} {1 \, {\rm kpc}} \right)^{\alpha_K}
\label{eq-fit_K}
\; \; .$$ The electron-density profiles of pNFW profiles in the same mass range correspond more closely to a shallow power law ($n_e \propto r^{-\zeta_1}$ with $\zeta_1 \approx 1.2$) at small radii and a steeper power law ($n_e \propto r^{-\zeta_2}$ with $\zeta_2 \approx 2.3$) at larger radii (see Figure \[fig-1\]). These two limiting power laws can be joined using the fitting formula $$n_e(r) = \left\{
\left[ n_1 \left( \frac {r} {1 \, {\rm kpc}} \right)^{-\zeta_1} \right]^{-2}
+ \left[ n_2 \left( \frac {r} {100 \, {\rm kpc}} \right)^{-\zeta_2} \right]^{-2}
\right\}^{-1/2}
\label{eq-fit_ne}
\; \; .$$ Together, fitting formulae (\[eq-fit\_K\]) and (\[eq-fit\_ne\]) determine the temperature profile via $kT(r) = K(r) n_e^{2/3}(r)$ and the thermal-pressure profile via $P = (\mu_e / \mu) K(r) n_e^{5/3} (r)$. Table \[table-Fitting\] gives the best-fitting coefficients for some representative pNFW profiles.
$v_c \, ( {\rm km \, s^{-1}})$ $M_{200} \, (M_\odot)$ $\min(t_{\rm cool}/t_{\rm ff})$ $Z/Z_\odot$ $K_1$ $\alpha_K$ $ n_1 \, ( {\rm cm^{-3}}) $ $\zeta_1$ $n_2 \, ( {\rm cm^{-3}}) $ $\zeta_2$
-------------------------------- ------------------------ --------------------------------- ---------------- ------- ------------ ----------------------------- ----------- ---------------------------- -----------
350 $8.0 \times 10^{12}$ 10 1.0 3.5 0.72 $8.4 \times 10^{-2}$ 1.2 $2.9 \times 10^{-4}$ 2.1
350 $8.0 \times 10^{12}$ 10 0.5 2.7 0.73 $1.2 \times 10^{-1}$ 1.2 $3.8 \times 10^{-4}$ 2.1
350 $8.0 \times 10^{12}$ 10 0.3 2.4 0.74 $1.5 \times 10^{-1}$ 1.2 $4.2 \times 10^{-4}$ 2.1
350 $8.0 \times 10^{12}$ 10 $Z_{\rm grad}$ 3.6 0.68 $8.6 \times 10^{-2}$ 1.1 $4.1 \times 10^{-4}$ 2.0
300 $5.1 \times 10^{12}$ 10 1.0 3.3 0.71 $5.6 \times 10^{-2}$ 1.2 $1.7 \times 10^{-4}$ 2.1
300 $5.1 \times 10^{12}$ 10 0.5 2.6 0.72 $8.0 \times 10^{-2}$ 1.2 $2.3 \times 10^{-4}$ 2.1
300 $5.1 \times 10^{12}$ 10 0.3 2.3 0.73 $9.6 \times 10^{-2}$ 1.2 $2.6 \times 10^{-4}$ 2.1
300 $5.1 \times 10^{12}$ 10 $Z_{\rm grad}$ 3.4 0.67 $5.8 \times 10^{-2}$ 1.1 $2.6 \times 10^{-4}$ 2.1
300 $5.1 \times 10^{12}$ 20 1.0 5.2 0.70 $2.9 \times 10^{-2}$ 1.1 $1.0 \times 10^{-4}$ 2.1
300 $5.1 \times 10^{12}$ 20 0.5 4.1 0.71 $4.3 \times 10^{-2}$ 1.1 $1.4 \times 10^{-4}$ 2.1
300 $5.1 \times 10^{12}$ 20 0.3 3.6 0.71 $5.1 \times 10^{-2}$ 1.2 $1.6 \times 10^{-4}$ 2.1
300 $5.1 \times 10^{12}$ 20 $Z_{\rm grad}$ 5.4 0.65 $3.0 \times 10^{-3}$ 1.1 $1.6 \times 10^{-4}$ 2.1
250 $2.9 \times 10^{12}$ 10 1.0 3.7 0.71 $2.8 \times 10^{-2}$ 1.2 $8.1 \times 10^{-5}$ 2.2
250 $2.9 \times 10^{12}$ 10 0.5 2.8 0.72 $4.2 \times 10^{-2}$ 1.2 $1.1 \times 10^{-4}$ 2.2
250 $2.9 \times 10^{12}$ 10 0.3 2.4 0.72 $5.1 \times 10^{-3}$ 1.2 $1.3 \times 10^{-4}$ 2.2
250 $2.9 \times 10^{12}$ 10 $Z_{\rm grad}$ 3.7 0.66 $2.9 \times 10^{-2}$ 1.1 $1.3 \times 10^{-4}$ 2.1
220 $2.0 \times 10^{12}$ 10 1.0 4.0 0.70 $1.7 \times 10^{-2}$ 1.2 $4.2 \times 10^{-5}$ 2.3
220 $2.0 \times 10^{12}$ 10 0.5 3.0 0.71 $2.5 \times 10^{-2}$ 1.2 $6.1 \times 10^{-5}$ 2.2
220 $2.0 \times 10^{12}$ 10 0.3 2.6 0.71 $3.1 \times 10^{-2}$ 1.2 $7.2 \times 10^{-5}$ 2.2
220 $2.0 \times 10^{12}$ 10 $Z_{\rm grad}$ 4.1 0.65 $1.8 \times 10^{-2}$ 1.1 $7.4 \times 10^{-5}$ 2.2
220 $2.0 \times 10^{12}$ 20 1.0 6.3 0.69 $8.6 \times 10^{-3}$ 1.1 $2.3 \times 10^{-5}$ 2.3
220 $2.0 \times 10^{12}$ 20 0.5 4.8 0.70 $1.3 \times 10^{-2}$ 1.1 $3.4 \times 10^{-5}$ 2.2
220 $2.0 \times 10^{12}$ 20 0.3 4.1 0.70 $1.6 \times 10^{-2}$ 1.1 $4.2 \times 10^{-5}$ 2.2
220 $2.0 \times 10^{12}$ 20 $Z_{\rm grad}$ 6.5 0.63 $9.2 \times 10^{-3}$ 1.1 $4.3 \times 10^{-4}$ 2.1
180 $1.1 \times 10^{12}$ 10 1.0 5.6 0.71 $5.4 \times 10^{-3}$ 1.2 $9.6 \times 10^{-6}$ 2.2
180 $1.1 \times 10^{12}$ 10 0.5 4.0 0.71 $8.7 \times 10^{-3}$ 1.2 $1.6 \times 10^{-5}$ 2.2
180 $1.1 \times 10^{12}$ 10 0.3 3.4 0.71 $1.1 \times 10^{-2}$ 1.2 $2.1 \times 10^{-5}$ 2.2
180 $1.1 \times 10^{12}$ 10 $Z_{\rm grad}$ 5.7 0.63 $5.9 \times 10^{-3}$ 1.1 $2.3 \times 10^{-5}$ 2.2
150 $6.3 \times 10^{11}$ 10 0.5 6.1 0.68 $2.8 \times 10^{-3}$ 1.2 $7.4 \times 10^{-6}$ 2.3
150 $6.3 \times 10^{11}$ 10 0.3 4.9 0.68 $3.4 \times 10^{-3}$ 1.1 $9.8 \times 10^{-6}$ 2.2
150 $6.3 \times 10^{11}$ 10 0.1 3.0 0.69 $8.1 \times 10^{-3}$ 1.2 $1.9 \times 10^{-5}$ 2.2
120 $3.2 \times 10^{11}$ 10 0.5 6.1 0.69 $1.4 \times 10^{-3}$ 1.2 $2.3 \times 10^{-6}$ 2.2
120 $3.2 \times 10^{11}$ 10 0.3 5.0 0.70 $1.9 \times 10^{-3}$ 1.2 $2.9 \times 10^{-6}$ 2.2
120 $3.2 \times 10^{11}$ 10 0.1 3.1 0.71 $3.7 \times 10^{-3}$ 1.2 $5.1 \times 10^{-6}$ 2.2
120 $3.2 \times 10^{11}$ 20 0.5 9.6 0.69 $7.0 \times 10^{-4}$ 1.2 $1.2 \times 10^{-6}$ 2.2
120 $3.2 \times 10^{11}$ 20 0.3 7.9 0.70 $9.5 \times 10^{-4}$ 1.2 $1.5 \times 10^{-6}$ 2.2
120 $3.2 \times 10^{11}$ 20 0.1 4.9 0.71 $1.9 \times 10^{-3}$ 1.2 $2.7 \times 10^{-6}$ 2.2
: pNFW Fitting Formula Coefficients (for $r_{\rm s} = 0.1 r_{200}$)
\[table-Fitting\]
natexlab\#1[\#1]{}
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[^1]: Technically, they are overstable, because they oscillate.
[^2]: The numerical value corresponds to $10^{51} \, {\rm erg}$ of supernova energy per $100 M_\odot$ of star formation.
|
---
abstract: 'We discuss the dynamical deposition of the Na atom, the Na$^+$ in and the Na$-6$ cluster on finite Ar clusters mocking up an infinite Ar surface. We analyze this scenario as a function of projectile initial kinetic energy and of the size the target cluster.'
address:
- '$^1$Laboratoire de Physique Théorique, UMR 5152, Université Paul Sabatier, 118 route de Narbonne, F-31062 Toulouse Cedex, France'
- '$^2$Institut für Theoretische Physik, Universität Erlangen Staudtstr. 7, D-91058 Erlangen, Germany'
author:
- 'P. M. Dinh$^1$, F. Fehrer$^2$, P.-G. Reinhard$^2$, and E. Suraud$^1$'
title: Size and charge effects on the deposition of Na on Ar
---
Introduction
============
Clusters on surfaces have motivated many studies over the past decades and still remain a topic of great interest [@Bru00], especially in relation to the synthesis of nanostructured surfaces [@ISSPIC9; @ISSPIC10; @ISSPIC11; @ISSPIC12]. Indeed, it turns out that it is possible to make a direct deposition of size selected clusters on a substrate [@Bin01; @Har00]. The deposition process may lead to a significant modification of the cluster, in terms of its electronic structure and ionic geometry. This is a consequence of the impact of the interface energy, the electronic band structure of the substrate, and the surface corrugation. These questions have already been widely investigated in great detail, especially from the structural point of view and both from the experimental [@Exp1; @Exp2; @Exp3] and theoretical [@BL; @CL; @HBL; @MH; @SurfPot1; @SurfPot2; @2harm; @IJMS] sides. The situation is somewhat different concerning the deposition process itself. Its theoretical description remains mostly limited to molecular dynamics (MD) approaches, which implies that a proper description of electronic degrees of freedom is missing. The reason for this defect is simple. The presence of a substrate makes the experimental handling of clusters easier but strongly complicates the theoretical description because of the huge number of degrees of freedom of the surface. The MD then provides the cheapest way to access, at least in a gross way, the dynamics of the substrate. It is nevertheless crucial to try to account for the surface’s electronic degrees of freedom, for example when non adiabatic processes are involved.
As a first step in the direction of a fully microscopic dynamical approach, one may consider relatively simple cluster/substrate combinations. This is for example the case of the deposit of a metal cluster on an insulator surface. The surface can then be included at a lower level of description, which simplifies the handling, as was, e.g., explored for the case of Na clusters on NaCl in [@IJMS; @Ipa04]. Cluster electrons were there described by means of Density Functional Theory and the coupling to the surface was achieved [*via*]{} an effective interface potential, itself tuned to [*ab-initio*]{} calculations [@MH]. Such an approach implies a total freezing of the surface itself, which sets severe limitations on its applicability. A somewhat better description of surface degrees of freedom in a still limited/simplified way can be achieved by considering again a rather inert substrate, but allowing for a minimum of dynamical response of the substrate. Such a model was recently proposed for describing sodium clusters embedded in rare gases [@Ger04b; @Feh06a; @Feh05c]. The method used was “hierarchical”, which is justified by the moderate interactions between cluster and surface. The interface can then be treated at a simpler level than the cluster’s degrees of freedom (microscopic treatment of cluster and classical treatment of environment with explicit account of dynamical polarizability effects), in the spirit of the coupled quantum-mechanical with molecular-mechanical method (QM/MM) often used in bio-chemistry [@Fie90a; @Gao96a; @Gre96a].
Such hierarchical methods, although much simpler than a fully microscopic approach, require sophisticated modeling and are thus restricted to finite systems. Nonetheless, the calculations in [@Feh05c; @Feh06a] were carried forth to a sufficiently large range of sizes to see the appearance of generic behaviors on the way towards the bulk. In the complementing case of deposit on a surface, we also consider finite substrates, as model cases for a surface. This is an acceptable compromise once the impact of the finiteness of the substrate has been properly analyzed. For a first exploration, we went one step further and restricted the present analysis to the even simpler case of the deposition of a single Na atom on a finite Ar cluster, adding one test case in which we shall explore the case of a finite Na cluster.
The goal of this paper is thus mostly the study of the dynamics of deposition of a sodium atom (projectile) on Ar clusters of various sizes Ar$_N$(target, $N=43,86$). We shall analyze the behaviors of both the atom and the cluster, especially as a function of deposit velocity and also consider size and charge effects (deposit of a Na$_6$ cluster and a Na$^+$ ion instead of a Na atom). The paper is organized as follows. Section \[sec:model\] gives a short presentation of the model used. A few more details on the model can be found in the appendix, where basic formulae and parameters are recalled. The following sections successively address the dependence on substrate size and on projectile velocity. We finally discuss the example of a true cluster deposit and of charge effects.
Model {#sec:model}
=====
The model was presented in detail in [@Feh05a] and we provide the basic formulae in the appendix. We recall briefly the basic ingredients. The Na cluster is described in terms of time-dependent local-density approximation for the electrons coupled to molecular dynamics for the ions (TDLDA-MD), a scheme which has been extensively validated for linear and non-linear dynamics in free metal clusters [@Rei03a; @Cal00]. The electron-ion interaction is treated by means of a soft, local pseudo-potentials [@Kue99]. Each Ar atom is described by two classical degrees-of-freedom: its center-of-mass and its electrical dipole moment. The explicit account of Ar dipoles allows to treat the polarizability of the atoms dynamically, with help of polarization potentials [@Dic58]. Atom-atom interactions are described by a standard Lennard-Jones potential. For the Ar-Na$^+$ interaction we employ effective potentials from the literature [@Rez95]. The electron-Ar core repulsion is modeled in the form proposed by [@Dup96], with a slight final readjustment of the parameters to the NaAr molecule as benchmark (bond length, binding energy and optical excitation). A Van der Waals interaction is also added and computed [*via*]{} the variance of dipole operators [@Ger04b; @Feh05a; @Dup96].
As explained in the appendix, the starting quantity is the total energy, constructed from the various pieces discussed above. The corresponding equations of motion can then be derived by standard variation, which leads to the time-dependent Kohn-Sham equations for the cluster electrons. One furthermore obtains Hamiltonian equations of motion for the classical degrees of freedom (Na$^+$ ions as well as Ar atom positions and dipoles). The initial condition is provided by the corresponding stationary solutions.
The definition of the Ar-Na configuration requires some specific handling. Indeed the Na atom can be initially placed above either an Ar atom or an interstitial site of the surface layer. In the case of Ar$_{43}$ (see Figure \[fig:NaAr43\]), the first option has been chosen, and for Ar$_{86}$, the second initial configuration has been used, see Figure \[fig:config\]. The Ar$_{86}$ is obtained from the stable free Ar$_{87}$ cluster where an outer Ar atom has been removed. The resulting Ar$_{86}$ is then rotated in order to present a flat surface to the impinging Na atom, this way simulating the flat interface provided by an infinite surface. The starting configuration for the deposit of Na$_6$ on Ar$_{43}$ is similar to that of the single Na atom. This means the top Na ion (the Na$_6$ is composed of a ring of 5 ions and an outer ion) is placed above an Ar atom.
The numerical solution proceeds with standard methods as detailed in [@Cal00]. The Kohn-Sham equations for the cluster electrons are solved using real space grid techniques. The time propagation proceeds using a time-splitting method. The stationary solution is attained by accelerated gradient iterations. We furthermore employ the cylindrically-averaged pseudo-potential scheme (CAPS) as introduced in [@Mon94a; @Mon95a], an approximation justified for the chosen test cases. In the following, the symmetry axis is denoted by the $z$ axis. It should nevertheless be noted that the dynamics of the Na$^+$ ions as well as that of the Ar atoms are treated in full 3D.
Dynamical deposition of Na on finite Ar clusters
================================================
An example
----------
We first consider, as an example, the case of the deposition of a Na atom on an Ar$_{43}$ cluster. This is illustrated in Figure \[fig:NaAr43\] where a few snapshots of the deposition process are presented. In that test case, the Ar cluster presents to the Na atom a rather small surface area of only 5 atoms and the the Na atom is initially positioned above an Ar atom (which shows strong core repulsion). These two features differ as compared to the case of Ar$_{86}$ illustrated in Figure \[fig:config\] and discussed later on in Section \[sec:size\]. They should [*a priori*]{} make an attachment of the Na sodium more difficult here. Still, the calculation shows that a faint binding takes place in this case within typically 3 ps. The Na atom loosely attaches to the Ar cluster surface, since it keeps on bouncing over 7 ps with a decreasing amplitude and around an average position of about 8 a$_0$, even a bit closer than the NaAr dimer bond length of 9.5 a$_0$. Mind that the initial Na kinetic energy $E_0$ was 13.6 meV, in between the 5 meV binding energy of the NaAr dimer and the 50 meV of the bonding in Ar bulk. Thus there is no surprise to observe the creation of a transient NaAr bond and a very slight rearrangement of the Ar cluster, which takes place over a time scale of order of 5 ps. Actually, the Na atom is accelerated during its fall and gets, before the hit, a kinetic energy almost three times higher than $E_0$ (this depends on its initial separation with the Ar first layer). Then after the hit, the extra kinetic energy is absorbed by the Ar cluster both at the side of kinetic energy (about 15 meV) and in terms of potential energy in its structure rearrangement. The residual kinetic energy of Ar atoms as well as the potential energy consumed in the Ar cluster rearrangement strongly depend on the available energy and the number of degrees of freedom. It is thus interesting to study the effect of variations of both these parameters.
Dependence on kinetic energy {#sec:Ekin}
----------------------------
We first consider the influence of the initial kinetic energy of the Na projectile. The results are displayed in Figure \[fig:NaAr86\] for the deposition of a Na atom on Ar$_{86}$ at various energies (E$_0$ = 13.6 to 870 meV). It should first be noted that, whatever the initial kinetic energy no electronic emission is observed. We shall thus restrict the discussion to ionic and atomic degrees of freedom. In the case of the two lowest impact energies, one observes some bounces, and finally a binding of the Na to the surface at a typical distance of about $8-10$ $a_0$ from the first layer. However for higher E$_{0}$, the Na is reflected by the Ar cluster, and no attachment of the Na is observed, although the Na is initially located above an interstitial position. This result can be confirmed by computing the Born-Oppenheimer surface of Na on an infinite Ar surface. The ground state surface exhibits a faint minimum around 7 $a_0$ above the Ar surface, which is fully compatible with our dynamical analysis on the finite Ar$_{86}$ cluster. Closer to the Ar surface, the Born-Oppenheimer surface exhibits a strong repulsion reflecting the core repulsion between Na and Ar. When the impact kinetic energy becomes too large the Na atom thus “misses” the faint minimum and directly explores the strongly repulsive part of the potential. The Na atom is then reflected by the cluster as observed in Figure \[fig:NaAr86\]. The energy threshold for neutral Na sticking seems to lie between 0.05 and 0.2 eV. Note that in this range of kinetic energies, the Ar cluster is not affected very much. This is particularly visible on the time dependence of its kinetic energy, which exhibits a rather smooth energy transfer from the Na to the Ar cluster. We also observe the propagation of a soft shock wave in the substrate. This participates to the slight heating of the Ar cluster (to a few tens of K), except for the highest energy where some deep Ar atoms are emitted because of the stronger wave propagating through the layers.
Dependence on Ar cluster size {#sec:size}
-----------------------------
We now consider the influence of the number of degrees of freedom on the capacity to dissipate the available energy. In practice, this amounts to test the influence of the Ar cluster size for constant initial kinetic energy of the Na atom. The comparison is presented in Figure \[fig:matrix\_size\] where we plot Na and Ar positions and kinetic energies as a function of time during the deposition process of a Na atom (with initial kinetic energy E$_0$ of 13.6 meV) on Ar$_{43}$ and Ar$_{86}$. As already mentioned, two main differences between both cases are to be noted. First the Na atom is initially above an Ar atom of the Ar$_{43}$, at a distance of 20 $a_0$. In the case of Ar$_{86}$, the Na atom is above an interstitial site and starts at a smaller distance from the Ar first layer, namely 15 $a_0$. Due to their different sizes, the effective Ar surface exhibits only 5 atoms in the case of Ar$_{43}$ but 12 Ar atoms for Ar$_{86}$.
Similar oscillating patterns are observed in both cases. 2 ps after the impact, almost all kinetic energy of the Na atom is transfered to the Ar cluster, while the Ar kinetic energy seems to reach an equilibrium and oscillates around a mean value slightly higher than the initial kinetic energy $E_0$. Mind that the transferred energy is distributed equally over all Ar atoms. This means that, as expected, the larger the target cluster size (thus the more available degrees of freedom), the smaller the relative energy shared per Ar atom, and so the more moderate the perturbation at their side. Note, furthermore, that more Na kinetic energy is available at the time of impact (maximum of Na kinetic energy in the right panels) for the case of Ar$_{86}$. This means that the Na atom is accelerated faster in that case as compared with Ar$_{43}$. The reason is that the larger cluster and its larger surface area provide more attraction from polarization potentials. Nonetheless, in both cases, the Na atom loosely binds to the surface, at about the same distance of about 8 $a_0$.
Example of a finite cluster deposit
===================================
In order to complement the analysis in terms of Ar cluster size, it is also interesting to consider the case of the deposition of a full Na cluster, instead of a single atom. This again enhances the number of degrees of freedom. We shall consider comparable impact kinetic energy per Na atom. In Figure \[fig:Na6deposit\] we consider an example of a deposition of a finite Na cluster (Na$_6$) on Ar$_{43}$. The initial kinetic energy of the cluster is $E_{\rm kin0}=13.6$ meV per Na atom. As in the case of a single Na atom (Figure \[fig:NaAr43\]) the pinning process proceeds stepwise with a slight bounce before the metal cluster finally attaches to the rare gas cluster. At variance with the case of a single atom, though, one can furthermore analyze the evolution of the shape of the deposited Na cluster. The considered Na$_6$ cluster is primarily strongly oblate, consisting of a pentagon of 5 Na atoms topped by one central atom. One can see that during the deposition process, the cluster shape is little affected. This might have been expected in view of the ideal “flat” shape of the Na$_6$ cluster which already presents a large contact area to the Ar surface. But we also found for other, and less favorable, geometries (e.g. the nearly spherical Na$_8$ cluster) that the cluster shape remains basically intact during deposition. The details of this scenario of course depend on the size of the Ar cluster target and on the initial kinetic energy of the impinging Na cluster, but qualitatively the example displayed in Figure \[fig:Na6deposit\] turns out to be quite typical. These results will be presented elsewhere [@wet].
Charge effects
==============
As a final point, we want now to analyze the influence of charge on the deposition process. We come back to the simple case of a single Na atom and consider the deposition of the corresponding Na$^+$ ion on Ar$_{86}$, at various initial kinetic energies. Born Oppenheimer calculations of Na$^+$ in contact to an Ar surface show that the attachment is much stronger (and closer to the surface) than in the case of the neutral species. This can be easily understood by remembering the key role played by the attractive Ar polarization potentials. The finite charge of the Na$^+$ ion strongly polarizes the surface and thus enhances the binding as compared to the neutral case. Of course short range repulsion remains present but will take the lead only on shorter distances. The attachment is thus expected to be stronger and closer to the surface.
Figure \[fig:NapAr86\] confirms the expectation. The Na$^+$ ion is practically swallowed by the Ar cluster and the Ar cluster itself undergoes stronger rearrangements. For the smallest $E_0$ presented in the top panels of Figure \[fig:NapAr86\], two of the six surface atoms are finally ejected from the Ar cluster (they are visible by the light lines going straight through Ar layers in the $z$-coordinate panel and having reached the lower end at 7 ps). The four other Ar atoms of the (strongly perturbed) surface remain bound to the whole cluster and participate to its strong rearrangement. In the case of intermediate $E_0$, the six Ar atoms are lifted from the first layer but still stick to the edges of the Ar cluster. The fact that the Na$^+$ goes deeper for $E_0=54.4$ meV seems to be due to the final ejection of an Ar atom. In the latter three cases, these six atoms get about 10% of the total Ar kinetic energy. Finally, for the highest $E_0$, the first layer just explodes after the impact. The six outmost Ar atoms absorb up to 50% of the vertical shock in term of kinetic energy transfer, and then follow a radial motion in an horizontal plane (see the thin horizontal lines around $z=0$ in the bottom left panel of Fig.\[fig:NapAr86\]). In all cases, the removal of the six atoms from the first Ar layer, either outside or at the edges of the Ar cluster, allows some Ar atoms deeper inside the substrate to move upwards, thus leaving a large vacancy, so that the Na$^+$ can penetrate even under the second layer. These results suggest that the number of layers under the Ar surface probably does not play an important role in the Na$^+$ inclusion. More important is the surface’s mobility for rearrangements.
Conclusion
==========
In this paper, we have presented results on the deposition of a Na atom, a Na$^+$ ion, and a Na$_6$ cluster on a dynamically polarizable Ar substrate represented by finite Ar clusters of various sizes. We have used time-dependent density-functional theory for the Na electrons coupled to molecular dynamics for the treatment of Na ions and Ar atoms. We have presented systematic results as a function of Ar cluster size and kinetic energy of the impinging Na atom. We have found that the neutral Na is not likely to penetrate into the Ar matrix and sticks to the Ar surface for initial kinetic energy lower than $\sim 0.2$ meV while being reflected for larger impact kinetic energies. In case of the positively charged Na$^+$, inclusion is observed, whatever the initial kinetic energy. The Ar matrix (or the finite Ar cluster) then undergoes strong perturbations and ejects one or more atoms to create a vacancy for the Na$^+$ inclusion. As a first exploratory example, we have also studied the deposition of a neutral Na$_6$ cluster. As for the neutral atom, we see also a binding to the surface and no penetration into the Ar substrate. Somewhat surprisingly, there is only little perturbation of the Na cluster internal structure. Continued systematic investigations on metal cluster deposition in Ar substrate are in progress.
Acknowledgments: This work was supported by the DFG, project nr. RE 322/10-1, the French-German exchange program PROCOPE nr. 07523TE, the CNRS Programme “Matériaux” (CPR-ISMIR), Institut Universitaire de France, the Huomboldt foundation and a Gay-Lussac price, and has benefited from the CALMIP (CALcul en MIdi-Pyrénées) computational facilities.
\[sec:enfundetail\]
The degrees of freedom of the model are:\
------------------------------------------- -------------------------------------------
$\{\varphi_n({\bf r}),n=1...N_{\rm el}\}$ wavefunctions of cluster electrons
$\{{\bf R}_I,I=1...N_{\rm ion}\}$ coordinates of cluster’s Na$^+$ ions
$\{{\bf R}_a,a=1...N_{\rm Ar}\}$ coordinates of Ar atoms (cores Ar$^{Q+}$)
$\{{\bf R'}_a,a=1...N_{\rm Ar}\}$ coordinates of the Ar valence clouds
------------------------------------------- -------------------------------------------
An Ar atom is described by two constituents with opposite charge, positive Ar core and negative Ar valence cloud, which allows a correct description of polarization dynamics. We associate a Gaussian charge charge distribution to both constituents having a width of the order of the 3p shell in Ar, in the spirit of [@Dup96]. The dynamical polarizability of the Na$^+$ ions is neglected and we treat them simply as charged point particles.
The total energy of the system is composed as: $$E_{\rm total}
=
E_{\rm Na cluster}
+
E_{\rm Ar}
+
E_{\rm coupl}
+
E_{\rm VdW}
\quad,$$ The energy of the Na cluster $E_{\rm Na cluster}$ consists out of TDLDA (with SIC) for the electrons, MD for ions, and a coupling of both by soft, local pseudo-potentials, see [@Cal00; @Rei03a] for details. The Ar system and its coupling to the clusters is described by $$\begin{aligned}
\label{eq:Na-cluster}\\
E_{\rm Ar}
&=&
\sum_a \frac{{\bf P}_a^2}{2M_{\rm Ar}}
+
\sum_a \frac{{{\bf P}'_{a}}^2}{2m_{\rm Ar}}
+
\frac{1}{2} k_{\rm Ar}\left({\bf R}'_{a}-{\bf R}_{a}\right)^2
\nonumber\\
&&
+
\sum_{a<a'}
\left[
\int d{\bf r}\rho_{{\rm Ar},a}({\bf r})
V^{\rm(pol)}_{{\rm Ar},a'}({\bf r})
+
V^{\rm(core)}_{\rm ArAr}({\bf R}_a - {\bf R}_{a'})
\right]
\quad,
\\
E_{\rm coupl}
&=&
\sum_{I,a}\left[
V^{\rm(pol)}_{{\rm Ar},a}({\bf R}_{I})
+
V'_{\rm NaAr}({\bf R}_I - {\bf R}_a)
\right]
\nonumber\\
&&
+
\int d{\bf r}\rho_{\rm el}({\bf r})\sum_a \left[
V^{\rm(pol)}_{{\rm Ar},a}({\bf r})
+
W_{\rm elAr}(|{\bf r}-{\bf R}_a|)
\right]
\quad,
\\
V^{\rm(pol)}_{{\rm Ar},a}({\bf r})
&=&
e^2{q_{\rm Ar}^{\mbox{}}}
\Big[
\frac{\mbox{erf}\left(|{\bf r}\!-\!{\bf R}^{\mbox{}}_a|
/\sigma_{\rm Ar}^{\mbox{}}\right)}
{|{\bf r}\!-\!{\bf R}^{\mbox{}}_a|}
-
\frac{\mbox{erf}\left(|{\bf r}\!-\!{\bf R}'_a|/\sigma_{\rm Ar}^{\mbox{}}\right)}
{|{\bf r}\!-\!{\bf R}'_a|}
\Big]
\quad,
\label{eq:Arpolpot}
\\
W_{\rm elAr}(r)
&=&
e^2\frac{A_{\rm el}}{1+e^{\beta_{\rm el}(r - r_{\rm el})}}
\label{eq:VArel}\\
V_{\rm ArAr}^{\rm (core)}(R)
&=&
e^2 A_{\rm Ar}\Bigg[
\left( \frac{R_{\rm Ar}}{R}\right)^{12}
-\left( \frac{R_{\rm Ar}}{R}\right)^{6}
\!\Bigg]
\label{eq:VArAr}
\\
V'_{\rm ArNa}(R)
&=&
e^2\Bigg[
A_{\rm Na} \frac{e^{-\beta_{\rm Na} R}}{R}
-
\frac{2}{1+e^{\alpha_{\rm Na}/R}}
\left(\frac{C_{\rm Na,6}}{R^6} + \frac{C_{\rm Na,8}}{R^8}\right)
\Bigg]
\label{eq:VpArNa}
\\
&&
4\pi \rho_{{\rm Ar},a}
=
\Delta V^{\rm(pol)}_{{\rm Ar},a}
\\
E_{\rm VdW}
&=&
e^2\frac{1}{2} \sum_a \alpha_a
\Big[
\frac{
\left(\int{d{\bf r} {\bf f}_a({\bf r}) \rho_{\rm el}({\bf r})}\right)^2
}{N_{\rm el}}
- \int{d{\bf r} {\bf f}_a({\bf r})^2 \rho_{\rm el}({\bf r})}
\Big]
\;,
\label{eq:EvdW}
\\
&&
{\bf f}_a({\bf r})
=
\nabla\frac{\mbox{erf}\left(|{\bf r}\!-\!{\bf R}^{\mbox{}}_a|
/\sigma_{\rm Ar}^{\mbox{}}\right)}
{|{\bf r}\!-\!{\bf R}^{\mbox{}}_a|}
\quad.
\label{eq:effdip}
\\
&&
\mbox{erf}(r)
=
\frac{2}{\sqrt{\pi}}\int_0^r dx\,e^{-x^2}
\quad.\end{aligned}$$ The various contributions are calibrated from independent sources, with a final fine tuning to the NaAr dimer modifying only the term $W_{\rm elAr}$. The parameters are summarized in the table. The third column of the table indicates the source for the parameters.
[|l|l|l|]{}
------------------------------------------------------------------------
$V^{\rm(pol)}_{{\rm Ar},a}$ & $q_{\rm Ar}
=
\frac{\alpha_{\rm Ar}m_{\rm el}\omega_0^2}{e^2}$ , $k_{\rm Ar}
=
\frac{e^2q_{\rm Ar}^2}{\alpha_{\rm Ar}}$ , $m_{\rm Ar}=q_{\rm Ar}m_{\rm el}$ & $\alpha_{\rm Ar}$=11.08$\,{\rm a}_0^3$\
------------------------------------------------------------------------
& $\sigma_{\rm RG}
=
\left(\alpha_{\rm Ar}\frac{4\pi}{3(2\pi)^{3/2}} \right)^{1/3}$ &\
------------------------------------------------------------------------
$W_{\rm elAr}$ & $A_{\rm el}$=0.47 , $\beta_{\rm el}$=1.6941/a$_0$ , $r_{\rm el}=$2.2 a$_0$ & fit to NaAr\
------------------------------------------------------------------------
$V^{\rm(core)}_{\rm ArAr}$ & $A_{\rm Ar}$=$1.367*10^{-3}$ Ry , $R_{\rm Ar}$=6.501 a$_0$ & fit to bulk Ar\
------------------------------------------------------------------------
$V'_{\rm ArNa}$ & $\beta_{\rm Na}$= 1.7624 a$_0^{-1}$ , $\alpha_{\rm Na}$= 1.815 a$_0$ , $A_{\rm Na}$= 334.85 &\
------------------------------------------------------------------------
& $C_{\rm{Na},6}$= 52.5 a$_0^6$ , $C_{\rm Na,8}$= 1383 a$_0^8$ & after [@Rez95]\
The (most important) polarization potentials are described by a valence electron cloud oscillating against the raregas core ion. Its parameters are: $q_{\rm Ar}$ the effective charge of valence cloud, $m_{\rm Ar}=q_{\rm Ar}m_{\rm el}$ the effective mass of valence cloud, $k_{\rm Ar}$ the restoring force for dipoles, and $\sigma_{\rm Ar}$ the width of the core and valence clouds. The $q_{\rm Ar}$ and $k_{\rm Ar}$ are adjusted to reproduce the dynamical polarizability $\alpha_D(\omega)$ of the Ar atom at low frequencies, namely the static limit $\alpha_D(\omega\!=\!0)$ and the second derivative of $\alpha''_D(\omega\!''=\!0)$. The width $\sigma_{\rm Ar}$ is determined consistently such that the restoring force from the folded Coulomb force (for small displacements) reproduces the spring constant $k_{\rm Ar}$.
The short range repulsion is provided by the various core potentials. For the Ar-Ar core interaction we employ a Lennard-Jones type potential with parameters reproducing binding properties of bulk Ar. The Na-Ar core potential is chosen according to [@Rez95], within properly avoiding double counting of the the dipole polarization-potential.
The pseudo-potential $W_{\rm elAr}$ for the electron-Ar core repulsion has been modeled according to the proposal of [@Dup96] with a final slight adjustment to the Na-Ar dimer, data taken from from [@Gro98] and [@Rho02a].
The Van-der-Waals energy $E_{\rm VdW}$ is a correlation from the dipole excitation in the Ar atom coupled with a dipole excitation in the cluster. We exploit that $\omega_{\rm Mie}\ll\Delta E_{\rm Ar}$ which simplifies the term to the variance of the dipole operator in the cluster, using again the regularized dipole operator ${\bf f}_a$ corresponding to the smoothened Ar charge distributions. The full dipole variance is simplified in terms of the local variance.
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|
---
abstract: 'A general family of $D$-dimensional, $K$-state cellular automata is proposed where the update rule is sequentially applied in each dimension. This includes the Biham–Middleton–Levine traffic model, which is a 2D cellular automaton with 3 states. Using computer simulations, we discover new properties of intermediate states for the BML model. We present some new 2D, 3-state cellular automata belonging to this family with application to percolation, annealing, biological membranes, and more. Many of these models exhibit sharp phase transitions, self organization, and interesting patterns.'
address: 'Carnegie Mellon University, Robotics Institute, 5000 Forbes Ave, Pittsburgh, PA 15213'
author:
- 'Daniel L. Lu'
title: Generalization of elementary cellular automata to a higher dimension family including the BML traffic model
---
Cellular automata; Transport phenomena; Phase transitions; Self-organization; BML model
Introduction
============
Cellular automata are useful in a variety of problems related to statistical mechanics, traffic flow, and so on. The simplest sort of cellular automaton is the *elementary cellular automaton* which is one-dimensional with two states. These have interesting behaviour such as chaos and Turing-completeness, and are described extensively in Wolfram’s pioneering paper [@wolfram] as well as later works [@nks; @survey].
The Biham–Middleton–Levine (BML) traffic model was first proposed in 1992 to study traffic flow [@bml], as a 2D analog of the elementary Rule 184. It consists of a rectangular lattice, with periodic boundary conditions, where each site may be empty, contain a red car, or a blue car. On each time step: all red cars synchronously attempt to move one step east if the site is empty; then all blue cars synchronously attempt to move one step south if the site is empty. For some parameter $p\in [0,1]$, the model is initialized by assigning to each site either a red car with probability $p/2$, or a blue car with probability $p/2$, or empty space otherwise. The model is interesting because, for small values of $p$ it self organizes into a free flow where cars move freely without ever stopping, whereas for large $p$ it converges to a global jam. It was initially believed that there is a sharp transition between these two phases, but in 2005 a stable intermediate phase was discovered for lattices of coprime dimensions [@raissa5], and then shown in 2008 to exist for square lattices [@raissa8].
The BML model’s simplicity and interesting behavior has inspired research that tweak every aspect of the model [@chowd], such as dimensionality [@chowd; @hku], lattice geometry [@hex], boundary conditions [@chowd; @nonorientable; @bmlr], the update rule [@chowd; @hku; @bmlr; @randomturn; @overpasses; @lanechange; @slowtostart], initialization [@chowd; @accidents; @initialization], and so on. Many other cellular automaton models were proposed for modelling traffic [@chowd].
Although the BML traffic model was described as an “analog” of Rule 184, no attempt was made to formulate a rigorous description of the analogy, or a general method of finding such analogs of other elementary rules [@bml]. We show that other elementary cellular automata can be generalized into two or more dimensions with arbitrarily many states, by sequentially applying the same rule in each dimension. Historically, work on two-dimensional cellular automata have mostly focused on the von Neumann or Moore neighborhood, resulting in a large neighborhood. By applying the same rule in each dimension sequentially, we only have a neighborhood of size 3 regardless of dimensionality. In this paper, we describe an extension of Wolfram’s naming scheme for elementary cellular automata to higher dimensions and states; then we describe a new discovery in the BML traffic model; finally we describe and analyze a variety of different cellular automata with interesting behaviors and resembling various physical systems.
Theory
======
In elementary cellular automata, which have one dimension and two states, the update rule is represented as a single lookup table, which stores the state of the next cell as a function of itself and its two neighbors. The contents of the lookup table, when interpreted as a binary integer, is called the Wolfram Code [@wolfram]. For example, Rule 184 is a model of one-dimensional traffic flow where there is one type of car that attempts to move right if an empty site exists.
111 110 101 100 011 010 001 000
----- ----- ----- ----- ----- ----- ----- -----
1 0 1 1 1 0 0 0
Likewise, for two-dimensional, cellular automata with three states, the next cell’s state can be stored as a function of itself and its neighbors. However, the von Neumann neighborhood has five sites (including the center), yielding a lookup table of size $3^5 = 243$. This is unwieldy and expensive. The Moore neighborhood has nine sites and is even bigger. Due to the large size of neighborhoods as a function of dimensionality, many two-dimensional cellular automata, such as the well-known Conway’s Game of Life, simply sum up the values of their neighbors (an approach known as *totalistic* cellular automata [@wolfram]).
For 2D cellular automata, our approach sequentially applies two updates, each of which only considers three sites (yielding a lookup table of $3^3 = 27$). That is, we alternatingly apply an update to each row, and then each column, and so on. For example, consider the BML traffic model. Let state 0 be the empty space, 1 be the car species whose turn it is to move, and 2 be the car species whose turn it is not to move. The BML update rule can be explained as “cars of type 1 move to the next cell if possible, then turn into cars of type 2; cars of type 2 turn into cars of type 1”. Then, the table is:
222 221 220 212 211 210 202 201 200
----- ----- ----- ----- ----- ----- ----- ----- -----
1 1 1 2 2 0 0 0 0
122 121 120 112 111 110 102 101 100
----- ----- ----- ----- ----- ----- ----- ----- -----
1 1 1 2 2 0 2 2 2
022 021 020 012 011 010 002 001 000
----- ----- ----- ----- ----- ----- ----- ----- -----
1 1 1 2 2 0 0 0 0
The rulestring of the BML traffic model, in base 3, is therefore 111220000111220222111220000. In base 10, it is the more concise 3922832263383. This generalizes trivially to higher dimensions and higher number of states. For $D$ dimensions and $K$ states, the length of the rulestring, written in base $K$, is of length $K^3$. The total number of rules is therefore $K^{(K^3)}$. For the rest of the paper we will focus on $D=2$, $K=3$.
The family of cellular automata we have described here is a special case of a broader class of cellular automata with *cyclic* rules. Some of the assumptions in elementary cellular automata regarding symmetry and equivalence no longer hold when the rules are applied in different dimensions in sequence. For example, in elementary cellular automata, flipping all the ones and zeros makes little difference but for the BML rule above, if we change all the ones to twos and twos to ones, an entirely different model is created (see Section \[sandflow\]).
By formulating the rulestring of such models as an integer, we can explore other interesting rules by randomly selecting one. In the following sections we discuss the BML model as well as other models with different rules. The vast majority of rules, when initialized randomly like the BML traffic model, appear to be entirely random. However, although we are sure many of these rules have complex and interesting properties such as universal computing, analysis of such rules is very difficult and we will instead focus on the ones exhibiting a visually obvious structure.
Intermediate phases in the BML traffic model with extreme mobility
==================================================================
Our formulation of the model yields an efficient implementation on modern commodity hardware, as is discussed in Section \[imp\]. While experimenting, we discovered the existence of very rare phases. In the BML traffic model, *mobility* is the mean speed of cars – equivalently, the ratio of cell births to cell population. We report the existence of periodic intermediate phases with very low mobility $v<0.1$ as well as very high mobility $v>0.9$. In previous studies demonstrating the existence of such phases [@raissa5; @raissa8], it was thought they have a mobility of around $0.5$. Our results with extreme mobility levels suggest that stable intermediate phases may exist regardless of mobility level. Some examples are shown in Figure \[bmlex\]. Out of 2628 simulations of the BML traffic model on a $128\times 128$ grid with $p=0.36$, 2539 resulted in a global jam. None exhibited the so-called “disordered intermediate” state described in [@raissa8]. Due to the low number of non-jamming results, is unclear what exactly the distribution of intermediate phases with respect to mobility is — a histogram is shown in Figure \[bmlhist\].
![Two realizations of BML traffic model on a $128\times 128$ grid, with $p=0.36$, after $10^8$ iterations. From left to right: mobility $v\approx 0.91$, $v\approx 0.27$, $v\approx 0.02$.[]{data-label="bmlex"}](bml_high.png "fig:"){width="3cm"} ![Two realizations of BML traffic model on a $128\times 128$ grid, with $p=0.36$, after $10^8$ iterations. From left to right: mobility $v\approx 0.91$, $v\approx 0.27$, $v\approx 0.02$.[]{data-label="bmlex"}](bml_mid.png "fig:"){width="3cm"} ![Two realizations of BML traffic model on a $128\times 128$ grid, with $p=0.36$, after $10^8$ iterations. From left to right: mobility $v\approx 0.91$, $v\approx 0.27$, $v\approx 0.02$.[]{data-label="bmlex"}](bml_low.png "fig:"){width="3cm"}
![Histogram of mobility, $v$, for the 87 out of 2628 simulations of the BML traffic model on a $128\times 128$ grid with $p=0.36$ which did not result in a global jam.[]{data-label="bmlhist"}](hist.pdf){width="5cm"}
Percolation {#sandflow}
===========
A percolation cellular automaton is obtained by switching the 1 and 2 in the rulestring for the BML traffic model. In decimal, the rule number is 7469071910973. The intuitive explanation is: there are are two species of particles, where red ones alternatingly attempt to move downwards and then rightwards, and blue ones stay still. This can be thought of as another analog of Rule 184. This can be used to model the percolation of fluids or granular particles as they flow around obstacles or are deposited onto arbitrary surfaces; or even human pedestrians as they attempt to walk diagonally downwards and rightwards while avoiding obstacles.
We perform some experiments where the state is initialized in the same way as the BML traffic model, for which results are shown in Figure \[sandfig\]. Like the BML traffic model, the red sand particles self-organize into streams that avoid obstacles when density is low; otherwise, they reach total jams. Unlike the BML traffic model, there does not seem to be a sharp transition between phases. Even when density is low, some particles can be trapped in small local jams. A plot is shown in Figure \[sandv\]. There appears to be no dependence on lattice size. A simple calculation inspired by mean field theory provides a loose upper bound on $v$: At speed $v$, the ratio of the number of static particles (both red and blue) to the total space is $\frac{(1-v)p}{2} + \frac{p}{2}$ static particles. The mean velocity $v$ is the probability that a red particle is not next to a static particle. Assuming red particles to be uniformly distributed yields the self-consistency equation: $$\begin{aligned}
v = 1 - \frac{(1-v)p}{2} - \frac{p}{2} \rightarrow v = 2\frac{p-1}{p-2}\end{aligned}$$ This calculation overestimates $v$, because it assumes the mobile particles are isotropically distributed whereas in fact only blue particles (obstacles) are; the red particles instead form streams or clump up into jams. The formation of these clumps or local jams is complex and possibly chaotic, and an analytical description is difficult. In fact, when observing the simulation for $p=0.5$, such jams often accumulate transiently at different locations before dissolving over the next iterations, only to converge far later.
Percolation theory gives another upper bound. From Figure \[sandv\], it appears that there is a threshold at around $p\approx0.6$ beyond which the model almost surely results in a global jam. For large lattice sizes, percolation theory states that when the density of obstacles exceeds $1-p_c$, where $p_c$ is the *percolation threshold* [@perc; @perc2], then the lattice is no longer connected. Since the density of obstacles in our model is $\frac{p}{2}$, this implies there is no path across the lattice for any red particle when $p > 2(1-p_c)$. From past results $p_c = 0.593$, meaning our system will almost surely jam for $p>0.815$.
![Rule 7469071910973: Percolation model on a $300\times 300$ grid, initialized in the same way as the BML traffic model, with $p=0.1, 0.5, 0.7$ respectively, steady state.[]{data-label="sandfig"}](sandflow10.png "fig:"){width="3cm"} ![Rule 7469071910973: Percolation model on a $300\times 300$ grid, initialized in the same way as the BML traffic model, with $p=0.1, 0.5, 0.7$ respectively, steady state.[]{data-label="sandfig"}](sandflow50.png "fig:"){width="3cm"} ![Rule 7469071910973: Percolation model on a $300\times 300$ grid, initialized in the same way as the BML traffic model, with $p=0.1, 0.5, 0.7$ respectively, steady state.[]{data-label="sandfig"}](sandflow70.png "fig:"){width="3cm"}
![Graph of mobility, $v$, with respect to density $p$ for Rule 7469071910973, on square lattices of size 64, 128, 256, 512, and 1024, after $10^5$ iterations.[]{data-label="sandv"}](sandv.pdf){width="8cm"}
Annealing
=========
Here we present a simple model demonstrating behavior similar to annealing and bearing some superficial resemblance to the square lattice Ising model. It is Rule 2828173986213, which has the following ternary representation:
222 221 220 212 211 210 202 201 200
----- ----- ----- ----- ----- ----- ----- ----- -----
1 0 1 0 0 0 1 0 1
122 121 120 112 111 110 102 101 100
----- ----- ----- ----- ----- ----- ----- ----- -----
0 0 0 0 2 2 0 2 2
022 021 020 012 011 010 002 001 000
----- ----- ----- ----- ----- ----- ----- ----- -----
1 0 1 0 2 2 1 2 0
Put simply, if a site’s neighborhood contains only cells of one colour, the site assumes that colour. Otherwise, the site becomes empty. When the model is seeded with some initial distribution of red and blue cells (same as in the BML model), these seeds aggressively expand to cover sites in their vicinities, and soon the lattice contains some red and blue regions separated by thin borders of empty cells. These meta-stable regions have boundaries closely approximating orthogonal polygons whose sides are approximately $45^\circ$ to the horizontal. A side of such a polygon can move in the direction of its normal with speed inversely proportional to its length. As shorter sides move rapidly and join with parallel sides to be longer and stabler, the complexity of each of these regions is reduced over time and the model appears to be annealing (Figure \[annealfig\]).
The reason why the polygonal sides move with speed inversely proportional to length is because they are not in fact *exactly* oriented $45^\circ$ to the horizontal. Instead, the side’s slope is often off by one unit. The one “flaw” in the side propagates along the side back and forth, at a constant speed; at the end of each orbit of the flaw, the side effectively moves one unit. Since the period of the flaw’s orbit is proportional to the length of the side, the side advances at a speed inversely proportional to the length.
Ultimately, the model converges to either red or blue, or some simple intermediate phase (e.g. half red, half blue and separated by parallel boundaries), independent of $p$. There does not appear to be any surprising phase change behavior for this model.
![Rule 2828173986213: Simple annealing model on a $300\times 300$ grid, initialized in the same way as the BML traffic model, with $p=0.5$. Shown here from left to right is the same realization after timesteps $t=0, 1, 10, 100, 1090$ respectively.[]{data-label="annealfig"}](2828173986213_0.png "fig:"){width="3cm"} ![Rule 2828173986213: Simple annealing model on a $300\times 300$ grid, initialized in the same way as the BML traffic model, with $p=0.5$. Shown here from left to right is the same realization after timesteps $t=0, 1, 10, 100, 1090$ respectively.[]{data-label="annealfig"}](2828173986213_1.png "fig:"){width="3cm"} ![Rule 2828173986213: Simple annealing model on a $300\times 300$ grid, initialized in the same way as the BML traffic model, with $p=0.5$. Shown here from left to right is the same realization after timesteps $t=0, 1, 10, 100, 1090$ respectively.[]{data-label="annealfig"}](2828173986213_10.png "fig:"){width="3cm"} ![Rule 2828173986213: Simple annealing model on a $300\times 300$ grid, initialized in the same way as the BML traffic model, with $p=0.5$. Shown here from left to right is the same realization after timesteps $t=0, 1, 10, 100, 1090$ respectively.[]{data-label="annealfig"}](2828173986213_100.png "fig:"){width="3cm"} ![Rule 2828173986213: Simple annealing model on a $300\times 300$ grid, initialized in the same way as the BML traffic model, with $p=0.5$. Shown here from left to right is the same realization after timesteps $t=0, 1, 10, 100, 1090$ respectively.[]{data-label="annealfig"}](2828173986213_1000.png "fig:"){width="3cm"}
Membrane-like models
====================
In the previous section, somewhat nontrivial behavior along boundaries separating different phases was briefly discussed. More complex boundaries can form between more complex phases.
For some rulestrings, when the state is initialized in the same way as the BML traffic model, the cellular automaton rapidly partitions itself into two or more different phases separated by a membrane-like frontier. Like biological membranes, this can move in time, allow particles to cross it, merge with other membranes, and so on. A consequence is that the membranes are often transient or meta-stable. Depending on the initial choice of $p$, the model may ultimately converge to one of the different phases.
One such model is Rule 152690720768, shown in Figure \[152690720768\]. This model was found by random rulestring generation and then visual inspection. Its microscopic behavior is too complex to be studied in detail here, so instead we perform experiments to determine its sensitivity to $p$. The model appears to have a sharp transition around $p=0.68$, above which the blue phase dominates and below which the white phase dominates (Figure \[membraneplot\]). Intriguingly, there are red cells sparsely interspersed throughout both phases.
![Rule 152690720768. Typical transient state on a $300\times 300$ lattice with $p=0.66$.[]{data-label="152690720768"}](membrane.png){height="3cm"}
![Rule 152690720768. A plot of the percentage of lattice occupied by blue cells after $10^5$ iterations, on a $1024\times 1024$ lattice. The transition appears to occur at around $p=0.68$.[]{data-label="membraneplot"}](membrane.pdf){width="8cm"}
![Rules 5882493049933, 1811177701721, 5340268068864, 3894972317834, 2403954491737, 400697024, 3245024084244, 456351711232, simulated on a $300\times 300$ lattice.[]{data-label="interesting"}](5882493049933.png "fig:"){width="3cm"} ![Rules 5882493049933, 1811177701721, 5340268068864, 3894972317834, 2403954491737, 400697024, 3245024084244, 456351711232, simulated on a $300\times 300$ lattice.[]{data-label="interesting"}](1811177701721.png "fig:"){width="3cm"} ![Rules 5882493049933, 1811177701721, 5340268068864, 3894972317834, 2403954491737, 400697024, 3245024084244, 456351711232, simulated on a $300\times 300$ lattice.[]{data-label="interesting"}](5340268068864.png "fig:"){width="3cm"} ![Rules 5882493049933, 1811177701721, 5340268068864, 3894972317834, 2403954491737, 400697024, 3245024084244, 456351711232, simulated on a $300\times 300$ lattice.[]{data-label="interesting"}](3894972317834.png "fig:"){width="3cm"}\
![Rules 5882493049933, 1811177701721, 5340268068864, 3894972317834, 2403954491737, 400697024, 3245024084244, 456351711232, simulated on a $300\times 300$ lattice.[]{data-label="interesting"}](2403954491737.png "fig:"){width="3cm"} ![Rules 5882493049933, 1811177701721, 5340268068864, 3894972317834, 2403954491737, 400697024, 3245024084244, 456351711232, simulated on a $300\times 300$ lattice.[]{data-label="interesting"}](400697024.png "fig:"){width="3cm"} ![Rules 5882493049933, 1811177701721, 5340268068864, 3894972317834, 2403954491737, 400697024, 3245024084244, 456351711232, simulated on a $300\times 300$ lattice.[]{data-label="interesting"}](3245024084244.png "fig:"){width="3cm"} ![Rules 5882493049933, 1811177701721, 5340268068864, 3894972317834, 2403954491737, 400697024, 3245024084244, 456351711232, simulated on a $300\times 300$ lattice.[]{data-label="interesting"}](456351711232.png "fig:"){width="3cm"}
Implementation {#imp}
==============
Study of cellular automata such as the BML traffic model is severely restricted by available computing power. For a moderately sized $128\times 128$ lattice, each time step requires on the order of $10^4$ site updates. Supposing the model takes up to $10^6$ time steps to converge, this already takes several minutes on a typical CPU assuming a typical $10^8$ updates per second. Past studies have used purpose-built hardware such MIT’s CAM8 architecture [@raissa5], for which development has unfortunately stopped in 2001 due to a disk crash caused by “mindless jerks” [@cam8]. Even then, it took a whole month for the simulations to run. Later researchers have implemented the BML traffic model on a graphics processing unit (GPU) [@nonorientable], which is suitable for the task due to the embarrassingly parallel nature of the model.
Our implementation also uses the GPU to process several rows of the lattice in parallel. Like [@nonorientable], our implementation uses the CUDA technology; the graphics card used in our experiments is a single NVIDIA GeForce GTX 970, which has 1664 CUDA cores [@gtx970].
As a minor note, in the implementation it is important to use a good random number generator for large simulations. Many common programming languages have a built-in pseudorandom number generator that has a period of $2^{32}$, far too small for generating $10^5$ simulations of size more than $10^4$ each. Our implementation uses the Mersenne Twister with a period of $2^{19937}-1$.
Conclusions
===========
The main contribution of this paper is the proposal of a family of cellular automata operating on a neighborhood of size 3 for arbitrary $D$ dimensions and $K$ states, which operates by sequentially applying the same update rule in each dimension. This family includes the widely-studied BML traffic model, for which we present new results regarding the mobility of intermediate states. Furthermore, we have briefly discussed the properties of some other cellular automata within this family, and their possible application to various problems including percolation and annealing. The advantages of using a cellular automaton for such problems include the simplicity of the model and ease of implementation.
Future work includes a more thorough exploration of such models (Figure \[interesting\]), as well as models with different numbers of dimensions and states. In particular, although this paper talks about the $D=2, K=3$ case, the simpler case of $D=2, K=2$ is yet to be explored. For this, there are only 256 rules, the same number as elementary cellular automata. For implementation, a possible idea for future optimization is the Gigantic Lookup Table optimization [@glut], which groups adjacent sites into a block that is updated all at once.
Our implementation is available open source at `https://bitbucket.org/dllu/bml-cuda/`.
An interactive visualization for alternative rules is available at `http://www.dllu.net/bml/`.
References
==========
[99]{} Wolfram, Stephen. “Statistical mechanics of cellular automata.” *Reviews of modern physics* 55.3 (1983): 601. Wolfram, Stephen. *A New Kind of Science*, Wolfram Media (2002). Ganguly, Niloy, et al. “A survey on cellular automata.” (2003). Biham, Ofer, A. Alan Middleton, and Dov Levine. “Self-organization and a dynamical transition in traffic-flow models.” *Physical Review A* 46.10 (1992): R6124. D’Souza, Raissa M. “Coexisting phases and lattice dependence of a cellular automaton model for traffic flow.” *Physical Review E* 71.6 (2005): 066112. Linesch, Nicholas J., and Raissa M. D’Souza. “Periodic states, local effects and coexistence in the BML traffic jam model.” *Physica A: Statistical Mechanics and its Applications* 387.24 (2008): 6170-6176. Lau, Chi-yung. *Numerical studies on a few cellular automation traffic models.* Diss. The University of Hong Kong (Pokfulam, Hong Kong), 2002. Chowdhury, Debashish, Ludger Santen, and Andreas Schadschneider. “Statistical physics of vehicular traffic and some related systems.” *Physics Reports* 329.4 (2000): 199-329. Vázquez, J. Carlos García, Salvador Rodríguez Gómez, and Fernando Sancho Caparrini. “Biham-Middleton-Levine Traffic Model in Two-Dimensional Hexagonal Lattice.” *Proceedings of the European Conference on Complex Systems* 2012. Springer International Publishing, 2014. Cámpora, Daniel, et al. “BML model on non-orientable surfaces.” *Physica A: Statistical Mechanics and its Applications* 389.16 (2010): 3290-3298. Ding, Zhong-Jun, et al. “Effect of randomization in the Biham–Middleton–Levine traffic flow model.” *Journal of Statistical Mechanics: Theory and Experiment* 2011.06 (2011): P06017. Nagatani, Takashi. “Effect of jam-avoiding turn on jamming transition in two-dimensional traffic flow model.” *Journal of the Physical Society of Japan* 63.4 (1994): 1228-1231. Nagatani, Takashi. “Effect of traffic accident on jamming transition in traffic-flow model.” *Journal of Physics A: Mathematical and General* 26.19 (1993): L1015. Ding, Zhong-Jun, et al. “Effect of overpasses in the Biham-Middleton-Levine traffic flow model with random and parallel update rule.” *Physical Review E* 88.2 (2013): 022809. Li, Qi-Lang, et al. “Effect of vehicles’ changing lanes in the Biham–Middleton–Levine traffic flow model.” *International Journal of Modern Physics C* (2014). Fukui, Minoru, and Yoshihiro Ishibashi. “Effect of reduced randomness on jam in a two-dimensional traffic model.” *Journal of Physical Society of Japan.* 65 (1996): 1871-1873. Huang, Wei, et al. “Effect of Car Length in the Biham-Middleton-Levine Traffic Model.” *Future control and automation.* Springer Berlin Heidelberg, 2012. 479-487. Sui, Qiao-Hong, et al. “Slow-to-start effect in two-dimensional traffic flow.” *Computer Physics Communications* 183.3 (2012): 547-551. Levinshtein, M. E., et al. “The relation between the critical exponents of percolation theory.” *Soviet Journal of Experimental and Theoretical Physics* 42 (1976): 197. Feng, Xiaomei, Youjin Deng, and Henk WJ Blöte. “Percolation transitions in two dimensions.” *Physical Review E* 78.3 (2008): 031136. “CAM8: Latest Developments”. MIT Laboratory for Computer Science.\
`http://www.ai.mit.edu/projects/im/cam8/latest.html`\
Accessed 2014-12-22. “GeForce GTX 970 | Specifications| GeForce”. NVIDIA.\
`http://www.geforce.com/hardware/desktop-gpus/geforce-gtx-970/specifications`\
Accessed 2014-12-22. Tyler, Tim. “Gigantic lookup table (GLUT)”.\
`http://cell-auto.com/optimisation/#glut`\
Accessed 2014-12-22.
|
---
abstract: 'Defects in silicon carbide (SiC) have emerged as a favorable platform for optically-active spin-based quantum technologies. Spin qubits exist in specific charge states of these defects, where the ability to control these states can provide enhanced spin-dependent readout and long-term charge stability of the qubits. We investigate this charge state control for two major spin qubits in 4H-SiC, the divacancy (VV) and silicon vacancy (), obtaining bidirectional optical charge conversion between the bright and dark states of these defects. We measure increased photoluminescence from VV ensembles by up to three orders of magnitude using near-ultraviolet excitation, depending on the substrate, and without degrading the electron spin coherence time. This charge conversion remains stable for hours at cryogenic temperatures, allowing spatial and persistent patterning of the relative charge state populations. We develop a comprehensive model of the defects and optical processes involved, offering a strong basis to improve material design and to develop quantum applications in SiC.'
author:
- Gary Wolfowicz
- 'Christopher P. Anderson'
- 'Andrew L. Yeats'
- 'Samuel J. Whiteley'
- Jens Niklas
- 'Oleg G. Poluektov'
- 'F. Joseph Heremans'
- 'David D. Awschalom'
bibliography:
- './library.bib'
title: 'Optical charge state control of spin defects in 4H-SiC'
---
Optically active color centers in wide bandgap semiconductors have shown considerable potential for a variety of spin-based quantum technologies, from quantum computing and quantum memories [@Waldherr2014] to nano-scale sensing [@Maze2008a; @Toyli2013; @Kucsko2013]. Spin defects in silicon carbide (SiC) in particular combine the optical properties required for single-spin measurements ([@Baranov2011; @Koehl2011; @Kraus2013; @Christle2014; @Widmann2015; @Christle2017]) with wafer-scale growth and silicon-like fabrication capabilities developed for high-power electronics. However, optimizing these systems for spin qubit applications requires an understanding of not only their spin and optical properties, as demonstrated in the negatively charged silicon vacancy () [@Janzen2009; @Baranov2011] and the neutral divacancy () [@Torpo2002; @Son2006; @Falk2013a; @Christle2017], but also an understanding of their charge properties.
Impurities in SiC and their charge states have been investigated for conventional electronics applications, as they play an important role in transport properties and in carrier compensation. Most studies involve deep level transient spectroscopy (DLTS) [@Booker2014; @Booker2016], electron spin resonance (ESR) [@Matsumoto1997; @Isoya2008] and density functional theory (DFT) [@Gali2012; @Gordon2015; @Weber2011] with a strong focus on the carbon vacancy () [@Umeda2005; @Son2012; @Booker2016]; fewer works have addressed and VV defects [@Umeda2009]. For the purpose of quantum information, it is desirable to understand the complete physics of the defects themselves, not just their influence on transport or other electrical characteristics of the substrate.
Here, we investigate the effect of optical illumination on the stability of the relevant (optically bright) charge states of VV and , the ability to control and convert these states between different charge levels, and the implications for quantum applications. We investigate these questions using a combination of techniques including photoluminescence (PL), optically-detected magnetic resonance (ODMR) and electron spin resonance (ESR). The VV and charge states are both stabilized to the and states required to observe PL, whose intensity can be enhanced by up to three orders of magnitude depending on the material (local defect concentrations and Fermi level). For VV in particular, we observe bidirectional charge conversion between the neutral (bright qubit state) and negative charge states using mainly near-ultraviolet (365-405 nm) and near-infrared (976 nm) light. This charge conversion is stable at cryogenic temperature and does not affect the ODMR contrast nor the electron spin coherence time, and can therefore be readily applied to increase PL emission from ensembles.
Charge state conversion can have multiple origins, including direct photoionization, free carrier recombination, and charge transfer between defects. In order to fully understand the involved processes, we measure the charge dynamics of VV, and nitrogen (N) under illumination, where N is the main dopant in our semi-insulating 4H-SiC samples. Excitation dependence with wavelengths ranging from 365 nm to 1310 nm were measured and simulated, offering a comprehensive picture of charge transfer between these defects. In particular, this allows us to identify that converts to the dark charge state under 976 nm illumination, while will convert to the dark charge state with above-bandgap light.
Control and understanding of these charge dynamics is crucial for maximizing spin qubit readout, choosing adequate background impurity concentrations in samples and optimizing designs of SiC nano-devices for quantum applications. Such methods have also been applied in the nitrogen-vacancy (NV) center in diamond for quantum optics applications [@Aslam2013], enabling for example reduced spectral diffusion [@Siyushev2013] or Stark tuning of the optical transitions through photoexcitation of trapped charges [@Bassett2011]. More exotic applications of charge dynamics include high density data storage [@Dhomkar2016], STED super-resolution imaging [@Han2010; @Chen2015] and charge quantum buses [@Doherty2016].
Results
=======
PL enhancement using UV illumination
------------------------------------
We initially observe a drastic increase in PL intensity of , by about 50 times, when continuously illuminating a semi-insulating 4H-SiC sample with a 405 nm (“UV”) laser diode, in addition to the 976 nm laser required for PL excitation. This is shown in (a) where the full PL spectrum for all the divacancies (PL1 to PL6 [@Koehl2011]) is taken with (blue) and without (black) 405 nm excitation. Both c-axis (PL1, PL2) and basal defects (PL3, PL4) show an increase in their PL intensity, with slight variation between defects, which we ascribe to charge conversion of the divacancy toward its observable neutral state. On the other hand, PL5 and PL6 remain completely unaffected, adding another unique feature to these currently unidentified defects on top of their strong room temperature PL emission. The PL enhancement with UV was observed in all semi-insulating wafers we measured, with gains ranging by a factor of 2 to 1000 (see Supplementary Figure 2), including samples obtained from separate commercial suppliers (Cree or Norstel), different growth batches, or even simply from separate positions within the same wafer. This strongly indicates an influence from the local environment, e.g. from the remaining concentration of N dopants or other impurities which is known to locally differ in as-grown wafers [@Jenny2004]. The PL intensity with UV remains fairly constant however from sample to sample.
![**Effect of near-bandgap illumination on 4H-SiC divacancies.** [**(a)**]{} PL spectrum with 976 nm excitation of the various divacancies in 4H-SiC, as designated in [@Koehl2011], without and with continuous illumination at 405 nm ($\approx 5$ mW optical power). All the observed PL lines except PL5 and PL6 are enhanced by the UV excitation, including both c-axis and basal defects. [**(b)**]{} Gain in the PL signal (integrated across PL1-4) as a function of excitation wavelength (energy) around the 4H-SiC bandgap (3.28 eV at 5 K [@Galeckas2002]). Power was normalized to 0.4 $\mu$W across the entire energy range. The onset of change in the curve is shifted from the bandgap energy due to absorption of longitudinal acoustic phonons (about 70-80 meV) [@Galeckas2002]. [**(c)**]{} Lifetime of the charge state after a 405 nm pulse at 6 K. No significant decay is observed after 12 hours. []{data-label="fig:PLspectrum"}](01-PLspectrum.pdf){width="\figwidth"}
In order to understand the effect of 405 nm illumination, and optimize the enhancement, the excitation wavelength is swept across the 4H-SiC bandgap energy (3.28 eV, 380 nm) as shown in (b). The PL gain slowly increases with excitation energy, and around 3.33-3.35 eV, slightly above the 4H-SiC bandgap (3.28 eV), it drastically turns up. This suggests two separate processes are altering the VV charge state from either or to : at low energies, we will see this is due to direct photoionization, while at high energies, the gain results from recombination of generated electron-hole pairs.
We now consider charge dynamics under illumination, starting from the stability of the conversion observed after UV excitation. As illustrated in (c), the system is initially pumped with 405 nm toward a high population (strong PL intensity), followed by a long delay to allow for relaxation and finally measurement using 976 nm. No change is observed over the course of 12 hours, a result largely expected for a deep defect at cryogenic temperature (6 K). More interestingly, the PL intensity always drops to a low level after turning off the UV excitation while 976 nm was continuously on. Combined with this long stability, this implies that the use of 976 nm to excite VV PL is simultaneously converting the VV out of its neutral charge state, toward a dark state (more details are given later on). This has significant consequences as wavelengths near 976 nm have been extensively used in recent PL- and ODMR-related works [@Christle2014; @Zargaleh2016; @Seo2016], owing to being close to the absorption maximum of the ground to excited state transition of , as well as being easily available commercially. These previous studies may therefore have been partially perturbed by charge conversion.
Illumination effects on spin properties
---------------------------------------
Above-bandgap excitation can be used to efficiently convert VV toward its neutral charge state, and more importantly drastically increases the PL intensity. For practical applications however, we verify this has no effect on the spin properties of . In (a), we first measure the ODMR contrast of PL2, i.e. the ratio of ODMR over PL intensity, which provides a direct measure of how the spin states may be affected during illumination. For these experiments, the 405 nm laser is replaced by a 365 nm (also called “UV”) light-emitting diode which is more efficient at charge conversion since it is above bandgap in energy. No difference in contrast is observed with or without 365 nm, and the charge conversion therefore does not significantly affect the spin state nor the readout mechanism. However, the signal-to-noise ratio improves by $\sim$70 times with illumination due to increased charge population. More details are given in the Supplementary Figure 6 and Supplementary Note 2 regarding the ODMR experiments presented here.
![**Charge conversion effect on the spin properties.** ODMR signals are given as relative photoluminescence intensities ($\Delta$PL) under microwave excitation. [**(a)**]{} CW-ODMR spectrum at 50 G and measured through a monochromator at the 1130.6 nm PL2 zero-phonon line to ensure no other contribution in the optical signal. The intensity is given as the ratio (i.e. contrast) between the ODMR and PL intensity, which remains constant with and without 365 nm illumination, indicating unchanged spin polarization and readout mechanisms. [**(b)**]{} Hahn echo decay experiment for PL2 at $\approx 400$ G, measured with pulsed-ODMR at 6 K. The 365 nm excitation is continuous throughout the sequence, resulting in a signal increase while the coherence time is unaffected. Decay with 976 nm excitation only was averaged 241 times more than the decay with also 365 nm illumination. Line (in black) is a stretched exponential fit (stretch factor $\approx$ 2) to the data. [**(c)**]{} CW-ODMR (PL2 at $\approx 400$ G) of a 4H-SiC sample with a 500 nm carbon-implant layer below the surface. The divacancies created at the layer are barely visible before 365 nm excitation. The implanted layer peak is also shifted from the bulk due to a magnetic field gradient across the sample. []{data-label="fig:SpinApp"}](02-ApplicationSpin){width="\figwidth"}
A second crucial property is the electron spin coherence of the defect. From (c), the charge stability at cryogenic temperature (6 K) is shown to be much longer than any coherence timescale [@Seo2016], however we check that even with constant 365 nm illumination ($\sim$0.2 mW) and corresponding electron-hole pair generation, the coherence time remains unaffected. At 400 G, the ensemble electron spin coherence is measured to be 0.7 ms and remains completely unaffected by either light or free carriers ((b)), while much longer averaging ($> \times 200$) was required to obtain similar signal-to-noise ratios without 365 nm illumination. This is not an obvious result as scattering or exchange interaction with free carriers can easily reduce or of defect spins [@Tyryshkin2012].
Until now, all measurements were realized on as-grown commercial wafers with naturally occurring impurity concentrations. However, carbon ion implantation or electron irradiation [@Falk2013] is often used to increase the PL intensity and to improve spatial resolution. The type of defects created by lattice damage during these processes cannot be well controlled however, though partially manageable using annealing, and the local Fermi level may shift away from obtaining a desired charge state. We test this with a 500 nm thick layer of implanted divacancies (see Methods section). When measuring the PL2 ODMR spectrum of this sample, as shown in (c), we obtain a broad “bulk” signal observable across the entire sample depth using simply 976 nm excitation. When 365 nm is turned on (with constant absorption over the sample depth), the bulk intensity increases as expected, but more importantly a narrower and more intense peak appears. The latter is assigned to the implanted layer which, being confined in depth, is less sensitive to inhomogeneity in the static magnetic field. Rabi experiments at the peak layer frequency yielded as expected an increased contrast (Supplementary Figure 7), demonstrating that the UV charge stabilization can be critical in such samples.
Charge state conversion
-----------------------
{width="\textwidth"}
We now consider in more depth the charge mechanisms within 4H-SiC, in particular the effect of 976 nm illumination which appears to convert VV toward a dark charge state ( or ). Since 976 nm is used both for PL excitation and causes charge conversion, it is necessary to separate the two contributions from the PL intensity, which can be achieved by looking at the conversion dynamics under pulsed light. This is realized using a three-pulse scheme: reset with either 976 nm or 365 nm, pump with a wavelength ranging from 365 nm to 1310 nm using various laser diodes, and measurement with 976 nm. Typical decay curves as a function of pumping duration are shown in (a) with different initial reset lasers, pump wavelength, pump power and temperatures.
In (b), fitted steady states and decay rates (see Methods section regarding the fitting) are plotted as a function of pump excitation wavelength. Steady-state intensities are all normalized by the steady-state PL intensity after UV pumping. A clear transition is observed between 940 nm and 976 nm, at about 1.3 eV, for both steady-state values and decay rates, with shorter wavelengths being increasingly more efficient at charge conversion toward . In addition, we measure a single wavelength at 1310 nm that tentatively suggests a second transition (between 976 nm and 1310 nm), where the charge state becomes insensitive to excitation (no observed decay). The wavelength transitions can be related to photoionization energies and, though the Franck-Condon shift is unknown here, to formation energies obtained from DFT calculations [@Gordon2015] and reproduced in (d). The divacancy defect in 4H-SiC has four stable charge states: $+$, $0$, $-$ and $2-$. The $(+/0)$ and $(0/-)$ transition levels are calculated to be, respectively, $\sim$ + 0.94 eV and $\sim$ - 1.1 eV, with and the valence and conduction band energies. Considering typical uncertainty in DFT calculations of 0.1 eV as well as Franck-Condon shift in the order of 0-0.3 eV (for for example [@Son2012]), this matches fairly well with our measured values and the fact that the (0/-)/940 nm transition is at higher energy than (0/+)/976 nm. We can therefore accredit our results to charge conversion between the and states.
We attempt to model the observed dynamics using the rate-equation model shown in (e), based on charge transfer between the divacancy and a trap of unknown origin. More details are given in both the Methods section and Supplementary Note 1. Simulated decays and their corresponding rates and steady states are shown in (a,b). Three experimental characteristics are nicely reproduced by the model here: i) the jump in charge conversion efficiency for above bandgap illumination, ii) the to transition fitted to be $E_{-0} = 1.295(5)$ eV and iii) the to transition roughly estimated at $E_{0-} = 1.00(7)$ eV (1150 to 1350 nm). For charge stability, an important parameter is the ratio of cross-sections between the electron ( $\rightarrow$ ) and hole ( $\rightarrow$ ) photoionization processes, which roughly follows the relation $K_{\sigma}\left(\frac{E-E_{-0}}{E-E_{0-}}\right)^{3/2}$ with $K_{\sigma} = 26 (8)$ (valid at or above $E_{-0}$). is therefore the stable charge state for any illumination above $\approx 1.3$ eV at cryogenic temperatures.
Finally, a temperature dependence of 976 nm pumping (365 nm reset) is taken between 5.5 K and 210 K, with corresponding steady states value shown in (c). Above 100 K, the effect of 976 nm pumping compared to 365 nm pumping is drastically reduced. The simulation is able to reproduce this feature owing to hole thermal emission from the trap, with a tentative activation energy between 0.05 and 0.15 eV depending on the fitting conditions, such as which temperatures to include in the data set.
In summary, we identify a sharp transition of the VV charge dynamics at around 1.3 eV (960 nm), corresponding to the ionization of to . Below 1.3 eV in energy, hole photoemission drives the charge state toward , while above, VV is preferentially in and remains stable for many hours after illumination. In addition, UV light above bandgap strongly drives the system toward .
Charge transfer between major defects
-------------------------------------
The experiments described previously made use of PL as a direct measurement of the divacancy neutral charge state, combined with photo-excitation to probe relevant energy levels as well as trapping or recombination dynamics. However, understanding all the major defects in 4H-SiC, not just the divacancy, is required to obtain a comprehensive picture of the sample behavior under illumination. While PL of can be measured, other important spin impurities such as or N are not photo-active, with no optical excited states in the bandgap. We thus turn toward electron spin resonance (ESR) to provide information on all the spin species.
A CW-ESR spectrum at X-band is shown in (a) with resonance peaks from PL1 to PL4 VV defect types, as well as a cluster of signals near the g-factor $g=2$, known to be from , and/or N [@Son2006; @Isoya2008]. In order to properly resolve some of these peaks, we subtract the ESR spectrum measured with 976 nm illumination from the spectrum obtained with either 940 nm or 365 nm illumination. The differential spectra, shown in (b), then correspond to possible charge transfer with VV which is extremely sensitive to these wavelengths (further considerations are discussed in the Methods section). The ESR peak intensities are given for the main identified defects in (c) (left). With 940 nm, two sets of resonances can be clearly assigned, the strongest due to ($T_{V2a}$ or $V2$ center) and the weaker from (k site). At this wavelength, undergoes photoionization to become , emitting an electron to the conduction band which is likely captured by , and resulting in an increase in . With 365 nm, large changes can be seen around $g=2$, possibly from free electrons and , though the peaks are too clustered to be resolved. On the side of $g=2$, appears much stronger while the peaks completely disappeared. While this may be due to carrier-induced spin relaxation, such behavior is also well explained by charge dynamics: N, initially in its neutral charge state due to either photoionization or thermal emission (shallow donor) before cooling down the sample, captures most of the generated electrons to give a high signal. The holes now in majority, are captured by the various deep defects, with being converted toward (high signal), toward (low signal), and possibly toward (high signal).
![**ESR at 15 K in semi-insulating 4H-SiC under illumination.** [**(a)**]{} CW-ESR spectrum measured at 9.7 GHz, and centered around g=2 ($\approx$ 3470 G, aligned to the c-axis). VV PL1-4 are highlighted in blue, while defects such as N, or are close to $g=2$ and highlighted in green. [**(b)**]{} Differential CW-ESR spectrum between either 940 nm and 976 nm excitation (left), or between 365 nm and 976 nm (right). Gaussian derivative lineshapes are simulated in color for known defects in 4H-SiC [@Isoya2008]. Their amplitudes only take into account transition probabilities, and not spin polarization or microwave saturation. [**(c)**]{} Normalized (per defect) CW-ESR intensity under 976 nm, 940 nm and 365 nm (left) and for different annealing condition of the sample (right). For , 365 nm is combined with 976 nm for spin polarization (and obtain enough signal). For the annealing dependence, the intensity was fitted under the best illumination condition for each defect, that is the maximum signal in the left panel. Annealed samples were only used in this panel ((c), right). []{data-label="fig:ESR"}](04-EPR){width="\figwidth"}
ESR measurements are often used to characterize the optimal annealing temperature for sample preparation. Changes in ESR intensity after sample annealing normally indicates variations in defect concentration, but can also be confused with a shift in the local Fermi level. After charge conversion, this second explanation is much less plausible. In (c), the ESR intensity under best illumination condition (highest signal for each defect) was tracked for N, , and VV for different annealing temperatures. Between 1000 C and 1400 C, the ESR signal of and VV significantly drops, which can be related to the defects becoming mobile followed by creation of multi-vacancies such as -- [@Gerstmann2003; @Zolnai2004; @Schmid2006; @Carlos2006].
![**Photo-dynamics of at 6 K.** Reset-pump-measure scheme similar to , but with 780 nm to excite PL in instead of 976 nm for . 365 nm reduces the PL intensity, likely from charge conversion to . Charge conversion was measured to be persistent without light on the experiment timescales. []{data-label="fig:Vsi"}](05-Vsi){width="\figwidth"}
The ESR experiments indicate a strong relationship between VV and , as they are both affected by the 940-976 nm transition and by similar annealing temperatures. With being also a photo-active qubit of interest, we directly measure its PL by exciting the sample with 780 nm (see Supplementary Figure 1) [@Baranov2011; @Embley2017]. Three-pulse experiments for are shown in with pumping using 365 nm, 976 nm, as well as 780 nm as a replacement for 940 nm which was impractical here. 365 nm pumping drastically decreases the intensity while both 976 nm and 780 nm convert back the charge state to , with 976 nm illumination being less effective. These observations are consistent with the ESR experiments. The charge conversion toward is ascribed to hole photo-emission, with the difference in conversion efficiency resulting from VV photo-emission. Indeed, VV emits holes under 976 nm and dominantly electrons under 780 nm, which has opposite charge effects for . For example, having more electrons with 780 nm excitation pushes faster toward its higher charge state .
Looking at formation energies for ([@Hornos2011; @Gordon2015], reproduced in Supplementary Figure 5), the $(+/0)$ transition is very shallow ( + 0-50 meV) and can be photo-excited at any wavelength while $(0/-)$ is near mid-gap ( + 1.3-1.5 eV). The temperature dependence and transient modeling in matches with the presence of as a shallow acceptor that can trap and re-emit holes. The activation energies from theory and simulation are similar (0.05-0.15 eV), though with very large uncertainties in both cases. For completeness, the $(-/2-)$ transition with a formation energy of - 0.6-0.8 eV is likely also excited at any wavelength below 976 nm, and therefore is more likely to be trapped in than in the $2-$ charge state. 780 nm illumination is therefore suitable for both PL excitation and charge stabilization.
![**Summary of charge transfer in semi-insulating 4H-SiC under various illumination conditions.** Strong transitions are shown by thicker arrows, and the steady-state population after illumination is approximatively represented by the grey area over each state. [**(a)**]{} Above-bandgap excitation and electron-hole generation. [**(b)**]{} Excitation above the photoionization transition ($\sim$1.3 eV). [**(c)**]{} Excitation below the photoionization transition, but above the hole emission. []{data-label="fig:Summary"}](06-Summary){width="\figwidth"}
The full VV//N charge conversion picture under illumination is finally summarized in for the three critical wavelengths explored in this work: 976 nm, 940 nm and 365 nm. was not taken into account due to lack of measurements, but it is likely trapped to or where it is too deep to be photo-excited ([@Hornos2011; @Gordon2015], reproduced in Supplementary Figure 5).
Summary of charge dynamics
--------------------------
Our overall summary of the charge conversion is as follows: i) Under 365 nm excitation and electron-hole generation, N dominantly traps electrons toward , while VV and capture the remaining holes to become and . ii) Below but close to 940 nm in wavelength, VV chiefly emits electrons and ends up in , while both captures those electrons and emits holes to become . N will be in an intermediate charge state as it absorbs both electron and holes, as well as being slighly photoionized. iii) At wavelengths higher than 976 nm, VV is converted to by hole emission; N and then both capture those holes and are photoionized, resulting in and a slow conversion toward .
The charge conversion and transfer mechanisms presented throughout this work should remain valid in most semi-insulating materials, where defects are in comparable concentrations. For n- or p-doped materials, impurities can of course still be photoionized, however electron-hole generation with above-bandgap light will likely set the local Fermi level to a different equilibrium than what is seen here.
Toward applications: charge patterning
--------------------------------------
To complete this study, we turn toward applications using our ability to control the VV charge state. In recent experiments in diamond [@Dhomkar2016], optical conversion between the NV$^{-}$ and NV$^{0}$ states was used to demonstrate the possibility of information storage by 3D patterning of the charge state. Because data can be both encoded in 3D as well as a gradient of charge conversion, high storage densities can theoretically be achieved. We present a similar demonstration of charge patterning in 4H-SiC, and though our experiments are realized at 6 K, offer the potential for storage across entire wafers compared to diamond. The VV charge conversion works relatively well up to 150-200 K, and may possibly be extended to room temperature with the adequate choice of material (dominant dopant or impurity concentration).
The patterning scheme is presented in (a) with: a UV (405 nm) pulse to initialize the sample toward , a write pulse with 976 nm to selectively obtain , and finally a short read pulse using 976 nm. The measurement pulse here weakly destroys the information due to undesired charge conversion, which is the main limitation to this technique. In (b), we test the spatial resolution of our setup by patterning a pixelized checkerboard design (left), first parallel to the sample plane (middle) and then in depth, orthogonal to the sample plane (right). Finally, for each pixel, we allow control over the amount of charge conversion, increasing the density of information that can be locally stored. This is demonstrated in (c) by patterning a 500 $\mu$m $\times$ 540 $\mu$m grey scale image parallel to the sample surface.
![**Spatial and amplitude control of the divacancy charge conversion.** [**(a)**]{} Imaging sequence for (b) and (c), with three sequential 2D sweeps: 1) Reset to high VV$^0$ concentration using 405 nm. 2) Write using 976 nm with varying duration for charge conversion back to a desired lower VV$^0$ concentration. 3) Read with a fast 976 nm pulse. [**(b)**]{} Pixel test of spatial control with, from left to right, the original pattern, the measured pattern in the X-Y plane (parallel to the sample surface) and the measured pattern in the X-Z plane (orthogonal to the sample surface). For X-Z, the intensity is normalized to the PL collection efficiency across the sample depth. [**(c)**]{} Amplitude control of the charge conversion using a gray scale image (left: original image, right: experiment). []{data-label="fig:Imaging"}](07-Imaging.pdf){width="\figwidth"}
Discussions
===========
In this work we have systematically investigated the charge properties of divacancies in semi-insulating 4H-SiC, as well as other relevant defects such as and N. Through optical excitation with wavelengths spanning from 365 to 1310 nm, VV was found to be stable in either its neutral or negative charge configuration. The photoionization energies of both and are fitted to be around 1.3 eV and 1 eV; in particular, the commonly used 976 nm excitation for PL measurements is found to be detrimental as it converts VV toward . Above-bandgap excitation efficiently reshuffles the charge states of all defects, with VV becoming bright () and becoming dark (). Overall, taking into account all impurities was necessary to obtain a complete picture of charge effects in these samples; such considerations are crucial for tuning wafer growth techniques, samples with implanted layers, surface impurities or for devices with complex electric potentials. Finally, we confirmed that these optical charge conversions drastically improve the PL intensity and do not impact in any way the spin properties (ODMR, coherence). Combined with recent studies [@Christle2017; @Fuchs2015] characterizing the spin and optical properties of VV or in 4H and 3C-SiC, this work on charge conversion/stabilization helps to complete the suite of techniques and technologies realized in NV centers in diamond for use in SiC, while allowing for novel applications such as optically controlling the charge of spins in electronic devices realized in SiC.
Methods
=======
Samples
-------
All measurements were performed on commercially available high-purity semi-insulating 4H-SiC diced wafers purchased from Cree [@Jenny2004], and using a scanning ODMR microscopy setup. Similar wafers have been used in other studies, with measured defect concentrations of N, VV, all in the order of $10^{14}-10^{16}$ cm$^{-3}$ [@Jenny2004; @Son2007; @Chandrashekhar2012]. For the implanted sample, a high energy carbon implant ($[^{12}\rm{C}] = 10^{13}\ \rm{cm}^{-2} , 190$ keV, 900C anneal for 40 mn) was used, resulting in a calculated (SRIM software) 500 nm thick layer of divacancies. For ODMR, the samples are fixed to a printed circuit board patterned with a coplanar waveguide for magnetic resonance, and mounted in a closed-cycle cryostat cooled down to 5-6 K (unless otherwise mentioned). PL, ODMR and ESR experiments were all realized on ensembles of defects.
PL and ODMR setups
------------------
For , the sample is excited with a 976 nm diode laser (40 mW at sample, focused with a 50X IR objective) and PL is measured with an InGaAs detector (1000-1300 nm after filtering). For , the sample is excited with a 780 nm diode laser (10 mW at sample) and PL is measured with a Si detector (850-950 nm after filtering, allowing simultaneous PL recording). Simplified schematics for the PL/ODMR setups are given in Supplemental Figure 1. All given optical powers were measured at the sample. Excitation spectra are recorded by inserting a monochromator immediately after a 100 W Xe white light source in the optical setup. Emission spectra or measurements at selective zero-phonon lines (ZPL) are recorded by inserting a monochromator before the detector. For the wavelength dependence, a set of laser diodes were successively collimated into a 300 $\mu$m multi-mode fiber and re-emitted into free space so as to ensure a constant spot position on the sample. It should also be noted there is no significant PL contribution from 405 nm or 365 nm illumination alone.
Transients and modeling
-----------------------
The three-pulse scheme used for the photo-dynamics requires careful choice of the 976 nm measurement pulse duration (0.1 ms) as it is necessary for exciting PL but can also change the charge state of VV. A long pulse would effectively smooth the decays and prevent good fitting at short times. The experimental decay rates and steady states are obtained from fitting with a stretched exponential function, with separate fitting parameters for each power, wavelength and temperature dependence. The actual decay curves are shown in Supplementary Figure 3, and all simulated lines in are from the rate-equation model. All details on this model are given in Supplementary Figure 4 and Supplementary Note 1, regarding e.g. simulation of the wavelength dependence (Grimmeiss model for deep trap [@Grimmeiss1975]) and the exact rate equations.
In total, 11 parameters are used for a simultaneous fit over a set of 70 decays curves, with the simulation results shown by the lines in (a), (b) and (c). Looking at the decays in (a), the fits are in excellent agreement in certain ranges (833 nm pumping) but do not account for all the charge dynamics as seen in the left figures. For 365 nm pumping, electron-hole pair generation dominates over all photoionization processes, and the free carrier concentration is determined by the recombination with all involved traps, not just VV. Hence for such a simple model, large discrepancies are expected. In addition, the simulation strictly considers a single defect while measuring an ensemble can easily smooth features in the decays, e.g. due to local variations in strain, charge, light intensity, etc.
Electron spin resonance
-----------------------
ESR experiments were realized on a X-band (dielectric resonator, 5 mm internal diameter) ELEXSYS E580 Bruker spectrometer at 15 K. In the differential experiments presented in , one important issue is the simultaneous effect of illumination on both spin (polarization, relaxation) and charge properties, and hence on the ESR intensity. The results presented may convolute both aspects, unlike the PL experiments which are clearly related to the charge state. Turning the lasers on and off would avoid this concern, however the signal was then simply too weak to obtain any information on N, or . The 940 and 976 nm excitation lasers are sufficiently close in energy to limit most effects but those related to the sharp photoionization transition in VV. In addition, these wavelengths are both in the VV absorption sideband, but close enough to see no appreciable differences in spin polarization due to inter-system crossing mechanisms. Similarly, they are also above the longest ZPL wavelength of (917 nm [@Baranov2011]), preventing any spin-polarization.
Acknowledgments
===============
We thank Adam Gali, Hosung Seo, Alexandre Bourassa and David Christle for discussions. G.W. acknowledges support from the University of Chicago/Advanced Institute for Materials Research (AIMR) Joint Research Center. A.L.Y, F.J H., and D.D.A. were supported by the US Department of Energy, Office of Science, Basic Energy Sciences, Materials Science and Engineering Division at Argonne National Laboratory. J.N. and O.G.P. were supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, Division of Chemical Sciences, Biosciences and Geosciences under Contract DE-AC02-06CH11357 at Argonne National Laboratory. C.P.A. was supported by the Department of Defense (DoD) through the National Defense Science and Engineering Graduate Fellowship (NDSEG) Program.
Author contributions
====================
G.W., C.P.A. and A.L.Y. performed the optical experiments. G.W., J.N., O.G.P. and F.J.H. performed the electron spin resonance experiments. S.J.W. and C.P.A. processed the annealed samples and A.L.Y. designed the implanted sample. All the authors contributed to analysis of the data, discussions and the production of the manuscript.
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abstract: 'Using an approach based on the time-dependent density-matrix renormalization group method, we study thermalization in spin chains locally coupled to an external bath. Our results provide evidence that quantum chaotic systems do thermalize, that is, they exhibit relaxation to an invariant ergodic state which, in the bulk, is well approximated by the grand canonical state. Moreover, the resulting ergodic state in the bulk does not depend on the details of the baths. On the other hand, for integrable systems we found that the invariant state in general depends on the bath and is different from the grand canonical state.'
author:
- Marko Žnidarič
- Tomaž Prosen
- Giuliano Benenti
- Giulio Casati
- Davide Rossini
title: 'Thermalization and ergodicity in one-dimensional many-body open quantum systems'
---
The emergence of canonical ensembles in quantum statistical mechanics from first principles is one of the key remaining old questions of theoretical physics. Even the definition of the temperature at the nano-scale poses a challenge [@Hartmann:04]. Namely, the main question is how to “derive” the canonical distribution? It has been realized that the canonical distribution is in a way “typical”: provided the overall system describing the environment plus a central system is in a generic pure state, the reduced state of the central system is with high probability canonical [@Gemmer:03]. However, how precisely the canonical distribution arises from dynamical laws, without a priori statistical assumptions, is still unclear. Motivation in the study of this fundamental aspect of nonequilibrium physics also comes from some recent experiments with ultracold bosonic gases, where absence of thermalization in closed, integrable, strongly correlated quantum systems has been observed [@kinoshita06].
For closed many-body systems, integrability is believed to play a crucial role in the relaxation to the [*Steady State*]{} (SS): the nonequilibrium dynamics of a chaotic system is expected to thermalize at the level of individual eigenstates [@ETH], as numerically observed in several physical models [@closed_nonint]. By contrast, for systems with non trivial integrals of motion, SSs usually carry memory of the initial conditions and are not canonical: maximizing the entropy while keeping the values of constants of motion fixed results in a generalized Gibbs ensemble [@Rigol]. Much less is known about the relaxation to the SS for [*open quantum systems*]{} [@henrich:05]; this is what we are going to address in this paper. We provide numerical evidence that, analogously to closed systems, the occurrence of thermalization is strictly related to system’s integrability, irrespective of the fine details of the baths. In particular we show that [*locally*]{} coupling a quantum chaotic many-body system to an environment is enough for a SS of the central system to be very close, in the bulk, to the canonical or grand canonical state (GCS). On the contrary, if the system is integrable, the constants of motion in general prevent thermalization and the form of the SS sensitively depends on the bath coupling operators. We show that the numerical description of an open quantum system in terms of a Lindblad equation with [*local*]{} coupling to the reservoirs is in some sense a [*computationally efficient, minimal model of thermalization*]{}. Such result paves the way for future simulations of quantum transport in large many-body quantum systems.
The time evolution for a generic state $\rho$ of an open quantum system can be described, under certain approximations, by a Lindblad master equation [@breuer:BOOK]: $$\frac{{\rm d}}{{\rm d}t}{\rho} = \frac{\ii}{\hbar} [ \rho, {\cal H} ]
+ {\hat {\cal L}}_{\rm B}\rho,
\label{eq:Lin}$$ where ${\cal H}$ is the Hamiltonian of the autonomous system, while the dissipation $\hat{\cal L}_{\rm B} = \gamma \sum_k \left( [ L_k \rho,L_k^{\dagger} ] +
[ L_k,\rho L_k^{\dagger} ] \right)$ is parametrized by certain Lindblad operators $L_k$ (hereafter we set $\hbar = k_B = 1$ and, unless noted otherwise, $\gamma=1$). The derivation of Eq. from first principles, i.e., from the Hamiltonian evolution of a system plus environment is rather tricky [@breuer:BOOK]; however it is the most general form of a completely positive, trace preserving, dynamical semi-group. Taking it for granted, we ask ourselves if, within this approximation, a [*finite*]{} many-body system can thermalize when coupled via some Lindblad operators $L_k$ acting only [*locally*]{} just on [*few*]{} degrees of freedom.
To elucidate the role covered by chaoticity in the thermalization process, we consider prototype one-dimensional spin-$1/2$-chain models with nearest neighbor interactions: ${\cal H} = \sum_{l=0}^{n-2} h_{l,l+1}$ ($h_{l,l+1}$ denoting the local energy density, and $n$ being the chain length). As we shall see, the chosen models exhibit a crossover from integrable to chaotic regime when a suitable parameter in their Hamiltonians is varied. With the term “chaotic” we refer, as usual, to a system whose bulk energy spectrum of highly excited levels obeys a random matrix statistics [@haake]; in particular, the level spacing statistics (LSS) $p(s)$ is well approximated by the Wigner-Dyson distribution $p_{\rm WD}(s)$ [@haake], whereas in an integrable system LSS typically turns out to be Poissonian, $p_{\rm P}(s)$.
We assume local coupling to the reservoirs, i.e., the dissipator $\hat{\cal L}_{\rm B}$ acts only on the $m$ ($\ll n$) leftmost $(l)$ and rightmost $(r)$ spins: $\hat{\cal L}_{\rm B}=
\hat{\cal L}^{l}_{\rm B}\otimes \hat{\openone}_{\rm bulk}
\otimes \hat{\cal L}^{r}_{\rm B}$. We construct $\hat{\cal L}_{\rm B}$ by generalizing the method discussed in Ref. [@JSTAT:09]. For this purpose, we first consider the GCS for the spin chain, $$\rho_{\cal G} (T,\mu) = Z^{-1}
\exp\left[-({\cal H} - \mu\, \Sigma^{\rm z})/T \right],
\label{eq:grandc}$$ where $\Sigma^{\rm z}=\sum_{l=0}^{n-1} \sigma_l^{\rm z}$ is the total magnetization \[$\sigma^\alpha_j$ ($\alpha={\rm x},{\rm y},{\rm z}$) being the Pauli operators for the $j$th spin\], $T$ the temperature, $\mu$ the “chemical potential”, and $Z = \tr{\left[\exp{(-({\cal H} - \mu\,\Sigma^{\rm z})/ T)}\right]}$ the partition function. Given a target temperature $T_{\rm targ}$ and a chemical potential $\mu_{\rm targ}$, the reduced $m$-spin target density matrix $\rho^{\lambda}_{\rm targ}$, $\lambda \in \{l,r\}$, is obtained after tracing $\rho_{\cal G} (T_{\rm targ},\mu_{\rm targ})$ over all but the $m$ leftmost/rightmost spins. We finally require that $\rho^{\lambda}_{\rm targ}$ is the unique eigenvector of $\hat{\cal L}^{\lambda}_{\rm B}$ with eigenvalue $0$, while all other eigenvalues are equal to $-1$. Such a choice produces, in absence of ${\cal H}$ and for a given spectral norm of $\hat{\cal L}^{\lambda}_{\rm B}$, the [*fastest*]{} convergence to $\rho^{\lambda}_{\rm targ}$ [@lindbladnote]. In the presence of ${\cal H}$ we obtain, for up to $n\approx 100$ spins, the SS solution of Eq. numerically by using a time-dependent Density Matrix Remormalization Group (tDMRG) method with a Matrix Product Operator (MPO) ansatz [@tDMRGreview].
In the following we are interested in the asymptotic state reached, independently of initial conditions, after a long time, $\rho_{\rm SS} \equiv \lim_{t \to \infty}\rho(t)$. In all simulations we carefully checked that the simulation time was long enough to reach convergence, which is exponential. Since Lindblad operators act only locally and $\rho_{\cal G}(T,\mu)$ is invariant for the unitary part of Eq. , $\rho_{\rm SS}$ cannot be equal to the GCS, unless it is also an eigenstate of the dissipator $\hat{\cal L}_{\rm B}$. In other words, one can have $\rho_{\rm SS}=\rho_{\cal G}(T,\mu)$ only if $\rho_{\cal G}(T,\mu)=\rho^{l}_{\rm targ}\otimes \rho_{\rm bulk}\otimes
\rho^{r}_{\rm targ}$, i.e., if the GCS is separable with respect to the border $m$ spins which are used in the coupling. Nevertheless for chaotic systems, as we shall see, sufficiently far from the boundaries the state is arbitrarily close to $\rho_{\cal G}(T,\mu)$, regardless of the entanglement with the coupled parts.
Let us start our numerical investigations by considering a spin-$1/2$ Ising chain in a tilted magnetic field, described by the energy density $$h_{l,l+1} = J_l \sigma_l^{\rm z} \sigma_{l+1}^{\rm z} +
\frac{b_{\rm x}}{2}(\sigma_{l}^{\rm x}+\sigma_{l+1}^{\rm x}) +
\frac{b_{\rm z}}{2}(\sigma_{l}^{\rm z}+\sigma_{l+1}^{\rm z}).
\label{eq:ising}$$ Its only conserved quantity is the total energy, therefore the expected invariant state is the canonical one $\rho_{\cal G}(T,0)$. To check thermalization, we solved the master equation for two different sets of parameters: (i) a transverse field $b_{\rm x}=1,b_{\rm z}=0$, for which the model is integrable and exhibits a Poissonian LSS; (ii) a tilted field $b_{\rm x}=1,b_{\rm z}=1$, for which it is chaotic with a Wigner-Dyson LSS [@PRE] (if not specified, we take $J_l = 1$ and couple two border spins, $m=2$). With the obtained $\rho_{\rm SS}$, we evaluated expectation values of several one- and two-spin observables in the bulk of the chain, and compared them to the theoretical ones as given by the canonical state $\rho_{\cal G}(T,0)$.
In the main plot of Fig. \[fig:xzCcti\] we show one-spin expectation values $\ave{\sigma^{\alpha}_{n/2}}=\tr{(\rho_{\rm SS} \, \sigma_{n/2}^{\alpha})}$ for the chaotic case: all numerical points fall on the curve given by theoretical expectation values for a canonical state. The same happens in the integrable Ising model. Such irrelevance of integrability is a peculiarity of certain few-body observables, similarly to what observed in a different context of out-of-equilibrium dynamics in closed systems [@rossini09]. Quite remarkably, we could not reach temperatures in the bulk below $\approx 1.7$ (see squares in Fig. \[fig:xzCcti\]), even by using very small $T_{\rm targ}\approx 0$. The reason resides in the already mentioned boundary effects due to entanglement between the boundary two spins and the bulk chain, which makes the cooling difficult. This must be contrasted with a zero attainable temperature in the case of separable states [@Giovannetti]. For entangled states though, our results show that to lower the minimal attainable temperature one has to reduce the effect of interaction at the boundaries which is responsible for entanglement. One way to do this is by switching on the interaction gently over a boundary layer of certain thickness $\tau$, $J_l=\sin{(\frac{l}{\tau}\frac{\pi}{2})}$ ($J_{n-2-l}=\sin{(\frac{l}{\tau}\frac{\pi}{2})}$), for $l=0,\ldots,\tau-1$, at the left (right) end and using a weaker coupling $\gamma$ (circles and triangles in Fig. \[fig:xzCcti\]).
To make comparison between $\rho_{\rm SS}$ and $\rho_{\cal G}(T,0)$ quantitative, we determined the “measured” temperature $T_{\rm meas}$ to which $\rho_{\rm SS}$ corresponds, which is in general different from $T_{\rm targ}$, due to boundary effects. Assuming that the SS is canonical in the bulk, one can extract $T_{\rm meas}$ by comparing observables that uniquely set the temperature. For Ising model , the energy density is sufficient, therefore we used the condition $\tr{[h_{n/2-1,n/2} \, \rho_{\rm SS}]} \equiv
\tr{[h_{n/2-1,n/2} \, \rho_{\cal G}(T_{\rm meas},0)]}$ to compute $T_{\rm meas}$. We then calculated theoretical expectation values of other observables, through $\rho_{\cal G}(T_{\rm meas},0)$; a comparison with the corresponding values for the reached SS may serve as an indicator of the [*quality of thermalization*]{}. In Fig. \[fig:Dxx\] we show differences between expectation values of $\sigma_l^{\rm x} \sigma_{l+1}^{\rm x}$, computed with $\rho_{\rm SS}$ and $\rho_{\cal G}(T_{\rm meas},0)$, for both chaotic and integrable Ising chains. A marked distinction between the two cases appears. First, in the chaotic model errors are much smaller than in the integrable one; second, switching $J_l$ gradually, which should decrease errors due to smaller boundary effects, in the integrable case even worsens the situation. The integrable Ising model therefore does not relax to a canonical state in the bulk. Similar results are obtained for other few-spin observables, as well as for the lowest moments of the energy distribution: we evaluated $\ave{[({\cal H}_6 - \ave{{\cal H}_6})/5]^p}$ ($p=2,...,5$ and ${\cal H}_6$ is the Hamiltonian of the 6 central spins) on the states $\rho_{\rm SS}$ and $\rho_{\cal G}(T_{\rm meas},0)$. In a chain of $n=40$ spins relative errors are never greater than $1 \%$ in the chaotic case, and are typically an order of magnitude larger in the integrable case.
![(Color online). SS (symbols) and GCS (full lines) expectation values of $q_j^{(4)}$ (left panel), $\sigma_j^{\rm z}$, $\sigma_j^{\rm z}\sigma_{j+1}^{\rm z}$, and $h_j$ (right panel) for the Heisenberg model with $n=89$ spins, $T_{\rm targ}=4$, $q_{\rm targ}=2$, $m=3$, $J_l=1$. In the left panel, “XX” and “XXZ” refer to two integrable cases without magnetic field (respectively at $\Delta = 0, \, 0.5$), “stagg.” to the chaotic case with $\Delta=0.5$ and period-3 staggered field with $B=2$. The curves in the right panel are for the chaotic case only. The GCS $\rho_{\cal G}(T_{\rm meas}=5.851,\mu_{\rm meas}=-0.534)$ for the chaotic case is obtained by matching $\ave{h_{3l+1,3l+2}}$ and $\ave{\sigma^{\rm z}_{3l+1}}$ for which lines are not shown in the right panel. []{data-label="fig:XXZstagg2"}](XXZstagg2_v2.eps)
To corroborate the importance of system’s integrability on the convergence to invariant statistical ensembles, we consider another prototype model of interacting spins: the Heisenberg XXZ chain in a magnetic field, described by the energy density $$h_{l,l+1} = J_l (\sigma_l^{\rm x} \sigma_{l+1}^{\rm x} +
\sigma_l^{\rm y} \sigma_{l+1}^{\rm y}+\Delta \sigma_l^{\rm z}
\sigma_{l+1}^{\rm z})+\frac{b_{l}}{2}\sigma_{l}^{\rm z} +
\frac{b_{l+1}}{2}\sigma_{l+1}^{\rm z}.
\label{eq:heis}$$ If the field is homogeneous the model is integrable and possesses, besides energy and magnetization, an infinite sequence of conserved quantities [@Grabowski]. On the other hand, integrability can be broken, e.g., simply by means of a period-3 staggered magnetic field, $b_{3k}=-B, \, b_{3k+1}=-B/2, \, b_{3k+2}=0$. In order to highlight the lack of thermalization in the integrable regime $B=0$, we target a non-Gibbsian state different from the GCS; namely, we use $\rho_{\rm non-{\cal G}}(T,q) \sim \exp{(-{\cal H}/T+q\, Q_4)}$, with $Q_4=-h_{0,1}-h_{n-2,n-1}+\sum_{l=0}^{n-4} q^{(4)}_l$, $q^{(4)}_l =
\sigma_l^{\rm x} \sigma_{l+1}^{\rm z} \sigma_{l+2}^{\rm z}\sigma_{l+3}^{\rm x} +
\sigma_l^{\rm y} \sigma_{l+1}^{\rm z} \sigma_{l+2}^{\rm z}\sigma_{l+3}^{\rm y}$, being a conserved charge for an open chain with $\Delta=0$ and $b_l=0$ [@Grabowski]. The idea is that, using a $q_{\rm targ}\ne 0$, in the integrable regime the SS exhibits strong deviations from the GCS, corresponding to $q=0$, while we expect chaotic dynamics to drive the bulk towards the GCS. Such expectation is confirmed by our numerical data. In Fig. \[fig:XXZstagg2\] we show the spatial dependence of various observables for integrable, as well as for chaotic cases. In the integrable cases deviations from the GCS expectations are large, while they become very small for a chaotic system. Analogously to the Ising model, we checked this statement also for other few-spin observables (we found that, in presence of chaos, the largest discrepancy among all the one- and two-spin observables amounts to $2\times 10^{-4}$); layered interactions in the integrable model do not help in thermalizing the system.
![(Color online). Relative differences $\Delta q^{(4)}_{\rm rel} = \Delta q^{(4)}(B)/\Delta q^{(4)}(0)$ in $q^{(4)}$ expectation values on the SS and the GCS evaluated in the bulk of the Heisenberg model with $\Delta = 0.5$, as the staggering strength $B$ is varied (full curves). Also shown is dependence of the $\eta$ function (dashed curves), characterizing the integrable-chaotic crossover. For both quantities the crossover takes place at smaller $B$ with increasing $n$. Inset: two examples of LSS in the integrable ($B = 0.01$) and chaotic ($B=1$) regimes; dashed lines denote Poissonian and Wigner-Dyson statistics [@haake], respectively.[]{data-label="fig:transition"}](transition2.eps)
A further confirmation of the role of integrability comes from a direct analysis of the quality of thermalization after gradually switching on the perturbation that drives the crossover from integrability to chaos: the longitudinal field $b_z$ in Eq. or the staggering intensity $B$ in Eq. . As shown in Fig. \[fig:transition\] for the Heisenberg model, such crossover is conveniently detected by the parameter $\eta\equiv
\int{|p(s)-p_{\rm WD}(s)|{\rm d}s}/\int{|p_{\rm P}(s)-p_{\rm WD}(s)|{\rm d}s}$; $\eta=1$ and $\eta=0$ correspond to Poissonian and Wigner-Dyson distributions, respectively. In the same figure we also plot deviations in $\ave{q^{(4)}}$ evaluated on the SS and on the corresponding GCS: $\Delta q^{(4)}=\tr{ [ q^{(4)}_l (\rho_{\rm SS}-\rho_{\cal G}
(T_{\rm meas},\mu_{\rm meas}))]}$ as the strength $B$ of the staggered magnetic field is increased. The progressive onset of chaos gradually improves the quality of thermalization, being $\Delta q^{(4)}$ a monotonic decreasing function of $B$. Moreover, the strength of the staggered field required to converge to the GC expectation value drops with the system size.
In conclusion, we have shown that, within the Lindblad equation formalism, coupling a one-dimensional quantum chaotic system locally to a bath results in a SS being equal to the invariant (grand)canonical state, far away from the coupled sites. In contrast, integrable systems do not thermalize and their SSs exhibit strong deviations from the (grand)canonical state, depending on the details of the coupling. The fact that for chaotic systems the SS does not depend on the details of the coupling, shows that very likely the same result would be obtained even for a harder-to-treat Hamiltonian evolution of a system plus environment or for higher dimensional systems. Our method should be applicable also to non-equilibrium situations. Indeed, by locally coupling a system to two or several baths at different values of temperature and chemical potentials, one should be able to efficiently control local thermalization. Thus, our results might open significant new perspectives in the simulation of quantum transport in many-body quantum systems in contact with thermal and chemical baths.
We thank V. Giovannetti for useful discussions. MŽ and TP are supported by the Program P1-0044, and the Grant J1-2208, of the Slovenian Research Agency.
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To construct such $\hat{\cal L}^{l}_{\rm B}$ (and similarly for $\hat{\cal L}^{r}_{\rm B}$), we first diagonalize $\rho^{l}_{\rm targ}$, thus getting eigenvalues $d_i$ and eigenvectors, $\rho^{l}_{\rm targ}=V^\dagger d V$. Then we obtain $\hat{\cal L}^{l}_{\rm B}$ after rotating the “diagonal” Lindblad superoperator $\hat{\cal L}^{\rm diag}_{\rm B}$, $\hat{\cal L}^{l}_{\rm B}=R \hat{\cal L}^{\rm diag}_{\rm B} R^{\rm T}$, with the orthogonal matrix $R_{\vec{\alpha},\vec{\beta}}=\tr{(V^\dagger \sigma^{\vec{\alpha}}\,
V \sigma^{\vec{\beta}})}/4$ ($\sigma^{\vec{\alpha}}=\sigma^{{\alpha_1}}\otimes
\cdots \otimes \sigma^{{\alpha_m}}$). A set of $4^m$ Lindblad operators leading to the diagonal superoperator $\hat{\cal L}^{\rm diag}_{\rm B}$ is given by $
L_{i_1 i_2 \ldots i_m} = \sqrt{\frac{d_k}{2\cdot 4^m}}\, r^{i_1}
\otimes r^{i_2} \otimes \cdots \otimes r^{i_m},
\label{eq:L}
$ with $i_j \in \{0,1,2,3\}$, $k=\sum_{j=1}^m{ (i_j\, \hbox{mod}\, 2 )\cdot 2^{j-1}}$, and $r^0 = \sigma^{\rm x}+\ii \sigma^{\rm y},r^1=\sigma^{\rm x} -
\ii \sigma^{\rm y},r^2=\mathbbm{1}+\sigma^{\rm z},r^3 =
\mathbbm{1}-\sigma^{\rm z}$.
See, e.g., A. J. Daley, J. Stat. Mech. P04005 (2004); F. Verstraete, Adv. Phys. [**57**]{}, 143 (2008). We implemented the propagator corresponding to Eq. in small time steps of length $dt=0.05$; for each of them we used a 4th-order Trotter expansion.
T. Prosen and M. Žnidarič, Phys. Rev. E [**75**]{}, 015202 (2007).
D. Rossini, Phys. Rev. Lett. [**102**]{}, 127204 (2009).
D. Burgarth and V. Giovannetti, Phys. Rev. Lett. [**99**]{}, 100501 (2007). We checked that, for separable ground states (e.g., in the ferromagnetic Heisenberg chain), boundary effects are absent and one can cool to $T=0$.
M. P. Grabowski and P. Mathieu, J. Phys. A [**29**]{}, 7635 (1996).
|
---
abstract: 'We establish a $q$-analog of our recent work on vertex representations and the McKay correspondence. For each finite group ${\Gamma}$ we construct a Fock space and associated vertex operators in terms of wreath products of $\Gamma\times \mathbb C^{\times}$ and the symmetric groups. An important special case is obtained when $\Gamma$ is a finite subgroup of $SU_2$, where our construction yields a group theoretic realization of the representations of the quantum affine and quantum toroidal algebras of $ADE$ type.'
address:
- 'Frenkel: Department of Mathematics, Yale University, New Haven, CT 06520'
- 'Jing: Department of Mathematics, North Carolina State University, Raleigh, NC 27695-8205 Mathematical Sciences Research Institute, 1000 Centennial Drive, Berkeley, CA 94720 '
- 'Wang: Department of Mathematics, North Carolina State University, Raleigh, NC 27695-8205 Department of Mathematics, Yale University, New Haven, CT 06520'
author:
- 'Igor B. Frenkel'
- Naihuan Jing
- Weiqiang Wang
title: Quantum vertex representations via finite groups and the McKay correspondence
---
[^1]
Introduction {#S:intro}
============
In our previous paper [@FJW] (see [@W; @FJW] for historical remarks and motivations) we have shown that the basic representation of an affine Lie algebra ${\widehat{\mathfrak g}}$ of ADE type can be constructed from a finite subgroup ${\Gamma}$ of $SU_2$ related to the Dynkin diagram of ${\widehat{\mathfrak g}}$ via the McKay correspondence. In particular, we have recovered a well-known construction [@FK; @Se] of the basic representation of ${\widehat{\mathfrak g}}$ from the root lattice $Q$ of the corresponding finite dimensional Lie algebra $\mathfrak g$. In fact our construction yields naturally the vertex representation of the toroidal Lie algebra ${\widehat{\widehat{\mathfrak g}}}$ which contains the affine Lie algebra as a distinguished subalgebra.
The main goal of the present paper is to $q$-deform our construction in [@FJW]. Again as in the undeformed case we will naturally obtain the earlier construction [@FJ] of the basic representation of the quantum affine algebra ${U_q(\widehat{\mathfrak g})}$ from the root lattice $Q$ and its generalization to the quantum toroidal algebra ${U_q(\widehat{\widehat{\mathfrak g}})}$ [@GKV] (also cf. [@Sa; @J3]). The $q$-deformation is achieved by replacing consistently the representation theory of ${\Gamma}$ by that of ${\Gamma\times \mathbb C^{\times}}$. The representation ring for ${\mathbb C^{\times}}$ is identified with the ring of Laurent polynomials $\mathbb C[q, q^{-1}]$ so that the formal variable $q$ corresponds to the natural one-dimensional representation of ${\mathbb C^{\times}}$. It turns out that rather complicated expressions for operators in Drinfeld realization of the quantum affine algebra ${U_q(\widehat{\mathfrak g})}$ and the quantum toroidal algebra ${U_q(\widehat{\widehat{\mathfrak g}})}$ follow instantly from the simple extra factor ${\mathbb C^{\times}}$. The idea to use representations of ${\mathbb C^{\times}}$ to obtain a $q$-deformation of the basic representation was mentioned in [@Gr] and is widely used in geometric constructions of representations (see e.g. [@CG]).
As in the previous paper [@FJW] we give the construction of quantum vertex operators starting from an arbitrary finite group ${\Gamma}$ and a self-dual virtual character ${\xi}$ of ${\Gamma\times \mathbb C^{\times}}$. Using the restriction and induction functors in representation theory of wreath products of ${\Gamma\times \mathbb C^{\times}}$ with the symmetric group $S_n$ for all $n$ we construct two “halves” of quantum vertex operators corresponding to any irreducible character ${\gamma}$ of ${\Gamma\times \mathbb C^{\times}}$. Then choosing an irreducible character of ${\mathbb C^{\times}}$, i.e. an integer power of $q$ we assemble both halves into one quantum vertex operator.
The special case when ${\Gamma}$ is a subgroup of $SU_2$ is important for the application to representation theory of ${U_q(\widehat{\mathfrak g})}$ and for relations [@W] to the theory of Hilbert schemes of points on surfaces. To recover the basic representation of ${U_q(\widehat{\mathfrak g})}$ we choose $${\xi}={\gamma}_0\otimes (q+q^{-1})-\pi\otimes 1_{{\mathbb C^{\times}}},$$ where ${\gamma}_0$ and $1_{{\mathbb C^{\times}}}$ are the trivial characters of ${\Gamma}$ and ${\mathbb C^{\times}}$ respectively, $q$ and $q^{-1}$ are the natural and its dual characters of ${\mathbb C^{\times}}$, and $\pi$ is the natural character of ${\Gamma}$ in $SU_2$. The fact that the quantum toroidal algebra intrinsically presents in our construction is an additional indication of its importance in representation theory of quantum affine algebras. Moreover when ${\Gamma}$ is cyclic of order $r+1$, $\pi\simeq {\gamma}\oplus {\gamma}^{-1}$, where ${\gamma}$ is the natural character of ${\Gamma}$, one can modify our virtual character ${\xi}$ with an extra parameter $p=q^k, k\in\mathbb Z$ by letting $${\xi}={\gamma}_0\otimes (q +q^{-1})-
({\gamma}\otimes p+{\gamma}^{-1}\otimes p^{-1}).$$ In the special case when $p=q^{\pm 1}$ the quantum vertex representation of the quantum toroidal algebra ${U_q(\widehat{\widehat{\mathfrak g}})}$ can be factored to the basic representation of the quantum affine algebra ${U_q(\widehat{\mathfrak g})}$. This is a $q$-analog of the factorization in the undeformed case, which exists for an arbitrary simply-laced affine Lie algebra.
To obtain the basic representations of quantum toroidal and affine algebras we only need the quantum vertex operators corresponding to irreducible representations of ${\Gamma}$ and their negatives in the Grothendieck ring of this group. We attach two halves of quantum vertex operators using the simplest nontrivial representations of ${\mathbb C^{\times}}$ namely $q$ and $q^{-1}$. Each of the two choices and only these two yield the basic representations of ${U_q(\widehat{\widehat{\mathfrak g}})}$ and ${U_q(\widehat{\mathfrak g})}$, in a perfect correspondence with the construction in [@FJ]. This choice of an irreducible character of ${\mathbb C^{\times}}$ is essentially the only freedom that exists in our construction of quantum vertex operators for the quantum affine and toroidal algebras and is fixed by comparison with the algebra relations. However it raises the question of constructing a “natural” quantum vertex operator corresponding to any virtual character ${\gamma}$ of ${\Gamma}$. This question is closely related to the well-known problem of finding a $q$-deformation of vertex operator algebras associated to the basic representation of an affine Lie algebra.
This paper is organized in a way similar to [@FJW]. In Sect. \[sect\_wreath\] we review the theory of wreath products of ${\Gamma}$ and extend it to ${\Gamma\times \mathbb C^{\times}}$. In Sect. \[sect\_weight\] we define the weighted bilinear form on ${\Gamma\times \mathbb C^{\times}}$ and its wreath products. In Sect. \[sect\_mckay\] we introduce two distinguished $q$-deformed weight functions associated to subgroups of $SU_2$. In Sect. \[sect\_heis\] we define the Heisenberg algebra associated to ${\Gamma}$ and the weighted bilinear form, and we construct its representation in a Fock space. In Sect. \[sect\_isom\] we establish the isometry between the representation ring of wreath products of ${\Gamma\times \mathbb C^{\times}}$ and the Fock space representation of the Heisenberg algebra. In Sect. \[sect\_vertex\] we construct quantum vertex operators acting on the representation ring of the wreath products. In Sect. \[sect\_ade\] we obtain the basic representations of quantum toroidal algebras and quantum affine algebras from representation theory of wreath products for ${\Gamma\times \mathbb C^{\times}}$.
Wreath products and vertex representations {#sect_wreath}
==========================================
The wreath product ${{\Gamma}_n}$
---------------------------------
Let $\Gamma$ be a finite group and $n$ a non-negative integer. The wreath product ${{\Gamma}_n}$ is the semidirect product of the $n$-th direct product ${\Gamma}^n={\Gamma}\times\cdots
\times{\Gamma}$ and the symmetric group $S_n$: $$\Gamma_n = \{(g, \sigma) | g=(g_1, \ldots, g_n)\in {\Gamma}^n,
\sigma\in S_n \}$$ with the group multiplication $$(g, \sigma)\cdot (h, \tau)=(g \, {\sigma} (h), \sigma \tau ) ,$$ where $S_n$ acts on ${\Gamma}^n$ by permuting the factors.
Let ${\Gamma}_*$ be the set of conjugacy classes of ${\Gamma}$ consisting of $c^0=\{1\}$, $c^1$, $\dots$, $c^r$ and ${\Gamma}^*$ be the set of $r+1$ irreducible characters: ${\gamma}_0, {\gamma}_1, \dots, {\gamma}_r$. Here we denote the trivial character of ${\Gamma}$ by ${\gamma}_0$. The order of the centralizer of an element in the conjugacy class $c$ is denoted by $\zeta_c$, so the order of the conjugacy class $c$ is $|c|=|{\Gamma}|/\zeta_c$, where $|{\Gamma}|$ is the order of ${\Gamma}$.
A partition ${\lambda}=({\lambda}_1, {\lambda}_2, \ldots, {\lambda}_l)$ is a decomposition of $n=|{\lambda}|={\lambda}_1+\cdots+{\lambda}_l$ with nonnegative integers: ${\lambda}_1\geq \dots \geq {\lambda}_l \geq 1$, where $l=l ({\lambda})$ is called the [*length*]{} of the partition ${\lambda}$ and ${\lambda}_i$ are called the [*parts*]{} of ${\lambda}$. Another notation for ${\lambda}$ is $${\lambda}=(1^{m_1}2^{m_2}\cdots)$$ with $m_i$ being the multiplicity of parts equal to $i$ in ${\lambda}$. Denote by $\mathcal P$ the set of all partitions of integers and by $\mathcal P(S)$ the set of all partition-valued functions on a set $S$. The weight of a partition-valued function $\rho=(\rho(s))_{s\in S}$ is defined to be $\|\rho\|=\sum_{s\in S}|\rho(s)|$. We also denote by $\mathcal P_n$ (resp. $\mathcal P_n(S)$) the subset of $\mathcal P$ (resp. $\mathcal P(S)$) of partitions with weight $n$.
Just as the conjugacy classes of $S_n$ are parameterized by partitions, the conjugacy classes of ${{\Gamma}_n}$ are parameterized by partition-valued functions on ${\Gamma}_*$. Let $x=(g, {\sigma})\in {\Gamma}_n$, where $g=(g_1, \ldots, g_n) \in {\Gamma}^n$ and ${\sigma}\in S_n$ is presented as a product of disjoint cycles. For each cycle $(i_1 i_2 \cdots i_k)$ of $\sigma$, we define the [*cycle-product*]{} element $g_{i_k} g_{i_{k -1}}
\cdots g_{i_1} \in \Gamma$, which is determined up to conjugacy in $\Gamma$ by $g$ and the cycle. For any conjugacy class $c\in {\Gamma}$ and each integer $i\geq 1$, the number of $i$-cycles in $\sigma$ whose cycle-product lies in $c$ will be denoted by $m_i(c)$. This gives rise to a partition $\rho(c)=(1^{m_1 (c)} 2^{m_2 (c)} \ldots )$ for $c \in {\Gamma}_*$. Thus we obtain a partition-valued function $\rho=( \rho (c))_{c \in {\Gamma}_*} \in {\mathcal P} ( {\Gamma}_*)$ such that $\|\rho\|=\sum_{i, c} i m_i(\rho(c)) =n$. This is called the [*type*]{} of the element $(g, \sigma)$. It is known [@M2] that two elements in the same conjugacy class have the same type and there exists a one-to-one correspondence between the sets $({{\Gamma}_n})_*$ and $\mathcal P_n({\Gamma}_*)$. We will freely say that $\rho$ is the type of the conjugacy class of ${{\Gamma}_n}$.
Given a class $c$ we denote by $c^{-1}$ the class $\{x^{-1}| x\in c\}$. For each $\rho\in \mathcal P({\Gamma}_*)$ we also associate the partition-valued function $$\overline{\rho}=(\rho(c^{-1}))_{c\in {\Gamma}_*}.$$
Given a partition $\lambda = (1^{m_1} 2^{m_2} \ldots )$, we denote by $$z_{{\lambda}} = \prod_{i\geq 1}i^{m_i}m_i!$$ the order of the centralizer of an element of cycle type ${\lambda}$ in $S_{|{\lambda}|}$. The order of the centralizer of an element $x = (g, \sigma) \in {\Gamma}_n$ of type $\rho=( \rho(c))_{ c \in {\Gamma}_*}$ is given by $$Z_{\rho}=\prod_{c\in {\Gamma}_*}z_{\rho(c)}\zeta_c^{l(\rho(c))}.$$
Grothendieck ring $R_{{\Gamma}\times C^{\times}}$
-------------------------------------------------
Let ${R_{\mathbb Z}(\Gamma)}$ be the $\mathbb Z$-lattice generated by ${\gamma}_i$, $i=0, \dots, r$, and $R({\Gamma})=\mathbb C\otimes{R_{\mathbb Z}(\Gamma)}$ be the space of complex class functions on the group ${\Gamma}$. In our previous work on the McKay correspondence and vertex representations [@W; @FJW] we studied the Grothendieck ring $
{R_{{\Gamma}}}= \bigoplus_{n\geq 0} R({\Gamma}_n).$ In the quantum case we need to add the ring $R(\mathbb C^{\times})$, the space of characters of $\mathbb C^{\times}=\{t\in\mathbb C|t\neq 0\}$.
Let $q$ be the irreducible character of ${\mathbb C^{\times}}$ that sends $t$ to itself. Then $R(\mathbb C^{\times})$ is spanned by irreducible multiplicative characters $q^n$, $n\in \mathbb Z$, where $$q^n(t)=t^n, \qquad t\in \mathbb C^{\times}.$$ Thus $R(\mathbb C^{\times})$ is identified with the ring $\mathbb C[q, q^{-1}]$, and we have $$R({\Gamma}\times \mathbb C^{\times})=R({\Gamma})\otimes R(\mathbb C^{\times}).$$
An elements of $R({\Gamma}\times \mathbb C^{\times})$ can be written as a finite sum: $$f=\sum_if_i\otimes q^{n_i}, \qquad f_i\in R({\Gamma}), n_i\in\mathbb Z.$$
We can also view $f$ as a function on ${\Gamma}$ with values in the ring of Laurent polynomials $\mathbb C[q, q^{-1}]$. In this case we will write $f^q$ to indicate the formal variable $q$, then $f^q(c)=\sum_if_i(c)q^{n_i}\in\mathbb C[q, q^{-1}]$. As a function on ${\Gamma}\times \mathbb C^{\times}$, we have $f(c, t)=\sum_if_i(c)t^{n_i}$.
Denote by ${R_{{\Gamma}\times \mathbb C^{\times}}}$ the following direct sum: $${R_{{\Gamma}\times \mathbb C^{\times}}}=\bigoplus_{n\geq 0} R({{\Gamma}_n}\times \mathbb C^{\times})
\simeq R_{{\Gamma}}\otimes \mathbb C[q, q^{-1}].$$
Hopf algebra structure on ${R_{{\Gamma}\times \mathbb C^{\times}}}$
-------------------------------------------------------------------
The multiplication $m$ in $\mathbb C^{\times}$ and the diagonal map $\mathbb C^{\times}\stackrel{d}{\longrightarrow}\mathbb C^{\times}\times \mathbb C^{\times}$ induce the Hopf algebra structure on $R(\mathbb C^{\times})$. $$\begin{aligned}
\label{E:hopf1}
m_{\mathbb C^{\times}}&: {R(\mathbb C^{\times})}\otimes {R(\mathbb C^{\times})}\stackrel{\cong }{\longrightarrow}
R({\mathbb C^{\times}}\times {\mathbb C^{\times}})
\stackrel{d^*}{\longrightarrow} R({\mathbb C^{\times}}),\\ \label{E:hopf2}
\Delta_{{\mathbb C^{\times}}}&: R({\mathbb C^{\times}}) \stackrel{m^*}{\longrightarrow}
R({\mathbb C^{\times}}\times {\mathbb C^{\times}})
\stackrel{\cong}{\longrightarrow}
R({\mathbb C^{\times}}) \otimes R({\mathbb C^{\times}}).\end{aligned}$$ In terms of the basis $\{q^n\}$ we have $$\begin{aligned}
q^i\cdot q^j&=q^{i+j},\\
\Delta(q^k)&=q^k\otimes q^k,\end{aligned}$$ where we abbreviate $\Delta_{{\mathbb C^{\times}}}$ by $\Delta$ and follow the convention of writing $a\cdot b=m_{{\mathbb C^{\times}}}(a\otimes b)$.
The antipode $S_{{\mathbb C^{\times}}}$ and the counit $\epsilon_{{\mathbb C^{\times}}}$ are given by $$S_{{\mathbb C^{\times}}}(q^n)=q^{-n}, \qquad \epsilon_{{\mathbb C^{\times}}}(q^n)=\delta_{n0}.$$
We extend the Hopf algebra structures on $R({\mathbb C^{\times}})$ and ${R_{{\Gamma}}}$ [@Z; @M2] into a Hopf algebra structure on ${R_{{\Gamma}\times \mathbb C^{\times}}}$ using a standard procedure in Hopf algebra [@A]. The multiplication and comultiplication are given by the respective composition of the following maps: $$\begin{aligned}
m &: R({{\Gamma}_n\times\mathbb C^{\times}}) \otimes R({{\Gamma}_m\times\mathbb C^{\times}})
\stackrel{\cong }{\longrightarrow} R({{\Gamma}_n\times\mathbb C^{\times}}\times {{\Gamma}_m\times\mathbb C^{\times}})\nonumber\\
&\qquad \stackrel{1\otimes m_{{\mathbb C^{\times}}}}{\longrightarrow}
R( {{{\Gamma}_n}\times{{\Gamma}_m}\times{\mathbb C^{\times}}})\stackrel{Ind\otimes 1}{\longrightarrow}
R( {\Gamma}_{n + m}\times{\mathbb C^{\times}});\\
\Delta&: R({{\Gamma}_n\times\mathbb C^{\times}}) \stackrel{Res\otimes 1}{\longrightarrow}
\oplus_{m=0}^nR( {\Gamma}_{n - m} \times {{\Gamma}_m\times\mathbb C^{\times}})\nonumber\\
&\qquad \stackrel{1\otimes \Delta_{{\mathbb C^{\times}}}}{\longrightarrow}
\oplus_{m=0}^n R( {\Gamma}_{n - m}\times {{\Gamma}_m\times\mathbb C^{\times}}\times {\mathbb C^{\times}})\nonumber\\
&\qquad \stackrel{\cong }{\longrightarrow}
\oplus_{m=0}^nR( {\Gamma}_{n - m}\times {\mathbb C^{\times}}) \otimes R({{\Gamma}_m\times\mathbb C^{\times}}),
\label{E:comult}\end{aligned}$$ where we have used the identification of $R({\mathbb C^{\times}}\times {\mathbb C^{\times}})$ with $R({\mathbb C^{\times}})\otimes R({\mathbb C^{\times}})$ in (\[E:hopf1\]-\[E:hopf2\]). Also $Ind: R({{\Gamma}_n}\times {{\Gamma}_m}) \longrightarrow R({\Gamma}_{n +m})$ denotes the induction functor and $Res: R({{\Gamma}_n}) \longrightarrow
R( {{\Gamma}}_{n - m} \times {{\Gamma}_m})$ denotes the restriction functor.
The antipode is given by $$S(f(g, t))=f(g^{-1}, t^{-1}), \qquad g\in {\Gamma}, t\in {\mathbb C^{\times}}.$$ In particular, $S({\gamma})(c)={\gamma}(c^{-1})$ for ${\gamma}\in{\Gamma}^*$. As we mentioned earlier, we may write $f\in {R_{{\Gamma}\times \mathbb C^{\times}}}$ as $$f^q(g)=\sum_if_i(g)q^{n_i},$$ Then $S(f^q)(g)=\sum_if_i(g^{-1})q^{-n_i}$.
The counit $\epsilon$ is defined by $$\epsilon(R({{\Gamma}_n\times\mathbb C^{\times}}))=0, \qquad\mbox{if \ \ } n\neq 0,$$ and $\epsilon$ on $R({\mathbb C^{\times}})$ is the counit of the Hopf algebra $R({\mathbb C^{\times}})$.
A weighted bilinear form on $R({{\Gamma}_n\times\mathbb C^{\times}})$ {#sect_weight}
=====================================================================
A standard bilinear form on ${R_{{\Gamma}\times \mathbb C^{\times}}}$
---------------------------------------------------------------------
Let $f, g\in R({\Gamma\times \mathbb C^{\times}})$ with $f=\sum_i f_i\otimes q^{n_i}$ and $g=\sum_i g_i\otimes q^{m_i}$. The $\mathbb C[q, q^{-1}]$-valued standard $\mathbb C$-bilinear form on $R({\Gamma\times \mathbb C^{\times}})$ is defined as $$\begin{aligned}
\langle f, g \rangle_{{\Gamma}}^q
&= \sum_{i,j} \langle f_i, g_j\rangle_{{\Gamma}}q^{n_i-m_j}\\
&= \sum_{i,j}\sum_{c\in {\Gamma}_*}\zeta_c^{-1}f_i(c)g_j(c^{-1})q^{n_i-m_j},\end{aligned}$$ where we recall that $c^{ -1}$ denotes the conjugacy class $\{ x^{ -1}| x \in c \}$ of ${\Gamma}$, and $\zeta_{c}$ is the order of the centralizer of the class $c$ in ${\Gamma}$. Sometimes we will also view the bilinear form as a function of $t\in{\mathbb C^{\times}}$: $$\langle f, g \rangle_{{\Gamma}}^q(t)
= \sum_{c \in \Gamma_*} \zeta_c^{ -1} f(c, t) S(g(c, t)).$$
The following is a direct consequence of the orthogonality of irreducible characters of ${\Gamma}$. $$\begin{aligned}
\langle {\gamma}_i\otimes q^k, {\gamma}_j \otimes q^l\rangle_{{\Gamma}}^q &= & \delta_{ij}
q^{k-l} , \nonumber \\
\sum_{ {\gamma}\in {\Gamma}^*} {\gamma}(c ') S({\gamma})( c)
&= & \delta_{c, c '} \zeta_c, \quad c, c ' \in {\Gamma}_*. \label{eq_orth}\end{aligned}$$
Let $\langle \ \ , \ \ \rangle_{{{\Gamma}_n}}^q$ be the $\mathbb C[q, q^{-1}]$-valued bilinear form on $R({{\Gamma}_n\times\mathbb C^{\times}})$. The $\mathbb C[q, q^{-1}]$-valued standard bilinear form in ${R_{{\Gamma}\times \mathbb C^{\times}}}$ is defined in terms of the bilinear form on $R( {{\Gamma}_n\times\mathbb C^{\times}})$ as follows: $$\langle u, v \rangle^q
= \sum_{ n \geq 0} \langle u_n, v_n \rangle_{{{\Gamma}_n}}^q,$$ where $u = \sum_n u_n$ and $v = \sum_n v_n$ with $u_n, v_n\in R({{\Gamma}_n}\times{\mathbb C^{\times}})$.
A weighted bilinear form on $R({\Gamma\times \mathbb C^{\times}})$
------------------------------------------------------------------
A class function ${\xi}\in R({\Gamma\times \mathbb C^{\times}})$ is called [self-dual]{} if for all $x\in{\Gamma}, t\in{\mathbb C^{\times}}$ $$\xi(x, t)=S(\xi(x, t)),$$ or equivalently $\xi^q(x)=\xi^{q^{-1}}(x^{-1})$.
We fix a self-dual class function $\xi$. The tensor product of two representations ${\gamma}$ and $\beta$ in $ R({\Gamma\times \mathbb C^{\times}})$ will be denoted by ${\gamma}*\beta $.
Let $a_{ij} \in \mathbb C[q, q^{-1}]$ be the (virtual) multiplicity of ${\gamma}_j$ in $ {\xi}* {\gamma}_i $, i.e., $$\begin{aligned}
\label{eq_tens}
{\xi}* {\gamma}_i
= \sum_{j =0}^r a_{ij} {\gamma}_j.\end{aligned}$$ We denote by $A^q$ the $ (r +1) \times (r +1)$ matrix $ ( a_{ij})_{0 \leq i,j \leq r}$.
Associated to $\xi$ we introduce the following weighted bilinear form $$\langle f, g \rangle_{{\xi}}^q = \langle {\xi}* f , g \rangle_{{\Gamma}}^q,
\quad f, g \in R( {\Gamma\times \mathbb C^{\times}}).$$ where we use the superscript $q$ to indicate the $q$-dependence. The superscript $q$ is often omitted if the $q$-variable in characters $f$ and $g$ is clear from the context. The explicit formula of the bilinear form is given as follows. $$\begin{aligned}
\langle f, g \rangle_{{\xi}}^q
&=&\frac 1{ |{\Gamma}|} \sum_{x\in {\Gamma}}{\xi}^q(x)f^q(x)g^{q^{-1}}(x^{-1})
\nonumber\\
& =& \sum_{c \in {\Gamma}_*} \zeta_c^{ -1} {\xi}^q(c) f^q(c) g^{q^{-1}}(c^{ -1}),
\label{eq_twist}\end{aligned}$$ which is the average of the character $ {\xi}* f * \overline{g}$ over ${\Gamma}$.
The self-duality of ${\xi}$ together with (\[eq\_twist\]) implies that $$a_{ij} = \overline{a_{ji}},$$ i.e. $A^q$ is a hermitian-like matrix with the bar action given by $\overline{q}=q^{-1}$.
The orthogonality (\[eq\_orth\]) implies that $$\label{E:qcartan}
a_{ij}=\langle {\gamma}_i, {\gamma}_j\rangle_{\xi}^q.$$
If ${\xi}$ is the trivial character ${\gamma}_0$, then the weighted bilinear form becomes the standard one on $ R( {\Gamma\times \mathbb C^{\times}})$.
A weighted bilinear form on $R( {{\Gamma}_n\times\mathbb C^{\times}})$
----------------------------------------------------------------------
Let $V$ be a ${\Gamma}\times {\mathbb C^{\times}}$-module which affords a character ${\gamma}$ in $R({\Gamma\times \mathbb C^{\times}})$. We can decompose $V$ as follows: $$V=\bigoplus_iV_i\otimes \mathbb C({k_i}),$$ where $V_i$ is a (virtual) ${\Gamma}$-module in $R({\Gamma})$ and $\mathbb C(k_i)$ is the one dimensional ${\mathbb C^{\times}}$-module afforded by the character $q^{k_i}$.
The $n$-th outer tensor product $V^{ \otimes n} $ of $V$ can be regarded naturally as a representation of the wreath product $({\Gamma}\times{\mathbb C^{\times}})_n$ via permutation of the factors and the usual direct product action. More precisely, note that ${{\Gamma}_n}\times {\mathbb C^{\times}}$ can be viewed as a subgroup of $({\Gamma}\times {\mathbb C^{\times}})_n$ by the diagonal inclusion from ${\mathbb C^{\times}}$ to $({\mathbb C^{\times}})^n$: $${{\Gamma}_n}\times{\mathbb C^{\times}}\longrightarrow ({{\Gamma}}^n\times{{\mathbb C^{\times}}}^n)\rtimes S_n=
({\Gamma}\times{\mathbb C^{\times}})_n.$$ This provides a natural ${{\Gamma}_n}\times{\mathbb C^{\times}}$-module structure on $V^{\otimes n}$. We denote its character by $\eta_n ( {\gamma})$. Explicitly we have $$\label{E:eta-action}
(g, \sigma, t).(v_1\otimes\cdots\otimes v_n)=
(g_{1}, t)v_{\sigma^{-1}(1)}\otimes\cdots
\otimes (g_{n}, t)v_{\sigma^{-1}(n)},$$ where $g=(g_1, \ldots, g_n)\in {\Gamma}^n$.
Let $\varepsilon_n$ be the (1-dimensional) sign representation of ${{\Gamma}_n}$ so that ${\Gamma}^n$ acts trivially while letting $S_n$ act as a sign representation. We denote by $\varepsilon_n ( {\gamma}) \in R({{\Gamma}_n\times\mathbb C^{\times}})$ the character of the tensor product of $\varepsilon_n\otimes 1$ and $V^{\otimes n}$.
The weighted bilinear form on $R( {{\Gamma}_n\times\mathbb C^{\times}})$ is now defined by $$\langle f, g\rangle_{{\xi}, {{\Gamma}_n}}^q =
\langle \eta_n ({\xi}) * f, g \rangle_{{{\Gamma}_n}}^q ,
\quad f, g \in R( {{\Gamma}_n\times\mathbb C^{\times}}).$$ We shall see in Corollary \[cor\_char\] that $\eta_n ({\xi})$ is self-dual if the class function ${\xi}$ is invariant under the antipode $S$. In such a case the matrix of the bilinear form $\langle \ , \ \rangle_{{\xi}}^q$ is equal to its adjoint (transpose and bar action).
We can naturally extend $\eta_n$ to a map from $R({\Gamma})\otimes q^k$ to $R({\Gamma\times \mathbb C^{\times}})$ as in the classical case (cf. [@W]). In particular, if $\beta$ and ${\gamma}$ are characters of representations $V$ and $W$ of ${\Gamma}$ respectively, then $$\begin{aligned}
\nonumber
&\eta_n (\beta\otimes q^k +{\gamma}\otimes q^l) \\ \label{eq_virt}
=\sum_{m =0}^n &Ind_{{\Gamma}_{n -m}\times {\mathbb C^{\times}}\times {{\Gamma}_m}\times {\mathbb C^{\times}}}^{{{\Gamma}_n\times\mathbb C^{\times}}}
[ \eta_{n -m} (\beta\otimes q^k) \otimes \eta_m ({\gamma}\otimes q^l) ],\\
&\eta_n (\beta\otimes q^k - {\gamma}\otimes q^l) \nonumber\\
=\sum_{m =0}^n &( -1)^m Ind_{{\Gamma}_{n -m}\times {\mathbb C^{\times}}\times {{\Gamma}_m}\times
{\mathbb C^{\times}}}^{{{\Gamma}_n\times\mathbb C^{\times}}}
[ \eta_{n -m} (\beta\otimes q^k) \otimes \varepsilon_m ({\gamma}\otimes q^l) ] . \label{eq_virt'}\end{aligned}$$
On ${R_{{\Gamma}\times \mathbb C^{\times}}}= \bigoplus_{n} R({{\Gamma}_n\times\mathbb C^{\times}})$ the weighted bilinear form is given by $$\langle u, v \rangle_{{\xi}}^q
= \sum_{ n \geq 0} \langle u_n, v_n \rangle_{{\xi}, {{\Gamma}_n}}^q$$ where $u = \sum_n u_n$ and $v = \sum_n v_n$ with $u_n, v_n\in R({{\Gamma}_n}\times{\mathbb C^{\times}})$.
The bilinear form $\langle \ , \ \rangle_{{\xi}}^q$ on ${R_{{\Gamma}\times \mathbb C^{\times}}}$ is $\mathbb C$-bilinear and takes values in $\mathbb C[q, q^{-1}]$. When $n =1$, it reduces to the weighted bilinear form defined on $R({\Gamma\times \mathbb C^{\times}})$.
We will often omit the superscript $q$ and use the notation $\langle \ , \ \rangle_{\xi}$ for the weighted bilinear form on $R_{{\Gamma\times \mathbb C^{\times}}}$.
Quantum McKay weights {#sect_mckay}
=====================
Quantum McKay correspondence
----------------------------
Let $d_i = {\gamma}_i (c^0)$ be the dimension of the irreducible representation of ${\Gamma}$ corresponding to the character ${\gamma}_i$.
The following generalizes a result of McKay [@Mc].
\[P:qMc\] For each class $c\in \Gamma_*$ the column vector $$v(c)= ( {\gamma}_0 (c), {\gamma}_1 (c), \ldots, {\gamma}_r (c) )^t$$ is an eigenvector of the $(r+1)\times(r+1)$-matrix $A^q=(\langle{\gamma}_i, {\gamma}_j\rangle_{\xi}^q)$ with eigenvalue $ {\xi}^q (c)$. In particular $(d_0, d_1, \ldots, d_r)$ is an eigenvector of $A^q$ with eigenvalue ${\xi}^q(c^0)$.
We compute directly that $$\begin{aligned}
\sum_{k=0}^r\langle{\gamma}_i, {\gamma}_k\rangle_{\xi}^q{\gamma}_k(c)&=\sum_k\sum_{c'\in\Gamma_*}
\zeta_{c'}^{-1}\xi^q(c'){\gamma}_i(c'){\gamma}_k({c'}^{-1}){\gamma}_k(c)\\
&=\sum_{c'\in\Gamma_*}\zeta_{c'}^{-1}\xi^q(c'){\gamma}_i(c')\sum_k
{\gamma}_k({c'}^{-1}){\gamma}_k(c)\\
&=\sum_{c'\in\Gamma_*}\zeta_{c'}^{-1}\xi^q(c'){\gamma}_i(c')\zeta_c\delta_{cc'}\\
&=\xi^q(c){\gamma}_i(c).\end{aligned}$$
Let $\pi$ be an irreducible faithful representation $\pi$ of ${\Gamma}$ of dimension $d$. For each integer $n$ we define the $q$-integer $[n]$ that can be viewed as a character of ${\mathbb C^{\times}}$ by $$[n]=\frac{q^n-q^{-n}}{q-q^{-1}}=q^{n-1}+q^{n-3}+\cdots +q^{-n+1}.$$ We take the following special class function $$\label{E:weight}
{\xi}={\gamma}_0\otimes [d] - \pi\otimes 1_{{\mathbb C^{\times}}},$$ where we have also used the symbol $\pi$ for the corresponding character, and $1_{{\mathbb C^{\times}}}=q^0$ is the trivial character of ${\mathbb C^{\times}}$.
\[P:nondeg\] The weighted bilinear form associated to (\[E:weight\]) is non-degenerate. If $\pi$ is an embedding of ${\Gamma}$ into $SU_d$ and $t\neq 1$ is a nonnegative real number, then the weighted bilinear form evaluated on $t$ is positive definite.
A simple fact of finite group theory says that $$\langle f, f\rangle_{\pi}\leq d\langle f, f\rangle.$$ Assume that $t\in\mathbb R_+$. Observe that $$t^{d-1}+t^{d-3}+\cdots+t^{-d+1}\geq d$$ and the equality holds if and only if $t=1$.
Let $A^q=(\langle{\gamma}_i, {\gamma}_j\rangle_{{\xi}}^q)=(a_{ij})$. Note that for any faithful representation $\pi$ of ${\Gamma}$ we have that $$\pi*{\gamma}_i=\sum_{j}c_{ij}{\gamma}_j, \qquad c_{ij}\in\mathbb N.$$ Then it follows that $$\begin{aligned}
\langle {\gamma}_i, {\gamma}_j \rangle_{{\xi}}^q(t)&=(t^{d-1}+t^{d-3}+\cdots+t^{-d+1})
\langle {\gamma}_i, {\gamma}_j \rangle -\langle {\gamma}_i, {\gamma}_j \rangle_{\pi}\\
&=[d](t)\delta_{ij}-c_{ij}=A^{1}+([d](t)-d)I.\end{aligned}$$ According to Steinberg (see e.g. [@FJW]), $A^1$ is positive semi-definite which generalizes McKay’s observation in the case of $d=2$. This implies that the eigenvalues of $A^q$ are $\geq [d](t)-d\geq 0$. Thus the matrix $A^q(t)$ is positive-definite when $t>0$ and $t\neq 1$.
We remark that when $|t|=1$ and $t$ is close to $1$, the signature of $A^q(t)$ is $(-1, 1, \ldots, 1)$ due to $[d](t)\leq d$.
The matrix $A^1$ is integral, and the entries of $A^q$ are the $q$-numbers of the corresponding entries in $A^1$ when $r\geq 2$.
Two quantum McKay weights {#S:Mcweights}
-------------------------
Let ${\Gamma}$ is a finite subgroup of $SU_2$ and we introduce the first distinguished self-dual class function $$\xi={\gamma}_0\otimes (q+q^{-1})-\pi\otimes 1_{{\mathbb C^{\times}}},$$ where $\pi$ is the character of the embedding of ${\Gamma}$ in $SU_2$.
The matrix of the weighted bilinear form $\langle \ , \ \rangle_{\xi}$ (cf. (\[E:qcartan\])) has the following entries: $$\label{E:qcartan1}
a_{ij}=\begin{cases} q+q^{-1}, & \mbox{if } i=j,\\
-1, & \mbox{if $\langle {\gamma}_i, {\gamma}_j\rangle_{\xi}^1=-1$,}\\
-2, & \mbox{if $\langle {\gamma}_i, {\gamma}_j\rangle_{\xi}^1=-2$ and
${\Gamma}={\mathbb Z}/2{\mathbb Z}$}\\
0, & \mbox{otherwise.}
\end{cases}$$
In particular when $q=1$ the matrix $(a_{ij}^1)$ coincides with the extended Cartan matrix of ADE type according to the five classes of finite subgroups of $SU_2$: the cyclic, binary dihedral, tetrahedral, octahedral, and icosahedral groups. McKay [@Mc] gave a direct correspondence between a finite subgroup of $SU_2$ and the affine Dynkin diagram $D$ of ADE type. Each irreducible character ${\gamma}_i$ corresponds to a vertex of $D$, and the number of edges between ${\gamma}_i$ and ${\gamma}_j$ ($i\neq j)$ is equal to $|\langle {\gamma}_i, {\gamma}_j\rangle_{\xi}^1|$, where $\langle {\gamma}_i, {\gamma}_j\rangle_{\xi}^1=a_{ij}^1$ are the entries of matrix $A^1$ of the weighted bilinear form $\langle \ , \ \rangle_{\xi}^1$. For this reason we will call our matrix $A^q=(a_{ij})=(\langle{\gamma}_i, {\gamma}_j\rangle_{\xi}^q)$ the quantum Cartan matrix.
Let ${\Gamma}$ be the cyclic subgroup of $SU_2$ of order $r+1$. We can introduce the second deformation parameter in the quantum Cartan matrix. Let ${\gamma}_i (i=0, \ldots, r)$ be the full set of irreducible characters of ${\mathbb C^{\times}}$ such that ${\gamma}_i*{\gamma}_j
={\gamma}_{i+j\mod r+1}$. The embedding of ${\Gamma}$ in $SU_2$ is given by $\pi={\gamma}_1+{\gamma}_r$.
For $p=q^k\in R({\mathbb C^{\times}})$ we let $$\xi=\xi^{q, p}
={\gamma}_0\otimes (q+q^{-1})-({\gamma}_1\otimes p+{\gamma}_{r}\otimes p^{-1}).$$ When $p=1$ the second choice reduces to the first choice in type $A$. This class function is self-dual since $S({\gamma}_i)={\gamma}_{r+1-i}, i=0, 1, \ldots, r$ and $S(q)=q^{-1}, S(p)=p^{-1}$.
It is easy to see that $$\label{E:qcartan2}
a_{ij}(q, p)=\langle{\gamma}_i, {\gamma}_j\rangle_{{\xi}}^{q, p}
=[2]\delta_{ij}-p\delta_{i+1, j}-p^{-1}\delta_{i-1, j}.$$ Thus the matrix of the weighted bilinear form $\langle \ , \ \rangle_{\xi}$ (cf. (\[E:qcartan\])) has the following form. $$\label{E:2varCartan}
\begin{pmatrix}
q+q^{-1} & -p & 0 & \cdots & -p^{-1} \\
-p^{-1} & q+q^{-1} & -p & \cdots & 0 \\
0 & -p^{-1} & q+q^{-1} & \cdots & 0 \\
\cdots & \cdots & \cdots & \cdots & \cdots \\
-p & 0 & \cdots & -p^{-1} & q+q^{-1}
\end{pmatrix}, \qquad \mbox{if } r\geq 2,$$ or $$\begin{pmatrix}
q+q^{-1} & -p-p^{-1}\\
-p-p^{-1} & q+q^{-1}
\end{pmatrix}, \qquad \mbox{if } r=1.$$
Note that when ${\Gamma}=1$, the matrix of the bilinear form $\langle \ \ , \ \
\rangle_{{\xi}}^{q, p}$ is $q+q^{-1}-p-p^{-1}$, which is degenerate when $q=p^{\pm 1}$.
We will call this matrix the $(q, p)$-Cartan matrix (of type A). The self-duality of ${\xi}^{q, p}$ transforms into the condition that the $(q, p)$-Cartan matrix is $*$-invariant, where the $*$ action is the composition of transpose and bar action. Namely, $a_{ij}(q, p)=a_{ji}(q^{-1}, p^{-1})$.
If $p\neq q^{\pm 1}$, then the bilinear form $\langle \ , \ \rangle_{{\xi}}^{q, p}$ is non-degenerate. If $p=q^{\pm 1}$, the bilinear form $\langle \ , \ \rangle_{{\xi}}^{q, p}$ is degenerate of rank $r$.
Let $A^{q, p}=(\langle {\gamma}_i, {\gamma}_j\rangle_{\xi}^{q, p})$ be the matrix of the bilinear form $\langle \ , \ \rangle_{{\xi}}^{q, p}$ and let $\omega$ be a $(r+1)$-th root of unity. Then ${\gamma}_i(c^j)=\omega^{ij}$ and ${\gamma}_i*{\gamma}_j={\gamma}_{i+j}$. From this and Proposition \[P:qMc\] we see that as a matrix over $\mathbb C[q, q^{-1}]$ the eigenvalues of $A^{p, q}$ are $q+q^{-1}-\omega^i p-\omega^{-i} p^{-1}$, $i=0, \ldots, r$. The function $q+q^{-1}-\omega^i p-\omega^{-i}p^{-1}\in R({\mathbb C^{\times}})$ is non-zero except when $i=0$ and $q=p^{\pm 1}$.
Quantum Heisenberg algebras and ${{\Gamma}_n}$ {#sect_heis}
==============================================
Heisenberg algebra ${\widehat{\mathfrak h}_{{\Gamma}, {\xi}}}$
--------------------------------------------------------------
Let ${\widehat{\mathfrak h}_{{\Gamma}, {\xi}}}$ be the infinite dimensional Heisenberg algebra over $\mathbb C[q, q^{-1}]$, associated with $\Gamma$ and ${\xi}\in R({\Gamma\times \mathbb C^{\times}})$, with generators $a_m(c), c\in{\Gamma}_*, m\in \mathbb Z$ and a central element $C$ subject to the following commutation relations: $$\label{eq_heis}
[a_m(c^{-1}), a_n(c')]
= m \delta_{m, -n} \delta_{c, c'}\zeta_{c}\xi_{q^m}(c)C,
\quad c, c ' \in {\Gamma}_*.$$
For $m\in \mathbb Z, {\gamma}\in {\Gamma}^*$ and $k\in \mathbb Z$ we define $$a_m( {\gamma}\otimes q^k )
= \sum_{c \in {\Gamma}_*} \zeta_c^{ -1}
{{\gamma}} (c) a_m(c)q^{mk}$$ and then extend it to $R({\Gamma\times \mathbb C^{\times}})$ linearly over $\mathbb C$. Thus we have for ${\gamma}\in R({\Gamma\times \mathbb C^{\times}})$ $$\label{eq_real}
a_m({\gamma})
= \sum_{c \in {\Gamma}_*} \zeta_c^{ -1}
{{\gamma}_{q^m} } (c) a_m(c).$$ In particular we have $a_m({\gamma}\otimes q^k)=a_m({\gamma})q^{mk}$.
It follows immediately from the orthogonality (\[eq\_orth\]) of the irreducible characters of ${\Gamma}$ that for each $c\in {\Gamma}_*$ $$a_{ m}( c) = \sum_{ {\gamma}\in {\Gamma}^*} S(\gamma(c)) a_m( {\gamma}).$$ Note that this formula is also valid if the summation runs through ${\Gamma}^*\otimes q^k$ with a fixed $k$.
\[prop\_orth\] The Heisenberg algebra ${\widehat{\mathfrak h}_{{\Gamma}, {\xi}}}$ has a new basis given by $a_{n}({\gamma})$ and $C$ ($ n\in \mathbb Z, {\gamma}\in {\Gamma}^*$) over $\mathbb C[q, q^{-1}]$ with the following relations: $$\label{E:heisen}
[ a_m({\gamma}), {a_n( {\gamma}' )}]=m\delta_{m, -n}\langle{\gamma}, {\gamma}'
\rangle_{\xi}^{q^m}C.$$
This is proved by a direct computation using Eqns. (\[eq\_heis\]), (\[eq\_twist\]) and (\[eq\_orth\]). $$\begin{aligned}
[ a_m({\gamma}), a_n( {\gamma}')]
& =& \sum_{c, c'\in {\Gamma}_*} \zeta_c^{-1}\zeta_{c'}^{-1}
{ {\gamma}} (c){\gamma}'(c')
[ a_m( c), a_n ({c'})] \\
& =& m \delta_{m, -n}
\sum_{c, c'\in {\Gamma}_*}\zeta_{c}^{-1}\zeta_{c'}^{-1}
{ {\gamma}} (c){\gamma}'(c') \delta_{c^{-1}, c'}\zeta_c\xi_{q^m}(c)C \\
& =& m \delta_{m, -n}
\sum_{c\in {\Gamma}_*}\zeta_{c}^{-1}
{ {\gamma}} (c){\gamma}'(c^{-1}) \xi_{q^m}(c)C \\
& =& m \delta_{m, -n} \langle{\gamma}, {\gamma}'\rangle_{\xi}^{q^m}C. \end{aligned}$$
Action of ${\widehat{\mathfrak h}_{{\Gamma}, {\xi}}}$ on the Space ${S_{{\Gamma\times \mathbb C^{\times}}}}$
------------------------------------------------------------------------------------------------------------
Let ${S_{{\Gamma\times \mathbb C^{\times}}}}$ be the symmetric algebra generated by $a_{-n}({\gamma}), n \in \mathbb N,
{\gamma}\in \Gamma_*$ over $\mathbb C[q, q^{-1}]$. We define $a_{-n}({\gamma}\otimes q^k)=a_{-n}({\gamma})q^{-kn}$ and the natural degree operator on the space ${S_{{\Gamma\times \mathbb C^{\times}}}}$ by $$\deg (a_{ -n}( {\gamma}\otimes q^k)) = n$$ which makes ${S_{{\Gamma\times \mathbb C^{\times}}}}$ into a $\mathbb Z_+$-graded algebra.
The space ${S_{{\Gamma\times \mathbb C^{\times}}}}$ affords a natural realization of the Heisenberg algebra ${\widehat{\mathfrak h}_{{\Gamma}, {\xi}}}$ with $C=1$. Since $a_{-n}({\gamma}\otimes q^k)=q^{-nk}a_{-n}({\gamma})$, it is enough to describe the action for $a_{-n}({\gamma})$. The central element $C$ acts as the identity operator. For $n>0$, $a_{-n}( {\gamma})$ act as multiplication operators on $ {S_{{\Gamma\times \mathbb C^{\times}}}}$. The element $a_n ({\gamma}), n \geq 0$ acts as a differential operator through contraction: $$\begin{aligned}
& a_n ({\gamma}). a_{-n_1}( \alpha_1) a_{-n_2} (\alpha_2)
\ldots a_{-n_k}( \alpha_k) \\
&= \sum_{i =1}^k
\langle {\gamma}, \alpha_i \rangle_{{\xi}}^{q^n}
a_{-n_1}( \alpha_1) a_{-n_2}(\alpha_2) \ldots
\check{a}_{-n_i}( \alpha_i) \ldots a_{-n_k}(\alpha_k ) .\end{aligned}$$ Here $n_i > 0, \alpha_i \in R({\Gamma})$ for $i =1, \ldots , k$, and $\check{a}_{-n_i}( \alpha_i)$ means the very term is deleted. In this case ${S_{{\Gamma\times \mathbb C^{\times}}}}$ is an irreducible representation of ${\widehat{\mathfrak h}_{{\Gamma}, {\xi}}}$ with the unit $1$ as the highest weight vector.
The bilinear form on ${S_{{\Gamma\times \mathbb C^{\times}}}}$
--------------------------------------------------------------
As a ${\widehat{\mathfrak h}_{{\Gamma}, {\xi}}}$-module, the space $ {S_{{\Gamma\times \mathbb C^{\times}}}}$ admits a bilinear form $\langle \ , \ \rangle_{{\xi}} '$ over $\mathbb C[q, q^{-1}]$ characterized by $$\begin{aligned}
\nonumber
\langle 1, 1\rangle_{\xi}'&=1,\\ \label{eq_form}
\langle au, v\rangle_{\xi}'&=\langle u, a^*v\rangle_{\xi}', \qquad a\in {\widehat{\mathfrak h}_{{\Gamma}, {\xi}}},\end{aligned}$$ with the adjoint map $*$ on ${\widehat{\mathfrak h}_{{\Gamma}, {\xi}}}$ given by $$\label{eq_hermit}
a_n({\gamma}\otimes q^k)^* = a_{-n}({\gamma}\otimes q^{k}),
\qquad n\in \mathbb Z.$$ Note that the adjoint map $*$ is a $\mathbb C$-linear anti-homomorphism of ${\widehat{\mathfrak h}_{{\Gamma}, {\xi}}}$, and $q^*=\overline q$. We still use the same symbol $*$ to denote the hermitian-like dual, since it clearly generalizes the $*$-action on the deformed Cartan matrix (\[E:2varCartan\]).
For any partition ${\lambda}=( {\lambda}_1, {\lambda}_2, \dots)$ and ${\gamma}\in {\Gamma}^*$, we define $$a_{-{\lambda}}( {\gamma}) = a_{-{\lambda}_1}( {\gamma})a_{ - {\lambda}_2}( {\gamma}) \dots .$$ For $\rho = ( \rho ({\gamma}) )_{ {\gamma}\in {\Gamma}^*} \in {\mathcal P}({\Gamma}^* )$, we define $$a_{ - \rho\otimes q^k } = q^{-k\|\rho\|}\prod_{{\gamma}\in {\Gamma}^*} a_{ - \rho ({\gamma})}({\gamma}).$$ It is clear that for a fixed $k\in \mathbb Z$ the elements $a_{ - \rho\otimes q^k},
\rho \in {\mathcal P}({\Gamma}^* )$ form a basis of ${S_{{\Gamma\times \mathbb C^{\times}}}}$ over $\mathbb C[q, q^{-1}]$.
Given a partition $ {\lambda}= ( {\lambda}_1, {\lambda}_2, \ldots )$ and $c \in {\Gamma}_*$, we define $$\begin{aligned}
a_{ - {\lambda}} (c\otimes q^k ) & =& q^{-k|{\lambda}|}
a_{ - {\lambda}_1}(c) a_{ - {\lambda}_2 } (c) \ldots, \\\end{aligned}$$ For any $\rho = ( \rho (c) )_{ c \in {\Gamma}_* } \in
\mathcal P ( {\Gamma}_* )$ and $k\in \mathbb Z$, we define $$a_{- \rho\otimes q^k}' = q^{-k\|\rho\|}
\prod_{ c \in {\Gamma}_*} a_{ - \rho (c)} (c).$$
It follows from Proposition \[prop\_orth\] that $$\begin{aligned}
\label{eq_inner}
\langle a_{ - \rho\otimes q^k}', {a_{ - \overline{\sigma}\otimes q^{l}}'}
\rangle_{{\xi}}'
= \delta_{\rho, \sigma}q^{\|\rho\|(l-k)}
Z_{\rho} \prod_{c \in {\Gamma}_*} \prod_{i\geq 1}{\xi}_{q^i}(c)^{m_i(\rho (c))},
\end{aligned}$$ where $\rho, \sigma \in \mathcal P ({\Gamma}_*)$. Note that $S(a_{-\rho\otimes q^k}')=a_{-\overline{\rho}\otimes q^{-k}}'$, where we recall that $\overline{\rho}
\in {\mathcal P}({\Gamma}_* )$ is the partition-valued function given by $c\mapsto \rho(c^{-1})$, $c\in {\Gamma}$.
The characteristic map as an isometry {#sect_isom}
=====================================
The characteristic map ${\mbox{ch}}$
------------------------------------
Let $\Psi : {{\Gamma}_n}\rightarrow {S_{{\Gamma\times \mathbb C^{\times}}}}$ be the map defined by $\Psi (x) = a_{ - \rho}'$ if $x \in {{\Gamma}_n}$ is of type $\rho$.
We define a $\mathbb C$-linear map $ch: {R_{{\Gamma}\times \mathbb C^{\times}}}\longrightarrow {S_{{\Gamma\times \mathbb C^{\times}}}}$ by letting $$\begin{aligned}
\nonumber
ch (f )
&= \langle f, \Psi \rangle_{{{\Gamma}_n}}\\
&= \sum_{\rho \in \mathcal P({\Gamma}_*)} Z_{\rho}^{-1} S(f(\rho))
a_{ - \rho}', \label{E:ch}\end{aligned}$$ where $f(\rho)\in \mathbb C[q, q^{-1}]$ is the value of $f$ at the elements of type $\rho$. The map $ch $ is called the [*characteristic map*]{}. This generalizes the definition of the characteristic map in the classical setting (cf. [@M2; @FJW]).
The space ${S_{{\Gamma\times \mathbb C^{\times}}}}$ can also be interpreted as follows. The element $a_{ -n } ({\gamma}), n >0 , {\gamma}\in {\Gamma}^*$ is identified as the $n$-th power sum in a sequence of variables $ y_{ g } = ( y_{i{\gamma}} )_{i \geq 1}$. By the commutativity among $a_{-n}({\gamma})$ (${\gamma}\in{\Gamma}^*, n>0$) and dimension counting it is clear that the space ${S_{{\Gamma\times \mathbb C^{\times}}}}$ is isomorphic with the space ${ {\Lambda}_{{\Gamma}}}$ of symmetric functions indexed by $ {\Gamma}^*$ tensored with $\mathbb C[q, q^{-1}]$ (cf. [@M2]).
Denote by $c_n (c \in {\Gamma}_*)$ the conjugacy class in ${{\Gamma}_n}$ of elements $(x, s) \in {{\Gamma}_n}$ such that $s$ is an $n$-cycle and $x \in c$. Denote by $\sigma_n (c\otimes q^k )$ the class function on ${{\Gamma}_n}\times{\mathbb C^{\times}}$ which takes values $n \zeta_ct^{-nk}$ (i.e. the order of the centralizer of an element in the class $c_n$ times $t^{-nk}$) on elements in the class $c_n\times t$ and $0$ elsewhere. For $\rho = \{ m_r (c) \}_{r \geq 1, c \in {\Gamma}_*}
\in \mathcal P_n ({\Gamma}^*)$ and $k\in\mathbb Z$, $$\sigma_{\rho\otimes q^k} = q^{-nk}
\prod_{r \geq 1, c \in {\Gamma}_*} \sigma_r (c)^{m_r (c)}$$ is the class function on ${{\Gamma}_n}\times{\mathbb C^{\times}}$ which takes value $Z_{\rho}t^{-nk}$ on the conjugacy class of type $\rho\times t$ and $0$ elsewhere. Given ${\gamma}\in {\Gamma}^*$ and $k\in \mathbb Z$, we denote by $\sigma_n ({\gamma}\otimes q^k)$ the class function on ${{\Gamma}_n\times\mathbb C^{\times}}$ which takes values $n {\gamma}(c)t^{-nk}$ on elements in the class $c_n\times t (c \in {\Gamma}_*)$ and $0$ elsewhere.
\[lem\_isom\] The map $ch$ sends $\sigma_{\rho\otimes q^k}$ to $a_{ - \rho\otimes q^k} '$. In particular, it sends $\sigma_n({\gamma}\otimes q^k )$ to $a_{ -n} ( {\gamma}\otimes q^k )$ in ${S_{{\Gamma\times \mathbb C^{\times}}}}$.
This is verified by the definition of $ch$ (\[E:ch\]) and the character values of $\sigma_n$ defined above.
Given ${\gamma}\in {\Gamma}^*$, the character value of $\eta_n({\gamma}\otimes q^k)$ on the conjugacy class $c_{\rho}$ of type $\rho=(\rho(c))_{c\in{\Gamma}_*}$ is given by $$\label{E:charval}
\eta_n({\gamma}\otimes q^k)(c_{\rho})
=\prod_{c\in {\Gamma}_*}{\gamma}(c)^{l(\rho(c))}q^{nk}.$$ In particular, we have $\eta_n({\gamma}\otimes q^k)=\eta_n({\gamma})q^{nk}$.
We first let $(g, \sigma)$ be an element of ${{\Gamma}_n}$ such that $\sigma$ is a cycle of length $n$, say $\sigma=(12\cdots n)$. Let $\{e_i\}$ be a basis of $V$, and ${\gamma}\otimes q^k$ is afforded by the action: $(h, t)e_j=\sum_{i}c_{ij}(h)t^ke_i$, where $h\in {\Gamma}$. We then have $$\begin{aligned}
&(g, \sigma, t).(e_{j_1}\otimes e_{j_2}\otimes \cdots\otimes e_{j_n})\\
&=(g_1, t)e_{j_n}\otimes (g_2, t)e_{j_1} \otimes \cdots\otimes (g_n, t)e_{j_{n-1}}\\
&=\sum_{i_1,\ldots, i_n}t^{kn} c_{i_{n}j_n}(g_1)c_{i_1j_1}(g_2)\cdots
c_{i_{n-1}j_{n-1}}(g_n)
e_{i_n}\otimes e_{i_1}\cdots\otimes e_{i_{n-1}}.\end{aligned}$$
It follows that $$\begin{aligned}
\eta_n ({\gamma}\otimes q^k) (c_{\rho}, t) &=& \mbox{trace }(g, \sigma, t)\\
&=& \sum_{j_1, \ldots, j_n} t^{kn}c_{j_1j_n}(g_1)c_{j_2j_1}(g_2)\cdots
c_{j_nj_{n-1}}(g_n)\\
& =& \mbox{trace } t^{kn}a(g_n) a(g_{n-1}) \ldots a(g_1) \\
& =& \mbox{trace } t^{kn}a(g_n g_{n -1} \ldots g_1) = {\gamma}(c)q^{kn}(t).
\end{aligned}$$
Given $x\times y \in {{\Gamma}_n}$ where $x \in {\Gamma}_r$ and $y \in {\Gamma}_{n -r}$, by (\[E:eta-action\]) we clearly have $$\eta_n ({\gamma}\otimes q^k) (x\times y, t) = \eta_n ({\gamma}\otimes q^k)
(x, t)
\eta_n ({\gamma}\otimes q^k)(y, t).$$ This immediately implies the formula.
A similar argument gives that $$\varepsilon_n ({\gamma}\otimes q^k ) ( x, t)
= (-1)^n \prod_{c\in {\Gamma}_*}
( - {\gamma}(c))^{l(\rho(c))}t^{nk},
\label{eq_signterm}$$ where $x$ is any element in the conjugacy class of type $\rho=(\rho(c))_{c\in{\Gamma}^*}$.
Formula (\[E:charval\]) is equivalent to the following: $$\label{E:charval''}
\eta_n({\gamma}\otimes q^k)(c_{\rho}, t)=\prod_{c\in {\Gamma}_*}
\prod_{i\geq 1}
({\gamma}\otimes q^k)(c, t^{i})^{m_i(\rho(c))}.$$
The following result allows us to extend the map from ${\gamma}\in{\Gamma}^*$ to $R({\Gamma}_n)$.
\[prop\_exp\] For any ${\gamma}\in R({\Gamma})$, we have $$\begin{aligned}
\sum\limits_{n \ge 0} {\mbox{ch}}( \eta_n( {\gamma}\otimes q^k ) ) z^n
&= \exp \Biggl( \sum_{ n \ge 1}
\frac 1n \, a_{-n}({\gamma})(q^{-k}z)^n \Biggr), \label{eq_exp} \\
\sum\limits_{n \ge 0} {\mbox{ch}}( \varepsilon_n( {\gamma}\otimes q^k ) ) z^n
&= \exp \Biggl( \sum_{ n \ge 1}
( -1)^{ n -1} \frac 1n \, a_{-n}({\gamma})(q^{-k}z)^n \Biggr).
\label{eq_sign}\end{aligned}$$
It follows from definition of ch (\[E:ch\]) and (\[E:charval”\]) that
$$\begin{aligned}
&&\sum\limits_{n \ge 0} {\mbox{ch}}( \eta_n( {\gamma}\otimes q^k ) ) z^n\\
&= & \sum_{\rho} Z_{\rho}^{ -1}
\prod_{c\in {\Gamma}_*}\prod_{i\geq 1}S({\gamma}_{q^{ik}} (c)^{m_i(\rho(c))})
a_{ -\rho (c) } z^{|| \rho||}q^{-||\rho||} \\
&= & \sum_{\rho} Z_{\rho}^{ -1}
\prod_{c\in {\Gamma}_*}{\gamma}(c)^{l(\rho(c))}
a_{ -\rho (c) } (q^{-k}z)^{|| \rho||} \\
&= & \prod_{c\in {\Gamma}_*} \Bigl ( \sum_{\lambda }
(\zeta_c^{ -1}{\gamma}(c) )^{l (\lambda)}
z_{\lambda}^{-1} a_{- \lambda} (c) (q^{-k}z)^{|\lambda|} \Bigr ) \\
&= & \exp \Biggl ( \sum\limits_{ n \geq 1}
\frac1n \sum\limits_{c \in {\Gamma}_*}
\zeta_c^{ -1} {\gamma}(c) a_{-n} (c) (q^{-k}z)^n \Biggl ) \\
&= & \exp \Biggl( \sum_{ n \ge 1}
\frac 1n \, a_{-n}({\gamma})(q^{-k}z)^n \Biggr).
\end{aligned}$$
Similarly we can prove (\[eq\_sign\]) using the following identity $$\begin{aligned}
\varepsilon_n ({\gamma}\otimes q^k ) ( x)
&= &(-1)^n \prod_{c\in {\Gamma}_*} \prod_{i\geq 1}
( - {\gamma}_{q^{ik}} (c))^{m_i(\rho(c))} \nonumber\\
&= &(-q^k)^n \prod_{c\in {\Gamma}_*} \prod_{i\geq 1}
( - {\gamma}(c))^{m_i(\rho(c))} \nonumber\\
&=&\varepsilon_n ({\gamma}) ( x)q^{nk}.
\label{eq_signterm'} \nonumber
\end{aligned}$$
The same argument as in the classical case (cf. [@FJW]) by using (\[eq\_virt\]) and (\[eq\_virt’\]) will show that the proposition holds for linear combination of simple characters such as ${\gamma}\otimes q^k-\beta\otimes q^k$, and thus it is true for any element ${\gamma}\otimes q^k$, where ${\gamma}\in R({\Gamma})$.
Comparing components we obtain $$\begin{aligned}
{\mbox{ch}}(\eta_n ({\gamma}\otimes q^k ) )
&=& \sum\limits_{{\lambda}}\frac {q^{-nk}}{z_{\lambda}}\,
a_{-\lambda}({\gamma}), \\
{\mbox{ch}}(\varepsilon_n ({\gamma}\otimes q^k ))
&=& \sum\limits_{{\lambda}}\frac {q^{-nk}}{z_{\lambda}}\,
( -1)^{ | {\lambda}| - l ( {\lambda})} a_{-\lambda}({\gamma}),\end{aligned}$$ where the sum runs over all partitions $\lambda$ of $n$.
\[cor\_char\] The formula (\[E:charval”\]) remains valid when ${\gamma}\otimes q^k$ is replaced by any element ${\xi}\in R({\Gamma\times \mathbb C^{\times}})$. In particular $\eta_n ({\xi})$ is self-dual provided that ${\xi}$ is invariant under the antipode $S$.
Isometry between ${R_{{\Gamma}\times \mathbb C^{\times}}}$ and ${S_{{\Gamma\times \mathbb C^{\times}}}}$
--------------------------------------------------------------------------------------------------------
The symmetric algebra ${S_{{\Gamma\times \mathbb C^{\times}}}}={ S_{\Gamma}}\otimes \mathbb C[q, q^{-1}]$ has the following Hopf algebra structure over $\mathbb C$. The multiplication is the usual one, and the comultiplication is given by $$\begin{aligned}
\Delta(q^k)&=q^k\otimes q^k\\
\Delta ( a_n ({\gamma}\otimes q^k ))
&=a_n({\gamma}\otimes q^k)\otimes q^{nk}+q^{nk}\otimes a_n({\gamma}\otimes q^k ),\end{aligned}$$ where ${\gamma}\in{\Gamma}^*$. The last formula is equivalent to the following: $$\label{E:coprod}
\Delta ( a_n (c\otimes q^k ))
=a_n(c\otimes q^k)\otimes q^{nk}+q^{nk}\otimes a_n
(c\otimes q^k ),$$ where $c\in{\Gamma}_*$. The antipode is given by $$\begin{aligned}
S(q^k)&=q^{-k},\\
S(a_n ({\gamma}\otimes q^k ))&=-a_n({\gamma}\otimes q^{-k})\end{aligned}$$ The antipode commutes with the adjoint (dual) map $*$: $$*^2=S^2=Id, \qquad S*=*S.$$
Recall that we have defined a Hopf algebra structure on ${R_{{\Gamma}\times \mathbb C^{\times}}}$ in Sect. \[sect\_wreath\].
\[P:isometry1\] The characteristic map $ {\mbox{ch}}: {R_{{\Gamma}\times \mathbb C^{\times}}}\longrightarrow {S_{{\Gamma\times \mathbb C^{\times}}}}$ is an isomorphism of Hopf algebras.
It follows immediately from the definition of the comultiplication in the both Hopf algebras (cf. (\[E:comult\]) and (\[E:coprod\])).
The comultiplication (\[E:coprod\]) is in fact induced from that of the classical case in [@FJW] and only works for $C=1$.
There is another coproduct called Drinfeld comultiplication $\Delta_D$ on the algebra ${S_{{\Gamma\times \mathbb C^{\times}}}}$ adjoined by a central element $q^c$. The formula on ${S_{{\Gamma\times \mathbb C^{\times}}}}$ at level $c$ is as follows [@J2]: $$\Delta_D(a_n({\gamma}))=a_n({\gamma})\otimes q^{|n|c/2}+q^{-|n|c/2}\otimes a_n({\gamma}).$$ We do not know a conceptual interpretation of the Drinfeld comultiplication in ${R_{{\Gamma}\times \mathbb C^{\times}}}$.
Recall that we have defined a bilinear form $\langle \ , \, \rangle_{{\xi}}$ on ${R_{{\Gamma}\times \mathbb C^{\times}}}$ and a bilinear form on ${S_{{\Gamma\times \mathbb C^{\times}}}}$ denoted by $\langle \ , \, \rangle_{{\xi}}'$, where ${\xi}$ is a self-dual class function. The following lemma is immediate from our definition of $\langle \ , \, \rangle_{{\xi}}'$ and the comultiplication $\Delta$.
The bilinear form $\langle \ , \, \rangle_{{\xi}} '$ on ${S_{{\Gamma\times \mathbb C^{\times}}}}$ can be characterized by the following two properties:
1). $\langle a_{ -n} (\beta\otimes q^k ), a_{ -m} ({\gamma}\otimes q^l )
\rangle_{{\xi}}^{'}
= \delta_{n, m} q^{n(l-k)}\langle \beta , {\gamma}\rangle_{{\xi}}^{'} ,
\quad \beta, {\gamma}\in {\Gamma}^*,$ $ k, l\in\mathbb Z.$
2). $ \langle f g , h \rangle_{{\xi}}^{'}
= \langle f \otimes g, \Delta h
\rangle_{{\xi}}^{'} ,$ where $f, g, h \in {S_{{\Gamma\times \mathbb C^{\times}}}}$, and the bilinear form on ${S_{{\Gamma\times \mathbb C^{\times}}}}\otimes {S_{{\Gamma\times \mathbb C^{\times}}}}$, is induced from $\langle \ , \ \rangle_{{\xi}}^{'}$ on ${S_{{\Gamma\times \mathbb C^{\times}}}}$.
\[th\_isometry\] The characteristic map is an isometry from the space $ ({R_{{\Gamma}\times \mathbb C^{\times}}}, \langle \ \ , \ \ \rangle_{{\xi}})$ to the space $ ({S_{{\Gamma\times \mathbb C^{\times}}}}, \langle \ \ , \ \ \rangle_{{\xi}}' )$.
By Corollary \[cor\_char\], the character value of $\eta_n ({\xi})$ at an element $x$ of type $\rho$ is $$\eta_n ({\xi}) ( x)= \prod_{c\in {\Gamma}_*}\prod_{i\geq 1} {\xi}_{q^i} (c)^{m_i(\rho(c))}.$$ Thus it follows from definition that $$\begin{aligned}
\langle \sigma_{ \rho\otimes q^k}, \sigma_{ \rho '\otimes q^l}
\rangle_{{\xi}}
& =& \sum_{\mu \in \mathcal P_n ({\Gamma}_*)}
Z_{\mu}^{ -1} q^{n(l-k)}{\xi}_{q} (c_{\mu }) \sigma_{\rho} (c_{\mu})
\sigma_{\rho '} (c_{\mu}) \\
& =& \delta_{\rho, \rho '}
Z_{\rho}^{ -1}q^{n(l-k)}{\xi}(c_{\rho}) Z_{\rho}Z_{\rho} \\
& =& \delta_{\rho, \rho '}
Z_{\rho} q^{n(l-k)}
\prod_{c \in {\Gamma}_*}\prod_{i\geq 1} {\xi}_{q^i} (c)^{m_i(\rho(c))}.
\end{aligned}$$
By Lemma \[lem\_isom\] and the formula (\[eq\_inner\]), we see that $$\langle \sigma_{ \rho\otimes q^k}, \sigma_{ \rho '\otimes q^l}
\rangle_{{\xi}}
= \langle a_{- \rho\otimes q^k}, a_{ - \rho '\otimes q^l} \rangle_{{\xi}} '
= \langle {\mbox{ch}}(\sigma_{ \rho\otimes q^k}),
{\mbox{ch}}(\sigma_{ \rho '\otimes q^l} ) \rangle_{{\xi}} '.$$ Since $\sigma_{\rho\otimes q^k}, \rho \in \mathcal P({\Gamma}_*)$ form a $\mathbb C$-basis of ${R_{{\Gamma}\times \mathbb C^{\times}}}$, we have shown that ${\mbox{ch}}: {R_{{\Gamma}\times \mathbb C^{\times}}}\longrightarrow {S_{{\Gamma\times \mathbb C^{\times}}}}$ is an isometry.
>From now on we will not distinguish the bilinear form $\langle \ , \ \rangle_{{\xi}}$ on ${R_{{\Gamma}\times \mathbb C^{\times}}}$ from the bilinear form $\langle \ , \ \rangle_{{\xi}}^{'}$ on ${S_{{\Gamma\times \mathbb C^{\times}}}}$.
Quantum vertex operators and ${R_{{\Gamma}\times \mathbb C^{\times}}}$ {#sect_vertex}
======================================================================
Vertex Operators and Heisenberg algebras in ${{\mathcal F}_{{\Gamma}\times \mathbb C^{\times}} }$
-------------------------------------------------------------------------------------------------
Let $Q$ be an integral lattice with basis ${\alpha}_i$, $i=0, 1, \ldots, r$ endowed with a symmetric bilinear form. As in the case of $q=1$ (cf. [@FK]), we fix a $2$-cocycle $\epsilon: Q\times Q \longrightarrow \mathbb C^{\times}$ such that $${\epsilon}({\alpha}, \beta)={\epsilon}(\beta, {\alpha})(-1)^{\langle \alpha, \beta \rangle +
\langle \alpha, \alpha \rangle \langle \beta , \beta \rangle}.$$ We remark that the cocycle can be constructed directly by prescribing the values of $({\alpha}_i, {\alpha}_j)\in \{\pm 1\}$ $(i<j)$.
Let $\xi$ be a self-dual virtual character in ${R_{{\Gamma}\times \mathbb C^{\times}}}$. Recall that the lattice ${R_{\mathbb Z}(\Gamma)}$ is a $\mathbb Z$-lattice under the bilinear form $\langle \ , \ \rangle_{{\xi}}^1$, here the superscript means $q=1$. For our purpose we will always associate a $2$-cocycle ${\epsilon}$ as in the previous subsection to the integral lattice $({R_{\mathbb Z}(\Gamma)}, \langle \ , \ \rangle_{{\xi}}^1 )$ (and its sublattices).
Let $\mathbb C[{R_{\mathbb Z}(\Gamma)}]$ be the group algebra generated by $e^{{\gamma}}$, ${\gamma}\in {R_{\mathbb Z}(\Gamma)}$. We introduce two special operators acting on $\mathbb C[ {R_{\mathbb Z}(\Gamma)}]$: A (${\epsilon}$-twisted) multiplication operator $e^{\alpha}$ defined by $$e^{\alpha }.e^{\beta } = {\epsilon}(\alpha, \beta) e^{\alpha +\beta},
\quad \alpha, \beta \in {R_{\mathbb Z}(\Gamma)},$$ and a differentiation operator ${\partial_{\alpha }}$ given by $$\begin{aligned}
{\partial_{{\alpha}}} e^{ \beta} =
\langle {\alpha}, \beta \rangle_{{\xi}}^1 e^{ \beta},
\quad \alpha, \beta \in {R_{\mathbb Z}(\Gamma)}.\end{aligned}$$ These two operators are then extended linearly to the space $$\label{E:fgc}
{{\mathcal F}_{{\Gamma}\times \mathbb C^{\times}} }= {R_{{\Gamma}\times \mathbb C^{\times}}}\otimes \mathbb C[{R_{\mathbb Z}(\Gamma)}]$$ by letting them act on the ${R_{{\Gamma}\times \mathbb C^{\times}}}$ part trivially.
We define the Hopf algebra structure on $\mathbb C[{R_{\mathbb Z}(\Gamma)}]$ and extend the Hopf algebra structure from ${R_{{\Gamma}\times \mathbb C^{\times}}}$ to ${{\mathcal F}_{{\Gamma}\times \mathbb C^{\times}} }$as follows. $$\Delta(e^{\alpha})=e^{\alpha}\otimes e^{\alpha},
\qquad S(e^{\alpha})=e^{-\alpha}.$$
The bilinear form $\langle\ , \ \rangle_{{\xi}}^q$ on ${R_{{\Gamma}\times \mathbb C^{\times}}}$ is extended to ${{\mathcal F}_{{\Gamma}\times \mathbb C^{\times}} }$ by $$\langle e^{\alpha}, e^{\beta}\rangle_{{\xi}}=\delta_{{\alpha},\beta}.$$
With respect to this extended bilinear form we have the $*$-action (adjoint action) on the operators $e^{\alpha}$ and ${\partial}_{\alpha}$: $$(e^{\alpha})^*=e^{-\alpha}, \qquad
(z^{{\partial}_{\alpha}})^*=z^{-{\partial}_{\alpha}}.$$
For each $k\in\mathbb Z$, we introduce the group theoretic operators $ H_{ \pm n}( {\gamma}\otimes q^k ),
E_{ \pm n} ( {\gamma}\otimes q^k), {\gamma}\in R({\Gamma}), n > 0 $ as the following compositions of maps: $$\begin{aligned}
H_{ -n} ( {\gamma}\otimes q^k ) &:&
R ( {{\Gamma}_m\times\mathbb C^{\times}})
\stackrel{ \eta_n ({\gamma}\otimes q^k) \otimes}{\longrightarrow}
R ( {{\Gamma}_n\times\mathbb C^{\times}}) \otimes R ( {{\Gamma}_m\times\mathbb C^{\times}})\\&&
\stackrel{ {Ind}\otimes m_{{\mathbb C^{\times}}} }{\longrightarrow}
R ( {\Gamma}_{n +m}\times{\mathbb C^{\times}}) \\
E_{ -n} ( {\gamma}\otimes q^k ) &:&
R ( {{\Gamma}_m\times\mathbb C^{\times}})
\stackrel{ \varepsilon_n ({\gamma}\otimes q^k) \otimes}{\longrightarrow}
R ( {{\Gamma}_n\times\mathbb C^{\times}}) \otimes R ( {{\Gamma}_m\times\mathbb C^{\times}})\\&&
\stackrel{ {Ind}\otimes m_{{\mathbb C^{\times}}} }{\longrightarrow}
R ( {\Gamma}_{n +m}\times{\mathbb C^{\times}}) \\
E_n ( {\gamma}\otimes q^k ) &:&
R ( {{\Gamma}_m\times\mathbb C^{\times}})
\stackrel{ {Res} }{\longrightarrow}
R( {{\Gamma}_n}) \otimes R( {\Gamma}_{m -n}\times{\mathbb C^{\times}})\\&&
\stackrel{ \langle \varepsilon_n ({\gamma}\otimes q^k),
\cdot \rangle_{{\xi}} }{\longrightarrow}
R ( {\Gamma}_{m -n}\times{\mathbb C^{\times}}) \\
H_n({\gamma}\otimes q^k ) &:&
R ( {{\Gamma}_m\times\mathbb C^{\times}})
\stackrel{ {Res} }{\longrightarrow}
R( {{\Gamma}_n}) \otimes R( {\Gamma}_{m -n}\times{\mathbb C^{\times}})\\&&
\stackrel{ \langle \eta_n ({\gamma}\otimes q^k),
\cdot \rangle_{{\xi}} }{\longrightarrow}
R ( {\Gamma}_{m -n}\times{\mathbb C^{\times}}) ,\end{aligned}$$ where $Res$ and $Ind$ are the restriction and induction functors in $R_{{\Gamma}}=\bigoplus_{n\geq 0}R({\Gamma}_n)$.
We introduce their generating functions in a formal variable $z$: $$\begin{aligned}
H_{\pm} ({\gamma}\otimes q^k, z) &=& \sum_{ n\geq 0} H_{ \mp n}
( {\gamma}\otimes q^k ) z^{\pm n}, \\
E_{\pm} ({\gamma}\otimes q^k, z) &=& \sum_{ n\geq 0} E_{ \mp n}
( {\gamma}\otimes q^k )( -z)^{\pm n}.\end{aligned}$$ We now define the vertex operators $Y_n ^{\pm}({\gamma}\otimes q^l, k)$ , ${\gamma}\in {\Gamma}^*$, $k, l\in \mathbb Z$, $n \in {\mathbb Z} + \langle {\gamma}, {\gamma}\rangle_{ {\xi}}^1 /2$ as follows. $$\begin{aligned}
\nonumber
Y ^{+}( {\gamma}\otimes q^l, k, z)
& = \sum\limits_{n \in
{\mathbb Z} + \langle {\gamma}, {\gamma}\rangle_{ {\xi}}^1 /2}
Y_n^{+}( \gamma\otimes q^l, k)
z^{ -n - \langle {\gamma}, {\gamma}\rangle_{ {\xi}}^1 /2}
\\
& = H_+ ({\gamma}\otimes q^l, z) E_- ({\gamma}\otimes q^{l-k} , z) e^{ {\gamma}}
(q^{-l}z)^{ \partial_{ {\gamma}}}, \label{eq_vo} \end{aligned}$$ $$\begin{aligned}
\nonumber
Y^-({\gamma}\otimes q^l, k, z)&=(Y^+({\gamma}\otimes q^l, k, z^{-1}))^*\\ \nonumber
&=\sum\limits_{n \in
{\mathbb Z} + \langle {\gamma}, {\gamma}\rangle_{ {\xi}}^1 /2}
Y_n^{-}( \gamma\otimes q^l, k)
z^{ -n - \langle {\gamma}, {\gamma}\rangle_{ {\xi}}^1 /2}\\
&=E_+ ({\gamma}\otimes q^{l-k}, z)
H_- ({\gamma}\otimes q^l , z) e^{ -{\gamma}}
(q^{-l}z)^{ -\partial_{ {\gamma}}}.\end{aligned}$$
One easily sees that the operators $Y_n^{\pm} ({\gamma}\otimes q^l, k)$ are well-defined operators acting on the space ${{\mathcal F}_{{\Gamma}\times \mathbb C^{\times}} }$.
We extend the $\mathbb Z_+$-gradation on ${R_{{\Gamma}\times \mathbb C^{\times}}}$ to a $\frac12\langle {\gamma}, {\gamma}\rangle_{{\xi}}^1 +
\mathbb Z_+$-gradation on ${{\mathcal F}_{{\Gamma}\times \mathbb C^{\times}} }$ by letting $$\begin{aligned}
\deg a_{ -n} ({\gamma}\otimes q^k ) = n , \quad
\deg e^{{\gamma}} = \frac12 \langle {\gamma}, {\gamma}\rangle_{{\xi}}^1 .\end{aligned}$$
We denote by ${\overline{R}_{ {\Gamma}\times \mathbb C^{\times}}}$ the subalgebra of ${R_{{\Gamma}\times \mathbb C^{\times}}}$ excluding the generators $a_n({\gamma}_0)$, $n\in \mathbb Z^{\times}$. The bilinear form $\langle \ , \ \rangle_{\xi}$ on $${\overline{\mathcal F}_{{\Gamma}\times \mathbb C^{\times}}}={\overline{R}_{ {\Gamma}\times \mathbb C^{\times}}}\otimes{\overline{R}_{\mathbb Z}({{\Gamma}})}$$ will be the restriction of $\langle \ , \ \rangle_{\xi}$ on ${{\mathcal F}_{{\Gamma}\times \mathbb C^{\times}} }$ to ${\overline{\mathcal F}_{{\Gamma}\times \mathbb C^{\times}}}$. In the case of the second choice of $\xi$ and $p=q^{\pm 1}$, the Fock space ${\overline{\mathcal F}_{{\Gamma}\times \mathbb C^{\times}}}$ can also be obtained as the quotient of ${{\mathcal F}_{{\Gamma}\times \mathbb C^{\times}} }$ modulo the radical of $\langle \ , \ \rangle_{\xi}$.
We define $ \widetilde{a}_{ -n} (\gamma\otimes q^k), n >0$ to be a map from ${R_{{\Gamma}\times \mathbb C^{\times}}}$ to itself by the following composition $$\begin{aligned}
R ({{\Gamma}_m\times\mathbb C^{\times}}) \stackrel{ \sigma_n ( {\gamma}\otimes q^k ) \otimes }{\longrightarrow}&
R({{\Gamma}_n\times\mathbb C^{\times}}) \otimes R ({{\Gamma}_m\times\mathbb C^{\times}}) \\
& \stackrel{{Ind\otimes m_{{\mathbb C^{\times}}}} }{\longrightarrow}
R ( {\Gamma}_{n +m}\times {\mathbb C^{\times}}).\end{aligned}$$ We also define $ \widetilde{a}_{ n} (\gamma\otimes q^k), n >0$ to be a map from ${R_{{\Gamma}\times \mathbb C^{\times}}}$ to itself as the composition $$\begin{aligned}
R ({{\Gamma}_m\times\mathbb C^{\times}}) \stackrel{ Res \otimes 1}{\longrightarrow}&
R({{\Gamma}_n\times\mathbb C^{\times}})\otimes R ( {{\Gamma}}_{m -n}\times {\mathbb C^{\times}})\\
&\stackrel{ \langle \sigma_n ( {\gamma}\otimes q^k), \cdot \rangle_{{\xi}}^q}{\longrightarrow}
R ( {{\Gamma}}_{m -n}\times{\mathbb C^{\times}}).\end{aligned}$$
The operators $\widetilde{a}_{n} (\gamma)$, ${\gamma}\in {\Gamma}^*, n\in\mathbb Z^{\times}$ satisfy the Heisenberg algebra relations (\[eq\_heis\]) with $C=1$.
This is similarly proved as for the classical setting in [@W].
Group theoretic interpretation of vertex operators
--------------------------------------------------
To compare the vertex operators $Y^{\pm}({\gamma}\otimes q^l, k, z)$ with the familiar vertex operators acting in the Fock space we introduce the space $${V_{{\Gamma}\times \mathbb C^{\times}}}= {S_{{\Gamma\times \mathbb C^{\times}}}}\otimes \mathbb C [ {R_{\mathbb Z}(\Gamma)}].$$
We extend the bilinear form $\langle \ , \ \rangle_{{\xi}}^q$ in ${S_{{\Gamma\times \mathbb C^{\times}}}}$ to the space ${V_{{\Gamma}\times \mathbb C^{\times}}}$ and also extend the $\mathbb Z_+$-gradation on ${S_{{\Gamma\times \mathbb C^{\times}}}}$ to a $\frac12 \mathbb Z_+$-gradation on ${V_{ {\Gamma}}}$.
We extend the characteristic map to the map $$ch: {{\mathcal F}_{{\Gamma}\times \mathbb C^{\times}} }\longrightarrow {V_{{\Gamma}\times \mathbb C^{\times}}}$$ by identity on ${R_{\mathbb Z}(\Gamma)}$. Then Proposition \[P:isometry1\] and Theorem \[th\_isometry\] imply that we have an isometric isomorphism of Hopf algebras. We can now identify the operators from the previous subsections with the operators constructed from the Heisenberg algebra.
\[T:characteristic\] For any ${\gamma}\in R({\Gamma})$ and $k\in\mathbb Z$, we have $$\begin{aligned}
\label{E:ch1}
{\mbox{ch}}\bigl ( H_+ ({\gamma}\otimes q^k, z) \bigl )
&=& \exp \biggl ( \sum\limits_{ n \ge 1} \frac 1n \,
a_{-n} ( {\gamma}) (q^{-k}z)^n \biggr ), \\ \label{E:ch2}
{\mbox{ch}}\bigl ( E_+ ({\gamma}\otimes q^k, z) \bigl )
&=& \exp\biggl ( -\sum\limits_{n\ge 1}\frac 1n \,
a_{-n}(\gamma)(q^{-k}z)^n\biggr ), \\ \label{E:ch3}
{\mbox{ch}}\bigl ( H_- ({\gamma}\otimes q^k , z) \bigl )
&=& \exp \biggl ( \sum\limits_{n \ge 1}\frac 1n \,
a_n({\gamma}) (q^{-k}z)^{-n}\biggr ), \\ \label{E:ch4}
{\mbox{ch}}\bigl ( E_- ({\gamma}\otimes q^k , z) \bigl )
&=& \exp\,\, \biggl ( -\sum\limits_{ n \ge 1}
\frac 1n \,{ a_n( {\gamma})} (q^{-k}z)^{ -n} \biggr ) .
\end{aligned}$$
The first and second identities were essentially established in Proposition \[prop\_exp\] together with Lemma \[lem\_isom\], where the components are viewed as operators acting on ${R_{{\Gamma}\times \mathbb C^{\times}}}$ or ${S_{{\Gamma\times \mathbb C^{\times}}}}$. Note that $a_n({\gamma}\otimes q^k)=a_n({\gamma})q^{kn}$.
We observe from definition that the adjoint $*$-action of $E_+ ({\gamma}\otimes q^k, z)$ and $H_- ({\gamma}\otimes q^k , z)$ with respect to the bilinear form $\langle \ , \ \rangle_{{\xi}}^q$ are $E_- ({\gamma}\otimes q^{k} , z^{-1})$ and $ H_- ({\gamma}\otimes q^{k} , z^{-1})$ respectively. The third and fourth identities are obtained by applying the adjoint action $*$ to the first two identities.
Replacing ${\gamma}$ by $-{\gamma}$ in (\[E:ch1\]) and (\[E:ch3\]) we obtain the equivalent formulas (\[E:ch2\]) and (\[E:ch4\]) respectively.
Applying the characteristic map to the vertex operators $Y^{\pm}({\gamma}, k, z)$, we obtain the following group theoretical explanation of vertex operators acting on the Fock space ${{\mathcal F}_{{\Gamma}\times \mathbb C^{\times}} }$.
\[T:vertexop\] For any ${\gamma}\in{R_{{\Gamma}}}$ and $k\in\mathbb Z$, we have $$\begin{aligned}
&Y^{+}( {\gamma}, k, z)\\
&= \exp \biggl ( \sum\limits_{ n \ge 1}
\frac 1n \, \widetilde{a}_{-n} ( {\gamma}) z^n \biggr ) \,
\exp \biggl ( -\sum\limits_{ n \ge 1}
\frac 1n \,{ \widetilde{a}_n( {\gamma})} q^{-kn} z^{ -n} \biggr )
e^{ {\gamma}} z^{ \partial_{{\gamma}}}\\
&=ch(H_+({\gamma}, z))ch(S(H_+({\gamma}\otimes q^{k}, z^{-1})^*))
e^{ {\gamma}} z^{ \partial_{{\gamma}}}, \end{aligned}$$
$$\begin{aligned}
&Y^{-}( {\gamma}, k, z)\\
&= \exp \biggl ( -\sum\limits_{ n \ge 1}
\frac 1n \, \widetilde{a}_{-n} ( {\gamma})q^{kn} z^n \biggr ) \,
\exp \biggl ( \sum\limits_{ n \ge 1}
\frac 1n \,{ \widetilde{a}_n( {\gamma})} z^{ -n} \biggr )
e^{ -{\gamma}} z^{-\partial_{{\gamma}}}\\
&=ch(S(H_+({\gamma}\otimes q^{k}, z^{-1})))
ch(H_+({\gamma}, z)^*)e^{-{\gamma}} z^{-\partial_{{\gamma}}}. \end{aligned}$$
We note that for ${\gamma}\in{\Gamma}^*, l\in\mathbb Z$ $$Y^{\pm}({\gamma}\otimes q^l, k, z)=Y^{\pm}({\gamma}, k, q^{-l}z).$$
It follows from Theorem \[T:vertexop\] that $$\begin{aligned}
&ch\big(Y^{\pm}({\gamma}, k, z)\big)=X^{\pm}({\gamma}, k, z)\\
&= \exp \biggl ( -\sum\limits_{ n \ge 1}
\frac 1n \, {a}_{-n} ( {\gamma})q^{n{(k\mp k)}/2} z^n \biggr )\\
&\qquad\times \exp \biggl ( \sum\limits_{ n \ge 1}
\frac 1n \,{ {a}_n( {\gamma})}q^{n{(-k\mp k)}/2} z^{ -n} \biggr )
e^{ \pm{\gamma}} z^{\pm\partial_{{\gamma}}}.\end{aligned}$$
In general the vertex operators $Y^{\pm}( {\gamma}, k, z)$ (for $k\in\mathbb Z$) generalize the vertex operators considered in [@J3] (for $k=\pm 1$). When $q=1$ they specialize to the vertex operators $Y^{\pm}( {\gamma}, z)$ studied in [@FJW].
Basic representations and the McKay correspondence {#sect_ade}
==================================================
Quantum toroidal algebras
-------------------------
Let $Q$ be the root lattice of an affine Lie algebra of simply laced type $A$, $D$, or $E$ with the invariant form $(\ \ |\ \ )$. The quantum toroidal algebra ${U_q(\widehat{\widehat{\mathfrak g}})}$ is the associative algebra generated by $x^{\pm}_{i}(n)$, $a_{i}(m)$, $q^{d}$, $q^c$ , $0\leq i\leq r, n, m\in \mathbb Z$ subject to the following relations [@GKV]: $$\begin{gathered}
q^{d}a_i(n)q^{-d}=q^na_i(n),
q^{d}x^{\pm}_i(n)q^{-d}=q^nx_i^{\pm}(n),\\
[a_i(m), a_j(n)]=\delta_{m,-n}\frac{[({\alpha}_i|{\alpha}_j)m]}{m}
\frac{q^{mc}-q^{-mc}}{q-q^{-1}},
\label{E:heisenberg}\\ \label{E:comm1}
[a_i(m), x_j^{\pm}(n)] =\pm \frac{[({\alpha}_i|{\alpha}_j)m]}{m}q^{\mp |m|c/2}
x_j^{\pm}(m+n),\\ \label{E:comm2}
(z-q^{\pm ({\alpha}_i, {\alpha}_j)}w)x_i^{\pm}(z)x_j^{\pm}(w)=x_j^{\pm}(w)x_i^{\pm}(z)(q^{\pm ({\alpha}_i, {\alpha}_j)}z-w), \\ \label{E:comm3}
[x^+_i(z), x^-_j(w)]
=\frac{\delta_{ij}
\{\delta(zw^{-1}q^{-c})\psi_i^+(wq^{c/2})-\delta(zw^{-1}q^c)
\psi_i^-(zq^{c/2})\}}{q-q^{-1}},\\ \label{E:comm4}
Sym_{z_1, \ldots z_N}\sum_{s=0}^{N=1-({\alpha}_i, {\alpha}_j)}(-1)^s\bmatrix N\\s\endbmatrix
x^{\pm}_i(z_1)\cdots x_i^{\pm}(z_s)\cdot\\ \nonumber
{\kern .5cm}\cdot x^{\pm}_j(w)x_i^{\pm}(z_{s+1})\cdots
x^{\pm}_i(z_N)=0, \quad\text{for}\quad ({\alpha}_i|{\alpha}_j)\leq 0, \end{gathered}$$ where the generators ${\alpha}(n)$ are related to $\psi^{\pm}_{i}(\pm n)$ via: $$\begin{gathered}
\label{E:comm5}
\psi_i^{\pm}(z)
=\sum_{n\geq 0}\psi_i^{\pm}(\pm n)z^{\mp n}=k_i^{\pm 1}exp(\pm(q-q^{-1})
\sum_{n>0}{\alpha}_i(\pm n)z^{\mp n}),\end{gathered}$$ and the Gaussian polynomial $$\bmatrix m\\ n\endbmatrix=\frac{[m]!}{[n]![m-n]!}, \qquad
[n]!=[n][n-1]\cdots [1].$$ The generating function of $x_n^{\pm}$ are defined by $$x_i^{\pm}(z)=\sum_{n\in \mathbb Z}x_i^{\pm}(n)z^{-n-1}, \qquad i=0, \ldots, r.$$
The quantum toroidal algebra contains a special subalgebra– the quantum affine algebra ${U_q(\widehat{\mathfrak g})}$, which is generated by simply omitting the generators associated to $i=0$. The relations are called the Drinfeld realization of the quantum affine algebras.
In the case of type $A$, the quantum toroidal algebra ${U_q(\widehat{\widehat{\mathfrak g}})}$ admits a further deformation ${U_{q, p}(\widehat{\widehat{\mathfrak g}})}$. Let $(b_{ij})$ be the skew-symmetric $(r+1)\times (r+1)$-matrix $$\label{E:skewsymm}
\pmatrix
0 & 1 & 0 & \cdots & 0 & -1\\
-1 & 0 & 1 & \cdots & 0 & 0\\
0 & -1 & 0 & \cdots & 0 & 0\\
\cdots & \cdots & \cdots & \cdots & \cdots & \cdots\\
0 & 0 & 0 & \cdots & 0 & 1\\
1 & 0 & 0 & \cdots & -1 & 0\endpmatrix.$$ The quantum toroidal algebra ${U_{q, p}(\widehat{\widehat{\mathfrak g}})}$ is the associative algebra generated by $x^{\pm}_{in}$, $a_i(m)$, $q^{d_1}$, $q^{d_2}$, $q^c$ , $0\leq i\leq r, m, n\in \mathbb Z$ subject to the following relations [@GKV; @VV]: $$\begin{gathered}
q^{d_1}a_i(n)q^{-d_1}=q^na_i(n),
q^{d_1}x^{\pm}_i(n)q^{-d_1}=q^nx_i^{\pm}(n),\\
q^{d_2}a_i(n)q^{-d_2}=a_i(n)\\
q^{d_2}x^{\pm}_i(n)q^{-d_2}
=q^{\pm\delta_{n0}}x_i^{\pm}(n),\\
[a_i(m), a_j(n)]=\delta_{m,-n}\frac{[({\alpha}_i|{\alpha}_j)m]}{m}
\frac{q^{mc}-q^{-mc}}{q-q^{-1}}p^{mb_{ij}},
\label{E:heisenberg2}\\ \label{E:comm1'}
[a_i(m), x_j^{\pm}(n)] =\pm \frac{[({\alpha}_i|{\alpha}_j)m]}{m}q^{\mp |m|c/2}
p^{mb_{ij}}
x_j^{\pm}(m+n),\\ \label{E:comm2'}
(p^{b_{ij}}z-q^{\pm ({\alpha}_i|{\alpha}_j)}w)x_i^{\pm}(z)x_j^{\pm}(w)
=x_j^{\pm}(w)x_i^{\pm}(z)(p^{b_{ij}}q^{\pm
({\alpha}_i|{\alpha}_j)}z-w), \\ \label{E:comm3'}
[x^+_i(z), x^-_j(w)]
=\frac{\delta_{ij}
\{\delta(zw^{-1}q^{-c})\psi_i^+(wq^{c/2})-\delta(zw^{-1}q^c)
\psi_i^-(zq^{c/2})\}}{q-q^{-1}},
\\ \label{E:comm4'}
Sym_{z_1, \ldots z_N}\sum_{s=0}^{N=1-({\alpha}_i|{\alpha}_j)}(-1)^s\bmatrix N\\s\endbmatrix
x^{\pm}_i(z_1)\cdots x_i^{\pm}(z_s)\cdot\\ \nonumber
{\kern .5cm}\cdot x^{\pm}_j(w)x_i^{\pm}(z_{s+1})\cdots
x^{\pm}_i(z_N)=0, \quad\text{for}\quad ({\alpha}_i|{\alpha}_j)\leq 0,
\nonumber\end{gathered}$$ where the generators $a_i(n)$ are related to $\psi^{\pm}_{i}(\pm m)$ via: $$\begin{gathered}
\label{E:comm5'}
\psi_i^{\pm}(z)
=\sum_{n\geq 0}\psi_i^{\pm}(\pm n)z^{\mp n}
=k_i^{\pm 1}exp(\pm(q-q^{-1})\sum_{n>0}{\alpha}_i(\pm
n)z^{\mp n}).\end{gathered}$$
We recall that the [*basic module*]{} of ${U_q(\widehat{\widehat{\mathfrak g}})}$ is the simple module generated by the highest weight vector $v_0$ such that $$\begin{aligned}
&a_i(n+1).v_0=0,
\qquad x^{\pm}_i(n).v_0=0, \qquad n\geq 0\\
& q^c.v_0=qv_0, \qquad q^d.v_0=v_0.\end{aligned}$$ We say a module is of level one if $q^c$ acts as $q$.
A new form of McKay correspondence
----------------------------------
In this subsection we let ${\Gamma}$ to be a finite subgroup of $SU_2$ and consider two distinguished choices of the class function ${\xi}$ in ${R_{{\Gamma}\times \mathbb C^{\times}}}$ introduced in Sect. \[S:Mcweights\].
First we consider $${\xi}={\gamma}_0\otimes (q+q^{-1}) - \pi\otimes 1_{{\mathbb C^{\times}}},$$ where $\pi$ is the character of the two-dimensional natural representation of ${\Gamma}$ in $SU_2$.
The Heisenberg algebra in this case has the following relations (cf. Prop. \[prop\_orth\] and (\[E:qcartan1\])). $$\label{E:heisen1}
[a_m({\gamma}_i), a_n({\gamma}_j)]=
\begin{cases}
m\delta_{m, -n}(q^m+q^{-m})C, & i=j\\
m\delta_{m, -n}a_{ij}^1C, & i\neq j
\end{cases},$$ where $a_{ij}^1$ are the entries of the affine Cartan matrix of ADE type (see (\[E:qcartan\]) at $d=2$).
When ${\Gamma}\neq \mathbb Z/2\mathbb Z$ or $1$, the relations (\[E:heisen1\]) can be simply written as follows: $$[a_m({\gamma}_i), a_n({\gamma}_j)]=m\delta_{m, -n}[a_{ij}]_{q^m}C.$$
Recall that the matrix $A^1 = (\langle {\gamma}_i, {\gamma}_j\rangle_{{\xi}}^1)
=(a_{ij}^1)_{0 \leq i,j \leq r}$ is the Cartan matrix for the corresponding affine Lie algebra [@Mc]. In particular $a_{ii}^1 =2$; $a_{ij}^1 =0$ or $-1$ when $i \neq j$ and ${\Gamma}\neq \mathbb Z / 2\mathbb Z$. In the case of ${\Gamma}= \mathbb Z / 2\mathbb Z$, $a_{01}^1 =a_{10}^1= -2$. Let $\mathfrak g$ (resp. $\hat{\mathfrak g}$) be the corresponding simple Lie algebra (resp. affine Lie algebra ) associated to the Cartan matrix $(a_{ij}^1)_{1\leq i, j\leq r}$ (resp. $A$). Note that the lattice ${R_{\mathbb Z}(\Gamma)}$ is even in this case.
We define the normal ordered product of vertex operators as follows. $$\begin{aligned}
&:Y^+({\gamma}_i, k, z)Y^+({\gamma}_j, k', w):\\
=&H_+({\gamma}_i, z)H({\gamma}_j,w)S(H_+({\gamma}_i\otimes q^k, z^{-1})^*
H_+({\gamma}_j\otimes q^{k'}, w^{-1})^*)\\
&\times e^{{\gamma}_i+{\gamma}_j}z^{\partial_{{\gamma}_i}}
w^{\partial_{{\gamma}_j}},\\
&:Y^+({\gamma}_i, k, z)Y^-({\gamma}_j, k', w):\\
=&H_+({\gamma}_i, z)H(-{\gamma}_j\otimes q^{-k'}, w)S(H_+({\gamma}_i\otimes q^k, z^{-1})^*
H_+(-{\gamma}_j\otimes q^{k'}, w^{-1})^*)\\
&\times e^{{\gamma}_i-{\gamma}_j}z^{\partial_{{\gamma}_i}}
w^{-\partial_{{\gamma}_j}}.\end{aligned}$$ Other normal ordered products are defined similarly.
We introduce for $a\in \mathbb R$ the following $q$-function: $$\begin{aligned}
\label{E:qfunc}
(1-z)_{q^{2}}^{a}&=\frac{(q^{-a+1}z;q^{2})_{\infty}}
{(q^{a+1}z;q^{2})_{\infty}}=exp\biggl(-\sum_{n=1}^{\infty}\frac{[an]}{n[n]}z^n
\biggl)\\
&=\sum_{m=0}^{\infty}
\begin{bmatrix} a\\ m\end{bmatrix} (-z)^m, \nonumber\end{aligned}$$ where we expand the power series using the $q$-binomial theorem and $$\begin{aligned}
\begin{bmatrix} a\\ m\end{bmatrix}&=\frac{(q^a-q^{-a})(q^{a-1}-q^{-a+1})\cdots (q^{a-m+1}-q^{-a+m-1})}{(q^m-q^{-m})(q^{m-1}-q^{-m+1})\cdots (q-q^{-1})},\\
(a; q)_{\infty}&=\prod_{n=0}^{\infty}(1-aq^n).\end{aligned}$$ When $a$ is a non-negative integer, $\bmatrix a\\ m\endbmatrix$ equals the Gaussian polynomial.
The identities in the following theorems are understood as usual by means of correlation functions (cf. e.g. [@FJ; @J1]).
\[th\_ope\] Let $\xi={\gamma}_0\otimes (q+q^{-1})-\pi\otimes 1_{{\mathbb C^{\times}}}$. Then the vertex operators $Y^{\pm}( {\gamma}_i, k, z), Y^{\pm}(-{\gamma}_j, k, z)$, ${\gamma}_i\in {\Gamma}^*, k\in\mathbb Z$ acting on the group theoretically defined Fock space ${{\mathcal F}_{{\Gamma}\times \mathbb C^{\times}} }$ satisfy the following relations.
$$\begin{aligned}
&& Y^{\pm}({\gamma}_i, k, z) Y^{\pm}({\gamma}_j, k, w)= {\epsilon}({\gamma}_i, {\gamma}_j)
:Y^{\pm}({\gamma}_i, k, z) Y^{\pm}({\gamma}_j, k, w):\\
&&\qquad\times\left\{
\begin{array}{cc}
1& \mbox{ $\langle {\gamma}_i, {\gamma}_j\rangle_{\xi}^1=0$}\\
(z-q^{\mp k}w)^{-1}& \mbox{ $\langle {\gamma}_i, {\gamma}_j\rangle_{\xi}^1=-1$}\\
(z-q^{\mp k-1}w)(z-q^{\mp k+1}w)&
\mbox{ $\langle {\gamma}_i, {\gamma}_j\rangle_{\xi}^1=2$}
\end{array},
\right.
\end{aligned}$$
$$\begin{aligned}
&& Y^{\pm}({\gamma}_i, k, z) Y^{\mp}({\gamma}_j, k, w) = {\epsilon}({\gamma}_i, {\gamma}_j)
:Y^{\pm}({\gamma}_i, k, z) Y^{\mp}({\gamma}_j, k, w):\\
&&\qquad\times\left\{
\begin{array}{cc}
1& \mbox{ $\langle {\gamma}_i, {\gamma}_j\rangle_{\xi}^1=0$}\\
(z-w)^{-1}& \mbox{ $\langle {\gamma}_i, {\gamma}_j\rangle_{\xi}^1=-1$}\\
(z-qw)(z-q^{-1}w)& \mbox{ $\langle {\gamma}_i, {\gamma}_j\rangle_{\xi}^1=2$}
\end{array},
\right.
\end{aligned}$$
$$\begin{aligned}
&& Y^{\pm}({\gamma}_i, k, z)Y^{\pm}(-{\gamma}_j, -k, w) \\
&&\qquad= {\epsilon}({\gamma}_i, {\gamma}_j)
:Y^{\pm}({\gamma}_i, k, z) Y^{\pm}(-{\gamma}_j, -k, w):\\
&&\qquad\times\left\{
\begin{array}{cc}
1& \mbox{ $\langle {\gamma}_i, {\gamma}_j\rangle_{\xi}^1=0$}\\
(z-q^{\mp k}w) & \mbox{ $\langle {\gamma}_i, {\gamma}_j\rangle_{\xi}^1=-1$}\\
(z-q^{\mp k-1}w)^{-1}(z-q^{\mp k+1}w)^{-1}
& \mbox{ $\langle {\gamma}_i, {\gamma}_j\rangle_{\xi}^1=2$}
\end{array},
\right.
\end{aligned}$$
$$\begin{aligned}
&& Y^{+}({\gamma}_i, k, z)Y^{-}(-{\gamma}_j, -k, w)\\
&&\qquad = {\epsilon}({\gamma}_i, {\gamma}_j)
:Y^{+}({\gamma}_i, k, z)Y^{-}(-{\gamma}_j, -k, w):\\
&&\qquad\times\left\{
\begin{array}{cc}
1& \mbox{ $\langle {\gamma}_i, {\gamma}_j\rangle_{\xi}^1=0$}\\
(z-q^{-2k}w)^{-1}& \mbox{ $\langle {\gamma}_i, {\gamma}_j\rangle_{\xi}^1=-1$}\\
(z-q^{-2k-1}w)(z-q^{-2k+1}w)& \mbox{ $\langle {\gamma}_i, {\gamma}_j\rangle_{\xi}^1=2$}
\end{array},
\right.
\end{aligned}$$
$$\begin{aligned}
&& Y^{-}({\gamma}_i, k, z)Y^{+}(-{\gamma}_j, -k, w)\\
&&\qquad= {\epsilon}({\gamma}_i, {\gamma}_j)
:Y^{-}({\gamma}_i, k, z)Y^{+}(-{\gamma}_j, -k, w):\\
&&\qquad\left\{
\begin{array}{cc}
1& \mbox{ $\langle {\gamma}_i, {\gamma}_j\rangle_{\xi}^1=0$}\\
(z-w)^{-1}& \mbox{ $\langle {\gamma}_i, {\gamma}_j\rangle_{\xi}^1=-1$}\\
(z-qw)(z-q^{-1}w)& \mbox{ $\langle {\gamma}_i, {\gamma}_j\rangle_{\xi}^1=2$}
\end{array}.
\right.
\end{aligned}$$
It is a routine computation to see that: $$\begin{aligned}
&& E_- ({\gamma}_i\otimes q^k, z) H_+ ({\gamma}_j\otimes q^l, w)\\
&=&H_+ ({\gamma}_j\otimes q^l, w)E_- ({\gamma}_i\otimes q^k, z)
(1-\frac wz q^{l-k})^{\langle {\gamma}_i, {\gamma}_j\rangle_{\xi}^1}_{q^2},\end{aligned}$$ where the $q$-analog of the power series $(1-x)_{q^2}^n$ is defined in (\[E:qfunc\]).
In particular, we have $$\begin{aligned}
(1-w/z)_{q^2}&=1-w/z,\\
(1-w/z)^2_{q^2}&=(1-q w/z)(1-q^{-1}w/z).\end{aligned}$$ Then the theorem is proved by observing that $z^{{\gamma}}e^{\partial_{\beta}}
=z^{\langle {\gamma}, \beta\rangle_{\xi}^1}e^{\partial_{\beta}}z^{{\gamma}}$.
Replacing the vertex operator $Y^{\pm}$ by $X^{\pm}$ via the characteristic map $ch$ in the above formulas, we get the corresponding formulas for vertex operators $X^{\pm}({\gamma}, k, z)$ acting on ${V_{{\Gamma}\times \mathbb C^{\times}}}$.
Now we consider the second distinguished class function $${\xi}^{q, p}={\gamma}_0\otimes (q+q^{-1})-({\gamma}_1\otimes p+{\gamma}_{r}\otimes p^{-1}),$$ when ${\Gamma}$ is a cyclic group of order $r+1$.
In this case the Heisenberg algebra (\[E:heisen\]) has the following relations according to Prop. \[prop\_orth\] and (\[E:qcartan2\]): $$\label{2ndheisen}
[ a_m({\gamma}_i), {a_n({\gamma}_j)}]=m\delta_{m, -n}
[a_{ij}^1]_{q^m}p^{m b_{ij}}C,$$ where $a_{ij}^1$ are the entries of the affine Cartan matrix of type A and $r\geq 2$. This is the same Heisenberg subalgebra ($c=1$) in ${U_{q, p}(\widehat{\widehat{\mathfrak g}})}$ provided that we identify $$a_i(n)=\frac{[n]}na_n({\gamma}_i).$$
Recall that $(b_{ij})$ is the skew-symmetric matrix given in (\[E:skewsymm\]). We need to slightly modify the definition of the middle term in the vertex operators. For each $i=0, 1, \ldots, r$ we define the modified operator $z^{\partial_{{\gamma}, p}}$ on the group algebra $\mathbb C[{R_{\mathbb Z}(\Gamma)}]$ by $$z^{\partial_{{\gamma}_i, p}}e^{\beta}
=z^{\langle{\gamma}_i, \beta\rangle_{\xi}^1}
p^{-\frac 12\sum_{j=1}^r\langle {\gamma}_i, m_j{\gamma}_j\rangle_{\xi}^1b_{ij}}e^{\beta},$$ where $\beta=\sum_{j}m_j{\gamma}_j\in {R_{\mathbb Z}(\Gamma)}$.
We then replace the operator $z^{\pm\partial_{{\gamma}_i}}$ in the definition of the vertex operators $Y^{\pm}({\gamma}_i, k, z)$ by the operator $z^{\pm\partial_{{\gamma}_i, p}}$. The formulas in Theorems \[T:vertexop\] remain true after the term $z^{\pm \partial}$ appearing in the formulas are modified accordingly.
The proof of the following theorem is similar to that of Theorem \[th\_ope\].
\[th\_ope1\] Let ${\Gamma}$ be a cyclic group of order $r+1$ and let $\xi={\gamma}_0\otimes (q+q^{-1})-({\gamma}_1\otimes p+{\gamma}_r\otimes p^{-1})$. The vertex operators $Y^{\pm}( {\gamma}_i, k, z)$ and $Y^{\pm}(-{\gamma}_i, k, z),
{\gamma}_i\in {\Gamma}^*$ acting on the group theoretically defined Fock space ${{\mathcal F}_{{\Gamma}\times \mathbb C^{\times}} }$ satisfy the following relations. $$\begin{aligned}
&& Y^{\pm}({\gamma}_i, k, z) Y^{\pm}({\gamma}_j, k, w)={\epsilon}({\gamma}_i, {\gamma}_j)
:Y^{\pm}({\gamma}_i, k, z) Y^{\pm}({\gamma}_j, k, w):\\
&&\qquad\left\{
\begin{array}{cc}
1& \mbox{ $\langle {\gamma}_i, {\gamma}_j\rangle_{\xi}^1=0$}\\
p^{-\frac 12b_{ij}}(z-q^{\mp k}p^{b_{ij}}w)^{-1}& \mbox{ $\langle {\gamma}_i, {\gamma}_j\rangle_{\xi}^1=-1$}\\
(z-q^{\mp k-1}w)(z-q^{\mp k+1}w)&
\mbox{ $\langle {\gamma}_i, {\gamma}_j\rangle_{\xi}^1=2$}
\end{array},
\right.
\end{aligned}$$ $$\begin{aligned}
&& Y^{\pm}({\gamma}_i, k, z) Y^{\mp}({\gamma}_j, k, w)= {\epsilon}({\gamma}_i, {\gamma}_j)
:Y^{\pm}({\gamma}_i, k, z) Y^{\mp}({\gamma}_j, k, w):\\
&&\qquad\left\{
\begin{array}{cc}
1& \mbox{ $\langle {\gamma}_i, {\gamma}_j\rangle_{\xi}^1=0$}\\
p^{-\frac 12b_{ij}}(z-p^{b_{ij}}w)^{-1}& \mbox{ $\langle {\gamma}_i, {\gamma}_j\rangle_{\xi}^1=-1$}\\
(z-qw)(z-q^{-1}w)& \mbox{ $\langle {\gamma}_i, {\gamma}_j\rangle_{\xi}^1=2$}
\end{array},
\right.
\end{aligned}$$
$$\begin{aligned}
&&Y^{\pm}({\gamma}_i, k, z)Y^{\pm}(-{\gamma}_j, -k, w) \\
&&\qquad ={\epsilon}({\gamma}_i, {\gamma}_j)
:Y^{\pm}({\gamma}_i, k, z) Y^{\pm}(-{\gamma}_j, -k, w):\\
&&\qquad\left\{
\begin{array}{cc}
1& \mbox{ $\langle {\gamma}_i, {\gamma}_j\rangle_{\xi}^1=0$}\\
p^{-\frac 12b_{ij}}(z-q^{\mp k}p^{b_{ij}}w) & \mbox{ $\langle {\gamma}_i, {\gamma}_j\rangle_{\xi}^1=-1$}\\
(z-q^{\mp k-1}w)^{-1}(z-q^{\mp k+1}w)^{-1}& \mbox{ $\langle {\gamma}_i, {\gamma}_j\rangle_{\xi}^1=2$}
\end{array},
\right.
\end{aligned}$$
$$\begin{aligned}
&& Y^{+}({\gamma}_i, k, z)Y^{-}(-{\gamma}_j, -k, w)\\
&&\qquad= {\epsilon}({\gamma}_i, {\gamma}_j)
:Y^{+}({\gamma}_i, k, z)Y^{-}(-{\gamma}_j, -k, w):\\
&&\qquad\left\{
\begin{array}{cc}
1& \mbox{ $\langle {\gamma}_i, {\gamma}_j\rangle_{\xi}^1=0$}\\
p^{-\frac 12b_{ij}}(z-q^{-2k}p^{b_{ij}}w)^{-1}& \mbox{ $\langle {\gamma}_i, {\gamma}_j\rangle_{\xi}^1=-1$}\\
(z-q^{-2k-1}w)(z-q^{-2k+1}w)& \mbox{ $\langle {\gamma}_i, {\gamma}_j\rangle_{\xi}^1=2$}
\end{array},
\right.
\end{aligned}$$
$$\begin{aligned}
&& Y^{-}({\gamma}_i, k, z)Y^{+}(-{\gamma}_j, -k, w)\\
&&\qquad = {\epsilon}({\gamma}_i, {\gamma}_j)
:Y^{-}({\gamma}_i, k, z)Y^{+}(-{\gamma}_j, -k, w):\\
&&\qquad\left\{
\begin{array}{cc}
1& \mbox{ $\langle {\gamma}_i, {\gamma}_j\rangle_{\xi}^1=0$}\\
p^{-\frac 12b_{ij}}(z-p^{b_{ij}}w)^{-1}& \mbox{ $\langle {\gamma}_i, {\gamma}_j\rangle_{\xi}^1=-1$}\\
(z-qw)(z-q^{-1}w)& \mbox{ $\langle {\gamma}_i, {\gamma}_j\rangle_{\xi}^1=2$}
\end{array}.
\right.
\end{aligned}$$
Replacing the vertex operators $Y^{\pm}$ by $X^{\pm}$ via the characteristic map $ch$ we obtain the corresponding results on the space ${V_{{\Gamma}\times \mathbb C^{\times}}}$.
Quantum vertex representations of ${U_q(\widehat{\widehat{\mathfrak g}})}$
--------------------------------------------------------------------------
For each $i=0$, $\dots$, $r$ let $$\widetilde{a_i}(n)=\frac{[n]}n a_n({\gamma}_i).$$ It follows from (\[E:heisen\]) and (\[E:heisen1\]) that $$\label{E:heisenberg1}
[\widetilde{a_i}(m), \widetilde{a_j}(n)]=\delta_{m, -n}\frac{[m\langle {\gamma}_i, {\gamma}_j\rangle_{\xi}^1]}m [m].$$ According to McKay, the bilinear form $\langle {\gamma}_i, {\gamma}_j\rangle_{\xi}^1$ is exactly the same as the invariant form $(\ |\ )$ of the root lattice of the affine Lie algebra ${\widehat{\mathfrak g}}$. This implies that the commutation relations (\[E:heisenberg1\]) are exactly the commutation relations (\[E:heisenberg\]) of the Heisenberg algebra in ${U_q(\widehat{\widehat{\mathfrak g}})}$ if we identify $\widetilde{a_i}(n)$ with ${a_i}(n)$. Thus the Fock space ${S_{{\Gamma\times \mathbb C^{\times}}}}$ is a level one representation for the Heisenberg subalgebra in ${U_q(\widehat{\widehat{\mathfrak g}})}$. Under the new variable (by identifying $a_i(n)$ with $\widetilde{a_i}(n)$) and after a $q$-shift we obtain that $$\begin{aligned}
&X^{+}({\gamma}_i\otimes q^{-k/2}, k, z)\\
&=\exp \biggl ( \sum\limits_{ n \ge 1}
\frac {a_{i} (-n )}{[n]} q^{kn/2}z^n \biggr ) \,
\exp \biggl ( -\sum\limits_{ n \ge 1}
\frac {a_i(n)}{[n]} q^{kn/2} z^{ -n} \biggr )
e^{ {\gamma}} z^{ \partial_{{\gamma}}} ,\\
&X^{-}({\gamma}_i\otimes q^{-k/2}, k, z)\\
&=\exp \biggl ( -\sum\limits_{ n \ge 1}
\frac {a_{i}(-n )}{[n]}q^{-kn/2} z^n \biggr ) \,
\exp \biggl ( \sum\limits_{ n \ge 1}
\frac {a_i(n)}{[n]} q^{-kn/2} z^{ -n} \biggr )
e^{ -{\gamma}} z^{ -\partial_{{\gamma}}}.\end{aligned}$$
The following theorem gives a $q$-deformation of the new form of McKay correspondence in [@FJW] and provides a direct connection from a finite subgroup ${\Gamma}$ of $SU_2$ to the quantum toroidal algebra ${U_q(\widehat{\widehat{\mathfrak g}})}$ of $ADE$ type.
\[T:quantum\] Given a finite subgroup ${\Gamma}$ of $SU_2$, each of the following correspondence gives a vertex representation of the quantum toroidal algebra ${U_q(\widehat{\widehat{\mathfrak g}})}$ on the Fock space ${{\mathcal F}_{{\Gamma}\times \mathbb C^{\times}} }$: $$\begin{aligned}
x_i^{\pm}(n)& \longrightarrow Y^{\pm}_n({\gamma}_i, -1), \\
a_i(n) &\longrightarrow \frac{[n]}n a_n({\gamma}_i), \qquad q^c\longrightarrow q ;\end{aligned}$$ or $$\begin{aligned}
x_i^{\pm}(n)& \longrightarrow Y^{\mp}_n(-{\gamma}_i, 1), \\
a_i(n) &\longrightarrow \frac{[n]}n a_n({\gamma}_i), \qquad q^c\longrightarrow q,\end{aligned}$$ where $i=0, \dots, r$, and $n\in\mathbb Z$.
Using the usual method of $q$-vertex operator calculus [@FJ; @J1] and Theorem \[th\_ope\] we see that the vertex operators $Y^{\pm}({\gamma}_i, \pm 1, z)$ satisfy relations (\[E:comm1\]), (\[E:comm2\]) and (\[E:comm4\]). Observe further that the above vertex operators at $k=\pm 1$ have the same form as those in the basic representations of the quantum affine algebras (see [@FJ]). Thus the relations (\[E:comm3\]) and (\[E:comm5\]) are also verified. For each fixed $k= 1$ or $-1$ we have shown that the operators $Y^{\pm}({\gamma}_i, \pm1, z)$ give a level one representation of the quantum toroidal algebra ${U_q(\widehat{\widehat{\mathfrak g}})}$ (see also [@Sa; @J3]).
Replacing $Y^{\pm}$ by $X^{\pm}$ in the above theorem, we obtain a vertex representation of ${U_q(\widehat{\widehat{\mathfrak g}})}$ in the space ${V_{{\Gamma}\times \mathbb C^{\times}}}$.
We can easily get the basic representation of the quantum affine algebra ${U_q(\widehat{\mathfrak g})}$ on a certain distinguished subspace of ${{\mathcal F}_{{\Gamma}\times \mathbb C^{\times}} }$.
Denote by ${\overline{S}_{{\Gamma}\times\mathbb C^{\times} }}$ the symmetric algebra generated by $a_{-n} ({\gamma}_i)$, $n >0$, $i =1, \ldots , r$ over $\mathbb C[q, q^{-1}]$. ${\overline{S}_{{\Gamma}\times\mathbb C^{\times} }}$ is isometric to ${\overline{R}_{ {\Gamma}\times \mathbb C^{\times}}}$.
We define $${\overline{\mathcal F}_{{\Gamma}\times \mathbb C^{\times}}}= {\overline{R}_{ {\Gamma}\times \mathbb C^{\times}}}\otimes \mathbb C [ {\overline{R}_{\mathbb Z}({{\Gamma}})}]
\cong {\overline{S}_{{\Gamma}\times\mathbb C^{\times} }}\otimes \mathbb C [ {\overline{R}_{\mathbb Z}({{\Gamma}})}].$$ The space ${V_{{\Gamma}\times \mathbb C^{\times}}}$ associated to the lattice ${R_{\mathbb Z}(\Gamma)}$ is isomorphic to the tensor product of the space ${\overline{R}_{ {\Gamma}\times \mathbb C^{\times}}}$ and ${R_{\mathbb Z}(\Gamma)}$ as well as the space associated to the rank $1$ lattice $\mathbb Z \alpha_0$.
Given a finite subgroup ${\Gamma}$ of $SU_2$, each of the following correspondence gives the basic representation of the quantum affine algebra ${U_q(\widehat{\mathfrak g})}$ on the Fock space ${\overline{\mathcal F}_{{\Gamma}\times \mathbb C^{\times}}}$: $$\begin{aligned}
x_i^{\pm}(n)& \longrightarrow Y^{\pm}_n({\gamma}_i, -1), \\
a_i(n) &\longrightarrow \frac{[n]}n a_n({\gamma}_i), \qquad q^c\longrightarrow q ;\end{aligned}$$ or $$\begin{aligned}
x_i^{\pm}(n)& \longrightarrow Y^{\mp}_n(-{\gamma}_i, 1), \\
a_i(n) &\longrightarrow \frac{[n]}n a_n({\gamma}_i), \qquad q^c\longrightarrow q,\end{aligned}$$ where $i=1, \dots, r$.
In the case of our second distinguished class function $$\xi^{q, p}={\gamma}_0\otimes (q+q^{-1})-
({\gamma}_1\otimes p+{\gamma}_r\otimes p^{-1}),$$ we need to consider the Fock space $${{\widetilde{\mathcal F}_{{\Gamma}\times \mathbb C^{\times}}}} = {R_{{\Gamma}\times \mathbb C^{\times}}}\otimes \mathbb C [ {R_{\mathbb Z}(\Gamma)}/ R^0_{\mathbb Z}]\cong
{S_{{\Gamma\times \mathbb C^{\times}}}}\otimes \mathbb C [{\overline{R}_{\mathbb Z}({{\Gamma}})}],$$ where $R^0_{\mathbb Z}$ is the radical of the bilinear form $\langle \ \ , \ \ \rangle_{{\xi}}^1$. The correspondence space for ${\widetilde{\mathcal F}_{{\Gamma}\times \mathbb C^{\times}}}$ under the characteristic map $ch$ will be denoted ${\widetilde{V}_{ {\Gamma}\times\mathbb C^{\times}}}$.
Using similar method as in the proof of Theorem \[T:quantum\] we derive the the following theorem.
Let ${\Gamma}$ be a cyclic group of order $r+1\geq 2$ and $p=q^{\pm 1}$. Each of the following correspondence gives the basic representation of ${U_q(\widehat{\widehat{\mathfrak g}})}$ on ${\widetilde{\mathcal F}_{{\Gamma}\times \mathbb C^{\times}}}$: $$\begin{aligned}
x_i^{\pm}(n)& \longrightarrow Y^{\pm}_n({\gamma}_i, -1), \\
a_i(n) &\longrightarrow \frac{[n]}n a_n({\gamma}_i), \qquad q^c\longrightarrow q ;\end{aligned}$$ or $$\begin{aligned}
x_i^{\pm}(n)& \longrightarrow Y^{\mp}_n(-{\gamma}_i, 1), \\
a_i(n) &\longrightarrow \frac{[n]}n a_n({\gamma}_i), \qquad q^c\longrightarrow q,\end{aligned}$$ where $i=0, \ldots, r$.
The algebraic picture obtained by replacing the vertex operator $Y^{\pm}$ by $X^{\pm}$ and ${{\mathcal F}_{{\Gamma}\times \mathbb C^{\times}} }$ by ${\widetilde{V}_{ {\Gamma}\times\mathbb C^{\times}}}$ in the above Theorem was given by Sato [@Sa].
This theorem partly shows why the two-parameter deformation for ${U_q(\widehat{\widehat{\mathfrak g}})}$ is only available in the case of type $A$. It also singles out the special case of $p=q^{\pm 1}$, where the matrix of the bilinear form $\langle \ , \ \rangle_{{\xi}}^{q, q^{\pm 1}}$ is semi-definite positive (see Sect. \[S:Mcweights\]) which permits the factorization of ${{\mathcal F}_{{\Gamma}\times \mathbb C^{\times}} }$ into ${\overline{\mathcal F}_{{\Gamma}\times \mathbb C^{\times}}}$.
We remark that our method can be generalized by replacing $R({\Gamma})$ by any finite dimensional Hopf algebra with a Haar measure. A more general deformation is obtained by replacing ${\mathbb C^{\times}}$ by any torsion-free abelian group. In another direction one can replace ${\mathbb C^{\times}}$ by its finite analog $\mathbb Z/r\mathbb Z$ to study ${U_q(\widehat{\mathfrak g})}$ at $r$th roots of unity.
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[^1]: I.F. is supported in part by NSF grant DMS-9700765. N.J. is supported in part by NSA grant MDA904-97-1-0062 and NSF grant DMS-9970493.
|
---
author:
- 'H. Schlattl, M Salaris, S. Cassisi'
- 'A. Weiss'
date: 'Received; accepted'
title: 'The surface carbon and nitrogen abundances in models of ultra metal-poor stars'
---
Introduction
============
Spectroscopic observations of ultra metal-poor (UMP) stars have disclosed that carbon and nitrogen are significantly overabundant with respect to Fe, compared to typical halo stars which show \[C/Fe\]$\approx$\[N/Fe\]$\approx$0.0. According to @beers:99 and @rbs:99 about 10% of the stars with \[Fe/H\]$\leq$$-$2.5 display \[C/Fe\]$\ge$1.0; this fraction rises to about 25% when \[Fe/H\]$\leq$$-$3.0. Also N appears to be overabundant by the same amount, while $\alpha$-elements show a ratio with respect to Fe that is typical of halo stars, i.e. \[$\alpha$/Fe\]$\approx$0.4. As discussed, e.g., by @nrb:97, it is difficult to explain these abundance ratios in terms of binary star evolution or ejecta from very massive Population III supernovae, which in particular produce few nitrogen but vast amounts of oxygen [@hewoo:02], so that the origin of the surface abundances of UMP stars is still shrouded in mystery.
Concerning the nature of the UMPs, different ideas may be followed, the first one being that they are true Pop III stars — i.e. of initially zero metallicity — with the observed heavy elements (Fe, etc.) resulting from pollution of just the envelope by other stars such as supernovae. In two recent papers (@wcss:00 [ Paper I], and @scsw:01 [ Paper II]) we have thus discussed in detail the evolution of low-mass zero-metal stars, including the effect of atomic diffusion, mass loss and surface metal pollution. One important result was that the small entropy barrier between the H- and He-rich regions allows the He-flash driven convective zone to penetrate the overlying H-rich layers at the tip of the red giant branch. The consequent inward migration of protons into high-temperature regions leads to an H-shell flash, which further increases the extension of the central convective region. At the late stages of this phase the convective envelope deepens and merges with the inner convective zone. The initially metal-free surface is therefore enriched by a large amount of matter processed in He- and H-burning reactions. In particular, very high C and N abundances follow. We denote this process of surface C and N enrichment as HElium Flash induced Mixing (HEFM).
star
------------------ -------------- -------------- ------------------ --------------- -------------- -------------
CS 22892-052[^1] 1.5$\pm$0.5 4850$\pm$100 $-$2.97$\pm$0.20 1.10$\pm$0.23 1.0$\pm$0.52 $\gtrsim$10
CS 22957-027[^2] 2.25$\pm$1.0 4839$\pm$130 $-$3.43$\pm$0.12 2.20$\pm$0.30 2.0$\pm$0.50 $\approx$10
CS 22948-027[^3] 1.0$\pm$0.3 4600$\pm$100 $-$2.57$\pm$0.23 2.0$\pm$0.18 1.8$\pm$0.24 $\approx$10
CS 22949-037[^4] 1.7$\pm$0.3 4900$\pm$100 $-$3.79$\pm$0.16 1.05$\pm$0.20 2.7$\pm$0.40 —
This scenario — already suggested by @fii:00 — could potentially explain the anomalous abundance pattern at the surface of UMP stars, since via this mechanism high abundances of C and N are produced in the post He-flash phases. In Paper II we have shown that an initially metal-free star of 0.82$\,M_{\odot}$, polluted by 0.0003$\,M_{\odot}$ of $Z$=0.02 material, undergoes HEFM and reproduces in its post He-flash phase approximately luminosity and surface gravity of two of the best studied UMP stars, namely CS 22892-052 and CS 22957-027. The stellar mass was chosen such that the age on the RGB of 13.7Gyr is compatible with the current estimates of the age of the universe, and with the age of CS 22892-052 estimated by @cpk:99 from nuclear chronology (15.6$\pm$4.6Gyr). The amount and composition of the polluting material was adjusted such that the observed surface \[Fe/H\] abundance (\[Fe/H\]$\approx$$-$3) was obtained and, as the result of the HEFM, \[C/Fe\] and \[N/Fe\] ratios considerably larger than zero resulted. However, the values of \[C/Fe\] and \[N/Fe\] predicted by the models are about 2 orders of magnitude higher than the observed ones; i.e., we predict \[C/Fe\]$\approx$\[N/Fe\]$\approx$4, while observations of these stars yield \[C/Fe\]$\approx$\[N/Fe\]$\approx$1–2 (see Table \[data\]). We note, however, that higher N overabundances are observed, e.g., in CS 22949-037 [@NRB:01].
An alternative explanation for the C and N enhancement in UMPs [is provided by HEFM]{} during the thermally pulsating asymptotic giant-branch phase (TP-AGB) which occurs in metal-free stars with $M$$\ga$1$\,M_\odot$ [@CCT; @fii:00]. However, age, surface gravity, and effective temperature of these AGB stars are more difficult to reconcile with those of the observed objects, which are therefore unlikely to be TP-AGB stars themselves. The idea in this case is that the observed low-mass UMP objects form binary systems with more massive AGB stars, which have transferred part of their C- and N-enriched envelope to their companion [@fii:00]. A potential problem of this hypothesis is that HEFM during the TP-AGB of intermediate-mass stars may enhance the surface oxygen content similar to carbon. [This feature can be found in models of @SLL:02, which however do not agree in this respect with computations of @CDLS:01]{}. An oxygen enhancement of \[O/Fe\]$\approx$1–2 might not necessarily be in contradiction to observations, as only very weak upper limits on the \[O/Fe\] ratio exist [@smcp:96; @nrb:97]. [But]{} the main problem of this scenario is that not all UMPs are in binary systems, and the orbital periods of those who are, are inconsistent with an AGB mass transfer paradigm [@PS:02]. Moreover, the origin of Fe in these stars is as uncertain as in our favoured scenario. It is presently not clear whether intermediate-mass stars with \[Fe/H\]$\approx$$-$3 undergo HEFM during the TP-AGB or rather evolve like ordinary Pop II objects. In the latter case the surface C, N, O, and Fe abundances of UMP stars would then be the result of mass transfer from their companion and the accretion of further material, e.g., a nearby supernova.
In summary, HEFM during the major He flash is presently the most promising astrophysical explanation for the high \[C,N/Fe\] ratios observed in UMP stars despite their failure of reproducing the absolute value of the C and N enrichment. Therefore we are studying to what extend HEFM occurs also in low-mass stars with initial $Z$$>$0, and whether those objects are showing the observed \[C,N/Fe\] ratios of CS 22892-052 and CS 22957-027. This extension of our model assumption is warranted by the results about star formation in primordial environments, which agree on the fact that the formation of (very) massive objects is strongly favoured in the absence of metals [@abn:02; @bcl:02]. If this indeed is the case, the UMPs would constitute the extreme end of Pop II, and had formed out of material already bearing the signature of individual supernova events [@shitsu:98].
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The final alternative for the origin of the carbon-rich UMPs is that they were already born with the abundances they now show. This scenario would receive support if single dwarfs of similar composition should definitely be identified. While one shifts the problem of identifying the source of the peculiar abundances to other stars, it is worthwhile, nevertheless, to investigate the evolution of such C-rich stars. For example, they are expected to reach standard RGB-tip luminosities and low gravities, which could be compared with those of observed objects. Stars of extremely low metallicity ($Z=0$ in particular), have a definitely shorter RGB, in contrast.
In addition to the three plausible models for the origin of the C-rich UMPs, we also investigate in this paper a physical problem associated with the HEFM. Due to the large uncertainties still affecting the treatment of convection in stellar models, we will assess the sensitivity of the HEFM process to different assumptions about the extension and mixing efficiency of convective regions in the stellar interiors.
The paper is structured as follows. Section 2 summarizes briefly the model computational techniques, input physics, and evolution of low-mass stars undergoing HEFM. Section 3 discusses the occurrence of HEFM and resulting surface abundances for different initial metallicities and various parameterizations of the stellar convective regions; Sect. 4 summarizes the results.
Modelling and evolution of stars undergoing HEFM during the He flash
====================================================================
All calculations (as the ones in Paper II) have been performed using the Garching stellar evolution code (@wsch:2000). The numerical details are presented in Paper II and will not be repeated here. The important feature of our code is the ability to follow in detail the evolution through the He flash considering simultaneously mixing and burning processes. Mixing is described by means of a time-dependent algorithm that treats convection as a fast diffusive process, where the diffusion constant is proportional to the convective velocity obtained from a convection theory. We have discussed in Paper II that the convective velocities predicted by both mixing-length and [@cm:91’s [-@cm:91]]{} theory lead to very similar results. In general, arbitrary changes by up to 2 orders of magnitude in the mixing speed do not affect appreciably the HEFM process.
Atomic diffusion is treated according to @tbl:94, opacities are from @ir:96 and @af:94, neutrino energy losses from @mki:85, and the equation of state (EOS) is a Saha EOS plus a simplified description of the degenerate electron gas in the core regions following @kw:90. Our reaction network follows the abundances of H, , , , , , , , , without assuming a priori equilibrium compositions for any of these chemical elements. The reaction rates for the $pp$- and CNO-burning have been taken from [@Adel:98], while the $\element[][12]C(\alpha,\gamma)\element[][16]O$-rate agrees with [@CF:85’s [-@CF:85]]{} value. The latter one is, in the relevant temperature range ($0.1 < T/(10^9\,\mathrm{K}) < 0.2$), about a factor of 2 higher than the rates of @CF:88 and about 30–50% higher than the most recent value of @KFJ:02.
Figure \[hr1\], as a reference, displays the evolutionary track of the model with 0.82$\,M_{\odot}$, $Z$=0, and surface chemical pollution mentioned in the introduction (similar to model B1 of Paper II), which experiences the HEFM.
The 0.0003$\,M_{\odot}$ of $Z$=0.02 polluting material are instantaneously accreted before the star reaches the zero-age main sequence; this approach maximizes the effect of accretion and diffusion effects. After this episode the star evolves along the main sequence producing energy through the $pp$-chain. Diffusion is never able to bring the surface metals deep enough to contribute to the burning.
At the sub-giant branch the core increases its temperature and density, and gradually 3-$\alpha$ reactions are setting in. In stars more massive than $\approx0.9\,M_\odot$ the central H content is not fully exhausted when sufficient C is produced ($X(\element[][12]C) \approx 10^{-10}$) to gain considerable nuclear energy from the CNO-cycle. As a consequence a thermal runaway through the CNO-cycle ensues, causing a characteristic blue loop in the H-R diagram. The consequent expansion of the inner regions reduces the efficiency of the 3-$\alpha$ process and CNO-cycle, terminating the runaway which lasts only about $10^7$ years. In less massive models, as the ones considered in this work, the central H is already exhausted before sufficient C can be produced, and thus no blue-loop occurs on the sub-giant branch.
Regardless of the occurrence of this CNO flash, all low-mass models settle on the RGB and evolve towards higher luminosities. While on the sub-giant branch their main energy source remains the $pp$-chain, the contribution of the CNO-cycle is increasing gradually with luminosity as more carbon is created at the inner tail of the H-burning shell by 3-$\alpha$ reactions. In a 0.8$\,M_\odot$ star about 50% of the H-burning energy star is produced by the CNO-cycle at the tip of the RGB.
At this stage the mass difference between the location of the maximum energy generation in the core and the He-core boundary is $\approx$0.29$\,M_{\odot}$. Soon after the onset of the He flash a convective shell develops, which reaches the H-rich matter in the envelope about 1 month after the flash (left arrow in Fig. \[conv1\]); as soon as H is carried into the interior hotter convective region, it starts to burn at a very high rate. Due to this extra energy input the upper boundary of the convective shell immediately moves closer to the surface (thus ingesting even more protons), while the He-burning rate is significantly reduced. The single inner convective region splits into two zones, one for the He- and one for the H-burning. The rapidly weakening He-burning shell can no longer support the underlying layers against contraction, and the released gravothermal energy forms a further convective region which disappears again after a few 1,000 years[^5].
About 500yr after the H ingestion the convective envelope deepens and merges with the convective region above the H-burning region (horizontal arrow in Fig. \[conv1\]). A huge amount of matter processed through H and He burning is brought to the surface enriched in He, C (produced during the He burning) and N (produced by the CNO cycle at the expenses of C). The resulting surface \[C/Fe\] and \[N/Fe\] ratios are, respectively, 4.1 and 4.3, while the surface oxygen abundance is practically unaltered with respect to the initial value (Table \[summ\]).
As a result of this dredge up of heavy elements (which happens on timescales of a few weeks) the envelope opacity is increased significantly; this produces an abrupt discontinuity in the effective temperature visible in Fig. \[hr1\]. At this stage, with a practically ceased He burning, the star behaves as a newborn RGB star, climbing its own RGB. The H shell produces the energy needed to support the star. Another He flash ensues at the end of this second RGB phase, this time with a smaller and less degenerate core. The flash is weaker and no HEFM happens. The subsequent evolution does not show any peculiar feature.
As a general rule, all the changes in the physical inputs and/or initial conditions that can contribute to an increase of the electron degeneracy and thus cause the location of the He-flash ignition to move closer to the border of the He core favour the occurrence of HEFM. Changes in the relative location as small as $\approx$0.02$\,M_{\odot}$ can make a difference between models undergoing HEFM and models which do not experience it. In Paper II we have shown how an increase of the initial He abundance, or an increase of the initial mass, or the inclusion of heavy element pollution at the stellar surface are disfavouring the onset of HEFM. On the other hand, the inclusion of element diffusion favours this process, while the inclusion of mass loss from the stellar surface does not affect at all the onset of the HEFM.
The influence of initial metallicity and convection
===================================================
In Paper II we have investigated the dependence of the HEFM process on several input-physics parameters and various assumptions adopted in computing stellar models; however, we have considered only stars with initial zero metallicity, plus some eventual metal pollution at the surface. It is worthwhile to investigate the dependence of the HEFM on the value of the initial stellar metallicity, given the unknown composition of these stars at the time of formation, as outlined in the introduction.
Because of the uncertainty in the treatment of convection in stellar structures, it is also important to verify how much the HEFM process is sensitive to different assumptions about the extension and mixing efficiency of convective regions in the stellar interior. In particular, we considered various amounts of overshooting from the canonical formal convective boundaries fixed by the Schwarzschild criterion.
In this section we will discuss these effects in theoretical models representative of CS 22892-052 and CS 22957-027.
Models with $Z$$>$0
-------------------
Models with initial $Y$=0.23 and increasing $Z$ have been computed in order to derive the value of the maximum initial metallicity which produces HEFM in a star with 0.82$\,M_{\odot}$. The heavy element mixture has been considered to be $\alpha$-enhanced, with \[$\alpha$/Fe\]=0.4; carbon and nitrogen were scaled solar. We obtain HEFM only for $Z<10^{-7}$, corresponding to \[Fe/H\]$<$$-$5.6. Since all observed \[Fe/H\]-abundances in UMPs so far are higher than $-4$, this already excludes the possibility that the C/N-anomalies were produced in stars with initial Fe-abundances as observed today. Due to the mixing of H-rich material into the H-depleted interior during the HEFM, the surface H abundance drops by about a factor of two, and the upper limit for the surface Fe abundance after this phase is \[Fe/H\]$<$$-$5.3, about 2 orders of magnitude smaller than the observed metallicities of CS 22892-052 and CS 22957-027. Whenever HEFM happens, the \[C/Fe\] and \[N/Fe\] ratios in the post He-flash phase are always of the order of 4, still about 2 orders of magnitude higher than the observed values.
This upper value for the metallicity is slightly increased if we take into account the revised plasma-neutrino rates by @hrw:94. In this case the limiting metallicity is $Z=10^{-6}$, corresponding to \[Fe/H\]=$-$4.3 after the HEFM, i.e., 1dex higher than in case of the old neutrino losses. The reason is that the degeneracy of the core increases and the He flash starts more off-centre. We also wish to notice here that, in spite of their different input physics and cruder treatment of the coupling between convection and burning, @fii:00 found a similar upper limit for the metallicity of a 0.80$\,M_{\odot}$ star undergoing HEFM, and similar values of the surface \[C,N/Fe\] ratios.
We have also tested the effect of implementing a new EOS by @i:2002, which covers the entire stellar structure in all relevant evolutionary phases, and closely reproduces the OPAL EOS [@rsi:96] in the common validity range [see, e.g., @s:2002]. The inclusion of this new EOS in the models does not modify the value of the largest critical heavy elements abundance for which the HEFM occurs. However, with this equation of state the stellar ages are slightly reduced (by about 0.5Gyr) with respect to our reference EOS, and therefore one has to choose a slightly lower mass to reproduce the observations, which might favour the occurrence of HEFM.
Therefore, we have computed a, what we call, ‘best model’ with an initial \[Fe/H\] equal to the observed one, and a mass which yields an age at the RGB tip well matching the age estimated for CS 22892-052 and CS 22957-027; we considered $M$=0.78$\,M_{\odot}$, $Z$=$2.0\times10^{-5}$ (which corresponds, including the $\alpha$ enhancement by 0.4dex, to \[Fe/H\]=$-$3.3), neutrino emission following @hrw:94, [@i:2002’s [-@i:2002]]{} EOS, plus atomic diffusion of H, He, and metals (which further reduces the ages by about 0.5Gyr). The corresponding evolutionary track is shown in Fig. \[hr1\], too (solid line). In spite of the reduced stellar mass the effect of the initial metallicity dominates, and *no* HEFM occurs. The location of the maximum energy generation at the flash is 0.30$\,M_{\odot}$ away from the boundary of the He core (solid line in Fig. \[core\]), a distance about 0.01$\,M_{\odot}$ larger than in the cases when HEFM occurs. We computed also the evolution of stars of 0.75$\,M_{\odot}$ and 0.82$\,M_{\odot}$ with the same initial composition as the 0.78$\,M_{\odot}$ star to span the age range determined for CS 22892-052 from nuclear chronology, not obtaining HEFM in any case.
Models including overshooting {#over}
-----------------------------
Models including overshooting have been computed in order to check its influence on both the onset of HEFM and the amount of C and N dredged up to the surface. Since overshooting increases the extension of convective regions, the occurrence of HEFM is in principle favoured. Our overshooting description follows @bhf:98 and it is modelled as an exponential diffusive process. The diffusion constant of the overshoot region decays exponentially outside the Schwarzschild convective boundary, starting from the value assumed by the convective diffusion coefficient at the convective boundary; the decay length is $F$ times the pressure scale height at the convective boundary. Hydrodynamical simulations of shallow convective envelopes in A stars by @fls:96 predict $F$=0.25$\pm$0.05, while for the overshooting from the convective envelopes of AGB stars @hbse:97 took $F=0.02$.
We have computed a series of models (adopting the same input physics as in Paper II) with different values for $F$, considering $M$=0.82$\,M_{\odot}$, $Y$=0.23 and $Z$=$2.0\times10^{-5}$ which corresponds to \[Fe/H\]=$-$3.3, approximately the \[Fe/H\] value determined for the two UMP stars under scrutiny. In these models the simpler EOS (Saha-EOS completed with degenerate electron gas in the deep interior) and the neutrino losses of [@mki:85] have been used.
We have first considered overshooting from all the convective boundaries, starting [from the main-sequence]{} phase. As a general result, we found that the inclusion of overshooting during the main sequence increases the lifetime of the small convective core which develops at the beginning of the main sequence, when the abundance gradually reaches nuclear equilibrium in the centre; this, in turn, causes an increase of the main-sequence lifetime and a decreased electron degeneracy in the He core along the RGB, which disfavours the occurrence of HEFM. Moreover, the increased lifetime would demand a larger stellar mass in order to obtain not too high stellar ages, and that would further reduce the probability for HEFM.
Models with $M$=0.82$\,M_{\odot}$ and F$=$0.03 do not show HEFM, and are continuing their evolution through the horizontal branch to the TP-AGB (dash-dotted line in Fig. \[hr1\]). During this phase the inter-shell convective zone is able to penetrate the H-rich envelope, similar to the HEFM process [@CCT; @fii:00]. The surface C and N abundances rise again by about 4dex, while O is increased by about 3dex. Hence, an O enrichment about one order of magnitude smaller than for C is expected in these stars, which implies that \[O/Fe\] should be at most 1 for the stars in Table \[data\]. Since no oxygen abundance could be determined in these stars, but only an upper limit of \[O/Fe\]$\la$0.6 [@smcp:96 for CS 22982-052], this scenario could provide a possible explanation for C-rich UMPs, too. However, surface gravity and temperature of these AGB stars are too low to be in agreement with the stars under scrutiny (Fig. \[hr1\]).
Higher values of $F$ do not lead to HEFM. In fact, for $F=0.1$ there is no He flash, but quiescent He-burning ignition and of course no HEFM.
In case overshooting is included only from the RGB phase onwards, the stellar lifetime is not affected, and one needs at least $F=0.17$ to get HEFM. This value is about 30% smaller as the one obtained by @fls:96 for A-star envelopes, but appears to be quite high compared to that needed in AGB stars [@hbse:97]; moreover, switching on overshooting only during the RGB phase appears to be entirely ad hoc; in addition, the resulting \[C/Fe\] and \[N/Fe\] ratios after the HEFM are again of the order of 4, i.e., too high with respect to the observations.
Models with reduced mixing efficiency {#mixeff}
-------------------------------------
[From]{} the computations performed until now, it appears that one of the main problem to the HEFM scenario is the too high C and N enrichment of the surface. One solution to obtain lower surface \[C,N/Fe\] ratios would be to have a smaller overlap between the convective region developing at the flash, and the overlying H-rich region. Smaller overlap implies less protons and therefore less processed C and N. This smaller overlap could be achieved by strongly reducing the diffusion coefficient associated to the convective mixing. Using standard mixing-length velocities, the crossing time of the convective shell is of the order of a few hours. A reduction of the “convective” diffusion coefficient is performed by simply reducing the multiplicative factor that links the coefficient itself to the convective velocity.
The upper boundary of the He-flash driven convective shell is at first order dictated by the energy of the flash and the location of the ignition point. However, after some protons are engulfed the increased energy production (due to the additional H burning) pushes the upper boundary further up, thus increasing the number of proton ingested. If the mixing efficiency is reduced, protons are mixed less deep and less energy is produced, with a smaller increase of the convective shell extension. This would cause a smaller final number of protons ingested and a lower N production.
In order to explore this scenario in more detail, we recomputed the polluted $Z$=0 model of Sect. 2, which experiences HEFM, decreasing the “convective” diffusion coefficient by a factor of $2\times10^4$. The extension of the convective regions are displayed in Fig. \[conv2\]. The upper boundary of the H-flash driven convective shell extends less into H-rich layers, and the location of the H-burning region is less deep; the thickness of the H convective shell before the dredge up is also thinner, implying a reduction of the amount of C dredged up with respect to the case with standard convective mixing. The final surface abundance ratios \[C/Fe\] and \[N/Fe\] are in this case 2.3 and 3.2 (Table \[summ\]), respectively, closer to the observed values.
The surface / ratio is also affected, being now increased to 5.6 compared to 4.5 which we obtain by using the standard mixing efficiency. This increased value is in slightly better agreement with the observed values of about 10. In spite of the much reduced mixing efficiency the dredge up still happens fast, on timescales of the order of $10^3$ years.
Discussion
==========
The results shown in the previous section highlight the fact that it is very difficult to reproduce the surface abundances of CS 22892-052 and CS 22957-027 from single-star evolution, at least in the case the observed surface abundances pattern is not primordial. To investigate this alternative, we have computed additionally the evolution of an 0.82$\,M_{\odot}$ star, with the same physics of our ‘best model’ and $Y$=0.23, but with $Z$=$6\times10^{-4}$ and \[C/Fe\]=\[N/Fe\]=2 (implying \[Fe/H\]=$-$3.3), in agreement with the observed abundances. The evolutionary track is shown in Fig. \[hr1\], while the He-core boundary before and at He ignition is plotted in Fig. \[core\]. As expected, this model does not experience HEFM (the He flash starts very deep in the core), so that the surface \[C/Fe\] and \[N/Fe\] ratios are not modified with respect to the initial values. Note that the effective temperature along the RGB is higher than for the case with HEFM, because of the much lower C-abundance. The location of the track fits reasonably well the observed gravities and effective temperatures, and thus, this model would be able to reproduce the observed properties of CS 22957-027. However, as discussed in Sect. 1, if this were a realistic model for the observed object, the source for the high carbon and nitrogen abundances still remains unknown, as it is not clear how to pollute the interstellar medium with matter showing these anomalous abundance ratios.
Fig. $f_D$ $[\frac{\element[][]{C}}{\element[][]{Fe}}]$ $[\frac{\element[][]{N}}{\element[][]{Fe}}]$ $[\frac{\element[][]{O}}{\element[][]{Fe}}]$ $[\frac{\element[][]{Fe}}{\element[][]{H}}]$ $\frac{\element[][12]{C}}{\element[][13]{C}}$ $\frac{\element[][13]{C}}{\element[][14]{N}}$
----------- -------------------- ---------------------------------------------- ---------------------------------------------- ---------------------------------------------- ---------------------------------------------- ----------------------------------------------- -----------------------------------------------
\[conv1\] 1 4.1 4.3 0.7 -3.4 4.5 0.36
\[conv2\] 5$\times$$10^{-5}$ 2.3 3.2 0.4 -3.4 5.6 0.07
: Surface abundances of initial metal-free 0.82$\,M_\odot$ stars polluted by $3\times10^{-4}$$\,M_\odot$ of alpha-enhanced (\[$\alpha$/Fe\]=0.4\]) material with $Z=0.02$ at the ZAMS. All models contain updated neutrino-losses [@hrw:94] and [@i:2002’s [-@i:2002]]{} EOS. The evolution of the convective regions after the flash are shown in the figures denoted in the first column. $f_D$ is the factor by which the standard diffusion efficiency has been multiplied in each model.\[summ\]
Thus, assuming that the observed surface composition is not primordial, the HEFM is presently the only way to get a surface C and N enhancement without invoking some kind of mass transfer from binary companions. [From]{} our tests in Sect. 2 there appear to be two main problems with the HEFM scenario:
Firstly, too much of C and N is transported to the surface after the HEFM, and secondly, no HEFM appears to occur for an initial metallicity that can match the observed \[Fe/H\] of the stars under scrutiny, unless surface heavy element pollution has been efficient, as discussed in .
Once again, we want to point out that, in spite of their different input physics and cruder treatment of the coupling between convection and burning, @fii:00 found similar upper limits for the metallicity of stars undergoing HEFM, and similar values of the \[C,N/Fe\] ratio for stellar mass and metallicity values similar to our ‘best model’.
Concerning the appearance of HEFM in stars with \[Fe/H\] in agreement with observations, one possible solution would be, as discussed in Sect.\[mixeff\], a completely ad hoc choice of overshooting efficiency. Another possibility is related to the H depletion during the dredge up of the HEFM products. During this phase the convective envelope reaches deeper regions involved in the HEFM, which are devoid of H. According to our models the surface \[Fe/H\] increases by $\approx$0.3dex because of the dilution of H. If the drop in hydrogen abundance is about five times higher, a surface \[Fe/H\] of about $-$3.3 could be obtained with an initial \[Fe/H\]=$-$4.3 corresponding to the upper limit of getting HEFM.
In order to achieve a larger H depletion, a smaller envelope thickness (in mass) is needed, i.e., a smaller H reservoir. We estimate that right before the flash ignition an envelope of only about 0.03$\,M_{\odot}$ is needed in order to reproduce the observed post-HEFM \[Fe/H\] surface values with our ’best model’; with this envelope mass the star would still be able to ignite He close to its RGB location (see, e.g., @cc:93), i.e., the effective temperature and gravity after the HEFM would still be in accordance with CS 22892-052 and CS 22957-027. Taking into account the age of the star of about 14Gyr, which fixes its initial mass to be about 0.8$\,M_\odot$, it would imply that about 0.27$\,M_{\odot}$ are lost during the evolution, a value which is not much higher than the average mass loss experienced by globular cluster stars (about 0.20$\,M_{\odot}$). The lack of metals in the outer layers of UMPs, however, should cause a strong decrease of mass-loss efficiency in comparison with more metal-rich stars, if radiative driven winds are the main source of mass loss. Nevertheless, due to our poor knowledge about mass-loss processes in RGB stars, this possibility cannot be completely ruled out.
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[^1]: data from [@nrb:97]
[^2]: C & N abundances from [@nrb:97l], $\mathrm{[Fe/H]}$, $g$, and $T_\mathrm{eff}$ from @bmb:98
[^3]: data from [@ANR:02]
[^4]: data from [@NRB:01]
[^5]: This convective region appeared also in the calculations presented in Paper II, but was not shown in the corresponding figure due do insufficient plot data.
|
---
abstract: 'Sufficiency of jets is a very important notion introduced by René Thom in order to establish the structural stability theory. The criteria for some sufficiency of jets are known as the Kuo condition and Thom type inequality, which are defined using the Kuo quantity and Thom quantity. Therefore these quantities are meaningful. In this paper we show the equivalence of Kuo and Thom quantities. Then we apply this result to the relative conditions to a given closed set.'
address:
- 'Institut de recherche Mathematique de Rennes, Université de Rennes 1, Campus Beaulieu, 35042 Rennes cedex, France'
- 'Department of Mathematics, Hyogo University of Teacher Education, Kato, Hyogo 673-1494, Japan'
author:
- Karim Bekka and Satoshi Koike
title: |
Equivalence of Kuo and Thom quantities\
for analytic functions
---
[^1]
Introduction
============
Let $f : ({\mathbb{R}}^n ,0) \to ({\mathbb{R}},0)$ be a $C^r$ function germ. The $r$-jet of $f$ at $0 \in {\mathbb{R}}^n$, $j^r f(0)$, has a unique polynomial representative $z$ of degree not exceeding $r$. We do not distinguish the $r$-jet $j^r f(0)$ and the polynomial representative $z$ here.
[**Kuiper-Kuo condition.**]{} There is a positive number $C > 0$ such that $$\| \operatorname{grad\,}z(x) \| \ge C \| x\|^{r-1}$$ holds in some neighbourhood of $0 \in {\mathbb{R}}^n.$
The Kuiper-Kuo condition is well-known as a criterion for $C^0$-sufficiency and $V$-sufficiency of $z$ in $C^r$ functions (N. Kuiper [@kuiper], T.-C. Kuo [@kuo1], J. Bochnak and S. Lojasiewicz [@bochnaklojasiewicz]). See §\[preli\] for the definitions of $C^0$-sufficiency and $V$-sufficiency of jet.
Let us recall the Kuo condition.
[**Kuo condition.**]{} There are positive numbers $C, \alpha, \bar w > 0$ such that $$\| \operatorname{grad\,}f(x) \| \ge C \| x\|^{r-1} \text{ in } {\mathcal H}_{r}(f; \bar w)
\cap \{\| x \| < \alpha\},$$ where ${\mathcal H}_{r}(f;\bar w) := \{ x \in \mathbb{R}^n : |f(x)| \le
\bar w \| x\|^{r}\}$ is the [*horn-neighbourhood of $f^{-1}(0)$ of degree $r$ and width $\bar{w}$*]{} (T.-C. Kuo [@kuo2]).
The Kuo condition is a criterion for $V$-sufficiency of $z$ in $C^r$ functions.
[**Condition ($\widetilde{K}$).**]{} There is a positive number $C > 0$ such that $$\| x\| \| \operatorname{grad\,}f(x) \| + |f(x)| \ge C \| x\|^r$$ holds in some neighbourhood of $0 \in {\mathbb{R}}^n.$
This condition is the Kuo condition in a different way. Therefore condition ($\widetilde{K}$) is also a criterion for $V$-sufficiency of $z$ in $C^r$ functions.
On the other hand, R. Thom formulated the following condition as a sufficient condition for $z$ to be $C^0$-sufficient in $C^r$-functions.
[**Thom type inequality.**]{} There are positive numbers $K, \beta > 0$ such that $$\sum_{i<j} \left| x_i \frac{\partial f}{\partial x_j} -
x_j \frac{\partial f}{\partial x_i} \right|^2 + |f(x)|^2 \geq K \| x\|^{2r}
\text{ for } \| x\| < \beta.$$
It is shown in [@bekkakoike1] that Thom type inequality condition is equivalent to the Kuiper-Kuo condition.
Throughout this paper, we denote by $\mathbb{N}$ the set of natural numbers in the sense of positive integers. Let $s \in {\mathbb{N}}\cup \{ \infty, \omega \}$, and let ${\mathcal E}_{[s]}(n,p)$ denote the set of $C^s$ map-germs : $({\mathbb{R}}^n,0)\to ({\mathbb{R}}^p,0)$.
Now we introduce the Kuo quantity $K_m$ and Thom quantity $T_m.$ The Thom quantity is a generalisation of the left side of Thom type inequality, and the Kuo quantity is a generalisation of the left side of a condition equivalent to condition ($\widetilde{K}$).
\[KTquantity\] Let $f\in {\mathcal E}_{[s]}(n,p)$, $n\geq p,$ and let $m \in {\mathbb{N}}$. Let us define two functions of the variable $x$: $$K_{m}(f,x) := \| x \|^m \sum_{1\leq i_1<\ldots<i_{p}\leq n}
\left| \det \left( \frac{D(f_1,\ldots, f_p)}{D(x_{i_1},\ldots,x_{i_{p}})}(x)
\right) \right|^m + \| f(x) \|^m$$ $$T_{m}(f,x) := \sum_{1\leq i_1<\ldots<i_{p+1}\leq n}
\left| \det \left(\frac{D(f_1,\ldots,
f_p,\rho)}{D(x_{i_1},\ldots,x_{i_{p+1}})}(x)\right) \right|^m
+ \| f(x) \|^m$$ where $\rho(x)=\| x\|^2.$ Note that $T_{m}(f,x) = \| f(x) \|^m$ in the case where $n = p$.
Related to the Kuo condition and Thom type inequality, we have shown the following result.
([@bekkakoike1]) Let $r \in {\mathbb{N}}$. For $f \in {\mathcal E}_{[r]}(n,p)$, $n \ge p$, the following conditions are equivalent.
\(1) There are positive numbers $C, \alpha > 0$ such that $K_2(f,x) \ge C \| x \|^{2r}$ for $\| x \| < \alpha .$
\(2) There are positive numbers $K, \beta > 0$ such that $T_2(f,x) \ge K \| x \|^{2r}$ for $\| x \| < \beta .$
The main purpose of this paper is to show the equivalence of the Kuo quantity and Thom quantity, which is a generalisation of the above result in a certain sense.
\[equivKT\] (Main Theorem). Let $f \in {\mathcal E}_{[\omega ]}(n,p)$, $n \ge p$. Then for any $m \in \mathbb{N},$ $$K_{m}(f,.)\thickapprox T_{m}(f,.).$$
Throughout this paper, we use the equivalence $\thickapprox$ in the following sense:
Let $f,g : U \to {\mathbb{R}}$ be non-negative functions, where $U \subset {\mathbb{R}}^N$ is an open neighbourhood of $0 \in {\mathbb{R}}^N$. If there are real numbers $K > 0$, $\delta > 0$ with $B_{\delta}(0) \subset U$ such that $f(x) \le K g(x)$ for any $x \in B_{\delta}(0)$, where $B_{\delta}(0)$ is a closed ball in ${\mathbb{R}}^N$ of radius $\delta$ centred at $0 \in {\mathbb{R}}^N$, then we write $f \precsim g$ (or $g \succsim f$). If $f \precsim g$ and $f \succsim g$, we write $f \thickapprox g$.
In the next section we mention the definitions of $C^0$-sufficiency and $V$-sufficiency of jets, and give the notion of the relative jet of a $C^s$ mapping to a given closed set ${\Sigma}$. We shall show our Main Theorem in §\[proof\], and apply the theorem to the relative conditions to a closed set ${\Sigma}$ in §\[application\].
Preliminaries {#preli}
=============
Sufficiency of jets {#sufficiency}
-------------------
Let $s \in {\mathbb{N}}\cup \{ \infty, \omega \}$. Let us recall ${\mathcal E}_{[s]}(n,p)$, the set of $C^s$ map-germs : $({\mathbb{R}}^n,0)\to ({\mathbb{R}}^p,0)$. Let $j^r f(0)$ denote the r-jet ($r \in {\mathbb{N}}$) of $f$ at $0 \in {\mathbb{R}}^n$ for $f \in {\mathcal E}_{[s]}(n,p)$, $s \ge r$, and let $J^r(n,p)$ denote the set of r-jets in ${\mathcal E}_{[s]}(n,p)$.
We say that $f,g\,\in {\mathcal E}_{[s]}(n,p)$ are $C^0$-[*equivalent*]{} (resp. $SV$-[*equivalent*]{}), if there exists a local homeomorphism $\sigma : ({\mathbb{R}}^n,0) \to ({\mathbb{R}}^n,0)$ such that $f = g \circ \sigma$ (resp. $\sigma (f^{-1}(0)) = g^{-1}(0)$). In addition, we say that $f,g\,\in {\mathcal E}_{[s]}(n,p)$ are $V$-[*equivalent*]{}, if $f^{-1}(0)$ is homeomorphic to $g^{-1}(0)$ as germs at $0\in \mathbb{R}^n$.
Let $w \in J^r(n,p).$ We call the $r$-jet $w$ $C^0$-[*sufficient*]{}, $SV$-[*sufficient*]{} and $V$-[*sufficient*]{} in $C^s$ mappings, $s \ge r$, if any two realisations $f$, $g\,\in {\mathcal E}_{[s]}(n,p)$ of $w,$ namely $j^rf(0) = j^rg(0)=w,$ are $C^0$-equivalent, $SV$-equivalent and $V$-equivalent, respectively.
Let us recall the Thom type inequality for $f \in {\mathcal E}_{[s]}(n,p)$, $n \ge p$ :
There are positive numbers $K, \alpha, \beta > 0$ such that $T_2(f,x) \ge K \| x \|^{\alpha}$ for $\| x \| < \beta .$
As mentioned in the Introduction, R. Thom considered this condition with $\alpha = 2r$ in the function case as a sufficient condition for $z = j^r(f)(0)$ to be $C^0$-sufficient in $C^r$ functions. On the other hand, he considered this condition in the mapping case as a sufficient condition for $SV$-sufficiency of jet.
The Kuo condition mentioned in the Introduction is a criterion for $V$-sufficiency of $z = j^r(f)(0)$ in $C^r$ functions. This condition is generalised to the mapping case, as a criterion for $V$-sufficiency of $z = j^r(f)(0)$ in $C^r$ mappings : $({\mathbb{R}}^n,0) \to ({\mathbb{R}}^p,0)$, $n \ge p$. For the details, see T.-C. Kuo [@kuo3].
Relative jet to a given closed set {#relativejet}
----------------------------------
Throughout this paper, let ${\Sigma}$ be a germ of a given closed subset of $ {\mathbb{R}}^n$ at $0 \in {\mathbb{R}}^n$ such that $0 \in {\Sigma}.$ Then we denote by $d(x,{\Sigma})$ the distance from a point $x \in {\mathbb{R}}^n$ to the subset ${\Sigma}.$
We consider on ${\mathcal E}_{[s]}(n,p)$ the following equivalence relation:
Two map-germs $f,g\,\in {\mathcal E}_{[s]}(n,p)$ are $r$-$\Sigma$-[*equivalent*]{}, denoted by $f\sim g$, if there exists a neighbourhood $U$ of $0$ in ${\mathbb{R}}^n$ such that the r-jet extensions of $f$ and $g$ satisfy $j^rf({\Sigma}\cap U)= j^rg({\Sigma}\cap U).$
We denote by $j^rf({\Sigma};0)$ the equivalence class of $f,$ and by $J^r_{{\Sigma}}(n,p)$ the quotient set ${\mathcal E}_{[s]}(n,p)/\sim.$
We can define the notions of $C^0$-sufficiency, $SV$-sufficiency and $V$-sufficiency of relative jets to ${\Sigma}$, similarly to in the non-relative case. In [@bekkakoike2] we gave criteria for the relative $r$-jet to be $C^0$-sufficient and $V$-sufficient in ${\mathcal E}_{[r]}(n,p)$ or ${\mathcal E}_{[r+1]}(n,p)$, using the relative Kuiper-Kuo condition and relative Kuo condition (or condition ($\widetilde{K}_{{\Sigma}}$)), respectively.
Proof of Main Theorem {#proof}
=====================
In this section we show the equivalence between the Kuo quantity $K_{m}$ and the Thom quantity $T_{m}$, namely our main theorem (Theorem \[equivKT\]).
Let $ord(\gamma(t))$ denote the order of $\gamma$ in $t$ for a $C^\omega$ function $\gamma: [0, \delta)\to \mathbb{R}.$
It is obvious that $K_{m}(f,.)\succsim T_{m}(f,.).$ Therefore we have to show the converse.
We first remark that if $x$ and $y$ are bigger than or equal to $0$, we have $$(x+y)^m \geq x^m + y^m \geq \frac{(x+y)^m}{2^m}.$$ It follows that $$K_{m}(f,x)\thickapprox v^{m}(x)+u^{m}(x)\thickapprox (h(x))^m$$ $$T_{m}(f,x)\thickapprox w^{m}(x)+u^{m}(x)\thickapprox (g(x))^m,$$ where $u(x)=\Vert f(x) \Vert$,
$v(x)=\Vert x\Vert \dis \sum_{1\leq i_1<\ldots<i_{p}\leq n}
\left| \det \left( \frac{D(f_1,\ldots, f_p)}{D(x_{i_1},\ldots,
x_{i_{p}})}(x)\right) \right| , $
$w(x)=\dis \sum_{1\leq i_1<\ldots<i_{p+1}\leq n}
\left| \det \left( \frac{D(f_1,\ldots, f_p,\rho)}{D(x_{i_1},\ldots,
x_{i_{p+1}})}(x)\right) \right|, (\text{where } \rho(x)=\| x\|^2),$
$h(x)=v(x)+u(x)$ and $g(x)=w(x)+u(x).$
Suppose now that $K_{m}(f,.)\precsim T_{m}(f,.)$ does not hold. Then by the curve selection lemma, there is a $C^\omega$ curve $\tilde\lambda=(\lambda, C): [0, \delta) \to \mathbb{R}^n\times \mathbb{R}$ with $\tilde\lambda(0)=(0,0)$ and $\tilde\lambda(t)\in
(\mathbb{R}^n \setminus \{0\}) \times \mathbb{R}^*,$ for $t\ne 0,$ such that $$\label{eq1}
(C(t))^m K_{m}(f,\lambda(t))>T_{m}(f,\lambda(t)).$$ We may write as: $$\label{eq2}
(C(t)(h \circ \lambda (t)))^m>(g \circ \lambda (t))^m .$$ Here we remark that the functions $g \circ \lambda , h \circ \lambda ,
u \circ \lambda ,v \circ \lambda$ and $w \circ \lambda$ are real analytic on $[0, \delta)$ and satisfying the conditions $$g \circ \lambda (0)=h \circ \lambda (0)=u \circ \lambda (0)=v \circ \lambda (0)
=w \circ \lambda (0)=0$$ and $$\lambda(t)\neq0,\quad C(t)>0,\quad h \circ \lambda (t)>0,\quad
g \circ \lambda (t)\geq0\quad\textrm{for}\quad 0<t<\delta$$
By , $C(t)(h \circ \lambda (t))>u \circ \lambda (t)$, $C(t)(h \circ \lambda (t))>w \circ \lambda (t)$ and $$v \circ \lambda (t)=h \circ \lambda (t)
-u \circ \lambda(t)\geq h \circ \lambda (t)(1-C(t)).$$ Then we have $$\begin{cases}
ord(C)+ord(h \circ \lambda )\leq ord(u \circ \lambda )\\
ord(C)+ord(h \circ \lambda )\leq ord(w \circ \lambda )\\
ord(v \circ \lambda )\leq ord(h \circ \lambda ).
\end{cases}
\label{eq3}$$ Note that we are not considering the second inequality in the case where $n = p$.
Let $\tilde\lambda$ be written as follows $\lambda_i(t)= a_1^{(i)}t^{\varepsilon_1(i)}+a_2^{(i)}t^{\varepsilon_2(i)}+ \ldots$
where $1\leq {\varepsilon_1(i)}<{\varepsilon_2(i)}<
\ldots$ and $\left \{\begin{array}{clcr}
a_1^{(i)} \ne 0& if & \lambda_i(t)\not\equiv 0\\
{\varepsilon_1(i)}=\infty& if & \lambda_i(t)\equiv 0\end{array}\right.$ $(1\leq i\leq n),$
$C(t)= u_1t^{b_1}+u_2t^{b_2}+ \ldots$ where $1\leq b_1<b_2<\ldots$ and $u_1\ne 0.$
Since condition is invariant under rotation, we can assume that $\varepsilon_1(1)<\varepsilon_1(i)$ for $i\ne 1.$
Let $f_j(\lambda(t))= d_1^{(j)}t^{q_1^{(j)}}+d_2^{(j)}t^{q_2^{(j)}}+ \ldots$, where $1\leq q_1^{(j)}<q_2^{(j)}< \ldots$ $(1\leq j\leq p).$ Then $$\frac{df_j \circ\lambda}{dt}(t)= q_1^{(j)}d_1^{(j)}t^{q_1^{(j)}-1}+q_2^{(j)}d_2^{(j)}t^{q_2^{(j)}-1}+ \ldots
\qquad(1\leq j\leq p).$$ It follows from that $$\label{eq4}
q_1^{(j)}\geq ord(C)+ord(h \circ \lambda )\qquad \text{
for all } j\in \{1,\ldots, p\}.$$
By again, we have
$$\label{eq5}
\varepsilon_1(1) +ord\left(\sum_{1\leq i_1<\ldots<i_p\leq n} \left|
\det \left( \frac{D(f_1,\ldots, f_p)}{D(x_{i_1},\ldots,
x_{i_p})}(\lambda(t))\right) \right|\right)\leq ord(h \circ \lambda ).$$
Therefore there is a $p$-tuple of integers $(k_1,\dots, k_p)$ with $1\leq k_1<\dots<k_p\leq n$ such that
$$\label{eq6}
\left \{\begin{array}{clcr}
ord(\vert \det \left( \frac{D(f_1,\ldots, f_p)}{D(x_{k_1},\ldots,
x_{k_p})}(\lambda(t))\right) \vert) \leq
ord(\vert \det \left( \frac{D(f_1,\ldots, f_p)}{D(x_{i_1},\ldots,
x_{i_p})}(\lambda(t))\right) \vert)\, \\
\text{for any}\,(i_1,\ldots, i_p), \ \ \text{and}\\
\qquad ord(\vert \det \left( \frac{D(f_1,\ldots, f_p)}{D(x_{k_1},
\ldots, x_{k_p})}(\lambda(t))\right) \vert) \leq
ord(h\circ \lambda ) -\varepsilon_1(1).
\end{array}\right.$$
We continue the proof of the converse, dividing it into two cases. We first consider the case where $n > p$. Then we have the following.
$k_{1}>1.$
Since $\dis\frac{df_j\circ\lambda}{dt}(t)= \sum _{i=1}^n{\frac{\partial f_j}{\partial x_i}}
(\lambda(t)){\frac{d\lambda_i}{dt}(t)},
\ (1\leq j\leq p),
$ we have
$$\left (\begin{array}{clcr}
\frac{df_1\circ\lambda}{dt}(t)\\
\vdots\\
\frac{df_p\circ\lambda}{dt}(t) \end{array}\right)
= {\frac{d\lambda_1}{dt}}(t)
\left (\begin{array}{clcr}
{\frac{\partial f_1}{\partial x_1}}(\lambda(t))\\
\vdots\\
{\frac{\partial f_p}{\partial x_1}}(\lambda(t)) \end{array}\right)
+\ldots+ {\frac{d\lambda_n}{dt}}(t)\left (\begin{array}{clcr}
{\frac{\partial f_1}{\partial x_n}}(\lambda(t))\\
\vdots\\
{\frac{\partial f_p}{\partial x_n}}(\lambda(t)) \end{array}\right).
\label{eq7}$$
Here we remark that, by
$$\label{eq8}
ord\left(\dis \frac{1}{\lambda'_1(t)}.\frac{df_j\circ\lambda}{dt}(t)\right) =q_1^{(j)}-\varepsilon_1(1) \geq ord(C)+ord(h\circ \lambda)-\varepsilon_1(1)
\quad (1\leq j\leq p),$$
and $$\label{eq9}
ord\left(\frac{\lambda'_i(t)}{\lambda'_1(t)}\right) \geq1
\qquad (2\leq i\leq n) .$$
Assume, by contradiction, that $k_{1}=1$ in . For simplicity, set $$A(t)=\left(\begin{array}{cccc}{\frac{\partial f_1}{\partial x_1}}(\lambda(t))&
{\frac{\partial f_1}{\partial x_{k_2}}}(\lambda(t))&\ldots&
{\frac{\partial f_1}{\partial x_{k_p}}}(\lambda(t))\\
\vdots & \vdots & & \vdots \\
{\frac{\partial f_p}{\partial x_1}}(\lambda(t))&
{\frac{\partial f_p}{\partial x_{k_2}}}(\lambda(t))&\ldots&
{\frac{\partial f_p}{\partial x_{k_p}}}(\lambda(t))\end{array}\right).$$ Then the determinant of the matrix $A(t)$ is the summation of determinants of the following matrices:
$$\begin{pmatrix}
\ldelim({3}{1cm}[$\frac{1}{\lambda'_1(t)}$]&\frac{df_1\circ\lambda}{dt}(t)&\rdelim){3}{0.4cm}[]
&{\frac{\partial f_1}{\partial x_{k_2}}}(\lambda(t))&\ldots&{\frac{\partial f_1}{\partial x_{k_p}}}(\lambda(t))\\
&\vdots&&\vdots&&\vdots\\
&\frac{df_p\circ\lambda}{dt}(t)&&{\frac{\partial f_p}{\partial x_{k_2}}}(\lambda(t))&\ldots&{\frac{\partial f_p}{\partial x_{k_p}}}(\lambda(t))
\end{pmatrix}
\label{eq10}$$
$$\begin{pmatrix}
\ldelim({3}{1.5cm}[$-\frac{\lambda'_i(t)}{\lambda'_1(t)}$]&
{\frac{\partial f_1}{\partial x_i}}(\lambda(t))&\rdelim){3}{0.4cm}[]
&{\frac{\partial f_1}{\partial x_{k_2}}}(\lambda(t))&\ldots&{\frac{\partial f_1}{\partial x_{k_p}}}(\lambda(t))\\
&\vdots&&\vdots&&\vdots\\
&{\frac{\partial f_p}{\partial x_i}}(\lambda(t))&&{\frac{\partial f_p}{\partial x_{k_2}}}(\lambda(t))&\ldots&{\frac{\partial f_p}{\partial x_{k_p}}}(\lambda(t))
\end{pmatrix}
\, \text{for } i\in\{2,\ldots, n\}.
\label{eq11}$$
By the order of the determinant of the matrix is bigger than or equal to $ord(C)+ord(h\circ \lambda)-\varepsilon_1(1)$, and by the order of the determinant of the matrix is bigger than the order of the determinant of the matrix . Therefore we have $$ord(\vert \det A(t)\vert)\geq ord(C)+ord(h \circ \lambda )-\varepsilon_1(1)>ord(h\circ \lambda)-\varepsilon_1(1)$$ which contradicts $\eqref{eq6}$. This completes the proof of the claim.
It follows from the Claim that there is a $p$-tuple $(k_1,\dots, k_p)$ with $1<k_1<\dots< k_p\leq n$ such that condition holds. Then $$ord\left(\left| \det \left( \frac{D(f_1,\ldots, f_p,\rho)}{D(x_{1},x_{k_1},
\ldots, x_{k_p})}(\lambda(t))\right) \right| \right) \leq
ord(\lambda) + ord\left(\left| \det \left( \frac{D(f_1,\ldots, f_p)}{D(x_{k_1},
\ldots, x_{k_p})}(\lambda(t)) \right) \right|\right)$$ $$\leq \varepsilon_1(1)+ord(h \circ \lambda ) -\varepsilon_1(1)
= ord(h \circ \lambda ).$$ This contradicts , and it follows that $K_{m}(f,.)\precsim T_{m}(f,.)$.
We next consider the case where $n = p$. Using a similar argument to the proof of the above Claim, we get the same contradiction for $$A(t)=\left(\begin{array}{cccc}{\frac{\partial f_1}{\partial x_1}}(\lambda(t))&
{\frac{\partial f_1}{\partial x_2}}(\lambda(t))&\ldots&
{\frac{\partial f_1}{\partial x_n}}(\lambda(t))\\
\vdots & \vdots & & \vdots \\
{\frac{\partial f_n}{\partial x_1}}(\lambda(t))&
{\frac{\partial f_n}{\partial x_2}}(\lambda(t))&\ldots&
{\frac{\partial f_n}{\partial x_n}}(\lambda(t))\end{array}\right).$$ Therefore it follows that $K_{m}(f,.)\precsim T_{m}(f,.)$, and this completes the proof.
The proof of Theorem \[equivKT\] uses essentially the curve selection lemma. Therefore it is not difficult to see that the results are still valid if we suppose only that $f$ is an arc-analytic and differentiable subanalytic map-germ; see [@kurdyka], [@hironaka] and [@bierstonemilman] for the notions and properties of subanalytic and arc-analytic functions.
Let $f = (f_1, f_2) : ({\mathbb{R}}^2,0) \to ({\mathbb{R}}^2,0)$ be a polynomial mapping defined by $\dis
f_1(x,y) = x - y^2, \ \ f_2(x,y) = x^2.
$ Then we have $
f_1(x,y)^2 + f_2(x,y)^2 = (x - y^2)^2 + x^4,
$ $\dis
\det \left( \frac{D(f_1, f_2)}{D(x,y)}((x,y))\right) = 4xy.
$ Therefore we have $$T_{2}(f,(x,y)) = (x - y^2)^2 + x^4, \ \
K_{2}(f,(x,y)) = 16(x^2 + y^2)x^2y^2 + (x - y^2)^2 + x^4.$$
To show that $T_{2}(f,(x,y)) \thickapprox K_{2}(f,(x,y)),$ we consider two cases.
In the case where $|x - y^2| \le \frac{1}{2} y^2$, we have $x\ge \frac{1}{2} y^2$. Therefore $64 x^4\geq 16x^2y^4$ and since for any constant $C> 65,$ $16x^4y^2=o((C-65)x^4)$ we get $$C T_{2}(f,(x,y)) \ge K_{2}(f,(x,y))$$ in a small neighbourhood of $(0,0) \in \mathbb{R}^2,$
In the case where $|x - y^2| \ge \frac{1}{2} y^2$ we can see that $$(x - y^2)^2 + x^4 \ge \frac{1}{4}y^4 + x^4 \ge
16x^2y^4 + 16x^4y^4 = 16(x^2 + y^2)x^2y^2$$ in a small neighbourhood of $(0,0) \in {\mathbb{R}}^2$.
Thus, for any constant $C> 65,$ we have $\dis
T_{2}(f,(x,y)) \le K_{2}(f,(x,y)) \le C T_{2}(f,(x,y))
$ in a small neighbourhood of $(0,0) \in {\mathbb{R}}^2,$ it follows that $T_{2}(f,(x,y)) \thickapprox K_{2}(f,(x,y)).$
Application to the relative case {#application}
================================
We now introduce some notion for a $C^r$-map germ $f:(\mathbb{R}^n,0)\to (\mathbb{R}^p,0)$ in order to extend to the relative case the previous equivalence defined in the non-relative case.
Let ${\Sigma}$ be a germ at $0 \in {\mathbb{R}}^n$ of closed set such that $0 \in {\Sigma}$. Given a map $g\in {\mathcal E}_{[r]}(n,p)$ with $j^rg({\Sigma};0) = j^rf({\Sigma};0)$. Let $f_t: (\mathbb{R}^n,0)\to (\mathbb{R}^p,0)$ denote the $C^r$ mapping defined by $$f_t(x)= f(x) + t(g(x)-f(x)) \text{ for }\, \,t\in [0,1].$$
\[rcompatibility\] A condition $(*)$ on a $C^r$ map $f$ is called ${\Sigma}$-$r$-[*compatible in the direction*]{} $g$, if $f_t$ satisfies condition $(*)$ for any $t\in [0,1].$ If condition $(*)$ is ${\Sigma}$-$r$-compatible in any direction $g\in {\mathcal E}_{[r]}(n,p)$ with $j^rg({\Sigma};0) = j^rf({\Sigma};0)$, we simply say condition $(*)$ is ${\Sigma}$-$r$-[*compatible*]{}.
Let $f: ( \mathbb{R}^n,0)\to (\mathbb{R}^p,0)$ be a $C^1$ map-germ, ${\Sigma}\subset \mathbb{R}^n$ be a germ of a closed set such that $0\in {\Sigma}$ and $r\in \mathbb{N}.$ For $m\in \mathbb{N}$, we introduce the following conditions: $$I^T_{r}(m):\quad \exists c,\delta>0\, \text{ such that } T_{m}(f,x)\geq c(d(x,{\Sigma}))^{rm}\, \text{ for } \|x\|<\delta ,$$ $$I^K_{r}(m):\quad \exists c,\delta>0\, \text{ such that }
K_{m}(f,x)\geq c(d(x,{\Sigma}))^{rm}\, \text{ for } \|x\|<\delta .$$
If $f$ is $C^{\omega},$ we have from Theorem \[equivKT\], for any $m\in \mathbb{N}$, $$I^T_{r}(m) \ \text{holds if and only if} \ I^K_{r}(m) \ \text{holds.}$$
Let ${\Sigma}$ be a germ at $0 \in {\mathbb{R}}^n$ of a closed set such that $0 \in {\Sigma}$. Let $r \in {\mathbb{N}}$, and let $f\in {\mathcal E}_{[r]}(n,p) $, $n\geq p$. Suppose that $j^rf({\Sigma},0)$ has a $C^\omega$ realisation. Then for any $m\in \mathbb{N}$, $$I^T_{r}(m) \text{ holds if and only if } I^K_{r}(m) \text{ holds. }$$
Let $g : ({\mathbb{R}}^n,0) \to ({\mathbb{R}}^p,0)$ be a $C^\omega$ realisation of $j^rf({\Sigma},0)$. From Theorem \[equivKT\], conditions $I^T_{r}(m)$ and $I^K_{r}(m)$ are equivalent for $g$. Therefore it suffices to show that conditions $I^T_{r}(m)$ and $I^K_{r}(m)$ are $r$-compatible. Let $f_{t}=f+th$ with $h = g - f.$ Then $\|h\|=o(d(.,{\Sigma})^{r}),$ $\|f_{t}\|\geq \|f\|-\|h\|$ and the expansion of the determinants give $$T_{m}(f_{t},x)= T_{m}(f,x)+o(d(x,{\Sigma}))^{rm}
\text{ and }
K_{m}(f_{t},x)= K_{m}(f,x)+o(d(x,{\Sigma}))^{rm}.$$ Thus the r-compatibilities of $I^T_{r}(m)$ and $I^K_{r}(m)$ follow.
As pointed out in [@bekkakoike2], any $r$-jet, $r \in {\mathbb{N}}$, has a unique polynomial realisation of degree not exceeding $r$ in the non-relative case, but some $r$-jets do not have even a $C^{\omega}$ realisation in the general relative case. Therefore, in the above theorem, the assumption that $j^rf({\Sigma},0)$ has a $C^\omega$ realisation makes sense.
Let ${\Sigma}$ be a germ at $0$ of a closed set. Let $r \in {\mathbb{N}}$, and let $f\in {\mathcal E}_{[r]}(n,p) $, $n\geq p$. Suppose that $j^rf({\Sigma},0)$ has a $C^\omega$ realisation. Then the following conditions are equivalent:
(1) There exists $m\in \mathbb{N}$ such that $I^T_{r}(m)$ holds
(2) For all $m\in \mathbb{N}$, $I^T_{r}(m)$ holds
(3) There exists $m\in \mathbb{N}$ such that $I^K_{r}(m)$ holds
(4) For all $m\in \mathbb{N}$, $I^K_{r}(m)$ holds
<!-- -->
1) It follows from the proof of Theorem \[equivKT\] that the equivalence between conditions $T_{m}$ and $K_{m}$ holds for any $ C^1$ map $f$ in a category where the analytic curve selection lemma is valid.
2) For $X_{1},\ldots,X_{l}\geq 0$ and a positive integer $m \in \mathbb{N}$, we have $$(X_{1}+\ldots+X_{l})^m \thickapprox X_{1}^m+\ldots+X_{l}^m.$$ Therefore we see that $K_{1}\thickapprox T_{1}$ if and only if for any $m\in \mathbb{N},\,\, K_{m}\thickapprox T_{m} .$
[99]{}
\#1
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[^1]: This research is partially supported by the Grant-in-Aid for Scientific Research (No. 26287011) of Ministry of Education, Science and Culture of Japan, and HUTE Short-Term Fellowship Program 2016.
|
---
author:
- Juan Galvis
- Eric Chung
- Yalchin Efendiev
- Wing Tat Leung
bibliography:
- 'references.bib'
- 'references1.bib'
title: 'On overlapping domain decomposition methods for high-contrast multiscale problems'
---
Summary
=======
We review some important ideas in the design and analysis of robust overlapping domain decomposition algorithms for high-contrast multiscale problems and propose a domain decomposition method better performance in terms of the number of iterations. The main novelty of our approaches is the construction of coarse spaces, which are computed using spectral information of local bilinear forms. We present several approaches to incorporate the spectral information into the coarse problem in order to obtain minimal coarse space dimension. We show that using these coarse spaces, we can obtain a domain decomposition preconditioner with the condition number independent of contrast and small scales. To minimize further the number of iterations until convergence, we use this minimal dimensional coarse spaces in a construction combining them with large overlap local problems that take advantage of the possibility of localizing global fields orthogonal to the coarse space. We obtain a condition number close to 1 for the new method. We discuss possible drawbacks and further extensions.
High-contrast problems. Introduction
====================================
The methods and algorithms, discussed in the paper, can be applied to various PDEs, even though we will focus on Darcy flow equations. Given $ { D}\subset \mathbb{R}^2$, $ { f:D \to \mathbb{R}}$ , and $ { g:\partial D\to \mathbb{R}}$, find $u:D\to \mathbb{R}$ such that $${\partial\over\partial x_i} \left( \kappa_{ij} {\partial u\over\partial x_j} \right) = f$$ with a suitable boundary condition, for instance $u=0$ on $\partial D$. The coefficient $\kappa_{ij}(x)=\kappa(x) \delta_{ij}$ represents the permeability of the porous media $D$. We focus on two-levels overlapping domain decomposition and use local spectral information in constructing “minimal” dimensional coarse spaces (MDCS). After some review on constructing MDCS and their use in overlapping domain decomposition preconditioners, we present an approach, which uses MDCS to minimize the condition number to a condition number closer to 1. This approach requires a large overlap (when comparted to coarse-grid size) and, thus, is more efficient for small size coarse grids. We present the numerical results and state our main theoretical result. We assume that there exists $\kappa_{\min}$ and $\kappa_{\max}$ with $
0< \kappa_{\min}\leq \kappa(x)\leq \kappa_{\max}$ for all $x\in D$. [**The coefficient $\kappa$ has a multiscale structure**]{} (significant local variations of $\kappa$ occur across $D$ at different scales). We also assume that [the coefficient $\kappa$ is a high-contrast coefficient]{} (the constrast is $ \eta={\kappa_{\max}}/{\kappa_{\min}}$). We assume that $\eta$ is large compared to the coarse-grid size.
It is well known that performance of numerical methods for high-contrast multiscale problems depends on $\eta$ and local variations of $\kappa$ across $D$. For classical finite element methods, the condition to obtain good approximation results is that the finite element mesh has to be fine enough to resolve the variations of the coefficient $\kappa$. Under these conditions, finite element approximation leads to the solution of very large (sparse) ill-conditioned problems (with the condition number scaling with $h^{-2}$ and $\eta$). Therefore, the performance of solvers depends on $\eta$ and local variations of $\kappa$ across $D$. This was observed in several works, e.g., [@ge09_1; @Graham1; @aarnes][^1].
Let $\mathcal{T}^h$ be a triangulation of the domain $D$, where $h$ is the size of typical element. We consider only the case of discretization by the classical finite element method $V=P_1(\mathcal{T}^h)$ of piecewise (bi)linear functions. Other discretizations can also be considered. The application of the finite element discretization leads to the solution of a very large ill-conditioned system $
Ax=b,
$ where $A$ is roughly of size $h^{-2}$ and the condition number of $A$ scales with $\eta$ and $h^{-2}$. In general, the main goal is to obtain an efficient good approximation of solution $u$. The two main solution strategies are: [**1. Choose $h$ sufficiently small and implement an iterative method**]{}. It is important to implement a preconditioner $M^{-1}$ to solve $M^{-1}Au=M^{-1}b$. Then, it is important to have the condition number of $M^{-1}A$ to be small and bounded independently of physical parameters, e.g., $\eta$ and the multiscale structure of $\kappa$. [**2. Solve a smaller dimensional linear system**]{} ($\mathcal{T}^H$ with [$H>h$]{}[^2] ) so that computations of solutions can be done efficiently. This usually involves the construction of a downscaling operator $R_0$ (from the coarse-scale to fine-scale $v_0\mapsto v$) and an upscaling operator (from fine-scale to coarse-scale, $v\mapsto v_0$) (or similar operators). Using these operators, the linear system $Au=b$ becomes a coarse linear system $A_0 u_0=b_0$ so that $R_0u_0$ or functionals of it can be computed. The main goal of this approach it to obtain a sub-grid capturing such that $|| u- R_0u_0||$ is small.
The rest of the paper will focus on the design of overlapping domain decomposition methods by constructing appropriate coarse spaces. First, we will review existing results, which construct minimal dimensional coarse spaces, such that the condition number of resulting preconditioner is independent of $\eta$. These coarse spaces use local spectral problems to extract the information, which can not be localized. This information is related to high-conductivity channels, which connect coarse-grid boundaries and important in domain decomposition preconditioners and multiscale simulations. Next, using these MDCS and oversampling ideas, we present a “hybrid” domain decomposition approach with a condition number close to 1 by appropriately selecting the oversampling size (i.e., overlapping size). We state our main result, discuss some limitations, and show a numerical example. We compare the results to some existing contrast-independent preconditioners.
Classical overlapping methods. Brief review
===========================================
We start with a non-overlapping decomposition $\{ D_i\}_{i=1}^{N_S}$ of the domain $D$ and obtain an overlapping decomposition $\{ D_i'\}_{i=1}^{N_S}$ by adding a layer of width $\delta$ around each non-overlapping subdomain. Let $A_j$ be the Dirichlet matrix corresponding to the overlapping subdomain $D_j'$. The one level method solves $M_1^{-1}A=M_1^{-1}b$ with $M^{-1}_1\sum_{j=1}^{N_S} R_j({A}_j)^{-1} R_j^T$ and the operators $R_j^T$, $j=1,\dots,N_S$, being the restriction to overlapping subdomain $D_j'$ operator and with the $R_j$ being the extension by zero (outside $D_j'$) operator. We have the bound $\mbox{Cond}(M^{-1}_1A)\leq C\left(1+{1}/{\delta H}\right)$. For high-contrast multiscale problems, it is known that $\displaystyle C \asymp \eta$.
Next, we introduce a coarse space, that is, a subspace $V_0\subset V$ of small dimension (when compared to the fine-grid finite element space V). We consider $A_0$ as the matrix form of the discretization of the equation related to subspace $V_0$. For simplicity of the presentation, let $A_0$ be the Galerkin projection of $A$ on the subspace $V_0$. That is $A_0=R_0AR_0^T$, where $R_0$ is a downscaling operator that converts coarse-space coordinates into fine-grid space coordinates. The two-levels preconditioner uses the coarse space and it is defined by $M^{-1}_2= R_0A_0^{-1}R_0^T+
\sum_{j=1}^{N_S} R_j ({A}_j)^{-1} R_j^T= R_0A_0^{-1}R_0^T+M_1^{-1}$. It is known that $\mbox{Cond}(M^{-1}A)\preceq
\eta
\left(1+{H}/{\delta}\right).$ The classical two-levels method is robust with respect to the number of subdomains but it is not robust with respect to $\eta$. The condition number estimates use Poincaré inequality and a small overlap trick; [@tw]. Without small overlap trick $\mbox{Cond}(M^{-1}A)\preceq \eta(1+H^2/\delta^2)$.
There were several works addressing the performance of classical domain decomposition algorithms for high-contrast problems. Many of these works considered simplified multiscale structures[^3], see e.g., [@tw] for some works by O. Widlund and his collaborators. We also mention the works by Sarkis and his collaborators, where they introduce the assumption of quasi-monotonicity [@MR1367653]. Sarkis also introduced the idea of using “extra” or additional basis functions as well as techniques that construct the coarse spaces using the overlapping decomposition (and not related to a coarse mesh); [@MR2099424]. Scheichl and Graham [@Graham1] and Hou and Aarnes [@aarnes], started a systematic study of the performance of classical overlaping domain decomposition methods for high-contrast problems. In their works, they used coarse spaces constructed using a coarse grid and special basis functions from the family of multiscale finite element methods. These authors designed two-levels domain decomposition methods that were robust (with respect to $\eta$) for special multiscale structures. None of the results available in the literature (before the method in papers [@ge09_1; @ge09_1reduceddim] was introduced) were robust for a coefficient not-aligned with the construction of the coarse space (i.e., not aligned either with the non-overlapping decomposion or the coarse mesh if any), i.e., the condition number of the resulting preconditioner is independent of $\eta$ for general multiscale coefficients.
Stable decomposition and eigenvalue problem. Review
===================================================
A main tool in obtaining condition number bounds is the construction of a stable decomposition of a global field. That is, if for all $v\in V=P^1_0(D,\mathcal{T}^h)$ there exists a decomposition $
v=v_0+\sum_{j=1}^{N_S} v_j$ with $v_0\in V_0$ and $ v_j\in V_j=P^1_0(D_j', \mathcal{T}^h)$, $j=1,\dots, N$, and $$\int_{D} \kappa |\nabla v_0|^2+\sum_{j=1}^{N_S}
\int_{D_j'}\kappa |\nabla v_j|^2 \leq C_0^2\int_{D}\kappa |\nabla v|^2$$ for $C_0>0$. Then, $\mbox{cond}(M_2^{-1}A)\leq c(\mathcal{T}^h,\mathcal{T}^H) C_0^2$. Existence of a suitable coarse interpolation $I_0:V\to V_0=\mbox{span}\{\Phi\}$ implies the stable decomposition above. Usually such stable decomposition is constructed as follows.
For the coarse part of the stable decomposition, we introduce a partition of unity $\{\chi_i\}$ subordinated to the coarse mesh (supp $\chi_i\subset \omega_i$ where $\omega_i$ is the coarse-block neighborhood of the coarse-node $x_i$). We begin by restricting the global field $v$ to $\omega_i$. For each coarse node neighborhood $\omega_i$, we identify local field that will contribute to the coarse space [$ I_0^{\omega_i}v $]{} so that the coarse space will be defined as $V_0= \mbox{Span}\{ \chi_i I_0^{\omega_i}v\}$. We assemble a coarse field as $v_0=I_0v=\sum_{i=1}^{N_S} \chi_i (I_0^{\omega_i}v )$. Note that in each block $ v-v_0=\sum_{i\in K} \chi_i (v- I_0^{\omega_i}v )$.
For the local parts of the stable decomposition, we introduce a partition of unity $\{\xi_j\}$ subordinated to the non-overlapping decomposition (supp $\xi_j\subset D_j'$). The local part of the stable decomposition is defined by $ v_j=\xi_j (v-v_0)$. For instance, to bound the energy of $v_j$, we have in each coarse-block $K$, $$\begin{aligned}
&&\int_{K} \kappa|\nabla v_j|^2
\preceq \int_{K} \kappa|\nabla \xi_j\left(\sum_{i\in K} \chi_i (v- I_0^{\omega_i}v )\right)|^2\\
&\preceq&\sum_{i\in K}\int_{K} \kappa(\xi_j\chi_i)^2|\nabla (v-I_0^{\omega_i}v)|^2
+{ \sum_{x_i\in K} \int_{K} \kappa|\nabla (\xi_j\chi_i)|^2| v-
I_0^{\omega_i}v |^2}.\end{aligned}$$
Adding up over $K$, we obtain, $$\begin{aligned}
\int_{D_j' } \kappa|\nabla v_j|^2 \preceq\sum_{i\in D_j'}\int_{D_j' } \kappa(\xi_j\chi_i)^2|\nabla (v-I_0^{\omega_i}v)|^2\\\
+{ \sum_{x_i\in \omega_j} \int_{D_j'} \kappa|\nabla (\xi_j\chi_i)|^2| v-
I_0^{\omega_i}v |^2}\\\end{aligned}$$ and we would like to bound the last term by $ C{ \int_{D_j' } \kappa |\nabla v|^2}$.
For simplicity of our presentation, we consider the case when the coarse elements coincide with the non-overlapping decomposition subdomains. That is, $D_j'=\omega_j$. In this case, we can replace $\xi$ by $\chi$ and replace $\nabla (\chi^2)$ by $\nabla \chi$ so that we need to bound $\sum_{x_i\in \omega_j} \int_{\omega_j} \kappa|\nabla \chi_i |^2| v-
I_0^{\omega_i}v |^2$. We refer to this design as [**coarse-grid based**]{}.
Similar analysis holds in the case when there is no coarse-grid and the coarse space is spanned by the partition of unity $\{\xi_j\}$. We can replace $\chi$ by $\xi$ and $\nabla (\xi^2)$ by $\nabla \xi$. In general these two partitions are not related (see Sec. \[secabs\]).
We now review the three main arguments to complete the required bound: 1) Poincaré inequality. 2) $L^\infty$ estimates. 3) Eigenvalue problem.
[**1. Poincaré inequality:**]{} Classical analysis uses Poincaré inequality to obtain the required bound above. That is, the inequality $ \frac{1}{H^2} \int_{\omega} (v-\bar{v})^2 \leq C \int_\omega |\nabla v|^2$ to obtain ${ \sum_{x_i\in \omega_j} \int_{\omega_j} \kappa|\nabla \chi_i |^2| v-
I_0^{\omega_i}v |^2}\preceq
{ \frac{1}{H^2} \int_{\omega_i}
\kappa| v- I_0^{\omega_i}v |^2} \preceq
C{ \int_{\omega_i} \kappa |\nabla v|^2}.$ In this case, $I_0^{\omega_i}v$ is the average of $v$ on the subdomain. For the case of high-contrast coefficients, $C$ depends on $\eta$, in general. For quasi-monotonic like coefficient it can be obtained that $C$ is independent of the contrast [@MR1367653]. We also mention [@ge09_1] for the case [ **locally connected high-contrast region**]{}. In this case $I_0^{\omega_i}v$ is a weighted average. From the argument given in [@ge09_1], it was clear that when the high-contrast regions break across the domain, defining only one average was not enough to obtain contrast independent constant in the Poincaré inequality.
[**2. $L^\infty$ estimates:**]{} Other idea is to use an $L^\infty$ estimate of the form $$\begin{aligned}
{ \sum_{x_i\in K}\int_{\omega_i} { \kappa|\nabla \chi_i|^2}| v-
I_0^{\omega_i}v |^2}
\preceq
\sum_{x_i\in K}{ || \kappa|\nabla \chi_i|^2 ||_\infty} { \int_{\omega_i} | v- I_0^{\omega_i}v |^2}.\end{aligned}$$ The idea in [@Graham1; @aarnes] was then to construct partition of unity such that $|| \kappa|\nabla \chi_i|^2 ||_\infty$ is bounded independent of the contrast and then to use classical Poincaré inequality estimates. Instead of minimizing the $L^\infty$, one can intuitively try to minimize $\int_K \kappa|\nabla \chi_i|^2 $. This works well when the multicale structure of the coefficient is confined withing the coarse blocks. For instance, for a coefficient and coarse-grid as depicted in Figure \[inclusions\] (left picture), we have that a two-level domain decomposition method can be proven to be robust with respect the value of the coefficient inside the inclusions. In fact, the coarse space spanned by classical multiscale basis functions with linear boundary conditions ($-\mbox{div}(\kappa \nabla \chi_i)=0$ in $K$ and linear on each edge of $\partial K$) is sufficient and the above proof works. Now consider the coefficient in Figure \[inclusions\] (center picture). For such cases, the boundary condition of the basis functions is important. In these cases, basis functions can be constructed such that the above argument can be carried on. Here, we can use multiscale basis functions with oscillatory boundary condition in its construction[^4].
For the coefficient in Figure \[inclusions\], right figure, the argument above using $L^\infty$ cannot be carried out unless we can work with larger support basis functions (as large as to include the high-contrast channels of the coefficient). If the support of the coarse basis function does not include the high-contrast region, then $|| \kappa|\nabla \chi_i|^2 ||_\infty$ increases with the contrast leading to non-robust two-level domain decomposition methods.
[**3. Eigenvalue problem.**]{} We can write $\displaystyle{ \sum_{x_j \in \omega_i}\int_{\omega_i} \kappa |\nabla \chi_j|^2 |v-
I_0^{\omega_i}v |^2}
\preceq
{ \frac{1}{H^2} \int_{\omega_i}
\kappa|(v- I_0^{\omega_i}v )|^2}\preceq C
{ \int_{\omega_i} \kappa |\nabla v|^2}$, where we need to justify the last inequality with constant independent of the contrast. The idea is then to consider the Rayleigh quotient, $$\mathcal{Q}(v):=\frac{\int_{\omega_i} \kappa |\nabla v|^2}{\int_{\omega_i} \kappa| v |^2}$$ with $v\in P^1(\omega_i)$. This quotient is related to an eigenvalue problem and we can define $ I_0^{\omega_i}v $ to be the projection on low modes of this quotient on $\omega_i$. The associated eigenproblem is given by $-\mbox{div}(k(x)\nabla \psi_\ell)=\lambda_\ell k(x)\psi_i$ in $\omega_i$ with homogeneous Neumann boundary condition for floating subdomains and a mixed homogeneous Neumann-Dirichlet condition for subdomains that touch the boundary. It turns out that the low part of the spectrum can be written as $\lambda_1\leq \lambda_2\leq ...\leq \lambda_L$ $<\lambda_{L+1}\leq ...$ where $\lambda_1 , ..., \lambda_L$ are small, asymptotically vanishing eigenvalues and $\lambda_L$ can be bounded below independently of the contrast. After identifying the local field $I_0^{\omega_i}v$, we then define the coarse space as $V_0=Span\{ I^h\chi_i \psi_j^{\omega_i} \}= Span\{ \Phi_i \}.$
[**Eigenvalue problem with multiscale partition of unity.**]{} Instead of the argument presented earlier, we can include the gradient of the partition of unity in the bounds (somehow similar to the ideas of $L^\infty$ bounds). We then need the following chain of inequalities, $\displaystyle { \int_{\omega_i}\underbrace{
\left( \sum_{x_j \in \omega_i} \kappa|\nabla \chi_j|^2\right)} _{\displaystyle :=H^{-2}\widetilde{\kappa}}|v-
I_0^{\omega_i}v |^2}
=
{ \frac{1}{H^2} \int_{\omega_i}
\widetilde{\kappa}|v- I_0^{\omega_i}v )|^2}
\preceq
{ \int_{\omega_i} \kappa |\nabla v|^2}.$ Here we have to consider Rayleigh quotient $\mathcal{Q}_{ms}(v):=\frac{\int_{\omega_i} \kappa |\nabla v|^2}{\int_{\omega_i} \widetilde{\kappa}| v |^2}$, $v\in P^1(\omega_i)$ and define $ I_0^{\omega_i}v $ as projection on low modes. Additional modes “complement” the initial space spanned by the partition of unity so that the resulting coarse space leads to robust methods with minimal dimension coarse spaces; [@ge09_1reduceddim].\
If we consider the two-level method with the (multiscale) spectral coarse space presented before, then $$\label{b1}
\mbox{cond}(M^{-1}A)\leq C(1+(H/\delta)^2),$$ where $C$ is independent of the contrast if enough eigenfunctions in each node neighborhood are selected for the construction of the coarse spaces. The constant $C$ and the resulting coarse-space dimension depend on the partition of unity (initial coarse-grid representation) used.
Abstract problem eigenvalue problems {#secabs}
------------------------------------
We consider an abstract variational problem, where the global bilinear form is obtained by assembling local bilinear forms. That is $a(u,v)=\sum_K a_K(R_Ku,R_Kv)$, where $a_K(u,v)$ is a bilinear form acting on functions with supports being the coarse block $K$. Define the subdomain bilinear form $a_{\omega_i}(u,v)=\sum_{K\subset \omega_i}a_K(u,v)$. We consider the abstract problem $$a(u,v)=F(v) \quad \mbox{ for all } v\in V.$$ We introduce $\{\chi_j\}$, a partition of unity subordianted to coarse-mesh blocks and $\{\xi_i\}$ a partition of unity subordianted to overlapping decomposition (not necessarily related in this subsection). We also define the “Mass” bilinear form (or energy of cut-off) $m_{\omega_i}$ and the Rayleigh quotient $\mathcal{Q}_{abs}$ by $$\displaystyle m_{\omega_i} (v,v) := \sum_{j\in \omega_i} a(\xi_i\chi_j v,\xi_i\chi_j v)\quad \mbox{ and } \quad \mathcal{Q}_{abs}(v):=\displaystyle \frac{a_{\omega_i} (v,v) }{m_{\omega_i} (v,v) }.$$
For the Darcy problem, we have $ m_{\omega_i} (v,v)
= \sum_{j\in \omega_i} \int_{\omega_i} \kappa |\nabla(\xi_i\chi_j v )|^2
\preceq \int_{\omega_i}\widetilde{ \kappa} |v|^2.$ The same analysis can be done by replacing the partition of unity function by partition of degree of freedom (PDoF). Let $\{\pmb\chi_j\}$ be PDoF subordianted to coarse mesh neighborhood and $\{\pmb\xi_i\}$ be PDoF subordianted to overlapping decomposition. As before, we define the cut-off bilinear form and quotient, $$\displaystyle m_{\omega_i} (v,v) := \sum_{j\in \omega_i} a(\pmb\xi_i\pmb\chi_j v,\pmb\xi_i\pmb\chi_j v) \quad \mbox{ and } \quad \mathcal{Q}_{abs2}(v):=\displaystyle \frac{a_{\omega_i} (v,v) }{m_{\omega_i} (v,v) }.$$ The previous construction alows applying the same design recursively and therefore to use the same ideas in a multilevel method. See [@eglw11].
Generalized Multiscale Finite Element Method (GMsFEM) eigenvalue problem
------------------------------------------------------------------------
We can consider the Rayleigh quotients presented before only in a suitable subspace that allows a good approximation of low modes. We call these subspace the snapshot spaces. Denote by $W_i$ the snapshot space corresponding to subdomain $\omega_i$, then we consider the Rayleigh quotient, $\displaystyle \mathcal{Q}_{gm}(v):= \frac{a_{\omega_i} (v,v) }{m_{\omega_i} (v,v) } \quad
\mbox{with $v\in W_i.$}$ The snapshot space can be obtained by dimension reduction techniques or similar computations. See [@egh12; @calo2016randomized]. For example, we can consider the following simple example. In each subdomain $\omega_i$, $i=1,\dots, N_S$: (1) Generate forcing terms $f_1,f_2,\dots,f_M$ randomly ($\int _{\omega_i}f_\ell=0$); (2) Compute the local solutions $ -\mbox{div}(\kappa \nabla u_\ell )=f_\ell $ with homogeneous Neumann boundary condition; (3) Generate $W_i=\mbox{span}\{ u_\ell \}\cup\{1\}$; (4) Consider $\mathcal{Q}_{gm}$ with $W_i$ in 3 and compute important modes.\
In Table 1, we see the results of using the local eigenvalue problem versus using the GMsFEM eigenvalue problem.
$\eta$ MS Full 8 rand. 15 rand
-------- ----- ------ --------- --------- -- --
$10^6$ 209 35 37 37
$10^9$ 346 38 44 38
: PCG iterations for different values $\eta$. Here $H=1/10$ with $h=1/200$. We use the GMsFEM eigevalue problem with $W_i=V_i$ (full local fine-grid space), column 2; $W_i$ spanned by $8$ random samples, column 4, and $W_i$ spanned by 15 samples, column 5. []{data-label="tab:perm1"}
Constrained coarse spaces, large overlaps, and DD
=================================================
In this section, we introduce a hybrid overlapping domain decomposition preconditioner. We use the coarse spaces constructed in [@chung2017constraint], which rely on minimal dimensional coarse spaces as discussed above. First, we construct local auxiliary basis functions following the minimal dimensional coarse spaces as discussed above. For each coarse-block $K\in \mathcal{T}^H$, we solve the eigenvalue problem with Rayleigh quotient $\mathcal{Q}_{ms}(v):=\frac{\int_{K} \kappa |\nabla v|^2}{\int_{K} \widehat{\kappa}| v |^2}$, where $\widehat{\kappa}=\kappa\sum_{j}|\nabla\chi_{j}|^{2}$. We assume $\lambda^{K}_{1}\leq\lambda^{K}_{2}\leq\dots$ and define the local auxiliary spaces by $$V_{aux}(K)=\text{span}\{\phi_{j}^{K}|1\leq j\leq L_K\} \mbox{ and } V_{aux}=\oplus_{K}V_{aux}(K).$$ Next, define a projection operator $\pi_K$ as the orthogonal projection on $V_{aux}$ with respect to the inner product $\int_K \widehat{\kappa} uv$ and $\pi_D= \oplus_{K}\pi_K$.
Let $K^+$ be obtained by adding $l$ layers of coarse elements to the coarse-block $K$. The coarse-grid multiscale basis $\psi_{j,ms}^{K}\in V(K^{+})=
P^1_0(K^{+})$ solve $$\int_{K^+}\!\!\kappa\nabla \psi_{j,ms}^{K} \nabla v+\int_{K^+}\!\!\!\!
\widehat{\kappa}\pi_D(\psi_{j,ms}^{K})\pi_D(v)=
\int_{K^+} \widehat{\kappa}\phi_{j}^{K} \pi_D (v),\;
\forall v\in V(K^{+}).$$ The coarse-grid multiscale space is defined as $V_{ms}=\text{span}\{\psi_{j,ms}^{(i)}\}.$
Before discussing the method using this coarse-grid space, we introduce some operators. We consider the (coarse solution) operator $A_{0,ms}^{-1}:L^{2}(\Omega)\mapsto V_{ms}$ by $$\int_D \nabla A_{0,ms}^{-1}(u) \nabla v=\int_{\Omega}uv\;\;\mbox{ for all } v\in V_{ms}$$ and the (local solutions) operators $A_{i,ms}^{-1}:L^{2}(\Omega)\mapsto V(\omega_{i}^{+})$ defined by, $$\int_{\omega_i}\kappa \nabla A_{i,ms}^{-1}(u_{i}) \nabla v+
\int_{\omega_i} \widehat{\kappa}\pi(A_{i}^{-1}(u_{i})) \pi(v)=\int_{\omega_{i}}\chi_{i}uv\;\;\mbox{ for all } v\in P^1(\omega_{i}^{+}),$$ where $\omega_{i}^{+}$ is obtained by enlarging $\omega_{i}$ by $k$ coarse-grid layers. Next, we can define the preconditioner[^5] $M$ by $$M^{-1}=(I-A_{0,ms}^{-1}A)\Big(\sum_{i}A_{i,ms}^{-1}\Big)(I-AA_{0,ms}^{-1})+A_{0,ms}^{-1}.$$ Note that this is a hybrid preconditioner as defined in [@tw]. Using some estimates in [@chung2017constraint], we can show the bound of the form, $$\label{b2}
\text{cond}(M^{-1}A)\leq\cfrac{1+C(1+\Lambda^{-1})^{\frac{1}{2}}E^{\frac{1}{2}}\max\{\tilde{\kappa}^{\frac{1}{2}}\}}{1-C(1+\Lambda^{-1})^{\frac{1}{2}}E^{\frac{1}{2}}\max\{\tilde{\kappa}^{\frac{1}{2}}\}}$$ where $E = 3(1+\Lambda^{-1}) \Big (1+(2(1+\Lambda^{-\frac{1}{2}}))^{-1}\Big)^{1-k}$, $C$ is a constant depend on the fine and coarse grid only and $\Lambda = \min_{K}{\lambda^{K}_{L_K+1}}$. See [@chung2017constraint] for the required estimates of the coarse space. The analysis of the local solvers of the hybrid method above will be presented elsewhere due to the page limitation. Here, we metion that the analysis do not use a stable decomposition so, in principle, a new family of robust method can be obtained. Moreover, we see that the condition number is close to $1$ if sufficient number of basis functions are selected (i.e., $\Lambda$ is not close to zero)[^6]. The overlap size usually involves several coarse-grid block sizes and thus, the method is effective when the coarse-grid sizes are small. We comment that taking the generous overlap $\delta=kH/2$ in , we get the bound $C(1+4/k^2)$ with $C$ independent of the contrast. The estimate (\[b2\]), on the other hand, gives a bound close to 1 if the oversampling is sufficiently large (e.g., the number of coarse-grid layers is related to $\log(\eta)$), which is due to the localization of global fields orthogonal to the coarse space.
Next, we present a numerical result and consider a problem with permeability $\kappa$ shown in Fig. \[fig:medium\]. The fine-grid mesh size $h$ and the coarse-grid mesh size are considered as $h=1/200$ and $H=1/20$. In Table \[tab:result\_CEM\], we present the number of iterations for using varying number of oversampling layers $k$ and value of the contrast $\eta$.
![The coarse mesh used in the numerical experiments. We highlight a coarse neighborhood and the results of adding 3 coarse-block layers to this neighborhood.[]{data-label="fig:coarse_mesh"}](mesh.pdf)
![The permeability $\kappa$ used in the numerical experiments. The grey regions indicate high-permeability region of order $\eta$ while the white regions indicates a low (order 1) permeability.[]{data-label="fig:medium"}](medium_2.pdf)
Number basis per $\omega$ k \# iter
--------------------------- --- ---------
3 3 3
3 4 2
3 5 2
3 6 1
: Number of iterations until convergence for the PCG with $H=1/20$, $h=1/200$ and $\text{tol}=1e-8$. Left: different number of oversampleing layers $k$ with $\eta=1e+4$. Right: different values of the contrast $\eta$ with $k=3$.[]{data-label="tab:result_CEM"}
Number basis per $\omega$ $\eta$ \# iter
--------------------------- -------- ---------
3 1e+3 3
3 1e+4 3
3 1e+5 3
: Number of iterations until convergence for the PCG with $H=1/20$, $h=1/200$ and $\text{tol}=1e-8$. Left: different number of oversampleing layers $k$ with $\eta=1e+4$. Right: different values of the contrast $\eta$ with $k=3$.[]{data-label="tab:result_CEM"}
We would like to emphasize that the proposed method has advantages if the coarse mesh size is not very coarse. In this case, the oversampled coarse regions are still sufficiently small and the coarse-grid solves can be relatively expensive. Consequently, one wants to minimize the number of coarse-grid solves in addition to local solves. In general, the proposed approach can be used in a multi-level setup, in particular, at the finest levels, while at the coarsest level, we can use original spectral basis functions proposed in [@ge09_1]. This is object of future research.
Conclusions
===========
In this paper, we give an overview of domain decomposition preconditioners for multiscale high-contrast problems. We emphasize the use of minimal dimensional coarse spaces in order to construct optimal preconditioners with the condition number independent of physical scales (contrast and spatial scales). We discuss various approaches in this direction. Furthermore, using these spaces and oversampling ideas, we design a new preconditioner with significant reduction in the number of iterations until convergence if oversampling regions are large enough (several coarse-grid blocks). We note that when using only minimal dimensional coarse spaces in additive Schwarz preconditioner with standard overlap size, we obtain around $19$ iterations. in the new method, our main goal is to reduce even further the number of iteration due to large coarse problem sizes. We obtained around 3 iteration until convergence for the new approach. A main point of the new methodology is that after removing the channels we are able to localize the remaining multiscale information via oversampling. Other interesting aspect of the new approach is that the bound can be obtained by estimating directly operator norms and do not require a stable decomposition.
[^1]: Due to the page limitation, only a few references are cited throughout.
[^2]: The coarse mesh does not necessarily resolve all the variations of $\kappa$
[^3]: These works usually assume some alignment between the coefficient heterogeneities and the initial non-overlapping decomposition
[^4]: We can include constructions of boundary conditions using $1D$ solution of the problem along the edges. Other choices include basis functions constructed using oversampling regions, energy minimizing partition of unity (global), constructions using limited global information (global), etc.
[^5]: Here we avoid restriction and extension operators for simplicity
[^6]: Having robust condition number close to 1 is important, specially in applications where the elliptic equation needs to be solved many times.
|
---
abstract: 'In this paper we are concerned with fully automatic and locally adaptive estimation of functions in a “signal $+$ noise”-model where the regression function may additionally be blurred by a linear operator, e.g. by a convolution. To this end, we introduce a general class of *statistical multiresolution estimators* and develop an algorithmic framework for computing those. By this we mean estimators that are defined as solutions of convex optimization problems with $\ell_\infty$-type constraints. We employ a combination of the alternating direction method of multipliers with Dykstra’s algorithm for computing orthogonal projections onto intersections of convex sets and prove numerical convergence. The capability of the proposed method is illustrated by various examples from imaging and signal detection.'
address:
- |
Institute for Mathematical Stochastics\
University of G[ö]{}ttingen\
Goldschmidtstra[ß]{}e 7, 37077 G[ö]{}ttingen
- |
Institute for Mathematical Stochastics\
University of G[ö]{}ttingen\
Goldschmidtstra[ß]{}e 7, 37077 G[ö]{}ttingen
- |
Institute for Mathematical Stochastics\
University of G[ö]{}ttingen\
Goldschmidtstra[ß]{}e 7, 37077 G[ö]{}ttingen\
and
- |
Max Planck Institute for Biophysical Chemistry\
Am Fa[ß]{}berg 11, 37077 G[ö]{}ttingen
author:
- Klaus Frick
- Philipp Marnitz
- Axel Munk
bibliography:
- 'literature.bib'
title: 'Statistical Multiresolution Dantzig Estimation in Imaging: Fundamental Concepts and Algorithmic Framework'
---
[^1]
Introduction {#intro}
============
In numerous applications, the relation of observable data $Y$ and the (unknown) signal of interest $u^0$ can be modeled as an inverse linear regression problem. We shall assume that the data $Y = \set{Y_{\vec\nu}}$ is sampled on the equidistant grid $X = \set{1,\dots,m}^d$, with $m,d\in \N$ and that $u^0\in U$ for some linear space $U$, such as the Euclidean space or a Sobolev class of functions. Hence the model can be formalized as $$\label{intro:lineqn}
Y_{\vec\nu} = (Ku^0)_{\vec\nu} + \eps_{\vec\nu},\quad \vec\nu \in X.$$ Here we assume that $\eps = \set{\eps_{\vec\nu}}_{\vec\nu \in X}$ are independent and identically distributed r.v. with $\E{\eps_{\vec\nu}} = 0$ and $\E{\eps_{\vec\nu}^2} = \sigma^2>0$ (white noise). Moreover, $K:U\ra (\R^{m})^d$ denotes a linear operator that encodes the functional relation between the quantities that are accessible by experiment and the underlying signal. Often the operator $K$ does not have a continuous inverse (or its inverse is ill-conditioned in a discrete setting, where $K$ is a matrix), that is estimation of $u^0$ given the data $Y$ is an *ill-posed problem*. As a consequence, estimators for $u^0$ can not be obtained by merely applying the inverse of $K$ to an estimator of $Ku^0$, in general. Instead, more sophisticated *statistical regularization* techniques have to be employed that, loosely speaking, are capable of simultaneously inverting $K$ and solving the regression problem.
The application we primarily have in mind is the reconstruction of low-dimensional signals (e.g. images) $u^0$ which are presumed to exhibit a strong neighborhood structure as it is characteristic for imaging or signal detection problems. These neighborhood relations are often modeled by prior smoothness or structural assumptions on $u^0$ (e.g. on the texture of an image).
The aim of this paper is twofold. First, we will introduce the broad class of *statistical multiresolution estimators (SMRE)*. We claim that numerous regularization techniques, that were recently proposed for different problems in various branches of applied mathematics and statistics, can be considered as special cases of these. Among others, this includes the *Dantzig selector* (see [@frick:BicRitTsy09; @frick:CanTao07; @frick:JamRadLv09] and references therein) that was recently proposed in the context of high dimensional statistics. Our prior focus, however, will be put on imaging problems and it will turn out that the aforementioned neighborhood relations can be modeled within our SMRE framework in a straightforward manner. This will result in *locally adaptive* and *fully automatic* image reconstruction methods.
The high intrinsic structure of the signals that are typically under consideration in imaging is in contrast to the usual situation in high-dimensional statistics. Here $u^0$ is usually assumed to be unstructured but to have a sparse representation with respect to some basis of $U$ (cf. [@frick:Tib94; @frick:CheDonSau01; @frick:CanTao07]). Consequently, the consistent estimation of $u^0$ is realized by minimizing a regularization functional which fosters sparsity, such as the $\ell_1$-norm of the coefficients, subject to an $\ell_\infty$-constraint on the coefficients of the residual, i.e. $$\label{intro:dantzig}
\inf_{u\in U}\norm{u}_1\quad\text{ s.t. }\quad \norm{K^*(Y - Ku)}_\infty\leq q.$$ In order to apply this approach for image reconstruction, two modifications become necessary: Often one aims to minimize other regularization functionals such as the total variation semi-norm (cf. [@frick:MamGee97; @frick:MohBerGolOsh11]) or Sobolev norms, say. Hence, we suggest to replace the $\ell_1$-norm in by a general convex functional $J$ that models the smoothness or texture information of signals or images (cf. [@frick:AujAubBlaCha05; @frick:Mey01]). Furthermore, we relax the $\ell_\infty$-constraint such that neighborhood relations of the image can be taken into account. This generalizes the Dantzig selector to this task in a natural way and obviously increases estimation efficientcy. As we will layout in Paragraph \[intro:algo\], this requires new algorithms to compute efficiently the resulting large scale optimization problem.
Statistical Multiresolution Estimation {#intro:smre}
--------------------------------------
We will now introduce the announced class of estimators. To this end, let $\S$ be some index set and $\mathcal{W} = \set{\omega^S~:~S\in\S}$ be a set of given weight-functions on the grid $X=\set{1,\ldots,m}^d$. A *statistical multiresolution estimator (SMRE)* (or *generalized Dantzig selector*), is defined as a solution of the constrained optimization problem $$\label{intro:smreeqn}
\inf_{u\in U} J(u)\quad\text{ s.t. }\quad \max_{S\in\S}
\abs{\sum_{\vec\nu \in X}
\omega^S_{\vec\nu}\left(\Lambda(Y-Ku)\right)_{\vec\nu}} \leq q.$$ Here, $J:U\ra \R$ denotes a regularization functional that incorporates a priori knowledge on the unknown signal $u^0$ (such as smoothness) and $\Lambda:(\R^m)^d\ra(\R^m)^d$ a possibly non-linear transformation. The constant $q$ can be considered as a universal *regularization parameter* that governs the trade-off between regularity and data-fit of the reconstruction. In most practical situations $q$ is chosen to be the $\alpha$-quantile $q_\alpha$ of the *multiresolution (MR) statistic* $T(\eps)$, where $T:(\R^m)^d\ra \R$ encodes the inequality constraint in , i.e. $$\label{intro:mrstateqn}
T(v) = \max_{S\in\S} \abs{\sum_{\vec\nu \in X}
\omega^S_{\vec\nu}\left(\Lambda(v)\right)_{\vec\nu}},\quad v\in(\R^m)^d.$$ To this end, we assume the distribution of $T(\eps)$ to be (approximately) known. This can either be obtained by simulations or in some cases the limiting distribution can even be derived explicitly. The regularization parameter $q$ then admits a sound statistical interpretation: Each solution $\hat
u_\alpha$ of satisfies $$\Prob\left( J(\hat u_\alpha) \leq J(u^0)\right) \geq \alpha$$ i.e. the estimator $\hat u_\alpha$ is *more regular* (in terms of $J$) than $u^0$ with a probability of at least $\alpha$. To see this simply observe that the true signal $u^0$ satisfies the constraint in with probability at least $\alpha$.
For a given estimator $\hat u$ of $u^0$, the set $\mathcal{W}$ is assumed to be rich enough in order to catch all relevant non-random signals that are visible in the residual $Y - K\hat u$. Then, the average function $$\label{intro:mean}
\mu_{S}(v) = \abs{\sum_{\vec\nu \in X}\omega^S_{\vec\nu}
\left(\Lambda(v)\right)_{\vec\nu}}$$ evaluated at $v = Y-K\hat u$ is supposed to be significantly larger than $q$ for at least one $\omega\in \mathcal{W}$, whenever $Y-K\hat u$ fails to resemble white noise. Put differently, the MR-statistic $T(Y-K\hat u)$ is bounded by $q$, whenever $Y - K\hat u$ is accepted as white noise according to the *resolution* provided by $\mathcal{W}$. In fact, this is a key observations that reveals numerous potential application areas of the estimation method . The examples we have in mind are mainly from *statistical signal detection and imaging*, where the index set $\S$ is typically chosen to be an overlapping (redundant) system of subsets of the grid $X$ and $\omega^S$ is the normalized indicator function on $S\in\S$. Consequently the inequality constraint in guarantees that the residual resembles white noise on all sets $S\in\S$. In other words, the SMRE approach in yields a reconstruction method that *locally adapts the amount of regularization* according to the underlying image features. We illustrate this in Section \[appl\] by various examples.
Summarizing, the optimization problem in amounts to choose the most parsimonious among all estimators $\hat u$ for which the residual $Y - K\hat u$ resembles white noise according to the statistic $T$. If $Y-K\hat u$ contains some non randon signal, i.e. $T(Y - K\hat u)$ is likely to be larger than $q$ and $u$ happens to lie outside the admissible domain of . Thus, the multi-resolution constraint prevents too parsimonious reconstructions due to the minimization of $J$.
Algorithmic Challenges and Related Work
---------------------------------------
### Multiresolution Methods
SMREs and related MR statistics have recently been studied in various contexts. We give a brief (but incomplete) overview.
Classical MR statistics are obtained from the general form in by setting $U = (\R^m)^d$ and $\Lambda = \id$. Moreover, one considers the system $\mathcal{W}$ to contain indicator functions on cubes. To be more precise, define the index set $\S$ to be the system of all $d$-dimensional cubes in $X$ and set $\omega^S =
\chi_S\slash \sqrt{\#S}.$ Then, the MR-statistic in reduces to $$T(v) = \max_{S\in\S}\frac{1}{\sqrt{\#S}}\abs{\sum_{\vec\nu \in S}
v_{\vec\nu}}.$$ This statistic was introduced in [@frick:SieYak00] (called scanning statistic there) in order to detect a signal against a noisy background. It was shown in [@frick:KabMun08] that $$\lim_{m\ra\infty} \frac{T(\eps)}{\sqrt{2d\log m}} = \sigma\quad\text{ a.s.}$$ If the system $\mathcal{S}$ is reduced to the set of all *dyadic* squares, then it was proved in [@frick:HotMarStiDavKabMun12] that (after suitable transformations) $T$ also converges weakly to the Gumbel distribution. There, the authors also established a method for locally adaptive image denoising employing linear diffusion equations with spatially varying diffusivity. SMREs have been studied recently for the case $d=1$ in [@frick:DavKovMei09] and [@frick:BoyKemLieMunWit09], where total-variation penalty and the number of jumps in piecewise constant regression were considered as regularization functional $J$, respectively. In [@frick:FriMarMun10] consistency and convergence rates for SMREs have been studied in a general Hilbert space setting.
SMREs with squared residuals, that is $\Lambda(v)_{\vec\nu} =
v_{\vec\nu}^2$, yield another class of estimators that have attracted much attention. Above all, the situation where $\S$ consists of the single set $X$ and $\omega^X$ is chosen to be the constant $1$ function is of special interest, since then reduces to the *penalized least square estimation*. In particular then can be rewritten into $$\label{intro:penleastsquare}
\inf_{u\in U} J(u) + \lambda \sum_{\vec\nu \in X} (Ku-Y)_{\vec\nu}^2$$ for a suitable multiplier $\lambda > 0$. If $J(u) = \norm{u}_1$ the LASSO estimator will result (cf. [@frick:Tib94]). Recently, also non-trivial choices of $\S$ were considered. In [@frick:BerCasRouSol03] $\S$ is chosen to consist of a partition of $G$ which is obtained beforehand by a Mumford-Shah segmentation. In [@frick:DonHinRin11], a subset $S\subset X$ is fixed and afterwards $\S$ is defined as the collection of all translates of $S$.
In [@frick:DueSpo01] MR-statistics are used for shape-constrained estimation based on testing qualitative hypothesis in nonparametric regression for $d=1$. Here, the weight functions $\omega^S$ incorporate qualitative features such as monotonicity or concavity. Similarly, MR-statistics are used in [@frick:DueWal08] in order to detect locations of local increase and decrease in density estimation. Much in the same spirit is the work in [@frick:DueJoh04] where multiscale sign tests are employed for computing confidence bands for isotonic median curves.
As mentioned previously, the *Dantzig-selector* [@frick:CanTao07] is also covered by the general SMRE framework in . To see this, set $U = \R^p$ (with typically $p\gg m$), $\Lambda = \id$ and define the weights $$\omega^S = K \chi_{S},\quad S\in \S.$$ Then, each solution of can be considered as a generalized Dantzig selector. The matrix $K\in\R^{m\times p}$ in this context is usually interpreted as *design matrix* of a high dimensional linear model. The classical Dantzig selector as introduced in [@frick:CanTao07] then results in the special case where $\S$ only consists of single-elemented subsets of $\set{1,\ldots,p}$ and $J$ is chosen to be the $\ell_1$-regularization functional $$J(u) = \norm{u}_1 = \sum_{i=1}^p \abs{u_i}.$$ Hence LASSO and Dantzig selector are uni-scale estimators which take into account the largest ($\S = \set{X}$) and smallest ($\S$ consists of all singletons in $\set{1,\ldots,p}$) scales, respectively. In this sense, they constitute two extreme cases of SMRE.
### Algorithmic Challenges {#intro:algo}
From a computational point of view, computing an SMRE amounts to solve the *constrained optimization problem* which can be rewritten into $$\label{intro:smreeqnp}
\inf_{u\in U} J(u)\quad\text{ s.t. }\quad \mu_{S}(Y - Ku)
\leq q, \; \forall(S\in\mathcal{S}).$$ We note that in practical applications the number of constraints in , that is the cardinality of the index set $\mathcal{S}$, can be quite large (in Section \[appl:denoising\] denoising of a $512\times
512$ image results in more than $6$ million inequalities). Moreover, the inequalities (even for the simplest case where $\Lambda = \id$) are mutually correlated. Both of these facts turn into a numerically challenging problem and standard approaches (such as interior point or conjugate gradient methods) perform far from satisfactorily.
The authors in [@frick:BerCasRouSol03; @frick:DonHinRin11; @frick:HotMarStiDavKabMun12] approach the numerical solution of by means of an analogon of with spatially dependent multiplier $\lambda\in(\R^m)^d$, i.e. $$\inf_{u\in U} J(u) + \sum_{\vec\nu \in X}\lambda_{\vec\nu}
(Ku-Y)_{\vec\nu}^2.$$ Starting from a (constant) initial parameter $\lambda = \lambda_0$, the parameter $\lambda$ is iteratively adjusted by increasing it in regions which were poorly reconstructed before according to the MR-statistic $T$. This approach strongly depends on the special structure of $\S$ that allows a straightforward identification of each set $S\in\S$ with a unique point in the grid $X$. Put differently, it is not clear how to modify this paradigm in order to solve for highly redundant systems $\S$ as we have it in mind.
Recently a general algorithmic framework was introduced in [@frick:BecCanGra10] for the solutions of large-scale convex cone problems $$\inf_{u\in U} J(u) \quad\text{ s.t. }\quad Ku-Y \in \mathcal{K}$$ where $\mathcal{K}$ is a convex cone in some Euclidean space. The approach was realized in the software package *Templates for First-Order Conic Solvers (TFOCS)* [^2]. The above formulation is very general and in order to recover one has to consider the cone $$\mathcal{K} = \set{(v,q)\in (\R^m)^d\times \R ~:~
\abs{\sum_{\vec\nu\in S} \Lambda(v)_{\vec\nu}} \leq q\;\forall(S\in\S)}$$ The approach in [@frick:BecCanGra10] employs the dual formulation of the problem $$\inf_{\xi \in V} J^*(K^* v) + \inner{Y}{v}\quad\text{ s.t. } v \in \mathcal{K}^*$$ which involves the computation of the dual cone $\mathcal{K}^*$ ($J^*$ denotes the Legendre-Fenchel dual of $J$). This approach is particularly appealing for the uni-scale Dantzig selector since in this situation the cone $\mathcal{K}$ coincides with the epi-graph of the $\ell^\infty$-norm and hence its dual cone is straightforward to compute (it is the epi-graph of the $\ell^1$-norm). As it is argued in [@frick:BecCanGra10], this approach is capable of computing Dantzig selectors for large scale problems in contrast to previous approaches such as standard linear programming techniques [@frick:CanTao07] or homotopy methods such as DASSO [@frick:JamRadLv09] or [@frick:Rom08]. As the authors stress, their approach works well in the case when $\mathcal{K}$ is the epi-graph of a norm for which the projections onto $\mathcal{K}^*$ are tractable and computationally efficient. However, for the applications we have in mind (such as locally adaptive imaging reconstruction), the approach in [@frick:BecCanGra10] is only of limited use: In contrast to the aforementioned epi-graphs, the large number of (strongly dependent) constraints in brings about a cone $\mathcal{K}$ that on the one hand exhibits a tremendous amount of faces compared to the dimension of the image space $\dim(H) = md$ and that on the other hand is no longer symmetric w.r.t. to the $q$-axis. Both of these facts turn the computation of dual cone $\mathcal{K}^*$ (or the projections onto it) into a most challenging problem, even in the simplest case when $\Lambda$ is linear.
The aim of this paper is to develop a general algorithmic framework that makes solutions of numerically accessible for many applications. In order to do so we propose to introduce a slack variable in and then use the *alternating direction method of multipliers*, an Uzawa-type algorithm that decomposes problem into a $J$-penalized least squares problem for the primal variable and a orthogonal projection problem on the feasible set of for the slack variable. This approach has the appealing effect that once an implementation for the projection problem is established, different regularization functionals $J$ can easily be employed without changing the backbone of the algorithm. Our work is much in the same spirit as [@frick:LuPonZha10], which considered an alternating direction method for the computation of the Dantzig selector recently. In this case the computation of the occurring orthogonal projections are available in closed form, whereas in our applications this is not the case due to the aforementioned dependencies.
In order to tackle the orthogonal projection problem we employ Dykstra’s projection method [@frick:BoyDyk86] which is capable of computing the projection onto the intersection of convex bodies by merely using the individual projections onto the latter. The efficiency of the proposed method hence increases considerably if the index set $\S$ can be decomposed into “few" partitions that contain indices of mutually independent inequalities in . In particular, by this approach we will be able to compute classical SMRE (as introduced in [@frick:DavKovMei09; @frick:FriMarMun10]) in $d=2$ space dimensions which to our knowledge has never been done so far. This puts us into the position to study the performance of such estimators compared with other benchmark methods in locally adaptive signal recovery (such as *adaptive weights smoothing* cf. [@frick:PolSpo00]). As it will turn out in Section \[appl\] it will outperform these visually as well as quantitatively.
Organization of the Paper
-------------------------
The paper is organized as follows: In Section \[impl\] we introduce a general algorithmic approach for computing SMREs. We will rewrite into a linearly constrained problem and compute a saddle point of the corresponding augmented Lagrangian by the alternating direction method of multipliers in Paragraph \[impl:deco\]. Under quite general assumption, we prove convergence of the algorithm in Theorem \[impl:alaconv\] and give some qualitative estimates for the iterates in Theorem \[impl:alaconvcor\]. One of the occurring minimization steps amounts to the computation of an orthogonal projection onto a convex set in Euclidean space. In Paragraph \[impl:proj\], this problem will be tackled by means of Dykstra’s projection algorithm introduced in [@frick:BoyDyk86]. Finally, we illustrate the performance of some particular instances of SMREs in Section \[appl\]: we study problems in nonparametric regression, image denoising and deconvolution of fluorescence microscopy images and compare our results to other methods by means of simulations.
Computational Methodology {#impl}
=========================
In this section we will address the question on how to solve the linearly constrained optimization problem . After discussing some notations and basic assumptions in Subsection \[review:assnot\], we will reformulate the problem in Paragraph \[impl:deco\] such that the alternating direction method of multipliers (ADMM), a Uzawa-type algorithm, can be employed as a solution method. As an effect, the task of computing a solution of is replaced by alternating
i) solving an unconstrained penalized least squares problem that is *independent of the MR-statistic $T$* and
ii) computing the orthogonal projection on a convex set in Euclidean space that is *independent of $J$*.
This reveals an appealing modular nature of our approach: The regularization functional $J$ can easily be replaced once a method for the projection problem is settled. For the latter we will propose an iterative projection algorithm in Paragraph \[impl:proj\] that was introduced by Boyle and Dykstra in [@frick:BoyDyk86].
Basic Assumptions and Notation {#review:assnot}
------------------------------
From now on, $X$ will stand for the $d$-dimensional grid $\set{1,\ldots,m}^d$ and agree upon $H = \R^X \simeq (\R^m)^d$ being the space of all real valued functions $v:X\ra \R$. Moreover, we assume that $\S$ denotes some index set and that $\mathcal{W} = \set{\omega^S~:~S\in\S}$ is a collection of elements in $H$. For two elements $v,w \in H$ we will use the standard inner product and norm $$\inner{v}{w} = \sum_{\vec\nu \in X} v_{\vec\nu} w_{\vec\nu}\quad \text{ and
}\quad \norm{v} = \sqrt{\inner{v}{v}}$$ respectively. Next, we assume that $\Lambda:H\ra H$ is continuous such that $\Lambda(0) = 0$ and that for all $S\in\S$ the mapping $$v\mapsto \inner{\omega^S}{\Lambda(v)}$$ is convex. With this notation, we can rewrite the average function in in the compact form $$\mu_{S}(v) = \abs{\inner{w^S}{\Lambda(v)}}.$$ We note, that it is not restricitve to consider more generaly $\Lambda:H\ra
\R^N$ with arbitrary $N\in\N$. This could e.g. be useful for augmenting the constraint set of with further constraints of different type. For the signal and image detection problems as studied in this paper, however, $\Lambda$ is always a pointwise transformation of the residuals. Hence, we will restrict our considerations on the case when $\Lambda:H\ra H$.
Furthermore, we define $U$ to be a separable Hilbert-space with inner product $\inner{\cdot}{\cdot}_U$ and induced norm $\norm{\cdot}_U$. The operator $K:U\ra
H$ is assumed to be linear and bounded and the functional $J:U\ra \R$ is convex and lower semi-continuous, that is $$\set{u_n}_{n\in\N}\subset U\text{ and } \lim_{n\ra\infty} u_n =: u\in U
\quad\Longrightarrow \quad J(u) \leq \liminf_{n\ra\infty}J(u_n).$$ Recall the definition of the MR-statistic in . Throughout this paper we will agree upon the following
\[review:assex\]
i) For all $y\in H$ there exists $u\in U$ such that $T(Ku-y)< q$.
ii) For all $y\in H$ and $c\in \R$ the set $$\set{u\in U ~:~ \max_{S\in \S}\mu_{S}(Ku-y)
+ J(u) \leq c}$$ is bounded.
Under Assumption \[review:assex\] it follows from standard techniques in convex optimization, that a solution of exists. As we will discuss in Section \[impl:deco\] it even follows that a saddle point of the corresponding Lagrangian exists (cf. Theorem \[impl:kktthm\] below). In this context Assumption \[review:assex\] i) is often referred to as *Slater’s constraint qualification* and is for instance satisfied if $K(U)$ is dense in $H$. Moreover, Assumption \[review:assex\] ii) will be needed in order to guarantee convergence of the algorithm for computing such a solution, as it is proposed in the upcoming section. This requirement is fulfilled if $J$ is coercive i.e. $$\lim_{\norm{u}_U\ra\infty} J(u)= \infty.$$ In many applications $U$ is some function space and $J$ a gradient based regularization method, such as the total variation semi-norm (cf. Section \[appl:denoising\]). Then a typical sufficient condition for Assumption \[review:assex\] ii) is that $K$ does not annihilate constant functions.
Alternating Direction Method of Multipliers {#impl:deco}
-------------------------------------------
By introducing a slack variable $v\in H$ we rewrite to the equivalent problem $$\label{impl:linconstr}
\inf_{u\in U, v\in H} J(u) + G(v) \quad\text{ subject to }\quad Ku +
v = Y.$$ Here, $G$ denotes the characteristic function on the feasible region $\mathcal{C}$ of , that is, $$\label{impl:feasible}
\mathcal{C} = \set{v\in H~:~ \mu_{S}(v) \leq q
\;\forall(S\in\S)}\quad \text{ and }\quad G(v) =
\begin{cases}
0 & \text{ if } v \in\mathcal{C} \\
+\infty & \text{ else}.
\end{cases}$$ Note that due to the assumptions on $\Lambda$, the set $\mathcal{C}$ is closed and convex. The technique of rewriting into is referred to as the *decomposition-coordination approach*, see e.g. Fortin & Glowinski [@frick:FG83 Chap. III]. There, Lagrangian multiplier methods are used for solving . To this end, we recall the definition of the *augmented Lagrangian* of Problem , that is $$\label{impl:lagr}
L_\lambda (u,v;p) = \frac{1}{2\lambda} \norm{Ku + v - Y}^2 + J(u) + G(v) -
\inner{p}{Ku + v - Y},\quad \lambda > 0.$$ The name stems from the fact that the ordinary Lagrangian $$L(u,v;p) = J(u) + G(v) - \inner{p}{Ku + v - Y}$$ is augmented by the quadratic penalty term $(2\lambda)^{-1} \norm{Ku + v - Y}^2$ that fosters the fulfillment of the linear constraints in . The *augmented Lagrangian method* consists in computing a saddle point $(\hat u, \hat v, \hat p)$ of $L_\lambda$, that is $$L_\lambda(\hat u,\hat v; p) \leq L_\lambda(\hat u, \hat v; \hat p) \leq
L_\lambda( u, v; \hat p),\quad \forall\left( (u,v,p)\in U\times H\times
H\right)$$ We note that each saddle point $(\hat u, \hat v, \hat p)$ of the augmented Lagrangian $L_\lambda$ is already a saddle point of $L$ and vice versa and that in either case the pair $(\hat u, \hat v)$ is a solution of (and thus $\hat u$ is a desired solution of ). This follows e.g. from [@frick:FG83 Chap 3. Thm. 2.1]. Sufficient conditions for the existence of saddle points are usually harder to come up with. Assumption \[review:assex\] summarizes a standard set of such conditions.
\[impl:kktthm\] Assume that Assumption \[review:assex\] holds. Then, there exists a saddle point $(\hat u, \hat v, \hat p)$ of $L_\lambda$.
According to [@frick:ET76 Chap. III, Prop. 3.1 and Prop. 4.2] a saddle point of $L$ exists, if there is an element $u_0\in U$ such that $G$ is continuous at $Ku_0-Y$ and that $$\label{impl:coercivity}
\lim_{\norm{u}_Q\ra\infty} J(u) + G(Ku-Y) = \infty.$$ According to Assumption \[review:assex\] i) and due to the continuity of $\Lambda$ the first requirement is clearly satisfied. Further, the coercivity assumption is a consequence of Assumption \[review:assex\] ii).
We will use the *Alternating Diretion Method of Multipliers (ADMM)* (cf. Algorithm \[impl:ala\]) as proposed in [@frick:FG83 Chap. III Sec. 3.2] for the computation of a saddle point of $L_\lambda$ (and hence of a solution of ): Successive minimization of the augmented Lagrangian $L_\lambda$ w.r.t. the first and second variable followed by an explicit step for maximizing w.r.t. the third variable is performed. Convergence of this method is established in Theorem \[impl:alaconv\] which is a generalization of [@frick:FG83 Chap. III Thm. 4.1]. We note that the proof, as presented in the Appendix \[app\] allows for *approximate* solution of the individual subproblems. For the sake of simplicity, we present the Algorithm in its exact form.
$Y\in H$ (data), $\lambda > 0$ (step size), $\tau \geq 0$ (tolerance). $(u[\tau],
v[\tau])$ is an approximate solution of computed in $k[\tau]$ iteration steps. $u_0\leftarrow \vec 0_{U}$ and $v_0 = p_0\leftarrow \vec 0_{H}$ $r \leftarrow \norm{Ku_0 + v_0 - Y}$ and $k \leftarrow 0$. $k\leftarrow k+1$. $v_k \leftarrow \tilde v$ where $\tilde v\in \mathcal{C}$ satisfies $$\label{ala:noise}
\norm{\tilde v - (Y + \lambda p_{k-1} - Ku_{k-1}) }^2 \leq \norm{v - (Y
+ \lambda p_{k-1} - Ku_{k-1}) }^2 \quad \forall(v\in \mathcal{C}).$$ $ u_k \leftarrow \tilde u$ where $\tilde u$ satisfies $$\label{ala:primal}
\frac{1}{2}\norm{ K\tilde u - (Y + \lambda p_{k-1} - v_k) }^2 + \lambda
J(\tilde u) \leq \frac{1}{2}\norm{ K u - (Y + \lambda p_{k-1} - v_k) }^2 + \lambda
J( u) \quad\forall(u\in U).$$ $p_k \leftarrow p_{k-1} - (K u_k + v_k -
Y)\slash \lambda$.$r\leftarrow \max(\norm{Ku_k + v_k - Y}, \norm{K(u_k-u_{k-1})})$. $u[\tau] \leftarrow u_k$ and $v[\tau] \leftarrow v_k$ and $k[\tau]
\leftarrow k$.
\[impl:alaconv\] Every sequence $\set{(u_k, v_k)}_{k\geq1}$ that is generated by Algorithm \[impl:ala\] is bounded in $U\times H$ and every weak cluster point is a solution of . Moreover, $$\sum_{k\in\N} \norm{Ku_k + v_k - Y}^2 + \norm{K(u_k - u_{k-1})}^2 < \infty.$$
i) Theorem \[impl:alaconv\] implies, that each weak cluster point of $\set{u_k}_{k\geq1}$ is a solution of . In particular, if the solution $u^\dagger$ of is unique (e.g. if $J$ is strictly convex), then $u_k \rightharpoonup u^\dagger$.
ii) Note in particular that is independent of the choice of $J$, while is independent of the multiresolution statistic being used. This decomposition gives the proposed method a neat modular appeal: once an efficient solution method for the projection problem is established (see e.g. Section \[impl:proj\]), the regularization functional $J$ in can easily be replaced by providing an algorithm for the penalized least squares problem . For most popular choices of $J$, problem is well studied and efficient computational methods are at hand (see [@frick:Vog02] for a extensive collection of algorithms and [@frick:KaiSom05] for an overview on MCMC methods).
For a given tolerance $\tau > 0$, Theorem \[impl:alaconv\] implies that Algorithm \[impl:ala\] terminates and outputs approximate solution $u[\tau]$ and $v[\tau]$ of . However, the breaking condition in Algorithm \[impl:ala\] merely guarantees that the linear constraint in is approximated sufficiently well. Moreover, we know from construction that $v[\tau] \in \mathcal{C}$, which implies $G(v[\tau]) = 0$. So, it remains to evaluate the validity of $u[\tau]$:
\[impl:alaconvcor\] Let $(\hat u, \hat v, \hat p) \in U\times H\times H$ be any saddle point of $L_\lambda$. Moreover, let $\tau > 0$ and $u[\tau]\in U$ be returend by Algorithm \[impl:ala\]. Then, $$0\leq J(u[\tau]) - J(\hat u) - \inner{K^*\hat p}{u[\tau] - \hat u }_U\leq
\tau\left( 6\norm{\hat p} + \frac{4\norm{K\hat
u} + 2\tau}{\lambda} \right) \quad\forall(\tau >
0).$$
The result in Theorem \[impl:alaconvcor\] shows how the accuracy of the approximate solution $u[\tau]$ depends on $\tau$. Moreover, it reveals that choosing a small step size $\lambda$ in Algorithm \[impl:ala\] possibly yields a slow decay of the objective functional $J$. However, it follows from the definition of $L_\lambda$ in that a small value for $\lambda$ fosters the linear constraint in .
\[impl:alaconvcortwo\] Let the assumtions of Theorem \[impl:alaconvcor\] be satisfied. Moreover, assume that $J$ is a quadratic functional, i.e. $J(u) =
\frac{1}{2}\norm{Lu}_V^2$, where $V$ is a further Hilbert-space and $L:U\supset
D \ra V$ is a linear, densely-defined and closed operator. Then $$\norm{L(u[\tau] - \hat u)} = \bigo(\sqrt{\tau}) \quad\forall(\tau > 0).$$
As already mentioned in the introduction, SMRE (i.e. finding solutions of ) reduces to the computation of *Dantzig selectors* for the particular setting $d=1$, $U = \R^p$ (with usually $p\gg m$) and $$J(u) = \norm{u}_{1}.$$ When applying Algorithm \[impl:ala\] the subproblem amounts to compute $$u_k \in \argmin_{u\in \R^p} \frac{1}{2}\norm{Ku - (Y+ \lambda p_{k-1} -
v_k)}^2 + \lambda
\norm{u}_{1}.$$ This is the well known *least absolute shrinkage and selection operator (LASSO)* estimator [@frick:Tib94]. For the classical Dantzig selector, one chooses $\S = \set{1,\ldots,p}$ and defines for $S\in\S$ the weight $\omega^S = K \chi_{\set{S}}$. Hence, the subproblem in this case consists in the orthonormal projection of $Y_k = Y+ \lambda p_{k-1} -
Ku_{k-1}$ onto the set $$\mathcal{C} = \set{v\in\R^m~:~ \abs{\sum_{1\leq j\leq m} \omega^S_j v_j}\leq
q\text{ for }1\leq S\leq p}.$$ The implications of Theorem \[impl:alaconvcor\] in the present case are in general rather weak. If the saddle point $\hat u$ is known to be $S$-sparse and when $K$ restricted to the support of $\hat u$ is injective, then it can be shown that $\abs{u[\tau] - \hat u}_{1} = \bigo(\tau)$.
We finally note that for this particular situation a slightly different decomposition than proposed in is favorable. To be more precise, define $\tilde K = K^T K$ and $\tilde Y = K^T Y$ and consider $$J(u) + \tilde G(v)\quad\text{ subject to }\quad \tilde K u - v = \tilde Y.$$ where $\tilde G$ is the characteristic function on the set $\set{v\in H~:~
\norm{v}_\infty \leq q}$. Algorithm \[impl:ala\] applied to this modified decomposition then results in the ADMM as introduced in [@frick:LuPonZha10]. In this case the projection in step has a closed from.
The Projection Problem {#impl:proj}
----------------------
Algorithm \[impl:ala\] resolves the constrained convex optimization problem into a quadratic program and an unconstrained optimization problem . The quadratic program in the $k$-th step of Algorithm \[impl:ala\] can be written as a projection: $$\label{impl:quadprob}
\inf_{v\in H}\norm{v - Y_k}^2
\quad\text{ subject to } \quad\mu_{S}(v) \leq q\;\forall(s\in \S)$$ where $Y_k = Y + \lambda p_{k-1} -Ku_{k-1}$. We reformulate the side conditions to $$\label{impl:sidecond}
v\in \mathcal{C} = \bigcap_{S\in \S} C_{S} \quad \text{ where
}
\quad C_{S} = \set{v \in H: \mu_{S}(v) \leq q}.$$ The sets $C_{S}$ are closed and convex and problem thus amounts to compute the projection $P_{\mathcal{C}}(Y_k)$ of $Y_k$ onto the intersection $\mathcal{C}$ of closed and convex sets. According to this interpretation, we use Dykstra’s projection algorithm as introduced in [@frick:BoyDyk86] to solve . This algorithm takes an element $v \in H$ and convex sets $D_1,\ldots,D_M \subset H$ as arguments. It then creates a sequence converging to the projection of $v$ onto the intersection of the $D_j$ by successively performing projections onto individual $D_j$’s. To this end, let $P_D(\cdot)$ denote the projection onto $D
\subset H$ and $S_D = P_D - \id$ be the corresponding projection step. Dykstra’s method is summarized in Algorithm \[impl:dyk\].
$h \in H$ (data), $D_1,\ldots,D_M \subset H$ (closed and convex sets) A sequence $\set{h_k}_{k\in\N}$ that converges strongly to $P_\mathcal{D}(h)$ where $\mathcal{D} = \bigcap_{j=1,\ldots,M} D_j$
$h_{0,0} \leftarrow h$ $h_{0,j} \leftarrow P_{D_j}(h_{0,j-1})$ and $Q_{0,j} \leftarrow
S_{D_j}(h_{0,j-1})$ $h_1\leftarrow h_{0,M}$ and $k \leftarrow 1$ $h_{k,0} \leftarrow h_k$ $h_{k,j} \leftarrow P_{D_j}(h_{k,j-1} - Q_{k-1,j})$ and $Q_{k,j}
\leftarrow S_{D_j}(h_{k,j-1} - Q_{k-1,j})$ $h_{k+1} \leftarrow h_{k,M}$ and $k \leftarrow k + 1$
A natural explanation of the algorithm in a primal-dual framework as well as a proof that the sequence $\set{h(M,k)}_{k\in\N}$ converges to $P_{\mathcal{D}}(h)$ in norm can be found in [@frick:GM89; @frick:DeuHun94]. For the case when $\mathcal{D}$ constitutes a polyhedron even explicit error estimates are at hand (cf. [@frick:Xu00]):
Let $\set{h_k}_{k\in\N}$ be the sequence generated by Algorithm \[impl:dyk\] and $P_\mathcal{D}(h)$ be the projection of the input $h$ onto $\mathcal{D}$. Then there exist constants $\rho > 0$ and $0\leq c< 1$ such that for all $k\in\N$ $$\norm{h_k - P_\mathcal{D}(h)} \leq \rho c^k.$$
\[impl:dyk\_conv\] The constant $c$ increases with the number $M$ of convex sets which intersection form the set $D$ that $h$ is to be projected on. The convergence rate therefore improves with decreasing $M$. For further details and estimates for the constants $\rho$ and $c$, we refer to [@frick:Xu00].
Note that application of Dykstra’s algorithm is particularly appealing if the projections $P_{D_j}$ can be easily computed or even stated explicitly, as it is the case in the following examples.
\[impl:projexpllin\] Assume that $\Lambda = \id$. Then the sets $C_{S}$ are the rectangular cylinders $$C_{S} = \set{v\in H~:~ \abs{\inner{\omega^S}{v}}\leq q}.$$ The projection can therefore be explicitly computed as $$%\label{impl:singleprojectionlin}
P_{C_{S}}(v) = \begin{cases}
v -
\sign\left(\inner{\omega^S}{v}\right)\frac{\omega^S}{\norm{\omega^S}}\left(\frac{\abs{\inner{\omega^S}{v}}}{\norm{\omega^S}}
- q \right) &
\text{ if }
\mu_{S}(v) >
q\\ v & \text{ else} \end{cases}.$$ The left image in Figure \[impl:figadmissible\] depicts an example for $\mathcal{C}$ for $H = \R^2$. For a detailed geometric interpretation of the MR-statistic we also refer to [@frick:Mil08].
\[impl:projexplquad\] Assume that $\Lambda(v)_{\vec\nu} = v_{\vec\nu}^2$. Then, it follows that $v\mapsto \inner{\omega^S}{\Lambda(v)}$ is convex if and only if $\omega^S_{\vec\nu}\geq 0$ for all $\vec\nu \in X$. In this case, the sets $C_{S}$ are elliptic cylinders $$C_{S} = \set{v\in H~:~ \sum_{\vec\nu \in X} \omega^S_{\vec\nu}
v_{\vec\nu}^2\leq q}.$$ Moreover, if $\omega^S_{\vec\nu} \in \set{0,1}$ for all $\vec\nu \in X$, then the projection $P_{C_S}$ can be explicitly computed as $$%\label{impl:singleprojectionquad}
P_{C_{S}}(v) = \begin{cases}
\frac{q}{\inner{\omega^S}{\Lambda(v)}} v \chi_{\set{\omega^S = 1}} + v
\chi_{\set{\omega^S=0}} &
\text{ if }
\mu_{S}(v) >
q\\ v & \text{ else} \end{cases}.$$ The right image in Figure \[impl:figadmissible\] depicts an example of $\mathcal{C}$ for $H = \R^2$.
[ ]{}
A first approach to use Dykstra’s algorithm to solve is to set $M = \#\S$ and identify $D_j$ with $C_{S}$ for all $j=1,\ldots,M$. In view of Remark \[impl:dyk\_conv\], however, it is clearly desirable to decrease the number $M$ of convex sets that enter Dykstra’s algorithm. In order to do so, we take a slightly more sophisticated approach than the one just presented. We partition the set $\S$ into $\S_1,\ldots,\S_M$ such that for all $S\not=\tilde S \in
\S_j$ $$\label{impl:orthogonal}
\omega^S \bot\, \omega^{\tilde S} \quad\text{ and }\quad
\frac{\partial}{\partial v_{\vec\nu}}\Lambda(\cdot)_{\tilde{\vec\nu}} \equiv
0,\forall(\vec\nu\in S, \tilde{\vec\nu}\in \tilde S)$$ and regroup $\set{C_{S}}_{s\in \S}$ into $\set{D_1,\ldots,D_M}$ with $$\label{impl:regroup}
D_j = \bigcap_{S \in \S_j} C_{S}.$$ Given the projections $P_{C_{S}}$, the projection onto $D_j$ can be easily computed: For $v\in H$ identify the set $$V_j = \set{S\in \S_j : \mu_{S}(v) > q}$$ of indices for which $v$ violates the side condition and set $$%\label{impl:projection}
P_{D_j}(v) = v- \sum_{S \in V_j} \left(P_{\mathcal{C}_{S}} -
\id\right) v$$
To keep $M$ small, we choose $\S_1 \subset \set{1,\ldots,N}$ as the biggest set such that holds for all $S,\tilde S \in \S_1$. We then choose $\S_2 \subset \S \setminus
\S_1$ with the same property and continue in this way until all indices are utilized. While this procedure does not necessarily result into $M$ being minimal with the desired property, it still yields a distinct reduction of $N$ in many practical situations. We will illustrate this approach for SMREs in imaging in Section \[appl\].
Applications {#appl}
============
In this section we will illustrate the capability of Algorithm \[impl:ala\] for computing SMREs in some practical situations: in Section \[appl:reg\] we will study a simply one-dimensional regression problem as it was also studied in [@frick:DavKovMei09], yet with a different penalty function $J$. In Section \[appl:denoising\] we illustrate how SMREs performs in image denoising. In both cases we compare our results to other methods. Finally, we will apply the SMRE technique to the problem of image deblurring in confocal fluorescence microscopy in Section \[appl:deblurring\].
Before we study the aforementioned examples, we clarify some common notation. We will henceforth assume that $U = H = (\R^m)^d$ with $d=1$ (Section \[appl:reg\]) and $d=2$ (Sections \[appl:denoising\] and \[appl:deblurring\]), respectively. Moreover, we will employ gradient based regularization functionals of the form $$\label{appl:tvpenalty}
J(u) = TV_p(u) := \frac{1}{p}\sum_{\vec\nu\in X} \abs{\D
u_{\vec\nu}}_2^p\quad\text{ with }p\in\set{1,2}$$ where $\abs{\cdot}_2$ is the Euclidean norm in $\R^d$ and $\D$ denotes the forward difference operator defined by $$(\D u_{\vec\nu})_i = \begin{cases}
u_{\vec\nu + \vec {e_i}} - u_{\vec\nu} & \text{ if }
1\leq \nu_i\leq n-1 \\
0 & \text{ else.}
\end{cases}$$
For the case $p = 2$ the minimization problem amounts to solve an implicit time step of the $d$-dimensional diffusion equation with initial value $(Y + \lambda p_{k-1} - v_k)$ and time step size $\lambda$. This can be solved by a simple (sparse) matrix inversion.
For the case $p =1$, $TV_1$ is better known as *total-variation semi-norm*. It was shown in [@frick:MamGee97] (see also [@frick:Gra07] for similar results in the continuous setting) that the *taut-string algorithm* (as introduced in [@frick:DK01]) constitutes an efficient solution method for in the case $d = 1$. In the general case $d\geq 1$, we employ the fixed point approach for solving the Euler-Lagrange equations for described in [@frick:DV97] (see also [@frick:Vog02 Chap. 8.2.4]). We finally note that the functional $TV_1$ fails to be differentiable; a fact that leads to serious numerical problems when trying to compute the Euler-Lagrange conditions for . Hence, we will use in our simulations a regularized version of $TV_1$ defined by $$\label{appl:tvreg}
TV_1^\beta(u) = \sum_{\vec\nu\in X} \sqrt{ (\D u_{\vec\nu})_i^2 + \beta^2}$$ for a small constant $\beta > 0$.
### Evaluation. {#evaluation. .unnumbered}
In order to evaluate the performance of SMREs, we will employ various distance measures between an estimator $\hat u$ and the true signal $u^0$. On the one hand, we will use standard measures such as *mean integrated squared error (MISE)* and the *mean integrated absolute error (MIAE)* which are given by $$\text{MISE} = \E{\frac{1}{m^d}\sum_{\vec\nu\in X} (\hat u_{\vec\nu} -
u^0_{\vec\nu})^2}\quad\text{ and }\quad
\text{MIAE} = \E{\frac{1}{m^d}\sum_{\vec\nu\in X} \abs{\hat u_{\vec\nu} -
u^0_{\vec\nu}}},$$ respectively. On the other hand, we also intend to measure how well an estimator $\hat u$ matches the “smoothness” of the true signal $u^0$, where smoothness is characterized by the regularization functional $J$. To this end, we introduce the *symmetric Bregman divergence* $$D^{\text{sym}}_J(\hat u,u^0) = \frac{1}{m^d}\sum_{\vec\nu\in X} \left(\nabla
J(\hat u)_{\vec\nu} - \nabla J(u^0)_{\vec\nu}\right)\left(\hat
u_{\vec\nu}-u^0_{\vec\nu}\right),$$ where $\nabla J$ denotes the gradient of the regularization functional $J$. Clearly, $D^{\text{sym}}_J(\hat u,u^0)$ is symmetric and since $J$ is assumed to be convex, also non-negative. However, the symmetric Bregman divergence usually does not satisfy the triangle inequality and hence in general does not define a (semi-) metric on $U$ [@frick:Csi91]. The following examples shed some light on how the Bregman divergence incorporates the functional $J$ in order to measures the distance of $\hat u$ and $u^0$.
\[appl:exbreg\] Let $J(u) = TV_p$ as in .
i) If $p=2$ , then $$D^{\text{sym}}_J(\hat u,u^0) = \sum_{\vec\nu\in X} \abs{\D \hat u_{\vec\nu} - \D
u^0_{\vec\nu}}_2^2.$$ In other words, the symmetric Bregman distance w.r.t. to $TV_2$ is the mean squared distance of the *derivatives* of $\hat u$ and $u^0$.
ii) If $p=1$, then $$\begin{split}
D^{\text{sym}}_J(\hat u, u^0) & = \frac{1}{m^2}\sum_{\vec\nu \in X}
\left(\frac{\D\hat u_{\vec\nu}}{\abs{\D \hat u_{\vec\nu}}} - \frac{\D
u^0_{\vec\nu}}{\abs{\D u^0_{\vec\nu}}}\right)\cdot\left(\D
\hat u_{\vec\nu} - \D u^0_{\vec\nu}\right) \\
& = \frac{1}{m^2}\sum_{\vec\nu \in X} \left(\abs{\D \hat u_{\vec\nu}}
+ \abs{\D u^0}\right)\left(1-\frac{\D\hat u_{\vec\nu}}{\abs{\D \hat
u_{\vec\nu}}} \cdot \frac{\D
u^0_{\vec\nu}}{\abs{\D u^0_{\vec\nu}}}\right) \\ & =
\frac{1}{m^2}\sum_{\vec\nu \in X} \left(\abs{\D \hat u_{\vec\nu}} + \abs{\D u^0}\right)(1-\cos \gamma_{\vec\nu}),
\end{split}$$ where $\gamma_{\vec\nu}$ denotes the angle between the level lines of $\hat
u$ and $u^0$ at the point $x_{\vec\nu}$. Put differently, the symmetric Bregman divergence w.r.t the total variation semi-norm $TV_1$ is small if for sufficiently many points $x_{\vec\nu}$ either both $\hat u$ and $u^0$ are constant in a neighborhood of $x_{\vec\nu}$ or the level lines of $\hat u$ and $u^0$ through $x_{\vec\nu}$ are parallel. In practice rather $TV_1^\beta$ in (for a small $\beta >
0$) instead of $TV_1$ is used in order to avoid singularities. Then, the above formulas are slightly more complicated.
We will use the mean symmetric Bregman divergence (MSB) given by $$\text{MSB} = \E{D^{\text{sym}}_J(\hat u,u^0)}$$ as an additional evaluation method. In all our simulations we approximate the expectations above by the empirical means of $500$ trials.
### Comparison with other methods. {#comparison-with-other-methods. .unnumbered}
We will compare the SMREs to other regression methods. Firstly, we will consider estimators obtained by the *global* penalized least squares method: $$\label{appl:rof}
\hat u(\lambda) := \argmin_{u\in H} \frac{1}{2} \sum_{\vec\nu\in
X}(u_{\vec\nu} - Y)^2 + \lambda J(u),\quad \lambda > 0.$$ In particular, we focus on estimators $\hat u (\lambda)$ that are closest (in some sense) to the true function $u^0$. We call such estimators *oracles*. We define the $\Ls{2}$- and Bregman-oracle by $\hat u_{\Ls{2}} = \hat
u(\lambda_2)$ and $\hat u_{\text{B}} = \hat u(\lambda_{\text{B}})$, where $$\lambda_2 := \E{\argmin_{\lambda > 0} \norm{ u^0 - \hat u(\lambda)}}
\quad\text{ and }\quad \lambda_{\text{B}} := \E{\argmin_{\lambda > 0}
D_J^{\text{sym}}(u^0,\hat u(\lambda))}$$ respectively. Of course, oracles are not available in practice, since the true signal $u^0$ is unknown. However, they represent ideal instances within the class of estimators given by that usually perform better than any data-driven parameter choices (such as cross-validation) and hence may serve as a reference.
Secondly, we also compare our approach to *adaptive weights smoothing (AWS)* [@frick:PolSpo00] which constitutes a benchmark technique for data-driven, spatially adaptive regression. We compute these estimators by means of the official R-package[^3] and denote them by $\hat u_{\text{aws}}^{\ker}$, where $\ker\in\set{\text{Gaussian},
\text{Triangle}}$ decodes the shape of the underlying regression kernel.
Non-parametric Regression {#appl:reg}
-------------------------
In this section we apply the SMRE technique to a nonparametric regression problem in $d=1$ dimensions, i.e. the noise model becomes $$\label{appl:regr}
Y_{\vec\nu} = u^0_{\vec\nu} + \eps_{\vec\nu}\quad \vec\nu=1,\ldots,m,$$ where we assume that $\eps_{\vec\nu}$ are independently and normally distributed r.v. with $\E{\eps_\nu} = 0$ and $\E{\eps_\nu^2}=\sigma^2$. The upper left image in Figure \[appl:fig:one\] depicts the true signal $u^0$ (solid line) and the data $Y$, with $m=1024$ and $\sigma = 0.5$. The application we have in mind with this example arises in NMR spectroscopy, where the NMR spectra provide structural information on the number and type of chemical entities in a molecule. In this context, we suggest to choose $J = TV_2$, since the true signal $u^0$ is rather smooth (see [@frick:DavKovMei09] for examples where $J$ is chosen to be the total variation of the first and second derivative).
Finally, we discuss the MR-statistic $T$ in . We choose $\Lambda = \id$ and the index set $\S$ to consist of all discrete intervals with side lengths ranging from $1$ to $100$. For an interval $S\in \S$ we set $\omega^S=(\#S)^{-1\slash 2}\chi_S$. Thus, each SMRE solves the constrained optimization problem $$\label{appl:regrsmre}
\inf_{u\in U} TV_2(u)\quad\text{ s.t. }\quad \frac{1}{\sqrt{\#
S}}\abs{\sum_{\vec\nu\in S} (Y-u)_{\vec\nu}} \leq q\quad\forall(S\in \S).$$ We choose $q$ to be the $\alpha$-quantile of the MR-statistic $T$, that is $$\label{appl:quantile}
q_\alpha = \inf\set{q\in\R ~:~ \Prob\left(\max_{S\in\S}\frac{1}{\sqrt{\#
S}}\abs{\sum_{\vec\nu\in S} \eps_{\vec\nu}}\leq q\right)\geq \alpha }\quad
\alpha\in (0,1).$$ We note that except for few special cases (cf. [@frick:HotMarStiDavKabMun12; @frick:Kab10]) closed form expressions for the distribution of the MT-statistic $T$ are usually not at hand. In practice one rather considers the empirical distribution of $T$ where the variance $\sigma^2$ can be estimated at a rate $\sqrt{md}$ (cf. [@frick:MunBisWagFre05]).
We will henceforth denote by $\hat u_\alpha$ a solution of with $q = q_\alpha$. As argued in Section \[intro:smre\], $\hat u_\alpha$ is smoother (i.e. has smaller value $TV_2$) than the true signal $u^0$ with a probability of at least $\alpha$ while it satisfies the constraint that the multiresolution statistic $T$ does not exceed $q_\alpha$. This is a sound statistical interpretation of the regularization parameter $\alpha$.
### Numerical results and simulations. {#numerical-results-and-simulations. .unnumbered}
In Figure \[appl:fig:one\] the oracles $\hat u_{\Ls{2}}$ and $\hat u_{\text{B}}$, the AWS-estimators $\hat u_{\text{aws}}^{\text{Triangle}}$ and and $\hat u_{\text{aws}}^{\text{Gauss}}$ as well as the SMRE $\hat u_{0.9}$, $\hat
u_{0.75}$ and $\hat u_{0.5}$ are depicted. It is evident that the SMRE matches the smoothness of the true object much better than the other estimators while the essential features of the signal (such as peak location and peak height) are preserved. In particular, almost no additional local extrema are generated by our approach which stays in obvious contrast to the other methods. Moreover, we point out that the SMRE are quite robust w.r.t. the choice of the confidence level $\alpha$.
We verify this behavior by a simulation study in Table \[appl:sim1\]. For different noise levels ($\sigma = 0.1, 0.3$ and $0,5$) we compare the MISE, MIAE and MSB. Additionally, we compute the *mean number of local maxima (MLM)* of $\hat u$ relative to the number of local maxima in $u^0$ (which is $11$). Here $\hat u$ is any of the above estimators. Note that the latter measure (similar to the MSB) takes into account the smoothness of the estimators where a value $\text{MLM}\gg 1$ indicates too many local maxima and hence a lack of regularity whereas $\text{MLM} < 1$ implies severe oversmoothing.
----------------------------------------- ------- ------- ------- -------- ------- ------- ------- --------
MISE MSB MIAE MLM MISE MSB MIAE MLM
$\hat u_{\Ls{2}}$ 0.009 0.008 0.071 11.881 0.046 0.027 0.156 10.915
$\hat u_{\text{B}}$ 0.009 0.007 0.070 11.700 0.048 0.026 0.149 10.359
$\hat u_{\text{aws}}^{\text{Triangle}}$ 0.007 0.007 0.048 2.551 0.040 0.035 0.112 3.053
$\hat u_{\text{aws}}^{\text{Gauss}}$ 0.054 0.040 0.068 1.971 0.062 0.041 0.107 2.230
$\hat u_{0.9}$ 0.008 0.004 0.047 1.336 0.056 0.019 0.127 1.273
$\hat u_{0.75}$ 0.007 0.004 0.044 1.342 0.050 0.018 0.121 1.290
$\hat u_{0.5}$ 0.007 0.004 0.043 1.366 0.046 0.017 0.116 1.290
----------------------------------------- ------- ------- ------- -------- ------- ------- ------- --------
: Simulation studies for one dimensional peak data set.[]{data-label="appl:sim1"}
----------------------------------------- ------- ------- ------- -------
MISE MSB MIAE MLM
$\hat u_{\Ls{2}}$ 0.091 0.037 0.213 9.860
$\hat u_{\text{B}}$ 0.094 0.036 0.206 9.135
$\hat u_{\text{aws}}^{\text{Triangle}}$ 0.078 0.058 0.162 3.141
$\hat u_{\text{aws}}^{\text{Gauss}}$ 0.079 0.043 0.149 2.330
$\hat u_{0.9}$ 0.134 0.034 0.207 1.194
$\hat u_{0.75}$ 0.120 0.032 0.196 1.241
$\hat u_{0.5}$ 0.109 0.030 0.186 1.238
----------------------------------------- ------- ------- ------- -------
: Simulation studies for one dimensional peak data set.[]{data-label="appl:sim1"}
As it becomes apparent from Table \[appl:sim1\], the SMREs are performing similarly well when compared to the reference estimators as far as the standard measures MISE and MIAE are concerned. For small noise levels ($\sigma
= 0.1$) SMREs even prove to be superior. The distance measures MSB and MLM, however, are significantly smaller for SMREs which indicates that these meet the smoothness of the true object $u^0$ much better than the reference estimators (cf. Example \[appl:exbreg\] i)). All in all, the simulation results confirm our visual impressions above.
### Implementation Details. {#implementation-details. .unnumbered}
The current index set $\S$ results in an overall number of constraints in of $$\# \S = \sum_{i=1}^{100} (1024 - i + 1) = 97450.$$ As pointed out in Section \[impl:proj\], the efficiency of Dykstra’s Algorithm can be increased by grouping independent side-conditions, that is side-conditions corresponding to intervals in $\S$ with empty intersection. For example, the system $\mathcal{S}$ can be grouped such that the intersection of the corresponding sets $D_1,\ldots,D_M$ in form $\mathcal{C}$ with $$M = \sum_{i=1}^{100} i = 5050.$$ In all our simulations we set $\tau = 10^{-4}$ and $\lambda = 1.0$ in Algorithm \[impl:ala\] which results in $k[\tau] \approx 100$ iterations and an overall computation time of approximately $20$ minutes for each SMRE. We note, however, that more than $95\%$ of the computation time is needed for the projection step and that a considerable speed up for the latter could be achieved by parallelization.
Image denoising {#appl:denoising}
---------------
In this section we apply the SMRE technique to the problem of image denoising, that is non-parametric regression in $d=2$ dimensions. In other words, we consider the noise model as in Section \[appl:reg\], where the index $\vec\nu$ ranges over the discrete square $\set{1,\ldots,m}^2$. In Figure \[appl:test\_images\] two typical examples for images $u^0$ and noisy observations $Y$ are depicted ($m=512$ and $\sigma=0.1$, where $u^0$ is scaled between $0$ (black) and $1$ (white)).
We will use the total-variation semi-norm $J=TV_1^\beta$ as regularization functional ($\beta = 10^{-8}$). Moreover, we choose $\Lambda$ to be defined as $$\label{appl:lambdasq}
\Lambda(v)_{\vec\nu} = v_{\vec\nu}^2,\quad \forall(\vec\nu\in {1,\ldots,m}^2).$$ The index set $\S$ is defined to be the collection of all discrete squares with side lengths ranging from $1$ to $25$ and we set $\omega^S = c_S\chi_S$ with yet to be defined constants $c_S$. Thus, each SMREs solves the constrained optimization problem $$\label{appl:regrsmre2d}
\inf_{u\in U} TV_1^\beta(u) \quad\text{ s.t. }\quad \sum_{\vec\nu\in S}
c_S (Y-u)_{\vec\nu}^2 \leq q \quad\forall(S\in \S).$$ We agree upon $q = 1$ and specify the constants $c_S$. To this end, compute for $s=1,\ldots,25$ the quantile values $$q_{\alpha,s} = \inf\set{q\in\R ~:~
\Prob\left(\max_{\substack{S\in\S \\ \#S = s}}\sum_{\vec\nu\in S}
\eps_{\vec\nu}^2\leq q\right)\geq 1-\alpha }\quad \alpha\in (0,1)$$ and set $c_S = q_{\alpha,\# S}^{-1}$. In other words, the definition of $c_S$ implies that the true signal $u^0$ satisfies the constraints in *for squares of a fixed side length $s$* with probability at least $\alpha$. We will henceforth denote by $\hat u_\alpha$ a solution of . We remark on this particular choice of the parameters $\omega_S$ below.
### Numerical results and simulations. {#numerical-results-and-simulations.-1 .unnumbered}
In Figures \[appl:resultscamera\] and \[appl:resultsroof\] the oracles $\hat u_{\Ls{2}}$ and $\hat u_{\text{B}}$, the AWS-estimators $\hat
u_{\text{aws}}^{\text{Triangle}}$ and $\hat u_{\text{aws}}^{\text{Gauss}}$ as well as the SMRE $\hat u_{0.9}$ are depicted (for the “cameraman” and “roof” test image respectively). It is rather obvious that the $\Ls{2}$-oracles are not favorable: although relevant details in the image are preserved, smooth parts (as e.g. the sky) still contain random structures. In contrast, the estimator $\hat u_{\text{aws}}^{\text{Gauss}}$ preserves smooth areas but looses essential details. The aws-estimator with triangular kernel performs much better, however, it gives piecewise constant reconstructions of smoothly varying portions of the image, which is clearly undesirable. The SMRE and the Bregman-oracle visually perform superior to the other methods. The good performance of the Bregman-oracle indicates that the symmetric Bregman distance is a good measure for comparing images. In contrast to the Bregman-oracle, the SMRE adapts the amount of smoothing to the underlying image structure: constant image areas are smoothed nicely (e.g. sky portions), while oscillating patterns (e.g. the grass part in the “cameraman” image or the roof tiles in the “roof” image) are recovered.
We evaluate the performance of the SMREs by means of a simulation study. To this end, we compute the MISE, MIAE and MSB and compare these values with the reference estimators. We note, however, that in particular the MISE and MIAE are not well suited in order to measure the distance of images for they are inconsistent with human eye perception. In [@frick:WanBovSheSim04] the *structural similarity index (SSIM)* was introduced for image quality assessment that takes into account luminance, contrast and structure of the images at the same time. We use the author’s implementation [^4] which is normalized such that the SSIM lies in the interval $[-1, 1]$ and is $1$ in case of a perfect match. We denote by MSSIM the empirical mean of the SSIM in our simulations.
In Table \[appl:sim2\] the simulation results are listed. A first striking fact is the good performance of the $\Ls{2}$-oracle w.r.t. the MISE and MIAE which is supposed to imply reconstruction properties superior to the other methods. Keeping in mind the visual comparison in Figures \[appl:resultscamera\] and \[appl:resultsroof\], however, this is rather questionable. On the other hand, it becomes evident that the $\Ls{2}$-oracle has a rather poor performance w.r.t. the MSB which is more suited for measuring image distances. It is therefore remarkable that the SMRE performs equally good as the Bregman-oracle which, in contrast to the SMRE, is not accessible (since $u^0$ is usually unknown). As far as the structural similarity measure MSSIM is concerned our approach proves to be superior to all others. Finally, the simulation results indicate that aws estimation is not favourable for denoising of natural images.
----------------------------------------- -------- -------- -------- -------- -------- -------- -------- --------
MISE MSB MIAE MSSIM MISE MSB MIAE MSSIM
$\hat u_{\Ls{2}}$ 0.0017 0.0314 0.0276 0.7739 0.0029 0.0499 0.0383 0.6700
$\hat u_{\text{B}}$ 0.0023 0.0256 0.0275 0.7995 0.0038 0.0405 0.0391 0.6607
$\hat u_{\text{aws}}^{\text{Triangle}}$ 0.0032 0.0482 0.0308 0.7657 0.0046 0.0702 0.0416 0.6205
$\hat u_{\text{aws}}^{\text{Gauss}}$ 0.0046 0.0470 0.0360 0.7284 0.0053 0.0686 0.0457 0.5668
$\hat u_{0.9}$ 0.0021 0.0252 0.0297 0.8024 0.0033 0.0374 0.0407 0.7003
----------------------------------------- -------- -------- -------- -------- -------- -------- -------- --------
: Simulation studies for the test images “cameraman” and “roof”.[]{data-label="appl:sim2"}
### Notes on the choice of $\Lambda$ and $\omega^S$. {#notes-on-the-choice-of-lambda-and-omegas. .unnumbered}
In general, a proper choice of the transformation $\Lambda$ and of the weight-functions $\omega^S$ can be achieved by including prior structural information on the true image to be estimated. Substantial parts of natural images, such as photographs, consists of oscillating patterns (as e.g. fabric, wood, hair, grass etc.). This becomes obvious in the standard test images depicted in Figure \[appl:test\_images\]. We claim that for signals that exhibit oscillating patterns, a quadratic transformation $\Lambda$ as in is favorable, since it yields (compared to the linear statistic studied in Section \[appl:reg\]) a larger power of the local test statistic on small scales.
In order to illustrate this, we simulate noisy observations $Y$ of the test images $u$ in Figure \[appl:test\_images\] as in with $\sigma
= 0.1$ and compute a *global* estimator $\hat u$ by computing a minimizer of the ROF-functional (with $\lambda = 0.1$). We intend to examine how well over-smoothed regions in $\hat u$ are detected by the MR-statistic $T(Y - \hat u)$ as in with two different average functions (cf. ) $$\mu_{1,S}(v) = \abs{\sum_{\vec\nu \in S} v_{\nu}} \quad\text{ and }\quad
\mu_{2,S}(v) = \sum_{\vec\nu \in S} v_{\nu}^2$$ respectively. For the sake of simplicity we restrict for the moment our considerations on the index set $\S$ of all $5\times 5$ sub-squares in $\set{1,\ldots,m}^2$. In Figure \[appl:local\_means\] the local means $\mu_{i,S}$ of the residuals $v = Y - \hat u$ for the “roof”-image are depicted. To be more precise, the center coordinate of each square $S\in\S$ is colored according to $\mu_{i,S}$, Hence, large values indicate locations where the estimator $\hat u$ is considered over-smoothed according to the statistic. It becomes visually clear that the localization of oversmoothed regions is better for $\mu_{2,S}$. This is a good motivation for incorporating the local means of the squared residuals in the SMRE model .
We finally comment on the choice of $c_S$. Since $\eps_{\vec\nu}$ are independent and normally distributed random variables, the (scaled) average function $$\sigma^{-2}\mu_S(\eps) = \sum_{\vec\nu \in S}
\left(\frac{\eps_{\vec\nu}}{\sigma}\right)^2$$ is $\chi^2$ distributed with $\# S$ degrees of freedom. Note that the distribution of $\sigma^{-2}\mu_S(\eps)$ is identical only for sets $S$ of the same scale $\# S$. As a consequence of this, it is likely that certain scales dominate the supremum in the MR-statistic $T$ which spoils the multiscale properties of our approach. As a way out, we compute normalizing constants *for each scale separately.*
An alternative approach would be to search for transformations that turn $\mu_S(\eps)$ into almost identically distributed random variables. Logarithmic and $p$-root transformations are often employed for this purpose (see e.g. [@frick:HawWix86]). This will be investigated separately.
### Implementation Details. {#implementation-details.-1 .unnumbered}
The current index set $\S$ results in an overall number of constraints in of $$\# \S = \sum_{i=1}^{25} (512-i+1)^2 = 6251300.$$ Again by grouping independent side-conditions, the system $\mathcal{S}$ can be grouped such that the intersection of the corresponding sets $D_1,\ldots,D_M$ in form $\mathcal{C}$ with $$M = \sum_{i=1}^{25} i^2 = 5525.$$ In all our simulations we set $\tau = 10^{-4}$ and $\lambda = 0.25$ in Algorithm \[impl:ala\] which results in $k[\tau] \approx 30$ iterations and a overall computation time of approximately $2$ hours for each SMRE. Hence, parallelization is clearly desirable in this case.
Deconvolution {#appl:deblurring}
-------------
Another interesting class of problems which can be approached by means of SMREs are deconvolution problems. To be more precise, we assume that $K$ is a convolution operator, that is $$(Ku)_{\vec\nu} = (k \ast u)_{\vec\nu} = \sum_{\vec m \in \R^d} k_{\vec\nu - \vec
m} u_{\vec m}$$ where $k$ is a square-summable kernel on the lattice $\Z^d$ and $u\in H$ is extended by zero-padding. We will focus on the situation where $k$ is a circular Gaussian kernel with standard deviation $\sigma$ given by $$\label{appl:defgauss}
k_{\vec\nu} = \frac{1}{(\sqrt{2 \pi} \sigma)^d} e^{-
\frac{\sum_{i=1}^d \nu_i^2}{2 \sigma^2}}.$$ With $Z = Y + \lambda p_{k-1}+v_k$, the primal step in Algorithm \[impl:ala\] amounts to solve $$u_k \leftarrow \argmin_{u\in H} \frac{1}{2}\sum_{\vec\nu\in
X}((Ku)_{\vec\nu} - Z_{\vec\nu})^2 + \lambda J(u),$$ where we choose $J$ to be as in and apply the techniques described in [@frick:Vog02] for the numerical solution.
In order to illustrate the performance of our approach in practical applications, we give an example from confocal microscopy, nowadays a standard technique in fluorescence microscopy (cf. [@frick:Paw06]). When recording images with this kind of microscope, the original object gets blurred by a Gaussian kernel (in first order). The observations (photon counts) can be modeled as a Poisson process, i.e. $$\label{appl:poiss}
Y_{\vec\nu} = \text{Poiss}((Ku^0)_{\vec\nu}),\quad \vec\nu \in X.$$
The image depicted in Figure \[appl:cytodata\] shows a recording of a PtK2 cell taken from the kidney of *potorous tridactylus*. Before the recording, the protein $\beta$-tubulin was tagged with a fluorescent marker such that it can be traced by the microscope. The image in \[appl:cytodata\] shows an area of $18\times 18$ $\tcmu \text{m}^2$ at a resolution of $798\times 798$ pixel. The point spread function of the optical system can be modeled as a Gaussian kernel with full width at half maximum of $230$nm, which corresponds to $\sigma = 4.3422$ in .
Note that does not fall immediately into the range of models covered by . We will adapt the present situation to the SMRE methodology described in Section \[intro\] by standardization and consider instead of the modified problem $$\label{appl:smreeqnpoiss}
\inf_{u\in U} J(u) \quad\text{ s.t.}\quad
T\left(\frac{Y-Ku}{\sqrt{Ku}}\right)\leq 1$$ where the division is understood pointwise. Clearly, the problem of finding a solution of is much more involved than solving for the constraints being *nonconvex*: firstly, the functional $G$ as defined in is nonconvex as a consequence of which the convergence result in Theorem \[impl:alaconv\] does not apply and secondly Dykstra’s projection algorithm as described in Section \[impl:dyk\] cannot be employed.
We propose the following ansatz in order to circumvent this problem: instead of projecting onto the intersection $\mathcal{C}$ of sets $C_S$ as described in , we now project in the $k$-th step of Algorithm \[impl:ala\] onto $$%\label{appl:poisproj}
\mathcal{C}_P[k] = \bigcap_{n=1}^N C_{P,S}[k] \quad \text{where} \quad
C_{P,S}[k] = \set{ v \in H~:~ \mu_{S}\left(v\slash \sqrt{Ku_k}\right)
\leq q}.$$ with a pointwise division by the square root of $K u_k$. Put differently, in the $k$-th step of Algorithm \[impl:ala\] we use the previous estimate $u_k$ of $u^0$ as a *lagged standardization* in order to approximate the constraints in . In fact, we use $\sqrt{\max(Ku_k,
\eps)}$ with a small number $\eps>0$ for standardization, in order to avoid instabilities.
We note that while with this modification Dykstra’s algorithm becomes applicable again, the projection problem now changes in each iteration step of Algorithm \[impl:ala\]. As a consequence, Theorem \[impl:alaconv\] does not hold anymore after this modification, either. So far, we have not come up with a similar convergence analysis.
We compute the SMRE $\hat u_{0.9}$ by employing Algorithm \[impl:ala\] with the modifications described above. As in the denoising examples in Section \[appl:denoising\] the index set $\S$ consists of all squares with the side-lengths $\set{1,\ldots,25}$ and we choose $\omega^S = \chi_S$ and $\Lambda =
\id$. We note, that this results in an overall number of $\#\S = 95~436~200$ inequality constraints. The constant $q$ are chosen as in , where we assume that $\eps_{\vec\nu}$ are independent and standard normally distributed r.v.
In Algorithm \[impl:ala\] we set $\lambda = 0.05$ and compute $100$ steps. We observe that after a few iterations ($\sim 15$) the error $\tau$ falls below $10^{-3}$ and almost stagnates thereafter, which is due to the fact that we do not increase the accuracy in the subroutines for and . Each iteration step in Algorithm \[impl:ala\] approximately takes $10$ minutes, where $90\%$ of the computation time is needed for . The result is depicted in Figure \[appl:cytoresult\].
The benefits of our method are twofold:
i) The amount of regularization is chosen in a *completely automatic way*. The only parameter to be selected is the level $\alpha$ in . Note that the parameter $\lambda$ in Algorithm \[impl:ala\] has no effect on the output (though it has an effect on the number of iterations needed and the numerical stability).
ii) The reconstruction has an appealing locally adaptive behavior which in the present example mainly concerns the gaps between the protein filaments: whereas the marked $\beta$-tubulin is concentrated in regions of basically one scale, the gaps in between actually make up the multiscale nature of the image.
In the present situation we are in the comfortable position to have a reference image at hand by means of which we can evaluate the result of our method: STED (STimulated Emission Depletion) microscopy constitutes a relatively new method, that is capable of recording images at a physically $5$-$10$ times higher resolution as confocal microscopy (see [@frick:Hel94; @frick:Hel07]). Hence a STED image of this object may serve as “gold standard” reference image.
Figure \[appl:cytosted\] depicts a STED recording of the PtK2 cell data set in Figure \[appl:cytodata\]. The comparison of the SMRE $\hat u_{0.9}$ with the STED recording in Figure \[appl:cytocomp\] shows that our SMRE technique chooses a reasonable amount of regularization: no artifacts due to under-regularization are generated and on the other hand almost all relevant geometrical features that are present in the high-resolution STED recording become visible in the reconstruction. In particular, we note that filament bifurcations (one such bifurcation is marked by a black box in Figure \[appl:cytocomp\]) become apparent in our reconstruction that are not visible in the recorded data.
Finally, we mention that aside to standardization, other transformations of the Poisson data could possibly be considered. For example *Anscombe’s transformation* is known to yield reasonable approximations to normality even for low Poisson-intensities and hence has a particular appeal for e.g. microscopy data with low photon-counts. We are currently investigating SMREs that employ Anscombe’s transform, where in particular the arising projection problems are challenging.
Conclusion and Outlook {#outlook}
======================
In this work, we propose a general estimation technique for nonparametric inverse regression problems in the white noise model based on the convex program . It amounts to finding a minimizer of a convex regularization functional $J(u)$ over a set of feasible estimators that satisfy the fidelty condition $T(Y-Ku)\leq q$, where $T$ is assumed to be the maximum over simple convex constraints and $q$ is some quantile of the statistic $T(\eps)$. Any such minimizer we call *statistical multiresolution estimator (SMRE)*. This approach covers well known uni-scale techniques, such as the Dantzig selector, but with a vast field of potentially new application areas, such as locally adaptive imaging. The particular appeal of the multi-scale generalization arises for those situations where a “neighboring relationship” within the signal can be employed to gain additional information by “averaging” neighboring residuals. We demonstrate in various examples that this improvement is drastic.
We approach the numerical solution of the problem by the ADMM (cf. Algorithm \[impl:ala\]) that decomposes the problem into two subproblems: A $J$-penalized least squares problem, independent of $T$, and an orthogonal projection problem onto the feasible set of that is independent of $J$. The first problem is well studied and for most typical choices of $J$ fast and reliable numerical approaches are at hand. The projection problem, however, is computational demanding, in particular for image denoising applications. We propose Dykstra’s cyclic projection method for its approximate solution. Finally, by extensive numerical experiments, we illustrate the performance of our estimation scheme (in nonparametric regression, image denoising and deblurring problems) and the applicability of our algorithmic approach.
Summarizing, this paper is meant to introduce a novel class of statistical estimators, to provide a general algorithmic approach for their numerical computation and to evaluate their performance by numerical simulations. The inherent questions on the asymptotic behaviour of these estimators (such as consistency, convergence rates or oracle inequalities) remain —to a large extent— unanswered. This opens an interesting area for future research.
A first attempt has been made in [@frick:FriMarMun10] where it is assumed that the model space $U\ni u^0$ is some Hilbert-space of real valued functions on some domain $\Omega$ and that $K:U\ra\L{2}$ is linear and bounded. The error model then has to be adapted accordingly. When $Y$ is a Gaussian process on $\L{2}$ with mean $Ku^0$ and variance $\sigma^2>0$, consistency and convergence rates for SMREs as $\sigma\ra 0^+$ have been proved in [@frick:FriMarMun10] for the case when $\Lambda = \id$. However, in order to extend these results to the present setting, one would rather work with a discrete sample of $Ku_0$ on the grid $X$ and then consider the case when the number of observations $N = md$ tends to infinity. The previous analysis in [@frick:FriMarMun10] indicates two major aspects that have to be considered in the asymptotic analysis for SMREs:
(a) As $N\ra\infty$ usually the cardinality of the index set $\mathcal{S}$ (and hence of the set of weight functions $\mathcal{W}$) gets unbounded. Thus, the mutliresolution statistic $T(\eps) = T_N(\eps)$ in is likely to degenerate unless it is properly normalized and $\W$ satisfies some *entropy condition*. In the linear case ($\Lambda = \id$) we utilized a result from [@frick:DueSpo01] that guarantees a.s. boundedness of $T_N(\eps)$.
(b) In order to derive convergence rates (or risk bounds) it is well known that the true signal $u^0$ has to satisfy some apriori regularity conditions. When using general convex regularization functionals $J$, this is usually expressed by the *source condition* $$K^* p^0 \in \partial J(u^0), \text{ for some } p^0 \in \L{2}.$$ Here $K^*$ denotes the adjoint of $K$ and $\partial J$ the (generalized) derivative of $J$. For example, if $J(u) = \frac{1}{2}\norm{u}^2$, then this conditions means that $u^0 \in \ran(K^*)$.
It would be of great interest to transfer and extend the results in [@frick:FriMarMun10] to the present situation. It is to be expected that (a) and (a) above are necessary assumptions for this purpose.
As stressed by the referees, other extensions are of interest and will be postponed to future work. In contrast to imaging, in many other applications the design $X$ is random, rather than fixed. In these situations an obvious way to extend our algorithmic framework would be to select suitable partitions $\S$ according to the design density, i.e. with finer resolution at locations with a high concentration of design points. It also remains an open issue how to extend the SMRE methodology to density estimation rather than regression, in particular in a deconvolution setup. For $d=1$ a first step in this direction has been taken in [@frick:DavKov04] and it will be of great interest to explore whether our approach allow this to be extended to $d\geq 2$.
Acknowledgement {#acknowledgement .unnumbered}
===============
K.F. and A.M. are supported by the DFG-SNF Research Group FOR916 *Statistical Regularization and Qualitative constraints*. P.M is supported by the BMBF project $03$MUPAH$6$ *INVERS*. A.M and P.M. are supported by the SFB755 *Photonic Imaging on the Nanoscale* and the SFB803 *Functionality Controlled by Organization in and between Membranes*.
We thank S. Hell, A. Egner and A. Schoenle (Department of NanoBiophotonics, Max Planck Institute for Biophysical Chemistry, G[ö]{}ttingen) for providing the microscopy data and L. D[ü]{}mbgen (University of Bern) for stimulating discussions. Finally, we are grateful to an associate editor and to an anonymous referee for their helpful comments.
Proofs {#app}
======
In this section we shall give the proofs of Theorems \[impl:alaconv\] and \[impl:alaconvcor\] as well as Corollary \[impl:alaconvcortwo\]. We note, that convergence of Algorithm \[impl:ala\] is a classical subject in optimization theory and a proof can e.g. be found in [@frick:FG83 Chap. III Thm. 4.1]. However, in order to apply these results, it is necessary that certain regularity conditions for $J$ hold, that are not realistic for our purposes (as e.g. in the case of total-variation regularization). The assertions of Theorems \[impl:alaconv\] and \[impl:alaconvcor\] are modifications of the standard results.
Moreover, we will allow for *approximate solution* of the subproblems and . To this end, we rewrite these two subproblems as variational inequalities, i.e. given $(u_{k-1}, v_{k-1},
p_{k-1})$ we find $(u_k, v_k, p_k)$ such that
$$\begin{gathered}
G(v) - G(v_k) + \lambda^{-1}\inner{Ku_{k-1} + v_k - Y - \lambda p_{k-1}}{v -
v_k} \geq -\eps_k,\;\forall v\in H \label{app:noise}\\
J(u) - J(u_k) + \lambda^{-1}\inner{Ku_k
+ v_k - Y - \lambda p_{k-1}}{Ku - Ku_k} \geq
-\delta_k,\;\forall u\in U \label{app:primal}\\
p_k = p_{k-1} - (Ku_k + v_k - Y)\slash \lambda\label{app:dual},\end{gathered}$$
where we assume that $\set{\eps_1,\eps_2,\ldots}$ and $\set{\delta_1,\delta_2,\ldots}$ are given sequences of positive numbers. Note that implies that $G(v_k) = 0$ and hence $v_k\in\mathcal{C}$ and that for $\eps_k = \delta_k = 0$ and are equivalent to and , respectively.
Finally, we remind the reader of the definition of the *subdifferential* (or generalized derivative) $\partial F$ of a convex function $F:V\ra \R$ on a real Hilbert-space $V$: $$\xi\in\partial F(v)\quad\Leftrightarrow\quad F(w)\geq F(v) +
\inner{\xi}{w-v}_V\;\forall(w\in V).$$ If $\xi\in\partial F(v)$, then $\xi$ is called *subgradient* of $F$ at $v$. It follows from [@frick:ET76 Chap III, Prop. 3.1 and Prop. 4.1] that the Lagrangian $L$ (and hence also the augmented Lagrangian $L_\lambda$) has a saddle-point $(\hat u, \hat v, \hat p)\in U\times H\times H$ if and only if $$\label{impl:kkt}
K\hat u + \hat v = Y,\quad K^*\hat p\in \partial J(\hat u)\quad\text{ and
}\quad \hat p\in \partial G(\hat v).$$ We will henceforth assume that $\set{(u_k, v_k, p_k)}_{k\in\N}$ is a sequence generated by iteratively repeating the steps - . Further, we introduce the notation $$\bar u_k := u_k - \hat u,\quad\bar v_k := v_k - \hat v\quad\text{ and }\quad
\bar p_k := p_k - \hat p.$$ We start with the following
\[app:lemma\] For all $k\geq 1$ we have that $$\begin{gathered}
\label{app:aux85}
\left( \norm{\bar p_{k-1}}^2 + \lambda^{-2} \norm{K\bar u_{k-1}}^2 \right) -
\left( \norm{\bar p_k}^2 + \lambda^{-2} \norm{K\bar u_k}^2 \right)
\\ \geq \lambda^{-2}\left( \norm{K\bar u_k + \bar v_k}^2 + \norm{K \bar
u_{k-1} - K \bar u_k}^2\right) - 2\lambda^{-2}(\delta_k + \delta_{k-1}) -
\delta_k - \eps_k\end{gathered}$$
The assertion follows by repeating the steps (5.6)-(5.25) in the proof of [@frick:FG83 Chap. III Thm. 4.1] after replacing (5.9) and (5.10) by and respectively.
We continue with the proof of Theorem \[impl:alaconv\]. More precisely, we prove the following generalized version
\[app:alaconv\] Assume that the sums $\sum_{k=1}^\infty \delta_k$ and $\sum_{k=1}^\infty \eps_k$ are finite. Then, the sequence $\set{(u_k, v_k)}_{k\geq1}$ is bounded in $U\times H$ and every weak cluster point is a solution of . Moreover, $$\sum_{k\in\N} \norm{Ku_k + v_k - Y}^2 + \norm{K(u_k - u_{k-1})}^2 < \infty.$$
Let $k\geq1$ and define $D =
\sum_{k=1}^\infty \delta_k$ and $E = \sum_{k=1}^\infty \eps_k$. Summing up Inequality over $k$ and keeping in mind that $K\bar u_k + \bar v_k = K u_k + v_k -Y$ and $K \bar u_{k-1} - K \bar u_k = K u_{k-1} - K u_k$ shows $$\begin{gathered}
%\label{app:aux9}
\sum_{k=1}^\infty \norm{K u_k + v_k - Y}^2 + \norm{K u_{k-1} - K u_k}^2
\\ \leq \lambda^2\norm{\hat p}^2 + \norm{K \hat u}^2 + (4\lambda^{-2}+1)D + E<
\infty\end{gathered}$$ where we have used that $\bar u_0 = \hat u$ and $\bar p_0 = \hat p$. Furthermore, it follows again from that $$\label{app:aux9}
\norm{\bar p_k}^2 + \lambda^{-2}\norm{K\bar u}^2 \leq \norm{\hat p}^2 +
\lambda^{-2}\norm{K\hat u}^2 + (4\lambda^{-2}+1)D + E < \infty$$ This together with the fact that $\norm{K u_k + v_k - Y} \ra 0$ shows that $$\max( \norm{Ku_k}, \norm{v_k}, \norm{p_k} ) = \bigo(1).$$ Together with the optimality condition for this in turn implies that for an arbitrary $u\in H$ $$\label{app:opt}
J(u_k) \leq J(u) + \lambda^{-1} \inner{K u_k + v_k - Y - \lambda
p_{k-1}}{K u - K u_k} + \delta_k = \bigo(1).$$ Summarizing, we find that $$\max_{S\in \S} \mu_{S}(Ku_k-Y)+J(u_k) \leq \max_{S\in
\S} \norm{\omega^S}\norm{\Lambda(Ku_k-Y)} + J(u_k) \leq c< \infty$$ for a suitably chosen constant $c\in\R$, since $\Lambda$ is supposed to be continuous. Thus, it follows from Assumption \[review:assex\] that $\set{u_k}_{k\in\N}$ is bounded and hence sequentially weakly compact. Now, let $(\tilde u, \tilde v, \tilde p)$ be a weak cluster point of $\set{(u_k,v_k,p_k)}_{k\in\N}$ and recall that $(\hat u, \hat v, \hat p)$ was assumed to be a saddle point of the augmented Lagrangian $L_\lambda$. Setting $u
= \hat u$ in thus results in $$\label{app:aux10}
\begin{split}
J(u_k) & \leq J(\hat u) + \lambda^{-1}\inner{K u_k + v_k - Y}{K\hat u
- K u_k} + \inner{p_{k-1}}{K u_k - K \hat u} + \delta_k\\
& = J(\hat u) +
\inner{p_{k-1}}{K u_k - K \hat u} + \smallo(1)
\end{split}$$ Using the relation $K\hat u + \hat v = Y$ we further find $$\begin{gathered}
\label{app:aux11}
\inner{p_{k-1}}{K u_k - K \hat u} = \inner{p_{k-1}}{K u_k - Y + \hat v} \\
= \inner{p_{k-1}}{K u_k +v_k - Y} - \inner{p_{k-1}}{ v_k - \hat v} =
\smallo(1) - \inner{p_{k-1}}{ v_k - \hat v}\end{gathered}$$ From the definition of $v_k$ in and from the fact that $\hat
v,v_k \in \mathcal{C}$ it follows that $$\inner{Y + \lambda p_{k-1} - (K u_{k-1} + v_k)}{\hat v - v_k} \leq \eps_k$$ which in turn implies that $$\begin{gathered}
\label{app:aux12}
- \inner{p_{k-1}}{ v_k - \hat v} \leq \lambda^{-1} \inner{Y - (K u_{k-1}
+ v_k)}{v_k- \hat v} + \eps_k \\ = \lambda^{-1} \inner{Y - (K u_{k}
+ v_k)}{v_k- \hat v} + \lambda^{-1} \inner{Ku_k - K u_{k-1}}{v_k- \hat v} +
\eps_k = \smallo(1)\end{gathered}$$ Combining , and gives $$\limsup_{k\ra\infty} J(u_k) \leq J(\hat u).$$ Now, choose a subsequence $\set{u_{\rho(k)}}_{k\in\N}$ such that $u_{\rho(k)}
\rightharpoonup \tilde u$. Since $J$ is convex and lower semi-continuous it is also weakly lower semi-continuous and hence the previous estimate yields $$J(\tilde u) \leq\liminf_{k\ra\infty} J(u_{\rho(k)}) \leq J(\hat u).$$ Moreover, we have that $v_{\rho(k)} \in \mathcal{C}$ for all $k\in\N$. Since $\mathcal{C}$ is closed we conclude that $\hat v \in \mathcal{C}$. Since $K\tilde u + \tilde v = Y$ this shows that $(\tilde u, \tilde v)$ solves and thus $J(\tilde u) = J(\hat u)$.
We proceed with the proof of Theorem \[impl:alaconvcor\]. Again we present a generalized version. To this end, let $D$ and $E$ be as in Proposition \[app:alaconv\].
\[app:alaconvcor\] There exists a constant $C = C(\lambda, \hat u, \hat v, \hat p, E, D)$ such that $$0\leq J(u[\tau]) - J(\hat u) - \inner{K^*\hat p}{u[\tau] - \hat u }_U\leq
C \tau + \delta_{k[\tau]} + \eps_{k[\tau]}\quad\forall(\tau >
0).$$
Define $B^2 = (4\lambda^{-1}+1)D + E$. Then it follows from that $$\begin{gathered}
\lambda^{-1}\norm{Ku_k - K\hat u} \leq \norm{\hat p} +
\lambda^{-1}\norm{K\hat u} + B \label{app:aux456} \\
\norm{p_k} \leq 2\norm{\hat p} + \lambda^{-1} \norm{K\hat u} +
B,\label{app:aux789}
\end{gathered}$$ where $(\hat u, \hat v, \hat p)$ is an arbitrary saddle point of $L_\lambda(u,v,p)$. Assume that $\tau > 0$ and that $k = k[\tau]$ is such that $$\max(\norm{K u_k + v_k - Y}, \norm{K u_{k-1} - K u_k}) \leq \tau.$$ Then, it follows from , , and and that $$\label{app:aux123}
\begin{split}
J(u_k) & \leq J(\hat u) + \lambda^{-1}\inner{K u_k + v_k - Y}{K\hat u
- K u_k} + \inner{p_{k-1}}{K u_k - K \hat u} + \delta_k\\
& \leq J(\hat u) + \tau(\norm{\hat p} +
\lambda^{-1}\norm{K\hat u} + B) + \norm{p_{k-1}}\tau + \inner{p_{k-1}}{\hat
v - v_k} + \delta_k \\ & \leq J(\hat u) + \tau(3\norm{\hat p} +
2\lambda^{-1}\norm{K\hat u} + 2B) + 2\lambda^{-1}\tau
\norm{v_k - \hat v} + \delta_k + \eps_k
\end{split}$$ After observing that $Ku_k - K\hat u + v_k -
\hat v = Y$ it follows that $\norm{v_k - \hat v} \leq \tau + \norm{Ku_k -
K\hat u}$ and combining and gives $$J(u_k) \leq J(\hat u) + \tau\left( 5\norm{\hat p} + 4\lambda^{-1}\norm{K\hat
u} + 4B + 2\lambda^{-1}\tau \right) + \delta_k + \eps_k.$$ Now, observe that from the definition of the subgradient and , it follows that $J(u_k) \geq J(\hat u) + \inner{K^*\hat p}{u_k-\hat u}$ and that $\inner{\hat p}{v_k -\hat v} \leq 0$. This and the fact that $K\hat u
+ \hat v = Y$ implies that $$\label{app:aux1000}
\begin{split}
0& \leq J(u_k) -J(\hat u) - \inner{K^*\hat p}{u_k-\hat u} \\
& = J(u_k) -J(\hat u) - \inner{\hat p}{Ku_k + v_k -Y} + \inner{\hat
p}{K\hat u + \hat v - Y} + \inner{\hat p}{v_k - \hat v} \\
& \leq J(u_k) -J(\hat u) + \norm{\hat p} \tau \\
& \leq \tau\left( 6\norm{\hat p} + 4\lambda^{-1}\norm{K\hat
u} + 4B + 2\lambda^{-1}\tau \right) + \delta_k + \eps_k.
\end{split}$$ This together with finally proves the first part of the assertion.
For the case when $D = E = 0$, it is seen from the proof of Proposition that the constant $C$ takes the simple form $$C = \tau\left( 6\norm{\hat p} + \frac{4\norm{K\hat
u} + 2\tau}{\lambda} \right).$$
Assume that $J(u) = \frac{1}{2}\norm{Lu}^2_V$. Then it follows (see e.g. [@frick:FriSch10 Lem. 2.4]) that the subdifferential $\partial J(\hat u)$ consists of the single element $L^*L \hat u$. Hence the extremality relations imply that $K^*\hat p = L^*L \hat u$. Now it is easy to observe that $$J(u_k) - J(\hat u ) - \inner{K^*\hat p}{u_k - \hat u} =
\frac{1}{2}\norm{L(u_k - \hat u)}^2_V.$$
[^1]: Correspondence to frick@math.uni-goettingen.de
[^2]: available at <http://tfocs.stanford.edu/>
[^3]: available at<http://cran.r-project.org/web/packages/aws/index.html>
[^4]: available at <https://www.ece.uwaterloo.ca/~z70wang/research/ssim/>
|
Algorithm and Theoretical Results {#sec:algo and results}
=================================
In this section, we present the new algorithm and its recovery guarantee. For ease of exposition, we assume all matrices are square (i.e., $m=n$), but emphasize that nothing is special about this assumption and all the results can be easily extended to rectangular matrices.
Proposed Algorithm {#subsec:proposed algorithms}
------------------
Alternating projections is a minimization approach that has been successfully used in many fields, including image processing [@wang2008new; @chan2000convergence; @o2007alternating], matrix completion [@keshavan2012efficient; @jain2013low; @hardt2013provable; @tannerwei2016asd], phase retrieval [@netrapalli2013phase; @cai2017fast; @zhang2017phase], and many others [@peters2009interference; @agarwal2014learning; @yu2016alternating; @pu2017complexity]. A non-convex algorithm based on alternating projections, namely AltProj, is presented in [@netrapalli2014non] for RPCA accompanied with a theoretical recovery guarantee. In each iteration, AltProj first updates $\bm{L}$ by projecting $\bm{D}-\bm{S}$ onto the space of rank-$r$ matrices, denoted $\mathcal{M}_r$, and then updates $\bm{S}$ by projecting $\bm{D}-\bm{L}$ onto the space of sparse matrices, denoted $\mathcal{S}$; see the left plot of Figure \[fig:illustration\] for an illustration. Regarding to the implementation of AltProj, the projection of a matrix onto the space of low rank matrices can be computed by the singular value decomposition (SVD) followed by truncating out small singular values, while the projection of a matrix onto the space of sparse matrices can be computed by the hard thresholding operator. As a non-convex algorithm which targets directly, AltProj is computationally much more efficient than solving the convex relaxation problem using semidefinite programming (SDP). However, when projecting $\bm{D}-\bm{S}$ onto the low rank matrix manifold, AltProj requires to compute the SVD of a full size matrix, which is computationally expensive. Inspired by the work in [@vandereycken2013low; @wei2016guarantees_completion; @wei2016guarantees_recovery], we propose an accelerated algorithm for RPCA, coined accelerated alternating projections (AccAltProj), to circumvent the high computational cost of the SVD. The new algorithm is able to reduce the per-iteration computational cost of AltProj significantly, while a theoretical guarantee can be similarly established.
Our algorithm consists of two phases: initialization and projections onto $\mathcal{M}_r$ and $\mathcal{S}$ alternatively. We begin our discussion with the second phase, which is described in Algorithm \[Algo:Algo1\]. For geometric comparison between AltProj and AccAltProj, see Figure \[fig:illustration\].
**Input:** $\bm{D}=\bm{L}+\bm{S}$: matrix to be split; $r$: rank of $\bm{L}$; $\epsilon$: target precision level; $\beta$: thresholding parameter; $\gamma$: target converge rate; $\mu$: incoherence parameter of $\bm{L}$. **Initialization** $k=0$ $\widetilde{\bm{L}}_{k}=\textnormal{Trim}(\bm{L}_{k},\mu)$ $\bm{L}_{k+1}=\mathcal{H}_r(\mathcal{P}_{\widetilde{T}_{k}}(\bm{D}-\bm{S}_{k}))$ $\zeta_{k+1}= \beta\left(\sigma_{r+1}\left(\mathcal{P}_{\widetilde{T}_{k}}(\bm{D}-\bm{S}_{k})\right) + \gamma^{k+1} \sigma_{1}\left(\mathcal{P}_{\widetilde{T}_{k}}(\bm{D}-\bm{S}_{k})\right)\right) $ $\bm{S}_{k+1}=\mathcal{T}_{\zeta_{k+1}}(\bm{D}-\bm{L}_{k+1})$ $k=k+1$ **Output:** $\bm{L}_k$, $\bm{S}_k$
**Input:** $\bm{L}=\bm{U}\bm{\Sigma} \bm{V}^T$: matrix to be trimmed; $\mu$: target incoherence level. $c_{\mu}=\sqrt{\frac{\mu r}{n}}$ $\bm{A}^{(i)}=\min\{1,\frac{c_{\mu}}{\|\bm{U}^{(i)}\|}\}\bm{U}^{(i)}$ $\bm{B}^{(j)}=\min\{1,\frac{c_{\mu}}{\|\bm{V}^{(j)}\|}\}\bm{V}^{(i)}$ **Output:** $\widetilde{\bm{L}}=\bm{A}\bm{\Sigma} \bm{B}$
Let $(\BL_k,\BS_k)$ be a pair of current estimates. At the $(k+1)^{th}$ iteration, AccAltProj first trims $\BL_k$ into an incoherent matrix $\widetilde{\bm{L}}_k$ using Algorithm \[Algo:Trim\]. Noting that $\widetilde{\bm{L}}_k$ is still a rank-$r$ matrix, so its left and right singular vectors define an $(2n-r)r$-dimensional subspace [@vandereycken2013low], $$\label{eq:tangent space tilde k}
\widetilde{T}_k=\{\widetilde{\bm{U}}_k\bm{A}^T+\bm{B}\widetilde{\bm{V}}_k^T ~|~\bm{A},\bm{B}\in\mathbb{R}^{n\times r} \},$$ where $\widetilde{\bm{L}}_k=\widetilde{\bm{U}}_k\widetilde{\bm{\Sigma}}_k\widetilde{\bm{V}}_k^T$ is the SVD of $\widetilde{\bm{L}}_k$[^1]. Given a matrix $\bm{Z}\in\mathbb{R}^{n\times n}$, it can be easily verified that the projections of $\bm{Z}$ onto the subspace $\widetilde{T}_k$ and its orthogonal complement are given by $$\label{eq:projection onto tangent space tilde k}
\mathcal{P}_{\widetilde{T}_k} \bm{Z}=\widetilde{\bm{U}}_k\widetilde{\bm{U}}_k^T\bm{Z}+\bm{Z}\widetilde{\bm{V}}_k\widetilde{\bm{V}}_k^T-\widetilde{\bm{U}}_k\widetilde{\bm{U}}_k^T\bm{Z}\widetilde{\bm{V}}_k\widetilde{\bm{V}}_k^T$$ and $$\label{eq:projection onto perpendicular space tilde k}
(\mathcal{I}-\mathcal{P}_{\widetilde{T}_k}) \bm{Z}=(\bm{I}-\widetilde{\bm{U}}_k\widetilde{\bm{U}}_k^T)\bm{Z}(\bm{I}-\widetilde{\bm{V}}_k\widetilde{\bm{V}}_k^T).$$ As stated previously, AltProj truncates the SVD of $\bm{D}-\bm{S}_k$ directly to get a new estimate of $\BL$. [ In contrast, AccAltProj first projects $\bm{D}-\bm{S}_k$ onto the low dimensional subspace $\widetilde{T}_k$, and then projects the intermediate matrix onto the rank-$r$ matrix manifold $\mathcal{M}_r$ using the truncated SVD.]{} That is, $$\bm{L}_{k+1}=\mathcal{H}_r(\mathcal{P}_{\widetilde{T}_{k}}(\bm{D}-\bm{S}_{k})),$$ where $\mathcal{H}_r$ computes the best rank-$r$ approximation of a matrix, $$\mathcal{H}_r(\bm{Z}):=\bm{Q}\bm{\Lambda}_r\bm{P}^T \textnormal{ where }\bm{Z}=\bm{Q\Lambda P}^T\textnormal{ is its SVD and $[\bm{\Lambda}_r]_{ii}:=\begin{cases} [\bm{\Lambda}]_{ii} & i\leq r\\ 0& \mbox{otherwise}. \end{cases}$}$$ Before proceeding, it is worth noting that the set of rank-$r$ matrices $\mathcal{M}_r$ form a smooth manifold of dimension $(2n-r)r$, and $\widetilde{T}_k$ is indeed the tangent space of $\mathcal{M}_r$ at $\widetilde{\bm{L}}_k$ [@vandereycken2013low]. Matrix manifold algorithms based on the tangent space of low dimensional spaces have been widely studied in the literature, see for example [@ngo2012scaled; @mishra2012riemannian; @vandereycken2013low; @mishra2014r3mc; @mishra2014fixed; @wei2016guarantees_completion; @wei2016guarantees_recovery] and references therein. In particular, we invite readers to explore the book [@absil2009optimization] for more details about the differential geometry ideas behind manifold algorithms.
One can see that a SVD is still needed to obtain the new estimate $\bm{L}_{k+1}$. Nevertheless, it can be computed in a very efficient way [@vandereycken2013low; @wei2016guarantees_completion; @wei2016guarantees_recovery]. Let $(\bm{I}-\widetilde{\bm{U}}_k\widetilde{\bm{U}}_k^T)(\bm{D}-\bm{S}_k)\widetilde{\bm{V}}_k=\bm{Q}_1\bm{R}_1$ and $(\bm{I}-\widetilde{\bm{V}}_k\widetilde{\bm{V}}_k^T)(\bm{D}-\bm{S}_k)\widetilde{\bm{U}}_k=\bm{Q}_2\bm{R}_2$ be the QR decompositions of $(\bm{I}-\widetilde{\bm{U}}_k\widetilde{\bm{U}}_k^T)(\bm{D}-\bm{S}_k)\widetilde{\bm{V}}_k$ and $(\bm{I}-\widetilde{\bm{V}}_k\widetilde{\bm{V}}_k^T)(\bm{D}-\bm{S}_k)\widetilde{\bm{U}}_k$, respectively. Note that $(\bm{I}-\widetilde{\bm{U}}_k\widetilde{\bm{U}}_k^T)(\bm{D}-\bm{S}_k)\widetilde{\bm{V}}_k$ and $(\bm{I}-\widetilde{\bm{V}}_k\widetilde{\bm{V}}_k^T)(\bm{D}-\bm{S}_k)\widetilde{\bm{U}}_k$ can be computed by one matrix-matrix subtraction between an $n\times n$ matrix and an $n\times n$ matrix, two matrix-matrix multiplications between an $n\times n$ matrix and an $n\times r$ matrix, and a few matrix-matrix multiplications between a $r\times n$ and an $n\times r$ or between an $n\times r$ matrix and a $r\times r$ matrix. Moreover, A little algebra gives $$\begin{aligned}
\mathcal{P}_{\widetilde{T}_{k}} (\bm{D}-\bm{S}_k) &= \widetilde{\bm{U}}_k\widetilde{\bm{U}}_k^T(\bm{D}-\bm{S}_k)+(\bm{D}-\bm{S}_k)\widetilde{\bm{V}}_k\widetilde{\bm{V}}_k^T-\widetilde{\bm{U}}_k\widetilde{\bm{U}}_k^T(\bm{D}-\bm{S}_k)\widetilde{\bm{V}}_k\widetilde{\bm{V}}_k^T \cr
&=\widetilde{\bm{U}}_k\widetilde{\bm{U}}_k^T(\bm{D}-\bm{S}_k)(\bm{I}-\widetilde{\bm{V}}_k\widetilde{\bm{V}}_k^T)+(\bm{I}-\widetilde{\bm{U}}_k\widetilde{\bm{U}}_k^T)(\bm{D}-\bm{S}_k)\widetilde{\bm{V}}_k\widetilde{\bm{V}}_k^T+\widetilde{\bm{U}}_k\widetilde{\bm{U}}_k^T(\bm{D}-\bm{S}_k)\widetilde{\bm{V}}_k\widetilde{\bm{V}}_k^T \cr
&=\widetilde{\bm{U}}_k\bm{R}_2^T\bm{Q}_2^T+\bm{Q}_1\bm{R}_1\widetilde{\bm{V}}_k^T+\widetilde{\bm{U}}_k\widetilde{\bm{U}}_k^T(\bm{D}-\bm{S}_k)\widetilde{\bm{V}}_k\widetilde{\bm{V}}_k^T \cr
&=\begin{bmatrix}\widetilde{\bm{U}}_k & \bm{Q}_1\end{bmatrix} \begin{bmatrix} \widetilde{\bm{U}}_k^T(\bm{D}-\bm{S}_k)\widetilde{\bm{V}}_k & \bm{R}_2^T \\ \bm{R}_1 & \bm{0} \end{bmatrix} \begin{bmatrix} \widetilde{\bm{V}}_k^T \\ \bm{Q}_2^T \end{bmatrix} \cr
&:= \begin{bmatrix}\widetilde{\bm{U}}_k & \bm{Q}_1\end{bmatrix} \bm{M}_k \begin{bmatrix} \widetilde{\bm{V}}_k^T \\ \bm{Q}_2^T \end{bmatrix},\end{aligned}$$ where the fourth line follows from the fact $\widetilde{\bm{U}}_k^T\bm{Q}_1=\widetilde{\bm{V}}_k^T\bm{Q}_2=\bm{0}$. Let $\bm{M}_k = \bm{U}_{M_k}\bm{\Sigma}_{M_k}\bm{V}_{M_k}^T$ be the SVD of $\bm{M}_k$, which can be computed using $O(r^3)$ flops since $\bm{M}_k$ is a $2r\times 2r$ matrix. Then the SVD of $\mathcal{P}_{\widetilde{T}_{k}} (\bm{D}-\bm{S}_k)=\widetilde{\bm{U}}_k\widetilde{\bm{\Sigma}}_k\widetilde{\bm{V}}_k^T$ can be computed by $$\widetilde{\bm{U}}_{k+1}=\begin{bmatrix}\widetilde{\bm{U}}_k & \bm{Q}_1\end{bmatrix}\bm{U}_{M_k},\quad \widetilde{\bm{\Sigma}}_{k+1}=\bm{\Sigma}_{M_k},\quad\textnormal{and}\quad \widetilde{\bm{V}}_{k+1}=\begin{bmatrix}\widetilde{\bm{V}}_k & \bm{Q}_2\end{bmatrix}\bm{V}_{M_k}$$ since both the matrices $\begin{bmatrix}\widetilde{\bm{U}}_k & \bm{Q}_1\end{bmatrix}$ and $\begin{bmatrix}\widetilde{\bm{V}}_k & \bm{Q}_2\end{bmatrix}$ are orthogonal. In summary, the overall computational costs of $\mathcal{H}_r(\mathcal{P}_{\widetilde{T}_{k}}(\bm{D}-\bm{S}_{k}))$ lie in one matrix-matrix subtraction between an $n\times n$ matrix and an $n\times n$ matrix, two matrix-matrix multiplications between an $n\times n$ matrix and an $n\times r$ matrix, the QR decomposition of two $n\times r$ matrices, an SVD of a $2r\times 2r$ matrix, and a few matrix-matrix multiplications between a $r\times n$ matrix and an $n\times r$ matrix or between an $n\times r$ matrix and a $r\times r$ matrix, leading to a total of $4n^2r+n^2+O(nr^2+r^3)$ flops. Thus, the dominant per iteration computational complexity of AccAltProj for updating the estimate of $\BL$ is the same as the novel gradient descent based approach introduced in [@yi2016fast]. In contrast, computing the best rank-$r$ approximation of a non-structured $n\times n$ matrix $\bm{D}-\bm{S}_k$ typically costs $O(n^2r)+n^2$ flops with a large hidden constant in front of $n^2r$.
After $\bm{L}_{k+1}$ is obtained, following the approach in [@netrapalli2014non], we apply the hard thresholding operator to update the estimate of the sparse matrix, $$\bm{S}_{k+1}=\mathcal{T}_{\zeta_{k+1}}(\bm{D}-\bm{L}_{k+1}),$$ where the thresholding operator $\mathcal{T}_{\zeta_{k+1}}$ is defined as $$[\mathcal{T}_{\zeta_{k+1}}\bm{Z}]_{ij} =
\begin{cases}
[\bm{Z}]_{ij} & |[\bm{Z}]_{ij}| >\zeta_{k+1}\\
0 & \mbox{otherwise}
\end{cases}$$ for any matrix $\bm{Z}\in\mathbb{R}^{m\times n}$. Notice that the thresholding value of $\zeta_{k+1}$ in Algorithm \[Algo:Algo1\] is chosen as $$\zeta_{k+1}= \beta\left(\sigma_{r+1}\left(\mathcal{P}_{\widetilde{T}_{k}}(\bm{D}-\bm{S}_{k})\right) + \gamma^{k+1} \sigma_{1}\left(\mathcal{P}_{\widetilde{T}_{k}}(\bm{D}-\bm{S}_{k})\right)\right) ,$$ which relies on a tuning parameter $\beta>0$, a convergence rate parameter $0\leq\gamma<1$, and the singular values of $\mathcal{P}_{\widetilde{T}_{k}} (\bm{D}-\bm{S}_k)$. Since we have already obtained all the singular values of $\mathcal{P}_{\widetilde{T}_{k}} (\bm{D}-\bm{S}_k)$ when computing $\bm{L}_{k+1}$, the extra cost of computing $\zeta_{k+1}$ is very marginal. Therefore, the cost of updating the estimate of $\BS$ is very low and insensitive to the sparsity of $\bm{S}$.
In this paper, a good initialization is achieved by two steps of modified AltProj when setting the input rank to $r$, see Algorithm \[Algo:Init1\]. With this initialization scheme, we can construct an initial guess that is sufficiently close to the ground truth and is inside the “basin of attraction” as detailed in the next subsection. Note that the thresholding parameter $\beta_{init}$ used in Algorithm \[Algo:Init1\] is different from that in Algorithm \[Algo:Algo1\].
**Input:** $\bm{D}=\bm{L}+\bm{S}$: matrix to be split; $r$: rank of $\bm{L}$; $\beta_{init}, \beta$: thresholding parameters. $\bm{L}_{-1}=\bm{0}$ $\zeta_{-1} = \beta_{init} \cdot \sigma_1^D$ $\bm{S}_{-1}=\mathcal{T}_{\zeta_{-1}}(\bm{D}-\bm{L}_{-1})$ $\bm{L}_0=\mathcal{H}_r(\bm{D}-\bm{S}_{-1})$ $\zeta_{0} = \beta \cdot \sigma_{1}(\bm{D}-\bm{S}_{-1})$ $\bm{S}_0=\mathcal{T}_{\zeta_{0}}(\bm{D}-\bm{L}_{0})$ **Output:** $\bm{L}_0$, $\bm{S}_0$
Theoretical Guarantee {#subsec:guaranteed results}
---------------------
In this subsection, we present the theoretical recovery guarantee of AccAltProj (Algorithm \[Algo:Algo1\] together with Algorithm \[Algo:Init1\]). The following theorem establishes the local convergence of AccAltProj.
\[thm:local convergence\] Let $\bm{L}\in\mathbb{R}^{n\times n}$ and $\bm{S}\in\mathbb{R}^{n\times n}$ be two symmetric matrices satisfying Assumptions and . If the initial guesses $\bm{L}_0$ and $\bm{S}_0$ obey the following conditions: $$\|\bm{L}-\bm{L}_0\|_2 \leq 8\alpha\mu r \sigma_1^L,\quad
\|\bm{S}-\bm{S}_0\|_\infty \leq \frac{\mu r}{n} \sigma_1^L,\quad \textnormal{and} \quad
supp(\bm{S}_0)\subset \Omega,$$ then the iterates of Algorithm \[Algo:Algo1\] with parameters $\beta =\frac{\mu r}{2n}$ and $\gamma\in\left(\frac{1}{\sqrt{12}},1\right)$ satisfy $$\|\bm{L}-\bm{L}_k\|_2 \leq 8\alpha\mu r \gamma^k\sigma_1^L,\quad
\|\bm{S}-\bm{S}_k\|_\infty \leq \frac{\mu r}{n} \gamma^k\sigma_1^L,\quad \textnormal{and} \quad
supp(\bm{S}_k)\subset \Omega.$$
The next theorem states that the initial guesses obtained from Algorithm \[Algo:Init1\] fulfill the conditions required in Theorem \[thm:local convergence\].
\[thm:initialization bound\] Let $\bm{L}\in\mathbb{R}^{n\times n}$ and $\bm{S}\in\mathbb{R}^{n\times n}$ be two symmetric matrices satisfying Assumptions and , respectively. If the thresholding parameters obey $\frac{\mu r\sigma_1^L}{n\sigma_1^D}\leq\beta_{init}\leq\frac{3\mu r\sigma_1^L}{n\sigma_1^D}$ and $\beta=\frac{{\mu r}}{2n}$, then the outputs of Algorithm \[Algo:Init1\] satisfy $$\|\bm{L}-\bm{L}_0\|_2 \leq 8\alpha\mu r \sigma_1^L,\quad
\|\bm{S}-\bm{S}_0\|_\infty \leq \frac{\mu r}{n} \sigma_1^L,\quad \textnormal{and} \quad
supp(\bm{S}_0)\subset \Omega.$$
The proofs of Theorems \[thm:local convergence\] and \[thm:initialization bound\] are presented in Section \[sec:proofs\]. The convergence of AccAltProj follows immediately by combining the above two theorems together.
For conciseness, the main theorems are stated for symmetric matrices. However, similar results can be established for nonsymmetric matrix recovery problems as they can be cast as problems with respect to symmetric augmented matrices, as suggested in [@netrapalli2014non]. Without loss of generality, assume $dm\leq n < (d+1)m$ for some $d\geq 1$ and construct $\overline{\BL}$ and $\overline{\BS}$ as $$\setstretch{1.5}
\overline{\BL}:=
\begin{bmatrix}\,
\smash{
\underbrace{
\begin{matrix}
\bm{0} &\cdots & \bm{0} \\
\vdots &\ddots &\vdots \\
\bm{0} &\cdots &\bm{0} \\
\BL^T &\cdots &\BL^T
\end{matrix}}_{d \textnormal{ times}} }
\begin{matrix}
~&\BL\\
~&\vdots\\
~&\BL\\
~&\bm{0}
\end{matrix}
\vphantom{
\begin{matrix}
\smash[b]{\vphantom{\Big|}}
0\\\vdots\\0\\0
\smash[t]{\vphantom{\Big|}}
\end{matrix}
}
\,\,\end{bmatrix}
\begin{matrix}
\left.
\vphantom{\begin{matrix}\BL\\\vdots\\\BL\end{matrix}}
\right\rbrace{\scriptstyle {d \textnormal{ times}}}\\~\\
\end{matrix}, \qquad
\overline{\BS}:=
\begin{bmatrix}\,
\smash{
\underbrace{
\begin{matrix}
\bm{0} &\cdots & \bm{0} \\
\vdots &\ddots &\vdots \\
\bm{0} &\cdots &\bm{0} \\
\BS^T &\cdots &\BS^T
\end{matrix}}_{d \textnormal{ times}} }
\begin{matrix}
~&\BS\\
~&\vdots\\
~&\BS\\
~&\bm{0}
\end{matrix}
\vphantom{
\begin{matrix}
\smash[b]{\vphantom{\Big|}}
0\\\vdots\\0\\0
\smash[t]{\vphantom{\Big|}}
\end{matrix}
}
\,\,\end{bmatrix}
\begin{matrix}
\left.
\vphantom{\begin{matrix}\BS\\\vdots\\\BS\end{matrix}}
\right\rbrace{\scriptstyle {d \textnormal{ times}}}\\~\\
\end{matrix}.$$\
\
Then it is not hard to see that $\overline{\BL}$ is $O(\mu)$-incoherent, and $\overline{\BS}$ is $O(\alpha)$-sparse, with the hidden constants being independent of $d$. Moreover, based on the connection between the SVD of the augmented matrix and that of the original one, it can be easily verified that at the $k^{th}$ iteration the estimates returned by AccAltProj with input $\overline{\BD}=\overline{\BL}+\overline{\BS}$ have the form $$\setstretch{1.5}
\overline{\BL}_k=
\begin{bmatrix}\,
\smash{
\underbrace{
\begin{matrix}
\bm{0} &\cdots & \bm{0} \\
\vdots &\ddots &\vdots \\
\bm{0} &\cdots &\bm{0} \\
\BL_k^T &\cdots &\BL_k^T
\end{matrix}}_{d \textnormal{ times}} }
\begin{matrix}
~&\BL_k\\
~&\vdots\\
~&\BL_k\\
~&\bm{0}
\end{matrix}
\vphantom{
\begin{matrix}
\smash[b]{\vphantom{\Big|}}
0\\\vdots\\0\\0
\smash[t]{\vphantom{\Big|}}
\end{matrix}
}
\,\,\end{bmatrix}
\begin{matrix}
\left.
\vphantom{\begin{matrix}\BL_k\\\vdots\\\BL_k\end{matrix}}
\right\rbrace{\scriptstyle {d \textnormal{ times}}}\\~\\
\end{matrix}, \qquad
\overline{\BS}_k=
\begin{bmatrix}\,
\smash{
\underbrace{
\begin{matrix}
\bm{0} &\cdots & \bm{0} \\
\vdots &\ddots &\vdots \\
\bm{0} &\cdots &\bm{0} \\
\BS_k^T &\cdots &\BS_k^T
\end{matrix}}_{d \textnormal{ times}} }
\begin{matrix}
~&\BS_k\\
~&\vdots\\
~&\BS_k\\
~&\bm{0}
\end{matrix}
\vphantom{
\begin{matrix}
\smash[b]{\vphantom{\Big|}}
0\\\vdots\\0\\0
\smash[t]{\vphantom{\Big|}}
\end{matrix}
}
\,\,\end{bmatrix}
\begin{matrix}
\left.
\vphantom{\begin{matrix}\BS_k\\\vdots\\\BS_k\end{matrix}}
\right\rbrace{\scriptstyle {d \textnormal{ times}}}\\~\\
\end{matrix},$$\
\
where $\BL_k,\BS_k$ are the the $k^{th}$ estimates returned by AccAltProj with input $\BD=\bm{L}+\bm{S}$.
Related Work {#subsec:related work}
------------
As mentioned earlier, convex relaxation based methods for RPCA have higher computational complexity and slower convergence rate which are not applicable for high dimensional problems. In fact, the convergence rate of the algorithm for computing the solution to the SDP formulation of RPCA [@candes2011robust; @chandrasekaran2011rank; @xu2010robust] is sub-linear with the per iteration computational complexity being $O(n^3)$. By contrast, AccAltProj only requires $O(\log(1/\epsilon))$ iterations to achieve an accuracy of $\epsilon$, and the dominant per iteration computational cost is $O(rn^2)$.
There have been many other algorithms which are designed to solve the non-convex RPCA problem directly. In [@WSLer2013], an alternating minimization algorithm was proposed for based on the factorization model of low rank matrices. However, only convergence to fixed points was established there. In [@gu2016low], the authors developed an alternating minimization algorithm for RPCA, which allows the sparsity level $\alpha$ to be $O(1/(\mu^{2/3}r^{2/3}n))$ for successful recovery, which is more stringent than our result when $r\ll n$. A projected gradient descent algorithm was proposed in [@chen2015fast] for the special case of positive semidefinite matrices based on the $\ell_1$-norm of each row of the underlying sparse matrix, which is not very practical.
In Table \[tab:algo compare\], we compare AccAltProj with the other two competitive non-convex algorithms for RPCA: AltProj from [@netrapalli2014non] and non-convex gradient descent (GD) from [@yi2016fast]. GD attempts to reconstruct the low rank matrix by minimizing an objective function which contains the prior knowledge of the sparse matrix. The table displays the computational complexity of each algorithm for updating the estimates of the low rank matrix and the sparse matrix, as well as the convergence rate and the theoretical tolerance for the number of non-zero entries in the sparse matrix.
From the table, we can see that AccAltProj achieves the same linear convergence rate as AltProj, which is faster than GD. Moreover, AccAltProj has the lowest per iteration computational complexity for updating both the estimates of $\BL$ and $\BS$ (ties with AltProj for updating the sparse part). It is worth emphasizing that the acceleration stage in AccAltProj which first projects $\bm{D}-\bm{S}_k$ onto a low dimensional subspace reduces the computational cost of the SVD in AltProj dramatically. Overall, AccAltProj will be substantially faster than AltProj and GD, as confirmed by our numerical simulations in next section. The table also shows that the theoretical sparsity level that can be tolerated by AccAltProj is lower than that of GD and AltProj. Our result looses an order in $r$ because we have replaced the spectral norm by the Frobenius norm when considering the reduction of the reconstruction error in terms of the spectral norm. In addition, the condition number of the target matrix appears in the theoretical result because the current version of AccAltProj deals with the fixed rank case which requires the initial guess is sufficiently close to the target matrix for the theoretical analysis. Nevertheless, we note that the sufficient condition regarding to $\alpha$ to guarantee the exact recovery of AccAltProj is highly pessimistic when compared with its empirical performance. Numerical investigations in next section show that AccAltProj can tolerate as large $\alpha$ as AltProj does under different energy levels.
[ |c||c|c|c|c| ]{} Algorithm & AccAltProj & AltProj & GD
Updating $\bm{S}$ & $\bm{O\left(n^2\right)}$ & $O\left(rn^2\right)$& $O\left(n^2+\alpha n^2\log(\alpha n)\right)$ Updating $\bm{L}$ & $\bm{O\left(rn^2\right)}$ & $O\left(r^2n^2\right)$ & $\bm{O\left(rn^2\right)}$ Tolerance of $\alpha$ & $O\left(\frac{1}{\max\{\mu r^2 \kappa^3,\mu^{1.5} r^2\kappa,\mu^2r^2\}}\right)$& $\bm{O\left(\frac{1}{\mu r}\right)}$& $O\left(\frac{1}{\max\{\mu r^{1.5}\kappa^{1.5},\mu r\kappa^2\}}\right)$ Iterations needed & $\bm{O\left(\log(\frac{1}{\epsilon})\right)}$ & $\bm{O\left(\log(\frac{1}{\epsilon})\right)}$ & $O\left(\kappa\log(\frac{1}{\epsilon})\right)$
[^1]: In practice, we only need the trimmed orthogonal matrices $\widetilde{\bm{U}}_k$ and $\widetilde{\bm{V}}_k$ for the projection $\mathcal{P}_{\widetilde{T}_k}$, and they can be computed efficiently via a QR decomposition. The entire matrix $\widetilde{\bm{L}}_k$ should never be formed in an efficient implementation of AccAltProj.
|
---
abstract: 'We study the possibility of generating tiny neutrino mass through a combination of type I and type II seesaw mechanism within the framework of an abelian extension of standard model. The model also provides a naturally stable dark matter candidate in terms of the lightest neutral component of a scalar doublet. We compute the relic abundance of such a dark matter candidate and also point out how the strength of type II seesaw term can affect the relic abundance of dark matter. Such a model which connects neutrino mass and dark matter abundance has the potential of being verified or ruled out in the ongoing neutrino, dark matter as well as accelerator experiments.'
author:
- Arnab Dasgupta
- Debasish Borah
title: Scalar Dark Matter with Type II Seesaw
---
Introduction
============
Recent discovery of the Higgs boson at the large hadron collider (LHC) experiment has established the standard model (SM) of particle physics as the most successful fundamental theory of nature. However, despite its phenomenological success, the SM fails to address many theoretical questions as well as observed phenomena. Three most important observed phenomena which the SM fails to explain are neutrino oscillations, matter-antimatter asymmetry and dark matter. Neutrino oscillation experiments in the last few years have provided convincing evidence in support of non-zero yet tiny neutrino masses [@PDG]. Recent neutrino oscillation experiments T2K [@T2K], Double ChooZ [@chooz], Daya-Bay [@daya] and RENO [@reno] have not only made the earlier predictions for neutrino parameters more precise, but also predicted non-zero value of the reactor mixing angle $\theta_{13}$. Matter-antimatter asymmetry of the Universe is encoded in the baryon to photon ratio measured by dedicated cosmology experiments like Wilkinson Mass Anisotropy Probe (WMAP), Planck etc. The latest data available from Planck mission constrain the baryon to photon ratio [@Planck13] as $$Y_B \simeq (6.065 \pm 0.090) \times 10^{-10}
\label{barasym}$$ Presence of dark matter in the Universe is very well established by astrophysics and cosmology experiments although the particle nature of dark matter in yet unknown. According to the Planck 2013 experimental data [@Planck13], $26.8\%$ of the energy density of the present Universe is composed of dark matter. The present abundance or relic density of dark matter is represented as $$\Omega_{\text{DM}} h^2 = 0.1187 \pm 0.0017
\label{dm_relic}$$ where $\Omega$ is the density parameter and $h = \text{(Hubble Parameter)}/100$ is a parameter of order unity.
Several interesting beyond standard model (BSM) frameworks have been proposed in the last few decades to explain each of these three observed phenomena in a natural way. Tiny neutrino masses can be explained by seesaw mechanisms which broadly fall into three types : type I [@ti], type II [@tii] and type III [@tiii]. Baryon asymmetry can be produced through the mechanism of leptogenesis which generates an asymmetry in the leptonic sector first and later converting it into baryon asymmetry through electroweak sphaleron transitions [@sphaleron]. The out of equilibrium CP violating decay of heavy Majorana neutrinos provides a natural way to create the required lepton asymmetry [@fukuyana]. There are however, other interesting ways to create baryon asymmetry: electroweak baryogenesis [@Anderson:1991zb], for example. The most well motivated and widely discussed particle dark matter (for a review, please see [@Jungman:1995df]) is the weakly interacting massive particle (WIMP) which interacts through weak and gravitational interactions and has mass typically around the electroweak scale. Weak interactions kept them in equilibrium with the hot plasma in the early Universe which at some point of time, becomes weaker than the expansion rate of the Universe leading to decoupling (or freeze-out) of WIMP. WIMP’s typically decouple when they are non-relativistic and hence known as the favorite cold dark matter (CDM) candidate.
Although the three observed phenomena discussed above could have completely different particle physics origin, it will be more interesting if they have a common origin or could be explained within the same particle physics model. Here we propose a model which has all the ingredients to explain these three observed phenomena naturally. We propose an abelian extension of SM (for a review of such models, please see [@langacker]) with a gauged $B-L$ symmetry. Neutrino mass can be explained by both type I and type II seesaw mechanisms. Some recent works related to the combination of type I and type II seesaw can be found in [@typeI+II]. Dark matter can be explained due to the existence of an additional Higgs doublet, naturally stable due to the choice of gauge charges under $U(1)_{B-L}$ symmetry. Unlike the conventional scalar doublet dark matter models, here we show how the origin neutrino mass can affect the dark matter phenomenology. Some recent works motivated by this idea of connecting neutrino mass and dark matter can be found in [@nuDM]. In supersymmetric frameworks, such scalar dark matter have been studied in terms of sneutrino dark matter in type I seesaw models [@susy1] as well as inverse seesaw models [@susy2]. We show that in our model, the dark matter abundance can be significantly altered due to the existence of a neutral scalar with mass slightly larger than the mass of dark matter, allowing the possibility of coannihilation. And interestingly, this mass splitting is found to be governed by the strength of type II seesaw term of neutrino mass in our model. We show that for sub-dominant type II seesaw term, dark matter relic abundance can get significantly affected due to coannihilation whereas for dominant type II seesaw, usual calculation of dark matter relic abundance follows incorporating self-annihilation of dark matter only.
This paper is organized as follows: in section \[model\], we outline our model with particle content and relevant interactions. In section \[numass\], we briefly discuss the origin of neutrino mass in our model. In section \[darkmatter\], we discuss the method of calculating dark matter relic abundance. In section \[results\], we discuss our results and finally conclude in section \[conclude\].
The Model {#model}
=========
We propose a $U(1)_{B-L}$ extension of the standard model with the particle content shown in table \[table1\]. Apart from the standard model fermions, three right handed neutrinos $\nu_R$ are added with lepton number $1$. This is in fact necessary to cancel the $U(1)_{B-L}$ anomalies. Among the scalars, $H$ is the standard model like Higgs responsible for giving mass to fermions and breaking electroweak gauge symmetry. The second Higgs doublet $\phi$ does not acquire vacuum expectation value (vev) and also has no coupling with the fermions. This will act like an inert doublet dark matter in our model whose stability is naturally guaranteed by the gauge symmetry. The scalar triplet $\Delta$ serves two purposes: its neutral component contributes to the light neutrino masses by acquiring a tiny $(\sim \text{eV})$ vev and also generates a mass splitting between the CP-even and CP-odd neutral scalars in the inert Higgs doublet $\phi$.
Particle $SU(3)_c \times SU(2)_L \times U(1)_Y$ $U(1)_{B-L}$
---------------- ---------------------------------------- ---------------
$ (u,d)_L $ $(3,2,\frac{1}{3})$ $\frac{1}{3}$
$ u_R $ $(\bar{3},1,\frac{4}{3})$ $\frac{1}{3}$
$ d_R $ $(\bar{3},1,-\frac{2}{3})$ $\frac{1}{3}$
$ (\nu, e)_L $ $(1,2,-1)$ $-1$
$e_R$ $(1,1,-2)$ $-1$
$\nu_R$ $(1,1,0)$ $-1$
$H$ $(1,2,1)$ $0$
$\phi$ $(1,2,1)$ $1$
$ \Delta$ $(1,3,2)$ $2$
$ S$ $(1,1,0)$ $2$
: Particle Content of the Model
\[table1\]
The Yukawa Lagrangian for the above particle content can be written as $$\begin{aligned}
\mathcal{L}_Y &=& Y_e \bar{L} H e_R + Y_{\nu}\bar{L}H^{\dagger}\nu_R+Y_d \bar{Q}Hd_R +Y_u \bar{Q}H^{\dagger}u_R \\
&& +Y_R S \nu_R \nu_R + f \Delta L L
\label{yuklag}\end{aligned}$$ The gauge symmetry of the model does not allow any coupling of the inert Higgs doublet $\phi$ with the fermions. The scalar Lagrangian of the model can be written as $$\begin{aligned}
\mathcal{L}_H &=& \frac{\lambda}{4} \left ( H^{\dagger i}H_i - \frac{v^2}{2} \right )^2 +m^2_1 (\phi^{\dagger i}\phi_i ) + \lambda_{\phi} (\phi^{\dagger i}\phi_i )^2 \\
&& + \lambda_1 (H^{\dagger i}H_i)(\phi^{\dagger j}\phi_j ) + \lambda_2 (H^{\dagger i}H_j)(\phi^{\dagger j}\phi_i )+ \mu_{\phi \Delta} (\phi \phi \Delta^{\dagger} + \phi^{\dagger} \phi^{\dagger} \Delta ) \\
&& + \lambda_3 (H H \Delta^{\dagger} S + H^{\dagger}H^{\dagger} \Delta S^{\dagger} ) + m^2_2 S^{\dagger}S + \lambda_S (S^{\dagger} S)^2 + m^2_{\Delta} \Delta^{\dagger}\Delta +\lambda_{\Delta} (\Delta^{\dagger}\Delta)^2\\
&& +\lambda_4 (H^{\dagger}H)(S^{\dagger}S)+\lambda_5 (\phi^{\dagger}\phi )( S^{\dagger}S)+ \lambda_6 (\Delta^{\dagger}\Delta )(S^{\dagger}S)+\lambda_7 (H^{\dagger}H)(\Delta^{\dagger}\Delta )+\lambda_8 (\phi^{\dagger}\phi )(\Delta^{\dagger}\Delta )\end{aligned}$$ Assuming that the inert doublet $\phi$ does not acquire any vev, the neutral scalar masses corresponding to the Higgs doublets $H, \phi$ can be written as $$m^2_h = \frac{1}{2}\lambda v^2$$ $$m^2_{H_0} = m^2_1 +\frac{1}{2}(\lambda_1+\lambda_2)v^2+2\mu_{\phi \Delta} v_L$$ $$m^2_{A_0} = m^2_1+\frac{1}{2}(\lambda_1+\lambda_2)v^2-2\mu_{\phi \Delta} v_L$$ where $m_h$ is the mass of SM like Higgs boson which is approximately 126 GeV and $v$ is the vev of the neutral component of SM like Higgs doublet $H$. The CP-even $(H_0)$ and CP-odd $(A_0)$ neutral components of inert doublet $\phi$ have a mass squared splitting proportional to $4\mu_{\phi \Delta} v_L$ where $v_L$ is the vev acquired by the neutral component of scalar triplet $\Delta$.
Neutrino Mass {#numass}
=============
Tiny neutrino mass can originate from both type I and type II seesaw mechanisms in our model. As seen from the Yukawa Lagrangian (\[yuklag\]), the right handed singlet neutrinos acquire a Majorana mass term after the $U(1)_{B-L}$ gauge symmetry gets spontaneously broken by the vev of the singlet scalar field $S$. The resulting type I seesaw formula for light neutrinos is given by the expression, $$m_{LL}^I=-m_{LR}M_{RR}^{-1}m_{LR}^{T}.$$ where $m_{LR} = Y_{\nu} v$ is the Dirac mass term of the neutrinos and $M_{RR} = Y_R\langle S \rangle =Y_R v_{BL}$ is the Majorana mass term of the right handed neutrinos. Demanding the light neutrinos to be of eV scale one needs $M_{RR}$ and hence $v_{BL}$ to be as high as $10^{14}$ GeV without any fine-tuning of dimensionless Yukawa couplings.
On the other hand, the type II seesaw contribution to neutrino mass comes from the term $f \Delta L L$ in the Yukawa Lagrangian (\[yuklag\]) if the neutral component of the scalar triplet $\delta^0$ acquires a tiny vev. The scalar triplet can be represented as $$\Delta =
\left(\begin{array}{cc}
\ \delta^+/\surd 2 & \delta^{++} \\
\ \delta^0 & -\delta^+/\surd 2
\end{array}\right) \nonumber$$ Minimizing the scalar potential gives the approximate value of $v_L$ as $$v_L = \frac{\lambda_3 v^2 \langle S \rangle}{m^2_{\Delta}}=\frac{\lambda_3 v^2 v_{BL}}{m^2_{\Delta}}
\label{vevvl}$$ where $\langle S \rangle = v_{BL}$ is the vev acquired by the singlet scalar field $S$ responsible for breaking $U(1)_{B-L}$ gauge symmetry spontaneously at high scale. Demanding the light neutrinos to be of eV scale one needs $m^2_{\Delta}/v_{BL}$ to be as high as $10^{14}$ GeV without any fine-tuning of dimensionless couplings.
Relic Abundance of Dark Matter {#darkmatter}
==============================
The relic abundance of a dark matter particle $\chi$ is given by the the Boltzmann equation $$\frac{dn_{\chi}}{dt}+3Hn_{\chi} = -\langle \sigma v \rangle (n^2_{\chi} -(n^{eqb}_{\chi})^2)$$ where $n_{\chi}$ is the number density of the dark matter particle $\chi$ and $n^{eqb}_{\chi}$ is the number density when $\chi$ was in thermal equilibrium. $H$ is the Hubble expansion rate of the Universe and $ \langle \sigma v \rangle $ is the thermally averaged annihilation cross section of the dark matter particle $\chi$. In terms of partial wave expansion $ \langle \sigma v \rangle = a +b v^2$. Numerical solution of the Boltzmann equation above gives [@Kolb:1990vq] $$\Omega_{\chi} h^2 \approx \frac{1.04 \times 10^9 x_F}{M_{Pl} \sqrt{g_*} (a+3b/x_F)}$$ where $x_F = m_{\chi}/T_F$, $T_F$ is the freeze-out temperature, $g_*$ is the number of relativistic degrees of freedom at the time of freeze-out. Dark matter particles with electroweak scale mass and couplings freeze out at temperatures approximately in the range $x_F \approx 20-30$. More generally, $x_F$ can be calculated from the relation $$x_F = \ln \frac{0.038g_{eff}m_{PL}m_{DM}<\sigma_{eff} v>}{g_*^{1/2}x_f^{1/2}}
\label{xf}$$ The expression for relic density again simplifies to [@Jungman:1995df] $$\Omega_{\chi} h^2 \approx \frac{3 \times 10^{-27} \text{cm}^3 \text{s}^{-1}}{\langle \sigma v \rangle}
\label{eq:relic}$$ The thermal averaged annihilation cross section $\langle \sigma v \rangle$ is given by [@Gondolo:1990dk] $$\langle \sigma v \rangle = \frac{1}{8m^4T K^2_2(m/T)} \int^{\infty}_{4m^2}\sigma (s-4m^2)\surd{s}K_1(\surd{s}/T) ds$$ where $K_i$’s are modified Bessel functions of order $i$, $m$ is the mass of Dark Matter particle and $T$ is the temperature.
Here we consider the neutral component of the scalar doublet $\phi$ as the dark matter candidate which is similar to the inert doublet model of dark matter discussed extensively in the literature [@Ma:2006km; @Barbieri:2006dq; @Majumdar:2006nt; @LopezHonorez:2006gr; @ictp; @borahcline]. We consider the lighter mass window for the scalar doublet dark matter $m_{DM} \leq M_W$, the W boson mass. Beyond the W boson mass threshold, the annihilation channel of scalar doublet dark matter into $W^+W^-$ pairs opens up reducing the relic abundance of dark matter below observed range for dark matter mass all the way upto around $500$ GeV. We note however, that there exists a region of parameter space $M_W < m_{DM} <160$ GeV which satisfy relic density bound if certain cancellations occur between several annihilation diagrams [@honorez1]. For the sake of simplicity, we stick to the low mass region $10 \; \text{GeV} < m_{DM} < M_W$ in this work. We also note that there are two neutral components in the doublet $\phi$, the lighter of which is stable and hence the dark matter candidate. If the mass difference between these neutral scalars $\Delta m = m_{A_0} -m_{H_0}$ is large compared to the freeze-out temperature $T_F$, then the next to lightest neutral scalar play no significant role in determination of dark matter relic density. However, if $\Delta m$ is of the order of freeze-out temperature then $A_0$ can be thermally produced and hence the coannihilations between $H_0$ and $A_0$ during the epoch of dark matter thermal annihilation can play a non-trivial role in determining the relic abundance of dark matter. The annihilation cross section of dark matter in such a case gets additional contributions from coannihilation between dark matter and next to lightest neutral component of scalar doublet $\phi$. This type of coannihilation effects on dark matter relic abundance were studied by several authors in [@Griest:1990kh; @coann_others]. Here we follow the analysis of [@Griest:1990kh] to calculate the effective annihilation cross section in such a case. The effective cross section can given as $$\begin{aligned}
\sigma_{eff} &= \sum_{i,j}^{N}\langle \sigma_{ij} v\rangle r_ir_j \nonumber \\
&= \sum_{i,j}^{N}\langle \sigma_{ij}v\rangle \frac{g_ig_j}{g^2_{eff}}(1+\Delta_i)^{3/2}(1+\Delta_j)^{3/2}e^{\big(-x_F(\Delta_i + \Delta_j)\big)} \nonumber \\\end{aligned}$$ where, $x_F = \frac{m_{DM}}{T}$ and $\Delta_i = \frac{m_i-m_{DM}}{m_{DM}}$ and $$\begin{aligned}
g_{eff} &= \sum_{i=1}^{N}g_i(1+\Delta_i)^{3/2}e^{-x_F\Delta_i}\end{aligned}$$ The thermally averaged cross section can be written as $$\begin{aligned}
\langle \sigma_{ij} v \rangle &= \frac{x_F}{8m^2_im^2_jm_{DM}K_2((m_i/m_{DM})x_F)K_2((m_j/m_{DM})x_F)} \times \nonumber \\
& \int^{\infty}_{(m_i+m_j)^2}ds \sigma_{ij}(s-2(m_i^2+m_j^2)) \sqrt{s}K_1(\sqrt{s}x_F/m_{DM}) \nonumber \\
\label{eq:thcs}\end{aligned}$$
In our model, the lightest neutral component of the scalar doublet $\phi$ is the dark matter candidate. We denote it as $H_0$ and the other neutral component is denoted as $A_0$. Since $A_0$ is heavier than $H_0$, it can always decay into $H_0$ and standard model particles (as shown in figure \[fig:decay1\]) depending on the mass difference. If the mass difference between $A_0$ and $H_0$ is small enough for $A_0$ to be thermally produced during the epoch of freeze-out then we have to compute both annihilation and coannihilation cross sections to determine the relic abundance. In the low mass regime $(m_{DM} < M_W)$, the self annihilation of either $H_0$ or $A_0$ into SM particles occur through standard model Higgs boson as shown in figure \[fig:feyn1\]. The corresponding annihilation cross section is given by $$\begin{aligned}
\sigma_{xx} &= \frac{|Y_f|^2|\lambda_x|^2}{16\pi s}
\frac{\left(s-4m^2_f\right)^{3/2}}{\sqrt{s-4m^2_x}(s-m^2_h + m^2_h\Gamma^2_h)^2}
\label{eq:crossH_0H_0}\end{aligned}$$ where $x\rightarrow H_0,A_0$, $\lambda_x$ is the coupling of $x$ with SM Higgs boson $h$ and $\lambda_f$ is the Yukawa coupling of fermions. $\Gamma_h = 4.15$ MeV is the SM Higgs decay width.
The coannihilation of $H_0$ and $A_0$ into SM particles can occur through a $Z$ boson exchange as shown in figure \[fig:feyn\]. The corresponding cross section is found to be $$\begin{aligned}
\sigma_{H_0A_0} &= \frac{1}{64 \pi^2 s}\sqrt{\frac{\left(s^2 - 4 m^{2}_fs\right)}{s^2-2(m^2_{H_0}+m^{2}_{A_0})s+(m^{2}_{H_0}-m^{2}_{A_0})^2}} \times \nonumber \\
& \frac{1}{4}\frac{g^4}{c^{4}_W}\frac{1}{\left[(s-m^{2}_z)^2 + m^{2}_z\Gamma^{2}_z\right]}\left[(a^{2}_f+b^{2}_f)\left((m^{2}_{H_0}-m^{2}_{A_0})^2 \right. \right.\nonumber \\
&- \left. \left. \frac{1}{3s}\left(s^2 - 2s(m^{2}_{H_0}+ m^{2}_{A_0}) + (m^{2}_{H_0}-m^{2}_{A_0})^2\right) (s-4m^{2}_f)c^{2}_\theta + (s-2m^{2}_f)(s-(m^2_{H_0}+m^{2}_{A_0}))\right ) \right. \nonumber \\
&+ \left. a_fb_fm^{2}_f(s-(m^{2}_{H_0} + m^{2}_{A_0}))\right]
\label{eq:crossH_0A_0}\end{aligned}$$ where $a_f = T^{f}_3 - s^{2}_WQ_f; \quad b_f = -s^{2}_WQ_f$. $\Gamma_z = 2.49$ GeV is the Z boson decay width.
We use these cross sections to compute the thermal averaged annihilation cross section given in equation (\[eq:thcs\]). Instead of assuming a particular value of $x_F$, we first numerically find out the value of $x_F$ which satisfies the following equation $$\begin{aligned}
e^{x_F} - \ln \frac{0.038g_{eff}m_{PL}m_{DM}<\sigma_{eff} v>}{g_*^{1/2}x_F^{1/2}} &= 0\end{aligned}$$ which is nothing but a simplified form of equation (\[xf\]). For a particular pair of $\lambda_{DM}$ and $m_{DM}$, we use this value of $x_F$ and compute the thermal averaged cross section$<\sigma_{eff} v>$ to be used for calculating relic abundance using equation (\[eq:relic\]).
We also calculate the lifetime of $A_0$ to make sure that $A_0$ is not long lived enough to play a role of dark matter in the present Universe. The decay width of $A_0$ is given by $$\begin{aligned}
\Gamma_{A_0} &= \int_{s_2}\int_{s_{3-}}^{s_{3+}} \frac{1}{32(2\pi)^3 m^3_{A_0} }f(s_2,s_3)ds_2 ds_3 \nonumber \\
f(s_2,s_3)&= \frac{N_cg^4}{4c^4_W}\bigg[(a_f^2+b_f^2)\big[(m^2_{A_0}-m^2_{H_0}-2m^2_f)^2-(s_2-s_3)^2-4a_fb_fm^2_f(m^2_{A_0}+m^2_{H_0}+s_3+s_2)\big]\bigg]\nonumber \\
&\times \frac{1}{\bigg[(4m^2_fm^2_{A_0}m^2_{H_0}-m^2_Z-s_3-s_2)^2-m^2_Z\Gamma^2_Z\bigg]} \end{aligned}$$ where, $$\begin{aligned}
s_{3\pm} &= m^2_f + m^2_{H_0} \frac{1}{s_2}\bigg[(m^2_{A_0}-s_2-m^2_f)(s_2-m^2_f+m^2_{H_0})\pm \lambda^{1/2}(s_2,m^2_{A_0},m^2_f)\lambda^{1/2}(s_2,m^2_f,m^2_{H_0})\bigg]\nonumber \\
s_2 &\in \bigg[(m_{H_0} + m_f)^2,(m_{A_0} - m_f)^2\bigg] \end{aligned}$$ Now if the difference $\Delta m = m_{A_0} - m_{H_0} $= 50 keV then $m_{A_0}$ will decay into $H_0$ and neutrinos only and its lifetime is $\Gamma_{A_0}^{-1}\hbar = 1.67962\times10^6$ s. If the difference is 5 MeV then $A_0$ can decay into up-quarks, electrons and neutrinos such that $\Gamma_{total} = \Gamma_u + \Gamma_e + \Gamma_{\nu}$ and therefore the lifetime will be $\Gamma_{total}^{-1}\hbar = 8.3659$ s. But if the difference is around 1 GeV then $A_0$ can decay into strange-quarks, down-quarks, up-quarks, muons, electrons and neutrinos such that $\Gamma_{total} = \Gamma_s + \Gamma_d + \Gamma_u+ \Gamma_{\mu} + \Gamma_e + \Gamma_{\nu}$ and therefore the lifetime will be $\Gamma_{total}^{-1}\hbar = 1.2 \times 10^{-11}$ s. WIMP dark matter typically freeze-out at temperature $T_F \sim m_{DM}/x_F$ where $x_f \sim 20-30$. This can roughly be taken to be the time corresponding to the electroweak scale $t_{EW} \sim 10^{-11}$ s. Thus, for all mass differences under consideration the lifetime of $A_0$ falls much below the present age of the Universe. Hence, the present dark matter relic density is totally contributed by the abundance of the lightest stable neutral scalar $H_0$.
![[Decay of $A_0$ into $H_0$ and two fermions (where the conventions are followed from [@Dreiner:2008tw] )]{}[]{data-label="fig:decay1"}](decayA0H0ff.pdf)
![[Self annihilation of $H_0(A_0)$ into two fermions (where the conventions are followed from [@Dreiner:2008tw] )]{}[]{data-label="fig:feyn1"}](feyn1.pdf)
![[Coannihilations of $H_0$ and $A_0$ into two fermions (where the conventions are followed from [@Dreiner:2008tw] ) ]{}[]{data-label="fig:feyn"}](feyn.pdf){width="100.00000%"}
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[c]{}\
Results
=======
We follow the approach and use the expressions discussed in the previous section to calculate the dark matter relic density. We first calculate the relic density of dark matter $H_0$ without considering coannihilation. We use the constraint on dark matter relic density (\[dm\_relic\]) and show the allowed parameter space in terms of $\lambda_{DM} = \lambda_1 + \lambda_2$, the dark matter-SM Higgs coupling and $m_{H_0}$, the dark matter mass. The results are shown as the red v-shaped region in figure \[fig:p20\]. We then allow coannihilation between $H_0$ and $A_0$ and show the allowed parameter space in the same $\lambda_{DM}-m_{H_0}$ plane. This corresponds to the black region in figure \[fig:p21\]. The plot \[fig:p21\] corresponds to mass splitting between $A_0$ and $H_0$: $\Delta m = 500$ keV.
In addition to the Planck 2013 constraints on dark matter relic density (\[dm\_relic\]), there is also a strict limit on the spin-independent dark matter-nucleon cross section coming from direct detection experiments, most recently from LUX experiment[@LUX]. The relevant scattering cross section in our model is given by [@Barbieri:2006dq] $$\sigma_{SI} = \frac{\lambda^2_{DM}f^2}{4\pi}\frac{\mu^2 m^2_n}{m^4_h m^2_{DM}}
\label{sigma_dd}$$ where $\mu = m_n m_{DM}/(m_n+m_{DM})$ is the DM-nucleon reduced mass. A recent estimate of the Higgs-nucleon coupling $f$ gives $f = 0.32$ [@Giedt:2009mr] although the full range of allowed values is $f=0.26-0.63$ [@mambrini]. We take the minimum upper limit on the dark matter-nucleon spin independent cross section from LUX experiment [@LUX] which is $7.6 \times 10^{-46} \; \text{cm}^2$ and show the exclusion line in figure \[fig:p20\], \[fig:p21\] with a label $\sigma^0_{SI}$. The exclusion line gets broaden as shown in blue in figure \[fig:p20\], \[fig:p21\] due to the uncertainty factor in Higgs-nucleon coupling. It should be noted that the spin independent scattering cross section written above (\[sigma\_dd\]) is only at tree level. Since dark matter in this model is part of a doublet under $SU(2)_L$, one-loop box diagrams involving two $W$ bosons or two $Z$ bosons can also give rise to direct detection cross section. As discussed in details by authors of [@oneloop], in the low mass regime of inert doublet dark matter, this one-loop correction is maximum in the resonance region $m_{DM} \sim m_h/2$. The ratio $\sigma_{SI} (\text{one-loop})/ \sigma_{SI} (\text{tree-level})$ is approximately $100$ in this region. We calculate the scattering cross section corresponding to one-loop diagrams mentioned in [@oneloop]. The final result not only depends upon $\lambda_{DM}$ but also on the charged Higgs mass. For simplicity, we take $m_{H^{\pm}} = m_{A_0}$. The one-loop result is shown as the exclusion line in figure \[fig:p20\], \[fig:p21\] with label $\sigma^0_{SI}+\sigma^1_{SI}$. Thus, in the absence of co-annihilation only a small region of the parameter space near the resonance is left from direct detection bound. In the presence of co-annihilation, more regions of parameter space gets allowed as seen from figure \[fig:p21\].
![[Forward scattering of $H_0 f(\overline{f})\rightarrow A_0f(\overline{f})$ through t-channel $Z$ boson (where the conventions are followed from [@Dreiner:2008tw] )]{}[]{data-label="fig:inelastic"}](H_0f-A_0f.pdf){width="100.00000%"}
Apart from the SM Higgs mediated scattering, there can be one more DM-nucleon scattering cross section due to the same interaction giving rise to coannihilation between $H_0$ and $A_0$. This corresponds to DM scattering off nuclei into $A_0$ through a Z boson exchange giving rise to an inelastic DM-nucleon scattering as shown in figure \[fig:inelastic\]. The cross-section for such a process can be calculated as $$\begin{aligned}
|\mathcal{M}_{H_0f\rightarrow A_0f}|^2 &= \frac{g^2}{4\cos^2 \theta_W}\frac{1}{(t-m^2_Z)^2-m^2_Z\Gamma^2_Z}\bigg{[}2(a^2_f+b^2_f)\bigg{[}\frac{1}{4}
(m^2_{H_0}-m^2_{A_0}+s-u)^2\nonumber \\
&+\frac{1}{2}(t-2m^2_f)(2(m^2_{H_0}m^2_{A_0}-t)\bigg{]}+4a_fb_fm^2_f(2(m^2_{A_0}+m^2_{H_0})-t)\bigg{]}
\nonumber \\
\frac{d\sigma}{d\Omega_{cm}} &= \frac{1}{64\pi^2s}\sqrt{\frac{s^2-2(m^2_{A_0}+m^2_f)s+(m^2_{A_0}-
m^2_f)^2}{s^2-2(m^2_{H_0}+m^2_f)s+(m^2_{H_0}-
m^2_f)^2}}|\mathcal{M}_{H_0f\rightarrow A_0f}|^2
\label{eq_cross_t}
\end{aligned}$$
Due to the strong Z boson coupling to DM in our model, such a scattering can give rise to an inelastic cross section which faces severe limits from direct detection experiments like Xenon100 [@Xenoninelastic]. Such inelastic dark matter within inert doublet dark matter was studied by authors in [@idmARINA]. They show that such inelastic dark matter scenario in inert doublet model is consistent with exclusion limits from direct detection experiments only when dark matter relic abundance is below the observed abundance. However, such a scattering process is kinematically forbidden if the mass difference between $H_0$ and $A_0$ is more than the kinetic energy of dark matter $H_0$. Taking the typical speed of WIMP dark matter to be $v \approx 270 \; \text{km/s} \sim 10^{-3}c$, the kinetic energy of a $100$ GeV WIMP is around $50$ keV. Thus for mass differences more than $50$ keV, dark matter relic abundance can get affected by $A_0$, but the direct detection cross section remain unaffected.
We also impose collider bounds by noting that the precision measurement of the $Z$ boson decay width at LEP I forbids the $Z$ boson decay channel $Z \rightarrow H_0 A_0$ which requires $m_{H_0}+m_{A_0} > m_Z \Rightarrow 2m_{H_0} > m_Z -\Delta m$. The excluded region $m_{H_0} < (m_Z-\Delta m)/2$ is shown as a pink shaded region in figure \[fig:p21\]. Apart from LEP I constraint on $Z$ decay width, LEP II constraints also rule out models satisfying $m_{H_0} < 80$ GeV, $m_{A_0} < 100$ GeV and $m_{A_0}-m_{H_0} > 8$ GeV [@Lundstrom:2008ai]. As we see from figure \[fig:p21\], the allowed region of $\lambda_{DM}-m_{DM}$ parameter space including coannihilation satisfy these constraints as the mass difference is not more than 8 GeV.
We further impose the constraint that invisible decay of the SM Higgs boson $h \rightarrow H_0 H_0$ do not dominate its decay width. Recent measurement of the SM Higgs properties constrain the invisible decay width to be below $30\%$ [@HiggsInv]. The invisible decay width is given by $$\Gamma_{\rm inv} = {\lambda^2_{DM} v^2\over 64 \pi m_h}
\sqrt{1-4\,m^2_{DM}/m^2_h}$$ We show the parameter space ruled out by this constraint on invisible decay width as the brown shaded region for $m_{H_0} < m_h/2$ in figure \[fig:p21\]. It can be easily seen that this imposes a weaker constraint than the DM direct detection constraint from LUX experiment.
In our analysis we have taken the mass differences between $A_0$ and $H_0$ to be $\Delta m = 500$ keV. As noted in section \[model\], the mass difference between $A_0$ and $H_0$ is given by $\Delta m^2 = 4\mu_{\phi \Delta} v_L$. Thus, for $\Delta m = 500$ keV and dominant type II seesaw such that $v_L = 0.1$ eV, the trilinear mass term $\mu_{\phi \Delta} \sim 600$ GeV. However, if we keep the trilinear mass term fixed at say, the $U(1)_{B-L}$ symmetry breaking scale, then different mass differences $\Delta m$ will correspond to different strengths of type II seesaw term. If we fix the trilinear mass term $\mu_{\phi \Delta}$ to be $10^9$ GeV say, then $\Delta m = 500 \; \text{keV}, 5\; \text{MeV}, 1 \; \text{GeV}$ will correspond to $v_L = 10^{-16}, 10^{-14}, 10^{-9} \; \text{GeV}$ respectively. The first two examples correspond to a case of sub-dominant type II seesaw similar to the ones discussed in [@typeI+II], whereas the third example $\Delta m = 1$ GeV corresponds to a type II dominant seesaw. Comparing this with equation (\[vevvl\]), one gets $$\frac{\lambda_3 v_{BL}}{m^2_{\Delta}} = 10^{-20}, 10^{-18}, 10^{-13} \; \text{GeV}^{-1}$$ respectively. This can be achieved by suitable adjustment of the symmetry breaking scales, the bare mass terms in the Lagrangian as well as the dimensionless couplings.
We note that, in conventional inert doublet dark matter model, this mass squared difference is $\lambda_{IDM} v^2$ where $\lambda_{IDM}$ is a dimensionless coupling. If we equate this mass difference to a few hundred keV, then the dimensionless coupling $\lambda_{IDM}$ has to be fine tuned to $10^{-12}-10^{-10}$. Such a fine tuning can be avoided in our model by suitably fixing the symmetry breaking scales and the bare mass terms in the Lagrangian.
Conclusion {#conclude}
==========
We have studied an abelian extension of SM with a $U(1)_{B-L}$ gauge symmetry. The model allows the existence of both type I and type II seesaw contributions to tiny neutrino masses. It also allows a naturally stable cold dark matter candidate: lightest neutral component of a scalar doublet $\phi$. Type II seesaw term is generated by the vev of a scalar triplet $\Delta$. We show that, in our model the vev of the scalar triplet not only decides the strength of type II seesaw term, but also the mass splitting between the neutral components of the scalar doublet $\phi$. If the vev is large (of the order of GeV, say), then mass splitting is large and hence the next to lightest neutral component $A_0$ plays no role in determining the relic abundance of $H_0$. However, if the vev is small such that the mass splitting is below $5-10\%$ of $m_{H_0}$, then $A_0$ can play a role in determining the relic abundance of $H_0$. In such a case, dark matter $(H_0)$ relic abundance gets affected due to coannihilation between these two neutral scalars. We compute the relic abundance of dark matter in both the cases: without and with coannihilation and show the change in parameter space. We incorporate the latest constraint on dark matter relic abundance from Planck 2013 data and show the allowed parameter space in the $\lambda_{DM}-m_{DM}$ plane, where $\lambda_{DM}$ is the dark matter SM Higgs coupling and $m_{DM}$ is the mass of dark matter $H_0$. We show the parameter space for mass splitting $\Delta m = 500 \; \text{keV}$. We point out that unlike the conventional inert doublet dark matter model, here we do not have to fine tune dimensionless couplings too much to get such small mass splittings between $A_0$ and $H_0$. This mass splitting can be naturally explained by the suitable adjustment of symmetry breaking scales and bare mass terms of the Lagrangian. It is interesting to note that for sub-dominant type II seesaw case, the dark matter relic abundance gets affected by coannihilation whereas for dominant type II seesaw case, usual dark matter relic abundance calculation applies taking into account of self-annihilations only.
We also take into account the constraint coming from dark matter direct detection experiments like LUX experiment on spin independent dark matter nucleon scattering. We incorporate the LEP I bound on $Z$ boson decay width which rules out the region $m_{H_0}+m_{A_0} < m_Z$. We then incorporate the constraint on invisible SM Higgs decay branching ratio from measurements done at LHC experiment. We show that after taking all these relevant constraints into account, there still remains viable parameter space which can account for dark matter as well as neutrino mass simultaneously. Thus our model not only gives rise to a natural dark matter candidate, but also provides a natural way to connect the dark matter relic abundance with the neutrino mass term from type II seesaw mechanism.
AD likes to thank Council of Scientific and Industrial Research, Govt. of India for financial support through Senior Research Fellowship (EMR No. 09/466(0125)/2010-EMR-I).
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|
---
abstract: 'We use the finite amplitude method (FAM), an efficient implementation of the quasiparticle random phase approximation, to compute beta-decay rates with Skyrme energy-density functionals for 3983 nuclei, essentially all the medium-mass and heavy isotopes on the neutron rich side of stability. We employ an extension of the FAM that treats odd-mass and odd-odd nuclear ground states in the equal filling approximation. Our rates are in reasonable agreement both with experimental data where available and with rates from other global calculations.'
author:
- 'E. M. Ney'
- 'J. Engel'
- 'N. Schunck'
title: 'Global Description of Beta Decay with the Axially-Deformed Skyrme Finite Amplitude Method: Extension to Odd-Mass and Odd-Odd Nuclei'
---
\[sec:intro\]Introduction
=========================
The origin of elements heavier than iron still remains an open question. Early work has shown that neutron capture in astrophysical processes is responsible for synthesizing those elements [@Burbidge1957; @Meyer1994]. Rapid neutron capture, through the “$r$-process,” is particularly interesting because its astrophysical site is still uncertain. The multi-messenger neutron star merger GW170817 [@Abbott2017] recently provided evidence that such events are the dominant source of $r$-process elements, but quantitative conclusions require more data. We need more reliable astrophysical simulations to connect future multi-messenger events with details of the underlying nucleosynthesis.
Abundances of $r$-process elements depend on a variety of nuclear properties, including masses, neutron-capture cross sections, photo-disintegration cross sections, fission yields, and beta-decay half-lives [@Horowitz2019]. Although some of these properties have been measured and tabulated [@ENSDF2019], the majority of nuclei relevant for the $r$-process are too unstable to be produced in the lab. Reliable $r$-process simulations thus require calculations in neutron-rich nuclei. Beta-decay half-lives are particularly important because they determine the overall timescale for neutron capture in the $r$-process [@Moller1997; @Engel1999] and affect the shape of the final abundance pattern [@Mumpower2014; @Shafer2016].
A variety of global beta-decay calculations exist, in the semi-gross theory [@Nakata1997], in a quasiparticle random-phase approximation (QRPA) plus macroscopic finite-range droplet model (FRDM) approach [@Moller1997; @Moller2003], in covariant density functional theory (DFT) [@Marketin2016], etc. DFT, covariant or not, is particularly attractive because it offers a self-consistent, microscopic framework for computing properties across the nuclear chart [@Ring2004; @Schunck2019]. For the calculation of beta decay in deformed superfluid nuclei, DFT amounts to the QRPA, built on a ground state produced by the Hartree-Fock-Bogoliubov (HFB) method, which incorporates pairing correlations into mean fields, all with density-dependent interactions.
In odd-mass and odd-odd nuclei (hereafter “odd” nuclei) pairing is “blocked” and the HFB ground-state contains a quasiparticle excitation [@Ring2004]. This complicates calculations because the ground-state is no longer invariant under time reversal [@bertsch2009a; @Schunck2010]. As a result additional approximations are often made in beta-decay calculations. Reference [@Homma1996], for example, treats one-quasiparticle states perturbatively, while Ref. [@Marketin2016] treats them as if they were zero-quasiparticle states. A more consistent way to approximate HFB blocked states while preserving time-reversal symmetry is through the equal filling approximation (EFA) [@Perez-Martin2008]. Numerous studies showed that the EFA is an excellent approximation to exact blocking [@duguet2001a; @bertsch2009; @Schunck2010]. Reference [@Shafer2016] recently developed a method to extend the EFA to the QRPA.
In this work we use the extension to carry out a global calculation of allowed and first-forbidden contributions to beta-minus decay in odd nuclei from near the valley of stability out to the neutron drip line. We use a global Skyrme density functional determined in Ref. [@Mustonen2016], thus extending that work, which was restricted to even-even nuclei, to all isotopes that play a role in the $r$-process.
This rest of this paper is as follows: Sec. \[sec:theory\] presents background for the finite amplitude method (FAM), which we use to compute QRPA strength functions, and its extension to the EFA. Section \[sec:method\] outlines some improvements to our implementation of the FAM since the work of Ref. [@Mustonen2016]. Section \[sec:results\] presents our results, compares them to those of other papers and to experiment, and addresses subtleties of the EFA-FAM. Section \[sec:conclusions\] contains concluding remarks.
\[sec:theory\]The proton-neutron finite amplitude method (pnFAM)
================================================================
\[ssec:pnfam\]The pnFAM for pure states
---------------------------------------
The QRPA linear-response function is the same as that from time-dependent HFB theory [@Ring2004]. One way of computing it is to diagonalize a set of matrices with dimension equal to that of the two-quasiparticle space. The construction of these matrices, which require two-body matrix elements of the potential, is time consuming in deformed nuclei. The FAM sidesteps the matrices, significantly speeding up the computation of linear response produced by energy-density functionals. Reference [@Nakatsukasa2007] first presented the FAM for the ordinary RPA, and Ref. [@Avogadro2011] did the same for the QRPA. Since then, the method has been used with covariant density functionals [@Niksic2013; @Liang2013; @Liang2014] and employed to compute transition strength in several contexts [@Inakura2009a; @Inakura2009; @Inakura2010; @Stoitsov2011; @Oishi2016].
Here we build on the work of Refs. [@Mustonen2014; @Mustonen2016; @Shafer2016], which used a charge-changing version of the FAM called the pnFAM together with the contour-integral method of Refs. [@Nakatsukasa2014; @Hinohara2015; @Hinohara2013; @Hinohara2015a] to compute beta-decay rates. A detailed account of the pnFAM and its application to beta decay appears in Ref. [@Mustonen2014]. Reference [@Shafer2016] used the EFA to extend the pnFAM to odd nuclei and compute beta-decay rates in the rare-earth nuclei that are important for $r$-process simulations. In order to highlight a few subtleties of the EFA-pnFAM, we recapitulate the main points of the theory here.
We begin with the time-dependent HFB equations $$\label{eq:tdhfb}
i \dot{\mathbb{R}}(t) = \big[\mathbb{H}[\mathbb{R}(t)]+\mathbb{F}(t),
\ \mathbb{R}(t) \big]\,.$$ Here, $\mathbb{R}$ is the generalized HFB density matrix, $\mathbb{H}$ is the HFB Hamiltonian matrix, and $\mathbb{F}$ is a matrix that represents a one-body time-dependent perturbation. The blackboard-bold letters indicate that these matrices are in the HFB quasiparticle basis, defined by the Bogoliubov transformation $\mathbb{W}$: $$\label{eq:hfb_W}
\mathbb{W} =
\begin{pmatrix}
U & V^* \\
V & U^*
\end{pmatrix}
\,,$$ where $U$ and $V$ are themselves matrices. In this basis the static ground-state Hamiltonian and the associated generalized density are diagonal: $$\label{eq:hfb_ground_state}
\mathbb{H}_{0} =
\begin{pmatrix}
E & 0 \\ 0 & -E
\end{pmatrix}
\,, \quad \mathbb{R}_0 =
\begin{pmatrix}
0 & 0 \\ 0 & 1
\end{pmatrix}
\,.$$
To first order in the perturbation $\mathbb{F}$, Eq. is $$\label{eq:linear_response}
i \dot{\delta \mathbb{R}}(t) = \big[\mathbb{H}_0, \delta \mathbb{R}(t) \big] +
\big[\delta\mathbb{H}(t) + \mathbb{F}(t), \mathbb{R}_0 \big] \,,$$ with $\delta \mathbb{R}(t) = \mathbb{R}(t) - \mathbb{R}_0$. If the perturbation is harmonic, the time-dependent quantities $\mathbb{F}(t)$, $\delta
\mathbb{H}(t)$, and $\delta \mathbb{R}(t)$ all take the form (e.g. for $\mathbb{F}$) $$\label{eq:time_dependent_quantities}
\begin{aligned}
\mathbb{F}(t) &= \mathbb{F}(\omega) e^{-i\omega t} + \mathbb{F}^\dagger(\omega)
e^{i\omega t} \\
\mathbb{F}(\omega) &=
\begin{pmatrix}
{F}^{11}(\omega) & {F}^{02}(\omega) \\
-{F}^{20}(\omega) & -{F}^{\overline{11}}(\omega)
\end{pmatrix}
\,.
\end{aligned}$$ We denote the perturbed density more specifically by $$\label{eq:density_response_omega}
\delta \mathbb{R}(\omega) =
\begin{pmatrix}
P(\omega) & X(\omega) \\
-Y(\omega) & -Q(\omega)
\end{pmatrix} \,.$$ When one substitutes Eqs. and into Eq. , the diagonal blocks $P$ and $Q$ vanish, and for a charge-changing external field only the proton-neutron matrix elements of the response are nonzero. These conditions lead to the pnFAM equations $$\label{eq:pnfam} \begin{aligned}
&\big( E_\pi + E_\nu - \omega \big) X_{\pi\nu}(\omega) = - \big( \delta
{H}^{20}_{\pi\nu}(\omega) + {F}^{20}_{\pi\nu}(\omega) \big) \\
&\big( E_\pi + E_\nu + \omega \big)\ Y_{\pi\nu}(\omega) = - \big( \delta
{H}^{02}_{\pi\nu}(\omega) + {F}^{02}_{\pi\nu}(\omega) \big) \,,
\end{aligned}$$ where the label $\pi$ denotes protons and the label $\nu$ denotes neutrons. The use of a finite-difference method to compute $\delta {H}$ is the source of the FAM’s speed. Because we do not consider mixing of protons and neutrons in the underlying HFB ground state, and because Skyrme functionals in use depend at most quadratically on charge-changing densities, the finite difference in the pnFAM reduces exactly to the evaluation of the Hamiltonian with the perturbed densities: $$\label{eq:hamiltonian_response}
\begin{aligned}
\delta \mathbb{H}^{(pn)} &= \lim_{\eta \to 0} \frac{1}{\eta} \bigg(
\mathbb{H} \Big[\mathbb{R}_0^{(pp,nn)} + \eta \delta\mathbb{R}^{(pn)} \Big]
- \mathbb{H} \Big[\mathbb{R}_0^{(pp,nn)} \Big] \bigg) \\
&= \mathbb{H} \Big[\delta\mathbb{R}^{(pn)} \Big] \,.
\end{aligned}$$
Once the FAM amplitudes $X$ and $Y$ are known, one can compute the strength function: $$\label{eq:fam_strength_1}
\frac{dB(F,\omega)}{d\omega} = -\frac{1}{\pi} \Im S(F, \omega) \,,$$ where $$\label{eq:fam_strength_2}
\begin{aligned}
S(F, \omega) &= \sum\limits_{\pi\nu} \big[ {F}^{20^*}_{\pi\nu}
X_{\pi\nu}(\omega) + {F}^{02^*}_{\pi\nu} Y_{\pi\nu}(\omega) \big]\\
&= - \sum\limits_n \bigg( \frac{\lvert \bra{n} \hat{F} \ket{0} \rvert^2}{\Omega_n -
\omega} + \frac{\lvert \bra{n} \hat{F}^\dagger \ket{0} \rvert^2}{\Omega_n + \omega}
\bigg) \,.
\end{aligned}$$
The FAM strength function has poles at QRPA excitation energies $\Omega_n$ with residues equal to the transition probabilities $\lvert \bra{n} \hat{F} \ket{0}
\rvert^2$. It also contains poles at $-\Omega_n$, with residues equal to the negative of transition probabilities for the conjugate operator $\lvert \bra{n}
\hat{F}^\dagger \ket{0} \rvert^2$. In beta-minus-decay calculations $\hat{F}$ contains the isospin lowering operator and $\hat{F}^\dagger$ contains the isospin raising operator; cf. Ref. [@Mustonen2014] for a list of the six allowed and first-forbidden operators. Thus, the poles with positive and negative residues correspond to beta-minus and beta-plus transitions, respectively. This point will become important in the EFA-pnFAM.
In practice we construct the strength function by solving the pnFAM equations separately for each of a large set of complex frequencies $\omega$. From Eqs. and , it is straightforward to show that each pole of $S(F, \omega)$ on the real axis contributes a Lorentzian of half-width $\gamma = \Im[\omega]$ to the strength function in the complex plane. The strength may be be calculated for a set of frequencies close to the real axis with a fixed half-width to mimic experimental strength measurements, or along a closed contour in the complex plane to calculate cumulative strength or decay rates.
\[ssec:ft\_pnfam\]The pnFAM for statistical ensembles
-----------------------------------------------------
Many HFB codes use the EFA to avoid the difficulties associated with the breaking of time-reversal symmetry [@Ring2004; @bertsch2009a] in odd nuclei. The originally *ad hoc* EFA can be understood as a special case of statistical HFB theory for an ensemble that is symmetric under time reversal [@Perez-Martin2008; @Schunck2010]. In systems with time-reversal symmetry, a state $\ket{\lambda}$ and its time-reversed partner $\ket{\overline{\lambda}}$ are degenerate, and the equal filling quasiparticle occupation probabilities, for axial but not spherical symmetry, are $$\label{eq:efa_occupations}
f_{\mu\nu} = \frac{1}{2} ( \delta_{\nu\lambda} + \delta_{\nu\overline{\lambda}})
\delta_{\mu\nu} \,.$$ In odd-odd nuclei, both the odd-proton and odd-neutron quasiparticles have non-zero occupation probabilities. Note that in this work, we do not consider neutron-proton pairing at the HFB level.
The statistical extension of the QRPA [@Sommermann1983] lets us use the FAM to treat excitations of HFB ensembles, taking into account at least partially the polarization of the even-even “core” by the odd nucleon. The EFA-FAM can be derived in the same way as the ordinary FAM, by promoting the ground-state generalized density matrix to a statistical density operator. Expectation values that, for example, define the particle densities, then become ensemble averages. The generalized HFB density matrix is no longer a projector and takes the more general form $$\label{eq:ft_density}
\widetilde{\mathbb{R}}_0 =
\begin{pmatrix}
f & 0 \\ 0 & 1-f
\end{pmatrix} \,.$$ In the usual finite-temperature theory, based on the grand canonical ensemble, the occupation probabilities are given by $f_{\mu\nu}= {(1+\exp(\beta
E_\mu))^{-1}} \delta_{\mu\nu}$ [@Goodman1981]. In the EFA we impose the occupation probabilities of Eq. .
To obtain the statistical pnFAM equations we simply replace the ground-state generalized density of Sec. \[ssec:pnfam\] with that of Eq.. The diagonal elements of the density response no longer vanish, and new statistical factors appear. Once again, for a charge-changing perturbation we need only the proton-neutron matrix elements, and so the statistical pnFAM equations are $$\begin{aligned}
&\big( E_\pi - E_\nu - \omega \big) P_{\pi\nu}(\omega) = - (f_\nu - f_\pi) \big(
\delta {H} + {F} \big)^{11}_{\pi\nu}(\omega) \\
&\big( E_\pi + E_\nu - \omega \big) X_{\pi\nu}(\omega) = - (1 - f_\pi - f_\nu)
\big( \delta {H} + {F} \big)^{20}_{\pi\nu}(\omega) \\
&\big( E_\pi + E_\nu + \omega \big) Y_{\pi\nu}(\omega) = - (1 - f_\pi - f_\nu)
\big( \delta {H} + {F} \big)^{02}_{\pi\nu}(\omega) \\
&\big( E_\pi - E_\nu + \omega \big) Q_{\pi\nu}(\omega) = - (f_\nu - f_\pi) \big(
\delta {H} + {F} \big)^{\overline{11}}_{\pi\nu}(\omega) \,.
\end{aligned}$$ The additional $P$ and $Q$ amplitudes arise because the non-zero occupation probabilities allow quasiparticles to be destroyed as well as created. The new transitions introduce an additional set of QRPA eigenvalues that contain quasiparticle-energy differences rather than sums [@Sommermann1983]. It is possible for these energy differences to be negative, indicating a transition to a state of lower energy. This does not mean, however, that the QRPA fails, as it does when the eigenvalues are imaginary. The statistical FAM strength has the same form as the usual strength in Eq. , but the residues become ensemble-averaged transition strengths, and $n$ runs over the expanded set of QRPA modes. More details on the EFA-FAM and a demonstration that it includes all necessary transitions for odd states, in the context of the particle-rotor model [@Bohr1998], appear in Ref. [@Shafer2016].
\[sec:method\]Computational method
==================================
\[ssec:groundstates\]HFB ground states and functional
-----------------------------------------------------
In obtaining our global set of half-lives, we introduce a number of small improvements to the procedure of Ref. [@Mustonen2016], in addition to the changes required to compute half-lives of odd nuclei. The first is in the determination of the HFB ground state/ensemble. To make sure that we identify the correct ground state, we perform three different calculations for each even-even nucleus by constraining the first ten iterations of the HFB solver to an oblate, spherical and prolate quadrupole shape before releasing the constraint. In contrast to Ref. [@Mustonen2016], which used a set of three fixed quadrupole constraints for all nuclei, we use the first-order mass-dependent relation [@Ring2004] $$\label{eq:Q2}
Q_2 = \frac{5}{100 \pi} \beta_2 A^{5/3} \,,$$ with values $\beta_2 = -0.2, 0.0, +0.2$. This procedure gives one, two or three different deformed minima, depending on the even-even nucleus. We then identify a number of candidate quasiparticle states within 1 MeV of the Fermi surface to block in the EFA. For odd-odd nuclei we consider all possible combinations of proton and neutron candidates. For every candidate (or candidate pair), we carry out the EFA on top of each available deformed even-even core, without constraints, and select the solution with the lowest energy. On occasion these are meta-stable super-deformed states, which we discard.
We use the Skyrme functional SKO$'$ [@Reinhard1999], which was found in Ref [@Mustonen2016] to give accurate $Q$-values across the nuclear chart. We fit the like-particle pairing strengths to the experimental pairing gaps of ten isotopes picked in a wide mass range $50 \le A \le 230$, and apply an ulta-violet cutoff of 60 MeV to the single particle space. For the pnFAM portion of the calculation we set the time-odd parameters and isoscalar pairing strength to the values determined in the fit “1A” of that reference. We therefore also use the same 16-shell deformed harmonic-oscillator basis that was used in the original fit. All HFB calculations are performed with the latest version of the <span style="font-variant:small-caps;">hfbtho</span> code [@perez2017].
\[ssec:betadecay\]Beta-decay half-lives
---------------------------------------
The next set of changes concerns the computation of the beta-decay half-lives, which is discussed in detail in Ref. [@Mustonen2014]. The procedure therein allows us to sum the phase-space-weighted strengths to all energetically allowed daughter states. For allowed transitions, we obtain the rate and half-life via $$\label{eq:halflife}
\lambda = \frac{\ln2}{\kappa} \sum\limits_n f(W_n) \lvert \bra{n} F \ket{0}
\rvert^2, \quad t_{1/2} = \frac{\ln2}{\lambda} \,,$$ where $\ket{n}$ is the $n^{\text{th}}$ state in the daughter nucleus, ${W_n=E_n/m_e c^2}$ is the energy, in units of electron mass, of the electron emitted during a transition to that state, and $\kappa = 6147.0 \pm 2.4 s$. To include first-forbidden transitions, we must consider a more complicated phase-space-weighted “shape factor” [@Mustonen2014]. We evaluate the right side of the first relation in Eq. by integrating the phase-space-weighted strength (Eq. ) along a circular complex energy contour [@Mustonen2014] that encloses all the poles below the decay $Q$-value. Because the phase-space integral $f(W_n)$ is not analytic, the authors of Ref. [@Mustonen2014] fit a polynomial to the integrals on the real axis, and analytically continued the polynomial. High-degree polynomials on evenly spaced grids, however, exhibit the Runge-phenomenon [@Runge1901], and can oscillate rapidly in the complex plane. We therefore elect here to use a rational function to interpolate the phase-space integrals on a 20-point Chebychev grid. Because the contour integrand is quite smooth, we use Gauss-Legendre quadrature to perform the contour integration.
The maximum QRPA energy relevant for beta decay defines the right bound of the circular energy contour. With the treatment of $Q$-values in Refs. [@Engel1999; @Mustonen2014; @Shafer2016], the energy released in the transition to the $n^{th}$ excited state in the daughter nucleus is $$\label{eq:Qval}
Q_{\beta}^{(n)} = \Delta M_{n-H} + \lambda_n - \lambda_p - \Omega_{n} \,,$$ where $\Delta M_{n-H}$ is the neutron-hydrogen mass difference, $\lambda_{p}$ and $\lambda_{n}$ are the proton and neutron HFB Fermi energies, and $\Omega_{n}
$ ($n\geq 1$) is the excitation energy of the $n^{\rm th}$ QRPA mode above the initial-nucleus ground state, after adjustment by the Fermi energies for the change in particle number. (Note that $\Omega_1$ is the “excitation energy” of the ground state of the daughter nucleus.) The maximum QRPA energy, which corresponds to an energy release of zero is then the excitation energy of, e.g., the daughter ground state plus the energy released in the transition to that state, $$E_{\rm max}^{\rm QRPA}
= Q_{\beta}^{(1)} + \Omega_{1}
= \Delta M_{n-H} + \lambda_n - \lambda_p \,.
\label{eq:eqrpamax}$$ and can be evaluated without knowing the daughter ground-state energy itself.
The left bound of the circular energy contour must still be chosen. It must be less than $\Omega_1$, which we do not know exactly, to include all relevant poles in the response. For even-even parent nuclei we can always choose it to be zero because pairing correlations always make $\Omega_1$ positive. For odd parent nuclei, however, $\Omega_1$ can be negative. If we neglect the effects of the QRPA residual interaction, we find explicitly that $$\label{eq:Egs}
\begin{aligned}
\Omega_1^{\text{even}} &\approx E_{\pi}^{\text{smallest}} + E_{\nu}^{\text{smallest}}\\
\Omega_1^{\text{n-odd}} &\approx E_{\pi}^{\text{smallest}} - E_{\nu}^{\text{blocked}}\\
\Omega_1^{\text{p-odd}} &\approx E_{\nu}^{\text{smallest}} - E_{\pi}^{\text{blocked}}\\
\Omega_1^{\text{odd-odd}} &\approx \text{min}\big[\Omega_1^{\text{p-odd}},\ \Omega_1^{\text{n-odd}} \big] \,.
\end{aligned}$$ The fact that $\Omega_1$ can be negative makes it difficult to choose the left bound. If we expand the contour arbitrarily, we risk including beta-plus poles with non-negligible negative strength[^1], but if we do not expand it enough, the QRPA residual interaction places $\Omega_1$ outside the contour. Because the pnFAM produces the strength function in Eq. directly, we do not have access to the underlying QRPA eigenvectors and therefore cannot separate beta-minus poles from beta-plus poles. Both the inclusion of beta-plus poles or the accidental exclusion of beta-minus poles at negative energies can cause the contour integration to artificially reduce the integrated (and phase-space-weighted) beta-minus strength, and therefore artificially increase the half-lives. For lack of a better prescription, we initially choose the left bound of the contour to be $$\label{eq:contour_left_bound}
E^{\text{QRPA}}_{\text{min}} =
\text{min}\big[0, \Omega_1] \,,$$ with $\Omega_1$ given by the approximations in Eq. , but correct the rates as described below when the contour integration appears to lead to errors.
{width="2\columnwidth"}
{width="2\columnwidth"}
\[sec:results\]Results
======================
\[ssec:odd\_corrections\] Half-lives and odd-nucleus subtleties
---------------------------------------------------------------
To carry out our calculations we bundle the HFB code <span style="font-variant:small-caps;">hfbtho</span> and the charge-changing FAM code <span style="font-variant:small-caps;">pnfam</span> together with a controlling python code called <span style="font-variant:small-caps;">p$_{\textsc{Y}}$nfam</span>. We calculate the beta-minus decay half-lives of nuclei on the neutron rich side of stability, from $Z=20$ to $Z=110$, out to the one-neutron drip line. The lightest nuclei in each isotopic chain are near $A=50$, and coincide with those used in the global even-even calculation of Ref. [@Mustonen2016]. We obtain 3983 ground states, 2998 of which are odd isotopes. Reference [@Mustonen2016], which included results to the two-neutron drip line, obtained 1387 even-even ground states with the same functional, versus our 985. Our computation consumed roughly 270,000 Xeon core hours.
Our results in even-even nuclei agree very closely with those of Ref. [@Mustonen2016], with a few improvements that can be attributed to our updated procedures. As mentioned in Sec. \[ssec:betadecay\], however, our contour-integration result may be inaccurate in odd nuclei if $\Omega_1$ is less than zero. To assess the validity of the contour integration, we calculate strength functions near the real axis. Though this is a more time-consuming calculation, it allows us to locate beta-minus and beta-plus poles, determine if there are errors in the contour integration, and decide how to correct incorrect half-lives.
We identify two subsets of nuclei, shown in Fig. \[fig:hl\_corrections\] panel a), for which we perform this additional calculation. The first, indicated by red circles, is a set of 224 odd nuclei that have decay rates significantly below the average for a given $Q$-value or that contain significant negative contributions. We refer to this set as “suspicious.” The second, shown with blue squares, is a random sample of 100 odd nuclei from the remaining population. Assuming that the probability of a half-life requiring correction is uniformly distributed, this sample size allows us to estimate the proportion of half-lives that require correction with a 10% margin of error at a 95% confidence level. We find that more than half of the examined lifetimes turn out to be correct, and those that are not contain errors of two types.
The first type, illustrated by the top panels of Fig. \[fig:contour\_integration\_failure\], can be corrected by simply shifting the left bound of the contour. In the figure, the original left bound (Eq. ) is the dashed vertical line, while the corrected left bound is the solid vertical line. There are two situations which cause this type of error. The first, similar to that shown in Fig. \[fig:contour\_integration\_failure\] panel b), occurs when the HFB estimate $\Omega_1$ is negative but the residual interaction moves it to a positive number $E < \lvert \Omega_1 \rvert$. This is corrected by placing the left bound at zero. The second, illustrated in panels a) and b) of Fig. \[fig:contour\_integration\_failure\], occurs when there exists a beta-minus (beta-plus) transition at negative (positive) energy, but either the corresponding beta-minus or beta-plus strength itself is negligible. This behavior occurs almost exclusively in odd nuclei adjacent to closed shells, where pairing vanishes and the transition that takes the parent farther from the closed shell is suppressed. These cases are corrected by shifting the contour to exclude (include) beta-minus (beta-plus) poles with negligible strength.
The second type of error, exemplified by panel c) of Fig. \[fig:contour\_integration\_failure\], is more difficult to correct. Two situations can give rise to this shape in the strength distribution: the existence of a non-negligible beta-minus pole at negative energy and an associated non-negligible beta-plus pole at positive energy, or, as in panels c) and d), the existence of poles at imaginary energies. To determine if any corrections are warranted, we pinpoint the location of the poles by calculating the strength parallel to the imaginary axis out to 1 MeV. We examine the strength in each multipole, and if the original contour integration contains any errors, we integrate along a contour that surrounds only the problematic poles (and only them) to determine the correction.
We identify 60 nuclei — 54 in the suspicious set and 6 in the random sample — that require only a simple adjustment of the contour, and 41 nuclei — 26 in the suspicious set and 15 in the random sample — that require more careful corrections (33 of which have an imaginary pole in at least one multipole). The results of correcting the half-lives appear in panel b) of Fig. \[fig:hl\_corrections\]. The amount of change is indicated by the black arrowheads. Most of the arrowheads lie hidden beneath the circles or squares, usually because the problems are in forbidden multipoles that contribute only a small amount to the rate. Only a few half-lives shrink by more than an order of magnitude, when a low-lying beta-minus transition is missing from the original contour. Some half-lives increase slightly after we remove positive contributions from imaginary poles. Orange triangles in panel b) correspond to nuclei with negative total decay rates that became positive after correction.
Our random sample suggests that about 6% of our unexamined results should be corrected simply, by shifting the left bound of the contour, and about 15% may require more intricate corrections. Only a single half-life in the random sample changes by more than 5%, however (it changes by 30%). Thus, the corrections to unverified half-lives are very likely small compared to the average error in our rates (see Fig. \[fig:bayesian\_result\]). Nuclei with half-lives that require significant correction very probably belong to the suspicious set that we have just analyzed.
Finally, we should mention that numerical error is an additional source of small negative contributions to rates. Both the HFB and FAM solutions contain numerical error from several sources, e.g., incomplete convergence, truncation, etc. These errors are compounded in the final strength function and amplified by the phase space. If a rate is very small, the contour integral that generates it can suffer from incomplete cancellation of large oscillations. In compiling our final table of half-lives, presented here as supplemental material [@supplemental], we break each rate into contributions from each multipole, set any negative contributions to zero, and re-sum. This procedure usually changes rates by less than $5\%$.
![\[fig:ensdf\_comparison\_corrections\] Same as panel b) of Fig. \[fig:hl\_corrections\] but compared with 2019 ENSDF data. Only odd nuclei are shown.](Fig3){width="1\columnwidth"}
In Fig. \[fig:ensdf\_comparison\_corrections\] we compare our final results with 2019 ENSDF experimental data [@ENSDF2019] for nuclei with experimental half-lives less than $10^{6}$ s. We highlight half-lives that are corrected, as in Fig. \[fig:hl\_corrections\] panel b), and find that corrections almost always improve the agreement with experiment. The majority of our data fall within one or two orders of magnitude of experiment for half-lives less than 1000 s. In the next section, we will quantify more rigorously the theoretical uncertainties associated with such calculations.
Figure \[fig:first\_forbidden\_contributions\] displays the contributions to decay rates of first-forbidden operators. We find, as do other groups, that first-forbidden contributions are important in many nuclei and observe competing effects: forbidden contributions scale with the nuclear radius and $Q$-value, becoming important in heavier nuclei far from stability, but they also become important near stability and closed shells where the allowed rate is very small and allowed contributions are suppressed.
![\[fig:first\_forbidden\_contributions\] First-forbidden contribution to the rates. ](Fig4){width="1\columnwidth"}
\[ssec:error\]Error analysis
----------------------------
One major challenge facing large scale calculations is the quantification of uncertainty. Most of the nuclei considered here are not experimentally accessible, and so we lack an experimental benchmark with which to evaluate our calculations. A simple way to deal with this challenge is to develop a *model* for the error. The model can be fit to data where available, and then extrapolated or interpolated to estimate errors for the remaining data. We use the simple model developed in Ref. [@Mustonen2016], which we summarize here. The error parameter of interest is, for the $i^{th}$ nucleus [@Moller2003], $$\label{eq:error_parameter_r}
r_i = \log_{10} \bigg(\frac{t_{\text{th}}}{t_{\text{exp}}}\bigg) \,.$$ To motivate a regression model for this parameter, we assume that there is a single dominant transition to a state near the daughter ground state, and that the forbidden shape factors depend much less on the $Q$-value than does the allowed phase space. These assumptions allow us to assign a single effective $Q$-value and shape factor $C_{\rm eff}$ to the decay; cf. [@Mustonen2014] for the definition of the shape factor $C$. Using $q_{\rm eff}$ to denote the effective $Q$-value in units of electron mass ($q_{\rm eff} = Q_{\rm eff} / m_e
c^2$), we model the error $r_i$ on the rate, as a function of the theoretical $Q$-value and charge of the daughter nucleus, as, $$r_i(q_{\text{g.s.}}^{\text{th}},Z_f) \approx c_{r_i} +
f_r(q_{\text{g.s.}}^{\text{th}}+1,Z_f) q_{r_i} \,,$$ where the errors in the effective shape factor, $c_{r}$, and the effective $Q$-value, $q_{r}$, are defined by $$c_r \equiv
\log_{10}\frac{C^{\text{exp}}_{\text{eff}}}{C^{\text{th}}_{\text{eff}}}, \quad
q_r \equiv \frac{q_{{\text{eff}}}^{{\text{exp}}} -
q_{\text{eff}}^{\text{th}}}{\ln10}\,,$$ and the $Q$-value dependence is carried by the phase space factor, $$f_r(q+1,Z_f) \equiv \frac{1}{f(q+1,Z_f)} \frac{df(q+1,Z_f)}{dq} \,.$$
Next, we assume that the $c_{r_i}$ and $q_{r_i}$, which depend on the nucleus $i$, are each normally distributed random variables with widths that are independent of the $Q$-value, and that the distributions for $c_{r_i}$ and $q_{r_i}$ contain a systematic bias that is independent of the nucleus and the $Q$-value. These assumptions allow us to write the error parameters for nucleus $i$ in the form $$\begin{aligned}
c_{r_i} &= b_c + \epsilon_{c}, \quad\epsilon_{c} \sim\mathcal{N}(0,\sigma_{c})\,, \\
q_{r_i} &= b_q + \epsilon_{q}, \quad\epsilon_{c} \sim\mathcal{N}(0,\sigma_{q})\,,
\end{aligned}$$ where $b_{c}, b_{q}, \sigma_{c}, \sigma_{q}$ are still undetermined parameters. Finally, since the assumptions of the model are best for large $Q$-values, we can make use of the Primakoff-Rosen approximation to the allowed phase space [@Suhonen2007], which lets us express $f_r(q+1,Z_f)$ as a simple rational function with no explicit dependence on the charge $Z_f$ of the daughter nucleus: $$\begin{gathered}
f_r(q+1,Z_f) \approx f_{r}^P(q) \\
\equiv \frac{5 (q+1)^4 - 20 (q+1) + 15}{(q+1)^5 - 10 (q+1)^2 + 15 (q+1) - 6} \,.\end{gathered}$$ We then end up with a one dimensional, non-linear error model with noise: $$r_i(q_{\text{g.s.}}^{\text{th}}) = b_c + f_{r}^{P}(q_{\text{g.s.}}^{\text{th}})b_{q}
+ \epsilon_i, \quad \epsilon_i \sim \mathcal{N}(0, \sigma_{r}) \,.$$ Because $c_{r}$ and $q_r$ are independent, their widths add in quadrature. We find, however, that only $\sigma_q$ is important and therefore take the width of the total noise term to be $$\sigma_r(q_{\text{g.s.}}^{\text{th}}) = \sqrt{\sigma_c^2 +
(f_r^P(q_{\text{g.s.}}^{\text{th}}) \sigma_q)^2} \approx
f_r^P(q_{\text{g.s.}}^{\text{th}}) \sigma_q \,.$$ That leaves three unknown parameters $b_c, b_q, \sigma_q$ to be determined.
To estimate the parameters, we use our own python adaptation of a Metropolis Monte Carlo code from Ref. [@bailer-jones2017] to sample the unnormalized Bayesian posterior distributions of ${\beta = \tan^{-1}{b}}$ and $\sigma_q$, with priors $$\label{eq:priors}
\begin{aligned}
P(\beta_{c}) &= P(\beta_{q}) = \frac{1}{2\pi} \\
P(\sigma_{q}) &\propto \log(\sigma_{q}) \,.
\end{aligned}$$ The sampling probability distribution is a multivariate Gaussian with a variance of $(0.02)^2$ for all three parameters. Following a burn-in period of $200,000$ steps, we retain every 100$^{\text{th}}$ iteration from the next million steps to reduce autocorrelation. From Gaussian kernel density estimates of the resulting distributions we esimate the most likely values to be $b_c=0.049$, $b_q=-0.082$, and $\sigma_q=1.807$. Figure \[fig:bayesian\_result\] shows the resulting confidence regions on top of our entire data set. We find hardly any bias, indicating that our half-lives are equally likely to be over- and under-predicted. The model is not reliable for very small $Q$-values, but for moderate to large $Q$-values it predicts that the majority of our calculated half-lives will differ from experiment by less than one order of magnitude. The data is slightly non-Gaussian, with the one and two standard deviation bands capturing $76\%$ and $94\%$ of the 718 data points, respectively.
![\[fig:bayesian\_result\] Bayesian fit to the bias function and one- and two-standard-deviation bands.](Fig5){width="1\columnwidth"}
\[ssec:comparisons\]Comparisons
-------------------------------
To evaluate our data where experimental values are unavailable, we compare our results to those of other global beta-decay calculations. The authors of Ref. [@Homma1996] (labeled “Homma” in Fig. \[fig:quality\_measures\]) conducted a microscopic pnQRPA calculation with schematic allowed and unique first-forbidden interactions, and treated odd nuclei perturbatively. Reference [@Nakata1997] (labeled “Nakata”) carried out a macroscopic calculation within the semi-gross theory. Reference [@Moller2003] (labeled Möller) combined microscopic and macroscopic approaches, using the finite-range droplet model for ground state properties, the pnQRPA with an empirical spreading for Gamow-Teller strength, and the gross theory for first-forbidden contributions. More recently, Ref. [@Costiris2009] (labeled “Costiris”) applied a neural network to predict half-lives. Finally, Ref. [@Marketin2016] (labeled “Marketin”) conducted a fully self-consistent covariant pnQRPA calculation with local fits to the isoscalar pairing strength, treating odd nuclei as if they were fully paired even nuclei with an odd number of nucleons on average.
![\[fig:quality\_measures\] Comparison of error-evaluation parameters among results of Refs. [@Marketin2016] (Marketin), [@Moller2003] (Möller), [@Costiris2009] (Costiris), [@Nakata1997] (Nakata), and [@Homma1996] (Homma).](Fig6){width="\columnwidth"}
{width="2\columnwidth"}
To compare our results to those of the other papers, we use the quality measures outlined, e.g., in Ref. [@Moller2003]: the mean $(M_r)$ and standard deviation $(\sigma_r)$ of the error parameter in Eq. , $$\label{eq:quality_measures}
M_r = \frac{1}{n} \sum\limits_{i=1}^n r_i, \quad \sigma_r = \bigg[ \frac{1}{n}
\sum\limits_{i=1}^n (r_i - M_r)^2 \bigg]^{1/2}\,.$$ We present these measures for the set of nuclei with experimental half-lives less than 1000 s, 100 s, 1 s, 0.5 s, 0.2 s, and 0.1 s. For Refs. [@Homma1996; @Nakata1997; @Costiris2009] we take the measures directly from the corresponding paper. References [@Moller2003; @Marketin2016] supplied their data set as supplemental material, and we recompute the quality measures with the more recent 2019 ENSDF experimental half-lives [@ENSDF2019]. Figure \[fig:quality\_measures\] summarizes the results. The differences in experimental data sets considered in each paper can be seen in part by noting the number of data points used to compute the quality measures. The errors for Ref. [@Marketin2016] are somewhat larger for long-lived isotopes than the values given in that paper because we include all the calculations in odd nuclei, while the authors excluded a few that they considered outliers. In general, our calculation is comparable in fidelity to the others. Unlike those, however, its treatment of odd nuclei is fully self-consistent, capturing in part the one-quasiparticle nature of such states through the EFA, and it uses a single energy functional with no local adjustments. Figure \[fig:nuclear\_chart\_hl\_comparisons\] compares all our results with those provided in Refs. [@Moller2003; @Marketin2016]. We generally predict longer half-lives than the other two models in heavier nuclei, and slightly shorter half-lives in lighter nuclei. The vast majority of our numbers fall within one order of magnitude of those of Ref. [@Moller2003]. Both we and Ref. [@Moller2003] predict significantly longer half-lives in heavy isotopes than does Ref. [@Marketin2016]. There do not appear to be any other significant systematic differences among the results.
\[sec:conclusions\]Conclusions
==============================
Using the statistical extension of the charge-changing finite amplitude method, we computed beta-decay half-lives of almost all odd-mass and odd-odd nuclei on the neutron-rich side of stability, in a fully microscopic and self-consistent way. The equal filling approximation allows us to retain time-reversal symmetry while sill largely including the effects of core polarization by the odd nucleon. We showed that in a few cases the EFA leads to the appearance of negative and even imaginary eigenvalues. Overall our half-lives are similar to those of other global calculations in reproducing experimental data. We supplemented these calculations with an estimate of theoretical uncertainties, which suggest that calculated half-lives fall within two orders of magnitude of experimental values for nuclei with $Q$-values greater than about 2 MeV. We also find, as do other groups, that first-forbidden contributions are important in many nuclei. We provided all the half-lives described here, along with associated ground-state properties, error estimates, and Gamow-Teller strength distributions, in the supplemental material [@supplemental].
We plan to extend our methods in several ways:
- We will use the statistical FAM with the grand canonical ensemble for finite temperature beta-decay calculations. Decay at non-zero temperature plays an important role in neutron-star mergers and core-collapse supernovae [@Langanke2000; @Langanke2001].
- We will improve the ability of the FAM to capture low-energy strength by including correlations beyond the QRPA. Although one must be careful in combining such correlations with density functionals, several procedures exist for doing so [@Gam15; @Robin16; @Niu2018]. An efficient implementation of an extension to the FAM would allow better global calculations.
- Finally, we will better treat the weak interaction. Here we restrict ourselves to the impulse approximation, neglecting many-body currents completely. Recent work shows that such currents account for a significant fraction of the quenching of Gamow-Teller strength [@Gysbers2019]. With an additional extension of the pnFAM we can take two-body currents into account.
Our calculations are also an important milestone in the development of a consistent description of the fission process within nuclear DFT [@schunck2016]. Although spontaneous fission-fragment half-lives, fragment distributions, and fragment excitation energies can already be computed in DFT, our work paves the way to for a description of the deexcitation of the fragments, including gamma emission and beta decay, within the same framework.
\[sec:acknowledgments\]Acknowledgments {#secacknowledgmentsacknowledgments .unnumbered}
======================================
Many thanks to M. Mustonen and T. Shafer, for guidance on the pnFAM, and to S. Guilliani for helpful discussions on beta decay. This work was supported in part by the Nuclear Computational Low Energy Initiative (NUCLEI) SciDAC-4 project under U.S. Department of Energy grant DE-SC0018223 and the FIRE collaboration. Some of the work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344. Computing support came from the Lawrence Livermore National Laboratory (LLNL) Institutional Computing Grand Challenge program.
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[^1]: Poles are symmetric around zero, so as soon as beta-minus strength appears at negative energy, some beta-plus strength (inverted in sign) appears at positive energy.
|
---
abstract: 'On 2018 July 28, GRB 180728A triggered *Swift* satellites and, soon after the determination of the redshift, we identified this source as a type II binary-driven hypernova (BdHN II) in our model. Consequently, we predicted the appearance time of its associated supernova (SN), which was later confirmed as SN 2018fip. A BdHN II originates in a binary composed of a carbon-oxygen core (CO$_{\rm core}$) undergoing SN, and the SN ejecta hypercritically accrete onto a companion neutron star (NS). From the time of the SN shock breakout to the time when the hypercritical accretion starts, we infer the binary separation $\simeq 3 \times 10^{10}$ cm. The accretion explains the prompt emission of isotropic energy $\simeq 3 \times 10^{51}$ erg, lasting $\sim 10$ s, and the accompanying observed blackbody emission from a thermal convective instability bubble. The new neutron star ($\nu$NS) originating from the SN powers the late afterglow from which a $\nu$NS initial spin of $2.5$ ms is inferred. We compare GRB 180728A with GRB 130427A, a type I binary-driven hypernova (BdHN I) with isotropic energy $> 10^{54}$ erg. For GRB 130427A we have inferred an initially closer binary separation of $\simeq 10^{10}$ cm, implying a higher accretion rate leading to the collapse of the NS companion with consequent black hole formation, and a faster, $1$ ms spinning $\nu$NS. In both cases, the optical spectra of the SNe are similar, and not correlated to the energy of the gamma-ray burst. We present three-dimensional smoothed-particle-hydrodynamic simulations and visualisations of the BdHNe I and II.'
author:
- 'Y. Wang'
- 'J. A. Rueda'
- 'R. Ruffini'
- 'C. Bianco'
- 'L. Becerra'
- 'L. Li'
- 'M. Karlica'
bibliography:
- '180728A.bib'
title: 'Two Predictions of supernova: GRB 130427A/SN 2013cq and GRB 180728A/SN 2018fip'
---
Introduction {#sec:1}
============
By the first minutes of data retrieved from Konus-*Wind*, *Swift*, *Fermi*, *AGILE* or other gamma-ray telescopes , and the determination of redshift by VLT/X-shooter, Gemini, NOT or other optical telescopes , it is possible to promptly and uniquely identify to which of the nine (9) subclasses of gamma-ray bursts (GRBs) a source belongs (See table \[tab:GRBsubclasses\] and the references therein). Consequently, it is possible to predict its further evolution, including the possible appearance time of an associated supernova (SN) expected in some of the GRB subclasses. This is what we have done in the case of GRB 130427A [@2015ApJ...798...10R; @2018ApJ...869..101R], and in the present case of GRB 180728A.
GRB 130427A is a BdHN I in our model, details in section \[sec:bdhn\] and in @2014ApJ...793L..36F [@2015PhRvL.115w1102F; @2015ApJ...812..100B; @2016ApJ...833..107B; @2018ApJ...852..120B]. The progenitor is a tight binary system, of orbital period $\sim 5$ min, composed of a carbon-oxygen core (CO$_{\rm core}$), undergoing a SN event, and a neutron star (NS) companion accreting the SN ejecta and finally collapsing to a black hole (BH). The involvement of a SN in BdHN I and the low redshift of $z=0.34$ [@2013GCN..14455...1L; @2013GCN..14478...1X; @2013GCN..14491...1F] enable us to predict that the optical signal of the SN will peak and be observed $\sim 2$ weeks after the GRB occurrence at the same position of the GRB [@2013GCN..14526...1R]. Indeed the SN was observed [@2013GCN..14646...1D; @2013ApJ...776...98X]. Details of GRB 130427A are given in section \[sec:130427A\].
The current GRB 180728A is a BdHN II in our model; it has the same progenitor as BdHN I, a binary composed of a CO$_{\rm core}$ and a NS companion, but with longer orbital period ($\gtrsim 10$ min), which is here determined for the first time. The CO$_{\rm core}$ undergoes SN explosion, the SN ejecta hypercritically accrete onto the companion NS. In view of the longer separation, the accretion rate is lower, it is not sufficient for the companion NS to reach the critical mass of BH. Since a SN is also involved in BdHN II and this source is located at low redshift $z=0.117$ [@GCN23055], its successful prediction and observation were also possible and it is summarised in section \[sec:180728A\_observation\]. From a time-resolved analysis of the data in section \[sec:picture\_and\_data\], we trace the physical evolution of the binary system. For the first time we observed a $2$ s signal evidencing the SN shockwave, namely the emergence of the SN shockwave from the outermost layers of the CO$_{\rm core}$ [see e.g. @1996snih.book.....A]. The SN ejecta expand and, after $10$ s, reach the companion NS inducing onto it a high accretion rate of about $10^{-3} M_\odot$ s$^{-1}$. Such a process lasts about $10$ s producing the prompt phenomena and an accompanying thermal component. The entire physical picture is described in section \[sec:picture\_of\_physics\], giving special attention to the new neutron star ($\nu$NS) originating from the SN. We explicitly show that the fast spinning $\nu$NS powers the afterglow emission by converting its rotational energy to synchrotron emission [see also @2018ApJ...869..101R], which has been never well considered in previous GRB models. We compare the initial properties of the $\nu$NS in GRB 130427A and in GRB 180728A, and derive that a $1$ ms $\nu$NS is formed in GRB 130427A while a $2.5$ ms $\nu$NS is formed in GRB 180728A. In section \[sec:neutron\_star\_charactristic\] we relate the very different energetic of the prompt emission to the orbital separation of the progenitors, which in turn determines the spin of the $\nu$NS, and the rest-frame luminosity afterglows. We simulate the accretion of the SN matter onto the NS companion in the tight binaries via three-dimensional (3D) smoothed-particle-hydrodynamic (SPH) simulations [@2018arXiv180304356B] that provide as well a visualisation of the BdHNe. The conclusions are given in section \[sec:conclusion\].
Binary-driven hypernova {#sec:bdhn}
=======================
Since the Beppo-SAX discovery of the spatial and temporal coincidence of a GRB and a SN , largely supported by many additional following events , a theoretical paradigm has been advanced for long GRBs based on a binary system [@2012ApJ...758L...7R]. It differs from the traditional theoretical interpretation of GRB which implicitly assumes that all GRBs originate from a BH with an ultra-relativistic jet emission .
Specifically, the binary system is composed by a CO$_{\rm core}$ and a NS companion in tight orbit. Following the onset of the SN, a hypercritical accretion process of the SN ejecta onto the NS occurs which markedly depends on the binary period of the progenitor [@2014ApJ...793L..36F; @2015PhRvL.115w1102F; @2015ApJ...812..100B; @2016ApJ...833..107B]. For short binary periods of the order of $5$ min the NS reaches the critical mass for gravitational collapse and forms a BH . For longer binary periods, the hypercritical accretion onto the NS is not sufficient to bring it to the critical mass and a more massive NS (MNS) is formed. These sources have been called BdHNe since the feedback of the GRB transforms the SN into a hypernova (HN) [@2017arXiv171205001R]. The former scenario of short orbital period is classified as BdHN type I (BdHN I), which leads to a binary system composed by the BH, generated by the collapse of the NS companion, and the $\nu$NS generated by the SN event. The latter scenario of longer orbital period is classified as BdNH type II (BdHN II), which leads to a binary NS system composed of the MNS and the $\nu$NS.
Having developed the theoretical treatment of such hypercritical process, and considering as well other binary systems with progenitors composed alternatively of CO$_{\rm core}$ and BH, to NS and white dwarf (WD), a general classification of GRBs has been developed; see @2016ApJ...832..136R and Table \[tab:GRBsubclasses\] for details. We report in the table estimates of the energetic, spectrum and different component of the prompt radiation, of the plateau, and all the intermediate phases, all the way to the final afterglow phase. The GRBs are divided in two main classes, the BdHNe, which cover the traditional long duration GRBs [@1993ApJ...405..273W; @Paczynski:1998ey], and the binary mergers, which are short-duration GRBs [@1986ApJ...308L..47G; @1986ApJ...308L..43P; @Eichler:1989jb]. There are currently nine subclasses in our model, the classification depends on the different compositions of the binary progenitors and outcomes, which are CO$_{\rm core}$ and compact objects as BH, NS, and WD. The same progenitors are possible to produce different outcomes, due to the different masses and binary separations.
--------------- ------ ----------------- ---------- -------------------- ------------- ------------------- -------------------------- ------------------- --
Class Type Previous Number *In-state* *Out-state* $E_{\rm p,i}$ $E_{\rm iso}$ $E_{\rm iso,Gev}$
Alias (MeV) (erg) (erg)
Binary Driven I BdHN $329$ CO$_{\rm core}$-NS $\nu$NS-BH $\sim0.2$–$2$ $\sim 10^{52}$–$10^{54}$ $\gtrsim 10^{52}$
Hypernova II XRF $(30)$ CO$_{\rm core}$-NS $\nu$NS-NS $\sim 0.01$–$0.2$ $\sim 10^{50}$–$10^{52}$ $-$
(BdHN) III HN $ (19) $ CO$_{\rm core}$-NS $\nu$NS-NS $\sim 0.01$ $\sim 10^{48}$–$10^{50}$ $-$
IV BH-SN $5$ CO$_{\rm core}$-BH $\nu$NS-BH $\gtrsim2$ $>10^{54}$ $\gtrsim 10^{53}$
I S-GRF $18$ NS-NS MNS $\sim0.2$–$2$ $\sim 10^{49}$–$10^{52}$ $-$
Binary II S-GRB $6$ NS-NS BH $\sim2$–$8$ $\sim 10^{52}$–$10^{53}$ $\gtrsim 10^{52}$
Merger III GRF $(1)$ NS-WD MNS $\sim0.2$–$2$ $\sim 10^{49}$–$10^{52}$ $-$
(BM) IV FB-KN$^{\star}$ $(1)$ WD-WD NS/MWD $ < 0.2$ $< 10^{51}$ $-$
V U-GRB $(0)$ NS-BH BH $\gtrsim2$ $>10^{52}$ $-$
--------------- ------ ----------------- ---------- -------------------- ------------- ------------------- -------------------------- ------------------- --
GRB 130427A as BdHN I {#sec:130427A}
=====================
GRB 130427A, as a BdHN I in our model, has been studied in our previous articles [@2015ApJ...798...10R; @2018ApJ...869..101R]. This long GRB is nearby ($z=0.314$) and energetic ($E_{iso} \sim 10^{54}$ erg) [@2013GCN..14455...1L; @2013GCN..14478...1X; @2013GCN..14491...1F; @2014Sci...343...48M]. It has overall the most comprehensive data to date, including the well observed $\gamma$-ray prompt emission [@2013GCN.14473....1V; @GCN14487], the full coverage of X-ray, optical and radio afterglow [@2013ApJ...779L...1K; @2014ApJ...781...37P; @2014Sci...343...38V; @2014MNRAS.444.3151V; @2014ApJ...792..115L; @2014MNRAS.440.2059A; @2017ApJ...837..116B], and the long observation of the ultra-high energy emission (UHE) [@2013ApJ...771L..13T; @2014Sci...343...42A; @2015ApJ...800...78A]. Also it has been theoretically well-studied, involving many interpretations, including: a black hole or a magnetar as the central engine [@2014MNRAS.439L..80B]; an unaccountable temporal spectral behaviors of the first $2.5$ s pulse by the traditional models [@2014Sci...343...51P]; the reverse-forward shock synchrotron model and its challenges in explaining the afterglow [@2013ApJ...776..119L; @2016ApJ...818..190F; @2016MNRAS.462.1111D; @2017Galax...5....6D]; the synchrotron or the inverse Compton origins for the ultra-high energy photons [@2013ApJ...773L..20L; @2013ApJ...776...95F; @2013MNRAS.436.3106P; @2014IJMPS..2860174T; @2014ApJ...789L..37V]; the missing of the neutrino detection and its interpretation [@2013ApJ...772L...4G; @2016MNRAS.458L..79J]. Our interpretation is alternative to the above traditional approach: 1) long GRBS are traditionally described as single systems while we assume a very specific binary systems as their progenitors. 2) The roles of the SN and of the $\nu$NS are there neglected, while they are essential in our approach as evidenced also in this article. 3) A central role in the energetics is traditionally attributed to the kinetic energy of ultra-relativistic blast waves extending from the prompt phase all the way to the late phase of the afterglow, in contrast to model-independent constraints observed in the mildly relativistic plateau and afterglow phases [@2015ApJ...798...10R; @2018ApJ...869..151R; @2018ApJ...869..101R]. In our approach the physics of the $e^+e^-$ plasma and its interaction with the SN ejecta as well as the pulsar-like behaviour of the $\nu$NS are central to the description from the prompt radiation to the late afterglow phases [@2018ApJ...852...53R]. One of the crucial aspects in our approach is the structure of the SN ejecta which, under the action of the hypercritical accretion process onto the NS companion and the binary interaction, becomes highly asymmetric. Such a new morphology of the SN ejecta has been made possible to be visualized thanks to a set of three-dimensional numerical simulations of BdHNe [@2014ApJ...793L..36F; @2016ApJ...833..107B; @2018arXiv180304356B].
On this ground, soon after the observational determination of the redshift [@2013GCN..14455...1L], by examining the detailed observations in the early days, we identified the BdHN origin of this source. On 2013 May 2, we made the prediction of the occurrence of SN 2013cq on GCN [@2013GCN..14526...1R quoted in appendix \[sec:gcns\]], which was duly observed in the optical band on 2013 May 13 [@2013GCN..14646...1D; @2013ApJ...776...98X].
To summarize our work on this GRB: in @2015ApJ...798...10R we presented the multiwavelength light curve evolution and interpreted them by a tight binary system with orbital separation $\sim 10^{10}$ cm. GRB 130427A has a very bright prompt $\gamma$-ray spike in the first $10$ s, then it decays, coinciding with the rising of the UHE (100 MeV$−-$100 GeV) emission. The UHE peaks at $\sim 20$ s, then gradually dims for some thousand seconds. Soft X-ray observations start from $195$ s, it has a steep decay then follows a normal power-law decay $\sim t^{-1.3}$. We evidenced the presence of a blackbody component in the soft X-ray data in the time-interval from $196$ s to $461$ s; within which the temperature decreases from $0.5$ keV to $0.1$ keV. The thermal component indicates an emitter expanding from $\sim 10^{12}$ cm to $\sim 10^{13}$ cm with velocity $\sim 0.8~c$. This mildly relativistic expansion from our model-independent inference contrasts with the traditional ultrarelativistic external shockwave interpretation [see e.g. @1998ApJ...497L..17S]. We attributed this thermal emission to the transparency of the SN ejecta outermost layer after being heated and accelerated by the energetic $e^+e^-$ plasma outflow of the GRB. The numerical simulations of this hydrodynamics process were presented in @2018ApJ...852...53R. As it is shown there, the resulting distance, velocity, and occurring time of this emission are all in agreement with the observations. Later in @2018ApJ...869..101R, we showed that the mildly relativistic ejecta can also account for the nonthermal component, in the early thousands of seconds powered by its kinetic energy, and afterward powered by the release of rotational energy of the millisecond-period $\nu$NS via a pulsar-like mechanism. The synchrotron emission well reproduces the observed optical and X-ray afterglow. A similar application of the $\nu$NS on GRB 180728A will be presented in section \[sec:afterglow\_from\_pulsar\], as well as the comparison to GRB 130427A.
Observation and Prediction {#sec:180728A_observation}
==========================
On 2018 July 28, we had the opportunity to make a prediction of the SN appearance in a BdHN II.
At 17:29:00 UT, On 2018 July 28, GRB 180728A triggered the *Swift*-BAT. The BAT light curve shows a small precursor and $\sim 10$ s later it was followed by a bright pulse of $\sim 20$ s duration [@GCN23046]. *Swift*-XRT did not slew to the position immediately due to the Earth limb, it began observing $1730.8$ s after the BAT trigger [@GCN23049]. The *Fermi*-GBM triggered and located GRB 180728A at 17:29:02.28 UT. The initial *Fermi*-LAT bore-sight angle at the GBM trigger time is 35 degrees, within the threshold of detecting GeV photons, but no GeV photon was found. The GBM light curve is similar to the one of Swift-BAT, consisting of a precursor and a bright pulse, the duration ($T_{90}$) is about $6.4$ s ($50$–$300$ keV) [@GCN23053]. A red continuum was detected by VLT/X-shooter and the absorption features of Mg II (3124, 3132), Mg I (3187), and Ca II (4395, 4434) were consistent with a redshift of $z=0.117$ [@GCN23055].
After the detection of the redshift, On 2018 July 31, we classify this GRB as an BdHN II in our model, based on its duration, peak energy, isotropic energy, and the existence of photons with energy $> 100$ MeV, criteria in table \[tab:GRBsubclasses\]. BdHN II involves the Type Ib/c supernova phenomenon, therefore, we predicted that a SN would appear at $14.7\pm 2.9$ days [@GCN23066] and be observed due to its low redshift. On 18 Augest 2018, @GCN23142 on behalf of the VLT/X-shooter team reported the discovery of the SN appearance, which was confirmed in @GCN23181. The text of these GCNs are reported in the appendix \[sec:gcns\]. The SN associated with GRB 180728A was named as SN 2018fip. Our prediction was confirmed.
We also predicted the supernova appearance in GRB 140206A [@2014GCN.15794....1R] and GRB 180720A [@GCN23019], but unfortunately the optical observation does not cover the expected time ($\sim 13$ days after the GRB trigger time) of the supernova appearance.
Data Analysis {#sec:picture_and_data}
=============
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GRB 180728A contains two spikes in the prompt emission observed by *Swift*-BAT, *Fermi*-GBM and Konus-*Wind* [@GCN23046; @GCN23053; @GCN23055]. In the following we defined our $t_0$ based on the trigger time of *Fermi*-GBM. The first spike, we name it as precursor, ranges from $-1.57$ s to $1.18$ s. And the second spike, which contains the majority of energy, rises at $8.72$ s, peaks at $11.50$ s, and fades at $22.54$ s, see figure \[fig:prompt\_all\_180728A\]. These time definitions are based on the count rate light curve observed by *Fermi*-GBM, and determined by applying the Bayesian block method [@1998ApJ...504..405S]. *Swift*-XRT started to observe $1730.8$ s after the BAT trigger, the luminosity of the X-ray afterglow follows a shallow decay with a power-law index $-0.56$ till $\sim 5000$ s, then a normal decay with a power-law index $-1.2$, which is a typical value [@2015ApJ...805...13L; @2018ApJS..234...26L].
Prompt Emission: Two Spikes {#sec:first_spike}
---------------------------
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The first spike, the precursor, shows a power-law spectrum with a power-law index $-2.31\pm 0.08$ in its $2.75$ s duration, shown in Fig. \[fig:prompt\_spectrum\] and in the appendix \[sec:model\_comparison\]. The averaged luminosity is $3.24^{+0.78}_{-0.55} \times 10^{49}$ erg s$^{-1}$, and the integrated energy gives $7.98^{+1.92}_{-1.34}\times 10^{49}$ erg in the energy range from $1$ keV to $10$ MeV, the Friedmann-Lemaitre-Robertson-Walker metric with the cosmological parameters from Planck mission [@2018arXiv180706209P][^1] are applied on computing the cosmological distance throughout the whole paper.
--------------- ---------------------------------------------- ---------------------------------------------- ------------------------ ---------------------
**Time** **Total Flux** **Thermal Flux** **Percentage** **Temperature**
(s) ($\text{erg}~ \text{s}^{-1} \text{cm}^{-2}$) ($\text{erg}~ \text{s}^{-1} \text{cm}^{-2}$) (keV)
8.72 - 10.80 $5.6^{+1.1}_{-0.9} \times 10^{-6}$ $4.1^{+3.2}_{-1.9} \times 10^{-7}$ $7.3^{+5.8}_{-3.7} \%$ $7.9^{+0.7}_{-0.7}$
10.80 - 12.30 $2.0^{+0.1}_{-0.1} \times 10^{-5}$ $7.1^{+6.0}_{-3.3} \times 10^{-7}$ $3.6^{+3.3}_{-1.6} \%$ $5.6^{+0.5}_{-0.5}$
--------------- ---------------------------------------------- ---------------------------------------------- ------------------------ ---------------------
The second spike rises $10.29$ s after the starting time of the first spike ($8.72$ s since the trigger time), lasts $13.82$ s and emits $2.73^{+0.11}_{-0.10}\times 10^{51}$ erg in the $1$ keV–$10$ MeV energy band, i.e. $84$ times more energetic than the first spike. The best fit of the spectrum is a Band function or a cutoff power-law, with an additional blackbody; see table \[tab:promptModel\] in the appendix \[sec:model\_comparison\] for the model comparison of the time resolved analysis and figure \[fig:prompt\_spectrum\] for the spectrum. We notice that the thermal component confidently exists in the second spike when the emission is luminous while, at times later than $12.30$ s, the confidence of the thermal component drops and a single cutoff power-law is enough to fit the spectrum. There could be many reasons for the missing thermal component at later times; for instance, the thermal component becomes less prominent and is covered by the non-thermal emission, or the thermal temperature cools to a value outside of the satellite energy band, or the thermal emission really disappears. In the present case the thermal blackbody component of temperature $\sim 7$ keV contributes $\sim 5\%$ to the total energy.
From the evolution of the thermal spectrum and the parameters presented in table \[tab:thermalFlux\], it is possible to determine the velocity and the radius of the system in a model-independent way. Following @2018ApJ...852...53R, we obtain that the radius in each of the two time intervals is $1.4^{+0.6}_{-0.4}\times 10^{10}$ cm and $4.3^{+0.9}_{-0.6} \times 10^{10}$ cm respectively, and the expanding velocity is $0.53^{+0.18}_{-0.15}~c$.
Supernova {#subsec:supernova}
---------
![Spectra comparison of three SNe: 1998bw, 2010bh, 2013cq, flux density is normalised at 10 parsec, data are retrieved from the Wiserep website (<https://wiserep.weizmann.ac.il>).[]{data-label="fig:sn2018fip"}](SNe){width="1.0\hsize"}
The optical signal of SN 2018fip associated with GRB 180728A was confirmed by the observations of the VLT telescope [@GCN23142; @GCN23181]. The SN 2018fip is identified as a Type Ic SN, its spectrum at $\sim 8$ days after the peak of the optical light curve matches with the Type Ic SN 2002ap [@2002ApJ...572L..61M], reported in @2018TNSCR1249....1S. In @GCN23142, there is the comparison of SN 2018fip with SN 1998bw and SN 2010bh, and in @2013ApJ...776...98X, there is the comparison of the SN associated with GRB 130427A, SN 2013cq, with SN 1998bw and SN 2010bh, associated with GRB 980425 and GRB 100316D [@2003ApJ...599L..95M], respectively. We show in figure \[fig:sn2018fip\] the spectral comparison of SN 1998bw, SN 2010bh, and SN 2013cq. We may conclude that the SNe are similar, regardless of the differences, e.g. in energetics ($\sim 10^{54}$ erg versus $\sim 10^{51}$ erg), of their associated GRBs (BdHN I versus BdHN II).
Physical Interpretation {#sec:picture_of_physics}
=======================
All the observations in section \[sec:picture\_and\_data\] can be well interpreted within the picture of a binary system initially composing a massive CO$_{\rm core}$ and a NS.
Prompt emission from a binary accretion system
----------------------------------------------
At a given time, the CO$_{\rm core}$ collapses forming a $\nu$NS at its center and producing a SN explosion. A strong shockwave is generated and emerges from the SN ejecta. A typical SN shockwave carries $\sim 10^{51}$ erg of kinetic energy [@1996snih.book.....A], which is partially converted into electromagnetic emission by sweeping the circumburst medium (CBM) with an efficiency of $\sim 10\%$ [see e.g. @2012SSRv..173..309B]. Therefore, the energy of $\sim 10^{50}$ erg is consistent with the total energy in the first spike. The electrons from the CBM are accelerated by the shockwave via the Fermi mechanism and emit synchrotron emission which explains the non-thermal emission with a power-law index $-2.31$ in the first spike.
The second spike with thermal component is a result of the SN ejecta accreting onto the companion NS. The distance of the binary separation can be estimated by the delay time between the two spikes, $\sim 10$ s. Since the outer shell of the SN ejecta moves at velocity $\sim 0.1~c$ [@2017AdAst2017E...5C], we estimate a binary separation $\approx 3\times 10^{10}$ cm. Following @2016ApJ...833..107B, the total mass accreted by the companion NS gives $\sim 10^{-2}~M_{\odot}$, which produces an emission of total energy $\sim 10^{51}$ erg, considering the accretion efficiency as $\sim 10\%$ [@1992apa..book.....F]. The majority of the mass is accreted in $\sim 10$ s, with an accretion rate $\sim 10^{-3}~M_{\odot}$ s$^{-1}$, therefore, a spike with luminosity $\sim 10^{50}$ erg s$^{-1}$ and duration $\sim 10$ s is produced, this estimation fits the second spike that observed well.
The time-resolved analysis of the blackbody components in the second spike indicate a mildly relativistic expanding source emitting thermal radiation. This emission is explained by the adiabatic expanding thermal outflow from the accretion region [@2006ApJ...646L.131F; @2009ApJ...699..409F]. The Rayleigh-Taylor convective instability acts during the initial accretion phase driving material away from the NS with a final velocity of the order of the speed of light. This material expands and cools, by assuming the spherically symmetric expansion, to a temperature [@1996ApJ...460..801F; @2016ApJ...833..107B] $$T = 6.84\, \left(\frac{S}{2.85}\right)^{-1}\,\left( \frac{r}{10^{10}\, {\rm cm}} \right)^{-1}\,{\rm keV},$$ where $S$ is the the entropy $$\begin{gathered}
S \approx 2.85 \left( \frac{M_{\rm NS}}{1.4\,M_\odot} \right)^{7/8}\left( \frac{\dot{M}_{\rm B}}{{10^{-3} M_\odot\,{\rm s}^{-1}}} \right)^{-1/4} \\ \times \left( \frac{r}{10^{10}\, {\rm cm}} \right)^{-3/8},\end{gathered}$$ in units of $k_B$ per nucleon. The system parameters in the above equations have been normalized to self-consistent values that fit the observational data, namely, the thermal emitter has a temperature $\sim 6$ keV, radius $\sim 10^{10}$ cm and expands with velocity $\sim 0.5~c$.
To have more details of the time-resolved evolution: for the two time bins in table \[tab:thermalFlux\], an expanding speed of $0.53$c gives the radius $1.4 \times 10^{10}$ cm and $4.3 \times 10^{10}$ cm respectively, as we fitted from the data. The luminosities are $2.11 \times 10^{50}$ erg/s and $7.56 \times 10^{50}$ erg/s respectively. If assuming the accretion efficiency is $10\%$, from the luminosity we obtain the accretion rate as $1.18 \times 10^{-3}$ $M_{\odot}$ s$^{-1}$ and $4.32 \times 10^{-3}$ $M_{\odot}$ s$^{-1}$. By applying the above two equations, the theoretical temperature is obtained to be $5.83 \pm 1.25$ keV and $3.93 \pm 0.39$ keV, the the thermal flux are $1.22 \pm 0.97 \times 10^{-7}$ $\text{erg}~ \text{s}^{-1} \text{cm}^{-2}$ and $2.37 \pm 0.85 \times 10^-7$ $\text{erg}~ \text{s}^{-1} \text{cm}^{-2}$ respectively. If we assume the accretion efficiency is $7\%$, following the same procedure, the theoretical temperature shall be $8.32 \pm 1.78$ keV and $5.61\pm 0.56$ keV, the thermal flux shall be $5.78 \pm 3.89 \times 10^{-7}$ $\text{erg}~ \text{s}^{-1} \text{cm}^{-2}$ and $7.17 \pm 2.51 \times 10^{-7}$ $\text{erg}~ \text{s}^{-1} \text{cm}^{-2}$ respectively. The observed value in table \[tab:thermalFlux\] are more consistent with the accretion efficiency of $7\%$.
The loss of rotational energy of the $\nu$NS, born after the SN explosion, powers the afterglow. This will be discussed in the next session.
Afterglow from the newly born pulsar {#sec:afterglow_from_pulsar}
------------------------------------
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We have applied the synchrotron model of mildly-relativistic outflow powered by the rotational energy of the $\nu$NS to GRB 130427A [@2018ApJ...869..101R]. From it we have inferred a $1$ ms $\nu$NS pulsar emitting dipole and quadrupole radiation. Here we summarise this procedure and apply it to GRB 180827A.
The late X-ray afterglow of GRB 180728A also shows a power-law decay of index $\sim -1.3$ which, as we show below, if powered by the pulsar implies the presence of a quadrupole magnetic field in addition to the traditional dipole one. The “magnetar” scenario with only a strong dipole field ($B_{dip} > 10^{14}$ G) is not capable to fit the late time afterglow . The dipole and quadrupole magnetic fields are adopted from @2015MNRAS.450..714P, where the magnetic field is cast into an expansion of vector spherical harmonics, each harmonic mode is defined by a set of the multipole order number $l$ and the azimuthal mode number $m$. The luminosity from a pure dipole ($l=1$) is $$L_{dip} = \frac{2}{3 c^3} \Omega^4 B_{dip}^2 R_{\rm NS}^6 \sin^2\chi_1,$$ and a pure quadrupole ($l=2$) is $$\begin{gathered}
L_{quad} = \frac{32}{135 c^5} \Omega^6 B_{quad}^2 R_{\rm NS}^8 \\ \times \sin^2\chi_1(\cos^2\chi_2+10\sin^2\chi_2),\end{gathered}$$ where $\chi_1$ and $\chi_2$ are the inclination angles of the magnetic moment, the different modes are easily separated by taking $\chi_1$ = 0 and any value of $\chi_2$ for $m = 0$, ($\chi_1$, $\chi_2$) = (90, 0) degrees for $m = 1$ and ($\chi_1$, $\chi_2$) = (90, 90) degrees for $m = 2$.
The observed luminosity is assumed to be equal to the spin-down luminosity as $$\begin{gathered}
\frac{dE}{dt} = -I \Omega \dot{\Omega } = - (L_{dip} + L_{quad}) \nonumber \\
= - \frac{2}{3 c^3} \Omega^4 B_{dip}^2 R_{\rm NS}^6 \sin^2\chi_1 \left(1+\eta^2 \frac{16}{45} \frac{R_{\rm NS}^2 \Omega^2}{c^2}\right),\end{gathered}$$ and $$\eta^2 = (\cos^2\chi_2+10\sin^2\chi_2) \frac{B_{quad}^2}{B_{dip}^2}.
\label{eq:eta}$$ where $I$ is the moment of inertia. The parameter $\eta$ relates to the ratio of quadrupole and dipole strength, $\eta = B_{quad}/B_{dip}$ for the $m=1$ mode, and $\eta = 3.16 \times B_{quad}/B_{dip}$ for the $m=2$ mode.
The bolometric luminosity is obtained by integrating the entire spectrum generated by the synchrotron model that fits the soft X-ray ($0.3$–$10$ keV) and the optical [see @2018ApJ...869..101R and figure 4 for example in]. Approximately the bolometric luminosity has a factor of $\sim 5$ times more luminous than the soft X-ray emission. In figure \[fig:B4E13\_2\_5ms\_luminosityb\], we show the bolometric luminosity light curve, the shape of the light curve is taken from the soft X-ray data since it offers the most complete time coverage.
We assume that the bolometric luminosity required from the synchrotron model is equal to the energy loss of the pulsar. The numerical fitting result shows that the BdHN II of GRB 180728A forms a pulsar with initial spin $P_0 = 2.5$ ms, which is slower than the pulsar of $P_0 = 1$ ms pulsar from the BdHN I of GRB 130427A. Both sources have similar dipole magnetic field $10^{12}$–$10^{13}$ G and a quadrupole component $\sim 30$–$100$ stronger ($\eta = 100$) than the dipole one. The strong quadrupole field dominates the emission in the early years while the dipole radiation starts to be prominent later when the spin decays. This is because the quadrupole emission is more sensitive to the spin period, as $\propto \Omega^6$, while the dipole is $\propto \Omega^4$. Therefore, the $\nu$NS shows up a dipole behaviour when observed today, since the quadrupole dominates a very small fraction ($\lesssim 10^{-5}$) of the pulsar lifetime.
A Consistent Picture and Visualisation {#sec:neutron_star_charactristic}
======================================
{width="0.475\hsize"}{width="0.49\hsize"}
In the previous sections, we have inferred the binary separation from the prompt emission, and the spin of the $\nu$NS from the afterglow data. In the following, we confirm the consistency of these findings by numerical simulations of these systems, and compare the commonalities and diversities of GRB 130427A and GRB 180728A as examples of BdHN I and BdHN II systems in our model, respectively.
From an observational point of view, GRB 130427A and GRB 180728A are both long GRBs, but they are very different in the energetic: GRB 130427A is one of the most energetic GRBs with isotropic energy more than $10^{54}$ erg, while GRB 180728A is in the order of $10^{51}$ erg, a thousand of times difference. GRB 130427A has observed the most significant ultra-high energy photons ($100$ MeV– $100$ GeV, hereafter we call GeV photons), it has the longest duration ($>1000$ s) of GeV emission, and it has the highest energy of a photon ever observed from a GRB. In constrast, GRB 180728A has no GeV emission detected. As for the afterglow, the X-ray afterglow of GRB 130427A is more luminous than GRB 180728A, but they both share a power-law decaying index $\sim -1.3$ after $10^4$ s. After more than $10$ days, in both GRB sites emerges the coincident optical signal of a type Ic SN, and the SNe spectra are almost identical as shown in section \[subsec:supernova\].
BdHN I and II have the same kind of binary progenitor, a binary composed of a CO$_{\rm core}$ and a companion NS, but the binary separation/period is different, being larger/longer for BdHN II.
The angular momentum conservation during the gravitational collapse of the pre-SN core that forms the $\nu$NS, i.e. $J_{\rm CO} = J_{\nu \rm NS}$, implies that the latter should be fast rotating, i.e.: $$\Omega_{\nu\rm NS} = \left(\frac{R_{\rm CO}}{R_{\nu\rm NS}}\right)^2\Omega_{\rm Fe}=\left(\frac{R_{\rm CO}}{R_{\nu\rm NS}}\right)^2\Omega_{\rm orb},$$ where $\Omega_{\rm orb} = 2\pi/P_{\rm orb} = \sqrt{G M_{\rm tot}/a_{\rm orb}^3}$ from the Kepler law, being $M_{\rm tot}=M_{\rm CO}+M_{\rm NS}$ the total mass of the binary before the SN explosion, and $M_{\rm CO} = M_{\nu\rm NS}+M_{\rm ej}$. We have assumed that the mass of the $\nu$NS is set by the mass of the iron core of the pre-SN CO$_{\rm core}$ and that it has a rotation period equal to the orbital period owing to tidal synchronization.
From the above we can see that $\nu$NS rotation period, $P_{\nu\rm NS}$ has a linear dependence with the orbital period, $P_{\rm orb}$. Therefore, the solution we have obtained for the rotation period of the $\nu$NS born in GRB 130427A ($P_{\nu\rm NS} \approx 1$ ms) and in the GRB 180728A ($P_{\nu\rm NS} \approx 2.5$ ms), see Fig. \[fig:B4E13\_2\_5ms\_luminosityb\], implies that the orbital period of the BdHN I would be a factor $\approx 2.5$ shorter than the one of the BdHN II. Based on this information, we seek for two systems in our simulations presented in @2018arXiv180304356B with the following properties: the same (or nearly) SN explosion energy, same pre-SN CO$_{\rm core}$ and initial NS companion mass, but different orbital periods, i.e. $P_{\rm II}/P_{\rm I}\approx 2.5$. The more compact binary leads to the BdHN I and the less compact one to the BdHN II and, by angular momentum conservation, they lead to the abovementioned $\nu$NSs.
We examine the results of the simulations for the pre-SN core of a $25~M_\odot$ zero-age main-sequence (ZAMS) progenitor and the initial mass of the NS companion $M_{\rm NS}=2~M_\odot$. A close look at Tables 2 and 7 in @2018arXiv180304356B show that, indeed, Model ‘25m1p08e’ with $P_{\rm orb} = 4.81$ min ($a_{\rm orb}\approx 1.35\times 10^{10}$ cm) and Model ‘25m3p1e’ with $P_{\rm orb} = 11.8$ min ($a_{\rm orb}\approx 2.61\times 10^{10}$ cm) give a consistent solution. In the Model ‘25m1p08e’ the NS companion reaches the critical mass (secular axisymmetric instability) and collapses to a BH; this model produces a BdHN I. In the Model ‘25m3p1e’ the NS companion does not reach the critical mass; this system produces an BdHN II. The system leading to the BdHN I remains bound after the explosion while, the one leading to the BdHN II, is disrupted. Concerning the $\nu$NS rotation period, adopting $R_{\rm CO}\sim 2.141\times 10^8$ cm [see Table 1 in @2018arXiv180304356B], $P_{\rm orb}\sim 4.81$ min and $P_{\rm orb}\sim 11.8$ min leads to $P_{\nu\rm NS} \sim 1$ ms and $2.45$ ms, respectively. We show in Fig. \[fig:SPHsimulation\] snapshots of the two simulations.
Conclusion {#sec:conclusion}
==========
The classification of GRBs in nine different subclasses allows us to identify the origin of a new GRB with known redshift from the observation of its evolution in the first hundred seconds. Then, we are able to predict the presence of an associated SN in the BdHN and its occurring time. We reviewed our previous successful prediction of a BdHN I in our model: GRB 130427A/SN 2013cq, and in this article, we presented our recent successful prediction of a BdHN II in our model: GRB 180728A/SN 2018fip.
The detailed observational data of GRB 180728A, for the first time, allowed us to follow the evolution of a BdHN II. The collapse of CO$_{\rm core}$ leads to a SN. We determine that the corresponding shockwave with energy $\sim 10^{51}$ erg emerges and produces the first $2$ s spike in the prompt emission. The SN ejecta expands and reaches at $\sim 3\times 10^{10}$ cm away from the NS companion. The accretion process starts with a rate $\sim 10^{-3}~M_{\odot}$ s$^{-1}$, the second powerful spike lasting $10$ s with luminosity $\sim 10^{50}$ erg s$^{-1}$, and a thermal component at temperature $\sim 7$ keV.
A $\nu$NS is formed from the SN. The role of $\nu$NS powering the afterglow has been evidenced in our study of GRB 130427A [@2018ApJ...869..101R]. This article emphasises its application on GRB 180728A. The $\nu$NS pulsar loses its rotational energy by dipole and quadrupole emission. In order to fit the observed afterglow data using a synchrotron model [@2018ApJ...869..101R], we require an initial $1$ ms spin pulsar for GRB 130427A, and a slower spin of $2.5$ ms for GRB 180728A. For close binary systems, the binary components are synchronised with the orbital period, from which we are able to obtain the orbital separation by inferring the CO$_{\rm core}$ period from the $\nu$NS one via angular momentum conservation. This second independent method leads to a value of the binary separation in remarkable agreement with the one inferred from the prompt emission, which shows the self-consistency of this picture. The SNe spectra observed in BdNH I and in BdHN II are similar, although the associated two GRBs markedly differ in energy. The SN acts as a catalyst; it triggers the GRB process. After losing a part of the ejecta mass by hypercritical accretion, the remaining SN ejecta are heated by the GRB emission, but the nuclear composition, which relates to the observed optical emission owing to the nuclear decay of nickel and cobalt [@1996snih.book.....A] is not influenced by such a GRB-SN interaction.
Besides providing the theoretical support of the BdHN I and II realisation, we have presented 3D SPH simulations that help in visualising the systems (see figure \[fig:SPHsimulation\]).
In short, we made a successful prediction of SN 2018fip associated with GRB 180728A based on our GRB classification that GRB 180728A belongs to BdHN II. The observations of the prompt emission and the afterglow portray, for the first time, a complete transitional stage of two binary stars. We emphasise the $\nu$NS from supernova playing a dominant role in the later afterglow, the comparison to GRB 130427A, a typical BdHN I in our model, is demonstrated and visualised.
The confirmation of the SN appearance, as well as the majority of this work were performed during the R.R. and Y.W.’s visit to the *Yau Mathematical Sciences Center* in Tsinghua University, Beijing. We greatly appreciate the kind hospitality of and the helpful discussion with Prof. Shing-Tung Yau. We also acknowledge Dr. Luca Izzo for discussions on the SNe treated in this work. We thank to the referee for the constructive comments that helped clarify many concepts and strengthen the time-resolved analysis.
GCNs {#sec:gcns}
====
**GCN 14526** - ***GRB 130427A: Prediction of supernova appearance***\
*The late x ray observations of GRB 130427A by Swift-XRT clearly evidence a pattern typical of a family of GRBs associated to supernova (SN) following the Induce Gravitational Collapse (IGC) paradigm . We assume that the luminosity of the possible SN associated to GRB 130427A would be the one of 1998bw, as found in the IGC sample described in . Assuming the intergalactic absorption in the I-band (which corresponds to the R-band rest-frame) and the intrinsic one, assuming a Milky Way type for the host galaxy, we obtain a magnitude expected for the peak of the SN of I = 22 - 23 occurring 13-15 days after the GRB trigger, namely between the 10th and the 12th of May 2013. Further optical and radio observations are encouraged.*
***GCN 23066 - GRB 180728A: A long GRB of the X-ray flash (XRF) subclass, expecting supernova appearance***
*GRB 180728A has $T_{90}=6.4$ s [@GCN23055], peak energy 142 (-15,+20) keV, and isotropic energy $E_{\rm iso} = (2.33 \pm 0.10) \times 10^{51}$ erg [@GCN23061]. It presents the typical characteristic of a subclass of long GRBs called X-ray flashes[^2] [XRFs, see @2016ApJ...832..136R], originating from a tight binary of a CO$_{\rm core}$ undergoing a supernova explosion in presence of a companion neutron star (NS) that hypercritically accretes part of the supernova matter. The outcome is a new binary composed by a more massive NS (MNS) and a newly born NS ($\nu$NS). Using the averaged observed value of the optical peak time of supernova [@2017AdAst2017E...5C], and considering the redshift $z=0.117$ [@GCN23055], a bright optical signal will peak at $14.7 \pm 2.9$ days after the trigger (12 August 2018, uncertainty from August 9th to August 15th) at the location of RA=253.56472 and DEC=-54.04451, with an uncertainty $0.43$ arcsec [@GCN23064]. The follow-up observations, especially the optical bands for the SN, as well as attention to binary NS pulsar behaviours in the X-ray afterglow emission, are recommended.*
***GCN 23142 - GRB 180728A: discovery of the associated supernova***
*... Up to now, we have observed at three epochs, specifically at 6.27, 9.32 and 12.28 days after the GRB trigger. The optical counterpart is visible in all epochs using the X-shooter acquisition camera in the g, r and z filters. We report a rebrightening of 0.5 $\pm$ 0.1 mag in the r band between 6.27 and 12.28 days. This is consistent with what is observed in many other lo 170827w-redshift GRBs, which in those cases is indicative of an emerging type Ic SN ...*
Data Fitting {#sec:data_fitting}
============
Data are fitted by applying the Monte Carlo Bayesian iterations using a Python package: The Multi-Mission Maximum Likelihood framework (3ML) [^3]. An example is shown in figure \[fig:datafitting\].
{width="\hsize"}
Model Comparison {#sec:model_comparison}
================
Spectra are fitted by Bayesian iterations. The AIC is preferred for comparing non-nested models, and BIC is preferred for nested models [@10.2307/2291091]. Log(likelihood) is adopted by the method of maximum likelihood ratio test which is treated as a reference of the model comparison [@10.2307/1912557]. Parameters are shown in table \[tab:promptModel\].
**Segment** **Time (s)** **Model** **Log(Likelihood)** **AIC** **BIC**
------------- --------------- ------------- --------------------- --------- ---------
Spike 1 -1.57 - 1.18 [PL]{} 430.03 864.17 869.59
Precursor CPL 430.03 866.27 874.35
(NaI7) Band 429.72 867.80 878.49
PL+BB 429.77 867.90 878.59
CPL+BB 429.77 870.08 883.36
Band+BB 429.61 871.98 887.79
Spike 2 8.72 - 10.80 PL 947.20 1898.46 1905.33
(NaI7+BGO1) CPL 838.91 1685.93 1696.22
Band 831.02 1670.21 1683.90
PL+BB 947.21 1902.59 1916.27
CPL+BB 827.67 1665.60 1682.66
[Band+BB]{} 823.90 1660.17 1680.59
10.80 - 12.30 PL 1334.10 2672.25 2679.13
CPL 809.83 1625.76 1636.05
Band 821.25 1650.68 1664.37
PL+BB 1334.10 2676.38 2690.06
[CPL+BB]{} 794.79 1599.85 1616.91
Band+BB 794.80 1599.86 1616.92
12.30 - 22.54 PL 1366.08 2736.23 2742.79
[CPL]{} 1216.52 2439.16 2448.97
Band 1366.43 2741.06 2754.09
PL+BB 1366.08 2740.37 2753.40
CPL+BB 1215.52 2443.35 2459.58
Band+BB 1366.63 2745.69 2765.11
[^1]: Hubble constant H0=($67.4\pm0.5$) km/s/Mpc, matter density parameter $\Omega_M = 0.315\pm0.007$.
[^2]: The previous name of BdHN I
[^3]: <https://github.com/giacomov/3ML>
|
---
abstract: 'This paper investigates the contact topology associated to symplectic log Calabi-Yau pairs $ (X,D,\omega ) $. We classify, up to toric equivalence, all circular spherical divisors with $ b^+\ge 1 $ that can be embedded symplectically into a symplectic rational surface and show they are all realized as symplectic log Calabi-Yau pairs if their complements are minimal. We apply such embeddability and rigidity results to determine the Stein fillability of all contact torus bundles induced as the boundaries of circular spherical divisors with $ b^+\ge 1 $. When $ D $ is negative definite, we give explicit betti number bounds for Stein fillings of its boundary contact torus bundle. Also we show that a large family of contact torus bundles are universally tight, generalizing a conjecture by Golla and Lisca.'
author:
- 'Tian-Jun Li, Cheuk Yu Mak, Jie Min'
bibliography:
- 'logCY-sequence.bib'
title: 'Symplectic log Calabi-Yau surfaces – contact aspects'
---
Introduction
============
Let $ X $ be a smooth rational surface and let $ D\subset X $ be an effective reduced anti-canonical divisor. Such pairs $ (X,D) $ are called anti-canonical pairs and has been extensively studied since Looijenga ([@Lo81]). The open surface $ X- D $ has log Kodaira dimension 0 and is called a log Calabi-Yau surface. Anti-canonical pairs also arise as resolutions of cusp singularities. Gross, Hacking and Keel studied the mirror symmetry aspects of anti-canonical pairs in [@GrHaKe11] and [@GrHaKe12]. In particular, they proved Looijenga’s conjecture on dual cusp singularities in [@GrHaKe11] and Torelli type results in [@GrHaKe12] conjectured by Friedman. In this direction, Pasceleff computed the symplectic cohomology of $ X- D $ in [@Pa13] and Keating proved homological mirror symmetry for a family of cusp singularities in [@Keating2018]. Such pairs were studied from a more symplectic point of view by the first and the second authors ([@LiMa16-deformation]), where they studied different notions of deformation equivalence and enumerated minimal models of symplectic log Calabi-Yau pairs.
The current paper serves as a sequel to [@LiMa16-deformation] and is devoted to the study of contact topology related to symplectic log Calabi-Yau pairs. We give proofs of many results previously announced in [@LiMa19-survey] and [@LiMi-ICCM]. In particular, we are interested in symplectic circular spherical divisors, which can be thought of as a local version of symplectic Looijenga pairs, in the sense that they are not required to be embedded in a closed symplectic 4-manifold. The embeddability and rigidity of such divisors are intimately related to the symplectic fillability and the topology of minimal symplectic fillings of their boundary torus bundles.
In this paper, a topological divisor refers to a connected configuration of finitely many closed embedded oriented smooth surfaces $D=C_1 \cup \dots \cup C_k$ in a smooth oriented 4 dimensional manifold $ X $ (possibly with boundary or non-compact), satisfying the following conditions: (1) each intersection between two components is positive and transversal, (2) no three components intersect at a common point, and (3) $ D $ does not intersect the boundary $ \partial X $. Since we are interested in the germ of a topological divisor, we usually omit $ X $ in the writing and just denote the divisor by $ D $. For each $ D $ denote by $N_D$ the neighborhood obtained by plumbing disk bundles over the components $ C_i $ and $Y_D=\partial N_D$ the plumbed 3-manifold oriented as the boundary of $ N_D $. They are well-defined up to orientation-preserving diffeomorphisms ([@Ne81-calculus]).
An intersection matrix is associated to each topological divisor. For a topological divisor $ D=\cup_{i=1}^k C_i $, we denote by $[C_i]$ the homology class of $C_i$ in $ H_2(X) $. Note that $H_2(N_D)$ is freely generated by $[C_i]$. The intersection matrix of $D$ is the $k$ by $k$ square matrix $Q_D=(s_{ij}=[C_i]\cdot [C_{j}])$, where $\cdot$ is used for any of the pairings $H_2(X) \times H_2(X), H^2(X) \times H_2(X), H^2(X) \times H^2(X, \partial X)$. Via the Lefschetz duality for $N_D$, the intersection matrix $Q_D$ can be identified with the natural homomorphism $Q_D: H_2(N_D)\to H_2(N_D, Y_D)$. We use homology and cohomology with ${\mathbb{Z}}$ coefficient unless otherwise specified.
For a symplectic 4-manifold $(X, \omega)$ a symplectic divisor is a topological divisor $D$ with each $C_i$ symplectic and having the orientation positive with respect to $\omega$.
A **symplectic log Calabi-Yau pair** $(X,D,\omega)$ is a closed symplectic 4-manifold $(X,\omega)$ together with a nonempty symplectic divisor $D=\cup C_i$ representing the Poincare dual of $c_1(X,\omega )$.
It’s an easy observation ([@LiMa19-survey]) that $ D $ is either a torus or a cycle of spheres. In the former case, $ (X,D,\omega ) $ is called an **elliptic log Calabi-Yau pair**. In the later case, it’s called a **symplectic Looijenga pair** and it can only happen when $ (X,\omega ) $ is rational. As a consequence, we have $ b^+(Q_D)=0 $ or $ 1 $. We also remark that symplectic log Calabi-Yau pairs have vanishing relative symplectic Kodaira dimension (cf. [@LiZh11-relative],[@LiMi-logkod]).
We call a topological divisor $ D $ consisting of a cycle of spheres a **circular spherical divisor** and a **symplectic circular spherical divisor** if such $ D $ is a symplectic divisor. For each circular spherical divisor $ D=\cup_{i=1}^k C_i $, we associate to it a self-intersection sequence $ \vec{s}(D)=(s_i=[C_i]^2)_{i=1}^{k} $. The orientation of $ D $ is a cyclic labeling up to permutation. So an oriented $ D $ can be described entirely by its self-intersection sequence up to cyclic permutation. Given any sequence $ \vec{s}=(s_i) $ with $ s_i\in {\mathbb{Z}}$, it can always be realized as the self-intersection sequence of a circular spherical divisor. So essentially there is no difference between a sequence and a circular spherical divisor (up to cyclic permutation) and we will not distinguish them in this paper. We also denote by $ (s) $ the topological divisor $ D=T $, where $ T $ is a torus with $ [T]^2=s $, and sometimes call it an elliptic sequence. This does not cause any confusion as we always require that a circular spherical divisor has length at least $ 2 $. The self-intersection sequence is a simplified version of the intersection matrix and we could define the $ b^+/b^-/b^0 $ of a divisor $ D $ or a sequence $ \vec{s} $ to be the $ b^+/b^-/b^0 $ of $ Q_D $, i.e. the number of positive/negative/zero eigenvalues.
A sequence $ \vec{s} $ is called **symplectically embeddable** if $ \vec{s}=\vec{s}(D) $ for a symplectic circular spherical divisor $ D $ which admits symplectic embedding into a closed symplectic 4-manifold $ (X,\omega ) $. A symplectically embeddable sequence $ \vec{s} $ is called a **rationally embeddable** if such $ (X,\omega ) $ can be chosen to be a symplectic rational surface, i.e. $ X\cong {\mathbb{C}}{\mathbb{P}}^2\# l\overline{{\mathbb{C}}{\mathbb{P}}}^2 $. A rationally embeddable sequence $ \vec{s} $ is called **anti-canonical** if $ \vec{s}=\vec{s}(D) $ for a symplectic Looijenga pair $ (X,D,\omega ) $. An anti-canonical sequence $ \vec{s} $ is called **rigid** if for any symplectic circular spherical divisor $ D$ in a rational $ (X,\omega ) $ with $ \vec{s}(D)=\vec{s} $ and $ X-D $ minimal, $ (X,D,\omega ) $ is a symplectic Looijenga pair. The complement of an anti-canonical $ D $ is by definition minimal. We prove, in particular, the converse is also true.
\[thm:embeddable=rigid\] For a sequence with $ b^+\ge 1 $, being symplectically embeddable, rationally embeddable, anti-canonical, and rigid are equivalent.
It is well-known that the boundary $ Y_D $ of a plumbing of a cycle of spheres $ D $ is a topological torus bundle (cf. [@Ne81-calculus]). Denote by $ T_A $ an oriented torus bundle over $ S^1 $ with monodromy $ A\in SL(2;{\mathbb{Z}}) $. A torus bundle $ T_A $ is called elliptic if $ |{{\rm tr}}A|<2 $, parabolic if $ |{{\rm tr}}A|=2 $ and hyperbolic if $ |{{\rm tr}}A|>2 $. If $ T_A $ is parabolic or hyperbolic, we call it positive (resp. negative) if $ {{\rm tr}}A $ is positive (resp. negative).
We can classify all symplectically embeddable sequences up to toric equivalence (see Definition \[def:toric eq\]) and list them according to the types of their boundary torus bundles.
\[thm:list\] A sequence with $ b^+\ge 1 $ is symplectically embeddable if and only if it is toric equivalent to one of the following:
1. $ (1,p) $ or $ (-1,-p) $ with $ p=1,2,3 $, in which case $ -Y_D $ is elliptic,
2. $ (1,1,p) $ with $ p\le 1 $, in which case $ -Y_D $ is positive parabolic,
3. $ (0,p) $ with $ p\le 4 $, in which case $ -Y_D $ is negative parabolic,
4. $ (1,p) $ with $ p\le -1 $ or $ (1,1-p_1,-p_2,\dots,-p_{l-1},1-p_l) $ blown-up (defined in Section \[subsection:rationally embeddable\]) with $ p_i\ge 2 $, $ l\ge 2 $, in which case $ -Y_D $ is negative hyperbolic.
The reason why we look at the negative boundary $ -Y_D $ comes from the contact point of view. As we will introduce in Proposition \[prop:unique-contact\], if $ D $ has a concave (resp. convex) plumbing $ P(D) $, there is a contact structure associated to $ D $ on the torus bundle as the negative (resp. positive) boundary of $ P(D) $, which we will denote by $ (-Y_D,\xi_D) $ (resp. $ (Y_D,\xi_D) $). This contact structure only depends on the topological divisor $ D $ and doesn’t vary with the symplectic structure $ \omega $ on $ P(D) $ ([@LiMa14-divisorcap], see also Proposition \[prop:unique-contact\]).
According to the existence of convex/concave plumbing, symplectic log Calabi-Yau pairs are separated into three groups. In the following, $Kod(Y, \xi)$ is the contact Kodaira dimension introduced in [@LiMa16-kodaira] (see Definition \[def:contact kod\]).
\[prop: convex-concave\] Let $(X,D,\omega)$ be a symplectic log Calabi-Yau pair, $Q_D$ the intersection matrix of $D$ and $(s_i)$ the self intersection sequence.
1. If $Q_D$ is negative definite, then $D$ admits [convex]{} neighborhoods and the induced contact 3-manifold $(Y_D, \xi_D)$ has $Kod(Y_D,\xi_D)\leq 0$.
2. If $b^+(Q_D)= 1$, up to local symplectic deformations, $D$ admits [concave]{} neighborhoods and the induced contact 3-manifold $(-Y_D, \xi_D)$ has $Kod(-Y_D,\xi_D)=-\infty$.
3. If $b^+(Q_D)=0$ and $Q_D$ is not negative definite, then it does not admit a regular neighborhood with contact boundary.
Symplectic fillability and Stein fillability of contact torus bundles have been extensively studied ([@BhOz14],[@DiGe01-fillability],[@El96-3torus],[@Et02-tight-not-fillable],[@Ga06-3handle],[@Li04-tight-not-fillable],[@VHM-thesis],[@DiLi18-torusbundle]). For a large family of contact torus bundles induced as divisor boundaries, Golla and Lisca investigated the topology of their Stein and minimal symplectic fillings in [@GoLi14]. In the case of elliptic log Calabi-Yau pairs, Ohta and Ono classified symplectic fillings of simple elliptic singularities up to symplectic deformation in [@OhOn03-simple-elliptic]. Using Theorem \[thm:embeddable=rigid\] and \[thm:list\], we determine the symplectic fillability of contact torus bundles $ (Y_D,\xi_D) $ for all circular spherical divisors $ D $ with $ b^+(Q_D)\ge 1 $ and study the topology of their fillings.
\[thm:embeddable=fillable\] Suppose $ D $ is a circular spherical divisor with $ b^+(Q_D)\ge 1 $. Then the following are equivalent
1. $ (-Y_D,\xi_D) $ is symplectic fillable,
2. $ (-Y_D,\xi_D) $ is Stein fillable,
3. $ D $ is symplectically embeddable, i.e. toric equivalent to one in Theorem \[thm:list\].
Furthermore, there are at most finitely many (Stein) minimal symplectic fillings of $(-Y_D, \xi_D)$ up to symplectic deformation, all having $b^+=0$ and $ c_1=0 $.
When a symplectic log Calabi-Yau pair $ (X,D,\omega ) $ has negative definite $ D $, it arises as a resolution of a cusp singularity and thus its boundary $ (Y_D,\xi_D ) $ is Stein fillable. We study the geography of its Stein fillings and give restrictions on its betti numbers and Euler number in Proposition \[prop:geography\].
Lastly, we investigate the universal tightness of $ \xi_D $ for a circular spherical divisor $ D $. Golla and Lisca showed in [@GoLi14] that a subfamily of the contact torus bundles they considered are universally tight. This led them to formulate the following conjecture.
\[conj:golla lisca\] Suppose $ D $ is a rationally embeddable circular spherical divisor with $ b^+(Q_D)=1 $, then $ (-Y_D,\xi_D) $ is universally tight.
This conjecture was confirmed for divisors with nonsingular intersection matrices by Ding and Li in [@DiLi18-torusbundle]. Both the results of Golla-Lisca and Ding-Li come from an extrinsic point of view and rely on the symplectic fillings of virtually overtwisted contact torus bundles.
In [@LiMi-ICCM], the first and the third authors approached this conjecture from an intrinsic angle, based on the Giroux correspondence between contact structures and open book decompositions. By proving the invariance of boundary contact structure under toric equivalence in Proposition \[prop:contact toric eq\], we extend the result in [@LiMi-ICCM] (see Theorem \[universally-tight\]) to the following.
\[thm:universally-tight\] Let $ D $ be a circular spherical divisor toric equivalent to one whose associated graph is non-negative (see Section \[section:universally tight\]), then $ (-Y_D,\xi_D) $ (not necessarily fillable) is universally tight, except possibly when $ -Y_D $ is parabolic torus bundle with monodromy $ \begin{pmatrix}
1 & n\\ 0 &1
\end{pmatrix}, n>0 $.
Because our approach is purely 3-dimensional in nature, our result differs from Conjecture \[conj:golla lisca\] in the sense that we don’t require $ D $ to be rationally embeddable. In fact, most contact structures we considered are not symplectic fillable, and thus cannot be studied by extrinsic methods.
**Acknowledgments**: This paper is inspired by the results and questions in Golla-Lisca [@GoLi14]. All authors are supported by NSF grant 1611680.
Topology of circular spherical divisors
=======================================
In this section we discuss several aspects of circular spherical divisors. We first introduce the notion of toric equivalence for topological divisors in Section \[section:toric eq\]. Then Section \[subsection:torus bundle\] reviews basic facts about their boundary torus bundles. Finally we give homological restrictions for a circular spherical divisor to be embedded in a closed manifold with $ b^+=1 $.
Toric equivalences {#section:toric eq}
------------------
\[def:toric eq\] For a topological divisor $ D=\cup C_i $, **toric blow-up** is the operation of adding a sphere component with self-intersection $ -1 $ between an adjacent pair of component $ C_i $ and $ C_{j} $ and changing the self-intersection of $ C_i $ and $ C_{j} $ by $ -1 $. **Toric blow-down** is the reverse operation.
$ D^0 $ and $ D^1 $ are **toric equivalent** if they are connected by toric blow-ups and toric blow-downs. $ D $ is said to be **toric minimal** if no component is an exceptional sphere (i.e. a component of self-intersection $ -1 $).
Note that toric blow-ups and blow-downs can be performed in the symplectic category by adding an extra parameter of symplectic area and are thus operations on symplectic divisors (see Section \[section:contact toric\]). Also note that we could keep toric blowing down a circular spherical divisor until either it becomes toric minimal or it has length $ 2 $. When a circular spherical divisor has length $ 2 $, we cannot further blow it down because it would result in a non-embedded sphere and thus not a topological divisor. So we exclude this case when we talk about toric blow-down.
\[lemma:toric eq topological\] The following are preserved under toric equivalence:
1. $D$ being a circular spherical divisor,
2. $ b^+(Q_D) $ and $ b^0(Q_D) $ (in particular the non-degeneracy of the intersection matrix $Q_D$),
3. the oriented diffeomorphism type of the plumbed 3-manifold $Y_D$.
\(1) is obvious and (3) is part of Proposition 2.1 in [@Ne81-calculus].
\(2) follows from a direct computation. Let $ D= (b_1,\dots,b_l) $ and $ D'=(-1,b_1-1,b_2,\dots,b_l-1) $ be its toric blow-up. Then the intersection matrix $ Q_{D'} $ is of the form $$\begin{pmatrix}
-1 & 1 & 0 & \dots & 1\\
1 & b_1 -1 & 1 &\dots & 0\\
0 & 1 & b_2 & \dots & \vdots \\
\vdots & & & & 1\\
1 & 0 & \dots & 1 & b_l-1
\end{pmatrix}.$$ It’s easy to see that by a change of basis, $ Q_{D'} $ is equivalent to $ (-1)\oplus Q_D $. So toric blow-up preserves $ b^+ $, $ b^0 $ and increases $ b^- $ by $ 1 $.
Here is an example to illustrate how a self-intersection $ 0 $ component in a circular spherical divisor can be used to balance the self-intersection numbers of the two sides by performing a toric blow-up and a toric blow-down.
\[eg: balancing self-intersection by $0$-sphere\] The following three cycles of spheres are toric equivalent: $$\begin{tikzpicture}
\node (x) at (0,0) [circle,fill,outer sep=5pt, scale=0.5] [label=above:$ 3 $]{};
\node (y) at (1,0) [circle,fill,outer sep=5pt, scale=0.5] [label=above:$ -2 $]{};
\node (z) at (1,-1) [circle,fill,outer sep=5pt, scale=0.5] [label=below:$ 0 $]{};
\draw (x) -- (y);\draw (y) -- (z);\draw (z) -- (x);
\end{tikzpicture}\quad
\begin{tikzpicture}
\node (x) at (0,0) [circle,fill,outer sep=5pt, scale=0.5] [label=above:$ 2 $]{};
\node (y) at (1,0) [circle,fill,outer sep=5pt, scale=0.5] [label=above:$ -2 $]{};
\node (z) at (1,-1) [circle,fill,outer sep=5pt, scale=0.5] [label=below:$ -1 $]{};
\node (w) at (0,-1) [circle,fill,outer sep=5pt, scale=0.5] [label=below:$ -1 $]{};
\draw (x) -- (y);\draw (y) -- (z);\draw (z) -- (w);\draw (x) to (w);
\end{tikzpicture}\quad
\begin{tikzpicture}
\node (x) at (0,0) [circle,fill,outer sep=5pt, scale=0.5] [label=above:$ 2 $]{};
\node (y) at (1,0) [circle,fill,outer sep=5pt, scale=0.5] [label=above:$ -1 $]{};
\node (z) at (0,-1) [circle,fill,outer sep=5pt, scale=0.5] [label=below:$ 0 $]{};
\draw (x) -- (y);\draw (y) -- (z);\draw (z) -- (x);
\end{tikzpicture}$$
We call such a move which changes a divisor $ (\dots,k,0,p,\dots) $ to a toric equivalent divisor $ (\dots,k-n,0,p+n,\dots) $ a **balancing move** based at the $ 0 $-sphere.
\[lem: not negative semi-definite => at least one non-negative\] Any circular spherical divisor is toric equivalent to a toric minimal one or one with sequence $(-1, p)$. If $D$ has sequence $(-1,p)$, then $Q_D$ is degenerate only if $p=-4$.
Suppose $D$ is a toric minimal cycle of spheres with sequence $(s_i)$. Then
1. $b^+(Q_D)\geq 1$ if and only if $s_i\geq 0$ for some $i$.
2. $Q_D$ is negative definite if $s_i\leq -2$ for all $i$ and less than $-2$ for some $i$. $Q_D$ is negative semi-definite but not negative definite if $s_i= -2$ for each $i$.
3. $Q_D$ is non-degenerate if either $s_1\geq 0$ and $s_i\leq -2$ for $i\geq 2$, or $s_1=s_2=0$ and $s_i\leq -2$ for $i\geq 3$.
By toric blow-down, any circular spherical divisor is toric equivalent to a toric minimal one or one of length $ 2 $. If $ D $ has length $ 2 $ and not toric minimal, then it is of the form $ (-1,p) $. Then $ \det(Q_D)=-p-4=0 $ only if $ p=-4 $.
\(1) and (2) are well-known (cf. Lemma 8.1 in [@Ne81-calculus], Lemma 2.5 in [@GaMa13-LF]).
To prove (3), by Lemma \[lem: property of continuous fraction\], we just need to show that the trace of the monodromy matrix is not equal to $2$. Notice that for $ k=2 $, $ Q_D=\begin{pmatrix}
s_1 & 1\\ 1 & s_2
\end{pmatrix} $, so $ Q_D $ is non-degenerate if $ s_1\ge 0 $ and $ s_2\le -2 $. For the following, we assume $ k\ge 3 $.
For this purpose, we recall the following observation in Lemma 5.2 in [@Ne81-calculus]: Suppose $t_i \le -2$ for all $i=1,\dots,k$ and $ p_j,q_j\in {\mathbb{Z}}$ defined recursively by $ p_{-1}=0,p_0=1,p_{i+1}=-t_{i+1}p_i-p_{i-1};q_{-1}=-1,q_0=0,q_{i+1}=-t_{i+1}q_i-q_{i-1} $. Then $A(-t_1,\dots,-t_j)= \begin{pmatrix}
p_j & q_j \\
-p_{j-1} & -q_{j-1}\end{pmatrix} $, with $p_j \ge p_{j-1}+1 \ge 0$, $q_j \ge q_{j-1}+1 \ge 0$ and $p_j \ge q_j+1\ge 0$ for all $j$. In [@Ne81-calculus], it was not claimed that $p_j \ge q_j+1$ but it is a standard fact and can be verified by induction.
We apply this observation to the chain of spheres with negative self-intersection. In the first case, the monodromy matrix is $$A(-s_1,-t_1\dots,-t_{k-1})= A(-t_1, \cdots, -t_{k-1}) \begin{pmatrix} -s_1 &1\\-1 &0\end{pmatrix} =\begin{pmatrix}
-s_1p_{k-1}-q_{k-1} & p_{k-1} \\
s_1p_{k-2}+q_{k-2} & -p_{k-2}\end{pmatrix},$$ with $s_1\geq 0$ and $p_k, q_k$ as in the observation above. The trace is $ -s_1p_{k-1}-q_{k-1}-p_{k-2}\le -q_{k-2}-p_{k-2}-1\le -2(q_{k-2}+1)\le -2 $, so not equal to $2$. In the second case, the monodromy matrix is $A(0,0, -t_1, \dots,-t_{k-2})= \begin{pmatrix}
-p_{k-2}
& -q_{k-2}\\
p_{k-3}
& q_{k-3} \end{pmatrix}
$. The trace is $ -p_{k-2}+q_{k-3}\le -p_{k-2}+q_{k-2}-1\le -2 $, so again cannot be $ 2 $.
Boundary torus bundles {#subsection:torus bundle}
----------------------
To describe the plumbed 3-manifold $Y_D$, we introduce the following matrix for a sequence of integers $(t_1, \cdots, t_k)$, $$A(t_1,\dots,t_k)= \begin{pmatrix}
t_k & 1 \\
-1 & 0 \end{pmatrix}
\begin{pmatrix}
t_{k-1} & 1 \\
-1 & 0 \end{pmatrix} \dots
\begin{pmatrix}
t_1 & 1 \\
-1 & 0 \end{pmatrix} \in SL_2({\mathbb{Z}}).$$
\[Theorem 6.1 in [@Ne81-calculus], Theorem 2.5 in [@GoLi14]\] \[lem: property of continuous fraction\] For a circular spherical divisor $D$ with self-intersection sequence $(s_1, ..., s_k)$, the plumbed 3-manifold $Y_D$ is the oriented torus bundle $T_A$ over $S^1$ with monodromy $A=A(-s_1,\dots,-s_k)$. The intersection matrix $Q_D$ is non-degenerate if the trace of $A(-s_1,\dots,-s_k)\ne 2$.
We now recall the classification of hyperbolic torus bundle over circle, which can be found in [@Ne81-calculus] (cf. [@GoLi14]).
All torus bundle over circle with hyperbolic monodromy $A$ and trace of $A$ greater than $2$ (resp. less than $-2$) is the oriented boundary of plumbing of circle of spheres (every intersection is positive) with self-intersections $(-b_1,\dots,-b_k)$ (resp. $(0,0,-b_1,\dots,-b_k)$) such that $b_i \ge 2$ for all $i$ and $b_i \ge 3$ for some $i$.
When ${{\rm tr}}A \ge 3$, the result is contained in Theorem 6.1 of [@Ne81-calculus]. When ${{\rm tr}}A \le -3$, the plumbing description in Theorem 6.1 of [@Ne81-calculus] has a negative intersection. One can blow up the negative intersection point as in Proposition 2.1 of [@Ne81-calculus] to get an equivalent plumbing with self-intersections $(1,-b_1+1,-b_2,\dots,-b_{k-1},-b_k+1)$ when $k \ge 2$ and $(1,-b_1+2)$ when $k=1$. By using the balancing move in Example \[eg: balancing self-intersection by $0$-sphere\], it is easy to see that another equivalent description is as oriented boundary of plumbing of circle of spheres with self-intersections $(0,0,-b_1,\dots,-b_k)$.
We would like to point out that if $Y$ is a hyperbolic torus bundle over a circle with monodromy $A$ and trace of $A$ greater than $2$, so is the monodromy of $-Y$ because the monodromy of $-Y$ is given by $A^{-1}$.
Homological restrictions when $ b^+(X)=1 $
------------------------------------------
In this subsection, we assume $ D=\cup C_i $ is a circular spherical divisor. Let $ r(D) $ denote the number of components of $ D $ and $r^{\geq 0}(D)$ the number of components with non-negative self-intersection. Here are some restrictions on homologous components of $D$.
\[lemma:homologous components\] For any $D$ embedded in a smooth 4-manifold $ X $, we have the following
1. At most three components are homologous in $X$. There are three homologous components only if $r(D)=3$.
2. There are a pair of homologous components only if $r(D)\leq 4$.
3. If $[C_i]=[C_{i+1}]$ for some $i$ then $r(D)=3,
s_i=s_{i+1}=1$, or $r(D)=2, s_i=s_{i+1}=2$.
Suppose there are $ m $ components homologous to $ a\in H_2(X) $ in $ D $. Note that $ a^2\in \{ 0, 1,2\} $ because the divisor has only one cycle and the components are required to intersect positively and transversally. If $ a^2=1 $, then these components are all adjacent. In order to form exactly one cycle, $ m $ is at most $ 3 $. In particular, when $ m=3 $, there cannot be other components, i.e. $ r(D)=3 $. If $ a^2=0 $, there is another component $ C $ intersecting all these components. Again in order to form exactly one cycle, $ m $ is at most $ 2 $. Similarly if $ a^2=2 $, they are adjacent and we must have $ m=r(D)=2 $. This proves (1) and (3).
Suppose $ C_i,C_j $ are a pair of homologous components in $ D $. If they are adjacent, then $ [C_i]\cdot [C_j]=1 $ or $ 2 $ and $ r(D)\le 3 $ by the above discussion. If they are not adjacent, then any other component intersecting $ C_i $ must also intersect $ C_j $. There must be exactly two such components to form a cycle. So $ r(D)=4 $ and this proves (2).
Note that the above restrictions hold locally. When $ X $ is closed with $b^+(X)=1$, there are various restrictions on components with non-negative self-intersection.
\[lem: non-negative components\] Suppose $D$ is embedded in a closed manifold $X$ with $b^+(X)=1$.
1. If $C_i$ and $C_j$ are not adjacent and $s_i\geq 0, s_j\geq 0$, then $[C_i]=\pm [C_j]$ and $s_i=s_j=0$.
2. $r^{\geq 0}(D)\leq 4$.
3. $r^{\geq 0}(D)=4$ only if $r(D)=4, s_i=0$ for each $i$ and $[C_1]=[C_3], [C_2]=[C_4]$.
4. Suppose $r(D)\geq 3$. If $s_i\geq 1, s_{i+1}\geq 1$ for some $i$, then $[C_i]=[C_{i+1}]$ and $s_i=s_{i+1}=1$. This is only possible when $r(D)=3$.
Since $b^+(X)=1$, by the light cone lemma (cf. [@McDuffSalamon1996]), any two disjoint components with non-negative self-intersection must be homologous up to sign and have self-intersection $0$.
\(2) and (3) follow from the (1).
For (4), we can assume the two spheres are $C_1$ and $C_2$. Since $r(D)\geq 3$, we have $[C_1]\cdot[C_2]=1$. By toric blowing up the intersection point between $C_1$ and $C_2$, we get two disjoint spheres with classes $ [C_1']=[C_1]-E $ and $ [C_2']=[C_2]-E $, where $ E $ is the exceptional class and $ [C_1']^2=[C_1]^2-1\ge 0 $, $ [C_2']^2=[C_2]^2-1\ge 0 $. Then by (1), we have $ [C_1']=[C_2'] $ with $ [C_1']^2=0 $ and thus $ [C_1]=[C_2] $ with $ [C_1]^2=1 $. The fact that $ r(D)=3 $ follows from (3) in Lemma \[lemma:homologous components\].
Note in (1) of Lemma \[lem: non-negative components\], if $ C_i $ and $ C_j $ are symplectic spheres, we would have $ [C_i]=[C_j] $.
\[lemma:topological cyclic\] Suppose $D$ has $ b^+(Q_D)= 1 $ and is embedded in a closed manifold $X$ with $b^+(X)=1$. Let $ k,p,p_1,p_2 $ be integers such that $ k\ge 0 $ and $ p,p_1,p_2<0 $. Up to cyclic permutation and orientation of $D$, we have the following.
1. If $r(D)\geq 5$, then $r^{\geq 0}(D)\leq 2$. When $r^{\geq 0}(D)=2$, $s_1\geq 0, s_2=0$.
2. If $r(D)=4$ and $r^{\geq 0}(D)\geq 3$, then $\vec{s}(D)=(k, 0, p, 0), k+p\leq 0$ and $ [C_2]=[C_4] $.
3. If $r(D)=4$ and $r^{\geq 0}(D)=2$, then the only possibilities of $\vec{s}(D)$ are\
(i) $(0, p_1, 0, p_2), [C_1]=[C_3]$,\
(ii) $(k, 0, p_1, p_2), p_1+p_2+k\leq 0$.
4. If $r(D)=3$ and $r^{\geq 0}(D)=3$, then the only possibilities of $\vec{s}(D)$ are\
(i) $(1, 1, 1), [C_1]=[C_2]=[C_3]$,\
(ii) $(1, 1, 0), [C_1]=[C_2]$,\
(iii) $(k, 0, 0)$, $ k\le 2 $.
5. If $r(D)=3$ and $r^{\geq 0}(D)=2$, then the only possibilities of $\vec{s}(D)$ are\
(i) $(1, 1, p), [C_1]=[C_2]$,\
(ii) $(k , 0, p), p+k\leq 2$.
6. If $r(D)=2$ and $r^{\geq 0}(D)=2$, then $\vec{s}(D)$ is one in family $ {\mathcal}{F}(2,2)= $\
$\{(4, 1), (4, 0), (3, 1), (3,0), (2, 2), (2, 1), (2, 0),
(1, 1), (1, 0), (0, 0)\}$.
7. If $r(D)=2$ and $r^{\geq 0}(D)=1$, then $\vec{s}(D)=(k, p)$.
8. If $r(D)=2$ and $r^{\geq 0}(D)=0$, then $\vec{s}(D)$ is one in family $ {\mathcal}{F}(2,0)= $\
$\{(-1, -1), (-1, -2), (-1, -3) \}$.
**Case (1):** Suppose $r(D)\geq 5$. If $r^{\geq 0}(D)\geq 3$ then two such components are not adjacent. But this is impossible due to the (1) of Lemma \[lem: non-negative components\] and the (3) of Lemma \[lemma:homologous components\]. Hence $r^{\geq 0}(D)\leq 2$ in this case. When $r^{\geq 0}(D)=2$, the two components must be adjacent by the same reasoning. The claim that one of them has self-intersection $0$ follows from (4) of Lemma \[lem: non-negative components\] and the (3) of Lemma \[lemma:homologous components\].
**Case (2):** The proof is similar when $r(D)=4$ and $r^{\geq 0}(D)\geq 3$. In this case two such components are not adjacent, say $C_2, C_4$. By the (1) of Lemma \[lem: non-negative components\], $[C_2]=[C_4]$, $s_2=0=s_4$.
**Case (3):** Suppose $r(D)=4$ and $r^{\geq 0}(D)=2$. If two such components are not adjacent, we can assume them to be $C_1, C_3$, which satisfy $[C_1]=[C_3]$ and $s_1=s_3=0$ by the (1) of Lemma \[lem: non-negative components\]. If the two components are adjacent, we can assume them to be $C_1, C_2$. Notice that $[C_1]\ne [C_2]$ due to the (3) of Lemma \[lemma:homologous components\]. Now it follows from the 4th bullet of Lemma \[lem: non-negative components\] that either $s_1=0$ or $s_2=0$.
**Case (4):** Suppose $r(D)=3=r^{\geq 0}(D)$. Since $s_i\geq 0$ for any $i$, It is easy to see (i), (ii), (iii) give all the possibilities by (4) of Lemma \[lem: non-negative components\]. It’s easily checked by hand that $ (k,0,0) $ has $ b^+\ge 2 $ when $ k\ge 3 $.
**Case (5):** If $r^{\geq 0}(D)=2$, apply (4) of Lemma \[lem: non-negative components\] to the pair of components $C_i, C_j$ with $s_i\geq 0, s_j\geq 0$.
**Case (6)(7)(8):** Suppose $r(D)=2$. Then we just check that the determinant of $Q_D=s_1 s_2-4 \leq 0$.
Symplectic log Calabi-Yau pairs
===============================
In this section, we review facts about the minimal models and deformation classes of symplectic log Calabi-Yau pairs studied in [@LiMa16-deformation].
Minimal models
--------------
We introduce another pair of operations on symplectic divisors used in the minimal reduction.
A [**non-toric blow-up**]{} of $D$ is the proper transform of a symplectic blow-up centered at a smooth point of $D$. A [**non-toric blow-down**]{} is the reverse operation which symplectically blows down an exceptional sphere not contained in $D$.
These operations preserve the log Calabi-Yau condition and have analogues in the holomorphic category.
A symplectic log Calabi-Yau pair $(X,D,\omega)$ is called minimal if $(X, \omega)$ is minimal, or $(X, D, \omega)$ is a symplectic Looijenga pair with $X=\mathbb{CP}^2 \# \overline{\mathbb{CP}^2}$.
Through a maximal sequence of non-toric blow-downs and then a maximal sequence of toric blow-downs, we get a toric minimal pair from any symplectic log Calabi-Yau pair $ (X,D,\omega ) $. Such a pair is actually minimal by [@Pi08] and is called a minimal model of $ (X,D,\omega ) $.
We recall here the homology types of minimal symplectic log Calabi-Yau pairs (modulo cyclic symmetry), all of them having length less than $5$.
\[thm:minimal model\] Any minimal symplectic log Calabi-Yau pair $ (X,D,\omega) $ has the same homology type as one of the following.
$\bullet$ Case $(A)$: $X$ is a symplectic ruled surface with base genus $1$. $D$ is a torus.
$\bullet$ Case $(B)$: $X=\mathbb{CP}^2$, $c_1=3h$.
$(B1)$ $D$ is a torus,
$(B2)$ $D$ consists of a $h-$sphere and a $2h-$sphere, or
$(B3)$ $D$ consists of three $h-$spheres.
$\bullet$ Case $(C)$: $X=S^2 \times S^2$, $c_1=2f_1+2f_2$, where $f_1$ and $f_2$ are the homology classes of the two factors.
$(C1)$ $D$ is a torus.
$(C2)$ $r(D)=2$ and $[C_1]=bf_1+f_2, [C_2]=(2-b)f_1+f_2$.
$(C3)$ $r(D)=3$ and $[C_1]=bf_1+f_2, [C_2]=f_2, [C_3]=(1-b)f_1+f_2$.
$(C4)$ $r(D)=4$ and $[C_1]=bf_1+f_2, [C_2]=f_1, [C_3]=-bf_1+f_2, [C_4]=f_1$.
The graphs in (C1), (C2), (C3) and (C4) are given respectively by $$\begin{tikzpicture}
\node (x) at (-0.5,0) [circle,fill,outer sep=5pt, scale=0.5] [label=above:$ 8 $]{};
\node (x1) at (1,0) [circle,fill,outer sep=5pt, scale=0.5] [label=above:$ 2b $]{};
\node (y1) at (2.5,0) [circle,fill,outer sep=5pt, scale=0.5] [label=above:$ 4-2b $]{};
\draw (x1) to[bend right] (y1);\draw (y1) to[bend right] (x1);
\node (x2) at (4,0) [circle,fill,outer sep=5pt, scale=0.5] [label=above:$ 2b $]{};
\node (y2) at (5.5,0) [circle,fill,outer sep=5pt, scale=0.5] [label=above:$ 0 $]{};
\node (z2) at (4,-1) [circle,fill,outer sep=5pt, scale=0.5] [label=below:$ 2-2b $]{};
\draw (x2) -- (y2);\draw (y2) -- (z2);\draw (z2) -- (x2);
\node (x3) at (7,0) [circle,fill,outer sep=5pt, scale=0.5] [label=above:$ 2b $]{};
\node (y3) at (8.5,0) [circle,fill,outer sep=5pt, scale=0.5] [label=above:$ 0 $]{};
\node (z3) at (8.5,-1) [circle,fill,outer sep=5pt, scale=0.5] [label=below:$ -2b $]{};
\node (w3) at (7,-1) [circle,fill,outer sep=5pt, scale=0.5] [label=below:$ 0 $]{};
\draw (x3) -- (y3);\draw (y3) -- (z3);\draw (z3) -- (w3);\draw (w3) to (x3);
\end{tikzpicture}$$
$\bullet$ Case $(D)$: $X=\mathbb{C}P^2 \# \overline{\mathbb{C}P^2}$, $c_1=f+2s$, where $f$ and $s$ are the fiber class and section class with $f\cdot f=0$, $f\cdot s=1$ and $s\cdot s=1$.
$(D1)$ $D$ cannot be a torus because it would not be minimal.
$(D2)$ $r(D)=2$, and either $([C_1],[C_2])=(af+s,(1-a)f+s)$ or $([C_1],[C_2])=(2s, f)$.
$(D3)$ $r(D)=3$ and $[C_1]=af+s, [C_2]=f, [C_3]=-af+s$.
$(D4)$ $r(D)=4$ and $[C_1]=af+s, [C_2]=f, [C_3]=-(a+1)f+s, [C_4]=f$.
The graphs in (D2), (D3) and (D4) are given respectively by $$\begin{tikzpicture}
\node (x) at (-2,0) [circle,fill,outer sep=5pt, scale=0.5] [label=above:$ 2a+1 $]{};
\node (y) at (-0.5,0) [circle,fill,outer sep=5pt, scale=0.5] [label=above:$ 3-2a $]{};
\draw (x) to [bend right] (y);\draw (y) to [bend right] (x);
\node (x1) at (1,0) [circle,fill,outer sep=5pt, scale=0.5] [label=above:$ 4 $]{};
\node (y1) at (2.5,0) [circle,fill,outer sep=5pt, scale=0.5] [label=above:$ 0 $]{};
\draw (x1) to[bend right] (y1);\draw (y1) to[bend right] (x1);
\node (x2) at (4,0) [circle,fill,outer sep=5pt, scale=0.5] [label=above:$ 2a+1 $]{};
\node (y2) at (5.5,0) [circle,fill,outer sep=5pt, scale=0.5] [label=above:$ 0 $]{};
\node (z2) at (4,-1) [circle,fill,outer sep=5pt, scale=0.5] [label=below:$ 1-2a $]{};
\draw (x2) -- (y2);\draw (y2) -- (z2);\draw (z2) -- (x2);
\node (x3) at (7,0) [circle,fill,outer sep=5pt, scale=0.5] [label=above:$ 2a+1 $]{};
\node (y3) at (8.5,0) [circle,fill,outer sep=5pt, scale=0.5] [label=above:$ 0 $]{};
\node (z3) at (8.5,-1) [circle,fill,outer sep=5pt, scale=0.5] [label=below:$ -2a-1 $]{};
\node (w3) at (7,-1) [circle,fill,outer sep=5pt, scale=0.5] [label=below:$ 0 $]{};
\draw (x3) -- (y3);\draw (y3) -- (z3);\draw (z3) -- (w3);\draw (w3) to (x3);
\end{tikzpicture}$$
Deformation classes
-------------------
In the holomorphic setting, two anti-canonical pairs are said to be (holomorphically) deformation equivalent if they are both isomorphic to fibers of a family of anti-canonical pairs over a connected base. Then we have the following finiteness result in [@Fr].
\[thm: finite kahler deformation\] There are only finitely many deformation types of anti-canonical pairs with the same self-intersection sequence.
In the symplectic world, various notions of equivalences have been introduced in the study of symplectic deformation classes of symplectic log Calabi-Yau pairs in [@LiMa16-deformation]. We recall their definitions here (See [@Sa13] for a thorough discussion of equivalence notions for symplectic manifolds).
Let $(X^0, D^0, \omega^0)$ and $(X^1, D^1, \omega^1)$ be two pairs of symplectic divisors with $r(D^0)=r(D^1)=k$. They are said to be **homologically equivalent** if there is an orientation preserving diffeomorphism $\Phi: X^0 \to X^1$ such that $\Phi_*[C^0_j]=[C^1_j]$ for all $j=1,\dots,k$. The homological equivalence is said to be **strict** if, in addition, $\Phi^*[\omega^1]=[\omega^0]$.
When $X^0=X^1$, they are said to be **symplectic homotopic** if $(D^0, \omega^0)$ and $(D^1, \omega^1)$ are connected by a family of symplectic divisors $(D^t, \omega^t)$, and they are further said to be **symplectic isotopic** if $\omega^t$ can be chosen to be a constant family.
$(X^0, D^0, \omega^0)$ and $(X^1, D^1, \omega^1)$ are said to be **symplectic deformation equivalent** if they are symplectic homotopic, up to an orientation preserving diffeomorphism. They are said to be **strictly symplectic deformation equivalent** if they are symplectic isotopic, up to an orientation preserving diffeomorphism.
In the holomorphic category, two anti-canonical pairs are deformation equivalent if they are homologically equivalent ([@Fr]). Similarly it was shown in [@LiMa16-deformation] that the symplectic deformations classes of symplectic log Calabi-Yau pairs are also completely determined by the homological information (the self-intersection sequence). So the list of homology types in Theorem \[thm:minimal model\] is actually the list of deformation classes of minimal models.
\[thm: symplectic deformation class=homology classes\] Two symplectic log Calabi-Yau pairs are symplectic deformation equivalent if they are homologically equivalent. In particular, each symplectic deformation class contains a Kähler pair.
Moreover, two symplectic log Calabi-Yau pairs are strictly symplectic deformation equivalent if they are strictly homologically equivalent.
The statement that each symplectic deformation class contains a Kähler pair is not stated in [@LiMa16-deformation] but it follows from the explanation in Section 4.2 of [@LiMa19-survey]. We would like to remark that Theorem \[thm: symplectic deformation class=homology classes\] also holds when $ D $ is a nodal sphere (using [@Bar99-nodal]) or a cuspidal sphere (using [@OhOn05-simple]).
Combining Theorem \[thm: symplectic deformation class=homology classes\] and \[thm: finite kahler deformation\], we obtain the following finiteness of symplectic deformation classes.
\[cor: finite deformation\] There are only finitely many symplectic deformation types of symplectic log Calabi-Yau pairs with the same self-intersection sequence.
By Theorem \[thm: symplectic deformation class=homology classes\], every symplectic deformation class contains a Kähler pair. The finiteness of Kähler Looijenga pairs follows directly from Theorem \[thm: finite kahler deformation\]. For elliptic symplectic log CY pairs, where the sequences are of length $1$, the finiteness is more straightforward – it follows from the finiteness of symplectic deformation types in the case of minimal pairs for each $(s)$, where $s=0, 8, 9$ (cf. Section 3 in [@LiMa16-deformation]), and the fact that there is only one way to blow up, up to deformation.
Embeddability and Rigidity of circular spherical divisors with $ b^+\ge 1 $ {#section:rigid}
===========================================================================
This section is devoted to the proof of Theorem \[thm:embeddable=rigid\] and \[thm:list\]. First we show that the several notions of sequences are indeed preserved under toric equivalence.
A sequence $ \vec{s} $ or a circular spherical divisor $ D $ being symplectically embeddable, rationally embeddable, anti-canonical or rigid is preserved under toric equivalence.
Since blow-up and blow-down are symplectic operations and blow-up/blow-down of a symplectic rational surface is still rational, it’s clear that being symplectically embeddable and rationally embeddable is preserved.
Let $ (X,D,\omega ) $ be a symplectic Looijenga pair. Blow up at a transverse intersection point of $ D $ to get $ (X',\omega') $ with the natural inclusion $ \iota_*:H_2(X;{\mathbb{Z}})\to H_2(X';{\mathbb{Z}}) $. Denote by $ D' $ the union of the proper transform of $ D $ and the exceptional curve $ E $. Then $ D' $ is a toric blow-up of $ D $ with $ [D']=[D]-[E] $. So $ [D']=\iota_*[D]-[E]=\iota_*(-K_X)-[E]=-K_{X'} $ and $ (X',D',\omega') $ is also a symplectic Looijenga pair. The proof for toric blow-down is similar.
Let $ D $ be rigid and $ D' $ be a toric blow-up of $ D $. Suppose $ D' $ is embedded in $ (X',\omega') $ as a symplectic circular spherical divisor such that $ X'-D' $ is minimal. Let $ E \subset D' $ be the exceptional sphere of the toric blow-up. Choose an almost complex structure $ J $ such that $ D' $ is $ J- $holomorphic, then we could blow down $ E $ to get $ D $ in $ (X,\omega ) $ such components of $ D $ still intersect positively and transversely. We claim that $ X- D $ is minimal. If not, then there must be a symplectic exceptional sphere $ E' $ in the complement of $ D $ and lifts to a symplectic exceptional sphere in the complement of $ D' $ in $ X' $. But $ X'- D' $ is minimal, contradiction. Since $ D $ is rigid and $ X-D $ is minimal, we have $ (X,D,\omega ) $ is a symplectic Looijenga pair. Then $ (X',D',\omega') $ is also a symplectic Looijenga pair by the previous paragraph and thus $ D' $ is rigid. The proof for toric blow-down is similar.
Notice that we have the following sequence of implications simply by their definitions.$$\text{symplectically embeddable} \Leftarrow \text{rationally embeddable} \Leftarrow \text{anti-canonical} \Leftarrow \text{rigid}$$ So to prove Theorem \[thm:embeddable=rigid\], it suffices to prove the converse for every arrow above. The proofs are distributed in Section \[subsection:symp embed = rational embed\] through \[subsection:anti-canonical = rigid\], as indicated in the titles. The proof of Theorem \[thm:list\] is contained in Section \[subsection:rationally embeddable\].
Maximal surfaces and pseudo-holomorphic curves {#section:maximal surface}
----------------------------------------------
In this subsection, we recall some useful notions in the theory of maximal surfaces ([@LiZh11-relative]) and pseudo-holomorphic curves in dimension 4 ([@McOp13-nongeneric]). These will be used frequently in the rest of the paper.
Suppose $ F\subset (M,\omega ) $ is a symplectically embedded surface without sphere components. $ F $ is called maximal if $ [F]\cdot E\neq 0 $ for any exceptional class $ E $.
The notion of maximal surfaces can be thought of as a relative version of minimality. The following lemma shows that positive genus surfaces have nice intersection property with exceptional classes.
\[lemma:J-holomorphic representative\] Let $ F $ be an embedded symplectic surface. Suppose the genus of each component $ F_i $ is positive. Then for any exceptional class $ e $, there exists an almost complex structure $ J $ such that both $ F $ and an embedded representative $ E $ of $ e $ are $ J- $holomorphic.
By positivity of intersection, we have $ [F]\cdot e\ge 0 $. In particular, $ [F]\cdot e=0 $ if and only if $ F $ and $ E $ are disjoint. As a consequence, for a symplectic surface with positive genus, being maximal is equivalent to having minimal complement.
McDuff and Opshtein studied the existence of embedded pseudo-holomorphic representatives relative to a pseudo-holomorphic normal crossing divisor. For exceptional classes, their result is a version of Lemma \[lemma:J-holomorphic representative\] for symplectic divisors.
Let $ D=\cup C_i $ be an $ \omega- $orthogonal symplectic divisor in $ (X,\omega ) $. An exceptional class $ e\in H_2(X;{\mathbb{Z}}) $ is called $ D- $good if $ e\cdot [C_i]\ge 0 $ for all $ i $.
\[lem: existence of nice J-sphere\](Theorem 1.2.7 of [@McOp13-nongeneric]) Let $D$ be an $\omega$-orthogonal symplectic divisor. There is a non-empty space $\mathcal{J}(D)$ of $\omega$-tamed almost complex structures making $D$ pseudo-holomorphic such that for any $ D- $good exceptional class $e$, there is a residual subset $\mathcal{J}(D,e) \subset \mathcal{J}(D)$ so that $e$ has an embedded $J$-holomorphic representative for all $J\in \mathcal{J}(D,e)$.
So for a $ D $-good exceptional class $ e $, we have $ e\cdot [D]\ge 0 $. In particular, $ e $ is $ D-$good if $ e\cdot [C_i]=0 $ for every component $ C_i $ of $ D $. So $ E $ is disjoint from $ D $ if and only if $ e\cdot [C_i]=0 $ for all $ i $. Denote by $ {\mathcal}{J}_\omega $ the space of $ \omega- $tamed almost complex structures on $ (X,\omega ) $. We recall the following definition from [@Li05-symplectic-surface-survey].
A homology class $ b\in H_2(X;{\mathbb{Z}}) $ is said to be stable if $ b $ is $ J $-effective for any $ J \in {\mathcal}{J}_\omega $, i.e. it can be represented by a $ J $-holomorphic curve.
In particular, by [@Mc90-structure] we see that any exceptional class $ e $ is stable. The following lemma says any symplectic surface class of non-negative self-intersection pairs non-negatively with any exceptional class.
Suppose $ a\in H_2(X;{\mathbb{Z}}) $ with $ a^2\ge 0 $ is realized by a connected embedded symplectic surface, then $ a\cdot b\ge 0 $ for any stable class $ b $.
Symplectically embeddable sequences are rationally embeddable {#subsection:symp embed = rational embed}
-------------------------------------------------------------
We start with the following observation on the embeddability of topological circular spherical divisors.
\[lemma:b\^+ leq 1\] A topological circular spherical divisor $ D $ cannot be embedded in a closed symplectic 4-manifold if $ b^+(Q_D)\ge 2 $.
Note that a divisor of form $ (-1,p) $ always has $ b^+\le 1 $ and can be excluded from our discussion. If $ D $ in closed symplectic 4-manifold $ (X,\omega) $ is not toric minimal, then by toric blow-downs, we can always get a toric minimal divisor $ D' $ embedded in $ (X',\omega') $ with $ b^+(Q_{D'})=b^+(Q_D)\ge 2 $ by Lemma \[lemma:toric eq topological\]. So we may assume $ D $ is toric minimal. Then there is at least one component $ C_i $ in $ D $ with $ C_i^2\ge 0 $ by Lemma \[lem: not negative semi-definite => at least one non-negative\], which implies $ (X,\omega ) $ is rational or ruled ([@Li99-smoothsphere]). This contradicts $ b^+(D)\ge 2 $ since $ b^+(X)=1 $ in this case.
As a result, there is no symplectically embeddable sequence with $ b^+\ge 2 $. In the rest of the section, we only need to consider circular spherical divisors with $ b^+= 1 $ in a closed symplectic manifold.
\[lemma:rational-embed\] Let $ D $ be a symplectic circular spherical divisor in $ (X,\omega ) $ with $ b^+(Q_D)= 1 $, then $ (X,\omega ) $ is rational.
Since being a symplectic rational surface is preserved under blow-up and blow-down, we could assume that $ D $ is toric minimal or of the form $ (-1,p) ,p>-4$. If $ s_i\ge 1 $ for some $ i $, then $ (X,\omega ) $ is rational as it contains a positive symplectic sphere ([@Mc90-structure]).
Now we assume $ D $ is toric minimal and $ s_i\le 0 $ for all $ i $. By Lemma \[lem: not negative semi-definite => at least one non-negative\], we must have $ s_i=0 $ for some $ i $ in order for $ b^+(Q_D)\ge 1 $ and thus $ (X,\omega ) $ must rational or ruled. Without loss of generality, we assume $ s_1=0 $. If $ X $ is irrational ruled with $ \pi:X\to B $, then $ [C_1] $ must be the fiber class. So $ [C_2] $ must contain a positive multiple of the section class and $ g(C_2)\ge g(B)\ge 1 $, which is contradiction.
Suppose $ D $ is of the form $ (-1,p) ,p>-4$. When $ p\ge 0 $, it follows from the same argument as above. The case $ (-1,\epsilon-2) $, $ \epsilon=-1,0,1 $, needs a different argument. Let $ D=C_1\cup C_2 $ with $ [C_1]^2=-1 $, $ [C_2]^2=\epsilon-2 $ and $ [C_1]\cdot [C_2]=2 $. Blow down $ C_1 $ to get $ X' $ such that $ X=X'\# \overline{{\mathbb{C}}{\mathbb{P}}}^2 $ and $ C_2 $ becomes an immersed nodal symplectic sphere $ C_2' $ with self-intersection $ 2+\epsilon $. Smoothing the singularity of $ C_2' $ we obtain a smoothly embedded symplectic torus $ T $ with self-intersection $ 2+\epsilon\ge 1 $. By Proposition 4.3 of [@LiMaYa14-CYcap], $ (X,\omega ) $ is rational or ruled. Suppose $ X $ is irrational ruled, then $ H_2(X;{\mathbb{Z}}) $ is generated by $ \{ f,s,e_1,\dots,e_k \} $ where $ f $ is the class of a fiber, $ s $ is the class of a section and $ e_i $’s are exceptional classes. Note that all exceptional classes in $ X $ are of the form $ e_i $ or $ f-e_i $. By Lemma 6.1 of [@seppi-li-wu-stability], we have $ [C_2]=bf+\sum \pm e_i $ and thus $ [C_1]\cdot [C_2]=\pm 1 $, which is a contradiction.
Rationally embeddable sequences {#subsection:rationally embeddable}
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In this section, we derive some restrictions on a symplectic circular spherical divisor embedded in a symplectic rational surface. In particular, we give a complete list of rationally embeddable sequences up to toric equivalence in Proposition \[prop:list\].
\[lemma:constraint on D\^2\] For a symplectic circular spherical divisor $D$ in a symplectic rational surface $(X, \omega)$ we have that $[D]^2\leq 9$.
By Proposition 3.14 in [@LiZh11-relative] and Theorem 6.10 in [@OhOn03-simple-elliptic], any maximal symplectic torus $T$ has $[T]^2\leq [K_{\omega}]^2 $, and if $[T]^2=9$ and $T$ is maximal, then $T$ is a torus in $X={\mathbb{C}}{\mathbb{P}}^2$. If the torus $ T $ is not maximal, we perform blow down away from $ T $ to get a maximal $T'\subset (X',\omega')$. Notice that $[T']^2\leq [K_{\omega'}]^2\leq 9$. Hence $[T]^2=[T']^2\leq 9$, and $[T]^2=9$ only if $X'={\mathbb{C}}{\mathbb{P}}^2$. Symplectically smooth out $D$ to obtain a symplectic torus $T$ and we have $ [T]=[D] $. Therefore we have the conclusion $[D]^2=[T]^2\leq 9$.
The above upper bound on $ [D]^2 $ becomes an upper bound on $ r(D) $ when $ s_i\ge -1 $ for all $ i $. Combined with the homological classification in Lemma \[lemma:topological cyclic\], we can determine exactly when such a sequence is rationally embeddable.
\[lemma:s\_i>=-1 embeddable\] Let $ \vec{s} $ be a sequence with $ s_i\ge -1 $ for all $ i $. Then it is rationally embeddable if and only if it is toric equivalent to $ (1,1,1) $ or $ (-1,k), -1\le k\le 5 $ or $ (1,k), 0\le k\le 4 $ or one in the family $ {\mathcal}{F}(2,2) $ of Lemma \[lemma:topological cyclic\] (6).
Denote by $ r $ the length of $ \vec{s} $ and $ D $ the divisor corresponding to $ \vec{s} $. In the case $ r\ge 3 $, we might assume the corresponding $ D $ is toric minimal and $ s_i\ge 0 $. Suppose $ D $ is embedded in symplectic rational surface $ (X,\omega) $, then $ 9\ge [D]^2=\sum_{i=1}^k(s_i+2)\ge 2r $, thus $ r\le 4 $. By Lemma \[lemma:topological cyclic\], we have that $ \vec{s} $ can be one of the following: $ (0,0,0,0), (1,1,1), (1,1,0),(k,0,0),0\le k\le 2 $. Note that $ (0,0,0,0) $ is toric equivalent to $ (1,0,-1,0) $ using the balancing move in Example \[eg: balancing self-intersection by $0$-sphere\] and thus toric equivalent to $ (1,1,1) $. Similarly, combining the balancing move and toric blow-down, we have that $ (1,1,0) $ is toric equivalent to $ (4,1) $ and $ (k,0,0) $ to $ (k+2,1) $. It’s easy to check they are indeed rationally embeddable as they can be realized as blow-ups of the minimal models in Theorem \[thm:minimal model\].
In the case $ r=2 $, we cannot assume $ D $ to be toric minimal. Again by Lemma \[lemma:topological cyclic\], we have that $ \vec{s} $ is $ (k,-1), -1\le k $ or one of $ (4, 1), (4, 0), (3, 1), (3,0), (2, 2), (2, 1), (2, 0),
(1, 1),\\ (1, 0), (0, 0) $. In the first case, we have $ 8\ge s(D)=k+3 $ because $ X $ contains at least one exceptional class. So $ k\le 5 $. They are all rationally embeddable as blow-ups of the minimal models.
Given two sequences $ \vec{s},\vec{s}' $ of length $ l $, we write $ \vec{s}\prec \vec{s}' $ if $ s_i\le s_i' $ for every $ 1\le i\le l $. And we say a sequence $ \vec{s} $ is **blown-up** if $ \vec{s}\prec \vec{d} $ for some toric blow-up $ \vec{d} $ of the sequence $ (1,1,1) $. Note that $ \vec{s}=(1,1-p_1,-p_2,\dots,-p_{l-1},1-p_l) $ being blown-up in our definition is equivalent to the dual cycle of $ (-p_1,\dots,-p_l) $ being embeddable in the sense of [@GoLi14].
\[lemma:hyperbolic sequence embeddable\] If $ \vec{s} $ is of the form $(1, -p_1+1, -p_2, ...., -p_{l-1}, -p_l+1)$ with $p_i\geq 2$ and $l\geq 2$, then it is rationally embeddable if and only if it is blown-up.
If $ \vec{s} $ is of such form and is blown-up, then $\vec{s} $ is rationally embeddable by Lemma 2.4 of [@GoLi14]. The converse was actually implicitly contained the proof of Theorem 3.1 in [@GoLi14] and we recall their argument here. Let $ D $ be embedded in a rational surface $ X $ with $ \vec{s}(D)=\vec{s}=(1,1-p_1,\dots,-p_{l-1},1-p_l) $, we blow up $ D $ to $ D' $ in $ X'={\mathbb{C}}{\mathbb{P}}^2\# M\overline{{\mathbb{C}}{\mathbb{P}}}^2 $ with $ M\ge 1 $ and $ \vec{s}(D')=(1,1-p_1,-p_2,\dots,-p_{l-1}-1,-1,-p_l) $. Let $ S' $ be the irreducible component in $ D' $ with $ [S']^2=-p_l $ and let $ \tilde{D}'=D'-S' $ be the symplectic string with intersection sequence $ (1,1-p_1,-p_2,\dots,-p_{l-1}-1,-1) $. By Theorem 4.2 of [@Lis08-lens], there is a sequence of symplectic blowdowns of $ X' $ to $ {\mathbb{C}}{\mathbb{P}}^2 $ such that $ \tilde{D}' $ blows down to the union of two lines $ l\cup l'\subset {\mathbb{C}}{\mathbb{P}}^2 $. During this process, $ S' $ blows down to a smoothly embedded symplectic sphere intersecting both $ l $ and $ l' $ exactly once, hence $ S' $ blows down to a line. So $ \vec{s} $ is a blow-up of the sequence $ (1,1,1) $ corresponding to three lines in $ {\mathbb{C}}{\mathbb{P}}^2 $, which exactly means $ \vec{s} $ is blown-up.
\[lemma:k+p=5\] $(5+p, -p)$ with $ p\ge -2 $ is not rationally embeddable if $ p\neq -1 $.
Suppose $ \vec{s}=(5+p,-p) $ is rationally embeddable and $ D $ is a symplectic circular spherical divisor in a symplectic rational surface $ X $ with $ s(D)=\vec{s} $. Observe that if $ p\ge 0 $, $ X $ cannot be $ {\mathbb{C}}{\mathbb{P}}^2 $ because there is no second homology class with negative self-intersection in $ {\mathbb{C}}{\mathbb{P}}^2 $. Also, $ (3,2) $ cannot be embedded into $ {\mathbb{C}}{\mathbb{P}}^2 $ either because there is no second homology class with self-intersection $ 2 $ or $ 3 $. So we could assume $ X={\mathbb{C}}{\mathbb{P}}^2\# l\overline{{\mathbb{C}}{\mathbb{P}}}^2 $ for some $ l\ge 1 $.
Now we use the standard form of sphere classes with positive square to show such configuration of symplectic spheres cannot be embedded in a symplectic rational surface.
**If $ C_1 $ is an odd sphere** with $[C_1]^2=2x+1\geq 3$, we use the $ {\mathbb{C}}{\mathbb{P}}^2\# l\overline{{\mathbb{C}}{\mathbb{P}}}^2 $ model with $ l\ge 1 $, where $ H_2(X;{\mathbb{Z}})={\mathbb{Z}}\{ h,e_1,\dots,e_l \} $. Then we have $$\begin{aligned}
&[C_1]= (x+1)h -x e_1,\\ &[C_2]=ah-be_1 - \sum_{i= 2}^l b_i e_i,\\
&[C_2]^2=a^2-b^2-\sum_{i=2}^l b_i^2=5-( 2x+1)=-2x+4,\\
&[C_1]\cdot [C_2]=(x+1)a-xb=2.
\end{aligned}$$ The equation with $ [C_2]^2$ implies that $a^2-b^2\geq -2x+4$. Any solution to $ (x+1)a-xb=0 $ is of the form $(a, b)=(ux+2, u(x+1)+2)$ for an integer $u$. We have that $$\begin{aligned}
a^2-b^2&=(ux+2)^2-(u(x+1)+2)^2\\&=-u(2ux+u+4)=(-2x-1)(u^2+\frac{4}{2x+1}u)=:F(u).
\end{aligned}$$ First we check that $ u\neq 0 $. If $ u=0 $, then $ (a,b)=(2,2) $. The adjunction formula for $g(C_2)=0$ is $(a-1)(a-2) -b(b-1)- \sum b_i(b_i-1)=0$. Since $y(y-1)\geq 0$ for any integer $y$, we have $$(a-1)(a-2)\ge b(b-1).$$ Clearly $(a,b)=(2,2)$ violates this inequality.
Since $0<\frac{4}{2x+1}<2$ the values of $(u^2+\frac{4}{2x+1}u)$ for an integer $u$ are smallest when $u=0$ or $-1$. We have $F(u)\leq F(-1)=-2x+3 < -2x+4$ for $u\leq -1$ and $F(u)\leq F(1)=-2x-5< -2x+4$ for $u\geq 1$. But this contradicts $ a^2-b^2\ge -2x+4 $.
**If $ C_1 $ is an even sphere** with $[C_1]^2=2x\geq 6$, we can use the $(S^2\times S^2)\# l \overline{{\mathbb{C}}{\mathbb{P}}}^2$ model with $l\geq 0$, where $ H_2(X;{\mathbb{Z}})={\mathbb{Z}}\{ s,f,e_1,\dots,e_l \} $. Then we have $$\begin{aligned}
&[C_1]= s+ x f,\\
&[C_2]=as+ bf - \sum_{j=1}^l b_j E_j,\\
&[C_2]^2=2ab-\sum_{j=1}^l b_j^2=5-2x,\\
&[C_1]\cdot [C_2]=ax+b=2.
\end{aligned}$$ The equation with $ [C_2]^2 $ implies that $ 2ab\geq -2x+5$. Any solution to $ax+b=2$ is of the form $(u, -ux+2)$ for an integer $u$. Similarly, we have that $$\begin{aligned}
2ab&=2u(-ux+2)\\ &=-2x u^2+4u=-2x(u^2+\frac{4}{2x}u)=:G(u).
\end{aligned}$$ Again we check that $ u\neq 0 $. If $ u=0 $, then $ (a,b)=(0,2) $. The adjunction formula for $g(C_2)=0$ is $-2(a-1)(b-1)- \sum b_i(b_i-1)=0$ Since $y(y-1)\geq 0$ for any integer $y$, we have $$(a-1)(b-1)\ge 0.$$ Clearly $(a,b)=(0,2)$ violates this inequality.
Since $0<\frac{4}{2x}<2$, we have $G(u)\leq G(-1)=-2x+4< -2x+5$ for $u\leq -1$ and $G(u)\leq G(1)=-2x-4< -2x+5$ for $ u\ge 1 $. But this contradicts $ 2ab\ge -2x+5 $.
\[prop:list\] Any rationally embeddable sequence with $ b^+\ge 1 $ is toric equivalent to one in the list:
1. $ (1,1,p) $ with $ p\le 1 $.
2. $ (1,p) $ with $ p\le 4 $.
3. $ (0,p) $ with $ p\le 4 $.
4. $ (1,1-p_1,-p_2,\dots,-p_{l-1},1-p_l) $ with $ p_i\ge 2 $, $ l\ge 2 $.
5. $ s_i\ge -1 $ for all $ i $.
6. $ (-1,-2) $, $ (-1,-3) $
The proof is a case-by-case analysis based on the length of the sequence. Since a rationally embeddable sequence must have $ b^+\le 1 $, Lemma \[lemma:topological cyclic\] applies here.
**Case 1**: $r(D)=2$.
- When $r^{\geq 0}(D)=2$, the pairs are listed in (6) of Lemma \[lemma:topological cyclic\]. They all belong to (5) of the list.
- When $r^{\geq 0}(D)=1$, they are of the forms $(k, p)$ with $ k\ge 0 $, $ q<0 $ and $k+p\leq 5$ by Lemma \[lemma:constraint on D\^2\]. If $ k+p=5 $, then by Lemma \[lemma:k+p=5\] the only rationally embeddable one is $ (1,4) $ belonging to (2). Now suppose $ k+p\le 4 $. If $ k\le 1 $, the sequence $ (k,p) $ belongs to (2) or (3) of the list. If $ k\ge 2 $, we can toric blow up the pair $ (C_1,C_2) $, and if necessary, apply successive toric blow-ups to the pairs of the proper transform of $ C_1 $ and the exceptional spheres to get $ \bar D $ with $ \bar s_1=1, \bar s_2 = -1, \bar s_i\le -2 $ for $ i\ge 3 $. Then $ \bar D $ belongs to (4) of the list.
- When $r^{\geq 0}(D)=0$, by (8) of Lemma \[lemma:topological cyclic\], there are only 3 spherical circular sequences with $b^+(Q_D)=1$: $(-1, -1), (-1, -2)$ and $(-1, -3)$. $ (-1,-1) $ belongs to (5) and $ (-1,-2),(-1,-3) $ belong to (6) of the list.
**Case 2**: $r(D)=3$ and $ D $ is toric minimal (if not, reduce to the $r=2$ case).
- When $r^{\geq 0}(D)=3$, it belongs to (5) of the list.
- When $r^{\geq 0}(D)=2$, by the 5th bullet of Lemma \[lemma:topological cyclic\], we can assume that $\vec{s}(D)=(1, 1, p\leq 1)$, or $\vec{s}(D)=(s_1\geq 0, 0, s_3)$ with $ s_1+s_3\leq 2$. The former case is already in the list. In the latter case, we can apply the balancing move as in Example \[eg: balancing self-intersection by $0$-sphere\] based at $ C_2 $ to decrease $ s_1 $ to $\bar s_1=0 $ and increase $ s_3 $ to $ \bar s_3=s_3+s_1 $. Denote the new divisor by $ \bar D $. By toric blowing up the pair $ (\bar C_1,\bar C_2) $ and contracting the proper transform of $ \bar C_1,\bar C_2 $, we can reduce the length of the divisor to 2. So $ D $ is toric equivalent to one in the list.
- When $r^{\geq 0}(D)=1$, if $s_1=1$, then it belongs to (4) of the list. If $s_1\geq 2$, toric blow up the pair $C_1, C_2$, and if necessary, apply successive toric blow-ups to the pairs of the proper transforms of $C_1$ and the exceptional spheres to get $\bar D$ with $\bar s_1=1, \bar s_2=-1, \bar s_i\leq -2$ for $i\geq 3$ (so $\bar D$ is not toric minimal), so $\bar D$ belongs to (4) of the list.
If $s_1=0$, apply the balancing move based at $C_1$ to increase $s_2$ to $\bar s_2=0$ (while decreasing $s_r$ to $\bar s_r-s_2$). Notice that $r^{\geq 0}(\bar D)=2, \bar s_1=\bar s_2=0$, and $\bar D$ toric minimal. We treated this case above.
**Case 3**: $r(D)=4$ and $D$ is toric minimal.
- When $r^{\geq 0}(D)=4$, it is $(0, 0, 0, 0)$ by (3) of Lemma \[lem: non-negative components\]. Note that any sequence of form $ (0,0,0,p) $ is toric equivalent to the sequence $ (1,1,p+1) $ by toric blowing up $ (C_1,C_2) $ and contracting the proper transforms of $ C_1,C_2 $.
- When $r^{\geq 0}(D)=3$, by (2) of Lemma \[lemma:topological cyclic\], $(s)=(k, 0, p, 0)$ with $k\geq 0, p<-k$. Using the balancing move based at $ C_2 $, $ (k,0,p,0) $ is toric equivalent to $ (0,0,k+p,0) $ and thus toric equivalent to $ (1,1,k+p+1) $.
- When $r^{\geq 0}(D)= 2$, by (3) of Lemma \[lemma:topological cyclic\] we have $(0, p_1, 0, p_2), p_i< 0$ or $(k\geq 0, 0, p_1, p_2),p_i<0, k+p_1+p_2\leq 0$.
For the case $(k, 0, p_1, p_2)$ we apply the balancing move based at $ C_2 $ to transform $ D $ to $\bar D$ with $\bar s_1=\bar s_2=0, \bar s_3=s_3+s_1=k+p_1\le 0, \bar s_4=s_4=p_2\leq -2$. By blowing up the pair $ (\bar C_1,\bar C_2) $ and contracting the proper transforms of $ \bar C_1,\bar C_2 $, we get a divisor $ \bar D' $ of length 3.
For the case $(0, p_1, 0, p_2), p_i<0$, using the balancing move based at $ C_3 $, it is toric equivalent to $ (0,0,0,p_1+p_2) $ and thus toric equivalent to $ (1,1,p_1+p_2+1) $.
- When $r^{\geq 0}(D)= 1$, if $s_1=1$ and $ s_i\le -2 $ for $ i\ge 2 $, then it belongs to (4) of the list.
If $s_1\geq 2$, toric blow up the pair $C_1, C_2$, and if necessary, apply successive toric blow-ups to the pairs of the proper transforms of $C_1$ and the exceptional spheres to get $\bar D$ with $\bar s_1=1, \bar s_2=-1, \bar s_i\leq -2$ for $i\geq 3$, so $\bar D$ belongs to (4) of the list.
If $s_1=0$, apply the balancing move based at $C_1$ to increase $s_2$ to $\bar s_2=0$ (while decreasing $s_r$ to $\bar s_r-s_2$). Notice that $r^{\geq 0}(\bar D)=2, \bar s_1=\bar s_2=0$, and $\bar D$ toric minimal. We have treated this case above.
**Case 4**: $ r(D)\geq 5$ and $ D $ is toric minimal.
This is proved by induction. Suppose we have proved the case $r(D)\leq k$ with some $k\geq 4$, where $D$ is not assumed to be toric minimal. The case $ r(D)=k+1 $ and $ D $ not toric minimal follows directly from induction hypothesis by toric blow-down. We will verify the case where $r(D)=k+1$ and $D$ toric minimal.
We have $r^{\geq 0}(D)\geq 1$ by (1) of Lemma \[lem: not negative semi-definite => at least one non-negative\] and we may assume that $s_1\geq 0$. By (1) of Lemma \[lemma:topological cyclic\], $r^{\geq 0}(D)\leq 2$.
- When $r^{\geq 0}(D)=2$, by (1) of Lemma \[lemma:topological cyclic\], we can assume that $s_1\geq s_2=0$. Apply the balancing move based at $C_2$ to transform to $\bar D$ with $\bar s_1=\bar s_2=0, \bar s_3=s_3+s_1, \bar s_r=s_r\leq -2$ for $ 4\le r\le k+1 $. Toric blow up the pair $(\bar C_1, \bar C_2)$ and then contract the proper transforms of $\bar C_1$ and $ \bar C_2$ to get $\bar D'$ with $\bar s_1'=1, \bar s_2'=\bar s_3+1, \bar s_{k}'=\bar s_{k+1}+1$. Since $ r(\bar D')=r(D)-1=k$, the induction hypothesis applies.
- Suppose $r^{\geq 0}(D)=1$ and we may assume $ s_1\ge 0 $. If $s_1=1$, then it belongs to (4) of the list. If $s_1\geq 2$, toric blow up the pair $C_1, C_2$, and if necessary, apply successive toric blow-ups to the pairs of the proper transforms of $C_1$ and the exceptional spheres to get $\bar D$ with $\bar s_1=1, \bar s_2=-1, \bar s_i\leq -2$ for $i\geq 3$, which belongs to (4) of the list.
If $s_1=0$, apply the balancing move based at $C_1$ to increase $s_2$ to $\bar s_2=0$ (while decreasing $s_{k+1}$ to $\bar s_{k+1}=s_{k+1}+s_2< -1$ as $ s_2,s_{k+1}< -1 $). Notice that $r^{\geq 0}(\bar D)=2, \bar s_1=\bar s_2=0$, and $\bar D$ is toric minimal. We have treated this case above.
The theorem is basically Proposition \[prop:list\] with the exception of a few cases. We are left to show that the list in Proposition \[prop:list\] is toric equivalent to the list in the theorem and every sequence in the theorem can indeed be symplectically embedded in a rational manifold. This is done by Lemma \[lemma:s\_i>=-1 embeddable\], Lemma \[lemma:hyperbolic sequence embeddable\] and simply observing the following toric equivalences (denoted by $ \sim $): $$\begin{aligned}
(-1,k)&\sim (1,-1,-2,\dots,-2) \text{ with $ k-1 $ number of $ -2 $}, k\ge 2\\
(1,4)&\sim (3,-1,0) \sim (1,1,0)\\
(2,2)&\sim (1,1,-1)
\end{aligned}$$ Their boundaries are classified by calculating the trace using Lemma \[lem: property of continuous fraction\] and the fact that $ -T_A=T_{A^{-1}} $.
Rationally embeddable sequences are anti-canonical
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The following is essentially due to Theorem 4.2 in [@Lis08-lens] and Theorem 3.1 in [@GoLi14].
\[lemma:hyperbolic sequence is anti-canonical\] If a rationally embeddable sequence is of the form $$(1, -p_1+1, -p_2, ...., -p_{l-1}, -p_l+1)$$with $p_i\geq 2$ and $l\geq 2$, then it is anti-canonical and rigid.
\[lemma:s\_i>=-1\] A rationally embeddable sequence $ \vec{s} $ with $ s_i\ge -1 $ for all $ i $ is anti-canonical and rigid.
Let $ D $ be a symplectic circular spherical divisor in a symplectic rational surface $ (X,\omega ) $ with $ \vec{s}(D)=\vec{s} $ such that its complement is minimal. We could symplectically smooth $ D $ to a symplectic torus $ T $ with $ [T]^2=[D]^2=\sum (s_i+2)>0 $. It suffices to prove $ T $ is maximal. Then by [@OhOn03-simple-elliptic], $ T $ actually represents $ c_1(X,\omega ) $, i.e. $ PD([D])=PD([T])=c_1(X,\omega ) $.
Suppose $ T $ is not maximal, then there exists exceptional class $ E $ such that $ [T]\cdot E =0$. Note that if $ s_i\ge 0 $, we have $ [C_i]\cdot E\ge 0 $. If $ s_i=-1 $, then $ [C_i] $ is an exceptional class. Note that $ [C_i]\neq E $ because $ [C_i]\cdot [D]=1 $. Then we have $ [C_i]\cdot E \ge 0 $ since both are stable classes and have positivity of intersection. So $ E $ is $ D- $good and there is an almost complex structure $ J $ such that $ D $ is $ J- $holomorphic and $ E $ has an embedded $ J- $holomorphic representative $ S $. Since $ [D]\cdot E=[T]\cdot E =0$, $ D $ and $ S $ are disjoint, contradicting the minimality of the complement of $ D $.
\[prop:anti-canonical\] Any rationally embeddable sequence with $b^+=1$ is anti-canonical.
It suffices to show that any rationally embeddable sequence listed in Proposition \[prop:list\] is anti-canonical.
\(4) is anti-canonical by Lemma \[lemma:hyperbolic sequence is anti-canonical\] and (5) is anti-canonical by Lemma \[lemma:s\_i>=-1\].
(1),(2) and (3) are realized respectively as non-toric blow-ups of minimal models (B3),(B2) and (D2) in Theorem \[thm:minimal model\].
\(6) is also realized as non-toric blow-ups of minimal models (B2) or (C2) or (D2).
Anti-canonical sequences are rigid {#subsection:anti-canonical = rigid}
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\[lemma:(-1,-3) rigid\] $ (-1,-3) $ is rigid.
Suppose $ D $ is a symplectic circular spherical divisor in a symplectic rational surface $ X $ with $ \vec{s}(D)=(-1,-3) $. Denote the components of $ D $ by $ A,B $, where $ A^2=-1,B^2=-3 $. Smooth $ D $ to get a symplectic torus $ T $ with $ T^2=0 $. Note that we must have $ K_X^2 \le 0 $. Otherwise the subspace in $ H_2(X;{\mathbb{Z}}) $ spanned by $ 3[A]+2[B] $ and $ -K_{X} $ has intersection form $ \begin{pmatrix}
3 & 1\\ 1 & (-K_{X})^2
\end{pmatrix} $ and is positive definite, contradicting $ b^+(X)=1 $. So we have that $ X $ must be $ {\mathbb{C}}{\mathbb{P}}^2\#l\overline{{\mathbb{C}}{\mathbb{P}}}^2 $ with $ l\ge 9 $.
When $ l=9 $, we have that $ T^2=K_{X}^2=0 $ and $ K_{X}\cdot T=0 $. By light cone lemma, we have $ T $ is proportional to $ -K_{X} $. Since $ T\cdot A=(A+B)\cdot A=1=-K_{X}\cdot A $, we actually have $ [D]=[T]=-K_{X} $, i.e. $ D $ is anti-canonical.
When $ l\ge 10 $, suppose $ T $ is maximal. By Proposition 3.14 of [@LiZh11-relative], we must have $ (K_{X}+T)^2\ge 0 $. However, $ (K_{X}+T)^2=K_{X}^2+2K_{X}\cdot T + T^2=K_{X}^2=9-l<0 $. So $ T $ cannot be maximal. By Theorem 3.21 of [@LiZh11-relative], exceptional classes orthogonal to $ [T] $ are pairwise orthogonal and above discussion implies that there are at least $ l-9 $ such exceptional classes. Denote them by $ Q_1,\dots ,Q_{l-9} $.
Now consider the blowup class $ \tilde{K}=K_{X}-Q_1-Q_2 -\dots - Q_{l-9} $. Then we have $$T^2=0, \quad (-\tilde{K})^2=0, \quad \text{ and }T\cdot (-\tilde{K})=0.$$ By the light cone lemma, we have $ T $ and $ -\tilde{K} $ are proportional. Pairing both with $ A $, we have $ T\cdot A=1=(-\tilde{K})\cdot A $. Therefore we conclude that $ T=-\tilde{K}=-K_{X_l}+Q_1+\dots + Q_{l-9} $.
Now we have $$\begin{aligned}
1=(A+B)\cdot A=T\cdot A=(-K_{X_l}+Q_1+\dots + Q_{l-9})\cdot A=1+\sum Q_i\cdot A.
\end{aligned}$$ Since both $ Q_i,A $ are stable classes, we must have $ Q_i\cdot A\ge 0 $ and thus $ Q_i\cdot A=0 $ for all $ i $. This implies $ Q_i\cdot B=Q_i\cdot (T-A)=0 $. So each $ Q_i $ is $ D- $good and has an embedded symplectic representative in the complement of $ D $.
So $ D $ has minimal complement only if $ l=9 $, where $ D $ is anti-canonical.
\[lemma:(-1,-2) rigid\] $ (-1,-2) $ is rigid.
The proof follows the exact same line as the proof for $ (-1,-3) $. Suppose $ D $ in a symplectic rational surface $ X $ is a symplectic circular spherical divisor with $ \vec{s}(D)=(-1,-2) $ and $ T $ is the symplectic torus we get from smoothing $ D $. Note that we must have $ K_X^2\le 1 $. Otherwise the subspace spanned by $ [T] $ and $ -K_{X} $ has intersection form $ \begin{pmatrix}
1 & 1\\ 1 & (-K_X)^2
\end{pmatrix} $ and is positive definite, contradicting $ b^+(X_l)=1 $. So $ X $ must be $ {\mathbb{C}}{\mathbb{P}}^2\#l\overline{{\mathbb{C}}{\mathbb{P}}}^2 $ with $ l\ge 8 $. We could show that $ D $ has minimal complement only if $ l=8 $. When $ D $ has minimal complement, the symplectic torus $ T $ we get from smoothing $ D $ is maximal and $ T^2=1>0 $. By [@OhOn03-simple-elliptic], $ T $ represents $ c_1(X,\omega ) $, i.e. $ PD([D])=c_1(X,\omega) $.
\[prop:rigid\] Anti-canonical sequences with $ b^+=1 $ are rigid.
It suffices to show anti-canonical sequences of forms listed in Proposition \[prop:list\] are rigid.
\(6) is rigid by Lemma \[lemma:(-1,-2) rigid\] and Lemma \[lemma:(-1,-3) rigid\].
\(5) is rigid by Lemma \[lemma:s\_i>=-1\].
\(4) is rigid by Lemma \[lemma:hyperbolic sequence is anti-canonical\].
\(3) follows from Theorem 3.5 in [@GoLi14]. The case of $p\leq -2$ in (2) follows from Theorem 3.1 in [@GoLi14], while the remaining sequences in (2) have $s_i\ge -1,\forall i$ and follows from (5).
Now we want to show the sequence $ (1,1,p) $ is rigid. By McDuff, $ (X,\omega ) $ is rational and it must be $ {\mathbb{C}}{\mathbb{P}}^2\# l\overline{{\mathbb{C}}{\mathbb{P}}}^2 $. By (4) and (5) of Lemma \[lemma:topological cyclic\], we may assume that $ [C_1]=[C_2]=h $ and $ [C_3]=h-\sum a_ie_i $, where $ \{h,e_1,\dots,e_l \} $ is a basis of $ H_2(X;{\mathbb{Z}}) $. Adjunction formula for $ C_3 $ says $ \sum (a_i^2-a_i)=0 $, which implies each $ a_i $ is $ 0 $ or $ 1 $. If $ a_k=0 $ for some $ k $, then $ [C_j]\cdot e_k=0 $ for $ j=1,2,3 $. Then $ e_k $ is $ D- $good and there is a symplectic sphere representing $ e_k $ in the complement of $ D $, contradicting the minimality.
Contact aspects
===============
Divisor neighborhood and contact structure
------------------------------------------
In this section, we review some results about the convexity of divisor neighborhoods and the induced contact structure on the boundary. Also, we recall the notion of Donaldson divisors, which will become useful later.
Let $ D $ be a symplectic divisor in symplectic 4-manifold $ (W,\omega ) $ (not necessarily closed). A closed regular neighborhood of $ D $ is called a plumbing of $ D $. A plumbing $ P(D) $ of $ D $ is called a **concave/convex plumbing** if it is a strong symplectic cap/filling of its boundary. A concave plumbing is also called a **divisor cap** of its boundary. Let $ Q_D $ be the intersection matrix of $ D $ and $ a=([C_i]\cdot [\omega])\in ({\mathbb{R}}_+)^k $ be the area vector of $ D $. A symplectic divisor $ D $ is said to satisfy the positive (resp. negative) **GS criterion** if there exists $ z\in ({\mathbb{R}}_+)^k $ (resp. $ ({\mathbb{R}}_{\le 0})^k $) such that $ Q_D z=a $.
The GS criterion provides a way to tell when the divisor neighborhood is convex or concave.
\[thm:divisorcap\] Let $ D\subset (W,\omega ) $ be an $ \omega $-orthogonal symplectic divisor. Then $ D $ has a concave (resp. convex) plumbing if $ (D,\omega ) $ satisfies the positive (resp. negative) GS criterion.
Note that a symplectic divisor can always be made $ \omega $-orthogonal by a local perturbation ([@Gom95-fibersum]). A necessary condition for $D$ to have concave or convex plumbing is $\omega$ being exact on the boundary $ Y_D $. To determine the exactness of $ \omega|_{Y_D} $, it suffices to check the following local criterion.
\[lem: non-degenerate intersection form\] $\omega|_{Y_D}$ is exact if and only if there is a solution for $z$ to the equation $Q_Dz=a$, where $a=([\omega]\cdot [C_1],\dots,[\omega]\cdot [C_k])$ is the area vector. In particular, this holds if $Q_D$ is non-degenerate.
One can also check by simple linear algebra that the above condition is preserved under toric equivalence. If $ \omega|_{Y_D} $ is exact, then there is the following dichotomy depending on whether $ D $ is negative definite.
\[thm:convex\] A negative definite symplectic divisor has a convex plumbing.
\[[@LiMa14-divisorcap]\] \[thm:concave\] Let $D \subset (W,\omega_0)$ be a symplectic divisor. If $Q_D$ is not negative definite and $\omega_0$ restricted to the boundary of $D$ is exact, then $\omega_0$ can be locally deformed through a family of symplectic forms $\omega_t$ on $W$ keeping $D$ symplectic and such that $(D,\omega_1)$ is a concave divisor.
Although the statement of Theorem \[thm:divisorcap\] concerns the ambient symplectic manifold $ (W,\omega ) $, it actually does not rely on it. Suppose $ D $ is only a topological divisor with intersection matrix $ Q_D $ such that there exists $ z,a $ satisfying the positive/negative GS criterion $ Q_D z=a $. Then Theorem \[thm:divisorcap\] actually constructs a compact concave/convex symplectic manifold $ (P(D),\omega(z)) $ such that $ D $ is $ \omega(z)- $orthogonal symplectic divisor in $ P(D) $ and $ a $ is the $ \omega(z)- $area vector of $ D $. The following uniqueness result implies that the symplectic structure $ \omega(z) $ may vary with $ z $ but the induced contact structure on the boundary only depends on $ D $.
\[prop:unique-contact\] Suppose $ D $ is an $ \omega- $orthogonal symplectic divisor which satisfies the positive/negative GS criterion. Then the contact structures induced on the boundary are contactomorphic, independent of choices made in the construction and independent of the symplectic structure $ \omega $, as long as $ (D,\omega ) $ satisfies positive/negative GS criterion.
Moreover, if $ D $ arises from resolving an isolated normal surface singularity, then the contact structure induced by the negative GS criterion is contactomorphic to the contact structure induced by the complex structure.
When $ D $ is negative definite, $ (Y_D,\xi_D ) $ is contactomorphic to the contact boundary of some isolated surface singularity (see Proposition \[prop:negative definite = stein fillable\]) and is called a Milnor fillable contact structure. A closed 3-manifold $ Y $ is called Milnor fillable if it carries a Milnor fillable contact structure. For every Milnor fillable $ Y $, there is a unique Milnor fillable contact structure ([@Caubel-Nemethi-Pampu2006]), i.e. the contact structure $ \xi_D $ only depends on the oriented homeomorphism type of $ Y_D $ instead of $ D $ when $ D $ is negative definite. In light of this uniqueness result, it is natural to ask if similar results hold when $ D $ has a concave neighborhood. The answer is no and the following counterexample is given in [@LiMa14-divisorcap].
Let $ D_1 $ be a single sphere with self-intersection $ 1 $ and $ D_2 $ be two spheres with self-intersections $ 1 $ and $ 2 $ intersecting at one point as follows. $$ \begin{tikzpicture}
\node (x) at (0,0) [circle,fill,outer sep=5pt, scale=0.5] [label=above: $ 1 $]{};
\node (y) at (2,0) [circle,fill,outer sep=5pt, scale=0.5] [label=above: $ 2 $]{};
\draw (x) to (y);
\end{tikzpicture}$$ Both divisors have a concave neighborhood. By [@Ne81-calculus], we can see that $ -Y_{D_1} $ and $ -Y_{D_2} $ are both orientation preserving homeomorphic to $ S^3 $. However, $ \xi_{D_1} $ is the unique tight contact structure on $ S^3 $ while $ \xi_{D_2} $ is overtwisted.
So far all the counterexamples we can construct consist of divisors with different $ b^+ $. So we refine our question to the following:
Suppose $ D_1 $ and $ D_2 $ are divisors with concave neighborhoods such that $ b^+(Q_{D_1})=b^+(Q_{D_2}) $ and $ -Y_{D_1}\cong -Y_{D_2} $. Is $ (-Y_{D_1},\xi_{D_1}) $ contactomorphic to $ (-Y_{D_2},\xi_{D_2}) $?
Note that $ D_1 $ and $ D_2 $ must be related by Neumann’s plumbing moves, including toric blow-ups/blow-downs and interior blow-ups/blow-downs (see Definition \[def:interior blow up\]). As a first step towards this question, we have the following proposition, whose proof is contained in the appendix.
\[prop:contact toric eq\] The contact structure induced by GS construction is invariant under toric blow-ups/blow-downs and interior blow-ups/blow-downs.
We also recall here the notions of maximal divisors and Donaldson divisors. They will be used to transit between minimal filling and minimal complement in Section \[section:fillability\].
For a symplectic divisor $ D=\cup C_i $, we extend the notion and also call the formal sum $ \tilde{D}=\sum z_iC_i $, $ z_i\in {\mathbb{Z}}$ a symplectic divisor. Then a symplectic divisor $ \tilde{D}=\sum z_iC_i $ in a closed symplectic 4-manifold $ (X,\omega ) $ is called a maximal divisor if $ [\tilde{D}]\cdot e>0 $ for any exceptional class $ e $.
Maximal divisors are natural extensions of maximal surfaces introduced in Section \[section:maximal surface\]. They are intimately related to divisor caps through the notion of Donaldson divisors.
Let $ \tilde{D}=\sum z_iC_i $ be a symplectic divisor in a symplectic cap $ (P,\omega) $ of $ (Y,\xi ) $. Then $ \tilde{D} $ is called a **Donaldson divisor** if $ [\tilde{D}]\in H^2(P;{\mathbb{Z}}) $ is the Lefschetz dual of $ c[(\omega,\alpha)] $ for some $ c>0 $, where $ \alpha $ is a contact form for $ \xi $ such that $ ([\omega,\alpha]) $ is a rational class.
\[lemma:divisor cap=donaldson\] Let $ (D,\omega ) $ be a symplectic divisor satisfying positive GS criterion $ Qz=a $ and $ P $ a concave plumbing of $ D $ with contact form $ \alpha $ on the boundary. Then $ PD([(\omega,\alpha)])=\sum z_i[C_i]\in H_2(P;{\mathbb}{R})$.
So for any divisor cap $ (P(D),\omega(z)) $, there is a Donaldson divisor $ \tilde{D}=\sum z_iC_i $. The importance of a Donaldson divisor lies in the fact that it is maximal when we close up the cap with a minimal symplectic filling.
\[lemma:donaldson=maximal\] Let $ (P,\omega_P) $ be a symplectic cap of $ (Y,\xi) $ and let $ \alpha_P $ be a contact 1-form for $ \xi $. Assume that $ [(\omega_P,\omega_P)] $ is a rational class and let $ [\tilde{D}] $ be the Lefschetz dual of $ c[(\omega_P,\alpha_P)] $ for some $ c>0 $. Then for any minimal symplectic filling $ (N,\omega_N ) $ of $ (Y,\xi ) $ and any symplectic exceptional class $ e $ in $ (X,\omega )=(P,\omega_P)\cup (N,\omega_N) $, we have $ [\tilde{D}]\cdot e>0 $.
Trichotomy
----------
To determine the convexity/concavity of plumbing of $ D $, we first study when $ \omega|_{Y_D} $ is exact using Lemma \[lem: non-degenerate intersection form\].
\[prop: exact\] For a symplectic log Calabi-Yau pair $(X, D, \omega)$, $\omega$ is exact on $Y_D$ if and only if $Q_D$ is negative definite or $b^+(Q_D)=1$.
The case that $D$ is a torus is clear. So we assume now that $D$ is a cycle of spheres. If $Q_D$ is negative definite then $\omega$ is exact on $Y_D$ by Lemma \[lem: non-degenerate intersection form\]. If $b^+(Q_D)=0$ but $Q_D$ is not negative definite, then by Lemma \[lem: not negative semi-definite => at least one non-negative\], $D$ is toric equivalent to a circle $D'$ of self-intersection $-2$ spheres and it is easy to check that for any $z \in \mathbb{R}^k$, $Q_{D'}z$ cannot have all entries being positive.
For the case $ b^+(Q_D)=1 $, it suffices to show $ \omega$ is exact on $ Y_D $ for all $ D $ listed in Theorem \[thm:list\]. By Lemma \[lem: property of continuous fraction\], we only need to look at the positive parabolic case, because $ Q_D $ is nondegenerate in all other cases.
Let $ a=(a_1,a_2,a_3)=([\omega]\cdot [C_1],[\omega]\cdot [C_2],[\omega]\cdot [C_3]) $ for the divisor $ (1,1,p) $. First if $ p=1 $, then by (4) of Lemma \[lemma:topological cyclic\], we have $ [C_1]=[C_2]=[C_3] $ and thus $ a_1=a_2=a_3 $. Then $ z_1=z_2=z_3=\dfrac{a_1}{3} $ is a solution to $ Q_Dz=a $. If $ p\neq 1 $, then again by (4) and (5) of Lemma \[lemma:topological cyclic\], we have $ [C_1]=[C_2] $ and thus $ a_1=a_2 $. Solving the equation $ Q_Dz=a $ is thus equivalent to solving $$\begin{pmatrix}
1&1\\1&p
\end{pmatrix}\begin{pmatrix}
z_1\\z_3
\end{pmatrix}=\begin{pmatrix}
a_1\\a_3
\end{pmatrix},$$ which is solvable as $ p\neq 1 $.
Generalizing the symplectic Kodaira dimension of a symplectic 4-manifold, the following contact Kodaira dimension for contact 3-manifolds was proposed in [@LiMa16-kodaira], based on the type of symplectic cap it admits.
\[def:contact kod\] Let $(W, \omega)$ be a concave symplectic 4-manifold with contact boundary $(Y, \xi)$. $(W, \omega)$ is called a Calabi-Yau cap of $(Y, \xi)$ if $c_1(W)$ is a torsion class, and it is called a uniruled cap of $(Y, \xi)$ if there is a contact primitive $\beta$ on the boundary such that $c_1(W)\cdot [(\omega, \beta)]>0$.
The contact Kodaira dimension of a contact 3-manifold $(Y, \xi)$ is defined in terms of uniruled caps and Calabi-Yau caps. Precisely, $Kod(Y, \xi)=-\infty$ if $(Y, \xi)$ has a uniruled cap, $Kod(Y, \xi)=0$ if it has a Calabi-Yau cap but no uniruled caps, $Kod(Y, \xi)=1$ if it has no Calabi-Yau caps or uniruled caps.
\[Proof of Theorem \[prop: convex-concave\]\] For Case (1), $Q_D$ is negative definite and hence there is a convex plumbing neighborhood $N_D$ with contact boundary $(Y_D, \xi_D)$ by Theorem \[thm:convex\]. Notice that $P=X-N_D$ is a symplectic cap of $Y_D$ with vanishing $c_1$, namely, it is a Calabi-Yau cap. It follows that $Kod(Y_D, \xi_D)\leq 0$.
For Case (2), it follows from Theorem \[thm:concave\] and Proposition \[prop: exact\] that, up to a local symplectic deformation, there is a concave plumbing neighborhood $N_D$ with contact boundary $(Y_D, \xi_D)$. Moreover, since $D$ is symplectic and represents $c_1(X)$, for any contact primitive $\alpha$ of $\omega|_{Y_D}$, we have $c_1(N_D)\cdot [(\omega,\alpha)]= c_1(X)|_{N_D} \cdot [(\omega,\alpha)] = D\cdot [(\omega, \alpha)]=D\cdot [\omega]>0$. Thus $N_D$ is a uniruled cap.
For Case (3), it follows from Proposition \[prop: exact\] that $\omega$ is not exact on $Y_D$.
Symplectic fillings when $ b^+=1 $ {#section:fillability}
----------------------------------
In this section we prove Theorem \[thm:embeddable=fillable\]. First we prove the equivalence between (1) and (3) in Theorem \[thm:embeddable=fillable\].
\[thm:fillability\] Let $ \vec{s} $ be a sequence and $ D $ be a circular spherical divisor with $ \vec{s}(D)=\vec{s} $ with $ b^+(Q_D)\ge 1 $. Then $ (-Y_D,\xi_D) $ is symplectic fillable if and only if $ \vec{s} $ is toric equivalent to one of Theorem \[thm:list\]. Moreover, all minimal symplectic fillings of such $ (-Y_D,\xi_D) $ has $ c_1=0 $ and $ b^+=0 $.
If $ \vec{s} $ is toric equivalent to one in Theorem \[thm:list\], then $ D $ admits an symplectic embedding into a symplectic rational surface $ (X,\omega ) $. By Theorem \[prop: convex-concave\], there is a concave symplectic neighborhood $ N_D $ of $ (D,\omega ) $ with contact boundary $ (-Y_D,\xi_D) $, then $ X-Int(N_D) $ is a symplectic filling of $ (-Y_D,\xi_D) $.
Now suppose $ (-Y_D,\xi_D ) $ is symplectic fillable and let $ (U,\omega_U ) $ be any minimal symplectic filling. Let $ z,a $ be a pair of vectors satisfy the positive GS criterion $ Q_Dz=a $. Then we have a divisor cap $ (P(D),\omega(z)) $ of $ (-Y_D,\xi_D) $. Glue $ (U,\omega_U ) $ with $ (P(D),\omega(z)) $ to get a closed manifold $ (X,\omega ) $. By Lemma \[lemma:b\^+ leq 1\] and \[lemma:rational-embed\], we have that $ b^+(Q_D)=1 $ and $ X $ is a rational surface and $ D $ is toric equivalent to one of Theorem \[thm:list\].
Let $ \tilde{D}=\sum z_iC_i $ be the Donaldson divisor in $ P $, then $ \tilde{D} $ is a maximal divisor in $ X$. For any exceptional class $ e $ in $ X $, we have $ \tilde{D}\cdot e>0 $. If $ X-D $ is not minimal, then there is an exceptional curve $ E \subset X-D $. Then $ [E]\cdot [C_i]=0 $ for all $ C_i $ in $ D $. So $ [\tilde{D}]\cdot [E]=\sum z_i [C_i]\cdot [E]=0 $, which contradicts the maximality of $ \tilde{D} $. So we conclude that $ X-D $ is minimal. Because $ D $ is a rationally embeddable circular divisor with $ b^+(Q_D)= 1 $, $ D $ is rigid and $ (X,D,\omega ) $ is a symplectic Looijenga pair. So we must have $ c_1=0 $ and $ b^+=0 $ for $ (N,\omega_N) $.
It turns out for symplectic log Calabi-Yau pairs, the contact boundary is not only symplectic fillable but also Stein fillable. The following theorem provides the crucial step from (3) to (2) in Theorem \[thm:embeddable=fillable\].
\[thm: as a support of ample line bundle\] For a symplectic Looijenga pair $(X, D, \omega)$ with $b^+(Q_D)= 1$, there exists a Kähler Looijenga pair $(\overline{X},\overline{D},\overline{\omega})$ in its symplectic deformation class such that $\overline{D}$ is the support of an ample line bundle. Then $(\overline{X}-\overline{D},\overline{\omega})$ provides a Stein filling of $ (-Y_D,\xi_D ) $ with $b^+=0$ and $c_1=0$.
Let $(X,D,\omega)$ be a symplectic Looijenga pair. By Theorem \[thm: symplectic deformation class=homology classes\] there is an holomorphic Looijenga pair $(\overline{X},\overline{D},\overline{\omega})$ symplectic deformation equivalent to $ (X,D,\omega ) $. By Theorem \[prop: convex-concave\] (ii), $\overline{\omega}$ is exact on $\partial P(\overline{D})$ and there is another symplectic form $ \overline{\omega}' $ deformation equivalent to $ \overline{\omega } $ such that $Q_D z' = a'$, where $ z'=(z_1',\dots,z_k')\in ({\mathbb{R}}_+)^k $ and $a'=([\overline{\omega}']\cdot [\overline{C_1}],\dots,[\overline{\omega}']\cdot [\overline{C_k}])$. It means that $\sum\limits_{i=1}^k z_i'[\overline{C_i}]$ pairs positively with all $[\overline{C_i}]$. Moreover, by Proposition 4.1 of [@GrHaKe12], we can choose a complex structure compatible with $\overline{\omega}$ such that $(\overline{X},\overline{D})$ is a generic pair. By adjunction formula and Hodge index theorem, any algebriac curve which does not intersect $\overline{D}$ is a self-intersection $-2$ rational curve or a self-intersection $0$ elliptic curve. There is no self-intersection $-2$ rational curve by the genericity of the pair $(\overline{X},\overline{D})$. We claim that there is no elliptic curve in the complement of $ \overline{D} $ of self-intersection $0$. Suppose there exists such elliptic curve $ T $. Since $ b^+(Q_D)=1 $, we can assume there is a self-intersection $0$ component $ \overline{C} $ of $\overline{D}$ (possibly after toric blow-down and non-toric blow-up). Using light cone lemma, we get that $ [T]=\lambda [\overline{C}] $ for some $ \lambda > 0 $. However, $ \overline{C} $ intersects other components of $ \overline{D} $ non-trivially, and so would $ T $, which is contradiction. Any algebraic curve that intersects $\overline{D}$ but not contained in $\overline{D}$ has positive pairing with $\sum_{i=1}^k z_i[\overline{C_i}]$. Also, by the choice of $\sum_{i=1}^k z_i[\overline{C_i}]$, it pairs positively with any irreducible curve in $\overline{D}$. Therefore, by Nakai-Moishezon criterion, $\sum_{i=1}^k z_i[\overline{C_i}]$ is an ample divisor and the support is $\overline{D}$. This finishes the proof.
\(1) is equivalent to (3) by Theorem \[thm:fillability\]. (2) obviously implies (1).
Since $ D $ being symplectically embeddable implies that $ D $ is anti-canonical, there is a symplectic Looijenga pair $ (X,D,\omega ) $. By Theorem \[thm: as a support of ample line bundle\], $ (-Y_D,\xi_D ) $ is Stein fillable.
As shown in the proof of Theorem \[thm:fillability\], every minimal symplectic filling gives rise to a symplectic Looijenga pair. Then the finiteness of minimal symplectic fillings follows from Corollary \[cor: finite deformation\]. The rest is also contained in Theorem \[thm:fillability\] and Theorem \[thm: as a support of ample line bundle\].
In [@GoLi14], Golla and Lisca have investigated a large family of such contact torus bundles $(-Y_D,\xi_D)$ arising from $D$ with $b^+(Q_D)\geq 1$. They proved that all Stein fillings of $(-Y_D, \xi_D)$ have $c_1=0$, $b_1=0$ and share the same $b_2$. Moreover, up to diffeomorphism, there are only finitely many Stein fillings, and there is a unique Stein filling if $|tr A|<2$. Here $A$ is the monodromy matrix of $Y_D$. Many of the results also hold for minimal symplectic fillings for this family.
A similar result for elliptic log Calabi-Yau pairs was obtained by Ohta and Ono in [@OhOn03-simple-elliptic]. Their results were stated for links of simple elliptic singularities, but actually concerns symplectic torus of positive self-intersection. We summarize their results as follows.
Let $ D $ be a torus with $ 0<[D]^2 \le 9 $ and $ [D]^2\neq 8 $, then the minimal symplectic filling of $ (-Y_D,\xi_D ) $ is unique up to diffeomorphism. In the case $ [D]^2=8 $, there are two diffeomorphism types of minimal symplectic fillings. All these minimal fillings have $ c_1=0 $ and $ b^+=0 $. When $ [D]^2\ge 10 $, there is a unique minimal symplectic filling up to diffeomorphism, but we don’t have $ c_1=0 $ in this case.
Geography of Stein fillings when $ Q_D $ is negative definite
-------------------------------------------------------------
A cusp singularity is the germ of an isolated, normal surface singularity such that the exceptional divisor of the minimal resolution is a cycle of smooth rational curves $D$ meeting transversely ([@EbWall1985]). When $ Q_D $ is negative definite, the symplectic log Calabi-Yau pair $ (X,D,\omega ) $ corresponds to resolutions of cusp singularities. Then we have $ (Y_D,\xi_D) $ is Milnor fillable and thus Stein fillable as a particular case of the following well-known result.
\[prop:negative definite = stein fillable\] Let $ D $ be a negative definite symplectic divisor in $ (X,\omega ) $ be symplectic Calabi-Yau and $ Q_D $ is negative definite. Then $ (Y_D,\xi_D ) $ is Stein fillable.
We include a brief argument here. Choose an almost complex structure $ J $ on $ X $ so that $ D $ is $ J- $holomorphic and $ J $ is integrable in a neighborhood $ N_D $ of $ D $ ([@Sikorav97]). Since $ D $ is negative definite, by the Mumford-Grauert criterion ([@Mumford1961],[@Grauert1962]), $ D $ arises as a resolution of an isolated surface singularity. So by Proposition \[prop:unique-contact\], $ \xi_D $ is contactomorphic to the Milnor fillable contact structure $ \xi_{can} $ induced by the complex tangencies. $ N_D $ is a holomorphic filling of $ (Y_D,\xi_{can}) $ and thus can be deformed to a Stein filling by [@BogOli97-stein].
Together with Theorems 1.3 and 1.8 in [@LiMaYa14-CYcap], Theorem \[prop: convex-concave\] has the following consequence. When $Q_D$ is negative definite, the Betti numbers of exact fillings of $(Y_D, \xi_D)$ are bounded. For elliptic log Calabi-Yau pairs, we have the following finiteness theorem.
\[Theorem 2 in [@OhOn03-simple-elliptic]\] Any simple elliptic singularity has either one or two minimal symplectic fillings up to diffeomorphism, arising either from a smoothing or the minimal resolution.
To study symplectic Looijenga pairs, we first give some information on the homology of a cycle of spheres in a symplectic rational surface $ X $.
\[cf. Theorem 2.5 and Theorem 3.1 in [@GoLi14]\] \[lem: homology of neighborhood\] Let $D$ be a cycle of spheres in a symplectic rational surface $X$ and $V=X-N_D$.
1. $H_2(N_D)={\mathbb{Z}}^{r(D)}=H^2(N_D), H_1(N_D)=H^1(N_D)={\mathbb{Z}}, H_3(N_D)=H^3(N_D)=0$.
2. $H_1(Y_D)\to H_1(N_D)$ is a surjection. If $Q_D$ is non-degenerate, then $b_1(Y_D)=1$ and the map $H_1(Y_D)\to H_1(N_D)$ has a finite kernel, $H_2(Y_D)=H^1(Y_D)={\mathbb{Z}}$ and the map $H_2(Y_D)\to H_2(N_D)$ is trivial.
3. Suppose $Q_D$ is non-degenerate and $b_1(X)=0$, then $b_1(V)=b_3(V)=0$, $b_2(V)=b_2(X)-r(D)-1$ and the map ${\mathbb{Z}}=H_2(Y_D)\to H_2(V)$ is injective.
The homology and cohomology of $N_D$ are straightforward to compute since $N_D$ deformation retracts to $D$.
The groups $H_i(Y_D)$ and the homomorphisms to $H_i(N_D)$ are computed via the long homology exact sequence of $(N_D, Y_D)$, the Lefschetz duality $H_i(N_D, Y_D)=H^{4-i}(N_D)$, the homology and cohomology of $N_D$ in the 1st bullet, and the interpretation of $Q_D$ as the restriction map $H_2(N_D)\to H_2(N_D, Y_D)$.
The homology of $V$ is computed via the Mayer-Vietoris sequence of the pair $(N_D, V)$. The vanishing of $b_1(V)$ follows from the portion $ H_1(Y_D) \to H_1(N_D) \oplus H_1(V) \to H_1(X)$, $b_1(X)=0$ and the surjection $H_1(Y_D)\to H_1(N_D)$. The vanishing of $b_3(V)$ follows from the portion $H_4(X)\cong H_3(Y_D) \to H_3(N_D) \oplus H_3(V) \to H_3(X)$ and $b_3(X)=0$. The formula for $b_2(V)$ and $H_2(Y_D)\to H_2(V)$ being injective follow from the portion $$H_3(X) \to H_2(Y_D)=\mathbb{Z} \to H_2(N_D)\oplus H_2(V)\to H_2(X)\to \hbox{Torsion group}$$ and the triviality of the map $H_2(Y_D)\to H_2(N_D)$.
We provide explicit Betti number bounds for Stein fillings below when $D$ is negative definite.
\[prop:geography\] Let $ (X,D,\omega ) $ be a symplectic Looijenga pair. Suppose that $D$ is toric minimal and negative definite and $V=X - N_D$. If $U$ is a Stein filling of $(Y_D,\xi_D)$, then $X_U=U \cup V$ has either $b^+=1$ or $3$, and $$b^+(X_U)=1+b^+(U)+b_2^0(U), \quad b_2^0(U)+b_1(U)=1.$$
- When $b^+(X_U)=1$, $X_U$ is rational or an integral homology Enriques surface, and $U$ is negative definite with $b_1(U)=1$. In this case $e(U)=b^-(U)$, where $e$ is the Euler number.
- When $b^+(X_U)=3$, $X_U$ is an integral homology $K3$, $(b_2^+(U),b_2^0(U),b_1(U))=(1,1,0)$ or $(2,0,1)$. In either case, $c_1(U)=0$ and $2\leq e(U)\leq 21$.
Since $U$ is Stein, we have $1=b_1(Y_D) \ge b_1(U)$. By the Mayer-Vietoris sequence of the pair $(U, V)$, we have the exact sequence $H_1(Y_D) \to H_1(U) \oplus H_1(V) \to H_1(X_U)\to 0$. By Lemma \[lem: homology of neighborhood\], $b_1(V)=0$, so $b_1(X_U) \leq 1$. Since $V$ is a Calabi-Yau cap, it follows from $b_1(X_U)\leq 1$ and Theorem in [@LiMaYa14-CYcap] that either (i). $b^+(X_U)=1, b_1(X_U)=0$ and $X_U$ is non-minimal rational or a minimal integral homology Enrique surface, or (ii). $b^+(X_U)=3, b_1(X_U)=0$ and $X_U$ is a minimal integral homology $K3$.
Since $b_3(X_U)=0, b_1(V)=0$, we have the exact sequence over ${\mathbb{Q}}$, $$\begin{aligned}
0 \to H_2(Y_D;{\mathbb{Q}})\cong \mathbb{Q} &\to H_2(U;{\mathbb{Q}})\oplus H_2(V;{\mathbb{Q}})\\ &\to H_2(X_U;{\mathbb{Q}})\to H_1(Y_D;{\mathbb{Q}})\cong {\mathbb{Q}}\to H_1(U;{\mathbb{Q}})\to 0.
\end{aligned}$$ Let $b_2^0(U)$ denote the dimension of the maximal isotropic subspace of $H_2(U;{\mathbb{Q}})$, which is the rank of the map ${\mathbb{Q}}\cong H_2(Y_D;{\mathbb{Q}})\to H_2(U;{\mathbb{Q}})$. Notice that $b_2^0(U)=0$ or $b_2^0(U)=1$, and $b_1(U)=0$ or $b_1(U)=1$. We claim that $b_2^0(U)+b_1(U)=1$.
First observe that $b_2^0(U)=1$ means that map ${\mathbb{Q}}=H_2(Y_D;{\mathbb{Q}})\to H_2(U;{\mathbb{Q}})$ is injective, and since the map $H_2(Y_D;{\mathbb{Q}})\to H_2(V;{\mathbb{Q}})$ is injective by Lemma \[lem: homology of neighborhood\], the map ${\mathbb{Q}}=H_2(Y_D;{\mathbb{Q}})\to H_2(U;{\mathbb{Q}})$ is injective if and only if the map ${\mathbb{Q}}=H_2(Y_D;{\mathbb{Q}})\to H_2(X_U;{\mathbb{Q}})$ is injective. Next observe that $b_1(U)=0$ if and only if the connecting homomorphism $H_2(X_U;{\mathbb{Q}})\to H_1(Y_D;{\mathbb{Q}})$ has rank $1$.
Finally observe that the map ${\mathbb{Q}}=H_2(Y_D;{\mathbb{Q}})\to H_2(X_U;{\mathbb{Q}})$ is injective if and only if the map $H_2(X_U;{\mathbb{Q}})\to H_1(Y_D;{\mathbb{Q}})={\mathbb{Q}}$ has rank $1$. We give a geometric argument. Take a smooth surface $S_1$ in $Y_D$ representing the generator of $H_2(Y_D)={\mathbb{Z}}$. Suppose there is a surface $S_2$ in $X_U$ such that $[S_2 \cap Y_D]\ne 0 \in H_1(Y_D;{\mathbb{Q}})$. Since $b_1(Y_D)=1$, viewed as classes of $Y_D$, $[S_1] \cdot [S_2 \cap Y_D] \neq 0$ by the Poincare duality for $Y_D$. Hence $[S_1] \cdot [S_2] \neq 0$ in $X_U$, which implies that $[S_1]\ne 0 \in H_2(X_U;{\mathbb{Q}})$. Reversing the argument proves the converse. The three observations together give the claim $b_2^0(U)+b_1(U)=1$.
Notice that the three observations also provide a decomposition of $H_2(X_U;{\mathbb{Q}})$. There are two cases depending on the whether the map ${\mathbb{Q}}=H_2(Y_D;{\mathbb{Q}})\to H_2(X_U;{\mathbb{Q}})$ has rank $0$ or $1$. When the rank is $ 0 $, then $H_2(X_U;{\mathbb{Q}})$ naturally decomposes as $$H_2(X_U;{\mathbb{Q}})\cong H_2(U;{\mathbb{Q}})/[S_1]\oplus H_2(V;{\mathbb{Q}})$$ and $b^{\pm}(X_U)=b^{\pm}(U)+b^{\pm}(V)$. When the rank is $1$, $H_2(X_U;{\mathbb{Q}})$ naturally decomposes as $$H_2(X_U;{\mathbb{Q}})\cong H_2(U;{\mathbb{Q}})/[S_1] \oplus H_2(P;{\mathbb{Q}})/[S_1]\oplus {\mathbb{Q}}[S_1] \oplus {\mathbb{Q}}[S_2].$$ The intersection pairing is non-degenerate on the orthogonal subspaces $H_2(U;{\mathbb{Q}})/[S_1]$ and $ H_2(V;{\mathbb{Q}})/[S_1]$, which implies that $b^{\pm}(X_U)= b^{\pm}(U)+b^{\pm}(V)+1$ since $[S_1]\cdot [S_1]=0$. Then $b^+(X_U)=1+b^+(U)+b_2^0(U)$ follows from the fact that $ b^+(V)=1 $. Finally, we compute the Euler number $e(U)$. Notice that $b_3(U)=b_4(U)=0, b_0(U)=1$. So $e(U)=1-b_1(U)+ b_2(U)$. When $ b^+(X_U)=1 $, we have $ b^+(U)=b_2^0(U)=0 $ and thus $ b_1(U)=1-b_2^0(U)=1 $. Then $ e(U)=b^-(U) $. When $b^+(X_U)=3$, there are two cases: $(b_2^+(U),b_2^0(U),b_1(U))=(1,1,0)$ or $(2,0,1)$, depending on the injectivity of map $ H_2(Y_D;{\mathbb{Q}})\to H_2(X_U;{\mathbb{Q}}) $. In the first case, $b_2(U)=1+1+b_2^-(U), b_1(U)=0$ so $e(U)=3+b_2^-(U)\geq 3$. In the second case, $b_2(U)=2+0+b_2^-(U), b_1(U)=1$ so $e(U)=2+b_2^-(U)\geq 2$. The inequality $e(U)\leq 21$ follows from $24=e(X_U)=e(U)+e(V)$, and $e(V) \geq 3$ (Lemma 4.3 in [@FM83]).
Finally, we discuss the potential implication of Proposition \[prop:geography\] for Stein fillings of cusp singularities. Each toric minimal, negative definite circular spherical divisor $D$ with $[D]^2\leq -2$ has a dual cycle $\check D$, with the property that the plumbed manifolds $Y_D$ and $Y_{\check D}$ are orientation reversing diffeomorphic (Theorem 7.1 in [@Ne81-calculus]). They corresponds to dual pairs of cusp singularities. Every pair of dual cycles embed in a Hirzebruch-Ionue surface as the only curves ([@Inoue77],[@EbWall1985]). Note that the dual cycle $ \check D $ may not be a circular spherical divisor.
A cusp singularity is called rational if its minimal resolution $ D $ is realized as the anti-canonical divisor of a rational surface. Looijenga proved that a cusp singularity is rational if the dual cusp singularity is smoothable. Conversely, Looijenga conjectured if $ (Y,D) $ is an anti-canonical pair, then the cusp singularity with minimal resolution $ \check D $ is smoothable. This conjecture was proved in [@GrHaKe11] via mirror symmetry and later in [@En] via integral-affine geometry.
Note that a smoothing of a cusp singularity gives a Stein filling of $ (Y_D,\xi_D) $ with $ b^+=1 $. In light of this, Proposition \[prop:geography\] provides some evidence to the following symplectic/contact analogue of the Looijenga conjecture.
If a cusp singularity does not have a rational dual, then it admits only negative definite Stein fillings.
Universally tight contact structures {#section:universally tight}
------------------------------------
We first introduce some notions of topological divisors. To each topological divisor $ D $, we can associate a decorated graph $ \Gamma=(V,E,\vec{g}=(g_i),\vec{s}=(s_i)) $ with each vertex $ v_i $ representing the embedded symplectic surface $ C_i $ and each edge connecting $ v_i,v_j $ corresponds to an intersection between $ C_i $ and $ C_j $. Each vertex $ v_i $ is weighted by the genus $ g_i=g(C_i) $ and self-intersection $ s_i=[C_i]^2 $. This generalizes the self-intersection sequence of a circular spherical divisor. Similarly, there is a one-to-one correspondence between decorated graphs and topological divisors.
By Proposition \[prop:unique-contact\], the boundary contact structure of a symplectic divisor neighborhood is independent of the symplectic form, i.e. area vector $ a $, as long as the positive/negative GS criterion is satisfied by some $ z $ and $ a $. So from the contact point of view, only the underlying topological divisor matters, which motivates the following definition.
Let $ \Gamma=(V,E,\vec{g},\vec{s}) $ be a decorated graph and denote by $ Q_\Gamma $ the intersection matrix of $ \Gamma $. We say $ \Gamma $ (or equivalently a topological divisor $ D $) satisfies the **positive GS criterion** if there exists $ z,a\in ({\mathbb{R}}_+)^k $ such that $ Q_\Gamma z=a $ and the **negative GS criterion** if $ Q_\Gamma z=-a $.
The following notions were introduced in [@etgu-ozbagci] and are intimately related to open book decompositions. For each vertex $ v_i $ of $ \Gamma $, denote by $ d_i $ the valence of $ v_i $. A decorated graph $ \Gamma $ is called **non-negative** if $ s_i+d_i\ge 0 $ for all vertex $ v_i $. Similarly we can define a decorated graph to be **non-positive**, **positive**, and **negative** in the obvious way. In addition, we would always require that $ s_i+d_i\neq 0 $ for some vertex $ v_i $ when we talk about non-positive and non-negative graphs. These notions extends naturally to topological divisors. Here is an easy observation.
\[lemma:toric non-negative\] A decorated graph being non-negative is preserved by toric blow-down.
It was shown in [@GaMa13-LF] that all non-positive graphs are actually negative definite and give convex symplectic plumbings. For non-negative graphs, we can show the following:
\[lemma:non-negative => positive GS\] All non-negative graphs satisfy the positive GS criterion.
Let $ \Gamma $ be a non-negative graph with $ k $ vertices and denote by $ Q_\Gamma=(Q_{ij}) $ the intersection matrix. We will find a pair of vectors $ z $ and $ a $ through an iterated perturbation process.
Start with $ z=(1,\dots,1)^T $ and $ a=Q_\Gamma z $. Since $ \Gamma $ is non-negative, we have $ a_j\ge 0 $ for all $ j $ and $ a_i>0 $ for some $ i $. So the index set $ I=\{ i|a_i>0 \} $ is nonempty.
Suppose $ a_l=0 $ and $ Q_{il}>0 $ for some $ i\in I $. Let $ z' $ be a new vector such that $ z_i'=z_i+\epsilon $ for some small positive $ \epsilon $ and $ z_j'=z_j $ for all other $ j $. Then we let $ a'=Q_\Gamma z' $ such that $ a'_j=a_j+\epsilon Q_{ji} $ for all $ j $. Since $ Q_{ji}\ge 0 $ for $ j\neq i $, we have $ a_j'\ge a_j $ for all $ j\neq i $. In particular, $ a_l'=a_l+\epsilon Q_{li}=\epsilon Q_{li}>0 $ as $ Q_{li}>0 $. For $ \epsilon $ small enough, we can also require that $ a_i'=a_i+\epsilon Q_{ii}>0 $. So we have $ I'=\{ i|a_i'>0 \}\supset I\cup \{ l \} $.
We could repeat the process using $ I',z',a' $ as the new $ I,z,a $. Since the graph $ \Gamma $ is finite, this process stops at some finite time and produces a pair of vectors $ z ,a\in ({\mathbb{R}}_+)^k $ such that $ Q_\Gamma z=a $.
When the graph associated to a divisor is non-positive, Gay and Stipsicz constructed an open book decomposition supporting the induced contact structure on the boundary of the divisor neighborhood ([@GaSt09]). This construction was extended to non-negative plumbing graphs in [@LiMi-ICCM]. Then using the open book decomposition, the classification results of Honda ([@honda2000]) and the explicit open book decomposition of Van-Horn-Morris ([@VHM-thesis]) were combined to prove the following result.
\[universally-tight\] Let $ D=\cup_{i=1}^k C_i $ be a circular spherical divisor satisfying the positive GS criterion and $ s_i=[C_i]^2\ge -2 $ for all $ i $. If in addition $ D $ satisfies
1. either $ k\ge 2 $ and $ s_i\ge 0 $ for some $ i $,
2. or $ k\ge 3 $ and $ s_i,s_j\ge -1 $ for some $ i\neq j $,
then $ (-Y_D,\xi_D) $ (not necessarily fillable) is universally tight, except possibly when $ -Y_D $ is parabolic torus bundle with monodromy $ \begin{pmatrix}
1 & n\\ 0 &1
\end{pmatrix}, n>0 $.
The positive GS criterion condition becomes redundant in light of Lemma \[lemma:non-negative => positive GS\]. By Proposition \[prop:contact toric eq\] and Lemma \[lemma:toric non-negative\], we may assume $ D $ has non-negative associated graph and is either toric minimal or of the form $ (-1,p), p\ge -2 $. In particular, we have $ s_i\ge -1 $ for some $ i $.
If $ D $ is toric minimal or $ (-1,p),p\ge 0 $, then we actually have $ s_i\ge 0 $ for some $ i $ and the result follows from (1) of Theorem \[universally-tight\]. The remaining case of $ (-1,-1) $ and $ (-1,-2) $ follows from Proposition 4.1 of [@GoLi14].
Appendix: Operations on divisors and invariance of contact structure
====================================================================
This appendix is devoted to the proof of Proposition \[prop:contact toric eq\] about the invariance of this contact structure under toric equivalence and interior blow-up/blow-down.
GS construction {#section:GS}
---------------
We briefly review the proof of Theorem \[thm:divisorcap\] as in [@GaSt09] and [@LiMa14-divisorcap]. The proof of Proposition \[prop:contact toric eq\] is based on this construction.
Recall that for each topological divisor $ D $, we can associate a decorated graph $ \Gamma=(V,E,\vec{g}=(g_i),\vec{s}=(s_i)) $ with each vertex $ v_i $ representing the embedded symplectic surface $ C_i $ and each edge connecting $ v_i,v_j $ corresponds to an intersection between $ C_i $ and $ C_j $. Each vertex $ v_i $ is weighted by the genus $ g_i=g(C_i) $ and self-intersection $ s_i=[C_i]^2 $. If $ (D,\omega) $ is a symplectic divisor, we can associate an augmented graph $ (\Gamma, a) $ by adding the area vector $ a=([\omega]\cdot [C_i])_{i=1}^k $.
For an augmented graph $ (\Gamma,a) $ and a vector $ z $ such that $ Q_\Gamma z=a $. Let $ z'=-\frac{1}{2\pi} z $ and fix a small $ \epsilon>0 $. For each vertex $ v_i $ and each edge $ e $ connecting to $ v_i $, we choose an integer $ s_{i,e} $ such that $ \sum_{e\in{\mathcal}{E}(v_i)} s_{i,e}=s_i $, where $ {\mathcal}{E}(v_i) $ denotes the set of edges $ e $ connecting to $ v_i $. Also, set $ x_{i,e}=-s_{i,e}z_i' - z_{j}' $, where $ v_j $ is the other vertex connected by $ e $.
Consider the first quadrant $ P=[0,\infty)^2\subset {\mathbb{R}}^2 $ and for some fixed $ \gamma $ and $ \delta $ let $ g:P\to [0,\infty) $ be a smooth function with level sets like in the following figure. So $ g(x,y)=x $ when $ y-x>\gamma $, $ g(x,y)=y $ when $ y-x<-\gamma $ and $ g $ is symmetric with respect to the line $ y=x $.
![Contour of function $ g(x,y) $[]{data-label="figure:g(x,y)"}](function-g)
The constants $ \gamma $ and $ \delta $ are chosen to be small enough so that for each vertex $ v_i $ and each edge $ e $ incident to $ v_i $, the line passing through $ (0,\epsilon ) $ with tangent vector $ (1,-s_{i,e}) $ should intersect $ g^{-1}(\delta ) $ in the region $ y-x>\gamma $. By symmetry, we also have the line passing through $ (\epsilon,0) $ with tangent vector $ (-s_{i,e},1) $ intersects $ g^{-1}(\delta) $ in the region $ y-x<-\gamma $.
For edge $ e $ connecting vertices $ v_i $ and $ v_j $, we can construct a local model $ (X_e,C_e,\omega_e,V_e,f_e) $ as follows. Let $ \mu:S^2\times S^2\to [z_i',z_i'+1]\times [z_j',z_j'+1] $ be the moment map of $ S^2\times S^2 $ onto its image. We set $ p_1 $, $ p_2 $ be the coordinates for $ [z_i',z_i'+1]$, $ [z_j',z_j'+1] $ and set $ q_1,q_2\in {\mathbb{R}}/2\pi{\mathbb{Z}}$ to be the corresponding fiber coordinates. Then $ \omega =dp_1\wedge dq_1 + dp_2\wedge dq_2 $ is the symplectic form on the preimage of the interior of the moment image. Let $ g_e(x,y)=g(x-z_i',y-z_j') $ and let $ R_e $ be the open subset of $ g_e^{-1}[0,\delta) $ between the line passing through $ (z_i',z_j'+2\epsilon) $ with tangent vector $ (1,-s_{i,e}) $ and the line passing through $ (z_j', z_i'+2\epsilon) $ with tangent vector $ (-s_{j,e},1) $. Let $ (X_e,\omega_e ) $ be the symplectic manifold given as the toric preimage $ \mu^{-1}(R_e) $. Let $ C_e=\mu_e^{-1}(\partial R_e) $, $ f_e=g_e\circ \mu_e $ and $ V_e $ be the Liouville vector field obtained by lifting the radial vector field $ p_1\partial_{p_1}+p_2\partial_{p_2} $ in $ {\mathbb{R}}^2 $.
(0,4.5) node (yaxis) \[above\] [$y$]{} |- (6,0) node (xaxis) \[right\] [$x$]{}; (1.5,0.9) coordinate (a\_1) – (4.5,0.9) coordinate (a\_2); (1.5,0.9) coordinate (b\_1) – (1.5,3.9) coordinate (b\_2); (3.2,1.8) coordinate (d\_1)– (5.4,1.8) coordinate (d\_2); (2.4,2.6) coordinate (e\_1)– (2.4,3) coordinate (e\_2); (d\_1) to \[bend left\] (e\_1); (1.7,3.3) node \[right\] [$R_{i,e} $]{}; (2.4,3) coordinate (j\_1)– (2.4,4.3) coordinate (j\_2);
(1.5,3.9) coordinate (f\_1)– (2.4,4.3) coordinate (f\_2); (1.5,2.4) coordinate (g\_1)– (2.4,2.8) coordinate (g\_2); (3,0.9) coordinate (h\_1)– (3.9,1.8) coordinate (h\_2); (3.8,1.3) node \[right\] [$R_{j,e} $]{}; (4.5,0.9) coordinate (i\_1)– (5.4,1.8) coordinate (i\_2); (c) at (intersection of a\_1–a\_2 and b\_1–b\_2); (yaxis |- c) node\[left\] [$z_{j}'$]{} -| (xaxis -| c) node\[below\] [$z_{i}'$]{}; (1.5,2.4) – (0,2.4) node\[left\] [$z_{j}'+\epsilon$]{}; (1.5,3.9) – (0,3.9) node\[left\] [$z_{j}'+2\epsilon$]{}; (3,0.9) – (3,0) node\[below\] [$z_{i}'+\epsilon$]{}; (4.5,0.9) – (4.5,0) node\[below\] [$z_{i}'+2\epsilon$]{};
Then for each vertex $ v_i $ with valence $ d_i $, we may associate a 5-tuple $ (X_i,C_i,\omega_i,V_i,f_i) $ as follows. Let $ g_i $ be the genus of $ v_i $ and $ \Sigma_i $ be a compact Riemann surface with genus $ g_i $ and $ d_i $ boundary components $ \partial_e\Sigma_i $ corresponding to each edge $ e $ connected to $ v_i $. We can find a symplectic form $ \beta_i $ and a Liouville vector field $ W_i $ on $ \Sigma_i $ such that there exists a collar neighborhood of $ \partial_e\Sigma_i $ parametrized as $ (x_{i,e}-2\epsilon,x_{i,e}-\epsilon]\times S^1 $ on which $ \beta_i=dt\wedge d\alpha $ and $ W_i=t\partial_t $. Then we define $ X_i=\Sigma_i\times D^2(\sqrt{2\delta}) $ and $ \omega_i=\beta_i + rdr\wedge d\theta $, where $ D^2(\rho ) $ is the disk of radius $ \rho $ and $ (r,\theta ) $ is the standard polar coordinate on the disk. We define $ f_i=\dfrac{r^2}{2} $, Liouville vector field $ V_i=W_i+(\dfrac{r}{2}+\dfrac{z_i'}{r})\partial_r $ and $ C_i=\Sigma_i-\partial \Sigma_i $.
Finally, the symplectic neighborhood $ (X,C,\omega,V,f) $ is constructed by gluing the local models together appropriately. Let $ R_{i,e} $ be the parallelogram in $ R_e $ cut out by the two lines with tangent vector $ (1,-s_{i,e}) $ passing through $ (z_i',z_j'+\epsilon) $ and $ (z_i',z_j'+2\epsilon) $ respectively. Similarly $ R_{j,e} $ is cut out by the two lines with tangent vector $ (-s_{j,e},1) $ passing through $ (z_j',z_i'+\epsilon) $ and $ (z_j',z_i'+2\epsilon) $ respectively. $ X_i $ can be glued to $ X_e $ by identifying $ \mu_e^{-1}(R_{i,e}) $ with $ (x_{i,e}-2\epsilon,x_{i,e}-\epsilon)\times S^1\times D^2(\sqrt{2\delta}) $. It’s easy to check that symplectic forms, functions and Liouville vector fields all match accordingly.
It’s easy to see that when $ (\Gamma,a) $ satisfies negative GS criterion, i.e. $ z\in ({\mathbb{R}}_-)^k $, the Liouville vector field $ V $ points outward along the boundary. So the glued 5-tuple $ (X,C,\omega,V,f) $ gives the desired convex neighborhood. And when $ (\Gamma,a) $ satisfies positive GS criterion, we have $ z\in ({\mathbb{R}}_+)^k $. Then we can choose $ t $ small enough such that $ V $ is inward pointing along the boundary of $ f^{-1}([0,t]) $, which gives a concave neighborhood. We would call this neighborhood the convex or concave plumbing of $ D $ and denote it by $ (P(D),\omega ) $.
In summary, given an augmented graph $ (\Gamma,a) $, a vector $ z $ satisfying positive/negative GS criterion and choices of parameters $ \epsilon , \delta ,t \in {\mathbb{R}}_+ $, $ \{s_{v,e}\in {\mathbb{Z}}|\sum_{e\in {\mathcal}{E}(v)} s_{v,e}=s_v \} $, $ g:[0,\infty )^2\to [0,\infty ) $, the above construction gives a symplectic plumbing $ (P(D),\omega ) $ with Liouville vector $ V $ along the boundary.
Contact structure and toric equivalence {#section:contact toric}
---------------------------------------
In this section we prove the first statement of Proposition \[prop:contact toric eq\]. We want to show that toric blow-up on the divisor doesn’t change the induced contact structure on boundary of plumbing. The construction in this section will be adapted a little to prove the second statement of Proposition \[prop:contact toric eq\] in the next section.
First we introduce the blow-up of an augmented graph, which is the symplectic version of toric blow-up. Consider the following local picture of an augmented graph (on the left), where each vertex is decorated by its self-intersection number, genus and symplectic area. The blow-up of this augmented graph with weight $ 2\pi a_0 $ is given on the right, which is the toric blow-up with areas specified in the graph. We call this an **augmented toric blow-up of edge $ e_0 $**. Similarly, the reverse operation is called an **augmented toric blow-down**.
$$\begin{tikzpicture}[baseline=0ex]
\node (x) at (0,0) [circle,fill,outer sep=5pt, scale=0.5] [label=above:$ {(s_1,g_1,a_1)} $][label=below:$ v_1 $] {};
\node (y) at (2,0) [circle,fill,outer sep=5pt,scale=0.5] [label=above:$ {(s_2,g_2,a_2)} $][label=below:$ v_2 $] {};
\draw (x) to (y);
\node at (1,0) [label=below:$ e_0 $]{};
\end{tikzpicture}
\quad\Longrightarrow\quad
\begin{tikzpicture}[baseline=0ex]
\node (x) at (0,0) [circle,fill,outer sep=5pt, scale=0.5] [label=above:$ {(s_1-1,g_1,a_1-2\pi a_0)} $][label=below:$ v_1 $] {};
\node (y) at (3,0) [circle,fill,outer sep=5pt, scale=0.5] [label=above:$ {(-1,0,2\pi a_0)} $][label=below:$ v_0 $] {};
\node (z) at (6,0) [circle,fill,outer sep=5pt, scale=0.5] [label=above:$ {(s_2-1,g_2,a_2-2\pi a_0)} $][label=below:$ v_2 $] {};
\node at (1.5,0) [label=below:$ e_1 $]{};
\node at (4.5,0) [label=below:$ e_2 $]{};
\draw (x) to (y);
\draw (z) to (y);
\end{tikzpicture}$$ Denote the original augmented graph by $ (\Gamma^{(1)},a^{(1)}) $ and the blown-up graph $ (\Gamma^{(2)},a^{(2)}) $. Note that $ Q_{\Gamma^{(2)}}z^{(2)}=a^{(2)} $ is still solvable after the augmented toric blow-up. If $ z^{(1)}=(z_1,z_2,\dots ) $ and $ a^{(1)}=(a_1,a_2,\dots ) $ satisfy $ Q_{\Gamma^{(1)}}z^{(1)}=z^{(1)} $, then after blow-up of area $ a_0 $, $ z^{(2)}=(z_1,z_1+z_2-2\pi a_0,z_2,\dots ) $ and $ a^{(2)}=(a_1-2\pi a_0,2\pi a_0,a_2-2\pi a_0,\dots) $ satisfy $ Q_{\Gamma^{(2)}}z^{(2)}=z^{(2)} $. So we could apply GS construction to both augmented graphs. In the following, we will denote the construction based on $ (\Gamma^{(1)},a^{(1)}) $ by GS-1 and denote the construction based on $ (\Gamma^{(2)},a^{(2)}) $ by GS-2.
For the choice of $ \{s_{v,e} \} $, note that the two graphs differ only near $ e_0 $. We could choose $ \{ s_{v,e} \} $ for GS-1 first and then choose the same $ \{s_{v,e}\} $ for all vertices and edges for GS-2, except the ones involved in the toric blow-up. We could choose $ s_{v_0,e_1}=0$, $s_{v_0,e_2}=-1 $ so that $ s_{v_0,e_1}+s_{v_0,e_2}=s_0=-1 $ and choose $ s_{v_1,e_1}=s_{v_1,e_0}-1 $, $ s_{v_2,e_2}=s_{v_2,e_0}-1 $. Then we have $ x_{v_1,e_1}=x_{v_1,e_0}-a_0 $, $ x_{v_2,e_2}=x_{v_2,e_0}-a_0 $, $ x_{v_0,e_1}=-z_1' $ and $ x_{v_0,e_2}=z_1'+a_0 $. The choice of other parameters will be specified later. Note that the choice of parameters won’t affect the boundary contact structure by Proposition \[prop:unique-contact\].
(0.5,4.5) node (yaxis) \[above\] [$y$]{} |- (6,0) node (xaxis) \[right\] [$x$]{};
(1.5,0.9) coordinate (a\_1) – (4.5,0.9) coordinate (a\_2); (1.5,0.9) coordinate (b\_1) – (1.5,3.9) coordinate (b\_2); (3.2,1.8) coordinate (d\_1)– (5.1,1.8) coordinate (d\_2); (2.4,2.6) coordinate (e\_1)– (2.4,4.5) coordinate (e\_2); (d\_1) to \[bend left\] (e\_1);
(b\_2) – (e\_2); (a\_2) – (d\_2); (1.5,2.4) coordinate (g\_1)– (2.4,3) coordinate (g\_2); (3,0.9) coordinate (h\_1)– (3.6,1.8) coordinate (h\_2); (3.7,1.3) node \[right\] [$R_{v_2,e_0}^{(1)} $]{}; (1.5,3.3) node \[right\] [$R_{v_1,e_0}^{(1)} $]{};
\(c) at (intersection of a\_1–a\_2 and b\_1–b\_2); (yaxis |- c) node\[left\] [$z_{2}'$]{} -| (xaxis -| c) node\[below\] [$z_{1}'$]{}; (1.5,2.4) – (0.5,2.4) node\[left\] [$z_{2}'+\epsilon^{(1)}$]{}; (1.5,3.9) – (0.5,3.9) node\[left\] [$z_{2}'+2\epsilon^{(1)}$]{}; (3,0.9) – (3,0);(2.6,0) node\[below\] [$z_{1}'+\epsilon^{(1)}$]{}; (4.5,0.9) – (4.5,0);(4.8,0) node\[below\] [$z_{1}'+2\epsilon^{(1)}$ ]{};
(1.5,2.7) – (3.3,0.9); (1.5,2.7) – (0.5,2.7) node\[left\] [$ z_2'+a_0 $]{}; (3.3,0.9) – (3.3,0); (3.6,0) node\[below\] [$ z_1'+a_0 $]{};
(e\_2) node\[right\] [$ \begin{pmatrix}
1\\ -s_{v_1,e_0}
\end{pmatrix} $]{}; (d\_2) node\[above\] [$ \begin{pmatrix}
-s_{v_2,e_0}\\1
\end{pmatrix} $]{};
[0.4]{}
(0,4.5) node (yaxis) \[above\] [$y$]{} |- (6,0) node (xaxis) \[right\] [$x$]{};
(1.5,0.9) coordinate (a\_1) – (4.5,0.9) coordinate (a\_2); (1.5,0.9) coordinate (b\_1) – (1.5,3.9) coordinate (b\_2); (2.9,1.5) coordinate (d\_1)– (4.5,1.5) coordinate (d\_2); (2.1,2.1) coordinate (e\_1)– (2.1,4.9) coordinate (e\_2); (d\_1) to \[bend left\] (e\_1);
(b\_2) – (e\_2); (a\_2) – (d\_2); (1.5,2.4) coordinate (g\_1)– (2.1,3.4) coordinate (g\_2); (3,0.9) coordinate (h\_1)– (3,1.5) coordinate (h\_2); (3.5,1.2) node \[right\] ; (1.4,3.7) node \[right\] ;
\(c) at (intersection of a\_1–a\_2 and b\_1–b\_2); (yaxis |- c) node\[left\] -| (xaxis -| c) node\[below\] ; (1.5,2.4) – (0,2.4) node\[left\] ; (1.5,3.9) – (0,3.9) node\[left\] ; (3,0.9) – (3,0) node\[below\] ; (4.5,0.9) – (4.5,0) node\[below\] ;
(2.1,4.7) node\[right\] ; (d\_2) node\[above\] ;
[0.4]{}
(0,4.5) node (yaxis) \[above\] [$y$]{} |- (6,0) node (xaxis) \[right\] [$x$]{};
(1.5,0.9) coordinate (a\_1) – (4.5,0.9) coordinate (a\_2); (1.5,0.9) coordinate (b\_1) – (1.5,3.9) coordinate (b\_2); (2.9,1.5) coordinate (d\_1)– (5.5,1.5) coordinate (d\_2); (2.1,2.1) coordinate (e\_1)– (2.1,4.5) coordinate (e\_2); (d\_1) to \[bend left\] (e\_1);
(b\_2) – (e\_2); (a\_2) – (d\_2); (1.5,2.4) coordinate (g\_1)– (2.1,3) coordinate (g\_2); (3,0.9) coordinate (h\_1)– (4,1.5) coordinate (h\_2); (3.8,1.1) node \[right\] ; (1.4,3.5) node \[right\] ;
\(c) at (intersection of a\_1–a\_2 and b\_1–b\_2); (yaxis |- c) node\[left\] -| (xaxis -| c);(1.3,0) node\[below\] ; (1.5,2.4) – (0,2.4) node\[left\] ; (1.5,3.9) – (0,3.9) node\[left\] ; (3,0.9) – (3,0) node\[below\] ; (4.5,0.9) – (4.5,0);(5,0) node\[below\] ;
(e\_2) node\[right\] ; (d\_2) node\[above\] ;
(0.5,4.5) node (yaxis) \[above\] [$y$]{} |- (6,0) node (xaxis) \[right\] [$x$]{};
(3,0.9) coordinate (a\_1) – (4.5,0.9) coordinate (a\_2); (1.5,2.4) coordinate (b\_1) – (1.5,3.9) coordinate (b\_2); (3.2,1.8) coordinate (d\_1)– (5.1,1.8) coordinate (d\_2); (2.4,2.6) coordinate (e\_1)– (2.4,4.5) coordinate (e\_2); (1.5,2.9) – (1.8,3.1); (3.5,0.9)–(3.7,1.2);
(d\_1) to \[bend left\] (e\_1);
(b\_2) – (e\_2); (a\_2) – (d\_2); (4,1) node \[right\] ; (1.45,3.5) node \[right\] ; (1.8,1.6) node \[left\] ; (2.3,1.1) node \[left\] ; (1.75,1.65)–(2.15,1.95); (2.25,1.15)–(2.55,1.5); (1.45,2.6) node\[right\] ; (3.2,0.9) node \[above\] ; (2.35,1.55) node \[above\] ;
(1.8,4.1)–(1.8,2.7); (4.7,1.2)–(3.3,1.2); (2,2.3)–(2.2,2.1); (2.5,1.8)–(2.7,1.6); (2.2,1.7)–(2.2,2.1); (2.5,1.4)–(2.5,1.8); (2.115,2.2)–(2.115,1.8); (2.615,1.7)–(2.615,1.3);
(2,2.3) to \[bend left\] (1.8,2.7); (3.3,1.2) to \[bend left\] (2.7,1.6); (2.1,2.2)–(2.6,1.7); (2.1,2.2)–(2.1,1.8); (2.6,1.7)–(2.6,1.3);
\(c) at (intersection of a\_1–a\_2 and b\_1–b\_2); (1.5,3.9) – (0.5,3.9) node\[left\] [$z_{2}'+2\epsilon^{(1)}$]{}; (1.5,2.9) –(0.5,2.9) node\[left\] [$ z_2'+a_0+\epsilon^{(2)} $]{};
(4.5,0.9) – (4.5,0);(4.8,0) node\[below\] [$z_{1}'+2\epsilon^{(1)}$]{}; (3.5,0.9)–(3.5,0);(3.8,0) node\[below\] [$ z_1'+a_0+\epsilon^{(2)} $]{};
(1.5,2.4) – (3,0.9); (1.5,2.4) – (0.5,2.4 ) node\[left\] [$ z_2'+a_0 $]{}; (3,0.9) – (3 ,0 ); (2.9,0) node\[below\] [$ z_1'+a_0 $]{};
(e\_2) node\[right\] [$ \begin{pmatrix}
1\\ -s_{v_1,e_0}
\end{pmatrix} $]{}; (d\_2) node\[above\] [$ \begin{pmatrix}
-s_{v_2,e_0}\\1
\end{pmatrix} $]{};
\[fig:blow-up together\]
(1.2,2.5) node (yaxis) \[above\] [$y$]{} |- (3,0.5) node (xaxis) \[right\] [$x$]{}; (1.5,1.9) – (2.5,0.9); (1.5,1.9) – (1,1.9 ) node\[left\] [$ z_2'+a_0 $]{}; (2.5,0.9) – (2.5 ,0.5);(2.65,0.5) node\[below\] [$ z_1'+a_0 $]{}; (2.3375,1.4775)–(1.6875,2.1275); (1.6875,2.1275)–(1.6875,1.7125); (2.3375,1.4775)–(2.3375,1.0625); (1.875,1.95)–(1.875,1.525); (2.175,1.65)–(2.175,1.225); (1.6,2.225)–(1.875,1.95); (2.5,1.325)–(2.175,1.65); (1.9,2.4) node \[right\] ; (2.3,2) node \[right\] ; (2,1.45) node \[above\] ; (1.95,2.35)–(1.8,1.8);(2.35,1.95)–(2.25,1.4);
(1.875,1.525) – (1.2,1.525 ) node\[left\] [$ z_2'+a_0-2\epsilon^{(2)} $]{}; (1.875,1.525) – (1.875,0.3 ) node\[below\] [$ z_1'+2\epsilon^{(2)} $]{}; (2.175,1.225) – (1.2 ,1.225) node\[left\] [$ z_2'+2\epsilon^{(2)} $]{}; (2.175,1.225) – (2.175 ,0.5) node\[below\] [$ z_1'+a_0-2\epsilon^{(2)} $]{}; (1.6875,1.7125) – (1.2,1.7125 ) node\[left\] [$ z_2'+a_0-\epsilon^{(2)} $]{}; (1.6875,1.7125) – (1.6875,0.5 ) node\[below\] [$ z_1'+\epsilon^{(2)} $]{}; (2.3375,1.0625) – (1.2,1.0625 ) node\[left\] [$ z_2'+\epsilon^{(2)} $]{}; (2.3375,1.0625) – (2.3375,0.3 );(2.4,0.3) node\[below\] [$ z_1'+a_0-\epsilon^{(2)} $]{};
In GS-1, the edge $ e_0 $ corresponds to the local model $ (X_{e_0}^{(1)},C_{e_0}^{(1)},\omega_{e_0}^{(1)},V_{e_0}^{(1)},f_{e_0}^{(1)}) $ with toric image $ R_{e_0}^{(1)} $ in Figure \[fig: GS construction after blow-up\]. The gluing region $ R_{v_1,e_0}^{(1)} $ is characterized by the vector $ \begin{pmatrix}
1 \\ -s_{v_1,e_0}
\end{pmatrix} $ and $ R_{v_2,e_0}^{(1)} $ is characterized by $ \begin{pmatrix}
-s_{v_2,e_0}\\ 1
\end{pmatrix} $.
In GS-2, the edge $ e_1 $ corresponds to the local model $ (X_{e_1}^{(2)},C_{e_1}^{(2)},\omega_{e_1}^{(2)},V_{e_1}^{(2)},f_{e_1}^{(2)}) $ with toric image $ R_{e_1}^{(2)} $ as in Figure \[fig:two edges\](a) with gluing region $ R_{v_1,e_1}^{(2)} $ characterized by vector $ \begin{pmatrix}
1\\ - s_{v_1,e_1}
\end{pmatrix}=\begin{pmatrix}
1\\ -s_{v_1,e_0}+1
\end{pmatrix} $ and $ R_{v_0,e_1}^{(2)} $ characterized by $ \begin{pmatrix}
-s_{v_0,e_1}\\1
\end{pmatrix} =\begin{pmatrix}
0 \\ 1
\end{pmatrix}$. Using the transformation $ \begin{pmatrix}
1 & 0\\-1 & 1
\end{pmatrix}\in GL(2,{\mathbb{Z}}) $, we could map $ R_{e_1}^{(2)} $ onto $ R_{e_1}^{(1)}$ in Figure \[fig:blow-up together\]. This gives a symplectomorphism $ \Phi_{e_1}:(\mu_{e_1}^{-1}(R_{e_1}^{(2)}),\omega_{e_1}^{(2)})\to (\mu_{e_0}^{-1}(R_{e_1}^{(1)}),\omega_{e_0}^{(1)} )$ and identifies the Liouville vector field $ V_{e_1}^{(2)} $ with $ V_{e_0}^{(1)} $. Similarly, the edge $ e_1 $ corresponds to the local model $ (X_{e_2}^{(2)},C_{e_2}^{(2)},\omega_{e_2}^{(2)},V_{e_2}^{(2)},f_{e_2}^{(2)}) $ with toric image $ R_{e_2}^{(2)} $ as in Figure \[fig:two edges\](b) with gluing region $ R_{v_0,e_2}^{(2)} $ characterized by $ \begin{pmatrix}
1\\ - s_{v_0,e_2}
\end{pmatrix}=\begin{pmatrix}
1\\ 1
\end{pmatrix} $ and $ R_{v_2,e_2}^{(2)} $ by $ \begin{pmatrix}
-s_{v_2,e_2}\\1
\end{pmatrix} =\begin{pmatrix}
-s_{v_2,e_0}+1 \\ 1
\end{pmatrix}$. Using the transformation $ \begin{pmatrix}
1 & -1\\0 & 1
\end{pmatrix}\in GL(2,{\mathbb{Z}}) $, we could map $ R_{e_2}^{(2)} $ onto $ R_{e_2}^{(1)}$ in Figure \[fig:blow-up together\]. This gives symplectomorphism $\Phi_{e_2}: (\mu_{e_2}^{-1}(R_{e_2}^{(2)}),\omega_{e_2}^{(2)})\to (\mu_{e_0}^{-1}(R_{e_2}^{(1)}),\omega_{e_0}^{(1)}) $, and identifies the Liouville vector field $ V_{e_2}^{(2)} $ with $ V_{e_0}^{(1)} $.
For vertex $ v_0 $, take $ X_{v_0}^{(2)}=[-z_1'-a_0+\epsilon^{(2)},-z_1'-\epsilon^{(2)}]\times S^1\times D^2_{\sqrt{2\delta^{(2)}}} $, $ \omega_{v_0}^{(2)}=dt\wedge d\alpha + rdr\wedge d\theta $ and $ V_{v_0}^{(2)}=t\partial_t+(\dfrac{r}{2}+\dfrac{z_0'}{r})\partial_r $. So we see that the local model $ (X_{v_0}^{(2)},C_{v_0}^{(2)},\omega_{v_0}^{(2)},V_{v_0}^{(2)},f_{v_0}^{(2)}) $ is exactly $ (\mu_{e_0}^{-1}(R_{v_0}^{(1)}),\mu_{e_0}^{-1}(L),\omega_{e_0}^{(1)},V_{e_0}^{(1)},f_{e_0}^{(1)}) $, where $ L $ is the line segment from point $ (z_1'+\epsilon^{(2)},z_2'+a_0-\epsilon^{(2)}) $ to $ (z_1'+a_0-\epsilon^{(2)},z_2'+\epsilon^{(2)}) $ in Figure \[fig:vertex-exceptional sphere\]. We can check the gluing of $ \mu^{-1}_{e_0}(R_{e_1}^{(1)}) $ with $\mu^{-1}_{e_0}( R_{v_0}^{(1)}) $ along $ \mu^{-1}_{e_0}(R_{v_0,e_1}^{(1)}) $ coincides with the gluing of $\mu^{-1}_{e_1}( R_{e_1}^{(2)}) $ with $ X_{v_0}^{(2)} $ along $ \mu^{-1}_{e_1}(R_{v_0,e_1}^{(2)}) $. Similarly, the gluing along $ \mu^{-1}_{e_0}(R_{v_0,e_2}^{(1)}) $ coincides with the gluing along $ \mu^{-1}_{e_2}(R_{v_0,e_2}^{(2)}) $. So the glued local model $ X_{e_1}^{(2)}\cup X_{v_0}^{(2)}\cup X_{e_2}^{(2)} $ is symplectomorphic to the preimage of the region $ R_{e_1}^{(1)}\cup R_{v_0}^{(1)}\cup R_{e_2}^{(1)} $ with Liouville vector fields identified.
Blow up the intersection point in $ P(D^{(1)}) $ corresponding to the edge $ e_0 $ symplectically with area $ 2\pi a_0 $ to get $ (P(D^{(1)})\# \overline{{\mathbb{C}}{\mathbb{P}}}^2,\omega^{bl}) $. This corresponds to cutting the corner from $ R_{e_0}^{(1)} $as shown in Figure \[fig GS construction\] and the resulting region is called $ R_{e_0}^{bl} $. Since blowing up an interior point doesn’t change the boundary, we have $ (Y_{D^{(1)}},\xi_{D^{(1)}})= \partial(P(D^{(1)}),\omega^{(1)}) \cong \partial(P(D^{(1)})\#\overline{{\mathbb{C}}{\mathbb{P}}}^2,\omega^{bl}) $.
Choose $ \delta^{(1)}, \delta^{(2)}, \epsilon^{(1)}, \epsilon^{(2)}, a_0 $ so that they satisfy $ \delta^{(2)}=\delta^{(1)}<\epsilon^{(2)}<\epsilon^{(1)} $, $ 2\epsilon^{(2)}<a_0<2\delta^{(1)} $ and $ a_0=2\epsilon^{(1)}-2\epsilon^{(2)} $. Such choice of $ a_0 $ ensures that there is enough area to blow-up and the interval in $ X_{v_0}^{(2)} $ is well defined. So the region $ R_{e_1}^{(1)}\cup R_{v_0}^{(1)}\cup R_{e_2}^{(1)} $ is embedded in $ R^{(1)}_{e_0} $. Since all other local models are the same for GS-1 and GS-2, by shrinking the region $ R^{(1)}_{e_0} $, we get an contact isotopy from $ (Y_{D^{(1)}},\xi_{D^{(1)}})\cong \partial(P(D^{(1)})\#\overline{{\mathbb{C}}{\mathbb{P}}}^2,\omega^{bl}) $ to $ (Y_{D^{(2)}},\xi_{D^{(2)}}) =\partial (P(D^{(2)}),\omega^{(2)}) $.
Half edge and blow-up of a vertex
---------------------------------
The construction outlined in Section \[section:GS\] actually only works for graphs with at least two vertices, but it can be modified to take care of the single vertex case. Now consider the augmented graph $ (\Gamma ,a) $ where $ \Gamma $ has only one vertex $ v $ decorated with genus $ g $ and self-intersection $ s $. As long as $ s\neq 0 $, there is always a solution $ z=\dfrac{a}{s} $.
According to GS construction, the vertex $ v $ corresponds to a local model $ (X_v,C_v,\omega_v,V_v,f_v) $. Here $ X_v=\Sigma_v\times D^2_{\sqrt{2\delta}} $ where $ \Sigma_v $ is a genus $ g $ surface with one boundary component. To close up and get a disk bundle over a closed genus $ g $ surface with Euler class $ s $, we need to glue $ X_v $ to a disk bundle over disk and add the suitable twisting. Consider the region $ R_{\tilde{e}} $ in Figure \[fig:half edge\], which is similar to the region $ R_{e} $ in Figure \[fig GS construction\] except we only have one gluing region $ R_{v,\tilde{e}} $. This region gives a local model $ (X_{\tilde{e}},C_{\tilde{e}},\omega_{\tilde{e}},V_{\tilde{e}},f_{\tilde{e}}) $ in the same way as the ordinary GS construction. Note that $ x_{\tilde{e}}=\mu_{\tilde{e}}^{-1}(R_{\tilde{e}})\cong D^2\times D^2_{\sqrt{2\delta}} $. Here the gluing region is specified by the vector $ \begin{pmatrix}
1 \\ -s
\end{pmatrix} $. By gluing these two local models, we get the desired disk bundle.
(0.5,4.5) node (yaxis) \[above\] [$y$]{} |- (6,0) node (xaxis) \[right\] [$x$]{};
(1.5,0) coordinate (b\_1) – (1.5,3) coordinate (b\_2); (2.4,0) coordinate (e\_1)– (2.4,3.6) coordinate (e\_2); (b\_2) – (e\_2); (1.5,1.5) coordinate (g\_1)– (2.4,2.1) coordinate (g\_2); (1.7,2.7) node \[right\] [$R_{v,\tilde{e}} $]{};
(1.5,0) node\[below\] [$ z' $]{}; (0.5,0) node\[left\][$ 0 $]{}; (1.5,1.5) – (0.5,1.5) node\[left\] [$\epsilon$]{}; (1.5,3) – (0.5,3) node\[left\] [$2\epsilon$]{}; (2.4,0) node\[below\] [$z'+\delta $]{}; (e\_2) node\[right\] [$ \begin{pmatrix}
1\\ -s
\end{pmatrix} $]{};
This region $ R_{\tilde{e}} $ works almost the same as an edge in ordinary GS construction and we call it a **half edge**, as shown below. $$\begin{tikzpicture}[baseline=0ex]
\node (x) at (0,0) [circle,fill,outer sep=5pt, scale=0.5] [label=above:$ {(s,g,a)} $][label=below:$ v $] {};
\end{tikzpicture}
\quad\Longleftrightarrow\quad
\begin{tikzpicture}[baseline=0ex]
\node (x) at (0,0) [circle,fill,outer sep=5pt, scale=0.5] [label=above:$ {(s,g,a)} $][label=below:$ v $] {};
\node (y) at (1,0) {};
\node at (1,0) [label=below:$ \tilde{e} $]{};
\draw (x) to (y);
\end{tikzpicture}$$ For any vertex $ v $ in an augmented graph $ (\Gamma,a) $, we have $ X_v\cong \Sigma_v \times D^2 $. Take any point $ p\in \Sigma_v $ and a small disk neighborhood $ D^2 $ of $ p $. This local neighborhood $ D^2\times D^2 $ can be regarded as the local model $ X_{\tilde{e}} $ corresponding to a half edge $ \tilde{e} $. Here we could choose the parameter $ s_{v,\tilde{e}}=0 $ so that $ s_{v,\tilde{e}}+\sum_{{\mathcal}{E}(v)} s_{v,e}=s_v $.
\[def:interior blow up\] For an augmented graph $ (\Gamma,a) $, let $ v $ be a vertex in $ \Gamma $. The following operation is called an **augmented interior blow-up** of vertex $ v $ with weight $ a_0 $, of which the reverse operation is also called an **augmented interior blow-down**. $$\begin{tikzpicture}[baseline=0ex]
\node (x) at (0,0) [circle,fill,outer sep=5pt, scale=0.5] [label=left:$ {(s,g,a)} $][label=below:$ v $] {};
\node (y) at (-1,1) [label=left:$ \dots $]{};
\node (z) at (-1,-1) [label=left:$ \dots $]{};
\draw (x) to (y);
\draw (x) to (z);
\end{tikzpicture}
\quad\Longrightarrow\quad
\begin{tikzpicture}[baseline=0ex]
\node (x) at (0,0) [circle,fill,outer sep=5pt, scale=0.5] [label=left:$ {(s-1,g,a-a_0)} $] {};
\node at (0,0.1) [label=below:$ v $]{};
\node (y) at (-1,1) [label=left:$ \dots $]{};
\node (z) at (-1,-1) [label=left:$ \dots $]{};
\draw (x) to (y);
\draw (x) to (z);
\node (e) at (2,0) [circle,fill,outer sep=5pt, scale=0.5] [label=above:$ {(-1,0,a_0)} $][label=below:$ v_0 $] {};
\draw (x) to (e);
\node (f) at (1,0) [label=below:$ \tilde{e} $]{};
\end{tikzpicture}$$ If we forget about the area $ a $, such operations can be performed on a decorated graph and thus on a topological divisor. In this case, they are called the **interior blow-up** of vertex $ v $ and the **interior blow-down**.
An augmented interior blow-up of vertex $ v $ can be regarded as the augmented toric blow-up of a half edge $ \tilde{e} $ stemming from $ v $ as shown in the following diagram, where the right arrow indicates an augmented toric blow-up of $ \tilde{e} $. $$\begin{tikzpicture}[baseline=0ex]
\node (x) at (0,0) [circle,fill,outer sep=5pt, scale=0.5] [label=left:$ {(s,g,a)} $] {};
\node (y) at (-1,1) [label=left:$ \dots $]{};
\node (z) at (-1,-1) [label=left:$ \dots $]{};
\draw (x) to (y);
\draw (z) to (x);
\end{tikzpicture}
\quad \Longleftrightarrow\quad
\begin{tikzpicture}[baseline=0ex]
\node (x) at (0,0) [circle,fill,outer sep=5pt, scale=0.5] [label=left:$ {(s,g,a)} $] {};
\node (y) at (-1,1) [label=left:$ \dots $]{};
\node (z) at (-1,-1) [label=left:$ \dots $]{};
\draw (x) to (y);
\draw (z) to (x);
\node (e) at (2,0){};
\node at (1,0) [label=below:$ \tilde{e} $]{};
\draw (e) to (x);
\end{tikzpicture}\Longrightarrow$$ $$\begin{tikzpicture}[baseline=0ex]
\node (x) at (0,0) [circle,fill,outer sep=5pt, scale=0.5] [label=left:$ {(s-1,g,a-a_0)} $] {};
\node (y) at (-1,1) [label=left:$ \dots $]{};
\node (z) at (-1,-1) [label=left:$ \dots $]{};
\draw (x) to (y);
\draw (z) to (x);
\node (e) at (1,0) [circle,fill,outer sep=5pt, scale=0.5] [label=above:$ {(-1,0,a_0)} $] {};
\draw (x) to (e);
\draw (e) to (2,0);
\node (f) at (1.5,0) [label=below:$ \tilde{e}' $]{};
\end{tikzpicture}
\quad\Longleftrightarrow\quad
\begin{tikzpicture}[baseline=0ex]
\node (x) at (0,0) [circle,fill,outer sep=5pt, scale=0.5] [label=left:$ {(s-1,g,a-a_0)} $] {};
\node (y) at (-1,1) [label=left:$ \dots $]{};
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The construction from Section \[section:contact toric\] also works for regions like $ R_{\tilde{e}} $ with a suitable choice of $ \epsilon $ and $ a_0 $. So we have that toric blowing up a half edge $ \tilde{e} $ doesn’t change its boundary contact structure. Thus we conclude that the boundary contact structure is invariant under interior blow-up of a vertex.
|
---
abstract: |
Motivated by the model theory of higher order logics, in [@cl1a] a certain kind of topological spaces had been introduced on ultraproducts. These spaces are called ultratopologies. Ultratopologies provide a natural extra topological structure for ultraproducts and using this extra structure in [@cl1a] some preservation and characterization theorems had been obtained for higher order logics.\
The purely topological properties of ultratopologies seem interesting on their own right. We started to study these properties in [@ut], where some questions remained open. Here we present the solutions of two such problems. More concretely we show that\
$(1)$ there are sequences of finite sets of pairwise different cardinalities such that in their certain ultraproducts there are homeomorphic ultratopologies and\
$(2)$ if $A$ is an infinite ultraproduct of finite sets then every ultratopology on $A$ contains a dense subset $D$ such that $|D| < |A|$.\
\
[*AMS Classification:* ]{} 03C20, 54A25, 54A99.\
[*Keywords:*]{} ultraproduct, ultratopology, dense set.
author:
- 'Gábor Sági[^1], Saharon Shelah[^2]'
title: On topological properties of ultraproducts of finite sets
---
\[section\] \[theorem\][Example]{} \[theorem\][Conjecture]{} \[theorem\][Examples]{} \[section\] \[theorem\][Fact]{} \[theorem\][Open problem]{}
\[theorem\][Corollary]{} \[theorem\][Definition]{} \[theorem\][Remark]{} \[theorem\][Lemma]{} \[theorem\] \[theorem\][Construction]{}
Introduction {#intro}
============
In first order model theory the ultraproduct construction can be applied rather often. this is because ultraproducts preserve the validity of first order formulas. It is also natural to ask, what connections can be proved between certain higher order formulas and ultraproducts of models of them.\
In [@cl1a] we answer related questions in terms of topological spaces which can be naturally associated to ultraproducts. These spaces are called ultratopologies and their definition can be found in [@cl1a] and also at the beginning of [@ut].\
Although ultratopologies were introduced from logical (model theoretical) reasons, these spaces can be interesting on their own right. In [@ut] a systematic investigation about these topological properties has been started. However, in [@ut] some problems remained open. In the present note we are dealing with two such problems.\
In Section \[hb\] we give a positive answer for Problem 5.2 of [@ut]: there are sequences of finite sets of pairwise different cardinalities such that their certain ultraproducts are still homeomorphic with respect to some carefully chosen ultratopologies. In fact, in Theorem \[discr\] below we show that if $U$ is a good ultrafilter and $A$ is any infinite ultraproduct of finite sets modulo $U$ then there is an ultratopology on $A$ in which the family of closed sets consists just the finite sebsets of $A$ and the whole $A$. From this the affirmative answer for Problem 5.2 of [@ut] can be immediately deduced. In Section \[ds\] we investigate the possible cardinalities of dense sets in ultratopologies, again on ultraproducts of finite sets. In Corollary \[dc\] we show that if ${\cal C}$ is any ultratopology on an infinite ultraproduct $A$ then the density of ${\cal C}$ is smaller then $|A|$, that is, one can always find a dense set whose cardinality is less than $|A|$.\
Throughout we use the following conventions. $I$ is a set and for every $i \in I$ $A_{i}$ is a set. Moreover $ A = \Pi_{i
\in I} A_{i} / U$ denotes the ultraproduct of $A_{i}$’s modulo an ultrafilter $U$.\
Every ordinal is the set of smaller ordinals and natural numbers are identified with finite ordinals. Throughout $\omega$ denotes the smallest infinite ordinal and $cf$ denotes the cofinality operation.\
In order to simplify notation, sometimes we will identify ${}^{k}(\Pi_{i \in I}A_{i})$ with $\Pi_{i \in I}{}^{k}A_{i}$ by the natural way, that is, $k$–tuples of sequences are identified with single sequences whose terms are $k$–tuples.\
If $X$ is a topological space and $A \subseteq X$ then $cl(A)$ denotes the closure of $A$. Suppose $k \in \omega$, $\langle
A_{i}: \ i \in I \rangle$ is a sequence of sets, $U$ is an ultrafilter on $I$ and $R_{i} \subseteq {}^{k}A$ is a given relation for every $i \in I$. Then the [*ultraproduct*]{} relation $\Pi_{i \in I} R_{i} /U$ is defined as follows.\
\
$ \Pi_{i \in I} R_{i} /U = \{ s/U \in ({}^{k} \Pi_{i \in I} A_{i}/U): \ \{ i \in I: \ s(i) \in R_{i} \} \in U \}$.
\
\
As we mentioned, we assume that the reader is familiar with the notions of “choice function”, “ultratopology”, “a point is close to a relation”, “$T(a,R)$”, etc. These notions were introduced in [@cl1a] and a short (but fairly complete) survey can be found at the beginning of [@ut].
Homeomorphisms between different ultraproducts {#hb}
==============================================
In [@ut] Problem 5.2 asks whether is it possible to choose ultrafilters $U_{1}$, $U_{2}$ and sequences of natural numbers $s = \langle n_{i}, \ i \in I \rangle$ and $ z = \langle m_{j}, \ j \in J \rangle$ so that\
$(i)$ $n_{i} \not= m_{j}$ for all $i \in I, j \in J$ and\
$(ii)$ for every $k \in \omega$ there are $k$–dimensional ultratopologies ${\cal C}_{k}$ in $\Pi_{i \in I} n_{i} / U_{1}$ and ${\cal D}_{k}$ in $\Pi_{j \in J} m_{j} / U_{2}$ such that ${\cal C}_{k}$ and ${\cal D}_{k}$ are homeomorphic?
We will give an affirmative answer. In fact, we prove the following theorem from which the above question can be easily answered.
\[discr\] Suppose $\langle n_{i}, \ i \in I \rangle$ is an infinite sequence of natural numbers and $U$ is a good ultrafilter on $I$ such that $A = \Pi_{i \in I} n_{i} / U$ is infinite. Then for every $k \in \omega$ there is a $k$–dimensional choice function on $A$ such that the family of closed sets in the induced ultratopology consists of the finite subsets of ${}^{k}A$ and ${}^{k}A$.
Let $c$ be any $k$–dimensional choice function on ${}^{k}A$. By modifying $c$, we will construct another choiche function $\hat{\ }$ which induces the required ultratopology. Let $E$ be the set of all triples $ \langle s, i, m \rangle$ where $s \in \Pi_{l \in I} {\cal P}({}^{k}n_{l})$, $i \in I$ such that $\Pi_{l \in I} s(l) / U$ is infinite and $m \in {}^{k}n_{i} -
s(i)$. It is easy to see that $|\Pi_{i \in I} {\cal
P}({}^{k}n_{i})| \leq |{}^{I} \omega| = 2^{|I|}$. Therefore $|E|
\leq 2^{|I|} \times |I| \times \aleph_{0} = 2^{|I|}$. Let $\{
\langle s_{\alpha}, i_{\alpha}, m_{\alpha} \rangle: \ \alpha <
2^{|I|} \}$ be an enumeration of $E$.\
By transfinite recursion we construct an injective function $f: E \rightarrow {}^{k}A$ such that for every $\langle
s, i, m \rangle \in E$ one has $f(\langle s, i, m \rangle) \in
\Pi_{l \in I} s(l) / U$. Suppose $f_{\beta}$ has already been defined on $ \{ \langle s_{\alpha}, i_{\alpha}, m_{\alpha}
\rangle: \ \alpha < \beta \}$ for all $\beta < \gamma \leq 2^{|I|}$ such that\
$\beta_{1} < \beta_{2} < \gamma \Rightarrow f_{\beta_{1}} \subseteq f_{\beta_{2}}$ and\
$|f_{\beta}| \leq |\beta|$.\
If $\gamma$ is a limit ordinal, then let $f_{\gamma} =
\cup_{\beta < \gamma} f_{\beta}$. Now suppose $\gamma$ is a successor ordinal, say $\gamma = \delta + 1$. Since $U$ is a good ultrafilter, by Theorem VI, 2.13 of [@clasth] it follows, that the cardinality of $B = \Pi_{l \in I} s(l) / U $ is $2^{|I|}$. Therefore there is an element $b \in B$ which is not in the range of $f_{\delta}$. Let $f_{\gamma}$ be $f_{\delta}
\cup \{ \langle \langle s_{\delta}, i_{\delta}, m_{\delta}
\rangle, b \rangle \}$. Clearly, $f = f_{2^{|I|}}$ is the required function.\
Now we construct a $k$–dimensional choice function $\hat{\ }$ as follows. If $a \not\in rng(f)$ then let $\hat{a} = c(a)$. Otherwise there is a unique $ e = \langle s, i, m \rangle \in E$ such that $f(e) = a$. Let $$\hat{a}(j) = \left\{ \begin{array}{ll}
c(a)(j) & \mbox{if $i \not= j$, } \\
m & \mbox{otherwise.}
\end{array}
\right.$$\
In this way we really defined a $k$–dimensional choice function on $A$. We claim that the closed sets of the induced ultratopology are exactly the finite subsets of ${}^{k}A$ and ${}^{k}A$.\
By Theorem 2.5 of [@ut] every ultratoplogy is $T_{1}$ therefore every finite subset of ${}^{k}A$ is closed. Let $F$ be an infinite closed subset of ${}^{k}A$ and suppose, seeking a contradiction, that there is an element\
\
$(*)$ $b \in {}^{k}A - F$.
\
\
By Corollary 2.2 of [@ut] $F$ is a decomposable relation, say $F = \Pi_{l \in I} s(l) / U$. Therefore, there is a $J \in U$ such that for every $i \in J$ one has $\hat{b}(i) \not\in s(i)$. Hence, for every $i \in J$ $e_{i} = \langle s, i, \hat{b}(i)
\rangle \in E$. By construction, for every $i \in J$ one has $f(e_{i}) \in F$ and $f(e_{i})\hat{\ }(i) = \hat{b}(i)$. This means that\
\
$\{ i \in I: \exists a \in F: \hat{a}(i) = \hat{b}(i) \} \supseteq J \in U$.
\
\
That is, $T(F,b) \in U$ (where $T$ is understood according to the new choice function $\hat{\ }$). Since we assumed that $F$ is closed, this implies $b \in F$ which contradicts to $(*)$.
There are ultrafilters $U_{1}$, $U_{2}$ (respectively, over $I$ and $J$) and sequences of natural numbers $s = \langle n_{i}, \ i
\in I \rangle$ and $ z = \langle m_{j}, \ j \in J \rangle$ so that\
$(i)$ $n_{i} \not= m_{j}$ for all $i \in I, j \in J$ and\
$(ii)$ for every $k \in \omega$ there are $k$–dimensional ultratopologies ${\cal C}_{k}$ in $\Pi_{i \in I} n_{i} / U_{1}$ and ${\cal D}_{k}$ in $\Pi_{j \in J} m_{j} / U_{2}$ such that ${\cal C}_{k}$ and ${\cal D}_{k}$ are homeomorphic.
Let $U_{1}, U_{2}$ be good ultrafilters and let $s$ and $z$ be arbitrary sequences of natural numbers satisfying the requirements of the corollary such that $|I| = |J|$ and both $A = \Pi_{i \in I} s_{i} / U_{1}$ and $B = \Pi_{j \in J} z_{j} / U_{2}$ are infinite. Let $ k \in \omega$ be arbitrary. By Theorem \[discr\] there are ultratopologies ${\cal C}_{k}$, ${\cal D}_{k}$, respectively on $A$ and $B$ such that\
the closed sets of ${\cal C}_{k}$ are exactly the finite subsets of ${}^{k}A$ and ${}^{k}A$ and\
the closed sets of ${\cal D}_{k}$ are exactly the finite subsets of ${}^{k}B$ and ${}^{k}B$.\
By Theorem VI, 2.13 of [@clasth] $|A|=|B| = 2^{|I|}$. Let $f: A \rightarrow B$ be any bijection. Then $f_{k}: {}^{k}A \rightarrow {}^{k}B$, $f(\langle a_{0},...,a_{k-1} \rangle = \langle f(a_{0}),...,f(a_{k-1}) \rangle$ is clearly a bijection from ${}^{k}A$ onto ${}^{k}B$ mapping finite subsets of ${}^{k}A$ to finite subsets of ${}^{k}B$. Thus, $f$ is the required homeomorphism.
cardinalities of dense sets {#ds}
===========================
Problem 5.3 (A) of [@ut] asks whether is it possible to choose a sequence of finite sets $s$ and an ultrafilter $U$ so that there is an utratopology on $A = \Pi_{i \in I} n_{i} / U$ in which every dense set has cardinality $|A|$. In this section we will show that this is impossible if $A$ is infinite. We start by a simple observation: every $k$-dimensional ultratopology is homeomorphic with an appropriate $1$–dimensional ultratopology.
\[dims\] Suppose ${\cal C}_{k}$ is a $k$–dimensional ultratopology on $A$. Then there is a $1$–dimensional ultratopology ${\cal D}$ which is homeomorphic to ${\cal C}$.
The idea is to identify $k$–tuples of sequences by sequences of $k$–tuples. By a slight abuse of notation, we will use this identification freely. Let $A = \Pi_{i \in I} A_{i} / U$ (here the $A_{i}$’s are not necessarily finite) and suppose $\hat{\ }$ is a $k$–dimensional choice function inducing ${\cal C}_{k}$. Let ${\cal B} = \Pi_{i \in I} {}^{k}A_{i} / U$. We define a $1$–dimensional choice function $c$ in $B$ as follows. If $s=\langle s_{i}: i \in I \rangle / U \in B$ then for each $j \in
k$ let $s^{j} = \langle s_{i}(j): \ i \in I \rangle / U$. Define $c(s) = \langle s^{0},...,s^{k-1} \rangle\hat{\ }$ and $\varphi:
{}^{k}A \rightarrow B$, $\varphi (\langle s^{0}/ U,...,s^{k-1}/U
\rangle) = \langle \langle s^{0}(i),...,s^{k-1}(i) \rangle: i \in
I \rangle / U$. Then clearly, $c$ is a $1$–dimensional choice function which induces an ultratopology ${\cal D}$ on $B$. Then for any $a \in {}^{k}A$ and $i \in I$ one has $\hat{a}(i) =
c(\varphi(a))(i)$. Now it is straightforward to check that $\varphi$ is a homeomorphism between ${\cal C}$ and ${\cal D}$.
Let ${\cal C}$ be an ultratopology on an ultraproduct $A = \Pi_{i \in I} n_{i} / U$ of finite sets. Suppose ${\cal C}$ can be induced by a choice function $\hat{\ }$. Let\
\
$G = \{ \langle i,m \rangle: \ i \in I, m \in n_{i}$ and $( \exists a \in A)(\hat{a}(i) = m) \}$
\
\
and for each $\langle i,m \rangle \in G$ let $a_{i,m} \in A$ be such that $\hat{a}_{i,m}(i)=m$. Clearly, if $I$ is infinite, then $|G| \leq |I|$. We claim that there is a dense subset $R$ of $A$ such that $|R| \leq |G|$. In fact, $R$ can be chosen to be $R =
\{ a_{i,m}: \ \langle i,m \rangle \in G \}$. To see this, suppose $a \in A$. Then for every $i \in I$ one has $\langle i,
\hat{a}(i) \rangle \in G$ and therefore $T(R,a) = I \in U$. Hence $cl(R) = A$, as desired.\
Now we are able to provide a negative answer for Problem 5.3 (A) of [@ut].
\[maint\] Suppose ${\cal C}$ is a $1$–dimensional ultratopology on an infinite ultraproduct $A = \Pi_{i \in I}
n_{i} / U$ where each $n_{i}$ is a finite set. Then $d({\cal C})
< |A|$.
Suppose, seeking a contradiction, that ${\cal C}$ is an ultratopology on $A$ such that the cardinality of every dense set in ${\cal C}$ is equal with $|A| = \kappa$. Using the notation just introduced in the remark before the theorem, $R$ is a dense subset of $A$ and therefore $|R| = |A|$. Let $<^{A}$ be a well ordering of $A$ (having order type $\kappa$). By transfinite recursion we define a sequence $\langle b_{i} \in
A: i < \kappa \rangle$ as follows. Assume $j < \kappa$ and $b_{l}$ has already been defined for every $l < j$. Let $W_{j} =
\{ b_{l}: l < j \}$ and let $V_{j} = \{ a \in A: T(W_{j},a) \in U
\}$. Since $j < \kappa$, $V_{j} \subseteq cl(W_{j}) \not= A$. If $j$ is an odd ordinal, then let $b_{j}$ be the $<^{A}$–first element of $R- W_{j}$. Otherwise let $b_{j}$ be the $<^{A}$–first element in $A-V_{j}$. Clearly, the following conditions are satisfied:\
$(i)$ for every $\langle i,m \rangle \in G$ there is a $j
< \kappa$ such that $\hat{b}_{j}(i) = m$, in fact, $R \subseteq \{b_{l}: \ l < \kappa \} = W_{\kappa}$.\
$(ii)$ for every $a \in A$ there is a $j < \kappa$ such that $a \in V_{j}$ (the smallest such $j$ will be denoted by $j_{a}$),\
$(iii)$ for every $j < \kappa$ there is an ordinal $j < s(j)$ such that $s(j) < \kappa$ and $b_{s(j)} \not\in cl(W_{j})$. (This is true because otherwise by $(i)$ one would have $cl(W_{j}) \supseteq cl(R) = A$ which is impossible since $|W_{j}| < \kappa$.)\
Now let $H = \{ i \in I: n_{i} = \{ \hat{b}_{j}(i): j < \kappa \}
\}$. We show that $H \in U$.\
Again, seeking a contradiction, assume $I-H \in U$. For every $i \in I-H$ let $e_{i} \in n_{i} - \{ \hat{b}_{j}(i): j <
\kappa \}$ be arbitrary, let $e = \langle e_{i}: i \in I-H
\rangle / U$ and let $O = \{i \in I: \hat{e}(i) = e_{i} \}$. Clearly, $O \in U$. In addition, if $i \in O \cap (I-H)$ then by $(i)$ there is a $j < \kappa$ such that $\hat{b}_{j}(i) =
\hat{e}(i) = e_{i}$ contradicting to the selection of $e_{i}$’s.\
For every $i \in H$ we introduce a binary relation $\prec^{i}$ as follows. If $n,m \in n_{i}$ then $n \prec^{i} m$ means that there is a $j \in \kappa$ such that $\hat{b}_{j}(i) =
n$ but for every $l \leq j$ $\hat{b}_{l}(i) \not= m$. Clearly, for each $i \in H$ the relation $\prec^{i}$ is irreflexive, transitive, and trichotome. Since $n_{i}$ is finite, for every $i
\in H$ there is an $\prec^{i}$–maximal element $y_{i} \in
n_{i}$. Let $y = \langle y_{i}: \ i \in H \rangle / U$ and let $K
= \{ i \in I: \hat{y}(i) = y_{i} \}$. Now by $(ii)$ and $(iii)$ $y \in V_{j_{y}}$ and $b_{s(j_{y})} \not\in cl(W_{j_{y}}) =
cl(V_{j_{y}})$. Thus, for every $i \in K \cap T(W_{j_{y}},y)$ one has\
\
$ (*) \indent \hat{y}(i) = y_{i} \in \{ \hat{b}_{l}(i): l < j_{y} \}.$
\
\
$cl(V_{j_{y}})$ is closed, therefore there is $L \in U$ such that for all $i \in L$ one has $\hat{b}_{s(j_{y})}(i) \not \in \{
\hat{b}_{l}(i): l < j_{y} \}$ and thus $\hat{b}_{l}(i) \prec^{i}
\hat{b}_{s(j_{y})}(i)$ for every $i \in L \cap H$ and for every $l < j_{y}$. Particularly, $(*)$ implies that if $i \in L \cap H
\cap K \cap T(W_{j_{y}},y)$ then $\hat{y}(i) = y_{i} \prec^{i}
\hat{b}_{s(j_{y})}(i)$ which is impossible since by construction, $y_{i}$ is the $\prec^{i}$–maximal element in $n_{i}$. This contradiction completes the proof.
Using Theorem \[dims\] the above results can be generalized to higher dimensional ultratopologies as well.
\[dc\] Let $k \in \omega$ be arbitrary and suppose ${\cal C}$ is a $k$–dimensional ultratopology on an infinite ultraproduct ${\cal A} = \Pi_{i \in I} n_{i} / U$ where each $n_{i}$ is a finite set. Then $d({\cal C}) < |A|$.
Assume, seeking a contradiction, that ${\cal C}$ is a $k$–dimensional ultratopology on $A$ whose density is $|A|$. Then, by Theorem \[dims\] there is a $1$–dimensional ultratopology with the above property, contradicting to Theorem \[maint\]
[99]{}
North–Holland, Amsterdam (1973). Math. Logic Quarterly, Vol. 48, No. 2, pp. 261–275, (2002). Accepted for publocation, Math. Logic Quarterly, (2004). North–Holland, Amsterdam (1990).
\
[^1]: Supported by Hungarian National Foundation for Scientific Research grant D042177.
[^2]: The second author would like to thank the Israel Science Foundation for partial support of this research (Grant no. 242/03). Publication 841.
|
---
abstract: 'A distorted black hole radiates gravitational waves in order to settle down in one of the geometries permitted by the no-hair theorem. During that relaxation phase, a characteristic damped ringing is generated. It can be theoretically constructed from the black hole quasinormal frequencies (which govern its oscillating behavior and its decay) and from the associated excitation factors (which determine intrinsically its amplitude) by carefully taking into account the source of the distortion. Here, by considering the Schwarzschild black hole in the framework of massive gravity, we show that the excitation factors have an unexpected strong resonant behavior leading to giant ringings which are, moreover, slowly decaying. Such extraordinary black hole ringings could be observed by the next generations of gravitational wave detectors and allow us to test the various massive gravity theories or their absence could be used to impose strong constraints on the graviton mass.'
author:
- Yves Décanini
- Antoine Folacci
- Mohamed Ould El Hadj
bibliography:
- 'Giant\_BH\_ringing.bib'
title: Giant black hole ringings induced by massive gravity
---
[*Introduction.*]{}— Gravitational waves, a major prediction of Einstein’s general relativity, should be observed directly in a near future by the next generations of gravitational wave detectors. Another fascinating prediction of Einstein’s theory, the existence of black holes (BHs), should be simultaneously confirmed. Indeed, if in its final stage the astrophysical process generating the observed gravitational radiation involves a distorted BH, the signal is then dominated, at intermediate time scales, by a characteristic damped ringing. It is due to the BH which radiates away all its distortions in the form of gravitational waves and relaxes toward a state permitted by the no-hair theorem. The frequencies and decay rates describing that ringing define the complex resonance spectrum of the BH. They are linked to its quasinormal modes (QNMs) [@Nollert:1999ji; @Kokkotas:1999bd; @Berti:2009kk; @Konoplya:2011qq], i.e., to those mode solutions of the wave equation which propagate inward at the horizon and outward at spatial infinity, and they can be considered as the BH fingerprint : they indicate beyond all doubt the existence of a horizon and they could be used to determine unambiguously the mass as well as the angular momentum of the BH.
These last years, generalizations of general relativity mediated by a massive spin-2 particle are the subject of intense activity (see Ref. [@Hinterbichler:2011tt] for a recent review). They have their source in the 70-year old Fierz-Pauli theory [@Fierz:1939ix]. They are motivated by purely theoretical considerations (the study of the deformations of general relativity with the graviton mass as deformation parameter) but they also arise from field theories in spacetimes with extra dimensions. Furthermore, and this is surely the main raison of their success, they could explain, without dark energy, the accelerated expansion of the present Universe. Of course, a hypothetical massive graviton is surely an ultralight particle (see Ref. [@Goldhaber:2008xy] for a description of the experimental constraints on the graviton mass but note that the mass limit strongly depends on the theory considered).
It is therefore natural to study BH perturbations and to reconsider gravitational radiation from BHs in massive gravity. It is only very recently that some works on this subject by Babichev and Fabbri [@Babichev:2013una] and by Brito, Cardoso and Pani [@Brito:2013wya; @Brito:2013yxa] have been achieved. They mainly discuss the fundamental problem of BH stability and therefore the existence of BHs in massive gravity. Of course, this depends on the model of gravity considered but, once we assume stability, it becomes really interesting to work on the structure of the signal emitted by a distorted BH with in mind the possibility to test, in a near future, the various massive spin-2 field theories with gravitational wave detectors.
Since the seventies, an increasing number of frequency- and time-domain studies dealing with massive fields propagating in BH spacetimes have highlighted the important modifications induced by the mass parameter which concern more or less directly the signal emitted by a distorted BH : (i) the resonance spectrum is enriched by the complex frequencies corresponding to quasibound states (see Refs. [@Deruelle:1974zy; @Damour:1976kh; @Zouros:1979iw; @Detweiler:1980uk] for important pioneering works and Ref. [@Brito:2013wya] for a recent study in massive gravity); (ii) as the mass parameter increases, the quasinormal frequencies migrate in the complex plane, a behavior observed numerically by various authors (see, e.g., Refs. [@Simone:1991wn; @Konoplya:2004wg] for pioneering works and Ref. [@Brito:2013wya] for a recent study in massive gravity) and analytically described recently [@Hod:2011zzd; @Decanini:2011eh]; (iii) at very late time (i.e., after the quasinormal ringing), the signal emitted by the relaxing BH is not described by the usual power-law tail behavior [@Price:1971fb] but, roughly speaking, by oscillations with a slowly-decaying amplitude (see, e.g., Ref. [@Burko:2004jn] as well as Ref. [@Hod:2013dka] for a recent study in massive gravity).
In a future article [@DFOEH1], we intend to discuss as fully as possible a new effect with amazing consequences : for massive bosonic fields in the Schwarzschild spacetime, the excitation factors of the QNMs have a strong resonant behavior which induces giant ringings. It is a totally unexpected effect. Indeed, until now it was assumed that, for massive fields, quasinormal ringings are less easily excited (see the introduction of Ref. [@Rosa:2011my] and references therein). We shall describe this effect numerically and confirm it analytically from semiclassical considerations based on the properties of the unstable circular geodesics on which a massive particle can orbit the BH.
In this letter, we only consider the massive spin-2 field because of its theoretical importance and due to the fascinating observational consequences that our results predict for such a field. We limit our study to the Fierz-Pauli theory in the Schwarzschild spacetime [@Brito:2013wya] which can be obtained, e.g., by linearization of the ghost-free bimetric theory of Hassan, Schmidt-May and von Strauss discussed in Ref. [@Hassan:2012wr] and which is inspired by the fundamental work of de Rham, Gabadadze and Tolley [@deRham:2010ik; @deRham:2010kj]. Furthermore, we mainly focus on the odd-parity $\ell=1$ mode of this field (similar results can be obtained for the other modes - see also Ref. [@DFOEH1]) and on the associated QNMs. Our letter is organized as follows. We first establish numerically the resonant behavior of the excitation factor of the $(\ell=1,n=0)$ QNM (and briefly discuss its overtones). It occurs in a large domain around a critical value of the mass parameter where the QNM is, in addition, weakly damped. As a consequence, it induces giant and slowly decaying ringings. These are constructed directly from the retarded Green function (an intrinsic point of view) and we compare them with the ringing generated by the odd-parity $(\ell=2,n=0)$ QNM of the massless theory which, in the context of Einstein gravity, provides one of the most important contribution to the BH ringing. With in mind observational considerations, it is necessary to check that a realistic perturbation describing the BH distortion does not neutralize the resonant effect previously discussed and to study ringings constructed from the quasinormal excitation coefficients (an extrinsic point of view) because they permit us to include the contribution of the perturbation into the BH response [@Berti:2006wq]. We then describe the perturbation by an initial value problem, an approach which has regularly provided interesting results [@Leaver:1986gd; @Andersson:1996cm; @Berti:2006wq]. We show that the excitation coefficient of the $(\ell=1,n=0)$ QNM also has a resonant behavior which still leads to giant and slowly decaying ringings. In a conclusion, we discuss some possible extensions of our work and its interest for gravitational wave astrophysics and theoretical physics.
Throughout this letter, we display our numerical results by using the dimensionless coupling constant ${\tilde \alpha}=2M\mu /{m_\mathrm{P}}^2$ (here $M$, $\mu$ and $m_\mathrm{P}= \sqrt{\hbar c /G} $ denote respectively the mass of the BH, the rest mass of the graviton and the Planck mass). We adopt units such that $\hbar = c = G = 1$ and assume a harmonic time dependence $\exp(-i\omega t)$ for the spin-2 field. We describe the exterior of the Schwarzschild BH by using both the radial coordinate $r \in ]2M,+\infty[$ and the so-called tortoise coordinate $r_\ast \in ]-\infty,+\infty[$ given by $r_\ast(r)=r+2M \ln[r/(2M)-1]$.
[*Resonant behavior of the quasinormal excitation factors and associated “intrinsic" giant ringings.*]{}— In Schwarzschild spacetime, the partial amplitude $\phi (t,r)$ describing the odd-parity $\ell=1$ mode of the massive spin-2 field satisfies [@Brito:2013wya] (the angular momentum index $\ell=1 $ will be, from now on, suppressed in all the formulas) $$\label{Phi_ell1}
\left[-\frac{\partial^2 }{\partial t^2}+\frac{\partial^2}{\partial r_\ast^2}-V(r) \right] \phi (t,r)=0$$ with the effective potential $V(r)$ given by $$\label{pot_RW_Schw}
V(r) = \left(1-\frac{2M}{r} \right) \left(\mu^2+
\frac{6}{r^2} -\frac{16M}{r^3}\right).$$ The associated retarded Green function can be written as $$\label{Gret_om}
G_\mathrm{ret}(t;r,r')=-\int_{-\infty +ic}^{+\infty +ic} \frac{d\omega}{2\pi} \frac{\phi^\mathrm{in}_{\omega}(r_<) \phi^\mathrm{up}_{\omega}(r_>)}{W (\omega)} e^{-i\omega t}$$ where $c>0$, $r_< =\mathrm{min} (r,r')$, $r_> =\mathrm{max} (r,r')$ and with $W (\omega)$ denoting the Wronskian of the functions $\phi^\mathrm{in}_{\omega}$ and $\phi^\mathrm{up}_{\omega}$. These two functions are linearly independent solutions of the Regge-Wheeler equation $$\label{RW}
\frac{d^2 \phi_{\omega}}{dr_\ast^2} + \left[ \omega^2 -V(r)\right] \phi_{\omega}=0.$$ When $\mathrm{Im} (\omega) > 0$, $\phi^\mathrm{in}_{\omega}$ is uniquely defined by its ingoing behavior at the event horizon, i.e., for $r_\ast \to -\infty$ $\phi^\mathrm{in}_{\omega} (r) \sim \exp[-i\omega r_\ast]$ and, at spatial infinity, i.e., for $r_\ast \to +\infty$, it has an asymptotic behavior of the form $$\begin{aligned}
\label{bc_2_in}
& & \phi^\mathrm{in}_{\omega}(r) \sim
\sqrt{ \frac{\omega}{p(\omega)}} \left[A^{(-)} (\omega) e^{-i[p(\omega)
r_\ast + [M\mu^2/p(\omega)] \ln(r/M)]}\right. \nonumber \\
& & \quad \quad \left. + A^{(+)} (\omega) e^{+i[p(\omega) r_\ast +
[M\mu^2/p(\omega)] \ln(r/M)]} \right].\end{aligned}$$ Similarly, $\phi^\mathrm{up}_{\omega}$ is uniquely defined by its outgoing behavior at spatial infinity, i.e., for $r_\ast \to +\infty$, $\phi^\mathrm{up}_{\omega} (r) \sim \sqrt{ \omega/p(\omega)} \exp \lbrace{+i[p(\omega) r_\ast +
[M\mu^2/p(\omega)] \ln(r/M)] \rbrace}$ and, at the horizon, i.e., for $r_\ast \to -\infty$ it has an asymptotic behavior of the form $$\label{bc_2_up}
\phi^\mathrm{up}_{\omega}(r) \sim
B^{(-)} (\omega) e^{-i\omega r_\ast} + B^{(+)} (\omega) e^{+i\omega r_\ast}.$$ Here $p(\omega)=\sqrt{\omega^2 - \mu^2 }$ denotes the “wave number" while $A^{(-)} (\omega)$, $A^{(+)} (\omega)$, $B^{(-)} (\omega)$ and $B^{(+)} (\omega)$ are complex amplitudes which, like the $\mathrm{in}$- and $\mathrm{up}$- modes, can be defined by analytic continuation in the full complex $\omega$-plane (or, more precisely, in a well-chosen multi-sheeted Riemann surface). By evaluating the Wronskian $W (\omega)$ at $r_\ast \to -\infty$ and $r_\ast \to +\infty$, we obtain $W (\omega) =2i\omega A^{(-)} (\omega) = 2i\omega B^{(+)} (\omega)$.
If the Wronskian $W (\omega)$ vanishes, the functions $\phi^\mathrm{in}_{\omega}$ and $\phi^\mathrm{up}_{\omega}$ are linearly dependent and propagate inward at the horizon and outward at spatial infinity, a behavior which defines the QNMs. The zeros of the Wronskian lying in the lower part of the complex $\omega$-plane are the frequencies of the $\ell=1$ QNMs. They are symmetrically distributed with respect to the imaginary $\omega$-axis. The contour of integration in Eq. (\[Gret\_om\]) may be deformed in order to capture them [@Leaver:1986gd]. By Cauchy’s Theorem, we can extract from the retarded Green function (\[Gret\_om\]) a residue series over the quasinormal frequencies $\omega_n$ lying in the fourth quadrant of the complex $\omega$-plane. We then obtain the contribution describing the BH ringing. It is given by $$\begin{aligned}
\label{Gret_ell_QNM}
&& G_\mathrm{ret}^\mathrm{QNM}(t;r,r')=2 \, \mathrm{Re} \left[ \sum_n {\cal B}_{n}
{\tilde \phi}_{\omega_n}(r) {\tilde \phi}_{\omega_n}(r') \right. \nonumber\\
&& \left. \vphantom{\sum_n} \times e^{-i\omega_n t + ip(\omega_n)r_\ast + ip(\omega_n)r'_\ast +
i[M\mu^2/p(\omega_n)] \ln(rr'/M^2)} \right]\end{aligned}$$ where $$\label{Excitation F}
{\cal B}_{n} = \left(\frac{1}{2 p(\omega)} \frac{A^{(+)} (\omega)}{\frac{dA^{(-)} (\omega)}{d\omega}} \right)_{\omega=\omega_n}$$ denotes the excitation factor corresponding to the complex frequency $\omega_n$. In Eq. (\[Gret\_ell\_QNM\]), the modes ${\tilde \phi}_{\omega_n}(r)$ are defined by normalizing the modes $\phi^\mathrm{in}_{\omega_n}(r)$ so that ${\tilde \phi}_{\omega_n}(r) \sim 1$ as $r \to +\infty $. In the sum, $n=0$ corresponds to the fundamental QNM (i.e., the least damped one) and $n=1,2,\dots $ to the overtones.
Quasinormal retarded Green functions such as (\[Gret\_ell\_QNM\]) do not provide physically relevant results at “early times" due to their exponentially divergent behavior as $t$ decreases. It is necessary to determine, from physical considerations, the time beyond which they can be used and such a time is the starting time $t_\mathrm{start}$ of the BH ringing. $t_\mathrm{start}$ can be “easily" obtained for massless fields (see, e.g., Ref. [@Berti:2006wq]). Indeed, we first note that the QNMs are semiclassically associated with the peak of the effective potential located close to $r_\ast \approx 0 $. Then, by assuming that the source at $r'_\ast$ and the observer at $r_\ast$ are far from the BH (i.e., that $r_\ast,r'_\ast \gg 2M$) we have $t_\mathrm{start} \approx r_\ast + r'_\ast$ (it is approximatively the time taken for the signal to travel from the source to the peak of the potential and then to reach the observer). For massive fields, the previous considerations must be slightly modified. We take into account the dispersive behavior of the QNMs and define $t_\mathrm{start}$ from group velocities. From the dispersion relation $p(\omega)=\sqrt{\omega^2 - \mu^2 }$, we can show that the group velocity corresponding to the quasinormal frequency $\omega_n$ is given by $v_\mathrm{g}=\mathrm{Re}[p(\omega_n)]/\mathrm{Re}[\omega_n]$. Because the peak of the effective potential still remains located close to $r_\ast \approx 0 $, we then obtain $t_\mathrm{start} \approx (r_\ast + r'_\ast)\mathrm{Re}[\omega_n]/\mathrm{Re}[p(\omega_n)]$ (here, we neglect the contribution of $[M\mu^2/p(\omega_n)] \ln(rr'/M^2)$).
In Fig. \[fig:OM\_n=0\], we display the effect of the graviton mass on $\omega_0$ (see also Fig. 2 in Ref. [@Brito:2013wya]) and in Fig. \[fig:B0\], we exhibit the strong resonant behavior of ${\cal B}_0$ occurring around the critical value ${\tilde \alpha} \approx 0.90$. The same kind of resonant behavior exists for the excitation factors ${\cal B}_n $ with $n \not= 0$ but the resonance amplitude decreases rapidly as the overtone index $n$ increases.
![\[fig:OM\_n=0\] Complex frequency $\omega_0$ of the odd-parity $(\ell=1,n=0)$ QNM.](Im2Mw_Re2Mw){height="2cm" width="8.5cm"}
![\[fig:B0\] Resonant behavior of the excitation factor ${\cal B}_0$ of the odd-parity $(\ell=1,n=0)$ QNM. The maximum of $|{\cal B}_0|$ occurs for the critical value ${\tilde \alpha} \approx 0.89757$; we then have $2M\omega_0 \approx 0.85969073 - 0.03878222 i$ and ${\cal B}_0 \approx 3.25237 + 19.28190 i$.](B0_Exfact){height="3.3cm" width="8.5cm"}
The resonant behavior of the excitation factor ${\cal B}_0$ occurring for masses in a range where the QNM is a long-lived mode (see Figs. \[fig:OM\_n=0\] and \[fig:B0\]) induces giant ringings which are, moreover, slowly or even very slowly decaying. In Fig. \[fig:IntrinsicRingings\], we plot, for two values of the graviton mass, the BH “intrinsic" ringings constructed from the quasinormal retarded Green function (\[Gret\_ell\_QNM\]) and we compare them to the ringing generated by the odd-parity $(\ell=2,n=0)$ QNM of the massless spin-2 field with quasinormal frequency $2M \omega_{20} \approx 0.74734337 - 0.17792463 i$ and with excitation factor ${\cal B}_{20} \approx 0.12690 + 0.02032 i$. Similar results can be obtained for various locations of the source and the observer. Giant BH ringings also exist for $n \not= 0$ but with less impressive characteristics.
[*Resonant behavior of the quasinormal excitation coefficients and associated “extrinsic" giant ringings.*]{}— In the previous section, we focussed on “intrinsic" ringings, i.e., on ringings directly constructed from the quasinormal retarded Green function and therefore depending only on the BH properties. Of course, with in mind astrophysical applications, it is now necessary to check that giant ringings also exist in the presence of a realistic perturbation or, in other words, that the convolution of the source of the perturbation with the retarded Green function does not modify, in a fundamental way, our results. This is a complex problem and, in this letter, we just discuss some of its elementary aspects.
Some years ago, dealing with the observability of quasinormal ringings, Andersson and Glampedakis associated to each QNM an effective amplitude $h_\mathrm{eff}$ achievable after matched filtering [@Andersson:1999wj; @Glampedakis:2001js]. For the massive QNMs considered here, it reads $$\label{Eff_Amp}
h_\mathrm{eff} \sim \mathrm{Re} \left[2 \sqrt{-\mathrm{Re}(\omega_n)/\mathrm{Im}(\omega_n)} \, p(\omega_n) {\cal B}_n \right].$$ We can use it in order to determine the effective amplitude of the ringings generated by the odd-parity $(\ell=1,n=0)$ QNM of massive gravity; we obtain $|h_\mathrm{eff}| \approx 54.74$ for ${\tilde \alpha}=0.89$ and $|h_\mathrm{eff}| \approx 13.02$ for ${\tilde \alpha}=1.05$. For the ringing generated by the odd-parity $(\ell=2,n=0)$ QNM of Einstein gravity we have $|h_\mathrm{eff}| \approx 0.40$. This is in perfect agreement with our previous results and suggests that giant BH ringings are astrophysically relevant and observable. However, it should be noted that Andersson-Glampedakis formula must be taken with a pinch of salt. As noted in Ref. [@Berti:2006wq], it seems helpful only if the quasinormal ringing is excited by localized initial data. So, it is necessary to consider a more general approach.
We now describe the BH perturbation by an initial value problem with Gaussian initial data. More precisely, we consider that, at $t=0$, the partial amplitude $\phi (t,r)$ governed by (\[Phi\_ell1\]) satisfies $$\label{Cauchy_data}
\phi (t=0,r)=\phi_0(r) \equiv \phi_0 \exp \left[-\frac{a^2}{(2M)^2} (r_\ast - \beta)^2 \right]$$ and $\partial_t\phi (t=0,r)=0$. By Green’s Theorem, we can show that the time evolution of $\phi (t,r)$ is described, for $t>0$, by $\phi (t,r)=\int \partial_t G_\mathrm{ret}(t;r,r') \phi_0(r') dr'_\ast.$ We can insert (\[Gret\_om\]) into this expression and deform again the contour of integration on $\omega$ in order to capture the contributions of the QNMs. We then isolate the BH ringing generated by the initial data: $$\begin{aligned}
\label{TimeEvolution_QNM}
&&\phi^\mathrm{QNM} (t,r)= 2 \, \mathrm{Re} \left[ \sum_n i\omega_n {\cal C}_n \right. \nonumber\\
&& \left. \vphantom{\sum_n} \qquad \times e^{-i\omega_n t+ip(\omega_n)r_\ast +
i[M\mu^2/p(\omega_n)] \ln(r/M)} \right].\end{aligned}$$ Here ${\cal C}_n$ denotes the excitation coefficient of the QNM with overtone index $n$. It takes explicitly into account the role of the BH perturbation and is given by $$\label{EC}
{\cal C}_n={\cal B}_n \int \frac{\phi_0(r')\phi^\mathrm{in}_{\omega_n}(r')}{\sqrt{\omega_n/p(\omega_n)}A^{(+)}(\omega_n)} dr'_\ast.$$
The excitation coefficients ${\cal C}_n$, like the excitation factors ${\cal B}_n$, have a resonant behavior but it is now more attenuated. Moreover, the maximum amplitude of the resonance is slightly shifted but still occurs for masses in a range where the QNM is a long-lived mode. In Fig. \[fig:C0\], we exhibit the strong resonant behavior of ${\cal C}_0$ for particular values of the parameters defining the initial data (\[Cauchy\_data\]). It occurs around the critical value ${\tilde \alpha} \approx 0.89$ and is rather similar to the behavior of the corresponding excitation factor ${\cal B}_0$. It should be noted that we have checked that it depends very little on the parameters defining the initial data (\[Cauchy\_data\]) (see Ref. [@DFOEH1] for a detailed study). Of course, for overtones, the resonance is more and more attenuated as the overtone index $n$ increases so, the ringings generated by the fundamental QNM are certainly the most interesting.
In Fig. \[fig:ExtrinsicRingings\], we plot, for the two values of the graviton mass considered in Fig. \[fig:IntrinsicRingings\], the BH “extrinsic" ringings defined by (\[TimeEvolution\_QNM\]) and we compare them to the ringing generated by the odd-parity $(\ell=2,n=0)$ QNM of the massless spin-2 field. This last one is constructed by noting that, for the same initial data, the excitation coefficient of the QNM is given by ${\cal C}_{20}/(2M) \approx 0.50761 - 0.29210 i$. Similar results as those displayed in Fig. \[fig:ExtrinsicRingings\] can be obtained for various values of the parameters defining the initial data (\[Cauchy\_data\]) and for various locations of the observer. Even if the role of the perturbation is taken into account, extraordinary BH ringings exist.
[*Conclusion.*]{}— In this letter, by considering the massive spin-2 field in Schwarzschild spacetime, we have pointed out a new effect in BH physics : the existence around particular values of the mass parameter of a strong resonant behavior for the excitation factors of the QNMs with, as a consequence, the existence of giant and slowly-decaying ringings. Such results are, in fact, a general feature of massive field theories in the Schwarzschild BH [@DFOEH1]. It would be interesting to study more realistic perturbations than the distortion described here by an initial value problem (e.g., the excitation of the BH by a particle falling radially or plunging), to consider alternative massive gravity theories and to extend our study to the Kerr BH. Finally, we would like to note that :
\(i) The Schwarzschild BH interacting with a massive spin-2 field is in general unstable [@Babichev:2013una; @Brito:2013wya] (see, however, Ref. [@Brito:2013yxa]). In the context of the theory considered here, the instability is due to the behavior of the propagating $\ell=0$ mode [@Brito:2013wya]. It is a “low-mass" instability which disappears for ${\tilde \alpha}$ above the threshold value ${\tilde \alpha}_c \approx 0.86$. So, the giant ringings predicted here occurring near and above the critical value ${\tilde \alpha} \approx 0.89$ are physically relevant.
\(ii) Even if the graviton is an ultralight particle, when it interacts with a supermassive BH, values of the coupling constant ${\tilde \alpha}$ leading to giant ringings can be easily reached. Indeed, supermassive BHs have their masses lying approximately between $10^6\, \mathrm{M}_{\odot}$ and $2\times 10^{10}\, \mathrm{M}_{\odot}$; so, if we assume that $\mu \approx 1.35\times 10^{-55} \, \mathrm{kg}$ (it seems to be the superior limit of the graviton mass in the framework of the ordinary Fierz-Pauli theory [@Finn:2001qi]), we have ${\tilde \alpha}$ lying approximately between $10^{-3}$ and $20$. As a consequence, due to the enormous number of supermassive BHs in the Universe, the extraordinary BH ringings discussed here could be observed by the next generations of gravitational wave detectors and used to test the various massive gravity theories or their absence could allow us to impose strong constraints on the graviton mass and to support, in a new way, Einstein’s general relativity.
[*Acknowledgments.*]{}— We wish to thank Andrei Belokogne for discussions and the “Collectivité Territoriale de Corse" for its support through the COMPA project.
|
---
abstract: 'Early recognition of abnormal rhythms in ECG signals is crucial for monitoring and diagnosing patients’ cardiac conditions, increasing the success rate of the treatment. Classifying abnormal rhythms into exact categories is very challenging due to the broad taxonomy of rhythms, noises and lack of large-scale real-world annotated data. Different from previous methods that utilize hand-crafted features or learn features from the original signal domain, we propose a novel ECG classification method by learning deep time-frequency representation and progressive decision fusion at different temporal scales in an end-to-end manner. First, the ECG wave signal is transformed into the time-frequency domain by using the Short-Time Fourier Transform. Next, several scale-specific deep convolutional neural networks are trained on ECG samples of a specific length. Finally, a progressive online decision fusion method is proposed to fuse decisions from the scale-specific models into a more accurate and stable one. Extensive experiments on both synthetic and real-world ECG datasets demonstrate the effectiveness and efficiency of the proposed method.'
address:
- 'School of Automation, Hangzhou Dianzi University'
- 'Department of Automation, University of Science and Technology of China'
- 'School of Electronics and Information, Hangzhou Dianzi University'
author:
- Jing Zhang
- Jing Tian
- Yang Cao
- Yuxiang Yang
- Xiaobin Xu
bibliography:
- 'KBS\_ECG.bib'
title: 'Deep Time-Frequency Representation and Progressive Decision Fusion for ECG Classification'
---
Decision-making,Electrocardiography,Fourier transforms,Neural networks
Introduction {#sec:Intro}
============
Electrocardiogram (ECG), which records the electrical depolarization-repolarization patterns of the heart’s electrical activity in the cardiac cycle, is widely used for monitoring or diagnosing patients’ cardiac conditions [@Giri2013Automated; @Acharya2016Automated; @kiranyaz2016real]. The diagnosis is usually made by well-trained and experienced cardiologists, which is laborious and expensive. Therefore, automatic monitoring and diagnosing systems are in great demand in clinics, community medical centers, and home health care programs. Although advances have been made in ECG filtering, detection and classification in the past decades [@kiranyaz2016real; @shyu2004using; @guler2005ecg; @mar2011optimization], it is still challenging for efficient and accurate ECG classification due to noises, various types of symptoms, and diversity between patients.
Before classification, a pre-processing filtering step is usually needed to remove a variety of noises from the ECG signal, including the power-line interference, base-line wander, muscle contraction noise, $etc$. Traditional approaches like low-pass filters and filter banks can reduce noise but may also lead some artifacts [@wu2009filtering]. Combining signal modeling and filtering together may alleviate this problem, but it is limited to a single type noise [@yan2010self; @blanco2008ecg]. Recently, different noise removal methods based on wavelet transform has been proposed by leveraging its superiority in multi-resolution signal analysis [@bhateja2013composite; @jenkal2016efficient; @poungponsri2013adaptive]. For instance, S. Poungponsri and X.H. Yu proposed a novel adaptive filtering approach based on wavelet transform and artificial neural networks that can efficiently removal different types of noises [@poungponsri2013adaptive].
For ECG classification, classical methods usually consist of two sequential modules: feature extraction and classifier training. Hand-crafted features are extracted in the time domain or frequency domain, including amplitudes, intervals, and higher-order statistics, $etc$. Various methods have been proposed such as filter banks [@afonso1999ecg], Kalman filter [@zeng2016inferring], Principal Component Analysis (PCA)[@Martis2013Characterization; @martis2012application], and wavelet transform (WT) [@ince2009generic; @jayachandran2010analysis; @daamouche2012wavelet; @shyu2004using; @garcia2016application]. Classifier models including Hidden Markov Models (HMM), Support Vector Machines (SVM) [@osowski2004support], Artificial Neural Networks (ANN) [@shyu2004using; @guler2005ecg; @ince2009generic; @barni2011privacy; @mar2011optimization; @wang2013ecg], and mixture-of-experts method [@hassan2017expert] have also been studied. Among them, a large number of methods are based on artificial neural networks due to its better modeling capacity. For example, L.Y. Shyu $et~al.$ propose a novel method for detecting Ventricular Premature Contraction (VPC) using the wavelet transform and fuzzy neural network [@shyu2004using]. By using the wavelet transform for QRS detection and VPC classification, their method has less computational complexity. I. Guler and E.D. Ubeyli propose to use a combined neural network model for ECG beat classification [@guler2005ecg]. Statistical features based on discrete wavelet transform are extracted and used as the input of first level networks. Then, sequential networks were trained using the outputs of the previous level networks as input. Unlike previous methods, T. Ince $et~al.$ propose a new method that uses a robust and generic ANN architecture and trains a patient-specific model with morphological wavelet transform features and temporal features for each patient [@ince2009generic]. Besides, some approaches have been proposed by combining several hand-crafted features to provide enhanced performance [@oster2015impact; @li2016signal]. Despite their usefulness, these methods have some common drawbacks: 1) the hand-crafted features rely on domain knowledge of experts and should be designed and tested carefully; 2) the classifier should have appropriate modeling capacity of such features; 2) The types of ECG signals are usually limited.
In the past few years, deep neural networks (DNN) have been widely used in many research fields and achieve remarkable performance. Recently, S. Kiranyaz $et~al.$ propose a 1-D convolutional neural network (CNN) for patient-specific ECG classification [@kiranyaz2016real]. They design a simple but effective network architecture and utilize 1-D convolutions to processing the ECG wave signal directly. G. Clifford $et~al.$ organized the PhysioNet/Computing in Cardiology Challenge 2017 for AF rhythm classification from a short single lead ECG recording. A large number of real-world ECG samples from patients are collected and annotated. It facilitates research on the challenging AF classification problem. Both hand-crafted feature-based methods and deep learning-based methods have been proposed [@hong2017encase; @zabihidetection; @teijeiro2017arrhythmia]. For example, S. Hong $et~al.$ propose an ensemble classifier based method by combining expert features and deep features [@hong2017encase]. T. Teijeiro $et~al.$ propose a combined two classifiers based method, $i.e.$, the first classifier evaluates the record globally using aggregated values for a set of high-level and clinically meaningful features, and the second classifier utilizes a Recurrent Neural Network fed with the individual features for each detected heartbeat [@teijeiro2017arrhythmia]. M. Zabihi $et~al.$ propose a hand-crafted feature extraction and selection method based on a random forest classifier [@zabihidetection].
In this paper, we propose a novel deep CNN based method for ECG classification by learning deep time-frequency representation and progressive decision fusion at different temporal scales in an end-to-end manner. Different from previous methods, 1) We first transform the original ECG signal into the time-frequency domain by Short-Time Fourier Transform (STFT). 2) Then, the time-frequency characteristics at different scales are learned by several scale-specific CNNs with 2-D convolutions. 3) Finally, we propose an online decision fusion method to fuse past and current decisions from different models into a more accurate one. We conducted extensive experiments on a synthetic ECG dataset consisting of 20 types of ECG signals and a real-world ECG dataset to validate the effectiveness of the proposed methods. The experimental results demonstrate its superiority over representative state-of-the-art methods.
Problem formulation {#sec:ProblemFormulation}
===================
Given a set of ECG signals and their corresponding labels, the target of a classification method is to predict their labels correctly. As depicted in Section \[sec:Intro\], it usually consists of two sequential modules: feature extraction and classifier training. Once the classifier is obtained, it can be used for unseen samples prediction, $i.e.$, testing phase. Mathematically, we denote the set of ECG wave signals as: $$X = \left\{ {\left( {{x_i},{y_i}} \right)\left| {i \in \Lambda } \right.} \right\},
\label{eq:ECGWaveSignal}$$ where ${x_i}$ is the $i^{th}$ sequence with $N$ samples : ${x_i} = {[{x_i}\left( 0 \right),{x_i}\left( 1 \right),...,{x_i}\left( {N - 1} \right)]^T} \in {R^N}$. ${y_i} \in \left\{ {0,..,C - 1} \right\}$ is the category of ${x_i}$, and $C$ is the number of total categories. $\Lambda $ is the index set of all samples. The feature extraction can be described as follows: $${f_i} = f\left( {{x_i},{\theta _f}} \right),
\label{eq:featureExtraction}$$ where ${f_i} \in {R^M}$ is the corresponding feature representation of signal ${x_i}$. Usually, the feature vector ${f_i}$ is more compact than the original signal ${x_i}$, $i.e.$, $M \ll N$. $f\left( { \cdot ,{\theta _f}} \right)$ is a mapping function from the original signal space to the feature space, and ${\theta _f}$ is the parameters associated with the mapping $f\left( \cdot \right)$. It is usually determined according to domain knowledge of experts and cross-validation. Given the feature representation, a classifier $g\left( { \cdot ,{\theta _g}} \right)$ predicts its category as follows: $${c_i} = g\left( {f\left( {{x_i},{\theta _f}} \right),{\theta _g}} \right),
\label{eq:classifier}$$ where ${\theta _g}$ is the parameters associated with the classifier $g\left( \cdot \right)$. ${c_i} \in \left\{ {0,..,C - 1} \right\}$ is the prediction. The frequently-used classifiers include SVM [@osowski2004support], ANN [@barni2011privacy; @mar2011optimization; @wang2013ecg], Random Forest, Deep CNN [@kiranyaz2016real], $etc$. Given the training samples, the training of a classifier can be formulated as an optimization problem of its parameter ${\theta _g}$ as follows: $$\theta _g^* = \mathop {\arg \min }\limits_{{\theta _g}} \sum\limits_{i \in {\Lambda _T}} {L\left( {g\left( {f\left( {{x_i},{\theta _f}} \right),{\theta _g}} \right),{y_i}} \right)},
\label{eq:thetaOpt}$$ where ${\Lambda _T}$ is the index set of training samples. $L\left( \cdot \right)$ is a loss function which depicts the loss of assigning a prediction category ${c_i}$ for a sample ${x_i}$ with label ${y_i}$, $e.g.$, margin loss in SVM model and cross-entropy loss in models of ANN or Random Forest.
{width="0.9\linewidth"}
{width="0.8\linewidth"}
For deep neural networks models, feature extraction (learning) and classifier training are integrated together in the neural network architecture as an end-to-end model. The parameters are optimized for training samples by using the error back propagation algorithm. Mathematically, it can be formulated as: $$\theta _h^* = \mathop {\arg \min }\limits_{{\theta _h}} \sum\limits_{i \in {\Lambda _T}} {L\left( {h\left( {{x_i},{\theta _h}} \right),{y_i}} \right)},
\label{eq:thetaOptDNN}$$ where $h\left( { \cdot ,{\theta _h}} \right)$ is the deep neural networks model with parameters ${\theta _h}$. For a modern deep neural networks architecture, $e.g.$, Deep CNN, it usually consists of many sequential layers like convolutional layers, pooling layer, nonlinear activation layer, and fully connected layer, $etc$. Therefore, $h\left( { \cdot ,{\theta _h}} \right)$ is a nonlinear mapping function with strong representation capacity and maps the original high-dimension input data to a low-dimension feature space, where features are more discriminative and compact.
The proposed approach {#sec:ProposedApproach}
=====================
Short-Time Fourier Transform {#subsec:STFT}
----------------------------
Although wave signals in the original time domain can be used as input of DNN to learn features, a time-frequency representation calculated within a short-window may be a better choice [@ullah2010dna]. Inspired by the work in speech recognition areas [@deng2013recent], where they show spectrogram features of speech are superior to Mel Frequency Cepstrum Coefficient (MFCC) with DNN, we first transform the original ECG wave signal into the time-frequency domain by using Short-Time Fourier Transform to obtain the ECG spectrogram representation. Mathematically, it can be described as follows: $${s_i}\left( {k,m} \right) = \sum\limits_{n = 0}^{N - 1} {{x_i}\left( n \right)w\left( {m - n} \right){e^{ - j\frac{{2\pi }}{N}kn}}},
\label{eq:STFT}$$ where $w( \cdot )$ is the window function, $e.g.$, Hamming window. ${s_i}\left( {k,m} \right)$ is the two-dimension spectrogram of ${x_i}$. Figure \[fig:waveSpectrogram\] shows some examples of spectrograms.
Architecture of the proposed CNN {#subsec:Architecture}
--------------------------------
[p[2.8cm]{}<p[1.5cm]{}<p[1.3cm]{}<p[0.9cm]{}<p[0.9cm]{}<p[0.9cm]{}<p[0.9cm]{}<]{} Network & Type & Input Size & Channels & Filter & Pad & stride\
\[0\][\*]{}[Proposed Network ($h1$)]{} & Conv1 & 1x32x4 & 32 & 3x3 & 1 & 1\
& Pool1 & 32x32x4 & - & 4x1 & 0 & (4,1)\
& Conv2 & 32x8x4 & 32 & 3x3 & 1 & 1\
& Pool2 & 32x8x4 & - & 4x2 & 0 & (4,2)\
& Fc3 & 32x2x2 & 64 & - & - & -\
& Fc4 & 64x1x1 & 20 & - & - & -\
& Params &\
& Complexity&\
\[0\][\*]{}[Network in [@kiranyaz2016real]]{} & Conv1 & 1x1x512 & 32 & 1x15 & (0,7) & (1,6)\
& Conv2 & 32x1x86 & 16 & 1x15 & (0,7) & (1,6)\
& Conv3 & 16x1x15 & 16 & 1x15 & (0,7) & (1,6)\
& Pool3 & 16x1x3 & - & 1x3 & 0 & 1\
& Fc4 & 16x1x1 & 10 & - & - & -\
& Fc5 & 10x1x1 & 20 & - & - & -\
& Params &\
& Complexity &\
\[tab:Architecture\]
Since we use the two-dimension spectrogram as input, we design a deep CNN architecture that involves 2-D convolutions. Specifically, the proposed architecture consists of 3 convolutional layers and 2 fully-connected layers. There is a max-pooling layer and a ReLU layer after the first two convolutional layers and a max-pooling layer after the last convolutional layer, respectively. Details are shown in Table \[tab:Architecture\]. As can be seen, it is quite light-weight with 18,976 parameters and 3.5x10$^5$ FLOPs. We also present the network architecture in [@kiranyaz2016real] as a comparison. Filter sizes and strides are adapted to the data length used in this paper. It can be seen that the proposed network has a comparable amount of parameters and computational cost with the one in [@kiranyaz2016real]. As will be shown in Section \[sec:Experiments\], the proposed method is computationally efficient and achieves a real-time performance even in an embedded device.
With the spectrogram ${s_i}$ as input, the CNN model predicts a probability vector ${p_i} = h\left( {{s_i},{\theta _h}} \right) \in {R^C}$ subjected to $\sum\limits_{c = 0}^{C - 1} {{p_{ic}}} = 1$ and ${p_{ic}} \ge 0$. Then the model parameter ${\theta _h}$ can be learned by minimizing the cross-entropy loss as follows: $$\begin{aligned}
{\theta _h^{*}} &= \mathop {\arg \min }\limits_{{\theta _h}} \sum\limits_{i \in {\Lambda _T}} {L\left( {h\left( {{s_i},{\theta _h}} \right),{y_i}} \right)} \\
&= \mathop {\arg \min }\limits_{{\theta _h}} - \sum\limits_{i \in {\Lambda _T}} {\sum\limits_{c = 0}^{C - 1} {{q_{ic}}\log \left( {{p_{ic}}} \right)} },
\end{aligned}
\label{eq:thetaOptCE}$$ where ${q_i} \in {R^C}$ is the one-hot vector of label ${y_i}$, $i.e.$, ${q_{ic}} = \delta \left( {{y_i},c} \right) \in \left\{ {0,1} \right\}$.
Usually, single beat is detected and classified [@guler2005ecg]. However, since long signals contain more beats given the sampling rate, prediction on it will be more accurate. In this paper, the length of each sample in the synthetic ECG dataset is 16384 at a sampling rate of 512Hz, which lasts 32s. We split each sample into sub-samples which have the same length of 512. Therefore, each sample contains about $\sim1$ beats. It is noteworthy that we do not explicitly extract the beat from the raw signal but use it as the input directly after the Short-Time Fourier Transform. Then, we train our CNN model on this dataset. Besides, to compare the performance of models for longer samples, we also split each sample into sub-samples of different lengths, $e.g.$, 2s, 4s, 8s, 16s. We use them to train our CNN models accordingly and denote all these scale-specific models as ${h_1} \sim {h_6}$, respectively. The width of the spectrogram is determined by the length of the wave signal given the window function, which varies for each model. Nevertheless, we use the same architecture for all models and change the pooling strides along columns accordingly while keeping the fully-connected layers fixed.
Optimization {#subsec:Optimization}
------------
The optimization of Eq. is not trivial since the objective function $L\left( \cdot \right)$ is non-linear and non-convex. Instead of using deterministic optimization methods [@jones1993lipschitzian; @sergeyev2018efficiency], we adopt the mini-batch Stochastic Gradient Descent (SGD) algorithm [@bottou1998online; @bottou2010large] in this paper. Mathematically, it can be formulated as: $$\theta _h^{t + 1} = \theta _h^t - {\gamma _t}\frac{1}{B}\sum\limits_{i = 1}^B {{\nabla _{{\theta _h}}}L\left( {h\left( {{s_i},{\theta _h}} \right),{y_i}} \right)},
\label{eq:sgd}$$ where $B$ is the number of training samples in each mini-batch, $\gamma _t$ is the learning rate at step $t$. It can be proved that as long as the learning rate $\gamma _t$ are small enough, the algorithm converges towards a local minimum of the empirical risk [@bottou1998online]. Specifically, to keep the optimization direction and prevent oscillations, we leverage the SGD algorithm with a momentum term [@rumelhart1988learning; @sutskever2013importance], $i.e.$, $$\Delta \theta _h^{t + 1} = \alpha \Delta \theta _h^t - {\gamma _t}\frac{1}{B}\sum\limits_{i = 1}^B {{\nabla _{{\theta _h}}}L\left( {h\left( {{s_i},{\theta _h}} \right),{y_i}} \right)},
\label{eq:sgd_momentum}$$ and $$\theta _h^{t + 1} = \theta _h^t + \Delta \theta _h^{t + 1},
\label{eq:sgd_momentum_update}$$ where $\Delta \theta _h$ is the momentum term and $\alpha$ is the momentum parameter. The setting of $\alpha$, $\gamma$, and $B$ will be presented in Section \[subsubsec:DatasetParams\].
Progressive online decision fusion {#subsec:OnlineFusion}
----------------------------------
For online testing, as the length of signal is growing, we can test it sequentially by using the scale-specific models. As illustrated in Figure \[fig:flowchart\], lower level models make decisions based on local patterns within short signals, while higher level models make decisions based on global patterns within long signals. These models can be seen as different experts focusing on different scales, whose decisions are complementary and could be fused as a more accurate and stable one [@zhang2015multi; @lu2016multilevel]. To this end, we propose a progressive online decision fusion method. Mathematically, it can be described as follows: $$\widetilde{h}\left( x \right) = \sum\limits_{s = 1}^{s = {s_l}} {{w_s}\left( {\frac{1}{{{k_s}}}\sum\limits_{k = 1}^{k = {k_s}} {{h_s}\left( {{x^{sk}}} \right)} } \right)},
\label{eq:fusion}$$ where $\widetilde{h}\left( \cdot \right)$ represents the fusion result, ${s_l} \in \left\{ {1,2,3,4,5,6} \right\}$ is the maximum level for a signal $x$ of specific length. ${x^{sk}}$ is the ${k^{th}}$ segment of $x$ for the ${s^{th}}$ level model ${h_s}$, and ${k_s}$ is the number of segments at $s^{th}$ level, $i.e.$, $k_s = 2^{s_l - s}$. For example, when the length of $x$ is 2048, ${s_l}$ will be 3, and ${k_1} \sim {k_3}$ will be 4, 2, 1, respectively. ${w_s}$ is the fusion weight of ${h_s}$ subjected to $\sum\limits_{s = 1}^{s = {s_l}} {{w_s}} = 1$. It can be seen from Eq. , decision at each segment at the same level is treated equally. It is reasonable since there is no prior knowledge favouring specific segment and the decision is made by the same model.
Assuming that the distribution of $h_s\left( \cdot \right)$ is independent from each other for any $s$, with a mean $\mu _s$ and variance $\sigma _s$, the expectation of Eq. can be derived as: $$\begin{aligned}
E\left [ \widetilde{h}\left( x \right) \right ] & = \sum\limits_{s = 1}^{s = {s_l}} {{w_s}\left( {\frac{1}{{{k_s}}}\sum\limits_{k = 1}^{k = {k_s}} {E\left [ {h_s}\left( {{x^{sk}}} \right) \right ] } } \right)} \\
& = \sum\limits_{s = 1}^{s = {s_l}} {{w_s}\left( {\frac{1}{{{k_s}}}\sum\limits_{k = 1}^{k = {k_s}} {\mu _s} } \right)} \\
& = \sum\limits_{s = 1}^{s = {s_l}} {{w_s \mu _s}}.
\end{aligned}
\label{eq:expectation}$$ For a given training sample $x_i$, $h_s\left( \cdot \right)$ shares the same training targets $y_i$. Therefore, $\mu _s$ should be the same for any $s$, denoted as $\mu$. Accordingly, we have the unbiased estimate $E\left [ \widetilde{h}\left( x \right) \right ] = \mu$. Similarly, we can derive the variance of $\widetilde{h}\left( x \right)$ as: $Var \left [ \widetilde{h}\left( x \right) \right ] = \sum\limits_{s = 1}^{s = {s_l}} {{\frac{w_s^2}{k_s} \sigma _s}}$. Usually, the variance $\sigma _s$ is decreased with the growth of sample length (scale $s$) since more “evidence” is accumulated. For instance, if we assume $\sigma _s = \frac{1}{k_s}\sigma$, then we will have $Var \left [ \widetilde{h}\left( x \right) \right ] = \frac{2^{2s_l + 2} -2^2}{2^{2s_l}s_l^2\left (2^2-1\right )} \sigma \le \sigma$ for uniform weights $w_s=\frac{1}{s_l}$, and $Var \left [ \widetilde{h}\left( x \right) \right ] = \frac{2^{4s_l + 2} -2^2}{2^{2s_l} \left (2^{s_l}-1\right )^2 \left ( 2^4-1 \right ) } \sigma \le \sigma$ for non-uniform weights defined in Eq. . Taking $s_l=6$ as an example, we will have $Var \left [ \widetilde{h}\left( x \right) \right ] \approx 0.037\sigma$ and $Var \left [ \widetilde{h}\left( x \right) \right ] \approx 0.2752\sigma$ for uniform and non-uniform weights, respectively. As can be seen, using a fusion decision will reduce the variance and get a more stable result.
Experiments {#sec:Experiments}
===========
Evaluation metrics {#subsec:evaluationMetrics}
------------------
First, we present the definition of the evaluation metrics used in this paper. Denoting the confusion matrix as $CM = \left[ {{c_{ij}}} \right]$, where $c_{ij}$ is number of samples belonging to the $i^{th}$ category but being predicted as the $j^{th}$ one, the Accuracy, Sensitivity, Specificity and F1 score can be calculated as follows.
$$Accuracy = {{\sum\limits_{i = 1}^{i = C} {{c_{ii}}} } \mathord{\left/
{\vphantom {{\sum\limits_{i = 1}^{i = C} {{c_{ii}}} } {\sum\limits_{i = 1}^{i = C} {\sum\limits_{j = 1}^{j = C} {{c_{ij}}} } }}} \right.
\kern-\nulldelimiterspace} {\sum\limits_{i = 1}^{i = C} {\sum\limits_{j = 1}^{j = C} {{c_{ij}}} } }}.
\label{eq:accuracyScore}$$
$$Sensitivit{y_i} = {{{c_{ii}}} \mathord{\left/
{\vphantom {{{c_{ii}}} {\sum\limits_{j = 1}^{j = C} {{c_{ij}}} }}} \right.
\kern-\nulldelimiterspace} {\sum\limits_{j = 1}^{j = C} {{c_{ij}}} }},
\label{eq:sensitivityScore}$$
where class $i$ represents the symptomatic classes, $e.g.$, RAF, FAF, $etc$.
$$Specificit{y_k} = {{{c_{kk}}} \mathord{\left/
{\vphantom {{{c_{kk}}} {\sum\limits_{j = 1}^{j = C} {{c_{kj}}} }}} \right.
\kern-\nulldelimiterspace} {\sum\limits_{j = 1}^{j = C} {{c_{kj}}} }},
\label{eq:specifictyScore}$$
where class $k$ represents the normal class.
$$F{1_i} = {{2{c_{ii}}} \mathord{\left/
{\vphantom {{2{c_{ii}}} {\left( {\sum\limits_{j = 1}^{j = C} {{c_{ij}}} + \sum\limits_{i = 1}^{i = C} {{c_{ij}}} } \right)}}} \right.
\kern-\nulldelimiterspace} {\left( {\sum\limits_{j = 1}^{j = C} {{c_{ij}}} + \sum\limits_{i = 1}^{i = C} {{c_{ij}}} } \right)}}.
\label{eq:F1Score}$$
$$F1 = \frac{1}{C}\sum\limits_{i = 1}^{i = C} {F{1_i}}.
\label{eq:meanF1Score}$$
---------------------------- ------------------- ------------------- ------------------- ------------------- ------------------- ------------------- -------------------
Methods RAF FAF AF SA AT ST PAC
SVM+FFT 0.95$\pm$0.02 0.73$\pm$0.01 0.77$\pm$0.08 **0.77$\pm$0.05** 0.85$\pm$0.02 0.78$\pm$0.10 0.76$\pm$0.08
1D CNN [@kiranyaz2016real] 0.96$\pm$0.01 0.92$\pm$0.04 0.84$\pm$0.06 0.39$\pm$0.03 0.90$\pm$0.01 0.96$\pm$0.02 0.20$\pm$0.09
Proposed 0.97$\pm$0.01 0.99$\pm$0.01 0.94$\pm$0.01 0.71$\pm$0.02 0.94$\pm$0.01 0.99$\pm$0.01 0.59$\pm$0.20
SVM+CNN Feature **0.98$\pm$0.01** **0.99$\pm$0.01** **0.97$\pm$0.01** 0.75$\pm$0.01 **0.97$\pm$0.01** **0.99$\pm$0.01** **0.82$\pm$0.04**
Methods VB VTr PVCCI VTa RVF FVF AVB-I
SVM+FFT 0.81$\pm$0.04 **0.71$\pm$0.10** 0.82$\pm$0.03 0.84$\pm$0.02 0.86$\pm$0.03 0.78$\pm$0.06 0.82$\pm$0.02
1D CNN [@kiranyaz2016real] 0.92$\pm$0.03 0.19$\pm$0.11 0.87$\pm$0.02 0.97$\pm$0.01 0.97$\pm$0.02 0.96$\pm$0.02 0.70$\pm$0.17
Proposed 0.93$\pm$0.01 0.18$\pm$0.19 0.89$\pm$0.03 0.99$\pm$0.01 0.99$\pm$0.01 0.98$\pm$0.01 0.77$\pm$0.15
SVM+CNN Feature **0.96$\pm$0.01** 0.34$\pm$0.12 **0.95$\pm$0.01** **1.00$\pm$0.00** **1.00$\pm$0.00** **0.99$\pm$0.01** **0.95$\pm$0.01**
Specificity
Methods AVB-II AVB-III RBBB LBBB PVC N
SVM+FFT 0.84$\pm$0.05 0.76$\pm$0.07 0.72$\pm$0.09 0.81$\pm$0.05 **0.76$\pm$0.10** 0.86$\pm$0.02
1D CNN [@kiranyaz2016real] 0.94$\pm$0.03 0.68$\pm$0.05 0.91$\pm$0.04 0.98$\pm$0.01 0.68$\pm$0.05 0.95$\pm$0.01
Proposed 0.92$\pm$0.03 0.86$\pm$0.04 0.95$\pm$0.01 0.98$\pm$0.01 0.72$\pm$0.05 0.96$\pm$0.01
SVM+CNN Feature **0.98$\pm$0.01** **0.93$\pm$0.03** **0.97$\pm$0.01** **0.98$\pm$0.01** 0.75$\pm$0.04 **0.98$\pm$0.01**
---------------------------- ------------------- ------------------- ------------------- ------------------- ------------------- ------------------- -------------------
\[tab:SensitivitySpecificityTrain\]
---------------------------- ------------------- ------------------- ------------------- ------------------- ------------------- ------------------- -------------------
Methods RAF FAF AF SA AT ST PAC
SVM+FFT 0.88$\pm$0.11 0.44$\pm$0.14 0.50$\pm$0.21 0.69$\pm$0.13 0.53$\pm$0.27 0.56$\pm$0.21 0.57$\pm$0.13
1D CNN [@kiranyaz2016real] 0.96$\pm$0.01 0.93$\pm$0.04 0.83$\pm$0.07 0.38$\pm$0.04 0.90$\pm$0.02 0.95$\pm$0.03 0.20$\pm$0.05
Proposed 0.98$\pm$0.01 0.99$\pm$0.01 0.95$\pm$0.03 0.72$\pm$0.02 0.94$\pm$0.02 0.96$\pm$0.03 0.62$\pm$0.22
SVM+CNN Feature 0.97$\pm$0.01 0.98$\pm$0.01 0.96$\pm$0.01 0.72$\pm$0.02 0.95$\pm$0.01 0.96$\pm$0.04 0.85$\pm$0.03
Proposed(Fusion) **0.99$\pm$0.01** **1.00$\pm$0.00** **0.99$\pm$0.01** **1.00$\pm$0.00** **1.00$\pm$0.00** **1.00$\pm$0.00** **1.00$\pm$0.00**
Methods VB VTr PVCCI VTa RVF FVF AVB-I
SVM+FFT 0.42$\pm$0.29 0.38$\pm$0.14 0.54$\pm$0.24 0.73$\pm$0.09 0.60$\pm$0.18 0.45$\pm$0.32 0.50$\pm$0.30
1D CNN [@kiranyaz2016real] 0.93$\pm$0.02 0.15$\pm$0.15 0.85$\pm$0.03 0.96$\pm$0.01 0.96$\pm$0.02 0.94$\pm$0.04 0.50$\pm$0.35
Proposed 0.94$\pm$0.01 0.32$\pm$0.23 0.91$\pm$0.02 0.99$\pm$0.01 0.98$\pm$0.02 0.99$\pm$0.01 0.56$\pm$0.30
SVM+CNN Feature 0.96$\pm$0.01 0.02$\pm$0.02 0.95$\pm$0.03 0.98$\pm$0.01 0.97$\pm$0.02 0.98$\pm$0.02 0.48$\pm$0.15
Proposed(Fusion) **1.00$\pm$0.00** **0.99$\pm$0.01** **0.99$\pm$0.01** **1.00$\pm$0.00** **1.00$\pm$0.00** **0.99$\pm$0.01** **0.86$\pm$0.20**
Specificity
Methods AVB-II AVB-III RBBB LBBB PVC N
SVM+FFT 0.66$\pm$0.14 0.56$\pm$0.21 0.46$\pm$0.07 0.48$\pm$0.33 0.40$\pm$0.24 0.60$\pm$0.18
1D CNN [@kiranyaz2016real] 0.93$\pm$0.08 0.64$\pm$0.07 0.89$\pm$0.10 0.98$\pm$0.01 0.69$\pm$0.05 0.95$\pm$0.02
Proposed 0.93$\pm$0.06 0.87$\pm$0.05 0.95$\pm$0.02 0.98$\pm$0.01 0.57$\pm$0.25 0.96$\pm$0.02
SVM+CNN Feature 0.95$\pm$0.05 0.89$\pm$0.05 0.95$\pm$0.02 0.98$\pm$0.01 0.57$\pm$0.22 0.97$\pm$0.02
Proposed(Fusion) **1.00$\pm$0.00** **0.95$\pm$0.05** **0.98$\pm$0.02** **1.00$\pm$0.00** **0.99$\pm$0.01** **0.99$\pm$0.01**
---------------------------- ------------------- ------------------- ------------------- ------------------- ------------------- ------------------- -------------------
\[tab:SensitivitySpecificityVal\]
Experiments on a synthetic ECG dataset {#subsec:syntheticExperiments}
--------------------------------------
### Dataset and parameter settings {#subsubsec:DatasetParams}
To verify the effectiveness of the proposed method, we construct a synthetic dataset by using an ECG simulator. The simulator can generate different types of ECG signals with different parameter settings. Generally, the ECG signals consist of four types of features, namely, trend, cycle, irregularities, and burst [@ullah2019modeling]. To cover these features, we choose 20 categories of ECG signals in this paper, which include Normal (N), Rough Atrial Fibrillation (RAF), Fine Atrial Fibrillation (FAF), Atrial Flutter (AF), Sinus Arrhythmia (SA), Atrial Tachycardia (AT), Supraventricular Tachycardia (ST), Premature Atrial Contraction (PAC), Ventricular Bigeminy (VB), Ventricular Trigeminy (VTr), Premature Ventricular Contraction Coupling Interval (PVCCI), Ventricular Tachycardia (VTa), Rough Ventricular Fibrillation (RVF), Fine Ventricular Fibrillation (FVF), Atrio-Ventricular Block I (AVB-I), Atrio-Ventricular Block (AVB-II), Atrio-Ventricular Block (AVB-III), Right Bundle Branch Block (RBBB), Left Bundle Branch Block (LBBB), and Premature Ventricular Contractions (PVC). There are a total of 2426 samples, about 120 samples per category. Each sample has a maximum length of 16384 points sampled at 512Hz. We split the sequence at a random position and merge them by changing their orders. In this way, we augment the dataset and simulate the time delay such that the model can capture the regularity and irregularity in the time series [@sharif2013fuzzy]. We use the 3-fold cross-validation to evaluate the proposed method.
Parameters are set as follows. We use the Hamming window of length 256 in Short-Time Fourier Transform and the overlap size is 128. The CNN model is trained in a total of 20,000 iterations with a batch size of 128. The learning rate decreases by half every 5,000 iterations from 0.01 to 6.25x$10^{-4}$. The momentum and the decay parameter are set to 0.9 and 5x$10^{-6}$, respectively. We implement the proposed method in CAFFE [@jia2014caffe] on a workstation with NVIDIA GTX Titan X GPUs if not specified.
### Comparisons with previous methods {#subsubsec:Comparisons}
We compare the performance of the proposed method with previous methods including SVM based on Fourier transform, the pilot Deep CNN method in [@kiranyaz2016real] which uses 1-D convolutions, and SVM based on the learned features of the proposed method. We report the sensitivity and specificity scores of different methods on both the training set and test set. We also report the average classification accuracy. The standard deviations of each index on the 3-fold cross-validation are also reported. Results are summarized in Table \[tab:SensitivitySpecificityTrain\], Table \[tab:SensitivitySpecificityVal\] and Table \[tab:Accuracy\].
Methods Training Set Test Set
---------------------------- ----------------- ---------------------
SVM+FFT 0.81($\pm$0.04) 0.56($\pm$0.12)
1D CNN [@kiranyaz2016real] 0.83($\pm$0.01) 0.81($\pm$0.04)
Proposed 0.88($\pm$0.03) 0.87($\pm$0.03)
SVM+CNN Feature 0.93($\pm$0.01) 0.87($\pm$0.02)
Proposed (Fusion) 0.99($\pm$0.01) **0.99($\pm$0.01)**
: Average classification accuracy of different methods on the training set and test test. Standard deviations (Std.) are listed in the brackets.
\[tab:Accuracy\]
It can be seen that the method in [@kiranyaz2016real] outperforms the traditional method using the FFT coefficients and SVM classifier. However, it is inferior to the proposed one which uses the 2-dimensional spectrogram as input, which benefits from the learned features of time-frequency characteristics. Besides, we use the learned features from the proposed method to train an SVM classifier. The results are denoted as “SVM+CNN Feature”. Compared with the SVM with FFT features, the performance of this classifier is significantly boosted. It demonstrates that the proposed method learns a more discriminative feature representation of the ECG signal. Interestingly, it is marginally better than the proposed CNN model which employs a linear classifier. It is reasonable since a more sophisticated nonlinear radial basis kernel is used in the SVM classifier. However, it shows a tendency toward overfitting $i.e.$, a larger gain on the training set.
Moreover, from Table \[tab:SensitivitySpecificityTrain\] and Table \[tab:SensitivitySpecificityVal\], we can find that categories of SA, PAC, VTr, AVB-I, and PVC are hard to be distinguished. We’ll shed light on the phenomenon by inspecting the learned features through the visualization technique and analyzing the confusion matrix between categories as follows.
### Analysis on learned features and confusion matrix between categories {#subsubsec:AnalysisFeatures}
First, we calculate the learned features from the penultimate layer for all test data. Then, we employ the t-Distributed Stochastic Neighbor Embedding (t-SNE) method proposed in [@maaten2008visualizing; @van2014accelerating] to visually inspect them. The visualization results are shown in Figure \[fig:featureProj\](a). As can be seen, some categories such as Normal(N), RVF, FVF, ST, RBBB, LBBB, PVCCI, VTa and RAF, are separated from other categories. However, some categories such as SA, PAC, PVC, VTr, AVB-I, and AVB-III, are overlapped with other categories as indicated by the red circles. We further plot them separately in Figure \[fig:featureProj\](b)-(e). For example, SA tends to be overlapped with AT and PVC, and PVC tends to be overlapped with PAC and VB. Nevertheless, they are separated from the Normal category, coinciding with the high specificity scores in Table \[tab:SensitivitySpecificityTrain\] and Table \[tab:SensitivitySpecificityVal\].
{width="0.9\linewidth"}
Besides, we also calculate the confusion matrix of the proposed method at the first level on the test set, which is shown in Figure \[Fig:ConfusionMatrix\](a). It is clear that some categories are overlapped with others, $e.g.$, SA, VTr, AVB-I, and PVC. The results are consistent with the visual inspection results in Figure \[fig:featureProj\].
### Online decision fusion performance {#subsubsec:OnlineDecisionFusion}
We test the fusion method in Section \[subsec:OnlineFusion\] at different levels ${s_l}$: 2, 3, 4, 5, and 6. Two kinds of fusion weights are compared: the uniform one and the one favouring high level models which is calculated as: $${w_s} = \frac{{{2^{s - 1}}}}{{{2^{{s_l}}} - 1}}.
\label{eq:weight}$$
Figure\[fig:accurayMultiScaleWholeAugFusion\] shows the classification accuracy of the proposed fusion method and the proposed single scale method at different levels. First, it can be seen that $h4$ achieves the best performance among all the single models at different levels. The reason may be that it makes a trade-off between data length and the number of model decisions. Compared with $h1$, the input data length is 16 times larger. Compared with $h6$, which only makes a single decision on the whole sequence, $h4$ can make 4 decisions from different scale-specific models and fuse them into a more accurate one.
Then, it can be seen that the fusion results are consistently better than the results of the single-scale model. The performance is improved consistently with the growth of data length. It validates the idea that fusing the complementary decisions from different models leads to a more accurate and stable one. Besides, using non-uniform weights does not provide any advantage over the uniform one. The non-uniform weight strategy favors the higher-level models than the decisions from the lower-level models. Though it is better than the single model, the gains are indeed very marginal. Especially at higher levels, the performance is largely dominated by the model at the highest level. Please refer to Section \[subsec:OnlineFusion\] for more details. In conclusion, the proposed online decision fusion method with uniform weights at level 6 achieves the best result. For example, the accuracy is boosted from 87% (single model at level 1) to 99%, and the standard deviation is reduced from 0.03 (single model at level 1) to 0.011. Its sensitivity and specificity scores are shown in Table \[tab:SensitivitySpecificityVal\], which shows a significant boost than other methods. The confusion matrix in Figure \[Fig:ConfusionMatrix\](b) shows the similar results. These results demonstrate the effectiveness of the proposed online decision fusion method.
Device
-------- ------ ------ ------ ------ ------ ------
Level GPU CPU GPU CPU GPU CPU
1 0.01 0.17 0.17 0.46 0.12 0.44
2 0.03 0.21 0.27 0.59 0.13 0.55
3 0.05 0.26 0.37 0.80 0.14 0.72
4 0.10 0.42 1.22 1.57 0.18 1.39
5 0.21 0.82 2.01 2.91 0.27 2.79
6 0.33 1.33 2.73 5.38 0.56 5.66
: Running times (millisecond, ms) of the proposed method at different settings.
\[tab:RunningTime\]
### Computational complexity and running time analysis {#subsubsec:ComputationalAnalysis}
We record the running times of the proposed method at GPU and CPU modes, respectively. Results are summarized in Table \[tab:RunningTime\]. As can be seen, the running time is only 0.33ms even if it is tested for the whole sequence (level 6). To further examine the computational efficiency of the proposed method, we test it on an NVIDIA Jetson TX2 embedded board. Again, the proposed method can achieve a real-time speed. Interestingly, the running times at GPU mode and CPU mode are comparable. We hypothesize that enlarging the batch size may make full advantage of the GPU acceleration. After enlarging the batch size 10 times, the superiority of GPU mode is significant. In conclusion, the proposed method is very efficient and promising to be integrated into a portable ECG monitor with limited computational resources.
Experiments on a real-world ECG dataset {#subsec:actualExperiments}
---------------------------------------
We also conducted extensive experiments on a real-world ECG dataset used in the 2017 PhysioNet/Computing in Cardiology Challenge [@clifford2017af]. The dataset is split into the training set, validation set, and test set. The training set contains 8,528 single-lead ECG recordings lasting from 9s to 60s. The validation set and test set contain 300 and 3,658 ECG recordings of similar lengths, respectively. The ECG recordings were sampled as 300 Hz. Each sample is labeled into four categories: Normal rhythm, AF rhythm, Other rhythm, and Noisy recordings. Only labels of the training set and validation set are publicly available. Some examples of the ECG waves are shown in Figure \[fig:physioNetExampleWaveforms\].
We train our model on the training set. Scores both on the training set and validation set are reported and compared with the top entries in the challenge. It is noteworthy that we add two more convolutional layers after the first and second convolutional layers in the network depicted in Table \[tab:Architecture\] such that it has a stronger representation capacity to handle the real-world ECG signals better. The number of convolutional filters and kernel sizes are the same as their preceding counterparts. The first fully-connected layer is kept the same. The output number of the last fully-connected layer is modified to be four to keep consistent with the number of categories. Each sample in the dataset is cropped or duplicated to have a length of 16,384. All other hyper-parameters are kept the same as the above experiments if not specified. We train the model at each level three times with random seeds and report the average scores and standard deviations.
![Examples of the ECG waveforms in PhysioNet dataset [@clifford2017af].[]{data-label="fig:physioNetExampleWaveforms"}](physioNet_example_waveforms){width="0.8\linewidth"}
### Comparisons of the proposed method and the top entries in the challenge {#subsubsec:onlineFusionPhysioNet}
![Visualization of the learned features from the proposed network on the PhysioNet dataset [@clifford2017af].[]{data-label="fig:featureProj_PhysioNet"}](val_aug2_featProj){width="0.5\linewidth"}
We report the results on this dataset in terms of mean accuracy and F1 score. To keep consistent with the evaluation protocol in [@clifford2017af], we report the average F1 scores for the first three categories. Besides, we also include the average F1 scores for all categories. As can be seen from Figure \[fig:onlineFusionAcc\_PhysioNet\] and Figure \[fig:onlineFusionF1\_PhysioNet\], the best results are achieved at level 4 ($h4$) and level 5 ($h5$) by the proposed online fusion method, $i.e.$, $100\% \pm 0\%$ classification accuracy and $100\% \pm 0\%$ F1 score. Meanwhile, the results of single models are also competitive. Best results are achieved at level 2 ($h2$) with a $99.44\% \pm 0.57\%$ accuracy and a $99.21\% \pm 0.81\%$ F1 score. It is consistent with the result in Section \[subsubsec:OnlineDecisionFusion\], where model $h4$ makes a trade-off between data length and the number of model decisions. The comparison results between the proposed method and the top entries in the challenge are listed in Table \[tab:Challenge-2017\]. The proposed methods achieve comparable or better results than the top entries on both the training set and validation set.
\[tab:Challenge-2017\]
### Analysis on learned features {#subsubsec:AnalysisFeatures_PhysioNet}
Similar to Section \[subsubsec:AnalysisFeatures\], we plot the learned features from model $h4$ on the validation set in Figure \[fig:featureProj\_PhysioNet\]. As can be seen, samples in each category are almost clustered together and separated from other clusters. For several samples in the category of ”Other rhythm”, they are near the clusters of ”Normal” and ”Noisy”. It implies that these samples are either with noise labels or hard cases which should be carefully handled.
In addition, we plot the spectrograms and their corresponding feature maps from $Conv1$, $Pool1$, $Conv2$ and $Pool2$ layers in Figure \[fig:featResponse\]. As can be seen, the first convolutional layer acts like a basic feature extractor which strengthens the informative parts in the spectrograms. Then, features in low and medium frequencies are pooled and contribute to the final classification. From the $Conv2$ feature maps, we can see that the proposed network generates strong responses in specific frequency zones and accumulate them along the temporal axis. By doing so and together with the online fusion, it learns effective and discriminative features to make an accurate classification.
Conclusion and future work {#sec:conclusion}
==========================
In this paper, we propose a novel deep CNN based method for ECG signal classification. It learns discriminative feature representation from the time-frequency domain by calculating the Short-Time Fourier Transform of the original wave signal. Besides, the proposed online decision fusion method fuses complementary decisions from different scale-specific models into a more accurate one. Extensive experiments on a synthetic 20-category ECG dataset and a real-world AF classification dataset demonstrate its effectiveness. Moreover, the proposed method is computationally efficient and promising to be integrated into a portable ECG monitor with limited computational resources. Future research may include: 1) devising or searching compact and efficient networks to handle complex real-world ECG data; 2) improving the online fusion by integrating both the decisions and learned features at different levels; 3) exploring the potential of the proposed method for nonlinear time series beyond ECG.
Acknowledgment {#acknowledgment .unnumbered}
==============
This work was partly supported by the National Natural Science Foundation of China (NSFC) under Grants 61806062, 61873077, and 61872327, the NSFC-Zhejiang Joint Fund for the Integration of Industrialization and Informatization under the Grant U1709215, the Fundamental Research Funds for the Central Universities under Grant WK2380000001, and the Zhejiang Province Key R&D Project under Grant 2019C03104.
|
---
abstract: |
**Context:** Computational diversity, i.e., the presence of a set of programs that all perform compatible services but that exhibit behavioral differences under certain conditions, is essential for fault tolerance and security.
**Objective:** We aim at proposing an approach for automatically assessing the presence of computational diversity. In this work, computationally diverse variants are defined as (i) sharing the same API, (ii) behaving the same according to an input-output based specification (a test-suite) and (iii) exhibiting observable differences when they run outside the specified input space.
**Method:** Our technique relies on test amplification. We propose source code transformations on test cases to explore the input domain and systematically sense the observation domain. We quantify computational diversity as the dissimilarity between observations on inputs that are outside the specified domain.
**Results:** We run our experiments on variants of classes from open-source, large and thoroughly tested Java classes. Our test amplification multiplies by ten the number of input points in the test suite and is effective at detecting software diversity.
**Conclusion:** The key insights of this study are: the systematic exploration of the observable output space of a class provides new insights about its degree of encapsulation; the behavioral diversity that we observe originates from areas of the code that are characterized by their flexibility (caching, checking, formatting, etc.).
author:
- |
Benoit Baudry$^o$, Simon Allier$^o$, Marcelino Rodriguez-Cancio$^{\dag\dag,o}$ and Martin Monperrus$^{\dag,o}$\
$^o$ Inria, France\
$^{\dag\dag}$ University of Rennes 1, France $^\dag$ University of Lille, France\
contact: benoit.baudry@inria.fr
bibliography:
- 'references.bib'
title: '[[DSpot]{}]{}: Test Amplification for Automatic Assessment of Computational Diversity'
---
copyrightspace
**KEYWORDS:** software diversity, software testing, test amplification, dynamic analysis.
Introduction {#sec:intro}
============
Computational diversity, i.e., the presence of a set of programs that all perform compatible services but that exhibit behavioral differences under certain conditions, is essential for fault tolerance and security [@avizienis85; @deswarte98; @donnell04; @franz10]. Consequently, it is of utmost importance to have systematic and efficient procedures to determine if a set of programs are computationally diverse.
Many works have tried to tackle this challenge, using input generation [@jiang09], static analysis [@kawaguchi2010conditional], or evolutionary testing [@yoo2012] and [@Carzaniga15] (concurrent of this work). Yet, having a reliable detection of computational diversity for large object-oriented programs is still a challenging endeavor.
In this paper, we propose an approach, called [[DSpot]{}]{}[^1], for assessing the presence of computational diversity, i.e., to determine if a set of program variants exhibit different behaviors under certain conditions. [[DSpot]{}]{}takes as input a test suite and a set of $n$ program variants. The $n$ variants have the same application programming interface (API) and they all pass the same test suite (i.e. they comply with the same executable specification). [[DSpot]{}]{}consists of two steps: (i) automatically transforming the test suite; and (ii) running this larger test suite, that we call “amplified test suite” on all variants to reveal visible differences in the computation.
The first step of [[DSpot]{}]{}is an original technique of test amplification [@xie06; @zhang2012; @pezze2013; @yoo2012]. Our key insight is to combine the automatic exploration of the input domain with the systematic sensing of the observation domain. The former is obtained by transforming the input values and method calls of the original test. The latter is the result of the analysis and transformation of the original assertions of the test suite, in order to observe the program state from as many observation points visible from the public API as possible. The second step of [[DSpot]{}]{}runs the augmented test suite on each variant. The observation points introduced during amplification generate new traces on the program state. If there exists a difference between the trace of a pair of variants, we say that these variants are computationally diverse. In other words, two variants are considered diverse if there exists at least one input outside the specified domain that triggers different behaviors on the variants which can be observed through the public API.
To evaluate the ability of [[DSpot]{}]{}at observing computational diversity, we consider open-source software applications. For each of them, we create program variants, and we manually check that they are computationally diverse, they form our ground truth. We then run [[DSpot]{}]{}for each program variant. Our experiments show that [[DSpot]{}]{}detects 100% of the computational diverse program variants. In the literature, the technique that is the most similar to test amplification is by Yoo and Harman [@yoo2012], called “test data regeneration” (TDR for short), we use it as baseline. We show that test suites amplified with [[DSpot]{}]{}detect twice more computationally diverse programs than TDR. In particular, we show that the new test input transformations that we propose bring a real added value with respect to TDR, to spot behavioral differences.
To sum up, our contributions are:
- an original set of test cases transformations for the automatic amplification of an object-oriented test suite.
- a validation of the ability of amplified test suites to spot computational diversity in variants of open-source large scale programs.
- a comparative evaluation against the closest related work [@yoo2012]
- original insights about the natural diversity of computation due to randomness and variety of runtime environments.
- a publicly available implementation [^2] and benchmark [^3].
The paper is organized as follows: section \[sec:background\] expands on the background and motivations for this work; section \[sec:approach\] describes the core technical contribution of the paper: the automatic amplification of test suites; section \[sec:eval\] presents our empirical findings about the amplification of real-world test suites and the assessment of diversity among program variants.
Background {#sec:background}
==========
In this paper, we are interested in computational diversity. Computational diversity is one kind of software diversity. Figure \[fig:view-diversity\] presents a high-level view of software diversity. Software diversity can be observed statically either on source or binary code. Computational diversity is the one that happens at runtime. The computational diversity we target in this paper is NVP-Diversity, which relates to N-version programming. It can be loosely defined as computational diversity that is visible at the module interface: different outputs for the same input.
![An High-level View of Software Diversity.[]{data-label="fig:view-diversity"}](SoftwareDiversityDSpot.pdf){width="0.85\columnwidth"}
public int subtract1(int a, int b) {
return a-b;
}
public int subtract2(int a, int b) throws OverFlowException {
BigInteger bigA = BigInteger.valueOf(a);
BigInteger bigB = BigInteger.valueOf(b);
BigInteger result = bigA.subtract(bigB);
if (result.lowerThan(Integer.MIN_VALUE))) {
throw new DoNotFitIn32BitException();
}
// the API requires an 32-bit integer value
return result.intValue();}}
N-version programming {#sec:nversion}
---------------------
In the Encyclopedia of Software Engineering, N-version programming is defined as “*a software structuring technique designed to permit software to be fault-tolerant, ie, able to operate and provide correct outputs despite the presence of faults*” [@knight1990n]. In N-version systems, $N$ variants of the same module, written by different teams, are executed in parallel. The faults are defined as an output of one or more variants that differ from the majority’s output. Let us consider a simple example with 2 programs, $p_1$ and $p_2$, if one observes a difference in the output for an input $x$ – $p_1(x) \neq p_2(x)$ – then a fault is detected.
Let us consider the example of Listing \[lst:wbbb\]. It shows two implementations of subtraction, which have been developed by two different teams: a typical N-version setup. `subtract1` simply uses the subtraction operator. `subtract2` is more complex, it leverages BigInteger objects to handle potential overflows.
The specification given to the two teams states that the expected input domain is $[-2^{16},2^{16}] \times [-2^{16},2^{16}]$. To that extent, both implementations are correct and equivalent. These two implementations are run in parallel in production using a N-version architecture.
If a production input is outside the specified input domain, e.g. `subtract1(2^{32}+1, 2)`, the behavior of both implementations is different and the overflow fault is detected.
NVP-Diversity {#sec:definition-diversity}
-------------
In this paper, we use the term NVP-Diversity to refer to the concept of computational diversity in N-version programming:
**Definition:** Two programs are NVP-diverse if and only if there exists at least one input for which the output is different.
Note that according to this definition, if two programs are equivalent on all inputs, they are not NVP-diverse.
In this work, we consider programs in mainstream object-oriented programming languages (our prototype handles Java software). In OO programs, there is no such thing, as “input” and “outputs”. This requires us to slightly modify our definition of NVP-Diversity.
Following [@harman2013comprehensive], we replace “input” by “stimuli” and “output” by “observation”. A stimuli is a sequence of method calls and their parameters on an object under test. An observation is a sequence of calls to specific methods, to query the state of an object (typically getter methods). The **input space** $\mathcal{I}$ of a class $P$ is the set of all possible stimuli for $P$. The **observation space** $\mathcal{O}$ is the set of all sets of observations.
Now, we can clearly define NVP-diversity for OO-programs.
**Definition:** Two classes are NVP-diverse if and only if there exists two respective instances that produce different observations for the same stimuli.
![Original and amplified points on the input and observation spaces of P.[]{data-label="fig:io-exploration"}](./io-spaces.pdf){width="0.85\columnwidth"}
Graphical Explanation {#sec:unspecified}
---------------------
The notion of NVP-diversity is directly related to activity of software testing as illustrated in figure \[fig:io-exploration\]. The first part of a test case, incl. creation of objects and method calls, constitutes the stimuli, i.e. a point in the program’s input space (black diamonds in the figure). An oracle in the form of an assertion invokes one method and compares the result to an expected value: this constitutes an observation point on the program state that has been reached when running the program with a specific stimuli, the observation points of a test suite are black circles in the right hand side of the figure. To this extent, we say that a test suite specifies a set of relations between points in the input and observation spaces.
Unspecified Input Space
-----------------------
In N-Version programming, by definition, the differences that are observed at runtime happen for unspecified inputs, which we call **the unspecified domain** for short. In this paper, we consider that the points that are not exercised by a test suite form the unspecified domain. They are the orange diamonds in the left-hand side of the figure.
Our Approach to Detect Computational Diversity {#sec:approach}
==============================================
We present [[DSpot]{}]{}, our approach to detect computational diversity. This approach is based on test suite amplification through automated transformations of test case code.
Overview {#sec:overview}
--------
The global flow of [[DSpot]{}]{}is illustrated in figure \[fig:overview\].
**Input:** [[DSpot]{}]{}takes as inputs a set of program variants $P_1\ldots P_n$, which all pass the same test suite $TS$. Conceptually, $P_x$ can be written in any programming language. There no assumption on the correctness or complexity of $P_x$, the only requirements is that they are all specified by the same test suite. In this paper, we consider unit tests, however, the approach can be straightforwardly extended to other kinds of tests such as integration tests.
**Output:** The output of [[DSpot]{}]{}is an answer to the question: are $P_1\ldots P_n$ NVP-diverse?
**Process:** First, [[DSpot]{}]{}amplifies the test suite to explore the unspecified input and observation spaces (as defined in Section \[sec:background\]). As illustrated in figure \[fig:io-exploration\], amplification generates new inputs and observations in the neighbourhood of the original points (new points are orange diamonds and green circles). This cartesian product of the amplified set of input and the complete set of observable points forms the amplified test suite $ATS$.
Also, Figure \[fig:overview\] shows the step “observation point selection”: this step removes the naturally random observations. Indeed, as discussed in more details further in the paper, some observations points produce diverse outputs between different runs of the same test case on the same program. This natural randomness comes from randomness in the computation and from specificities of the execution environment (addresses, file system, etc).
![An overview of [[DSpot]{}]{}: a decision procedure for automatically assessing the presence of NVP-diversity.[]{data-label="fig:overview"}](./diverse-sosies.pdf){width="0.8\columnwidth"}
Once [[DSpot]{}]{}has generated an amplified test suite, it runs it on a pair of program variants to compare their visible behavior, as captured by the observation points. If some points reveal different values on each variant, they are considered as computationally diverse.
Test Suite Transformations
--------------------------
Our approach for amplifying test suites systematically explores the neighbourhood of the input and observation points of the original test suite. In this section we discuss the different transformations we perform for test suite amplification and algorithm \[alg:amplification\] summarizes that procedure.
### Exploring the Input Space
**Literals and statement manipulation:** The first step of amplification consists in transforming all test cases in the test suite with the following test case transformations. Those transformations operate on literals and statements:
Transforming literals:
: given a test case $tc$, we run the following transformations for every literal value: a String value is transformed in three ways: remove, add a random character, and replace a random character by another one; a numerical value $i$ is transformed in four ways: $i+1$, $i-1$, $i\times2$, $i\div2$; a boolean value is replaced by the opposite value. These transformations are performed at line \[alg:line:literals\] of algorithm \[alg:amplification\].
Transforming statement:
: given a test case $tc$, for every statement $s$ in $tc$ we generate two test cases: one test case in which we remove s and another one in which we duplicate s. These transformations are performed at line \[alg:line:statements\] of algorithm \[alg:amplification\].
Given the transformations described above, the transformation process has the following characteristics: (i) each time we transform a variable in the original test suite, we generate a new test case (i.e., we do not ‘stack’ the transformations on a single test case); (ii) the amplification process is exhaustive: given $s$ the number of String values, $n$ the number of numerical values, $b$ the number of booleans and $st$ the number of statements in an original test suite $TS$, [[DSpot]{}]{}produces an amplified test suite ATS of size: $|ATS|=s*3+n*4+b+st*2$.
These transformations, especially the one on statements, can produce test cases that cannot be executed (e.g., removing a call to `add` before a `remove` on a list). In our experiments, this accounted for approximately 10% of the amplified test cases.
**Assertion removal:** The second step of amplification consists of removing all assertions from the test cases (line \[alg:line:statements\] of algorithm \[alg:line:asserts\]). The rationale is that the original assertions are here to verify the correctness, which is not the goal of the generated test cases. Their goal is to assess computational differences. Indeed, assertions that were specified for test case $ts$ in the original test suite are most probably meaningless for a test case that is variant of $ts$. When removing assertions, we are cautious to keep method calls that can be passed as a parameter of an `assert` method. We analyze the code of the whole test suite to find all assertions using the following heuristic: an assertion is a call to a method which name contains either `assert` or `fail` and which is provided by the JUnit framework. If one parameter of the assertion is a method call, we extract it, then we remove the assertion. In the final amplified test suite, we keep the original test case, but also remove its assertion.
Listing \[new\_TCs\] illustrates the generation of two new test cases. The first test method `testEntrySetRemoveChangesMap()` is the original one, slightly simplified for sake of presentation. The second one `testEntrySetRemoveChangesMap_Add`, duplicates the statement `entrySet.remove` and does not contain the assertion anymore. The third test method `testEntrySetRemoveChangesMap_DataMutator` replaces the numerical value 0 by 1.
public void testEntrySetRemove() { // #1
...
for (int i = 0; i < sampleKeys.length; i++) {
entrySet.remove(new DefaultMapEntry<K, V>(sampleKeys[i], sampleValues[i]));
assertFalse(
"Entry should have been removed from the underlying map.",
getMap().containsKey(sampleKeys[i]));
} // end for
... }
public void testEntrySetRemove_Add() { // #2
...
// call duplication
entrySet.remove(new DefaultMapEntry<K, V>(sampleKeys[i], sampleValues[i]));
entrySet.remove(new DefaultMapEntry<K, V>(sampleKeys[i], sampleValues[i]));
getMap().containsKey(sampleKeys[i]);
... }
public void testEntrySetRemove_Data() { // #3
...
// integer increment
// int i = 0 -> int i = 1
for (int i = 1 ; i < (sampleKeys.length) ; i++) {
entrySet.remove(new DefaultMapEntry<K, V>(sampleKeys[i], sampleValues[i]));
getMap().containsKey(sampleKeys[i]);
} // end for
... }
### Adding Observation Points
Our gaol is to observe different observable behaviors between a program and variants of this program. Consequently, we need observation points on the program state. We do this by enhancing all the test cases in $ATS$ with observation points(line \[alg:line:observations\] of algorithm \[alg:line:asserts\]). These points are responsible for collecting pieces of information about the program state during or after the execution of the test case. In this context, an observation point is a call to a public method, which result is logged in an execution trace.
For each object $o$ in the original test case ($o$ can be part of an assertion or a local variable of the test case), we do the following:
- we look for all getter methods in the class of $o$ (i.e., methods which name starts with `get`, that takes no parameter and whose return type is not void, and methods which name starts with `is` and return a boolean value) and call each of them. We also collect the values of all public fields.
- if the `toString` method is redefined for the class of $o$, we call it (we ignore the hashcode that can be returned by `toString`)
- if the original assertion included a method call on $o$, we include this method call as an observation point.
[lp[5.5cm]{}p[3cm]{}XXXX]{} **Project** & **Purpose** & **Class** & **LOC** & **\#tests** & **** & **\#variants**\
commons-codec & Data encoding & Base64 & 255 &72 & 98% & 12\
commons-collections & Collection library & TreeBidiMap& 1202 & 111 & 92% & 133\
commons-io & Input/output helpers & FileUtils& 1195 & 221 & 82%& 44\
commons-lang & General purpose helpers (e.g. String) & StringUtils& 2247 & 233 & 99%& 22\
guava & Collection library & HashBiMap & 525 & 35 & 91%& 3\
gson & Json library & Gson & 554 & 684& 89%& 145\
JGit & Java implementation of GIT& CommitCommand & 433 & 138 & 81% & 113\
\[tab:dataset\]
**Filtering observation points:** This introspective process provides a large number of observation points. Yet, we have noted in our pilot experiments that some of the values that we monitor change from one execution to another. For instance, the identifier of the current thread changes between two executions. In Java, `Thread.currentThread().getId()` is an observation point that always needs to be discarded for instance.
If we keep those naturally varying observation points, [[DSpot]{}]{}would say that two variants are different while the observed difference would be due to randomness. This would be spurious results that are irrelevant for computational diversity assessment. Consequently, we discard certain observation points as follows. We instrument the amplified tests $ATS$ with all observation points. Then, we run $ATS$ 30 times on $P_x$, and repeat these 30 runs on three different machines. All observation points for which at least one value varies between at least two runs are filtered out (line \[alg:line:observations\] of algorithm \[alg:line:filter\]).
To sum up, [[DSpot]{}]{}produces an amplified test suite $ATS$ that contains more test cases than the original one in which we have injected observation points in all test cases.
Detecting and Measuring the Visible Computational Diversity
-----------------------------------------------------------
The final step of [[DSpot]{}]{}, runs the amplified test suite on pairs of program variants. Given $P_1$ and $P_2$, the number of observation points which have a different values on each variant accounts for visible computational diversity. When we compare a set of variants, we use the mean number of differences over each pair of variants.
Implementation
--------------
Our prototype implementation amplifies Java source code [^4]. The test suites are expected to be written using the JUnit testing framework, which is the \#1 testing framework for Java. It uses Spoon [@spoon] to manipulate the source code in order to create the amplified test cases. [[DSpot]{}]{}is able to amplify a test suite within minutes.
The main challenges for the implementation of [[DSpot]{}]{}were as follows: handle the many different situations that occur in real-world large test suites (use different versions of JUnit, modularize the code of the test suite itself, implement new types of assertions, etc.); handle large traces for comparison of computation (as we will see in the next section, we collect hundreds of thousands observations on each variant); spot the natural randomness in test case execution to prevent false positives in the assessment of computational diversity.
Evaluation {#sec:eval}
==========
To evaluate whether [[DSpot]{}]{}is capable of detecting computational diversity, we set up a novel empirical protocol and apply it on large-scale Java programs. Our guiding research question is: **Is [[DSpot]{}]{}capable of identifying realistic large scale programs that are computationally diverse?**
Protocol {#sec:protocol}
--------
First, we take large open-source Java programs that are equipped with good test suites. Second, we forge variants of those programs using a technique from our previous work [@baudry14]. We call the variants sosie programs [^5].
\[def:sosie\] **Sosie** (noun). Given a program $P$, a test suite $TS$ for $P$ and a program transformation $T$, a variant $P'$=$T(P)$ is a sosie of $P$ if the two following conditions hold 1) there is at least one test case in $TS$ that executes the part of $P$ that is modified by $T$ 2) all test cases in $TS$ pass on $P'$.
Given an initial program, we synthesize sosies with source code transformations that are based on the modification of the abstract syntax tree (AST). As previous work [@legoues12; @Schulte13], we consider three families of transformation that manipulate statement nodes of the AST: 1) remove a node in the AST (Delete); 2) adds a node just after another one (Add); 3) replaces a node by another one, e.g. a statement node is replaced by another statement (Replace). For “Add” and “Replace”, the **transplantation point** refers to where a statement is inserted, the **transplant statement** refers to the statement that is copied and inserted and both transplantation and transplant points are in the same AST (we do not synthesize new code, nor take code from other programs). We consider transplant statements that manipulate variables of the same type as the transplantation point, and we bind the names of variables in the transplant to names that are in the namespace of the transplantation point. We call these transformations *Steroid* transformations, and more details are available in our previous work [@baudry14].
Once we have generated sosie programs, we manually select a set of sosies that indeed expose some computational diversity. Third, we amplify the original test suites using our approach and also using a baseline technique by Yoo and Harman [@yoo2012] presented in \[sec:baseline\]. Finally, we run both amplified test suites and measure the proportion of variants (sosies) that are detected as computationally different. We also collect additional metrics to further qualify the effectiveness of [[DSpot]{}]{}.
Dataset
-------
We build a dataset of subject programs for performing our experiments. The inclusion criteria are the following: 1) the subject program must be real-world software; 2) the subject program must be written in Java; 3) the subject program’s test suite must use the JUnit testing framework ; 4) the subject program must have a good test suite (a statement coverage higher than 80%).
This results in Apache Commons Math, Apache Commons Lang, Apache Commons Collections, Apache Commons Codec and Google GSON and Guava. The dominance of Apache projects is due to the fact that they are among the very rare organizations with a very strong development discipline.
In addition, we aim at running the whole experiments in less than one day (24 hours). Consequently we take a single class for each of those projects as well as all the test cases that exercise it at least once.
Table \[tab:dataset\] provides the descriptive statistics of our dataset. It gives the subject program identifier, its purpose, the class we consider, the class’ number of lines of code (LOC), the number of tests that execute at least once one method of the class under consideration, the statement coverage and the total number of program variants we consider (excluding the original program). We see that this benchmark covers different domains, such as data encoding and collections, and is only composed of well-tested classes. In total, there are between 12 and 145 computationally diverse variants of each program to be detected. This variation comes from the relative difficulty of manually forging computationally diverse variants depending on the project.
[l|XX|XXX|X|p[0.7cm]{}]{} & & & &\
& \#TC & \#assert or obs.& \#TC exec. & \#assert or obs. exec. & \#disc. obs. & \# branch cov. & \# path cov.\
codec & 72 & 509 & 72 & 3528 & & 124 & 1245\
codec-DSpot & 672 ($\times$9) & 10597 ($\times$20) & 672 & 16920 & 12 & 126 & 12461\
collections & 111 & 433 & 768 & 7035 & & 223 & 376\
collections-DSpot & 1291 ($\times$12) & 14772 ($\times$34) & 9202 & 973096& 0 & 224 & 465\
io & 221 & 1330 & 262 & 1346 & & 366 & 246\
io-DSpot & 2518 ($\times$11)& 20408 ($\times$15) & 2661 & 209911& 54313 & 373 & 287\
lang & 233 & 2206 & 233 & 2266 & & 1014 & 797\
lang-DSpot & 988 ($\times$4) & 12854 ($\times$6) & 12854 & 57856 & 18 & 1015 & 901\
guava & 35 & 84 & 14110 & 20190 & & 60 & 77\
guava-DSpot & 625 ($\times$18) & 6834 ($\times$81) & 624656& 9464 & 0 & 60 & 77\
gson & 684 & 1125 & 671 & 1127 & & 106 & 84\
gson-DSpot & 4992 ($\times$7) & 26869 ($\times$24) & 4772 & 167150& 144 & 108 & 137\
JGit &138 & 176 & 138 & 185 & & 75 & 1284\
JGit-DSpot & 2152 ($\times$16) & 90828 ($\times$516)& 2089 & 92856 & 13377 & 75 & 1735\
\[tab:test-amplification\]
Baseline {#sec:baseline}
--------
In the area of test suite amplification, the work by Yoo and Harman [@yoo2012] is the most closely related to our approach. Their technique is designed for augmenting input space coverage but can be directly applied to detecting computational diversity. Their algorithm, called test data regeneration – TDR for short – is based on four transformations on numerical values in test cases: data shifting ($\lambda x.x+1$ and $\lambda x.x-1$ ) and data scaling (multiply or divide the value by 2) and a hill-climbing algorithm based on the number of fitness function evaluations. They consider that a test case calls a single function, their implementation deals only with numerical functions and they consider the numerical output of that function as the only observation point. In our experiment, we reimplemented the transformations on numerical values since the tool used by Yoo is not available. We remove the hill-climbing part since it is not relevant in our case. Analytically, the key differences between [[DSpot]{}]{}and TDR are: TDR stacks mutliple transformations together; [[DSpot]{}]{}has more new transformation operators on test cases: [[DSpot]{}]{}considers a richer observation space based on arbitrary data types and sequences of method calls.
Research Questions
------------------
We first examine the results of our test amplification procedure
**RQ1a: what is the number of generated test cases?** We want to know whether our transformation operators on test cases enable us to create many different new test cases, i.e. new points in the input space. Since [[DSpot]{}]{}systematically explores all neighbors according to the transformation operators, we measure the number of generated test cases to answer this basic research question.
**RQ1b: what is the number of additional observation points?** In addition to creating new input points, [[DSpot]{}]{}creates new observation points. We want to know the order of magnitude of the number of those new observation points. To have a clear explanation, we start by performing only observation point amplification (without input point amplification) and count the total number of observations. We compare this number with the initial number of assertions, which exactly corresponds to the original observation points.
Then, we evaluate the ability of the amplified test suite to assess computational diversity.
**RQ2a: does [[DSpot]{}]{}identify more computationally diverse programs than TDR?** Now, we want to compare our technique with the related work. We count the number of variants that are identified as computationally different using [[DSpot]{}]{}and TDR. The one with with the highest value is better.
**RQ2b: does the efficiency of [[DSpot]{}]{}come from the new inputs or the new observations?** [[DSpot]{}]{}stacks two techniques: the amplification of the input space and the amplification of the observation space. To study their impact in isolation, we count the number of computationally diverse program variants that are detected by the original input points equipped with new observation points and by the amplified set of input points with the original observations.
The last research questions digs deeper in the analysis of amplified test cases and computationally diverse variants.
**RQ3a: What is the number of natural randomness in computation?** Recall that [[DSpot]{}]{}removes some observation points that naturally varies even on the same program. This phenomenon is due to the natural randomness of computation. To answer this question quantitatively, we count the number of discarded observation points, to answer it quantitatively, we discuss one case study.
**RQ3b: what is the richness of computational diversity?** Now, we really understand the reasons behind the computational diversity we observe. We take a random sample of three pairs of computationally diverse program variants and analyze them. We discuss our findings.
Empirical Results
-----------------
We now discuss the empirical results obtained on applying [[DSpot]{}]{}on our dataset.
[lXXXXXX]{} & \#variants detected by [[DSpot]{}]{}& \#variants detected by TDR & input space effect & observation space effect & mean \# of divergences\
commons-codec & 12/12 & 10/12 &12/12 & 10/12 & 21.9\
commons-collections & 133/133 & 133/133 & 133/133 & 133/133 & 5207.9\
commons-io & 44/44 & 18/44 & 42/44 & 18/44 & 405.5\
commons-lang & 22/22 & 0/22 & 10/22 & 0/22 & 22.9\
guava & 3/3 & 0/3 & 0/3 & 3/3 & 2\
gson &145/145 &0/145 & 134/145 & 0/145 & 801.5\
jgit &113/113 &0/113 & 113/113 & 0/113 & 1565.4\
\[tab:diversity-measures\]
### \# of Generated Test Cases
Table \[tab:test-amplification\] presents the key statistics of the amplification process. The lines of these table go by pair: one that provides data for one subject program and the following one that provides the same data gathered with the test suite amplified by [[DSpot]{}]{}. Columns from 2 to 5 are organized in two groups: the first group gives a static view on the test suites (e.g. how many test methods are declared); the second group draws a dynamic picture of the test suites under study (e.g. how many assertions are executed).
Indeed, in real, large-scale programs, test cases are modular. Some test cases are used multiple times because they are called by other test cases. For instance, a test case that specifies a contract on a collection is called when testing all implementations of collections (ArrayList, LinkedList, etc.). We call them *generic tests*.
Let’s first concentrate on the static values. Column 2 gives the number of test cases in the original and amplified test suites, while column 3 gives the number of assertions in the original test suites and the number of observations in the amplified.
One can see that our amplification process is massive. We create between 4x and 12x more test cases than the original test suites. For instance, the test suite considered for commons.codec contains 72 test cases. [[DSpot]{}]{}produces an amplified test suite that contains 672 test methods: 9x more than the original test suite. The original test suite observes the state of the program with 509 assertions, while [[DSpot]{}]{}employs 10597 observations points to detect computational differences.
Let us now consider the dynamic part of the table. Column 4 gives the number of tests executed (\#TC exec.) and column 5 the number of assertions executed or the number of observation points executed. Column 6 gives the number of the discarded observation points because of natural variations (discussed in more details in section \[sec:natural:randomness\]). As we can see, the number of generated tests (\#ATC exec.) is impacted by amplification. For instance, for commons.collection there are 1291 tests in the amplified test suite, but altogether, 9202 test cases are executed. The reason is that we synthesize new test cases that use other generic test methods. Consequently, this increases the number of executed generic test methods, which is included in our count.
Our test case transformations yield a rich exploration of the input space. Columns 7 to 11 of Table \[tab:test-amplification\] provide deeper insigths about the synthesized test cases. Colum 7 gives the branch coverage of the original test suites and the amplified ones (lines with \*-DSPOT identifiers). While original test suites have a very high branch coverage rate, yet, DSpot is still able to generate new teststhat cover a few previously uncovered branches. For instance, the amplified test suite for commons-io/FileUtils reaches 7 branches that were not executed by the original test suite. Meanwhile, the original test suite for guava/HashBiMap already covers 90% of the branches and DSpot did not generate test cases that cover new branches.
The richness of the amplified test suite is also revealed in the last column of the table (path coverage): it provides the cumulative number of different paths executed by the test suite in all methods under test. The amplified test suites cover much more paths than the original ones, which means that they trigger a much wider set of executions of the class under test than the original test suites. For instance, for Guava, the total number of different paths covered in the methods under test increases from 84 to 137. This means that, while the amplified test suite does not cover many new branches, it executes the parts that were already covered in many novel ways, increasing the diversity of executions that are tested. There is one extreme case in the `encode` method of commons-codec[^6]: the original test suite covers 780 different paths in this method, while the amplified test suite covers 11356 different paths. This phenomenon is due to the complex control flow of the method and to the fact that its behavior directly depends on the value of an array of bytes that takes many new values in the amplified test suite.
The amplification process is massive and produces rich new input points: the number of declared and executed test cases and the diversity of executions from test cases increase.
### \# of Generated Observation Points
Now we focus on the observation points. The fourth column of Table \[tab:test-amplification\] gives the number of assertions in original test suite. This corresponds to the number of locations where the tester specifies expected values about the state of the program execution. The fifth column, gives the number of observation points in the amplified test suite. We do not call them assertions since they do not contain an expected value, i.e., there is no oracle. Recall that we use those observation points to compare the behavior of two program variants in order to assess the computational diversity.
As we can see, we observe the program state on many more observation points than the original assertions. As discussed in Section \[sec:definition-diversity\], those observations points use the API of the program under consideration, hence allow to reveal visible and exploitable computational diversity. However, this number also encompasses the observation points on the new generated test cases.
If we look at the dynamic perspective (second part of Table \[tab:test-amplification\]), one observes the same phenomenon as for test cases and assertions, there are many more points actually observed during test execution than statically declared ones. The reasons are identical, many observations points are in generic test methods that are executed several times, or are within loops in test code.
These results validate our initial intuition that a test suite only covers a small portion of the observation space. It is possible to observe the program state from many other observation points.
### Effectiveness
We want to assess whether our method is effective for identifying computationally diverse program variants. As golden truth, we have the forged variants for which we know that they are NVP-diverse (see Section \[sec:protocol\]), their numbers are given in the descriptive Table \[tab:dataset\]. The benchmark is publicly available at <http://diversify-project.eu/data/>.
We run [[DSpot]{}]{}and TDR to see whether those two techniques are able to detect the computationally diverse programs. Table \[tab:diversity-measures\] gives the results of this evaluation. The first column contains the name of the subject program. The second column gives the number of variants detected by [[DSpot]{}]{}. The third column gives the number of variants detected by TDR. The last three columns explore more in depth whether computational diversity is reveales by new input points or new observation points or both, we will come back to them later.
As we can see, [[DSpot]{}]{}is capable of detecting all computationally diverse variants of our benchmark. On the contrary, the baseline technique, TDR, is always worse. Either it detects only a fraction of them (e.g. 10/12 for commons.codec) or even not at all. The reason is that TDR, as originally proposed by Yoo and Harman, focuses on simple programs with shallow input spaces (one single method with integer arguments). On the contrary, [[DSpot]{}]{}is designed to handle rich input spaces, incl. constructor calls, method invocations and strings. This has a direct impact on the effectiveness of detecting computational diversity in program variants.
Our technique is based on two insights: the amplification of the input space and the amplification of the observation space. We now want to understand the impact of each of them. To do so, we disable one or the other kind of amplification and measure the number of detected variants. The result of this experiment is given in the last two columns of Table \[tab:diversity-measures\]. Column “input space effect” gives the number of variants that are detected only by the exploration of the input space (i.e. by observing the program state only with the observation method used in the original assertions). Column “observation space effect” gives the number of variants that are detected only by the exploration of the observation space (i.e. by observing the result of method calls on the objects involved in the test). For instance, for commons-codec, all variants (12/12) are detected by exploring the input space, and 10/12 are detected by exploring the observation space. This means that 10 of them are detected are detected either by one exploration or the other one. On the contrary for guava, only the exploration of the observation space enables [[DSpot]{}]{}to detect the three computationally diverse variants of our benchmark.
By comparing columns “input space effect” and “observation space effect” one sees that our two explorations are not mutually exclusive and are complementary. Some variants are detected by both kinds of exploration (as in the case of commons-codec). For some subjects, only the exploration of the input space is effective (e.g. commons-lang), while for others (guava), this is the opposite. Globally, the exploration of the input space is more efficient, most variants are detected this way.
Let us now consider the last column of Table \[tab:diversity-measures\]. It gives the mean number of observation points for which we observe a difference between the original program and the variant to be detected. For instance, among the 12 variants for commons.codec, there is on average 21.9 observation points for which there is a difference. Those numbers are high, showing that the observation points are not independent. Many of the methods we call to observe the program state inspect a different facet of the same state. For instance, in a list, the methods `isEmpty()` and `size` are semantically correlated.
The systematic exploration of the input and the observation spaces is effective at detecting behavioral diversity between program variants.
### Natural Randomness of Computation {#sec:natural:randomness}
When experimenting with [[DSpot]{}]{}on real programs, we noticed that some observation points naturally vary even when running the same test case several times on the same program. For instance, a hashcode that takes into account a random salt can be different between two runs of the same test case. We call this effect, the “natural randomness” of test case execution.
We distinguish two kinds of natural variations in the execution of test suites. First, some observation points vary *over time* when the test case is executed several times on the same environment (same machine, OS, etc.). This is the case for the hashcode example. Second, some observation points vary *depending on the execution environment*. For instance, if one adds an observation point on a file name, the path name convention is different on Unix and Windows systems. If method `getAbsolutePath` is an observation point, it may return `/tmp/foo.txt` on Unix and `C:\tmp\foo.txt` on Windows. While this first example is pure randomness, the second only refers to variations in the runtime environment.
void testCanonicalEmptyCollectionExists() {
if (((supportsEmptyCollections()) && (isTestSerialization())) && (!(skipSerializedCanonicalTests()))) {
Object object = makeObject();
if (object instanceof Serializable) {
String name = getCanonicalEmptyCollectionName(object);
File f = new java.io.File(name);
// observation on f
Logger.logAssertArgument(f.getCanonicalPath());
Logger.logAssertArgument(f.getAbsolutePath());
.....
}}
}
Interestingly, this natural randomness is not problematic in the case of the original test suites, because it remains below the level of observation of the oracles (the test suite assertions in JUnit test suites). However, in our case, if one keeps an observation point that is impacted by some natural randomness, this would produce a **false positive** for computational diversity detection. Hence, as explained in Section \[sec:approach\], one phase of [[DSpot]{}]{}consists in detecting the natural randomness first and discarding the impacting observation points.
Our experimental protocol enables us to quantify the number of discarded observation points. The 6th column of Table \[tab:test-amplification\] gives this number. For instance, for commons-codec, [[DSpot]{}]{}detects 12 observation points that naturally vary. This column shows two interesting facts. First, there is a large variation in the number of discarded observation points, it goes up to 54313 for commons-io. This case, together with JGIT (the last line), is due to the heavy dependency of the library on the underlying file system (commons-io is about IO – hence file systems –operations, JGIT is about manipulating GIT versioning repositories that are also stored on the local file system).
Second, there are two subject programs (commons-collections and guava) for which we discard no points at all. In those programs, [[DSpot]{}]{}does not detect a single point that naturally varies by running 100 times the test suite on three different operating systems. The reasons is that the API of those subject programs does not allow to inspect the internals of the program state up to the naturally varying parts (e.g. the memory addresses). We consider this good as this, it shows that the encapsulation is good: more than providing an intuitive API, more than providing a protection against future changes, *it also completely encapsulates the natural randomness of the computation*.
Let us now consider a case study. Listing \[monitor\_ex\] shows an example of an amplified test with observation points for Apache Commons Collection. There are 12 observation methods that can be called on the object `f` instance of `File` (11 getter methods and `toString`). The figure shows two getter methods that return different values from one run to another (there are 5 getter methods with that kind of behavior for a `File` object). We ignore these observation points when comparing the original program with the variants.
The systematic exploration of the observable output space provides new insights about the degree of encapsulation of a class. When a class gives public access to variables that naturally vary, there is a risk that when used in oracles, they result in flaky test cases.
### Nature of Computational Diversity {#sec:nature-computational-diversity}
Now we want to understand more in depth the nature of the NVP-diversity we are observing. Let us discuss three case studies.
//original program
void writeStringToFile(File file, String data, Charset encoding, boolean append) throws IOException {
OutputStream out = null;
out = openOutputStream(file, append);
IOUtils.write(data, out, encoding);
out.close(); }
// variant
void writeStringToFile(File file, String data, Charset encoding, boolean append) throws IOException {
OutputStream out = null;
out = new FileOutputStream(file, append);
IOUtils.write(data, out, encoding);
out.close(); }
void testCopyDirectoryPreserveDates() {
try {
File sourceFile = new File(sourceDirectory, "hello/txt");
FileUtils.writeStringToFile(sourceFile, "HELLO WORLD", "UTF8");
catch (Exception e) {
DSpot.observe(e.getMessage());
}
}
Listing \[lst:roci3\] shows two variants of the `writeStringToFile()` method of Apache Commons IO. The original program calls `openOutputStream`, which checks different things about the file name, while the variant directly calls the constructor of `FileOutputStream`. These two variants behave differently outside the specified domain: in case `writeStringToFile()` is called with an invalid file name, the original program handles it, while the variant throws a `FileNotFoundException`. Our test transformation operator on String values produces such a file name, as shown in the test case of listing \[lst:roci3test\]: a “.” is changed into a star “/”. This made the file name an invalid one. Running this test on the variant results in a `FileNotFoundException`.
// Original program
void toJson(Object src, Type typeOfSrc, JsonWriter writer){
writer.setSerializeNulls(oldSerializeNulls); } }
//variant
void toJson(Object src, Type typeOfSrc, JsonWriter writer){
writer.setIndent(" ")
} }
public void testWriteMixedStreamed_remove534() throws IOException {
%* {\color{gray} \emph{...} } *)
gson.toJson(RED_MIATA, Car.class, jsonWriter);
jsonWriter.endArray();
Logger.logAssertArgument(com.google.gson.MixedStreamTest.CARS_JSON);
Logger.logAssertArgument(stringWriter.toString());
%* {\color{gray} \emph{...} } *)
}
Let us now consider listing \[lst:roci2\], which shows two variants of the `toJson()` method from the Google Gson library. The last statement of the original method is replaced by another one: instead of setting the serialization format of the `writer` it set the indent format. Each variant creates a JSon with slightly different formats, and none of these formatting decisions are part of the specified domain (and actually, specifying the exact formatting of the JSon String could be considered as over-specification). The diversity among variants is detected by the test cases displayed in figure \[lst:roci2test\], which adds an observation point (a call to `toString()`) on instances of `StringWriter`, which are modified by `toJson()`.
The next case study is in listing \[lst:roci4\]: two variants of the method `decode()` in the `Base64` class of the Apache Commons Codec library. The original program has a `switch-case` statement in which case 1 execute a break. An original comment by the programmers indicates that it is probably impossible. The test case in listing \[lst:ampli\_test4\] amplifies one of the original test case with a mutation on the String value in the `encodedInt3` variable (the original String has an additional ‘$\backslash$’ character, removed by the “remove character” transformation). The amplification on the observation points adds multiple observations points. The single observation point shown in the listing is the one that detects computational diversity: it calls the static `decodeInteger()` method which returns 1 on the original program and 0 on the variant. In addition to validating our approach, this example anecdotally answers the question of the programmer, case 1 is possible, it can be triggered from the API.
These three case examples are meant to give the reader a better idea of how DSpot was able to detect the variants. We discuss how augmented test cases reveal this diversity (both with amplified inputs and observation points). We illustrate three categories of code variations that maintain the expected functionality as specified in the test suite, but still induce diversity (different checks on input, different formatting, different handling of special cases).
The diversity that we observe originates from areas of the code that are characterized by their flexibility (caching, checking, formatting, etc.). These areas are very close to the concept of *forgiving region* proposed by Martin Rinard [@Rinard12].
Threats to Validity
-------------------
// Original program
void decode(final byte[] in, int inPos, final int inAvail, final Context context) {
switch (context.modulus) {
case 0 : // impossible, as excluded above
case 1 : // 6 bits - ignore entirely
// not currently tested; perhaps it is impossible?
break;
}
// variant
void decode(final byte[] in, int inPos, final int inAvail, final Context context) {
switch (context.modulus) {
case 0 : // impossible, as excluded above
case 1 :
}
@Test
void testCodeInteger3_literalMutation222() {
String encodedInt3 =
"FKIhdgaG5LGKiEtF1vHy4f3y700zaD6QwDS3IrNVGzNp2"
+ "rY+1LFWTK6D44AyiC1n8uWz1itkYMZF0aKDK0Yjg==";
Logger.logAssertArgument(Base64.decodeInteger(encodedInt3.getBytes(Charsets.UTF_8)));
}}
[[DSpot]{}]{}is able to effectively detect NVP-diversity using test suite amplification. Our experimental results are subject to the following threats.
First, this experiment is highly computational, a bug in our evaluation code may invalidate our findings. However, since we have manually checked a sample of cases (the case studies of Section \[sec:natural:randomness\] and Section \[sec:nature-computational-diversity\]) we have a high confidence in our results. Our implementation is publicly available [^7].
Second, we have forged the computationally diverse program variants. Eventually, as shown on Table \[tab:diversity-measures\], our technique [[DSpot]{}]{}is able to detect them all. The reason is that we had a bias towards our technique when forging those variants. This is true for all self-made evaluations. This threat on the results of the comparative evaluation against TDR is mitigated by the analytical comparison of the two approaches. Both the input space and the output space of TDR (respectively an integer tuple and a returned value) are simpler and less powerful than our amplification technique.
Third, our experiments consider one programming language (Java) and different application domains. To further assess the external validity of our results, new experiments are required on different technologies and more application domains.
Related work {#sec:related}
============
The work presented is related to two main areas: the identification of similarities or diversity in source code and the automatic augmentation of test suites.
**Computational diversity** The recent work by Carzaniga et al. [@Carzaniga15] has a similar intent as ours: automatically identifying dissimilarities in the execution of code fragments that are functionally similar. They use random test cases generated by Evosuite to get execution traces and log the internals of the execution (executed code and the read/write operations on data). The main difference with our work is that they assess computational diversity and with random testing instead of test amplification.
Koopman and DeVale [@koopman1999] aim at quantifying the diversity among a set of implementations of the POSIX operating system, with respect to their responses to exceptional conditions. Diversity quantification in this context is used to detect which versions of POSIX provide the most different failure profiles and should thus be assembled to ensure fault tolerance. Their approach relies on Ballista to generate millions of input data and the outputs are analyzed to quantify the difference. This is an example of diversity assessment with intensive fuzz testing and observation points on crashing states.
Many other works look for semantic equivalence or diversity through static or dynamic analysis. Gabel and Su [@gabel10] investigate the level of granularity at which diversity emerges in source code. Their main finding is that, for sequences up to 40 tokens, there is a lot of redundancy. Beyond this (of course fuzzy) threshold, the diversity and uniqueness of source code appears. Higo and Kusumoto [@higo14] investigate the interplay between structural similarity, vocabulary similarity and method name similarity, to assess functional similarity between methods in Java programs. They show that many contextual factors influence the ability of these similarity measures to spot functional similarity (e.g., the number of methods that share the same name, or the fact that two methods with similar structure are in the same class or not). Jiang and Su [@jiang09] extract code fragments of a given length and randomly generate input data for these snippets. Then, they identify the snippets that produce the same output values (which are considered functionally equivalent, w.r.t the set of random test inputs). They show that this method identifies redundancies that static clone detection does not find. Kawaguchi and colleagues [@kawaguchi2010conditional] focus on the introduction of changes that break the interface behavior. They also use a notion of partial equivalence, where “two versions of a program need only be semantically equivalent under a subset of all inputs”. Gao and colleagues [@gao2008] propose a graph-based analysis to identify semantic differences in binary code. This work is based on the extraction of call graphs and control flow graphs of both variants and on comparisons between these graphs in order to spot the semantic variations. Person and colleagues [@person2008] developed differential symbolic execution, which can be used to detect and characterize behavioral differences between program versions.
**Test suite amplification** In the area of test suite amplification, the work by Yoo and Harman [@yoo2012] is the most closely related to our approach, and we used as the baseline for computational diversity assessment. They amplify test suites only with transformations on integer values, while we also transform boolean and String literals, as well as statements test cases. Yoo and Harman also have two additional parameters for test case transformation: the interaction level that determines the number of simultaneous transformation on the same test case, and the search radius that bounds their search process when trying to improve the effectiveness of augmented test suites. Their original intent is to increase the input space coverage to improve test effectiveness. They do not handle the oracle problem in that work.
Xie [@xie06] augments test suites for Java program with new test cases that are automatically generated and he automatically generates assertions for these new test cases, which can check for regression errors. Harder et al. [@Harder03] propose to retrieve *operational abstractions*, i.e., invariant properties that hold for a set of test cases. These abstractions are then used to compute operational differences, which detects diversity among a set of test cases (and not among a set of implementations as in our case). While the authors mention that operational differencing can be used to augment a test suite, the generation of new test cases is out of this work’s scope. Zhang and Elbaum [@zhang2012] focus on test cases that verify error handling code. Instead of directly amplifying the test cases as we propose, they transform the program under test: they instrument the target program by mocking the external resource that can throw exceptions, which allow them to amplify the space of exceptional behaviors exposed to the test cases. Pezze et al. [@pezze2013] use the information provided in unit test cases about object creation and initialization to build composite test cases that focus on interactions between classes. Their main result is that the new test cases find faults that could not be revealed by the unit test cases that provided the basic material for the synthesis of composite test cases. Xu et al. [@xu2011hybrid] refer to “test suite augmentation” as the following process: in case a program P evolves into P’, identify the parts of P’ that need new test cases and generate these tests. They combine concolic and search-based test generation to automate this process. This hybrid approach is more effective than each technique separately, but with increased costs. Dallmeier et al. [@dallmeier2010] automatically amplify test suites by adding and removing method calls in JUnit test cases. Their objective is to produce test cases that cover a wider set of execution states than the original test suite in order to improve the quality of models reverse engineered from the code.
Conclusion {#sec:conclusion}
==========
In this paper, we have presented [[DSpot]{}]{}, a novel technique for detecting one kind of computational diversity between a pair of programs. This technique is based on test suite amplification: the automatic transformation of the original test suite. [[DSpot]{}]{}uses two kinds of transformations, for respectively exploring new points in the program’s input space and exploring new observation points on the execution state. after execution with the given input points.
Our evaluation on large open-source projects shows that test suites amplified by [[DSpot]{}]{}are capable of assessing computational diversity and that our amplification strategy is better than the closest related work, a technique called TDR by Yoo and Harman [@yoo2012]. We have also presented a deep qualitative analysis of our empirical findings. Behind the performance of [[DSpot]{}]{}, our results shed an original light on the specified and unspecified parts of real-world test suites and the natural randomness of computation.
This opens avenues for future work. There is a relation between the natural randomness of computation and the so-called flaky tests (those tests that occasionally fail). To use, the assertions of the flaky tests are at the border of the natural undeterministic parts of the execution: sometimes they hit it, sometimes they don’t. With such a view, we imagine an approach that characterizes this limit and proposes an automatic refactoring of the flaky tests so that they get farther from the limit of the natural randomness and enter again into the good, old and reassuring world of determinism.
Acknowledgements
================
This work is partially supported by the EU FP7-ICT-2011-9 No. 600654 DIVERSIFY project.
[^1]: [[DSpot]{}]{}stands for **d**iversity **spot**ter
[^2]: <http://diversify-project.github.io/test-suite-amplification.html>
[^3]: [http://diversify-project. eu/data/](http://diversify-project. eu/data/)
[^4]: the prototype is available here: <http://diversify-project.github.io/test-suite-amplification.html>
[^5]: The word sosie is a French word that literally means “look alike”
[^6]: line 331 in the Base64 class <https://github.com/apache/commons-codec/blob/ca8968be63712c1dcce006a6d6ee9ddcef0e0a51/src/main/java/org/apache/commons/codec/binary/Base64.java>
[^7]: <http://diversify-project.github.io/test-suite-amplification.html>
|
---
abstract: 'We introduce a new quasi-isometry invariant of metric spaces called the hyperbolic dimension, ${\operatorname{hypdim}}$, which is a version of the Gromov’s asymptotic dimension, ${\operatorname{asdim}}$. One always has ${\operatorname{hypdim}}\le{\operatorname{asdim}}$, however, unlike the asymptotic dimension, ${\operatorname{hypdim}}{\mathbb{R}}^n=0$ for every Euclidean space ${\mathbb{R}}^n$ (while ${\operatorname{asdim}}{\mathbb{R}}^n=n$). This invariant possesses usual properties of dimension like monotonicity and product theorems. Our main result says that the hyperbolic dimension of any Gromov hyperbolic space $X$ (with mild restrictions) is at least the topological dimension of the boundary at infinity plus 1, ${\operatorname{hypdim}}X\ge\dim{{\partial}_{\infty}}X+1$. As an application we obtain that there is no quasi-isometric embedding of the real hyperbolic space ${\operatorname{H}}^n$ into the metric product of $n-1$ metric trees stabilized by any Euclidean factor, $T_1\times\dots\times T_{n-1}\times{\mathbb{R}}^m$, $m\ge 0$.'
author:
- 'Sergei Buyalo[^1] & Viktor Schroeder[^2]'
title: Hyperbolic dimension of metric spaces
---
Introduction
============
We introduce a new quasi-isometry invariant of metric spaces called the hyperbolic dimension, ${\operatorname{hypdim}}$, which is a version of the Gromov’s asymptotic dimension, ${\operatorname{asdim}}$. One always has ${\operatorname{hypdim}}\le{\operatorname{asdim}}$, however, unlike the asymptotic dimension, ${\operatorname{hypdim}}{\mathbb{R}}^n=0$ for every Euclidean space ${\mathbb{R}}^n$ (while ${\operatorname{asdim}}{\mathbb{R}}^n=n$). This invariant possesses usual properties of dimension like monotonicity and product theorems. To formulate our main result, we recall that a metric space $X$ has [*bounded growth at some scale*]{}, if for some constants $r$, $R$ with $R>r>0$, and $N\in{\mathbb{N}}$ every ball of radius $R$ in $X$ can be covered by $N$ balls of radius $r$, see [@BoS].
\[Thm:main\] Let $X$ be a geodesic Gromov hyperbolic space, which has bounded growth at some scale and whose boundary at infinity ${{\partial}_{\infty}}X$ is infinite. Then $${\operatorname{hypdim}}X\ge\dim{{\partial}_{\infty}}X+1.$$
As an application we obtain.
\[Cor:nonemb\] For every $n\ge 2$ there is no quasi-isometric embedding ${\operatorname{H}}^n\to T_1\times\dots\times T_{n-1}\times{\mathbb{R}}^m$ of the real $n$-dimensional hyperbolic space ${\operatorname{H}}^n$ into the product of $n-1$ trees stabilized by any Euclidean factor ${\mathbb{R}}^m$, $m\ge 0$.
For $n=2$ this result has been proved in [@BS2] by a different method.
In [@BS2] we have constructed for every $n\ge 2$ a quasi-isometric embedding of ${\operatorname{H}}^n$ into the $n$-fold product of homogeneous trees whose vertices have an infinite (countable) degree and whose edges have length 1. By Corollary \[Cor:nonemb\], that embedding is optimal with respect to the number of tree-factors even if we allow the stabilization by Euclidean factors. Furthermore, it follows that ${\operatorname{hypdim}}{\operatorname{H}}^n=n$ for every $n\ge 2$.
In [@JS] it was shown that for every $n$ there exists a right angled Gromov hyperbolic Coxeter group ${\Gamma}_n$ with virtual cohomological dimension and coloring number equal to $n$. In [@DS], a bilipschitz embedding $f_n:X_n\to T\times\dots\times T$ of the Cayley graph $X_n$ of ${\Gamma}_n$ into the $n$-fold product of an arbitrary exponentially branching tree $T$ has been constructed. The Cayley graph of any Gromov hyperbolic group satisfies the conditions of Theorem \[Thm:main\], and $\dim{{\partial}_{\infty}}X_n=\dim{{\partial}_{\infty}}{\Gamma}_n=n-1$ by a result from [@BM](every finitely generated Coxeter group is virtually torsion free). Thus Theorem \[Thm:main\] implies (see Theorem \[Thm:nonemb\] below) that the embedding $f_n$ is optimal w.r.t. the number of tree-factors even if we allow the stabilization by Euclidean factors.
[**Hyperbolic dimension versus subexponential corank**]{}
In our earlier paper [@BS1] we introduced another quasi-isometry invariant of metric spaces called subexponential corank. This invariant gives an upper bound for the topological dimension of a Gromov hyperbolic space which can be quasi-isometrically embedded into a given metric space $X$, ${\operatorname{rank}}_h(X)\le{\operatorname{corank}}(X)$. Thus ${\operatorname{corank}}$ is a useful tool for finding obstacles to such embeddings, and it works perfectly well in many cases, see [@BS1] for details. However, not in all, e.g., for quasi-isometric embeddings ${\operatorname{H}}^n\to T_1\times\dots\times T_k\times{\mathbb{R}}^m$ it gives only $k\ge n-2$, while ${\operatorname{hypdim}}$ gives optimal $k\ge n-1$ by Corollary \[Cor:nonemb\]. This drawback of ${\operatorname{corank}}$ is closely related to that ${\operatorname{corank}}(T)=1>0=\dim{{\partial}_{\infty}}T$ for every metric tree $T$, while ${\operatorname{corank}}(X)=\dim{{\partial}_{\infty}}X$ for every CAT($-1$) Hadamard manifold $X$.
On the other hand, ${\operatorname{hypdim}}$, which is perfect for product of trees, is much harder to compute than ${\operatorname{corank}}$. For example, we do not even know the precise value of ${\operatorname{hypdim}}({\operatorname{H}}^2\times{\operatorname{H}}^2)$ (it must be 3 or 4). We also do not see any direct way to compare ${\operatorname{corank}}$ and ${\operatorname{hypdim}}$. At the present stage of knowledge, it looks like that these two invariants are in a sense independent, and each of them works perfectly well in its own range while failing in the other.
[**Structure of the paper.**]{} In section \[sect:ubg\] we introduce and discuss properties of a class of metric spaces with uniformly bounded growth rate, UBG-spaces, which is a key ingredient of the hyperbolic dimension. The main result here is Proposition \[Pro:endubg\], which is an important step in the proof of Theorem \[Thm:main\].
In section \[sect:threedef\] we give three definitions of the hyperbolic dimension following the standard line of the topological dimension theory, and prove their equivalence. It is convenient to use different definitions in different situations. In section \[sect:properties\] we discuss properties of the hyperbolic dimension and prove monotonicity and product theorems. Section \[sect:proof\] is devoted to the proof of Theorem \[Thm:main\] and Corollary \[Cor:nonemb\].
Here we briefly recall some notions which are used in the body of the paper. We denote by $|x-x'|$ the distance in a metric space $X$ between $x$, $x'\in X$. A map $f:X\to Y$ between metric spaces is quasi-isometric if for some ${\Lambda}\ge 1$, ${\lambda}\ge 0$ the estimates $$\frac{1}{{\Lambda}}|x-x'|-{\lambda}\le|f(x)-f(x')|\le{\Lambda}|x-x'|+{\lambda},$$ hold for every $x$, $x'\in X$. In this case we also say that $f$ is $({\Lambda},{\lambda})$-quasi-isometric. A metric space is geodesic if every two its points are connected by a geodesic. By a CAT($-1$)-space we mean a complete, geodesic space whose triangles are thinner than the comparison triangles in the real hyperbolic plane ${\operatorname{H}}^2$.
[**Acknowledgment.**]{} The first author is pleased to acknowledge the hospitality and the support of the University of Zürich where this research has been carried out.
Spaces with uniformly bounded growth rate {#sect:ubg}
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Here we introduce a class of metric spaces which is a key ingredient of the notion of the hyperbolic dimension.
Definition and properties
-------------------------
We say that a metric space $X$ has uniformly bounded growth rate (or is an UBG-space) if for every $\rho>0$ there exist $N\in{\mathbb{N}}$ and $r_0>0$ so that every ball of radius $r\ge r_0$ in $X$ contains at most $N$ points which are $\rho r$-separated.
Equivalently, $X$ is UBG if for every ${\sigma}>1$ there exist $N$ and $r_0$ such that every ball of radius ${\sigma}r$ in $X$ with $r\ge r_0$ can be covered by $N$ balls of radius $r$. Or $X$ is UBG if for every $\rho>0$ there is $N$ such that every ball of radius 1 in the scaled ${\lambda}X$ for all sufficiently small ${\lambda}>0$ contains at most $N$ points which are $\rho$-separated.
The notion of UBG-spaces is close to the notion of [*asymptotically doubling*]{} spaces, where a metric space $X$ is asymptotically doubling if for some positive constants $N$ and $r_0$ every ball in $X$ of radius $r\ge r_0$ can be covered by at most $N$ balls of radius $r/2$. One can show essentially by the same argument as in [@BoS p.295] that if a geodesic space $X$ is asymptotically doubling then it is UBG. However, the assumption that $X$ is geodesic is too restrictive for our purposes. UBG-spaces typically appear in our work as preimages of some sets under quasi-isometric maps, and there is no reason for them to be geodesic. In other words, we study and use UBG-spaces as tools rather than in their own right.
For technical reason, it is convenient to characterize an UBG-space by two functions. We let ${\mathcal{N}}$ be the set of functions $N:(0,1)\to{\mathbb{N}}$ and ${\mathcal{R}}$ be the set of functions $R:(0,1)\to[0,\infty)$. Then the definition above says that $X$ is UBG if for some functions $N\in{\mathcal{N}}$ and $R\in{\mathcal{R}}$ and for every $\rho\in(0,1)$ every ball of radius $r\ge R(\rho)$ in $X$ contains at most $N(\rho)$ points which are $\rho r$-separated. In this case we say that $X$ is $(N,R)$-bounded. We also say that $X$ is $N$-bounded if it is $(N,R)$-bounded for some $R\in{\mathcal{R}}$.
1\. Any Euclidean space ${\mathbb{R}}^n$ is $(N,R)$-bounded for $N(\rho)\asymp\rho^{-n}$ and $R(\rho)\equiv 0$.
The basic example of UBG-spaces is this.
2\. Let $B$ be a bounded metric space. Then the metric product $X=B\times{\mathbb{R}}^n$ is $(N,R)$-bounded for $N(\rho)\asymp\rho^{-n}$ and $R(\rho)\ge\frac{2{\operatorname{diam}}B}{\rho}$. We emphasize that in this example the function $N$ counting the number of separated points is actually independent of $B$, while the function $R$ describing the corresponding scales tends to infinity as ${\operatorname{diam}}B\to\infty$ if one takes as $B$, say, an ${\mathbb{R}}$-tree.
Two functions $N=N(\rho)$ and $R=R(\rho)$ are included in the definition of UBG-spaces instead of fixing some $\rho\in(0,1)$ for the purpose to make this notion quasi-isometry invariant.
\[Lem:indubg\] Let $f:X\to Y$ be a $({\Lambda},{\lambda})$-quasi-isometric map, where ${\Lambda}\ge 1$, ${\lambda}\ge 0$. Assume that $Y$ is $(N,R)$-bounded for some $N\in{\mathcal{N}}$ and $R\in{\mathcal{R}}$. Then $X$ is $(N',R')$-bounded for $N'(\rho)=N(\rho/2{\Lambda}^2)$ and $R'(\rho)=\max\{\frac{1}{{\Lambda}}(R(\rho)-{\lambda}),
\frac{{\lambda}}{{\Lambda}}(1+\frac{1}{\rho})\}$.
Fix $\rho\in (0,1/2{\Lambda}^2)$. Then for all $r\ge R(\rho)$ every ball $B_r{\subset}Y$ contains at most $N(\rho)$ $\rho r$-separated points. We have $f(B_{r'}){\subset}B_{{\Lambda}r'+{\lambda}}$ for every ball $B_{r'}{\subset}X$, hence, for $r'\ge\frac{1}{{\Lambda}}(R(\rho)-{\lambda})$ the set $f(B_{r'})$ contains at most $N(\rho)$ ${\sigma}$-separated points, ${\sigma}=\rho({\Lambda}r'+{\lambda})$. Take ${\sigma}'={\Lambda}({\sigma}+{\lambda})$. If $x$, $x'\in X$ are ${\sigma}'$-separated, then $f(x)$, $f(x')\in Y$ are ${\sigma}$-separated. It follows that the ball $B_{r'}$ itself contains at most $N(\rho)$ ${\sigma}'$-separated points. We put $R'(\rho)=\max\{\frac{1}{{\Lambda}}(R(\rho)-{\lambda}),
\frac{{\lambda}}{{\Lambda}}(1+\frac{1}{\rho})\}$, $\rho'=2{\Lambda}^2\rho$. Hence, for $r'\ge R'(\rho)$ we have ${\sigma}'\le 2{\Lambda}^2\rho r'=\rho'r'$, and $\rho'r'$-separated points are certainly ${\sigma}'$-separated. Then for every $r'\ge R'(\rho)$ every ball $B_{r'}{\subset}X$ contains at most $N(\rho)$ $\rho'r'$-separated points, and the space $X$ is $(N',R')$-bounded, where $N'(\rho)=N(\rho/2{\Lambda}^2)$ and $R'$ as above.
\[Cor:ubgqi\] The property of a metric space to have a uniformly bounded growth rate is a quasi-isometry invariant.
\[Lem:ubgpol\] Every UBG space $Y$ has a polynomial growth rate, that is, there exists $k=k(Y)>0$ such that for every (sufficiently large) ${\delta}$ every ball of radius $r$ in $Y$ contains at most $d\cdot r^k$ ${\delta}$-separated points provided $r$ is sufficiently large, where the constant $d>0$ depends only on $Y$ and ${\delta}$.
We fix $\rho\in(1/2,1)$ and take ${\delta}$ so that ${\delta}'=\frac{2\rho-1}{\rho}{\delta}\ge R(\rho)$. Then every ball $B_{{\delta}'}{\subset}Y$ of radius ${\delta}'$ contains at most $N=N(\rho)$ points which are ${\delta}'$-separated.
Take $r\ge{\delta}$. Then $r>R(\rho)$, and any ball $B_r{\subset}Y$ contains at most $N$ points which are $\rho r$-separated. Take a maximal $\rho r$-separated subset in $B_r$. Then the balls $B_{\rho r}$ centered at its points cover $B_r$, and their number is at most $N$. Applying this argument to every $B_{\rho r}$ from the covering of a fixed $B_r$ and proceeding by induction, we obtain that $B_r$ contains at most $N^q$ points which are $\rho^qr$-separated, provided $\rho^qr\ge{\delta}'$. There is $q\in{\mathbb{N}}$ with $\rho^qr<{\delta}\le\rho^{q-1}r$. For this $q$ the condition $\rho^qr\ge{\delta}'$ is fulfilled, thus every $B_r$ contains at most $N^q\le d\cdot r^k$ points which are ${\delta}$-separated, where $k=\ln N/\ln\frac{1}{\rho}$, $d=(\rho{\delta})^{-k}$.
\[Lem:ubgprod\] Let $X$, $Y$ be UBG-spaces. Then $Z=X\times Y$ is an UBG-space.
The proof is straightforward, if one uses the covering definition of UBG.
UBG-subsets in a CAT($-1$)-space {#subsect:catubg}
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Let $X$ be a CAT($-1$)-space. We fix a base point $x_0\in X$ and define the angle metric ${\angle_{\infty}}$ in the boundary at infinity ${{\partial}_{\infty}}X$ as follows. Given $\xi$, $\xi'\in{{\partial}_{\infty}}X$, we consider the unit speed geodesic rays $c_{\xi}$, $c_{\xi'}$ from $x_0$ to $\xi$, $\xi'$ respectively, and put $${\angle_{\infty}}(\xi,\xi')=\lim_{s\to\infty}
\angle({\overline}{c}_{\xi}(s){\overline}o\ {\overline}{c}_{\xi'}(s)),$$ where $\angle({\overline}{c}_{\xi}(s){\overline}o\ {\overline}{c}_{\xi'}(s))$ is the angle at ${\overline}o$ of the comparison triangle in ${\operatorname{H}}^2$ for the triangle $c_{\xi}(s)x_0c_{\xi'}(s)$. From the hyperbolic geometry we know that $$\tan\left(\frac{1}{4}{\angle_{\infty}}(\xi,\xi')\right)=
e^{-{\operatorname{dist}}({\overline}o,{\overline}\xi\,{\overline}\xi')},$$ where ${\overline}\xi$, ${\overline}\xi'\in{{\partial}_{\infty}}{\operatorname{H}}^2$ satisfy $\angle_{{\overline}o}({\overline}\xi,{\overline}\xi')={\angle_{\infty}}(\xi,\xi')$, and ${\overline}\xi\,{\overline}\xi'$ is the geodesic in ${\operatorname{H}}^2$ with the end points at infinity ${\overline}\xi$, ${\overline}\xi'$. Thus ${\angle_{\infty}}(\xi,\xi')\le 4e^{-{\operatorname{dist}}({\overline}o,{\overline}\xi\,{\overline}\xi')}$.
[*The shadow*]{} of a set $A{\subset}X$ is a subset ${\operatorname{sh}}(A){\subset}{{\partial}_{\infty}}X$ which consists of the ends $\xi$ of all rays $x_0\xi$ intersecting $A$ (so ${\operatorname{sh}}(x_0)={{\partial}_{\infty}}X$). Given ${\delta}>0$ we define [*the angle ${\delta}$-measure*]{} of $A$, $\angle_{\delta}A$, as $$\angle_{\delta}A=\inf_{{\mathcal{C}}}\sum_{B\in{\mathcal{C}}}{\operatorname{diam}}({\operatorname{sh}}(B)),$$ where the infimum is taken over all coverings ${\mathcal{C}}$ of $A$ by balls of radius $\ge{\delta}$ in $X$.
\[Lem:ubgshadow\] Given functions $N\in{\mathcal{N}}$, $R\in{\mathcal{R}}$, for every sufficiently large ${\delta}$ there is a positive constant $C$ depending only on $\rho\in(1/2,1)$, $N(\rho)$, $R(\rho)$ and ${\delta}$ such that if a subset $A{\subset}X$ is $(N,R)$-bounded and ${\operatorname{dist}}(x_0,A)\ge c>{\delta}$, then $$\angle_{\delta}A\le C\cdot e^{-c/2},$$
As in the proof of Lemma \[Lem:ubgpol\], we fix $\rho\in(1/2,1)$ and take ${\delta}\ge\frac{\rho}{2\rho-1}R(\rho)$. Then, by Lemma \[Lem:ubgpol\], every ball $B_r{\subset}X$ with $r>{\delta}$ contains at most $dr^k$ points of $A$ which are ${\delta}$ separated, where $k$ depends on $\rho$, $N(\rho)$, and $d=(\rho{\delta})^{-k}$. Furthermore, since $k$ is independent of ${\delta}$, we can assume that $e^{c/2}\ge(c+1)^k$ for each $c>{\delta}$.
Take a maximal ${\delta}$-separated subset $A'{\subset}A$. Then $A{\subset}\cup_{a\in A'}B_{\delta}(a)$. For any ball $B_{\delta}(a)$, $a\in A'$, consider $\xi$, $\xi'\in{\operatorname{sh}}(B_{\delta}(a))$ with ${\angle_{\infty}}(\xi,\xi')={\operatorname{diam}}({\operatorname{sh}}(B_{\delta}(a)))$. Then ${\operatorname{diam}}({\operatorname{sh}}(B_{\delta}(a)))\le 4e^{-{\operatorname{dist}}({\overline}o,{\overline}\xi\,{\overline}\xi')}$ in notation introduced above. We take $x\in x_0\xi\cap B_{\delta}(a)$, $x'\in x_0\xi'\cap B_{\delta}(a)$ and consider the piecewise geodesic curve ${\gamma}$ in $X$, which consists of the geodesic rays $x\xi$, $x'\xi'$ and the segment $xx'$. The curve ${\gamma}$, as well as the geodesic $\xi\xi'$, connects in $X$ the points $\xi$, $\xi'$, and ${\operatorname{dist}}(x_0,{\gamma})\ge{\operatorname{dist}}(x_0,B_{\delta}(a))=|x_0-a|-{\delta}$. Furthermore, ${\operatorname{dist}}(x_0,{\gamma})\le{\operatorname{dist}}({\overline}o,{\overline}\xi\,{\overline}\xi')$ by comparison with ${\operatorname{H}}^2$. Thus $${\operatorname{diam}}({\operatorname{sh}}(B_{\delta}(a)))\le 4e^{{\delta}-|x_0-a|}$$ and $$\angle_{\delta}A\le\sum_{a\in A'}{\operatorname{diam}}({\operatorname{sh}}(B_{\delta}(a))).$$ Since $c>{\delta}$, for each $\tau\ge c+1$, the number of points from $A'$ whose distances to $x_0$ lie in the interval $[\tau-1,\tau)$ is $\le d\cdot\tau^k$. Thus we have $$\begin{aligned}
\angle_{\delta}A&\le& 4e^{{\delta}}\sum_{a\in A'}e^{-|x_0-a|}\le
4de^{{\delta}}\left(\sum_{q=0}^{\infty}(c+q+1)^ke^{-q}\right)e^{-c}\\
&\le&4de^{{\delta}}\left(\sum_{q=0}^{\infty}(q+1)^ke^{-q}\right)(c+1)^ke^{-c}
\le C\cdot e^{-c/2}\end{aligned}$$ by the choice of $c$.
A cut property of UBG-subsets
-----------------------------
A subset $A$ of a metric space $X$ is said to be roughly connected if for some ${\sigma}>0$ and for every $a$, $a'\in A$ there is a sequence $a_0=a,\dots,a_k=a'$ in $A$ with $|a_i-a_{i-1}|\le{\sigma}$, $i=1,\dots,k$. Such a sequence is called a rough or a ${\sigma}$-path between $a$ and $a'$, and we also say that $A$ is ${\sigma}$-connected. One says that a roughly connected subset $A$ of a geodesic space $X$ is [*cut-quasi-convex,*]{} if there is $c>0$ such that for every $a$, $a'\in A$ and every $x\in aa'$, every rough path in $A$ between $a$, $a'$ intersects the ball $B_c(x){\subset}X$ of radius $c$ centered at $x$. In this case we also say that $A$ is $c$-cut-convex, and $c$ is called [*the cut radius*]{} of $A$. This property, obviously, implies that every geodesic segment $aa'{\subset}X$ with the end points $a$, $a'\in A$ lies in the $c$-neighborhood of $A$, i.e., the set $A$ is in particular quasi-convex. This justifies our terminology.
\[Pro:cutubg\] Assume that a ${\sigma}$-connected subset $A$ of CAT($-1$)-space $X$ is $(N,R)$-bounded for some functions $N\in{\mathcal{N}}$, $R\in{\mathcal{R}}$. Then $A$ is cut-quasi-convex, and the cut radius $c$ depends only on $\rho\in(1/2,1)$, $N(\rho)$, $R(\rho)$ and ${\sigma}$, $c=c(\rho,N(\rho),R(\rho),{\sigma})$.
Fix $\rho\in(1/2,1)$ and take a sufficiently large ${\delta}$ provided by Lemma \[Lem:ubgshadow\]. Furthermore, we can assume that ${\delta}\ge{\sigma}$. Next, we take $c'>{\delta}$ such that $C\cdot e^{-c'/2}<\pi$, where $C$ is the constant from Lemma \[Lem:ubgshadow\], and put $c=c'+{\delta}$. Then $c=c(\rho,N(\rho),R(\rho),{\sigma})$.
Assume that for some $a$, $a'\in A$ there is a ${\sigma}$-path ${\gamma}$ in $A$ between $a$, $a'$ which misses the ball $B_c(x_0){\subset}X$ for some $x_0\in aa'$. Then $A'=\cup_{b\in{\gamma}}B_{\delta}(b)$ is a connected subset in $X$ containing the points $a$, $a'$, thus $\angle_{\delta}A'\ge\pi$ for every ${\delta}>0$.
On the other hand, ${\operatorname{dist}}(x_0,A')\ge c'$, and we have $\angle_{\delta}A\le C\cdot e^{-c'/2}<\pi$ by Lemma \[Lem:ubgshadow\]. This is a contradiction, and hence $A$ is cut-quasi-convex with the cut radius $c$.
\[Cor:hypubg\] If a ball $B_r$ of radius $r$ in a CAT($-1$)-space $X$ is $(N,R)$-bounded for some $N\in{\mathcal{N}}$, $R\in{\mathcal{R}}$, then $r\le c$ for some constant $c=c(\rho,N(\rho),R(\rho))$, $\rho\in(1/2,1)$.
Discussing the cut-quasi-convex property, we have used so far only the fact that any UBG-space has a polynomial growth rate (actually, a subexponential growth rate suffices). In what follows, the next Proposition plays a key role, and we use in it the whole power of the definition of UBG-spaces.
\[Pro:endubg\] Let $A$ be an $N$-bounded (for some function $N\in{\mathcal{N}}$), subset in a CAT($-1$)-space $X$. Then for every ${\sigma}>0$ the union ${{\partial}_{\infty}}A_{\sigma}$ of the boundaries at infinity of ${\sigma}$-connected components of $A$ contains at most $M$ points, where $M<\infty$ depends only on the function $N$.
Fix $x_0\in X$. It follows from the cut-quasi-convex property of any roughly connected component of $A$ that a tail of every geodesic ray $x_0\xi{\subset}X$, $\xi\in{{\partial}_{\infty}}A_{\sigma}$, lies in the $c$-neighborhood of $A$ for some $c>0$ depending only on the bounding parameters for $A$ and ${\sigma}$. Thus for every $\rho\in(0,1)$ and for all sufficiently large $r$ the ball $B_r(x_0){\subset}X$ contains at least as much $\rho r$-separated points from $A$ as the cardinality of ${{\partial}_{\infty}}A_{\sigma}$. Hence, the claim.
Three definitions of the hyperbolic dimension {#sect:threedef}
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The hyperbolic dimension is a variation of the Gromov’s asymptotic dimension (see [@Gr 1.E]) with only difference that we take as “small” the UBG-sets. Here we give three equivalent definitions of the hyperbolic dimension following the standard line of topological dimension theory. As in [@BD], we find convenient to use the Lebesgue number of a covering in our definitions instead of $d$-multiplicity as in [@Gr 1.E].
Recall that the Lebesgue number of a covering ${\mathcal{U}}$ of a metric space $X$ is the maximal radius $L({\mathcal{U}})$ such that any (open) ball in $X$ of that radius is contained in some element of the covering, $$L({\mathcal{U}})=\inf_{x\in X}\max{\{{\operatorname{dist}}(x,X{\setminus}U):\,\text{$U\in{\mathcal{U}}$}\}}.$$
Given $N\in{\mathcal{N}}$, a covering ${\mathcal{U}}$ of a metric space $X$ is said to be [*uniformly $N$-bounded,*]{} if
- there is a function $R\in{\mathcal{R}}$ such that every element of the covering is $(N,R)$-bounded;
- any finite union of elements of the covering is $N$-bounded.
\[Rem:boundcover\] To get a better grip on the second property, which is rather strong, assume that $X$ is a CAT($-1$)-space. Then by Proposition \[Pro:endubg\], the boundary at infinity ${{\partial}_{\infty}}A_{{\sigma}}{\subset}{{\partial}_{\infty}}X$ of roughly connected components of any finite union $A=\cup_iU_i$ of elements $U_i\in{\mathcal{U}}$ has the cardinality bounded above independently of the number of the elements.
First definition
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The hyperbolic dimension of a metric space $X$, ${\operatorname{hypdim}}_1 X$, is the minimal $n$ such that for every $d>0$ there are a function $N\in{\mathcal{N}}$ and a covering of $X$ by $n+1$ subsets $X_j=\cup_{{\alpha}}X_{j{\alpha}}$, $j=1,\dots,n+1$ such that
- $X_{j{\alpha}}\cap X_{j{\alpha}'}={\emptyset}$ for every $j=1,\dots,n+1$ and all ${\alpha}\neq{\alpha}'$;
- the covering $\{X_{j{\alpha}}\}$ of $X$ is uniformly $N$-bounded and its Lebesgue number is $\ge d$.
Second definition
-----------------
The hyperbolic dimension of a metric space $X$, ${\operatorname{hypdim}}_2 X$, is the minimal $n$ such that for every $d>0$ there are a function $N\in{\mathcal{N}}$ and a covering ${\mathcal{U}}$ of $X$ with Lebesgue number $L({\mathcal{U}})\ge d$ and multiplicity $\le n+1$, which is uniformly $N$-bounded.
Clearly, ${\operatorname{hypdim}}_2X\le{\operatorname{hypdim}}_1X$.
Third definition
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Given a index set $J$, we let $R^J$ be the Euclidean space of functions $J\to{\mathbb{R}}$ with finite support, i.e., $x\in{\mathbb{R}}^J$ iff only finitely many coordinates $x_j=x(j)$ are not zero. Then the distance is well defined by $$|x-x'|^2=\sum_{j\in J}(x_j-x_j')^2.$$ Let ${\Delta}^J{\subset}{\mathbb{R}}^J$ be the standard simplex, i.e., $x\in{\Delta}^J$ iff $x_j\ge 0$ for all $j\in J$ and $\sum_{j\in J}x_j=1$.
A metric in $n$-dimensional simplicial complex $P$ is said to be [*uniform*]{} if $P$ is isometric to a subcomplex of ${\Delta}^J{\subset}{\mathbb{R}}^J$ for some index set $J$. Every simplex ${\sigma}{\subset}P$ is then isometric to the standard $k$-simplex ${\Delta}^k{\subset}{\mathbb{R}}^{k+1}$, $k=\dim{\sigma}$ (so, for a finite $J$, $\dim{\Delta}^J=|J|-1$). For every simplicial polyhedron $P$ there is the canonical embedding $u:P\to{\Delta}^J$, where $J$ is the vertex set of $P$, which is affine on every simplex. Its image $P'=u(P)$ is called [*the uniformization*]{} of $P$, and $u$ is the uniformization map.
The hyperbolic dimension of a metric space $X$, ${\operatorname{hypdim}}_3X$, is the minimal $n$ such that for every ${\lambda}>0$ there are a function $N\in{\mathcal{N}}$ and a ${\lambda}$-Lipschitz map $p:X\to P$ into a uniform $n$-dimensional simplicial polyhedron $P$, for which the covering ${\{p^{-1}({\operatorname{st}}_v):\,\text{$v\in P$}\}}$ of $X$ by preimages of the open stars ${\operatorname{st}}_v{\subset}P$ of the vertices of $P$ is uniformly $N$-bounded.
Equivalence of the definitions
------------------------------
\[Pro:equivhypdim\] For every metric space $X$ we have $${\operatorname{hypdim}}_1X={\operatorname{hypdim}}_2X={\operatorname{hypdim}}_3X.$$
We have already mentioned that ${\operatorname{hypdim}}_2X\le{\operatorname{hypdim}}_1X$ easily follows from the definitions. The proof of ${\operatorname{hypdim}}_3X\le{\operatorname{hypdim}}_2X$ is fairly standard (see [@Gr 1.E$_1$], [@BD Propositions 1,2]). Denote $n={\operatorname{hypdim}}_2X$. Given $d>0$, we have a function $N\in{\mathcal{N}}$ and a uniformly $N$-bounded covering ${\mathcal{U}}=\{U_j\}_{j\in J}$ of $X$ with multiplicity $\le n+1$ and Lebesgue number $L({\mathcal{U}})\ge d$. Using these data we construct a ${\lambda}$-Lipschitz map $p:X\to{\Delta}^J$ with ${\lambda}\le\frac{(n+2)^2}{d}$, whose image lies in a $n$-dimensional subpolyhedron $P{\subset}{\Delta}^J$, as follows.
Given $j\in J$, we define $q_j:X\to{\mathbb{R}}$ by $q_j(x)=\min\{d,{\operatorname{dist}}(x,X{\setminus}U_j)\}$. Then $\sum_{j\in J}q_j(x)\ge d$ for every $x\in X$. Furthermore, $q_j(x')\le q_j(x)+|x-x'|$ and $$\sum_{j\in J}q_j(x')\le\sum_{j\in J}q_j(x)+(n+1)|x-x'|,$$ because in each sum there are at most $n+1$ nonzero summands. Using this one obtains $$\frac{1}{\sum_{j\in J}q_j(x')}\le\frac{1}{\sum_{j\in J}q_j(x)}
+\frac{(n+1)|x-x'|}{d\cdot\sum_{j\in J}q_j(x)}.$$ Now, we put $p_j(x)=q_j(x)/\sum_{j\in J}q_j(x)$. Then abbreviating ${\Sigma}=\sum_{j\in J}q_j(x)$, ${\Sigma}'=\sum_{j\in J}q_j(x')$ we obtain $$p_j(x')-p_j(x)=\frac{q_j(x')-q_j(x)}{{\Sigma}'}
+\left(\frac{1}{{\Sigma}'}
-\frac{1}{{\Sigma}}\right)q_j(x)\le\frac{n+2}{d}|x-x'|.$$ Finally, for $p=\{p_j\}_{j\in J}$ we have $$|p(x')-p(x)|^2=\sum_{j\in J}(p_j(x')-p_j(x))^2
\le\frac{(n+2)^2(2n+2)}{d^2}|x-x'|^2,$$ hence $p:X\to{\Delta}^J$ is ${\lambda}$-Lipschitz with ${\lambda}\le\frac{(n+2)^2}{d}$. Since for every $x\in X$ there are at most $n+1$ nonzero coordinates $p_j(x)$, $j\in J$, the image $p(X)$ lies in a $n$-dimensional subpolyhedron $P{\subset}{\Delta}^J$. For each vertex $v\in P$ the preimage of its open star $p^{-1}({\operatorname{st}}_v){\subset}X$ is contained in some element of the covering ${\mathcal{U}}$. Thus the covering ${\{p^{-1}({\operatorname{st}}_v):\,\text{$v\in P$}\}}$ of $X$ is uniformly $N$-bounded. It follows that ${\operatorname{hypdim}}_3X\le{\operatorname{hypdim}}_2X$.
To prove that ${\operatorname{hypdim}}_1X\le{\operatorname{hypdim}}_3X$, we assume that for every ${\lambda}>0$ there are a function $N\in{\mathcal{N}}$ and a ${\lambda}$-Lipschitz map $p:X\to P$ into a uniform $n$-dimensional simplicial polyhedron $P$, for which the covering ${\{p^{-1}({\operatorname{st}}_v):\,\text{$v\in P$}\}}$ of $X$ is uniformly $N$-bounded.
Every $j$-dimensional simplex ${\sigma}{\subset}P$ is matched by its barycenter, which is the vertex $v_{\sigma}$ in the first barycentric subdivision ${\operatorname{ba}}P$ of $P$. Let $P_j$ be the union of the open starts $P_{j{\sigma}}$ of ${\operatorname{ba}}P$ of all $v_{\sigma}$ with $\dim{\sigma}=j$, $$P_j=\bigcup_{\dim{\sigma}=j}P_{j{\sigma}}.$$ Then $P_{j{\sigma}}\cap P_{j{\sigma}'}={\emptyset}$ for ${\sigma}\neq{\sigma}'$ and $P=\cup_{j=0}^nP_j$. Now, we put $X_{j{\sigma}}=p^{-1}(P_{j{\sigma}})$, $X_j=p^{-1}(P_j)$. Then $X=\cup_{j=0}^nX_j$ and $X_{j{\sigma}}\cap X_{j{\sigma}'}={\emptyset}$ for every $j=0,\dots,n$ and all ${\sigma}\neq{\sigma}'$. Furthermore, the stars $P_{j{\sigma}}$ of ${\operatorname{ba}}P$ are contained in appropriate open stars of $P$, thus the covering $\{X_{j{\sigma}}\}$ of $X$ is uniformly $N$-bounded. Since the polyhedron $P$ is uniform, there is a lower bound $l_n>0$ for the Lebesgue number of the covering $\{P_{j{\sigma}}\}$. Therefore, the Lebesgue number of $\{X_{j{\sigma}}\}$ is at least $l_n/{\lambda}$. This shows that ${\operatorname{hypdim}}_1X\le{\operatorname{hypdim}}_3X$.
From now on, we use notation ${\operatorname{hypdim}}X$ for the common value $${\operatorname{hypdim}}_1X={\operatorname{hypdim}}_2X={\operatorname{hypdim}}_3X.$$
Properties of the hyperbolic dimension {#sect:properties}
======================================
The following two properties of the hyperbolic dimension are obvious from the definitions:
- ${\operatorname{hypdim}}X=0$ for every UBG-space $X$.
- ${\operatorname{hypdim}}X\le{\operatorname{asdim}}X$ for every metric space $X$.
(The asymptotic dimension of a metric space $X$, ${\operatorname{asdim}}X$, is defined precisely in the same way as the hyperbolic dimension taking instead of uniformly $N$-bounded coverings, coverings by sets with uniformly bounded diameter, see [@Gr], [@BD]). The last inequality can be strong. For example, ${\operatorname{asdim}}{\mathbb{R}}^n=n$ for every $n\ge 0$, while ${\operatorname{hypdim}}{\mathbb{R}}^n=0$ since ${\mathbb{R}}^n$ is an UBG-space.
Monotonicity of the hyperbolic dimension
----------------------------------------
The following simple but important monotonicity theorem implies, in particular, that the hyperbolic dimension is a quasi-isometry invariant of a metric space.
\[Thm:monhypdim\] Let $f:X\to X'$ be a quasi-isometric map between metric spaces $X$, $X'$. Then $${\operatorname{hypdim}}X\le{\operatorname{hypdim}}X'.$$
The proof is straightforward using the second definition of the hyperbolic dimension and Lemma \[Lem:indubg\].
\[Cor:qinv\] The hyperbolic dimension is a quasi-isometry invariant of metric spaces.
Product Theorem for the hyperbolic dimension {#subsect:product}
--------------------------------------------
While simplicial complexes appear as nerves of coverings, they are not convenient for the proof of the Product Theorem for the hyperbolic dimension. Instead, we use cubical complexes. Thus we first describe relation between simplicial and cubical complexes.
Given a index set $J$, one defines $Q^J={\{x\in{\mathbb{R}}^J:\,\text{$\max_{j\in J}x_j=1,x_j\ge 0$}\}}$. There is a canonical homeomorphism $\pi_J:Q^J\to{\Delta}^J$ (the radial projection) given by $$\pi_J(x)=\frac{x}{\sum_{j\in J}x_j}$$ for every $x\in Q^J$.
\[Lem:lipi\] Restricted to any $n$-dimensional coordinate subspace ${\mathbb{R}}^n{\subset}{\mathbb{R}}^J$ the map $\pi_J$ is Lipschitz with the Lipschitz constant $\le n+1$.
For $x$, $x'\in Q^J\cap{\mathbb{R}}^n$ we have $|x_j-x_j'|\le|x-x'|$ for every $j\in J$ and $\sum_{j\in J}|x_j-x_j'|\le n|x-x'|$, $\sum_{j\in J}x_j\ge 1$. Using this, one easily obtains $|\pi_J(x)-\pi_J(x')|\le(n+1)|x-x'|$, and the claim follows.
The inverse homeomorphism ${\omega}_J=\pi_J^{-1}:{\Delta}^J\to Q^J$ is given by ${\omega}_J(x)={\lambda}(x)x$, where ${\lambda}(x)=(\max_{j\in J}x_j)^{-1}$.
\[Lem:lipinv\] Restricted to any $n$-dimensional coordinate subspace ${\mathbb{R}}^n{\subset}{\mathbb{R}}^J$ the map ${\omega}_J$ is Lipschitz with the Lipschitz constant $\le n(1+\sqrt n)$.
For $x$, $x'\in{\Delta}^J\cap{\mathbb{R}}^n$ we have $|x_j-x_j'|\le|x-x'|$ for every $j\in J$ and $$|\max_{j\in J}x_j-\max_{j\in J}x_j'|\le
\max_{j\in J}|x_j-x_j'|\le|x-x'|,$$ $\max_{j\in J}x_j$, $\max_{j\in J}x_j'\ge 1/n$. Using this, we easily obtain $|{\omega}_J(x)-{\omega}_J(x')|\le(n+n\sqrt n)|x-x'|$, and the claim follows.
Faces of dimension $k\ge 0$ of $Q^J$ are defined as the closures of its subsets where exactly $k$ coordinates have values in $(0,1)$. They are isometric to the unit cube in ${\mathbb{R}}^k$, and every such a face is mapped by $\pi_J$ onto the union of simplices of the first barycentric subdivision ${\operatorname{ba}}{\Delta}^J$. Vice versa, for every vertex of a $k$-dimensional simplex ${\Delta}^k{\subset}{\Delta}^J$ its (closed) star in the first barycentric subdivision ${\operatorname{ba}}{\Delta}^k$ is mapped by ${\omega}_J$ homeomorphically onto a $k$-dimensional face of $Q^J$. Therefore, the image ${\omega}_J({\Delta}^k){\subset}Q^J$ consists of $k+1$ cubical $k$-faces. As a corollary of this and Lemmas \[Lem:lipi\], \[Lem:lipinv\], we obtain.
\[Lem:simplcub\] For any $n$-dimensional subcomplex $P{\subset}{\Delta}^J$, the image $P'={\omega}_J(P)$ is a $n$-dimensional cubical subcomplex of $Q^J$, and ${\omega}_J:P\to P'$ is a Lipschitz homeomorphism with the Lipschitz constant depending only on $n$.
Conversely, for any $n$-dimensional subcomplex $P'{\subset}Q^J$, the image $P=\pi_J(P'){\subset}{\Delta}^J$ is a $n$-dimensional simplicial subcomplex of ${\operatorname{ba}}{\Delta}^J$, and $\pi_J:P'\to P$ is a Lipschitz homeomorphism with the Lipschitz constant depending only on $n$.
Now, we prove the product Theorem for the hyperbolic dimension.
\[Thm:prodhypdim\] For any metric spaces $X_1$, $X_2$, we have $${\operatorname{hypdim}}(X_1\times X_2)\le{\operatorname{hypdim}}X_1+{\operatorname{hypdim}}X_2.$$
We use the third definition of the hyperbolic dimension. Given ${\lambda}_k>0$ there are a function $N_k\in{\mathcal{N}}$ and a ${\lambda}_k$-Lipschitz map $p_k:X_k\to P_k{\subset}{\Delta}^{J_k}$ into $n_k$-dimensional uniform simplicial polyhedron $P_k$, where $n_k={\operatorname{hypdim}}X_k$, such that the covering ${\{p_k^{-1}({\operatorname{st}}_v):\,\text{$v\in P_k$}\}}$ of $X_k$ is uniformly $N_k$-bounded, $k=1,2$. We assume that $n_1$, $n_2<\infty$, since otherwise there is nothing to prove.
By Lemma \[Lem:simplcub\], $P_k'={\omega}_k(P_k){\subset}Q^{J_k}$ is a $n_k$-dimensional cubical subcomplex of $Q^{J_k}$, and the homeomorphism ${\omega}_k={\omega}_{J_k}:P_k\to P_k'$ is Lipschitz with the Lipschitz constant depending only on $n_k$, $k=1,2$. Then ${\omega}_k\circ p_k:X_k\to P_k'$ is Lipschitz with Lipschitz constant $\le{\operatorname{const}}(n_k)\cdot{\lambda}_k$. Furthermore, the covering ${\{({\omega}_k\circ p_k)^{-1}({\operatorname{st}}_w):\,\text{$w\in P_k'$}\}}$ of $X_k$ by preimages of the open cubical stars ${\operatorname{st}}_w$ of the vertices of $P_k'$ is uniformly $N_k$-bounded, because every such a star ${\operatorname{st}}_w{\subset}P_k'$ lies in ${\omega}_k({\operatorname{st}}_v)$ for an appropriate vertex $v\in P_k$, $k=1,2$.
Let $J=J_1\cup J_2$ be the disjoint union. We define $p:X_1\times X_2\to{\mathbb{R}}^J$ by $p(x_1,x_2)=({\omega}_1\circ p_1(x_1),{\omega}_2\circ p_2(x_2))$ for every $(x_1,x_2)\in X_1\times X_2$. Then $p(X_1\times X_2){\subset}P'$, where $P'=P_1'\times P_2'{\subset}Q^J$ is a cubical subcomplex of dimension $n_1+n_2$, and the map $p$ is Lipschitz with Lipschitz constant ${\operatorname{Lip}}(p)\le{\operatorname{const}}(n_1,n_2)\cdot\max\{{\lambda}_1,{\lambda}_2\}$. Using Lemma \[Lem:ubgprod\] one easily checks that the covering ${\{p^{-1}({\operatorname{st}}_w):\,\text{$w\in P'$}\}}$ of $X_1\times X_2$ by preimages of the open cubical stars is uniformly $N$-bounded for some function $N\in{\mathcal{N}}$ depending only on $N_1$, $N_2$. Applying the homeomorphism $\pi_J:Q^J\to{\Delta}^J$, we obtain a simplicial subcomplex $P=\pi_J(P'){\subset}{\operatorname{ba}}{\Delta}^J$ and a $\mu$-Lipschitz map $\pi_J\circ p:X_1\times X_2\to P$, $\mu\le{\operatorname{const}}(n_1,n_2)\cdot\max\{{\lambda}_1,{\lambda}_2\}$, with required $N$-boundedness property. It remains to compose this map with a Lipschitz simplicial homeomorphism sending $P$ into a simplicial subcomplex of ${\Delta}^{J'}$, where $J'$ is the set of all nonempty finite subsets in $J$.
Proof of Theorem \[Thm:main\] {#sect:proof}
=============================
d-multiplicity of a covering {#subsect:dmult}
----------------------------
Fix $d>0$. Recall that the $d$-multiplicity of a covering ${\mathcal{U}}$ of a metric space $X$ is $\le n$, if no ball of radius $d$ in $X$ meets more than $n$ elements of the covering (see [@Gr 1.E]). One can define an auxiliary hyperbolic dimension ${\operatorname{hypdim}}'X$ as a minimal $n$ such that for every $d>0$ there are a function $N\in{\mathcal{N}}$ and a uniformly $N$-bounded covering ${\mathcal{U}}$ of $X$ with $d$-multiplicity $\le n+1$.
\[Lem:ahypdim\] For every metric space $X$ we have ${\operatorname{hypdim}}'X\le{\operatorname{hypdim}}X$.
Let ${\mathcal{U}}=\{U_j\}_{j\in J}$ be a uniformly $N$-bounded covering of $X$ with multiplicity $\le n+1$, $n={\operatorname{hypdim}}X$, and Lebesgue number $L({\mathcal{U}})\ge 2d$ for some function $N\in{\mathcal{N}}$ and some $d>0$. Following [@BD Assertion 1], we define $V_j=U_j{\setminus}D_d(X{\setminus}U_j)$, $j\in J$, where $D_d(A)$ is the metric $d$-neighborhood of $A{\subset}X$. Since $L({\mathcal{U}})\ge 2d$, the collection ${\mathcal{V}}=\{V_j\}_{j\in J}$ is still a covering of $X$. Moreover, ${\mathcal{V}}$ is uniformly $N$-bounded because $V_j{\subset}U_j$ for every $j\in J$. Furthermore, if a ball $B_d(x){\subset}X$ meets some $V_j$, then $x\in U_j$. Thus the $d$-multiplicity of ${\mathcal{V}}$ is $\le n+1$. Hence, the claim.
Ends of uniformly $N$-bounded coverings of $X$
----------------------------------------------
Recall that a metric space $X$ has bounded geometry if there are $\rho_X>0$ and a function $M_X:(0,\infty)\to(0,\infty)$ such that every ball $B_r{\subset}X$ of radius $r>0$ contains at most $M_X(r)$ points which are $\rho_X$-separated.
\[Lem:boundend\] Let $X$ be a CAT($-1$)-space with bounded geometry, and let ${\mathcal{U}}$ be a uniformly $N$-bounded covering of $X$ with finite $d$-multiplicity for a function $N\in{\mathcal{N}}$ and a sufficiently large $d$. Furthermore, assume that the elements of ${\mathcal{U}}$ are ${\sigma}$-connected for some ${\sigma}\ge 10d$. Then ${{\partial}_{\infty}}U{\subset}{{\partial}_{\infty}}X$ is finite for every $U\in{\mathcal{U}}$ and the number of different elements of ${\mathcal{U}}$ with infinite diameter is finite.
Since the covering ${\mathcal{U}}$ is uniformly $N$-bounded, there is a function $R\in{\mathcal{R}}$ such that every element of the covering is $(N,R)$-bounded. Then by Proposition \[Pro:cutubg\] every $U\in{\mathcal{U}}$ is cut-quasi-convex with cut radius $c>0$ depending only on $N$, $R$ and ${\sigma}$. By Proposition \[Pro:endubg\] there is an upper bound $M<\infty$ for the cardinality of the union ${{\partial}_{\infty}}A_{\sigma}$ of the boundaries at infinity of ${\sigma}$-connected components of any $N$-bounded $A{\subset}X$. In particular, $|{{\partial}_{\infty}}U|\le M$ for every $U\in{\mathcal{U}}$.
We assume that $X$ has $(\rho_X,M_X)$-bounded geometry, and that $d$-multiplicity of ${\mathcal{U}}$ is $\le n$ for $d\ge\rho=\rho_X$. Then we put $M_0=(n+1)\cdot M\cdot M_X(2c)$ and assume that there are $\ge M_0$ different elements of the covering ${\mathcal{U}}$ with infinite diameter. Since every finite union of elements of ${\mathcal{U}}$ is $N$-bounded, there is $\xi\in{{\partial}_{\infty}}X$ which is a common point of ${{\partial}_{\infty}}U$ for at least $M_0/M$ different elements $U\in{\mathcal{U}}$.
We fix $x_0\in X$ and consider the geodesic ray $x_0\xi{\subset}X$. By the cut-quasi-convex property of $U$, a tail $x_1\xi{\subset}x_0\xi$ lies in the $c$-neighborhood of every selected above $U$. Thus for every $x\in x_1\xi$ the ball $B_{2c}(x)$ intersects at least $(n+1)\cdot M_X(2c)$ different elements of the covering. On the other hand, $B_{2c}(x)$ can be covered by $\le M_X(2c)$ balls of radius $\rho$. Hence, there is a ball $B_\rho(x'){\subset}X$ which intersects at least $n+1$ different elements of the covering. This is a contradiction, because $d\ge\rho$ and $d$-multiplicity of the covering is $\le n$. Thus there is at most $M_0-1$ different elements of ${\mathcal{U}}$ with infinite diameter.
Radial contraction
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\[Lem:coveray\] Let ${\mathcal{W}}$ be a locally finite collection of subsets of finite diameter in the ray $[0,\infty)$. Then there is a 1-Lipschitz homeomorphism $f:[0,\infty)\to [0,\infty)$ such that every set $f(W)$, $W\in{\mathcal{W}}$ has diameter at most 1.
We let $r_0=[0,\infty)$ and $V_1$ be the set of all $W\in{\mathcal{W}}_0={\mathcal{W}}$ which intersect the initial segment of length 1 of $r_0$. Since the collection ${\mathcal{W}}$ is locally finite, $V_1$ is finite. Then $t_1=\sup{\{t\in r_0:\,\text{$t\in W\in V_1$}\}}$ is finite, and we define ${\varphi}_1:r_0\to r_0$ by ${\varphi}_1(t)=t$ for $t\in[0,1]$, and for $t\ge 1$ by ${\varphi}_1(t)=t$ if $t_1\le 2$, ${\varphi}_1(t)=\frac{1}{t_1-1}(t-1)+1$ otherwise. Then ${\varphi}_1$ is a 1-Lipschitz homeomorphism, and the diameter ${\operatorname{diam}}({\varphi}_1(W))\le 2$ for every $W\in V_1$.
We put $f_1={\varphi}_1$ and note that ${\mathcal{W}}_1=f_1({\mathcal{W}})\cap r_1$ is a locally finite collection of subsets in the ray $r_1=[1,\infty)$. Now, we apply the same procedure to the ray $r_1$ and the collection ${\mathcal{W}}_1$ extending the resulting 1-Lipschitz homeomorphism ${\varphi}_2:r_1\to r_1$ to $r_0$ by the identity and putting $f_2={\varphi}_2\circ f_1$.
Repeating we obtain a stabilizing sequence of 1-Lipschitz homeomorphisms $f_n={\varphi}_n\circ f_{n-1}:r_0\to r_0$, $f_n=f_{n-1}$ on $[0,n]$, and for which ${\operatorname{diam}}(f_n(W))\le 2$ for all $W\in{\mathcal{W}}$ intersecting $[0,n]$. Composing the limit homeomorphism $\lim_{n\to\infty}f_n$ with the homothety ${\lambda}(t)=t/2$, we obtain a required homeomorphism $f$.
Cone over the boundary at infinity
----------------------------------
The following estimate from below is the major step in the proof of Theorem \[Thm:main\].
\[Pro:hypdimcone\] Let $X{\subset}{\operatorname{H}}^n$, $n\ge 2$, be the convex hull of a compact infinite $Z{\subset}{{\partial}_{\infty}}{\operatorname{H}}^n$. Then ${\operatorname{hypdim}}X\ge\dim Z+1$.
For the proof we need some preparations. We fix a base point $o\in X$ and define the cone over $Z$, ${\operatorname{Co}}(Z){\subset}X$, as the union of all geodesic rays emanating from $o$ towards $Z$. Note that ${\operatorname{Co}}(Z)$ with the metric induced from ${\operatorname{H}}^n$ in general is neither CAT($-1$) nor even geodesic space as for example in the case $\dim Z=0$. On the other hand, ${\operatorname{Co}}(Z)$ is cobounded in $X$ (see [@BoS Proposition 10.1(2)]), that is ${\operatorname{dist}}(x,{\operatorname{Co}}(Z))\le{\sigma}_0$ for some ${\sigma}_0>0$ and every $x\in X$. In what follows, we also use the angle metric in ${{\partial}_{\infty}}{\operatorname{H}}^n$ based at $o$.
Next, we consider the annulus ${\operatorname{An}}(Z){\subset}{\operatorname{Co}}(Z)$, which consists of all $x\in{\operatorname{Co}}(Z)$ with $1\le|x-o|\le 2$. Clearly, ${\operatorname{An}}(Z)$ is homeomorphic to $Z\times I$, $I=[0,1]$. According to a well known result from the dimension theory (see [@Al]), the topological dimension $$\dim{\operatorname{An}}(Z)=\dim Z+1 \tag{$\ast$}.$$ We also need the following simple fact from the dimension theory.
\[Lem:remove\] Let $Z$ be an infinite compact metric space. Then for every finite $A{\subset}Z$ and all sufficiently small ${\varepsilon}>0$ we have $\dim(Z{\setminus}D_{\varepsilon}(A))=\dim Z$, where $D_{\varepsilon}(A)$ is the open ${\varepsilon}$-neighborhood of $A$.
Since $Z$ is infinite, one can assume that $n=\dim Z>0$. Let $Y{\subset}Z$ be the subset which consists of all points $z\in Z$ at which $Z$ has dimension $n$. It is well known (see [@HW Ch.IV.5]) that $\dim Y=n$, thus $Y$ cannot be finite, and $Y\not{\subset}D_{\varepsilon}(A)$ for all sufficiently small ${\varepsilon}>0$. The claim follows.
The space ${\operatorname{H}}^n$ and hence its subspace $X$ certainly have a bounded geometry. Let $\rho_{{\operatorname{H}}^n}>0$ and $M_{{\operatorname{H}}^n}:(0,\infty)\to(0,\infty)$ be the corresponding bounding parameters. Assume that ${\operatorname{hypdim}}X<\dim Z+1$. Then, moreover, ${\operatorname{hypdim}}'X<\dim Z+1$ (see sect. \[subsect:dmult\]). Thus for $d\ge\rho_{{\operatorname{H}}^n}$ there are a function $N\in{\mathcal{N}}$ and a uniformly $N$-bounded covering ${\mathcal{U}}$ of $X$ with $d$-multiplicity $\le\dim Z+1$. We fix ${\sigma}\ge 10d$ and note that taking ${\sigma}$-connected components of ${\mathcal{U}}$ changes neither $N$-boundedness nor $d$-multiplicity. Thus we can assume W.L.G. that the elements of ${\mathcal{U}}$ are ${\sigma}$-connected. By Lemma \[Lem:boundend\] there are only finitely many elements $U\in{\mathcal{U}}$ with infinite diameter, and the boundary at infinity each of them is finite.
We let ${\mathcal{V}}$ be the set of ${\sigma}$-connected components of the induced covering ${\mathcal{U}}\cap{\operatorname{Co}}(Z)$ of ${\operatorname{Co}}(Z)$. Then ${\mathcal{V}}$ is uniformly $N$-bounded and its $d$-multiplicity is $\le\dim Z+1$. The covering ${\mathcal{V}}$ of ${\operatorname{Co}}(Z)$ can be represented as the disjoint union ${\mathcal{V}}={\mathcal{V}}_0\cup{\mathcal{V}}_\infty$, where ${\mathcal{V}}_0$ consists of all $V\in{\mathcal{V}}$ with finite diameter, and respectively ${\mathcal{V}}_\infty$ consists of all $V\in{\mathcal{V}}$ with infinite diameter.
There are natural polar coordinates $x=(z,t)$, $z\in Z$, $t\ge 0$, in ${\operatorname{Co}}(Z)$. Every 1-Lipschitz homeomorphism $f:[0,\infty)\to[0,\infty)$ induces a 1-Lipschitz homeomorphism $F:{\operatorname{Co}}(Z)\to{\operatorname{Co}}(Z)$ by $F(z,t)=(z,f(t))$, which does not change the visual diameter ${\operatorname{diam}}({\operatorname{sh}}(A))$ of any $A{\subset}{\operatorname{Co}}(Z)$.
Given $V\in{\mathcal{V}}_0$ let $W=W_V{\subset}[0,\infty)$ be the ray projection of $V$, i.e., $$W={\{t\ge 0:\,\text{$(z,t)\in V\ \text{for some}\
z\in Z$}\}}.$$ Then ${\operatorname{diam}}W<\infty$ and the collection ${\mathcal{W}}={\{W_V:\,\text{$V\in{\mathcal{V}}_0$}\}}$ is locally finite in $[0,\infty)$ since $Z$ is compact and ${\mathcal{V}}_0$ is locally finite. By radial contraction Lemma \[Lem:coveray\] there is 1-Lipschitz homeomorphism $f:[0,\infty)\to[0,\infty)$ such that every set $f(W){\subset}[0,\infty)$, $W\in{\mathcal{W}}$, has diameter at most 1. For the induced homeomorphism $F:{\operatorname{Co}}(Z)\to{\operatorname{Co}}(Z)$ we denote by ${\mathcal{V}}^1$ the covering $F({\mathcal{V}})={\{F(V):\,\text{$V\in{\mathcal{V}}$}\}}$ of ${\operatorname{Co}}(Z)$. Then ${\mathcal{V}}^1={\mathcal{V}}_0^1\cup{\mathcal{V}}_{\infty}^1$ for ${\mathcal{V}}_0^1=F({\mathcal{V}}_0)$, ${\mathcal{V}}_{\infty}^1=F({\mathcal{V}}_\infty)$, and every $V\in{\mathcal{V}}_0^1$ lies in some annulus of width 1 centered at $o$.
Consider the sequence of contracting homeomorphisms $F_k:{\operatorname{Co}}(Z)\to{\operatorname{Co}}(Z)$ given by $F_k(z,t)=(z,\frac{1}{k}t)$, $(z,t)\in{\operatorname{Co}}(Z)$, $k\in{\mathbb{N}}$, and the corresponding sequence of coverings ${\mathcal{V}}^k={\mathcal{V}}_0^k\cup{\mathcal{V}}_{\infty}^k$, where ${\mathcal{V}}^k=F_k({\mathcal{V}}^1)$. Using Lemma \[Lem:ubgshadow\] and the fact that $F_k$ does not change the visual diameter, one easily obtains that the diameter of the elements from ${\operatorname{An}}(Z)\cap{\mathcal{V}}_0^k$ vanishes as $k\to\infty$.
On the other hand, the angle measure of $U{\setminus}B_c(o)$, $U\in{\mathcal{U}}$, is exponentially small in $c$ by Lemma \[Lem:ubgshadow\]. It follows that for every ${\varepsilon}>0$ the shadow of $U{\setminus}B_c(o)$ is contained in the ${\varepsilon}$-neighborhood of the finite set ${{\partial}_{\infty}}U$, if $c$ is chosen sufficiently large, ${\operatorname{sh}}(U{\setminus}B_c(o)){\subset}D_{\varepsilon}({{\partial}_{\infty}}U){\subset}{{\partial}_{\infty}}{\operatorname{H}}^n$. Thus for each $V\in{\mathcal{V}}_\infty$ the sequence $F_k\circ F(V)\cap{\operatorname{An}}(Z)$ converges to a finite union of segments $(z,[1,2]){\subset}{\operatorname{An}}(Z)$, $z\in Z_V$, as $k\to\infty$, where $Z_V{\subset}{{\partial}_{\infty}}U$ for $U\in{\mathcal{U}}$, $U\supset V$. Since there are only finitely many elements $U\in{\mathcal{U}}$ with infinite diameter, and every $V\in{\mathcal{V}}_\infty$ is contained in one of them, there is a finite $A{\subset}Z$ such that $Z_V{\subset}A$ for every $V\in{\mathcal{V}}_\infty$. Moreover, for every ${\varepsilon}>0$ there is $k_{\varepsilon}\in{\mathbb{N}}$ such that the compact set ${\operatorname{An}}_{\varepsilon}(Z)={\operatorname{An}}(Z{\setminus}D_{\varepsilon}(A))$ misses any $V\in{\mathcal{V}}_{\infty}^k$, and thus it is covered by elements of ${\mathcal{V}}_0^k$, $k\ge k_{\varepsilon}$. The multiplicity of the covering ${\mathcal{V}}_0^k$ of ${\operatorname{An}}_{\varepsilon}(Z)$ is $\le\dim Z+1$, and the diameter of its elements vanishes as $k\to\infty$. Thus $\dim{\operatorname{An}}_{\varepsilon}(Z)\le\dim Z$. However, by ($\ast$) and Lemma \[Lem:remove\], $\dim{\operatorname{An}}_{\varepsilon}(Z)=\dim Z+1$ for all sufficiently small ${\varepsilon}>0$. This is a contradiction.
Proofs of Theorem \[Thm:main\] and Corollary \[Cor:nonemb\]
-----------------------------------------------------------
By [@BoS Theorem 1.1], the space $X$ is roughly similar to a convex subset of ${\operatorname{H}}^n$ for some integer $n$. Actually, it is proved there that ${{\partial}_{\infty}}X$ is homeomorphic to a compact $Z{\subset}{\operatorname{H}}^n$ such that $X$ is roughly similar to the convex hull of $Z$ in ${\operatorname{H}}^n$. Thus we can assume that $X$ is the convex hull of some compact, infinite $Z{\subset}{\operatorname{H}}^n$. Now, Proposition \[Pro:hypdimcone\] completes the proof.
[*Proof of Corollary \[Cor:nonemb\].*]{} We actually prove a more general result.
\[Thm:nonemb\] Assume that there is a quasi-isometric embedding $$f:X\to T_1\times\dots\times T_k\times Y,$$ of a Gromov hyperbolic space $X$, satisfying the conditions of Theorem \[Thm:main\], into the $k$-fold product of metric trees $T_1,\dots,T_k$ stabilized by an UBG-factor $Y$. Then $k\ge\dim{{\partial}_{\infty}}X+1$.
First, we note that ${\operatorname{hypdim}}Y=0$. Next, the asymptotic dimension of every metric tree $T$ is at most 1, ${\operatorname{asdim}}T\le 1$, see [@DJ Proposition 4]. Therefore, ${\operatorname{hypdim}}T\le 1$, and by the Product Theorem the hyperbolic dimension of the target space is at most $k$. On the other hand, ${\operatorname{hypdim}}X\ge\dim{{\partial}_{\infty}}X+1$ by Theorem \[Thm:main\]. Now, the required estimate follows from mononicity of ${\operatorname{hypdim}}$, see Theorem \[Thm:monhypdim\].
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T. Januszkiewicz and J. Swiatkowski, [*Hyperbolic Coxeter groups*]{}, Comment. Math. Helv. 78 (2003), 555–583.
Sergei Buyalo,11em= Viktor Schroeder,\
St. Petersburg Dept. of Steklov Institut für Mathematik, Universität\
Math. Institute RAS, Fontanka 27, Zürich, Winterthurer Strasse 190,\
191023 St. Petersburg, Russia CH-8057 Zürich, Switzerland\
[sbuyalo@pdmi.ras.ru]{}\
[^1]: Supported by RFFI Grant 02-01-00090, CRDF Grant RM1-2381-ST-02 and SNF Grant 20-668 33.01
[^2]: Supported by Swiss National Science Foundation
|
---
abstract: 'We give a new proof of Braden’s theorem ([@Br]) about *hyperbolic restrictions* of constructible sheaves/D-modules. The main geometric ingredient in the proof is a 1-parameter family that degenerates a given scheme $Z$ equipped with a $\BG_m$-action to the product of the attractor and repeller loci.'
author:
- 'V. Drinfeld and D. Gaitsgory'
title: On a theorem of Braden
---
Introduction {#introduction .unnumbered}
============
Given a scheme $Z$ (or algebraic space) of finite type over a field $k$ of characteristic 0, let $\Dmod(Z)$ denote the DG category of D-modules on it.
If $f:Z_1\to Z_2$ is a morphism of such schemes one has the de Rham direct image functor $f_{\bullet}:\Dmod(Z_1)\to \Dmod(Z_2)$ and the $!$-pullback functor $f^!:\Dmod(Z_2)\to \Dmod(Z_1)$. One also has the *partially* defined functor $f^\bullet:\Dmod(Z_2)\to \Dmod(Z_1)$, left adjoint to $f_{\bullet}\,$.
Suppose now that $Z$ is equipped with an action of the group $\BG_m\,$. Let $Z^+$ (resp., $Z^-$) denote the corresponding attractor (resp., repeller) locus, see Sects \[ss:attr\] and \[ss:repeller\] for the definitions. Let $Z^0$ denote the locus of $\BG_m$-fixed points.
Consider the diagram $$\label{e:square with arrow prev}
\xy
(30,0)*+{Z^+}="Y";
(0,-30)*+{Z^-}="Z";
(30,-30)*+{Z.}="W";
(-5,5)*+{Z^0}="U";
{\ar@{->}_{p^-} "Z";"W"};
{\ar@{->}^{p^+} "Y";"W"};
{\ar@{->}^{i^-} "U";"Z"};
{\ar@{->}_{i^+} "U";"Y"};
\endxy$$
Let $\Dmod(Z)^{\BG_m\on{-mon}}\subset \Dmod(Z)$ be the full subcategory consisting of $\BG_m$-monodromic[^1] objects. In the context of D-modules, Braden’s theorem [@Br] (inspired by a result[^2] from [@GM]) says that the composed functors $$(i^+)^\bullet\circ (p^+)^! \text{ and } (i^-)^!\circ (p^-)^\bullet, \quad \Dmod(Z)\to \Dmod(Z^0)$$ are both defined [^3] on objects of $\Dmod(Z)^{\BG_m\on{-mon}}$ and we have a canonical isomorphism $$\label{e:Braden preview}
(i^+)^\bullet\circ (p^+)^!|_{\Dmod(Z)^{\BG_m\on{-mon}}} \simeq (i^-)^!\circ (p^-)^\bullet|_{\Dmod(Z)^{\BG_m\on{-mon}}}\;\; .$$
In his paper [@Br], T. Braden formulated and proved his theorem assuming that $Z$ is a normal algebraic variety. Although his formulation is enough for practical purposes, we prefer to formulate and prove this theorem for *algebraic spaces* of finite type over a field (without any normality or separateness conditions).
In this more general context, the representability of the functors defining the attractor $Z^+$ (and other related spaces such as $\wt{Z}$ from [Sect. \[sss:behind\]]{} below) is no longer obvious; it is established in [@Dr].
Braden’s theorem is hugely important in geometric representation theory.
Here is a typical application in the context of Lusztig’s theory of induction and restriction of character sheaves.
Take $Z=G$, a connected reductive group. Let $P\subset G$ be a parabolic, and let $P^-$ be an opposite parabolic, so that $M:=P\cap P^-$ identifies with the Levi quotient of both $P$ and $P^-$. Denote the corresponding closed embeddings by $$M\overset{i^+}\hookrightarrow P\overset{p^+}\hookrightarrow G \text{ and }
M\overset{i^-}\hookrightarrow P^-\overset{p^-}\hookrightarrow G.$$
Then the claim is that we have a canonical isomorphism of functors $$\Dmod(G)^{\on{Ad}_G\on{-mon}}\to \Dmod(M)^{\on{Ad}_M\on{-mon}}$$ between the corresponding categories of $\on{Ad}$-monodromic D-modules: $$\label{e:restr}
(i^+)^\bullet\circ (p^+)^!\simeq (i^-)^!\circ (p^-)^\bullet.$$
The proof is immediate from : the corresponding $\BG_m$-action is the adjoint action corresponding to a co-character $\BG_m\to M$, which maps to the center of $M$ and is dominant regular with respect to $P$. [^4]
For other applications of Braden’s theorem see [@Ach], [@AC], [@AM], [@Bi-Br], [@GH], [@Ly1], [@Ly2], [@MV], [@Nak].
The goal of this paper is to give an alternative proof of Braden’s theorem. The reason for our decision to publish it is that
\(a) the new proof gives another point of view on “what Braden’s theorem is really about";
\(b) a slight modification of the new proof of Braden’s theorem allows to prove a new result in the geometric theory of automorphic forms, see [@DrGa3 Thm. 1.2.5].
Let us explain the idea of the new proof.
Let us complete the diagram to $$\label{e:square with arrow prev enh}
\xy
(30,0)*+{Z^+}="Y";
(0,-30)*+{Z^-}="Z";
(30,-30)*+{Z,}="W";
(-5,5)*+{Z^0}="U";
{\ar@{->}_{p^-} "Z";"W"};
{\ar@{->}^{p^+} "Y";"W"};
{\ar@{->}^{i^-} "U";"Z"};
{\ar@{->}_{i^+} "U";"Y"};
{\ar@<-1.3ex>_{q^+} "Y";"U"};
{\ar@<1.3ex>^{q^-} "Z";"U"};
\endxy$$ where $$q^+:Z^+\to Z^0 \text{ and } q^-:Z^-\to Z^0$$ are the corresponding contraction maps.
First, we observe that the functors $(i^+)^\bullet$ and $(i^-)^!$, when restricted to the corresponding monodromic categories, are isomorphic to $(q^+)_\bullet$ and $(q^-)_!$, respectively. Hence, the isomorphism can be rewritten as $$\label{e:Braden preview q}
(q^+)_\bullet\circ (p^+)^!|_{\Dmod(Z)^{\BG_m\on{-mon}}} \simeq (q^-)_!\circ (p^-)^\bullet|_{\Dmod(Z)^{\BG_m\on{-mon}}}.$$
Next we observe that the functor $(q^-)_!\circ (p^-)^\bullet$ is the left adjoint functor of $(p^-)_\bullet\circ (q^-)^!$. Hence, *the isomorphism can be restated as the assertion that the functors $$(q^+)_\bullet\circ (p^+)^!|_{\Dmod(Z)^{\BG_m\on{-mon}}} \text{ and } (p^-)_\bullet\circ (q^-)^!|_{\Dmod(Z)^{\BG_m\on{-mon}}}$$ form an adjoint pair.*
\[sss:behind\] In turns out that the co-unit for this adjunction, i.e., the map $$\label{e:counit preview}
(q^+)_\bullet\circ (p^+)^!\circ (p^-)_\bullet\circ (q^-)^!\to \on{Id}_{\Dmod(Z^0)}\, ,$$ is easy to write down (just as in the original form of Braden’s theorem, a map in one direction is obvious).
The crux of the new proof consists of writing down the unit for the adjunction, i.e., the corresponding map $$\label{e:unit preview}
\on{Id}_{\Dmod(Z)^{\BG_m\on{-mon}}}\to (p^-)_\bullet\circ (q^-)^! \circ (q^+)_\bullet\circ (p^+)^!|_{\Dmod(Z)^{\BG_m\on{-mon}}}\, .$$
The map comes from a certain geometric construction described in [Sect. \[s:deg\]]{}. Namely, we construct a $1$-parameter “family" [^5] of schemes (resp., algebraic spaces) $\wt{Z}_t$ mapping to $Z\times Z$ (here $t\in \BA^1$) such that for $t\ne 0$ the scheme (resp., algebraic space) $\wt{Z}_t$ is the graph of the map $t:Z{\buildrel{\sim}\over{\longrightarrow}}Z$, and $\wt{Z}_0$ is isomorphic to $Z^+\underset{Z^0}\times Z^-$.
\[ss:other\]
This paper is written in the context of D-modules on schemes (or more generally, algebraic spaces of finite type) over a field $k$ of characteristic 0.
However, Braden’s theorem can be stated in other sheaf-theoretic contexts, where the role of the DG category $\Dmod(Z)$ is played by a certain triangulated category $D(Z)$. The two other contexts that we have in mind are as follows:
\(i) $k$ is any field, and $D(Z)$ is the derived category of ${\mathbb Q}_\ell$-sheaves with constructible cohomologies,
\(ii) $k=\BC$, and $D(Z)$ is the derived category of sheaves of $R$-modules with constructible cohomologies (where $R$ is any ring).
In these two contexts the new proof of Braden’s theorem presented in this article goes through with the following modifications:
First, the functors $f^\bullet$ and $f_!$ are always defined, so one should not worry about pro-categories.[^6]
Second, the definition of the $G$-monodromic category $D(Z)^{G\on{-mon}}$ (where $G$ is any algebraic group, e.g., the group $\BG_m$) should be slightly different from the definition of $\Dmod(Z)^{G\on{-mon}}$ given in [Sect. \[sss:mon\]]{}.
Namely, $D(Z)^{G\on{-mon}}$ should be defined as the full subcategory of $D(Z)$ *strongly generated* by the essential image of the pullback functor $D(Z/G)\to D(Z)$ (i.e., its objects are those objects of $D(Z)$ that can be obtained from objects lying in the image of the above pullback functor by a *finite* iteration of the procedure of taking the cone of a morphism).
In Sects. \[s:actions\] and \[s:deg\] we will work over an arbitrary ground field $k$, and in Sects. \[s:Braden1\]-\[s:Verifying\] we will assume that $k$ has characteristic $0$ (because we will be working with D-modules).
In this article all schemes, algebraic spaces, and stacks are assumed to be “classical" (as opposed to derived).
When working with D-modules, our conventions follow those of [@DrGa1 Sects. 5 and 6]. The only notational difference is that for a morphism $f:Z_1\to Z_2$, we will denote the direct image functor $\Dmod(Z_1)\to \Dmod(Z_2)$ by $f_\bullet$ (instead of $f_{\dr,*}$), and similarly for the left adjoint, $f^\bullet$ (instead of $f^*_{\dr}$).
Given an an algebraic space (or stack) $Z$ of finite type over a field $k$ of characteristic 0, we let $\Dmod(Z)$ denote the DG category of D-modules on it. Our conventions regarding DG categories follow those of [@DrGa1 Sect. 0.6].
On the other hand, the reader may prefer to replace each time the DG category $\Dmod(Z)$ by its homotopy category, which is a triangulated category.[^7] Then the formulations and proofs of the main results of this article will remain valid. Moreover, once we know that the morphism in the triangulated setting is the co-unit of an adjunction, it follows that the same is true in the DG setting.
In Appendix \[s:pro\] we define the notion of pro-completion $\on{Pro}(\bC)$ of a DG category $\bC$.
The reader who prefers to stay in the triangulated world, can replace it by the category of all covariant triangulated functors from the homotopy category $\on{Ho}(\bC)$ to the homotopy category of complexes of $k$-vector spaces. (Note that the category of such functors is not necessarily triangulated, but this is of no consequence for us.)
Sects. \[s:actions\]-\[s:deg\] are devoted to the geometry of $\BG_m$-actions on algebraic spaces.
Let $Z$ be an algebraic space of finite type over the ground field $k$, equipped with a $\BG_m$-action. In [Sect. \[s:actions\]]{} we define the attractor $Z^+$ and the repeller $Z^-$ by $$\label{e:attr&repel}
Z^+:=\GMaps(\BA^1,Z), \quad\quad Z^-:=\GMaps(\BA^1_-\, ,Z),$$ where $\GMaps$ stands for the space of $\BG_m$-equivariant maps and $\BA^1_-:=\BP^1-\{\infty\}$ (or equivalently, $\BA^1_-$ is the affine line equipped with the $\BG_m$-action opposite to the usual one). The basic facts on $Z^\pm$ are formulated in [Sect. \[s:actions\]]{}; the proofs of the more difficult statements are given in [@Dr].
As was already mentioned in [Sect. \[sss:behind\]]{}, in the proof of Braden’s theorem we use a certain 1-parameter family of algebraic spaces $\wt{Z}_t$, $t\in\BA^1$. These spaces are defined and studied in [Sect. \[s:deg\]]{}. The definition is formally similar to : namely, $$\wt{Z}_t:=\GMaps (\BX_t\,, Z),$$ where $\BX_t$ is the hyperbola $\tau_1\cdot \tau_2=t$ and the action of $\lambda\in\BG_m$ on $\BX_t$ is defined by $$\tilde\tau_1=\lambda\cdot \tau_1\, , \quad\quad\tilde\tau_2=\lambda^{-1}\cdot\tau_2\, .$$ Note that $\BX_0$ is the union of the two coordinate axes, which meet at the origin; accordingly, $\wt{Z}_0$ identifies with $Z^+\underset{Z^0}\times Z^-$ (as promised in [Sect. \[sss:behind\]]{}).
In [Sect. \[s:Braden1\]]{} we first state Braden’s theorem in its original formulation, and then reformulate it as a statement that certain two functors are adjoint (with the specified co-unit of the adjunction).
In [Sect. \[s:unit\]]{} we carry out the main step in the proof of [Theorem \[t:Braden adj\]]{} by constructing the unit morphism for the adjunction.
The geometric input in the construction of the unit is the family $t\rightsquigarrow \wt{Z}_t$ mentioned above. The input from the theory of D-modules is the *specialization map* $$\on{Sp}_\CK:\CK_1\to \CK_0\, ,$$ where $\CK$ is a $\BG_m$-monodromic object in $\Dmod(\BA^1\times \CY)$ (for any algebraic space/stack $\CY$), and where $\CK_1$ and $\CK_0$ are the !-restrictions of $\CK$ to $\{1\}\times \CY$ and $\{0\}\times \CY$, respectively. The map $\on{Sp}_\CK$ is a simplified version of the specialization map that goes from nearby cycles to the !-fiber.
In [Sect. \[s:Verifying\]]{} we show that the unit and co-unit indeed satisfy the adjunction property.
In Appendix \[s:pro\] we define the notion of pro-completion $\on{Pro}(\bC)$ of a DG category $\bC$.
We thank A. Beilinson, T. Braden, J. Konarski, and A. J. Sommese for helpful discussions.
The research of V. D. is partially supported by NSF grants DMS-1001660 and DMS-1303100. The research of D. G. is partially supported by NSF grant DMS-1063470.
Geometry of $\BG_m$-actions: fixed points, attractors, and repellers {#s:actions}
====================================================================
In this section we review the theory of action of the multiplicative group $\BG_m$ on a scheme or algebraic space $Z$. Specifically, we are concerned with the fixed-point locus, denoted by $Z^0$, as well as the attractor/repeller spaces, denoted by $Z^+$ and $Z^-$, respectively.
The main results of this section are [Proposition \[p:Z\^0closed\]]{} (which says that the fixed-point locus is closed), Theorem \[t:attractors\] (which ensures representability of attractor/repeller sets), and [Proposition \[p:Cartesian\]]{} (the latter is used in the construction of the unit of the adjunction given in [Sect. \[sss:defining co-unit\]]{}).
In the case of a scheme equipped with a locally linear $\BG_m$-action these results are well known (in a slightly different language).
\[ss:k spaces\]
We fix a field $k$ (of any characteristic). By a $k$-*space* (or simply *space*) we mean a contravariant functor $Z$ from the category of affine schemes to that of sets which is a sheaf for the fpqc topology. Instead of $Z(\Spec(R))$ we write simply $Z(R)$; in other words, we consider $Z$ as a covariant functor on the category of $k$-algebras.
Note that for any scheme $S$ we have $Z(S)=\Maps (S,Z)$, where $\Maps$ stands for the set of morphisms between spaces. Usually we prefer to write $\Maps (S,Z)$ rather than $Z(S)$.
We write $\on{pt}:=\Spec(k)$.
General spaces will appear only as “intermediate" objects. For us, the really geometric objects are *algebraic spaces* over $k$. We will be using the definition of algebraic space from [@LM] (which goes back to M. Artin). [^8]
Any quasi-separated scheme (in particular, any scheme of finite type) is an algebraic space. The reader may prefer to restrict his attention to schemes, and even to separated schemes, as this will cover most of the cases of interest to which the main the result of this paper, i.e., [Theorem \[t:braden original\]]{}, is applied.
Note that in the definition of spaces, instead of considering affine schemes as “test schemes", one can consider algebraic spaces (any fpqc sheaf on the category of affine schemes uniquely extends to an fpqc sheaf on the category of algebraic spaces).
A morphism of spaces $f:Z_1\to Z_2$ is said to be a *monomorphism* if the corresponding map $$\Maps (S,Z_1)\to\Maps (S,Z_2)$$ is injective for any scheme $S$. In particular, this applies if $Z_1$ and $Z_2$ are algebraic spaces. It is known that a morphism *of finite type* between schemes (or algebraic spaces) is a monomorphism if and only if each of its geometric fibers is a reduced scheme with at most one point.
A morphism of algebraic spaces is said to be *unramified* if it has locally finite presentation and its geometric fibers are finite and reduced.
\[ss:GMaps\]
Let $Z_1$ and $Z_2$ be spaces. We define the space $\MMaps(Z_1,Z_2)$ by $$\Maps(S,\MMaps(Z_1,Z_2)):=\Maps(S\times Z_1,Z_2)$$ (the right-hand side is easily seen to be an fpqc sheaf with respect to $S$).
Let $Z_1,Z_2$ be spaces equipped with an action of $\BG_m$. Then we define the space $\GMaps(Z_1,Z_2)$ as follows: for any scheme $S$, $$\Maps (S,\GMaps(Z_1,Z_2)):=\Maps (S\times Z_1,Z_2)^{\BG_m}$$ (the right-hand side is again easily seen to be an fpqc sheaf with respect to $S$).
The action of $\BG_m$ on $Z_2$ induces a $\BG_m$-action on $\GMaps(Z_1,Z_2)$.
Note that even if $Z_1$ and $Z_2$ are schemes, the space $\GMaps(Z_1,Z_2)$ does not have to be a scheme (or an algebraic space), in general.
\[ss:fixed\_points\]
Let $Z$ be a space equipped with an action of $\BG_m$. Then we set $$Z^0:=\GMaps(\on{pt},Z).$$
Note that $Z^0$ is a subspace of $Z$ because $\Maps (S,Z^0)=\Maps (S,Z)^{\BG_m}$ is a subset of $\Maps (S,Z)$.
$Z^0$ is called *the subspace of fixed points* of $Z$.
We have the following result:
\[p:Z\^0closed\] If $Z$ is an algebraic space (resp. scheme) of finite type then so is $Z^0$. Moreover, the morphism $Z^0\to Z$ is a closed embedding.
The assertion of the proposition is nearly tautological if $Z$ is separated. This case will suffice for most of the cases of interest to which the main result of this paper applies.
The proof in general is given in [@Dr Prop. 1.2.2]. It is not difficult; the only surprise is that $Z^0\subset Z$ is closed even if $Z$ is not separated. (Explanation in characteristic zero: $Z^0$ is the subspace of zeros of the vector field on $Z$ corresponding to the $\BG_m$-action.)
\[ex:fixed-affine\] Suppose that $Z$ is an affine scheme $\Spec(A)$. A $\BG_m$-action on $Z$ is the same as a $\BZ$-grading on $A$. Namely, the $n$-th component of $A$ consists of $f\in \Gamma(Z,\CO_Z)$ such that $f(\lambda\cdot z)=\lambda^n\cdot f(z)$.
It is easy to see that $Z^0=\Spec(A^0)$, where $A^0$ is the maximal graded quotient algebra of $A$ concentrated in degree 0 (in other words, $A^0$ is the quotient of $A$ by the ideal generated by homogeneous elements of non-zero degree).
\[ss:attr\]
Let $Z$ be a space equipped with an action of $\BG_m$. Then we set $$\label{e:attr}
Z^+:=\GMaps(\BA^1,Z),$$ where $\BG_m$ acts on $\BA^1$ by dilations.
$Z^+$ is called the *attractor* of $Z$.
\[sss:structures\]
\(i) $\BA^1$ is a monoid with respect to multiplication. The action of $\BA^1$ on itself induces an $\BA^1$-action on $Z^+$, which extends the $\BG_m$-action defined in [Sect. \[ss:GMaps\]]{}.
\(ii) Restricting a morphism $\BA^1\times S\to Z$ to $\{1\}\times S$ one gets a morphism $S\to Z$. Thus we get a $\BG_m$-equivariant morphism $p^+:Z^+\to Z$.
Note that if $Z$ is *separated* (i.e., the diagonal morphism $Z\to Z\times Z$ is a closed embedding), then $p^+:Z^+\to Z$ is a *mono*morphism. To see this, it suffices to interpret $p^+$ as the composition $$\GMaps (\BA^1,Z)\to \GMaps (\BG_m,Z)=Z.$$
Thus if $Z$ is separated then $p^+$ identifies $Z^+(S)$ with the subset of those points $f:S\to Z$ for which the map $S\times \BG_m\to Z$, defined by $(s,t)\mapsto t\cdot f(s)$, extends to a map $S\times \BA^1\to Z$; informally, the limit $$\label{e:limit}
\underset{t\to 0}{lim}\,\, t\cdot z$$ should exist.
\(iii) Recall that $Z^0=\GMaps (\on{pt}, Z)$. The $\BG_m$-equivariant maps $0:\on{pt}\to\BA^1$ and $\BA^1\to \on{pt}$ induce the maps $$q^+:Z^+\to Z^0 \text{ and } i^+:Z^0\to Z^+,$$ such that $q^+\circ i^+=\id_{Z^0}$, and the composition $p^+\circ i^+$ is equal to the canonical embedding $Z^0{\hookrightarrow}Z$.
Note that if $Z$ is separated then for $z\in Z^+(S)\subset Z(S)$ the point $q^+(S)$ is the limit .
\[sss:contracting\]
Let $Z$ be a separated space. Then it is clear that if a $\BG_m$-action on $Z$ can be extended to an action of the monoid $\BA^1$ then such an extension is unique. In this case we will say that that the $\BG_m$-action is *contracting*.
\[p:contracting\] Let $Z$ be a separated space of finite type equipped with a $\BG_m$-action. The morphism $p^+:Z^+\to Z$ is an isomorphism if and only if the $\BG_m$-action on $Z$ is contracting.
The “only if" assertion follows from [Sect. \[sss:structures\]]{}(i). For the “if" assertion, we note that the $\BA^1$-action on $Z$ defines a morphism $g:Z\to Z^+$ such that the composition of the maps $$\label{e:mono-epi}
Z\overset{g}\longrightarrow Z^+\overset{p^+}\longrightarrow Z$$ equals $\id_Z$. Since the map $p^+$ is a monomorphism (see [Sect. \[sss:structures\]]{}(ii)), the assertion follows.
\[r:contracting\] In [@Dr Prop. 1.4.15] it will be shown that if $Z$ is an *algebraic* space of finite type, then the assertion of [Proposition \[p:contracting\]]{} remains valid even if $Z$ is not separated: i.e., $p^+$ is an isomorphism if and only if the $\BG_m$-action on $Z$ can be extended to an $\BA^1$-action; moreover, such an extension is unique.
\[sss:attractors-affine\]
Suppose that $Z$ is affine, i.e., $Z=\Spec(A)$, where $A$ is a $\BZ$-graded commutative algebra. It is easy to see that in this case $Z^+$ is represented by the affine scheme $\Spec(A^+)$, where $A^+$ is the maximal $\BZ^{\geq 0}$-graded quotient algebra of $A$ (in other words, the quotient of $A$ by the ideal generated by by all homogeneous elements of $A$ of strictly negative degrees).
By Example \[ex:fixed-affine\], $Z^0=\Spec(A^0)$, where $A^0$ is the maximal graded quotient algebra of $A$ (or equivalently, of $A^+$) concentrated in degree 0. Since the algebra $A^+$ is $\BZ^{\geq 0}$-graded, $A^0$ identifies with the $0$-th graded component of $A^+$. Thus we obtain the homomorphisms $A^0{\hookrightarrow}A^+{\twoheadrightarrow}A^0$. They correspond to the morphisms $$Z^0\overset{\;\;q^+}\longleftarrow Z^+\overset{\;\;i^+}\longleftarrow Z^0\,.$$
We have:
\[l:U\^+\] Let $Z$ be a space equipped with a $\BG_m$-action, and let $Y\subset Z$ be a $\BG_m$-stable open subspace.
[*(i)*]{} Suppose that $Y\to Z$ is an open embedding. Then the subspace $Y^+\subset Z^+$ equals $(q^+)^{-1}(Y^0)$, where $q^+$ is the natural morphism $Z^+\to Z^0$.
[*(ii)*]{} Suppose that $Y\to Z$ is a closed embedding. Then the subspace $Y^+\subset Z^+$ equals $(p^+)^{-1}(Y)$, where $q^+$ is the natural morphism $Z^+\to Z$.
Let $Y\to Z$ be an open embedding. For any test scheme $S$, we have to show that if $$f:S\times \BA^1\to Z$$ is a $\BG_m$-equivariant morphism such that $\{0\}\times S\subset f^{-1}(Y)$ then $f^{-1}(Y)=S\times \BA^1$. This is clear because $f^{-1}(Y)\subset S\times \BA^1$ is open and $\BG_m$-stable.
Let $Y\to Z$ be a closed embedding. An $S$-point of $(p^+)^{-1}(Y)$ is a $\BG_m$-equivariant morphism $f:S\times \BA^1\to Y$ such that $S\times \BG_m\subset f^{-1}(Y)$. Since $f^{-1}(Y)$ is closed in $S\times \BA^1$ this implies that $f^{-1}(Y)=S\times \BA^1$, i.e., $f(S\times \BA^1)\subset Y$.
\[ss:Results\_attractors\]
We have the following assertion:
\[t:attractors\] Let $Z$ be an algebraic space of finite type equipped with a $\BG_m$-action. Then
*(i)* $Z^+$ is an algebraic space of finite type;
*(ii)* The morphism $q^+:Z^+\to Z^0$ is affine.
The proof of this theorem is given in [@Dr Thm. 1.4.2]. Here we will prove a particular case (see [Sect. \[sss:loc lin\]]{}), sufficient for most of the cases of interest to which the main result of this paper applies.
Combining [Theorem \[t:attractors\]]{} with [Proposition \[p:Z\^0closed\]]{}, we obtain:
*(i)* If $Z$ is a separated algebraic space of finite type then so is $Z^+$.
*(ii)* If $Z$ is a scheme of finite type then so is $Z^+$.
Follows from Theorem \[t:attractors\](ii) because by [Proposition \[p:Z\^0closed\]]{}, $Z^0$ is a closed subspace of $Z$.
\[sss:loc lin\]
\[d:locally linear\] An action of $\BG_m$ on a scheme $Z$ is said to be *locally linear* if $Z$ can be covered by open affine subschemes preserved by the $\BG_m$-action.
\[r:q-proj loc lin\]
Suppose that $Z$ admits a $\BG_m$-equivariant locally closed embedding into a projective space $\BP(V)$, where $\BG_m$ acts linearly on $V$. Then the action of $\BG_m$ is locally linear.
For this reason, locally linear actions include most of the cases of interest that come up in practice.
\[r:locally linear\] If $k$ is *algebraically closed* and $Z_{\red}$ is a *normal* separated[^9] scheme of finite type over $k$, then by a theorem of H. Sumihiro, *any action of $\BG_m$ on $Z$ is locally linear*. (The proof of this theorem is contained in [@Sum] and also in [@KKMS p.20-23] and [@KKLV].)
Let us prove [Theorem \[t:attractors\]]{} in the locally linear case on a scheme. First, we note that [Lemma \[l:U\^+\]]{}(i) reduces the assertion to the case when $Z$ is affine. In the latter case, the assertion is manifest from [Sect. \[sss:attractors-affine\]]{}.
\[ss:further contractors\]
The results of this subsection are included for completeness; they will not be used for the proof of the main theorem of this paper.
We let $Z$ be an algebraic space of finite type equipped with a $\BG_m$-action.
We have:
\[p: p\^+\]
[*(i)*]{} If $Z$ is separated then $p^+:Z^+\to Z$ is a monomorphism.
[*(ii)*]{} If $Z$ is an affine scheme then $p^+:Z^+\to Z$ is a closed embedding.
[*(iii)*]{} If $Z$ is proper then each geometric fiber of $p^+:Z^+\to Z$ is reduced and has exactly one geometric point.
[*(iv)*]{} The fiber of $p^+:Z^+\to Z$ over any geometric point of $Z^0\subset Z$ is reduced and has exactly one geometric point.
Point (i) has been proved in [Sect. \[sss:structures\]]{}(ii). Point (ii) is manifest from [Sect. \[sss:attractors-affine\]]{}. Point (iii) follows from point (i) and the fact that any morphism from $\BA^1-\{ 0\}$ to a proper scheme extends to the whole $\BA^1$.
After base change, point (iv) is equivalent to the following lemma:
\[l:constant\] If $f:\BA^1\to Z$ is a $\BG_m$-equivariant morphism such that $f(1)\in Z^0$ then $f$ is constant.
The map $\on{pt}\to Z$, corresponding to $f(1)\in Z(k)$ is a closed embedding (whether or not $Z$ is separated). Hence, the assertion follows from [Lemma \[l:U\^+\]]{}(ii).
\[ex:P\^1\] Let $Z$ be the projective line $\BP^1$ equipped with the usual action of $\BG_m\,$. Then $p^+:Z^+\to Z$ is the canonical morphism $\BA^1\sqcup\{\infty\}\to\BP^1$. In particular, $p^+$ is *not a locally closed embedding.*
\[e:P\^1 glued\] Let $Z$ be the curve obtained from $\BP^1$ by gluing $0$ with $\infty$. Equip $Z$ with the $\BG_m$-action induced by the usual action on $\BP^1$. The map $\BP^1\to Z$ induces a map $(\BP^1)^+\to Z^+$. It is easy to see that the composed map $$\BA^1\hookrightarrow (\BP^1)^+\to Z^+$$ is an isomorphism $\BA^1\simeq Z^+$.
Suppose that the action of $\BG_m$ is locally linear. Then [Proposition \[p: p\^+\]]{}(ii) and [Lemma \[l:U\^+\]]{} imply that the map $p^+$ is, *Zariski locally on the source*, a locally closed embedding.
Note, however, that is is not the case in general, as can be seen from Example \[e:P\^1 glued\].
In the example of $\BP^1$, the restriction of $p^+:Z^+\to Z$ to each connected component of $Z^+$ *is* a locally closed embedding. This turns out to be true in a surprisingly large class of situations (but there are also important examples when this is false):
Let $Z$ be a separated scheme over an algebraically closed field $k$ equipped with a $\BG_m$-action. Then each of the following conditions ensures that the restriction of $p^+:Z^+\to Z$ to each connected component [^10] of $Z^+$ is a locally closed embedding:
[*(i)*]{} $Z$ is smooth;
[*(ii)*]{} $Z$ is normal and quasi-projective;
[*(iii)*]{} $Z$ admits a $\BG_m$-equivariant locally closed embedding into a projective space $\BP(V)$, where $\BG_m$ acts linearly on $V$.
Case (i) is due to A. Bia[ł]{}ynicki-Birula [@Bia]. Case (iii) immediately follows from the easy case $Z=\BP(V)$. Case (ii) turns out to be a particular case of (iii) because by Theorem 1 from [@Sum], if $Z$ is normal and quasi-projective then it admits a $\BG_m$-equivariant locally closed embedding into a projective space.
In case (i) the condition that $Z$ be a scheme (rather than an algebraic space) is essential, as shown by A. J. Sommese [@Som].
In case (ii) the quasi-projectivity condition is essential, as shown by J. Konarski [@Kon] using a method developed by J. Jurkiewicz [@Ju1; @Ju2]. In this example $Z$ is a 3-dimensional toric variety which is proper but not projective; it is constructed by drawing a certain picture on a 2-sphere, see the last page of [@Kon].
In case (ii) normality is clearly essential, see Example \[e:P\^1 glued\].
The results of this subsection are included for the sake of completeness and will not be needed for the sequel.
We let $Z$ be an algebraic space of finite type, equipped with an action of $\BG_m$.
First, we have:
\[l:TZ0\] For any $z\in Z^0$ the tangent space[^11] $T_zZ^0\subset T_zZ$ equals $(T_zZ)^{\BG_m}$.
We can assume that the residue field of $z$ equals $k$ (otherwise do base change). Then compute $T_zZ^0$ in terms of morphisms $\Spec k[\varepsilon ]/(\varepsilon^2 )\to Z^0$.
Next we claim:
\[p:unrami\] Let $Z$ be an algebraic space of finite type equipped with a $\BG_m$-action. Then the map $p^+:Z^+\to Z$ is unramified.
We can assume that $k$ is algebraically closed. Then we have to check that for any $\zeta\in Z^+(k)$ the map of tangent spaces $$\label{e:differential}
T_{\zeta}Z^+\to T_{p^+({\zeta})}Z$$ induced by $p^+:Z^+\to Z$ is injective. Let $f:\BA^1\to Z$ be the $\BG_m$-equivariant morphism corresponding to $\zeta$. Then $$\label{e:TZ+l}
T_{\zeta}Z^+=\Hom_{\BG_m}(f^*(\Omega^1_Z),\CO_{\BA^1}),$$ and the map assigns to a $\BG_m$-equivariant morphism $\varphi :f^*(\Omega^1_Z)\to\CO_{\BA^1}$ the corresponding map between fibers at $1\in\BA^1$. So the kernel of consists of those $\varphi\in\Hom_{\BG_m}(f^*(\Omega^1_Z),\CO_{\BA^1})$ for which $\varphi |_{\BA^1-\{ 0\}}=0$. This implies that $\varphi =0$ because $\CO_{\BA^1}$ has no nonzero sections supported at $0\in\BA^1$.
Finally, we claim:
\[p:smoothness\] Suppose that $Z$ is smooth. Then $Z^0$ and $Z^+$ are smooth. Moreover, the morphism $q^+:Z^+\to Z^0$ is smooth.
We will only prove that $q^+$ is smooth. (Smoothness of $Z^0$ can be proved similarly, and smoothness of $Z^+$ follows.)
It suffices to check that $q^+$ is formally smooth. Let $R$ be a $k$-algebra and $\bar R=R/I$, where $I\subset R$ is an ideal with $I^2=0$. Let $\bar f :\Spec(\bar R)\times \BA^1\to Z$ be a $\BG_m$-equivariant morphism and let $\bar f_0:\Spec(\bar R)\to Z^0$ denote the restriction of $\bar f$ to $$\Spec(\bar R)\overset{\on{id}\times \{0\}}\hookrightarrow \Spec(\bar R)\times \BA^1\,.$$ Let $\varphi :\Spec(R)\to Z^0$ be a morphism extending $\bar f_0\,$. We have to extend $\bar f$ to a $\BG_m$-equivariant morphism $f :\Spec(R)\times\BA^1\to Z$ so that $f_0=\varphi$, where $f_0:=f|_{\Spec(R)}$.
Since $Z$ is smooth, we can find a *not-necessarily equivariant* morphism $f :\Spec(\bar R)\times \BA^1\to Z$ extending $\bar f$ with $f_0=\varphi$. Then standard arguments show that the obstruction to existence of a $\BG_m$-equivariant $f$ with the required properties belongs to $$H^1(\BG_m\, ,M), \quad
M:=H^0(\Spec(\bar R)\times\BA^1\, ,\bar f^*(\Theta_Z)\otimes\CJ )\underset{\bar R}\otimes I,$$ where $\Theta_Z$ is the tangent bundle of $Z$ and $\CJ\subset\CO_{\Spec(R)\times\BA^1}$ is the ideal of the zero section. But $H^1$ of $\BG_m$ with coefficients in any $\BG_m$-module is zero.
\[ss:repeller\]
Set $\BA^1_-:=\BP^1-\{\infty\}$; this is a monoid with respect to multiplication containing $\BG_m$ as a subgroup. One has an isomorphism of monoids $$\label{e:inversion}
\BA^1{\buildrel{\sim}\over{\longrightarrow}}\BA^1_-\, , \quad\quad t\mapsto t^{-1}.$$
Given a space $Z$, equipped with a $\BG_m$-action, we set $$\label{e:repel}
Z^-:=\GMaps(\BA^1_-,Z).$$
$Z^-$ is called the *repeller* of $Z$.
Just as in [Sect. \[sss:structures\]]{} one defines an action of the monoid $\BA^1_-$ on $Z^-$ extending the action of $\BG_m\,$, a $\BG_m$-equivariant morphism $p^-:Z^-\to Z$, and $\BA^1_-$-equivariant morphisms $q^-:Z^-\to Z^0$ and $i^-:Z^0\to Z^-$ (where $Z^0$ is equipped with the trivial $\BA^1_-$-action). One has $q^-\circ i^-=\id_{Z^0}\,$, and the composition $p^-\circ i^-$ is equal to the canonical embedding $Z^0{\hookrightarrow}Z$.
Using the isomorphism , one can identify $Z^-$ with the attractor for the inverse action of $\BG_m$ on $Z$ (this identification is $\BG_m$-*anti*-equivariant). Thus the results on attractors from Sects. \[sss:attractors-affine\] and \[ss:Results\_attractors\] imply similar results for repellers.
In particular, if $Z$ is the spectrum of a $\BZ$-graded algebra $A$ then $Z^-$ canonically identifies with $\Spec (A^-)$, where $A^-$ is the maximal $\BZ_-$-graded quotient algebra of $A$.
In this subsection we let $Z$ be an algebraic space of finite type, equipped with an action of $\BG_m$.
First, we claim:
\[l:closed\] The morphisms $i^{\pm}:Z^0\to Z^{\pm}$ are closed embeddings.
It suffices to consider $i^+$. By Theorem \[t:attractors\](ii), the morphism $q^+:Z^+\to Z^0$ is separated. One has $q^+\circ i^+=\id_{Z^0}\,$. So $i^+$ is a closed embedding.
Now consider the fiber product $Z^+\underset{Z}\times Z^-$ formed using the maps $p^{\pm}:Z^{\pm}\to Z$.
\[p:Cartesian\] The map $$\label{e:open-closed}
j:=(i^+,i^-):Z^0\to Z^+\underset{Z}\times Z^-$$ is both an open embedding and a closed one (i.e., is the embedding of a union of some connected components).
If $Z$ is affine then $j$ is an isomorphism (this immediately follows from the explicit description of $Z^{\pm}$ in the affine case, see Sects. \[sss:attractors-affine\] and \[ss:repeller\]).
In general, $j$ is not necessarily an isomorphism. To see this, note that by and , we have $$\label{e:fiberprod}
Z^+\underset{Z}\times Z^-=\GMaps(\BP^1,Z)$$ (where $\BP^1$ is equipped with the usual $\BG_m$-action), and a $\BG_m$-equivariant map $\BP^1\to Z$ does not have to be constant, in general.
We will give a proof in the case when $Z$ is a scheme; the case of an arbitrary algebraic space is treated in [@Dr Prop. 1.6.2].
Writing $j$ as $$Z^0=Z^0\underset{Z}\times Z^0\,\overset{(i^+,i^-)}\longrightarrow\, Z^+\underset{Z}\times Z^-,$$ and using [Lemma \[l:closed\]]{}, we see that $j$ is a closed embedding.
To prove that $j$ is an open embedding, we note that the following diagram is Cartesian:
$$\CD
Z^0 @>{\sim}>> \MMaps^{\BG_m}(\on{pt},Z) @>>> \MMaps(\on{pt},Z) \\
@V{j}VV @VVV @VVV \\
Z^+\underset{Z}\times Z^- @>{\sim}>> \MMaps^{\BG_m}(\BP^1,Z) @>>> \MMaps(\BP^1,Z).
\endCD$$
Now, the required result follows from the next (easy) lemma:
For a scheme $Z$, the map $$Z=\MMaps(\on{pt},Z) \to \MMaps(\BP^1,Z)$$ induced by the projection $\BP^1\to \on{pt}$ is an open embedding.
\[c:contractive\]
[*(i)*]{} If the map $p^+:Z^+\to Z$ is an isomorphism then so are the maps $Z^0\overset{i^-}\longrightarrow Z^-\overset{q^-}\longrightarrow Z^0$.
[*(ii)*]{} If the map $p^-:Z^-\to Z$ is an isomorphism then so are the maps $Z^0\overset{i^+}\longrightarrow Z^+\overset{q^+}\longrightarrow Z^0$.
Let us prove (ii). By [Proposition \[p:Cartesian\]]{}, the morphism $i^+:Z^0\to Z^+$ is an open embedding. It remains to show that any point $\zeta\in Z^+$ is contained in $i^+(Z^0)$. Set $$U_{\zeta}:=\{t\in\BA^1\,|\, t\cdot\zeta\in i^+(Z^0)\}.$$ We have to show that $1\in U_{\zeta}$. But $U_{\zeta}$ is an open $\BG_m$-stable subset of $\BA^1$ containing $0$, so $U_{\zeta}=\BA^1$.
Geometry of $\BG_m$-actions: the key construction {#s:deg}
=================================================
We keep the conventions and notation of [Sect. \[s:actions\]]{}. The goal of this section is, given an algebraic space $Z$ equipped with a $\BG_m$-action, to study a certain canonically defined algebraic space $\wt{Z}$, equipped with a morphism $\wt{Z}\to\BA^1\times Z\times Z$, such that for $t\in\BA^1-\{0\}$ the fiber $\wt{Z}_t$ equals the graph of the map $t:Z{\buildrel{\sim}\over{\longrightarrow}}Z$, and the fiber $\wt{Z}_0\,$, corresponding to $t=0$, equals $Z^+\underset{Z^0}\times Z^-$.
As was mentioned in [Sect. \[sss:behind\]]{}, the space $\wt{Z}$ is the main geometric ingredient in the proof of [Theorem \[t:Braden adj\]]{}. However, the reader can skip this section now and return to it when the time comes.
The main points of this section are following. In [Sect. \[ss:tilde Z\]]{} we define the space $\wt{Z}$ and the main pieces of structure on it (e.g., the morphism $\wt{p}:\wt{Z}\to\BA^1\times Z\times Z$ and the action of $\BG_m\times \BG_m$ on $\wt{Z}$). In [Sect. \[ss:repr of inter\]]{} we address the question of representability of $\wt{Z}$. In [Sect. \[ss:fiber products\]]{} we prove [Proposition \[p:2open embeddings\]]{}, which plays a key role in Sect. \[s:Verifying\].
\[ss:hyperb\]
\[sss:family of hyperbolas\] Set $\BX:=\BA^2=\Spec k[\tau_1,\tau_2]$. Throughout the paper equip $\BX$ with the structure of a scheme over $\BA^1$, defined by the map $$\BA^2\to\BA^1 ,\quad (\tau_1,\tau_2)\mapsto \tau_1\cdot \tau_2\, .$$
Let $\BX_t$ denote the fiber of $\BX$ over $t\in\BA^1$; in other words, $\BX_t\subset\BA^2$ is the curve defined by the equation $\tau_1\cdot \tau_2=t$. If $t\ne 0$ then $\BX_t$ is a hyperbola, while $\BX_0$ is the “coordinate cross" $\tau_1\cdot \tau_2=0$.
One has $\BX_0=\BX_0^+\cup\BX_0^-$, where $$\label{e:X0+-}
\BX_0^+:=\{(\tau_1,\tau_2)\in\BA^2\,|\, \tau_2=0\}\, , \quad \BX_0^-:=\{(\tau_1,\tau_2)\in\BA^2\,|\, \tau_1=0\}\, .$$
\[sss:X\_S\] For any scheme $S$ over $\BA^1$ set $$\BX_S:=\BX \underset{\BA^1}\times S\, .$$ If $S=\Spec(R)$ we usually write $\BX_R$ instead of $\BX_S\,$.
\[sss:structure X\] We will need the following pieces of structure on $\BX$:
[(i)]{} The projection $\BX\to \BA^1$ admits two canonically defined sections: $$\label{e:two sections}
\sigma_1(t)=(1,t) \text{ and } \sigma_2(t)=(t,1).$$
[(ii)]{} The scheme $\BX$ carries a tautological action of the monoid $\BA^1\times \BA^1$: $$(\lambda_1,\lambda_2)\cdot (\tau_1,\tau_2)=(\lambda_1\cdot \tau_1,\lambda_2\cdot \tau_2)\,.$$
This action covers the action of $\BA^1\times \BA^1$ on $\BA^1$, given by the product map $\BA^1\times \BA^1\to \BA^1$ and the tautological action of $\BA^1$ on itself.
[(iii)]{} In particular, we obtain an action of $\BG_m\times \BG_m$ on $\BX$.
This action covers the action of $\BG_m\times \BG_m$ on $\BA^1$, given by the product map $\BG_m\times \BG_m\to \BG_m$ and the tautological action of $\BG_m$ on $\BA^1$.
\[sss:theaction\] Consider the action of the anti-diagonal copy of $\BG_m$ on the scheme $\BX$ from [Sect. \[sss:structure X\]]{}(iii). That is, $$\label{e:hyperbolic}
\lambda\cdot (\tau_1,\tau_2):=(\lambda\cdot \tau_1,\; \lambda{}^{-1}\cdot \tau_2).$$
This action preserves the morphism $\BX\to\BA^1$, so for any scheme $S$ over $\BA^1$ one obtains an action of $\BG_m$ on $\BX_S\,$.
\[r:theaction\] If $S$ is over $\BA^1-\{ 0\}$, then $\BX_S$ is $\BG_m$-equivariantly isomorphic to $S\times \BG_m$ by means of either of the maps $\sigma_1$ or $\sigma_2$.
\[ss:tilde Z\]
Given a space $Z$ equipped with a $\BG_m$-action, define $\wt{Z}$ to be the following space over $\BA^1$: for any scheme $S$ over $\BA^1$ we set $$\Maps_{\BA^1} (S, \wt{Z}):=\Maps (\BX_S,Z)^{\BG_m},$$ where $\BX_S$ is acted on by $\BG_m$ as in [Sect. \[sss:theaction\]]{}.
In other words, for any scheme $S$, an $S$-point of $\wt{Z}$ is a pair consisting of a morphism $S\to\BA^1$ and a $\BG_m$-equivariant morphism $\BX_S\to Z$.
Note that for any $t\in\BA^1(k)$ the fiber $\wt{Z}_t$ has the following description: $$\wt{Z}_t=\GMaps (\BX_t\, ,Z).$$
\[sss:tilde p\]
The sections $\sigma_1$ and $\sigma_2$ (see [Sect. \[sss:structure X\]]{}(i)) define morphisms $$\pi_1:\wt{Z}\to Z \text{ and } \pi_2:\wt{Z}\to Z,$$ respectively.
Let $$\label{e:tilde p}
\wt{p}:\wt{Z}\to \BA^1\times Z\times Z$$ denote the morphism whose first component is the tautological projection $\wt{Z}\to \BA^1$, and the second and the third components are $\pi_1$ and $\pi_2$, respectively.
\[sss:action of G\_m\^2\]
Note that the action of the group $\BG_m\times \BG_m$ on $\BX$ from [Sect. \[sss:structure X\]]{}(iii) gives rise to an action of $\BG_m\times \BG_m$ on $\wt{Z}$. This action has the following properties:
[(i)]{} It is compatible with the canonical map $\wt{Z}\to \BA^1$ via the multiplication map $\BG_m\times \BG_m\to \BG_m$ and the *inverse* of the canonical action of $\BG_m$ on $\BA^1$.
[(ii)]{} It is compatible with $\pi_1:\wt{Z}\to Z$ via the projection on the first factor $\BG_m\times \BG_m\to \BG_m$ and the initial action of $\BG_m$ on $Z$.
[(iii)]{} It is compatible with $\pi_2:\wt{Z}\to Z$ via the projection on the second factor $\BG_m\times \BG_m\to \BG_m$ and the *inverse* of the initial action of $\BG_m$ on $Z$.
\[sss:anti-diagonal\] Restricting to the *anti-diagonal* copy of $\BG_m\subset \BG_m\times \BG_m$ (i.e., $\lambda\mapsto (\lambda,\lambda^{-1})$), we obtain an action of $\BG_m$ on $\wt{Z}$. (It is the same action as the one induced by the initial action of $\BG_m$ on $Z$). This $\BG_m$-action on $\wt{Z}$ preserves the projection $\wt{Z}\to \BA^1$.
Both maps $\pi_1$ and $\pi_2$ are $\BG_m$-equivariant.
For $t\in \BA^1$ let $$\label{e:tilde p_t}
\wt{p}_t:\wt{Z}_t\to Z\times Z$$ denote the morphism induced by (as before, $\wt{Z}_t$ stands for the fiber of $\wt{Z}$ over $t$).
It is clear that $(\wt{Z}_1,\wt{p}_1)$ identifies with $(Z,\Delta_Z:Z\to Z\times Z)$. More generally, for $t\in \BA^1-\{0\}$ the pair $(\wt{Z}_t,\wt{p}_t)$ identifies with the graph of the map $Z\to Z$ given by the action of $t\in \BG_m\,$.
More precisely, we have:
\[p:outside 0\] The morphism induces an isomorphism between $$\BG_m\underset{\BA^1}\times \wt{Z}$$ and the graph of the action morphism $\BG_m\times Z\to Z$.
We are now going to construct a canonical morphism $$\label{e:0 fiber}
\wt{Z}_0\to Z^+\underset{Z^0}\times Z^-.$$
Recall that $\wt{Z}_0=\GMaps (\BX_0\, ,Z)$ and $\BX_0=\BX_0^+\cup\BX_0^-$, where $\BX_0^+$ and $\BX_0^-$ are defined by formula . One has $\BG_m$-equivariant isomorphisms $$\BA^1{\buildrel{\sim}\over{\longrightarrow}}\BX_0^+, \; \; s\mapsto (s,0); \quad\quad\quad \BA^1_-{\buildrel{\sim}\over{\longrightarrow}}\BX_0^-, \; \; s\mapsto (0,s^{-1}).$$ They define a morphism $$\wt{Z}_0=\GMaps (\BX_0\, ,Z)\to\GMaps (\BX_0^+ ,Z){\buildrel{\sim}\over{\longrightarrow}}\GMaps (\BA^1 ,Z)=Z^+$$ and a similar morphism $\wt{Z}_0\to Z^-$.
By construction, the following diagram commutes: $$\label{e:over Z times Z}
\xymatrix{
\wt{Z}_0 \ar[d]^{}\ar[r]^{\wt{p}_0}& Z\times Z\\\
Z^+\underset{Z^0}\times Z^-\ar@{->}[r]^{}&Z^+\times Z^-.\ar[u]_{{p^+}\times {p^-}}
}$$
We now claim:
\[p:tilde Z\_0\] Let $Z$ be a scheme. Then the map is an isomorphism.
Follows from the fact that for an affine scheme $S$, the diagram $$\CD
S\times \on{pt} @>>> S\times \BX_0^+ \\
@VVV @VVV \\
S\times \BX_0^- @>>> S\times \BX_0
\endCD$$ is a push-out diagram *in the category of schemes*.
\[r:tilde Z\_0\] In [@Dr Prop. 2.1.11] it is shown that the map is an isomorphism more generally when $Z$ is an algebraic space.
\[r:tilde Z\_0’\] Combining the isomorphism with the isomorphism $\wt{Z}_1\simeq Z$, we can interpret $\wt{Z}$ as an $\BA^1$-family[^12] of spaces interpolating between $Z$ and its “degeneration" $Z^+\underset{Z^0}\times Z^-$. Hence, the title of the subsection.
We have:
\[p:closed and open\]
[*(i)*]{} Let $Y\subset Z$ be a $\BG_m$-stable closed subspace. Then the diagram $$\xymatrix{
\wt{Y} \ar[d]_{\wt{p}_Y} \ar[r]^{}& \wt{Z} \ar[d]^{\wt{p}_Z}\\\
\BA^1\times Y\times Y\;\ar@{^{(}->}[r]^{}&\BA^1\times Z\times Z
}$$ is Cartesian. In particular, the morphism $\wt{Y}\to\wt{Z}$ is a closed embedding.
[*(ii)*]{} Let $Y\subset Z$ be a $\BG_m$-stable open subspace. Then the above diagram identifies $\wt{Y}$ with an open subspace of the fiber product $$\wt{Z}\underset{\BA^1\times Z\times Z}\times (\BA^1\times Y\times Y)\, .$$ In particular, the morphism $\wt{Y}\to\wt{Z}$ is an open embedding.
Set $$\oBX:=\BX-\{0\},$$ where $0\in \BX$ is the zero in $\BX=\BA^2$. For $S\to \BA^1$, set $\oBX_S:=\BX_S\underset{\BX}\times \oBX$.
\(i) Let $S$ be a scheme over $\BA^1$ and $f:\BX_S\to Z$ a $\BG_m$-equivariant morphism. Formula defines two sections of the map $\BX_S\to S$. We have to show that if $f$ maps these sections to $Y\subset Z$ then $f(\BX_S)\subset Y$. By $\BG_m$-equivariance, we have $$f(\oBX_S)\subset Y\,.$$
Since $\oBX_S$ is schematically dense in $\BX_S$ this implies that $f(\BX_S)\subset Y$.
\(ii) Just as before, we have a $\BG_m$-equivariant morphism $f:\BX_S\to Z$ such that $f(\oBX)\subset Y$. The problem is now to show that the set $$\{ s\in S\,|\,\BX_s\subset f^{-1}(Y)\}$$ is open in $S\,$.
The complement of this set equals $\pr_S(\BX_S -f^{-1}(Y))$, where $\pr_S :\BX_S\to S$ is the projection. The set $\pr_S(\BX_S -f^{-1}(Y))$ is closed in $S$ because $\BX_S -f^{-1}(Y)$ is a closed subset of $\BX_S -\oBX_S$, while the morphism $\BX_S -\oBX_S\to S$ is closed (in fact, it is a closed embedding).
Next we claim:
\[sss:props tilde p\] Let $Z$ be separated. Then the map $$\wt{p}:\wt{Z}\to \BA^1\times Z\times Z$$ is a monomorphism.
As before, set $\oBX:=\BX-\{0\}$, where $0\in \BX$ is the zero in $\BX=\BA^2$. Given a map $S\to \wt{Z}$, the corresponding $\BG_m$-equivariant map $$\oBX_S\to Z$$ is completely determined by the composition $$S\to \wt{Z} \overset{\wt{p}}\longrightarrow \BA^1\times Z\times Z\,.$$
Now use the fact that $\oBX_S$ is schematically dense in $\BX_S$.
If $Z$ is separated then so is $\wt{Z}$.
We are going to prove:
\[p:2new tilde\] Assume that $Z$ is an affine scheme of finite type. Then the morphism $\wt{p}:\wt{Z}\to \BA^1\times Z\times Z$ is a closed embedding. In particular, $\wt{Z}$ is an affine scheme of finite type.
If $Z$ is a closed subscheme of an affine scheme $Z'$ and the proposition holds for $Z'$ then it holds for $Z$ by [Proposition \[p:closed and open\]]{}(i). So we are reduced to the case that $Z$ is a finite-dimensional vector space equipped with a linear $\BG_m$-action.
If the proposition holds for affine schemes $Z_1$ and $Z_2$ then it holds for $Z_1\times Z_2\,$. So we are reduced to the case that $Z=\BA^1$ and $\lambda\in\BG_m$ acts on $\BA^1$ as multiplication by $\lambda^n$, $n\in\BZ$.
In this case it is straightforward to compute $\wt{Z}$ directly. In particular, one checks that $\wt{p}$ identifies $\wt{Z}$ with the closed subscheme of $\BA^1\times Z\times Z$ defined by the equation $x_2=t^n\cdot x_1$ if $n\ge 0$ and by the equation $x_1=t^{-n}\cdot x_2$ if $n\le 0$ (here $t,x_1,x_2$ are the coordinates on $\BA^1\times Z\times Z=\BA^3$).
\[ss:repr of inter\]
We have the following assertion, which is proved in [@Dr Thm. 2.2.2 and Prop. 2.2.3]:
\[t:tildeZ\] Let $Z$ be an algebraic space (resp., scheme) of finite type equipped with a $\BG_m$-action. Then $\wt{Z}$ is an algebraic space (resp., scheme) of finite type.
Below we will give a proof in the case when $Z$ is a scheme and the action of $\BG_m$ on $Z$ is locally linear. This case will be sufficient for the applications in this paper.
By assumption, $Z$ can be covered by open affine $\BG_m$-stable subschemes $U_i$. By [Proposition \[p:2new tilde\]]{}, each $\wt{U}_i$ is an affine scheme of finite type. By [Proposition \[p:closed and open\]]{}(ii), for each $i$ the canonical morphism $\wt{U}_i\to\wt{Z}$ is an open embedding. It remains to show that $\wt{Z}$ is covered by the open subschemes $\wt{U}_i$.
It suffices to check that for each $t\in\BA^1$ the fiber $\wt{Z}_t$ is covered by the open subschemes $(\wt{U}_i)_t$. For $t\ne 0$ this is clear from [Proposition \[p:outside 0\]]{}. It remains to consider the case $t=0$.
By [Proposition \[p:tilde Z\_0\]]{}, $\wt{Z}_0=Z^+\underset{Z^0}\times Z^-$. So a point of $\wt{Z}_0$ is a pair $(z^+,z^-)\in Z^+\times Z^-$ such that $q^+(z^+)=q^-(z^-)$. The point $q^+(z^+)=q^-(z^-)$ is contained in some $U_i\,$. By Lemma \[l:U\^+\](i), we have $z^+,z^-\in U_i\,$. So our point $(z^+,z^-)\in\wt{Z}_0$ belongs to $ (\wt{U}_i)_0\,$.
Let $Z$ be an algebraic space of finite type, and assume that the $\BG_m$-action on $Z$ is contracting, i.e., the $\BG_m$-action can be extended to an action of the monoid $\BA^1$ (recall that such an extension is unique, see [Sect. \[sss:contracting\]]{} including [Remark \[r:contracting\]]{}). In this case we claim:
\[p:2contracting\]
[*(i)*]{} The morphism $\wt{p}:\wt{Z}\to \BA^1\times Z\times Z$ identifies $\wt{Z}$ with the graph of the $\BA^1$-action on $Z$; in particular, the composition $$\label{e:first iso}
\wt{Z}\overset{\wt{p}}\longrightarrow\BA^1\times Z\times Z\to\BA^1\times Z\times \on{pt}=\BA^1\times Z$$ is an isomorphism.
[*(ii)*]{} The inverse of is the morphism $$\label{e:beta}
Z\times \BA^1\to\wt{Z},$$ corresponding to the $\BG_m$-equivariant map $Z\times \BX\to Z$, defined by $$(z,\tau_1,\tau_2)\mapsto\tau_1\cdot z\, ,\quad\quad (\tau_1,\tau_2)\in\BX\, ,\; z\in Z.$$
Let $\alpha :\wt{Z}\to\BA^1\times Z$ denote the composition and $\beta :\BA^1\times Z\to\wt{Z}$ the morphism . It is easy to see that $\alpha\circ\beta=\id$.
In order to prove that $\beta\circ \alpha=\id$, it is enough to show that $\alpha$ is a monomorphism. By [Theorem \[t:tildeZ\]]{}, we are dealing with a morphism between algebraic spaces of finite type, so being a monomorphism is a fiber-wise condition. Thus, it suffices to show that $\alpha$ induces an isomorphism between fibers over any $t\in\BA^1$.
For $t\ne 0$ this follows from [Proposition \[p:outside 0\]]{}. If $t=0$ then by [Proposition \[p:tilde Z\_0\]]{} (resp., Remark \[r:tilde Z\_0\] in the case of algebraic spaces), the morphism in question is the composition $$Z^+\underset{Z^0}\times Z^-\to Z^+\overset{p^+}\longrightarrow Z\,.$$ By [Proposition \[p:contracting\]]{} (resp., Remark \[r:contracting\] in the non-separated case), $p^+$ is an isomorphism, and the projection $q^-:Z^-\to Z^0$ is also an isomorphism by [Corollary \[c:contractive\]]{}(i).
From [Proposition \[p:2contracting\]]{} we formally obtain the following one:
\[p:dilating\] Let $Z$ be an algebraic space, and assume that the *inverse* of the $\BG_m$-action on $Z$ is contracting. Then:
[*(i)*]{} the morphism $\wt{p}:\wt{Z}\to \BA^1\times Z\times Z$ is a monomorphism, which identifies $\wt{Z}$ with $$\{(t,z_1,z_2\,)\in\BA^1\times Z\times Z\,|\, z_1=t^{-1}\cdot z_2\,\};$$ in particular, the composition $$\label{e:second iso}
\wt{Z}\overset{\wt{p}}\longrightarrow\BA^1\times Z\times Z\to\BA^1\times \on{pt}\times Z=\BA^1\times Z$$ is an isomorphism.
[*(ii)*]{} The inverse of is the morphism $$\label{e:2beta}
Z\times \BA^1\to\wt{Z},$$ corresponding to the $\BG_m$-equivariant map $Z\times \BX\to Z$, defined by $$(z,\tau_1,\tau_2)\mapsto\tau_2^{-1}\cdot z\, ,\quad\quad (\tau_1,\tau_2)\in\BX\, ,\; z\in Z.$$
The material in this subsection is included for completeness and will not be used in the sequel.
Throughout this subsection, $Z$ will be be an algebraic space of finite type equipped with a $\BG_m$-action.
We claim:
\[p:2smoothness\] If $Z$ is smooth then the canonical morphism $\wt{Z}\to\BA^1$ is smooth.
It suffices to check formal smoothness. We proceed just as in the proof of [Proposition \[p:smoothness\]]{}. Let $R$ be a $k$-algebra equipped with a morphism $\Spec(R)\to\BA^1$. Let $\bar R=R/I$, where $I\subset R$ is an ideal with $I^2=0$. Let $\bar f\in\Maps (\BX_{\bar R}\, , Z)^{\BG_m}$. We have to show that $\bar f$ can be lifted to an element of $\Maps (\BX_R\, , Z)^{\BG_m}$. Since $\BX_R$ is affine and $Z$ is smooth there is no obstruction to lifting $\bar f$ to an element of $\Maps (\BX_R, Z)$. The standard arguments show that the obstruction to existence of a $\BG_m$-equivariant lift is in $H^1(\BG_m\, ,M)$, where $M:=H^0(\BX_{\bar R}\, ,\bar f^*(\Theta_Z))\underset{\bar R}\otimes I$ and $\Theta_Z$ is the tangent bundle of $Z$. But $H^1$ of $\BG_m$ with coefficients in any $\BG_m$-module is zero.
Let $Z$ be affine. In this case, by [Proposition \[p:2new tilde\]]{}, the morphism $\wt{p}$ identifies $\wt{Z}$ with the closed subscheme $\wt{p}(\wt{Z})\subset\BA^1\times Z\times Z$. By [Proposition \[p:outside 0\]]{}, the intersection of $\wt{p}(\wt{Z})$ with the open subscheme $$\BG_m\times Z\times Z\subset \BA^1\times Z\times Z$$ is equal to the graph of the action map $\BG_m\times Z\to Z$. Hence, $\wt{Z}$ contains the closure of the graph in $\BA^1\times Z\times Z$.
\[r:nonflat\] In general, this containment is not an equality. E.g., this happens if $Z$ is the hypersurface in $\BA^{2n}$ defined by the equation $x_1\cdot y_1+\ldots x_n\cdot y_n=0$ and the $\BG_m$-action on $Z$ is defined by $\lambda(x_1\, ,\dots,x_n\, ,y_1\, ,\ldots, y_n)=
(\lambda\cdot x_1\, ,\ldots,\lambda\cdot x_n\, ,\lambda^{-1}\cdot y_1\, ,\ldots,\lambda^{-1}\cdot y_n)$.
However, one has the following:
If $Z$ is affine and smooth then $$\wt{p}(\wt{Z})=\overline{\Gamma},$$ where $\Gamma\subset\BG_m\times Z\times Z$ is the graph of of the action map $\BG_m\times Z\to Z$ and $\overline{\Gamma}$ denotes its scheme-theoretical closure in $ \BA^1\times Z\times Z\,$.
Indeed, this immediately follows from [Proposition \[p:2smoothness\]]{}.
We claim:
\[p:props tilde p’\] The morphism $\wt{p}:\wt{Z}\to \BA^1\times Z\times Z$ is unramified.
The morphism $\wt{p}$ is of finite presentation (because $\wt{Z}$ and $ \BA^1\times Z\times Z$ have finite type over $k$). It remains to check the condition on the geometric fibers of $\wt{p}$. Over $\BA^1-\{0\}$, it follows from [Proposition \[p:outside 0\]]{}. Over $0\in \BA^1$ it follows from [Proposition \[p:unrami\]]{} combined with [Proposition \[p:tilde Z\_0\]]{} (for schemes) and Remark \[r:tilde Z\_0\] (for arbitrary algebraic spaces).
Recall that according to [Proposition \[p:2new tilde\]]{}, if $Z$ is affine, the map $\wt{p}$ is a closed embedding.
Note, however, that if $Z$ is the projective line $\BP^1$ equipped with the usual $\BG_m$-action then the map $\wt{p}:\wt{Z}\to \BA^1\times Z\times Z$ is *not a closed* embedding (because, e.g., the scheme $\wt{Z}_0=Z^+\underset{Z^0}\times Z^-$ is not proper).
We have the following assertion:
\[p:Pn\] Let $Z$ be a projective space $\BP^n$ equipped with an arbitrary $\BG_m$-action. Then the morphism $\wt{p}:\wt{Z}\to \BA^1\times Z\times Z$ is a locally closed embedding.
For a suitable coordinate system in $\BP^n$, the $\BG_m$-action is given by $$\lambda *(z_0:\ldots :z_n)=(\lambda^{m_0}\cdot z_0: \ldots :\lambda^{m_n}\cdot z_n),\quad \lambda\in\BG_m \, .$$ Let $U_i\subset Z=\BP^n$ denote the open subset defined by the condition $z_i\ne 0$. It is affine, so by [Proposition \[p:2new tilde\]]{}, the canonical morphism $\wt{U}_i\to\BA^1\times U_i\times U_i$ is a closed embedding. Thus to finish the proof of the proposition, it suffices to show that $\wt{p}^{-1}(\BA^1\times U_i\times U_i)=\wt{U}_i\,$. By [Proposition \[p:outside 0\]]{}, $\wt{p}^{-1}(\BG_m\times U_i\times U_i)=\BG_m\underset{\BA^1}\times \wt{U}_i\,$. So it remains to prove that the morphism $\wt{p}_0:\wt{Z}_0\to Z\times Z$ has the following property: $(\wt{p}_0)^{-1}(U_i\times U_i)=(\wt{U}_i)_0\,$. Identifying $\wt{Z}_0$ with $Z^+\underset{Z^0}\times Z^-$ and using [Lemma \[l:U\^+\]]{}(i), we see that it remains to prove the following lemma:
\[l:Pn\] Let $z^+,z^-\in\BP^n$. Suppose that $$\lim_{\lambda\to 0}\lambda*z^+=\lim_{\lambda\to\infty }\lambda*z^-=\zeta\, .$$ If $z^+,z^-\in\ U_i$ then $\zeta\in U_i\,$.
Write $z^+=(z^+_0:\ldots :z^+_n)$, $z^-=(z^-_0:\ldots :z^-_n)$, $\zeta =(\zeta_0:\ldots :\zeta_n)$. We have $z^{\pm}_i\ne 0$, and the problem is to show that $\zeta_i\ne 0$.
Suppose that $\zeta_i= 0$. Choose $j$ so that $\zeta_j\ne 0\,$. Then $z^{\pm}_j\ne 0$ and $$\lim_{\lambda\to 0}\lambda^{m_i-m_j}\cdot (z_i/z_j)=\zeta_i/\zeta_j=0, \quad
\lim_{\lambda\to \infty}\lambda^{m_i-m_j}\cdot (z_i/z_j)=\zeta_i/\zeta_j=0\, .$$ This means that $m_i>m_j$ and $m_i<m_j$ at the same time, which is impossible.
As a corollary of [Proposition \[p:Pn\]]{}, combined with [Proposition \[p:closed and open\]]{}, we obtain that if $Z$ admits a $\BG_m$-equivariant locally closed embedding into a projective space $\BP(V)$, where $\BG_m$ acts linearly on $V$, then the morphism $\wt{p}:\wt{Z}\to \BA^1\times Z\times Z$ is a locally closed embedding. (Recall, however, that the map $p^{\pm}:Z^{\pm}\to Z$ is typically not a locally closed embedding, see Example \[ex:P\^1\].)
More generally, suppose that the $\BG_m$-action on $Z$ is locally linear. Then the proof of [Theorem \[t:tildeZ\]]{} shows that in this case the map $\wt{p}$ is, *Zariski locally on the source*, a locally closed embedding.
However, even this is not the case for a general $Z$:
Consider the curve obtained from $\BP^1$ by gluing $0$ with $\infty$. Equip $Z$ with the $\BG_m$-action induced by the usual action on $\BP^1$. Then $\wt{p}:\wt{Z}\to \BA^1\times Z\times Z$ is not a locally closed embedding, locally on the source. In fact, already $\wt{p}_0:\wt{Z}_0\to Z\times Z$ is not a locally closed embedding locally on the source (because the maps $p^{\pm}:Z^{\pm}\to Z$ are not).
\[ss:fiber products\]
In this subsection we let $Z$ be an algebraic space of finite type, equipped with an action of $\BG_m$.
In [Sect. \[sss:tilde p\]]{} we defined morphisms $\pi_1,\pi_2:\wt{Z}\to Z$. In [Sect. \[s:Verifying\]]{} we will need to consider the fiber product $$\label{e:fibered1}
Z^-\underset{Z}\times \wt{Z}\, ,$$ formed using $\pi_1:\wt{Z}\to Z$, and the fiber product $$\label{e:fibered2}
\wt{Z}\underset{Z}\times Z^+ \, ,$$ formed using $\pi_2:\wt{Z}\to Z$.
\[sss:defining the 2 maps\] Consider the composition $$\label{e:embedding1}
\BA^1\times Z^+\to\wt{Z^+}=\wt{Z^+}\underset{Z^+}\times Z^+ \to \wt{Z}\underset{Z}\times Z^+,$$ where the first arrow is the morphism for the space $Z^+$ and the second arrow is induced by the morphism $p^+:Z^+\to Z$. Consider also the similar composition $$\label{e:embedding2}
\BA^1\times Z^-\to\wt{Z^-}=Z^-\underset{Z^-}\times\wt{Z^-} \to Z^-\underset{Z}\times\wt{Z},$$ where the first arrow is the morphism for the space $Z^-$. In [Sect. \[s:Verifying\]]{} we will need the following result.
\[p:2open embeddings\] The compositions and are open embeddings.
Note that unlike the situation of [Proposition \[p:Cartesian\]]{}, these embeddings are usually not closed.
By Propositions \[p:2contracting\] and \[p:dilating\], the maps $\BA^1\times Z^+\to\wt{Z^+}$ and $\BA^1\times Z^-\to\wt{Z^-}$ are isomorphisms, so [Proposition \[p:2open embeddings\]]{} means that the morphisms $$\wt{Z^+}\to\wt{Z}\underset{Z}\times Z^+ ,\quad \wt{Z^-}\to Z^-\underset{Z}\times\wt{Z}$$ are open embeddings.
In the course of the proof of [Proposition \[p:2open embeddings\]]{} we will see that if $Z$ is affine, then the maps and are isomorphisms.
We will prove [Proposition \[p:2open embeddings\]]{} assuming that the action of $\BG_m$ on $Z$ is locally linear. The general case is considered in [@Dr Prop. 3.1.3].
We will show that is an open embedding. The case of is similar.
First, [Proposition \[p:closed and open\]]{}(ii) and [Lemma \[l:U\^+\]]{}(i) allow to reduce the assertion to the case when $Z$ is affine. In the affine case we will show that the map is an isomorphism.
Next, it follows from [Proposition \[p:closed and open\]]{}(i) and [Lemma \[l:U\^+\]]{}(ii) that if $Z\to Z'$ is a closed embedding and is an isomorphism for $Z'$, then it is also an isomorphism for $Z$. This reduces the assertion to the case when $Z$ is a vector space equipped with a linear action of $\BG_m$.
Third, it is easy to see that if $Z=Z_1\times Z_2$, and is an isomorphism for $Z_1$ and $Z_2$, then it is an isomorphism for $Z$. This reduces the assertion further to the case when either the action of $\BG_m$ on $Z$ or its inverse is contracting.
Suppose that the action is contracting. In this case $Z^+\simeq Z$ by [Proposition \[p:contracting\]]{}, and under this identification the map $$\wt{Z^+}\underset{Z^+}\times Z^+ \to \wt{Z}\underset{Z}\times Z^+,$$ appearing in , is the identity map.
Suppose that the inverse of the given $\BG_m$-action on $Z$ is contracting. By [Corollary \[c:contractive\]]{}(ii), we can identify $Z^+\simeq Z^0$, and by [Proposition \[p:dilating\]]{}, $\wt{Z}\simeq \BA^1\times Z$. Under these identifications, the map is the identity map $$\BA^1\times Z^0\to (\BA^1\times Z)\underset{Z}\times Z^0\simeq \BA^1\times Z^0\,.$$
Braden’s theorem {#s:Braden1}
================
From now on we will assume that the ground field $k$ has characteristic 0 (because we will be working with D-modules).
The goal of this section is to state Braden’s theorem ([Theorem \[t:braden original\]]{}) in the context of D-modules, and reduce it to another statement ([Theorem \[t:Braden adj\]]{}) that says that certain two functors are adjoint.
Braden’s theorem applies to any algebraic space $Z$ of finite type over $k$, equipped with an action of $\BG_m$. The reader may prefer to restrict his attention to the case of $Z$ being a scheme or even a separated scheme.
Furthermore, because of Remark \[r:q-proj loc lin\], for most applications, it is sufficient to consider the case when the $\BG_m$-action on $Z$ is locally linear, which would make the present paper self-contained, as the main technical results in Sects. \[s:actions\]-\[s:deg\] were proved only in this case.
\[ss:original statement\]
\[sss:mon\]
Let $G$ be an algebraic group. If $Z$ is an algebraic space of finite type equipped with a $G$-action, then $$\Dmod(Z)^{G\on{-mon}}\subset \Dmod(Z)$$ stands for the full subcategory generated by the essential image of the pullback functor $\Dmod(Z/G)\to \Dmod(Z)$, where $Z/G$ denotes the quotient *stack*. Here one can use either the $!$- or the $\bullet$-pullback: this makes no difference as the morphism $Z\to Z/\BG_m$ is smooth, and hence the two pullback functors differ by the cohomological shift by $2\cdot\dim(G)$.
Note that if the $G$-action is trivial then $\Dmod(Z)^{G\on{-mon}}=\Dmod(Z)$ (because the morphism $Z\to Z/G$ admits a section).
From now on let $Z$ be an algebraic space of finite type equipped with a $\BG_m$-action. Consider the commutative diagram $$\label{e:square with arrow}
\xy
(0,0)*+{Z^+\underset{Z}\times Z^-}="X";
(30,0)*+{Z^+}="Y";
(0,-30)*+{Z^-}="Z";
(30,-30)*+{Z.}="W";
(-20,20)*+{Z^0}="U";
{\ar@{->}_{p^-} "Z";"W"};
{\ar@{->}^{p^+} "Y";"W"};
{\ar@{->}_{'p^-} "X";"Y"};
{\ar@{->}^{'p^+} "X";"Z"};
{\ar@{->}_{j} "U";"X"};
{\ar@{->}_{i^-} "U";"Z"};
{\ar@{->}^{i^+} "U";"Y"};
\endxy$$ (The definitions of $Z^0$, $Z^{\pm}$, $i^{\pm}$, and $p^{\pm}$ were given in Sects. \[ss:fixed\_points\], \[ss:attr\], and \[ss:repeller\].)
Recall that by [Proposition \[p:Cartesian\]]{}, the morphism $j:Z^0\to Z^+\underset{Z}\times Z^-$ is an open embedding (and also a closed one).
We consider the categories $$\Dmod(Z)^{\BG_m\on{-mon}},\,\, \Dmod(Z^+)^{\BG_m\on{-mon}},\,\, \Dmod(Z^-)^{\BG_m\on{-mon}}$$ and $$\Dmod(Z^0)^{\BG_m\on{-mon}}=\Dmod(Z^0).$$
Consider the functors $$(p^+)^!:\Dmod(Z)^{\BG_m\on{-mon}}\to \Dmod(Z^+)^{\BG_m\on{-mon}} \text{ and }
(i^-)^!:\Dmod(Z^-)^{\BG_m\on{-mon}}\to \Dmod(Z^0).$$
The formalism of pro-categories (see Appendix \[s:pro\]) also provides the functors
$$(p^-)^\bullet: \Dmod(Z)^{\BG_m\on{-mon}}\to \on{Pro}(\Dmod(Z^-)^{\BG_m\on{-mon}})$$ and $$(i^+)^\bullet: \Dmod(Z^+)^{\BG_m\on{-mon}}\to \on{Pro}(\Dmod(Z^0)),$$ left adjoint in the sense of Sect. A.3 to $$(p^-)_\bullet:\Dmod(Z^-)^{\BG_m\on{-mon}}\to \Dmod(Z)^{\BG_m\on{-mon}}$$ and $$(i^+)_\bullet: \Dmod(Z^0)\to \Dmod(Z^+)^{\BG_m\on{-mon}},$$ respectively.
Consider the composed functors $$(i^+)^\bullet\circ (p^+)^! \text{ and } (i^-)^!\circ (p^-)^\bullet,\quad
\Dmod(Z)^{\BG_m\on{-mon}}\to \on{Pro}(\Dmod(Z^0)).$$
They are called the functors of *hyperbolic restriction*.
We claim that there is a canonical natural transformation $$\label{e:Braden trans}
(i^+)^\bullet\circ (p^+)^! \to (i^-)^!\circ (p^-)^\bullet.$$ Namely, the natural transformation is obtained via the $((i^+)^\bullet,(i^+)_\bullet)$-adjunction from the natural transformation $$\label{e:the natural transformation}
(p^+)^! \to (i^+)_\bullet\circ (i^-)^!\circ (p^-)^\bullet,$$ defined in terms of diagram as follows.
Note that since $j:Z^0\to Z^+\underset{Z}\times Z^-$ is an *open embedding* (see [Proposition \[p:Cartesian\]]{}), the functor $j^!$ is left adjoint to $j_\bullet\,$. Now define the morphism to be the composition $$\begin{gathered}
(p^+)^!\to (p^+)^!\circ (p^-)_\bullet \circ (p^-)^\bullet \simeq ({}'p^-)_\bullet\circ ({}'p^+)^!\circ (p^-)^\bullet\to \\
\to ({}'p^-)_\bullet\circ j_\bullet\circ j^! \circ ({}'p^+)^! \circ (p^-)^\bullet
\simeq (i^+)_\bullet\circ (i^-)^!\circ (p^-)^\bullet,\end{gathered}$$ where $(p^+)^!\circ (p^-)_\bullet\simeq ({}'p^-)_\bullet\circ ({}'p^+)^!$ is the base change isomorphism and the map $$\on{Id}\to j_\bullet\circ j^!$$ comes from the $(j^!,j_\bullet)$-adjunction.
We are now ready to state Braden’s theorem:
\[t:braden original\] The functors $$(i^+)^\bullet\circ (p^+)^! \text{ and } (i^-)^!\circ (p^-)^\bullet, \quad \Dmod(Z)^{\BG_m\on{-mon}}\to \on{Pro}(\Dmod(Z^0))$$ take values in $\Dmod(Z^0)\subset \on{Pro}(\Dmod(Z^0))$ and the map is an isomophism.
As we will see in [Sect. \[sss:contr\]]{}, the fact that the functor $(i^+)^\bullet\circ (p^+)^!$ takes values in $\Dmod(Z^0)\subset \on{Pro}(\Dmod(Z^0))$ is easy to prove. The fact that the functor $(i^-)^!\circ (p^-)^\bullet$ takes values in $\Dmod(Z^0)$ will follow *a posteriori* from the isomorphism with $(i^+)^\bullet\circ (p^+)^!$.
Assume for a moment that the $\BG_m$-action on $Z$ extends[^13] to an action of the monoid $\BA^1$. (Informally, this means that the $\BG_m$-action on $Z$ contracts it onto the fixed point locus $Z^0$.)
\[p:simple Braden\] In the above situation we have the following:
[*(a)*]{} The left adjoint $i^\bullet:\Dmod(Z)\to \on{Pro}(\Dmod(Z^0))$ of $i_\bullet$ sends $\Dmod(Z)^{\BG_m\on{-mon}}$ to $\Dmod(Z^0)$, and we have a canonical isomorphism $$i^\bullet|_{\Dmod(Z)^{\BG_m\on{-mon}}}\simeq q_\bullet|_{\Dmod(Z)^{\BG_m\on{-mon}}} \; .$$ More precisely, for each $\CF\in \Dmod(Z)^{\BG_m\on{-mon}}$ the natural map $$q_\bullet(\CF)\to q_\bullet\circ i_\bullet\circ i^\bullet(\CF)=(q\circ i)_\bullet\circ i^\bullet(\CF)=i^\bullet(\CF)$$ is an isomorphism.
[*(b)*]{} The left adjoint $q_!:\Dmod(Z)\to \on{Pro}(\Dmod(Z^0))$ of $q^!$ sends $\Dmod(Z)^{\BG_m\on{-mon}}$ to $\Dmod(Z^0)$, and we have a canonical isomorphism $$q_!|_{\Dmod(Z)^{\BG_m\on{-mon}}}\simeq i^!|_{\Dmod(Z)^{\BG_m\on{-mon}}} \; .$$ More precisely, for each $\CF\in \Dmod(Z)^{\BG_m\on{-mon}}$ the natural map $$i^!(\CF)\to i^!\circ q^!\circ q_!(\CF) =(q\circ i)^!\circ q_!(\CF) =q_! (\CF)$$ is an isomorphism.
For the proof see [@DrGa2 Theorem C.5.3].
Note that we can reformulate point (a) of [Proposition \[p:simple Braden\]]{} above as the statement that the (iso)morphism $$q_\bullet\circ i_\bullet\to \on{Id}_{\Dmod(Z^0)}$$ defines the co-unit of an adjunction between $$q_\bullet:\Dmod(Z)^{\BG_m\on{-mon}}\rightleftarrows \Dmod(Z^0):i_\bullet.$$
Similarly, point (b) of [Proposition \[p:simple Braden\]]{} can be reformulated as the statement that the (iso)morphism $$i^!\circ q^! \to \on{Id}_{\Dmod(Z^0)}$$ defines the co-unit of an adjunction between $$i^!:\Dmod(Z)^{\BG_m\on{-mon}} \rightleftarrows \Dmod(Z^0):q^!.$$
\[ss:Reformulation of Braden\]
\[sss:contr\]
We return to the set-up of [Theorem \[t:braden original\]]{}. By [Proposition \[p:simple Braden\]]{}, we obtain canonical isomorphisms $$(i^+)^\bullet\simeq (q^+)_\bullet \text{ and } (i^-)^!\simeq (q^-)_!.$$
In particular, we obtain that the functor $$(i^+)^\bullet\circ (p^+)^!\simeq (q^+)_\bullet \circ (p^+)^!$$ sends $\Dmod(Z)^{\BG_m\on{-mon}}$ to $\Dmod(Z^0)$.
In addition, we see that the functor $$(i^-)^!\circ (p^-)^\bullet\simeq (q^-)_!\circ (p^-)^\bullet$$ is the left adjoint functor to $(p^-)_\bullet\circ (q^-)^!$.
\[sss:defining co-unit\]
Now define a natural transformation $$\label{e:Braden co-unit}
\left((q^+)_\bullet \circ (p^+)^!\right)\circ \left((p^-)_\bullet\circ (q^-)^!\right)\to \on{Id}_{\Dmod(Z^0)}$$ to be the composition $$\begin{gathered}
(q^+)_\bullet \circ (p^+)^! \circ (p^-)_\bullet\circ (q^-)^!\simeq
(q^+)_\bullet \circ ({}'p^-)_\bullet\circ ({}'p^+)^! \circ (q^-)^! \to \\
\to (q^+)_\bullet \circ ({}'p^-)_\bullet\circ j_\bullet\circ j^! \circ ({}'p^+)^! \circ (q^-)^!
\simeq (q^+)_\bullet \circ (i^+)_\bullet \circ (i^-)^!\circ (q^-)^! \simeq \\
\simeq (q^+\circ i^+)_\bullet\circ (q^-\circ i^-)^!=
\on{Id}_{\Dmod(Z^0)}.\end{gathered}$$
The above natural transformation corresponds to the diagram
$$\xy
(-20,0)*+{Z}="X";
(20,0)*+{Z^0.}="Y";
(0,20)*+{Z^+}="Z";
(-40,20)*+{Z^-}="W";
(-60,0)*+{Z^0}="U";
(-20,40)*+{Z^-\underset{Z}\times Z^+}="V";
(-20,85)*+{Z^0}="T";
{\ar@{->}^{p^+} "Z";"X"};
{\ar@{->}_{q^+} "Z";"Y"};
{\ar@{->}_{p^-} "W";"X"};
{\ar@{->}^{q^-} "W";"U"};
{\ar@{->}_{'p^-} "V";"Z"};
{\ar@{->}^{'p^+} "V";"W"};
{\ar@{->}_{j} "T";"V"};
{\ar@{->}^{i^-} "T";"W"};
{\ar@{->}_{i^+} "T";"Z"};
{\ar@{->}^{\on{id}} "T";"Y"};
{\ar@{->}_{\on{id}} "T";"U"};
\endxy$$
The natural transformation gives rise to (and is determined by) a natural transformation $$\label{e:reform Braden trans}
(q^+)_\bullet \circ (p^+)^!\to \left((p^-)_\bullet\circ (q^-)^!\right)^L\simeq (q^-)_!\circ (p^-)^\bullet.$$ Here $\left((p^-)_\bullet\circ (q^-)^!\right)^L$ denotes the left adjoint of $(p^-)_\bullet\circ (q^-)^!$ in the sense of Sect. A.3.
It follows by diagram chase that the following diagram of natural transfomations commutes: $$\CD
(q^+)_\bullet \circ (p^+)^! @>{\text{\eqref{e:reform Braden trans}}}>> (q^-)_!\circ (p^-)^\bullet \\
@A{\sim}AA @A{\sim}AA \\
(i^+)^\bullet \circ (p^+)^! @>{\text{\eqref{e:Braden trans}}}>> (i^-)^!\circ (p^-)^\bullet
\endCD$$
Hence, the assertion of [Theorem \[t:braden original\]]{} follows from the next one:
\[t:Braden adj\] The natural transformation is the co-unit of an adjunction for the functors $$(q^+)_\bullet \circ (p^+)^!:\Dmod(Z)^{\BG_m\on{-mon}}\rightleftarrows \Dmod(Z^0):(p^-)_\bullet\circ (q^-)^!$$
\[ss:equivariant version\]
\[sss:Consider now\]
Consider now the stacks $$\CZ:=Z/\BG_m\, ,\,\, \CZ^0:=Z^0/\BG_m\, ,\,\, \CZ^{\pm}:=Z^{\pm}/\BG_m$$ and the morphisms $$\sfp^{\pm}:\CZ^{\pm}\to \CZ\, ,\,\,\sfq^{\pm}:\CZ^{\pm}\to\CZ^0$$induced by the morphisms $$p^{\pm}:Z^{\pm}\to Z ,\,\,q^{\pm}:Z^{\pm}\to Z^0$$from Sects. \[sss:structures\] and \[ss:repeller\].
The construction of the natural transformation can be rendered verbatim to produce a natural transformation $$\label{e:Braden co-unit equiv}
\left((\sfq^+)_\bullet \circ (\sfp^+)^!\right)\circ \left((\sfp^-)_\bullet\circ (\sfq^-)^!\right)\to \on{Id}_{\Dmod(\CZ^0)}.$$
We will prove the following version of [Theorem \[t:Braden adj\]]{}:
\[t:Braden adj equiv\] The natural transformation is the co-unit of an adjunction for the functors $$(\sfq^+)_\bullet \circ (\sfp^+)^!:\Dmod(\CZ)\rightleftarrows \Dmod(\CZ^0):(\sfp^-)_\bullet\circ (\sfq^-)^!$$
Let us prove that [Theorem \[t:Braden adj equiv\]]{} implies [Theorem \[t:Braden adj\]]{}.
We need to show that for $\CM\in \Dmod(Z)^{\BG_m\on{-mon}}$ and $\CN\in \Dmod(Z^0)^{\BG_m\on{-mon}}$, the map $$\Hom_{\Dmod(Z)^{\BG_m\on{-mon}}}\left(\CM,(p^-)_\bullet\circ (q^-)^!(\CN)\right)\to
\Hom_{\Dmod(Z^0)^{\BG_m\on{-mon}}}\left((q^+)_\bullet \circ (p^+)^!(\CM),\CN\right),$$ induced by , is an isomorphism.
By the definition of $\Dmod(Z)^{\BG_m\on{-mon}}$, we can assume that $\CM$ is the $\bullet$-pullback of some $\CM'\in \Dmod(\CZ)$. Let $\CN'$ denote the $\bullet$-direct image of $\CN$ under the canonical map $Z^0\to \CZ^0$.
Since all the maps $Z\to \CZ$, $Z^0\to \CZ^0$ and $Z^{\pm}\to \CZ^{\pm}$ are smooth, we have the following commutative diagram (with the vertical arrows being isomorphisms by adjunction): $$\CD
\Hom\left(\CM,(p^-)_\bullet\circ (q^-)^!(\CN)\right) @>>>
\Hom\left((q^+)_\bullet \circ (p^+)^!(\CM),\CN\right) \\
@V{\sim}VV @VV{\sim}V \\
\Hom\left(\CM',(\sfp^-)_\bullet\circ (\sfq^-)^!(\CN')\right) @>>>
\Hom\left((\sfq^+)_\bullet \circ (\sfp^+)^!(\CM'),\CN'\right).
\endCD$$
Hence, if the bottom horizontal arrow is an isomorphism, then so is the top one.
Construction of the unit {#s:unit}
========================
In this section we will perform the main step in the proof of [Theorem \[t:Braden adj equiv\]]{}; namely, we will construct the *unit* for the adjunction between the functors $(\sfq^+)_\bullet \circ (\sfp^+)^!$ and $(\sfp^-)_\bullet\circ (\sfq^-)^!$.
\[ss:specialization\]
In this subsection we describe the general set-up for the specialization map. The concrete situation in which this set-up will be applied is described in Sects. \[sss:concrete situation\]-\[sss:concrete\] below.
Let $\CY$ be an algebraic *stack* [^14] of finite type. Consider the stack $\BA^1\times \CY$, and let $\iota_1$ and $\iota_0$ be the maps $\CY\to \BA^1 \times \CY$ corresponding to the points $1$ and $0$ of $\BA^1$, respectively. Let $\pi$ denote the projection $\BA^1\times \CY\to \CY$.
Let $\CK$ be an object of $\Dmod(\BA^1\times \CY)^{\BG_m\on{-mon}}$, where $$\Dmod(\BA^1\times \CY)^{\BG_m\on{-mon}}\subset \Dmod(\BA^1\times \CY)$$ is the full subcategory generated by the essential image of the pullback functor $$\Dmod\left((\BA^1/\BG_m)\times \CY\right)\to \Dmod(\BA^1\times \CY).$$
Set $$\CK_1:=\iota_1^!(\CK), \quad\quad \CK_0:=\iota_0^!(\CK).$$
We are going to construct a canonical map $$\label{e:specialization}
\on{Sp}_\CK:\CK_1\to \CK_0\, ,$$ which will depend functorially on $\CK$. We will call it the *specialization map*.
The map is a simplified version of the specialization map that goes from the nearby cycles functor to the !-fiber.
First, note that [Proposition \[p:simple Braden\]]{}(b) and the definition of the category $\Dmod(-)$ for an algebraic stack [^15] imply that the functor $\pi_!$, left adjoint to $\pi^!:\Dmod(\CY)\to \Dmod(\BA^1\times \CY)$, is defined on the subcategory $\Dmod(\BA^1\times \CY)^{\BG_m\on{-mon}}$, and the natural transformation $$\iota_0^!\to \iota_0^!\circ \pi^!\circ \pi_!\simeq \pi_!$$ is an isomorphism.
Now, we construct the natural transformation as $$\iota_1^!(\CK)\simeq \pi_!\circ (\iota_1)_!\circ \iota_1^!(\CK) \to \pi_!(\CK)\simeq \iota_0^!(\CK),$$ where the morphism $\pi_!\circ (\iota_1)_!\circ \iota_1^!(\CK)\to \pi_!(\CK)$ comes from the $((\iota_1)_!,\iota_1^!)$-adjunction. Note that the functor $(\iota_1)_!$ is well-defined because $\iota_1$ is a closed embedding.
\[sss:specialization for constant\]
It is easy to see that if $\CK=\omega_{\BA^1}\times \CK_\CY$ for some $\CY\in \Dmod(\CY)$, then the map is the identity endomorphism of $$\iota_1^!(\CK)\simeq \CK_\CY\simeq \iota_0^!(\CK).$$
\[sss:functoriality of specialization\]
It is also easy to see from the construction that the natural transformation is functorial with respect to maps between algebraic stacks in the following sense.
Let $f:\CY'\to \CY$ be a map. Then for $\CK':=(\id_{\BA^1}\times f)^!(\CK)$ the diagram $$\CD
\CK'_1 @>{\on{Sp}_{\CK'}}>> \CK'_0 \\
@A{\sim}AA @AA{\sim}A \\
f^!(\CK_1) @>{f^!(\on{Sp}_\CK)}>> f^!(\CK_0) \\
\endCD$$ commutes.
Let now $f$ be representable and quasi-compact. Then for $\CK'\in \Dmod(\BA^1\times \CY')^{\BG_m\on{-mon}}$ and $$\CK:=(\id_{\BA^1}\times f)_\bullet(\CK'),$$ the diagram $$\CD
\CK_1 @>{\on{Sp}_\CK}>> \CK_0 \\
@V{\sim}VV @VV{\sim}V \\
f_\bullet(\CK'_1) @>{f_\bullet(\on{Sp}_{\CK'})}>> f_\bullet(\CK'_0) \\
\endCD$$ also commutes.
\[ss:kernels\]
According to [@DrGa1 Definition 1.1.8], an algebraic stack of finite type over $k$ is said to be QCA if the automorphism groups of its geometric points are affine.
If $f:\CY\to \CY'$ is a morphism between QCA stacks then one has a canonically defined functor $$f_\blacktriangle:\Dmod(\CY)\to \Dmod(\CY')$$ defined in [@DrGa1 Sect. 9.3].
The functor $f_\blacktriangle$ is a “renormalized version" of the usual functor $f_\bullet$ of de Rham direct image (see [@DrGa1 Sect. 7.4]). The problem with the functor $f_\bullet$ is that it is very poorly behaved unless the morphism $f$ is representable [^16]. For example, it fails to satisfy the projection formula and *is not compatible with compositions*, see [@DrGa1 Sect. 7.5] for more details. The functor $f_\blacktriangle$ cures all these drawbacks, and it equals the usual functor $f_\bullet$ if $f$ is representable.
\[sss:functors and kernels\]
Let $\CY_1$ and $\CY_2$ be QCA algebraic stacks. For an object $\CQ\in \Dmod(\CY_1\times \CY_2)$, consider the functor $$\sF_\CQ:\Dmod(\CY_1)\to \Dmod(\CY_2),\quad \CM\mapsto (\on{pr}_2)_\blacktriangle(\on{pr}_1^!(\CM)\sotimes \CQ),$$ where $\on{pr}_i:\CY_1\times \CY_2\to \CY_i$ are the two projections, and $\sotimes$ is the usual tensor product on the category of D-modules.
We will refer to $\CQ$ as the *kernel* of the functor $\sF_\CQ$.
In fact, it follows from [@DrGa1 Corollary 8.3.4] that the assignment $\CQ\rightsquigarrow \sF_\CQ$ defines an equivalence between the category $\Dmod(\CY_1\times \CY_2)$ and the DG category of *continuous* [^17] functors $\Dmod(\CY_1)\to \Dmod(\CY_2)$.
For example, if $\CY_1=\CY_2=\CY$, then for $$\CQ:=(\Delta_\CY)_\blacktriangle(\omega_\CY)\in \Dmod(\CY\times \CY)$$ the corresponding functor $\sF_\CQ$ is the identity functor on $\Dmod(\CY)$. Here $\omega_{\CY}\in \Dmod(\CY)$ denotes the dualizing complex on a stack $\CY$.
\[sss:corr\]
More generally, let $$\label{e:corr}
\xy
(-15,0)*+{\CY_1}="X";
(15,0)*+{\CY_2}="Y";
(0,15)*+{\CY_0}="Z";
{\ar@{->}_{f_1} "Z";"X"};
{\ar@{->}^{f_2} "Z";"Y"};
\endxy$$ be a diagram of QCA algebraic stacks. Set $$\CQ:=(f_1\times f_2)_\blacktriangle(\omega_{\CY_0})\in \Dmod(\CY_1\times \CY_2).$$
Then, by the projection formula, the functor $\sF_\CQ$ identifies with $(f_2)_\blacktriangle\circ (f_1)^!$.
\[sss:two routes\]
The reader who is reluctant to use the (potentially unfamiliar) functor $f_\blacktriangle$ can proceed along either of the following two routes:
\(i) The usual functor of direct image $f_\bullet$ is well-behaved when restricted to the subcategory $\Dmod(\CY)^+$ of bounded below (=eventually coconnective) objects. It is easy to see that working with this subcategory would be sufficient for the proof of [Theorem \[t:Braden adj equiv\]]{}. [^18]
This strategy can be used in order to adapt the proof of [Theorem \[t:Braden adj equiv\]]{} to the context of $\ell$-adic sheaves.
\(ii) One can use the following assertion.
Suppose that the morphism $f_2:\CY_0\to\CY_2$ is representable. Then
[*(i)*]{} The kernel $\CQ:=(f_1\times f_2)_\blacktriangle(\omega_{\CY_0})$ is canonically isomorphic to $(f_1\times f_2)_\bullet(\omega_{\CY_0})$;
[*(ii)*]{} The functor $$\sF_\CQ:\Dmod(\CY_1)\to \Dmod(\CY_2),\quad
\CM\mapsto (\on{pr}_2)_\blacktriangle(\on{pr}_1^!(\CM)\sotimes \CQ)\simeq (f_2)_\blacktriangle\circ (f_1)^! (\CM )$$ is canonically isomorphic to the functor $$\CM\mapsto (\on{pr}_2)_\bullet(\on{pr}_1^!(\CM)\sotimes \CQ).$$
Since $f_2:\CY_0\to\CY_2$ is representable, so is the morphism $f_1\times f_2:\CY_0\to\CY_1\times\CY_2\,$. This implies (i).
We have canonical isomorphisms $$\on{pr}_1^!(\CM)\sotimes \CQ\simeq (f_1\times f_2)_\blacktriangle (f_1^!(\CM ))\simeq (f_1\times f_2)_\bullet (f_1^!(\CM ))$$ (the first one holds by projection formula and the second because $f_1\times f_2$ is representable). So $$(\on{pr}_2)_\bullet(\on{pr}_1^!(\CM)\sotimes \CQ)\simeq
((\on{pr}_2)_\bullet\circ (f_1\times f_2)_\bullet) (f_1^!(\CM )).$$ One also has $$\sF_\CQ (\CM )\simeq (f_2)_\blacktriangle\circ (f_1)^! (\CM )\simeq (f_2)_\bullet (f_1^!(\CM ))=
((\on{pr}_2)\circ (f_1\times f_2))_\bullet (f_1^!(\CM )).$$
Finally, the fact that $f_1\times f_2$ is representable (see [@DrGa1 Proposition 7.5.7] [^19]) implies that $$((\on{pr}_2)\circ (f_1\times f_2))_\bullet\simeq (\on{pr}_2)_\bullet\circ (f_1\times f_2)_\bullet\,.$$
\[sss:reformulating\] In [Sect. \[sss:Consider now\]]{} we introduced the stacks $$\CZ:=Z/\BG_m\, ,\,\, \CZ^0:=Z^0/\BG_m\, ,\,\, \CZ^{\pm}:=Z^{\pm}/\BG_m$$ and the morphisms $\sfp^{\pm}:\CZ^{\pm}\to \CZ\,$, $\sfq^{\pm}:\CZ^{\pm}\to\CZ^0.$ Now consider the diagram $$\label{e:unit diagram}
\xy
(-20,0)*+{\CZ^0}="X";
(20,0)*+{\CZ.}="Y";
(0,20)*+{\CZ^-}="Z";
(-40,20)*+{\CZ^+}="W";
(-60,0)*+{\CZ}="U";
(-20,40)*+{\CZ^+\underset{\CZ^0}\times \CZ^-}="V";
{\ar@{->}_{\sfq^-} "Z";"X"};
{\ar@{->}^{\sfp^-} "Z";"Y"};
{\ar@{->}^{\sfq^+} "W";"X"};
{\ar@{->}_{\sfp^+} "W";"U"};
{\ar@{->}^{'\sfq^+} "V";"Z"};
{\ar@{->}_{'\sfq^-} "V";"W"};
\endxy$$ Our goal is to construct a canonical morphism from $\on{Id}_{\Dmod(\CZ)}$ to the composed functor $$\label{e:other comp}
\left((\sfp^-)_\bullet\circ (\sfq^-)^!\right)\circ \left((\sfq^+)_\bullet \circ (\sfp^+)^!\right):\Dmod(\CZ)\to \Dmod(\CZ).$$ The good news is that all morphisms in diagram are representable. In particular, $\sfp^-$ and $\sfq^+$ are representable, so $(\sfp^-)_\bullet=(\sfp^-)_\blacktriangle$ and $(\sfq^+)_\bullet =(\sfq^+)_\blacktriangle\,$.
Thus, the problem is to construct a canonical morphism from $\on{Id}_{\Dmod(\CZ)}$ to the composed functor $$\label{e:non-dangerous}
\left((\sfp^-)_\blacktriangle\circ (\sfq^-)^!\right)\circ \left((\sfq^+)_\blacktriangle \circ (\sfp^+)^!\right):\Dmod(\CZ)\to \Dmod(\CZ).$$ Using base change[^20], we further identify the functor with $$\label{e:other comp triangle}
(\sfp^-\circ {}'\sfq^+)_\blacktriangle\circ (\sfp^+\circ {}'\sfq^-)^!,$$ where $'\sfq^+$ and $'\sfq^-$ are as in diagram .
\[sss:q0&1\]
Set $$\CQ_0:=(\sfp^+\times \sfp^-)_\blacktriangle(\omega_{\CZ^+\underset{\CZ^0}\times \CZ^-})\in \Dmod(\CZ\times \CZ).$$
Then the functor (and, hence, ) is canonically isomorphic to $\sF_{\CQ_0}\,$.
The identity functor $\Dmod(\CZ)\to \Dmod(\CZ)$ equals $\sF_{\CQ_1}$, where $$\CQ_1:=(\Delta_\CZ)_\blacktriangle(\omega_\CZ)\in \Dmod(\CZ\times \CZ).$$
In [Sect. \[ss:constructing map of kernels\]]{} we will construct a canonical map $$\label{e:map on kernels}
\CQ_1\to \CQ_0\, .$$
By Sects. \[sss:reformulating\]-\[sss:q0&1\] and \[sss:functors and kernels\], the map of kernels induces a natural transformation $$\label{e:Braden unit}
\on{Id}_{\Dmod(\CZ)}\to \left((\sfp^-)_\blacktriangle\circ (\sfq^-)^!\right)\circ \left((\sfq^+)_\blacktriangle \circ (\sfp^+)^!\right)$$ between the corresponding functors.
In [Sect. \[s:Verifying\]]{} we will prove that the natural transformations and satisfy the properties of unit and co-unit of an adjunction between the functors $(\sfq^+)_\blacktriangle \circ (\sfp^+)^!$ and $(\sfp^-)_\blacktriangle\circ (\sfq^-)^!$.
\[ss:constructing map of kernels\] We will first define an object $$\CQ\in\Dmod(\BA^1\times \CZ\times \CZ)^{\BG_m\on{-mon}}\, ,$$ which “interpolates" between $\CQ_1$ and $\CQ_0$. We will then define to be the specialization morphism $\on{Sp}_\CQ\,$.
\[sss:concrete situation\]
Recall the algebraic space $\wt{Z}$ from [Sect. \[s:deg\]]{} and set $$\wt\CZ:=\wt{Z}/\BG_m,\quad \wt\CZ_t:=\wt{Z}_t/\BG_m\simeq \wt\CZ\underset{\BA^1}\times \{t\}$$ (the action of $\BG_m$ on $\wt\CZ$ was defined in [Sect. \[sss:anti-diagonal\]]{}).
Consider the morphisms $$\wt\sfp :\wt\CZ\to \BA^1\times \CZ\times \CZ \,\,\,\text{ and } \,\,\,\wt\sfp_t :\wt\CZ_t\to \CZ\times \CZ$$ induced by the maps and , respectively.
Set $$\CQ:=\wt\sfp_\blacktriangle(\omega_{\wt\CZ})\in \Dmod(\BA^1\times \CZ\times \CZ).$$
\[sss:Q mon\]
We claim that $\CQ$ belongs to the subcategory $\Dmod(\BA^1\times \CZ\times \CZ)^{\BG_m\on{-mon}}$.
In fact, we claim that $\CQ$ is the pullback of a canonically defined object of the category $\Dmod(\BA^1/\BG_m\times \CZ\times \CZ)$. Indeed, this follows from the existence of the Cartesian diagram
$$\CD
\wt\CZ @>{=}>> \wt{Z}/\BG_m @>>> \wt{Z}/\BG_m\times \BG_m \\
@V{\wt\sfp}VV @V{\wt{p}/\BG_m}VV @VVV \\
\BA^1\times \CZ \times \CZ @>{=}>> \BA^1\times Z/\BG_m\times Z/\BG_m @>>> \BA^1/\BG_m\times Z/\BG_m\times Z/\BG_m,
\endCD$$ where $\BG_m\times \BG_m$ acts on $\wt{Z}$ as in [Sect. \[sss:action of G\_m\^2\]]{}.
\[sss:concrete\]
Recall that the pair $(\wt{Z}_1,\wt{p}_1)$ identifies with $(Z,\Delta_Z)$, and the pair $(\wt{Z}_0,\wt{p}_0)$ identifies with $(Z^+\underset{Z^0}\times Z^-,p^+\times p^-)$.
Therefore, the pair $(\wt\CZ_1,\wt\sfp_1)$ identifies with $(\CZ,\Delta_\CZ)$, and the pair $(\wt\CZ_0,\wt\sfp_0)$ identifies with $(\CZ^+\underset{\CZ^0}\times \CZ^-,\sfp^+\times \sfp^-)$.
Hence, by base change, the objects $\CQ_1$ and $\CQ_0$ from [Sect. \[sss:q0&1\]]{} identify with the !-restrictions of $\CQ$ to $$\{1\}\times \CZ\times \CZ\to \BA^1\times \CZ\times \CZ \text{ and }
\{0\}\times \CZ\times \CZ\to \BA^1\times \CZ\times \CZ,$$ respectively.
Now, the sought-for map is given by the map $\on{Sp}_\CQ$ of .
Verifying the adjunction properties {#s:Verifying}
===================================
In [Sect. \[sss:Consider now\]]{} we introduced the stacks $$\CZ:=Z/\BG_m\, ,\,\, \CZ^0:=Z^0/\BG_m\, ,\,\, \CZ^{\pm}:=Z^{\pm}/\BG_m$$ and the morphisms $\sfp^{\pm}:\CZ^{\pm}\to \CZ\,$, $\sfq^{\pm}:\CZ^{\pm}\to\CZ^0.$
In Sects. \[ss:Reformulation of Braden\]-\[ss:equivariant version\] we constructed a natural transformation $$\left((\sfq^+)_\bullet \circ (\sfp^+)^!\right)\circ \left((\sfp^-)_\bullet\circ (\sfq^-)^!\right)\to \on{Id}_{\Dmod(\CZ^0)}
\, ,$$ see formula . Since the morphisms $\sfp^-$ and $\sfq^+$ are representable we have $(\sfp^-)_\bullet=(\sfp^-)_\blacktriangle$ and $(\sfq^+)_\bullet =(\sfq^+)_\blacktriangle\,$. So the above natural transformation can be rewritten as a natural transformation $$\label{e:2Braden co-unit}
\left((\sfq^+)_\blacktriangle \circ (\sfp^+)^!\right)\circ \left((\sfp^-)_\blacktriangle\circ (\sfq^-)^!\right)\to \on{Id}_{\Dmod(\CZ^0)}\, .$$
In [Sect. \[s:unit\]]{} we constructed a natural transformation $$\label{e:2Braden unit}
\on{Id}_{\Dmod(\CZ)}\to \left((\sfp^-)_\blacktriangle\circ (\sfq^-)^!\right)\circ \left((\sfq^+)_\blacktriangle \circ (\sfp^+)^!\right),$$ see formula .
To prove Theorem \[t:Braden adj equiv\], it suffices to show that the compositions $$\label{e:first composition}
(\sfp^-)_\blacktriangle\circ (\sfq^-)^!\to \left((\sfp^-)_\blacktriangle\circ (\sfq^-)^!\right)\circ \left((\sfq^+)_\blacktriangle \circ
(\sfp^+)^!\right)\circ \left((\sfp^-)_\blacktriangle\circ (\sfq^-)^!\right)\to (\sfp^-)_\blacktriangle\circ (\sfq^-)^!$$ and $$\label{e:second composition}
(\sfq^+)_\blacktriangle \circ (\sfp^+)^!\to \left((\sfq^+)_\blacktriangle \circ
(\sfp^+)^!\right)\circ \left((\sfp^-)_\blacktriangle\circ (\sfq^-)^!\right)\circ \left((\sfq^+)_\blacktriangle \circ (\sfp^+)^!\right)\to
(\sfq^+)_\blacktriangle \circ (\sfp^+)^!$$ corresponding to and are isomorphic to [^21] the identity morphisms.
We will do so for the composition . The case of is similar and will be left to the reader.
The key point of the proof is [Sect. \[sss:key\]]{}, which relies on the geometric [Proposition \[p:2open embeddings\]]{}. More precisely, we use the part of [Proposition \[p:2open embeddings\]]{} about $Z^-$. To treat the composition , one has to use the part of [Proposition \[p:2open embeddings\]]{} about $Z^+$.
We will use the notation $$\label{e:BIG}
\BIG:=\left((\sfp^-)_\blacktriangle\circ (\sfq^-)^!\right)\circ \left((\sfq^+)_\blacktriangle \circ (\sfp^+)^!\right)\circ
\left((\sfp^-)_\blacktriangle\circ (\sfq^-)^!\right).$$
By base change, $\BIG$ is given by pull-push along the following diagram:
$$\label{e:comp diag 1}
\xy
(-20,0)*+{\CZ^0}="X";
(20,0)*+{\CZ\, .}="Y";
(0,20)*+{\CZ^-}="Z";
(-40,20)*+{\CZ^+}="W";
(-60,0)*+{\CZ}="U";
(-20,40)*+{\CZ^+\underset{\CZ^0}\times \CZ^-}="V";
(-100,0)*+{\CZ^0}="T";
(-80,20)*+{\CZ^-}="S";
(-60,40)*+{\CZ^-\underset{\CZ}\times \CZ^+}="R";
(-40,60)*+{\CZ^-\underset{\CZ}\times \CZ^+\underset{\CZ^0}\times \CZ^-}="Q";
{\ar@{->}_{\sfq^-} "Z";"X"};
{\ar@{->}^{\sfp^-} "Z";"Y"};
{\ar@{->}^{\sfq^+} "W";"X"};
{\ar@{->}_{\sfp^+} "W";"U"};
{\ar@{->} "V";"Z"};
{\ar@{->} "V";"W"};
{\ar@{->}_{\sfq^-} "S";"T"};
{\ar@{->}^{\sfp^-} "S";"U"};
{\ar@{->} "R";"S"};
{\ar@{->} "R";"W"};
{\ar@{->} "Q";"R"};
{\ar@{->} "Q";"V"};
\endxy$$
Set $$\label{e:tilde Z-}
\wt{Z}^-:=Z^-\underset{Z}\times \wt{Z}\, ,$$ where the fiber product is formed using the composition $$\wt{Z} \overset{\wt{p}}\longrightarrow \BA^1\times Z\times Z\to Z\times Z \overset{\on{pr}_1}\longrightarrow Z$$ (i.e., the morphism $\pi_1:\wt{Z}\to Z$ from [Sect. \[sss:tilde p\]]{}).
For $t\in \BA^1$ set $\wt{Z}^-_t:=Z^-\underset{Z}\times \wt{Z}_t\,$.
Let $\wt{p}^-:\wt{Z}^-\to \BA^1\times Z^-\times Z$ denote the map obtained by base change from $$\wt{p}:\wt{Z} \to \BA^1\times Z\times Z\,.$$ Let $$r: \wt{Z}^-\to \BA^1\times Z^0\times Z$$ denote the composition of $\wt{p}^-:\wt{Z}^-\to \BA^1\times Z^-\times Z$ with the morphism $$\id_{\BA^1}\times q^-\times \id_Z:\BA^1\times Z^-\times Z\to\BA^1\times Z^0\times Z.$$ Let $r_t:\wt{Z}^-_t\to Z^0\times Z$ denote the morphism induced by $r: \wt{Z}^-\to \BA^1\times Z^0\times Z\,$. \[sss:stacky notation\] Recall that $$\wt\CZ:=\wt{Z}/\BG_m,\quad \wt\CZ_t:=\wt{Z}_t/\BG_m\simeq \wt\CZ\underset{\BA^1}\times \{t\}\, ,$$ where $\wt{Z}$ is the algebraic space from [Sect. \[s:deg\]]{} (the action of $\BG_m$ on $\wt\CZ$ was defined in [Sect. \[sss:anti-diagonal\]]{}).
Set $$\wt\CZ^-:=\CZ^-\underset{\CZ}\times \wt{\CZ}=\wt Z^-/\BG_m\, ,\quad\quad
\wt\CZ^-_t:=\CZ^-\underset{\CZ}\times \wt{\CZ}_t=\wt Z^-_t/\BG_m\,.$$
Let $$\sfr:\wt\CZ^-\to \BA^1\times \CZ^0\times \CZ\, ,\quad \sfr_t:\wt\CZ^-_t\to \CZ^0\times \CZ$$ be the morphisms induced by $r: \wt{Z}^-\to \BA^1\times Z^0\times Z$ and $r_t:\wt{Z}^-_0\to Z^0\times Z$, respectively. In particular, we have the morphisms $\sfr_0$ and $\sfr_1$ corresponding to $t=0$ and $t=1$.
\[sss:small diagram\] By [Proposition \[p:tilde Z\_0\]]{}, we have an isomorphism $\wt{Z}_0{\buildrel{\sim}\over{\longrightarrow}}Z^+\underset{Z^0}\times Z^-$. The corresponding isomorphism $$\wt{Z}^-_0:=Z^-\underset{Z}\times \wt{Z}_0{\buildrel{\sim}\over{\longrightarrow}}Z^-\underset{Z}\times Z^+\underset{Z^0}\times Z^-$$ induces an isomorphism $$\wt\CZ^-_0\simeq \CZ^-\underset{\CZ}\times \CZ^+\underset{\CZ^0}\times \CZ^-.$$
Thus the upper term of diagram is $\wt\CZ^-_0$. The compositions $$\CZ^-\underset{\CZ}\times \CZ^+\underset{\CZ^0}\times \CZ^-\to \CZ^-\underset{\CZ}\times \CZ^+\to
\CZ^-\overset{\, \sfq^-}\longrightarrow \CZ^0 \quad \mbox{and}\quad
$$ \^-\^+\^-\^+\^-\^- from diagram are equal, respectively, to the compositions $$\wt{\CZ}^-_0\overset{\sfr_0}\longrightarrow \CZ^0\times \CZ \overset{\on{pr}_1}\longrightarrow \CZ^0
$$ \^-\_0\^0 (the morphism $\sfr_0$ was defined in [Sect. \[sss:stacky notation\]]{}).
Hence, the functor $\BIG$ is given by pull-push along the diagram
$$\label{e:comp diag B}
\xy
(-20,0)*+{\CZ^0}="X";
(20,0)*+{\CZ\, .}="Y";
(0,20)*+{\wt\CZ^-_0}="Z";
{\ar@{->}_{\on{pr}_1\circ \sfr_0} "Z";"X"};
{\ar@{->}^{\on{pr}_2\circ \sfr_0} "Z";"Y"};
\endxy$$
\[ss:nat trans via kernels\] The goal of this subsection is to describe the natural transformations $$\BIG\to (\sfp^-)_\blacktriangle\circ (\sfq^-)^! \text{ and } (\sfp^-)_\blacktriangle\circ (\sfq^-)^! \to \BIG$$ at the level of kernels.
Set $$\CS:=\sfr_\blacktriangle (\omega_{\wt\CZ^-})\in \Dmod(\BA^1\times \CZ^0\times \CZ),$$ where $\sfr:\wt\CZ^-\to \BA^1\times \CZ^0\times \CZ$ was defined in [Sect. \[sss:stacky notation\]]{}.
As in [Sect. \[sss:Q mon\]]{}, one shows that $$\CS\in \Dmod(\BA^1\times \CZ^0\times \CZ)^{\BG_m\on{-mon}}\, .$$
Set also $$\CS_0:=(\sfr_0)_\blacktriangle (\omega_{\wt\CZ^-_0})\in \Dmod(\CZ^0\times \CZ),\quad \quad
\CS_1:=(\sfr_1)_\blacktriangle (\omega_{\wt\CZ^-_1})\in \Dmod(\CZ^0\times \CZ).$$ By [Sect. \[sss:small diagram\]]{}, the functor $\Phi$ identifies with $\sF_{\CS_0}\,$.
Now set $$\CT:=(\sfq^-\times \sfp^-)_\blacktriangle(\omega_{\CZ^-}).$$ We have $$(\sfp^-)_\blacktriangle\circ (\sfq^-)^! \simeq \sF_{\CT}\, .$$
\[sss:j tilde\]
Recall the open embedding $$j:Z^0\hookrightarrow Z^-\underset{Z}\times Z^+,$$ see [Proposition \[p:Cartesian\]]{}.
Let $j^-$ denote the corresponding open embedding $$Z^-\hookrightarrow Z^-\underset{Z}\times Z^+\underset{Z^0}\times Z^-\simeq Z^-\underset{Z}\times \wt{Z}_0=: \wt{Z}^-_0,$$ obtained by base change.
Let $\sfj^-$ denote the corresponding open embedding $$\CZ^-\hookrightarrow \wt\CZ^-_0.$$
Note that the composition $$\CZ^-\overset{\,\sfj^-}\hookrightarrow \wt\CZ^-_0\overset{\sfr_0}\longrightarrow \CZ^0\times \CZ$$ equals $\sfq^-\times \sfp^-$.
\[sss:1at the level of kernels\]
Recall that the morphism $\BIG\to(\sfp^-)_\blacktriangle\circ (\sfq^-)^!$ comes from the morphism $$\left((\sfq^+)_\blacktriangle \circ (\sfp^+)^!\right)\circ \left((\sfp^-)_\blacktriangle\circ (\sfq^-)^!\right)\to \on{Id}_{\Dmod(\CZ^0)}$$ constructed in Sects. \[ss:Reformulation of Braden\]-\[ss:equivariant version\]. By construction, the natural transformation $$\BIG\to(\sfp^-)_\blacktriangle\circ (\sfq^-)^!$$ corresponds to the map of kernels $$\label{e:S_0 to T}
\CS_0\to \CT$$ equal to the composition $$\CS_0:=(\sfr_0)_\blacktriangle (\omega_{\wt\CZ^-_0})\to
(\sfr_0)_\blacktriangle\circ \sfj^-_\blacktriangle (\omega_{\CZ^-}) {\buildrel{\sim}\over{\longrightarrow}}(\sfq^-\times \sfp^-)_\blacktriangle(\omega_{\CZ^-})=:\CT,$$ where the first arrow comes from $$\omega_{\wt\CZ^-_0}\to \sfj^-_\bullet\circ (\sfj^-)^\bullet(\omega_{\wt\CZ^-_0})\simeq \sfj^-_\bullet(\omega_{\CZ^-})\simeq
\sfj^-_\blacktriangle(\omega_{\CZ^-}).$$
The (tautological) identification $\wt{Z}_1\simeq Z$ defines an identification $$\label{e:tautological identification}
\wt\CZ^-_1\simeq \CZ^-,$$ so that the morphism $\sfr_1:\CZ^-_1\rightarrow \CZ^0\times \CZ$ identifies with $\sfq^-\times \sfp^-$.
Hence, we obtain a tautological identification $$\label{e:T to S_1}
\CT\simeq \CS_1\, .$$
\[sss:2at the level of kernels\]
The map $\on{Sp}_\CS$ of defines a canonical map $$\label{e:S_1 to S_0}
\CS_1\to \CS_0\, .$$
By [Sect. \[sss:functoriality of specialization\]]{}, the natural transformation $(\sfp^-)_\blacktriangle\circ (\sfq^-)^!\to\BIG$ comes from the map $$\label{e:T to S_0}
\CT\to \CS_1\to \CS_0\, ,$$ equal to the composition of and .
Thus, in order to prove that the composition is the identity map, it suffices to show that the composed map $$\label{e:composed kernels}
\CT\to \CS_1\to \CS_0\to \CT$$ is the identity map on $\CT$.
Recall the open embedding $$j^-:Z^-\hookrightarrow \wt{Z}^-_0$$ introduced in [Sect. \[sss:j tilde\]]{}.
Let $\overset{\circ}{\wt{Z}}{}^-$ denote the open subset of $\wt{Z}^-$ obtained by removing the closed subset $$\left(\wt{Z}^-_0-Z^-\right)\subset \wt{Z}^-_0\subset \wt{Z}^-.$$
Let $\overset{\circ}{\wt\CZ}{}^-$ denote the corresponding open substack of $\wt\CZ^-$. Let $\overset{\circ}{\wt\CZ}{}^-_t$ denote the fiber of $\overset{\circ}{\wt\CZ}{}^-$ over $t\in \BA^1$.
By definition, the open embedding $$\sfj^-:\CZ^-\hookrightarrow \wt\CZ^-_0$$ defines an *isomorphism* $$\label{e:fiber at 0 open}
\CZ^-{\buildrel{\sim}\over{\longrightarrow}}\overset{\circ}{\wt\CZ}{}^-_0.$$
Note that the isomorphism $\CZ^-{\buildrel{\sim}\over{\longrightarrow}}\wt\CZ{}^-_1$ of still defines an isomorphism $$\label{e:fiber at 1 open}
\CZ^-{\buildrel{\sim}\over{\longrightarrow}}\overset{\circ}{\wt\CZ}{}^-_1.$$
Let $$\osfr:\overset{\circ}{\wt\CZ}{}^-\to \BA^1\times \CZ^0\times \CZ\quad \text{ and } \quad
\osfr_t:\overset{\circ}{\wt\CZ}{}^-\to \CZ^0\times \CZ$$ denote the morphisms induced by the maps $\sfr$ and $\sfr_t$ from [Sect. \[sss:stacky notation\]]{}.
Set $$\oCS:=\osfr_\blacktriangle(\omega_{\overset{\circ}{\wt\CZ}{}^-}),$$ and also $$\oCS_0:=(\osfr_0)_\blacktriangle(\omega_{\overset{\circ}{\wt\CZ}{}^-_0})\quad \text{ and } \quad
\oCS_1:=(\osfr_1)_\blacktriangle(\omega_{\overset{\circ}{\wt\CZ}{}^-_1}).$$
The open embedding $\overset{\circ}{\wt\CZ}{}^-\hookrightarrow \wt\CZ^-$ gives rise to the maps $$\CS\to \oCS,\quad \CS_0\to \oCS_0, \quad \CS_1\to \oCS_1\, .$$
As in Sects. \[sss:1at the level of kernels\]-\[sss:2at the level of kernels\], we have the natural transformations $$\label{e:composed kernels open}
\CT\to \oCS_1\to \oCS_0\to \CT.$$
Moreover, the diagram $$\CD
\CT @>>> \CS_1 @>>> \CS_0 @>>> \CT \\
@V{\id}VV @VVV @VVV @VV{\id}V \\
\CT @>>> \oCS_1 @>>> \oCS_0 @>>> \CT
\endCD$$ commutes.
Hence, in order to show that the composed map is the identity map, *it suffices to show that the composed map is the identity map.* We will do this in the next subsection.
\[sss:key\]
Recall now the open embedding $$\BA^1\times Z^-\to Z^-\underset{Z}\times \wt{Z}=:\wt{Z}^-$$ of .
By definition, it induces an isomorphism $$\BA^1\times Z^-\simeq \overset{\circ}{\wt{Z}}{}^-.$$
Dividing by the action of $\BG_m$, we obtain an isomorphism $$\label{e:key}
\BA^1\times \CZ^-\simeq \overset{\circ}{\wt\CZ}{}^-.$$
Under this identification, we have:
- The map $\osfr:\overset{\circ}{\wt\CZ}{}^-\to \BA^1\times \CZ^0\times \CZ$ identifies with the map $\BA^1\times \CZ^-\to\BA^1\times \CZ^0\times \CZ$ induced by $\id_{\BA^1}:\BA^1\to \BA^1$ and $ (\sfq^-\times \sfp^-): \CZ^-\to \CZ^0\times \CZ\,$.
- The isomorphism $\CZ^-{\buildrel{\sim}\over{\longrightarrow}}\overset{\circ}{\wt\CZ}{}^-_1$ of corresponds to the identity map $$\CZ^-\to (\BA^1\times \CZ^-)\underset{\BA^1}\times \{1\}\simeq \CZ^-.$$
- The isomorphism $\CZ^-{\buildrel{\sim}\over{\longrightarrow}}\overset{\circ}{\wt\CZ}{}^-_0$ of corresponds to the identity map $$\CZ^-\to (\BA^1\times \CZ^-)\underset{\BA^1}\times \{0\}\simeq \CZ^-.$$
Hence, we obtain that the composition identifies with $$\CT\simeq \iota_1^!(\omega_{\BA^1}\boxtimes \CT) \overset{\on{Sp}}\longrightarrow \iota_0^!(\omega_{\BA^1}\boxtimes \CT) \simeq \CT,$$ where $\on{Sp}:=\on{Sp}_{\omega_{\BA^1}\boxtimes \CT}$ is the specialization map for the object $$\omega_{\BA^1}\boxtimes \CT \in \Dmod(\BA^1\times \CZ^0\times \CZ).$$
The fact that the above map is the identity map on $\CT$ follows from [Sect. \[sss:specialization for constant\]]{}.
Pro-categories {#s:pro}
==============
[**A.1.**]{} For a DG category $\bC$ let $\on{Pro}(\bC)$ denote its pro-completion, thought of as the DG category opposite to that of covariant exact functors $\bC\to \Vect$, where $\Vect$ denotes the DG category of complexes of $k$-vector spaces. [^22]
Yoneda embedding defines a fully faithful functor $\bC\to \on{Pro}(\bC)$. Any object in $\on{Pro}(\bC)$ can be written as a filtered limit (taken in $\on{Pro}(\bC)$) of co-representable functors.
[**A.2.**]{} A functor $\sF:\bC'\to \bC''$ between DG categories induces a functor denoted also by $\sF$ $$\on{Pro}(\bC')\to \on{Pro}(\bC'')$$ by applying the *right Kan extension* of the functor $$\bC'\overset{\sF}\longrightarrow \bC''\hookrightarrow \on{Pro}(\bC'')$$ along the embedding $\bC'\to \on{Pro}(\bC')$.
The same construction can be phrased as follows: for $\wt\bc'\in \on{Pro}(\bC')$, thought of as a functor $\bC'\to \Vect$, the object $\sF(\wt\bc')$, thought of as a functor $\bC''\to \Vect$, is the *left Kan extension* of $\wt\bc'$ along the functor $\sF:\bC'\to \bC''$.
Explicitly, if $\wt\bc\in \on{Pro}(\bC')$ is written as $\underset{i\in I}{\underset{\longleftarrow}{lim}}\, \bc_i$ with $\bc_i\in \bC'$, then $$\sF(\wt\bc)\simeq \underset{i\in I}{\underset{\longleftarrow}{lim}}\, \sF(\bc_i),$$ as objects of $\on{Pro}(\bC'')$ and $$\sF(\wt\bc)\simeq \underset{i\in I}{\underset{\longrightarrow}{lim}}\, \CMaps_{\bC''}(\sF(\bc_i),-),$$ as functors $\bC''\to \Vect$.
[**A.3.**]{} Let $\sG:\bC'\to \bC''$ be a functor between DG categories. We can speak of its left adjoint $\sG^L$ as a functor $\bC''\to \on{Pro}(\bC')$. Namely, for $\bc''\in \bC''$ the object $\sG^L(\bc'')\in \on{Pro}(\bC')$, thought of as a functor $\bC'\to \Vect$ is given by $$(\sG^L(\bc''))(\bc')=\CMaps_{\bC''}(\bc'',\sG(\bc')).$$
[**A.4.**]{} We let the same symbol $\sG^L$ also denote the functor $\on{Pro}(\bC'')\to \on{Pro}(\bC')$ obtained as the right Kan extension of $\sG^L:\bC''\to \on{Pro}(\bC')$ along $\bC''\hookrightarrow \on{Pro}(\bC'')$.
The functor $\sG^L$ is the left adjoint of the functor $\sG:\on{Pro}(\bC')\to \on{Pro}(\bC'')$.
We can also think of $\sG^L$ as follows: for $\wt\bc''\in \on{Pro}(\bC'')$, thought of as a functor $\bC''\to \Vect$, the object $\sG^L(\wt\bc'')$, thought of as a functor $\bC'\to \Vect$ is given by $$(\sG^L(\wt\bc''))(\bc')=\wt\bc''(\sG(\bc')).$$
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[^1]: The definition of $\BG_m$-monodromic object is recalled in Subsect. \[sss:mon\].
[^2]: In [@GM] M. Goresky and R. MacPherson work in a purely topological setting. They work with correspondences rather than torus actions. According to [@GM Prop. 9.2], under a certain condition (which is satisfied if the correspondence comes from a $\BG_m$-action) one has $A_4^\bullet{\buildrel{\sim}\over{\longrightarrow}}A_5^\bullet\,$, where $A_i^\bullet$ is defined in [@GM Prop. 4.5]. This is the proptotype of Braden’s theorem.
[^3]: The issue here is that in the context of D-modules the $\bullet$-pullback functor is only partially defined.
[^4]: Unfortunately, it seems that this particular proof of the isomorphism , although very simple and well-known in the folklore, does not appear in the published literature.
[^5]: The quotation marks are due to the fact that this “family" is not flat, in general. If $Z$ is affine then each $\wt{Z}_t$ is a closed subscheme of $Z\times Z$. If $Z$ is separated, then for each $t$ the map $\wt{Z}_t\to Z\times Z$ is a monomorphism (but not necesaarily a locally closed embedding).
[^6]: Pro-categories are considered in Appendix \[s:pro\].
[^7]: However, the most natural approach to *constructing* the triangulated category of D-modules on an algebraic stack is to construct the corresponding DG category first, as is done in [@DrGa1].
[^8]: In particular, quasi-separatedness is included into the definition of algebraic space. Thus the quotient $\BA^1/\BZ$ (where the discrete group $\BZ$ acts by translations) is *not* an algebraic space.
[^9]: We do not know if separateness is really necessary in Sumihiro’s theorem.
[^10]: Using the $\BA^1$-action on $Z^+$, it is easy to see that each connected component of $Z^+$ is the preimage of a connected component of $Z^0$ with respect to the map $q^+:Z^+\to Z^0\,$.
[^11]: We define the tangent space by $T_zZ:=(T_z^*Z)^*$, where $T_z^*Z$ is the fiber of $\Omega^1_{Z/k}$ at $z$. (The equality $T_z^*Z=m_z/m_z^2$ holds *if the residue field of $z$ is finite and separable* over $k$.)
[^12]: In general, this “family" is not flat, see the example from [Remark \[r:nonflat\]]{}.
[^13]: By Remark \[r:contracting\], such extension is unique if it exists.
[^14]: We use the conventions from [@DrGa1 Sect. 1.1] for algebraic stacks. We refer the reader to [@DrGa1 Sect. 6] for a review of the DG category of D-modules on algebraic stacks of finite type.
[^15]: According to [@DrGa1 Sect. 6.1.1], an object of $\Dmod(\CY )$ is a “compatible collection" of objects of $\Dmod(S )$ for all schemes $S$ of finite type mapping to $\CY$.
[^16]: Or, more generally, *safe* in the sense of [@DrGa1 Definition 10.2.2].
[^17]: Recall that a functor between cocomplete DG categories is said to be continuous if it commutes with arbitrary direct sums.
[^18]: Note, however, that if one redefines the assignment $\CQ\rightsquigarrow \sF_\CQ$ using $(\on{pr}_2)_\bullet$ instead of $(\on{pr}_2)_\blacktriangle$ then one obtains a *different* functor, even when evaluated on $\Dmod(\CY_1)^+$.
[^19]: For any composable morphisms $g,g'$ between stacks one has a morphism $g_\bullet\circ g'_\bullet\to (g\circ g')_\bullet\,$, which is *not necessarily an isomorphism*. However, it is an isomorphism if $g'$ is representable. In [@DrGa1 Proposition 7.5.7] this is proved if $g'$ is schematic, but the same proof applies if $g'$ is only representable.
[^20]: Since we have switched to the renormalized direct images, we can apply base change and do other standard manipulations.
[^21]: In the future we will skip the words “isomorphic to" in similar situations. (This is a slight abuse of language since we work with the DG categories of D-modules rather than with their homotopy categories.)
[^22]: A way to deal with set-theoretical difficulties is to require that our functors commute with $\kappa$-filtered colimits for some cardinal $\kappa$, see [@Lur Def. 5.3.1.7].
|
---
author:
- 'Jose Beltrán Jiménez$^{a,b}$, Jose A. R. Cembranos$^c$ and Jose M. Sánchez Velázquez$^c$'
title: 'On scalar and vector fields coupled to the energy-momentum tensor'
---
IFT-UAM/CSIC-18-031
Introduction
============
General Relativity (GR) is the standard framework to describe the gravitational interaction and, after more than a century since its inception, it still stands out as the most compelling candidate owed to its excellent agreement with observations on a wide regime of scales [@Will:2014kxa]. From a theoretical viewpoint, GR can be regarded as the theory describing an interaction mediated by a massless spin 2 particle. The very masslessness of this particle together with explicit Lorentz invariance makes it to naturally couple to the energy-momentum tensor and, since it also carries energy-momentum, consistency dictates that it needs to present self-interactions. This requirement has sometimes led to regard gravity as a theory for a spin 2 particle that is consistently coupled to its own energy-momentum tensor so that the total energy-momentum tensor is the source of the gravitational field[^1]. These interactions can be constructed order by order following the usual Noether procedure (see for instance [@Ortin]) and one obtains an infinite series of terms. One could attempt to re-sum the series directly or to use Deser’s procedure [@Deser:1969wk] of introducing auxiliary fields so that the construction of the interactions ends at the first iteration. Either way, GR arises as the full non-linear theory and the equivalence principle together with diffeomorphism symmetry come along in a natural way (see also [@Padmanabhan:2004xk; @Butcher:2009ta; @Deser:2009fq; @Barcelo:2014mua] for some recent related works discussing in detail the bootstrapping procedure).
In this work, we intend to develop a family of theories for scalar and vector fields following a similar bootstrapping approach as the one leading to GR, i.e., by prescribing a coupling to the energy-momentum tensor that remains at the full non-linear level. Unlike the case of gravity where the coupling to the energy-momentum tensor comes motivated from the requirement of maintaining gauge invariance (so it is a true consistency requirement rather than a prescription), in our case there is no [*necessity*]{} to have a [*consistent*]{} coupling to the energy-momentum tensor nor self-couplings of this form. However, the construction of theories whose interactions are universally described in terms of the energy-momentum tensor (as to fulfill some form of equivalence principle) is an alluring question in relation with gravitational phenomena. Let us remind that, starting from Newton’s law, the simplest (perhaps naive) relativistic completion is to promote the gravitational potential to a scalar field. However, the most leading order coupling of the scalar to the energy-momentum tensor is through its trace and, therefore, there is no bending of light. This is a major obstacle for this simple theory of gravity based on a scalar field since the bending of light is a paramount feature of the gravitational interaction. Nevertheless, the problem of finding a theory for a scalar field that couples in a self-consistent manner to the energy-momentum tensor is an interesting problem on its own that has already been considered in the literature [@Kraichnan:1955zz; @Freund:1969hh; @Deser:1970zzb; @Sami:2002se]. Here we will extend those results for the case of more general couplings for a scalar field (adding a shift symmetry that leads to derivative couplings) and explore the case of vector fields coupled to the energy-momentum tensor.
The paper is organised as follows: We start by briefly reviewing and re-obtaining known results for a scalar field coupled to the trace of the energy-momentum tensor. We then extend the results to incorporate a shift-symmetry for the scalar in the coupling to the energy-momentum tensor, what leads to a theory for a derivatively coupled scalar. After obtaining results in the second order formalism, we turn to discuss the construction of the full non-linear theories in the first order formalism, where the resummation procedures can be simplified. We end the scalar field case by considering couplings to matter. After working out the scalar field case, we consider theories for a vector field coupled to the energy-momentum tensor. We will devote Sec. \[sec:Superpotentials\] to discuss the role played by superpotential terms and Sec. \[Sec:EffectiveMetrics\] to present a procedure to obtain the interactions from a generating functional defined in terms of an effective metric. In Sec. \[Sec:Phenomenology\] we give constraints obtained from several phenomenological probes and we conclude in Sec. \[Sec:Discussion\] with a discussion of our results.
Scalar gravity {#Sec:ScalarGravity}
==============
We will start our study with the simplest case of a scalar field theory that couples to the trace of the energy-momentum tensor to recover known results for scalar gravity. Then, we will extend these results to include derivative interactions that typically arise from disformal couplings. Such couplings will be the natural ones when imposing a shift symmetry for the scalar field, as usually happens for Goldstone bosons. We will also consider the problem from a first order point of view. Finally, couplings to matter, both derivative and non-derivative, will be constructed.
Self-interactions for scalar gravity {#sec:Conformalscalar}
------------------------------------
Let us begin our tour on theories coupled to the energy-momentum tensor from a scalar field and focus on the self-coupling problem neglecting other fields, i.e., we will look for consistent couplings of the scalar field to its own energy-momentum tensor. Firstly, we need to properly define our procedure. Our starting point will be the action for a free massive scalar field given by \_[(0)]{}=12\^4x(\_\^-m\^2\^2 ). The goal now is to add self-interactions of the scalar field through couplings to the energy-momentum tensor. This can be done in two ways, either by imposing a coupling of the scalar field to its own energy-momentum tensor at the level of the action or by imposing its energy-momentum tensor to be the source in the field equations. We will solve both cases for completeness and to show the important differences that arise in both procedures at the non-linear level. Let us start by adding an interaction of the scalar to the energy-momentum tensor of the free field in the action as follows \_[(1)]{}=-\^4xT\_[(0)]{} \[eq:Firstcouplingphi\] with ${M_{\rm sc}}$ some mass scale determining the strength of the interaction and $T_{(0)}$ the trace of the energy-momentum tensor of the free scalar field, i.e., the one associated to ${\mathcal{S}}_{(0)}$. We encounter here the usual ambiguity due to the different available definitions for the energy-momentum tensor that differ either by a term of the form $\partial_\alpha \Theta^{[\alpha\mu]\nu}$ with $\Theta^{[\alpha\mu]\nu}$ some super-potential antisymmetric in the first pair of indices so that it is off-shell divergenceless, or by a term proportional to the field equations (or more generally, any rank-2 tensor whose divergence vanishes on-shell). In both cases, the form of the added piece guarantees that all of the related energy-momentum tensors give the same Lorentz generators, i.e., they carry the same total energy and momentum. We will consider in more detail the role of such boundary terms in Sec. \[sec:Superpotentials\] and, until then, we will adopt the Hilbert prescription to compute the energy-momentum tensor in terms of a functional derivative with respect to an auxiliary metric tensor as follows[^2] T\^(-)\_[\_=\_]{}, where in the action we need to replace $\eta_{\mu\nu}\to \gamma_{\mu\nu}$ with $\gamma_{\mu\nu}$ some background (Lorentzian) metric and $\gamma$ its determinant. This definition has the advantage of directly providing a symmetric and gauge-invariant (in case of fields with spin and/or internal gauge symmetries) energy-momentum tensor. In general, this does not happen for the canonical energy-momentum tensor obtained from Noether’s theorem, although the Belinfante-Rosenfeld procedure [@BelinfanteRosenfeld] allows to [*correct*]{} it and transform it into one with the desired properties [^3]. For the scalar field theory we are considering, the energy-momentum tensor is T\_[(0)]{}\^=\^\^-\^\_, which is also the one obtained as Noether current so that the above discussion is not relevant here. However, in the subsequent sections dealing with vector fields this will be important since the canonical and Hilbert energy-momentum tensors differ.
After settling the ambiguity in the energy-momentum tensor, we can now write the first order corrected action for ${\mathcal{S}}_{(0)}$ by incorporating the coupling (\[eq:Firstcouplingphi\]), so we obtain \_[(0)]{}+\_[(1)]{}=12\^4x. As usual, when we introduce the coupling of the scalar field to the energy-momentum tensor, the new energy-momentum tensor of the whole action acquires a new contribution and, therefore, the coupling $-\frac{\varphi}{M_{sc}}T$ receives additional corrections that will contribute to order $1/{M_{\rm sc}}^2$. The added $1/{M_{\rm sc}}^2$ interaction will again add a new correction that will contribute an order $1/{M_{\rm sc}}^3$ term and so on. This iterative process will continue indefinitely so we end up with a construction of the interactions as a perturbative expansion in powers of $\varphi/{M_{\rm sc}}$ and, thus, we obtain an infinite series whose resummation will give the final desired action. The iterative process for the case at hand gives the following expansion for the first few terms: =12\^4x. \[eq:PertScalar1\] It is not difficult to identify that we obtain the first terms of a geometric progression with ratios $2\varphi/{M_{\rm sc}}$ and $4\varphi/{M_{\rm sc}}$ which can then be easily resummed. One can confirm this by realising that a term $(\varphi/{M_{\rm sc}})^n\partial_\mu\varphi\partial^\mu\varphi$ gives a correction $2(\varphi/{M_{\rm sc}})^{n+1}\partial_\mu\varphi\partial^\mu\varphi$, while a term $(\varphi/{M_{\rm sc}})^{m+2}$ introduces a correction $4(\varphi/{M_{\rm sc}})^{m+3}$. Then, the resummed series will be given by =12\^4x, \[Actionscalar1\] with ()=\_[n=0]{}\^(2)\^n=,()=\_[n=0]{}\^(4)\^n=. \[resconformalscalar\] This recovers the results in [@Sami:2002se] in the corresponding limits. Technically, the geometric series only converges for[^4] $2\varphi<{M_{\rm sc}}$, but the final result can be extended to values $2\varphi>{M_{\rm sc}}$, barring the potential poles at $2\varphi/{M_{\rm sc}}=1$ and $4\varphi/{M_{\rm sc}}=1$ that occur for positive values of the scalar field, assuming ${M_{\rm sc}}>0$, while for $\varphi<0$ the functions are analytic. Let us also notice that, had we started with an arbitrary potential for the scalar field $V(\varphi)$ instead of a mass term, the corresponding final action would have resulted in a re-dressed potential with the same factor, i.e., the effect of the interactions on the potential would be $V(\varphi)\to {\mathcal{U}}(\varphi) V(\varphi)$ and, as a particular case, if we start with a constant potential $V_0$ corresponding to a cosmological constant, the same re-dressing will take place so that the cosmological constant becomes a $\varphi-$dependent quantity. In any case, we find it more natural to start with a mass term in compliance with the prescribed procedure of generating the interactions through the coupling to the energy-momentum tensor, i.e., the natural starting point is the free theory.
An alternative way to resum the series that will be very useful in the less obvious cases that we will consider later is to notice that the resulting perturbative expansion (\[eq:PertScalar1\]) allows to guess the final form of the action to be of the form (\[Actionscalar1\]). Then, we can impose the desired form of our interactions to the energy-momentum tensor so that the full non-linear action must satisfy =\^4x(12()\_\^-()V())=\^4x(12\_\^-V()-T), \[Actionscalar2\] with $T$ the trace of the energy-momentum tensor of the full action, i.e., T=-()\_\^+4V. We have also included here an arbitrary potential for generality. Thus, we will need to have =\^4x(12\_\^-V)=\^4x\[Actionscalar3\] from which we can recover the solutions for ${\mathcal{K}}$ and ${\mathcal{U}}$ given in (\[resconformalscalar\]). Notice that this method allows to obtain the final action without relying on the convergence of the perturbative series and, thus, the aforementioned extension of the resummed series is justified. As a final remark, it is not difficult to see that, had we started with a coupling to an arbitrary function of $\varphi$ of the form $-f(\varphi/{M_{\rm sc}})T_{(0)}$, the final result would be the same with the replacement $\varphi/{M_{\rm sc}}\to f(\varphi/{M_{\rm sc}})$ in the final form of the function ${\mathcal{K}}$ and ${\mathcal{U}}$, recovering that way the results of [@Sami:2002se].
We have then obtained the action for a scalar field coupled to its own energy-momentum tensor at the level of the action. However, as we mentioned above, we can alternatively impose the trace of the energy-momentum tensor to be the source of the scalar field equations, i.e., the full theory must lead to equations of motion satisfying (+m\^2)=- T, \[equationscalar1\] again with $T$ the total energy-momentum tensor of the scalar field. Before proceeding to solve this case, let us comment on some important differences with respect to the gravitational case involving a spin-2 field. The above equation is perfectly consistent at first order, i.e., we could simply add $T_{(0)}$ on the RHS, so we already have a consistent theory and there is no need to include higher order corrections. This is in high contrast with the construction in standard gravity where the Bianchi identities for the spin-2 field (consequence of the required gauge symmetry) are incompatible with the conservation of the energy-momentum tensor and one must add higher order corrections to have consistent equations of motion. For the scalar gravity case, although not imposed by the consistency of the equations, we can extend the construction in an analogous manner and impose that the source of the equation is not given in terms of the energy-momentum tensor of the free scalar field, but the total energy-momentum tensor. As before, we could proceed order by order to find the interactions, but we will directly resort to guess the final action to be of the form given in (\[Actionscalar1\]) and obtain the required form of the functions ${\mathcal{K}}$ and ${\mathcal{U}}$ for the field equations to be of the form given in (\[equationscalar1\]). For the sake of generality, we will consider a general bare potential $V(\varphi)$ instead of a simple mass term. By varying (\[Actionscalar1\]) w.r.t. the scalar field we obtain =-()\^2- that must be compared with the prescribed form of the field equation +V’=- T=. \[equationscalar\] Thus, we see that the functions ${\mathcal{K}}$ and ${\mathcal{U}}$ must satisfy the following equations ’=-,’+(-)=. The solution for ${\mathcal{K}}$ can be straightforwardly obtained to be = \[eq:solKsource\] where we have chosen the integration constant so that ${\mathcal{K}}(0)=1$, i.e., we absorbed ${\mathcal{K}}(0)$ into the normalization of the free field. It might look surprising that the solution for ${\mathcal{K}}$ in this case is related to (\[resconformalscalar\]) by a change of sign of $\varphi$. This could have been anticipated by noticing that the construction of the theory so that $T$ appears as a source of the field equations requires an extra minus sign with respect to the coupling at the level of the action to compensate for the one introduced by varying the action. Thus, the two series only differ by this extra $(-1)^n$ factor in the series that results in the overall change of sign of $\varphi$.
From the obtained equations we see that the solution for ${\mathcal{U}}$ depends on the form of the bare potential $V(\varphi)$. We can solve the equation for an arbitrary potential and the solution is given by =with $C_1$ an integration constant that must be chosen so that ${\mathcal{U}}(0)=1$. If $V'(\varphi)\neq0$, we need to set $C_1=0$. Remarkably, if we take a quadratic bare potential corresponding to adding a mass for the scalar field (which is the most natural choice if we start from a free theory), the above solution reduces to ${\mathcal{U}}=1$. In that case, the resummed action reads =\^4x, which is the action already obtained by Freund and Nambu in [@Freund:1969hh], and which reduces to Nordstrøm’s theory in the massless limit. We have obtained here the solution for the more general case with an arbitrary bare potential, in which case the solution for ${\mathcal{U}}$ depends on the form of the potential. Finally, if we have $V'=0$, the starting action contains a cosmological constant $V_0$ and the obtained solution for ${\mathcal{U}}(\varphi)$ gives the scalar field re-dressing of the cosmological constant, which is $(1+2\varphi/{M_{\rm sc}})^2$ obtained after setting $C_1=V_0$ as it corresponds to have ${\mathcal{U}}(0)=1$. We will re-obtain this result in Sec. \[Sec:Scalarmattercoupling\] when studying couplings to matter fields.
Derivatively coupled scalar gravity {#Sec:ScalarDerivative}
-----------------------------------
After warming up with the simplest coupling of the scalar field to the trace of its own energy-momentum tensor, we will now look at interactions enjoying a shift symmetry, what happens for instance in models where the scalar arises as a Goldstone boson, a paradigmatic case in gravity theories being branons, that are associated to the breaking of translations in extra dimensions [@DoMa]. This additional symmetry imposes that the scalar field must couple derivatively to the energy-momentum tensor and this further imposes that the leading order interaction must be quadratic in the scalar field, i.e., we will have a coupling of the form $\partial_\mu\varphi\partial_\nu\varphi T^{\mu\nu}$. This is also the interaction arising in theories with disformal couplings [@Disformal]. Although an exact shift symmetry is only compatible with a massless scalar field, we will leave a mass term for the sake of generality (and which could arise from a softly breaking of the shift symmetry). In fact, again and for the sake of generality, we will consider a general bare potential term. Then, the action with the first order correction arising from the derivative coupling to the energy-momentum tensor in this case is given by \_[(0)]{}+\_[(1)]{}=\^4x, with ${M_{\rm sd}}$ some mass scale. It is interesting to notice that now the coupling is suppressed by ${M_{\rm sd}}^{-4}$ so that the leading order interaction corresponds to a dimension 8 operator, unlike in the previous non-derivative coupling whose leading order was a dimension 5 operator. As before, the added interaction will contribute to the energy-momentum tensor so that the interaction needs to be corrected. If we proceed with this iterative process, we find the expansion $$\begin{aligned}
{\mathcal{S}}_\varphi=&\int{\mathrm{d}}^4x\left[\frac12\Big(1+X+3X^2+15X^3+\cdots \Big)\partial_\mu\varphi\partial^\mu\varphi-\Big(1-X-X^2-3X^3 \cdots \Big)V(\varphi) \right].
\label{eq:Pert1}\end{aligned}$$ where we have defined $X\equiv(\partial\varphi)^2/{M_{\rm sd}}^4$. Again, we could obtain the general term of the generated series and eventually resum it. However, it is easier to use an Ansatz for the resummed action by noticing that, from (\[eq:Pert1\]), we can guess the final form of the action to be \_=\^4x\[eq:AnsatzScalar\] with ${\mathcal{K}}(X)$ and ${\mathcal{U}}(X)$ some functions to be determined from our prescribed couplings. Thus, by imposing that the final action must satisfy \_=\^4xwith $T_{\mu\nu}$ the total energy-momentum tensor, we obtain the following relation: \_&=&\^4x\
&=&\^4x,where the prime stands for derivative w.r.t. its argument. Thus, the functions ${\mathcal{K}}(X)$ and ${\mathcal{U}}(X)$ will be determined by the following first order differential equations (X)&=&1+X(X)+2X\^2’(X),\
(X)&=&1-X(X)+2X\^2’(X). \[eq:DerivativeScalar\] We have thus reduced the problem of resuming the series to solving the above differential equations. The existence of solutions for these differential equations will guarantee the convergence (as well as the possible analytic extensions) of the perturbative series. Although not important for us here, it is possible to obtain the explicit analytic solutions as &=&-[[Ei]{}]{}\_[1/2]{}(-1/(2X))\
&=&-[[Ei]{}]{}\_[-1/2]{}(-1/(2X)). where ${{\rm Ei}}_n(x)$ stands for the exponential integral function of order $n$ and we have chosen the integration constants in order to have a well-defined solution for $X\to0$. In principle, one might think that boundary conditions must be imposed so that ${\mathcal{K}}(0)={\mathcal{U}}(0)=1$. However, these boundary conditions are actually satisfied by all solutions of the above equations since they are hardwired in the own definition of the functions ${\mathcal{K}}$ and ${\mathcal{U}}$ through the perturbative series. The way to select the right solution is thus by imposing regularity at the origin $X=0$. Even this condition is not sufficient to select one single solution and this is related to the fact that the perturbative series must be interpreted as an asymptotic expansion[^5], rather than a proper series expansion. In fact, it is not difficult to check that the perturbative series is divergent, as it is expected for asymptotic expansions. Thus, the above solution is actually one of many different possible solutions. We will find these equations often and we will defer a more detailed discussion of some of their features to the Appendix \[Appendix\].
So far we have focused on the coupling $\partial_\mu\varphi \partial_\nu \varphi T^{\mu\nu}$, but, at this order, we can be more general and allow for another interaction of the same dimension so that the first correction becomes \_[(1)]{}=\^4x(b\_1\_\_+b\_2\_\^\_) T\^\_[(0)]{}, where $b_1$ and $b_2$ are two arbitrary dimensionless parameters, one of which could actually be absorbed into ${M_{\rm sd}}$, but we prefer to leave it explicitly to keep track of the two different interactions. The previous case then reduces to $b_2=0$, which is special in that the coupling does not depend on the metric and, as we will see in Sec. \[Sec:EffectiveMetrics\], this has interesting consequences in some constructions. For this more general coupling, the perturbative series is $$\begin{aligned}
{\mathcal{S}}_\varphi=&\int{\mathrm{d}}^4x\left[\frac12\Big(1+(b_1-2b_2)X+3b_1(b_1-2b_2)X^2+3b_1(b_1-2b_2)(5b_1+2b_2)X^3+\cdots \Big)\partial_\mu\varphi\partial^\mu\varphi\right.\nonumber\\
&\left. -\Big(1-(b_1+4b_2)\left(X+(b_1-2b_2)X^2+3b_1(b_1-2b_2)X^3 \cdots \right)\Big)V(\varphi) \right].
\label{eq:Pert2}\end{aligned}$$ To resum the series we can follow the same procedure as before using the same Ansatz for the resummed action as in (\[eq:AnsatzScalar\]), in which case we obtain that the following relation must hold: &=&\^4x\
&=&\^4x\
&=&\^4x. Thus, the equations to be satisfied in this case are &=&1+(b\_1-2b\_2)X+2(b\_1+b\_2)X\^2’,\
&=&1-(b\_1+4b\_2)X+2(b\_1+b\_2)X\^2’. \[Eq:eqsderivativecoupling\] The additional freedom to choose the relation between the two free parameters $b_1$ and $b_2$ allows now to straightforwardly obtain some particularly interesting solutions. Firstly, for $b_1+b_2=0$, the equations become algebraic and the unique solution is given by (X)=(X)=. This particular choice of parameters that make the equations algebraic is remarkable because it precisely corresponds to coupling the energy-momentum tensor to the orthogonal projector to the gradient of the scalar field $\eta_{\mu\nu}-\partial_\mu\varphi \partial_\nu\varphi/(\partial\varphi)^2$. On the other hand, we can see from the perturbative series (\[eq:Pert2\]) that the condition $b_1-2b_2=0$ cancels all the corrections to the kinetic term and this can also be seen from the differential equations where it is apparent that, for those parameters, ${\mathcal{K}}=1$ is the corresponding solution. Moreover, for that choice of parameters, we see from the perturbative expansion that ${\mathcal{U}}=1-6b_2X$, which can be confirmed to be the solution of the equation for ${\mathcal{U}}$ with $b_1=2b_2$. Likewise, for $b_1+4b_2=0$, all the corrections to the potential vanish and only the kinetic term is modified. It is worth mentioning that the iterative procedure used to construct the interactions also allows to obtain polynomial solutions of arbitrarily higher order by appropriately choosing the parameters. All these interesting possibilities are explained in more detail in the Appendix \[Appendix\].
Finally, let us notice that a constant potential $V_0$ that amounts to introducing a cosmological constant in the free action leads to a re-dressing of the cosmological constant analogous to what we found above for the non-derivative coupling, but with the crucial difference that now the cosmological constant becomes kinetically re-dressed in the full theory.
In the general case we see that we obtain a particular class of K-essence theories where the $\varphi$-dependence is entirely given by the starting potential, but it receives a kinetic-dependent re-dressing. On the other hand, if we start with an exact shift symmetry, given that the interactions do not break it, the resulting theory reduces to a particular class of $P(X)$ theories.
First order formalism {#Sec:ScalarFirstOrder}
---------------------
In the previous section we have looked at the theory for a scalar field that is derivatively coupled to its own energy-momentum tensor. The problem was reduced to solving a couple of differential equations expressed in (\[eq:DerivativeScalar\]). Here we will explore the same problem but from the first order formalism perspective. In the case of non-abelian gauge fields and also in the case of gravity, the first order formalism has proven to significantly simplify the problem since the iterative process ends at the first iteration [@Deser:1969wk]. For non-abelian theories, the first order formalism solves the self-coupling problem in one step instead of the four iterations required in the Lagrangian formalism. In the case of gravity, the simplification is even greater since it reduces the infinite iterations of the self-coupling problem to only one. This is in fact the route used by Deser to obtain the resummed action for the self-couplings of the graviton [@Deser:1969wk]. The significant simplifications in these cases encourage us to consider the construction of the theories with our prescription in the first order formalism in order to explore if analogous simplifications take place. As a matter of fact, the first order formalism for scalar gravitation was already explored in [@Deser:1970zzb] for the massless theory and with a conformal coupling so that the trace of the energy-momentum tensor appears as the source of the scalar field. It was then shown the equivalence of the resulting action with Nordstrøm’s theory of gravity and the massless limit of the theory obtained by Freund and Nambu [@Freund:1969hh] with the first order formalism (which we reproduced and extended above). We will use this formalism for the theories with derivative couplings to the energy-momentum tensor, what in the first order formalism means couplings to the canonical momentum.
The first thing we need to clarify is how we are going to define the theory in the first order formalism. The starting free theory for a massive scalar field $\varphi$ can be described by the following first order action: \_[(0)]{}=\^4x, with $\pi^\mu$ the corresponding momentum in phase space. Upon variations with respect to the momentum $\pi^\mu$ and the scalar field we obtain the usual Hamilton equations $\partial_\mu\pi^\mu+m^2\varphi=0$ and $\pi_\mu=\partial_\mu\varphi$, which combined gives the desired equation $(\Box+m^2)\varphi=0$. At the lowest order we then prescribe a coupling to the energy-momentum tensor as[^6] \_[(0)]{}+\_[(1)]{}=\^4xwith $T_{(0)}^{\mu\nu}$ the energy-momentum tensor corresponding to the free theory ${\mathcal{S}}_{(0)}$. This is the form that we will also require for the final theory replacing $T_{(0)}^{\mu\nu}$ by the total energy-momentum tensor. Following the same reasoning as in the previous section, the final theory should admit an Ansatz of the following form: =\^4xwhere the Hamiltonian[^7] ${\mathcal{H}}$ will be some function of the phase space coordinates. Lorentz invariance imposes that the momentum can only enter through its norm. As in the Lagrangian formalism, the energy-momentum tensor admits several definitions that differ by a super-potential term or quantities vanishing on-shell. As before, we shall resort to the Hilbert energy-momentum tensor. In this approach, one needs to specify the tensorial character of the fields, which are usually assumed to be true tensors. In some cases, it is however more convenient to assume that some fields actually transform as tensorial densities. In the Deser construction, assuming that the graviton is a tensorial density simplifies the computations. At the classical level and on-shell, assuming different weights only results in terms that vanish on-shell in the energy-momentum tensor[^8]. In the present case, it is convenient to assume that $\pi^\mu$ is a tensorial density of weight 1 such that $p^\mu=\pi^\mu/\sqrt{-\gamma}$ is a tensor of zero weight. The advantage of using this variable is twofold: on one hand, this tensorial weight for $\pi^\mu$ makes $\pi^\mu\partial_\mu\varphi$ already a weight-0 scalar without the need to introduce the $\sqrt{-\gamma}$ in the volume element and, consequently, this term will not contribute to the energy-momentum tensor. On the other hand, the variation of the Hamiltonian ${\mathcal{H}}$ with respect to the auxiliary metric $\gamma_{\mu\nu}$ gives (p\^2)=p\^2=-\^2$\gamma^{\mu\nu}-\frac{\pi^\mu\pi^\nu}{\pi^2}$\_ which is proportional to the orthogonal projector to the momentum and, thus, although it does contribute to the energy-momentum tensor, it will not contribute to the interaction $T^{\mu\nu}\pi_\mu \pi_\nu$. To derive the above variation, we have taken into account that the Hamiltonian remains a scalar after the covariantisation and, because of the assumed weight of $\pi^\mu$, it will become a function of ${\mathcal{H}}(\varphi,\pi^2)\to{\mathcal{H}}\big(\varphi,p^2\big)={\mathcal{H}}\big(\varphi,\pi^2/\vert\gamma\vert\big)$. The covariantised action then reads =\^4xand the total energy-momentum tensor computed with the described prescription is given by T\^=-2\^2(\^-)+\^. \[Eq:Tfirstorder\] If we compare with the canonical energy-momentum tensor $\Theta^{\mu\nu}=2\frac{\partial{\mathcal{H}}}{\partial \pi^2}\pi^\mu\pi^\nu-{\mathcal{L}}\eta^{\mu\nu}$, we see that the difference is ${\mathcal{H}}-2\pi^2\frac{\partial{\mathcal{H}}}{\partial \pi^2}-{\mathcal{L}}$, which vanishes upon use of the Hamilton equation $\partial_\mu\varphi=\frac{\partial{\mathcal{H}}}{\partial \pi^\mu}$ and the relation between the Hamiltonian and the Lagrangian via a Legendre transformation ${\mathcal{L}}=\pi^\mu\partial_\mu\varphi-{\mathcal{H}}$. The final action must therefore satisfy the relation &=&\^4x=\^4x\
&=&\^4xwhere we see that the chosen weight for $\pi^\mu$ has greatly simplified the interaction term that simply reduces to $\pi^2{\mathcal{H}}$. The above relation then leads to the algebraic equation =12(\^2+m\^2\^2)-\^2so that the Hamiltonian of the desired action will be given by =12. We see that the use of the first order formalism has substantially simplified the resolution of the problem since we do not encounter differential equations. Needless to say that the solutions obtained in both first and second order formalisms are different. This apparent ambiguity in the resulting theory as obtained with the first or the second order formalism actually reflects the ambiguity in the definition of the energy-momentum tensor because, as discussed above, the energy-momentum tensor Eq. (\[Eq:Tfirstorder\]) differs from the one used in the second order formalism by a term that vanishes on-shell (see also Footnote \[F7\]). Expressing the theory obtained here in the second order formalism is not very illuminating so we will not give it, although it would be straightforward to do it. Let us finally notice that, if the leading order term in the Hamiltonian for the limit $\pi^2/{M_{\rm sd}}^4\ll1$ is assumed to be ${\mathcal{H}}_0$, then it is not difficult to see that the full Hamiltonian will be =. i.e., the procedure simply re-dresses the seed Hamiltonian with the factor $(1+\pi^2/{M_{\rm sd}}^4)^{-1}$. One interesting property of the resulting theory is that the Hamiltonian density ${\mathcal{H}}$ for the massless case saturates to the scale ${M_{\rm sd}}^4$ at large momenta.
Coupling to matter fields {#Sec:Scalarmattercoupling}
-------------------------
In the previous subsections we have found the action for the self-interacting scalar field through its own energy-momentum tensor, both with ultra-local and derivative couplings. Now we will turn our analysis to the couplings of the scalar field with other matter fields following the same philosophy, i.e., the scalar will couple to the energy-momentum tensor of matter fields. For simplicity, we will only consider the case of a matter sector described by a scalar field $\chi$. The derivative couplings will be the same as we will obtain for the vector field couplings to matter that will be treated in the next section so that, in order not to unnecessarily repeat the derivation, we will not give it here and discuss it in Sec. \[Sec:VectorMatter\]. Thus, we will only deal with the conformal couplings so that our starting action for the proxy scalar field $\chi$ including the first order coupling to $\varphi$ is \_[,(0)]{}+\_[,(1)]{}=\^4xwhere $T_{\chi,(0)}$ is the trace of the energy-momentum tensor of the proxy field and $W(\chi)$ the corresponding potential. Going on in the iterative process yields the following series for the total action \_=\^4 x. This expression has the form of a geometric series so it is straightforward to resum it yielding the final action for scalar gravity algebraically coupled to matter: =\^4x (12()\_\^-()W()) with ()=\_[n=0]{}\^(2)\^n=,()=\_[n=0]{}\^(4)\^n=. Not very surprisingly, we obtain the same result as for the self-couplings of the scalar field, i.e., both the kinetic and potential terms get re-dressed by the same factors as we found in Sec. \[sec:Conformalscalar\] for $\varphi$. If the matter sector consists of a cosmological constant, which would correspond to a constant scalar field in the above solutions, the coupling procedure gives rise to an additional modification of the $\varphi$ potential or, equivalently, the cosmological constant becomes a $\varphi$-dependent quantity, which is also the result that we anticipated in Sec \[sec:Conformalscalar\] for a constant potential of $\varphi$. Some phenomenological consequences of this mechanism were explored in a cosmological context in [@Sami:2002se].
As in the self-coupling case, we could have introduced the coupling so that the trace of the energy-momentum tensor appears as a source of the scalar field equations. In order to obtain the theory with the required property, we shall follow the procedure of assuming the following action for the scalar gravity field $\varphi$ and the scalar proxy field $\chi$: =\^4 x (12()\_\^-()V()+12()\_\^-()W()). \[eq:Schi\] with ${\mathcal{K}}$, ${\mathcal{U}}$, $\tilde{\alpha}$ and $\tilde{\beta}$ some functions of $\varphi$ that will be determined from our requirement and $W(\chi)$ is some potential for the scalar $\chi$. In our Ansatz we have included the self-interactions of the scalar field encoded in ${\mathcal{K}}$ and ${\mathcal{U}}$. Since this sector was resolved above, we will focus here on the couplings to $\chi$ so that $\tilde{\alpha}$ and $\tilde{\beta}$ are the functions to be determined by imposing the $\varphi$ field equations be of the form +V’()=-T. The trace of the total energy-momentum tensor derived from (\[eq:Schi\]) is given by T=4() V()+4() W()-() \_\^-() \_\^and hence we obtain that the equation of motion must be of the form +V’()=-(4() V()+4() W()-() \_\^-() \_\^). \[eq:phichi1\] On the other hand, varying (\[eq:Schi\]) with respect to $\varphi$ yields =-(12’()\_\^-(() V())’+12’()\_\^-’()W()). \[eq:phichi2\] Comparing (\[eq:phichi1\]) and (\[eq:phichi2\]) will give the equations that must be satisfied by the functions in our Ansatz for the action. The $\varphi$ sector has already been solved in the previous subsection, so we will only pay attention to the $\chi$ sector now. Then, we see that the functions $\tilde{\alpha}$ and $\tilde{\beta}$ must satisfy the following equations =,=. The solution for these equations, taking into account the functional form of ${\mathcal{K}}(\varphi)$ given in \[eq:solKsource\], is then = 1+, = (1+)\^2 that coincides with the expression given in [@Freund:1969hh]. We see again that, although both procedures give the same leading order coupling to matter for the scalar field, the full theory crucially depends on whether the coupling is imposed at the level of the action or the equations. If we consider again the case of a cosmological constant as the matter sector, we see that its re-dressing with the scalar field will be different in both cases. It could be interesting to explore the differences with respect to the analysis performed in [@Sami:2002se], where the coupling was assumed to occur at the level of the action.
Vector gravity
==============
After having revisited and extended the case of a scalar field coupled to the energy-momentum tensor, we now turn to the case of a vector field. Since vectors present a richer structure than scalars due to the possibility of having a gauge invariance or not depending on whether the vector field is massless of massive, we will distinguish between gauge invariant couplings and non-gauge invariant couplings. For the latter, the existence of a decoupling limit where the dominant interactions correspond to those of the longitudinal mode will lead to a resemblance between some of the interactions obtained here and those of the derivatively coupled scalar studied above.
Self-coupled Proca field
------------------------
Analogously to the scalar field case, our starting point will be the action for a massive vector field given by the Proca action[^9] \_[(0)]{}=\^4x(-14F\_F\^+12m\^2A\^2) where $F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu$, $A^2\equiv A_\mu A^\mu$ and $m^2$ is the mass of the vector field. The energy-momentum tensor of this field is given by T\_[(0)]{}\^=-F\^F\^\^+\^F\_F\^-\^A\^2+m\^2 A\^A\^. Unlike the case of the scalar field, this energy-momentum tensor does not coincide with the canonical one obtained from Noether’s theorem and, thus, the Belinfante-Rosenfeld procedure would be needed to obtain a symmetric energy-momentum tensor, showing the importance of the choice in the definition of the energy-momentum tensor in the general case.
Along the lines of the procedure carried out in the previous sections, we will now introduce self-interactions of the vector field by coupling it to its energy-momentum tensor, so the first correction will be \_[(1)]{}=\^4xA\_A\_T\_[(0)]{}\^ with ${M_{\rm vc}}^2$ the corresponding coupling scale. In this case, the leading order interaction corresponds to a dimension 6 operator. Since this interaction will also contribute to the energy-momentum tensor, we will need to add yet another correction as in the previous cases, resulting in an infinite series in $A^2/{M_{\rm vc}}^2$ that reads: =\^4xwhere $Y\equiv A^2/{M_{\rm vc}}^2$. Again, to resum the iterative process we will use a guessed form for the full action. The above perturbative series makes clear that the final form of the action will take the form =\^4x\
where the functions $\alpha$, $\beta$ and ${\mathcal{U}}$ will be obtained by imposing the desired form of the interactions through the total energy-momentum tensor, i.e., we need to have &=&\^4x\
&=&\^4x\
&=&\^4x. Thus, the coupling functions in the resummed action need to satisfy the following first order differential equations &=& 1-Y+2Y\^2’,\
&=&+3Y+2 Y\^2’,\
&=&1+Y+2Y\^2’ . These equations are of the same form as the ones obtained for the derivatively coupled scalar field and the solutions will also present similar features. For instance, the perturbative series will need to be interpreted as asymptotic expansions of the solutions of the above equations. Furthermore, the integration constants that determine the desired solution are already implemented in the equations so we need to impose regularity at the origin, but this only selects a unique solution for a given semi-axis, either $Y>0$ or $Y<0$, that can then be matched to an infinite family of solutions in the complementary semi-axis (see Appendix \[Appendix\]). The particular form of the solutions is not specially relevant for us here (although it will be relevant for practical applications), but we will only remark that they correspond to the class of theories for a vector field that is quadratic in the field strength or, in other words, in which the field strength only enters linearly in the equations.
As for the derivative couplings of the scalar field, we can be more general and allow for a coupling of the form \_[(1)]{}=\^4x(b\_1A\_A\_+b\_2A\^2\_) T\_[(0)]{}\^. The iterative process in this case gives rise to =\^4x.We can again use our Ansatz for the final action to obtain that the differential equations to be satisfied are &=& 1-b\_1Y+2(b\_1+b\_2)Y\^2’\
&=&b\_1+(3b\_1+2b\_2)Y+2(b\_1+b\_2) Y\^2’\
&=&1+(b\_1-2b\_2)Y+2(b\_1+b\_2)Y\^2’ . Remarkably, we see from the perturbative expansion that the case $b_1=0$ exactly cancels all the corrections to the kinetic part so that $\alpha=1$, $\beta=0$ and only the potential sector is modified. It is not difficult to check that this is indeed a solution to the above differential equations. We can also see here again that the choice $b_1+b_2=0$, which corresponds to a coupling to the orthogonal projector to the vector field given by $\eta_{\mu\nu}-A_\mu A_\nu/A^2$, reduces the equations to a set of algebraic equations whose solution is &=&,\
&=&=,\
&=&. These solutions show that $\alpha$ and $\beta$ present different analytic properties depending on whether the field configuration is timelike or spacelike, but, in any case, $\beta$ has a pole for $\vert b_1 Y\vert =1$ so that it seems reasonable to demand $\vert b_1Y\vert<1$ for this particular solution. The properties of these equations are similar to the ones we found in Sec. \[Sec:ScalarDerivative\] for the derivatively coupled case and, in fact, the equation for ${\mathcal{U}}$ here is the same as the equation for ${\mathcal{K}}$ in (\[Eq:eqsderivativecoupling\]), which is of course no coincidence. Thus, the more detailed discussion given in Appendix \[Appendix\] also applies and, in particular, it will also be possible to obtain polynomial solutions by appropriately choosing the parameters, owed to the recursive procedure used to construct the interactions.
From our general solution we can also analyse what happens if our starting free theory is simply a Maxwell field, i.e., $m^2=0$. In that case, only the terms containing $\alpha$ and $\beta$ will have an effect and, in fact, they will provide the vector field with a mass around non-trivial backgrounds of $F_{\mu\nu}$, signaling that the number of perturbative propagating polarisations will depend on the background configuration. However, the lack of a gauge symmetry in the full theory makes a vanishing bare mass seem like an unnatural choice.
So far we have solved the problem of a vector field coupled to its own energy-momentum tensor at the full non-linear level in the action. We could also follow the procedure of finding the theory such that the energy momentum-tensor is the source of the vector field equations of motion. This is analogous to the case of scalar gravity considered above and also the procedure that leads to GR for the spin-2 case. However, a crucial difference arises for the vector field[^10] case owed to the fact that the leading order interaction is quadratic in the vector field and, thus, the energy-momentum tensor cannot act as a source of the vector field equations. We have different possibilities then on how to generalise the linear coupling to the energy-momentum tensor to the full theory at the level of the field equations. Because of the lack of a clear criterion at this point and the existence of several different inequivalent possibilities of carrying out this procedure, we will not pursue it further here and we will content ourselves with the analysis of the construction of the theories where the coupling occurs at the level of the action. We will simply mention that a possibility to get this difficulty around is by breaking Lorentz invariance. If we introduce some fixed vector $u^\mu$ (that could be identified for instance with some vev of the vector field), then we could construct our interactions as e.g. $A_\mu T^{\mu\nu}u_\nu$ so that $T^{\mu\nu}u_\nu$ would act as the source of the vector field equations and, then, we could extend this result at the non-linear level.
Derivative gauge-invariant self-couplings
-----------------------------------------
In the previous section we have studied the case where the vector field couples to its energy-momentum tensor without imposing gauge invariance. For a scalar field, the equations of motion do not contain any off-shell conserved current derived from some Bianchi identities, and this makes a crucial difference with respect to the vector field case where we do have the Bianchi identities derived from the $U(1)$ symmetry of the Maxwell Lagrangian. Even if we break the gauge symmetry by adding a mass term, the off-shell current leads to a constraint equation that must be satisfied. In the theories obtained in the previous section by coupling $A_\mu$ to the energy-momentum tensor, we also obtain a constraint equation by taking the divergence of the corresponding field equations. This constraint is actually the responsible for keeping three propagating degrees of freedom in the theory. In fact, starting from a massless vector field with its gauge invariance, the couplings actually generate a mass term around non-trivial backgrounds, increasing that way the number of polarisations. In this section, we will aim to re-consider our construction by maintaining the $U(1)$ gauge symmetry of Maxwell theory also in the couplings of the vector field to its own energy-momentum tensor. This also resembles somewhat the extension of the scalar field case to include derivative couplings arising from imposing a shift symmetry, although with crucial differences, for instance the scalar field interactions are based on a global symmetry while the ones considered in this section will be dictated by a gauge symmetry.
After the above clarifications on the procedure that we will follow in this section, we can write our starting action describing a massless vector field \_[(0)]{}=-14\^4x F\_ F\^, and add interactions to its own energy-momentum tensor respecting the $U(1)$ symmetry. At the lowest order, these interactions can be written as \_[(1)]{}=\^4x (b\_1F\_ F\_\^+b\_2 F\_ F\^\_)T\^\_[(0)]{} with ${M_F}$ the corresponding coupling scale, $b_{1,2}$ dimensionless parameters and the Maxwell energy-momentum tensor given by T\_[(0)]{}\^=-F\^F\^\_+\^F\_F\^. Before proceeding further, let us digress on the form of the introduced interactions. We have followed the easiest possible path of adding couplings that trivially respect the original $U(1)$ symmetry by using the already [*gauge invariant*]{} field strength $F_{\mu\nu}$. This is along the lines of the Pauli interaction term for a charged fermion $\psi$ given by $\bar{\psi}[\gamma^\mu,\gamma^\nu]\psi F_{\mu\nu}$, where gauge invariance is trivially realised. Of course, this non-renormalisable term is within the effective field theory, but the usual renormalisable coupling $A_\mu J^\mu$ gives the leading order interaction. This is somehow analogous to existing constructions for the spin-2 case where one seeks for a consistent coupling of the graviton to the energy-momentum tensor respecting the original linearised diffeomorphisms invariance. Besides the usual coupling $h_{\mu\nu}T^{\mu\nu}$ that respects the symmetry upon conservation of $T^{\mu\nu}$, the realisation of the linearised diffeomorphisms can be achieved in two different ways, namely: one can either introduce a coupling of the form $h_{\mu\nu} P^{\mu\nu\alpha\beta} T_{\alpha\beta}$ with $P^{\mu\nu\alpha\beta}$ an identically divergenceless projector $\partial_\mu P^{\mu\nu\alpha\beta}=0$ (see for instance [@Deser:1968zza]), or one can add a coupling of the matter fields to an exactly gauge invariant quantity. The second approach is possible by using that the linearised Riemann tensor $R^{\rm L}_{\mu\nu\alpha\beta}(h)$ is [*gauge invariant*]{}, and not only covariant (see the seminal Wald’s paper [@Wald:1986bj] and [@Hertzberg:2016djj] for interesting discussions on these alternative couplings). This is the analogous construction scheme we are following here. One could have also tried to follow another approach and tried to construct non-derivative couplings with a field dependent realisation of the original $U(1)$ symmetry whose lowest order in fields is given by the usual transformation law. The non-linear completion of gauge symmetry can correspond either to the original $U(1)$ symmetry up to some field redefinition or to a genuine non-linear completion, what will likely require transformations involving higher derivatives of the gauge parameter (see for instance [@Wald:1986bj]). This interesting path will not be pursued further here and we will focus on the simplest case.
After briefly discussing some alternatives for gauge invariant couplings, we will proceed with the construction considered here. The iterative process in this case leads to the perturbative series &=&-14\^4xwhere $Z\equiv F_{\alpha\beta}F^{\alpha\beta}/{M_F}^4$, $\tilde{Z}\equiv F_{\alpha\beta}\tilde{F}^{\alpha\beta}/{M_F}^4$ and $\tilde{F}^{\mu\nu}=\frac12\epsilon^{\mu\nu\alpha\beta}F_{\alpha\beta}$ is the dual of the field strength. Moreover, we have used the identity $4F^\mu{}_\nu F^\nu{}_\rho F^\rho{}_\sigma F^\sigma{}_\mu=2(F_{\mu\nu}F^{\mu\nu})^2+(F_{\mu\nu}\tilde{F}^{\mu\nu})^2$. This perturbative series is not straightforwardly resummed, so we will follow the alternative procedure of making an Ansatz for the resummed action. Since we are maintaining gauge invariance the resulting action must be a function of the two independent Lorentz and gauge invariants, i.e., the action must take the form =M\_F\^4\^4x (Z,), with ${\mathcal{K}}$ a function to be determined. Notice that our prescribed interactions do not break parity so that ${\mathcal{K}}$ will need to be an even function of $\tilde{Z}$. Given our requirement of a coupling to the energy-momentum tensor, the action also needs to take the form $$\begin{aligned}
{\mathcal{S}}& =\int {\mathrm{d}}^4x\[-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\frac{1}{M_F^4}\(b_1 F_{\mu\alpha}F_\nu{}^\alpha+b_2F_{\alpha\beta}F^{\alpha\beta}\eta_{\mu\nu}\)T^{\mu\nu}\]\notag\\
& = M_F^4\int{\mathrm{d}}^4 x\[-\frac{Z}{4}-(b_1+4b_2)Z({\mathcal{K}}-\tilde{Z}{\mathcal{K}}_{\tilde{Z}})+(2(b_1+2b_2)Z^2+b_1\tilde{Z}^2){\mathcal{K}}_Z\].\end{aligned}$$ From these two expressions we conclude that ${\mathcal{K}}(Z,\tilde{Z})$ must satisfy the partial differential equation: =--(b\_1+4b\_2)Z(-\_)+\_Z. \[Eq:ZZt\] We can easily check that for $b_1=0$ the perturbative series does not generate any interaction, which is nothing but a reflection of the conformal invariance of the Maxwell Lagrangian in 4 dimensions that leads to a traceless energy-momentum tensor. If we consider that case, the above equation reduces to ${\mathcal{K}}=-Z/4+4b_2(-{\mathcal{K}}+Z{\mathcal{K}}_Z+\tilde{Z}{\mathcal{K}}_{\tilde{Z}})$, that has the Maxwell Lagrangian as a regular solution. On the other hand, for $3b_1+4b_2=0$, we see that only the first corrections in the perturbative expansion remains so that one expects the full solution to take the simple polynomial form ${\mathcal{K}}=-1/4 \big[(1+b_1 Z) Z + b_1 \tilde{Z}^2\big]$. One can readily check that this is indeed the solution of Eq. (\[Eq:ZZt\]) with $b_2=-3b_1/(4b_4)$. As in the previous sections dealing with derivatively coupled scalars or non-gauge invariant couplings for vectors, the existence of this polynomial solution is a direct consequence of the recursive procedure generating the interactions and, thus, we could also choose the parameters $b_1$ and $b_2$ as to have some higher order polynomial solutions for the gauge invariant theories obtained here.
First order formalism {#first-order-formalism}
---------------------
In the previous section we have looked at the theory for a vector field that is coupled to its own energy-momentum tensor in a gauge invariant way. We have just seen that the gauge-invariant coupling leads to a partial differential equation that is, in general, not easy to solve so we will now consider the problem from the first order formalism perspective, hoping that it will simplify the resulting equations, as it happens in other contexts. Unfortunately, we will see that this does not seem to be the case here. The starting free theory for a $U(1)$ invariant vector field can be described in the first order formalism by the action \_[(0)]{}=\^4x. Upon variations with respect to the momentum $\Pi^{\mu\nu}$ and the vector field, we obtain the usual De Donder-Weyl-Hamilton equations $\Pi_{\mu\nu}=\partial_\nu A_\mu-\partial_\mu A_\nu=-F_{\mu\nu}$ and $\partial_\mu\Pi^{\mu\nu}=0$ respectively, that reproduce the usual Maxwell equations $\partial_\mu F^{\mu\nu}=0$. By integrating out the momentum, we reproduce the usual Maxwell theory in the second order formalism. At the lowest order we then prescribe the self-interaction be of the form \_[(0)]{}+\_[(1)]{}=\^4xwith $T_{(0)}^{\mu\nu}$ the energy-momentum tensor corresponding to the free theory ${\mathcal{S}}_{(0)}$. This is the form that we will require for the final theory replacing $T_{(0)}^{\mu\nu}$ by the total energy-momentum tensor. Following the same reasoning as in the previous section, the final theory should be described by the following action =\^4x\[Eq:SAnsatzfirstorder\] where the Hamiltonian ${\mathcal{H}}$ must be a function of the momentum due to gauge invariance and, furthermore, Lorentz invariance imposes that the dependence must be through the only two independent Lorentz scalars, namely $Z=\Pi^{\mu\nu}\Pi_{\mu\nu}/{M_F}^4$ and $\tilde{Z}=\Pi^{\mu\nu}\tilde{\Pi}_{\mu\nu}/{M_F}^4$. Parity will also impose it to be an even function of $\tilde{Z}$. In order to proceed to compute the energy-momentum tensor, we need to choose the tensorial character of the phase space variables and, as we discussed in Sec. \[Sec:ScalarFirstOrder\], this may lead to substantial simplifications. As in Sec. \[Sec:ScalarFirstOrder\], let us assume that $\Pi^{\mu\nu}$ is a density so that $P^{\mu\nu}=\Pi^{\mu\nu}/\sqrt{-\gamma}$ is a zero-weight tensor. This has the advantage that the term $\Pi^{\mu\nu}\partial_{[\mu}A_{\nu]}$ in (\[Eq:SAnsatzfirstorder\]) will not contribute to $T^{\mu\nu}$. The total energy-momentum tensor is thus given by T\^=\^+4\^\^\_\[Eq:Tvectorfirstorder\] so that the interaction term reads (b\_1 \_\_\^+b\_2\^\_\_)T\^=(b\_1+4b\_2)Z(-)+. From this expression it is already apparent that the resummation will necessarily involve the resolution of a partial differential equation as in the first order formalism case. In fact, the resulting equation will be of the same type and, consequently, resorting to the first order formalism does not lead to any simplification. One may think that another choice of the weight for the momentum could lead to some simplifications, but that is not the case and, in fact, choosing an arbitrary weight leads to the same result. To show this more explicitly, let us assume that the momentum has an arbitrary weight $w$ so that $P^{\mu\nu}=(\sqrt{-\gamma})^{-w}\Pi^{\mu\nu}$ is a tensor of zero weight. Then, the variation of the Hamiltonian with respect to the auxiliary metric will give =-(w\^2\^-2\^\^\_)\_+12(1-2w)\^\_\^\_ where we have taken into account that the Hamiltonian ${\mathcal{H}}$ becomes a function ${\mathcal{H}}\big(Z,\tilde{Z})\to{\mathcal{H}}\big(\vert\gamma\vert^{-w/2}Z,\vert\gamma\vert^{-w/2}\tilde{Z}\big)$ after our covariantisation choice. The total energy-momentum tensor can be readily computed to be T\^=\^+4\^\^\_ . \[Eq:Tgeneralw\] It is easy to see that this expression directly gives the energy-momentum tensor of a Maxwell field with ${\mathcal{H}}=-\frac14\Pi^2$ if we set $w=0$ and integrate out the momentum by using $\Pi^{\mu\nu}=-F^{\mu\nu}$. For the general case, we need to express the energy-momentum tensor in phase space variables. From the equation for $\Pi^{\mu\nu}$ we have \_[\[]{}A\_[\]]{}==2\_+\_. If we insert this expression into (\[Eq:Tgeneralw\]) we obtain T\^=\^+4\^\^\_, which does not depend on the weight and, therefore, it is exactly the same that we obtained in (\[Eq:Tvectorfirstorder\]). Of course, this is not very surprising, since, as a consequence of the argument in footnote \[F7\], different choices of weights only result in quantities that vanish on-shell. In this case, the use of the equation of motion of $\Pi^{\mu\nu}$ in order to express $T^{\mu\nu}$ in phase space variables precisely corresponds to the mentioned on-shell difference.
For completeness, let us give the resulting equation in this case =-Z-(b\_1+4b\_2)Z(-)-. We see that, as advertised, the first order formalism does not seem to give any advantage with respect to the second order formalism in this case. It may be that there is some clever choice of phase space coordinates that does reduce the difficulty of the problem.
Coupling to matter fields {#Sec:VectorMatter}
-------------------------
We will end our study of the vector field case by considering couplings to matter fields through the energy-momentum tensor. As in Sec. \[Sec:Scalarmattercoupling\] we will take a scalar field $\chi$ as a proxy for the matter. Moreover, as we mentioned in $\ref{Sec:Scalarmattercoupling}$, the results obtained here will also give how the scalar field $\varphi$ couples to matter fields when the interactions follow our prescription. Thus, our starting action will now contain the additional term \_[,(1)]{}=\^4xwhere $T^{\mu\nu}$ includes the energy-momentum of the own vector field plus the contribution coming from the scalar field. The iterative process applied to this case yields the following expansion $$\begin{aligned}
{\mathcal{S}}_\chi=&\int{\mathrm{d}}^4x\left[\frac12\mathcal{K}^{\mu\nu}\partial_\mu\chi\partial_\nu\chi -\left(1-(b_1+4b_2)\left(Y+(b_1-2b_2)Y^2+3b_1(b_1-2b_2)Y^3 \cdots\right) \right)W(\chi) \right]\end{aligned}$$ where we have defined \^&&\^\
&&+2b\_1(1+2(b\_1-b\_2)Y+(9b\_1\^2-8b\_1b\_2-4b\_2\^2)Y\^2 )A\^A\^. In this case, our Ansatz for the resummed action is \_=\^4xso we have that the action reads $$\begin{aligned}
{\mathcal{S}}_\chi&=\int{\mathrm{d}}^4x\left[\frac12\partial_\mu\chi\partial^\mu\chi-V(\chi)+\frac{1}{{M_{\rm vd}}^2}(b_1A_\mu A_\nu+b_2 A^2\eta_{\mu\nu}) T^{\mu\nu}\right]\nonumber\\
&=\int{\mathrm{d}}^4x\left[\frac12\left(1-(b_1+2b_2)YC+2(b_1+b_2)Y^2C'\right)\partial_\mu\chi\partial^\mu\chi\right.\nonumber\\
&\hspace{2cm}+\frac12\left(2b_1C+3b_1YD+2(b_1+b_2)Y^2D' \right)A^\mu A^\nu\partial_\mu\chi\partial_\nu\chi\nonumber\\
&\hspace{2cm}\left.-\left(1+(b_1+4b_2)YU-2(b_1+b_2)Y^2U' \right)W(\chi)\right]\end{aligned}$$ and, therefore, the functions $C$, $D$ and $U$ should be the solutions of C&=&1-(b\_1+2b\_2)YC+2(b\_1+b\_2)Y\^2C’,\
D&=&2b\_1C+3b\_1YD+2(b\_1+b\_2)Y\^2D’ ,\
U&=&1+(b\_1+4b\_2)YU-2(b\_1+b\_2)Y\^2U’ . We see again that when coupling to the orthogonal projector, i.e., $b_2=-b_1$, the equations become algebraic and the solution is C&=&,\
D&=&,\
U&=&. The form of the equations are similar to the ones found in the precedent sections so we will not repeat once again the same discussion, but obviously the same types of solutions will exist in this case. Let us however mention that the same results for the matter coupling can be obtained for the derivatively coupled scalar field upon the replacement $A_\mu\to\partial_\mu\varphi$.
Superpotential terms {#sec:Superpotentials}
====================
In the previous sections we have considered the energy-momentum tensor obtained from the usual prescription of coupling it to gravity and taking variational derivatives with respect to the metric. However, as we already explained above, the energy-momentum tensor (as any usual Noether current) admits the addition of super-potential terms with vanishing divergence either identically or on-shell. This freedom in the definition of the energy-momentum tensor can be used to [*improve*]{} the canonical energy-momentum tensor in special cases. For instance, theories involving spin 1 fields give non-symmetric canonical energy-momentum tensors that can be symmetrised by adding suitable superpotentials given in terms of the generators of the corresponding Lorentz representation (the Belinfante-Rosenfeld procedure). If the theory has a gauge symmetry, one can also add super-potential terms (on-shell divergenceless this time) to obtain a gauge invariant energy-momentum tensor[^11] obtaining then the energy-momentum tensor that results from the Rosenfeld prescription. Theories featuring scale invariance admit yet another improvement to make the energy-momentum tensor traceless[^12]. This traceless energy-momentum tensor is not the one obtained from the Hilbert prescription (nor with the Belinfante-Rosenfeld procedure) upon minimal coupling to gravity, but one needs to add a non-minimal coupling to the curvature, which simply tells us that the iterative coupling procedure to gravity starting from the improved energy-momentum tensor gives rise to non-minimal couplings. This example illustrates how considering different super-potential terms can result in different theories for the full action. In this section we will briefly discuss this point within our constructions for the self-interactions of scalar and vector fields to their own energy-momentum tensors.
Let us start with the scalar field and consider a particular family of terms that lead to interesting results. Thes lowest order object that is identically divergence-free is given by X\_[1]{}\^=\^-\^\^, where we have introduced the factor ${M_{\rm sd}}$ to match the dimension of an energy-momentum tensor for $X_1^{\mu\nu}$. This corresponds to a super-potential term of the form $\partial_\alpha(\partial^{[\alpha}\eta^{\mu]\nu})$. Now we want to study the effect on the full theory of adding this boundary term to the energy-momentum tensor. Notice that this object will give rise to an operator of lower dimensionality than the coupling to the energy-momentum tensor and, therefore, the added correction will be suppressed by one less power of the corresponding scale. In the case of the conformal coupling, we can see that the interaction $\varphi X^\mu{}_\mu$ simply amounts to a re-scaling of the kinetic term, so we will move directly to the derivative coupling. In that case, the first order correction is given by \_[(1)]{}=\_\_X\_[1]{}\^==()\^2where we have integrated by parts and dropped a total derivative in the last term. We can recognize here the cubic Galileon Lagrangian [@Nicolis:2008in] that we have obtained by simply following our coupling prescription to the identically conserved object $X_{1}^{\mu\nu}$ that one can legitimately add to the energy-momentum tensor. By iterating the process we obtain the following perturbative expansion for the Lagrangian: =(1+3X+15X\^2+105X\^3+). We can now use that, for an arbitrary function ${\mathcal{G}}(X)$ and upon integration by parts, we have (X)\_\_\^\^=12[M\_[sd]{}]{}\^4\^\_(X)-12(X)with ${M_{\rm sd}}^4\tilde{{\mathcal{G}}}'(X)={\mathcal{G}}(X)$, to express the final Lagrangian as =32(1+52X+X\^2+X\^3+). This Lagrangian is a particular case of the so-called KGB models [@Deffayet:2010qz] with a shift symmetry. Remarkably, we have obtained this Lagrangian from our construction, which can be understood in a similar fashion to the generation of non-minimal couplings in the case of gravity.
We can also obtain a [*resummed*]{} action by adding the superpotential term to the full theory so that its Lagrangian should read =\_0+(T\^+X\^\_1) being $T^{\mu\nu}$ the total energy-momentum tensor, i.e., the one computed from ${\mathcal{L}}$ by means of the Hilbert prescription. For the sake of generality, we have added here the more general coupling involving $b_1$ and $b_2$ as discussed in the precedent sections. In the above Lagrangian we have also included the term ${\mathcal{L}}_0$ that corresponds to the [*free*]{} Lagrangian, i.e., the part that survives in the decoupling limit ${M_{\rm sd}}\rightarrow\infty$. This term will not affect the resummation of the interactions generated from the term with $X^{\mu\nu}_1$ so we do not need to consider it here (we already solved that part in the previous sections). Now we can notice that the interactions generated from the superpotential terms are of the form ${\mathcal{L}}={M_{\rm sd}}G(X)\Box\varphi$, up to boundary terms, so we need to have \_X=[M\_[sd]{}]{}G(X)=(X\_[1]{}\^+T\_X\^) being $T_X^{\mu\nu}$ the total energy-momentum tensor of ${\mathcal{L}}_X$. By computing this energy-momentum tensor we finally get \_X=[M\_[sd]{}]{}G(X)=[M\_[sd]{}]{}with $\tilde{G}'(X)=XG'(X)$ and we have integrated by parts. From this relation we then see that the following equation must hold G(X)=32X+2(b\_1+b\_2)X\^2G’(X)+(b\_1-2b\_2)(X). Notice that this is an integro-differential equation, but we can easily transform it into the ordinary differential equation 2(b\_1+b\_2)XF’(X)+(3b\_1-1X)F(X)+32=0 with $F(X)\equiv XG'(X)$. As usual, the coupling to the orthogonal projector, i.e., $b_1=-b_2$ reduces the equations to a set of algebraic equations, although in this case an integration to go from $F$ to $G$ is necessary. In that case we obtain $F=-\frac32(3b_1-1/X)^{-1}$ so that $G=-1/(2b_1)\log(1-3b_1 X)$. In this case, the integration constant is irrelevant because it corresponds to a total derivative in the Lagrangian.
We have considered the simplest of the identically divergence-free superpotential terms, but we could consider the whole family given (in arbitrary dimension $d$) by X\_[n]{}\^[M\_[sd]{}]{}\^[d-3n]{}\^[\_1\_n\_[n+1]{}\_[d-1]{}]{}\^[\_1\_n]{}\_[\_[n+1]{}\_[d-1]{}]{}\_[\_1]{}\_[\_1]{}\_[\_n]{}\_[\_n]{}which can be seen to be trivially divergence-free by virtue of the antisymmetry of the Levi-Civita tensor. These higher order superpotentials would generate higher order versions of the Galileon Lagrangians. At the first order, each $X_n^{\mu\nu}$ will obviously generate the $n$-th Galileon Lagrangian (in fact, Galileon fields are precisely defined by coupling the gradients of the scalar field to identically divergen-free objects), while the higher order corrections will eventually produce sub-classes of shift-symmetric generalised Galileons fields.
Similar results to those found for the scalar field can be obtained by adding superpotential terms in the case of vector fields. We can consider the lowest order and identically conserved object Y\_1\^=[A]{}\^-\^A\^which corresponds to a superpotential of the form $\Theta^{\alpha\mu\nu}=2A^{[\alpha}\eta^{\mu]\nu}$. Unlike in the scalar field case, this superpotential is not symmetric in the last two indices. While in the scalar field case, a non-derivative coupling did not produce new terms, for the vector field case already algebraic couplings generate new interesting interactions (as expected because the non-derivative coupling for the vector is related to the derivative coupling for the scalar). The leading order interaction by coupling the vector to this superpotential term gives \_[(0)]{}=A\_A\_Y\^\_1=32 A\^2[A]{}where we have integrated by parts and dropped a total derivative term. We see that, as expected, we recover the cubic vector Galileon Lagrangian [@VectorGalileon]. It is remarkable that this interaction corresponds to a dimension four operator which means that it is not suppressed by the scale ${M_{\rm vc}}$ or, in other words, it will survive in the decoupling limit ${M_{\rm vc}}\rightarrow\infty$. It is not surprising the resemblance of this interaction with the case of the derivatively coupled scalar previously discussed and it is not difficult to convince oneself that the resummation will lead to the same type of equations. In addition, very much like in the scalar field case, there are higher order superpotential terms that can be constructed for the vector field case, but, given the similarities with the scalar field couplings, we will not give more details here. A more interesting class of super-potential terms would be those respecting the $U(1)$ gauge invariance. However, it is known that there are no Galileon-like interactions for abelian vector fields [@Deffayet:2013tca] so that we do not expect to find anything crucially new by adding gauge invariant superpotential terms, but similar interactions to the ones already worked out.
Effective metrics and generating functionals {#Sec:EffectiveMetrics}
============================================
In the precedent sections we have studied the coupling of scalar and vector fields to the energy-momentum tensor and how this can be generalised to the full theory. The definition that we have mostly considered for the energy-momentum tensor is the Hilbert prescription, although we have briefly commented on some interesting consequences of considering superpotential terms in the previous section. Since the Hilbert prescription gives the energy-momentum tensor as a functional derivative with respect to some fiducial metric, a natural question is to what extent the full theory can be expressed in terms of an effective metric. In this section we intend to briefly discuss this aspect with special emphasis in the cases considered throughout this work.
For the clarity of our construction, let us go back to the beginning and consider again the case of a conformally coupled scalar field considered in \[sec:Conformalscalar\]. The starting point there was a scalar field coupled to the trace of the energy-momentum tensor as[^13] \_[int]{}\^[(1)]{}=T\_[(0)]{}. It is not difficult to see that this interaction can be conveniently written as the following functional derivative \_[int]{}\^[(1)]{}=\_[J=0]{} where the zeroth order action ${\mathcal{S}}_{(0)}$ is evaluated on the conformal effective metric $h_{\mu\nu}=\exp(-2J\varphi)\eta_{\mu\nu}$ with $J$ an external field[^14]. The equivalence of the two expressions can be easily checked by using the chain rule: \_[J=0]{}=( )\_[J=0]{}. It is now immediate to re-obtain the results of Sec. \[sec:Conformalscalar\] as well as obtaining generalisations. Let us assume that the initial action is a linear combination of homogeneous functions of the metric so we have \_[(0)]{}\[\_\]=\_i\^[w\_i]{} \_[(0),i]{}\[\_\], with $\lambda$ some parameter and $w_i$ the degree of homogeneity of the corresponding term ${\mathcal{S}}_{(0),i}$. Then, the functional derivative can be straightforwardly computed as \_[J=0]{}=\_i\^4x(-2w\_iJ)\_[(0),i]{}\_[J=0]{}=\_i(-2w\_i)\_[(0),i]{}. Since all the dependence on the metric in this first order correction is again in ${\mathcal{L}}_{(0),i}$ we have, for this specific case, that the $n$-th order interaction will be given by \_[int]{}\^[(n)]{}=\_[J=0]{}=\_[J=0]{}=\_i(-2w\_i)\^n\_[(0),i]{} so that the resummed Lagrangian for the term of degree $w_i$ is \_[(i)]{}=\_[n=0]{}\^\_[J=0]{}=\_[(0),i]{}=\_[(0),i]{}. This exactly reproduces the results of Sec. \[sec:Conformalscalar\] since the kinetic term of the scalar has degree $1$ while any potential term for a scalar field has degree $2$. Notice that for a term of zero degree there is no correction, in accordance with the fact that the energy-momentum tensor of a zeroth weight (i.e. conformally invariant) is traceless.
After warming up with the simplest example (which in fact allows for a full resolution of the problem) we can proceed to develop a general framework. Let us consider a leading order interaction to the energy-momentum tensor of the general form \_\^[(1)]{}=\_ T\^ where $\Omega_{\mu\nu}$ is some rank-2 tensor built out of the field which we want to couple to $T^{\mu\nu}$, be it the scalar or the vector under consideration throughout this work. Then, it is easy to see that the generating effective metric will be given by h\_=\_ \[eq:defheff\] with $\omega_{\mu\nu}{}^{\alpha\beta}$ some rank-4 tensor satisfying $\omega_{\mu\nu}{}^{\alpha\beta}\eta_{\alpha\beta}=\Omega_{\mu\nu}$. This effective metric indeed generates the desired interaction from the functional derivative \_\^[(1)]{}=\_[J=0]{}, while the higher order interactions are simply \_\^[(n)]{}=\_[J=0]{}. This general construction can be straightforwardly applied to the cases that we have considered in the precedent sections. The coupling of the scalar field corresponds to taking $\omega_{\mu\nu}{}^{\alpha\beta}$ proportional to the identity in the space of rank-4 tensors. The algebraic vector field coupling corresponds to $\omega_{\mu\nu}{}^{\alpha\beta}=\frac14\(b_1A_\mu A_\nu+b_2A^2\eta_{\mu\nu}\) \eta^{\alpha\beta}$, while the gauge invariant coupling is generated by $\omega_{\mu\nu}{}^{\alpha\beta}=b_1 F_\mu{}^\alpha F_\nu{}^\beta+\frac14 b_2 F^2\eta_{\mu\nu}\eta^{\alpha\beta}$.
An interesting case is when $\Omega_{\mu\nu}$ does not depend on $\eta_{\mu\nu}$, which happens when $\omega_{\mu\nu}{}^{\alpha \beta}$ is linear in the inverse metric. To see why this is particularly interesting, let us obtain the second correction to the original action, that will be given, in general, by \^[(2)]{}&=&()\_[J=0]{}= \_[J=0]{}\
&=&\_[J=0]{}=-2\_[J=0]{}.where we have used the chain rule and the definition of the generating metric (\[eq:defheff\]). Now, the last expression is substantially simplified if $\Omega_{\mu\nu}=\omega_{\mu\nu}{}^{\alpha\beta} \eta_{\alpha\beta}$ does not depend on the metric $\eta_{\mu\nu}$ because, in that case, $\omega_{\mu\nu}{}^{\alpha\beta} \eta_{\alpha\beta}$ will not acquire a dependence on $J$ so we obtain \^[(2)]{}=(-2)\^2\_\^ \_ \_\^ \_()\_[J=0]{}. It is then straightforward to iterate this process and arrive at the following expression for the final Lagrangian \_=\_[n=0]{}\^(-2)\^n\_[J=0]{}\_[\_1\_1]{}\_[\_n\_n]{}. This (asymptotic) expansion reminds of a Taylor expansion, though without the required $1/n!$. It is not difficult to motivate the appearance of the missing factorial by simply imposing that the variation should appear at the level of the field equations instead of the action, similarly to what we did for the scalar field case at the end of Sec. \[sec:Conformalscalar\]. In that case, the resulting series will be \_=\_[n=0]{}\^\_[\_1\_1]{}\_[\_n\_n]{}. \[eq:TotalLeffmetric\] which admits a straightforward resummation as the Taylor expansion of the seed action ${\mathcal{S}}_{(0)}$ around $2\Omega_{\mu\nu}$, i.e., the total action will be ${\mathcal{S}}={\mathcal{S}}_0[\eta_{\mu\nu}-2\Omega_{\mu\nu}]$. Of course, this reproduces the well-known result for gravity when $\Omega_{\mu\nu}$ is identified with the metric perturbation $-2h_{\mu\nu}$. We see here that the same applies when a scalar field is derivatively coupled to matter fields as $\partial_\mu\varphi\partial_\nu\varphi T^{\mu\nu}$ or a vector field is coupled as $A_\mu A_\nu T^{\mu\nu}$. In both cases, the resulting coupling to matter fields is through a disformal metric $g_{\mu\nu}=\eta_{\mu\nu}-2A_\mu A_\nu$ and $g_{\mu\nu}=\eta_{\mu\nu}-2\partial_\mu \varphi \partial_\nu\varphi$ respectively. This result shows that the couplings of the precedent sections with $b_2=0$ are indeed special because they satisfy the above condition of having the corresponding $\Omega_{\mu\nu}$ independent of the metric and, therefore, admit a resummation in terms of an effective metric.
Finally, let us notice that the expression (\[eq:TotalLeffmetric\]) also serves as a starting point to explore a class of “non-metric” theories, i.e., theories where the coupling is not through an effective metric, analogous to the theories explored in [@Blanchet:1992rx] as non-metric departures from GR.
Phenomenology {#Sec:Phenomenology}
=============
The phenomenology associated with the different scalar and vector couplings discussed along this work is very rich and depends on the particular term. The best known case corresponds to the scalar coupling to the trace of the energy momentum tensor. These types of couplings leads to the standard Jordan-Brans-Dicke framework and dilaton structures. In this section, we would like to summarize the main phenomenology related to disformal scalars and vectors. In particular, we would like to focus on the phenomenology that is shared by both fields due to the equivalence theorem. For instance, for the vectorial case, we will focus in the regime where its mass is negligible with respect to the physical energy involved in the process. Indeed, following an effective theory approach, we can work in the perturbation limit. In such a case, the dimension 6 interaction term can be written as: $$\mathcal{L}_V = \frac{1}{{M_{\rm vc}}^2}A_\mu A_\nu T^{\mu\nu}\;.
\label{eq:vcoupling}$$ This term dominates the distinctive phenomenology associated to the vector fields discussed along this work. This type of interaction corresponds also to the vector modes associated to the metric in theories with additional spatial dimensions [@DoMa]. The phenomenology of these [*graviphotons*]{} have been studied for the brane world scenario in different works under the name of [*brane vectors*]{} [@Clark; @Clark:2008fn; @Clark:2008zw]. In this framework, there is a Higg-like mechanism that provides mass to the graviphotons. At high energies with respect to such a mass, the longitudinal mode of these vectors can be identified with the branons $\varphi$, the scalar degree of freedom associated to the fluctuation of the brane along the extra dimensions [@DoMa]. Within this regime, the experimental signatures of these vectors can be computed by the branon disformal coupling [@Cembranos:2016jun]: $$\mathcal{L}_D = \frac{1}{m^2\,{M_{\rm vc}}^2}\partial_\mu \varphi \partial_\nu \varphi T^{\mu\nu}\;,
\label{eq:coupling}$$ where $m$ is the mass of the vector field. A detailed analysis of the constraints to the above interaction has been developed in the context of branon fields [@Sundrum; @DoMa; @BSky; @Alcaraz:2002iu; @thesis]. This term dominates the distinctive observational signatures of disformal scalars. Not only potential searches in colliders have been studied in different works [@strumia; @Alcaraz:2002iu], but also astrophysical constraints have been analyzed, such as the ones coming from cooling of stelar objects [@Kugo; @CDM; @CDM2] or the associated to the relic abundances of this type of massive vectors [@Clark; @Clark:2008fn].
At high energies, the vector fields can be searched by analyzing Lagrangian (\[eq:coupling\]). In this case, the vector mass times the vector coupling is the combination of parameters which suffers the constraints from present data. On the contrary, for the disformal scalars, the constraints apply directly to the disformal scalar coupling. They may be detected at the Large Hadron Collider (LHC) or in a future generation of accelerators [@Alcaraz:2002iu; @Brax:2012ie; @Brax:2014vva; @Clark; @Coll; @L3; @Cembranos:2004jp; @LHCDirect; @Landsberg:2015pka; @Khachatryan:2014rwa; @BWRad]. For the case of the LHC, the most sensitive production process is gluon fusion giving a gluon in addition to a pair of longitudinal vectors or disformal scalars; and the quark-gluon interaction giving a quark and a pair of the commented particles. These processes contribute to the monojet $J$ plus transverse missing momentum and energy signal. An additional process is the quark-antiquark annihilation giving a photon and the mentioned pair of new modes. In such a case, the signal is a single photon in addition to the transverse missing momentum and energy. The cross-section of the subprocesses were computed in Refs. [@Cembranos:2004jp] and [@LHCDirect]. The analysis of the single photon channel is simpler and cleaner but the monojet channel is more sensitive.
In addition to these processes, there are other complementarity constraints on the same combination of parameters corresponding to other collider data. A summary of these analyses can be found in Table \[tabHad\] [@DoMa; @BW2; @Clark; @Coll; @Cembranos:2004jp; @LHCDirect]. In this Table, the limits coming from HERA, LEP-II and Tevatron are compared with the present restrictions from LHC running at a centre of mass energy (c.m.e.) of 8 TeV and the prospects for the LHC running at 14 TeV c.m.e. with full luminosity. Other missing transverse momentum and energy processes, such as those related to the monolepton channel [@Brax:2014vva], are also potential signatures of the models developed in this work. In the same reference, the authors discuss other different phenomenological signatures, but they are subdominant due to the important dependence of the interaction with the energy.
On the other hand, it has been shown that the new modes under study introduce radiative corrections, which generate new couplings among SM particles, which can be described by an effective Lagrangian. Although the study of such processes demands the introduction of new parameters, they can provide interesting effects in electroweak precision observables, anomalous magnetic moments, or SM four particle interactions [@BWRad].
------------------------------------------------------------------------------------------------
Experiment $\sqrt{s}$(TeV) ${\mathcal L}$(pb$^{-1}$) $\sqrt{m{M_{\rm vc}}}$(GeV)
-------------------- ----------------- --------------------------- -----------------------------
HERA$^{\,1} 0.3 110 19
$
Tevatron-II$^{\,1} 2.0 $10^3$ 304
$
Tevatron-II$^{\,2} 2.0 $10^3$ 285
$
LEP-II$^{\,2} 0.2 600 214
$
LHC$^{\,2} 8 $19.6\times10^{3}$ 523
$
LHC$^{\,1} 14 $10^5$ 1278
$
LHC$^{\,2} 14 $10^5$ 948
$
------------------------------------------------------------------------------------------------
: Summary table for the phenomenology of vectors and disformal scalars coupled to the energy-momentum tensor at colliders. Monojet and single photon analyses are labeled by the upper indices ${1,2}$, respectively. Present bounds and prospects for the LHC [@Landsberg:2015pka; @Khachatryan:2014rwa; @Cembranos:2004jp; @LHCDirect] are compared with constraints from LEP [@Alcaraz:2002iu; @L3], HERA and Tevatron [@Cembranos:2004jp]. $\sqrt{s}$ means the centre of mass energy associated with the total process; ${\mathcal L}$ denotes the total integrated luminosity; $\sqrt{m{M_{\rm vc}}}$ is the constraint at the $95\;\%$ confidence level by assuming a very light vector (in the limit $m\rightarrow0$). The effective coupling is not valid for energy scales $\Lambda^2\gtrsim 8\pi\sqrt{2} m {M_{\rm vc}}$ [@BWRad]. []{data-label="tabHad"}
We can also compute the thermal relic abundance corresponding to these new fields by assuming they are stable [@CDM; @CDM2; @Clark:2008fn]. The larger the coupling scale $M$, the weaker the annihilating cross-section into SM particle-antiparticle pairs, and the larger the relic abundance. This is the expected conclusion since the sooner the decoupling occurs, the larger the relic abundance is. Therefore the cosmological restrictions related to the relic abundance are complementary to those coming from particle accelerators. Indeed, a constraint such as $\Omega_{D} < {\mathcal{O}}(0.1)$ means a lower limit for the value of the cross-sections in contrast with the upper limits commented above from non observation at colliders. If we assume that the DM halo of the Milky Way has an important amount of these new vectors or scalars, its flux on the Earth could be sufficiently large to be measured in direct detection experiments. These experiments measure the rate $R$, and energy $E_{R}$ of nuclear recoils. These constraints depend on different astrophysical assumptions as it has been discussed in different analyses [@CDM; @CDM2; @Clark:2008fn; @Cembranos:2016jun; @LHCDirect].
If the abundance of these new particles is significant, they cannot only be detected by direct detection experiments, but also by indirect ones. In fact, a pair of vectors or scalars can annihilate into ordinary particles such as leptons, gauge bosons, quarks or Higgs bosons. Their annihilations from different astrophysical regions produce fluxes of cosmic rays. Depending on the characteristics of these fluxes, they may be discriminated from the background. After the annihilation and propagation, the particle species that can be potentially detected by different detectors are gamma rays, neutrinos and antimatter (fundamentally, positrons and antiprotons). In particular, gamma rays and neutrinos have the advantage of maintaining their original trajectory. Indeed, this analysis are more sensitive for the detection of these signatures [@vivi].
On the other hand, there are astrophysical observations that are able to constraint the parameter space of the new fields studied in this work independently of their abundance. For example, one of the most successful predictions of the standard cosmological model is the relative abundances of primordial elements. These abundances are sensitive to several cosmological parameters and were used in Refs. [@CDM] and [@CDM2] in order to constrain the number of light fields. These restrictions apply in this case since the new particles will behave as dark radiation for small enough masses. For instance, the production of $^4$He increases with an increasing rate of the expansion $H$ and the Hubble parameter depends on the total amount of radiation. However, these restrictions are typically important for relatively strong couplings. On the contrary, if the new modes decouple above the QCD phase transition, $ m {M_{\rm vc}}\sim 10000 $ GeV$^2$, the limit increases so much that the restrictions become extremely weak [@CDM2]. Different astrophysical bounds can be obtained from modifications of cooling processes in stellar objects like supernovae [@Kugo; @CDM; @CDM2; @Brax:2014vva]. These processes take place by energy loosing through light particles such as photons and neutrinos. However, if the mass of the new particles is low enough, these new particles are expected to carry a fraction of this energy, depending on their mass and the coupling to the SM fields. These constraints are restricting up to masses of order of the GeV. For heavier fields, the limits on the coupling disappear due to the short value of the mean free path of the vector particle inside the stellar object [@CDM2]. A summery of these astrophysical and cosmological constraints for a particular model of a single disformal scalar described by the branon Lagrangian can be seen in Fig. \[FigBranons\].
In the previous paragraphs, we have summarized the phenomenology associated to the abundance of these new vectors and disformal scalars by assuming it was generated by the thermal decoupling process in an expanding universe. However, if the reheating temperature $T_{RH}$ after inflation was sufficiently low, then these new fields were never in thermal equilibrium with the primordial plasma [@nonthermal]. However, still there is the possibility for them to be produced non-thermally, very much in the same ways as axions [@axions] or other bosonic degrees of freedom [@Cembranos:2012kk; @santos]. This possibility modifies some of the previous astrophysical signatures, since they have typically associated a much lower mass. In particular, the potential isotropies related to the coherent relic density of the new vectors could constitute a very distinctive signature [@santos].
{width="12cm"}
Discussion {#Sec:Discussion}
==========
In this work we have explored the construction of theories involving scalar and vector fields following a procedure inspired by gravity so that the fields are coupled to the total energy-momentum tensor. This defining property is not free from ambiguities arising from the different definitions of the energy-momentum tensor that we fix by using the Hilbert prescription. Another ambiguity present in the construction comes from requiring a coupling at the level of the action or a coupling such that it is the source of field equations. We have mostly employed the former, although we have commented upon this subject and considered the latter in some specific cases.
We have started by reviewing some known results that exist in the literature concerning scalar fields. We have then extended these results to theories where the coupling of the scalar to the energy-momentum tensor respects a shift symmetry, i.e., the scalar couples derivatively. We have reduced the resummation of the infinite perturbative series to the resolution of some differential equations, whose properties we explain in some detail in Appendix \[Appendix\]. We have nevertheless highlighted some particularly remarkable parameters choices where the solutions can be readily obtained as well as the existence of polynomial solutions. The difficulty encountered in obtaining the full theory within the second order formalism encouraged us to consider the problem from a first order perspective, where it is known that the self-coupling procedure simplifies substantially. We have indeed confirmed that this is also the case for the problem at hand and we managed to reduce the resumation to the resolution of algebraic instead of differential equations. We ended our tour on scalar fields by exploring couplings to matter fields taking a scalar as a proxy.
After considering scalar fields, we delved into vector fields coupled to the energy-momentum tensor. This case offers a richer spectrum of results owed to the possibility of having a gauge symmetry. We have first constructed the theory for a vector coupled to the energy-momentum tensor without taking care of the gauge invariance. We have then considered gauge-invariant couplings where the $U(1)$ symmetry is realised in the usual way in the vector field so that the couplings are through the field strength of the vector. We have however commented on the interesting possibility of having the $U(1)$ symmetry non-linearly realised in the vector field so that the resulting action could explicitly contain the vector not necessarily through $F_{\mu\nu}$, but still keeping two propagating dof’s for the vector. This path would be interesting to pursue further, specially if it can be linked to the conservation of the energy-momentum tensor (or some related conserved current) so that the theories are closer to the gravity case, i.e., the consistent couplings are imposed by symmetries rather than by a somewhat ad-hoc prescription. Similarly to the scalar field theories, the resummed actions for the vector fields reduced to solving some differential equations with similar features. However, unlike for theories with scalars, the first order formalism applied to the vector field case did not result in any simplification of the iterative problem. Finally, we also considered couplings to a scalar as a proxy for the matter fields. Although we did not consider other types of fields throughout this work, it would not be difficult to include higher spin fields.
After constructing the actions for our self-interacting fields through the energy-momentum tensor, we have discussed the impact of superpotential terms arising from the ambiguity in the energy-momentum tensor. We have shown that these superpotential terms lead to the generation of Galileon interactions for both, the scalar and the vector.
Given the motivation from gravity theories that triggered our study and the prescribed couplings to the energy-momentum tensor, we have explored generating functionals defined in terms of an effective metric. This method allowed us to re-derive in a simpler way some of the results in the previous sections and give a general procedure to generate all the interactions. Moreover, we have shown that the generating functional procedure greatly simplifies if the leading order correction to the energy-momentum tensor does not depend on the metric. If that is the case, the series can be regarded as a Taylor expansion and the final resummed action is simply the original action coupled to an effective metric.
Finally, we have considered the phenomenology of these type of theories. The possible experimental signatures associated with the scalar and vector couplings discussed along this work is very rich and depends on the particular term under study. We have focused on the phenomenology related to disformal scalars and vectors. In fact, at high energies, the longitudinally polarized vector and the disformal scalar are related by the equivalence theorem. Following an effective theory approach, we can work in the perturbation limit. In such a case, monojet and single photon analysis at the LHC are the most sensitive signatures, constraining the dimensional couplings at the TeV scale. On the other hand, both models can support viable dark matter candidates, offering a broad range of astrophysical and cosmological observable possibilities depending of their stability. Note that given the quadratic character of the leading order interaction for the derivative couplings of the scalar or the non-gauge invariant couplings of the vector, the corresponding force will decay faster than the Newtonian behaviour $1/r^2$. However, if there is some vev for the fields, be it $\langle \partial_\mu\varphi\rangle$ or $\langle A_\mu\rangle$ for the scalar and the vector respectively, then we can recover the Newtonian-like force with a coupling constant determined by the vev of the field. Indeed the non-linear nature of the interactions may give rise to screening mechanisms, with interesting phenomenology for dark energy models.
An interesting question that we have not addressed in this work is the relation of our resulting actions with geometrical frameworks. We have already given some arguments in Sec. \[Sec:EffectiveMetrics\] as to what extent our constructed theories could be regarded as couplings to an effective metric. For the massless scalar field case, it is known that the resulting action can be interpreted in terms of the Ricci scalar of a metric conformally related to the Minkowski metric (which is nothing but Nordstrøm’s theory). It would be interesting to check if analogous results exist for the derivatively coupled case and/or the vector field case, for instance in terms of curvature objects of disformal metrics. Perhaps a suited framework for these theories would be the arena provided by Weyl geometries or generalised Weyl geometries which naturally contain an additional vector field, which could also be reduced to the gradient of a scalar, in which case it is known as Weyl integrable spacetimes or WIST. Another extension of our work can be associated with the analysis of multi-field theories and the study of the role played by internal symmetries within these constructions.
We thank Pepe Aranda for useful discussions. This work is supported by the MINECO (Spain) projects FIS2014-52837-P and FIS2016-78859-P (AEI/FEDER). JBJ acknowledges the support of the Spanish MINECO’s “Centro de Excelencia Severo Ochoa” Programme under grant SEV-2016-0597. JMSV acknowledges the support of Universidad Complutense de Madrid through the predoctoral grant CT27/16.
Discussion of the recurrent differential equations {#Appendix}
==================================================
In this appendix we will discuss the differential equation that we have recurrently obtained throughout this paper. The equation can be expressed as 2x\^2F’(x)+(px-1)F(x)+1=0 \[Eq:master\] with $p$ a constant. Taking for instance our first encounter with this type of equation in (\[Eq:eqsderivativecoupling\]), that we reproduce here for convenience &=&1+(b\_1-2b\_2)X+2(b\_1+b\_2)X\^2’,\
&=&1-(b\_1+4b\_2)X+2(b\_1+b\_2)X\^2’, \[Eq:eqsderivativecoupling2\] it is easy to check that they can be brought into the form (\[Eq:master\]) by simply rescaling $X\to X/(b_1+b_2)$, which is always legitimate except for $(b_1+b_2)=0$, in which case, as we already discussed, the equations become algebraic. The values for the constant parameter $p$ are then $p=(b_1-2b_2)/(b_1+b_2)$ and $p=-(b_1+4b_2)/(b_1+b_2)$ for ${\mathcal{K}}$ and ${\mathcal{U}}$ respectively. It is clear that $x=0$ is a singular point of the equation. If we evaluate the equation at $x=0$ we find $F(0)=1$, as it should since the series of the functions satisfying this equation start at 1. Furthermore, by taking subsequent derivatives of the equation and evaluating at $x=0$, we can reproduce the corresponding perturbative series, which is in turn an asymptotic expansion. Alternatively, we can seek for solutions in the form of a power series $F=\sum_{n=0}F_nx^n$. By substituting this series in the equation we find that $F_0=1$ and the following recurrent formula for the coefficients with $n{\geqslant}1$: F\_n=F\_[n-1]{}, n1, \[Eq:recursive\] which can be readily solved as F\_n=\_[j=1]{}\^n,n1. \[Eq:Fn\] From this general solution we see that $F_1=p$ so that if we choose the parameters such that $p=0$, then the solution is simply $F=1$ because then only $F_0$ remains non-vanishing. This result of course corresponds to the choice of parameters that cancelled either the corrections to the kinetic terms or to the potential discussed throughout this work. In particular, we see that $p=0$ corresponds to $b_1=2b_2$ and $b_1=-4b_4$ in (\[Eq:eqsderivativecoupling2\]), as we already found in Sec. \[Sec:ScalarDerivative\].
The fact that we can write the general solution in terms of a recursive relation is due to the iterative procedure prescribed to built our actions and this in turn has noteworthy consequence, namely, we can choose parameters such that the solution becomes polynomial. Since the coefficients $F_n$ are recursively given by (\[Eq:recursive\]), if we have that $F_{r+1}=0$ for some $r$, then $F_{n}=0$ for $n>r$ and, thus, the solution is a polynomial of degree $r$. The result commented in the previous paragraph that $F=1$ for $p=0$ is a particular case of this general result with $r=0$. In general, if we want the solution to be a polynomial of degree $r$ we will need to impose $p+2r=0$. As an illustrative example, if we want ${\mathcal{K}}$ and ${\mathcal{U}}$ in (\[Eq:eqsderivativecoupling2\]) to be polynomials of degree $r$ and $s$ respectively, we need to choose the parameters $b_1$ and $b_2$ satisfying =-2r,=2s. These equations do not admit a general solution for arbitrary $r$ and $s$. If we eliminate for instance $b_1$, we end up with the equation $b_2(1+r-s)=0$ (for $r$ and $s$ integers), that imposes the relation $s=1+r$ and leaves $b_2$ as a free parameter. The solution for $b_1$ is then $b_1=2b_2(1-r)/(1+2r)$. Thus, in this particular case, if we want both functions to become polynomials, they cannot be of arbitrary degree, but ${\mathcal{U}}$ must be one degree higher than ${\mathcal{K}}$. For $r=0$, we have $s=1$ and the corresponding theory has $b_1=2b_2$, i.e., for those parameters ${\mathcal{K}}$ is constant and ${\mathcal{U}}$ is polynomial of degree 1, as we found in Sec. \[Sec:ScalarDerivative\].
If we do not impose the solutions to be polynomials, we can do better than the solution expressed as the (asymptotic) series with general coefficients given in (\[Eq:Fn\]) and obtain the general solution of the equation in terms of known functions, which can be expressed as F(x)=e\^[-1/(2x)]{}x\^[-p]{}with $C$ a constant that must be chosen in order to have a well-defined solution at $x=0$. The above solution can also be expressed in terms of exponential integral functions as follows F(x)=Cx\^[-p/2]{}e\^[-1/(2x)]{}-e\^[-1/(2x)]{}[[Ei]{}]{}\_[p/2]{}(-1/(2x)). Since the exponential integral has the asymptotic expansion ${{\rm Ei}}_n(x)\sim e^{-x}/x$ for large $x$, the second term in the above solution is regular at the origin and, thus, in order to have a regular solution we must impose $C=0$ so that the desired solution is finally F(x)=-[[Ei]{}]{}\_[p/2]{}(-1/(2x)). The exponential integral presents a branch cut for the negative axis. In the final solution, it must be interpreted as the real part of the analytic continuation to the complex plane.
{width="8cm"} {width="8cm"}
We can gain some more insights on the equation by considering the equivalent autonomous system (t)&=&(1-px(t))F(t)-1\
(t)&=&2x\^2(t) where the dot means derivative with respect to $t$. The integral curves of this autonomous system are the solutions of our original equation. The only fix point is given by $x=0$, $F=1$ and the associated eigenvalues are $0$ and $1$. Moreover, it is also easy to see that the $x$-axis is indeed a separatrix, ($\dot{x}=0$ on that axis) so the only solution that can cross it must necessarily pass through $(F=1,x=0)$. The fact that there is a negative eigenvalue indicates the existence of a centre (slow) manifold. We will not provide a detailed analysis of the general properties of the associated autonomous system, but we will simply enumerate some important properties that are illustrated in Fig. \[FigA\]. We can see that there is only one trajectory than can smoothly pass through $x=0$ so the requirement to have a well-defined behaviour for $x\to0$ singles out one solution on the left half plane. However, since the trajectories approach $x=1$ along a centre manifold from the right half-plane, one can, in principle, match the regular solution coming from the left to any of the existing solutions in the right half-plane.
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[^1]: Let us note however that the actual crucial requirement is to maintain the gauge symmetry at the non-linear level.
[^2]: This definition is not free from subtleties either since one also needs to choose a covariantisation procedure and the tensorial character of the fields. We will assume a minimal coupling prescription for the covariantisation and that all the fields keep their tensorial character as in the original Minkowski space.
[^3]: See also for a method to construct an energy-momentum tensor that can be interpreted as a generalisation of the Belinfante-Rosenfeld procedure.
[^4]: The convergence of the generated perturbative series will be a recurrent issue throughout this work. In fact, most of the obtained series will need to be interpreted as asymptotic series of the true underlying theory. We will discuss this issue in more detail in due time.
[^5]: In fact, the solution resembles one of the paradigmatic examples of asymptotic expansion $e^{-1/t}{{\rm Ei}}(1/t)=\sum_{n=0}^\infty n! t^{n+1}$.
[^6]: Of course, we could have also added a term $\pi^2 T$, but the considered coupling will be enough to show how the use of the first order formalism leads to simpler non-differential equations.
[^7]: Let us stress that this Hamiltonian function will not give, in general, the energy of the system, although that is the case for homogeneous configurations.
[^8]: If we re-define a given field $\Phi$ with a metric-dependent change of variables $\Phi\rightarrow \Phi'=\Phi'(\Phi,\gamma_{\mu\nu})$ (as it happens when the re-definition corresponds to a change in the tensorial weight of the field), we have the following relation for the variation of the action =()\_[’]{}+ .Since the second term on the RHS vanishes on the field equations of $\Phi'$ we obtain that both energy-momentum tensors coincide on-shell.\[F7\]
[^9]: Of course we could consider an arbitrary potential, but a mass term is the natural choice if we really assume that we start with a free theory.
[^10]: The discussion presented here also applies to the case of derivative couplings for the scalar field.
[^11]: Let us recall the Weinberg-Witten theorem here that prevents the construction of a Lorentz covariant and gauge invariant energy-momentum tensor for particles with spin ${\geqslant}2$.
[^12]: With only scale invariance the trace of the energy-momentum tensor is given by the divergence of a vector and only when the theory exhibits full conformal invariance the energy-momentum tensor can be made traceless. Since theories that are scale invariant are also conformally invariant, we do not make a distinction here.
[^13]: In order to alleviate the notation in this section we will drop all the scales used in the precedent sections.
[^14]: We have chosen here the exponential for the conformal factor for simplicity, but any conformal factor $\Omega(J)$ satisfying $\Omega(0)=1$ and $\Omega'(0)=-2\varphi$ would do the job.
|
---
abstract: 'Starting with a many-atom master equation of a kinetic, restricted solid-on-solid (KRSOS) model with external material deposition, we investigate nonlinear aspects of the passage to a mesoscale description for a crystal surface in 1+1 dimensions. This latter description focuses on the motion of an atomic line defect (i.e. a step), which is defined via appropriate statistical average over KRSOS microstates. Near thermodynamic equilibrium and for low enough supersaturation, we show that this mesoscale picture is reasonably faithful to the Burton-Cabrera-Frank (BCF) step-flow model. More specifically, we invoke a maximum principle in conjunction with asymptotic error estimates to derive the elements of the BCF model: (i) a diffusion equation for the density of adsorbed adatoms; (ii) a step velocity law; and (iii) a linear relation for the mass flux of adatoms at the step. In this vein, we also provide a criterion by which the adatom flux remains linear in supersaturation, suggesting a range of non-equilibrium conditions under which the BCF model remains valid. Lastly, we make use of kinetic Monte Carlo simulations to numerically study effects that drive the system to higher supersaturations – e.g. deposition of material onto the surface from above. Using these results, we describe empirical corrections to the BCF model that amount to a nonlinear relation for the adatom flux at the step edge.'
address:
- 'Department of Mathematics, University of Maryland, College Park, MD 20742'
- 'National Institute of Standards and Technology, Gaithersburg, MD 20899'
- 'Institute for Physical Science and Technology, University of Maryland, College Park, MD 20742'
- 'Center for Scientific Computation and Mathematical Modeling, University of Maryland, College Park, MD 20742'
author:
- 'Joshua P. Schneider'
- 'Paul N. Patrone'
- Dionisios Margetis
title: |
Towards a high-supersaturation theory of crystal growth: Nonlinear one-step flow model\
in 1+1 dimensions
---
,
,
Crystal surface; Burton-Cabrera-Frank (BCF) model; Line defects; Atom hopping; kinetic Monte Carlo; Mesoscale Limit
81.10.Aj; 68.55.-a; 68.35.Md; 64.60.De
Introduction {#sec:intro}
============
A critical task in materials science has been to understand how atomic defects on crystal surfaces form and evolve, so that one can assess the stability of novel nanostructures and small devices. Below the roughening transition, the growth of nanostructures is mediated by the motion of atomic line defects which resemble steps [@BCF51; @PimpinelliVillain98; @JeongWilliams99]. These steps act as sources and sinks for adsorbed atoms (adatoms). A goal is to understand the dynamics of steps, which can in turn elucidate the morphological evolution of crystal surfaces at large scales [@Misbah10]. Studies of the Burton-Cabrera-Frank (BCF) model of step flow [@BCF51] and its variants [@JeongWilliams99] have long played a key role in achieving this goal.
The BCF model [@BCF51] was originally formulated as a phenomenological mesoscale theory for [*near-equilibrium*]{} processes that cause steps to move via mass conservation. The principal ingredients are: (i) a step velocity law; (ii) the adatom diffusion on terraces, i.e. nanoscale domains bounded by steps; and (iii) a [*linear*]{} kinetic law for the density of flux at steps. This model has been enriched to the point where it is now used to describe step flow under an apparently wide range of kinetics on a crystal surface. Nonetheless, basic aspects of step flow remain poorly understood. Recently, it was indicated that BCF theory in 1+1 dimensions can be interpreted as a dilute limit of the adatom gas [@PatroneEinsteinMargetis14; @LuLiuMargetis14]. In this context, the relative deviation (supersaturation) of the adatom density from its equilibrium value is a variable of importance. An emerging question is the following: How can [*corrections*]{} to the BCF theory be described consistently with atomistic dynamics on a crystal lattice?
The goals of this paper are twofold. First, we seek to demonstrate the validity of the BCF model via appropriate selection of atomistic parameters. Second, we wish to describe empirical corrections to a BCF-like model of one-step flow in correspondence to atomistic dynamics in 1+1 dimensions. We carry out (i) a formal analysis based on a master equation for a kinetic restricted solid-on-solid (KRSOS) model, in the spirit of [@PatroneEinsteinMargetis14]; and (ii) numerical computations via kinetic Monte Carlo (KMC) simulations of the KRSOS model. Our main results in this study are summarized as follows.
- Assuming that an arbitrary yet finite number of particle states contribute to the system evolution, we derive a formal power-series expansion for the steady-state solution to a master equation with external material deposition.
- We prove a “maximum principle” for a master equation including material deposition onto the surface from above.
- By a microscopic averaging in the time-dependent setting, we derive exact expressions for the adatom flux at the step edge. Our formulas separate the linear kinetic law of the BCF model, which is exact in the dilute limit, from higher-order corrections that result from many-particle states.
- Heuristically, we show how the BCF-type model results as the mesoscale limit of the master equation at low enough supersaturations.
- We find bounds for possible deviations, which are expressed in terms of discrete averages over special microscale configurations, from the linear kinetic law for adatom fluxes at the step and from the diffusion equation on the terrace. Our derivation makes use of estimates resulting from our “maximum principle”.
- By using KMC simulations, we empirically determine these supersaturation-driven corrections to the BCF theory, particularly a nonlinear (quadratic) relation for the adatom mass flux at the step edge. Furthermore, we empirically determine a condition by which the linear kinetic law for adatom flux at the step is reasonably accurate (see Remark 7 in Section \[sec:Corrections\]).
We assume that the reader is familiar with the basic concepts of epitaxial growth. For broad reviews on the subject, see, e.g., [@PimpinelliVillain98; @JeongWilliams99; @Misbah10].
Approach and key outcomes {#subsec:Intro-approach}
-------------------------
Our starting point is a master equation for the probability density of atomistic configurations. This description, which includes external material deposition with rate (atoms per unit time) $F$, is an extension of the deposition-free master equation for adatoms invoked in [@PatroneMargetis14; @PatroneEinsteinMargetis14]. Our equation embodies generic microscopic rules of the KRSOS model for: adatom hopping on terraces (or, the microscopic process for surface diffusion); and detachment and attachment of atoms at the step edge according to detailed balance. The corresponding mathematical formalism also accounts for events that do not conserve the total mass, or adatom number, as a result of the nonzero $F$. A simplifying assumption inherent to our formalism is that adatoms do not form bonds away from the step edge and do not interact with each other electrostatically or elastically or by indirect (through-substrate) mechanisms; hence, adatom-adatom correlations originate solely from kinetics.
Our atomistic model allows for the analytical derivation of an exact, closed-form expression for the equilibrium concentration of adatoms on the terrace if $F=0$. Furthermore, for arbitrary $F$, we derive formulas for the adatom fluxes at step edges via formal analysis of the master equation. In this vein, kinetic contributions to the adatom mass flux are manifestly connected to atomistic variables.
For low enough supersaturation, we formally show that the adatom motion gives rise to the familiar linear kinetic relation characteristic of BCF-type models. On the other hand, at high supersaturations, nonlinear corrections to this relation must be taken into account. By use of KMC simulations, we demonstrate this nonlinearity and numerically estimate the dependence of the coefficients of the emergent nonlinear kinetic relation on the attachment-detachment and deposition rates of the atomistic processes.
Specifically, within the mesoscale picture of a crystal surface, the adatom fluxes, $J_\pm$, toward the step edge are empirically described by an expansion of the form $$\label{nlkr_bcf}
J_\pm \approx c^{eq}\sum_{n=1}^{N_*} \kappa_\pm^{(n)} \,\bar{\sigma}_{\pm}^n~.$$ In this relation, $c^{eq}$ is the equilibrium adatom concentration at the step edge (for $F=0$); $\mathcal C_\pm$ is the adatom concentration at the upper ($-$) or lower ($+$) terrace; $\bar{\sigma}_{\pm}=\mathcal C_{\pm}/c^{eq}-1$ is the corresponding supersaturation; and $\kappa_{\pm}^{(n)}$ are kinetic coefficients that in principle depend on atomistic rates. Note that only the $n=1$ term in is present in BCF-type models [@JeongWilliams99]. In the present work, we provide evidence for the origin of nonlinear terms in ; and numerically compute the coefficients $\kappa_{\pm}^{(n)}$ for $n=1,\,2$ in certain typical cases of atomistic dynamics. A related outcome of our numerical computations is an empirical condition for the validity of the linear kinetic law.
Past works {#subsec:Intro-Past}
----------
It is worthwhile placing our work in the appropriate context of past literature, e.g. [@PatroneEinsteinMargetis14; @PatroneMargetis14; @LuLiuMargetis14; @Zhao05; @AckermanEvans11; @Zhaoetal15; @Saum09; @Zangwill92]. Our goal here is to explore the connection of an atomistic model to mesoscale descriptions of crystals. This theme bears resemblance to the main objectives of [@PatroneEinsteinMargetis14; @PatroneMargetis14; @LuLiuMargetis14; @Zhao05]. However, in these works [@PatroneEinsteinMargetis14; @PatroneMargetis14; @LuLiuMargetis14; @Zhao05] the authors’ attention focuses on near-equilibrium processes, whereas in the present paper we study the kinetic regime in which external material deposition tends to drive the system away from equilibrium. A similar task is undertaken in [@PatroneCaflischDM_12], albeit via a (coarse-grained) “terrace-step-kink” model. We herein avoid [*a priori*]{} approximations associated with the diluteness of the adatom system which is a key, explicit assumption in [@PatroneEinsteinMargetis14; @PatroneMargetis14; @LuLiuMargetis14].
Our study here has a perspective distinct from that of [@AckermanEvans11; @Zhaoetal15; @Saum09] in which extensive computations are carried out in 2+1 dimensions. In particular, in [@AckermanEvans11; @Zhaoetal15] the authors derive a set of refined boundary conditions at the step edge that depend on the local environment on the basis of a discrete diffusion equation with a [*fixed*]{} step position. In [@Saum09] only numerical comparisons of KMC simulations to aspects of the BCF model are shown. A different view is adopted in [@Zangwill92] where a high-dimensional master equation is reduced to a Langevin-type equation for height columns on the crystal lattice. We should also mention the probabilistic approach in [@MarzuolaWeare13], which addresses the passage from an atomistic description within a solid-on-solid model to a [*fully*]{} continuum picture.
A discussion that the boundary condition involving the mass flux toward the step edge may exhibit a nonlinear behavior as a function of the adatom density can be found in [@VoigtBalykov2006a; @CermelliJabbour2005]. Specifically, in [@VoigtBalykov2006a] the authors carry out numerical simulations of a “terrace-step-kink” model that reveal a nonlinear dependence of adatom fluxes on the supersaturation, $\bar{\sigma}_\pm$; and relate this behavior to the step-continuum thermodynamic approach of [@CermelliJabbour2005].
In the present treatment, we point out such a nonlinearity at the mesoscale from a kinetic atomistic perspective, in an effort to avoid continuum thermodynamic principles. By recourse to atomistic mechanisms, we argue that nonlinear terms in the boundary conditions for the mass flux naturally emerge as the system is driven farther from equilibrium; cf. . Building on our results, our long-term goal is to address phenomena in 2D settings, for which more complicated atomistic models are necessary [@Caflischetal_99].
Limitations {#subsec:Intro-Limitations}
------------
Our work has several limitations. To start with, our atomistic model is one dimensional (1D). This simplification has some unphysical consequences. For example, it leaves out surface features intimately connected to the effect of step stiffness [@BCF51]. Step meandering, an important 2D effect, is completely absent in our approach. Furthermore, we do not account for nucleation, which at low enough temperatures is known to cause deviations from the usual kinetic law for the step velocity [@Shitara93]. In a related vein, we are unable to adequately model advection at the atomistic scale and derive corresponding terms in the continuum limit. In our 1D setting, island formation dictates that equilibrium cannot be established; thus we exclude bonding between adatoms [@PatroneMargetis14]. Furthermore, we focus on the motion of a single step. This consideration leaves out elastic and entropic step-step interactions [@JeongWilliams99; @PimpinelliVillain98]. Admittedly, our version of a microscopic master equation is simplified since it forms a direct extension, by addition of external deposition, of the description in [@PatroneMargetis14]. This model is deemed suitable for low and moderate adatom densities. Thus, our formalism may not entirely capture the full range of effects arising at high supersaturation. Our analysis, which focuses on averages of microscopic variables such as the number of adatoms per lattice site, leaves unexplored the issue of stochastic fluctuations in step motion [@LuLiuMargetis14].
Outline of paper {#subsec:Intro-Outline}
----------------
The remainder of the paper is organized as follows. Section \[sec:ReviewModels\] provides a review of the main models: the BCF-type model of step flow in 1D (Section \[subsec:BCFmodel\]), which offers the usual mesoscale picture phenomenologically; and the atomistic KRSOS model (Section \[subsec:SOSmodel\]), which is the starting point and basis of our analysis. In Section \[sec:Max-Steady\_SOS\], we introduce a maximum principle and calculate the steady-state solution for a master equation of the KRSOS model with material deposition onto the surface from above. A discrete version of the BCF model is developed in Section \[sec:DiscreteBCF\]. The BCF-type model corresponding to our atomistic dynamics is formally derived in Section \[sec:BCFregime\]. In Section \[sec:Corrections\], we characterize high-supersaturation corrections for the mass flux at the step edge; and propose a more general kinetic relation for mesoscale step flow models. Finally, Section \[sec:Discussion\] contains a summary and discussion of our results.
[**Notation.**]{} We write $f = \mathcal O(g)$ ($f =o(g)$) to imply that $|f/g|$ is bounded by a nonzero constant (approaches zero) in a prescribed limit. For ease in notation, we write $f=\mathcal O(h,g)$ to imply a relation of the form $f=\mathcal O(h)+\mathcal O(g)$. We use the symbol $\lesssim$ to denote boundedness up to the constant factor. The symbol $\mathbbold{R}$ denotes the set of reals.
Background: Mesoscale and atomistic models {#sec:ReviewModels}
==========================================
In this section, we describe ingredients of the mesoscale and atomistic models, which form the core of our paper. In particular, we review basics of the BCF model, and introduce the master equation of atomistic dynamics with material deposition onto the surface from above.
Mesoscale: BCF model {#subsec:BCFmodel}
--------------------
By phenomenological principles, the BCF model treats adatoms and the crystal surface in a continuum fashion in the lateral direction, yet retains the atomistic detail of the crystal in the vertical direction [@BCF51; @JeongWilliams99]. This approach makes use of an adatom concentration, $\rho$, on each terrace in the laboratory frame. Accordingly, the BCF theory is comprised of the following major elements: (a) A step velocity law, which expresses mass conservation for adatoms; (b) a diffusion equation for $\rho$; and (c) a linear kinetic relation for the adatom flux normal to the step edge. To simplify the model without losing sight of the essential physics, we assume that the desorption and evaporation of atoms is negligible. (It should be noted, however, that desorption is treated by BCF [@BCF51].)
$\begin{array}{c}
\includegraphics[width=\textwidth]{BCF_Picture.pdf}
\end{array}$
The geometry of a step ajoining two terraces is depicted in Fig. \[fig:bcf\_picture\]. The upper ($-$) terrace, on the left of step edge, and lower ($+$) terrace, on the right of step edge, differ in height by $a'$, an atomic length. In this view, the adatoms are represented by the concentration field $\rho(x,t)$. Let $\varsigma(t)$ be the position of the step edge. We apply screw-periodic boundary conditions in the spatial coordinate, $x$.
Now consider the motion of the step. The step velocity, $v(t)=\dot \varsigma(t)$, is determined by mass conservation: $$\label{eq:step_velocity_bcf}
v = \frac{\Omega}{a'} \left( J_- - J_+ \right)~,$$ where $J_\pm$ denotes the $x$-directed mass flux at the step edge on the upper ($-$) or lower ($+$) terrace, $\Omega = aa'$ is the atomic area, and $a$ is the lattice spacing in the lateral ($x$-) direction.
For later algebraic convenience, we define $\hat{x}:=x-\varsigma(t)$ which is the coordinate relative to the step edge. On each terrace, the variable $\mathcal C(\hat{x},t)=\rho(x,t)$ satisfies the diffusion equation [@BCF51] $$\label{eq:diffusion_eq_bcf}
\frac{\partial \mathcal{C}}{\partial t} = \mathcal{D} \frac{\partial^2 \mathcal{C}}{\partial {\hat{x}}^2} + v\frac{\partial \mathcal{C}}{\partial \hat{x}} + \mathcal{F}~,$$ where $\mathcal{D}$ is the macroscopic adatom diffusivity and $\mathcal{F}$ is the mesoscopic external deposition flux [@PimpinelliVillain98; @JeongWilliams99]. Note the presence of the advection term, $v (\partial\mathcal C/\partial \hat{x})$, on the right-hand side of ; this term originates from $\partial\rho/\partial t$ in the corresponding diffusion equation for $\rho(x,t)$, viz., $\partial\rho/\partial t=\mathcal D (\partial^2 \mathcal \rho/\partial x^2)+\mathcal F$. Thus, the flux at the step edge consistent with Fick’s law is $$\label{eq:J-Fick_BCF}
J_\pm=-\mathcal D (\partial \mathcal C/\partial \hat{x})_\pm -v \mathcal C_\pm=-\mathcal D (\partial \rho/\partial x)_\pm -v \rho_\pm~.$$ The remaining ingredient of the BCF model is a set of boundary conditions for $\mathcal C$, or $\rho$, at the step edge through the mass flux, $J(x,t)$. BCF originally introduced Dirichlet boundary conditions, by which the restriction $\mathcal C_\pm$ of $\mathcal C$ at the step edge is set equal to an equilibrium value, $c^{eq}$ [@BCF51]. Later on, a Robin boundary condition was imposed [@Chernov61] via the linear version of condition . Note that the Robin boundary condition is typically a linear relation between the solution of a partial differential equation and its normal derivative at a free boundary. This condition was later improved by incorporation of the Ehrlich-Schwoebel barrier [@EhrlichHudda66; @SchwoebelShipsey66]; see [@PimpinelliVillain98]. The linear kinetic relation for the mass flux at the step is $$\label{eq:lkr_bcf}
J_\pm =\mp \kappa_\pm \left(\mathcal{C}_\pm - c^{eq}\right)~,$$ where $\kappa_\pm$ describes the rate of attachment/detachment of atoms at the step in the presence of an Ehrlich-Schwoebel barrier.
In [@VoigtBalykov2006a], numerical simulations based on a “terrace-step-kink” model suggest a nonlinear dependence of $J_\pm$ on $\mathcal{C}_\pm - c^{eq}$. The authors argue that this can be explained by the thermodynamic approach of [@CermelliJabbour2005]. The observation of such a nonlinear effect motivates us to conjecture a generalized relation of form , where the terms corresponding to $n\ge 2$ account for far-from-equilibrium, high-supersaturation corrections to the traditional linear kinetic law . In Section \[sec:Corrections\], we provide evidence for that emerges from kinetic aspects of our simplified 1D atomistic model.
Microscale: KRSOS model {#subsec:SOSmodel}
-----------------------
At the microscale, we consider a simple-cubic crystal surface with a single step [@PatroneMargetis14]. The surface consists of distinct height columns on an 1D lattice of lateral spacing $a$, with total length $L=Na$; see Fig. \[fig:sos\_rates\]. We consider $L=\mathcal O(1)$ as $a\to 0$, e.g., by setting $L=1$. Screw-periodic boundary conditions are applied in the $x$-direction.
Atoms of the top layer that have two in-plane nearest neighbors are [*step atoms*]{} [@PatroneMargetis14]; these atoms are immobile in our model. In contrast, the atom of the step edge, which lies at one end of the top layer, has a single in-plane nearest neighbor and is referred to as an [*edge atom*]{}; it may detach from the step and move to one of the adjacent terraces. By this picture, the adatoms are movable atoms that are neither edge atoms nor step atoms [@PatroneMargetis14].
In our model, we do not allow islands to form; thus, if any two adatoms become nearest neighbors on a terrace, they do not form a bond with each other. Adatoms are free to diffuse across the surface until they reach the step, which acts as a sink or source of them. Externally deposited atoms are assumed to become adatoms on the terrace instantly, and may not attach to the step directly [@LuLiuMargetis14].
### On atomistic processes and system representation {#sssec:tr-multi}
Our model is characterized by transitions between discrete configurations of adatoms. The total mass of these configurations is not conserved if $F\neq 0$. The transitions are controlled by $F$ as well as by Arrhenius rates which have the form $\nu\exp[-E/(k_BT)]$, where $\nu$ is an attempt frequency, $T$ is the absolute temperature, $k_B$ is Boltzmann’s constant, and $E$ is an appropriate activation energy; see [@PatroneMargetis14]. These kinetic rates correspond to atomistic processes of surface diffusion and attachment/detachment at the step.
The basic processes allowed by our 1D atomistic model are shown in Fig. \[fig:sos\_rates\]. The requisite atomistic rates can be described as follows [@PatroneMargetis14]. First, the rate $D=\nu \exp[-E_h/(k_B T)]$ accounts for unbiased adatom hopping on terraces, sufficiently away from a step edge. The extra factor $\phi_\pm = \exp[-E_\pm/(k_BT)]$ expresses additional energy barriers, $E_{\pm}$, corresponding to adatom attachment to the step edge from the lower ($+$) or upper ($-$) terrace. Lastly, the factor $k=\exp[-E_b/(k_BT)]$ accounts for the extra energy, $E_b$, that is necessary for the breaking of the edge-atom bonds with step atoms so that the atom detaches from the step edge.
$\begin{array}{c}
\includegraphics[width=\textwidth]{SOS_Rates.pdf}
\end{array}$
In principle, our 1D system can have an (countably) infinite number of adatom configurations under the influence of a nonzero deposition, $F$. Following [@PatroneMargetis14], we are compelled to represent such atomistic configurations by [*multisets*]{}. A multiset, $\boldsymbol{\alpha}$, is an unordered list whose entries correspond to the positions of adatoms on the 1D lattice; in particular, $\boldsymbol{\alpha} = \{\}$ expresses a configuration that is void of adatoms. Departing slightly from [@PatroneMargetis14], we use multisets $\boldsymbol{\alpha}$ containing the [*Lagrangian coordinates*]{} of adatoms. For notational clarity, the indices $i$ and $j$ reference lattice sites in a fixed (Eulerian) coordinate system, whereas $\hat{\imath}$ and $\hat{\jmath}$ label sites in a (Lagrangian) coordinate frame relative to the step. Accordingly, repeated entries in $\boldsymbol{\alpha}$ indicate multiple adatoms occupying the same lattice site. For example, the system configuration represented by $\boldsymbol{\alpha} = \{\hat{\imath},\hat{\jmath},\hat{\jmath}\}$ has one adatom at site $\hat{\imath}$ and two adatoms at site $\hat{\jmath}$. The number of adatoms correponding to $\boldsymbol{\alpha}$ is simply $|\boldsymbol{\alpha}|$, the cardinality of the multiset.
In connecting the atomistic model to one-step flow with $F=0$ in 1D, our approach relies on explicitly determining the position of the step edge at time $t>0$ from the number of adatoms, $|\boldsymbol{\alpha}|$, and the initial adatom configuration [@PatroneMargetis14]. This is a consequence of mass conservation. If $F\neq 0$, however, more information is needed in order to track the step edge: At every atomistic transition, one must account for the atoms deposited from above.
For our purposes, a system representation that allows this bookeeping results from using an integer, $m$, in addition to using $\boldsymbol{\alpha}$. This $m$ is the total [*mass*]{}, or number of atoms, of the system. Thus, if $m_0$ is the initial mass then $m-m_0$ measures the overall mass increase because of external deposition. Finally, given the initial position of the site to the right of the step edge, $s_0$, in a fixed (Eulerian) coordinate system, we can explicitly track the step for all time $t>0$.
[**Definition 1.**]{} (Representation of atomistic system.) The pair $(\boldsymbol{\alpha}, m)$ defines the [*state*]{} of the atomistic system: the multiset $\boldsymbol\alpha$ expresses the adatom configuration and the index $m$ is an integer that counts the overall mass of the system. Thus, if $m_0$ is the initial mass then $m-m_0$ counts how many adatoms are deposited on the surface from above.
[**Definition 2.**]{} (Discrete step position.) For each state $(\boldsymbol{\alpha}, m)$, the discrete [*step position*]{} in Eulerian coordinates is $s(\boldsymbol{\alpha}, m) = s_0-|\boldsymbol{\alpha}|+m-m_0$. For fixed mass $m$, the step position is uniquely determined from the number of adatoms $|\boldsymbol{\alpha}|$ and the initial position of the site to immediately to the right of the step edge, $s_0$. Accordingly, $s(\boldsymbol{\alpha}, m)$ also references the site to the right of the step edge; see Figure \[fig:sos\_rates\].
### Master equation {#sssec:MasterEqn}
The KRSOS model is characterized by a time-dependent probability density function (PDF), $p_{\boldsymbol{\alpha},m}(t)$, defined over the domain of discrete states $(\boldsymbol{\alpha}, m)$. Accordingly, the time evolution of the system is described by the master equation $$\label{eq:master_eq}
\dot{p}_{\boldsymbol{\alpha},m}(t) = \sum_{\boldsymbol{\alpha}',m'} T_{(\boldsymbol{\alpha},m),(\boldsymbol{\alpha}',m')} p_{\boldsymbol{\alpha}',m'}(t)~,$$ under given initial data, $p_{\boldsymbol\alpha, m}(0)$. In the above, $T_{(\boldsymbol{\alpha},m),(\boldsymbol{\alpha}',m')}$ expresses the overall transition of the system from state $(\boldsymbol{\alpha}',m')$ to state $(\boldsymbol{\alpha},m)$. Evidently, master equation (\[eq:master\_eq\]) governs a Markov process with countably infinite states.
Next, we describe the rates $T_{(\boldsymbol{\alpha},m),(\boldsymbol{\alpha}',m')}$. The nonzero transition rates obey the following rules:
\[eq:transition\_rates\] $$\begin{aligned}
T_{(\boldsymbol{\alpha},m),(\boldsymbol{\alpha}',m')} &= D, & \mbox{ if } m=m' \mbox{ and } |\boldsymbol{\alpha}| = |\boldsymbol{\alpha}'| \mbox{ and } \left|\boldsymbol{\alpha}\setminus\boldsymbol{\alpha}'\right| = 1 \nonumber\\
& & \mbox{ and } \Big| ||\boldsymbol{\alpha}\setminus\boldsymbol{\alpha}'|| - ||\boldsymbol{\alpha}'\setminus\boldsymbol{\alpha}||\Big| = 1; \\
T_{(\boldsymbol{\alpha},m),(\boldsymbol{\alpha}',m')} &= D\phi_\pm, & \mbox{ if } m=m' \mbox{ and } |\boldsymbol{\alpha}| = |\boldsymbol{\alpha}'| - 1 \nonumber\\
& & \mbox{ and } \boldsymbol{\alpha}'\setminus\boldsymbol{\tilde{\alpha}} = \{\pm 1\}; \\
T_{(\boldsymbol{\alpha},m),(\boldsymbol{\alpha}',m')} &= Dk\phi_\pm, & \mbox{ if } m=m' \mbox{ and } |\boldsymbol{\alpha}| = |\boldsymbol{\alpha}'| + 1 \nonumber\\
& & \mbox{ and } \boldsymbol{\alpha}\setminus\boldsymbol{\tilde{\alpha}'} = \{\pm 1\}; \\
T_{(\boldsymbol{\alpha},m),(\boldsymbol{\alpha}',m')} &= \frac{F}{N-1}, & \mbox{ if } m=m'+1 \mbox{ and } |\boldsymbol{\alpha}| = |\boldsymbol{\alpha}'| + 1 \nonumber\\
& & \mbox{ and } |\boldsymbol{\alpha}\setminus\boldsymbol{\alpha}'| = 1;\end{aligned}$$ and, so that probability is conserved, $$\label{eq:transitions_probability_conservation}
T_{(\boldsymbol{\alpha}',m'),(\boldsymbol{\alpha}',m')} =\quad -\sum\limits_{(\boldsymbol{\alpha},m)\atop (\boldsymbol{\alpha},m)\neq (\boldsymbol{\alpha}',m')} T_{(\boldsymbol{\alpha},m),(\boldsymbol{\alpha}',m')} \qquad \mbox{for all } (\boldsymbol{\alpha}',m'). \qquad$$
All transition rates not listed in (\[eq:transition\_rates\]) are zero. Here, we introduce the multiset difference $\boldsymbol{\alpha}\setminus\boldsymbol{\alpha}'$, which itself is a multiset containing the elements in $\boldsymbol{\alpha}$ that are not in $\boldsymbol{\alpha}'$, counting multiplicity. For example, $\{\hat{\imath},\hat{\jmath},\hat{\jmath}\}\setminus\{\hat{\jmath}\} = \{\hat{\imath},\hat{\jmath}\}$. Additionally, the symbol $||\cdot||$ indicates the $\ell^p$-norm with $p\geq 1$, and we define the “multiset increment operation” as $\boldsymbol{\tilde{\alpha}} = \{\hat{\imath}+1|\mbox{ for all } \hat{\imath}\in \boldsymbol{\alpha}\}$, i.e. the set $\boldsymbol{\tilde{\alpha}}$ is just $\boldsymbol{\alpha}$ after each element has been incremented by one. Setting $F=0$ in (\[eq:transition\_rates\]) reduces to the master equation governing surface relaxation [@PatroneMargetis14].
Among the transition rates that are zero, notable examples include
\[eq:transition\_zero\_rates\] $$\begin{aligned}
T_{(\boldsymbol{\alpha},m),(\boldsymbol{\alpha}',m')} &= 0, & \mbox{ if } m=m' \mbox{ and } |\boldsymbol{\alpha}| < |\boldsymbol{\alpha}'| - 1 \mbox{ or } |\boldsymbol{\alpha}| > |\boldsymbol{\alpha}'| + 1; \\
T_{(\boldsymbol{\alpha},m),(\boldsymbol{\alpha}',m')} &= 0, & \mbox{ if } m=m' \mbox{ and } |\boldsymbol{\alpha}| = |\boldsymbol{\alpha}'|+1 \nonumber\\
& & \mbox{ and } -1 \in \boldsymbol{\alpha}'; \\
T_{(\boldsymbol{\alpha},m),(\boldsymbol{\alpha}',m')} &= 0, & \mbox{ if } m=m'+1 \mbox{ and } |\boldsymbol{\alpha}| \leq |\boldsymbol{\alpha}'|.\end{aligned}$$
Equation (\[eq:transition\_zero\_rates\]a) indicates that no more than one atom may attach to or detach from the step in a single transition. Equation (\[eq:transition\_zero\_rates\]b) asserts that no atoms may detach if the site directly above the edge atom is occupied, and (\[eq:transition\_zero\_rates\]c) prevents atoms from being deposited at $s(\boldsymbol{\alpha},m)$. Note that the transitions described in (\[eq:transition\_rates\]a)-(\[eq:transition\_rates\]c), along with (\[eq:transition\_zero\_rates\]a) and (\[eq:transition\_zero\_rates\]b) are subject to detailed balance [@LuLiuMargetis14; @PatroneMargetis14].
The master equation (\[eq:master\_eq\]) along with transition rates (\[eq:transition\_rates\]) and (\[eq:transition\_zero\_rates\]) completely govern the full mass-dependent microscale model. For some of our purposes, notably the maximum principle of Section \[sec:Max-Steady\_SOS\] and its application, we require an alternate version of equation (\[eq:master\_eq\]) that describes the evolution of a [*marginalized*]{} PDF, $p_{\boldsymbol{\alpha}}(t)$, where the mass variable has been summed.
[**Definition 3.**]{} (Marginal probability density function.) The marginal probability density is $$\label{eq:marginal_pdf}
p_{\boldsymbol{\alpha}}(t) = \sum_m p_{\boldsymbol{\alpha},m}(t),$$ where $p_{\boldsymbol{\alpha},m}(t)$ satisfies (\[eq:master\_eq\]) with transition rates (\[eq:transition\_rates\]) and (\[eq:transition\_zero\_rates\]).
The marginal PDF in Definition 3 satisfies what will be referred to as the marginalized master equation, found by summing over the mass variable $m$ on both sides of (\[eq:master\_eq\]). It is important to note that a sum over $m$ on the right-hand side of master equation (\[eq:master\_eq\]) involves careful consideration of rules (\[eq:transition\_rates\]) since the transition rates $T_{(\boldsymbol{\alpha},m),(\boldsymbol{\alpha}',m')}$ depend on $m$ in addition to the PDF $p_{\boldsymbol{\alpha},m}(t)$.
Next, we give the marginalized master equation and rules for the associated transition rates. The master equation for the marginalized PDF of Definition 3 is $$\begin{aligned}
\label{eq:master_eq_marginal}
\dot{p}_{\boldsymbol{\alpha}}(t)
&= \sum_{\boldsymbol{\alpha}'} T_{\boldsymbol{\alpha},\boldsymbol{\alpha}'} p_{\boldsymbol{\alpha}'}(t) \nonumber\\
&= D\sum_{\boldsymbol{\alpha}'} \left[ A_{\boldsymbol{\alpha},\boldsymbol{\alpha}'} + \epsilon B_{\boldsymbol{\alpha},\boldsymbol{\alpha}'} \right] p_{\boldsymbol{\alpha}'}(t)~.\end{aligned}$$ Here, $A_{\boldsymbol{\alpha},\boldsymbol{\alpha}'}$ accounts for the atomistic processes of attachment and detachment at the step edge, and atom hopping on each terrace, as described in [@PatroneMargetis14]; and $B_{\boldsymbol{\alpha},\boldsymbol{\alpha}'}$ together with the non-dimensional parameter $\epsilon \equiv F/D$ account for material deposition onto the surface from above. In (\[eq:master\_eq\_marginal\]), $\epsilon$ plays the role of a P' eclet number, measuring the deposition rate relative to terrace diffusion. Note that the symbol $R$ has previously been used for the inverse ratio, $R=D/F$, in part of the physics literature [@AmarFamily95].
The scaled, non-zero transition rates $A_{\boldsymbol{\alpha},\boldsymbol{\alpha}'}$ and $B_{\boldsymbol{\alpha},\boldsymbol{\alpha}'}$ can be described by rules similar to those in (\[eq:transition\_rates\]), viz.
$$\begin{aligned}
\label{eq:transition_rates_marginal}
A_{\boldsymbol{\alpha},\boldsymbol{\alpha}'}=& 1~, \qquad\ \mbox{if } |\boldsymbol{\alpha}| = |\boldsymbol{\alpha}'| \mbox{ and } \left|\boldsymbol{\alpha}\setminus\boldsymbol{\alpha}'\right| = 1 \mbox{ and } \Big| ||\boldsymbol{\alpha}\setminus\boldsymbol{\alpha}'|| - ||\boldsymbol{\alpha}'\setminus\boldsymbol{\alpha}||\Big| = 1~; \nonumber \\
A_{\boldsymbol{\alpha},\boldsymbol{\alpha}'}=& \phi_\pm~, \quad\ \ \mbox{if }\ |\boldsymbol{\alpha}| = |\boldsymbol{\alpha}'| - 1 \mbox{ and } \boldsymbol{\alpha}'\setminus\boldsymbol{\tilde{\alpha}} = \{\pm 1\}~; \nonumber\\
A_{\boldsymbol{\alpha},\boldsymbol{\alpha}'} =& k\phi_\pm~,\quad\, \mbox{if }\ |\boldsymbol{\alpha}| = |\boldsymbol{\alpha}'| + 1 \mbox{ and } \boldsymbol{\alpha}\setminus\boldsymbol{\tilde{\alpha}}' = \{\pm 1\}~; \nonumber\\
B_{\boldsymbol{\alpha},\boldsymbol{\alpha}'}=& \frac{1}{N-1}~,\quad \mbox{if } |\boldsymbol{\alpha}| = |\boldsymbol{\alpha}'| + 1 \mbox{ and } \left|\boldsymbol{\alpha}\setminus\boldsymbol{\alpha}'\right| = 1; \nonumber\\
A_{\boldsymbol{\alpha}',\boldsymbol{\alpha}'}=& -\sum\limits_{\boldsymbol{\alpha}\atop \boldsymbol{\alpha}\neq \boldsymbol{\alpha}'} A_{\boldsymbol{\alpha},\boldsymbol{\alpha}'}~,\quad
B_{\boldsymbol{\alpha}',\boldsymbol{\alpha}'}= -\sum\limits_{\boldsymbol{\alpha}\atop \boldsymbol{\alpha}\neq \boldsymbol{\alpha}'} B_{\boldsymbol{\alpha},\boldsymbol{\alpha}'} \quad \mbox{for all } \boldsymbol{\alpha}'~. \end{aligned}$$
In the spirit of [@PatroneMargetis14], one may view master equation as a kinetic hierarchy of coupled particle equations for adatoms. For fixed number of adatoms, $|\boldsymbol{\alpha}|=n$, a combinatorial argument entails that there are $$\label{eq:num_states}
\omega(n)=\left( \begin{array}{c} n+N-2 \\ n \end{array} \right)$$ distinct atomistic configurations on the 1D lattice of size $N$ ($N\ge 2$). By , $\ln\omega(n)$ grows as $\mathcal O(n)$ for $n\gg 1$ with $n=\mathcal O(N)$. Evidently, situations with high enough supersaturation imply the contribution to $p_{\boldsymbol{\alpha}}$ from states that have many adatoms, i.e., large $|\boldsymbol{\alpha}|$. In [@PatroneMargetis14; @PatroneEinsteinMargetis14] this complication is avoided by restricting attention to the dilute regime, in which the dynamics of are dominated primarily by $0$- and $1$-particle states. In the present work, we aim to account for higher particle states. Unless we state otherwise, we assume that the number of adatoms cannot exceed a certain bound, $M$: $|\boldsymbol{\alpha}|\le M$, where $M$ is a fixed yet arbitrary positive integer.
KRSOS model: Maximum principle and steady state solution {#sec:Max-Steady_SOS}
========================================================
In this section, we point out a significant property of the KRSOS model. First, we state and prove a maximum principle for marginalized master equation , which forms an extension of the maximum principle in [@PatroneMargetis14] (Section \[subsec:max\]). This principle relies on the presumed existence of a steady-state solution. For $F=0$, the steady-state solution is known to correspond to the equilibrium distribution of adatoms [@PatroneMargetis14]. Here, we derive an exact, closed-form expression for this equilibrium solution via the canonical ensemble [@Huang] (Section \[subsec:EquilibriumNoDep\]). We also discuss the steady-state solution of for sufficiently small $F$, $F\neq 0$, by assuming that a finite number of particle states contribute to the governing kinetic hierarchy (Section \[subsec:SteadyStateSOS\]).
Maximum principle for master equation {#subsec:max}
-------------------------------------
Now consider marginalized master equation with $\epsilon\ge 0$.
[**Proposition 1.**]{} [*If a non-trivial steady-state solution, $p_{\boldsymbol\alpha}^{ss}$, of exists, then any solution $p_{\boldsymbol\alpha}(t)$ satisfies*]{} $$\label{eq:maximum_principle}
\max_{\boldsymbol{\alpha}} \frac{p_{\boldsymbol{\alpha}}(t)}{p_{\boldsymbol{\alpha}}^{ss}} \leq \max_{\boldsymbol{\alpha}} \frac{p_{\boldsymbol{\alpha}}(0)}{p_{\boldsymbol{\alpha}}^{ss}}~,\quad t>0~.$$
We proceed to prove Proposition 1 by invoking the identity $$\label{eq:ss_condition}
\sum_{\boldsymbol{\alpha}'} T_{\boldsymbol{\alpha},\boldsymbol{\alpha}'} p_{\boldsymbol{\alpha}'}^{ss} = 0~.$$
[*Proof of Proposition 1.*]{} Equation can be written as $$\begin{aligned}
\label{eq:mp_algebra}
\dot{p}_{\boldsymbol{\alpha}}(t)
&= \sum_{\boldsymbol{\alpha}'} T_{\boldsymbol{\alpha},\boldsymbol{\alpha}'} p_{\boldsymbol{\alpha}',m'}(t) \notag \\
&= T_{\boldsymbol{\alpha},\boldsymbol{\alpha}} p_{\boldsymbol{\alpha}}(t) + \sum_{\boldsymbol{\alpha}' \neq \boldsymbol{\alpha}} T_{\boldsymbol{\alpha},\boldsymbol{\alpha}'} p_{\boldsymbol{\alpha}'}(t) \notag \\
&= T_{\boldsymbol{\alpha},\boldsymbol{\alpha}} p_{\boldsymbol{\alpha}}^{ss} \frac{p_{\boldsymbol{\alpha}}(t)}{p_{\boldsymbol{\alpha}}^{ss}} + \sum_{\boldsymbol{\alpha}' \neq \boldsymbol{\alpha}} T_{\boldsymbol{\alpha},\boldsymbol{\alpha}'} p_{\boldsymbol{\alpha}'}^{ss} \frac{p_{\boldsymbol{\alpha}'}(t)}{p_{\boldsymbol{\alpha}'}^{ss}} \notag \\
&= \sum_{\boldsymbol{\alpha}' \neq \boldsymbol{\alpha}} T_{\boldsymbol{\alpha},\boldsymbol{\alpha}'} p_{\boldsymbol{\alpha}'}^{ss} \left\{ \frac{p_{\boldsymbol{\alpha}'}(t)}{p_{\boldsymbol{\alpha}'}^{ss}} -
\frac{p_{\boldsymbol{\alpha}}(t)}{p_{\boldsymbol{\alpha}}^{ss}} \right\}~.\end{aligned}$$ Note that $T_{\boldsymbol{\alpha},\boldsymbol{\alpha}'} p_{\boldsymbol{\alpha}'}^{ss} \geq 0$ for all $\boldsymbol{\alpha}' \neq \boldsymbol{\alpha}$. Thus, the sign of $\dot{p}_{\boldsymbol{\alpha}}(t)$ is determined by the quantity in brackets. In particular, if $\boldsymbol{\alpha}$ maximizes (minimizes) $p_{\boldsymbol{\alpha}'}(t)/p_{\boldsymbol{\alpha}'}^{ss}$ over all $\boldsymbol{\alpha}'$, then $\dot{p}_{\boldsymbol{\alpha}}(t) \leq 0$ ($\dot{p}_{\boldsymbol{\alpha}}(t) \geq 0$). This assertion implies the desired maximum principle (and corresponding minimum principle), thus concluding the proof. $\square$
Proposition 1 states that, relative to the steady state, the solution $p_\alpha(t)$ of (\[eq:master\_eq\_marginal\]) will never deviate more than the initial data. In other words, the system cannot be driven away from the steady state distribution.
[**Remark 1.**]{} If the initial data $p_{\boldsymbol{\alpha}}(0)$ satisfies $$\label{eq:inital_data_near_eq}
\max_{\boldsymbol{\alpha}} \frac{p_{\boldsymbol{\alpha}}(0)}{p_{\boldsymbol{\alpha}}^{ss}} \leq C~,$$ for a parameter-independent constant $C$, then Proposition 1 implies $p_{\boldsymbol{\alpha}}(t) \lesssim p_{\boldsymbol{\alpha}}^{ss}$ for all $t>0$. This property will enable us to estimate certain averages in Section \[sec:DiscreteBCF\].
[**Remark 2.**]{} In general, Proposition 1 cannot be applied to the full mass-dependent master equation (\[eq:master\_eq\]) since the assumption of existence of a steady-state, and therefore equation (\[eq:ss\_condition\]), is violated when $\epsilon>0$. To see this, consider (\[eq:master\_eq\]) after marginalizing in $\boldsymbol{\alpha}$. The resulting equation, $$\label{eq:mass_master_eq}
\dot{p}_{m}(t) = F\left[p_{m-1}(t) - p_{m}(t)\right],$$ subject to $p_0(0) = 1$, is satisfied by the Poisson distribution $p_m(t) = \frac{(Ft)^m e^{-Ft}}{m!}$, for which $p_m(t)=0$ for any $m=\mathcal O(1)$ in the limit as $t\to \infty$. Thus, no non-trivial steady state exists.
Equilibrium distribution, $F=0$ {#subsec:EquilibriumNoDep}
-------------------------------
Next, we discuss the case without external deposition, $F=0$ ($\epsilon=0$). For this purpose we will assume that the system is initially in a state whose mass is $m_0$ with probability one. Then, the use of mass index $m$ for states of the KRSOS model is unnecessary since $m=m_0$ for all time $t>0$. In this situation, the full master equation (\[eq:master\_eq\]) is equivalent to the marginalized master equation (\[eq:master\_eq\_marginal\]). With this in mind, we will refer to the latter equation in the case $\epsilon=0$.
When external deposition is absent, master equation has an equilibrium solution, $p_{\boldsymbol{\alpha}}^{eq}$, by Kolmogorov’s criterion [@PatroneMargetis14]. This equilibrium solution was given in [@PatroneMargetis14] by recourse to the canonical ensemble of statistical mechanics. We will follow the same approach and improve the result of [@PatroneMargetis14] by representing $p_{\boldsymbol{\alpha}}^{eq}$ in simple closed form.
Recall that $Dk\phi_\pm$ is the detachment rate, where $k = \exp[-E_b/(k_BT)]$ and $E_b$ measures the energy of the adatom resulting from detachment of an edge atom.
We apply the formalism of the canonical ensemble to particle states of our system [@Huang]. In the KRSOS model, the energy of each adatom configuration, $\boldsymbol{\alpha}$, is simply $|\boldsymbol{\alpha}|E_b$. By applying the Boltzmann-Gibbs distribution at equilibrium, we can assert that the probability of having $n$ adatoms is ${\rm Prob}\{|\boldsymbol{\alpha}|=n\}=c\, \exp[-nE_b/(k_BT)]$ where $c$ is a normalization constant. Consequently, the partition function, $Z$, for adatoms is computed as $$\begin{aligned}
\label{eq:partition_function}
Z =& \sum\limits_{\boldsymbol{\alpha}} e^{-|\boldsymbol{\alpha}|E_b/(k_BT)}\nonumber\\
=& \sum_{n=0}^\infty \omega(n) \,e^{-nE_b/(k_BT)}=\frac{1}{(1-k)^{N-1}}~,\end{aligned}$$ by using and the binomial expansion. Thus, the equilibrium solution is $$\label{eq:p_eq}
p_{\boldsymbol{\alpha}}^{eq} = \frac{1}{Z} e^{-|\boldsymbol{\alpha}|E_b/T}
= (1-k)^{N-1} k^{|\boldsymbol{\alpha}|}~.$$ Notice that $p_{\boldsymbol{\alpha}}^{eq} \propto k^{|\boldsymbol{\alpha}|}$ if $k\ll 1$; cf. [@PatroneMargetis14].
Steady state of atomistic model with $F>0$ {#subsec:SteadyStateSOS}
------------------------------------------
In this subsection, we discuss a plausible steady-state solution of our KRSOS model for sufficiently small $\epsilon$ ($\epsilon=F/D$). To this end, we employ a formal argument based on the assumption that only a finite number of particle (adatom) states contribute to the system evolution; $|\boldsymbol\alpha|\le M$ for arbitrary yet fixed $M$.
In contrast to the case with $F=0$, for which the Kolmogorov criterion holds and entails the existence of the equilibrium solution [@PatroneMargetis14], microscopic reversibility is lost for nonzero deposition flux, $F>0$. In fact, by use of a known kinetic model, in Appendix \[app:BirthDeath\] we show that no steady-state solution of master equation exists in the kinetic regime with $\epsilon > \phi_+ + \phi_-$. This latter condition implies that the external deposition rate is larger than the rate of attachment; consequently, from a physical viewpoint, a steady accumulation of adatoms on the terrace can occur for long enough times.
In the remainder of this section, we make the conjecture (but do not prove) that a finite number of particle states contribute to the system evolution. Accordingly, we restrict attention to the kinetic regime with sufficiently small $\epsilon$ ($\epsilon\ll 1$). Our conjecture is favored by KMC simulations, a sample of which is shown in Fig. \[fig:simulation\_mp\_states\].
$\begin{array}{ccc}
\includegraphics[width=0.34\textwidth]{NumberofAdatoms_20th_plot_InterDep-eps-converted-to.pdf} &
\includegraphics[width=0.34\textwidth]{NumberofAdatoms_60th_plot_InterDep-eps-converted-to.pdf} &
\includegraphics[width=0.34\textwidth]{NumberofAdatoms_100th_plot_InterDep-eps-converted-to.pdf}\\
\mathbf{(a)} & \mathbf{(b)} & \mathbf{(c)}
\end{array}$
By enforcement of restriction $|\boldsymbol{\alpha}| \leq M$, master equation reduces to $$\begin{aligned}
\label{eq:truncated_master_equation}
\dot{\mathbf{p}}^\epsilon(t) &= \mathfrak T\mathbf{p}^\epsilon(t) \notag \\
&= D(\mathfrak A+\epsilon \mathfrak B)\mathbf{p}^\epsilon(t)~,\end{aligned}$$ where $\mathbf{p}^\epsilon$ is the $\epsilon$-dependent vector of dimension $\Omega(M):=\sum_{n=0}^M \omega(n)=(M+N-1)!/[M!\,(N-1)!]$ formed by $p_{\boldsymbol{\alpha}}$, and the $\mathfrak T$ matrix is split into the (finite-dimensional) attachment/detachment matrix, $\mathfrak A$, and deposition matrix, $\epsilon\,\mathfrak B$, in correspondence to the $A_{\boldsymbol\alpha,\boldsymbol\alpha'}$ and $\epsilon B_{\boldsymbol\alpha,\boldsymbol\alpha'}$ of . We impose the $\epsilon$-independent initial data $\mathbf p(0)=:\mathbf p_0$.
The approximation of the full microscopic model by offers two obvious advantages. First, the time-dependent solution of can be expressed conveniently in terms of a matrix exponential. Second, the unique steady-state distribution of (\[eq:truncated\_master\_equation\]) always exists; it is the normalized eigenvector of the $\mathfrak T$ matrix with zero eigenvalue[@VanKampen2007].
We proceed to formally express the steady-state solution, $\mathbf p^{ss,\epsilon}$, of as an appropriate series expansion in $\epsilon$. This task can be carried out in several ways; for example, through the conversion of to a Volterra integral equation, an approach that we choose to apply here. By treating $\epsilon D \mathfrak B \mathbf p^\epsilon(t)$ as a forcing term in , the method of variation of parameters yields $$\label{eq:integral_eq}
\mathbf{p}^\epsilon(t) = \Phi(t)\, \left[ \mathbf{p}_0 + \epsilon D\int_0^t \Phi^{-1}(t') \mathfrak B \mathbf{p}^\epsilon(t') dt' \right]~,$$ where $\Phi(t):=\exp(D\mathfrak A\,t)$. We mention in passing that, by the usual theory of Volterra equations, has a unique solution locally in time [@Tricomi1985].
The matrix $\mathfrak A$ is diagonalizable because it corresponds to the transition matrix of a Markov process satisfying detailed balance [@VanKampen2007]. Thus, we apply the decomposition $\mathfrak A = V\Lambda V^{-1}$ where $\Lambda = \mbox{diag}\{\lambda_j\}_{j=1}^{\Omega(M)}$, $\{\lambda_j\}$ are the (non-dimensional) eigenvalues of $\mathfrak A$, and $V$ is a matrix whose column vectors are the respective eigenvectors. Let $\{\lambda_j\}_{j=1}^{\Omega(M)}$ be ordered, $0=\lambda_1>\lambda_2\ge \cdots\ge \lambda_{\Omega(M)}$ [@VanKampen2007]. By $\Phi(t) = Ve^{Dt\Lambda}V^{-1}$, we have $\Phi^{-1}(t) = Ve^{-Dt\Lambda}V^{-1}$. Hence, is recast to $$\label{eq:integral_eq_conv}
\mathbf{p}^\epsilon(t) = Ve^{Dt\Lambda}V^{-1} \mathbf{p}_0 + \epsilon D\int_0^t Ve^{D(t-t')\Lambda} V^{-1} \mathfrak B \mathbf{p}^\epsilon(t') dt'~.$$ At this stage, a formula for $\mathbf p^\epsilon(t)$ ensues by standard methods. We resort to the Laplace transform, $\widehat{\mathbf{p}}^\epsilon(s)=\int_0^\infty e^{-st}\,\mathbf p^\epsilon(t)\,dt$, of $\mathbf p^\epsilon(t)$ with ${\rm Re} s> c >0$ for some suitable positive number $c$; by , we directly obtain $$\label{eq:laplace_trans_p}
\widehat{\mathbf{p}}^\epsilon(s) = \left[I - \epsilon D \,V\mathfrak D(s)V^{-1}\mathfrak B\right]^{-1} V\mathfrak D(s)V^{-1}\mathbf{p}_0~,$$ where $\mathfrak D(s) := \mbox{diag}\{(s-D\lambda_j)^{-1}\}_{j=1}^{\Omega(M)}$ and $I$ is the unit matrix. In , we assume that $\epsilon$ is small enough so that the requisite inverse matrix makes sense.
The next step in this approach is to compute the inverse transform of . However, in principle, this choice requires carrying out in the right half of the $s$-plane the inversion of the matrix $I-\epsilon D\, V\mathfrak D(s) V^{-1}\mathfrak B$ for arbitrary $M$. This task is considerably simplified for the steady-state solution, $\mathbf p^{ss,\epsilon}$, in the limit $t\to\infty$, as shown in Appendix \[app:Asymptotics\]. The resulting formula reads $$\label{eq:p_ss}
\mathbf{p}^{ss,\epsilon} = \mathbf{p}^{0} + \sum_{l=1}^\infty \left( -\epsilon \mathfrak A^\dagger \mathfrak B \right)^l \mathbf{p}^0~.$$ In the above, $\mathbf{p}^0$ corresponds to the equilibrium solution in the absence of external deposition ($\epsilon=0$), and $\mathfrak A^\dagger$ denotes the Moore-Penrose pseudoinverse of $\mathfrak A$. Equation (\[eq:p\_ss\]) indicates the relative contributions of external deposition and diffusion/attachment/detachment processes to the steady state of the hypothetical $M$-particle KRSOS model underlying this calculation.
Our heuristics leave several open questions regarding the meaning of for large particle number $M$. For instance, the behavior with $M$ of the bound for $\epsilon$ needed for convergence has not been addressed. A related issue is to estimate the error by the truncation of series , after a finite number of terms are summed. We expect that ceases to be meaningful as $M\to\infty$.
Averaging: Discrete mesoscale model {#sec:DiscreteBCF}
===================================
In this section, we heuristically show how discrete variables that form averages of microscale quantities on the lattice are plausibly related to mesoscale BCF-type observables of physical interest. In particular, diffusion equation (\[eq:diffusion\_eq\_bcf\]) and step velocity law (\[eq:step\_velocity\_bcf\]) have clear discrete counterparts. A noteworthy finding of our approach is a set of discrete boundary conditions at the step which [*partially*]{} agree with linear kinetic relation . In fact, our exact formulas for discrete fluxes at the left and right of the step edge manifest corrections to .
Definitions of basic averages {#subsec:Averages}
-----------------------------
We now define the average step position, $\varsigma(t)$, in terms of probabilities $p_{\boldsymbol{\alpha},m}(t)$.
[**Definition 4.**]{} The average step position is $$\label{eq:varsigma}
\varsigma(t) = a\sum\limits_{\boldsymbol{\alpha},m} s(\boldsymbol{\alpha},m) p_{\boldsymbol{\alpha},m}(t)~,$$ where $s(\boldsymbol{\alpha},m)$, given in Definition 2, is in an integer that denotes the site directly to the right of the step edge in the fixed reference frame of the 1D lattice.
Note that, in Definition 4, $\big||\boldsymbol{\alpha}|-(m-m_0)\big|$ is the number of adatoms that are exchanged with the step edge and, thus, solely contribute to step motion.
Next, we define the adatom number per lattice site, which plays the role of the adatom density in the mesoscale picture. We use the following two interrelated variables. (i) The density, $c_{\hat{\jmath}}(t)$, of adatoms [*relative*]{} to the step, where $\hat{\jmath}$ counts the lattice sites to the [*right*]{} of the step. This $c_{\hat{\jmath}}(t)$ is a Lagrangian variable. (ii) The Eulerian density, $\rho_j(t)$, at site $j$ of the fixed 1D lattice.
\[eq:density\] [**Definition 5.**]{} (i) The Lagrangian-type adatom density is defined by $$\begin{aligned}
\label{eq:c_j}
c_{\hat{\jmath}}(t) &= \sum\limits_{\boldsymbol{\alpha},m} \nu_{\hat{\jmath}}(\boldsymbol{\alpha}) p_{\boldsymbol{\alpha},m}(t)/a \nonumber\\
&= \sum\limits_{\boldsymbol{\alpha}} \nu_{\hat{\jmath}}(\boldsymbol{\alpha}) p_{\boldsymbol{\alpha}}(t)/a,\end{aligned}$$ where $\nu_{\hat{\jmath}}(\boldsymbol{\alpha})$ is the number of adatoms at site $\hat{\jmath}$ for a system with adatom configuration $\boldsymbol{\alpha}$.
\(ii) The Eulerian adatom density is $$\label{eq:rho_j}
\rho_j(t) = \sum\limits_{\boldsymbol{\alpha},m} \nu_{j-s(\boldsymbol{\alpha},m)}(\boldsymbol{\alpha}) p_{\boldsymbol{\alpha},m}(t)/a~,$$ where $j-s(\boldsymbol{\alpha},m)$ is the Lagrangian coordinate corresponding to Eulerian $j$.
Regarding Definition 5, it is important to note that $\nu_{\hat{\jmath}}(\boldsymbol{\alpha})$ is a factor counting the number of instances of $\hat{\jmath}$ in multiset $\boldsymbol{\alpha}$. Since both $\hat{\jmath}$ and $\boldsymbol{\alpha}$ use the same coordinate system, $\nu_{\hat{\jmath}}(\boldsymbol{\alpha})$ is independent of $m$, allowing for $c_{\hat{\jmath}}(t)$ to be expressed in terms of the marginalized PDF $p_{\boldsymbol{\alpha}}(t)$. In contrast, the definition of $\rho_j(t)$ cannot be written in terms of $p_{\boldsymbol{\alpha}}(t)$ because of the mass dependence in the index of $\nu_{j-s(\boldsymbol{\alpha},m)}(\boldsymbol{\alpha})$. The variable $c_{\hat{\jmath}}(t)$ is most useful in discrete equations for fluxes and boundary conditions at the step (Section \[subsec:gen\_flux\]). On the other hand, $\rho_{j}(t)$ is more convenient to use in the derivation of the discrete diffusion equation, at lattice sites sufficiently away from the step edge (Section \[subsec:discr\_eq\]).
[**Remark 3.**]{} For $F=0$, the equilibrium adatom density, $c^{eq}$, at any lattice site can be computed by (\[eq:p\_eq\]) and (\[eq:c\_j\]); or, alternatively, directly from partition function (\[eq:partition\_function\]) since the number of adatoms is constant everywhere on the terrace: $$\label{eq:ceq_def}
c^{eq} = \frac{\langle n \rangle}{(N-1)a} = \frac{k/a}{1-k}~,$$ where $\langle n \rangle$ denotes the average total number of adatoms.
We now proceed to define the adatom fluxes at the step edge by virtue of our rules for atomistic transitions (Section \[sssec:MasterEqn\]).
[**Definition 6.**]{} The flux $J_\pm$ on the right ($+$, lower terrace) or left ($-$, upper terrace) of the step edge is $$\begin{aligned}
\label{eq:J_pm_def}
J_\pm(t) =& \mp \sum_{\boldsymbol{\alpha},m} \mathbbold{1}(\nu_{-1}(\boldsymbol{\alpha})=0) \notag\\
&\times \left[ T_{(\boldsymbol{\alpha}_\pm,m),(\boldsymbol{\alpha},m)} p_{\boldsymbol{\alpha},m}(t) -
T_{(\boldsymbol{\alpha},m),(\boldsymbol{\alpha}_\pm,m)} p_{\boldsymbol{\alpha}_\pm,m}(t) \right]~.\end{aligned}$$ Here, indicator function $\mathbbold{1}(\cdot)$ is one if its argument is true and zero otherwise, and $\boldsymbol{\alpha}_\pm = \boldsymbol{\tilde\alpha} \cup \{\pm 1\}$ denotes the adatom configuration resulting from a rightward ($+$) or leftward ($-$) detachment. The factor $\mathbbold{1}(\nu_{-1}(\boldsymbol{\alpha})=0)$ excludes configurations that involve adatoms on top of the edge atom, i.e., those configurations for which detachment is forbidden and are inaccessible via attachment events. Additionally, the external deposition of adatoms does not contribute to the mass flux $J_\pm$, and therefore only a single value of $m$ enters (\[eq:J\_pm\_def\]); see (\[eq:transition\_zero\_rates\]).
Generalized discrete flux at step edge {#subsec:gen_flux}
--------------------------------------
In this section, we derive an exact expression for the discrete flux at the step edge. This expression forms the basis for characterizing [*corrections*]{} to linear kinetic law in the discrete setting.
By manipulating (\[eq:J\_pm\_def\]), we directly find a formula for the flux in terms of atomistic parameters, the differences $c_{\pm 1}(t) - c^{eq}$, and other discrete averages, viz., $$\begin{aligned}
\label{eq:J_+_algebra}
J_+(t) =& \sum_{\boldsymbol{\alpha},m} \mathbbold{1}(\nu_{-1}(\boldsymbol{\alpha})=0)
\left[ T_{(\boldsymbol{\alpha}_+,m),(\boldsymbol{\alpha},m)} p_{\boldsymbol{\alpha},m}(t) -
T_{(\boldsymbol{\alpha},m),(\boldsymbol{\alpha}_+,m)} p_{\boldsymbol{\alpha}_+,m}(t) \right] \notag \\
=& Dk\phi_+ \sum_{\boldsymbol{\alpha},m} \mathbbold{1}(\nu_{-1}(\boldsymbol{\alpha})=0) p_{\boldsymbol{\alpha},m}(t) - D\phi_+ \sum_{\boldsymbol{\alpha},m} \mathbbold{1}(\nu_{1}(\boldsymbol{\alpha})=1) p_{\boldsymbol{\alpha},m}(t) \notag \\
=& Dk\phi_+ \sum_{\boldsymbol{\alpha}} \mathbbold{1}(\nu_{-1}(\boldsymbol{\alpha})=0) p_{\boldsymbol{\alpha}}(t) - D\phi_+ \sum_{\boldsymbol{\alpha}} \mathbbold{1}(\nu_{1}(\boldsymbol{\alpha})=1) p_{\boldsymbol{\alpha}}(t) \notag \\
=& Dk\phi_+ \left[ 1 - \sum_{\boldsymbol{\alpha}} \mathbbold{1}(\nu_{-1}(\boldsymbol{\alpha})>0) p_{\boldsymbol{\alpha}}(t) \right] \notag \\
&\quad- D\phi_+ \sum_{\boldsymbol{\alpha}} \mathbbold{1}(\nu_{1}(\boldsymbol{\alpha})=1) \nu_{1}(\boldsymbol{\alpha}) p_{\boldsymbol{\alpha}}(t) \notag \\
=& D\phi_+a \left[ \frac{k/a}{1-k}(1-k) - k\sum_{\boldsymbol{\alpha}} \mathbbold{1}(\nu_{-1}(\boldsymbol{\alpha})>0) p_{\boldsymbol{\alpha}}(t)/a \right] \notag \\
&\quad- D\phi_+a \left[ c_1(t) - \sum_{\boldsymbol{\alpha}} \mathbbold{1}(\nu_{1}(\boldsymbol{\alpha})>1) \nu_{1}(\boldsymbol{\alpha}) p_{\boldsymbol{\alpha}}(t)/a \right] \notag \\
=& -D\phi_+a\left[ c_1(t) - c^{eq} \right] - D\phi_+a f_+(t)~.\end{aligned}$$ In the above, the second equality results from substitution of the transition rates ; the third equality results from summing over the mass variable; the fourth equality makes use of the complement rule, $\mathbbold{1}(\nu_{-1}(\boldsymbol{\alpha})=0) = 1-\mathbbold{1}(\nu_{-1}(\boldsymbol{\alpha})>0)$; and the fifth equality involves adding and subtracting the last sum to make $c_1(t)$ appear. Similar steps can be used to derive the corresponding formula for $J_-(t)$. Together, these fluxes can be written as $$\label{eq:J_pm}
J_\pm(t) = \mp D\phi_\pm a\left[ c_{\pm 1}(t) - c^{eq} \right] \mp D\phi_\pm a f_\pm(t)~.$$ In , the first term on the right-hand side is the discrete analog of the linear kinetic relation of the BCF model. We invoke the definitions for $c_{\pm 1}(t)$ and $c^{eq}$ according to (\[eq:c\_j\]) and (\[eq:ceq\_def\]), respectively; and define
\[eq:f-flux\] $$\begin{aligned}
\label{eq:f_+}
f_+(t) &= k\biggl[c^{eq} + \sum_{\boldsymbol{\alpha}} \mathbbold{1}(\nu_{-1}(\boldsymbol{\alpha})>0) p_{\boldsymbol{\alpha}}(t)/a \biggr] \notag \\
&\quad - \sum_{\boldsymbol{\alpha}} \mathbbold{1}(\nu_{1}(\boldsymbol{\alpha})>1) \nu_{1}(\boldsymbol{\alpha}) p_{\boldsymbol{\alpha}}(t)/a\end{aligned}$$ and $$\begin{aligned}
\label{eq:f_-}
f_-(t) &= k\left[c^{eq} + \sum_{\boldsymbol{\alpha}} \mathbbold{1}(\nu_{-1}(\boldsymbol{\alpha})>0) p_{\boldsymbol{\alpha}}(t)/a \right] \notag \\
&\quad - \sum_{\boldsymbol{\alpha}} \mathbbold{1}(\nu_{1}(\boldsymbol{\alpha})>0) \nu_{-1}(\boldsymbol{\alpha}) p_{\boldsymbol{\alpha}}(t)/a \notag \\
&\quad - \sum_{\boldsymbol{\alpha}} \mathbbold{1}(\nu_{1}(\boldsymbol{\alpha})=0) \mathbbold{1}(\nu_{-1}(\boldsymbol{\alpha})>1)\notag\\ &\qquad \times \left[\nu_{-1}(\boldsymbol{\alpha})-1\right] p_{\boldsymbol{\alpha}}(t)/a~.\end{aligned}$$
Equations signify corrections to the discrete analog of the linear kinetic law of the BCF model; cf. and . The corresponding [*corrective*]{} fluxes, $f_\pm$, measure the frequency by which the atomistic system visits configurations that forbid detachment of the edge atom or attachment of an adatom from the right ($+$) or left ($-$) of the step edge. We expect that the magnitudes of correction terms $f_\pm$ are negligibly small in the appropriate low-density regime for adatoms [@PatroneEinsteinMargetis14; @PatroneMargetis14].
In Section \[sec:BCFregime\], we find bounds for the above corrective fluxes, $f_\pm$. In particular, we describe systematically a parameter regime for $k$ and $F$ in which these corrections are negligible. In contrast, if $f_\pm(t)$ are important then the discrete fluxes $J_\pm(t)$ may no longer have a linear dependence on $c_{\pm1}(t)$, which in turn signifies the onset of high-supersaturation behavior; see Section \[sec:Corrections\].
Discrete diffusion and step velocity law {#subsec:discr_eq}
----------------------------------------
In this section, we formally describe the change of the adatom density and average step position with time. In addition, we complement with a mass-transport relation between discrete fluxes and densities near the step edge, which forms a discrete analog of Fick’s law for diffusion. What we find are equations for discrete diffusion and discrete versions of Fick’s law that have corrections terms resulting from atomistic configurations with multiple adatoms at the same lattice site. First, by Definition 5 and master equation (\[eq:master\_eq\]), the time evolution of $\rho_j(t)$ is described by $$\begin{aligned}
\label{eq:rho_j_dot_master_eqn}
\dot{\rho}_j(t) &= \sum\limits_{\boldsymbol{\alpha},m} \nu_{j-s(\boldsymbol{\alpha},m)}(\boldsymbol{\alpha})
\sum\limits_{(\boldsymbol{\alpha}',m') \atop \ne (\boldsymbol{\alpha},m)} \left[
T_{(\boldsymbol{\alpha},m),(\boldsymbol{\alpha}',m')} p_{\boldsymbol{\alpha}',m'}(t) -
T_{(\boldsymbol{\alpha}',m'),(\boldsymbol{\alpha},m)} p_{\boldsymbol{\alpha},m}(t) \right]/a \nonumber\\
&= \sum\limits_{\boldsymbol{\alpha},m} \sum\limits_{\boldsymbol{\alpha}',m'} \left[\nu_{j-s(\boldsymbol{\alpha},m)}(\boldsymbol{\alpha}) - \nu_{j-s(\boldsymbol{\alpha}',m')}(\boldsymbol{\alpha}') \right] T_{(\boldsymbol{\alpha},m),(\boldsymbol{\alpha}',m')} p_{\boldsymbol{\alpha}',m'}(t)/a.\end{aligned}$$ In (\[eq:rho\_j\_dot\_master\_eqn\]) we invoke property (\[eq:transitions\_probability\_conservation\]) of the transition rates to make the difference $\nu_{j-s(\boldsymbol{\alpha},m)}(\boldsymbol{\alpha})-\nu_{j-s(\boldsymbol{\alpha}',m')}(\boldsymbol{\alpha}')$ appear. By identifying values of this difference, most of which are zero, it is possible to simplify our formula for $\dot{\rho}_j(t)$ and express the right-hand side of (\[eq:rho\_j\_dot\_master\_eqn\]) in terms of known averages.
[**Remark 4.**]{} It should be noted that the derivation of equation (\[eq:rho\_j\_dot\_master\_eqn\]) is independent of our definition of $\nu_{\hat{\jmath}}(\boldsymbol{\alpha})$. Equations similar to (\[eq:rho\_j\_dot\_master\_eqn\]) can be found using general properties of master equations describing Markov processes; namely that transitions must conserve probability. In particular, the above also applies if $\nu_{\hat{\jmath}}(\boldsymbol{\alpha})$ is replaced with $\mathbbold{1}(\nu_{\hat{\jmath}}(\boldsymbol{\alpha})>0)$, which amounts to using the definition of adatom density in [@PatroneMargetis14].
Now, by transition rates (\[eq:transition\_rates\]) and our definition of $\nu_{\hat{\jmath}}(\boldsymbol{\alpha})$, we write equation (\[eq:rho\_j\_dot\_master\_eqn\]) as [@Schneider2016] $$\begin{aligned}
\label{eq:rho_j_dot}
\dot{\rho}_j(t)
&= D\left[ \rho_{j-1}(t) -2\rho_{j}(t)+\rho_{j+1}(t) \right] + \frac{F}{(N-1)a} \notag\\
&- D\left[ R_{j-1}(t) -2R_{j}(t)+R_{j+1}(t) \right] \notag\\
&- \sum\limits_{\boldsymbol{\alpha},m} \bigg\{ \delta_{j,s(\boldsymbol{\alpha},m)} \Big[ D\mathbbold{1}(\nu_{-1}(\boldsymbol{\alpha})>0) + D\mathbbold{1}(\nu_{1}(\boldsymbol{\alpha})>0) + \frac{F}{N-1} \Big] \notag\\
&\quad\; -\delta_{j,s(\boldsymbol{\alpha},m)-1} \Big[ D\mathbbold{1}(\nu_{-1}(\boldsymbol{\alpha})>0) + Dk\phi_- \mathbbold{1}(\nu_{-1}(\boldsymbol{\alpha})=0) \notag\\
&\qquad\qquad\qquad\qquad- D\phi_-\mathbbold{1}(\nu_{1}(\boldsymbol{\alpha})=0)\mathbbold{1}(\nu_{-1}(\boldsymbol{\alpha})>0) \Big] \notag \\
&\quad\; - \delta_{j,s(\boldsymbol{\alpha},m)+1} \Big[ D\mathbbold{1}(\nu_{1}(\boldsymbol{\alpha})>0) + Dk\phi_+\mathbbold{1}(\nu_{-1}(\boldsymbol{\alpha})=0) \notag\\
&\qquad\qquad\qquad\qquad - D\phi_+\mathbbold{1}(\nu_{1}(\boldsymbol{\alpha})=1) \Big] \bigg\} p_{\boldsymbol{\alpha},m}(t)/a~.\end{aligned}$$ A few remarks on are in order. The first two lines include a diffusion-type second-order difference scheme for $\rho_j(t)$ and the accompanying correction to discrete diffusion, respectively. The third line of (\[eq:rho\_j\_dot\]) removes certain terms that do not contribute when $j=s(\boldsymbol{\alpha},m)$; no atoms are deposited to the step edge, for example. The remaining terms express boundary conditions at the right ($j=s(\boldsymbol{\alpha},m)+1$) or left ($j=s(\boldsymbol{\alpha},m)-1$) of the step; they contribute only when $j$ is in the vicinity of the step via the Kronecker delta functions. The high-occupation correction terms, $R_j(t)$, are defined as $$\label{eq:diffusion_corrections}
R_j(t) = \sum\limits_{\boldsymbol{\alpha},m} \left[ \nu_{j-s(\boldsymbol{\alpha},m)}(\boldsymbol{\alpha}) - \mathbbold{1}(\nu_{j-s(\boldsymbol{\alpha},m)}(\boldsymbol{\alpha})>0) \right]p_{\boldsymbol{\alpha},m}(t)/a~.$$ Equation (\[eq:diffusion\_corrections\]) measures discrete corrections related to certain correlated motion of adatoms. In particular, these corrections to discrete diffusion arise from the fact that the KRSOS model includes constant adatom hopping rates, regardless of the number of adatoms present at a given lattice site. In effect, atomistic configurations with multiple adatoms at the same lattice site introduce interactions between particles since only one is able to move. This high-occupancy effect can be seen in (\[eq:diffusion\_corrections\]) since $\nu_{j-s(\boldsymbol{\alpha},m)}(\boldsymbol{\alpha}) - \mathbbold{1}(\nu_{j-s(\boldsymbol{\alpha},m)}(\boldsymbol{\alpha})>0) \ne 0$ when two or more adatoms are at site $j$.
The fourth and fifth lines of suggest that
\[eq:discrete\_bc\_rl\_steps\] $$\begin{aligned}
\label{eq:discrete_bc_l_steps}
D\sum\limits_{\boldsymbol{\alpha},m} & \Big[ \mathbbold{1}(\nu_{-2}(\boldsymbol{\alpha})>0) - \mathbbold{1}(\nu_{-1} (\boldsymbol{\alpha})>0) \notag \\
&\quad + k\phi_- \mathbbold{1}(\nu_{-1}(\boldsymbol{\alpha})=0) - \phi_- \mathbbold{1}(\nu_{1}(\boldsymbol{\alpha})=0) \mathbbold{1}(\nu_{-1}(\boldsymbol{\alpha})>0) \notag \\
&= D\left[ c_{-2}(t) - c_{-1}(t) \right] - a^{-1}J_-(t) - D\left[ \hat{R}_{-2}(t) - \hat{R}_{-1}(t) \right]~,\end{aligned}$$ where $J_-$ is defined by (\[eq:J\_pm\_def\]). The analogous result can be reached for the last two lines of (\[eq:rho\_j\_dot\]) yielding the corresponding condition for $J_+$, at the left of the step edge: $$\begin{aligned}
\label{eq:discrete_bc_r_steps}
D\sum\limits_{\boldsymbol{\alpha},m} & \Big[ \mathbbold{1}(\nu_{2}(\boldsymbol{\alpha})>0) - \mathbbold{1}(\nu_{1} (\boldsymbol{\alpha})>0) \notag \\
&\quad + k\phi_+ \mathbbold{1}(\nu_{-1}(\boldsymbol{\alpha})=0) - \phi_+ \mathbbold{1}(\nu_{1}(\boldsymbol{\alpha})=1) \notag \\
&= D\left[ c_{2}(t) - c_{1}(t) \right] + a^{-1}J_+(t) - D\left[ \hat{R}_{2}(t) - \hat{R}_{1}(t) \right]~.\end{aligned}$$
It should be noted that the high-occupation terms $\hat{R}_{\hat{\jmath}}(t)$ found in equations (\[eq:discrete\_bc\_rl\_steps\]) are corrections of the same origin as (\[eq:diffusion\_corrections\]), but represented in Lagrangian coordinates. These are $$\label{eq:diffusion_corrections_lagrangian}
\hat{R}_{\hat{\jmath}}(t) = \sum\limits_{\boldsymbol{\alpha}} \left[ \nu_{\hat{\jmath}}(\boldsymbol{\alpha}) - \mathbbold{1}(\nu_{\hat{\jmath}}(\boldsymbol{\alpha})>0) \right]p_{\boldsymbol{\alpha}}(t)/a~.$$ In view of , we can now extract the discrete boundary terms at the step edge by setting $j=s(\boldsymbol\alpha,m)\pm1$ ($\hat{\jmath}=\pm 1$) in . Hence, we find $$\label{eq:discrete_bc}
J_\pm(t) = \mp Da \left[ c_{\pm 2}(t) - c_{\pm 1}(t) \right] \mp Da \left[ \hat{R}_{\pm 2}(t) - \hat{R}_{\pm 1}(t) \right]~.$$ Equations are a semi-discrete version of Fick’s law including corrections to diffusive fluxes at each side of the step edge.
[**Remark 5.**]{} In contrast to (\[eq:J-Fick\_BCF\]), the advection term at the step edge does not appear in . We have not been able to derive this term from the atomistic model. We attribute this inability to the fact that certain atomistic processes at the step edge are forbidden; see (\[eq:transition\_zero\_rates\]a) and (\[eq:transition\_zero\_rates\]b). In effect, these forbidden processes can cause adatoms to pile up in front of the step edge, thereby conserving mass at that atomistic scale. In Appendix \[app:Advection\], we develop a plausibility argument on the basis of the atomistic model for the appearance of the advection term, $\dot{\varsigma}~\partial \mathcal{C}/\partial \hat{x}$, away from the step edge; cf. (\[eq:J-Fick\_BCF\]).
We now provide the anticipated mass conservation statement that involves the average step velocity, $\dot\varsigma(t)$; see Definition 4. This average is explicitly expressed by use of master equation , ensemble average for $\varsigma(t)$, and formula for fluxes $J_\pm$. It can be shown that [@Schneider2016] $$\label{eq:step_velocity_discrete}
\dot{\varsigma}(t) = a\left[ J_-(t)-J_+(t) \right]~.$$ We omit the details for derivation of this result.
In summary, motion laws and form discrete analogs to BCF-type equations -, notwithstanding advection. The correction terms $R_j(t)$ and $\hat{R}_{\hat{\jmath}}(t)$, respectively appearing in equations and , are negligible if the system parameters are suitably scaled with the lattice spacing, $a$ [@PatroneMargetis14]. A systematic analysis of these corrections is given in Section \[sec:BCFregime\].
Continuum step flow with estimates for discrete corrections {#sec:BCFregime}
===========================================================
In this section, we systematically derive the continuum step-flow equations of the discrete mesoscale model (Section \[sec:DiscreteBCF\]) in the limit where the lateral lattice spacing, $a$, approaches zero, and the microscale kinetic parameters scale properly with $a$. To investigate the behavior of discrete corrections with the kinetic parameters $k$ and $F$, we determine $L^\infty$-bounds for certain corrections via the “maximum principle”, Proposition 1. Our estimates apply to (i) the correction terms $f_\pm(t)$ found in fluxes , and (ii) high-occupation corrections to discrete density, e.g. the terms $\hat{R}_{\hat{\jmath}}(t)$, described in Section \[subsec:discr\_eq\].
Our formal argument forms an extension of the deposition-free (with $F=0$) case [@PatroneMargetis14] under the hypothesis that only a finite number of particle states contribute to system evolution; $|\boldsymbol\alpha|\le M$. Consequently, as $a\downarrow 0$ we extract a set of BCF-type equations for the moving step along with error estimates for the emerging diffusion equation and the linear kinetic law at the step edge.
Bounds for discrete corrections {#subsec:BCFregime-bounds}
-------------------------------
The first part of our program can be outlined as follows. First, in view of Remark 1 for the initial data $p_{\boldsymbol\alpha}(0)$, we estimate $f_\pm(t)$ and $\hat{R}_{\hat{\jmath}}(t)$ in terms of the marginal steady-state solution $p_{\boldsymbol\alpha}^{ss,\epsilon}$ of the master equation. Second, by invoking the $\epsilon$-series expansion of Section \[subsec:SteadyStateSOS\] for $p_{\boldsymbol\alpha}^{ss,\epsilon}$, we derive the desired estimates for $f_\pm(t)$ and $\hat{R}_{\hat{\jmath}}(t)$; these signify corrections to the linear kinetic law for the adatom fluxes, $J_\pm$, and high-occupation corrections to the discrete diffusion equation on the terrace, respectively. This second stage of our derivation of bounds on discrete corrections makes use of the equilibrium distribution $p_{\boldsymbol\alpha}^{eq}$, equation (\[eq:p\_eq\]), for $\epsilon^0$-terms in our formal expansion, and the $\epsilon^1$-terms found in (\[eq:p\_ss\]). In principle, higher order terms can also be computed, but are neglected in our analysis.
[**Proposition 2.**]{} [*The corrective fluxes (\[eq:f-flux\]) at the step edge satisfy the estimate $$\label{eq:f_pm_estimate}
f_\pm(t) = \mathcal{O}\left(\frac{k}{1-k}\,\frac{k}{a},\, \frac{\epsilon N}{(1+\phi)a}\right)~,$$ where $\phi = \phi_+$ or $\phi_-$. Similarly, the high-occupation corrections (\[eq:diffusion\_corrections\_lagrangian\]) to densities satisfy $$\label{eq:R_j_estimate}
\hat{R}_{\hat{\jmath}}(t) = \mathcal{O}\left(\frac{k}{1-k}\,\frac{k}{a},\, \frac{\epsilon N}{a}\right)~.$$ In these estimates, the constants entering the respective bounds are independent of time and parameters of the problem.*]{}
Thus, if we assume that $k/a=\mathcal{O}(1)$, $\phi \leq \mathcal{O}(1)$ [@PatroneMargetis14; @LuLiuMargetis14] and $\epsilon < \mathcal{O}(a^2)$, we can assert that $f_\pm(t)$ can be neglected when compared to the linear-in-density part of the discrete flux $J_\pm(t)$. Furthermore, the corrections $\hat{R}_{\hat{\jmath}}(t)$ are small compared to density $c_{\hat{\jmath}}(t)$, and therefore $R_j(t) \ll \rho_j(t)$ as well. These controlled approximations reveal a kinetic regime in which equations (\[eq:J\_pm\]), (\[eq:discrete\_bc\]), and the discrete diffusion equation in (\[eq:rho\_j\_dot\]) reduce to discrete versions of linear kinetic law (\[eq:lkr\_bcf\]), Fick’s law (\[eq:J-Fick\_BCF\]) and the continuum diffusion equation for density. The remainder of this subsection is devoted to proving Proposition 2. Note that the mass dependence of $f_\pm(t)$ and $\hat{R}_{\hat{\jmath}}(t)$ has already been summed out; see equations (\[eq:f-flux\]) and (\[eq:diffusion\_corrections\_lagrangian\]).
[*Proof of Proposition 2*]{}. We proceed to derive and (\[eq:R\_j\_estimate\]) through heuristics. Define
\[eq:f\_ss\] $$\begin{aligned}
\tilde f_+^{ss,\epsilon} &:= k\left[c^{eq} + \sum_{\boldsymbol{\alpha}} \mathbbold{1}(\nu_{-1}(\boldsymbol{\alpha})>0) p_{\boldsymbol{\alpha}}^{ss,\epsilon}/a \right] \notag \\
&\quad +\sum_{\boldsymbol{\alpha}} \mathbbold{1}(\nu_{1}(\boldsymbol{\alpha})>1) \nu_{1}(\boldsymbol{\alpha}) p_{\boldsymbol{\alpha}}^{ss,\epsilon}/a~, \label{eq:f_+_ss} \\
\tilde f_-^{ss,\epsilon} &:= k\left[c^{eq} + \sum_{\boldsymbol{\alpha}} \mathbbold{1}(\nu_{-1}(\boldsymbol{\alpha})>0) p_{\boldsymbol{\alpha}}^{ss,\epsilon}/a \right] \notag \\
&\quad + \sum_{\boldsymbol{\alpha}} \mathbbold{1}(\nu_{1}(\boldsymbol{\alpha})>0) \nu_{-1}(\boldsymbol{\alpha}) p_{\boldsymbol{\alpha}}^{ss,\epsilon}/a \notag \\
&\quad + \sum_{\boldsymbol{\alpha}} \mathbbold{1}(\nu_{1}(\boldsymbol{\alpha})=0) \mathbbold{1}(\nu_{-1}(\boldsymbol{\alpha})>1) \left[\nu_{-1}(\boldsymbol{\alpha})-1\right] p_{\boldsymbol{\alpha}}^{ss,\epsilon}/a~\mbox{, and } \label{eq:f_-_ss} \\
\hat{R}_{\hat{\jmath}}^{ss,\epsilon} &:= \sum\limits_{\boldsymbol{\alpha}} \left[ \nu_{\hat{\jmath}}(\boldsymbol{\alpha}) - \mathbbold{1}(\nu_{\hat{\jmath}}(\boldsymbol{\alpha})>0) \right] p_{\boldsymbol{\alpha}}^{ss,\epsilon}/a~.\end{aligned}$$
By Remark 1 and exact formulas and (\[eq:diffusion\_corrections\_lagrangian\]) for $f_\pm(t)$ and $\hat{R}_{\hat{\jmath}}(t)$, respectively, we have $$\label{eq:f-ineq}
|f_\pm(t)|\lesssim \tilde f_\pm^{ss,\epsilon}~\mbox{, and }\ |\hat{R}_{\hat{\jmath}}(t)|\lesssim \hat{R}_{\hat{\jmath}}^{ss,\epsilon}~,~t>0~.$$ Inequalities (\[eq:f-ineq\]) are not particularly useful since they do not explicitly manifest the dependence on the kinetic parameters of interest. We need to use some results from Section \[subsec:SteadyStateSOS\] in order to refine these estimates.
In correspondence to for the truncated system, we write $p_{\boldsymbol{\alpha}}^{ss,\epsilon} = p_{\boldsymbol{\alpha}}^{ss,(0)} + \epsilon p_{\boldsymbol{\alpha}}^{ss,(1)} + \mathcal{O}(\epsilon^2)$, where $p_{\boldsymbol{\alpha}}^{ss,(l)}$ is the $l$-th order term of the underlying series expansion in $\epsilon=F/D$; in particular, $p_{\boldsymbol{\alpha}}^{ss,(0)} = p_{\boldsymbol{\alpha}}^{eq}$ is the zeroth-order contribution to the steady state. The constant entering the term $\mathcal O(\epsilon^2)$ may depend on parameters of the problem but is immaterial for our purposes. We will neglect terms with $l\ge 2$ in the $\epsilon$-expansion for $p_{\boldsymbol{\alpha}}^{ss,\epsilon}$.
By inspection of , we define the following sums.
\[eq:S\_n\] $$\begin{aligned}
S_1^{(l)} &:= \sum_{\boldsymbol{\alpha}} \mathbbold{1}(\nu_{-1}(\boldsymbol{\alpha})>0) p_{\boldsymbol{\alpha}}^{ss,(l)}~, \label{eq:s1_n} \\
S_2^{(l)} &:= \sum_{\boldsymbol{\alpha}} \mathbbold{1}(\nu_{1}(\boldsymbol{\alpha})>1) \nu_{1}(\boldsymbol{\alpha}) p_{\boldsymbol{\alpha}}^{ss,(l)}~, \label{eq:s2_n} \\
S_3^{(l)} &:= \sum_{\boldsymbol{\alpha}} \mathbbold{1}(\nu_{1}(\boldsymbol{\alpha})>0) \nu_{-1}(\boldsymbol{\alpha}) p_{\boldsymbol{\alpha}}^{ss,(l)}~, \label{eq:s3_n} \\
S_4^{(l)} &:= \sum_{\boldsymbol{\alpha}} \mathbbold{1}(\nu_{1}(\boldsymbol{\alpha})=0) \mathbbold{1}(\nu_{-1}(\boldsymbol{\alpha})>1) \left[\nu_{-1}(\boldsymbol{\alpha})-1\right] p_{\boldsymbol{\alpha}}^{ss,(l)}~, \label{eq:s4_n} \\
S_5^{(l)} &:= \sum\limits_{\boldsymbol{\alpha}} \left[ \nu_{\hat{\jmath}}(\boldsymbol{\alpha}) - \mathbbold{1}(\nu_{\hat{\jmath}}(\boldsymbol{\alpha})>0) \right] p_{\boldsymbol{\alpha}}^{ss,(l)}~. \label{eq:s5_n}\end{aligned}$$
Accordingly, formulas (\[eq:f\_ss\]) are recast to the forms
\[eq:f\_pm\_ss\_exp\] $$\begin{aligned}
\tilde f_+^{ss,\epsilon} &= k c^{eq} + \frac{k}{a} \left[ S_1^{(0)} + \epsilon S_1^{(1)} \right] + \frac{1}{a} \left[ S_2^{(0)} + \epsilon S_2^{(1)} \right] + \mathcal{O}(\epsilon^2)~, \label{eq:f_pm_ss_exp+} \\
\tilde f_-^{ss,\epsilon} &= kc^{eq} + \frac{k}{a}\left[ S_1^{(0)} + \epsilon S_1^{(1)} \right] +\frac{1}{a} \left[ S_3^{(0)} + \epsilon S_3^{(1)} \right] \notag \\
&\quad + \frac{1}{a} \left[ S_4^{(0)} + \epsilon S_4^{(1)} \right] + \mathcal{O}(\epsilon^2)~.\label{eq:f_pm_ss_exp-}\end{aligned}$$
First, we compute $S_i^{(0)}$ ($i=1,\,2,\,3,\,4,\,5$), which amount to contributions from the equilibrium solution of the master equation, for $F=0$. By (\[eq:p\_eq\]) and $0<k<1$, we write $$\begin{aligned}
\label{s1_0}
S_1^{(0)} &= \frac{1}{Z} \sum_{\boldsymbol{\alpha}} \mathbbold{1}(\nu_{-1}(\boldsymbol{\alpha})>0) k^{|\boldsymbol{\alpha}|} \notag \\
&= \frac{1}{Z} \sum_{n=1}^\infty \left( \begin{array}{c} n+N-3 \\ n-1 \end{array} \right) k^n = k~.\end{aligned}$$ The binomial coefficient in is the number of $n$-particle configurations with at least one adatom in the site immediately to the left of the step. Similarly, we have $$\begin{aligned}
\label{s2_0}
S_2^{(0)} &= \frac{1}{Z} \sum_{\boldsymbol{\alpha}} \mathbbold{1}(\nu_{1}(\boldsymbol{\alpha})>1) \nu_{1}(\boldsymbol{\alpha}) k^{|\boldsymbol{\alpha}|} \notag \\
&= \frac{1}{Z} \sum_{l=2}^\infty l k^l \sum_{n=l}^\infty \left( \begin{array}{c} n-l+N-3 \\ n-l \end{array} \right) k^{n-l} \notag \\
&= (1-k)\sum_{l=2}^\infty lk^l = \frac{2k^2-k^3}{1-k}~,\end{aligned}$$ $$\begin{aligned}
\label{s3_0}
S_3^{(0)} &= \frac{1}{Z} \sum_{\boldsymbol{\alpha}} \mathbbold{1}(\nu_{1}(\boldsymbol{\alpha})>0) \nu_{-1}(\boldsymbol{\alpha}) k^{|\boldsymbol{\alpha}|} \notag \\
&= \frac{k}{Z} \sum_{l=1}^\infty l k^l \sum_{n=l+1}^\infty \left( \begin{array}{c} n-l-1+N-3 \\ n-l-1 \end{array} \right) k^{n-l-1} \notag \\
&= k (1-k) \sum_{l=1}^\infty l k^l = \frac{k^2}{1-k}~,\end{aligned}$$ $$\begin{aligned}
\label{s4_0}
S_4^{(0)} &= \frac{1}{Z} \sum_{\boldsymbol{\alpha}} \mathbbold{1}(\nu_{1}(\boldsymbol{\alpha})=0) \mathbbold{1}(\nu_{-1}(\boldsymbol{\alpha})>1) \left[\nu_{-1}(\boldsymbol{\alpha})-1\right] k^{|\boldsymbol{\alpha}|} \notag \\
&= \frac{k}{Z} \sum_{l=2}^\infty (l-1)k^{l-1} \sum_{n=l}^\infty \left( \begin{array}{c} n-l+N-4 \\ n-l \end{array} \right)k^{n-l} \notag \\
&= k(1-k)^2 \sum_{l=1}^\infty lk^l = k^2~,\end{aligned}$$ and $$\begin{aligned}
\label{s5_0}
S_5^{(0)} &= \frac{1}{Z} \sum\limits_{\boldsymbol{\alpha}} \left[ \nu_{\hat{\jmath}}(\boldsymbol{\alpha}) - \mathbbold{1}(\nu_{\hat{\jmath}}(\boldsymbol{\alpha})>0) \right] k^{|\boldsymbol{\alpha}|} \notag \\
&= \frac{1}{Z} \sum_{l=2}^\infty (l-1) k^l \sum_{n=l}^\infty \left( \begin{array}{c} n-l+N-3 \\ n-l \end{array} \right) k^{n-l} \notag \\
&= k(1-k)\sum_{l=1}^\infty l k^l = \frac{k^2}{1-k}~.\end{aligned}$$ We have followed the convention that the index $l$ is used to account for restrictions on states coming from indicator functions and the index $n$ replaces the number of adatoms, $|\boldsymbol{\alpha}|$.
All that remains is to compute terms proportional to $\epsilon$. This task calls for estimating the sums $S_i^{(l)}$, defined in , for $l=1$. Hence, we would need to invoke the first-order steady-state solution, $p_{\boldsymbol{\alpha}}^{ss,(1)}$, of marginalized master equation (\[eq:master\_eq\_marginal\]). As alluded to in Section \[subsec:SteadyStateSOS\], we do not have, strictly speaking, a simple closed-form solution. Instead, by restricting attention to a finite number of particle states ($|\boldsymbol\alpha|\le M$), we provide approximations for the requisite (infinite) sums by finite sums via expansion (\[eq:p\_ss\]). This approximation amounts to replacing the sums $S_i^{(1)}$ with quadratic forms of appropriately defined vectors, $\mathbf{y}_i\in \mathbbold{R}^{\Omega(M)}$; thus, we write $S_i^{(1)} \approx \mathbf{y}_i^T \mathfrak A^\dagger \mathbf{z}$, where the vectors $\mathbf{y}_i$ have entries that correspond to the indicator functions and/or $\nu_{\pm1}(\boldsymbol{\alpha})$ according to (\[eq:S\_n\]) and $\mathbf{z} = -\mathfrak B \mathbf{p}^0$.
Next, we obtain estimates for $S_i^{(1)}$ as follows (see Section \[subsec:SteadyStateSOS\]): $$\begin{aligned}
\label{eq:si_1}
|S_i^{(1)}| &\approx |\mathbf{y}_i^T \mathfrak A^\dagger \mathbf{z}| \notag\\
&= \left|\left[\left(\frac{\mathbf{y}_i^T}{\mathbf{y}_i^T\mathbf{y}_i}\right)^T\right]^\dagger \mathfrak A^\dagger \left[\frac{\mathbf{z}^T}{\mathbf{z}^T\mathbf{z}}\right]^\dagger\right| \notag\\
&= \left[ \frac{\left|\mathbf{z}^T \mathfrak A \mathbf{y}_i\right|}{\mathbf{y}_i^T\mathbf{y}_i \, \mathbf{z}^T\mathbf{z}} \right]^\dagger
= \frac{\mathbf{y}_i^T\mathbf{y}_i \, \mathbf{z}^T\mathbf{z}}{\left|\mathbf{z}^T \mathfrak A \mathbf{y}_i\right|} \notag\\
&\lesssim \left| \mathbf{z}^T \mathfrak A \mathbf{y}_i \right|^{-1}~,\end{aligned}$$ whenever $\mathbf{z}^T \mathfrak A \mathbf{y}_i \ne 0$. The above calculation uses several properties of the Moore-Penrose pseudoinverse, i.e. $\mathbf{y}^\dagger = \mathbf{y}^T/(\mathbf{y}^T\mathbf{y})$ provided $\mathbf{y}\ne \mathbf{0}$ (second line), $(\mathfrak A\mathfrak B)^\dagger = \mathfrak B^\dagger\mathfrak A^\dagger$ (third line), and the pseudoinverse of a nonzero constant is its multiplicative inverse (fourth line). The last line of (\[eq:si\_1\]) results from observing that the numerator results in a fixed constant for each $i$. Since $\mathfrak A$, $\mathfrak B$, $\mathbf{p}^0$, and $\mathbf{y}_i$ are known, (\[eq:si\_1\]) is a computable estimate. After some linear algebra, we find $$\label{eq:si_1_values}
|S_i^{(1)}| \lesssim \left\{ \begin{array}{ll}
\frac{N}{1+\phi}~, & i=1,\,2,\,3,\,4, \\
N~, & i=5;
\end{array} \right.$$ in the above, $\phi = \phi_+$ or $\phi_-$.
In summary, if the initial data of the atomistic system is near the steady state in the sense of (\[eq:inital\_data\_near\_eq\]), then maximum principle (\[eq:maximum\_principle\]) implies . Inequalities along with – yield estimates and (\[eq:R\_j\_estimate\]) for $0<k<1$ and sufficiently small $\epsilon$. This statement concludes our heuristic proof of Proposition 2. $\square$
[**Remark 6.**]{} The estimates in Proposition 2 are based upon several assumptions and approximations, including: (i) application of the maximum principle (Proposition 1); (ii) truncation of marginalized master equation for $\epsilon>0$; (iii) asymptotics for the Laplace transform and formal power-series expansion for steady-steady distribution, $p^{ss}_{\boldsymbol{\alpha}}$; and (iv) $L^\infty$-bounds for correction terms (\[eq:f-flux\]) and (\[eq:diffusion\_corrections\_lagrangian\]). Consequently, estimates (\[eq:f\_pm\_estimate\]) and (\[eq:R\_j\_estimate\]) are not expected to be optimal. In particular, the bounds involving $\epsilon$ can likely be improved.
BCF-type model as a scaling limit {#subsec:BCF-lim}
---------------------------------
By Proposition 2, we are now able to shed light on the scaling limit of the atomistic model, which leads to the BCF-type description of Section \[subsec:BCFmodel\]. In view of (\[eq:R\_j\_estimate\]), let us impose $$\label{eq:suff-conds}
k= \mathcal O(a)~,\quad \epsilon = \mathcal O(a^3)~,$$ which ensure that high-occupancy corrections $\hat{R}_{\hat{\jmath}}(t)$ and $R_j(t)$ are small compared to the corresponding densities. On the other hand, we assume $\rho_j(t) \to \rho(x,t)$ and $c_{\hat{\jmath}}(t) \to \mathcal{C}(\hat{x},t)$ as the lattice parameter approaches zero, $a \to 0$. The Taylor expansion of $\rho(x,t)$ about $x=ja$ yields $$\frac{\rho_{j-1}(t) - 2\rho_j(t) + \rho_{j+1}(t)}{a^2}=\frac{\partial^2 \rho}{\partial x^2}(ja,t) + \mathcal{O}(a^2)~.$$ By the usual notion of macroscopic diffusion, it is natural to set $\mathcal D:=Da^2=\mathcal{O}(1)$, the surface diffusivity; and $\mathcal F:=\frac{F}{(N-1)a}$, the deposition flux per unit length [@PatroneMargetis14; @LuLiuMargetis14]. Thus, the first two lines of (\[eq:rho\_j\_dot\]), with (\[eq:suff-conds\]), imply $$\label{eq:diffusion_eq_eulerian}
\frac{\partial \rho}{\partial t} = \mathcal{D} \frac{\partial^2 \rho}{\partial x^2} - \mathcal{D} \frac{\partial^2 \mathcal{R}}{\partial x^2} + \mathcal{F}~,$$ as $a \to 0$, where $\mathcal{R}(x,t)$ is the continuum limit of $R_j(t)$. By (\[eq:suff-conds\]), the terms $\mathcal{D}~\partial^2 \mathcal{R}/\partial x^2$ and $\mathcal{F}$ are $\mathcal{O}(a)$, and hence are negligible to leading order. With exception of the deposition term, the resulting leading-order equation for $\rho(x,t)$ is the anticipated diffusion equation in Eulerian coordinates; cf. (\[eq:diffusion\_eq\_bcf\]). By sharpening estimates for $\epsilon$ in Proposition 2, $\mathcal{F}$ may subsequently appear as a leading-order term.
Furthermore, the observation that $\hat{R}_{\hat{\jmath}}(t) = \mathcal{O}(a)$, enables us to obtain Fick’s law of diffusion at the step edge by virtue of . By Taylor expanding the Lagrangian adatom density, $\mathcal{C}$, about $\hat{x}=\hat{\jmath}a$ with $\hat{\jmath}=0$ for the right ($+$) side of the step and $\hat{\jmath}=N-1$ for the left ($-$) side of the step as $a\to 0$, we obtain the formulas
\[eq:ficks\_law\_step\] $$\begin{aligned}
J_+(t) &= -\mathcal{D} \left.\frac{\partial \mathcal{C}}{\partial \hat{x}}\right|_{\hat{x}=0^+} + \mathcal O(a)~,\label{eq:ficks_law_r} \\
J_-(t) &= -\mathcal{D} \left.\frac{\partial \mathcal{C}}{\partial \hat{x}}\right|_{\hat{x}=L^-} + \mathcal O(a)~, \label{eq:ficks_law_l}\end{aligned}$$
where $\hat{x}=\hat{\jmath}a$ and $L=Na=\mathcal O(1)$ by virtue of the screw-periodic boundary conditions. In the above, we used the expansions $c_2(t)-c_1(t)=a [(\partial\mathcal C/\partial \hat{x})|_{x=0^+}+\mathcal O(a)]$ and $c_{-2}(t)-c_{-1}(t)=a[-(\partial\mathcal C/\partial \hat{x})|_{x=L^-}+\mathcal O(a)]$.
Equations (\[eq:ficks\_law\_step\]) need to be complemented with kinetic boundary conditions at the step edge. Hence, we now apply with and . First, we set $\kappa_\pm:=D\phi_\pm a=\mathcal O(1)$ [@PatroneMargetis14; @LuLiuMargetis14]. Second, by inspection of estimate , we assume $\phi_\pm \leq \mathcal{O}(1)$ along with (\[eq:suff-conds\]). Consequently, we can assert that $|f_\pm(t)|= \mathcal O(a)$. Thus, by we obtain
\[eq:lkr\_coarse\] $$\begin{aligned}
J_+(t)&= -\kappa_+ \left[\mathcal{C}(0^+,t) - c^{eq}\right]+\mathcal O(a)~, \label{eq:lkr_coarse_r} \\
J_-(t)&= \kappa_- \left[\mathcal{C}(L^-,t) - c^{eq}\right]+\mathcal O(a)~, \label{eq:lkr_coarse_lr}\end{aligned}$$
as $a \to 0$; recall that $c^{eq}=(k/a)(1-k)^{-1}\approx k/a$ if $k=\mathcal O(a)$ [@PatroneMargetis14].
The last component of the BCF model that emerges from the discrete equations is the step velocity law. This law is provided by (\[eq:step\_velocity\_discrete\]); in this equation, the factor multiplying the difference in the adatom flux across the step edge equals $\Omega/a'$. Thus, (\[eq:step\_velocity\_discrete\]) is precisely (\[eq:step\_velocity\_bcf\]) pertaining to the BCF model.
Corrections to BCF linear kinetic relation: A numerical study {#sec:Corrections}
=============================================================
In this section, we carry out KMC simulations to illustrate the behavior of adatom fluxes at the step edge for high enough supersaturations, defined as $\sigma_\pm = c_{\pm1}/c^{eq}-1$. We demonstrate that in this regime of high detachment or deposition flux, described in more detail below, the boundary condition for the adatom fluxes at the step edge can deviate significantly from linear kinetic relation ; thus, in principle the corrective fluxes, $f_\pm(t)$, may [*not*]{} be negligible. Since we have been unable to express these contributions, $f_\pm$, in terms of mesoscale quantities such as the adatom density, KMC simulations remain our primary tool for describing the high-supersaturation effects. For a discussion of this point, see Section \[sec:Discussion\].
First, we make an educated yet empirical attempt to outline plausible conditions by which the conventional BCF model, particularly the linear kinetic relation for the flux, becomes questionable. By estimate (\[eq:f\_pm\_estimate\]) of Proposition 2, the microscale system may no longer be modeled in accord with the BCF theory if, for example: $k=O(1)$; or, $k=\mathcal O(a)$ and $\epsilon=\mathcal O(a)$. These conditions indicate situations in which high supersaturation may occur, because of large enough detachment rate at the step edge or high enough deposition onto the surface from above.
$\begin{array}{ccc}
\includegraphics[width=0.34\textwidth]{FluxVsSigma_NoDep-eps-converted-to.pdf} &
\includegraphics[width=0.34\textwidth]{FluxVsSigma_LowDep6-eps-converted-to.pdf} &
\includegraphics[width=0.34\textwidth]{FluxVsSigma_InterDep-eps-converted-to.pdf}\\
\mathbf{(a)} & \mathbf{(b)} & \mathbf{(c)}
\end{array}$
Figure \[fig:lkr\] depicts the adatom flux on the right of the step edge under conditions that enable the system to remain close to thermodynamic equilibrium, i.e., for sufficiently small detachment rate or external deposition rate. In these cases, the supersaturation has small values. Linear kinetic law , with neglect of the corrective flux, $f_+$, is found on average to provide a reasonably accurate approximation for the adatom flux at the step edge.
The remaining plots of this section depict situations in which the adatom flux on the right of the step edge may deviate from linear kinetic law , and thus $f_{+}$ may become significant. In particular, Figures \[fig:no\_lkr\_k\] and \[fig:no\_lkr\_F\] reveal the behavior of the adatom flux versus supersaturation on the right of the step for large enough $k$ or $F$, respectively. The deviation from the conventional linear behavior predicted by is manifested differently in each case.
Let us consider the high-detachment rate cases with zero deposition, as these are depicted in Figure \[fig:no\_lkr\_k\]. If the supersaturation is sufficiently close to zero, when the flux is small enough, then the flux is approximately linear with supersaturation but with a slope that can be different from the value $D\phi_+ a$ predicted by kinetic law . Farther away from equilibrium, the dependence of adatom flux on supersaturation evidently becomes nonlinear. This nonlinear behavior becomes more pronounced for larger $k$.
$\begin{array}{ccc}
\includegraphics[width=0.33\textwidth]{FluxVsSigma_InterKappa08-eps-converted-to.pdf} &
\includegraphics[width=0.33\textwidth]{FluxVsSigma_InterKappa06-eps-converted-to.pdf} &
\includegraphics[width=0.33\textwidth]{FluxVsSigma_HighKappa-eps-converted-to.pdf}\\
\mathbf{(a)} & \mathbf{(b)} & \mathbf{(c)}
\end{array}$
Next, consider a small detachment factor, $k$, but large deposition rate $F$; see Figure \[fig:no\_lkr\_F\]. We observe that for the smallest value of $F$ used in these plots \[Figure \[fig:no\_lkr\_F\](a)\], the flux computed via KMC simulations agrees reasonably well with linear kinetic law for a wide range of values for the supersaturation, $\sigma_+$. For larger values of $F$ \[Figures \[fig:no\_lkr\_F\](b), (c)\], the flux remains linear in the density with a slope equal to the predicted value, $D\phi_+ a$, as the density approaches its equilibrium value. However, as the supersaturation increases in magnitude, the nonlinear dependence of the flux is noticeable and becomes more pronounced with increasing varied parameter, $F$. It is worth noting that an increase in the deposition rate $F$ used in KMC simulations beyond the one used in Figure \[fig:no\_lkr\_F\](c), even by a factor of two, drastically alters the long-time behavior of the system: apparently, no steady state can be established for sufficiently large $F$.
$\begin{array}{ccc}
\includegraphics[width=0.33\textwidth]{FluxVsSigma_HighDep-eps-converted-to.pdf} &
\includegraphics[width=0.34\textwidth]{FluxVsSigma_HighDep2-eps-converted-to.pdf} &
\includegraphics[width=0.33\textwidth]{FluxVsSigma_HighDep4-eps-converted-to.pdf}\\
\mathbf{(a)} & \mathbf{(b)} & \mathbf{(c)}
\end{array}$
As described above, the high-$k$ and high-$F$ cases of Figures \[fig:no\_lkr\_k\] and \[fig:no\_lkr\_F\], respectively, differ in the way that the effect of corrective flux $f_+$ is manifested in the observed value of the flux [*near equilibrium*]{}. Let us make an effort to discuss the origin of this behavior in the context of the atomistic model by resorting to formula (\[eq:f\_+\]). The first line in these formulas contains the prefactor $k$ along with a sum over states with one or more adatoms in the lattice site corresponding to the edge atom. This set of configurations does [*not*]{} allow for atom detachment; thus, according to this contribution to $f_+$, the change of the flux with supersaturation should be suppressed. This prediction should explain the behavior of the slope of the flux versus supersaturation as shown in Figure \[fig:no\_lkr\_k\]. The remaining terms in (\[eq:f\_+\]) come from two- or higher-particle states, which furnish significant contributions if $k$ or $F$ is sufficiently large. These remaining corrections account for configurations in which attachment is inhibited, thus causing an overall increase of the flux out of the step. This prediction is consistent with Figure \[fig:no\_lkr\_F\].
We have been unable to explicitly express the corrective fluxes, $f_\pm(t)$, as a function of adatom densities $c_{\pm 1}(t)$ on the basis of the analytical model. In order to quantify the nonlinear behavior of the flux near the step edge, we fit the fluxes computed by KMC simulations to polynomials of $\sigma_+ = c_{1}/c^{eq}-1$. Figure \[fig:fitted\_flux\] shows the fitted flux in two cases where deviations are significant: High $k$ with small $F$; and high $F$ with small $k$. In each case, a quadratic polynomial of supersaturation appears to capture adequately the behavior of the flux versus supersaturation.
$\begin{array}{cc}
\includegraphics[width=0.4\textwidth]{FluxVsSigma_highkappa_quadratic_fit-eps-converted-to.pdf} &
\includegraphics[width=0.4\textwidth]{FluxVsSigma_highdep4_quadratic_fit-eps-converted-to.pdf}\\
\mathbf{(a)} & \mathbf{(b)}
\end{array}$
We conclude that a linear kinetic relation for the adatom flux at the step edge in principle does [*not*]{} suffice to capture the full range of phenomena displayed by the atomistic solid-on-solid model. Instead, it is more reasonable to propose a discrete expansion of the form $$\label{eq:nlkr_discrete}
\mp J_\pm \approx c^{eq}\sum_{n=1}^{N_*} B_\pm^{(n)} \sigma_\pm^n~,$$ where the number, $N_*$, would be speculated empirically. At the mesoscale, the corresponding [*nonlinear kinetic relation*]{} for flux at the step edge is provided by (\[nlkr\_bcf\]). A systematic derivation of this relation from the atomistic model is still elusive.
[**Remark 7.**]{} Based on our KMC results, we expect that reasonably reduces to conventional linear kinetic relation (\[eq:lkr\_bcf\]) of the BCF model if $$\label{eq:cond-micro}
\mathcal A:=k+\frac{\epsilon}{\phi_++\phi_-}\ll 1~.$$ This empirical criterion appears less restrictive on $\epsilon$ than estimates (\[eq:f\_pm\_estimate\]) and (\[eq:R\_j\_estimate\]), suggesting that the bounds in Proposition 2 may be improved; see Remark 6. Accordingly, if $\mathcal A$ is large enough, then (nonlinear) terms with $n\ge 2$ should become significant.
In our KMC simulations, we observe that the linear kinetic relation for the adatom flux is reasonably accurate if the quantity $\mathcal A$ of (\[eq:cond-micro\]) does not exceed $0.01$.
Discussion and conclusion {#sec:Discussion}
=========================
Starting with an idealized atomistic solid-on-solid-type model in 1D, we studied the mesoscale description of the kinetics of a single step with some emphasis on the relation between the adatom flux and density at the step edge. Our approach relied on a combination of a master equation for adatom states, in the context of an analytical model, and KMC simulations in 1D. A noteworthy result was our heuristic derivation from the master equation of exact formulas for the adatom flux; these indicate the physical origin, in terms of atomistic transitions, of corrections to the linear kinetic relation of the BCF model. Furthermore, by using a “maximum principle” inherent to the master equation, we estimated the aforementioned corrections for small lattice spacing. By KMC simulations, we observed deviations of the behavior of the flux from the conventional (linear-in-density) prediction of the BCF model.
The master equation approach in this paper forms a nontrivial extension of the analytical model invoked in [@PatroneMargetis14; @PatroneEinsteinMargetis14]. By including material deposition onto the surface from above, we accounted for adatom states that do not conserve the total mass of the system. We formally showed that the system has a steady state only for sufficiently small external deposition flux.
Our atomistic model, despite its inability to include truly 2D effects, captures a few basic elements of diffusion processes on crystal surfaces below the roughening transition. In particular, our model describes hopping of adatoms; and, most importantly, attachment/detachment of atoms at the step in the absence of kinks. The 1D character of the model, however, poses a few severe limitations. For example, nucleation cannot be included in our analysis. Because of such limitations, more work is needed in order to connect atomistic processes to the wealth of realistic phenomena accompanying crystal evolution below the roughening transition.
A possible criticism of our approach concerns our analysis about the structure of the corrective fluxes, $f_\pm$. This structure appears to be specific to the 1D character of our model. Admittedly, in our approach these correction terms are only associated with rules for attachment and detachment of atoms at the step from/to the terrace. Thus, we leave out the effect of kinks, which is expected to partially alter the mass flux since kink sites can act as local sources or sinks for atoms [@Caflischetal_99]. Because of kinks, the local curvature of the step is expected to affect the equilibrium adatom density in a 2D mesoscale setting, giving rise to the step “stiffness” [@deGennes_68; @Einstein_03]. To derive this effect from a fully atomistic model remains an open problem [@MargetisCaflisch_08]. In spite of these complications, it is reasonable to expect that our 1D model captures features of 2D step motion if kinks are sufficiently far apart from each other, that is, if the kink density is small.
In this vein, it is natural to ask: Would it be possible to physically improve our one-step atomistic model by retaining its 1D character yet enriching it with effects that become significant for low [*and*]{} high supersaturations? A possibility is to account for pair correlations of adatoms due to their energetic interactions. The next stage in this direction would be to consider two steps by including entropic and elastic step-step interactions in the modeling. This task requires appropriate discretization of elastic effects on the lattice [@Saitoetal_01; @Baskaranetal_15]. Another scenario that could be examined in the 1D setting is the process of step permeability, which thus far is modeled phenomenologically at the mesoscale [@OzdemirZangwill_92]. Since our model does not describe processes by which adatoms may pass from one side of the step to the other without attaching to the step edge, it seems that permeability is not inherent in our treatment. It is possible that permeability emerges from elastic effects since these induce long-range correlations.
In this work, we focused on averages for the adatom density, step position, and adatom flux, motivated by the known structure of the BCF model. Hence, we have not addressed the stochastic fluctuations arising in step motion. These fluctuations are often significant [@JeongWilliams99]. We believe that this stochastic effect should be more pronounced at high enough supersaturations. The step fluctuations are known to be intimately connected to step stiffness [@Einstein_03]; thus, a 2D atomistic model would be a natural starting point for their systematic in-depth study.
Acknowledgments {#acknowledgments .unnumbered}
===============
The authors wish to thank Professors T. L. Einstein, R. V. Kohn, E. Lubetzky and J. D. Weeks for illuminating discussions, as well as T. J. Burns and W. F. Mitchell for helpful reviews of this manuscript. The research of the first author (JPS) and third author (DM) was supported by NSF DMS-1412769 at the University of Maryland.
On birth-death processes {#app:BirthDeath}
========================
In this appendix, we use a toy model for the birth-death Markov process (see, for example, [@Kelly2011]) to indicate that no steady state of the master equation exists for large enough external deposition flux, $F$.
A birth-death process is a Markov process with infinitely many states, labeled by $n=0,\,1,\,\dots$, for which transitions are only allowed between the $n$-th and $(n+1)$-th states. Our discrete KRSOS model has this structure if we classify configurations $\boldsymbol{\alpha}$ by the number of adatoms on the terrace, i.e., by the cardinality $|\boldsymbol{\alpha}|=n$ for $n=0,\,1,\,\dots$. For our purposes, the detachment and deposition-from-above events correspond to births while attachment events correspond to deaths.
$\begin{array}{c}
\includegraphics[width=\textwidth]{Birth_death.pdf}
\end{array}$
Let $\gamma_n$ denote the rate for the transition from the $n$-particle state to the $(n+1)$-particle state (the process of “birth”) and $\theta_n$ denote the rate for the transition from the $n$-particle state to the $(n-1)$-particle state (“death”); see Figure \[fig:birth\_death\]. If $p_n^{ss}$ is the steady-state probability of the $n$-particle configuration, we have the following balance equations. $$\begin{aligned}
\label{eq:birth_death}
\gamma_n p_n^{ss} &= \theta_{n+1} p_{n+1}^{ss}~, \notag \\
\gamma_0 p_0^{ss} &= \theta_{1} p_{1}^{ss}~.\end{aligned}$$ Solving (\[eq:birth\_death\]) for $p_n^{ss}$, we find the formula $$\label{eq:p_n_eq}
p_n^{ss} = \frac{\gamma_{n}\gamma_{n-1}\cdots\gamma_{0}}{\theta_{n+1}\theta_{n}\cdots\theta_{1}} p_{0}^{ss},$$ which of course must satisfy the normalization constraint $\sum\limits_{n=0}^\infty p_n^{ss} = 1$. This constraint can be written as $$\label{eq:norm_condition}
p_0^{ss}\sum_{n=0}^\infty \frac{\gamma_{n}\gamma_{n-1}\cdots\gamma_{0}}{\theta_{n+1}\theta_{n}\cdots\theta_{1}} < \infty~.$$ In other words, if condition (\[eq:norm\_condition\]) does not hold, the probabilities in are not normalizable and, thus, no steady state exists.
In our setting of epitaxial growth in 1D, the rates $\gamma_n$ and $\theta_n$ are determined by the transition rates for attachment, detachment and deposition. These rates satisfy the following equations. $$\begin{aligned}
\label{eq:births}
\gamma_n &= \frac{1}{p_n^{ss}}\sum\limits_{\boldsymbol{\alpha} \atop |\boldsymbol{\alpha}|=n} \left\{ F + Dk(\phi_+ + \phi_-)\mathbbold{1}(\nu_{-1}(\boldsymbol{\alpha})=0) \right\}p_{\boldsymbol{\alpha}}^{ss} \notag \\
&= F + Dk(\phi_++\phi_-)a_n~,\end{aligned}$$ $$\begin{aligned}
\label{eq:deaths}
\theta_n &= \frac{1}{p_n^{ss}}\sum\limits_{\boldsymbol{\alpha} \atop |\boldsymbol{\alpha}|=n} \left\{ D\phi_+\mathbbold{1}\left(\nu_1(\boldsymbol{\alpha})=1\right) + D\phi_-\mathbbold{1}\left(\nu_1(\boldsymbol{\alpha})=0\right) \right.\notag\\
&\quad \times \left. \mathbbold{1}\left(\nu_1(\boldsymbol{\alpha})>0\right) \right\}p_{\boldsymbol{\alpha}}^{ss} \notag \\
&= D\phi_+b_n + D\phi_-d_n~,\end{aligned}$$ where $a_n$, $b_n$ and $d_n$ are the probabilities of $n$-particle configurations that forbid detachment, attachment from the right, and attachment from the left, respectively. By (\[eq:births\]) and (\[eq:deaths\]), the ratio of birth and death rates is bounded below, i.e., we have the inequality $$\label{eq:bd_lower_bound}
\frac{\gamma_n}{\theta_{n+1}} \geq \frac{F}{D(\phi_+ + \phi_-)}\quad (n=0,\,1,\,\ldots)~.$$ We deduce that the condition $F>D(\phi_+ + \phi_-)$ implies that no steady state may exist because normalization condition (\[eq:norm\_condition\]) is violated. In the context of the 1D epitaxial system, having $F>D(\phi_+ + \phi_-)$ means that the deposition rate is faster than the attachment rate. In this case, the number of particles on the terrace constantly grows and, hence, no steady state can be established.
On steady-state solution: Asymptotics of inverse Laplace transform {#app:Asymptotics}
==================================================================
In this appendix, we derive formula (\[eq:p\_ss\]) for the steady-state probability density. Our derivation relies on the following ideas. (i) We assume that $\epsilon=F/D$ is sufficiently small so that $[I-\epsilon V\mathfrak D(s)V^{-1}\mathfrak B]^{-1}$ exists and we can write this matrix as $\sum_{n=0}^\infty [\epsilon V\mathfrak D(s)V^{-1}\mathfrak B]^n$; and (ii) the quantity $\mathbf{p}^{ss}$, which is the limit of $\mathbf{p}(t)$ as $t \to\infty$, comes from contributions to the inverse Laplace transform of (\[eq:laplace\_trans\_p\]) corresponding to the pole at $s=\lambda_1=0$ in the Laplace complex ($s$-) domain. Recall that $\mathfrak D(s) := \mbox{diag}\{(s-D\lambda_j)^{-1}\}_{j=1}^{\Omega(M)}$, where $\lambda_j$ are the eigenvalues of matrix $\mathfrak A$ of the deposition-free problem. For a review of basic techniques in computing inverse Laplace transforms, which we do not elaborate on here, see [@Schiff2013].
Let us now elaborate on these ideas. From (i) we may write $\mathbf{p}(t)$ as $$\begin{aligned}
\label{eq:p_laplace_integral}
\mathbf{p}(t) &= \frac{1}{2\pi i} \int_{\gamma-i\infty}^{\gamma+i\infty} \sum\limits_{n=0}^\infty \left(\epsilon V\mathfrak D(s)V^{-1}\mathfrak B\right)^n V\mathfrak D(s)V^{-1}\mathbf{p}(0)e^{st}ds\notag \\
&= \frac{1}{2\pi i} \sum\limits_{n=0}^\infty \epsilon^n V \mathcal{I}_n(t) V^{-1}\mathbf{p}(0)~,\end{aligned}$$ where $\gamma$ is a positive constant and the integrals $\mathcal{I}_n(t)$ are defined by $$\label{eq:complex_integral}
\mathcal{I}_n(t) = \int_{\gamma-i\infty}^{\gamma+i\infty} \left(\mathfrak D(s)\tilde{\mathfrak B}\right)^n \mathfrak D(s)\,e^{st}ds~;\quad \tilde{\mathfrak B}:= V^{-1}\mathfrak B V~.$$
The integrand in (\[eq:complex\_integral\]) is a matrix resulting from the product $(\mathfrak D\tilde{\mathfrak B})^n \mathfrak D$, whose entries have the form $$\label{eq:complex_integrand}
\left[(\mathfrak D\tilde{\mathfrak B})^n \mathfrak D\right]_{ij} = \frac{ \tilde{b}_{ik_1}\tilde{b}_{k_1k_2}\cdots\tilde{b}_{k_{n-2}k_{n-1}}\tilde{b}_{k_{n-1}j} }{ (s-\lambda_i)(s-\lambda_{k_1})\cdots(s-\lambda_{k_{n-1}})(s-\lambda_j) }~;$$ $\tilde b_{kl}$ are appropriate coefficients independent of the Laplace variable, $s$.
By (ii) above, since we seek the steady-state solution $\mathbf{p}^{ss}$, the main contribution to the integral comes from the pole at $s=\lambda_1=0$. The only terms which include $(s-\lambda_1)^{-1}$ are those with $j=1$ in (\[eq:complex\_integrand\]). This conclusion can be reached after simplification in the algebra, which can be described as follows. The rows of the matrix $V^{-1}$ contain the left-eigenvectors of matrix $\mathfrak A$. Because of conservation of probability, by which the column sums of $\mathfrak A$ and $\mathfrak B$ are zero, we must have $\left(V^{-1}\right)_{1i} = \left(V^{-1}\right)_{1j}$ for all $i,j$. The matrix $V^{-1}\mathfrak B$, which forms the left matrix product in the $\tilde{\mathfrak B}$ defined in , must obey $$\label{eq:b_property}
\sum_i \left(V^{-1}\right)_{1i} \left(\mathfrak B\right)_{ij} = 0 \quad \mbox{for all } j.$$ Hence, the right-hand side of (\[eq:complex\_integrand\]) vanishes whenever any of the indices $i,k_1,k_2,\dots k_{n-1}$ is equal to unity. Consequently, the steady-state contributions to integral (\[eq:complex\_integral\]) come only from simple poles at $s=\lambda_1$, specifically the terms in (\[eq:complex\_integrand\]) for which $j=1$.
If $n=0$, integral (\[eq:complex\_integral\]) equals $e^{Dt\Lambda}$, the inverse transform of $\mathfrak D(s)$. If $n>0$ an asymptotic expansion for $\mathcal{I}_n(t)$ as $t\to\infty$ may be computed using the residue theorem, viz., $$\begin{aligned}
\label{eq:complex_integral_computation}
\left[\mathcal{I}_n(t)\right]_{ij}
&= \int_{\gamma-i\infty}^{\gamma+i\infty} \left[(\mathfrak D\tilde{\mathfrak B})^n \mathfrak D\right]_{ij}e^{st}ds \notag \\
&= \int_{\gamma-i\infty}^{\gamma+i\infty} \frac{ \tilde{b}_{ik_1}\tilde{b}_{k_1k_2}\cdots\tilde{b}_{k_{n-2}k_{n-1}}\tilde{b}_{k_{n-1}j} }{ (s-\lambda_i)(s-\lambda_{k_1})\cdots(s-\lambda_{k_{n-1}})(s-\lambda_j) }e^{st}ds \notag \\
&\approx 2\pi i \sum_{i,\,k_1,\,k_2,\,\dots,\, k_{n-1}\neq1} \frac{ \tilde{b}_{ik_1}\tilde{b}_{k_1k_2}\cdots\tilde{b}_{k_{n-2}k_{n-1}}\tilde{b}_{k_{n-1}1} }{ (-\lambda_i)(-\lambda_{k_1})\cdots(-\lambda_{k_{n-1}}) } \notag \\
&= \left\{ \begin{array}{ll}
2\pi i \left[\left( -\Lambda^\dagger\tilde{\mathfrak B} \right)^n\right]_{i1}~, & j=1~, \\
0~, & j>1~,
\end{array} \right. \end{aligned}$$ as $t\to\infty$. In the above calculation, the symbol $\approx$ implies that the respective result of contour integration leaves out contributions from poles other than $s=\lambda_1$ in the limit $t\to\infty$. For the same reason, all entries in the matrix $\mathcal{I}_n(t)$ other than the first column are asymptotically small and are neglected.
Finally, by substitution of asymptotic formula into (\[eq:p\_laplace\_integral\]), we compute the steady-state probability distribution as $$\begin{aligned}
\label{eq:p_ss_asymp}
\mathbf{p}^{ss,\epsilon} &= \sum_{n=0}^\infty \epsilon^n V\left( -\Lambda^\dagger\tilde{\mathfrak B} \right)^n [\mathbf{e}_1,\mathbf{0},\mathbf{0},\dots,\mathbf{0}] V^{-1}\mathbf{p}(0) \notag \\
&= \sum_{n=0}^\infty \left( -\epsilon \mathfrak A^\dagger \mathfrak B \right)^n \mathbf{p}^0~,\end{aligned}$$ which leads to . In the above, we used the definition $\tilde{\mathfrak B} = V^{-1}\mathfrak BV$ along with $\mathfrak A^\dagger = V\Lambda^\dagger V^{-1}$. Equation (\[eq:p\_ss\_asymp\]) is written in terms of the equilibrium distribution, $\mathbf{p}^0$, which satisfies (\[eq:truncated\_master\_equation\]) when $\epsilon = 0$. This distribution can be derived in the same way as (\[eq:p\_eq\]), or computed as $\mathbf{p}^0 = \lim_{t\to\infty} \exp(D\mathfrak{A}t)\mathbf{p}(0) = V[\mathbf{e}_1,\mathbf{0},\mathbf{0},\dots,\mathbf{0}]V^{-1}\mathbf{p}(0)$, where $\mathbf{e}_1$ is the $\Omega(M)$-dimensional vector $\mathbf{e}_1 = (1,0,0,\dots,0)^{T}$.
On the extraction of advection from a microscopic average {#app:Advection}
=========================================================
In this appendix, we develop a plausibility argument for the extraction of the continuum-scale advection term, $-v \partial_{\hat{x}}\mathcal C(\hat{x},t)$, which enters diffusion equation (\[eq:diffusion\_eq\_bcf\]), from the atomistic model ($\partial_x=\partial/\partial x$). The derivation of estimates for corrections entering our formula lie beyond our scope. Our argument provides a heuristic reconciliation of continuum-scale advection with the atomistic and probabilistic perspectives of the master equation approach followed in our work. We will invoke the notation $v=\dot \varsigma$ for the average step velocity; recall that $\varsigma=\varsigma(t)$ is the average step position.
Consider the Eulerian adatom density of (\[eq:rho\_j\]), Definition 5. First, note that the corresponding sum can be conveniently rewritten as $$\label{eq:rho-j-alt}
\rho_j(t)=\sum_{n\in \mathbb{Z}}\;\sum_{(\boldsymbol{\alpha},m)\in \mathfrak{S}(n)}\nu_{j-s_0+n}({\boldsymbol\alpha})\,p(\boldsymbol{\alpha},m; t)/a~,$$ where $|n|$ counts the total number of adatoms detached from ($n>0$) or attached to ($n<0$) the step edge and, thus, determines the microscopic position, $s_0-n$, of the step on the lattice; $p(\boldsymbol{\alpha},m; t):=p_{\boldsymbol{\alpha},m}(t)$ for notational convenience; and $\mathfrak{S}(n):=\{(\boldsymbol{\alpha},m)\,\big|\,|\boldsymbol{\alpha}|=n+(m-m_0)\}$, the set of all allowed values of $(\boldsymbol{\alpha},m)$ for fixed $n$.
In order to extract the advection term sufficiently away from the step edge, we take into account the decomposition of $p(\boldsymbol{\alpha},m; t)$ into products of the form $p(\boldsymbol{\alpha},m \big| n ;t)\,\wp(n;t)$. In this product, $\wp(n;t)$ is the probability that the microscopic step lies at the lattice site $s_0-n$ at time $t$, and $p(\boldsymbol{\alpha},m\big| n; t)$ is the conditional probability for state $(\boldsymbol{\alpha},m)$ to occur [*given*]{} that the step edge is at site $s_0-n$. Hence, is recast to the formula $$\label{eq:rho-j-cond}
\rho_j(t)=\sum_n \wp(n;t)\sum_{(\boldsymbol{\alpha},m)\in\mathfrak{S}(n)}\nu_{j-s_0+n}({\boldsymbol\alpha})\,p(\boldsymbol{\alpha},m\big| n;t)/a~,$$ for fixed $j$. Clearly, the right-hand side of becomes the discrete Lagrangian density $c_{\hat{\jmath}}(t)$ if $j-s_0+n$ under the summation sign is replaced by the index $\hat{\jmath}$; cf. (26a) in Definition 5.
At this stage, by inspection of , we define $$c(\hat{\jmath} \big| \aleph;t):=\sum_{(\boldsymbol{\alpha},m)\in\mathfrak{S}(\aleph)}\nu_{j-s_0+\aleph}({\boldsymbol\alpha})\,p(\boldsymbol{\alpha},m\big| \aleph;t)/a~.$$ This formula expresses the (conditional) Lagrangian adatom density at fixed site $\hat{\jmath}$ [*given*]{} that the step position is at site $s_0-\aleph$. Here, $\aleph$ is the discrete [*random variable*]{} with values $n\in\mathbb{Z}$ that represents the number of adatoms detached from the step edge. Accordingly, we compute $$\label{eq:Lagran-time-deriv}
\frac{dc_{\hat{\jmath}}(t)}{dt}=\sum_n \dot{\wp}(n;t) c(\hat{\jmath}\big| n;t)+\langle \partial_t{c}(\hat{\jmath}\big|\aleph;t)\rangle,$$ where $\langle f(\aleph;t)\rangle$ is the expectation of the random variable $f(\aleph;t)$ under the probability distribution $\wp(n;t)$, viz., $\langle f(\aleph;t)\rangle:=\sum_n\wp(n;t) f(n;t)$, with $f(~\cdot~;t)=\partial_tc(\hat{\jmath}\big| \,\cdot\, ;t)$.
Next, we show that plausibly generates a discrete version of the anticipated advection term at long times. For this purpose, we assume that the density of adatoms is sufficiently low, and thereby hypothesize that $\wp(n;t)$ is well approximated by the Poisson distribution with parameter $\varsigma(t)/a$ and $t\gg F^{-1}$; cf. (17) with $\varsigma(t)=aFt$. Hence, we write $\dot{\wp}(n;t) \approx [\dot{\varsigma}(t)/a]\{\wp(n-1;t)-\wp(n;t)\}$, bearing in mind that correction terms neglected in this formula should account for finite times and the effect of higher adatom numbers per site, controlled by $k$ and $F$. By applying summation by parts in the screw-periodic setting of our system we obtain $$\begin{aligned}
\label{eq:adv-discr}
\sum_n \dot{\wp}(n;t) c(\hat{\jmath}\big| n;t)&\approx [\dot{\varsigma}(t)/a]\sum_n \wp(n;t)\{c(\hat{\jmath}\big| n+1;t)-c(\hat{\jmath}\big| n;t)\}\notag\\
&= \dot{\varsigma}(t) a^{-1} \left\{\langle c(\hat{\jmath}\big| \aleph+1;t)\rangle-\langle c(\hat{\jmath}\big| \aleph;t)\rangle \right\}~.\end{aligned}$$ Were it true that $\langle c(\hat{\jmath}\big| \aleph+\ell;t) \rangle \approx \langle c(\hat{\jmath}-\ell\big| \aleph;t) \rangle$ for any integer $\ell$, expressing the translation invariance of the adatom system relative to the step edge, would imply $$\sum_n \dot{\wp}(n;t) c(\hat{\jmath}\big| n;t) \approx -\dot{\varsigma}(t) a^{-1} \left\{\langle c(\hat{\jmath}\big| \aleph;t)\rangle-\langle c(\hat{\jmath}-1\big| \aleph;t)\rangle \right\}~,$$ which approaches $-\dot{\varsigma}~\partial_{\hat{x}}\mathcal C(\hat{x},t)$ as $a\downarrow 0$. We leave it as an open question to what extent this approximation is true.
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---
abstract: 'A statistical multistream description of quantum plasmas is formulated, using the Wigner–Poisson system as dynamical equations. A linear stability analysis of this system is carried out, and it is shown that a Landau-like damping of plane wave perturbations occurs due to the broadening of the background Wigner function that arises as a consequence of statistical variations of the wave function phase. The Landau-like damping is shown to suppress instabilities of the one- and two-stream type.'
author:
- 'Dan Anderson, Björn Hall, Mietek Lisak and Mattias Marklund'
title: Statistical Effects in the Multistream Model for Quantum Plasmas
---
Introduction
============
It has recently been pointed out [@Haas-Manfredi-Feix] that the persistent trend towards increased miniaturization of electronic devices implies that quantum effects will become important also for certain transport processes, for which so far classical models have been sufficient. An example of such a generalized transport equation, in the form the Schrödinger–Poisson equation was analyzed in Ref.[@Haas-Manfredi-Feix]. This analysis of a quantum plasma is based on the hydrodynamic formulation of the Schrödinger–Poisson system, where macroscopic plasma quantities such as density and average velocity are introduced. However, the analysis does not take into account statistical (or kinetic) effects associated with the finite width of the probability distribution function. Kinetic effects are well-known in plasma physics, where they may lead to the phenomenon of Landau damping.
The possibilities of using a general approach based on the Wigner function [@Wigner; @Moyal] was commented upon in Ref.[@Haas-Manfredi-Feix], but only a simpler approach based on macroscopic quantities was used. Obviously, in doing so the possibilities of Landau-damping like effects are lost. In fact, the possibility of obtaining Landau damping is also mentioned in Ref.[@Haas-Manfredi-Feix], although in connection with a possible generalization to the multi-stream case, in accordance with the classical picture of Dawson [@Dawson]. Particular attention was given to the classical one- and two stream instabilities in a cold plasma and it was shown that the main quantum effect on the wave propagation could be characterized as a generalized dispersion.
However, recently much attention, within the nonlinear optics community, has been devoted to effects of partial wave incoherence e.g.in the form of phase noise on a constant amplitude wave [@Hall-etal; @Christodoulides-etal; @Mitchell-etal]. In particular, it has been shown in Ref. [@Hall-etal], where the Wigner transform was introduced as a means to study the modulational instability of an optical plane wave, that the phase noise gives rise to a Landau-like damping effect on the one stream modulational instability. It is the purpose of the present work to generalize the analysis made in Ref. [@Haas-Manfredi-Feix] by analyzing the properties of the one- and two-stream instabilities in a quantum plasma using the Wigner formalism and including the effect of phase noise developed in Ref. [@Hall-etal]. The results clearly show the suppressing effect on the instabilities due to the Landau-like damping effect caused by the phase noise of the Wigner function.
Quantum statistical dynamics
============================
In non-relativistic many-body problems, the Wigner transformation is a useful means to derive equations describing the quantum statistical dynamics of the system of interest. Thus, one is able to generalize the classical Vlasov equation to a quantum mechanical regime, in the sense that the dynamical equation for the Wigner function describes particles moving in a self-consistent force field and in such a way that the evolution equation for the Wigner function takes the form of its classical analogue in the limit $\hbar
\rightarrow 0$.
Haas et al. [@Haas-Manfredi-Feix] have considered the dynamics of a quantum plasma described by the nonlinear Schrödinger–Poisson system of equations:
\[eq:Schrodinger-Poisson\] $$\begin{aligned}
i\hbar\frac{\partial\psi_i}{\partial t} +
\frac{\hbar^2}{2m}\frac{\partial^2\psi_i}{\partial x^2} +
e\phi\psi_i = 0 \ , \label{eq:Schrodinger} \\
\frac{\partial^2\phi}{\partial x^2} = \frac{e}{\varepsilon_0}
\left( \sum_{i = 1}^N \langle|\psi_i|^2\rangle - n_0 \right) \ ,
\label{eq:Poisson} \end{aligned}$$
where $i = 1, ..., N$ numbers the electrons as described by pure states, with $\psi_i$ being the wave function for each such state; $\phi(x,t)$ is the electrostatic potential, while $m$ and $-e$ are the mass and charge of the electrons, respectively. The fixed ion background has the density $n_0$. Following Ref. [@Hall-etal], we have introduced the Klimontovich statistical average, denoting it by $\langle\cdot\rangle$. The statistical averaging becomes important when the wave function contains e.g. a stochastically varying phase [@Hall-etal].
In Ref. [@Haas-Manfredi-Feix], the one-stream and two-stream models have been investigated and the dispersion relation for the two-stream instability was derived, showing an appearance of a new, purely quantum branch. We note that the analysis presented in Ref. [@Haas-Manfredi-Feix] is based on the hydrodynamic formulation of the system (\[eq:Schrodinger-Poisson\]), where macroscopic plasma quantities, such as density and average velocity, are introduced. However, this type of analysis does not take into account statistical properties of the wave function that may lead to a broadening of the probability distribution function. In fact, such effects may give rise to a Landau-like damping both in the case of the single-stream and two-stream instabilities.
In order to take the statistical effects into account, it is convenient to introduce the Wigner distribution function $W_i(x, t; p)$, corresponding to the wave function $\psi_i(x, t)$, as
$$\label{eq:Wigner}
W_i(x, t; p) = \frac{1}{2\pi\hbar} \int_{-\infty}^{+\infty} dy \,
\exp(ipy/\hbar) \langle \psi_i^*(x + y/2, t)\psi_i(x - y/2, t) \rangle\ ,$$
which has the property $$\int_{-\infty}^{+\infty} dp \, W_i(x, t; p) =
\langle|\psi_i(x,t)|^2\rangle \ .$$
Using Eq. (\[eq:Wigner\]), Eq. (\[eq:Schrodinger\]) can be formulated as a kinetic equation for the Wigner distribution, viz the Wigner–Moyal equation $$\label{eq:Wignereq}
\left(\frac{\partial}{\partial t} +
\frac{p}{m}\frac{\partial}{\partial x}\right)W_i +
\frac{2e}{\hbar}\phi\sin\left(
\frac{\hbar}{2}\frac{\stackrel{\leftarrow}{\partial}}{\partial x}
\frac{\stackrel{\rightarrow}{\partial}}{\partial p} \right)W_i = 0 \
,$$ where the sine-operator is defined in terms of its Taylor expansion. Correspondingly, Eq. (\[eq:Poisson\]) can be rewritten as $$\label{eq:Poisson2}
\frac{\partial^2\phi}{\partial x^2} = \frac{e}{\varepsilon_0}\left(
\sum_{i = 1}^{N}\int_{-\infty}^{+\infty} dp \, W_i -n_0 \right) \
.$$ Clearly, an equilibrium solution of Eqs. (\[eq:Wignereq\]) and (\[eq:Poisson\]) is $\phi = 0$ and $W_i = W_{i0}(p)$.
In order to study the modulational stability of the system (\[eq:Wignereq\])–(\[eq:Poisson2\]), we introduce a small perturbation according to
$$\begin{aligned}
W_i(x,t;p) = W_{i0}(p) + \widetilde{W}_i\exp[i(Kx - \Omega t)] \ ,
\\
\phi(x, t) = \tilde{\phi}\exp[i(Kx - \Omega t)] \ , \end{aligned}$$
where $|\widetilde{W}_i| \ll |W_{i0}|$ and $K$ and $\Omega$ are the wave number and frequency of the perturbation, respectively. The fact that the background distribution $W_{i0}$ is assumed to be only a function of $p$ corresponds to the assumption of a plane wave function with constant amplitude, but with a stochastically varying phase, the characteristic properties of which are expressed by $W_{i0}(p)$. Linearizing Eqs.(\[eq:Wignereq\]) and (\[eq:Poisson2\]), we obtain
\[eq:Wigner-Poisson\] $$\begin{aligned}
-i\left(\Omega - \frac{p}{m}K\right)\widetilde{W}_i +
\frac{2e}{\hbar}\tilde{\phi} \sin\left( \frac{i\hbar K}{2}
\frac{\stackrel{\rightarrow}{\partial}}{\partial p} \right) W_{i0}
= 0 \ , \\
-K^2\tilde{\phi} = \frac{e}{\varepsilon_0}\sum_{i = 1}^{N}
\int_{-\infty}^{+\infty} dp \, \widetilde{W}_i \ ,\end{aligned}$$
where $\tilde{\phi}$ is the potential perturbation. Note that the fact that the unperturbed potential $\phi$ is $\phi_0 = 0$ means that $$\sum_{i = 1}^{N}
\int_{-\infty}^{+\infty} dp \, W_{i0} = n_0 \ .$$
Eliminating $\tilde{\phi}$ in Eqs. (\[eq:Wigner-Poisson\]), we obtain the dispersion relation $$\label{eq:disprel}
\frac{2ie^2m}{\varepsilon_0\hbar K^3} \sum_{i = 1}^{N}
\int_{-\infty}^{+\infty} dp \, \frac{1}{p - m\Omega/K}
\sin\left( \frac{i\hbar K}{2}
\frac{\stackrel{\rightarrow}{\partial}}{\partial p} \right)W_{i0} +
1 = 0\ .$$ Using the fact that $$\begin{aligned}
&& 2\sin\left(\frac{i\hbar K}{2}
\frac{\stackrel{\rightarrow}{\partial}}{\partial p}
\right)W_{i0}(p) \nonumber \\
&& \quad\qquad = i\left[ W_{i0}(p + \hbar K/2) - W_{i0}(p - \hbar
K/2) \right] \ ,\end{aligned}$$ relation (\[eq:disprel\]) can be written in the form $$\label{eq:disprel2}
1 = \frac{e^2m}{\varepsilon_0\hbar K^3}\sum_{i = 1}^{N}
\int_{-\infty}^{+\infty} dp \, \frac{W_{i0}(p + \hbar K/2)
- W_{i0}(p - \hbar K/2)}{p - \Omega m/K} \ .$$ Note that the pole $p = \Omega m/K$ gives rise to both a principal part and an imaginary residue, as in the classical analysis of Landau damping in plasma physics.
Let us now consider the cases of one-stream and two-stream plasmas.
One-stream plasma
-----------------
The dispersion relation (\[eq:disprel2\]) reduces to $$\label{eq:onestreamdisp}
1 = \frac{e^2m}{\varepsilon_0\hbar K^3}
\int_{-\infty}^{+\infty} dp \, \frac{W_{0}(p + \hbar K/2)
- W_{0}(p - \hbar K/2)}{p - \Omega m/K} \ ,$$ where $W_0 \equiv W_{10}$. For a one-component Wigner spectrum with a deterministic phase, i.e. a monoenergetic beam, $W_0(p)$ is given by $$W_0(p) = n_0 \delta(p - p_0) \ ,$$ which corresponds to a monochromatic plane wave function with constant amplitude and phase. Equation (\[eq:onestreamdisp\]) then yields $$1 = \frac{n_0e^2m}{\varepsilon_0 K^2}\frac{1}{(p_0 - \Omega m/K)^2 -
\hbar^2K^2/4} \ ,$$ i.e., $$\label{eq:onestreamfluid}
(\Omega - v_0K)^2 = \omega_p^2 + \frac{\hbar^2K^4}{4m^2} \ ,$$ where $v_0 \equiv p_0/m$ and $\omega_p^2 \equiv
n_0e^2/m\varepsilon_0$. The expression (\[eq:onestreamfluid\]) is exactly the same as the one obtained in Ref.[@Haas-Manfredi-Feix]. It shows that quantum effects give rise to wave dispersion for short wave-lengths.
Let us now assume that the phase $\varphi(x)$ of the wave function $\psi_0$ varies stochastically, and that the corresponding correlation function is given by $$\langle e^{-i\varphi(x + y/2)}e^{i\varphi(x - y/2)} \rangle =
e^{-p_T|y|} \ .$$ This corresponds to the Lorentzian spectrum $$W_0(p) = \frac{n_0}{\pi}\frac{p_T}{(p - p_0)^2 + p_T^2} \ ,$$ and the dispersion relation (\[eq:onestreamdisp\]) now yields $$\Omega - \frac{p_0}{m}K = \left( \omega_p^2 + \frac{\hbar^2K^4}{4m^2}
\right)^{1/2} - i\frac{p_T}{m}K \
,$$ This result implies a completely new effect, a Landau-like damping due to the width of the spectral distribution describing the stochastic variation of the phase, i.e. due to the partial incoherence of the beam. Furthermore, the damping effect increases with increasing incoherence, i.e. with increasing $p_T$.
Two-stream plasma
-----------------
According to Eq.(\[eq:disprel2\]), the dispersion relation becomes $$\begin{aligned}
1 &=& \frac{e^2m}{\varepsilon_0\hbar K^3}\int_{-\infty}^{+\infty} dp
\, \left[ \frac{W_{10}(p + \hbar K/2) - W_{10}(p - \hbar K/2)}{p -
\Omega m/K} \right. \nonumber \\
&&\qquad \left. + \frac{W_{20}(p + \hbar K/2) - W_{20}(p - \hbar
K/2)}{p - \Omega m/K} \right ] \ .
\label{eq:twostreamdisp}\end{aligned}$$ For monochromatic beams with $$W_{j0}(p) = n_{0j} \delta(p - p_{0j}) \ ; \ j = 1, 2 \ ,$$ we get from Eq. (\[eq:twostreamdisp\]) $$\begin{aligned}
1 &=& \frac{\omega_{p1}^2}{(\Omega - p_{01}K/m)^2 -
\hbar^2K^4/4m^2} \nonumber \\
&&\qquad + \frac{\omega_{p2}^2}{(\Omega - p_{02}K/m)^2 -
\hbar^2K^4/4m^2} \ ,
\label{eq:twostreamfluid} \end{aligned}$$ where $\omega_{pj}^2 = e^2n_{0j}/\varepsilon_0m$ and $n_{01} + n_{02}
= n_0$. If we follow Ref. [@Haas-Manfredi-Feix] and consider the symmetric case where $n_{01}
= n_{02} = n_0/2$, $p_{01} = -p_{02} \equiv p_0$, we obtain $$\begin{aligned}
&& \bar{\Omega}^4 - \left( 1 + 2\bar{K}^2 + \frac{H^2\bar{K}^4}{2}
\right)\bar{\Omega}^2 \nonumber \\
&&\, - \bar{K}^2\left( 1 -
\frac{H^2\bar{K}^2}{4}\right) \left( 1 - \bar{K}^2 +
\frac{H^2\bar{K}^4}{4}\right) = 0
\label{eq:twostreamfluid2}\end{aligned}$$ from Eq. (\[eq:twostreamfluid\]). Here we have introduced dimensionless variables according to $$\label{eq:dimlessvar}
\bar{\Omega} = \Omega/\omega_{p0} \ , \quad \bar{K} =
p_0K/\omega_{p0}m \ , \quad H = \hbar\omega_{p0}m/p_0^2 \ .$$ Equation (\[eq:twostreamfluid2\]) is identical to the result obtained from the hydrodynamical theory, as in Ref. [@Haas-Manfredi-Feix]. The solution of Eq.(\[eq:twostreamfluid2\]) is $$\label{eq:constraint1}
\bar{\Omega}^2 = \frac{1}{2} + \bar{K}^2 + \frac{H^2\bar{K}^4}{4}
\pm \frac{1}{2}\sqrt{1 + 8\bar{K}^2 + 4H^2\bar{K}^6} \ ,$$ which implies $\bar{\Omega}^2 < 0$ and concomitant instability if $$\label{eq:constraint2}
(H^2\bar{K}^2 - 4)(H^2\bar{K}^4 - 4\bar{K}^2 + 4) < 0 \ .$$ This condition can be written $$1 - \frac{1}{\bar{K}^2} < \frac{H^2\bar{K}^2}{4} < 1 \ ,$$ which reduces to the well-known two-stream instability result $K^2 < 1$ in the classical limit $H \rightarrow 0$.
However, we infer from Eq. (\[eq:constraint2\]) that the quantum effect has a subtle influence on the instability. Equation (\[eq:constraint2\]) implies instability when the following condition is satisfied in $(\bar{K}, H)$ space, viz $$H_-^2(\bar{K}) \equiv \frac{4}{\bar{K}^2}\left( 1 -
\frac{1}{\bar{K}^2} \right) < H^2
< \frac{4}{\bar{K}^2} \equiv H_+^2(\bar{K}) \ .$$ A qualitative plot of this is given in Fig. \[fig1\] (a similar figure and discussion was given in Ref. [@Haas-Manfredi-Feix], but for later reference we present the figure and a discussion related to it).
Figure \[fig1\] implies that when $H=0$, instability occurs only for $0 <
\bar{K} < 1$. However, when $H \neq 0$, a more complicated picture emerges. In fact, as is seen from Fig. \[fig1\], the quantum effect plays both a stabilizing and a destabilizing role. For $H > 1$, instability occurs for all $\bar{K}$ such that $0 \leq \bar{K} \leq K_+(H) \equiv
2/H$. Thus, for $1 \leq H \leq 2$, the region of instability is increased, whereas for $H \geq 2$ it is decreased as compared to the case $H = 0$.
For $H < 1$, instability occurs in two $K$-bands, viz $0 \leq \bar{K}
\leq K_-^{(1)}(H)$ and $K_-^{(2)}(H) \leq \bar{K} \leq K_+(H)$, where $K_-^{(1,2)}(H)$ are the two solutions of the equation $1 -
1/\bar{K}^2 = H^2\bar{K}^2/4$, i.e.
$$\begin{aligned}
K_-^{(1)}(H) = \frac{2}{H^2}\left( 1 + \sqrt{1 - H^2} \right) \, ,
\\
K_-^{(2)}(H) = \frac{2}{H^2}\left( 1 - \sqrt{1 - H^2} \right) \, .\end{aligned}$$
For all values of $H < 1$, this implies a larger range of unstable wave numbers as compared to the classical case $H = 0$.
Let us now assume that the unperturbed Wigner distributions have Lorentzian form, in analogy to the case of a one-stream plasma, i.e. $$W_{j0}(p) = \frac{n_{0j}}{\pi}\frac{p_{Tj}}{(p - p_{0j})^2 +
p_{Tj}^2} \ ; \ j = 1, 2 \ .$$ From Eq. (\[eq:twostreamdisp\]) we then obtain $$\begin{aligned}
1 &=& \frac{\omega_{p1}^2}{[\Omega - (p_{01} - ip_{T1})K/m]^2 -
\hbar^2K^4/4m^2} \nonumber \\
&& + \frac{\omega_{p2}^2}{[\Omega - (p_{02} -
ip_{T2})K/m]^2 - \hbar^2K^4/4m^2} \ .\end{aligned}$$ Following Ref. [@Haas-Manfredi-Feix], we consider the case when $p_{01} = -p_{02} \equiv p_0$ and $n_{01} = n_{02} = n_0/2$, while for the statistical broadening we assume $p_{T1} = p_{T2} \equiv p_T$. Using the dimensionless variables given by (\[eq:dimlessvar\]), we get $$\label{eq:special}
(\bar{\Omega} + i\alpha\bar{K})^2 =
\frac{1}{2} + \bar{K}^2 + \frac{H^2\bar{K}^4}{4}
\pm \frac{1}{2}\sqrt{1 + 8\bar{K}^2 + 4H^2\bar{K}^6} \ ,$$ where we have introduced the relative broadening $\alpha \equiv
p_T/p_0$. Thus, in the limit $p_T \rightarrow 0$, we regain the result of Eq. (\[eq:constraint2\]) and Ref. [@Haas-Manfredi-Feix]. However, in the previously unstable region we now obtain $$\begin{aligned}
{\rm Im}(\bar{\Omega}) &=& -\alpha\bar{K} + \left[ \frac12\left( 1 +
8\bar{K}^2 + 4H^2\bar{K}^6 \right)^{1/2} \right.\nonumber \\
&&\qquad\qquad\quad \left. - \frac{1}{2} - \bar{K}^2 -
\frac{H^2\bar{K}^4}{4} \right]^{1/2} \ .\end{aligned}$$ Again, the broadening $\alpha$ tends to suppress the growth, and the condition ${\rm Im}(\bar{\Omega}) > 0$ is now given by $$\begin{aligned}
\alpha &<& \frac{1}{\bar{K}}%
\left[ \frac12\left( 1 +
8\bar{K}^2 + 4H^2\bar{K}^6 \right)^{1/2} \right. \nonumber \\
&& \qquad\qquad \left. - \frac12 - \bar{K}^2 - \frac{H^2\bar{K}^4}{4} \right]^{1/2}
.\end{aligned}$$
In the classical limit $H \rightarrow 0$, the region of unstable $\bar{K}$-values is reduced to $\bar{K} < K_c$ by the damping effect, where $$K_c = \frac{\sqrt{1 - \alpha^2}}{1 + \alpha^2} < 1 \ .$$ Clearly, for $\alpha \geq 1$, no instability is possible for any $\bar{K}$. Another illustration of this is the small-$\bar{K}$ expansion of the growth rate, which reads $${\rm Im}(\bar{\Omega}) \simeq (1 - \alpha)\bar{K}$$
The stabilizing influence of $\alpha$ in the general case of $H \neq
0$ can be inferred as follows:\
Consider first the case of small $\bar{K}$, while keeping $H^2\bar{K}^2/4 \sim {\mathscr O}(1)$, i.e. we investigate the growth rate close to the stability boundary. In this limit we obtain $${\rm Im}(\bar{\Omega}) \simeq \left( \sqrt{1 - \frac{H^2\bar{K}^2}{4}}
- \alpha \right)\bar{K} \ ,$$ which clearly shows the stabilizing effect of the damping. In particular, the stability threshold is now given by $$H = \frac{2}{\bar{K}}\sqrt{1 - \alpha^2} \ ,$$ Qualitatively this implies a lowering of the upper threshold curve for small $\bar{K}$ and a concomitant decrease of the region of instability.
Consider next the limit $\bar{K} \gg 1$, while still assuming $H^2\bar{K}^2/4 \sim {\mathscr O}(1)$, i.e. we examine the effects of the damping on the narrow instability region, see Fig.\[fig1\]. Introduce the notation $$\Delta h \equiv 1 - \frac{H^2\bar{K}^2}{4} \ .$$ The growth rate can then be written as $${\rm Im}(\bar{\Omega}) \simeq -\alpha\bar{K} + \sqrt{\Delta h\left(
\frac{1}{2} - \bar{K}^2\Delta h \right)} \ ,$$ and the stability thresholds become determined by $$\Delta h = \frac{1}{4\bar{K}^2} \pm \sqrt{\frac{1}{16\bar{K}^4} -
\alpha^2} \ .$$ When $\alpha = 0$, we regain the previous limit curves $\Delta h = 0$ and $\Delta h = 1/(2\bar{K}^2)$. The effect of a nonzero $\alpha$ is to narrow the instability region and to terminate it at the finite wave number $\bar{K} = 1/(2\sqrt{\alpha})$. For increasing $\alpha$, the unstable region decreases and, as in the case of small $\bar{K}$, we expect the instability to be essentially quenched for $\alpha \gtrsim 1$ .
Discussion
==========
In this work, we have presented an analysis of a multi-stream quantum plasma, including the effect of phase noise in the beam wave function. As compared the fluid description of a quantum plasma used in Ref. [@Haas-Manfredi-Feix], the present analysis is based on the quantum mechanical Wigner formalism. The phase noise, or partial incoherence, of the beam wave functions is shown to give rise to a Landau-like damping effect, which tends to suppress the instabilities occurring in both the one- and two beam cases. The damping rate increases with increasing degree of incoherence as expressed by the width of the probability distribution function for the phase noise. The physical origin of this damping effect is the non-coherent properties of the beam wave function as opposed to the wave-particle interaction characteristic of the conventional Landau damping. The new Landau-like effect is not a true wave damping, but a conservative rearrangement of the spectrum of the beam wave function. This phenomenon has recently attracted considerable interest, both theoretically[@Hall-etal; @Christodoulides-etal] and experimentally [@Mitchell-etal], within the area of nonlinear optics, where it has been shown to suppress the modulational and self-focusing instabilities [@Bang-Edmundson-Krolikowski; @Soljacic-etal; @Anastassion-etal], e.g. for optical beams in nonlinear photo-refractive media. The present work is the first attempt to extend this theory to a quantum plasma.
[99]{} F. Haas, G. Manfredi and M. Feix, Phys. Rev. E **62** 2763 (2000).
E. P. Wigner, Phys. Rev. **40**, 749 (1932).
J. E. Moyal, Proc. Cambridge Philos. Soc. **45**, 99 (1949).
J. Dawson, Phys. Fluids **4**, 869 (1961).
B. Hall et al. Statistical Theory for Incoherent Light Propagation in Nonlinear Media, preprint.
D. N. Christodoulides et al., Phys. Rev. Lett. **78**, 646 (1997).
M. Mitchell et al., Phys. Rev. Lett. **77**, 490 (1996).
O. bang, D. Edmundson and W. Królikowski, Phys. Rev. Lett. **83**, 5479 (1999).
M. Soljacic et al., Phys. Rev. Lett. **84**, 467 (2000).
C. Anastassion et al., Phys. Rev. Lett. **85**, 4888 (2000).
|
---
abstract: 'Current 6D object pose estimation methods usually require a 3D model for each object. These methods also require additional training in order to incorporate new objects. As a result, they are difficult to scale to a large number of objects and cannot be directly applied to unseen objects. In this work, we propose a novel framework for 6D pose estimation of unseen objects. We design an end-to-end neural network that reconstructs a latent 3D representation of an object using a small number of reference views of the object. Using the learned 3D representation, the network is able to render the object from arbitrary views. Using this neural renderer, we directly optimize for pose given an input image. By training our network with a large number of 3D shapes for reconstruction and rendering, our network generalizes well to unseen objects. We present a new dataset for unseen object pose estimation–MOPED. We evaluate the performance of our method for unseen object pose estimation on MOPED as well as the ModelNet dataset.'
author:
- |
Keunhong Park\
University of Washington\
Seattle, WA\
[kpar@cs.washington.edu]{}
- |
Arsalan Mousavian\
NVIDIA\
Seattle, WA\
[amousavian@nvidia.com]{}
- |
Yu Xiang\
NVIDIA\
Seattle, WA\
[yux@nvidia.com]{}
- |
Dieter Fox\
NVIDIA\
Seattle, WA\
[dieterf@nvidia.com]{}
bibliography:
- 'references.bib'
title: |
LatentFusion: End-to-End Differentiable Reconstruction and Rendering\
for Unseen Object Pose Estimation
---
|
---
abstract: 'We present a lattice Boltzmann algorithm based on an underlying free energy that allows the simulation of the dynamics of a multicomponent system with an arbitrary number of components. The thermodynamic properties, such as the chemical potential of each component and the pressure of the overall system, are incorporated in the model. We derived a symmetrical convection diffusion equation for each component as well as the Navier Stokes equation and continuity equation for the overall system. The algorithm was verified through simulations of binary and ternary systems. The equilibrium concentrations of components of binary and ternary systems simulated with our algorithm agree well with theoretical expectations.'
author:
- Qun Li
- 'A.J. Wagner'
title: 'A Symmetric Free Energy Based Multi-Component Lattice Boltzmann Method'
---
Introduction
============
Multicomponent systems are of great theoretical and practical importance. An example of an important ternary system, that inspired the current paper, is the formation of polymer membranes through immersion precipitation [@akthakul]. In this process a polymer-solvent mixture is brought in contact with a non-solvent. As the non-solvent diffuses into the mixture, the mixture phase-separates, leaving behind a complex polymer morphology which depends strongly on the processing conditions. The dependence of the morphology on the parameters of the system is as yet poorly understood. Preliminary lattice Boltzmann simulations of this system exist [@akthakul]. However, this work did not recover the correct drift diffusion equation. A general fully consistent lattice Boltzmann algorithm with an underlying free energy to simulate multicomponent systems is still lacking. This paper strives to bring us a step nearer to achieving this goal.
There are several previous lattice Boltzmann methods for the simulation of multi-component systems. There are three main roots for these approaches. There are those derived from the Rothmann-Keller approach [@RK; @Reis] that attempt to maximally phase-separate the different components. A second approach by Shan and Chen is based on mimicking the microscopic interactions [@Shan1; @Shan2; @shanx] and a third approach after Swift, Orlandini and Yeomans [@osborn; @orlandini] is based on an underlying free energy. All of these have different challenges. Since we are interested in the thermodynamics of phase-separation we find it convenient to work with a method based on a free energy. This allows us to easily identify the chemical potentials of the components. This is convenient since the gradients of the chemical potentials drive the phase separation as well as the subsequent phase-ordering.
The challenge for the LB simulation of a multicomponent system lies in the fact that momentum conservation is only valid for the overall system but not for each component separately, and diffusion occurs in the components. For a binary system of components $A$ and $B$ with densities $\rho^A$ and $\rho^B$, the simulation usually traces the evolution of the total density $\rho^A + \rho^B$ and the density difference $\rho^A -
\rho^B$ [@orlandini]. Although this scheme is successful in the simulation of a binary system [@orlandini; @wagner2001], its generalization for the LB simulations of systems with an arbitrary number of components is asymmetric. For instance, to simulate a ternary system of components $A$, $B$, and $C$ with densities $\rho^A$, $\rho^B$ and $\rho^C$, the total density of the system, $\rho^A + \rho^B + \rho^C$, should be traced, and the other two densities to be traced may be chosen as, e.g., $\rho^B$ and $\rho^A - \rho^C$ [@lamura]. This approach is likely to be asymmetric because the three components are treated differently as is the case of Lamura’s model [@lamura]. If an LB method is not symmetric, it will lose generality an will only be adequate for special applications. In this paper, we established a multicomponent lattice Boltzmann method based on an underlying free energy that is manifestly symmetric.
Macroscopic Equations for Multicomponent System
===============================================
The equation of motion for a multicomponent system are given by the continuity and Navier-Stokes equations for the overall system and a drift diffusion equation for each component separately. The continuity equation is given by $$\partial_t \rho + \nabla \cdot {\bf J} = 0 , \label{continuityEQ} \\$$ where $ \rho $ is the mass density of the fluid, ${\bf J}$ is the mass flux which is given by ${\bf J} \equiv \rho \, {\bf u}$, and ${\bf u}$ is the macroscopic velocity of the fluid. The Navier-Stokes equation describes the conservation of momentum: $$\partial_t(\rho u_\alpha) + \partial_\beta(\rho u_\alpha u_\beta)
= -\partial_\beta P_{\alpha \beta}+ \partial_\beta \sigma_{\alpha \beta} + \rho F_{\alpha} , \label{NSEeq}$$ where $P_{\alpha\beta}$ and $\sigma_{\alpha \beta}$ are the pressure and viscous stress tensors respectively, ${F_\alpha}$ is the component $\alpha$ of an external force on a unit mass in a unit volume, and the Einstein summation convention is used. For Newtonian fluids, the viscous stress tensor is given by $$\sigma_{\alpha \beta} = \eta \left( \partial_\beta u_\alpha + \partial_\alpha u_\beta - \frac{d}{2}\delta_{\alpha \beta}
\nabla \cdot {\mathbf u} \right) + \mu_B \delta_{\alpha \beta} \nabla \cdot {\mathbf u},
\label{stresstensor}$$ where $\eta$ is the shear viscosity, and $\mu_B$ is bulk viscosity, and $d$ is the spacial dimension of the system.
Free energy, chemical potential, and pressure are key thermodynamic concepts to understand the phase behavior of a system. The chemical potential of each component can be obtained by a functional derivative as $$\mu^{\sigma} = \frac{\delta {\mathcal F}}{\delta n^\sigma} \label{chemicalpotential},$$ where $\mu^\sigma$ is the chemical potential of component $\sigma$; $n^\sigma$ is the number density of component $\sigma$; and $\mathcal F$ is the total free energy of the system.
The pressure in a bulk phase in equilibrium is given by $$p = \sum_\sigma n^\sigma \mu^\sigma - \psi . \label{ppp}$$ The pressure tensor is determined by two constraints: $ P_{\alpha \beta} = p
\delta_{\alpha \beta}$ in the bulk and $\Delta P_{\alpha \beta} = \sum_\sigma n^\sigma \nabla \mu^\sigma$ everywhere.
In multicomponent systems, there are two mechanisms for mass transport: convection and diffusion. Convection is the flow of the overall fluid, while diffusion occurs where the average velocities of components are different. The velocity of the overall fluid is a macroscopic quantity because it is conserved, but the average velocities of the components are not. The macroscopic velocity of the fluid $\mathbf{u}$ can be expressed in terms of the density $\rho^\sigma$ and velocity $u^\sigma$ of each component in the form of $${\bf u} \equiv \frac{ \sum_\sigma \rho^\sigma {\bf u}^\sigma}{ \sum_\sigma \rho^\sigma}. \label{uaverage}$$ With the notation $$\Delta {\bf u}^\sigma \equiv {\bf u}^\sigma - {\bf u}, \label{dsl}$$ the flux of each component can be divided into a convection part ${\bf
J}^{\sigma c}$ and a diffusion part ${\bf J}^{\sigma d}$: $${\bf J}^\sigma \equiv \rho^\sigma {\bf u}^\sigma = \rho^\sigma ( {\bf u} + \Delta {\bf u}^\sigma )
= {\bf J}^{\sigma c} + {\bf J}^{\sigma d}. \label{jjj}$$ Because mass conservation still holds for each component, the continuity equation for each component is valid: $$\partial_t \rho^\sigma + \nabla \cdot {\bf J}^\sigma = 0 . \label{onecon}$$ Substituting Eq. (\[jjj\]) into Eq. (\[onecon\]), the convection diffusion equation for a component can be obtained. $$\partial_t \rho^\sigma + \nabla \cdot {\bf J}^{\sigma c} = - \nabla \cdot {\bf J}^{\sigma d} . \label{condieq}$$ From Eqs. (\[uaverage\]) and (\[dsl\]), we see that $$\sum_\sigma {\bf J}^{\sigma d} = 0, \label{abab}$$ which ensures the recovery of the continuity equation for the overall system. The diffusion process between two components is related to the difference of the chemical potential of the two components, which is also called the exchange chemical potential [@jones]. Recognizing that the gradient of the exchange chemical potential determines the diffusion processes, we obtain a first order approximation for the diffusion flux of one component into all other components as $${\bf J}^{\sigma d} = - \sum_{\sigma '} M^{\sigma \sigma '} \nabla ( \mu^\sigma - \mu^{\sigma'}) ,
\label{akak}$$ where $\sigma$ and $\sigma '$ enumerate the components; $\mu^\sigma$ and $\mu^{\sigma '}$ are the chemical potentials of components $\sigma$ and $\sigma '$; and $M^{\sigma \sigma '}$ is a symmetric positive definite mobility tensor.
A simple model for the diffusion process assumes that a diffusion flux between two components is proportional to the overall density and the concentration of each component. Then mobility tensor can be expressed as $$M^{\sigma \sigma '} = k^{\sigma \sigma '} \frac{\rho^\sigma \rho^{\sigma '}}{\rho} ,
\label{mss}$$ where $k^{\sigma \sigma '}$ is the constant diffusion coefficient between components $\sigma$ and $\sigma '$. It depends on components but is independent of the total densities and concentration of each component. Substituting Eq. (\[mss\]) into Eq. (\[akak\]), we have $${\bf J}^{\sigma d} = - \sum_{\sigma '} k^{\sigma \sigma '} \frac{\rho^\sigma \rho^{\sigma '}}{\rho} \nabla ( \mu^\sigma - \mu^{\sigma'}).
\label{bfjj}$$ Substituting Eq. (\[bfjj\]) into Eq. (\[condieq\]), the general form of a convection diffusion equation is obtained as $$\partial_t \rho^\sigma + \nabla (\rho^\sigma {\bf u}) =
\nabla \sum_{\sigma '} k^{\sigma \sigma '} \frac{\rho^\sigma \rho^{\sigma '}}{\rho}
\nabla ( \mu^\sigma - \mu^{\sigma '}) .
\label{gfcd}$$
Lattice Boltzmann for Multicomponent System
===========================================
To simulate a multicomponent fluid using LB we set up a LB equation for each component. The LBE for a component $\sigma$ of a multicomponent system is given by $$\begin{aligned}
&& f^\sigma_i ({\bf r} + {\bf v_i} \Delta t , t + \Delta t) - f^\sigma_i ({\bf r}, t)\nonumber \\
&=& \Delta t \left[ \frac{1}{\tau}
\Big(f^{\sigma e}_i ({\bf r}, t) - f^{\sigma}_i ({\bf r},t) \Big) + F^\sigma_i \right], \label{ssLB}\end{aligned}$$ where $f^\sigma_i ({\bf r},t )$ is the particle distribution function with velocity ${\bf v}_i$ for component $\sigma$, $f^{\sigma e}({\bf r},t)$ is its equilibrium distribution and $F^\sigma_i$ is the forcing term of component $\sigma$ due to the mean potential field generated by the interaction of the component $\sigma$ with the other components. The main task in setting up this lattice Boltzmann method is to determine the correct form of the forcing term $F^\sigma_i$ which will recover the convection diffusion equation (\[gfcd\]).
The density of each component and the total density are given by $$\begin{aligned}
\rho^\sigma &=& \sum_i f_i^\sigma, \\
\rho &=& \sum_\sigma \rho^\sigma.\end{aligned}$$ The average velocity of one component $\sigma$ and the overall fluid can be defined as $$\begin{aligned}
\rho^\sigma u^\sigma_\alpha & \equiv & \sum_i f^\sigma v_{i\alpha} ,\\
\rho u_\alpha & \equiv & \sum_\sigma \rho^\sigma u_{\alpha}^\sigma ,\end{aligned}$$ where $u^\sigma_\alpha $ is the average velocity of the component $\sigma $, and $u_\alpha
$ is the average velocity of the overall fluid.
The moments of equilibrium distributions for one component are chosen to be $$\begin{aligned}
\sum_i f^{\sigma e}_i &=& \rho^\sigma \nonumber, \\
\sum_i f^{\sigma e}_i v_{i \alpha} & = & \rho^\sigma u_\alpha, \nonumber \\
\sum_i f^{\sigma e} v_{i\alpha} v_{i\beta} & = &\frac{\rho^\sigma}{3} \delta_{\alpha \beta}
+ \rho^\sigma u_\alpha u_\beta, \nonumber\\
\sum_i f^{\sigma e} v_{i\alpha} v_{i\beta} v_{i\gamma} & = &
\frac{\rho^\sigma}{3}(u_\alpha \delta_{\alpha \beta} + u_\beta \delta_{\alpha \gamma}
+ u_\gamma \delta_{\alpha \beta} )\end{aligned}$$ The moments for the forcing terms of one component are $$\begin{aligned}
\sum_i F^\sigma_i &=& 0 \label{ff1} \\
\sum_i F^\sigma_i v_{i\alpha} &=& \rho^\sigma a^\sigma_\alpha, \label{ff2}\\
\sum_i F^\sigma_i v_{i\alpha} v_{i\beta} &=& \rho^\sigma (a^\sigma_\alpha u^\sigma_\beta
+ a^\sigma_\beta u^\sigma_\alpha), \label{ff3} \\
\sum_i F^\sigma_i v_{i\alpha} v_{i\beta} v_{i\gamma} &=& \frac{1}{3} \rho^\sigma
( a^\sigma_\alpha \delta_{\beta \gamma} + a^\sigma_\beta \delta_{\alpha \beta} + a^\sigma_\gamma \delta_{\alpha \beta} ) .\end{aligned}$$ To utilize the analysis of the one component system we can establish a LB equation for the total density by defining $$\begin{aligned}
\sum_\sigma f_i^\sigma &=& f_i, \nonumber\\
\sum_\sigma F_i^\sigma &=& F_i, \nonumber\\
\sum_\sigma \rho^\sigma a^\sigma_\alpha &=& \rho a_\alpha . \end{aligned}$$ Similar to the counterparts of the one-component system, the moments for the overall equilibrium distribution function are given by $$\begin{aligned}
\sum_i f^e_i &=& \rho ,\nonumber\\
\sum_i f^e_i v_{i\alpha} &=& \rho u_\alpha , \nonumber \\
\sum_i f_i^e v_{i\alpha}v_{i\beta} &=& \frac{1}{3} \rho \delta_{\alpha \beta} + \rho u_\alpha u_\beta , \nonumber\\
\sum_i f_i^e v_{i\alpha} v_{i\beta} v_{i\gamma} &=& \frac{1}{3} \rho (u_\alpha \delta_{\beta \gamma} + u_\beta \delta_{\alpha \gamma}
+ u_\gamma \delta_{\alpha \beta}) \nonumber \\
&& + \rho u_\alpha u_\beta u_\gamma + Q_{\alpha \beta \gamma}. \end{aligned}$$ The moments for the overall force terms are then given by $$\begin{aligned}
\sum_i F_i &=& 0 , \nonumber \\
\sum_i F_i v_{i\alpha} &=& \rho a_\alpha, \nonumber \end{aligned}$$ Using Eq. (\[ff3\]), we obtain $$\begin{aligned}
&& \sum_i F_i v_{i\alpha} v_{i\beta} \nonumber \\
&=& \sum_\sigma ( a_{\alpha}^\sigma u_{\beta}^\sigma + a_{\beta}u_{\alpha}^\sigma) \nonumber \\
&=& \rho (a_\alpha u_\beta + a_\beta u_\alpha) + \sum_\sigma ( a_\alpha^\sigma \Delta u_{\beta}^\sigma
+ a_{\beta}^\sigma \Delta u_{\alpha}^\sigma), \label{olso}\end{aligned}$$ where the second term of Eq. (\[olso\]) is of a higher order smallness than the first terms, and therefore does not enter the hydrodynamic equations to second order. For the third moment we have $$\sum_i F_i v_{i\alpha} v_{i\beta} v_{i\gamma} = \frac{1}{3} \rho
(a_\alpha \delta_{\beta \gamma} + a_\beta \delta_{\alpha \beta} + a_\gamma \delta_{\alpha \beta} ) .$$ By summing Eq. (\[ssLB\]) over $\sigma$, an effective LB equation for the total density is $$\begin{aligned}
&&f_i ({\bf r} + {\bf v}_i \Delta t , t + \Delta t) - f_i ({\bf r}, t) \nonumber \\
&=& \Delta t \left[ \frac{1}{\tau} \Big(f^e_i ({\bf r}, t) - f_i ({\bf r},t) \Big) + F_i \right] , \label{sgLB}\end{aligned}$$ This is identical to the LB equation for a system of one component. Therefore, the continuity equation and the Navier Stokes equation of the overall fluid of a multicomponent system are recovered as $$\partial_t \rho + \partial_\alpha ( \rho U_\alpha ) = 0 + O (\epsilon^3), \label{scon}$$ where $U_\alpha \equiv u_\alpha + a_\alpha \Delta t / 2$ is the macroscopic velocity of the fluid. The Navier Stokes equation for the overall fluid is: $$\begin{aligned}
&& \partial_t (\rho U_\beta) + \partial_\alpha ( \rho U_\alpha U_\beta) \nonumber \\
&=& - \partial_\alpha \left( \frac{1}{3} \rho \delta_{\alpha \beta} \right) \nonumber \\
&&+ \partial_\alpha \left( \frac{w}{3} \rho (\partial_\alpha U_\beta + \partial_\beta U_\alpha) \right) + \rho a_\beta \nonumber \\
& & - w \partial_\gamma \partial_\alpha \rho u_\alpha u_\beta u_\gamma .
\label{snse} \end{aligned}$$ To recover the convection diffusion equation of each component, we performed a Taylor expansion on the left of Eq. (\[ssLB\]) to second order: $$\begin{aligned}
&& \Delta t (\partial_t + v_{i\alpha} \partial_\alpha) f^\sigma_i
+ \frac{(\Delta t )^2}{2} (\partial_t + v_{i\alpha} \partial_\alpha)^2 f^\sigma_i
+ O(\epsilon^3) \nonumber \\
&=& \Delta t \left( \frac{1}{\tau} (f^{\sigma e}_i - f^\sigma_i ) +F^\sigma_i \right) . \label{s2LB}\end{aligned}$$ Because of the recursive nature of Eq. (\[s2LB\]), $f^\sigma_i$ can be expressed by $f_i^{\sigma e }$ and derivatives of $f^{\sigma e}_i$ as $$\begin{aligned}
f^\sigma_i &=& f_i^{\sigma e} + \tau F^\sigma_i \nonumber \\
&&- \tau (\partial_t + v_{i\alpha}\partial_\alpha )
(f_i^{\sigma e} + \tau F^\sigma_i) + O(\epsilon^2) .
\label{bfe}\end{aligned}$$ Substituting Eq. (\[bfe\]) into the left side of Eq. (\[s2LB\]) we obtain $$\begin{aligned}
&&(\partial_t + v_{i\alpha} \partial_\alpha) (f^{\sigma e}_i + \tau F^\sigma_i) \nonumber \\
&-& w (\partial_t + v_{i\alpha} \partial_\alpha)^2 (f^{\sigma e}_i + \tau F^\sigma_i)
+ O(\epsilon^3) . \nonumber \\
&=& \frac{1}{\tau} (f^{\sigma e}_i + \tau F^\sigma_i - f^\sigma_i ). \label{s3LB}\end{aligned}$$ Summing Eq. (\[s3LB\]) over $i$ gives, $$\begin{aligned}
&& \partial_t \rho^\sigma + \partial_\alpha (\rho^\sigma u_\alpha) + \tau \partial_\alpha ( \rho^\sigma a^\sigma_\alpha) \nonumber\\
&-& w \sum_i (\partial_t + v_{i\alpha} \partial_\alpha)^2 (f^{\sigma e}_i + \tau F^\sigma_i) = O(\epsilon^3)
\label{sgao}\end{aligned}$$ The first moment of $f_i^e$ and $f_i^{\sigma e}$ are not identical, and the continuity equation cannot be obtained. Eq. (\[sgao\]) shows that $\partial_t \rho^\sigma + \partial_\alpha (\rho^\sigma u_\alpha) + \tau \partial_\alpha ( \rho^\sigma a^\sigma_\alpha)$ is of order $O(\epsilon^2)$, and $F^\sigma_i $ is of order $O(\epsilon)$. Therefore $\partial_t \rho^\sigma + \partial_\alpha
(\rho^\sigma u_\alpha) $ is of order $O (\epsilon^2)$, and we get $$\begin{aligned}
&& w \sum_i (\partial_t + v_{i\alpha} \partial_\alpha)^2 (f^{\sigma e}_i + \tau F^\sigma_i)\\
&=& w \partial_\beta \left[ \partial_t (\rho^\sigma u_\beta ) + \partial_\alpha (\frac{\rho^\sigma}{3}
+ \rho^\sigma u_\alpha u_\beta ) \right] + O(\epsilon^3) .\end{aligned}$$ So Eq. (\[sgao\]) can be simplified to $$\begin{aligned}
&& \partial_t \rho^\sigma + \partial_\alpha (\rho^\sigma u_\alpha) + \tau \partial_\alpha \rho^\sigma a^\sigma_\alpha + O(\epsilon^3)\nonumber \\
&&- w \partial_\beta \left[ \partial_t (\rho^\sigma u_\beta)
+ \partial_\alpha (\frac{\rho^\sigma}{3}
+ \rho^\sigma u_\alpha u_\beta ) \right] = 0 . \label{ssim}\end{aligned}$$ Eqs. (\[snse\]) yields $$\partial_t U_\beta = - U_\alpha \partial_\alpha U_\beta - \frac{1}{\rho}
\partial_\alpha \left( \frac{\rho}{3} \delta_{\alpha \beta}
\right) + a_\beta + O(\epsilon^2) .
\label{stu}$$ From Eq. (\[sgao\]) it follows that $$\partial_t \rho^\sigma = - \partial_\alpha ( \rho^\sigma U_\alpha ) + O(\epsilon^2).
\label{sxx}$$ Inserting Eqs. (\[stu\]) and (\[sxx\]) into Eq. (\[ssim\]) we get $$\begin{aligned}
\partial_t ( \rho^\sigma u_\beta ) = &-& \partial_\alpha (\rho^\sigma U_\alpha U_\beta)
- \frac{\rho^\sigma}{\rho} \partial_\alpha \left( \frac{\rho}{3} \delta_{\alpha \beta} \right) \nonumber \\
&+&\rho^\sigma a_\beta + O(\epsilon^2) .
\label{jeep}\end{aligned}$$ Substituting Eq. (\[jeep\]) into Eq. (\[ssim\]) results in $$\begin{aligned}
%&&
\partial_t \rho^\sigma + \partial_\alpha (\rho^\sigma U_\alpha)
% \nonumber \\
&=& \partial_\alpha \bigg[ \tau \rho^\sigma a_\alpha- \tau \rho^\sigma a^\sigma_\alpha
+ w \partial_\alpha \left( \frac{\rho^\sigma}{3} \right)
\nonumber \\&&
- w \frac{\rho^\sigma}{\rho} \partial_\alpha \left( \frac{\rho}{3} \right) \bigg] ,
\label{geq}\end{aligned}$$ From this we deduce that the correct form of the forcing term is $$\begin{aligned}
\rho^\sigma a_\alpha^\sigma &\equiv& - \frac{w}{\tau} \rho^\sigma \partial_\alpha ( \mu^\sigma - \frac{1}{3} \ln \rho^\sigma ) \nonumber \\
&=& - \frac{w}{\tau} \left( \rho^\sigma \partial_\alpha \mu^\sigma - \frac{1}{3} \partial_\alpha \rho^\sigma \right) ,
\label{definef} \end{aligned}$$ where the coefficient is $\frac{w}{\tau} = 1- \frac{\Delta t}{2 \tau} $. This coefficient approaches 1 as $\Delta t$ approaches 0, as one would expect from the continuum limit. This constitutes the main result of this paper. Plugging Eq. (\[definef\]) into Eq. (\[geq\]), we then obtain the convection diffusion equation $$\begin{aligned}
%&&
\partial_t \rho^\sigma + \partial_\alpha (\rho^\sigma U_\alpha )
%\nonumber \\
&=& \partial_\alpha \left[ w \sum_{\sigma'} \frac{\rho^\sigma\rho^{\sigma'}}{\rho} \partial_\alpha ( \mu^\sigma - \mu^{\sigma'}) \right] .
\label{woche}\end{aligned}$$
The diffusion flux of component $\sigma$ is $$J^{\sigma d} = - w \sum_{\sigma'} \frac{\rho^\sigma\rho^{\sigma'}}{\rho} \partial_\alpha ( \mu^\sigma - \mu^{\sigma'}).
\label{gogogo}$$ So that the $w$ in Eq. (\[gogogo\]) is equivalent to $k^{\sigma \sigma'}$ in Eq. (\[bfjj\]).
Numerical validation
====================
We examined the equilibrium behavior of phase separated binary and ternary systems. We used the Flory-Huggins free which is a very popular model to study polymer solutions. It is given by $$\begin{aligned}
\mathcal{F} &=& \int \left( - \sum_\sigma \theta n^\sigma m^\sigma
+ \sum_\sigma n^\sigma \theta \ln \phi^\sigma \right. \nonumber \\
&& \qquad + \left. \sum_\sigma\sum_{\sigma'} \frac{1}{2} \chi^{\sigma \sigma'} \theta m^\sigma n^\sigma
\phi^{\sigma'} \right) dV ,
\label{lgf}\end{aligned}$$ where $m^\sigma$ is the polymerization of the component $\sigma$, $n^\sigma$ is its number density, and $\phi^\sigma$ is its volume fraction. It is defined as $$\phi^\sigma = \frac{m^\sigma n^\sigma}{\sum_\sigma ( m^\sigma n^\sigma)} = \frac{\rho^\sigma}{\rho},$$ where $ \rho^\sigma $ is the mer density of component $\sigma$ and $ \rho$ is the mer density of the system, which is a constant in the Flory-Huggins model. To validate our algorithm we compared the binodal lines obtained by our algorithm to the theoretical ones obtained by minimizing the free energy. We used the interfacial tension parameter $\kappa =0$ in all our LB simulations of binary and ternary systems, because there is an intrinsic surface tension in the LB simulation due to higher order terms [@wagner2006a], which did not appear explicitly in the second order Taylor expansion presented in this paper. Since we are only evaluating the phase-behavior here we use a one dimensional model known as D1Q3. This model has the velocity set $\{v_i\} =
\{-1,0,1\}$. This is an important test since all other frequently used higher dimensional model have this D1Q3 model as a projection.
We consider two binary systems: a monomer system with $m^A = 1$ and $m^B=1$, and a polymer system with $m^A = 10$ and $m^B =1$. For both systems, the total density was $\rho = 100$. Throughout this paper we choose the self interaction parameters to vanish: $\chi^{\sigma\sigma}=0$. The critical volume fractions for the monomer system are $\phi^A =0.5$ and $\phi^B = 0.5$ and for the polymer systems are $\phi^A =0.24$ and $\phi^B =
0.76$. To induce phase separation a small sinusoidal perturbation $\Delta\phi(x)$ was added in the initial conditions. The amplitude of the perturbation is 0.1 and its wavelength is the lattice size. The initial volume fraction of component A is given by $\phi^A (x) = \phi^{A0} +
\Delta \phi (x)$. The initial volume fraction of component B is given by $\phi^B
(x) = \phi^{B0} - \Delta \phi (x)$.
The monomer system was simulated with different inverse relaxation times. In Figure \[binary\_binodal\] we show results for $ 1/\tau =
0.7$, and $0.9$. We see that the equilibrium densities have only a very slight dependence on the relaxation time, although the range of stability depends noticeably on the relaxation time. The polymer system was simulated with only one inverse relaxation time of $ 1/ \tau =
0.9$. Starting from the critical point and increasing the $\chi^{AB}$ value for each initial condition until the simulations were numerically unstable, we obtained a pair of binodal points for each initial condition. The system reached a stable state after about 5000 time steps. The measurement were taken after 50000 time steps to be sure that an equilibrium state had been reached.
For the polymer system, Figure \[LBchem\] shows the comparison of the total density, and the volume fractions and chemical potentials of each component to the corresponding theoretical values. The total density of a system in equilibrium by LB is essentially constant with a variation of $\Delta\rho/\rho<10^{-5}$. The volume fractions of each component in the LB simulation agree well with the theoretical values. The chemical potential of each component by the LB simulation was very close to the theoretical value. The chemical potential $\mu_A$, corresponding to the polymer component, varied slightly with a difference for the bulk values of about $2\cdot10^{-2}$ and a variation in the interface of about $4\cdot10^{-2}$. This is the underlying reason for the small deviation from the theoretically predicted concentration. The potential $\mu_B$ was nearly constant with a variation of less than $10^{-4}$ in the bulk and a variation of about $10^{-3}$ at the interface. For large values of $\chi^{AB}$ this discrepancy increases leading to the noticeable variation of the equilibrium densities of the polymer system as shown in Figure \[binary\_binodal\].
We also performed LB simulations with two ternary systems: a monomer system with $m^A = 1$, $m^B=1$, and $m^C=1$ and a polymer system with $m^A =
10$, $m^B =1$, and $m^C =1$. The $\chi$ parameters for both systems were $\chi^{AB} =3$, $\chi^{AC} = 0.5$, and $\chi^{BC} = 0.2$. The other $\chi$ parameters were zero. The inverse relaxation time constant for both simulations was $1/ \tau = 0.9$. The critical point for the monomer system was $\phi^{A,cr} = 0.32$, $\phi^{B,cr} = 0.32$, and $\phi^{C,cr} = 0.36$. The critical point for the polymer system was $\phi^{A,cr} = 0.14$, $\phi^{B,cr} = 0.11$, and $\phi^{C,cr}
=0.75$. The initial state of each simulation were set from the critical points towards the end point ($\phi^A = 0.5$, $\phi^B = 0.5$, $\phi^C
=0$). Initially a small sinusoidal wave perturbation $\Delta \phi$ of an amplitude of 0.1 and wavelength of the lattice size was superimposed on the initial volume fraction of the A component. This perturbation was subtracted from the B component and the C component was constant. I performed a LB simulation for each set of initial volume fractions and obtained the volume fractions of the two phases in the equilibrium state, resulting in two binodal points. The simulation reached a stable state after about 20,000 time steps. The measurements were taken after 200,000 time steps to make sure the equilibrium state was reached.
Figure \[triLBm10phase\] shows the comparison of the binodal points by LB simulation to the theoretical binodal lines of both systems. The binodal points obtained by the LB simulation agree fairly well with the theoretical binodal lines for the monomer and polymer systems. The simulation becomes unstable when $\phi^A$ is close to zero, i.e. when one component is nearly depleted. In this region the simulation results also deviate noticeably from the theoretical binodal lines. Immediately near the critical point, the evolution of the system becomes extremely slow so the slight deviation between the binodal points obtained through the LB simulation and the theoretical ones probably indicates that the LB simulation was not yet fully equilibrated.
For the polymer system Figure \[triLBm10ch\] shows a comparison of the volume fractions and chemical potentials of each component. The total density of the system is again nearly constant with variation of less than $\Delta\rho/\rho<10^{-4}$ in the bulk. At the interface there is a small variation of $\Delta\rho/ \rho<10^{-2}$. The volume fractions of each phase in the simulation were very close to their theoretical values. The chemical potential of component A was slightly different in two phases with a variation of about $2\cdot10^{-2}$, while the chemical potentials of components B and C were much closer in the two phases with a variation of less than $10^{-4}$.
Outlook
=======
We have presented a general lattice Boltzmann algorithm for systems with an arbitrary number of components which is based on an underlying free energy. In this algorithm the key thermodynamic quantities such as the chemical potentials of the components are immediately accessible. It is also manifestly symmetric for all components. We tested the equilibrium behavior of the new algorithm for two and three component systems in each case examining both the case of monomer and polymer mixtures with an underlying Flory-Huggins free energy. We obtained to expected phase-diagrams to good accuracy and the chemical potentials were constant to good accuracy for the monomer systems. Polymer systems were more challenging to simulate but still obtained acceptable results for $m=10$. Higher polymerizations, however, become increasingly difficult to realize with the current algorithm.
There are three directions in which we hope to extend this algorithm in the future. The current algorithm does not allow for component dependent mobility. We are working on developing an algorithm that can recover an arbitrary mobility tensor $\kappa^{\sigma\sigma'}$. The chemical potential is only approximately constant. Recent progress for liquid-gas systems [@wagner2006a] makes us hopeful that we will be able to ensure that the chemical potential is constant up to machine accuracy. And lastly we hope to extend to model so that it can simulate polymer systems with significantly larger polymerization.
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|
---
abstract: |
As it is known, finitely presented quivers correspond to Dynkin graphs (Gabriel, 1972) and tame quivers – to extended Dynkin graphs (Donovan and Freislich, Nazarova, 1973). In the article “Locally scalar reresentations of graphs in the category of Hilbers spaces” (Func. Anal. and Appl., 2005) authors showed the way to tranfer these results to Hilbert spaces, constructed Coxeter functors and proved an analogue of Gabriel theorem fol locally scalar (orthoscalar in the sequel) representations (up to the unitary equivalence).
The category of orthoscalar representations of a quiver can be considered as a subcategory in the category of all representations (over a field $\mathbb C$). In the present paper we study the connection between indecomposable orthoscalar representations in the subcategory and in the category of all representations.
For the quivers, corresponded to extended Dynkin graphs, orthoscalar representations which cannot be obtained from the simplest by Coxeter functors (regular representations) are classified.
address:
- Институт математики НАН Украины
- Институт математики НАН Украины
- Институт математики НАН Украины
author:
-
- 'Kruglyak S.A.'
- 'Nazarova L.A.'
title: |
Orthoscalar representations of quivers on the\
category of Hilberts spaces. II.
---
Введение {#введение .unnumbered}
========
Более 30-ти лет назад (см. [@Roi75]) в работах И.М. Гельфанда, В.А. Пономарева, П. Габриеля и авторов было показано, что ряд проблем линейной алгебры (возникшие в теории представлений алгебр, теории групп и модулей Хариш—Чандра) допускают содержательное изучение как на наивном языке приведения наборов матриц теми или иными допустимыми преобразованиями, так и в категорно-функторных терминах. Отметим представления колчанов и частично упорядоченных множеств. Было доказано, в частности, что конечно-представимые (ручные) колчаны соответствуют графам (расширенным графам) Дынкина.
В [@KruRoi05] был указан путь перенесения этих результатов на представления колчанов в гильбертовых пространствах, построены функторы Кокстера и доказан аналог теоремы Габриеля [@Gabriel72] для локально-скалярных представлений колчанов (в дальнейшем мы заменим термин “локально-скалярные представления” [@KruRoi05] на термин “ортоскалярные представления” , считая его более удачным). В [@KruRoi05] доказано, что у колчанов, соответствующих расширенным графам Дынкина, размерности неразложимых ортоскалярных представлений не ограничены в совокупности (как следует из [@KruRabSam02]-[@AlbOstSam06] расширенные графы Дынкина среди не конечно представимых и только они не имеют бесконечномерных неразложимых ортоскалярных представлений). Мы даём описание неразложимых ортоскалярных представлений для расширенных графов Дынкина (см. теоремы 1-3), указывая не только на их сходство, но и на отличие от неразложимых представлений таких графов в линейных пространствах (см. замечание 2).
О неразложимости в категории представлений колчана и её подкатегории ортоскалярных представлений
================================================================================================
Напомним некоторые определения и факты (см. [@KruNazRoi06]) об ортоскалярных представлениях колчанов. Колчан $Q$ с множеством вершин $Q_v$, $|Q_v|=N$, и множеством стрелок $Q_a$ называется *разделённым*, если $Q_v=\overset{\circ}{Q}\bigsqcup
\overset{\bullet}{Q}$, и для любой $\alpha \in Q_a$ её начало $t_\alpha \in \overset{\circ}{Q}$ и конец $h_\alpha \in
\overset{\bullet}{Q}$. Колчан $Q$ *однократный*, если при $\alpha \neq \beta$ либо $t_\alpha \neq t_\beta$, либо $h_\alpha
\neq h_\beta$. Вершины из $\overset{\circ}{Q}$ будем называть *четными*, из $\overset{\bullet}{Q}$ — *нечётными*.
Пусть $m=\big|\overset{\bullet}{Q}\big|$, $n=\big|\overset{\circ}{Q}\big|$, $\overset{\bullet}{Q}=\{i_1,i_2,\ldots,i_m\}$, $\overset{\circ}{Q}=\{j_1,i_2,\ldots,j_n\}$. Представление $T$ колчана $Q$ ставит в соответствие вершине $i\in Q_v$ конечномерное линейное пространство $T(i)$, а стрелке $\alpha: j \rightarrow i, \
\alpha \in Q_a$ линейное отображение $T_{ij}: T(j) \rightarrow
T(i)$.
Представление $T$ однократного разделённого колчана при фиксированных базисах пространств $T(i)$, $i\in Q_v$, можно ассоциировать с матрицей, разделённой на $m$ горизонтальных и $n$ вертикальных полос, т.е. блочной матрицей $$T=\big[ T_{i_l,j_k} \big]_{k=\overline{1,n},\ l=\overline{1,m}}.$$ При этом будем считать, что $T_{i_l,j_k}=0$, если не существует $\alpha \in Q_a$ такой, что $t_\alpha=j_k, h_\alpha=i_l$.
Пусть $\textrm{Rep}Q$ — категория представлений колчана $Q$, объекты которой есть представления, а морфизм представления $T$ в представление $\widetilde T$ определяется как семейство линейных отображений $C=\{C_i\}_{i\in Q_v}$,$C_i: T(i) \rightarrow
\widetilde T(i)$, таких, что для каждой $\alpha \in Q_a$ c $t_\alpha=j$, $h_\alpha = i$ диаграмма $$\label{CD}
\begin{CD}
T(j) @>T(\alpha)>> T(i)\\
@VVC_jV @VVC_iV \\
\widetilde T(j) @>\widetilde T(\alpha)>> \widetilde T(i)
\end{CD}$$ коммутативна, т.е. $C_iT_{ij}=\widetilde T_{ij}C_j$.
Пусть представления $T, \widetilde T$ заданы матрицами $$T=\big[ T_{i_l,j_k} \big]_{k=\overline{1,n},\ l=\overline{1,m}}
\quad
\mbox{и} \quad
\widetilde T=\big[\widetilde T_{i_l,j_k} \big]_{k=\overline{1,n},\ l=\overline{1,m}}$$ Введём матрицы $A=\textrm{diag}\{C_{i_1},C_{i_2},\ldots,C_{i_m}\}$, $B=\textrm{diag}\{C_{j_1},C_{j_2},\ldots,C_{j_n}\}$. Тогда из коммутативности диаграмм (\[CD\]) следует $$\label{CD2}
AT=\widetilde{T}B.$$ Будем в дальнейшем говорить, что $C=(A,B).$
Пусть $\mathcal H$ — категория унитарных (конечномерных гильбертовых) пространств. Обозначим через $\textrm{Rep}(Q,\mathcal
H)$ подкатегорию в $\textrm{Rep}Q$, объекты которой есть представления $T$, для которых $T(i)$ — унитарные пространства $(i\in Q_v)$, а морфизмы $C:T\rightarrow \widetilde T$ — те из морфизмов в $\textrm{Rep}Q$, для которых, кроме (\[CD\]), коммутативными будут и диаграммы $$\label{CD3}
\begin{CD}
T(j) @<T(\alpha)^*<< T(i)\\
@VVC_jV @VVC_iV \\
\widetilde T(j) @<\widetilde T(\alpha)^*<< \widetilde T(i)
\end{CD}$$ т.е. будут выполняться равенства $$\label{CD4}
AT=\widetilde{T}B, \quad BT^*=\widetilde{T}^*A$$
Представления $T,\ \widetilde T$ из $\textrm{Rep}Q$ (соотв., из $\textrm{Rep}(Q,\mathcal H)$) эквивалентны в $\textrm{Rep}Q$ (соотв., в $\textrm{Rep}(Q,\mathcal H)$), если найдётся обратимый морфизм $C:T\rightarrow \widetilde T$. Можно показать, что $T$ и $\widetilde T$ эквивалентны в $\textrm{Rep}(Q,\mathcal H)$ тогда и только тогда, когда они унитарно эквивалентны (см., например, [@Roi79]), т.е. обратимый морфизм можно выбрать состоящим из унитарных матриц $C_i$.
Обозначим $\overrightarrow{T_i}=\big[T_{i,j_1};T_{i,j_2};\ldots;T_{i,j_n}\big]$, $T_j^\downarrow=\left[\begin{array}{c}
T_{i_1,j} \\
T_{i_2,j} \\
\vdots \\
T_{i_m,j}
\end{array}\right].$
Представление $T$ разделенного однократного колчана $Q$ из категории $\textrm{Rep}(Q,\mathcal H)$ назовём *ортоскалярным*, если каждому $i\in Q_v$ сопоставлено вещественное неотрицательное число $\chi_i$, и выполняются следующие условия: $$\begin{aligned}
\label{OS1}
\overrightarrow{T_i}\cdot \overrightarrow{T_i}^*&=\chi_i I_i
\quad
\mbox{при} \ \ i \in \overset{\bullet}{Q}, \\ \label{OS}
T_j^{\downarrow *}\cdot T_j^{\downarrow}&=\chi_j I_j \quad
\mbox{при} \ \ j \in \overset{\circ}{Q},\end{aligned}$$ здесь $I_i$ — матрица единичного оператора в $T(i)$. Ортоскалярной будем называть и матрицу представления $T$.
В определении представлений колчана $Q$ в $\mathcal H$ можно было бы отказатся от конечномерности пространств $T(i)$, расматривая и бесконечномерные представления.
Будем говорить, что $Q$ (ортоскалярно) *конечнопредставим* в $\mathcal H$, если все его ортоскалярные представления распадаются в прямую сумму (конечную либо бесконечную) конечномерных представлений, размерности неразложимых представлений ограничены в совокупности и в каждой размерности число неразложимых представлений с данным характером конечно.
Определим категорию $\textrm{Rep}_{os}(Q,\mathcal H)$ как полную подкатегорию в $\textrm{Rep}(Q,\mathcal H)$, объекты которой есть ортоскалярные представления колчана $Q$.
Обозначим через $\mathbb R^{G_v}$ линейное вещественное протранство из наборов $x=(x_i)$ действительных чиел $x_i\ (i\in G_v)$, элементы $x$ из $\mathbb R^{G_v}$ будем называть $G$-векторами.
Ортоскалярному представлению $T$ разделённого однократного колчана $Q$ сопоставим два $N$-мерных $G$-вектора ($N=m+n$): *размерность* $d=\{d(j)\}_{j\in Q_v}$ представления $T$, где $d(j)=\dim T(j)$, и *характер* $\chi=\{\chi(j)\}_{j\in Q_v}, \
\chi(j)=\chi_j$ определены выше (см. (\[OS1\]), (\[OS\])). Два ортоскалярных представления $T$ и $\widetilde T$ эквивалентны, если существуют такие унитарные матрицы $U=\textrm{diag}\{U_{i_1},U_{i_2},\ldots,U_{i_m}\}$ и $V=\textrm{diag}\{V_{j_1},V_{j_2},\ldots,V_{j_n}\}$, что $$\begin{aligned}
\label{OS2}
UT&=\widetilde TV, \quad \mbox{или} \\
\widetilde{T}_{i_l,j_k}&=U_{i_l}T_{i_l,j_k}V_{j_k}^*\end{aligned}$$
Представление $T$ будем называть *шуровским* (*brick*) в категории $\textrm{Rep}Q$ (соответственно $\textrm{Rep}(Q,\mathcal
H)$, $\textrm{Rep}_{os}(Q,\mathcal H)$), если его кольцо эндоморфизмов в этой категории одномерно (изоморфно $\mathbb C$). Очевидно, если $T$ — шуровское представление, то $T$ неразложимо (в соответствующей категории). Если $T$ — неразложимое в категории $\textrm{Rep}(Q,\mathcal H)$, то оно в ней шуровское. Действительно алгебра $\mathfrak{A}=\textrm{End}T$ есть конечномерная $*$-алгебра. Если $C=(A,B) \in \textrm{End}T$ то $C^*=(A^*,B^*)$. Если $C \in
\textrm{Rad} \mathfrak{A}$, то $CC^*=(AA^*,BB^*) \in \textrm{Rad}
\mathfrak{A}$ и $CC^*$ — нильпотентный элемент, поэтому $AA^*,
BB^*$ нильпотентные и положительные операторы. Значит, $C=(0,0)$, и алгебра $\mathfrak{A}$ полупростая. С другой стороны, алгебра $\mathfrak{A}$ — локальна, как алгебра эндоморфизмов неразложимого представления. Значит, $\mathfrak{A}\simeq\mathbb C$.
Представление $T$ колчана $Q$ называется *точным* если $T(i)\neq 0$ при всех $i \in Q_v$. *Носителем* представления $T$ называется множество $Q_v^T=\{i\in Q_v \ |\ T(i)\neq 0\}$. Характер представления определён однозначно на носителе $Q_v^T$ представления (и неоднозначно вне носителя). Если $Q_v^T=Q_v$, то характер ортоскалярного представления определён однозначно и обозначается $\chi_T$, в общем случае обозначим через $\{\chi_T\}$ множество всех характеров представления $T$. Ясно, что если $T$ и $\widetilde T$ унитарно эквивалентны, то $\{\chi_T\}=\{\chi_{\widetilde T}\}.$
Категория $\textrm{Rep}_{os}(Q,\mathcal H)$ есть (неполная) подкатегория категории $\textrm{Rep}(Q)$. Установим связь между неразложимыми объектами этих категорий. Основной вопрос: останется ли неразложимое ортоскалярное представление неразложимым в категории $\textrm{Rep}(Q)$, и поскольку, как мы отмечали, неразложимые в $\textrm{Rep}_{os}(Q,\mathcal H)$ представления есть шуровские, останутся ли они шуровскими в категории $\textrm{Rep}(Q)$?
Второй пункт нижеприведенного утверждения доказан в [@KruNazRoi06].
\[T1\] Пусть $Q$ разделённый однократный колчан.
1. Если $T$, $\widetilde T \in \textrm{Rep}_{os}(Q,\mathcal H)$ — представления с одинаковым характером и $T$ эквивалентно $\widetilde
T$ в $\textrm{Rep}(Q)$, то $T$ эквивалентно $\widetilde T$ в $\textrm{Rep}_{os}(Q,\mathcal H)$;
2. Если $T$ — неразложимое представление в $\textrm{Rep}_{os}(Q,\mathcal
H)$ то $T$ — неразложимое, более того, шуровское и в $\textrm{Rep}(Q)$.
Доказательство теоремы \[T1\] опирается на следующие вспомогательные утверждения.
Пусть $C=(A,B)$ — морфизм представления $T$ в $\widetilde T$ в категории $\textrm{Rep}(Q)$, т.е. выполняется равенство $$AT=\widetilde{T}B,$$ и $A, B$ есть унитарные отображения. Тогда $C=(A,B)$ есть морфизм представления $T$ в представление $\widetilde T$ и в категории $\textrm{Rep}_{os}(Q,\mathcal H)$ т.е. выполняются и равенства $$BT^*=\widetilde{T}^*A.$$
Из (\[CD2\]) следует $T^*A^*=B^*\widetilde T^*$ или, учитывая унитарность $A$ и $B$, имеем $T^*A^{-1}=B^{-1}\widetilde T^*$. Поэтому $$BT^*=\widetilde{T}^*A.$$
Пусть $C=(A,B)$ морфизм представления $T$ в себя (эндоморфизм представления $T$) в категории $\textrm{Rep}(Q)$, т.е. $$\label{CD7}
AT=TB$$ и $A,\ B$ — самосопряжённые операторы. Тогда $C=(A,B)$ есть эндоморфизм представления $T$ и в категории $\textrm{Rep}(Q, \mathcal
H)$, т.е. и $$\label{CD8}
AT^*=T^*B.$$
Действительно, (\[CD8\]) получается из (\[CD7\]) операцией сопряжения.
\[l3\] ([@KruNazRoi06]) Пусть $Z=\big[z_{ij}\big]_{i=1,m,\ j=1,n}$, $W=\big[w_{ij}\big]_{i=1,m,\
j=1,n}$ — матрицы над полем $\mathbb C$, имеющие одинаковые положительные длины ($|\overrightarrow{x}|$, $|y^\downarrow|$) соответствующих срок и соответствующих столбцов. Пусть $A=diag\{a_1,a_2,\ldots,a_m\}$, $B=diag\{b_1,b_2,\ldots,b_n\}$ — матрицы над $\mathbb R$, $a_i>0,\ b_j>0$ при $i=1,m$, $j=1,n$, и пусть $AZ=WB$. Тогда $Z=W$.
Обозначим через $K$ число ненулевых элементов в каждой из матриц $Z, W$. Доказательство леммы \[l3\] проводится ([@KruNazRoi06]) индукцией по тройкам чисел $(m,n,K);$ считая, что $(m_1,n_1,K_1)<(m_2,n_2,K_2)$, если $m_1\leq m_2$, $n_1\leq n_2$ и хотя бы одно неравенство строгое, либо если $m_1=m_2$, $n_1=n_2$, но $K_1<K_2$. Базу индукции получаем при $m=1$, либо $n=1$, либо $K=\max(m,n)$.
*Доказательство теоремы \[T1\].*
\(a) Пусть $C=(A,B)$ осуществляет эквивалентность представлений $T$, $\widetilde T$ из $\textrm{Rep}_{os}(Q,\mathcal H)$ в категории $\textrm{Rep}(Q)$, т.е. $AT=\widetilde TB$ ($A$ и $B$ обратимые матрицы). Пусть $A=XU$, $B=VY$ — полярные разложения матриц $A$ и $B$, где $U, V$ — унитарны, $X, Y$ — положительные невырожденые матрицы (при этом можно считать, что $U,V,X,Y$ имеют такую же блочно-диагональную структуру, как $A$ и $B$). Пусть, кроме того, $$X=U_1^*\widetilde X U_1, \quad Y=V_1\widetilde Y V_1^*,$$ где $U_1, V_1$ - унитарные матрицы, $\widetilde X, \widetilde Y$ — диагональные матрицы с положительными числами на диагонали. Тогда $$\label{CD9}
U_1^*\widetilde X U_1 U T= \widetilde T V V_1 \widetilde Y V_1^*$$ или $$\widetilde X(U_1 U T V_1)= (U_1\widetilde T V V_1) \widetilde Y.$$ Так как длины соответствующих строк и столбцов матриц $U_1UTV_1$ и $U_1\widetilde T V V_1$ равны в силу предположений, то по лемме 3 $$U_1 U T V_1= U_1\widetilde T V V_1,$$ а тогда, после сокращений, $$\label{CD10}
UT=\widetilde T V,$$ и представления $T$ и $\widetilde T$ эквивалентны в $\textrm{Rep}_{os}(Q,\mathcal H)$ по лемме 1.
\(b) Пусть $T$-шуровское представление в $\textrm{Rep}_{os}(Q,\mathcal H)$ и $C=(A,B)\in \textrm{End} T$ в $\textrm{Rep}(Q)$. Можно считать, что $A$ и $B$ невырождены (прибавляя в случае необходимости к $A$ и $B$ подходящее кратное единичного оператора). Тогда по (\[CD10\]) при $T=\widetilde T$ из того, что $(U,V)$ есть эндоморфизм представления $T$ в $\textrm{Rep}_{os}(Q,\mathcal H)$, следует что $U$ и $V$ кратны единичному оператору (с одним и тем же скаляром в качестве сомножителя). Сокращая равенство (\[CD9\]) на этот скаляр, мы получим $$XT=TY,$$ где $X$ и $Y$ — самосопряжённые операторы, а тогда по лемме 2 из-за шуровости $T$ в $\textrm{Rep}_{os}(Q,\mathcal H)$ операторы $X$ и $Y$ скалярны (с одним и тем же скаляром); следовательно, такими будут и операторы $A=XU$ и $B=VY$. Значит $T$ — неразложимое шуровское представление в $\textrm{Rep}(Q)$.
Размерности неразложимых ортоскалярных представлений
====================================================
С колчаном $Q$ связана *форма Титса* $q(x)$ на $\mathbb
R^{Q_v}$: если $x\in \mathbb R^{Q_v}$, то $$q(x)=\sum\limits_{i\in Q_v} x_i^2-\sum\limits_{\alpha \in Q_a}
x_{t_\alpha} x_{h_\alpha}.$$ Из работы [@Kac82] следует, что размерности неразложимых (в $\textrm{Rep}(Q)$) представлений колчана $Q$ совпадают с положительными *корнями* соответствующего графа $G=G(Q)$, причём, для графов Дынкина и расширенных графов Дынкина такие корни совпадают в точности с решениями уравнений $q(x)=1$ и $q(x)=0, x\in
\mathbb Z_{+}^{Q_v}$ (здесь $\mathbb Z_{+}^{Q_v}=\{x\in \mathbb
Z^{Q_v} | x\neq0, x_i\geq0\}$). Корни $x$ при $q(x)=1$ называются *действительными* а при $q(x)=0$ — *мнимыми*. Мнимые корни кратны минимальному мнимому положительному корню $\delta=\delta_{G}$.
Фиксируем нумерацию вершин в $\overset{\bullet}{Q}=\{i_1,i_2,\ldots,i_m\}$, и $\overset{\circ}{Q}=\{j_1,i_2,\ldots,j_n\}$ (для определенности будем считать что узловая точка, если она одна, лежит в $\overset{\bullet}{Q}$; будем вершины из $\overset{\circ}{Q}$ обозначать также как $i_{m+k}=j_k, \ k\in \overline{i,n}$). Пусть $x\in \mathbb R^{Q_v}$, $x_k=x(i_k)$, при $k\in \overline{1,m+n}$, $c$—преобразование Кокстера на $\mathbb R^{Q_v}$, $c=\sigma_{i_{n+m}}\cdots\sigma_{i_2}\sigma_{i_1}$, $(\sigma_{i_k}(x))_k=-x_k+\sum\limits_{l,i_l-i_k}x_l$, $(\sigma_{i_k}(x))_l=x_l$ при $l\neq k$. Ясно, что $\sigma_i^2=\textrm{id}$ при $i\in \overline{1,n+m}$. Поэтому $c^{-1}=\sigma_{i_1}\sigma_{i_2}\cdots\sigma_{i_{n+m}}$. Будем также пользоваться обозначениями $\overset{\bullet}{c}=\sigma_{i_{n}}\cdots\sigma_{i_2}\sigma_{i_1}$, $\overset{\circ}{c}=\sigma_{i_{m+n}}\cdots\sigma_{i_{m+2}}\sigma_{i_{m+1}}$ и называть эти преобразования отражениями Кокстера (${\overset{\bullet}{c}{}^2}=\textrm{id}$, $\overset{\circ}{c}
{}^2=\textrm{id}$). Обозначим $\overset{\bullet}{c}_k=\underbrace{\ldots
\overset{\bullet}{c}\overset{\circ}{c}\overset{\bullet}{c}}_{k \
\mbox{\Small раз}}$, $\overset{\circ}{c}_k=\underbrace{\ldots
\overset{\circ}{c}\overset{\bullet}{c}\overset{\circ}{c}}_{k \
\mbox{\Small раз}}$, $k \in \mathbb N$. Вектор $x \in \mathbb
R_{+}^{G_v}$ *регулярен*, если $c^t(x) \in \mathbb R_{+}^{G_v}$ при любом $t \in \mathbb Z$ и *сингулярен* в противном случае (терминология восходит к [@GelPon70]).
Представление $T$ колчана $Q$ *сингулярно*, если $T$ неразложимо, конечномерно и его размерность $d$ — сингулярный вектор; $T$ *регулярно*, если $T$ неразложимо, конечномерно и не сингулярно.
Пусть $\delta_G=(\delta_1,\ldots,\delta_m,\delta_{m+1},\ldots,\delta_{m+n})$. Построим линейную форму $$L_{G}(x)=\sum\limits_{i_k\in \overset{\bullet}{G}} \delta_k x_k-
\sum\limits_{i_{m+k}\in \overset{\circ}{G}} \delta_{m+k}
x_{m+k}, \quad x\in \mathbb R^{G_v}.$$
Справедливо утверждение (см., например, [@RedRoi04]): пусть $G$ — расширенный граф Дынкина; для того чтобы корень $x\in \mathbb
R^{G_v}$ был сингулярен, необходимо и достаточно, чтобы $L_G(x)\neq
0$.
Отметим, что у графа Дынкина все корни дейтвительны и сингулярны, у расширенного графа Дынкина все мнимые корни регулярные, а вот действительные корни разбиваются на два сорта (действительные сингулярные и действительные регулярные). Для колачана $Q$, граф которого $G$ есть расширенный граф Дынкина, справедливо утверждение:
(cм. [@CrawBoe]) Если $T$ есть неразложимое в $\textrm{Rep}(Q)$ регулярное представление колчана $Q$, граф $G$ которого есть расширенный граф Дынкина, то $T$ — шуровское представление тогда и только тогда когда $\dim T=d\leq\delta_G$.
Докажем следующее утверждение:
Пусть $G$ — расширенный граф Дынкина, $d\in \mathbb Z_{+}^G$ корень графа $G$, удовлетворяющий условию $d<\sigma_G$. Тогда либо $d$ — неточный корень (одна из координат нулевая), либо получается из некоторого неточного корня $\tilde d$ применением отражений Кокстера ${\overset{\bullet}{c}}$ и ${\overset{\circ}{c}}$.
Достаточно показать, что такой точный корень можно отражениями Кокстера преобразовать в неточный корень $\widetilde d.$
Рассмотрим граф $\widetilde D_n$
(140,50)
(0,40) (20,20) (50,20) (60,18) (70,18) (80,18) (90,20) (120,20) (140,40) (140,0) (0,0)
(0,40)[(1,-1)[20]{}]{} (0,0)[(1,1)[20]{}]{} (20,20)[(1,0)[30]{}]{}(90,20)[(1,0)[30]{}]{} (140,40)[(-1,-1)[20]{}]{} (140,0)[(-1,1)[20]{}]{}
(-15,40)[$a_1$]{} (-15,0)[$a_2$]{} (20,25)[$z_1$]{} (50,25)[$z_2$]{} (90,25)[$z_{n-4}$]{} (125,20)[$z_{n-3}$]{} (155,40)[$b_1$]{} (155,0)[$b_2$]{}
Пусть $d=(d_{a_1},d_{a_2},d_{z_1},\ldots,d_{z_{n-3}},d_{b_1},d_{b_2}).$ Как известно, $$\delta_{\widetilde{D}_n}=\begin{array}{c}
1 \\
1
\end{array}
2\ 2\ \ldots\ 2\ 2 \begin{array}{c}
1 \\
1
\end{array}$$ (координаты вектора располагаются так же, как соответствующие вершины в графе). Так как $d$ — точный корень, то $d_{a_1}=d_{a_2}=d_{b_1}=d_{b_2}=1$. Назовём *отмеченным* первый индекс $k$, для которого $x_{z_k}=1$ (т.е. $x_{z_1}=x_{z_2}=\ldots=x_{z_{k-1}}=2)$). Пусть, для определённости, $z_{k-1} \in \overset{\bullet}{G}$. Тогда у $\overset{\bullet}{c}d$ координаты $x_{a_1},x_{a_2},x_{z_1},\ldots,x_{z_{k-2}}$ совпадают с соответствующими координатами вектора $d$, а $x_{z_{k-1}}=1$, т.е. отмеченный индекс уменьшается на единицу. Применяя к $\overset{\bullet}{c}d$ преобразование $\overset{\circ}{c}$, продолжим этот процесс. В результате мы придём к вектору, у которого отмеченный индекс равен 1, а стало быть на следующем шаге получим вектор, у которого $x_{a_1}=x_{a_2}=0$, это и будет искомый неточный вектор $\tilde
d$.
Для графов $\widetilde E_6$, $\widetilde E_7$, $\widetilde E_8$ результат можно получить аналогичными рассуждениями или, в крайнем случае, прямым перебором (положительных корней $d$ таких, что $d<\delta_G$, конечное число).
$\delta_{\widetilde
A_n}=(1,1,\ldots,1)$, так что из условия $d<\delta_{\widetilde
A_n}$ следует, что одна из координат нулевая и $d$ есть неточный вектор.
Функторы Кокстера и ортоскалярные представления, размерности которых есть действительные корни
==============================================================================================
Пусть $g\in Q_v$ и $\Pi_g$ — простейшее представление колчана $Q:
\Pi_g(g)=\mathbb C$, $\Pi_g(i)=0$ при $i\neq g,\ i\in Q_v$. Ясно, что если $f_g$ — характер представления $\Pi_g$, то $f_g(g)=0$; будем предполагать, что $f_g(i)>0$ при $i\neq g$.
Обозначим через $\textrm{Rep}(Q,d,\chi)$ полную подкатегорию неразложимых представлений из $\textrm{Rep}_{os}(Q,\mathcal H)$ с фиксироваными размерностью $d$ и характером $\chi$. Если в $\textrm{Rep}(Q,d,\chi)$ не входит простейшее представление, то $\chi(i)>0$ для $d(i)\neq 0$ (в силу неразложимости представлений).
В [@KruRoi05] были введены *функторы отражений Кокстера* $\overset{\bullet}{F}$ и $\overset{\circ}{F}$: $$\begin{split}
\overset{\circ}{F}: \textrm{Rep}(Q,d,\chi) \rightarrow
\textrm{Rep}(Q,\overset{\circ}{c}(d),\overset{\circ}{\chi}),
\\
\overset{\bullet}{F}: \textrm{Rep}(Q,d,\chi) \rightarrow
\textrm{Rep}(Q,\overset{\bullet}{c}(d),\overset{\bullet}{\chi}),
\end{split}$$ где
$$\label{CD11}
\begin{split}
\overset{\circ}{d}(i)=\overset{\circ}{c}(d)(i)=
\left\{\begin{array}{c}
-d(i)+\sum\limits_{j:j-i}d(j) \quad \mbox{при} \ i\in \overset{\circ}{Q} \\
d(i) \quad \mbox{при} \ i\in \overset{\bullet}{Q}
\end{array} \right.,\\
\overset{\bullet}{d}(i)=\overset{\bullet}{c}(d)(i)=
\left\{\begin{array}{c}
-d(i)+\sum\limits_{j:j-i}d(j) \quad \mbox{при} \ i\in \overset{\bullet}{Q} \\
d(i) \quad \mbox{при} \ i\in \overset{\circ}{Q}
\end{array} \right.,
\end{split}$$
$$\label{CD12}
\begin{split}
\overset{\circ}{\chi}(i)=
\left\{\begin{array}{c}
-\chi(i)+\sum\limits_{j:j-i}\chi(j) \quad \mbox{при} \ i\in \overset{\bullet}{Q_v^T} \\
\chi(i) \quad \mbox{при} \ i\notin \overset{\bullet}{Q_v^T}
\end{array} \right.,\\
\overset{\bullet}{\chi}(i)=
\left\{\begin{array}{c}
-\chi(i)+\sum\limits_{j:j-i}\chi(j) \quad \mbox{при} \ i\in \overset{\circ}{Q_v^T} \\
\chi(i) \quad \mbox{при} \ i\notin \overset{\circ}{Q_v^T}
\end{array} \right.,\\
\end{split}$$
(здесь $\overset{\bullet}{Q_v^T}=Q_v^T\cap\overset{\bullet}{Q}$ и $\overset{\circ}{Q_v^T}=Q_v^T\cap\overset{\circ}{Q}$).
$\overset{\bullet}{F}$ и $\overset{\circ}{F}$ есть функторы эквивалентности категорий, и можно проверить, что их двукратное применение приводит к функтору, эквивалентному тождественному.
В дальнейшем будем пользоваться обозначениями $\overset{\circ}{F}_k=\underbrace{\ldots
\overset{\circ}{F}\overset{\bullet}{F}\overset{\circ}{F}}_{k \
\mbox{\Small раз}}$, $\overset{\bullet}{F}_k=\underbrace{\ldots
\overset{\bullet}{F}\overset{\circ}{F}\overset{\bullet}{F}}_{k \
\mbox{\Small раз}}$.
В [@KruRoi05] функторы отражений Кокстера вводятся в предположении, что граф $G$ колчана $Q$ не содержит циклов. Конструкция функторов $\overset{\bullet}{F}$, $\overset{\circ}{F}$ практически без изменений переносится на все однократные разделённые колчаны, и для них имеют место формулы (\[CD11\]-\[CD12\]).
Пусть $T\in \textrm{Rep}_{os}(Q,\mathcal H)$, $Q_v^T$ — носитель, $d_T=d_T(i)$ — размерность, $\{\chi_T=\chi_T(i)\}$ — характер представления $T$ (напомним, что вне $Q_v^T$ он определён неоднозначно). Будем придавать $\chi_T(i)$ при $i\in Q_v \setminus
Q_v^T$ произвольные положительные значения, таким образом $\chi_T$ будет зависеть от $|Q_v|-|Q_v^T|=r$ положительных параметров.
Докажем теорему,
Если $Q$ есть разделённый однократный колчан, граф $G$ которого есть граф Дынкина, либо расширенный граф Дынкина, $d=\{d_i\}_{i\in Q_v}$ его *точный* (т.е. $d_i>0$) *действительный корень*, то неразложымые ортоскалярные представления колчана $Q$ в размерности $d$ зависят от $|Q_v|-1$ положительных параметров.
a\) Пусть $d$ — действительный сингулярный корень графа $G$ и $T$ — неразложимое ортоскалярное представление размерности $d$, $d$ получается из некоторого простейшего корня $d_g=(0,0,\ldots,1,0,\ldots,0)$ отражениями Кокстера $\overset{\circ}{c}, \overset{\bullet}{c}$ (1 соответствует вершине $g$). Пусть, для определённости, $g\in\overset{\bullet}{G}$. Тогда для некоторого $k\in \mathbb
N$, $d=\overset{\circ}{c}_kd_g$, а $T=\overset{\circ}{F}_k(\Pi_g)$. $\Pi_g$ зависит от $|Q_v|-1$ положительных параметров (положительных значений характера в вершинах $i\neq g$). От них же зависят скалярные операторы $\overrightarrow{T}\cdot\overrightarrow{T}^*$ при $i\in \overset{\bullet}{Q}$ и $T_j^{\downarrow*}\cdot T_j^{\downarrow}$ при $j\in
\overset{\circ}{Q}$ в ненулевых пространствах. Соответствие между наборами параметров и неразложимыми ортоскалярными представлениями в размерности $d$ взаимнооднозначное в силу обратимости преобразований $\overset{\circ}{c}$, $\overset{\bullet}{c}$, $\overset{\circ}{\chi}$, $\overset{\bullet}{\chi}$ и $\overset{\circ}{F}$, $\overset{\bullet}{F}$.
б) Пусть $d$ — действительный регулярный корень графа $G$ (для расширенного графа Дынкина) и $T$ — неразложимое ортоскалярное представление размерности $d$. Если $d$ — точный корень, то по лемме 5 он получается из некоторого неточного корня $\tilde
d$ отражениями Кокстера $\overset{\circ}{c}$, $\overset{\bullet}{c}$. Пусть (для определённости) $d=\overset{\circ}{c}_k \tilde
d$. Ортоскалярное представление $\widetilde T$ в размерности $\tilde d$ можно рассматривать как точное представление колчана, соответствующего графу Дынкина. Если $Q_v^{\widetilde T}$ — носитель представления $\widetilde T$, то $\widetilde T$ зависит по вышеизложенному от $|Q_v^{\widetilde T}|-1$ положительных параметров. Значения характера представления $\widetilde T$ в точках $Q_v \setminus Q_v^{\widetilde T}$ положим равными произвольным положительным числам: $T=\overset{\circ}{F}_k(\widetilde T)$, представление $T$ зависит от $|Q_v \setminus Q_v^{\widetilde T}|+|Q_v^{\widetilde T}|-1=|Q_v|-1$ положительных параметров.
Ортоскалярные представления, размерности которых есть мнимые корни
==================================================================
Пусть $Q$ — разделённый однократный колчан, граф которого $G$ есть расширенный граф Дынкина. Из теоремы 1 и леммы 4 следует, что единственный мнимый корень, который может быть размерностью неразложимого ортоскалярного представления колчана $Q$ есть $\delta_G$.
Докажем теорему аналогичную теореме 2, но относящуюся к мнимым корням графа $G$ (заметим, что в случае $\chi_T=\delta_G$ для графов $\widetilde D_4, \widetilde E_6, \widetilde E_7$ на другом языке и другими методами унитарная классификация неразложимых представлений получена в [@OstSam99]-[@Ost04]).
\[T3\] Если $Q$ есть разделённый однократный колчан, граф $G$ которого есть расширенный граф Дынкина, то в размерности $\delta_G$ неразложимые ортоскалярные представления колчана $Q$ зависят от $|Q_v|+1$ положительных параметров.
Теорему докажем фактическим описанием всех неразложимых представлений указанных колчанов в размерностях $\delta_G$.
a\) Пусть $G=\widetilde A_n$ — цикл с $n$ вершинами. Из предположения о разделенности и однократности колчана следует, что $n$ чётное и $n\!\geq4$. $\delta_{\widetilde
A_n}=(1,1,\ldots,1).$ Представление $T$ в размерности $\delta_{\widetilde A_n}$ задаётся $n$ ненулевым комплексными числами. Переходом к унитарноэквивалентному представлению $n-1$ число можно сделать положительным ($|Q_v|-1$ параметр); допустимые преобразования, не меняющие значения этих параметров, не меняют и $n$–е число (зависящее от 2-х положительных параметров — аргумента и модуля). Таким образом, $T$ зависит от $n$ параметров, принимающих произвольные положительные значения, и одного параметра $\phi$, принимающего значения на отрезке $(0,2\pi]$, т.е. от $|Q_v|+1=n+1$ параметров.
b\) Пусть $G=\widetilde D_n \ (n\geq4)$:
(240,40) (0,30) (20,20) (50,20) (60,18) (70,18) (80,18) (90,20) (120,20) (140,30) (140,10) (0,10)
(0,30)[(2,-1)[20]{}]{} (0,10)[(2,1)[20]{}]{} (20,20)[(1,0)[30]{}]{}(90,20)[(1,0)[30]{}]{} (140,30)[(-2,-1)[20]{}]{} (140,10)[(-2,1)[20]{}]{}
(-15,30)[$a_1$]{} (-15,10)[$a_2$]{} (20,25)[$c_1$]{} (45,25)[$c_2$]{} (80,25)[$c_{n-4}$]{} (110,25)[$c_{n-3}$]{} (145,30)[$b_1$]{} (145,10)[$b_2$]{}
(155,20)
(165,20)[$ \delta_{\widetilde D_n}=\begin{array}{c}
1 \\
1
\end{array}
2\ 2\ \ldots\ 2\ 2 \begin{array}{c}
1 \\
1
\end{array}$]{}
Пусть $T$ — неразложимое ортоскалярное представление размерности $\delta_{\widetilde D_n}$ соответствующего разделённого колчана $Q$. Для определённости, будем считать, что $c_1 \in \overset{\bullet}{Q}$.
Для удобства введём обозначения и нумерацию матриц представления следующим образом: $$\begin{split}
A_1&=T_{c_1,a_1}, \ A_2=T_{c_1,a_2}, \ X_1=T_{c_1,c_2}, \
X_2=T_{c_3,c_2},\ \ldots \\
B_1&=T_{c_{n-3},b_1}, \ B_2=T_{c_{n-3},b_2} \quad \mbox{при}\ \
n \ \
\mbox{чётном и} \\
B_1&=T_{b_1,c_{n-3}}, \ B_2=T_{b_2,c_{n-3}} \quad \mbox{при} \ \
n \ \
\mbox{нечётном.}
\end{split}$$ Рассмотрим случай чётного $n$. При $n=4$ ортоскалярные неразложымые представления графа $Q$ в размерности $\delta_{\widetilde D_4}$ зависят (см. [@KruNazRoi06]) от $6=|Q_v|+1$ положительных параметров.
Пусть $n>4$. Допустимыми (унитарными) преобразованиями матриц представления матрицы $X_1, X_2,\ldots,X_{n-4}$ можно диагонализировать с неотрицательными числами на диагоналях. Пусть $A=[A_1|A_2]$, $X_i=\left[
\begin{array}{cc}
x_i & 0 \\
0 & y_i \\
\end{array}
\right]$, $B=\left[
\begin{array}{c|c}
B_1 & B_2 \\
\end{array}
\right]
$, $i=\overline{1,n-4}$. Тогда из условий ортоскалярности представления мы легко получаем, что матрицы $AA^*$ и $BB^*$ также диагональны. Пусть $$\label{CD13}
AA^*=\left[
\begin{array}{cc}
x^2_0 & 0 \\
0 & y^2_0 \\
\end{array}
\right], \quad
BB^*=\left[
\begin{array}{cc}
x^2_{n-3} & 0 \\
0 & y^2_{n-3} \\
\end{array}
\right].$$ Тогда условия ортоскалярности выглядят так: $$\left[
\begin{array}{cc}
x^2_0 & 0 \\
0 & y^2_0 \\
\end{array}
\right]+
\left[
\begin{array}{cc}
x^2_{i+1} & 0 \\
0 & y^2_{i+1} \\
\end{array}
\right]=
\chi_{c_{i+1}}I_{c_{i+1}}, \quad i
\in \overline{0,n-4},$$ и поэтому $
x_i^2+x_{i+1}^2=y_i^2+y_{i+1}^2,
$ а значит $$\label{CD14}
y_{i+1}^2=x_{i+1}^2+(-1)^{i}(x_0^2-y_0^2), \quad i\in
\overline{0,n-4}.$$ Если $x_0^2=y_0^2$ (либо $x_j^2=y_j^2$ для любого фиксированого $j$), то из (\[CD14\]) следует, что $x_i^2=y_i^2$ для $i \in
\overline{0,n-3}$. В этом случае матрица $$\label{CD15}
\left[
\begin{array}{c|c|c|c}
A_1 & A_2 & B_1 & B_2
\end{array}
\right]$$ задаёт ортоскалярное представление $\widehat{T}$ колчана $\widehat{Q}$, соответствующего графу $\widetilde D_4$, и при переходе к эквивалентному представлению колчана $Q$, не меняющему приведенного вида матриц $X_i$, мы получаем переход к представлению колчана $\widehat{Q}$, эквивалентному представлению $\widehat{T}$. Такие представления колчана $\widehat{Q}$, как мы отметили выше, зависят не более, чем от 6 параметров. Соответствующие представления колчана $Q$ (учитывая параметры $x_1, x_2,\ldots,x_{n-4}$ и (\[CD14\])) зависят не более чем от $6+(n-4)=n+2=|Q_v|+1$ параметров.
Пусть $x_i\neq y_i$ при $i\in \overline{0,n-3}$ и $(U_i)_{i\in G_v}$ осуществляют эквивалентность представления $T$ c ортоскалярным представлением $\widetilde T$, для которого $\widetilde X_i=X_i$, $i\in \overline{0,n-4}$ ($U_i$ — унитарные матрицы). В этом случае легко убедится, что $$U_i=\left[
\begin{array}{c c}
u_i & 0 \\
0 & v_i \\
\end{array}
\right]$$
Такими преобразованиями в матрице $ \left[
\begin{array}{c|c}
A & B
\end{array}
\right]=\left[
\begin{array}{c|c|c|c}
a_{11} & a_{12} & b_{11} &
b_{12}\\
a_{21} & a_{22} & b_{21} &
b_{22}\\
\end{array}
\right]$ можно все елементы, кроме 2-х, для определённости, $b_{21}$ и $b_{22}$, сделать вещественными, и параметризовать следующим образом: $$[A|B]=\left[
\begin{array}{c|c|c|c}
x_0\cos\phi_1 & x_0\sin\phi_1 & x_{n-3}\cos\phi_2 &
x_{n-3}\sin\phi_2\\
y_0\sin\phi_1 & -y_0\cos\phi_1 & y_{n-3}\sin\phi_2e^{i\theta} &
-y_{n-3}\cos\phi_2e^{i\theta}\\
\end{array}
\right]$$ (здесь мы воспользовались тем, что $$AA^*=\left[
\begin{array}{cc}
x^2_0 & 0 \\
0 & y^2_0 \\
\end{array}
\right], \quad \mbox{и} \quad
BB^*=\left[
\begin{array}{cc}
x^2_{n-3} & 0 \\
0 & y^2_{n-3} \\
\end{array}
\right] ).$$
Такие представления зависят от $n+2=|Q_v|+1$ положительных параметров $x_0,\ x_1,\ \ldots,x_{n-3},\ y_0,\ \phi_1,\ \phi_2,\
\theta$.
Случай нечётного $n$ изучается аналогично с заменой матрицы $
\left[
\begin{array}{c|c|c|c}
A_1 & A_2 & B_1 & B_2
\end{array}
\right]
$ на матрицу $
\left[
\begin{array}{c|c|c|c}
A_1 & A_2 & B_1^* & B_2^*
\end{array}
\right]
$
с) Пусть $G=\widetilde E_6$; соответствующий разделенный колчан $Q$ имеет вид
(160,80)
(0,0) (30,0) (60,0) (90,0) (120,0) (60,30) (60,60)
(28,0)[(-1,0)[27]{}]{} (32,0)[(1,0)[27]{}]{} (88,0)[(-1,0)[27]{}]{} (92,0)[(1,0)[27]{}]{} (60,28)[(0,-1)[27]{}]{} (60,32)[(0,1)[27]{}]{}
(0,-10)[$a_1$]{} (30,-10)[$a_2$]{} (60,-10)[$z$]{} (90,-10)[$c_2$]{} (120,-10)[$c_1$]{} (65,25)[$b_2$]{} (65,55)[$b_1$]{}, (140,0)
Пусть $T$ — неразложимое ортоскалярное представление колчана $Q$ в размерности $\delta_{\widetilde E_6}$, $$T=\begin{array}{|c|c|c|}
\hline
0 & 0 & C_1 \\
\hline
0 & B_1 & 0 \\
\hline
A_1 & 0 & 0 \\
\hline
A_2 & B_2 & C_2\\
\hline
\end{array}$$ Здесь $
A_1=T_{a_1,a_2}; \ B_1=T_{b_1,b_2};\ C_1=T_{c_1,c_2}; \
A_2=T_{z,a_2}; \ B_2=T_{z,b_2}; \ C_2=T_{z,c_2}.
$ Матрица $A_1$ имеет размерность $1\times2$. Унитарными преобразованиями столбцов привёдем её к виду $
A_1=[0\ \ x_0], \quad x_0>0.
$ Унитарными преобразованиями строк матрицу $A_2=\left[
\begin{array}{cc}
a_{11} & a_{12} \\
a_{21} & a_{22} \\
a_{31} & a_{32} \\
\end{array}
\right]$ можно привести к виду $A_2=\left[
\begin{array}{cc}
x_1 & a_{12} \\
0 & a_{22} \\
0 & a_{32} \\
\end{array}
\right], \quad x_1>0 $ и из ортогональности столбцов матрицы $\begin{array}{|c|}
\hline A_1 \\
\hline A_2 \\
\hline
\end{array}$ следует $a_{12}=0$. $$D=\left[
\begin{array}{c|c|c}
A_2 & B_2 & C_2 \\
\end{array}
\right]=
\left[
\begin{array}{cc|cc|cc}
x_1 & 0 & b_{11} & b_{12} & c_{11} & c_{12} \\
0 & a_{22} & b_{21} & b_{22} & c_{21} & c_{22} \\
0 & a_{32} & b_{31} & b_{32} & c_{31} & c_{32} \\
\end{array}
\right].$$ Унитарными преобразованиями столбцов (не меняя $A_2$) можно добиться, чтобы $b_{11}=0$, $c_{11}=0$; $b_{12}\neq0$, $c_{12}\neq0$, иначе, можно проверить, представление разложимо; допустимыми преобразованиями можно сделать $b_{12}$ и $c_{12}$ положительными. Унитарными преобразованиями двух последних строк матрицы $D$ можно сделать $b_{22}=0$, а тогда из ортогональности первых двух строк получим, что $c_{22}=0$. Итак, $$D=\left[
\begin{array}{c|c|c}
A_2 & B_2 & C_2 \\
\end{array}
\right]=
\left[
\begin{array}{cc|cc|cc}
x_1 & 0 & 0 & b_{12} & 0 & c_{12} \\
0 & a_{22} & b_{21} & 0 & c_{21} & 0 \\
0 & a_{32} & b_{31} & b_{32} & c_{31} & c_{32} \\
\end{array}
\right].$$
Значения характера $\chi_i$ $(i\in Q_v)$ связаны соотношением $$\chi_{a_1}+\chi_{b_1}+\chi_{c_1}+3\chi_{x}=2(\chi_{a_2}+\chi_{b_2}+\chi_{c_2})$$ (следует сумму квадратов модулей матричных элементов матрицы $T$ подсчитать двумя способами — сначала складывая эти квадраты модулей по строкам, а потом по столбцам). Это дает возможность перейти к нормированному представлению со значением характера $\chi_z=1$, поделив все матрицы представления на одно и то же число $\sqrt{\chi_z}$ (нормировка представления не меняет минимального числа параметров, от которых зависит представление).
Вектор $(x_1,b_{12},c_{12})$, где $x_1>0,\ b_{12}>0,\ c_{12}>0$ и $x_1^2+b_{12}^2+c_{12}^2=1$, может быть параметризован следующим образом $$(x_1,b_{12},c_{12})=(\sin\psi_1,
\sin\phi_1\cos\psi_1,\cos\phi_1\cos\psi_1),$$ где $0<\phi_1<\frac{\pi}{2}$, $0<\psi_1<\frac{\pi}{2}$. Унитарными преобразованими столбцов элементы $a_{22}, b_{21}, c_{21}$ могут быть сделаны неотрицательными. Так как $a_{22}^2+
b_{21}^2+c_{21}^2=1$, то вектор $(a_{22}, b_{21}, c_{21})$ может быть параметризован следующим образом: $$(a_{22},b_{21},c_{21})=(\cos\phi_2 \cos\psi_2,
\sin\phi_2\cos\psi_2,\sin\psi_2),$$ где $0\leq\phi_2\leq\frac{\pi}{2}$, $0\leq\phi_2\leq\frac{\pi}{2}$, $\phi_2\cdot\psi_2\neq 0$ (иначе представление разложимо).
Умножением 3-й строки матрицы $D$ на элемент вида $e^{i\phi}$ (унитарное преобразование строк) сделаем элемент $c_{31}$ вещественным неотрицательным $$a_{32}\overline{a}_{32}+b_{31}\overline{b}_{31}+
b_{32}\overline{b}_{32}+c_{31}^2+
c_{32}\overline{c}_{32}=1$$ Пусть $$\label{CD16}
a_{32}\overline{a}_{32}+b_{31}\overline{b}_{31}+c_{31}^2=\sin^2\psi_3,
\ \
b_{32}\overline{b}_{32}+c_{31}^2=\cos^2\psi_3.$$ Учитывая ортогональность 1-й и 3-й строки матрицы $D$ и (\[CD16\]), вектор $(b_{32},c_{33})$ может быть пропараметризован так: $$(b_{32},c_{33})=(-\cos\phi_1\cos\psi_3 e^{i\theta_1},\sin\psi_1\cos\psi_3
e^{i\theta_1})$$ Учитывая (\[CD16\]) вектор $(a_{32}, b_{31}, c_{31})$ может быть пропараметризован так: $$(a_{32}, b_{31}, c_{31})=(\cos\phi_3\cos\psi_4\sin\phi_3 e^{i\theta_2},
\sin\phi_3\cos\psi_4\sin\phi_3
e^{i\theta_3},
\sin\psi_4\sin\psi_3).$$
Таким образом, в матрице $D$, которую мы будем называть *основой представления*, $$\begin{split}
A_2&=
\left[
\begin{array}{cc}
\sin\psi_1 & 0\\
0 & \cos\phi_2\cos\psi_2\\
0 & \cos\phi_3\cos\psi_4\sin\psi_3 e^{i\theta_2} \\
\end{array}
\right]
\\
B_2&=
\left[
\begin{array}{cc}
0 & \sin\phi_1\cos\psi_1\\
\sin\phi_2\cos\psi_2 & 0\\
\sin\phi_3\cos\psi_4\sin\psi_3 e^{i\theta_3} & -\cos\phi_1\cos\psi_3e^{i\theta_1} \\
\end{array}
\right] \\
C_2&=
\left[
\begin{array}{cc}
0 & \cos\phi_1\\
\sin\phi_2 & 0\\
\sin\psi_4\sin\psi_3 & \sin\phi_1\cos\psi_3e^{i\theta_1} \\
\end{array}
\right]
\end{split}$$ Следовательно, основа представления, зависит от вещественных параметров $$\label{CD17}
\phi_1, \ \phi_2, \ \phi_3,\ \psi_1, \ \psi_2, \ \psi_3, \
\psi_4, \ \theta_1, \ \theta_2, \ \theta_3.$$ Из условий ортоскалярности cледует, что строки матрицы $D$ должны быть ортонормированы. Из ортогональности 2-й и 3-й строки имеем соотношение: $$\begin{split}
&\cos\phi_2\cos\phi_3\cos\psi_2\sin\psi_3\cos\psi_4e^{i\theta_2}+
\sin\phi_2\sin\phi_3\cos\psi_2\sin\psi_3\cos\psi_4e^{i\theta_3}+\\&+
\sin\psi_2\sin\psi_3\sin\psi_4=0,
\end{split}$$ которому соответствует 2 “вещественных” соотношения — равенство нулю вещественной и мнимой части. Следовательно, среди параметров (\[CD17\]) только $8=|Q_v|+1$ независимых.
Покажем, что основой $D$ ортоскалярное представление $T$ определяется однозначно.
$$\left[\begin{array}{c}
A_1 \\
A_2 \\
\end{array}\right]=
\begin{array}{|cc|}
\hline 0 & x_0 \\
\hline x_1 & 0 \\
0 & a_{22} \\
0 & a_{32}\\
\hline
\end{array}\ , \quad x_0>0.$$
Из равенства длин столбцов следует $x_0^2+a_{22}^2+a_{32}\overline{a_{32}}=x_1^2$, так что положительное число $x_0$ по основе представления определяется однозначно. $$\left[\begin{array}{c}
B_1 \\
B_2 \\
\end{array}\right]=
\begin{array}{|cc|}
\hline b_{01} & b_{02} \\
\hline 0 & b_{12} \\
b_{21} & 0 \\
b_{31} & b_{32}\\
\hline
\end{array}\ , \quad
\left[\begin{array}{c}
C_1 \\
C_2 \\
\end{array}\right]=
\begin{array}{|cc|}
\hline c_{01} & c_{02} \\
\hline 0 & c_{12} \\
c_{21} & 0 \\
c_{31} & c_{32}\\
\hline
\end{array}$$
Унитарными преобразованиями строк элементы $b_{01}$ и $c_{01}$ можно сделать вещественными неотрицательными: $$b_{01}=y_{0}, \quad b_{02}=y_1e^{i\theta}.$$ Из условий ортоскалярности вытекает, что $$\begin{split}
b_{01}^2+b_{21}^2+|b_{31}|^2&=|b_{02}|^2+b_{12}^2+|b_{32}|^2,
\quad \mbox{так что} \\
b_{01}^2-|b_{02}|^2&=b_{21}^2+|b_{32}|^2-b_{21}^2-|b_{31}|^2=s.
\\
b_{01}b_{02}+\overline{b_{31}}b_{32}&=0, \quad \mbox{так что} \\
b_{01}b_{02}=b_{01}|b_{02}|e^{i\theta}&=\sin\phi_3\cos\psi_4\sin\psi_3\cos\phi_1\cos\psi_3e^{i(\theta_1-\theta_3)}.\\
\end{split}$$
Поэтому $\theta=\theta_1-\theta_3$ и $$\begin{split}
b_{01}^2\cdot(-|b_{02}|^2)=-\sin^2\phi_3\cos^2\psi_4\sin^2\psi_3\cos^2\phi_1\cos^2\psi_3=-t
\end{split}$$ Стало быть числа $b_{01}^2$ и $-|b_{02}|^2$ есть корни квадратного уравнения $$z^2-s\cdot z-t=0,\quad t>0.$$ Уравнение имеет два вещественных корня разного знака, таким образом $|b_{01}|$, $|b_{02}|$ и $\theta$ определяются основой предствления однозначно.
Аналогично по основе представления однозначно определяются и числа $c_{01}$, $c_{02}$. Стало быть в размерности $\delta_{\widetilde
E_6}$ неразложимые ортоскалярные представления зависят не более чем от $|Q_v|+1$, а неразложымые ортоскалярные представления общего положения в точности от $8=|Q_v|+1$ параметров.
d\) Пусть $G=\widetilde E_7$; соответствующий разделённый колчан имеет вид
(200,40)
(0,0) (30,0) (60,0) (90,0) (120,0) (150,0) (180,0) (90,30)
(2,0)[(1,0)[27]{}]{} (58,0)[(-1,0)[27]{}]{} (62,0)[(1,0)[27]{}]{} (90,28)[(0,-1)[27]{}]{} (118,0)[(-1,0)[27]{}]{} (122,0)[(1,0)[27]{}]{} (178,0)[(-1,0)[27]{}]{}
(0,-10)[$a_1$]{} (30,-10)[$a_2$]{} (60,-10)[$a_3$]{} (90,-10)[$z$]{} (120,-10)[$b_3$]{} (150,-10)[$b_2$]{} (180,-10)[$b_1$]{} (95,30)[$c_1$]{}
$
\begin{array}{ccccccccc}
& & & & 2 & & & \\
\mbox{Как извесно,}\quad \delta_{\widetilde E_7}= & 1 & 2 & 3 & 4 & 3 & 2& 1
\end{array}
$
Пусть $T$ — неразложимое ортоскалярное представление колчана $Q$ в размерности $\delta_{\widetilde E_7}$, $$\begin{gathered}
T=
\left[\begin{array}{ccccc}
A_{11} & A_{12} & 0 & 0 & 0 \\
0 & A_{22} & A_{23} & A_{24} & 0 \\
0 & 0 & 0 & A_{34} & A_{35} \\
\end{array}\right]\\
=
\left[\begin{array}{c|c|c|c|c}
\begin{array}{c}
a_{11} \\
a_{21}
\end{array}
& \begin{array}{ccc}
a_{12} & a_{13} & a_{14} \\
a_{21} & a_{22} & a_{24}
\end{array}
& 0 & 0 & 0 \\
\hline
0 & \begin{array}{ccc}
a_{32} & a_{33} & a_{34} \\
a_{42} & a_{43} & a_{44} \\
a_{52} & a_{53} & a_{54} \\
a_{62} & a_{63} & a_{64}
\end{array}
& \begin{array}{cc}
a_{35} & a_{36} \\
a_{45} & a_{46} \\
a_{55} & a_{56} \\
a_{65} & a_{66}
\end{array}
& \begin{array}{ccc}
a_{37} & a_{38} & a_{39} \\
a_{47} & a_{48} & a_{49} \\
a_{57} & a_{58} & a_{59} \\
a_{67} & a_{68} & a_{69}
\end{array} & 0 \\
\hline
0 & 0 & 0 & \begin{array}{ccc}
a_{77} & a_{78} & a_{79} \\
a_{87} & a_{88} & a_{89}
\end{array} & \begin{array}{c}
a_{7,10} \\
a_{8,10}
\end{array}\\
\end{array}\right] \ ,\end{gathered}$$
$$\begin{split}
A_{11}&=T_{a_2,a_1}; \ A_{12}=T_{a_2,a_3}; \ A_{22}=T_{z,a_3}; \
A_{23}=T_{z,c_1}; \\
A_{24}&=T_{z,b_3}; \ \ \; A_{34}=T_{b_2,b_3}; \ A_{35}=T_{b_2,b_1}.
\end{split}$$
Допустимыми унитарными преобразованиями строк и столбцов некоторые из элементов матрицы $T$ преобразуем в нули, некоторые из елементов станут равными нулю в силу ортогональности строк и столбцов внутри полос.
Символ $\overline{0}_k$ на каком-либо место матрицы $T$ будет означать, что на этом месте может буть получен ноль на $k$-м шаге унитарными преобразованиями столбцов вертикальной полосы, $0|_k$ — строк горизонтальной полосы, $\overrightarrow{{0}}_k$ — ноль получен в силу ортогональности столбцов вертикальной полосы, ${0\!\!\downarrow_k}$ — в силу ортогональности строк. При этом делая нули на $k$-м шаге, мы не “портим” нули, полученные ранее. Умножением строк и столбцов на числа вида $e^{i\phi}$ (унитарные преобразования) некоторые из элементов можно сделать вещественными неотрицательными, и мы на это укажем прямо в матрице ($a_{ij}\geq0$ либо $a_{ij}\leq0$). Мы указываем на строгое сравнение ($a_{ij}>0$ либо $a_{ij}<0$), если из равенства элемента $a_{ij}$ нулю следует разложимость представления. В результате матрицу $T$ можно привести к виду
$$\begin{gathered}
\left[
\begin{array}{ccc}
A_{11} & A_{12} & 0 \\
0 & A_{22} & A_{23} \\
0 & 0 & 0 \\
\end{array}
\right]= \\
\left[\begin{array}{c|c|c}
\begin{array}{c}
a_{11}\geq 0 \\
a_{21}\geq 0 \\
\end{array}
& \begin{array}{ccc}
0|_{10} & a_{13}\geq0 & a_{14}\leq0 \\
a_{22}>0 & a_{23}\leq0 & \overrightarrow{{0}}_{11}
\end{array}
&\begin{array}{cc}
0\quad & \quad 0 \\
0 \quad & \quad 0 \\
\end{array} \\
\hline
\begin{array}{c}
0 \\ 0 \\0 \\0 \\
\end{array} & \begin{array}{ccc}
a_{32}>0 & \overline{0}_3 & \overline{0}_3 \\
a_{42}>0 & a_{43}\geq0 & \overline{0}_7 \\
0|_4 & a_{53}\geq0 & a_{54}\geq0 \\
0|_4 & 0|_8 & a_{64}\geq0
\end{array}
& \begin{array}{cc}
a_{35}>0 & \overline{0}_3 \\
a_{45}<0 & \overrightarrow{{0}}_6 \\
0\!\!\downarrow_5 & a_{56}\geq0 \\
0\!\!\downarrow_5 & a_{66}
\end{array} \\
\hline
\begin{array}{cc}
0 \\
0 \\
\end{array} & \begin{array}{ccc}
0\quad \quad \quad & 0 & \quad \ \quad 0 \\
0\quad \quad \quad & 0 & \quad \ \quad 0 \\
\end{array} & \begin{array}{cc}
0\quad &\quad 0 \\
0\quad &\quad 0 \\
\end{array}
\end{array}\right]\end{gathered}$$
$$\left[
\begin{array}{cc}
0 & 0 \\
A_{24} & 0 \\
A_{34} & A_{35} \\
\end{array}
\right]=\left[
\begin{array}{c|c}
\begin{array}{ccc}
0 \quad \; & \quad\; 0 &\quad \quad 0 \\
0 \quad \; & \quad\; 0 &\quad \quad 0 \\
\end{array} & \begin{array}{c}
0 \\
0 \\
\end{array} \\
\hline
\begin{array}{ccc}
\overrightarrow{{0}}_3 & \overrightarrow{{0}}_3 & a_{39} \\
a_{47}>0 & \overline{0}_7 & 0|_2 \\
a_{57}\leq0 & a_{58}\geq0 & 0|_2 \\
0\!\!\downarrow_9 & a_{68} & 0|_2
\end{array} & \begin{array}{c}
0 \\
0 \\
0 \\
0
\end{array} \\
\hline
\begin{array}{ccc}
0|_{10} & a_{78}\geq0 & \overline{0}_1 \\
a_{87}\geq0 & a_{88}\geq0 & \overline{0}_1 \\
\end{array} & \begin{array}{c}
a_{7,10}\geq0 \\
a_{8,10}\leq0 \\
\end{array}\\
\end{array}\right]$$ Нормируем представление $T$ (будем считать, что $\chi_z=1$). Тогда матрицу $D=\left[
\begin{array}{ccc}
A_{22} & A_{23} & A_{24} \\
\end{array}
\right]$ (будем называеть ее основой представления) можно параметризировать следующим образом: $$\begin{gathered}
A_{22}=\left[
\begin{array}{ccc}
\cos\phi_1\cos\psi_1 & 0 & 0 \\
\sin\phi_1\cos\psi_2 & \cos\phi_2\sin\psi_2 & 0 \\
0 & \sin\phi_2\cos\psi_4 & \cos\phi_3\cos\psi_3\sin\psi_4\\
0 & 0 & \sin\phi_4\sin\psi_5 \\
\end{array}
\right] \\
A_{23}=
\left[
\begin{array}{cc}
\sin\phi_1\cos\psi_1 & 0 \\
-\cos\phi_1\cos\psi_2 & 0 \\
0 & \sin\phi_3\cos\psi_3\sin\psi_4\\
0 & \cos\phi_4\sin\psi_5e^{i\theta_1}\\
\end{array}
\right]\\
A_{24}=
\left[
\begin{array}{ccc}
0 & 0 & \sin\psi_1 \\
\sin\phi_2\sin\psi_2 & 0 & 0 \\
-\cos\phi_2\cos\psi_4 & \sin\psi_3\sin\phi_4 & 0 \\
0 & \cos\psi_5e^{i\theta_2} & 0 \\
\end{array}
\right]\end{gathered}$$
Строки матрицы $D$ ортонормированы, если дополнительно выполняется соотношение (означающее ортогональность 3-й и 4-й строк): $$\label{CD19}
\begin{split}
\cos\phi_3\sin\phi_4\cos\psi_3\sin\psi_4\sin\psi_5&+\sin\phi_3\cos\phi_4\cos\psi_3\sin\psi_4\sin\psi_5e^{i\theta_1}\\
&+\sin\psi_3\sin\psi_4\cos\psi_5e^{i\theta_2}=0.
\end{split}$$ Это соотношение эквивалентно двум вещественным, так что из 11 параметров $\phi_1-\phi_4,\ \psi_1-\psi_5,\ \theta_1,\ \theta_2$ независимыми являются $9=|Q_v|+1$.
Легко видеть, что по основе $D$ представления $T$ матрицы $A_{11}$, $A_{12}$, $A_{34}$, $A_{35}$ (ранее частично приведенные) находятся однозначно.
Так из ортоскалярности столбцов матрицы $\left[
\begin{array}{c}
A_{24} \\
A_{34} \\
\end{array}
\right]$ (т.е. из равенства $A_{24}^*A_{24}+ A_{34}^*A_{34}=\chi_{b_3}I_{b_3}$) следует, что $$\begin{split}
&a_{87}^2+a_{47}^2+a_{54}^2=a_{39}^2\\
&a_{87}a_{88}+a_{54}a_{58}=0\\
&a_{78}^2+a_{88}^2+a_{58}^2+a_{68}\overline{a_{68}}=a_{39}^2.
\end{split}$$ Из этих равенств последовательно и однозначно находятся элементы $a_{87}$, $a_{88}$, $a_{78}$. Из ортоскалярности строк матрицы $\left[
\begin{array}{cc}
A_{34} &
A_{35} \\
\end{array}
\right]$ (т.е. из равенства $A_{34}A_{34}^*+
A_{35}A_{35}^*=\chi_{b_2}I_{b_2}$) следует, что $$\begin{split}
&a_{78}^2+a_{7,10}^2=a_{87}^2+a_{88}^2+a_{8,10}^2\\
&a_{78}a_{88}+a_{7,10}a_{8,10}=0,
\end{split}$$ откуда однозначно находятся элементы $a_{7,10}$ и $a_{8,10}$.
Параметризуем столбцы матрицы $\left[
\begin{array}{c}
A_{12} \\
A_{22} \\
\end{array}
\right]$ следующим образом: $$\left[
\begin{array}{c}
A_{12} \\
A_{22} \\
\end{array}
\right]=
\left[
\begin{array}{ccc}
0 & x\sin\alpha_2\sin\beta_2 & -x\sin\beta_3 \\
x\sin\beta_1 & -x\cos\alpha_2\sin\beta_2 & 0 \\
\hline
x\cos\alpha_1\cos\beta_1 & 0 & 0 \\
x\sin\alpha_1\cos\beta_1 & x\sin\alpha_3\cos\beta_2 & 0 \\
0 & x\cos\alpha_3\cos\beta_2 & x\sin\alpha_4\cos\beta_3 \\
0 & 0 & x\cos\alpha_4\cos\beta_3
\end{array}
\right]\ ,$$ здесь $x>0$.
Покажем, что элементы матрицы $A_{12}$ элементами матрицы $A_{22}$ определяются однозначно. Так как $x\cos\alpha_1\cos\beta_1=a_{32}$, $x\sin\alpha_1\cos\beta_1=a_{42}$, то $x^2\cos^2\beta_1=a_{32}^2+a_{42}^2$. Аналогично, $x^2\cos^2\beta_2=a_{43}^2+a_{53}^2$ и $x^2\cos^2\beta_3=a_{54}^2+a_{64}^2$. Поэтому $$\begin{split}
x\sin\beta_1&=\sqrt{x^2-(a_{32}^2+a_{42}^2)} \\
x\sin\beta_2&=\sqrt{x^2-(a_{43}^2+a_{53}^2)} \\
x\sin\beta_3&=\sqrt{x^2-(a_{54}^2+a_{64}^2)} \\
\end{split}$$ Из ортогональности столбцов матрицы $\left[
\begin{array}{c}
A_{12} \\
A_{22} \\
\end{array}
\right]$ следует $$\begin{split}
-x^2\sin\beta_1\sin\beta_2\cos\alpha_2+a_{42}a_{43}&=0 \\
-x^2\sin\beta_2\sin\beta_3\sin\alpha_2+a_{53}a_{54}&=0
\end{split}$$ Поэтому $$\begin{split}
\cos\alpha_2&=\frac{a_{42}a_{43}}{\sqrt{\big[x^2-(a_{32}^2+a_{42}^2)\big]\cdot\big[x^2-(a_{43}^2+a_{53}^2)\big]}}
\\
\sin\alpha_2&=\frac{a_{53}a_{54}}{\sqrt{\big[x^2-(a_{54}^2+a_{64}^2)\big]\cdot\big[x^2-(a_{43}^2+a_{53}^2)\big]}}
\end{split}$$
Значение $x=\chi_{a_3}$ определяется по основе представления однозначно: если $A_{22}$ унитарно эквивалентна матрице $\textrm{diag}\{r_1,r_2,r_3\}$, $r_i\geq0$, то $x=\max\limits_{i}r_i$ (это легко получить, диагонализируя матрицы $A_{12}$, $A_{22}$ унитарными преобразованиями, при этом существенно используется ортоскалярность представления). Элементы $a_{11}$, $a_{21}$ однозначно находятся по матрице $A_{12}$ (аналогично тому, как находятся єлементы $a_{7,10}$, $a_{8,10}$ по матрице $A_{34}$).
e\) Пусть $G=\widetilde E_8$; соответствующий разделенный колчан имеет вид
(220,40)
(0,0) (30,0) (60,0) (90,0) (120,0) (150,0) (180,0) (150,30) (210,0)
(2,0)[(1,0)[27]{}]{} (58,0)[(-1,0)[27]{}]{} (62,0)[(1,0)[27]{}]{} (150,28)[(0,-1)[27]{}]{} (118,0)[(-1,0)[27]{}]{} (122,0)[(1,0)[27]{}]{} (178,0)[(-1,0)[27]{}]{} (182,0)[(1,0)[27]{}]{}
(0,-10)[$a_1$]{} (30,-10)[$a_2$]{} (60,-10)[$a_3$]{} (90,-10)[$a_4$]{} (120,-10)[$a_5$]{} (150,-10)[$z$]{} (180,-10)[$b_2$]{} (210,-10)[$b_1$]{} (155,30)[$c_1$]{}
Как извесно, $$\begin{array}{cccccccccc}
& & & & & & 3 & &\\
\delta_{\widetilde E_8}= & 1 & 2 & 3 & 4 & 5 & 6 & 4 & 2
\end{array}$$
Пусть $T$ — неразложимое ортоскалярное представление колчана $Q$ в размерности $\delta_{\widetilde E_8}$, $$T=\left[\begin{array}{ccccc}
A_{11} & A_{12} & 0 & 0 & 0 \\
0 & A_{22} & A_{23} & 0 & 0 \\
0 & 0 & A_{33} & A_{34} & A_{35} \\
0 & 0 & 0 & 0 & A_{45} \\
\end{array}
\right],$$ где $$\begin{split}
A_{11}&=T_{a_2,a_1}; \ A_{12}=T_{a_2,a_3};\ A_{22}=T_{a_4,a_3}; \ A_{23}=T_{a_4,a_5};\\
A_{33}&=T_{z,a_5}; \ A_{34}=T_{z,c_1}; \ A_{35}=T_{z,b_2}; \
A_{45}=T_{b_1,b_2}.
\end{split}$$
Допустимыми унитарными преобразованиями строк и столбцов некоторые мз елементов матрицы $T$ преобразуем в нули, некоторые из елементов станут равными нулю в силу ортогональность строк и стоблцов внутри полос. Следуя предыдущим договорённостям об обозначениях, матрицу $T$ можно привести к виду, при котором
$$\begin{gathered}
\left[
\begin{array}{ccc}
A_{11} & A_{12} \\
0 & A_{22} \\
\end{array}
\right]= \left[
\begin{array}{c|ccc}
a_{11}>0 & a_{12}>0 & \overrightarrow{{0}}_{26} & \overrightarrow{{0}}_{26}\\
0\!\!\downarrow_{27} & O|_{25} & a_{23}>0 & a_{24}<0\\
\hline
0 & a_{32}>0 & \overline{0}_{21} & \overline{0}_{21}\\
0 & 0\!\!\downarrow & a_{43}>0 & \overline{0}_{23} \\
0 & 0\!\!\downarrow & a_{53}>0 & a_{54}>0 \\
0 & 0\!\!\downarrow & 0\!\!\downarrow_{24} & a_{64}>0 \\
\end{array}
\right]
\\
\left[
\begin{array}{c}
A_{23} \\
A_{33} \\
\end{array}
\right]= \left[
\begin{array}{ccccc}
a_{35}>0 & \overrightarrow{{0}}_{17} & \overrightarrow{{0}}_{17} & \overrightarrow{{0}}_{17} & \overrightarrow{{0}}_{17} \\
0|_{16} & a_{46}>0 & a_{47}<0 & \overrightarrow{{0}}_{19} & \overrightarrow{{0}}_{19} \\
0|_{16} & 0|_{18} & a_{57}>0 & a_{58}<0 & \overrightarrow{{0}}_{21} \\
0|_{16} & 0|_{18} & 0|_{20} & a_{59}>0 & a_{69}<0 \\
\hline
a_{75}>0 & \overrightarrow{{0}}_2 & \overrightarrow{{0}}_2 & \overrightarrow{{0}}_2 & \overrightarrow{{0}}_2 \\
0|_1 & a_{86}\geq0 & \overline{0}_6 & \overline{0}_6 & \overline{0}_6 \\
0|_1 & a_{96}\geq0 & a_{97}\geq0 & \overline{0}_7 & \overline{0}_7 \\
0|_1 & 0|_7 & a_{10,7}\geq0 & a_{10,8}\geq0 & \overline{0}_{11} \\
0|_1 & 0|_7 & 0|_{10} & a_{11,8}\geq0 & a_{11,9}\geq0 \\
0|_1 & 0|_7 & 0|_{10} & a_{12,9}\geq0 & a_{12,9}\geq0
\end{array}
\right] \end{gathered}$$
$$\begin{gathered}
\left[
\begin{array}{cc}
A_{34} & A_{35}\\
0 & A_{45}\\
\end{array}
\right]=\\
\left[
\begin{array}{ccc|ccccc}
a_{7,10}>0 & \overline{0}_2 & \overline{0}_2 & a_{7,13}>0 & \overline{0}_2 & \overline{0}_2 & \overline{0}_2 \\
a_{8,10}>0 & \overrightarrow{{0}}_6 &
\overrightarrow{{0}}_6
& a_{8,13}<0 & a_{8,14}>0 & \overrightarrow{{0}}_6 & \overrightarrow{{0}}_6 \\
0|_3 & \!\!\!\! a_{9,11}>0 & \overline{0}_7 & 0\!\!\downarrow_4 & a_{9,14}<0 & \overrightarrow{{0}}_9 & \overrightarrow{{0}}_9 \\
0|_3 & \!\!\!\! a_{10,11}<0 & \overline{0}_{13} & 0\!\!\downarrow_4 & 0\!\!\downarrow_8 & a_{10,15}>0 & \overline{0}_{14} \\
0|_3 & 0\!\!\downarrow_{14} &\!\!\!\! a_{11,12}\geq0 & 0\!\!\downarrow_4 & 0\!\!\downarrow_8 & a_{11,15}<0 & a_{11,16}\geq0 \\
0|_3 & 0\!\!\downarrow_{14} &\!\!\!\! a_{12,12} & 0\!\!\downarrow_4 & 0\!\!\downarrow_8 & 0\!\!\downarrow_{15} &
a_{12,16} \\
\hline
0 & 0 &0 & a_{13,13}>0 & a_{13,14}>0 & 0_5 & 0_5 \\
0 & 0 &0 & 0\!\!\downarrow_5 & 0\!\!\downarrow_5 & a_{14,15}>0 & a_{14,16}>0 \\
\end{array}
\right]\end{gathered}$$
Основу представления $D=\left[
\begin{array}{ccc}
A_{33} & A_{34} & A_{35} \\
\end{array}
\right]$ можно параметризовать следующим образом $$\begin{split}
a_{75}&=\sin\psi_1, \ a_{7,10}=\cos\phi_1\cos\psi_1, \
a_{7,13}=\sin\phi_1\cos\psi_1 \\
a_{86}&=\sin\phi_2\sin\psi_2, \ a_{8,10}=\sin\phi_1\cos\psi_2,\
a_{8,13}=-\cos\phi_1\cos\psi_2,\\ &a_{8,14}=\cos\phi_2\sin\psi_2\\
a_{96}&=\cos\phi_2\sin\psi_3, \ a_{97}=\sin\phi_3\cos\psi_3, \
a_{9,11}=\cos\phi_3\cos\psi_3; \\
&a_{9,14}=-\sin\phi_2\sin\psi_3
\end{split}$$ $$\begin{split}
a_{10,7}&=\cos\phi_3\sin\psi_4; \ a_{10,8}=\sin\phi_4\cos\psi_4;
\ a_{10,11}=-\sin\phi_3\sin\psi_4; \\
&a_{10,15}=-\cos\phi_4\cos\psi_4 \\
a_{11,8}&=\cos\phi_4\cos\psi_4\cos\psi_5; \
a_{11,9}=\cos\phi_5\sin\psi_{5}; \
a_{11,12}=\sin\phi_5\sin\psi_5; \\
&a_{11,15}=\sin\phi_4\cos\psi_4\cos\psi_5;\
a_{11,16}=\sin\psi_4\cos\psi_5 \\
a_{12,9}&=\cos\phi_6\cos\psi_6;\
a_{12,12}=\sin\phi_6\cos\psi_6e^{i\theta_1}; \
a_{12,16}=-\sin\psi_6e^{i\theta_2}.
\end{split}$$
Строки матрицы $D$ ортонормированы, если дополнительно выполняется соотношение (означающее ортогональность 5-й и 6-й строк): $$\begin{split}
\cos\phi_5\cos\phi_6\sin\psi_5\cos\psi_6&+\sin\phi_5\sin\phi_6\sin\psi_5\cos\psi_6e^{i\theta_1}\\
&-\sin\psi_4\cos\psi_5\sin\psi_6e^{i\theta_2}=0.
\end{split}$$
Это соотношение эквивалентно двум вещественным.
Кроме того из ортоскалярности представления следует равенство длин столбцов (как векторов в унитарном пространстве) матрицы $A_{34}$, поэтому имеем еще 2 соотношения: $$\begin{split}
\cos^2\phi_1\cos^2\psi_1+\sin^2\phi_1\cos^2\psi_2&=\cos^2\phi_3\cos^2\psi_3+\sin^2\phi_3\sin^2\psi_4\\
\mbox{и} \quad
\cos^2\phi_1\cos^2\psi_1+\sin^2\phi_1\cos^2\psi_2&=\sin^2\phi_5\sin^2\psi_5+\sin^2\phi_6\cos^2\psi_6.
\end{split}$$ Таким образом из 14 параметров $\phi_1-\phi_6, \ \psi_1-\psi_6,\
\theta_1,\ \theta_2$ независимыми являются $10=|Q_v|+1$.
Также, как и в предыдущих случаях, легко показать, что по основе $D$ представления $T$ матрицы $A_{11},\ A_{12},\ A_{22},\ A_{23},\
A_{45}$ (ранее частично приведенные) находятся однозначно.
Неразложимые представления расширенных графом Дынкина в категории линейных пространств изучены в [@Naz73],[@DonFre73].
Как и для категории линейных пространств, размерности неразложимых ортоскалярных представлений этих графов не ограничены в совокупности. В размерности, являющейся действительным корнем графа, при фиксированном характере, неразложимое представление единственно. Для размерностей, являющихся мнимыми корнями графа, неразложимые ортоскалярные представления, в отличие от линейно-алгебраического случая, существуют только в размерности совпадающей с минимальным положительным мнимым корнем, но также зависят, при фиксированном характере, от одного (комплексного) параметра.
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Ройтер А.В. Боксы с инволюцией // в кн.: Представления и квадратичные формы. 155, АН УССС, Ин-т мат., Киев. – 1979. – c. 124–126.
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---
abstract: 'Understanding the transport properties of a porous medium from a knowledge of its microstructure is a problem of great interest in the physical, chemical and biological sciences. Using a first-passage time method, we compute the mean survival time $\tau$ of a Brownian particle among perfectly absorbing traps for a wide class of triply-periodic porous media, including minimal surfaces. We find that the porous medium with an interface that is the Schwartz P minimal surface maximizes the mean survival time among this class. This adds to the growing evidence of the multifunctional optimality of this bicontinuous porous medium. We conjecture that the mean survival time (like the fluid permeability) is maximized for triply periodic porous media with a simply connected pore space at porosity $\phi=1/2$ by the structure that globally optimizes the specific surface. We also compute pore-size statistics of the model microstructures in order to ascertain the validity of a “universal curve" for the mean survival time for these porous media. This represents the first nontrivial statistical characterization of triply periodic minimal surfaces.'
author:
- 'Jana Gevertz$^1$ and S. Torquato$^{1,2,3,4,5}$'
title: Mean survival times of absorbing triply periodic minimal surfaces
---
INTRODUCTION
============
Fluid-saturated porous media are ubiquitous in nature (e.g., geological and biological media) and in synthetic situations (e.g., filters, cements and foams) [@torquato02; @Sa03]. Understanding the transport properties of a porous medium from a knowledge of its microstructure is a subject that spans many fields [@torquato02; @Sa03; @Co96; @Ka02; @Ol04; @Ha07]. In particular, physical phenomena involving simultaneous diffusion and reaction in porous media abound in the physical and biological sciences [@torquato02; @Sa03; @Be93]. Considerable attention has been devoted to instances in which diffusion occurs in the pore region of the porous medium with a “trap” region whose interface can absorb the diffusing species via a surface reaction. Examples are found in widely different processes, such as heterogeneous catalysis, fluorescence quenching, cell metabolism, ligand binding in proteins, migration of atoms and defects in solids, and crystal growth [@torquato02]. A key parameter in such processes is the [*mean survival time*]{} $\tau$, which gives the average lifetime of the diffusing species before it gets trapped.
It is noteworthy that while there has been a significant amount of progress made on the determination of the structures that optimize a variety of transport and mechanical properties of porous media [@torquato02; @hashin; @kohn], there have been no studies that have attempted to find the optimal isotropic structures for the mean survival time $\tau$ [@To04]. It has been recently demonstrated that triply periodic two-phase bicontinuous composites with interfaces that are the Schwartz P and D minimal surfaces are not only geometrically extremal, but are also extremal for simultaneous transport of heat and electricity [@torquato02b; @compet; @Si07]. A minimal surface is one that is locally area minimizing i.e., every point has zero mean curvature. Triply periodic minimal surfaces are minimal surfaces that are periodic in all three coordinate directions. An important subclass of triply-periodic minimal surfaces are those that partition space into two disjoint but intertwining regions that are simultaneously continuous (i.e., bicontinuous) [@anderson90; @jung07]. Examples of such surfaces include the Schwartz primitive (P), the Schwartz diamond (D), and the Schoen gyroid (G) surfaces \[see Fig. \[fig:1\]\]; each disjoint region has volume fraction equal to 1/2. Such triply periodic minimal surfaces arise in a variety of systems, including nanocomposites [@nano], micellar materials [@micelle], block copolymers [@block] as well as lipid-water systems and certain cell membranes [@mem1; @mem2; @mem3; @mem4].
![\[fig:1\] (Color online) Unit cells of three different minimal surfaces. (a) Schwartz P surface, (b) Schwartz D surface, (c) Schoen G surface. Image is adapted from Ref. [@jung05].](figure1.eps){width="4.5in"}
The multifunctionality of such two-phase systems has been further established by showing that they are also extremal when a competition is set up between the effective bulk modulus and electrical (or thermal) conductivity of the bicontinuous composite [@torquato04]. Jung and Torquato [@jung05] also explored the multifunctionality of the three minimal surfaces shown in Fig. \[fig:1\] with respect to Stokes (slow viscous) flow. The simulations conducted by Jung and Torquato revealed that the Schwartz P porous medium has the largest fluid permeability $k$ among a class of structures studied with porosity $\phi=1/2$. Further, the fluid permeability was found to be inversely proportional to the specific surface $s$ (interface area per unit volume). This led the authors to conjecture that the maximal fluid permeability for a triply periodic porous medium with a simply connected pore space [@simply] at a porosity $\phi$=1/2 is achieved by the structure that globally minimizes the specific surface [@jung05].
In this manuscript, we explore whether the mean survival time $\tau$ of a Brownian particle among perfectly absorbing traps is maximized by the Schwartz P structure among a wide class of triply-periodic porous media based on the class of six models studied in Ref. [@jung05]. Further, we go beyond considering just a representative medium from each class of models and explore several of the models over a [*range*]{} of parameter values, as will be further explained below. This inquiry is also motivated by a cross-property bound that rigorously links the fluid permeability $k$ to the mean survival time $\tau$ [@torquato90]: $$k \le \tau.
\label{k-tau}$$ It should be noted that the inequality of (\[k-tau\]) becomes an equality for transport in parallel cylindrical tube bundles of arbitrary cross-sectional geometry [@torquato90; @tube]. It is also a relatively tight bound (when appropriately scaled) for transport around distributions of inclusions [@torquato90]. It is clear that the permeability $k$ will be maximized for a simply connected pore space (e.g., the presence of dead ends are undesirable because they would not contribute to the flow). The inequality (\[k-tau\]) and the results of Ref. [@jung05] suggest that $\tau$ is maximized by the same simply connected microstructure that maximizes $k$ for $\phi=1/2$.
We will also test whether our calculations for $\tau$ collapse on to a “universal" scaling relation for the mean survival time [@torquato97]. This requires us to compute the pore-size density functions for the triply periodic surfaces considered in this paper.
In Sec. II, we define terminologies and give a precise statement of the problem. In Sec. III, we describe the first-passage time technique that we utilize to compute the mean survival time for a wide class of triply periodic porous **media** that are generally bicontinuous. Section IV reports our finding for $\tau$. In Sec. V, we discuss a universal scaling relation for $\tau$, report pore-size statistics for the triply periodic porous media studied here, and ascertain the applicability of the universal curve for these structures. Finally, in Sec. VI, we make concluding remarks and discuss the ramifications of our results.
Definitions and Problem Statement
=================================
The mean survival time $\tau$ arises in steady-state diffusion of reactants in a trap-free pore region $\mathcal{V}_1$ with diffusion coefficient $D$ among static traps with a unit rate of production of the reactants per unit pore volume [@torquato02]. When the reactants come in contact with the pore-trap interface $\partial\mathcal{V}$, they get absorbed. Using homogenization theory [@torquato02], it has been shown that $\tau$, the average time traveled before a diffusing particle gets trapped, is given by: $$\tau = \frac{{\displaystyle}\langle u \rangle}{{\displaystyle}\phi D},$$ where $u(\mathbf{r})$ is the scaled concentration field of reactants, which satisfies the diffusion equation $$\Delta u = -1 \quad \text{ in }\mathcal{V}_1$$ $$u = 0 \quad \text{ on } \partial\mathcal{V},$$ and $\Delta$ is the Laplacian operator.
Following Ref. [@jung05], we compute $\tau$ for a wide class triply periodic structures: the Schwartz P, Schwartz D and Schoen G minimal surfaces \[Fig. \[fig:1\]\], a cubic pore-square channel model \[Fig. \[fig:2\](a)\], and a spherical pore-circular channel model \[Fig. \[fig:2\](b)\], and an array of spherical traps arranged on a simple cubic lattice [@To92] (which of course is not bicontinuous). The pore-channel models each contain two parameters, $a$ and $b$, that enable one to control the relative size of the pores (either spherical or cubic in shape) to the size of the channels (either circular or square cross-sections) [@jung05]. The parameter $a$ determines channel width/diameter, whereas $b$ determines the width/diameter of the pore. This class of models is bicontinuous provided that $a>0$. In the current investigation, we compute $\tau$ for the pore-channel models using a wider range of values for $a$ and $b$ than used in Ref. [@jung05], while keeping $\phi$ fixed at 1/2.
![\[fig:2\] (Color online) Unit cells of two different pore-channel models. (a) Cubic pore-square channel and its cross-section. (b) Spherical pore-circular channel and its cross section. Image is adapted from Ref. [@jung05].](figure2.eps){width="4.in"}
FIRST-PASSAGE TIME METHOD FOR COMPUTING MEAN SURVIVAL TIME
==========================================================
The mean survival time $\tau$ for the aforementioned triply periodic porous media could be calculated using standard random-walk techniques that simulate the detailed zig-zag trajectory of a Brownian particle [@Le89]. We instead compute $\tau$ for these microstructures using an efficient first-passage time algorithm [@torquato89]. The fundamental periodic cell is taken to be a cube of side length unity.
The first-passage time algorithm is implemented by applying the following set of rules:
1. Introduce a Brownian particle into a random position in the trap-free phase.
2. While the walker is sufficiently far from the two-phase interface, construct the largest sphere of radius $R_i$ centered at the Brownian particle that just touches the two-phase interface.
3. In one step, the walker jumps to a random point on the surface of the sphere \[Fig. \[fig:FPT\]\], with an average hitting time of: $$\bar{t} = R_i^2/(6D).$$
4. Repeat steps (2)-(3) until the walker is within some prescribed small distance $\delta$ (taken to be $10^{-8}$ in our simulations) from the two-phase interface.
5. Repeat steps (1)-(4) for many random walkers and calculate the mean survival time from the following equation: $$\tau = \frac{{\displaystyle}\left\langle \sum_i R_i^2 \right\rangle}{{\displaystyle}6D},$$ where angular brackets denote an ensemble average.
The mean survival time $\tau$ of the pore-channel models are determined using this algorithm. However, since we only have discrete (not continuous) representations of the minimal surfaces [@jung05], the discrete analog of the algorithm is utilized in these cases [@torquato99]. In the discrete case, first-passage cubes of length $2R_i$ are utilized instead of first-passage spheres, and the average time it takes to move to the surface of a first-passage cube is $\bar{t} \approx 0.225R_i^2/D$.
![\[fig:FPT\] (Color online) Two-dimensional depiction of continuous first-passage time method applied to the spherical pore-circular channel model. The Brownian particle walks, by jumping to the surface of the largest possible first-passage sphere at each step, until it gets trapped at the two-phase interface (maroon boundary layer, which appears as dark grey if the image is being viewed in black and white).](figure3.eps){width="4.0in"}
RESULTS FOR THE MEAN SURVIVAL TIMES
===================================
The value of $\tau$ for each of the aforementioned models is given in Table \[tab:table1\], along with the fluid permeability and specific surface. All fluid permeability measurements (if available), as well as the specific surface measurements for the minimal surfaces, and the pore-channel models with $b=0$ are taken from Ref. [@jung05]. We see that $\tau$ always obeys the rigorous bound specified by (1). We also see that a simply connected pore phase is a crucial topological feature required to achieve large mean survival times at a porosity $\phi=1/2$. However, a simply connected pore phase is not a sufficient condition, as evidenced by the relatively small mean survival times associated with the Schwartz D and Schoen G minimal surfaces. These two minimal surfaces have rather large specific surfaces and hence serve as efficient traps for the diffusing Brownian particles. Indeed, we see that the survival times are inversely proportional to the corresponding specific surfaces for all of the bicontinuous structures. The pore spaces of structures with large $\tau$ are expected to be simply connected and therefore it would not be unreasonable for the survival time to be inversely proportional to the specific surface in these instances. As hypothesized, we find that among the structures examined, the Schwartz P porous medium maximizes the mean survival time. To get some idea of the fluctuations about the average $\tau$ values, we computed the associated variance $\sigma_{\tau}^2$ for trapping for the three minimal surfaces and found that $\sigma_{\tau}^2=0.00051, 0.00013$ and $0.00005$ for the Schwartz P, Schoen G and Schwartz D surfaces, respectively.
**Structure** $\boldsymbol{\tau}$ $\mathbf{k}$ $\mathbf{s}$
------------------------------- --------------------- -------------- --------------
Schwartz P 0.0173950 0.0034765 2.3705
Schoen G 0.0093266 0.0022889 3.1284
Schwartz D 0.0060414 0.0014397 3.9011
Cubic-pore channel
($a=0.25$; $b=0$) 0.0139289 0.0030744 3.0000
($a=0.1324$; $b=0.25$) 0.0123813 0.0005310 3.8360
($a=0.0781$; $b=0.3125$) 0.0122259 0.0000948 3.9254
($a=0$; $b=0.3969$) 0.0127852 - 3.7804
Sphere-pore channel
($a=0.2836$; $b=0$) 0.0167934 0.0034596 2.63990
($a=0.1480$; $b=0.2794$) 0.0164995 - 2.73649
($a=0.0908$; $b=0.3633$) 0.0161733 - 2.93057
($a=0$; $b=0.4924$) 0.0161093 - 3.04681
Spherical trap ($\phi = 0.5$) 0.0139640 0.0030591 3.0780
: \[tab:table1\] Mean survival time $\tau$, fluid permeability $k$, and specific surface $s$ of triply periodic structures. All quantities are made dimensionless using the side length of the unit cell.
Given that both the mean survival time $\tau$ and fluid permeability $k$ are made dimensionless with the side length of the unit cell, one might ask why these dimensionless bulk properties are two to three orders of magnitude smaller than unity? This is a well-known behavior for such transport properties of porous media, optimal or not [@torquato02; @Sa03]. For example, the fluid permeability (average of the velocity field) can be regarded to be the “effective pore channel area of the dynamically connected" part of the pore space. For non-simply connected pore spaces, there generally will be regions that contain fluid but do not actively contribute to the flow (not dynamically connected). Moreover, because of the no-slip condition, the velocity only becomes significantly large sufficiently away from the pore-solid interface for general porous media. Thus, these two effects conspire to make the effective area of a pore channel, i.e., the fluid permeability $k$, considerably less than the geometric pore channel sizes. (This effect is even true for the simple case of flow in a tube; see Ref. [@tube]). Similar arguments apply to the mean survival time. The perfectly absorbing boundary condition at the pore-solid interface (i.e., zero concentration field) means that the concentration field becomes appreciably large sufficiently away from the pore-solid interface, which results in a mean survival time (average of the concentration field) that is generally several orders of magnitude smaller than dictated by the largest pore dimensions.
Pore-Size Functions and Universal Scaling Relation for Survival Time
=====================================================================
Pore-Size Function
------------------
Porous media whose interfaces are triply periodic minimal surfaces apparently have remarkable macroscopic properties. Nonetheless, these structures have yet to be statistically characterized using nontrivial descriptors. Here we present pore-size functions for these structures as well as the triply periodic circular channels. This is motivated by the fact that the pore-size density function $P(\delta)$ arises in rigorous lower bounds on $\tau$ [@torquato02] as well as a universal curve for $\tau$ [@torquato97], described below. The quantity $P(\delta)d\delta$ gives the probability that a randomly chosen point in the pore region lies at a distance between $\delta$ and $\delta + d\delta$ from the nearest point on the interface [@torquato02].
We employ the following algorithm to determine the pore-size function $P(\delta)$ [@coker95]:
1. Choose a random location in the pore phase.
2. Find the radius of the largest sphere centered at the above point that just touches the two-phase interface.
3. Repeat steps 1-2 for many random locations and create a list of radii.
4. After sampling sufficiently, bin the sphere radii. Divide the number of radii in each bin by the total number of radii to determine $P(\delta)$.
The interpolated functional form of $P(\delta)$ is given in Eq. (\[SchwartzP\]), (\[SchwartzD\]), (\[SchoenG\]) and (\[circ\_channel\]) for the Schwartz P, Schwartz D, Schoen G minimal surfaces, and the circular channel model (spherical pore circular-channel model with $b$ = 0), respectively \[see Fig. \[fig:pore\_size\]\]: $$\begin{aligned}
P(\delta) = \left\{ \begin{array}{cr}
-149\delta^5 + 166\delta^4 - 61.2\delta^3 + 8.2\delta^2 -
0.696\delta + 0.164, & \textrm{if $\delta < 0.419$} \\
0, & \textrm{otherwise}
\end{array} \right.
\label{SchwartzP}\end{aligned}$$
$$\begin{aligned}
P(\delta) = \left\{ \begin{array}{cr}
2238\delta^5 - 523\delta^4 - 44.1\delta^3 + 12.8\delta^2 -
1.55\delta + 0.275, & \textrm{if $\delta < 0.202$} \\
0, & \textrm{otherwise}
\end{array} \right.
\label{SchwartzD}\end{aligned}$$
$$\begin{aligned}
P(\delta) = \left\{ \begin{array}{cr}
-1388\delta^5 + 897\delta^4 - 225\delta^3 + 23.4\delta^2 -
1.47\delta + 0.223, & \textrm{if $\delta < 0.234$} \\
0, & \textrm{otherwise}
\end{array} \right.
\label{SchoenG}\end{aligned}$$
$$\begin{aligned}
P(\delta) = \left\{ \begin{array}{cr}
-162\delta^5 + 173\delta^4 - 59.9\delta^3 + 7.57\delta^2 -
0.764\delta + 0.176, & \textrm{if $\delta < 0.234$} \\
0, & \textrm{otherwise}.
\end{array} \right.
\label{circ_channel}\end{aligned}$$
For the spherical pore (spherical pore circular-channel model with $a$ = 0), the pore size density function is known exactly:
$$\begin{aligned}
P(\delta) = \left\{ \begin{array}{cr}
\frac{{\displaystyle}3(a-\delta)^2}{{\displaystyle}a^3}, & \textrm{if $\delta < a$} \\
0, & \textrm{otherwise.}
\end{array} \right.\end{aligned}$$
![\[fig:pore\_size\] (Color online) Pore size density function data generated from the presented algorithm for the three minimal surfaces and the circular-channel model, along with the best-fit fifth degree polynomial.](figure4.eps){width="5.0in"}
Universal Curve
---------------
Based on rigorous lower bounds on the survival time [@torquato02], the following “universal" scaling relation for $\tau$ has been found to apply to a wide class of microstructures and range of porosities [@torquato97]: $$\frac{{\displaystyle}\tau}{\tau_0} = \frac{{\displaystyle}8}{{\displaystyle}5}x + \frac{{\displaystyle}8}{{\displaystyle}7}x^2,
\label{universal}$$ where $\tau_0 = 3\phi_2/(D\phi s^2)$, $x = \langle \delta \rangle ^2/(\tau_0 D)$, and $\langle \delta \rangle$ is the mean pore size, defined by the first moment of the pore-size density function $P(\delta)$ [@torquato02]: $$\langle \delta \rangle = \int_0^\infty \delta P(\delta)\;d\delta.
\label{pore}$$
To explore the robustness of the universal curve, we check if the reported mean survival time data for a subset of the triply periodic surfaces falls on the curve. In order to compare the reported $\tau$ results to Eq. (\[universal\]), we determine $\langle \delta \rangle$ from the pore-size density function $P(\delta)$ results given above. Figure \[fig:universal\] reveals that the scaled survival times of the minimal surfaces fall relatively close to the universal curve. The corresponding values for the other bicontinuous structures considered here are close to the minimal-surface values and hence are not shown in the figure.
![\[fig:universal\] (Color online) The dimensionless mean survival time $\tau/\tau_0$ versus the scaled mean pore size squared, $\langle \delta \rangle^2/(\tau_0 D)$, for various two-phase media. The solid curve is the universal scaling relation (\[universal\]) [@torquato97]. ](figure5.eps){width="4.5in"}
CONCLUSIONS and DISCUSSION
==========================
To conclude, we have shown that a simply connected pore phase is a crucial topological feature required to achieve large mean survival times at a porosity $\phi=1/2$. However, a simply connected pore phase is not a sufficient condition feature; one must also have interfaces with small specific surface $s$. Indeed, we found that the survival times are inversely proportional to the corresponding specific surfaces for all of the triply periodic structures considered here. Nonetheless, the pore spaces of structures with large survival times are expected to be simply connected and therefore it would not be unreasonable for $\tau$ to be inversely proportional to the specific surface in these instances. We have demonstrated, as hypothesized, that the Schwartz P porous medium maximizes the mean survival time among the set of twelve triply-periodic structures considered here. This lends further evidence to the multifunctional optimality of the Schwartz P minimal surface, making this structure of great practical value to guide the design of new materials with a host of desirable bulk properties. Moreover, scaled survival times of the minimal surfaces fall relatively close to the universal curve (\[universal\]) found in Ref. [@torquato97].
Based on our findings, we conjecture that the mean survival time (like the fluid permeability) is maximized for triply-periodic porous media with a simply connected pore space at porosity $\phi=1/2$ by the structure the globally optimizes the specific surface. The verification of this conjecture remains an outstanding open question. This extremal problem falls in the general class of [*isoperimetric problems*]{}, which are notoriously difficult to solve. A prototypical isoperimetric example is Kelvin’s problem: the determination of the space-filling arrangement of closed cells of equal volume that minimizes the surface area. Although it is believed that the Weaire-Phelan structure [@We94] is an excellent solution to Kelvin’s problem, there is no proof that it is a globally optimal one. Our conjecture is also likely a difficult one to prove.
In this regard, it is noteworthy that an original goal of Ref. [@jung07] was to show that the triply-periodic surface with minimal specific surface $s$ at porosity $\phi= 1/2$ is the Schwartz P surface. While numerical simulations provided empirical evidence supporting this proposition, the authors of Ref. [@jung07] could not prove it rigorously. However, they were able to show that the Schwartz P, Schwartz D, and Schoen G minimal surfaces are local minima of the specific surface area $s$ at fixed volume fraction $\phi=1/2$. Thus, the question of the global optimality of the Schwartz P surface (i.e., minimal total interface surface area or specific surface $s$), is an open question for future investigation.
ACKNOWLEDGMENTS {#acknowledgments .unnumbered}
===============
S. T. thanks the Institute for Advanced Study for its hospitality during his stay there. This work was supported by the Office of Basic Energy Sciences, U.S. Department of Energy, under Grant No. DE-FG02-04-ER46108.
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|
---
abstract: 'In this article, we propose and study several discrete versions of homogeneous and inhomogeneous one-dimensional Fokker-Planck equations. In particular, for these discretizations of velocity and space, we prove the exponential convergence to the equilibrium of the solutions, for time-continuous equations as well as for time-discrete equations. Our method uses new types of discrete Poincaré inequalities for a “two-direction” discretization of the derivative in velocity. For the inhomogeneous problem, we adapt hypocoercive methods to the discrete cases.'
address:
- 'Équipe MEPHYSTO, Inria, 40 avenue Halley, 59650 Villeneuve d’Ascq'
- 'Laboratoire de Mathématiques J. Leray, UMR 6629 du CNRS, Université de Nantes, 2, rue de la Houssinière, 44322 Nantes Cedex 03, France'
- 'Fédération de Mathématiques FR3487, CentraleSupélec, 3 rue Joliot-Curie, 91190 Gif-sur-Yvette'
author:
- Guillaume Dujardin
- Frédéric Hérau
- Pauline Lafitte
title: 'Coercivity, hypocoercivity, exponential time decay and simulations for discrete Fokker-Planck equations'
---
[^1]
Introduction
============
In this article we study the long time behavior of the solutions of discrete versions of the following [*inhomogeneous*]{}[^2] Fokker–Planck equation $$\label{eq:IHFPF} {\D_t} F + v {\D_x} F - {\D_v}({\D_v}+v) F = 0, \qquad
F|_{t=0} = F^0,$$ where $F= F(t,x,v)$ with $t\geq 0$, $x$ in the one-dimensional torus $\T$, and $v \in \R$. In general, this problem is set with $F^0\in L^1 (\T \times \R, \dx\,\dv)$ with norm $1$, non-negative, and one looks for solutions of with values in the same set at all time $t\geq 0$.
To begin with, we study discretizations of the much simpler [ *homogeneous*]{}[^3] Fokker–Planck equation, set [*a priori*]{} in $L^1 (\dv)$ $$\label{eq:HFPF} {\D_t} F - {\D_v} ({\D_v}+v) F = 0, \qquad F|_{t=0} =
F^0,$$ where $F= F(t,v)$ is unknown for $t>0$ and $ v \in \R$. In particular, we use this equation to introduce a first discretization of the operator ${\D_v}$ in Section \[sec:homogeneous\], that we later generalize to the inhomogeneous case in Section \[sec:eqinhomo\].
We include in this paper the theoretical study of these discretizations of the two equations above when the one-dimensional velocity variable $v$ stays in a bounded symmetric interval of the form $(-\vmax,\vmax)$ for some $\vmax>0$. In this case, these equations are supplemented with homogeneous boundary conditions at $v=\pm\vmax$ in the form $({\D_v} + v)F(\cdot,\cdot,\pm\vmax)=0$. As in the unbounded velocity case, we first introduce a discretization of the operator ${\D_v}$ in Section \[sec:homobounded\] that we later generalize to the inhomogeneous case in Section \[sec:eqinhomoboundedvelocity\].
All sections but the Introduction share the same structure. We first recall the statements for the continuous solutions of the continuous equation, as well as the continuous tools that allow to prove the results in the continuous setting: one usually works in a Hilbertian subspace of $L^1$, uses the equilibrium of the equation to write a rescaled equation, and derives the exponential convergence of the continuous solutions to equilibrium using estimates on well-adapted entropies. Then, we introduce discretized operators together with a functional framework dedicated to the equation at hand and we introduce the analogous tools that allow to mimic the continuous setting and prove the exponential convergence to equilibrium for the [*discretized*]{} equations, in space, time and velocity. The main goal of this article is to introduce and analyze these discretizations to obtain full proofs of exponential convergences to equilibrium for discretizations of homogeneous as well as inhomogeneous Fokker–Planck equations. At the end of Sections \[sec:homobounded\] and \[sec:eqinhomoboundedvelocity\], we provide the reader with numerical results that illustrate our theoretical analysis.
As in the continuous cases, our analysis starts with discrete equilibrium for the discretized equations, that are analogous to the continuous Maxwellian $$\label{eq:gaussianintro} \mu(v) = c \exp^{-v^2/2},$$ (where $c$ is a positive normalization constant) which is an equilibrium state for the continuous equations and . Part of the discretization and, more importantly, the functional framework, use deeply the discrete equilibrium. This allows in particular to obtain fundamental functional inequalities as the discrete level, such as the Poincaré–Wirtinger inequality which reads for the homogeneous unbounded continuous case $$\int_\R g^2 \mu \dv \leq \int_\R ({\D_v} g)^2 \mu \dv, \qquad \textrm{ when } \qquad
\int_\R g \mu \dv = 0.$$ In all cases, this type of inequalities, together with adapted commutation relations for the discretized operators, and mass-preservation properties, allows for entropy dissipation control, which in the end yields exponential convergence to equilibrium.
We propose and analyze several schemes in this paper but we present in this introduction the two main ones and the corresponding results. We postpone to the end of this introduction the references to the other schemes and results.
The first scheme is an implicit Euler method in time for discretization of the inhomogeneous Fokker–Planck equation set on the unbounded velocity domain $\R$. We consider the following discretization of $\R^+ \times \T \times \R$. For a fixed $\dth>0$ we discretize the half line $\R^+$ by setting for all $n\in\N$, $t_n = n\dth$. For a sequence $(G^n)_{n\in \N}$, the discretization $D_t$ of the time-derivation operator ${\D_t}$ is defined by $$(D_t G)^n = \frac{G^{n+1}- G^n}{\dth}, \qquad n \in \N.$$ For a small fixed $\dvh >0$, we discretize the real line $\R$ by setting for all $i\in\Z$, $v_i = i\dvh$ and we work (concerning velocity only) in the set $$\ell^1(\Z,\dvh) = \set{ G \in \R^{\Z} \ | \ \sum_{ i \in \Z} \abs{G_{i}} \dvh <
\infty },$$ with the naturally associated norm. We consider the following “two-direction” discretization of the derivation operator in velocity: For $G \in \ell^1(\Z,\dvh)$, we define $\Dv G\in \ell^1(\Z^*,\dvh)$ by the following formulas $$\label{eq:defderivintro} (\Dv G)_i = \frac{G_{i+1}-G_{i}}{\dvh}
\textrm{ for } i<0, \qquad (\Dv G)_i = \frac{G_i-G_{i-1}}{\dvh} \textrm{ for }
i>0.$$ For $G\in \ell^1(\Z,\dvh)$ or $G\in \ell^1(\Z^*,\dvh)$ we define also $v G$ by $ (v G)_i = v_i G_i $ (either for $i\in \Z$ or $i \in \Z^*$ depending on the framework we work in)[^4]. The discretized Maxwellian ${\muh = (\muh)_{i \in \Z}}$, analogous of the continuous one is defined by $$\muh_i = \frac{c_\dvh}{ \prod_{\ell=0}^{|i|} (1+ v_\ell\dvh )}, \qquad i \in \Z.$$ It satisfies $(\Dv + v) \muh = 0$, just as $\mu$ solves $({\D_v}+v)\mu=0$. Since we shall later work in a Hilbertian framework, we introduce the formal adjoint $\Dvs$ of the velocity derivation operator $\Dv$. For $G \in \ell^1(\Z^*,\dvh)$, we define $\Dvs G \in \ell^1(\Z, \dvh)$ by the following formulas[^5] $$\begin{split}
& (\Dvs G)_i = \frac{G_{i}-G_{i-1}}{\dvh} \textrm{ for } i<0, \qquad \qquad
(\Dvs G)_i = \frac{G_{i+1}-G_i}{\dvh} \textrm{ for } i>0,
\\
& \qquad \qquad \textrm{ and } \qquad \qquad (\Dvs G)_0 =
\frac{G_1-G_{-1}}{\dvh}.
\end{split}$$ In order to discretize the one dimensional torus $\T$, we denote by $\dxh >0$ the step of the uniform discretization of $\T$ into $N\in\N^*$ sub-intervals, and we denote by $\jjj = \Z/ N\Z$ the corresponding finite set of indices. In what follows, the index $i \in \Z$ will always refer to the velocity variable and the index $j \in \jjj$ to the space variable. The discretized derivation-in-space operator $\Dx$ is defined by the following centered scheme : for $G = (G_{j})_{j\in \jjj}$ we set $$(D_x G)_{j} = \frac{G_{j+1} - G_{j-1}}{2\dxh}, \qquad j\in \jjj.$$ We now extend the definitions above to sequences with indices in $\jjj\times\Z$, in the sense that the velocity index $j$ plays no role in the definition of $\Dx$ and the space index $i$ plays no role in the definition of $v,\Dv,\Dvs$ and $\muh$. The discrete mass of a sequence $G \in \ell^1(\jjj \times \Z)$ is defined by $$m(G) = \dxh\dvh \sum_{j \in \jjj, i \in \Z} G_{j,i} .$$ The first discretized version of that we consider in this Introduction is the following implicit Euler scheme with unknown $(F_n)_{n\in\N} \in (\ell^1(\jjj \times \Z))^\N$: $$\label{eq:eulerimplicite} F^{n+1} = F^n - \dth \sep{ v \Dx
F^{n+1} + \Dvs (\Dv + v) F^{n+1}} = 0, \qquad F^0 \in \ell^1(\jjj \times
\Z).$$ Before stating our main result for the solutions of this last equation, we introduce two adapted Hilbertian spaces and an adapted entropy functional. First, we define using the discretized equilibrium $\muh$ the two spaces $$\Ldeuxmudvdxh = \set{ g \in \R^{\jjj \times \Z} \ | \ \dxh\dvh\sum_{j \in \jjj,
i \in \Z} \sep{g_{j,i}}^2 \muh_i < \infty },$$ and $$\Ldeuxmudvdxs = \set{ h \in \R^{\jjj \times \Z^*} \ | \ \dxh\dvh\sum_{j \in
\jjj, i \in \Z^*} \sep{g_{j,i}}^2 \mus_i < \infty },$$ where $\mus$ is a “two-direction” translation of $\muh$ to be precised later. We denote the naturally associated norms respectively by $\norm{\cdot}$ and $\norms{\cdot}$. Note that there is a natural injection $\mu \Ldeuxmudvdxh \hookrightarrow \ell^1(\jjj\times \Z)$. Second, we define the following modified Fisher information, for all doubly indexed sequence $G$, $$\eeed(G) = \norm{ \frac{G}{\muh}}^2 + \norms{ \Dv \sep{\frac{G}{\muh}}}^2 +
\norm{ \Dx \sep{\frac{G}{\muh}}}^2.$$ The main result concerning the scheme is the following.
\[thm:eulerimplicite\] For all $\dvh>0$, $\dxh>0$ and $\dth>0$, the problem is well-posed in the space of finite Fisher information and the scheme preserves the mass. Besides, there exists explicit positive constants $\kappa_\delta$, $C_\delta$ and $\dvh_0$ such that for all $\dvh <\dvh_0$, $\dxh>0$ and $\dth>0$, for all $F^0$ of mass $1$ such that $\eeed(F^0) <\infty$, the corresponding solution $(F^n)_{n\in\N}$ of satisfies for all $n\geq 0$, $$\eeed(F^n-\muh) \leq C_\delta
(1+2\dth \kappa_\delta)^{-n} \eeed(F^0-\muh).$$
In the theorem, well-posedness means that the corresponding discrete semi-group is well defined in the space of finite Fisher information. Note that there is no Courant-Friedrichs-Lewy (CFL) stability condition linking the numerical parameters $\dth$, $\dvh$ and $\dxh$ (the scheme is implicit). The whole theorem is proved in Section \[subsec:eqinhomototalementdiscretisee\] using tools developed in the preceding sections and briefly introduced above. Note that, as a direct corollary, we straightforwardly get the exponential trend of a solution $(F^n)_{n \in \N}$ to the equilibrium $\muh$:
\[cor:decrexpdiscretnonborne\] Consider the constants $\kappa_\delta$, $C_\delta$ and $\dvh_0$ given by Theorem \[thm:eulerimplicite\]. Then for all $\dth >0$ there exists $\kappa_\dth>0$ explicit with $\lim_{\dth \rightarrow 0} \kappa_\dth = \kappa_\delta$ such that for all $\dvh <\dvh_0$, all $\dxh >0$, all $F^0$ of mass $1$ such that $\eeed(F^0) <\infty$, the solution $(F^n)_{n\in\N}$ of satisfies for all $n\geq 0$, $$\eeed(F^n-\muh) \leq C_\delta \exp^{-2 \kappa_\dth n\dth} \eeed(F^0-\muh).$$
The second discretization scheme we emphasize in this introduction is explicit and deals with Equation set on a finite velocity domain $(-\vmax, \vmax)$. The main reason for proposing this scheme is that numerical simulations we will present in Sections \[sec:homobounded\] and \[sec:eqinhomoboundedvelocity\] are only possible with a finite set of indices in all variables.
Our aim is now to discretize the following equation $$\begin{split}
& {\D_t} F + v {\D_x} F - {\D_v}({\D_v}+v) F = 0, \qquad F|_{t=0} = F^0, \\
& \qquad \qquad ({\D_v} + v)F|_{\pm \vmax} = 0,
\end{split}$$ where $F= F(t,x,v)$ with $t\geq 0$, $x \in \T$ and $v \in I = (-\vmax, \vmax)$, and $F^0\in L^1 (\T \times I, \dx\,\dv)$ is fixed. For all $t>0$, the unknown $F(t,\cdot, \cdot)$ is in $L^1 (\T \times I, \dx\,\dv)$. We keep the notations and definitions for the time and space discrete derivatives and we change to a finite setting the definition of the velocity one. The discretization in velocity is the following: For a positive integer $\imax$, we define the set of indices $$\iii = \set{-\imax + 1, -\imax + 2, \cdots, -1, 0, 1, \cdots, \imax-2, \imax
-1}.$$ Note for further use that the boundary indices $\pm\imax$ do not belong to the full set $\iii$ of indices. We set $\dvh=\vmax/\imax$ and for all $i\in\iii$, $v_i = i\dvh$. We also set $v_{\pm\imax} = \pm \vmax$. The new discrete Maxwellian $\muh \in \R^\iii$ is defined by $$\muh_i = \frac{c_\dvh}{ \prod_{\ell=0}^{|i|} (1+ v_\ell\dvh )}, \qquad i \in \iii,$$ where the normalization constant $c_\dvh$ is defined such that $\dvh \sum_{i\in \iii} \muh_i= 1$. For the sake of simplicity, we will keep the same notation $\muh$ as in the unbounded velocity case. Note also that we do not need to define the discrete Maxwellian $\muh$ at the boundary indices $\pm \imax$. We work in the following in the space ${\ell^1(\iii,\dvh)}$ of all finite real sequences $g = (g_i)_{i\in \iii}$ with the norm $\dvh\sum_{i \in \iii} \abs{g_i}$. As we did above in the infinite velocity case, we introduce another set of shifted indices and another discrete Maxwellian. We set $$\iiis = \set{-\imax , -\imax + 1, \cdots, -2, -1, 1, 2, \cdots, \imax-1, \imax},$$ and define $\mus \in {\ell^1(\iiis, \dvh)}$ by for all $ i\in\iiis$, $$\mus_i = \muh_{i+1} \textrm{ for } i<0, \qquad \mus_i = \muh_{i-1} \textrm{ for
} i>0.$$ We consider the discrete derivation operators $\Dv$ and $\Dvs$ that are the same as is the unbounded case except at the boundary where we impose a discrete Neumann condition. A good framework is the following: we define $\Dv : \ell^1(\iii, \dvh) \longrightarrow \ell^1(\iiis, \dvh) $ for all $G \in \ell^1(\iii, \dvh)$ by $$\label{eq:defderivn}
\begin{split}
& (\Dv G)_i = \frac{G_{i+1}-G_{i}}{\dvh} \textrm{ when } -\imax+1\leq i \leq -1, \\
&
(\Dv G)_i = \frac{G_i-G_{i-1}}{\dvh} \textrm{ when } 1 \leq i \leq \imax-1, \\
& ((\Dv + v)G)_{\pm \imax} \defegal \muh \Dv \sep{ \frac{G}{\muh}}_{\pm
\imax} = 0.
\end{split}$$ The last condition defines only implicitly both the derivation and the multiplication at index $\pm \imax$. For $G\in \ell^1(\iii)$ or $G\in \ell^1(\iiis)$ we define also $v G$ by $ (v G)_i = v_i G_i $ (either for $i\in \iii$ or $i \in \iiis$ depending on the framework we work in, and without ambiguity). Similarly, we define $\Dvs : \ell^1(\iiis, \dvh) \longrightarrow \ell^1(\iii, \dvh) $ for all $H \in \ell^1(\iii, \dvh)$ by[^6] $$\begin{split}
& (\Dvs H)_i = \frac{H_{i}-H_{i-1}}{\dvh} \textrm{ when } -\imax+1 \leq i< -1, \\
& (\Dvs H)_i = \frac{H_{i+1}-H_i}{\dvh} \textrm{ when } 1 \leq i \leq \imax
-1,
\\
& (\Dvs H)_0 = \frac{H_1-H_{-1}}{\dvh}.
\end{split}$$ As in the unbounded case, we define the mass of a sequence $G \in \ell^1(\jjj \times \iii)$ by $$m(G) = \dxh\dvh \sum_{j \in \jjj, i \in \iii} G_{j,i}.$$ The second discretized version of is the following explicit Euler scheme with unknown $F \in (\ell^1(\jjj \times \iii))^\N$: $$\label{eq:eulerexplicite}
\begin{split}
& F^{n+1} = F^n - \dth \sep{ v \Dx F^{n} + \Dvs (\Dv + v) F^{n}} = 0,
\qquad F^0 \in \ell^1(\jjj \times \iii), \\
\end{split}$$ where we note that the Neumann type boundary condition is now included in the definition of the derivation operator $\Dv$ in . We work with the following Hilbertian structures on $\R^{\jjj \times \iii}$ and $\R^{\jjj \times \iiis}$: $$\Ldeuxmudvdxh = \set{ g \in \R^{\jjj \times \iii} \ | \ \dxh\dvh\sum_{j \in
\jjj, i \in \iii} \sep{g_{j,i}}^2 \muh_i < \infty },$$ and $$\Ldeuxmudvdxs = \set{ h \in \R^{\jjj \times \iiis} \ | \ \dxh\dvh\sum_{j \in
\jjj, i \in \iiis} \sep{g_{j,i}}^2 \mus_i < \infty },$$ with the naturally associated norms again denoted respectively by $\norm{\cdot}$ and $\norms{\cdot}$. There is again a natural injection $\mu \Ldeuxmudvdxh \hookrightarrow \ell^1(\jjj\times \iii)$. We define the same modified Fisher information as in the unbounded case but in this new framework $$\eeed(G) = \norm{ \frac{G}{\muh}}^2 + \norms{ \Dv \sep{\frac{G}{\muh}}}^2 +
\norm{ \Dx \sep{\frac{G}{\muh}}}^2\label{eq:mfi}.$$
For the scheme , the well-posedness for all $\dth>0$ is granted since we are in a finite dimensional setting. Since the scheme is explicit, a CFL type condition is needed. For that purpose, we introduce the following CFL constant $$\bcfl = \max \set{ 1, 4 \frac{1+ \dvh \vmax}{\dvh^2},4 \frac{1+ \dvh
\vmax}{\dxh^2},4\frac{\vmax^2}{\dxh^2}}.$$ The main result in this explicit in time and bounded in velocity inhomogeneous setting is the following
\[thm:eulerexplicite\] The scheme preserves the mass. Besides, there exists explicit positive constants $\kappa_\delta$, $C_\delta$, $\dvh_0$ and $C_{\rm CFL}$ such that for all $\dvh\in(0,\dvh_0)$ and $\dxh>0$, for all $F^0$ of mass $1$ such that $\eeed(F^0) <\infty$, for all $\dth>0$ satisfying the CFL condition $C_{\rm CFL}\bcfl\dth<1$, the solution $(F^n)_{n\in\N}$ of the scheme satisfies for all $n\in\N$, $$\eeed(F^n-\muh) \leq C_\delta
(1-2\dth \kappa_\delta)^{n} \eeed(F^0-\muh).$$
The values of the explicit constants are given in Theorem \[thm:decrexpeulerexpldiscr\] in Section \[sec:eqinhomoboundedvelocity\]. Note that, as a direct corollary, using an asymptotic development of the logarithm, we straightforwardly get the exponential trend of a solution $(F^n)_{n \in \N}$ to the equilibrium $\muh$:
Consider the constants $\kappa_\delta$, $C_\delta$, $\dvh_0$ and $C_{CFL}$ given by Theorem \[thm:eulerexplicite\]. For all $\dvh\in(0,\dvh_0)$ and $\dxh>0$, for all $\dth >0$ satisfying the CFL condition $C_{\rm CFL}\bcfl\dth<1$, there exists $\kappa_\dth>0$ explicit with $\lim_{\dth \rightarrow 0} \kappa_\dth = \kappa_\delta$ such that for all $F^0$ of mass $1$ such that $\eeed(F^0) <\infty$, the solution $(F^n)_{n\in\N}$ of satisfies for all $n\in\N$, $$\eeed(F^n-\muh) \leq C_\delta \exp^{-2 \kappa_\dth n\dth} \eeed(F^0-\muh).$$
As was already stated, the main goal of our paper is to propose and analyze hypocoercive numerical schemes for inhomogeneous kinetic equations, for which one can prove exponential in time return to the equilibrium. In the literature, one can find theoretical results either about numerical schemes for homogeneous kinetic equations, built upon coercivity for discrete models, or about exact solutions of inhomogeneous equations, built upon hypocoercivity techniques. In this paper, we want to tackle both problems at the same time and prove theoretical results on exponential time return to equilibrium for discrete [ *and*]{} inhomogeneous kinetic equations. Up to our knowledge, these are the first theoretical results dealing with the two difficulties at the same time.
Concerning the simpler homogeneous kinetic equations, the question of finding efficient schemes has a long story and deep recent developments. Let us mention a few results that are already known in these directions. One can find this kind of problems for example in [@CC70] for the linear homogeneous Fokker-Planck equation in a fully discrete setting. More recently, schemes have been proposed for nonlinear degenerate parabolic equations that numerically preserve the exponential trend to equilibrium (see for example [@BCF12] for a finite volume scheme which works numerically even for nonlinear problems). This question has also been addressed numerically together with that of the order of the schemes, for nonlinear diffusion and kinetic equations [*e.g.*]{} in [@PR16]. In particular, it is known that, even for the [*linear*]{} Fokker-Planck equation, “wrong” discretizations lead to “wrong” qualitative behaviour of the schemes in long time. So-called spectral methods are also proposed (see for example recent developments for the Boltzmann equation in [@AGT16]), with the drawback that they do not ensure the non-negativity of the solutions. Let us also mention the recent paper [@Filbet2017], where a finite volume scheme is introduced for a class of boundary-driven convection-diffusion equations on bounded domains. The question of the long-time behaviour of the scheme is addressed using the relative entropy structure.
Concerning inhomogeneous kinetic (continuous) equations, the so-called hypocoercive theory is now rather well understood with various results concerning many models. In this direction, first results on linear models were obtained in [@Her06], [@MN06] [@Vil09] or [@DMS15]. They were in fact adapted on the very abstract theory of hypoellipticity of Kohn or (type II hypoelliptic operators) of Hörmander that explain in particular the regularization of such degenerate parabolic equations. The cornerstone of the theory is that, although the drift $v.\nabla_x$ is degenerate (at $v=0$ in particular), one commutator with the velocity gradient erases the degeneracy : $[\nabla_v, v.\nabla_x] = \nabla_x$. The main feature of the hypocoercive theory is that this commutation miracle leads also to exponential return to the equilibrium (independantly of the regularization property). One other feature is that it can be enlarged to collision kernels even without diffusive velocity kernel and to many other inhomogeneous kinetic models systems (see e.g. [@Vil09; @BDMMS17] or the introduction course [@Her17a]).
Concerning the numerical analysis of inhomogeneous kinetic equations, we mention the paper [@PZ16] where the Kolmogorov equation is discretized in order to get short time estimates, following the short time continuous “hypocoercive” strategy proposed in [@Her07]. However, the corresponding scheme is not asymptotically stable and no notion of equilibrium or long-time behaviour is proposed there. This paper was anyway a source of inspiration of the present work (see also point 4 in Section \[sec:ccl\] here for further interactions between the two articles). We also mention the work on the Kolmogorov–Fokker–Planck equation carried out in [@FosterLoheacTran2017], where a time-splitting technique based on self-similarity properties is used for solutions that decay like inverse powers of the time.
In this article we show that the hypocoercive theory is sufficiently robust to indeed give exponential time decay of partially or fully discretized inhomogeneous equations. This is done here in the case of the Fokker-Planck equation in one dimension. We cover fully discretized as well as semi-discretized situations. We propose, for each setting, for the first time up to our knowledge, a full proof of exponential convergence towards equilibrium for the corresponding solutions. Once again these proofs use discrete analogues to the continuous tools, such as the Poincaré inequality and the hypocoercive techniques. Even for the simple homogeneous setting, to our knowledge, the (optimal) discrete Poincaré inequality with a weight is new (see Proposition \[prop:poindiscrete\]) in both bounded and unbounded cases.
We hope that this approach can be generalized to various multi-dimensional kinetic models of the form ${\D_t} u + Pu=0$, with $P$ hypocoercive. One aim would be to write a systematic “black box scheme” theorem with $P= X_0 - L$ where $L$ is the collision kernel (independently studied in velocity variable only) and $X_0$ the drift, as proposed in e.g. [@DMS15] in the continuous case. In this sense a lot of work has to be done. Of course we also hope that our scheme approach can be used to predict some results for more complex situations including non-linear inhomogeneous ones.
The outline of this article is the following. In the second section, we deal with the homogeneous equation in time and velocity only, with velocity varying in the full real line. We first recall the continuous framework in a very simplified and concise way. Then, we adapt it to semi-discrete and fully discrete cases. In particular, we focus on the homogeneous case and we state a new discrete Poincaré inequality with the discrete Gaussian weight $\muh$.
In the third section, we deal with the full inhomogeneous case , and propose a concise version of the continuous results. Then, we adapt these results to several discretized versions of the equation: the semi-discrete in time case, the implicit semi-discrete in space and velocity case, ending with the full implicit discrete case corresponding to Theorem \[thm:eulerimplicite\]. In particular, we develop discrete versions of the commutation Lemmas at the core of the (continuous) hypocoercive method.
In the fourth section, we focus on the homogeneous case set on a bounded velocity domain. We only deal with the continuous and the explicit fully discrete case. Once again, a new Poincaré inequality is proposed. Moreover, a CFL condition appears.
In the fifth section, we consider the inhomogeneous problem set on a bounded velocity domain. We first present the continuous case. Then, we propose the study of the fully discrete case with an Euler explicit scheme leading to Theorem \[thm:eulerexplicite\].
In the appendix, we propose some comments and possible generalizations, as well as a table summarizing the main results concerning discrete commutators.
The homogeneous equation {#sec:homogeneous}
========================
The continuous time-velocity setting {#subsec:homogeneouscontinuous}
------------------------------------
We start by recalling the main features of the continuous equation set on the unbounded domain $\R$. These features will have discrete analogues described in the next subsection.
Since we are interested in the long time behavior and the trend to the equilibrium, we start by checking what the good equilibrium states are. We first look at the continuous homogeneous equation . We say that a function $\mu (v)$ is an equilibrium if $-{\D_v}({\D_v}+v) \mu(v) = 0$. The first idea is to suppose only that $({\D_v} + v) \mu(v) = 0$ which leads to $$\label{eq:Maxwellian}
\mu(v) = \frac{1}{\sqrt{2\pi}} \exp^{-v^2/2},$$ if we impose in addition that $\mu\geq 0$ is $L^1(\dv)$-normalized.
A standard strategy in statistical mechanics is then to build an adapted functional framework (a subspace of $L^1(\dv)$) where non-negativity of the collision operator $-{\D_v}({\D_v}+v) $ is conserved. A standard choice is then to take $F (t,\cdot)\in \mu \Ldeuxmudv \hookrightarrow L^1(\dv)$ where $\mudv = \mu(v) \dv$. We check then that operator $-{\D_v}({\D_v}+v) $ is self-adjoint in $\mu \Ldeuxmudv$, with compact resolvent. Therefore it has discrete spectrum and $0$ is a single eigenvalue associated with the eigenfunction $\mu$. In fact, this result can be easily checked using the following change of unknown, which will be of deep and constant use through out this article.
We pose for the following $F= \mu+\mu f$ and call $f$ the rescaled density. With this new unknown function, and in the new adapted framework, the equation writes $$\label{eq:HFPf} {\D_t} f +(- {\D_v}+v) {\D_v} f = 0 , \qquad f|_{t=0} =
f^0,$$ where $f= f(t,\cdot) \in \Ldeuxmudv \hookrightarrow L^1(\mu \dv)$. The non-negativity of the collision kernel is then direct to verify: in $\Ldeuxmudv$ with the associated scalar product we have ${\D_v}^* = (-{\D_v}+ v)$ and therefore for all $g\in\Hunmudv$ with $(- {\D_v}+v){\D_v} g\in\Ldeuxmudv$, $$\seq{(- {\D_v}+v){\D_v} g, g}_{\Ldeuxmudv} = \norm{{\D_v} g}^2_{\Ldeuxmudv} = \int_\R
|{\D_v} g|^2 \mudv.$$ it is easy to check that operator $P=(- {\D_v}+v){\D_v}$ is maximal accretive ([@HN04]) with domain $D(P) = \set{ g \in \Ldeuxmudv \ | \ (- {\D_v}+v){\D_v} g\in\Ldeuxmudv}$ and using the Hille–Yosida Theorem, one obtains at once the existence and uniqueness of the solution $f$ of in $\ccc^1(\R^+, \Ldeuxmudv) \cap \ccc^0(\R^+, D(P))$ for all $f^0\in D(P)$, and that the problem is also well-posed in $\ccc^0(\R^+, \Ldeuxmudv)$ in the sense of distributions. From the preceding equality, for $g\in \Ldeuxmudv$, $$(- {\D_v}+v) {\D_v} g = 0 \Longleftrightarrow {\D_v} g = 0 \Longleftrightarrow
g\text{ is constant,}$$ and therefore the constants are the only equilibria of the equation . Note that in this $L^2$ framework, the conservation of mass is obtained by integrating equation against the constant function $1$ in $\Ldeuxmudv$ to obtain for all $t\geq 0$, $$\label{eq:consmass}
\seq{f(t)} \defegal \int_\R f(t,v) \mu (v)\dv = \seq{f(t),1}_{\Ldeuxmudv} = \seq{f^0}.$$ In that case a system with null mass corresponds to a rescaled density $f$ such that $f \perp 1$ in $\Ldeuxmudv$. Note that Equation is also well posed in $\Hunmudv$ thanks to the Hille–Yosida Theorem again, and that it yields a unique solution in $\ccc^1(\R^+, \Hunmudv) \cap \ccc^0(\R^+, D_{\Hunmudv}(P))$ for all $f^0\in \Hunmudv$, where $D_\Hunmudv(P)$ is the domain of $P= (-{\D_v} + v) {\D_v}$ in $\Hunmudv$. Of course, this solution coincides with the one with values in $\Ldeuxmudv$ when $f^0\in \Hunmudv$.
One of the main tools in the study of the return to equilibrium for Fokker–Planck equations is the Poincaré inequality. There are many ways of proving it (including the compact resolvent property) but one direct way, well adapted to a coming discretization, can be inspired by the original proof by Poincaré in the flat case.
\[lem:Poincarecontinu\] For all $g \in \Hunmudv$, we have $$\norm{g-\seq{g}}^2_\Ldeuxmudv \leq \norm{{\D_v} g}_\Ldeuxmudv^2.$$
Replacing if necessary $g$ by $g-\seq{g}$, it is sufficient to prove the result for $\seq{g}=0$. In the following, we denote for simplicity $g(v) = g$, $g(v') = g'$, $\mu(v) = \mu $ and $\mu(v') = \mu'$. We first note that $$\begin{split}
\int_\R g^2\mudv & = \frac{1}{2} \iint_{\R^2} (g'-g)^2 \mudv \mudvp,
\end{split}$$ since $2 \iint g g' \mudv \mudvp = 2 \int g\mudv \int g' \mudvp = 0$. Using that $g'-g= \int_v^{v'} {\D_v} g (w) \dw$ we can write $$\begin{gathered}
\int_\R g^2\mudv
= \frac{1}{2} \iint_{\R^2} \sep{ \int_v^{v'} {\D_v} g (w) \dw}^2 \mudv \mudvp \\
\leq \frac{1}{2} \iint_{\R^2} \sep{ \int_{v}^{v'} \abs{{\D_v} g (w)}^2
\dw}(v'-v) \mudv \mudvp
\end{gathered}$$ where we used the Cauchy–Schwarz inequality in the flat space. Let us denote by $\G$ an anti-derivative of $\abs{{\D_v} g}^2$, for example this one : $\G(v) = \int_{0}^v \abs{{\D_v} g (w)}^2 \dw$. We have then $$\label{eq:eqintermedpoincarecontinu}
\begin{split}
& \int_\R g^2\mudv \\
& \leq \frac{1}{2} \iint_{\R^2} \sep{ \G'-\G}(v'-v) \mudv \mudvp
= \frac{1}{2} \iint_{\R^2} \sep{ \G'-\G}(v'-v)\mu \mu' \dv \dv' \\
& = \frac{1}{2} \sep{ \iint_{\R^2} \G'v'\mudv \mudvp + \iint_{\R^2} \G v
\mudv \mudvp -\iint_{\R^2} \G v' \mudv\mudvp
-\iint_{\R^2} \G' v \mudv \mudvp } \\
& = \int_{\R} \G v \mudv,
\end{split}$$ where we used the Fubini Theorem and the fact that $\int v \mudv = 0$ and $\int \mudv = 1$ (and their counterparts in variable $v'$). At this point, it is sufficient to note that ${\D_v} \mu = -v \mu$ and perform an integration by parts to obtain with the inequality above, $$\begin{split}
\int_\R g^2\mudv & \leq \int_\R (\G v \mu) \dv = - \int_\R \G ({\D_v} \mu) \dv
=\int_\R ({\D_v} \G) \mudv = \int_\R |{\D_v} g|^2 \mudv.
\end{split}$$ The proof is complete.
A direct consequence of this Poincaré inequality is the exponential convergence to the equilibrium in the space $\Ldeuxmudv$ of the solution $f$ of , that we prove below. In Section \[sec:inhomc\], we will use an entropy formulation to prove the exponential convergence to the equilibrium of the solutions of the [ *inhomogeneous*]{} Fokker–Planck equation. For this reason, we decide to adopt the same framework in this section, devoted to the (simpler) homogeneous case. We define the two following entropies for $g\in \Ldeuxmudv$ and $g\in\Hunmudv$ respectively : $$\fff(g) = \norm{g}_\Ldeuxmudv^2, \qquad \ggg(g) = \norm{g}_\Ldeuxmudv^2 +
\norm{{\D_v} g}_\Ldeuxmudv^2.$$ Note that these entropies are exactly the squared norms of $g$ in $\Ldeuxmudv$ and $\Hunmudv$ respectively. To keep notations short, in the remaining of this section, we denote by $\norm{\cdot}$ the $\Ldeuxmudv$ norm. The exponential convergence to the equilibrium of the solutions of is stated in the following easy Theorem.
\[thm:exponentialtrendtoequilibrium\] Let $f^0 \in \Ldeuxmudv$ such that $\seq{f^0} = 0$ and let $f$ be the solution in $\ccc^0(\R^+,\Ldeuxmudv)$ of (in the semi-group sense). Then $\seq{f(t)} = 0$ for all $t\geq 0$, and we have $$\label{eq:decrFhomo}
\forall t\geq 0,\qquad
\fff(f(t)) \leq \exp^{-2t}\fff(f^0).$$
If in addition $f^0 \in \Hunmudv$, then $f \in \ccc^0(\R^+, \Hunmudv)$ and we have $$\label{eq:decrGhomo}
\forall t\geq 0,\qquad
\ggg(f(t)) \leq \exp^{-t}\ggg(f^0).$$
We first recall that operator $P = (-{\D_v} + v) {\D_v}$ is the generator of a semi-group of contractions in both $\Ldeuxmudv$ and $\Hunmudv$. This is direct to check that $\Hunmudv$ is dense in $\Ldeuxmudv$ and that when both defined, the solutions of the heat problem ${\D_t} f + P f = 0$ coincide. In the following, we therefore focus on the $\Hunmudv$ case corresponding to solutions with finite modified entropy $\ggg$.
We denote by $D_{\Hunmudv}(P) $ the domain of $P$ in $\Hunmudv$. We note again that $D_{\Hunmudv}(P) $ is dense in $\Hunmudv $, and we consider a solution $f$ of which satisfies $$f \in \ccc^1(\R^+, \Hunmudv) \cap \ccc^0(\R^+, D_{\Hunmudv}(P)).$$ All the computations below are therefore authorized. The main inequalities and are then consequences of the above mentioned density properties and of the definition of a bounded semi-group.
We compute the time derivative of the corresponding entropies along the exact solution $f$ of . Using , we have for all $t\geq 0$, $\seq{f(t)} = \seq{f^0} = 0$. For the first entropy, we have $$\ddt \fff(f) = -2 \seq{ (-{\D_v}+v){\D_v} f , f} = -2 \norm{{\D_v} f}^2 \leq -2
\norm{f}^2 = -2 \fff(f),$$ where we used the Poincaré Lemma \[lem:Poincarecontinu\]. This directly gives . For the second entropy $\ggg$, we do the same: $$\begin{split}
\ddt \ggg(f) & = -2 \seq{ (-{\D_v}+v){\D_v} f , f} -2
\seq{ {\D_v}(-{\D_v}+v){\D_v} f , {\D_v} f} \\
& = -2 \norm{{\D_v} f}^2 - 2 \norm{(-{\D_v} +v){\D_v} f}^2 \\
& \leq - \norm{f}^2 - \norm{{\D_v} f}^2 - 2 \norm{(-{\D_v} +v){\D_v} f}^2 \leq -
\ggg(f),
\end{split}$$ where we used the following splitting : $2 \norm{{\D_v} f}^2 \geq \norm{{\D_v} f}^2 + \norm{f}^2$, obtained again with Lemma \[lem:Poincarecontinu\]. We therefore get the result . The proof is complete.
The following corollary is then straightforward, as a reformulation of the preceding Theorem.
\[cor:exponentialtrendtoequilibrium\] Let $f^0 \in \Ldeuxmudv$ and let $f$ be the solution in $\ccc^0(\R^+, \Ldeuxmudv)$ of . Then for all $t\geq 0$, $$\norm{f(t)- \seq{f^0}}_\Ldeuxmudv \leq \exp^{-t}\norm{f^0- \seq{f^0}}_\Ldeuxmudv.$$ If in addition $f^0 \in \Hunmudv$ then $f \in \ccc^0(\R^+, \Hunmudv)$ and we have for all $t\geq 0$, $$\norm{f(t)- \seq{f^0}}_\Hunmudv \leq \exp^{-\frac{t}{2}}\norm{f^0-
\seq{f^0}}_\Hunmudv.$$
Discretizing the velocity variable {#subsec:homsd}
----------------------------------
In the discrete and semi-discrete cases, the main difficulty is to find a suitable discretization of the equation that will mimic the qualitative asymptotic properties of the continuous equation, see [*e.g*]{} Theorem \[thm:exponentialtrendtoequilibrium\]. In particular, one has to decide how to discretize the differential operators in $v$. For a small fixed $\dvh >0$, we discretize the real line $\R_v$ by setting for all $i\in\Z$, $v_i = i\dvh$.
We work now step by step, and look first at what could be a suitable equilibrium state $\muh$ replacing $\mu$ in the continuous case. As in the continuous case, $\muh$ has to satisfy elementary structural properties. The first ones are to be positive and to be normalized in the (discrete) probability space $\ell^1(\Z, \dvh)$ which means $$\label{eq:normalizationdemudv}
\norm{\muh}_{\ell^1(\Z, \dvh)} = \dvh \sum_i \muh_i = 1.$$ Mimicking the continuous case, we also require $\muh$ to be even and to satisfy the equation $(\Dv + v)\muh = 0$ where $\Dv$ is a discretization of ${\D_v}$ and $v$ stands for the sequence $( v_i)_{i\in \Z}$ or by extension the multiplication term by term by it. A good choice for $\Dv$ leading to this property is the following :
\[def:dv\] Let $G \in \ell^1(\Z,\dvh)$, we define $\Dv G\in \ell^1(\Z^*,\dvh)$ by the following formulas $$(\Dv G)_i = \frac{G_{i+1}-G_{i}}{\dvh} \textrm{ for } i<0, \qquad (\Dv G)_i =
\frac{G_i-G_{i-1}}{\dvh} \textrm{ for } i>0,$$ and $v G\in \ell^1(\Z^*,\dvh)$ by $$(v G)_i = v_i G_i \textrm{ for } i\neq 0,$$ when this series is absolutely convergent.
With this definition, solving the equation $(\Dv + v)\muh = 0$ leads to the following proposition.
\[lem:muh\] Assume $\dvh>0$ is fixed. Then there exists a unique positive, $\ell^1(\Z, \dvh)$ - normalized, solution $\nu$ of $(\Dv + v)\nu = 0$. We denote this solution by $\muh$. There exists a unique positive constant $c_\dvh$ such that $$\muh_i = \frac{c_\dvh}{ \prod_{\ell=0}^{|i|} (1+ v_\ell\dvh )}, \qquad i \in \Z.$$ Moreover, $\muh$ is even.
Note that the discrete Maxwellian $\muh$ converges to the continuous Maxwellian $\mu$ defined in when $\dvh$ tends to 0 in the following sense : $${\sup_{i\in\Z}} |\muh_i-\mu(v_i)| \underset{\dvh\to 0}{\longrightarrow} 0.$$
The proof is a direct computation. The fundamental equations term by term solved by $\muh$ are indeed $$\label{eq:relationsmudv}
\left\{
\begin{array}{ll}
\dfrac{\muh_i -\muh_{i-1} }{\dvh} + v_i \muh_i = 0 & \quad \textrm{ for } i>0 \\
\dfrac{\muh_{i+1} -\muh_{i} }{\dvh} + v_i \muh_i =0 & \quad \textrm{ for } i<0,
\end{array}
\right.$$ which give the expression of $\muh$ up to a normalization constant.
With the discretization $\Dv+v$ of the operator ${\D_v}+v$ above, we propose the following discretization $-\Dvs$ of $-{\D_v}$ so that the discretized version of , with operator $\Pd= -\Dvs(\Dv+v)$, has a non-negative collision kernel.
\[def:dvs\] Let $G \in \ell^1(\Z^*, \dvh)$, we define $\Dvs G \in \ell^1(\Z, \dvh)$ by the following formulas $$\begin{split}
& (\Dvs G)_i = \frac{G_{i}-G_{i-1}}{\dvh} \textrm{ for } i<0, \qquad
\qquad (\Dvs G)_i = \frac{G_{i+1}-G_i}{\dvh} \textrm{ for } i>0
\\
& \qquad \qquad \textrm{ and } \qquad \qquad (\Dvs G)_0 =
\frac{G_1-G_{-1}}{\dvh},
\end{split}$$ (be careful, there is no mistake in the denominator of $(\Dvs G)_0$). We also define the operator $\vs$ from $\ell^1(\Z^*, \dvh)$ to $\ell^1(\Z, \dvh)$ by setting for $G\in \ell^1(\Z^*, \dvh)$, $$\forall i\neq 0,\quad (\vs G)_i=v_i G_i \qquad \text{and} \qquad
(\vs G)_0=0.$$
We are now in position to define a good discretization of the main equation and the adapted discretized framework.
For a given $F^0\in\ell^1(\Z,\dvh)$, we shall say that a function $F \in \ccc^0(\R^+, \ell^1(\Z, \dvh))$ satisfies the (flat) semi-discrete homogeneous Fokker–Planck equation if $$\label{eq:DHFPF} {\D_t} F - \Dvs(\Dv+v) F = 0, \qquad F|_{t=0} =
F^0,$$ in the sense of distributions.
As in the continuous case, we perform the change of unknown, thanks to the discrete equilibrium state $\muh$: $G = \muh g$ so that $$G \in \ell^1(\Z, \dvh) \Longleftrightarrow g \in \ell^1(\Z, \muh \dvh).$$ Let us perform this change of unknown in the differential operator $-\Dvs(\Dv+v)$. For $i>0$, we have $$\begin{split}
((\Dv+v)G)_i & = ((\Dv+v)\muh g)_i
= \frac{\muh_i g_i -\muh_{i-1} g_{i-1}}{\dvh} + v_i \muh_i g_i \\
& = \Bigg(\underbrace{ \frac{\muh_i -\muh_{i-1} }{\dvh} + v_i \muh_i
}_{=0}\Bigg) g_i +\muh_{i-1} \frac{ g_i - g_{i-1}}{\dvh} = \muh_{i-1} (\Dv
g)_i.
\end{split}$$ Similarly, we find for $i<0$, $$\begin{split}
((\Dv+v)G)_i & = ((\Dv+v)\muh g)_i
= \frac{\muh_{i+1} g_{i+1} -\muh_{i} g_{i}}{\dvh} + v_i \muh_i g_i \\
& = \Bigg( \underbrace{\frac{\muh_{i+1} -\muh_{i} }{\dvh} + v_i \muh_i
}_{=0}\Bigg) g_i +\muh_{i+1} \frac{ g_{i+1} - g_{i}}{\dvh} = \muh_{i+1} (\Dv
g)_i.
\end{split}$$ From the computation above, we get that $$\label{eq:double}
\begin{split}
-\Dvs((\Dv+v)G) & = \muh (-\Dvs + \vs) \Dv g,
\end{split}$$ Therefore, for any $F\in\ccc^0(\R^+, \ell^1(\Z, \dvh))$, setting for all $t\geq 0$, $f(t,\cdot)=(F(t,\cdot)-\muh)/\muh$, we have $${\D_t} F - \Dvs(\Dv+v) F = \muh ({\D_t} f + ( - \Dvs +\vs) \Dv f),$$ where we recall that the multiplication is done term by term. This computation motivates the definition of the following rescaled equation.
For a given $f^0\in\ell^1(\Z,\muh\dvh)$, we shall say that a function $f \in \ccc^0(\R^+, \ell^1(\Z, \muh\dvh))$ satisfies the (scaled) semi-discrete homogeneous Fokker–Planck equation if $$\label{eq:DHFPf} {\D_t} f + ( - \Dvs +\vs) \Dv f = 0, \qquad
f|_{t=0} = f^0,$$ in the sense of distributions.
With the definitions and computations above, $F$ is a solution of the flat semi-discrete Fokker–Planck equation if and only if $f$ defined by $F = \muh + \muh f$ is a solution of the scaled semi-discrete Fokker–Planck equation .
Just as we recalled in the continuous velocity setting in Section \[sec:homogeneous\], the next step in the discrete velocity setting is to find a suitable subspace of $\ell^1(\Z, \muh\dvh)$, with a Hilbertian structure, in which the non-negativity property of the collision operator is satisfied. We mimic the continuous case and choose the space $ \ell^2(\Z, \muh \dvh) \hookrightarrow \ell^1(\Z, \mu^h \dvh)$ denoted for short $\Ldeuxmudvh$.
\[def:homd\] We define the space $\Ldeuxmudvh$ to be the Hilbertian subspace of $\R^\Z$ of sequences $g$ such that $$\norm{ g}_{\Ldeuxmudvh }^2 \defegal \dvh \sum_{i\in \Z} (g_i)^2 \muh_i <\infty.$$ This defines a Hilbertian norm, and the related scalar product will be denoted by $\seq{ \cdot, \cdot}$. For $g \in \Ldeuxmudvh$, we also define $$\seq{g} \defegal \sum_{i\in \Z} g_i \mu_i^h \dvh = \seq{g, 1}_{\Ldeuxmudvh },$$ the mean of $g$ (with respect to this weighted scalar product).
In order to give achieve a useful functional framework for the (scaled) homogeneous Fokker-Planck equation in this discrete velocity setting, we introduce now a shifted Maxwellian $\mus \in \ell^1(\Z^*, \dvh)$ and a new suitable Hilbert subspace that appears naturally in the computations:
\[def:homds\] Let us define $\mus \in \ell^1(\Z^*, \dvh)$ by $$\mus_i = \muh_{i+1} \textrm{ for } i<0, \qquad \mus_i = \muh_{i-1} \textrm{ for
} i>0.$$ We define the space $\Ldeuxmudvs$ to be the subspace of $\R^{\Z^*}$ of sequences $g \in \ell^1(\Z^*, \mus\dvh)$ such that $$\norm{g}_{\Ldeuxmudvs }^2 \defegal \dvh \sum_{i \in \Z^*} (g_i)^2 \mus_i
<\infty.$$ This defines a Hilbertian norm, and the related scalar product will be denoted by $\seq{ \cdot, \cdot}_\sharp$. Eventually, we define $$\Hunmudvh = \set{ g \in \Ldeuxmudvh, \textrm{ s.t. } \Dv g \in \Ldeuxmudvs }.$$
In contrast to the classical finite differences setting where the discretizations of ${\D_v}$ give rise to [*bounded*]{} linear operators (with continuity constants of size $1/\dvh$), the above definition makes $D_v$ an [*unbounded*]{} linear operator from $\Ldeuxmudvh$ to $\Ldeuxmudvs$, with domain $\Hunmudvh$. Moreover, the multiplication operator $\vs$ is a [ *bounded*]{} linear operator from $\Ldeuxmudvs$ to $\Ldeuxmudvh$, with constant of size $1/\dvh$.
We now summarize the structural properties of Equation and the involved operator in the following Proposition:
\[prop:hstruct\] The following properties hold true for all $\dvh>0$.
1. Let us consider $ \Pd = ( - \Dvs +\vs) \Dv$ with domain $$D(\Pd) =\set{g \in \Ldeuxmudvh, \ | \ ( - \Dvs +\vs) \Dv f \in \Ldeuxmudvh}.$$ Then $\Pd$ is self-adjoint non-negative with dense domain and is maximal accretive in $\Ldeuxmudvh $. Moreover, for all $h \in \Ldeuxmudvs$, $g \in \Ldeuxmudvh $ for which it makes sense $$\label{eq:of} \seq{ ( - \Dvs +\vs) h, g} = \seq{h, \Dv g}_\sharp,
\quad \textrm{ and } \quad
\seq{ ( - \Dvs +\vs) \Dv g, g} = \norm{\Dv g}_\Ldeuxmudvs^2.$$
2. For an initial data $f^0
\in D(\Pd)$, there exists a unique solution of in $\ccc^1(\R^+,
\Ldeuxmudvh) \cap \ccc^0(\R^+,
D(\Pd))$, and the associated semi-group naturally defines a solution in $\ccc^0(\R^+, \Ldeuxmudvh)$ when $f^0\in\Ldeuxmudvh$.
3. The preceding properties remain true if we consider operator $\Pd$ in $\Hunmudvh$ with domain $D_\Hunmudvh(\Pd)$. In particular it defines a unique solution of in $\ccc^1(\R^+,
\Hunmudvh) \cap \ccc^0(\R^+,D_\Hunmudvh(\Pd)) $ if $f^0 \in
D_\Hunmudvh(\Pd)$ and a semi-group solution $f \in \ccc^0(\R^+,
\Hunmudvh)$ if $f^0 \in \Hunmudvh$.
4. Constant sequences are the only equilibrium states of equation and the evolution preserves the mass $\seq{f(t)}
= \seq{f^0}$ for all $t\geq 0$.
The proof of the second equality in is a direct consequence of the first equality there, and leads directly to the self-adjointness and the non-negativity of $( - \Dvs +\vs) \Dv$.
The proof of the first equality in is very similar to the one of but we propose it for completeness. We write for $h \in \Ldeuxmudvs$ and $g \in \Ldeuxmudvh$ with finite supports $$\label{eq:calcof}
\begin{split}
\dvh^{-1} \seq{ ( - \Dvs +\vs) h, g} & = \sum_i ((-\Dvs +\vs) h)_i g_i \mu_i \\
& = \sum_{i>0} ((-\Dvs +\vs) h)_i g_i \mu_i -(\Dvs h)_0 g_0 \mu_0
+\sum_{i<0} ((-\Dvs +\vs) h)_i g_i \mu_i
\end{split}$$ The first term in the last right hand side of reads $$\begin{split}
& \sum_{i>0} ((-\Dvs +\vs) h)_i g_i \mu_i \\
& = \sum_{i>0} \sep{ -\frac{h_{i+1} - h_i}{\dvh} + v_i h_i} g_i \mu_i \\
& = \sum_{i>0} h_i \sep{ \frac{-g_{i-1}\mu_{i-1} + g_i \mu_i }{\dvh}
+ v_i g_i \mu_i} + \frac{h_1 g_0}{\dvh} \mu_0 \\
& = \sum_{i>0} h_i g_i \sep{ \frac{-\mu_{i-1} + \mu_i }{\dvh} + v_i
\mu_i}
+ \sum_{i>0} h_i \sep{ - \frac{g_{i-1}- g_i}{\dvh}}\mu_{i-1} + \frac{h_1 g_0}{\dvh} \mu_0 \\
& = \sum_{i>0} h_i (\Dv g)_i \mu_{i-1} + \frac{h_1 g_0}{\dvh} \mu_0,
\end{split}$$ where for the last equality we used the fact that $(\Dv+ v)\muh = 0$. Similarly for the third term in the last right hand side of , we get $$\begin{split}
& \sum_{i<0} ((-\Dvs +\vs) h)_i g_i \mu_i \\
& = \sum_{i<0} \sep{ -\frac{h_{i} - h_{i-1}}{\dvh} + v_i h_i} g_i \mu_i \\
& = \sum_{i<0} h_i \sep{ \frac{-g_i \mu_i +g_{i+1}\mu_{i+1} }{\dvh}
+ v_i g_i \mu_i} - \frac{h_{-1} g_0}{\dvh} \mu_0 \\
& = \sum_{i<0} h_i g_i \sep{ \frac{-\mu_{i} + \mu_{i+1} }{\dvh}
+ v_i \mu_i} + \sum_{i<0} h_i \sep{ - \frac{g_{i+1}- g_i}{\dvh}}\mu_{i+1}
- \frac{h_{-1} g_0}{\dvh} \mu_0 \\
& = \sum_{i<0} h_i (\Dv g)_i \mu_{i+1} - \frac{h_{-1} g_0}{\dvh} \mu_0.
\end{split}$$ The center term in is then $$-(\Dvs h) g_0 \mu_0 = -\frac{ h_1- h_{-1}}{\dvh} g_0 \mu_0.$$ Therefore the sum of the 3 terms in the last right hand side of reads $$\dvh^{-1} \seq{ ( - \Dvs +\vs) h, g} = \sum_{i>0} h_i (\Dv g)_i \mu_{i-1} +
\sum_{i<0} h_i (\Dv g)_i \mu_{i+1} = \dvh^{-1} \seqs{h, \Dv g},$$ since the boundary terms disappear. This is the first equality in .
Concerning the functional analysis and existence of solutions, we observe that the maximal accretivity of $( - \Dvs +\vs) \Dv$ in both $\Ldeuxmudvh$ and $\Hunmudvh$ is then direct to get. In particular, the non-negativity in $\Hunmudvh$ follows from the following identity for $g \in D_\Hunmudvh(\Pd)$: $$\seq{ \Dv( - \Dvs +\vs) \Dv g, \Dv g} = \norm{( - \Dvs +\vs)\Dv g}_\Ldeuxmudv^2
\geq 0.$$ The fact that the equation is well-posed is then a direct consequence of the Hille–Yosida Theorem. The fact that constant sequences are the only equilibrium solutions comes from the fact that for any solution $f\in \ccc^1(\R^+, \Hunmudvh)$, $$\ddt \norm{f}^2 = - \norms{\Dv f}^2,$$ and the preservation of mass comes from the fact that $${\D_t} \seq{f} = \seq{(-\Dv +v)\Dv f, 1} = \seq{\Dv f, \Dv 1} = 0,$$ for any solution $f$ such that $f^0 \in D(\Pd)$, and then in general by density of $D(\Pd)$ in $\Ldeuxmudvh$. The proof is complete.
As in the continuous case, the Poincaré inequality is a fundamental tool to prove the exponential convergence of the solution. It appears that such an inequality is true with $\norm{ \cdot }_{\Ldeuxmudvs }^2$ in the right-hand side, even though the index $0$ is missing in the definition of this norm.
\[prop:poindiscrete\] Let $g\in\Hunmudvh$. Then, $$\norm{g-\seq{g}}^2_{\Ldeuxmudvh } \leq \norm{ \Dv g}_{\Ldeuxmudvs }^2.$$
We essentially follow the proof of the continuous case done before in Section \[subsec:homogeneouscontinuous\]. Let us take $g\in \Hunmudvh$. Replacing if necessary $g$ by $g-\seq{g}$, it is sufficient to prove the result for $\seq{g}=0$. We first note that, with the normalization of $\muh$, we have $$\begin{split}
\dvh^{-1} \norm{g}^2 = \sum_i g_i^2 \muh_i & = \frac{\dvh}{2} \sum_{i,j} (
g_j-g_i)^2 \muh_i \muh_j = \dvh \sum_{i<j} ( g_j-g_i)^2 \muh_i \muh_j,
\end{split}$$ since $2 \sum_{i,j} g_i g_j \muh_i \muh_j = 2 \sum_i g_i \muh_i \sum_j g_j \muh_j =
0$ implies that the diagonal terms are zero. Now for $i<j$, we can write the telescopic sum $$g_j-g_i = \sum_{\ell=i+1}^j (g_\ell-g_{\ell-1}),$$ so that $$\label{eq:interm}
\begin{split}
\dvh^{-1}\sum_i g_i^2 \muh_i & = \sum_{i < j} \sep{ \sum_{\ell=i+1}^j
(g_\ell-g_{\ell-1}) }^2 \muh_i \muh_j \leq \sum_{i < j} \sep{
\sum_{\ell=i+1}^j (g_\ell-g_{\ell-1})^2 } (j-i) \muh_i \muh_j,
\end{split}$$ where we used the discrete flat Cauchy–Schwarz inequality. Let us now introduce $\G$ a discrete anti-derivative of $(g_\ell-g_{\ell-1})^2$, for example this one: $$\G_j = - \sum_{\ell=j+1}^{-1} (g_\ell-g_{\ell-1})^2 \textrm{ for } j\leq -1,
\qquad \G_j = \sum_{\ell=0}^j (g_\ell-g_{\ell-1})^2 \textrm{ for } j\geq 0,$$ so that for all $i<j$ we have $\G_j - \G_i = \sum_{\ell=i+1}^{j} (g_\ell-g_{\ell-1})^2$. We infer from $$\begin{split}
\dvh^{-1}\sum_i g_i^2 \muh_i & \leq \sum_{i < j} \sep{ \G_j-\G_i } (j-i)
\muh_i \muh_j = \frac{1}{2} \sum_{i,j} \sep{ \G_j-\G_i } (j-i) \muh_i
\muh_j,
\end{split}$$ where in the last equality we used that $\sep{\G_j-\G_i } (j-i) = \sep{\G_i-\G_j } (i-j)$ and the fact that the diagonal terms vanish. We can now split the last sum into four parts: $$\begin{split}
\dvh^{-1}\sum_i g_i^2 \muh_i & \leq \frac{1}{2} \sep{
\sum_{i,j} \G_j j \muh_i \muh_j +\sum_{i,j} \G_i i \muh_i \muh_j
-\sum_{i,j}
\G_i j \muh_i \muh_j - \sum_{i,j} \G_j i \muh_i \muh_j} \\
& \leq \dvh^{-1} \sum_{i} \G_i i \muh_i = \dvh^{-1}\sum_{i \neq 0} \G_i i
\muh_i,
\end{split}$$ where we used the discrete Fubini Theorem and the fact that $\sum_{j} j\muh_j = 0$ and $\dvh\sum_j \muh_j = 1$ (and their counterparts in variable $i$), by parity and normalization of $\muh$. The last step is to perform a discrete integration by part (Abel transform) using deeply the functional equation satisfied by $\muh$ that we recall now : $$i\muh_i = -\frac{\muh_i- \muh_{i-1}}{\dvh^2} \textrm{ for } i>0, \qquad i\muh_i
= -\frac{\muh_{i+1} - \muh_{i}}{\dvh^2} \textrm{ for } i<0.$$ We therefore get $$\begin{split}
\sum_{i \neq 0} \G_i i \muh_i& = \sum_{i > 0} \G_i i \muh_i
+ \sum_{i < 0} \G_i i \muh_i\\
& = - \sum_{i > 0} \G_i \frac{\muh_i- \muh_{i-1}}{\dvh^2}
- \sum_{i <0 } \G_i \frac{\muh_{i+1} - \muh_{i}}{\dvh^2} \\
& = - \sum_{i > 0} \frac{ \G_i - \G_{i+1}}{\dvh^2} \muh_i +
\frac{\G_1}{\dvh^2} \muh_0 - \sum_{i < 0} \frac{ \G_{i-1} - \G_{i}}{\dvh^2}
\muh_i - \frac{\G_{-1}}{\dvh^2} \muh_0.
\end{split}$$ Now, using the definition of $\G$ and in particular the fact that $$\G_1 - \G_{-1} = (g_{1}- g_0)^2 + (g_{0}- g_{-1})^2,$$ we obtain $$\label{eq:sommemirac}
\begin{split}
\sum_{i \neq 0} \G_i i \muh_i
& = \sum_{i > 0} \sep{ \frac{ g_{i+1}- g_i}{\dvh}}^2 \muh_i
+ \sum_{i < 0} \sep{ \frac{ g_{i}- g_{i-1}}{\dvh}}^2 \muh_i \\
& \qquad \qquad \qquad \qquad + \sep{\frac{ g_{1}- g_0}{\dvh}}^2 \muh_0 +
\sep{\frac{ g_{0}- g_{-1}}{\dvh}}^2
\muh_0 \\
& = \dvh^{-1} \norm{ \Dv g}_{\Ldeuxmudvs }^2
\end{split}$$ and therefore $ \norm{g}_{\Ldeuxmudvh }^2 \leq \norm{\Dv g}_{\Ldeuxmudvs }^2$. The proof is complete.
We can now study the exponential convergence to the equilibrium in the spaces $\Ldeuxmudvh$ and $\Hunmudvh$ of the solution $f$ of , for $f^0\in \Ldeuxmudvh$ and $f^0\in \Hunmudvh$ respectively. As in the continuous case of Section \[subsec:homogeneouscontinuous\], we propose two different entropies well-adapted to the coming discretization case: $$\fffd(g) = \norm{g}_{\Ldeuxmudvh }^2, \qquad \gggd(g) = \norm{g}_{\Ldeuxmudvh
}^2 + \norm{\Dv g}_{\Ldeuxmudvs}^2,$$ defined for $g\in\Ldeuxmudvh$ and $g\in\Hunmudvh$ respectively.
Our result for the exponential convergence to equilibrium of the exact solution of the discrete evolution equation is the following.
Let $f^0 \in \Ldeuxmudvh$ such that $\seq{f^0} = 0$ and let $f$ be the solution of (in the semi-group sense) in $\ccc^0(\R^+, \Ldeuxmudvh)$ with initial data $f^0$. Then for all $t\geq0$, $$\fffd(f(t)) \leq \exp^{-2t}\fffd(f^0).$$ If in addition $f^0 \in \Hunmudvh$ and $f$ is the semi-group solution in $f \in \ccc^0(\R^+, \Hunmudvh)$, then for all $t\geq 0$ $$\gggd(f(t)) \leq \exp^{-t}\gggd(f^0).$$
We follow the steps of the proof of Theorem \[thm:exponentialtrendtoequilibrium\]. In particular we take $f^0 \in D_\Hunmudvh(\Pd)$ in all the computations below, so that the computations and differentiations below are authorized, and the Theorem is then a consequence of the density of $D_\Hunmudvh(\Pd)$ in $\Ldeuxmudvh$ or $\Hunmudvh$.
For the first entropy, we have, using , , and Proposition \[prop:poindiscrete\], $$\ddt \fffd(f) = -2 \seq{ (-\Dv^\sharp+\vs)D_v f , f}
= -2 \norm{\Dv f}_{\Ldeuxmudvs }^2 \leq -2 \norm{f}^2
= -2 \fffd(f).$$ Now we deal with the second entropy $\gggd$. We use the discrete Poincaré inequality of Proposition \[prop:poindiscrete\] and the same splitting $$2 \norm{\Dv f}_{\Ldeuxmudvs }^2 = \norm{\Dv f}_{\Ldeuxmudvs }^2 + \norm{\Dv
f}_{\Ldeuxmudvs }^2 \geq \norm{\Dv f}_{\Ldeuxmudvs }^2 + \norm{f}^2,$$ as in the proof of Theorem \[thm:exponentialtrendtoequilibrium\]. We get next from equations and $$\begin{split}
\ddt \gggd(f) & = -2 \seq{ (-\Dvs+\vs) \Dv f , f}_{\Ldeuxmudvh }
-2 \seq{ \Dv(-\Dvs+\vs) \Dv f , \Dv f}_{\Ldeuxmudvs } \\
& = -2 \norm{\Dv f}_{\Ldeuxmudvs }^2 - 2 \norm{(-\Dvs+\vs) \Dv f}_{\Ldeuxmudvh }^2\\
& \leq - \norm{f}_{\Ldeuxmudvh }^2 - \norm{\Dv f}_{\Ldeuxmudvs }^2 - 2
\norm{(-\Dvs+\vs) \Dv f}_{\Ldeuxmudvh }^2 \leq - \gggd(f).
\end{split}$$ The proof is complete.
As in the Corollary \[cor:exponentialtrendtoequilibrium\] we therefore immediately get
Let $f^0 \in \Ldeuxmudvh$ and let $f$ be the solution of in $\ccc^0(\R^+, \Ldeuxmudvh)$ with initial data $f^0$. Then for all $t\geq 0$, $$\norm{f(t) - \seq{f^0}}_\Ldeuxmudvh \leq \exp^{-t}\norm{f^0 - \seq{f^0}}_\Ldeuxmudvh.$$ If in addition $f^0 \in \Hunmudvh$ then $f \in \ccc^0(\R^+, \Hunmudvh)$ and we have $$\norm{f(t) - \seq{f^0}}_\Hunmudvh \leq \exp^{-\frac{t}{2}}\norm{f^0 -
\seq{f^0}}_\Hunmudvh.$$
Remark on the full discretization {#subsec:fulld}
---------------------------------
A full discretization of the preceding equation is of course possible, using the velocity discretization introduced in this section, and, for example the implicit Euler scheme $$f^{n} = f^{n+1} -\dth ( - \Dvs +\vs) \Dv f^{n+1}.$$ In order to describe the long time behavior of such a fully discretized scheme, the functional framework introduced in this Section can be used, and similar arguments work to obtain exponential convergence to equilibrium[^7]. We do not present in this paper the corresponding statements and results since they are actually not difficult to obtain, and may be thought as very simple versions of the results of the following sections. Indeed, we shall focus on the discretization on the full inhomogeneous equation in Section \[sec:eqinhomo\] and on the discretization of the homogeneous and inhomogeneous equations and on a bounded velocity domain with Neumann conditions (in velocity) in Sections \[sec:homobounded\] and \[sec:eqinhomoboundedvelocity\].
The inhomogeneous equation in space, velocity and time {#sec:eqinhomo}
======================================================
In this Section, we deal with the inhomogeneous equation with velocity domain $\R$ and its discretized versions. We present the fully continuous analysis in the first subsection. Then, we study in Subsection \[subsec:discretentcontinuenxv\] the semi-discretization in time by the implicit Euler scheme. Afterwards, we focus in Subsection \[sec:inhomsd\] on the semi-discretization in space and velocity only. In particular, we introduce part of the material that will be needed in the final study of the fully-discretized implicit Euler scheme which is considered in Subsection \[subsec:eqinhomototalementdiscretisee\], where we prove Theorem \[thm:eulerimplicite\].
The fully continuous analysis {#sec:inhomc}
-----------------------------
In this subsection we recall briefly now standard results about the original inhomogeneous Fokker-Planck equation with unknown $F(t,x,v)$ with $(t,x,v) \in\R^+\times\T\times\R$ and where $\T=[0,1]_{\rm per}$. The equation reads $${\D_t} F+v{\D_x}F-{\D_v} ({\D_v}+v)F=0, \qquad F|_{t=0} = F^0,$$ We assume that the initial density $F^0$ is non-negative, in $L^1(\T\times\R)$, and satisfies $\int_{\T\times\R}F^0 \dx\dv=1$. We directly check that $(x,v) \longmapsto \mu(v) $ is an equilibrium of the equation, and we shall continue to denote this function $\mu$ (in the sense that it is now a constant function w.r.t. the variable $x$). As in the homogeneous case, it is convenient to work in the subspace $\mu \Ldeuxmudvdx \hookrightarrow L^1(\dv\dx)$ and take benefit of the associated Hilbertian structure. We therefore pose for the following $f=(F-\mu)/\mu$, and we perform here the analysis for $f\in \Ldeuxmudvdx$ as we did in $\Ldeuxmudv$ in the homogeneous case in Section \[sec:homogeneous\]. The rescaled equation writes $$\label{eq:rescaled}
{\D_t} f+v{\D_x}f+(-{\D_v}+v){\D_v} f=0, \qquad \qquad f|_{t=0} = f^0.$$ The non-negativity of the associated operator $P = v{\D_x}+(-{\D_v}+v){\D_v} $ is straightforward since $v{\D_x}$ is skew-adjoint in $\Ldeuxmudvdx$. The maximal accretivity of this operator in $\Ldeuxmudvdx$ or $\Hunmudvdx$ is not so easy and we refer for example to [@HelN04]. As in the homogeneous case, using the Hille–Yosida Theorem, this implies that for an initial datum $f^0 \in D(P)$ (resp. $D_\Hunmudvdx(P)$) there exists a unique solution in $\ccc^1( \R^+, \Ldeuxmudvdx) \cap \ccc^0( \R^+, D(P)) $ (resp. $\ccc^1(\R^+, \Hunmudvdx) \cap \ccc^0( \R^+, D_\Hunmudvdx(P) )$. As before we will call semi-group solution the function in $\ccc^0(\R^+, \Ldeuxmudvdx)$ (resp. $\ccc^0(\R^+, \Hunmudvdx)$) given by the semi-group associated to $P$ with the suitable domain.
From now on, the norms and scalar products without subscript are taken in $\Ldeuxmudvdx$.
As in the homogeneous case, we shall define an entropy adapted to the $\Hunmudvdx$ framework. Its exponential decay, however, is a bit more difficult to prove in the inhomogeneous case. As consequence of the maximal accretivity, we first note that, for $f^0\in D_\Hunmudvdx(P)$, along the corresponding solution of , we have $$\ddt \norm{f}^2 = -2 \seq{ v{\D_x} + (-{\D_v}+v){\D_v} f, f} = -2 \norm{D_v f}^2 \leq 0,$$ so that $g\mapsto \norm{g}^2$ is an entropy of the system. Such an inequality is nevertheless not strong or precise enough to get an exponential decay. In order to prepare for the discrete cases in the next sections, we again introduce and recall a particularly simple entropy leading to the result.
For $C>D>E>1$ to be precised later, the modified entropy is defined for $g\in \Hunmudvdx$ by $$\label{eq:entropyfunc}
\hhh(g) \defegal C\norm{g}^2+D\norm{{\D_v} g}^2+E\seq{{\D_v} g,{\D_x} g}+\norm{{\D_x}g}^2.$$ We will show later that for well chosen $C,D,E$, $t\mapsto \hhh(f(t))$ is exponentially decreasing when $f$ solves the rescaled equation with initial datum $f^0\in \Hunmudvdx$. As a norm in $\Hunmudvdx$ we choose the standard one defined for $g\in \Hunmudvdx$ by $$\norm{g}_\Hunmudvdx \defegal \left(\norm{g}^2 + \norm{{\D_v} g}^2
+ \norm{{\D_x} g}^2\right)^\frac12.$$ We first prove that $\sqrt{\hhh}$ is equivalent to the $\Hunmudvdx$-norm.
\[lem:equiv\] Assume $C>D>E>1$ are given such that $E^2<D$. For all $g\in \Hunmudvdx$, one has $$\dfrac{1}{2}\norm{g}_{\Hunmudvdx}^2\leq\hhh(g)\leq 2C\norm{g}_{\Hunmudvdx}^2.$$
Using a standard Cauchy–Schwarz–Young inequality, we observe that $$2\abs{E\seq{{\D_v} g,{\D_x} g}}\leq E^2\norm{{\D_v} g}^2+\norm{{\D_x} g}^2,$$ which implies for all $g\in \Hunmudvdx$ $$\begin{gathered}
\underbrace{C}_{1/2\leq}\norm{g}^2
+\underbrace{(D-E^2/2)}_{1/2\leq D/2\leq}\norm{{\D_v} g}^2
+\dfrac{1}{2}\norm{{\D_x}g}^2 \\
\leq \hhh(g)\leq
C\norm{g}^2+\underbrace{(D+E^2/2)}_{\leq D+D/2\leq 3C/2\leq
2C}\norm{{\D_v} g}^2+\underbrace{3/2}_{\leq 3C/2\leq 2C} \norm{{\D_x}g}^2,\end{gathered}$$ which in turn implies the result since $E^2<D$.
As in the homogeneous case, one of the main ingredients to prove the exponential decay is again a Poincaré inequality, which is essentially obtained by tensorizing the one in velocity with the one in space. In the following, we denote the mean of $g\in \Ldeuxmudvdx$ with respect to all variables by $$\seq{g} \defegal \iint g(x,v) \mudv\dx.$$
\[lem:fullPoincarecontinu\] For all $g \in \Hunmudvdx$, we have $$\norm{g-\seq{g}}^2 \leq \norm{{\D_v} g}^2 + \norm{{\D_x} g}^2.$$
Replacing if necessary $g$ by $g-\seq{g}$, it is sufficient to prove the result for $\seq{g}=0$. For convenience, we introduce $\rho: x\mapsto \int g(x,\cdot) \mudv$, the macroscopic density of probability. Recall the standard Poincaré inequality in space only $$\norm{\rho}^2 \leq \frac{1}{4\pi^2}\norm{{\D_x} \rho}^2\leq \norm{{\D_x} \rho}^2,$$ which is a consequence of the the fact that the torus $\T$ is compact and the fact that $\int \rho \dx = \iint g \mu \dv\dx = 0$ (note that the proof of this last Poincaré inequality is very standard and could be done following the method employed in the proof of Lemma \[lem:Poincarecontinu\]). Now we observe that orthogonal projection properties and Fubini Theorem imply $$\norm{\rho}^2_\Ldeuxdx \leq \norm{g}^2
\qquad \text{and}\qquad
\norm{{\D_x}\rho}^2_\Ldeuxdx \leq \norm{{\D_x} g}^2,$$ since $(x,v) \mapsto \rho(x)$ (resp. $(x,v) \mapsto {\D_x}\rho(x)$) is the orthogonal projection of $g$ (resp. ${\D_x} g$) onto the closed space $$\set{ (x,v) \longmapsto \phi(x) \ | \ \phi \in \Ldeuxdx},$$ and $\norm{\phi \otimes 1 }_{} = \norm{\phi}_\Ldeuxdx$ for all $\phi \in \Ldeuxdx$ since we are in probability spaces (there is a natural injection $\Ldeuxdx \hookrightarrow \Ldeuxmudvdx$ of norm $1$). Using the Fubini Theorem again, we also directly get from Lemma \[lem:Poincarecontinu\] that $$\norm{g- \rho\otimes 1}_{}^2 \leq \norm{{\D_v} g}_{}^2.$$ We therefore can write, using orthogonal projection properties again, that $$\begin{split}
\norm{g}_{}^2 & = \norm{g- \rho\otimes 1}_{}^2 + \norm{ \rho\otimes 1}_{}^2 \\
& = \norm{g- \rho\otimes 1}_{}^2 + \norm{\rho}_\Ldeuxdx^2 \\
& \leq \norm{{\D_v} g}_{}^2 + \norm{{\D_x} \rho}_\Ldeuxdx^2 \\
& \leq \norm{{\D_v} g}_{}^2 + \norm{{\D_x} g}_{}^2.
\end{split}$$ The proof is complete.
For convenience, we will sometimes denote in the following $$\Xzero = v {\D_x},$$ so that the equation satisfied by $f$ is ${\D_t} f = -\Xzero f -(-{\D_v} +v) {\D_v} f$. We shall use intensively the fact that $\Xzero$ is skew-adjoint and the formal adjoint of $(-{\D_v} + v)$ is ${\D_v}$, together with the commutation relations $$\label{eq:commrel}
\adf{{\D_v}, X_0} = {\D_x}, \qquad \adf{{\D_x}, X_0} = 0,
\qquad\text{and}\qquad
\adf{{\D_v},(-{\D_v}+v)}=1.$$
\[thm:decrexp\] Assume that $C>D>E>1$ satisfy $E^2<D$ and $(2D+E)^2<2C$. Let $ f^0 \in \Hunmudvdx$ such that $\seq{f^0} = 0$ and let $f$ be the solution (in the semi-group sense) in $\ccc^0(\R^+, \Hunmudvdx)$ of Equation . Then for all $t\geq 0$, $$\hhh(f(t))\leq \hhh(f^0)\exp^{-2\kappa t}.$$ with $2\kappa = \frac{E}{8C}$.
We suppose that $f^0 \in D_\Hunmudvdx(P)$ and we consider the corresponding solution $f$ of in $\ccc^1(\R^+, \Hunmudvdx) \cap \ccc^0(\R^+, D_\Hunmudvdx(P))$ with initial datum $f^0$. The theorem for a general $f^0\in\Hunmudvdx$ is then a consequence of the density of $D_\Hunmudvdx(P)$ in $\Hunmudvdx$.
We compute separately the time derivatives of the four terms defining $\hhh(f(t))$. Omitting the dependence on $t$, the time derivative of the first term in $\hhh(f(t))$ reads $$\begin{aligned}
\ddt \norm{f}^2_{\Ldeuxmudvdx}&
=2\seq{{\D_t}f,f}=-2\underbrace{\seq{\Xzero f,f}}_{=0}-2\seq{(-{\D_v}+v){\D_v} f,f}\\
&=-2\norm{{\D_v} f}_{\Ldeuxmudvdx}^2.
\end{aligned}$$ The second term writes $$\begin{aligned}
\ddt \norm{{\D_v} f}^2_{\Ldeuxmudvdx}&=2\seq{{\D_v}({\D_t}f),{\D_v} f}\\
&=-2\seq{{\D_v}(\Xzero f+(-{\D_v}+v){\D_v} f),{\D_v} f}\\
&=-2\underbrace{\seq{\Xzero {\D_v} f,{\D_v} f}}_{=0}
-2\seq{\adf{{\D_v},\Xzero }f,{\D_v} f}-2\seq{{\D_v}(-{\D_v}+v){\D_v} f,{\D_v} f}.\end{aligned}$$ We again use the fact that $\Xzero $ is a skew-adjoint operator in $\Ldeuxmudvdx$ and the fundamental relation $\adf{{\D_v},\Xzero }={\D_x}$ and we get $$\ddt \norm{{\D_v} f}^2_{\Ldeuxmudvdx}=-2 \seq{{\D_x}f,{\D_v} f}-2\norm{(-{\D_v}+v){\D_v} f}^2.$$ The time derivative of the third term can be computed as follows $$\begin{aligned}
\ddt \seq{{\D_x}f,{\D_v} f}
= & -\seq{{\D_x}(\Xzero f+(-{\D_v}+v){\D_v} f),{\D_v} f}
-\seq{{\D_x}f,{\D_v}(\Xzero f+(-{\D_v}+v){\D_v} f)}\\
= & -\seq{ {\D_x} \Xzero f,{\D_v} f}
-\seq{ {\D_x} f,{\D_v} \Xzero f} \qquad (I) \\
& -\seq{{\D_x} (-{\D_v}+v){\D_v} f, {\D_v} f}
-\seq{{\D_x} f, {\D_v} (-{\D_v}+v){\D_v} f}. \qquad (II)\end{aligned}$$ For the term $(I)$ we use the fact that $\Xzero$ is skew-adjoint and the commutation relations to obtain $$\begin{aligned}
(I) = & \underbrace{-\seq{ \Xzero {\D_x} f,{\D_v} f}
-\seq{ {\D_x} f, \Xzero {\D_v} f}}_{0} -\seq{ {\D_x} f,
\adf{ {\D_v}, \Xzero} f} = - \norm{{\D_x} f}^2.\end{aligned}$$ For the term $(II)$ we use that the adjoint of ${\D_v}$ is $-({\D_v} +v)$ and the one of ${\D_x}$ is $-{\D_x}$ and we get $$\begin{aligned}
(II) = & \seq{ (-{\D_v}+v){\D_v} f,{\D_x} {\D_v} f}
+ \seq{{\D_v} (-{\D_v}+v) f,{\D_x}{\D_v} f} \\
= & 2\seq{ (-{\D_v}+v){\D_v} f,{\D_x} {\D_v} f} +
\seq{\adf{ {\D_v} , (-{\D_v}+v)} f,{\D_x}{\D_v} f}.\end{aligned}$$ Now the commutation relation yields $$\begin{aligned}
(II) = & 2\seq{ (-{\D_v}+v){\D_v} f,{\D_x} {\D_v} f} +
\seq{ f,{\D_x}{\D_v} f} \\
= & 2\seq{ (-{\D_v}+v){\D_v} f,{\D_x} {\D_v} f} -
\seq{ {\D_x} f,{\D_v} f}.\end{aligned}$$ Form the preceding estimates on $(I)$ and $(II)$ we therefore have $$\ddt \seq{{\D_x}f,{\D_v} f}=-\norm{{\D_x}f}^2+2
\seq{(-{\D_v}+v){\D_v} f,{\D_x}{\D_v} f}-\seq{{\D_x}f,{\D_v} f}.$$ Finally, observing that ${\D_x} f$ also solves , we obtain for the last term of $\hhh(f(t))$ the same estimate as the one we obtained for the first term: $$\ddt \norm{{\D_x}f}^2_{\Ldeuxmudvdx}=-2\norm{{\D_v} {\D_x} f}_{\Ldeuxmudvdx}^2.$$ Eventually, we obtain $$\begin{aligned}
\ddt
\hhh(f)&=-2C\norm{{\D_v} f}^2-2D\norm{(-{\D_v}+v){\D_v} f}^2
-E\norm{{\D_x}f}^2-2\norm{{\D_x}{\D_v} f}^2 \nonumber\\
\label{eq:decrentropycontinuenonborne}
&\qquad -(2D+E)\seq{{\D_x}f,{\D_v} f}+2E\seq{(-{\D_v}+v){\D_v} f,{\D_x}{\D_v} f}.\end{aligned}$$
Only the last two terms above do not have a sign [*a priori*]{}. Using the Cauchy–Schwarz–Young inequality, we observe that $$\abs{(2D+E)\seq{{\D_x}f,{\D_v} f}}\leq \dfrac{1}{2}\norm{{\D_x}f}^2
+\frac{(2D+E)^2}{2}\norm{{\D_v} f}^2,$$ and $$\abs{2E\seq{(-{\D_v}+v){\D_v} f,{\D_x}{\D_v} f}}\leq
\norm{{\D_x}{\D_v} f}^2 +E^2 \norm{(-{\D_v}+v){\D_v} f}^2.$$ Therefore, assuming again that $1<E<D<C$, $E^2<D$ and $(2D+E)^2<2C$, we get $$\ddt \hhh(f)\leq
-C\norm{{\D_v} f}^2-(E-1/2)\norm{{\D_x}f}^2\leq -\dfrac{E}{2}(\norm{{\D_v} f}^2+\norm{{\D_x}f}^2).$$ Using the Poincaré inequality in space-velocity proven in Lemma \[lem:fullPoincarecontinu\] with constant $1$, we derive $$\begin{aligned}
-\dfrac{E}{2}(\norm{{\D_v} f}^2+\norm{{\D_x}f}^2)
&\leq -\dfrac{E}{4}(\norm{{\D_v} f}^2+\norm{{\D_x}f}^2)-\dfrac{E}{4}\norm{f}^2
\leq-\dfrac{E}{4}\dfrac{1}{2C}\hhh(f),\end{aligned}$$ using eventually the equivalence property proven in Lemma \[lem:equiv\]. We therefore have with $2\kappa = E/8C$ $$\ddt \hhh(f)\leq - 2\kappa \hhh(f),$$ and Theorem \[thm:decrexp\] is a consequence of the Gronwall Lemma. The proof is complete.
Let $C>D>E>1$ be chosen as in Theorem \[thm:decrexp\], and pose $\kappa = E/(16C)$. Let $f^0\in \Hunmudvdx$ such that $\seq{f^0} = 0$ and let $f$ be the semi-group solution in $\ccc^0(\R^+, \Hunmudvdx)$ of equation . Then for all $t\geq 0$, we have $$\norm{f(t)}_{\Hunmudvdx}\leq 2\sqrt{C} \exp^{-\kappa t} \norm{f^0 }_{\Hunmudvdx}.$$
Choose $C>D>E>1$ as in Theorem \[thm:decrexp\] and set $\kappa =E/(16C)$. We apply Theorem \[thm:decrexp\] and Proposition \[lem:equiv\] to $f$ and we obtain for all $t\geq 0$, $$\norm{f(t)}^2_{\Hunmudvdx} \leq 2 \hhh(f(t))
\leq 2 \exp^{-2\kappa t} \hhh(f^0)
\leq 4 C \exp^{-2\kappa t}\norm{f^0}^2_\Hunmudvdx.$$ The proof is complete.
The semi-discretization in time {#subsec:discretentcontinuenxv}
-------------------------------
In order to solve Equation numerically, we consider the one-step implicit Euler method. We introduce the time step $\dth>0$ supposed to be small.
We shall say that a sequence $(f^n)_{n\in\N} \in (\Ldeuxmudvdx)^\N$ (resp. $(\Hunmudvdx)^\N$) satisfies the (scaled) time-discrete inhomogeneous Fokker-Planck equation if for a given $f^0$ in $\Ldeuxmudvdx$ (resp. $\Hunmudvdx$), for all $n \in \N$, $$\label{eq:eulerimpl}
f^{n+1} = f^n - \dth (\Xzero f^{n+1}+(-{\D_v}+v){\D_v} f^{n+1}),$$ for some $\dth>0$.
The main goal of this section is to prove that this numerical scheme has the same asymptotic behavior as that of the exact flow, in the sense that it satisfies a discrete analogue of Theorem \[thm:decrexp\] (see Theorem \[thm:decrexpeulerimpl\]).
We first check that this implicit scheme is well posed.
\[prop:stabiliteetmoyennepreservee\] For all given initial condition $f^0$ in $\Ldeuxmudvdx$ (resp. $\Hunmudvdx$), and all $\dth>0$, there exists a unique solution $f \in (\Ldeuxmudvdx)^\N$ (resp. $(\Hunmudvdx)^\N$) of the time-discrete evolution equation . Moreover it satisfies for all $n\in \N$, $$\norm{f^n} \leq \norm{f^0}, \qquad \seq{f^n} = \seq{f^0}.$$
Let us denote $P = \Xzero +(-{\D_v}+v) {\D_v}$. Then equation writes $$(\Id + \dth P) f^{n+1} = f^n.$$ The linear operator $P$ is maximal accretive in $\Ldeuxmudvdx$ (resp. $\Hunmudvdx$, see [@HelN04]), so that the resolvent $(\Id + \dth P)^{-1}$ is a well defined operator in $\Ldeuxmudvdx$ (resp. $\Hunmudvdx$) of norm $1$. This implies the well-posedness and the uniform boundedness of the norms of the functions $f^n$ with respect to $n$. Similarly to the continuous case, we have in addition $$\seq{f^{n+1}} = \seq{f^n} + \dth \iint \sep{ \Xzero f^{n+1} + (-{\D_v}+v) {\D_v}
f^{n+1}} \mudvdx = \seq{f^n} + 0 = \seq{f^0},$$ by integration by parts. The proof is complete.
In order to prove the exponential (discrete-)time decay of the solutions in Theorem \[thm:decrexpeulerimpl\], similar to the exponential decay of the continuous solutions (Theorem \[thm:decrexp\]), we shall examine the behaviour of the same entropy $\hhh$ defined in along numerical solutions of .
\[lem:CSplus\] Assume $C>D>E>1$ and $E^2<D$. Let us introduce the bilinear map $\varphi$ defined for all $g,\gtilde\in \Hunmudvdx$ by $$\varphi(g,\gtilde)=C\seq{g,\gtilde}+D\seq{{\D_v} g,{\D_v} \gtilde}
+ \frac{E}{2}\seq{{\D_x}g,{\D_v}\gtilde}+\frac{E}{2}\seq{{\D_v} g,{\D_x}
\gtilde}
+ \seq{{\D_x} g,{\D_x} \gtilde}.$$ Then $\phi$ defines a scalar product in $\Hunmudvdx$ and the associated norm is $\sqrt{\hhh(\cdot)}$. In particular one has $$|\varphi(g,\gtilde)| \leq \sqrt{\hhh(g)}\sqrt{\hhh(\gtilde)}
\leq \frac{1}{2} \hhh(g) + \frac{1}{2} \hhh(\gtilde).$$
The map $\varphi$ is bilinear and symmetric on $\Hunmudvdx$. It is positive definite on $\Hunmudvdx$ provided $E^2<D$ using Proposition \[lem:equiv\]. In particular, it is non-negative and one has the Cauchy–Schwarz’ inequality $$\forall g,\gtilde\in \Hunmudvdx,\qquad |\varphi(g,\gtilde)|
\leq\sqrt{\mathcal H(g)}\sqrt{\mathcal H(\gtilde)}.$$ The last inequality is just another Young’s inequality.
We now state the main Theorem of this section.
\[thm:decrexpeulerimpl\] Assume that $C>D>E>1$ satisfy $E^2<D$ and $(2D+E)^2<2C$. For all $\dth>0$ and $f^0\in \Hunmudvdx$, we denote by $(f^n)_{n\in\N}$ the sequence solution of the implicit Euler scheme . If $\seq{f^0}=0$, then $$\forall n\in\N,\qquad \hhh(f^n)\leq (1+ 2 \kappa \dth)^{-n} \hhh(f^0).$$ with $\kappa = E/(16C)$.
In addition, for all $\dth >0$ there exists $k>0$ (explicit) with $\lim_{\dth \rightarrow 0} k = \kappa$ such that $$\forall n\in\N,\qquad \hhh(f^n)\leq \hhh(f^0)\exp^{-2 k n\dth}.$$
Using Proposition \[prop:stabiliteetmoyennepreservee\], the sequence $(f^n)_{n\in\N}$ satisfies for all $n\in\N$ $\seq{f^n} = \seq{f^0} =0$. Fix $n\in\N$. We evaluate the four terms in the definition of $\mathcal H(f^{n+1})$ as follows. Taking the $\Ldeuxmudvdx$-scalar product of relation with $f^{n+1}$ yields $$\norm{f^{n+1}}^2 = \seq{f^n,f^{n+1}}-\dth\seq{\Xzero f^{n+1},f^{n+1}}
- \dth \seq{(-{\D_v}+v){\D_v} f^{n+1},f^{n+1}}.$$ The first term in $\dth$ above vanishes by skew-adjointness of the operator $\Xzero $. The second term in $\dth$ above can be rewritten to obtain $$\label{eq:estim1eulerimpl}
\norm{f^{n+1}}^2 = \seq{f^n,f^{n+1}}
- \dth \norm{{\D_v} f^{n+1}}^2,$$ since $-{\D_v}+v$ is the formal adjoint of ${\D_v}$. Differentiating relation with respect to $v$ and taking the $\Ldeuxmudvdx$-scalar product with ${\D_v} f^{n+1}$ allows to write $$\begin{gathered}
\norm{{\D_v} f^{n+1}}^2 = \seq{{\D_v} f^n,{\D_v} f^{n+1}}
-\dth\seq{\Xzero {\D_v} f^{n+1},{\D_v} f^{n+1}} \\
- \dth \seq{{\D_x} f^{n+1},{\D_v} f^{n+1}} - \dth
\seq{{\D_v}(-{\D_v}+v){\D_v} f^{n+1},{\D_v} f^{n+1}}.
\end{gathered}$$
As before, the skew-adjointness of $\Xzero $ makes the first term in $\dth$ vanish. The third term in $\dth$ can be rewritten as before so that $$\label{eq:estim2eulerimpl}
\norm{{\D_v} f^{n+1}}^2 = \seq{{\D_v} f^n,{\D_v} f^{n+1}}
- \dth \seq{{\D_x} f^{n+1},{\D_v} f^{n+1}}
- \dth \norm{(-{\D_v}+v){\D_v} f^{n+1}}^2.$$
For the third term in $\mathcal H(f^{n+1})$, we first compute ${\D_v} f^{n+1}$ with and take its $\Ldeuxmudvdx$-scalar product with ${\D_x} f^{n+1}$ to write $$\begin{aligned}
\lefteqn{\seq{{\D_v} f^{n+1},{\D_x} f^{n+1}} = } \\
& & \seq{{\D_v} f^n,{\D_x} f^{n+1}}
- \dth \seq{\Xzero {\D_v} f^{n+1},{\D_x} f^{n+1}}
- \dth \seq{{\D_x}f^{n+1},{\D_x} f^{n+1}}\\
& & - \dth \seq{{\D_v} (-{\D_v}+v){\D_v} f^{n+1},{\D_x} f^{n+1}}.\end{aligned}$$ Using that $[{\D_v},(-{\D_v}+v)]=1$, we obtain $$\begin{aligned}
\lefteqn{\seq{{\D_v} f^{n+1},{\D_x} f^{n+1}} =}\\
& & \seq{{\D_v} f^n,{\D_x} f^{n+1}}
- \dth \seq{\Xzero {\D_v} f^{n+1},{\D_x} f^{n+1}}
- \dth \norm{{\D_x}f^{n+1}}^2\\
& & - \dth \seq{{\D_v} f^{n+1},{\D_x} f^{n+1}}
- \dth \seq{(-{\D_v}+v)\partial^2_v f^{n+1},{\D_x} f^{n+1}}.\end{aligned}$$ Then, we compute ${\D_x} f^{n+1}$ with and take its $\Ldeuxmudvdx$-scalar product with ${\D_v} f^{n+1}$ to write $$\seq{{\D_x} f^{n+1},{\D_v} f^{n+1}} =
\seq{{\D_x} f^n,{\D_v} f^{n+1}}
- \dth \seq{v{\D_x} ^2 f^{n+1},{\D_v} f^{n+1}}
- \dth \seq{{\D_x}(-{\D_v}+v){\D_v} f^{n+1},{\D_v} f^{n+1}}.$$
Summing up the last two identities yields $$\begin{aligned}
\lefteqn{\seq{{\D_v} f^{n+1},{\D_x} f^{n+1}}
+ \seq{{\D_x} f^{n+1},{\D_v} f^{n+1}}=}\\
& & \seq{{\D_v} f^n,{\D_x} f^{n+1}} + \seq{{\D_x} f^n,{\D_v} f^{n+1}} \\
& & - \dth \norm{{\D_x}f^{n+1}}^2 - \dth \seq{{\D_v} f^{n+1},{\D_x} f^{n+1}}\\
& & - \dth \seq{\Xzero {\D_v} f^{n+1},{\D_x} f^{n+1}}
- \dth \seq{(-{\D_v}+v)\partial^2_v f^{n+1},{\D_x} f^{n+1}}\\
& & + \dth \seq{{\D_x} f^{n+1},\Xzero {\D_v} f^{n+1}}
- \dth \seq{{\D_x}(-{\D_v}+v){\D_v} f^{n+1},{\D_v} f^{n+1}}.\end{aligned}$$ Using the skew-adjointness of ${\D_x}$ and the fact that $(-{\D_v}+v)^\star={\D_v}$ twice, we obtain $$\begin{aligned}
\lefteqn{\seq{{\D_v} f^{n+1},{\D_x} f^{n+1}}
+ \seq{{\D_x} f^{n+1},{\D_v} f^{n+1}}=\seq{{\D_v} f^n,{\D_x} f^{n+1}}
+ \seq{{\D_x} f^n,{\D_v} f^{n+1}} }\label{eq:estim3eulerimpl}\\
& & - \dth \norm{{\D_x}f^{n+1}}^2 - \dth \seq{{\D_v} f^{n+1},{\D_x} f^{n+1}}
+2 \dth \seq{(-{\D_v}+v){\D_v} f^{n+1},{\D_x} {\D_v} f^{n+1}}
\nonumber.\end{aligned}$$ For the last term in $\mathcal H(f^{n+1})$, we compute the $\Ldeuxmudvdx$-scalar product of ${\D_x} f^{n+1}$ computed with relation with ${\D_x} f^{n+1}$. This yields directly using the skew-adjointness of ${\D_x}$ and the fact that $(-{\D_v}+v)^\star={\D_v}$, $$\label{eq:estim4eulerimpl}
\norm{{\D_x} f^{n+1}}^2 = \seq{{\D_x} f^{n},{\D_x} f^{n+1}}
- \dth \norm{{\D_v} {\D_x} f^{n+1}}^2.$$ Summing up relations , , and with respective coefficients $C$, $D$, $E/2$ and $1$, we obtain $$\begin{aligned}
\lefteqn{\mathcal H(f^{n+1}) = \varphi(f^{n},f^{n+1})
-\dth \left( C \norm{{\D_v} f^{n+1}}^2
+ \left(D+\frac{E}{2}\right)\seq{{\D_x} f^{n+1},{\D_v} f^{n+1}}\right.}\\
& & \left.
+ D \norm{(-{\D_v} + v ){\D_v} f^{n+1}}^2
+ \frac{E}{2}\norm{{\D_x} f^{n+1}}^2 - E\seq{(-{\D_v}+v){\D_v} f^{n+1},{\D_v}{\D_x}
f^{n+1}} + \norm{{\D_v}{\D_x} f^{n+1}}^2\right).\end{aligned}$$
Using Lemma \[lem:CSplus\], we may write $$\begin{aligned}
\lefteqn{\mathcal H(f^{n+1}) \leq \frac{1}{2}\mathcal H(f^n)
+\frac{1}{2}\mathcal H(f^{n+1})
-\dth \left( C \norm{{\D_v} f^{n+1}}^2
+ \left(D+\frac{E}{2}\right)\seq{{\D_x} f^{n+1},{\D_v} f^{n+1}}\right.}\\
& & \left.
+ D \norm{(-{\D_v} + v ){\D_v} f^{n+1}}^2
+ \frac{E}{2}\norm{{\D_x} f^{n+1}}^2 - E\seq{(-{\D_v}+v){\D_v} f^{n+1},{\D_v}{\D_x}
f^{n+1}}
+ \norm{{\D_v}{\D_x} f^{n+1}}^2\right).\end{aligned}$$
This relation is to be compared with the (time)-continuous one . The very same estimates as that used in the end of the proof of Theorem \[thm:decrexp\], with $f$ replaced with $f^{n+1}$, ensure that $$\mathcal H(f^{n+1}) \leq \mathcal H(f^n) - \dth\frac{E}{4}\frac{1}{2C} \mathcal H(f^{n+1}).$$ This gives by induction $$\forall n\in\N,\qquad \mathcal H(f^{n}) \leq (1+ 2\kappa \dth)^{-n}\mathcal
H(f^0).$$ Using a Taylor development of the exponential function we get Theorem \[thm:decrexpeulerimpl\].
As in the continuous case, we can state as a corollary of the preceding Theorem the exponential decay in $\Hunmudvdx$ norm, which is a direct consequence of the equivalence of the norms $\sqrt{\hhh(\cdot)}$ and $\norm{\cdot}_\Hunmudvdx$ stated in Lemma \[lem:equiv\].
Let $C>D>E>1$ be chosen as in Theorem \[thm:decrexpeulerimpl\]. Let $\kappa$ be defined as in the same Theorem. For all $\dth >0$ there exists $\kappa_\dth >0$ (explicit) with $\lim_{\dth \rightarrow 0} \kappa_\dth = \kappa$ such that for all $f^0\in \Hunmudvdx$ with $\seq{f^0} = 0$, the sequence solution $(f^n)_{n\in\N}$ of the implicit Euler scheme satisfies for all $n\in \N$, $$\norm{f^n}_\Hunmudvdx \leq 2 \sqrt{C} \exp^{- \kappa_\dth n\dth} \norm{f^0}_\Hunmudvdx.$$
The semi-discretization in space and velocity {#sec:inhomsd}
---------------------------------------------
In this subsection we are interested in the semi-discretized equation in space and velocity. The time is a continuous variable again.
We denote by $\dxh >0$ the step of the uniform discretization of the torus $\T$ into $N$ subintervals, and denote $\jjj = \Z/ N\Z$ the finite set of indices of the discretization in $x \in \T$. In what follows, the index $i \in \Z$ will always refer to the velocity variable and the index $j \in \jjj$ to the space variable. As mentioned in the introduction, the derivation-in-space discretized operator is defined by the following centered scheme
For a sequence $G = (G_{i,j})_{i\in \Z, j\in \jjj}$ we define $\Dx G$ by $$\forall i \in \Z, j\in \jjj, \qquad (\Dx G)_{j,i} = \frac{G_{j+1,i} -
G_{j-1,i}}{2\dxh}.$$ For a sequence $G = (G_{i,j})_{i\in \Z^*, j\in \jjj}$ we define $\Dx G$ by $$\forall i \in \Z^*, j\in \jjj, \qquad (\Dx G)_{j,i} = \frac{G_{j+1,i} -
G_{j-1,i}}{2\dxh}.$$ Depending on the context, we will use the first definition or the other. Similarly, we will keep on writing $v$ the pointwise multiplication by $v_i$ from the set of sequences indexed by $\jjj\times\Z$ to itself [**and**]{} from the set of sequences indexed by $\jjj\times\Z^*$ to itself depending on the context. However, we use the notation $\vs$ from Subsection \[subsec:homsd\] (see Definition \[def:dvs\], and add $j\in\jjj$ as a parameter) of the pointwise multiplication operator by $v_i$ from the set of sequences indexed by $\jjj\times\Z^*$ to the set of sequences indexed by $\jjj\times\Z$.
Concerning the velocity discretization, we stick on the one corresponding to the homogeneous case introduced in Subsection \[subsec:homsd\]. The definition of $\Dv$, $\Dvs$, $\muh$ and $\mus$ are the same (with the space index $j$ playing the role of a parameter) as in Definitions \[def:dv\], \[def:dvs\], \[def:homd\] and \[def:homds\].
The original semi-discretized equation that we consider is $${\D_t}F+v \Dx F-\Dvs(\Dv+v)F=0, \qquad F|_{t=0} = F^0,$$ where $F^0\in \ell^1(\jjj\times\Z)$ is a non-negative function with $\norm{F^0}_{\ell^1(\jjj\times\Z)}=1$ and the unknown $F$ is such that for all $t>0$, $F(t) \in \ell^1(\jjj\times\Z)$. As in Section \[subsec:homsd\], we rather work with the rescaled function $f$ defined by $$F = \muh + \muh f,$$ where $\muh$ is the Maxwellian introduced in Lemma \[lem:muh\], now considered as a function of both $i$ and $j$. In that case for all $t>0$ we have the equivalence $$F\in \ell^1(\jjj\times\Z, \dvh \dxh) \Longleftrightarrow f \in
\ell^1(\jjj\times\Z, \muh \dvh \dxh).$$
Referring again to the homogeneous setting studied in Section \[sec:homogeneous\], we introduce the definition of a solution of the (scaled) semi-discretized equation that we will study in this subsection.
We shall say that a function $f \in \ccc^0( \R^+, \ell^1(\jjj\times\Z, \muh \dvh \dxh))$ satisfies the (scaled) semi-discrete inhomogeneous Fokker-Planck equation if $$\label{eq:FPISD}
{\D_t}f+v \Dx f+(-\Dvs+\vs)\Dv f=0,$$ in the sense of distributions
As in the homogeneous case of Section \[sec:homogeneous\], we work in Hilbertian subspaces of $\ell^1(\jjj\times\Z, \muh \dvh \dxh)$ that we introduce below.
We define the space $\Ldeuxmudvdxh$ to be the Hilbertian subspace of $\R^{\jjj\times\Z}$ made of sequences $f $ such that $$\norm{ f}_{\Ldeuxmudvdxh}^2 \defegal \sum_{j\in \jjj, i\in \Z} (f_{j,i})^2
\muh_i \dvh \dxh <\infty.$$ This defines a Hilbertian norm, and the related scalar product will be denoted by $\seq{ \cdot, \cdot}$. For $f \in \Ldeuxmudvdxh$, we define the mean of $f$ (with respect to this weighted scalar product in both velocity and space) as $$\seq{f} \defegal \sum_{j\in \jjj, i\in \Z} f_{j,i} \muh_i \dvh \dxh = \seq{f,
1}.$$ We define the space $\Ldeuxmudvdxs$ to be the Hilbertian subspace of $\R^{\jjj\times\Z^*}$ made of sequences $g $ such that $$\norm{ g}_{\Ldeuxmudvdxs }^2 \defegal \sum_{j\in \jjj, i\in \Z^*} (g_{j,i})^2
\mus_i \dvh \dxh <\infty.$$ This defines also a Hilbertian norm, and the related scalar product will be denoted by $\seqs{ \cdot, \cdot}$. Eventually we define $$\Hunmudvdxh = \set{ f \in \Ldeuxmudvdxh, \textrm{ s.t. } \Dv f \in
\Ldeuxmudvdxs, \ \Dx f \in \Ldeuxmudvdxh },$$ with the norm $$\norm{f}^2_\Hunmudvdxh = \norm{f}^2_\Ldeuxmudvdxh + \norm{\Dv f}^2_\Ldeuxmudvdxs
+ \norm{\Dx f}^2_\Ldeuxmudvdxh.$$
We define the operator $\Pd$ involved in Equation by $$\Pd= \Xzerod + (-\Dvs+\vs)\Dv,$$ with $\Xzerod = v \Dx : \Ldeuxmudvdxh \hookrightarrow \Ldeuxmudvdxh$ defined by $$( \Xzerod f)_{j,i} = (v \Dx f)_{j,i} \textrm{ when } i\neq 0, \qquad ( \Xzerod
f)_{j,0} =0.$$
This way, Equation reads ${\D_t} f+ \Pd f = 0$. We summarize the structural properties of and of the operator $\Pd$ in the following Proposition. From now on and for the rest of this subsection, we work in $\Ldeuxmudvdxh$ and denote (when no ambiguity happens) the corresponding norm $\norm{\cdot}$ without subscript. Similarly $\norms{\cdot}$ stands for the norm in $\Ldeuxmudvdxs$.
We have
1. The operator $\Pd = \Xzerod + ( - \Dvs +\vs) \Dv$ on $\Ldeuxmudvdxh$ equipped with its graph domain $D(\Pd)$ is maximal accretive in $\Ldeuxmudvdxh$.
2. The Operator $( - \Dvs +\vs) \Dv$ is formally self-adjoint and the operator $\Xzerod$ is formally skew-adjoint in $\Ldeuxmudvdxh$. Moreover, for all $g \in \Ldeuxmudvdxh$, $h \in \Ldeuxmudvdxs $ for which it makes sense $$\begin{aligned}
\label{eq:ofin}
\seq{ ( - \Dvs +\vs) h, g} = \seq{h, \Dv g}_\sharp, \\
\label{eq:ofin2}
\seq{\Pd g,g} = \seq{ ( - \Dvs +\vs) \Dv g, g} = \norms{\Dv g}^2.
\end{aligned}$$
3. For an initial data $f^0
\in
D(\Pd)$, there exists a unique solution of in $\ccc^1(\R^+,
\Ldeuxmudvdxh) \cap \ccc^0(\R^+,
D(\Pd))$, and the associated semi-group naturally defines a solution in $\ccc^0(\R^+, \Ldeuxmudvdxh)$ for all $f^0\in \Ldeuxmudvdxh$.
4. The preceding properties remain true if we consider the operator $\Pd$ in $\Hunmudvdxh$ with domain $D_\Hunmudvdxh(\Pd)$. In particular it defines a unique solution of in $\ccc^1(\R^+,
\Hunmudvdxh) \cap \ccc^0(\R^+,D_\Hunmudvdxh(\Pd)) $ if $f^0 \in
D_\Hunmudvdxh(\Pd)$ and a semi-group solution $f \in \ccc^0(\R^+,
\Hunmudvdxh)$ if $f^0 \in \Hunmudvdxh$.
5. Constant sequences are the only equilibrium states of equation and the evolution preserves the mass $\seq{f(t)}
= \seq{f^0}$ for all $t\geq 0$.
The maximal accretivity can be proven using the same scheme of proof as in the continuous case and we won’t do it here, referring to [@HelN04]. The skew-adjointedness of $\Xzerod$ is clear since we chose a centered scheme in space. The properties stated in and are direct consequences of the homogeneous analysis (see Proposition \[prop:hstruct\]). The well-posedness is then a direct consequence of Hille–Yosida’s Theorem. In particular, we can check that if $f$ is a solution in $\ccc^1(\R^+, \Ldeuxmudvdxh)$ $$\ddt \norm{f}^2 = -2 \seq{\Pd f, f} = -2 \norm{\Dv f}_\sharp^2 \leq 0,$$ so that the $\Ldeuxmudvdxh$ norm is non-increasing. For the last point, we first infer that if $f$ is a stationary solution then $$\ddt \norm{f}^2 = -2 \norm{\Dv f}_\sharp^2 =0 \Longrightarrow \Dv f = 0.$$ Introducing the macroscopic density $\rho$ defined for all $j\in\jjj$ by $\rho_j
= \dvh \sum_{i \in \Z} f_{j,i}\muh_i$, the fact that $\Dv
f=0$ yields that for all $(j,i)\in\jjj\times\Z$, $f_{j,i} =
\rho_j$. Then, the equation $\Xzerod f = 0$ implies that $\rho_j$ does not depend on $j\in\jjj$ and we summarize this with $$f_{j,i} = \rho_j = \seq{f} = \seq{f^0}, \qquad \forall (j,i) \in \jjj \times \Z,$$ so that constant sequences are the only equilibrium states of the equation. The remaining parts of the proof follow the ones of the continuous case. The proof is complete.
For later use, we introduce the operator $S=\Dv v - v \Dv$ from $\Ldeuxmudvdxh$ to $\Ldeuxmudvdxs$, where the first operator $v$ is the pointwise multiplication by $v_i$ at each $(j,i)\in\jjj\times\Z$ and the second one is the pointwise multiplication by $v_i$ at each $(j,i)\in\jjj\times\Z^*$. The operator $S$ will essentially play the role of $\adf{{\\Dv},v}$ in the continuous case. We observe that $S$ is a shift operator in the velocity variable and we have the following lemma:
\[lem:constcontS\] Operator $S : \Ldeuxmudvdxh \hookrightarrow \Ldeuxmudvdxs$ satisfies the following: for all $g\in \Ldeuxmudvdxh$, we have for all $j\in \jjj$, $$(S g)_{j,i} = g_{j,i+1} \textrm{ for } i \leq -1, \qquad (S g)_{j,i} = g_{j,
i-1} \textrm{ for } i \geq 1,$$ and $$\norm{g}^2\leq \norms{S g}^2\leq 2\norm{g}^2.$$
Let $g\in\Ldeuxmudvdxh$. We first compute $\Dv v g$ (where the multiplication operator $v$ is supposed to be defined from $\Ldeuxmudvdxh $ to $\Ldeuxmudvdxh$). We omit for convenience the index $j\in\jjj$ in the computations. We have $$(\Dv(v g))_{i} = \dfrac{v_{i+1}g_{i+1}-v_{i}g_{i}}{\dvh} \textrm{ for } i \leq
-1, \qquad (\Dv(v g))_{i} = \dfrac{v_i g_{i}-v_{i-1}g_{i-1}}{\dvh} \textrm{ for }
i \geq 1.$$ Similarly we compute $v \Dv g$ (where the multiplication operator $v$ is now supposed to be defined from $ \Ldeuxmudvdxs $ to $ \Ldeuxmudvdxs$): $$(v \Dv g)_{i} = v_i\dfrac{g_{i+1}-g_{i}}{\dvh} \textrm{ for } i \leq -1, \qquad
(v \Dv g)_{i} = v_i\dfrac{g_{i}-g_{i-1}}{\dvh} \textrm{ for } i \geq 1.$$ Comparing the two preceding results gives the expression of $S g$. We now compute the norms using the definition of $\mus$ and get $$\begin{aligned}
(\dvh\dxh)^{-1} \norms{S g}^2&
=\sum_{j\in\jjj,i\leq -1}g_{j,i+1}^2\mus_i + \sum_{j\in\jjj,i\geq 1}g_{j,i-1}^2\mus_i\\
&=\sum_{j\in\jjj,i\leq 0}g_{j,i}^2\muh_i + \sum_{j\in\jjj,i\geq 0}g_{j,i}^2\muh_i\\
&=(\dvh\dxh)^{-1} \norm{g}^2+\muh_0\sum_{j\in\jjj}g_{j,0}^2.
\end{aligned}$$ This last term is one of the terms (the centered one) in the definition of the norm in $\Ldeuxmudvdxh$, and we therefore get $$\norm{g}^2\leq \norms{S g}^2 \leq 2 \norm{g}^2.$$ The proof is complete.
We define the operator $S^\sharp:\Ldeuxmudvdxs\rightarrow\Ldeuxmudvdxh$ to be the adjoint of the operator $S$, [*i.e.*]{} satisfying the relation $$\forall (g,h)\in \Ldeuxmudvdxh\times\Ldeuxmudvdxs,\qquad
\seq{S g,h}_{\sharp}=\seq{g,S^\sharp h}.$$
This is again a shift operator in the velocity variable, but it is not injective, and we have the following lemma
\[lem:constcontSsharp\] Operator $\Ss : \Ldeuxmudvdxs \hookrightarrow \Ldeuxmudvdxh$ satisfies the following: For $h\in\Ldeuxmudvdxs$, we have for all $j\in\jjj$, $$(\Ss h)_{j,i} = h_{j,i-1} \textrm{ for } i\leq -1, \qquad (\Ss h)_{j,0} =
h_{j,-1}+h_{j,1},\qquad (\Ss h)_{j,i} = h_{j,i+1} \textrm{ for } i\geq 1.$$ Moreover, for all $h \in\Ldeuxmudvdxs$, we have $$\norm{S^\sharp h}^2 \leq 4 \norm{h}_\sharp^2.$$
The proof is straightforward, using similar tools as in the one of Lemma \[lem:constcontS\].
In order to apply a procedure similar to the one we used in the continuous inhomogeneous case in Section \[sec:inhomc\], we introduce the following modified entropy defined for $g\in \Hunmudvdxh$ by $$\hhhd(g) \defegal C\norm{g}^2+D\norms{\Dv g}^2+E\seqs{\Dv g,S\Dx g}+\norm{\Dx g}^2,$$ for well chosen $C>D>E>1$ to be defined later. We will show in a moment that for these parameters, $t\mapsto \hhhd(f(t))$ is exponentially decreasing in time when $f$ is the semi-group solution of the scaled inhomogeneous Fokker-Planck equation with initial datum $f^0\in\Hunmudvdxh$ of zero mean. Before doing this, we compare this entropy $\hhhd$ with the usual $\Hunmudvdxh$ norm.
\[lem:equivd\] If $2E^2<D$ then for all $g\in \Hunmudvdxh$, $$\dfrac{1}{2}\norm{g}_{\Hunmudvdxh}^2\leq\hhhd(g)\leq 2C\norm{g}_{\Hunmudvdxh}^2.$$
We stick to the proof of Lemma \[lem:equiv\]. Let $g\in \Hunmudvdxh$. We use the Cauchy–Schwarz–Young inequality and observe that $$2\abs{E\seq{\Dv g,S\Dx g}_\sharp}\leq 2E^2\norm{\Dv g}_\sharp^2
+\frac{1}{2}\norms{S\Dx g}^2 \leq 2E^2\norm{\Dv g}_\sharp^2+\norm{\Dx g}^2,$$ where we used Lemma \[lem:constcontS\] for the last inequality. This implies $$\begin{gathered}
\underbrace{C}_{1/2\leq}\norm{g}^2 +\underbrace{(D-E^2)}_{1/2\leq
E^2\leq}\norm{\Dv g}_\sharp^2
+\dfrac{1}{2}\norm{\Dx g}^2 \\
\leq \hhhd(g)\leq C\norm{g}^2+\underbrace{(D+E^2)}_{\leq D+D/2\leq 3C/2\leq
2C}\norm{\Dv g}_\sharp^2+\underbrace{3C/2}_{\leq 2C} \norm{\Dx g}^2,
\end{gathered}$$ which implies the result since $2E^2<D$.
As in the continuous case, we need a full Poincaré inequality in space and velocity. We first note that, for functions $\rho$ of the space variable $j\in\jjj$ only, provided that $N=\#\jjj$ is odd (which is assumed from now on in this paper), the Poincaré inequality $$\label{eq:Poincarediscretxseulement}
\norm{\rho-\seq{\rho}}^2_\Ldeuxdxh \leq \norm{\Dx \rho}^2_\Ldeuxdxh,$$ is standard (and easy to reproduce following the proof of Lemma \[lem:Poincarecontinu\]), where $$\label{eq:defL2dxh}
\norm{\rho}^2_\Ldeuxdxh = \sum_{j\in \jjj} \rho_j^2 \dxh,$$ is the standard norm on the discretized torus, $$\seq{\rho} = \sum_{j\in \jjj} \rho_j \dxh,$$ is the mean of $\rho$ and $\Dx$ is the centered finite difference derivation operator defined above. In particular, for $g\in \Ldeuxmudvdxh$, one can apply to the macroscopic density $\rho$ of $g$ defined of $j\in\jjj$ by $\rho_j = \dvh \sum_{i} g_{j,i}\muh_i$. The fully discrete Poincaré inequality of Lemma \[lem:fullPoincarediscret\] is then a consequence of Proposition \[prop:poindiscrete\] in velocity only (following the proof of the continuous case stated in Lemma \[lem:Poincarecontinu\]).
\[lem:fullPoincarediscret\] For all $g \in \Hunmudvdxh$, we have $$\norm{g-\seq{g}}^2_\Ldeuxmudvdxh \leq \norm{\Dv g}_\Ldeuxmudvdxs^2 +
\norm{\Dx g}_\Ldeuxmudvdxh^2.$$
Replacing if necessary $g$ by $g-\seq{g}$, it is sufficient to prove the result for $\seq{g}=0$. We observe that Parseval’s formula and discrete Fubini’s theorem imply $$\norm{\rho}^2_\Ldeuxdxh \leq \norm{g}^2_\Ldeuxmudvdxh \qquad \text{and} \qquad
\norm{\Dx\rho}^2_\Ldeuxdxh \leq \norm{\Dx g}^2_\Ldeuxmudvdxh,$$ since $(j,i) \rightarrow \rho_j$ (resp. $(j,i) \rightarrow (\Dx\rho)_j$) is the orthogonal projection of $g$ (resp. $\Dx g$) onto the closed space $$\set{ (j,i) \longmapsto \phi_j \ | \ \phi \in \Ldeuxdxh},$$ and $\norm{\phi \otimes 1 }_\Ldeuxmudvdxh = \norm{\phi}_\Ldeuxdxh$ for all $\phi \in \Ldeuxdxh$ since we are in probability spaces. We note here the natural injection $\Ldeuxdxh \hookrightarrow \Ldeuxmudvdxh$ of norm $1$. Using the discrete Fubini theorem again, we also directly get from Proposition \[prop:poindiscrete\] that $$\norm{g- \rho\otimes 1}_\Ldeuxmudvdxh^2 \leq \norm{\Dv g}_\Ldeuxmudvdxs^2.$$ Using Parseval’s formula again yields $$\begin{split}
\norm{g}_\Ldeuxmudvdxh^2 & = \norm{g- \rho\otimes 1}_\Ldeuxmudvdxh^2 + \norm{ \rho\otimes 1}_\Ldeuxmudvdxh^2 \\
& = \norm{g- \rho\otimes 1}_\Ldeuxmudvdxh^2 + \norm{\rho}_\Ldeuxdxh^2 \\
& \leq \norm{\Dv g}_\Ldeuxmudvdxs^2 + \norm{\Dx \rho}_\Ldeuxdxh^2 \\
& \leq \norm{\Dv g}_\Ldeuxmudvdxs^2 + \norm{\Dx g}_\Ldeuxmudvdxh^2.
\end{split}$$ The proof is complete.
We can now state the main Theorem of this subsection concerning the exponential return to equilibrium of solutions of Equation .
\[thm:decrexpd\] There exists $C>D>E>1$, $\dvh_0>0$ and $\kappa_d>0$ explicit such that the following holds: For all $ f^0 \in \Hunmudvdxh$ such that $\seq{f^0} = 0$, the solution $f$ (in the semi-group sense) in $\ccc^0(\R^+, \Hunmudvdxh)$ of Equation with initial data $f^0$ satisfies $$\hhhd(f(t))\leq \hhhd(f^0)\exp^{-2\kappa_d t},$$ for all $t\geq 0$, $\dvh\in(0,\dvh_0)$ and $\dxh>0$.
(of Theorem \[thm:decrexpd\] – 1/4) We divide the proof in four parts, and we insert technical lemmas in between those parts, so that the reader may understand why new discrete operators are introduced and studied, as the computations go. Let us consider $f$ the solution in $\ccc^1(\R^+, D_\Hunmudvdxh(\Pd))$ with initial data $f^0 \in D_\Hunmudvdxh(\Pd)$. This choice allows all the computations done below, and Theorem \[thm:decrexpd\] will be a direct consequence of the density of $D_\Hunmudvdxh(\Pd)$ in $\Hunmudvdxh$ and the boundedness of the associated semi-group.
As in the continuous case, we shall differentiate w.r.t. time the four terms appearing in the definition of $\hhhd$. The derivatives of the 1st, 2nd and 4th term are fairly easy to estimate, as we will see below. The more intricate estimate of the derivative of the 3rd term will require Lemmas \[lem:triple\], \[lem:sb\] and \[lem:estimdelta\].
For the derivative of the first term in $\hhhd$, we compute $$\begin{aligned}
\ddt\norm{f}^2&=2\seq{f,-v \Dx f-(-\Dvs+\vs)\Dv f}\\
&=-2\seq{f,-v \Dx f}-2\seq{f,-(-\Dvs+\vs)\Dv f}.
\end{aligned}$$ Using the fact that $v \Dx$ is skew-adjoint in $\Ldeuxmudvdxh$ and the identity derived from , we obtain $$\label{eq:premier} \ddt \norm{f}^2=-2\norms{\Dv f}^2.$$ The second term of the time derivative can be computed as follows: $$\begin{aligned}
\ddt\norms{\Dv f}^2&=2\seqs{\Dv\left(-v \Dx-(-\Dvs+\vs)\Dv\right) f,\Dv f} \nonumber \\
&=-2\seqs{\Dv(v \Dx)f,\Dv f}-2\seqs{\Dv(-\Dvs+\vs)\Dv f,\Dv f} \nonumber \\
&=-2\seqs{\underbrace{\adf{\Dv,v \Dx}}_{=\adf{\Dv,v}\Dx=S\Dx}f,\Dv f}
-2\underbrace{\seqs{v \Dx \Dv f,\Dv f}}_{=0}-2\underbrace{\norm{(-\Dvs+\vs)\Dv f}^2}_{\mbox{using
} \eqref{eq:ofin2}} \nonumber \\
&=-2\seqs{S\Dx f,\Dv f}-2\norm{(-\Dvs+\vs)\Dv f}^2.
\label{eq:second}
\end{aligned}$$ The time derivative of the last term in $\hhh(f)$ is $$\label{eq:quatrieme}
\ddt \norm{\Dx f}^2=-2\norms{\Dv \Dx f}^2.$$ since $\Dx$ commutes with the full operator.
All the difficulties are concentrated in the third term. We are going to need a few lemmas in order to be able to write the time-derivative of that third term in . After that, we will get back to the proof of the Theorem by expressing the time-derivative of $t\mapsto \hhhd(f(t))$ in using an entropy-dissipation term. We will need a last lemma (Lemma \[lem:dissdiss\]) to estimate the entropy-dissipation term before getting to the end of the proof of Theorem \[thm:decrexpd\].
In order to prepare the computations, we state and prove two lemmas concerning discrete commutators.
\[lem:triple\] We have $$\Dv (-\Dv^\sharp + \vs) S - S(-\Dv^\sharp+\vs) \Dv = S+ \sigma,$$ where $\sigma$ is the singular operator from $\Ldeuxmudvdxh$ to $\Ldeuxmudvdxs$ defined for $g\in \Ldeuxmudvdxh$ by $$\qquad (\sigma g)_{j, -1} = \frac{g_{j,1}-g_{j,0}}{\dvh^2}, \qquad (\sigma
g)_{j, 1} = -\frac{g_{j,0}-g_{j,-1}}{\dvh^2} \qquad \textrm{ and } (\sigma
g)_{j,i}=0 \ \textrm{ for } \ |i| \geq 2,$$ for all $ j \in \jjj$.
We postpone the proof of this computational lemma to the end of the paper, where Table \[tab:comm\] summarizes all the computations of commutators.
The second lemma of commutation type is the following
\[lem:sb\] We define the operator $S^b : \Ldeuxmudvdxh \longrightarrow \Ldeuxmudvdxs$ by $$\begin{aligned}
& (S^b g)_{j,i} = g_{j,i+1} \textrm{ for } i\leq -1 \qquad \qquad
(S^b g)_{j,i} = -g_{j,i-1} \textrm{ for } i\geq 1,\end{aligned}$$ for all $g \in
\Ldeuxmudvdxh$ and $ j \in \jjj$. Then we have $$S\Dx v \Dx g -v \Dx S\Dx g= \dvh S^b \Dx^2 g.$$ Moreover $$\norm{g}^2 \leq \norm{S^\flat g}_\sharp^2 \leq 2 \norm{g}^2.$$
We postpone the proof to the table at the end of the paper (see Table \[tab:comm\]).
(of Theorem \[thm:decrexpd\] – 2/4) We go on with the proof of Theorem \[thm:decrexpd\], and we recall that we consider a solution $f \in \ccc^1(\R^+, D_\Hunmudvdxh(P))$. We want to estimate the derivative of the third term defining $\hhhd(f(t))$. Let us compute
$$\begin{aligned}
& \ddt\seq{S\Dx f,\Dv f}_\sharp \\
& = -\seqs{S\Dx (\Xzerod f+(-\Dvs +\vs)\Dv f),\Dv f}-
\seqs{S\Dx f,\Dv (\Xzerod f+(-\Dvs +\vs)\Dv f)}\\
& = -\seqs{S\Dx \Xzerod f,\Dv f} -\seqs{S\Dx f,\Dv \Xzerod f} \qquad (I) \\
&\qquad - \seqs{S\Dx (-\Dvs +\vs)\Dv f,\Dv f)} - \seqs{S\Dx f,\Dv (-\Dvs
+\vs)\Dv f)}.
\qquad (II)
\end{aligned}$$
We first deal with the sum (I) in the previous equality. Using Lemma \[lem:sb\] and $[\Dv, \Xzerod ] = S\Dx$ we get $$\begin{aligned}
(I) =& -\seqs{\Xzerod S\Dx f,\Dv f} -\seqs{S\Dx f,\Xzerod \Dv f} \\
& - \dvh \seqs{S^b \Dx^2 f, \Dv f} - \norms{S\Dx f}^2 \\
=& - \dvh \seqs{S^b \Dx^2 f, \Dv f} - \norms{S\Dx f}^2,
\end{aligned}$$ where we used that the first two terms compensate by skew-adjunction of $\Xzerod $. Using that $\Dx$ is skew-adjoint and commutes with $S^b$ we get $$(I) = \dvh \seqs{S^b \Dx f, \Dx \Dv f} - \norms{S\Dx f}^2.$$
Now we deal with the term (II). We first use that the adjoint of $\Dv$ is $(-\Dv^\sharp+\vs)$ two times and we get $$\begin{aligned}
(II) &= -\seqs{S\Dx (-\Dvs+\vs)\Dv f,\Dv f} -\seqs{\Dv(-\Dvs+\vs) S\Dx f,\Dv f}.\end{aligned}$$ Now from Lemma \[lem:triple\] applied to the second term we get $$\begin{aligned}
(II) &= -2\seqs{S\Dx (-\Dvs+\vs)\Dv f,\Dv f} -\seqs{ S\Dx f,\Dv f}- \seqs{\sigma \Dx f, \Dv f}.\end{aligned}$$ We used also that $\Dx$ commutes with all operators. This yields $$\begin{aligned}
(II) &= 2 \seq{(-\Dvs+\vs)\Dv f,S^\sharp \Dx \Dv f} -\seqs{ S\Dx f,\Dv f}
- \seqs{\sigma \Dx f, \Dv f}.\end{aligned}$$
Using the relations above for $(I)$ and $(II)$, we get eventually for the derivative of the third term: $$\begin{aligned}
\label{eq:troisieme}
& \ddt\seq{S\Dx f,\Dv f}_\sharp \\
&= - \norms{S\Dx f}^2+ \dvh \seqs{S^b \Dx f, \Dv \Dx f} \nonumber\\
& \qquad \qquad + 2 \seq{(-\Dvs+\vs)\Dv f,S^\sharp \Dx \Dv f} -\seqs{ S\Dx
f,\Dv f}
- \seqs{\sigma \Dx f, \Dv f}.\nonumber\end{aligned}$$
The first term in this sum has a sign. All the other terms except the last one are easy to deal with, as in the continuous case. The last one involving $\sigma$ is more involved since it seems to be singular. Anyway, it can also be controlled as shown in this last lemma.
\[lem:estimdelta\] For all $\varepsilon>0$ and $g \in \Ldeuxmudvdxh$ we have $$\label{eq:estimsigma}
\abs{ \seqs{\sigma \Dx g, \Dv g} } \leq \frac{1}{\varepsilon}
\norm{(-\Dvs+\vs)\Dv g}^2 + \varepsilon
\norms{ \Dv \Dx g}^2.$$
For all $j\in\jjj$, the contribution to the scalar product in the right-hand side of reduces to two terms according to the expression of $\sigma$ (see Lemma \[lem:triple\]). We denote by $\seq{ ., .}_\Ldeuxdxh $ the scalar product in the variable $j$ only, associated to the norm defined in . In the computations below, we omit for convenience the subscript $j$ corresponding to the space discretization. We have $$\begin{aligned}
\seqs{\sigma \Dx g, \Dv g} =
& \seq{\frac{\Dx g_1-\Dx g_0}{\dvh^2}, \frac{g_0-g_{-1}}{\dvh}}_\Ldeuxdxh
\mu_0 \dvh
- \seq{\frac{\Dx g_0-\Dx g_{-1}}{\dvh^2}, \frac{g_1-g_{0}}{\dvh}}_\Ldeuxdxh \mu_0 \dvh.\end{aligned}$$ Using that $\Dx $ is skew-adjoint (or using an Abel transform in $j$), we get $$\begin{aligned}
\seqs{\sigma \Dx g, \Dv g} =
& 2 \seq{\frac{\Dx g_1-\Dx g_0}{\dvh^2}, \frac{g_0-g_{-1}}{\dvh}}_\Ldeuxdxh \mu_0 \dvh.\end{aligned}$$ For convenience, we set $\GPLUS= \frac{g_1- g_0}{\dvh}=(\Dv g)_{1}$ and $\GMOINS = \frac{g_0-g_{-1}}{\dvh} = (\Dv g)_{-1}$. We have then $$\begin{aligned}
\seqs{\sigma \Dx g, \Dv g} =
& 2 \seq{ \frac{\Dx \GPLUS}{\dvh}, \GMOINS}_\Ldeuxdxh \mu_0 \dvh \\
= & 2 \seq{ \Dx \frac{ \GPLUS-\GMOINS}{\dvh}, \GMOINS}_\Ldeuxdxh \mu_0 \dvh
+ \frac{2}{\dvh} \seq{ \Dx \GMOINS, \GMOINS}_\Ldeuxdxh \mu_0 \dvh.\end{aligned}$$ The last term is zero and we therefore get, using a last integration by part for the first term $$\begin{aligned}
\seqs{\sigma \Dx g, \Dv g}
= & - 2 \seq{ \frac{ \GPLUS-\GMOINS}{\dvh}, \Dx \GMOINS}_\Ldeuxdxh \mu_0 \dvh.\end{aligned}$$ We observe that $$\frac{ \GPLUS-\GMOINS}{\dvh} = \frac{ \frac{g_1-g_0}{\dvh}
- \frac{g_0-g_{-1}}{\dvh}}{\dvh} = (\Dvs \Dv g)_0 = -((-\Dvs +\vs) \Dv g)_0.$$ Hence, for all $\varepsilon>0$ $$\begin{aligned}
\abs{\seqs{\sigma \Dx g, \Dv g}} \leq
& 2 \norm{ ((-\Dvs +\vs) \Dv g)_0}_\Ldeuxdxh \norm{({\Dx}\Dv g)_{-1}}_\Ldeuxdxh \mu_0 \dvh \\
\leq
& \frac{1}{\varepsilon}
\norm{ ((-\Dvs +\vs) \Dv g)_0}^2_\Ldeuxdxh \mu_0 \dvh + \varepsilon
\norm{({\Dx}\Dv g)_{-1}}^2_\Ldeuxdxh \mu_0 \dvh \\
\leq & \frac{1}{\varepsilon}
\norm{(-\Dvs+\vs)\Dv g}^2 +
\varepsilon
\norms{ \Dv \Dx g}^2.\end{aligned}$$ The proof of the Lemma is complete.
(of Theorem \[thm:decrexpd\] – 3/4) Now we come back to the proof of Theorem \[thm:decrexpd\]. We consider all the four relations , , and , and we multiply the first one by $C$, the second one by $D$, the third more involved one by $E$, and we get by addition $$\begin{aligned}
\ddt \hhhd(f(t)) & = & -2 \left( C \norm{\Dv f}_\sharp^2
+ D \seq{S\Dx f,\Dv f}_\sharp
+ D \norm{(-\Dvs+\vs)\Dv f}^2 \right.\nonumber\\
&& \left. +\frac{E}{2} \norm{S\Dx f}_\sharp^2
- \frac{E}{2} \dvh \seq{S^b \Dx f,\Dx \Dv f}_\sharp \right. \nonumber\\
&& \left. -E \seq{(-\Dvs+\vs)\Dv f,S^\sharp \Dx \Dv f}\right.\nonumber\\
&& \left.
+\frac{E}{2} \seq{S\Dx f,\Dv f}_\sharp
+\frac{E}{2}\seq{\sigma \Dx f,\Dv f}_\sharp
+\norm{\Dx \Dv f}_\sharp^2 \right) \nonumber \\
&& \qquad \defegal -2 \dddd(f). \label{eq:defdcde}\end{aligned}$$
The term $2\dddd(f) $ is the discrete entropy-dissipation term and we prove that it can be bounded below (so that, in particular, it has a sign) for well chosen parameters $C$, $D$, and $E$. This is the goal of the following lemma.
\[lem:dissdiss\] There exists constants $C>D>E>1$ and $\dvh_0>0$ such that for all $g \in \Hunmudvdxh$, $\dvh\leq \dvh_0$ and $\dxh>0$, $$\label{eq:entropydissipationineq}
\dddd(g) \geq \kappa_d \hhhd(g),$$ with $\kappa_d = 1/(4C)$. Moreover, it is sufficient for the constants above to satisfy relations - to come to ensure that the result above hold.
Grouping terms and estimating the big parentheses in , we obtain first for all $\theta>0$, $$\begin{aligned}
& \dddd(g) \geq &
C \norm{\Dv g}_\sharp^2
+ \left(D+\frac{E}{2}\right) \seq{S\Dx g,\Dv g}_\sharp
+ D \norm{(-\Dvs+v)\Dv g}^2 \\
&& +\frac{E}{2} \norm{S\Dx g}_\sharp^2
- \frac{1}{2}\norm{S^b \Dx g}_\sharp^2
- \frac{1}{2}\frac{E^2\dvh^2}{4} \norm{\Dx \Dv g}_\sharp^2 \\
&& - \frac{1}{2} \frac{1}{\theta}E^2\norm{(-\Dvs+v)\Dv g}^2
- \frac{1}{2} \theta \norm{S^\sharp \Dx \Dv g}^2\\
&&
-\frac{E}{2}\left|\seq{\sigma \Dx g,\Dv g}_\sharp\right|
+\norm{\Dx \Dv g}_\sharp^2.\end{aligned}$$ Using the continuity constants of $S$, $S^\sharp$ and $S^\flat$ (see Lemmas \[lem:constcontS\], \[lem:constcontSsharp\] and \[lem:sb\]), as well as Lemma \[lem:estimdelta\], we obtain for all $\varepsilon>0$,
$$\begin{aligned}
& \dddd(g) \geq &
C \norm{\Dv g}_\sharp^2
- \frac{1}{2}\norm{S\Dx g}_\sharp^2
- \frac{\left(D+\frac{E}{2}\right)^2}{2} \norm{\Dv g}_\sharp^2 \\
&& + D \norm{(-\Dvs+v)\Dv g}^2 \\
&& +\frac{E}{2} \norm{\Dx g}^2
- \norm{\Dx g}^2
- \frac{1}{2}\frac{E^2\dvh^2}{4} \norm{\Dx \Dv g}_\sharp^2 \\
&& - \frac{1}{2} \frac{1}{\theta} E^2 \norm{(-\Dvs+v)\Dv g}^2
- 2 \theta \norm{\Dx \Dv g}_\sharp^2\\
&&
-\varepsilon\frac{E}{2}\norm{\Dx \Dv g}_\sharp^2
- \frac{1}{\varepsilon}\frac{E}{2}
\norm{(-\Dvs+\vs)\Dv g}^2
+\norm{\Dx \Dv g}_\sharp^2.\end{aligned}$$
Using again the continuity constant of $S$ from Lemma \[lem:constcontS\] and grouping terms, we find $$\begin{aligned}
& \dddd(g) \geq &
\left(C- \frac{\left(D+\frac{E}{2}\right)^2}{2}\right) \norm{\Dv g}_\sharp^2
+\left(\frac{E}{2}-2\right) \norm{\Dx g}^2
\nonumber\\
& &
+ \left( D - \frac{1}{2} \frac{1}{\theta} E^2 - \frac{1}{\varepsilon}\frac{E}{2} \right)
\norm{(-\Dvs+v)\Dv g}^2
\nonumber\\
& &
+\left( 1
-\varepsilon\frac{E}{2}
- \frac{1}{2}\frac{E^2\dvh^2}{4}
- 2 \theta
\right)
\norm{\Dx \Dv g}_\sharp^2.\end{aligned}$$
Let us now discuss the existence of a set of constants that achieve the functional inequality . First, we fix $$\label{eq:choixE}
E\geq 6.$$ Then, we can choose $\theta$, $\eps$ and $\dvh_0>0$ such that $$\theta=1/8, \qquad \varepsilon=1/(4E), \qquad \dvh_0^2E^2/8\leq 1/8,$$ so that we obtain that for all $\dvh\leq \dvh_0$ $$1
-\varepsilon\frac{E}{2}
- \frac{1}{2}\frac{E^2\dvh^2}{4}
- 2 \theta\geq 1/2.$$ Then, we can choose $D$ big enough to ensure that $$\label{eq:choixD}
D - \frac{1}{2} \frac{1}{\theta} E^2 - \frac{1}{\varepsilon}\frac{E}{2} \geq 1 \qquad {\rm and} \qquad D>2E^2.$$ Eventually, we choose $C$ big enough to ensure that $$\label{eq:choixC}
C- \frac{\left(D+\frac{E}{2}\right)^2}{2}\geq 1.$$
When all these constraints are fulfilled, we get that $$\label{eq:pourplustard}
\dddd(g) \geq \norm{\Dv g}_\sharp^2 + \norm{\Dx g}^2+\norm{(-\Dvs+v)\Dv g}^2
+\frac{1}{2} \norm{ \Dv \Dx g}_\sharp^2.$$ Using now the Poincaré estimate from Lemma \[lem:fullPoincarediscret\] applied to half of the right-hand-side of the last inequality, we get $$\dddd(g) \geq \frac{1}{2} \norm{\Dv g}_\sharp^2 + \frac{1}{2}\norm{\Dx g}^2
+ \frac{1}{2}\norm{g}^2.$$ Since $D>2E^2$ by , Lemma \[lem:equivd\] about the equivalence of the $\Hunmudvdxh$ and the $\hhhd$ norms ensures that $$\dddd(g) \geq \frac{1}{4C}\hhhd(g).$$
(of Theorem \[thm:decrexpd\] – 4/4) Provided $C>D>E>1$ are chosen as above, we have along the solution $f$ of the discrete scaled Fokker-Planck equation with zero mean, with the estimates above and in particular $$\ddt \hhhd(f(t)) \leq -2 \dddd(f) \leq -2 \kappa_d \hhhd(f(t)).$$ Gronwall’s lemma gives directly the result of Theorem \[thm:decrexpd\]. This completes the proof.
The full discretization and proof of Theorem \[thm:eulerimplicite\] {#subsec:eqinhomototalementdiscretisee}
--------------------------------------------------------------------
In this subsection we prove Theorem \[thm:eulerimplicite\], which will be a direct consequence of Theorem \[thm:decrexpeulerimpldiscr\] below. We directly work on the scaled sequence $f$ defined by $F = \muh + \muh f$ where $F$ satisfies .
We shall say that a sequence $f = (f^n)_{n \in \N} \in (\ell^1(\jjj\times \Z, \muh\dvh\dxh))^\N$ satisfies the scaled fully discrete implicit inhomogeneous Fokker-Planck equation if, for some $\dth>0$, $$\label{eq:eulerimpldiscr}
\forall n\in\N,\qquad f^{n+1} = f^n
- \dth (v \Dx f^{n+1}+(-\Dvs+\vs) \Dv f^{n+1}).$$
As in all the previous cases, we can check that constant sequences are the only equilibrium states of this equation, and that the mass conservation property is satisfied: $$\forall n\in\N,\qquad \seq{f^n} = \seq{f^0},$$ where we use all the notations and definitions of Subsection \[subsec:discretentcontinuenxv\], and in particular work in $\Ldeuxmudvdxh$ or $\Hunmudvdxh$.
In Subsection \[subsec:discretentcontinuenxv\], we proved a time-discrete result (Theorem \[thm:decrexpeulerimpl\]) for the solutions in the continuous (in space and velocity) setting , in accordance with the behaviour of the exact solutions (Theorem \[thm:decrexp\]). The goal of this section is to prove a similar time-discrete result for the solutions of the implicit Euler scheme , in accordance with the result (Theorem \[thm:decrexpd\]) for the exact solutions of in the discrete (in velocity and space) setting.
As in the semi-discrete case, we shall work with the modified entropy defined by $$\hhhd(g) = C \norm{g}^2 + D \norms{\Dv g}^2+ E \seqs{\Dv g, S\Dx g} + \norm{\Dx g}^2,$$ for well chosen $C > D > E >1$ to be defined later. Under the condition $2 E^2 <D$, Lemma \[lem:equivd\] holds. We denote by $\phid$ the polar form associated to $\hhhd$ defined for $g,\gtilde\in \Hunmudvdxh$ by $$\phid(g,\gtilde)=C\seq{g,\gtilde}+D\seq{\Dv g,\Dv \gtilde}_\sharp
+ \frac{E}{2}\left(\seq{S\Dx g,\Dv\gtilde}_\sharp
+\seq{\Dv g,S \Dx \gtilde}_\sharp\right)
+ \seq{\Dx g,\Dx \gtilde},$$ and recall that the Cauchy–Schwarz–Young inequality holds and reads $$\label{eq:CSY}
|\phid(g,\gtilde)| \leq \sqrt{\hhhd(g)}\sqrt{\hhhd(\gtilde)}
\leq \frac{1}{2} \hhhd(g) + \frac{1}{2} \hhhd(\gtilde),$$ just as in the continuous (in space and velocity) case (see Lemma \[lem:CSplus\]).
The main result of this section (leading directly to Theorem \[thm:eulerimplicite\] in the introduction) is the following theorem.
\[thm:decrexpeulerimpldiscr\] Assume $C>D>E>1$, $\dvh_0>0$ and $\kappa_d$ are chosen as in Theorem \[thm:decrexpd\]. Then for all $ f^0 \in \Hunmudvdxh$, for all $\dth>0$, $\dvh\in(0,\dvh_0)$, and $\dxh>0$, the problem with initial datum $f^0$ is well-posed in $\Hunmudvdxh$. Suppose in addition that $\seq{f^0} = 0$ and let $(f^n)_{n\in\N}$ denote the sequence solution of Equation with initial datum $f^0$, we have in this case for all $n\geq 0$, $$\hhhd(f^n)\leq (1+ 2\kappa_d \dth)^{-n} \hhhd(f^0).$$
Doing just as we did at the end of the proof of Theorem \[thm:decrexpeulerimpl\] for continuous space and velocity variables, the result above implies first, exponential convergence to $0$ with respect to the discrete time of $(\hhhd(f^n)_{n\in\N})$ and second, exponential convergence of $(f^n)_{n\in\N}$ to its mean in $\Hunmudvdxh$ for all $f^0\in\Hunmudvdxh$. This allows to prove Corollary from Theorem \[thm:eulerimplicite\].
Let $f^0 \in \Hunmudvdxh$ and consider in this space the unbounded operator $\Pd = v \Dx +(-\Dvs+v) \Dv $ with domain $D_{\Hunmudvdxh} (\Pd)$. It was mentioned in the preceding section that this operator is maximal accretive. Let us fix $\dth>0$. Equation reads for all $n\in\N$, $$f^{n+1} = (Id + \dth \Pd)^{-1} f^n.$$ This relation gives sense to the a unique sequence solution $f= (f^n)_{n\in \N} \in \Hunmudvdxh$ by induction since $(Id + \dth \Pd)^{-1} : \Hunmudvdxh \longrightarrow
D_{\Hunmudvdxh} (P) \hookrightarrow \Hunmudvdxh$.
Assume now that $\seq{f^0}=0$. By induction, we directly get that for all $n\in\N$, $\seq{f^n}=0$. We fix now $n\in\N$ and compute the four terms appearing in the definition of $\hhhd(f^{n+1})$ before estimating their sum. We start by computing the $\Ldeuxmudvdxh$-scalar product of $f^{n+1}$ with itself using relation on the left to obtain $$\begin{aligned}
\norm{f^{n+1}}^2 & = & \seq{f^n,f^{n+1}} - \dth \underbrace{\seq{v \Dx f^{n+1},f^{n+1}}}_{=0}
- \dth \seq{(-\Dvs+\vs) \Dv f^{n+1},f^{n+1}}\nonumber\\
\label{eq:estim1discr}
& = & \seq{f^n,f^{n+1}} - \dth \norm{\Dv f^{n+1}}_\sharp^2,\end{aligned}$$ using .
Next, we compute $\Ldeuxmudvdxs$-scalar product of $\Dv f^{n+1}$ with itself using relation on the left to obtain $$\begin{aligned}
\lefteqn{\norm{\Dv f^{n+1}}^2_\sharp = }\\
& & \seq{\Dv f^n,\Dv f^{n+1}}_\sharp - \dth \seq{\Dv v \Dx f^{n+1},\Dv f^{n+1}}_\sharp
- \dth \seq{\Dv(-\Dvs+\vs) \Dv f^{n+1},\Dv f^{n+1}}_\sharp.\\\end{aligned}$$ The first term in $\dth$ can be rewritten as $$\begin{aligned}
- \dth \seq{\Dv v \Dx f^{n+1},\Dv f^{n+1}}_\sharp & = &
-\dth \seq{\adf{\Dv,v \Dx}f^{n+1},\Dv f^{n+1}}_\sharp
-\dth \underbrace{\seq{v \Dx \Dv f^{n+1},\Dv f^{n+1}}_\sharp}_{=0}\\
& = & -\dth \seq{S\Dx f^{n+1},\Dv f^{n+1}}_\sharp,\end{aligned}$$ thanks to the definition of $S$. The second term in $\dth$ becomes, using \[eq:ofin\], $$- \dth \seq{\Dv(-\Dvs+\vs) \Dv f^{n+1},\Dv f^{n+1}}_\sharp
= - \dth \norm{(-\Dvs+\vs)\Dv f^{n+1}}^2.$$ We infer, for the second term in $\hhhd (f^{n+1})$, $$\label{eq:estim2discr}
\norm{\Dv f^{n+1}}^2_\sharp =
\seq{\Dv f^n,\Dv f^{n+1}}_\sharp - \dth \seq{S\Dx f^{n+1},\Dv f^{n+1}}_\sharp
- \dth \norm{(-\Dvs+\vs)\Dv f^{n+1}}^2.$$
For the third term in $\hhhd(f^{n+1})$, we compute $2\seq{S\Dx f^{n+1},\Dv f^{n+1}}_\sharp$ using relation once on the left and once on the right to obtain $$\begin{aligned}
\lefteqn{2\seq{S\Dx f^{n+1},\Dv f^{n+1}}_\sharp = }\\
& & \seq{S\Dx f^{n},\Dv f^{n+1}}_\sharp + \seq{S\Dx f^{n+1},\Dv f^{n}}_\sharp\\
& & -\dth\left(
\seq{S\Dx v \Dx f^{n+1},\Dv f^{n+1}}_\sharp + \seq{S\Dx f^{n+1},\Dv v \Dx f^{n+1}}_\sharp
\right)\\
& & -\dth\left(
\seq{S\Dx(-\Dvs+\vs)\Dv f^{n+1},\Dv f^{n+1}}_\sharp
+ \seq{S\Dx f^{n+1},\Dv(-\Dvs+\vs)\Dv f^{n+1}}_\sharp
\right).\end{aligned}$$ The two terms in $\dth$ above can be computed just as terms $(I)$ and $(II)$ in the proof of Theorem \[thm:decrexpd\] (with $f$ there replaced by $f^{n+1}$ here) to get as in $$\begin{aligned}
\lefteqn{2\seq{S\Dx f^{n+1},\Dv f^{n+1}}_\sharp = } \nonumber\\
& & \seq{S\Dx f^{n},\Dv f^{n+1}}_\sharp + \seq{S\Dx f^{n+1},\Dv f^{n+1}}_\sharp\nonumber\\
& & -\dth\left(
\norm{S\Dx f^{n+1}}_\sharp^2 - \dvh\seq{S^b \Dx f^{n+1},\Dx \Dv f^{n+1}}_\sharp
\right)\nonumber\\
& & -\dth\left(
-2 \seq{(-\Dvs+\vs)\Dv f^{n+1},S^\sharp \Dx \Dv f^{n+1}} \right. \nonumber\\
& & \left. \qquad \qquad \qquad + \seq{S\Dx f^{n+1},\Dv f^{n+1}}_\sharp
+\seq{\sigma \Dx f^{n+1},\Dv f^{n+1}}_\sharp
\right),\label{eq:estim3discr}\end{aligned}$$ where we used Lemmas \[lem:triple\] and \[lem:sb\].
For the last term in $\hhhd(f^{n+1})$, we compute as for , $$\label{eq:estim4discr}
\norm{\Dx f^{n+1}}^2 = \seq{\Dx f^n,\Dx f^{n+1}} - \dth \norm{\Dx \Dv f^{n+1}}_\sharp^2.$$
Summing up the four identities , , and , multiplied respectively by $C$, $D$, $E/2$ and $1$, we infer that $$\begin{aligned}
\lefteqn{\hhhd(f^{n+1}) = \phid(f^n,f^{n+1})}\\
& & -\dth\left[
C \norm{\Dv f^{n+1}}_\sharp^2
+ D \seq{S\Dx f^{n+1},\Dv f^{n+1}}_\sharp
+ D \norm{(-\Dvs+\vs)\Dv f^{n+1}}^2 \right. \\
& & \qquad \qquad \left. +\frac{E}{2} \norm{S\Dx f^{n+1}}_\sharp^2 -
\frac{E}{2} \dvh\seq{S^b \Dx f^{n+1},\Dx \Dv f^{n+1}}_\sharp \right.\\
& & \qquad \qquad \left.
-E \seq{(-\Dvs+\vs)\Dv f^{n+1},S^\sharp \Dx \Dv f^{n+1}}
+\frac{E}{2} \seq{S\Dx f^{n+1},\Dv f^{n+1}}_\sharp \right.\\
& & \qquad \qquad \qquad \qquad \left.
+\frac{E}{2}\seq{\sigma \Dx f^{n+1},\Dv f^{n+1}}_\sharp
+\norm{\Dx \Dv f^{n+1}}_\sharp^2
\right].\end{aligned}$$
We recognize here inside square brackets exactly the same term as the one in parentheses defining $\dddd(f)$ in with $f^{n+1}$ here instead of $f$ there, so that the preceding identity reads $$\hhhd(f^{n+1}) = \phid(f^n,f^{n+1}) -\dth \dddd(f^{n+1}).$$ Using Lemma \[lem:dissdiss\] we therefore get that for $C$, $D$, $E$ and $\dvh_0$ be chosen as in -, we have $$\hhhd(f^{n+1}) = \phid(f^n,f^{n+1}) -\dth \kappa_d \hhhd(f^{n+1}),$$ with $\kappa_d = 1/(4C)$.
Using Cauchy–Schwarz–Young with the scalar product $\phid$ (see ), we obtain for all $n\in\N$, $$\hhhd(f^{n+1}) \leq \frac{1}{2} \hhhd(f^{n+1}) + \frac{1}{2} \hhhd (f^n)
-\dth \kappa_d \hhhd(f^{n+1}),$$ which yields for all $n\in\N$, $$\hhhd(f^{n+1}) \leq \hhhd (f^n)
-2\kappa_d \dth \hhhd(f^{n+1}),$$ which implies $$\hhhd(f^{n}) \leq (1+ 2\dth \kappa_d)^{-n} \hhhd (f^0).$$ This concludes the proof of the theorem.
The homogeneous equation on bounded velocity domains {#sec:homobounded}
====================================================
In this Section, we study a discretization of the homogeneous Fokker–Planck equation with velocity variable confined in the interval $I = (-\vmax,\vmax)$, where $\vmax>0$ is given. We first briefly treat the fully continuous case, and then we focus on the fully discrete explicit case : this is possible since only a finite number of points of discretization are needed (in contrast to the case where $v$ was on the whole real line in the preceding sections). The choice of discretization is again made to ensure exponential convergence to the equilibrium and the functional framework is built using the natural Maxwellian (stationary solution of the problem, again denoted $\muh$ below).
In this section, we also prepare the study of the inhomogeneous equation in Section \[sec:eqinhomoboundedvelocity\]. Part of the material is very similar to the one developed in Section \[sec:homogeneous\] and we will sometimes refer to there.
Note that the functional spaces in space and velocity introduced and used in Sections \[sec:homobounded\] and \[sec:eqinhomoboundedvelocity\] are finite dimensional. We will however specify norms on these spaces and constants for (continuous) linear operators between such spaces, to emphasize the behaviour of those norms and constants when the discretization parameters $\dvh$ and $\dxh$ tend to $0$.
The fully continuous case {#sec:homoboundedcontinuous}
-------------------------
We consider here the case where the velocity domain is an interval $$I = (-\vmax,\vmax), \qquad \vmax>0,$$ and focus on the fully continuous case. We thus need a boundary condition and choose a homogeneous Neumann one, to ensure total mass conservation. Our new problem is thus $$\label{eq:IHFPFb}
{\D_t} F - {\D_v}({\D_v}+v) F = 0, \qquad F|_{t=0} = F^0, \qquad (({\D_v}+v)F)(\pm\vmax)=0.$$
The initial density $F^0$ is a non-negative function from $I$ to $\R^+$ such that $\int_I F^0(v) \dv = 1$. The function $$I \ni v \mapsto \frac{1}{\sqrt{2\pi}} \exp^{-v^2/2},$$ is a continuous equilibrium of , but we need to renormalize it in $L^1(I,\dv)$. We keep the same notation as in the first sections of this paper and we define this normalized equilibrium $$\mu(v)=\dfrac{\exp^{-v^2/2}}{\displaystyle\int_{I}\exp^{-w^2/2}\dw}.$$
In the same way as in the unbounded velocity domain cases, we pose $F=\mu+ \mu f$, and the rescaled density $f$ solves equivalently $$\label{eq:HFPfb}
{\D_t} f +(- {\D_v}+v) {\D_v} f = 0 , \qquad f|_{t=0} = f^0, \qquad {\D_v} f(\pm\vmax)=0.$$
We work with the following adapted functional spaces: We introduce the space $\Ldeuxmudvb$ and it subspace $\Hunmudvb = \set{ g \in \Ldeuxmudvb, \ {\D_v} g \in \Ldeuxmudvb}$. We again denote $\int_I g(v) \mudv$ by $\seq{g}$.
As in the continuous homogeneous case (see Section \[sec:homogeneous\] for example), the main ingredient in the proof of the convergence to the equilibrium is the Poincaré inequality, that we prove now.
\[lem:Poincarecontinub\] For all $g \in \Hunmudvb$ with , we have $$\norm{g-\seq{g}}^2_\Ldeuxmudvb \leq \norm{{\D_v} g}_\Ldeuxmudvb^2.$$
The proof follows exactly the same lines as in the full space case described in Lemma \[lem:Poincarecontinu\]. We take $g \in \Ldeuxmudvb$ and assume that $\seq{g}=0$. The first steps of the proof are exactly the same as that of the proof of Lemma \[lem:Poincarecontinu\], changing $\R$ in $I$ until relation there. Note that we again use strongly Fubini Theorem and the fact that $\int_I v \mudv = 0$ and $\int_I \mudv = 1$ (and their counterparts in variable $v'$). We therefore have $$\begin{split}
& \int_I g^2\mudv = \int_{I} \G v \mudv,
\end{split}$$ where we have set as before $\G(v) = \int_{0}^v \abs{{\D_v} g (w)}^2 \dw$ for $|v| \leq \vmax$. Using that ${\D_v} \mu = -v \mu$ and an integration by part, we get $$\begin{split}
\norm{g}_\Ldeuxmudvb^2 & = \int_{(-\vmax,\vmax)} \G\, v\, \mudv \\
&= - \int_{(-\vmax,\vmax)} \G\, ({\D_v} \mu)\, \dv
\\
& =- [\G\mu]_{-\vmax}^{\vmax} + \int_{(-\vmax,\vmax)} {\D_v} \G\, \mudv\\
&=-\mu(\vmax)\int_{-\vmax}^{\vmax}\abs{{\D_v} g}^2+ \int_{(-\vmax,\vmax)}
\abs{{\D_v} g}^2 \mudv\\
& \leq \norm{{\D_v} g}_\Ldeuxmudvb^2.
\end{split}$$ The proof is complete.
Now we can state the main result concerning the convergence to the equilibrium for Equation . We consider the operator $P = (- {\D_v}+v) {\D_v}$ with domain $$D(P) = \set{ g \in \Ldeuxmudvb, \ (- {\D_v}+v) {\D_v} g \in \Ldeuxmudvb,
\ {\D_v} g (\pm \vmax) =0},$$ which corresponds to the operator with Neumann conditions. Note that constant functions are in $D(P)$. Equation reads ${\D_t} f+ P f = 0$ and we define the two following entropies for $g\in \Ldeuxmudvb$ and $g\in\Hunmudvb$ respectively :
$$\fff(g) = \norm{g}_\Ldeuxmudvb^2, \qquad \ggg(g) = \norm{g}_\Ldeuxmudvb^2
+ \norm{{\D_v} g}_\Ldeuxmudvb^2.\label{eq:fgh}$$
The following result holds
\[thm:exponentialtrendtoequilibriumb\] Let $f^0 \in \Ldeuxmudvb$. The Cauchy problem has a unique solution $f$ in $\ccc^0(\R^+, \Ldeuxmudvb)$. If $f^0$ is such that $\seq{f^0} = 0$, then $\seq{f(t)} = 0$ for all $t\geq 0$ and we have $$\forall t\geq 0,\qquad
\fff(f(t)) \leq \exp^{-2t}\fff(f^0).$$
If in addition $f^0 \in \Hunmudvb$, then $f \in \ccc^0(\R^+, \Hunmudvb)$ and we have $$\forall t\geq 0,\qquad
\ggg(f(t)) \leq \exp^{-t}\ggg(f^0).$$
The proof follows exactly the lines of the proof of Theorem \[thm:exponentialtrendtoequilibrium\]. The existence part is insured by the Hille–Yosida theorem again (either in $\Ldeuxmudvb$ or in $\Hunmudvb$). As in the unbounded case, the key points are the fact that the operator $P = (-{\D_v}+v){\D_v}$ is self-adjoint on $\Ldeuxmudvb$ with Neumann boundary condition and the Poincaré inequality (Lemma \[lem:Poincarecontinub\]).
The full discretization with discrete Neumann conditions {#subsec:homsdb}
--------------------------------------------------------
As in the unbounded case, we discretize the interval of velocities $I = (-\vmax, \vmax)$ and the equation with boundary condition by introducing an operator $\Dv$. This indeed yields a discretization of the rescaled equation .
For a fixed positive integer $\imax$, we set $$\label{eq:defdv}
\dvh=\frac{\vmax}{\imax},$$ and $$\iii = \set{-\imax + 1, -\imax + 2, \cdots, -1, 0, 1, \cdots, \imax-2, \imax -1}.$$ Moreover, we define $$\label{eq:defvib}
\forall i\in\iii, \quad v_i = i\dvh, \qquad v_{\pm\imax} = \pm \vmax.$$ Note for further use that the boundary indices $\pm\imax$ do not belong to the full set $\iii$ of indices. The new discrete Maxwellian $\muh \in \R^\iii$, is defined by $$\label{eq:defMaxwellboundedvelocity}
\muh_i = \frac{c_\dvh}{ \prod_{\ell=0}^{|i|} (1+ v_\ell\dvh )}, \qquad i \in \iii,$$ where the normalization constant $c_\dvh>0$ is defined such that $\dvh\sum_{i\in \iii} \muh_i= 1$. This definition is consistent with the Definition \[def:dvb\] of the operator $\Dv$ in the sense that it satisfies . For the sake of simplicity, we will keep the same notation as in the unbounded velocity case. Note again that we do not need to define the Maxwellian at the boundary indices $\pm \imax$.
We work in the following in the space ${\ell^1(\iii, \muh \dvh)}$ of all finite sequences $g = (g_i)_{i\in \iii}$ with the norm $\dvh\sum_{i \in \iii} \abs{g_i}\muh_i$. We note that $$\label{eq:normalizationmuhboundeddomain}
\norm{1}_{\ell^1(\iii, \muh \dvh)} = \norm{\muh}_{\ell^1(\iii, \dvh)} = 1.$$
For the analysis to come, we introduce another set of indices and a new Maxwellian $\mus$. We set $$\iiis = \set{-\imax , -\imax + 1, \cdots, -2, -1, 1, 2, \cdots, \imax-1, \imax}
=\left(\iii\setminus\{0\}\right)\cup \{\pm\imax\},$$ and define $\mus \in {\ell^1(\iiis, \dvh)}$ for all $ i\in\iiis$ by, $$\label{eq:defmusb}
\mus_i = \muh_{i+1} \textrm{ for } i<0, \qquad \mus_i = \muh_{i-1} \textrm{ for } i>0.$$ We now adapt to this finite case of indices the definitions of the discrete derivation given in the unbounded velocity case (see there Definitions \[def:dv\] and \[def:dvs\]).
\[def:dvb\] Let $g \in \ell^1(\iii,\muh \dvh)$, we define $\Dv g\in \ell^1(\iiis, \mus \dvh)$ by the following formulas for $i \in \iiis$, $$\begin{split}
& (\Dv g)_i = \frac{g_{i+1}-g_{i}}{\dvh} \textrm{ when } -\imax+1\leq i \leq -1, \qquad
(\Dv g)_i = \frac{g_i-g_{i-1}}{\dvh} \textrm{ when } 1 \leq i \leq \imax-1, \\
& \qquad \qquad \qquad \qquad \textrm{ and } \qquad \qquad (\Dv g)_{\pm \imax} = 0,
\end{split}$$ and $v g\in \ell^1(\iiis,\muh \dvh)$ by $$(v g)_i = v_i g_i \ \textrm{ for }\ 1\leq |i| \leq \imax-1
\qquad
\textrm{ and } \qquad (v g)_{\pm \imax} = v_{\pm \imax} g_{\pm(\imax-1)}.$$ Similarly for $h \in \ell^1(\iiis, \mus \dvh)$, we define $\Dvs h \in \ell^1(\iii, \muh \dvh)$ by the following formulas for all $i \in \iii$, $$\begin{split}
& (\Dvs h)_i = \frac{h_{i}-h_{i-1}}{\dvh} \textrm{ when } -\imax+1 \leq i<
-1, \qquad (\Dvs h)_i = \frac{h_{i+1}-h_i}{\dvh} \textrm{ when } 1 \leq i
\leq \imax -1
\\
& \qquad \qquad \qquad \textrm{ and } \qquad \qquad (\Dvs h)_0 =
\frac{h_1-h_{-1}}{\dvh}.
\end{split}$$ For $h\in \ell^1(\iiis, \muh \dvh)$, we also define $\vs g \in \ell^1(\iii, \dvh)$ by $$\forall i \in \iii \setminus \set{0} ,\quad (\vs h)_i=v_i h_i \qquad \text{and} \qquad
(\vs g)_0=0.$$
Looking at the proof of Lemma \[lem:muh\], we directly check that with this definition we have $$\label{eq:okmuh}
\forall i\in\iii\setminus\{0\},\qquad
[(\Dv + v)\muh]_i = 0,$$ The definition of the derivative at the boundary points (always $0$) is nevertheless adapted to the scaled equation. This is not in contradiction with the preceding equality which occurs only in $\iii\setminus\{0\}$. We write below the (rescaled) fully discrete homogeneous Fokker–Planck equation, noting that the discrete Neumann conditions are included in the definition of $\Dv$.
We shall say that a sequence $f = (f^n)_{n\in \N} \in (\ell^1(\iii, \muh\dvh))^\N$ satisfies the (scaled) full discrete explicit homogeneous Fokker–Planck equation with initial data $f^0$ if $$\label{eq:DHFPfb}
\forall n\in\N,\qquad
f^{n+1} = f^n -\dth ( - \Dvs +\vs) \Dv f^n,$$ for some $\dth>0$.
In order to solve this equation, we build Hilbertian norms on $\R^\iii$ and $\R^\iiis$, taking into account the conservation of mass and insuring the non-negativity of the associated operator.
We denote by $\Ldeuxmudvh$ the space $\R^\iii$ endowed with the Hilbertian norm $$\norm{ g}_{\Ldeuxmudvh }^2 \defegal \dvh \sum_{i\in \iii} (g_i)^2 \muh_i.$$ The related scalar product is denoted by $\seq{ \cdot, \cdot}$. For $g \in \Ldeuxmudvh$, we also define $ \seq{g} \defegal \sum_{i\in \iii} g_i \mu_i^h \dvh = \seq{g, 1}_{\Ldeuxmudvh }, $ the mean of $g$. Similarly, we denote by $\Ldeuxmudvs$ the space $$\Ldeuxmudvs = \set{ g \in \R^\iiis, g_{\pm \imax} = 0},$$ endowed with the Hilbertian norm $$\norm{g}_{\Ldeuxmudvs }^2 \defegal \dvh \sum_{i \in \iiis} (g_i)^2 \mus_i,$$ and the related scalar product is denoted by $\seq{ \cdot, \cdot}_\sharp$. We denote by $\Hunmudvh$ the space $\R^\iii$ endowed with the norm $$\norm{g}_{\Hunmudvh}^2=\norm{g}_{\Ldeuxmudvh}^2+\norm{\Dv g}_{\Ldeuxmudvs}^2.$$
We introduce the associated operator with discrete Neumann conditions and its functional and structural properties.
Let $\dvh$ be defined by and $\dth>0$ be given and sufficiently small.
1. We have $\Dv : \Ldeuxmudvhb \rightarrow \Ldeuxmudvsb$ and $\Dvs : \Ldeuxmudvsb \rightarrow \Ldeuxmudvhb$ and $P = ( - \Dvs +\vs) \Dv$ is a bounded operator on $\Ldeuxmudvhb $.
2. For all $h \in \Ldeuxmudvsb$, $g \in \Ldeuxmudvhb $ we have $$\label{eq:ofb}
\seq{ ( - \Dvs +\vs) h, g} = \seq{h, \Dv g}_\sharp,
\qquad \textrm{ and } \qquad
\seq{ ( - \Dvs +\vs) \Dv h, h} = \norm{\Dv h}_\Ldeuxmudvs^2.$$
3. For an initial data $f^0 \in \Ldeuxmudvh$, there exists a unique solution of in $ (\Ldeuxmudvh)^\N$.
4. Constant sequences are the only equilibrium states of Equation .
5. The mass is conserved by the discrete evolution, [*i.e.*]{} for all $n\in \N$, $\seq{f^n} = \seq{f^0}$.
The linear operator $P$ is a mapping from the finite dimensional linear space $\Ldeuxmudvhb$ to itself. Hence it is bounded. The proof of the second equality in is a direct consequence of the first equality, and leads directly to the self-adjointness and the non-negativity of $(
- \Dvs +\vs) \Dv$. The (maximal) accretivity of $( - \Dvs +\vs)
\Dv$ in both $\Ldeuxmudvh$ and $\Hunmudvh$ is easy to get (perhaps adding a constant to the operator). The fact that the equation is well-posed is a direct consequence of the fact that the scheme is explicit. The fact that constant sequences are the only equilibrium solutions is an easy consequence of the second identity in .
Due to its importance in the functional framework we give a complete proof of the first equality in although it is very similar to the one of . We write for $h \in \Ldeuxmudvsb$ and $g \in \Ldeuxmudvhb$ $$\label{eq:calcofb}
\begin{split}
& \dvh^{-1} \seq{ ( - \Dvs +\vs) h, g} \\
& = \sum_{i \in \iii} ((-\Dvs +\vs) h)_i g_i \mu_i \\
& = \sum_{1 \leq i \leq \imax -1} ((-\Dvs +\vs) h)_i g_i \mu_i -(\Dvs h)_0 g_0
\mu_0 +\sum_{-\imax + 1 \leq i \leq -1} ((-\Dvs +\vs) h)_i g_i \mu_i.
\end{split}$$ For the first term in the right-hand side of , we have $$\begin{split}
& \sum_{1 \leq i \leq \imax -1} ((-\Dvs +\vs) h)_i g_i \mu_i \\
& = \sum_{1 \leq i \leq \imax -1} \sep{ -\frac{h_{i+1} - h_i}{\dvh} + v_i h_i} g_i \mu_i \\
& = \sum_{1 \leq i \leq \imax -1} h_i \sep{ \frac{-g_{i-1}\mu_{i-1} + g_i
\mu_i }{\dvh} + v_i g_i \mu_i} + \frac{h_1 g_0}{\dvh} \mu_0 -
\frac{h_{\imax} g_{\imax-1}}{\dvh} \mu_{\imax-1}.
\end{split}$$ Since $h \in \Ldeuxmudvsb$ we have $h_{\imax}=0$. Therefore we have $$\begin{split}
& \sum_{1 \leq i \leq \imax -1} ((-\Dvs +\vs) h)_i g_i \mu_i \\
& = \sum_{1 \leq i \leq \imax -1} h_i g_i \sep{ \frac{-\mu_{i-1} + \mu_i
}{\dvh} + v_i \mu_i} + \sum_{1 \leq i \leq \imax -1} h_i
\sep{ - \frac{g_{i-1}- g_i}{\dvh}}\mu_{i-1} + \frac{h_1 g_0}{\dvh} \mu_0 \\
& = \sum_{1 \leq i \leq \imax -1} h_i (\Dv g)_i \mu_{i-1} + \frac{h_1 g_0}{\dvh} \mu_0 \\
& = \sum_{1 \leq i \leq \imax} h_i (\Dv g)_i \mus_{i} + \frac{h_1 g_0}{\dvh}
\mu_0,
\end{split}$$ where we used , the definition of $\mus$, and again the fact that $h_{\imax} = 0$. Similarly we get $$\begin{split}
& \sum_{-\imax + 1 \leq i \leq -1} ((-\Dvs +\vs) h)_i g_i \mu_i
= \sum_{-\imax \leq i \leq -1} h_i (\Dv g)_i \mus_{i} - \frac{h_{-1} g_0}{\dvh} \mu_0.
\end{split}$$ The center term in the right-hand side of is $-(\Dvs h) g_0 \mu_0 = -\frac{ h_1- h_{-1}}{\dvh} g_0 \mu_0$, so that we have $$\dvh^{-1} \seq{ ( - \Dvs +\vs) h, g} = \sum_{i \in \iiis} h_i (\Dv g)_i \mus_{i}
= \dvh^{-1} \seqs{h, \Dv g},$$ since the boundary terms around $0$ disappear. This is the first equality in and the proof is complete.
As in the cases with unbounded velocity domains (see Sections \[sec:homogeneous\] and \[sec:eqinhomo\]), in continuous or discretized settings, and as in the case with bounded velocity domain in the continuous setting (see Lemma \[lem:Poincarecontinub\]), the Poincaré inequality is a fundamental tool to obtain the convergence of the solution, and we give below a version for the bounded velocity case adapted to the velocity discretization above.
\[prop:poindiscreteb\] Let $\dvh>0$ be defined as in , and let $g\in\Ldeuxmudvh$. Then, $$\norm{g-\seq{g}}^2_{\Ldeuxmudvh } \leq \norm{ \Dv g}_{\Ldeuxmudvs }^2.$$
Although part of the proof is similar to the proofs of previous Poincaré inequalities in this paper, we give a complete proof, following the lines of the one of Proposition \[prop:poindiscrete\]. This is to illustrate how our choice of discretization of the bounded velocity domain allows to obtain this fundamental inequality. We take $g\in \Ldeuxmudvh$ with $\seq{g}=0$ (note that the boundary conditions are preserved by addition of a constant). We have with the normalization convention $$\begin{split}
\dvh^{-1} \norm{g}^2_\Ldeuxmudvh = \sum_{-\imax < i <\imax} g_i^2 \muh_i
&= \frac{\dvh}{2} \sum_{-\imax < i,j < \imax} ( g_j-g_i)^2 \muh_i \muh_j \\
& = \dvh\sum_{-\imax < i<j <\imax} ( g_j-g_i)^2 \muh_i \muh_j,
\end{split}$$ since $2 \sum_{-\imax < i,j < \imax} g_i g_j \muh_i \muh_j = 2 \sum_{-\imax < i < \imax} g_i \muh_i \sum_{-\imax < j < \imax} g_j \muh_j
= 0$. For $i<j$, we can write the telescopic sum $$g_j-g_i = \sum_{\ell=i+1}^j (g_\ell-g_{\ell-1}),$$ so that $$\begin{split}
\dvh^{-1}\sum_{-\imax < i <\imax} g_i^2 \muh_i &
= \sum_{{-\imax < i <j<\imax}} \sep{ \sum_{\ell=i+1}^j (g_\ell-g_{\ell-1}) }^2 \muh_i \muh_j \\
& \leq \sum_{{-\imax < i <j<\imax}} \sep{ \sum_{\ell=i+1}^j (g_\ell-g_{\ell-1})^2 } (j-i) \muh_i \muh_j,
\end{split}$$ where we used the discrete flat Cauchy–Schwarz inequality. Let us now introduce $\G$ the discrete anti-derivative of $(g_\ell-g_{\ell-1})^2$, given by $$\G_j = - \sum_{\ell=j+1}^{-1} (g_\ell-g_{\ell-1})^2 \textrm{ for } j\leq -1,
\qquad \G_j = \sum_{\ell=0}^j (g_\ell-g_{\ell-1})^2 \textrm{ for } j\geq 0,$$ we get (exactly as after ) that $$\begin{split}
\dvh^{-1}\sum_{-\imax < i <\imax} g_i^2 \muh_i &
= \dvh^{-1}\sum_{{-\imax < i <\imax}} \G_i i \muh_i
= \dvh^{-1}\sum_{{-\imax < i <\imax}, i\neq 0} \G_i i \muh_i,
\end{split}$$ where we used the fact that $\sum_{{-\imax < j <\imax}} j\muh_j = 0$ and $\sum_{-\imax < i <\imax} \muh_j = \dvh^{-1}$. The last step is to perform a discrete integration by part using deeply the functional equation satisfied by $\muh$ and taking here the boundary terms. We write using that functional property of $\muh$, $$\begin{split}
\sum_{{-\imax+1 \leq i \leq \imax-1 }, \ i \neq 0} \G_i i \muh_i
& = \sum_{1 \leq i \leq \imax-1 } \G_i i \muh_i
+ \sum_{-\imax+1 \leq i \leq -1 } \G_i i \muh_i\\
& = - \sum_{1 \leq i \leq \imax-1} \G_i \frac{\muh_i- \muh_{i-1}}{\dvh^2}
- \sum_{-\imax+1 \leq i \leq -1 } \G_i \frac{\muh_{i+1} - \muh_{i}}{\dvh^2} \\
& = - \sum_{1 \leq i \leq \imax-2} \frac{ \G_i - \G_{i+1}}{\dvh^2} \muh_i
+ \frac{\G_1}{\dvh^2} \muh_0 - \frac{\G_{\imax-1}}{\dvh^2} \muh_{\imax-1} \\
& \qquad -
\sum_{-\imax+2 \leq i \leq -1} \frac{ \G_{i-1} - \G_{i}}{\dvh^2} \muh_i
- \frac{\G_{-1}}{\dvh^2} \muh_0 + \frac{\G_{-\imax+1}}{\dvh^2} \muh_{\imax-1}.
\end{split}$$ Now, using the definition of $\G$ and in particular the fact that $$\G_1 - \G_{-1} = (g_{1}- g_0)^2 + (g_{0}- g_{-1})^2,$$ we obtain as in but with the additional boundary terms $$\label{eq:finaldeuxbordsb}
\begin{split}
& \sum_{{-\imax+1 \leq i \leq \imax-1 }, \ i \neq 0} \G_i i \muh_i
= \dvh^{-1} \norm{ \Dv g}_{\Ldeuxmudvs }^2 - \sep{ \frac{\G_{\imax-1}}{\dvh^2} -
\frac{\G_{-\imax+1}}{\dvh^2}} \muh_{\imax-1}.
\end{split}$$ Now we have by definition of the anti-derivative $\G$, $$\begin{split}
\sep{\frac{\G_{\imax-1}}{\dvh^2} -
\frac{\G_{-\imax+1}}{\dvh^2}} \muh_{\imax-1}
= \frac{\G_{\imax-1} -\G_{-\imax+1}}{\dvh^2} \muh_{\imax-1} \\
= \sep{ \sum_{l=-\imax +2}^{\imax-1} (g_l-g_{l-1})^2} \muh_{\imax-1} \geq 0.
\end{split}$$ since this term is non-negative we get from $$\begin{split}
\sum_{{-\imax+1 \leq i \leq \imax-1 }, \ i \neq 0} \G_i i \muh_i
& \leq \dvh^{-1} \norm{ \Dv g}_{\Ldeuxmudvs }^2.
\end{split}$$ The proof is complete.
Before stating the main result of this subsection, we estimate the norm of the operator $\Dvs+\vs$ from $\Ldeuxmudvs$ to $\Ldeuxmudvh$.
\[lem:adjDvsplusvs\] Let $\dvh$ be defined in . We have for all $g \in \Ldeuxmudvs$, $$\label{eq:bornedvsv}
\norm{(-\Dvs+\vs) g}_\Ldeuxmudvh^2 \leq \frac{4(1+ \dvh \vmax)}{\dvh^2} \norm{g}^2_\Ldeuxmudvs.$$
The operator $(-\Dvs +v)$ is bounded from $\Ldeuxmudvs$ to $\Ldeuxmudvh$ since it is a linear mapping between finite dimensional normed spaces. Note that it is equivalent to estimate the norm of its adjoint $
\Dv : \Ldeuxmudvh \longrightarrow \Ldeuxmudvs
$. For this, we consider $1 \leq j \leq \imax$ and recall that $\mus_j = \muh_{j- 1} = (1 + v_j\dvh ) \muh_j$ from definitions and , where $v_j = j \dvh$ by definition . By symmetry, we infer that $$\forall j\in\iiis,\qquad
0\leq \mus_j \leq (1 + \dvh \abs{v_j}) \muh_j \leq (1 + \dvh \vmax)\muh_j.$$ On the other hand, for $j\in\{1,\cdots,\imax\}$, $$\abs{(\Dv g)_j}^2 \leq \frac{2}{\dvh^2} \sep{ \abs{g_j}^2 + \abs{g_{j - 1}}^2 }.$$ Similar estimates hold for $ -\imax \leq j \leq -1$ with $j-1$ replaced by $j+1$ in the last inequality. Using these results we get for $g \in \Ldeuxmudvh$ that $$\begin{split}
\dvh^{-1}\norm{\Dv g}_\Ldeuxmudvs^2
& = \sum_{i=-\imax+1, \ i\neq 0}^{\imax-1} \abs{(\Dv g)_i}^2 \mus_i \\
& \leq \frac{4}{\dvh^2} \sum_{i=-\imax+1}^{\imax-1} \abs{g_i}^2 (1 + \dvh \vmax) \muh_i,
\end{split}$$ which implies $$\label{eq:bornedv}
\begin{split}
\norm{\Dv g}_\Ldeuxmudvs^2 \leq \frac{4(1+ \dvh \vmax)}{\dvh^2} \norm{g}^2_\Ldeuxmudvh.
\end{split}$$ Therefore, by adjunction, we have .
We give below the result about the exponential trend to the equilibrium in the $\Ldeuxmudvh$ and $\Hunmudvh$ norms of the solution $(f^n)_{n\in\N}$ of the explicit Euler scheme . As in the continuous and unbounded cases we look at the following two entropies
$$\fffd(g) \defegal \norm{g}_{\Ldeuxmudvh }^2, \qquad \gggd(g) \defegal \norm{g}_{\Ldeuxmudvh
}^2
+ \norm{\Dv g}_{\Ldeuxmudvs}^2,\label{eq:deffgh}$$
defined for $g\in\R^\iii$. The second entropy is called the Fisher information. The result is the following.
\[thm:mainexplhom\] Let $\dvh>0$ be defined by and set $$\cfl \defegal \frac{4 (1+ \dvh \vmax)}{\dvh^2}.$$ Suppose that $\dth>0$ is such that the following CLF condition holds $$\label{eq:CFLhomog}
\dth \cfl <1,$$ and set $\kappa=1-\dth\cfl$. For all $f^0 \in \Ldeuxmudvh$ such that $\seq{f^0} = 0$, we denote by $(f^n)_{n\in\N}$ the solution of in $ (\Ldeuxmudvh)^\N$ with initial data $f^0$. We have for all $n\in \N$, $$\fffd(f^n) \leq (1-2\kappa \dth)^n \fffd(f^0),$$ and $$\gggd(f^n) \leq (1-\kappa \dth)^n \gggd(f^0).$$
The scheme is well-defined and one has for all $n\in\N$, $\seq{f^n}=0$ by induction. We look at the explicit scheme for some $n\in\N$ $$\label{eq:explib}
f^{n+1} = f^n -\dth (-\Dvs +\vs)\Dv f^n,$$ and we prove below the following estimate $$\label{eq:ineqb}
\norm{f^{n+1}}^2_\Ldeuxmudvh \leq \norm{f^n}^2_\Ldeuxmudvh
-2\dth \norm{\Dv f^n}_\Ldeuxmudvs^2 + 2 \dth^2 \norm{ (-\Dvs +\vs)\Dv f^n}^2_\Ldeuxmudvh.$$ For this, we first take the scalar product of with $f^{n+1}$. We get successively $$\begin{split}
& \norm{f^{n+1}}^2_\Ldeuxmudvh \\
& = \seq{ f^n, f^{n+1} } - \dth \seq{ (-\Dvs +\vs)\Dv f^n, f^{n+1}}_\Ldeuxmudvh \\
& = \seq{ f^n, f^{n+1} } - \dth \seq{ \Dv f^n, \Dv f^{n+1}}_\Ldeuxmudvs \\
& \leq \frac{1}{2} \norm{f^n}^2_\Ldeuxmudvh + \frac{1}{2}
\norm{f^{n+1}}^2_\Ldeuxmudvh -\dth \norm{\Dv f^{n}}^2_\Ldeuxmudvs
- \dth \seq{ \Dv f^n, \Dv \sep{f^{n+1}-f^n}}_\Ldeuxmudvs \\
& \leq \frac{1}{2} \norm{f^n}^2_\Ldeuxmudvh + \frac{1}{2}
\norm{f^{n+1}}^2_\Ldeuxmudvh -\dth \norm{\Dv f^{n}}^2_\Ldeuxmudvs
+ \dth^2 \seq{ \Dv f^n, \Dv (-\Dvs +\vs)\Dv f^n}_\Ldeuxmudvs \\
& \leq \frac{1}{2} \norm{f^n}^2_\Ldeuxmudvh + \frac{1}{2}
\norm{f^{n+1}}^2_\Ldeuxmudvh -\dth \norm{\Dv f^{n}}^2_\Ldeuxmudvs + \dth^2
\norm{ (-\Dvs +\vs) \Dv f^n}^2_\Ldeuxmudvh.
\end{split}$$ where we used again to obtain the terms in $\dth^2$, and we also used . Multiplying the preceding inequality by $2$ gives then . Using Lemma \[lem:adjDvsplusvs\] with $g = \Dv f^n$ in the last term of , we obtain $$\label{eq:ineqbter} \norm{f^{n+1}}^2_\Ldeuxmudvh \leq
\norm{f^n}^2_\Ldeuxmudvh -2\dth \norm{\Dv f^n}_\Ldeuxmudvs^2 + 2 \dth^2
\frac{4(1+ \dvh \vmax)}{\dvh^2} \norm{ \Dv f^n}^2_\Ldeuxmudvs.$$ Using the CFL condition and the definition of $\kappa$ given in the statement of the theorem, we infer $$\label{eq:ineqbquatinterm}
\norm{f^{n+1}}^2_\Ldeuxmudvh \leq \norm{f^n}^2_\Ldeuxmudvh -2\dth \kappa \norm{\Dv f^n}_\Ldeuxmudvs^2.$$ Using the discrete Poincaré inequality of Proposition \[prop:poindiscreteb\], this implies $$\norm{f^{n+1}}^2_\Ldeuxmudvh \leq \norm{f^n}^2_\Ldeuxmudvh -2\dth \kappa
\norm{ f^n}_\Ldeuxmudvh^2 = (1-2\kappa \dth) \norm{ f^n}^2,$$ so that by induction $$\norm{ f^n}_\Ldeuxmudvh^2 = \fffd(f^n) \leq (1-2\kappa \dth)^n \fffd(f^0).$$ This proves the result for the first entropy $\fffd$.
For the second entropy $\gggd$, we fix $n\in\N$ and we need to get an estimate on $\norm{\Dv f^{n+1}}_\Ldeuxmudvs^2$. Therefore, we apply the operator $\Dv$ to , which yields $$\Dv f^{n+1} = \Dv f^n -\dth \Dv (-\Dvs +\vs)\Dv f^n.$$ Following exactly the same method as in the proof of with $\Dv f$ instead of $f$ and operator $\Dv (-\Dvs +\vs)$ instead of $(-\Dvs +\vs)\Dv$, we get $$\begin{gathered}
\norm{\Dv f^{n+1}}^2_\Ldeuxmudvs \leq \norm{\Dv f^n}^2_\Ldeuxmudvs -2\dth \norm{(-\Dvs+\vs)\Dv f^n}_\Ldeuxmudvh^2 \\ + 2 \dth^2 \norm{ \Dv(-\Dvs +\vs)\Dv f^n}^2_\Ldeuxmudvs.\end{gathered}$$ Using the explicit bound of $\Dv$ given in (at the end of the proof of Lemma \[lem:adjDvsplusvs\]), we have $$\begin{gathered}
\norm{\Dv f^{n+1}}^2_\Ldeuxmudvs \leq \norm{\Dv f^n}^2_\Ldeuxmudvs -2\dth \norm{(-\Dvs+\vs)\Dv f^n}_\Ldeuxmudvh^2 \\
+ 2 \dth^2 \frac{4(1+ \dvh \vmax)}{\dvh^2} \norm{ (-\Dvs+\vs) \Dv f^n}^2_\Ldeuxmudvh,\end{gathered}$$ so that under the CFL condition , we get $$\norm{\Dv f^{n+1}}^2_\Ldeuxmudvs \leq \norm{\Dv f^n}^2_\Ldeuxmudvs -2\dth \kappa \norm{(-\Dvs+\vs)\Dv f^n}_\Ldeuxmudvh^2.$$ In particular, we have $$\label{eq:ineqbquatdv}
\norm{\Dv f^{n+1}}^2_\Ldeuxmudvs \leq \norm{\Dv f^n}^2_\Ldeuxmudvs.$$ Using and the discrete Poincaré inequality of Proposition \[prop:poindiscreteb\], we obtain $$\begin{split}
\norm{f^{n+1}}^2_\Ldeuxmudvh & \leq \norm{f^n}^2_\Ldeuxmudvh -2\dth \kappa \norm{\Dv f^n}_\Ldeuxmudvs^2 \\
& \leq \norm{f^n}^2_\Ldeuxmudvh -\dth \kappa \norm{\Dv f^n}_\Ldeuxmudvs^2
-\dth \kappa \norm{\Dv f^n}_\Ldeuxmudvs^2 \\
& \leq \norm{f^n}^2_\Ldeuxmudvh -\dth \kappa \norm{\Dv f^n}_\Ldeuxmudvs^2
-\dth \kappa \norm{f^n}_\Ldeuxmudvh^2.
\end{split}$$ Adding this inequality and yields $$\begin{split}
\gggd(f^{n+1}) & = \norm{f^{n+1}}^2_\Ldeuxmudvh + \norm{\Dv f^{n+1}}^2_\Ldeuxmudvs \\
& \leq \norm{f^n}^2_\Ldeuxmudvh + \norm{\Dv f^{n}}^2_\Ldeuxmudvs
-\dth \kappa \norm{f^n}_\Ldeuxmudvh^2 -\dth \kappa \norm{\Dv f^n}_\Ldeuxmudvs^2 \\
& \leq (1-\dth \kappa) \gggd(f^{n}),
\end{split}$$ so that by induction $$\gggd(f^n) \leq (1-\kappa \dth)^n \gggd(f^0).$$ The proof is complete.
Numerical results
-----------------
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[0.48]{}
[0.48]{}
[0.48]{}
This subsection is devoted to the numerical results obtained through the explicit discretization of .
The quantities of interest here are $\fffd$ and $\gggd$, defined in . According to Theorem \[thm:mainexplhom\], they are expected to decrease geometrically fast. The tests that are presented here aim at illustrating this fact in two cases:
- the initial datum is a step function (see Figure \[fig:hom1\]-(A)). The logarithms of the entropy $\fffd$ and of the Fisher information $\gggd$ decrease linearly fast (see Figure \[fig:hom1\]-(B)), with a rate that is close to $2$, as can be seen in Figures \[fig:hom1\]-(C). The exponential decrease is consistent with Theorem \[thm:mainexplhom\], and the rate being close to $2$ is consistent with Theorem \[thm:exponentialtrendtoequilibriumb\] for $\fffd$, and shows the bound to be optimal, and better than expected for $\gggd$.
- the initial datum is a random function (see Figure \[fig:hom2\]-(A)) The logarithms of the entropy $\fffd$ and of the Fisher information $\gggd$ decrease linearly fast (see Figure \[fig:hom2\]-(B)), with a rate that is close to $2$, as can be seen in Figures \[fig:hom2\]-(C). Again, the exponential decrease is consistent with Theorem \[thm:mainexplhom\], and the rate being going to $2$ is consistent with Theorem \[thm:exponentialtrendtoequilibriumb\] for $\fffd$ and $\gggd$.
Comparing the two previous test cases, we get a hint that there is a very fast regularizing effect in short time, as noted in [@PZ16]. The second initial datum is way less smooth that the first one and the range of the decrease rate is a lot larger in the second case. A perspective of our work would be to investigate the change of slope in Figure \[fig:hom2\]-(B).
The inhomogeneous equation on bounded velocity domains {#sec:eqinhomoboundedvelocity}
======================================================
This section is devoted to the analysis of the inhomogeneous Fokker-Planck equation on bounded velocity domains, in the fully discretized setting, meaning discretized in velocity, in space and in time. The main result is the exponential convergence to equilibrium of numerical solutions stated in Theorem \[thm:decrexpeulerexpldiscr\]. We first recall briefly in Section \[sec:inhomcb\] the statements for the continuous equation set on a bounded velocity domain. Next, we study in Section \[sec:inhomobounded\] a full discretization by an explicit Euler scheme in time, by an extension of the operators $\Dv$ and $\Dvs$ introduced in Section \[sec:homobounded\] in velocity to this inhomogeneous case, and a space discretization operator $\Dx$ similar to the one introduced in the unbounded velocity inhomogeneous case in Section \[sec:inhomsd\]. In this context, we prove our main result : Theorem \[thm:decrexpeulerexpldiscr\]. We conclude with numerical simulations carried out using this numerical scheme.
The fully continuous analysis {#sec:inhomcb}
-----------------------------
In order to prepare the fully discrete inhomogeneous case in the next subsection, we briefly show how to extend the results of Section \[sec:inhomc\] for the inhomogeneous equation on an [*unbounded*]{} velocity domain to the case of a [*bounded*]{} velocity domain.
In this bounded-velocity setting, we stick to the notations introduced in Section \[sec:homoboundedcontinuous\] for the homogeneous case. In particular the velocity domain is $I = (-\vmax,\vmax)$ for some $\vmax>0$. We propose a suitable functional framework for the following inhomogeneous Fokker–Planck equation with unknown $F(t,x,v)$ where $(t,x,v) \in \R^+ \times\T\times I$ $$\label{eq:IHFPFbinhomo}
{\D_t} F+v{\D_x}F-{\D_v} ({\D_v}+v)F=0, \qquad F|_{t=0} = F^0, \qquad (({\D_v}+v)F)(\cdot,\cdot, \pm\vmax)=0.$$ The initial datum $F^0$ is a non-negative function of $L^1(\T\times I,\dx\dv)$ with $\int_{\T\times I} F^0(x,v)\dx \dv=1$. The Maxwellian function $$\mu(x,v)=\dfrac{\exp^{-v^2/2}}{\displaystyle\int_{I}\exp^{-w^2/2}\dw},$$ is a continuous equilibrium of , normalized in $L^1(\T \times I,\dx \dv)$. As we did for the unbounded velocity domain case in Section \[sec:inhomc\], we pose $F=\mu+\mu f$, and the rescaled density $f$ solves $$\label{eq:IHFPfb}
{\D_t} f + v{\D_x} f+(- {\D_v}+v) {\D_v} f = 0 , \qquad f|_{t=0} = f^0, \qquad {\D_v} f(\cdot,\cdot, \pm\vmax)=0.$$ We introduce the corresponding functional space $\Ldeuxmudvdxb$ and its subspace $$\Hunmudvdxb \defegal \set{ g \in \Ldeuxmudvdxb, \ {\D_v} g \in \Ldeuxmudvdxb}.$$ For $g\in L^1(\T \times I, \mu\dx \dv)$, we denote its $(x,v)$-mean by $\seq{g} = \iint_{\T\times I} g(v) \mudv\dx$. From now on, the norms and scalar products without subscript are taken in $\Ldeuxmudvdxb$. In these spaces, we have again a Poincaré inequality (see Lemma \[lem:fullPoincarecontinub\] below). The proof of that inequality follows exactly the lines of the one for the continuous, inhomogeneous, unbounded-velocity case presented in Lemma \[lem:fullPoincarecontinu\] (but using the homogeneous Poincaré inequality on bounded velocity domain of Lemma \[lem:Poincarecontinub\] as a tool, instead of the homogeneous Poincaré inequality on unbounded velocity domain (Lemma \[lem:Poincarecontinu\])):
\[lem:fullPoincarecontinub\] For all $g \in \Hunmudvdx$, we have $$\norm{g-\seq{g}}^2 \leq \norm{{\D_v} g}^2 + \norm{{\D_x} g}^2.$$
In order to state the main result concerning the convergence to the equilibrium for the solutions of Equation in Theorem \[thm:decrexpb\], we introduce a little more functional framework. We consider the operator $P = v{\D_x} + (- {\D_v}+v) {\D_v}$ with domain $$D(P) = \set{ g \in \Ldeuxmudvdxb,
\ (v{\D_x} + (- {\D_v}+v) {\D_v} )g \in \Ldeuxmudvdxb, \ {\D_v} g (\cdot,\pm \vmax) =0},$$ which corresponds to the evolution operator in with Neumann conditions in velocity. Note that constant functions are in $D(P)$. Equation reads then ${\D_t} f+ P f = 0$ with initial condition $f(0,\cdot,\cdot)=f^0$.
The non-negativity of the operator $P$ is straightforward since $v{\D_x}$ is skew-adjoint in $\Ldeuxmudvdxb$. The maximal accretivity of this operator in $\Ldeuxmudvdxb$ or $\Hunmudvdxb$ is not so easy and we refer for example to [@HelN04]. As in the unbounded velocity case, using the Hille–Yosida Theorem, this implies that for an initial datum $f^0 \in \Ldeuxmudvdxb$ (resp. $\Hunmudvdxb$) there exists a unique solution in $\ccc^0( \R^+, \Ldeuxmudvdxb)$ (resp. $\ccc^0(\R^+, \Hunmudvdxb$). Moreover, for if $f^0 \in D(P)$ (resp. $D_\Hunmudvdxb(P)$), there exists a unique solution in $\ccc^1( \R^+, \Ldeuxmudvdxb)$ (resp. $\ccc^1(\R^+, \Hunmudvdxb$).
As a norm in $\Hunmudvdx$ we choose the standard, the square of which is defined for $g\in \Hunmudvdxb$ by $$\norm{g}_\Hunmudvdxb^2 = \norm{g}^2 + \norm{{\D_v} g}^2
+ \norm{{\D_x} g}^2.$$
As in the unbounded velocity case for Section \[sec:eqinhomo\], we shall define a modified entropy adapted to the $\Hunmudvdxb$ framework. For $C>D>E>1$ to be precised later, it is defined for $g\in \Hunmudvdxb$ by $$\hhh(g) = C\norm{g}^2+D\norm{{\D_v} g}^2+E\seq{{\D_v} g,{\D_x} g}+\norm{{\D_x}g}^2.$$
Following exactly the proof of Lemma \[lem:equiv\] we again check that
If $E^2<D$ then for all $g\in \Hunmudvdxb$, $$\dfrac{1}{2}\norm{g}_{\Hunmudvdxb}^2\leq\hhh(g)\leq 2C\norm{g}_{\Hunmudvdxb}^2.$$
The main result is then the following theorem, the proof of which is exactly the same as that of Theorem \[thm:decrexp\]
\[thm:decrexpb\] Assume that $C>D>E>1$ satisfy $E^2<D$ and $(2D+E)^2<2C$. Let $ f^0 \in \Hunmudvdxb$ such that $\seq{f^0} = 0$ and let $f$ be the solution in $\ccc^0(\R^+, \Hunmudvdxb)$ of Equation . Then for all $t\geq 0$, $$\hhh(f(t))\leq \hhh(f^0)\exp^{-2\kappa t}.$$ with $2\kappa = \frac{E}{8C}$.
The following corollary is also similar to the one proposed after the proof of Theorem \[thm:decrexp\].
\[cor:decrexpb\] Let $C>D>E>1$ be chosen as in Theorem \[thm:decrexp\], and pose $\kappa = E/(16C)$. Let $f^0\in \Hunmudvdxb$ and let $f$ be the solution in $\ccc^0(\R^+, \Hunmudvdxb)$ of Equation . Then for all $t\geq 0$, we have $$\norm{f(t) - \seq{f^0}}_{\Hunmudvdxb}\leq 2\sqrt{C} \exp^{-\kappa t} \norm{f^0 - \seq{f^0}}_{\Hunmudvdxb}.$$
The full discretization and proof of Theorem \[thm:eulerexplicite\] {#sec:inhomobounded}
-------------------------------------------------------------------
As we did in the unbounded case, we want to discretize the velocity domain $I = (-\vmax, \vmax)$ and the equation and boundary conditions of .
Concerning the discretization of the velocity variable, we use the very same definitions introduced in Subsection \[subsec:homsdb\] in the homogeneous setting for $\imax$, $\dvh$, the sets $\iii$ and $\iiis$, the operators $\Dv$, $\Dvs$, $v$ and $\vs$, the discretized Maxwellians $\muh$ and $\mus$ (see [*e.g.*]{} Definition \[def:dvb\]). For these operators, the space index $j$ plays the role of a parameter.
Concerning the discretization of the space periodic domain $\T$, we pick from Section \[sec:inhomsd\] the definitions and notations. We denote $\dxh >0$ the (uniform) step of discretization of the torus $\T$ into $N$ intervals, and denote $\jjj = \Z/ N\Z$ the finite set of indices of the discretization in $x \in \T$. In what follows, the index $i \in \iii$ will always refer to the velocity variable and the index $j \in \jjj$ to the space variable. In particular, for a sequence $f = (f_{i,j})_{i\in \Z, j\in \jjj}$ the derivative-in-space $\Dx f$ is then defined by the following centered scheme $$\forall i \in \iii, j\in \jjj, \qquad (\Dx f)_{j,i} = \frac{f_{j+1,i} - f_{j-1,i}}{2\dxh}.$$
Our goal is to introduce a [*discrete*]{} functional framework that allows to conclude to qualitatively correct asymptotic behaviour for the numerical schemes in Theorem \[thm:decrexpeulerexpldiscr\], by mimicking the proofs of the results recalled in Section \[sec:inhomcb\] for the [*continuous*]{} inhomogeneous equation on bounded velocity domain. Before introducing the time-discretization, we equip $\R^{\jjj\times\iii}$ with the $\ell^1(\jjj\times\iii, \muh \dvh \dxh)$ norm and we introduce adapted Hilbertian norms.
We denote by $\Ldeuxmudvdxh$ the space $\R^{\jjj\times\iii}$ made of finite sequences $g$ and set $$\norm{ g}_{\Ldeuxmudvdxh }^2 \defegal \dvh \dxh \sum_{j\in \jjj, i\in \iii} (g_{j,i})^2 \muh_i .$$ This defines a squared Hilbertian norm, and the related scalar product will be denoted by $\seq{ \cdot, \cdot}$. For $g \in \Ldeuxmudvdxh$, we also define the mean $$\seq{g} \defegal \dvh \dxh\sum_{j\in \jjj, i\in \iii} g_{j,i} \muh_i = \seq{g, 1}
,$$ of $g$ (with respect to this weighted scalar product in both velocity and space). We define the space $\Ldeuxmudvdxs$ to be $\R^{\jjj\times\iiis}$ endowed with the Hilbertian norm defined for $h\in \R^{\jjj\times\iiis}$ by its square $$\norm{ h}_{\Ldeuxmudvdxs }^2 \defegal \dvh \dxh \sum_{j\in \jjj, i\in \iiis} (h_{j,i})^2 \mus_i .$$ The related scalar product will be denoted by $\seqs{ \cdot, \cdot}$. Eventually we define $\Hunmudvdxh$ to be the space $\Ldeuxmudvdxh=\R^{\jjj\times\iii}$ with the Hilbertian norm defined by its square for $g\in\R^{\iii\times\jjj}$ as $$\norm{g}^2_\Hunmudvdxh \defegal \norm{g}^2_\Ldeuxmudvdxh + \norm{\Dv g}^2_\Ldeuxmudvdxs +
\norm{\Dx g}^2_\Ldeuxmudvdxh.$$
We define the operator $\Pd$ involved in the discretized rescaled Fokker–Planck equation by $$\Pd = \Xzerod + (-\Dvs+\vs)\Dv$$ with $\Xzerod = v \Dx : \Ldeuxmudvdxh \hookrightarrow \Ldeuxmudvdxh$ defined by for $i\in \iii$ by $$( \Xzerod g)_{j,i} = (v \Dx g)_{i,j} \textrm{ when } i\neq 0, \qquad ( \Xzerod g)_{j,0} =0.$$
The discretized version of the rescaled equation is therefore the linear ODE set in $\R^{\jjj\times\iii}$ that reads $$\label{eq:inhomocontinuoustimediscreteFPbounded}
{\D_t} f+ \Pd f = 0.$$ We now summarize the structural properties of \[eq:inhomocontinuoustimediscreteFPbounded\] and of the operator $\Pd$ in the following Proposition. From now on and for the rest of this subsection, we work in $\Ldeuxmudvdxh$ and denote (when no ambiguity happens) the corresponding norm $\norm{\cdot}$ without subscript. Similarly $\norms{\cdot}$ stands for the norm in $\Ldeuxmudvdxs$.
We have
1. The operator $( - \Dvs +\vs) \Dv$ is self-adjoint and the operator $\Xzerod$ is skew-adjoint in $\Ldeuxmudvdxh$. Moreover, for all $g \in \Ldeuxmudvdxh$, $h \in \Ldeuxmudvdxs $, we have $$\begin{aligned}
\seq{ ( - \Dvs +\vs) h, g} = \seq{h, \Dv g}_\sharp, \\
\label{eq:ofinb2}
\seq{\Pd g,g} = \seq{ ( - \Dvs +\vs) \Dv g, g} = \norm{\Dv g}_\sharp^2.\end{aligned}$$
2. Constant functions are the only equilibrium states of equation and we have the conservation of mass property : for all $t\geq 0$, $\seq{f(t)} = \seq{f^0}$.
We pick from Section \[sec:inhomsd\] the definitions of the operators $S$, $S^\sharp$ and $S^\flat$ as well as the results and embeddings given in Lemmas \[lem:constcontS\] and \[lem:constcontSsharp\] with the velocity set of index $\Z$ or $\Z^*$ there replaced here by $\iii$ or $\iiis$ respectively. Note that the spaces $\Ldeuxmudvdxh$ and $\Ldeuxmudvdxs$ here are exactly adapted to the inherent shift defining $S$, $S^\sharp$ and $S^\flat$. Moreover, it is clear that the commutations lemmas \[lem:triple\], \[lem:sb\] and \[lem:estimdelta\] remain true thanks to our choice of indices $\iii$, $\iiis$ and the functional associated spaces of the current section.
We pick from the same section \[sec:inhomsd\] the definition of the following modified entropy defined for $g\in \Hunmudvdxh$ by $$\hhhd(g) = C\norm{g}^2+D\norms{\Dv g}^2+E\seqs{\Dv g,S\Dx g}+\norm{\Dx g}^2,$$ for well chosen $C>D>E>1$ to be defined later. Lemma \[lem:equivd\] remains true in the bounded-velocity discretized context this section and we have again with the same proof as there.
If $2E^2<D$ then for all $g\in \Hunmudvdxh$, $$\label{eq:equivdb}
\dfrac{1}{2}\norm{g}_{\Hunmudvdxh}^2\leq\hhhd(g)\leq 2C\norm{g}_{\Hunmudvdxh}^2.$$
Provided that $2E^2<D$, the modified entropy $\hhhd$ defines a Hilbertian norm on $\R^{\jjj\times\iii}$, associated with the following polar form $$\phid(g,\gtilde)=C\seq{g,\gtilde}+D\seq{\Dv g,\Dv \gtilde}_\sharp
+ \frac{E}{2}\left(\seq{S\Dx g,\Dv \gtilde}_\sharp+\seq{\Dv g,S \Dx \gtilde}_\sharp\right)
+ \seq{\Dx g,\Dx \gtilde},$$ defined for $g$, $\gtilde \in \R^{\jjj\times\iii}$. The Cauchy–Schwarz–Young inequality holds true $$\label{eq:CSYbounded}
|\phid(g,\gtilde)| \leq \sqrt{\hhhd(g)}\sqrt{\hhhd(\gtilde)}
\leq \frac{1}{2} \hhhd(g) + \frac{1}{2} \hhhd(\gtilde),$$ for all $g,\tilde g$. Moreover, the Poincaré inequality in space holds true as well. First, in the form of in the discretized space variable, and then, following exactly the lines of the proof of Lemma \[lem:fullPoincarediscret\], in the form of the following Lemma.
\[lem:fullPoincarediscretb\] For all $g \in \Hunmudvdxh$, we have $$\norm{g-\seq{g}}^2_\Ldeuxmudvdxh \leq \norm{\Dv g}_\Ldeuxmudvdxs^2 + \norm{\Dx g}_\Ldeuxmudvdxh^2.$$
The discretization in time of the rescaled inhomogeneous discretized Fokker–Planck equation that we consider is given by the following explicit scheme
We shall say that a sequence $f = (f^n)_{n \in \N} \in (\Ldeuxmudvdxh)^\N$ satisfies the scaled fully discrete explicit inhomogeneous Fokker-Planck equation if for some $\dth>0$ and all $n\in\N$, $$\label{eq:eulerexpldiscrinhomo}
f^{n+1} = f^n - \dth (v \Dx f^{n}+(-\Dvs+\vs) \Dv f^{n}).$$
As in all the previous cases, we can check that constant sequences are the only equilibrium states of this equation, and that the mass conservation property is satisfied: for all $n\in\N$, $\seq{f^n} = \seq{f^0}$.
Before getting to the main result of this section in Theorem \[thm:decrexpeulerexpldiscr\], we state the following Lemma, which provides us with explicit bounds on the norms of the linear continuous operators in the discrete equation .
\[lem:CFLinhomo\] Let us define $$\label{eq:defabc}
\quad a^2 = 4 \frac{1+ \dvh \vmax}{\dvh^2},
\qquad b^2 = 4 \frac{1+ \dvh \vmax}{\dxh^2},
\qquad c^2 = 4\frac{\vmax^2}{\dxh^2},$$ and set $$\bcfl = \max \set{ 1, a^2,b^2,c^2}.$$ Then we have for all $g \in \Ldeuxmudvdxh$ and $h \in \Ldeuxmudvdxs$ $$\label{eq:estimnorms}
\begin{split}
\norms{\Dv g} \leq a \norm{g}, \qquad \norms{S \Dx g} & \leq b \norm{g},
\qquad \norm{ \Dx g} \leq b \norm{g}, \\
\norm{\Xzerod g} \leq c \norm{g}, \qquad & \norms{\Xzerod h} \leq c \norms{h}.
\end{split}$$
Let us first prove now . We first note that the inequality is already proven in . The proof of the second one follows exactly the same proof. For the third one, we directly have by triangular inequality that $$\norm{ \Dx g} \leq \frac{2}{\dxh} \norm{g} \leq b \norm{g}.$$ For the inequalities involving $\Xzerod$, we just note that operator multiplication by $v$ is bounded with bound $\vmax$ and use the bound for $\Dx$ above, which yields directly the result.
We can now state the main Theorem of this subsection concerning the exponential trend to equilibrium of solutions of Equation .
\[thm:decrexpeulerexpldiscr\] Assume $C>D>E>1$ and $\dvh_0\in(0,1)$ are chosen as in Theorem \[thm:decrexpd\] and set $$\bcfl = \max \set{ 1, 4 \frac{1+ \dvh \vmax}{\dvh^2}, 4 \frac{1+ \dvh
\vmax}{\dxh^2},4\frac{\vmax^2}{\dxh^2}}.$$ For all $\dvh\in (0,\dvh_0)$, $\dxh>0$, $f^0 \in \Hunmudvdxh$ such that $\seq{f^0} = 0$, and $\dth>0$ satisfying the CFL condition $$\label{eq:condcflinhom}
4 (C+ 4D + 9E +2) \dth \bcfl (1+\vmax^2)< 1,$$ the solution $(f^n)_{n\in\N}$ of the discretized inhomogeneous Fokker–Planck equation in $ (\Hunmudvdxh)^\N$ with initial data $f^0$ satisfies $$\forall n\in\N,\qquad \hhhd(f^n) \leq (1-2\kappa \dth)^n \hhhd(f^0),$$ where $\kappa>0$ is such that $4C\kappa = 1-4(C+ 4D + 9E +2) (1+\vmax^2)\dth \bcfl$.
Fix $\dvh\in(0,\dvh_0)$, $\dxh>0$ and $\dth>0$ as in the hypotheses. Let $f^0\in \Hunmudvdxh$ with zero mean. Denote by $(f^n)_{n\in\N}$ the sequence in $\R^{\jjj\times\iii}$ provided by the explicit Euler scheme for which we recall that $n\in\N$, $\seq{f^n}=0$. We fix $n\in\N$ and as in the proof of Theorem \[thm:decrexpeulerimpldiscr\], we compute the four terms appearing in the definition of $\hhhd(f^{n+1})$ before estimating their sum. For this, we extensively use the computations done there and in the proof of Theorem \[thm:mainexplhom\]. Our method is the following : bound every term in $\hhhd(f^{n+1})$ by a sum of three terms of order $0$, $1$ and $2$ in $\dth$. Then, sum up the inequalities after multiplication by $C$, $D$, $E$, and $1$. Recognize $\dddd(f^n)$ in the sum of terms of order $1$, then transform the sum of the terms of order $2$ into a of order $1$ using the CFL condition that can be integrated in the preceding term of order $1$ thanks to a version of adapted to this bounded velocity context. Eventually, conclude using the Cauchy–Schwarz–Young inequality .
First, we compute the squared $\Ldeuxmudvdxh$-norm of $f^{n+1}$ using relation twice. This yields
$$\begin{aligned}
\lefteqn{\norm{f^{n+1}}^2}\nonumber\\
& = & \seq{f^n,f^{n+1}} -\dth \seq{\Pd f^n,f^{n+1}}\nonumber\\
& = & \seq{f^n,f^{n+1}} -\dth \seq{\Pd f^n,f^{n}}+\dth^2\seq{\Pd f^n,\Pd f^n}\nonumber\\
& = & \seq{f^n,f^{n+1}} - \dth\norm{\Dv f^n}^2_\sharp
+\dth^2\rrrd_1(f^n),
\label{eq:ineqbi}
\end{aligned}$$
using for the term in $\dth$ and defining $$\rrrd_1(f^n)=\norm{\Pd f^n}^2,$$ for the term in $\dth^2$.
For the second term in the definition of the discrete entropy $\hhhd$, we compute the squared $\Ldeuxmudvdxs$-norm of $\Dv f^{n+1}$ using relation twice. This yields $$\label{eq:ineqbii}
\begin{split}
\lefteqn{\norm{\Dv f^{n+1}}^2_\sharp }\\
& = \seq{\Dv f^n,\Dv f^{n+1}}_\sharp-
\dth\seq{\Dv v\Dx f^n,\Dv f^{n+1}}_\sharp - \dth\seq{\Dv
(-\Dvs+\vs)\Dv f^n,\Dv f^{n+1}}_\sharp\\
& = \seq{\Dv f^n,\Dv f^{n+1}}_\sharp-\dth\seq{S\Dx f^n,\Dv f^{n+1}}_\sharp
-\dth\seq{v\Dv\Dx f^n,\Dv f^{n+1}}_\sharp\\
& - \dth \seq{(-\Dvs+\vs)\Dv f^{n},(-\Dvs+\vs)\Dv f^{n+1}}\\
& = \seq{\Dv f^n,\Dv f^{n+1}}_\sharp-\dth\seq{S\Dx f^n,\Dv f^{n}}_\sharp
+\dth^2 \seq{S\Dx f^n,\Dv \Pd f^{n}}_\sharp\\
& -\dth\underbrace{\seq{v\Dx\Dv f^n,\Dv f^{n}}_\sharp}_{=0}
+\dth^2\seq{v\Dv\Dx f^n,\Dv \Pd f^{n}}_\sharp\\
& - \dth \seq{(-\Dvs+\vs)\Dv f^{n},(-\Dvs+\vs)\Dv f^{n}}\\
&
+ \dth^2 \seq{(-\Dvs+\vs)\Dv f^{n},(-\Dvs+\vs)\Dv \Pd f^{n}}\\
& = \seq{\Dv f^n,\Dv f^{n+1}}_\sharp
-\dth\left(
\seq{S\Dx f^n,\Dv f^{n}}_\sharp
+ \norm{(-\Dvs+\vs)\Dv f^n}^2
\right)+\dth^2 \rrrd_2(f^n),
\end{split}$$ where we have set $$\begin{aligned}
\rrrd_2(f^n) &= &
\seq{S\Dx f^n,\Dv \Pd f^{n}}_\sharp
+
\seq{v\Dv\Dx f^n,\Dv \Pd f^{n}}_\sharp\\
& & +
\seq{(-\Dvs+\vs)\Dv f^{n},(-\Dvs+\vs)\Dv \Pd f^{n}}.\end{aligned}$$
For the third term in $\hhhd(f^{n+1})$, we take advantage of the computations carried out in Section \[sec:eqinhomo\] for the unbounded in velocity, inhomogeneous, semi-discretized and implicit case. In particular, we have as in the following relation (with $f^n$ here instead of $f^{n+1}$ there in the right-hand side), by using the definition of the explicit Euler scheme twice $$\begin{aligned}
\lefteqn{2\seq{S\Dx f^{n+1},\Dv f^{n+1}}_\sharp = }\\
& & \seq{S\Dx f^{n},\Dv f^{n+1}}_\sharp + \seq{S\Dx f^{n+1},\Dv f^{n}}_\sharp\\
& & -\dth\left(
\seq{S\Dx v \Dx f^{n},\Dv f^{n+1}}_\sharp + \seq{S\Dx f^{n+1},\Dv v \Dx f^{n}}_\sharp
\right)\\
& & -\dth\left(
\seq{S\Dx(-\Dvs+\vs)\Dv f^{n},\Dv f^{n+1}}_\sharp
+ \seq{S\Dx f^{n+1},\Dv(-\Dvs+\vs)\Dv f^{n}}
\right).\end{aligned}$$
Using again Equation to replace $f^{n+1}$ in the terms in $\dth$ above, we get $$\label{eq:termdt}
\begin{split}
\lefteqn{ 2\seq{S\Dx f^{n+1},\Dv f^{n+1}}_\sharp=}\\
& \seq{S\Dx f^{n},\Dv f^{n+1}}_\sharp + \seq{S\Dx f^{n+1},\Dv f^{n}}_\sharp\\
& -\dth\left(
\seq{S\Dx v \Dx f^{n},\Dv f^{n}}_\sharp + \seq{S\Dx f^{n},\Dv v \Dx f^{n}}_\sharp
\right)\\
& -\dth\left(
\seq{S\Dx(-\Dvs+\vs)\Dv f^{n},\Dv f^{n}}_\sharp
+ \seq{S\Dx f^{n},\Dv(-\Dvs+\vs)\Dv f^{n}}_\sharp \right)\\
& + \dth^2 \rrrd(f^n),
\end{split}$$ where $\rrrd_3(f^n) $ is given by $$\begin{split}
\rrrd_3(f^n) = & \seq{S\Dx \Xzerod f^{n},\Dv (\Xzerod+ (-\Dvs+\vs)\Dv)) f^{n}}_\sharp
\\
& + \seq{S\Dx (\Xzerod+ (-\Dvs+\vs)\Dv)) f^{n},\Dv \Xzerod f^{n}}_\sharp \\
& +
\seq{S\Dx(-\Dvs+\vs)\Dv f^{n},\Dv (\Xzerod+ (-\Dvs+\vs)\Dv)) f^{n}}_\sharp \\
& + \seq{S\Dx (\Xzerod+ (-\Dvs+\vs)\Dv)) f^{n},\Dv(-\Dvs+\vs)\Dv f^{n}}_\sharp.
\end{split}$$
The two terms in $\dth$ in can be computed just as terms $(I)$ and $(II)$ in the proof of Theorem \[thm:decrexpd\] (with $f$ there replaced by $f^{n}$ here ) and we obtain $$\label{eq:ineqbiii}
\begin{split}
\lefteqn{2\seq{S\Dx f^{n+1},\Dv f^{n+1}}_\sharp =}\\
& \seq{S\Dx f^{n},\Dv f^{n+1}}_\sharp + \seq{S\Dx f^{n+1},\Dv f^{n}}_\sharp\\
& -\dth\left(
\norm{S\Dx f^{n}}_\sharp^2 - \dvh\seq{S^b \Dx f^{n},\Dx \Dv f^{n}}_\sharp
\right)\\
& + 2 \dth
\seq{(-\Dvs+v)\Dv f^{n},S^\sharp \Dx \Dv f^{n}} \\
& -\dth\left( \seq{S\Dx f^{n},\Dv f^{n}}_\sharp+\seq{\sigma \Dx f^{n},\Dv f^{n}}_\sharp
\right) + \dth^2 \rrrd_3(f^n),
\end{split}$$ where we used adapted versions of Lemmas \[lem:triple\] and \[lem:sb\].
Since $\Dx$ commutes with itself and with $(-\Dvs+\vs)\Dv$, the sequence $(\Dx f^n)_{n\in\N}$ also solves the recursion relation . Adapting our the computation that led to above, we infer that the last term in $\hhhd(f^{n+1})$ satisfies $$\begin{aligned}
\lefteqn{\norm{\Dx f^{n+1}}^2}\nonumber\\
& \leq & \seq{\Dx f^n,\Dx f^{n+1}} - \dth\norm{\Dv \Dx f^n}^2_\sharp
+\dth^2\rrrd_1(\Dx f^n).
\label{eq:ineqbiiii}
\end{aligned}$$
Summing up the four identities , , and , multiplied respectively by $C$, $D$, $E/2$ and $1$, we infer that $$\label{eq:bilanentropy}
\begin{split}
& \hhhd(f^{n+1}) = \phid(f^n,f^{n+1})\\
& -\dth\left[ C \norm{\Dv f^{n}}_\sharp^2 + D \seq{S\Dx f^{n},\Dv
f^{n}}_\sharp + D \norm{(-\Dvs+v)\Dv f^{n}}^2
+\frac{E}{2} \norm{S\Dx f^{n}}_\sharp^2 \right.\\
& \qquad \left. - \frac{E}{2} \dvh\seq{S^b \Dx f^{n},\Dx \Dv f^{n}}_\sharp
- E \seq{(-\Dvs+v)\Dv f^{n},S^\sharp \Dx \Dv f^{n}}\right.\\
& \qquad \left. +\frac{E}{2} \seq{S\Dx f^{n},\Dv f^{n}}_\sharp
+\frac{E}{2}\seq{\sigma \Dx f^{n},\Dv f^{n}}_\sharp +\norm{\Dx \Dv
f^{n}}_\sharp^2
\right] \\
& \qquad + \dth^2 \left(C\rrrd_1(f^n)+ D\rrrd_2(f^n) + \frac{E}{2}\rrrd_3(f^n)
+\rrrd_1(\Dx f^n)\right).
\end{split}$$
We recognize here in square brackets in the same term as the one defining $\dddd(f)$ in with $f^{n}$ here instead of $f$ there, and in our bounded velocity context. It remains to show how to handle the terms in $\dth^2$ in using the CFL condition . To do so, we set for all $g\in\Ldeuxmudvdxh$ $$M(g) = \norm{g}^2_\Hunmudvdxh + \norm{(-\Dvs+\vs)\Dv g}^2 + \norm{\Dv\Dx g}_\sharp^2.$$ Note that, in view of relation adapted to our bounded velocity setting and of the Poincaré inequality of Lemma \[lem:fullPoincarediscretb\], we have for all $g$ with zero mean $$\label{eq:propMD}
M(g) \leq 2 \dddd(g).$$ For the rest of the proof, we use the constants $a$, $b$ and $c$ defined in in Lemma . For the term in , we have $$\label{eq:estimR1}
\begin{split}
|\rrrd_1(f^n)| & \leq 2 (\norm{\Xzerod f^n}^2+\norm{(-\Dvs+\vs)\Dv f^n}^2)\\
& \leq 2 \left(c^2\norm{f^n}^2+\norm{(-\Dvs+\vs)\Dv f^n}^2\right)\\
& \leq 2\bcfl \left(\norm{f^n}^2+\norm{(-\Dvs+\vs)\Dv f^n}^2\right) \\
& \leq 2\bcfl M(f^n),
\end{split}$$ since $\bcfl$ is greater than 1. For the term in $\dth^2$ in , we have first $$\begin{aligned}
\lefteqn{\left| \seq{S\Dx f^n,\Dv \Pd f^{n}}_\sharp \right|}\\
& \leq & \frac12\left(
\norm{S\Dx f^n}_\sharp^2 + \norm{\Dv \Pd f^n}_\sharp^2
\right)\\
& \leq & \frac{b^2}{2}\norm{f^n}^2 + a^2 \left(\norm{v\Dx f^n}^2
+\norm{(-\Dvs+\vs)\Dv f^n}^2 \right)\\
& \leq & (a^2+b^2)(1+\vmax^2)
\left(\norm{f^n}^2 + \norm{\Dx f^n}^2 +\norm{(-\Dvs+\vs)\Dv f^n}^2 \right)\\
& \leq & 2\bcfl (1+\vmax^2) M(f^n).\end{aligned}$$ Second, we have $$\begin{aligned}
\lefteqn{\left|\seq{v\Dv\Dx f^n,\Dv \Pd f^{n}}_\sharp\right|}\\
& \leq & \frac12\left(
\vmax^2 \norm{\Dv\Dx f^n}_\sharp^2 + a^2 \norm{(\Xzerod + (-\Dvs+\vs)\Dv) f^n}^2
\right)\\
& \leq & \frac{\vmax^2}{2} \norm{\Dv\Dx f^n}_\sharp^2
+ a^2 \left(\norm{v\Dx f^n}^2 + \norm{(-\Dvs+\vs)\Dv f^n}^2\right)\\
& \leq & (1+a^2)(1+\vmax^2) \left(
\norm{\Dv\Dx f^n}_\sharp^2 + \norm{\Dx f^n}^2 + \norm{(-\Dvs+\vs)\Dv f^n}^2
\right)\\
& \leq & 2\bcfl (1+\vmax^2) M(f^n).\end{aligned}$$ Third, we have $$\begin{aligned}
\lefteqn{\left|\seq{(-\Dvs+\vs)\Dv f^{n},(-\Dvs+\vs)\Dv \Pd f^{n}}\right|}\\
& \leq &
\norm{(-\Dvs+\vs)\Dv f^{n}} \left(\norm{(-\Dvs+\vs)\Dv \Xzerod f^{n}}
+ \norm{(-\Dvs+\vs)\Dv (-\Dvs+\vs)\Dv f^{n}}\right)\\
& \leq &
a^2
\norm{(-\Dvs+\vs)\Dv f^{n}} \left(
\norm{\Xzerod f^{n}} + \norm{(-\Dvs+\vs)\Dv f^{n}}
\right)\\
& \leq &
a^2
\left(
\vmax \norm{(-\Dvs+\vs)\Dv f^{n}}
\norm{\Dx f^{n}}
+ \norm{(-\Dvs+\vs)\Dv f^{n}}^2
\right)\\
& \leq &
2 (1+a^2)(1+\vmax^2) M(f^n)\\
& \leq &
4 \bcfl (1+\vmax^2) M(f^n).\end{aligned}$$ In the end, we get $$\label{eq:estimR2}
\left|\rrrd_2(f^n)\right| \leq 8 \bcfl (1+\vmax^2) M(f^n).$$
Let us get now to the third remainder term $\rrrd_3(f^n)$. One has first $$\begin{aligned}
\lefteqn{\left|\seq{S\Dx \Xzerod f^n,\Dv (\Xzerod+ (-\Dvs+v)\Dv)) f^n}_\sharp\right|} \\
& = & \left| \seq{\Xzerod S\Dx f^n+ \dvh \Dx S^\flat \Dx f^n, \Xzerod \Dv f^n
+ S \Dx f^n + \Dv (-\Dvs+v)\Dv) f^n}_\sharp\right|,\end{aligned}$$ where we used that $\adf{\Dv, \Xzerod} = S \Dx$ for the second term in the scalar product and $S \Dx \Xzerod = \Xzerod S\Dx + \dvh \Dx S^\flat \Dx$ for the first one. Noting that the operator norm of $S$ is equal to the one of $S^\flat$ we therefore get that $$\begin{aligned}
\lefteqn{\left|\seq{S\Dx \Xzerod f^n,\Dv (\Xzerod+ (-\Dvs+\vs)\Dv)) f^n}_\sharp\right|} \\
& \leq & \sep{ \norms{\Xzerod S\Dx f^n}+ \dvh \norms{\Dx S^\flat \Dx f^n}}
\sep{\norms{\Xzerod \Dv f^n} + \norms{S \Dx f^n} + \norms{\Dv (-\Dvs+\vs)\Dv f^n} } \\
& \leq & \sep{ c \norms{ S\Dx f^n}+ b \norms{ S \Dx f^n}} \sep{c \norms{ \Dv
f^n}
+ \norms{S \Dx f^n} + a \norm{(-\Dvs+\vs)\Dv f^n} } \\
& \leq &(c+b)(c+1+a) \sep{\norms{ \Dv f^n}^2 + \norms{S \Dx f^n}^2
+ \norm{(-\Dvs+\vs)\Dv f^n}^2 }\\
& \leq & 12\bcfl M(f^n).\end{aligned}$$ Similarly, we get $$\begin{aligned}
\lefteqn{\left|\seq{S\Dx (\Xzerod+ (-\Dvs+v)\Dv)) f^n,\Dv \Xzerod f^n}_\sharp\right|} \\
& = & \left|\seq{\Xzerod S\Dx f^n+ \dvh \Dx S^\flat \Dx f^n + S\Dx (-\Dvs+v)\Dv
f^n,
\Xzerod \Dv f^n + S \Dx f^n} \right|\\
& \leq & \sep{ \norms{\Xzerod S\Dx f^n}+ \dvh \norms{S^\flat \Dx \Dx f^n} +
\norms{S\Dx (-\Dvs+v)\Dv f^n}}\sep{ \norms{\Xzerod \Dv f^n} + \norms{S \Dx f^n}} \\
& \leq & \sep{ c\norms{ S\Dx f^n}+ b \dvh \norms{ S \Dx f^n}
+ b \norms{ (-\Dvs+v)\Dv f^n}}\sep{ c \norms{ \Dv f^n} + \norms{S \Dx f^n}} \\
& \leq & (c + 2b)(c+1) \sep{\norms{ \Dv f^n}^2 + \norms{S \Dx f^n}^2
+ \norm{(-\Dvs+v)\Dv f^n}^2 }\\
& \leq & 12\bcfl M(f^n) .\end{aligned}$$ The same type of estimates also yields $$\begin{aligned}
\lefteqn{\left|\seq{S\Dx(-\Dvs+v)\Dv f^n,\Dv (\Xzerod+ (-\Dvs+v)\Dv))
f^n}_\sharp\right|} \\
& \leq & b ( 2b +a) \sep{\norms{ \Dv f^n}^2 + \norms{S \Dx f^n}^2
+ \norm{(-\Dvs+v)\Dv f^n}^2 }\\
& \leq & 6 \bcfl M(f^n),\end{aligned}$$ and $$\begin{aligned}
\lefteqn{\left|\seq{S\Dx (\Xzerod+ (-\Dvs+v)\Dv)) f^n,\Dv(-\Dvs+v)\Dv f^n}\right|} \\
& \leq & 3 b a \sep{\norms{ \Dv f^n}^2 + \norms{S \Dx f^n}^2
+ \norm{(-\Dvs+v)\Dv f^n}^2 }\\
& \leq & 6 \bcfl M(f^n).\end{aligned}$$ Adding the last four inequalities yields by triangle inequality $$\label{eq:estimR3}
\rrrd_3(f^n) \leq 36 \bcfl M(f^n).$$
For the last remainder term, one may write $$\begin{aligned}
\label{eq:estimR4}
|R_1(\Dx f^n)| & \leq & 2 (\norm{\Xzerod \Dx f^n}^2+\norm{(-\Dvs+\vs)\Dv \Dx f^n}^2)\\
& \leq & 2 \left(c^2\norm{\Dx f^n}^2+a^2\norm{\Dx f^n}^2\right)\\
& \leq & 2\bcfl \left(\norm{\Dx f^n}^2+\norm{\Dx f^n}^2\right) \\
& \leq & 4\bcfl M(f^n),\end{aligned}$$
From , , and , we infer that the term in $\dth^2$ in can be bounded as follows: $$\begin{aligned}
\lefteqn{
\left|C\rrrd_1(f^n)+ D\rrrd_2(f^n) + \frac{E}{2}\rrrd_3(f^n) +\rrrd_1(\Dx f^n)
\right|
}\\
&\leq & \bcfl (1+\vmax^2)\left(2C + 8D+ 18E + 4\right) M(f^n).\end{aligned}$$ In view of , since $f^n$ has zero mean, we infer that $$\begin{aligned}
\lefteqn{
\left|C\rrrd_1(f^n)+ D\rrrd_2(f^n) + \frac{E}{2}\rrrd_3(f^n) +\rrrd_1(\Dx f^n)
\right|
}\\
&\leq & 4\bcfl (1+\vmax^2)\left(C + 4D+ 9E + 2\right) \ddd(f^n).\end{aligned}$$
Using the inequality above, we rewrite in the form $$\hhhd(f^{n+1}) \leq \phid(f^n,f^{n+1}) -\dth \left(1-
\dth 4 (C+ 4D + 9E +2) \bcfl (1+\vmax^2)\right)\dddd(f^{n}).$$
Using the CFL condition and the definition of $\kappa$ in the statement of Theorem \[thm:decrexpeulerexpldiscr\], we obtain from and the last inequality that $$\hhhd(f^{n+1}) \leq \phid(f^n,f^{n+1}) - 4C \kappa \dth \dddd(f^{n}).$$ Using a version of Lemma \[lem:dissdiss\] adapted to our finite velocity context, we get that for $C$, $D$, $E$ and $\dvh_0\in(0,1)$ chosen as in -, we have $4C \dddd(f^{n}) \geq \hhh(f^{n})$ so that $$\hhhd(f^{n+1}) \leq \phid(f^n,f^{n+1}) - \kappa \dth \hhhd(f^{n}).$$
Using the fact Cauchy–Schwarz–Young inequality for $\phid$, we infer that for all $n\in\N$, $$\hhhd(f^{n+1}) \leq \frac{1}{2} \hhhd(f^{n+1}) + \frac{1}{2} \hhhd (f^n)
-\dth \kappa \hhhd(f^{n}),$$ which yields for all $n\in\N$, $$\hhhd(f^{n+1}) \leq (1-2\kappa \dth) \hhhd (f^n),$$ which implies by induction that for all $n\in\N$, $$\hhhd(f^{n}) \leq (1- 2\dth \kappa)^{n} \hhhd (f^0).$$ This concludes the proof of Theorem \[thm:decrexpeulerexpldiscr\].
As noted for the homogeneous equation in bounded velocity domain at the beginning of Section \[sec:homobounded\], the functional spaces $\Ldeuxmudvdxh$, $\Ldeuxmudvdxs$ and $\Hunmudvdxh$ associated to the discretization in space and velocity of the inhomogeneous equation are finite dimensional in this bounded velocity setting. Hence, linear operators are continuous. The next Lemma provides us with estimates on the norms of the linear differential operators at hand, that will be helpful to establish the result (Theorem \[thm:decrexpeulerexpldiscr\]) on the long time behaviour of the solutions of the explicit Euler scheme under CFL condition.
Numerical results
-----------------
[.4]{} ![Numerical simulations of Scheme with a random function as initial datum[]{data-label="fig:inhom1"}](evolfinhom_t-0_cas-\casinhom.png "fig:"){height="6cm"}
[.4]{}
[.4]{} ![Numerical simulations of Scheme with a $(x,v)$-radial function as initial datum[]{data-label="fig:inhom2"}](evolfinhom_t-0_cas-\casinhom.png "fig:"){height="6cm"}
[.4]{}
We now turn to the implementation of the forward Euler discretization of the inhomogeneous equation on a bounded domain in $v$ and a periodic domain in $x$.
In reference to the homogeneous case, we define the Fisher information as $$\gggd(g)\defegal \norm{g}^2 + \norm{{\Dv} g}^2
+ \norm{{\Dx} g}^2$$ that we know thanks to to be equivalent to $\hhhd$ and we recall that $$\fffd(g)= \norm{g}^2.$$ According to Theorem \[thm:eulerexplicite\], they are expected to decrease geometrically fast. The tests that are presented here aim at illustrating this fact in two cases:
- the initial datum is a random function in $(x,v)$, with a Gaussian envelope in $v$ (see Figure \[fig:inhom1\]-(A)). The logarithms of the entropy $\fffd$ and of the Fisher information $\gggd$ decrease linearly fast (see Figure \[fig:inhom1\]-(B)), with a rate that goes to $2$, as can be seen in Figures \[fig:inhom1\]-(C). The exponential decrease is consistent with Theorem \[thm:eulerexplicite\], and the rates are consistent with Theorem \[thm:decrexpb\] and Corollary \[cor:decrexpb\].
- the initial datum is a radial function in $(x,v)$ (see Figure \[fig:inhom2\]-(A)). The logarithms of the entropy $\fffd$ and of the Fisher information $\gggd$ decrease linearly fast (see Figure \[fig:inhom2\]-(B)), with a rate that is larger than $3$, as can be seen in Figures \[fig:inhom2\]-(C). The Fisher information also seems to decrease in a faster way than the entropy in short time.
Again, comparing the two previous test cases, we get a hint that there is a very fast regularizing effect in short time, as noted in [@PZ16]. The second initial datum is a kind of 1d test case because of its radial nature. A perspective of our work would be to investigate the change of slope at $t=1$ in Figure \[fig:inhom2\]-(B). Also, the rate seen on the right-hand side of Figures \[fig:inhom2\]-(C) is concave, whereas its behavior as shown to be convex in all three other tests. We believe it is also something worth investigating.
Generalizations and Remarks {#sec:ccl}
===========================
In Sections \[sec:homogeneous\] to \[sec:eqinhomoboundedvelocity\] we proposed several schemes conserving the basic properties of kinetic equations. Many direct generalizations are possible, and we list below some of them among other considerations concerning the proofs and results.
1. This is clear that the preceding results have their $d$-dimensional counterparts, quasi-straightforwardly in the unbounded case or even for bounded velocity (tensorized) domains. We did not give the corresponding statements in order not to hide the main features of our analysis.
2. Concerning the space variable, direct generalization are also possible, since a careful study of the proofs shows that in fact we just need the following assumptions concerning the $\Dx$ derivative:
1. $\Dx$ is (formally) skew-adjoint,
2. $\norm{\Dx \phi} \geq c_p \norm{ \phi - \seq{\phi}}$ (Poincaré inequality).
Note that in particular the full discrete Poincaré inequalities presented in Propositions \[lem:fullPoincarecontinu\], \[lem:fullPoincarediscret\] or \[lem:fullPoincarediscret\] remain true.
3. We did not show in details the maximal accretivity of the associated operators in the inhomogeneous discrete case (Subsections \[sec:inhomsd\] and \[subsec:eqinhomototalementdiscretisee\]). We just mention that the proof of the continuous case given e.g. in [@HN04 Proposition 5.5] can be easily adapted, without even the use of hypoellipticity results since we are in a discrete setting. A direct consequence of the maximal accretivity of operator $P$ with domain $D(P) \subset H$ in a is that this operator leads to a natural semi-group correctly defining the solution $F(t)$ of ${\D_t} F + P F = 0$ for initial data even in $H$. This procedure is employed many times in this article with $H = \Ldeuxmudv$, $H = \Ldeuxmudvdx$, $H=\Hunmudv$, $H=\Hunmudvdx$ etc... and their discrete counterparts (both in the unbounded or bounded velocity setting).
4. In this paper, we presented a $H^1$ approach (and not an $L^2$ one, except in the homogeneous case). Indeed this allows to work only with local operators and their finite differences counterparts leading to low numerical cost. This could be interesting to see how to extend the result to the $L^2$ framework. Anyway, merging the results of [@PZ16] in short time (to be adapted to our schemes) and the results would give indeed the full convergence to the equilibrium in $L^2$ for inhomogeneous models.
5. We did not focus on the preservation of the non-negativity of the numerical solutions by the schemes we introduced. However, this preservation is straightforward at least in the homogeneous case, for the explicit methods (convexity arguments) as well as for implicit methods (monotonicity arguments).
6. We did not also prove in details to what extend the Neumann problems of Sections \[sec:homobounded\] and \[sec:eqinhomoboundedvelocity\] are good approximations of the the unbounded ones presented in Sections \[sec:homogeneous\] and \[sec:eqinhomo\]. This kind of considerations is standard in semi-classical analysis and could be done using resolvent identity type procedures, as is done e.g. in the study of the tunnelling effect e.g. in [@DS99].
7. As a by-product of our analysis, the discrete schemes proposed in the preceding sections are naturally asymptotically stable: this is a direct consequence of the trend to the equilibrium. They also clearly are consistent by construction and therefore convergent.
As natural but not straightforward generalizations, we mention the ones below that are the subject of coming works.
We showed in this paper several Poincaré inequalities, and perhaps the first and more surprising one is the one given in Proposition \[prop:poindiscrete\]. One interesting direction is to study the corresponding log-Sobolev inequality in this discrete context, and the consequences on the exponential decay using standard entropy-entropy dissipation techniques (see e.g. [@Vil09]).
In this paper we focused on the Fokker-Planck operator, and the definition of the velocity derivatives takes deeply into account what corresponds to incoming and outgoing particles (corresponding to indices positive or negative in ). A natural extension would be to check how this can be extended to the Landau collision kernel case, which also involves derivatives, in order to keep positivity and self-adjointness properties. In fact it could be also interesting to look at the current two-direction method also for other collision kernels such as linearized Boltzmann or BGK ones.
Appendix : Commutation identities {#appendix-commutation-identities .unnumbered}
=================================
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[^1]: G.D. is supported by the Inria project-team MEPHYSTO and the Labex CEMPI (ANR-11-LABX-0007-01). F.H. is supported by the grant “NOSEVOL” ANR-2011-BS01019-01.
[^2]: [*ie*]{} involving the space variable $x$ and the velocity variable $v$
[^3]: [*ie*]{} involving the variable $v$ but not the variable $x$
[^4]: Note that, in these definitions, the range of indices of the image $\Dv G$ is $\Z^*$ and not $\Z$, in order to keep into account the natural shift induced by the “two-direction” definition of $\Dv$.
[^5]: We emphasize the fact that there is no mistake in the denominator of $(\Dvs G)_0$.
[^6]: Once again, there is no typo in the formula defining $(\Dvs H)_0$.
[^7]: Note anyway that the explicit Euler scheme $$f^{n+1} = f^n -\dth ( - \Dvs +\vs) \Dv f^n ,$$ is not well posed due to the fact that the discretized operator $( - \Dvs +\vs) \Dv $ is not bounded.
|
---
abstract: 'A formula for the structure constants of the multiplication of Schubert classes is obtained in [@GK19]. In this note, we prove analogous formulae for the Chern–Schwartz–MacPherson (CSM) classes and Segre–Schwartz–MacPherson (SSM) classes of Schubert cells in the flag variety. By the equivalence between the CSM classes and the stable basis elements for the cotangent bundle of the flag variety, a formula for the structure constants for the latter is also deduced.'
address: 'Department of Mathematics, University of Toronto, Toronto, ON, Canada'
author:
- Changjian Su
title: Structure constants for Chern classes of Schubert cells
---
Introduction
============
In the equivariant cohomology of a flag variety, there is a natural basis given by the fundamental classes of Schubert varieties. It is well known that the structure constants of multiplication of this basis (or its dual basis) enjoy a positivity property [@G01]. This also holds in the equivariant K theory of the flag varieties [@B02; @AGM11; @K17].
However, a manifestly positive formula for the structure constants (and its equivariant K theory analogue) is only completely known for the Grassmannians and 2-step partial flag variety, see [@KT03; @Bu02; @BKPT16; @KZJ17]. In certain special cases, these structure constants are just the localizations of the basis elements, which are given by positive formulae [@B99; @AJS94]. Recently, a manifestly polynomial formula is found by Goldin and Knutson [@GK19] both in the equivariant cohomology and equivariant K theory of the complete flag varieties.
Knutson communicated to the author that, with Zinn-Justin, they can produce puzzle formulae for 3-step and 4-step partial flag varieties [@KZJ]. Instead of considering the Schubert classes in the flag variety, they consider quotient of stable basis elements by the zero section class in the equivariant cohomology of the cotangent bundle of the partial flag variety, which has a natural $\bbC^*$ action by dilating the cotangent fibers.
The stable basis (or stable envelope) is introduced by Maulik and Okounkov [@MO19] in their work on quantum cohomology of Nakajima quiver varieties. Using the stable envelope, they constructed geometrical solutions, called R matrices, of the Yang–Baxter equations. Through the general RTT formalism [@MO19 Section 5.2], a Yangian can be constructed from these geometric R matrices. By its very definition, the Yangian acts on the equivariant cohomology of the Nakajiama varieties, generalizing earlier constructions of Varagnolo [@V00] via correspondences.
The stable basis is not only defined for Nakajima quiver varieties, it is also defined for a large class of varieties called symplectic resolutions, among which the cotangent bundle of the flag variety is the most classical example. The stable basis for the cotangent bundle is studied in [@S17; @RTV15]. By convolution [@CG10] and the result in [@L89], the graded affine Hecke algebra acts on the cohomology of the cotangent bundle of the flag variety, and it is shown in [@S17] that the stable basis elements are permuted by the Hecke operators.
The graded affine Hecke algebra also appears in the work of Aluffi and Mihalcea [@AM16] on the Chern–Schwartz–MacPherson (CSM) class of Schubert cells. The CSM class theory is a Chern class theory for singular varieties, which was constructed by MacPherson [@M74]. In the case of the flag variety, the CSM class of a Schubert cell was conjectured by Aluffi and Mihalcea [@AM09; @AM16] to be a non-negative linear combination of the Schubert classes. The Grassmannian case is proved by Huh [@H16].
Aluffi and Mihalcea show that the CSM classes of the Schubert cells are also permuted by the Hecke operators [@AM16]. Thus, the pullback of the stable basis elements to the flag variety are identified with the CSM classes of the Schubert cells, see [@AMSS17; @RV15]. Besides, the stable basis elements are related to the characteristic cycles of regular holonomic $\calD$ modules on the flag varieties, which are effective by definition. This observation is used in [@AMSS17] to prove the non-equivariant positivity conjecture of Aluffi and Mihalcea for any (partial) flag varieties. We refer interested readers to the survey papers [@O15; @O18; @SZ19; @S] for more applications of the stable envelopes to representation theory and enumerative geometry problems.
Now we have identified the numerator of the classes considered by Knutson and Zinn-Justin [@KZJ], the stable basis elements, with the CSM classes. On the other hand, if we pullback the class of the zero section in the $\bbC^*$-equivariant cohomology of the cotangent bundle, and set the $\bbC^*$-equivariant parameter to 1, we get the total Chern class of the flag variety up to a sign. Thus, the classes considered in *loc. cit.* can be identified with the quotient of the CSM classes of the Schubert cells by the total Chern class of the flag variety, which are called the Segre–Schwartz–MacPherson (SSM) classes of the Schubert cells, see [@AMSS19a]. Under the non-degenerate Poincaré pairing on the equivariant cohomology of the flag variety, the CSM classes and SSM classes are dual to each other, just as the usual Schubert classes and the opposite ones.
The main Theorem of this note is a formula for the structure constants of the SSM classes of the Schubert cells. To state it, let us introduce some notation. Let $G$ be a complex Lie group with Borel subgroup $B$ and maximal torus $T$. For any $w$ in the Weyl grop $W$, let $Y(w)^\circ:=B^-wB/B\subset G/B$ be the opposite Schubert cell in the flag variety, where $B^-$ is the opposite Borel subgroup. The SSM classes are denoted by ${{s^T_{\text{SM}}}}(Y(w)^\circ)$, see Section \[sec:Chern\]. Let $c_{u,v}^w$ be the structure constants of $\{{{s^T_{\text{SM}}}}(Y(w)^\circ)|w\in W\}$. For any simple root $\alpha$, let $\partial_\alpha$ denote the following operator on $H_T^*(\operatorname{pt})=\bbC[\ft]$: $$\partial_\alpha(f)=\frac{f-s_\alpha(f)}{\alpha},$$ where $s_\alpha(f)$ is the usual Weyl group action on $f\in H_T^*(\operatorname{pt})$. Let $T^\vee_\alpha:=\partial_\alpha+s_\alpha\in \operatorname{End}_\bbC H_T^*(\operatorname{pt})$. Extend naturally these operators to the fraction field $\operatorname{Frac}H_T^*(pt)$. The formula is
\[thm:main\] For any $u,v,w\in W$, let $Q$ be a reduced word for $w$. Then $$c_{u,v}^w=\sum_{\substack{R, S\subset Q,\\ \prod R=u,\prod S=v}}\left(\prod_{q\in Q}\frac{\alpha_q^{[q\in R\cap S]}}{1+\alpha_q}s_q(-T^\vee_q)^{[q\notin R\cup S]}\right)\cdot 1\in \operatorname{Frac}H_T^*(pt),$$ where the exponent $``[\sigma]"$ is 1 if the statement $\sigma$ is true, 0 otherwise.
This is generalized to the partial flag variety case in Theorem \[thm:Pcase\]. In the non-equivariant limit, this formula computes the topological Euler characteristic of the intersection of three Schubert cells in general positions, see Equation . By [@Sch17 Theorem 1.2], these non-equivariant limit constants also compute the non-equivariant SSM/CSM classes of Richardson cells in terms of SSM/CSM classes of the Schubert cells.
Since the CSM classes behave well under pushforward, while the SSM classes behave well under pullback (see Lemma \[lem:relations\]), the proof of Theorem \[thm:main\] can not be applied to the CSM classes directly. Nonetheless, using the relation between the CSM classes and the SSM classes, we can have a formula for the structure constants for the CSM classes of Schubert cells in the complete flag variety, see Theorem \[thm:csm\]. By the equivalence between the CSM classes and the stable basis for the cotangent bundle of the flag variety [@AMSS17; @RV15], we also get the structure constants for the stable basis, see Theorem \[thm:stablecstru\]. However, these does not generalize to the partial flag variety case.
Goldin and Knutson [@GK19] also have a formula in the equivariant K theory of the flag variety. The K-theoretic generalization of the CSM class (resp. SSM class) is the motivic Chern class (resp. Segre motivic Chern class) [@BSY], and the motivic Chern classes of the Schubert cells are also related to the K theory stable basis elements [@O17; @SZZ17; @AMSS19b; @FRW18; @SZ19]. Unfortunately, the author fails to generalize Theorem \[thm:main\] to the case of Segre motivic Chern classes, due to the fact that the Hecke operators in equivariant K theory satisfy a quadratic relation, see Remark \[rem:k\] for more details. Nonetheless, most of the results in Sections \[sec:BS\] and \[sec:proof\] can be naturally extended to the equivariant K theory.
This note is structured as follows. In Section \[sec:Chern\], we introduce the CSM/SSM classes and recall their basic properties. In Section \[sec:BS\], we focus on Bott–Samelson varieties and relate the CSM classes of Schubert cells to the CSM classes of the cells in the Bott–Samelson varieties. By this relation, the structure constants for the SSM classes of Schubert cells are certain linear combinations of the structure constants for some basis in the equivariant cohomology of the Bott–Samelson varieties, and the main theorem is reduced to Theorem \[thm:structure2\], which is proved by induction in Section \[sec:proof\]. Finally, the result is extended to the parabolic case in Section \[sec:P\].
[*Acknowledgments.*]{} The author thanks A. Knutson and A. Yong for discussions. Special thanks go to L. Mihalcea for providing the proof of Theorem \[thm:csm\].
Notation. {#notation. .unnumbered}
---------
Let $G$ be a complex Lie group, with a Borel subgroup $B$ and a maximal torus $T\subset B$. Let $\ft$ be the Lie algebra of the maximal torus. Then the equivariant cohomology of a point $H_T^*(\operatorname{pt})$ is identified with $\bbC[\ft]$. Let $R^+$ denote the roots in $B$, and $W$ be the Weyl group with Bruhat order $\leq$ and the longest element $w_0$. For any root $\alpha$, let us use $\alpha>0$ to denote $\alpha\in R^+$. For any $w\in W$, let $X(w)^\circ=BwB/B$ and $Y(w)^\circ=B^-wB/B$ be the Schubert cells in the flag variety $G/B$. Let $X(w)=\overline{X(w)^\circ}$ and $Y(w)=\overline{Y(w)^\circ}$ be the Schubert varieties.
Chern classes of Schubert cells {#sec:Chern}
===============================
In this section, we first recall the definitions of Chern–Schwartz–MacPherson (CSM) classes and Segre–Schwartz–MacPherson (SSM) classes. Then we consider the case of the flag variety, and recall basic properties of the CSM and SSM classes of the Schubert cells.
Preliminaries
-------------
Let us first recall the definition of Chern–Schwartz–MacPherson classes. For any quasi projective variety $X$ over $\bbC$, let $\calF(X)$ denote the group of constructible functions on $X$, i.e., $\calF(X)$ consists of functions $\varphi = \sum_Z c_Z {1\hskip-3.5pt1}_Z$, where the sum is over a finite set of constructible subsets $Z \subset X$, ${1\hskip-3.5pt1}_Z$ is the characteristic function of $Z$ and $c_Z \in \bbZ$ are integers. If $f:Y\rightarrow X$ is a proper morphism, we can define a pushforward $f_*:\calF(Y)\rightarrow \calF(X)$ by setting $f_*({1\hskip-3.5pt1}_Z)(p)=\chi(f^{-1}(p)\cap Z)$, where $Z\subset Y$ is a locally closed subvariety, $p\in X$, and $\chi$ is the topological Euler characteristic. According to a conjecture attributed to Deligne and Grothendieck, there is a unique natural transformation $c_*: \calF \to H_*$ from the functor $\calF$ of constructible functions on a complex quasi projective variety to the homology functor, such that if $X$ is smooth then $c_*({1\hskip-3.5pt1}_X)=c(TX)\cap [X]$, where $c(TX)$ is the total Chern class. The naturality of $c_*$ means that it commutes with proper pushforward. This conjecture was proved by MacPherson [@M74]. The class $c_*({1\hskip-3.5pt1}_X)$ for possibly singular $X$ was shown to coincide with a class defined earlier by M.-H. Schwartz [@S65a; @S65b]. For any constructible subset $Z\subset X$, we call the class $c_{SM}(Z):=c_*({1\hskip-3.5pt1}_Z)\in H_*(X)$ the *Chern–Schwartz–MacPherson* (CSM) class of $Z$ in $X$. If $X$ is smooth, we call ${{s_{\text{SM}}}}(Z):=\frac{c_*({1\hskip-3.5pt1}_Z)}{c_*({1\hskip-3.5pt1}_X)}\in H_*(X)$ the *Segre–Schwartz–MacPherson* (SSM) class of $Z$ in $X$ [^1].
The theory of CSM classes was later extended to the equivariant setting by Ohmoto [@O06]. Assume that $X$ has a $T$ action. A group $\calF^T(X)$ of [*equivariant*]{} constructible functions is defined by Ohmoto in [@O06 Section 2]. We recall the main properties that we need:
1. If $Z \subseteq X$ is a constructible set which is invariant under the $T$-action, its characteristic function ${1\hskip-3.5pt1}_Z$ is an element of $\calF^T(X)$. We will denote by $\calF_{inv}^{T}(X)$ the subgroup of $\calF^{T}(X)$ consisting of $T$-invariant constructible functions on $X$. (The group $\calF^T(X)$ also contains other elements, but this will be immaterial for us.)
2. Every proper $T$-equivariant morphism $f: Y \to X$ of algebraic varieties induces a homomorphism $f_*^T: \calF^T(X) \to \calF^T(Y)$. The restriction of $f_*^T$ to $\calF_{inv}^{T}(X)$ coincides with the ordinary push-forward $f_*$ of constructible functions. See [@O06 Section 2.6].
Ohmoto proves [@O06 Theorem 1.1] that there is an equivariant version of MacPherson transformation $$c_*^T: \calF^T(X) \to H_*^T(X)$$ that satisfies $c_*^T( {1\hskip-3.5pt1}_X) = c^T(TX) \cap [X]_T$ if $X$ is a non-singular variety, and that is functorial with respect to proper push-forwards. The last statement means that for all proper $T$-equivariant morphisms $Y\to X$ the following diagram commutes: $$\xymatrix{
\calF^T(Y) \ar[r]^{c_*^T} \ar[d]_{f_*^T} & H_*^T(Y) \ar[d]^{f_*^T} \\
\calF^T(X) \ar[r]^{c_*^T} & H_*^T(X)
}$$
Let $Z$ be a $T$-invariant constructible subset of $X$. We denote by ${{c^T_{\text{SM}}}}(Z):=c_*^T({1\hskip-3.5pt1}_{Z}) {~\in H_*^T(X)}$ the $T$-[*equivariant Chern–Schwartz–MacPherson (CSM) class*]{} of $Z$. If $X$ is smooth, we denote by ${{s^T_{\text{SM}}}}(Z):=\frac{c_*^T({1\hskip-3.5pt1}_{Z})}{c^T({1\hskip-3.5pt1}_X)} {~\in H_*^T(X)}$ the $T$-[*equivariant Segre–Schwartz–MacPherson (SSM) class*]{} of $Z$.
CSM and SSM classes of Schubert cells
-------------------------------------
In this section, we recall some basic properties of the CSM and SSM classes of the Schubert cells [@AM16; @AMSS17].
### Schubert classes
The maximal torus $T$ acts on the flag variety $G/B$ by left multiplication. Since $G/B$ is smooth and projective, we identity the $T$-equivariant homology group $H_*^T(G/B)$ with the $T$-equivariant cohomolgoy $H_T^*(G/B)$, which is a module over $H_T^*(\operatorname{pt})=\bbC[\ft]$. Therefore, we regard the CSM and SSM classes in $H_T^*(G/B)$. The torus fixed points on $G/B$ are in one-to-one correspondence with the Weyl group. For any $w\in W$, $wB\in G/B$ is the corresponding fixed point. For any $\gamma\in H_T^*(G/B)$, let $\gamma|_w\in H_T^*(\operatorname{pt})$ denote the restriction of $\gamma$ to the fixed point $wB$.
There is a non-degenerate Poincaré pairing $\langle-,-\rangle$ on $H_T^*(G/B)$. A natural basis for $H_T^*(G/B)$ is formed by the Schubert classes $\{[X(w)]|w\in W]\}$, with dual basis the opposite Schubert classes $\{[Y(w)]|w\in W\}$. I.e., $\langle[X(w)],[Y(u)]\rangle=\delta_{w,u}$ for any $w,u\in W$. It is well known that the structure constants for the multiplication of the basis $\{[Y(w)]|w\in W\}$ is non-negative [@G01]. However, a manifestly positive formula for the structure constants (and its equivariant K theory analogue) is only known in special cases, see [@KT03; @Bu02; @KZJ17; @B02; @B99; @AJS94]. Nevertheless, a manifestly polynomial formula is obtained in [@GK19].
### Hecke action {#sec:hecke}
For any simple root $\alpha_i$, let $P_i$ be the minimal parabolic subgroup containing the Bore subgroup $B$. Let $\pi_i:G/B\rightarrow G/P_i$ denote the projection. Then the divided difference operator is $\partial_i:=\pi_i^*\pi_{i*}\in \operatorname{End}_{H_T^*(\operatorname{pt})}H^*_T(G/B)$. On $H^*_T(G/B)$, we also have a right Weyl group action induced by the fibration $G/T\rightarrow G/B$ and the right Weyl group action on $G/T$. For any torus weight $\lambda$, let $\calL_\lambda:=G\times_B\bbC_\lambda\in \operatorname{Pic}_T(G/B)$. Then as operators on $H^*_T(G/B)$, we have [@BGG73] $$s_i=\operatorname{id}+c_1^T(\calL_{\alpha_i})\partial_i.$$ Following [@AM16; @AMSS17], define [^2] $$\calT_i:=\partial_i-s_i, \textit{\quad and \quad} \calT^\vee_i:=\partial_i+s_i.$$ Then they are adjoint to each other, see [@AMSS17 Lemma 5.2]. I.e., for any $\gamma_1,\gamma_2\in H_T^*(G/B)$, $\langle\calT_i(\gamma_1),\gamma_2\rangle=\langle\gamma_1,\calT^\vee_i(\gamma_2)\rangle$. Besides, it is proved in [@AM16 Proposition 4.1] that the $\calT_i$’s satisfy the usual relation in the Weyl group. By adjointness and nondegeneracy of the pairing, the $\calT^\vee_i$’s also satisfy the same relations. Thus, for any $w\in W$, we can form $\calT_w$ and $\calT^\vee_w$. These operators $\calT_w$’s (or $\calT^\vee_w$’s) together with the multiplication by the first Chern classes $c_1^T(\calL_\lambda)$ give an action of the degenerate affine Hecke algebra on $H^*_T(G/B)$.
### CSM and SSM classes {#sec:CSMSSM}
For any $w\in W$, the CSM class of the Schubert cell $X(w)^\circ$ is described very easily by the Hecke operators as follows ([@AM16 Corollary 4.2]) $$\label{equ:csmhecke}
{{c^T_{\text{SM}}}}(X(w)^\circ)=\calT_{w^{-1}}([X(id)]),$$ where $[X(id)]$ is the point class $[B]\in H_T^*(G/B)$. Expand the CSM classes in the Schubert classes $${{c^T_{\text{SM}}}}(X(w)^\circ)=\sum_{u\leq w}c^T(u;w)[X(u)]\in H_T^*(G/B),$$ where $c^T(u;w)\in H_T^*(\operatorname{pt})$. The leading coefficient is $c^T(w;w)=\prod_{\alpha>0,w\alpha<0}(1-w\alpha)$, see [@AM16 Proposition 6.5]. Therefore, the transition matrix between the CSM classes of the Schubert cells and the Schubert varieties are triangular with non-zero diagonals. Thus, $\{{{c^T_{\text{SM}}}}(X(w)^\circ)|w\in W\}$ is a basis for the localized equivariant cohomology $H_T^*(G/B)_{\operatorname{loc}}:=H_T^*(G/B)\otimes_{H_T^*(\operatorname{pt})}\operatorname{Frac}H_T^*(\operatorname{pt})$, where $\operatorname{Frac}H_T^*(\operatorname{pt})$ is the fraction field of $H_T^*(\operatorname{pt})$. Since the non-equivariant limit of the leading coefficients $c(w;w)=1$, the non-equivariant CSM classes of Schubert cells $\{{{c_{\text{SM}}}}(X(w)^\circ)|w\in W\}$ forms a basis for the cohomolgoy $H^*(G/B)$.
The CSM classes of the Schubert cells can be identified with the Maulik and Okounkov’s stable basis elements (see Section \[sec:stable\]) for the cotangent bundle of the flag variety, which are related to representation of the Lie algebra of $G$, see [@MO19; @S17; @AMSS17; @SZ19]. This type of relation plays an important role in the proof of the non-equivariant case of the positivity conjecture of Aluffi–Mihalcea. I.e., it is conjectured in [@AM16] that $$c^T(u;w)\in \bbZ_{\geq 0}[\alpha|\alpha>0].$$ The non-equivariant case is proved in [@AMSS17].
Recall the SSM class of a Schubert cell $Y(w)^\circ$ is defined by $${{s^T_{\text{SM}}}}(Y(w)^\circ):=\frac{{{c^T_{\text{SM}}}}(Y(w)^\circ)}{c^T(T(G/B))}\in H_T^*(G/B)_{\operatorname{loc}}.$$ Since $c^T(T(G/B))\cup c^T(T^*(G/B))=\prod_{\alpha>0}(1-\alpha^2)\in H^*(G/B)$, see [@AMSS17 Lemma 8.1], $${{s^T_{\text{SM}}}}(Y(w)^\circ)=\frac{{{c^T_{\text{SM}}}}(Y(w)^\circ)c^T(T^*(G/B))}{\prod_{\alpha>0}(1-\alpha^2)}\in \frac{1}{\prod_{\alpha>0}(1-\alpha^2)}H_T^*(G/B).$$ It is easy to see that $\{{{s^T_{\text{SM}}}}(Y(w)^\circ)|w\in W\}$ is a basis for the localized equivariant cohomology $H_T^*(G/B)_{\operatorname{loc}}$, and in the non-equivariant cohomology, $${{s_{\text{SM}}}}(Y(w)^\circ)={{c_{\text{SM}}}}(Y(w)^\circ)\cup c(T^*(G/B))=[Y(w)]+\cdots,$$ where $\cdots$ denotes some element in $H^{> 2\ell(w)}(G/B)$. Therefore, the non equivariant SSM classes $\{{{s_{\text{SM}}}}(Y(w)^\circ)|w\in W\}$ form a basis for $H^*(G/B)$.
By [@AMSS17 Theorem 7.3] (after the specialization $\hbar=1$), we have [^3] $$\label{equ:ssmTvee}
{{s^T_{\text{SM}}}}(Y(w)^\circ)=\frac{1}{\prod_{\alpha>0} (1+\alpha)}{{c^{T,\vee}_{\text{SM}}}}(Y(w)^\circ).$$ Here ${{c^{T,\vee}_{\text{SM}}}}(Y(w)^\circ)$ is defined to be $\calT^\vee_{w^{-1}w_0}([Y(w_0)])$, where $[Y(w_0)]$ is the point class $[w_0B]\in H_T^*(G/B)$, see [@AMSS17 Definition 5.3].
The CSM classes and the SSM classes are dual to each other, i.e., for any $w,u\in W$, we have ([@AMSS17 Theorem 9.4]) $$\label{equ:duality}
\langle{{c^T_{\text{SM}}}}(X(w)^\circ), {{s^T_{\text{SM}}}}(Y(u)^\circ)\rangle=\delta_{w,u}.$$
Structure constants for the SSM classes
---------------------------------------
Define the structure constants for the multiplication of the SSM classes by the following formula $$\label{equ:structure1}
{{s^T_{\text{SM}}}}(Y(u)^\circ)\cup {{s^T_{\text{SM}}}}(Y(v)^\circ)=\sum_w c_{u,v}^w{{s^T_{\text{SM}}}}(Y(w)^\circ)\in H_T^*(G/B)_{\operatorname{loc}},$$ where $c_{u,v}^w\in \operatorname{Frac}H_T^*(\operatorname{pt})$. It is easy to see that $w\geq u,v$ in the above summand, and $$c_{u,w}^w={{s^T_{\text{SM}}}}(Y(u)^\circ)|_w.$$
Fix a reduced word $Q=s_{\alpha_{1}}s_{\alpha_{2}}\cdots s_{\alpha_{l}}$ for $w$. For any subword $R\subset Q$, let $\prod R$ denote the product of the simple reflections in $R$. Since a simple reflection can appear more than twice in the a $Q$, we will use the notation $s_1--$ and $--s_1$ to denote the two different subwords in $Q=s_1s_2s_1$. The localization of the CSM classes is ([@AMSS17 Corollary 6.7] with $\hbar=1$) $$\label{equ:csmlocalization}
{{c^T_{\text{SM}}}}(Y(u)^\circ)|_w=\prod_{\alpha>0,w\alpha>0}(1-w\alpha)\sum_{\substack{R\subset Q,\\\prod R=u}}\prod_{i\in R}(\prod_{j\in Q, j< i}s_j) \alpha_i.$$ Therefore, $$\label{equ:restriction}
c_{u,w}^w=\frac{{{c^T_{\text{SM}}}}(Y(u)^\circ)|_w}{c^T(T(G/B))|_w}=\frac{\sum_{\substack{R\subset Q,\\\prod R=u}}\prod_{i\in R}(\prod_{j\in Q, j< i}s_j) \alpha_i}{\prod_{\alpha>0,w\alpha<0}(1-w\alpha)}.$$
By Equation , we have $$c_{u,v}^w=\langle{{s^T_{\text{SM}}}}(Y(u)^\circ)\cup {{s^T_{\text{SM}}}}(Y(v)^\circ), {{c^T_{\text{SM}}}}(X(w)^\circ)\rangle.$$ Taking the non-equivariant limit and using [@Sch17 Theorem 1.2], we get $$\label{equ:nonequiv}
c_{u,v}^w=\chi\left(Y(u)^\circ\cap gY(v)^\circ\cap hX(w)^\circ\right),$$ where $\chi$ denotes the topological Euler characteristic, and $g,h\in G$, such that $Y(u)^\circ$, $gY(v)^\circ$ and $hX(w)^\circ$ intersect transversally. Here we also used the fact that for any constructible function $\varphi$ on $G/B$, $\int_{G/B} c_*(\varphi)=\chi(G/B,\varphi)$, which follows directly from the functoriality of the MacPherson transformation $c_*$ applied to the morphism $G/B\rightarrow \operatorname{pt}$. Besides, in the non-equivariant limit, the left hand side of Equation is the SSM class of a Richardson cell by [@Sch17 Theorem 1.2]. Thus, these constants $c_{u,v}^w$ are the expansion coefficients of the SSM (resp. CSM) classes of the Richardson celles in the SSM (resp. CSM) classes of the Schubert cells.
For any simple root $\alpha$, let $\partial_\alpha$ denote the following operator on $H_T^*(\operatorname{pt})=\bbC[\ft]$: $$\partial_\alpha(f)=\frac{f-s_\alpha(f)}{\alpha},$$ where $s_\alpha(f)$ is the usual Weyl group action on $f\in H_T^*(\operatorname{pt})$. Define $T^\vee_\alpha:=\partial_\alpha+s_\alpha\in \operatorname{End}_\bbC H_T^*(\operatorname{pt})$. Extend naturally these operators to the fraction field $\operatorname{Frac}H_T^*(pt)$.
The main theorem of this note is the following formula for the structure constants $c_{u,v}^w$.
\[thm:structure1\] For any $u,v,w\in W$, let $Q$ be a reduced word for $w$. Then $$c_{u,v}^w=\sum_{\substack{R, S\subset Q,\\ \prod R=u,\prod S=v}}\left(\prod_{q\in Q}\frac{\alpha_q^{[q\in R\cap S]}}{1+\alpha_q}s_q(-T^\vee_q)^{[q\notin R\cup S]}\right)\cdot 1,$$ where the exponent $``[\sigma]"$ is 1 if the statement $\sigma$ is true, 0 otherwise.
1. It is easy to check that when $v=w$, the above formula is the same as the one in Equation .
2. In [@GK19], the authors also obtain formulae for the K theory structure constants. The analogue of the CSM/SSM classes in K theory are the motivic Chern classes and Segre motivic Chern classes, see [@BSY; @AMSS19b]. It is interesting to generalize the above formula for the Segre motivic Chern classes.
Let $G=\operatorname{SL}(3,\bbC)$ with simple roots $\alpha_1,\alpha_2$. Let $\alpha_3:=\alpha_1+\alpha_2$ be the non-simple root. Consider the case $w=s_1s_2s_1$, $u=s_1$ and $v=s_2$. Let $Q=s_1s_2s_1$. Then $R$ can be $s_1--$ or $--s_1$, while $S=-s_2-$. The $c_{u,v}^w$ is computed as follows $$\begin{aligned}
c_{u,v}^w=&\left(\frac{1}{1+\alpha_1}s_1\frac{1}{1+\alpha_2}s_2\frac{1}{1+\alpha_1}s_1(-T^\vee_1)\right)\cdot 1\\
&+\left(\frac{1}{1+\alpha_1}s_1(-T^\vee_1)\frac{1}{1+\alpha_2}s_2\frac{1}{1+\alpha_1}s_1\right)\cdot 1\\
=&-\frac{1}{(1+\alpha_1)(1+\alpha_2)(1+\alpha_3)}\\
&-\frac{1}{(1+\alpha_1)(1+\alpha_2)(1+\alpha_3)}\\
=&-\frac{2}{(1+\alpha_1)(1+\alpha_2)(1+\alpha_3)}.\end{aligned}$$
Structure constants for the CSM classes
---------------------------------------
In this section, we give a formula for the structure constants for the CSM classes.
Define the structure constants $d_{u,v}^w$ by $$\label{equ:structure2}
{{c^T_{\text{SM}}}}(Y(u)^\circ)\cup {{c^T_{\text{SM}}}}(Y(v)^\circ)=\sum_w d_{u,v}^w{{c^T_{\text{SM}}}}(Y(w)^\circ)\in H_T^*(G/B)_{loc},$$ where $d_{u,v}^w\in \operatorname{Frac}H_T^*(\operatorname{pt})$. In particular, $d_{u,w}^w={{c^T_{\text{SM}}}}(Y(u)^\circ)|_w$.
Define a $\bbC$-linear map $\varphi:H_T^*(G/B)\rightarrow H_T^*(G/B)$ by $\varphi(\gamma)=(-1)^i\gamma$ for any $\gamma\in H^{2i}_T(G/B)$. This is an algebra automorphism of $H_T^*(G/B)$. By definition, for any degree $i$ homogeneous polynomial $f\in H_T^{2i}(\operatorname{pt})$ on $\ft$, $\varphi(f)=(-1)^if$. Extend $\varphi$ to $\operatorname{Frac}H_T^*(\operatorname{pt})$ by $\varphi(\frac{f}{g})=\frac{\varphi(f)}{\varphi(g)}$ for any $f,g\in H_T^*(\operatorname{pt})$. Thus, we can also extend $\varphi$ to $H_T^*(G/B)_{loc}$.
Then the structure constants $d_{u,v}^w$ is related to the $c_{u,v}^w$ in Equation as follows.
\[thm:csm\] For any $u,v,w\in W$, we have $$d_{u,v}^w=(-1)^{\ell(u)+\ell(v)-\ell(w)}\varphi(c_{u,v}^w)\prod_{\alpha>0}(1-\alpha)\in \operatorname{Frac}H_T^*(\operatorname{pt}).$$ In particular, in the non-equivariant case, $d_{u,v}^w=(-1)^{\ell(u)+\ell(v)-\ell(w)}c_{u,v}^w$.
Using Equations , and the equality $$\prod_{\alpha>0,w\alpha>0}(1-w\alpha)\prod_{\alpha>0,w\alpha<0}(1+w\alpha)=\prod_{\alpha>0}(1-\alpha),$$ it is easy to check directly the Theorem when $v=w$.
[^4] By [@AMSS17 Proposition 5.4] (after setting $\hbar=1$), $${{c^{T,\vee}_{\text{SM}}}}(Y(w)^\circ)=(-1)^{\ell(w)}\sum_k(-1)^{\dim G/B-k}{{c^T_{\text{SM}}}}(Y(w)^\circ)_k,$$ where ${{c^T_{\text{SM}}}}(Y(w)^\circ)_k\in H_{2k}^T(X)=H_T^{2\dim G/B-2k}(G/B)$ is the degree $2\dim G/B-2k$ component of ${{c^T_{\text{SM}}}}(Y(w)^\circ)$ defined by $${{c^T_{\text{SM}}}}(Y(w)^\circ)=\sum_k {{c^T_{\text{SM}}}}(Y(w)^\circ)_k.$$ Therefore, applying $\varphi$ to Equation , we get $$\varphi({{s^T_{\text{SM}}}}(Y(w)^\circ))=\frac{(-1)^{\ell(w)}}{\prod_{\alpha>0} (1-\alpha)}{{c^T_{\text{SM}}}}(Y(w)^\circ).$$ Applying the automorphism $\varphi$ to Equation and comparing with Equation , we get $$d_{u,v}^w=(-1)^{\ell(u)+\ell(v)-\ell(w)}\varphi(c_{u,v}^w)\prod_{\alpha>0}(1-\alpha).$$
Structure constants for the stable basis for $T^*(G/B)$ {#sec:stable}
-------------------------------------------------------
In this section, we give a formula for the structure constants for the stable basis in the cotangent bundle of the complete flag variety $G/B$.
Let us first recall the definition of the stable basis. The torus $\bbC^*$ acts on $T^*(G/B)$ by scaling the cotangent fiber by a character of $-\hbar$, and it acts trivially on the zero section $G/B$. The fixed points $(T^*(G/B))^{T\times \bbC^*}$ are in one-to-one correspondence with the Weyl group, and they all lie in the zero section. For any $w\in W$, $(wB,0)$ is the corresponding fixed point. For any $\gamma\in H_{T\times \bbC^*}^*(T^*(G/B))$, let $\gamma|_w\in H_{T\times \bbC^*}^*(\operatorname{pt})=\bbC[\ft][\hbar]$ denote the pullback of $\gamma$ to the fixed point $(wB,0)$. On $ H_{T\times \bbC^*}^*(T^*(G/B))$, there is a non-degenerate Poincaré pairing $\langle-,-\rangle_{T^*(G/B)}$ defined by localization as follows $$\langle\gamma_1,\gamma_2\rangle_{T^*(G/B)}=\sum_w\frac{\gamma_1|_w\gamma_2|_w}{\prod_{\alpha>0}(-w\alpha)(w\alpha-\hbar)}\in \operatorname{Frac}H_{T\times \bbC^*}^*(\operatorname{pt}),$$ where $\gamma_1,\gamma_2\in H_{T\times \bbC^*}^*(T^*(G/B))$. For any $w\in W$, let $T_{Y(w)^\circ}^*(G/B)$ denote the conormal bundle of the opposite Schubert cell $Y(w)^\circ$ inside $G/B$. Then the stable basis is given by
There exist unique $T\times \bbC^*$-equivariant Lagrangian cycles $\{\operatorname{stab}_-(w)\,|\, w\in W\}$ in $T^*(G/B)$ which satisfy the following properties:
1. $supp (\operatorname{stab}_-(w))\subset \bigcup_{u\geq w} \overline{T_{Y(u)^\circ}^*(G/B)}$;
2. $\operatorname{stab}_-(w)|_w= \prod\limits_{\alpha>0,w\alpha>0}(w\alpha-\hbar)\prod\limits_{\alpha>0,w\alpha<0}w\alpha$;
3. $\operatorname{stab}_-(w)|_u$ is divisible by $\hbar$, for any $u>w$ in the Bruhat order.
From the first and second properties, it follows that $\{\operatorname{stab}_-(w)\,|\, w\in W\}$ is a basis, which is called the *stable basis*, for the localized equivariant cohomology $H_{T\times \bbC^*}^*(T^*(G/B))_{\operatorname{loc}}$. Here the negative sign $-$ denotes the anti-dominant Weyl chamber. For the dominant Weyl chamber $+$, there is also a stable basis $\{\operatorname{stab}_+(w)\,|\, w\in W\}$, where $\operatorname{stab}_+(w)$ is supported on $\bigcup_{u\leq w} \overline{T_{X(u)^\circ}^*(G/B)}$. Moreover, these two bases are dual to each other, i.e., for any $w,u\in W$, $\langle\operatorname{stab}_-(w),\operatorname{stab}_+(u)\rangle_{T^*(G/B)}=(-1)^{\dim G/B}\delta_{w,u}$, see [@S17 Remark 2.2(3)].
Define the structure constants $e_{u,v}^w$ by the formula $$\label{equ:stablestructre}
\operatorname{stab}_-(u)\cup \operatorname{stab}_-(v)=\sum_w e_{u,v}^w\operatorname{stab}_-(w).$$ By the duality, $$e_{u,v}^w=(-1)^{\dim G/B}\langle\operatorname{stab}_-(u)\cup \operatorname{stab}_-(v),\operatorname{stab}_+(w)\rangle_{T^*(G/B)}\in \operatorname{Frac}H_{T\times\bbC^*}^*(\operatorname{pt}).$$ By the second property of the stable basis, the intersection of the support for $\operatorname{stab}_-(u)$ and $\operatorname{stab}_+(w)$ is proper. Therefore, $e_{u,v}^w=\langle\operatorname{stab}_-(u)\cup \operatorname{stab}_-(v),\operatorname{stab}_+(w)\rangle_{T^*(G/B)}$ lies in the non-localized cohomology ring $H_{T\times\bbC^*}^*(\operatorname{pt})$. Since the stable basis elements are given by Lagrangian cycles, each of them lives in the cohomology degree $2\dim G/B$. Thus, a degree count shows $e_{u,v}^w\in H_{T\times\bbC^*}^*(\operatorname{pt})=\bbC[\ft][\hbar]$ is a homogeneous polynomial of degree $2\dim X$. Therefore, we can recover the homogeneous polynomial $e_{u,v}^w$ from its specialization $e_{u,v}^w|_{\hbar=1}\in H_T^*(\operatorname{pt})$, which will be related to the structure constants $d_{u,v}^w$ in Equation .
Let $\iota:G/B\hookrightarrow T^*(G/B)$ denote the inclusion of the zero section. By [@AMSS17 Proposition 6.9(ii)], we have $$\label{equ:stabecsm}
\iota^*(\operatorname{stab}_-(w))|_{\hbar=1}=(-1)^{\dim G/B}{{c^T_{\text{SM}}}}(Y(w)^\circ).$$ Therefore, applying $\iota^*$ to Equation , letting $\hbar=1$ and comparing with Equation , we get
\[thm:stablecstru\] For any $u,v,w\in W$, we have $$e_{u,v}^w|_{\hbar=1}=(-1)^{\dim G/B}d_{u,v}^w.$$
However, using the current method, we can not obtain similar results for the cotangent bundle of partial flag varieties. The case of $T^*\operatorname{Gr}(k,n)$ is studied in [@C17].
Bott–Samelson varieties {#sec:BS}
=======================
The proof of the main Theorem \[thm:structure1\] uses the Bott–Samelson variety. In this section, we introduce this variety and its basic properties, and reduce the main theorem to a formula for the structure constants for some basis in the equivariant cohomology of the Bott–Samelson variety. In the remaining parts of this note, we fix a word $Q=s_{\alpha_{1}}s_{\alpha_{2}}\cdots s_{\alpha_{l}}$ (not necessarily reduced) in simple reflections.
Definition and properties
-------------------------
Recall that the Bott–Samelson variety associated to the word $Q$ is $$\operatorname{BS}^Q:=P_{\alpha_{1}}\times_B P_{\alpha_{2}}\times_B\cdots \times_B P_{\alpha_{l}}/B,$$ where $P_{\alpha_{k}}$ are the minimal parabolic subgroups corresponding to $\alpha_{k}$, and the quotient is by the equivalence relation given by $(g_1,g_2,\cdots g_l)\sim (g_1b_1,b_1^{-1}g_2b_2,\cdots, b_{l-1}^{-1}g_lb_l)$ for any $g_k\in P_{k}$ and $b_k\in B$. Let $[g_1,g_2,\cdots g_l]$ denote the resulting equivalence class in $\operatorname{BS}^Q$.
The maximal torus $T$ acts by left multiplication on $\operatorname{BS}^Q$, and there are $2^{\#Q}$ fixed points. To be more specific, the set of sequences $\{(g_1,g_2,\cdots g_l)\in P_{\alpha_{1}}\times P_{\alpha_{2}}\times \cdots \times P_{\alpha_{k}}|g_{k}\in \{1,s_{k}\}\}$ map bijectively to the fixed points set $(\operatorname{BS}^Q)^T$. Thus, we can index the fixed points by subwords $J\subset\{s_{1},s_{2},\cdots s_{l}\}$ of $Q$. For a subword $J\subset Q$, the torus weights of the tangent space $T_J\operatorname{BS}^Q$ is ([@M07 Lemma 1(iii)]) $$\{(\prod_{j\in J,j\leq i}s_j) (-\alpha_i)|1\leq i\leq l\}.$$
The Bott–Samelson variety $\operatorname{BS}^Q$ has a cell decomposition as follows, see [@M07 Section 3.2]. For any subword $J\subset Q$, we have a submanifold $$\operatorname{BS}^J:=\{[g_1,g_2,\cdots g_l]\in \operatorname{BS}^Q| g_k\in B \textit{ if } k\notin J\}\subset \operatorname{BS}^Q,$$ which can be identified with the Bott–Samelson variety for the word $J$. It has an open dense cell $$\operatorname{BS}^J_\circ:=\operatorname{BS}^J\setminus \cup_{S\subsetneq J}\operatorname{BS}^S,$$ which contains a unique fixed point corresponding to the subword $J\subset Q$. From definition, it is easy to see that $$\label{equ:cell}
\operatorname{BS}^J=\sqcup_{S\subset J}\operatorname{BS}_\circ^S.$$
The equivariant cohomology $H_T^*(\operatorname{BS}^Q)$ has a natural basis given by the fundamental classes of the sub Bott–Samelson varieties $\{[\operatorname{BS}^J]|J\subset Q\}$. Let $\langle-,-\rangle_{\operatorname{BS}^Q}$ denote the non-degenerate Poincaré pairing on $H_T^*(\operatorname{BS}^Q)$. By the Atiyah–Bott localization theorem and the above description of the weights at torus fixed point, we have the following formula for the pairing.
For any $\gamma_1,\gamma_2\in H_T^*(\operatorname{BS}^Q)$, we have $$\langle\gamma_1,\gamma_2\rangle_{\operatorname{BS}^Q}=\sum_{J\subset Q}\frac{\gamma_1|_J \gamma_2|_J}{\prod_{i\in Q}(\prod_{j\in J,j\leq i}s_j) (-\alpha_i)},$$ where $\gamma_i|_J$ denotes the restriction of $\gamma_i$ to the fixed point $J$.
Let $\pi:\operatorname{BS}^Q\rightarrow G/B$ be the natural map $$\pi([g_1,g_2,\cdots g_l])=(\prod_ig_i)B/B.$$ The image is the Schubert variety $X(\tilde{\prod} Q)$, where $\tilde{\prod} Q\in W$ is the Demazure product, see [@GK19]. If $Q$ is reduced, then $\tilde{\prod} Q=\prod Q\in W$, $\operatorname{BS}^Q$ is a resolution of singularity for $X(\prod Q)$ and $\pi_*([\operatorname{BS}^Q])=[X(\prod Q)]$. Otherwise, $\pi_*([\operatorname{BS}^Q])=0$.
CSM classes of cells and dual basis
-----------------------------------
For each subword $J\subset Q$, we have the equivariant CSM classes ${{c^T_{\text{SM}}}}(\operatorname{BS}^J_\circ)\in H_T^*(\operatorname{BS}^Q)$. By the same reason as in Section \[sec:CSMSSM\], $\{{{c^T_{\text{SM}}}}(\operatorname{BS}^J_\circ)|J\subset Q\}$ is a basis for the localized equivariant cohomology $H_T^*(\operatorname{BS}^Q)_{\operatorname{loc}}$. Using the non-degenerate pairing $\langle-,-\rangle_{\operatorname{BS}^Q}$, we can get a dual basis, denoted by $\{T_J|J\subset Q\}$. I.e., for any two subwords $J,R\subset Q$, $$\label{equ:duality2}
\langle{{c^T_{\text{SM}}}}(\operatorname{BS}^J_\circ),T_R\rangle_{\operatorname{BS}^Q}=\delta_{R,J}.$$ Since the restriction ${{c^T_{\text{SM}}}}(\operatorname{BS}^J_\circ)|_S=0$ if $S\nsubseteq J$, $T_R|_S=0$ if $R\nsubseteq S$.
By the cell decomposition in Equation , we get $${{c^T_{\text{SM}}}}(\operatorname{BS}^J)=\sum_{S\subset J}{{c^T_{\text{SM}}}}(\operatorname{BS}^S_\circ).$$ Hence, $$\label{equ:closure}
\langle{{c^T_{\text{SM}}}}(\operatorname{BS}^J),T_S\rangle_{\operatorname{BS}^Q}=\left\{\begin{array}{cc}
1 &, \textit{ if } S\subset J\\
0 &, \textit{ otherwise }
\end{array}\right.$$ Moreover, we have the following localization formula for ${{c^T_{\text{SM}}}}(\operatorname{BS}^J)$.
\[lem:cellrestriction\] For any $R\subset J$, we have $${{c^T_{\text{SM}}}}(\operatorname{BS}^J)|_R=\prod_{j\in J}\left(1+(\prod_{\substack{r\in R\\ r\leq j}}s_r)(-\alpha_j)\right)\prod_{q\in Q\setminus J}(\prod_{\substack{r\in R\\ r\leq q}}s_r)(-\alpha_q).$$
Let $\iota$ denote the inclusion of $\operatorname{BS}^J$ into $\operatorname{BS}^Q$, which is a proper map. Then by the functoriality and normalization properties of the MacPherson transformation $c_*^T$, we get $${{c^T_{\text{SM}}}}(\operatorname{BS}^J)=\iota_{*}({{c^T_{\text{SM}}}}(\operatorname{BS}^J)_J)=\iota_{*}(c^T(T(\operatorname{BS}^J)),$$ Here, ${{c^T_{\text{SM}}}}(\operatorname{BS}^J)_J$ denotes the CSM classes of $\operatorname{BS}^J$ in $H^*_T(\operatorname{BS}^J)$, which equals $c^T(T(\operatorname{BS}^J)$ by the normalization condition. Let $N_{\operatorname{BS}^J/\operatorname{BS}^Q}$ denote the normal bundle of $\operatorname{BS}^J$ inside $\operatorname{BS}^Q$. Then $$\begin{aligned}
{{c^T_{\text{SM}}}}(\operatorname{BS}^J)|_R&=c^T(T_R(\operatorname{BS}^J))e^T(N_{\operatorname{BS}^J/\operatorname{BS}^Q}|_R)\\
&=\prod_{j\in J}\left(1+(\prod_{\substack{r\in R\\ r\leq j}}s_r)(-\alpha_j)\right)\prod_{q\in Q\setminus J}(\prod_{\substack{r\in R\\ r\leq q}}s_r)(-\alpha_q),\end{aligned}$$ where $e^T(N_{\operatorname{BS}^J/\operatorname{BS}^Q}|_R)$ denotes the $T$-equivariant Euler class of the fiber at $R$ of the normal bundle $N_{\operatorname{BS}^J/\operatorname{BS}^Q}$.
These CSM classes of cells in $\operatorname{BS}^Q$ and the dual classes are related to the CSM and SSM classes of Schubert cells by the following lemma.
\[lem:relations\] For any subword $J\subset Q$, we have $$\pi_*({{c^T_{\text{SM}}}}(\operatorname{BS}^J_\circ))={{c^T_{\text{SM}}}}(X(\prod J)^\circ),$$ and $$\pi^*({{s^T_{\text{SM}}}}(Y(w)^\circ))=\sum_{J\subset Q, \prod J=w} T_J.$$
\[rem:k\]
1. The CSM classes behaves well with respect to proper pushforward, while the SSM classes behaves well with respect to pullback. If we switch the position between the CSM and SSM classes, such formulae would not hold. This is the reason why the method of Goldin and Knutson [@GK19] can only be applied to the SSM classes, not the CSM classes.
2. These formulae can be generalized to equivariant K theory, with the CSM (resp. SSM) classes replaced by the motivic Chern (resp. Segre motivic Chern) classes, see [@AMSS19b; @MW]. However, since the corresponding Hecke operators in the equivariant K theory satisfy the quadratic relation $(T_s+1)(T_s-q)=0$ instead of the simple relation $T_s^2=1$, the formulae would be much more complicated. For example, there will many terms in first pushforward formula. This prevents the author from computing the structure constants for the Segre motivic Chern classes for the Schubert cells.
Let $J=s_{\alpha_{i_1}}s_{\alpha_{i_2}}\cdots s_{\alpha_{i_k}}\subset Q$. By [@AM16 Lemma 3.1 and Theorem 3.3], we get $$\pi_*({{c^T_{\text{SM}}}}(\operatorname{BS}^J_\circ))=\calT_{i_k}\Bigg(\pi_*\bigg({{c^T_{\text{SM}}}}(\operatorname{BS}^{s_{\alpha_{i_1}}s_{\alpha_{i_2}}\cdots s_{\alpha_{i_{k-1}}}}_\circ)\bigg)\Bigg).$$ Hence, $$\pi_*({{c^T_{\text{SM}}}}(\operatorname{BS}^J_\circ))=\calT_{i_k}\circ\cdots \circ\calT_{i_2}\circ\calT_{i_1}([X(\operatorname{id})])=\calT_{(\prod J)^{-1}}([X(\operatorname{id})]),$$ where in the second equality, we used the fact that the operators $\calT_i$’s defined in Section \[sec:hecke\] satisfy the usual Weyl group relation, see [@AM16 Proposition 4.1]. Then the first statement follows from Equation .
For the second one, we have $$\begin{aligned}
\pi^*({{s^T_{\text{SM}}}}(Y(w)^\circ))&=\sum_{J\subset Q}\langle\pi^*({{s^T_{\text{SM}}}}(Y(w)^\circ)),{{c^T_{\text{SM}}}}(\operatorname{BS}^J_\circ)\rangle_{\operatorname{BS}^Q}T_J\\
&=\sum_{J\subset Q}\langle {{s^T_{\text{SM}}}}(Y(w)^\circ),\pi_*({{c^T_{\text{SM}}}}(\operatorname{BS}^J_\circ))\rangle T_J\\
&=\sum_{J\subset Q}\langle {{s^T_{\text{SM}}}}(Y(w)^\circ),{{c^T_{\text{SM}}}}(X(\prod J)^\circ)\rangle T_J\\
&=\sum_{J\subset Q, \prod J=w} T_J.\end{aligned}$$ Here the first and the last equalities follows from Equations and , the second one follows from the projection formula, and the third one follows from the first equality in this lemma.
Structure constants for the dual basis
--------------------------------------
Define the structure constants for the dual basis $\{T_J|J\subset Q\}$ in $H_T^*(\operatorname{BS}^Q)_{\operatorname{loc}}$ by $$\label{equ:defstructure2}
T_R T_S=\sum_{J}b_{R,S}^J T_J\in H_T^*(\operatorname{BS}^Q)_{\operatorname{loc}},$$ where $b_{R,S}^J\in \operatorname{Frac}H_T^*(\operatorname{pt})$ and $R,S\subset J$. Taking the coefficients of $T_R$ on both sides, we get $T_S|_R=b_{R,S}^R$, which is given in Proposition \[prop:restriction\].
These structure constants are related to those $c_{u,v}^w$ in Equation by the following lemma.
Assume $Q$ is a reduced word, with product $\prod Q=w\in W$. Then $$c_{u,v}^w=\sum_{\substack{R,S\subset Q,\\ \prod R=u,\prod S=v}}b_{R,S}^Q.$$
Applying $\pi^*$ to Equation and using Lemma \[lem:relations\], we get $$\begin{aligned}
&\pi^*({{s^T_{\text{SM}}}}(Y(u)^\circ))\cup \pi^*({{s^T_{\text{SM}}}}(Y(v)^\circ))\\
=&(\sum_{R\subset Q, \prod R=u} T_R)(\sum_{S\subset Q, \prod S=v} T_S)\\
=&\sum_{\substack{R\subset Q, \prod R=u\\S\subset Q, \prod S=v}}\sum_Jb_{R,S}^J T_J\\
=&\sum_z c_{u,v}^z\pi^*({{s^T_{\text{SM}}}}(Y(z)^\circ))\\
=&\sum_z c_{u,v}^z \sum_{J\subset Q, \prod S=z} T_J.\end{aligned}$$ Taking the coefficient of $T_Q$ on both sides, we get $$c_{u,v}^w=\sum_{\substack{R,S\subset Q,\\ \prod R=u,\prod S=v}}b_{R,S}^Q.$$
Therefore, the main Theorem \[thm:structure1\] is reduced to the following
\[thm:structure2\] For any word $Q$ (not necessarily reduced) with subwords $R,S\subset Q$, we have $$b_{R,S}^Q=\left(\prod_{q\in Q}\frac{\alpha_q^{[q\in R\cap S]}}{1+\alpha_q}s_q(-T^\vee_q)^{[q\notin R\cup S]}\right)\cdot 1\in \operatorname{Frac}H_T^*(pt).$$
Proof of Theorem \[thm:structure2\] {#sec:proof}
===================================
In this section, we will first give a localization formula for the dual basis $\{T_J|J\subset Q\}$. Theorem \[thm:structure2\] will follow by an induction argument on the number of elements in the word $Q$.
Localization of the dual basis
------------------------------
First of all, we have
\[lem:pairingwithclosure\] For any $\gamma\in H_T^*(\operatorname{BS}^Q)$, $$\langle\gamma, {{c^T_{\text{SM}}}}(\operatorname{BS}^J)\rangle_{\operatorname{BS}^Q}=\sum_{R\subset J}\gamma|_R\prod_{j\in J}\frac{1+(\prod_{\substack{r\in R\\ r\leq j}}s_r)(-\alpha_j)}{(\prod_{\substack{r\in R\\ r\leq j}}s_r)(-\alpha_j)}.$$
By the Atiyah–Bott localization formula and Lemma \[lem:cellrestriction\], we have $$\begin{aligned}
\langle\gamma, {{c^T_{\text{SM}}}}(\operatorname{BS}^J)\rangle_{\operatorname{BS}^Q}&=\sum_{R\subset Q}\frac{\gamma|_R {{c^T_{\text{SM}}}}(\operatorname{BS}^J)|_R}{e^T(T_R\operatorname{BS}^Q)}\\
&=\sum_{R\subset J}\gamma|_R\prod_{j\in J}\frac{1+(\prod_{\substack{r\in R\\ r\leq j}}s_r)(-\alpha_j)}{(\prod_{\substack{r\in R\\ r\leq j}}s_r)(-\alpha_j)}.\end{aligned}$$
The localizaiton of the dual basis is given by the following formula.
\[prop:restriction\] For any subwords $S\subset R\subset Q$, $$T_S|_R=\left(\prod_{r\in R}\frac{\alpha_r^{[r\in S]}}{1+\alpha_r}s_r\right)\cdot 1\in \operatorname{Frac}H_T^*(pt).$$
By the localization theorem, let us denote the classes $T_S'\in H_T^*(\operatorname{BS}^Q)$ defined by the localization conditions $$T'_S|_R=\left\{\begin{array}{cc}
\left(\prod_{r\in R}\frac{\alpha_r^{[r\in S]}}{1+\alpha_r}s_r\right)\cdot 1 & \textit{ if }S\subset R\\
0 & \textit{ otherwise }.
\end{array}\right.$$ By Equation , we only need to show $$\langle T'_S,{{c^T_{\text{SM}}}}(\operatorname{BS}^J)\rangle_{\operatorname{BS}^Q}=\left\{\begin{array}{cc}
1, & \textit{ if } S\subset J,\\
0, &\textit{ otherwise }.
\end{array}\right.$$ Since $$T'_S|_R=\left(\prod_{r\in R}\frac{\alpha_r^{[r\in S]}}{1+\alpha_r}s_r\right)\cdot 1= \frac{\prod_{s\in S}(\prod_{\substack{r\in R\\ r\leq s}}s_r)(-\alpha_s)}{\prod_{r\in R}\left(1+(\prod_{\substack{r'\in R\\ r'\leq r}}s_{r'})(-\alpha_r)\right)},$$ Lemma \[lem:pairingwithclosure\] gives, $$\begin{aligned}
&\langle T'_S,{{c^T_{\text{SM}}}}(\operatorname{BS}^J)\rangle_{\operatorname{BS}^Q}\\
=&\sum_{S\subset R\subset J}T_S'|_R\prod_{j\in J}\frac{1+(\prod_{\substack{r\in R\\ r\leq j}}s_r)(-\alpha_j)}{(\prod_{\substack{r\in R\\ r\leq j}}s_r)(-\alpha_j)}\\
=&\sum_{S\subset R\subset J}\frac{\prod_{j\in J\setminus R}\left(1+(\prod_{\substack{r\in R\\ r\leq j}}s_r)(-\alpha_j)\right)}{\prod_{j\in J\setminus S}(\prod_{\substack{r\in R\\ r\leq j}}s_r)(-\alpha_j)}.\end{aligned}$$ Thus, it suffices to check that for any $S\subset J\subset Q$, $$\label{equ:reducedstep}
\sum_{S\subset R\subset J}\frac{\prod_{j\in J\setminus R}\left(1+(\prod_{\substack{r\in R\\ r\leq j}}s_r)(-\alpha_j)\right)}{\prod_{j\in J\setminus S}(\prod_{\substack{r\in R\\ r\leq j}}s_r)(-\alpha_j)}=1.$$ We prove it by induction on the number of words in $J$. If $J=\emptyset$, it is obvious. Now assume $\#J\geq 1$, and Equation is true for any $J'$ such that $\#J'<\#J$. Suppose $s_\alpha$ is the first word in $J$ and $J=s_\alpha J_0$. We consider the following two cases according to whether $S$ contains $s_\alpha$ as its first word or not.
[*Case I:*]{} $S$ contains $s_\alpha$ as the first word. Then for all the $R$ such that $S\subset R\subset J$, $s_\alpha$ will also be the first word for $R$. Let $S=s_\alpha S_0$ and $R=s_\alpha R_0$. Then $$\begin{aligned}
&\sum_{S\subset R\subset J}\frac{\prod_{j\in J\setminus R}\left(1+(\prod_{\substack{r\in R\\ r\leq j}}s_r)(-\alpha_j)\right)}{\prod_{j\in J\setminus S}(\prod_{\substack{r\in R\\ r\leq j}}s_r)(-\alpha_j)}\\
=&s_\alpha\left(\sum_{S_0\subset R_0\subset J_0}\frac{\prod_{j\in J_0\setminus R_0}\left(1+(\prod_{\substack{r\in R_0\\ r\leq j}}s_r)(-\alpha_j)\right)}{\prod_{j\in J_0\setminus S_0}(\prod_{\substack{r\in R_0\\ r\leq j}}s_r)(-\alpha_j)}\right)\\
=&s_\alpha(1)\\
=&1,\end{aligned}$$ where the second equality follows from induction step.
[*Case II:*]{} $s_\alpha$ is not the first word in $S$. We separate the sum in Equation into two terms as follows $$\begin{aligned}
&\sum_{S\subset R\subset J}\frac{\prod_{j\in J\setminus R}\left(1+(\prod_{\substack{r\in R\\ r\leq j}}s_r)(-\alpha_j)\right)}{\prod_{j\in J\setminus S}(\prod_{\substack{r\in R\\ r\leq j}}s_r)(-\alpha_j)}\\
=&\sum_{\substack{S\subset R\subset J\\ s_\alpha\in R}}\frac{\prod_{j\in J\setminus R}\left(1+(\prod_{\substack{r\in R\\ r\leq j}}s_r)(-\alpha_j)\right)}{\prod_{j\in J\setminus S}(\prod_{\substack{r\in R\\ r\leq j}}s_r)(-\alpha_j)}+\sum_{\substack{S\subset R\subset J\\ s_\alpha\notin R}}\frac{\prod_{j\in J\setminus R}\left(1+(\prod_{\substack{r\in R\\ r\leq j}}s_r)(-\alpha_j)\right)}{\prod_{j\in J\setminus S}(\prod_{\substack{r\in R\\ r\leq j}}s_r)(-\alpha_j)}\\
=&\frac{1}{\alpha}s_\alpha(1)+\frac{1-\alpha}{-\alpha}\\
=&1,\end{aligned}$$ where the second equality follows from induction. This finishes the proof of Equation .
Suppose $s_\alpha$ is the first word in $R$ and $R=s_\alpha R_0$. Then an immediate corollary is
\[cor:reflections\]
1. If $s_\alpha$ is the first word in $S$ and $S=s_\alpha S_0$, then $$T_S|_R=\frac{\alpha}{1+\alpha}s_\alpha (T_{S_0}|_{R_0}).$$
2. If $s_\alpha$ is not the first word in $S$, then $$T_S|_R=\frac{1}{1+\alpha}s_\alpha (T_{S}|_{R_0}).$$
Proof of Theorem \[thm:structure2\] {#proof-of-theorem-thmstructure2}
-----------------------------------
In this section, we prove Theorem \[thm:structure2\] by induction on $\#Q$.
Restricting both sides of Equation to the fixed point $Q$, we get $$T_R|_Q T_S|_Q=\sum_{R,S\subset J\subset Q}b_{R,S}^J T_J|_Q.$$ Therefore, $$\label{equ:recursion}
b_{R,S}^Q=\frac{T_R|_Q T_S|_Q-\sum_{R,S\subset J\subsetneq Q}b_{R,S}^J T_J|_Q}{T_Q|_Q}.$$
*Proof of Theorem \[thm:structure2\]* We prove the Theorem by induction on $\#Q$. If $Q=\emptyset$, it is obvious. Assume the theorem is true for any $Q$ with $\#Q=k\geq 0$. Now assume $\#Q=k+1$, the first word is $s_\alpha$ and $Q=s_\alpha Q_0$. Consider different cases for $R,S$ containing, or not containing the first letter of $Q$.
*Case I*: Suppose neither $R$ or $S$ contains the first word $s_\alpha$ of $Q$. Then we have
The sum in Equation can be rewritten as $$\sum_{R,S\subset J\subsetneq Q}b_{R,S}^J T_J|_Q=b_{R,S}^{Q_0}\frac{1}{1+\alpha}s_\alpha
(T_{Q_0}|_{Q_0})+\sum_{R,S\subset J\subsetneq Q_0}\frac{1}{(1+\alpha)^2}s_\alpha(b_{R,S}^J)s_\alpha(T_{J}|_{Q_0}).$$
We separate the left hand side into two parts as follows $$\begin{aligned}
&\sum_{R,S\subset J\subsetneq Q}b_{R,S}^J T_J|_Q\\
=&\sum_{\substack{R,S\subset J\subsetneq Q\\ s_\alpha\notin J}}b_{R,S}^J T_J|_Q+\sum_{\substack{R,S\subset J\subsetneq Q\\ s_\alpha\in J}}b_{R,S}^J T_J|_Q\\
=&\sum_{R,S\subset J\subset Q_0}b_{R,S}^J T_J|_{s_\alpha Q_0}+\sum_{R,S\subset J_0\subsetneq Q_0}b_{R,S}^{s_\alpha J_0} T_{s_\alpha J_0}|_{s_\alpha Q_0}\\
=&\left(b_{R,S}^{Q_0} \frac{1}{1+\alpha}s_\alpha(T_{Q_0}|_{Q_0})+\frac{1}{1+\alpha}\sum_{R,S\subset J\subsetneq Q_0}b_{R,S}^J s_\alpha(T_J|_{ Q_0})\right)\\
&+\left(\frac{\alpha}{1+\alpha}\sum_{R,S\subset J_0\subsetneq Q_0}\frac{1}{1+\alpha}s_\alpha(-T_\alpha^\vee)(b_{R,S}^{J_0}) s_\alpha(T_{J_0}|_{Q_0})\right)\\
=&b_{R,S}^{Q_0}\frac{1}{1+\alpha}s_\alpha
(T_{Q_0}|_{Q_0})+\sum_{R,S\subset J\subsetneq Q_0}\frac{1}{(1+\alpha)^2}s_\alpha(b_{R,S}^J)s_\alpha(T_{J}|_{Q_0}).\end{aligned}$$ Here the third equality follows from Corollary \[cor:reflections\] and the induction step for $s_\alpha J_0\subsetneq Q$, and the last equality follows from the formula $s_\alpha(-T_\alpha^\vee)=s_\alpha(-\partial_\alpha-s_\alpha)=-s_\alpha \frac{1}{\alpha}(1-s_\alpha)-1=\frac{1}{\alpha}s_\alpha-\frac{1+\alpha}{\alpha}$.
Plugging this into Equation , we get $$\begin{aligned}
b_{R,S}^Q&=\frac{T_R|_Q T_S|_Q-\sum_{R,S\subset J\subsetneq Q}b_{R,S}^J T_J|_Q}{T_Q|_Q}\\
&=\frac{\frac{1}{(1+\alpha)^2}s_\alpha(T_R|_{Q_0} T_S|_{Q_0})-b_{R,S}^{Q_0}\frac{1}{1+\alpha}s_\alpha(T_{Q_0}|_{Q_0})-\sum_{R,S\subset J\subsetneq Q_0}\frac{1}{(1+\alpha)^2}s_\alpha(b_{R,S}^J)s_\alpha(T_{J}|_{Q_0})}{\frac{\alpha}{1+\alpha}s_\alpha(T_{Q_0}|_{Q_0})}\\
&=\frac{1}{\alpha(1+\alpha)}s_\alpha\left(\frac{T_R|_{Q_0} T_S|_{Q_0}-\sum_{R,S\subset J\subsetneq Q_0}b_{R,S}^JT_{J}|_{Q_0})}{T_{Q_0}|_{Q_0}}\right)-\frac{1}{\alpha}b_{R,S}^{Q_0}\\
&=\frac{1}{\alpha(1+\alpha)}s_\alpha(b_{R,S}^{Q_0})-\frac{1}{\alpha}b_{R,S}^{Q_0}\\
&=\frac{1}{1+\alpha}s_\alpha(-T^\vee_\alpha)(b_{R,S}^{Q_0}),\end{aligned}$$ as desired.
*Case II*: Suppose both $R$ or $S$ contain the first word $s_\alpha$ of $Q$. Let $R=s_\alpha R_0$ and $S=S_\alpha S_0$. Every $J$ satisfying $R,S\subset J\subset Q$ also contains $s_\alpha$ as its first word. Let $J=s_\alpha J_0$. Then by induction step for $J\subsetneq Q$ and Corollary \[cor:reflections\], we have $$\begin{aligned}
\sum_{R,S\subset J\subsetneq Q}b_{R,S}^J T_J|_Q&=\sum_{R_0,S_0\subset J_0\subsetneq Q_0}b_{s_\alpha R_0,s_\alpha S_0}^{s_\alpha J_0} T_{s_\alpha J_0}|_{s_\alpha Q_0}\\
&=\sum_{R_0,S_0\subset J_0\subsetneq Q_0}\frac{\alpha}{1+\alpha}s_\alpha(b_{R_0, S_0}^{J_0})\frac{\alpha}{1+\alpha} s_\alpha(T_{ J_0}|_{Q_0}).\end{aligned}$$ Putting this into Equation , we get $$\begin{aligned}
b_{R,S}^Q&=\frac{T_R|_Q T_S|_Q-\sum_{R,S\subset J\subsetneq Q}b_{R,S}^J T_J|_Q}{T_Q|_Q}\\
&=\frac{\alpha}{1+\alpha}s_\alpha\left(\frac{T_{R_0}|_{Q_0} T_{S_0}|_{Q_0}-\sum_{R_0,S_0\subset J_0\subsetneq Q_0}b_{R_0,S_0}^{J_0} T_{J_0}|_{Q_0}}{T_{Q_0}|_{Q_0}}\right)\\
&=\frac{\alpha}{1+\alpha}s_\alpha(b_{R,S}^{Q_0}),\end{aligned}$$ as desired.
As $b_{R,S}^Q=b_{S,R}^Q$, we are left with the last case.
*Case III*: Suppose $R=s_\alpha R_0$, while $S$ does not begin with $s_\alpha$. Every $J$ satisfying $R,S\subset J\subset Q$ also contains $s_\alpha$ as its first word. Let $J=s_\alpha J_0$. Then by induction step for $J\subsetneq Q$ and Corollary \[cor:reflections\], we have $$\begin{aligned}
\sum_{R,S\subset J\subsetneq Q}b_{R,S}^J T_J|_Q&=\sum_{R_0,S\subset J_0\subsetneq Q_0}b_{s_\alpha R_0,S}^{s_\alpha J_0} T_{s_\alpha J_0}|_{s_\alpha Q_0}\\
&=\sum_{R_0,S\subset J_0\subsetneq Q_0}\frac{1}{1+\alpha}s_\alpha(b_{R_0, S}^{J_0})\frac{\alpha}{1+\alpha} s_\alpha(T_{ J_0}|_{Q_0}).\end{aligned}$$ Plugging this into Equation and using Corollary \[cor:reflections\], we get $$\begin{aligned}
b_{R,S}^Q&=\frac{T_R|_Q T_S|_Q-\sum_{R,S\subset J\subsetneq Q}b_{R,S}^J T_J|_Q}{T_Q|_Q}\\
&=\frac{1}{1+\alpha}s_\alpha\left(\frac{T_{R_0}|_{Q_0} T_{S}|_{Q_0}-\sum_{R_0,S\subset J_0\subsetneq Q_0}b_{R_0,S}^{J_0} T_{J_0}|_{Q_0}}{T_{Q_0}|_{Q_0}}\right)\\
&=\frac{1}{1+\alpha}s_\alpha(b_{R,S}^{Q_0}).\end{aligned}$$ This finishes the proof in all cases.
Parabolic case {#sec:P}
==============
In this section, we generalize Theorem \[thm:structure1\] to the parabolic case.
Let $P$ be a parabolic subgroup containing the Borel subgroup $B$. Let $W_P$ be the Weyl group of a Levi subgroup of $P$, and $W^P=W/W_P$ be the set of minimal length representatives. For any $w\in W^P$, let $X(wW_P)^\circ$ (resp. $Y(wW_P)^\circ$) denote the Schubert cell $BwP/P$ (resp. $B^{-}wP/P$). Let $p:G/B\rightarrow G/B$ be the natural projection.
The properties of the CSM/SSM classes of $G/P$ are given by
\[lem:BP\]
1. For any $w\in W$, $$p_*({{c^T_{\text{SM}}}}(X(w)^\circ))={{c^T_{\text{SM}}}}(X(wW_P)^\circ)\in H^*_T(G/P).$$
2. For any $w\in W^P$, $$p^*({{s^T_{\text{SM}}}}(Y(wW_P)^\circ))=\sum_{x\in W_P}{{s^T_{\text{SM}}}}(Y(wx)^\circ)\in H^*_T(G/B).$$
3. For any $u,w\in W^P$, $$\langle{{c^T_{\text{SM}}}}(X(wW_P)^\circ),{{s^T_{\text{SM}}}}(Y(uW_P)^\circ)\rangle=\delta_{w,u}.$$
The first one follows from the functoriality of the MacPherson transformation. The second one follows from the Verdier–Riemann–Roch Theorem (see [@AMSS17 Theorem 9.2, or Equation (37)]). The last one is [@AMSS17 Theorem 9.4].
Define the structure constants for the SSM classes as follow: $${{s^T_{\text{SM}}}}(Y(uW_P)^\circ)\cup{{s^T_{\text{SM}}}}(Y(vW_P)^\circ)=\sum_{w\in W^P}c_{u,v}^w(P){{s^T_{\text{SM}}}}(Y(wW_P)^\circ)\in H_T^*(G/P)_{\operatorname{loc}},$$ where $u,v\in W^P$ and $c_{u,v}^w(P)\in \operatorname{Frac}H_T^*(\operatorname{pt})$. Applying $p^*$ to the above equation and using Lemma \[lem:BP\], we get $$(\sum_{x\in W_P} {{s^T_{\text{SM}}}}(Y(ux)^\circ))\cup (\sum_{y\in W_P} {{s^T_{\text{SM}}}}(Y(vy)^\circ))=\sum_{w\in W^P}c_{u,v}^w(P)\sum_{x\in W_P} {{s^T_{\text{SM}}}}(Y(wx)^\circ).$$ For any $w\in W^P$, taking the coefficient of ${{s^T_{\text{SM}}}}(Y(w)^\circ)$ on both sides, we obtain
\[thm:Pcase\] For any $u,v,w\in W^P$, $$c_{u,v}^w(P)=\sum_{x\in W_P,y\in W_P}c_{ux,vy}^w.$$
It was communicated by A. Knutson [@KZJ] to the author that the structure constants for the SSM classes of the Schubert cells in a partial flag variety can be used to deduce some puzzle formulae for the Schubert structure constants in 2/3/4-step partial flag varieties in type A, generalizing the earlier works [@KT03; @Bu02; @BKPT16]. In fact, the authors in [@KZJ] consider the quotient of the stable basis elements [@MO19] by the class of the zero section in the equivariant cohomology of the cotangent bundle of the flag variety, which has a natural $\bbC^*$ action by scaling the cotangent fibers. The pullback of the stable basis elements to the flag variety are the CSM classes of the Schubert cells, see Equation . Besides, the pullback of the $\bbC^*$-equvivariant class of the zero section is the total Chern class of the flag variety (after setting the $\bbC^*$-equvivariant parameter to 1 and up to a sign). Thus, the classes considered in [@KZJ] are precisely the SSM classes of the Schubert cells. This is one of my motivations for this work.
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[^1]: If $X$ is not smooth, we can embed $X$ into a smooth ambient space, and use the total Chern class of the ambient space to define the SSM classes, see [@AMSS19a].
[^2]: These operators are the $\hbar=1$ specialization of $L_i$ and $L^\vee_i$ in [@AMSS17 Section 5.2].
[^3]: Here we have used the fact $(-1)^{\dim G/B}e^{T\times \bbC^*}(T^*(G/B))|_{\hbar=1}=c^T(T(G/B))$, see the proof of Corollary 7.4 in *loc. cit.*.
[^4]: The author thanks L. Mihalcea for pointing out this proof.
|
---
abstract: 'Let $1 \le m \le n$. We prove various results about the chessboard complex ${\mathsf{M}_{m,n}}$, which is the simplicial complex of matchings in the complete bipartite graph $K_{m,n}$. First, we demonstrate that there is nonvanishing $3$-torsion in $\tilde{H}_d({\mathsf{M}_{m,n}};{\mathbb{Z}})$ whenever $\frac{m+n-4}{3} \le d \le m-4$ and whenever $6 \le m < n$ and $d=m-3$. Combining this result with theorems due to Friedman and Hanlon and to [[Shareshian]{}]{} and Wachs, we characterize all triples $(m,n,d)$ satisfying $\tilde{H}_{d}({\mathsf{M}_{m,n}};{\mathbb{Z}}) \neq 0$. Second, for each $k \ge 0$, we show that there is a polynomial $f_k(a,b)$ of degree $3k$ such that the dimension of $\tilde{H}_{k+a+2b-2}({\mathsf{M}_{k+a+3b-1,k+2a+3b-1}};{\mathbb{Z}}_3)$, viewed as a vector space over ${\mathbb{Z}}_3$, is at most $f_k(a,b)$ for all $a \ge 0$ and $b \ge k+2$. Third, we give a computer-free proof that $\tilde{H}_2({\mathsf{M}_{5,5}};{\mathbb{Z}}) \cong {\mathbb{Z}}_3$. Several proofs are based on a new long exact sequence relating the homology of a certain subcomplex of ${\mathsf{M}_{m,n}}$ to the homology of ${\mathsf{M}_{m-2,n-1}}$ and ${\mathsf{M}_{m-2,n-3}}$.'
address: 'Department of Mathematics, KTH, 10044 Stockholm, Sweden'
author:
- Jakob Jonsson
title: 'On the $3$-torsion Part of the Homology of the Chessboard Complex'
---
[^1]
This is a preprint version of a paper published in [*Annals of Combinatorics*]{} [**14**]{} (2010), No. 4, 487–505.
Introduction
============
Given a family $\Delta$ of graphs on a fixed vertex set, we identify each member of $\Delta$ with its edge set. In particular, if $\Delta$ is closed under deletion of edges, then $\Delta$ is an abstract simplicial complex.
A [*matching*]{} in a simple graph $G$ is a subset $\sigma$ of the edge set of $G$ such that no vertex appears in more than one edge in $\sigma$. Let ${\mathsf{M}_{{}}}(G)$ be the family of matchings in $G$; ${\mathsf{M}_{{}}}(G)$ is a simplicial complex. We write ${\mathsf{M}_{n}} = {\mathsf{M}_{{}}}(K_n)$ and ${\mathsf{M}_{m,n}} = {\mathsf{M}_{{}}}(K_{m,n})$, where $K_n$ is the complete graph on the vertex set $[n] = \{1, \ldots, n\}$ and $K_{m,n}$ is the complete bipartite graph with block sizes $m$ and $n$. ${\mathsf{M}_{n}}$ is the [*matching complex*]{} and ${\mathsf{M}_{m,n}}$ is the [*chessboard complex*]{}.
The topology of ${\mathsf{M}_{n}}$, ${\mathsf{M}_{m,n}}$, and related complexes has been subject to analysis in a number of theses [@Andersen; @Dong; @Garst; @thesis; @Kara; @Kson] and papers [@Ath; @BBLSW; @BLVZ; @Bouc; @DongWachs; @FH; @KRW; @RR; @ShWa; @Ziegvert]; see Wachs [@Wachs] for an excellent survey and further references.
Despite the simplicity of the definition, the homology of the matching complex ${\mathsf{M}_{n}}$ and the chessboard complex ${\mathsf{M}_{m,n}}$ turns out to have a complicated structure. The rational homology is well-understood and easy to describe thanks to beautiful results due to Bouc [@Bouc] and Friedman and Hanlon [@FH], but very little is known about the integral homology and the homology over finite fields.
A previous paper [@bettimatch] contains a number of results about the integral homology of the matching complex ${\mathsf{M}_{n}}$. The purpose of the present paper is to extend a few of these results to the chessboard complex ${\mathsf{M}_{m,n}}$.
For $1 \le m \le n$, define $$\nu_{m,n} = \min \{m-1, \lceil \mbox{$\frac{m+n-4}{3}$}\rceil\}
= \left\{
\begin{array}{ll}
\lceil \frac{m+n-4}{3}\rceil & \mbox{if } m \le n \le 2m+1; \\
m-1 & \mbox{if } n \ge 2m-1.
\end{array}
\right.$$ Note that $\lceil \frac{m+n-4}{3}\rceil = m-1$ for $2m-1 \le n \le 2m+1$. By a theorem due to [[Shareshian]{}]{} and Wachs [@ShWa], ${\mathsf{M}_{m,n}}$ contains nonvanishing homology in degree $\nu_{m,n}$ for all $m,n \ge 1$ except $(m,n) = (1,1)$. Previously, Friedman and Hanlon demonstrated that this bottom nonvanishing homology group is finite if and only if $m \le n \le
2m-5$ and $(m,n) \notin \{(6,6),(7,7),(8,9)\}$.
To settle their theorem, [[Shareshian]{}]{} and Wachs demonstrated that $\tilde{H}_{\nu_{m,n}}({\mathsf{M}_{m,n}};{\mathbb{Z}})$ contains nonvanishing $3$-torsion whenever the group is finite. One of our main results provides upper bounds on the rank of the $3$-torsion part. Specifically, in Section \[boundschess-sec\], we prove the following:
For each $k \ge 0$, $a \ge 0$, and $b \ge k+2$, we have that $\dim \tilde{H}_{k+a+2b-2}({\mathsf{M}_{k+a+3b-1,k+2a+3b-1}};{\mathbb{Z}}_3)$ is bounded by a polynomial in $a$ and $b$ of degree $3k$. \[boundsintro-thm\]
An equivalent way of expressing Theorem \[boundsintro-thm\] is to say that $$\dim \tilde{H}_{d}({\mathsf{M}_{m,n}};{\mathbb{Z}}_3) \le f_{3d-m-n+4}(n-m,m-d-1)$$ whenever $m \le n \le 2m-5$ and $\frac{m+n-4}{3} \le d \le \frac{2m+n-7}{4}$, where $f_k$ is a polynomial of degree $3k$ for each $k$. The bound in Theorem \[boundsintro-thm\] remains true over any coefficient field.
Note that we express the theorem in terms of the following transformation: $$\left\{
\begin{array}{ccrcrcrcl}
k &\!\!=&\!\! -m&\!\!-&\!\!n&\!\!+&\!\! 3d&\!\!+&\!\!4
\\
a &\!\!=&\!\! -m&\!\!+&\!\!n
\\
b &\!\!=&\!\! m&\!\! &\!\! &\!\!-&\!\!d &\!\! - &\!\!1
\end{array}
\right. \Leftrightarrow
\left\{
\begin{array}{ccrcrcrcl}
m &\!\!=&\!\! k&\!\!+&\!\!a&\!\!+&\!\!3b&\!\!-&\!\!1 \\
n &\!\!=&\!\! k&\!\!+&\!\!2a&\!\!+&\!\!3b&\!\!-&\!\!1 \\
d &\!\!=&\!\! k&\!\!+&\!\!a&\!\!+&\!\!2b&\!\!-&\!\!2.
\end{array}
\right.
\label{mnd2kab-eq}$$ Assuming that $m \le n$, each of the three new variables measures the difference between two important parameters:
- For $m \le n \le 2m+1$, we have that $k$ measures the difference between the degree $d$ and the bottom degree in which ${\mathsf{M}_{m,n}}$ has nonvanishing homology; $\frac{k}{3} = d - \frac{m+n-4}{3}$.
- $a$ is the difference between the block sizes $n$ and $m$.
- $b$ is the difference between $\dim {\mathsf{M}_{m,n}} = m-1$ and $d$.
To establish Theorem \[boundsintro-thm\], we introduce two new long exact sequences; see Sections \[exseq00-G-11-sec\] and \[exseqG-21-23-sec\]. These two sequences involve the subcomplex ${\Gamma_{m,n}}$ of ${\mathsf{M}_{m,n}}$ obtained by fixing a vertex in the block of size $n$ and removing all but two of the edges that are incident to this vertex. Our first sequence is very simple and relates the homology of ${\mathsf{M}_{m,n}}$ to that of ${\Gamma_{m,n}}$ and ${\mathsf{M}_{m-1,n-1}}$. Our second sequence is more complicated and relates ${\Gamma_{m,n}}$ to ${\mathsf{M}_{m-2,n-1}}$ and ${\mathsf{M}_{m-2,n-3}}$. Combining the two sequences and “cancelling out” ${\Gamma_{m,n}}$, we obtain a bound on the dimension of the ${\mathbb{Z}}_3$-homology of ${\mathsf{M}_{m,n}}$ in terms of ${\mathsf{M}_{m-1,n-1}}$, ${\mathsf{M}_{m-2,n-1}}$, and ${\mathsf{M}_{m-2,n-3}}$. By an induction argument, we obtain Theorem \[boundsintro-thm\].
For $k=0$, Theorem \[boundsintro-thm\] says that $\dim \tilde{H}_{\nu_{m,n}}({\mathsf{M}_{m,n}};{\mathbb{Z}}_3)$ is bounded by a constant whenever $m \le n \le 2m-5$ and $m+n \equiv 1 \pmod{3}$. Indeed, [[Shareshian]{}]{} and Wachs [@ShWa] proved that $\tilde{H}_{\nu_{m,n}}({\mathsf{M}_{m,n}};{\mathbb{Z}}) \cong {\mathbb{Z}}_3$ for any $m$ and $n$ satisfying these equations. Their proof was by induction on $m+n$ and relied on a computer calculation of $\tilde{H}_2({\mathsf{M}_{5,5}};{\mathbb{Z}})$. In Section \[bottomchess-sec\], we provide a computer-free proof that $\tilde{H}_2({\mathsf{M}_{5,5}};{\mathbb{Z}}) \cong {\mathbb{Z}}_3$, again using the exact sequences from Sections \[exseq00-G-11-sec\] and \[exseqG-21-23-sec\].
In Section \[higherchess-sec\], we use results about the matching complex ${\mathsf{M}_{n}}$ from a previous paper [@bettimatch] to extend [[Shareshian]{}]{} and Wachs’ $3$-torsion result to higher-degree groups:
For $m+1 \le n \le 2m-5$, there is $3$-torsion in $\tilde{H}_{d}({\mathsf{M}_{m,n}}; {\mathbb{Z}})$ whenever $\frac{m+n-4}{3} \le d \le m-3$. There is also $3$-torsion in $\tilde{H}_{d}({\mathsf{M}_{m,m}}; {\mathbb{Z}})$ whenever $\frac{2m-4}{3} \le d \le m-4$. \[nonvan3intro-thm\]
Note that $m+1 \le n \le 2m-5$ and $\frac{m+n-4}{3} \le d \le m-3$ if and only if $k\ge 0$, $a \ge 1$, and $b \ge 2$, where $k$, $a$, and $b$ are defined as in (\[mnd2kab-eq\]). Moreover, $m=n$ and $\frac{2m-4}{3} \le d \le m-4$ if and only if $k\ge 0$, $a = 0$, and $b \ge 3$.
Our proof of Theorem \[nonvan3intro-thm\] relies on properties of the top homology group of ${\mathsf{M}_{k,k+1}}$ for different values of $k$; this group was of importance also in the work of [[Shareshian]{}]{} and Wachs [@ShWa].
Thanks to Theorem \[nonvan3intro-thm\] and Friedman and Hanlon’s formula for the rational homology [@FH], we may characterize those $(d,m,n)$ satisfying $\tilde{H}_d({\mathsf{M}_{m,n}};{\mathbb{Z}})
\neq 0$:
For $1 \le m \le n$, we have that $\tilde{H}_d({\mathsf{M}_{m,n}};{\mathbb{Z}})$ is nonzero if and only if either of the following is true:
- $\lceil\frac{m+n-4}{3}\rceil \le d \le m-2$. Equivalently, $k\ge
0$, $a \ge 0$, and $b \ge 1$.
- $d= m-1$ and $n \ge m+1$. Equivalently, $k\ge 2-a$, $a \ge
1$, and $b = 0$.
Again, see Section \[higherchess-sec\] for details.
The problem of detecting $p$-torsion in the homology of $\tilde{M}_{m,n}$ for $p \neq 3$ remains open. In this context, we may mention that there is $5$-torsion in the homology of the matching complex ${\mathsf{M}_{14}}$ [@m14]. By computer calculations [@moretorsion], further $p$-torsion is known to exist for $p \in \{5, 7, 11, 13\}$.
Notation {#basic-sec}
--------
We identify the two parts of the graph $K_{m,n}$ with the two sets $[m] = \{1, 2, \ldots, m\}$ and $[{\overline{n}}] = \{{\overline{1}}, {\overline{2}}, \ldots,
{\overline{n}}\}$. The latter set should be interpreted as a disjoint copy of $[n]$; hence each edge is of the form $i{\overline{j}}$, where $i \in [m]$ and $j \in [n]$. Sometimes, it will be useful to view ${\mathsf{M}_{m,n}}$ as a subcomplex of the matching complex ${\mathsf{M}_{m+n}}$ on the complete graph $K_{m+n}$. In such situations, we identify the vertex ${\overline{j}}$ in $K_{m,n}$ with the vertex $m+j$ in $K_{m+n}$ for each $j \in [n]$.
For finite sets $S$ and $T$, we let ${\mathsf{M}_{S,T}}$ denote the matching complex on the complete bipartite graph with blocks $S$ and $T$, viewed as disjoint sets in the manner described above. In particular, ${\mathsf{M}_{{[m}},[n]}] = {\mathsf{M}_{m,n}}$. For integers $a \le b$, we write $[a,b] = \{a, a+1, \ldots, b-1, b\}$.
The [*join*]{} of two families of sets $\Delta$ and $\Sigma$, assumed to be defined on disjoint ground sets, is the family $\Delta {*}\Sigma = \{ \delta \cup \sigma: \delta \in \Delta,
\sigma \in \Sigma\}$.
Whenever we discuss the homology of a simplicial complex or the relative homology of a pair of simplicial complexes, we mean reduced simplicial homology. For a simplicial complex $\Sigma$ and a coefficient ring ${\mathbb{F}}$, we let $e_0 \wedge \cdots \wedge e_d$ denote a generator of $\tilde{C}_d(\Sigma; {\mathbb{F}})$ corresponding to the set $\{e_0,
\ldots, e_d\} \in \Sigma$. Given a cycle $z$ in a chain group $\tilde{C}_d(\Sigma;
{\mathbb{F}})$, whenever we talk about $z$ as an element in the induced homology group $\tilde{H}_d(\Sigma;{\mathbb{F}})$, we really mean the homology class of $z$.
We will often consider pairs of complexes $(\Gamma, \Delta)$ such that $\Gamma \setminus \Delta$ is a union of families of the form $$\Sigma = \{\sigma\} {*}{\mathsf{M}_{S,T}},$$ where $\sigma = \{e_1, \ldots, e_s\}$ is a set of pairwise disjoint edges of the form $i{\overline{j}}$, and where $S$ and $T$ are subsets of $[m]$ and $[n]$, respectively, such that $S \cap
e_i = {\overline{T}} \cap e_i = \emptyset$ for each $i$. We may write the chain complex of $\Sigma$ as $$\tilde{C}_d(\Sigma;{\mathbb{F}}) = (e_1 \wedge \cdots \wedge e_s){\mathbb{F}}{\otimes}_{\mathbb{F}}\tilde{C}_{d-|\sigma|}({\mathsf{M}_{S,T}};{\mathbb{F}}),$$ defining the boundary operator as $$\partial(e_1 \wedge \cdots \wedge e_s {\otimes}_{\mathbb{F}}c)
= (-1)^s e_1 \wedge \cdots \wedge e_s {\otimes}_{\mathbb{F}}\partial(c).$$ For simplicity, we will often suppress ${\mathbb{F}}$ from notation. For example, by some abuse of notation, we will write $$e_1 \wedge \cdots \wedge e_s {\otimes}\tilde{C}_{d-|\sigma|}({\mathsf{M}_{S,T}}) =
(e_1 \wedge \cdots \wedge e_s){\mathbb{F}}{\otimes}_{\mathbb{F}}\tilde{C}_{d-|\sigma|}({\mathsf{M}_{S,T}};{\mathbb{F}}).$$
We say that a cycle $z$ in $\tilde{C}_{d-1}({\mathsf{M}_{m,n}};{\mathbb{F}})$ has [*type*]{} ${\genfrac{[}{]}{0pt}{}{m_1,n_1}{d_1}} \wedge \cdots \wedge
{\genfrac{[}{]}{0pt}{}{m_s,n_s}{d_s}}$ if there are partitions $[m] = \bigcup_{i=1}^s S_i$ and $[n] = \bigcup_{i=1}^s T_i$ such that $|S_i| = m_i$ and $|T_i| = n_i$ and such that $z = z_1 \wedge \cdots \wedge z_s$, where $z_i$ is a cycle in $\tilde{C}_{d_i-1}({\mathsf{M}_{S_i,T_i}};{\mathbb{F}})$ for each $i$.
Two classical results {#classic}
---------------------
Before proceeding, we list two classical results pertaining to the topology of the chessboard complex ${\mathsf{M}_{m,n}}$.
For $m,n \ge 1$, ${\mathsf{M}_{m,n}}$ is $(\nu_{m,n}-1)$-connected. \[smallest-thm\]
Indeed, the $\nu_{m,n}$-skeleton of ${\mathsf{M}_{m,n}}$ is vertex decomposable [@Ziegvert]. Garst [@Garst] settled the case $n \ge 2m-1$ in Theorem \[smallest-thm\]. As already mentioned in the introduction, there is nonvanishing homology in degree $\nu_{m,n}$ for all $(m,n) \neq (1,1)$; see Section \[bottomchess-sec\] for details.
The transformation (\[mnd2kab-eq\]) maps the set $\{(m,n,\nu_{m,n}) : 1 \le m \le n\}$ to the set of triples $(k,a,b)$ satisfying either of the following:
- $k \in \{0,1,2\}$, $a \ge 0$, and $b \ge 1$ (corresponding to $d = \lceil \frac{m+n-4}{3} \rceil$ and $m \le n \le 2m-2$).
- $2-a \le k \le 2$ and $b = 0$ (corresponding to $0 \le d = m-1$ and $n \ge 2m-1$).
Friedman and Hanlon [@FH] established a formula for the rational homology of ${\mathsf{M}_{m,n}}$; see Wachs [@Wachs] for an overview. For our purposes, the most important consequence is the following result:
For $1 \le m \le n$, we have that $\tilde{H}_{d}({\mathsf{M}_{m,n}}; {\mathbb{Z}})$ is infinite if and only if $(m-d-1)(n-d-1) \le d+1$, $m \ge d+1$, and $n \ge d+2$. In particular, $\tilde{H}_{\nu_{m,n}}({\mathsf{M}_{m,n}}; {\mathbb{Z}})$ is finite if and only if $n \le 2m-5$ and $(m,n) \notin
\{(6,6),(7,7),(8,9)\}$. \[chessfinite-thm\]
With $k$, $a$, and $b$ defined as in (\[mnd2kab-eq\]), the conditions $1 \le m \le n$, $(m-d-1)(n-d-1) \le d+1 \le m$, and $n \ge d+2$ are equivalent to $$b(a+b) \le k + a + 2b - 1 \Longleftrightarrow
(b-1)(a+b-1) \le k,$$ $a \ge 0$, $b \ge 0$, $a+b \ge 1$, and $k+a+3b \ge 2$. Moreover, the conditions $d = \nu_{m,n}$, $m \le n \le 2m-5$, and $(m,n) \notin \{(6,6),(7,7),(8,9)\}$ are equivalent to $k \in \{0,1,2\}$, $a \ge 0$, $b \ge 2$, and $(k,a,b) \notin
\{(1,0,2), (2, 0, 2), (2, 1, 2)\}$.
Four long exact sequences
=========================
We present four long exact sequences relating different families of chessboard complexes. In this paper, we will only use the third and the fourth sequences; we list the other two sequences for reference. Throughout this section, we consider an arbitrary coefficient ring ${\mathbb{F}}$, which we suppress from notation for convenience.
Long exact sequence relating ${\mathsf{M}_{m,n}}$, ${\mathsf{M}_{m,n-1}}$, and ${\mathsf{M}_{m-1,n-1}}$ {#exseq00-01-11-sec}
-------------------------------------------------------------------------------------------------------
Define $$P^{m-1,n-1}_{d} =
\bigoplus_{s=1}^m s{\overline{1}} {\otimes}\tilde{H}_{d}({\mathsf{M}_{{[m}}\setminus
\{s\},[2,n]}]).$$ For each $m \ge 1$ and $n \ge 1$, we have a long exact sequence $$\begin{CD}
& & & & \cdots @>>>
P^{m-1,n-1}_{d} \\
@>>>
\tilde{H}_{d}({\mathsf{M}_{m,n-1}}) @>>>
\tilde{H}_{d}({\mathsf{M}_{m,n}}) @>>>
P^{m-1,n-1}_{d-1} \\
@>>>
\tilde{H}_{d-1}({\mathsf{M}_{m,n-1}}) @>>> \cdots .
\end{CD}$$
This is the long exact sequence for the pair $({\mathsf{M}_{m,n}}, {\mathsf{M}_{m,n-1}})$.
We refer to this sequence as the [*00-01-11 sequence*]{}, thereby indicating that the sequence relates ${\mathsf{M}_{m-0,n-0}}$, ${\mathsf{M}_{m-0,n-1}}$, and ${\mathsf{M}_{m-1,n-1}}$. Note that the sequence is asymmetric in $m$ and $n$; swapping the indices, we obtain an exact sequence relating ${\mathsf{M}_{m,n}}$, ${\mathsf{M}_{m-1,n}}$, and ${\mathsf{M}_{m-1,n-1}}$.
Long exact sequence relating ${\mathsf{M}_{m,n}}$, ${\mathsf{M}_{m-1,n-2}}$, ${\mathsf{M}_{m-2,n-1}}$, and ${\mathsf{M}_{m-2,n-2}}$ {#exseq00-12-21-22-sec}
-----------------------------------------------------------------------------------------------------------------------------------
Define $$\begin{aligned}
P^{m-1,n-2}_{d} &=&
\bigoplus_{t=2}^n 1{\overline{t}} {\otimes}\tilde{H}_{d}({\mathsf{M}_{{[2,m}},[2,n]\setminus
\{t\}}]); \\
Q^{m-2,n-1}_{d} &=&
\bigoplus_{s=2}^m s{\overline{1}}
{\otimes}\tilde{H}_{d}({\mathsf{M}_{{[2,m}}\setminus\{s\},[2,n]}]); \\
R^{m-2,n-2}_{d} &=&
\bigoplus_{s=2}^m \bigoplus_{t=2}^n
1{\overline{t}} \wedge s{\overline{1}} {\otimes}\tilde{H}_{d}({\mathsf{M}_{{[2,m}} \setminus \{s\},[2,n] \setminus \{t\}}]).
\end{aligned}$$ For each $m \ge 2$ and $n \ge 2$, we have a long exact sequence $$\begin{CD}
& & & & \cdots @>>>
R^{m-2,n-2}_{d-1} \\
@>>>
P^{m-1,n-2}_{d-1}
\oplus
Q^{m-2,n-1}_{d-1}
@>>>
\tilde{H}_{d}({\mathsf{M}_{m,n}}) @>>>
R^{m-2,n-2}_{d-2} \\
@>>>
P^{m-1,n-2}_{d-2}
\oplus
Q^{m-2,n-1}_{d-2} @>>> \cdots .
\end{CD}$$
We refer to this sequence as the [*00-12-21-22 sequence*]{}. The sequence played an important part in [[Shareshian]{}]{} and Wachs’ analysis [@ShWa] of the bottom nonvanishing homology of ${\mathsf{M}_{m,n}}$. Note that the sequence is symmetric in $m$ and $n$.
Long exact sequence relating ${\mathsf{M}_{m,n}}$, ${\Gamma_{m,n}}$, and ${\mathsf{M}_{m-1,n-1}}$ {#exseq00-G-11-sec}
-------------------------------------------------------------------------------------------------
The sequence in this section is very similar, but not identical, to the 00-01-11 sequence in Section \[exseq00-01-11-sec\]. Define $$\label{Gamma-eq}
{\Gamma_{m,n}} = \{\sigma \in {\mathsf{M}_{m,n}}: s{\overline{1}} \notin \sigma \mbox{
for } s \in [3,m]\}.$$
Define $$\hat{P}^{m-1,n-1}_{d} =
\bigoplus_{s=3}^m s{\overline{1}} {\otimes}\tilde{H}_{d}({\mathsf{M}_{{[m}}\setminus
\{s\},[2,n]}]);$$ note that this definition differs from that in Section [\[exseq00-01-11-sec\]]{}. For each $m \ge 1$ and $n \ge 1$, we have a long exact sequence $$\begin{CD}
& & & & \cdots @>>>
\hat{P}^{m-1,n-1}_{d} \\
@>>>
\tilde{H}_{d}({\Gamma_{m,n}}) @>>>
\tilde{H}_{d}({\mathsf{M}_{m,n}}) @>>>
\hat{P}^{m-1,n-1}_{d-1} \\
@>>>
\tilde{H}_{d-1}({\Gamma_{m,n}}) @>>> \cdots .
\end{CD}$$
This is the long exact sequence for the pair $({\mathsf{M}_{m,n}},{\Gamma_{m,n}})$.
We refer to this sequence as the [*00-$\mathit{\Gamma}$-11 sequence*]{}. Note that the sequence is asymmetric in $m$ and $n$.
Long exact sequence relating ${\Gamma_{m,n}}$, ${\mathsf{M}_{m-2,n-1}}$, and ${\mathsf{M}_{m-2,n-3}}$ {#exseqG-21-23-sec}
-----------------------------------------------------------------------------------------------------
Recall the definition of ${\Gamma_{m,n}}$ from (\[Gamma-eq\]).
Write $$\begin{aligned}
Q^{m-2,n-1}_{d} &=&
(1{\overline{1}} - 2{\overline{1}}) {\otimes}\tilde{H}_{d}({\mathsf{M}_{{[3,m}},[2,n]}]); \\
R^{m-2,n-3}_{d} &=&
\bigoplus_{s \neq t \in [2,n]}
1{\overline{s}}\wedge 2{\overline{t}} {\otimes}\tilde{H}_{d}({\mathsf{M}_{{[3,m}},[2,n] \setminus \{s,t\}}]).
\end{aligned}$$ For each $m \ge 2$ and $n \ge 3$, we have a long exact sequence $$\begin{CD}
& & & & \cdots @>>>
R^{m-2,n-3}_{d-1} \\
@>\varphi^*>>
Q^{m-2,n-1}_{d-1} @>\iota^*>>
\tilde{H}_{d}({\Gamma_{m,n}}) @>>>
R^{m-2,n-3}_{d-2} \\
@>>>
Q^{m-2,n-1}_{d-2} @>>> \cdots ,
\end{CD}$$ where $\varphi^*$ is induced by the map $\varphi$ defined by $$\varphi(1{\overline{s}} \wedge 2{\overline{t}}{\otimes}x)
= (1{\overline{1}} - 2{\overline{1}}) {\otimes}x.$$ and $\iota^*$ is induced by the natural map $\iota((1{\overline{1}} - 2{\overline{1}}) {\otimes}x) = (1{\overline{1}} - 2{\overline{1}}) \wedge
x$. \[exseqG-21-23-thm\]
Define a filtration $$\Delta^0_{m,n} \subset
\Delta^1_{m,n} \subset
\Delta^2_{m,n} = {\Gamma_{m,n}}$$ as follows:
- $\Delta^2_{m,n} = {\Gamma_{m,n}}$.
- $\Delta^1_{m,n}$ is the subcomplex of $\Delta^2_{m,n}$ obtained by removing all faces containing $\{1{\overline{s}},2{\overline{t}}\}$ for some $s,t \in [2,n]$.
- $\Delta^0_{m,n}$ is the subcomplex of $\Delta^1_{m,n}$ obtained by removing the elements $1{\overline{2}}, \ldots, 1{\overline{n}}$ and $2{\overline{2}}, \ldots, 2{\overline{n}}$.
Writing $\Delta^{-1}_{m,n} = \emptyset$, let us examine $\Delta^{i}_{m,n} \setminus
\Delta^{i-1}_{m,n}$ for $i = 0,1,2$.
$\bullet$ $i=0$. Note that $$\Delta^0_{m,n} = {\mathsf{M}_{2,1}} {*}{\mathsf{M}_{{[3,m}},[2,n]}]
\cong
{\mathsf{M}_{2,1}} {*}{\mathsf{M}_{m-2,n-1}}.$$ As a consequence, $$\tilde{H}_d(\Delta^0_{m,n})
\cong (1{\overline{1}} -2{\overline{1}}) {\otimes}\tilde{H}_{d-1}({\mathsf{M}_{{[3,m}},[2,n]}])
= Q_{d-1}^{m-2,n-1}.$$
$\bullet$ $i=1$. Observe that $$\Delta^1_{m,n} \setminus \Delta^0_{m,n}
= \bigcup_{a=1}^2\bigcup_{u=2}^n \{\{a{\overline{u}}\}\}
{*}{\mathsf{M}_{\{3-a\},\{1\}}} {*}{\mathsf{M}_{{[3,m}},[2,n] \setminus
\{u\}}].$$ It follows that $$\tilde{H}_d(\Delta^1_{m,n},\Delta^0_{m,n}) =
\bigoplus_{a,u}
a{\overline{u}} {\otimes}\tilde{H}_{d-1}({\mathsf{M}_{\{3-a\},\{1\}}} {*}{\mathsf{M}_{{[3,m}},[2,n] \setminus
\{u\}}]) = 0;$$ ${\mathsf{M}_{\{3-a\},\{1\}}} \cong {\mathsf{M}_{1,1}}$ is a point. In particular, $\tilde{H}_d(\Delta^1_{m,n}) \cong
\tilde{H}_d(\Delta^0_{m,n})$.
$\bullet$ $i=2$. We have that $$\Delta^2_{m,n} \setminus \Delta^1_{m,n}
=
\bigcup_{s,t \in [2,n]} \{\{1{\overline{s}},2{\overline{t}}\}\} {*}{\mathsf{M}_{{[3,m}},[2,n] \setminus
\{s,t\}}];$$ we may hence conclude that $$\tilde{H}_d(\Delta^2_{m,n},\Delta^1_{m,n}) =
\bigoplus_{s,t} 1{\overline{s}} \wedge 2{\overline{t}} {\otimes}\tilde{H}_{d-2}({\mathsf{M}_{{[3,m}},[2,n] \setminus
\{s,t\}}]) = R_{d-1}^{m-2,n-3}.$$ By the long exact sequence for the pair $(\Delta^2_{m,n},\Delta^1_{m,n})$, it remains to prove that the induced map $\varphi^*$ has properties as stated in the theorem. Now, in the long exact sequence for $(\Delta^2_{m,n},\Delta^1_{m,n})$, the induced boundary map from $\tilde{H}_{d+1}(\Delta^2_{m,n},\Delta^1_{m,n})$ to $\tilde{H}_{d}(\Delta^1_{m,n})$ maps the element $1{\overline{s}} \wedge 2{\overline{t}}{\otimes}z$ to $(2{\overline{t}} - 1{\overline{s}}) {\otimes}z$. Since $$(2{\overline{t}} - 1{\overline{s}}) {\otimes}z-\partial((1{\overline{1}} \wedge 2{\overline{t}}
+ 1{\overline{s}}\wedge 2{\overline{1}})
{\otimes}z) = (1{\overline{1}} - 2{\overline{1}}) {\otimes}z,$$ we are done.
We refer to the sequence in Theorem \[exseqG-21-23-thm\] as the [*$\mathit{\Gamma}$-21-23 sequence*]{}. Note that the sequence is asymmetric in $m$ and $n$.
Bottom nonvanishing homology {#bottomchess-sec}
============================
Using the long exact sequences in Sections \[exseq00-G-11-sec\] and \[exseqG-21-23-sec\], we give a computer-free proof that $\tilde{H}_{2}({\mathsf{M}_{5,5}};{\mathbb{Z}})$ is a group of size three. While the proof is complicated, our hope is that it may provide at least some insight into the structure of ${\mathsf{M}_{5,5}}$ and related chessboard complexes.
We have that $\tilde{H}_{2}({\mathsf{M}_{5,5}};{\mathbb{Z}}) \cong {\mathbb{Z}}_3$. \[m55-thm\]
First, we examine ${\mathsf{M}_{3,4}}$; for alignment with later parts of the proof, we consider ${\mathsf{M}_{{[3,5}},[2,5]}]$, thereby shifting the first index two steps and the second index one step. The long exact 00-$\Gamma$-11 sequence from Section \[exseq00-G-11-sec\] becomes $$\begin{CD}
0 @>>>
\tilde{H}_2({\Gamma_{{[3,5}},[2,5]}]) @>>>
\tilde{H}_2({\mathsf{M}_{{[3,5}},[2,5]}]) @>\omega^*>>
5{\overline{2}}{\otimes}\tilde{H}_1({\mathsf{M}_{{[3,4}},[3,5]}]) \\
@>>>
\tilde{H}_1({\Gamma_{{[3,5}},[2,5]}]) @>\iota^*>>
\tilde{H}_1({\mathsf{M}_{{[3,5}},[2,5]}]) @>>>
0.
\end{CD}$$ As [[Shareshian]{}]{} and Wachs observed [@ShWa §6], the complex ${\mathsf{M}_{m,m+1}}$ is an orientable pseudomanifold of dimension $m-1$. In particular, ${\mathsf{M}_{{[3,5}},[2,5]}]$ and ${\mathsf{M}_{{[3,4}},[3,5]}]$ are orientable pseudomanifolds of dimensions $2$ and $1$, respectively. Moreover, the top homology group of ${\mathsf{M}_{{[3,5}},[2,5]}]$ is generated by $$z = \sum_{\pi \in {\mathfrak{S}_{[2,5]}}}
{{\rm sgn}}(\pi) \cdot 3{\overline{\pi(3)}}\wedge 4{\overline{\pi(4)}} \wedge
5{\overline{\pi(5)}},$$ and the top homology group of ${\mathsf{M}_{{[3,4}},[3,5]}]$ is generated by $$z' = \sum_{\pi \in {\mathfrak{S}_{[3,5]}}}
{{\rm sgn}}(\pi) \cdot 3{\overline{\pi(3)}}\wedge 4{\overline{\pi(4)}}.$$ Since $\omega^*(z) = -z'$, the map $\omega^*$ is an isomorphism. As a consequence, the map $\iota^*$ induced by the natural inclusion map is also an isomorphism.
The long exact $\Gamma$-21-23 sequence for ${\Gamma_{{[3,5}},[2,5]}]$ from Section \[exseqG-21-23-sec\] becomes $$\begin{CD}
0 @>>>
(3{\overline{2}} - 4{\overline{2}}){\otimes}\tilde{H}_0({\mathsf{M}_{{\{5\},[3,5}}}])
@>\iota^*>>
\tilde{H}_1({\Gamma_{{[3,5}},[2,5]}]) @>>>
0,
\end{CD}$$ which yields that each of $\tilde{H}_1({\Gamma_{{[3,5}},[2,5]}])$ and $\tilde{H}_1({\mathsf{M}_{{[3,5}},[2,5]}])$ is generated by $e_i = (3{\overline{2}} - 4{\overline{2}}) \wedge (5{\overline{3}}-5{\overline{i}})$ for $i \in \{4,5\}$.
Now, consider ${\mathsf{M}_{5,5}}$. The tail end of the $\Gamma$-21-23 sequence is $$\begin{CD}
& & \displaystyle{\bigoplus_{s,t}}\ 1{\overline{s}} \wedge 2{\overline{t}}{\otimes}\tilde{H}_1({\mathsf{M}_{{[3,5}},[2,5]\setminus \{s,t\}}]) \\
@>\varphi^*>>
(1{\overline{1}}-2{\overline{1}}){\otimes}\tilde{H}_1({\mathsf{M}_{{[3,5}},[2,5]}])
@>\iota^*>> \tilde{H}_2({\Gamma_{5,5}}) \rightarrow
0,
\end{CD}$$ where the first sum ranges over all pairs of distinct elements $s,t \in [2,5]$. Writing $\{s,t,u,v\} = [2,5]$, we note that $\tilde{H}_1({\mathsf{M}_{{[3,5}},[2,5]\setminus
\{s,t\}}]) = \tilde{H}_1({\mathsf{M}_{{[3,5}},\{u,v\}}])$ is generated by the cycle $$z_{uv} = 3{\overline{u}} \wedge 4{\overline{v}} + 4{\overline{v}} \wedge 5{\overline{u}} + 5{\overline{u}}
\wedge 3{\overline{v}} + 3{\overline{v}} \wedge 4{\overline{u}} + 4{\overline{u}} \wedge 5{\overline{v}} +
5{\overline{v}} \wedge 3{\overline{u}}.$$ By Theorem \[exseqG-21-23-thm\], $\varphi^*$ maps $1{\overline{s}} \wedge 2{\overline{t}}{\otimes}z_{uv}$ to $(1{\overline{1}} - 2{\overline{1}}) {\otimes}z_{uv}$. Since $z_{uv} = z_{vu}$, we conclude that the image under $\varphi^*$ is generated by the six cycles $z_{23},z_{24},z_{25}, z_{34},z_{35},z_{45}$.
In $\tilde{H}_1({\mathsf{M}_{{[3,5}},[2,5]}])$, we have that $z_{st} = z_{uv}$, because $z_{st} - z_{uv}$ equals the boundary of $$\begin{aligned}
\gamma &=& 3{\overline{u}} \wedge 5{\overline{s}} \wedge 4{\overline{v}}
-
5{\overline{s}} \wedge 4{\overline{v}} \wedge 3{\overline{t}}
+
4{\overline{v}} \wedge 3{\overline{t}} \wedge 5{\overline{u}}
-
3{\overline{t}} \wedge 5{\overline{u}} \wedge 4{\overline{s}} \\
&+&
5{\overline{u}} \wedge 4{\overline{s}} \wedge 3{\overline{v}}
-
4{\overline{s}} \wedge 3{\overline{v}} \wedge 5{\overline{t}}
+
3{\overline{v}} \wedge 5{\overline{t}} \wedge 4{\overline{u}}
-
5{\overline{t}} \wedge 4{\overline{u}} \wedge 3{\overline{s}} \\
&+&
4{\overline{u}} \wedge 3{\overline{s}} \wedge 5{\overline{v}}
-
3{\overline{s}} \wedge 5{\overline{v}} \wedge 4{\overline{t}}
+
5{\overline{v}} \wedge 4{\overline{t}} \wedge 3{\overline{u}}
-
4{\overline{t}} \wedge 3{\overline{u}} \wedge 5{\overline{s}}.
\end{aligned}$$ Namely, $\gamma$ is of the form $a_1 \wedge a_2
\wedge a_3 - a_2
\wedge a_3 \wedge a_4 + \cdots - a_{12} \wedge a_1 \wedge a_2$, which yields the boundary $-a_1\wedge a_3 + a_2 \wedge a_4 -
\cdots + a_{12} \wedge a_2$. As a consequence, the image under $\varphi^*$ is generated by the three cycles $z_{34},z_{35},z_{45}$.
Assume that $s = 2$ and $\{t,u,v\} = \{3,4,5\}$ and write $$\begin{aligned}
w_{uv} &=&
5{\overline{u}} \wedge 4{\overline{s}} \wedge 3{\overline{v}}
- 4{\overline{s}} \wedge 3{\overline{v}} \wedge 5{\overline{t}}
+ 3{\overline{v}} \wedge 5{\overline{t}} \wedge 4{\overline{u}} \\
&-& 5{\overline{t}} \wedge 4{\overline{u}} \wedge 3{\overline{s}}
+ 4{\overline{u}} \wedge 3{\overline{s}} \wedge 5{\overline{v}}.
\end{aligned}$$ We obtain that $$\begin{aligned}
\partial(w_{uv}+w_{vu}) &=&
(5{\overline{u}} \wedge 4{\overline{s}} - 5{\overline{u}} \wedge 3{\overline{v}} + 4{\overline{s}} \wedge
5{\overline{t}}
- 3{\overline{v}} \wedge 4{\overline{u}} + 5{\overline{t}} \wedge 3{\overline{s}} \\
&-& 4{\overline{u}} \wedge 5{\overline{v}} + 3{\overline{s}} \wedge 5{\overline{v}}) +
(5{\overline{v}} \wedge 4{\overline{s}} - 5{\overline{v}} \wedge 3{\overline{u}}
+ 4{\overline{s}} \wedge 5{\overline{t}}\\
&-& 3{\overline{u}} \wedge 4{\overline{v}} + 5{\overline{t}} \wedge 3{\overline{s}}
- 4{\overline{v}} \wedge 5{\overline{u}} + 3{\overline{s}} \wedge 5{\overline{u}})\\
&=&
(4{\overline{s}} - 3{\overline{s}}) \wedge (2\cdot 5{\overline{t}} - 5{\overline{u}} - 5{\overline{v}})
- z_{uv}.
\end{aligned}$$ Since $s=2$, it follows that $z_{uv}$ is equal to either $-e_4 -
e_5$, $2e_4-e_5$, or $-e_4+2e_5$ in $\tilde{H}_1({\mathsf{M}_{{[3,5}}[2,5]}])$ depending on the values of $t$, $u$, and $v$.
We conclude that the set $\{\varphi^*(1{\overline{s}}\wedge 2{\overline{t}}
{\otimes}z_{uv}) : \{s,t,u,v\} = [2,5]\}$ generates the subgroup $\{ (1{\overline{1}}-2{\overline{1}}) {\otimes}(ae_4+be_5) :
a-b \equiv 0 \pmod{3}\}$ of $(1{\overline{1}}-2{\overline{1}}) {\otimes}\tilde{H}_1({\mathsf{M}_{{[3,5}},[2,5]}])$. As a consequence, $\tilde{H}_2({\Gamma_{5,5}}) \cong {\mathbb{Z}}_3$, and $$\rho = (1{\overline{1}}-2{\overline{1}}) \wedge (3{\overline{2}}-4{\overline{2}}) \wedge
(5{\overline{3}}-5{\overline{4}})$$ is a generator for this group. Swapping ${\overline{3}}$ and ${\overline{4}}$, we obtain $-\rho$; we obtain the same result if we swap $3$ and $4$ or if we swap $1$ and $2$. Hence, by symmetry, the group $$T = {\mathfrak{S}_{\{1,2\}}} \times {\mathfrak{S}_{\{3,4,5\}}} \times
{\mathfrak{S}_{\{{\overline{2}},{\overline{3}},{\overline{4}},{\overline{5}}\}}}$$ acts on $\tilde{H}_2({\Gamma_{5,5}}) \cong {\mathbb{Z}}_3$ by $\pi(\rho) = {{\rm sgn}}(\pi) \cdot \rho$.
It remains to prove that $\tilde{H}_2({\Gamma_{5,5}}) \cong
\tilde{H}_2({\mathsf{M}_{5,5}})$. For this, consider the tail end of the 00-$\Gamma$-11 sequence from Section \[exseq00-G-11-sec\]: $$\begin{CD}
\displaystyle{\bigoplus_{x=3}^5}\ x{\overline{1}}{\otimes}\tilde{H}_2({\mathsf{M}_{{[5}}\setminus \{x\},[2,5]}])
@>\psi^*>>
\tilde{H}_2({\Gamma_{5,5}}) @>>>
\tilde{H}_2({\mathsf{M}_{5,5}}) \rightarrow
0
\end{CD}$$ By a result due to [[Shareshian]{}]{} and Wachs [@ShWa Lemma 5.9], we have that $\tilde{H}_2({\mathsf{M}_{{[5}}\setminus \{x\},[2,5]}]) \cong
\tilde{H}_2({\mathsf{M}_{4,4}})$ is generated by cycles of type ${\genfrac{[}{]}{0pt}{}{3,2}{2}} \wedge {\genfrac{[}{]}{0pt}{}{1,2}{1}}$ and cycles of type ${\genfrac{[}{]}{0pt}{}{2,3}{2}} \wedge {\genfrac{[}{]}{0pt}{}{2,1}{1}}$; recall notation from Section \[basic-sec\]. By properties of $\psi^*$, we need only prove that any such cycle vanishes in $\tilde{H}_2({\Gamma_{5,5}})$ whenever $x \in [3,5]$.
$\bullet$ A cycle of the first type is of the form $z = \lambda \cdot \gamma \wedge (d{\overline{u}}-d{\overline{v}})$, where $\lambda$ is a constant scalar, $$\gamma = a{\overline{s}} \wedge b{\overline{t}} + b{\overline{t}} \wedge c{\overline{s}} + c{\overline{s}}
\wedge a{\overline{t}} + a{\overline{t}} \wedge b{\overline{s}} + b{\overline{s}} \wedge c{\overline{t}} +
c{\overline{t}} \wedge a{\overline{s}},$$ $\{a,b,c,d\} = [5] \setminus \{x\}$, and $\{s,t,u,v\} = [2,5]$. By the above discussion, swapping ${\overline{s}}$ and ${\overline{t}}$ in $z$ should yield $-z$, but obviously the same swap in $\gamma$ again yields $\gamma$, which implies that $z = - z$; hence $z=0$.
$\bullet$ A cycle of the second type is of the form $z = \lambda \cdot \gamma \wedge (c{\overline{v}}-d{\overline{v}})$, where $\lambda$ is a constant scalar, say $\lambda = 1$, and $$\gamma = a{\overline{s}} \wedge b{\overline{t}} + b{\overline{t}} \wedge a{\overline{u}} + a{\overline{u}}
\wedge b{\overline{s}} + b{\overline{s}} \wedge a{\overline{t}} + a{\overline{t}} \wedge b{\overline{u}} +
b{\overline{u}} \wedge a{\overline{s}};$$ again $\{a,b,c,d\} = [5] \setminus \{x\}$ and $\{s,t,u,v\} = [2,5]$. If $\{a,b\} \subset [3,5]$, then we may swap $a$ and $b$ and again conclude that $z = -z$; the same argument applies if $\{a,b\} =
\{1,2\}$. For the remaining case, we may assume that $c \in [1,2]$ and $d \in [3,5]$. Swapping $d$ and $x$ yields $-z = \gamma \wedge (c{\overline{v}}-x{\overline{v}})$; recall that $x \in
[3,5]$. As a consequence, $$2z = z-(-z) = \gamma \wedge (x{\overline{v}}-d{\overline{v}}) =
\partial(c{\overline{1}} \wedge \gamma \wedge (x{\overline{v}}-d{\overline{v}}));$$ hence $z$ is again zero. Namely, since $c \in [1,2]$, we have that $c{\overline{1}}$ is an element in ${\Gamma_{5,5}}$. As a consequence, $\psi^*$ is the zero map as desired.
By Theorems \[smallest-thm\] and \[chessfinite-thm\], the connectivity degree of ${\mathsf{M}_{m,n}}$ is exactly $\nu_{m,n}-1$ whenever $n \ge 2m-4$ or $(m,n) \in
\{(6,6),(7,7),(8,9)\}$. As mentioned in the introduction, [[Shareshian]{}]{} and Wachs [@ShWa] extended this result to all $(m,n) \neq (1,1)$, thereby settling a conjecture due to Björner [[et al$.$]{}]{} [@BLVZ]:
If $m \le n \le 2m-5$ and $(m,n) \neq (8,9)$, then there is nonvanishing $3$-torsion in $\tilde{H}_{\nu_{m,n}}({\mathsf{M}_{m,n}}; {\mathbb{Z}})$. If in addition $m+n \equiv 1 \pmod{3}$, then $\tilde{H}_{\nu_{m,n}}({\mathsf{M}_{m,n}}; {\mathbb{Z}}) \cong {\mathbb{Z}}_3$. \[chesstorsion-thm\]
By Theorem \[chessallover-thm\] in Section \[higherchess-sec\], there is nonvanshing $3$-torsion also in $\tilde{H}_{\nu_{8,9}}({\mathsf{M}_{8,9}}; {\mathbb{Z}})$; in that theorem, choose $(k,a,b) = (2,1,2)$.
[**\[Table 1\]**]{}
In fact, [[Shareshian]{}]{} and Wachs provided much more specific information about the exponent of $\tilde{H}_{\nu_{m,n}}({\mathsf{M}_{m,n}};
{\mathbb{Z}})$; see Table \[chessexp-fig\].
The group $\tilde{H}_{\nu_{m,n}}({\mathsf{M}_{m,n}}; {\mathbb{Z}})$ is torsion-free if and only if $n \ge 2m-4$. \[chesstorsion1-conj\]
The conjecture is known to be true in all cases but $n=2m-4$ and $n=2m-3$; [[Shareshian]{}]{} and Wachs [@ShWa] settled the case $n=2m-2$.
For all $(m,n) \neq (1,1)$, we have that $\tilde{H}_{\nu_{m,n}}({\mathsf{M}_{m,n}}; {\mathbb{Z}})$ is nonzero. \[chesstorsion-cor\]
Higher-degree homology {#generalchess-sec}
======================
In Section \[higherchess-sec\], we detect $3$-torsion in higher-degree homology groups of ${\mathsf{M}_{m,n}}$. In Section \[boundschess-sec\], we proceed with upper bounds on the dimension of the homology over ${\mathbb{Z}}_3$.
$3$-torsion in higher-degree homology groups {#higherchess-sec}
--------------------------------------------
This section builds on work previously published in the author’s thesis [@thesis; @lmnthesis]. Fix $n,d \ge 0$ and let $\gamma$ be an element in $\tilde{H}_{d-1}({\mathsf{M}_{n}};{\mathbb{Z}})$; note that we consider the matching complex ${\mathsf{M}_{n}}$. For each $k \ge 0$, define a map $$\left\{
\begin{array}{l}
\theta_k : \tilde{H}_{k-1}({\mathsf{M}_{k,k+1}};{\mathbb{Z}})
\rightarrow \tilde{H}_{k-1+d}({\mathsf{M}_{2k+1+n}};{\mathbb{Z}}) \\
\theta_k(z) = z \wedge \gamma^{(2k+1)},
\end{array}
\right.$$ where we obtain $\gamma^{(2k+1)}$ from $\gamma$ by replacing each occurrence of the vertex $i$ with $i+2k+1$ for every $i \in [n]$.
For any prime $p$, we have that $\theta_k$ induces a homomorphism $$\theta_{k} {\otimes}_{\mathbb{Z}}\iota_p : \tilde{H}_{k-1}({\mathsf{M}_{k,k+1}};{\mathbb{Z}})
{\otimes}_{\mathbb{Z}}{\mathbb{Z}}_p \rightarrow
\tilde{H}_{k-1+d}({\mathsf{M}_{2k+1+n}};{\mathbb{Z}}) {\otimes}_{\mathbb{Z}}{\mathbb{Z}}_p,$$ where $\iota_p : {\mathbb{Z}}_p \rightarrow {\mathbb{Z}}_p$ is the identity. The following result about the matching complex is a special case of a more general result from a previous paper [@bettimatch].
Fix $k_0 \ge 0$. With notation and assumptions as above, if $\theta_{k_0} {\otimes}_{\mathbb{Z}}\iota_p$ is a monomorphism, then $\theta_{k} {\otimes}_{\mathbb{Z}}\iota_p$ is a monomorphism for each $k \ge k_0$. \[torsionallover-thm\]
As alluded to in the proof of Theorem \[m55-thm\] in Section \[bottomchess-sec\], we have that ${\mathsf{M}_{k,k+1}}$ is an orientable pseudomanifold of dimension $k-1$; hence $\tilde{H}_{k-1}({\mathsf{M}_{k,k+1}};{\mathbb{Z}}) \cong {\mathbb{Z}}$. [[Shareshian]{}]{} and Wachs [@ShWa §6] observed that this group is generated by the cycle $$z_{k,k+1} = \sum_{\pi \in {\mathfrak{S}_{[k+1]}}}
{{\rm sgn}}(\pi) \cdot 1{\overline{\pi(1)}}\wedge \cdots \wedge
k{\overline{\pi(k)}}.$$ Note that the sum is over all permutations on $k+1$ elements. Theorem \[torsionallover-thm\] implies the following result.
With notation and assumptions as in Theorem [\[torsionallover-thm\]]{}, if $(z_{k_0,k_0+1} \wedge \gamma^{(2k_0+1)}) {\otimes}1$ is nonzero in $\tilde{H}_{k_0-1+d}({\mathsf{M}_{2k_0+1+n}};{\mathbb{Z}}) {\otimes}{\mathbb{Z}}_p$, then $(z_{k,k+1} \wedge \gamma^{(2k+1)}) {\otimes}1$ is nonzero in $\tilde{H}_{k-1+d}({\mathsf{M}_{2k+1+n}};{\mathbb{Z}}) {\otimes}{\mathbb{Z}}_p$ for all $k \ge k_0$. \[torsionallover-cor\]
We will also need a result about the bottom nonvanishing homology of the matching complex. Define $$\begin{aligned}
\nonumber
\gamma_{3r} &=& (12-23) \wedge (45-56) \wedge (78-89) \\
& & \wedge \cdots \wedge ((3r-2)(3r-1)-(3r-1)(3r));
\label{gamman-eq}
\end{aligned}$$ this is a cycle in both $\tilde{C}_{r-1}({\mathsf{M}_{3r}}; {\mathbb{Z}})$ and $\tilde{C}_{r-1}({\mathsf{M}_{3r+1}}; {\mathbb{Z}})$.
For $r \ge 2$, we have that $\tilde{H}_{r-1}({\mathsf{M}_{3r+1}};{\mathbb{Z}}) \cong {\mathbb{Z}}_3$. Moreover, this group is generated by $\gamma_{3r}$ and hence by any element obtained from $\gamma_{3r}$ by permuting the underlying vertex set. \[bouctor-thm\]
Assume that $m+n \equiv 0 \pmod{3}$ and $m \le n \le 2m$. Define the cycle $\gamma_{m,n}$ in $\tilde{H}_{\nu_{m,n}}({\mathsf{M}_{m,n}}; {\mathbb{Z}})$ recursively as follows, the base case being $\gamma_{1,2} = 1{\overline{1}}-1{\overline{2}}$: $$\gamma_{m,n} =
\left\{
\begin{array}{ll}
\gamma_{m-1,n-2} \wedge (m({\overline{n-1}})-m{\overline{n}}) & \mbox{if } m<n; \\
\gamma_{m-2,n-1} \wedge ((m-1){\overline{n}}-m{\overline{n}}) & \mbox{if } m=n.
\end{array}
\right.
\label{gammamn-eq}$$ For $n > m$, we define $\gamma_{n,m}$ by replacing $i{\overline{j}}$ with $j{\overline{i}}$ in $\gamma_{m,n}$ for each $i \in [m]$ and $j \in [n]$.
Recall that $\nu_{m,n} = \frac{m+n-4}{3}$ whenever $m \le n \le 2m-2$.
There is $3$-torsion in $\tilde{H}_{d}({\mathsf{M}_{m,n}}; {\mathbb{Z}})$ whenever $$\left\{
\begin{array}{ccl}
m+1 \le n \le 2m-5 \\ \\
\left\lceil\frac{m+n-4}{3}\right\rceil \le d \le m-3
\end{array}
\right.
\Longleftrightarrow
\left\{
\begin{array}{ccl}
k &\ge& 0\\
a &\ge& 1\\
b &\ge& 2,
\end{array}
\right.$$ where $k$, $a$, and $b$ are defined as in [(\[mnd2kab-eq\])]{}. Moreover, there is $3$-torsion in $\tilde{H}_{d}({\mathsf{M}_{m,m}}; {\mathbb{Z}})$ whenever $$\left\lceil\frac{2m-4}{3}\right\rceil \le d \le m-4
\Longleftrightarrow
\left\{
\begin{array}{ccl}
k &\ge& 0\\
a &=& 0 \\
b &\ge& 3.
\end{array}
\right.$$ \[chessallover-thm\]
Assume that $k \ge 0$, $a \ge 1$, and $b \ge 2$. Writing $m_0 = a+3b-2$ and $n_0 = 2a+3b-3$, we have the inequalities $$a+3b-2 \le 2a+3b-3 \le 2a+6b-9
\Longleftrightarrow
m_0 \le n_0 \le 2m_0-5.
\label{ab-eq}$$
Note that $m_0+n_0 = 3a+6b-5 \equiv 1 \pmod{3}$. Define $$w_{k+1} = z_{k+1,k+2} \wedge \gamma_{m_0,n_0-1}^{(k+1,k+2)},$$ where we obtain $\gamma_{m_0,n_0-1}^{(k+1,k+2)}$ from the cycle $\gamma_{m_0,n_0-1}$ defined in (\[gammamn-eq\]) by replacing $i{\overline{j}}$ with $(i+k+1)({\overline{j+k+2}})$. View $\gamma_{m_0,n_0-1}$ as an element in the homology of ${\mathsf{M}_{m_0,n_0}}$. Since $z_{k+1,k+2}$ has type ${\genfrac{[}{]}{0pt}{}{k+1,k+2}{k+1}}$ and since $\gamma_{m_0,n_0-1}$ has type ${\genfrac{[}{]}{0pt}{}{a+3b-2,2a+3b-3}{a+2b-2}}$ (or rather ${\genfrac{[}{]}{0pt}{}{a+3b-2,2a+3b-4}{a+2b-2}} \wedge {\genfrac{[}{]}{0pt}{}{0,1}{0}}$), we obtain that $w_{k+1}$ has type $${\genfrac{[}{]}{0pt}{}{k+1+a+3b-2,k+2+2a+3b-3}{k+1+a+2b-2}} =
{\genfrac{[}{]}{0pt}{}{m,n}{d+1}};$$ hence we may view $w_{k+1}$ as an element in $\tilde{H}_d({\mathsf{M}_{m,n}};{\mathbb{Z}})$.
Choosing $k=0$, we obtain that $$w_1 = z_{1,2} \wedge \gamma_{m_0,n_0-1}^{(1,2)}.$$ We claim that $w_1$ has order three when viewed as an element in $$\tilde{H}_{\frac{m_0+n_0-1}{3}}({\mathsf{M}_{m_0+n_0+3}};{\mathbb{Z}}) =
\tilde{H}_{a+2b-2}({\mathsf{M}_{3a+6b-2}};{\mathbb{Z}}).$$ Namely, we may relabel the vertices to transform $w_1$ into the cycle $\gamma_{m_0+n_0+2}$ defined in (\[gamman-eq\]). Since $m_0 + n_0 + 3 \ge 13$, Theorem \[bouctor-thm\] yields the claim.
Applying Corollary \[torsionallover-cor\], we conclude that $w_{k+1} {\otimes}1$ is a nonzero element in the group $\tilde{H}_{k+a+2b-2}({\mathsf{M}_{2k+3a+6b-2}};{\mathbb{Z}}) {\otimes}{\mathbb{Z}}_3 = \tilde{H}_{d}({\mathsf{M}_{m+n}};{\mathbb{Z}}) {\otimes}{\mathbb{Z}}_3$ for every $k \ge 0$. As a consequence, $w_{k+1} {\otimes}1$ is nonzero also in $$\tilde{H}_{k+a+2b-2}({\mathsf{M}_{k+a+3b-1,k+2a+3b-1}};{\mathbb{Z}}) {\otimes}{\mathbb{Z}}_3 = \tilde{H}_d({\mathsf{M}_{m,n}};{\mathbb{Z}}) {\otimes}{\mathbb{Z}}_3$$ for every $k \ge 1$. Since $\tilde{H}_{a+b-3}({\mathsf{M}_{m_0,n_0}};{\mathbb{Z}})$ is an elementary $3$-group by Theorem \[chesstorsion-thm\] and (\[ab-eq\]), the order of $\gamma_{m_0,n_0-1}$ in $\tilde{H}_{r}({\mathsf{M}_{m_0,n_0}};{\mathbb{Z}})$ is three. It follows that the order of $w_{k+1}$ in $\tilde{H}_d({\mathsf{M}_{m,n}};{\mathbb{Z}})$ is three as well.
The remaining case is $m=n$, in which case the upper bound on $d$ is $m-4$ rather than $m-3$. Since $a=0$, we get $$\left\{
\begin{array}{rcrcrcrcr}
k & = & -2m & + & 3d & + & 4 \\
b & = & m & - & d & - & 1
\end{array}
\right.
\Leftrightarrow
\left\{
\begin{array}{rcrcrcrcl}
m & = & k & + & 3 b & - & 1 \\
d & = & k & + & 2 b & - & 2.
\end{array}
\right.$$ Clearly, $k \ge 0$ and $b \ge 3$.
Consider the cycle $w_{k+1} = z_{k+1,k+2} \wedge
\gamma_{3b-2,3b-4}^{(k+1,k+2)}$. By Corollary \[torsionallover-cor\], $w_{k+1} {\otimes}1$ is nonzero in $\tilde{H}_{k+2b-2}({\mathsf{M}_{2k+6b-2}};{\mathbb{Z}}) {\otimes}{\mathbb{Z}}_3$. Namely, up to the names of the vertices, $w_1$ coincides with $\gamma_{6b-3}$ in (\[gamman-eq\]), which is a nonzero element of order three in the group $\tilde{H}_{2b-2}({\mathsf{M}_{6b-2}};{\mathbb{Z}})$ by Theorem \[bouctor-thm\]; $b \ge 3$. We conclude that $w_{k+1} {\otimes}1$ is a nonzero element in $\tilde{H}_{k+2b-2}({\mathsf{M}_{k+3b-1,k+3b-1}};{\mathbb{Z}})
{\otimes}{\mathbb{Z}}_3 = \tilde{H}_d({\mathsf{M}_{m,m}};{\mathbb{Z}}) {\otimes}{\mathbb{Z}}_3$. Since $3b-3 \ge 6$, we have that $\gamma_{3b-2,3b-4}$ must have order three in $\tilde{H}_{2b-3}({\mathsf{M}_{3b-2,3b-3}};{\mathbb{Z}})$; apply Theorem \[chesstorsion-thm\]. This implies that the same must be true for $w_{k+1}$ in $\tilde{H}_d({\mathsf{M}_{m,m}};{\mathbb{Z}})$.
The group $\tilde{H}_{5}({\mathsf{M}_{8,9}}; {\mathbb{Z}}) =
\tilde{H}_{\nu_{8,9}}({\mathsf{M}_{8,9}}; {\mathbb{Z}})$ contains nonvanishing $3$-torsion. As a consequence, there is nonvanishing $3$-torsion in $\tilde{H}_{\nu_{m,n}}({\mathsf{M}_{m,n}}; {\mathbb{Z}})$ whenever $m \le n \le 2m-5$.
The first statement is a consequence, of Theorem \[chessallover-thm\]; choose $k=2$, $a=1$, and $b=2$. For the second statement, apply Theorem \[chesstorsion-thm\].
For $1 \le m \le n$, the group $\tilde{H}_{d}({\mathsf{M}_{m,n}}; {\mathbb{Z}})$ is nonzero if and only if either $$\left\lceil\frac{m+n-4}{3}\right\rceil \le d \le m-2
\Longleftrightarrow
\left\{
\begin{array}{ccl}
k &\ge& 0\\
a &\ge& 0\\
b &\ge& 1
\end{array}
\right.$$ or $$\left\{
\begin{array}{ccl}
m &\ge& 1\\
n &\ge& m+1\\
d &=& m-1
\end{array}
\right.
\Longleftrightarrow
\left\{
\begin{array}{ccl}
k &\ge& 2-a\\
a &\ge& 1\\
b &=& 0,
\end{array}
\right.$$ where $k$, $a$, and $b$ are defined as in [(\[mnd2kab-eq\])]{}. \[chesshomology-thm\]
For homology to exist, we certainly must have that $b \ge 0$, and we restrict to $a \ge 0$ by assumption. Moreover, $b = 0$ means that $d = m-1$, in which case there is homology only if $m \le n-1$, hence $a \ge 1$ and $k+a \ge 2$; for the latter inequality, recall that we restrict our attention to $m \ge 1$. Finally, $k < 0$ reduces to the case $b = 0$, because we then have homology only if $n \ge 2m+2$ and $d=m-1$; apply Theorem \[smallest-thm\].
For the other direction, Theorem \[chessallover-thm\] yields that we only need to consider the following cases:
$\bullet$ $k \ge 0$, $a=0$, and $b=2$. By Theorem \[chessfinite-thm\], we have infinite homology for $a=0$ and $b=2$ if and only if $k
\ge (b-1)(a+b-1) = a+1 = 1$. The remaining case is $(k,a,b) =
(0,0,2) \Longleftrightarrow (m,n,d) = (5,5,2)$, in which case we have nonzero homology by Theorem \[m55-thm\].
$\bullet$ $k \ge 0$, $a \ge 0$, and $b = 1$. This time, Theorem \[chessfinite-thm\] yields infinite homology for $a \ge 0$ and $b=1$ as soon as $k \ge 0$.
$\bullet$ $k \ge 2-a$, $a \ge 1$, and $b=0$. By yet another application of Theorem \[chessfinite-thm\], we have infinite homology for $b=0$ whenever $a \ge 1$, $k \ge 1-a$, and $k+a \ge 2$. Since the third inequality implies the second, we are done.
For $1 \le m \le n$, the group $\tilde{H}_{d}({\mathsf{M}_{m,n}}; {\mathbb{Z}})$ contains $3$-torsion if and only if $$\left\{
\begin{array}{ccl}
m \le n \le 2m-5 \\ \\
\left\lceil\frac{m+n-4}{3}\right\rceil \le d \le m-3
\end{array}
\right.
\Longleftrightarrow
\left\{
\begin{array}{ccl}
k &\ge& 0\\
a &\ge& 0\\
b &\ge& 2.
\end{array}
\right.$$ \[chesstorsion2-conj\]
Note that Conjecture \[chesstorsion2-conj\] implies Conjecture \[chesstorsion1-conj\]. Conjecture \[chesstorsion2-conj\] remains unsettled in the following cases:
- $d=m-2$: $9 \le m+2 \le n \le 2m-3$. Equivalently, $k \ge 1$, $a
\ge 2$, and $b = 1$. Conjecture: There is no $3$-torsion.
- $d=m-3$: $8 \le m = n$. Equivalently, $k \ge 3$, $a = 0$, and $b = 2$. Conjecture: There is $3$-torsion.
The conjecture is fully settled for $n = m+1$ and $n \ge
2m-2$; see [[Shareshian]{}]{} and Wachs [@ShWa] for the case $n=2m-2$, and use Theorem \[smallest-thm\] for the case $n \ge
2m-1$. For the case $n=m+1$, we have that $\tilde{H}_{m-2}({\mathsf{M}_{m,m+1}};{\mathbb{Z}})$ is torsion-free, because ${\mathsf{M}_{m,m+1}}$ is an orientable pseudomanifold; see Spanier [@Spanier Ex. 4.E.2].
Bounds on the homology over ${\mathbb{Z}}_3$ {#boundschess-sec}
--------------------------------------------
Fix a field ${\mathbb{F}}$ and let $$\begin{aligned}
{\beta_{d}^{m,n}} &=& \dim_{\mathbb{F}}\tilde{H}_d({\mathsf{M}_{m,n}};{\mathbb{F}});\\
{\alpha_{d}^{m,n}} &=& \dim_{\mathbb{F}}\tilde{H}_d({\Gamma_{m,n}};{\mathbb{F}});
\end{aligned}$$ ${\Gamma_{m,n}}$ is defined as in (\[Gamma-eq\]).
For each $m \ge 2$ and $n \ge 3$, we have that $${\beta_{d}^{m,n}} \le
{\beta_{d-1}^{m-2,n-1}}
+ (m-2) {\beta_{d-1}^{m-1,n-1}}
+ 2\mbox{$\binom{n-1}{2}$}{\beta_{d-2}^{m-2,n-3}}.$$ Thus, by symmetry, $${\beta_{d}^{m,n}} \le
{\beta_{d-1}^{m-1,n-2}} +
(n-2) {\beta_{d-1}^{m-1,n-1}} +
2\mbox{$\binom{m-1}{2}$}{\beta_{d-2}^{m-3,n-2}}$$ whenever $m \ge 3$ and $n \ge 2$. \[dmtchess-lem\]
By the long exact 00-$\Gamma$-11 sequence in Section \[exseq00-G-11-sec\], we have that $${\beta_{d}^{m,n}} \le {\alpha_{d}^{m,n}} + (m-2) {\beta_{d-1}^{m-1,n-1}}.$$ Moreover, the long exact $\Gamma$-21-23 sequence in Section \[exseqG-21-23-sec\] yields the inequality $${\alpha_{d}^{m,n}} \le {\beta_{d-1}^{m-2,n-1}} +
2\mbox{$\binom{n-1}{2}$} {\beta_{d-2}^{m-2,n-3}}.$$ Summing, we obtain the desired inequality.
Define $\hat{\beta}_{k}^{a,b} = {\beta_{d}^{m,n}}$, where $k$, $a$, and $b$ are defined as in (\[mnd2kab-eq\]). We may rewrite the second inequality in Lemma \[dmtchess-lem\] as follows:
We have that $$\hat{\beta}_{k}^{a,b} \le
\hat{\beta}_{k}^{a-1,b} + (k+2a+3b-3)\hat{\beta}_{k-1}^{a,b} +
2\mbox{$\binom{k+a+3b-2}{2}$}\hat{\beta}_{k-1}^{a+1,b-1}
\label{hatbetachess-eq}$$ for $k \ge 0$, $a \ge 0$, and $b \ge 2$. \[dmtchess-cor\]
With ${\mathbb{F}}= {\mathbb{Z}}_3$ and $d = \nu_{m,n}$, the second bound in Lemma [\[dmtchess-lem\]]{} is sharp whenever $m \le n \le 2m-5$, $m+n \equiv 1 \pmod{3}$, and $(m,n) \neq (5,5)$. Equivalently, the bound is sharp whenever $k = 0$, $a \ge 0$, $b
\ge 2$, and $(k,a,b) \neq (0,0,2)$, where $k$, $a$, and $b$ are defined as in [(\[mnd2kab-eq\])]{}. \[basecase-thm\]
Since $\hat{\beta}^{a,b}_0 = 1$ for $a \ge 0$ and $b \ge 2$ by Theorem \[chesstorsion-thm\], it suffices to prove that $$\hat{\beta}_{0}^{a-1,b} + (2a+3b-3)\hat{\beta}_{-1}^{a,b} +
2\mbox{$\binom{a+3b-2}{2}$}\hat{\beta}_{-1}^{a+1,b-1} = 1
\label{basecase-eq}$$ for all $a$ and $b$ as in the theorem; apply Corollary \[dmtchess-cor\]. Clearly, $\hat{\beta}_{0}^{a-1,b} = 1$; when $a=0$, use the fact that $\hat{\beta}_{0}^{-1,b} = \hat{\beta}_{0}^{1,b-1}$. Moreover, Theorem \[smallest-thm\] yields that $\hat{\beta}_{-1}^{a,b} =
0$ whenever $a \ge 0$ and $b \ge 1$. As a consequence, we are done.
For each $k \ge 0$, there is a polynomial $f_k(a,b)$ of degree $3k$ such that $\hat{\beta}^{a,b}_{k} \le f_{k}(a,b)$ whenever $a \ge 0$ and $b \ge k+2$ and such that $$f_k(a,b) = \frac{1}{3^kk!}\left((a+3b)^3-9b^3\right)^{k} +
\epsilon_k(a,b)$$ for some polynomial $\epsilon_k(a,b)$ of degree at most $3k-1$. Equivalently, $${\beta_{d}^{m,n}} \le f_{3d-m-n+4}(n-m,m-d-1)$$ for $m \le n \le 2m-5$ and $\frac{m+n-4}{3} \le d \le \frac{2m+n-7}{4}$. \[bettichess-thm\]
The case $k=0$ is a consequence of Theorem \[chesstorsion-thm\]. Assume that $k \ge 1$ and $b > k+2$.
First, assume that $a > 0$. Induction and Corollary \[dmtchess-cor\] yield that $$\begin{aligned}
\hat{\beta}_{k}^{a,b} -
\hat{\beta}_{k}^{a-1,b} &\le&
(k+2a+3b-3)f_{k-1}(a,b) \nonumber\\
&+&
2\mbox{$\binom{k+a+3b-2}{2}$} f_{k-1}(a+1,b-1),
\label{hatbetasimpchess2-eq}
\end{aligned}$$ where $f_{k-1}$ is a polynomial with properties as in the theorem. The right-hand side is of the form $$g_k(a,b) =
\frac{1}{3^{k-1}(k-1)!}\left((a+3b)^3-9b^3\right)^{k-1}(a+3b)^2 +
h_k(a,b),$$ where $h_k(a,b)$ is a polynomial of degree at most $3k-2$. Now, $$\begin{aligned}
& &
\frac{1}{3^{k-1} (k-1)!}\left((a+3b)^3-9b^3\right)^{k-1}(a+3b)^2
\\
&=&
\frac{1}{3^{k-1} (k-1)!} \sum_{\ell=0}^{k-1} \binom{k-1}{l}
(a+3b)^{3k-3\ell-1} (-9b^3)^{\ell}.
\end{aligned}$$ Summing over $a$, we obtain that $$\hat{\beta}_{k}^{a,b} \le
\hat{\beta}_{k}^{0,b} + \sum_{i=1}^a g_k(i,b).$$ The right-hand side is a polynomial in $a$ and $b$ with dominating term $$\begin{aligned}
& &
\frac{1}{3^{k-1} (k-1)!} \sum_{\ell=0}^{k-1} \binom{k-1}{l}
\frac{(a+3b)^{3k-3\ell}-(3b)^{3k-3\ell}}{3k-3\ell}
(-9b^3)^{\ell} \nonumber \\
&=&
\frac{1}{3^{k} k!} \sum_{\ell=0}^{k} \binom{k}{l}
\left(((a+3b)^3)^{k-\ell} - (27b^3)^{k-\ell}\right)
(-9b^3)^{\ell}
\nonumber \\
&=&
\frac{1}{3^{k} k!} \left((a+3b)^3-9b^3\right)^k
- \frac{1}{3^{k} k!}(18b^3)^{k}. \label{tailbterm-eq}
\end{aligned}$$
Proceeding with $\hat{\beta}_{k}^{0,b}$ for $b \ge k+3$, note that $\hat{\beta}_{k}^{-1,b} = \hat{\beta}_{k}^{1,b-1}$. As a consequence, $$\begin{aligned}
\hat{\beta}_{k}^{0,b} &\le&
\hat{\beta}_{k}^{1,b-1} + (k+3b-3)\hat{\beta}_{k-1}^{0,b} +
2\mbox{$\binom{k+3b-2}{2}$}\hat{\beta}_{k-1}^{1,b-1} \\
&\le& \hat{\beta}_{k}^{0,b-1} +
(k+3b-4)\hat{\beta}_{k-1}^{1,b-1} +
2\mbox{$\binom{k+3b-4}{2}$}\hat{\beta}_{k-1}^{2,b-2}\\
&+& (k+3b-3)\hat{\beta}_{k-1}^{0,b} +
2\mbox{$\binom{k+3b-2}{2}$}\hat{\beta}_{k-1}^{1,b-1}.
\end{aligned}$$ Using induction, we conclude that $$\begin{aligned}
\hat{\beta}_{k}^{0,b} &\le&
\hat{\beta}_{k}^{0,b-1} +
9b^2f_{k-1}(2,b-2) +
9b^2f_{k-1}(1,b-1) +O(b^{3k-2}) \\
&=& 18b^2 \frac{(18b^3)^{k-1}}{3^{k-1}(k-1)!} +O(b^{3k-2})
= \frac{18^kb^{3k-1}}{3^{k-1}(k-1)!} +O(b^{3k-2}),
\end{aligned}$$ where $f_{k-1}$ is a polynomial with properties as in the theorem. Summing over $b$, we may conclude that $\hat{\beta}_{k}^{0,b}$ is bounded by a polynomial in $b$ with dominating term $\frac{18^kb^{3k}}{3^{k}k!}$. Combined with (\[tailbterm-eq\]), this yields the theorem.
Acknowledgments {#acknowledgments .unnumbered}
===============
I thank an anonymous referee for several useful comments. This research was carried out at the Technische Universität Berlin and at the Massachusetts Institute of Technology in Cambridge, MA.
[99]{} J. L. Andersen, *Determinantal Rings Associated with Symmetric Matrices: a Counterexample*, PhD Thesis, University of Minnesota, 1992. , [Decompositions and connectivity of matching and chessboard complexes]{}, [*Discrete Comput. Geom.*]{} [**31**]{} (2004), no. 3, 395–403. E. Babson, A. Björner, S. Linusson, J. [[Shareshian]{}]{} and V. Welker, [Complexes of not $i$-connected graphs]{}, [*Topology*]{} [**38**]{} (1999), no. 2, [271–299]{}. A. Björner, L. Lovász, S. T. Vrećica and R. T. [[$\check{\mathrm{Z}}$ivaljević]{}]{}, [Chessboard complexes and matching complexes]{}, *J. London Math. Soc. (2)* **49** (1994), 25–39. S. Bouc, [Homologie de certains ensembles de 2-sous-groupes des groupes symétriques]{}, [*J. Algebra*]{} **150** (1992), 187–205. , [*The Topology of Bounded Degree Graph Complexes and Finite Free Resolutions*]{}, PhD Thesis, [University of Minnesota]{}, 2001. , [Combinatorial Laplacian of the matching complex]{}, [*Electronic J. Combin.*]{} **9** (2002), no. 1, R17. J. Friedman and P. Hanlon, On the Betti numbers of chessboard complexes, [*J. Algebraic Combin.*]{} [**8**]{} (1998), 193–203. P. F. Garst, [*Cohen-Macaulay complexes and group actions*]{}, Ph.D. Thesis, University of Wisconsin-Madison, 1979. J. Jonsson, *Simplicial Complexes of Graphs*, Doctoral Thesis, KTH, 2005. J. Jonsson, [*Simplicial Complexes of Graphs*]{}, Lecture Notes in Mathematics 1928, Springer, 2008. J. Jonsson, Exact sequences for the homology of the matching complex, [*J. Combin. Theory Ser. A*]{} [**115**]{} (2008), 1504–1526. J. Jonsson, Five-torsion in the homology of the matching complex on $14$ vertices, [ *J. Algebraic Combin.*]{} [**29**]{} (2009), no. 1, 81–90. J. Jonsson, More torsion in the homology of the matching complex, submitted. , [Homology of complexes of degree one graphs]{}, PhD Thesis, [Stanford University]{}, 1994. D. B. Karaguezian, V. Reiner and M. L. Wachs, Matching complexes, bounded degree graph complexes and weight spaces of $GL_n$-complexes, [*J. Algebra*]{} [**239**]{} (2001), 77–92. , [*Propriétés homotopiques du complexe de [Quillen]{} du groupe symétrique*]{}, PhD Thesis, [Université de Lausanne]{}, 2000. , [Minimal resolutions and homology of chessboard and matching complexes]{}, [*J. Algebraic Combin.*]{} [**11**]{} (2000), [135–154]{}. J. [[Shareshian]{}]{} and M. L. Wachs, Torsion in the matching complex and chessboard complex, [*Adv. Math.*]{} [**212**]{} (2007), no. 2, 525–570. E.H. Spanier, [*Algebraic Topology*]{}, McGraw-Hill, 1966. M. L. Wachs, [Topology of matching, chessboard and general bounded degree graph complexes]{}, *Alg. Universalis* **49** (2003), no. 4, 345–385. G. M. Ziegler, Shellability of chessboard complexes, [*Israel J. Math.*]{} **87** (1994), 97–110.
$2m-n$ Restriction $\epsilon_{m,n}$ $k$ $a$ $b$
--------- ------------- ---------------------------------- ----- --------- -------
$5$ $3$ $0$ $\ge 0$ $2$
$6$ $m \ge 7$ divides $\epsilon_{7,8}$ $1$ $\ge 1$
$7$ $m \ge 9$ divides $\epsilon_{9,11}$ $2$ $\ge 2$
$8$ $3$ $0$ $\ge 0$ $3$
$9$ divides $\gcd(9,\epsilon_{9,9})$ $1$ $\ge 0$
$10$ $m=10$ multiple of $3$ $2$ $=0$
$m \ge 11$ divides $\epsilon_{7,8}$ $\ge 1$
$11+3t$ $t \ge 0$ $3$ $0$ $\ge 0$ $4+t$
$12+3t$ divides $\gcd(9,\epsilon_{9,9})$ $1$
$13+3t$ $2$
: The exponent $\epsilon_{m,n}$ of $\tilde{H}_{\nu_{m,n}}({\mathsf{M}_{m,n}}; {\mathbb{Z}})$ for $m \le n \le 2m-5$ and $(m,n) \not\in
\{(6,6),(7,7),(8,9)\}$. On the right we give the values $k$, $a$, and $b$ defined as in (\[mnd2kab-eq\]).
\[chessexp-fig\]
[^1]: Research supported by European Graduate Program “Combinatorics, Geometry, and Computation”, DFG-GRK 588/2.
|
---
abstract: 'We report novel magnetotransport properties of the low temperature Fermi liquid in SrTiO$_{3-\emph{x}}$ single crystals. The classical limit dominates the magnetotransport properties for a magnetic field perpendicular to the sample surface and consequently a magnetic-field induced resistivity minimum emerges. While for the field applied in plane and normal to the current, the linear magnetoresistance (MR) starting from small fields ($<$ 0.5 T) appears. The large anisotropy in the transverse MRs reveals the strong surface interlayer scattering due to the large gradient of oxygen vacancy concentration from the surface to the interior of SrTiO$_{3-\emph{x}}$ single crystals. Moreover, the linear MR in our case was likely due to the inhomogeneity of oxygen vacancies and oxygen vacancy clusters, which could provide experimental evidences for the unusual quantum linear MR proposed by Abrikosov \[A. A. Abrikosov, Phys. Rev. B **58**, 2788 (1998)\].'
author:
- 'Z. Q. Liu (Zhiqi Liu)$^{1,2}$'
- 'W. M. Lü$^{1,3}$'
- 'X. Wang$^{1,2}$'
- 'Z. Huang$^{1}$'
- 'A. Annadi$^{1,2}$'
- 'S. W. Zeng$^{1,2}$'
- 'T. Venkatesan$^{1,2,3}$'
- 'Ariando$^{1,2}$'
title: 'Magnetic-field induced resistivity minimum with in-plane linear magnetoresistance of the Fermi liquid in SrTiO$_{3-\emph{x}}$ single crystals'
---
Introduction
============
SrTiO$_{3}$ (STO) is one of the most important workhorses in oxide electronics. Recently, a two-dimensional electron gas \[1,2\] and electronic phase separation \[3\] have been demonstrated to emerge on the bare STO single crystal surface. Understanding the electronic properties of STO under different oxidation states is therefore crucial to reveal the origin of these novel phenomena and to use STO in electronic devices. Generally, STO is a nonpolar band insulator with an indirect bandgap of $\sim$3.27 eV \[4\] and a large dielectric constant *$\varepsilon$$_{r}$* \[5\]. A semiconducting (or metallic) phase of STO can be obtained by reduction \[6\], chemical doping \[7\] or photo-carrier injection \[8\], with a high carrier mobility ($>$10,000 cm$^{2}$V$^{-1}$s$^{-1}$) at low temperatures, a large density-of-states effective mass *m$_{D}$* = 5$\sim$6*m*$_{0}$ \[9,10\] and a large cyclotron mass *m$_{c}$* = 1.5$\sim$2.9*m*$_{0}$ \[10\], where *m*$_{0}$ is the electron rest mass. The high mobility carriers allow the observation of magnetic quantum effects like Shubnikov-de Haas oscillation. However, the additional conditions \[11\] *E$_{F}$* = ($\hbar$$^{2}$/2*m$_{D}$*)(3$\pi$$^{2}$*n*)$^{2/3}$ $\gg$ *kT* and $\hbar$*w$_{c}$* = $\hbar$*eB/m$_{c}$* $\gg$ *kT* for a pronounced effect of quantization on magnetotransport have also to be considered, where *E$_{F}$* is the Fermi energy, *n* the carrier density, *B* the applied magnetic field and *w$_{c}$* = *eB/m$_{c}$* the cyclotron frequency. Taking *m$_{c}$* = 2*m*$_{0}$ for example, the magnetic energy $\hbar$*w$_{c}$* at 9 T exceeds the thermal energy *kT* below $\sim$6 K.
![\[fig1\] (Color online) Temperature dependence of (a) resistivity (*$\rho$-T*), (b) carrier density (*n-T*) and (c) mobility (*$\mu$-T*) of a SrTiO$_{3}$ single crytal reduced in $\sim$10$^{-7}$ Torr vacuum at 950 $^\circ$C for 1 h. Inset of (a): linear fitting of *T*$^{2}$ dependence of the resistivity. The carrier density *n* in (b) was averaged over the entire crystal thickness.](liu_fig1.pdf){width="3.4in"}
In this paper we report electrical and magnetotransport studies of SrTiO$_{3-\emph{x}}$ single crystals ($5\times5\times0.5$ mm$^3$) which were reduced in $\sim$10$^{-7}$ Torr at 950$^\circ$C \[6,12\] for different times. The electrical contacts were made by wire bonding using aluminum wires. All the transport measurements were performed in a Quantum Design Physical Property Measurement System.
Electrical Properties
=====================
The temperature dependence of resistivity (*$\rho$-T*), carrier density (*n-T*) (*n* is an average carrier density value over the entire crystal thickness) and mobility (*$\mu$-T*) of samples reduced for 1 h is depicted in Fig.1, showing a metallic behaviour over the whole temperature range from 300 to 2 K (Fig. 1(a)). The origin of this metallic behavior can be understood in terms of the Mott criterion \[13\]. The critical carrier density for a metal-insulator transition is given by the Mott critical carrier density *n$_{c}$* $\approx$ (0.25/*a$^{\ast}$*)$^{3}$, where *a$^{\ast}$* = 4$\pi$*$\varepsilon$$_{r}$$\varepsilon$*$_{0}$$\hbar$$^{2}$/*m$_{D}$e*$^{2}$ is the effective Bohr radius and *$\varepsilon$*$_{0}$ the vacuum permittivity. The measured carrier density at 300 K is $\sim$1.5$\times$10$^{18}$ cm$^{-3}$, which is more than three times *n$_{c}$* $\sim$ 4.9$\times$10$^{17}$ cm$^{-3}$, considering the room temperature *$\varepsilon$$_{r}$* $\approx$ 300 and *m$_{D}$* $\approx$ 5*m*$_{0}$ for STO. Interestingly, the resistivity from 2 up to $\sim$80 K exhibits an obvious behavior of a strongly correlated Fermi liquid as fitted in the inset of Fig. 1(a), reminiscent of the normal state of electron-doped cuprate superconductors \[14\] and the *p*-wave superconductor Sr$_{2}$RuO$_{4}$ \[15\], noting that semiconducting STO is also superconducting at very low temperatures \[12\]. Moreover, the common Fermi liquid origin of the *T*$^{2}$ resistivity and superconductivity of *n*-type SrTiO$_{3}$ has been elaborately discussed by Marel *et al*. \[16\].
The detailed *n-T* curve determined by Hall measurements is shown in Fig. 1(b). At high temperatures, the carrier density slightly fluctuates; however, it increases with decreasing temperature at low temperatures especially between 100 and 10 K. This unexpected behavior is unphysical from the viewpoint of thermal activation. In fact, this behavior was also observed previously in semiconducting STO single crystals \[7,8\] and LaAlO$_{3}$/STO heterostructures grown at low oxygen pressures \[3,17\]. Therefore there should be another intrinsic mechanism affecting the carrier density.
One unique property of STO is that its *$\varepsilon$$_{r}$* increases with lowering temperature (especially from $\sim$100 K) and saturates at 4 K because of the quantum-mechanical stabilization of the paraelectric phase \[5\]. It seems plausible to assume that in SrTiO$_{3-\emph{x}}$ part of carriers are trapped by the Coulomb potentials of the majority of positively charged defects due to the strongly ionic nature of the lattice. As the *$\varepsilon$$_{r}$* increases, the Coulomb potentials will be suppressed due to dielectric screening since the screened Coulomb potential \[18\] is inversely proportional to *$\varepsilon$$_{r}$*. Hence the increase of *$\varepsilon$$_{r}$* could serve as a kind of detrapping mechanism and consequently account for the increase of carrier density in SrTiO$_{3-\emph{x}}$ at low temperatures.
The *$\mu$-T* curve is plotted in Fig. 1(c) on a logarithmic scale. The high mobility, up to $\sim$11,000 cm$^{2}$V$^{-1}$s$^{-1}$ at 2 K, decreases with temperature rapidly and varies in accordance with certain power laws above 30 K, where the scattering of electrons by polar optical phonons dominates and results in $\mu$ $\propto$ *T$^{-\beta}$* \[19\]. The effect of the structural phase transition in STO at $\sim$105 K on the mobility can be apparently seen from the linear fittings owing to the variation in the activation of phonon modes. Below 30 K, the scattering of electrons is dominated by ionized defect scattering and electron-electron Umklapp scattering because of the Fermi liquid behavior \[15\]. However, the electron-electron scattering should be enhanced with increasing carrier density, so the ionized defect scattering is the more pronounced mechanism for the continuously increasing mobility similarly due to the dielectric screening of ionized scattering potentials \[7\]. Finally, the trend to saturation appears from 5 K possibly corresponding to the quantum paraelectric phase in STO.
![\[fig2\] (Color online) (a) *$\rho$-T* curves under different magnetic fields. (b) Extracted resistivity minimum temperature from (a) versus magnetic field. (c) Large field Hall effect at 2 K from -5 to 5 T.](liu_fig2.pdf){width="3.4in"}
Magnetotransport Properties
===========================
The *$\rho$-T* curves under different magnetic fields perpendicular to the surface are shown in Fig. 2(a). The transverse MR effect ($\triangle\rho$/$\rho$(0) = \[$\rho$(B)- $\rho$(0)\]/$\rho$(0)) is notable only below $\sim$50 K and always positive. Intriguingly, the *$\rho$-T* curves under sufficiently strong magnetic fields ($\geq$2.5 T) exhibit a resistivity minimum at a temperature *T$_{min}$*, which increases with *B* monotonically as plotted in Fig. 2(b). The resistivity minimum cannot pertain to a Kondo effect or weak localization since they are inherently antagonistic towards magnetic fields. The antilocalization effect is also ruled out since it is typically very small (of the order of few percents). One possible origin for this behaviour could be magnetic field induced carrier freeze-out due to the considerable shrinking of electron wave functions if the magnetic field strength is much larger than the Coulomb forces \[20\]. The magnetic freeze-out would cause an increase of Hall coefficient. To examine this, Hall measurements was performed up to 5 T at 2 K. However, the observed linear Hall effect indicates that large magnetic fields are not affecting the carrier density at all as seen in Fig. 2(c).
To explore the origin of the magnetic-field induced resistivity minimum, the out-of-plane transverse MR was measured up to 9 T. As shown in Fig. 3(a), both the 2 and 10 K MR curves exhibit an obvious quadratic shape. For simplicity, we analyze our data using the single band picture, where the quadratic MR can be described by the classical orbital scattering. The Fermi energy at 2 K is *E$_{F}$* $\approx$ 1.08 meV and more than $\sim$6*kT*(2 K) taking *m$_{D}$* $\approx$ 5*m*$_{0}$ and *n* as the average carrier density over the entire crystal thickness at 2 K as shown in Fig. 1(b), corresponding to a degenerate gas state (*E$_{F}$* $\geq$ *kT*). Indeed, the carrier density of the surface region should be larger than the average value due to the inhomogeneity of oxygen vacancies (this will be discussed later), which would thus lead to an even larger *E$_{F}$*. However, no signature of quantum oscillations in MR was seen in spite of the high mobility. This could be because both the *m$_{D}$* and *m$_{c}$* have been largely enhanced from strong electron correlations of the Fermi liquid and eventually the initially assumed degenerate gas actually exists in a non-degenerate (*E$_{F}$* $\leq$ *kT*)“liquid" state. In addition, the Shubnikov-de Haas oscillation was observed typically at lower temperatures below 2 K \[10\].
The MR of 9 T at 2 K is extremely large, more than 2,900% and decreases to $\sim$1,200% at 10 K. The classical transverse orbital scattering can be described by $\triangle\rho$/$\rho$(0) = $\alpha$$^{2}$$\mu$$^{2}$*B*$^{2}$, where $\alpha$ is a material dependent constant. By fitting the MR curves of 2 and 10 K, the average value of $\alpha$ is obtained to be $\sim$0.32, which is comparable to the $\alpha$ $\sim$ 0.38 in *n*-type InSb \[21\]. The mobility of 2 K is larger than that of 10 K and therefore the MR of 2 K is far larger as shown in Fig. 3(a). Under a sufficiently large *B*, the MR difference between 2 and 10 K is so large that the overall resistivity $\rho$(*B,T*) = $\rho$(0,*T*) + $\alpha$$\mu$$^{2}$*B*$^{2}$$\rho$(0,*T*) at 2 K can become larger than that of 10 K although $\rho$(0,2 K) is smaller than $\rho$(0,10 K) in the normal metallic state. In this way, the intriguing resistivity minimum under a large magnetic field can be understood.
 (a) Out-of-plane magnetoresistance (MR) at 2 and 10 K up to 9 T. Inset: schematic of the measurement geometry. (b) Magnetic field dependence of the resistivity (*$\rho$-B*) for 2 and 10 K up to 5 T. ](liu_fig3.pdf){width="2.3in"}
To further confirm the origin of this resistivity minimum, the magnetic field dependence of resistivity (*$\rho$-B*) at 2 and 10 K are compared in Fig. 3(b). There is an evident crossover between the two *$\rho$-B* curves at $\sim$2.5 T, above which the resistivity at 2 K exceeds that at 10 K. The crossover field is consistent with the critical *B* in Fig. 2(a). The low temperature resistivity $\rho$(*B,T*) = $\rho$(0,*T*) + $\alpha$$\mu$$^{2}$*B*$^{2}$$\rho$(0,*T*) can be readily simulated up to 80 K by considering the *T*$^{2}$ dependence of $\rho$(0,*T*) and mathematically representing the mobility below 30 K with an exponential fitting. The simulated results (not shown) were consistent with the curves in Fig. 2(a), which suggests that the above resistivity comparison is not only true between 2 and 10 K but also valid for other temperatures. Finally we conclude that the magnetic field induced resistivity minimum originates from the extremely large MR and its pronounced increase with decreasing temperature at low temperatures. The large MR is achieved by the fairly high $\mu$, sufficiently large *B* and the possible stabilization of the classical limit by strong electron correlations of the Fermi liquid.
Similar behavior was also observed in SrTiO$_{3-\emph{x}}$ single crystals reduced for 2 h, which possess a larger room temperature carrier density and also a high mobility at 2 K. The Fermi liquid behavior, *i.e.* the *T*$^{2}$ dependence of the resistivity, also exists but up to $\sim$65 K. Nevertheless, as the reducing time was prolonged to 8 h for a STO single crystal, the room temperature carrier density reaches $\sim$6$\times$10$^{18}$ cm$^{-3}$ and consequently the mobility at 2 K is only $\sim$2,500 cm$^{2}$V$^{-1}$s$^{-1}$. As a result there is no observable resistivity minimum even under a 9 T magnetic field.
The condition for the strong-field region $\hbar$*w$_{c}$* $\gg$ *kT*, in which most of the carriers are in the lowest Landau magnetic quantum level, was coined as the “quantum limit" \[22\]. The theoretical analyses \[23-25\] indicate that for both degenerate and nondegenerate statistics the transverse MR has a quadratic field dependence in the classical low-field case with $\hbar$*w$_{c}$* $\ll$ *kT* but a linear dependence in the quantum limit. Moreover, the other criterion *n* $\ll$ (eB/$\hbar$)$^{3/2}$ \[25\] should also be fulfilled for the usual quantum linear MR as observed in high mobility InSb \[21,26\], Ge \[27\], graphite \[28\] and also recently in the topological insulator Bi$_{2}$Se$_{3}$ \[29\]. As shown in Fig. 4(a), the MR curves exhibit highly linear field dependence while the *B* is applied in plane and transverse to the current. The magnitude of the in-plane MR is on average $\sim$92% at 2 K and $\sim$66% at 10 K under 9 T, which are far smaller than the out-of-plane values. Accordingly neither the crossover in the *$\rho$-B* curves (inset of Fig. 4(a)) nor the magnetic field induced resistivity minimum was observed. In our case, the linear MR starts from a very small field $<$ 0.5 T as seen from Fig. 4(a), which in turn corresponds to a critical carrier density of $\sim$1.36$\times$10$^{17}$ cm$^{-3}$ for the usual quantum linear MR. The average carrier density at 2 K in our case is already $\sim$1.85$\times$10$^{18}$ cm$^{-3}$ and more than one order of magnitude larger than the critical carrier density. In this case, the linear MR may be out of the usual quantum linear MR picture \[30\].
![\[fig4\] (Color online) (a) In-plane transverse MR at 2 K and 10 K up to 9 T. The upper and lower insets are the corresponding *$\rho$-B* curves of the two temperatures and the schematic of measurement geometry respectively. (b) The parameter $\frac{\rho(\emph{B})\cdot\emph{kT}}{\rho(0)\cdot\mu\emph{$_{B}$}\emph{B}}$ plotted as a function of magnetic field.](liu_fig4.pdf){width="2.3in"}
On the other hand, the linear MR starting from small fields was previously also observed in nonstoichiometric silver chalcogenides Ag$_{2+\emph{x}}$Se and Ag$_{2+\emph{x}}$Te by Xu *et al.* \[31\]. In that case, the criteria for the usual quantum linear MR can also not be fulfilled. However, Abrikosov \[25,32\] proposed another model of quantum linear MR for the Ag$_{2+\emph{x}}$Se and Ag$_{2+\emph{x}}$Te case with two assumptions, *i.e.*, (1) the substance is inhomogenous, consisting of clusters of excess silver atoms with a high electron density, surrounded by a medium with a much lower electron density; (2) in this medium the electron energy spectrum is close to a gapless semiconductor with a linear dependence of energy on momentum, and thereby explained all the data in \[31\] satisfactorily. Our scenario seems quite close to the above case since the inhomogeneity of oxygen vacancies in SrTiO$_{3-\emph{x}}$ single crystals has been a long-standing issue \[33,34\]. Moreover, the oxygen vacancy clustering in SrTiO$_{3-\emph{x}}$ has been well studied both theoretically by Cuong *et al*. \[35\] and experimentally by Muller *et al.* \[36\]. Similar to clusters of excess silver atoms in nonstoichiometric silver chalcogenides, oxygen vacancy clusters have a high electron density. Thus, the linear MR in our case could be another example for the unusual quantum linear MR induced by inhomogeneities and conductive clusters.
Considering the further spin-orbital splitting of the Landau magnetic levels under a magnetic field *B*, the energy scale $\mu$*$_{B}$B* is involved, where $\mu$*$_{B}$* is the Bohr magneton. The dimensionless parameter $\frac{\rho(\emph{B})\cdot\emph{kT}}{\rho(0)\cdot\mu\emph{$_{B}$}\emph{B}}$ is theoretically predicted to be relatively independent of temperature for the quantum limit as illustrated in \[26\]. It is plotted in Fig. 4(b) as a function of field for two different temperatures and the parameter is approaching a constant value. This further supports the idea that the linear in-plane transverse MR is intrinsically a kind of quantum MR.
In early studies on the transport properties of semiconducting STO, STO single crystals were thermally reduced in vacuum and high temperature, which are similar to what we used, for quite a long time (typically up to 10 days \[6\]) to achieve uniform oxygen vacancy distribution. However, the oxygen vacancy distribution obtained by that long time reduction was found \[34\] to be still not ideally uniform over the entire thickness of single crystals. Furthermore, the O$_{vac}$ in short-time (like 1 h in our case) reduced STO single crystals are therefore far from uniform leading to a large gradient in O$_{vac}$ concentration \[34\] and thus also in carrier density \[37\] from the surface to the interior. Typically the oxygen vacancy doping can increase carrier density but on the other hand can decrease mobility in SrTiO$_{3-\emph{x}}$ due to elctron-impurity scattering, enhanced electron-electron scattering and suppressed screening effect. However, the lowering of mobility by oxygen vacancies is not as significant as the increase of carrier density by oxygen vacancy doping as can be seen from the electronic transport property studies of SrTiO$_{3-\emph{x}}$ single crystals by Frederikse *et al.* \[6\] and SrTiO$_{3-\emph{x}}$ films by Ohtomo and Hwang \[38\]. Thus, the higher oxygen vacancy concentration results in smaller resistivity and accordingly the conducting channel.
Consequently the measured transport data should only well represent the properties of the surface layers for this kind of samples. The highly anisotropic transport properties of the Fermi liquid can therefore be understood by the strong surface interlayer scattering due to the large inhomogeneity of O$_{vac}$ and accordingly the density-of-states along the out-of-plane direction. Even though the free carriers in our STO samples are not only confined to the very top surface, the sharp gradient of the carriers with the largest concentration at the top surface can result in the highly anisotropic transport properties.
Conclusions
===========
In conclusion, we studied the electrical and magnetotransport properties of SrTiO$_{3-\emph{x}}$ single crystals. It was found that the Fermi liquid exists at low temperatures and the dielectric constant of STO plays an important role in carrier density and mobility. A magnetic-field induced resistivity minimum was explored to originate from the high mobility and the possible strengthening of the classical limit by mass enhancement of strong electron correlations. The linear in-plane transverse MR, potential for linear MR sensors, was observed and attributed to the unusual quantum linear MR due to the inhomogeneity of oxygen vacancies and also oxygen vacancy clustering. The large anisotropy in transverse MRs reveals the strong surface interlayer scattering due to the inhomogeneity of O$_{vac}$ at the surface of SrTiO$_{3-\emph{x}}$. By this work, we demonstrate a potential route to quantum linear MR in virtue of inhomogeneities.
We thank the National Research Foundation (NRF) Singapore under the Competitive Research Program (CRP) “Tailoring Oxide Electronics by Atomic Control” NRF2008NRF-CRP002-024, National University of Singapore (NUS) cross-faculty grant and FRC (ARF Grant No. R-144-000-278-112) for financial support.
[99]{}
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|
Jan B. Gutowski$^1$, and W. A. Sabra$^2$\
.6cm
$^1$*Department of Mathematics, King’s College London.\
Strand, London WC2R 2LS\
United Kingdom\
E-mail: jan.gutowski@kcl.ac.uk*
\
$^2$*Centre for Advanced Mathematical Sciences and Physics Department,\
American University of Beirut, Lebanon\
E-mail: ws00@aub.edu.lb*
**Abstract**
> Recent results on the relation between hyper-Kähler geometry with Torsion and solutions admitting Killing spinors in minimal de sitter supergravity are extended to more general supergravity models with vector multiplets.
Introduction
============
The strong relation between complex geometry and supersymmetry has been known for some time by now. It was observed first by Zumino [@Zumino:1979et] that demanding $N=2$ supersymmetry on a two-dimensional non-linear sigma model puts the restriction that the target space metric must be described by a Kähler manifold. Extending the supersymmetry to $N=4$, the target manifold then becomes a hyper-Kähler manifold [@AlvarezGaume:1981hm]. The Wess-Zumino-Witten couplings [@Witten:1983ar] in the non-linear sigma model can be interpreted as torsion potentials from the target space viewpoint [@Curtright:1984dz; @Braaten:1985is]. Thus, it is natural to expect that adding such couplings will lead to Kähler and hyper-Kähler torsion (HKT) target space geometries [@howepaphkt; @hullhkt; @hull:1984; @hull:1986].
In string compactifications, demanding that the four dimensional low energy action has $N=1$ supersymmetry forces the six dimensional compact manifold to be a Calabi-Yau 3-fold [@Candelas:1985en]. Another connection between complex geometry and supersymmetry was also revealed in the study of the moduli space metric of supersymmetric electrically charged five-dimensional black holes which was found to be described by a HKT manifold [@gibbonshkt; @multihkt].
More recently, Kähler and hyper-Kähler geometry also arise in connection with the study of supersymmetric solutions in supergravity theories. For instance, the four-dimensional base spaces of time-like supersymmetric solutions of ungauged and gauged five dimensional supergravity are given, respectively, by a hyper-Kähler and Kähler manifold [@Gauntlett:2002nw; @Gauntlett:2003fk].
The embedding of cosmological Einstein gravity in a supergravity theory is allowed provided that the cosmological constant is either vanishing or negative. However, in the case of a positive cosmological constant, the concept of fake supergravity can be introduced as a solution generating technique. In this case, a Killing spinor equation is obtained from the analytic continuation of the equation resulting from the vanishing of the gravitini supersymmetry variation in the corresponding theory with negative cosmological constant. Recently, the programme of the classification of all solutions admitting (pseudo-)Killing spinors in de Sitter supergravity theories was initiated in [@hkt5]. There it was shown that the base space of time-like solutions of five dimensional de Sitter supergravity is given by four dimensional HKT geometry. Moreover, solutions admitting null Killing vectors were later analysed in [@gtnull5d] where it was found that those solutions are related to a one-parameter family of Gauduchon-Tod spaces [@gaudtod]. Our present work is the generalisation of the results of [@hkt5] to five dimensional supergravity models with scalar fields which could be of relevance to cosmological models.
This paper is organized as follows. Section two contains a brief description of the models under study and the analysis of the fake gravitino and dilatino Killing spinor equations of the five dimensional de Sitter supergravity with vector multiplets. The general structure of the pseudo-supersymmetric solutions, admitting Killing spinors that give rise to a timelike vector field, is obtained. In section three and four we provide some examples and in section five we give some final remarks.
*Note added*: At the time of completion of this work, we have become aware of the work of [@maeda], in which a subclass of the pseudo-supersymmetric solutions with a hyper-Kähler base space was examined. However, in general the base space is hyper-Kähler with torsion, and the classification constructed in this paper describes the most general pseudo-supersymmetric solution.
Fake $N=2$ supergravity and Killing Spinors
===========================================
The model we will be considering in our present work is $N=2$, $D=5$ gauged supergravity coupled to abelian vector multiplets [@gunaydin] whose bosonic action is given by $$S={\frac{1}{16\pi G}}\int \left( R+2g^{2}{\mathcal{V}}\right)
{\mathcal{\ast }}1-Q_{IJ}\left( dX^{I}\wedge \star dX^{J}+F^{I}\wedge \ast
F^{J}\right) -{\frac{C_{IJK}}{6}}F^{I}\wedge F^{J}\wedge A^{K}
\label{action}$$ where $I,J,K$ take values $1,\ldots ,n$ and $F^{I}=dA^{I}$ are the two-forms representing gauge field strengths (one of the gauge fields corresponds to the graviphoton). The constants $C_{IJK}$ are symmetric in $IJK$, we will assume that $Q_{IJ}$ is invertible, with inverse $Q^{IJ}$. The $X^{I}$ are scalar fields subject to the constraint $${\frac{1}{6}}C_{IJK}X^{I}X^{J}X^{K}=X_{I}X^{I}=1\,. \label{eqn:conda}$$ The fields $X^{I}$ can thus be regarded as being functions of $n-1$ unconstrained scalars $\phi ^{r}$. We list some useful relations $$\begin{aligned}
Q_{IJ} &=&{\frac{9}{2}}X_{I}X_{J}-{\frac{1}{2}}C_{IJK}X^{K} \nonumber \\
\text{ \ \ \ }Q_{IJ}X^{J} &=&{\frac{3}{2}}X_{I}\,,\qquad Q_{IJ}dX^{J}=
-{\frac{3}{2}}dX_{I}\ , \nonumber \\
{\mathcal{V}} &=&9V_{I}V_{J}(X^{I}X^{J}-{\frac{1}{2}}Q^{IJ}) \ ,\end{aligned}$$ here $V_{I}$ are constants.
Fake supergravity theory is obtained by sending $g^{2}$ to $-g^{2}$ in the above action. We start out analysis of pseudo-supersymmetric de-Sitter solutions by examining the fake gravitino Killing spinor equation:
$$\left[ \nabla _{M}-{\frac{i}{8}}\Gamma _{M}H_{N_{1}N_{2}}\Gamma
^{N_{1}N_{2}}+{\frac{3i}{4}}H_{M}{}^{N}\Gamma _{N}-
g(\frac{i}{2}X\Gamma _{M}-\frac{3}{2}A{}_{M})\right] \epsilon =0, \label{fakekilling}$$
where we have defined $$V_{I}X^{I}=X,\text{ \ \ \ \ \ }V_{I}A^{I}{}_{M}=A{}_{M},\text{ \ \ \ \ \ \ }
X_{I}F^{I}{}_{MN}=H_{MN}.$$
We shall analyse the solutions of the Killing spinor equations using spinorial geometry techniques originally developed to analyse supersymmetric solutions in ten and eleven dimensional supergravity [@papadopd11; @papadopiib], and which have since been used to analyse a large variety of supersymmetric solutions in numerous theories. For de Sitter supergravity in five-dimensions, one takes the space of Dirac spinors to be the space of complexified forms on $\mathbb{R}^{2}$, which are spanned over $\mathbb{C}$ by $\{1,e_{1},e_{2},e_{12}\}$ where $e_{12}=e_{1}\wedge e_{2}$. The action of complexified $\Gamma $-matrices on these spinors is given by
$$\begin{aligned}
\Gamma _{\alpha } &=&\sqrt{2}e_{\alpha }\wedge \ , \nonumber \\
\Gamma _{\bar{\alpha}} &=&\sqrt{2}i_{e_{\alpha }}\ ,\end{aligned}$$
for $\alpha =1,2$, and $\Gamma _{0}$ satisfies $$\Gamma _{0}1=-i1,\quad \Gamma _{0}e^{12}=-ie^{12},\quad \Gamma
_{0}e^{j}=ie^{j}\ \ j=1,2\ .$$
The spacetime metric has signature $(-,+,+,+,+)$ and is written in the following basis $$ds^{2}=-(\mathbf{e}^{0})^{2}+2\delta _{\alpha \bar{\beta}}\mathbf{e}^{\alpha
}\mathbf{e}^{\bar{\beta}}\ . \label{metricform}$$ The $Spin(4,1)$ gauge transformations can be used to fix the Killing spinor to take the form $\epsilon =f1$. Moreover, we can set $f=1,$ using the $\mathbb{R}$ transformation [@hkt5]
$$\epsilon \rightarrow e^{\lambda }\epsilon ,\qquad V_{I}A^{I}\rightarrow
V_{I}A^{I}-{\frac{2}{3g}}d\lambda ,$$
which leaves the Killing spinor equation invariant$.$ With all this information, we obtain from (\[fakekilling\]), the following conditions:
$$\begin{aligned}
H_{\alpha }^{\text{ \ }\alpha }+2{g}X-6{g}A{}_{0}
-2\Omega _{0,\alpha }^{\text{ \ \ \ \ \ }\alpha } &=&0, \notag \\
H_{0\alpha }-\Omega _{0,0\alpha } &=&0, \notag \\
\left( \Omega _{0,\bar{\alpha}\bar{\beta}}
-{\frac{1}{2}}H_{\bar{\alpha}\bar{\beta}}\right) \epsilon ^{\bar{\alpha}\bar{\beta}} &=&0, \notag \\
\frac{1}{2}\Omega _{\beta ,\alpha }^{\text{ \ \ \ \ }\alpha }
+{\frac{3}{4}}H_{0\beta }+\frac{3g}{2}A{}_{\beta } &=&0, \notag \\
\Omega _{\alpha ,0\bar{\beta}}+\frac{1}{2}H_{\mu }^{\text{ \ }\mu }\delta
_{\alpha \bar{\beta}}-{\frac{3}{2}}H_{\alpha \bar{\beta}}+{g}X\delta
_{\alpha \bar{\beta}} &=&0, \notag \\
\Omega _{\beta ,\bar{\mu}\bar{\nu}}\epsilon ^{\bar{\mu}\bar{\nu}}+H^{0\mu
}\epsilon _{\alpha \mu } &=&0, \notag \\
\Omega _{\bar{\alpha},0\bar{\beta}}-\frac{1}{2}H_{\bar{\alpha}\bar{\beta}}
&=&0, \notag \\
\Omega _{\bar{\beta},\mu }{}^\mu+\frac{1}{2}H_{0\bar{\beta}}
+3gA{}_{\bar{\beta}} &=&0, \notag \\
\Omega _{\bar{\beta},\bar{\mu}\bar{\nu}}\epsilon ^{\bar{\mu}\bar{\nu}} &=&0.
\label{gravitini}\end{aligned}$$
The above equations then imply:
$$\begin{aligned}
A{}_{0} &=&\frac{X}{3}, \notag \\
A{}_{\alpha } &=&-\frac{1}{3g}\Omega _{0,0\alpha },
\notag \\
H_{0\alpha } &=&\Omega _{0,0\alpha }, \notag \\
H_{\alpha \bar{\beta}} &=&\frac{2}{3}\left(
\Omega _{\alpha ,0\bar{\beta}}+\Omega_{\mu,0}{}^\mu \delta _{\alpha \bar{\beta}}
+3{g}X\delta _{\alpha \bar{\beta}}\right) , \notag \\
H_{\bar{\alpha}\bar{\beta}} &=&2\Omega _{\bar{\alpha},0\bar{\beta}}^{\text{
\ \ \ \ }} \ , \label{grav}\end{aligned}$$
together with the purely geometric constraints
$$\begin{aligned}
\Omega _{\lbrack 0,\alpha ]\beta } &=&0, \notag \\
\Omega _{0,\mu }^{\text{ \ \ \ \ \ }\mu }-\Omega _{\mu ,0}^{\text{ \ \ \ \ \
}\mu }-2{g}X &=&0, \notag \\
\Omega _{(\alpha ,\mid 0\mid \bar{\beta})} &=&
-{g}X\delta _{\alpha \bar{\beta}} \ , \label{geometryone}\end{aligned}$$
and
$$\begin{aligned}
\Omega _{\alpha ,\mu \nu } &=&0, \notag \\
\Omega _{\alpha ,\beta }^{\text{ \ \ \ \ }\beta }+\frac{1}{2}\Omega
_{0,0\alpha } &=&0, \notag \\
\Omega _{\alpha ,\bar{\mu}\bar{\nu}}
-\frac{1}{2}\delta _{\alpha \bar{\mu}}\Omega _{0,0\bar{\nu}}
+\frac{1}{2}\delta _{\alpha \bar{\nu}}\Omega _{0,0\bar{\mu}} &=&0. \label{geometrytwo}\end{aligned}$$
The dilatino Killing spinor equation is given by $$\left( (F_{MN}^{I}-X^{I}H_{MN})\Gamma ^{MN}-2i\nabla _{M}X^{I}\Gamma
^{M}-4gV_{J}(X^{I}X^{J}-{\frac{3}{2}}Q^{IJ})\right) \epsilon =0$$ which, on setting $\epsilon =1$, implies
$$\begin{aligned}
F^{I}{}_{\alpha }{}^{\alpha } &=&X^{I}H_{\alpha }{}^{\alpha }, \notag \\
F^{I}{}_{0\alpha } &=&X^{I}H{}_{0\alpha }-\partial _{\alpha }X^{I}, \notag
\\
F^{I}{}_{\alpha \beta } &=&X^{I}H_{\alpha \beta } \ , \notag \\
\partial _{0}X^{I} &=&2g(X^{I}V_{J}X^{J}-{\frac{3}{2}}Q^{IJ}V_{J}) \ .
\label{gaugino}\end{aligned}$$
To proceed, we examine the conditions implied by ([\[geometryone\]]{}) and ([\[geometrytwo\]]{}). Define the 1-form $V=\mathbf{e}^{0}$, and introduce a $t $ coordinate such that the dual vector field is $V=-{\frac{\partial }{\partial t}}$. We also introduce the real coordinates $x^{m}$, for $m=1,2,3,4 $. The vielbein is then given by $$\mathbf{e}^{0}=dt+\omega _{m}dx^{m},\text{ \ \ }\mathbf{e}^{\alpha }=
\mathbf{e}^{\alpha }{}_{m}dx^{m}\ .$$
From (\[geometryone\]), it can be easily demonstrated that
$$\begin{aligned}
\left( \mathcal{L}_{V}\mathbf{e}^{\alpha }\right) _{\bar{\beta}} &=&0,
\notag \\
\left( \mathcal{L}_{V}\mathbf{e}^{\alpha }\right) _{\beta } &=&\left( \Omega
_{0,\text{ \ }\beta }^{\text{ \ \ }\alpha }-
\Omega _{\beta ,\text{ \ }0}^{\text{ \ \ }\alpha }+\frac{1}{2}
\left( \Omega _{0,\mu }^{\text{ \ \ \ \ \ }\mu }
+\Omega _{\mu ,}{}^{\mu }{}_{0}\right) \delta ^{\alpha }{}_{\beta}\right)
-{g}X\delta ^{\alpha }{}_{\beta }.\end{aligned}$$
The quantity $$\Omega _{0,\text{ \ }\beta }^{\text{ \ \ }\alpha }
-\Omega _{\beta ,\text{ \ }0}^{\text{ \ \ }\alpha }+\frac{1}{2}
\left( \Omega _{0,\mu }^{\text{ \ \ \ \ \ }\mu }+\Omega _{\mu ,}{}^{\mu }{}_{0}\right) \delta ^{\alpha }{}_{\beta }$$ is anti-Hermitian and traceless (i.e. $\in su(2)$) and as such it can be gauged away by applying a $SU(2)\subset Spin(4,1)$ gauge transformation to the Killing spinors, which leaves $1$ invariant. In this gauge, $$\mathcal{L}_{V}\mathbf{e}^{\alpha }=-{g}X\mathbf{e}^{\alpha }.$$ Define ${\hat{\mathbf{e}}}^{\alpha }$ by $$\mathbf{e}^{\alpha }=G{\hat{\mathbf{e}}}^{\alpha }\ ,$$ with $$\frac{\partial _{t}G}{G}=gX,$$ then $$\mathcal{L}_{V}{\hat{\mathbf{e}}}^{\alpha }=0.$$ In what follows we introduce the base manifold $B$ with the $t$-independent metric $$ds_{B}^{2}=2\delta _{\alpha \bar{\beta}}{\hat{\mathbf{e}}}^{\alpha }
{\hat{\mathbf{e}}}^{\bar{\beta}}\ .$$ Let us denote the spin connections on the manifold $B$ by ${\hat{\Omega}}$ and rewrite the conditions in (\[geometrytwo\]) in terms of ${\hat{\Omega}}
$. The third condition in ([\[geometrytwo\]]{}) can be written as $$2{\hat{\Omega}}_{\alpha ,\bar{\mu}\bar{\nu}}-2G^{-1}
\big({\tilde{\partial}}_{\hat{\bar{\mu}}}G\delta _{\alpha \bar{\nu}}-
{\tilde{\partial}}_{\hat{\bar{\nu}}}G\delta _{\alpha \bar{\mu}}\big)-G^{-2}
\big(\delta _{\alpha \bar{\mu}}\partial _{t}(G^{2}\omega _{\hat{\bar{\nu}}})
-\delta _{\alpha \bar{\nu}}\partial _{t}(G^{2}\omega _{\hat{\bar{\mu}}})\big)=0.$$ Contracting with $\delta ^{\alpha \bar{\nu}}$ we obtain
$$\partial _{t}\left( G^{2}\omega _{{\hat{\bar{\mu}}}}\right) =
-2G^{2}\hat{\Omega}_{\alpha ,\bar{\mu}}^{\text{ \ \ \ \ }\alpha }
+\left( {{\tilde{d}}}G^{2}\right) _{\hat{\bar{\mu}}} \label{ansar}$$
which gives
$$\omega_{{\hat{\bar{\mu}}}}=M\hat{\Omega}_{\alpha ,\bar{\mu}}^{\text{ \ \ \ \
\ }\alpha }+G^{-2} (\mathcal{Q} + {\tilde{d}} \int G^2)_{{\hat{\bar{\mu}}}}$$
with
$$\partial _{t} {\mathcal{Q}}_{{\hat{\bar{\mu}}}}=0,\text{ \ \ \ \ \ }M=
-\frac{2}{G^{2}} \int G^{2} \ .$$
Therefore we can write
$$\omega =M\mathcal{P}+G^{-2}(\mathcal{Q} + \tilde{d} \int G^2)\ ,$$
where $$\mathcal{P}\equiv {\hat{\Omega}}_{{\bar{\beta}},\alpha }{}^{\bar{\beta}}{\hat{\mathbf{e}}}^{\alpha }
+{\hat{\Omega}}_{{\beta },{\bar{\alpha}}}{}^{{\beta }}{\hat{\mathbf{e}}}^{\bar{\alpha}}=
\mathcal{P}_{m}dx^{m}\ ,
\label{ppot}$$ and $\mathcal{Q}$ are 1-forms on the base manifold $B$ satisfying $$\mathcal{L}_{V}\mathcal{Q}=0\ ,\text{ \ \ \ }\mathcal{L}_{V}\mathcal{P}=0.$$
The remaining two conditions in (\[geometrytwo\]) give
$$\hat{\Omega}_{\alpha ,\mu \nu }=0 \ , \label{b1}$$
and $$\hat{\Omega}_{\alpha ,\mu }^{\text{ \ \ \ \ }\mu }-\hat{\Omega}_{\bar{\mu}
,\alpha }^{\text{ \ \ \ \ \ }\bar{\mu}}=0. \label{b2}$$
It is convenient to define the almost hypercomplex structure
$$\begin{aligned}
\mathbf{J}^{1} &=&{\hat{\mathbf{e}}}^{1}\wedge {\hat{\mathbf{e}}}^{2}
+{\hat{\mathbf{e}}}^{\bar{1}}\wedge {\hat{\mathbf{e}}}^{\bar{2}}\ , \notag \\
\mathbf{J}^{2} &=&i{\hat{\mathbf{e}}}^{1}\wedge {\hat{\mathbf{e}}}^{\bar{1}}
+i{\hat{\mathbf{e}}}^{2}\wedge {\hat{\mathbf{e}}}^{\bar{2}}\ , \notag \\
\mathbf{J}^{3} &=&-i{\hat{\mathbf{e}}}^{1}\wedge {\hat{\mathbf{e}}}^{2}
+i{\hat{\mathbf{e}}}^{\bar{1}}\wedge {\hat{\mathbf{e}}}^{\bar{2}}\ ,\end{aligned}$$
where $\mathbf{J}^i$ satisfy the algebra of the imaginary unit quaternions. The conditions ([\[b1\]]{}) and ([\[b2\]]{}) are then equivalent to
$$d\mathbf{J}^{i}=-2\mathcal{P}\wedge \mathbf{J}^{i}\ ,\qquad i=1,2,3\ ,
\label{hyperhermit}$$
where ${\mathcal{P}}$ is given by ([\[ppot\]]{}). The condition ([\[hyperhermit\]]{}) implies that the base $B$ is hyper-Kähler with torsion (HKT), i.e. $$\nabla ^{+}\mathbf{J}^{i}=0\ ,$$ where the connection of the covariant derivative $\nabla ^{+}$ is given by $$\Gamma ^{(+)}{}^{i}{}_{jk}=\{{}_{jk}^{i}\}+\Theta^{i}{}_{jk}\ ,$$ and where $\Theta$ is the torsion 3-form on $B$ given by $$\Theta=\star _{4}\mathcal{P}\ ,$$ where we take the volume form on the base space $B$ to be $-{\frac{1 }{2}}
\mathbf{J}^1 \wedge \mathbf{J}^1$, in this convention $\mathbf{J}^i$ are anti-self-dual. Note that ([\[hyperhermit\]]{}) implies that $$\begin{aligned}
d {\mathcal{P}}\wedge \mathbf{J}^i=0\end{aligned}$$ for $i=1,2,3$. Equivalently, the anti-self-dual projection of $d {\mathcal{P}}$ vanishes, $$\begin{aligned}
(d {\mathcal{P}})^-=0 \ .\end{aligned}$$
The constraints (\[gravitini\]) give for the gauge fields
$$\label{vx1a}
A=V_{I}A{}^{I}= -{\frac{2 }{3g}} G^{-1} dG + X \mathbf{e}^0 + {\frac{2 }{3g}}
{\mathcal{P}}$$
and $$\begin{aligned}
\label{vx1b}
H = d \mathbf{e}^0 + \Psi\end{aligned}$$ where $\Psi$ is a traceless (1,1) form on $B$, i.e. $\Psi$ is a self-dual 2-form on $B$, with $$\begin{aligned}
\label{vx1c}
\Psi = {\frac{4 }{3}} \big( G^{-2} \int G^2 \big) d {\mathcal{P}} -{\frac{2
}{3}} G^{-2} \big( d {\mathcal{Q}} +2 {\mathcal{Q}} \wedge {\mathcal{P}}
\big)^+\end{aligned}$$ where here ${}^+$ denotes the self-dual projection onto the base space $B$. Next we consider the conditions obtained from ([\[gaugino\]]{}). The first three conditions imply that $$\begin{aligned}
F^I= d \big( X^I \mathbf{e}^0 \big) + \Psi^I\end{aligned}$$ where $\Psi^I$ are closed, $t$-independent self-dual 2-forms on $B$, satisfying, as a consequence of ([\[vx1a\]]{}), $$\begin{aligned}
\label{vx2}
V_I \Psi^I = {\frac{2 }{3g}} d {\mathcal{P}}\end{aligned}$$ and as a consequence of ([\[vx1b\]]{}) $$\begin{aligned}
\label{vx3}
X_I \Psi^I = \Psi\end{aligned}$$ where $\Psi$ is given by ([\[vx1c\]]{}).
The final condition in ([\[gaugino\]]{}) implies that $$\begin{aligned}
\label{vx4}
X_I = 2g \big( G^{-2} \int G^2 \big) V_I + G^{-2} Z_I , \qquad \partial_t
Z_I =0\end{aligned}$$ where $Z_I$ are $t$-independent functions on $B$.
On substituting ([\[vx4\]]{}) into ([\[vx3\]]{}), and making use of ([\[vx2\]]{}) one obtains $$Z_{I}\Psi ^{I}=-{\frac{2}{3}}\big(d{\mathcal{Q}}+2{\mathcal{Q}}\wedge
{\mathcal{P}}\big)^{+} \ .$$
Next, it is convenient to make a co-ordinate transformation to simplify the solution, and define $$u=\int G^{2}.$$
The metric, gauge field strengths and scalars are then given by
$$\begin{aligned}
ds^{2} &=&-G^{-4}\big(du-2u{\mathcal{P}}+{\mathcal{Q}}\big)^{2}
+G^{2}ds_{B}^{2}, \notag \label{ssol1} \\
F^{I} &=&d\bigg(G^{-2}X^{I}\big(du-2u{\mathcal{P}}+{\mathcal{Q}}\big)\bigg)
+\Psi ^{I}, \notag \\
X_{I} &=&G^{-2}\big(2guV_{I}+Z_{I}\big).\end{aligned}$$
where $ds_{B}^{2}$ is the $u$-independent metric on a (strong) HKT manifold $B$. ${\mathcal{P}}$ is a $u$-independent 1-form on $B$ satisfying ([\[hyperhermit\]]{}) and $d{\mathcal{P}}$ is a self-dual 2-form on $B$. ${\mathcal{Q}}$ is another $u$-independent 1-form on $B$, $Z_{I}$ are $u$-independent functions on $B$, and $\Psi ^{I}$ are closed self-dual, $u$-independent 2-forms on $B$ satisfying $$V_{I}\Psi ^{I}={\frac{2}{3g}}d{\mathcal{P}} \label{ssol1v1}$$ and $$Z_{I}\Psi ^{I}=-{\frac{2}{3}}\big(d{\mathcal{Q}}+2{\mathcal{Q}}\wedge
{\mathcal{P}}\big)^{+}. \label{ssol1v2}$$
It remains to consider the gauge field equations; one finds, after some computation, that $${\hat{\nabla}}^{i}\big(-{\frac{3}{2}}dZ_{I}+3Z_{I}{\mathcal{P}}
+3gV_{I}{\mathcal{Q}}\big)_{i}+{\frac{1}{8}}C_{IJK}\Psi _{ij}^{J}\Psi ^{Kij}=0
\label{geq1}$$ where ${\hat{\nabla}}$ denotes the Levi-Civita connection of $B$, and here all indices are frame indices on $B$. We remark that, as a consequence of the integrability conditions examined in Appendix B of [@gshalf], the Killing spinor equations, together with the gauge field equations and Bianchi identity are sufficient to imply that the Einstein and the scalar field equations hold automatically, without any further constraint.
Note that the solution ([\[ssol1\]]{}) together with the conditions in ([\[ssol1v1\]]{}), ([\[ssol1v2\]]{}) and ([\[geq1\]]{}) are invariant under the conformal re-scaling $$ds_{B}^{2}=e^{-2h}ds_{B}^{\prime 2} \label{conf1}$$ where $h$ is a $u$-independent function, together with the re-definitions $$u=e^{2h}u^{\prime },\qquad {\mathcal{P}}={\mathcal{P}}^{\prime }+dh,
\qquad {\mathcal{Q}}=e^{2h}{\mathcal{Q}}^{\prime },\qquad Z_{I}=e^{2h}Z_{I}^{\prime },
\qquad G= e^h G^\prime \ .
\label{conf2}$$ A HKT manifold is called *strong* HKT if the associated torsion $\Theta $ is closed, or equivalently $$d\star _{4}{\mathcal{P}}=0$$ where $\star _{4}$ denotes the Hodge dual on $B$. For the solutions under consideration here, by making an appropriate conformal transformation as described above, one can without loss of generality take $B$ to be a strong HKT manifold.
In order to recover the solutions for the minimal theory determined in [@hkt5], one sets $$C_{111}={\frac{2}{\sqrt{3}}},\qquad X^{1}=\sqrt{3},\qquad X_{1}=
{\frac{1}{\sqrt{3}}}$$ and hence we set $$V_{1}={\frac{1}{\sqrt{3}}},\qquad Z_{1}=0$$ In addition, one has $$G=e^{gt},\qquad g=-{\frac{\chi }{2\sqrt{3}}},\qquad \Psi ^{1}=
-{\frac{4}{\chi }}d{\mathcal{P}},\qquad F^{1}=2F$$ where $F$ is the Maxwell field strength of the minimal theory.
It is also useful to consider a co-ordinate transformation of the form $$u^{\prime }=u-\Theta$$ where the function $\Theta $ does not depend on $u$, and set $${\mathcal{Q}}^{\prime }={\mathcal{Q}}-2\Theta {\mathcal{P}}+d\Theta ,\qquad
Z_{I}^{\prime }=Z_{I}+2g\Theta V_{I} \ .$$ Under these transformations, the solution given in ([\[ssol1\]]{}), ([[ssol1v1]{}]{}), ([\[ssol1v2\]]{}) , together with the gauge equations ([[geq1]{}]{}) are invariant. It is clear that one can always choose the function $\Theta $ such that $$d\star _{4}{\mathcal{Q}}^{\prime }=0$$ and one can therefore work in a gauge for which *both* ${\mathcal{P}}$ and ${\mathcal{Q}}$ are co-closed. It should however be noted that the gauge in which ${\mathcal{Q}}$ is co-closed is not the same gauge in which the solutions to the minimal theory are constructed as described in [@hkt5]; this is because the minimal theory gauge has $Z_{1}=0$ and $G$ is a function only of $t$. One cannot in general make a gauge transformation of the form described above and keep $Z_{1}^{\prime }=0$ as well. In what follows it will be most convenient to work with the gauge choice for which $$d\star _{4}{\mathcal{P}}=d\star _{4}{\mathcal{Q}}=0\ .$$
Solutions with a tri-holomorphic isometry
=========================================
It is straightforward to analyse the case when the base manifold $B$ is strong HKT and admits a tri-holomorphic isometry, which we denote by ${\frac{\partial }{\partial x^{5}}}$, and we take this isometry to be a symmetry of the full solution. Such base spaces have been classified in [@gaudtod; @papadmon; @chavetodvalent], and the metric on $B$ is given by $$ds_{B}^{2}=W^{-1}\big(dx^{5}+\varphi \big)^{2}+Wds_{E}^{2} \label{tri1}$$ where $E$ is a constrained 3-dimensional Einstein-Weyl geometry, consisting of a $x^{5}$-independent 3-metric $\gamma _{ij}$, a $x^{5}$-independent 1-form $\alpha $ on $E$, and an $x^{5}$-independent scalar $\alpha _{0}$ on $E$, satisfying $$\star _{E}d\alpha =-d\alpha _{0}-\alpha _{0}\alpha ,\qquad d\star _{E}\alpha
=0$$ where $\star _{E}$ denotes the Hodge dual on $E$, and the Ricci tensor of $E$ satisfies $${}^{(E)}R_{ij}+\nabla _{(i}\alpha _{j)}+\alpha _{i}\alpha _{j}=\gamma _{ij}
({\frac{1}{2}}\alpha _{0}^{2}+\alpha ^{\ell }\alpha _{\ell })$$ where here $\nabla $ denotes the Levi-Civita connection of $E$, and $\varphi
$ is a $x^{5}$ independent 1-form on $E$ satisfying $$\star _{E}d\varphi =dW+W\alpha$$ and the function $W$ does not depend on $x^{5}$. The volume form of $B$, $\epsilon _{B}$, and the volume form of $E$, $d\mathrm{vol}_{E}$ are related by $$\epsilon _{B}=W(dx^{5}+\varphi )\wedge d\mathrm{vol}_{E}.$$ The torsion of $B$ is determined by ${\mathcal{P}}$, with $${\mathcal{P}}=-{\frac{\alpha _{0}}{2W}}(dx^{5}+\varphi )-{\frac{1}{2}}\alpha$$ which is co-closed as a consequence of the previous conditions. We further remark that the functions $W$, $\alpha _{0}$ satisfy $$(\Delta _{E}+\alpha ^{i}\nabla _{i})W=(\Delta _{E}+\alpha ^{i}\nabla
_{i})\alpha _{0}=0$$ where $\Delta _{E}$ is the Laplacian on $E$.
To proceed further with the analysis, note that self-duality of $\Psi ^{I}$, together with the requirement that $d\Psi ^{I}=0$, are equivalent to $$\Psi ^{I}=-{\frac{1}{2}}(dx^{5}+\varphi )\wedge d\big(W^{-1}K^{I}\big)-
{\frac{1}{2}}W\star _{E}d\big(W^{-1}K^{I}\big)$$ where $K^{I}$ are $x^{5}$-independent functions on $E$ satisfying $$(\Delta _{E}+\alpha ^{i}\nabla _{i})K^{I}=0.$$ The condition ([\[ssol1v1\]]{}) constrains the $K^{I}$ via $$V_{I}K^{I}={\frac{1}{3g}}\alpha _{0}+kW \label{ssol3a2}$$ for constant $k$. Next, it is straightforward to solve the gauge equation ([\[geq1\]]{}) to find $$Z_{I}={\frac{1}{24}}C_{IJK}W^{-1}K^{J}K^{K}+L_{I}$$ where $L_{I}$ are $x^{5}$-independent functions on $E$ satisfying $$(\Delta _{E}+\alpha ^{i}\nabla _{i})L_{I}=0.$$ Finally we solve for the 1-form ${\mathcal{Q}}$. We decompose this 1-form as $${\mathcal{Q}}=Q_{5}(dx^{5}+\chi )+{\tilde{Q}}$$ where the function $Q_{5}$ does not depend on $x^{5}$, and ${\tilde{Q}}$ is a $x^{5}$-independent 1-form on $E$. The condition $d\star _{B}{\mathcal{Q}}
=0$ implies that $$d\star _{E}\tilde{Q}=0$$ and the condition ([\[ssol1v2\]]{}), after some manipulation, implies that $$Q_{5}=-{\frac{1}{48}}W^{-2}C_{IJK}K^{I}K^{J}K^{K}-{\frac{3}{4}}
W^{-1}L_{I}K^{I}+M$$ where $M$ is a $x^{5}$-independent function on $E$ satisfying $$(\Delta _{E}+\alpha ^{i}\nabla _{i})M=0$$ and ${\tilde{Q}}$ also must satisfy $$d{\tilde{Q}}+\alpha \wedge {\tilde{Q}}+\alpha _{0}\star _{E}{\tilde{Q}}
=W\star _{E}dM-M\star _{E}dW-{\frac{3}{4}}\big(K^{I}\star
_{E}dL_{I}-L_{I}\star _{E}dK^{I}\big).$$
Solutions with a Conformally Hyper-Kähler base
==============================================
Suppose that the base space $B$ is conformally hyper-Kähler. Then ${\mathcal{P}}$ is closed, and using the conformal transformation described in ([\[conf1\]]{}) and ([\[conf2\]]{}), one can without loss of generality set ${\mathcal{P}}=0$, i.e. one can take $B$ to be a hyper-Kähler manifold, which we denote by $HK$. We shall also work in a gauge for which ${\mathcal{Q}}$ is co-closed on $HK$, as described previously. Hence the solution can be written as $$\begin{aligned}
ds^{2} &=&-G^{-4}\big(du+{\mathcal{Q}}\big)^{2}+G^{2}ds_{HK}^{2}, \notag
\label{ssol1b} \\
F^{I} &=&d\bigg(G^{-2}X^{I}\big(du+{\mathcal{Q}}\big)\bigg)+\Psi ^{I},
\notag \\
X_{I} &=&G^{-2}\big(2guV_{I}+Z_{I}\big),\end{aligned}$$ where $ds_{HK}^{2}$ is the $u$-independent metric on a hyper-Kähler manifold $HK$, ${\mathcal{Q}}$ is a $u$-independent 1-form on $HK$, $Z_{I}$ are $u$-independent functions on $HK$, and $\Psi ^{I}$ are self-dual, $u$-independent 2-forms on $HK$ satisfying $$d\Psi ^{I}=0 \label{ssol2a}$$ and $$V_{I}\Psi ^{I}=0 \label{ssol2b}$$ and $$Z_{I}\Psi ^{I}=-{\frac{2}{3}}\big(d{\mathcal{Q}}\big)^{+} \label{ssol2c}$$ and $$d\star _{4}{\mathcal{Q}}=0 \label{ssol2d}$$ and $$d\star _{4}dZ_{I}+{\frac{1}{6}}C_{IJK}\Psi ^{J}\wedge \Psi ^{K}=0 \ .
\label{ssol2e}$$
In order to recover the special case for which the base space is hyper-Kähler with a triholomorphic isometry, i.e. a Gibbons-Hawking manifold, for which the triholomorphic isometry is a symmetry of the full solution, one takes the analysis of the previous section and sets $E=\mathbb{R}^{3}$, $\alpha _{0}=0$, $\alpha =0$, with $W=H$ where $H$ is a harmonic function on $\mathbb{R}^{3}$ and $\varphi $ is a $x^{5}$-independent 1-form on $\mathbb{R}^{3}$ satisfying $$d\varphi =\star _{\mathbb{R}^{3}}dH$$ and the remaining functions $K^{I}$, $L_{I}$, $M$ which are used in the construction of the solution are also harmonic functions on $\mathbb{R}^{3}$. We remark that one can also allow $Z_{I}$ to depend linearly on $x^{5}$ by taking $$Z_{I}={\frac{1}{24}}C_{IJK}H^{-1}K^{J}K^{K}+L_{I}+cV_{I}x^{5}$$ for constant $c$, with $\Psi ^{I}$, ${\mathcal{Q}}$ unchanged (and $H$, $K^{I}$, $L_{I}$, $M$ still $x^{5}$-independent). It is straightforward to show that adding such a term linear in $x^{5}$ to $Z_{I}$ does not give any contribution to the LHS of the conditions ([\[ssol2c\]]{}) and ([\[ssol2e\]]{}).
The black hole solutions found in [@liusabra; @Klemm:2000gh; @Behrndt:2003cx] are a special case of the solutions found here, for which all the harmonic functions depend only on $r$, and hence have poles only at $r=0$.
Final Remarks
=============
In this paper we have studied timelike solutions admitting Killing spinors of five dimensional de Sitter supergravity with Abelian vector multiplets. The four dimensional base space of these solutions was found to be given by a four dimensional HKT geometry. In our present work we have also described two special classes of solution. First, we considered the case when the HKT manifold admits a tri-holomorphic Killing vector field. Then we considered the case for which the HKT manifold is conformally hyper-Kähler. The conformally hyper-Kähler class of solutions includes all previously constructed solutions in the literature as special cases. It would be of great interest to construct new solutions in the non-conformally hyper-Kähler case, as these might be of relevance to black hole physics and cosmology. It would also be particularly interesting to determine whether there exist regular (pseudo) supersymmetric black ring solutions in de Sitter supergravity. Finally, a continuation of our present work is to study the solutions of the null case for the theories considered here and possibly generalising these results to de Sitter supergravity in other dimensions. Work along these directions is in progress.
Acknowledgments {#acknowledgments .unnumbered}
===============
The work of WS was supported in part by the National Science Foundation under grant number PHY-0903134. JG is supported by the EPSRC grant EP/F069774/1.
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|
---
abstract: 'We study the spectrum and eigenstates of the quantum discrete Bose-Hubbard Hamiltonian in a finite one-dimensional lattice containing two bosons. The interaction between the bosons leads to an algebraic localization of the modified extended states in the normal mode space of the noninteracting system. Weight functions of the eigenstates in the space of normal modes are computed by using numerical diagonalization and perturbation theory. We find that staggered states do not compactify in the dilute limit for large chains.'
author:
- 'Jean Pierre Nguenang$^{1,2}$'
- 'R. A. Pinto$^{1}$'
- 'Sergej Flach$^{1}$'
title: 'Quantum q-breathers in a finite Bose-Hubbard chain: The case of two interacting bosons'
---
Introduction
============
The study of discrete breathers in different physical systems has had remarkable developments during the last two decades [@FlachPhysRep295; @physicstoday; @Sievers; @AubryPhysicaD103]. These excitations are generic time-periodic and spatially localized solutions of the underlying classical Hamiltonian lattice with translational invariance. Their spatial profiles localize exponentially for short-range interaction. Recent experimental observations of breathers in various systems include such different cases as bond excitations in molecules, lattice vibrations and spin excitations in solids, electronic currents in coupled Josephson junctions, light propagation in interacting optical waveguides, cantilever vibrations in micromechanical arrays, cold atom dynamics in Bose-Einstein condensates loaded on optical lattices, among others [@SchwarzPRL83; @SatoNature; @SwansonPRL82; @TriasPRL84; @BinderPRL84; @EisenbergPRL81; @FleischerNature422; @SatoPRL90; @EiermannPRL92]. In many cases quantum dynamics is important. Quantum breathers consist of superpositions of nearly degenerate many-quanta bound states, with very long times to tunnel from one lattice site to another [@Fleurov; @ScottPhysLettA119; @BernsteinNonlin3; @BernsteinPhysicaD68; @WrightPhysicaD69; @Wang; @Aubry; @Flach1; @AubryPhysicaD103; @Fleurov1998; @kalosakas2; @Dorignac2004; @Eilbeck2004; @Pinto; @Schulman; @Schulman2006; @Proville2006]. Remarkably quantum breathers, though being extended states in a translationally invariant system, are characterized by exponentially localized weight functions, in full analogy to their classical counterparts.
Recently the application of these ideas to normal mode space allowed to explain many facettes of the Fermi-Pasta-Ulam (FPU) paradox [@Fermi], which consists of the nonequipartition of energy among the linear normal modes in a nonlinear chain. There the energy is localized around the initial normal mode which is excited. Introducing the notion of q-breathers [@FlachPRL95; @IvanchenkoPRL2006; @FlachPRE2006], which are time-periodic excitations localized in the normal mode space, the FPU paradox and some related problems were successfully explained. Despite the fact that the interaction in normal mode space is long-ranged, it is selective and purely nonlinear, thus q-breathers localize exponentially in normal mode space.
In this paper we address the properties of quantum q-breathers. We study a one-dimensional quantum lattice problem with two quanta. By defining an appropriate weight function in the normal mode space we explore the localization properties of the eigenstates of the system. We observe localization of the weight function as a function of the wave number, which we interprete as a signature of quantum q-breather excitations. By using a numerical diagonalization of the Hamiltonian and nondegenerate perturbation theory we find algebraic decay of the weight function in the normal mode space, at variance to the exponential decay found for q-breathers in the case of a classical nonlinear system. Another intriguing difference is based on the interference effects of two interacting quanta. For the general case the quantum q-breather states approach the noninteracting eigenstates in the dilute limit of large chains. However, states with Bloch momentum close to $\pm \pi$ keep their finite localization in that limit.
In section II we describe the model and introduce the basis we use to write down the Hamiltonian matrix. In section III we review results on the properties of two-quanta bound states - the simplest versions of a quantum breather. In section IV we consider the case of extended states. We introduce a weight function to describe localization in the normal mode space, and obtain analytical results using perturbation theory. We present our numerical results obtained by diagonalization of the Hamiltonian matrix, comparing them to analytical estimations. We conclude in section V.
The model
=========
We study a one-dimensional periodic lattice with $f$ sites described by the Bose-Hubbard (BH) model. This is a quantum version of the discrete nonlinear Schrödinger equation, which has been used to describe a great variety of systems [@Scott1]. The BH Hamiltonian is given by [@Eilbeck94] $$\label{eq:hamiltonian}
\hat{H} = \hat{H}_0 + \gamma\hat{H}_1,$$ where $$\hat{H}_0=-\sum_{j=1}^f b_j^+(b_{j-1}+b_{j+1}),$$ $$\hat{H}_1 = -\frac{1}{2}\sum_{j=1}^f b_j^+b_j^+b_jb_j.$$ Here $ b_j^+$ and $b_j$ are the bosonic creation and annihilation operators which satisfy the commutation relations $[b_i,b_j^{+}]=\delta_{ij}$, $[b_i^{+},b_j^{+}]=[b_i,b_j]=0$. $\gamma$ is the parameter controlling the strength of the interaction, and the chain of length $f$ is subject to periodic boundary conditions. The chain is translational invariant and the Hamiltonian (\[eq:hamiltonian\]) commutes with the number operator $\hat{N}=\sum_{j=1}^f{b_j^+b_j}$, whose eigenvalue is denoted by $n$. We consider the simplest non-trivial case of $n=2$. It is of direct relevance to studies and observations of bound two-vibron states [@Cohen1969; @Kimball; @Richter1988; @GuyotSionnest1991; @Dai1994; @Chin1995; @Jakob1996; @JakobPr75; @Pouthier2003JCP; @Okuyama2001; @Pouthier2003PRE; @Edler2004]
In order to describe the quantum states, we use a number state basis [@Eilbeck94] $|\Phi_n\rangle=|n_1,n_2,...,n_f\rangle$, where $n_i$ represents the number of bosons at site i $(n=\sum{n_i})$. As an example $|0200000\rangle$ corresponds to a state with two bosons on the second site and zero bosons elsewhere. For a given number of bosons each eigenstate is a linear combination of number states with fixed $n$. In addition to the number of quanta $n$ there are $n-1$ further quantum numbers which define the relative distance between the bosons. For $n=2$ that reduces to defining one further relative distance $j-1$ between the two quanta, which can take $(f+1)/2$ different values in our case: $$|\Psi_2\rangle=\sum_{j=1}^{\frac{f+1}{2}}v_j|\Phi_2^j\rangle.$$ Due to translation invariance the eigenstates of $\hat{H}$ are also eigenstates of the translational operator $\hat{T}$, where $\tau=\exp(ik)$ is its eigenvalue with $k=2\pi\nu/f$ being the Bloch wave number and $\nu\in [-(f-1)/2,
(f-1)/2]$. Due to periodic boundary conditions $\hat{T}|n_1,n_2,\cdots,n_f\rangle=|n_f,n_1,n_2,\cdots,n_{f-1}\rangle$. For the sake of simplicity we deal with an odd number of sites $f$. Thus we can construct number states which are also Bloch states: $$|\Phi_2^j\rangle=\frac{1}{\sqrt{f}}
\sum_{s=1}^f\Big(\frac{\hat{T}}{\tau}\Big)^{s-1}|1\underbrace{0\cdots0}_{j-1}1\cdots\rangle
.$$ With this basis we can derive the eigenenergies for each given Bloch wave number $k$ from $\hat{H}_k|\Psi_n\rangle=E|\Psi_n\rangle$ after computing the eigenvalues of the matrix with the same structure as in [@Eilbeck94] for the case of the BH system: $$\begin{aligned}
\label{matrix}
\hat{H}_k = -\left( \begin{array}{cccccc}
\gamma & q\sqrt{2} & & & & \\
q^*\sqrt{2} & 0 & q & & & \\
& q^* & 0 & q & & \\
& & \ddots & \ddots &\ddots & \\
& & & q^* & 0 & q \\
& & & & q^* & p
\end{array} \right),\end{aligned}$$ with $q=1+\tau$ and $p=\tau^{-(f+1)/2} +
\tau^{-(f-1)/2}$. By varying the Bloch wavenumber in its irreducible range, we obtain the eigenenergy spectrum shown in Fig.\[spectrum\].
Bound states: Localization in real space
========================================
In Fig.\[spectrum\](a-c) we show that as the interaction parameter is increasing, an isolated ground state eigenvalue $E_2(k)$ appears for each $k$ that corresponds to a bound state [@Eilbeck94]. For this isolated ground state there is a high probability of finding two quanta on the same site. In the limit $f\to\infty$ the bound state eigenvalue has the analytical expression [@Eilbeck94; @Eilbeck03] : $$E_2(k) = -\sqrt{\gamma^2 + 16\cos^2{k/2}},$$ and the corresponding (unnormalized) eigenvector $\mathbf{v}=(v_1,v_2,\ldots)$ is [@Eilbeck03] $$\mathbf{v} = \left( \frac{1}{\sqrt{2}},\mu,\mu^2,\mu^3,\ldots \right),$$ where $$\mu = -\frac{(\gamma + E_2(k))e^{ik/2}}{4\cos(k/2)}.$$ A suitable weight function of this isolated ground state has the form: $$C_{j} \equiv |v_j|^2 = |\mu|^{2(j-1)} = e^{2\lambda (j-1)}, \; \; j>1,$$ where $C_1=1/2$ and $\lambda =\ln|\mu|$, Since $|\mu|^2<1$ for $\gamma \neq 0$, the weight function shows exponential decay when the distance between the two bosons increases. That result corresponds to the exponential localization of classical discrete breathers [@FlachPhysRep295; @physicstoday; @Sievers; @AubryPhysicaD103]. However note that for $|k| \rightarrow \pi$ we have $\mu \rightarrow 0$ independently on the value of $\gamma \neq 0$. Thus one obtains compact localization. Note that it is said that a state is compact in a certain basis, if it occupies a certain subspace, but has exactly zero overlap with the rest.
The compact localization for $|k| \rightarrow \pi$ is not observed in the classical limit, and relies on the fact that the Schrödinger equation is a linear wave equation which admits (destructive) interference effects.
![\[spectrum\]Energy spectrum of the Bose-Hubbard model for different values of the interaction $\gamma$: (a) $\gamma=0.1$, (b) $\gamma=1.0$, and (c) $\gamma=10$. Here $f=101$.](Figure1.eps){width="2.in"}
Quantum q-breathers: localization in normal mode space
=======================================================
All the other states (except the bound state) form the two quanta continuum. Their energies for $\gamma=0$ correspond to the sum of two single particle energies with the constraint that the sum of their momenta equals the Bloch momentum $k$. One arrives at $$E_{k,k_1}^0 = -2[\cos(k_1)+\cos(k_1+k)],$$ where $k_1=\pi\nu_1/[(f+1)/2]-k/2$ is the conjugated momentum of the relative coordinate (distance) of both quanta and $\nu_1 = 1,\ldots,(f+1)/2$. $E_{k,k_1}^0$ has a finite spread at fixed $k$ (see Fig.\[spectrum\]). However for $k=\pm\pi$ the spectrum becomes degenerate. Thus for $|k\pm\pi|\ll 1$ the eigenenergies are very close (almost degenerate). Remarkably the bounds of the spectrum for $\gamma\neq 0$ are very well described by the $\gamma=0$ result. Increasing $\gamma$ at fixed $k$, the eigenenergies will slightly move, but never cross. Thus a continuation of an eigenstate at $\gamma=0$ to $\gamma\neq 0$ will preserve its relative ordering with respect to the other eigenenergies.
For $\gamma\neq 0$ these quantum q-breather states will be deformed. In analogy to the study of the fate of normal modes in classical nonlinear systems [@FlachPRL95; @IvanchenkoPRL2006; @FlachPRE2006], we will study the changes of the two-quanta continuum. For finite $f$ and $\gamma$ the new states will be spread in the basis of the $\gamma=0$ continuum. For $f\to\infty$ one expects that the new states become again identical with the $\gamma=0$ states, since the two quanta will meet on the lattice with less probability as $f$ increases. Thus we will test the compactification of the new states in the $\gamma=0$ eigenstate basis both for $\gamma\to 0$ and for $f\to\infty$.
We compute the weight functions in normal mode space in order to probe the signature of quantum q-breathers. For this purpose we start by using perturbation theory to set up these weight functions, where $H_1$ is the perturbation. We fix the Bloch momentum $k$, and choose an eigenstate $|\Psi_{\tilde{k}_1}^0\rangle$ of the unperturbed case $\gamma=0$. Upon increase of $\gamma$ it becomes a new eigenstate $|\Psi_{\tilde{k}_1}\rangle$, which will have overlap with several eigenstates of the $\gamma=0$ case. We expand the eigenfunction of the perturbed system to the first order approximation: $$|\Psi_{\tilde{k}_1}\rangle=|\Psi_{\tilde{k}_1}^0\rangle+\gamma\sum_{{{k}_1^{\prime}}\neq
\tilde{k}_1}\frac{\langle\Psi_{k_1^{\prime}}^{0}|\hat{H}_1|\Psi_{\tilde{k}_1}^0\rangle}{{E_{\tilde{k}_1}^0}-E_{{k}_1^{\prime}}^0}|\Psi_{{k}_1^{\prime}}^0\rangle.$$ The perturbation of strength $\gamma$ is local in the matrix representation (\[matrix\]), thus the relevant perturbation parameter is $\gamma/f$. This has to be compared to the typical spacing of unperturbed eigenenergies. For Bloch wave numbers far from $\pm \pi$ the spacing is of order $1/f$, so the approximation should work for $\gamma < 1$. For Bloch wave numbers close to $\pm \pi$ the approximation breaks down if $\gamma \geq \pi -|k|$.
The off-diagonal ($k_1\neq\tilde{k}_1$) weight function at the first order is given by : $$C(k_1;\tilde{k}_1) \equiv |\langle\Psi_{k_1}^0|\Psi_{\tilde{k}_1}\rangle|^2 =
\frac{|\langle\Psi_{k_1}^0|\hat{H}_1|\Psi_{\tilde{k}_1}^0\rangle|^2}{|E_{\tilde{k}_1}^0-E_{{k}_1}^0|^2}.$$ $E_{k_1}^0$ and $E_{\tilde{k}_1}^0$ are the eigenenergies of the unperturbed system. With $\Delta=k_1-\tilde{k}_1$ the weight function can be rewritten in the following form
$$\label{eq:correlation}
C(k_1;\tilde{k}_1) = \frac{A^2\gamma^2}
{64(f+1)^2\cos^2(\frac{k}{2})\sin^2(\frac{\Delta}{2})\Big[\sin(\frac{2\tilde{k}_1+k}{2})\cos(\frac{\Delta}{2})
+ \cos(\frac{2\tilde{k}_1+k}{2})\sin(\frac{\Delta}{2})\Big]^2}, \; \; \;
k_1\neq\tilde{k}_1,$$
where $A$ is a constant. For $\gamma= 0$, $|\Psi_{\tilde{k}_1}\rangle =
|\Psi_{\tilde{k}_1}^0\rangle$, and the weight function is compact. For $|\Delta|\ll 1$ $$C(k_1;\tilde{k}_1) \approx \frac{\gamma^2}
{(f+1)^2\frac{{\Delta}^2}{2}\cos^2(\frac{k}{2})\Big[\sin(\frac{2\tilde{k}_1+k}{2})
+\frac{\Delta}{2}\cos(\frac{2\tilde{k}_1+k}{2})\Big]^2}, \; \; \;
k_1\neq\tilde{k}_1.$$
From this formula we obtain several interesting results. First of all, the decay of the weight function with increasing $\Delta$ means that we have localization in normal mode space. For $2\tilde{k}_1 + k\neq 0,2\pi$, we have $$C\sim \frac{\gamma^2}{(f+1)^2}\frac{1}{\Delta^2}.$$ We find algebraic decay $\sim 1/\Delta^2$ of the weight function, and for $\gamma\to 0$ or $f\to\infty$ the weight function compactifies. For $2\tilde{k}_1 + k = 0,2\pi$, we have $$C\sim \frac{\gamma^2}{(f+1)^2}\frac{1}{\Delta^4}.$$ Here we find algebraic decay $\sim 1/\Delta^4$ of the weight function, that also compactifies when $\gamma\to 0$ or $f\to\infty$. Finally, for $k$ close to $\pm\pi$ and large $f$, $C\sim \gamma^2/\Delta^2$. Thus we find that the $f$-dependence drops out for staggered states $|k\pm\pi|\ll 1$, and these states do not compactify for $f\to\infty$. That is a remarkable quantum interference property, since both simple intuition (see above) and classical theory predict the opposite.
In Fig.\[gcorr\] we show numerical results obtained by diagonalization of the Hamiltonian for different values of $\gamma$.
![\[gcorr\]Weight function for different values of the interaction $\gamma$. Here f=101, $k=0$, and $\tilde{k}_1=\frac{2}{3}\pi$. Dashed lines are results using the formula (\[eq:correlation\]) with $A^2=3.8$.](Figure2.eps){width="3.3in"}
We find localization in normal mode space, which can be interpreted as a quantum q-breather. When increasing $\gamma$ the quantum q-breather becomes less localized, and for large values of the interaction (from $\gamma=10$ on) results do not change. The dashed lines are the results using the formula (\[eq:correlation\]) with $A^2=3.8$, value that was obtained by fitting the numerical results for the lowest $\gamma$ ($=0.001$). We can see good agreement with numerical results up to $\gamma=1$, beyond which perturbation theory does not fit anymore. In Fig.\[kcorr\], we show that the weight function is more localized for $k=0$ and less localized for $k\to -\pi$.
![\[kcorr\]Weight function for different values of the Bloch wave number $k$. Here $\gamma=0.1$, $f=101$, and $\tilde{k}_1+k/2=\frac{2}{3}\pi$. Dashed lines are results using the formula (\[eq:correlation\]) with $A^2=3.8$.](Figure3.eps){width="3.3in"}
While probing the influence of the size of the nonlinear quantum lattice on the localization phenomenon, we find in Fig.\[scorr\] that as the size increases the states compactify as we expected. In Fig.\[scorrlog\] we see the $1/\Delta^2$ decay for eigenstates fulfilling $2\tilde{k}_1+k\neq 0,2\pi$ ($k=0$), and in Fig.\[scorrlog2\] the $1/\Delta^4$ decay for eigenstates fulfilling $2\tilde{k}_1+k=0,2\pi$ (also $k=0$). Both results agree with the analytical results using perturbation theory.
![\[scorr\].Weight function for different sizes of the system. Here $\gamma=0.1$, $k=0$, and $\tilde{k}_1\approx \frac{2}{3}\pi$ for all curves. Dashed lines are results using the formula (\[eq:correlation\]) with $A^2=3.8$.](Figure4.eps){width="3.3in"}
![\[scorrlog\]The same as in Fig.\[scorr\] in log-log scale.](Figure5.eps){width="3.3in"}
![\[scorrlog2\]Weight function for eigenstates with different $\tilde{k}_1$. Here $\gamma=0.1$, $f=101$, and $k=0$.](Figure6.eps){width="3.3in"}
In Fig.\[scorrind\] we observe the predicted independence of the localization phenomenon from the size of the system when $k$ is close to $-\pi$. It is interesting that in this case the weight function does not compactify in the dilute limit $f\to\infty$ as one would expect from simple grounds. The reason is that the larger $f$, the closer we can tune the Bloch wave number to $\pm \pi$, where the perturbation expansion breaks down.
![\[scorrind\]Weight function for different sizes of the system close to the band edge $k=-\pi$. Here $\gamma=0.001$ and $\tilde{k}_1+k/2=\frac{2}{3}\pi$. The dashed line is the result using the formula (\[eq:correlation\]) with $A^2=3.8$.](Figure7.eps){width="3.3in"}
Conclusions
===========
In this work we studied the properties of quantum q-breathers in a one-dimensional chain containing two quanta modeled by the Bose-Hubbard Hamiltonian. To explore localization phenomena in this system we computed appropriate weight functions of the eigenstates in the normal mode space using both perturbation theory and numerical diagonalization. We observe localization of these weight functions, that is interpreted as a signature of quantum q-breathers. The localization is stronger when the size of the system increases. Unlike the classical case where the localization is exponential, here we found algebraic localization. This is a long range behavior, which follows from the fact that the interaction $\gamma$ induces a linear perturbation of the eigenstates which is local in real space, and also local in the matrix representation in (\[matrix\]). That induces a mean-field type interaction between the normal modes, and naturally leads to algebraic localization. Note that the matrix (\[matrix\]) is formally analogous to a semi-infinite tight-binding chain with a defect at one end. Nevertheless it appears in our context when starting with a translationally invariant system, but with many-particle states which include interaction.
Since the effective interaction strength drops in the dilute limit of large chains, we observe stronger localization (except for the case of staggered states). The crucial difference to the classical model is, that while the linear classical dynamics coincides with the single particle quantum problem, nonlinearity in the classical model effectively deforms the single particle dynamics (and adds many other features like chaos etc). The interaction in the quantum problem takes the wave function into the new Hilbert space of many-body wave functions, which is still a linear space, but higher dimensional. Another feature of that quantum interaction is the fact that staggered states do not compactify in the dilute limit of large chains. That property is based on the interference of quantum states, and is not observed in the corresponding classical nonlinear equation. A similar (yet weaker) signature of quantum interference is the observed change of the power of the algebraic decay from two (generic) to four when choosing particular values of the wave number $k_1$, which depend on the Bloch wave number $k$. And yet another signature of quantum q-breathers is the fact that they keep a finite localization in the limit $\gamma \rightarrow \infty$ as seen in Fig.\[gcorr\], and at variance to their classical counterparts, which turn from exponentially localized to completely delocalized in that limit. The reason is that in this strong interaction limit extended two-boson states correspond to their noninteracting counterparts which are projected onto the basis space which does not contain doubly occupied chain sites, while strong nonlinearity in the classical problem completely deforms periodic orbits of the noninteracting system.
We are aware of the fact that the quantum problem studied here is a linear one (in terms of differential equations). Its correspondence to a classical nonlinear system can be observed in the limit of many bosons when treating the many particle quantum states within a Hartree approximation, which projects onto product states. Often the classical description is also achieved using suitable (e.g. coherent state) representations. The presented results have an unambigous meaning in the chosen basis of the noninteracting system. Yet they will of course in general depend on the chosen basis. Therefore it remains a puzzling question, how to restore exponential localization of classical q-breathers from the algebraic decay of quantum q-breathers with two bosons, in the limit of larger numbers of bosons. The fate of quantum q-breathers in higher dimensional lattices is another interesting open question, which will be left to future work. Dr Jean Pierre Nguenang acknowledges the warm hospitality of the Max Planck Institute for the Physics of Complex Systems in Dresden. This work was supported by the DFG (grant No. FL200/8) and by the ESF network-programme AQDJJ.
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---
abstract: 'The absence of a characteristic momentum scale in the pseudo-potential description of atomic interactions in ultracold (two-component Fermi) gases is known to lead to divergences in perturbation theory. Here we show that they also plague the calculation of the dynamics of the total energy following a quantum quench. A procedure to remove the divergences is devised, which provides finite answers for the time-evolution of the total energy after a quench in which the interaction strength is ramped up according to an arbitrary protocol. An important result of this analysis is the time evolution of the asymptotic tail of the momentum distribution (related to Tan’s contact) to leading order in the scattering length. Explicit expressions for the dynamics of the total energy and the contact for a linear interaction ramp are obtained, as a function of the interaction ramp time in the crossover from the sudden quench to the adiabatic limit are reported. In sudden quench limit, the contact, following a rapid oscillation, reaches a stationary value which is different from the equilibrium one. In the adiabatic limit, the contact grows quadratically in time and later saturates to its equilibrium value for the final value of the scattering length.'
author:
- 'Chen-How Huang'
- 'Miguel A. Cazalilla'
bibliography:
- 'cite.bib'
title: |
Total Energy Dynamics and Asymptotics of the Momentum Distribution\
Following an Interaction Quench in a Two-component Fermi Gas
---
Introduction
============
The single-channel model [@LHYpseudopotential] provides a compact, single-parameter description of interactions in ultracold gases. Within this model, interactions in a two-component Fermi gas are described by the following term in the Hamiltonian: $$\hat{V}=\frac{g}{2\Omega}
\sum_{{\boldsymbol{pkqr}}}\sum_{\sigma\neq\alpha}c^{\dagger}_{{\boldsymbol{p}}\sigma}c^{\dagger}_{{\boldsymbol{k}}\alpha}c_{{\boldsymbol{q}}\alpha}
c_{{\boldsymbol{r}}\sigma} \delta_{{\boldsymbol{p}}+{\boldsymbol{k}},{\boldsymbol{q}}+{\boldsymbol{r}}},\label{eq:U}$$ where $c_{{\boldsymbol{p}}\sigma}$ ($c^{\dag}_{{\boldsymbol{p}}\sigma}$) are fermion destruction (creation) operators obeying $\{c_{{\boldsymbol{p}}\sigma},c^{\dag}_{{\boldsymbol{k}}\alpha}\} = \delta_{{\boldsymbol{pk}}} \delta_{\sigma \alpha}$ ($\sigma,\alpha = \uparrow,\downarrow$) and $\{c_{{\boldsymbol{p}}\sigma},c_{{\boldsymbol{k}}\alpha}\} = 0$. However, in terms of the parameter $g$, it appears as if Eq. holds for arbitrarily large particle momentum exchange, ${\boldsymbol{K}} = {\boldsymbol{p}}-{\boldsymbol{r}}$. In other words, the interaction in lacks of a characteristic momentum scale, which nevertheless exists for real interactions but depends on the microscopic details of the two-atom potential.
The lack of a characteristic momentum scale has important consequences for the calculation of physical properties using the single-channel model. For instance, the perturbation series for the ground state energy is plagued with divergences arising from the behavior of integrals at high momenta (see e.g. [@abrikosov1975methods; @pathria1996statistical]). In addition, for momenta smaller than the inverse effective range (i.e. $p\ll R^{-1}$ [^1]), the momentum distribution, $n_{p\sigma} = \langle c^{\dag}_{{\boldsymbol{p}}\sigma} c_{{\boldsymbol{p}}\sigma}\rangle$ exhibits a $p^{-4}$ tail at large momenta $p\gg k_F$ ($k_F$ is the Fermi momentum). This renders divergent the kinetic energy, $E_{\text{kin}}=\sum_{p\sigma} n_{p\sigma} \epsilon_p$ [@1/k4tail; @shinatan_contact] ($\epsilon_p = p^2/2m$ is the single-particle dispersion and $m$ is the atom mass). In connection to this problem, Tan [@shinatan_contact] has recently shown that the total energy can be entirely written in terms of the momentum distribution $n_{k\sigma}$ and a parameter, $C$, which he termed ‘contact’. The latter controls the asymptotic behavior $\sim p^{-4}$ of the momentum distribution $n_{p\sigma}$.
In many-body perturbation theory, the removal (renormalization) of the divergences described above begins by recognizing that the coupling $g$ in Eq. is not physical and must be replaced by the measurable $s$-wave scattering length, $a_s$. Thus, Eq. can be used to compute the two-particle scattering amplitude (see Eq. \[eq:tmatrix\] below). We can parametrize the latter in terms of the $s$-wave phase shift $\delta_s(K)$. In the limit of small momentum exchange of the colliding particles ${\boldsymbol{K}}={\boldsymbol{p}}-{\boldsymbol{r}}$, the phase shift obeys: $$K \cot\delta_s(K) = -\frac{1}{a_s} + \frac{K^2R}{2}+\cdots$$ This expression allows to relate the coupling $g$ in Eq. to the scattering length $a_s$ ($R$ is the effective range). An equivalent treatment relies on perturbation theory to relate $g$ to $a_s$ through the following expression for the two-particle $T$-matrix [@abrikosov1975methods; @pathria1996statistical]: $$T_{\sigma\alpha}({\boldsymbol{k}},{\boldsymbol{p}}) = g + \frac{2g^2}{\Omega}\sum_{{\boldsymbol{qr}}} \frac{\delta_{{\boldsymbol{p}}+{\boldsymbol{k}},{\boldsymbol{q}}+{\boldsymbol{r}}}}{E_{pkqr}} + O(g^3) = \frac{4\pi a_s}{m}.
\label{eq:tmatrix}$$ In the above equation, $\sigma \neq \alpha$ and $E_{pkqr} = \epsilon_{p}+\epsilon_{k}-\epsilon_{q}-\epsilon_{r}$.
In this paper, we study how to remove the divergences that appear in the expressions describing the nonequilibrium dynamics of the total energy following an interaction quantum quench. We consider a fairly general quench in which the system evolution is dictated by the Hamiltonian: $$H(t) = H_0 + S(t) V,$$ where $H_0 =\sum_{{\boldsymbol{p}}\sigma} \epsilon_p c^{\dag}_{{\boldsymbol{p}}\sigma} c_{{\boldsymbol{p}}\sigma}$ is the kinetic energy and $V$ is given in Eq. . The function $S(t)$ describes the experimental protocol followed to turn on the interaction starting from the non-interacting system (accounting for the effects of a weak initial interaction can be also done perturbatively and will be reported elsewhere [@unpub]). We will see that the divergences that plague the calculation of the ground state energy in equilibrium are also present in the perturbative calculation of the total energy dynamics. Specializing to the case of a linear ramp, where $$S(t)=\theta(t)\left[\theta(T_r-t)\frac{t}{T_r}+\theta(t-T_r)\right],\label{eq:ramp_protocol}$$ we obtain the time-evolution of the total energy and the momentum distribution as a function of ramp time $T_r$. Tuning $T_r$ allows us to study the crossover from the sudden (i.e. $T_r\to 0$) to the quasi-adiabatic (i.e. $T_r\to +\infty$) limit.
In order to compute the total energy dynamics, we also compute the instantaneous momentum distribution, i.e. $$\begin{aligned}
n_{p\sigma}(t)=\mathrm{Tr} \: \left[ \rho(t) c^{\dagger}_{{\boldsymbol{p}}\sigma}c_{{\boldsymbol{p}}\sigma} \right],\end{aligned}$$ where $\rho(t)$ describes the state of the system at time $t$ (see Sec. \[sec:appa\] for details). The perturbative expansion for this quantity does not contain any divergent integrals. However, the leading perturbative corrections appearing at $O(g^2)$ behave as $n_{p\sigma}\sim p^{-4}$ at large momentum $p$. Likewise, the dynamics of the total energy is obtained from: $$\begin{aligned}
E_{\text{tot}}(t)&=\mathrm{Tr} \: \left[ \rho(t) H(t)\right] \notag \\
&= \mathrm{Tr} \: \left[ \rho(t) H_0\right] + S(t)
\mathrm{Tr} \: \left[ \rho(t) V\right],\end{aligned}$$ where the first term in the second line is the instantaneous kinetic energy, i.e. $$E_{\mathrm{kin}}(t) = \mathrm{Tr} \: \left[ \rho(t) H_0 \right] = \sum_{{\boldsymbol{p}}\sigma} \epsilon_{p} n_{p\sigma}(t).$$ The high momentum tail $\sim p^{-4}$ of the instantaneous momentum distribution $n_{p\sigma}(t)$ at $p\gg k_F$ renders $E_{\mathrm{kin}}(t)$ divergent, as in the equilibrium case. Interestingly, when the coupling $g$ is replaced by the scattering length $a_s$ by the renormalization method described in Sec. \[sec:re\], the divergence $E_{\mathrm{kin}}(t)$ is cancelled by an additional contribution from the interaction energy $E_{\mathrm{int}}(t) = S(t)
\mathrm{Tr} \: \left[ \rho(t) V\right]$. This cancellation parallels the similar cancellation happening in equilibrium and allows us to isolate the leading term in perturbative expansion of the tail of the instantaneous momentum distribution. For a linear interaction ramp, we find that the definition of Tan’s contact requires some care in order to fully capture the crossover behavior of the tail from the sudden to the adiabatic limit.
Our results also allow to explore the dynamics of the two-component Fermi gas following linear ramp in the interaction strength. A subject of particular interest in this situation is existence of a pre-thermalized regime [@Moeckel2008; @Moeckel2009; @nessi_shorttime2014; @nopreth2d_2014; @prethandth_2009_eckstein; @Nessi_glass_2015; @prethandth_silva_2013; @turnable_integrablity_2014; @nearintegrable_Lagen2016; @Moeckel2010; @spinprethermal2017; @prethandth_mitra_2013; @PhysRevLett.97.156403; @preth_th_luttinger_2016; @prethspinchain_Gong2013; @GGEpreth_2011_kollar; @Miguel2016; @spin_prethandth_2015; @nearintegrable_alba_2017; @preth_spin_short_2018]. Thus, we have explored the existence of a pre-thermalized regime as a function of the ramp time. Previous studies on pre-thermalization in ultracold Fermi gases have focused on the behavior of the momentum distribution near the Fermi momentum $k_F$ at zero temperature [@Moeckel2008; @Moeckel2009; @nessi_shorttime2014; @nopreth2d_2014; @prethandth_2009_eckstein; @Nessi_glass_2015; @GGEpreth_2011_kollar; @Miguel2016; @spin_prethandth_2015]. It was concluded that the persistence of a discontinuity at $k_F$ at the same time that the total energy has reached its final (thermalized) value [@PhysRevLett.93.142002] characterizes the pre-thermalized regime. Here we report results for the dynamics of the full momentum distribution at finite temperatures as well as large momenta, which can provide a more experimentally accessible way to characterize the pre-thermalized regime. This is because in realistic systems, the discontinuity of the momentum distribution at $k_F$ is absent due to finite temperature effects and trap confinement. On the other hand, as we argue below, the dynamics of the full momentum distribution at finite temperature and its asymptotic behavior at high momenta contains a great deal of useful information about the pre-thermalized regime.
The rest of this article is organized as follows. In section \[sec:appa\], we give the derivation of observables. In section \[sec:re\], we discuss the renormalization procedure in and out of equilibrium. In section \[sec:ramp\], we specialize to the a linear ramp quench and study the dynamics of total energy and the momentum distribution to show the emergence of pre-thermalization in short time. In section \[sec:contact\], we derive the contact for a ramp of interaction in the interaction strength and discuss its dynamics and relation to pre-thermalization. In Section \[sec:conclu\], we summarize our results and present the conclusions of this work. Throughout, we use units where $\hbar=1$ and $k_B=1$.
Evolution of Observables {#sec:appa}
========================
![\[1\] Closed-time contour $C$. Times $\tau$ and $\bar{\tau}$ lie on the time ordered and anti-time ordered branches, respectively. $\tau$ is earlier than $\bar{\tau}$ in contour ordering. The turning point $t$ is the time of at which observable in which we are interested is evaluated. $|\Phi_0\rangle$ is the initial state.[]{data-label="contour"}](C2.pdf){width="0.9\columnwidth"}
In what follows, we use perturbation theory to obtain the short to intermediate time dynamics of the system as the interaction is quenched. To leading order in the scattering length, $a_s$, this treatment is valid when the strength of interaction being quenched is small. Indeed, earlier work [@Moeckel2008; @nessi_shorttime2014; @Miguel2016; @Moeckel2009; @Moeckel2010; @perturb_prethandth_2016; @PhysRevB.92.235135; @GGEpreth_2011_kollar; @perturb_werner2013; @kollarperturbation_2013], it has been established that pre-thermalization is accessible through perturbation theory in the quenched interaction. We calculated the observables to second order in the quench interaction. The valid time scale therefore fulfills the condition $E_F t\sim (k_Fa_s)^{-2} \ll (k_Fa_s)^{-3}$, wheres $E_F = k^2_F/2m$ is the Fermi energy. This requires $k_F a_s \ll 1$ in order to apply to hold at long times.
In this section, we provide the details of the derivation of the time evolution of the observables considered in this work. In the interaction picture, the evolution of an observable can be calculated from the following expression, which is amenable to perturbative expansion: $$\begin{aligned}
\langle O(t)\rangle&=\frac{\langle \mathcal{T}[e^{-i \int_C dt\: \tilde{V}(t)} O(t)]\rangle}{\langle \mathcal{T}[e^{-i \int_C dt\: \tilde{V}(t)}] \rangle}\label{eq:opert}.\end{aligned}$$ In Eq. , $\tilde{V}(t)=e^{i H_0 t}Ve^{-i H_0 t}S(t)$ is the quench interaction in interaction picture, and $O = c^{\dag}_{{\boldsymbol{p}}\sigma}c_{{\boldsymbol{p}}\sigma},V,\ldots $ stands for the observable of interest. Time $t$ lies on the closed contour $C$ shown in Fig. \[contour\] and $\mathcal{T}$ is the time-ordering symbol on $C$. Since we are working with a closed time contour, trictly speaking, the denominator of Eq. equals unity. However, it is needed when expanding in powers of $\tilde{V}(t)$ in order to cancel disconnected Feynman graphs. The expectation values are computed according to $\langle ... \rangle = \text{Tr}[ e^{-\beta H_0} ...]$, which is the thermal average with respect to an non-interacting initial Hamiltonian at absolute temperature $T = \beta^{-1}$.
Expanding Eq. in powers of the quenched interaction $\tilde{V}(t)$ yields: $$\begin{aligned}
O(t)&=\langle O\rangle\notag -i\int_C dt_1 \langle \mathcal{T}\left[ \tilde{V}(t_1)O(t) \right] \rangle_c \notag \\
&\quad + \frac{(-i)^2}{2!}\int_C dt_1 dt_2 \, \langle \mathcal{T} \left[ \tilde{V}(t_1)\tilde{V}(t_2)O(t) \right] \rangle_c\notag \\
&\quad + \cdots\end{aligned}$$ As mentioned above, we shall only take into account the fully connected contributions (denoted by $\langle \ldots \rangle_c$ in the expression above) resulting from the application of Wick’s theorem. When using the latter, there are four possible choices of time arguments for the fermion propagator: $$i G_{p\sigma}(t_1,t_2)= \langle \mathcal{T} \left[ c_{{\boldsymbol{p}}\sigma}(t_1)c_{{\boldsymbol{p}}\sigma}^{\dagger}(t_2)\right]\rangle, \label{eq:g0}$$ where $t_1$ and $t_2$ can be either in the $\tau$ or $\bar{\tau}$ branches of $C$. The free fermion propagator can be written in matrix form as follows: $$\begin{aligned}
&\mathcal{G}_{p\sigma}(a,b)=
\begin{pmatrix}
iG^{T}_{p\sigma}(a,b)&iG^{<}_{p\sigma}(a,\bar{b})\\
iG^{>}_{p\sigma}(\bar{a},b)&iG^{\tilde{T}}_{p\sigma}(\bar{a},\bar{b})
\end{pmatrix}.\label{eq:g}\end{aligned}$$ Using $c_{{\boldsymbol{p}}\sigma}(t)=c_{{\boldsymbol{p}}\sigma}e^{-i\epsilon_{p}t}$, the elements of the above matrix can be evaluated to yield: $$\begin{aligned}
i G_{p\sigma}^{<}(t_1,\bar{t}_2)&= -n_{p\sigma}^0e^{i\epsilon_p(t_2-t_1)},\label{eq:g1}\\
i G_{p\sigma}^{>}(\bar{t}_1,t_2)&=(1- n^0_{p\sigma})e^{i\epsilon_p(t_2-t_1)},\label{eq:g2}\\
i G^{T}_{p\sigma}(t_1,t_2)&=
\theta(t_1-t_2) iG_{p\sigma}^{>}(\bar{t}_1,t_2)\notag\\
&+\theta(t_2-t_1)iG^{<}_{p\sigma}(t_1,\bar{t}_2),\label{eq:g3}\\
i G^{\tilde{T}}_{p\sigma}(\bar{t}_1,\bar{t}_2)&=
\theta(t_2-t_1) iG^{>}_{p\sigma}(\bar{t}_1,t_2)\notag\\
&\quad +\theta(t_1-t_2)iG^{<}_{p\sigma}(t_1,\bar{t}_2).\label{eq:g4}\end{aligned}$$
Instantaneous momentum distribution {#sec:md}
-----------------------------------
![First and second order diagram for momentum distribution. $\tilde{V}(t)$ is the quench interaction and its time dependence can be chosen to be on the two time branches. Therefore, we need to consider four types of green function from the contour ordering.[]{data-label="fig:feyn1"}](md.pdf){width="\columnwidth"}
Let us first consider the evolution of the momentum distribution. The latter is obtained from the Eq. by setting $O = \hat{n}_{p\sigma} = c^{\dag}_{p\sigma} c_{p\sigma}$. Hence, expanding the evolution operator up to second order in the quenched interaction, the following expression is obtained: $$\begin{gathered}
n_{p\sigma}(t)=\langle \hat{n}_{p\sigma}\rangle\notag -i\int_C dt_1 \langle \mathcal{T}\left[ \tilde{V}(t_1)\hat{n}_{p\sigma}(t) \right] \rangle_c \\
+\frac{(-i)^2}{2!}\int_C dt_1 dt_2 \, \langle \mathcal{T} \left[ \tilde{V}(t_1)\tilde{V}(t_2)\hat{n}_{p\sigma}(t) \right] \rangle_c
+ \cdots \end{gathered}$$ The expectation values in the above expressions can be represented diagrammatically. In the closed time contour, we can express the expectation value using the self-energy and propagator matrices, which to second order in the quenched interaction yield: $$\begin{gathered}
n_{p\sigma}(t) = \langle \hat{n}_{p\sigma}\rangle + \int_C dt_1\: \mathcal{G}_{p\sigma}(t,t_1) \Sigma^{(1)}_{p\sigma}(t_1)\mathcal{G}_{p\sigma}(t_1,t) \\
+ \int_C
dt_1 \int_C dt_2 \: \mathcal{G}_{p\sigma}(t,t_2)\:
\Sigma^{(2)}_{p\sigma}(t_2,t_1) \mathcal{G}_{p\sigma}(t_1,t)
+\ldots \\
\label{eq:n}\end{gathered}$$ The propagator $\mathcal{G}_{p\sigma}(a,b)$ is defined in Eq. , and $\Sigma^{(1)}_{p\sigma}(t_1)$, $\Sigma^{(2)}_{p\sigma}(t_2,t_1)$ can be found using diagrams (see below). For the calculation of equal-time expectation values, we choose the time argument of the observable (i.e. $t$) to lie slightly before the turning point of the contour $C$, which is on the time ordered (i.e. $\tau$) branch. In this case, the fermion propagators must be obtained from Eq. , which yields: $$\mathcal{G}_{p\sigma}(t,b)=e^{-i\epsilon_{p}(t-b)}\begin{pmatrix}
1-n^0_{p\sigma}&-n^0_{p\sigma}\\0&0\end{pmatrix}
\label{eq:Gg1},$$ where the non-vanishing entries correspond to either $b$ lying before or after $t$ on the contour $C$. Similarly, $$\mathcal{G}_{p\sigma}(a,t)=
e^{-i\epsilon_{p}(a-t)}\begin{pmatrix} -n^0_{p\sigma}&0\\1-n^0_{p\sigma}&0\end{pmatrix}.
\label{eq:G2}$$ The two non-zero entries in the above matrix correspond to $a$ lying before or after $t$ on the contour $C$.
The self-energy can be calculated from the diagrams shown in Fig. \[fig:feyn1\] and the propagators, Eqs. to . Thus, to first order in $V(t)$, we obtain: $$\begin{aligned}
&\Sigma^{(1)}_{\sigma}=\frac{g}{2\Omega}
\sum_{{\boldsymbol{k}}}n^0_{k,-\sigma}.\label{eq:s1}\end{aligned}$$ Combining the matrix from Eq. , the propagators, Eq. and Eq. , and the self-energy for the first order correction, Eq. , we obtain that first order correction to the instantaneous momentum distribution vanishes, i.e. $n_{p\sigma}^{(1)}(t)=0$. Ultimately, this is a consequence of the initial state being an eigenstate of the occupation operator $\hat{n}_{p\sigma} = c^{\dag}_{{\boldsymbol{p}}\sigma} c_{{\boldsymbol{p}}\sigma}$.
At second order in the interaction, we need to use the following self-energy matrix, which contains four different combinations of the time arguments $(t_1,t_2)$ on the two branches of the contour $C$: $$\begin{aligned}
&\Sigma^{(2)}_{p\sigma}(b,a)=-\frac{2 g^2}{\Omega^2}S(t_1)S(t_2) \sum_{{\boldsymbol{kqr}}}\delta_{{\boldsymbol{p}}+{\boldsymbol{k}},{\boldsymbol{q}}+{\boldsymbol{r}}}\notag \\
&\times
\begin{pmatrix}\bar{\Sigma}_{\sigma}^{(2,T)}(b,a)&\bar{\Sigma}_{\sigma}^{(2,>)}(\bar{b},a)\\\bar{\Sigma}_{\sigma}^{(2,<)}(b,\bar{a}) &\bar{\Sigma}_{\sigma}^{(2,\tilde{T})}(\bar{b},\bar{a})\end{pmatrix}.\end{aligned}$$ Evaluating the diagrams in Fig. \[fig:feyn1\] using the propagators in Eqs. to , we obtain the following expressions for the elements of the self-energy matrix: $$\begin{aligned}
\bar{\Sigma}_{\sigma}^{(2,<)}(t_2,\bar{t}_1)&=i^3G^{<}_{k\alpha}(t_2,\bar{t}_1)G^{>}_{q\sigma}(\bar{t}_1,t_2)G^{>}_{r\alpha}(\bar{t}_1,t_2),\notag\\
&=-(1-n^0_{q\sigma})(1-n^0_{r\alpha}) n^0_{k\alpha}e^{i(t_1-t_2)(\epsilon_{q}+\epsilon_{r}-\epsilon_{k})} ,\label{s1}\\
\bar{\Sigma}^{(2,>)}(\bar{t}_2,t_1)&=i^3G^{>}_{k\alpha}(\bar{t}_2,t_1)G^{<}_{q\sigma}(t_1,\bar{t}_2)G^{<}_{r\alpha}(t_1,\bar{t}_2),\notag \\
&=n^0_{q\sigma}n^0_{r\alpha}(1-n^0_{k\alpha})e^{i(t_1-t_2)(\epsilon_{q}+\epsilon_{r}-\epsilon_{k})},\label{s2}\\
\bar{\Sigma}^{(2,T)}(t_2,t_1)&=i^3G^{T}_{k\alpha}(t_2,t_1)G^{T}_{q\sigma}(t_1,t_2)G^T_{r\alpha}(t_1,t_2),\notag\\
=\theta(t_2&-t_1)\Sigma^{(2,>)}(t_2,\bar{t}_1)+\theta(t_1-t_2)\Sigma^{(2,<)}(\bar{t}_2,t_1),\label{s3} \\
\bar{\Sigma}^{(2,\tilde{T})}(\bar{t}_2,\bar{t}_1)&=i^3G^{\bar{T}}_{k\alpha}(\bar{t}_2,\bar{t}_1)G^{\bar{T}}_{q\sigma}(\bar{t}_1,\bar{t}_2)G^{\bar{T}}_{r\alpha}(\bar{t}_1,\bar{t}_2),\notag\\
=\theta(t_1&-t_2)\Sigma^{(2,>)}(t_2,\bar{t}_1)+\theta(t_2-t_1)\Sigma^{(2,<)}(\bar{t}_2,t_1).\label{s4}\end{aligned}$$ Combining Eq. , the propagators (cf. Eq. \[eq:g1\] to Eq. \[eq:g4\]) and the second order corrections to the self-energy, Eqs. to , we arrive at: $$n^{(2)}_{p\sigma}(g,t) =-\frac{2g^2}{\Omega^2}\sum_{{\boldsymbol{kqr}}} A^{\sigma}_{pkqr} F^{(2)}(E_{pkqr},t) \delta_{{\boldsymbol{p}}+{\boldsymbol{k}},{\boldsymbol{q}}+{\boldsymbol{r}}},\label{eq:nk2}$$ where $A^{\sigma}_{pkqr}$ denotes the function: $$\begin{gathered}
A^{\sigma}_{pkqr} = \sum_{\alpha\neq\sigma}\left[n^0_{p\sigma}n^0_{k\alpha}(1-n^0_{q\sigma})(1-n^0_{r\alpha}) \right. \\
\left. \quad -(1-n^0_{p\sigma})(1-n^0_{k\alpha})n_{q\sigma}^0n_{r\alpha}^0\right].\end{gathered}$$ In Eq. , $E_{pkqr}=\epsilon_p+\epsilon_k-\epsilon_q-\epsilon_r$, and the function $$F^{(2)}(E,t) = \int\limits_{-\infty}^{t} dt_1 \int\limits_{-\infty}^{t} dt_2\: S(t_1)S(t_2) \: e^{i E(t_1-t_2)} \label{eq:f2}$$ has been introduced. Note that $F^{(2)}(E,t)$ depends on the explicit form of $S(t)$ (see Sec. \[sec:ramp\] and Appendix \[sec:F\] for the form of this function in a number of important limiting cases).
For $p > k_F$, the above expression, Eq. , reduces to $$\begin{gathered}
n^{(2)}_{p\sigma}(g,t) =\frac{2g^2}{\Omega^2}\sum_{{\boldsymbol{kqr}},\sigma} (1 -n_{k,-\sigma}) n_{q,\sigma}n_{r,-\sigma}\\
\times F^{(2)}(E_{pkqr},t) \delta_{{\boldsymbol{p}}+{\boldsymbol{k}},{\boldsymbol{q}}+{\boldsymbol{r}}}.\end{gathered}$$ Furthermore, momentum conservation requires that ${\boldsymbol{k}} = {\boldsymbol{q}}+{\boldsymbol{r}} - {\boldsymbol{p}}$. Since the occupation factors in this expression force $q, r \leq k_F$, it follows that $|{\boldsymbol{q}}+{\boldsymbol{r}}| \leq 2 k_F$. Thus, for $p\gg k_F$, $k \gg k_F$, and the above expression simplifies to $$\begin{gathered}
n^{(2)}_{p\gg k_F,\sigma}(g, t) =\frac{2g^2}{\Omega^2}\sum_{{\boldsymbol{kqr}}} n_{q\sigma}n_{r,-\sigma} \delta_{{\boldsymbol{p}}+{\boldsymbol{k}},{\boldsymbol{q}}+{\boldsymbol{r}}}\\
\times F^{(2)}(E_{pkqr},t).\label{eq:nplarge}\end{gathered}$$ It will be shown in Sec. \[sec:contact\] that this expression leads to a $\sim p^{-4}$ tail of the momentum distribution, which renders the kinetic energy divergent.
Unrenormalized kinetic energy {#sec:Ekin}
-----------------------------
From the above result for the instantaneous momentum distribution, we can derive the second order correction to the kinetic energy: $$\begin{gathered}
\delta E_{\mathrm{kin}}^{(2)}(q,t)=\sum_{p,\sigma}\epsilon_p n^{(2)}_{p\sigma}(t)\\
=-\frac{2g^2}{\Omega^2}\sum_{{\boldsymbol{pkqr}},\sigma} \epsilon_p A^{\sigma}_{pkqr} F^{(2)}(E_{pkqr},t)\delta_{{\boldsymbol{k}}+{\boldsymbol{p}},{\boldsymbol{q}}+{\boldsymbol{r}}} \\
=\frac{-g^2}{2\Omega^2}\sum_{{\boldsymbol{pkqr}},\sigma} A^{\sigma}_{pkqr} E_{pkqr}F^{(2)}(E_{pkqr},t)\delta_{{\boldsymbol{k}}+{\boldsymbol{p}},{\boldsymbol{q}}+{\boldsymbol{r}}}
\label{eq:Ek}\end{gathered}$$ where, in the last line, we have used that $\sum_{\sigma}A^{\sigma}_{pkqr} = \sum_{\sigma}A^{\sigma}_{kpqr} = - \sum_{\sigma}A^{\sigma}_{qrpk} = -\sum_{\sigma}A^{\sigma}_{rqpk} $ and $F^{(2)}(-E,t) = F^{(2)}(E,t)$. Notice that in two extreme limits, i.e. the sudden and adiabatic limits $F^{(2)}(E,t) \sim 1/E$ for $E\to +\infty$ (see Appendix \[sec:F\]). Thus, the above expression is divergent, as in equilibrium (the expression of the adiabatic limit coincides with the equilibrium one). Generically, for any other quench protocol function $S(t)$ between these two extreme limits, we expect the same behavior and the above expression to remain divergent, as it is also confirmed for a linear ramp quench, see Sec. \[sec:ramp\].
Unrenormalized interaction energy {#sec:Eint}
---------------------------------
![Feynman diagrams for the first- and second-order corrections to the interaction energy.[]{data-label="fig:intdiag"}](Vi.pdf){width="\columnwidth"}
The interaction energy can be obtained from Eq. by setting $O = \tilde{V}(t)$. Expanding this expression to second order in $g$ yields $$\begin{gathered}
\delta E_{\mathrm{int}}(g,t)=\langle \Phi_0|\tilde{V}(t)| \Phi_0\rangle\\
-i\int_C dt_1 \: \langle \Phi_0| \mathcal{T}\left[\tilde{V}(t_1)\tilde{V}(t)\right]|\Phi_0\rangle_c + \cdots \end{gathered}$$ The calculation in this case proceeds in a similar fashion to the one in previous section. Evaluating the Feynman diagrams in Fig. \[fig:intdiag\], we obtain: $$\begin{aligned}
\delta E_{\mathrm{int}}(t)&=V^{(1)}(t) + V^{(2)}(t) + \cdots,\\
V^{(1)}(t) &= \frac{g}{2\Omega}S(t)\sum_{\alpha\neq\sigma}
\sum_{kp}n^0_{k\sigma}n^0_{p\alpha}\label{E1},\\
V^{(2)}(t) &= \int_Cd\tau_1\: S(\tau_1)
\Lambda^{(2)}(\tau_1;t).
\label{eq:int}\end{aligned}$$ Note that the first order term, $V^{(1)}(t)$, Eq. , is proportional to the function $S(t)$. The second order term, $V^{(2)}(t)$ can be obtained from the matrix: $$\begin{aligned}
\Lambda^{(2)}(t_1,t)&=\begin{pmatrix}\Lambda^{(2,T)}(t_1,t)&\Lambda^{(2,<)}(t_1,\bar{t})\\
\Lambda^{(2,>)}(\bar{t}_1,t)&\Lambda^{(2,\tilde{T})}(\bar{t}_1,\bar{t})\end{pmatrix},\\
&=\begin{pmatrix}\Lambda^{(2,T)}(t_1,t)&0\\
\Lambda^{(2,>)}(\bar{t}_1,t)&0\end{pmatrix}.\end{aligned}$$ In the last line, we have used that $t$ lies on the time-ordered branch of the contour $C$. Using the free propagator (cf. Eqs. \[eq:g1\] to \[eq:g4\]), the second order term becomes: $$\begin{aligned}
&\Lambda^{(2)}(t_1,t)=\frac{-ig^2 S(t)}{\Omega^2}\sum_{{\boldsymbol{pkqr}}}e^{iE_{pkqr}(t_1-t)}\\
&\times \begin{pmatrix}
(1-n^0_{p\sigma})(1-n^0_{k\alpha})n^0_{q\sigma}n^0_{r\alpha}&0 \\
n^0_{p\sigma}n^0_{k\alpha}(1-n^0_{q\sigma})(1-n^0_{r\alpha})&0\end{pmatrix}.\label{eq:L2}
\end{aligned}$$ From the above expression, using Eq. , we obtain the second order correction to the interaction energy: $$\begin{gathered}
\delta E_{\mathrm{int}}^{(2)}(g,t)=S(t)\int_C dt_1\: \Lambda^{(2)}(t,t_1) \: S(t_1) \\
=\frac{g^2 S(t)}{\Omega^2}\sum_{{\boldsymbol{pkqr}},\sigma} A^{\sigma}_{pkqr}F^{(1)}(E_{pkqr},t)\delta_{{\boldsymbol{k}}+{\boldsymbol{p}},{\boldsymbol{q}}+{\boldsymbol{r}}}.\label{eq:eint}\end{gathered}$$ In the last line, we have introduced the function $$F^{(1)}(E,t)=\int_{-\infty}^{t} \sin\left[E(t-t_1)\right] S(t) dt_1, \label{eq:f1}$$ which depends on the function $S(t)$ that defines the quench protocol. The sine function in Eq. appears after swapping the dummy momenta around in order to symmetrize the integrand, that is, after using: $$\begin{aligned}
\sum_{{\boldsymbol{pkqr}}}f_{{\boldsymbol{pkqr}}}=\frac{1}{2}\sum_{{\boldsymbol{pkqr}}}\left[ f_{{\boldsymbol{pkqr}}}+f_{{\boldsymbol{qrpk}}} \right].\end{aligned}$$ Again, notice that that in the sudden and adiabatic limits, $F^{(1)}(E,t) \sim 1/E$ for $E\to +\infty$ (see Appendix \[sec:F\]), which means that Eq. is divergent, as in equilibrium (the expression of the adiabatic limit coincides with the equilibrium result, see Appendix. \[sec:F\]). Generically, for any other quench protocol function $S(t)$ between these two extreme limits, we expect the above expression to remain divergent. This is explicitly confirmed for a linear ramp protocol in Sec. \[sec:ramp\].
Unrenormalized total energy {#sec:Etot}
---------------------------
From the results obtained in previous sections for the interaction and kinetic energy, we can obtain the dynamics of the total energy for a quench in which the interaction is switched on according to an arbitrary protocol described by $S(t)$. The first order correction is: $$\delta E^{(1)}_{\mathrm{tot}}(g,t)=\frac{g}{2\Omega}\sum_{{\boldsymbol{pk}},\sigma\neq\alpha} n_{p\sigma}n_{k\alpha},$$ and the second order correction reads: $$\begin{aligned}
\delta E^{(2)}_{\mathrm{tot}}(g,t)&=\delta E^{(2)}_{\mathrm{kin}}(g,t)+\delta E^{(2)}_{\mathrm{int}}(g,t),\notag\\
&=\frac{g^2}{\Omega^2}\sum_{{\boldsymbol{pkqr}},\sigma} A^{\sigma}_{pkqr} \: F_{\mathrm{tot}}(E_{pkqr},t) \delta_{{\boldsymbol{p}}+{\boldsymbol{k}},{\boldsymbol{q}}+{\boldsymbol{r}}}, \notag\\
&= \frac{4g^2}{\Omega^2}\sum_{pkqr}n^0_{p,\uparrow}n^0_{k,\downarrow}(1-n^0_{q,\uparrow})(1-n^0_{r,\downarrow})\notag\\&\qquad\times F_{\mathrm{tot}}(E_{pkqr},t) \delta_{{\boldsymbol{p}}+{\boldsymbol{k}},{\boldsymbol{q}}+{\boldsymbol{r}}}
\label{eq:Etott}\end{aligned}$$ where, in the last line, we have introduced the function: $$\begin{aligned}
F_{\mathrm{tot}}(E,t)=S(t)F^{(1)}(E,t)-\frac{E}{2}F^{(2)}(E,t).\label{eq:F}\end{aligned}$$ and used that $F_{\mathrm{tot}}(-E,t) = -F_{\mathrm{tot}}(E,t)$ to obtain the expression in the last line of . Furthermore, notice that, as shown in Appendix \[sec:F\], $F_{\mathrm{tot}}(E,t)$ vanishes in the limit of a sudden quench. This ensures that the second order correction to the total energy vanishes, as required by energy conservation. See Eq. in Appendix \[sec:F\] and Sec. \[sec:ramp\] for a more in depth discussion of this point.
Elimination of divergences {#sec:re}
==========================
As we have mentioned in the Introduction, perturbation theory in the powers of the *bare* interaction $V\propto g$ (cf. Eq. \[eq:U\]) yields an expression for the equilibrium total energy containing divergent integrals [@Abrikosov1965; @Parthia2011]. The divergences appear at second in the coupling $g$ and their elimination, i.e. their ‘renormalization’, is possible by realizing that the *unphysical* $g$ must be replaced by the *physical* $s$-wave scattering length, $a_s$. The latter is related to $g$ via the two-particle scattering amplitude (i.e. the $T$-matrix), see Eq. .
The same divergences reappear in the perturbative expansion for the dynamics of the total energy obtained in previous sections. Indeed, as we show by explicit calculation below, the kinetic energy is divergent because the leading order \[i.e. $O(g^2)$\] correction to instantaneous momentum distribution behaves as $\sim p^{-4}$ for $p\gg k_F$. Hence, $E_{\mathrm{kin}}(t) = \sum_{p\sigma} \epsilon_p n_{p\sigma}(t)$ is divergent at all times $t$. Similar divergences appear in the expressions for the interaction energy at $O(g^2)$ as well.
However, in the nonequilibrium case, it is difficult to rely on the $T$-matrix because defining the latter requires the introduction of asymptotic scattering states. Those states are well defined for interactions that are switched on and off adiabatically, as it is assumed in equilibrium, but this becomes difficult when the interaction changes (sometimes, very rapidly) in time as it is the case of our study. Thus, the generalization of the renormalization procedure employed in equilibrium to the problem of interest here is not straightforward. Instead, we show below that the renormalization procedure can be carried out by computing the perturbative corrections to the evolution of the total two-particle energy.
Renormalization in equilibrium
------------------------------
Let us begin by reviewing how the divergences are eliminated in equilibrium case. As mentioned above, we shall compute the shift of the total energy for two particles to relate $g$ to the $s$-wave scattering length, $a_s$ [@pathria1996statistical]. This approach reproduces the well-known equilibrium results based on the scattering matrix approach [@pathria1996statistical; @Abrikosov1965]. The shift to the total energy for two-particles is defined by: $$\begin{aligned}
\label{eq:eq}
\delta E^{2-\mathrm{body}}_{\mathrm{tot}}
({\boldsymbol{p}}\sigma,{\boldsymbol{k}}\alpha)&= E^{2-\mathrm{body}}_{\mathrm{kin}} + E^{2-\mathrm{body}}_{\mathrm{int}}\notag\\
&=\frac{\langle \Psi_{{\boldsymbol{kp}},\sigma\alpha} | \left( H_0+V \right) | \Psi_{{\boldsymbol{kp}},\sigma\alpha}\rangle}{\langle \Psi_{{\boldsymbol{kp}},\sigma\alpha} | \Psi_{{\boldsymbol{kp}},\sigma\alpha}\rangle}\notag \\
&\quad -\langle\Psi^{0}_{{\boldsymbol{kp}},\sigma\alpha}| H_0 | \Psi^{0}_{{\boldsymbol{kp}},\sigma\alpha}\rangle,\end{aligned}$$ where $|\Psi^{0}_{{\boldsymbol{kp}},\sigma\alpha}\rangle=c^{\dag}_{{\boldsymbol{p}}\sigma}c^{\dag}_{{\boldsymbol{k}}\alpha}|0\rangle$ describes a state of two free particles with $\sigma \neq \alpha$, and $|\Psi_{{\boldsymbol{pk}},\sigma\alpha}\rangle$ is the two-particle state perturbed by the interactions. Using time-independent perturbation theory, $$\begin{aligned}
|\Psi_{{\boldsymbol{kp}},\sigma\alpha}\rangle &=|\Psi^0_{{\boldsymbol{kp}},\sigma\alpha}\rangle +\sum_{{\boldsymbol{qr}}\neq {\boldsymbol{kp}}} |\Psi^{0}_{{\boldsymbol{qr}},\sigma\alpha}\rangle \notag\\
& \times \frac{\langle\Psi^{0}_{{\boldsymbol{qr}},\sigma\alpha} |V|\Psi^0_{{\boldsymbol{kp}},\sigma\alpha}\rangle}{E_{pkqr}}+O(V^2),\notag\\
&=|\Psi^0_{{\boldsymbol{kp}},\sigma\alpha}\rangle +\frac{g}{\Omega}\sum_{{\boldsymbol{qr}}} \frac{\delta_{{\boldsymbol{p}}+{\boldsymbol{k}},{\boldsymbol{q}}+{\boldsymbol{r}}} }{E_{pkqr}} \left[ |\Psi^{0}_{{\boldsymbol{qr}},\sigma\alpha}\rangle \right. \notag\\
&\left.\qquad + |\Psi^{0}_{{\boldsymbol{qr}},\alpha \sigma}\rangle \right] + O(g^2),
\label{eq:2bodystate}\end{aligned}$$ where $E_{kpqr} = \epsilon_k + \epsilon_p - \epsilon_q - \epsilon_r$. Hence, the shift of the kinetic energy is $$\begin{aligned}
E^{2-\mathrm{body}}_{\mathrm{kin}} &=
\frac{\langle \Psi_{{\boldsymbol{kp}},\sigma\alpha} | H_0 | \Psi_{{\boldsymbol{kp}},\sigma\alpha}\rangle}{\langle \Psi_{{\boldsymbol{kp}},\sigma\alpha} | \Psi_{{\boldsymbol{kp}},\sigma\alpha}\rangle}-\langle\Psi^{0}_{{\boldsymbol{kp}},\sigma\alpha}| H_0 | \Psi^{0}_{{\boldsymbol{kp}},\sigma\alpha}\rangle \notag\\
&= -\frac{2g^2}{\Omega^2}\sum_{{\boldsymbol{qr}}\neq{\boldsymbol{kp}}} \frac{\delta_{{\boldsymbol{k}}+{\boldsymbol{p}},{\boldsymbol{q}}+{\boldsymbol{r}}}}{E_{pkqr}} + O(g^3).\end{aligned}$$ To the same order in $g$, the shift to the interaction energy reads: $$\begin{aligned}
E^{2-\mathrm{body}}_{\mathrm{int}} &=
\frac{\langle \Psi_{{\boldsymbol{kp}},\sigma\alpha} | V | \Psi_{{\boldsymbol{kp}},\sigma\alpha}\rangle}{\langle \Psi_{{\boldsymbol{kp}},\sigma\alpha} | \Psi_{{\boldsymbol{kp}},\sigma\alpha}\rangle}\notag \\
&= \frac{g}{\Omega} + \frac{4g^2}{\Omega^2} \sum_{{\boldsymbol{qr}}\neq {\boldsymbol{kp}}} \frac{\delta_{{\boldsymbol{k}}+{\boldsymbol{p}},{\boldsymbol{q}}+{\boldsymbol{r}}}}{E_{pkqr}} + O(g^3).\label{eq:perint}\end{aligned}$$ Thus, the total energy shift is $$\begin{aligned}
\delta E^{2-\mathrm{body}}_{\mathrm{tot}} &= E^{2-\mathrm{body}}_{\mathrm{kin}} + E^{2-\mathrm{body}}_{\mathrm{int}} \notag \\
&= \frac{g}{\Omega} + \frac{2g^2}{\Omega^2} \sum_{{\boldsymbol{qr}}} \frac{\delta_{{\boldsymbol{k}}+{\boldsymbol{p}},{\boldsymbol{q}}+{\boldsymbol{r}}}}{E_{pkqr}} + O(g^3).
\label{eq:pertres}\end{aligned}$$ Let us define the physical scattering amplitude by requiring that: $$\begin{aligned}
\delta E^{2\mathrm{-body}}_{\mathrm{tot}}(p\sigma,k\alpha)=\frac{4\pi a_s}{m \Omega},\label{eq:as}\end{aligned}$$ after equating it to Eq. , we arrive at Eq. . Inverting the series gives the coupling $g$ in terms of the scattering length $$\begin{aligned}
g=\frac{4\pi a_s}{m}-\frac{2}{\Omega}\left(\frac{4\pi a_s}{m}\right)^2 \sum_{qr}\frac{\delta_{{\boldsymbol{p}}+{\boldsymbol{k}},{\boldsymbol{q}}+{\boldsymbol{r}}}}{E_{pkqr}}+O(a_s^3),
\label{eq:invertg}\end{aligned}$$ which allows to remove the divergences in the expressions for the many-particle ground state energy [@abrikosov1975methods; @pathria1996statistical]. Note that the same result can be obtained directly from the perturbative expression for the *total* energy shift . However, we have chosen this more cumbersome method in order to emphasize that using the interaction energy instead would introduce in Eq. a spurious factor of two (compare Eqs. and ), which would not remove the divergences.
Nonequilibrium renormalization {#sec:reneq}
------------------------------
Let us now turn to the nonequilibrium situation. In the calculation described in Sec. \[sec:appa\], the evolution of the total energy shift is obtained in a perturbative series in the coupling $g$: $$\begin{aligned}
\delta E_{\mathrm{tot}}(g,t)=E_{\mathrm{tot}}^{(1)}(g,t)+E_{\mathrm{tot}}^{(2)}(g,t)+ \cdots\end{aligned}$$ where $E^{(n)}_{\mathrm{tot}}(g,t) = O(g^n)$. However, this expansion contains divergent integrals. Our goal in this section is to remove the divergences, which can be carried out after replacing $g$ by $a_s$, in a procedure similar to the one outlined above for the equilibrium case.
Adding up the first and second order corrections obtained in Sec. \[sec:appa\] yields: $$\begin{gathered}
\delta E_{\mathrm{tot}}(g,t)=\frac{g }{\Omega}\sum_{{\boldsymbol{pk}}}n_{p,\uparrow}^0n_{k,\downarrow}^0 S(t)\\
+\frac{4g^2}{\Omega^2}\sum_{{\boldsymbol{pkqr}}}n^0_{p,\uparrow}n^0_{k,\downarrow}
(1-n^0_{q,\uparrow}) (1-n^0_{r,\downarrow}) \delta_{{\boldsymbol{p}}+{\boldsymbol{k}},{\boldsymbol{q}}+{\boldsymbol{r}}}\\
\times F_{\mathrm{tot}}(E_{pkqr},t) +O(g^3).\label{eq:Etot}\end{gathered}$$ For two-particles, the evolution of the total energy shift can be obtained in a similar fashion, which leads to: $$\begin{gathered}
\delta E^{\mathrm{2-body}}_{\mathrm{tot}}({\boldsymbol{p}}\sigma,{\boldsymbol{k}}\alpha,t)=\frac{gS(t)}{\Omega}+\frac{4g^2}{\Omega^2}\sum_{{\boldsymbol{qr}}}
\delta_{{\boldsymbol{p}}+{\boldsymbol{k}},{\boldsymbol{q}}+{\boldsymbol{r}}}\\
\times F_{\mathrm{tot}}(E_{\mathrm{pkqr}},t) +O(g^3).\end{gathered}$$ Next, we require that: $$\delta E^{\mathrm{2-body}}_{\mathrm{tot}}({\boldsymbol{p}}\sigma,{\boldsymbol{k}}\alpha,t) =
\frac{4\pi a_s }{m} \frac{S(t)}{\Omega}.$$ Inverting the series will gives the coupling $g$ in terms of the scattering length [^2]: $$\begin{aligned}
g(t)S(t)&=\left(\frac{4\pi a_s}{m}\right) S(t) -\frac{4}{\Omega}\sum_{{\boldsymbol{qr}}}\left(\frac{4\pi a_s}{m}\right)^2 \delta_{{\boldsymbol{p}}+{\boldsymbol{k}},{\boldsymbol{q}}+{\boldsymbol{r}}} \notag \\
& \times F_{\mathrm{tot}}(E_{pkqr},t) +O(a_s^3),
\label{eq:gc}\end{aligned}$$ In the limit where the interaction is switched adiabatically, notice that $F_{\text{tot}}(E)=2E^{-1}$ (see Appendix \[sec:F\]) and thus we recovers the equilibrium result, Eq. .
{width="\textwidth"}
Substituting Eq. into the first order correction for total energy shift, Eq. , yields the following correction: $$\begin{aligned}
\Delta_{\mathrm{tot}}(t)=\frac{4}{\Omega}\left(\frac{4\pi a_s}{m}\right)^2\sum_{{\boldsymbol{pkqr}}}n_{p,\uparrow}^0n_{k,\downarrow}^0 \delta_{{\boldsymbol{p}}+{\boldsymbol{k}},{\boldsymbol{q}}+{\boldsymbol{r}}}\notag\\
\times F_{\mathrm{tot}}(E_{pkqr},t),
\end{aligned}$$ which is second order in the scattering length $a_s$ and divergent. However, this divergence exactly cancels the one already present in the second order correction to the energy, i.e. $E_{\mathrm{tot}}^{(2)}(g^2,t)$ (after replacing $g$ by $(4\pi a_s/m)$). Thus, the finite expression for the dynamics of the total energy to second order in $a_s$ is : $$\begin{gathered}
\delta E_{\mathrm{tot}}\left(g = \frac{4\pi a_s}{m},t\right)-\Delta_{\mathrm{tot}}(t)=\left(\frac{4\pi a_s}{m}\right)\frac{ S(t)}{\Omega}\sum_{{\boldsymbol{pk}}}n_{p,\uparrow}^0n_{k,\downarrow}^0\\
-2\left(\frac{4\pi a_s}{m}\right)^2\sum_{{\boldsymbol{pkqr}}}n^0_{p,\uparrow}n^0_{k,\downarrow}\left(n^0_{q,\uparrow}+n^0_{r,\downarrow}\right) \delta_{{\boldsymbol{p}}+{\boldsymbol{k}},{\boldsymbol{q}}+{\boldsymbol{r}}}\\
\times F_{\mathrm{tot}}(E_{pkqr},t) +O(a_s^3),\label{eq:Etot1}\end{gathered}$$ Furthermore, according to the dependence on $S(t)$ and time, we can split $\Delta_{\mathrm{tot}}(t)$ into the sum of $\Delta_{\mathrm{kin}}(t) \sim F^{(2)}(E,t)$ and $\Delta_{\mathrm{int}}(t)\sim F^{(1)}(E,t)$. Explicitly, $$\begin{aligned}
\Delta_{\mathrm{kin}}(t)&= -\frac{4}{\Omega^2}\left(\frac{4\pi a_s}{m}\right)^2\sum_{{\boldsymbol{pkqr}}}n^0_{p\uparrow}n^0_{k\downarrow}\delta_{{\boldsymbol{p}}+{\boldsymbol{k}},{\boldsymbol{q}}+{\boldsymbol{r}}}\notag\\
&\times \frac{E_{pkqr}}{2}F^{(2)}(E_{pkqr},t),\label{eq:dk}\\
\Delta_{\mathrm{int}}(t) &=\frac{4}{\Omega^2}\left(\frac{4\pi a_s}{m}\right)^2\sum_{{\boldsymbol{pkqr}}}n^0_{p\uparrow}n^0_{k\downarrow}\delta_{{\boldsymbol{p}}+{\boldsymbol{k}},{\boldsymbol{q}}+{\boldsymbol{r}}}\notag\\
&\times S(t) F^{(1)}(E_{pkqr},t).\label{eq:di}\end{aligned}$$ Symmetrizing the dependence on momenta ${\boldsymbol{p}},{\boldsymbol{k}},{\boldsymbol{q}}$ and ${\boldsymbol{r}}$ of the integrand with the help of $F^{(2)}(-E,t) = F^{(2)}(E,t)$, $\Delta_{\text{kin}}(t)$ can be written as $$\Delta_{\text{kin}}(t) = \sum_{{\boldsymbol{k}}} \epsilon_k \delta n_{k}(t),\label{eq:renkin}$$ where $$\begin{aligned}
\delta n_k (t) &=-\frac{4}{\Omega^2}\left(\frac{4\pi a_s}{m}\right)^2\sum_{{\boldsymbol{pqr}}}\left[n^0_{p\uparrow}n^0_{k\downarrow}-n^0_{q\uparrow}n^0_{r\downarrow}\right]\delta_{{\boldsymbol{p}}+{\boldsymbol{k}},{\boldsymbol{q}}+{\boldsymbol{r}}}\notag\\
&\times F^{(2)}(E_{pkqr},t) \label{eq:dn}\end{aligned}$$ After setting $g = 4\pi a_s/m$ in Eq. and subtracting $\Delta_{\text{kin}}(t)$, the divergence in the kinetic energy is cancelled and it is possible to obtain a finite result for the dynamics of the kinetic energy. Furthermore, for $p \gg k_F$ (compare Eq. and Eq. after setting $g = 4\pi a_s/m$), we obtain: $$\delta n_{p\gg k_F}(t) = \sum_{\sigma} n^{(2)}_{p\gg k_F,\sigma}\left(g = \frac{4\pi a_s}{m},t\right). \label{eq:asymnp}$$ This explains why adding $-\Delta_{\mathrm{kin}}(t)$ to the kinetic energy cancels the divergence which arises from the behavior of the instantaneous momentum distribution at large momenta. Indeed, this result provides the basis for our analysis of the dynamics of Tan’s contact following the interaction quench, which is presented in Sec. \[sec:contact\]. In the following section, we shall evaluate the above expressions for a linear ramp of the interaction strength, for which $S(t)$ is given by Eq. .
Results for a linear ramp quench {#sec:ramp}
================================
In this section, we obtain explicit results for quenches in which the interaction strength is ramped up linearly in time and $S(t)$ is given by Eq. . As we have seen above, the first order correction to the total energy is just the expectation value of $V$ in the initial state (cf. Eq. \[eq:U\]) multiplied by $S(t)$. At second order in the scattering length $a_s$, using Eq. and evaluating the time integrals for the functions $F^{(1)}(t)$ (cf. Eq. \[eq:f1\]) and $F^{(2)}(t)$ (cf. Eq. \[eq:f2\]) yields the following expression for the time dependence function, $F_{\mathrm{tot}}(E,t) = F_{\mathrm{tot}}(E,T_r,t)$: $$\begin{aligned}
F_{\mathrm{tot}}(E,T_r,t)
&=G(Et,E T_r) F_{\mathrm{tot}}^{\mathrm{eq}}(E),
\label{eq:Ftot}\\
G(x,y) &= \frac{
\theta(x-y) H(y)+\theta(y-x) H(x) }{(2 y)^2},\notag\\
H(x) &= x^2-4\sin^2\left(\frac{x}{2}\right).
\end{aligned}$$ For the purpose of the discussion below, we have introduced a scaling function, $G(x,y)$, along with $F_{\mathrm{tot}}^{\text{eq}}(E)=E^{-1}$. The latter is the form that $F_{\mathrm{tot}}(E,t)$ takes in the adiabatic limit (see Appendix \[sec:F\]).
Let us next consider the form of $F_{\mathrm{tot}}(E,T_r,t)$ in specific limiting cases. For a sudden quench, $T_r \to 0$ and $G(Et,ET_r) = 0$. Therefore, $$\begin{aligned}
F^{\mathrm{sudden}}_{\mathrm{tot}}(E,t)=\lim_{T_r\to 0} F_{\mathrm{tot}}(E,T_r,t)=0,
\end{aligned}$$ which implies that, to second order, the shift to total energy vanishes. Indeed, this is expected from energy conservation in the limit of a sudden quench. To this this, let us compute the total energy directly for a sudden quench: $$\begin{aligned}
E^{\mathrm{sudden}}_{\mathrm{tot}}(t>0)&=\langle e^{iHt}H(0^+) e^{-iHt}\rangle,\notag\\
&=\langle H(0^+) \rangle,\notag\\
&=E^{(0)}_{\text{tot}} +\langle V \rangle,\label{eq:enc}
\end{aligned}$$ where $E^{(0)}_{\text{tot}}= \langle H_0 \rangle$ is the total energy before the quench (and $\langle \ldots \rangle$ is the average over the initial state). In other words, the dynamics of the total energy in the sudden quench limit simply amounts to a constant energy shift happening at $t = 0$. The shift is the expectation value of the interaction in the initial state, which is first order in $a_s$. Hence, not only the $O(a^2_s)$ term vanishes (in agreement with our findings) but so do all higher order corrections. However, in the adiabatic limit where $T_r\to +\infty$, we have $$\begin{aligned}
F_{\mathrm{tot}}^{\mathrm{adiabatic}}(E,t)&=\lim_{T_r\to +\infty} F_{\mathrm{tot}}(E,T_r,t),\notag\\
&=\left[\theta(T_r-t)\frac{t^2}{T_r^2}+\theta(t-T_r)\right]F^{\mathrm{eq}}_{\mathrm{tot}}(E),\notag\\
&=S(t,T_r)^2F_{\text{tot}}^{\text{eq}}(E).\label{eq:Framp}\end{aligned}$$ This means that at times $t < T_s$, there is a transient during which energy grows quadratically in time. The growth saturates to the equilibrium value for $t >T_r$, i.e. the interaction strength reaches its full value.
We have numerically evaluated the momentum integrals and obtained the evolution the second-order correction to the total energy and the instantaneous momentum distribution following a linear ramp in the interaction strength for a uniform Fermi gas in three dimensions. The left panel of Fig. \[fig:ramp\] shows the time evolution of second order correction to the total energy for different ramp times $T_r$. As the sudden quench limit where $T_r\to 0$ is approached, the second order energy shift vanishes, as explained above. In the opposite limit, as the ramp time $T_r$ increases, second-order energy shift saturates at a value that approaches the equilibrium energy shift after initially growing quadratically at short times.
Concerning the dynamics of the instantaneous momentum distribution, as discussed in Sec. \[sec:md\], it is described by the function $F^{(2)}(E,T_r, t)$. For a linear ramp in the interaction strength, this function can be written as follows (see Appendix \[sec:F\] for the details) : $$F^{(2)}_{\text{ramp}}(E,T_r,t) =G^{(2)}(Et,ET_r)F^{(2)}_{\mathrm{eq}}(E),
\label{eq:F2}$$ where $$\begin{aligned}
G^{(2)}(x,y) &= \frac{1}{x^2}\left\{2+\theta(y-x)\left[x^2-2\cos(x) \right. \right. \notag\\
&\left. \left. +2 x \sin(y-x)- 2x \sin(y) \right] \right.\notag\\
&\left.+\theta(x-y)\biggl[y^2-2\cos(y)-2y \sin(y)\biggr]\right\} \label{eq:gs2} \end{aligned}$$ is a scaling function of the dimensionless variables $x = ET_r$ and $y = E t$ and $F^{(2)}_{\text{eq}}(E)=1/E^2$ is the equilibrium value of $F^{(2)}(E,t)$. Thus, as the sudden quench limit where $T_r \to 0$ is approached $$\begin{gathered}
F^{(2)}(E,T_r, t) = \left[ 4 \sin^2\left(\frac{E t}{2}\right) + E T_r \sin(E t) \right. \\
\left. + O(T^2_r) \right] F^{(2)}_{\text{eq}}(E). \label{eq:gs2st}\end{gathered}$$ Therefore, we see that the second order correction to instantaneous momentum distribution reaches a stationary value that obeys the following relation: $$\lim_{t\to +\infty} \lim_{T_r\to 0} n^{(2)}_{k\sigma}(t,T_r)= n^{(2,\textrm{st})}_{k} = 2 n^{(2,\mathrm{eq})}_{k\sigma},$$ where $n^{(2,\mathrm{eq})}_{k\sigma}$ is the second order correction to the equilibrium momentum distribution of the Fermi gas with interaction strength equal to $g = 4\pi a_s/m$. This result can be regarded as a generalization of the relationship obtained in Ref. [@Moeckel2008; @Moeckel2009] between the discontinuity of the momentum distribution at Fermi surface i.e. $Z(t) = \lim_{\delta\to 0} \left[ n_{k_F+\delta,\sigma}(t) - n_{k_F-\delta,\sigma}(t) \right]$ in the pre-thermalized regime and in equilibrium: $$1-Z_{\mathrm{st}}=2(1-Z_{\mathrm{eq}}),\label{eq:zst}$$ where $Z_{\mathrm{st}}$ and $Z_{\mathrm{eq}}$ stand for the stationary (i.e. pre-thermalized) and equilibrium values of $Z$, respectively.
On the other hand, in the adiabatic limit where $E_F T_r \gg 1$, we have $$F^{(2)}(E,T_r, t) = \left[1 - \frac{2 \sin(E t)}{E T_r} + O(T^{-2}_r) \right] F^{(2)}_{\text{eq}}(E)$$ Hence, the long time behavior of the instantaneous momentum distribution approaches the equilibrium momentum distribution with an interaction strength $g = 4\pi a_s/m$, i.e. $$\lim_{t\to +\infty} \lim_{T_r \to +\infty} n^{(2)}_{k\sigma}(t,T_r) = n^{(2,\textrm{eq})}_{k\sigma},$$ as expected.
In the right-hand panel of Fig. \[fig:ramp\], we have plotted the second-order correction to the instantaneous momentum distribution $n^{(2)}_{p,\sigma}(t)$ for $\sigma = \uparrow$. For reasons of experimental interest, we consider an initial (non-interacting) state at finite temperature $T = 0.1 T_F$, where $T_F = E_F = k^2_F/2m$ is the Fermi temperature (in $\hbar= k_B = 1$ units), rather than the zero-temperature ground state, as considered in previous studies [@Moeckel2008; @GGEpreth_2011_kollar; @nessi_shorttime2014]. Notice that the momentum distribution reaches a stationary value for $t/T_r\gg 1$. This follows from the asymptotic behavior of $F^{(2)}(E,T_r, t)$ for $t/T_r \gg 1$: $$F^{(2)}(E,T_r,t\gg T_r) \simeq \left[ 1 + \frac{4 \sin^2(E T_r/2)}{E^2T^2_r} \right] F^{(2)}_{\textrm{eq}}(E).$$ Thus, for any finite $T_r\neq 0$, the long time limit of instantaneous momentum distribution for $k \sim k_F$ (see dashed line in Fig. \[fig:ramp\]) differs from the equilibrium value $n^{(2,\textrm{eq})}_{k}$.
Dynamics of the contact {#sec:contact}
=======================
![(color line) Log-log plot of the stationary value of the second order correction to the momentum distribution $\delta n_{st}(k,T_r)$ derived from the renormalization procedure explained in Sec. \[sec:re\], for $E_{\mathrm{F}}t=20$. The high-momentum $\sim k^{-4}$ tail is displayed (dashed line). Note that the curves for the sudden quench limit $T_r\to 0$ and $T_r E_F = 6$ exhibit the same asymptote for $k/k_F \gg 1$. Closer inspection reveals a crossover of the $T_r E_F = 6$ curve between two different asymptotes. See explanation in Sec. \[sec:inbtw\]. The interaction strength is chosen as $k_{\mathrm{F}} a_s=0.02$ and $T/T_F=0.1$.[]{data-label="fig:asympt"}](asymptoticm.pdf){width="\columnwidth"}
In the previous section, we have studied the behavior of the perturbative corrections to the total energy and the instantaneous momentum distribution $n_{k\sigma}(t)$ for $k \sim k_F$ ($k_F$ being the Fermi momentum). In this section, we turn our attention to the asymptotic behavior of the latter for $k \gg k_F$. In this regime, $n_{k\sigma}(t)$ exhibits a $k^{-4}$ decay, similar to its behavior in equilibrium [@shinatan_contact]. However, the pre-factor (the non-equilibrium version of Tan’s contact) depends both on time, $t$ and the energy, $\epsilon_k = k^2/2m$.
In connection with the asymptotic behavior of the instantaneous momentum distribution, in Sec. \[sec:reneq\] we have shown that the correction that removes of the divergences from the perturbative expression for the total energy factorizes into a kinetic and an interaction-energy contribution. The correction to the kinetic energy, $\Delta_{\mathrm{kin}}(t)$ can be written in terms of the function $\delta n_k(t)$ (see Eq. and following equations). As shown in Sec. \[sec:reneq\], this function also yields the asymptotic behavior of the instantaneous momentum distribution, $\sum_{\sigma} n_{k\sigma}(t)$ for $k\gg p_F$ (see Eq. \[eq:asymnp\]). Hence, it is possible to extract the dynamics of Tan’s contact by studying the asymptotic behavior of Eq. . To this end, let us rewrite $\delta n_k(t)$ as follows: $$\begin{gathered}
\delta n_k(T_r,t)=\frac{4}{\Omega^2}\left(\frac{4\pi a_s}{m}\right)^2\sum_{{\boldsymbol{qr}}}\left[n_{q\sigma}^0n^0_{r\alpha}-n^0_{k\sigma}n^0_{|{\boldsymbol{q}}+{\boldsymbol{r}}-{\boldsymbol{k}}|,\alpha}\right]\\
\times
F^{(2)}_{\text{ramp}}\left[\frac{k^2}{m}\left( 1+\frac{{\boldsymbol{q}}\cdot{\boldsymbol{r}}}{k^2}-\frac{{\boldsymbol{k}}\cdot({\boldsymbol{q}}+{\boldsymbol{r}})}{k^2}\right),T_r,t\right],\label{eq:Cneq}\end{gathered}$$ where have implemented momentum conservation by requiring that ${\boldsymbol{p}} = {\boldsymbol{q}}+{\boldsymbol{r}}-{\boldsymbol{k}}$. In order to evaluate the above integrals numerically in the large $k$ limit, it is convenient to parametrize [@Doggen:2015_contact] ${\boldsymbol{q}}=({\boldsymbol{p}}+{\boldsymbol{k}})/2+{\boldsymbol{s}}$, ${\boldsymbol{r}}=({\boldsymbol{p}}+{\boldsymbol{k}})/2-{\boldsymbol{s}}$. The result of the numerical evaluation of $\delta n_k(T_r,t)$ is shown in Figure \[fig:asympt\] as a function of $k/k_F$ for $t = 20 E^{-1}_{\mathrm{F}}$. For this time, $\delta n_k(T_r,t)$ has reached a stationary value. At large $k/k_F$, we observe that it exhibits $1/k^4$ dependence. However, the detailed asymptotic behavior of $\delta n_k(T_r,t)$ is quite rich and also depends on the ramp time $T_r$, as shown in Fig. \[fig:asympt\]. Indeed, a careful comparison the results for $T_r = 6 E^{-1}_{\mathrm{F}}$ and $T_r \to 0$ is worth here. For the former value of $T_r$, the asymptotic $\sim k^{-4}$ behavior is reached for smaller values of $k/k_F$ than for the $T_r\to 0$ results, which approach the sudden quench. In addition, close inspection of the curves shows that $\delta n_k(k,T_r \to 0)$ first approaches the $k^{-4}$ with a different coefficient, and only at large $k/k_F$ finally converges to the same asymptote as $\delta n_k(T_r = 6 E^{-1}_F,t)$. We explain this behavior below.
Since the asymptotic behavior of the $\delta n_k(T_r,t)$ appears to be rather complex, a naïve generalization of Tan’s contact from the equilibrium case, i.e. $$C_{\mathrm{Tan}}(t)=\lim_{k\to \infty}k^4\delta n_{k}(t), \label{eq:ctrt}$$ does not suffice to fully capture it. Thus, in order to illustrate this point, let us analyze the asymptotic behavior of the instantaneous momentum distribution in the sudden quench and adiabatic limits.
Sudden quench limit
-------------------
{width="\textwidth"}
In the sudden quench limit, using the following property (cf. Eqs. \[eq:gs2\] and Eq. \[eq:gs2st\]): $$\lim_{ET_r\to0}G^{(2)}(Et,ET_r)=4\sin^2(Et/2),$$ we obtain the following behavior for the $k^{-4}$ asymptote: $$\begin{aligned}
C^{\mathrm{sudden}}_{\mathrm{neq}}(t\gg1/\epsilon_k)&=\lim_{k\to\infty}\lim_{\epsilon_kT_r\to0}k^4\delta n(k,t\gg1/\epsilon_k),\notag\\
&=\frac{4}{\Omega^2}\left(4\pi a_s\right)^2\sum_{q,r}n_q^0n_r^0,\notag\\
&= 2C_{\mathrm{eq}},\end{aligned}$$ where $C_{\mathrm{eq}}$ is the equilibrium contact calculated to leading order in perturbation theory (see Ref. [@Doggen:2015_contact] and below). The above relation is similar to the relation obeyed in the pre-thermalized regime by the discontinuity of the momentum distribution at the Fermi momentum, Eq. .
Adiabatic limit
---------------
In the adiabatic limit where $\epsilon_kT_r\to\infty$, we obtain the following expression for the asymptote: $$\begin{aligned}
C_{\mathrm{neq}}^{\mathrm{adiabatic}}(t,T_r)&=\lim_{k\to\infty}\lim_{\epsilon_k T_r\to\infty} k^4\delta n_k(T_r,t),\notag\\
&=\left[\theta(T_r-t)\frac{t^2}{T_r^2}+\theta(t-T_r)\right]C_{\mathrm{eq}},\notag\\
&=\left[S(t,T_r)\right]^2 C_{\mathrm{eq}},\label{eq:Cramp}\end{aligned}$$ Thus, like the total energy energy there is a transient for $t <T_r$ where it grows quadratically in time. However, after the interaction reaches its full value at $t \ge T_r$, it saturates to the equilibrium value, $C_{\mathrm{eq}}$, in the steady state.
In between {#sec:inbtw}
----------
Finally, let us consider the general case of a finite ramp time $T_r > 0$. The left panel in Fig. \[fig:contact\] shows the time evolution of the asymptote of $k^4\delta n(k,T_r,t)$ normalized to the equilibrium contact for different ramp times, $T_r$ (time is measured in $\epsilon^{-1}_k$ units). At short times, the asymptote of $k^4\delta n(k,T_r,t)$ oscillates with a period $t_{\mathrm{os}}\sim 2 \left(k/k_{F}\right) \epsilon^{-1}_k$. As $k\to\infty$ the oscillation dies out and, at large $\epsilon_k t$, the asymptote of $\delta n_k(T_r,t)/C_{\mathrm{eq}}$ approaches a constant. The value of the latter varies between $1$ and $2$, depending on the value of $T_r$. The dependence on $T_r$ is shown on the right panel of Fig. \[fig:contact\], which illustrates how the asymptotic behavior of $k^4 \delta n_k(T_r,t)$ crosses over from the adiabatic limit where $$k^4 \delta n_k(T_r,t)\to C_{\mathrm{eq}}$$ to the sudden-quench limit where $$k^4 \delta n_k(T_r,t) \to 2 C_{\mathrm{eq}}.$$ The crossover happens for $T_r \sim \epsilon^{-1}_k$. The explanation for this behavior is as follows: For large but fixed $k\gg k_F$, any ramp of the interaction strength in a time much longer than $\epsilon^{-1}_k$ is regarded by the fermions at momentum ${\boldsymbol{k}}$ as adiabatic and therefore the behavior of asymptote approaches the adiabatic limit. However, for $T_r \ll \epsilon^{-1}_k$, the fermions at momentum ${\boldsymbol{k}}$ experience the quench as sudden, and therefore the asymptote approaches the sudden limit.
Conclusions {#sec:conclu}
===========
In conclusion, we have discussed the renormalization of total energy obtained from the single-channel model in the context of interaction quenches. The resulting expressions are finite and, when evaluated for a linear ramp in the interaction strength starting from an non-interacting state in a two-component Fermi gas, allow us to study the crossover from the sudden-quench to the adiabatic limit. In addition, we have also studied the behavior of instantaneous momentum distribution. Thus, we found signatures of the pre-thermalization emerging at short to intermediate times after a linear ramp of the interaction. We have shown that the pre-thermalization signatures persist even at finite temperatures and they are also visible in the high-momentum tail of the distribution function. These results are important for the experimental characterization of this nonequilibirum state.
We have analyzed the dynamics of the high momentum tail of the instantaneous momentum distribution, which is related to the non-equilibrium dynamics of the Tan’s contact. Thus, we have uncovered an interesting crossover from adiabatic to sudden-quench dynamics in the asymptotic behavior of the momentum distribution as a function of the ramp time and the momentum scale of the fermions at the tail. Although explicit results were obtained only for a specific quench protocol that assumes a linear ramp of the interaction strength, they should also hold for more general quenches provided the ramp time $T_r$ is replaced by the relevant switching-on time scale of the quench protocol.
The results reported in this work can be experimentally verified by preparing a three-dimensional two-component Fermi gas with an accessible broad Feshbach resonance in a non-interacting state, and quenching it to an interacting state. The total energy dynamics can be accurately measured from the time-of-flight images obtained by turning the scattering length to zero before releasing the gas from the trap. In addition, the renormalization method described here can be readily applied to the computation of other interesting nonequilibrium properties such as the full work distribution. In addition, in future work it would be interesting to extending it beyond the second order in the scattering length.
We thank Y. Takahashi and Y. Takasu, and S.-Z. Zhang for useful discussions and commennts. We acknowledge support from the Ministry of Science and Technology of Taiwan through grants 102-2112-M-007-024-MY5 and 107-2112-M-007-021-MY5, as well as from the National Center for Theoretical Sciences (NCTS, Taiwan).
Time dependence of $F^{(1)}, F^{(2)}$ and $F_{\mathrm{tot}}$ {#sec:F}
============================================================
In the previous sections, we have used a number of results for the functions $F^{(1,2)}(E,t)$, $F_{\mathrm{tot}}(E,t)$. In this section, we obtain their form for a linear ramp as well as several other important limiting cases, namely the sudden-quench and adiabatic limits. Let us recall that $$\begin{aligned}
F_{\mathrm{tot}}(E,t)&= S(t) F^{(1)}(E,t)-\frac{E}{2}F^{(2)}(E,t)\label{eq:Fapp},\end{aligned}$$ describes the time dependence for the total energy, and $$\begin{aligned}
F^{(1)}(E,t)&=\int\limits_{-\infty}^{t} \sin\left[E(t-t_1)\right] S(t) dt_1,\\
F^{(2)}(E,t)&=-\int\limits_{-\infty}^{t}dt_1 \int\limits_{-\infty}^{t} dt_2\: S(t_1)S(t_2) e^{iE(t_2-t_1)}\end{aligned}$$ describe the time dependence for interaction and kinetic energy (momentum), respectively. For a ramp quench, setting $S(t,T_r)=\theta(t)[\theta(t-T_r)+\theta(T_r-t)t/T_r]$, we can derive: $$\begin{aligned}
F^{(1)}_{\text{ramp}}(E&,T_r,t)=\frac{1}{E^2T_r}\biggl\{\theta(T_r-t)\biggl[Et-\sin(Et)\biggr]\\
&+\theta(t-T_r)\biggl[ET_r-\sin(Et)+\sin(E(t-T_r)\biggr]
\biggr\},\label{eq:Fr1}\\
F^{(2)}_{\text{ramp}}(E&,T_r,t)=\frac{1}{E^4T_r^2}\biggl\{2+\theta(t-T_r)\biggl[E^2T_r^2\notag\\
& -2\cos(ET_r) +2ET_r[\sin(E(t-T_r))-\sin(Et)]\biggr]\notag\\
&+\theta(T_r-t)\biggl[E^2t^2-2\cos(Et)-2Et \sin(Et)\biggr]\biggr\},
\label{eq:Fr2}\end{aligned}$$ They describe the time dependence of total energy shift: $$\begin{aligned}
F_{\mathrm{tot}}(E,T_r,t)&=\frac{1}{2E^3 T_r^2}\biggl\{\theta(t-T_r)\biggl[E^2T_r^2-\sin^2(ET_r/2)\biggr] \notag\\
& \qquad+\theta(T_r-t)\biggl[E^2t^2-4\sin^2(Et/2)\biggr]\biggr\}\label{eq:Ftotapp}.
\end{aligned}$$ For the the sudden quench $S(t)=\theta(t)$. We found the two time dependent functions $F^{(1)}(E,t)$ and $F^{(2)}(E,t)$: $$\begin{aligned}
F^{(1)}_{\text{sudden}}(E,t)&=\frac{2\sin^2\left(Et/2\right)}{E},\label{eq:FS1}\\
F^{(2)}_{\text{sudden}}(E,t)&=\frac{4\sin^2\left(Et/2\right)}{E^2},\label{eq:FS2}\end{aligned}$$ from which, the function that controls the time dependence of total energy in the sudden quench limit: $$\begin{aligned}
F_{\mathrm{tot}}^{\mathrm{sudden}}(E,t>0)&= F^{(1)}(E,t)-\frac{E}{2}F^{(2)}(E,t) \notag\\
&=0,\label{eq:zero}\end{aligned}$$ as required by the conservation of total energy (see discussion in Sec. \[sec:ramp\], Eq. \[eq:enc\]). In the equilibrium limit, using $S(t)=e^{-\eta |t|}$ with $\eta\to 0^+$ to describe the adiabatic switching of the interaction, we obtain: $$\begin{aligned}
F^{(1)}_{\mathrm{eq}}(E,t)&=\frac{1}{E},\label{eq:FS1a}\\
F^{(2)}_{\mathrm{eq}}(E,t)&=\frac{1}{E^2},\label{eq:FS2a}\end{aligned}$$ which lead to the equilibrium results. Hence, $$\begin{aligned}
F^{\mathrm{eq}}_{\mathrm{tot}}\left(E,t\right)&=\frac{1}{2E}. \label{eq:ftot_adiabatic}\end{aligned}$$
[^1]: This condition obeyed by most ultracold gases even in the vicinity of broad Feshbach resonances where $|a_s|$ diverges
[^2]: Strictly speaking the coupling $g$ should be (weakly) time dependent. This is reminiscent of the equilibrium situation where Eq. \[eq:invertg\] would require $g$ to be (weakly) energy dependent.
|
---
abstract: 'We say that an action of a countable discrete inverse semigroup on a locally compact Hausdorff space is amenable if its groupoid of germs is amenable in the sense of Anantharaman-Delaroche and Renault. We then show that for a given inverse semigroup ${\mathcal{S}}$, the action of ${\mathcal{S}}$ on its spectrum is amenable if and only if every action of ${\mathcal{S}}$ is amenable.'
author:
- 'Ruy Exel[^1] Charles Starling[^2]'
bibliography:
- 'C:/Users/Charles/Dropbox/Research/bibtex.bib'
title: Amenable actions of inverse semigroups
---
Introduction
============
There are numerous ways to generalize the notion of a group. One such generalization is that of an [*inverse semigroup*]{}, that is, a semigroup ${\mathcal{S}}$ for which each $s\in{\mathcal{S}}$ has a unique “inverse” $s^*$, in the sense that $ss^*s = s$ and $s^*ss^* = s^*$. In this note, we address the question of what it means for an inverse semigroup to be amenable.
Amenability of discrete groups is an active and lively area of research. There are many equivalent definitions for what it means for a group to be amenable, and so those who attempt to define amenability for inverse semigroups have had many potential definitions to choose from. As discussed in [@Pa99] and [@Mi10], some of the more familiar notions of amenability for groups, such as the existence of a left translation-invariant mean, produce unsatisfactory answers when applied to inverse semigroups. The definition of group amenability that motivates this work is given by the following equivalent statements for a discrete group $G$:
- $G$ is amenable;
- the action of $G$ on a point is amenable;
- every action of $G$ on a locally compact Hausdorff space is amenable;
(see [@AD02 Example 2.7] for the definitions and details).
In generalizing the above, we first define what it means for an action of an inverse semigroup to be amenable, Definition \[amenableactiondef\]. Essentially, an action is amenable if its [*groupoid of germs*]{} is amenable in the sense of [@AR00]. We then show (Theorem \[maintheorem\]) that (b) and (c) above are equivalent for an inverse semigroup when “a point” in (b) is replaced with “its spectrum”. This change is natural for two reasons. The first is that if a group is viewed as an inverse semigroup, then its spectrum is a point. The second is that the action of an inverse semigroup on a point may not be well-defined. In this way, we believe that amenability of the action of a inverse semigroup on its spectrum is a good candidate for the definition of amenability.
In [@Mi10], it is argued that [*weak containment*]{} is a natural notion of amenability for an inverse semigroup. The conditions of Theorem \[maintheorem\] imply weak containment (see Remark \[weakcontainment\]), however a recent example of Willett [@Wi15] suggests that the converse situation might be more delicate.
This short note is organized as follows. Before proving our main result, in Section 2 we define the notion of a [*$d$-bijective groupoid homomorphism*]{} and show that the existence of such a map from a groupoid ${\mathcal{G}}$ into an amenable groupoid implies that ${\mathcal{G}}$ is also amenable. This is used in proving our main result in Section 3; to prove Theorem \[maintheorem\] we construct a $d$-bijective map between the two relevant groupoids.
Amenable groupoids and $d$-bijective homomorphisms
==================================================
In this section we prove a preliminary result about amenability of étale groupoids. After giving the necessary background, we define a certain type of groupoid homomorphism, such that if we have such a map from a groupoid ${\mathcal{G}}$ to an amenable groupoid, then ${\mathcal{G}}$ must also be amenable.
Recall that a [*groupoid*]{} is a set ${\mathcal{G}}$ with a distinguished subset ${\mathcal{G}}^{(2)} \subset {\mathcal{G}}\times {\mathcal{G}}$, called the set of [*composable pairs*]{}, a product map ${\mathcal{G}}^{(2)} \to {\mathcal{G}}$ with $(\gamma, \eta)\mapsto \gamma\eta$, and an inverse map from ${\mathcal{G}}$ to ${\mathcal{G}}$ with $\gamma \mapsto \gamma^{-1}$ such that
1. $(\gamma^{-1})^{-1} = \gamma$ for all $\gamma\in {\mathcal{G}}$,
2. If $(\gamma, \eta), (\eta, \nu)\in {\mathcal{G}}^{(2)}$, then $(\gamma\eta,\nu), (\gamma, \eta\nu)\in {\mathcal{G}}^{(2)}$ and $(\gamma\eta)\nu = \gamma(\eta\nu)$,
3. $(\gamma, \gamma^{-1}), (\gamma^{-1},\gamma)\in {\mathcal{G}}^{(2)}$, and $\gamma^{-1}\gamma\eta = \eta$, $\xi\gamma\gamma^{-1} = \xi$ for all $\eta, \xi$ with $(\gamma, \eta), (\xi,\gamma) \in {\mathcal{G}}^{(2)}$.
The set of [*units*]{} of ${\mathcal{G}}$ is the subset ${\mathcal{G}}^{(0)}$ of elements of the form $\gamma\gamma^{-1}$. The maps $r: {\mathcal{G}}\to {\mathcal{G}}^{(0)}$ and $d:{\mathcal{G}}\to {\mathcal{G}}^{(0)}$ given by $$r(\gamma) = \gamma\gamma^{-1}, \hspace{1cm} d(\gamma) = \gamma^{-1}\gamma$$ are called the [*range*]{} and [*source*]{} maps respectively. One sees that $(\gamma, \eta)\in {\mathcal{G}}^{(2)}$ is equivalent to $r(\eta) = d(\gamma)$. One thinks of a groupoid ${\mathcal{G}}$ as a set of “arrows” between elements of ${\mathcal{G}}^{(0)}$. An arrow $\gamma$ “starts” at $d(\gamma)$ and “ends” at $r(\gamma)$.
A map $\varphi: {\mathcal{G}}\to \mathcal{H}$ between groupoids is called a [*groupoid homomorphism*]{} if $(\gamma, \eta)\in {\mathcal{G}}^{(2)}$ implies that $(\varphi(\gamma), \varphi(\eta))\in \mathcal{H}^{(2)}$ and $\varphi(\gamma\eta) = \varphi(\gamma)\varphi(\eta)$. A short calculation shows that this implies that $\varphi(\gamma^{-1}) = \varphi(\gamma)^{-1}$, and so $\varphi({\mathcal{G}}^{(0)}) \subset \mathcal{H}^{(0)}$, $r\circ\varphi = \varphi\circ r$, and $d\circ\varphi = \varphi \circ d$.
A [*topological groupoid*]{} is a groupoid which is a topological space where the inverse and product maps are continuous, where we are considering ${\mathcal{G}}^{(2)}$ with the product topology inherited from ${\mathcal{G}}\times{\mathcal{G}}$. A topological groupoid is called [*étale*]{} if it is locally compact, its unit space is Hausdorff, and the range and source maps are local homeomorphisms. These properties imply that ${\mathcal{G}}^{(0)}$ is open. Furthermore, in a second countable étale groupoid, the spaces $${\mathcal{G}}_x : = d^{-1}(x), \hspace{1cm} {\mathcal{G}}^x := r^{-1}(x)$$ are discrete for all $x\in {\mathcal{G}}^{(0)}$. We note that an étale groupoid may not be Hausdorff, even though we always assume the unit space is.
\#1[(\#1)]{}
The following theorem from [@AR00 Corollary 3.3.8] will be used as our definition of amenability for a second countable étale groupoid.
\[amenablegroupoid\] Let $\mathcal{G}$ be a second countable étale groupoid. The following are equivalent:
1. $\mathcal{G}$ is amenable.
2. There exists a sequence $(g_n)$ of Borel functions on ${\mathcal{G}}$ such that
1. the function $x\mapsto \sum_{r(\gamma) = x}|g_n(\gamma)|$ is bounded;
2. for all $x\in \mathcal{G}^{(0)}$ and $n\in{\mathbb{N}}$ we have $\sum_{r(\gamma) = x}g_n(\gamma) = 1$; and
3. for all $\gamma\in\mathcal{G}$ the sequence $$\sum_{r(\gamma) = r(\eta)}\left|g_n(\gamma^{-1}\eta) - g_n(\eta)\right|$$ converges to 0 with $n$.
3. There exists a sequence $(f_n)$ of Borel functions on ${\mathcal{G}}$ such that
1. the function $x\mapsto \sum_{d(\gamma) = x}|f_n(\gamma)|$ is bounded;
2. for all $x\in \mathcal{G}^{(0)}$ and $n\in{\mathbb{N}}$ we have $\sum_{d(\gamma) = x}f_n(\gamma) = 1$; and
3. for all $\gamma\in\mathcal{G}$ the sequence $$\sum_{d(\gamma) = d(\eta)}\left|f_n(\eta\gamma^{-1}) - f_n(\eta)\right|$$ converges to 0 with $n$.
(1)$\Leftrightarrow$(2) is [@AR00 Corollary 3.3.8], and (2)$\Leftrightarrow$(3) follows by composing the given functions with the groupoid inverse map and redefining variables.
We now define a type of groupoid homomorphism which arises naturally when considering inverse semigroup actions.
\[dbijectivedef\] Let ${\mathcal{G}}$ and $\mathcal{H}$ be groupoids. We say that a groupoid homomorphism $\varphi:
{\mathcal{G}}\to \mathcal{H}$ is [*$d$-bijective*]{} (or [*source-bijective*]{}) if for all $x\in {\mathcal{G}}^{(0)}$, the restriction $\varphi: {\mathcal{G}}_x \to \mathcal{H}_{\varphi(x)}$ is bijective. We will say that $\varphi$ is [*$r$-bijective*]{} (or [*range-bijective*]{}) if for all $x\in {\mathcal{G}}^{(0)}$, the restriction $\varphi: {\mathcal{G}}^x \to \mathcal{H}^{\varphi(x)}$ is bijective.
The definition of a groupoid is symmetric with respect to the range and source map, so the following should not be surprising.
A map $\varphi:{\mathcal{G}}\to \mathcal{H}$ is $d$-bijective if and only if it is $r$-bijective.
Take $x\in {\mathcal{G}}^{(0)}$. From the definitions, one sees that $({\mathcal{G}}_x)^{-1} = \{\gamma^{-1}\mid \gamma\in {\mathcal{G}}_x\} = {\mathcal{G}}^x$. Because groupoid homomorphisms commute with the inverse map, it is also clear that $\left.\varphi\right|_{{\mathcal{G}}_x}\circ ^{-1} = \left.\varphi\right|_{{\mathcal{G}}^x}$. If $\varphi: {\mathcal{G}}_x \to \mathcal{H}_{\varphi(x)}$ is bijective, then so is $\left.\varphi\right|_{{\mathcal{G}}_x}\circ ^{-1} = \left.\varphi\right|_{{\mathcal{G}}^x}$. The other direction is analogous.
\[dbijectiveamenable\] Suppose that ${\mathcal{G}}$ and $\mathcal{H}$ are étale groupoids, and that ${\varphi}: {\mathcal{G}}\to \mathcal{H}$ is a $d$-bijective Borel map. If $\mathcal{H}$ is amenable, then so is ${\mathcal{G}}$.
Suppose that we have $\varphi: {\mathcal{G}}\to \mathcal{H}$ as in the statement, and that $(f_n)$ is a sequence of functions on $\mathcal{H}$ as in Theorem \[amenablegroupoid\].3. For each $n\in {\mathbb{N}}$, define $h_n$ on ${\mathcal{G}}$ by $$h_n := f_n\circ \varphi.$$ Because $\varphi$ and the $f_n$ Borel, the $h_n$ are Borel as well. Take $x\in {\mathcal{G}}^{(0)}$, and calculate $$\begin{aligned}
\sum_{\gamma\in {\mathcal{G}}_x}h_n(\gamma) &= & \sum_{\gamma\in {\mathcal{G}}_x} f_n(\varphi(\gamma))\\
& =&\sum_{\nu\in \mathcal{H}_{\varphi(x)}} f_n(\nu)\\
&=& 1.\end{aligned}$$ The second line above is due to the fact that $\varphi$ is a bijection between ${\mathcal{G}}_x$ and $\mathcal{H}_{\varphi(x)}$. Hence, the $h_n$ satisfy Theorem \[amenablegroupoid\].3(b). A similar calculation shows that the $h_n$ satisfy Theorem \[amenablegroupoid\].3(a).
Finally, fix $\gamma\in {\mathcal{G}}$ and let $x = d(\gamma)$. We calculate $$\begin{aligned}
\sum_{\eta\in {\mathcal{G}}_x}\left|h_n(\eta\gamma^{-1}) - h_n(\eta)\right| &=&\sum_{\eta\in {\mathcal{G}}_x}\left|f_n(\varphi(\eta\gamma^{-1})) - f_n(\varphi(\eta))\right|\\
&=& \sum_{\varphi(\eta)\in \mathcal{H}_{\varphi(x)}} \left|f_n(\varphi(\eta)\varphi(\gamma^{-1})) - f_n(\varphi(\eta))\right|\\
&=& \sum_{\nu\in \mathcal{H}_{\varphi(x)}} \left|f_n(\nu\varphi(\gamma^{-1})) - f_n(\nu)\right|\\
&\stackrel{n\to \infty}{\to}& 0.\end{aligned}$$ Again, the third line is due to the fact that $\varphi$ is a bijection between ${\mathcal{G}}_x$ and $\mathcal{H}_{\varphi(x)}$. Thus, the $h_n$ satisfy Theorem \[amenablegroupoid\].3(c) and we are done.
We are grateful to the referee for pointing out that the above can also be seen as a corollary of [@AR00 Theorem 5.3.14].
Amenable inverse semigroup actions
==================================
In this section we define inverse semigroups and their actions. To each action of an inverse semigroup one may associate an étale groupoid, and we will say that an action is amenable if the groupoid associated to it is amenable in the sense of the last section. We then use Proposition \[dbijectiveamenable\] to prove our main result: that if a certain universal action of an inverse semigroup is amenable, then all of its actions are amenable.
By an [*inverse semigroup*]{} we mean a semigroup ${\mathcal{S}}$ such that for each $s\in{\mathcal{S}}$ there is a unique $s^*\in{\mathcal{S}}$ such that $s^*ss^* = s^*$ and $ss^*s = s$. For convenience, we will also assume that there is an element $0\in{\mathcal{S}}$ such that $0s = s0 = 0$ for all $s\in{\mathcal{S}}$. If a given inverse semigroup does not have a zero element (such as in the case of a group), one may simply adjoin a zero element to it and extend the multiplication in the obvious way – the resulting semigroup is an inverse semigroup with zero. An element $e\in {\mathcal{S}}$ is called an [*idempotent*]{} if $e^2 = e$. For each $s\in {\mathcal{S}}$, the elements $s^*s$ and $ss^*$ are idempotents. The set of idempotents forms a commutative subsemigroup of ${\mathcal{S}}$. From now on we will denote by $E$ the set of idempotent elements of an inverse semigroup without making specific reference to the inverse semigroup. Any inverse semigroup we consider is assumed to be countable and discrete.
Let $X$ be a set. Then the [*symmetric inverse monoid*]{} on $X$ is the set $\mathcal{I}(X)$ of bijections between subsets of $X$: $$\mathcal{I}(X) = \{ f: U\to V\mid U, V\subset X, f\text{ bijective}\}.$$ This becomes an inverse semigroup when given the operation of composition of functions on largest domain possible. If $f, g\in \mathcal{I}(X)$ are such that the range of $g$ does not intersect the domain of $f$, then the product $fg$ is the empty function, which is the zero element of $\mathcal{I}(X)$.
We now recall the definition of this note’s principal object of study.
An [*action*]{} of an inverse semigroup ${\mathcal{S}}$ on a locally compact Hausdorff space $X$ is a semigroup homomorphism $\alpha: {\mathcal{S}}\to \mathcal{I}(X)$ such that for each $s$, the map $\alpha_s$ is continuous and its domain is open in $X$, and the union of all the domains of the $\theta_s$ coincides with $X$. We also require that $\alpha_0$ is the empty map on $X$. If $\alpha$ is an action of ${\mathcal{S}}$ on $X$, we will write $\alpha: {\mathcal{S}}\curvearrowright X$.
We note that for an action $\alpha: {\mathcal{S}}\curvearrowright X$ and an idempotent $e\in E$, the map $\alpha_e$ is necessarily the identity map on its domain, which we denote $D_e^\alpha\subset X$. Furthermore, for $s\in {\mathcal{S}}$, the domain of the function $\alpha_s$ coincides with the domain of the idempotent $\alpha_{s^*s}$, and so we write $\alpha_s: D^\alpha_{s^*s}\to D^\alpha_{ss^*}$. For each $s\in {\mathcal{S}}$, the inverse of $\alpha_s$ is $\alpha_{s^*}$ which is continuous by definition, and so each $\alpha_s$ is a homeomorphism.
Given an action $\alpha$ of an inverse semigroup ${\mathcal{S}}$ on a space $X$ we can form a groupoid which encodes the action. The [*groupoid of germs*]{} for such an action is $$\label{germs}
{\mathcal{G}}(\alpha) = \{ [s,x] \mid s\in{\mathcal{S}}, x\in D^\alpha_{s^*s}\}$$ where two elements $[s,x]$ and $[t,y]$ are equal if and only if $x=y$ and there exists $e\in E$ such that $x\in D^\alpha_e$ and $se = te$. The groupoid operations are given by $$[s,x]^{-1} = [s^*, \alpha_s(x)], \hspace{0.5cm} r([s,x]) = \alpha_s(x), \hspace{0.5cm} d([s,x]) = x, \hspace{0.5cm} [t, \alpha_s(x)][s,x] = [ts, x].$$ For $s\in {\mathcal{S}}$ and an open set $U\subset D_{s^*s}^\alpha$, let $$\Theta(s, U) = \{[s, x]\mid x\in U\}.$$ Sets of this form generate a topology on ${\mathcal{G}}(\alpha)$, and under this topology ${\mathcal{G}}(\alpha)$ is étale, and because ${\mathcal{S}}$ is countable, ${\mathcal{G}}(\alpha)$ is second countable. For a more detailed discussion of inverse semigroup actions and groupoids of germs, the interested reader is directed to [@Ex08 §4].
Given this construction, we make the following definition.
\[amenableactiondef\] We say that an action $\alpha$ of an inverse semigroup ${\mathcal{S}}$ on a locally compact Hausdorff space $X$ is [*amenable*]{} if the groupoid of germs ${\mathcal{G}}(\alpha)$ is amenable.
An inverse semigroup ${\mathcal{S}}$ acts on an intrinsic space built from a natural order structure on its idempotent set. For $e,
f\in E$, we write $e\leqslant f$ if $ef = e$; this defines a partial order on $E$. A [*filter*]{} in $E$ is a nonempty subset $\xi\subset E$ which does not contain the zero element, is closed under the product, and such that if $e\in \xi$ and $f\in E$ with $e\leqslant f$, then $f\in \xi$. The set of all filters will be denoted ${\widehat E_0}$, and can be viewed as a subset of the product space $\{0,1\}^E$. We give ${\widehat E_0}$ the relative topology from this space. If $X, Y\subset E$ are finite subsets of $E$, define $$U(X, Y) = \{ \xi\in
{\widehat E_0}\mid X\subset \xi, Y\cap \xi = \varnothing\}.$$ The collection of such sets forms a basis for the topology on ${\widehat E_0}$. The space ${\widehat E_0}$ is called the [*spectrum*]{} of ${\mathcal{S}}$.
We now define the intrinsic action $\theta$ of ${\mathcal{S}}$ on ${\widehat E_0}$. For $e\in E$, let $$D_e^\theta:= U(\{e\}, \varnothing) = \{\xi\in {\widehat E_0}\mid e\in \xi\}$$ and define $\theta_s: D^\theta_{s^*s}\to D^\theta_{ss^*}$ by $$\theta_s(\xi) = \{e\in E\mid sfs^*\leqslant e\text{ for some }f\in \xi\}.$$ The groupoid of germs for this action ${\mathcal{G}}(\theta)$ is sometimes called the [*universal groupoid*]{} for ${\mathcal{S}}$. This groupoid was defined in [@Pa99].
We can now state our main theorem.
\[maintheorem\] Let ${\mathcal{S}}$ be an inverse semigroup. Then the following are equivalent.
1. The canonical action $\theta: {\mathcal{S}}\curvearrowright {\widehat E_0}$ is amenable.
2. Every action of ${\mathcal{S}}$ on a locally compact Hausdorff space is amenable.
Of course, the difficult part of the above is to prove that the groupoid of germs of a given action $\alpha$ is amenable assuming that ${\mathcal{G}}(\theta)$ is. To do this, for any action $\alpha:{\mathcal{S}}\curvearrowright X$ we produce a $d$-bijective map from ${\mathcal{G}}(\alpha)$ to ${\mathcal{G}}(\theta)$ and appeal to Proposition \[dbijectiveamenable\].
[**For the duration of this paper, we fix an inverse semigroup ${\mathcal{S}}$ and an action $\alpha:{\mathcal{S}}\curvearrowright X$ of ${\mathcal{S}}$.**]{}
For each $x\in X$, the relation $s\sim_x t$ if and only if $[s,x] = [t,x]$ is an equivalence relation on the set of all $s$ in ${\mathcal{S}}$ such that $x\in D^\alpha_{s^*s}$. The equivalence class of an element $s$ will be denoted $[s]_x^\alpha$ and the set of all equivalence classes will be denoted $[{\mathcal{S}}]^\alpha_x$. The set $[{\mathcal{S}}]^\alpha_x$ can be thought of as a partition of the set $\{s\in{\mathcal{S}}\mid x\in D^\alpha_{s^*s}\}$.
We define a map $\rho: X\to {\widehat E_0}$ by $$\label{rhodef}
\rho(x) = \{ e\in E\mid x\in D^\alpha_e\}.$$ It is immediate that for each $x\in X$, the set $\rho(x)$ is a filter in $E$. We also have the following facts about $\rho$:
\[rhofacts\] Let $\rho:X \to {\widehat E_0}$ be as in . Then:
1. For all $x\in X$ and $e\in E$, $x\in D_e^\alpha$ if and only if $\rho(x)\in D_e^\theta$.
2. For all $x\in X$, we have $\{s\in{\mathcal{S}}\mid x\in D_{s^*s}^\alpha\} = \{s\in{\mathcal{S}}\mid \rho(x)\in D_{s^*s}^\theta\}$.
3. For all $x\in X$ and $s, t$ in the set referred to above, we have that $[s]_x^\alpha = [t]_x^\alpha$ if and only if $[s]_{\rho(x)}^\theta = [t]_{\rho(x)}^\theta$.
4. For all $x\in X$ and $s\in {\mathcal{S}}$ with $x\in D_{s^*s}^\alpha$, we have that $\rho(\alpha_s(x)) = \theta_s(\rho(x))$.
<!-- -->
1. For $x\in X$, $x\in D_e^\alpha$ if and only if $e\in \rho(x)$, which is equivalent to saying that $\rho(x)\in D_e^\theta$.
2. This is a direct consequence of 1.
3. Take $x\in X$ and $s, t\in {\mathcal{S}}$ such that $x\in D^\alpha_{s^*s}\cap D^\alpha_{t^*t}$. Then $[s, x] = [t, x]$ if and only if there exists $e\in E$ such that $x\in D^\alpha_e$ and $se = te$, which by 1 is equivalent to $x\in D^\theta_e$ and $se = te$.
4. If $e\in \rho(\alpha_s(x))$, then $\alpha_s(x)\in D_e^\alpha$, which implies that $x\in D_{s^*es}^\alpha$, and so $\rho(x)\in D_{s^*es}^\theta$. Hence, $\theta_s(\rho(x)) \in D_{ss^*ess^*}^\theta\subset D_{e}^\theta$, so we have that $e\in \theta_s(\rho(x))$.
Conversely, if $e\in \theta_s(\rho(x))$, then there is an idempotent $f$ such that $x\in D_f^\alpha$ and $sfs^* \leqslant e$, which is to say $sfs^*e = sfs^*$. This implies that $D_{sfs^*}^\alpha\subset D_e^\alpha$, and so $\alpha_s(x)\in D_{e}^\alpha$, whence $e\in \rho(\alpha_s(x))$.
The map $\rho$ induces a map $\tilde\rho: {\mathcal{G}}(\alpha) \to {\mathcal{G}}(\theta)$ defined by $\tilde\rho([s,x]) = [s, \rho(x)]$. This map is well-defined by Lemma \[rhofacts\].3.
\[rhohomo\] The map $\tilde\rho: {\mathcal{G}}(\alpha) \to {\mathcal{G}}(\theta)$ is a $d$-bijective groupoid homomorphism.
We first check that $\tilde\rho$ is a groupoid homomorphism. We only have $([s,x], [t,y])\in {\mathcal{G}}(\alpha)^{(2)}$ if $y = \alpha_{t^*}(x)$. We calculate $$\begin{aligned}
\tilde\rho([s,x]) & = & [s, \rho(x)],\\
\tilde\rho([t,\alpha_{t^*}(x)]) & = & [t, \rho(\alpha_{t^*}(x))],\\
&=& [t, \theta_{t^*}(\rho(x))],\end{aligned}$$ and so $(\tilde\rho([s,x]), \tilde\rho([t,y]))\in {\mathcal{G}}(\theta)^{(2)}$. Furthermore, $$\begin{aligned}
\tilde\rho([s,x])\tilde\rho([t,\alpha_{t^*}(x)]) & = & [s, \rho(x)][t, \theta_{t^*}(\rho(x))]\\
& = & [st, \rho(\alpha_{t^*}(x))],\\
&=& \tilde\rho([st,\alpha_{t^*}(x)]),\\
&=& \tilde\rho([s,x][t,\alpha_{t^*}(x)]),\end{aligned}$$ whence $\tilde\rho$ is a groupoid homomorphism.
Now we show that $\tilde\rho: {\mathcal{G}}(\alpha)_x\to {\mathcal{G}}(\theta)_{\rho(x)}$ is a bijection for all $x\in X$. If $[s, \rho(x)]\in {\mathcal{G}}(\theta)_{\rho(x)}$, then $\tilde\rho([s, x]) = [s, \rho(x)]$, and so $\tilde\rho$ is surjective. Now, take $[s, x], [t, x]\in {\mathcal{G}}(\alpha)_x$, and suppose that $[s, \rho(x)]= [t, \rho(x)]$. This implies that $[s]^\theta_{\rho(x)} = [t]^\theta_{\rho(x)}$, which by Lemma \[rhofacts\].3 is equivalent to $[s]^\alpha_x = [t]^\alpha_x$, which gives us that $[s, x] = [t, x]$. Hence, $\tilde\rho:{\mathcal{G}}(\alpha)_x\to {\mathcal{G}}(\theta)_{\rho(x)}$ is bijective.
\[borellemma\] The maps $\rho: X\to {\widehat E_0}$ and $\tilde\rho: {\mathcal{G}}(\alpha) \to {\mathcal{G}}(\theta)$ are Borel maps.
Sets of the form $D^\theta_e$ together with their complements form a subbasis for the topology on ${\widehat E_0}$. Since the Borel sets form a $\sigma$-algebra, so we need only check that for all $e\in E$ the set $\rho^{-1}(D^\theta_e)$ is Borel. A short calculation shows that $\rho^{-1}(D^\theta_e) = D_e^\alpha$.
Now, suppose $s\in {\mathcal{S}}$ and we have $e\in E$ such that $D_e^\theta\subset D_{s^*s}^\theta$. Then $$\tilde\rho^{-1}(\Theta(s, D_e^\theta)) =\Theta(s,D_e^\alpha)$$ which is an open set. Furthermore, $$\tilde\rho^{-1}(\Theta(s, D_{s^*s}^\theta\setminus D_e^\theta)) =\Theta(s,D_{s^*s}^\alpha)\setminus\Theta(s,D_e^\alpha)$$ which is a Borel set. Sets of these types generate the topology of ${\mathcal{G}}(\theta)$, so $\tilde\rho$ is Borel.
We need only prove that (a)$\Rightarrow$(b), because (b)$\Rightarrow$(a) is obvious. That (a)$\Rightarrow$(b) follows from Proposition \[dbijectiveamenable\], Lemma \[rhohomo\], and Lemma \[borellemma\].
We close with two remarks regarding our result.
\[weakcontainment\] To an étale groupoid ${\mathcal{G}}$ one can associate C\*-algebras $C^*({\mathcal{G}})$ and $C^*_r({\mathcal{G}})$, called the [*C\*-algebra of ${\mathcal{G}}$*]{} and the [*reduced C\*-algebra of ${\mathcal{G}}$*]{} respectively. In this work we are not concerned with the specifics (and the interested reader is directed to [@R80] for more details), but there is always a surjective $*$-homomorphism $\lambda: C^*({\mathcal{G}})\to C^*_r({\mathcal{G}})$. If ${\mathcal{G}}$ is amenable, then $\lambda$ is an isomorphism, [@AR00 Proposition 6.1.8].
[*Weak containment*]{} for an inverse semigroup was defined in [@DP85], and in [@Mi10] it was argued that it is a good candidate for the definition of amenability for inverse semigroups. It is true that an inverse semigroup ${\mathcal{S}}$ with universal action $\theta$ satisfies weak containment if and only if the map $\lambda: C^*({\mathcal{G}}(\theta))\to C^*_r({\mathcal{G}}(\theta))$ is an isomorphism, see [@Pa99 Theorem 4.4.2]. Hence, the equivalent conditions of Theorem \[maintheorem\] imply weak containment, though it is not known whether the converse holds, see [@AR00 Remark 6.1.9] and [@Wi15].
\[tightcounterexample\] Another intrinsic action of an inverse semigroup ${\mathcal{S}}$ is that on a subspace of ${\widehat E_0}$, called the [*tight spectrum*]{} of ${\mathcal{S}}$. One considers in ${\widehat E_0}$ the subset of all [*ultrafilters*]{}, that is, the filters which are not properly contained in another filter. The tight spectrum is then the closure in ${\widehat E_0}$ of the set of all ultrafilters, and is denoted ${\widehat E_{\text{tight}}}$. The action of ${\mathcal{S}}$ on ${\widehat E_0}$ restricts to an action on ${\widehat E_{\text{tight}}}$, and the resulting groupoid of germs is called the [*tight groupoid*]{} of ${\mathcal{S}}$ and is denoted ${\mathcal{G}_{\text{tight}}}({\mathcal{S}})$. For details of this construction, the reader is directed to [@Ex08].
Our thought when setting out to investigate amenability of inverse semigroup actions was that perhaps the following entry could be added to Theorem \[maintheorem\]:
1. The canonical action $\theta: {\mathcal{S}}\curvearrowright {\widehat E_{\text{tight}}}$ is amenable.
This however is not true, as evidenced by the following counterexample which was relayed to us by Benjamin Steinberg.
Let $G$ be some discrete nonamenable group (such as the free group on two elements), and let $G^0$ denote the inverse semigroup obtained by adjoining an ad-hoc zero element $0$ to $G$. Now, let ${\mathcal{S}}$ be the inverse semigroup obtained by adjoining a further ad-hoc zero element $0'$ to $G^0$. The set of idempotents for this inverse semigroup is $E = \{1_G, 0, 0'\}$, and $${\widehat E_0}= \{\{1_G, 0\}, \{1_G\}\}$$ $${\widehat E_{\text{tight}}}= \{\{1_G, 0\}\}.$$ Let $\xi = \{1_G, 0\}$ so that ${\widehat E_{\text{tight}}}= \{\xi\}$. Suppose we have two germs $[s, \xi], [t, \xi]$ in ${\mathcal{G}_{\text{tight}}}({\mathcal{S}})$. We note that neither $s$ nor $t$ can be equal to $0'$. Since $0\in\xi$, $\xi\in D^\theta_0$, and $s0 = t0 = 0$, we have $[s, \xi] =[t, \xi]$ and so ${\mathcal{G}_{\text{tight}}}({\mathcal{S}})$ is the trivial (one-point) groupoid, hence amenable. However, ${\mathcal{G}}(\theta)$ is the union of the nonamenable group $G$ with a single point, and so is not amenable.
[**Acknowledgment:**]{} We are thankful to Benjamin Steinberg for relaying to us the example in Remark \[tightcounterexample\], and to the referee for a careful reading.
[^1]: Partially supported by CNPq (Brazil).
[^2]: Supported by CNPq (Brazil).
|
[ <span style="font-variant:small-caps;">Charges of dyons in ${ {\cal N} }=2$ supersymmetric\
gauge theory</span>]{}
[<span style="font-variant:small-caps;">Michael Yu. Kuchiev</span>]{}
\
[kmy@phys.unsw.edu.au]{}\
[ABSTRACT]{}
Expressions for electric and magnetic charges of dyons, which become massless in the strong-coupling limit of the supersymmetric ${ {\cal N} }=2$ gauge theory with an arbitrary gauge group are presented. Transitions into different vacua of the ${ {\cal N} }=1$ gauge theory, when the ${ {\cal N} }=2$ supersymmetry is broken explicitly to the ${ {\cal N} }=1$ case, are discussed. The existence of a minimal set of light dyons, which are necessary to describe this transition, is established. The total number of these dyons equals the product of the rank and dual Coxeter number of the gauge group. A conjecture, which states that this minimal set incorporates all possible light dyons, is discussed. A relation of dyon charges with monodromies at weak and strong couplings is outlined and comparison with known charges of dyons for particular gauge groups is made.
Introduction {#intro}
============
The properties of dyons, which become massless in the strong-coupling limit in the pure ${ {\cal N} }=2$ gauge theory described by the Seiberg-Witten solution are discussed. Explicit simple expressions for magnetic and electric charges of these dyons are written for an arbitrary gauge group. The total number of dyons is shown to depend on two parameters that govern the gauge algebra, its dual Coxeter number and its rank. The number of different massless dyons is shown to be related to the Witten index, which equals the number of different vacua in the ${ {\cal N} }=1$ supersymmetric gauge theory.
The Seiberg-Witten solution for the ${ {\cal N} }=2$ supersymmetric gauge theory [@Seiberg:1994rs; @Seiberg:1994aj] exploited the idea of S-duality, which expresses physics of strong-coupling phenomena in terms of the weakly coupled light dyons, thus providing an exact description of low-energy properties of the theory for an arbitrary coupling constant. This approach was described with the help of the algebraic curve, which gives the prepotential as an analytical function of the scalar field, as was discussed for the $SU(2)$ gauge group in [@Seiberg:1994rs; @Seiberg:1994aj]. The idea was extended to cover pure gauge theory with other gauge groups, gauge theory with matter, as well as used to study a number of related new phenomena in the ${ {\cal N} }=2$ supersymmetric gauge theory \[3-43\]. For the classical gauge groups ($A,B,C,D$ series) the curve, which describes the solution is widely believed to be hyperelliptic, though [@Martinec:1995by] suggested the non-hyperelliptic description for all gauge groups, which is based on the analogy with the integrable systems. Exceptional groups ($G,F,E$) prove to be more complicated for an analysis, see discussion in [@Abolhasani:1996ik; @Landsteiner:1996ut; @Alishahiha:2003hj].
The prepotential derived from this analysis provides a way to establish the magnetic and electric charges of light dyons, which were presented explicitly for the simplest gauge groups, including $SU(2)$ [@Seiberg:1994rs], $SU(3)$ [@Klemm:1995wp], $SU(4)$ and $G_2$ [@Hollowood:1997pp]. Clearly, the charges of light dyons are very interesting by themselves, which inspires their study for a general gauge group. This issue was addressed in [@Hollowood:1997pp], which based analysis on [@Martinec:1995by] that related the Seiberg-Witten solution to the spectral curve of an integrable system. The work [@Hollowood:1997pp] provided a general procedure to derive the charges of the dyons, though the expressions found were involving.
The present work considers light dyons using basic properties of the theory directly, avoiding references to the curve that governs the theory. It is known from [@Seiberg:1994rs] that light dyons describe the explicit breaking of the ${ {\cal N} }=2$ supersymmetry down to ${ {\cal N} }=1$ supersymmetry. Therefore matching the known fundamental properties of ${ {\cal N} }=2$ and ${ {\cal N} }=1$ supersymmetric gauge theories one can extract information related to properties of light dyons. The work is divided into two parts. Sections \[N=2\] - \[monopole\] summarize basic properties of the supersymmetric ${ {\cal N} }=1,2$ gauge theories. Sections \[chiral\] - \[conc\] derive and discuss expressions for the charges of dyons.
Supersymmetric ${ {\cal N} }=2$ gauge theories {#N=2}
===============================================
The supersymmetric ${ {\cal N} }=2$ gauge theory includes the scalar field $A$, two chiral spinors $\psi$ and $\lambda$, where the latter represents the gaugino, and the gauge field $v_\mu$, all in the adjoint representation of a gauge group, which is a simple Lie group $\mathsf{G}$ with an algebra $\mathsf{g}$ [@Sohnius:1985qm]. The energy of the scalar field turns zero provided this field has a coordinate independent value that lies in the Cartan subalagebra ${ \mathsf{g}_{\,\mathrm{C}} }$ of the gauge algebra, $A\in { \mathsf{g}_{\,\mathrm{C}} }\subset \mathsf{g}$, and satisfies $\Re \,(A)\propto \Im\,(A)$. Thus, the scalar field can develop an expectation value in the vacuum, which makes the vacuum state degenerate, the moduli space is given by ${ \mathsf{g}_{\,\mathrm{C}} }$, and and the scalar field in the vacuum, $A\in { \mathsf{g}_{\,\mathrm{C}} }$, can be treated as an $r$-dimensional vector, $A\equiv (A_1,\dots,A_r)$.
The vacuum expectation value of the scalar field breaks the gauge symmetry spontaneously. Generically, the symmetry is broken down to $r$ products of gauge $U(1)$, $G \rightarrow U(1)\times \cdots \times U(1)$, where $r$ is the rank of the algebra $\mathsf{g}$. There also remains unbroken a discrete group of gauge transformations, which comprises the Weyl group of $\mathsf{g}$, as discussed below. In the perturbation theory region this gauge breaking generates masses for all degrees of freedom, except for those that correspond to the $r$ unbroken $U(1)$ gauge symmetries. As a result, there are $r$ massless gauge fields in the theory, which are similar to photons; each such photon $v_\mu$ is accompanied by the corresponding massless fields $A,\psi$ and $\lambda$, which all have no electric charges and belong to the Cartan subalgebra ${ \mathsf{g}_{\,\mathrm{C}} }$. The low-energy properties of the theory are described by one function, the prepotential ${ {\cal F} }$, which is a holomorphic function of the scalar field ${ {\cal F} }={ {\cal F} }(A)$, as was argued in [@Seiberg:1988ur]. For a strong scalar field the coupling constant is weak. In this case one can write the prepotential explicitly in a simple form $$\begin{aligned}
{ {\cal F} }(A)\simeq \frac{i}{8\pi}\,\sum_\alpha \,(\alpha \cdot A)^2 \ln \frac{(\alpha \cdot A)^2~ }{\Lambda^2}~.
\label{FPTh}\end{aligned}$$ Here summation runs over all roots $\alpha$ of the algebra $\mathsf{g}$, the dot-product refers to the usual scalar product in an $r$-dimensional space, and $\Lambda$ is the conventional cut-off parameter. The limit of strong scalar field implies $|A^2|\gg \Lambda^2$. The notation $V^2\equiv V\cdot V$ for any $r$-dimensional vector $V$ is used throughout. Calculating the sum in (\[FPTh\]) with the logarithmic accuracy, i.e. presuming that $\ln(\alpha\cdot A)^2 \approx \ln A^2$, which can be done when the scalar field is not close to a wall of the Weyl camera, one writes $$\begin{aligned}
{ {\cal F} }(A)\approx \frac{i}{4\pi}\, { h^{\vee} }\,A^2\ln \frac{A^2}{\Lambda^2}~,
\label{Flog}\end{aligned}$$ where ${ h^{\vee} }$ is the dual Coxeter number of the algebra $\mathsf{g}$. For relevant properties of simple Lie algebras see [@Bourbaki:2002; @Di-Francesco:1997; @Slansky:1981yr]. [^1] Deriving (\[Flog\]) the following identity was used $$\begin{aligned}
\sum_{\alpha}\,\alpha_i\,\alpha_j\,=\,2\,{ h^{\vee} }\,\delta_{ij}~.
\label{ahv}\end{aligned}$$ Here and below the subscripts $i,j=1,\,\dots\,r$ refer to the Cartesian components of $r$-dimensional vectors. Equation (\[ahv\]) is valid provided the roots are normalized conventionally, with large roots satisfying $\alpha^2=2$.
In [@Seiberg:1994rs] it is explained that the dual field ${ A_\mathrm{D} }$, which is defined by $$\begin{aligned}
{ A_\mathrm{D} }=\frac{\partial { {\cal F} }(A)} {\partial A}~,
\label{AD}\end{aligned}$$ plays a major role in the problem. Similarly to $A$, this dual field is an $r$-dimensional vector. Since ${ {\cal F} }$ is holomorphic, the dual field is a holomorphic function of $A$ as well. In the weak coupling limit (\[FPTh\]) this function reads $$\begin{aligned}
A_\mathrm{D} \simeq \frac{i}{4\pi}\,\sum_\alpha \,\alpha \,(\alpha \cdot A)
\left( \ln \frac{(\alpha \cdot A)^2}{\Lambda^2}+1\right) \approx
\frac{i}{2\pi} \, { h^{\vee} }\,A \, \ln \frac{A^2}{\Lambda^2}~,
\label{APTh}\end{aligned}$$ where the last equality is written in the large-logarithm approximation introduced in (\[Flog\]). Effective coupling constants, which govern low-energy properties of the theory, are represented by a $r\times r$ matrix $\tau$, $$\begin{aligned}
\tau_{ij}\,\equiv \,
\left(\frac{\theta}{2\pi} +4\pi i\, g^{-2}\right)_{ij}\,=\,
\frac{\partial {{ A_\mathrm{D} }}_i }{\partial A_j} ~.
\label{tau}\end{aligned}$$ Here $\theta$ and $g$ are the $r\times r$ matrices of theta-angles and proper coupling constants respectively. Equations (\[APTh\]), (\[tau\]) show that in the weak coupling region $$\begin{aligned}
\tau_{ij}\approx \frac{i}{2\pi}\, { h^{\vee} }\ln \frac{A^2}{\Lambda^2}\,\,\delta_{ij}~.
\label{tauPTh}\end{aligned}$$ This equality implies that for the weak coupling there is only one coupling constant, whose asymptotic behavior is governed by the coefficient $b$ of the Gell-Mann - Low beta-function that equals $$\begin{aligned}
b=2 { h^{\vee} }\,,
\label{b=2}\end{aligned}$$ in accord with expectations for the ${ {\cal N} }=2$ supersymmetric gauge theory, see e.g. [@Peskin:1997qi].
The spontaneous breaking of the gauge symmetry keeps intact a set of discrete gauge transformations, the Weyl group $\mathsf{W}$ of the algebra $\mathsf{g}$, which comprises reflections in hyperplanes orthogonal to roots. Such a reflection transforms any $r$-dimensional vector $V$ via $V\rightarrow V'=\rho_\alpha V$, where $\rho_\alpha$ is an $r\times r$ matrix $$\begin{aligned}
\rho_\alpha\,=\,
1-\alpha \,\otimes\,{ \alpha^{\vee} }~,
\label{R}\end{aligned}$$ Here $\alpha^\vee$ indicates a coroot $$\begin{aligned}
\alpha^\vee=\,\frac{2 }{\,\alpha^2}\,\,\alpha
\label{av}\end{aligned}$$ and a conventional notation for $r\times r$ matrices $(1)_{ij}=\delta_{ij}$, $(\,\alpha \,\otimes \,{ \alpha^{\vee} })_{ij}= \alpha_{i}\,{ \alpha^{\vee} }_{j}$ is employed throughout (being complemented below by an obvious $(0)_{ij}=0$). It is convenient [@Seiberg:1994rs] to introduce the $2r$-dimensional vector
$$\varPhi\,=\,\begin{pmatrix} ~{ A_\mathrm{D} }\\ A \end{pmatrix}~.
\label{Phi}$$
Under the transformation of $A$ and ${ A_\mathrm{D} }$ by $\rho_\alpha \in \mathsf{W}$ this vector is transformed according to $$\begin{aligned}
&\Phi
\rightarrow \Phi' \, = \,P_\alpha \,\Phi~,
\label{Rcl}
\\
& P_{\,\alpha}\,=\,\begin{pmatrix} \rho_\alpha &0 \\ 0 & \rho_\alpha \end{pmatrix}~.
\label{Rrho} \end{aligned}$$ All entries in the $2\times 2$ block matrix here are $r\times r$ matrices, similar notation is used below.
However, if one considers a continuous transformation of the scalar field, which starts from $A$ and ends at $\rho_\alpha A$, then (\[APTh\]) shows that there arises an additional contribution to ${ A_\mathrm{D} }$, which can be interpreted as a quantum correction. It is proportional to the variation of the logarithmic function that acquires $\pm\,i\pi$ when the scalar field crosses the wall of the Weyl camera, which is orthogonal to $\alpha$. The sign of this variation depends on the way the crossing is fulfilled. Taking this variation as $-i\pi$ (we return to this point considering (\[NewR\]) below) one can write the monodromy, which was suggested in [@Danielsson:1995is], that describes the transformation of the scalar field along the path considered $$\begin{aligned}
\Phi\,
\rightarrow \,\Phi' \,=\,
{R}_{\,\alpha}\,\Phi~,
\label{Rq}\end{aligned}$$ where ${R}_\alpha$ is $$\begin{aligned}
{R}_\alpha &\,=\,\begin{pmatrix} \rho_\alpha & \alpha \otimes \alpha \\ 0 & \rho_\alpha \end{pmatrix}\,=\,P_\alpha\,T_\alpha~.
\label{Ra}\end{aligned}$$ Here $P_\alpha$ is the Weyl reflection (\[Rrho\]) and $T_\alpha$ is the matrix $$T_\alpha \,=\,
\begin{pmatrix} 1 & -\alpha \otimes \alpha \\ 0 & ~~~1 \end{pmatrix}~.
\label{Ta}$$ It was shown in [@Danielsson:1995is] that the set of $r$ matrices $R_\alpha$ defined in (\[Ra\]) generates the Brieskorn braid group.
There are global symmetries important for the Seiberg-Witten solution. One of them represents the global $SU(2)$ symmetry, which is specific to the ${ {\cal N} }=2$ supersymmetry. Transformations from this $SU(2)$ group treat $\psi$ and $\lambda$ as a doublet, leaving the scalar and vector fields $A,v_\mu$ invariant. There is also a chiral $U(1)$ symmetry, which on the classical level manifests itself via the following transformations of the fields $$\begin{aligned}
\label{U(1)}
& \vartheta \rightarrow e^{i\gamma} \vartheta~,\quad
\psi \rightarrow e^{i\gamma}\psi~,\quad
\lambda \rightarrow e^{i\gamma} \lambda~,&
\\
\label{2gammaA}
& A \rightarrow e^{2i\gamma} A~.\end{aligned}$$ Here $\vartheta$ is a conventional anti-commuting variable of the ${ {\cal N} }=1$ superspace [@Wess:1991]. Quantum corrections break this symmetry to $Z_{\,4{ h^{\vee} }}$, forcing the phase $\gamma$ in (\[U(1)\]), (\[2gammaA\]) to take only discrete values $$\begin{aligned}
\gamma=2\pi\,\frac {m} {4{ h^{\vee} }}~,\quad m=0,1,\,\dots\,,4 { h^{\vee} }-1~.
\label{gamma}\end{aligned}$$ The effects, which lead to this restriction on $\gamma$ can be attributed to the variation of the $\theta$-angle of the theory, which takes place due to the chiral transformation (\[U(1)\]). This variation reads $\Delta \,\theta \,=\,4{ h^{\vee} }\gamma$. The symmetry persists provided this variation is an integer of $2\pi$, $$\Delta \,\theta\,=\,4{ h^{\vee} }\gamma\,=\,2\pi m~,
\label{Delta}$$ which leads to (\[gamma\]). Overall, the global symmetry is $\left( SU(2)\times {Z}_{\,4{ h^{\vee} }}\right)/{Z}_2$, the divisor ${Z}_2$ eliminates double-counting of the center of $SU(2)$, which is also present in ${Z}_{\,4{ h^{\vee} }}$.
The transformation of the scalar field $A$ in (\[2gammaA\]) is accompanied by the transformation of the dual field ${ A_\mathrm{D} }$. Using (\[APTh\]) one finds $$\begin{aligned}
{ A_\mathrm{D} }\rightarrow { A_\mathrm{D} }'= \exp(\,2i\gamma\,) \left({ A_\mathrm{D} }-\frac{\gamma}{\pi}\,\sum_\alpha\,(\alpha\cdot A)\,\alpha \right) =
\exp\left( \,\pi i/{ h^{\vee} }\, \right) \left({ A_\mathrm{D} }-A \right)\,.
\label{ADZ}\end{aligned}$$ Here the second term in the big parentheses originates from the logarithmic function in (\[APTh\]). The last identity in (\[ADZ\]) is written using (\[ahv\]) and assuming $m=1$ in (\[gamma\]). Equation (\[ADZ\]) shows that the defining element of the chiral ${Z}_{\,4{ h^{\vee} }}$ symmetry manifests itself via the following transformation of the scalar field $$\begin{aligned}
\Phi\,\rightarrow \,
\Phi' \, = \,
\exp\left( \,\pi i/{ h^{\vee} }\right) M\,\Phi~,
\label{ADgamma}\end{aligned}$$ where the $2r\times 2r$ matrix $M$ reads $$M\,=\,\begin{pmatrix} 1 & \!-1 \\ 0 & \,~1 \end{pmatrix}~.
\label{M}$$ This transformation can be considered as a monodromy that arises when the phase $\gamma$ is treated as a continuous variable that varies from $\gamma=0$ to the value $\gamma=2\pi/{ h^{\vee} }$ allowed by (\[gamma\]).
Explicit forms for the monodromies in (\[Rq\]) and (\[ADgamma\]) were presented in the weak coupling limit. However, the prepotential, and therefore the dual field remain holomorphic functions of the scalar field even in the strong-coupling region. Consequently, a continuous variation of the scalar field is accompanied by a continuous variation of the dual field (provided no singularities are crossed). This implies that the discrete-valued matrices on the right-hand sides of (\[Rq\]),(\[ADgamma\]) do not change, if we consider both these transformations as monodromies under continuous variations of the scalar field and the path along which the transformation is defined. Thus, the transformations in (\[Rq\]),(\[ADgamma\]) remain well defined even when the strong-coupling region is considered.
Supersymmetric ${ {\cal N} }=1$ gauge theory {#N=1}
=============================================
The supersymmetric ${ {\cal N} }=1$ gauge theory includes the gauge field $ v_\mu $ and gaugino $ \lambda $ [@Wess:1991]. On the classical level the theory possesses the global chiral $U(1)$ symmetry $$\begin{aligned}
\vartheta\rightarrow e^{i\delta}\,\vartheta~,\quad
\lambda \rightarrow e^{i\delta}\,\lambda~.
\label{chiralN1}\end{aligned}$$ Quantum corrections break the chiral symmetry to ${Z}_{\,2{ h^{\vee} }}$. The latter group is defined by transformations in (\[chiralN1\]), in which the phase takes the following discrete values $$\begin{aligned}
\delta\,=\,2\pi\,\frac{k} {2{ h^{\vee} }}~,\quad k=0,1,\,\dots\,2{ h^{\vee} }-1~.
\label{delta}\end{aligned}$$ The resulting ${Z}_{\,2{ h^{\vee} }}$ is further broken spontaneously down to ${Z}_{\,{ h^{\vee} }}$. Overall, the breaking of the chiral symmetry follows the pattern $$\begin{aligned}
U(1)\,\rightarrow \, {Z}_{\,2{ h^{\vee} }}\,
\rightarrow \, {Z}_{\,{ h^{\vee} }}~.
\label{chiralN=1}\end{aligned}$$ This pattern manifests itself through the gaugino condensate $\langle\lambda\lambda\, \rangle$, which is present in the vacuum, $\langle\lambda\lambda\, \rangle\ne0$, see e.g. [@Peskin:1997qi; @Shifman:1999mv] for discussion and references. According to (\[delta\]), (\[chiralN=1\]) the ${Z}_{\,{ h^{\vee} }}$ global symmetry results in a transformation of the gaugino condensate $$\begin{aligned}
\langle\lambda\lambda\, \rangle\,\rightarrow \,\exp\left(\, 2\pi i /{ h^{\vee} }\right)\,\langle\lambda\lambda\, \rangle~.
\label{ll}\end{aligned}$$ It is commonly believed that the symmetry breaking (\[chiralN=1\]) and the presence of the gaugino condensate generate a mass gap. It is also presumed that there is no vacuum degeneracy, except for the one specified in (\[ll\]), which makes the phase of the gaugino condensate a convenient marker for the vacuum. Equation (\[ll\]) shows that shifts of this phase by $2\pi k/{ h^{\vee} },~k=0,1,\,\dots \,{ h^{\vee} }-1$ constitute ${Z}_{\,{ h^{\vee} }}$. Different values of the phase mark different vacua. The vacuum reveals therefore the ${ h^{\vee} }$-fold degeneracy. This assessment complies with calculations based on the Witten index $I_\mathrm{\,W}$ [@Witten:1982df], which counts the difference between the number of bosonic $n_\mathrm{b}$ and fermionic $n_\mathrm{f}$ zero-energy states $$\begin{aligned}
I_\mathrm{\,W}=n_\mathrm{b}-n_\mathrm{f}~,
\label{IW}\end{aligned}$$ and is designed to be invariant under continuum transformations of parameters of the theory. The calculations of [@Witten:1982df; @Witten:1997bs] based on this property of the index showed that for the supersymmetric ${ {\cal N} }=1$ gauge theory the index reads $$\begin{aligned}
I_\mathrm{\,W}\,=\,{ h^{\vee} }~.
\label{I=h}\end{aligned}$$ For the pure gauge theory it is presumed that $n_\mathrm{f}=0$. Equation (\[I=h\]) implies therefore that there are precisely ${ h^{\vee} }$ vacua, in accord with the symmetry breaking pattern (\[chiralN=1\]) and chiral transformations of the gaugino condensate (\[ll\]).
Monopoles and dyons {#monopole}
===================
The Seiberg-Witten solution expresses the low-energy properties of the ${ {\cal N} }=2$ supersymmetric theory at strong coupling in terms of light monopoles and dyons. The magnetic $g$ and electric $q$ charges of a dyon describe its interaction with $r$ different electro-magnetic fields that are present in the theory. Therefore both $g$ and $q$ are described by $r$-dimensional vectors. Moreover, the electric and magnetic charges lie in the lattice of roots and coroots of the algebra $\mathsf{g}$ respectively, see the review [@Weinberg:2006rq], $$\begin{aligned}
\label{qdirect}
q &\,=\,\sum_{\alpha \,\in\,\, \Delta} \,m_\alpha\,\alpha~,\quad m_\alpha \in {Z}~, \\
\label{gdual}
g &\,=\,\sum_{\alpha \,\in \,\,\Delta} \,n_\alpha\,\alpha^\vee~,\quad n_\alpha \in {Z}~.\end{aligned}$$ Equation (\[qdirect\]) complies with the fact that in the pure gauge theory all fields are in the adjoint representation of $\mathsf{g}$, which is associated with the lattice of roots. From (\[qdirect\]) one derives (\[gdual\]) by applying the Dirac-Schwinger-Zwanziger quantization condition, which states that magnetic and electric charges $g_1,~q_1$ and $g_2,~q_2$ of any two dyons satisfy $$g_1\cdot q_2-g_2\cdot q_1\,\in \,{Z}~,
\label{dsz}$$ which can be conveniently casted into $$\begin{aligned}
{ {\cal G} }_1\, \Omega\,\,{ {\cal G} }_{\,2}^{\,T} \,\equiv \,(\,g_1,\,q_1\,) \,
\begin{pmatrix} ~~0 & 1~ \\ -1 & 0~ \end{pmatrix}\,
\begin{pmatrix} g_2 \\ q_2 \end{pmatrix}\,
\in {Z}~.
\label{DSZ}\end{aligned}$$ Here $${ {\cal G} }=(\,g,\,q\,)
\label{Ggq}$$ represents magnetic and electric charges of a dyon, and the identity in (\[DSZ\]) employs the symplectic metric $\Omega$ $$\Omega\,=\,\begin{pmatrix} ~~0 & 1~ \\ -1 & 0~ \end{pmatrix}~.
\label{Omega}$$ Equations (\[qdirect\]),(\[gdual\]) comply with (\[DSZ\]) since any two roots $\alpha,\beta \in \mathsf{g}$ satisfy $\alpha^\vee\cdot\beta \in {Z}\,$. It suffices to limit summation in (\[qdirect\]),(\[gdual\]) to simple roots, which is specified as $\alpha\in \Delta$, where $\Delta$ is a set of $r$ simple roots of $\mathsf{g}$.
The dyons in the Seiberg-Witten solution are presumed to be BPS states [@Bogomolny:1975de; @Prasad:1975kr]. Consequently, as was noted in [@Witten:1978mh], the mass $m_{{ {\cal G} }}$ of a dyon with the charge ${ {\cal G} }=(g,q)$ is related to the central charge ${ {\cal Z} }_{\,{ {\cal G} }}$ $$\begin{aligned}
\label{MZ}
m_{\,{ {\cal G} }} & \,=\, 2^{\,1/2} \,\,|\,{ {\cal Z} }_{\,{ {\cal G} }}\,|~, \\
\label{Z}
{ {\cal Z} }_{\,{ {\cal G} }} & \,=\, g \cdot { A_\mathrm{D} }+q\cdot A \,\equiv \,{ {\cal G} }\,\varPhi~.
\end{aligned}$$
Chiral transformations for dyon charges {#chiral}
=======================================
Our goal is to study [ light]{} dyons, i.e. those dyons that become massless at an appropriate value of the vacuum expectation value of the scalar field. In the vicinity of this value the mass of the light dyon is small. When a dyon becomes massless, one can consider breaking the ${ {\cal N} }=2$ supersymmetry to the ${ {\cal N} }=1$ case by introducing a mass term for the fields $A$ and $\psi$ that comprise the ${ {\cal N} }=1$ chiral multiplet $\Phi$. As a result the vacuum of the ${ {\cal N} }=1$ supersymmetric theory arises, in which a condensate of dyons is developed [@Seiberg:1994rs]. This phenomenon has an important impact on the problem by making all fields massive. Thus, the dyons, which are initially created within the ${ {\cal N} }=2$ supersymmetry can explain the origin of the mass gap for the ${ {\cal N} }=1$ theory. The light dyons become massless in the strong-coupling region of the ${ {\cal N} }=2$ theory, but the S-duality implies that their behavior is governed by the dual coupling constant, which is weak in this region, making theoretical framework reliable.
These arguments due to Seiberg and Witten [@Seiberg:1994rs] show that massless dyons bridge properties of the ${ {\cal N} }=2$ and ${ {\cal N} }=1$ supersymmetric gauge theories. This phenomenon, which was discussed in detail in [@Seiberg:1994rs] for an example of the $SU(2)$ gauge group, will be considered in the present work as a general feature of the Seiberg-Witten solution valid for an arbitrary gauge group. Let us use this property of the problem as an opportunity to study dyons by applying known properties of ${ {\cal N} }=2$ and ${ {\cal N} }=1$ supersymmetric theories. Take some light dyon with the charge ${ {\cal G} }$ defined by (\[Ggq\]). Assume that a condensate of dyons is developed, in which this particular dyon is contributing when the ${ {\cal N} }=2$ supersymmetry is broken to the ${ {\cal N} }=1$ case (we will argue below that several dyons necessarily condense together). The resulting condensate of dyons describes some vacuum of the ${ {\cal N} }=1$ theory, call this vacuum state $|\,0\, \rangle$. Clearly this vacuum is not unique, (\[I=h\]) shows that there are other vacua, ${ h^{\vee} }$ of them overall. Let us assume we make a transition from $|\,0\, \rangle$ into some other vacuum $|\,0\,'\, \rangle$, $|\,0\, \rangle\rightarrow |\,0\,'\, \rangle$. Within the framework of ${ {\cal N} }=1$ supersymmetry this transition can be associated with a discrete chiral transformation in (\[ll\]).
Consider now the same transformation $|\,0\, \rangle\rightarrow |\,0\,' \rangle$ in terms of dyons. The vacuum $|\,0\, \rangle$ was presumed to arise due to condensation of dyons having charge ${ {\cal G} }$. Similarly, we should expect that a different vacuum state $|\,0\,' \rangle$ is created due to condensation of dyons with some different charge ${ {\cal G} }\,'$. In other words, a transition from one vacuum state to another, $|\,0\, \rangle\rightarrow |0\,' \rangle$, should be accompanied by a transformation of the dyon charge, from the initial charge ${ {\cal G} }$ to the different one ${ {\cal G} }\,'$, ${ {\cal G} }\rightarrow{ {\cal G} }\,'$. We know that transitions between different vacua $|\,0\, \rangle\rightarrow |\,0\,' \rangle$ in the ${ {\cal N} }=1$ theory are classified by the symmetry group $Z_{\,{ h^{\vee} }}$ that arises from the discrete chiral transformations, as per (\[chiralN=1\]),(\[ll\]). Consequently, we are to expect that a similar symmetry group should classify related transformations ${ {\cal G} }\rightarrow{ {\cal G} }\,'$ of the dyon charges.
Let us specify this symmetry group. Since the dyons are created within the framework of ${ {\cal N} }=2$ supersymmetry, we need to look at discrete symmetries available in the ${ {\cal N} }=2$ case. The transitions between different vacua of the ${ {\cal N} }=1$ theory are generated by the chiral transformations. One has to expect therefore that similar chiral transformations are responsible for the transformations between different charges of dyons. Thus, there should exist a connection between the discrete $Z_{\,{ h^{\vee} }}$ symmetry, which governs transformations between different vacua of the ${ {\cal N} }=1$ supersymmetric theory, and the chiral $Z_{\,4{ h^{\vee} }}$ symmetry of the ${ {\cal N} }=2$ supersymmetry.
The action of the $Z_{\,4{ h^{\vee} }}$ symmetry on the charge of a dyon can be derived from (\[Z\]) by re-interpreting the chiral transformation of the scalar field in this equation in terms of a transformation of the dyon charge, see (\[G->GM\]) below. Equation (\[2gammaA\]) states that the scalar field has charge 2 under chiral transformations. We conclude that the influence of the chiral group $Z_{\,4{ h^{\vee} }}$ on the charge of a dyon can manifest itself only through a smaller group $Z_{\,2{ h^{\vee} }}$. This latter group matches perfectly the full discrete chiral group $Z_{\,2{ h^{\vee} }}$ of the ${ {\cal N} }=1$ supersymmetry. Thus, we observe a correspondence between chiral transformations of the dyon charges within the ${ {\cal N} }=2$ supersymmetry and the discrete chiral symmetry of the ${ {\cal N} }=1$ theory.
The vacuum of ${ {\cal N} }=1$ theory breaks the chiral group $Z_{\,2{ h^{\vee} }}$ spontaneously to $Z_{\,{ h^{\vee} }}$, see (\[chiralN=1\]),(\[ll\]). This breaking effectively eliminates the difference between the fields $\lambda$ and $-\lambda$ in the sense that both these fields give the same contribution to the gaugino condensate $\langle \lambda \lambda\rangle$ in the vacuum. For future reference let us cast this statement, i.e. the fact that the fields $\lambda$ and $-\lambda$ give the same contribution to the gaugino condensate, into $$\lambda\,\equiv -\lambda~.
\label{la-la}$$ In other words, a spontaneous breaking $Z_{\,2{ h^{\vee} }}\rightarrow Z_{\,{ h^{\vee} }}$ takes place because the elements $1$ and $-1$ of the group $Z_{\,2{ h^{\vee} }}$, which is taken in the multiplicative notation, are identified modulo $Z_{\,2}$ $$Z_{\,{ h^{\vee} }}\, =\, Z_{\,2{ h^{\vee} }}/Z_{\,2}~.
\label{1-1}$$ Since we presume that the structure of the ${ {\cal N} }=1$ theory is reproduced by the condensates of light dyons, we have to admit also that the pattern of the symmetry breaking described by (\[1-1\]) should govern the behavior of condensates of light dyons as well. This means that the identity $1\,\equiv -1$, which is acknowledged in (\[1-1\]) for vacua of the ${ {\cal N} }=1$ theory, should be applicable for the condensates of light dyons as well. The properties of light dyons are governed by the scalar field, thus prompting this identity to be implemented in terms of the scalar field. Since $\lambda$ and $A$ belong to the same ${ {\cal N} }=2$ hypermultiplet, there exits only one option, namely we have to admit that (\[la-la\]),(\[1-1\]), when applied within the framework of the ${ {\cal N} }=2$ supersymmetric theory imply that the two values of the scalar field, $A$ and $-A$, should be identical $$A\equiv -A~.
\label{A-A}$$ This identity should govern the behavior of light dyons. Suppose that the field $A$ makes some dyons massless. Consider the condensate of these dyons, which is related to some vacuum of the ${ {\cal N} }=1$ supersymmetric gauge theory, when the ${ {\cal N} }=2$ supersymmetric gauge theory is broken to the ${ {\cal N} }=1$ case. Then (\[A-A\]) is to be understood as a statement that the field $-A$ also produces massless dyons, and the condensate created by these dyons is identical to the condensate created when the value of the scalar field equals $A$. The identity between the two condensates leads to the identity of the related two vacua of the ${ {\cal N} }=1$ theory, in accord with (\[la-la\]). Thus, (\[la-la\]) and (\[A-A\]) present two different points of view on the same spontaneous breaking of symmetry. Equation (\[la-la\]) describes it in terms of the vacuum state of the ${ {\cal N} }=1$ supersymmetric gauge theory, while (\[A-A\]) looks at it from the perspective of the ${ {\cal N} }=2$ supersymmetry, where the dyons are defined.
From (\[A-A\]) one deduces that when the condensate of dyons arises then the group $Z_{\,2{ h^{\vee} }}$, which governs chiral transformations of the scalar field, is broken spontaneously to $Z_{\,{ h^{\vee} }}$, in accord with (\[1-1\]). This discussion can be summed up as the following pattern of the symmetry breaking $$U(1)\rightarrow Z_{\,4{ h^{\vee} }}\rightarrow Z_{\,2{ h^{\vee} }}\rightarrow Z_{\,{ h^{\vee} }}~,
\label{chiralN=2}$$ which describes how the chiral symmetry of the ${ {\cal N} }=2$ gauge theory is implemented for light dyons. The first step here shows a conventional breaking of the classical chiral symmetry by quantum effects, which was mentioned in Section \[N=2\]. The second step is justifies by the fact that the chiral transformations of the charges of dyons are related to the scalar field that has charge 2. The last, third step is valid for light dyons in view of (\[A-A\]). Note the resemblance between (\[chiralN=2\]) and (\[chiralN=1\]).
We conclude that the subgroup $Z_{\,{ h^{\vee} }}$ of the chiral group, which describes transformations between different vacua for the ${ {\cal N} }=1$ supersymmetry, $|\,0\, \rangle \rightarrow |\,0\,'\, \rangle$, is matched in the ${ {\cal N} }=2$ supersymmetric theory by a similar subgroup $Z_{\,{ h^{\vee} }}$, which describes transformations between the charges of dyons, and which also originates in the chiral symmetry.
Let us calculate charges of dyons in different condensates, which comprise different vacua of the ${ {\cal N} }=1$ gauge theory. The discussion above shows that it suffices to apply chiral transformations of the ${ {\cal N} }=2$ supersymmetry to the charge of one light dyon. Consider the transformation of the mass $ m_{\,{ {\cal G} }}$ of a dyon under the chiral transformation of the scalar field. From (\[ADgamma\]) one deduces $$m_{\,{ {\cal G} }}\,=\,2^{\,1/2} \,\,\left| \,\,{ {\cal G} }\,\Phi\,\right| \rightarrow \,m_{\,{ {\cal G} }\,'}\, =\,
2^{\,1/2} \,\,
\left|\,\,{ {\cal G} }\,\Phi\,'\,\right|\,=\,
2^{\,1/2} \,\,
\left|\,\,{ {\cal G} }\, M \,\Phi\,\right|~.
\label{mm}$$ We can now interpret (\[mm\]) as the definition of the chiral transformation for the charge of the light dyon, thus deriving $${ {\cal G} }\, \rightarrow \, { {\cal G} }\,'\,=\,{ {\cal G} }\,M~,
\label{G->GM}$$ or more explicitly $$(g,q) \rightarrow (g',q')\,=\,(g,q-g)~.
\label{gqgq}$$ This transformation complies with the Witten effect [@Witten:1979ey], which states that the magnetic charge $g$ gives a contribution $\delta q$ to the electric charge of a dyon $$\delta q\,=\,-g\,\theta/(2\pi)~,
\label{eq:}$$ where $\theta$ is the theta-angle. The effect is often discussed for the pure electromagnetic theory, whereas in the case considered there are $r$ electromagnetic theories present simultaneously, but this minor generalization brings no complications since different electromagnetic fields do not interact. The chiral transformation, which changes the charge of a dyon, results in the variation of the $\theta$-angle by $2\pi$, see (\[Delta\]), forcing thus a change of the electric charge of $\delta q=-g$, in accord with (\[gqgq\]).
Applying the transformation (\[G->GM\]) $m$ times to the dyon with the charge ${ {\cal G} }$, and taking into account also inversed transformations we find a set of dyons with charges
\[Gm\] $$\begin{aligned}
\label{Mm}
{ {\cal G} }\,'\,&=\,{ {\cal G} }\,M^{\,m}~,\\
\label{qm}
(\,g\,'\,,\,q\,'\,)\,& =\,(\,g,\,q-m\,g\,)~. \end{aligned}$$
Consider a restriction on the integer $m$ here that stems from (\[A-A\]). Equation (\[2gammaA\]) shows that in order to fulfill the transformation $A \rightarrow -A$ the integer $m$ in (\[gamma\]), and consequently in (\[Mm\]),(\[qm\]), should satisfy $m={ h^{\vee} }$. According to (\[A-A\]) the transformation $A \rightarrow -A$ does not change the vacuum of the ${ {\cal N} }=1$ supersymmetric gauge theory. In other words, a variation of $m$ in (\[Mm\]),(\[qm\]) by ${ h^{\vee} }$ keeps the vacuum of the ${ {\cal N} }=1$ theory intact. We conclude that in order to describe different vacua of the ${ {\cal N} }=1$ theory it suffices to take $m$ in (\[Mm\]),(\[qm\]) modulo ${ h^{\vee} }$ $$m \in Z_{\,{ h^{\vee} }}~.
\label{mZh}$$ As an illustration of the validity of (\[Gm\]), (\[mZh\]) recall the $SU(2)$ gauge group [@Seiberg:1994rs], when two dyons with the charges $$\begin{aligned}
\frac{{ {\cal G} }_0}{2^{\,1/2}}\,=\,(1,0)~,\quad \frac{{ {\cal G} }_{1}}{2^{\,1/2}}\,=(1,-1)
\label{SU2}\end{aligned}$$ play a major role. Note that $\mathsf{G}=SU(2)$ implies $r=1$, $\alpha=2^{\,1/2}$, ${ h^{\vee} }=2$, and $Z_{\,{ h^{\vee} }}=Z_{\,2}$, and that notation in (\[SU2\]) follows definitions given in (\[qdirect\]),(\[gdual\]) and (\[Ggq\]), which prompt the coefficient $2^{\,1/2}=\alpha$. One observes that the necessity of having two dyons, as well as values of their charges in (\[SU2\]) for $\mathsf{G}\,=SU(2)$ comply with (\[Gm\]),(\[mZh\]).
Minimal set of light dyons {#minimal}
==========================
Let us find a minimal set of light dyons, including in this set only those dyons that are absolutely necessary to describe any possible condensate of dyons, which is created when the ${ {\cal N} }=2$ supersymmetry is broken down to the ${ {\cal N} }=1$ case. To construct this minimal set start from the simplest dyon, a monopole. Thus presume that there exists a light monopole with the charge ${ {\cal G} }= ({ \alpha^{\vee} },0)$, $\alpha \in \Delta$. In this case (\[qm\]) shows that there exist a series of light dyons with charges ${ {\cal G} }_{\,\alpha, \,m}$, $${ {\cal G} }_{\,\alpha, \,m} \, = \,(\,{ \alpha^{\vee} },-m \,{ \alpha^{\vee} })~,\quad m\,\in\,Z_{\,{ h^{\vee} }}~,
\label{Gam}$$ where ${ \alpha^{\vee} }$ is a fixed simple coroot. The monopole charge is reproduced here when $m=0$.
Assume now that the monopole with the charge ${ {\cal G} }_{\,\alpha, 0}$ participates in the creation of a condensate, which arises when the ${ {\cal N} }=2$ supersymmetry is broken down to ${ {\cal N} }=1$ supersymmetry. According to [@Seiberg:1994rs] the condensation of dyons is responsible for creation of a vacuum of the ${ {\cal N} }=1$ supersymmetric gauge theory. It was shown in Section \[chiral\] that other dyons with charges given in (\[Gam\]) arise due to chiral transformations. Therefore, each such chiral transformation brings a condensate of monopoles into some other condensate, which is created with the help of a dyon with the charge ${ {\cal G} }_{\,\alpha, \,m}$. Overall, as (\[mZh\]) states, there are ${ h^{\vee} }$ different dyons described by the series (\[Gam\]). Correspondingly, there are ${ h^{\vee} }$ different condensates of dyons, which in turn correspond to ${ h^{\vee} }$ different vacua of the ${ {\cal N} }=1$ supersymmetric gauge theory.
The existence of the mass gap in the vacuum of the ${ {\cal N} }=1$ supersymmetric gauge theory puts further restriction on the condensate of dyons. Consider a light dyon with the charge ${ {\cal G} }_{\,\alpha, \,m}$. Consider further a condensate, in which this dyon plays a role. For the sake of an argument presume firstly that this condensate includes only dyons with the charge ${ {\cal G} }_{\,\alpha, \,m}$, i.e. there are no other dyons with different charges in this condensate. However, this presumption runs into contradiction. The dyons of charge ${ {\cal G} }_{\,\alpha, \,m}$ interact with only those degrees of freedom, ‘dual photons’ and their superpatners, that originate from the gauge $U(1)$-group specified by the vector $\alpha$ that defines the charge ${ {\cal G} }_{\,\alpha, \,m}$ of the dyon. All other degrees of freedom, which are related to other $r-1$ available gauge $U(1)$-groups, do not interact with these dyons. Consequently a hypothetical condensate constructed from the dyons with the charge ${ {\cal G} }_{\,\alpha, \,m}$ is not able to break these $r-1$ gauge $U(1)$ symmetries, thus leaving the corresponding $r-1$ degrees of freedom massless. This contradicts the existence of the mass gap in the ${ {\cal N} }=1$ supersymmetric gauge theory.
To remedy this problem one has to admit that the proper condensate of dyons, which describes the breaking of the supersymmetry ${ {\cal N} }=2\rightarrow { {\cal N} }=1$, is created by several dyons with different charges, which become massless in the ${ {\cal N} }=2$ gauge theory simultaneously and which simultaneously go into the condensate state, when the ${ {\cal N} }=2$ supersymmetry is broken, ${ {\cal N} }=2\rightarrow { {\cal N} }=1$ . Then any given degree of freedom among the $r$ available gauge $U(1)$-symmetries would interact with some dyon in this condensate. As a result, this condensate is able to break all $r$ dual gauge $U(1)$-symmetries producing the mass gap, in accord with properties of the supersymmetric ${ {\cal N} }=1$ gauge theory.
We see that there should exist $r$ different massless dyons, which participate in the creation of the necessary condensate. Let us verify that the condensate is produced by $r$ dyons, which have charges $${ {\cal G} }_{\,\alpha, \,m} \, = \,(\,{ \alpha^{\vee} },-m \,{ \alpha^{\vee} })~,\quad \alpha \, \in \,\Delta~,
\label{aid}$$ where $m$ is fixed. A first test for this assessment gives the Dirac-Schwinger-Zwanziger quantization condition. The condensate constructed from a set of $r$ dyons can be described by conventional methods only if the dyons are mutually local [@t'Hooft:1981ht]. This means that the quantization condition (\[DSZ\]) for any two dyons from this set should read $${ {\cal G} }_1\, \Omega\,\,{ {\cal G} }_{\,2}^{\,T} \,=\,0~.
\label{zero}$$ The charges in (\[aid\]) clearly satisfy this condition $${ {\cal G} }_{\,\alpha, \,m} \,\Omega ~\,{ {\cal G} }_{\,\beta, \,m} ^{\,T} =0~, \quad\quad\alpha,\beta\,\in\,\,\Delta~.
\label{ab}$$ To compare (\[aid\]), (\[zero\]) with the known results remember that according to [@Klemm:1995wp], which studied the case of $\mathsf{G}=SU(3)$, two mutually local dyons can become massless simultaneously, in agreement with (\[aid\]) and (\[zero\]) since $\mathsf{G}=SU(3)$ implies ${ h^{\vee} }=2$. We will return to this comparison in more detail in Section \[comp\]. Condition (\[zero\]) specifies mutually local light dyons. There are known particular situations, which arise near the cusps, when several mutually nonlocal dyons become massless simultaneously, as was discussed in [@Argyres:1995jj], though these cases will remain outside the scope of the present work.
One more important test for the fact that (\[aid\]) gives charges of those light dyons that create the condensate when the ${ {\cal N} }=2$ supersymmetry is broken, ${ {\cal N} }=2\rightarrow{ {\cal N} }=1$, is provided by the Weyl symmetry. Equations (\[R\]), (\[Rcl\]),(\[Rrho\]) define the Weyl reflection for the scalar field. Combining them with (\[MZ\]),(\[Z\]) for the mass of dyons, one defines the Weyl reflection for the charges of dyons. Thus, one finds that the Weyl reflection for the charge ${ {\cal G} }$ has an expected form $${ {\cal G} }\,\rightarrow \,{ {\cal G} }'\,=\,{ {\cal G} }\,P_\beta~,
\label{GG'}$$ where the reflection $P_\beta$ is taken in the hyperplane orthogonal to the root $\beta$. Applying the transformation (\[GG’\]) to the charge ${ {\cal G} }={ {\cal G} }_{\,\alpha, \,m}$ one derives from (\[GG’\]),(\[Rrho\]),(\[R\]) $$\begin{aligned}
& { {\cal G} }_{\,\alpha, \,m}~\rightarrow ~{ {\cal G} }_{\,\alpha, \,m}\,P_\beta\,=\, { {\cal G} }_{\,\alpha',\,m} ~,\\
& \alpha' \,=\,\rho_{\beta}\,\alpha \, =\, \alpha-\beta\,\,(\,\beta^\vee \cdot \alpha\,)~.
\label{GGrho}\end{aligned}$$ Several Weyl reflections allow one to transform the charge ${ {\cal G} }_{\,\alpha, \,m}$ into any other charge ${ {\cal G} }_{\,\alpha',\,m}$, $\alpha'\in \Delta$, provided the lengths of roots $\alpha$ and $\alpha'$ are same, $\alpha'{\,^2}=\alpha^2$. It follows from this that if the dyon with the charge ${ {\cal G} }_{\,\alpha, \,m}$ is light, i.e. it becomes massless for some value of the scalar field, then all other dyons with the charges ${ {\cal G} }_{\,\alpha',\,m}$, $\alpha'\in \Delta$, $\alpha'{\,^2}=\alpha^2$, are also light, i.e. become massless for some values of the scalar field. In general case these values of the scalar field are different for different light dyons.
For simply-laced $\mathsf{g}$ this discussion shows that all dyons with charges specified by (\[aid\]) are light ones. Moreover, this statement remains valid for nonsimply-laced $\mathsf{g}$ as well. Assume that there is one massless dyon, whose charge satisfies (\[aid\]) and equals ${ {\cal G} }_{\alpha,\,m}$. Assume, for example, that the simple root $\alpha$ is long, $\alpha^2=2$. Then we already know that there exist also massless dyons with charges ${ {\cal G} }_{\alpha',\,m}$, where $\alpha'$ are simple large roots, $\alpha'\in\Delta$, $\alpha'^2=2$. We also know that the total number of massless dyons should be $r$. Therefore there must exist additional dyons. We can assume that their magnetic charges equal $\beta^\vee$, where $\beta$ is any simple short root, $\beta\in\Delta$, $\beta^2<2$. Let us call the electric charges of these dyons $q_\beta$. Thus the total charges of the additional dyons are presumed to be ${ {\cal G} }_\beta=(\beta^\vee,q_\beta)$. Let us verify that these charges comply with (\[aid\]), which allows all these dyons to participate in the condensate that is responsible for the ${ {\cal N} }=2\rightarrow { {\cal N} }=1$ supersymmetry breaking. In other words let us verify that ${ {\cal G} }_\beta={ {\cal G} }_{\beta,\,m}$. Consider the quantization conditions (\[zero\]) for all available $r$ massless dyons, which read $$\begin{aligned}
&{ \alpha^{\vee} }\cdot(q_\beta+m \beta^\vee)\,=\,0~,
\label{bb}
\\
&\beta^\vee\cdot q_\gamma-\gamma^\vee\cdot q_\beta\,=\,0~,
\label{gg}\end{aligned}$$ where $\alpha,\beta,\gamma\in\Delta$ are simple roots, $\alpha$ is a long root, while $\beta,\gamma$ are short roots. An obvious solution to (\[bb\]),(\[gg\]) is $q_\beta=-m \beta^\vee$, which complies with (\[aid\]). To see that this is the only possible solution introduce the electric charges $x_\beta$ via $q_\beta=-m\beta^\vee+x_\beta$. Then (\[bb\]) gives $$\alpha\cdot x_\beta=0~.
\label{xx}$$ Fixing $\beta$ here and running $\alpha$ over the set of all long simple roots one immediately concludes that $x_\beta=0$, which leads to the desired identity ${ {\cal G} }_\beta={ {\cal G} }_{\beta,\,m}$. Thus we verified that if there is one massless dyon with the charge ${ {\cal G} }_{\alpha,\,m}$, then there exist a set of $r$ dyons with charges satisfying (\[aid\]).
Combining (\[Gam\]),(\[aid\]) one concludes that in order to describe the breaking of the supersymmetry ${ {\cal N} }=2\rightarrow{ {\cal N} }=1$ in terms of light dyons it is necessary to consider a particular set of dyons, call it the minimal set, which have the following charges $${ {\cal G} }_{\,\alpha, \,m}=(\,{ \alpha^{\vee} },-m\,{ \alpha^{\vee} }\,)~, \quad \quad\quad \alpha \, \in \,\, \Delta~,\quad m\,\in\,Z_{\,{ h^{\vee} }}~.
\label{ma}$$ For a given $m$ this set includes a subset of $r$ dyons. Each one of them becomes massless at some value of the scalar field. Generically these values are different for different dyons. However, there should exist a particular value of the scalar field that makes all these $r$ dyons massless simultaneously. A necessity for this phenomenon follows from the fact that the vacuum of the ${ {\cal N} }=1$ supersymmetric gauge theory is obviously invariant under the Weyl reflection. Consequently, the condensate of dyons that describes a transition ${ {\cal N} }=2\rightarrow { {\cal N} }=1$ should be invariant under the Weyl reflections as well. This condition can only be satisfied if all $r$ dyons participate in the condensate becoming massless simultaneously.
Different values of $m$ in (\[ma\]) correspond to different vacua of the ${ {\cal N} }=1$ theory. The number of different vacua of the ${ {\cal N} }=1$ supersymmetric gauge theory equals its Witten index specified in (\[I=h\]), $I_\mathrm{\,W}={ h^{\vee} }$. A shift of $m$ in the set (\[ma\]) is generated by the chiral transformation of the dyon charge, which is described by (\[G->GM\]),(\[gqgq\]). This transition is matched by the chiral transformation of the gaugino condensate in (\[ll\]).
Remember that generically (\[qdirect\]) states that electric charges are allowed to reside anywhere in the lattice of roots. Equation (\[ma\]) is more assertive, stating that for the dyons considered the electric charges are collinear to their magnetic charges and lie in the lattice of coroots, which for nonsimply-laced gauge algebras incorporates fewer points than the lattice of roots.
The minimal set of dyons (\[ma\]) allows different representations. The definition of the set of simple roots $\Delta$ depends on the choice of the basis in the $r$-dimensional space of roots. Taking a different basis, one gets a different minimal set of light dyons. [^2] An expansion of this construction can be made using transformations of charges of light dyons given by a symplectic, integer valued matrix $${ {\cal G} }\, \rightarrow\, { {\cal G} }^{\,'}\,=\,{ {\cal G} }\,M~,\quad M\,\in\, Sp\,(2r,Z)~,
\label{MIII}$$ which is in line with the idea known as the democracy of dyons. Considering this transformation, one needs to transform simultaneously the subset of those $r$ dyons, which are massless at some particular value of the scalar field. We will use this option in Section \[comp\] to verify that a set of light dyons, which at first sight looks different from the minimal set, can be made compliant with (\[ma\]).
Let us briefly repeat the arguments, which allow one to unfold the minimal set of light dyons specified in (\[ma\]) starting from one monopole. Take a light monopole with the charge ${ {\cal G} }_{\,\alpha, \,0}$. Using the Weyl reflections verify that there should exist also light monopoles with the charges ${ {\cal G} }_{\,\beta,\,0}$, $\beta\in\Delta$, overall $r$ of them. Applying the chiral transformations to the charges ${ {\cal G} }_{\,\beta,\,0}$ reproduce the full set of ${ h^{\vee} }\, r$ charges of light dyons in (\[ma\]). One learns that ${ h^{\vee} }\, r$ is the minimal number of light dyons necessary to describe the condensates of dyons responsible for the breaking of the supersymmetry ${ {\cal N} }=2\rightarrow{ {\cal N} }=1$.
These arguments make it tempting to presume that the minimal set of light dyons specified by (\[ma\]) incorporates all light dyons. For future reference let us call this assumption the minimal hypothesis, or conjecture. Let us reiterate the facts that support its validity. The presented derivation of the minimal set of dyons uses only basic, fundamental symmetries of the theory, the discrete chiral symmetry and the Weyl symmetry. The minimal set as able to reproduce the transition from the ${ {\cal N} }=2$ supersymmetric gauge theory to the ${ {\cal N} }=1$ case, describing important features of the ${ {\cal N} }=1$ supersymmetric gauge theory, the mass gap and the number of vacua.
Strong and weak coupling monodromies {#braid}
====================================
It was argued in Refs. [@Klemm:1994qj; @Argyres:1995jj; @Klemm:1995wp] that every light dyon generates a monodromy $$M({ {\cal G} })=1+\Omega\,{ {\cal G} }^{\,T}\otimes\,{ {\cal G} }\,=\,
\begin{pmatrix} ~~1+q\otimes g & \quad\, q\otimes q \\ \quad~\! -g\otimes g & 1-g\otimes q~~\end{pmatrix}~,
\label{M(G)}$$ where ${ {\cal G} }=(g,q)$ is the dyon charge. This monodromy satisfies the duality condition $$M({ {\cal G} })\,\Omega\, M({ {\cal G} })^T\,=\,\Omega~.
\label{dua}$$ Combined with (\[M(G)\]) it implies $M({ {\cal G} })\in Sp\,(2r,Z)$. The charge ${\mathcal G}$ of the dyon is an eigenvector of this monodromy with the eigenvalue 1 $${ {\cal G} }\,M({ {\cal G} })\,=\,{ {\cal G} }~.
\label{eigen1}$$ To illustrate validity of (\[M(G)\]) remember the gauge group $\mathsf{G}=SU(2)$ discussed in [@Seiberg:1994rs], which has two light dyons with the charges defined in (\[SU2\]). According to (\[M(G)\]) these charges generate the following monodromies $$\begin{aligned}
M(\,{ {\cal G} }_0\,) =
\begin{pmatrix} ~~\,1 & 0~ \\ -2 & 1~\end{pmatrix}~,\quad\quad
M(\,{ {\cal G} }_{1}\,) =\begin{pmatrix} \,-1 & 2~ \\ \,-2 & 3~\end{pmatrix}~.
\label{M01}\end{aligned}$$ For $\mathsf{G}=SU(2)$ there is only one simple root $\alpha$, and consequently only one matrix ${R}\equiv {R}_\alpha$, which represents a monodromy at weak coupling (\[Ra\]) $${R}\,=\,
\begin{pmatrix} -1 & ~~2\, \\ ~~\,0 & -1 \,\end{pmatrix}~.
\label{Ra2}$$ Equations (\[M01\]),(\[Ra2\]) imply an important equality $$M(\,{ {\cal G} }_0\,)\,M(\,{ {\cal G} }_{1}\,)\,=\,{R}~,
\label{MMR}$$ that was casted as $M_1M_{-1}=M_\infty$ in [@Seiberg:1994rs], which matches the monodromies related to light dyons at strong coupling with the monodromy at weak coupling.
Consider a generalization of (\[MMR\]) for an arbitrary gauge group. The monodromy (\[M(G)\]) for a light dyon with the charge ${ {\cal G} }_{\,\alpha \,m}$ reads $$M\left(\,{ {\cal G} }_{\,\alpha, \,m}\,\right)\,\equiv\,
M_{\,\alpha,\,m}\,=\,
\begin{pmatrix} ~1-m \,{ \alpha^{\vee} }\otimes \,{ \alpha^{\vee} }&\quad m^2\, { \alpha^{\vee} }\otimes \,{ \alpha^{\vee} }\\ \quad\ ~\,\, -{ \alpha^{\vee} }\otimes \,{ \alpha^{\vee} }& 1+m\;{ \alpha^{\vee} }\otimes \,{ \alpha^{\vee} }~\end{pmatrix}~.
\label{finds}$$ If $\alpha$ is a long root, $\alpha^2=2$, then (\[finds\]) implies that $$M_{\,\alpha,\,m}\,M_{\,\alpha,\,m+1}\,=\,
{R}_{\,\alpha}~,
\label{lon}$$ which rephrases (\[MMR\]). This means that for any simply laced $\mathsf{g}$ (\[lon\]) gives the desired relation between the monodromies at strong and weak couplings.
For nonsimply laced gauge groups we consider below two possibilities. The first one presumes that the minimal hypothesis, which was formulated at the end of Section \[minimal\], is valid. We will see that in this case (\[lon\]) needs to be modified for short roots. The other option is to discard the minimal hypothesis extending the minimal set of light dyons. Then the condition that matches the strong and weak coupling monodromies can be presented in a form similar to (\[lon\]).
### Minimal hypothesis and basic monodromies for light dyons {#min-basic}
Consider the first opportunity, assuming that the minimal hypothesis is valid, i.e. the minimal set specified in (\[ma\]) exhausts the list of light dyons. In this case it is convenient to introduce a new matrix $\mathcal{M}_{\,\alpha,\,m}\,\in\,Sp\,(\,2r,Z\,)$ $$\mathcal{M}_{\,\alpha,\,m}=\bfone+\frac{1}{\nu_\alpha}(M_{\,\alpha,\,m}-\bfone)
=
\begin{pmatrix} 1-m \,\alpha \otimes \,{ \alpha^{\vee} }& \quad ~m^2\, \alpha\otimes \,{ \alpha^{\vee} }\\
\quad~\,-\alpha \otimes \,{ \alpha^{\vee} }& 1+ m\,~\alpha\otimes \,{ \alpha^{\vee} }~\end{pmatrix}\,.
\label{Mnu}$$ Here $\bfone$ is the $2r\times 2r$ unity matrix, and $$\nu_\alpha\,=\,2/\alpha^2\,=\,1,2,3~,
\label{123}$$ which gives $\nu_\alpha=1$, for long roots, $\nu_\alpha=2$ for short roots of any algebra except $G_2$, and $\nu_\alpha=3$ for short roots of $G_2$. Equation (\[finds\]) implies that $({M}_{\,\alpha,\,m}-\bfone)^2=0$, which leads to $$\big(\,\mathcal{M}_{\,\alpha,\,m}\,\big)^{\nu_\alpha}\,=\,{M}_{\,\alpha,\,m}~.
\label{M^nu}$$ The matrix $\mathcal{M}_{\,\alpha,\,m}$ possesses the following properties $$\begin{aligned}
\mathcal{M}_{\,\alpha,\,m}\,\Omega\, \mathcal{M}_{\,\alpha,\,m}^T\,=\,\Omega~,
\label{duaNew}\\
{ {\cal G} }_{\,\alpha,\, m}\,\mathcal{M}_{\,\alpha,\,m}\,=\,{ {\cal G} }_{\,\alpha, \,m}~,
\label{GM=G}\end{aligned}$$ which are similar to (\[dua\]),(\[eigen1\]) that define the conventional monodromy related to a light dyon. Equations (\[duaNew\]),(\[Mnu\]) imply $\mathcal{M}_{\,\alpha,\,m}\in
Sp\,(2r,Z)$. Let us call $\mathcal{M}_{\,\alpha,\,m}$ the basic monodromy related to the light dyon; it is basic in the sense that the conventional monodromy ${M}_{\,\alpha,\,m}$ equals its integer power, as per (\[M\^nu\]). In particular, for a long root $\alpha$ they coincide, $\mathcal{M}_{\,\alpha,\,m}={M}_{\,\alpha,\,m}$.
From (\[Mnu\]) one finds that $\mathcal{M}_{\,\alpha,\,m}$ satisfies the following identity $$\begin{aligned}
\mathcal{M}_{\,\alpha,\,m}\,\,\mathcal{M}_{\,\alpha,\,m+1}\,=\,
\mathcal{R}_{\,\alpha}~.
\label{MnuMnuR}\end{aligned}$$ Here the matrix $$\mathcal{R}_{\,\alpha}\,=\,\begin{pmatrix}
~\rho_\alpha & ~\alpha \,\otimes\,{ \alpha^{\vee} }~\\ ~0 & \rho_\alpha
\end{pmatrix} \,=\,P_\alpha\,\big(\,T_\alpha\,\big)^{\nu_\alpha}\,=\,R_{\,\alpha}\,\big(\,T_{\,\alpha}\,\big)^{\,\nu_\alpha-1}~,
\label{NewR}$$ where $P_\alpha$ and $T_\alpha$ are defined in (\[Rrho\]),(\[Ta\]). The matrix $\mathcal{R}_{\,\alpha}$ has a similarity with the matrix of the monodromy $R_{\,\alpha}$ defined in (\[Ra\]). The difference between them is in an integer coefficient in front of the nondiagonal matrix element in the $2\times 2$ block matrices, which equals $\alpha\otimes\alpha$ in (\[Ra\]), and $\alpha\otimes{ \alpha^{\vee} }=\nu_{\alpha}\, \alpha\otimes\alpha$ in (\[NewR\]). For a long root $\alpha$ this difference disappears, $\mathcal{R}_\alpha = {R}_\alpha $
Consider a short root $\alpha$ for a nonsimply laced $\mathsf{g}$, when $\mathcal{R}_\alpha \ne {R}_\alpha $. Remember that discussing the monodromy $R_{\,\alpha}$ for the scalar field $\Phi$ we used a path that connects $A$ with $\rho_\alpha A$. Equation (\[Ra\]) was written presuming the simplest path, which crosses only once the wall of the Weyl camera that is orthogonal to $\alpha$. This path gives the contribution $-\pi i$ to the logarithmic function in (\[APTh\]), which leads to the nondiagonal term $ \alpha \otimes \alpha$ in the block-matrix defining $R_{\,\alpha}$ in (\[Ra\]). One can modify the path that connects $A$ with $\rho_\alpha A$ in such a way as to reproduce $\mathcal{R}_\alpha $ instead. Assume that $\alpha$ is a short root of $\mathsf{G}$, $\mathsf{G}\neq G_2$, when $\nu_\alpha=2$. Consider the path that starts from $A$ and crosses the same wall of the Weyl camera twice, forward and backward. Presume that this double crossing results in the combined variation of the logarithmic function $-2\pi i$. Assume the path returns back to the starting point $A$. After that take the Weyl reflection $P_\alpha$, which brings the path to the desired final point $\rho_\alpha A$. This definition of the path leads to the monodromy $\mathcal{R}_{\,\alpha}$ for the scalar field $\Phi$. Similarly, when $\mathsf{G}=G_2$ and $\nu_\alpha=3$, consider the path which runs from $A$ to $\rho_\alpha\,A$ crossing the wall of the camera trice, twice forward and once backward, giving the total contribution $-3\pi i$ to the logarithmic function. This path again leads to the monodromy $\mathcal{R}_{\,\alpha}$ in (\[NewR\]). We see that $\mathcal{R}_{\,\alpha}$ and $R_{\,\alpha}$ describe the same property of the system, the monodromy at weak coupling, which is defined by the path along which the variation of the scalar field is taken. The difference between them is related to the way this path is defined for short roots.
We conclude that embracing the minimal hypothesis, one can rely on (\[MnuMnuR\]), which generalizes (\[MMR\]) and provides a match between the monodromies at strong and weak couplings. The existence of this match supports the validity of the minimal hypothesis. However, to make this support stronger, one needs to verify that the basic monodromies are compatible with the Seiberg-Witten solution.
### Including more light dyons {#extension}
Consider the alternative option. Let us extend the set of dyons presuming that alongside the minimal set (\[ma\]) there exist also additional light dyons with the charges $${ {\cal G} }{\,'}_{\!\!\alpha,\,m}=(\,{ \alpha^{\vee} },\,-m\,{ \alpha^{\vee} }\!-\alpha\,)~.
\label{sho}$$ For a long root $\alpha$, ${ \alpha^{\vee} }=\alpha$, this assumption does not change the set of light dyons because ${ {\cal G} }{\,'}_{\!\!\alpha,\,m}=\,{ {\cal G} }_{\,\alpha,\,m+1}$. Therefore for the simply laced $\mathsf{g}$ (\[sho\]) does not define new charges. However, for short roots $\alpha$ (\[sho\]) makes a presumption that there exist light dyons with new charges, thus enlarging the set of light dyons. From (\[M(G)\]) one derives $$M\left({ {\cal G} }{\,'}_{\!\!\alpha, \,m}\right)
\equiv
M^{\,'}_{\alpha, \,m}=
\begin{pmatrix} 1-(\,m{ \alpha^{\vee} }+\alpha\,)\, \otimes \,{ \alpha^{\vee} }& (\,m{ \alpha^{\vee} }+\alpha\,)\otimes \,(\,m{ \alpha^{\vee} }+\alpha\,)~
\\ \quad\quad\quad\quad\quad -{ \alpha^{\vee} }\otimes \,{ \alpha^{\vee} }& \quad~~~1+{ \alpha^{\vee} }\,\otimes \,(\,m{ \alpha^{\vee} }+\alpha\,)~\end{pmatrix},
\label{M'}$$ and consequently finds $$M_{\,\alpha,\,m}\,M{\,'}_{\!\!\alpha,\,m}\,=\,{R}_{\,\alpha}~.
\label{MM'}$$ This relation generalizes (\[MMR\]) for an arbitrary gauge group by using only conventional monodromies $M({ {\cal G} })$, though this construction presumes that the minimal set of dyons is extended.
For the gauge group $G_2$, there exists an additional ambiguity. Alongside the dyons with charges given by (\[sho\]) one can presume also/instead that there exist light dyons with charges $${ {\cal G} }{\,''}_{\!\!\alpha,\,m}=(\,{ \alpha^{\vee} },\,-m\,{ \alpha^{\vee} }+\alpha\,)~,
\label{G''}$$ which differ from ${ {\cal G} }{\,'}_{\alpha,\,n}$ for any values of $m,n$, ${ {\cal G} }{\,''}_{\!\!\alpha,\,m}\neq
{ {\cal G} }{\,'}_{\!\alpha,\,m}$ due to the mere fact that short roots of $G_{\,2}$ satisfy ${ \alpha^{\vee} }=3\alpha$. [^3] Introducing $$M\left({ {\cal G} }{\,''}_{\!\!\alpha, \,m}\right)\,=\,M{\,''}_{\!\!\alpha,\,m}~,
\label{M''}$$ one can write $$M{\,''}_{\!\!\alpha,\,m}\,M_{\,\alpha,\,m}\,=\,{R}_{\,\alpha}~,
\label{M''M}$$ which gives another possible generalization of (\[MMR\]).
Summarizing, we discussed in Subsections \[min-basic\], \[extension\] two options, which allow one to write a relation between the monodromies at strong and weak couplings. One of them presumes that the minimal set exhausts the list of light dyons, which leads to (\[MnuMnuR\]) written in terms of the basic monodromies (\[Mnu\]). Alternatively, one can rely on conventional monodromies for light dyons in (\[MM’\]), but then the list of light dyons should go beyond the minimal set. The difference between these two opportunities manifests itself only for nonsimply laced gauge algebras. To establish which one of the two available options takes place, a more detailed study is necessary.
Comparison with known results {#comp}
=============================
In order to test (\[ma\]) let us discuss several facts, which have been known for dyon charges previously.
Gauge group $SU(2)$
-------------------
For $\mathsf{G}=SU(2)$, which was discussed in [@Seiberg:1994rs], (\[SU2\]) shows that the number of light dyons as well as their charges comply with (\[ma\]).
Gauge group $SU(3)$
-------------------
For $\mathsf{G}=SU(3)$ [@Klemm:1995wp] found the following charges of light dyons [^4]
\[G16\] $$\begin{aligned}
{ {\cal G} }_1 &\,=\,&(\alpha_1,\,-\alpha_1)\,,
\label{G1}
\\
{ {\cal G} }_2 &\,=\,&(\alpha_1,\,0)\,,
\label{G2}
\\
{ {\cal G} }_3 &\,=\,&(\alpha_2,\,0)\,,
\label{G3}
\\
{ {\cal G} }_4 &\,=\,&(\alpha_2,\,\alpha_2)\,,
\label{G4}
\\
{ {\cal G} }_5 &\,=\,&(-\alpha_1-\alpha_2,\,\alpha_1)\,,
\label{G5}
\\
{ {\cal G} }_6 &\,=\,&(-\alpha_1-\alpha_2,-\alpha_2)\,.
\label{G6}\end{aligned}$$
The roots $\alpha_i,~i=1,2$ are labeled conventionally (same is true for the roots of other groups discussed below). To clarify notation let us present them in the Cartesian coordinates $(x,y)$ in the following form $$\alpha_1\,=\,\frac{1}{\sqrt{2}} \,\,(\,1,-\sqrt{3}\,)~,
\quad
\alpha_2\,=\,\frac{1}{\sqrt{2}} \,\,(\,1,\,\sqrt{3})~.
\label{alpha}$$ Consider the Weyl reflection of the charge ${ {\cal G} }_1$ in the plain that is orthogonal to the root ${\alpha_1+\alpha_2}$, which is accompanied by a change of sign of the charge of the dyon (by taking the anti-dyon). The result reads $${ {\cal G} }_1=(\alpha_1,\,-\alpha_1) \,\rightarrow \,-\,{ {\cal G} }_1\,P_{\alpha_1+\,\alpha_2}\,=\,
(\alpha_2,\,-\alpha_2)\,\equiv \,{ {\cal G} }^{\,'}~.
\label{G1->}$$ Make a similar transformation for the charge ${ {\cal G} }_4$ $${ {\cal G} }_4=(\alpha_2,\,\alpha_2) \,\rightarrow \,-\,{ {\cal G} }_4\,P_{\alpha_1+\,\alpha_2}\,=\,
(\alpha_1,\,\alpha_1)\,\equiv \,{ {\cal G} }^{\,''}~.
\label{G2->}$$ Observe that dyons with charges
\[123456final\] $$\begin{aligned}
& \{\,{ {\cal G} }_1,\,{ {\cal G} }_2,\,{ {\cal G} }^{\,''}\}\,=\,{ {\cal G} }_{\,\alpha_1,\,m}~,
\label{126}
\\
&\{\,{ {\cal G} }^{\,'},\,{ {\cal G} }_3,\,{ {\cal G} }_4\,\}\,=\,{ {\cal G} }_{\,\alpha_2,\,m}~,
\label{634}\end{aligned}$$
comply with the formula for ${ {\cal G} }_{\,\alpha,\,m}$ in (\[ma\]) in which $\alpha=\alpha_1,\alpha_2$ and $m\in Z_{3}$.
At a glance the charges ${ {\cal G} }_5$ and ${ {\cal G} }_6$ in (\[G5\]),(\[G6\]) deviate from predictions of (\[ma\]), but there is a way to transform these charges making them identical to ${ {\cal G} }^{\,''}$ and ${ {\cal G} }^{\,'}$ respectively. In [@Klemm:1995wp] it is stated that pairs of mutually local dyons can become massless simultaneously, which agrees with the discussion in Section \[minimal\]. Consequently the dyons with the charges ${ {\cal G} }_1$ and ${ {\cal G} }_6$ become massless simultaneously. Consider the $6\times 6$ matrix $M$ $$M=
\,\begin{pmatrix}
~1 & 0 \\
-g\otimes g &1
\end{pmatrix} ~.
\label{M1}$$ where $g$ is a vector $$g\,=\,\frac{1}{\sqrt{3}}\,\,\alpha_1 \,+ \frac{2}{\sqrt{3}}\,\,\alpha_2
\,=\,\frac{1}{\sqrt{2}} \,(\,\sqrt{3}\, ,\,1\,)~.
\label{g1}$$ Clearly, $M\in Sp\,(6,Z)$. This fact allows us to apply (\[MIII\]) to the charges ${ {\cal G} }_1$ and ${ {\cal G} }_6$, transforming them as follows $$\begin{aligned}
& { {\cal G} }_1\,\rightarrow { {\cal G} }_1\,M\,=\,{ {\cal G} }_1~, \\
& { {\cal G} }_6\,\rightarrow { {\cal G} }_6\,M\,=\,{ {\cal G} }^{\,'}~.
\label{16}\end{aligned}$$ These equations show that instead of a pair of mutually local light dyons with the charges ${ {\cal G} }_1,\,{ {\cal G} }_6$ one can consider light dyons with the charges ${ {\cal G} }_1,\,{ {\cal G} }^{\,''}$. Similarly, instead of the mutually local light dyons with charges ${ {\cal G} }_4,\,{ {\cal G} }_5$ one can consider the light dyons with charges ${ {\cal G} }_4,{ {\cal G} }^{\,'}$. The necessary transformation is $$\begin{aligned}
& { {\cal G} }_4\,\rightarrow { {\cal G} }_4\,\tilde{M}\,=\,{ {\cal G} }_4~, \\
& { {\cal G} }_5\,\rightarrow { {\cal G} }_5\,\tilde{M}\,=\,{ {\cal G} }^{\,''}~.
\label{45}\end{aligned}$$ Here $$\begin{aligned}
& \tilde{M}=
\,\begin{pmatrix}
~1 & 0 \\
\tilde{g}\otimes \tilde{g} &1
\end{pmatrix}\,\in\,Sp\,(6,Z) ~,
\label{M2}
\\
& \tilde g\,=\,\frac{2}{\sqrt{3}}\,\,\alpha_1 \,+ \frac{1}{\sqrt{3}}\,\,\alpha_2
\,=\,\frac{1}{\sqrt{2}} \,\,(\,\sqrt{3}\,,\,-1\,)~.
\label{g2}\end{aligned}$$ The net effect of (\[45\]),(\[16\]) is a transformation of the charges ${ {\cal G} }_5, { {\cal G} }_6$ into ${ {\cal G} }^{\,''},\,{ {\cal G} }^{\,'}$. As a result the set of light dyons (\[G16\]) is transformed into the set (\[123456final\]), whose charges comply with (\[ma\]). The total number of these dyons, six, also agrees with (\[ma\]). The fact that pairs of mutually local dyons can become massless simultaneously is once again in line with the discussion in Section \[minimal\].
Dyon charges and integrable systems
-----------------------------------
The charges of dyons were discussed in [@Hollowood:1997pp] using the results of [@Martinec:1995by], which related the Seiberg-Witten solution to the spectral curve of a periodic Toda lattice. It was found in [@Hollowood:1997pp] that there exist two series of dyons with charges ${ {\cal G} }^{(1)}_{\,\alpha}$ and ${ {\cal G} }^{(2)}_{\,\alpha}$, $\alpha\in \Delta$,
\[G1G2\] $$\begin{aligned}
{ {\cal G} }^{(1)}_{\,\alpha}&\,=\,(\,{ \alpha^{\vee} },\,p_\alpha \,\alpha\,)~,
\label{G1in}
\\
{ {\cal G} }^{(2)}_{\,\alpha}&\,=\,(\,{ \alpha^{\vee} },\,(p_\alpha+1) \,\alpha\,)~.
\label{G2in}\end{aligned}$$
Here $p_\alpha$ are integers, which should be found from equations formulated in [@Hollowood:1997pp]. These equations are involving, and their general solution has not been presented. There is a similarity between (\[G1G2\]) and (\[ma\]), in that in both cases the electric and magnetic charges are collinear. There are also distinctions. Equation (\[ma\]) specifies that the only restriction on the integer $m$ is $m\in Z_{\,{ h^{\vee} }}$, and that $m$ is a factor in front of the coroot ${ \alpha^{\vee} }$, whereas $p_\alpha$, which plays a similar role, is a factor in front of the root $\alpha$. For more detail see Subsections \[su4\], \[g22\].
Gauge group $SU(4)$ {#su4}
-------------------
For $\mathsf{G}=SU(4)$ it was found in [@Hollowood:1997pp] that
\[G12123\] $$\begin{aligned}
{ {\cal G} }^{(1)}_{\alpha_1}&=(\alpha_1,m\,\alpha_1),~\,
{ {\cal G} }^{(1)}_{\alpha_2}=(\alpha_2,(m+1)\alpha_2),~\,
{ {\cal G} }^{(1)}_{\alpha_3}=(\alpha_3,(m+2)\alpha_3),\\
{ {\cal G} }^{(2)}_{\alpha}&={ {\cal G} }^{(1)}_{\alpha}+(\,0,\,\alpha),\quad\quad
\alpha= \alpha_1,\,\alpha_2,\,\alpha_3.\end{aligned}$$
These charges look similar to the ones in (\[ma\]). There is a subtlety though. Equations (\[G12123\]) presume that $m\,\in\,Z$, while (\[ma\]) states that $m\,\in\,Z_{\,{ h^{\vee} }}=Z_{4}$; ${ h^{\vee} }=4$ for $\mathsf{G}=SU(4)$. A possible explanation for this distinction is that the present work aims at finding the smallest possible number of dyons that are necessary to describe the ${ {\cal N} }=2$ gauge theory, while [@Hollowood:1997pp] did not pursue this goal.
Gauge group $G_2$ {#g22}
-----------------
For $\mathsf{G}\,=\,G_2$ [@Hollowood:1997pp] found that there are dyons with charges
\[G212\] $$\begin{aligned}
{ {\cal G} }^{(1)}_{\alpha_1}&=({ \alpha^{\vee} }_1,m \alpha_1),\quad\quad~~~ { {\cal G} }^{(1)}_{\alpha_2}=({ \alpha^{\vee} }_2,(3m\!-\!1)\alpha_2)=({ \alpha^{\vee} }_2,m{ \alpha^{\vee} }_2\!-\!\alpha_2),
\\
{ {\cal G} }^{(2)}_{\alpha_1}&=({ \alpha^{\vee} }_1,(m+1) \alpha_1),~\, { {\cal G} }^{(2)}_{\alpha_2}=({ \alpha^{\vee} }_2,3m\alpha_2)=({ \alpha^{\vee} }_2,m{ \alpha^{\vee} }_2).
\label{G12}\end{aligned}$$
It is taken into account here that ${ \alpha^{\vee} }_2=3\alpha_2$. Note a similarity of ${ {\cal G} }^{(1)}_{\,\alpha_1}$, ${ {\cal G} }^{(2)}_{\,\alpha_1}$ and ${ {\cal G} }^{(2)}_{\,\alpha_2}$ with ${ {\cal G} }_{\,\alpha,\,m}$ in (\[ma\]), though again an integer $m$ is understood differently, $m\in Z$ in (\[G212\]), while $m\in Z_{\,{ h^{\vee} }}=Z_4$ in (\[ma\]), remember ${ h^{\vee} }=4$ for $\mathsf{G}=G_2$. The charge ${ {\cal G} }^{(1)}_{\,\alpha_2}$ does not fit into the minimal set given by (\[ma\]), but it complies with an extension of the minimal set discussed in (\[sho\]). [^5] This latter fact seems to indicate that for $\mathsf{G}=G_2$ the minimal hypothesis proves erroneous, though further study of this point is necessary.
Conclusions {#conc}
===========
It is shown that in the ${ {\cal N} }=2$ supersymmetric gauge theory there exists a minimal set of light dyons with charges specified by (\[ma\]). This conclusion is derived from symmetries of the ${ {\cal N} }=2$ and ${ {\cal N} }=1$ supersymmetric gauge theories. One of them is the discrete chiral symmetry. Another is the Weyl symmetry, which is a remnant of the broken gauge symmetry. The minimal set includes ${ h^{\vee} }\,r$ light dyons. Here ${ h^{\vee} }$ is the dual Coxeter number, which counts the number of vacua of the ${ {\cal N} }=1$ supersymmetric gauge theory and also equals the Witten index for this theory, while $r$ is the rank of the gauge group.
Since the minimal set of dyons accounts for the fundamental symmetries of the problem, it is tempting to conjecture that this set provides a complete list of all light dyons, which are necessary to describe the properties of the ${ {\cal N} }=2$ supersymmetric gauge theory. The validity of this conjecture was verified using the condition that matches monodromies at weak and strong couplings. The conjecture passes the test provided the Seiberg-Witten solution incorporates the so called basic monodromies, which are related to conventional strong-coupling monodromies by (\[M\^nu\]). This fact makes it interesting to check whether the basic monodromies are present in the Seiberg-Witten solution. The conjecture was also tested by comparison with the known charges of dyons for several simplest gauge groups. The unitary groups $SU(n)$ for $n=2,3,4$ were found to comply with the conjecture, while for $G_2$ the validity of the conjecture is not evident, though this latter case needs to be studied further.
A financial support of the Australian Research Council is acknowledged.
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[^1]: The dual Coxeter number ${ h^{\vee} }={ h^{\vee} }(\mathsf{g})$ of the algebra $\mathsf{g}$, the eigenvalue of the quadratic Casimir operator in the adjoint representation $C_2(\mathsf{g})$, and the Dynkin index of the adjoint representation $\chi_\mathrm{adj}(\mathsf{g})$ are all related, $2{ h^{\vee} }(\mathsf{g})=C_2(\mathsf{g})=\chi_\mathrm{adj}(\mathsf{g})$, see e.g. [@Di-Francesco:1997], Eqs.(13.128),(13.134).
[^2]: This implies that if $\alpha$ is any root, $\alpha\in \mathsf{g}$, then the dyon with the charge ${ {\cal G} }_{\,\alpha,\,m}$ is also light.
[^3]: This ambiguity does not manifest itself for other nonsimply laced gauge algebras, since their small roots satisfy ${ \alpha^{\vee} }=2\alpha$ leading to ${ {\cal G} }{\,''}_{\!\!\alpha,\,m+1}={ {\cal G} }{\,'}_{\!\!\alpha,\,m}$.
[^4]: Equation (4.20) of [@Klemm:1995wp] gives the magnetic and electric charges in the simple root basis and Dynkin basis respectively.
[^5]: The ambiguity for $\mathsf{G}=G_2$ mentioned in (\[sho\]),(\[G”\]) of the present work does not seem to be present in equations (6.10) and (5.12) of [@Hollowood:1997pp], from which (\[G212\]) of the present work are derived.
|
---
author:
- 'J. Iglesias-Páramo'
- 'A. Boselli'
- 'G. Gavazzi'
- Antonio Zaccardo
date: 'Received ...; accepted ...'
title: 'Tracing the star formation history of cluster galaxies using the H$\alpha$/UV flux ratio'
---
Introduction
============
A number of environmental mechanisms able of affecting significantly the evolution of galaxies in rich clusters have been proposed in the literature: gas stripping by ram pressure (Gunn & Gott 1972; Abadi, Moore & Bower 1999), galaxy-galaxy harassment in close encounters (Moore et al. 1996), tidal stirring by the cluster potential (Byrd & Valtonen 1990; Fujita 1998).
These mechanisms should produce morphological disturbances, gas removal and, on long timescales, significant quenching of the star formation rates (SFRs) of galaxies due to “fuel” exhaustion (see Gavazzi et al. 2002a). However galaxy-galaxy interactions might also enhance the star formation in gas-rich systems, both in their nuclei and disks, as it has been observed at several wavelengths (Larson & Tinsley 1978; Kennicutt et al. 1987; Hummel et al. 1987; Soifer et al. 1984), although this [**enhancement**]{} might be mild (Bergvall et al. 2003). The dynamical interaction of galaxies with the IGM can also produce an enhancement in the galaxies SFR by ram pressure (Fujita & Nagashima 1999).
A conclusive evidence for a separate evolution of galaxies in clusters is offered by the Butcher-Oemler effect (Butcher & Oemler 1978), i.e. distant ($z \sim 0.3$) clusters show a larger fraction of blue galaxies than nearby clusters. Several follow-up studies (Couch & Sharples 1987; Barger et al. 1996; Couch et al. 1994; Poggianti et al. 1999; Balogh et al. 1999) brought to today’s accepted scenario that clusters are continuously accreting galaxies from their neighborhood, with the accretion rate increasing with look-back time.
Several observable quantities have been proposed as reliable estimators of the SFRs of galaxies (Kennicutt 1998; Rosa-González et al. 2002): H$\alpha$, UV, radio continuum and Far-IR luminosities. Among these, we focus our analysis on the H$\alpha$ and UV luminosities. The H$\alpha$ luminosity comes from stars more massive than 10M$_{\odot}$ and it traces the SFR in the last $\leq
10^{7}$ yr. The UV luminosity at 2000 Å comes from stars more massive than 1.5M$_{\odot}$ and it can be used as an indicator of the SFR in the last $\approx 10^{8}$ yr, under the condition that it stayed approximately constant during this period. The two quantities combined, in other words the ratio $f$(H$\alpha$)/$f$(UV), should give us a “clock” suitable for telling if the SFR was constant over the last $10^{8}$ yr.
The present paper is aimed at studying the role of the cluster environment on the star formation histories of cluster galaxies by using the $f$(H$\alpha$)/$f$(UV) ratio for a sample of galaxies in four nearby clusters: Virgo, Coma, Abell 1367 and Cancer. This analysis relies on the multifrequency database that we collected so far for a large sample of galaxies in nearby cluster and we made available to the community through the GOLDmine WEB site (Gavazzi et al. 2003). Beside the H$\alpha$ and UV data which are directly used for computing the two SFR indicators, other corollary data (e.g. Near-IR, Far-IR, H[i]{} fluxes and optical spectroscopy) are used throughout this paper. These corollary data play a fundamental role in the determination of the dust extinction at UV wavelengths (through the FIR/UV ratio, e.g. Buat & Xu 1996) and at H$\alpha$ (from the Balmer decrement, e.g. Lequeux et al. 1981).
The galaxy sample is presented in Section 2. The observed vs. expected $f$(H$\alpha$)/$f$(UV) ratio for cluster galaxies is discussed in Section 3. In Section 4 we discuss the limitations and the potentiality of the method applied in this preliminary analysis. A brief summary of the results is presented in Section 5. Details about the estimate of the birthrate parameter $b$ are given in Appendix A. A second appendix contains a detailed analysis of the observational uncertainty affecting the $f$(H$\alpha$)/$f$(UV) ratio.
The sample of cluster galaxies
==============================
The sample analyzed in this work includes late-type galaxies (morphological type later than Sa) belonging to four nearby clusters: Virgo, Coma, A1367 and Cancer. Among Virgo galaxies we selected all objects in the Virgo Cluster Catalogue (VCC, Binggeli et al. 1985 with $m_{\mbox{\scriptsize pg}}$ $\leq$ 18) and for Coma, A1367 and Cancer all galaxies in the Zwicky Catalogue (CGCG, Zwicky et al. 1961-1968 with $m_{\mbox{\scriptsize pg}}$ $\leq$ 15.7). The accuracy of the morphological classification is excellent for the Virgo galaxies (Binggeli et al. 1985; 1993). Because of the larger distances, the morphology of galaxies belonging to the other surveyed regions suffers from an uncertainty of about 1.5 Hubble type bins. We assume a distance of 17 Mpc for the members (and possible members) of Virgo cluster A, 22 Mpc for Virgo cluster B, 32 Mpc for objects in the M and W clouds (see Gavazzi et al. 1999). Members of the Cancer, Coma and A1367 clusters are assumed at distances of 62.6, 86.6 and 92 Mpc respectively. Isolated galaxies in the Coma supercluster are assumed at their redshift distance adopting $H_{0}$ = 75 km s$^{-1}$ Mpc$^{-1}$.
The observational dataset
-------------------------
The photometric and spectroscopic data necessary for carrying out the present analysis (taken from the GOLDmine database: http://goldmine.mib.infn.it/; Gavazzi et al. 2003) are the following:
1. H$\alpha$ fluxes, necessary to determine the present ($\leq$ 10$^7$ years), massive SFR (Kennicutt 1998). H$\alpha$+\[N[ii]{}\] fluxes have been obtained from imaging (Iglesias-Paramo et al. 2002; Boselli & Gavazzi 2002; Boselli et al. 2002; Gavazzi et al. 2002b, and references therein): they are integrated values and, contrary to many other samples used for similar analysis, they do not suffer from aperture biases. The estimated error on the H$\alpha$+\[N[ii]{}\] flux is $\sim$ 15%.
2. UV (2000 Å) fluxes, useful to compute the intermediate age ($\leq 3
\times 10^{8}$ years) star formation activity (Buat et al. 1987). The UV data are taken from the FAUST (Lampton et al. 1990) and the FOCA (Milliard et al. 1991) experiments. For the sake of consistency with our previous works, we transformed UV magnitudes taken at 1650 Å by Deharveng et al. (1994) to 2000 Å assuming a constant colour index $m_{2000} =
m_{1650} + 0.2$ mag (see Boselli et al. 2003). These are total magnitudes, determined by integrating the UV emission down to the weakest detectable isophote. The estimated error on the UV magnitude is 0.3 mag in general, but it ranges from 0.2 mag for bright galaxies to 0.5 mag for weak sources observed in frames with larger than average calibration uncertainties.
3. Far-IR (60, 100 $\mu$m) fluxes, for obtaining an accurate UV extinction correction (Buat et al. 2002; Boselli et al. 2003). Far-IR at 60 and 100 $\mu$m integrated flux densities from the IRAS survey are taken mainly from the IRAS FSC (Moshir et al. 1989). Three galaxies are not detected at one of these two IRAS bands and an upper limit is estimated to the flux: VCC 1725, CGCG 119-053 and CGCG 097-062. In addition, no IRAS data are available for VCC 1699; instead, ISO data were used for this galaxy. The conversion between the ISO and IRAS fluxes was taken from Boselli et al. (2003). Typical uncertainties on the Far-IR data are $\sim$ 15%.
4. Long slit integrated spectroscopy with detected H$\alpha$ and H$\beta$ lines, necessary for the determination of the Balmer decrement. Long slit, drift-scan mode spectra were obtained by (Gavazzi et al. 2003b) by drifting the slit over the whole galaxy disk, as in Kennicutt (1992). These are intermediate ($\lambda/\Delta\lambda \sim$ 1000) resolution spectra in the range ($3600 - 7200$ Å). The accuracy on the determination of the line intensities is $\approx$ 10% for H$\alpha$ and H$\beta$ and $\approx$ 15% for \[N[ii]{}\]$\lambda\lambda$6548,6584Å.
Due to these strong observational constraints the final sample is restricted to 98 galaxies. Because of the above selection criteria, and in particular owing to the condition that galaxies must be detected in H$\beta$, our sample might be affected by observational biases that will be discussed in a subsequent section.
Further corollary data, when available, are used to provide information on the evolutionary state of the sample galaxies:
1. In order to quantify the degree of perturbation by the cluster-IGM, we use the H[i]{} deficiency parameter, defined as the logarithm of the ratio of the observed H[i]{} mass to the average H[i]{} mass of isolated objects of similar morphological type and linear size (Haynes & Giovanelli 1984). Galaxies with $def(\mbox{H{\sc i}}) < 0.4$ are considered as unperturbed objects.
2. The asymmetry of the H[i]{} profile of the individual galaxies was also included in our analysis as an indicator of interactions in the last $\simeq 10^{8}$ yr, as done in Gavazzi (1989). A galaxy with a line of sight inclination $>$ 30$^\circ$ is considered asymmetric in H[i]{} if its profile deviates significantly from the expected two-horns profile typical of unperturbed inclined galaxies, i.e. if the peak of the lowest horn is smaller than 50 % that of the highest one. By definition, this indicator is meaningless for face-on galaxies where the profile is a single horn. H[i]{} profiles were taken from Giovanelli & Haynes (1985), Bothun et al. (1985), Chincarini et al. (1983), Helou et al. (1984), Gavazzi (1989), Haynes & Giovanelli (1986), Hoffman et al. (1989), Schneider et al. (1990) and Bravo-Alfaro et al. (2001).
3. Near-IR total $H$-band magnitudes are derived consistently with Gavazzi & Boselli (1996) for most of the galaxies, with an accuracy of 10%. For galaxies VCC 318, 459, 664, 971, 1189, 1575, 1678, 1699 and 1929, with no $H$-band magnitude available, it was derived from the $K$-band magnitude adopting $(H - K) = 0.25$ on average. The $H$-band magnitude for VCC 552 and 1091 was taken from the 2MASS All-Sky Extended Source Catalog (XSC).
$H$-band luminosities are required, together with the H$\alpha$ ones, to estimate the birthrate parameter, $b$, defined as (Kennicutt et al. 1994): $$b = \frac{\mbox{SFR}(t)}{\left< \mbox{SFR}(t') \right>}$$ where SFR($t$) is the SFR at the present epoch and $\left< \mbox{SFR}(t')
\right>$ is the average SFR over the galaxy lifetime. If we model the SF history of normal galaxies with a delayed exponential law, called “a la Sandage” (Gavazzi et al. 2002a), a value of the birthrate parameter $b_{\mbox{\scriptsize model}}$ can be estimated. On the other hand, an observational value of the birthrate parameter $b_{\mbox{\scriptsize obs}}$ can be obtained from the H$\alpha$ and $H$-band luminosities (see Boselli et al. 2001 and Appendix A for details about the calculation of $b_{\mbox{\scriptsize model}}$ and $b_{\mbox{\scriptsize obs}}$). As the cluster environment can alter the galaxies’ SFH, the ratio $b_{\mbox{\scriptsize obs}}/b_{\mbox{\scriptsize model}}$ should reflect the deviation of the real SFH from the analytical one, and thus it should provide us with an estimate of the effect of the environment on cluster galaxies.
The basic quantities used in this analysis are listed in Table \[tabla\], arranged as follows:\
Col. (1): Galaxy name.\
Col. (2): $\log$ of the H$\alpha$ flux corrected for dust extinction and \[N[ii]{}\] contamination as described in Section 2.2.1., in erg s$^{-1}$ cm$^{-2}$.\
Col. (3): $\log$ of the UV flux corrected for extinction as described in Section 2.2.2., in erg s$^{-1}$ cm$^{-2}$ Å$^{-1}$.\
Col. (4): $\log$ of the $H$-band luminosity.\
Col. (5): H[i]{} deficiency parameter.\
Col. (6): Asymmetry of the H[i]{} profile: “S” = symmetric, “A” = asymmetric and “?”= unclassified H[i]{} profile.
Extinction Correction
---------------------
### H$\alpha$$+$\[N[ii]{}\] fluxes
H$\alpha$$+$\[N[ii]{}\] fluxes have been corrected for dust extinction and \[N[ii]{}\] contamination as in Buat et al. (2002). The integrated spectra, available for all galaxies, have been used to estimate the H$\alpha$/\[N[ii]{}\] line ratio and the Balmer decrement. On all spectra we were able to measure the underlying Balmer absorption at H$\beta$. This measurement is absolutely necessary for an accurate determination of the Balmer decrement in intermediate star forming galaxies, where the underlying absorption is comparable to the emission line. The average H$\beta$ equivalent width in absorption in our sample is 4.75 Å.
Despite the fact that the two \[N[ii]{}\] forbidden lines are close to H$\alpha$, the triplet was successfully deblended in most cases by fitting three gaussian components to the ensemble of the three lines or taking advantage of the fact that the ratio \[N[ii]{}\]$\lambda$6548/$\lambda$6584 is approximately constant. For those galaxies for which the \[N[ii]{}\]$\lambda$6548 Å emission line was not detected, we used the theoretical relationship \[N[ii]{}\]$\lambda$6548$+$$\lambda$6584 Å $=$ 1.33 $\times$ \[N[ii]{}\]$\lambda$6584 Å (Osterbrock 1989).
Given the proximity of the \[N[ii]{}\] doublet, deblending the underlying Balmer absorption at H$\alpha$ results impossible. Since on average the equivalent width in absorption at H$\alpha$ is expected similar to within few percent to that of the H$\beta$ (see Charlot & Longhetti 2001 and references therein) its inclusion should result in a negligible correction to the relatively strong H$\alpha$ line. Therefore no correction for underlying absorption at H$\alpha$ was applied.
### UV fluxes
UV fluxes have been corrected for galactic extinction according to Burstein & Heiles (1982) and for internal extinction assuming the recipe of Boselli et al. (2003), based on the Far-IR to UV flux ratio. This correction is, at present, the most accurate and less model dependent, being mostly independent on the geometry, on the SFH of galaxies and on the assumed extinction law.
For the three galaxies with available fluxes at only one of the IRAS bands (60 or 100$\mu$m), the flux in the undetected IRAS band was estimated using the templates SED of galaxies of similar luminosity given in Boselli et al. (2003).
The H$\alpha$/UV ratio of star forming galaxies
===============================================
Gavazzi et al. (2002a) showed that the time evolution of optically selected galaxies of the Virgo cluster can be reproduced assuming an universal IMF (Salpeter) and a SFH “a la Sandage”. This form represents a “delayed exponential” SFH whose analytical representation as a function of time $t$ (where $t$ is the age of the galaxy) is: $$\label{sandage}
\mbox{SFR}(t,\tau) \propto (t/\tau)^{2} e^{-t^{2}/2 \tau^{2}}$$ As described in Gavazzi et al. (2002a; see their fig. 5), the temporal evolution of this family of functions is a delayed rise of the SFR up to a maximum (at $t=\sqrt{2} \tau$), followed by an exponential decrease. Both the delay time and the steepness of the decay are regulated by a single parameter $\tau$. The parameter $\tau$ was found to scale with the $H$-band luminosity, or in other words that the SFR of galaxies at a given time is determined by its $H$-band luminosity. The values of $\tau$ found for our sample galaxies range from $3.5 \leq \tau \leq 8.5$ Gyr for normal spirals and $\tau \geq 8.5$ for star forming dwarf galaxies of types Im and BCDs.
For any galaxy whose spectral energy distribution (SED) is known, the knowledge of SFR($t$) allows to predict the expected value (at any time $t$) of any observable quantity $A$ once we know its time evolution by integrating over the lifetime of the galaxy: $$A_{\mbox{\scriptsize exp}}(t) = \int_{0}^{t} \mbox{SFR}(t',\tau) A(t' - t) dt'$$ where $A_{\mbox{\scriptsize exp}}(t)$ is the expected value of the variable $A$ at time $t$ and $t=0$ corresponds to the epoch of galaxy formation.
Assuming eq. \[sandage\] for the SFH, we show in fig. \[ha\_uv\_t\] the time evolution of the $f$(H$\alpha$)/$f$(UV) ratio for different values of $\tau$. The Starburst99 (Leitherer et al. 1999) code was used, assuming a Salpeter IMF and solar metallicity. As the plot shows, the ratio $f$(H$\alpha$)/$f$(UV) shows a steep decrease in the first 1 Gyr of evolution for any $\tau$. Between 1 and 13 Gyr the ratio $f$(H$\alpha$)/$f$(UV) continues to decrease for $\tau = 1$ Gyr which is typical of the brightest elliptical galaxies (see Gavazzi et al. 2002a). For $\tau \geq 3$ Gyr, appropriate for normal spirals and star forming dwarfs such as those analyzed in our work, the ratio $f$(H$\alpha$)/$f$(UV) remains almost constant for $t \geq 10^{9}$ yr. Thus, if spiral galaxies follow a time evolution “a la Sandage” as in eq. \[sandage\] (i.e., an almost constant SFR over the last $\approx 10^{8}$ yr), we expect $\log$ $f$(H$\alpha$)/$f$(UV) $\approx 1.43$ at the present time, according to the Starburst99 code and assuming solar metallicity and a Salpeter IMF between $0.1$ and 100M$_{\odot}$. A good agreement with this value is found when comparing the stability of this result to previous values reported in the literature assuming realistic SFHs and similar IMFs and metallicities – 1.37 (Kennicutt et al. 1998), 1.42 (Madau et al. (1998) and 1.51 (Boselli et al. 2001) – so, it will be used as the reference value in the subsequent analysis.
The histogram of the observed $f$(H$\alpha$)/$f$(UV) ratio for our sample galaxies in fig. \[histo\_ha\_uv\] shows an almost symmetric distribution centered at $\log
f(\mbox{H}\alpha)/f(\mbox{UV}) = 1.17$ ($\sigma = 0.25$ dex), significantly lower than the expected value for a SFH “a la Sandage” ($\log f(\mbox{H}\alpha)/f(\mbox{UV}) = 1.43$, indicated by the dashed line in the plot).
The dispersion of the $f$(H$\alpha$)/$f$(UV) distribution is consistent with that expected from the observational uncertainties, as shown in Appendix B. The systematic difference between the average observed value and the model prediction ($0.27$ dex) can hardly be explained by systematic errors in the calibration of the data and of the models. The nature of this difference, which we believe real, is discussed in what follows.
Variable IMF
------------
The $f$(H$\alpha$)/$f$(UV) ratio depends on the assumed IMF. Changing the slope and $M_{\mbox{\scriptsize up}}$ of the IMF results in changes in the relative numbers of the high-to-low mass stars as summarized in Table \[hauvmodels\]. This table shows, for instance, the dependence of the $f$(H$\alpha$)/$f$(UV) ratio on the IMF, assuming instantaneous bursts and constant star formation during $10^{6}$, $10^{7}$ and $10^{8}$ yr (Starburst99 models). Three IMFs were chosen: Salpeter ($\alpha = -2.35$ and $M_{\mbox{\scriptsize up}} = 100$M$_{\odot}$), truncated Salpeter ($\alpha = -2.35$ and $M_{\mbox{\scriptsize up}} = 30$M$_{\odot}$), and Miller-Scalo ($\alpha = -3.30$ and $M_{\mbox{\scriptsize up}} = 100$M$_{\odot}$; Miller & Scalo 1979). Not unexpectedly the Salpeter IMF gives the highest $f$(H$\alpha$)/$f$(UV) ratio, since it corresponds to the highest high-to-low mass stars fraction. Changing the IMF produces changes of $f$(H$\alpha$)/$f$(UV) of the order of $\pm 0.25$ dex for the constant SFR case. These results are quite stable against the use of different population synthesis models: using similar initial conditions, Starburst99 yields values of $f$(H$\alpha$)/$f$(UV) $\approx 0.07$ dex larger than PEGASE2.
From the observational point of view there is no compelling evidence for a non universal IMF in galaxies. The lack of any relationship between the $f$(H$\alpha$)/$f$(UV) ratio and the morphological type or luminosity, as shown in fig. \[ha\_uv\_type\], justifies the use of the same IMF for all classes of galaxies. Moreover several studies indicate that the Salpeter IMF for $M_{*} \geq 3$M$_{\odot}$ is adequate for several nearby galaxies and for the Galaxy (Sakhibov & Smirnov 2000; Massey 1998; Massey & Hunter 1998; see however Figer et al. 1999; Eisenhauer et al. 1998).
Variable metallicity
--------------------
The $f$(H$\alpha$)/$f$(UV) ratio also depends on the metallicity of galaxies, as shown in Table \[hauvmodels\]. The dispersion due to metallicity is maximum for an instantaneous burst and it decreases in the case of constant star formation over the last $10^{8}$ yr. The contribution of metallicity to the dispersion of the $f$(H$\alpha$)/$f$(UV) distribution should be however minor since the metallicities of our sample galaxies range from Z$_{\odot}$ to $0.1$Z$_{\odot}$ (see Gavazzi et al. 2002a). We expect the dispersion of the $f$(H$\alpha$)/$f$(UV) distribution due to metallicity to be $\pm 0.04$ dex around the mean theoretical value of $\log f(\mbox{H}\alpha)/f(\mbox{UV})$, this result being independent on the adopted populations synthesis model. Thus, the systematic difference between the observed $f$(H$\alpha$)/$f$(UV) distribution and the theoretical value can hardly be ascribed to the different metallicities of the sample galaxies.
Escaping of Lyman continuum photons
-----------------------------------
A non negligible fraction of the Lyman continuum photons can escape galaxies without ionizing hydrogen atoms. This effect would produce an overall shift of the $f$(H$\alpha$)/$f$(UV) distribution towards lower values. Indirect estimates of the escape fraction of Lyman continuum photons from H[ii]{} regions determined from the ionization of the diffuse gas by Zurita et al. (2000) led to values of $\sim$ 50% in spiral discs. [**However, as pointed out by these authors, this has to be taken as an upper limit to the photons which escape from the galaxy since many of these ionizing photons will be absorbed by the diffuse interstellar medium so they will not escape.**]{} Using a similar technique, Bland-Hawthorn & Maloney (1999) estimated, from the H$\alpha$ emission of the Magellanic stream, that the escape fraction of the Milky Way is $\approx 6$%. More direct measurements (i.e. based on the observation of the Lyman continuum photons and not on the effect of the ionization), less dependent on geometrical effects, have shown that the escape of Lyman continuum photons from nearby starburst galaxies into the intergalactic medium is probably less than $\approx 10$% (e.g. Leitherer et al. 1995, Heckman et al. 2001, Deharveng et al. 2001). This effect is expected to be even less important in normal galaxies than in starbursts, thus it can be discarded as the main responsible for the $f$(H$\alpha$)/$f$(UV) discrepancy.
Absorption of Lyman continuum photons by dust
---------------------------------------------
Models of galaxy evolution usually assume that all Lyman continuum photons produce the ionization of one hydrogen atom, contributing to the H$\alpha$ flux. However, if dust is mixed with gas in the star formation regions, only a fraction $f'$ of the Lyman continuum photons will encounter an hydrogen atom, the remaining $(1 - f')$ being absorbed by the dust grains mixed with the ionized gas. This effect, proposed by Inoue et al. (2000) should be properly taken into account to evaluate the energy budget of the star formation regions, thus to calibrate the SFRs of galaxies from Far-IR fluxes. Moreover it also produces a significant shift of the observed $f$(H$\alpha$)/$f$(UV) ratio with respect to the model predictions. It has been shown by Hirashita et al. (2001) that the absorption of UV photons by dust should not depend much on metallicity, so we can safely assume that this effect will affect in a similar manner all galaxies in our sample. An average value of $f' \approx 0.57$ was found by Hirashita et al. (2003) for a sample of galaxies similar to ours, assuming approximately constant SFRs over the last $\approx
10^{8}$ yr. When applying this result to our sample of galaxies we obtain an almost perfect agreement between the observed and expected values of the $f$(H$\alpha$)/$f$(UV) ratio.
Non constant SFRs \[nonconsta\]
-------------------------------
Galaxies with normal (i.e. “a la Sandage”) SFH, for any $\tau$ (fig. 1) can be assumed to have “constant” star formation over the last $10^{8}$ yr. In these conditions the expected $f$(H$\alpha$)/$f$(UV) is almost constant. However, Table \[hauvmodels\] shows that for fixed IMFs or metallicities, non negligible differences of $f$(H$\alpha$)/$f$(UV) are found for different SFHs. It is then worthwhile to evaluate the consequences of a non-constant SFH on the $f$(H$\alpha$)/$f$(UV) ratio, which was shown in Table \[hauvmodels\] to produce variations on this quantity. A non-constant SFH cannot be discarded if bursts of star formation occurred along the evolution of galaxies. Such events are very likely to have taken place in clusters of galaxies because of galaxy-galaxy and galaxy-IGM interactions.
Fig. \[mode\] shows the effect on the $f$(H$\alpha$)/$f$(UV) ratio of instantaneous bursts of star formation superposed to the normal evolution assumed “a la Sandage”, with different values of $\tau$. We have represented bursts of intensity 10 and 100 times the expected SFRs for each value of $\tau$. One important point is that the changes in the $f$(H$\alpha$)/$f$(UV) ratio are insensitive to $\tau$. The plot shows a significant increase of $f$(H$\alpha$)/$f$(UV) due to the production of stars with $M \geq 8$M$_{\odot}$ responsible of the H$\alpha$ emission in the first $3 \times 10^{6}$ yr (region [**a**]{}), followed by a steep decrease as the burst fades away (region [**b**]{}). Some 10$^{8}$ yr later, $f$(H$\alpha$)/$f$(UV) recovers its value previous to the burst (region [**c**]{}). The amplitude of both the increase and the decrease of $f$(H$\alpha$)/$f$(UV) is larger for stronger bursts. We remark that values of $f$(H$\alpha$)/$f$(UV) significantly lower than the one predicted by models for constant SFR (end of region [**b**]{} in the plot) are reached only in the case of bursts of intensities $\geq 10$ times the normal current SFRs of galaxies. The presence of the strong burst of star formation can thus account for both a shift and an increase of the dispersion of the $f$(H$\alpha$)/$f$(UV) distribution.
The temporal dependence of the star formation induced by galaxy interactions is far more complex that just a single instantaneous burst (see Noguchi 1991 and Mihos et al. 1991,1992). The period over which the star formation is enhanced can last for about $10^{8}$ yr. To show the influence of a more complex pattern of star formation on the $f$(H$\alpha$)/$f$(UV) ratio we show in fig. \[mode3\] the $f$(H$\alpha$)/$f$(UV) evolution for a burst of $10^{8}$ yr duration, overimposed to a normal evolution SFH. In this case, after the first $10^{7}$ yr, the $f$(H$\alpha$)/$f$(UV) ratio decreases slowly with time and, by $10^{8}$ yr, it converges to the value for normal galaxies.
Once we know the effect of a single star formation burst on the $f$(H$\alpha$)/$f$(UV) ratio of a single galaxy, we simulate the expected distributions of $f$(H$\alpha$)/$f$(UV) for a population of galaxies following a SFH “a la Sandage”, with several overimposed star formation episodes randomly distributed in time. Three parameters are let free in each simulation:
- the time over which all galaxies experience a burst of star formation: $3 \times 10^{6}$, $10^{8}$ and $10^{9}$ yr, coincident with the timescales of three environmental mechanisms acting on cluster galaxies,
- the duration of the bursts: instantaneous and $10^{8}$ yr,
- the maximum intensity of the burst: 10, 100 and 1000 times the expected SFR for galaxies following an evolution “a la Sandage” at $t = 13$ Gyr.
In order to reproduce a more realistic variety of burst intensities we also produced simulations in which the maximum intensity of the bursts was randomly chosen between 0 and a certain value (namely 10, 100 and 1000 times the normal SFR). By combining the above cases we have reproduced the values of the $f$(H$\alpha$)/$f$(UV) ratio for 36 scenarios. An error budget consistent with the one of our dataset (detailed in Appendix B) was included in the simulated H$\alpha$ and UV fluxes. For each scenario up to 100 simulations were run. The comparison of the resulting $f$(H$\alpha$)/$f$(UV) distributions with the observed one are reported in Table \[tabsimu\].
Let us first summarize the scenarios with instantaneous star formation bursts:
- For scenarios 1 to 6, where the star formation episodes are spread along the last $3 \times 10^{6}$ yr, the average $\log$$f$(H$\alpha$)/$f$(UV) increases from the theoretical value 1.43, up to 1.91 times. All these scenarios produce distributions of $\log$$f$(H$\alpha$)/$f$(UV) non consistent with the observed one, as reflected by the unacceptably high $\chi_{\mbox{\scriptsize n}}^{2}$.
- Scenarios 7 to 12, which correspond to star formation episodes spread along $10^{8}$ yr, show average values of $f$(H$\alpha$)/$f$(UV) lower than the expected value (strongly depending on the intensity of the star formation episodes). This result is expected since, as shown in fig. \[mode\], after a strong star formation burst the value of $\log$$f$(H$\alpha$)/$f$(UV) is below the expected value and it takes about $10^{8}$ yr to recover. It is remarkable that scenarios 9 and 12, which corresponds to star formation bursts of the order of 100 (1000) times the current SFRs, produce $f$(H$\alpha$)/$f$(UV) distributions consistent with the observed one.
- Finally, scenarios 13 to 18, for which the star formation episodes are spread along $10^{9}$ yr, yield average values of $f$(H$\alpha$)/$f$(UV) slightly lower than the theoretical one. This means that the effect of an instantaneous burst of star formation is shorter than $10^{9}$ yr. None of these scenarios provide $f$(H$\alpha$)/$f$(UV) consistent with the observed one.
Concerning the scenarios with bursts of $10^{8}$ yr duration:
- The behavior of scenarios 19 to 30 is similar to that of scenarios 1 to 6, that is, their average $f$(H$\alpha$)/$f$(UV) ratio is enhanced with respect to the value corresponding to normal galaxies and they show a high average value of $\chi_{\mbox{\scriptsize n}}^{2}$.
- Scenarios 31 to 36 behave like scenarios 7 to 12, with only one of them (namely scenario 33) been fairly consistent with the observed distribution.
Summarizing, we note that bursts of intensities about 100 times the expected SFRs for normal galaxies are required to obtain a $f$(H$\alpha$)/$f$(UV) distribution consistent with the one observed in our sample galaxies, under the assumption that non constant star formation is the only mechanism governing the $f$(H$\alpha$)/$f$(UV) ratio.
Discussion
==========
In the previous section we explored some physical mechanisms that could possibly explain the inconsistency between the observed distribution of $f$(H$\alpha$)/$f$(UV) and the theoretical value. We reached the conclusion that, while the dispersion is consistent with the observational uncertainties, the difference between the average observed value of $f(\mbox{H}\alpha)/f(\mbox{UV})$ and the theoretical value is real and might have physical implications. Of all the explored possibilities only two seem able to reproduce the observed $f$(H$\alpha$)/$f$(UV) distribution, namely: non constant SFRs over the last $10^{8}$ yr and the absorption of Lyman continuum photons by dust within star forming regions.
The non constant SFR hypothesis has been used by Sullivan et al. (2000, 2001) to explain the discrepancy between the observed and the theoretical H$\alpha$ and UV fluxes in a sample of UV selected galaxies. For galaxies in clusters, where interactions are likely to take place, the “non constant” star formation scenario seems the realistic one. However, since not all cluster galaxies are affected by the environment in the same way, we split our sample in several subsamples in order study the behavior of the $f$(H$\alpha$)/$f$(UV) distributions for galaxies in various evolutionary stages:
- Galaxies showing clear morphological disturbances are known to be experiencing recent interactions with close neighbors or with the IGM, and in most cases an enhancement of their SFRs is reflected on their H$\alpha$ fluxes (timescale for production of Lyman continuum photons $\leq 10^{7}$ yr). Three galaxies of our sample belong to this category: CGCG 097-073 and CGCG 097-079 (Gavazzi et al. 2001a) and CGCG 097-087 (Gavazzi et al. 2001b). These galaxies will be referred hereafter as the [**[*“interacting”*]{}**]{} subsample.
- Galaxies with asymmetric H[i]{} profiles are known to have experienced interactions on timescales of $\simeq 5 \times 10^{8}$ yr (Gavazzi 1989), corresponding to the timescale necessary for redistributing the neutral gas throughout the disk. In our sample, these are the galaxies labeled “A” in last column of Table \[tabla\]. Hereafter, we will refer to them as the [**[*“asymmetric”*]{}**]{} subsample[^1]. Given that the timescale for removing the H[i]{} asymmetries is usually larger than the timescale over which the effects of the interactions are apparent (i.e. close galaxy-galaxy interactions), the enhancement of the SFRs for these galaxies is expected to be lower than for the [*“interacting”*]{} ones.
- Another measure of the interaction with the environment is provided by the HI deficiency parameter. As galaxies approach the cluster center they loose their peripheral gas envelope due to ram-pressure stripping, preventing their subsequent star formation. The timescale for this process is $\simeq 10^{9}$ yr, which approximately corresponds to the cluster crossing time. We consider as [**[*“deficient”*]{}**]{} galaxies those with $def(\mbox{H{\sc i}}) \geq 0.4$. We exclude from this subsample deficient galaxies with asymmetric profiles, in order separate the effects of H[i]{} deficiency from interactions.
- Finally, we define a [**[*“reference”*]{}**]{} sample of galaxies for which no traces of interaction with the cluster environment are found: they have a normal H[i]{} content (i.e., $def(\mbox{H{\sc
i}}) < 0.4$) and do not show neither clear signatures of interactions nor asymmetric H[i]{} profiles. These galaxies will be considered hereafter as “normal” galaxies.
Table \[tabmedia\] lists the average values of the $f$(H$\alpha$)/$f$(UV) and $b_{\mbox{\scriptsize obs}}/b_{\mbox{\scriptsize model}}$ ratios for each different subsample. Figs. \[ha\_uv\_hidef\] and \[dbirth\_hidef\] show the $f$(H$\alpha$)/$f$(UV) and $b_{\mbox{\scriptsize obs}}/b_{\mbox{\scriptsize model}}$ ratios of the individual galaxies of the three subsamples vs. their H[i]{} deficiency. The [*“reference”*]{} galaxies show $\left< \log b_{\mbox{\scriptsize obs}}/b_{\mbox{\scriptsize model}} \right> = 0$, meaning that their recent star formation activity coincides with the expected one. In addition, $\left< \log f(\mbox{H}\alpha)/f(\mbox{UV}) \right> = 1.11$, which does not correspond to the theoretical value of 1.43 predicted by synthesis models. Given that these galaxies are selected for their normal H[i]{} content and no traces of interactions, we take this value as a reference value for normal star forming galaxies. Absorption of $\simeq 45$% Lyman continuum photons by dust within H[ii]{} regions should account for the discrepancy between the observed and theoretical value of the $f$(H$\alpha$)/$f$(UV) ratio for normal galaxies.
Moving on to galaxies perturbed by the cluster environment, we find that the [*“interacting”*]{} and [*“asymmetric”*]{} galaxies show values of $\log$$f$(H$\alpha$)/$f$(UV) 0.14 dex higher than [*“reference”*]{} galaxies. As we showed in Section 3.5, the presence of star formation bursts is likely to produce such an enhancement. From the $b_{\mbox{\scriptsize obs}}/b_{\mbox{\scriptsize model}}$ ratio we estimate that the intensity of the star formation activity is at present 3.5 times higher for the [*“interacting”*]{} galaxies compared to the [*asymmetric*]{} and the [*“reference”*]{} ones. Given that the [*“interacting”*]{} galaxies are presently undergoing an interaction, the age of the burst is $\approx 10^{6}$ yr, thus the increase of the star formation activity is maximal (see fig. \[mode\]). We thus expect that, consistently with model predictions (Fujita 1998), galaxy–galaxy or galaxy–IGM interactions in clusters can induce bursts of star formation able to increase by up to a factor of $\approx 4$ the expected SFR of normal late-type galaxies.
Finally, we analyze the behavior of the [*“deficient”*]{} galaxies. These galaxies have been shown to have lower than expected star formation activity as measured by the $b$ parameter (Boselli et al. 2001). However, we find for them higher $f$(H$\alpha$)/$f$(UV) and of $b_{\mbox{\scriptsize obs}}/b_{\mbox{\scriptsize model}}$ than for the [*“reference”*]{} galaxies. This apparent contradiction might be due to selection effects. To illustrate this point we show in Table \[bias\] the average values of $EW(\mbox{H}\alpha + \mbox{[N{\sc ii}]})$ for subsamples of galaxies satisfying the various observational constraints, separately for deficient and non deficient galaxies. It appears that, as more observational constraints are applied, the resulting average $EW(\mbox{H}\alpha + \mbox{[N{\sc ii}]})$ tends to increase, biasing towards more actively star forming galaxies. For the non deficient galaxies, this bias affects the average $EW(\mbox{H}\alpha + \mbox{[N{\sc ii}]})$ by less than 30%. For the deficient galaxies, by imposing the condition for H$\beta$ line detection, the estimate of $EW(\mbox{H}\alpha + \mbox{[N{\sc ii}]})$ results doubled. The [*“reference”*]{}, [*“interacting”*]{} and [*“asymmetric”*]{} samples are less affected by this selection bias because they contain non-deficient objects.
Conclusions
===========
The $f$(H$\alpha$)/$f$(UV) ratio of cluster galaxies is analyzed in this paper as a promising tool to estimate if the the star formation history of galaxies has remained constant on timescales of $\simeq 10^{8}$ yr. The observed $f$(H$\alpha$)/$f$(UV) distribution is compared to the one predicted by models of galaxies, assuming a continuum SFH. The dispersion of the observed $f$(H$\alpha$)/$f$(UV) distribution is consistent with the one expected from the observational uncertainties. We find a systematic negative difference between the average observed value and the model predictions. We discuss some mechanisms that could possibly produce such an observed difference and we highlight the two most likely ones: the absorption and, in a minor way, the escape of Lyman continuum photons and the occurrence of star formation bursts overimposed to a smooth SFH.
The $f$(H$\alpha$)/$f$(UV) distribution is considered for different galaxy subsamples, each of them comprising galaxies in different evolutionary stages, possibly induced by the cluster environment. The [*“reference”*]{} unperturbed galaxies have $f$(H$\alpha$)/$f$(UV) lower by 0.34 dex on average than the one predicted by the models. We suggest that absorption (and to a lesser extent escape) of Lyman continuum photons causes the observed discrepancy. We estimate that about 45% of the Lyman continuum photons are absorbed by dust in the star forming regions before ionization, consistently with the estimate of Hirashita et al. (2003) on similar objects.
When galaxies with signatures of recent or past interactions with the cluster environment ([*“interacting”*]{} and [*“asymmetric”*]{}) are considered, we find that their $f$(H$\alpha$)/$f$(UV) ratio is slightly higher than the one of [*“reference”*]{} galaxies. Even though the absorption of Lyman continuum photons is taken into account, the observed $f$(H$\alpha$)/$f$(UV) ratio can be reconciled to the predicted one only assuming that these objects underwent bursts of star formation of intensity $\sim$ 100 times larger than normal, as intense as Arp 220. Objects of this kind are however not presently observed in nearby clusters.
The present observational uncertainties on both the H$\alpha$ and UV fluxes are still too large to allow disentangling the effects of recent star formation bursts from those of absorption of Lyman continuum photons. However we stress the potentiality of the proposed H$\alpha$/UV method for studying the recent history of star formation in late type galaxies, once improvements in modeling the radiation transfer through the dust in star forming regions will be achieved and more precise UV and Far-IR photometry will be available. This will soon become a reality after the GALEX and ASTRO-F experiments will perform their all sky surveys, providing $\Delta f(\mbox{UV})$ and $\Delta
f(\mbox{Far-IR}) \approx 10$%.
We thank Veronique Buat and Jean Michel Deharveng for interesting comments and suggestions. JIP acknowledges the Fifth Framework Program of the EU for a Marie Curie Postdoctoral Fellowship. This research has made use of the NASA/IPAC Extragalactic Database (NED) and of the NASA/ IPAC Infrared Science Archive, which are operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration. We also acknowledge the GOLD Mine Database, operated by the Universitá degli Studi di Milano-Bicocca.
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$$
[lccccc]{} Name & $\log f(\mbox{H}\alpha)$ & $\log f(\mbox{UV})$ & $\log L_{\mbox{\scriptsize H}}$ & $def(\mbox{H{\sc i}})$ & H[i]{} asymm.\
& (erg s$^{-1}$ cm$^{-2}$) & (erg s$^{-1}$ cm$^{-2}$ Å$^{-1}$) & (L$_{\odot}$) & &\
VCC 25 & $-$11.59 & $-$12.75 & 10.39 & $-$0.21 & ?\
VCC 66 & $-$11.48 & $-$12.89 & 10.22 & $-$0.20 & ?\
VCC 89 & $-$11.74 & $-$12.78 & 10.65 & $-$0.03 & S\
VCC 92 & $-$11.21 & $-$12.33 & 10.99 & 0.33 & ?\
VCC 131 & $-$12.60 & $-$13.30 & 9.52 & 0.09 & S\
VCC 157 & $-$11.31 & $-$12.65 & 10.48 & 0.61 & S\
VCC 221 & $-$11.94 & $-$13.12 & 9.88 & 0.41 & S\
VCC 307 & $-$10.73 & $-$11.95 & 10.94 & 0.01 & S\
VCC 318 & $-$12.48 & $-$13.68 & 9.18 & $-$0.13 & S\
VCC 382 & $-$12.05 & $-$12.59 & 10.65 & $-$0.32 & S\
VCC 459 & $-$12.58 & $-$13.61 & 8.73 & $-$0.07 & S\
VCC 491 & $-$11.75 & $-$12.89 & 9.42 & $-$0.29 & S\
VCC 508 & $-$10.72 & $-$11.86 & 10.98 & $-$0.06 & S\
VCC 552 & $-$12.23 & $-$13.29 & 8.99 & $-$0.42 & S\
VCC 664 & $-$12.17 & $-$13.38 & 8.92 & 0.62 & S\
VCC 667 & $-$12.75 & $-$13.71 & 9.78 & 0.58 & S\
VCC 692 & $-$12.50 & $-$13.36 & 9.64 & 0.66 & S\
VCC 699 & $-$12.37 & $-$13.31 & 9.71 & 0.19 & S\
VCC 787 & $-$12.47 & $-$13.35 & 9.65 & 0.26 & S\
VCC 801 & $-$11.58 & $-$13.12 & 9.97 & $-$0.62 & S\
VCC 827 & $-$12.10 & $-$13.15 & 10.14 & 0.08 & S\
VCC 836 & $-$11.51 & $-$12.57 & 10.54 & 0.69 & S\
VCC 849 & $-$12.11 & $-$13.36 & 9.79 & 0.41 & S\
VCC 851 & $-$12.18 & $-$13.79 & 9.80 & 0.23 & A\
VCC 865 & $-$11.86 & $-$13.12 & 9.69 & 0.38 & S\
VCC 873 & $-$11.15 & $-$12.79 & 10.39 & 0.63 & S\
VCC 905 & $-$12.63 & $-$13.59 & 9.66 & 0.35 & S\
VCC 912 & $-$12.19 & $-$13.26 & 9.88 & 0.99 & S\
VCC 921 & $-$11.89 & $-$13.05 & 9.64 & 0.59 & S\
VCC 938 & $-$12.09 & $-$13.26 & 9.74 & 0.36 & S\
VCC 939 & $-$12.32 & $-$13.18 & 9.87 & 0.24 & S\
VCC 957 & $-$11.64 & $-$12.90 & 9.91 & 0.02 & ?\
VCC 971 & $-$12.40 & $-$13.55 & 9.50 & 0.20 & ?\
VCC 979 & $-$11.98 & $-$13.14 & 10.45 & 1.17 & S\
VCC 980 & $-$12.52 & $-$13.55 & 8.77 & 0.67 & ?\
VCC 1002 & $-$11.65 & $-$13.21 & 10.22 & 0.47 & S\
VCC 1091 & $-$12.22 & $-$13.47 & 8.86 & $-$0.35 & S\
VCC 1118 & $-$12.08 & $-$13.27 & 10.08 & 0.51 & S\
VCC 1189 & $-$12.55 & $-$13.65 & 9.25 & 0.34 & A\
VCC 1193 & $-$12.43 & $-$13.79 & 9.28 & $-$0.05 & S\
VCC 1205 & $-$12.41 & $-$13.02 & 9.73 & $-$0.03 & S\
VCC 1290 & $-$12.20 & $-$13.18 & 9.91 & 0.05 & S\
VCC 1379 & $-$12.09 & $-$13.07 & 9.84 & 0.15 & S\
VCC 1393 & $-$12.22 & $-$13.32 & 9.47 & 0.23 & S\
VCC 1401 & $-$10.76 & $-$12.20 & 11.18 & 0.55 & S\
VCC 1450 & $-$12.01 & $-$12.99 & 9.47 & 0.54 & S\
VCC 1508 & $-$11.59 & $-$12.75 & 9.93 & $-$0.26 & S\
VCC 1516 & $-$11.77 & $-$13.19 & 9.85 & 0.80 & S\
VCC 1532 & $-$12.35 & $-$13.44 & 9.55 & 0.82 & S\
VCC 1554 & $-$11.28 & $-$12.56 & 9.90 & $-$0.37 & S\
$$\end{table}
\clearpage
\addtocounter{table}{-1}
\begin{table}
\caption[]{Continued.}
% \label{tabla}$$
[lccccc]{} Name & $\log f(\mbox{H}\alpha)$ & $\log f(\mbox{UV})$ & $\log L_{\mbox{\scriptsize H}}$ & $def(\mbox{H{\sc i}})$ & H[i]{} asymm.\
& (erg s$^{-1}$ cm$^{-2}$) & (erg s$^{-1}$ cm$^{-2}$ Å$^{-1}$) & (L$_{\odot}$) & &\
VCC 1575 & $-$11.08 & $-$12.27 & 10.76 & 0.19 & S\
VCC 1588 & $-$12.04 & $-$13.05 & 10.05 & 0.68 & S\
VCC 1678 & $-$12.45 & $-$13.62 & 8.81 & $-$0.06 & S\
VCC 1699 & $-$12.90 & $-$13.75 & 8.83 & 0.04 & S\
VCC 1725 & $-$12.70 & $-$13.74 & 8.97 & 0.55 & S\
VCC 1811 & $-$12.31 & $-$13.21 & 9.85 & 0.23 & S\
VCC 1929 & $-$12.63 & $-$13.34 & 9.51 & 0.35 & S\
VCC 1943 & $-$11.43 & $-$12.95 & 10.26 & 0.25 & S\
VCC 1972 & $-$11.32 & $-$12.66 & 10.50 & 0.27 & S\
VCC 1987 & $-$11.23 & $-$12.31 & 10.66 & $-$0.29 & A\
VCC 2058 & $-$11.15 & $-$12.91 & 10.35 & 0.90 & S\
CGCG 043-034 & $-$11.42 & $-$12.86 & 9.79 & $-$0.29 & S\
CGCG 043-071 & $-$11.32 & $-$12.68 & 10.12 & $-$0.75 & S\
CGCG 043-093 & $-$11.43 & $-$12.64 & 10.27 & $-$0.07 & S\
CGCG 097-062 & $-$12.97 & $-$14.30 & 10.10 & 0.31 & A\
CGCG 097-068 & $-$11.74 & $-$13.43 & 10.78 & $-$0.14 & S\
CGCG 097-073 & $-$12.77 & $-$14.07 & 10.00 & 0.16 & A\
CGCG 097-079 & $-$12.70 & $-$13.87 & 10.02 & 0.25 & A\
CGCG 097-087 & $-$11.97 & $-$13.25 & 10.88 & 0.19 & A\
CGCG 097-091 & $-$12.69 & $-$13.67 & 10.85 & $-$0.18 & S\
CGCG 097-120 & $-$12.13 & $-$13.75 & 11.06 & 0.90 & S\
CGCG 100-004 & $-$11.33 & $-$12.60 & 10.52 & $-$0.24 & S\
CGCG 119-029 & $-$12.13 & $-$13.34 & 10.75 & $-$0.30 & S\
CGCG 119-041 & $-$12.83 & $-$13.74 & 10.51 & 0.30 & S\
CGCG 119-043 & $-$12.70 & $-$13.94 & 10.07 & 0.29 & S\
CGCG 119-046 & $-$12.15 & $-$13.39 & 10.32 & $-$0.22 & S\
CGCG 119-047 & $-$12.42 & $-$13.43 & 10.42 & $-$0.61 & S\
CGCG 119-053 & $-$13.00 & $-$13.98 & 10.12 & $-$0.37 & S\
CGCG 119-054 & $-$12.44 & $-$13.91 & 10.70 & — & ?\
CGCG 119-059 & $-$13.13 & $-$13.93 & 9.66 & 0.14 & S\
CGCG 119-068 & $-$12.66 & $-$13.84 & 10.40 & $-$0.27 & S\
CGCG 119-085 & $-$13.61 & $-$14.19 & 10.38 & $-$0.17 & S\
CGCG 127-049 & $-$12.52 & $-$13.88 & 10.51 & 0.32 & S\
CGCG 160-020 & $-$13.00 & $-$13.82 & 9.98 & 0.27 & S\
CGCG 160-026 & $-$12.96 & $-$14.11 & 10.31 & 0.23 & A\
CGCG 160-055 & $-$12.56 & $-$13.41 & 10.96 & 0.49 & A\
CGCG 160-058 & $-$12.65 & $-$14.00 & 10.58 & 0.40 & S\
CGCG 160-067 & $-$12.62 & $-$13.86 & 10.06 & $-$0.05 & S\
CGCG 160-076 & $-$12.88 & $-$14.08 & 9.87 & $-$0.35 & S\
CGCG 160-086 & $-$12.84 & $-$14.09 & 10.05 & 0.76 & ?\
CGCG 160-088 & $-$12.06 & $-$14.04 & 10.88 & 0.42 & S\
CGCG 160-106 & $-$12.55 & $-$13.90 & 10.99 & 0.54 & ?\
CGCG 160-108 & $-$12.92 & $-$14.09 & 10.16 & — & ?\
CGCG 160-128 & $-$12.73 & $-$13.91 & 9.86 & — & ?\
CGCG 160-139 & $-$12.60 & $-$13.81 & 10.02 & $-$0.19 & A\
CGCG 160-213 & $-$12.72 & $-$13.81 & 10.10 & — & ?\
CGCG 160-252 & $-$12.49 & $-$13.55 & 10.38 & 0.56 & S\
CGCG 160-260 & $-$12.47 & $-$13.64 & 11.21 & 0.81 & S\
$$\end{table}
\clearpage
\begin{table}
\caption[]{Dependence of the $f$(H$\alpha$)/$f$(UV) ratio on the IMF parameters, metallicity and star formation history: (1) Evolutionary synthesis code; (2) Metallicity; (3) IMF slope; (4)
Lower limit for the IMF; (5) Upper limit for the IMF; (6) Time interval over which the SFR is considered constant; (7) $\log$ of the $f$(H$\alpha$)/$f$(UV) ratio.}
\label{hauvmodels}$$
[lcccccc]{} Source & $Z$ & IMF & $M_{\mbox{\scriptsize low}}$ & $M_{\mbox{\scriptsize up}}$ & $t$ & $\log$ $f$(H$\alpha$)/$f$(UV)\
PEGASE2 & 0.0004 & Salpeter & 1 & 100 & $10^{7}$ & 1.79\
PEGASE2 & 0.004 & Salpeter & 1 & 100 & $10^{7}$ & 1.69\
PEGASE2 & 0.02 & Salpeter & 1 & 100 & $10^{7}$ & 1.51\
PEGASE2 & 0.05 & Salpeter & 1 & 100 & $10^{7}$ & 1.35\
Starburst99 & 0.001 & Salpeter & 1 & 100 & $10^{7}$ & 1.75\
Starburst99 & 0.004 & Salpeter & 1 & 100 & $10^{7}$ & 1.67\
Starburst99 & 0.008 & Salpeter & 1 & 100 & $10^{7}$ & 1.63\
Starburst99 & 0.020 & Salpeter & 1 & 100 & $10^{7}$ & 1.57\
Starburst99 & 0.040 & Salpeter & 1 & 100 & $10^{7}$ & 1.51\
PEGASE2 & 0.0004 & Salpeter & 1 & 100 & $10^{8}$ & 1.54\
PEGASE2 & 0.004 & Salpeter & 1 & 100 & $10^{8}$ & 1.46\
PEGASE2 & 0.02 & Salpeter & 1 & 100 & $10^{8}$ & 1.33\
PEGASE2 & 0.05 & Salpeter & 1 & 100 & $10^{8}$ & 1.20\
Starburst99 & 0.001 & Salpeter & 1 & 100 & $10^{8}$ & 1.50\
Starburst99 & 0.004 & Salpeter & 1 & 100 & $10^{8}$ & 1.44\
Starburst99 & 0.008 & Salpeter & 1 & 100 & $10^{8}$ & 1.42\
Starburst99 & 0.020 & Salpeter & 1 & 100 & $10^{8}$ & 1.38\
Starburst99 & 0.040 & Salpeter & 1 & 100 & $10^{8}$ & 1.34\
Starburst99 & 0.001 & Salpeter & 1 & 30 & $10^{8}$ & 1.04\
Starburst99 & 0.004 & Salpeter & 1 & 30 & $10^{8}$ & 0.93\
Starburst99 & 0.008 & Salpeter & 1 & 30 & $10^{8}$ & 0.88\
Starburst99 & 0.020 & Salpeter & 1 & 30 & $10^{8}$ & 0.80\
Starburst99 & 0.040 & Salpeter & 1 & 30 & $10^{8}$ & 0.78\
Starburst99 & 0.001 & Miller-Scalo & 1 & 100 & $10^{8}$ & 0.95\
Starburst99 & 0.004 & Miller-Scalo & 1 & 100 & $10^{8}$ & 0.89\
Starburst99 & 0.008 & Miller-Scalo & 1 & 100 & $10^{8}$ & 0.86\
Starburst99 & 0.020 & Miller-Scalo & 1 & 100 & $10^{8}$ & 0.82\
Starburst99 & 0.040 & Miller-Scalo & 1 & 100 & $10^{8}$ & 0.78\
$$\end{table}
\clearpage
\begin{table}
\caption[]{The simulated scenarios for
instantaneous bursts of star formation: (1) Identificator of the model; (2) Time interval over which all simulated galaxies
experience a burst of star formation (yr); (3) Duration of the burst (yr); (4) Maximum intensity of the burst in units of the
expected SFR of galaxies following an evolution ``a la Sandage'' at $t = 13$~Gyr; (5) Average $\chi^{2}_{\mbox{\scriptsize n}}$
between the observed and each of the simulated distributions; (6) Average value of $f$(H$\alpha$)/$f$(UV) for 100
simulated distributions; (7) Average of $\sigma$$f$(H$\alpha$)/$f$(UV) for 100
simulated distributions.}
\label{tabsimu}$$
[rcccccc]{} Id. & $\Delta t$ & Duration & Intensity & $\left<\chi^{2}_{\mbox{\scriptsize n}}\right>$ & $\left<\log
f(\mbox{H}\alpha)/f(\mbox{UV})\right>$ & $\left< \sigma
\right>$\
1 & $3 \times 10^{6}$ & Inst. & 10 for all galaxies & $7.39\pm0.98$ & 1.71 & 0.22\
2 & $3 \times 10^{6}$ & Inst. & Random between 0 and 10 & $6.20\pm0.98$ & 1.58 & 0.26\
3 & $3 \times 10^{6}$ & Inst. & 100 for all galaxies & $6.17\pm1.44$ & 1.90 & 0.23\
4 & $3 \times 10^{6}$ & Inst. & Random between 0 and 100 & $6.94\pm1.13$ & 1.81 & 0.24\
5 & $3 \times 10^{6}$ & Inst. & 1000 for all galaxies & $6.04\pm1.47$ & 1.91 & 0.23\
6 & $3 \times 10^{6}$ & Inst. & Random between 0 and 1000 & $5.88\pm1.25$ & 1.91 & 0.26\
7 & $10^{8}$ & Inst. & 10 for all galaxies & $2.62\pm0.70$ & 1.41 & 0.26\
8 & $10^{8}$ & Inst. & Random between 0 and 10 & $2.71\pm0.61$ & 1.39 & 0.25\
9 & $10^{8}$ & Inst. & 100 for all galaxies & $0.87\pm0.38$ & 1.25 & 0.28\
10 & $10^{8}$ & Inst. & Random between 0 and 100 & $1.53\pm0.48$ & 1.37 & 0.26\
11 & $10^{8}$ & Inst. & 1000 for all galaxies & $3.03\pm0.74$ & 0.92 & 0.42\
12 & $10^{8}$ & Inst. & Random between 0 and 1000 & $1.03\pm0.38$ & 1.12 & 0.39\
13 & $10^{9}$ & Inst. & 10 for all galaxies & $2.67\pm0.62$ & 1.35 & 0.23\
14 & $10^{9}$ & Inst. & Random between 0 and 10 & $2.84\pm0.66$ & 1.41 & 0.22\
15 & $10^{9}$ & Inst. & 100 for all galaxies & $2.46\pm0.66$ & 1.35 & 0.23\
16 & $10^{9}$ & Inst. & Random between 0 and 100 & $2.68\pm0.74$ & 1.34 & 0.23\
17 & $10^{9}$ & Inst. & 1000 for all galaxies & $1.59\pm0.53$ & 1.31 & 0.34\
18 & $10^{9}$ & Inst. & Random between 0 and 1000 & $1.84\pm0.45$ & 1.39 & 0.27\
19 & $3 \times 10^{6}$ & $10^{8}$ & 10 for all galaxies & $7.53\pm0.96$ & 1.73 & 0.26\
20 & $3 \times 10^{6}$ & $10^{8}$ & Random between 0 and 10 & $4.88\pm0.78$ & 1.53 & 0.27\
21 & $3 \times 10^{6}$ & $10^{8}$ & 100 for all galaxies & $6.35\pm1.17$ & 1.88 & 0.25\
22 & $3 \times 10^{6}$ & $10^{8}$ & Random between 0 and 100 & $6.87\pm1.23$ & 1.82 & 0.24\
23 & $3 \times 10^{6}$ & $10^{8}$ & 1000 for all galaxies & $6.05\pm1.31$ & 1.91 & 0.24\
24 & $3 \times 10^{6}$ & $10^{8}$ & Random between 0 and 1000 & $6.04\pm1.42$ & 1.92 & 0.24\
25 & $10^{8}$ & $10^{8}$ & 10 for all galaxies & $5.48\pm0.83$ & 1.53 & 0.24\
26 & $10^{8}$ & $10^{8}$ & Random between 0 and 10 & $4.98\pm0.78$ & 1.50 & 0.23\
27 & $10^{8}$ & $10^{8}$ & 100 for all galaxies & $5.84\pm0.79$ & 1.55 & 0.22\
28 & $10^{8}$ & $10^{8}$ & Random between 0 and 100 & $5.68\pm0.82$ & 1.53 & 0.23\
29 & $10^{8}$ & $10^{8}$ & 1000 for all galaxies & $5.62\pm0.82$ & 1.59 & 0.27\
30 & $10^{8}$ & $10^{8}$ & Random between 0 and 1000 & $5.58\pm0.88$ & 1.57 & 0.26\
31 & $10^{9}$ & $10^{8}$ & 10 for all galaxies & $1.87\pm0.51$ & 1.37 & 0.26\
32 & $10^{9}$ & $10^{8}$ & Random between 0 and 10 & $2.33\pm0.58$ & 1.38 & 0.25\
33 & $10^{9}$ & $10^{8}$ & 100 for all galaxies & $1.11\pm0.40$ & 1.09 & 0.47\
34 & $10^{9}$ & $10^{8}$ & Random between 0 and 100 & $1.28\pm0.42$ & 1.19 & 0.37\
35 & $10^{9}$ & $10^{8}$ & 1000 for all galaxies & $1.94\pm0.50$ & 0.68 & 0.70\
36 & $10^{9}$ & $10^{8}$ & Random between 0 and 1000 & $1.31\pm0.40$ & 0.88 & 0.66\
$$\end{table}
\clearpage
\begin{table}
\caption[]{Averaged values of $f$(H$\alpha$)/$f$(UV) and $\log
b_{\mbox{\scriptsize obs}} /b_{\mbox{\scriptsize model}}$ for the various analyzed subsamples. Numbers in
parenthesis correspond to one standard deviation.
}
\label{tabmedia}$$
[lcccc]{} Subsample & Num. gal. & $\left< def(\mbox{H{\sc i}}) \right>$ & $\left<\log
f(\mbox{H}\alpha)/f(\mbox{UV})\right>$ & $\left<\log
b_{\mbox{\scriptsize obs}}/b_{\mbox{\scriptsize model}}\right>$\
& [**57**]{} & [**$-$0.03(0.28)**]{} & [**1.11(0.24)**]{} & [**0.00(0.41)**]{}\
[*Interacting*]{} & 3 & 0.20(0.05) & 1.25(0.07) & 0.55(0.41)\
[*Asymmetric*]{} & 6 & 0.11(0.27) & 1.24(0.20) & $0.01$(0.34)\
[*Deficient*]{} & 27 & 0.66(0.18) & 1.25(0.27) & $0.07$(0.50)\
$$
$$
[lccc]{} Observational &\
constraint &\
& All & $def(\mbox{H{\sc i}}) < 0.4$ & $def(\mbox{H{\sc i}}) \geq 0.4$\
None & 17 & 22 & 10\
FIR det. & 19 & 24 & 11\
UV det. & 21 & 28 & 12\
FIR & UV det. & 23 & 31 & 13\
H$\beta$ det. & 27 & 29 & 20\
$$
![ Evolution of the $f$(H$\alpha$)/$f$(UV) ratio with time for galaxies with SFH “a la Sandage”, for values of $\tau = 1,
3.2, 5$ and $15$ Gyr. The Salpeter IMF and solar metallicity are assumed. []{data-label="ha_uv_t"}](fig1.eps){width="17cm"}
![ Histogram of the observed $f$(H$\alpha$)/$f$(UV) ratio for galaxies in our sample. The dashed line corresponds to the average expected value for evolutionary models “a la Sandage”. []{data-label="histo_ha_uv"}](fig2.eps){width="17cm"}
![ $f$(H$\alpha$)/$f$(UV) ratio vs. the Hubble type for all galaxies in our sample. A random value between $-0.4$ and 0.4 has been added to each numerical type to avoid overplotting. []{data-label="ha_uv_type"}](fig3.eps){width="17cm"}
![ Effect of instantaneous bursts of star formation on the $f$(H$\alpha$)/$f$(UV) ratio over a normal evolution “a la Sandage”. The thick continuous line represents unperturbed evolution “a la Sandage” for $3.2 \leq
\tau \leq 15$ Gyr (the thickness of the line accounts for the dispersion of the models). The Salpeter IMF and solar metallicity are assumed. The X axis gives the age of the instantaneous burst, assuming galaxies 13 Gyr old. The dashed (dot dashed) lines correspond to star formation bursts of intensities 10 (100) times the corresponding “a la Sandage” SFR at $t
= 13$ Gyr for $\tau = 3.2$ and $15$ Gyr. []{data-label="mode"}](fig4.eps){width="17cm"}
![ Same as fig. \[mode\] for a burst of $10^{8}$ yr duration. []{data-label="mode3"}](fig5.eps){width="17cm"}
![ Histograms of $f$(H$\alpha$)/$f$(UV) for six different simulations of scenarios 9 and 12 (dashed lines). The observed histogram is given with solid lines.[]{data-label="dt10"}](fig6a.eps "fig:"){width="8.5cm"} ![ Histograms of $f$(H$\alpha$)/$f$(UV) for six different simulations of scenarios 9 and 12 (dashed lines). The observed histogram is given with solid lines.[]{data-label="dt10"}](fig6b.eps "fig:"){width="8.5cm"} ![ Histograms of $f$(H$\alpha$)/$f$(UV) for six different simulations of scenarios 9 and 12 (dashed lines). The observed histogram is given with solid lines.[]{data-label="dt10"}](fig6c.eps "fig:"){width="8.5cm"} ![ Histograms of $f$(H$\alpha$)/$f$(UV) for six different simulations of scenarios 9 and 12 (dashed lines). The observed histogram is given with solid lines.[]{data-label="dt10"}](fig6d.eps "fig:"){width="8.5cm"} ![ Histograms of $f$(H$\alpha$)/$f$(UV) for six different simulations of scenarios 9 and 12 (dashed lines). The observed histogram is given with solid lines.[]{data-label="dt10"}](fig6e.eps "fig:"){width="8.5cm"} ![ Histograms of $f$(H$\alpha$)/$f$(UV) for six different simulations of scenarios 9 and 12 (dashed lines). The observed histogram is given with solid lines.[]{data-label="dt10"}](fig6f.eps "fig:"){width="8.5cm"}
![ The relationship between $\log$ $f$(H$\alpha$)/$f$(UV) and the H[i]{} deficiency. Interacting galaxies are marked with filled dots. Galaxies with asymmetric H[i]{} profiles are labeled with “A”. The short-dashed horizontal line corresponds to the average value of $f$(H$\alpha$)/$f$(UV) for the reference sample. The dashed vertical line corresponds to $def(\mbox{H{\sc i}}) = 0.4$: plusses with $def$(H[i]{}) $\leq 0.4$ represent the reference sample.[]{data-label="ha_uv_hidef"}](fig7.eps){width="17cm"}
![ The relationship between $\log b_{\mbox{\scriptsize obs}}/b_{\mbox{\scriptsize model}}$ and the H[i]{} deficiency. The dashed vertical line corresponds to corresponds to $def(\mbox{H{\sc i}}) = 0.4$. Symbols as in fig. \[ha\_uv\_hidef\].[]{data-label="dbirth_hidef"}](fig8.eps){width="17cm"}
The intensity of the star formation bursts
==========================================
The $b$ parameter is the ratio of the recent to the total SFRs over the whole life of a galaxy as defined by Kennicutt et al. (1994). If the SFR of a galaxy as a function of time is known, then: $$\label{b}
b = \frac{\mbox{SFR}(t,\tau) \times t}{\int_{0}^{t} \mbox{SFR}(t',\tau)dt'}$$ where $t$ is the current epoch and the galaxies are assumed to be formed at $t' = 0$. If a simple exponential SF history is assumed: $$\mbox{SFR}(t,\tau) = \mbox{SFR}_{0} e^{-t/\tau}
\label{exp}$$ the $b$ parameter can be expressed, following Boselli et al. (2001) as: $$b_{\mbox{\scriptsize model}} = \frac{t \times e^{-t/\tau}}{\tau (1 - e^{-t/\tau})}
\label{bmod}$$ These authors also report an empirical relationship: $$\label{tau_h}
\log L_{\mbox{\scriptsize H}} = -2.5 \times \log \tau +12$$ that, together with eq. \[bmod\], provides the link between the $b$ parameter and the $H$-band luminosity of a galaxy, in the case of an exponential SFH of eq. \[exp\]. An independent way to obtain the value of $b$ from purely observational considerations is as following Boselli et al. (2001): $$b_{\mbox{\scriptsize obs}} = \left( \frac{L_{\mbox{\scriptsize H}\alpha}}{10^{41}}\right)
\times 0.26 \times \left( \frac{t}{L_{\mbox{H\scriptsize }}} \right)$$ where $t$ is as in eq. \[b\] in yr, and $L_{\mbox{\scriptsize
H}\alpha}$ and $L_{\mbox{\scriptsize H}}$ are the H$\alpha$ and $H$-band luminosities respectively.
The comparison of $b$ obtained from the average empirical relationship between $\tau$ and $L_{\mbox{\scriptsize H}}$ (i.e. $b_{\mbox{\scriptsize model}}$), and from $L_{\mbox{\scriptsize H}}$ and $L_{\mbox{\scriptsize H}\alpha}$ (i.e. $b_{\mbox{\scriptsize obs}}$) should reflect the deviations from a smooth evolution on timescales of the order of $3 \times 10^{6}$ yr.
The error budget
================
This appendix is aimed at estimating the total error budget of the $f$(H$\alpha$)/$f$(UV) ratio as computed from our data. We adopt the following expression for the $f$(H$\alpha$)/$f$(UV) ratio: $$\log f(\mbox{H}\alpha)/f(\mbox{UV}) = \log f_{0}(\mbox{H}\alpha) \pm
\Delta f_{0}(\mbox{H}\alpha) -
\log \left[ 1 + \frac{I(\mbox{H}\alpha)}{I(\mbox{[N{\sc
ii}]})} \times \frac{1 \pm\Delta I(\mbox{H}\alpha)}{1 \pm\Delta I(\mbox{[N{\sc
ii}]})} \right] - \left(\frac{1}{0.335} - 1 \right) \times$$ $$\times \log \left[
\frac{I(\mbox{H}\alpha)}{I(\mbox{H}\beta)}
\frac{1}{2.87} \times
\frac{1 \pm \Delta I(\mbox{H}\alpha)}{1 \pm
\Delta I(\mbox{H}\beta_{\mbox{\scriptsize emi}} \pm \Delta I(\mbox{H}\beta_{\mbox{\scriptsize abs}})} \right] -
\log f_{0}(\mbox{UV}) \pm \Delta f_{0}(\mbox{UV}) -$$ $$-0.466 -
\log \left[ \frac{f_{0}(\mbox{Far-IR})}{f_{0}(\mbox{UV})} \times \frac{1 \pm \Delta
f_{0}(\mbox{Far-IR})}{1 \pm \Delta f_{0}(\mbox{UV})} \right] -
0.433 \times \log \left[
\frac{f_{0}(\mbox{Far-IR})}{f_{0}(\mbox{UV})} \times \frac{1 \pm \Delta
f_{0}(\mbox{Far-IR})}{1 \pm \Delta f_{0}(\mbox{UV})} \right]^{2}
\label{ecuerror}$$ where,
- $f_{0}$(H$\alpha$), $f$(Far-IR) and $f_{0}$(UV) are the measured integrated luminosities from imaging data in the corresponding passbands,
- $\Delta f_{0}(\mbox{H}\alpha)$, $\Delta f_{0}(\mbox{Far-IR})$ and $\Delta
f_{0}(\mbox{UV})$ are the uncertainties of the H$\alpha$, Far-IR and UV fluxes,
- $I$(H$\alpha$), $I$(H$\beta$) and $I$(\[N[ii]{}\]) are the fluxes of the corresponding emission lines as measured from the optical spectra,
- $\Delta I$(H$\alpha$), $\Delta I$(H$\beta_{\mbox{\scriptsize emi}}$) and $\Delta I$(\[N[ii]{}\]) are the uncertainties on the fluxes of the corresponding emission lines,
- $\Delta I$(H$\beta_{\mbox{\scriptsize abs}}$) is the uncertainty on the flux of the H$\beta$ absorption line.
The formula used to derive the extinction at 2000 Å was taken from Buat et al. (1999). In order to estimate our total error budget, we run Monte-Carlo simulations of the distribution of 56 values with the error budget shown in eq. \[ecuerror\]. The individual sources of uncertainty were assumed to follow a gaussian distribution. The error sources are listed in Table \[formuerr\]. For our simulations we assumed typical values of $f_{0}(\mbox{Far-IR})/f_{0}(\mbox{UV}) = 1$, $I(\mbox{H}\alpha)/I(\mbox{H}\beta) = 3$ and $I(\mbox{[N{\sc
ii}]})/I(\mbox{H}\alpha) = 0.2$. The simulated distributions turned out to be fairly symmetric with typical dispersions of $\sigma = 0.27
\pm 0.03$ dex. The centers of the distributions showed typical variations of $\pm 0.03$ dex.
$$
[lc]{} Uncertainty source & Estimated\
$\Delta f_{0}(\mbox{H}\alpha)$ & 15%\
$\Delta f_{0}(\mbox{Far-IR})$ & 15%\
$\Delta f_{0}(\mbox{UV})$ & 20%\
$\Delta I$(H$\alpha$) & 10%\
$\Delta I$(H$\beta_{\mbox{\scriptsize emi}}$) & 10%\
$\Delta I$(N\[[ii]{}\]) & 15%\
$\Delta I$(H$\beta_{\mbox{\scriptsize abs}}$) & 20%\
$$
[^1]: In order to avoid confusion, we do not include the galaxies from the [*“interacting”*]{} subsample in the [*“asymmetric”*]{} subsample, although these three galaxies show an asymmetric profile.
|
---
abstract: 'This paper develops the singular perturbation theory for a particular discrete-time nonlinear system which models the saturating inductor buck converter using cycle-by-cycle digital control.'
author:
- '[^1]'
title: 'Singular Perturbation Theory for a Finite-Dimensional, Discrete-Time Chi Nonlinear System'
---
Introduction
============
Singular perturbation theory [@tikhonov1952] is a well-known method for studying nonlinear systems with two well-separated time scales. The $\chi$ nonlinear system is a finite-dimensional, discrete-time nonlinear system which can be used to model a broad class of power electronics systems.
Although there exist several discrete versions of singular perturbation theory in the literature [@tzuu1999tac; @bouyekhf1996reduced; @litkouhi1984infinite; @kafri1996stability; @bouyekhf1997analysis], to the best of authors’ knowledge, there is no such theorem which can be directly applied to the $\chi$ nonlinear system. Therefore, in this paper, we establish singular perturbation theory for the $\chi$ nonlinear system.\
System description
==================
The $\chi$ *nonlinear system* is a finite-dimensional, discrete-time nonlinear system whose state-space representation can be expressed as $$\begin{aligned}
\label{orgsysmodel1} x[n+1] &= x[n] + f(\mu z[n]), \\
\label{orgsysmodel2} z[n+1] &= g(x[n], z[n], \mu z[n]),\end{aligned}$$ where $\mu$ is called [*perturbation parameter*]{}.\
Direct quote from [@xiao2019buck]:
4em We assume the following assumptions hold for all $(n,x,z) \in [0,\infty) \times D_x \times D_z$ containing the origin for some domain $D_x \subset \mathbb{R}^n$ and $D_z \subset \mathbb{R}^m$:
the matrix $\partial g / \partial z - I_{m}$ is invertible;
the function $f$ and $g$ are Lipschitz in $(x,z)$ with Lipschitz constant $L_f$ and $L_g$;
$f(0) = 0$, $g(0,0,0) = 0$.
From assumption (a) above and the implicit function theorem [@edwards2012advanced], the equation $z = g(x, z, 0)$ has explicit solution $z = h(x)$. We assume the function $h$ is Lipschitz in $x$ with Lipschitz constant $L_h$. We define the *reduced model* by $$\begin{aligned}
\label{redumodel1}
x_s[n+1] &= x_s[n] + f(\mu h(x_s[n])), \\
z_s[n] &= h(x_s[n]).\end{aligned}$$ The reduced model describes trajectories of $x$ and $z$ which an observer sees in the slow time frame when $\mu$ approaches 0. We define the *boundary-layer model* by $$\begin{aligned}
\label{boundumodel1}
y[n+1] =& g(x_s[n], y[n] \nonumber \\
&+ h(x_s[n]),0) - h(x_s[n]).\end{aligned}$$ where $y[n] = z_f[n] - z_s[n]$. The $z_f[n]$ is the trajectory of $z$ which an observer sees in the fast time frame when $\mu$ approaches 0. The boundary-layer model describes the difference between the trajectory of $z$ which an observer sees in the fast time frame and that in the slow time frame. We note that $y(n) = 0$ is a solution for the boundary-layer model.
Theory
======
The following Theorem \[Theorem:TractoryConvergence\] shows the relationship between the trajectory of the original system and that of the reduced model as well as the boundary-layer model.\
Direct quote from [@xiao2019buck]:
\[Theorem:TractoryConvergence\]
4em If $x_s = 0$ is an exponentially stable equilibrium of system (\[redumodel1\]) and $y = 0$ is an exponentially stable equilibrium of system (\[boundumodel1\]), uniform in $x_s$, then there exists a positive constant $\mu^*$ such that for all $n \ge 0$ and $0< \mu \le \mu^{*}$, the singular perturbation problem (\[orgsysmodel1\]) and (\[orgsysmodel2\]) has a unique solution $x[n,\mu]$, $z[n, \mu]$ on $[0,\infty)$, and $ x[n, \mu] - x_s[n, \mu] = O(\mu), z[n,\mu] - h(x_s[n,\mu]) - y[n] = O(\mu)$ hold uniformly for $n \in [0, \infty)$, where $x_s[n, \mu]$ and $y[n]$ are the solutions of the system (\[redumodel1\]) and system (\[boundumodel1\]). Furthermore, there exists $n_1 > 0$, such that $z[n,\mu] - h(x_s[n]) = O(\mu)$ holds uniformly for $n \in [n_1, \infty)$.
We use mathematic induction to prove [^2] $$\begin{aligned}
\label{eqn:approxx}
\norm{x[n]-x_s[n]} \le \lambda \mu.\end{aligned}$$ Equation (\[eqn:approxx\]) holds when $n=1$ because $$\begin{aligned}
\norm{x[1]-x_s[1]} = \norm{f(\mu z_0)} \le L \norm{z_0} \mu.\end{aligned}$$ We suppose that there exists $\lambda$ satisfying $\norm{x[n]-x_s[n]} \le \lambda \mu $ when $n = k$. The following derivations prove (\[eqn:approxx\]) when $n=k+1$: $$\begin{aligned}
&\norm{x[k+1]-x_s[k+1]} \nonumber \\
= & \, \norm{x[k] + f(\mu z[k])-x_s[k] - f(\mu x_s[k])} \nonumber \\
\le & \, \norm{x[k]-x_s[k]} + L_f \mu \norm{z[k] - x_s[k]} \nonumber \\
\le & \, \lambda \mu + L_f \mu \norm{y(k)} \nonumber \\
= & \, (\lambda + L_f \beta ) \mu.\end{aligned}$$ By defining $ \lambda^* = (\lambda + L_f \beta)$, we show $\norm{x[k+1]-x_s[k+1]} = O(\mu) $.\
By mathematic induction, we conclude that $$\begin{aligned}
\norm{x[n] - x_s[n]} = O(\mu).\end{aligned}$$ Then we prove $z[n,\mu] - h(x_s[n]) - y[n] = O(\mu)$. $$\begin{aligned}
&\norm {z[n] - h(x_s[n]) - y[n]} \nonumber \\
= \; & \|g(x[n-1], z[n-1], \mu(z[n-1]) - h(x_s[n]) \nonumber \\
& - g(x_s[n-1], y[n-1] \nonumber \\
& + h(x_s[n-1]),0) + h(x_s[n-1])\| \nonumber \\
\le \; & \| g(x[n-1], z[n-1], \mu(z[n-1]) \nonumber \\
&- g(x[n-1], z[n-1], 0) \| \nonumber \\
\; & + \; \norm{g(x[n-1], z[n-1], 0) - g(x_s[n-1], z[n-1], 0)} \nonumber \\
\; & + \; \norm{ h(x_s[n-1]) - h(x_s[n])} \nonumber \\
\le \; &L_g \mu \norm{z[n-1]} + L_g \norm{x[n-1]-x_s[n-1]} \nonumber \\
& + L_h L_f \mu \norm{x_s[n-1]} \nonumber \\
\le \; &L_g \mu \; \left(\norm{x_s[n-1]}+ \norm{y[n-1]}\right) \nonumber \\
& +L_g \norm{x[n-1]-x_s[n-1]} + L_h L_f \mu \norm{x_s[n-1]} \nonumber \\
\le \; & \left(L_g( \lambda + \alpha + \beta ) + L_hL_f\alpha \right)\mu.\end{aligned}$$ We conclude that $$\begin{aligned}
z[n,\mu] - h(x_s[n]) - y[n] = O(\mu).\end{aligned}$$ Finally, we prove that there exists $n_1 > 0$, such that $$\begin{aligned}
z[n] - h(x[n]) = O(\mu)\end{aligned}$$ holds uniformly for $n \in [n_1, \infty)$.\
Let $n_1 = -\text{ln}(\mu)/\theta$, for all $n > n_1 $, $$\begin{aligned}
\norm{y[n]} \le \norm{y[n_1]} = \epsilon \norm{y[0]} e^{\theta n_1} = \epsilon \norm{y[0]} \mu.\end{aligned}$$ We proved $\norm{y[n]} = O(\mu)$ for all $n > n_1 $. Therefore $ z[n] - h(x[n]) = O(\mu)$ holds uniformly for $n \in [n_1, \infty)$.
The following Theorem \[Theorem:Stability\] shows the relationship between the stability of the original system and that of the reduced model as well as the boundary-layer model.
\[Theorem:Stability\] There exists $\mu^* > 0$ such that for all $\mu \le \mu^*$, then $x = 0$, $z = 0$ is an exponentially stable equilibrium of the singular perturbation problem (\[orgsysmodel1\]) and (\[orgsysmodel2\]).
The exponential stability of the $x = 0$ of system (\[redumodel1\]) is equivalent to $$\begin{aligned}
\label{eqn:lyapfuncx1} &\gamma_1 \norm{x}^2 \le V(x) \le \gamma_2 \norm{x}^2, \\
\label{eqn:lyapfuncx2} &V(x[n+1]) \le \sigma_1 V(x[n]),\end{aligned}$$ where $\gamma_1>0$, $\gamma_2>0$, $0 < \sigma_1 < 1$. The exponential stability of the equilibrium $y = 0$ of system (\[boundumodel1\]) requires the following equations to hold uniformly in $x$ $$\begin{aligned}
\label{eqn:lyapfuncy1} &\gamma_3 \norm{y}^2 \le W(y) \le \gamma_4 \norm{y}^2,\\
\label{eqn:lyapfuncy2} &W(y[n+1]) \le \sigma_2 W(y[n]),\end{aligned}$$ where $\gamma_3>0$, $\gamma_4>0$, $0 < \sigma_2 < 1$.\
We use $\nu = V + W$ as a Lyapunov function candidate for the system (\[redumodel1\]) and (\[boundumodel1\]). From (\[eqn:lyapfuncx1\]), (\[eqn:lyapfuncx2\]), (\[eqn:lyapfuncy1\]) and (\[eqn:lyapfuncy2\]) $$\begin{aligned}
\label{eqn:lyapfuncz1} &\text{min}\{\gamma_1, \gamma_3\} \norm{[x,y]}^2 \le \nu(x,y)\le \text{max}\{\gamma_2, \gamma_4\} \norm{[x,y]}^2, \\
\label{eqn:lyapfuncz2} &\nu(x[n+1], y[n+1]) \le \text{max}\{\sigma_1, \sigma_2\}\nu(x[n], y[n]).\end{aligned}$$ From (\[eqn:lyapfuncz1\]) and (\[eqn:lyapfuncz2\]), the equilibrium $[x,y] = 0$ of the systems (\[redumodel1\]) and (\[boundumodel1\]) is exponentially stable. Considering Theorem \[Theorem:TractoryConvergence\], we can conclude that the equilibrium $[x,y] = 0$ of the systems (\[orgsysmodel1\]) and (\[orgsysmodel2\]) is exponentially stable. A more rigorous proof can be performed by following the same methods as the proofs of Proposition 8.1 and Proposition 8.2 in [@bof2018lyapunov].
Conclusion
==========
The theoretical contribution of this paper is developing singular perturbation theory for the $\chi$ nonlinear system.
[1]{}
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[^1]: \*Xiaofan Cui and Al-Thaddeus Avestruz are with the Department of Electrical Engineering and Computer Science, University of Michigan, Ann Arbor, MI 48109, USA cuixf@umich.edu, avestruz@umich.edu.
[^2]: In this paper, we assume the 2-norm $\|\cdot\| = \|\cdot\|_2$.
|
---
abstract: 'The infrared spectra of starburst galaxies are dominated by the low-excitation lines of \[NeII\] and \[SIII\], and the stellar populations deduced from these spectra appear to lack stars larger than about 35 M$_\odot$. The only exceptions to this result until now were low metallicity dwarf galaxies. We report our analysis of the mid-infrared spectra obtained with IRS on Spitzer of the starburst galaxy NGC 1222 (Mkn 603). NGC 1222 is a large spheroidal galaxy with a starburst nucleus that is a compact radio and infrared source, and its infrared emission is dominated by the \[NeIII\] line. This is the first starburst of solar or near-solar metallicity, known to us, which is dominated by the high-excitation lines and which is a likely host of high mass stars. We model the emission with several different assumptions as to the spatial distibution of the high- and low-excitation lines and find that the upper mass cutoff in this galaxy is 40 M$_\odot$–100 M$_\odot$.'
author:
- 'Sara C. Beck, Jean L. Turner & Jenna Kloosterman'
title: 'The Extraordinary Infrared Spectrum of NGC 1222 (Mkn 603)'
---
Introduction
============
Star formation in starburst galaxies is often concentrated in compact, highly obscured regions. It is difficult to measure the stellar content of these sources. The stars cannot be studied directly, but only through the nebulae they excite. The proper use of nebular diagnostics has been controversial. In a study of all the galaxies with infrared spectra in the literature as of 2003, @RR04 concluded that the young stellar populations in galaxies of solar metallicity have relatively low $\rm T_{eff}$ and that their IMFs, if Salpeterian, cannot extend to 100 M$_\odot$. They suggested that the upper mass cutoffs are $\lesssim$ 40 M$_\odot$. Dwarf galaxies of much lower than solar metallicity may have higher mass stars; masses as high as 60 M$_\odot$ are needed to explain the spectra of the three such galaxies in their sample. This apparent lack of high mass stars in starbursts is a major puzzle of star formation studies. Is it real? Or is the apparent lack due to observational bias or problems in interpretation? We report here on the infrared spectra of the unusual S0 galaxy NGC 1222 (Markarian 603; D=34 Mpc ($\rm h_o/71~$Mpc)$^{-1}$.) This galaxy is remarkable because it is only slightly, if at all, less metal-rich than our Galaxy, yet has the mid-infrared spectral signature of the most massive, hot stars, including a high ratio of IRAS 60–100 flux. Our analysis of the galaxy is based on ground-based high-resolution maps of the radio and infrared continuum and near-infrared spectra, and on mid-infrared spectra obtained with the IRS Spectrometer on the Spitzer Space Telescope. We model the mid-infrared emission from this galaxy as arising in the superposition of a compact and bright nebula that contains the youngest stars and produces the high-excitation ionic lines, and a “cooler" spectrum region of more extended ($\gtrsim$500 pc) emission. The line ratios are compared to the results of STARBURST99 stellar populations and MAPPINGS and CLOUDY photoionization models. The models are assumption-rich and the results are only limits. However, even with these caveats we can say that this galaxy has the highest upper mass cutoff yet known for a galaxy that is not a metal-poor dwarf.
In the next section we describe the observations and how we model the galaxy, in the Analysis section the STARBURST99 and photoionization grids, and in the Discussion, our conclusions.
Observations
============
Previous Work
-------------
@PB93 obtained optical spectra and images of the Markarian galaxy, NGC 1222. The galaxy has the optical appearance of a triple system. They found that the main source has a starburst nucleus of high surface brightness, superimposed on a fainter elliptical component of approximately 13 kpc by 9 kpc extent to the 25 mag arcsec$^{-2}$ isophote. Within 2 (20 kpc) of the nucleus are two compact sources of high surface brightness, C1 and C2, which they argue are dwarf galaxies interacting with the main galaxy. In spite of the proximity of the compact components, the overall isophotal morphology, even at large galactocentric radii, remains fairly symmetric. The optical spectra show that all three sources are photo-ionized, rather than shock excited, and that the main source has the characteristic spectrum of a starburst. @PB93 found elemental abundances of O, Ne and N to be $\rm 12+log[x/H] ~8.57\pm0.09, 8.53\pm0.026$ and $7.35\pm0.17$ respectively. @CDD01 obtained K-band spectra of NGC 1222 and report the Br$\,\gamma$ flux of the galaxy in 2 apertures: $\rm 23.5\pm0.8\times10^{-15}erg\,s^{-1}~cm^{-2}$ in a large beam of 3$\times $9 and $16.9\pm 0.4$ in a 3$ \times$ 3 beam, also consistent with a bright, compact starburst with an additional extended ionized component. H$\alpha$ + \[NII\] images of NGC 1222 show that the ionized gas is brightest at the locations of the compact continuum sources [@AHM90].
NGC 1222 is a strong infrared source and is in the Revised Bright Galaxy Sample [@2003AJ....126.1607S]. Its IRAS fluxes are 0.59, 2.28, 13.06 and 15.41 Jy at 12, 25, 60 and 100 respectively; it has the high ratio of 60/100 flux typical of active starbursts. @E96 searched for CO in NGC 1222 and did not detect any, so the molecular gas mass is not directly detected. However, dust emission in NGC 1222 has been detected at $850\mu$m by @2000MNRAS.315..115D using SCUBA and it has been mapped in $HI$ with the VLA by @2004MNRAS.351..362T The logs of the atomic gas and dust masses are 9.38 and 6.66; NGC 1222 is at the low end of the sample for both quantities.
Radio Continuum Maps
--------------------
The spatial distribution of the starburst is a key element of the interpretation of the mid-infrared spectra, and NRAO VLA[^1] maps allow us to map the structure of the starburst. We show radio maps at 6 cm (C band) and 1.3 cm (K band) obtained from the VLA archives in Figures 1a and 1b. The 1.3 cm image is shown superimposed on the 6 cm in Figure 1c. Details of the observations are in Table 1. The beam sizes of the observations differ by about a factor of two; for the overlay in Figure 1c, the maps have been convolved to the same 1.75$\times$ 1.2 beam. The VLA maps are sensitive to structures less than $\sim$15 in size.
In the 6 cm map, there is a secondary source about 10 east of the main source, and there is a $3\sigma$ feature in K-band at the same location. Comparing the radio data to the only image of @PB93, we associate the second radio source with their C1 object, which they identify as a separate interacting system. We see no radio counterpart to their C2 to the southwest of the main source, although it is a bright H$\alpha$ + \[NII\] source [@AHM90]. @C90 find a size of 11 $\times 8$ at 20 cm and gives a total 20 cm flux of 55 mJy in 18 and 39.7 in 5. The C1 source may appear as an extension in their map. The position of our radio source agrees with that of @SW86 20 cm map and is about 10 off the quoted optical source position, but that position is quoted with large ($\pm5$) uncertainties.
The radio emission at 1.3 cm comes almost entirely from a bright source about 4(620 pc) in diameter; the 1.3 cm emission has an unusual shape which could be an incomplete ring, or two bright peaks not completely separated. The 6 cm emission has the same bright source with structure consistent with the 1.3 cm shape; the beam size at 6 cm is almost twice that at 1.3 cm so the same detail cannot be seen. The 6 cm map also shows extended emission, mostly in the east-west direction. The radio maps are consistent with the $11.7\mu$m image in the Keck/LWS images of Turner, Gorjian & Beck 2007, which also shows a bright, concentrated source extended EW. Neither C1 nor C2 fall in the field of the LWS observations.
Comparison of the VLA maps at 6 and 1.3 cm, and with single dish fluxes, reveals much about the spatial distribution of the radio emission and the starburst. The galaxy looks very different in the two images, so there are clearly variations in spectral index and sources of the radio emission. The expected sources of radio emission in starburst galaxies are nonthermal synchrotron emission, with spectral index, $\alpha \sim -0.7$ to $-0.8$, and thermal free-free emission from HII regions, with $\alpha \sim -0.1$ ($S\propto \nu^\alpha$). The 1.3 cm source has flux, found using the AIPS program TVSTAT, of $S_{1.3\,cm}=14 \pm 2$ mJy in 28 square arcseconds. The central source at 6 cm, excluding the extended emission components, has $S_{6\,cm}=14 \pm 4$ mJy in the same region. The central source is thus a bright 1.3 cm emitter, roughly as bright at 1.3 cm as at 6 cm, and the radio emission in the central region is mostly if not entirely thermal. Single dish fluxes measured by @marx94 with the Effelsberg 100 m telescope are 30.7 $\pm 3$ mJy at 6 cm and 14.8 mJy $\pm 3$ at 2.8 cm; the 100 m beamsizes are 145 at 6 cm and 69 at 2.8 cm. Comparison of the total mapped flux in our 6 cm VLA map to the Effelberg fluxes indicates that there is a significant extended component to the 6 cm emission that is not detected in our VLA maps. This extended component appears to be dominated by nonthermal synchrotron emission, based on the single dish spectral index of -0.9. By contrast, the 1.3 cm VLA flux is essentially equal to the 2.8 cm Effelsberg flux. Since optically thin thermal free-free emission from $H_{II}$ regions has a nearly flat spectrum, with a $\nu^{-0.1}$ spectrum, then the VLA map must detect nearly all of the 1.3 cm emission in this galaxy, and the source must be largely free-free emission. Although a strong thermal radio source, this central source is not the brightest source in the image of H$\alpha$ + \[NII\] image of @AHM90, probably because of extinction ($\S 3.2$).
Thus the high frequency radio fluxes indicate that the 1.3 cm VLA map of NGC 1222 is mostly thermal free-free emission from HII regions, and that this emission is confined to the source shown in the map of Figure 1b. In the VLA maps, the 1.3 cm emission in NGC 1222 appears even stronger relative to the 6 cm emission than the slight negative spectral decrement of thermal emission would require. This source may be one of those, common in extreme starbursts, where the radio spectrum rises instead of falling at high frequencies because lower frequencies have significant optical depth [@THB98; @BTK00; @J01].
The inner region of NGC 1222 contains a large population of young ionizing stars. If we assume the 1.3 cm emission in the VLA maps is entirely thermal free-free emission, the observed flux implies $\rm N_{Lyc}$ of $\rm1.45\times10^{54}~s^{-1}$ for the central 3. This is the ionization equivalent of $1.45\times10^5$ standard O7 stars.
Spitzer IRS Spectra
-------------------
NGC 1222 was observed with the IRS on Spitzer in staring mode as part of the IRS Standard Spectra program, Program 14, for which J. Houck was Principal Investigator . The middle-infrared spectral diagnostics we need for this study fall in the bandpass of the Short-Hi module, which covers the wavelength range 9.9-19.6 with spectral resolution of $\approx 600$. The NGC 1222 data were obtained in two positions separated by about 3 roughly parallel to the slit. Their full Spitzer identifiers are in the Table; here they will be called Spectrum 1 and Spectrum 2. Spectrum 1 is shown in Figure 2. The line fluxes obtained for the diagnostic lines, for each spectrum, are shown in Table 2. We extracted the spectra from the post-BCD data provided by the Spitzer pipeline. IRS data requires different treatment for extended and point sources. Since NGC 1222 is neither a point source nor a flat and smooth extended source, we follow the SINGS group [@seth06], who use the extended source option in the SPICE package to obtain line over continuum fluxes in clumpy galaxies. They have shown that where the flux calibration can be constrained by other data, the extended source calibration is usually accurate to within 20%. The uncertainty in the fluxes and the line ratios is dominated by the calibration difficulties: since the correction for spatial structure depends strongly on the wavelength, both the relative and the absolute calibration are affected. We assume 20% as the working uncertainty for all the lines.
The nomimal position of Spectrum 2 is 03$^h$ 08$^m$56.7$^s$, $-$25718 and of Spectrum 1 03$^h$ 08$^m$ 56.9$^s$, $-$2 57 20; the former agrees better with the radio position. However, the beam size in the IRS is large, on these scales, and also depends on wavelength: the beam is around $3\arcsec$ at $10\mu$m and $6\arcsec$ at $20\mu$m, so the positions actually overlap for most of the wavelength range. Spectrum 2 has somewhat stronger lines than Spectrum 1 but the difference is less than the calibration uncertainty in all cases, and the line ratios on which the analysis in the next section depends differ by less than $10\%$ between the two observations. We have therefore taken the average of the line ratios for each observation and work with that number for simplicity.
Analysis
========
Spatial Stucture
----------------
The ground-based radio, K-band and $11.7\mu$m continuum data give a consistent picture of NGC 1222: the young stars are concentrated into a central source no larger than 3 diameter, $\sim 470$pc, which produces most of the infrared and radio emission in the galaxy. This is the source called the starburst nucleus by @CDD01 and by @PB93. We do not know if it is the kinematic nucleus of the galaxy, as there is little dynamic information, but it is clearly the dominant region of star formation and the actual starburst. (Note that the Spitzer Short-Hi observations do not cover the optical and 6 cm source C1, and we cannot judge whether it is also an RISN or some other kind of emission region.) There are more young stars in a less dense region 2-3 times as large as the compact source. The phenomenon of a very concentrated group of young stars with strong radio and dominant infrared emission is a common one in starburst galaxies; we have called such sources in other galaxies RISN [Radio-Infrared Super-Nebulae @B02].
How are we to analyse the infrared spectra in light of the observed spatial structure of the galaxy? The suite of infrared diagnostic lines can be for convenience thought of as the high-excitation lines, \[NeIII\] and \[SIV\], and the low-excitation lines \[NeII\] and \[SIII\]. (The \[ArII\] and \[ArIII\] fine-structure lines at 6.99 and 8.99 are also useful diagnostics, but do not fall in the wavelength region measured by the Spitzer high resolution modules, and cannot be seen clearly in the low-resolution results). The homo- and hetero-nuclear line ratios give a measure of the relative strength in the total radiation field of the hard photons that create the ions of $Ne^{++}$ and $S^{+3}$ and the softer photons responsible for the other ions. From this measure of the ionization it is possible to work backwards to stellar populations that could produce that ionizing field. This is an active area of research and there are several programs and approaches, comprehensively reviewed in @RR04.
For the line ratios to be meaningful they must compare lines from the same source. The galaxy appears in both the radio and optical maps to have a compact main source embedded in more diffuse emission. The IRS slit is large enough to encompass most of the total emission, compact and diffuse together (Figure 1). If the high- and low-excitation lines are distributed differently between the compact and diffuse components, the global line ratios will be subject to misinterpretation. This phenomenon has been observed in other galaxies, particularly those that are forming super star clusters, such as NGC 5253 [@C99]. It is also what we should expect, from the nature of the IMF.
In any IMF the most massive stars will be only a small fraction of the total by number. To have enough very massive stars that they significantly affect the spectrum, the total number of stars must be very large. So the most massive stars will be found in NGC 1222, if at all, in the compact source, which as shown above must contain tens of thousands of O stars. A star must be larger than 40-45 $M_\odot$ to excite as much \[NeIII\] as \[NeII\] emission. \[NeII\], in contrast, is excited by the more common late-O and early B stars: even a B0 star of only 20$M_\odot$ can produce strong \[NeII\] emission. These lower mass stars are a much larger fraction of the total by number and will be common even in the sparser stellar population of the diffuse source.
So we argue that almost all the \[NeIII\] should be assigned to the compact 3 source visible in the VLA 1.3 cm image. But what about the \[NeII\] and the sulfur lines? What fraction of the line strengths should be attributed to the compact source and what to the extended? We note that @CDD01 found an excess of $\rm 6.6 \times 10^{-15} erg\,s^{-1}\, cm^{-2}$ of Br$\,\gamma$ emission in their large beam over that in their small. Their small beam being comparable in size to the compact source, and their larger beam to the entire Spitzer slit, we take this extended component of emission and estimate the \[NeII\] flux that should be produced by that quantity of ionized gas. We use the neon abundance found by @PB93 of $3.34\times10^{-4}$ relative to H and assume that all the neon is singly ionized; the latter is the default case in normal [[H[ii]{}]{}]{} regions. With these assumptions the extended component is expected to produce a \[NeII\] line of 1.8 Jy strength. Which equals, within the calculational uncertainties, the entire \[NeII\] line flux seen by Spitzer! This argues that the low-excitation line of \[NeII\] is almost entirely produced in the extended emission and that the contribution of the compact source is negligible.
So NGC 1222 appears to resemble NGC 5253 [@C99], where all the \[NeIII\] flux seen in the much larger beam of ISO proved to be emitted by a single compact source, and all the \[NeII\] from an extended emission region. How will the other lines track? We don’t know enough of the structure of the ionization field to treat this fully, but can get a crude and approximate answer from the ionization potentials of the sulfur ions. $S^{++}$ can only exist in the $Ne^+$ zone, but $S^{+3}$ can coexist with $Ne^+$ as well as with $Ne^{++}$. So the extended diffuse emission could produce some part of the \[SIV\] as well as essentially all of the \[SIII\]. In NGC 5253, this was not the case, and all the \[SIV\] was produced by the compact source, but in II Zw 40 (another low-metallicity dwarf) as much as 30% of the SIV may come from outside the compact source [@MH06].
Since we do not know the spatial distribution of the mid-infrared lines, we therefore work with the line ratios we would derive for the following three cases. These cases should cover all possibilities for the distribution of the mid-infrared lines in NGC 1222.
[*Model 1:* ]{} no spatial separation; the lines are co-extensive in origin. We know from the $Br\gamma$ results and all the arguments above that this model is unrealistic. Nevertheless, we present it as an extreme lower limit to the stellar temperatures and masses.
[*Model 2:*]{} the most stratified model; the high and low-excitation lines are essentially disjoint. In this case NGC 1222 would resemble NGC 5253, where all the low-excitation lines were from the diffuse component. For the calculations we assume that $90\%$ of the \[NeIIII\] and \[SIV\] are from the compact source and $90\%$ of the \[NeII\] and SIII\] from the diffuse. We chose $90\%$ because the remainder it leaves is consistent with the weakest Spitzer lines and the calculated \[NeII\] flux. This will give the hottest and most massive stars. It is also, we think, the most realistic model.
[*Model 3:*]{} an intermediate case, found by applying the solar neon abundance rather than the higher value found by @PB93 to the gas producing the extended $Br\gamma$ emission. This gives 0.96 Jy of \[NeII\] in the extended emission, for a model in which 60% of the low excitation lines \[SIII\] and \[NeII\] are in the extended source and 40% in the compact. In this model as in model 2, $90\%$ of the high-excitation lines are from the compact source; the distributions of \[NeII\] and \[SIII\] are changed.
These physical models are simple but they are consistent with everything we know about the source and about similar star formation regions and they will show how the stellar types deduced depend on the spatial distribution. Line ratios for the models are given in Tables 3 and 4.
Extinction Effects
------------------
In the above discussion we used the observed line strengths without correction for extinction. Based on optical spectra, @PB93 found the maximum reddening to be 0.4 mag, for a visual extinction of 1.2 mag. The $Br\gamma/$radio ratio tells a different story; the $Br\gamma$ fluxes are considerably lower than the value predicted from the radio continuum flux, which for $T_e=7500$ K and no extinction at $Br\gamma$ would give $\rm 1.6\times10^{-13}erg\,s^{-1}\, cm^{-2}$ in 3$\times$3. The discrepancy with the observed gives 2.4 mag of extinction at B$\,\gamma$ and 19 magnitudes $A_v$. The different extinctions derived from different markers are common in star formation regions, where the optical emission region and the infrared sources can be physically distinct, and where much of the extinction is produced within the HII regions themselves.
An obscuration of 19 magnitudes is typical of very dense compact $H_{II}$ regions. If we accept that value for the region producing the infrared lines in NGC 1222 we will have to consider the effect on our line ratios. We follow @G02 and take $A_{SIV}=0.7 A_k$, $A_{NeII}=0.28 A_k$, $A_{NeIII}=0.22 A_k$, and $A_{SIII}=0.32 A_k$. The extinctions at the \[NeII\], \[NeIII\], and \[SIII\] lines are so similar that even with this high figure for the total extinction the line ratios will change by only about 20%. The uncertainty introduced by the extinction is thus less that that due to the spatial structure, the metal content, and the instrument calibration. The problem is more severe for \[SIV\], where the line fluxes would be increased by a factor of 4.8. Correcting the \[SIV\]/\[SIII\] and \[SIV\]/\[NeII\] ratios for this extinction would increase them by factors of 2.3 and 2.5, respectively. This extinction factor would move the line ratios calculated for model 1 close to those of model 3, and those of model 3 close to model 2.
The distribution of obscuring matter in NGC 1222 is probably not uniform, just as the emission is not uniform. The extinction may be higher towards the compact source and lower in the extended emission region. Until the galaxy is mapped with high resolution in the infrared, the above results are only approximate.
Photoionization Models
----------------------
Deducing the stellar population from the observed infrared lines is a complicated process. The line ratios are compared to those predicted by models, and the inputs into the models, which include the time dependance of the star formation process (i.e., is it an instantaneous burst or continuous), the mass function and mass limits of the burst, the stellar atmospheres, and the metal content of the stars and of the nebula. Much work has gone into the establishment and testing of the photoionization models and improvements continue, especially on the stellar atmospheres, the inclusion of stellar winds, and the inclusion of the Wolf-Rayet stage. The models are constantly changing and improving.
We use the STARBURST99 software and data package [@L99] to model the ionizing spectra produced by different candidate stellar populations, and the MAPPINGS code [@K07] to calculate the output spectrum. This method gives results generally in good agreement [@RR04] with output spectra calculated by CLOUDY [@F06] from ionizing spectra generated by STARBURST99. The models are most limited, or least realistic, in their geometrical simplicity—real star formation regions are not likely to be spherical, symmetric or uniformly filled. In NGC 1222, as in almost all galaxies, we know that there is complex structure but it can’t be observed directly. So we use the default geometrical assumptions in the photoionization code of sphericity, uniform filling, and isobaric density structure with $P/k=10^5$ and mean temperature $10^4$. The ionization parameter is set by the STARBURST99 output luminosity and the density. The models compare bursts of $10^6$ M$_{\odot}$ with a Kroupa IMF with power-law exponents 1.3 for 0.1 to 0.5 M$_\odot$ and 2.3 for higher masses. We assume Pauldrach/Hillier stellar atmospheres, Geneva evolution tracks with high mass-loss rates, and dust of standard depletions. We vary the parameters 1) upper mass limit of the IMF, which may be refered to as the mass and 2) the age. We also ran models for different values of metal content.
The infrared diagnostic lines are very sensitive to the metal content of the source. Starbursts with lower metal content will have hotter nebulae than others of the same mass parameters and ages but with more metals. This is partly because stars with fewer metals, and thus less line-blanketing, will have hotter spectra than metal rich stars of the same mass, and partly because the ionized nebula cool by emitting metal lines. Until now, the starbursts in which \[NeIII\] is clearly stronger than \[NeII\] have been low metal-content blue dwarf galaxies: NGC 5253, II Zw 40, and NGC 55. The metal content of NGC 1222, in contrast, is close to Galactic: @PB93 describe it as “only slightly, if at all deficient", with \[O/H\] $70\%$ of solar, \[Ne/H\] almost twice solar, and only \[N/H\], at 35% of solar, really deficient. Other elements were not measured. STARBURST99 offers stellar atmospheres in a range of metallicities, but the only realistic options for this galaxy were 0.4 solar, solar or twice solar. We used solar value as it is closest to the reported. The metal content of the nebula can be adjusted in the codes and we used gas with the stated abundances of O, Ne and N and the solar abundance of all others.
A weakness of the models is that the stellar atmosphere abundance cannot be fine-tuned, and if the stars in NGC 1222 have metal content significantly lower than solar, they will produce more high-excitation gas than predicted and the stellar types derived from the models may be misleadingly high. We tested the possible impact of this by running several models with stellar atmospheres of 0.4 solar metallicity. If the nebular gas abundance patterns were the same, the two stellar atmosphere models produced line ratios within 15% of each other. So unless the stellar metal abundances are radically different from the estimated, this will not strongly affect the results.
The Most Massive Stars in the Starburst
=======================================
The Compact Source
------------------
In Figure 3 we show the inferred line ratios we would observe for the three models of spatial distribution, overplotted on simulation results. For Model 1, which treats the whole galaxy as one uniform source of all mid-infrared lines, and will give the coolest, least massive stars, the \[NeIII\]/\[NeII\] ratio can be fit by age less than $~2.2$ Myr and an upper mass limit of 35 M$_{\odot}$, or alternatively, age 2.2–3.2 Myr and any mass above 40 M$_{\odot}$. In Model 2, which we favor because of the radio morphology, the extreme concentration model in which nearly all the low excitation emission comes from extended ($>$500 pc) gas, only an upper limit of $\gtrsim 50-70~M_{\odot}$ stars and age less than 2 Myr can match the results. For Model 3, the intermediate case in which 40% of the low excitation \[SIII\] and \[NeII\] flux is emitted by the nuclear region, the neon line ratio gives an upper mass limit around 40M$_{\odot}$ for age less than 2 Myr or up to 100 M$_{\odot}$ if in the 2–3 Myr range where the ionization is coolest.
\[NeIII\]/\[NeII\] has two clear advantages over the other line ratios we can form with the Spitzer data: it is homonuclear, and the two lines have very similar, very low, extinction (the line ratio does not have to be corrected for extinction at the current level of analysis). The \[SIV\] line is very close to the silicate absorption feature and has the highest extinction of the useful infrared lines, and the sulfur line ratio, since it covers the greatest wavelength range, is also the most susceptible to the wavelength dependance of the extended source calibration. With these warnings, the \[SIV\]/\[SIII\] line ratio without extinction correction is best fit by, in Model 1: masses less than 30 M$_{\odot}$ and ages less than 3 Myr; Model 2: masses greater than 50 M$_{\odot}$ and ages less than 2.5 Myr, and Model 3: masses of 30 M$_{\odot}$ and ages less than 3 Myr. Extinction corrections move these results towards higher masses; the corrections in section 3.2 will give mass limits of about 30, 40, and $>$70 M$_{\odot}$ for the three spatial models.
The \[SIV\]/\[NeII\] line ratio can be affected both by the significant extinction and by abundance. The simulations show that model 1 can be fit with 30–40 M$_{\odot}$ stars less than 2 Myr old, model 2 by $>$50 M$_{\odot}$, and model 3 by a mass $>$40 M$_{\odot}$. The extinction correction results moves the best fit to 40 and 70 M$_{\odot}$ for models 1 and 3 and more than 100 M$_{\odot}$ for model 2.
Note that in many cases the line ratios can agree with either a very early age, or a time later than about 3.5 Myr when stars have entered the Wolf-Rayet phase, which produces a hard ionization field. We did not consider the second stage because the Wolf-Rayet feature has not been seen in NGC 1222. But the high extinction and the paucity of deep optical spectra in this galaxy mean that Wolf-Rayet stars have not been strictly ruled out, and the possibility should be kept in mind.
The Diffuse Emission
--------------------
The \[NeIII\]/\[NeII\] ratio in the diffuse region, although based on assumptions, is plausible because the \[NeIII\] is almost certain to be strongly concentrated and because the \[NeII\] flux is derived in a fairly straightforward way. The \[SIII\] cannot be so easily calculated because the abundance of sulfur is not known. With these considerations, the line ratios agree with a relatively low upper mass limit of 30 M$_\odot$, and age less than 4 Myr. In this regime, Models 2 and 3 fit the same photoionization grids equally well.
Discussion
===========
Deducing the stellar population of a star cluster from the nebular line it excites is never straightforward; we must find the best values of mass and age with only three line ratios observed. In NGC 1222 it is even more complex, because of the large slit on Spitzer which combined flux from the compact source and from the extended emission, so that we must dis-entangle the spatial distribution from the observed line ratios. We have presented many models and must now consider which are the most physical and probable.
That the compact source in NGC 1222 must contain most of the \[NeIII\] and \[SIV\], and that the extended Br$\,\gamma$ emission must be associated with some \[NeII\] and \[SIII\], is certain. Reality is thus somewhere between our Models 2 and 3 (the most concentrated and intermediate cases). Among the line ratios, we give the highest weight to \[NeIII\]/\[NeII\] as it is homonuclear and not strongly affected by extinction. (The \[SIV\]/\[SIII\] ratio compares two lines whose critical densities are significantly different, $3.7\times10^4$ for \[SIV\] and $9\times10^3$ for \[SIII\], both within the range of densities expected for compact star formation regions, in addition to the considerable extinction correction.) So the best-fitting models are those with upper mass limit between 40 M$_\odot$ and 100 M$_{\odot}$ and age less than 2 Myr. These results cannot be better refined until high resolution observations of the infrared lines, which would directly measure the contributions of the compact and diffuse component, can be carried out.
Comparision to Other Simulation Results
---------------------------------------
We compare our model results to the CLOUDY simulations of @RR04. They used the same STARBURST99 input spectra but CLOUDY instead of the Mappings photoionization code; in Figure 11 of their paper it may be seen that for high masses and solar metallicity they agree very well. We show in Figure 4 the line ratios as a function of time for CLOUDY simulations and solar metallicity, taken from that paper. The \[NeIII\]/\[NeII\] line ratio of NGC 1222 is so high that even the unrealistic, least concentrated Model 1 would require an upper mass limit of $\sim 50~M_{\odot}$, and for the other models even 100 M$_{\odot}$ would not provide enough ionization. The \[SIV\]/\[SIII\] and \[SIV\]/\[NeII\] ratios behave similarly.
The MAPPINGS models of this paper and the CLOUDY models of @RR04 used different parameters and structures for the ionized regions, but it is apparent that the nebular metallicity is the driving factor that makes the results so different. If the metallicity of NGC 1222 were 0.2 solar, it would be like NGC 5253, which has similar line ratios, and which (Figure 4 of Rigby & Rieke) can be fit with M$_{upper}>40$M$_{\odot}$ . In a solar metallicity system, the line ratios of NGC 1222 can only be fit by truly extreme stellar populations. In the moderately sub-solar metal models which are most appropriate for the galaxy, the stellar population deduced is unusual but not extreme.
Conclusions
===========
NGC 1222 is a remarkable galaxy. While there are uncertainties in the models we used for analyzing ionized gas emission, the results are clear and not model-dependent: NGC 1222 has the highest excitation nebular spectrum and the highest deduced upper mass limit for the starburst stellar mass function of any galaxy yet studied that is not a metal-poor blue dwarf. NGC 1222 has about the same infrared diagnostic line ratios as the iconic metal-poor dwarf NGC 5253, but NGC 1222 has near-solar metallicity. It is unique among currently studied infrared galaxies.
The infrared-line ratios of NGC 1222 would be extreme, for a galaxy of near-solar metallicity, even without the extra factor added by our spatial modeling. With the models, the line ratios inferred are unique. That the inferred line ratios do resemble those observed in low-metal dwarfs suggests that perhaps the infrared emission is generated in a low-metallicity region. It would explain the results if, for example, NGC 1222 is actually a low-metal content dwarf that has merged with a system of higher metal content, and if @PB93 measured their optical spectra in an area heavily contaminated with metals from the second system. The fact that there is a bright companion within 2 kpc of the central source of NGC 1222 indicates that the galaxy is currently experiencing a significant interaction. It would also would be consistent with another anomaly of NGC 1222: its low molecular gas content [attempts to measure the CO have only set upper limits: @C92; @E96]. Starburst galaxies are usually rich in molecular gas unless they are metal-poor dwarfs. But NGC 1222 cannot be a mis-classified dwarf galaxy, because it is too big and too bright. It has roughly half the infrared flux of NGC 5253, yet is 10 times further away. Its total luminosity puts it well outside the range of a dwarf galaxy.
Higher spatial resolution measurements of the mid-infrared lines, specifically \[NeII\] and \[SIV\] which can be observed from the ground with sub-arcsecond resolution, would permit us to see directly the compact and extended components, and to determine how much extinction affects each. Higher resolution maps in the radio would find the spatial structure. The source cannot be a single cluster: at 1.3 cm its diameter is over 500 pc and its ionization is equivalent to $1.4\times10^5$ O7 stars, twenty times that of the “supernebula" in NGC 5253. If the ionization is produced by stars with a Salpeter IMF and a lower mass cutoff of 1 M$_\odot$ the total mass in stars is $2-3 \times10^7$ M$_\odot$. This exceeds by a large factor the most massive star clusters known. The high 1.3 cm flux raises the possibility that there are very compact, dense, rising-spectrum sources in NGC 1222. This would put it in the class of the most extreme and youngest starbursts, sources that are currently not well understood.
Why is this galaxy so different from most solar metallicity galaxies? The answer may be in the two companions which it is either absorbing or otherwise interacting with. A merger with one companion usually results in an intense burst of star formation. Perhaps in a double merger the process becomes especially extreme.
Acknowledgements
================
This research has also made use of the NASA/IPAC Extragalactic Database (NED) and the NASA/ IPAC Infrared Science Archive (IRSA) which are operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration. We are grateful to Dr. V. Gorjian for IRS advice and to Dr. J. Rigby for use of the CLOUDY figures and helpful discussions.
[*Facilities:*]{} , .
Armus, L., Heckman, T. M., & Miley, G. K. 1990, , 364, 471 Beck, S. C., Turner, J. L., & Kovo, O. 2000, , 120, 244 Beck, S. C., Turner, J. L., Langland-Shula, L. E., Meier, D. S., Crosthwaite, L. P., & Gorjian, V. 2002, , 124, 2516 Chini, R., Krugel, E. & Steppe, H. 1992, , 255, 87 Coziol, R., Doyon, R., & Demers, S. 2001, , 325, 1081 Condon, J. J., Helou, G., Sanders. D. B. & Soifer, B. T. 1990, , 73, 359 Crowther, P., Beck, S., Willis, A., Conti, P., Morris, P., & Sutherland, R. 1999, , 304, 654 Dunne, L., Eales, S., Edmunds, M., Ivison, R., Alexander, P. & Clements, D. L. 2000, , 315, 115 Elfhag, T., Booth, R. S., Hoglund, B., Johansson, L. E. B., & Sandqvist, Aa. 1996, , 115, 439 Ferland, G. J. 2006, Hazy, A Brief Introduction to Cloudy 06.02, University of Kentucky Internal Report Giveon, U., Morriset, C., & Sternberg, A. 2002, , 392, 501 Johnson, K., Kobulnicky, H., Massey, P. & Conti, P. 2001, , 559, 864 Leitherer, C., et al. 1999, , 123, 3 Kewley, L., Snijders, N., Groves, B., Dopita, M., Evans, I., Sutherland, R., Binette, L., Allen, M., & Leitherer, C., 2007, in preparation Martin-Hernandez, N., Schaerer, D., Peeters, E., Tielens, A. G. G. M., Sauvage, M. 2006, å, 455, 853 Marx, M., Kruegel, E., Klein, U., & Wielebinski, R. 1994, , 281, 718 Petrosian, A. R., & Burenkov, A. N.,1993, , 279, 21 Rigby, J., & Rieke, G. 2004, , 606, 237 Sanders, D. B., Mazzarella, J. M., Kim, D.-C., Surace, J. A., & Soifer, B. T. 2003, , 126, 1607 Seth, A., 2006, Ph.D. Thesis, University of Washington Sramek, R. A. & Weedman, D. W., 1986, , 302, 640 Thomas, H., Alexander, P., Clemens. M., Green, D., Dunne, J. & Eales, S. 2004, , 351, 362 Turner, J. L., Beck, S. C., & Ho, P. T. P. 2000, , 532, L109 Turner, J. L., Ho, P. T. P., & Beck, S. C. 1998, , 116, 212
[lcccc]{}
C, 5 & AS286 & 11/23/87& ${1.75\times1.2}$ & 0.05\
K, 22 & AT309 & 6/24/05& ${0.96\times0.53}$ & 0.17\
[lcccc]{} 9071872-0006-5-E1394294 & 1.7 & 1.35 & 0.32 & 1.1\
9071872-0007-5-E1394295 & 1.72 & 1.6 & 0.38 & 1.37\
[lccc]{} Model 1 & 1.17 & 0.28 & 0.24\
Model 2 & 10.4 & 2.6 & 2.1\
Model 3 & 2.6 & 0.63 & 0.53\
[lccc]{}
Model 1 & 1.13 & 0.25 & 0.26\
Model 2 & 0.14 & 0.02 & 0.026\
Model 3 & 0.19 & 0.04 & 0.04\
[^1]: The National Radio Astronomy Observatory is a facility of the National Science Foundation, operated under cooperative agreement by Associated Universities, Inc.
|
---
abstract: 'We prove two dichotomy results for detecting long paths as patterns in a given graph. The [[NP]{}]{}-hard problem [Longest Induced Path]{} is to determine the longest induced path in a graph. The [[NP]{}]{}-hard problem [Longest Path Contractibility]{} is to determine the longest path to which a graph can be contracted to. By combining known results with new results we completely classify the computational complexity of both problems for $H$-free graphs. Our main focus is on the second problem, for which we design a general contractibility technique that enables us to reduce the problem to a matching problem.'
author:
- Walter Kern
- 'Daniël Paulusma[^1]'
title: 'Contracting to a Longest Path in $H$-Free Graphs'
---
Introduction {#s-intro}
============
The [Hamiltonian Path]{} problem, which is to decide if a graph has a hamiltonian path, is one of the best-known problems in Computer Science and Mathematics. A more general variant of this problem is that of determining the length of a longest path in a graph. Its decision version [Longest Path]{} is equivalent to deciding if a graph can be modified into the $k$-vertex path $P_k$ for some given integer $k$ by using vertex and edge deletions. Note that an alternative formulation of [Hamilton Path]{} is that of deciding if a graph can be modified into a path (which must be $P_n$) by using only edge deletions. As such, these problems belong to a wide range of graph modification problems where we seek to modify a given graph $G$ into some graph $F$ from some specified family of graphs ${\cal F}$ by using some prescribed set of graph operations. As [Hamiltonian Path]{} is [[NP]{}]{}-complete (see [@GJ79]), [Longest Path]{} is [[NP]{}]{}-complete as well. The same holds for the problem [Longest Induced Path]{} [@GJ79], which is to decide if a graph $G$ contains an induced path of length at least $k$, that is, if $G$ can be modified into a path $P_k$ for some given integer $k$ by using only vertex deletions.
Here we mainly focus on the variant of the above two problems corresponding to another central graph operation, namely edge contraction. This variant plays a role in many graph-theoretic problems, in particular [Hamilton Path]{} [@HV78; @HV81]. The *contraction* of an edge $uv$ of a graph $G$ deletes the vertices $u$ and $v$ and replaces them by a new vertex made adjacent to precisely those vertices that were adjacent to $u$ or $v$ in $G$ (without introducing self-loops or multiple edges). A graph $G$ contains a graph $G'$ as a [*contraction*]{} if $G$ can be modified into $G'$ by a sequence of edge contractions.
[.99]{} <span style="font-variant:small-caps;">[Longest Path Contractibility]{}</span>\
----------------- -------------------------------------------------------
* Instance:* [a connected graph $G$ and a positive integer $k$.]{}
*Question:* [does $G$ contain $P_k$ as a contraction?]{}
----------------- -------------------------------------------------------
The [Longest Path Contractibility]{} problem is [[NP]{}]{}-complete as well [@BV87]. Due to the computational hardness of [Longest Path]{}, [Longest Induced Path]{} and [Longest Path Contractibility]{} it is natural to restrict the input to special graph classes. We briefly discuss some known complexity results for the three problems under input restrictions.
A common property of most of the studied graph classes is that they are [*hereditary*]{}, that is, they are closed under vertex deletion. As such, they can be characterized by a family of forbidden induced subgraphs. In particular, a graph is [*$H$-free*]{} if it does not contain a graph $H$ as an induced subgraph, and a graph class is [*monogenic*]{} if it consists of all $H$-free graphs for some graph $H$. Hereditary graph classes defined by a small number of forbidden induced subgraphs, such as monogenic graph classes, are well studied, as evidenced by studies on (algorithmic and structural) decomposition theorems (e.g. for bull-free graphs [@Ch12] or claw-free graphs [@CS05; @HMLW11]) and surveys for specific graph problems (e.g. for [Colouring]{} [@GJPS17; @RS04]).
All the known [[NP]{}]{}-hardness results for [Hamiltonian Path]{} carry over to [Longest Path]{}. For instance, it is known that [Hamiltonian Path]{} is [[NP]{}]{}-complete for chordal bipartite graphs and strongly chordal split graphs [@Mu96], line graphs [@Be81] and planar graphs [@GJT76]. Unlike for [Hamiltonian Path]{}, there are only a few hereditary graph classes for which the [Longest Path]{} problem is known to be polynomial-time solvable; see, for example [@UU07]. In particular, [Longest Path]{} is polynomial-time solvable for circular-arc graphs [@MB14], distance-hereditary graphs [@GHK13], and cocomparability graphs [@IN13; @MC12]. The latter result generalized the corresponding results for bipartite permutation graphs [@UV07] and interval graphs [@IMN11]. The few graph classes for which the [Longest Induced Path]{} problem is known to be polynomial-time solvable include the classes of $k$-chordal graphs [@Ga02; @IOY08], AT-free graphs [@KMT03], graphs of bounded clique-width [@CMR00] (see also [@KMT03]) and graphs of bounded mim-width [@JKT17]. Finding a longest induced path in an $n$-dimensional hypercube is known as the [Snake-in-the-Box]{} problem [@Ka58], which has been well studied.[^2]
Unlike the [Longest Path]{} and [Longest Induced Path]{} problems, [Longest Path Contractibility]{} is [[NP]{}]{}-complete even for [*fixed*]{} $k$ (that is, $k$ is not part of the input). In order to explain this, let $F$-[Contractibility]{} be the problem of deciding if a graph $G$ contains some fixed graph $F$ as a contraction. The complexity classification of $F$-[Contractibility]{} is still open (see [@BV87; @LPW08; @LPW08b; @HKPST12]), but Brouwer and Veldman [@BV87] showed that already $P_4$-[Contractibility]{} and $C_4$-[Contractibility]{} are [[NP]{}]{}-complete (where $C_k$ denotes the $k$-vertex cycle). In fact, $P_4$-[Contractibility]{} problem is [[NP]{}]{}-complete even for $P_6$-free graphs [@HPW09], whereas Heggernes et al. [@HHLP14] showed that $P_6$-[Contractibility]{} is [[NP]{}]{}-complete for bipartite graphs.[^3] The latter result was improved to $k=5$ in [@DP17]. Moreover, $P_7$-[Contractibility]{} is [[NP]{}]{}-complete for line graphs [@FKP13]. Hence, [Longest Path Contractibility]{} is [[NP]{}]{}-complete for all these graph classes as well. On the positive side, [Longest Path Contractibility]{} is polynomial-time solvable for $P_5$-free graphs [@HPW09].
Our interest in the [Longest Induced Path]{} problem also stems from a close relationship to a vertex partition problem, which played a central role in the graph minor project of Robertson and Seymour [@RS95], as we will explain.
Our Results {#our-results .unnumbered}
-----------
We first give a dichotomy for [Longest Induced Path]{} using known results for [Hamiltonian Path]{} and some straightforward observations (see Section \[s-pre\] for a proof). Our main result is a dichotomy for [Longest Path Contractibility]{}. We use ‘+’ to denote the disjoint union of two graphs, and a [*linear forest*]{} is the disjoint union of one or more paths.
\[t-main0\] Let $H$ be a graph. If $H$ is a linear forest, then [Longest Induced Path]{} restricted to $H$-free graphs is polynomial-time solvable; otherwise it is [[NP]{}]{}-complete.
\[t-main\] Let $H$ be a graph. If $H$ is an induced subgraph of $P_2+P_4$, $P_1+P_2+P_3$, $P_1+P_5$ or $sP_1+P_4$ for some $s\geq 0$, then [Longest Path Contractibility]{} restricted to $H$-free graphs is polynomial-time solvable; otherwise it is [[NP]{}]{}-complete.
Theorem \[t-main\] shows that [Longest Path Contractibility]{} is polynomial-time solvable for $H$-free graphs only for some specific linear forests $H$. This is in contrast to the situation for [Longest Induced Path]{}, as shown by Theorem \[t-main0\]. To extend the aforementioned results from [@DP17; @FKP13; @HHLP14; @HPW09] for [Longest Path Contractibility]{} to the full classification given in Theorem \[t-main\] we do as follows.
First, in Section \[s-poly\], we prove the four new polynomial-time solvable cases of Theorem \[t-main\]. In each of these cases $H$ is a linear forest, and proving these cases requires the most of our analysis.[^4] Every linear forest $H$ is $P_r$-free for some suitable value of $r$ and $P_r$-free graphs do not contain $P_r$ as a contraction. Hence, it suffices to prove that for each $1\leq k\leq r-1$, the $P_k$-[Contractibility]{} problem is polynomial-time solvable for $H$-free graphs for each of the four linear forests listed in Theorem \[t-main\]. In fact, as $P_3$-[Contractibility]{} is trivial, we only have to consider the cases where $4\leq k\leq r-1$. Our general technique for doing this is based on transforming an instance of $P_k$-[Contractibility]{} for $k\geq 5$ into a polynomial number of instances of $P_{k-1}$-[Contractibility]{} until $k=4$.
For $k=4$ we cannot apply this transformation, as this case - as we outline below - is closely related to the $2$-[Disjoint Connected Subgraphs]{} problem. This problem takes as input a triple $(G,Z_1,Z_2)$, where $G$ is a graph with two disjoint subsets $Z_1$ and $Z_2$ of $V(G)$. It asks if $V(G)\setminus (Z_1\cup Z_2)$ has a partition into sets $S_1$ and $S_2$, such that $Z_1\cup S_1$ and $Z_2\cup S_2$ induce connected subgraphs of $G$. Robertson and Seymour [@RS95] proved that the more general problem $k$-[Disjoint Connected Subgraphs]{} (for $k$ subsets $Z_i$) is polynomial-time solvable as long as the union of the sets $Z_i$ has constant size.[^5] However, in our context, $Z_1$ and $Z_2$ may have arbitrarily large size. In that case, $2$-[Disjoint Connected Subgraphs]{} is [[NP]{}]{}-complete even if $|Z_1|=2$ (and only $Z_2$ is large) [@HPW09].
To work around this obstacle, we use the fact [@HPW09] that the two outer vertices of the $P_4$, to which the input graph $G$ must be contracted, may correspond to single vertices $u$ and $v$ of $G$. We then “guess” $u$ and $v$ to obtain an instance $(G-\{u,v\},N(u),N(v))$ of $2$-[Disjoint Subgraphs]{}. That is, we seek for a partition of $(V(G)\setminus \{u,v\})\setminus ((N_u)\cup N(v))$ into sets $S_u$ and $S_v$, such that $N(u)\cup S_u$ and $N(v)\cup S_v$ are connected. The latter implies that we can contract these two sets to single vertices corresponding to the two middle vertices of the $P_4$.
After guessing $u$ and $v$ we exploit their presence, together with the $H$-freeness of $G$, for an extensive analysis of the structure of $S_u$ and $S_v$ of a potential solution $(S_u,S_v)$. To this end we introduce in Section \[s-p4\] some general terminology and first show how to check in general for solutions in which the part of $S_u$ or $S_v$ that ensures connectivity of $N(u)\cup S_u$ or $N(v)\cup S_v$, respectively, has bounded size. We call such solutions constant. If we do not find a constant solution, then we exploit their absence. For the more involved cases we show that in this way we can branch to a polynomial number of instances of a standard matching problem.
In Section \[s-hard\] we prove the new [[NP]{}]{}-completeness results. In particular, we prove that $P_k$-[Contractibility]{}, for some suitable value of $k$, is [[NP]{}]{}-complete for bipartite graphs of large girth, strengthening the known result for bipartite graphs of [@HHLP14].
In Section \[s-classification\] we show how to combine our new polynomial-time and [[NP]{}]{}-hardness results with the known [[NP]{}]{}-completeness results for $K_{1,3}$-free graphs [@FKP13] and $P_6$-free graphs [@HPW09] in order to obtain Theorem \[t-main\].
In Section \[s-cycle\], we briefly discuss the cycle variant of our problem, called the [Longest Cycle Contractibility]{} problem [@Bl82; @Ha99; @Ha02]. Its complexity classification for $H$-free graphs is still incomplete, but we show that it differs from the classification of [Longest Path Contractibility]{} for $H$-free graphs.
In Section \[s-con\] we pose some open problems. In particular, the complexity classification of [Longest Path]{} is still open for $H$-free graphs, and we describe the state-of-art for this problem.
Preliminaries {#s-pre}
=============
In Section \[s-gt\] we give some general graph-theoretic terminology and a helpful lemma for $P_4$-free graphs. In Section \[s-t1\] we give a short proof of Theorem \[t-main0\]. In Section \[l-et\] we give some terminology related to edge contractions.
General Terminology and a Lemma for $P_4$-Free Graphs {#s-gt}
-----------------------------------------------------
We consider finite undirected graphs with no self-loops. Let $G=(V,E$ be a graph. Let $S\subseteq V$. Then $G[S]=(S,\{uv\in E\; |\; u,v\in S\})$ denotes the subgraph of $G$ [*induced*]{} by $S$. We say that $S$ is [*connected*]{} if $G[S]$ is connected. We may write $G-S=G[V\setminus S]$. The *neighbourhood* of $v\in V$ is the set $N(v)=\{u\; |\; uv\in E\}$ and the [*closed neighbourhood*]{} is $N[v]=N(v)\cup \{v\}$. The [*length*]{} of a path $P$ is its number of edges. The [*distance*]{} $\operatorname{dist}_G(u,v)$ between vertices $u$ and $v$ is the length of a shortest path between them. Two disjoint sets $S, T\subset V$ are [*adjacent*]{} if there is at least one edge between them; $S$ and $T$ are [*(anti)complete*]{} to each other if every vertex of $S$ is (non)adjacent to every vertex of $T$. The set $S$ [*covers*]{} $T$ if every vertex of $T$ has a neighbour in $S$. The [*subdivision*]{} of an edge $e=uv$ in $G$ replaces $e$ by a new vertex $w$ and two new edges $uw$ and $wv$.
A graph $G$ is [*$H$-free*]{} for some other graph $H$ if $G$ does not contain $H$ as an induced subgraph. For a set $H_1,\ldots,H_p$ of graphs, $G$ is [*$(H_1,\ldots,H_p)$-free*]{} if $G$ is $H_i$-free for $i=1,\ldots,p$. A graph is [*complete bipartite*]{} if it consists of a single vertex or its vertex set can be partitioned into two independent sets $A$ and $B$ that are complete to each other. The [*claw*]{} $K_{1,3}$ is the complete bipartite graph with $|A|=1$ and $|B|=3$. The graph $K_n$ is the complete graph on $n$ vertices.
The [*disjoint union*]{} $G_1+\nobreak G_2$ of two vertex-disjoint graphs $G_1$ and $G_2$ is the graph $(V(G_1)\cup V(G_2), E(G_1)\cup E(G_2))$; the disjoint union of $r$ copies of a graph $G$ is denoted $rG$. A [*forest*]{} is a graph with no cycles. A [*linear forest*]{} is a forest of maximum degree at most 2, that is, a disjoint union of one or more paths. The [*join*]{} operation $\times$ adds an edge between every vertex of $G_1$ and every vertex of $G_2$. A graph $G$ is a [*cograph*]{} if $G$ can be generated from $K_1$ by a sequence of join and disjoint union operations. A graph is a cograph if and only if it is $P_4$-free (see, e.g., [@BLS99]). The following well-known lemma follows from this fact and the definition of a cograph. In particular, to prove that a connected $P_4$-free graph $G$ has a spanning complete bipartite graph with partition classes $A$ and $B$, we can do as follows: take the complement $\overline{G}=(V,\{uv\; |\; uv\not \in E\; \mbox{and}\; u\neq v\}$ of $G$ and put the vertex set of one connected component of $\overline{G}$ in $A$ and all the other vertices of $\overline{G}$ in $B$.
\[l-p4\] Every connected $P_4$-free graph on at least two vertices has a spanning complete bipartite subgraph, which can be found in polynomial time.
We remind the reader of the following notions.The [*girth*]{} of a graph $G$ that is not a forest is the number of vertices in a shortest induced cycle of $G$. The [*line graph*]{} $L(G)$ of a graph $G=(V,E)$ has $E$ as vertex set and there is an edge between two vertices $e_1$ and $e_2$ of $L(G)$ if and only if $e_1$ and $e_2$ have a common end-vertex in $G$. Every line graph is readily seen to be $K_{1,3}$-free.
The Proof of Theorem \[t-main0\] {#s-t1}
--------------------------------
We now present a short proof for Theorem \[t-main0\]. We start with the following lemma.
\[l-girthpath\] Let $p\geq 3$ be some constant. Then [Longest Induced Path]{} is [[NP]{}]{}-complete for graphs of girth at least $p$.
We reduce from [Hamiltonian Path]{}. Let $G$ be a graph on $n$ vertices. We subdivide each edge $e$ of $G$ exactly once and denote the set of new vertices $v_e$ by $V'$. We denote the resulting graph by $G'$ and note that $G'$ is bipartite with partition classes $V$ and $V'$. We claim that $G$ has a Hamiltonian path if and only if $G'$ has an induced path of length $2n-2$.
First suppose that $G$ has a Hamiltonian path $u_1u_2\cdots u_n$. Then the path on vertices $u_1, v_{u_1u_2}, u_2,
\ldots, v_{u_{n-1}u_n}, u_n$ is an induced path of length $2n-2$ in $G'$. Now suppose that $G'$ has an induced path $P'$ of length $2n-2$. Then either $P'$ starts and finished with a vertex of $V$, or $P'$ starts and finishes with a vertex of $V'$. In the first case $P'$ contains $n$ vertices of $G$, so $P$ contains all vertices $u_1,\ldots,u_n$ of $G$, say in this order. Then $u_1u_2\cdots u_n$ is a Hamiltonian path of $G$. In the second case $P'$ contains $n-1$ vertices of $V$, say vertices $u_1,\ldots,u_{n-1}$ in that order. As $P'$ is an induced path and vertices of $V'$ are only adjacent to vertices of $V$, this means that the end-vertices of $P'$ are both adjacent to $u_n$. Hence, we find that $u_1u_2\cdots u_n$ is a Hamiltonian path of $G$ (and the same holds for $u_nu_1\cdots u_{n-1}$).
We note that the girth of $G'$ is twice the girth of $G$. Hence, we obtain the result by applying this trick sufficiently many times.
We also need the following lemma.
\[l-line\] The [Longest Induced Path]{} problem is [[NP]{}]{}-complete for line graphs.
We reduce from [Hamiltonian Path]{}. Let $G=(V,E)$ be a graph on $n$ vertices. We construct the line graph $L(G)$ of $G$. We claim that $G$ has a Hamiltonian path if and only if $L(G)$ has an induced path on $n-1$ vertices. First suppose that $P$ is a Hamiltonian path in $G$. Then the edges of $P$ form an induced path of length $n-1$ in $L(G)$. Now suppose that $L(G)$ has an induced path $\tilde{P}$ on $n-1$ vertices. Let $e_1, \dots, e_{n-1}$ be the $n-1$ edges of $\tilde{P}$ in that order. As $\tilde{P}$ is induced in $L(G)$, no two edges $e_i$ and $e_j$ with $i<j$ have a vertex $v \in V$ in common unless $j=i+1$. Hence, $P=\{e_1, \dots, e_{n-1}\}$ must be a Hamiltonian path in $G$.
We are now ready to prove Theorem \[t-main0\].
[**Theorem \[t-main0\]. (restated)**]{} [*Let $H$ be a graph. If $H$ is a linear forest, then [Longest Induced Path]{} restricted to $H$-free graphs is polynomial-time solvable; otherwise it is [[NP]{}]{}-complete.*]{}
Let $G$ be an $H$-free graph. First suppose that $H$ is a linear forest. Then there exists a constant $k$ such that $H$ is an induced subgraph of $P_k$. This means that the length of a longest induced path of $G$ is at most $k-1$. Hence, we can determine a longest path in $G$ in $O(n^{k-1})$ time by brute force.
Now suppose that $H$ is not a linear forest. First assume that $H$ contains a cycle. Let $g$ be the girth of $H$. We set $p=g+1$. Then the class of $H$-free graphs contains the class of graphs of girth at least $p$. Hence, we can use Lemma \[l-girthpath\] to find that [Longest Induced Path]{} is [[NP]{}]{}-complete for $H$-free graphs. Now assume that $H$ contains no cycle. As $H$ is not a linear forest, $H$ must be a forest with at least one vertex of degree at least 3. Then the class of $H$-free graphs contains the class of $K_{1,3}$-free graphs. Recall that every line graph is $K_{1,3}$-free. Hence, the class of line graphs is contained in the class of $H$-free graphs. Then we can use Lemma \[l-line\] to find that [Longest Induced Path]{} is [[NP]{}]{}-complete for $H$-free graphs.
Terminology Related to Edge Contractions {#l-et}
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Recall that the contraction of an edge $uv$ of a graph $G$ is the operation that deletes $u$ and $v$ from $G$ and replaces them by a new vertex made adjacent to precisely those vertices that were adjacent to $u$ or $v$ in $G$ (without introducing self-loops or multiple edges). We denote the graph obtained from a graph $G$ by contracting $e=uv$ by $G/e$. We may denote the resulting vertex by $u$ (or $v$) again and say that we [*contracted $e$*]{} on $u$ (or $e$ on $v$).
Recall also that a graph $G$ contains a graph $H$ as a contraction if $G$ can be modified into $H$ via a sequence of edge contractions. Alternatively, a graph $G$ contains a graph $H$ as a contraction if and only if for every vertex $x\in V(H)$ there exists a nonempty subset $W(x)\subseteq V(G)$ of vertices in $G$ such that:
- $W(x)$ is connected;
- the set ${\cal W}=\{W(x)\; |\; x\in V_H\}$ is a partition of $V(G)$; and
- for every $x_i,x_j\in V(H)$, $W(x_i)$ and $W(x_j)$ are adjacent in $G$ if and only if $x_i$ and $x_j$ are adjacent in $H$.
By contracting the vertices in each $W(x)$ to a single vertex we obtain the graph $H$. The set $W(x)$ is called an $H$-[*witness bag*]{} of $G$ for $x$. The set ${\cal W}$ is called an [*$H$-witness structure*]{} of $G$ (which does not have to be unique). A pair of (non-adjacent) vertices $(u,v)$ of a graph $G$ is $P_k$-*suitable* for some integer $k\geq 3$ if and only if $G$ has a $P_k$-witness structure ${\cal W}$ with $W(p_1)=\{u\}$ and $W(p_k)=\{v\}$, where $P_k=p_1\dots p_k$; see Figure \[f-p4witness\] for an example.
![Two $P_4$-witness structures of a graph; the grey vertices form a $P_4$-suitable pair [@HPW09].[]{data-label="f-p4witness"}](p4witness.pdf)
The following known lemma shows why $P_k$-suitable pairs are of importance.
\[l-outer\] For $k\geq 3$, a graph $G$ contains $P_k$ as a contraction if and only if $G$ has a $P_k$-suitable pair.
Lemma \[l-outer\] leads to the following auxiliary problem, where $k\geq 3$ is a fixed integer, that is, $k$ is not part of the input.
[.99]{} <span style="font-variant:small-caps;">[$P_k$-Suitability]{}</span>\
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* Instance:* [a connected graph $G$ and two non-adjacent vertices $u,v$.]{}
*Question:* [is $(u,v)$ a $P_k$-suitable pair?]{}
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The next, known observation follows from the fact that $P_k$-[Contractibility]{} is trivial for $k\leq 2$, whereas for $k=3$ we can use Lemma \[l-outer\] combined with the observation that $P_3$-[Suitability]{} is polynomial-time solvable (two non-adjacent vertices $u$, $v$ form a $P_3$-suitable pair in a connected graph $G$ if and only if $G-\{u,v\}$ is connected).
\[l-trivial\] For $k\leq 3$, [$P_k$-Contractibility]{} can be solved in polynomial time.
We now show the following lemma, which will be helpful for proving our results.
\[l-reduce\] Let $k\geq 4$ and let $(G,u,v)$ be an instance of $P_k$-[Suitability]{} with $u$ and $v$ at distance $d > k$. Let $P$ be a shortest path from $u$ to $v$. Then $(G,u,v)$ can be reduced in polynomial time to $d-2$ instances $(G/e,u,v)$, one for each edge $e\in E(P)$ that is not incident to $u$ and $v$, with $\operatorname{dist}(u,v) =d-1$, such that $(G,u,v)$ is a yes-instance if and only if at least one of the new instances $(G/e,u,v)$ is a yes-instance of $P_k$-[Suitability]{}.
First suppose that $(G,u,v)$ is a yes-instance of $P_k$-[Suitability]{}. Then $G$ has a $P_k$-witness structure ${\cal W}$ with $W(p_1)=\{u\}$ and $W(p_k)=\{v\}$. As $d\geq k$, at least one bag of ${\cal W}$ will contain both end-vertices of an edge $e$ of $P$. Then contracting $e$ yields a $P_{k}$-witness structure ${\cal W}'$ for $(G/e,u,v)$ with $W'(p_1)=\{u\}$ and $W'(p_k)=\{v\}$. As $W(p_1)$ and $W(p_k)$ only contain $u$ and $v$, respectively, the end-vertices of $e$ belong to some bag $W(p_i)$ with $2\leq i \leq k-1$. Hence, $e$ is not incident to $u$ and $v$.
Now suppose that $P$ contains an edge $e$ not incident to $u$ and $v$ such that $(G/e,u,v)$ is a yes-instance of $P_k$-[Suitability]{}. Then $G/e$ has a $P_k$-witness structure ${\cal W}'$ with $W'(p_1)=\{u\}$ and $W'(p_k)=\{v\}$. Let $e=st$ and say that we contracted $e$ on $s$. As $e$ is not incident to $u$ and $v$, we find that $\{s,t\}\cap \{u,v\}=\emptyset$. Hence, $s$ belongs to some bag $W'(p_i)$ with $2\leq i \leq k-1$. Then in $W'(p_i)$ we uncontract $e$ (so the new bag will contain both $s$ and $t$). This yields a $P_k$-witness structure ${\cal W}$ of $G$ with $W(p_1)=\{u\}$ and $W(p_k)=\{v\}$.
In our polynomial-time algorithms for constructing $P_k$-witness structures (to prove Theorem \[t-main\]) we put vertices in certain sets that we then try to extend to $P_k$-witness bags (possibly via branching) and we will often apply the following rule:
[Contraction Rule.]{} If two adjacent vertices $s$ and $t$ end up in the same bag of some potential $P_k$-witness structure, then contract the edge $st$.
For a graph $G=(V,E)$, we say that we [*apply the [Contraction Rule]{} on some set*]{} $U\subseteq V$ if we contract every edge in $G[U]$. The advantage of applying this rule is that we obtain a smaller instance and that we can exploit the fact that the resulting set $G[U]$ has become independent.
It is easy to construct examples that show that a class of $H$-free graphs is not closed under contraction if $H$ contains a vertex of degree at least 3 or a cycle. However, all polynomial-time solvable cases of Theorem \[t-main\] involve forbidding a linear forest $H$. The following known lemma, which is readily seen, shows that the [Contraction Rule]{} does preserve $H$-freeness as long as $H$ is a linear forest. Hence, we can safely apply the rule in our proofs of the polynomial-time solvable cases of Theorem \[t-main\].
\[l-contract\] Let $H$ be a linear forest and let $G$ be an $H$-free graph. Then the graph obtained from $G$ after contracting an edge is also $H$-free.
The Polynomial-Time Solvable Cases of Theorem \[t-main\] {#s-poly}
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In this section we prove that [Longest Path Contractibility]{} is polynomial-time solvable for $H$-free graphs if $H=P_2+P_4$ (Section \[s-p2p4\]), $H=P_1+P_2+P_3$ (Section \[s-p1p2p3\]), $H=P_1+P_5$ (Section \[s-p1p5\]) and $H=sP_1+P_4$ for every integer $s\geq 0$ (Section \[s-sp1p4\]). To solve [Longest Path Contractibility]{} in each of these cases we will eventually check if the input graph can be contracted to $P_4$. This turns out to be the hardest situation to deal with in our proofs. Due to Lemma \[l-outer\], we can solve it by checking for each pair of distinct vertices $u$, $v$ with $N(u)\cap N(v)=\emptyset$ if $(G,u,v)$ is a yes-instance of $P_4$-[Suitability]{} (note that for any other pair $u$, $v$, we have that $(G,u,v)$ is a no-instance of $P_4$-[Suitability]{}). In Section \[s-p4\] we first provide a general framework by introducing some additional terminology and one general result for solving $P_4$-[Suitability]{}.
On Contracting a Graph to $\mathbf{P_4}$ {#s-p4}
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Let $(G,u,v)$ be an instance of $P_4$-[Suitability]{}. For every $P_4$-witness structure of $G$ with $W(p_1)=\{u\}$ and $W(p_4)=\{v\}$ (if it exists), every neighbour of $u$ belongs to $W(p_2)$ and every neighbour of $v$ belongs to $W(p_3)$. Throughout our proofs we let $T(u,v)=V(G)\setminus (N[u]\cup N[v])$ denote the set of remaining vertices of $G$, which still need to be placed in either $W(p_2)$ or $W(p_3)$. We write $T=T(u,v)$ if no confusion is possible. We say that a partition $(S_u,S_v)$ of $T$ is a [*solution*]{} for $(G,u,v)$ if $N(u)\cup S_u$ and $N(v)\cup S_v$ are both connected. Hence, a solution $(S_u,S_v)$ for $(G,u,v)$ corresponds to a $P_4$-witness structure ${\cal W}$ of $G$, where $W(p_1)=\{u\}$, $W(p_2)=N(u)\cup S_u$, $W(p_3)=N(v)\cup S_v$ and $W(p_4)=\{p_4\}$. A solution $(S_u,S_v)$ for $(G,u,v)$ is [*$\alpha$-constant*]{} for some constant $\alpha\geq 0$ if the following holds: either $S_u$ contains a set $S_u'$ of size $|S_u'|\leq\alpha$ such that $N(u)\cup S_u'$ is connected, or $S_v$ contains a set $S_v'$ of size $|S_v'|\leq\alpha$ such that $N(v)\cup S_v'$ is connected. We prove the following lemma.
\[l-constant\] Let $(G,u,v)$ be an instance of $P_4$-[Suitability]{}. For every constant $\alpha\geq 0$, it is possible to check in $O(n^{\alpha+2})$ time if $(G,u,v)$ has an $\alpha$-constant solution.
We first do the following check for vertex $u$. For each set $S$ of size $|S|\leq \alpha$ we check if $N(u)\cup S$ is connected and if every vertex of $N(v)$ is in the same connected component $D$ of the subgraph of $G$ induced by $(T\setminus S)\cup N(v)$. If so, then we put all vertices of $T \setminus V(D)$ in $S_u$ and all vertices of $T\cap V(D)$ in $S_v$. As $G$ is connected, this yields a solution $(S_u,S_v)$ for $(G,u,v)$. This takes $O(n^2)$ time for each set $S$. As the number of sets $S$ is $O(n^\alpha)$, the total running time is $O(n^{\alpha+2})$. We can do the same check in $O(n^{\alpha+2})$ time for vertex $v$. This proves the lemma.
Let $(S_u,S_v)$ be a solution for an instance $(G,u,v)$ of $P_4$-[Suitability]{} that is not $7$-constant (the value $\alpha=7$ comes from our proofs). If $G[S_u]$ and $G[S_v]$ each contain at least one edge, then $(S_u,S_v)$ is [*double-sided*]{}. If exactly one of $G[S_u]$, $G[S_v]$ contains an edge, then $(S_u,S_v)$ is [*single-sided*]{}. If both $S_u$ and $S_v$ are independent sets, then $(S_u,S_v)$ is [*independent*]{}.
The Case $\mathbf{H=P_2+P_4}$ {#s-p2p4}
-----------------------------
We now show that [Longest Path Contractibility]{} is polynomial-time solvable for $(P_2+P_4)$-free graphs. As mentioned, we will do so via the auxiliary problem $P_k$-[Suitability]{}. We first give, in Lemma \[l-top4\], a polynomial-time algorithm for $P_4$-[Suitability]{} for $(P_2+P_4)$-free graphs. This is the most involved part of our algorithm. As such, we start with an outline of this algorithm.
[*Outline of the $P_4$-Suitability Algorithm for $(P_2+P_4)$-free graphs.*]{}\
We first observe that for an instance $(G,u,v)$, we may assume that $u$ and $v$ are of distance at least 3, and consequently, $N(u)\cap N(v)=\emptyset$, and moreover we may assume that $N(u)$ and $N(v)$ are independent. Recall that $T=V(G)\setminus (N[u]\cup N[v])$. To get a handle on the adjacencies between $T$ and $V(G)\setminus T$ we will apply a (constant) number of branching procedures. For example, we will prove in this way that $G[T]$ may be assumed to be $P_4$-free. Each time we branch we obtain, in polynomial time, a polynomial number of new, smaller instances of [$P_4$-Suitability]{} satisfying additional helpful constraints, such that the original instance is a yes-instance if and only if at least one of the new instances is a yes-instance. We then consider each new instance separately. That is, we either solve, in polynomial time, the problem for each new instance or create a polynomial number of new and even smaller instances via some further branching.
Our first goal is to check if $(G,u,v)$ has an $7$-constant solution. If so then we are done. Otherwise we prove that the absence of $7$-constant solutions implies that $(G,u,v)$ has no double-sided solution either. Hence, it remains to test if $(G,u,v)$ has a single-sided solution or an independent solution. We check single-sidedness with respect to $u$ and $v$ independently. We show that in both cases this leads either to a solution or to a polynomial number of smaller instances, for which we only need to check if they have an independent solution. This will enable us to branch in such a way that afterwards we may assume that $T$ is an independent set and that the solution we are looking for is equivalent to finding a star cover of $N(u)$ and $N(v)$ with centers in $T$. The latter problem reduces to a matching problem, which we can solve in polynomial time.
\[l-top4\] $P_4$-[Suitability]{} can be solved in polynomial time for $(P_2+P_4)$-free graphs.
Let $(G,u,v)$ be an instance of $P_4$-[Suitability]{}, where $G$ is a connected $(P_2+P_4)$-free graph. We may assume without loss of generality that $u$ and $v$ are of distance at least 3, that is, $u$ and $v$ are non-adjacent and $N(u)\cap N(v)=\emptyset$; otherwise $(G,u,v)$ is a no-instance.
Recall that $T=V(G)\setminus (N[u]\cup N[v])$. Recall also that we are looking for a partition $(S_u,S_v)$ of $T$ that is a solution for $(G,u,v)$, that is, $N(u)\cup S_u$ and $N(v)\cup S_v$ must both be connected. In order to do so we will construct partial solutions $(S_u',S_v')$, which we try to extend to a solution $(S_u,S_v)$ for $(G,u,v)$. We use the [Contraction Rule]{} from Section \[s-pre\] on $N(u)\cup S_u'$ and $N(v)\cup S_v'$, so that these two sets will become independent. By Lemma \[l-contract\], the resulting graph will always be $(P_2+P_4)$-free. For simplicity, we will denote the resulting instance by $(G,u,v)$ again. After applying the [Contraction Rule]{} the size of the set $T$ will be reduced if a vertex $t\in T$ was involved in an edge contraction with a vertex from $N(u)$ or $N(v)$. In that case we say that we [*contracted $t$ away*]{}.
At the beginning of our algorithm, $S_u'=S_v'=\emptyset$, and we start by applying the [Contraction Rule]{} on $N(u)$ and $N(v)$. This leads to the following claim.
[[*\[c-ind\] $N(u)$ and $N(v)$ are independent sets.*]{}\
]{}\
[**Phase 1: Exploiting the structure of $\mathbf{G[T]}$**]{}
In the first phase of our algorithm, we will look into the structure of $G[T]$. Suppose $G[T]$ contains an induced $P_4$ on vertices $a_1$, $a_2$, $a_3$, $a_4$. If there exists a vertex $t\in N(u)$ not adjacent to any vertex of $\{a_1,a_2,a_3,a_4\}$, then $\{u,t\}\cup \{a_1,a_2,a_3,a_4\}$ induces a $P_2+P_4$ in $G$, a contradiction. Hence, $\{a_1,a_2,a_3,a_4\}$ must cover $N(u)$. Similarly, $\{a_1,a_2,a_3,a_4\}$ must cover $N(v)$. Suppose $G[T]$ has another induced $P_4$ on vertices $\{b_1,b_2,b_3,b_4\}$ such that $\{a_1,a_2,a_3,a_4\}\cap \{b_1,b_2,b_3,b_4\}=\emptyset$. By the same arguments, $\{b_1,b_2,b_3,b_4\}$ also covers $N(u)$ and $N(v)$. This means that $N(u)\cup \{a_1,a_2,a_3,a_4\}$ and $N(v)\cup \{b_1,b_2,b_3,b_4\}$ are both connected. We put each remaining vertex of $T$ into either $S_u$ or $S_v$ (which is possible, as $G$ is connected). This yields a ($4$-constant) solution for $(G,u,v)$.
From now on, assume that $G[T]$ contains no induced copy of $P_4$ that is vertex-disjoint from $a_1a_2a_3a_4$ (so, every other induced $P_4$ in $G[T]$ contains at least one vertex of $\{a_1,a_2,a_3,a_4\}$). Below we will branch into $O(n^{16})$ smaller instances in which $G[T]$ is $P_4$-free, such that $(G,u,v)$ has a solution if and only if at least one of these new instances has a solution.
[**Branching I**]{} ($O(n^{16})$ branches)\
We branch by considering every possibility for each $a_i$ $(1\leq i\leq 4$) to go into either $S_u$ or $S_v$ for some solution $(S_u,S_v)$ of $(G,u,v)$ (if it exists). We do this vertex by vertex leading to a total of $2^4$ branches. Suppose we decide to put $a_i$ in $S_u$. If $a_i$ is adjacent to a vertex of $N(u)$, then we apply the [Contraction Rule]{} on $N(u)\cup \{a_i\}$ to contract $a_i$ away. If $a_i$ is not adjacent to any vertex of $N(u)$, then we do as follows. For each solution $(S_u,S_v)$ with $a_i\in S_u$, there must exist a shortest path $P_i$ in $G[N(u)\cup S_u]$ from $a_i$ to a vertex of $N(u)$ (as $N(u)\cup S_u$ is connected). As $G$ is $(P_2+P_4)$-free, $G$ is $P_7$-free. Hence, $P_i$ must have at most six vertices and thus at most four inner vertices. We consider all possibilities of choosing at most four vertices of $T$ to belong to $S_u$ as inner vertices of $P_i$. As we may need to do this for $i=1,\ldots,4$, the above leads to a total of $O(n^{16})$ additional branches.
For each branch we do as follows. For $i=1,\ldots,4$ we apply the [Contraction Rule]{} on $N(u)\cup \{a_i\} \cup V(P_i)$ to contract $a_i$ and the vertices of $V(P_i)$ away. We denote the resulting instance by $(G,u,v)$ again. Note that the property (Claim \[c-ind\]) that $N(u)$ and $N(v)$ are independent sets is maintained. Moreover, as every induced $P_4$ in $G[T]$ contained at least one vertex of $\{a_1,\ldots,a_4\}$, the following claim holds now as well.
[[*\[c-p4free\] $G[T]$ is $P_4$-free.*]{}\
]{}\
We now prove the following claim.
[[*\[c-either\] Let $(S_u,S_v)$ be a solution for $(G,u,v)$ that is not $7$-constant. Let $t,x_1,x_2$ be three vertices of $T$ with $tx_1\notin E(G)$, $tx_2\notin E(G)$ and $x_1x_2\in E(G)$. If $t,x_1,x_2$ are in $S_u$, then every neighbour of $t$ in $N(u)$ is adjacent to at least one of $x_1,x_2$. If $t,x_1,x_2$ are in $S_v$, then every neighbour of $t$ in $N(v)$ is adjacent to at least one of $x_1,x_2$.*]{}\
]{}\
[*Proof of Claim \[c-either\].*]{} We assume without loss of generality that $t,x_1,x_2$ belong to $S_u$. Suppose $t$ has a neighbour $w\in N(u)$ that is not adjacent to $x_1$ and $x_2$. Suppose there exists a vertex $w' \in N(u)$ not adjacent to any of $t,x_1,x_2$. Then, as $N(u)$ is independent by Claim \[c-ind\], $\{x_1,x_2\}\cup \{w',u,w,t\}$ is an induced $P_2+P_4$, a contradiction. Hence, $\{t,x_1,x_2\}$ covers $N(u)$. As $N(u)\cup S_u$ is connected, $G[N(u)\cup S_u]$ contains a shortest path $P$ from $t$ to $x_1$. As $G$ is $(P_2+P_4)$-free, $G$ is $P_7$-free. Hence, $P$ has at most four inner vertices (possibly including $x_2$). As $V(P)\cup \{x_2\} \cup N(u)$ is connected and $|V(P) \cup \{x_2\}|\leq 7$, we find that $(S_u,S_v)$ is a $7$-constant solution, a contradiction. This proves the claim.
We will use the above claim at several places in our proof, including in the next stage.
[**Phase 2: Excluding 7-constant solutions and double-sided solutions**]{}
We first show that we may exclude double-sided solutions if we have no $7$-constant solutions.
[[*\[c-double\]If $(G,u,v)$ has a double-sided solution, then $(G,u,v)$ also has a $7$-constant solution.*]{}\
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[*Proof of Claim \[c-double\].*]{} For contradiction, assume that $(G,u,v)$ has a double-sided solution $(S_u,S_v)$ but no $7$-constant solution. By definition, $G[S_u]$ and $G[S_v]$ contain some edges $x_1x_2$ and $x_1'x_2'$, respectively. Then $N(u)$ must contain a vertex $w$ that is not adjacent to $x_1$ and $x_2$; otherwise the two vertices $x_1,x_2$, which belong to $S_u$, cover $N(u)$ and this would imply that $(S_u,S_v)$ is a $2$-constant solution, and thus also a $7$-constant solution. As $N(u)\cup S_u$ is connected and $N(u)$ is an independent set by Claim \[c-ind\], set $S_u$ must contain a vertex $t$ that is adjacent to $w$. Then, by Claim \[c-either\], vertex $t$ must be adjacent to at least one of $x_1,x_2$, say $x_1$. For the same reason, $S_v$ contains a vertex $t'$ that is adjacent to at least one of $x_1',x_2'$, say $x_1'$, and to some vertex $w'\in N(v)$ that is not adjacent to $x_1'$ and $x_2'$.
Let $y\in N(v)$. If no vertex of $\{t,x_1,x_2\}$ is adjacent to $y$, then $\{v,y\}\cup \{u,w,t,x_1\}$ is an induced $P_2+P_4$ in $G$, unless $wy\in E(G)$. However, in that case $\{x_1,x_2\}\cup \{u,w,y,v\}$ is an induced $P_2+P_4$, a contradiction. Hence, $\{t,x_1,x_2\}$ covers $N(v)$. For the same reason we find that $\{t',x_1',x_2'\}$ covers $N(u)$. Then $(G,u,v)$ has a $3$-constant solution $(S_u^*,S_v^*)$ (which is $7$-constant by definition) with $\{t',x_1',x_2'\}\subseteq S_u^*$ and $\{t,x_1,x_2\}\subseteq S_v^*$, a contradiction. This proves the claim.
Recall that a solution $(S_u,S_v)$ for $(G,u,v)$ is single-sided if exactly one of $G[S_u]$, $G[S_v]$ contains an edge and independent if $S_u,S_v$ are both independent sets. We now do as follows. First we check in polynomial time if $(G,u,v)$ has a $7$-constant solution by using Lemma \[l-constant\]. If so, then we are done. From now on assume that $(G,u,v)$ has no $7$-constant solution. Then, by Claim \[c-double\] it follows that $(G,u,v)$ has no double-sided solution.
From the above, it remains to check if $(G,u,v)$ has a single-sided solution or an independent solution. If $(G,u,v)$ has a single-sided solution $(S_u,S_v)$ that is not independent, then either $S_u$ or $S_v$ is independent. Our algorithm will first look for a solution $(S_u,S_v)$ where $S_u$ is independent. We say that it is doing a [*$u$-feasibility check*]{}. If afterwards we have not found a solution $(S_u,S_v)$ where $S_u$ is independent, then our algorithm will repeat the same steps but now under the assumption that the set $S_v$ is independent. That is, in that case our algorithm will perform a [*$v$-feasibility check*]{}.
[**Phase 3: Doing a $\mathbf{u}$-feasibility check**]{}
We start by exploring the structure of a solution $(S_u,S_v)$ that is either single-sided or independent, and where $S_u$ is an independent set. As $S_u$ and $N(u)$ are both independent sets, $G[N(u)\cup S_u]$ is a connected bipartite graph. Hence, $S_u$ contains a set $S_u^*$, such that $S_u^*$ covers $N(u)$. We assume that $S_u^*$ has minimum size. Then each $s\in S_u^*$ has a nonempty set $Q(s)$ of neighbours in $N(u)$ that are not adjacent to any vertex in $S_u^*\setminus \{s\}$; otherwise we can remove $s$ from $S_u^*$, contradicting our assumption that $S_u^*$ has minimum size. We call the vertices of $Q(s)$ the [*private*]{} neighbours of $s$ with respect to $S_u^*$.
We note that $N(u)\cup S_u^*$ does not have to be connected. However, as $(G,u,v)$ has no $7$-constant solution, and thus no $1$-constant solution, we find that $S_u^*$ has size at least 2. We may assume that there is no vertex $t\in S_u\setminus S_u^*$, such that $N(t)\cap N(u)$ strictly contains $N(s)\cap N(u)$ for some $s\in S_u^*$ (otherwise we put $t$ in $S_u^*$ instead of $s$). Let $Q_u$ be the union of all private neighbour sets $Q(s)$ ($s\in S_u^*$). As $|S_u^*|\geq 2$, we observe that $G[Q_u\cup S_u^*]$ is the disjoint union of a set of at least two stars whose centers belong to $S_u^*$.
First suppose that $N(u)\setminus Q_u=\emptyset$. As $N(u)\cup S_u$ is connected and $G[Q_u\cup S_u^*]$ is the disjoint union of at least two stars, there exists a vertex $t\in S_u\setminus S_u^*$ that is adjacent to vertices $z\in Q(s)$ and $z'\in Q(s')$ for two distinct vertices $s,s'\in S_u^*$. As $N(u)\setminus Q_u=\emptyset$, we find that $Q(s)=N(s)\cap N(u)$. By our choice of $S_u^*$, this means that $Q(s)$ contains at least one vertex $w$ that is not adjacent to $t$. Similarly, $Q(s')$ contains a vertex $w'$ that is not adjacent to $t$. By the definition of $Q(s)$ and $Q(s')$, we find that $w$ and $z$ are not adjacent to $s'$, and $w'$ is not adjacent to $s$. Then $\{w',s'\}\cup \{w,s,z,t\}$ is an induced $P_2+P_4$ of $G$, a contradiction.
From the above we find that $N(u)\setminus Q_u\neq \emptyset$. Let $y\in N(u)\setminus Q_u$. As $S_u^*$ covers $N(u)$ and $y\notin Q_u$, we find that $y$ must be adjacent to at least two vertices $s,s'\in S_u^*$. Suppose $y$ is not adjacent to some vertex $s^*\in S_u^*$. Let $z\in Q(s)$ and $z^*\in Q(s^*)$. By the definition of the sets $Q(s)$ and $Q(s^*)$, we find that $z$ is not adjacent to $s'$ and $s^*$ and that $z^*$ is not adjacent to $s$ and $s'$. In particular it holds that $z\neq z^*$. Then $\{s^*,z^*\}\cup \{z,s,y,s'\}$ is an induced $P_2+P_4$ in $G$, a contradiction. Hence, $y$ must be adjacent to all of $S_u^*$, that is, $N(u)\setminus Q_u$ must be complete to $S_u^*$. Note that this implies that $N(u)\cup S_u^*$ is connected.
To summarize, if $(G,u,v)$ has a solution $(S_u,S_v)$ in which $S_u$ is an independent set, then the following holds for such a solution $(S_u,S_v)$:
- The set $S_u$ contains a subset $S_u^*$ of size at least 2 that covers $N(u)$, such that each vertex in $S_u^*$ has a nonempty set $Q(s)$ of private neighbours with respect to $S_u^*$, and moreover, the set $N(u)\setminus Q_u$, where $Q_u=\bigcup_{s\in S_u^*}Q(s)$, is nonempty and complete to $S_u^*$.
*Remark.* We emphasize that $S_u^*$ is unknown to the algorithm, as we constructed it from the unknown $S_u$, and consequently, our algorithm does not know (yet) the sets $Q(s)$.
[**Phase 3a: Reducing $\mathbf{N(u)\setminus Q_u}$ to a single vertex $\mathbf{w_u}$**]{}
We will now branch into a polynomial number of smaller instances, in which $N(u)\setminus Q_u$ consists of just one single vertex $w_u$. As we will show below, we can even identify $w_u$ and $Q_u$ for each of these new instances. Again, we will ensure that if one of these new instances has a solution, then $(G,u,v)$ has as solution. If none of these new instances has a solution, then $(G,u,v)$ may still have a solution $(S_u,S_v)$. However, in that case $S_u$ is not an independent set, while $S_v$ must be an independent set. As mentioned, we will check this by doing a $v$-feasibility check as soon as we have finished the $u$-feasibility check.
[**Branching II**]{} ($O(n^4)$ branches)\
We will determine exactly those vertices of $N(u)$ that belong to $Q_u$ via some branching, under the assumption that $(G,u,v)$ has a solution $(S_u,S_v)$, where $S_u$ is independent, that satisfies (P). By (P), $S_u^*$ consists of at least two (non-adjacent) vertices $s$ and $s'$. Let $w\in Q(s)$ and $w'\in Q(s')$. We branch by considering all possible choices of choosing these four vertices. This leads to $O(n^4)$ branches, which we each process in the way described below.
If we selected $s$ and $s'$ correctly, then $s,s'$ belong to an independent set $S_u$ that together with $S_v=T\setminus S_u$ forms a solution for $(G,u,v)$ that is not $7$-constant. This implies that $\{s,s'\}$ does not cover $N(u)$. Hence, we can pick a vertex $w^*\in N(u)\setminus \{w,w'\}$. If $w^*$ is adjacent to both $s$ and $s'$, then $w^*$ must belong to $N(u)\setminus Q_u$. In the other case, that is, if $w^*$ is adjacent to at most one of $s,s'$, then $w^*$ must belong to $Q_u$. Hence, we have identified in polynomial time the (potential) sets $Q_u$ and $N(u)\setminus Q_u$. Moreover, by applying the [Contraction Rule]{} on $N(u)\cup \{s,s'\}$ we can contract $s$ and $s'$ away. This also contracts all of $N(u)\setminus Q_u$ into a single vertex which, as we mentioned above, we denote by $w_u$. Thus $w_u$ is complete to $S_u^*$.
We denote the resulting instance by $(G,u,v)$ again. We also let $T_1=N(w_u)\cap T$ and $T_2=T\setminus T_1$. Note that $S_u^*\subseteq T_1$. As $S_u^*$ covers $N(u)$ and every vertex of $S_u^*$ is adjacent to $w_u$, we find that $N(u) \cup S_u^*$ is connected. Due to the latter and because every vertex of $T_2$ is not in $S_u^*$, we may put without loss of generality every vertex $t\in T_2$ with a neighbour in $N(v)$ in $S_v$. That is, we may contract such a vertex $t$ away by applying the [Contraction Rule]{} on $N(v)\cup \{t\}$. By the same reason, we may contract every edge between two vertices in $T_2$. Hence, we have proven the following claim.
[[*\[c-t2\]$T_2$ is an independent set that is anticomplete to $N(v)$.*]{}\
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Note that by definition, no vertex of $T_2$ is adjacent to $w_u$ either. In a later stage we will modify $T_2$ and this property may no longer hold. However, we will always maintain the properties that $T_2$ is independent and anticomplete to $N(v)$.
By Lemma \[l-constant\] we check in polynomial time if $(G,u,v)$ has a $7$-constant solution. If so, then we are done. From now on suppose that $(G,u,v)$ has no $7$-constant solution. Recall that we are still looking for a single-sided or independent solution $(S_u,S_v)$, where $S_u$ is an independent set. We first show that we can modify $G$ in polynomial time such that afterwards $G[T]$ is $(K_3+P_1)$-free.
Suppose $G[T]$ contains an induced $K_3+P_1$, say with vertices $x_1,x_2,x_3,y$ and edges $x_1x_2$, $x_2x_3$, $x_3x_1$. Consider a solution $(S_u,S_v)$ for $(G,u,v)$, where $S_u$ is an independent set. Recall that we already checked on $7$-constant solutions. Hence, $(S_u,S_v)$ is not $7$-constant. As $S_u$ is an independent set, at least two of $x_1,x_2,x_3$, say $x_1,x_2$, must belong to $S_v$. Then $(S_v \cap (N[x_1]\cup N[x_2]))\cup N(v)$ is connected; otherwise, as $S_v\cup N(v)$ is connected by definition, there would exist a vertex $t\in S_v\setminus (N[x_1]\cup N[x_2])$ with a neighbour in $N(v)$ that is not adjacent to $x_1$ and $x_2$, contradicting Claim \[c-either\]. As $y$ does not belong to $N[x_1]\cup N[x_2]$, this means that $y$ is not needed for $S_v$.
From the above we can do as follows. If $y$ has a neighbour in $N(u)$, then we contract $y$ away by applying the [Contraction Rule]{} on $N(u)\cup \{y\}$. Otherwise, if $y$ has no neighbour in $N(u)$, then $y\in T_2$. As $N(u)\cup S_u^*$ is connected for some set $S^*_u \subseteq S_u\cap T_1$, this means that $y$ is not needed for $S_u$ either. Hence, we may contract the edge between $y$ and an arbitrary neighbour of $y$ (as $G$ is connected, $y$ has at least one such neighbour). We apply this rule, in polynomial time, for every induced copy of $K_3+P_1$ in $G[T]$. Note that Claim \[c-t2\] is maintained and that in the end the following claim holds.
[[*\[c-t1\]$G[T]$ is $(K_3+P_1)$-free.*]{}\
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We will now do some further branching to obtain $O(n)$ smaller instances in which $G[T_1]$ is $K_3$-free, such that the following holds. If one of these new instances has a solution, then $(G,u,v)$ has as solution. If none of these new instances has a solution, then $(G,u,v)$ may still have a solution $(S_u,S_v)$, but in that case $S_u$ is not an independent set while $S_v$ must be an independent set; this will be verified when we do the $v$-feasibility check.
[**Branching III**]{} ($O(n)$ branches)\
We consider all possibilities of putting one vertex $t\in T_1$ in $S_u$. This leads to $O(n)$ branches. For each branch we do as follows. As $t$ is adjacent to $w_u$ (because $t\in T_1$), we can contract $t$ away using the [Contraction Rule]{} on $N(u)\cup \{t\}$. As $S_u$ is independent, every neighbour $t'$ of $t$ in $T_1$ must go to $S_v$. If such a neighbour $t'$ is adjacent to a vertex of $N(v)$, this means that we may contract $t'$ away by using the [Contraction Rule]{} on $N(v)\cup \{t'\}$. If $t'$ has no neighbour in $N(v)$, then we put $t' $ in $T_2$. By the [Contraction Rule]{} we may contract all edges between $t'$ and its neighbours in $T_2$, such that $T_2$ is an independent set again that is anticomplete to $N(v)$, so Claim \[c-t2\] is still valid (but $T_2$ may now contain vertices adjacent to $w_u$). We denote the resulting instance by $(G,u,v)$ again. As $G[T]$, and consequently, $G[T_1]$ is $(K_3+P_1)$-free due to Claim \[c-t1\], we find afterwards that the following holds for each branch.
[[*\[c-t1b\]$G[T_1]$ is $K_3$-free.*]{}\
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By Lemma \[l-constant\] we check in polynomial time if $(G,u,v)$ has an $7$-constant solution. If so, then we are done. From now on assume that $(G,u,v)$ has no $7$-constant solution. Note that $(G,u,v)$ has no double-sided solution either, as then the original instance has a double-sided solution, which we already ruled out (alternatively, apply Claim \[c-double\]). We will focus on the following task (recall that a solution $(S_u,S_v)$ for $(G,u,v)$ is independent if both $S_u$ and $S_v$ are independent sets).
[**Phase 3b: Looking for independent solutions after branching**]{}
We will now branch to $O(n^5)$ smaller instances for which the goal is to find an independent solution. As before, if one of the newly created instances has a solution, then $(G,u,v)$ has as solution. If none of these new instances has a solution, then $(G,u,v)$ may still have a solution $(S_u,S_v)$. However, in that case $S_u$ is not independent and $S_v$ must be an independent set. This will be verified when we do the $v$-feasibility check.
We say that an instance $(G,u,v)$ satisfies the $(*)$-property if the following holds:
$(*)$ If $(G,u,v)$ has a solution $(S_u,S_v)$ where $S_u$ is an independent set, then $(G,u,v)$ has an independent solution.
Let $D_1,\ldots,D_q$ be the connected components of $G[T]$ for some $q\geq 1$. First suppose that every $D_i$ consists of a single vertex. Then $G[T]$ is an independent set. Hence, any solution for $(G,u,v)$ will be independent. We conclude that $(*)$ holds already. Now suppose that at least one of $D_1,\ldots,D_q$, say $D_1$, has more than one vertex.
We first consider the case where another $D_i$, say $D_2$, also has more than one vertex. We claim that $(*)$ is again satisfied already. In order to see this, assume that $(G,u,v)$ has a solution $(S_u,S_v)$, where $S_u$ is an independent set, but $S_v$ contains two adjacent vertices $x_1$ and $x_2$. We assume without loss of generality that $x_1$ and $x_2$ belong to $D_2$. Hence, $V(D_1)$ is anticomplete to $\{x_1,x_2\}$. Suppose $D_1\cap S_v\neq \emptyset$. Let $t\in D_1\cap S_v$. Recall that $(S_u,S_v)$ is not a $7$-constant solution, as $(G,u,v)$ does not have such solutions. Then, by Claim \[c-either\], we find that $S_v$ has a set $S_v'$ that contains $x_1,x_2$ but not $t$, such that every vertex of $N(v)$ is adjacent to a vertex of $S_v'$ and $S_v'\cup N(v)$ is connected. Hence, we may put $t$ into $S_u$. Similarly, we may put every other vertex of $D_1\cap S_v$ into $S_u$. As $T_2$ is an independent set by Claim \[c-t2\], at least one vertex of $D_1$ belongs to $T_1$ and is thus adjacent to $w_u\in N(v)$ by definition. This means that $(S_u\cup (V(D_1)\cap S_v),S_v\setminus V(D_1))$ is another solution for $(G,u,v)$. However, this solution is double-sided, a contradiction. So, from now on, we assume that $D_1$ contains more than one vertex and that $D_2,\ldots,D_q$ each have exactly one vertex.
Recall that $T_2$ is an independent set that is anticomplete to $N(v)$ due to Claim \[c-t2\]. Suppose $t\in T_2$ does not belong to $D_1$. Then $t$ is an isolated vertex of $G[T]$ that is not adjacent to any vertex of $N(v)$. As $G$ is connected, $t$ is adjacent to at least one vertex of $N(u)$. We apply the [Contraction Rule]{} on $N(u)\cup \{t\}$ to contract $t$ away. Afterwards, we find that every vertex of $T_2$ must belong to $D_1$.
Let $B_1,\ldots,B_p$ be the connected components of $G[T_1\cap V(D_1)]$ for some $p\geq 1$. By Claim \[c-p4free\], $G[T]$, and thus $G[T_1\cap V(D_1)]$, is $P_4$-free (note that we only contracted edges during the branching and thus maintained $P_4$-freeness due to Lemma \[l-contract\]). As $G[T_1]$ is also $K_3$-free by Claim \[c-t1\], each $B_i$ is a complete bipartite graph on one or more vertices due to Lemma \[l-p4\].
First suppose that $p=1$. Recall that $T_2$ is an independent set by Claim \[c-t2\] that belongs to $D_1$. In this case we show how to branch into $O(n^2)$ new and smaller instances, such that $(G,u,v))$ has a solution $(S_u,S_v)$, in which $S_u$ is an independent set, if and only if one of these new instances has such a solution. Moreover, each new instance will have the property that either $(*)$ has been obtained or $p\geq 2$ holds.
[**Branching IV**]{} ($O(n^2)$ branches)\
We consider each possibility of choosing one vertex $t \in B_1$ to be placed in $S_u$. This leads to $O(n)$ branches. In each branch we contract $t$ away by the [Contraction Rule]{} on $N(u)\cup \{t\}$ (note that $tw_u\in E(G)$, as $t\in T_1$). Since $S_u$ is an independent set, we must place all neighbours of $t$ in $S_v$. In order to contract these neighbours away using the [Contraction Rule]{} we may need to branch once more by considering every vertex that is in $B_1$ and that has at least one neighbour in $N(v)$. This leads to $O(n)$ additional branches. Hence, the total number of branches for this stage is $O(n^2)$. For each branch we observe that $T_1$ has become an independent set (as the vertices in the components $D_2,\ldots, D_q$ form an independent set as well). By applying the [Contraction Rule]{} we ensure that $T_2$ is an independent set that remains anticomplete to $N(v)$.
First suppose that $T_1\cap V(D_1)$ consists of a single vertex $t^*$. If $T_2\neq \emptyset$, then we do as follows. Recall that we are looking for a solution $(S_u,S_v)$ with $T_2\subseteq S_v$. As $N(v)\cup S_v$ must be connected but $T_2$ is anticomplete to $N(v)$ by Claim \[c-t2\], vertex $t^*$ must be placed into $S_v$. Hence, if $t^*$ is not adjacent to a vertex in $N(v)$, we discard the branch. Otherwise we contract $T_2\cup \{t^*\}$ away by applying the [Contraction Rule]{} on $N(v)\cup T_2\cup \{t^*\}$. Hence, we obtained $T_2=\emptyset$. As $T_1$ is an independent set, this means that $(*)$ holds.
Now suppose that $T_1\cap V(D_1)$ consists of more than one vertex. As $T_1$ is an independent set, this means that $G[T_1\cap V(D_1)]$ has $p\geq 2$ connected components. Hence, we have arrived in the case where $p\geq 2$. We denote the resulting instance by $(G,u,v,)$ again, and we let also $B_1,\ldots,B_p$ denote the connected components of $G[T_1\cap V(D_1)]$ again.
From the above we are now in the situation where $(G,u,v)$ is an instance for which $p\geq 2$ holds. By Lemma \[l-p4\] and because $D_1$ is connected and $P_4$-free, $D_1$ has a spanning complete bipartite graph $B^*$. As $p\geq 2$, all vertices of $V(B_1)\cup \cdots \cup V(B_p)$ belong to the same partition class of $B^*$. By definition, these vertices are in $T_1$. Hence, as $T_2$ is an independent set in $D_1$, all vertices of $T_2$ form the other bipartition class of $B^*$. Consequently, $T_2$ is complete to $T_1\cap V(D_1)$. We will do some branching.
[**Branching V**]{} ($O(n)$ branches)\
Every vertex of $T_2$ will belong to $S_v$ in any solution $(S_u,S_v)$ where $S_u$ is an independent set, but without having any neighbours in $N(v)$ due to Claim \[c-t2\]. This means that $S_v$ contains at least one vertex $t$ of $V(D_1)\cap T_1$. We branch by considering all possibilities of choosing this vertex $t$. Indeed, as $T_2$ is complete to $T_1$, it suffices to check single vertices $t\in T_1$ that have a neighbour in $N(v)$. This leads to $O(n)$ branches. For each branch we do as follows. We contract the vertices of $T_2\cup \{t\}$ away using the [Contraction Rule]{} on $N(v)\cup T_2\cup \{t\}$. We denote the resulting instance by $(G,u,v)$ and observe that $T_2=\emptyset$, so $T=T_1$.
Note that $G[T]=G[T_1]$ now consists of connected components $B_1',\ldots,B_{p'}'$ for some $p'\geq 1$, where each $B_i'$ is a complete bipartite graph. If every $B_i'$ consists of a single vertex, then $G[T]$ is an independent set. Hence, any solution for $(G,u,v)$ will be independent. We conclude that $(*)$ holds. Now suppose that at least one of $B_1',\ldots,B_{p'}'$, say $B_1'$, has more than one vertex. If another $B_i'$, say $B_2'$, also has more than one vertex, then $(*)$ is also satisfied already. We can show this in the same way as before, namely when we proved this for the sets $D_1,\ldots,D_q$. From now on we may assume that $B_1'$ consists of more than one vertex and that $B_2',\ldots,B_{p'}'$ have only one vertex. So, in particular, $B_1'$ is a complete bipartite graph on at least two vertices. We will do some branching.
[**Branching VI**]{} ($O(n^2)$ branches)\
We consider each possibility of choosing one vertex $t \in B_1'$ to be placed in $S_u$. This leads to $O(n)$ branches. In each branch we contract $t$ away by applying the [Contraction Rule]{} on $N(u)\cup \{t\}$ (note that $tw_u\in E(G)$, as $t\in T_1$, so we can indeed do this). Since $S_u$ is an independent set, we must place all neighbours of $t$ in $S_v$. In order to contract these neighbours away using the [Contraction Rule]{} we proceed as follows. All neighbours of $t$ that are adjacent to $N(v)$ we can contract away by applying the [Contraction Rule]{} on $N(v)\cup \{t\}$. If all neighbours of $t$ disappeared this way, this yields $T=T_1$, an independent set as required. Otherwise, we need to include another vertex of $B'_1$ into $S_v$. So we branch on the $O(n)$ vertices $t' \in B_1'$ that are adjacent to $v$. Contracting such a $t'$ away makes all other neighbours of $t$ adjacent to $N(v)$ and we can contract them away. In any case, eventually we will end up with $T=T_1$ being independent. Consequently, $S_v$ must be an independent set for any solution $(S_u,S_v)$ where $S_u$ is an independent set. This means that we achieved $(*)$.
If we have not yet found a solution, then by achieving $(*)$, as shown above, we have reduced the problem to $O(n^5)$ instances, for which we search for an independent solution. We consider these new instances one by one. For simplicity, we denote the instance under consideration by $(G,u,v)$ again.
[**Phase 3c: Searching for private solutions**]{}
In this phase we introduce a new type of independent solution that we call private. In order to define this notion, we first describe our branching procedure which will get us to this new notion.
[**Branching VII.**]{} ($O(n^4)$ branches)\
First we process $N(v)$ in the same way as we did for $N(u)$ in Branching II. That is, in polynomial time via $O(n^4)$ branches, we find a partition of $N(v)$ into a set $Q_v$ of private neighbours and a vertex $w_v$ that will be complete to $S_v$. To be more specific, if $(G,u,v)$ has a solution $(S_u,S_v)$ in which $S_u$ and $S_v$ are independent sets, then the following holds for such a solution $(S_u,S_v)$:
- The independent set $S_u$ contains a subset $S_u^*$ of size at least 2 that covers $N(u)$, such that each $s\in S_u^*$ has a nonempty set $Q_u(s)$ of private neighbours with respect to $S_u^*$, and moreover, the set $N(u)\setminus Q_u$, where $Q_u=\bigcup Q_u(s)$, consists of a single vertex $w_u$ that is complete to $S_u^*$.
- The independent set $S_v$ contains a subset $S_v^*$ of size at least 2 that covers $N(v)$, such that each $s\in S_v^*$ has a nonempty set $Q_v(s)$ of private neighbours with respect to $S_v^*$, and moreover, the set $N(v)\setminus Q_v$, where $Q_v=\bigcup Q_v(s)$, consists of a single vertex $w_v$ that is complete to $S_v^*$.
We call an independent solution $(S_u,S_v)$ satisfying (P1) and (P2) a *private solution*. We emphasize that by now all branches are guaranteed to have private solutions or no solutions at all. Thus in what follows we will only search for private solutions. While doing this we may modify the instance $(G,u,v)$, but we will always ensure that private solutions are pertained. In particular, if we contract a vertex $t \in S_u^*$ to $w_u$ using the [Contraction Rule]{} on $N(u)\cup \{t\}$, then this leads to a private solution $(S_u,S_v)$ with $t \notin S_u^*$. Then all private neighbours of $t$ become adjacent to $w_u$ and, by the [Contraction Rule]{}, they get contracted to $w_u$ as well. However, if $t \notin S_u^*$, then contracting $t$ to $w_u$ will make the neighbours of $t$ in $N(u)$ adjacent to $w_u$ and the [Contraction Rule]{} contracts these to $w_u$. As a consequence, some vertices in $S_u^*$ may have no private neighbours in $N(u)$ and hence leave $S_u^*$. If this reduces $|S_u^*|$ to $1$, then we will notice this by checking for $1$-constant solutions, which takes polynomial time due to Lemma \[l-constant\]. If we find a $1$-constant solution, then we stop and conclude that our original instance is a yes-instance. Otherwise, we know that $|S_u^*| \ge 2$, and hence private solutions pertain (should such solutions exist at all). In the remainder, we will perform this test implicitly whenever we apply the [Contraction Rule]{}.
We now prove the following two claims.
[[*\[c-both\]Every vertex of $T$ is adjacent to both $w_u$ and $w_v$.*]{}\
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*Proof of Claim \[c-both\].* Consider a vertex $t\in T$. First suppose that $t\in T$ is neither adjacent to $w_u$ nor to $w_v$. As $G$ is connected, $t$ will be adjacent to some other vertex in $S_u$ or $S_v$ in every solution $(S_u,S_v)$. Hence, $(G,u,v)$ has no independent solutions, and thus no private solutions, and we can discard the branch. From now on assume that every vertex in $T$ is adjacent to at least one of $w_u$, $w_v$. If $t\in T$ is adjacent to only $w_u$ and not to $w_v$, then by the same argument we must apply the [Contraction Rule]{} on $N(u)\cup \{t\}$. Similarly, if $t\in T$ is adjacent to only $w_v$ and not to $w_u$, then we must apply the [Contraction Rule]{} on $N(v)\cup \{t\}$. We discard a branch whenever two adjacent vertices in $T$ were involved in an edge contraction with some neighbour in $N(u)$, or with some neighbour in $N(v)$.
[[*\[c-bipartite\] If $(G,u,v)$ has a private solution, then $G[T]$ must be the disjoint union of one or more complete bipartite graphs.*]{}\
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[*Proof of Claim \[c-bipartite\].*]{} If $G[T]$ is not bipartite, then $(G,u,v)$ has no independent solution $(S_u,S_v)$, as $T=S_u\cup S_v$. Hence, $(G,u,v)$ has no private solution. Assume that $G[T]$ is bipartite. By Claim \[c-p4free\], $G[T]$ is $P_4$-free. Then the claim follows by Lemma \[l-p4\].
By Claim \[c-bipartite\] we may assume that $G[T]$ is the disjoint union of one or more complete bipartite graphs; otherwise we discard the branch.
We now prove that $T$ can be changed into an independent set via some branching. Suppose $T$ is not an independent set yet. Let $B_1,\ldots,B_r$, for some $r\geq 1$, denote the connected components of $G[T]$ that have at least one edge (note that $G[T]$ may also contain some isolated vertices). By Claim \[c-bipartite\], every $B_i$ is complete bipartite.
[[*\[c-4\] If $(G,u,v)$ has a private solution, then $r\leq 3$.*]{}\
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*Proof of Claim \[c-4\].* Assume that $r\geq 4$. We will prove that $(G,u,v)$ has no private solution. Suppose that $T$ contains four connected components with edges, say $B_1,\ldots, B_4$, for which the following holds: $V(B_1)$ covers some subset $A_1^u\subseteq N(u)$ and $V(B_2)$ covers some subset $A_2^u\subseteq N(u)$, such that $A_1^u\setminus A_2^u \neq \emptyset$, whereas $V(B_3)$ covers some subset $A_3^v\subseteq N(v)$ and $V(B_4)$ covers some subset $A_4^v\subseteq N(v)$, such that $A_3^v\setminus A_4^v \neq \emptyset$. Let $w\in A_1^u\setminus A_2^u$, say $w$ is adjacent to vertex $s$ of $B_1$ (and not to any vertex of $B_2$). Let $x_1$ and $x_2$ be two adjacent vertices of $B_2$, which exist as $B_2$ contains an edge. Suppose $V(B_1)\cup V(B_2)$ does not cover $N(u)$. Then there exists a vertex $w'$ that has no neighbour in $V(B_1)\cup V(B_2)$. However, then $\{x_1,x_2\}\cup \{s,w,u,w'\}$ is an induced $P_2+P_4$ of $G$, a contradiction. Hence, $V(B_1) \cup V(B_2)$ covers $N(u)$. Similarly, we find that $V(B_3)\cup V(B_4)$ covers $N(v)$. Then, as each vertex of $T$ is adjacent to both $w_u$ and $w_v$, we find that $G[N(u)\cup V(B_1)\cup V(B_2)]$ and $G[N(u)\cup V(B_3)\cup V(B_4)]$ are connected. This is not possible, as then the original instance has a double-sided solution, which we already ruled out after Claim \[c-double\].
If two sets from $V(B_1),\ldots,V(B_r)$, say $V(B_1)$ and $V(B_2)$, cover the same subset of $N(u)$ *and* the same subset of $N(v)$, then we can apply the [Contraction Rule]{} on $N(u)\cup V(B_1)$ and on $N(v)\cup V(B_2)$ to find that the original instance has a double-sided solution if it has a solution. However, as we already ruled this out, this is not possible either.
Now consider the sets $B_1$ and $B_2$. From the above, we deduce the following. We may assume without loss of generality that $V(B_1)$ and $V(B_2)$ cover different subsets of $N(u)$. This implies that $V(B_3),\ldots, V(B_r)$ all cover the same subset $A$ of $N(v)$. We can also apply the above on $B_1$ and $B_3$ to find that $B_1$ and $B_3$ must either cover different subsets of $N(u)$ or different subsets of $N(v)$. Suppose $B_1$ and $B_3$ cover different subsets of $N(v)$. Then, again from the above, $B_2, B_4,\ldots, B_r$ must cover the same subset of $N(u)$. As $B_1$ and $B_2$ cover different subsets of $N(u)$, this means that $B_1$ and $B_4$ cover different subsets of $N(u)$. This implies that $B_2$ must cover the same set $A$ as $B_4,\ldots,B_r$. As $B_3$ covers $A$ as well, this means that $B_2$ and $B_3$ cover the same subset of $N(v)$. Hence, they must cover different subsets of $N(u)$. However, the latter implies that $B_1$ and $B_4$ cover the same subset of $N(v)$. As $B_4$ covers $A$, just like $B_3$, we find that $B_1$ and $B_3$ cover the same subset $A$ of $N(v)$, a contradiction. Hence, $B_1$ and $B_3$ cover the same subset of $N(v)$, namely $A$, and by symmetry the same holds for $B_2$.
As $V(B_1)$ covers $A_1^u$ and $V(B_2)$ covers $A_2^u$ such that $A_1^u\setminus A_2^u \neq \emptyset$, we can use the same arguments as before to deduce that $V(B_1)$ and $V(B_2)$ must cover $N(u)$. We put the vertices of $B_1$ and $B_2$ into $S_u$ and the vertices of $B_i$ for $i\geq 3$ plus all other (isolated) vertices of $T$ into $S_v$. If $(S_u,S_v)$ is a solution for $(G,u,v)$, then the original solution has a double-sided solution, which we already ruled out. Hence, there exists a vertex $z$ of $N(v)$ that is not adjacent to any vertex of $T\setminus (V(B_1)\cup V(B_2))$. However, as every $V(B_i)$ covers the same subset $A$ of $N(v)$, no vertex of $B_1$ and $B_2$ is adjacent to $z$ either. This implies that $(G,u,v)$ is a no-instance, meaning that we can discard this branch.
By Claim \[c-4\] we may assume that $r\leq 3$, that is, $G[T]$ has at most three connected components $B_i$ with an edge; otherwise we discard the branch. As $r\leq 3$, we can now do some branching to obtain $O(1)$ smaller instances in which $T$ is an independent set, such that $(G,u,v)$ has a private solution if and only if at least one of these new instances has a private solution.
[**Branching VIII**]{} ($O(1)$ branches)\
For $i=1,\ldots, r$, let $Y_i$ and $Z_i$ be the bipartition classes of $B_i$. Let $1\leq i\leq r$. As $S_u$ and $S_v$ must be independent sets and every $B_i$ is complete bipartite, either $Y_i$ belongs to $S_u$ and $Z_i$ belongs to $S_v$, or the other way around. We branch by considering both possibilities. We do this for each $i\in \{1,\ldots, r\}$. This leads to $2^r\leq 2^3$ branches, as $r\leq 3$ due to Claim \[c-4\]. In each branch we apply the [Contraction Rule]{} to contract $Y_i$ and $Z_i$ away (note that here our remark about pertaining private solutions applies). We consider every resulting instance separately. We denote such an instance again by $(G,u,v)$, for which we have proven the following claim.
[[*\[c-indep\] $T$ is an independent set.*]{}\
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We now continue as follows. As $T$ is an independent set by Claim \[c-indep\], the sets $S_u$ and $S_v$ of any solution $(S_u,S_v)$ will be independent (should $(G,u,v)$ have a solution). Recall also that $\{w_u,w_v\}$ is complete to $T$ by Claim \[c-both\]. We are looking for a private solution $(S_u,S_v)$, which we recall is an independent solution for which sets $S_u^*$ and $S_v^*$ exist so that (P1) and (P2) are satisfied. We make the following observation. Let $R=T\setminus (S_u^*\cup S_v^*)$ be the set of all other vertices of $T$. Consider a vertex $z\in R$. We note that if $z\in S_u$, then $(S_u\setminus \{z\},S_v\cup \{z\})$ is also a solution for $(G,u,v)$; this follows from (P1) and (P2) and the fact that $w_v$ is adjacent to $z\in T$. Similarly, if $z\in S_v$, then $(S_u\cup \{z\},S_v\setminus \{z\})$ is a solution as well.
We prove the following four claims.
[[*\[c-atleast\] Let $w\in N(u)\cup N(v)$. Then we may assume without loss of generality that $w$ is adjacent to at least two vertices of $T$.*]{}\
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[*Proof of Claim \[c-atleast\].*]{} Suppose $N(u)$ or $N(v)$, say $N(u)$, contains a vertex $w$ that is adjacent to at most one vertex of $T$. If $w$ has no neighbours in $T$, then $(G,u,v)$ has no solution and we discard the branch. Suppose $w$ has exactly one neighbour $z\in T$. Then $z$ belongs to $S_u^*$ for every (private) solution $(S_u,S_v)$ of $(G,u,v)$ (assuming $(G,u,v)$ is a yes-instance). Hence, we may apply the [Contraction Rule]{} on $N(u)\cup \{z\}$. We apply this operation exhaustively, while pertaining private solutions as before.
It may happen that in this process it turns out that two vertices $z,z'$ both belong to $S_u^*$ for every (private) solution $(S_u,S_v)$ of $(G,u,v)$, while they share a neighbour in $N(u)\setminus \{w_u\}$. This contradicts (P1). Hence, in this case we find that $(G,u,v)$ does not have a private solution and we may discard the branch. Otherwise, in the end, we have obtained in polynomial time an instance with the desired property. As we ensure that private solutions pertain, the size of $S_u^*$ remains at least $2$.
[[*\[c-notall\] Let $z\in T$. Then we may assume without loss of generality that $z$ is non-adjacent to at least one vertex of $N(u)$ and to at least one vertex of $N(v)$.*]{}\
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[*Proof of Claim \[c-notall\].*]{} Suppose $z\in T$ is adjacent to all vertices of $N(u)$ or to all vertices of $N(v)$, say to all vertices of $N(u)$. Then we can check in polynomial time if $T\setminus \{z\}$ covers $N(v)$. If so, then $\{z,T\setminus \{z\}$ is a ($1$-constant) solution of $(G,u,v)$ and we can stop. Otherwise $z$ must belong to $S_v$ for any solution $(S_u,S_v)$ of $(G,u,v)$. In that case we may apply the [Contraction Rule]{} on $N(v)\cup \{z\}$. We apply this operation exhaustively (we again recall that we ensure that private solutions pertain by checking for $1$-constant solutions). Moreover, it may happen that during this process two vertices $z,z'$ will end up in the same set $S_u$ or $S_v$ for any private solution $(S_u,S_v)$, while sharing a neighbour in $N(u)\setminus \{w_u\}$. As the sets $S_u$ and $S_v$ are independent in a private solution $(S_u,S_v)$, this means that $(G,u,v)$ does not have a private solution and we may discard the branch. Otherwise, in the end, we have obtained in polynomial time an instance with the desired property.
[[*\[c-three\] Let $s$ and $t$ be any two distinct vertices of $T$. Then we may assume without loss of generality that either $N(u)\cap N(s)\cap N(t)=\{w_u\}$; or $N(u)\cap N(s)=N(u)\cap N(t)$; or $\{s,t\}$ covers $N(u)$. Similarly, we may assume without loss of generality that either $N(v)\cap N(s)\cap N(t)=\{w_v\}$; or $N(v)\cap N(s)=N(v)\cap N(t)$; $\{s,t\}$ covers $N(v)$.*]{}\
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[*Proof of Claim \[c-three\].*]{} By symmetry it suffices to prove only the first statement. Assume $T$ contains two vertices $s$ and $t$, for which there exist distinct vertices $w\in (N(u)\setminus \{w_u\})\cap N(s)\cap N(t)$; $w'\in (N(u)\cap N(s))\setminus N(t)$ and $w''\in N(u)\setminus (N(s)\cup N(t))$. Note that $w_u\notin \{w,w',w''\}$. Recall that in this stage we are looking for private solutions for $(G,u,v)$. Consider an arbitrary private solution $(S_u,S_v)$ (if it exists). Then $w''\in Q_u(z)$ for some $z\in S_u^*$. Note that $z\notin \{s,t\}$, as neither $s$ nor $t$ is adjacent to $w''$.
The above means that $z$ must be adjacent to at least one of $w,w'$, as otherwise the set $\{w'',z\}\cup \{w',s,w,t\}$ induces a $P_2+P_4$ in $G$, which is not possible. Hence, at least one of $w$ or $w'$ will be a private neighbour of $z$, that is, will belong to $Q_u(z)$. As $s$ is adjacent to both $w$ and $w'$ and $N(u)\setminus Q_u=\{w_u\}$ (see property (P1) of the definition of a private solution), this means that $s$ does not belong to $S_u^*$. We conclude that $s$ belongs to $R=T\setminus (S_u^*\cup S_v^*)$ or to $S_v^*$ for any private solution $(S_u,S_v)$ of $(G,u,v)$. As $s$ is adjacent to $w_v\in N(v)$, we may therefore apply the [Contraction Rule]{} on $N(v)\cup \{s\}$, ensuring persistence of private solutions (in case there are any) in the usual way. We do this exhaustively, and in the end we find that the claim holds. Note that we obtained this situation in polynomial time.
[[*\[c-wu\] Let $s$ and $t$ be any two distinct vertices of $T$ that together cover $N(u)$. Then there exists a nonempty set $A(v)\subseteq N(v)$ that is complete to $\{s,t\}$ and anticomplete to $T\setminus \{s,t\}$, or $(G,u,v)$ has a $2$-constant solution. The same holds for $u$ and $v$ interchanged.*]{}\
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[*Proof of Claim \[c-wu\].*]{} Assume without loss of generality that $\{s,t\}$ covers $N(u)$. Then we find that $(\{s,t\}, T\setminus \{s,t\})$ is a $2$-constant solution unless $N(v)$ contains a nonempty set $A(v)$ that is anticomplete to $T\setminus \{s,t\}$). By Claim \[c-atleast\] we find that $A(v)$ is complete to $\{s,t\}$.
We will use Claims \[c-atleast\]–\[c-wu\] to prove the following claim.
[[*\[c-notboth\] Let $s$ and $t$ be two distinct vertices in $T$ such that $\{s,t\}$ covers $N(u) \cup N(v)$. Then $(G,u,v)$ has a $2$-constant solution.*]{}\
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[*Proof of Claim \[c-notboth\].*]{} Assume that $(G,u,v)$ has no $2$-constant solution. Then by Claim \[c-wu\], there is a nonempty set $A(u)\subseteq N(u)$ that is complete to $\{s,t\}$ and anticomplete to $T\setminus \{s,t\}$. Similarly, there exists a nonempty set $A(v)\subseteq N(v)$ that is complete to $\{s,t\}$ and anticomplete to $T\setminus \{s,t\}$. By Claim \[c-notall\] we find that $N(u)$ contains a vertex $w$ that is not adjacent to $s$. As $\{s,t\}$ covers $N(u)$, this means that $w$ is adjacent to $t$. By Claim \[c-atleast\] we find that $w$ is adjacent to some vertex $s'\in T\setminus \{s,t\}$. As $s'$ is anticomplete to $A(u)$, Claim \[c-three\] tells us that $\{s',t\}$ covers $N(u)$. By the same argument, there exists a vertex $t'$ such that $\{s,t'\}$ covers $N(v)$. Putting $s',t$ in $S_u$ and $s,t'$ in $S_v$ (together with all other vertices of $T$) yields a $2$-constant solution $(S_u,S_v)$ of $(G,u,v)$. This is a contradiction.
We continue as follows. By Lemma \[l-constant\] we check in polynomial time if $(G,u,v)$ has a $2$-constant solution. If so, then we are done. Otherwise, we obtain the following claim, which immediately follows from Claim \[c-notboth\] and the fact that if one pair of vertices of $T$ covers $N(u)$ and another pair covers $N(v)$, then we obtained a $2$-constant solution.
[[*\[c-onlyr\] We may assume without loss of generality that every pair of (distinct) vertices $\{s,t\}$ in $T$ does not cover $N(u)$; hence, $\{s,t\}$ may only cover $N(v)$.*]{}\
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We call a pair of vertices $s,t$ of $T$ a [*$2$-pair*]{} if $\{s,t\}$ covers $N(v)$. Let $T_v$ be the set of vertices of $T$ involved in a 2-pair. We continue by proving the following claim.
[[*\[c-2pair\] Every vertex of $T_v$ belongs to exactly one $2$-pair.*]{}\
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[*Proof of Claim \[c-2pair\].*]{} Let $s\in T_v$. By definition, $s$ belongs to at least one $2$-pair. For contradiction, suppose that $s$ belongs to more than one $2$-pair. Then there exist vertices $t_1$, $t_2$ in $T_v$, such that $\{s,t_1\}$ and $\{s,t_2\}$ both cover $N(v)$. As $(G,u,v)$ has no $2$-constant solution, $N(u)$ contains a nonempty set $A_1(u)\subseteq N(u)$ that is complete to $\{s,t_1\}$ and anticomplete to $T\setminus \{s,t_1\}$ due to Claim \[c-wu\]. By the same claim, $N(u)$ contains a nonempty set $A_2(u)\subseteq N(u)$ that is complete to $\{s,t_2\}$ and anticomplete to $T\setminus \{s,t_2\}$. Let $w_1\in A_1(u)$ and $w_2\in A_2(u)$; note that $w_2\neq w_u$. Then $w_1$ is adjacent to $s$ but not to $t_2$, whereas $w_2\neq w_u$ is a common neighbour of $s$ and $t_2$. As $\{s,t_2\}$ does not cover $N(u)$ due to Claim \[c-onlyr\], this contradicts Claim \[c-three\].
We next prove that actually $T_v = \emptyset$. Suppose that $T_v\neq \emptyset$. Let $(s,t) \in T_v$. By Claim \[c-wu\], there exists a nonempty subset $A(u)$ of $N(u)$ that is complete to $\{s,t\}$ and anticomplete to $T\setminus \{s,t\}$. As $(G,u,v)$ has no $2$-constant solution, $s$ and $t$ do not cover all of $N(u)$. By Claim \[c-notall\], we find that $s$ is not adjacent to some vertex $w\in N(v)$. As $(s,t)$ is a 2-pair, $t$ is adjacent to $w$. By Claim \[c-atleast\], we find that $w$ is adjacent to a vertex $z\in T\setminus \{s,t\}$. From Claim \[c-2pair\] it follows that $(t,z)$ is not a 2-pair, so $t$ and $z$ do not cover all of $N(v)$. By Claim \[c-three\] and the fact that $t$ and $z$ have a common neighbour different from $w_v$, namely $w$, this means that $t$ and $z$ are adjacent to the same neighbours in $N(v)$. However, then $(s,z)$ is 2-pair, contradicting Claim \[c-2pair\]. This means that we have indeed proven the following claim.
[[*\[c-tempty\] $T_v=\emptyset$.*]{}\
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[**Phase 3d: Translating the problem into a matching problem**]{}
We are now ready to translate the instance $(G,u,v)$ into an instance of a matching problem. Recall that $w_u$ and $w_v$ are the vertices in $N(u)$ and $N(v)$ that are complete to $T$. By Claims \[c-three\] and \[c-tempty\] we can partition $N(u)\setminus \{w_u\}$ into sets $N_1(u)\cup \dots \cup N_q(u)$ for some $q\geq 1$ such that two vertices of $N(u)$ have the same set of neighbours in $T$ if and only if they both belong to $N_i(u)$ for some $i\in \{1,\ldots,q\}$. Similarly, we can partition $N(v)\setminus \{w_v\}$ into sets $N_1(v)\cup \dots \cup N_r(v)$ for some $r\geq 1$ such that two vertices of $N(v)$ have the same set of neighbours in $T$ if and only if they both belong to $N_i(v)$ for some $i\in \{1,\ldots,r\}$. We may remove all but one vertex of each $N_h(u)$ and each $N_i(v)$ to obtain an equivalent instance, which we denote by $(G,u,v)$ again.
Let $G'$ be the graph obtained from $G$ after removing the vertices $u,v,w_u,w_v$ and every edge between a vertex of $N(u)$ and a vertex of $N(v)$. Note that $G'$ is bipartite with partition classes $(N(u)\setminus \{w_u\})\cup (N(v)\setminus \{w_v\})$ and $T$. It remains to compute a maximum matching $M$ in $G'$. We can do this by using the Hopcroft-Karp algorithm, which runs in $O(m\sqrt{n})$-time on bipartite graphs with $n$ vertices and $m$ edges. If $|M|=|N(u)|+|N(v)|-2$, then each vertex in $(N(u)\setminus \{w_u\})\cup (N(v)\setminus \{w_v\})$ is incident to an edge of $M$, and hence, we found a (private) solution for $(G,u,v)$. If $|M|<|N(u)|+|N(v)|-2$, then $(G,u,v)$ has no (private) solution, and we discard the branch.
The above concludes the description of the $u$-feasibility check. If we found a branch with a solution, then we translate it in polynomial time to a solution for the original instance. Otherwise we perform Phase 4.
[**Phase 4: Doing a $\mathbf{v}$-feasibility check**]{}
As mentioned, our algorithm now does a $v$-feasibility check, that is, it checks for the existence of a solution $(S_u,S_v)$, where $S_v$ is an independent set and $G[S_u]$ may contain edges. As we can repeat exactly the same steps as in Phase 3, this phase takes polynomial time as well. This concludes the description of our algorithm.
The correctness of our algorithm follows from the above description. We now analyze its run-time. The branching is done in eight stages, namely Branching I-VIII and yields a total number of $O(n^{30})$ branches. As explained in each step above, processing each branch created in Branching I-VI until we start branching again takes polynomial time. Checking for $1$-constant solutions to ensure survival of private solutions takes constant time as well. Moreover, processing each of the branches created in Branch VII takes polynomial time as well. We conclude that the total running time of our algorithm is polynomial.
Via Lemma \[l-reduce\] and a reduction to $P_4$-[Suitability]{} we obtain:
\[l-top5\]$P_5$-[Suitability]{} can be solved in polynomial time for $(P_2+P_4)$-free graphs.
Let $(G,u,v)$ be an instance of $P_5$-[Suitability]{}, where $G$ is a connected $(P_2+P_4)$-free graph. We may assume without loss of generality that $u$ and $v$ are of distance at least 4 from each other, as otherwise $(G,u,v)$ is a no-instance. By the [Contraction Rule]{} and Lemma \[l-contract\] we may also assume without loss of generality that $N(u)$ and $N(v)$ are both independent sets; otherwise if, say, $G[N(u)]$ contains an edge $e$, then we contract $e$ to obtain an equivalent but smaller instance $(G',u,v)$, where $G'$ is also $(P_2+P_4)$-free due to Lemma \[l-contract\].
First suppose $|N(u)|=1$, say $N(u)=\{u'\}$ for some $u'\in V(G)$. Then we solve $P_4$-[Suitability]{} on instance $(G-u,u',v)$. We can do this in polynomial time due to Lemma \[l-top4\].
Now suppose $|N(u)|\geq 2$. Note that $\operatorname{dist}(u,v) \leq 5$, as $G$ is $P_7$-free. By Lemma \[l-reduce\] we may assume that $\operatorname{dist}(u,v)=4$. We will explore the structure of the $P_5$-witness bags $W(p_2)$ and $W(p_3)$ should they exist. Let $Z$ be the set that consists of all vertices $z$ with $\operatorname{dist}(u,z)=\operatorname{dist}(z,v)=2$. Then $Z$ must be a subset of $W(p_3)$. As $N(u)$ is not connected, $W(p_2)$ must contain at least one other vertex $s$ adjacent to some vertex $t \in N(u)$. Suppose $s$ is non-adjacent to some other vertex $t'\in N(u)$. Let $w$ be a neighbour of $v$. As $s \in W(p_2)$ and $w \in W(p_4)$, we find that $s$ and $w$ are not adjacent. Then the set $\{v,w\}\cup \{s,t,u,t'\}$ induces a $P_2+P_4$ in $G$, a contradiction. Hence, $s$ is adjacent to every vertex of $N(u)$. We consider all possibilities of choosing vertex $s$ from the set $V(G)\setminus (N[u]\cup N[v]\cup Z)$. This leads to $O(n)$ branches. In each branch we contract the set $N(u)\cup \{s\}$ to a single vertex $u'$. Let $G'$ be the resulting graph. Then we solve $P_4$-[Suitability]{} on instance $(G',u',v)$. As $G'$ is $(P_2+P_4)$-free by Lemma \[l-contract\], we can do this in polynomial time due to Lemma \[l-top4\].
From the above we conclude that we can check in polynomial time if $(u,v)$ is a $P_5$-suitable pair of $G$.
We use Lemma \[l-top5\] to prove Lemma \[l-top6\].
\[l-top6\]$P_6$-[Suitability]{} can be solved in polynomial time for $(P_2+P_4)$-free graphs.
Let $(G,u,v)$ be an instance of $P_6$-[Suitability]{}, where $G$ is a connected $(P_2+P_4)$-free graph. We may assume without loss of generality that $u$ and $v$ are of distance at least 5 from each other, as otherwise $(G,u,v)$ is a no-instance. We may also assume without loss of generality that $N(u)$ and $N(v)$ are both independent sets; otherwise if, say, $G[N(u)]$ contains an edge $e$, then we contract $e$ to obtain an equivalent but smaller instance $(G',u,v)$, where $G'$ is also $(P_2+P_4)$-free due to Lemma \[l-contract\].
First suppose $|N(u)|=1$, say $N(u)=\{u'\}$ for some $u'\in V(G)$. Then we solve $P_5$-[Suitability]{} on instance $(G-u,u',v)$. We can do this in polynomial time due to Lemma \[l-top5\].
Now suppose $|N(u)|\geq 2$. We assume $W(p_1)=\{u\}$ and we will explore the structure of the $P_6$-witness bag $W(p_2)$ should it exist. As $N(u)$ is not connected, $W(p_2)$ must contain at least one other vertex $s$. Suppose that $s$ is adjacent to some vertex $t\in N(u)$ and non-adjacent to some other vertex $t'\in N(u)$. Let $w$ be a neighbour of $v$. Then the set $\{v,w\}\cup \{s,t,u,t'\}$ induces a $P_2+P_4$ in $G$, a contradiction. Hence, $s$ is adjacent to every vertex of $N(u)$. We consider all possibilities of choosing vertex $s$ from the set $V(G)\setminus (N[u]\cup N[v])$. This leads to $O(n)$ branches. In each branch we contract the set $N(u)\cup \{s\}$ to a single vertex $u'$. Let $G'$ be the resulting graph. Then we solve $P_5$-[Suitability]{} on instance $(G',u',v)$. As $G'$ is $(P_2+P_4)$-free by Lemma \[l-contract\], we can do this in polynomial time due to Lemma \[l-top5\].
From the above we conclude that we can check in polynomial time if $(u,v)$ is a $P_6$-suitable pair of $G$.
We now combine Lemmas \[l-outer\] and \[l-trivial\] with Lemmas \[l-top4\]–\[l-top6\] to obtain the following theorem.
\[t-p2p4\] The [Longest Path Contractibility]{} problem is polynomial-time solvable for $(P_2+P_4)$-free graphs.
Let $G$ be a connected $(P_2+P_4)$-free graph. We may assume without loss of generality that $G$ has at least one edge. Note that $G$ is $P_7$-free. Hence, $G$ does not contain $P_7$ as a a contraction. By combining Lemmas \[l-top4\]–\[l-top6\] with Lemma \[l-outer\] we can check in polynomial time if $G$ contains $P_k$ as a contraction for $k=6,5,4$. If not, then we check if $G$ contains $P_3$ as a contraction by using Lemma \[l-trivial\] combined with Lemma \[l-outer\]. If not then, as $G$ has an edge, $P_2$ is the longest path to which $G$ can be contracted to.
The Case $\mathbf{H=P_1+P_2+P_3}$ {#s-p1p2p3}
---------------------------------
We will prove that [Longest Path Contractibility]{} is polynomial-time solvable for $(P_1+P_2+P_3)$-free graphs.
We will start by showing that $P_4$-[Suitability]{} is polynomial-time solvable for $(P_1+P_2+P_3)$-free graphs. The proof of this result uses similar but more simple arguments than the proof of Lemma \[l-top4\].
\[l-top4c\] The $P_4$-[Suitability]{} problem can be solved in polynomial time for $(P_1+P_2+P_3)$-free graphs.
Let $(G,u,v)$ be an instance of $P_4$-[Suitability]{}, where $G$ is a connected $(P_1+P_2+P_3)$-free graph. We may assume without loss of generality that $u$ and $v$ are of distance at least 3, that is, $u$ and $v$ are non-adjacent and $N(u)\cap N(v)=\emptyset$; otherwise $(G,u,v)$ is a no-instance.
Recall that $T=V(G)\setminus (N[u]\cup N[v])$. Recall also that we are looking for a partition $(S_u,S_v)$ of $T$ that is a solution for $(G,u,v)$, that is, $N(u)\cup S_u$ and $N(v)\cup S_v$ must both be connected. In order to do so we will construct partial solutions $(S_u',S_v')$, which we try to extend to a solution $(S_u,S_v)$ for $(G,u,v)$. We use the [Contraction Rule]{} from Section \[s-pre\] on $S_u'$ and $S_v'$, so that these sets will become independent. By Lemma \[l-contract\], the resulting graph will always be $(P_1+P_2+P_3)$-free. For simplicity, we denote the resulting instance by $(G,u,v)$ again. After applying the [Contraction Rule]{} the size of the set $T$ may be reduced by at least one. As before, if $t\in T$ was involved in an edge contraction with a vertex from $N(u)$ or $N(v)$ when applying the rule, then we say that we contracted $t$ away.
We start by applying the [Contraction Rule]{} on $N(u)$ and $N(v)$. This leads to the following claim.
[[*\[c-ind2\] $N(u)$ and $N(v)$ are independent sets.*]{}\
]{}\
We now check if $(G,u,v)$ has an $8$-constant solution, which we can check in polynomial time due to Lemma \[l-constant\]. If so, then $(G,u,v)$ is a yes-answer and we stop. From now on suppose that $(G,u,v)$ has no $8$-constant solution. Then we prove the following claim (recall that a solution $(S_u,S_v)$ is independent if $S_u$ and $S_v$ are independent sets).
[[*\[c-ind3\] Every solution of $(G,u,v)$ is independent (if $(G,u,v)$ has solutions).*]{}\
]{}\
[*Proof of Claim \[c-ind3\].*]{} Let $(S_u,S_v)$ be a solution for $(G,u,v)$ that is not independent, say $s,t$ belong to $S_u$ with $st\in E(G)$. If $\{s,t\}$ is anticomplete to a set of two neighbours $w,w'$ of $u$, then $\{v\}\cup \{s,t\}\cup \{w,u,w'\}$ is an induced $P_1+P_2+P_3$ of $G$, a contradiction. Hence, $\{s,t\}$ covers all but at most one vertex of $N(u)$. Suppose that $\{s,t\}$ covers $N(u)$, Then, as $s$ and $t$ are adjacent in $G$, we find that $(S_u,S_v)$ is a $2$-constant solution and thus a $8$-constant solution, which is not possible. Hence, $N(u)$ contains a unique vertex $w$ that is not adjacent to $s$ and $t$, but that is adjacent to some $z\in T\setminus \{s,t\}$. As $G$ is $(P_1+P_2+P_3)$-free, $G$ is $P_8$-free. Then $G[N(u)\cup S_u]$ contains a path $P$ on at most seven vertices from $s$ to $z$. The path $P$, together with vertex $t$ that may not be on $P$, shows that $(S_u,S_v)$ is a $8$-constant solution, a contradiction.
We will now analyze the structure of an independent solution $(S_u,S_v)$. As $S_u$ and $N(u)$ are both independent sets, $G[N(u)\cup S_u]$ is a connected bipartite graph. Hence, $S_u$ contains a set $S_u^*$, such that $S_u^*$ covers $N(u)$. We assume that $S_u^*$ has minimum size. Then each $s\in S_u^*$ has a nonempty set $Q(s)$ of vertices in $N(u)$ that are not adjacent to any vertex in $S_u^*\setminus \{s\}$; otherwise we can remove $s$ from $S_u^*$, contradicting our assumption that $S_u^*$ has minimum size. We call the vertices of $Q(s)$ the [*private*]{} neighbours of $s\in S_u^*$ with respect to $S_u^*$.
As $(G,u,v)$ has no $8$-constant solution, and thus no $1$-constant solution, we find that $S_u^*$ has size at least 2. Suppose $Q(s)$ contains at least two private neighbours $w_1,w_2$ of some vertex $s\in S_u^*$. As $|S_u^*|\geq 2$, there exists a vertex $s'\in S_u^*$ with $s'\neq s$. Let $w_3\in Q(s')$. Then $\{v\}\cup \{w_3,s'\}\cup \{w_1,s,w_2\}$ is an induced $P_1+P_2+P_3$ of $G$, a contradiction. Hence, each set $Q(s)$ has size 1. We denote the unique vertex of $Q(s)$ by $w_u^s$. So, $w_u^s$ is adjacent to $s$ but not to any other vertex from $S_u^*$. Let $Q_u$ be the set of all vertices $w_u^s$. Then $G[Q_u\cup S_u^*]$ is the disjoint union of $|S_u^*|$ edges.
We claim that the set $N(u)\setminus Q_u$ is complete to $S_u^*$. In order to see this, let $w\in N(u)\setminus Q_u$. By definition, $w$ is adjacent to at least two vertices $s_1,s_2$ of $S_u^*$. For contradiction, assume that $w$ is not adjacent to some vertex $s_3\in S_u^*$. Then $\{v\} \cup \{s_3,w_u^{s_3}\}\cup \{s_1,w,s_2\}$ induces a $P_1+P_2+P_3$ in $G$, which is not possible.
As $G[N(u)\cup S_u]$ is connected as well and $S_u$ is an independent set, every vertex $t\in S_u\setminus S_u^*$ must be adjacent to at least one vertex of $N(u)$. However, we claim that every vertex of $S_u\setminus S_u^*$ is adjacent to at most one vertex of $Q_u$. For contradiction, assume that $S_u\setminus S_u^*$ contains a vertex $t$ that is adjacent to two vertices of $Q_u$, say to $w_u^s$ and $w_u^{s'}$ for some $s,s'\in S_u^*$ with $s\neq s'$. Recall that $S_u$ is independent. Consequently, if $t$ is non-adjacent to $w_u^{s''}$ for some $s''\in S_u^*\setminus \{s,s'\}$, then $G$ contains an induced $P_1+P_2+P_3$ with vertex set $\{v\} \cup \{w_u^{s''},s''\} \cup \{w_u^s,t,w_u^{s'}\}$, a contradiction. Hence, $t$ is adjacent to every vertex of $Q_u$. If $t$ is adjacent to every vertex of $N(u)$, then $(G,u,v)$ has a $1$-constant solution, and this an $8$-constant solution, which we ruled out already. Hence, the set $N(u)\setminus N(t)$ is nonempty. As $t$ is adjacent to every vertex of $Q_u$, the set $N(u)\setminus N(t)$ is a subset of $N(u)\setminus Q_u$. Recall that $N(u)\setminus Q_u$ is complete to $S_u^*$. Hence, $N(u)\setminus N(t)$ is complete to $S_u^*$. Let $s\in S_u^*$. Then $\{s,t\}$ covers $N(u)$, and moreover $G[N(u)\cup \{s,t\}]$ is connected. This means that $(G,u,v)$ has a $2$-constant solution and thus an $8$-constant solution, which is not possible. We conclude that every vertex of $S_u\setminus S_u^*$ is adjacent to at most one vertex of $Q_u$.
Finally, we prove that $N(u)\setminus Q_u$ is nonempty. For contradiction, assume that $N(u)\setminus Q_u$ is empty. Then $N(u)=Q_u$. As $G[Q_u\cup S_u^*]$ is the disjoint union of a number of edges, and $G[N(u)\cup S_u]$ is connected, there must exist a vertex $t\in S_u\setminus S_u^*$ that is adjacent to at least two vertices of $Q_u$. However, we proved above that this is not possible. We conclude that $N(u)\setminus Q_u$ is nonempty.
We can deduce all the claims above with respect to $v$ as well. To summarize, any independent solution $(S_u,S_v)$ for $(G,u,v)$ satisfies the following two properties:
- The independent set $S_u$ contains a subset $S_u^*$ of size at least 2 that covers $N(u)$, such that each vertex $s\in S_u^*$ has exactly one private neighbour $w_u^s$ in $N(u)$ with respect to $S_u^*$, and moreover, the set $N(u)\setminus Q_u$, where $Q_u=\{w_u^s\; |\; s\in S_u^*\}$, is nonempty and complete to $S_u^*$, and every vertex of $S_u\setminus S_u^*$ is adjacent to at most one vertex of $Q_u$ and to at least one vertex of $N(u)\setminus Q_u$.
- The independent set $S_v$ contains a subset $S_v^*$ of size at least 2 that covers $N(v)$, such that each vertex in $s\in S_v^*$ has exactly one private neighbour $w_v^s$ in $N(v)$ with respect to $S_v^*$, and moreover, the set $N(v)\setminus Q_v$, where $Q_v=\{w_v^s\; |\; s\in S_v^*\}$, is nonempty and complete to $S_v^*$, and every vertex of $S_v\setminus S_v^*$ is adjacent to at most one vertex of $Q_v$ and to at least one vertex of $N(u)\setminus Q_v$.
*Remark.* We emphasize that $S_u^*$ and $S_v^*$ are unknown to the algorithm, as we constructed it from the unknown sets $S_u$ and $S_v$, and consequently our algorithm does not know (yet) the sets $Q_u$ and $Q_v$.
We will now branch into $O(n^8)$ smaller instances in which $N(u)\setminus Q_u$ and $N(u)\setminus Q_v$ consist of just one single vertex $w_u$ and $w_v$, respectively, such that $(G,u,v)$ has an independent solution if and only if at least one of the new instances has an independent solution. Moreover, we will be able to identify $w_u$ and $w_v$, and consequently, the sets $Q_u$ and $Q_v$, in polynomial time.
[**Branching**]{} ($O(n^8)$ branches)\
We will determine exactly those vertices of $N(u)$ that belong to $Q_u$ via some branching, under the assumption that $(G,u,v)$ has an independent solution $(S_u,S_v)$ that satisfies (P1) and (P2). By (P1), $S_u^*$ consists of at least two (non-adjacent) vertices $s$ and $s'$. By (P2), $S_v^*$ consists of at least two (non-adjacent) vertices $t$ and $t'$. We branch by considering all possible choices of choosing these four vertices together with their private neighbours $w_u^s$, $w_u^{s'}$, $w_v^{t}$, $w_v^{t'}$ (which are unique by (P1) and (P2)). This leads to $O(n^8)$ branches.
For each branch we do as follows. We discard the branch in which $G[\{s,s',w_u^s,w_u^{s'}\}]$ and $G[\{t,t',w_v^t,w_v^{t'}\}]$ are not both isomorphic to $2P_2$. We put a vertex $y\in N(u)$ in $N(u)\setminus Q_u$ if and only if $y$ is a common neighbour of $s$ and $s'$. This gives us the set $Q_u$. We obtain the set $Q_v$ in the same way. If there exists a vertex in $Q_u\setminus \{w_u^s,w_u^{s'}\}$ that is adjacent to one of $s,s'$, then we discard the branch. We also discard the branch if there exists a vertex in $Q_v\setminus \{w_v^t,w_v^{t'}\}$ that is adjacent to one of $t,t'$. Moreover, by applying the [Contraction Rule]{} on $N(u)\cup \{s,s'\}$ we can contract $s$ and $s'$ away. This contracts all vertices of $N(u)\setminus Q_u$ into a single vertex which we denote by $w_u$ due to (P1). Similarly, we branch $t$ and $t'$ away and this leads to the contraction of $N(v)\setminus Q_v$ into a single vertex $w_v$ due to (P2). Note that we have identified $w_u$ and $w_v$ in polynomial time. We denote the resulting instance by $(G,u,v)$ again.
Consider a vertex $z\in T$. Firs suppose that $z$ is not adjacent to $w_u$. Then $z$ does not belong to $S_u$ in any independent solution $(S_u,S_v)$ for $(G,u,v)$ by (P1).Hence $z$ must belong to $S_v$ for any independent solution $(S_u,S_v)$ for $(G,u,v)$. However, (P2) tells us that If $z$ is not adjacent to $w_v$, then $z$ cannot belong to the set $S_v$ of any independent solution $(S_u,S_v)$ for $(G,u,v)$. Hence, in that case we must discard the branch. Otherwise, that is, if $z$ is adjacent to $w_v$, then we check the following. If $z$ has two neighbours in $N(v)\setminus \{w_v\}$, then $z$ does not belong to $S_v$ in any independent solution $(S_u,S_v)$ for $(G,u,v)$ due to (P2). Hence, we will discard the branch. If $z$ is adjacent to at most one vertex of $N_v\setminus \{w_v\}$, then we apply the [Contraction Rule]{} on $N(v)\cup \{z\}$ to contract $z$ away. As a side effect, the possible neighbour of $z$ in $N_v\setminus \{w_v\}$ will be contracted away as well. Now suppose that $z$ is not adjacent to $w_v$. Then we perform the same operation with respect to $u$. We apply this operation exhaustively on both $u$ and $v$. This takes polynomial time. In the end we either discarded the branch or have found a new instance, which we also denote by $(G,u,v)$ again, in which every vertex of $T$ is adjacent to $w_u$ and to $w_v$.
Consider again a vertex $z\in T$. If $z$ is adjacent to only $w_u$ and $w_v$ and to at most one other vertex $w$ in $N(u)\cup N(v)$, then we apply the [Contraction Rule]{} on $G[N(u)\cup \{z\}]$ (if $w\in N(u)$) or $G[N(v)\cup \{z\}]$ (in the other two cases) in order to contract $z$ away. As a side effect, the possible other neighbour of $z$ in $(N(u)\cup N(v)) \setminus \{w_u,w_v\}$ will be contracted away as well. If $z$ is adjacent to more than one vertex of $N(u)\setminus \{w_u\}$, then $z$ does not belong to $S_u$ in any independent solution $(S_u,S_v)$ for $(G,u,v)$. We check if $z$ is adjacent to more than one vertex of $N(v)\setminus \{w_v\}$. If so, then $z$ does not belong to $S_v$ in any independent solution $(S_u,S_v)$ for $(G,u,v)$. In that case we will discard the branch. Otherwise we will apply the [Contraction Rule]{} on $N(v)\cup \{z\}$ to contract $z$ away. Again, as a side effect, the possible neighbour of $z$ in $N(v)\setminus \{w_v\}$ will be contracted away as well. If $z$ is adjacent to more than one vertex of $N(u)\setminus \{w_v\}$, we perform a similar operation with respect to $u$. We apply this rule exhaustively. This takes polynomial time. In the end we find that every vertex of $T$ is adjacent to $w_u$ and $w_v$ and to exactly one vertex of $Q_u$ and to exactly one vertex of $Q_v$.
We now remove all edges of $G[T]$. We also remove $w_u$ and $w_v$ from the graph. This yields a bipartite graph $G'$ with partition classes $N(u)\cup N(v)\setminus \{w_u,w_v\}$ and $T$. It remains to compute a maximum matching $M$ in $G'$. We can do this by using the Hopcroft-Karp algorithm [@HK73], which runs in $O(m\sqrt{n})$-time on bipartite graphs with $n$ vertices and $m$ edges. If $|M|=|N(u)|+|N(v)|-2$ then we found a solution for $(G,u,v)$; otherwise we discard the branch. Note that we did not explicitly forbid that two adjacent vertices of $T$ ended up in $S_u$ or two adjacent vertices of $T$ ended up in $S_v$: we have ruled out the existence of such solutions already (but they would still be perfectly acceptable if they did exist).
As mentioned, we translate a solution found for some branch into a solution for the original instance. We can do so in polynomial time. If we find no yes-answer for the instance of any branch, then we conclude that the original instance has no solution.
The correctness of our algorithm follows from the above description. We now analyze its run-time. There is only one branching procedure, which yields a total number of $O(n^{8})$ branches. As explained above, processing each branch takes polynomial time. In particular, checking for $8$-constant solutions takes polynomial time due to Lemma \[l-constant\]. We conclude that the total running time of our algorithm is polynomial.
We proceed in the same way as in the case where $H=P_2+P_4$. That is, we will use Lemma \[l-top4c\] to prove Lemma \[l-top5c\]. Then we use Lemma \[l-top5c\] to prove Lemma \[l-top6c\], and we use Lemma \[l-top6c\] to prove Lemma \[l-top7c\].
\[l-top5c\] The $P_5$-[Suitability]{} problem can be solved in polynomial time for $(P_1+P_2+P_3)$-free graphs.
Let $(G,u,v)$ be an instance of $P_5$-[Suitability]{}, where $G$ is a connected $(P_1+P_2+P_3)$-free graph. We may assume without loss of generality that $u$ and $v$ are of distance at least 4 from each other, as otherwise $(G,u,v)$ is a no-instance. We may also assume without loss of generality that $N(u)$ and $N(v)$ are independent sets; otherwise, say $N(u)$ contains an edge, we apply the [Contraction Rule]{} on $N(u)$ to obtain an equivalent but smaller instance $(G',u,v)$, where $G'$ is also $(P_1+P_2+P_3)$-free due to Lemma \[l-contract\].
If $N(u)$ consists of exactly one vertex $u'$, then we can instead solve $P_5$-[Suitability]{} on instance $(G-u,u',v)$. By Lemma \[l-top5c\] this takes polynomial time. Hence, we may assume that $N(u)$, and for the same reason, $N(v)$ have size at least 2.
By Lemma \[l-reduce\] we may assume that $\operatorname{dist}(u,v)=4$. Let $M$ consist of all vertices of $G$ that are of distance 2 from $u$ and of distance 2 from $v$. Note that $M\neq \emptyset$, as $\operatorname{dist}(u,v)=4$. Moreover, if $G$ has a $P_5$-witness structure ${\cal W}$ with $W(p_1)=\{u\}$ and $W(p_5)=\{v\}$, then $M\subseteq W(p_3)$ must hold.
Let $z,z'$ be two vertices in $N(v)$. Suppose $x\notin M\cup \{u\}$ is adjacent to $w\in N(u)$ but not to $w'\in N(u)$. As $x$ is not in $M$ and adjacent to $w\in N(u)$, we find that $x$ is not adjacent to $z$ and $z'$. However, then $\{w'\}\cup \{w,x\}\cup \{z,v,z'\}$ induces a $P_1+P_2+P_3$ in $G$, a contradiction. Hence, every vertex not in $M\cup \{u\}$ is either complete to $N(u)$ or anticomplete to $N(u)$. This means that if $G$ has a $P_5$-witness structure ${\cal W}$ with $W(p_1)=\{u\}$ and $W(p_5)=\{v\}$, then the following holds: $W(p_2)\setminus N(u)$ contains a vertex $s$, such that $N(u)\cup \{s\}$ is connected.
We now branch by considering all possibilities of choosing this vertex $s$; note that we only have to consider vertices of $G$ that are of distance 2 from $u$ and that are not in $M$. This leads to $O(n)$ branches. We consider each branch separately, as follows. First we contract all edges in $G[N(u)\cup \{s\}]$. If this does not yield a single vertex $u'$, then we discard the branch. Otherwise we let $G'$ be the resulting graph. The graph $G'-u'$ consists of at least two connected components, one of which consists of vertex $u$, and the other one contains $v$ and $N(v)$. We contract away the vertices of any other connected component $D$ of $G'-u'$ by applying the [Contraction Rule]{} on $\{u'\}\cup V(D)$. It remains to check if $(G'-u,u',v)$ is a yes-instance of $P_4$-[Suitability]{}. We can do this in polynomial time via Lemma \[l-top4c\]. As there are $O(n)$ branches, the total running time of our algorithm is polynomial.
\[l-top6c\] The $P_6$-[Suitability]{} problem can be solved in polynomial time for $(P_1+P_2+P_3)$-free graphs.
Let $(G,u,v)$ be an instance of $P_6$-[Suitability]{}, where $G$ is a connected $(P_1+P_2+P_3)$-free graph. We may assume without loss of generality that $u$ and $v$ are of distance at least 5 from each other, as otherwise $(G,u,v)$ is a no-instance. We may also assume without loss of generality that $N(u)$ and $N(v)$ are independent sets; otherwise, say $N(u)$ contains an edge, we apply the [Contraction Rule]{} on $N(u)$ to obtain an equivalent but smaller instance $(G',u,v)$, where $G'$ is also $(P_1+P_2+P_3)$-free due to Lemma \[l-contract\].
If $N(u)$ consist of exactly one vertex $u'$, then we can instead solve $P_5$-[Suitability]{} on instance $(G-u,u',v)$. By Lemma \[l-top5c\] this takes polynomial time. Hence, we may assume that $N(u)$, and for the same reason, $N(v)$ are independent sets of size at least 2. Let $z,z'$ be two vertices in $N(v)$. Suppose $x\notin N(u)\cup \{u\}$ is adjacent to $w\in N(u)$ but not to $w'\in N(u)$. Then $\{w'\}\cup \{w,x\}\cup \{z,v,z'\}$ induces a $P_1+P_2+P_3$ in $G$, a contradiction. Hence, every vertex not in $N(u)\cup \{u\}$ is either complete to $N(u)$ or anticomplete to $N(u)$. This means that if $G$ has a $P_5$-witness structure ${\cal W}$ with $W(p_1)=\{u\}$ and $W(p_5)=\{v\}$, then the following holds: $W(p_2)\setminus N(u)$ contains a vertex $s$, such that $N(u)\cup \{s\}$ is connected.
We now branch by considering all possibilities of choosing this vertex $s$. This leads to $O(n)$ branches. We consider each branch separately, as follows. First we contract all edges in $G[N(u)\cup \{s\}]$. If this does not yield a single vertex $u'$, then we discard the branch. Otherwise we let $G'$ be the resulting graph. The graph $G'-u'$ consists of at least two connected components, one of which consists of vertex $u$, and the other one contains $v$ and $N(v)$. We contract away the vertices of any other connected component $D$ of $G'-u'$ by applying the [Contraction Rule]{} on $\{u'\}\cup V(D)$. It remains to check if $(G'-u,u',v)$ is a yes-instance of $P_4$-[Suitability]{}. We can do this in polynomial time via Lemma \[l-top5c\]. As there are $O(n)$ branches, the total running time of our algorithm is polynomial.
\[l-top7c\] The $P_7$-[Suitability]{} problem can be solved in polynomial time for $(P_1+P_2+P_3)$-free graphs.
Let $(G,u,v)$ be an instance of $P_7$-[Suitability]{}, where $G$ is a connected $(P_1+P_2+P_3)$-free graph. We may assume without loss of generality that $u$ and $v$ are of distance at least 6 from each other, as otherwise $(G,u,v)$ is a no-instance. Note that in fact $u$ and $v$ are of distance exactly 6 from each other, as otherwise $G$ contains an induced $P_1+P_2+P_3$. We may also assume without loss of generality that $N(u)$ is an independent set; otherwise we apply the [Contraction Rule]{} on $N(u)$ to obtain an equivalent but smaller instance $(G',u,v)$, where $G'$ is also $(P_1+P_2+P_3)$-free due to Lemma \[l-contract\].
Suppose $N(u)$ contains two vertices $w$ and $w'$. As $u$ and $v$ are of distance 6 from each other, there exists a vertex $y$ with $\operatorname{dist}(u,y)=\operatorname{dist}(v,y)=3$. Let $z\in N(v)$. Then the set $\{y\}\cup \{v,z\}\cup \{w,u,w'\}$ induces a $P_1+P_2+P_3$ in $G$, a contradiction. Hence, $N(u)$ consist of exactly one vertex $u'$. We can therefore solve $P_6$-[Suitability]{} on instance $(G-u,u',v)$. By Lemma \[l-top6c\] this takes polynomial time.
We are now ready to prove the main result of Section \[s-p1p2p3\].
\[t-p1p2p3\] The [Longest Path Contractibility]{} problem is polynomial-time solvable for $(P_1+P_2+P_3)$-free graphs.
Let $G$ be a connected $(P_1+P_2+P_3)$-free graph. We may assume without loss of generality that $G$ has at least one edge. Then $G$ is $P_8$-free. Hence, $G$ does not contain $P_8$ as a a contraction. By combining Lemmas \[l-top4c\]–\[l-top7c\] with Lemma \[l-outer\] we can check in polynomial time if $G$ contains $P_k$ as a contraction for $k=7,6,5,4$. If not, then we check if $G$ contains $P_3$ as a contraction by using Lemma \[l-trivial\] combined with Lemma \[l-outer\]. If not then, as $G$ has an edge, $P_2$ is the longest path to which $G$ can be contracted to.
The Case $\mathbf{H=P_1+P_5}$ {#s-p1p5}
-----------------------------
We will prove that [Longest Path Contractibility]{} is polynomial-time solvable for $(P_1+P_5)$-free graphs. This result extends a corresponding result of [@HPW09] for $P_5$-free graphs. Its proof is based on the same but slightly generalized arguments as the result for $P_5$-free graphs and comes down to the following lemma.
\[l-all\] Let $k\geq 4$ and let $G$ be a $(P_1+P_5)$-free graph with a $P_k$-suitable pair $(u,v)$ such that $N(u)$ is an independent set. Then $G$ has a $P_k$-witness structure ${\cal W}$ with $W(p_1)=\{u\}$ and $W(p_k)=\{v\}$, for which the following holds: $W(p_2)\setminus N(u)$ contains a set $S$ of size at most $2$ such that $N(u)\cup S$ is connected.
As $(u,v)$ is a $P_k$-suitable pair, $G$ has a $P_k$-witness structure ${\cal W}$ with $W(p_1)=\{u\}$ and $W(p_k)=\{v\}$. For contradiction, assume that $W(p_2)\setminus N(u)$ contains no set $S$ of size at most 2 such that $N(u)\cup S$ is connected. Then $W(p_2)\setminus N(u)$ contains at least three vertices $x_1$, $x_2$, $x_3$ such that one of the following holds:
- for $i=1,2,3$, vertex $x_i$ is adjacent to some vertex $w_i\in N(u)$ with $w_i\notin N(x_h)\cup N(x_j)$, where $\{h,i,j\}=\{1,2,3\}$; or
- $N(u)\subseteq N(x_1)\cup N(x_2)$, but $G[N(u)\cup \{x_1\}\cup \{x_2\}]$ is not connected.
First assume that (i) holds. Recall that $N(u)$ is an independent set. Then $x_1x_2 \in E(G)$, as otherwise the set $\{v\}\cup \{x_1,w_1,u,w_2,x_2\}$ induces a $P_1+P_5$ in $G$, which is not possible. However, now the set $\{v\}\cup \{w_3,u,w_2,x_2,x_1\}$ induces a $P_1+P_5$ in $G$, a contradiction. Hence, (i) cannot hold. Now assume that (ii) holds. As $(G,u,v)$ has no $1$-constant solution, $x_1$ has a neighbour $w_1\in N(u)$ not adjacent to $x_2$ and $x_2$ has a neighbour $w_2\in N(u)$ not adjacent to $x_1$. As $G[N(u)\cup \{x_1\}\cup \{x_2\}]$ is not connected but $N(u)\subseteq N(x_1)\cup N(x_2)$, we have that $x_1x_2\notin E(G)$ However, then the set $\{v\}\cup \{x_1,w_1,u,w_2,x_2\}$ induces a $P_1+P_5$ in $G$, a contradiction. Hence (ii) does not hold either, a contradiction.
As a consequence of Lemma \[l-all\], we get that $P_4$-[Suitability]{} is easy and that $P_k$-[Suitability]{} reduces to $P_4$-[Suitability]{}, as we will see.
\[l-top4b\] The $P_4$-[Suitability]{} problem can be solved in polynomial time for $(P_1+P_5)$-free graphs.
Let $(G,u,v)$ be an instance of $P_4$-[Suitability]{}, where $G$ is a connected $(P_1+P_5)$-free graph. We may assume without loss of generality that $u$ and $v$ are of distance at least 3 from each other, as otherwise $(G,u,v)$ is a no-instance. We may also assume without loss of generality that $N(u)$ is an independent set; otherwise we apply the [Contraction Rule]{} on $N(u)$ to obtain an equivalent but smaller instance $(G',u,v)$, where $G'$ is also $(P_1+P_5)$-free due to Lemma \[l-contract\]. By Lemma \[l-all\] we find that if $(G,u,v)$ has a solution, then $G$ has a $2$-constant solution. We can check the latter in $O(n^4)$ time by Lemma \[l-constant\].
\[l-top5b\] The $P_5$-[Suitability]{} problem can be solved in $O(n^6)$ time for $(P_1+P_5)$-free graphs.
Let $(G,u,v)$ be an instance of $P_5$-[Suitability]{}, where $G$ is a connected $(P_1+P_5)$-free graph. We may assume without loss of generality that $u$ and $v$ are of distance at least 4 from each other, as otherwise $(G,u,v)$ is a no-instance. We may also assume without loss of generality that $N(u)$ is an independent set; otherwise we apply the [Contraction Rule]{} on $N(u)$ to obtain an equivalent but smaller instance $(G',u,v)$, where $G'$ is also $(P_1+P_5)$-free due to Lemma \[l-contract\].
If $(u,v)$ is a $P_5$-suitable pair, then by Lemma \[l-all\], $G$ has a $P_5$-witness structure ${\cal W}$ with $W(p_1)=\{u\}$ and $W(p_5)=\{v\}$, for which the following holds: $W(p_2)\setminus N(u)$ contains a set $S$ of size at most $2$, such that $N(u)\cup S$ is connected.
We now branch by considering all possibilities of choosing this set $S$. This leads to $O(n^2)$ branches. We consider each branch separately, as follows. First we contract all edges in $G[N(u)\cup S]$. If this does not yield a single vertex $u'$, then we discard the branch. Otherwise we let $G'$ be the resulting graph. The graph $G'-u'$ consists of at least two connected components, one of which consists of vertex $u$, and the other one contains $v$ and $N(v)$. If there are more components in $G'-u'$ than these two, we contract each such component $D$ to $u'$ by applying the [Contraction Rule]{} on $\{u'\}\cup V(D)$. It remains to check if $(G'-u,u',v)$ is a yes-instance of $P_4$-[Suitability]{}. We can do this in $O(n^4)$ time via Lemma \[l-top4b\]. As there are $O(n^2)$ branches, the total running time of our algorithm is $O(n^6)$.
\[l-top6b\] The $P_6$-[Suitability]{} problem can be solved in $O(n^8)$ time for $(P_1+P_5)$-free graphs.
We reduce $P_6$-[Suitability]{} to $P_5$-[Suitability]{} in exactly the same way we reduced $P_5$-[Suitability]{} to $P_4$-[Suitability]{} in the proof of Lemma \[l-top5b\]. This leads to $O(n^2)$ branches. For each branch we apply Lemma \[l-top5b\], which takes $O(n^6)$ time. Hence the total running time of $O(n^8)$.
We are now ready to prove the main result of Section \[s-p1p5\].
\[t-p1p5\] The [Longest Path Contractibility]{} problem is polynomial-time solvable for $(P_1+P_5)$-free graphs.
Let $G$ be a connected $(P_1+P_5)$-free graph. We may assume without loss of generality that $G$ has at least one edge. Note that $G$ is $P_7$-free. Hence, $G$ does not contain $P_7$ as a a contraction. By combining Lemmas \[l-top4b\]–\[l-top6b\] with Lemma \[l-outer\] we can check in polynomial time if $G$ contains $P_k$ as a contraction for $k=6,5,4$. If not, then we check if $G$ contains $P_3$ as a contraction by using Lemma \[l-trivial\] combined with Lemma \[l-outer\]. If not then, as $G$ has an edge, $P_2$ is the longest path to which $G$ can be contracted to.
The Case $\mathbf{H=sP_1+P_4}$ {#s-sp1p4}
------------------------------
We adopt/extend the notation from Section \[s-p4\]. Let $(G,u,v)$ be an instance of $P_k$-[Suitability]{} with $k \ge 4$. A *solution* is a witness structure $\mathcal{W}=\{W(p_1), \dots,
W(p_k)\}$ with $W(p_1)=\{u\}, W(p_2)=N(u) \cup S_u , W(p_{k-1}) = N(v) \cup S_v$ and $W(p_k)=\{v\}$. We let $T:= V\setminus
(N[u] \cup N[v])$. Thus $S_u$ and $S_v$ are disjoint subsets of $T$ such that $N(u) \cup S_u$ and $N(v) \cup S_v$ are connected. As in Section \[s-p4\], we call a solution $\alpha$-constant if there exists a subset $S_u'\subseteq S_u$ with $N(u) \cup S_u'$ connected and $|S_u'| \le \alpha$, or there exists $S_v'\subseteq S_v$ with $N(v)\cup S_v'$ connected and $|S_v' |\le \alpha$.
Let $(\{u\}, N(u) \cup S_u, \dots)$ be a solution and $S'_u \subseteq S_u$ such that $N(u) \cup S'_u$ is connected. We define the *closure* $\overline{S'_u}$ of $S'_u$ as the set of all vertices in $S_u$ that are connected to $v$ in $G$ only via $N(u) \cup S'_u$.
\[l-clos\] Let $(\{u\},N(u) \cup S_u, W(p_3), \dots, W(p_{k-1}),\{v\})$ be a solution for an instance $(G,u,v)$ of $P_k$-[Suitability]{} for some $k\geq 4$. If $S_u' \subseteq S_u$ such that $N(u) \cup S'_u$ is connected, then $(\{u\},N(u) \cup \overline{S'_u}, W(p_3)\cup S_u \setminus \overline{S'_u}, \dots,W(p_{k-1}),\{v\})$ is also a solution for $(G,u,v)$.
We check the three properties for witness structures. All bags in the new partition are mutually disjoint. Connectedness of $W(p_3)\cup S_u \setminus \overline{S'_u}$: Any $s \in S_u\setminus\overline{S'_u}$ is joined to $v$ by a path $P$ that does not pass through $N(u) \cup S'_u$ (by definition). Moreover, $P$ does not hit $\overline{S'_u}$ since from there; by definition, we cannot reach $v$ without passing through $N(u) \cup S'_u$. Since vertices in $W(p_2)=N(u)\cup S_u$ are only adjacent to vertices in $W(p_1) \cup W(p_2) \cup W(p_3)$, we find that $P$ must be contained in $S_u \setminus \overline{S'_u}$ until it eventually reaches $W(p_3)$. Connectedness of $W(p_3) \cup S_u \setminus \overline{S'_u}$ follows.
As it turns out, it suffices to search for $\alpha$-constant solutions:
\[l-constps\] An instance $(G,u,v)$ of $P_k$-[Suitability]{}, where $G$ is $(sP_1+P_4)$-free, has a solution if and only if it has an $\alpha$-constant solution, where $\alpha=(s+2)(2s+4)$.
We may, as usual, assume that $N(u)$ is independent (otherwise we apply the [Contraction Rule]{} to $N(u)$ without any effect on $S_u$ in solutions $(\{u\}, N(u) \cup S_u, \dots)$). First suppose that $(G,u,v)$ has an $\alpha$-constant solution. Then obviously $(G,u,v)$ has a solution.
Now suppose that $(G,u,v)$ has a solution $(\{u\}, N(u) \cup S_u, \dots)$. Let $t \in S_u$ and let $S_u^* \subseteq S_u$ be a minimum size subset that covers (e.g., dominates) $N(u)$. Each $z \in S_u^*$ is connected to $t$ by some path $P_z \subseteq S_u$. Since $G$ is $(sP_1+P_4)$-free, $P_z$ has at most $2s+4$ vertices. Hence, $S_v':= \bigcup_{z \in S_u^*} P_z$ has size at most $|S_u^*|(2s+4)$ and is connected and covers $N(u)$ (as $S_u^*$ does). Then $(G,u,v)$ is an $\alpha$-constant solution, unless $|S_u'| > \alpha$. From now on suppose that $|S_u'| > \alpha$, so in particular $|S_u^*| > s+2$.
We show that $S_u^*$ is independent. For contradiction, assume that $z,z'\in S_u^*$ are adjacent. Let $w,w'\in N(u)$ be private neighbours of $z,z'$, resp. Then $wzz'w'$ induces a $P_4$. Since $G$ is $(sP_1+P_4)$-free, $\{z,z'\}$ must cover almost all vertices in $N(u)$ (which may be assumed independent) except at most $s-1$ vertices, say, $w_1, \dots, w_{s-1}$. Thus a minimum size cover $S_u^*$ of $N(u)$ has at most $s+1$ vertices ($z,z'$ and at most $s-1$ others covering $w_1, \dots, w_{s-1}$), contradicting the fact that $|S_u^*| > s+2$.
Next we prove that any two vertices $z,z'\in S_u^*$ cover disjoint sets in $N(u)$. For contradiction, assume that $z,z'\in S_u^*$ have a common neighbour $w \in N(u)$. Since $z\in S_u^*$ also has a private neighbour $w' \in N(u)$, we find an induced $P_4=w'zwz'$ and conclude that $\{z, z'\}$ must cover all but at most $s-1$ vertices in $N(u)$, a contradiction again.
From the above we conclude that $S_u^* \cup N(u)$ is a disjoint union of stars. Recall that $|S_u^*| > s+2> 1$. Therefore, to be connected, $S_u$ must contain a vertex $t \in S_u \setminus S_u^*$ connecting two vertices $z,z'\in S_u^*$. Let again $w$ be a private neighbour of $z$ in $N(u)$. Then $wztz'$ is a $P_4$, implying that $\{t,z,z'\}$ must cover all but $s-1$ vertices in $N(u)$, leading to a contradiction as before. Summarizing, we have shown that $(S_u, \dots)$ is an $\alpha$-constant solution.
Combining Lemmas \[l-clos\] and \[l-constps\] gives the desired result:
\[t-sp1p4\] For every constant $s\geq 0$, the [Longest Path Contractibility]{} problem is polynomial-time solvable for $(sP_1+P_4)$-free graphs.
By Lemma \[l-constps\] we may focus on $\alpha$-constant solutions, If $(G,u,v)$ has an $\alpha$-constant solution $(\{u\}, N(u) \cup S_u, \dots)$ with $S_u'\subseteq S_u$ of size at most $\alpha$, we may guess this set $S_u'$ and extend it to its closure $\overline{S'_u}$ (by adding all vertices that are connected to the rest of the graph only through $N(u)\cup S_u'$ using Lemma \[l-clos\]) in time $O(n^{\alpha+2})$. We may then contract $\{u\} \cup \overline{S'_u}$, thereby reducing $P_k$-[Suitability]{} to $P_{k-1}$-[Suitability]{}. Since $G$ is $(sP_1+P_4)$-free, we may assume that $k \le 2s+4$. (If $k\geq 2s+5$, then every instance $(G,u,v)$ where $G$ is $(sP_1+P_4)$-free is a no-instance of $P_k$-[Suitability]{}). Thus only $2s$ such reductions are required.
The NP-Complete Cases of Theorem \[t-main\] {#s-hard}
===========================================
In this section we prove the new [[NP]{}]{}-complete cases of Theorem \[t-main\].
A [*hypergraph*]{} ${\cal H}$ is a pair $(Q,{\mathcal S})$, where $Q=\{q_1,\ldots,q_m\}$ is a set of $m$ [*elements*]{} and ${\mathcal S}=\{S_1,\ldots,S_n\}$ is a set of $n$ [*hyperedges*]{}, which are subsets of $Q$. A [*$2$-colouring*]{} of ${\cal H}$ is a partition of $Q$ into two (nonempty) sets $Q_1$ and $Q_2$ with $Q_1\cap S_j \ne\emptyset$ and $Q_2 \cap S_j \ne\emptyset$ for each $S_j$. This leads to the following decision problem.
[.99]{} <span style="font-variant:small-caps;">[Hypergraph 2-Colourability]{}</span>\
----------------- -----------------------------------------
* Instance:* [a hypergraph ${\cal H}$.]{}
*Question:* [does ${\cal H}$ have a 2-colouring?]{}
----------------- -----------------------------------------
Note that [Hypergraph 2-Colourability]{} is [[NP]{}]{}-complete even for hypergraphs ${\cal H}$ with $S_i\ne \emptyset$ for $1\leq i\leq n$ and $S_n=Q$. By a reduction from [Hypergraph 2-Colourability]{}, Brouwer and Veldman [@BV87] proved that $P_4$-[Contractibility]{} is [[NP]{}]{}-complete. That is, from a hypergraph ${\cal H}$ they built a graph $G_{\cal H}$, such that ${\cal H}$ has a $2$-colouring if and only if $G_{\cal H}$ has $P_4$ as a contraction. We first recall the graph $G_{\cal H}$ from [@BV87], which was obtained from a hypergraph ${\cal H}$ with $S_i\ne \emptyset$ for $1\leq i\leq n$ and $S_n=Q$ (see Figure \[f-hypergraph\] for an example).
- Construct the [*incidence graph*]{} of $(Q,{\mathcal S})$, which is the bipartite graph with partition classes $Q$ and ${\cal S}$ and an edge between two vertices $q_i$ and $S_j$ if and only if $q_i\in S_j$.
- Add a set ${\mathcal S}'=\{S_1',\ldots,S_n'\}$ of $n$ new vertices, where we call $S_j'$ the [*copy*]{} of $S_j$.
- For $i=1,\ldots m$ and $j=1,\ldots, n$, add an edge between $q_i$ and $S_j'$ if and only if $q_i\in S_j$.
- For $j=1,\ldots, n$ and $\ell=1,\ldots,n$, add an edge between $S_j$ and $S_\ell'$, so the subgraph induced by ${\cal S}\cup {\cal S}'$ will be complete bipartite.
- For $h=1,\ldots, m$ and $i=1,\ldots,m$, add an edge between $q_h$ and $q_i$, so $Q$ will be a clique.
- Add two new vertices $t_1$ and $t_2$.
- For $j=1,\ldots,n$, add an edge between $t_1$ and $S_j$, and between $t_2$ and $S_j'$.
![An example of a graph $G_{\cal H}$ for some hypergraph ${\cal H}$ [@LPW08].[]{data-label="f-hypergraph"}](hypergraph.pdf)
As mentioned, Brouwer and Veldman [@BV87] proved the following.
\[l-bv87\] A hypergraph ${\cal H}$ has a $2$-colouring if and only if $G_{\cal H}$ has $P_4$ as a contraction.
A [*split graph*]{} is a graph whose vertex set can be partitioned into two (possibly empty) sets $K$ and $I$, where $K$ is a clique and $I$ is an independent set. It is well known that a graph is split if and only if it is $(2P_2,C_4,C_5)$-free [@FH77]. Let ${\cal H}$ be a hypergraph. Observe that the subgraphs of $G_{\cal H}$ induced by $Q\cup {\cal S}$ and $Q\cup {\cal S}'$, respectively, are split graphs. Hence, we make the following observation.
\[l-split\] Let ${\cal H}$ be a hypergraph. Then the subgraphs of $G_{\cal H}$ induced by $Q\cup {\cal S}$ and $Q\cup {\cal S}'$, respectively, are $(2P_2,C_4,C_5)$-free.
We will need the following known lemma from [@HPW09].
\[l-p6\] Let ${\cal H}$ be a hypergraph. Then the graph $G_{\cal H}$ is $P_6$-free.
We complement Lemma \[l-p6\] with the following lemma.
\[l-freefree\] Let ${\cal H}$ be a hypergraph. Then the graph $G_{\cal H}$ is $(2P_1+2P_2,3P_2, 2P_3)$-free.
We will prove that $G=G_{\cal H}$ is $(2P_1+2P_2,3P_2, 2P_3)$-free by considering each graph in $\{2P_1+2P_2,3P_2, 2P_3\}$ separately.
[**$\mathbf{(2P_1+2P_2)}$-freeness.**]{} For contradiction, assume that $G$ contains a subgraph $H$ isomorphic to $2P_1+2P_2$. Let $D_1$ and $D_2$ be the two connected components of $H$ that contain an edge. Let $x$ and $y$ denote the two isolated vertices of $H$. First suppose that one of $x,y$, say $x$, belongs to ${\cal S}\cup {\cal S}'$, say to ${\cal S}$. Then $D_1$ and $D_2$ do not contain $t_1$ and also do not contain any vertex from ${\cal S}'$. The latter implies that $D_1$ and $D_2$ cannot contain vertex $t_2$ either. Hence, $D_1$ and $D_2$ only contain vertices from ${\cal S}\cup Q$, contradicting Lemma \[l-split\]. Hence, $x$ and $y$ must both belong to $Q\cup \{t_1,t_2\}$. Suppose one of them, say $x$, belongs to $Q$. Then $D_1$ and $D_2$ do not contain any vertices from $Q$ and thus only contain vertices from ${\cal S}\cup {\cal S}'\cup \{t_1,t_2\}$. However, $G[{\cal S}\cup {\cal S}'\cup \{t_1,t_2\}]$ is complete bipartite, and thus $2P_2$-free, a contradiction. We thus found that $\{x,y\}=\{t_1,t_2\}$. Then $D_1$ and $D_2$ may not contain any vertices from ${\cal S}\cup {\cal S}'$. Consequently, $D_1$ and $D_2$ only contain vertices from $Q$. This is not possible, as $Q$ is a clique. We conclude that $G$ is $(2P_1+2P_2)$-free.
[**$\mathbf{3P_2}$-freeness.**]{} For contradiction, assume that $G$ contains a subgraph $H$ isomorphic to $3P_2$. Let $D_1,D_2,D_3$ be the three connected components of $H$. Suppose one of $D_1,D_2,D_3$, say $D_1$, contains a vertex from ${\cal S}\cup {\cal S}'$, say $D_1$ contains a vertex from ${\cal S}$. Then $D_2$ and $D_3$ do not contain $t_1$ and also do not contain any vertex from ${\cal S}'$. The latter implies that $D_2$ and $D_3$ cannot contain vertex $t_2$ either. Hence, $D_2$ and $D_3$ only contain vertices from ${\cal S}\cup Q$, contradicting Lemma \[l-split\]. This means that $H$ contains no vertex from ${\cal S}\cup {\cal S}'$. Consequently, $H$ does not contain $t_1$ and $t_2$ either. However, then $H$ consists of vertices from $Q$ only. This is not possible, as $Q$ is a clique. We conclude that $G$ is $3P_2$-free.
[**$\mathbf{2P_3}$-freeness.**]{} For contradiction, assume that $G$ contains a subgraph $H$ isomorphic to $2P_3$. Let $D_1$ and $D_2$ be the two connected components of $H$. Suppose one of $D_1,D_2$, say $D_1$, contains a vertex from $Q$. As $Q$ is a clique, this means that $D_1$ must contain at least one vertex of ${\cal S}\cup {\cal S}'$, say $D_1$ contains a vertex of ${\cal S}$. Then $D_2$ cannot contain any vertex from $Q\cup \{t_1\}$ or from ${\cal S}'$. The latter implies that $D_2$ does not contain $t_2$ either. Hence, $D_2$ only contains vertices from ${\cal S}$. This is not possible, as ${\cal S}$ is an independent set. We conclude that neither $D_1$ nor $D_2$ contains a vertex from $Q$. Hence, $H$ only contains vertices from ${\cal S}\cup {\cal S}'\cup \{t_1,t_2\}$. However, $G[{\cal S}\cup {\cal S}'\cup \{t_1, t_2\}]$ is complete bipartite, and thus $2P_3$-free, a contradiction. We conclude that $G$ is $2P_3$-free.
It is readily seen that $P_4$-[Contractibility]{} belongs to [[NP]{}]{}. Hence, we obtain the following result from Lemmas \[l-bv87\], \[l-p6\], and \[l-freefree\].
\[t-hard\] [$P_4$-Contractibility]{} is [[NP]{}]{}-complete for $(2P_1+2P_2,3P_2, 2P_3,P_6)$-free graphs.
By modifying the graph $G_{\cal H}$ we prove the next theorem.
\[t-girth\] Let $p\geq 4$ be some constant. Then [$P_{2p}$-Contractibility]{} is [[NP]{}]{}-complete for bipartite graphs of girth at least $p$.
We assume without loss of generality that $p$ is even. We reduce again from [Hypergraph $2$-Colouring]{}, using a suitable subdivision of the graph $G_{\mathcal{H}}$ in order to satisfy the bipartiteness and girth constraints. Let $\mathcal{H}$ be a hypergraph. We first construct $G_{\mathcal{H}}$. We then subdivide edges in $G_{\mathcal{H}}$ as follows. The edge $S_nS'_n$ is *not* subdivided. All other edges $S_iS'_j$ are subdivided by an *even* number of vertices, namely by $p-2$, each. All edges joining $Q$ to $\mathcal{S'}$ are also subdivided $p-2$ times. So each of these edges becomes a path of odd length ($p-1$). Edges joining $Q$ to itself or to $\mathcal{S}$ are subdivided by an *odd* number of vertices, namely $p-1$, each. So each of these edges becomes a path of even length ($p$). In addition, we attach paths of length $p-2$, one to each of $t_1$ and $t_2$. Denote these paths by $P_i$ with end-vertices $t_i$ and, say, $\bar{t}_i, i=1,2$. Call the resulting graph $\bar{G}_{\mathcal{H}}$. In what follows we will denote the paths of length $p-1$ or $p$ obtained by subdividing an edge $xy$ in $G_{\mathcal{H}}$ by $\overline{xy}$.
The distance between $\bar{t}_1$ and $\bar{t}_2$ in $\bar{G}_{\mathcal{H}}$ equals $2(p-2)+3=2p-1$. The unique shortest path is given by $P=(P_1,S_n,S'_n,P_2)$. No other pair of vertices in $\bar{G}_{\mathcal{H}}$ is this far apart. (For example, the distance between $S_i \in \mathcal{S}$ and $q \in Q\backslash S_i$ equals $p$ or $2+p$, the length of the path via $t_1$ and $S_n$.)
It is straightforward to check that $\bar{G}_{\mathcal{H}}$ is bipartite: Any path joining $\mathcal{S}$ to $\mathcal{S'}$ has odd length. Hence, there are no odd cycles that hit both $\mathcal{S}$ and $\mathcal{S'}$. Similarly, all paths joining $Q$ and $\mathcal{S}'$ have odd length. So there cannot be any odd cycle in the subgraph induced by $Q\cup \mathcal{S}'$. The same argument applies to cycles in the subgraph induced by $Q \cup \mathcal{S}$. Here, again, all paths between $Q$ and $\mathcal{S}$ have the same parity (this time even). This shows that $\bar{G}_{\mathcal{H}}$ is indeed bipartite. The graph $\bar{G}_{\mathcal{H}}$ also has girth at least $p$. (Recall that any subdivided edge became a path of length at least $p-1$.) It remains to prove that ${\cal H}$ has a 2-colouring if and only if $\bar{G}_{\mathcal{H}}$ contains $P_{2p}$ as a contraction.
First suppose that ${\cal H}$ has a 2-colouring $(Q_1,Q_2)$. Then define a contraction of $\bar{G}_{\mathcal{H}}$ to $P$ with corresponding witness structure $\{W(x), x \in P\}$ as follows. For each $(i,j) \neq (n,n)$ pick any subdivision vertex $v_{ij} \in \overline{S_iS'_j}$ and let $P_{ij}$ denote the (vertices of the) subpath of $\overline{S_iS_j'}$ from $S_i$ to $v_{ij}$. Similarly, let $P'_{ij}$ denote the (vertices of) $\overline{S_iS_j'}\backslash P_{ij}$, the “other half” of the path from $S_i$ to $S_j'$. Now the witnesses can be defined as follows: $$\begin{aligned}
\nonumber
W(t)&= \{t\} ~\text{for} ~t \in P_1 \\ \nonumber
W(S_n)& = \bigcup_i \{\overline{S_iq}~|~q \in S_i\cap Q_1\} \cup \bigcup_{q,q'\in Q_1}\overline{qq'} \cup
\bigcup_{(i,j)\neq(n,n)} P_{ij}\\ \nonumber
W(S'_n)& = \bigcup_i \{\overline{S'_iq}~|~q \in S'_i\cap Q_2\} \cup \bigcup_{q,q'\in Q_2}\overline{qq'} \cup
\bigcup_{(i,j)\neq(n,n)} P'_{ij}\\ \nonumber
W(t)&= \{t\} ~\text{for} ~t \in P_2 \\ \nonumber\end{aligned}$$ To check correctness, we verify the three conditions for witnesses (observing that disjointness of the bags $W(x)$ is obvious).
- $W(S_n)$ is connected: Indeed, all of $P_{ij}$ is connected to $S_i$ and this (as we assume $Q=Q_1\cup Q_2$ is a $2$-colouring of $\mathcal{H}$) contains some $q \in Q_1$, so $\overline{S_iq}$ joins $S_i$ to $q$. The latter, in turn, is joined to $S_n$. The same arguments apply to $W(S'_n)$.
- Any two consecutive bags $W(x)$ and $W(y)$ (that is, when $x$ and $y$ are neighbours in $P$) are adjacent: Indeed, $t_1$ is adjacent to $S_n$, $S_n$ is adjacent to $S'_n$, and $S'_n$ is adjacent to $t_2$.
- If $x$ and $y$ are non-adjacent in $P$, then $W(x)$ and $W(y)$ are non-adjacent in $\bar{G}_{\mathcal{H}}$: Indeed, $t_1$ is only adjacent to $W(S_n)$ and this in turn is only adjacent to $W(S'_n)$ and $t_1$.
Thus, indeed, $\bar{G}_{\mathcal{H}}$ contains $P_{2p}$ as a contraction.
Now suppose that $\bar{G}_{\mathcal{H}}$ contains $P_{2p}$ as a contraction. Since $\bar{t}_1$ and $\bar{t}_2$ are the only vertices at distance $2p-1$ in $\bar{G}_{\mathcal{H}}$, the only possibility is that $\bar{G}_{\mathcal{H}}$ contracts to $P=(P_1,S_n, S'_n, P_2)$. Let $\{W(x), x\in P\}$ be a corresponding witness structure.
*Claim 1:*\
(i) $S_i \in W(t_1)\cup W(S_n)$ and $S'_i \in W(t_2)\cup W(S'_n)$ for $i=1, \dots, n$.\
(ii) $q \in W(t_1) \cup W(S_n)\cup W(S'_n) \cup W(t_2)$ for all $q \in Q$.\
*Proof of Claim 1.* In order to have all $W(x), x \in P$ connected, the subdivision vertices on $\overline{S_iS'_j}$ must belong to the same bags $W(x)$ as either $S_i$ or $S'_j$. The vertices $S_i$ and $S'_j$, however, must be in different (adjacent) bags: Indeed, $S_i, S'_j\in W(x)$ would imply that both $t_1$ and $t_2$ were adjacent to (or contained in) $W(x)$, contradicting the third condition for witness structures. The same argument shows that $S_i$ must either be in $W(t_1)$ or an adjacent bag, that is, in $W(S_n)$ or in $W(t)$, where $t$ is the unique neighbour of $t_1$ in $P_1$. The latter, however, is impossible: If $S_i \in W(t)$, then $S_i$ must be connected to $t$ within $W(t)$. But the only path joining $S_i$ to $t$ in $\bar{G}_{\mathcal{H}}$ runs through $t_1$, which does not belong to $W(t)$. Thus, indeed, (i) follows.
Part (ii) can be proved in the same way: If $q\in W(t)$ with $t \in P_1\backslash \{t_1\}$, then $q$ should be connected to $t$ within $W(t)$. But, again, the only path connecting $q$ and $t$ runs through $t_1$, a contradiction.
We claim that the partition $Q=Q_1 \cup Q_2$ given by $Q_1= Q \cap (W(t_1)\cup W(S_n))$ and $Q_2:= Q \cap (W(t_2) \cup W(S'_n))$ is a $2$-colouring of ${\cal H}$. That is, we will show that each $S_i \in \mathcal{S}$ contains some $q \in Q_1$ and, similarly, each $S'_i \in \mathcal{S'}$ contains some $q \in Q_2$.
Let $S_i \in \mathcal{S}$. From Claim 1 it follows that $S_i \in W(t_1)\cup W(S_n)$. For each $q \in S_i$ we follow the path $\overline{S_iq}$ from $S_i$ to $q$ in $\bar{G}_{\mathcal{H}}$. Let $v$ be the last vertex on this path that belongs to $W(t_1)\cup W(S_n)$. If $v=q$, then $q \in Q \cap (W(t_1)\cup W(S_n)) =Q_1$ and we are done. Hence, assume $v\neq q$. Then, in particular, $q \notin W(t_1)\cup W(S_n)$. From Claim 1 we know that $q \in W(t_2) \cup W(S'_n)$. The path $\overline{S_iq}$ starts in $S_i \in W(t_1) \cup W(S_n)$ and ends in $q \in W(t_2)\cup W(S'_n)$. Since only $W(S_n)$ and $W(S'_n)$ are adjacent, this path must eventually pass from $W(S_n)$ to $W(S'_n)$ for the last time. Hence, $v\in W(S_n)$ and, therefore, must be connected to $S_n$ within $W(S_n)$. As $v$ is a subdivision vertex on $\overline{S_iq}$, this connection can only be via $S_i$ or $q$. But $q$ is not in $W(S_n)$, so the connection must be via $S_i$ and we conclude that $S_i \in W(S_n)$. Hence, $S_i$ must be connected to $S_n$ within $W(S_n)$. The only paths in $\bar{G}_{\mathcal{H}}$ connecting $S_i$ to $S_n$ run through either $t_1$ (which does *not* belong to $W(S_n)$) or some $S'_j$ (which also does not belong to $W(S_n)$) or - the last possibility - some $\tilde{q} \in S_i$. Hence, indeed, at least one such $\tilde{q} \in S_i$ must belong to $W(S_n)$. But then $\tilde{q} \in Q_1$ (by definition of $Q_1$), as required.
As a consequence of Theorem \[t-girth\], [Longest Path Contractibility]{} is [[NP]{}]{}-complete for bipartite graphs of arbitrarily large girth. This strengthens the corresponding result for bipartite graphs, which following from a result of [@HHLP14]. For our dichotomy result we need the following consequence of Theorem \[t-girth\].
\[c-girth\] Let $H$ be a graph that has a cycle. Then [Longest Path Contractibility]{} is [[NP]{}]{}-complete for $H$-free graphs.
Let $g$ be the girth of $H$. We set $p=g+1$ and note that the class of $H$-free graphs contains the class of graphs of girth at least $p$. Hence, we can apply Theorem \[t-girth\].
The Proof of Theorem \[t-main\] {#s-classification}
===============================
We will use the following result from [@FKP13] as a lemma (in fact this result holds even for line graphs which form a subclass of the class of $K_{1,3}$-free graphs).
\[l-claw\] The $P_7$-[Contractibility]{} problem is [[NP]{}]{}-complete for $K_{1,3}$-free graphs.
By using the results from the previous sections and the above result we can now prove our classification theorem.
[**Theorem \[t-main\]. (restated)**]{} [*Let $H$ be a graph. If $H$ is an induced subgraph of $P_1+P_5$, $P_1+P_2+P_3$, $P_2+P_4$ or $sP_1+P_4$ for some $s\geq 0$, then [Longest Path Contractibility]{} restricted to $H$-free graphs is polynomial-time solvable; otherwise it is [[NP]{}]{}-complete.*]{}
If $H$ is an induced subgraph of $P_1+P_5$, $P_1+P_2+P_3$, $P_2+P_4$ or $sP_1+P_4$ for some $s\geq 0$, then we use Theorems \[t-p2p4\]–\[t-sp1p4\] to find that [Longest Path Contractibility]{} is polynomial-time solvable for $H$-free graphs. From now on suppose $H$ is not of this form. Below we will prove that in that case [Longest Contractibility]{} is [[NP]{}]{}-complete for $H$-free graphs.
If $H$ contains a cycle, then we apply Corollary \[c-girth\] to prove that [Longest Path Contractibility]{} is [[NP]{}]{}-complete for $H$-free graphs. Assume that $H$ is a forest. If $H$ has a vertex of degree at least 3, then the class of $H$-free graphs contains the class of $K_{1,3}$-free graphs. Hence, we can apply Lemma \[l-claw\] to find that [Longest Path Contractibility]{} is [[NP]{}]{}-complete for $H$-free graphs.
From now on we assume that $H$ is a linear forest. As $H$ is not an induced subgraph of $sP_1+P_4$, we find that $H$ contains at least one edge. We distinguish three cases.
[**Case 1.**]{} The number of connected components of $H$ is at least 3.\
First suppose that at least three connected components of $H$ contain an edge. Then $H$ contains an induced $3P_2$. This means we may apply Theorem \[t-hard\] to find that [Longest Path Contractibility]{} is [[NP]{}]{}-complete for $H$-free graphs.
Now suppose that exactly two connected components of $H$ contain an edge. If $H$ contains at least four connected components, then $H$ contains an induced $2P_1+2P_2$. This means we may apply Theorem \[t-hard\] to find that [Longest Path Contractibility]{} is [[NP]{}]{}-complete for $H$-free graphs. Hence, $H=P_1+P_r+P_s$ for some $2\leq r\leq s$. If $s\geq 4$, then $H$ contains an induced $2P_1+2P_2$. This means we may apply Theorem \[t-hard\] to find that [Longest Path Contractibility]{} is [[NP]{}]{}-complete for $H$-free graphs. If $s=3$ and $r=2$, then $H=P_1+P_2+P_3$, a contradiction. If $s=3$ and $r=3$, then $H$ contains an induced $2P_3$. This means we may apply Theorem \[t-hard\] to find that [Longest Path Contractibility]{} is [[NP]{}]{}-complete for $H$-free graphs. Hence, $s=2$, and thus $r=2$. Then $H=P_1+2P_2$ is an induced subgraph of $P_1+P_5$, a contradiction.
Finally suppose that exactly one connected component of $H$ contains an edge. Then $H=sP_1+P_r$ for some $r\geq 2$. As $H$ is not an induced subgraph of $sP_1+P_4$, we find that $r\geq 5$. If $r\geq 6$, then $H$ contains an induced $P_6$. This means we may apply Theorem \[t-hard\] to find that [Longest Path Contractibility]{} is [[NP]{}]{}-complete for $H$-free graphs. Hence, $r=5$. As $H\neq P_1+P_5$, we find that $s\geq 2$. Then $H=sP_1+P_5$ contains an induced $2P_1+2P_2$. This means we may apply Theorem \[t-hard\] to find that [Longest Path Contractibility]{} is [[NP]{}]{}-complete for $H$-free graphs.
[**Case 2.**]{} The number of connected components of $H$ is exactly 2.\
Then $H=P_r+P_s$ for some $r$ and $s$ with $1\leq r\leq s$. If $r\geq 3$ then $H$ contains an induced $2P_3$, and if $s\geq 6$ then $H$ contains an induced $P_6$. In both cases we apply Theorem \[t-hard\] to find that [Longest Path Contractibility]{} is [[NP]{}]{}-complete for $H$-free graphs. From now on assume that $r\leq 2$ and $s\leq 5$. If $s\leq 4$, then $H$ is an induced subgraph of $P_2+P_4$, a contradiction. Hence, $s=5$. If $r=1$, then $H=P_1+P_5$, a contradiction. Thus $r=2$. Then $H$ contains an induced $3P_2$. This means we may apply Theorem \[t-hard\] to find that [Longest Path Contractibility]{} is [[NP]{}]{}-complete for $H$-free graphs.
[**Case 3.**]{} The number of connected components of $H$ is exactly 1.\
If $H=P_r$ for some $r\leq 5$, then we use Theorem \[t-p1p5\]. Otherwise $P_6$ is an induced subgraph of $H$, and we use Theorem \[t-hard\].
Longest Cycle Contractibility {#s-cycle}
=============================
The length of of a longest cycle a graph $G$ can be contracted to is called the [*co-circularity*]{} [@Bl82] or [*cyclicity*]{} [@Ha99] of $G$. This leads to the following decision problem.
[.99]{} <span style="font-variant:small-caps;">[Longest Cycle Contractibility]{}</span>\
----------------- -----------------------------------------------
* Instance:* [a connected graph $G$ and an integer $k$.]{}
*Question:* [does $G$ contain $C_k$ as a contraction?]{}
----------------- -----------------------------------------------
Hammack proved that [Longest Cycle Contractibility]{} is [[NP]{}]{}-complete for general graphs [@Ha02] but polynomial-time solvable for planar graphs [@Ha99]. It is also known that $C_6$-[Contractibility]{}, and thus [Longest Cycle Contractibility]{}, is [[NP]{}]{}-complete for $K_{1,3}$-free graphs [@FKP13] and bipartite graphs [@DP17], and thus for $C_r$-free graphs if $r$ is odd. The purpose of this section is to show that the complexities of [Longest Cycle Contractibility]{} and [Longest Path Contractibility]{} may not coincide on $H$-free graphs.
For a given hypergraph ${\cal H}=(Q,{\cal S})$ we first construct the graph $G_{\cal H}$ as before. We then add an edge between vertices $t_1$ and $t_2$. This yields the graph $G_{\cal H}'$. We need the following result from [@BV87].
\[l-bv87b\] A hypergraph ${\cal H}$ has a $2$-colouring if and only if $G_{\cal H}'$ has $C_4$ as a contraction.
We now prove the following lemma.
\[l-p2p4again\] Let ${\cal H}$ be a hypergraph. Then the graph $G_{\cal H}'$ is $(P_2+P_4)$-free.
For contradiction, assume that $G_{\cal H}'$ contains a subgraph $H$ isomorphic to $P_2+P_4$. Let $D_1$ be the connected component of $H$ on two vertices, and let $D_2$ be the connected components of $H$ on four vertices. As the subgraph of $G_{\cal H}'$ induced by ${\cal S}\cup {\cal S}'\cup \{t_1,t_2\}$ is complete bipartite and thus $P_4$-free, we find that $D_2$ must contains a vertex of $Q$. As $Q$ is a clique, $D_2$ must also contain at least two vertices from ${\cal S}\cup {\cal S}'$. We may without loss of generality assume that $D_2$ contains a vertex from ${\cal S}$. This means that $D_1$ contains no vertex from $Q\cup {\cal S}'\cup \{t_1\}$. Hence, $D_1$ only contains vertices of ${\cal S}\cup \{t_2\}$. As the latter set is independent, this is not possible. We conclude that $G_{\cal H}'$ is $(P_2+P_4)$-free.
We note that, in line with our polynomial-time result of [Longest Path Contractibility]{} for $(P_2+P_4)$-free graphs (Theorem \[t-p2p4\]), the graph $G_{\cal H}$ may not be $(P_2+P_4)$-free: as $t_1$ and $t_2$ are not adjacent in $G_{\cal H}$, two vertices of $Q$ together with vertices $t_1,S_j,S_\ell',t_2$ may form an induced $P_2+P_4$ in $G_{\cal H}$.
It is readily seen that $C_4$-[Contractibility]{} belongs to [[NP]{}]{}. Hence, we obtain the following result from Lemmas \[l-bv87b\] and \[l-p2p4again\].
\[t-cycle\] [$C_4$-Contractibility]{} is [[NP]{}]{}-complete for $(P_2+P_4)$-free graphs.
Theorem \[t-cycle\] has the following consequence.
\[c-cycle\] [Longest Cycle Contractibility]{} is [[NP]{}]{}-complete for $(P_2+P_4)$-free graphs.
Recall that [Longest Path Contractibility]{} is polynomial-time solvable for $(P_2+P_4)$-free graphs by Theorem \[t-main\]. Hence, combining this result with Corollary \[c-cycle\] shows that the two problems behave differently on $(P_2+P_4)$-free graphs.
Conclusions {#s-con}
===========
We completely classified the complexities of [Longest Induced Path]{} and [Longest Path Contractibility]{} problem for $H$-free graphs. Such a classification is still open for [Longest Path]{} and below we briefly present the state of art.
A graph is [*chordal bipartite*]{} if it is bipartite and every induced cycle has length 4. In other words, a graph is chordal bipartite if and only if it is $(C_3,C_5,C_6,\ldots)$-free. A direct consequence of the [[NP]{}]{}-hardness of [Hamiltonian Path]{} for chordal bipartite graphs and strongly chordal split graphs [@Mu96], or equivalently, strongly chordal $(2P_2,C_4,C_5)$-free graphs [@FH77] is that [Hamiltonian Path]{}, and therefore, [Longest Path]{} is [[NP]{}]{}-complete for $H$-free graphs if $H$ has a cycle or contains an induced $2P_2$. The [[NP]{}]{}-hardness of [Hamiltonian Path]{} for line graphs [@Be81], and thus for $K_{1,3}$-free graphs, implies the same result for $H$-free graphs if $H$ is a forest with a vertex of degree at least 3. On the positive side, [Longest Path]{} is polynomial-time solvable for $P_4$-free graphs due to the corresponding result for its superclass of cocomparability graphs [@IN13; @MC12]. This leaves open the following cases.
Determine the computational complexity of [Longest Path]{} for $H$-free graphs when:
- $H=sP_1+P_r$ for $3\leq r\leq 4$ and $s\geq 1$
- $H=sP_1+P_2$ for $s\geq 2$
- $H=sP_1$ for $s\geq 3$.
We showed that the complexities of [Longest Cycle Contractibility]{} and [Longest Path Contractibility]{} do not coincide for $H$-free graphs. However, the complexity of [Longest Cycle Contractibility]{} for $H$-free graphs has not been settled yet. For instance, if $H$ is a cycle, the cases $H=C_4$ and $H=C_6$ are still open.
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[^1]: Supported by Research Project Grant RPG-2016-258 from the Leverhulme Trust.
[^2]: The complexity status of [Snake-in-the Box]{} is still open. A table of world records for small values of $n$ can be found at http://ai1.ai.uga.edu/sib/sibwiki/doku.php/records.
[^3]: In [@HHLP14], the problem is equivalently formulated as a graph modification problem: can a graph $G$ be modified into a graph $F$ from some specified family ${\cal F}$ by using at most $\ell$ edge contractions for some given integer $\ell\geq 0$? We refer to for instance [@ALSZ17; @AST17; @AH83; @BGHP14; @CG13; @Ep09; @GHP13; @GM13; @HHLLP14; @HHLP13; @LMS13; @MOR13; @WAN81; @WAN83], for both classical and fixed-parameter tractibility results for various families ${\cal F}$ including the family of paths, which form the focus in this paper.
[^4]: This is in line with research for other graph problems restricted to $H$-free graphs. In fact, classes of $H$-free graphs, where $H$ is a linear forest are still poorly understood. There is a whole range of graph problems, e.g. [Independent Set]{}, [$3$-Colouring]{}, [Feedback Vertex Set]{}, [Odd Cycle Transversal]{}, and [Dominating Induced Matching]{}, for which it is not known if they are [[NP]{}]{}-complete on $P_k$-free graphs for some integer $k$, such that they are [[NP]{}]{}-complete on $P_k$-free graphs (see [@BDFJP]).
[^5]: If every $Z_i$ has size 2, then we obtain the well-known [$k$-Disjoint Paths]{} problem.
|
---
abstract: 'We prove that every stationary polyhedral varifold minimizes area in the following senses: (1) its area cannot be decreased by a one-to-one Lipschitz ambient deformation that coincides with the identity outside of a compact set, and (2) it is the varifold associated to a mass-minimizing flat chain with coefficients in a certain metric abelian group.'
address: |
Department of Mathematics\
Stanford University\
Stanford, CA 94305
author:
- Brian White
date: 'November 29, 2019. Revised January 11, 2020.'
nocite:
- '[@pedrosa-ritore]'
- '[@hoffman-wei]'
title: Stationary polyhedral varifolds minimize area
---
NOTE: After this paper was written, I learned that (1) above was proved by Choe [@choe] and that (2) was proved by Morgan [@morgan]. Hence this note should be viewed as an exposition of their results. (Perhaps the result in §7 is new.)
Introduction
============
The tangent cone to any $2$-dimensional minimal variety (i.e., stationary varifold with a positive lower bound on density) is polyhedral (because the link is a stationary $1$-varifold in the unit sphere, and thus is a finite union of geodesic arcs [@allard-almgren].) If the variety is area-minimizing in some sense, then the cone is also area-minimizing. Thus it is natural to ask: which $2$-dimensional stationary cones, or, more generally, which $m$-dimensional stationary polyhedral cones, are area-minimizing in some sense?
In her celebrated 1976 paper [@taylor-soap] about soap films, Jean Taylor classified all two-dimensional, multiplicity-one polyhedral cones in ${\mathbf{R}}^3$ that minimize area in the following sense: the area (without counting multiplicity) cannot be decreased by a compactly supported Lipschitz deformation. The Lipschitz deformation need not be one-to-one. She showed that there are only three such cones: the plane, the union of three halflplanes meeting at equal angles along their common edge, and the cone over a regular tetrahedron. However, there are many other stationary polyhedral cones. For example, in ${\mathbf{R}}^3$, there are seven other $2$-dimensional, multiplicity-one stationary cones that have, away from the vertex, only triple-junction-type singularities [@taylor-soap]\*[pp. 501, 502]{}. Taylor’s theorem leaves open the possibility that those other cones could be area-minimizing in some other sense. For example, one could ask
1. Which stationary polyhedral cones minimize area with respect to deformations by compactly supported Lipschitz homeomorphisms of the ambient space?
2. Which stationary polyhedral cones can be assigned orientations and multiplicities in some coefficient group in such that a way that the cone is mass-minimizing as a flat chain for that coefficent group?
In this paper, we show that [**all**]{} stationary polyhedral cones have both of those properties. Indeed, the properties hold for every stationary polyhedral varifold, whether or not conical.
In particular, we prove
\[ONE\] Suppose that $V$ is an $m$-dimensional, rectifiable varifold in an open subset $U$ of ${\mathbf{R}}^N$. Suppose that $V$ is stationary, that $V$ is supported in a finite union of affine $m$-planes, and that $V$ has finite mass. Then $${\operatorname{M}}(V) \le {\operatorname{M}}(f_\#V)$$ for every Lipschitz homeomorphism $f: U\to U$ such that $\{x: \phi(x)\ne x\}$ has compact closure in $U$.
Here ${\operatorname{M}}(V)$ is the total mass of the varifold $V$. (Thus ${\operatorname{M}}(V)$ is $\mu_V({\mathbf{R}}^N)$ in the notation of [@simon-book] or $\|V\|({\mathbf{R}}^N)$ in the notation of [@allard].)
Theorem \[ONE\] is a consequence of the following:
\[TWO\] Suppose that $\Gamma$ is a closed subset of ${\mathbf{R}}^N$ consisting of a finite union of $(m-1)$-dimensional polyhedra. Suppose that $V$ is an $m$-dimensional, compactly supported, rectifiable varifold in ${\mathbf{R}}^N$ such that
1. $V$ is supported in a finite union of affine $m$-planes, and
2. $V$ is stationary in ${\mathbf{R}}^N\setminus \Gamma$.
Then there is a metric abelian group $G$ and a polyhedral $m$-chain $A$ with coefficients in $G$ such that
1. $V$ is the varifold associated to $A$.
2. $\partial A$ is supported in $\Gamma$.
3. $A$ is mass-mininimizing: if $A'$ is any other flat $m$-chain (with coefficients in $G$) such that $\partial A'=\partial A$, then ${\operatorname{M}}(A)\le {\operatorname{M}}(A')$.
We can choose the coefficient group $G$ to be a certain finite-dimensional Euclidean space (namely $\Lambda_n{\mathbf{R}}^N$) with its associated norm. Alternatively, we can choose $G$ to be a discrete group with the property: $$\label{property}
\text{ $\{g\in G: |g|\le \lambda\}$ is finite for every $\lambda<\infty$}. \tag{*}$$ If the varifold is an integral varifold, then we can also require that $|g|$ is an integer for every $g\in G$.
(Theorem \[ONE\] and Theorem \[TWO\] with $G=\Lambda_m{\mathbf{R}}^N$ are proved in §\[main-section\]. Theorem \[changing-theorem\] in §\[changing-section\] shows how one can then construct from $\Lambda_n{\mathbf{R}}^N$ a suitable coefficient group that has Property .)
For either choice of coefficient group, the resulting space of flat chains satisfies the compactness theorem: given any sequence of flat chains supported in a compact set with mass and boundary mass bounded above, there is a subsequence that converges in the flat topology. However, if $G$ is a normed vector space over ${\mathbf{R}}$, then there will be finite-mass flat chains with coefficients in $G$ that fail to be rectifiable. On the other hand, if the coefficient group has Property , then every finite mass flat chain is rectifiable. (These assertions about rectifiablity and lack of rectifiability follow from [@white-rectifiability]\*[Theorem 7.1]{}.) Thus coefficient groups with Property provide a nice setting for the Plateau problem: mass minimizing surfaces (for any boundary) exist, and they must be rectifiable. This paper shows that every stationary polyhedral cone arises as the varifold associated to a mass-minimizing cone for such a coefficient group.
This paper is meant to be largely self-contained. See [@white-fluids] for a brief overview of flat chain with coefficients in a metric abelian group, or Fleming’s original paper [@fleming] for a thorough treatment.
I would like to thank Giovanni Alberti for stimulating discussions on the topic of this paper.
Abstract Calibrations
=====================
Let ${\mathcal C}$ be a collection (e.g., of surfaces), let ${\operatorname{M}}:{\mathcal C}\to{\mathbf{R}}$ be a real-valued function (e.g., “mass" or “weighted area"), and let $\sim$ be an equivalence relation on ${\mathcal C}$ (e.g, the relation “is homologous to".) An [**abstract calibration**]{} on $({\mathcal C},M,\sim)$ is a function $$\Phi: {\mathcal C}\to {\mathbf{R}}$$ such that
1. $\Phi$ is constant on each equivalence class, and
2. $\Phi(S)\le {\operatorname{M}}(S)$ for every $S\in{\mathcal C}$.
If $\Phi(S)={\operatorname{M}}(S)$, we say that $S$ is [**calibrated**]{} by $\Phi$.
Abstract calibrations are of interest because of the following trivial theorem:
Suppose that $\Phi:{\mathcal C}\to{\mathbf{R}}$ is a calibration on $({\mathcal C},{\operatorname{M}},\sim)$ and that $S\in {\mathcal C}$ is calibrated by $\Phi$. Then $S$ minimizes ${\operatorname{M}}$ in its equivalence class: $${\operatorname{M}}(S)\le {\operatorname{M}}(S') \quad\text{for all $S' \sim S$}.$$ Furthermore, if $S'\sim S$, then $S'$ minimizes ${\operatorname{M}}$ in its equivalence class if and only if $S'$ is also calibrated by $\Phi$.
Now suppose that ${\mathcal C}$ is an abelian group and that $\mathcal{B}$ is a subgroup. (Think of ${\mathcal C}$ as a collection of $m$-chains and ${\mathcal B}$ as the chains in ${\mathcal C}$ that are boundaries of $(m+1)$-chains.) Then we have the equivalence relation on ${\mathcal C}$ given by $$S \sim S' \quad\text{if and only if}\quad S-S' \in {\mathcal B}.$$ Now suppose that $$\Phi: {\mathcal C}\to{\mathbf{R}}$$ is a homomorphism that vanishes on $\mathcal{B}$. Then $\Phi$ is constant on equivalence classes for the equivalence relation $\sim$. Thus if $\Phi(\cdot)\le {\operatorname{M}}(\cdot)$, then $\Phi$ is an [**abstract calibration**]{} for $({\mathcal C},{\operatorname{M}}, \sim)$. In that case, we will call $\Phi$ a (generalized) [**calibration**]{}.
Polyhedral Chains
=================
Fix an ambient Euclidean space ${\mathbf{R}}^N$ and an abelian group $(G,+)$ with a norm $|\cdot|$, i.e., with a function $$|\cdot|: G\to{\mathbf{R}}$$ such that the distance function $\operatorname{d}(g,g'):= |g-g'|$ makes $G$ into a metric space.
A [**formal polyhedral $m$-chain**]{} (in ${\mathbf{R}}^N$, with coefficients in $G$) is a formal sum $$\sum_{i=1}^k g_i \sigma_i$$ where $g_i\in G$ and $\sigma_i$ is an oriented $m$-dimensional polyhedron in $E$. Let ${\mathcal{P}}_m^\textnormal{formal}={\mathcal{P}}_m^\textnormal{formal}(G)$ be the abelian group of formal polyhedral $m$-chains in ${\mathbf{R}}^N$ with coefficients in $G$.
Consider the equivalence relation on formal polyhedral chains generated by $$g\sigma \equiv g\sigma' + g \sigma''$$ if $\sigma'$ and $\sigma''$ are obtained from $\sigma$ by subdivision, and by $$g\sigma \equiv - g\tilde \sigma$$ is $\tilde \sigma$ is obtained from $\sigma$ by reversing the orientation.
A [**polyhedral $m$-chain**]{} is an equivalence class in ${\mathcal{P}}_m^\textnormal{formal}$ under this equivalence relation. We let ${\mathcal{P}}_m={\mathcal{P}}_m(G)$ be the abelian group of polyhedral $m$-chains with coefficients in $G$.
If $\sum_{i=1}^k g_i \sigma_i$ is a formal polyhedral $m$-chain, we let $$\sum_{i=1}^k g_i [\sigma_i]$$ denote the corresponding equivalence class, i.e., the corresponding polyhedral chain. Every polyhedral chain has a representation $\sum_ig_i[\sigma_i]$ in which the $\sigma_i$ are non-overlapping. For such a representation, the mass (or weighted area) of the chain is given by $${\operatorname{M}}\left(\sum g_i[\sigma_i]\right) := \sum |g_i|\,{\mathcal{H}}^m(\sigma_i).$$
A Canonical Calibration {#canonical-section}
=======================
In this section, we fix nonnegative integers $m$ and $N$ with $m<N$. Let $G$ be the additive group $\Lambda_m{\mathbf{R}}^N$ of $m$-vectors in ${\mathbf{R}}^N$. We give $\Lambda_m{\mathbf{R}}^N$ the Euclidean norm. That is, we give $\Lambda_m{\mathbf{R}}^N$ the norm associated to the Euclidean structure for which the $m$-vectors ${\mathbf e}_{i_1}\wedge \dots \wedge {\mathbf e}_{i_m} (i_1<i_2<\dots < i_m)$ form an orthonormal basis of $\Lambda_m{\mathbf{R}}^N$.
(For the study of non-rectifiable chains, it would probably be better to use the mass norm [@federer-fleming]\*[page 461]{} on $\Lambda_n{\mathbf{R}}^N$. However, in this paper we are primarily interested in rectifiable chains, and whether one uses the Euclidean norm or the mass norm does not affect the masses of rectifiable chains.)
If $\sigma$ is an oriented $m$-dimensional polyhedron in ${\mathbf{R}}^N$, we let $\eta(\sigma)$ be the corresponding simple unit $m$-vector.
Now define a map $$\Phi: {\mathcal{P}}_m^\textnormal{formal}(G) \to {\mathbf{R}}$$ by $$\Phi\left(\sum_i g_i \sigma_i \right) = \sum_i (g_i\cdot \eta(\sigma_i)) {\mathcal{H}}^m{\sigma_i}.$$ Note that $\Phi$ is an additive homomorphism.
If $\tilde \sigma$ is obtained from $\sigma$ by reversing the orientation, then $\eta(\tilde \sigma)=-\eta(\sigma)$, so $$(-g)\cdot\eta(\tilde \sigma) = g \cdot \sigma,$$ and therefore $$\Phi(-g\tilde \sigma) = \Phi(g\sigma).$$ Similarly, $$\Phi(g\sigma) = \Phi(g\sigma') + \Phi(g\sigma'')$$ if $\sigma'$ and $\sigma''$ are obtained by subdividing $\sigma$.
Hence $\Phi$ induces a well-defined homomorphism $$\begin{aligned}
&\Phi: {\mathcal{P}}_m(G) \to {\mathbf{R}}, \\
&\Phi\left(\sum_i g_i[\sigma_i]\right) = \sum (g_i\cdot \eta(\sigma_i)) {\mathcal{H}}^m(\sigma_i).\end{aligned}$$
\[calibration-criterion\] Suppose that $G=\Lambda_m{\mathbf{R}}^N$ and that $$A = \sum g_i [\sigma_i]$$ is a polyhedral chain in ${\mathcal{P}}_m(G)$, where the $\sigma_i$ are non-overlapping. Then $$\Phi(A)\le {\operatorname{M}}(A),$$ with equality if and only if each $g_i$ is a nonnegative scalar multiple of $\eta(\sigma_i)$.
By Cauchy-Schwartz, $g_i\cdot\eta(\sigma_i)\le |g_i|$, with equality if and only if $g_i$ is a nonnegative scalar multiple of $\eta(\sigma_i)$. Hence $$\begin{aligned}
\Phi(A)
&= \sum (g_i\cdot\eta(\sigma_i)) {\mathcal{H}}^m(\sigma_i) \\
&\le \sum |g_i| {\mathcal{H}}^m(\sigma_i) \\
&= {\operatorname{M}}(A).\end{aligned}$$ with equality if and only if each $g_i$ is a nonnegative scalar multiple of $\eta(\sigma_i)$.
\[stokes-theorem\] Suppose that $G=\Lambda_m{\mathbf{R}}^N$ and that $A\in {\mathcal{P}}_m(G)$ is the boundary of an $(m+1)$-chain. Then $\Phi(A)=0$.
Let $g\in G$, let $\tau$ be an oriented $(m+1)$-dimensional polyhedron in $E$, and let $\sigma_1,\dots,\sigma_k$ be the $m$-dimensional faces of $\tau$ with the induced orientations. It suffices to show that $$\label{zero}
\Phi\left(\sum_i g [\sigma_i]\right) = 0.$$ Since any $m$-vector is a sum of simple $m$-vectors, it suffices to prove when $g$ is a simple $m$-vector. Let $V$ be the oriented $m$-plane associated to $g$. Let $\omega$ be the volume form on $V$, $\Pi: {\mathbf{R}}^N\to V$ be orthogonal projection, and let $\Omega = \Pi^\#\omega$. Note that $$\Phi(g[\sigma_i]) = |g| \int_{\sigma_i}\Omega.$$ Thus $$\begin{aligned}
\Phi\left(\sum_i g[\sigma_i]\right)
&= |g|\sum \int_{\sigma_i}\Omega \\
&= |g| \int_{\partial \tau}\Omega \\
&= 0\end{aligned}$$ by Stokes Theorem (since $\Omega$ is constant and therefore $d\Omega=0$.)
Recall that the [**flat norm**]{} of $A\in {\mathcal{P}}_m(G)$ is given by $${\mathcal F}(A) = \inf_{Q\in {\mathcal{P}}_{m+1}(G)} ({\operatorname{M}}(A + \partial Q) + {\operatorname{M}}(Q)),$$ which is trivially $\le {\operatorname{M}}(A)$.
\[Phi-bound-theorem\] If $G=\Lambda_m{\mathbf{R}}^N$ and $A\in {\mathcal{P}}_m(G)$, then $$\Phi(A) \le {\mathcal F}(A) \le {\operatorname{M}}(A),$$ where ${\mathcal F}(A)$ is the flat norm of $A$.
Let $Q$ be a polyhedral $(m+1)$-chain. By Theorem \[stokes-theorem\], $$\begin{aligned}
\Phi(A)
&= \Phi(A + \partial Q) \\
&\le {\operatorname{M}}(A+\partial Q) \\
&\le {\operatorname{M}}(A+\partial Q) + {\operatorname{M}}(Q).\end{aligned}$$ Hence $$\Phi(A) \le \inf_Q ({\operatorname{M}}(A+\partial Q) + {\operatorname{M}}(Q)) = {\mathcal F}(A).$$
The map $\Phi$ extends continuously to an additive homomorphism $\Phi: {\mathcal F}_m(G)\to {\mathbf{R}}$ such that $$\Phi(A)\le {\mathcal F}(A)\le {\operatorname{M}}(A)$$ for every flat $m$-chain $A$.
The corollary follows immediately from the theorem since ${\mathcal F}_m(G)$ is the metric space completion of ${\mathcal{P}}_m(G)$ with respect the flat norm.
Polyhedral Varifolds
====================
If $\sigma$ is an $m$-dimensional polyhedron in ${\mathbf{R}}^N$, we let ${\operatorname{var}}(\sigma)$ be the associated $m$-dimensional, multiplicity-$1$ rectifiable varifold, i.e., the rectifiable varifold whose associated Radon measure is ${\mathcal{H}}^m\llcorner \sigma$. An $m$-dimensional [**polyhedral varifold**]{} in ${\mathbf{R}}^N$ is a varifold of the form $$\sum_{i=1}^k c_i {\operatorname{var}}(\sigma_i),$$ where each $\sigma_i$ is a polyhedron and each $c_i\ge 0$. By subdividing, we can assume that if $i\ne j$, then $\sigma_i\cap\sigma_j$ is either empty or is a common face of $\sigma_i$ and $\sigma_j$ of dimension $<m$.
As in §\[canonical-section\], we let $G=\Lambda_m{\mathbf{R}}^N$. If $\sigma$ is an oriented $m$-dimensional polyhedron in ${\mathbf{R}}^N$, we let $\eta(\sigma)$ be the simple unit $m$-vector associated with the orientation. Let $\left<\sigma\right>$ be the polyhedral $m$-chain in ${\mathcal{P}}_m(G)$ given by $$\left<\sigma\right> = \eta(\sigma) [\sigma].$$ Note that if $\tilde \sigma$ is obtained from $\sigma$ by reversing the orientation, then ${\left<}\tilde \sigma{\right>}={\left<}\sigma{\right>}$. Thus given an unoriented $m$-dimensional polyhedron $\sigma$, we have a well-defined polyhedral $m$-chain ${\left<}\sigma{\right>}$ in ${\mathcal{P}}_m(G)$.
If $$V = \sum_{i=1}^k c_i {\operatorname{var}}(\sigma_i)$$ is an $m$-dimensional polyhedral varifold, we let $${\left<}V{\right>}= \sum_{i=1}^k c_i {\left<}\sigma_i{\right>}.$$ If we give the $\sigma_i$ orientations, then $${\left<}V{\right>}= \sum_{i=1}^k c_i \eta(\sigma_i)[\sigma_i],$$ from which it follows (by Theorem \[calibration-criterion\]) that $V$ is calibrated by $\Phi$. Note that ${\operatorname{M}}({\left<}V{\right>})={\operatorname{M}}(V)$. Thus ${\left<}\,\cdot\,{\right>}$ is a mass-preserving homomorphism from the additive semigroup of $m$-dimensional polyhedral varifolds to the additive group ${\mathcal{P}}_m(G)$ of $m$-dimensional polyhedral chains.
\[boundary-support\] Let $V$ be an $m$-dimensional polyhedral varifold in ${\mathbf{R}}^N$, and let $\Gamma$ be the union of some of the $(m-1)$-dimensional faces of polyhedra in $V$. Then $V$ is stationary in ${\mathbf{R}}^N\setminus \Gamma$ if and only if $\partial {\left<}V{\right>}$ is supported in $\Gamma$.
Since the result is essentially local, it suffices to prove it for $$V = \sum_i c_i{\operatorname{var}}(\sigma_i)$$ where the $\sigma_i$ are polyhedra with a common $(m-1)$-dimensional face $\tau$ and where $\Gamma$ is the union of the faces $\ne \tau$ of the various $\sigma_i$.
Give $\tau$ an orientation, and then give each $\sigma_i$ the orientation that induces the chosen orientation on $\tau$. Let $\eta(\tau)$ be the simple unit $(m-1)$-vector associated to the orientation of $\tau$ and let $\eta(\sigma_i)$ be the simple unit $m$-vector associated with the orientation of $\sigma_i$. Then $$\label{nu-equation}
\eta(\sigma_i) = \eta(\tau)\wedge \nu_i,$$ where $\nu_i$ is the unit normal to $\tau$ that lies in the $m$-plane containing $\sigma_i$ and that points out from $\sigma_i$.
Note that $$\label{stationary-equation}
\begin{gathered}
\text{$V$ is stationary in $\Gamma^c$} \\
\iff
\sum c_i \nu_i = 0 \\
\iff
\sum c_i\eta(\sigma_i)=0.
\end{gathered}$$ On the other hand, $$\begin{aligned}
( \partial {\left<}{\operatorname{var}}(\sigma_i){\right>}) \llcorner \Gamma^c
&=
(\partial \eta(\sigma_i) [\sigma_i])\llcorner \Gamma^c \\
&=
\eta(\sigma_i) [\tau],\end{aligned}$$ so $$\label{boundary-equation}
( \partial {\left<}V{\right>}) \llcorner \Gamma^c
=
\left( \sum c_i \eta(\sigma_i) \right) [\tau].$$ The theorem follows immediately from and .
The Main Theorem {#main-section}
================
\[main-theorem\] Let $G=\Lambda_m{\mathbf{R}}^N$. If $V$ is an $m$-dimensional polyhedral varifold, then ${\left<}V{\right>}$ is a mass-minimizing polyhedral chain in ${\mathcal{P}}_m(G)$. If $\Gamma$ is closed set (such as the union of some of the faces of polyhedra in $V$) and if $V$ is stationary in $\Gamma^c$, then $\partial {\left<}V{\right>}$ is supported in $\Gamma$ and $${\operatorname{M}}(V) \le {\operatorname{M}}(\phi_\#V)$$ for any lipschitz homeomorphism $\phi: {\mathbf{R}}^N\to{\mathbf{R}}^N$ such that $\phi(x)=x$ for all $x\in \Gamma$.
By construction, ${\left<}V{\right>}$ is calibrated by $\Phi$. Hence ${\left<}V{\right>}$ is homologically mass-minimizing. Since the ambient space ${\mathbf{R}}^N$ is homologically trivial, that means that ${\left<}V{\right>}$ minimizes mass, i.e., that ${\operatorname{M}}({\left<}V{\right>})\le {\operatorname{M}}(A)$ for any flat chain $A$ in ${\mathcal F}_m(G)$ with $\partial A=\partial {\left<}V{\right>}$.
Now suppose that $V$ is stationary in $\Gamma^c$. Then $\partial {\left<}V{\right>}$ is supported in $\Gamma$ (by Theorem \[boundary-support\]), so $${\operatorname{M}}({\left<}V{\right>}) \le {\operatorname{M}}( f_\# {\left<}V{\right>})$$ for any Lipschitz map $f:{\mathbf{R}}^N\to{\mathbf{R}}^N$ such that $f(x)=x$ for all $x\in \Gamma$. Now ${\operatorname{M}}({\left<}V{\right>})={\operatorname{M}}(V)$, and, if $f$ is one-to-one, then ${\operatorname{M}}(f_\#{\left<}V{\right>}) = {\operatorname{M}}(f_\#V)$. Thus $${\operatorname{M}}(V) \le {\operatorname{M}}(f_\#V).$$
Changing the Coefficient Group {#changing-section}
==============================
\[changing-theorem\] Let $G$ be an abelian group with norm $|\cdot|_G$. Let $S=\{g_1,\dots,g_k\}$ be a finite subset of $G$, and let $H$ be the subgroup of $G$ generated by $S$. Define a norm $|\cdot|_H$ on $H$ by $$|g|_H = \min \left\{ \sum_{i=1}^k |n_i|\,|g_i|: g = \sum_{i=1}^k n_i g_i \right\},$$ where the $n_i$ are integers. Suppose that $A$ is a rectifiable flat chain in ${\mathcal F}_m(G)$ and that all of its multiplicities are in $S$. Then $A$ may also be regarded as a rectifiable flat chain in ${\mathcal F}_m(H)$ (with the same mass). Furthermore, if $A$ is mass-minimizing in ${\mathcal F}_m(G)$, then it is also mass-minimizing in ${\mathcal F}_m(H)$.
Note that $\{g\in H: |g|_H\le \lambda\}$ is finite if $\lambda<\infty$. Note also that $|\cdot|_H$ is the largest norm on $H$ such that $|g_i|_H=|g_i|_G$ for each of the generators $g_1,\dots,g_k$.
Let ${\operatorname{M}}_G$ and ${\mathcal F}_G$ denote mass and flat norm on chains (polyhedral or flat) with coefficients in the group $G$ with the norm $|\cdot|_G$. Let ${\operatorname{M}}_H$ and ${\mathcal F}_H$ denote mass and flat norm on chains with coefficients in the group $H$ with the norm $|\cdot|_H$. Since ${\operatorname{M}}_H\ge {\operatorname{M}}_G$ on ${\mathcal{P}}_m(H)$, it follows that ${\mathcal F}_H\ge {\mathcal F}_G$ on ${\mathcal{P}}_m(H)$. Consequently, the inclusion $${\mathcal{P}}_m(H) \subset {\mathcal{P}}_m(G)$$ extends to an inclusion $${\mathcal F}_m(H) \subset {\mathcal F}_m(G)$$ such that ${\operatorname{M}}_G(T)\le {\operatorname{M}}_H(T)$ and ${\mathcal F}_G(T)\le {\mathcal F}_H(T)$ for $T\in {\mathcal F}_m(H)$.
Now suppose that $A$ is an ${\operatorname{M}}_G$-minimizing rectifiable chain in ${\mathcal F}_m(G)$ and that the multiplicities of $A$ lie in $S$. Then $A$ is also a rectifiable chain in ${\mathcal F}_m(H)$, and ${\operatorname{M}}_H(A)={\operatorname{M}}_G(A)$. If $A'$ is a chain in ${\mathcal F}_m(H)$ with $\partial A'=\partial A$, then $${\operatorname{M}}_G(A)\le{\operatorname{M}}_G(A')$$ since $A$ is ${\operatorname{M}}_G$-minimizing. Since ${\operatorname{M}}_G(A)={\operatorname{M}}_H(A)$ and ${\operatorname{M}}_G(A')\le {\operatorname{M}}_H(A')$, it follows that $${\operatorname{M}}_H(A) \le {\operatorname{M}}_H(A').$$ Hence $A$ is ${\operatorname{M}}_H$-minimizing in ${\mathcal F}_m(H)$.
|
---
abstract: 'We present an accurate model of the muon-induced background in the DAMA/LIBRA experiment. Our work challenges proposed mechanisms which seek to explain the observed DAMA signal modulation with muon-induced backgrounds. Muon generation and transport are performed using the MUSIC/MUSUN code, and subsequent interactions in the vicinity of the DAMA detector cavern are simulated with [Geant4]{}. We estimate the total muon-induced neutron flux in the detector cavern to be $\Phi_n^\nu = 1.0\times10^{-9}$ cm$^{-2}$ s$^{-1}$. We predict $3.49\times10^{-5}$ counts/day/kg/keV, which accounts for less than $0.3\%$ of the DAMA signal modulation amplitude.'
address: 'Department of Physics and Astronomy, University of Sheffield, Sheffield S3 7RH, UK'
author:
- 'J. Klinger, V. A. Kudryavtsev'
bibliography:
- 'muneu-dama-v5.bib'
title: 'Muon-induced neutrons do not explain the DAMA data'
---
***[=]{}***
Introduction
============
The [DAMA/LIBRA ]{}experiment [@Bernabei2008297] is a highly radiopure NaI(Tl) scintillation detector located at the Gran Sasso National Laboratory (LNGS) which aims to measure the annual modulation signature of dark matter particles [@Bernabei:2009du; @doi:10.1142/S0217751X13300226]. Both the [DAMA/LIBRA ]{}experiment and the first generation DAMA/NaI experiment reported the observation of an approximately annual variation in the number of events observed in the 2-6 keV energy range with a combined significance of approximately 9.3 $\sigma$ [@Bernabei:2013xsa]. If the observed signal modulation is to be explained by the elastic scattering of dark matter particles, it would require that the interaction cross-section of dark matter with nucleons, and the mass of dark matter particles, to be within values that are already excluded by other experiments [@PhysRevD.90.091701; @PhysRevLett.106.131302; @PhysRevD.86.051701; @PhysRevLett.112.091303; @PhysRevLett.112.241302; @PhysRevLett.107.051301; @PhysRevLett.109.181301].
One mechanism that has been proposed in order to explain the DAMA signal modulation is the production of neutrons due to the scattering of cosmogenic muons in the material surrounding the detector [@Blum; @Ralston; @Nygren]. The cosmogenic muon-induced-neutron flux ${\Phi_n^\mu }$ is expected to have an annual variation related to the mean air temperature above the surface of the Earth that affects the muon flux ${\Phi^{\mu} }$ at the surface, and hence underground. This proposal has been disputed for a number of reasons [@doi:10.1142/S0217751X13300226], notably as the annual variation of cosmogenic muons is approximately 30 days out of phase with the DAMA signal [@PhysRevD.85.063505; @Fernandez-Martinez:1443567; @DAMAmuon; @DAMAmuon2].
An extension to this mechanism has also been proposed, which introduces the possibility of a contribution to the total neutron flux from the interactions of solar neutrinos [@PhysRevLett.113.081302]. The solar neutrino-induced neutron flux ${\Phi_n^\nu }$ is also expected to have an annual modulation, due the eccentricity of the Earth’s orbit about the Sun. It is shown that the phase of ${\Phi_n^\nu }$ can shift the phase of the total neutron flux relative to ${\Phi_n^\mu }$.
One should note that most explanations focus on the phase, rather than the amplitude, of the modulation. A rough estimate of the modulated rate of muon-induced neutrons ${R_n^\mu }$ in the [DAMA/LIBRA ]{}experiment can be calculated as: $${R_n^\mu }= {S^\mu }\frac{{\Phi_n^\mu }A t }{ m} \approx 4.6\times10^{-5} \text{~events~/~day~/~kg}$$ where ${\Phi_n^\mu }$ is taken from previous estimates at [LNGS ]{}[@Wulandari:2004bj; @Persiani], ${S^\mu }$ is the amplitude of the muon flux modulation [@selvi2009; @Bellini:2012te] (relative to ${\Phi^{\mu} }$), $A$ is the active surface area perpendicular to ${\Phi^{\mu} }$ of one [DAMA/LIBRA ]{}detector module, $t$ is the number of seconds in a day and $m$ is the active mass of one detector module. This rate is three orders of magnitude less than the modulated DAMA signal, even before the acceptance for muon-induced events in the DAMA analysis is considered.
Despite the latter calculation, and previous work [@doi:10.1142/S0217751X13300226; @DAMAmuon; @DAMAmuon2], the recent paper [@PhysRevLett.113.081302] claims that the DAMA signal can partly be explained by muons. The estimate presented by Davis in Ref. [@PhysRevLett.113.081302], includes only a very approximate calculation of the amplitude of muon- and neutrino-induced signals. The calculation of ${\Phi_n^\mu }$ contradicts the estimates from Refs. [@doi:10.1142/S0217751X13300226; @DAMAmuon; @DAMAmuon2]. The calculation neglects that most neutrons will be accompanied a showering muon, and would therefore not be accepted by DAMA/LIBRA. Also, the value of the mean free path for neutrons in rock that is used (taken from Ref. [@PhysRevD.86.054001]), is actually for liquid argon; which has approximately half of the density of [LNGS ]{}rock. Ref. [@PhysRevLett.113.229001] has additionally shown that the required ${\Phi_n^\nu }$ is at least six orders of magnitude too large.
It is evident from such contradictions that a full Monte Carlo simulation of the muon-induced background in the [DAMA/LIBRA ]{}experiment is required. This would be able to model ${\Phi_n^\mu }$, the detector response and the DAMA event acceptance, and also any enhancement of ${\Phi_n^\mu }$ due the high-$Z$ shielding used by DAMA/LIBRA.
In this paper, we present an accurate calculation of ${\Phi_n^\mu }$ and the total muon-induced background in order to show that the DAMA signal modulation cannot be explained by any muon-induced mechanism. We perform a full simulation of the [DAMA/LIBRA ]{}apparatus and detector shielding in [Geant4.9.6]{} [@Agostinelli2003250], with muons transported to the DAMA cavern in [LNGS ]{}using the MUSIC/MUSUN code [@Kudryavtsev2009339; @Antonioli1997357]. In total, we simulate twenty years of muon-induced data.
Simulation framework {#sim}
====================
We perform the simulation of particle propagation in two stages. In the first stage, only muon transportation from the surface of the Earth down to an underground site is considered and secondary particles [^1] are neglected. In the second stage, the transport and interactions of all particles (including secondary particles) are fully simulated through the material surrounding the [DAMA/LIBRA ]{}apparatus.
Muon transport simulation
-------------------------
The first stage of the simulation is performed using the MUSIC muon transport code [@Kudryavtsev2009339; @Antonioli1997357]. The MUSIC code propagates muons from the surface of the Earth through a uniform rock of density $\rho = 2.71$ g cm$^{-3}$ and records energy distributions of muons at different depths. The MUSUN code [@Kudryavtsev2009339] calculates muon spectra from the modified Gaisser’s parameterisation that takes into account the curvature of the Earth and muon lifetime, convoluted with the slant depth distribution at LNGS. This parameterisation has been previously shown to have a good fit to LVD data [@Kudryavtsev2009339]. The MUSUN code subsequently samples muons on the surface of a cuboid with a height of 35 m and perpendicular dimensions of 20 m $\times$ 40 m. This cuboid includes most of the corridor where the [DAMA/LIBRA ]{}experiment is located and a few meters of rock around it.
[DAMA/LIBRA ]{}detector simulation {#sim2}
----------------------------------
The second stage of the simulation is performed using [Geant4.9.6]{} [@Agostinelli2003250]. The [Geant4.9.6]{} [shielding]{} physics list has been used, and we include the muon-nuclear interaction process. The interactions of low-energy neutrons ($< 20$ MeV) are described by high-precision data-driven models [@endf]. Previous studies [@Araujo2008471; @Persiani; @1748-0221-6-05-P05005; @Schmidt201328; @Reichhart201367] have validated the simulation of neutron production, transport and detection against data. The level of agreement is better than a factor of two.
In this phase of the simulation, all primary and secondary particles are transported from the surface of the cuboid until all surviving particles have propagated outside of the cuboid volume. The cuboid is modeled as [LNGS ]{}rock with a density $\rho = 2.71$ g cm$^{-3}$ and a chemical composition as described in Ref. [@Wulandari2004313]. A corridor (‘cavern’) is positioned within the cuboid, such that there is 10 m of [LNGS ]{}rock overburden, and otherwise 5 m of [LNGS ]{}rock surrounding the cavern walls and floor.
The [DAMA/LIBRA ]{}detector housing is placed halfway along the length of the cavern, adjacent to a cavern wall. The housing is composed of [LNGS ]{}concrete with density $\rho = 2.50$ g cm$^{-3}$ and a chemical composition as described in Ref. [@Wulandari2004313]. The [DAMA/LIBRA ]{}apparatus and detector housing are described in Ref. [@Bernabei2008297]. There are a number of concentric layers of shielding surrounding the [DAMA/LIBRA ]{}detector. Extending outwards from the detector, we model 10 cm of copper, 15 cm of lead, 1.5 mm of cadmium, 50 cm of polyethylene and 1 m of [LNGS ]{}concrete.
We model each of the 25 [DAMA/LIBRA ]{}detector modules, containing a central cuboidal crystal composed of NaI, in addition to light-guides and photomultiplier tubes [@Bernabei2008297]. The dimensions of each module, including a further 2 mm of copper shielding, are $10.6\times10.6\times66.2$ cm$^3$. The 25 modules are placed in a $5\times5$ arrangement in the vertical and width dimensions of the cavern.
The muon-induced neutron flux {#flux}
=============================
-- -------- ----------- ----------- -----------------------------------------------
Cavern $> 0$ MeV $> 1$ MeV $> 1$ MeV (**[\*]{})\
This study & (1) & 10 & 4.0 & 5.0\
This study & (2) & 7.6 & 5.8 & 10\
Wulandari et al. & (2) & No data & 4.3 & 8.5\
Persiani & (3) & 7.2 & 2.7 & No data\
**
-- -------- ----------- ----------- -----------------------------------------------
: A comparison of ${\Phi_n^\mu }$ (in units of $10^{-10}$ cm$^{-2}$ s$^{-1}$) predicted by this study, Wulandari et al. [@Wulandari:2004bj] and Persiani [@Persiani]. The column titled ‘Cavern’ indicates the three distinct cavern geometries used: (1) in this study; (2) by Wulandari et al. and (3) by Persiani. The range of considered neutron energies is shown, and ‘(\*)’ indicates that back-scattered neutrons are included.
\[tab:nuflux\]
In the stage of simulation described in Section \[sim2\], muons and secondary particles are transported through 10 m of [LNGS ]{}rock above the DAMA cavern, and also through 5 m on the sides and underside of the cavern. Integrating over the surface area of the cavern, our simulation predicts ${\Phi_n^\mu }= 1.0\times10^{-9}$ cm$^{-2}$ s$^{-1}$, excluding back-scattering [^2].
In Table \[tab:nuflux\], we compare our result to the simulation of Wulandari et al. [@Wulandari:2004bj], which is performed using FLUKA [@fluka], and of Persiani [@Persiani], which is performed using [Geant4.9.3]{} and MUSIC/MUSUN. Integrating over all neutron energies, our results are consistent within about 30% of the previous estimates. We additionally demonstrate a dependance of ${\Phi_n^\mu }$ on the dimensions of the cavern by scaling ${\Phi_n^\mu }$ to the cavern proportions used by Wulandari et al. (compare the first two rows in Table \[tab:nuflux\]). We attribute this to the different fluxes and energy spectra of vertical and inclined muons. High-$Z$ materials in the detector shielding (lead and copper) will lead to an enhancement of ${\Phi_n^\mu }$ which could, potentially, contribute to the modulated signal. Figure \[fig:neutronFlux\] shows ${\Phi_n^\mu }$ as a function of neutron energy, as predicted in the [LNGS ]{}cavern and after all particles are propagated through the various layers of the [DAMA/LIBRA ]{}shielding. It is shown that ${\Phi_n^\mu }$ increases by a factor of $> 5$ due to the shielding. As we will discuss in Section \[results\], this enhancement of ${\Phi_n^\mu }$ is still insufficient to explain the DAMA data.
![The distribution of ${\Phi_n^\mu }$ as a function of neutron energy for neutrons entering the cavern in which [DAMA/LIBRA ]{}is situated (solid lines) and entering the [DAMA/LIBRA ]{}detector modules after all shielding is traversed (dashed line). The distributions excluding and including back-scattered neutrons are shown in black and red respectively.[]{data-label="fig:neutronFlux"}](neutronFlux-eps-converted-to.pdf){width="\linewidth"}
Analysis {#results}
========
In our simulation, each detector module is treated independently and all information due to an energy deposition from any particle in the NaI crystal volumes is recorded. DAMA categorises events as being [[*single-hit*]{} ]{}or [[*multiple-hit*]{} ]{}if the event has an energy deposit in only a single crystal or in multiple crystals respectively. The DAMA signal region, in which the modulated signal is observed, is then defined only for [[*single-hit*]{} ]{}events with a total energy deposit (${E_{\text{Dep}} }$) in the range 2-6 keV.
In the following sections, we will show that the number of muon-induced events entering the [DAMA/LIBRA ]{}signal region is too low to explain the signal modulation.
Resolution and quenching factors
--------------------------------
We model the [DAMA/LIBRA ]{}experimental resolution by applying a Gaussian smearing to the sum of all energy deposited in each crystal, using resolution parameters provided by Ref. [@Bernabei2008297]. As [Geant4]{} does not account for the quenching of energy depositions in nuclear recoils, we apply correction factors obtained from previous studies [@Tovey:1998ex; @Spooner:1994ca; @1748-0221-3-06-P06003; @Gerbier:1998dm; @Simon:2002cw]. Figure \[fig:neutronenergy\] shows the energy spectrum of all crystals with energy depositions, after corrections factors are applied.
![The energy spectrum for all crystals with energy depositions, with resolution and nuclear recoil quenching factors applied. The equivalent of twenty years of muon-induced data is presented.[]{data-label="fig:neutronenergy"}](eDeplog_no_cuts_resolution_quenching-eps-converted-to.pdf){width="\linewidth"}
Single-hit and multiple-hit events {#singlehit}
----------------------------------
An important detail that is neglected in previous attempts [@Blum; @Ralston; @Nygren; @PhysRevLett.113.081302] to explain the DAMA signal modulation with muon-induced backgrounds is the acceptance for [[*single-hit*]{} ]{}events. Figure \[fig:ncrystals\] shows the number of crystals in events with ${E_{\text{Dep}} }\geq 2$ keV per crystal, and in which at least one crystal in the event has a total energy absorption of 2-6 keV. It is shown that $< 9\%$ of these events are [[*single-hit*]{} ]{}events, which suppresses any enhancement of ${\Phi_n^\mu }$ due to interactions in the detector shielding.
![The distribution of the hit multiplicity in events with ${E_{\text{Dep}} }\geq 2$ keV per crystal. At least one crystal has a total energy absorption in the range 2-6 keV. The equivalent of twenty years of muon-induced data is presented.[]{data-label="fig:ncrystals"}](nCrystals_2to6keV-eps-converted-to.pdf){width="\linewidth"}
Events in the signal region {#signalregion}
---------------------------
In this section, we present the number of muon-induced [[*single-hit*]{} ]{}events predicted by our simulation. The distribution of the energy deposited in crystals in [[*single-hit*]{} ]{}events is shown in Figure \[sig:signalregion\]. For ${E_{\text{Dep}} }< 20$ keV, the muon-induced background is dominated by isolated neutrons.
In the range 2-6 keV there are 245 muon-induced events predicted over a period equivalent to twenty years. The total sensitive mass of the [DAMA/LIBRA ]{}detector is 242.5 kg, therefore we predict the rate of muon-induced events in this energy range to be $3.49\times10^{-5}$ counts / day / kg / keV with approximately 6% statistical uncertainty. We estimate the systematic uncertainty to be approximately 30% by comparing different predictions of ${\Phi_n^\mu }$, as presented in Section \[flux\].
We are able to compare our prediction to the conservative estimate presented in Ref. [@DAMAmuon; @DAMAmuon2] which is in agreement with our results. The calculated event rate accounts for $\sim 0.3\%$ of the modulation amplitude reported by DAMA of $(1.12 \pm 0.12)\times10^{-2}$ counts / day / kg / keV [@Bernabei:2013xsa]. It is clear from this comparison that, even if the systematic uncertainty is bigger than our estimates, no muon-induced background can be used to explain the observed signal modulation. Our simulations (Figures \[fig:ncrystals\] and \[sig:signalregion\]) also show that, if muon-induced backgrounds could explain the DAMA data, one should expect a non-negligible modulation of the muon-induced background above 6 keV, as well as for events with multiple hits, which is not seen by DAMA.
![The distribution of the total energy deposition in [[*single-hit*]{} ]{}events. The blue dashed line and the black solid line show the sum of all energy depositions before and after energy resolution is considered, respectively. The stacked colored bars indicate the relative fraction of all events (before energy resolution is considered) attributed to events in which only neutrons deposit energy (yellow) and other events (blue). The equivalent of twenty years of muon-induced data is presented.[]{data-label="sig:signalregion"}](eDep_single_hit-eps-converted-to.pdf){width="\linewidth"}
Discussion
==========
In this section we will argue that muon-induced neutrons cannot explain the DAMA data, even before any estimate of ${\Phi_n^\mu }$ is performed.
We start the discussion in a general way, by considering any possible source of modulated signal, including dark matter, as has been done in Ref. [@Kudryavtsev201091]. The measured rate of events at [DAMA/LIBRA ]{}is clearly dominated by radioactive background above 6 keV, which imposes a strict limit on any interpretation of the modulated signal. This radioactive background is almost flat at low energies [@Kudryavtsev201091], with the exception of a peak from $^{40}$K at about 3 keV, which agrees with the DAMA measurements. To preserve the shape (‘flatness’) of the radioactive background in the region 2-6 keV, the total signal should be small and hence, the modulated fraction of the signal should be large. As an example, the measured modulated signal rate of 0.019 counts / day / kg / keV at 2-3 keV, assumed to be 5% of the total (average) signal, will give the total signal rate of 0.38 counts / day / kg / keV. This is already a significant fraction of the total measured rate at 2-3 keV (about 30%), requiring the radioactive background rate to drop by 30% at this energy whilst maintaining a flat background above 6 keV. No model of radioactivity predicts a dip in the background below 6 keV [@Kudryavtsev201091].
Let us now consider muon-induced backgrounds within this context. We assume that ${\Phi_n^\mu }$ and ${\Phi^{\mu} }$ are modulated in a similar way, linked to the mean muon energy at [LNGS ]{}[@Kudryavtsev2003688]. The LVD [@selvi2009] and Borexino [@Bellini:2012te] experiments have observed a muon flux modulation in the range of 1.3-1.5% of the total ${\Phi^{\mu} }$. If the modulated signal in DAMA is due to a muon-induced effect, then the total rate of this ‘effect’ will be $0.0112 / 0.014 \approx 0.8$ counts / day / kg / keV. This is approximately equal to the total rate of $\sim 1$ counts / day / kg / keV observed by DAMA in the 2-6 keV energy range [@Bernabei:2013xsa]. The effect is more dramatic in the 2-3 keV energy range, where the modulated signal is approximately 0.0190 counts / day / kg / keV [@Bernabei:2013xsa]. This would imply a total muon-induced background of $0.0190 / 0.014 \approx 1.4$ counts / day / kg / keV, which is higher that the total rate of events observed by DAMA. This is excluded by radioactivity models [@Kudryavtsev201091]. It is clear from the latter discussion that for any explanation of the DAMA signal to be consistent with the measured spectrum of events, it must satisfy the following qualitative criteria:
- The amplitude of the effect must be very small compared to the DAMA event rate.
- The modulation amplitude of the effect must not be much smaller than the average amplitude of the effect.
- Any effect not satisfying the latter two criteria implies that there is a new model of suppressed radioactivity in the region 2-6 keV, that does not apply above 6 keV.
- The modulation of the effect must only affect [[*single-hit*]{} ]{}events, whilst disregarding [[*multiple-hit*]{} ]{}events.
- The explanation must simultaneously predict the phase and the period of the modulation.
An explanation which incorporates muon-induced backgrounds cannot satisfy these criteria.
Conclusions
===========
We have presented an accurate simulation of the muon-induced background in the [DAMA/LIBRA ]{}experiment, in response to proposals to explain the observed DAMA signal modulation with muon-induced neutrons. We have performed a full simulation of the [DAMA/LIBRA ]{}apparatus, shielding and detector housing using [Geant4.9.6]{}.
We have calculated the muon-induced neutron flux in [LNGS ]{}to be ${\Phi_n^\mu }= 1.0\times10^{-9}$ cm$^{-2}$ s$^{-1}$ (without back-scattering), which is consistent with previous simulations. After selecting events which satisfy the DAMA signal region criteria, our simulation predicts a background rate of $3.49\times10^{-5}$ counts / day / kg / keV. This accounts for approximately $0.3\%$ of the modulation amplitude. We find that one would expect a non-negligible modulation of muon-induced background above 6 keV, as well as for events with multiple hits, which is not seen by DAMA.
We conclude from our study that muon-induced neutrons are unable explain the DAMA data. Furthermore, a large signal event rate, independently of the source of this signal, is inconsistent with radioactive background models.
[^1]: We define ‘primary’ particles as those which are present at the surface at the Earth, and ‘secondary’ particles as those which are subsequently produced.
[^2]: We define the back-scattered neutron flux as including independent counts from neutrons that re-enter the cavern due to scattering in the surrounding rock.
|
---
abstract: 'Nowadays, deep learning technology is growing faster and shows dramatic performance in computer vision. However, it turns out that the deep learning model is highly vulnerable to small perturbation called an *adversarial attack*. So far, although many of the *defense mechanism* has been proposed to mitigate the effect of the adversarial attack, all of them are under rigorous assumptions. However, our approach is not tied up any assumptions since our insight stems from the *Tensor Decomposition*. In this paper, we experimentally demonstrated that decomposing the tensor would be an effective countermeasure against several adversarial attacks. We conducted experiments with well-known benchmarks such as MNIST, CIFAR-10, and ImageNet dataset. Our experimental results show that this simple method has capable of having attack resilience and robustness against adversarial attacks. To the best of our knowledge, this is the first approach to leverage the tensor decomposition as a defense mechanism. We hope that leveraging the tensor decomposition becomes a universal approach to solve inherent corner cases of deep learning models.'
author:
- 'Seungju Cho, Tae Joon Jun, Mingu Kang, Daeyoung Kim'
bibliography:
- 'egbib.bib'
title: Applying Tensor Decomposition for the Robustness against Adversarial Attack
---
Introduction
============
Over the past several years, advances in deep neural networks (DNNs) have widely expanded the ability of what the machine can deal with. Especially, DNNs have achieved remarkable successes for image classification [@krizhevsky2012imagenet; @sermanet2013overfeat] and it even goes beyond human capability [@he2016deep]. With this performance, deep learning technology has started to be applied to various areas. However, some papers [@szegedy2013intriguing; @goodfellow2014explaining; @carlini2017towards; @kurakin2016adversarial; @kurakin2016adversarial_scale; @moosavi2016deepfool; @Eykholt_2018_CVPR; @Chen2017EADEA] proved that even DNNs can be easily fooled by small changes to input that is imperceptible to a human eye. According to these studies, carefully crafted perturbations to the vision-based applications can induce systems to behave in unexpected ways. Indeed, this is small enough to be inconspicuous, but some researches show that its influence might be more than expected since even state-of-the-art models get an almost zero-classification accuracy under [@carlini2017towards].
Considering the deep learning models do not hesitate whenever judge the output, it might cause crucial accidents. For instance, Fig. \[fig:AdvExample\] represents the adversarial examples in the image classification task. Although all of the images can be seen as an ostrich by human visible intuition, deep learning model outputs clearly different labels due to lack of such intuition. From a more theoretical perspective, misclassification occurs when the adversarial perturbations cross the decision boundary, but the existing classifier has no such intuition that can ward it off. Now, the corner case of DNNs which have been alluded to adversarial attacks is getting pervasive and being more sophisticated. As a result, the vulnerability of adversarial attacks hinders its adoption for some safety-critical system and also security-sensitive application, including an autonomous-driving car [@Eykholt_2018_CVPR; @sitawarin2018darts].
Since the advent of such adversarial attacks, many researchers or vendors have paid significant attention to adversarial examples. This is because they might not want to go through all the risks of their applications or models. They might as well choose to verify the robustness rather than take risks. However, the resistance against adversarial examples renders another challenge, for no method can be a cure-all against adversarial attacks. To make up for corner cases of DNNs, several studies [@goodfellow2014explaining; @szegedy2013intriguing; @meng2017magnet; @papernot2016distillation; @song2017pixeldefend; @jia2019comdefend; @Liao_2018_CVPR; @xie2019feature] have proposed the defense mechanism against adversarial attacks to mitigate the potential of the risk by adversary. These defense mechanisms can be viewed as two main approaches: (1) changing the model itself, which can improve the robustness by training with adversarial examples, e.g., *adversarial training* [@goodfellow2014explaining; @szegedy2013intriguing; @tramer2017ensemble], (2) preprocessing the inputs to diminish the effect of adversarial noise, e.g., *Magnet, Comdefend, PixelDefend, Defense-GAN, HGD, etc.* [@meng2017magnet; @jia2019comdefend; @song2017pixeldefend; @samangouei2018defense; @Liao_2018_CVPR; @xie2019feature]. However, (1) are designed to deal with specific adversarial attack strategies in mind, so generalization is likely to be restricted. It implies that the models using this method may be vulnerable to another attack optimized with such attack strategies. On the other hand, (2) utilize models with a vast amount of legitimate data to purify the inputs itself, instead of assuming some attack strategies. The main point of (2) is to measure the distance between the inputs and manifold of the legitimate images, and then approximate or guide the adversarial images closer to the manifold of the legitimate images. To purify the inputs, therefore, a well-generalized model should be required to assure the performance. Given that the adversarial images occupy the low probability region trained with legitimate samples [@song2017pixeldefend], poorly generalized models cannot ensure that the aforementioned approaches get a good result. Also, another attack technique might be created by an adversary who knows the model’s structure. In a nutshell, as these approaches are likely to be a temporary expedient, the universal defense approaches which can cover a myriad of risk should be explored. Here, we propose a novel intuition for deep learning models, which can make the model universally robust. To the best of our knowledge, this is the first approach to explore the defense in terms of the universal point of view. Our approach leverages the potential power of the tensor decomposition to diminish the effect of adversarial noise by using the reconstructed images as an input of a deep learning model. The reconstructed inputs are fed into the classifier, and we experimentally demonstrate that such simple preprocessing could be an effective countermeasure against the adversarial attack. On MNIST, a degradation of top-1 accuracy on adversarial example is less than 1% against four adversarial attacks and less than 10 % on CIFAR-10 and ImageNet. This result outperforms recent defense mechanisms [@Liao_2018_CVPR; @jia2019comdefend]. To ensure that deep learning applications extend their potential of utilization toward other domains, it would be better to take into account the robustness of those applications. If you want to avoid cherry-picking doubt and make the model more general across a variety of risks including adversarial attacks, our insight would be an interesting candidate. Our contribution is as follows:
1. **High Compatibility.** Our approach leverages the tensor decomposition for preprocessing the inputs which might have been affected by the adversary. We do not assume anything such as attack strategies or classifiers, just use an input as a reconstructed input by using the tensor decomposition method, which indicates that our proposed method can be relatively easy to utilize and be applied to whatever the classifier is.
2. **Efficient Engineering Complexity.** As we mentioned above, tensor decomposition just depends on what the input it is, so it is not tied up with the attack strategies or classifiers. Therefore, we do not need to focus on how the model classifies the input since tensor decomposition is free from the model dependency. It implies that retraining the model or augmenting the training data could no longer be required. It requires only processing time to reconstruct inputs. Even more, the processing time is negligible.
3. **Integrity of the inputs.** When it comes to the reconstruction process, some information that in charge of the important role might be lost. Although state-of-the-art approaches [@Liao_2018_CVPR; @jia2019comdefend] have gotten remarkable performance, their proposed model degrades the performance with even the clean images. This is some kind of a trade-off. It thus makes it difficult to apply defense mechanisms. However, tensor decomposition could incur less adverse effects on the clean images, and ensure its performance even at the high-dimension dataset such as ImageNet.
Related work
============
#### **Adversarial Attacks.**
Szegedy *et al.* found the existence of adversarial perturbation that breaks the image classifier thorough solving adversarial optimization problem [@szegedy2013intriguing; @goodfellow2014explaining]. They show the model accuracy is dropped even though the perturbed image looks similar to human eyes. Goodfellow *et al.* [@goodfellow2014explaining] uses the sign of the gradient of input with respect to the loss function of the target model. This method is called Fast Gradient Sign Method (FGSM) since it updates input once. With a similar idea, [@kurakin2016adversarial] uses FGSM in an iterative way. Chen *et al.* [@Chen2017EADEA] leverages $L_{1}$ distortion to generate effective adversarial examples and improve the attack transferability which refers to the attack success rate using the adversarial examples which come from the substitute models. In other words, high transferability implies that the performance of the target model might depreciate even without the knowledge about the model, i.e., *black box attack*. Carlini and Wagner [@carlini2017towards] changes the optimization problem defined in [@szegedy2013intriguing] for achieving more powerful attack. [@moosavi2016deepfool] measures the minimum size required for the attack. They approximate the decision boundary of the model and update input repeatably until the model misclassifies it. Besides the image classification task, [@Eykholt_2018_CVPR; @sitawarin2018darts] demonstrated that adversarial attacks can be applied beyond the digital space, so security concerns could arise in even physical space such as the autonomous-driving car.
#### **Adversarial Defense.**
To counter adversarial attacks, some works trained the model with adversarial examples to ensure that the model has a resilience against those adversarial examples, which have been called *adversarial training*. During the process of training, they generate adversarial images for improving the performance. Although it works, it depends on the particular adversarial data used in the training process. For instance, [@kurakin2016adversarial_scale] shows their approach is robust in the simple attack, but not in a more sophisticated attack. In addition, it has an engineering penalty since it requires retraining the model. If it takes longer to create an adversarial example, it will take more time to retrain the model. Instead of using the data augmentation, methods to change the model itself were also proposed [@papernot2016distillation]. They change the objective function of the problem for obtaining the robustness. However, this approach also has to retrain the model, so it also boils down to increasing the engineering complexity.
In recent years, several papers [@meng2017magnet; @jia2019comdefend; @song2017pixeldefend; @Liao_2018_CVPR; @xie2019feature; @samangouei2018defense] preprocess the inputs before putting into the classifier. They propose the model which serves the direction to approximate the distribution of the adversarial images as close as possible to the decision boundary. All of the methods require a well-generalized defense model, so the even clean images could be affected when the model is poorly generalized. It results in damage to the integrity of the inputs. To guarantee the integrity of the model, all of them require well-generalized classifiers to detect if the input is adversarial or approximate the manifold of legitimate samples. Our approach is similar to those approaches in terms of the preprocessing, yet differentiation is our method does not need any premises, including the detector or well-generalized models. Consequently, our method does not hurt the performance in terms of integrity.
Background
==========
Adversarial Attack
------------------
Basically, all of the attacks use the gradient of data with respect to the loss function of the target model. In this section, we briefly review the basic method of adversarial attack.
**Fast Gradient Sign Method (FGSM):** The *FGSM* is proposed by [@goodfellow2014explaining]. It is a simple and effective attack method. The image $X$ is perturbed as follows. $$X = X + \epsilon \cdot | sign(\triangledown_{X} l(X,y_{true})) |$$ Where $\epsilon$ is a magnitude of noise and $l(X,y_{true})$ is a loss with respect to the true label of the image. It adjusts input X by adding a sign of the gradient of X. It increases the loss function of the target model so that the model misjudges the adjusted input. Since it updates input X once, it is also called *single-step* method.
**Basic Iterative Method (BIM):** The *BIM* is a repetitive version of FGSM [@kurakin2016adversarial]. it is a more powerful attack method compared to the FGSM. And it is also called *Iterative FGSM*. It uses the following equations: $$X_{t+1} = \text{clip}_{X,\epsilon}(X_{t} + \alpha \cdot | sign(\triangledown_{X} l(X_{t},y_{true})) |)$$ Where $X_{0} = X$, $\alpha$ is a step size for adjusting $X_t$ , and clip function ensures that $X_t \in (X-\epsilon,X+\epsilon )$ for all $t$. It is also called *multi-step* method. Here $\alpha$ is the $1$ in the scale of 0 to 255 in the original paper.
**DeepFool:** DeepFool attack approximates the decision boundary of a classifier, and measure the minimal perturbations that are sufficient to fool the classifier [@moosavi2016deepfool]. For the affine multiclass classifier, they calculate the distance $d$ as follows. $$d = \frac{|f_k(x) - f_l(x)|}{w'} ~ \text{where}~ w' = \|\nabla f_k(x) - \nabla f_l(x) \|_2$$ Here $f_k$ and $f_l$ are classifier for $k$ and $l$-th class. Similar to BIM, they update $x$ in an iterative way. For nonlinear classifiers, they approximate the linear boundary and find the distance to fool the nonlinear classifier.
**Carlini & Wagner (C&W):** Carlini and Wagner [@carlini2017towards] define an optimizaion problem to find an adversarial example. They define following optimizaion formulation. $$\text{min}_{\delta} ~ D(x,x+\delta) + c \cdot f(x+\delta)$$
Here $D$ is a distance metric to measure the distance between the clean image and adversarial image. $f(\cdot)$ is an objective function to control the result of original classifier $C$. C&W attack is one of the most powerful attacks in this literature. We visualize the adversarial image generated by each method in Fig. \[fig:advs\]
[0.19]{} ![Visualization of adversarial examples on each method. The pre-trained Resnet101 [@he2016deep] model classifies clean image as a *monarch*, image with FGSM as a *sea slug*, image with BIM as a *Doberman*, image with DeepFool as a *hornbill* and image with C&W as a *longicorn*. However, we can check that theses images seem to the same in the human eye[]{data-label="fig:advs"}](Images/clean.png "fig:"){width="\textwidth"}
[0.19]{} ![Visualization of adversarial examples on each method. The pre-trained Resnet101 [@he2016deep] model classifies clean image as a *monarch*, image with FGSM as a *sea slug*, image with BIM as a *Doberman*, image with DeepFool as a *hornbill* and image with C&W as a *longicorn*. However, we can check that theses images seem to the same in the human eye[]{data-label="fig:advs"}](Images/FGSM.png "fig:"){width="\textwidth"}
[0.19]{} ![Visualization of adversarial examples on each method. The pre-trained Resnet101 [@he2016deep] model classifies clean image as a *monarch*, image with FGSM as a *sea slug*, image with BIM as a *Doberman*, image with DeepFool as a *hornbill* and image with C&W as a *longicorn*. However, we can check that theses images seem to the same in the human eye[]{data-label="fig:advs"}](Images/bim.png "fig:"){width="\textwidth"}
[0.19]{} ![Visualization of adversarial examples on each method. The pre-trained Resnet101 [@he2016deep] model classifies clean image as a *monarch*, image with FGSM as a *sea slug*, image with BIM as a *Doberman*, image with DeepFool as a *hornbill* and image with C&W as a *longicorn*. However, we can check that theses images seem to the same in the human eye[]{data-label="fig:advs"}](Images/deepfool.png "fig:"){width="\textwidth"}
[0.19]{} ![Visualization of adversarial examples on each method. The pre-trained Resnet101 [@he2016deep] model classifies clean image as a *monarch*, image with FGSM as a *sea slug*, image with BIM as a *Doberman*, image with DeepFool as a *hornbill* and image with C&W as a *longicorn*. However, we can check that theses images seem to the same in the human eye[]{data-label="fig:advs"}](Images/carlini.png "fig:"){width="\textwidth"}
Tensor Decomposition
--------------------
A *tensor* is a multi-dimensional array. For instance, the color image is a tensor consists of height, width, and the color channel. A tensor decomposition method decomposes a tensor into low dimension tensors. The *CANDECOMP/PARAFC* [@carroll1970analysis; @harshman1970foundations] decomposition approximates a tensor $X$ as a sum of the outer product of the tensor belonging to each dimension as Fig. \[fig:cp\]. We refer to this as a *CP* decomposition.
Let $X \in \mathbb{R}^{I \times J \times K}$, $a_i \in \mathbb{R}^{I}, b_i \in \mathbb{R}^{J} \ \text{and}\ c_i \in \mathbb{R}^{K}$ for $i = 1,\dots,r$. Here $r$ is the number of components and it is a hyperparameter. If $r$ is small, tensor$X$ is approximated into low dimension tensor. So we call deciding $r$ as choosing the dimension for convenience. Then $x_{ijk}$ is approximated as follows. $$x_{ijk} \approx \sum_{l = 1}^{r} a_{il}b_{jl}c_{kl}$$
The *Tucker* decomposition [@tucker1963implications; @tucker1966some] is another way to decompose a tensor. It is a kind of higher order principal component analysis [@kolda2009tensor]. It decomposes tensor as a core tensor and factor tensors as Fig. \[fig:tucker\].
Let $X \in \mathbb{R}^{I \times J \times K}$, $A \in \mathbb{R}^{I \times P}, B \in \mathbb{R}^{J \times Q} , C \in \mathbb{R}^{I \times R} \text{and} \ G \in \mathbb{R}^{P \times Q \times R} $ for $i = 1,\dots,I, j = 1,\dots,J,k = 1,\dots,K$. Here the size of the core tensor $P,Q$ and $R$ are the number of the components and it is hyperparameter. When the size of the core tensor is fixed, the size of the factor tensor is decided according to the size of the core tensor. Similar to the number of the components of *CP* decomposition, we call deciding the size of the core tensor as a choosing the dimension. Then $x_{ijk}$ is approximated as follows.
$$x_{ijk} \approx \sum_{p = 1}^{P}\sum_{q = 1}^{Q}\sum_{r = 1}^{R} g_{pqr}a_{ip}b_{jq}c_{kr}$$
Method
======
To mitigate the effect of the adversarial attacks, our insight stems from the tensor decomposition. In this section, all the paragraphs that describe revolve around how we can apply this magic, i.e., tensor decomposition, as a defense mechanism against the adversarial attacks.
Tensor decomposition as a preprocessing
---------------------------------------
As you can see in Fig. \[fig:advs\], adversarial examples are too sophisticated to be recognized by human senses, including well-generalized deep learning models. To prevent the potential threat, our model simply uses the reconstructed image from the tensor decomposition as an input. We conjecture that the effect of the adversarial perturbations could be reduced by approximating the tensors toward the low dimension. To cast light on our hypothesis, we conduct brief experiments based on the visual sense. Fig. \[fig:noise\] represents the examples of the noise. Intuitively, adversarial noise crafted by the FGSM seems to distinguishable from others, while the rest of them are relatively similar to each other. In other words, the tensor decomposition can transform adversarial noise into random noise, e.g., gaussian noise. We can say that the tensor decomposition has an ability to purify the adversarial noise in this regard. Even though such random noise might also degrade the performance, it would be no worse than original adversarial noise considering they are crafted by adversarial intend.
[0.24]{} ![Visualization of example noises. (a) is the adversarial noise crafted by FGSM. (b) is the gaussian noise, (c),(d) are reconstructed images by using *Tucker*, *CP* decomposition respectively[]{data-label="fig:noise"}](Images/adv_noise.png "fig:"){width="\textwidth"}
[0.24]{} ![Visualization of example noises. (a) is the adversarial noise crafted by FGSM. (b) is the gaussian noise, (c),(d) are reconstructed images by using *Tucker*, *CP* decomposition respectively[]{data-label="fig:noise"}](Images/gaussian.png "fig:"){width="\textwidth"}
[0.24]{} ![Visualization of example noises. (a) is the adversarial noise crafted by FGSM. (b) is the gaussian noise, (c),(d) are reconstructed images by using *Tucker*, *CP* decomposition respectively[]{data-label="fig:noise"}](Images/cp_noise.png "fig:"){width="\textwidth"}
[0.24]{} ![Visualization of example noises. (a) is the adversarial noise crafted by FGSM. (b) is the gaussian noise, (c),(d) are reconstructed images by using *Tucker*, *CP* decomposition respectively[]{data-label="fig:noise"}](Images/tucker.png "fig:"){width="\textwidth"}
![We show a change of reconstructed images using *CP* and *Tucker* decomposition by varying the value of the dimension (The value is getting higher order by left $\rightarrow$ right)[]{data-label="fig:gradation"}](Images/cp_gradation.png){width="\textwidth"}
![We show a change of reconstructed images using *CP* and *Tucker* decomposition by varying the value of the dimension (The value is getting higher order by left $\rightarrow$ right)[]{data-label="fig:gradation"}](Images/tucker_gradation.png){width="\textwidth"}
Based on this light, we utilize *CP* and *Tucker* decomposition methods to verify how effective these methods actually are under various attack strategies. Before the main experiments, both methods require to set the dimension, e.g., $r$ for the *CP* and $P,Q$, $R$ for the *Tucker*. As the dimension of the tensor increases, the quality of the reconstructed image gets better as shown in Fig. \[fig:gradation\]. The high quality of the reconstructed image is not always better, however, so the dimension needs to be decided in a heuristic manner. We thus studied the ablation study to find out the effective hyperparameters under *CP* and *Tucker* and consider two kinds of factors to decide the hyperparameters, accuracy and time complexity. We randomly sampled 1,000 images from CIFAR-10, and then generate the adversarial images by applying FGSM, BIM, DeepFool, and C&W, respectively. As follow, those images are reconstructed by *CP* and *Tucker* decomposition. Finally, we measured the accuracy and time complexity using the reconstructed images.
[0.48]{} ![ (a) Accuracy with respect to dimension (b) Mean time for processing single image with respect to dimension[]{data-label="fig:graph"}](Images/accuracy.png "fig:"){width="\textwidth"}
[0.48]{} ![ (a) Accuracy with respect to dimension (b) Mean time for processing single image with respect to dimension[]{data-label="fig:graph"}](Images/time.png "fig:"){width="\textwidth"}
The result summarized in Fig. \[fig:graph\]. While accuracy increased as dimensions increased by the middle, accuracy tends to decrease gradually. And the processing time increases as dimension increases. So we decide to use 40 % of the original dimension. For instance, the size of the image is 32 by 32 in the CIFAR-10 dataset. In the case of *CP* decomposition, the size of the three tensors in Fig. \[fig:cp\] will be 32,32 and 3. So we choose rank $r = 8$. We use a similar argument when choosing the size of the core tensor of *Tucker* decomposition. In particular, we don’t compress the channel dimension $3$ in *Tucker* decomposition since we don’t want to lose color information. Thus, the size of the core tensor of the *Tucker* decomposition is in the form of height, width and $3$. We use an open-source library [@kossaifi2019tensorly] for each decomposition method.
Denoise Autoencoder as a supplement
-----------------------------------
We should consider one more thing before putting the input value into the classifier. When it comes to reconstructing the images, we should consider the loss of information as it might affect the classification result. If the input is clean images, it would work even worse. To diminish the adverse effect from that point, we add denoise autoencoder into the procedure. Our method is based on a coarse-to-fine approach. Through the reconstructed inputs from decomposed tensors, we remove the coarse-grained adversarial features. We expect that some fine-grained features that might be lost by the coarse-grained approach—which is more likely to occur in high-dimension, could be compensated pass through the denoise autoencoder. Equipped with this approach, we set up the denoise autoencoder architecture as follows. The numerical value in Table \[table:autoencoder\] stands for input channel and output channel respectively. And the filter size is $3 \times 3$.
---------- ---- ---- ---------- ---- ---- -- -- -- -- -- -- --
MNIST CIFAR
Conv,ELU 1 2 Conv,ELU 3 6
Conv,ELU 2 4 Conv,ELU 6 12
Conv,ELU 4 8 Conv,ELU 12 24
Conv,ELU 8 16 Conv,ELU 24 48
Conv,ELU 16 32 Conv,ELU 48 96
Conv,ELU 32 16 Conv,ELU 96 48
Conv,ELU 16 8 Conv,ELU 48 24
Conv,ELU 8 4 Conv,ELU 24 12
Conv,ELU 4 2 Conv,ELU 12 6
Conv,ELU 2 1 Conv,ELU 6 3
---------- ---- ---- ---------- ---- ---- -- -- -- -- -- -- --
: Architecture of autoencoder trained with each dataset[]{data-label="table:autoencoder"}
We utilize the CIFAR-10 images at the RGB scale and MNIST image at the grayscale. We set the learning rate to $1e-4$ and used Adam [@kingma2014adam] as an optimizer. And we use mean square error (MSE) for loss function. For both models, we train autoencoder for 10 epochs. We henceforth denote autoencoder as *AE*.
Overall architecture
--------------------
We describe the overall architecture in detail. Fig. \[fig:architecture\] represents the overall flow of our proposed method. First, we approximate the input image via the tensor decomposition method. The inputs could be adversarial images or clean images. Our method does not spend time deciding whether the input is adversarial or not. That’s the reason why our model does not require a well-generalized model. In other words, whatever the input is, our model splits the input into several tensors based on *CP* or *Tucker* decomposition, and then reconstruct them. As follows, the reconstructed images are passed through the denoise autoencoder, which might compensate for losing the information that may in charge of an important role in that image. Note that our method does not have a model dependency, so it can be applied in conjunction with every classifier.
Experiment
==========
Dataset
-------
For the MNIST, CIFAR-10 data, we test on the full test data, which are composed of 10,000 images on MNIST and 50,000 images on CIFAR-10. For ImageNet, we select randomly 1,000 images as similar setting [@jia2019comdefend; @kurakin2018adversarial]. Since our proposed method decomposes the input whichever clean image or adversarial image, we tested on both clean images and adversarial images.
Evaluation
----------
We measure the top-1 accuracy on clean images and adversarial images on each dataset. For MNIST, we use a simple model consists of two convolutional layers. For CIFAR-10 and ImageNet, we basically use pre-trained Resnet101 [@he2016deep]. In particular, for CIFAR-10, we finetune pre-trained Resnet101 for 10 classes. We use FGSM, BIM, DeepFool and C&W attack methods. For the distance metric, a related research area mainly uses $L_{\infty}$ and $L_2$ norm [@carrara2018adversarial; @jia2019comdefend]. In detail, we use $L_{\infty}$ for FGSM, BIM, and DeepFool attack. And for the C&W attack, we use a $L_2$ metric. We try to find small perturbation when applying the adversarial attack since the noise is visible when the perturbation is not small enough. We generate adversarial images by using open source library Foolbox [@rauber2017foolbox]. In detail, we try 100 epsilons from 0 to 1 for FGSM. For BIM, we set 5 as a number of iteration. And for DeepFool, we set 50 as a maximum number of steps and set 50 as a maximum iteration number of C&W. And we measure the pre-processing time for calculating the additional time consuming for the proposed method. Also, we compare our results to other state-of-the-art defense models. For a fair comparison, we compare the ratio between the accuracy of the clean image and the adversarial image since the accuracy of a clean image is a little bit different depending on the setting.
Results
-------
We achieve remarkably high accuracy against adversarial attacks. In most cases, the *CP* is better than *Tucker* decomposition method. In some case of ImageNet dataset, *Tucker* decomposition method is better than *CP*. For instance, when attack with FGSM and C&W method, the result was the best by using *Tucker* decomposition. And in the case of clean images, the accuracy reduction was about 1% on all datasets. It means that we do not harm the original model in a normal case which is the input image is clean. The autoencoder is highly effective on MNIST dataset. Although the autoencoder does not have much effect on clean images, it improves the performance of various adversarial attacks on MNIST dataset. In addition to MNIST dataset, there have been small performance improvements for other datasets by using the denoise autoencoder. The numerical results are summarized in Table \[table:MNIST\], \[table:CIFAR\] and \[table:Imagenet\].
------- ----------- ------- ------- ------- ---------- -------
Model Method Clean FGSM BIM DeepFool C&W
Original 99.06 0.00 0.21 0.00 0.00
CP 99.01 71.62 95.42 81.68 88.29
CP+AE 98.64 96.15 98.88 98.83 98.6
Tucker 99.06 71.95 91.81 79.32 84.52
Tucker+AE 98.06 95.26 98.65 98.54 98.05
------- ----------- ------- ------- ------- ---------- -------
: Top-1 accuracy of each method on MNIST dataset[]{data-label="table:MNIST"}
------- ----------- ------- ------- ------- ---------- -------
Model Method Clean FGSM BIM DeepFool C&W
Original 98.87 0.0 0.0 0.0 0.0
CP 97.96 92.52 92.49 94.67 93.63
CP+AE 98.11 92.88 93.21 94.93 94
Tucker 95.99 87.74 87.68 90.73 90.57
Tucker+AE 96.00 88.44 88.56 91.28 91.2
------- ----------- ------- ------- ------- ---------- -------
: Top-1 accuracy of each method on CIFAR-10 dataset[]{data-label="table:CIFAR"}
------- ----------- ------- ------- ------- ---------- ------
Model Method Clean FGSM BIM DeepFool C&W
Original 77.6 0.0 0.0 0.0 0.0
CP 76.3 61.2 75.4 75.7 61.2
CP+AE 74.8 62.9 75.00 75.5 62.1
Tucker 76.4 65.00 73.4 73.8 65.3
Tucker+AE 74.7 65.6 73.7 74.7 66
------- ----------- ------- ------- ------- ---------- ------
: Top-1 accuracy of each method on ImageNet dataset[]{data-label="table:Imagenet"}
Even the DeepFool and C&W attacks are more accurate and powerful attack compared to the FGSM and BIM attacks, the accuracy after decomposition is higher than the case of FGSM and BIM attacks.
Comparison with other defense methods
-------------------------------------
We measure the ratio of accuracy on clean images and adversarial images generated by FGSM, BIM, DeepFool and C&W attack for a fair comparison. Here the $L_{\infty}$ is restricted to $8$ in $255$ scale. The defense ratio is defined as follows. $$\textit{Defense ratio} = \frac{\textit{Top-1 accuracy on adversarial images}}{\textit{Top-1 accuracy on clean images}}$$ We compare the performance of recent defense methods, HGD [@Liao_2018_CVPR] and Comdefend [@jia2019comdefend]. Fig. \[fig:bar\] shows the results. We select Resnet101 [@he2016deep] and Inception V3 (IncV3) [@szegedy2016rethinking] as base model. And we tested on 1,000 images from the ImageNet data. Our methods outperform in all cases compared to two recent defense methods. This result verifies that our method is effective. Moreover, our method does not depend on attack methods and the target model classifier, thus it can be easily combined with every model.
[0.48]{} ![ (a) Defense ratio of each attack method on Resnet101 (b) Defense ratio of each attack method on IncV3. Note that our method outperforms other defense methods[]{data-label="fig:bar"}](Images/bar_1.png "fig:"){width="\textwidth"}
[0.48]{} ![ (a) Defense ratio of each attack method on Resnet101 (b) Defense ratio of each attack method on IncV3. Note that our method outperforms other defense methods[]{data-label="fig:bar"}](Images/bar_2.png "fig:"){width="\textwidth"}
Time analysis
-------------
We measure the preprocessing time of each method. We pick randomly 1,000 images in MNIST,CIFAR-10, and ImageNet. And we calculate the average processing time per image. In most case, the *CP* decomposition requires more time compared to *Tucker* decomposition. In the case of MNIST and CIFAR-10, the time required to reconstruction is similar in both cases. However, In the case of ImageNet, the *CP* decomposition takes about 10 times more than the *Tucker* method. Table \[table:time\] summarizes preprocessing time of each dataset on each method.
[\*5c]{} dataset&CP&CP+AE&Tucker&Tucker + AE\
MNIST & 0.005 & 0.006 & 0.003 & 0.004\
CIFAR-10 & 0.1052 & 0.1161 & 0.01268 & 0.015\
ImageNet & 1.07 & 1.18 & 0.1566 & 0.17\
White box scenario
------------------
In the white box scenario, we should assume the adversary knows full defense mechanism according to [@carlini2017adversarial]. In our method, note that the input image is always decomposed and reconstructed, and the decomposed components are always started from the random tensor. In detail, the component tensors of each decomposition method initialized to random tensor and then trained to approximate the original tensor. Thus, the input is always random tensor and the original image is a label itself like unsupervised learning. Therefore, there are no fixed weights, so the adversary can not generate adversarial examples concerning the tensor decomposition method. Hence, our propose method is robust on the white box attack scenario.
Conclusions
===========
In this work, we verify the tensor decomposition is a simple and powerful method for purifying the adversarial perturbation. When we combine denoise autoencoder with the tensor decomposition method, the proposed method achieves higher accuracy against adversarial attacks. We experiment with our method against various adversarial attacks such as DeepFool and C&W attacks and discuss why this method is robust in the white box scenario.
Our intuition applying tensor decomposition into the adversarial attack is as follows. Since the adversarial perturbation is so small that it is hard to catch a difference, such a small perturbation would be removed by approximating the image tensor using low dimensional tensors. Since there is no straightforward algorithm to choose the dimension of the component tensors of the *CP* and *Tucker* decomposition, finding the best dimension remains for future work. Also, establishing a theoretical base why tensor decomposition is robust against adversarial attack is left to our future work.
|
---
abstract: 'This paper analyzes the Krylov convergence rate of a Helmholtz problem preconditioned with Multigrid. The multigrid [method]{} is applied to the Helmholtz problem formulated on a complex contour and uses [GMRES as a smoother substitute]{} at each level. A one-dimensional model is analyzed both in a continuous and discrete way. It is shown that the Krylov convergence rate of the continuous problem is independent of the wave number. The discrete problem, however, can deviate significantly from this bound due to a pitchfork in the spectrum. It is further shown in numerical experiments that the convergence rate of the Krylov method approaches the continuous bound as the grid distance $h$ gets small.'
author:
- 'B. Reps and W. Vanroose[^1]'
bibliography:
- 'bib\_revision\_arxiv.bib'
date: October 2011
title: Analyzing the wave number dependency of the convergence rate of a multigrid preconditioned Krylov method for the Helmholtz equation with an absorbing layer
---
[*Dept. Mathematics and Computer Science, Universiteit Antwerpen, Middelheimlaan 1, 2020 Antwerpen, Belgium*]{}
Introduction
============
The Helmholtz equation plays a central role in seismic imaging, electromagnetic scattering and many other applications. For $x \in
\Omega \subset \mathbb{R}^d$ the equation reads $$Hu\left({\mathbf}{x}\right) \equiv \left[-\triangle - k\left({\mathbf}{x}\right)^2\right] u\left({\mathbf}{x}\right) = f({\mathbf}{x}) \,,$$ where the wave number $k({\mathbf}{x})$ depends on the coordinates ${\mathbf}{x}$ and can model, for example, the change in refractive index of the material through which electromagnetic waves are propagating, $f({\mathbf}{x})$ models the source of the waves and $-\triangle$ is the $d$-dimensional negative Laplacian.
[In theory many applications need to be solved on an infinite domain, yet in practice a numerical solution method must truncate the domain in some way. Therefore, on the finite computational domain the equation is solved with outgoing wave conditions on the artificially introduced boundaries]{}. Over the years many good outgoing wave boundary conditions have been proposed such as *Perfectly Matched Layers* (PML) [@B94; @CW94]. [This leads to a spectrum of the operator where each eigenvalue has an imaginary part to represent the damping of the outgoing waves on the exterior layers]{}. In a similar way in the physics and chemistry literature absorbing boundary conditions based on *Exterior Complex Scaling* (ECS) [@AC71; @BC71; @S79] are used for example in break-up problems [@mccurdy2004solving].
After discretization the Helmholtz problem becomes a linear system $H_hu_h=f_h$. Due to the [negative shift, with a magnitude that depends on the wave number $k$,]{} the matrix $A$ is indefinite. Indeed, the wave number shifts the spectrum of $-\triangle$, which is positive definite, to the left. The eigenvalues corresponding to the smooth modes can get close to zero or have a negative real part. These spectral properties lead to a large condition number and iterative methods perform poorly. Efficient preconditioners for the negative Laplacian, such as a multigrid method, fail when applied to the Helmholtz problem. A recent review of the difficulties of iterative methods for the Helmholtz problem is given in [@gander2010].
[An important step in the improvement of the iterative solution of the Helmholtz problem was taken by Bayliss, Goldstein and Turkel [@BGT83; @BGT85] with the shifted Laplacian preconditioner. Instead of approximately inverting the original Helmholtz operator with e.g. ILU or a few multigrid cycles as a preconditioning step, the Laplacian or positively shifted Laplacian is used as a preconditioner. This positive definite operator serves as an approximation of the Helmholtz operator and can be efficiently inverted with standard iterative methods. A significant extension of this idea was made by the introduction of the *complex shifted Laplacian*, a related Helmholtz problem with a complex-valued wave number, which makes a better preconditioner and can still be efficiently solved with multigrid [@EVO04; @EVO06].]{} The complex-valued wave number prevents that eigenvalues of the preconditioner come close to zero at any level of the multigrid hierarchy. This is particularly useful in the coarse grid correction where diverging [numerical]{} resonances can appear when an eigenvalue of a coarse level approaches the origin [@Elman].
In [@JCP-paper] it was shown that scaling the wave number with a complex value has the same effect as scaling the grid distance with a complex value. As a result the Helmholtz problem can be efficiently preconditioned by a Helmholtz operator discretized on a complex-valued grid. This might be of interest for problems where complex-valued grid distances are already used to implement the absorbing boundary conditions. This is the case for ECS [@mccurdy2004solving] or PML [@CW94].
The introduction of complex wave numbers (or grids) avoids the appearance of resonances, however, it does not prevent traditional smoothers like $\omega$-Jacobi or Gauss-Seidel to be unstable for the smooth modes. [In this paper we are interested in developing a matrix-free method, though we mention that also ILU smoothers can be unstable for similar reasons. In [@polynomialsmoother] we analyze GMRES as a replacement of the traditional smoothers when multigrid is applied to a preconditioning operator based on complex-valued grids. Numerical experiments show that only a few GMRES iterations are needed at every level, as opposed to the results in [@Elman] where multigrid was used to invert the original Helmholtz operator which requires more GMRES iterations at some intermediate levels.]{}
Note that for the complex shifted Laplacian preconditioner the complex shift has a parameter. The choice of the parameter is analyzed in [@vanGijzen2007spectral] and [@osei2010preconditioning].
We mention other promising preconditioning techniques such as Moving Perfectly Matched Layers [@mpml], a transformation that turns the Helmholtz problem into a reaction-advection-diffusion problem [@HM11], application of separation-of-variables [@plessix], algebraic multilevel methods [@boll], the wave-ray method [@brandt1997wave; @livshits2006accuracy], and combined complex shifted Laplacian and deflation [@sheikh2009fast].
[In this paper we focus the analysis on the wave number dependency of the convergence behavior of a preconditioned Krylov subspace method.]{} The preconditioning [operator]{} is the Helmholtz operator discretized on a complex-valued grid and it is inverted with multigrid using GMRES as a smoother substitute as suggested in [@Elman] and [@polynomialsmoother]. The paper starts with a review of a one-dimensional continuous model problem in Section \[sec:modelproblem\]. For this problem the eigenvalues of the preconditioned [operator]{} are derived analytically and we find that the Krylov convergence rate should be independent of the wave number in Section \[sec:continuouseigenvaluesprecon\]. The discrete problem, discussed in Section \[sec:discrete\], however, does not have this bound. We explain the origin of this deviation and give estimates for the different regions in the convergence as a function of the wave number $k$. In Section \[sec:numerical\] we illustrate the theory with numerical examples.
Model problem {#sec:modelproblem}
=============
In this section we formulate a one-dimensional Helmholtz model problem that is representative for higher dimensional problems that arise in many applications. It is a Helmholtz problem with a constant wave number $k$ on the domain $\Omega=[0,1]\in\mathbb{R}$, $$\label{eq:helm0}
\begin{cases}
Hu(x)\equiv \left[-\frac{d^2}{dx^2} -k^2\right]u(x) &= f(x), \quad \forall x \in (0,1);\\
u(0) = 0;\\
u(1) = \mbox{outgoing wave,}
\end{cases}$$ with a zero Dirichlet boundary condition on the left boundary $x=0$ and an outgoing wave boundary condition on the right boundary $x=1$. The right hand side $f(x)$ represents a localized source term.
The Helmholtz problem with ECS
------------------------------
The outgoing wave boundary condition in is implemented with exterior complex scaling (ECS) [@mccurdy2004solving], an equivalent formulation of the PML technique by Bérenger [@B94]. Therefore the domain is extended to $\Omega\cup\Gamma=[0,1]\cup(1,R]\in\mathbb{R}$ after which a complex coordinate transformation is defined as, $$\label{eq:domain}
z(x) = \left\{
\begin{array}{ll}
x, & \hbox{$x \in [0,1]$;} \\
1+(x-1)e^{\imath{\theta_{\gamma}}}, & \hbox{$x \in (1,R]$.}
\end{array}
\right.$$ We write $R_z=z(R)\in\mathbb{C}$ for the new complex right boundary. This results in the domain $\Omega\cup\Gamma_z=[0,1]\cup(1,R_z]\in\mathbb{C}$ that is the union of the original real domain $[0,1]$ and a complex line connecting the point $1$ to $R_z$, see Figure \[fig:domain\]. In this paper we use linear complex scaling by simply rotating the absorbing layer over an angle ${\theta_{\gamma}}$ in the complex plane, but smoother transitions, with a non-constant angle, are also possible. Posing a zero Dirichlet boundary condition in $R_z$ implies an outgoing wave in the original right boundary $x=1$ [@CW94].
The Helmholtz problem translates into $$\label{eq:helm}
\begin{cases}
Hu(z)\equiv \left[-\frac{d^2}{dz^2} -k^2\right]u(z) &= f(z), \quad \forall z \in (0,1]\cup(1,R_z];\\
u(0) = u(R_z) = 0,\\
\end{cases}$$ with homogeneous Dirichlet boundary conditions at $z(0)=0$ and $z(R)=R_z$. Note that the source term $f(z)$ was assumed to vanish outside $[0,1]$.
We define the ECS grid on the complex stretched domain , $$\label{eq:ecsgrid}
(z_j)_{0\leq j\leq n+m} =
\begin{cases}
j h, & (0\leq j\leq n);\\
1+(j-n){h_{\gamma}}, & (n+1\leq j\leq n+m),
\end{cases}$$ that consists of $n$ intervals of the grid distance [$h=1/n$]{} followed by $m$ intervals of the complex grid distance [${h_{\gamma}}=(R-1)e^{{\imath}{\theta_{\gamma}}}/m$]{} for the complex contour as illustrated in Figure \[fig:domain\]. We discretize the second derivative operator on the grid with the Shortley-Weller finite difference scheme for non-uniform grids $$\label{eq:shortwell}
\frac{d^2u}{dz^2}(z_j) \approx
\frac{2}{h_{j-1}+h_j}\left(\frac{1}{h_{j-1}}u_{j-1}-\left(\frac{1}{h_{j-1}}+\frac{1}{h_j}\right)u_j
+\frac{1}{h_j}u_{j+1}\right)$$ in grid point $j$, where $h_{j-1}$ and $h_j$ are the left and right grid distance respectively, and may belong either to the $h$ category or to the ${h_{\gamma}}$ category. The result is a linear system of equations $$\label{eq:discretehelm}
H_hu_h \equiv (-L_h-k^2I_h)u_h = f_h,$$ with a unique solution $u_h$ that approximates the continuous solution $u$ of the Helmholtz equation . The higher dimensional Laplacian $\triangle$ is then constructed with Kronecker products of this one-dimensional discrete Laplacian matrix $L_h$.
![The domain of the model problem on the continuous domain (top) and the discrete grid (bottom). \[fig:domain\]](domain "fig:"){width="90.00000%"}\
![The domain of the model problem on the continuous domain (top) and the discrete grid (bottom). \[fig:domain\]](discretedomain "fig:"){width="90.00000%"}
Spectrum of the discretization matrix {#ssec:specdiscretehelm}
-------------------------------------
For the one-dimensional model problem the solution $u_h$ of is easily found with an exact inversion of the tridiagonal matrix $H_h$ in Equation . As the bandwidth of the matrix grows with the dimension of the problem, so does the computational cost of direct methods. Iterative methods need to be used instead, such as multigrid and Krylov subspace solvers. The one-dimensional model has been analyzed in [@JCP-paper] in order to help in configuring these methods efficiently. More specifically their performance depends on the position of the eigenvalues of the matrix $H_h$ in the complex plane. Define $\gamma = \frac{h_\gamma}{h}$, then the eigenvalues of $-L_h$ are the solutions of $$\label{eq:eigcond}
F(t) \equiv \frac{\tan(2n p(t))}{\tan(2m q(t))}+\frac{\cos(p(t))}{\cos(q(t))} = 0,$$ with $p(t)=\frac{1}{2}\arccos(1-\frac{t}{2}h^2)$, $q(t)=\frac{1}{2}\arccos(1-\frac{t}{2}\gamma^2
h^2)$.\
Figure \[fig:pitchfork\] shows that the spectrum ($\bullet$) of $-L_h$ has a typical pitchfork shape. It is bounded in the complex plane by a triangle $\widehat{t_0t_1t_2}$ described by the points $t_0=0$, $t_1=4/h^2$ and $t_2=4/{h_{\gamma}}^2$. Starting in the origin $t_0=0$ we find eigenvalues along the complex line $\rho e^{-2\imath{\theta_{\alpha}}}$ with $\rho>0$, where ${\theta_{\alpha}}$ is the argument of $R_z$. [It was shown in [@JCP-paper] that these]{} eigenvalues approximate the smallest eigenvalues ($\times$) of the continuous Laplacian operator $-\triangle$ that will be derived in Section \[sec:continuouseigenvalues\]. They correspond to the smoothest modes spread over the entire ECS domain [and we will therefore call them the *smooth* eigenvalues]{}. At a certain point $t_b$ the line of smooth eigenvalues splits up into two branches. One pronounced complex branch [consists of eigenvalues with associated eigenvectors that have their largest components at indices $n\leq j \leq n+m$. Since these eigenvectors have nearly-zero components at indices $1\leq j \leq n-1$ that correspond to the interior real region of the grid in , we say that they are mainly located on the complex contour of the domain $\Gamma_z=[1,R_z]$. Whereas the other branch of eigenvalues in the spectrum lies closer to the real axis and corresponds to eigenvectors with their largest components at indices $1\leq j \leq n-1$, in other words, they are located on the real interior domain $\Omega=[0,1]$, see also Figure \[fig:outereigvec\].]{} Together with the line of smooth eigenvalues the latter branch causes potential numerical problems as they lie close to the real axis around the points $t_0=0$ and $t_1=4/h^2$. For the entire Helmholtz operator $H_h$ in with a constant wave number $k$ the pitchfork shaped spectrum, and the bounding triangle, is shifted in the negative real direction over a distance $k^2$.
![The eigenvalues of the Laplacian discretized on the ECS domain ($\bullet$) lie along a pitchfork shaped figure in the lower half of the complex plane, close to the eigenvalues of the Laplacian restricted to the interior real domain ($\triangleleft$) and the complex contour ($\triangleright$) respectively. The smallest eigenvalues approximate the eigenvalues of the continuous Laplacian ($\times$), until they split up in a point $t_b$ ([$\circ$]{}), into two branches with limiting points $t_1 = 4/h^2$ and $t_2 = 4/({h_{\gamma}})^2$. Eigenvalues accumulate near the real axis around $0$ and $t_1$.[]{data-label="fig:pitchfork"}](pitchfork){width="\textwidth"}
As the higher dimensional Helmholtz problems are constructed with Kronecker products, these results on the spectrum of the discretization matrix $H_h$ are easily extended. Every eigenvalue $\lambda$ of the $d$-dimensional Laplacian is a sum of eigenvalues $\lambda^{(l)}$ of the one-dimensional cases, $\lambda = \sum_{l=1}^{d}{\lambda^{(l)}}$. This allows us to stick the discussion to the basic academic one-dimensional model problem. Note that real applications may require more carefully engineered domains with e.g. smoother complex stretching, higher order discretization methods or an absorbing ECS layer on both sides of the domain. These generalizations might have an effect on the eigenvalues of the discretization matrix, but the main topology remains a bounded pitchfork shaped spectrum with the smoothest eigenvalues aligned, close to the continuous case.
If we assume that the spectrum of the Helmholtz discretization matrix lies inside the triangle $\widehat{t_0t_1t_2}-k^2$ in the complex plane, it is straightforward to see the main issues for iterative methods. First of all the size of the triangle grows as $h^{-2}$ which can be expected with the Laplacian operator involved. This bad conditioning destroys the efficiency of Krylov subspace methods. It would not necessarily be an issue for a multigrid method, however another difficulty is the indefiniteness of the matrix. The negative Helmholtz shift $-k^2$ drives the upper branch of the pitchfork closer towards or even past the origin. This makes the coarse grid correction in multigrid highly unstable due to a possible numerical resonance at a coarser level as was reported in [@JCP-paper; @polynomialsmoother]. A common solution is a preconditioned Krylov subspace method where another matrix $M_h$ is defined such that, $$M_h^{-1}H_hu_h=M_h^{-1}f_h,$$ can be easily solved instead. The preconditioning matrix $M_h$ is chosen such that it is efficiently invertible with a fast multigrid method and such that the preconditioned matrix $M_h^{-1}H_h$ is well conditioned, that is, its eigenvalues are clustered around $1$ away from the origin. The complex shifted Laplacian $M^{CSL} = -\triangle -\beta k^2$ has been a successful choice introduced by Erlangga [@EVO04] for Sommerfeld radiation conditions. Simply shifting the Laplacian downwards into the complex plane fixes the coarse grid correction in multigrid. In [@JCP-paper] this idea was used with ECS boundary conditions, together with the introduction of the closely related complex stretched grid (CSG) operator $M^{CSG}$, that is constructed by discretizing the original Helmholtz operator $-\triangle -k^2$ on a different complex stretched domain, $$\label{eq:domaincsg}
z(x) = \left\{
\begin{array}{ll}
xe^{\imath{\theta_{\beta}}}, & \hbox{$x \in [0,1]$;} \\
e^{\imath{\theta_{\beta}}}+(x-1)e^{\imath{\theta_{\gamma}}}, & \hbox{$x \in (1,R]$.}
\end{array}
\right.$$ This domain is complex scaled over $e^{\imath{\theta_{\beta}}}$ in the interior region $[0,1]$; the exterior complex contour has the same scaling $e^{\imath{\theta_{\gamma}}}$, see Figure \[fig:domaincsg\]. The spectrum of the discretized operator $M_h^{CSG}=L_h^{CSG}-k^2I_h$ is pitchfork shaped as the original Helmholtz operator $H_h=L_h-k^2I_h$, but with the troublesome upper branch deeper in the complex plane, see Figure \[fig:pitchfork\_csg\]. Indeed, back scaling the entire preconditioning domain over the inner angle ${\theta_{\beta}}$ with $e^{-\imath{\theta_{\beta}}}$ returns a regular ECS domain with a real interior region and an ECS layer with a reduced angle ${\theta_{\gamma}}-{\theta_{\beta}}$. Discretizing the Laplace operator on this latter ECS domain gives the scaled matrix $e^{2\imath{\theta_{\beta}}}L_h^{CSG}-k^2I_h$ and so $M_h^{CSG}=L_h^{CSG}-k^2I_h$ must have a pitchfork shaped spectrum too, though somewhat more narrow and rotated away from the real axis over an angle $-2{\theta_{\beta}}$. Similar to the complex shift $\beta$ in $M^{CSL}$, the exact choice of the interior scaling angle ${\theta_{\beta}}$ determines the performance of multigrid on the preconditioner versus the overall convergence rate of the preconditioned Krylov subspace method. In [@polynomialsmoother] [GMRES is suggested as a smoother substitute in]{} multigrid which permits small angles ${\theta_{\beta}}$ for the preconditioner $M^{CSG}$. This improves the Krylov subspace convergence significantly. The goal of this paper is to have a better understanding of the spectrum of the preconditioned operator $M_h^{-1}H_h$, where eventually $M_h=M_h^{CSG}$ will be inverted with a multigrid method.
![The domain of the CSG preconditioner (solid line) differs from the original ECS domain (dashed line) in the interior region where it is scaled into the complex plane by $e^{\imath\theta_\beta}$. The exterior complex contour has the same scaling $e^{\imath\theta_\gamma}$ as the original ECS domain.[]{data-label="fig:domaincsg"}](ComplexDomain.png){width="90.00000%"}
![The spectrum ($\bullet$) of the Laplacian discretized on the preconditioning domain in Figure \[fig:domaincsg\] is pitchfork shaped too, but with the upper branch rotated away from the real axis.[]{data-label="fig:pitchfork_csg"}](pitchfork_csg){width="\textwidth"}
Eigenvalues of the 1D Laplacian on the complex domain {#sec:continuouseigenvalues}
=====================================================
In this section we discuss the eigenvalues of the Helmholtz problem formulated on an ECS domain as in Figure \[fig:domain\]. To this aim we first consider the Laplacian on a one-dimensional stretched domain $$\label{eq:contour}
z(x) = \int_0^x q(t) dt,$$ in the complex plane. [In order to simplify the discussion in this section we purposely use the integral representation for the ECS domain, as opposed to the formulation in .]{} We are interested in eigenmodes [of the Laplacian]{}, $$-\frac{d^2}{dz^2} u_j = \lambda_j u_j,$$ with Dirichlet boundary conditions $u_j(0)=0$ and $u_j(z(R))=0$. For the remainder of this discussion we will drop the subscript $j$ on $u$ and $\lambda$. After applying the chain rule, the equation becomes $$\left[-\frac{1}{q(x)}\frac{d}{dx} \frac{1}{q(x)} \frac{d}{dx} -\lambda\right] u(x)=0,$$ with $u(0)=0$ and $u(R)=0$.
[The domain is described by Equation with $$\label{eq:ecsdomain}
q(x) = \begin{cases}
1 \quad & 0 \le x \le r,\\
\gamma \equiv e^{{\imath}{\theta_{\gamma}}} \quad& r< x \le R,
\end{cases}$$ where $r=1$ for the model problem in .]{} We then have a second order ODE with constant coefficients on $[0,r]$ and on $(r,R]$ and the solutions can be written as a linear combination of two fundamental solution. We denote with $u_1(x)$ the solution on the first interval and $u_2(x)$ the solution on the second interval. In the point $r$ the solutions of the both subdomains need to be matched with the conditions $$\begin{cases}
u_1(r) &= u_2(r), \\
\lim_{\epsilon \rightarrow 0}\frac{1}{q(r-\epsilon)}u_1^\prime(r-\epsilon) &= \lim_{\epsilon \rightarrow 0} \frac{1}{q(r+\epsilon)}u_2^\prime(r+\epsilon),
\end{cases}$$ where the jump condition on the derivative expresses that $u(z(x))$ needs to be continuously differentiable along the transformed domain $z(x)$. Solving the equation on each subdomain with boundary conditions $u_1(0)=0$ and $u_2(R)=0$ leads to $$\begin{cases}
u_1(x) = A\sin(x\sqrt{\lambda}), & 0\le x \le r; \\
u_2(x) = B\sin((x-R)\gamma\sqrt{\lambda}), & r \le x \le R,
\end{cases}$$ where $A$ and $B$ are unknown coefficients. The solutions $u_1$ and $u_2$ have to fulfill the matching condition in $r$ $$u_1(r) = u_2(r) \quad \text{and}\quad u_1^{\prime}(r) = \frac{1}{\gamma} u_2^\prime(r).$$ After setting $A=1$, which can be done without loss of generality, and eliminating $B$ with the first condition, the second equation becomes $$\cos\left(r\sqrt{\lambda}\right) - \frac{\sin\left(r\sqrt{\lambda}\right)\cos\left(\left(r-R\right)\gamma\sqrt{\lambda}\right)}{\sin\left(\left(r-R\right)\gamma \sqrt{\lambda} \right)} = 0.$$ After some trigoniometry this leads to the condition $$\sin\left(\sqrt{\lambda}((R-r)\gamma + r) \right)=0,$$ thus to the eigenvalues $$\lambda_j = \left(\frac{j \pi}{\left(R-r\right) \gamma + r}\right)^2,$$ for $j\in\mathbb{N}_0$.
Note that $r + (R-r)\gamma=R_z=z(R)$ is the end point of the complex contour on which we have solved the [ODE]{}. In this point we have enforced Dirichlet boundary condition. [Therefore these eigenvalues belong to eigenmodes that are standing waves]{} on the domain $[0,R_z]$. The eigenvalues are independent of the details of the complex contour. If we would have taken a more complicated contour[, with e.g. quadratical scaling instead of linear scaling over a constant angle ${\theta_{\gamma}}$ as in ,]{} the eigenvalues would be the same.
Note that the discrete problem, discussed in Section \[sec:modelproblem\], only approximates the first few eigenmodes along this line. Then, at a certain point along the line, the spectrum of the discrete operator will bifurcate into two branches as shown in Figure \[fig:pitchfork\]. This branch point will be discussed in Section \[ssec:branchpoint\].
The spectrum of the Helmholtz problem with constant wave number $k$ is now $$\label{eq:eigenvalues}
\lambda_j(k) = \left(\frac{j \pi}{R_z}\right)^2 -k^2.$$ with $j\in\mathbb{N}_0$. These are the eigenvalues of the Laplacian shifted over $-k^2$.
Eigenvalues of the preconditioned problem on the 1D domain {#sec:continuouseigenvaluesprecon}
==========================================================
Let us now look at the eigenvalues of the preconditioning operator $M^{CSG}$. It is defined on a domain described by $$\label{eq:csgdomain}
p(x) = \begin{cases}
\beta \equiv e^{{\imath}{\theta_{\beta}}}, & 0 \le x \le r;\\
\gamma \equiv e^{{\imath}{\theta_{\gamma}}}, & r < x \le R.
\end{cases}$$ Let us denote the end point of this complex domain as $\tilde{R}_z =
\int_0^R p(t)dt$. The eigenvalues of the preconditioning operator are described by $j^2\pi^2/\tilde{R}_z^2$, using the results of the previous section. So both for the original problem as for the preconditioned operator we have that the eigenvalues lie on a straight line in the complex plane.
Let us assume that for each $j$ the eigenvectors for the domains defined by $p$ and $q$ are the same. Then the eigenvalues $\mu$ of the preconditioned operator $(M^{CSG})^{-1}H$ can be approximated by $$\mu_j = \frac{ j^2 \pi^2/R_z^2 -k^2}{j^2 \pi^2/\tilde{R}^2_z -k^2}.$$ This can be rewritten as $$\label{eq:eigMHapprox}
\mu_j = \frac{\tilde{R}^2_z}{R^2_z}\frac{ j^2 \pi^2/k^2 -R_z^2}{j^2 \pi^2/k^2 -\tilde{R}_z^2},$$ [which is the evaluation of a linear fractional transformation $LF:\mathbb{C}\to\mathbb{C}:w\mapsto\frac{\tilde{R}^2_z}{R^2_z}\frac{w -R_z^2}{w -\tilde{R}_z^2}$ in the points $j^2 \pi^2/k^2\in\mathbb{R}$ with $j\in\mathbb{N}_0$. It is a known property in complex analysis that $LF$ maps lines to lines or circles in the complex plane. In this case]{} we find that the eigenvalues $\mu_j$ form a circle in the complex plane with radius $$\frac{|\tilde{R}^2_z|}{|R^2_z|}\left|\frac{R_z-\tilde{R}_z}{2 \Im(\tilde{R}_z)}\right|.$$ This circle does not include the origin. Note that the radius of the circle is independent of the wave number $k$.
However, as the next discussion will show, the assumption that the eigenvectors for a given $j$ on the $p$ and $q$ domains are the same is invalid [because the eigenvalues of $(M^{CSG})^{-1}H$ are in fact different from $\mu_j$ in . Indeed, in order to]{} understand the spectrum we have to solve the eigenvalues of the operator $$\begin{aligned}
\label{eq:MHcontinu}
\left(-\frac{1}{p(x)}\frac{d}{dx}\frac{1}{p(x)}\frac{d}{dx} -k^2 \right)^{-1} \left(-\frac{1}{q(x)}\frac{d}{dx}\frac{1}{q(x)}\frac{d}{dx}-k^2 \right)u(x) = \mu u(x),\end{aligned}$$ which is a generalized eigenvalue problem $$\left(-\frac{1}{q(x)}\frac{d}{dx}\frac{1}{q(x)}\frac{d}{dx}-k^2 \right)u(x) = \mu \left(-\frac{1}{p(x)}\frac{d}{dx}\frac{1}{p(x)}\frac{d}{dx} -k^2 \right)u(x)$$ that becomes, after reordering $$\begin{aligned}
&&\left[ -\frac{1}{q(x)}\frac{d}{dx}\frac{1}{q(x)}\frac{d}{dx} + \mu\frac{1}{p(x)}\frac{d}{dx}\frac{1}{p(x)}\frac{d}{dx} -(1-\mu)k^2 \right]u(x) = 0,\\
&\Leftrightarrow & \left[ -\frac{1}{\tilde{q}(x)}\frac{d}{dx}\frac{1}{\tilde{q}(x)}\frac{d}{dx} -(1-\mu)k^2 \right]u(x) = 0,\end{aligned}$$ with $1/\tilde{q}(x)=\sqrt{1/q(x)^2-\mu/p(x)^2}$.
\[prop:eigMHcontinu\] For the model problems with domains defined by [$q(r)$ in and $p(r)$ in ]{}, the eigenvalues of the operator in are $$\mu_j = \frac{s_j^2 -1 }{s_j^2/\beta^2- 1}$$ with [ $$s_j = \frac{1}{r}\left(\frac{j\pi}{k} - \gamma(R-r)\right) .$$ ]{}
The piecewise constant $p$ and $q$ again lead to a second order ODE with constant coefficients for $u_1(x)$ on the interval $[0,r]$ and for $u_2(x)$ on the interval $[r,R]$. $$\begin{cases}
- (1 - \mu \frac{1}{\beta^2}) \frac{d^2}{dx^2} u_1 -(1-\mu)k^2 u_1 = 0 & \quad 0 \le x \le r,\\
-(1 - \mu) \frac{1}{\gamma^2} \frac{d^2}{dx^2} u_2 -(1-\mu)k^2 u_2 = 0 &\quad r \le x \le R,
\end{cases}$$ with $u_1(0)=0$ and $u_2(R)=0$. The solutions $u_1$ and $u_2$ are $$\begin{cases}
u_1(x) = A\sin\left(k\beta\sqrt{\frac{1-\mu}{\beta^2-\mu}} x \right) &\quad 0 \le x \le r,\\
u_2(x) = B\sin\left(k\gamma(x-R) \right) &\quad r \le x \le R,\\
\end{cases}$$ that need to be matched by the conditions $$\begin{cases}
u_1(r) &= u_2(r);\\
\lim_{\epsilon \rightarrow 0}\frac{1}{\tilde{q}(r-\epsilon)}u_1^\prime(r-\epsilon) &= \lim_{\epsilon \rightarrow 0} \frac{1}{\tilde{q}(r+\epsilon)}u_2^\prime(r+\epsilon).
\end{cases}$$ Without loss of generality we can choose $A=1$. Requiring continuity of the solution in $r$ leads to $$B = \frac{\sin\left( k \beta\sqrt{(1-\mu)/(\beta^2-\mu)} r\right)}{\sin\left(k\gamma(r-R)\right)}.$$ Inserting this in the matching condition for the derivatives leads, after some trigoniometry, to $$\sin\left(k\beta\sqrt{\frac{1 - \mu}{\beta^2 - \mu}} r + k\gamma\left(R-r\right) \right) = 0.$$ The eigenvalues are the solutions of $$k\beta\sqrt{\frac{1 - \mu}{\beta^2 - \mu}} r + k\gamma\left(R-r\right) = j \pi$$ and we find that $$\mu_j = \frac{s_j^2 -1 }{s_j^2/\beta^2- 1},$$ where $$s_j = \frac{1}{r}\left(\frac{j\pi}{k} - \gamma(R-r)\right)$$
\[cor:parametric\] The eigenvalues of the preconditioned operator $(M^{CSG})^{-1}H$ lie on a parametric curve $t:[-\Re(\eta), \infty) \rightarrow \mathbb{C}$ that maps $t$ to $$\frac{(t-{\imath}\Im(\eta))^2-1}{(t-{\imath}\Im(\eta))^2/\beta^2 - 1},$$ with $\eta = \gamma(R/r-1)$. When $\Im(\gamma)=0$, the curve [lies on a circle]{} through $0$, [$\beta^2$]{} and $1$.
Splitting $s_j$ [in Proposition \[prop:eigMHcontinu\]]{} into a real and imaginary part leads to $$s_j = \frac{1}{r}\left(\frac{j\pi}{k} - \Re\left(\gamma\right)(R-r)\right) - {\imath}\frac{1}{r}\Im\left(\gamma\right)(R-r).$$ Define $\eta = \gamma(R/r-1)$, then for each $j\in\mathbb{N}_0$ there is a $t \in [-\Re(\eta), +\infty)\subset\mathbb{R}$ such that $s_j = t -{\imath}(R/r-1)\Im\left(\gamma\right)$. [In the limit when there is no exterior complex scaling, i.e. $\Im(\gamma)=0$, we can reparametrize by $\hat{t}=t^2\in\mathbb{R}$ and so the curve reduces to a linear fractional transformation. It follows that the real line is mapped to a circle through the points $0$, [$\beta^2$]{} and $1$.]{}
It is important to note that changing the wave number $k$ does not alter the parametric curve. Indeed, changing $k$ only modifies the real part of $s_j$ which leads to a different particular choice $t$ that gives the position of the eigenvalue $\mu_j$ on the curve. This means that there is an upper bound for the condition number $\kappa=\frac{\max_j|\mu_j|}{\min_j|\mu_j|}$ of the preconditioned problem that is independent of $k$ [and suggests a fast convergence of the preconditioned Krylov subspace method]{}.
The spread of the eigenvalues on the parametric curve can change as a function of $k$. First note that in the limit $j\rightarrow \infty$ the [eigenvalues $\mu_j$]{} go to $\beta^2$. In a similar way, the eigenvalues will accumulate near $\beta^2$ as $k \rightarrow 0$. In the other case, as $k$ [gets larger]{}, the smoothest eigenvalues $\mu_j$ with $j\ll k$ will go to the other end of the curve, $$\lim_{j/k\to0}\mu_j = \frac{\gamma^2(R/r-1)^2-1}{\gamma^2(R/r-1)^2/\beta^2-1}\approx 1,$$ [leading to a spectrum that is completely spread over the curve.]{}
[We clearly observe this behavior in Figure \[fig:eigMA\] where the spectrum of the preconditioned system is plotted for different values of $k$ for the one-dimensional model problem with $r=1$, $R=1.25$, outer ECS angle ${\theta_{\gamma}}=\frac{\pi}{6}$, inner angle for the preconditioner ${\theta_{\beta}}=0.18\approx\frac{\pi}{17}$. The circles mark the first $80$ eigenvalues of the continuous problem, as given by Proposition \[prop:eigMHcontinu\]. They lie on the parametric curve derived in Corollary \[cor:parametric\] which is visualized with a solid line. For a small wave number $k=0.4$, in the upper left subfigure, all eigenvalues $\mu_j$, with $1\leq j\leq80$ lie close to $\beta^2$. In the upper right plot with $k=6.4$ we see that $\mu_{1}$ and $\mu_{2}$ almost reach the other end of the curve, while the remaining eigenvalues $\mu_j$ with $j\geq3$ still lie closer to $\beta^2$. Finally in the lower two subfigures the eigenvalues are more spread along the curve as $k$ grows larger.]{}
In a similar way the 2D eigenvalues, or for any higher dimension, will be bounded by a parametric curve that is independent of the wave number. In contrast to the one-dimensional case the 2D spectrum will fill up the region bounded by the curve with eigenvalues.
Discrete operator {#sec:discrete}
=================
Deviations from the continuous problem {#ssec:deviation}
--------------------------------------
However, when the problem is discretized, with for example finite differences, the Krylov convergence rate can differ significantly from the bounds predicted by the analysis of the continuous problem in Section \[sec:continuouseigenvalues\] and \[sec:continuouseigenvaluesprecon\].
[ Indeed, [Figure \[fig:gmres\_iterations\] shows]{} the number of GMRES iterations to solve the Helmholtz problem with discretization matrix $H_h$, preconditioned with the complex shifted grid matrix $M_h^{CSG}$ that is exactly inverted, as a function of the wave number $k$. The results are for a 1D (solid line) and a 2D (dashed line) problem with $80$ grid points per dimension, $n=64$ interior grid points and $m=16$ additional points for the ECS layer. For the 2D problem we recognize three regions in the convergence rate. First, between $k=0$ and $k\approx16.4$, the number of iterations is ramped up from a few to about $18$. The second region is between $k\approx16.4$ and $k\approx21$, where the number of iterations remains constant. Finally from $k=21$ on the number of iteration rises again. This is in contrast to the analysis of the continuous problem that predicts a convergence rate that is independent of $k$, once it is large enough. These observations can be explained with the help of the spectrum of the discrete preconditioned system $M_h^{-1}H_h$ of the one-dimensional problem shown in Figure \[fig:eigMA\]. ]{}
[ For the wave numbers in the first region, $0\leq k\leq16.4$, the spectrum of the discrete problem ($\bullet$) is a good approximation for the spectrum of the continuous continuous problem ($\circ$), except for one or two spurious eigenvalues. As a consequence, the spectrum lies along the parametric curve (solid line). The initial growth in the convergence behavior is due to the fact that the eigenvalues are accumulated near $\beta^2$ for small $k$ and start spreading over the curve towards $1$ for larger $k$, leading to an increasing condition number $\kappa = \frac{\max_j{|\mu_j|}}{\min_j{|\mu_j|}}$ where $1\leq j\leq80$. In Figure \[fig:conditionnumber\] the condition number is plotted as a function of the wave number $k$. Indeed, first the condition number is close to $1$ until eigenvalue $\mu_{80}$ moves significantly along the curve reaching the point with the smallest possible absolute value for $k\approx2.5$. The next eigenvalue $\mu_{79}$ gives rise to a second peak around $k\approx5$ when it reaches that same minimal point on the curve. As the spectrum starts spreading more equally over the curve, see e.g. lower left subfigure in Figure \[fig:eigMA\], the condition number seems to converge to $\kappa\approx 2.5$ and the number of GMRES iterations stagnates around $18$ for the second region $k\approx16.4$ to $k\approx21$ on Figure \[fig:gmres\_iterations\]. ]{}
[ However, from $k\approx16.4$ on the discrete condition number starts behaving differently from what the continuous eigenvalues predict. Around $k\approx21$ it even grows above the upper bound given by the curve. This is because the eigenvalues of the discrete preconditioned system start deviating from the continuous spectrum, see e.g. lower right subfigure in Figure \[fig:eigMA\]. Whereas the continuous eigenvalues remain on the curve, the discrete spectrum grows outside the curve from a certain critical wave number. This divergence between the continuous and the discrete preconditioned spectrum finds its origin in the spectrum of the original Helmholtz operator. At the end of Section \[sec:continuouseigenvalues\] we mentioned that only the first few eigenvalues $\lambda_j$ in of the continuous Helmholtz operator $H$ are well approximated by the eigenvalues of the discretization matrix $H_h$ given by . In Figure \[fig:pitchfork\] we see that the continuous spectrum ($\times$) lies along a line in the complex plane, whereas the discrete spectrum ($\bullet$) branches from this line at a certain point $t_b\in\mathbb{C}$. The location of this point will be an indicator for the start of the third region in the convergence behavior of GMRES. ]{} [ For the 2D problem the three different convergence regions are more pronounced than for the 1D problem because the region bounded by the curve gets filled with extra eigenvalues. ]{}
Predicting the branch point {#ssec:branchpoint}
---------------------------
Next we derive an explicit formula for the [approximate]{} position of the branch point in the spectrum of the discretization matrix, see Figure \[fig:pitchfork\]. From this point on, the eigenvalues of the discrete problem start to deviate significantly from the eigenvalues of the continuous problem. This branch point will predict from which $k$ on we can expect a rising cost of the numerical solution method. A surprising result of this section is that this branch point does not shift with the order of the discretization.
\[prop:branchpoint\] Consider a one-dimensional grid as in defined on an ECS domain consisting of two parts, $[0,r]$ for the interior with $n$ grid points, and the complex interval $[r,R_z]=[r,z(R)]$ for the exterior part with $m$ grid points. The smallest eigenvalues of the negative Laplacian discretized on this ECS grid $-L_h$, with zero Dirichlet conditions at the boundaries $0$ and $R_z$, lie along the complex line $$\begin{aligned}
t(\rho) = \left(\frac{\rho}{R_z}\right)^2, \quad \mbox{with }\rho>0,\end{aligned}$$ close to the eigenvalues of the continuous operator $t_j = \left(\frac{j\pi}{R_z}\right)^2$ with $j\in\mathbb{N}_0$. For larger eigenvalues the spectrum of $-L_h$ splits into two branches around the point $t_b = \left(\frac{\rho_b}{R_z}\right)^2$ with $$\begin{aligned}
\rho_b = \frac{|R_z|^2}{r\Im(R_z)}W\left(4n\Im(R_z)\left|\frac{1}{R_z}\sqrt{\frac{R-r}{R_z-R}}\right|\right)\end{aligned}$$ where $R_z=z(R)$ and $W(.)$ is the Lambert-W function.
The spectrum of the Laplacian discretized on an ECS grid has a pitchfork shape. In order to find the point where the pitchfork splits we start from the condition that has the eigenvalues of the discrete Laplacian $-L_h$ as solutions. In the rest of this discussion we will consider the scaled Laplacian $L = -h^2L_h$. The eigenvalues of $L$ are again the solutions of , but with $p(t)=\frac{1}{2}\arccos(1-\frac{t}{2})$, $q(t)=\frac{1}{2}\arccos(1-\frac{t}{2}\gamma^2)$ instead. The condition is equivalent to $$\begin{aligned}
\label{eq:eigcond1}
F_1(t) \equiv \sin(2n p(t))\cos(2m q(t))\cos(q(t))+\cos(p(t))\cos(2n p(t))\sin(2m q(t)) = 0,\end{aligned}$$
Since we are interested in the smallest eigenvalues of $L$ we can use the approximate condition $$\begin{aligned}
F_2(t) \equiv \sin\left(\left(n+m\gamma\right)\sqrt{t}\right)-\frac{1}{2}\sin\left(n\sqrt{t}\right)\cos\left(m \gamma\sqrt{t}\right)\tan\left(\frac{\sqrt{t}}{2}\right)\sqrt{t}\varepsilon = 0,\end{aligned}$$ with $\varepsilon=\gamma-1$.\
This is easily derived from by using the Taylor series $p(t) = \frac{\sqrt{t}}{2} +\mathcal{O}(|t|^{3/2})$ and $q(t) = \gamma\frac{\sqrt{t}}{2} +\mathcal{O}(|t|^{3/2})$, for $|t| \ll 1$ and substituting $\gamma \equiv 1+\varepsilon$. Moreover, since $|\varepsilon|<1$ for realistic exterior complex scaling with an ECS angle $\theta_\gamma<\frac{\pi}{4}$, we have used $\cos\left(\gamma\frac{\sqrt{t}}{2}\right) = \cos\left(\frac{\sqrt{t}}{2}\right) -\frac{1}{2}\sin\left(\frac{\sqrt{t}}{2}\right)\sqrt{t}\varepsilon +\mathcal{O}(|t\varepsilon^2|)$.\
We will now look at the evaluation of function $F_2$ along the complex line $$t(\rho) = \left(\frac{\rho}{n + m\gamma}\right)^2 \quad \mbox{with }\rho>0.$$ It returns real numbers between $-1$ and $1$ for the first term of $F_2$, and complex numbers for the second term. The latter term is small, for small $\rho$, and thus the roots of $F_2$ will approximately be the roots $t_j=\left(\frac{j\pi}{n + m\gamma}\right)^2$, with $j\in\mathbb{N}_0$, of the first term. The eigenvalues will branch from the line $t(\rho)$ when the second term of $F_2$ becomes more important. This is when, $$\begin{aligned}
& |\frac{1}{2}\sin\left(n\frac{\rho}{n+m\gamma}\right)\cos\left(m\gamma\frac{\rho}{n+m\gamma}\right)\tan\left(\frac{\rho}{2(n + m\gamma)}\right)\frac{\rho}{n + m\gamma}\varepsilon| \approx 1, \\
\Leftrightarrow& |\frac{\varepsilon}{8}\sin\left(\rho\left(1-\frac{2m\gamma}{n+m\gamma}\right)\right)\left(\frac{\rho}{n + m\gamma}\right)^2| \approx 1,\end{aligned}$$ and after using the identity $R_z=r+(R-r)\gamma$, $$\begin{aligned}
\Leftrightarrow& |\frac{\varepsilon}{8}\sin\left(\rho\left(\frac{2(R_z-r)}{R_z}-1\right)\right)\left(\frac{\rho h}{R_z}\right)^2| \approx 1, \\
\Leftrightarrow& \frac{h|\sqrt{\varepsilon}|}{4r|R_z\Im(\frac{1}{R_z})|} \rho r|\Im(\frac{1}{R_z})| e^{\rho r|\Im(\frac{1}{R_z})|} \approx 1, \\
\Leftrightarrow& \rho \approx \frac{W(c)}{r|\Im(\frac{1}{R_z})|},\end{aligned}$$ where $W(c)$ is the Lambert-W function, [defined such that $c=W(c)e^{W(c)}$,]{} and evaluated in $$c=\frac{4r|R_z\Im(\frac{1}{R_z})|}{h|\sqrt{\varepsilon}|}= 4n\Im(R_z)\left|\frac{1}{R_z}\sqrt{\frac{R-r}{R_z-R}}\right|,$$ with $\varepsilon = \gamma-1=\frac{R_z-R}{R-r}$. The point $t_b$ along the line $t(\rho)$ where the pitchfork splits into two branches is now approximately given by $$\begin{aligned}
& \rho_b = \frac{W(c)}{r|\Im(\frac{1}{R_z})|} = \frac{|R_z|^2}{r\Im(R_z)}W\left(4n\Im(R_z)\left|\frac{1}{R_z}\sqrt{\frac{R-r}{R_z-R}}\right|\right),\\
\Rightarrow & t_b = \left(\frac{\rho_b}{n+m\gamma}\right)^2 = \left(\frac{\rho_b h}{R_z}\right)^2.\end{aligned}$$ So for the eigenvalues of the unscaled operator $L$ we have $t_b = \left(\frac{\rho_b}{R_z}\right)^2$.
The point $t_b=\left(\frac{\rho_b}{R_z}\right)^2$ predicts the point in the spectrum of the discrete Laplacian where the pitchfork splits. The smallest eigenvalues lie close to $\frac{j^2\pi^2}{R_z^2}$, with $j\in\mathbb{N}_0$ such that $j\pi\leq\rho_b$. This is illustrated in Figure \[fig:splitpoint\] where the 32 smallest eigenvalues are plotted for three different grid sizes $n$, together with the branch point predictions $t_b$, for the ECS domain $[0,r]\cup(r,R_z]=[0,1]\cup(1,1+0.25e^{\imath\pi/6}]$.
![The 32 smallest eigenvalues ($\bullet$) of the discretized Laplacian for $n=32$, $n=512$ and $n=8192$. The branch point $t_b$ ([$\circ$]{}) in the spectrum moves further in the complex plane as predicted by the formula in Proposition \[prop:branchpoint\].[]{data-label="fig:splitpoint"}](splitpoint){width="\textwidth"}
Figure \[fig:splitpoint\_abs\] shows the distance to the origin $|t_b|$ of the predicted branch point as a function of the interior grid size $n$, the other domain parameters are fixed. The branch point was also detected experimentally by measuring the deviation from the line $t(\rho) =
\left(\frac{\rho}{R_z}\right)^2$ with $\rho>0$. As the grid size $n$ increases, the tail of the pitchfork grows proportional to the square of the Lambert W-function, $|t_b| \sim W(n)^2$, and not according to the order of discretization.
![The absolute value of the branch point $t_b$ of the pitchfork in the spectrum of the discretized Laplacian for $n=2^{j}$ with $j=5,\ldots,20$. As predicted by the formula ($\bullet$) in Proposition \[prop:branchpoint\], measured experimentally ($\circ$) and with a higher order scheme in the turning point $r$ ($\square$). As the grid size $n$ increases, the length of the typical line of smooth eigenvalues grows proportional to the square of the Lambert W-function, $|t_b| \sim W(n)^2$, and not as the order of the discretization.[]{data-label="fig:splitpoint_abs"}](splitpoint_abs){width="\textwidth"}
[ The spectrum of the indefinite Helmholtz discretization matrix $H_h$ is achieved by shifting the pitchfork spectrum of the Laplacian to the left in the complex plane over a distance determined by the wave number $k$. We can now predict the value $k$ for which the eigenvalues of the discrete preconditioned system $M_h^{-1}H_h$ start growing outside the parametric curve of the continuous preconditioned spectrum. Since $t_b$ marks the point in the pitchfork shaped spectrum of $H_h$ where the badly-approximated continuous eigenvalues lie on the right hand side, we expect this to happen when $t_b$ is of the same order of magnitude as the smooth eigenvalues, this means when $|t_b|=k^2$. In Figures \[fig:gmres\_iterations\] and \[fig:conditionnumber\] the critical wave number $k_b\equiv\sqrt{|t_b|}=17.9327$ is marked. Indeed, for wave numbers $k\leq k_b$ the eigenvalues of the continuous preconditioned operator lead to a good prediction of the effective discrete condition number and the rise in GMRES iterations is a direct consequence of the spreading along the parametric curve. After a short region of stagnation where the curve is densely filled with eigenvalues, the number of iterations increases again, however, this time due to the growing deviaton of the discrete eigenvalues from the continuous eigenvalues. In the next section we will study the convergence behavior for wave numbers in this third region with numerical experiments. ]{}
Numerical experiments {#sec:numerical}
=====================
In this section we analyze the performance of a Krylov subspace method applied to a Helmholtz problem with constant wave number $k$ in a square domain $\Omega=(0,1)^2$ with a centered point source $$\label{eq:helm2D}
f(x,y) =
\begin{cases}
1,\quad \mbox{if } x=1/2=y,\\
0,\quad \mbox{elsewhere}.
\end{cases}$$ This two-dimensional problem is built with Kronecker products of the one-dimensional model problem with outgoing wave boundary conditions in every direction. All boundaries are therefore extended with an ECS layer with an angle ${\theta_{\gamma}}= \frac{\pi}{6}$ to absorb outgoing waves. The results from the previous sections are still useful if we take into account that the spectrum of the two-dimensional operator is the set of all possible sums of two eigenvalues of the one-dimensional case. This means the spectrum of the discretization matrix $H_h$ looks like a sum of pitchforks now, as discussed in Section \[ssec:deviation\], with three points close to the real axis $t=-k^2$, $t=\frac{4}{h^2}-k^2$ and $t=\frac{8}{h^2}-k^2$. These eigenvalues correspond respectively to the smoothest mode on the domain, the eigenmode which is oscillatory in one dimension and smooth in the other and finally a mode that is oscillatory in both dimensions.
The complex stretched grid preconditioning matrix $M_h^{CSG}$ is constructed by discretizing the problem on a complex stretched grid with a small inner angle ${\theta_{\beta}}= 0.18 \approx \frac{\pi}{17}$. This ensures that for every level in the multigrid hierarchy the eigenvalues are bounded away from zero. The angle for the outer ECS layers is kept at ${\theta_{\gamma}}=
\frac{\pi}{6}$ as in the original Helmholtz problem as illustrated in Figure \[fig:domaincsg\]. For a detailed description of the spectrum of this preconditioning matrix we refer to [@JCP-paper] and [@polynomialsmoother].
As a consequence the preconditioning matrix $M_h^{CSG}$ can efficiently be inverted with a multigrid method with either [three steps of GMRES as a smoother substitute, denoted as GMRES($3$), or a specific polynomial smoother as suggested in [@polynomialsmoother]]{}. However, because the smoother can differ each application the actual preconditioner is not the same in every outer Krylov step. Therefore FGMRES, the flexible GMRES method [@SaadFGMRES], is used as outer Krylov subspace methods. We discuss the performance of preconditioned FGMRES before convergence to a residual norm of order $10^{-6}$. The preconditioning matrix $M_h^{CSG}$ is approximately inverted with one V(1,1)-cycle with GMRES($3$) as smoother. The experiments are all run in <span style="font-variant:small-caps;">Matlab</span> on two quad core Intel Xeon CPUs (E5462 @ 2.80GHz). Figure \[fig:krylov\_fgmres\] shows the convergence results for preconditioned FGMRES to solve the 2D Helmholtz problem with a residual norm below $10^{-6}$ for wave numbers ranging from $k=15$ to $k=180$. For these wave numbers the smoothest eigenvalues of the discretization matrices have a negative real part. Each curve shows the same experiment for a fixed grid size $n$ in one dimension, the grid size of the ECS layer is related as $m=n/4$. Just like the wave numbers the grid sizes are purposely chosen over a wide range as well, from $n=16$ to $n=2048$ in one dimension, in order to expose the full effect of the preconditioning on the convergence behavior of FGMRES. A discussion on the physical accuracy of the grid sizes lies not within the scope of this analysis. As we are merely interested in the convergence behavior of Krylov subspace methods we explain these curves as a function of the increasing wave number $k$.
For each grid size both the convergence rate and the number of iterations grow initially as a function of $k$ up to a peak where $k\approx\frac{2}{h}=2n$. This corresponds to the first critical point $t=\frac{4}{h^2}-k^2$ where the pitchfork in the spectrum of $H_h$ nearly touches the real axis. These eigenvalues correspond to eigenmodes that oscillate rapidly in one direction while they are smooth in the other. There is now an eigenvalue of $H_h$ near the origin and the preconditioned system $M_h^{-1}H_h$ obviously suffers from this too. As the wave number $k$ increases more, the pitchfork is shifted further to the left in the complex plane. The convergence improves slightly until the second critical point $t=\frac{8}{h^2}-k^2$ comes too close to the origin for $k\approx\frac{2\sqrt{2}}{h}=2\sqrt{2}n$. After this, the spectrum has completely shifted into the negative real part of the complex plane, making the Helmholtz matrix negative definite. This is observed on the curves as a sudden improvement in convergence. As a reference these experiments were repeated for the smallest grid sizes with regular GMRES and an exact inversion of the preconditioner $M_h^{CSG}$, in order to eliminate the effect of the approximate multigrid inverse. In Figure \[fig:krylov\_gmres\_exact\] the described convergence behavior is then more pronounced.
\
\
Only for the two smallest grid sizes this behavior lies completely within the tested range of wave numbers $k$. Indeed, in Table \[tab:criticalk\] the critical wave numbers are $k_1=\frac{2}{h}=2n$ and $k_2=\frac{2\sqrt{2}}{h}=2\sqrt{2}n$ are listed for the different grid sizes in the experiments. The first critical wave number $k_1$ has the worst performance because the spectrum of $H_h$ reaches its most extreme indefiniteness, with the pitchfork perfectly spread over the negative and positive real part of the lower half of the complex plane. For larger $k$ the spectrum tends more to negative definiteness. The second peak is right before it turns completely negative definite in the second critical wave number $k_2$ after which the convergence rate obviously drops drastically.
$n$ 16 32 64 128 256 512 1024 2048
-------------------- ------ ------ ------- ------- ------- -------- -------- --------
$k_1$ 32 64 128 256 512 1048 2048 4096
$k_2$ 45.3 90.5 181.0 362.0 724.1 1448.2 2896.3 5792.6
$k_b=\sqrt{|t_b|}$ 12.8 13.8 20.3 26.2 35.7 40.6 50.0 53.1
: The critical wave numbers $k_1$ and $k_2$ for the discrete Helmholtz problem with constant $k$ for different interior grid sizes $n$. For these values of $k$ the preconditioned Krylov method reaches its worst performance as we see in Figures \[fig:krylov\_fgmres\] and \[fig:krylov\_gmres\_exact\]. The absolute value of the branch point indicates a wave number $k_b=\sqrt{|t_b|}$ that marks the end of an early plateau in the convergence behavior discussed in Section \[ssec:deviation\].[]{data-label="tab:criticalk"}
For a realistic physical solution the grid size should be large enough in order to represent the wave accurately. Higher wave numbers $k$ require finer meshes [@babuska2000pollution]. This means that in practical circumstances only the region on the curve long before the first peak is important, where the spectrum is only slightly negative definite. In that region we still profit from the fact that for $k\rightarrow 0$ the eigenvalues of the preconditioner accumulate into a single point (see Section \[ssec:deviation\]). There we see a rise and the stagnation of the number of iterations into a plateau with the branch point $t_b$ as indicator. However, on Figures \[fig:krylov\_fgmres\] and \[fig:krylov\_gmres\_exact\] the range of wave numbers $k$ is too wide to clearly uncover this initial effect as in Figure \[fig:gmres\_iterations\].
Discussion and Conclusions
==========================
In this paper we have analyzed the convergence rate of a multigrid preconditioned Krylov solver for Helmholtz problems with absorbing boundary conditions. The multigrid method inverts a preconditioner that is a Helmholtz operator discretized on a complex-valued grid rather than on a real grid. This preconditioner is comparable to the complex shifted Laplacian. The multigrid method uses GMRES($3$) as a smoother at each level.
To understand the Krylov convergence, we have proposed a model problem with a Dirichlet boundary condition on one side and an outgoing wave boundary condition at the other. The outgoing boundary condition is implemented with exterior complex scaling (ECS) that extends the domain with a complex-valued contour. This model problem is representative for the implementation of absorbing boundary layers in various applications such as ECS, which is often used in chemistry and physics or PMLs, which are frequently used in engineering.
We have analyzed this model both in a continuous and a discrete way. For the continuous problem we have found that the spectrum of the preconditioned operator lies on a curve in the complex plane which is bounded away from zero. This leads to an expected Krylov convergence rate that is bounded for all wave numbers. For small wave numbers the convergence rate is faster since the eigenvalues accumulate to a single point.
In the discrete problem the spectrum behaves similarly to the continuous problem for small wave numbers. However, for larger wave numbers the spectrum can deviate significantly. This finds it origin in the properties of the discrete Helmholtz operator that has a pitchfork in the spectrum, where only one of the arms of the pitchfork approximates the spectrum of the continuous operator. These deviations destroy the nice convergence expectations given by the continuous operator. The distance to the origin of the predicted point $t_b$ where the spectrum bifurcates grows only very slowly as a function of the number of grid points. Numerical experiments show that also varying wave numbers result in similar pitchfork shaped spectra with branching points that can still be fairly well estimated using Proposition \[prop:branchpoint\].
As a rule of thumb the Krylov convergence is bounded when $k^2$ is smaller than the absolute value of the branch point $t_b$. There we can expect a bounded convergence rate. For wave numbers $k$ larger than this branch point the number of iterations rises until $k^2\approx 4/h^2$, where the number of iterations is maximal. We have a second but milder peak at $k^2\approx 8/h^2$. From then on the spectrum is negative definite and the convergence rate drops rapidly and only a few iterations are required to solve the system.
We conclude that there is no overall $k$-independent convergence rate, yet the number of iterations diminishes as the number of grid points is increased. To further improve the convergence rate of the iterative method it is possible to engineer the parameters of the absorbing layer as a function of the wave number. For large wave numbers $k$ we do not need a large ECS grid or a large rotation angle to absorb the wave. Adapting the parameters can reduce the number of iterations to solve the problem.
Although the current analysis is for constant wave numbers $k$, we believe many results will still be valid when the wave number varies over space. Indeed, a space-dependent $k(x)$ will only affect the smoothest eigenvalues while the extreme values that determine the diverging behavior depend on the grid distance and remain the same.
There still remain important challenges in the development of an efficient solver for the Helmholtz problem with space-dependent wave numbers based on complex stretched domains. In numerical experiments with strongly space-dependent wave numbers that allow evanescent waves we have seen serious deteriorations of the convergence rate [@polynomialsmoother]. This is caused by the multigrid coarse grid correction on levels too coarse to resolve the evanascent waves. This is a subject of future research.
Acknowledgement {#acknowledgement .unnumbered}
===============
This research has been funded by the *Fonds voor Wetenschappelijk Onderzoek (FWO)* by the project G.0174.08 and *Krediet aan navorser* 1.5.145.10.\
[The authors would like to thank Hisham bin Zubair for sharing a multigrid implementation.]{}
[^1]: Email: [wim.vanroose@ua.ac.be]{}
|
---
abstract: 'We have computed Wilsonian effective action in a simple model with spontaneously broken chiral parity. We have computed Wilsonian running of relevant parameters which makes it possible to discuss in a consistent manner the issues of fine-tuning and stability of the scalar potential. This has been compared with the standard picture based on running. Since Wilsonian running includes automatically integration of heavy degrees of freedom, the running differs markedly from the version. However, similar behaviour can be observed: scalar mass squared parameter and the quartic coupling can change sign from a positive to a negative one due to running which causes spontaneous symmetry breaking or an instability in the renormalizable part of the potential for a given range of scales. However, care must be taken when drawing conclusions, because of the truncation of higher dimension operators. Taking scalar field’s amplitude near the cut-off $\Lambda$ may cancel suppression due to the scale and only suppression due to small couplings partially justifies truncation in this region. Also, when taking the cut-off higher, to include larger amplitudes of the fields, higher-order irrelevant operators, whose coefficients grow with scale, may affect the conclusion about stability. The running allows one to resume relatively easily a class of operators corresponding to large logarithms to form RGE improved effective potential valid over a huge range of scales. In the Wilsonian approach this would correspond to following the running of a large number of irrelevant operators, which is technically problematic. As for the issue of fine-tuning, since in the Wilsonian approach power-law terms are not subtracted, one can clearly observe the quadratic sensitivity of fine-tuning measure to the change of the cut-off scale. The Wilsonian version of the radiative symmetry breaking mechanism has been described.'
author:
- |
Tomasz Krajewski[^1] Zygmunt Lalak[^2]\
Institute of Theoretical Physics, Faculty of Physics, University of Warsaw\
ul. Pasteura 5, Warsaw, Poland
bibliography:
- 'FTVSWEA.bib'
title: 'Fine-tuning and vacuum stability in Wilsonian effective action'
---
Introduction
============
The recent discovery of the Higgs boson at the Large Hadron Collider [@Aad:2012tfa; @Chatrchyan:2012ufa] promotes the question about protection of the electroweak breaking scale to one of the most puzzling problems of fundamental physics. The observed compatibility of properties of the newly observed particle with predictions coming from Standard Model additionally strengthens tension between standard theoretical reasoning which results in prediction of new physics near the electroweak scale and reality. Neither supersymmetry nor composite Higgs models, perhaps most attractive solutions to hierarchy problem, are favoured by the observed value of Higgs mass [@Giardino:2012ww; @Carmi:2012in; @Plehn:2012iz]. Moreover, production and decay rates have not provided unambiguous evidence for new physics.
This situation strengthens the need of revisiting the naturalness principle which have been used as a guide for model building, since its formulation in the late 1970s and early 1980s [@Susskind:1978ms; @'tHooft:1979bh; @Veltman:1980mj]. Numerous authors [@Aoki:2012xs; @Farina:2013mla; @Jegerlehner:2013cta; @Jegerlehner:2013nna; @Bian:2013xra; @deGouvea:2014xba; @Bar-Shalom:2014taa] propose new definitions of naturalness. Our goal is less ambitious. We shall try to state clearly a treatment of fine-tuning based on Wilsonian effective action and corresponding Wilsonian renormalization group. Idea of Wilsonian effective action is close to the intuitive understanding of cutoff regularization. In standard discussion based on quadratic divergences the artificial meaning of a scale of effective theory is given to the regularization parameter $\Lambda$. This effects in the regularization dependence of this kind of analysis. On the contrary, in the Wilsonian method high energy modes are integrated out in a self consistent, regularization independent way and effective theory has a well-defined effective action. Moreover, this treatment is universal and depends very weakly on a preferred UV completion (quantum gravity, string theory, etc.). Main impact on effective action from states with masses greater than the scale of the effective theory can be parametrized by values of couplings of the Wilsonian effective action. Further corrections are highly suppressed as far as heavy masses are separated from the scale of the effective theory.
Given a model where vacuum expectation value of a scalar field can be generated with quantum corrections we can also show how the stability of the effective action looks like from the point of view of Wilsonian running. This has been compared with the standard picture based on running. Since Wilsonian running includes automatically integration of heavy degrees of freedom, the running differs markedly from the version. Nevertheless, similar behaviour can be observed: scalar mass squared parameter and the quartic coupling can change sign from a positive to a negative one due to running. This causes spontaneous symmetry breaking or an instability in the renormalizable part of the potential for a given range of scales. However, care must be taken when drawing conclusions, because of the truncation of higher dimension operators.
While simple cut-off analysis of scalar field models has been performed earlier, the goal of the present note is to use consistently Wilsonian approach and to make clear comparison with the discussion based on the Gell-Mann-Low running.
The paper is organized as follows. In Section \[model\] we specify the model and define truncation. We argue in Subsection \[Lagrangian\] that this model should present behaviour similar to that known from the SM. In Subsection \[truncation\] we show in what sense RGEs for chosen truncation can be thought as an analogue of 1-loop running. In Section \[RGE\] we present calculated RGEs. Their numerical solution is discussed in Section \[numerical\]. Section \[fine\_tuning\] is dedicated to numerical estimation of fine-tuning of parameters of Wilsonian effective action. In Section \[stability\] we discuss the issue of radiative stability of the effective action and in \[conclusion\] we summarize our results. Appendix \[Wilsonian\] contains brief introduction to Wilsonian RGE and Functional Renormalization Group (FRG) methods. The derivatives of loop integrals used during calculations are given in Appendix \[loop\_integrals\]. In Appendix \[matching\] we discuss matching conditions which give parameters of Wilsonian effective action in terms of measurable quantities.
Basic features of the model\[model\]
====================================
Couplings \[Lagrangian\]
------------------------
For the sake of clarity we consider a simple model that exhibits certain interesting features of the SM. The model consists of a massless Dirac fermion $\Psi$ which couples via Yukawa interaction to a real scalar field $\Phi$ with a quartic self-coupling. This Lagrangian takes the form: $$\mathcal{L}=i \overline{\Psi} \slashed{\partial} \Psi + \frac{1}{2} \partial_\mu \Phi \partial^\mu \Phi - \frac{1}{2} M^2 \Phi^2 - Y \Phi \overline{\Psi} \Psi - \frac{\lambda}{4!} {\Phi}^4. \label{Lagrangian_density}$$ The above Lagrangian is symmetric under (chiral) $\mathbb{Z}_2$ which acts on $\Phi$ as $\Phi\to-\Phi$ and on $\Psi$ as $\Psi \to \gamma^5 \Psi$. We consider the case of non-zero vacuum expectation value for the field $\Phi$, which breaks this symmetry spontaneously. In broken symmetry phase the Lagrangian density will take the form: $$\mathcal{L}=i \overline{\Psi} \slashed{\partial} \Psi - m \overline{\Psi} \Psi + \frac{1}{2} \partial_\mu \Phi \partial^\mu \Phi - \frac{1}{2} M^2 \Phi^2 - Y \Phi \overline{\Psi} \Psi -\frac{g}{3!} \Phi^3 - \frac{\lambda}{4!} {\Phi}^4. \label{lagriangian_density_broken}$$ The fermion $\Psi$ allows one to model top quark coupling to Higgs boson, which is known to give the main contribution to quadratic divergences in the mass of the SM scalar and to high-scale instability of the quartic coupling. This model was previously investigated with methods of FRG in [@Clark:1992jr; @Clark:1994ya; @Gies:2013fua] in order to estimate non-perturbative bound on Higgs boson mass. The same issue was discussed in [@Gies:2014xha] with a slightly different Lagrangian. In [@Branchina:2005tu] stability of potential was discussed with the help of the naive cut-off procedure.
Perturbative derivation of RGE\[truncation\]
--------------------------------------------
We have calculated Wilsonian renormalization group equations at the lowest non-trivial order. The Wilsonian action can include an infinite number of non-renormalizable operators, however they are suppressed at the low cut-off scale. Hence our truncation contains the following operators: $$\mathcal{L}_{\Lambda}=i \overline{\Psi}_{\Lambda} \slashed{\partial} \Psi_{\Lambda} + \frac{1}{2} \partial_\mu \Phi_{\Lambda} \partial^\mu \Phi_{\Lambda} - \frac{1}{2} M^2_{\Lambda} {\Phi_{\Lambda}}^2 - Y_{\Lambda} \Phi_{\Lambda} \overline{\Psi}_{\Lambda} \Psi_{\Lambda} - \frac{\lambda_{\Lambda}}{4!} {\Phi_{\Lambda}}^4. \label{truncation_Lagrangian}$$ We define Wilson coefficients $M^2_{\Lambda}$, $Y_{\Lambda}$ and $\lambda_{\Lambda}$ as values of respectively two-, three- and four- point Green’s functions at the kinematic point with vanishing external momenta. We use following graphics for cutoff propagator of the scalar field $\Phi$ $$\parbox{20mm}{
\begin{fmffile}{heavy_scalar_propagator}
\fmfset{arrow_len}{3mm}
\fmfset{thick}{thin}
\begin{fmfgraph*}(20,10)
\fmfleft{i}
\fmfright{o}
\fmf{dbl_dots}{i,o}
\end{fmfgraph*}
\end{fmffile}
} = \frac{\theta_0(p^2-\Lambda^2)}{p^2+M_\Lambda^2}$$ and the fermionic field $\Psi$ $$\parbox{20mm}{
\begin{fmffile}{heavy_fermion_propagator}
\fmfset{arrow_len}{3mm}
\fmfset{thick}{thin}
\begin{fmfgraph*}(20,10)
\fmfleft{i}
\fmfright{o}
\fmf{heavy}{i,o}
\end{fmfgraph*}
\end{fmffile}
} = \frac{\theta_0(p^2-\Lambda^2)}{\slashed{p}+m_\Lambda}.$$
In the diagrams lines
[light\_scalar\_propagator]{}
(20,10)
and
[light\_fermion\_propagator]{}
(20,10)
represent respectively scalar and fermionic low-energy modes.
Lowest order calculation
------------------------
The RGE that we obtain can be thought of as an analogue of 1-loop type RGE. To see the relation more precisely, let us for a moment assume that we add an operator $\Phi^6$ to truncation . Its lowest-order contribution to $\beta$-function for $\lambda$ comes from diagrams with loop propagator starting and ending in the same vertex. For a theory with bare Lagrangian of the form the operator $\Phi^6$ must be, in standard treatment, generated by loop diagrams. Diagrams given in Fig. \[scalar\_loop\_dim\_6\] and Fig. \[fermion\_loop\_dim\_6\] show the lowest order contribution to Wilsonian coefficient of the $\Phi^6$ operator.
These contributions are respectively proportional to ${\lambda}^3$ and ${Y}^6$. Lowest order contribution to $\beta$-function for the coupling $\lambda$ given by the $\Phi^6$ operator comes from loop diagram shown in Fig. \[beta\_dim\_6\].
[dim\_6]{}
(32,32)
Combining diagrams from Figs. \[loop\_dim\_6\] and \[beta\_dim\_6\] we obtain 2-loop diagrams given in Fig. \[full\_dim\_6\].
To sum up, operator $\Phi^6$ is generated at one loop level and the second loop is needed to obtain contribution to the $\beta$-function for $\lambda$. The lowest order non-trivial contributions to type $\beta$-function are of the order ${\lambda}^2$ and ${Y}^4$. The contribution to Wilsonian RGE generated by the operator $\Phi^6$ appears at the 2-loop level in the type RGE.
On the other hand, we neglected all higher dimension operators with derivatives, for example $\partial_\mu \Phi \partial^\mu \Phi \Phi^2$. It is an easy task to check that the diagram from the Fig. \[momentum\_dependent\] gives momentum-dependent[^3] contribution to scalar self-energy.
[Z1-fermion-fermion]{}
(32,20)
More precisely, contribution coming from this diagram depends on momentum squared $p^2$ logarithmically. To recover this dependence from the effective action one needs to consider an infinite number of operators of the form $\Box^n \Phi^2 \colon n=2, \dots$. All these operators are irrelevant near the Gaussian fixed point (the free theory).
Flow equations\[RGE\]
=====================
Calculating RGE using Mathematica
---------------------------------
Our procedure of calculating $\beta$ functions is as follows:
1. Draw all 1PI and counterterms diagrams with certain number of external fields and write down formal expressions for them.
2. Expand loop integrals in series of external momenta and choose interesting terms (for vertex corrections we set external momenta to zero).
3. Represent loop integrals as standard Passarino–Veltman [@Passarino:1978jh] functions.
4. Express Passarino–Veltman functions in terms of functions $I_N$ with IR cutoff $\Lambda$ introduced in [@Bilal:2007ne].
5. Differentiate result with respect to $\Lambda$. This step gives expressions in terms of derivatives of $I_N$ functions.
6. Substitute derivatives of $I_N$ functions by expressions given in Appendix \[loop\_integrals\].
We have assumed such a procedure for two reasons. Firstly, by calculating effective action before differentiation with respect to the cutoff $\Lambda$ we avoid problems connected with a sharp cutoff derivative (however, one must perform detailed calculation of derivatives with respect to external momenta). Secondly, this algorithm is easy to implement using FeynArts [@Hahn:2000kx] and FeynCalc [@Mertig:1990an]. FeynArts can easily generate 1-loop 1PI diagrams for the Lagrangian density, which we use as an input for FeynRules [@Alloul:2013bka]. FeynCalc without any modification of the source code calculates Passarino–Veltman representation of diagrams generated by FeynArts. Finally we substitute (using Mathematica package modelled after ANT package [@Angel:2013hla]) Passarino–Veltman integrals by their derivatives with respect to cutoff $\Lambda$ which we calculated in advance. Calculations performed in Mathematica provide a crosscheck for calculations made by hand.
Using this producer differences between Wilsonian effective action treatment and standard one based on cut-off regularization are easy to observe and origin of terms in Wilsonian RGE coming from different diagrams can be determined.
RGE for the model \[RGE\_eq\]
-----------------------------
As we discuss in the Appendix \[flow\_equations\] it is convenient to express the Wilsonian RGE in terms of dimensionless parameters. We use dimensionless parameters $\nu_{\Lambda}:=\frac{v_{\Lambda}}{{\Lambda}}$, ${\Omega_{\Lambda}}^2:=\frac{{M_{\Lambda}}^2}{{\Lambda}^2}$, $\omega_{\Lambda}:=\frac{m_{\Lambda}}{{\Lambda}}$ and $\gamma_\Lambda:= \frac{g_\Lambda}{{\Lambda}}$. We define $v_{\Lambda}$ by the requirement that the shift $\Phi\mapsto\Phi-v_{\Lambda}$ gives effective action without $\mathbb{Z}_2$-odd terms.
Derivatives with respect to the scale $\Lambda$ of the diagrams from Figs. \[v\_broken\], \[Z1\_broken\], \[Z2\_broken\], \[y\_broken\], \[g3\_broken\] and \[lambda\_broken\] give flow equations respectively for vacuum expectation value of $\Phi$ field, masses of scalar $\Phi$ and fermion $\Psi$, Yukawa coupling, Wilson coefficient $g$ for the operator $\Phi^3$ and the coupling $\lambda$. In addition we determine the scaling of fields from diagrams given in Figs. \[Z1\_broken\] and \[Z2\_broken\]. The RGE read as follows:
[Z2-fermion-scalar]{}
(32,20)
$$\begin{gathered}
{\Lambda} \frac{d \nu_{\Lambda}}{d{\Lambda}}=-{\nu_{\Lambda}}+\frac{1}{{\Omega_{\Lambda}}^2}\left[\frac{8Y_{\Lambda}}{(4 \pi )^2} \frac{\omega_{\Lambda}}{(1+{\omega_{\Lambda}}^2)}-\frac{{{\gamma}_{\Lambda}}}{(4 \pi )^2} \frac{1}{(1+{\Omega_{\Lambda}}^2)}\right]\\
-\frac{\nu_{\Lambda}}{2}\left[ 4\frac{Y_{\Lambda}^2}{(4 \pi) ^2} \frac{\left(3-{\omega_{\Lambda}}^2\right) \left(1+4{\omega_{\Lambda}}^2\right) }{3 \left({\omega_{\Lambda}}^2+1\right)^3}{\Omega_{\Lambda}}^2+\frac{{{\gamma}_{\Lambda}}^2}{3(4 \pi) ^2}\frac{2{\Omega_{\Lambda}}^2-1}{(1+{\Omega_{\Lambda}}^2)^3}\right].\end{gathered}$$
$$\begin{gathered}
{\Lambda} \frac{d \omega_{\Lambda}}{d{\Lambda}}=-{\omega_{\Lambda}}-\frac{2Y_{\Lambda}^2\omega_{\Lambda}}{(4 \pi )^2} \left[\frac{1}{(1+{\Omega_{\Lambda}}^2) \left(1+{\omega_{\Lambda}}^2\right)}-\frac{1}{(1+{\Omega_{\Lambda}}^2)
\left({\Omega_{\Lambda}}^2-{\omega_{\Lambda}}^2\right)}\right.\\
\left.+\frac{1}{\left({\Omega_{\Lambda}}^2-{\omega_{\Lambda}}^2\right)^2}\log
\left(\frac{1+{\Omega_{\Lambda}}^2}{1+{\omega_{\Lambda}}^2}\right)\right]
-Y_{\Lambda} \left[\frac{8Y_{\Lambda}}{(4 \pi )^2} \frac{\omega_{\Lambda}}{(1+{\omega_{\Lambda}}^2)}-\frac{{{\gamma}_{\Lambda}}}{(4 \pi )^2} \frac{1}{(1+{\Omega_{\Lambda}}^2)})\right]\end{gathered}$$
$$\begin{gathered}
{\Lambda} \frac{d {\Omega_{\Lambda}}^2}{d{\Lambda}}=-2 {\Omega_{\Lambda}}^2+4\frac{Y_{\Lambda}^2}{(4 \pi) ^2} \left[\frac{\left(3-{\omega_{\Lambda}}^2\right) \left(1+4{\omega_{\Lambda}}^2\right) }{3 \left({\omega_{\Lambda}}^2+1\right)^3}{\Omega_{\Lambda}}^2-\frac{2{\omega_{\Lambda}}^2-2} {\left({\omega_{\Lambda}}^2+1\right)^2}\right]\\
+\frac{\lambda_{\Lambda}}{(4 \pi) ^2}\frac{1}{ ({\Omega_{\Lambda}}^2+1)}+\frac{{{\gamma}_{\Lambda}}^2}{(4 \pi) ^2}\left[\frac{2}{3}\frac{1}{1+{\Omega_{\Lambda}}^2}-\frac{2}{3}\frac{1}{(1+{\Omega_{\Lambda}}^2)^2}+\frac{{\Omega_{\Lambda}}^2}{(1+{\Omega_{\Lambda}}^2)^3}\right]\\
-{\gamma}_{\Lambda} \left[\frac{8Y_{\Lambda}}{(4 \pi )^2} \frac{\omega_{\Lambda}}{(1+{\omega_{\Lambda}}^2)}-\frac{{{\gamma}_{\Lambda}}}{(4 \pi )^2} \frac{1}{(1+{\Omega_{\Lambda}}^2)})\right]\end{gathered}$$
$$\begin{gathered}
{\Lambda} \frac{d Y_{\Lambda}}{d{\Lambda}}=\frac{Y_{\Lambda}^3}{(4 \pi) ^2} \left[\frac{{\omega_{\Lambda}}^2-1}{ \left({\omega_{\Lambda}}^2+1\right)^2 ({\Omega_{\Lambda}}^2+1)}-\frac{8}{3\left( \omega_{\Lambda}^2+1\right)}+\frac{38}{3 \left(\omega_{\Lambda}^2+1\right)^2}-\frac{6}{3 \left(\omega_{\Lambda}^2+1\right)^3}\right.\\
+ \left.\frac{2}{\left({\Omega_{\Lambda}}^2-{\omega_{\Lambda}}^2\right)^2} \log \left(\frac{{\Omega_{\Lambda}}^2+1}{{\omega_{\Lambda}}^2+1}\right)-\frac{2}{({\Omega_{\Lambda}}^2+1)\left({\Omega_{\Lambda}}^2-{\omega_{\Lambda}}^2\right)}\right]\\
+ \frac{{{\gamma}_{\Lambda}^2} {Y_{\Lambda}} }{(4 \pi) ^2}\left[\frac{1}{3} \frac{1}{\left({\Omega_{\Lambda}}^2+1\right)^2} -\frac{1}{2}\frac{1}{\left({\Omega_{\Lambda}}^2+1\right)^3}\right]-\frac{2{{\gamma}_{\Lambda}}Y_{\Lambda}^2}{(4 \pi) ^2}\frac{\omega_{\Lambda}}{(1+{\Omega_{\Lambda}}^2)(1+{\omega_{\Lambda}}^2)}\end{gathered}$$
$$\begin{gathered}
{\Lambda} \frac{d {{\gamma}_{\Lambda}}}{d{\Lambda}}=-{\gamma}_{\Lambda}+\frac{3{{\gamma}_{\Lambda}} \lambda_{\Lambda}}{(4 \pi) ^2}\frac{1}{{\Omega_{\Lambda}}^2+1}-\frac{7{{\gamma}_{\Lambda}}^3}{2(4 \pi) ^2}\frac{1}{({\Omega_{\Lambda}}^2+1)^3}-\frac{Y_{\Lambda}^3 \omega_{\Lambda}}{(4 \pi) ^2}\frac{16\left(3-{\omega_{\Lambda}}^2\right)}{({\omega_{\Lambda}}^2+1)^3}\\
+\frac{{{\gamma}_{\Lambda}} Y_{\Lambda}^2}{(4 \pi) ^2} \frac{2 \left(3-{\omega_{\Lambda}}^2\right)(1+4{\omega_{\Lambda}}^2)}{\left({\omega_{\Lambda}}^2+1\right)^3}
-\lambda_{\Lambda} \left[\frac{8Y_{\Lambda}}{(4 \pi )^2} \frac{\omega_{\Lambda}}{(1+{\omega_{\Lambda}}^2)}-\frac{{{\gamma}_{\Lambda}}}{(4 \pi )^2} \frac{1}{(1+{\Omega_{\Lambda}}^2)})\right]\end{gathered}$$
$$\begin{gathered}
{\Lambda} \frac{d \lambda_{\Lambda}}{d{\Lambda}}=3 \frac{\lambda_{\Lambda}^2}{(4 \pi )^2}\frac{1}{(1+{\Omega_{\Lambda}}^2)^2}- \frac{2 Y_{\Lambda}^4}{(4 \pi )^2}\left[\frac{1}{\left(1+{\omega_{\Lambda}}^2\right)^2}-8\frac{{\omega_{\Lambda}}^2}{\left(1+{\omega_{\Lambda}}^2\right)^4}\right]
+\frac{8}{3} \frac{\lambda_{\Lambda} Y_{\Lambda}^2 }{(4 \pi )^2} \frac{ \left(3-{\omega_{\Lambda}}^2\right)(1+4{\omega_{\Lambda}}^2)}{\left(1+{\omega_{\Lambda}}^2\right)^3}\\
+\frac{6{{\gamma}_{\Lambda}}^4}{(4 \pi )^2}\frac{1}{({\Omega_{\Lambda}}^2+1)^4}+\frac{2{{\gamma}_{\Lambda}^2} \lambda_{\Lambda}}{(4 \pi )^2}\left[\frac{2}{3}\frac{1}{({\Omega_{\Lambda}}^2+1)^2}-\frac{7}{({\Omega_{\Lambda}}^2+1)^3}\right].\end{gathered}$$
Numerical solutions of RGE\[numerical\]
=======================================
The RGE have been solved numerically. We used matching conditions given in Appendix \[matching\] to compute initial conditions for RGE at $\Lambda=100$ (we use units of GeV through the paper) in terms of physical quantities (see Appendix \[matching\] for definitions) with renormalization scale $\mu=100$. The example solution with values: $m_{ph}=174$, $M_{ph}=125$, $\lambda_{ph}=0.2$, $v_{ph}=264$, ${g}_{ph}=52.8$ and $Y_{ph}=1$ is presented in the Fig. \[numerical\_solution\]. Double-logarithmic plot in the Fig. \[numerical\_solution\] shows parameters of the effective Wilsonian action as functions of the scale $\Lambda$. Orange line represents Yukawa coupling $Y_\Lambda$ which runs typically rather slowly. Gray line corresponds to the quartic coupling $\lambda_{\Lambda}$. This coupling runs faster, because of the contribution from the fermionic loop. Couplings $\omega_\Lambda$ (green line), $\Omega^2_\Lambda$ (red line) and ${\gamma}_\Lambda$ (cyan line) for low values of $\Lambda$ run like relevant couplings due to rescaling, but after reaching scales of the order of the masses, they change their behaviour to a slow running near constant value. The behaviour of the above couplings for high vales of $\Lambda$ is caused by the quadratic divergencies (or more precisely by the same diagrams which generate quadratic divergences). The same behaviour is manifested by vacuum expectation value $\nu_\Lambda$ plotted as the blue line.
![Example of numerical solution of RGE corresponding to: $m_{ph}=174$, $M_{ph}=125$, $\lambda_{ph}=0.2$, $v_{ph}=264$, ${g}_{ph}=52.8$ and $Y_{ph}=1$.\[numerical\_solution\]](plot.pdf){width="\textwidth"}
Solutions with different initial conditions have the same qualitative behaviour. In the Fig. \[numerical\_variation\] we plotted families of solutions with a single parameter varied: the solution with a physical quantity multiplied by $\frac{3}{4}$ and another one with the same parameter multiplied by $\frac{4}{3}$. Reference solution has been plotted as well. Fig. \[numerical\_omega\] shows solutions with initial conditions $m_{ph}=\frac{3}{4}174$ and $m_{ph}=\frac{4}{3}174$. For solutions presented in Figs. \[numerical\_Omega\], \[numerical\_g\] and \[numerical\_l\] we have respectively changed $M_{ph}$, ${g}_{ph}$ and $\lambda_{ph}$. In all plots we use the same colors as in Fig. \[numerical\_solution\] to indicate analogous parameters.
The important observation is that all presented solutions give values very close to each other for $\Lambda$ of the order of $10^6$. Small change of parameters in effective action at high scale generates physical parameters different by orders of magnitude, since the solutions corresponding to different low-scale parameters run very closely to each other when the scale grows. This is the sign that a fine-tuning of parameters in effective action at high scales is required in order to get the prescribed values of physical observables.
![Numerical solution of type RGE for discussed model corresponding to: $m_{ph}=174$, $M_{ph}=125$, $\lambda_{ph}=0.2$, $v_{ph}=264$, ${g}_{ph}=52.8$ and $Y_{ph}=0.5$.\[plot\_Gell-Mann–Low\]](plot_GL.pdf){width="\textwidth"}
The example of numerical solution for type RGE for the same theory is given in the Fig. \[plot\_Gell-Mann–Low\]. Comparing the Fig. \[plot\_Gell-Mann–Low\] with the Fig. \[numerical\_solution\] one finds that the flow of parameters of Wilsonian effective action is much more complicated than the running in method. One should note that Wilsonian RGE accommodate decoupling of massive particles i.e. corrections from particles with masses grater than $\Lambda$ are strongly suppressed.
Fine-tuning\[fine\_tuning\]
===========================
The standard measure $\Delta_{c_i}$ of fine-tuning with respect to the variable $c_i$ is defined as $$\Delta_{c_i} = \frac{\partial \log v^2}{\partial \log {c_i}^2},\label{finetuning_definition}$$ where $c_i$ is a coupling in the model and $v$ is the vacuum expectation value of the field which breaks symmetry spontaneously (here - chiral parity). As a measure of fine-tuning of the whole model we take [@Ellis:1986yg; @Barbieri:1987fn; @Ghilencea:2012gz]: $$\Delta = \left( \sum_i {\Delta_{c_i}}^2\right)^{\frac{1}{2}}.\label{finetuning_definition_2}$$ We have computed $\Delta_{c_i}$ for parameters of the effective action as functions of scale $\Lambda$.
Unfortunately, the effective action for $\Lambda=0$ cannot be obtained by direct numerical integration of RGE. The left-hand side of RGE given in Section \[RGE\] can be rewritten as $$\Lambda \frac{d c_i}{d \Lambda} = \frac{d c_i}{d \log
\left(\Lambda/\Lambda_0\right)},$$ where $c_i$ is a dimensionless parameter and $\Lambda_0$ is the scale at which initial conditions are set. For the purpose of numerical integration couplings are functions of $t:=\log \left(\Lambda/\Lambda_0\right)$. The point $\Lambda=0$ corresponds to the limit $t \to -\infty$ which cannot be reached using numerical methods. For that reason we approximated $v_\Lambda$ for $\Lambda=0$ ($v_0$), by the value at $\Lambda=10^{-4}$ i.e. $v_{10^{-4}}$. We used $\Lambda=10^{-4}$, because this turns out to be the lowest scale which gives $\nu_\Lambda$ safe from numerical errors. On the other hand, $v_\Lambda$ changes very slowly between $\Lambda=1$ and $\Lambda=10^{-4}$, so $v_{10^{-4}}$ should be good approximation for $v_0$. To sum up, we have computed fine-tuning measure by taking numerical derivatives of $\nu_{\Lambda}$ with respect to dimensionless parameters $\omega_\Lambda$, $\Omega^2_\Lambda$, ${\gamma}_\Lambda$, $\lambda_\Lambda$, $Y_\Lambda$ over the range of scales $10<\Lambda<10^6$. Figs. \[finetuning\_omega\], \[finetuning\_Omega\], \[finetuning\_g\], \[finetuning\_l\] and \[finetuning\_Y\] show respectively $\left| \frac{\partial \log \nu^2_{10^{-4}}}{\partial \log{\omega_\Lambda}^2} \right|$, $\left| \frac{\partial \log \nu^2_{10^{-4}}}{\partial \log {\Omega^2_\Lambda}^2} \right|$, $\left| \frac{\partial \log \nu^2_{10^{-4}}}{\partial \log {{\gamma}_\Lambda}^2} \right|$, $\left| \frac{\partial \log \nu^2_{10^{-4}}}{\partial \log{\lambda_\Lambda}^2} \right|$ and $\left| \frac{\partial \log \nu^2_{10^{-4}}}{\partial \log {Y_\Lambda}^2} \right|$.
The spikes visible in the Fig. \[numerical\_finetuning\] are points where derivatives change their signs. Due to logarithmic scale and finite resolution of plots the zeros of derivatives cannot be correctly depicted and are represented in Fig. \[numerical\_finetuning\] as a finite spikes. Even if one of the derivatives vanishes, the others are typically non-zero and stays non-zero and smooth. In Fig. \[finetuning\_all\] the measure as a function of scale $\Lambda$ is shown. The power function $\propto \Lambda^p$ which has been fitted to fine-tuning curve is shown in red in each plot in the Fig. \[numerical\_finetuning\]. The fitted powers $p$ are given in Table \[power\]. The power-like functions have been fitted over the interval $10^{3} \leq \Lambda \leq 10^{6} $ (that is above assumed mass thresholds). The reason is the visible change of the behaviour of the flow of parameters below $10^{3}$. On the other hand, the flows above $10^{3}$ can be smoothly extrapolated to arbitrarily high scales.
---------- ------------------ -------------------- --------------- ------------------- ------------- ----------
$\omega_\Lambda$ $\Omega^2_\Lambda$ ${g}_\Lambda$ $\lambda_\Lambda$ $Y_\Lambda$ combined
p 2.24 2.19 2.20 2.20 2.16 2.19
$\sigma$ 0.05 0.04 0.04 0.04 0.04 0.02
---------- ------------------ -------------------- --------------- ------------------- ------------- ----------
: Estimated powers $p$ and their standard deviations $\sigma$ from the fits to the fine-tuning measures .\[power\]
Vacuum stability\[stability\]
=============================
An interesting issue is the question of spontaneous symmetry breaking and stability of the potential seen from the point of view of the Wilsonian approach. In this approach one starts with a bare action at a high scale and keeps integrating out consecutive shells of momenta, or coarse-graining, to obtain effective action at lower scales. Eventually, the vacuum structure should emerge in the infra-red limit.
In the model studied in this paper one can try to answer the question whether the $\mathbb{Z}_2$ symmetry of can be broken by radiative corrections to the scalar mass parameter. What one finds is that for the low values of $\Omega_{\Lambda}^2$ and high values of Yukawa coupling $Y_{\Lambda}$ in the effective action at high $\Lambda$, scalar mass-squared parameter can flow to a negative value at low $\Lambda$. The change of sign of $M_{\Lambda}^2$ indicates that stable vacuum of the theory must have non-zero vacuum expectation value of scalar field $\Phi$ (as long as the quartic coupling stays positive). Moreover, quartic scalar coupling $\lambda_{\Lambda}$ can run negative for higher $\Lambda$ which shows similar behaviour as the one observed in the type running (Fig. \[plot\_Gell-Mann–Low\]) known from Standard Model. In the context of SM the zero of quartic self-coupling is usually considered as an indication of instability of the electroweak vacuum and of the existence of a second minimum of the scalar potential. In the Wilsonian approach however, simple analysis based on quartic coupling alone is insufficient, because higher dimension operators with higher powers of the scalar field $\Phi$, which we suppressed in our truncation , may dominate scalar potential for large values of $\Phi$. The impact coming from higher dimension operators was previously investigated in [@Gies:2013fua; @Gies:2014xha] and [@Lalak:2014qua]. The results presented in [@Gies:2013fua] validate our observation that quartic coupling constant $\lambda$ can be driven negative in UV by RGE flow. On the other hand analysis of running of higher dimension operators presented in [@Gies:2013fua] supports the statement that higher dimension operators stabilise scalar potential. This is at odds with the explicit calculation of [@Lalak:2014qua] where examples with instabilities induced by higher order operators have been given. To draw strong conclusions one needs a procedure of resummation of possibly large contributions to scalar potential coming from operators with all higher powers of $\Phi$. However, the observed instability of Wilsonian quartic coupling may be seen as an indication of a crossover behaviour at higher scales, since in the region of $\langle \Phi \rangle$ comparable to $\Lambda$ and well below the UV cutoff, one still expects higher-dimension operators to be suppressed with respect to the quartic one by powers of small couplings, since well below the UV cutoff the coefficients of higher-order operators should be dominated by “renormalizable” couplings.
The example of a solution demonstrating such features is plotted in the Fig. \[numerical\_instability\]. For this solution scalar mass parameter $\Omega_{\Lambda}^2$ vanishes at he scale $\Lambda=2.67 \times 10^4$ and quartic coupling $\lambda_{\Lambda}$ has a zero at $\Lambda = 1.07 \times 10^6$. While investigating features of this solution one can notice a strong dependence of the scale of symmetry breaking on the value of Yukawa coupling $Y_{\Lambda}$. This fine-tuning problem makes one choose very precisely the initial condition for Yukawa coupling in order to make the symmetry breaking scale low.
![Example of numerical solution of RGE in which radiative symmetry breaking takes place. Plot corresponds to values $Y_{\Lambda}=1.461$, $M_{\Lambda}^2= 5 \times 10^{10}$, $\lambda_{\Lambda}=0.1$ and $v_{\Lambda}={g}_{\Lambda}=m_{\Lambda}=0$ at $\Lambda=10^6$.\[numerical\_instability\]](plotstability.pdf){width="\textwidth"}
The issue of spontaneous symmetry breaking can be studied with the help of the Fig. \[numerical\_Omegavslambda\] in which numerical solutions with different initial conditions are projected on the plane spanned by $\lambda_{\Lambda}$ and $\Omega^2_{\Lambda}$. All solutions given there have $Y_{\Lambda}=1.461$, $\lambda_{\Lambda}=0.1$ and $v_{\Lambda}={g}_{\Lambda}=m_{\Lambda}=0$ as a initial conditions set at $\Lambda=10^6$, but initial value of $\Omega^2_{\Lambda}$ varying. From the Fig. \[numerical\_Omegavslambda\] one can see that if for any scale $\Lambda$ couplings will be lower than certain critical value $\Omega^2_{cr}$ then $\Omega^2$ will run negative in the IR and chiral parity will be spontaneously broken. Moreover as can be seen in the critical value $\Omega^2_{cr}$ is rather sensitive to Yukawa coupling $Y$. From the behaviour shown in the Fig. \[numerical\_OmegavsY\] one concludes that $\Omega^2_{cr}$ decreases when the value of $Y$ increases and for any value of $\Omega^2$ there exist a critical value of Yukawa coupling $Y_{cr}$. Once $Y_{cr}$ is exceeded, the radiative spontaneous symmetry breaking appears. Hence fourth quadrant of the Fig. \[numerical\_OmegavsY\] gives direct evidence of the Coleman–Weinberg mechanism at work.
![Flow of $\lambda_{\Lambda}$ and $\Omega^2_{\Lambda}$ in the range of parameters where spontaneous symmetry breaking takes place. Plot corresponds to $Y_{\Lambda}=1.461$, $\lambda_{\Lambda}=0.1$, $v_{\Lambda}={g}_{\Lambda}=m_{\Lambda}=0$ and varying $M_{\Lambda}^2$ at $\Lambda=10^6$. Moving along the lines in the direction of arrows corresponds to decreasing scale $\Lambda$.\[numerical\_Omegavslambda\]](plotblack1.pdf){width="\textwidth"}
![Flow of $Y_{\Lambda}$ and $\Omega^2_{\Lambda}$ in the range of parameters where spontaneous symmetry breaking takes place. Plot corresponds to $\lambda_{\Lambda}=0.1$, $M_{\Lambda}^2= 5 \times 10^{10}$, $v_{\Lambda}={g}_{\Lambda}=m_{\Lambda}=0$ and varying $Y_{\Lambda}$ at $\Lambda=10^6$. Moving along the lines in the direction of arrows corresponds to decreasing scale $\Lambda$.\[numerical\_OmegavsY\]](plotblack2.pdf){width="\textwidth"}
Conclusions\[conclusion\]
=========================
In this paper we have used Wilsonian effective action to investigate fine-tuning and vacuum stability in a simple model exhibiting spontaneous breaking of a discrete symmetry and large fermionic radiative corrections which are able to destabilise quartic scalar self-coupling. Regulator independence of Wilsonian RG provides consistent and well-defined procedure to analyse the issue of quadratic divergences. In the simplified model simulating certain features of SM the Wilsonian renormalization group equations have been studied. We have explained in what sense the truncation adopted in the calculations corresponds to 1-loop running. In fact, in both cases the approximations used correspond to lowest-order quantum effects within each renormalization scheme.
Numerical solutions of RGE have revealed interesting behaviour, caused by the same diagrams that generate quadratic divergences. An operator relevant near Gaussian fixed point (for example mass parameter for scalar particles) can run like marginal or even irrelevant operator, rather than decrease with growing scale. Furthermore solutions for different physical quantities flow close to each other with increasing scale. The flow in the direction of some common value indicates severe fine-tuning. In such a situation small changes of boundary values of parameters at high scale produce very different vacuum expectation values for scalar field and other measurable quantities at low energies. We have estimated fine-tuning as a function of scale of the effective theory. For all parameters the adopted measure of fine-tuning grows rapidly. Power-like functions fitted to the obtained fine-tuning curves grow faster than $\Lambda^2$ over the range of scales taken into account in the study.
It should be stressed that Wilsonian RGEs, in contrast to running, accommodate automatically decoupling of heavy particles. As noticed in Section \[numerical\], the contribution to flow coming from particles with masses $M_{heavy}$ greater than the scale $\Lambda$ of the effective action is strongly suppressed. The main contributions to the interactions generated by heavy states are integrated out during calculation of the effective action for $\Lambda \ll M_{heavy}$ and are included in the effective Wilson coefficients. These properties of Wilsonian RGE explain why fine-tuning of Wilson parameters is so interesting. Let us imagine a more fundamental theory (say theory A) in which the SM is embedded. If one calculates in theory A effective action for the scale $\Lambda$ below, but not very much, the lowest mass of the particles from the New Physics sector, one obtains certain values of the Wilson coefficients $c^{A}$. On the other hand one can extrapolate the flow obtained from the SM to the scale $\Lambda$ and calculate the Wilson coefficients $c^{SM}$. Couplings computed in both ways should match, that is $c^{A}_{\Lambda} =c^{SM}_{\Lambda}$. If the couplings $c^A$ are different from $c^{SM}$ at the level of fine-tuning $\Delta c$, that is $c^{A}_{\Lambda} (1 \pm \Delta c ) = c^{SM}_{\Lambda}$, theory A will produce IR effective action completely different from the SM.
We have studied the issue of spontaneous symmetry breaking due to radiative corrections in the Wilsonian framework. We have demonstrated that there exists a critical value $\Omega^2_{cr}$ below which $\Omega^2$ runs negative in the IR and symmetry becomes spontaneously broken. Moreover, critical value $\Omega^2_{cr}$ is sensitive to Yukawa coupling $Y$. One can see that $\Omega^2_{cr}$ decreases when the value of $Y$ increases and for any value of $\Omega^2$ there exist a critical value of Yukawa coupling $Y_{cr}$. Once $Y_{cr}$ is exceeded, the radiative spontaneous symmetry breaking appears, which is a direct evidence of Coleman–Weinberg mechanism at work.
[**Acknowledgements**]{}
This work has been supported by National Science Center under research grant DEC-2012/04/A/ST2/00099and partially under research grant DEC-2011/01/M/ST2/02466. Authors (Z.L.) are grateful to the Mainz Institute for Theoretical Physics (MITP) for its hospitality and its partial support during the completion of this work.
Wilsonian RGE\[Wilsonian\]
==========================
The aim of this section is to present brief introduction to the methods of Wilsonian RGE and to set the notation. We will define most of the terms that we used in the text of this paper. Method of discrete Renormalization Group Equations has been presented in Wilson’s and Kogut’s review [@Wilson:1973jj], a short introduction can be found in book of Peskin and Schroeder [@0201503972]. RGE for Legendre effective action have been derived in [@Wetterich:1992yh; @Bonini:1992vh; @Morris:1993qb]. Interesting formal developments on FRG (Functional Renormalisation Group) are presented in [@Rosten:2010vm]. Introductory review on FRG in gauge theories has been given in [@Gies:2006wv]. Further references can be found in the book [@Kopietz:2010zz] as well as in [@Polonyi:2001se] and [@Bagnuls:2000ae].
Wilsonian effective action\[Wilsonian\_effective\_action\]
----------------------------------------------------------
The fundamental object of QFT is generating functional $\mathcal{Z}[J]$ which is given in path integral formalism by formal expression[^4] $$\mathcal{Z}[J] = \int \mathcal{D} \Phi e^{-S_E[\Phi] + \int d^4 x_E J \Phi},\label{generating_functional}$$ where $S_E$ is Euclidean action for the $\Phi$ field(s) and $J$ is source(s). We can rewrite in equivalent form using Fourier transform[^5] $\hat{\Phi}$ $$\mathcal{Z}[\hat{J}] = \int \mathcal{D} \hat{\Phi} e^{-S_E[\hat{\Phi}] + \int \frac{d^4 p }{(2 \pi)^4} \hat{J} \hat{\Phi}} = \int \prod_p d \hat{\Phi} (p) e^{-S_E[\hat{\Phi}(p)] + \int \frac{d^4 p }{(2 \pi)^4} \hat{J} (p) \hat{\Phi}(p)}. \label{generating_functional_momentum}$$ We introduce projection operator $$\left( b_\Lambda \hat{F} \right) (p) = \theta_0 (p^2 - \Lambda^2) \hat{F} (p),$$ where $\theta_0$ is defined as follows $$\theta_0 (x) = \left\{\begin{array}{ll} 1 & \textrm{for } x > 0 \\ 0 & \textrm{for } x \le 0\end{array} \right. . \label{Heaviside}$$ We can divide integration variables $\hat{\Phi} (p)$ into two classes: $$\begin{aligned}
\hat{\Phi}_{\le} (p) &= (1- b_\Lambda) \hat{\Phi}(p), & \hat{\Phi}_> (p)&=b_\Lambda \hat{\Phi}(p).\end{aligned}$$ It is easy to find that $$\prod_p d \hat{\Phi}(p)=\left(\prod_{p \colon p^2 \le \Lambda^2} d \hat{\Phi}_{\le}(p)\right) \left(\prod_{p' \colon p'^2 > \Lambda^2} d \hat{\Phi}_>(p')\right).$$ Action $S_E[\hat{\Phi}]$ can be rewritten as $$S_E[ \hat{\Phi}_{\le} + \hat{\Phi}_>] = S_E[ \hat{\Phi}_{\le}]+S_E[ \hat{\Phi}_>]+\mathcal{S}_E[ \hat{\Phi}_{\le},\hat{\Phi}_>],$$ where we denoted by $\mathcal{S}_E[ \hat{\Phi}_{\le},\hat{\Phi}_>]$ the part which depends on both $ \hat{\Phi}_{\le}$ and $\hat{\Phi}_>$. In this notation generating functional $\mathcal{Z}[\hat{J}]$ takes form $$\begin{gathered}
\mathcal{Z}[ \hat{J}_{\le},\hat{J}_>] = \int \!\!\! \prod_{p \colon p^2 \le \Lambda^2} d \hat{\Phi}_{\le}(p) e^{-S_E[ \hat{\Phi}_{\le}(p)] + \int \frac{d^4 p }{(2 \pi)^4} \hat{J}_{\le}(p) \hat{\Phi}_{\le}(p)}\\
\cdot \int \!\!\! \prod_{p' \colon p'^2 > \Lambda^2} d \hat{\Phi}_>(p') e^{-S_E[\hat{\Phi}_>(p') ]-\mathcal{S}_E[\hat{\Phi}_{\le}(p),\hat{\Phi}_>(p') ] + \int \frac{d^4 p }{(2 \pi)^4} \hat{J}_>(p') \hat{\Phi}_>(p') }\end{gathered}$$ where we have used orthogonality of modes with different momenta and have defined $$\begin{aligned}
\hat{J}_{\le} (p) &= (1- b_\Lambda) \hat{J}(p), & \hat{J}_> (p)&=b_\Lambda \hat{J}(p).\end{aligned}$$ Let us now imagine for a moment that we integrate over all $\hat{\Phi}_>$ in generating functional $\mathcal{Z}[ \hat{J}_{\le},\hat{J}_>]$. Then we will obtain $$\mathcal{Z}[ \hat{J}_{\le},\hat{J}_>] = \int \!\!\! \prod_{p \colon p^2 \le \Lambda^2} d \hat{\Phi}_{\le}(p) e^{-S_E[\hat{\Phi}_{\le}]+\int \frac{d^4 p }{(2 \pi)^4} \hat{J}_{\le}(p) \hat{\Phi}_{\le}(p)} \mathcal{Z}_> [\hat{J}_>].$$ If we restricts ourselves to generating functional $\mathcal{Z}_{\le}[\hat{J}_{\le}]$ for Greens functions of low-energy modes $\hat{\Phi}_{\le}$ then it will get form: $$\mathcal{Z}_{\le}[\hat{J}_{\le}] = \int \!\!\! \prod_{p \colon p^2 \le \Lambda^2} d \hat{\Phi}_{\le} (p) e^{-S_E[\hat{\Phi}_{\le}]+\log \mathcal{Z}_> [0] +\int \frac{d^4 p }{(2 \pi)^4} \hat{J}_{\le}(p) \hat{\Phi}_{\le}(p)}. \label{generating_functional_integrated}$$ Action $S_\Lambda [\hat{\Phi}_{\le}] := S_E[\hat{\Phi}_{\le}]-\log \mathcal{Z}_> [0]$ is called Wilsonian effective action for scale $\Lambda$.
Flow equations\[flow\_equations\]
---------------------------------
We define generating functional $W_\Lambda [\hat{J}]$ by equation $$e^{W_\Lambda[\hat{J}]}:= \int \!\!\! \prod_{p \colon p^2 > \Lambda^2} d \hat{\Phi}_{>} (p) e^{-S_E[\hat{\Phi}_{\le}+\hat{\Phi}_>] +\int \frac{d^4 p }{(2 \pi)^4} \hat{J}(p) \left( \hat{\Phi}_{\le}(p) + \hat{\Phi}_{>}(p)\right)}.\label{W_definition}$$ If $S_E$ can be decomposed as $$S_E[\hat{\varphi}] =: S_{I}[\hat{\varphi}] + S_{G}[\hat{\varphi}] \label{interaction}$$ with some arbitrary interaction part $S_{I}$, and a Gaussian part of the general form $$S_{G}[\hat{\varphi}]=:- \int \frac{d^4 p}{(2 \pi)^4} \frac{1}{2}\hat{\varphi}(-p) {G}^{-1} \hat{\varphi}(p). \label{Gaussian}$$ then it can be showed that following definition $$e^{W_\Lambda[\hat{J}]} = \int \prod_p d \hat{\Phi} (p) e^{-S_I[\hat{\Phi}(p)] + \int \frac{d^4 p}{(2 \pi)^4}\left[ \frac{1}{2} \hat{\Phi} (-p) \theta_0 (p^2-\Lambda^2) {G}^{-1}(p) \hat{\Phi}(p) + \hat{J} (p) \hat{\Phi}(p)\right]}. \label{effective_partition}$$ is equivalent to .
For calculating $\beta$-functions, it is convenient to consider Legendre transform $\Gamma_\Lambda[\phi]$ of generating functional $W_\Lambda[J]$. Before we define Wilsonian RGE we should reduce redundant degrees of freedom in $\Gamma_\Lambda [\phi]$. Firstly it can happen that $$\left.\frac{\delta \Gamma_\Lambda}{\delta \phi} \right|_{\phi=0} \neq 0.$$ In such a case we shift the the field $\phi\mapsto \phi_0 + \varphi$ by the solution $\phi_0$ of equation $\left.\frac{\delta \Gamma_\Lambda}{\delta \phi} \right|_{\phi=\phi_0} = 0$. Secondly there are redundant degree of freedom corresponding to the rescaling of the field $\varphi$. We assume that $\varphi$ has canonical kinetic term in $\Gamma_\Lambda[\varphi]$. Wilsonian RGE are differential equations describing change of $\Gamma_\Lambda[\varphi]$ due to the change of scale $\Lambda$.
Legendre effective action $\Gamma_\Lambda[\varphi]$ can be expanded in Taylor series in powers of field $\varphi$ $$\Gamma_\Lambda[\hat{\varphi}] = \sum_{n \in \mathbb{N}} \frac{1}{n!} \hat{\varphi}^n \frac{\delta^n}{\delta \hat{\varphi}^n} \Gamma_\Lambda.$$ Each coefficient of Taylor expansion can be expanded in derivatives of $\varphi$ $$\frac{\delta}{\delta \hat{\varphi}(p_1)} \cdots \frac{\delta}{\delta \hat{\varphi}(p_n)} \Gamma_\Lambda= \left(\prod_{i=1}^n \int \frac{d^4 p_i}{(2 \pi)^4}\right) \delta \left(\sum_{i=1}^n p_i\right) \sum_{i_1, \dots i_n \in \mathbb{N}} C^{(n)}_{i_1, \dots, i_n} p_1^{i_1} \cdots p_n^{i_n}.$$ Coefficients $C^{(n)}_i$ are called Wilson coefficients. It is convenient to use dimensionless parameters $$c^{(n)}_i =C^{(n)}_i \Lambda^{-\dim C^{(n)}_i}$$ where $\dim C^{(n)}_i$ is canonical dimension of the coefficient. RGE expressed in terms of dimensionless parameters is dynamical system.
Legendre effective action $\Gamma_\Lambda[\varphi]$ in the limit $\Lambda \to \infty$ is called classical action. This limit usually does not exist in strict mathematical sense, because Wilson coefficients are typically divergent when $\Lambda \to \infty$. Classical action for theories that are not finite should be thought as a formal expression which must be renormalized.
Perturbation theory
-------------------
Wilsonian effective action and flow equations can be approximately derived in perturbation theory and expressed by Feynman diagrams. Starting with the Wilsonian effective action $W_\Lambda[\hat{J}]$ (so Legendre effective action $\Gamma_\Lambda[\varphi]$ too) can be computed in perturbation theory in a manner analogous to computation of the generating functional for connected Green’s functions $W[\hat{J}]$ (1PI effective action $\Gamma_{\textrm{1PI}}[\varphi]$), but with a propagator substituted by a propagator with a cutoff: $$G_\Lambda (p) := \theta_0 (p^2-\Lambda^2) G (p).$$ Legendre effective action $\Gamma_\Lambda[\varphi]$, like the 1PI effective action $\Gamma_{\textrm{1PI}}[\varphi]$, has usually infinitely many terms. Wilsonian RGE for a generic theory is the set of infinitely many coupled equations for infinitely many couplings which rarely can be solved exactly. For practical purposes one typically have to use certain approximation strategy. The most common consists in truncation of effective action and in considering only a subset of Wilson coefficients which are relevant for the problem in question.
Derivatives of loop integrals with IR cutoff\[loop\_integrals\]
===============================================================
During calculation of RGE we used following derivatives of integrals $I_N(R)$ defined in [@Bilal:2007ne]. $$\frac{\partial}{\partial \Lambda} I_1(R) = \Lambda^2 \frac{1}{\Lambda} \frac{-2i}{(4 \pi)^2} \frac{1}{1+\frac{R}{\Lambda^2}}+{\ensuremath{\operatorname{\mathcal{O}}\left(\varepsilon\right)}}$$ $$\frac{\partial}{\partial \Lambda} I_2(R) = \frac{1}{\Lambda} \frac{-2i}{(4 \pi)^2} \frac{1}{(1+\frac{R}{\Lambda^2})^2}+{\ensuremath{\operatorname{\mathcal{O}}\left(\varepsilon\right)}}$$ $$\frac{\partial}{\partial \Lambda} I_N(R) = \Lambda^{-2(N-2)} \frac{1}{\Lambda} \frac{-2i}{(4 \pi)^2}\frac{1}{(1+\frac{R}{\Lambda^2})^N}+{\ensuremath{\operatorname{\mathcal{O}}\left(\varepsilon\right)}}$$
1-loop matching conditions\[matching\]
======================================
We used 1-loop matching conditions in order to express Wilsonian coefficients of the effective action at the matching scale $\Lambda$ by physical quantities. The later can be formally obtained from the effective action in the limit of the scale $\Lambda \to 0$, which however is out of reach of numerical methods as discussed in Section \[fine\_tuning\].
FeynArts/FeynCalc packages for Mathematica were employed to calculate matching conditions. In addition we used ANT package modified to become compatible with FeynCalc. Moreover, we added infinite terms listed in [@Angel:2013hla] to the expressions in this package in order to check consistency of our calculation. We have written an analogous package that contains loop integrals in dimensional regularization with IR cutoff.
We expressed parameters of the effective action in terms of physical vacuum expectation value $v_{ph}$ of $\Phi$ field, pole masses $m_{ph}$, $M_{ph}$ of respectively $\Psi$ and $\Phi$ fields. We have defined Yukawa coupling $Y_{ph}$ and trilinear coupling ${g}_{ph}$ as a 1PI part of scattering amplitude at kinematic point with incoming momentum square equal to $\mu^2$ and outgoing momenta squares equal to $\mu^2$ and $0$. In definition of $Y_{ph}$ we assumed that outgoing fermion has non-zero momentum. Coupling $\lambda_{ph}$ is defined at kinematic point with Mandelstam variables equal to $\mu^2$.
Diagrams that contribute to matching conditions are discussed in Section \[RGE\] and are presented in the Figs. from \[v\_broken\] to \[lambda\_broken\].
[^1]: Tomasz.Krajewski@fuw.edu.pl
[^2]: Zygmunt.Lalak@fuw.edu.pl
[^3]: Diagram from the Fig. \[momentum\_dependent\] depends on external momentum of a scalar if we integrate over all modes or just a part of fermionic modes in the loop.
[^4]: It is convenient to use Euclidean space in the context of Wilsonian RGE, so we will assume for a moment that we use Euclidean formulation of QFT.
[^5]: We will denote Fourier transform of field $F(x)$ as $\hat{F}(p)$.
|
---
abstract: |
We discuss the asymptotic lower bound on the inner radius of nodal domains that arise from Laplacian eigenfunctions $ {\varphi_{\lambda}}$ on a closed Riemannian manifold $ (M,g) $.
First, in the real-analytic case we present an improvement of the currently best known bounds, due to Mangoubi ([@Man1]). Furthermore, using recent results of Hezari ([@Hezari], [@Hezari2]) we obtain $ \log $-type improvements in the case of negative curvature and improved bounds for $ (M,g) $ possessing an ergodic geodesic flow.
Second, we discuss the relation between the distribution of the $ L^2 $ norm of an eigenfunction $ {\varphi_{\lambda}}$ and the inner radius of the corresponding nodal domains. In the spirit of [@Colding-Minicozzi] and [@Jacobson-Mangoubi], we consider a covering of good cubes and show that, if a nodal domain is sufficiently well covered by good cubes, then its inner radius is large.
author:
- Bogdan Georgiev
title: On the lower bound of the inner radius of nodal domains
---
Introduction {#sec:Intro}
============
Let $ M $ be a closed Riemannian manifold of dimension $n \geq 3$ with metric $ g $ and denote by ${\varphi_{\lambda}}$ an eigenfunction of the Laplacian ${\Delta}$ of $ M $, corresponding to the eigenvalue $\lambda$. Assume that $ \| {\varphi_{\lambda}}\|_{L^2} = 1 $. We are interested in the geometry of nodal domains in the high-energy limit, i.e. as $ \lambda \rightarrow \infty $. For a readable and far-reaching survey we refer to [@Z] and [@Z2].
By a result of Dan Mangoubi ([@Man1]), it is known that for a nodal domain $ \Omega_\lambda $, corresponding to $ {\varphi_{\lambda}}$, the following asymptotic estimate holds: $$\label{eq:Asymptotic-Bounds-Inner-Radius}
\frac{c_1}{\lambda^{\frac{n-1}{4}+\frac{1}{2n}}} \leq \operatorname{inrad}(\Omega_\lambda) \leq \frac{c_2}{\sqrt{\lambda}},$$ where $ c_{1,2} $ depend on $ (M, g) $.
In particular, the asymptotic estimates are sharp in the case of a Riemannian surface, i.e. the inner radius of a nodal domain is comparable to the wavelength $\frac{1}{\sqrt{\lambda}} $. A natural question is whether the mentioned lower bound is optimal also for higher dimensions.
Our first result concerns an improvement in the real-analytic case.
\[thm:Inradius-Real-Analytic\] Let $ (M,g) $ be a real-analytic closed manifold of dimension at least $ 3 $. Let $ {\varphi_{\lambda}}$ be an eigenfunction of the Laplace operator $ \Delta $ and $ \Omega_\lambda $ be a nodal domain of $ {\varphi_{\lambda}}$. Then, there exist constants $ c_1 $ and $ c_2 $ which depend only on $ (M,g) $, such that $$\frac{c_1}{\lambda} \leq \operatorname{inrad}(\Omega_\lambda) \leq \frac{c_2}{\sqrt{\lambda}}$$ Moreover, if $ {\varphi_{\lambda}}$ is positive (resp. negative) on $ \Omega_\lambda $, then a ball of this radius can be inscribed within a wavelength distance to a point where $ {\varphi_{\lambda}}$ achieves its maximum (resp. minimum) on $ \Omega_\lambda $.
We note that Theorem \[thm:Inradius-Real-Analytic\] improves Mangoubi’s estimates for dimensions $ n \geq 5 $.
The argument consists of two ingredients.
First, we observe that one can almost inscribe a wavelength ball in the nodal domain up to a small in volume error set. In fact, a well-known result due to Lieb ([@L]) states that for arbitrary domains $ \Omega $ in $ \mathbb{R}^n $ one can find almost inscribed balls of radius $ \frac{1}{\sqrt{\lambda_1(\Omega)}} $. Furthermore, we refer to [@MS] for a result in this spirit stated in terms of capacities.
In [@Georgiev-Mukherjee] we were able to obtain a refinement of Lieb’s result, specifying the location where a ball of wavelength size can almost be inscribed, as well as the way the error set grows in volume nearby. In particular, wavelength balls can almost be inscribed at points where $ {\varphi_{\lambda}}$ achieves $ \| {\varphi_{\lambda}}\|_{L^\infty(\Omega)} $.
Second, one would like to somehow rule out the error set that may enter in the almost inscribed ball near a point of maximum $ x_0 \in \Omega_\lambda $. One way to argue is as follows. Being in the real-analytic setting, eigenfunctions resemble polynomials of degree $ \sqrt{\lambda} $. This observation was utilized in the works of Donnelly-Fefferman ([@DF2]) and Jakobson-Mangoubi ([@Jacobson-Mangoubi]). What is more, if one takes the unit cube and subdivide it into wavelength-sized small cubes, then these polynomials will be close to their average on most of the small cubes. This implies that the growth of eigenfunctions is controlled on most wavelength-smaller cubes. Now, roughly speaking, we start from a wavelength cube at $ x_0 $ and rescale to the unit cube $ I^n $. Further, $ I^n $ is subdivided into wavelength cubes $ Q_\nu $ and hence most of them will be good. But, if the error set intersects each $ Q_\nu $ deeply it will gain sufficient volume to contradict the volume decay of the first step. This means that there is a $ Q_\nu $ which is not deeply intersected by the error set.
Moreover, utilizing some recent results of Hezari ([@Hezari]) we get that, if one assumes in addition that $ (M,g) $ is negatively curved, then the inradius improves by a factor of $ \log \lambda $. A similar argument works also for $ (M,g) $ with ergodic geodesic flow.
In this note, we also investigate the effect of the moderate growth of $ {\varphi_{\lambda}}$ on a nodal domain’s inner radius. To this end, we exploit a covering by good/bad cubes, inspired by [@Colding-Minicozzi] and [@Jacobson-Mangoubi]. Let us fix a finite atlas $(U_i, \phi_i)$ of $ M $, such that the transition maps are bounded in $C^1$-norm and the metric on each chart domain $U_i$ is comparable to the Euclidean metric in $\mathbb{R}^n$: $$\label{eq:Metric-Comparability}
\frac{1}{4} e_i \leq g \leq 4 e_i,$$ where we have denoted $ e_i := \phi_i^{*}e $ with $ e $ being the standard Euclidean metric.
We can arrange that $M$ is covered by cubes $K_i \subseteq U_i $, where we decompose $K_i$ into small cubes $K_{ij}$ of size $h$ (to be determined later). Throughout we will denote by $\delta K_{ij}$ the concentric cube, scaled by some fixed scaling factor $ \delta > 1 $.
\[def:Good-Bad-Cubes\] A cube $K_{ij}$ is called $\gamma$-good, if $$\int_{\delta K_{ij}} {\varphi_{\lambda}}^2 \leq \gamma \int_{K_{ij}} {\varphi_{\lambda}}^2.$$ Otherwise, we say that $K_{ij}$ is $\gamma$-bad. We also denote by $\Gamma$ the union of all good cubes (i.e. the good set) and $\Xi := M \backslash \Gamma $.
We have the following
\[th:Main-Theorem\]
Let $ (M,g) $ be a smooth closed Riemannian manifold of dimension at least $ 3 $. Let $ \Omega_\lambda $ be a fixed nodal domain of the eigenfunction $ {\varphi_{\lambda}}$.
Then $$\operatorname{inrad}(\Omega_\lambda) \geq C \gamma^{\frac{2-n}{n}} \tau^{\frac{1}{2}} \lambda^{-\frac{1}{2}},$$ where $ \tau := \int_{\Omega_\lambda \cap \Gamma} \space {\varphi_{\lambda}}^2 / \int_{\Omega_\lambda} {\varphi_{\lambda}}^2 $ and $ C = C(M,g) $.
Roughly, Theorem \[th:Main-Theorem\] implies that, if the bulk of the $ L^2 $ norm over the nodal domain is contained in good cubes, then the nodal domain possesses large inner radius.
In Section \[sec:Comments\] we deduce the following corollaries:
\[cor:Energy-Estimate\] Let $ (M,g) $ be a smooth closed Riemannian manifold of dimension at least $ 3 $. For a nodal domain $ \Omega_\lambda $ of $ {\varphi_{\lambda}}$, one has $$\operatorname{inrad}(\Omega_\lambda) \geq C \frac{\| {\varphi_{\lambda}}\|_{L^2(\Omega_\lambda)}^{\frac{2(n-2)}{n}}}{\sqrt{\lambda}},$$ with $ C = C(M, g)$.
Note that the inequality in Corollary \[cor:Energy-Estimate\] is useful only in dimensions $ 3 $ and $ 4 $, as an application of the standard Hölder inequality gives: $$\operatorname{inrad}(\Omega_\lambda) \geq C \frac{\| {\varphi_{\lambda}}\|_{L^2(\Omega_\lambda)}}{\sqrt{\lambda}},$$ which is sharper in higher dimensions.
Moreover, we note that as a by-product we obtain
\[cor:Fat-Nodal-Domain\] Let $ (M,g) $ be a smooth closed Riemannian manifold of dimension at least $ 3 $. There exists a nodal domain of ${\varphi_{\lambda}}$, denoted by ${\Omega_{\varphi_{\lambda}}^*}$, such that $$\operatorname{inrad}({\Omega_{\varphi_{\lambda}}^*}) \asymp \frac{1}{\sqrt{\lambda}},$$ In other words, there exist constants $ C_1, C_2 $, depending on $ (M, g) $, such that $$\frac{C_1}{\sqrt{\lambda}} \leq \operatorname{inrad}({\Omega_{\varphi_{\lambda}}^*}) \leq \frac{C_2}{\sqrt{\lambda}}$$ for $ \lambda $ large enough.
As communicated by Dan Mangoubi, Corollary \[cor:Fat-Nodal-Domain\] also follows directly by looking at a point, where $ {\varphi_{\lambda}}$ achieves its maximum over $ M $, and further using standard elliptic estimates. Indeed, by rescaling we may assume that $ {\varphi_{\lambda}}(x_0) = \| {\varphi_{\lambda}}\|_{L^\infty(M)} = 1 $. Elliptic estimates then imply that $ \| \nabla {\varphi_{\lambda}}\|_{L^\infty(M)} \leq C \sqrt{\lambda} $ which shows that there is a wavelength inscribed ball at $ x_0 $.
We note that the scaling factor $ \delta > 0 $ also enters in the constants above - however, in our discussion it is fixed and later explicitly set as $ \delta := 16\sqrt{n} $ for technical reasons.
The plan for the rest of this note goes as follows.
In Section \[sec:Real-Analytic-Case\] we provide the details behind the proof of Theorem \[thm:Inradius-Real-Analytic\], following the plan outlined above. We end the Section by discussing the case of $ (M,g) $-negatively curved.
In Section \[sec:Asymmetry-History\] we recall the essential steps behind the lower bound in (\[eq:Asymptotic-Bounds-Inner-Radius\]). Roughly speaking, one cuts a nodal domain $\Omega_\lambda$ into small cubes of size $\operatorname{inrad}(\Omega_\lambda)$. Then, among these, one finds a special cube $ K_{i_0 j_0} $, over which the Rayleigh quotient is carefully estimated. Here an essential role is played by how one compares the volumes of $ \{{\varphi_{\lambda}}> 0 \}$ and $ \{ {\varphi_{\lambda}}< 0 \}$ in the special cube (also known as asymmetry estimates).
The motivation behind Theorem \[th:Main-Theorem\] comes from the question, whether in the special cube $ K_{i_0 j_0} $ one has $ \operatorname{Vol}(\{{\varphi_{\lambda}}> 0 \}) \sim \operatorname{Vol}(\{{\varphi_{\lambda}}< 0 \})$. Having this would imply that the inner radius is comparable to the wavelength $\frac{1}{\sqrt{\lambda}} $, i.e. the optimal asymptotic bound.
We also introduce a covering by small good and bad cubes, which arises when one investigates the way the local $L^2$-norm of an eigenfunction ${\varphi_{\lambda}}$ grows (w.r.t. the domain of integration). Good cubes represent places of controlled $L^2$-norm growth. The motivation behind this consideration is the fact that the volumes of the positivity and negativity set of ${\varphi_{\lambda}}$ in a good cube are comparable, i.e. their ratio is bounded by constants ([@Colding-Minicozzi]).
In Section \[sec:Proof-Main\] we show the statement of Theorem \[th:Main-Theorem\] and its Corollaries $ \ref{cor:Fat-Nodal-Domain} $ and $ \ref{cor:Energy-Estimate} $.
We end the discussion by making some further comments in Section \[sec:Comments\].
Acknowledgements
----------------
I am grateful to Henrik Matthiesen, Mayukh Mukherjee, Alexander Logunov, Dan Mangoubi, Hamid Hezari and Steve Zelditch for their encouragement, comments and valuable remarks. I would also like to thank my supervisor Werner Ballmann for the constant support and all the knowledge.
The inradius of nodal domains in the real analytic case {#sec:Real-Analytic-Case}
=======================================================
We consider the case of a real analytic manifold $ (M,g) $ of dimension at least $ 3 $. A main insight in this situation is that polynomials approximate eigenfunctions sufficiently well, i.e. an eigenfunction $ {\varphi_{\lambda}}$ exhibits a behaviour, which is similar to that of a polynomial of degree $ \sqrt{\lambda} $. We refer to the papers of Donnelly-Fefferman [@DF2] and Jakobson-Mangoubi [@Jacobson-Mangoubi] for a vivid illustration of this observation.
We now prove Theorem \[thm:Inradius-Real-Analytic\].
Let us assume without loss of generality that $ {\varphi_{\lambda}}$ is positive on $ \Omega_\lambda $ and let $ x_0 \in \Omega_\lambda $ be a point where $ {\varphi_{\lambda}}$ reaches a maximum on $ \Omega_\lambda $. First, an examination of the proof of Theorem $ 1.3 $ of [@Georgiev-Mukherjee], gives $$\label{eq:Large-Volume-near-Max-Point}
\frac{\operatorname{Vol}(B_{\frac{r}{\sqrt{\lambda}}} (x_0) \cap \Omega_\lambda ) }{\operatorname{Vol}(B_{\frac{r}{\sqrt{\lambda}}} (x_0))} > 1 - c r^{\frac{2n}{n-2}},$$
where $ r $ is sufficiently small. This quantitatively means that the more we shrink the ball $ B_{\frac{r}{\sqrt{\lambda}}} (x_0) $, the “more inscribed” it becomes having a small error set, outside of the nodal domain $ \Omega_\lambda $. We remark that the existence of such an almost inscribed wavelength ball could also be deduced from the corresponding statement in $ \mathbb{R}^n $ (cf. [@L]) and a partition of unity argument. However, the result in [@Georgiev-Mukherjee] specifies also the location of such a ball, i.e. at a point $ x_0 $ where $ \phi_\lambda $ achieves a maximum.
We would like to take $r$ sufficiently small (to be determined later), and then rescale the ball $B_{\frac{r}{\sqrt{\lambda}}}$ to the unit ball $B_1$ in $\mathbb{R}^n$. We will denote the rescaled eigenfunction on the unit ball by ${\varphi_{\lambda}^{loc}}$.
We now recall and adapt results from [@Jacobson-Mangoubi]. First, we recall the following (cf. (Proposition 4.1,[@Jacobson-Mangoubi]))
\[lem:Preferred-Cube\] Let $ (M,g) $ be real-analytic. There exists a cube $Q \subseteq B_1$, which does not depend on ${\varphi_{\lambda}^{loc}}$ and $\lambda$, and has the following property: suppose $\delta > 0$ is taken, so that $\delta \leq \frac{C_1}{\sqrt{\lambda}}$. We decompose $Q$ into smaller cubes $Q_\nu$ with sides in the interval $(\delta, 2\delta)$. Then, there exists a subset of $E_\epsilon \subseteq Q$ of measure $| E_\epsilon | \leq C_2 \epsilon \sqrt{\lambda} \delta $, so that $$\frac{1}{C_3(\epsilon)} \leq \frac{({\varphi_{\lambda}^{loc}}(x))^2}{Av_{(Q_\nu)_x} ({\varphi_{\lambda}^{loc}})^2} \leq C_3(\epsilon), \quad \forall x \in Q \backslash E_\epsilon,$$ with $C_3(\epsilon) \rightarrow \infty $ as $\epsilon \rightarrow 0$. The constants $C_1, C_2, C_3$ do not depend on ${\varphi_{\lambda}}$ and $\lambda$. The notation $ Av_{{(Q_\nu)}_x} F$ denotes the average of $ F $ over a cube $ Q_\nu $ which contains $ x $.
Lemma \[lem:Preferred-Cube\] stems from the corresponding result for holomorphic functions $ F $ (cf. Propositions 3.2 and 3.7, [@Jacobson-Mangoubi]), defined on the unit cube $ I^n $ and satisfying the following growth condition: $$\label{eq:Holomorphic-Growth}
\sup_{B_2^n} |F| \leq |F(0)|e^{C\sqrt{\lambda}},$$ where $ B_2^n = B_2 \times \dots \times B_2 \subseteq \mathbb{C}^n $ denotes the $ 2 $-polydisk. To transfer the statement to the local eigenfunction $ {\varphi_{\lambda}^{loc}}$ one observes that $ {\varphi_{\lambda}^{loc}}$ admits a holomorphic continuation on a ball, whose size does not depend on $ \lambda $ (Lemma 7.1, [@DF2]). Here one uses the fact that on a wavelength scale $ {\varphi_{\lambda}^{loc}}$ is almost harmonic, i.e. it is a solution to slight perturbation of the standard Laplace equation. In other words, $ {\varphi_{\lambda}^{loc}}$ is holomorphically extended to a ball $ \| z \| \leq \rho $, where $ \rho $ does not depend on $ \lambda, {\varphi_{\lambda}^{loc}}$. Moreover, $$\label{eq:holo-continuation}
\sup_{\| z \| \leq \rho} |{\varphi_{\lambda}^{loc}}(z)| \leq C \sup_{B_1} |{\varphi_{\lambda}^{loc}}(x)|,$$ where $ C $ again does not depend on $ \lambda, {\varphi_{\lambda}^{loc}}$. In this direction we also remark that, in general, $ {\varphi_{\lambda}}$ extends analytically to a uniform Grauert tube of $ M $, having radius which is independent of $ \lambda $ - we refer to Sections $ 1 $ and $ 11 $, $ \cite{Z3} $.
We now subdivide the cube $ Q $ into small cubes $ Q_\nu $ for which Lemma \[lem:Preferred-Cube\] holds.
$ Q_\nu $ is called $ E_\epsilon $-*good*, if $$\frac{| E_\epsilon \cap Q_\nu |}{|Q_\nu|} < 10^{-2n}\omega_n,$$ where $ \omega_n $ denotes the volume of the unit ball in $ \mathbb{R}^n $. Otherwise, $ Q_\nu $ is $ E_\epsilon $-*bad*.
It turns out that the $ E_\epsilon $-good cubes $ Q_\nu $ are not that different from the good cubes, defined in Section \[sec:Asymmetry-History\] (cf. also Lemma 5.3, [@Jacobson-Mangoubi]). We have
\[lem:Good-Ball\] Let $ Q_\nu $ be an $ E_\epsilon $-good cube. Let $ B \subseteq 2B \subseteq Q_\nu $ be a ball centered somewhere in $ \frac{1}{2} Q_\nu $, whose size is comparable to the size of $ Q_\nu $. Then $$\frac{\int_{2B} ({\varphi_{\lambda}^{loc}})^2}{\int_B ({\varphi_{\lambda}^{loc}})^2} \leq \tilde{C_1} C_3(\epsilon),$$ where $ C_3(\epsilon) $ comes from Lemma \[lem:Preferred-Cube\] and $ \tilde{C_1} $ depends only on the dimension $ n $.
The proportion of bad cubes to all cubes is smaller than $ \tilde{C_2} |E_\epsilon| $, where $ \tilde{C_2} $ depends only on the dimension.
By fixing $ \epsilon $ sufficiently small, the previous two Lemmata imply that we can arrange that $ 90\% $ of the small cubes $ Q_\nu $ to be $ E_\epsilon $-good. Moreover, each $ E_\epsilon $-good cube possesses a comparable inscribed ball, having a fixed growth exponent $ C_\epsilon $.
Now, let us define the error set (or “spike”) $ S := B_1 \backslash \Omega_\lambda $. The volume decay (\[eq:Large-Volume-near-Max-Point\]) gives $$\label{eq:Spike-Estimate}
\frac{|S|}{|B_1|} \leq c r^{\frac{2n}{n-2}}.$$ Now, suppose that $ S $ intersects each small $ E_\epsilon $-good cube $\frac{1}{2} Q_\nu $. Otherwise, there will be an inscribed cube of radius $ \frac{C}{{\lambda}} $ in the nodal domain $ \Omega_\lambda $ and the claim follows.
There are two cases:
1. Suppose that in a $ E_\epsilon $-good cube $ Q_\nu $ the nodal set does not intersect $ \frac{1}{2}Q_\nu $. This means that $ \frac{1}{2} Q_\nu \subseteq S $, hence $$\frac{|S \cap Q_\nu|}{|Q_\nu|} \geq \frac{1}{2^n}.$$
2. Suppose that the nodal set intersects $ \frac{1}{2}Q_\nu $. Since $ Q_\nu $ is $ E_\epsilon $-good, we can find a ball $ \tilde{B} $ as in Lemma \[lem:Good-Ball\], so that its center lies on the nodal set. Elliptic estimates (cf. Proposition \[prop:Controlled-Asymmetry\] below and Proposition 5.4, [@Jacobson-Mangoubi]) imply that $$\frac{|\{ {\varphi_{\lambda}^{loc}}< 0 \} \cap \tilde{B}|}{|\tilde{B}|} \geq C.$$ By definition $ \{ {\varphi_{\lambda}^{loc}}< 0 \} \cap \tilde{B} \subseteq S \cap \tilde{B} $, so we get $$\frac{|S \cap Q_\nu|}{|Q_\nu|} \geq C.$$
Summing up the two cases over all $ E_\epsilon $-good cubes, we see that $$\frac{| S\cap Q|}{|Q|} \geq C.$$
By using the estimate (\[eq:Spike-Estimate\]) and selecting $ r $ sufficiently small, we arrive at a contradiction. This means that there is a $ E_\epsilon $-good cube $ Q_\nu $, so that $ \frac{1}{2} Q_\nu \subseteq \Omega_\lambda $
We note that the statement extends also to points $ x_0 $ at which the eigenfunction almost reaches its maximum on $ \Omega_\lambda $ in the sense, that $$C {\varphi_{\lambda}}(x_0) \geq \| {\varphi_{\lambda}}\|_{L^\infty(\Omega_\lambda)},$$ for some fixed constant $ C > 0 $. In particular, if there are multiple “almost-maximum” points $ x_0 $, there should be an inscribed ball of radius $ \frac{1}{\lambda} $ near each of them.
Let us observe that the estimates essentially depend on the growth of $ {\varphi_{\lambda}}$ at $ x_0 $. We have used the upper bound $ C\sqrt{\lambda} $ on the doubling exponent in the worst possible scenario as shown by Donnelly-Fefferman. It is believed that $ {\varphi_{\lambda}}$ rarely exhibits such an extremal growth. If the growth is better, this allows to take larger cubes $ Q_\nu $ and the bound on the inner radius improves. In particular, a constant growth implies the existence of a wavelength inscribed ball.
To conclude this section, let us briefly mention that the case of $ (M, g) $ being a closed smooth Riemannian manifold could be treated by a combinatorial approach. We address these issues in a further note.
The quantum ergodic case
------------------------
First, we mention some recent results of H. Hezari ([@Hezari] and [@Hezari2]), addressing quantum ergodic sequences of eigenfunctions. Let us assume that $ (M,g) $ is a closed Riemannian manifold with negative sectional curvature. Let $ ({{{\varphi_{\lambda}}}_i}) $ be any orthonormal basis of $ L^2(M) $, where $ ({{{\varphi_{\lambda}}}_i}) $ are eigenfunctions with eigenvalues $ \lambda_i $. Then, for a given $ \epsilon > 0 $, there exists a density-one subsequence $ S_\epsilon $, so that $$\label{eq:Hezari-Improvement}
a_1 (\log \lambda_j)^{\frac{(n-1)(n-2)}{4n^2} - \epsilon} \lambda_j^{-\frac{1}{2} - \frac{(n-1)(n-2)}{4n}} \leq \operatorname{inrad}(\Omega_\lambda)$$
We refer to [@Hezari] for further details.
The heart of Hezari’s argument lies in observing that growth exponents (i.e. doubling exponents) improve, provided that eigenfunctions equidistribute at small scales (cf. [@Hezari2]). More precisely, if we assume that for some small $ r > \frac{1}{\sqrt{\lambda}} $, we have $$K_1 r^n \leq \int_{B_r(x)} |{\varphi_{\lambda}}|^2 \leq K_2 r^n,$$ for $ K_1, K_2 $ fixed constants and all geodesic balls $ B_r(x) $, then $$\log \left( \frac{\sup_{B_{2s}(x)} |{\varphi_{\lambda}}|^2}{\sup_{B_s(x)} |{\varphi_{\lambda}}|^2} \right) \leq C r \sqrt{\lambda}.$$ Here the statement holds for all $ s $ smaller than $ 10r $. In particular, in the negatively curved setting, results of $ \cite{Hezari-Riviere} $ give that $ r $ above could be taken as $ (\log \lambda)^{-k} $ for any $ k \in (0, \frac{1}{2n}) $. Further, plugging the improved growth bound into Mangoubi’s argument (outlined in Section \[sec:Asymmetry-History\]) gives the improved estimate (\[eq:Hezari-Improvement\]).
Now, let us assume that $ (M,g) $ is real-analytic and negatively curved. Then, Hezari’s improved growth bound affects the estimate (\[eq:Holomorphic-Growth\]) and Lemma \[lem:Preferred-Cube\], showing that we may subdivide the preferred cube into smaller cubes $ Q_\nu $ of size $ \frac{(\log \lambda)^k}{\sqrt{\lambda}} $, most of which will be good as prescribed by the Lemma. Continuing the argument in the proof of Theorem \[thm:Inradius-Real-Analytic\], we arrive at $$\operatorname{inrad}(\Omega_\lambda) \geq \frac{C (\log \lambda)^k}{\lambda},$$ where $ k $ could be taken as any number in $ (0, \frac{1}{2n}) $.
Inradius estimates via asymmetry {#sec:Asymmetry-History}
================================
From now on we will be considering a closed smooth Riemannian manifold $ (M,g) $. Let us consider an eigenfunction $ {\varphi_{\lambda}}$ and an associated nodal domain $\Omega_\lambda \subseteq M$.
First, we recall the main ideas behind the asymptotic lower bound:
$$\frac{c_1}{\lambda^{\frac{n-1}{4}+\frac{1}{2n}}} \leq \operatorname{inrad}(\Omega_\lambda),$$
as outlined in $ \cite{Man1} $.
We denote by $\psi$ the restriction of $ {\varphi_{\lambda}}$ to $ \Omega_\lambda $, extended by $ 0 $ to $ M $. Then $ \psi $ realizes the first Dirichlet eigenvalue of $\Omega_\lambda$, i.e. $$\frac{\int_{\Omega_\lambda} |\nabla \psi|^2 d\operatorname{Vol}}{\int_{\Omega_\lambda} | \psi|^2 d\operatorname{Vol}} = \lambda_1(\Omega_\lambda) = \lambda.$$
We may assume that $\phi_i(U_i)$ is a cube $K_i$, whose edges are parallel to the coordinate axes, and we can further cut it into small non-overlapping small cubes $K_{ij} \subseteq K_i$ of appropriately selected side length $h$, comparable to $\operatorname{inrad}(\Omega_\lambda)$.
Having this construction in mind, we define *local Rayleigh quotients*:
A local Rayleigh, associated to the eigenfunction $\psi$ and decomposition $\{ K_{ij} \}$ as above, is the quantity $$R_{ij}(\psi) := \frac{\int_{\phi_i^{-1}(K_{ij})} | \nabla \psi|^2 d\operatorname{Vol}}{\int_{\phi_i^{-1}(K_{ij})} | \psi|^2 d\operatorname{Vol}}.$$
The main idea is to find a specific small cube $K_{i_0 j_0}$, so that $R_{i_0 j_0}$ is bounded in a suitable way from above and from below. To be more precise, one has the following:
\[prop:Reyleigh-Bounds\] Having the decomposition $\{ K_{ij} \}$ and eigenfunction $\psi$ as above, there exists a cube $K_{i_0 j_0}$, such that:
$$\label{eq:Medium-Rayleigh}
C \frac{\left( \lambda^{\frac{1-n}{2}} \right)^{1-\frac{2}{n}}}{h^2} \leq R_{i_0 j_0} \leq m \lambda_1(\Omega_\lambda) = m \lambda,$$
where $C = C(M,g)$ is a constant and $m$ denotes the number of the charts $\{( U_i, \phi_i) \}$.
Note that (\[eq:Medium-Rayleigh\]) leads to the asymptotic lower bound on $ \operatorname{inrad}(\Omega_\lambda) $, because of the choice $ h \sim \operatorname{inrad}(\Omega_\lambda) $.
\[Proof of Proposition \[prop:Reyleigh-Bounds\]\] First, there exists a cube $K_{i_0 j_0}$, for which the upper bound in (\[eq:Medium-Rayleigh\]) holds, since otherwise one has a contradiction with the definition of $ \psi $ by summing over all small cubes.
Note that we may assume w.l.o.g. that $ \psi $ is negative on $ \Omega_\lambda $ and then, by definition, one gets $$\label{eq:Psi-Eigenf}
\operatorname{Vol}(\{ \psi = 0 \}\cap K_{i_0 j_0}) \geq \operatorname{Vol}(\{ {\varphi_{\lambda}}>0 \}\cap K_{i_0 j_0}),$$
One now proceeds to bound $ R_{i_0 j_0} $ from below, using consecutively a Poincare inequality and a capacity estimate (Sections 10.1.2 and 2.2.3, respectively, from [@Maz]):
$$\begin{gathered}
\label{eq:Poincare-Inequalities}
R_{i_0 j_0} \geq C \frac{\operatorname{cap}_2(\{ \psi = 0 \} \cap K_{i_0 j_0}, 2 K_{i_0 j_0})}{h^n} \geq \tilde{C} \frac{\left( \operatorname{Vol}(\{ \psi = 0 \}\cap K_{i_0 j_0}) \right)^\frac{n-2}{n}}{h^n} \geq \\
\geq \tilde{C} \frac{\left( \operatorname{Vol}(\{ {\varphi_{\lambda}}>0 \}\cap K_{i_0 j_0}) \right)^\frac{n-2}{n}}{h^n}.
\end{gathered}$$
with $ C, \tilde{C} $ depending on $ M $.
Thus, it is natural to search for a lower bound on the volume of the positivity set of $ {\varphi_{\lambda}}$ contained in $ K_{i_0 j_0} $.
By using maximum principles for elliptic PDE in combination with a growth bound, due to Donnelly-Fefferman ([@DF2]), D. Mangoubi showed ([@Man1]) that the following asymmetry estimate holds: $$\frac{\operatorname{Vol}(\{ {\varphi_{\lambda}}>0 \}\cap K_{i_0 j_0})}{\operatorname{Vol}( K_{i_0 j_0})} \geq \frac{C}{\lambda^{\frac{n-1}{2}}}.$$ Noting that $ \operatorname{Vol}( K_{i_0 j_0}) \sim h^n$ and putting the above chain of inequalities together yields the claim.
A few comments are in order. First, note that an improvement of the asymmetry of nodal domains leads to a direct improvement of the lower bound for the inner radius. Moreover, it has been suggested (e.g. [@NPS]) that the asymmetry of nodal domains may be better than $\frac{C}{\lambda^{(n-1)/2}}$.
Second, in some sense it has been shown (e.g. [@DF2]; Lemma $ 4 $ in [@Colding-Minicozzi]), that there are many small cubes where the asymmetry of a nodal domain is already far better (in fact asymptotically optimal). In other words, one should expect that on a lot of small cubes $ \operatorname{Vol}(\{{\varphi_{\lambda}}> 0 \}) \sim \operatorname{Vol}(\{{\varphi_{\lambda}}< 0 \})$.
These remarks suggest that, in order to improve the lower inradius bound, it would be desirable to exhibit a better asymmetry of $\Omega_\lambda$ in the specific cube $K_{i_0 j_0}$.
Good cubes and bad cubes {#sec:Cubes}
------------------------
Let us fix an eigenfunction ${\varphi_{\lambda}}$ of ${\Delta}$ with eigenvalue $\lambda$ and $\| {\varphi_{\lambda}}\|_{L^2(M)} = 1$.
As above we consider the finite atlas $\{ (U_i, \phi_i) \}$ of $ M $, and arrange that $M$ is covered by cubes $K_i \subseteq U_i $, where $K_i$ is decomposed into small cubes $K_{ij}$ of size $h$ (to be determined later). Again denoting by $\delta K_{ij}$ the concentric cube, scaled by some fixed scaling factor $ \delta > 1 $, we may also assume that $\delta K_{ij} \subseteq U_i$.
We note that the metric $g$ is comparable to the Euclidean one on each cube $K_i$ and, moreover, each point $x \in M$ is contained in at most $\kappa_\delta$ of the concentric cubes $\delta K_{ij}$, where $\kappa_\delta$ is some constant, not depending on the chosen cube size $h$.
In the light of Definition \[def:Good-Bad-Cubes\], we have that the covering $ K_{ij} $ is divided into good and bad cubes.
First, we note that the covering is robust, in the sense that the good cubes can be arranged to consume most of the $ L^2 $ norm. Essentially, by using the definition one can show:
\[lem:L2-norm-lower-bound\] We have $$\int_{\Gamma} {\varphi_{\lambda}}^2 \geq 1 - \frac{\kappa_\delta}{\gamma}.$$
Using $\| {\varphi_{\lambda}}\|_{L^2} = 1$, we have $$\begin{gathered}
\int_{\Gamma} {\varphi_{\lambda}}^2 \geq 1 - \int_{\Xi} {\varphi_{\lambda}}^2 \geq 1 - \sum_{K_{ij}\text{-bad}} \int_{K_{ij}} {\varphi_{\lambda}}^2 \geq 1 - \sum_{K_{ij}\text{-bad}} \frac{1}{\gamma} \int_{\delta K_{ij}} {\varphi_{\lambda}}^2 \geq \\ \geq 1 - \frac{\kappa_\delta}{\gamma} \int_M {\varphi_{\lambda}}^2 = 1 - \frac{\kappa_\delta}{\gamma}.\\\end{gathered}$$
Again, we note that without any dependence on $ \lambda $ or the size of the small cubes $h$ one is able to control how big (in $ L^2 $ sense) the good set $ \Gamma $ is.
Proof of Theorem \[th:Main-Theorem\] {#sec:Proof-Main}
====================================
We now show how the portion of the $ L^2 $ norm over a nodal domain, occupied by good cubes, gives a lower bound on the inner radius. Roughly, having a lot of good cubes over a nodal domain increases the chance that $ K_{i_0 j_0} $ is a good one. As in the method of Mangoubi, sketched in Section \[sec:Asymmetry-History\], we find a special small cube and estimate the corresponding Rayleigh quotient in a similar way. However, we will have the advantage that the special cube is also good, which would lead to optimal asymmetry.
\[Proof of Theorem \[th:Main-Theorem\]\]
\[cl:The-Good-Cube\] There exists a good cube $K_{i_0 j_0}$, such that $$\label{eq:The-Good-Cube-Rayleigh}
R_{\left(\delta K_{i_0 j_0} \right)} (\psi) \leq \frac{{\kappa_\delta}}{\tau} \lambda_1(\Omega_\lambda),$$ where $ \psi $ is defined as in Section \[sec:Asymmetry-History\] and $R_{\left(\delta K_{i_0 j_0} \right)} (\psi)$ denotes the local Rayleigh quotient w.r.t. the cube $\delta K_{i_0 j_0}$. As above, $ \kappa_\delta $ denotes the maximal number of cubes $ \delta K_{ij} $ that can intersect at a given point.
First, let us denote by $\delta \Gamma$ the union of all good cubes scaled by a factor of $\delta > 1$. Assuming the contrary, we get: $$\begin{gathered}
\int_{\Omega_\lambda} |\nabla \psi|^2 \geq \int_{\Omega_\lambda \cap \delta\Gamma} |\nabla \psi|^2 \geq \frac{1}{\kappa_\delta} \sum_{K_{ij}\text{-good}} \int_{\Omega_\lambda \cap \delta K_{ij}} |\nabla \psi|^2 > \\
> \frac{1}{\tau}\lambda_1(\Omega_\lambda) \sum_{K_{ij}\text{-good}} \int_{\Omega_\lambda \cap \delta K_{ij}} |\psi|^2 \geq \frac{1}{\tau}\lambda_1(\Omega_\lambda)\int_{\Gamma} | \psi|^2 \geq \\
\geq \lambda_1(\Omega_\lambda) \int_{\Omega_\lambda} |\psi|^2.\\
\end{gathered}$$ Hence, a contradiction with the definition of $\psi$.
This means that in the cube $\delta K_{i_0 j_0}$ we have the upper bound from (\[eq:Medium-Rayleigh\]). However, we have the advantage that $K_{i_0 j_0}$ is $\gamma$-good - this implies that the asymmetry and the geometry of the nodal set is under control.
From now on let us fix $ \delta := 16\sqrt{n} $. The following proposition is similar to Proposition 1 in [@Colding-Minicozzi] and Proposition 5.4 in [@Jacobson-Mangoubi]. We supply the technical details for completeness:
\[prop:Controlled-Asymmetry\] Let $\gamma, \rho > 1$ be given. Then there exists $\Lambda > 0$, such that, if one takes the cube size $r \leq \frac{\rho}{\sqrt{\lambda}}$ for $\lambda \geq \Lambda$ and assumes that ${\varphi_{\lambda}}$ vanishes somewhere in $\frac{1}{2} K_{i_0 j_0}$, then $$\frac{\operatorname{Vol}(\{{\varphi_{\lambda}}> 0\} \cap \delta K_{i_0 j_0})}{\operatorname{Vol}(\delta K_{i_0 j_0})} \geq \frac{C}{\gamma^2},$$ where $C$ depends on $ n, \rho, (M, g)$.
The same holds for the negativity set. Hence the asymmetry of $\Omega_\lambda$ in $\delta K_{i_0 j_0}$ is bounded below by the constant $C/\gamma^2 > 0$, which essentially depends on the good/bad growth condition and not on $\lambda$.
(of Proposition) We denote by $ K_r(p) $ the cube of edge size $ r $ centered at $ p $, whose edges a parallel to the coordinate axes. We also denote by $ B_r(p) $ a metric ball (w.r.t the metric $ g $) of radius $ r $ centered at $ p $. Let us assume that $ K_r(p) := K_{i_0 j_0} $. By the metric comparability (\[eq:Metric-Comparability\]), we have: $$B_{\frac{r}{4}} \subseteq K_r(p) \subseteq B_{\sqrt{n}r}(p).$$
Recall the following generalization of the mean value principle (Lemma 5, [@Colding-Minicozzi]):
\[lem:Generalized-Mean-Value\] There exists $ R = R(M, g) > 0 $, such that if $ r \leq R $ and $ {\varphi_{\lambda}}(p) = 0 $, then $$\left| \int_{B_r(p)} {\varphi_{\lambda}}\right| \leq \frac{1}{3} \int_{B_r(p)} |{\varphi_{\lambda}}|.$$
By assumption, there exists a point $ q \in \frac{1}{2} K_{i_0 j_0}, {\varphi_{\lambda}}(q) =0 $, so the lemma, in combination with metric comparability, implies that $$\int_{K_r(q)} |{\varphi_{\lambda}}| \leq \int_{B_{\sqrt{n}r}(q)} |{\varphi_{\lambda}}| \leq 3 \int_{B_{\sqrt{n}r}(q)} {\varphi_{\lambda}}^+ \leq 3 \int_{K_{4\sqrt{n}r}(q)} {\varphi_{\lambda}}^+,$$ where $ {\varphi_{\lambda}}^+, {\varphi_{\lambda}}^- $ respectively denote the positive and negative part of $ {\varphi_{\lambda}}$.
Hence, $$\begin{gathered}
\label{eq:Inequality-Chain1}
\frac{1}{9}\left( \int_{K_{2r}(q) } |{\varphi_{\lambda}}| \right)^2 \leq \left( \int_{K_{8\sqrt{n}r}(q)} {\varphi_{\lambda}}^+ \right)^2 \leq \\
\leq \operatorname{Vol}(K_{8\sqrt{n}r}(q) \cap \{ {\varphi_{\lambda}}> 0 \}) \int_{K_{16\sqrt{n}r}(q)} {\varphi_{\lambda}}^2, \\
\end{gathered}$$ where we have used the Cauchy-Schwartz inequality.
We estimate further the integral from the last expression: $$\begin{gathered}
\label{eq:Inequality-Chain2}
\left( \int_{K_{16\sqrt{n}r}(q)} {\varphi_{\lambda}}^2 \right)^2 \leq
\gamma^2 \left( \int_{K_{r}(p)} {\varphi_{\lambda}}^2 \right)^2 \leq \\ \leq \gamma^2 \left( \int_{K_{2r}(q)} |{\varphi_{\lambda}}| |{\varphi_{\lambda}}| \right)^2 \leq \gamma^2 \sup_{K_{2r}(q)} {\varphi_{\lambda}}^2 \left( \int_{K_{2r}(q)} |{\varphi_{\lambda}}| \right)^2.
\end{gathered}$$
Note that, since $ r $ is comparable to wavelength, elliptic estimates (Theorem 1.2, [@Li-Schoen]) imply: $$\label{eq:Inequality-Chain3}
\sup_{K_{2r}(q)} {\varphi_{\lambda}}^2 \leq \sup_{B_{2\sqrt{n}r}(q)} {\varphi_{\lambda}}^2 \leq C_0 r^{-n} \int_{B_{4\sqrt{n}r}(q)} {\varphi_{\lambda}}^2 \leq C_0 r^{-n} \int_{K_{16\sqrt{n}r}(q)} {\varphi_{\lambda}}^2,$$ where $ C_0 = C_0(M, g, \rho, n) $.
Plugging (\[eq:Inequality-Chain3\]) into (\[eq:Inequality-Chain2\]), one gets $$\int_{K_{16\sqrt{n}r}(q)} {\varphi_{\lambda}}^2 \leq C_0 \gamma^2 r^{-n} \left( \int_{K_{2r}(q)} |{\varphi_{\lambda}}| \right)^2$$ and in combination with (\[eq:Inequality-Chain1\]) this yields $$\frac{r^n}{9C_0 \gamma^2} \leq \operatorname{Vol}(\{ {\varphi_{\lambda}}> 0 \} \cap K_{8\sqrt{n}r}(q) ) \leq \operatorname{Vol}(\{{\varphi_{\lambda}}> 0\} \cap \delta K_{i_0 j_0}).$$ The statement of the proposition follows after dividing by $ \operatorname{Vol}( \delta K_{i_0 j_0}) \leq C_1 r^n$.
One may also exhibit a version of Lemma \[lem:Generalized-Mean-Value\] for cubes, thus making some of the constants better. However, using balls and comparability as above suffices for our purposes.
To finish the proof of the main statement, we put together the latter observations.
Again, we consider an atlas and cube decomposition as above. Following [@Man1], we fix the size of the small cube-grid $$h := 8 \max_i r_i,$$ where $r_i$ denotes the inner radius of $\Omega_\lambda$ in the chart $(U_i, \phi_i)$ with respect to the Euclidean metric.
We consider the cube $ K_{i_0 j_0} $, prescribed by Claim \[cl:The-Good-Cube\]. The choice of $ h $ ensures that $ {\varphi_{\lambda}}(q) = 0 $ for some $ q \in \frac{1}{2} K_{i_0 j_0} $. Then the conditions of Proposition \[prop:Controlled-Asymmetry\] are satisfied. This means that $$\operatorname{Vol}(\{\psi = 0\} \cap \delta K_{i_0 j_0}) \geq \operatorname{Vol}(\{{\varphi_{\lambda}}> 0\} \cap \delta K_{i_0 j_0}) \geq \frac{C}{\gamma^2} h^n.$$
We plug the latter in the Poincare and capacity estimates (\[eq:Poincare-Inequalities\]) and recall that (\[eq:The-Good-Cube-Rayleigh\]) holds. We get $$C \frac{1}{h^ 2} \left(\frac{1}{\gamma^2}\right)^{\frac{n-2}{n}}\leq \frac{{\kappa_\delta}}{\tau} \lambda_1(\Omega_\lambda).$$ A rearrangement gives $$h \geq C \left[ \sqrt{\frac{\tau}{\kappa_\delta}} \left( \frac{1}{\gamma}\right)^{\frac{n-2}{n}} \right] \frac{1}{\sqrt{\lambda}}.$$ The proof finishes by recalling that $ h \leq 8\operatorname{inrad}(\Omega_\lambda) $ by assumption.
Some further comments and corollaries {#sec:Comments}
=====================================
Let us briefly explain the Corollaries \[cor:Fat-Nodal-Domain\] and \[cor:Energy-Estimate\].
\[Proof of Corollary \[cor:Fat-Nodal-Domain\]\]
Let us fix $ \gamma := 4 \kappa_\delta $. A simple summation argument, yields
\[cl:Nice-Nodal-Domain\] There exists a nodal domain ${\Omega_{\varphi_{\lambda}}^*}$, such that $$\int_{\Gamma \cap {\Omega_{\varphi_{\lambda}}^*}} {\varphi_{\lambda}}^2 \geq 3 \int_{\Xi \cap {\Omega_{\varphi_{\lambda}}^*}} {\varphi_{\lambda}}^2.$$ In particular, $$\int_{\Gamma} (\psi^*)^2 \geq 3 \int_{\Xi} (\psi^*)^2,$$ where $\psi^*$ is the function, which realizes $\lambda_1({\Omega_{\varphi_{\lambda}}^*})$, extended by zero outside ${\Omega_{\varphi_{\lambda}}^*}$.
Indeed, assuming the contrary and summing over all nodal domains, one gets a contradiction with Lemma \[lem:L2-norm-lower-bound\] and the fact that $\| {\varphi_{\lambda}}\|_{L^2} = 1$.
Now, apply Theorem \[th:Main-Theorem\] with $ {\Omega_{\varphi_{\lambda}}^*}$.
We now prove the energy inequality. The idea is just to tailor $ \gamma $ along $ \Omega_\lambda $.
\[Proof of Corollary \[cor:Energy-Estimate\]\]
In the light of Lemma \[lem:L2-norm-lower-bound\], we just take $$\gamma := \frac{4 \kappa_\delta}{\| {\varphi_{\lambda}}\|_{L^2(\Omega_\lambda)}^2},$$ thus having $$\int_{\Gamma} {\varphi_{\lambda}}^2 \geq 1 - \frac{\| {\varphi_{\lambda}}\|_{L^2(\Omega_\lambda)}^2}{4}.$$ This ensures that $\Omega_\lambda$ satisfies the condition of Theorem \[th:Main-Theorem\] with $ \tau = 1 / 4 $ and the prescribed $ \gamma $. So, the claim follows from Theorem \[th:Main-Theorem\].
In particular, $$\left( \operatorname{inrad}(\Omega_\lambda) \sqrt{\lambda} \right)^{\frac{2n}{n-2}} \geq C \| {\varphi_{\lambda}}\|_{L^2(\Omega_\lambda)}$$ and summing over all nodal domains yields $$\sum_{\Omega_\lambda} \operatorname{inrad}(\Omega_\lambda)^{\frac{2n}{n-2}} \geq \frac{C}{\lambda^{\frac{n}{2n-4}}},$$ with the constant $C$ being better than the constant $C_1$, appearing in Theorem \[th:Main-Theorem\]. This allows one to obtain an estimate on the generalized mean with exponent $ \frac{n}{n-2} $ of all the inner radii corresponding to different nodal domains.
Note that the main obstruction against the application of Theorem \[th:Main-Theorem\] is the fact that one needs to know that the $L^2$-norm of ${\varphi_{\lambda}}$ over $\Omega_\lambda$ is mainly contained in good cubes and this should be uniform w.r.t. $\lambda $ (or at least conveniently controlled).
As further questions one might ask whether a dissipation of the bad cubes is to be expected in some special cases (e.g. the case of ergodic geodesic flow) - that is, is it true that a nodal domain should have a well-distributed $ L^2 $ norm in the sense of Theorem $ \ref{th:Main-Theorem} $?
A relaxed version of this question is, of course, a probabilistic statement of the kind - a significant amount of nodal domains should enjoy the property of having well-distributed $ L^2 $ norm.
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Introduction
============
In this paper a *new* class of *solvable* $N$-body problems is identified. They describe an arbitrary number $N$ of unit-mass point particles whose time-evolution, generally taking place in the *complex* plane, is characterized by *Newtonian* equations of motion “of goldfish type” (acceleration equal force, with specific velocity-dependent one-body and two-body forces, see below) featuring several arbitrary coupling constants. The *solvable* character of these models is demonstrated by the possibility to ascertain their time evolution by purely *algebraic* operations. In particular it is shown below that their initial-value problems are solved by finding the eigenvalues of a time-dependent $N\times N$ matrix $U(t) $ explicitly defined in terms of the initial positions and velocities of the $N$ particles. Some of these models are *asymptotically isochronous*, i.e. in the remote future they become completely periodic with a period $T$ independent of the initial data (up to exponentially vanishing corrections). Alternative formulations of these models, obtained by changing the dependent variables from the $N$ zeros of a monic polynomial of degree $N$ to its $N$ coefficients, are also exhibited.
The main idea to obtain these results is to identify the $N$ coordinates $z_{n}(t) $ characterizing the positions of the particles of the $N$-body problem with the $N$ eigenvalues of an $N\times N$ matrix $U(t) $, itself evolving according to a *solvable* (or *integrable*) matrix ODE. This technique was invented long ago by Olshanetsky and Perelomov [@OP1; @OP2; @OP3; @OP4] and has been much exploited subsequently, identifying thereby many *solvable* (or * integrable*) many-body problems: see for instance [@C2001] (in particular its Section 2.1.3.2 entitled “The technique of solution of Olshanetsky and Perelomov (OP)”) and [@C2008] (in particular its Section 4.2.2 entitled “Goldfishing”), and the more recent papers [@CC1; @CC2; @CC3; @CC4; @CC5; @CC6; @CC7; @CC8; @CC9; @CC10; @CC11; @CC12; @CC13; @CC14; @CC15; @CC16; @CC17]. The present paper provides *new* results of this kind, by taking as point of departure a *solvable* ODE characterizing the time-evolution of the $N\times N$ matrix $U(t) $ which is different or more general than those previously employed to this end. These results are treated in the following section, including its subsections and Appendices \[appendixA\] and \[appendixB\] where all the equations solved in this paper are listed (hence, the reader wishing to get an immediate glance at them may immediately jump to these appendices). The last section, entitled “Outlook”, tersely outlines further developments whose treatment is postponed to subsequent papers.
*Solvable* $\boldsymbol{N}$-body problems {#section2}
=========================================
In this section we describe two classes of *solvable* $N$-body problems. The models of the first class are not new, hence their treatment is not elaborated beyond their identification; several models of the second class are *new,* hence they are fully dealt with.
In Subsection \[section2.1\] we introduce a system of two matrix first-order ordinary differential equations (ODEs) characterizing the time-evolution of the two $N\times N$ matrices $U\equiv U(t) $ and $V\equiv V (
t ) $, and we indicate how the corresponding initial-value problem can be explicitly solved.
In Subsection \[section2.2\] we show how – via the introduction of two appropriate *ansätze* – the $N$ eigenvalues $z_{n}(t) $ of the matrix $U(t) $ can be identified with the $N$ coordinates of $N$ unit-mass point particles whose time-evolution, generally taking place in the *complex* plane, is characterized by *Newtonian* equations of motion (“acceleration equal force”, with nonlinear one-body and two-body forces). These models are thereby shown to be *solvable*. The first *ansatz* yields various models whose solvability was already known; hence their treatment is limited to the derivation of their equations of motion. The second *ansatz* yields several *new* models – as well as several previously known models – characterized by equations of motion with specific velocity-dependent one-body and two-body forces “of goldfish type” featuring several arbitrary coupling constants. They are all listed in Appendix \[appendixA\]. The alternative class of many-body models obtained by changing the dependent variables from the $N$ zeros of a monic polynomial of degree $N$ to its $N$ coefficients is discussed in Subsection \[section2.3\]; the corresponding equations of motion are listed in Appendix \[appendixB\].
As it is well known (see for instance [@C2001], in particular Chapter 4, entitled “Solvable and/or integrable many-body problems in the plane, obtained by complexification”), models such as those described below, which describe the motions of $N$ points in the *complex* $z$-plane, can be reformulated as $N$-body models describing the motion in a plane of $N$ point-particles, the positions of which are identified by *real* 2-vectors in that plane. While we leave the elaboration of this connection to the interested reader, we feel justified by it to refer to our findings as describing “physical” (if not necessarily “realistic”) many-body problems in the plane.
A *solvable* system of two matrix ODEs {#section2.1}
--------------------------------------
In this subsection we discuss the following system of two matrix ODEs, satisfied by the two $N\times N$ matrices $U$ and $V$: $$\begin{gathered}
\dot{U}=\alpha U+\beta U^{2}+\gamma V+\eta ( U V+V U ) ,\qquad
\dot{V}=\rho V . \label{EqUV}\end{gathered}$$
[**Notation.**]{} Upper-case Latin letters generally denote $N\times N$ matrices (unless otherwise indicated), with the (scalar!) $N$ being throughout an arbitrary *positive integer*. Here and below the matrices are time-dependent (unless otherwise indicated), in particular $U\equiv U(t)$, $V\equiv V(t) $. Lower case letters generally denote scalars. The $5$ scalar parameters $\alpha $, $\beta $, $\gamma $, $\eta $, $\rho $ are time-independent but *a priori* arbitrary, until specific restrictions on their values are explicitly mentioned. Superimposed dots always indicate time differentiations.
The factor $\beta $ multiplying the term $U^{2}$ in the right-hand side of the first of the two matrix ODEs (\[EqUV\]) could be eliminated (i.e., replaced by unity) by rescaling $U$(and by correspondingly replacing $\gamma $ with an adjusted value, say $\tilde{\gamma})$; then the factor $\eta $ multiplying the term $ ( U V+V U )$, or the (adjusted) factor $\tilde{\gamma}$ multiplying the term $V$, could also be eliminated (i.e., replaced by unity) by rescaling $V $. It is preferable not to do so in order to keep open the possibility to set one or the other of these parameters, $\eta $ or $\gamma $, to zero (but we will never set *both* of them to zero, to avoid decoupling the time evolution of $U(t) $ from that of $V(t)$). Note moreover that the introduction of a constant scalar term (implicitly multiplied by the $N\times N$ unit matrix $I$) in the right-hand side of the first equation would amount – up to a redefinition of other parameters – to adding to the matrix $U$ the $N\times N$ unit matrix $I$ multiplied by a new (constant) parameter, implying just a constant shift of all the eigenvalues of the matrix $U$, a trivial change not worth pursuing; while the introduction of two *different* parameters in front of the two terms $UV$ and $VU$ could be eliminated by adding, in the right-hand side of the first of the two matrix ODEs (\[EqUV\]), the commutator $[ U,V]
$ of $U$ and $V$ times a convenient parameter, without any effect on the eigenvalues of $U$. If some of the parameters in the matrix ODEs (\[EqUV\]) vanish this model may reduce to one of those that have been previously treated, see [@C2008] (in particular its Section 4.2.2 entitled “Goldfishing”) and the more recent papers [@CC1; @CC2; @CC3; @CC4; @CC5; @CC6; @CC7; @CC8; @CC9; @CC10; @CC11; @CC12; @CC13; @CC14; @CC15; @CC16; @CC17]; we do not discard these models in the following, since we consider in any case worthwhile to exploit the unified treatment provided by the present approach based on (\[EqUV\]). As for the possibility of replacing the right-hand side of the second matrix ODE (\[EqUV\]) with a more general function of the matrix $V$, it is a possibility whose exploration is postponed to a future investigation (see Section \[section3\]).
*Solution of the system of two matrix ODEs* . Clearly the solution of the second of the two matrix ODEs (\[EqUV\]) reads$$\begin{gathered}
V(t) =V_{0} \exp ( \rho t ) , \qquad V_{0}\equiv V (0). \label{SolvV}\end{gathered}$$
To solve the first of the two matrix ODEs (\[EqUV\]) it is convenient to set $$\begin{gathered}
U=-\frac{\eta }{\beta } V- ( \beta F ) ^{-1} \dot{F} , \label{UVF}\end{gathered}$$ obtaining thereby for the $N\times N$ matrix $F\equiv F(t) $ the following second-order *linear* matrix ODE:
\[ODEF\] $$\begin{gathered}
\ddot{F}-\alpha \dot{F}+F W=0 , \label{EqF}
\\
W= ( -\alpha \eta +\beta \gamma +\eta \rho ) V-\eta
^{2} V^{2} . \label{W}\end{gathered}$$
It is then plain via (\[SolvV\]) that the *general solution* of this matrix ODE reads $$\begin{gathered}
F(t) =F_{+} f_{+} ( t;V_{0} ) +F_{-} f_{-} (t;V_{0} ) , \label{Ft}\end{gathered}$$ where $F_{\pm }$ are two *arbitrary constant* $N\times N$ matrices and the two *scalar* functions $f_{\pm } ( t;v ) $ (which of course become $N\times N$ *matrices* when the *scalar* $v$ is replaced by the $N\times N$ *matrix* $V_{0}$, see (\[Ft\])) are two independent solutions of the *scalar* second-order *linear* ODE$$\begin{gathered}
\ddot{f}-\alpha \dot{f}+\big[ ( -\alpha \eta +\beta \gamma +\eta
\rho ) v \exp ( \rho t ) -\eta ^{2} v^{2} \exp (
2 \rho t ) \big] f=0 . \label{ODEf}\end{gathered}$$ And it is easily seen that two independent solutions of this ODE are given by the following formulas:
\[ff\] $$\begin{gathered}
f_{+} ( t;v ) =\exp \left[ -\frac{\eta v}{\rho } \exp ( \rho
t ) \right] \Phi \left( -\frac{\beta \gamma }{2 \eta \rho };1-\frac{\alpha }{\rho };\frac{2\eta v}{\rho }\exp ( \rho t) \right),
\label{f+}
\\
f_{-} ( t;v ) = \exp \left[ -\frac{\eta v}{\rho } \exp (
\rho t ) \right] \exp ( \alpha t ) \Phi \left( \frac{2 \alpha \eta -\beta \gamma }{2 \eta \rho };1+
\frac{\alpha }{\rho };\frac{2 \eta v}{\rho } \exp ( \rho t )
\right) .\end{gathered}$$
Here the (scalar!) function $\Phi ( a;c;z) $ is the confluent hypergeometric function (see, for instance, [@HTF1]).
Note that the formula (\[UVF\]) with (\[Ft\]) implies the following *explicit solution* of the *initial-value problem* for the matrix $U(t) $:
\[ExplSolUt\] $$\begin{gathered}
U(t) =-\frac{1}{\beta }\left\{ \eta V_{0}\exp ( \rho
t ) +\big[ f_{+} ( t;V_{0} ) +C~f_{-} ( t;V_{0} ) \big] ^{-1} \big[ \dot{f}_{+} ( t;V_{0} ) +C \dot{f}_{-} (
t;V_{0} ) \big] \right\} , \label{Ut}\end{gathered}$$ with the time-independent matrix $C$ defined in terms of the initial data $U( 0) $ and $V_{0}\equiv V( 0) $ (see (\[SolvV\])) as follows: $$\begin{gathered}
C = -\big\{ f_{+} ( 0;V_{0} ) [ \beta U ( 0 )
+\eta V_{0} ] +\dot{f}_{+} ( 0;V_{0} ) \big\}
\big\{ f_{-} ( 0;V_{0} ) [ \beta U ( 0 )
+\eta V_{0} ] +\dot{f}_{-} ( 0;V_{0} ) \big\} ^{-1}\! .\!\!\!\end{gathered}$$
\[remark2.2\] The fact that the expression (\[UVF\]) (with (\[SolvV\]) and (\[ODEF\])) of the matrix $U(t) $ satisfies (\[EqUV\]), and likewise that the functions $f_{\pm } ( t;v ) $ (see (\[ff\])) satisfy the ODE (\[ODEf\]), can be easily verified. Of course these formulas are valid for *generic* values of the parameters that appear in them, excluding special cases – such as vanishing values of $\rho $ or $\eta $ (see (\[ff\])) or of $\beta $ (see (\[Ut\])), or values of $\alpha $ and $\rho $ such that $\alpha /\rho $ is an *integer* (in which case the two functions $f_{\pm }\left( t;v\right) $ do *not* provide two *independent* solutions of the ODE (\[ODEf\]): see Section 6.7 of [@HTF1]) – which are clearly problematic (although the formulas may in these cases be reinterpreted via appropriate limiting procedures). We ignore hereafter these issues, even when listing below *solvable* $N$-body models a few of which belong to these problematic cases. Indeed in this paper we mainly limit our consideration to the *identification* of *solvable* $N$-body problems; these findings open the way to obtaining a rather detailed understanding of their actual behaviors (see for instance the following Remark \[remark2.3\]), but such analyses exceed the scope of this paper: they should be done on a case-by-case basis, and shall perhaps be postponed to the moment when some of these $N$-body models evoke a specific, theoretical or applicative, interest.
\[remark2.3\] If the parameter $\rho $ is purely *imaginary* and the parameter $\alpha $ is *real* and *negative*,
$$\begin{gathered}
\rho =\frac{2 \pi \text{\textbf{i}}}{T} , \qquad \alpha <0 \label{rhoT}\end{gathered}$$
(of course with $T$ *real* and *nonvanishing,* and **i** the *imaginary unit*, **i**$^{2}=-1$), then clearly the matrix $U(t) $ (see (\[ExplSolUt\]) with (\[ff\])) is *asymptotically* *isochronous* (i.e., *asymptotically periodic* with the period $T$ independent of the initial data): indeed in this case, as $t\rightarrow +\infty $, $$\begin{gathered}
U(t) =U_{+}(t) +O [ \exp ( \alpha t ) ]\end{gathered}$$with $U_{+}(t) $ given by the formula (\[ExplSolUt\]) with $C=0,$ $$\begin{gathered}
U_{+}(t) =-\frac{1}{\beta }\left\{ \eta V_{0}\exp ( \rho
t ) + [ f_{+} ( t;V_{0} ) ] ^{-1} [ \dot{f}
_{+} ( t;V_{0} ) ] \right\} ,\end{gathered}$$hence (see (\[rhoT\]) and (\[f+\])) it is *periodic* with period $T$, $$\begin{gathered}
U_{+} ( t+T ) =U_{+}(t) .\end{gathered}$$
This observation entails of course the property of *asymptotic isochrony* for the *solvable* $N$-body models identified below featuring parameters $\alpha $ and $\rho $ which satisfy the conditions ([rhoT]{}).
Identification of *solvable* many-body models {#section2.2}
---------------------------------------------
The starting point is to introduce the $N$ eigenvalues $z_{n}(t)
$ of the matrix $U(t) ,$ and the corresponding *diagonalizing* matrix $R(t) ,$ by setting
\[UVM\] $$\begin{gathered}
U(t) =R(t) Z(t) [ R (
t ) ] ^{-1} , \qquad Z(t) =\text{diag} [ z_{n} (
t ) ] . \label{UZ}\end{gathered}$$Here and hereafter indices such as $n$, $m$, $\ell$, $k$ run over the integers from $1$ to $N$ (unless otherwise indicated).
Likewise we set$$\begin{gathered}
V(t) =R(t) Y(t) [ R (
t ) ] ^{-1} , \qquad Y_{nn}(t) =y_{n}(t) ,
\label{VY}\end{gathered}$$and we introduce the $N\times N$ matrix $M(t) $ by setting $$\begin{gathered}
M(t) = [ R(t) ] ^{-1} \dot{R} (
t ) , \qquad M_{nn}(t) =\mu _{n}(t) . \label{M}\end{gathered}$$
\[remark2.4\] The diagonalizing matrix $R(t) $ is defined up to right-multiplication by an *arbitrary diagonal* $N\times N$ matrix $D(t) $, since the replacement of $R(t) $ by $\tilde{R}(t) =R(t) D(t) $ does not affect (\[UZ\]). But it changes $M(t) $ (see (\[M\])) into $\tilde{M}(t) = [ \tilde{R}(t) ] ^{-1}
\dot{\tilde{R}}(t) $ implying the following change of its diagonal elements: $\mu _{n}(t) \Longrightarrow \tilde{\mu }_{n}(t) =\mu _{n}(t) +\dot{d}_{n}(t)
/d_{n}(t) $, where the $N$ quantities $d_{n}(t) $ are the, *a priori* arbitrary, elements of the diagonal matrix $D(t) $. Hence we retain the privilege to assign at our convenience (see below) the diagonal elements $\mu _{n}(t) $ of the matrix $M(t) $.
It is now easily seen that via these assignments (\[UVM\]) the two matrix ODEs (\[EqUV\]) get rewritten as follows: $$\begin{gathered}
\dot{Z}+ [ M,Z ] =\alpha Z+\beta Z^{2}+\gamma Y+\eta ( Z
Y+Y Z ) , \qquad \dot{Y}+ [ M,Y ] =\rho Y . \label{EqZY}\end{gathered}$$ Here and hereafter the notation $ [ A,B ] $ denotes the commutator of the two matrices $A$ and $B$: $ [ A,B ] \equiv A B-B A$.
Let us now look, componentwise, at the *diagonal* and *off-diagonal* elements of these two matrix ODEs.
The *diagonal* part of the first of the two ODEs (\[EqZY\]) reads
$$\begin{gathered}
\dot{z}_{n}=\alpha z_{n}+\beta z_{n}^{2}+\gamma y_{n}+2 \eta z_{n} y_{n} , \label{zndot}\end{gathered}$$
implying $$\begin{gathered}
y_{n}=\frac{\dot{z}_{n}-\alpha z_{n}-\beta z_{n}^{2}}{\gamma +2 \eta z_{n}
} . \label{yn}\end{gathered}$$
The *off-diagonal* part of the first of the two ODEs (\[EqZY\]) reads
$$\begin{gathered}
-\left( z_{n}-z_{m}\right) M_{nm}= [ \gamma +\eta (
z_{n}+z_{m} ) ] Y_{nm} , \qquad n\neq m ,\end{gathered}$$
implying$$\begin{gathered}
M_{nm}=-\left[ \frac{\gamma +\eta \left( z_{n}+z_{m}\right) }{z_{n}-z_{m}}\right] Y_{nm} , \qquad n\neq m . \label{Mnm}\end{gathered}$$
The *diagonal* part of the second of the two ODEs (\[EqZY\]) reads
$$\begin{gathered}
\dot{y}_{n}=\rho y_{n}+\sum_{\ell =1, \ell \neq n}^{N}\left( Y_{n\ell
} M_{\ell n}-M_{n\ell } Y_{\ell n}\right) ,\end{gathered}$$
implying, via (\[Mnm\]), $$\begin{gathered}
\dot{y}_{n}=\rho y_{n}+2 \sum_{\ell =1, \ell \neq n}^{N}\left\{ Y_{n\ell
} Y_{\ell n} \left[ \frac{\gamma +\eta ( z_{n}+z_{\ell }) }{z_{n}-z_{\ell }}\right] \right\} . \label{yndot}\end{gathered}$$
Hence, by time-differentiation of (\[zndot\]) we see via (\[yn\]) and (\[yndot\]) that the coordinates $z_{n}(t) $ satisfy the following system of $N$ equations of motion of *Newtonian* type (“acceleration equal force”, with one-body and two-body forces): $$\begin{gathered}
\ddot{z}_{n}=-\alpha \rho z_{n}-\beta \rho z_{n}^{2}+ ( \alpha
+\rho ) \dot{z}_{n}+2 \beta \dot{z}_{n} z_{n}
+\frac{2 \eta \dot{z}_{n} \left( \dot{z}_{n}-\alpha z_{n}-\beta
z_{n}^{2}\right) }{\gamma +2 \eta z_{n}} \nonumber \\
\hphantom{\ddot{z}_{n}=}{}
+2 ( \gamma +2 \eta z_{n} ) \sum_{\ell =1, \ell \neq
n}^{N}\left\{ Y_{n\ell } Y_{\ell n} \left[ \frac{\gamma +\eta (
z_{n}+z_{\ell } ) }{z_{n}-z_{\ell }}\right] \right\} .
\label{zndotdot}\end{gathered}$$
But in these equations of motion the role of “two-body coupling constants” is played by the quantities $Y_{n\ell } Y_{\ell n}$ (with $\ell \neq n$) which are in fact *time-dependent*. Indeed the time evolution of the *off-diagonal* elements $Y_{nm}$ of the matrix $Y$ is determined by the *off-diagonal* part of the second of the two ODEs (\[EqZY\]) which componentwise read
$$\begin{gathered}
\dot{Y}_{nm}=\rho Y_{nm}+\sum_{k=1}^{N}\left(
Y_{nk} M_{km}-M_{nk} Y_{km}\right) , \qquad n\neq m ,\end{gathered}$$
yielding, via (\[VY\]), (\[M\]), (\[yn\]), (\[Mnm\]) and a bit of algebra, $$\begin{gathered}
\frac{\dot{Y}_{nm}}{Y_{nm}}=\alpha +\rho +\beta ( z_{n}+z_{m} ) -
\frac{\dot{z}_{n}-\dot{z}_{m}}{z_{n}-z_{m}}
+\frac{\eta \left( \dot{z}_{n}-\alpha z_{n}-\beta z_{n}^{2}\right) }{
\gamma +2 \eta z_{n}}+\frac{\eta \left( \dot{z}_{m}-\alpha z_{m}-\beta
z_{m}^{2}\right) }{\gamma +2 \eta z_{m}} \nonumber \\
\hphantom{\frac{\dot{Y}_{nm}}{Y_{nm}}=}{}
-\mu _{n}+\mu _{m} +\sum_{\ell =1, \ell \neq n,m}^{N}\left\{ \frac{Y_{n\ell } Y_{\ell m}}{Y_{nm}} \left[ 2 \eta + ( \gamma +2 \eta z_{\ell } ) \left( \frac{1}{z_{n}-z_{\ell }}+\frac{1}{z_{m}-z_{\ell }}\right) \right] \right\} , \nonumber\\
n\neq m . \label{Ynmdot}\end{gathered}$$
So, in order that (\[zndotdot\]) become the $N$ *Newtonian* equations of motion of a genuine $N$-body problem one must either provide some “physical interpretation” for the quantities $Y_{n\ell }$ (with $\ell
\neq n$) – possibly in terms of internal degrees of freedom, an alternative we do not pursue in this paper – or find a way to “get rid” of these quantities, i.e. find a way to express them – if at all possible – via the $N$ coordinates $z_{m}\equiv z_{m}(t) $ and possibly also the $N$ velocities $\dot{z}_{m}\equiv \dot{z}_{m}(t) $. Previous experience [@C2001; @C2008; @CC1; @CC2; @CC3; @CC4; @CC5; @CC6; @CC7; @CC8; @CC9; @CC10; @CC11; @CC12; @CC13; @CC14; @CC15; @CC16; @CC17] suggest that two types of *ansätze* are the appropriate starting points to try and achieve this goal.
### First ansatz
The first *ansatz* reads $$\begin{gathered}
Y_{nm}=\frac{( g_{n} g_{m}) ^{1/2}}{z_{n}-z_{m}} ,\qquad n\neq m ,
\label{Ans1}\end{gathered}$$ where we reserve the option to make a convenient assignment for the $N$ functions $g_{n}$ of the coordinate $z_{n}$, $g_{n}\equiv g_{n} (
z_{n} ) \equiv g_{n} [ z_{n}(t) ] $.
The insertion of this *ansatz* in (\[Ynmdot\]) yields, after a bit of trivial algebra, $$\begin{gathered}
\frac{\dot{g}_{n}}{2 g_{n}}-\frac{\alpha +\rho }{2}-\beta z_{n}-\frac{\eta
\left( \dot{z}_{n}-\alpha z_{n}-\beta z_{n}^{2}\right) }{\gamma +2 \eta
z_{n}}+ ( ( n\rightarrow m ) ) \nonumber \\
\qquad {} =\frac{g_{n} ( \gamma +2 \eta z_{m} ) -g_{m} ( \gamma
+2 \eta z_{n} ) }{( z_{n}-z_{m}) ^{2}}-\mu _{n}+\mu _{m}
\nonumber \\
\qquad\quad{} +\sum_{\ell =1, \ell \neq n}^{N}\left\{ g_{\ell } \left[ \frac{\gamma
+2 \eta z_{n}}{( z_{n}-z_{\ell }) ^{2}}\right] \right\}
-\sum_{\ell =1, \ell \neq m}^{N}\left\{ g_{\ell } \left[ \frac{\gamma
+2 \eta z_{m}}{( z_{m}-z_{\ell }) ^{2}}\right] \right\}
, \qquad n\neq m . \label{EqAns1}\end{gathered}$$ Here and throughout the notation $+ ( ( n\rightarrow m )
) $ indicates the addition of whatever comes before it, with the index $n$ replaced by $m$.
We now take advantage of the freedom (see Remark \[remark2.4\]) to assign the diagonal elements $\mu _{n}$ of the matrix $M$ by setting $$\begin{gathered}
\mu _{n}=\sum_{\ell =1, \ell \neq n}^{N}\left\{ g_{\ell } \left[ \frac{\gamma +2 \eta z_{n}}{ ( z_{n}-z_{\ell } ) ^{2}}\right] \right\} ,
\label{mun}\end{gathered}$$ and we moreover make the assignment$$\begin{gathered}
g_{n}=g ( \gamma +2 \eta z_{n} ) , \label{gn}\end{gathered}$$with $g$ an arbitrary constant (i.e., $\dot{g}=0$). Thereby the system of $N ( N-1 ) $ equations (\[EqAns1\]) gets reduced to the following, much simpler, system of only $N$ algebraic equations: $$\begin{gathered}
-\frac{\alpha +\rho }{2}-\beta z_{n}+\frac{\eta \left( \alpha z_{n}+\beta
z_{n}^{2}\right) }{\gamma +2 \eta z_{n}}=0 ,\end{gathered}$$which amounts to the following $3$ equations (recall that we exclude the uninteresting possibility that $\gamma $ and $\eta $ both vanish):
$$\begin{gathered}
\beta \eta =0 ,
\\
\beta \gamma +\eta \rho =0 ,
\\
\gamma \left( \alpha +\rho \right) =0 .\end{gathered}$$
This boils down to the following $3$ possibilities:
$$\begin{gathered}
\alpha =\beta =\rho =0 ,\end{gathered}$$
or $$\begin{gathered}
\beta =\gamma =\rho =0 ,\end{gathered}$$ or$$\begin{gathered}
\beta =\eta =0 , \qquad \rho =-\alpha .\end{gathered}$$
The corresponding *solvable* $N$-body models – obtained by inserting (\[Ans1\]) with (\[gn\]) and with these assignments of the parameters in the *Newtonian* equations of motion (\[zndotdot\]) – read, in the first 2 of these 3 cases – after conveniently setting
$$\begin{gathered}
\gamma +2 \eta z_{n}(t) =\exp [ 2 c \zeta _{n} (
t ) ]\end{gathered}$$
with $c$ an arbitrary nonvanishing constant – as follows: $$\begin{gathered}
\ddot{\zeta}_{n}=\left( \frac{g^{2} \eta ^{4}}{c^{2}}\right) \frac{d}{d \zeta _{n}}\sum_{\ell =1, \ell \neq n}^{N}\sinh ^{-2} [ c ( \zeta
_{n}-\zeta _{\ell } ) ] ;\end{gathered}$$
while in the third of these 3 cases they read $$\begin{gathered}
\ddot{z}_{n}=\alpha ^{2} z_{n}+g^{2} \gamma ^{4} \frac{d}{d z_{n}}\sum_{\ell
=1, \ell \neq n}^{N} ( z_{n}-z_{\ell } ) ^{-2} .\end{gathered}$$ But these are well-known *solvable* $N$-body problems, see for instance [@C2001]. Hence we conclude that from the matrix system ([EqUV]{}) *no* *new* *solvable* many-body models are obtained via the *ansatz* (\[Ans1\]).
### Second ansatz
We proceed then to consider a second *ansatz*, reading $$\begin{gathered}
Y_{nm}=\left[ g_{n} g_{m} \left( \dot{z}_{n}+f_{n}\right) \left( \dot{z}_{m}+f_{m}\right) \right] ^{1/2} , \qquad n\neq m , \label{Ans2}\end{gathered}$$with $g_{n}$ and $f_{n}$ functions of the coordinate $z_{n}$ that we reserve to assign later: $g_{n}\equiv g_{n} ( z_{n} ) \equiv g_{n} [
z_{n}(t) ] $, $f_{n}\equiv f_{n} ( z_{n} ) \equiv
f_{n} [ z_{n}(t) ] $. Its insertion in (\[Ynmdot\]) yields, again after a bit of trivial algebra and now the assignment $\mu
_{n}=0$ (again justified by Remark \[remark2.4\]), the following system of $N ( N-1 ) $ second-order ODEs:$$\begin{gathered}
\frac{1}{2} \left( \frac{\ddot{z}_{n}+\dot{f}_{n}}{\dot{z}_{n}+f_{n}}+\frac{
\dot{g}_{n}}{g_{n}}-\alpha -\rho \right) -\beta z_{n}-\eta \frac{\dot{z}
_{n}-\alpha z_{n}-\beta z_{n}^{2}}{\gamma +2 \eta z_{n}} \nonumber \\
\qquad{} +\eta ( \dot{z}_{n}+f_{n} ) g_{n}-\sum_{\ell =1, \ell \neq
n}^{N}\left\{ ( \dot{z}_{\ell }+f_{\ell } ) g_{\ell } \left[
\frac{\gamma +\eta ( z_{n}+z_{\ell } ) }{z_{n}-z_{\ell }}\right]
\right\} \nonumber \\
\qquad{} -\frac{\dot{z}_{n} [ g_{n} ( \gamma +2 \eta z_{n} ) -1 ]
+f_{n} g_{n} ( \gamma +2 \eta z_{n} ) }{z_{n}-z_{m}}+ (
( n\rightarrow m ) ) =0 , \qquad
n\neq m . \label{EqAns2}\end{gathered}$$
To reduce this system we clearly must now set
\[fg\] $$\begin{gathered}
g_{n}=\frac{1}{\gamma +2 \eta z_{n}} , \label{g}
\\
f_{n}=f^{( 0) }+f^{( 1) } z_{n}+f^{( 2)
} z_{n}^{2} , \label{fn}\end{gathered}$$
where $f^{( 0) }$, $f^{( 1) }$, $f^{( 2) }$ are $3$ *constant* parameters that we reserve to assign later. Thereby the system of $N ( N-1 ) $ ODEs (\[EqAns2\]) gets transformed into the following system of (only) $N$ ODEs: $$\begin{gathered}
\ddot{z}_{n}-2 ( \dot{z}_{n}+f_{n} ) \sum_{\ell =1, \ell \neq
n}^{N}\left\{ \frac{ ( \dot{z}_{\ell }+f_{\ell } ) [ \gamma
+\eta ( z_{n}+z_{\ell } ) ] }{( z_{n}-z_{\ell } )
( \gamma +2 \eta z_{\ell } ) }\right\} \nonumber \\
\qquad{} = \left[ \alpha +\rho +2 \beta z_{n}+\frac{2 \eta \left( \dot{z}
_{n}-\alpha z_{n}-\beta z_{n}^{2}-f_{n}\right) }{\gamma +2 \eta z_{n}}
\right] ( \dot{z}_{n}+f_{n} )
+\big( f^{( 1) }+2 f^{( 2) } z_{n}\big) f_{n} .\!\!\!
\label{EqAns2bis}\end{gathered}$$ Note that to write this system in more compact form we employed a mixed notation, using sometimes the functions $f_{n}$ or $f_{\ell }$ instead of their explicit expressions, see (\[fn\]).
To complete the task of ascertaining for which values of the (so far arbitrary) $8$ *constant* parameters $\alpha $, $\beta $, $\gamma $, $\eta $, $\rho $, $f^{( 0) }$, $f^{( 1) }$, $f^{(
2) }$ the system of $N( N-1) $ ODEs (\[EqAns2bis\]) is satisfied we must utilize the equations of motion (\[zndotdot\]) which, via the *ansatz* (\[Ans2\]) with (\[fg\]), now read $$\begin{gathered}
\ddot{z}_{n}-2 ( \dot{z}_{n}+f_{n} ) \sum_{\ell =1, \ell \neq
n}^{N}\left\{ \frac{( \dot{z}_{\ell }+f_{\ell }) [ \gamma
+\eta ( z_{n}+z_{\ell }) ] }{( z_{n}-z_{\ell })
( \gamma +2 \eta z_{\ell }) }\right\} \nonumber \\
\qquad{} = -\alpha \rho z_{n}-\beta \rho z_{n}^{2}+ ( \alpha +\rho )
\dot{z}_{n}+2 \beta \dot{z}_{n} z_{n}
+\frac{2 \eta \dot{z}_{n} \left( \dot{z}_{n}-\alpha z_{n}-\beta
z_{n}^{2}\right) }{\gamma +2 \eta z_{n}} . \label{EqMotGold1}\end{gathered}$$ Comparison of this system of $N$ ODEs to the system (\[EqAns2bis\]) yields the following system of $N$ algebraic equations (note that – as it were, “miraculously” – the velocities $\dot{z}_{n}$ have disappeared from these equations): $$\begin{gathered}
\alpha \rho z_{n}+\beta \rho z_{n}^{2}+\big[ \alpha +\rho +f^{(
1) }+2 \big( \beta +f^{( 2) }\big) z_{n}\big] f_{n}
=\frac{2 \eta \left( \alpha z_{n}+\beta z_{n}^{2}+f_{n}\right) f_{n}}{\gamma +2 \eta z_{n}} . \label{Cond2}\end{gathered}$$
It is now clear that, in order to satisfy this system – identically, i.e.for any values of the coordinates $z_{n}$ – one must set either
\[case(i)\] $$\begin{gathered}
\text{case~$(i)$}: \ f_{n}= ( a+b z_{n} ) ( \gamma
+2 \eta z_{n} ) \label{fab1}\end{gathered}$$ implying (see (\[fn\])) $$\begin{gathered}
\text{case $(i)$}: \ f^{( 0) }=a \gamma , \qquad f^{(1) }=2 a \eta +b \gamma , \qquad f^{( 2) }=2 b \eta ,\end{gathered}$$
or
\[case(ii)\] $$\begin{gathered}
\text{case $(ii)$}: \ f_{n}=-\alpha z_{n}-\beta z_{n}^{2}+ (
a+b z_{n} ) ( \gamma +2 \eta z_{n} ) \label{fab2}\end{gathered}$$ implying (see (\[fn\])) $$\begin{gathered}
\text{case $(ii)$}: \ f^{( 0) }=a \gamma , \qquad f^{(1) }=2 a \eta +b \gamma -\alpha , \qquad f^{( 2) }=2 b \eta
-\beta .\end{gathered}$$
So, in both cases, we hereafter only retain the freedom to assign the $2$ constants $a$, $b$ rather than the $3$ constants $f^{( 0) }$, $ f^{( 1) }$, $ f^{( 2) }$.
Clearly in case $(i)$ we get the following system of $N$ algebraic equations:
$$\begin{gathered}
\alpha \rho z_{n}+\beta \rho z_{n}^{2}+ [ \alpha +\rho +2 a \eta
+b \gamma +2 ( \beta +2 b \eta ) z_{n} ]
\left[ a \gamma + ( 2 a \eta +b \gamma ) z_{n}+2 b \eta
z_{n}^{2}\right] \nonumber \\
\qquad =2 \eta \left[ \alpha z_{n}+\beta z_{n}^{2}+ ( a+b z_{n} )
( \gamma +2 \eta z_{n} ) \right] ( a+b z_{n} ) ,\end{gathered}$$
and likewise in case $(ii)$ $$\begin{gathered}
\alpha \rho z_{n}+\beta \rho z_{n}^{2}+ [ \rho +2 a \eta +b \gamma
+4 b \eta z_{n} ]
\left[ a \gamma + ( 2 a \eta +b \gamma -\alpha ) z_{n}+ (
2 b \eta -\beta ) z_{n}^{2}\right] \nonumber \\
\qquad =2 \eta \left[ -\alpha z_{n}-\beta z_{n}^{2}+ ( a+b z_{n} )
( \gamma +2 \eta z_{n} ) \right] ( a+b z_{n} ) .\end{gathered}$$
Hence in case $(i)$ the following set of $4$ nonlinear algebraic equations must be satisfied by the $7$ parameters $a$, $b$, $\alpha$, $\beta$, $\gamma$, $\eta $, $\rho $:
\[case1\] $$\begin{gathered}
b \eta ( \beta +2 b \eta ) =0 ,
\\
( \beta +2 b \eta ) ( \rho +2 a \eta +2 b \gamma )
=0 ,
\\
( \alpha +2 a \eta +b \gamma ) ( \rho +b \gamma )
+2 a ( \beta +b \eta ) \gamma =0 ,
\\
a ( \alpha +\rho +b \gamma ) \gamma =0 .\end{gathered}$$
Likewise in case $(ii)$ the following set of $4$ nonlinear algebraic equations must be satisfied by the $7$ parameters $a$, $b$, $\alpha$, $\beta$, $\gamma$, $\eta $, $\rho $:
\[case2\] $$\begin{gathered}
b \eta ( \beta -2 b \eta ) =0 ,
\\
b [ 2 \eta ( 2 a \eta +2 b \gamma -\alpha +\rho ) -\beta
\gamma ] =0 ,
\\
2 a \eta \rho +b \gamma ( 4 a \eta +b \gamma -\alpha +\rho )
=0 ,
\\
a ( \rho +b \gamma ) \gamma =0 .\end{gathered}$$
Then a trivial if rather tedious computation yields, in case $(i)$ respectively in case $(ii)$, the solutions reported in Table 2.1 respectively in Table 2.2 (note that for $\alpha =\beta =\eta
=0$ these two cases coincide, so the corresponding results are only included in Table 2.2).
\[table2.1\]
$\left\vert
\begin{array}{cccccccc}
\# & a & b & \alpha & \beta & \gamma & \eta & \rho \\
1 & 0 & 0 & \ast & \ast & \ast & \ast & 0 \\
2 & \ast & 0 & \ast & 0 & 0 & \ast & 0 \\
3 & \ast & 0 & -2 a \eta & 0 & 0 & \ast & \ast \\
4 & \ast & 0 & -2 a \eta & 0 & \ast & \ast & 2 a \eta \\
5 & \ast & 0 & -2 a \eta & \ast & 0 & \ast & -2 a \eta \\
6 & \ast & 0 & 2 a \eta & 4 a \eta ^{2}/\gamma & \ast & \ast & -2 a \eta \\
7 & 0 & \ast & -b \gamma & 0 & \ast & 0 & \ast \\
8 & \ast & \ast & -b \gamma & 0 & \ast & 0 & 0 \\
9 & 0 & \ast & \ast & 0 & \ast & 0 & -b \gamma \\
10 & 0 & \ast & -b \gamma & \ast & \ast & 0 & -2 b \gamma \\
11 & \ast & \ast & b \gamma & b^{2}\gamma /a & \ast & 0 & -2 b \gamma \\
12 & 0 & \ast & -b \gamma & -2 b \eta & \ast & \ast & \ast \\
13 & 0 & \ast & \ast & -2 b \eta & \ast & \ast & -b \gamma \\
14 & \ast & \ast & \ast & -2 b \eta & 0 & \ast & 0 \\
15 & \ast & \ast & -2 a \eta & -2 b \eta & 0 & \ast & \ast \\
16 & -b \gamma & \ast & \ast & -2 b \eta & \ast & \ast & -\alpha -b \gamma
\\
17 & \ast & \ast & -2 a \eta & -2 b \eta & \ast & \ast & -\alpha -b \gamma\end{array}\right\vert $
[This table indicates the $17$ sets of values to be assigned to the $7$ parameters $a$, $b$, $\alpha$, $\beta$, $\gamma$, $\eta $, $\rho $ in order to satisfy the $4$ equations characterizing case $(i)$, see (\[case1\]). Cases with $\gamma
=\eta =0$ are excluded. Asterisks indicate that the corresponding parameters can be assigned freely. Note that each of the $7$ lines $1$, $12$, $13$, $14 $, $15$, $16$, $17$ assigns values to only $3$ of the $7$ parameters, while each of the other $10$ lines assigns values to $4$ of the $7$ parameters.]{}
[**Table 2.2**]{} \[table2.2\]
$\left\vert
\begin{array}{cccccccc}
\# & a & b & \alpha & \beta & \gamma & \eta & \rho \\
1 & 0 & 0 & \ast & \ast & \ast & \ast & \ast \\
2 & \ast & 0 & \ast & \ast & \ast & \ast & 0 \\
3 & 0 & \ast & b \gamma +\rho & 0 & \ast & 0 & \ast \\
4 & \ast & \ast & 0 & 0 & \ast & 0 & -b \gamma \\
5 & 0 & \ast & b \gamma +\rho & 2 b \eta & \ast & \ast & \ast \\
6 & 0 & \ast & \rho & 2 b \eta & 0 & \ast & \ast \\
7 & \ast & \ast & 2 a \eta & 2 b \eta & \ast & \ast & -b \gamma\end{array}\right\vert $
[This table indicates the $7$ sets of values to be assigned to the $7$ parameters $a$, $b$, $\alpha$, $\beta$, $\gamma$, $\eta $, $\rho $ in order to satisfy the $4$ equations characterizing case $(ii)$, see (\[case2\]). Cases with $\gamma
=\eta =0$ are excluded. Asterisks indicate that the corresponding parameters can be assigned freely. Note that each of the lines $1$ and $2$ assigns values to only $2$ of the $7$ parameters, lines $3$, $4$ and $6$ assign values to $4$ of the $7$ parameters, and lines $5$ and $7$ assign values to $3$ of the $7$ parameters.]{}
We therefore conclude that the many-body models of goldfish type characterized by the *Newtonian* equations of motion (\[EqMotGold1\]) (or, equivalently, (\[EqAns2bis\])) are *solvable* provided either the quantities $f_{n}$ are expressed by (\[fab1\]) and the $7$ parameters $a$, $b$, $\alpha$, $\beta$, $\gamma$, $\eta $, $\rho $ are consistent with Table 2.1 or the quantities $f_{n}$ are expressed by (\[fab2\]) and the $7$ parameters $a$, $b$, $\alpha$, $\beta$, $\gamma$, $\eta $, $\rho $ are consistent with Table 2.2. There are therefore altogether $24$ *solvable* models. Some of these models are however *not new* (in particular, when some parameters vanish); moreover in some cases the solution (\[ExplSolUt\]) with (\[ff\]) of the matrix equation (\[EqUV\]) is only valid in a limiting sense (see Remark \[remark2.2\]). We leave these issues to whoever will be interested – possibly in specific, theoretical or applicative, contexts – in more detailed investigations of anyone of these models; our focus in this paper is rather in the unified treatment, and the simultaneous display (see the appendices below), of all of them (except for the elimination, as already mentioned, of the cases with $\gamma =\eta =0$).
It is convenient to write the corresponding equations of motion in two different ways, depending whether the parameter $\eta $ does or does not vanish.
If the parameter $\eta $ vanishes, $\eta =0$, the equations of motion read as follows: $$\begin{gathered}
\ddot{z}_{n} = -\alpha \rho z_{n}-\beta \rho z_{n}^{2}+( \alpha
+\rho ) \dot{z}_{n}+2 \beta \dot{z}_{n} z_{n} +2 ( \dot{z}_{n}+f_{n}) \sum_{\ell =1, \ell \neq n}^{N}\left[
\frac{\left( \dot{z}_{\ell }+f_{\ell }\right) }{( z_{n}-z_{\ell
}) }\right] \label{EqMotznEtaEqZero}\end{gathered}$$ with the following assignments corresponding respectively to case $(i)$ and case $(ii)$.
In case $(i)$ $$\begin{gathered}
f_{n}=a \gamma +b \gamma z_{n} \label{fnEtaEqZeroCase(i)}\end{gathered}$$and the $7$ parameters $a$, $b$, $\alpha$, $\beta$, $\gamma$, $\eta$, $\rho $ are restricted according to Table 2.3 (being the relevant subcase of Table 2.1).
\[table2.3\]
$\left\vert
\begin{array}{cccccccc}
\# & a & b & \alpha & \beta & \gamma & \eta & \rho \\
1 & 0 & \ast & -b \gamma & 0 & \ast & 0 & \ast \\
2 & \ast & \ast & -b \gamma & 0 & \ast & 0 & 0 \\
3 & 0 & \ast & \ast & 0 & \ast & 0 & -b \gamma \\
4 & 0 & \ast & -b \gamma & \ast & \ast & 0 & -2 b \gamma \\
5 & \ast & \ast & b \gamma & b^{2}\gamma /a & \ast & 0 & -2 b \gamma\end{array}\right\vert $
[This table indicates the $5$ sets of values to be assigned to the $7$ parameters $a$, $b$, $\alpha$, $\beta$, $\gamma$, $\eta$, $\rho $ in (\[EqMotznEtaEqZero\]) with (\[fnEtaEqZeroCase(i)\]). Asterisks indicate that the corresponding parameters can be assigned freely. Note that in every case there are $3$ free parameters.]{}
In case $(ii)$ $$\begin{gathered}
f_{n}=a \gamma +\left( b \gamma -\alpha \right) z_{n}-\beta z_{n}^{2}
\label{fnEtaEqZeroCase(ii)}\end{gathered}$$and the $6$ parameters $a$, $b$, $\alpha$, $\beta$, $\gamma$, $\rho $ are restricted according to Table 2.4 (being the relevant subcase of Table 2.2).
\[table2.4\]
$\left\vert
\begin{array}{cccccccc}
\# & a & b & \alpha & \beta & \gamma & \eta & \rho \\
1 & 0 & \ast & b \gamma +\rho & 0 & \ast & 0 & \ast \\
2 & \ast & \ast & 0 & 0 & \ast & 0 & -b \gamma\end{array}\right\vert $
[This table indicates the set of values to be assigned to the $7$ parameters $a$, $b$, $\alpha$, $\beta$, $\gamma$, $\eta$, $\rho $ in (\[EqMotznEtaEqZero\]) with (\[fnEtaEqZeroCase(ii)\]). Asterisks indicate that the corresponding parameters can be assigned freely: in the second line $a$, $b$, and $\gamma $ are $3$ free parameters, in the first line $b$, $\gamma $ and $\rho $ are $3$ free parameters.]{}
If the parameter $\eta $ does *not* vanish, $\eta \neq 0$, the equations of motion are more conveniently written in terms of the dependent variables $$\begin{gathered}
x_{n}(t) \equiv z_{n}(t) +\frac{\gamma }{2 \eta } ,\end{gathered}$$ reading then as follows:
\[EqMotznEtaNotZero\] $$\begin{gathered}
\ddot{x}_{n} = \frac{\dot{x}_{n}^{2}}{x_{n}}+\lambda \frac{\dot{x}_{n}}{x_{n}}+\beta \dot{x}_{n} x_{n}+\rho \left( \dot{x}_{n}+\lambda -\mu
x_{n}-\beta x_{n}^{2}\right) \nonumber \\
\hphantom{\ddot{x}_{n} =}{}
+ ( \dot{x}_{n}+f_{n} ) \sum_{\ell =1, \ell \neq n}^{N}\left[
\frac{ ( \dot{x}_{\ell }+f_{\ell } ) ( x_{n}+x_{\ell } )
}{( x_{n}-x_{\ell }) x_{\ell }}\right]\end{gathered}$$ or equivalently $$\begin{gathered}
\ddot{x}_{n}=\big( \lambda -2 f^{(0) }\big) \frac{\dot{x}_{n}}{x_{n}}+[ ( N-2) f^{(1) }+\rho ]
\dot{x}_{n}+\big( -2 f^{(2) }+\beta \big) \dot{x}_{n} x_{n}
-\frac{\big( f^{(0) }\big) ^{2}}{x_{n}}+\rho \lambda\nonumber\\
\hphantom{\ddot{x}_{n}=}{}
+ ( N-2 ) f^{(0) } f^{(1) }
+\big[ {-}\rho \mu + ( N-1 ) \big( f^{(1) }\big)
^{2}-2 f^{(0) } f^{(2) }\big] x_{n} \nonumber \\
\hphantom{\ddot{x}_{n}=}{}
+\big[ {-}\beta \rho + ( N-2 ) f^{(1) } f^{(2) }\big] x_{n}^{2}-\big( f^{(2) }\big)
^{2} x_{n}^{3}
+ ( \dot{x}_{n}+f_{n} ) \sum_{k=1}^{N}\left[ \frac{\big( \dot{x}_{k}+f^{(0) }\big) }{x_{k}}+f^{(2) } x_{k}\right]
\nonumber \\
\hphantom{\ddot{x}_{n}=}{}
+2 \sum_{\ell =1, \ell \neq n}^{N}\left[ \frac{( \dot{x}_{n}+f_{n}) ( \dot{x}_{\ell }+f_{\ell }) }{(
x_{n}-x_{\ell }) }\right]\end{gathered}$$ with $$\begin{gathered}
\lambda =\frac{( 2 \alpha \eta -\beta \gamma ) \gamma }{4 \eta
^{2}} , \qquad \mu =\alpha -\frac{\beta \gamma }{\eta } \qquad \text{so that} \qquad \mu
^{2}=\alpha ^{2}-\frac{\lambda }{4} , \label{landamu}\end{gathered}$$
and with the following assignments corresponding respectively to case $(i)$ and case $(ii)$.
In case $(i)$ $$\begin{gathered}
f_{n}=( 2 a \eta -b \gamma ) x_{n}+2 b \eta x_{n}^{2}
\label{fnEtaEqNotZeroCase(i)}\end{gathered}$$and the $7$ parameters $a$, $b$, $\alpha$, $\beta$, $\gamma$, $\eta $, $\rho $ are restricted according to Table 2.5 (being the relevant subcase of Table 2.1).
\[table2.5\]
$\left\vert
\begin{array}{cccccccc}
\# & a & b & \alpha & \beta & \gamma & \eta & \rho \\
1 & 0 & 0 & \ast & \ast & \ast & \ast & 0 \\
2 & \ast & 0 & \ast & 0 & 0 & \ast & 0 \\
3 & \ast & 0 & -2 a \eta & 0 & 0 & \ast & \ast \\
4 & \ast & 0 & -2 a \eta & 0 & \ast & \ast & 2 a \eta \\
5 & \ast & 0 & -2 a \eta & \ast & 0 & \ast & -2 a \eta \\
6 & \ast & 0 & 2 a \eta & 4 a \eta ^{2}/\gamma & \ast & \ast & -2 a \eta \\
7 & 0 & \ast & -b \gamma & -2 b \eta & \ast & \ast & \ast \\
8 & 0 & \ast & \ast & -2 b \eta & \ast & \ast & -b \gamma \\
9 & \ast & \ast & \ast & -2 b \eta & 0 & \ast & 0 \\
10 & \ast & \ast & -2 a \eta & -2 b \eta & 0 & \ast & \ast \\
11 & -b \gamma & \ast & \ast & -2 b \eta & \ast & \ast & -\alpha -b \gamma
\\
12 & \ast & \ast & -2 a \eta & -2 b \eta & \ast & \ast & -\alpha -b \gamma\end{array}\right\vert $
[This table indicates the $12$ sets of values to be assigned to the $7$ parameters $a$, $b$, $\alpha$, $\beta$, $\gamma$, $\eta $, $\rho $ in (\[EqMotznEtaNotZero\]) with (\[fnEtaEqNotZeroCase(i)\]). Asterisks indicate that the corresponding parameters can be assigned freely (with $\eta \neq 0$). Note that each of the $7$ lines $1$, 7, 8, 9, 10, 11, 12 assigns values to only $3$ of the $7 $ parameters, while the other $5$ assign values to $4$ of the $7$ parameters.]{}
In case $(ii)$ $$\begin{gathered}
f_{n}=\frac{( 2 \alpha \eta -\beta \gamma ) \gamma }{4 \eta
^{2}}+\left( 2 a \eta -b \gamma -\alpha +\frac{\beta \gamma }{\eta }\right)
x_{n}+( 2 b \eta -\beta ) x_{n}^{2}
\label{fnEtaEqNotZeroCase(ii)}\end{gathered}$$ and the $7$ parameters $a$, $b$, $\alpha$, $\beta$, $\gamma$, $\eta $, $\rho $ are restricted according to Table 2.6 (being the relevant subcase of Table 2.2):
\[table2.6\]
$\left\vert
\begin{array}{cccccccc}
\# & a & b & \alpha & \beta & \gamma & \eta & \rho \\
1 & 0 & 0 & \ast & \ast & \ast & \ast & \ast \\
2 & \ast & 0 & \ast & \ast & \ast & \ast & 0 \\
3 & 0 & \ast & b \gamma +\rho & 2 b \eta & \ast & \ast & \ast \\
4 & 0 & \ast & \rho & 2 b \eta & 0 & \ast & \ast \\
5 & \ast & \ast & 2 a \eta & 2 b \eta & \ast & \ast & -b \gamma\end{array}\right\vert $
[This table indicates the $5$ sets of values to be assigned to the $7$ parameters $a$, $b$, $\alpha$, $\beta$, $\gamma$, $\eta$, $\rho $ in (\[EqMotznEtaNotZero\]) with (\[fnEtaEqNotZeroCase(ii)\]). Asterisks indicate that the corresponding parameters can be assigned freely (with $\eta \neq 0$). Note that each of the first $2$ lines assigns values to only $2$ of the $7$ parameters, while the third and fifth lines assign values to $3$ of the $7$ parameters, and the fourth line assign values to $4$ of the $7$ parameters.]{}
These $24$ *Newtonian* equations of motion are exhibited in Appendix \[appendixA\]. Their *solvable* character is of course implied by the fact that the $N$ coordinates $z_{n}(t) $ coincide with the $N$ eigenvalues of the matrix $U(t) $ (see (\[UZ\])). To ascertain the behavior of these solutions $z_{n}(t) $ one must in each case take account of the restrictions on the parameters characterizing these models, as detailed above, which are of course also relevant in order to identify the corresponding evolution of the matrix $U(t) $: as implied by inserting in the explicit formula (\[ExplSolUt\]) with (\[ff\]) – in addition to the parameters $\alpha $, $\beta $, $\gamma$, $\eta $, $\rho $ associated with the *solvable* $N $-body model under consideration – the expressions of the *initial* values $U(0) $ and $V(0) \equiv V_{0}$ of the matrices $U(t) $ and $V(t) $ in terms of the $N$ *initial* values $z_{n}(0) $ of the $N$ coordinates and the $N$ *initial* values $\dot{z}_{n}(0) $ of the $N$ velocities. To obtain these expressions it is useful to note that it is possible – and convenient – to assume that the *diagonalizing* matrix $R(t) $ (see (\[UVM\])) is *initially* just the $N\times N$ *unit* matrix $I$, $$\begin{gathered}
R(0) =I ,\end{gathered}$$implying *initially* (see (\[UVM\]))
$$\begin{gathered}
U(0) =\text{diag} [ z_{n}(0) ]
, \qquad U_{nm}(0) =\delta _{nm} z_{n}(0) ,
\label{Uzero}
\\
V_{0}\equiv V(0) =Y(0) , \qquad V_{nm}(0)
=\delta _{nm} y_{n}(0) + ( 1-\delta _{nm} )
Y_{nm}(0) . \label{Vzero}\end{gathered}$$
The first of these two formulas provides the explicit expression of $U (0) $ in terms of the initial data $z_{n}(0) $.
In the second formula the *initial* values $y_{n}(0) $ of the *diagonal* elements of the matrix $V_{0}$ in terms of the initial coordinates $z_{n}(0) $ and velocities $\dot{z}_{n}(0) $ of the $N$ particles read $$\begin{gathered}
y_{n}(0) =\frac{\dot{z}_{n}(0) -\alpha z_{n} (0) -\beta z_{n}^{2}(0) }{\gamma +2 \eta z_{n}(0) } \label{ynzero}\end{gathered}$$(see (\[yn\])), while the *off-diagonal* elements $Y_{nm}(0) $ (with $n\neq m$) of the matrix $V_{0}$ are given by the *ansatz* (\[Ans2\]) yielding $$\begin{gathered}
Y_{nm}(0) =\big\{ g_{n}(0) g_{m}(0)
[ \dot{z}_{n}(0) +f_{n}(0) ] [ \dot{z}_{m}(0) +f_{m}(0) ] \big\}
^{1/2} , \qquad n\neq m , \label{Ynmzerob}\end{gathered}$$with the quantities $g_{n}(0) $ and $f_{n}(0) $ given by the formulas (see (\[fg\]))$$\begin{gathered}
g_{n}(0) =\frac{1}{\gamma +2 \eta z_{n}(0) }
, \qquad f_{n}=f^{(0) }+f^{(1) } z_{n}(0)
+f^{(2) } z_{n}^{2}(0) ,\end{gathered}$$
with the appropriate assignments of the parameters $\gamma $, $\eta $, $f^{(0) }$, $f^{(1) }$ and $f^{(2) }$ characterizing the various *solvable* many-body models, see above (in particular for $f^{(0) }$, $f^{(1) }$ and $f^{(2) }$ see (\[fnEtaEqZeroCase(i)\]) or (\[fnEtaEqZeroCase(ii)\]) or (\[fnEtaEqNotZeroCase(i)\]) or (\[fnEtaEqNotZeroCase(ii)\]), as appropriate).
Note moreover that the *dyadic* character of the *off-diagonal* part of the matrix $V_{0}=Y(0)$, see (\[Ynmzerob\]), implies a simplification when one must compute functions of this matrix $V_{0}$ such as those appearing in the explicit expression (\[ExplSolUt\]) with (\[ff\]) of $U(t) $; this simplification becomes particularly significant when the matrix $V_{0}=Y(0) $ is altogether *dyadic*, $Y_{nm}(0) =v_{n} v_{m}$, since for any *dyadic* matrix, say $X_{nm}=x_{n} x_{m}$, there holds the simple formula $$\begin{gathered}
\varphi ( X ) =\varphi (0) I+\frac{\varphi (
x ) -\varphi (0) }{x} X , \qquad x^{2}=\sum_{k=1}^{N}x_{n}^{2} ,\end{gathered}$$where $\varphi ( x ) $ is any (scalar) function for which the (matrix) expression $\varphi ( X ) $ makes good sense. This simplification clearly happens iff
$$\begin{gathered}
f_{n}=-\alpha z_{n}-\beta z_{n}^{2} ,\end{gathered}$$
as implied by (\[yn\]) and (\[Ans2\]) with (\[fg\]), hence in case $(i)$ whenever (see (\[case(i)\]) and Table 2.1)$$\begin{gathered}
a \gamma =0 , \qquad 2 a \eta +b \gamma =-\alpha , \qquad 2 b \eta =-\beta ,\end{gathered}$$and in case $(ii)$ whenever (see (\[case(ii)\]) and Table 2.2) $$\begin{gathered}
a=b=0 .\end{gathered}$$
*Special cases and their $($autonomous$)$ isochronous variants*. Certain special models among those identified above as *solvable* can be *isochronized* via the following change of dependent and independent variables, $$\begin{gathered}
z_{n}(t) =\exp ( \text{\textbf{i}} \sigma \omega t )
\zeta _{n}\left( \tau \right) , \qquad \tau =\frac{\exp ( \text{\textbf{i}} \omega t ) -1}{\text{\textbf{i}} \omega } . \label{trick}\end{gathered}$$ Here the quantities $\zeta _{n} ( \tau ) $ are assumed to satisfy the *Newtonian* equations written above, see (\[EqMotznEtaEqZero\]) with (\[fnEtaEqZeroCase(i)\]) or (\[fnEtaEqZeroCase(ii)\]), of course with the new (*complex*) independent variable $\tau $ replacing the time $t$; $\omega $ is an *arbitrary* *real* (for definiteness, *positive*) *constant* to which we associate the period $$\begin{gathered}
T=\frac{2 \pi }{\omega } ; \label{T}\end{gathered}$$and the number $\sigma $ is adjusted so as to produce, for the dependent variables $z_{n}\equiv z_{n}(t) $ (with the *real* independent variable $t$ interpreted as “time”) *autonomous* equations of motion (the special models providing the starting points for the application of this trick being appropriately selected to allow such an outcome). Since the application of this trick is by now quite standard (see, for instance, Section 2.1 entitled “The trick” of [@C2008]), we dispense here from any detailed discussion of this approach and limit ourselves to reporting the results.
This trick is only applicable to (\[EqMotznEtaEqZero\]) with (\[fnEtaEqZeroCase(i)\]) in the very special cases with $\alpha =\rho =0$ and either $\gamma =0$ or $a=b=0$ (as long as one is only interested in getting *autonomous* equations of motion). Then the assignment $\sigma =1$ yields the *isochronous* equations of motion
\[Isozn\] $$\begin{gathered}
\ddot{z}_{n} =3 \text{\textbf{i}} \omega \dot{z}_{n}+2 \omega
^{2} z_{n}+2 \beta z_{n} ( \dot{z}_{n}-\text{\textbf{i}} \omega
z_{n} ) +2 ( \dot{z}_{n}-\text{\textbf{i}} \omega z_{n} ) \sum_{\ell
=1,\, \ell \neq n}^{N}\left[ \frac{ ( \dot{z}_{\ell }-\text{\textbf{i}} \omega z_{\ell } ) }{z_{n}-z_{\ell }}\right] .\end{gathered}$$
Likewise the application of this trick to (\[EqMotznEtaEqZero\]) with (\[fnEtaEqZeroCase(ii)\]), again in the very special cases with $\alpha =\rho
=0 $ and either $\gamma =0$ or $a=b=0$, and again with the assignment $\sigma =1 $, yields the *isochronous* equations of motion$$\begin{gathered}
\ddot{z}_{n} = 3 \text{\textbf{i}} \omega \dot{z}_{n}+2 \omega
^{2} z_{n}+2 \beta z_{n} ( \dot{z}_{n}-\text{\textbf{i}} \omega
z_{n} ) \nonumber \\
\hphantom{\ddot{z}_{n} =}{}
+2 ( \dot{z}_{n}-\text{\textbf{i}} \omega z_{n}-\beta
z_{n}^{2} ) \sum_{\ell =1, \ell \neq n}^{N}\left[ \frac{( \dot{z}
_{\ell }-\text{\textbf{i}} \omega z_{\ell }-\beta z_{\ell }^{2}) }{z_{n}-z_{\ell }}\right] .\end{gathered}$$
Neither one of these two models is however new: see Examples 4.2.2-6 and 4.2.2-7 in [@C2008].
An analogous treatment applied, *mutatis mutandis*, to ([EqMotznEtaNotZero]{}) with (\[fnEtaEqNotZeroCase(i)\]) in the special case with $\rho =0$, $2 a \eta =b \gamma $ and $2 \alpha \eta =\beta \gamma $ (implying $\lambda =0$), yields (again via the assignment $\sigma =1$) the *isochronous* equations of motion $$\begin{gathered}
\ddot{x}_{n}=\text{\textbf{i}} \omega \dot{x}_{n}+\omega ^{2} x_{n}+\frac{\dot{x}_{n}^{2}}{x_{n}}+\beta x_{n} ( \dot{x}_{n}-\text{\textbf{i}} \omega x_{n}) +\left( \dot{x}_{n}-\text{\textbf{i}} \omega x_{n}+2 b \eta
x_{n}^{2}\right) \nonumber \\
\hphantom{\ddot{x}_{n}=}{}
\times \sum_{\ell =1, \ell \neq n}^{N}\left[ \frac{( \dot{x}_{\ell }-
\text{\textbf{i}} \omega x_{\ell }+2 b \eta x_{\ell }^{2}) (
x_{n}+x_{\ell }) }{( x_{n}-x_{\ell }) x_{\ell }}\right] .\end{gathered}$$
Likewise an analogous treatment applied to (\[EqMotznEtaNotZero\]) with (\[fnEtaEqNotZeroCase(ii)\]) in the special case with $\rho =0$ and either $2 \alpha \eta =\beta \gamma $ (implying $\lambda =0$) and $\alpha
=-2 a \eta +b \gamma $ or $\gamma =0$ (implying $\lambda =0$) and $\alpha
=2 a \eta $, yields (again via the assignment $\sigma =1$) the same *isochronous* equations of motion (up to the, merely notational, replacement of $2 b \eta $ with $2 b \eta -\beta $).
While finally this treatment applied, with $\sigma =-1$, to (\[EqMotznEtaNotZero\]) with (\[fnEtaEqNotZeroCase(ii)\]) in the special case with $b=\beta =\rho =0$ and $\alpha =2 a \eta $ yields the *isochronous* equations of motion $$\begin{gathered}
\ddot{x}_{n}=\text{\textbf{i}} \omega \dot{x}_{n}-\omega ^{2} x_{n}+\frac{\dot{x}_{n}^{2}}{x_{n}}+a \gamma \frac{\dot{x}_{n}}{x_{n}}+a \gamma \text{\textbf{i}} \omega \nonumber \\
\hphantom{\ddot{x}_{n}=}{}
+ ( \dot{x}_{n}+\text{\textbf{i}} \omega x_{n}+a \gamma )
\sum_{\ell =1, \ell \neq n}^{N}\left[ \frac{ ( \dot{x}_{\ell }+\text{\textbf{i}} \omega x_{\ell }+a \gamma ) ( x_{n}+x_{\ell
} ) }{( x_{n}-x_{\ell }) x_{\ell }}\right] .\end{gathered}$$
A related class of *solvable* many-body models {#section2.3}
----------------------------------------------
In this subsection we consider the *Newtonian* equations of motion that obtain by identifying the $N$ dependent variables of the models discussed above as the $N$ *zeros* of a monic (time-dependent) polynomial of degree $N$, and by then focussing on the time-evolution of the $N$ *coefficients* of this polynomial. It is again convenient to treat separately the two cases with $\eta =0$ and with $\eta \neq 0$.
In the $\eta =0$ case the starting point are the equations of motion (\[EqMotznEtaEqZero\]) with (\[fnEtaEqZeroCase(i)\]) or (\[fnEtaEqZeroCase(ii)\]). We then introduce the time-dependent (monic) polynomial $\psi ( z,t) $ whose zeros are the $N$ eigenvalues $z_{n}(t) $ of the $N\times N$ matrix $U(t) $:
\[Psi\] $$\begin{gathered}
\psi ( z,t ) =\det [ z I-U(t) ] ,
\\
\psi \left( z,t\right) =\prod_{n=1}^{N} [ z-z_{n}(t) ]
=z^{N}+\sum_{m=1}^{N} \big[ c_{m}(t) z^{N-m} \big] .
\label{psipsi}\end{gathered}$$ The last of these formulas introduces the $N$ coefficients $c_{m}\equiv
c_{m}(t) $ of the monic polynomial $\psi ( z,t ) $; of course it implies that these coefficients are related to the zeros $z_{n}(t) $ as follows: $$\begin{gathered}
c_{1}=-\sum_{n=1}^{N}z_{n} , \qquad c_{2}=\sum_{n,m=1; n>m}^{N}z_{n} z_{m} ,
\label{c12}\end{gathered}$$
and so on.
The fact that the initial-value problem associated with the time evolution of the $N$ coordinates $z_{n}$ can be *solved* by *algebraic* operations implies that the same *solvable* character can be attributed to the time evolution of the monic polynomial $\psi (
z,t ) $ and of the $N$ coefficients $c_{m}(t) $. The procedure to obtain the equations of motion satisfied by the $N$ coefficients $c_{m}(t) $ from the $N$ equations of motion satisfied by the $N$ zeros is tedious but standard; a key role in this development are the identities reported, for instance, in Appendix A of [@C2008] (but note that there are two misprints in these formulas: in equation (A.8k) the term $ ( N+1 ) $ inside the square brackets should instead read $ ( N-3 ) $; in equation (A.8l) the term $N^{2}$ inside the square brackets should instead read $N ( N-2 ) $ – these misprints have been corrected in the recent paperback version of this monograph [@C2008]). Here we limit our presentation to reporting the final result.
The equation characterizing the time evolution of the monic polynomial $\psi
( z,t ) $ implied by the *Newtonian* equations of motion (\[EqMotznEtaEqZero\]) with (\[fn\]) reads as follows: $$\begin{gathered}
\psi _{tt}-2 \big( f^{(0) }+f^{(1) } z+f^{(2) } z^{2}\big) \psi _{zt}+\big( p^{(0) }+p^{(1) } z\big) \psi _{t} \nonumber\\
\qquad{}
+\big( q_{2}^{(0) }+q_{2}^{(1) } z+q_{2}^{(2) } z^{2}+q_{2}^{( 3) } z^{3}+q_{2}^{( 4)
} z^{4}\big) \psi _{zz} \nonumber \\
\qquad{} +\big( q_{1}^{(0) }+q_{1}^{(1) } z+q_{1}^{(2) } z^{2}+q_{1}^{( 3) } z^{3}\big) \psi _{z}
+\big( q_{0}^{(0) }+q_{0}^{(1) } z+q_{0}^{(2) } z^{2}\big) \psi =0 , \label{PDEpsi}\end{gathered}$$ with
\[pq\] $$\begin{gathered}
p^{(0) }=-\alpha -\rho +2 ( N-1 ) f^{(1)
}-2 f^{(2) } c_{1} , \qquad p^{(1) }=2 \big[ {-}\beta
+ ( N-2 ) f^{(2) }\big] ;
\\
q_{2}^{(0) } = \big( f^{(0) }\big)
^{2} , \qquad q_{2}^{(1) }=2 f^{(0) } f^{(1)
} , \qquad q_{2}^{(2) }=2 f^{(0) } f^{(2)
}+\big( f^{(1) }\big) ^{2} , \nonumber \\
q_{2}^{(3) } = 2 f^{(1) } f^{(2)
} , \qquad q_{2}^{(4) }=\big( f^{(2) }\big) ^{2} ;
\\
q_{1}^{(0) } = -2 (N-1) f^{(0)
} f^{(1) }+2 f^{(0) } f^{(2) } c_{1} ,
\nonumber \\
q_{1}^{(1) } = -\alpha \rho -2 (N-2) f^{(0) } f^{(2) }+2 f^{(1) } f^{(2)
} c_{1} -2 (N-1) \big( f^{(1) }\big) ^{2} ,
\nonumber \\
q_{1}^{(2) } = -\beta \rho -2 ( 2 N-3 ) f^{ (1) } f^{(2) }+2 \big( f^{(2) }\big)
^{2} c_{1} , \nonumber \\
q_{1}^{(3) } = -2 (N-2) \big( f^{(2)
}\big) ^{2} ;
\\
q_{0}^{(0) } = N \alpha \rho -2 N f^{(0)
} f^{(2) }+N (N-1) \big( f^{(1)
}\big) ^{2} \nonumber \\
\hphantom{q_{0}^{(0) } =}{}
-\big[ \beta \rho +2 (N-1) f^{(1) } f^{(2) }\big] c_{1}+2 \big( \beta +f^{(2) }\big) \dot{c}_{1}+2 \big( f^{(2) }\big) ^{2} c_{2} , \nonumber \\
q_{0}^{(1) } = N \beta \rho +2 N (N-2) f^{(1) } f^{(2) }-2 (N-1) \big( f^{(2) }\big) ^{2} c_{1} , \nonumber \\
q_{0}^{(2) } = N ( N-3 ) \big( f^{(2)
}\big) ^{2} ,\end{gathered}$$
where of course the quantities $f^{(0) }$, $f^{(1)
}$, $f^{(2) }$ should be expressed in terms of the other parameters as implied by (\[fn\]) with (\[fnEtaEqZeroCase(i)\]) or (\[fnEtaEqZeroCase(ii)\]), as the case may be. Note that, while this equation, (\[PDEpsi\]), satisfied by the function $\psi ( z,t ) $ (where of course subscripted variables denote partial differentiations) might seem a *linear PDE*, it is in fact a *nonlinear functional equation*, because some of its coefficients, see (\[pq\]), depend on the quantities $c_{1}$ and $c_{2}$ which themselves depend on $\psi $, indeed clearly (see (\[Psi\]))
$$\begin{gathered}
c_{m}\equiv c_{m}(t) =\frac{\psi ^{ ( N-m ) }(0,t) }{( N-m) !} ,\end{gathered}$$
where we used the shorthand notation $\psi ^{( j) }(z,t) $ to denote the $j$-th partial derivative with respect to the variable $z$ of $\psi ( z,t) $, $$\begin{gathered}
\psi ^{( j) }( z,t) \equiv \frac{\partial ^{j} \psi
( z,t) }{\partial z^{j}} , \qquad j=1,2,\dots .\end{gathered}$$
Likewise, the equation characterizing the time evolution of the monic polynomial $\phi \left( x,t\right) $ implied by the *Newtonian* equations of motion (\[EqMotznEtaNotZero\]) via the following assignment (analogous to (\[psipsi\])), $$\begin{gathered}
\phi ( x,t ) =\prod_{n=1}^{N} [ x-x_{n}(t) ]
=x^{N}+\sum_{m=1}^{N}\big[ c_{m}(t) x^{N-m}\big] ,
\label{chi}\end{gathered}$$ reads $$\begin{gathered}
\phi _{tt}-2 \big( f^{(0) }+f^{(1) } x+f^{(2) } x^{2}\big) \phi _{xt}+\left( \frac{p^{( -1) }}{x}
+p^{(0) }+p^{(1) } x\right) \phi _{t} \nonumber
\\
\qquad+\big( q_{2}^{(0) }+q_{2}^{(1) } x+q_{2}^{(2) } x^{2}+q_{2}^{(3) } x^{3}+q_{2}^{(4)
} x^{4}\big) \phi _{xx} \\
\qquad{}
+\left( \frac{q_{1}^{( -1) }}{x}+q_{1}^{(0)
}+q_{1}^{(1) } x+q_{1}^{(2) } x^{2}+q_{1}^{(3) } x^{3}\right) \phi _{x} \nonumber \\
\qquad{}+\left( \frac{q_{0}^{( -1) }}{x}+q_{0}^{(0)
}+q_{0}^{(1) } x+q_{0}^{(2) } x^{2}\right) \phi
=0, \label{PDEchi}\end{gathered}$$ now with
\[pqchi\] $$\begin{gathered}
p^{\left( -1\right) } = 2 f^{(0) }-\lambda , \qquad
p^{(0) } = -\rho +N f^{(1) }-f^{(2)
} c_{1}+f^{(0) } \frac{c_{N-1}}{c_{N}}-\frac{\dot{c}_{N}}{c_{N}} , \nonumber \\
p^{(1) } = 2 (N-1) f^{(2) }-\beta ;
\\
q_{2}^{(0) } = \big( f^{(0) }\big)
^{2} , \qquad q_{2}^{(1) }=2 f^{(0) } f^{(1)
} , \qquad q_{2}^{(2) }=2 f^{(0) } f^{(2)
}+\big( f^{(1) }\big) ^{2} , \nonumber \\
q_{2}^{(3) } = 2 f^{(1) } f^{(2)
} , \qquad q_{2}^{(4) }=\big( f^{(2) }\big) ^{2} ;
\\
q_{1}^{( -1) } = -\big( f^{(0) }\big) ^{2} ,
\qquad
q_{1}^{(0) } = \rho \lambda -N f^{(0) } f^{(1) }+f^{(0) } f^{(2) } c_{1}+f^{(0) } \frac{\dot{c}_{N}}{c_{N}}-\big( f^{(0) }\big) ^{2} \frac{c_{N-1}}{c_{N}} , \nonumber \\
q_{1}^{(1) } = -\rho \mu -2 (N-1) f^{(0) } f^{(2) }+f^{(1) } f^{(2)
} c_{1}-(N-1) \big( f^{(1) }\big) ^{2}
+f^{(1) } \frac{\dot{c}_{N}}{c_{N}}-f^{(0)
} f^{(1) } \frac{c_{N-1}}{c_{N}} , \nonumber \\
q_{1}^{(2) } = -\beta \rho - ( 3 N-4 ) f^{(1 ) } f^{(2) }
+\big( f^{(2) }\big) ^{2} c_{1}+f^{(2) } \frac{\dot{c}_{N}}{c_{N}}-f^{(0) } f^{(2) } \frac{c_{N-1}}{c_{N}} , \nonumber \\
q_{1}^{(3) } = - ( 2 N-3 ) \big( f^{(2)
}\big) ^{2} ;
\\
q_{0}^{( -1) } = \big( \lambda -2 f^{(0) }\big) \frac{\dot{c}_{N}}{c_{N}}+\big( f^{(0) }\big) ^{2} \frac{c_{N-1}}{c_{N}} , \nonumber \\
q_{0}^{(0) } = -\beta \rho c_{1}+\beta \dot{c}_{1}+\big(
f^{(2) } c_{1}-N f^{(1) }\big) \frac{\dot{c}_{N}}{c_{N}}
+f^{(0) } \big( N f^{(1) }-f^{(2)
} c_{1}\big) \frac{c_{N-1}}{c_{N}} , \nonumber \\
q_{0}^{(1) } = N \beta \rho +N (N-2) f^{(1) } f^{(2) }-(N-1) \big( f^{(2)
}\big) ^{2} c_{1} -N f^{(2) } \frac{\dot{c}_{N}}{c_{N}}+f^{(0)
} f^{(2) } \frac{c_{N-1}}{c_{N}} , \nonumber \\
q_{0}^{(2) } = N (N-2) \big( f^{(2)
}\big) ^{2} .\end{gathered}$$
The equations of motion of *Newtonian* type satisfied by the $N$ coefficients $c_{m}(t) $ which obtain from (\[PDEpsi\]) hence correspond to the *Newtonian* equations of motion (\[EqMotznEtaEqZero\]) read as follows: $$\begin{gathered}
\ddot{c}_{m}-2 ( N-m+1 ) f^{(0) } \dot{c}_{m-1}+\big[
-2 ( N-m ) f^{(1) }+p^{(0) }\big] \dot{c}_{m} \nonumber\\
\qquad+\big[ {-}2 ( N-m-1 ) f^{(2) }+p^{(1) }\big] \dot{c}_{m+1}
+ ( N-m+2 ) ( N-m+1 ) q_{2}^{(0) } c_{m-2}
\nonumber \\
\qquad{}+ ( N-m+1 ) \big[ ( N-m ) q_{2}^{(1)
}+q_{1}^{(0) }\big] c_{m-1} \nonumber \\
\qquad{}
+\big\{ ( N-m ) \big[ ( N-m-1 ) q_{2}^{(2) }+q_{1}^{(1) }\big] +q_{0}^{(0) }\big\}
c_{m} \nonumber \\
\qquad{}
+\big\{ ( N-m-1 ) \big[ ( N-m-2 ) q_{2}^{(3) }+q_{1}^{(2) }\big] +q_{0}^{(1) }\big\}
c_{m+1} \nonumber \\
\qquad {} +\big\{ ( N-m-2 ) \big[ ( N-m-3 ) q_{2}^{(4) }+q_{1}^{(3) }\big] +q_{0}^{(2) }\big\}
c_{m+2}=0 , \label{Eqcm}\end{gathered}$$ where $c_{n}$ vanishes for $n<0$ and for $n>N$ while $c_{0}=1$ (see (\[Psi\])), and of course the coefficients $p^{( j) }$ and $q_{k}^{( j) }$ are defined by (\[pq\]). Again, this system of ODEs might seem *linear*, but it is in fact *nonlinear* because some of its coefficients depend on the dependent variables $c_{1}$ and $c_{2}$, see (\[pq\]). The more explicit version of these equations of motion that obtain by expressing the various coefficients in terms of the free parameters are listed in Appendix \[appendixB\]. They are of course just as *solvable* as the *Newtonian* equations of motion satisfied by the $N$ coordinates $x_{n}(t) $, see Appendix \[appendixA\], to which they correspond via (\[Psi\]); and in particular whenever the parameter $\rho $ is *imaginary* and the parameter $\alpha $ is *real* and *negative* they are *asymptotically isochronous* with period $T$ (see Remark \[remark2.3\]).
Likewise, the equations of motion of *Newtonian* type satisfied by the $N$ coefficients $c_{m}(t) $ which obtain from (\[PDEchi\]) hence correspond to the *Newtonian* equations of motion (\[EqMotznEtaNotZero\]) read as follows: $$\begin{gathered}
\ddot{c}_{m}+\big[ p^{( -1) }-2 ( N-m+1) f^{(0) }\big] \dot{c}_{m-1}
+\big[ {-}2 ( N-m ) f^{(1) }+p^{(0) }
\big] \dot{c}_{m}\nonumber\\
\qquad{}+\big[ {-}2 ( N-m-1 ) f^{(2)
}+p^{(1) }\big] \dot{c}_{m+1}
+ ( N-m+2 ) \big[ ( N-m+1 ) q_{2}^{(0)
}+q_{1}^{ ( -1 ) }\big] c_{m-2} \nonumber \\
\qquad {}
+\big\{ ( N-m+1 ) \big[ ( N-m ) q_{2}^{(1) }+q_{1}^{(0) }\big] +q_{0}^{( -1)
}\big\} c_{m-1} \nonumber \\
\qquad {}
+\big\{ ( N-m ) \big[ ( N-m-1 ) q_{2}^{(2) }+q_{1}^{(1) }\big] +q_{0}^{(0) }\big\}
c_{m} \nonumber \\
\qquad{}
+\big\{ ( N-m-1 ) \big[ ( N-m-2 ) q_{2}^{(3) }+q_{1}^{(2) }\big] +q_{0}^{(1) }\big\}
c_{m+1} \nonumber \\
\qquad{}
+\big\{ ( N-m-2 ) \big[ ( N-m-3 ) q_{2}^{ (4) }+q_{1}^{(3) }\big] +q_{0}^{(2) }\big\}
c_{m+2}=0 , \label{Eqcm2}\end{gathered}$$ where of course again $c_{n}$ vanishes for $n<0$ and for $n>N$ while $c_{0}=1 $ (see (\[chi\])) and of course the coefficients $p^{(j)}$ and $q_{k}^{(j) }$ are now defined by (\[pqchi\]). Again, this system of ODEs might seem *linear*, but it is in fact *nonlinear* because some of its coefficients depend on the dependent variables $c_{1}$, $c_{N-1}$ and $c_{N}$, see (\[pqchi\]). The more explicit version of these equations of motion that obtains by expressing the various coefficients in terms of the free parameters are listed in Appendix \[appendixB\]. They are of course just as *solvable* as the *Newtonian* equations of motion satisfied by the $N$ coordinates $x_{n}(t) $, see Appendix \[appendixA\], to which they correspond via (\[chi\]); and in particular whenever the parameter $\rho $ is *imaginary* and the parameter $\alpha $ is *real* and *negative* they are *asymptotically isochronous* with period $T$ (see Remark \[remark2.3\]).
*Special cases and their (autonomous) isochronous variants*. Certain special models among those identified above (in this subsection) as *solvable* can be *isochronized* by an analogous trick to that employed at the end of the preceding Subsection \[section2.2\]. One route to this end takes as starting point the *isochronized* systems of *Newtonian* equations of motion (\[Isozn\]) and applies to them the same procedure employed above to obtain the equations of motion (\[Eqcm\]) with (\[pq\]) and (\[Eqcm2\]) with (\[pqchi\]). An equivalent procedure is to apply to certain special subcases of these systems of ODEs, (\[Eqcm\]) with (\[pq\]) and (\[Eqcm2\]) with (\[pqchi\]), the following change of dependent and independent variables:$$\begin{gathered}
c_{m}(t) =\exp \left( \text{\textbf{i} }\sigma m \omega
t\right) \chi _{m}\left( \tau \right) , \tau =\frac{\exp \left( \text{\textbf{i}} \omega t\right) -1}{\text{\textbf{i}} \omega } .\end{gathered}$$Here the quantities $\chi _{n}\left( \tau \right) $ are assumed to satisfy the systems of ODEs written above, see (\[Eqcm\]) with (\[pq\]) and ([Eqcm2]{}) with (\[pqchi\]), of course with the new (*complex*) independent variable $\tau $ replacing the time $t$; $\omega $ is an *arbitrary* *real* (for definiteness, *positive*) *constant* to which we associate the period $T,$ see (\[T\]); and the number $\sigma $ is adjusted so as to produce *autonomous* ODEs for the new dependent variables $c_{m}\equiv c_{m}(t) $ (with the *real* independent variable $t$ interpreted as “time”: the special models providing the starting points for the application of this trick being appropriately selected in order to allow such an outcome). Since the application of this trick is quite standard, we dispense here from any detailed discussion of this approach and limit ourselves to reporting the results.
The ODEs that follow from (\[Eqcm\]) (with $\alpha =\rho =\eta =0,$ $f^{(0) }=f^{(1) }=f^{(2) }=0$ and $\sigma =1$) read as follows:
\[Isocm\] $$\begin{gathered}
\ddot{c}_{m} = \text{\textbf{i}} (2 m+1) \omega \dot{c}_{m}-\big[2 \beta (
\dot{c}_{1}-\text{\textbf{i}} \omega c_{1})-m (m+1) \omega ^{2}\big] c_{m}
\nonumber \\
\hphantom{\ddot{c}_{m} =}{}
+2 \beta \dot{c}_{m+1}-\text{\textbf{i}} 2 \beta (m+1) \omega c_{m+1} ;\end{gathered}$$ those that follow from (\[Eqcm\]) (with $\alpha =\rho =\eta =0$, $f^{(0) }=f^{(1) }=0$, $f^{(2) }=-\beta $ and $\sigma =1$) read instead as follows $$\begin{gathered}
\ddot{c}_{m} = [\text{\textbf{i}} (2 m+1) \omega -2 \beta c_{1}] \dot{c} _{m} +
\big[m (m+1) \omega ^{2}+\text{\textbf{i}} 2 m \beta \omega c_{1}-2 \beta
^{2} c_{2}\big] c_{m} \nonumber \\
\hphantom{\ddot{c}_{m} =}{}
+2 m \dot{c}_{m+1}+2 m \big[\beta ^{2} c_{1}-\text{\textbf{i}} (m+1) \omega
\big] c_{m+1} -(m-1) (m+2) \beta ^{2} c_{m+2} .\end{gathered}$$
Neither one of these two *isochronous* many-body problems is new.
The analogous results that follows from (\[Eqcm2\]) are instead generally *new*. There are then two sets of cases. The first set of *isochronous* models obtain from the assignment $\sigma =1$ and read
\[Isocm1\] $$\begin{gathered}
\ddot{c}_{m}=\left[-\text{\textbf{i}} (N-2 m-1) \omega +f^{(2)} c_{1}+\frac{\dot{c}_{N}}{c_{N}}\right] \dot{c}_{m}
+\big(\beta -2 m f^{(2)}\big) \dot{c}_{m+1}\nonumber\\
\hphantom{\ddot{c}_{m}=}{}
+\left[-m (N-m-1) \omega ^{2}+(N-m) \omega
f^{(2)} c_{1}
-\frac{\dot{c}_{N}}{c_{N}} \big(\text{\textbf{i}} m \omega +f^{(2)} c_{1}+\beta
(\text{\textbf{i}} \omega c_{1}-\dot{c}_{1})\big)\right] c_{m} \nonumber \\
\hphantom{\ddot{c}_{m}=}{}
+\left[ -\text{\textbf{i}}(N-2 m) (m+1) \omega f^{(2)}-\text{\textbf{i}}
(m+1) \omega \beta
+m \big(f^{(2)}\big)^{2} c_{1}+(m+1) f^{(2)} \frac{\dot{c}_{N}}{c_{N}}\right] c_{m+1} \nonumber \\
\hphantom{\ddot{c}_{m}=}{}
-m (m+2) \big(f^{(2)}\big)^{2} c_{m+2} ,\end{gathered}$$ with the following restriction on the parameters: $$\begin{gathered}
\rho =\lambda =f^{(0)}=f^{(1)}=0 .\end{gathered}$$
The second set of *isochronous* models obtain from the assignment $\sigma =-1$ and read
\[Isocm2\] $$\begin{gathered}
\ddot{c}_{m}=(2 N-2 m+1) \lambda \dot{c}_{m-1}
+\left[\text{\textbf{i}} (N-2 m+1) \omega +\frac{\dot{c}_{N}}{c_{N}}-\lambda \frac{c_{N-1}}{c_{N}}\right] \dot{c}_{m}\nonumber \\
\hphantom{\ddot{c}_{m}=}{}
-(N-m) (N-m+2) \lambda ^{2} c_{m-2}
+\bigg[ \text{\textbf{i}} (m-1) (2 N-2 m+1) \omega \nonumber \\
\hphantom{\ddot{c}_{m}=}{}
+(N-m+1) (N-m-1) \lambda \left(\lambda \frac{c_{N-1}}{c_{N}}-\text{\textbf{i}} N \omega -\frac{\dot{c}_{N}}{c_{N}}\right)\bigg] c_{m-1}
\nonumber \\
\hphantom{\ddot{c}_{m}=}{}
+\left[-m (N-m+1) \omega ^{2}+\text{\textbf{i}} m \omega \left(\frac{\dot{c}_{N}}{c_{N}}-\lambda \frac{c_{N-1}}{c_{N}}\right)\right] c_{m} ,\end{gathered}$$ with the following restrictions on the parameters: $$\begin{gathered}
b=\beta =\rho =f^{(1)}=f^{(2)}=0 , \qquad \alpha =2 a \eta , \qquad f^{(0)}=\lambda
=a \gamma .\end{gathered}$$
Outlook {#section3}
=======
Results analogous, but somewhat more general, than those reported in this paper can be obtained by an analogous treatment based on a somewhat more general – but still *solvable* – system of two $N\times N$ matrix ODEs than (\[EqUV\]), such as, for instance, $$\begin{gathered}
\dot{U}=\alpha U+\beta U^{2}+\gamma V+\eta ( U V+V U ) , \qquad
\dot{V}=\rho _{0}+\rho V +\rho _{2} V^{2} ,\end{gathered}$$ which contains the $2$ additional scalar constants $\rho _{0}$ and $\rho
_{2} $ (and clearly reduces to (\[EqUV\]) for $\rho _{0}=\rho _{2}=0$). These developments will be reported in subsequent papers.
Finally, let us recall that *Diophantine* findings can be obtained from a *nonlinear* *autonomous* *isochronous* dynamical system by investigating its behavior in the *infinitesimal vicinity* of its equilibria. The relevant equations of motion become then generally *linear*, but they of course retain the properties to be *autonomous* and *isochronous*. For a system of *linear* *autonomous* ODEs, the property of *isochrony* implies that *all* the eigenvalues of the matrix of its coefficients are *integer numbers* (up to a common rescaling factor). When the *linear* system describes the behavior of a *nonlinear* *autonomous* system in the *infinitesimal vicinity* of its equilibria, these matrices can generally be *explicitly* computed in terms of the values at equilibrium of the dependent variables of the original, *nonlinear* model. In this manner nontrivial *Diophantine* findings and conjectures have been discovered and proposed: see for instance the review of such developments in Appendix C (entitled “Diophantine findings and conjectures”) of [@C2008]. Analogous results obtained by applying this approach to the *isochronous* systems of *autonomous nonlinear* ODEs introduced above – and in subsequent papers – will be reported if they turn out to be novel and interesting.
First appendix {#appendixA}
==============
In this appendix we list the $24$ *Newtonian* equations of motion whose *solvable* character has been demonstrated in this paper. In each case the parameters they feature (such as $a$, $b$, $\alpha$, $\beta$, $\gamma$, $\eta$, $\rho $, as the case may be) are *arbitrary constants*; the (assigned) values of the other ones of these parameters (which also characterize the time-evolution of the solutions of these equations, see (\[ExplSolUt\])), are also reported. Let us emphasize that if the parameter $\rho $ is an *imaginary* number and the parameter $\alpha $ is *real* and *negative*, the corresponding many-body problem is *asymptotically isochronous* with period $T$, see Remark \[remark2.3\]; and that *isochronous* many-body models are characterized by the $4$ *Newtonian* equations of motion (\[Isozn\]) displayed at the end of Subsection \[section2.2\]. Let us also mention again that the equations of motion reported below are *not* all new; in particular *not* new are clearly those whose corresponding equations of motion in the following Appendix \[appendixB\] are *linear*.
$\eta =0$, case $(i)$ ($5$ models, corresponding to Table 2.3):
\(1) $a=\beta =0$, $\alpha =-b \gamma$, $f_{n}=-\alpha z_{n}$:
$$\begin{gathered}
\ddot{z}_{n}=-\alpha \rho z_{n}+ ( \alpha +\rho ) \dot{z}
_{n}+2 ( \dot{z}_{n}-\alpha z_{n} ) \sum_{\ell =1,\,\ell \neq
n}^{N}\left( \frac{\dot{z}_{\ell }-\alpha z_{\ell }}{z_{n}-z_{\ell }}
\right) ;\end{gathered}$$
\(2) $\alpha =-b \gamma$, $\beta =\rho =0$, $f_{n}=- ( a/b )
\alpha -\alpha z_{n}$: $$\begin{gathered}
\ddot{z}_{n}=-\alpha \dot{z}_{n}+2 \left( \dot{z}_{n}-\frac{a \alpha }{b}
-\alpha z_{n}\right) \sum_{\ell =1,\, \ell \neq n}^{N}\left( \frac{\dot{z}
_{\ell }-a \alpha /b-\alpha z_{\ell }}{z_{n}-z_{\ell }}\right) ;\end{gathered}$$
\(3) $a=\beta =0$, $\rho =-b \gamma$, $f_{n}=-\rho z_{n}$: $$\begin{gathered}
\ddot{z}_{n}=-\alpha \rho z_{n}+ ( \alpha +\rho ) \dot{z}
_{n}+2 ( \dot{z}_{n}-\rho z_{n} ) \sum_{\ell =1,\, \ell \neq
n}^{N}\left( \frac{\dot{z}_{\ell }-\rho z_{\ell }}{z_{n}- z_{\ell }}\right)
;\end{gathered}$$
\(4) $a=0$, $\alpha =-b \gamma$, $\rho =-2 b \gamma =2 \alpha$, $f_{n}=-\alpha z_{n}$: $$\begin{gathered}
\ddot{z}_{n}=-2 \alpha ^{2} z_{n}-2 \alpha \beta z_{n}^{2}+3 \alpha \dot{z}_{n}+ 2 \beta \dot{z}_{n}z_{n}+2 ( \dot{z}_{n}-\alpha z_{n} )
\sum_{\ell =1,\ell \neq n}^{N}\left( \frac{\dot{z}_{\ell }- \alpha z_{\ell
}}{z_{n}-z_{\ell }}\right) ;\end{gathered}$$
\(5) $\alpha =b \gamma$, $\beta =b^{2} \gamma /a=b \alpha /a$, $\rho
=-2 b \gamma =-2 \alpha$, $f_{n}=a \alpha /b+\alpha z_{n}$: $$\begin{gathered}
\ddot{z}_{n}=2 \alpha ^{2} z_{n}+\frac{2 b}{a} \alpha ^{2} z_{n}^{2}+\frac{2 b}{a} \alpha \dot{z}_{n} z_{n}-\alpha \dot{z}_{n} \nonumber \\
\hphantom{\ddot{z}_{n}=}{}
+2 \left( \dot{z}_{n}+\alpha \frac{a}{b}+\alpha z_{n}\right) \sum_{\ell
=1,\ell \neq n}^{N}\left( \frac{\dot{z}_{\ell }+a \alpha /b+ \alpha z_{\ell
}}{z_{n}-z_{\ell }}\right) .\end{gathered}$$
$\eta =0$, case $(ii)$ ($2$ models, corresponding to Table 2.4):
\(1) $a=\beta =0$, $\alpha =b \gamma +\rho$, $f_{n}=-\rho z_{n}$:
$$\begin{gathered}
\ddot{z}_{n}=- ( b \gamma +\rho ) \rho z_{n}+ ( b \gamma
+2 \rho ) \dot{z}_{n}+2 ( \dot{z}_{n}-\rho z_{n} )
\sum_{\ell =1,\, \ell \neq n}^{N}\left( \frac{\dot{z}_{\ell }-\rho z_{\ell }}{z_{n}-z_{\ell }}\right) ;\end{gathered}$$
\(2) $\alpha =\beta =0$, $\rho =-b \gamma$, $f_{n}=a \gamma +b \gamma z_{n}$: $$\begin{gathered}
\ddot{z}_{n}=-b \gamma \dot{z}_{n}+2 ( \dot{z}_{n}+ a \gamma
+ b \gamma z_{n} ) \sum_{\ell =1,\, \ell \neq n}^{N}\left( \frac{\dot{z} _{\ell }+a \gamma + b \gamma z_{\ell }}{z_{n}-z_{\ell }}\right) .\end{gathered}$$
$\eta \neq 0$, case $(i)$ ($12$ models, corresponding to Table 2.5):
\(1) $a=b=\rho =0$, $f_{n}=0$, $ \lambda =(2 \alpha \eta - \beta \gamma
) \gamma / ( 4 \eta ^{2} )$, $ \mu =\alpha -\beta \gamma /\eta$:
$$\begin{gathered}
\ddot{x}_{n}=\frac{\dot{x}_{n}^{2}}{x_{n}}+\lambda \frac{\dot{x}_{n}}{x_{n}}
+\beta \dot{x}_{n} x_{n}+\dot{x}_{n} \sum_{\ell =1,\, \ell \neq n}^{N}\left[
\frac{\dot{x}_{\ell } ( x_{n}+x_{\ell }) }{( x_{n}-x_{\ell
}) x_{\ell }}\right] ;\end{gathered}$$
\(2) $b=\beta =\gamma =\rho =0$, $f_{n}=2 a \eta x_{n}$, $ \lambda =0$, $\mu =\alpha $: $$\begin{gathered}
\ddot{x}_{n}=\frac{\dot{x}_{n}^{2}}{x_{n}}+( \dot{x}_{n}+2 a \eta
x_{n}) \sum_{\ell =1,\, \ell \neq n}^{N}\left[ \frac{ ( \dot{x}
_{\ell }+ 2 a \eta x_{\ell } ) ( x_{n}+x_{\ell } ) }{ (
x_{n}-x_{\ell } ) x_{\ell }}\right] ;\end{gathered}$$
\(3) $b=\beta =\gamma =0$, $\alpha =-2 a \eta$, $f_{n}=-\alpha x_{n}$, $\lambda =0$, $\mu =\alpha $: $$\begin{gathered}
\ddot{x}_{n}=\frac{\dot{x}_{n}^{2}}{x_{n}}+\rho ( \dot{x}_{n}-\alpha
x_{n} ) + ( \dot{x}_{n}-\alpha x_{n} ) \sum_{\ell =1,\, \ell
\neq n}^{N}\left[ \frac{(\dot{x}_{\ell }-\alpha x_{\ell })(x_{n}+x_{\ell })}{(x_{n}-x_{\ell }) x_{\ell }}\right] ;\end{gathered}$$
\(4) $b=\beta =0$, $\alpha =-2 a \eta$, $\rho =-\alpha$, $f_{n}=-\alpha
x_{n}$, $ \lambda =\alpha \gamma / ( 2 \eta )$, $ \mu
=\alpha $: $$\begin{gathered}
\ddot{x}_{n}=\frac{\dot{x}_{n}^{2}}{x_{n}}+\frac{\alpha \gamma }{2 \eta }
\frac{\dot{x}_{n}}{x_{n}}-\alpha \left( \dot{x}_{n}+\frac{\alpha \gamma }{
2 \eta }-\alpha x_{n}\right) \nonumber\\
\hphantom{\ddot{x}_{n}=}{}
+(\dot{x}_{n}-\alpha x_{n}) \sum_{\ell =1,\, \ell \neq n}^{N}\left[ \frac{
( \dot{x}_{\ell }-\alpha x_{\ell } ) ( x_{n}+x_{\ell
} ) }{( x_{n}-x_{\ell }) x_{\ell }}\right] ;\end{gathered}$$
\(5) $b=\gamma =0$, $\alpha =\rho =-2 a \eta$, $f_{n}=-\alpha x_{n}$, $\lambda =0$, $\mu =\alpha $: $$\begin{gathered}
\ddot{x}_{n}=\frac{\dot{x}_{n}^{2}}{x_{n}}+\beta \dot{x}_{n} x_{n}+\alpha
\left( \dot{x}_{n}-\alpha x_{n}-\beta x_{n}^{2}\right) \nonumber\\
\hphantom{\ddot{x}_{n}=}{}
+ ( \dot{x}_{n}-\alpha x_{n} ) \sum_{\ell =1,\, \ell \neq n}^{N}
\left[ \frac{ ( \dot{x}_{\ell }-\alpha x_{\ell } ) (
x_{n}+x_{\ell } ) }{ ( x_{n}-x_{\ell } ) x_{\ell }}\right] ;\end{gathered}$$
\(6) $b=0$, $\alpha =2 a \eta$, $\beta =4 a \eta ^{2}/\gamma =2 \eta \alpha
/\gamma$, $\rho =-\alpha$, $ f_{n}=\alpha z_{n}$, $ \lambda =0$, $ \mu =-\alpha $: $$\begin{gathered}
\ddot{x}_{n}=\frac{\dot{x}_{n}^{2}}{x_{n}}+\frac{2 \alpha \eta }{\gamma } \dot{x}_{n} x_{n}-\alpha \left( \dot{x}_{n}+\alpha x_{n}-\frac{2 \alpha
\eta }{\gamma } x_{n}^{2}\right) \nonumber \\
\hphantom{\ddot{x}_{n}=}{}
+ ( \dot{x}_{n}+\alpha x_{n} ) \sum_{\ell =1,\, \ell \neq n}^{N}
\left[ \frac{ ( \dot{x}_{\ell }+\alpha x_{\ell } ) (
x_{n}+x_{\ell } ) }{ ( x_{n}-x_{\ell } ) x_{\ell }}\right] ;\end{gathered}$$
\(7) $a=0$, $ \alpha =-b \gamma$, $ \beta =-2 b \eta$, $f_{n}=\alpha
x_{n}-\beta x_{n}^{2}$, $\lambda =0$, $\mu =-\alpha $: $$\begin{gathered}
\ddot{x}_{n}=\frac{\dot{x}_{n}^{2}}{x_{n}}+\beta \dot{x}_{n} x_{n}+\rho
\left( \dot{x}_{n}+\alpha x_{n}+\beta x_{n}^{2}\right) \nonumber \\
\hphantom{\ddot{x}_{n}=}{}
+\left( \dot{x}_{n}+\alpha x_{n}-\beta x_{n}^{2}\right)
\sum_{\ell =1,\, \ell \neq n}^{N}\left[ \frac{\left( \dot{x}_{\ell
}+\alpha x_{\ell }-\beta x_{\ell }^{2}\right) ( x_{n}+x_{\ell
} ) }{ ( x_{n}-x_{\ell } ) x_{\ell }}\right] ;\end{gathered}$$
\(8) $a=0$, $\beta =-2 b \eta$, $\rho =-b \gamma$, $ f_{n}=-b \gamma
x_{n}+2 b \eta x_{n}^{2}$, $ \lambda =(\alpha + b \gamma ) \gamma
/ ( 2 \eta )$, $\mu =\alpha +2 b\gamma $: $$\begin{gathered}
\ddot{x}_{n}=\frac{\dot{x}_{n}^{2}}{x_{n}}+\frac{\left( \alpha + b \gamma
\right) \gamma }{2 \eta } \frac{\dot{x}_{n}}{x_{n}}-2 b \eta \dot{x}
_{n} x_{n}
-b \gamma \left[ \dot{x}_{n}+\frac{ ( \alpha +b \gamma ) \gamma
}{2 \eta }- ( \alpha +2 b \gamma ) x_{n}+ 2 b \eta x_{n}^{2}
\right] \nonumber \\
\hphantom{\ddot{x}_{n}=}{}
+\left( \dot{x}_{n}-b \gamma x_{n}+2 b \eta x_{n}^{2}\right)
\sum_{\ell =1,\, \ell \neq n}^{N}\left[ \frac{\left( \dot{x}_{\ell
}-b \gamma x_{\ell }+2 b \eta x_{\ell }^{2}\right) ( x_{n}+x_{\ell
} ) }{(x_{n}-x_{\ell }) x_{\ell }}\right] ;\end{gathered}$$
\(9) $\gamma =\rho =0$, $\beta =2 b \eta$, $f_{n}=2 a \eta x_{n}+2 b \eta
x_{n}^{2}$, $\lambda =0$, $\mu =\alpha $: $$\begin{gathered}
\ddot{x}_{n}=\frac{\dot{x}_{n}^{2}}{x_{n}}-2 b \eta \dot{x}_{n} x_{n}
+\left( \dot{x}_{n}+2 a \eta x_{n}+2 b \eta x_{n}^{2}\right) \nonumber\\
\hphantom{\ddot{x}_{n}=}{}\times
\sum_{\ell =1,\, \ell \neq n}^{N}\left[ \frac{\left( \dot{x}_{\ell
}+ 2 a \eta x_{\ell }+ 2 b \eta x_{\ell }^{2}\right) ( x_{n}+x_{\ell
} ) }{ ( x_{n}-x_{\ell } ) x_{\ell }}\right] ;\end{gathered}$$
\(10) $\gamma =0$, $\alpha =-2 a \eta$, $\beta =-2 b \eta$, $f_{n}=-\alpha
x_{n}-\beta x_{n}^{2}$, $\lambda =0$, $\mu =\alpha $: $$\begin{gathered}
\ddot{x}_{n}=\frac{\dot{x}_{n}^{2}}{x_{n}}+\beta \dot{x}_{n} x_{n}+\rho
\left( \dot{x}_{n}-\alpha x_{n}-\beta x_{n}^{2}\right) \nonumber \\
\hphantom{\ddot{x}_{n}=}{}
+\left( \dot{x}_{n}-\alpha x_{n}-\beta x_{n}^{2}\right) \sum_{\ell
=1,\, \ell \neq n}^{N}\left[ \frac{\left( \dot{x}_{\ell }-\alpha x_{\ell
}-\beta x_{\ell }^{2}\right) ( x_{n}+x_{\ell } ) }{ (
x_{n}-x_{\ell } ) x_{\ell }}\right] ;\end{gathered}$$
\(11) $a=-b \gamma$, $\beta =-2 b \eta$, $\rho =-\alpha -b \gamma$, $f_{n}=- ( 2 \eta +1 ) b \gamma x_{n}+2 b \eta x_{n}^{2}$, $\lambda = ( \alpha +b \gamma ) \gamma / ( 2 \eta )$, $\mu =\alpha +2 b \gamma $: $$\begin{gathered}
\ddot{x}_{n}=\frac{\dot{x}_{n}^{2}}{x_{n}}+\frac{ ( \alpha +b \gamma
) \gamma }{2 \eta } \frac{\dot{x}_{n}}{x_{n}}-2 b \eta \dot{x}_{n} x_{n} - ( \alpha +b \gamma ) \left[ \dot{x}_{n}+\frac{ ( \alpha
+b \gamma ) \gamma }{2 \eta }- ( \alpha + 2 b \gamma )
x_{n}+2 b \eta x_{n}^{2}\right] \nonumber \\
\hphantom{\ddot{x}_{n}=}{}
+\left[ \dot{x}_{n}- ( 2 \eta +1 ) b \gamma x_{n}+2 b \eta
x_{n}^{2}\right] \nonumber\\
\hphantom{\ddot{x}_{n}=}{}\times
\sum_{\ell =1,\, \ell \neq n}^{N}\left\{ \frac{\left[ \dot{x}_{\ell
}- ( 2 \eta +1 ) b \gamma x_{\ell }+2 b \eta x_{\ell }^{2}\right]
( x_{n}+x_{\ell } ) }{ ( x_{n}-x_{\ell } ) x_{\ell }}
\right\} ;\end{gathered}$$
\(12) $\alpha =-2 a \eta$, $ \beta =-2 b \eta$, $ \rho =-\alpha
-b \gamma$, $ f_{n}= ( 2 a \eta -b \gamma ) x_{n}+2 b \eta
x_{n}^{2}$, $ \lambda = ( -2 a \eta + b \gamma ) \gamma
/ ( 2 \eta )$, $ \mu =-2 a \eta +2 b \gamma $: $$\begin{gathered}
\ddot{x}_{n}=\frac{\dot{x}_{n}^{2}}{x_{n}}+\frac{ ( b \gamma -2 a \eta
) \gamma }{2 \eta } \frac{\dot{x}_{n}}{x_{n}}-2 b \eta \dot{x}_{n} x_{n} \nonumber \\
\hphantom{\ddot{x}_{n}=}{}
- ( \alpha +b \gamma ) \left[ \dot{x}_{n}+\frac{ ( b \gamma
- 2 a \eta ) \gamma }{2 \eta }+ ( 2 a \eta -2 b \gamma )
x_{n}+2 b \eta x_{n}^{2}\right] \nonumber \\
\hphantom{\ddot{x}_{n}=}{}
+\left[ \dot{x}_{n}+ ( 2 a \eta -b \gamma ) x_{n}+2 b \eta
x_{n}^{2}\right] \nonumber \\
\hphantom{\ddot{x}_{n}=}{}\times
\sum_{\ell =1,\, \ell \neq n}^{N}\left\{ \frac{\left[ \dot{x}_{\ell
}+ ( 2 a \eta -b \gamma ) x_{\ell }+2 b \eta x_{\ell }^{2}\right]
( x_{n}+x_{\ell } ) }{ ( x_{n}-x_{\ell } ) x_{\ell }}\right\} .\end{gathered}$$
$\eta \neq 0$, case $(ii)$ ($5$ models, corresponding to Table 2.6):
\(1) $a=b=0$, $f_{n}=\lambda -\mu x_{n}-\beta x_{n}^{2}$, $ \lambda
= ( 2 \alpha \eta -\beta \gamma ) \gamma / ( 4 \eta
^{2} )$, $\mu =\alpha -\beta \gamma /\eta $:
$$\begin{gathered}
\ddot{x}_{n}=\frac{\dot{x}_{n}^{2}}{x_{n}}+\lambda \frac{\dot{x}_{n}}{x_{n}}+\beta \dot{x}_{n} x_{n}+\rho \left( \dot{x}_{n}+\lambda -\mu
x_{n}-\beta x_{n}^{2}\right) \nonumber \\
\hphantom{\ddot{x}_{n}=}{}
+\left( \dot{x}_{n}+\lambda -\mu x_{n}-\beta x_{n}^{2}\right)
\sum_{\ell =1,\, \ell \neq n}^{N}\left[ \frac{\left( \dot{x}_{\ell
}+\lambda -\mu x_{\ell }-\beta x_{\ell }^{2}\right) ( x_{n}+x_{\ell
} ) }{( x_{n}-x_{\ell } ) x_{\ell }}\right] ;\end{gathered}$$
\(2) $b=\rho =0$, $f_{n}=\lambda + ( 2 a \eta -\mu ) x_{n}-\beta
x_{n}^{2}$, $\lambda = ( 2 \alpha \eta -\beta \gamma )
\gamma / ( 4 \eta ^{2} )$, $ \mu =\alpha -\beta \gamma /\eta $: $$\begin{gathered}
\ddot{x}_{n}=\frac{\dot{x}_{n}^{2}}{x_{n}}+\lambda \frac{\dot{x}_{n}}{x_{n}}+\beta \dot{x}_{n} x_{n}
+\left[ \dot{x}_{n}+\lambda + ( 2 a \eta -\mu ) x_{n}-\beta
x_{n}^{2}\right] \nonumber \\
\hphantom{\ddot{x}_{n}=}{}
\sum_{\ell =1,\, \ell \neq n}^{N}\left\{ \frac{\left[\dot{x}_{\ell }+\lambda
+ ( 2 a \eta -\mu ) x_{\ell }-\beta x_{\ell }^{2}\right] (
x_{n}+x_{\ell } ) }{ ( x_{n}-x_{\ell } ) x_{\ell }}\right\}
;\end{gathered}$$
\(3) $a=0$, $\alpha =b \gamma +\rho$, $\beta =2 b \eta$, $f_{n}=\lambda
-\rho x_{n}$, $\lambda = ( \rho -b \gamma ) \gamma / (
2 \eta )$, $ \mu =\rho -b \gamma $: $$\begin{gathered}
\ddot{x}_{n}=\frac{\dot{x}_{n}^{2}}{x_{n}}+\lambda \frac{\dot{x}_{n}}{x_{n}}+2 b \eta \dot{x}_{n} x_{n}
+\rho \left[ \dot{x}_{n}+\lambda- ( \rho -b \gamma )
x_{n}-2 b \eta x_{n}^{2}\right] \nonumber \\
\hphantom{\ddot{x}_{n}=}{}
+ ( \dot{x}_{n}+\lambda -\rho x_{n} )
\sum_{\ell =1,\, \ell \neq n}^{N}\left[ \frac{ ( \dot{x}_{\ell
}+\lambda -\rho x_{\ell } ) ( x_{n}+x_{\ell } ) }{(x_{n}-x_{\ell }) x_{\ell }}\right] ;\end{gathered}$$
\(4) $a=\gamma =0$, $\alpha =\rho$, $\beta =2 b \eta$, $f_{n}=-\alpha
x_{n}$, $\lambda =0$, $\mu =\alpha $: $$\begin{gathered}
\ddot{x}_{n}=\frac{\dot{x}_{n}^{2}}{x_{n}}+\beta \dot{x}_{n} x_{n}+\alpha
\left( \dot{x}_{n}-\alpha x_{n}-\beta x_{n}^{2}\right) \nonumber \\
\hphantom{\ddot{x}_{n}=}{}
+ ( \dot{x}_{n}-\alpha x_{n} ) \sum_{\ell =1,\, \ell \neq n}^{N}\left[ \frac{ ( \dot{x}_{\ell }-\alpha x_{\ell } ) (
x_{n}+x_{\ell } ) }{ ( x_{n}-x_{\ell } ) x_{\ell }}\right] ;\end{gathered}$$
\(5) $\alpha =2 a \eta$, $\beta =2 b \eta$, $\rho =-b \gamma
$, $f_{n}=\lambda +b \gamma x_{n}$, $ \lambda = ( 2 \alpha \eta
-\beta \gamma ) \gamma / ( 2 \eta )$, $ \mu =2 a \eta
-2 b \gamma $: $$\begin{gathered}
\ddot{x}_{n}=\frac{\dot{x}_{n}^{2}}{x_{n}}+\lambda \frac{\dot{x}_{n}}{x_{n}}+2 b \eta \dot{x}_{n} x_{n}
-b \gamma \left[ \dot{x}_{n}+\lambda- ( 2 \alpha \eta -\beta \gamma
) x_{n}-2 b \eta x_{n}^{2}\right] \nonumber \\
\hphantom{\ddot{x}_{n}=}{}
+ ( \dot{x}_{n}+\lambda+b \gamma x_{n} ) \sum_{\ell =1,\ell
\neq n}^{N}\left[ \frac{ ( \dot{x}_{\ell }+\lambda +b \gamma x_{\ell
} ) ( x_{n}+x_{\ell } ) }{ ( x_{n}-x_{\ell } )
x_{\ell }}\right] .\end{gathered}$$
Second appendix {#appendixB}
===============
In this appendix we list the second series of $24$ *Newtonian* equations of motion whose *solvable* character has been demonstrated in this paper; they correspond to those reported in Appendix \[appendixA\] via the transformation among the $N$ zeros $z_{n}$ and the $N$ coefficients $c_{m}$ of a monic polynomial, see (\[Psi\]) and (\[chi\]). In each case the parameters they feature (such as $a$, $b$, $\alpha$, $\beta$, $\gamma$, $\eta$, $\rho $, as the case may be) are *arbitrary constants*; the (assigned) values of the other ones of these parameters (which also characterize the time-evolution of the solutions of these equations, see (\[ExplSolUt\])), are also reported. Let us emphasize that if the parameter $\rho $ is an *imaginary* number and the parameter $\alpha $ is *real* and *negative*, the corresponding many-body problem is *asymptotically isochronous* with period $T$, see Remark \[remark2.3\]; and that *isochronous* many-body models are characterized by the $4$ *Newtonian* equations of motion (\[Isocm\]), (\[Isocm1\]) and (\[Isocm2\]) displayed at the end of Subsection \[section2.3\]. Let us also mention again that the equations of motion reported below are *not* all new; in particular all those that are *linear* are of course *not* new.
Let us recall that it is always assumed that $c_{n}=0$ for $n<0$ and for $n>N$, and $c_{0}=1 $.
$\eta =0$, case $(i)$ ($5$ models, corresponding to Table 2.3):
\(1) $a=\beta =0$, $\alpha =-b \gamma$:
$$\begin{gathered}
\ddot{c}_{m}+[(1-2m) \alpha -\rho ] \dot{c}_{m}+m \alpha [(m-1) \alpha
+\rho ] c_{m}=0 ;\end{gathered}$$
\(2) $\beta =\rho =0$, $\alpha =-b \gamma$: $$\begin{gathered}
\ddot{c}_{m}-2(N-m-1) a \gamma \dot{c}_{m-1}+(2 m-1) b \gamma \dot{c}_{m}
+(N-m+2) (N-m+1) a^{2} \gamma ^{2} c_{m-2}
\nonumber \\
\qquad{}
-2(N-m+1) (m-1) a b \gamma ^{2} c_{m-1}+m (m-1) b^{2} \gamma ^{2}=0;\end{gathered}$$
\(3) $\alpha =\beta =0$, $\rho =-b \gamma$: $$\begin{gathered}
\ddot{c}_{m}+[(1-2 m ) \rho -\alpha ] \dot{c}_{m}+m \rho [(m-1) \rho
+\alpha ] c_{m}=0 ;\end{gathered}$$
\(4) $a=0, \alpha =-b \gamma$, $\rho =2 \alpha$: $$\begin{gathered}
\ddot{c}_{m}-(2 m+1) \alpha \dot{c}_{m}-2 \beta \dot{c}_{m+1}
+[m (m+1) \alpha ^{2}-2 \alpha \beta c_{1}+2 \beta \dot{c}_{1}] c_{m} \nonumber\\
\qquad{}
+2 (m+1) \alpha \beta c_{m+1}=0 ;\end{gathered}$$
\(5) $\alpha =b \gamma$, $\beta =b^{2} \gamma / a$, $\rho =-2 b \gamma$: $$\begin{gathered}
\ddot{c}_{m}-2 (N-m+1) a \gamma \dot{c}_{m-1}+(2 m-1) b \gamma \dot{c}_{m}-
\frac{2 b^{2} \gamma }{a} \dot{c}_{m+1} \nonumber \\
\qquad{} +(N-m+2) (N-m+1) a^{2} \gamma ^{2} c_{m-2}
-2(m-1) (N-m+1) a b \gamma ^{2} c_{m-1} \nonumber \\
\qquad{}
+\left[m (m-3) b^{2} \gamma ^{2}+\frac{2 b^{3} \gamma ^{2}}{a} c_{1}+\frac{2 b^{2} \gamma }{a} \dot{c}_{1}\right] c_{m}
-(m+1) \frac{2 b^{3} \gamma ^{2}}{a} c_{m+1}=0 .\end{gathered}$$
$\eta =0$, case $(ii)$ ($2$ models, corresponding to Table 2.4):
\(1) $\alpha =\beta =0$, $\rho =-b \gamma$:
$$\begin{gathered}
\ddot{c}_{m}-2 (N-m+1) a \gamma \dot{c}_{m-1}+(2 m-1) b \gamma \dot{c}_{m}
+(N-m+2) (N-m+1) a^{2} \gamma ^{2} c_{m-2} \nonumber \\
\qquad {} -2 (m-1) (N-m+1) a b \gamma ^{2} c_{m-1}+m (m-1) b^{2} \gamma ^{2} c_{m}=0 ;\end{gathered}$$
\(2) $a=\beta =0, \alpha =b \gamma +\rho :$ $$\begin{gathered}
\ddot{c}_{m}-(2 m \rho +b \gamma ) \dot{c}_{m}+\left[m b \gamma \rho
-\left(N^{2}-m^{2}\right) \rho ^{2}\right] c_{m}=0 .\end{gathered}$$
$\eta \neq 0$, case $(i)$ ($12$ models, corresponding to Table 2.5):
\(1) $a=b=\rho =0$, $\lambda = [ (2 \alpha \eta -\beta \gamma ) \gamma
] / ( 2 \eta ) ^{2}$, $\mu =\alpha -\beta \gamma / \eta$:
$$\begin{gathered}
\ddot{c}_{m}-\lambda \dot{c}_{m-1}-\frac{\dot{c}_{N}}{c_{N}} \dot{c}_{m}-\beta \dot{c}_{m+1}+\lambda \frac{\dot{c}_{N}}{c_{N}} c_{m-1}+\beta \dot{c}_{1} c_{m}=0 ;\end{gathered}$$
\(2) $b=\beta =\gamma =\rho =0$, $\lambda =0$, $\mu =\alpha$: $$\begin{gathered}
\ddot{c}_{m}-\left[(2 N-4 m) a \eta -\frac{\dot{c}_{N}}{c_{N}}\right] \dot{c}
_{m}-2 m a \eta \left(2 a \eta +\frac{\dot{c}_{N}}{c_{N}}\right) c_{m}=0 ;\end{gathered}$$
\(3) $b=\beta =\gamma =0$, $\alpha =-2 a \eta$, $\lambda =0$, $\mu =\alpha$: $$\begin{gathered}
\ddot{c}_{m}+\left[(N-2 m) \alpha -\rho -\frac{\dot{c}_{N}}{c_{N}}\right] \dot{c}_{m}
-\left[m (N-m) \alpha ^{2}+(N-m) \alpha \rho -m \alpha \frac{\dot{c}_{N}}{c_{N}} \right] c_{m}=0 ;\end{gathered}$$
\(4) $b=\beta =0$, $\alpha =-2 a \eta$, $\rho =2 a \eta =-\alpha$, $\lambda
=-a \gamma$, $\mu =\alpha$: $$\begin{gathered}
\ddot{c}_{m}+a \gamma \dot{c}_{m-1}-\left[(N-2 m+1) \alpha +\frac{\dot{c}_{N}}{c_{N}}\right] \dot{c}_{m}
+a \gamma \left[(N-m-1) \alpha -\frac{\dot{c}_{N}}{c_{N}}\right] c_{m-1} \nonumber
\\
\qquad{} +\left[-(N-m) \alpha ^{2}+2 (N-m) \alpha ^{2}+m \alpha \frac{\dot{c}_{N}}{c_{N}}\right] c_{m}=0 ;\end{gathered}$$
\(5) $b=\gamma =0$, $\alpha =\rho =-2 a \eta$, $\lambda =0$, $\mu =\alpha$: $$\begin{gathered}
\ddot{c}_{m}+\left[(N-2 m-1) \alpha -\frac{\dot{c}_{N}}{c_{N}}\right] \dot{c}_{m}
-\left[(m+1) (N-m) \alpha ^{2}+m \alpha \frac{\dot{c}_{N}}{c_{N}}+\alpha \beta
c_{1}-\beta \dot{c}_{1}\right] c_{m} \nonumber \\
\qquad{} +(m+1) \alpha \beta c_{m+1}=0 ;\end{gathered}$$
\(6) $b=0, \alpha =2 a \eta$, $\beta =4 a \eta ^{2} / \gamma$, $\rho =-2 a \eta$, $\lambda =0$, $\mu =-2 a \eta$: $$\begin{gathered}
\ddot{c}_{m}-\left[2(N-2 m+1) a \eta +\frac{\dot{c}_{N}}{c_{N}}\right] \dot{c}_{m}-
\frac{4 a \eta ^{2}}{\gamma } \dot{c}_{m+1} \\
\qquad{} -2 a \eta \left[2 (m-1) (N-m) a \eta +m \frac{\dot{c}_{N}}{c_{N}}-\frac{4 a \eta
^{2}}{\gamma } c_{1}-\frac{2 \eta }{\gamma } \dot{c}_{1}\right] c_{m}
-(m+1) \frac{8 a^{2} \eta ^{3}}{\gamma } c_{m+1}=0 ; \nonumber\end{gathered}$$
\(7) $a=0, \alpha =-b \gamma$, $\beta =-2 b \eta$, $\lambda =0$, $\mu =-\alpha$: $$\begin{gathered}
\ddot{c}_{m}-\left[(N-2 m) \alpha +\rho +\beta c_{1}+\frac{\dot{c}_{N}}{c_{N}}\right]
\dot{c}_{m}-(2 m+1) \beta \dot{c}_{m+1} \nonumber \\
\qquad {} -\left[m (N-m) \alpha ^{2}-(N-m) \alpha \rho +(N-m) \alpha \beta c_{1}
+(\beta c_{1}+m \alpha ) \frac{\dot{c}_{N}}{c_{N}}+\beta \rho c_{1}-\beta
\dot{c}_{1}\right] c_{m} \nonumber \\
\qquad{}
-\left\{(m+1) \beta \rho -[2 N^{2}-(3 m+5) N+4 m+4] \alpha \beta
+m \beta ^{2} c_{1}+(m+1) \beta \frac{\dot{c}_{N}}{c_{N}}\right\} c_{m+1}
\nonumber \\
\qquad{}
-m (m+2) \beta ^{2} c_{m+2}=0 ;\end{gathered}$$
\(8) $a=0$, $\beta =-2 b \eta$, $\rho =-b \gamma$, $\lambda =(\alpha +b \gamma
) \gamma / ( 2 \eta )$, $\mu =\alpha +2 b \gamma$: $$\begin{gathered}
\ddot{c}_{m}-\lambda \dot{c}_{m-1}+\left[(N-2 m+1) b \gamma -2 b \eta c_{1}-\frac{\dot{c}_{N}}{c_{N}}\right] \dot{c}_{m}
+4 (m-1) \dot{c}_{m+1}\nonumber \\
\qquad +\lambda \left[-(N-m+1) b \gamma +\frac{\dot{c}_{N}}{c_{N}}\right] c_{m-1}
-\bigg[(m+4) (N-m) b^{2} \gamma ^{2}+2 (2 N-2 m+1) b^{2} \gamma \eta c_{1}
\nonumber \\
\qquad{} -(N-m) b \alpha \gamma +2 b \eta \dot{c}_{1}+(m b \gamma -2 b \eta c_{1}) \frac{\dot{c}_{N}}{c_{N}}\bigg] c_{m} \nonumber \\
\qquad +2 \left[(m+1) (N-2 m+1) b^{2} \gamma \eta
-2 m b^{2} \eta ^{2} c_{1}-(m+1) b \eta \frac{\dot{c}_{N}}{c_{N}}\right] c_{m+1}\nonumber\\
\qquad{}+4 m (m+2) c_{m+2}=0 ;\end{gathered}$$
\(9) $\gamma =\rho =0$, $\beta =-2 b \eta$, $\lambda =0$, $\mu =\alpha$: $$\begin{gathered}
\ddot{c}_{m}-\left[2 (N-2 m)+2 b \eta c_{1}+\frac{\dot{c}_{N}}{c_{N}}\right] \dot{c}_{m}+2 (2 m+1) b \eta \dot{c}_{m+1} \nonumber \\
\qquad{} +\left[-4 m (N-m) a^{2} \eta ^{2}+4 (N-m) a b \eta ^{2} c_{1}+\beta \dot{c}_{1}
+2 (b \eta c_{1}-m a \eta ) \frac{\dot{c}_{N}}{c_{N}}\right] c_{m} \nonumber
\\
\qquad {} -2 \left[2 (m+1) (N-2 m) a b \eta ^{2}+2 m b^{2} \eta ^{2} c_{1}+(m+1) b \eta \frac{\dot{c}_{N}}{c_{N}}\right] c_{m+1} \nonumber \\
\qquad{} +4 m (m+2) c_{m+2}=0 ;\end{gathered}$$
\(10) $\gamma =0$, $\alpha =-2 a \eta$, $\beta =-2 b \eta$, $\lambda =0$, $\mu
=\alpha$: $$\begin{gathered}
\ddot{c}_{m}+\left[(N-2 m) \alpha -\rho +\beta c_{1}-\frac{\dot{c}_{N}}{c_{N}}\right] \dot{c}_{m}-(2 m+1) \beta \dot{c}_{m+1} \nonumber \\
\qquad {} -\left[m (N-m) \alpha ^{2}+(N-m) \alpha \rho -(N-m) \alpha \beta c_{1}+\beta
\rho c_{1}
-\beta \dot{c}_{1}+(\beta c_{1}-m \alpha ) \frac{\dot{c}_{N}}{c_{N}}\right] c_{m}
\nonumber \\
\qquad{} -\left[(m+1) (N-2 m) \alpha \beta -(m+1) \beta \rho +m \beta
^{2} c_{1}-(m+1) \beta \frac{\dot{c}_{N}}{c_{N}}\right] c_{m+1} \nonumber \\
\qquad {} +m (m+2) c_{m+2}=0 ;\end{gathered}$$
\(11) $a=-b \gamma$, $\beta =-2 b \eta$, $\rho =-\alpha -b \gamma$, $\lambda
=(\alpha +b \gamma ) \gamma / ( 2 \eta )$, $\mu =\alpha
+2 b \gamma$: $$\begin{gathered}
\ddot{c}_{m}-\lambda \dot{c}_{m-1}
+\left\{[2 (N-2 m) \eta +N-2 m+1] b \gamma +\alpha -2 b \eta c_{1}-\frac{\dot{c}_{N}}{c_{N}}\right\} \dot{c}_{m} \nonumber \\
\qquad {} +2 (2 m+1) \dot{c}_{m+1}+\lambda \left[\frac{\dot{c}_{N}}{c_{N}}-(N-m+1) (\alpha
+b \gamma )\right] c_{m-1} \nonumber \\
\qquad{} -\bigg\{m (N-m) b^{2} \gamma ^{2} (2 \eta +1)^{2}\!-(N-m) (\alpha +b \gamma
) (\alpha +2 b \gamma ) +2 (N-m) b^{2} \gamma \eta (2 \eta +1) c_{1} \nonumber \\
\qquad{}+2 b \eta (\alpha +b \gamma
) c_{1}+2 b \eta \dot{c}_{1}
-[2 b \eta c_{1}+m b \gamma (2 \eta +1)] \frac{\dot{c}_{N}}{c_{N}}\bigg\} c_{m}
\nonumber \\
\qquad{} +2 \bigg[(m+1) (N-2 m) b^{2} \gamma \eta (2 \eta +1)+(m+1) b \eta (\alpha
+b \gamma )
-2 m b^{2} \eta ^{2} c_{1}\nonumber\\
\qquad{} -(m+1) b \eta \frac{\dot{c}_{N}}{c_{N}}\bigg] c_{m+1}+m (m+2) c_{m+2}=0 ;\end{gathered}$$
\(12) $\alpha =-2 a \eta$, $\beta =-2 b \eta$, $\rho =-\alpha -b \gamma
=2 a \eta -b \gamma$, $\lambda =(b \gamma -2 a \eta ) \gamma / \left( 2 \eta
\right)$, $\mu =2 b \gamma -2 a \eta$: $$\begin{gathered}
\ddot{c}-\lambda \dot{c}_{m-1}-\left[(N-2 m+1) (2 a \eta -b \gamma )+2 b \eta
c_{1}+\frac{\dot{c}_{N}}{c_{N}}\right] \dot{c}_{m} \nonumber \\
\qquad{} +2 (2 m+1) b \eta \dot{c}_{m+1}+\lambda \left[(N-m-1) (2 a \eta -b \gamma )+\frac{\dot{c}_{N}}{c_{N}}\right] c_{m-1} \nonumber \\
\qquad {} +\bigg\{-m (N-m) (2 a \eta -b \gamma )^{2}+2 (N-m+1) (2 a \eta -b \gamma ) b \eta
c_{1} \nonumber \\
\qquad{} +(N-m) (2 a \eta -b \gamma ) (2 a \eta -2 b \gamma )-2 b \eta \dot{c}_{1}
+[2 b \eta c_{1}-m (2 a \eta -b \gamma )] \frac{\dot{c}_{N}}{c_{N}}\bigg\} c_{m}
\nonumber \\
\qquad{} -2 \left[(m+1) (N-2 m+1) (2 a \eta -b \gamma ) b \eta
+2 m b^{2} \eta ^{2} c_{1}+(m+1) b \eta \frac{\dot{c}_{N}}{c_{N}}\right] c_{m+1}\nonumber\\
\qquad{}
+4 m (m+2) b^{2} \eta ^{2} c_{m+2}=0 .\end{gathered}$$
$\eta \neq 0$, case $(ii)$ ($5$ models, corresponding to Table 2.6):
\(1) $a=b=0$, $\lambda = [ (2 \alpha \eta -\beta \gamma ) \gamma ]
/ ( 2 \eta ) ^{2}$, $\mu =\alpha -\beta \gamma / \eta$:
$$\begin{gathered}
\ddot{c}_{m}-(2 N-2 m+1) \lambda \dot{c}_{m-1}
+\left[(N-2 m) \mu -\rho +\beta c_{1}+\lambda \frac{c_{N-1}}{c_{N}}-\frac{\dot{c}_{N}}{c_{N}}\right] \dot{c}_{m} \nonumber \\
\qquad{} -(2 m+1) \beta \dot{c}_{m+1}+(N-m+2) (N-m) \lambda ^{2} c_{m-2} -\bigg[(N-2 m) (N-m+1) \lambda \mu
\nonumber \\
\qquad{} -(N-m+1) \rho \lambda
+(N-m+1) \beta \lambda c_{1}+\lambda \frac{\dot{c}_{N}}{c_{N}}-\lambda
^{2} \frac{c_{N-1}}{c_{N}}\bigg] c_{m-1} \nonumber \\
\qquad{} +\bigg[m (N-m) \big(2 \beta \lambda -\mu ^{2}\big)+(\beta c_{1}-m \mu ) \left(\lambda \frac{c_{N-1}}{c_{N}}-\frac{\dot{c}_{N}}{c_{N}}\right) \nonumber \\
\qquad -(N-m) \rho \mu +(N-m) \beta \mu c_{1}-\beta \rho c_{1}+\beta \dot{c}_{1}\bigg] c_{m} \nonumber \\
\qquad {} -\bigg[(m+1) (N-2 m) \beta \mu +(m+1) \beta \rho +m \beta
^{2} c_{1}-(m+1) \beta \frac{\dot{c}_{N}}{c_{N}} \nonumber \\
\qquad{} -(N-m-2) \beta \lambda \frac{c_{N-1}}{c_{N}}\bigg] c_{m+1}+m (m+2) \beta
^{2} c_{m+2}=0 ;\end{gathered}$$
\(2) $b=\rho =0$, $\lambda = [ (2 \alpha \eta -\beta \gamma ) \gamma
] / ( 2 \eta ) ^{2}$, $\mu =\alpha -\beta \gamma / \eta$: $$\begin{gathered}
\ddot{c}_{m}-(2 N-2 m+1) \lambda \dot{c}_{m-1}
+\left[(N-2 m) (\mu -2 a \eta )+\beta c_{1}+\lambda \frac{c_{N-1}}{c_{N}}-\frac{\dot{c}_{N}}{c_{N}}\right] \dot{c}_{m} \nonumber \\
\qquad{} -(2 m+1) \beta \dot{c}_{m+1}+(N-m) (N-m+2) \lambda ^{2} c_{m-2} \nonumber\\
\qquad{}
+\bigg[(N-2 m) (N-m+1) \lambda (2 a \eta -\mu )-(N-m+1) \beta \lambda c_{1}
\nonumber \\
\qquad{}+(N-m) \lambda \frac{\dot{c}_{N}}{c_{N}}-(N-m) \lambda ^{2} \frac{c_{N-1}}{c_{N}}\bigg]c_{m-1}
+\bigg\{m (N-m) \big[2 \beta \lambda -(2 a \eta -\mu )^{2}\big]\nonumber\\
\qquad {}-(N-m) (2 a \eta -\mu
) \beta c_{1}+\beta \dot{c}_{1}
+[\beta c_{1}+m (2 a \eta -\mu )] \left(\lambda \frac{c_{N-1}}{c_{N}}-\frac{\dot{c}_{N}}{c_{N}}\right)\bigg\} c_{m} \nonumber \\
\qquad{} -\bigg[(m+1) (N-2 m) \beta \mu +(m+1) \beta \rho +m \beta ^{2} c_{1}
\nonumber \\
\qquad{}-(m+1) \beta \frac{\dot{c}_{N}}{c_{N}}-(N-m-2) \beta \lambda \frac{c_{N-1}}{c_{N}}\bigg] c_{m+1}
+m (m+2) \beta ^{2} c_{m+2}=0 ;\end{gathered}$$
\(3) $a=0$, $\alpha =b \gamma +\rho$, $\beta =2 b \eta$, $\lambda =\gamma \rho
/ ( 2 \eta )$, $\mu =\rho -b \gamma$: $$\begin{gathered}
\ddot{c}_{m}-(2 N-2 m+1) \lambda \dot{c}_{m-1}-\left[(N-2 m+1) \rho -\lambda \frac{c_{N-1}}{c_{N}}+\frac{\dot{c}_{N}}{c_{N}}\right] \dot{c}_{m} \nonumber \\
\qquad{} -2 b \eta \dot{c}_{m+1}+(N-m) (N-m+2) \lambda ^{2} c_{m-2} \nonumber \\
\qquad{} +\left[(N-m+1) (N-2 m+1) \lambda \rho +(N-m) \lambda \left(\frac{\dot{c}_{N}}{c_{N}}-\lambda \frac{c_{N-1}}{c_{N}}\right)\right] c_{m-1} \nonumber \\
\qquad {}-\bigg[(m+1) (N-m) \rho ^{2}-(N-m) b \gamma \rho +m \rho \frac{\dot{c}_{N}}{c_{N}}-m \lambda \rho \frac{c_{N-1}}{c_{N}} \nonumber \\
\qquad{} -2 b \eta \rho c_{1}-2 b \eta \dot{c}_{1}\bigg] c_{m}+2 (m+1) b \eta \rho c_{m+1}=0 ;\end{gathered}$$
\(4) $a=\gamma =0$, $\alpha =\rho$, $\beta =2 b \eta$, $\lambda =0$, $\mu =\alpha$: $$\begin{gathered}
\ddot{c}_{m}+\left[(N-2 m-1) \alpha -\frac{\dot{c}_{N}}{c_{N}}\right] \dot{c}_{m}-\beta
\dot{c}_{m+1} \nonumber \\
\qquad {} -\left[(m+1) (N-m) \alpha ^{2}-m \alpha \frac{\dot{c}_{N}}{c_{N}}+\alpha \beta
c_{1}-\beta \dot{c}_{1}\right] c_{m} +(m+1) \alpha \beta c_{m+1}=0 ;\end{gathered}$$
\(5) $\alpha =2 a \eta$, $\beta =2 b \eta$, $\rho =-b \gamma$, $\lambda = [
(2 a \eta -b \gamma ) \gamma ] / ( 2 \eta )$, $\mu
=2 a \eta -2 b \gamma$: $$\begin{gathered}
\ddot{c}_{m}-(2 N-2 m+1) \lambda \dot{c}_{m-1}-\left[(N-2 m-1) b \gamma -\lambda
\frac{c_{N-1}}{c_{N}}+\frac{\dot{c}_{N}}{c_{N}}\right] \dot{c}_{m} \nonumber
\\
\qquad{} -\left[(N+2 m+1) (N-m+1) b \gamma \lambda +(N-m) \lambda \left(\lambda \frac{c_{N-1}}{c_{N}}-\frac{\dot{c}_{N}}{c_{N}}\right)\right] c_{m-1} \nonumber \\
\qquad {} +\bigg[-(m+2) (N-m) b^{2} \gamma ^{2}+m b \gamma \left(\lambda \frac{c_{N-1}}{c_{N}}-\frac{\dot{c}_{N}}{c_{N}}\right) \nonumber \\
\qquad{} +2 (N-m) a b \gamma \eta +2 b^{2} \gamma \eta c_{1}+2 b \eta \dot{c}_{1}\bigg] c_{m}
-2 (m+1) b^{2} \gamma \eta c_{m+1}=0 .\end{gathered}$$
Acknowledgements {#acknowledgements .unnumbered}
----------------
We would like to thank an unknown referee whose intervention allowed us to eliminate a mistake contained in the original version of our paper.
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[^1]: A paperback edition of this monograph has been published by Oxford University Press in September 2012: it coincides with the hardback version, except for the corrections of several misprints, and the addition of a two-page “Preface to the paperback edition” reporting a number of relevant new references.
|
---
abstract: 'We study the pair production of new heavy leptons within a new $U(1)''$ symmetry extension of the Standard Model. Because of the new symmetry, the production and decay modes of the new heavy leptons would be different from those of three families of the standard model. The pair production cross sections depending on the mixing parameter and the mass of heavy leptons have been calculated for the center of mass energies of $0.5$ TeV, $1$ TeV and $3$ TeV. The accessible ranges of the parameters have been obtained for different luminosity projections at linear colliders. We find the sensitivity to the range of mixing parameter $-1<x<1$ for the mass range $M_{l''}<800$ GeV at $\sqrt{s}=3$ TeV and $L_{int}=100$ fb$^{-1}$'
author:
- 'V. Ari'
- 'O. Çakir'
- 'S. Kuday'
title: 'Pair Production of New Heavy Leptons with $U(1)''$ Charge at Linear Colliders'
---
Introduction
============
The standard model (SM) of particle physics has been shown to successfully describe fundamental particle interactions. However, some of the problems (such as symmetry breaking, dark matter, flavor and CP violation) do not find adequate answers within the standard model of three fermion families, and there are considerable interest in new heavy fermions to be searched at high energy colliders. On the quark sector, the direct searches performed by the ATLAS collaboration excludes the heavy quark masses $m_{b'}\lesssim670$ GeV and $m_{t'}\lesssim656$ GeV [@ATLAS], and CMS collaborations have ruled out new heavy down-type and up-type quarks with masses $m_{b'}\lesssim611$ GeV and $m_{t'}\lesssim570$ GeV [@CMS]. In these experimental bounds, the branching of decay $b'\to tW$ or $t'\to bW$ is assumed to be unity; relaxing them leads to slightly weaker bounds as discussed in [@Flacco2010]. On the lepton sector, the LEP measurement of the $Z$ boson width strongly supports the fact there are only three light active neutrinos [@Nakamura2010]. This measurement leads to a mass constraint for a new heavy stable neutrino of mass $m_{\nu'}>45$ GeV. In addition, direct production searches at LEPII establish a lower limit of the order of $100$ GeV for a new charged lepton ($l'$) and unstable neutrino. It is also straightforward to check that the recent discovery of Higgs boson with mass of $125$ GeV tune the new neutrino mass $m_{\nu'}>m_{H}/2$. Furthermore, the off-diagonal lepton mixings are required to be smaller than 0.115 consistent with the recent global fits [@Lacker2010].
One of the simplest extensions of the SM is to include an $U(1)'$ gauge symmetry. After the $U(1)'$ symmetry breaking, there could remain a residual discrete symmetry as emphasized in Ref. [@Soni2013], and this would cause the lightest new fermion to be stable. The discrete symmetry for the new model could protect new heavy fermions from the SM fermions to explain $Z$ boson width measurement at LEP. The model could provide a stable particle for dark matter candidate, a new source of CP violation, baryon$-$lepton number conservation. The particle spectrum of the model [@Soni2013] will include, in addition to the SM particles, an extra heavy fermion family, an extra Higgs doublet, a Higgs singlet as well as an extra gauge boson ($Z'$). Since the SM fermions have vanishing $U(1)'$ charges, the new gauge boson $Z'$ cannot form a dilepton or dijet signal besides the $Z-Z'$ mixing effects which is constrained to be tiny by the precise $Z$ measurements at LEP [@Langacker2009]. Recently, the most stringent limits on a heavy neutral gauge boson $Z'_{S}$ with the same universal couplings to fermions as the $Z$ boson (sequential model) have been set using measurements from ATLAS [@ATLAS_2013_ZP] and CMS [@CMS_2012_ZP], translated to a lower bound on the mass $m_{Z'}>2.79$ TeV and $2.96$ TeV, respectively.
In this work, we use the new heavy fermion model accompanied by an $U(1)'$ symmetry under which only the heavy fermions have nonzero charge. The heavy charged lepton pair production at Linear Colliders (LC) through the process $e^{+}e^{-}\to l^{'+}l^{'-}$ and subsequent decay of $l^{'-}\to\nu'W^{-}$ ending up with a stable heavy neutrino have been examined at linear collider energies $\sqrt{s}=0.5$ TeV for the International Linear Collider (ILC) [@ILC2013], $1$ TeV for its upgrade and $3$ TeV for Compact Linear Collider (CLIC) [@CLIC2013].
Heavy Leptons
=============
The interactions of new heavy leptons ($l'$,$\nu'$) can be described by the following Lagrangian
$$\begin{aligned}
L' & = & -g_{e}\bar{l'}\gamma^{\mu}l'A_{\mu}-\frac{g_{z}}{2}\bar{l'}\gamma^{\mu}(c_{V}^{l'}-c_{A}^{l'}\gamma^{5})l'Z_{\mu}-\frac{g_{z}}{2}\bar{\nu'}\gamma^{\mu}(c_{V}^{\nu'}-c_{A}^{\nu'}\gamma^{5})\nu'Z_{\mu}\\
& & \frac{g_{z}}{2}\bar{l'}\gamma^{\mu}(c'{}_{V}^{l'}-c'{}_{A}^{l'}\gamma^{5})l'Z'_{\mu}-\frac{g_{z}}{2}\bar{\nu'}\gamma^{\mu}(c'{}_{V}^{\nu'}-c'{}_{A}^{\nu'}\gamma^{5})\nu'Z'_{\mu}\\
& & -\frac{g_{w}}{2\sqrt{2}}U_{\nu'l'}\bar{l'}\gamma^{\mu}(1-\gamma^{5})\nu'W_{\mu}+H.c.\end{aligned}$$
where $g_{e}$, $g_{W}$ and $g_{Z}$ are the electromagnetic, weak-charged and weak-neutral coupling constants, respectively. The $A_{\mu}$, $W_{\mu}$ and $Z_{\mu}$ are the fields for photon, $W$ boson and $Z$ boson, respectively. The field $Z'_{\mu}$ is for the new $Z'$ boson. The $U_{\nu'l'}$ is the mixing element for the charged current coupling of heavy leptons. Here, we consider a long-lived neutrino with unit mixing element. The $c_{V}$ and $c_{A}$ are vector and axial-vector couplings with the $Z$ boson. The $c'_{V}$ and $c'_{A}$ are vector and axial-vector couplings with the new $Z'$ boson. These couplings can be expressed as $c_{V}={\cal Q}_{L}+{\cal Q}_{R}$ and $c_{A}={\cal Q}_{L}-{\cal Q}_{R}$ with the left and right handed fermion gauge charges ${\cal Q}_{L}$ and ${\cal Q}_{R}$, respectively. The $U(1)'$ charge is defined as ${\cal Q}=(B-L)+xY$ with the mixing parameter $x$. In the model, the SM fermions are not charged under the $U(1)'$ while the new fermions have the gauge charges as shown in Table \[tab:tab1\].
Field $U(1)'$ charge $c_{V}$ $c_{A}$
------------ ---------------- --------- ---------
$t'_{L}$ $1/3+x$
$t'_{R}$ $1/3+4x$
$b'_{L}$ $1/3+x$
$b'_{R}$ $1/3-2x$
$l'_{L}$ $-1-3x$
$l'_{R}$ $-1-6x$
$\nu'_{L}$ $-1-3x$
$\nu'_{R}$ $-1$
: The $U(1)'$ charges of new heavy fermions with a new parity. \[tab:tab1\]
As it can be seen from Table \[tab:tab1\], the key values for the parameter $x$ can be calculated as follows: for quarks $c_{V}^{t'}(c_{V}^{b'})$ vanishes when $x=-2/15(2/3)$, respectively; and for leptons $c_{V}^{l'}(c_{V}^{\nu'})$ vanishes when $x=-2/9(-2/3)$, which are in the range of $-2/3<x<2/3$. A specific value of the mixing parameter $x=0$ corresponds to a vector-type coupling with the $Z$ boson. Here, we assume that charged heavy lepton decays through the process $l'\to\nu'+W^{-}$ within a large range of parameters (mixing and mass). If the mass difference between the heavy lepton and neutrino is small enough, then the heavy lepton decay may result in missing transverse energy and off-shell $W$ boson.
Cross Sections
==============
The pair production of heavy charged leptons can be performed through the process $e^{+}e^{-}\to l^{'+}l^{'-}$ and subsequent decays of $l^{'-}\to\nu'W^{-}$ and $l^{'+}\to\bar{\nu}'W^{+}$ ending up with a pair of stable heavy neutrinos and a pair of $W$ bosons. The contributing diagrams for the signal are shown in Figure \[fig:fig1\]. The calculation of the cross sections for the signal and background is performed using CalcHEP [@Belyaev2013] with the implementation of new heavy leptons and their interactions.
![Diagrams contributing to the pair production process at linear colliders. \[fig:fig1\]](fig1)
At $\sqrt{s}=0.5$ TeV, the cross sections for the signal of heavy lepton pairs with mass $M_{l'}=200$ GeV are given as $8.72$, $1.38$, $0.34$, $1.06$, $3.52$, $7.74$ and $21.40$ pb for the parameter $x$ values $-1.0$, $-0.5$, $-0.25$, $0.0$, $0.25$, $0.50$ and $1.0$, respectively. The $e^{+}e^{-}$ collider (ILC) with $\sqrt{s}=0.5$ TeV, has the potential up to the kinematical range ($m_{l',\nu'}\leq250$ GeV) for the direct production of new heavy lepton pairs. However, the CLIC with multi-TeV extends the mass range for the new heavy leptons. At the center of mass energies of the linear colliders with $\sqrt{s}=1$ TeV and $\sqrt{s}=3$ TeV, the cross sections for the signal are shown in Figure \[fig:fig2\] and Figure \[fig:fig3\] , respectively. It is clear from Figures \[fig:fig2\] and \[fig:fig3\], the cross section has a minimum around $x=-0.25$. At the center of mass energy $\sqrt{s}=1$ (3) TeV, we calculate the change in the cross sections as $\Delta\sigma\simeq0.5$ ($0.05$) pb for a change in mixing parameter $|\Delta x|=0.25$ around the minimum. The signal cross sections depending on the heavy lepton mass and mixing parameter are given in Table \[tab:tab2\] for $\sqrt{s}=1$ TeV and in Table \[tab:tab3\] for $\sqrt{s}=3$ TeV.
![The pair production cross sections for the process $e^{+}e^{-}\to l'^{+}l'^{-}$ depending on the parameter $x$ for different mass values of $M_{l'}=200$, $300$ and $400$ GeV at the center of mass energy $\sqrt{s}=1$ TeV. \[fig:fig2\]](fig2)
![The same as Fig. \[fig:fig2\], but for the mass values of $M_{l'}=200$, $400$ and $800$ GeV at $\sqrt{s}=3$ TeV. \[fig:fig3\]](fig3)
--------------- -------- --------------------- --------------------- --------------------- -------- -------- --------
$M_{l'}(GeV)$ $-1.0$ $-0.5$ $-0.25$ $0.0$ $0.25$ $0.5$ $1.0$
$200$ $2.82$ $4.71\times10^{-1}$ $1.21\times10^{-1}$ $3.19\times10^{-1}$ $1.07$ $2.36$ $6.59$
$300$ $2.59$ $4.24\times10^{-1}$ $1.09\times10^{-1}$ $3.04\times10^{-1}$ $1.01$ $2.23$ $6.19$
$400$ $2.07$ $3.32\times10^{-1}$ $8.55\times10^{-2}$ $2.55\times10^{-1}$ $0.84$ $1.84$ $5.10$
--------------- -------- --------------------- --------------------- --------------------- -------- -------- --------
: Cross sections (pb) for the signal process $e^{+}e^{-}\to l'^{+}l'^{-}$ at $\sqrt{s}=1$ TeV depending on the heavy lepton mass $M_{l'}$ and the parameter $x$. \[tab:tab2\]
--------------- --------------------- --------------------- --------------------- --------------------- --------------------- --------------------- ---------------------
$M_{l'}(GeV)$ $-1.0$ $-0.5$ $-0.25$ $0.0$ $0.25$ $0.5$ $1.0$
$200$ $3.21\times10^{-1}$ $5.47\times10^{-2}$ $1.42\times10^{-2}$ $3.55\times10^{-2}$ $1.19\times10^{-1}$ $2.64\times10^{-1}$ $7.39\times10^{-1}$
$300$ $3.20\times10^{-1}$ $5.43\times10^{-2}$ $1.40\times10^{-2}$ $3.54\times10^{-2}$ $1.19\times10^{-1}$ $2.63\times10^{-1}$ $7.37\times10^{-1}$
$400$ $3.17\times10^{-1}$ $5.37\times10^{-2}$ $1.39\times10^{-2}$ $3.54\times10^{-2}$ $1.18\times10^{-1}$ $2.62\times10^{-1}$ $7.34\times10^{-1}$
$500$ $3.13\times10^{-1}$ $5.28\times10^{-2}$ $1.37\times10^{-2}$ $3.53\times10^{-2}$ $1.18\times10^{-1}$ $2.61\times10^{-1}$ $7.29\times10^{-1}$
$600$ $3.09\times10^{-1}$ $5.18\times10^{-2}$ $1.34\times10^{-2}$ $3.51\times10^{-2}$ $1.17\times10^{-1}$ $2.59\times10^{-1}$ $7.22\times10^{-1}$
$700$ $3.02\times10^{-1}$ $5.04\times10^{-2}$ $1.31\times10^{-2}$ $3.48\times10^{-2}$ $1.16\times10^{-1}$ $2.55\times10^{-1}$ $7.12\times10^{-1}$
$800$ $2.94\times10^{-1}$ $4.87\times10^{-2}$ $1.26\times10^{-2}$ $3.43\times10^{-2}$ $1.14\times10^{-1}$ $2.51\times10^{-1}$ $6.98\times10^{-1}$
--------------- --------------------- --------------------- --------------------- --------------------- --------------------- --------------------- ---------------------
: Cross sections (pb) for the signal process $e^{+}e^{-}\to l'^{+}l'^{-}$ at $\sqrt{s}=3$ TeV depending on the heavy lepton mass $M_{l'}$ and the parameter $x$. \[tab:tab3\]
The cross sections for the background processes contributing to the lepton+dijet and missing transverse energy (MET) in the the final state are given in Table \[tab:tab4\]. The cross sections for the processes can be multiplied by the corresponding branching ratios, such as $BR(W^{+}\to l^{+}\nu)=0.22$, $BR(W^{+}\to2j)=0.68$, $BR(Z\to\nu\bar{\nu})=0.20$ and $BR(h\to ZZ^{*})=0.016$ to obtain the cross sections for the interested final state.
Cross sections (pb) $\sqrt{s}=0.5$ TeV $\sqrt{s}=1$ TeV $\sqrt{s}=3$ TeV
---------------------------------------- --------------------- --------------------- ---------------------
$e^{+}e^{-}\to W^{+}W^{-}Z$ $4.39\times10^{-2}$ $6.52\times10^{-2}$ $3.82\times10^{-2}$
$e^{+}e^{-}\to W^{+}W^{-}h$ $5.85\times10^{-3}$ $4.21\times10^{-3}$ $1.25\times10^{-3}$
$e^{+}e^{-}\to W^{+}W^{-}$ $7.68\times10^{0}$ $2.87\times10^{0}$ $4.98\times10^{-1}$
$e^{+}e^{-}\to W^{+}W^{-}\nu\bar{\nu}$ $1.09\times10^{-2}$ $3.38\times10^{-2}$ $1.47\times10^{-1}$
$l^{\pm}+2j+\mbox{MET}$ $1.15\times10^{0}$ $4.36\times10^{-1}$ $9.76\times10^{-2}$
: Cross sections (pb) for the dominant background contributing to the final state $l^{\pm}+2j+\mbox{MET}$. \[tab:tab4\]
Analysis
========
We take into account the final state containing $l^{\pm}+2j+$MET. The transverse momentum and pseudo-rapidity distributions of the final state lepton from the signal and background process are given in Fig. \[fig:fig4\] and \[fig:fig5\], respectively. It is seen from Fig. \[fig:fig4\] that a transverse momentum cut $p_{T}>15$ GeV on the lepton will be required for the acceptance. However, higher $p_{T}$ cut will not improve the signal to background ratio in the analysis. Furthermore, the lepton pseudo-rapidity cut of $-1.5<\eta<1.5$ can be used to suppress the relevant background as seen from Fig. \[fig:fig5\]. The difference between the pseudo-rapidity distributions for the signal and background is more pronounced at higher center of mass energies such as $\sqrt{s}=1$ TeV and $3$ TeV. The cross section for the main background process $e^{+}e^{-}\to W^{+}W^{-}$ with $W^{\pm}\to l^{\pm}\bar{\nu}_{l}$ and $W^{\mp}\to q\bar{q}'$ will be reduced by $60\%$, $75\%$ and $90\%$, after applying these cuts, depending on the center of mass energies $\sqrt{s}=0.5$, $1$ and $3$ TeV, respectively. Moreover, the other background process $e^{+}e^{-}\to W^{+}W^{-}Z$ (where $W^{\pm}\to l^{\pm}\bar{\nu}_{l}$, $W^{\mp}\to q\bar{q}'$ and $Z\to\nu_{l}\bar{\nu}_{l}$) will give contribution with a reduction in cross section by $50\%$ after the cuts.
![The transverse momentum distribution of the final state muon ($\mu^{-}$) from the signal process $e^{+}e^{-}\to W^{+}\mu^{-}\bar{\nu}_{\mu}\nu'\bar{\nu}'$ with the parameters $x=1$ and $M_{l'}=200$ GeV, and from the background process $e^{+}e^{-}\to W^{+}\mu^{-}\bar{\nu}_{\mu}$ at $\sqrt{s}=500$ GeV. \[fig:fig4\]](fig4)
![The pseudo-rapidty distribution of the final state muon ($\mu^{-}$) from the signal process $e^{+}e^{-}\to W^{+}\mu^{-}\bar{\nu}_{\mu}\nu'\bar{\nu}'$ with the parameters $x=1$ and $M_{l'}=200$ GeV and from background process $e^{+}e^{-}\to W^{+}\mu^{-}\bar{\nu}_{\mu}$ at $\sqrt{s}=500$ GeV. \[fig:fig5\]](fig5)
In order to present the potential of the linear colliders to search for heavy lepton signal, we use a brief statistical significance analysis with $S/\sqrt{B}$. We find accessible ranges of the mixing parameter and the mass of heavy leptons depending on the integrated luminosity at different center of mass energies. We present the exclusion plot ($95\%$ C.L.) for the heavy lepton mass $M_{l'}$ and mixing parameter $x$ at $\sqrt{s}=1$ TeV and $L_{int}=20$, $100$, $1000$ pb$^{-1}$ and $10$ fb$^{-1}$ in Fig. \[fig:fig6\]. Fig. \[fig:fig7\] presents the integrated luminosity needed to exclude ($95\%$ C.L.) the ranges of the parameter $x$ at $\sqrt{s}=1$ TeV depending on the heavy lepton masses of $350$, $375$, $400$ and $450$ GeV.
![Exclusion plot ($95\%$ C.L.) for the heavy lepton mass $M_{l'}$ and mixing parameter $x$ at $\sqrt{s}=1$ TeV and different integrated luminosities: thick (red) line, dashed (green) line, dotted (blue) line and dot-dashed (magenta) line corresponds to $L_{int}=20$, $100$, $1000$ pb$^{-1}$ and $10$ fb$^{-1}$, respectively. \[fig:fig6\]](fig6)
![The integrated luminosity needed to exclude ($95\%$ C.L.) the ranges of the parameter $x$ at $\sqrt{s}=1$ TeV depending on the heavy lepton masses: thick (red) line, dashed (green) line, dotted (blue) line and dot-dashed (magenta) line corresponds to the mass values $M_{l'}=350$, $375$, $400$ and $450$ GeV, respectively. \[fig:fig7\]](fig7)
In Fig. \[fig:fig8\], we present the exclusion plot ($95\%$ C.L.) for the heavy lepton mass $M_{l'}$ and mixing parameter $x$ at $\sqrt{s}=3$ TeV and $L_{int}=5$, $10$, $30$ and $100$ fb$^{-1}$. Fig. \[fig:fig9\] presents the integrated luminosity needed to exclude ($95\%$ C.L.) the ranges of the parameter $x$ at $\sqrt{s}=3$ TeV depending on the heavy lepton masses of $200$, $400$, $500$ and $600$ GeV.
![Exclusion plot ($95\%$ C.L.) for the heavy lepton mass $M_{l'}$ and mixing parameter $x$ at $\sqrt{s}=3$ TeV and different integrated luminosities: thick (red) line, dashed (green) line, dotted (blue) line and dot-dashed (magenta) line corresponds to $L_{int}=5$, $10$, $30$ and $100$ fb$^{-1}$, respectively. \[fig:fig8\]](fig8)
![The integrated luminosity needed to exclude ($95\%$ C.L.) the ranges of the parameter $x$ at $\sqrt{s}=3$ TeV depending on the heavy lepton masses: thick (red) line, dashed (green) line, dotted (blue) line and dot-dashed (magenta) line corresponds to $M_{l'}=200$, $400$, $500$ and $600$ GeV, respectively. \[fig:fig9\]](fig9)
It can be seen from the contour plots Fig. \[fig:fig6\] and Fig. \[fig:fig8\], the accessible range of the parameters are defined outside of the contour lines depending on the luminosity of the collider. For example, at $\sqrt{s}=1$ TeV and $L_{int}=100$ fb$^{-1}$ the search can be performed beyond the range of mixing parameter $-0.4<x<0.1$ for given $M_{l'}=400$ GeV, while at $\sqrt{s}=3$ TeV and $L_{int}=10$ fb$^{-1}$ the inaccessible range becomes only $-0.3<x<-0.1$ for given $M_{l'}=600$ GeV. We find the ranges of parameters which can be accessed beyond $-0.4<x<0.1$ and $-0.25<x<-0.2$ for $M_{l'}=400$ GeV at the center of mass energies $\sqrt{s}=1$ TeV and $3$ TeV at $L_{int}=10$ fb$^{-1}$, respectively. However, the whole region of interest of the parameter $x$ can be accessed for $M_{l'}<800$ GeV at $\sqrt{s}=3$ TeV and $L_{int}=100$ fb$^{-1}$.
Conclusion
==========
The charged $l'^{\pm}$ lepton will have a clear signature at linear colliders. It is shown that the accessible ranges of the parameters of new heavy leptons can be searched through the process $e^{+}e^{-}\to l'^{+}l'^{-}$ with their subsequent decays $l'^{\pm}\to\overset{(-)}{\nu}'+W^{\pm}$. In this work, the lepton+dijet+missing transverse energy ($l^{\pm}+2j+$MET) final states for the signal and background have been taken into account to find the luminosity required to search for the new heavy lepton mass and mixing parameter. If the LHC find clues about the new physics models, the linear collider at TeV scale can enhance the accessible range of parameter space from these models.
The work of O.C. and S.K. is supported in part by the Ministry of Development (also formerly called State Planning Organization - DPT) project under Grant No. DPT2006K-120470.
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---
abstract: 'Scott’s information systems provide a categorically equivalent, intensional description of Scott domains and continuous functions. Following a well established pattern in denotational semantics, we define a linear version of information systems, providing a model of intuitionistic linear logic (a new-Seely category), with a “set-theoretic" interpretation of exponentials that recovers Scott continuous functions via the co-Kleisli construction. From a domain theoretic point of view, linear information systems are equivalent to prime algebraic Scott domains, which in turn generalize prime algebraic lattices, already known to provide a model of classical linear logic.'
author:
- '$\textrm{A.~Bucciarelli}^{\dagger}$'
- '$\textrm{A.~Carraro}^{\circ\dagger}$'
- '$\textrm{T.~Ehrhard}^{\dagger}$'
- '$\textrm{A.~Salibra}^{\circ}$'
title: On Linear Information Systems
---
Introduction
============
The ccc of Scott domains and continuous functions, which we call $\Sd$, is the paradigmatic framework for denotational semantics of programming languages. In that area, much effort has been spent in studying more “concrete" structures for representing domains.
At the end of the 70’s G. Kahn and G. D. Plotkin [@Kahn78] developed a theory of concrete domains together with a representation of them in terms of concrete data structures. In the early 80’s G. Berry and P.-L. Curien [@Berry82] defined a ccc of concrete data structures and sequential algorithms on them. At the same time Scott [@Scott82] also developed a representation theory for Scott domains which led him to the definition of information systems; these structures, together with the so-called approximable relations, form the ccc $\Inf$, which is equivalent to $\Sd$.
So it was clear that many categories of “higher-level" structures such as domains had equivalent descriptions in terms of “lower-level" structures, such as concrete data structures and information systems, which are collectively called webs.
At the end of the 80’s, J.-Y. Girard [@Girard86] discovered linear logic starting from a semantical investigation of the second-order lambda calculus. His seminal work on the semantics of linear logic proofs [@Girard87; @Girard88], introduced a category of webs, the coherence spaces, equivalent to the category of coherent qualitative domains and stable maps between them. Coherence spaces form a $\ast$-autonomous and thus a model of classical Linear Logic (LL).
From the early 90’s on, there has been a wealth of categorical models of linear logic, arising from different areas: we mention here S. Abramsky and R. Jagadeesan’s games [@Abramsky92], Curien’s sequential data structures [@Curien94], G. Winskel’s event structures [@Winskel88; @Zhang92], and Winskel and Plotkin’s bistructures [@Winskel94] whose associated co-Kleisli category is equivalent to a full-sub-ccc of Berry’s category of bidomains [@Berry79].
Remarkably, all the above-mentioned models lie outside Scott semantics.
Despite the observation made by M. Barr’s in 1979 [@Barr79] that the category of complete lattices and linear maps is $\ast$-autonomous, it was a common belief in the Linear Logic community that the standard Scott semantics could not provide models of classical LL, until 1994, when M. Huth showed [@Huth94] that the category $\Palglat$ of prime-algebraic complete lattices and lub-preserving maps is $\ast$-autonomous and its associated ccc $\Palglat^!$ (the co-Kleisli category of the “!" comonad) is a full-sub-ccc of $\Cpo$. A few years later, Winskel rediscovered the same model in a semantical investigation of concurrency [@Winskel99; @Winskel04]: indeed he showed that the category $\Scottl$ whose objects are preordered sets and the morphisms are functions from downward closed to downward closed subsets which preserve arbitrary unions is $\ast$-autonomous; this category is equivalent to Huth’s. T. Ehrhard [@Ehrhard09] continues this investigation and shows that the extensional collapse of the category $\Rel^!$, where $!$ is a comonad based on multi-sets over the category $\Rel$ of sets and relations, is the category $\Scottl^!$ and that both are new-Seely categories (in the sense of Bierman [@Bierman95]).
Summing up, there are several categorical models of LL in Scott semantics. In this paper we provide a representation of these models as webs. Our starting point are information systems, of which we provide a linear variant together with linear approximable relations: such data form the category $\Infl$, which we prove to be a symmetric monoidal closed category; $\Infl$ is equivalent to the category $\Psd$ of prime algebraic Scott domains and has as full-sub-categories $\Inflfull$ (equivalent to the category $\Palglat$) and, ultimately, $\Rel$.
We define a comonad $!$ over $\Infl$, based on sets rather than multi-sets, which makes $\Infl$ a new-Seely category and hence a model of intuitionistic MELL: our approach is different from that of [@Ehrhard09] in that our comonad is not an endofunctor of $\Rel$; we don’t need to consider multisets exactly because we work in the bigger category $\Infl$. We also notice that $\Inflfull$ is the largest $\ast$-autonomous full-subcategory of $\Infl$ and that $\Infl^!$ is a full sub-ccc of $\Inf$.
The category of linear informations systems
===========================================
Let $A$ be a set. We adopt the following conventions: letters $\ga, \gb, \gc, \ldots$ are used for elements of $A$; letters $a, b, c, \ldots$ are used for elements of $\cP_\rf(A)$; letters $x, y, z, \ldots$ are used for arbitrary elements of $\cP(A)$.
A *linear information system* (LIS, for short) is a triple $\cA= (A, \Con, \vdash)$, where contains all singletons and $\vdash \ \subseteq A \times A$ satisfies the axioms listed below.
- if $a \in \Con$ and $\forall \gb \in b.\exists \ga \in a.\ \ga \vdash \gb$, then $b \in \Con$
- $\ga \vdash \ga$
- if $\ga \vdash \gb \vdash \gc$, then $\ga \vdash \gc$
The set $A$ is called the *web* of $\cA$ and its elements are called *tokens*. Let $\cA$, $\cB$ be two LISs. A relation $R \subseteq A \times B$ is *linear approximable* if for all $\ga,\ga' \in A$ and all $\gb,\gb' \in B$:
- if $a \in \Con_A$ and $\forall \gb \in b.\exists \ga \in a.\ (\ga, \gb) \in R$, then $b \in \Con_B$
- if $\ga' \vdash_A \ga \ R \ \gb \vdash_B \gb'$, then $(\ga', \gb') \in R$
Linear approximable relations compose as usual: $S \circ R = R; S = \{(\ga,\gc) \in A \times C : \exists \gb \in B.\ \ga \ R \ \gb \ S \ \gc\}$.
We call $\Infl$ the category with LISs as objects and linear approximable relations as morphisms. We reserve the name $\Inflfull$ for the full-subcategory of $\Infl$ whose objects are exactly those LISs $\cA$ for which $\Con_A = \cP_\rf(A)$. It is not difficult to see that the category $\Rel$ of sets and relations is a full-subcategory of $\Inflfull$, consisting of exactly those LISs $\cA$ for which $\Con_A = \cP_\rf(A)$ and .
The cartesian structure of $\Infl$
----------------------------------
We now define the cartesian product and coproduct in the category $\Infl$, which model the additive connectives of linear logic.
Let $\cA_1$, $\cA_2$ be LISs. Define the LIS $\cA_1 \binampersand \cA_2 = (A_1 \binampersand A_2, \Con, \vdash)$ as follows:
- $A_1 \binampersand A_2 = A_1 \uplus A_2$
- $\{(i_1,\gc_1), \ldots, (i_m, \gc_m)\} \in \Con$ iff $\{\gc_j : j \in [1,m],\ i_j = 1\} \in \Con_{A_1}$ and $\{\gc_j : j \in [1,m],\ i_j = 2\} \in \Con_{A_2}$
- $(i, \gc) \vdash (j,\gc')$ iff $i = j$ and $\gc \vdash_{A_i} \gc'$
The projections $\pi_i \in \Infl(\cA_1 \binampersand \cA_2, \cA_i)$ are given by , $i = 1,2$. For and , the pairing is given by\
Define $\top = (\emptyset, \emptyset, \emptyset)$. The LIS $\top$ is the terminal object of $\Infl$, since for any LIS $\cA$, the only morphism $R \in \Infl(\cA, \top)$ is $\emptyset$.
Let $\cA_1$, $\cA_2$ be LISs. Define the LIS $\cA_1 \oplus \cA_2 = (A_1 \oplus A_2, \Con, \vdash)$ as follows:
- $A_1 \oplus A_2 = A_1 \uplus A_2$
- $\{(i_1,\gc_1), \ldots, (i_m, \gc_m)\} \in \Con$ iff $\{\gc_j : j \in [1,m],\ i_j = 1\} \in \Con_{A_1}$ and $\{\gc_j : j \in [1,m],\ i_j = 2\} \in \Con_{A_2}$
- $(i, \gc) \vdash (j,\gc')$ iff $i = j$ and $\gc \vdash_{A_i} \gc'$
Therefore cartesian products and coproducts coincide. The injections $\iota_i \in \Infl(\cA_i, \cA_1 \oplus \cA_2)$ are obtained reversing the projections, so that , $i = 1,2$. For and , the “co-pairing" is given by\
Define $\mathbf{0} = \top = (\emptyset, \emptyset, \emptyset)$: this LIS is also the initial object of $\Infl$ and the unit of the coproduct.
The monoidal closed structure of $\Infl$
----------------------------------------
We now define the tensor product and its dual in the category $\Infl$, which model the multiplicative connectives of linear logic.
Let $\cA$, $\cB$ be LISs. Define the LIS $\cA \otimes \cB = (A \otimes B, \Con, \vdash)$ as follows:
- $A \otimes B = A \times B$
- $\{(\ga_1, \gb_1), \ldots, (\ga_m, \gb_m)\} \in \Con$ iff $\{\ga_1, \ldots, \ga_m\} \in \Con_A$ and $\{\gb_1, \ldots, \gb_m\} \in \Con_B$
- $(\ga, \gb) \vdash (\ga',\gb')$ iff $\ga \vdash_A \ga'$ and $\gb \vdash_B \gb'$
For $R \in \Infl(\cA, \cC)$ and $S \in \Infl(\cB, \cD)$, $R \otimes S \in \Infl(\cA \otimes \cB, \cC \otimes \cD)$ is given by\
It is easy to check that\
is a bifunctor and that it is a symmetric tensor product, with natural isomorphisms
- $\phi_{\cA,\cB,\cC}^\otimes : \cA \otimes (\cB \otimes \cC) \to (\cA \otimes \cB) \otimes \cC$ given by $$\phi_{\cA,\cB,\cC}^\otimes = \{( (\ga,(\gb,\gc)),((\ga',\gb'),\gc') ) : \ga \vdash_A \ga',\ \gb \vdash_B \gb',\ \gc \vdash_C \gc' \}$$
- $\sigma_{\cA,\cB}^\otimes: \cA \otimes \cB \to \cB \otimes \cA$ given by $\sigma_{\cA,\cB}^\otimes = \{((\ga,\gb), (\gb',\ga')) : \ga \vdash_A \ga',\ \gb \vdash_B \gb'\}$
- $\rho_\cA^\otimes: \cA \otimes \mathbf{1} \to \cA$ given by $ \rho_\cA^\otimes = \{((\ga, \ast),\ga') : \ga \vdash_A \ga'\}$
- $\lambda_\cA^\otimes: \mathbf{1} \otimes \cA \to \cA$ given by $\lambda_\cA^\otimes = \{((\ast, \ga),\ga') : \ga \vdash_A \ga'\}$
Define $\mathbf{1} = (\{\ast\}, \{\emptyset, \{\ast\}\},\{(\ast, \ast)\})$. The LIS $\mathbf{1}$ is the unit of the tensor product.
The above data make $\Infl$ a symmetric monoidal category. As for the cartesian structure, also the dual of the tensor product, $\bindnasrepma$, coincides with the tensor in this category and thus the respective units $\bot$ and $\mathbf{1}$ are equal. So we take $\bot = (\{\ast\}, \{\emptyset, \{\ast\}\},\{(\ast, \ast)\})$ to be also the unit of $\bindnasrepma$.
We now proceed to define the exponential objects of $\Infl$.
Let $\cA$, $\cB$ be LISs. Define the LIS $\cA \multimap \cB = (A \multimap B, \Con, \vdash)$ as follows:
- $A \multimap B = A \times B$
- $\{(\ga_1, \gb_1), \ldots, (\ga_m, \gb_m)\} \in \Con$ iff for all $J \subseteq [1,m]$, $\{\ga_j : j \in J\} \in \Con_A$ implies $\{\gb_j : j \in J\} \in \Con_B$
- $(\ga,\gb) \vdash (\ga',\gb')$ iff $\ga' \vdash_A \ga$ and $\gb \vdash_B \gb'$
Define a natural isomorphism $\cur: \Infl(\cA \otimes \cC, \cB) \to \Infl(\cC, \cA \multimap \cB)$, (the linear currying) as $\cur(R) = \{(\gc, (\ga, \gb)) : ((\ga, \gc), \gb) \in R \}$. Define also the (linear) evaluation morphism as .
The above data make $\Infl$ a symmetric monoidal closed category, and thus a model of intuitionistic MLL proofs. The category $\Infl$ is however a rather degenerate model since the multiplicative connectives $\bindnasrepma$ and $\otimes$ coincide as well as the additives $\binampersand$ and $\oplus$.
We now briefly discuss the issue of duality in the category $\Infl$. Let $\cA$ be a LIS and consider the LIS $\cA \multimap \bot$; an explicit description of such object is as follows:
- $A \multimap \bot = A \times \{\ast\}$
- $\Con_{A \multimap \bot} = \cP_\rf(A \times \{\ast\})$
- $(\ga, \ast) \vdash_{A \multimap \bot} (\gb, \ast)$ iff $\gb \vdash_A \ga$
Therefore $\bot$ is not a dualizing object in $\Infl$, but it is so in $\Inflfull$, where the family of arrows $\partial_\cA: \cA \to (\cA \multimap \bot) \multimap \bot$ defined by $\partial_\cA = \{(\ga,((\ga',\ast),\ast)) : \ga \vdash_A \ga'\}$, for each LIS $\cA$, is a natural isomorphism. In other words $\Inflfull$ is the largest $\ast$-autonomous full-subcategory of $\Infl$.
$\Infl$ is a new-Seely category
-------------------------------
In this section we define a comonad $!$ over $\Infl$ and prove that it gives a symmetric strong monoidal functor. Finally we prove that $\Infl$ is a new-Seely category and thus a model of intuitionistic MELL proofs, in which the exponential modality $!$ has a set-theoretic interpretation. For categorical notions we refer to Melliès [@Mellies].
Let $\cA$ be a LIS. Define the LIS $! \cA = (! A, \Con, \vdash)$ as follows:
- $! A = \Con_A$
- $\{a_1, \ldots, a_k\} \in \Con$ iff $\cup_{i=1}^{k} a_i \in \Con_A$
- $a \vdash b$ iff $\forall \gb \in b.\exists \ga \in a.\ \ga \vdash_A \gb$
Note that for $X \in !!A$ and $a \in !A$ we have $X \vdash_{!!A} \{b\}$ implies $\cup X \vdash_{!A} b$ but not viceversa. As an example, consider that $\{\alpha, \beta\} \vdash_{!A} \{\alpha, \beta\}$ but in general not $\{\{\alpha\}, \{\beta\}\} \vdash_{!!A} \{\{\alpha, \beta\}\}$.
Let $R \in \Infl(\cA, \cB)$. Define $! R \in \Infl(! \cA, ! \cB)$ as It is an easy matter to verify that is a functor. Moreover “$!$" is a comonad with digging defined by and dereliction defined by
As a matter of fact “$!$" is also a monad if endowed with natural transformations *codigging* defined by and *codereliction* defined by This is due to the fact that iff and iff .
This also shows a further symmetry: for each object $!\cA$,
- the digging morphism is subsumed by the codereliction morphism in the sense that\
, since for $X \in !!A$ and $a \in !A$ we have iff ;
- the codigging morphism is subsumed by the dereliction morphism in the sense that\
.
The forthcoming lemma shows that “$!$" is a symmetric strong monoidal functor. Before proving this, we shall explicit the symmetric monoidal structure $(\Infl, \binampersand, \top)$ involved in the proof:
- $\phi_{\cA,\cB,\cC}^\binampersand : \cA \binampersand (\cB \binampersand \cC) \to (\cA \binampersand \cB) \binampersand \cC$ given by $$\begin{aligned}
\phi_{\cA,\cB,\cC}^\binampersand & = & \big\{\big((1,\ga), (1,(1,\ga'))\big) : \ga \vdash_{A} \ga' \big\} \cup \big\{\big((2,(1,\gb)), (1,(2,\gb'))\big) : \gb \vdash_{B} \gb' \big\} \cup \\
& & \cup \big\{\big((2,(2,\gc)), (2,\gc')\big) : \gc \vdash_{C} \gc' \big\}\end{aligned}$$
- $\sigma_{\cA,\cB}^\binampersand: \cA \binampersand \cB \to \cB \binampersand \cA$ given by $$\sigma_{\cA,\cB}^\binampersand = \big\{\big((1,\ga), (2,\ga')\big) : \ga \vdash_{A} \ga' \big\} \cup \big\{\big((2,\gb), (1,\gb')\big) : \gb \vdash_{B} \gb' \big\}$$
- $\rho_\cA^\binampersand: \cA \binampersand \top \to \cA$ given by $\rho_\cA^\binampersand = \big\{\big((1,\ga),\ga' \big) : \ga \vdash_A \ga' \big\}$
- $\lambda_\cA^\binampersand: \top \binampersand \cA \to \cA$ given by $\lambda_\cA^\binampersand = \big\{\big((2,\ga),\ga' \big) : \ga \vdash_A \ga' \big\}$
\[SSMF\] The functor $!: (\Infl, \binampersand, \top) \to (\Infl, \otimes, \mathbf{1})$ is symmetric strong monoidal.
We give the natural isomorphisms $\textsf{m}_{\cA,\cB}:\ ! \cA \otimes ! \cB \cong !(\cA \binampersand \cB)$ and $\textsf{n}: \mathbf{1} \cong ! \top$ making $!$ a symmetric strong monoidal functor. Define
- $\textsf{n} = \{(\ast, \{\emptyset\})\}$
- $\textsf{m}_{\cA,\cB} = \big\{\big( (a,b), \{(1,\ga') : \ga' \in a' \} \cup \{(2,\gb') : \gb' \in b' \} \big) : a,a' \in \ ! A,\ b,b' \in \ ! B,\ a \vdash_{! A} a',\ b \vdash_{! B} b' \big\}$
We now proceed by verifying the commutation of the required diagrams. First observe that both $\textsf{m}_{\cA,\cB} \otimes \id_{! \cC} ; \textsf{m}_{\cA \binampersand \cB, \cC} ; ! \phi_{\cA,\cB,\cC}^\binampersand$ and $\phi_{\cA,\cB,\cC}^\otimes ; \textsf{id}_{! \cA} \otimes \textsf{m}_{\cB,\cC} ; \textsf{m}_{\cA ,\cB \binampersand \cC}$ are equal to the set of all pairs $$\Big( ((a,b),c), \{(1,(1, \alpha_1)), \ldots, (1,(1, \alpha_{n_1})), (2,(1, \beta_1)), \ldots, (2, (1,\beta_{n_2})), (2,(2, \gamma_1)), \ldots, (2,(2, \gamma_{n_3}))\} \Big)$$ where $a \vdash_{!A} \{\alpha_1, \ldots, \alpha_{n_1}\}$, $b \vdash_{!B} \{\beta_1, \ldots, \beta_{n_2}\}$, and $c \vdash_{!C} \{\gamma_1, \ldots, \gamma_{n_3}\}$. Therefore the diagram
$$\xymatrix{
(! \cA \otimes ! \cB) \otimes ! \cC \ar[r]^{\phi_{\cA, \cB, \cC}^\otimes} \ar[d]_{\textsf{m}_{\cA,\cB} \otimes \textsf{id}_{! \cC}} & ! \cA \otimes (! \cB \otimes ! \cC) \ar[d]^{\textsf{id}_{! \cA} \otimes \textsf{m}_{\cB,\cC}} \\
! (\cA \binampersand \cB) \otimes ! \cC \ar[d]_{\textsf{m}_{\cA \binampersand \cB,\cC}} & ! \cA \otimes ! (\cB \binampersand \cC) \ar[d]^{\textsf{m}_{\cA, \cB \binampersand \cC}} \\
! ((\cA \binampersand \cB) \binampersand \cC) \ar[r]^{! \phi_{\cA, \cB, \cC}^\binampersand} & ! (\cA \binampersand (\cB \binampersand \cC)) \\
}$$ commutes. Finally the “units" and the “symmetry" diagrams $$\xymatrix{
! \cA \otimes \mathbf{1} \ar[r]^{\rho_{!\cA}^\otimes} \ar[d]_{\textsf{id}_{! \cA} \otimes \textsf{n}} & ! \cA & & \mathbf{1} \otimes ! \cB \ar[r]^{\lambda_{!\cB}^\otimes} \ar[d]_{\textsf{n} \otimes \textsf{id}_{! \cB}} & ! \cB & & !\cA \otimes !\cB \ar[r]^{\sigma_{!\cA,!\cB}^\otimes} \ar[d]_{\textsf{m}_{\cA,\cB}} & !\cB \otimes !\cA \ar[d]^{\textsf{m}_{\cB,\cA}} \\
! \cA \otimes ! \top \ar[r]^{\textsf{m}_{\cA, \top}} & ! (\cA \binampersand \top) \ar[u]_{! \rho_\cA^\binampersand} & & ! \top \otimes ! \cB \ar[r]^{\textsf{m}_{\top, \cB}} & ! (\top \binampersand \cB) \ar[u]_{! \lambda_\cB^\binampersand} & & !(\cA \binampersand \cB) \ar[r]^{!\sigma_{\cA,\cB}^\binampersand} & ! (\cB \binampersand \cA)
}$$ all commute because
- $\id_{! \cA} \otimes \textsf{n} ; \textsf{m}_{\cA, \top} ; !\rho_\cA^\binampersand = \{((a, \ast),a') : a \vdash_{! A} a'\} = \rho_{!\cA}^\otimes$,
- $\textsf{n} \otimes \textsf{id}_{! \cB} ; \textsf{m}_{\top, \cB} ; !\lambda_\cB^\binampersand = \{((\ast, b),b') : b \vdash_{! B} b'\} = \lambda_{!\cB}^\otimes$,
- both $\sigma_{!\cA,!\cB}^\otimes;\textsf{m}_{\cB,\cA}$ and $\textsf{m}_{\cA,\cB}; !\sigma_{\cA,\cB}^\binampersand$ equal the morphism\
$\big\{\big( (a,b), \{(1,\gb') : \gb' \in b' \} \cup \{(2,\ga') : \ga' \in a' \} \big) : b' \in \ ! B,\ a' \in \ ! A,\ b \vdash_{! B} b',\ a \vdash_{! A} a' \big\}$.
$\Infl$ is a new-Seely category.
By Lemma \[SSMF\], $!$ is a symmetric strong monoidal functor; it remains to check the coherence diagram in the definition of new-Seely category. Indeed we have: $$\begin{aligned}
\dig_\cA \otimes \dig_\cB ; \textsf{m}_{! \cA,! \cB} & = & \big\{\big( (a,b) , \{(1,a') : a' \in X'\} \cup \{(2,b') : b' \in Y'\} \big) : a \vdash_{! A} \cup X',\ b \vdash_{! B} \cup Y' \big\} \\
& = & \textsf{m}_{\cA,\cB} ; \dig_{\cA \binampersand \cB}; ! \langle ! \pi_1, ! \pi_2 \rangle\end{aligned}$$ So that the following diagram commutes. $$\xymatrix{
! \cA \otimes ! \cB \ar[r]^{\textsf{m}_{\cA, \cB}} \ar[dd]_{\dig_\cA \otimes \dig_\cB} & ! (\cA \binampersand \cB) \ar[d]^{\dig_{\cA \binampersand \cB}} \\
& !! (\cA \binampersand \cB) \ar[d]^{! \langle ! \pi_1, ! \pi_2 \rangle} \\
!! \cA \otimes !! \cB \ar[r]^{\textsf{m}_{!\cA, !\cB}} & ! (! \cA \binampersand !\cB)
}$$
A representation theorem
------------------------
Not surprisingly, $\Infl$ turns out to be equivalent to the category of prime algebraic Scott domains and linear continuous functions. In this section, we outline this equivalence.
An element $p$ of a Scott domain $\cD$ is *prime* if, whenever $B \subseteq_\rf D$ is upper bounded and $p \leq \vee B$, there exists $b \in B$ such that $p \leq b$. The domain $\cD$ itself is *prime algebraic* if every element of $D$ is the least upper bound of the set of prime elements below it. The set of prime elements of $\cD$ is denoted by $\Prime{D}$.
A *linear function* between two prime algebraic Scott domains is a Scott continuos function that commutes with all (existing) least upper bounds. The category of prime algebraic Scott domains and linear function is denoted by $\Psd$.
A *point* of an information system $\cA$ is a subset $x \subseteq A$ satisfying the following two properties:
- if $u \subseteq_\rf x$ then $u \in \Con$ ($x$ is *finitely consistent*)
- if $\alpha\in x$ and $\alpha \vdash \alpha'$, then $\alpha' \in x$ ($x$ is *closed w.r.t.* $\vdash$)
We are now able to relate linear information systems to the corresponding categories of domains.
\[functors-for-equivalence\] Given $f,\cD,\cE, R ,\cA, \cB$ such that $f \in \Psd(\cD,\cE)$ and $R \in \Infl(\cA, \cB)$, we define:
- $\cA^+$ is the set of points of $\cA$ ordered by inclusion.
- $R^+(x) = \{\beta \in B \mid \exists \alpha \in x.\ (\alpha,\beta)\in R\}$
- $\cD^- = (\Prime{D},\Con,\vdash)$ where $a \in \Con$ iff $a$ is upper bounded and $p \vdash p'$ iff $p'\leq_\cD p$
- $f^- = \{(p,p') \in \Prime{D} \times \Prime{E} \mid f(p) \geq_\cD p'\}$
\[equivalence\] The functors $(\_ )^+,(\_ )^-$ define an equivalence between the categories $\Infl$ and $\Psd$.
It is an easy task to check that $(\_ )^+: \Infl \to \Psd$ and ,$(\_ )^-: \Psd \to \Infl$ are indeed full and faithful functors and the two composite endofunctors $(-)^- \circ (-)^+$ and $(-)^+ \circ (-)^-$ are naturally isomorphic to the identity functor of $\Infl$ and $\Psd$, respectively. Moreover every hom-set $\Infl(\cA, \cB)$, ordered by inclusion of relations, is a prime algebraic Scott domain and $\Infl(\cA, \cB) = (\cA \multimap \cB)^+$, so that the functors $(-)^-$ and $(-)^+$ preserve exponentials, i.e. $(\cA \multimap \cB)^+ \cong \Psd(\cA^+, \cB^+)$ in the category $\Psd$ and $\Psd(\D, \E)^- \cong \D^- \multimap \E^-$ in the category $\Infl$. Finally the two functors also preserve products, since $(\cA \binampersand \cB)^+ \cong \cA^+ \times \cB^+$ in the category $\Psd$ and $(\D \times \E)^- \cong \D^- \binampersand \E^-$ in the category $\Infl$.
In view of the categorical equivalence stated in Theorem \[equivalence\], Proposition \[co-Kleisli\] shows how to recover Scott-continuous functions from linear ones. This equivalence specializes to an equivalence between $\Inflfull$ and the category $\Palglat$ of prime algebraic lattices and linear continuos functions.
Classical versus linear information systems
===========================================
In the previous section we have treated a categorical equivalence explaining how linear information systems constitute a representation for prime algebraic Scott domains. Along the same lines Scott domains have an appealing representation as information systems introduced by Dana Scott in [@Scott82]. More recently [@Spreen08] a more general class of information systems have also been axiomatized, namely that of continuous information systems , defined in order to constitute a representation for continuous domains.
An information system consists of a set of tokens, over which are imposed an entailment and a consistency relation; it determines a Scott domain with elements those sets of tokens which are consistent and closed with respect to the entailment relation; the ordering is again just set inclusion. Vice versa a Scott domain defines an information system through its compact elements. In this section we review the basic notions on information systems.
Information systems are then organized in a category equivalent to that of Scott domains and continuous maps, $\Sd$, but more “concrete” and easier to work with under many respects. For example they have been used in [@Salibra09] to show that there is no reflexive object in $\Sd$ whose theory is exactly the least extensional lambda theory $\gl\gb\eta$.
In this section we explain the relation between the “classical" Scott’s information systems and linear information systems.
Originally ([@Scott82]) an information system (IS, for short) is a triple $\cA= (A, \Con, \vdash)$, where contains all singletons and $\vdash \ \subseteq \Con \times \Con$ satisfies the axioms listed below.
- if $a \in \Con$ and $a \vdash b$, then $b \in \Con$
- if $a' \subseteq a$, then $a \vdash a'$
- if $a \vdash b \vdash c$, then $a \vdash c$
A relation $R \subseteq \Con_A \times \Con_B$ is *approximable* if:
- if $a \in \Con_A$ and $a \ R \ b$, then $b \in \Con_B$
- if $a' \vdash_A a \ R \ b \vdash_B b'$, then $a'\ R\ b'$
Clearly the every approximable relation, included $\vdash$, is completely determined by tokens on the right-hand side in the sense that $a \ R \ b$ iff $\forall \beta \in b.\ a \ R \ \{\beta\}$. Hence we shall identify each approximable relation $R$ with its *trace* $\{(a, \beta) : (a, \{\beta\}) \in R \}$.
We call $\Inf$ the category with ISs as objects and approximable relations as morphisms. It is well-known that $\Inf(\cA, \cB)$, ordered by inclusion of relations, is a Scott domain. Let us recall the definition of exponentials in $\Inf$ ([@Larsen91]).
Let $\cA$, $\cB$ be ISs. Define the IS $\cA \Rightarrow \cB = (A \Rightarrow B, \Con, \vdash)$ as follows:
- $A \Rightarrow B = \Con_A \times B$
- $\{(a_1, \beta_1), \ldots, (a_m, \beta_m)\} \in \Con$ iff for all $J \subseteq [1,m]$, $\cup \{a_j : j \in J\} \in \Con_A$ implies $\{\beta_j : j \in J\} \in \Con_B$
- $\{(a_1, \beta_1), \ldots, (a_m,\beta_m)\} \vdash (a', \beta')$ iff $\{\beta_j : a' \vdash a_j,\ 1 \leq j \leq m \} \vdash_B \beta'$
Similarly to the functors given in Definition \[functors-for-equivalence\], there are functors $(\_ )^\bullet: \Inf \to \Sd$ and which define another equivalence of categories. In particular for a given IS $\cA$, $\cA^\bullet$ is the collection of all subsets $x \subseteq A$ satisfying the following two properties:
- if $u \subseteq_\rf x$ then $u \in \Con$ ($x$ is *finitely consistent*)
- if $ a \subseteq x$ and $a \vdash \alpha'$, then $\alpha' \in x$ ($x$ is *closed w.r.t.* $\vdash$)
Again we have that $\Inf(\cA, \cB) = (\cA \Rightarrow \cB)^\bullet$. The categorical equivalence stated in Theorem \[equivalence\] mirrors perfectly the equivalence between the categories $\Inf$ and $\Sd$, so that the definition of linear information system and linear approximable relation is exactly what is required in order to capture the passage from the category of Scott domains and continuous functions to that of prime algebraic Scott domains and linear functions.
In fact both linear information systems and linear approximable relations can be seen as particular information systems and approximable relations, respectively, exactly as prime algebraic Scott domains are Scott domains and linear functions are continuous functions. The next proposition, based on this fact, shows that again, following a well-established pattern, the comonad $!$ allows to recover non-linear approximable relations form linear ones. As usual, we denote by $\Infl^!$ the co-Kleisli category of the comonad $!$ over $\Infl$.
\[co-Kleisli\] $\Infl^!$ is a full-sub-ccc of $\Inf$.
Let $\cA,\cB$ be LISs and let $\cA \Rightarrow \cB$ be the exponential object formed in the category $\Inf$: $\cA \Rightarrow \cB$ is a linear information system and it is an easy matter to see that $\cA \Rightarrow \cB = !\cA \multimap \cB$. Moreover $\cC^+ = \cC^\bullet$, for any LIS $\cC$, and thus .
The space $\Infl(\cA, \cB)$ of clearly embeds into $\Inf(\cA, \cB)$ (exactly as the space of linear functions embeds into that of continuous functions). The embedding is given by the map $\varphi: \Infl(\cA, \cB) \hookrightarrow \Inf(\cA, \cB)$ given by $\varphi(R) = \{(a, \beta) : \exists \alpha \in a.\ \alpha \ R \ \beta \}$. In other words the linear approximable relations are elements of $\Inf(\cA, \cB)$, i.e. exactly those approximable relations $S$ for which $(a, \beta) \in S$ iff $\exists \alpha \in a.\ (\alpha, \beta) \in S$. This is the analogue of the condition, dealing with preservation of existing suprema, that isolates linear functions between Scott domains among the continuous ones.
Conclusions and future work
===========================
In this paper we defined the category $\Infl$, whose objects and arrows result from a linearization of Scott’s information systems. We show this category to be symmetric monoidal closed and thus a model of MLL. We moreover prove that $\Infl$ is a new-Seely category, with a “set-theoretic” interpretation of exponentials via a comonad $!$; this is made possible by the presence of the entailment relation, which is always non-trivial in objects of the form $!\cA$: even if the entailment $\vdash_A$ is the equality we have that $\Infl(!\cA, \cA) \subset \cP(!\cA \times \cA)$ and this rules out Ehrhard’s counterexample for the naturality of dereliction. In the purely relational model of classical MELL, $\Rel$, the use of multisets is needed.
Indeed a comonad based on multi-sets, let’s say $\dag$, can be defined in our framework too, yelding a different co-Kleisli category. A similar situation arises in the framework of the coherence spaces model of LL. In that case Barreiro and Ehrhard [@Barreiro97] proved that the extensional collapse of the hierarchy of simple types associated to the multi-set interpretation [*is*]{} the hierarchy associated to the “set” interpretation. This means in particular that, as models of the simply typed $ \lambda$-calculus, the former discriminates finerly than the latter, being sensitive, for instance, to the number of occurrences of a variable in a term. It is likely that, in a similar way, $\Infl^!$ is the extensional collapse of $\Infl^\dag$.
The categories $\Infl$ and $\Inflfull$ may be themselves compared using the same paradigm. Trivializing the consitency relation boils down to add points to the underlying domains; is that another instance of extensional collapse situation? In the case of the simple types hierarchy over the booleans, the Scott model is actually the extensional collapse of the lattice-theoretic one [@Bucciarelli96].
Summing up it appears that, by tuning the linear information systems in different ways, one obtains different frameworks for the interpretation of proofs, whose inter-connections remain to be investigated.
This is, in our opinion, the main advantage of the approach, with respect to the existing descriptions of the Scott continuous models of linear logic [@Barr79; @Huth94; @Huth94b; @Winskel99].
Linear information systems are close to several classes of webbed models of the pure the $\lambda$-calculus: they generalize Berline’s [*preordered sets with coherence*]{} [@Berline00], where a set of tokens is consistent if and only if its elements are pairwise coherent. We plan to investigate whether such a generalisation is useful for studying the models of the $\lambda$-calulus in $\Psd$. Actually, one of our motivations was to settle a representation theory for a larger class of cartesian closed categories, whith $\Rel^!$ as a particular case, in order to provide a tool for investigating “non-standard”[^1] models. We get $\Rel$ as a full subcategory of $\Infl$, but the bunch of axioms on information-like structures making $\Rel^!$ an instance of the co-Kleisli construction remains to be found.
Another original motivation for investigating linear information systems, that we are pursuing, was the definition of a framework suitable for the interpretation of Boudol’s $\lambda$-calculus with resources [@Boudol93].
Finally we point out another research direction, suggested by the work of Ehrhard and Regnier ([@Ehrhard06], for example). In Köthe spaces [@Ehrhard02] as well as in finiteness spaces [@Ehrhard05], linear logic formulae are interpreted as topological vector spaces, and proofs of linear logic as linear continuous maps between these spaces. Then exponentials appear as “symmetric tensor algebra" constructions [@Blute93]. In the models considered there, linear maps from $!X$ to $Y$ can be seen as “analytic functions" (that is, functions definable by a power series) from the vector space $X$ to the vector space $Y$ and therefore can be differentiated. Classically, the derivative of a function $f: X \to Y$ is a function $f: X \to (X \multimap Y)$ such that for each $x \in X$, the linear function $f'(x)$ (the derivative of $f$ at point $x$) is the “best linear approximation" of the function $X \to Y$ which maps $u \in X$ to $f(x + u) \in Y$ (the general definition is local). In the analytic case, differentiation turns a linear function $f: !X \to Y$ into a linear function $f': !X \to (X \multimap Y)$, that is, $f': (!X \otimes X) \to Y$. It turns out that $f'$ can be obtained from $f$ by composing it (as a linear function from $!X$ to $Y$) on the left with a particular linear morphism $d: (!X \otimes X) \to !X$. This morphism itself can be defined in terms of more primitive operations on $!X$.
This can certainly be done also in the category $\Infl$ following for [@Hyland03], for example. However the induced differential combinator in this case does not reflect the idea of approximation typical of Scott semantics. The purpose we have in mind is to investigate the possibility of symmetric tensor bialgebra constructions, in the category $\Infl$, giving rise in the equivalent category $\Psd$ to a reasonable notion of derivative and compatible with the usual idea of approximation in Scott semantics.
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[^1]: Let us call “standard” a model of the $\lambda$-calculus that is an instance of one of the “main” semantics: continuous, stable, strongly stable.
|
---
abstract: 'Using time-resolved $1s$-$2p$ excitonic Lyman spectroscopy, we study the orthoexciton-to-paraexcitons transfer, following the creation of a high density population of ultracold $1s$ orthoexcitons by resonant two-photon excitation with femtosecond pulses. An observed fast exciton-density dependent conversion rate is attributed to spin exchange between pairs of orthoexcitons. Implication of these results on the feasibility of BEC of paraexcitons in Cu$_2$O is discussed.'
author:
- 'M. Kubouchi'
- 'K. Yoshioka'
- 'R. Shimano'
- 'A. Mysyrowicz'
- 'M. Kuwata-Gonokami'
title: 'Study of ortho-to-paraexciton conversion in Cu$_2$O by excitonic Lyman spectroscopy'
---
The observation of Bose-Einstein condensation (BEC) of neutral atoms, more than 70 years after its prediction, constitutes a major advance in physics of the last decade [@rb; @ketterle]. Ensembles of ultracold atoms with high controllability of density, temperature, and interaction strength reveals new aspects of many body quantum phenomena. In particular, by applying a magnetic field one can control the sign and magnitude of the scattering length between atoms. This allows probing the low temperature transition between collective pairing of fermionic atoms with attractive interaction and BEC of molecularlike bosonic entities [@jin]. Excitons, composite particles in semiconductors made of fermions, may provide another system particularly well suited for the study of this transition in a slightly different context. With increasing densities, the fluid should evolve continuously from a dilute Bose-Einstein condensate of excitons into a dielectric superfluid consisting of a BCS-like degenerate two-component Fermi system with Coulomb attraction[@keldysh; @conte]. Several recent experimental results have renewed interest in the search of BEC in photo excited semiconductors[@butov].
It has been long recognized that Cu$_2$O provides a material with unique advantages for the observation of BEC of excitons [@mysyrowicz]. Because of the positive parity of the valence and conduction band minima at the center of the Brillouin zone, their radiative recombination is forbidden in the dipole approximation, conferring a long radiative lifetime to the $n=1$ exciton. The $n=1$ exciton level is split by exchange interaction in a triply degenerate orthoexciton state (symmetry $\Gamma _5^+$; 2.033 eV at 2 K), and a lower lying singly degenerate paraexciton ($\Gamma _2^+$; 2.021 eV at 2 K) which is optically inactive to all orders. Several experiments have shown intriguing results in luminescence and transport suggesting the occurrence of a degenerate excitonic quantum fluid at high densities and low temperatures in this material [@fortin; @snoke]. However, a controversy has been raised on the actual density of excitonic particles created by optical pumping. Based on luminescence absolute quantum efficiency measurement, it has been claimed that a Auger effect with giant cross section destroys excitons when the density exceeds 10$^{15}$ cm$^{-3}$, preventing BEC [@ohara]. On the other hand, other authors have shown that another process, orthoexciton-paraexciton conversion by spin exchange is much more probable at low temperature [@prb61_16619]. Since this last effect does not destroy excitons, but merely increases the orthoexciton-paraexciton conversion rate, it does not prevent BEC. To resolve this controversy, it is therefore of prime importance to explore the ortho-para conversion rate as a function of density. More generally it is important to study the dynamics of paraexcitons in order to optimize the pumping conditions to achieve BEC. One of the major difficulties in this respect was the lack up to now of a sensitive spectroscopic method to probe optically inactive paraexcitons.
In this Letter, we use a new spectroscopic approach to study the dynamics of orthoexciton-paraexciton conversion in Cu$_2$O. The technique consists of probing the transition from the populated $1s$ to the $2p$ state, with a midinfrared (MIR) light beam. MIR excitonic spectroscopy is the counterpart of Lyman spectroscopy in atomic hydrogen [@haken; @jcs45_949; @johnsen]. It is especially well suited for the study of paraexcitons in Cu$_2$O because the $1s$-$2p$ transition is allowed even if the $1s$ paraexciton is optically inactive. With the use of a short pump pulse to excite orthoexciton and a weak MIR short probe light pulse, one can follow the gradual buildup of the paraexciton population from ortho-para conversion and its subsequent decay. Since the dipole matrix element for $1s$-$2p$ transition is known, one can extract the density of paraexcitons from the strength of the Lyman absorption. This has to be contrasted with luminescence data, where exciton density estimates rely on measurements of absolute radiative quantum efficiency, a notoriously difficult task. In addition, the lineshape of Lyman absorption yields precise information on the energy distribution of $1s$ excitons. In particular, Johnsen and Kavoulakis pointed out that is should show a characteristic abrupt change when excitonic BEC occurs [@johnsen].
To follow the density and temperature evolution of the orthoexcitons and paraexcitons starting with a well controlled situation, we first generate orthoexcitons by resonant two-photon absorption (TPA), using an ultrashort laser pulse. Drawing an analogy with the two-photon excitation of biexcitons in CuCl with ultrashort laser pulses [@cucl], we note that the created orthoexcitons have a very low initial temperature despite the large laser bandwidth, because of the small group velocity dispersion at the TPA laser frequency ($\hbar \omega =1.0164$ eV). In fact, the initial momentum spread of the generated orthoexcitons is even much smaller than in the case of biexcitons in CuCl. A conservative estimate yields an initial orthoexciton temperature much less than $10^{-3}$ K.
![\[figure1\] Experimental set up for time-resolved pump-probe spectroscopy of excitonic Lyman transitions. The 150 fs duration pump pulse is obtained from a Ti:sapphire laser and optical parametric amplifier. The pump pulse wavelength can be tuned around 1220 or 600 nm. The midinfrared probe pulse, of same duration is obtained by parametric down conversion. It is tunable around 10 $\mu$m. On the right hand side, the energy diagram of the relevant levels is shown.](figure1_0.eps)
![\[figure2\] (a) Lyman absorption recorded in a 170 $\mu$m thick single crystal of Cu$_2$O held at 4.2 K at different delays from the pump beam. The pump beam tuned at 1220 nm (two-photon resonant excitation of the orthoexciton) is incident along a \[100\] crystal axis. (b) Same as in (a), except for the pump wavelength, now tuned at 600 nm (non-resonant one photon excitation of orthoexcitons).](figure2_2.eps)
The experimental setup and a relevant energy diagram are shown in Fig. \[figure1\]. A 150 fs laser pulse tunable around 1220 nm provides the pump source. A tunable light pulse in the MIR around 10 $\mu$m provides the weak probe source. The 170 $\mu$m thick single crystal platelets were cooled by contact with a liquid Helium bath. Unless otherwise specified, the pump laser was propagating along a \[100\] crystal axis.
Representative Lyman absorption spectra are shown in Fig. \[figure2\] at various pump-probe delays ranging between $-10$ and 800 ps. Figure \[figure2\](a) is obtained with two-photon resonant excitation using a pump pulse energy of 1.0 $\mu$J focused on a spot diameter of 400 $\mu$m (0.81 mJ/cm$^2$), corresponding to an intensity of 5.4 GW/cm$^2$ and an esimated initial orthoexciton density of about 10$^{16}$ cm$^{-3}$ assuming a two-photon absorption coefficient of $\beta = 0.001$ cm/MW [@jolk]. Figure \[figure2\](b) is obtained by tuning the pump pulse to 600 nm, inside the phonon-assisted absorption edge of the $n=1$ orthoexciton. Because the exciton hyperfine splitting is large (12 meV) for the $n=1$ level and negligibly small (much smaller than the experimental limit of the spectral resolution 0.3 meV) for the $n=2$ level, one can record simultaneously the Lyman transition for orthoexcitons and paraexcitons. Under both pumping conditions, two lines appear around 116 meV and 129 meV, exactly where the $1s$-$2p$ Lyman transitions of the ortho- and paraexcitons are expected [@kubouchi].
We have carefully verified that the appearance of a signal at the position of the paraexciton line at 129 meV in Fig. \[figure2\](a) is not simply due to the direct creation of paraexcitons by the pump pulse, using the following measurements. First, it was verified that both lines at 116 meV and 129 meV disappear if the pump frequency is tuned off orthoexciton resonance. Secondly, we have measured their polarization dependence as a function of the pump polarization vector, for two different direction of propagation with respect to the crystal axes (see Fig. \[figure3\]). As mentioned before, the two-photon absorption to the paraexciton is forbidden so that no polarization dependence can be assigned to such a transition. One expect, for the orthoexciton state $\Gamma_5^+$, a dependence of the two-photon absorption with the polarization angle of the form [@inoue; @bader]: $$\begin{aligned}
\Delta \alpha \propto \sin ^2 2\theta,\end{aligned}$$ if $\boldsymbol{k}\parallel$ \[100\], and $$\begin{aligned}
\Delta \alpha \propto \sin ^2 2\theta + \sin ^4 \theta,\end{aligned}$$ for $\boldsymbol{k}\parallel$ \[110\]. Where $\Delta \alpha$ is the induced absorption, $\theta$ is the angle between the polarization vector and the crystal axis \[001\] and the $\boldsymbol{k}$ vector points along the laser beam direction. The observed behaviour for both lines at 116 meV and 129 meV are the same, as shown in Fig. \[figure3\], and it exhibits the polarization dependence expected for a two-photon transition to the $\Gamma_5^+$ orthoexciton state. Both experiments therefore indicate that orthoexcitons are first created by the pump source, and subsequently decay into paraexcitons.
![\[figure3\] Lyman absorption of orthoexciton (open circles) measured as a function of angle between the laser polarization and the crystal axis \[001\], for two crystal orientations, $\boldsymbol{k}\parallel$ \[100\] (a) and $\boldsymbol{k}\parallel$ \[110\] (b). The Lyman absorption of paraexciton \[closed triangles: (c),(d)\] exhibits the same dependence.](figure3_2.eps)
The temporal evolution of the shapes of the two Lyman absorption lines shown in Fig. \[figure2\] reflects the dynamics of the distribution functions of orthoexcitons and paraexcitons. At long delay, $\Delta t >$ 200 ps, the lines acquire a width corresponding to the lattice temperature both under one-photon and two-photon pumping. At early times, however, there is a significant difference, depending on the type of excitation (one or two photon). In the one-photon excitation, the lines are broader and shifted to higher energies \[see Fig. \[figure2\](b)\]. This signals a higher excitonic effective temperature, due to the excess energy delivered to the exciton gas in the pumping process. With resonant two-photon excitation \[see Fig. \[figure2\](a)\], the phase space compression scheme confers an initial effective temperature to the orthoexciton gas well below that of the lattice, as already mentioned [@cucl].
![\[figure4\] (a) Kinetics of the Lyman absorption of ortho- (open circles) and paraexcitons (closed triangles) following two-photon resonant excitation of the orthoexciton line. (b) The expanded trace at early time shows the fast rise of the paraexciton population. Crystal temperature is 4.2 K. The excitation density is 0.81 mJ/cm$^2$](Layout2.eps)
![\[figure5\] (a) Variation of the increase of the Lyman absorption of paraexcitons for various pump intensities. The increase of absorption is fitted to an exponential, as shown. (b) The exponential values obtained at different pump laser intensities are shown. The line obeys the relation: $\tau^{-1}=a + C_{\text{exp}} n_{\text{ex}}$, with $\ a=2.7 \pm 1.4 \ \mathrm{ns}^{-1}$ and $C_{\text{exp}}= (2.8 \pm 0.4)\times 10^{-15}\ \mathrm{cm}^{3}$/ns.](Layout5.eps)
The time evolution of the ortho- and paraexcitons densities is shown in more details within a small \[Fig. \[figure4\](b)\] and large \[Fig. \[figure4\](a)\] time interval under two-photon resonant excitation of $1s$ orthoexcitons with a pump pulse energy of 0.81 mJ/cm$^2$. One can observe a very fast rise of the line at 129 meV, in less than 1 ps. A different kinetics is observed on a longer time scale. The line population keeps increasing, but at a slower rate while the orthoexciton line at 116 meV decays with a time constant of the order of 1 ns. Figure \[figure5\] shows the variation of the paraexciton Lyman absorption at 129 meV on a long time scale at higher pump intensities. Its rise time in the interval 2-100 ps becomes faster when the pump intensity is increased \[Fig. \[figure5\](a)\]. It can be approximately fitted by an exponential, with a laser intensity dependent time constant, as shown in Fig. \[figure5\](b).
To discuss the kinetics, we distinguish three characteristic times: picoseconds, 100 ps, and nanoseconds. We start with the slowest process which have been recently discussed comprehensively [@jang]. From the temperature dependence of the exciton luminescence kinetics, Jang *et al.* have shown that ortho-para conversion occurs via the participation of a transverse acoustic phonon. This mechanism is relatively slow, with a characteristic conversion time of the order of nanoseconds. It is independent of particle density but increases with temperature. The slow decay rate of orthoexcitons with a nanoseconds time constant seen in Fig. \[figure4\], accompanied by a buildup of paraexcitons at a similar rate is consistent with this process.
We next consider behaviour in the range of 100 ps, where we observe the excitation density dependent increase of the area of the paraexciton Lyman line as shown in Fig. \[figure5\]. We can exclude Auger-type process with loss of particles since the area of orthoexciton and paraexciton signals are almost conserved as we find in Fig. \[figure4\]. Kavoulakis and Mysyrowicz have proposed a fast orthoexciton-paraexciton conversion effective at high exciton densities and low temperature [@prb61_16619]. It corresponds to an electron spin exchange between two orthoexcitons in a relative singlet spin configuration, resulting in their conversion in two paraexcitons. Such a mechanism scales like $$\begin{aligned}
\frac{dn_{o}}{dt} = - C {n_o}^2,\end{aligned}$$ where $n_o$ is the orthoexciton density and the constant $C$ is evaluated to be of the order of $\sim 5 \times 10^{-16} \ \mathrm{cm^{3}/ns}.$ From the spectral area of its Lyman absorption line, we estimate the orthoexciton density $$\begin{aligned}
n _{o} = \frac{\hbar c \sqrt{\varepsilon _b}}{4 \pi ^2\Delta E_{1s \text{-}2p}|\mu _{1s \text{-}2p}|^2}
\int_0^{\infty}\Delta \alpha _o(E)\text{d}E .\end{aligned}$$ Where $\Delta E_{1s \text{-}2p}$ and $\mu _{1s \text{-}2p}$ are the transition energy and dipole moment of $1s$-$2p$ transition. The background dielectric function $\varepsilon _b$ in the frequency of Lyman transition is estimated to be about 7. In our previous paper [@kubouchi], we took a $1s$-$2p$ dipole moment of 6.3 $e$Å$\ $by direct analogy with hydrogen atom [@artoni]. For the yellow series of excitons in Cu$_2$O, we need to take into account the central cell correction effects which reduces the overlap between $1s$ and $2p$ exciton wave function yielding the revised dipole moment of 1.64 $e$Å[@baym; @jorger]. The exponential values obtained from the experiment at different pump densities are plotted in Fig. \[figure5\](b) as a function of the orthoexciton density estimated from the above formula. From the slope shown in Fig. \[figure5\](b), we obtain the constant $C_{\text{exp}} = (2.8 \pm 0.4) \times 10^{-15} \ \text{cm}^3$/ns, a factor 6 larger than the prediction of Ref.[@prb61_16619]. The spin exchange mechanism therefore provides a convincing scenario to explain the rapid orthoexciton-paraexciton conversion occurring on a 100 ps time scale, when the orthoexciton density exceeds 10$^{15}$ cm$^{-3}$.
We finally address the initial kinetics, when a narrow line at 129 meV is seen at an orthoexciton density $< 10^{15}$ cm$^{-3}$ \[Fig. \[figure4\](b)\]. The spin exchange process is not expected to contribute significantly in such a low density regime. We identify the fast response as being due to the contribution of the $1s$-$3p$ orthoexciton transition, which accidentally lies close to the $1s$-$2p$ paraexciton line. In order to confirm this interpretation, excperiments should be performed with better spectral resolution at early times, in order to resolve the higher order term $1s$-$4p$ of the Lyman series.
In conclusion, we have demonstrated a scheme to detect the build up of paraexcitons following the creation of an ultracold orthoexciton population in Cu$_2$O. Orthoexciton-paraexciton conversion by spin exchange between pairs of orthoexcitons has been detected. Metastable biexcitons could resonantly enhance the scattering similar to the Feshbach resonance of cold atoms. The observed enhanced production of spin forbidden paraexcitons from cold orthoexcitons provides a unique opportunity to reach BEC states of excitons.
The authors are grateful to S. Nobuki for the experimental support. The authors are also grateful to T. Tayagaki, N. Naka, and Yu. P. Svirko for stimulating discussions. This work is partly supported by the Grant-in-Aid for Scientific Research (S) from Japan Society for the Promotion of Science (JSPS).
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abstract: 'Magnetic fields are believed to play an important role in the evolution of molecular clouds, from their large scale structure to dense cores, protostellar envelopes, and protoplanetary disks. How important is unclear, and whether magnetic fields are the dominant force driving star formation at any scale is also unclear. In this review we examine the observational data which address these questions, with particular emphasis on high angular resolution observations. Unfortunately the data do not clarify the situation. It is clear that the fields are important, but to what degree we don’t yet know. Observations to date have been limited by the sensitivity of available telescopes and instrumentation. In the future ALMA and the SKA in particular should provide great advances in observational studies of magnetic fields, and we discuss which observations are most desirable when they become available.'
author:
- 'Tyler L. Bourke$^1$ & Alyssa A. Goodman$^2$'
title: Magnetic Fields in Molecular Clouds
---
Are Magnetic Fields Important?
==============================
Magnetic fields are believed to play an important role in the evolution of molecular clouds, and hence star formation. However, despite progess on both the observational and theoretical fronts in recent years, many questions remain to be answered. Fundamental questions include (1) what is the dominant mechanism driving star formation, magnetic fields or turbulence, and (2) how important are magnetic fields at different stages in the star formation process?
For most of the past two decades the prevailing picture for the evolution of a dense molecular cloud core to form a protostar has been one of “quasi-static” evolution of a strongly magnetised core through ambipolar diffusion, over a time scale $>\!>$ the free-fall time, $t_{f\!f}$, leading eventually to inside-out collapse onto the central region (Shu et al. 1987, 1999; Mouschovias & Ciolek 1999). Recently a new theory has emerged, where the molecular clouds themselves are short-lived phenomena (liftimes a few $t_{f\!f}$ at most) and the star formation process is dynamical from the outset. In this picture supersonic turbulence is the dominant force in controlling the evolution of clouds and cores, and regulates the star formation rate (see reviews by Mac Low & Klessen 2003, Vázquez-Semadeni 2004, and reference therein).
The quasi-static model implies that cloud cores should be strongly magnetically subcritical (i.e., static magnetic fields are strong enough to provide support against gravity for $t >\!> t_{f\!f}$). In addition to supercritical cores (where the magnetic field provides no support), the dynamical model is able to accommodate approximately critical or slightly subcritical cores, as they will quickly evolve into supercritical cores and collapse. However, measurements of magnetic field strengths (and hence magnetic flux-to-mass ratios) in cores, which are needed to discrimate between the two models, are very difficult to obtain. The only practical tool available for directly measuring the field strength is the Zeeman effect (Crutcher 1999, 2004). Unfortunately molecules that are sensitive to the Zeeman effect either have transitions that are too high in frequency (e.g., CN at 113.5 GHz) or are too weak (CCS at 22-46 GHz) for current instrumentation, or don’t sample the densest gas in low-mass cores (OH at 1.6 GHz), to provide a large set of measurements of the field strength.
In the less dense regions of molecular clouds around the cores, where a number of Zeeman measurements have been made (mainly using OH), both with single dish radiotelescopes and interferometers, the results suggest that clouds are approximately critical (Figure 1; Bourke et al. 2001; Crutcher 1999, 2004; see also Myers & Goodman 1988), which does not seem to favour the quasi-static model of magnetic field support. However, measurements of field strengths in dense cores ($n \geq 10^5$ cm$^{-3}$) are required.
The direction and dispersion of polarization vectors tracing magnetic field lines can theoretically be used to discriminate between the quasi-static and turbulent models (Matthews et al. 2001, 2002, 2004; Matthews & Wilson 2002a,b; Henning et al. 2001; Wolf et al. 2003; Crutcher 2004). If the field is sufficiently strong then a small dispersion is predicted, and field lines should lie approximately parallel to a dense core’s minor axis, or show an hourglass morphology if it has evolved toward the collapse phase, when observed with high angular resolution. In the turbulence driven model field lines may appear regular on the larger size scales traced by the filaments in which the cores are embedded, but are not expected to show any regular pattern on the size scale of the cores themselves. Of course, reality is never this simple. Observations of cores and filaments in polarized dust emission sometimes show the regular pattern expected in the strong field case, but with sufficient dispersion to imply that turbulence is important, and with mean direction close to but not parallel to the cores’ minor axes, as expected in the quasi-static model (however, this last point can be explained as a projection effect - Basu 2000).
At present the data do not strongly support either model. In fact the data can be explained with either model with a sufficient number of caveats! Clearly real clouds and cores have both magnetic fields and turbulence, and the dominant mechanism may be region dependent, e.g., MHD turbulence on large scales and ambipolar diffusion on small scales. Until a large number of magnetic field measurements are made in the densest regions of cores, and the full three dimensional field structure of clouds and cores can be determined, deciding between the two will be difficult.
In the following sections we look in more detail at the observations that lead us to these conclusions, in particular those made with “high angular resolution”, which for this field does not automatically imply interferometers. We also examine what future observations are required to make progress in this field.
Observational Overview - High Resolution Observations
=====================================================
Due to the difficultly in making magnetic field measurements with current interferometers, this section on high resolution observations of magnetic fields includes Zeeman and dust polarization measurements made with sub-arcminute beams on single dish telescopes (i.e., IRAM 30-m; JCMT 15-m).
Magnetic fields in molecular clouds can be probed by a number of means. The Zeeman effect has been used to determine the line-of-sight field strength ($B_{los}$) in thermal lines, both in emission and absorption (Crutcher 1999; Bourke et al. 2001; and references therein). Masers are also potential tools, but the uncertainty in determining the physical conditions and sizes of the masing regions make it difficult to interpret the results. Polarimetry of aligned dust grains is the other major observational technique, which can be used to map the morphology of the field in the plane of the sky, and to estimate the field strength ($B_p$) using the modified Chandrasekhar-Fermi (C-F) method (Ostriker et al.2001; Padoan et al. 2001; Heitsch et al. 2001a). Studies have been conducted in both the near-infrared, the far-infrared, and more recently in the (sub)millimetre. As discussed in the following sections, each method has a number of shortcomings and full information on the 3D structure of magnetic fields in molecular clouds are lacking in most cases. Possible methods for obtain such information are discussed in §3.
Zeeman Measurements
-------------------
Most observations of the Zeeman effect in molecular clouds have been performed using single dish telescopes observing the OH transitions at 1.6 GHz, with a few studies using the VLA. Other secure detections of the Zeeman effect have come from HI, CN and excited transitions of OH (Güsten et al. 1994).
High resolution Zeeman observations of the densest regions of nearby low mass cores are almost non-existent. CCS observations of the chemically evolved, heavily depleted starless core L1498 (Levin et al. 2001) and the chemically young L1521E (Shinnaga et al. 1999; see also Aikawa 2004) have been made with sub-arcminute resolution. Shinnaga et al. claim a detection with $B_{los} = 160\pm46 \mu$G using the 45 GHz transition, and state that this value is larger than the critical value (which unfortunately they do not give). This result requires confirmation, preferably using other CCS transitions at lower frequencies.
The Zeeman effect has been observed in the CN 1-0 transition at 113 GHz with the IRAM 30-m (23$''$ beam) by Crutcher et al. (1999) toward four cores associated with the high mass star-forming regions OMC1, DR21OH and M17SW. As the [*frequency offset*]{} due to the Zeeman effect is [*independent*]{} of the line frequency, whereas the [*Doppler broadened line width*]{} is [*proportional*]{} to the line frequency, the ratio of the Zeeman effect to the line width decreases as the frequency of the line increases. This makes observations of the Zeeman effect at mm wavelengths much less sensitive than at cm wavelengths, hence the paucity of observations with this potentially rewarding transition. Because CN 1-0 consists of a number of hyperfine components with different Zeeman effects, the detections are fairly secure. Crutcher et al. conclude that the cores observed are supercritical by a factor 2–3.
VLA Zeeman observations with beam sizes $5-60''$ in the HI 21 cm and OH 18 cm transitions have been performed toward a number of high-mass star forming regions (e.g., W3 – Roberts et al. 1993; S106 – Roberts et al.1995; DR21 – Roberts et al. 1997; NGC 2024 – Crutcher et al. 1999; M17 – Brogan et al. 1999, 2001; NGC 6334 – Sarma et al. 2000). In many cases it is claimed that the HI and/or OH emission is in fact tracing high density ($>10^4$ cm$^{-3}$) gas in a PDR interface between the ionized and molecular regions. Line-of-sight field strengths up to 0.5 mG have been observed in most regions, with large variations across the mapped areas (Figure 2). In NGC 6334 the field actually changes sign, while in other regions it drops to 0 in places, indicating significant changes in direction of the field. This result, and the similar result in M17, could explain the lack of detection in single dish Zeeman observations in these sources (Bourke et al. 2001). In all these studies it is inferred that the magnetic field is either approximately critical (W3, S106, NGC 6334) or supercritical by a factor 2–3 (NGC 2024, M17). There is certainly no clear evidence for a subcritical cloud in any observations of the Zeeman effect alone.
Dust and Spectral Line Polarization
-----------------------------------
Some years ago it was hoped that polarized background starlight observed in the infrared would be a useful probe of the denser regions of molecular clouds, but the percentage polarization was not observed to increase as expected (Goodman et al. 1995). More recently the polarized thermal emission in the far-infrared and (sub)millimetre regions has been used to infer the field direction in the plane of the sky from a number of clouds and cores (Dotson et al. 2000; Davis et al. 2000; Matthews et al.2001, 2002, 2004; Matthews & Wilson 2002a,b; Henning et al. 2001; Wolf et al. 2003; Crutcher et al. 2004). Interestingly these studies also find that the percentage polarization does not increase toward the regions of maximum intensity (and hence density). In fact a [*decrease*]{} in percentage polarization is observed in most cases. The likely explanation in most cases is poor grain alignment due to spherical grain growth (“bad grains” – Goodman et al. 1995; Lazarian et al. 1997; Padoan et al.2001).
The far infrared observations of high mass star forming regions using the KAO have been discussed elsewhere (Dotson et al. 2000 and references therein). All the KAO maps show regular structures (but not necessarily uniform) with depolarization at the highest intensities. It would be useful to apply more recent analysis techniques to these data, for example the modified C-F method.
Two instruments have been used to obtain most of the (sub)millimetre results published at this time – the SCUBA polarimeter at 850 $\mu$m, and BIMA at 3 and 1 mm. Observations have also been made with OVRO (Akeson & Carlstrom 1997) and the 350 $\mu$m polarimeter HERTZ on the CSO (Schleuning et al. 2000; Houdé et al. 2002).
Matthews et al. (2004) have combined SCUBA and BIMA observations of Orion and Perseus (B1), comparing the large and small scale field directions. They find that in Orion the field direction in the cores is similar to that of the filaments in which they are embedded, but in B1 the cores show different orientations. The reason for these differences is unclear, but they could imply that B1 is globally supported by turbulence, with local density enhancements able to undergo collapse, whereas Orion is not turbulently supported (although its turbulence is “greater”), resulting in more ordered field lines on all scales (Heitsch et al. 2001b; Mac Low & Klessen 2003). The relevant physical parameters need to be evaluated to examine this (e.g., mass-to-flux ratio, turbulent line width, virial terms). A SCUBA map of the Serpens region, containing a number of protostars, was presented by Davis et al. (2000). In the NW cluster the field shows some degree of regularity, but in the presumably older SE there is a large dispersion in field direction between the protostars, possibly suggesting the field becomes less important at the core size scale as star formation progresses.
Recent mapping studies of individual low mass starless cores (Ward-Thompson et al. 2000; Crutcher et al. 2004) and protostars (Henning et al. 2001; Wolf et al. 2003; Valleé et al. 2003) have been made with SCUBA. All these results show depolarization at the highest intensities (Padoan et al. 2001). In starless cores the fields are uniform, but not aligned with the core minor axes, displaying offsets of up to 30, and all cores are found to be supercritical (or at least are not clearly subcritical), with field strengths inferred using the modified C-F method ($B_p \sim$50-150 $\mu$G). In protostellar cores the fields do not appear to be as uniform (possibly due to outflow disruption), and no clear preferred orientation with respect to either the outflows or cores is evident, although there is some suggestion the field lines are either aligned parallel or perpendicular to the outflow axis on a case by case basis. Field strengths estimated via the modified C-F method are typically 100–200 $\mu$G, but unfortunately are not compared to the critical values.
Interferometric studies at mm wavelengths have been made with both OVRO (Akeson & Carlstrom 1997) and BIMA (Rao et al. 1998; Lai et al. 2002, 2003a), of both low and high mass protostars. As the OVRO observations only produced a couple of measurements per field, we concentrate on the BIMA results. Rao et al. (1998) observed Orion-KL at 3 and 1 mm with $\sim$5$''$ beams. A relatively ordered field was observed except near the position of IRc2, where the field direction changed by 90. This is explained as the effect of the outflow on the dust grains causing alignment due to streaming motions (the “Gold” effect – Lazarian 1997). Lai et al. (2002) observed NGC 2024 FIR 5 at 1 mm with 2$''$ resolution. A uniform field was observed, with a slow change in position angle, which they modelled as due to an hourglass shaped morphology. They claim the pattern is due to contraction of the psuedo-disk perpendicular to the field (e.g., Galli & Shu 1993). Applying the modified C-F method they infer a field strength in the plane of the sky of $\sim$3.5 mG. This is extremely large compared to the Zeeman OH result of 65 $\mu$G for the line-of-sight component. If the field is not close to the plane of the sky, as these numbers would suggest, then the result could be explained as due to beam averaging of the OH data (60$''$) or the OH data does not trace the same high density region as the 1 mm dust emission. Another explanation is the modified C-F method is not applicable in this case.
Linear polarization of CO emission has been observed with BIMA toward the low mass protostar NGC 1333 IRAS 4A (Girart et al. 1999; Figure 3), and the high mass protostellar region DR21(OH) (Lai et al. 2003a). Girart et al. find that the dust polarization pattern toward IRAS 4A is consistent with a pinch (hourglass) configuration. Linear polarization of CO is detected mainly toward the redshifted outflow lobe. The polarization of spectral lines is predicted to be either parallel or perpendicular to the field, depending on a number of factors (Goldreich & Kylafis 1982). Girart et al. argue that in this case the field is parallel to the polarization vectors (which are perpendicular to the dust polarization vectors), and speculate that the magnetic field is bending the outflow (Figure 3). In the high mass protostar DR21(OH) Lai et al. also infer that the field traced by the linearly polarized CO emission is parallel to the field, by comparison with the polarized dust emission. Polarized CO emission is observed over a larger area than the dust, allowing the field morphology to be inferred over a similar area. Lai et al. deduce that the two dust continuum peaks (MM1 & MM2) are condensations within a magnetic flux tube. Applying the modified C-F method field strengths of $B_p \sim 1$ mG are inferred, about twice that found for $B_{los}$ through CN Zeeman observations (Crutcher et al. 1999). Combining these results implies that the field is pointed toward us at an angle of $\sim$30to the line of sight.
Probing the Magnetic Field in 3D
================================
For many years observational studies of magnetic fields in molecular clouds were restricted to probing the line-of-sight component via the Zeeman effect in thermal (non-maser) lines, or the plane of the sky component via dust polarization (and more recently linearly polarized spectral lines). A method to combine Zeeman observations with polarized background starlight was proposed by Myers & Goodman (1991; see also Goodman & Heiles 1994) to deduce the 3D field structure. However, this method has not been commonly used, perhaps a result of concerns about the usefulness of background polarized starlight in molecular clouds (depolarization; lack of stars), and the lack of Zeeman observations over the large angular sizes required at that time to obtain a sufficiently large number of stars for polarization studies. Recent theoretical simulations have shown that a modified C-F method can be used to infer the plane of the sky field strength under certain conditions (see Ostriker et al. 2001; Padoan et al. 2001; Heitsch et al. 2001a for specifics), and when applying this method to observational data care must be taken to ensure these conditions are met. If the field probed by the Zeeman effect and the dust polarization is believed to arise from a common region, then combining the Zeeman and C-F methods enables the full field strength and its angle to be determined. This has been attempted by Lai et al. (2003b) for DR21(OH) using CN 1-0 Zeeman data (Crutcher et al. 1999). Unfortunately the densities probed by the dust continuum ($>10^5$ cm$^{-3}$) are not in general probed by OH, the molecular most used in Zeeman observations. Potential Zeeman sensitive molecules that probe these densities (CN, CCS, CCH, SO) are discussed later.
Another technique for probing the 3-D field structure in the weakly ionized regions of molecular clouds (i.e., dense cores) involves the use of Zeeman data, dust polarimetry, and measurements of the ratio of ion-to-neutral line widths (Houdé et al. 2002). The advantage of this technique over the simple one describe above is that all the quantities needed are measured directly by observations, unlike the modified C-F method. Houdé et al. used this method to infer the structure of the field in M17, and more recently Lai et al. (2003b) have used the technique to infer the field in DR21(OH) at high angular resolution, combining their BIMA data described above with new OVRO data. Although Lai et al. find that the angle of the field to our line of sight is unchanged from their earlier estimate, the field strength is significantly lower (0.4 mG cf. 1 mG), which they suggest is a result of overestimating the field strength using the modified C-F method, due to smoothing of the polarization dispersion in the BIMA data.
With the next generation of interferometers combining great improvements in sensitivity with high angular resolution (ALMA, SKA), it may be possible to use these techniques to determine the full field structure and strengths in a more representative sample of molecular clouds and cores.
Unanswered Questions & Future Directions
========================================
As stated at the beginning of this review there are two fundamental questions into which we would like to gain insight: (1) what is the dominant mechanism driving star formation, and (2) how important are magnetic fields at different stages in the star formation process?
The results discussed here unfortunately are inconclusive to answer (1). Most of the Zeeman observations and many of the polarization studies have been toward regions that are already forming stars, where the magnetic field should not be dominant. So it is no surprise that this is the result found through observations. In the few observations of starless cores, the observations again suggest the field is important though not dominant (Crutcher et al. 2004). However, the isolated starless cores L183 and L1544 show spectral signatures of inward motions (Lee et al.2001), and so appear to be at an advanced stage of evolution just prior to collapse.
For core evolution most of the results suggest that by the time the protostellar stage is reached magnetic fields are not energetically dominant. But they are still important, as shown by the many observational examples of uniform and ordered polarization patterns, and in some cases the hourglass-like morphologies which might suggest core contraction due to ambipolar diffusion, as least during the inital stages of evolution toward forming a protostar.
At present the observations are insufficient to address (2). The observations of high mass regions suffer either from lack of resolution, even with interferometers, or don’t sample the diffuse parts of molecular clouds which are more representative of the overall cloud than those regions that have obtained sufficient density to form stars and are therefore bright enough to be well detected by Zeeman or polarization studies. Studies of low mass regions suffer from similar problems, and in addition do not probe every evolutionary stage, from chemically young protostellar cores (Aikawa 2004), through to protoplanetary disks (Dutrey 2004; Wilner 2004).
New instruments will help us to obtain a little more knowledge on both these issues. In particular we highlight the importance of ALMA in dust polarization and linearly polarized emission line studies at (sub)mm wavelengths, and the SKA for Zeeman studies.
In order to make progress on question (1) we need Zeeman measurements throughout molecular clouds, which could be provided by OH Zeeman observations with the SKA of lines which are too weak to provide detections with existing telescopes. If the modified C-F method can be tested more thoroughly in simulations and if it is applied correctly to observational data, it may be a useful tool to provide information on the 3D field when used with Zeeman data. These observations will not be easy. The technique of Myers & Goodman (1991) to combine Zeeman and background polarized starlight observations to infer the 3D field structure should be re-examined as a tool to probe the lower density regions of molecular clouds (which contain the bulk of the material), particularly with today’s 10-m class optical telescopes and infrared array cameras.
We would like to know the field strengths within dense protostellar cores before the onset of star formation. Observations using ALMA of CN 1-0 at 113.5 GHz, CCH at 85 GHz, and SO at 30 and 100 GHz, and the SKA of CCS at 11 and 22 GHz (Bel & Leroy 1989, 1998; Shinnaga & Yamamoto 2000), may provide these data, particular if the cores are carefully selected so that these molecules are not selectively depleted (e.g., L1521E). In such cores the method of Houdé et al. may provide information on the full 3D structure of the field. We would also like to know this information for protostellar cores at both the Class 0 and Class I phases, to examine the importance of the field during protostellar evolution.
In order to understand the high resolution observations of dust polarization, the observed depolarized at high intensities needs to be explained quantitatively. Further, numerical simulations of turbulence dominant molecular clouds that include magnetic fields need to be able to resolve protostellar cores and their fields, and not just treat them as sink particles, for comparison with observations (Vázquez-Semadeni 2004).
The new generation of interferometers (CARMA, ALMA and SKA) will provide the sensitivity and resolution required to make progress on two questions where essentially no observational information exists on the magnetic field – how important are magnetic fields in protostellar disks (disk viscosity, angular momentum transport; Hartmann 1998), and what drives and collimates protostellar outflows and jets (X-wind – Shang 2004; Disk wind – Königl & Pudritz 2000). Zeeman observations of some or all of CN, CCH, CCS, and SO in thermal emission will be very important in the study of protostellar disks (Qi 2000; Dutrey 2004) and envelopes (van Dishoeck & Blake 1998; Aikawa 2004), in particular if the full field can be inferred by combining these data with linear polarization studies of dust and CO. Polarization observations at size scales of a few AU or better may help to decide which mechanism is responsible for launching and collimating protosteller jets. These observations will still be difficult even with ALMA et al.
So our final words are directly to those planning the construction of ALMA and the SKA:
- Please provide ALMA with good polarimeters for dust and spectral line polarization studies at submm wavelengths, and the capability for Zeeman observations down to 30 GHz.
- Please push the upper frequency of the SKA to at least 24 GHz, to allow for Zeeman observations of thermal CCS (22 GHz), of water masers (22 GHz), and non-Zeeman observations of the important inversion transitions of ammonia near 24 GHz. Please pay particular attention to the polarization characteristics of potential array designs.
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|
---
author:
- 'D. I. Plokhov[^1]'
- 'A. I. Popov'
- 'A. K. Zvezdin[^2]'
title: |
Macroscopic quantum dynamics of toroidal moment\
in Ising-type rare-earth clusters
---
Introduction
============
Magnetic nanoclusters are in the focus of intense research because they exhibit remarkable quantum properties such as macroscopic quantum tunneling (MQT) of magnetization, molecular bistability, quantum interference, Berry phase effects, etc. Besides the obvious fundamental significance, their investigation is of great importance for the fronts of nanoelectronics, namely molecular spintronics [@Boga; @Sanv; @Meie; @Sonc], as well as for full-scale quantum computer development in the aspect of overcoming of non-scalability and difficulties in qubit state control.
In this respect, magnetic molecular nanoclusters with spin chirality (Cu$_3$, V$_{15}$, Dy$_3$, etc.) are of special interest. The extra degree of freedom can be used for qubit coding. It was shown by the example of Cu$_3$ magnetic cluster [@Loss] that the chirality is the source of a spin-electric effect whereby the spin states of the cluster is driven by an electric field, which is better localized than magnetic one [@Nowa].
The alternative possibility is the clusters with the non-magnetic ground state. It is pointed [@Mila] that getting rid of the spin makes the qubit very little sensitive to magnetic noise, but creates at the same time a potential problem for measurement since there is no Zeeman coupling of the qubit to an external magnetic field. One of possible solutions to this problem relies on the presence of an orbital moment associated to the chirality [@Mila].
Recent experimental and theoretical studies [@Luzo; @Chib; @EPLT] of a dysprosium based triangular cluster demonstrate the possibility of chirality control with an electric current (or just with crossed electric and magnetic fields) via interaction with toroidal moment, which is a natural characteristic of chirality [@EPLT]. It is remarkable that the cluster is of zero magnetic moment in the ground state in spite of the fact that there is a certain order in spin arrangement in it. The molecules with toroidal moment were first implied in [@Ceul]. Recently, such a molecule (namely Dy$_3$) was synthesized [@Tang] and experimentally investigated [@Luzo; @Chib]. Other examples of molecules with toroidal moment are nanocluster V$_{15}$ [@V-15] and probably nanocluster Cu$_3$ [@Loss]. Toroidal ordering in crystals is reviewed in [@Kopa].
Today, the quantum properties of chiral molecular clusters with toroidal moment is lacking of systematic research. In the present paper, the intriguing question of macroscopic quantum tunneling of toroidal moment (spin chirality) is considered for the first time. We predict the existence of Rabi oscillations in the clusters, which provides the good evidence for the MQT. The current driven dynamics is also considered, both in equilibrium and in the frames of the Landau-Zener-Stückelberg tunneling model. Due to the coupling between the toroidal moment and the current, it is possible to distinguish between states with opposite chirality, which is important for the purpose of qubit coding.
The model
=========
We consider a spin ring, a system of $N$ non-Kramers rare-earth ions located in the apices of a regular polygon. The Hamiltonian of the rare-earth polygonal cluster reads as follows $$\label{hamt}
{\cal H} = \sum_{i=1}^{N} {\cal H}_{CF}^{(i)} + {\cal H}_{INT} + g_J \mu_B \sum_{i=1}^{N} {\bf H} {\bf J}_i, \\$$ where ${\cal H}_{CF}^{(i)}$ is the operator of the crystal field at the $i$-th ion location, ${\bf J}_i$ is the total angular momentum of the $i$-th ion, ${\bf H}$ is the external magnetic field strength, and ${\cal H}_{INT}$ is the Hamiltonian of the dipole and exchange interactions of the rare-earth ions in the cluster.
![\[spin\] The spin structure of a triangular rare-earth molecular cluster and the local easy axes orientation in respect of the laboratory $XYZ$ reference frame. The thicker arrows represent the spins of the rare-earth ions in the molecule in the state with toroidal moment $T_Z = + 3 T_0$.](triangle.fig)
The ground state of rare-earth ions in compounds is formed mainly due to the influence of a crystal field and often has a strong anisotropy of the magnetic moment. For example, the ground state of Dy$^{3+}$ ions in Dy$_3$ cluster is very close Kramers doublet $| M_J = \pm 15/2 \rangle$ [@Luzo; @Chib; @EPLT] and responds only to the $z_i$ local component of an external magnetic field (see fig. 1). The first excited state is separated from the ground one by the energy of 200 cm$^{-1}$. The wave functions of the excited state are close to $| M_J = \pm 13/2 \rangle$. We can therefore conclude that the environment of rare-earth ions is almost axially symmetric. In the case of non-Kramers ions, small asymmetrical perturbations remove degeneration, thus making the ground state be quasi-doublet, i.e. the two close singlets with splitting small compared with distance from the higher levels. According to the Griffiths theorem [@Grif], the magnetic moments of the ions can be directed only along a specific axis, namely the local $z_i$-axes.
Let $| a_i \rangle$ and $| b_i \rangle$ be the eigenstates of the $i$-th (hereafter $i = 1, ..., N$) ion in the crystal field in respect of the local axes. The $z_i$-axes are perpendicular to the bisectors of the $N$-gon. The abscissa axes $x_i$ are then directed along the bisectors at the apices of the $N$-gon. Projection of the Hamiltonian in eq. (\[hamt\]) on the Hilbert space with functions $| \chi^{(i)}_{\pm} \rangle = (| a_i \rangle \pm i | b_i \rangle) / \sqrt{2}$ constituting the basic set yields the effective Hamiltonian $$\label{heff}
{\cal H}_{EFF} = - \frac{j}{2} \sum_{i \ne k}^{N} \sigma_{iz} \sigma_{kz}
- \frac{\Delta}{2} \sum_{i = 1}^{N} \sigma_{ix}
- \sum_{i = 1}^{N} \tilde{\mu_i} H_{z_i} \sigma_{iz},$$ where $\sigma_x$, $\sigma_y$, and $\sigma_z$ stand for the Pauli matrices, $j$ is the exchange interaction constant, $\Delta$ is the energy gap between the singlet levels $| a_i \rangle$ and $| b_i \rangle$ (as a rule $\Delta << j$), and $\tilde{\mu_i} = g_J \mu_B {\mathop{\mathrm{Im}}\nolimits}\langle a_i | J_{z_i} | b_i \rangle$. Usually, $| \chi_{\pm}^{(i)} \rangle \approx | M_J = \pm J \rangle \equiv | \pm \rangle$. It will be shown below that eq. (\[heff\]) is valid not only for non-Kramers ions, but for Kramers ions as well. In the latter case, the splitting $\Delta$ can be produced by an external magnetic field $H$, and $\Delta \rightarrow 0$ at $H \rightarrow 0$.
There are $2^N$ possible orderings of the ion spins in the rare-earth polygonal cluster. The wave functions of the cluster can be written as $$\label{ords}
\chi_n = \prod_{i=1}^{N} | \sigma_{n_i} \rangle,$$ where $\sigma_{n_i}$ stands for the sign (“$+$” or “$-$”) of the $i$-th ion ($i = 1, ..., N$) spin projection onto the local $z_i$ axis in the $n$-th state ($n = 1, ..., 2^N$).
The spin orderings can be characterized in terms of spin chirality. It is clear that spins in the states $\chi_n$ ($n = 1, ..., 2^{N-1}$) and their conjugates $\chi_n$ ($n = 2^{N-1} + 1, ..., 2^N$) are inversely twisted, i.e. the states have opposite chirality. The natural physical quantity associated with spin chirality in this case is the $T$-odd polar vector of the anapole (toroidal) moment, which corresponds to the first term of the toroidal family in the multipole expansion of an arbitrary electric current distribution. We would remind that the toroidal moment operator in the case of localized magnetic ions can be defined as $${\bf T} = \frac{1}{2} g_J \mu_B \sum_{i=1}^{N} \left[ {\bf r}_i \times {\bf J}_i \right],$$ where ${\bf r}_i$ is the radius-vector connecting the center of the $N$-gon with the $i$-th apex (all $r_i = r_0$).
Hereafter we deal with the dimensionless values of the toroidal moment $\tau = T_Z / T_0$, where $T_0 = \frac{1}{2} g_J \mu_B r_0 J$. Obviously, the toroidal moment of the system in a $\chi_n$ state is $\tau = q_{+} - q_{-}$, where $q_{\pm}$ is the number of “$+$”s and “$-$”s in the $\chi_n$ state. The values of the toroidal moments for conjugate states are different in sign.
Macroscopic quantum tunneling of the toroidal moment
====================================================
In the limiting case $\Delta \rightarrow 0$ (i.e. $\Delta << j$) the ground state is $| + ... + \rangle$ and $| - ... - \rangle$, which is doubly degenerated. Such a situation brings up the actual question on the macroscopic quantum coherence (MQC), i.e. the cluster ground state degeneration removal, and on the macroscopic quantum tunneling (MQT) of the toroidal moment, for instance, from the state $| T_Z = + T \rangle$ to the state $| T_Z = - T \rangle$, where $T = N T_0 + {\cal O}((\Delta/j)^2)$ is the value of the toroidal moment at the finite quantity $\Delta$. Both questions come to calculation of the imaginary-time ($\tau = it$, $\tau \in [ 0; \tau_0 ]$) transition amplitude between the states, which can be treated in terms of the path integral [@Feyn] $$\left\langle + T \left| \exp \left( \frac{{\cal H} \tau_0}{\hbar} \right) \right| - T \right\rangle
= \int \exp \left( - \frac{S_E}{\hbar} \right) D\theta D\varphi,$$ where $D\theta D\varphi$ is the integrating measure, $$S_E = - \int_{0}^{\tau_0} {\cal L} (-i\tau) d\tau$$ is the Euclidean action, and ${\cal L}$ is the Lagrangian of the system. The wave functions of the rare-earth polygonal cluster in the considered Hilbert space, which is the direct product of the state spaces of each of the $N$ cluster ions, can be presented as $$\label{psif}
| \psi \rangle = \prod_{i=1}^{N}
\left( \alpha(\theta_i,\varphi_i) | \chi_{+}^{(i)} \rangle + \beta(\theta_i,\varphi_i) | \chi_{-}^{(i)} \rangle \right),$$ where $\alpha(\theta,\varphi) = \cos \frac{\theta}{2}$ and $\beta(\theta,\varphi) = \sin \frac{\theta}{2} \cdot e^{-i\varphi}$ are the Cayley-Klein parameters depending on polar $\theta_i$ and azimuthal $\varphi_i$ angles of the ${\bf \sigma}_i$ in the $i$-th local coordinate system with polar axis directed along the $i$-th easy axis.
The quantum dynamical equation of motion for the toroidal moment reads $$\label{dtdt}
i \hbar \frac{d {\bf T}}{d t} = \left[ {\bf T}, {\cal H} \right],$$ where the Hamilton operator ${\cal H}$ is defined by eq. (\[hamt\]). We suppose that the dynamics of ${\bf T} (t)$ is mainly conditional on the spins of the cluster ions, i.e. ${\bf r}_i (t) = const$. If one averages out eq. (\[dtdt\]) with the wave function from eq. (\[psif\]), one obtains $$\label{dndt}
\frac{\hbar}{2} \frac{d {\bf n}_i}{d t} = \left[ {\bf n}_i \times \frac{\partial E}{\partial {\bf n}_i} \right],$$ where ${\bf n}_i = \{ \sin\theta_i\cos\varphi_i; \sin\theta_i\sin\varphi_i; \cos\theta_i \}$ are the unit vectors and $$E = \langle {\cal H} \rangle = - \frac{\Delta}{2} \sum_{i=1}^{N} n_{x_i} - \frac{j}{2} \sum_{i \ne k}^{N} n_{z_i} n_{z_k}$$ is the energy of the system at the zero temperature.
It is crucial for the MQT to take place that there are coherent macroscopic processes, which naturally means that $\theta_1 = ... = \theta_N \equiv \theta$ and $\varphi_1 = ... = \varphi_N \equiv \varphi$. The symmetry of the system is not affected unless the external magnetic field has a component directed in the plane of the rare-earth polygon. In terms of the $(\theta, \varphi)$ variable set eqs. (\[dndt\]) take the form $$\begin{aligned}
\label{eksy}
\hbar \dot{\varphi} = 4 j \cos \theta - \Delta \cot \theta \cos \varphi, \\ \nonumber
\hbar \dot{\theta} = - \Delta \sin \varphi,
\end{aligned}$$ which are the Lagrange-Euler equations of the Lagrangian $$\label{lagr}
{\cal L} = \frac{N \hbar}{2} (1 - \cos \theta) \dot{\varphi} + N j \cos^2\theta + \frac{N \Delta}{2} \sin\theta \cos\varphi.$$
The first term in eq. (\[lagr\]) is the Wess-Zumino one. Generally, the term is calculated making use of well-known formula for the Berry phase and interpreted as a contribution conditional on the gauge field of the Dirac monopole located at the point ${\bf n} = (0; 0; 0)$. It is interesting that the Isingness of the considered system do not have an impact on the form of the Wess-Zumino term, i.e. on the interaction between the molecule and the geometry field of the Dirac monopole, as opposed to its interaction with the external field.
The probability of the tunneling per time unit in the quasi-classical limit is determined [@Legg] by the expression $$P = A \exp(-B),$$ where $B = 2 S_E / \hbar$ is the Gamov constant, and $$S_E = \int \left( -i \frac{3\hbar}{2} (1 - \cos \theta) \frac{d\varphi}{d\tau} + E(\theta, \varphi) \right) d\tau$$ is the Euclidean action calculated on the instanton tunneling trajectory. The pre-exponential factor $A \sim \omega_0$ [@Chud], where $\omega_0$ is the instanton frequency (see below).
In order to calculate the tunneling trajectory and the corresponding contribution to the action we switch analytically in eq. (\[eksy\]) and eq. (\[lagr\]) to the imaginary time $\tau = i t$, and note that eqs. (\[eksy\]) have the first integral $$\label{imot}
j \cos^2\theta + \frac{\Delta}{2} \sin\theta \cos\varphi = const.$$
Now we consider the stationary points and the transition solution (“bounce trajectory”) of the eqs. (\[eksy\]). Points $\varphi = 0$ and $\cos\theta = \pm \sigma_0 = \pm \sqrt{1 - (\Delta / 4j)^2}$ are the stationary points of the system, which are actually the terminal points of the tunneling between states with toroidal moment $T_Z = \pm T = \pm N T_0 \cdot \sigma_0$. Point $\theta = \pi / 2$ and $\varphi = 0$ is the minimum point of the inverted energy.
Eliminating the $\varphi$-variable from the system of eqs. (\[eksy\]), we come to the instanton equation $$\label{imag}
\frac{d \cos \theta}{d \tau} = \frac{2 j}{\hbar} \left( \sigma_0^2 - \cos^2\theta \right).$$ Integrating the equation and taking into account the initial conditions $\cos \theta |_{\tau = \pm \infty} = \pm \sigma_0$, we get the instanton solution $$\cos \theta = \sigma_0 \cdot \tanh \left( \frac{2 j \sigma_0 \tau}{\hbar} \right),$$ whence it follows that the instanton frequency $\omega_0$ is $$\omega_0 = \frac{4 j \sigma_0}{\hbar}.$$ The time dependence of the $\varphi$-variable is then specified by the motion integral, see eq. (\[imot\]): $$\cos \varphi = \frac{2 - \sigma_0^2 - \cos^2 \theta}{2 \sqrt{1-\sigma_0^2} \sin\theta}.$$ It should be noted that Euclidean angle $\varphi$ takes on imaginary values. Substituting the obtained explicit dependencies $\theta = \theta(\tau)$ and $\varphi = \varphi(\tau)$ into eq. (\[lagr\]) for the system Lagrangian and calculating the action $S_E$ on the tunneling trajectory, we obtain for the Gamov constant $$\nonumber
B = \frac{2 S_E}{\hbar} = \frac{2 N j}{\hbar} \int_{-\infty}^{+\infty}
\frac{\cos^2\theta \cdot (\sigma_0^2 - \cos^2\theta)}{1 - \cos^2\theta} d\tau,$$ which finally comes to $$\label{gamc}
B = N \left( \ln \frac{1+\sigma_0}{1-\sigma_0} - 2 \sigma_0 \right).$$
It is seen that $B \cong 2 N \sigma_0^3 / 3$ if $\sigma_0 \rightarrow 0$, and also $B \rightarrow \infty$ if $\sigma_0 \rightarrow 1$. The tunneling frequency is $\nu \sim \frac{\omega_0}{\pi} e^{-B}$. It is determined by the $\sigma_0$, the value of the effective toroidal moment. If the exchange interaction between the cluster ions is strong enough ($j >> \Delta$), then $\sigma_0 \rightarrow 1$, and the tunneling probability vanishes.
The quantum mechanism of the tunneling dominates over thermal-activated processes of the spin reorientation at low enough temperatures. The temperature dependence of the speed of the latter is $\sim \exp(-U_0/kT)$, where $U_0$ is the height of the energy barrier. The crossover temperature $T^{\ast}$, at which both factors are of equal significance is defined by the condition $B = U_0 / kT$, therefore $$\label{temc}
T^{\ast} = \frac{2 N j}{k B(\sigma_0)} \cdot
\left( 1 - \frac{\sigma_0^2}{2} - \sqrt{1 - \sigma_0^2} \right).$$ It follows from the eq. (\[temc\]) that $T^{\ast} \rightarrow 0$ if $\sigma_0 \rightarrow 0$.
Rare-earth triangular clusters
==============================
The archetype of the noncollinear Ising model is a recently synthesized molecular dysprosium triangle [@Luzo; @Tang]. The standard, based on the path integral, quasiclassical technique to handle a problem of macroscopic quantum tunneling [@Feyn; @Hemm; @Chut; @Menz; @Legg] is only qualitatively suitable for rare-earth triangles due to low spin of the system. To get quantitative results we have to deal with the solutions of the Schroedinger equation in the relative Hilbert space.
There are eight possible orderings of the ion spins in the rare-earth triangular cluster, which are $\chi_1 = | + + + \rangle$, $\chi_2 = | - + \: + \rangle$, $\chi_3 = | + - \: + \rangle$, $\chi_4 = | + + \: - \rangle$, and $\chi_n = \chi_{n - 4} (+ \leftrightarrow -)$ for $n = 5, 6, 7, 8$, see eq. (\[ords\]).
In the basis of the eight vectors $$\begin{array}{cc}
\psi_1 = \chi_1, & \psi_2 = (\chi_2 + \chi_3 + \chi_4) / \sqrt{3}, \\
\psi_3 = (2\chi_2 - \chi_3 - \chi_4) / \sqrt{6}, & \psi_4 = (\chi_3 - \chi_4) / \sqrt{2},
\end{array}$$ ($\psi_{n+4}$ are obtained as $\psi_n$ by the replacement $\chi_n \rightarrow \chi_{n+4}$ for $n = 1, 2, 3, 4$), the matrix of the Hamiltonian in eq. (\[heff\]) in a zero magnetic field reads $$\label{8xx8}
{\cal H} = \left( \begin{array}{cc}
{\cal H}_0 & {\cal H}_{\Delta} \\
{\cal H}_{\Delta} & {\cal H}_0
\end{array} \right),$$ where $${\cal H}_0 = \left( \begin{array}{cccc}
-3j/4 & -\Delta\sqrt{3}/2 & 0 & 0 \\
-\Delta\sqrt{3}/2 & j/4 & 0 & 0 \\
0 & 0 & j/4 & 0 \\
0 & 0 & 0 & j/4
\end{array} \right),$$ and $${\cal H}_{\Delta} = \left( \begin{array}{cccc}
0 & 0 & 0 & 0 \\
0 & -\Delta & 0 & 0 \\
0 & 0 & \Delta/2 & 0 \\
0 & 0 & 0 & \Delta/2
\end{array} \right).$$
It is possible to obtain the eigenvalues and eigenstates of the Hamiltonian, ${\cal H} \Phi_n = E_n \Phi_n$ ($n = 1, ..., 8$). The relevant explicit expressions are given in table \[eigs\]. Clearly, the eigenvectors $\Phi_n$ ($n = 1,
..., 8$) are an orthonormal set.
1 2 3 4 5 6 7 8
----------------------- ----------------------- ---------------------- ---------------------- ---------------------- ---------------------- ----------------------- -----------------------
$\varepsilon_{+}(-x)$ $\varepsilon_{-}(-x)$ $\varepsilon_{+}(x)$ $\varepsilon_{-}(x)$ $\frac{1 + 2x}{4}$ $\frac{1 + 2x}{4}$ $\frac{1 - 2x}{4}$ $\frac{1 - 2x}{4}$
$ f_{-}(-x)$ $f_{+}(-x)$ $ f_{-}(x)$ $ f_{+}(x)$ 0 0 0 0
$-f_{+}(-x)$ $f_{-}(-x)$ $-f_{+}(x)$ $ f_{-}(x)$ 0 0 0 0
0 0 0 0 0 $\frac{1}{\sqrt{2}}$ 0 $\frac{1}{\sqrt{2}}$
0 0 0 0 $\frac{1}{\sqrt{2}}$ 0 $\frac{1}{\sqrt{2}}$ 0
$ f_{-}(-x)$ $f_{+}(-x)$ $-f_{-}(x)$ $-f_{+}(x)$ 0 0 0 0
$-f_{-}(-x)$ $f_{-}(-x)$ $ f_{+}(x)$ $-f_{-}(x)$ 0 0 0 0
0 0 0 0 0 $\frac{1}{\sqrt{2}}$ 0 $-\frac{1}{\sqrt{2}}$
0 0 0 0 $\frac{1}{\sqrt{2}}$ 0 $-\frac{1}{\sqrt{2}}$ 0
The eigenstate with the lowest energy is $\Phi_2$, which is the superposition of the states close to the states with $| \tau = \pm 3 \rangle$, i.e. $\Phi_2 = \left( \psi_{+} + \psi_{-} \right) / \sqrt{2}$, where we make use of $\psi_{\pm} = \sqrt{2} \left( f_{+} (-x) \cdot \psi_{1,5} + f_{-} (-x) \cdot \psi_{2,4} \right)$. The expected values of the toroidal moment in these states are $$\label{taux}
\langle \psi_{\pm} | \hat{\tau} | \psi_{\pm} \rangle = \pm \left( 2 + \frac{1-x}{\sqrt{1-2x+4x^2}} \right).$$
Supposed that initially the system is in the $| \psi_{+} \rangle$-state, one can obtain the probability $P(t)$ to find the system in the $| \psi_{-} \rangle$-state by the time moment of $t$ $$\begin{aligned}
\label{rabi}
P(t) = \left| \langle \psi_{-} | \psi_{t} \rangle \right|^2 = \nonumber \\
= \left| \sum_{n=1}^{8} \langle \Phi_n | \psi_{-} \rangle \langle \psi_{+} | \Phi_n \rangle
\exp \left( -\frac{i}{\hbar} E_n t \right) \right|^2,
\end{aligned}$$ where $| \psi_t \rangle = \exp \left( - \frac{i}{\hbar} {\cal H} t \right) | \psi_{+} \rangle $.
The probability function $P(t)$ oscillates (see fig. \[oscs\]), thus giving the clear evidence of the macroscopic quantum tunneling of the toroidal moment in the system. If the level splitting $\Delta$ is small compared to the exchange constant $j$, the oscillation frequency depends on parameter $x = \Delta / j$ as $\nu = \alpha x^3$ with $\alpha = 48$ GHz, so for the typical value $\tau = \nu^{-1} \sim 1$ ms [@Luzo] we have $x \sim 0.003$, i.e. $\Delta \sim 0.03$ cm$^{-1}$ if $j = 10$ cm$^{-1}$. The expected value of the toroidal moment given by eq. (\[taux\]) is very close to $\pm 3$, namely $\langle \hat{\tau} \rangle \approx \pm 3 (1 - x^2 / 2) \sim \pm (3 - 1 \cdot 10^{-5})$.
![\[oscs\] The Rabi-type oscillations between states $| \tau \approx \pm 3 \rangle $, the longer period corresponding to $\Delta = 0.003 j$ spans over eight shorter periods corresponding to the doubled splitting $\Delta = 0.006 j$, thus showing the cubical oscillation period dependence on the splitting.](Graph01.jpg)
Let us now consider the influence of magnetic and electric fields on the MQT of the toroidal moment. The states of a rare-earth ion of the cluster in the crystal field are $| 1 \rangle \equiv | a \rangle$, $| 2 \rangle \equiv | b \rangle$, and $| k \rangle$, the energy levels are $-\Delta / 2$, $\Delta / 2$, and $W_k \sim 100$ cm$^{-1}$ respectively. In the general case, the projection of the Hamiltonian ${\cal V} = {\cal H}_{CF} + {\cal H}_Z$ (with the Zeeman part ${\cal H}_Z = g \mu_B {\bf H J}$) onto the quasidoublet states gives $\tilde{V}_{ij} = V_{ij} - \sum_k V_{ik} V_{kj} / W_k$, $i, j = 1, 2$.
In the case of the magnetic field applied perpendicular to the plane of the ion triangle $$\label{tvij}
\tilde{V}_{11} = -\frac{\Delta}{2} - q_1 H^2, \ \tilde{V}_{22} = \frac{\Delta}{2} - q_2 H^2, \ \tilde{V}_{12} = \tilde{V}_{21} = 0,$$ where $q_{1,2} = g_J^2 \mu_B^2 \sum_k \frac{1}{W_k} \langle 1,2 | J_y | k \rangle \langle k | J_y | 1,2 \rangle$.
The symmetry of the system is not broken in the perpendicular field, which, as it follows from eq. (\[tvij\]), results in the renormalization of the splitting $\Delta$ only $$\Delta \rightarrow \tilde{\Delta} = \Delta + (q_1 - q_2) H^2.$$
Moreover, in the case of Kramers (dysprosium) ions, such a field produces a small splitting $\Delta \sim g_y H$ of the ground state due to the smallness of $g_y$, thus removing the degeneration. Since the typical value of relaxation time is 1 ms [@Luzo], the splitting $\Delta$ is estimated at 0.03 cm$^{-1}$. The factor $g_y$ then equals to 0.06 in a 1 T magnetic field. This way, the theory developed is valid not only for non-Kramers but also for Kramers Ising-type ions as well.
An external static electric field perpendicular to the plane of the ion triangle has no influence on the considered system, because the MEE properties of the rare-earth triangular clusters are caused by the field, directed in the plane of the ion triangle [@EPLT].
The current driven dynamics
===========================
Let us consider now the interaction between the toroidal moment and an external electric current. To describe the interaction we put the term $\hat{V} = \frac{4\pi}{c} {\bf j} \hat{{\bf T}}$ into the Hamiltonian in eq. (\[8xx8\]). The electric current ${\bf j}$ could be the displacement current ${\bf j} = \frac{1}{4\pi} \frac{\partial {\bf E}}{\partial t}$. If the electric field strength ${\bf E}$ linearly depends on the time and is directed along the $Z$-axis, then $\hat{V} = \frac{v}{c} \cdot T_0 \cdot \hat{\tau}$, where $v = \partial E_Z / \partial t$. The $\hat{\tau}$-operator matrix has the diagonal form $\tau = (+3,+1,+1,+1,-3,-1,-1,-1)$ in the basis of $\psi_n$ ($n = 1, ..., 8$). The Hamiltonian ${\cal H} + \hat{V}$ of the system can also be diagonalized, the eigenvalues $E^{(j)}_n$ and eigenvectors $\Phi^{(j)}_n$ ($n = 1, ..., 8$) are not given here due to their unhandiness. The energies of the two low-lying levels could be approximated (for $x << 1$) in the vicinity of the avoided level crossing at $j_z = 0$ as $$\label{lowe}
E_{\pm} (x,j_z) = E_0 \mp \sqrt{\left( \frac{4\pi}{c} \cdot 3T_0 \cdot j_z \right)^2 + \left( \frac{\delta}{2} \right)^2},$$ where the crossing energy is $E_0 = - \frac{3}{4} j$ and the avoided level splitting is $\delta = \frac{3}{2} j x^3$.
The toroidal moment is a conjugate variable against the electric current density and can be found at zero temperature as $$\tau (x) = \frac{c}{4 \pi T_0} \frac{\partial E_{-} (x,j_z)}{\partial j_z}.$$ The plot of the relative (equilibrium) dependence is given in fig. \[tmeqlz\] (curve 1). It is seen, that the current changes the direction of the spin twisting to the opposite, thus reversing the toroidal moment of the system in a way of the relatively sharp jump of $\Delta \tau = 6$. The anapole moment reversal, which could be called a reanapolization, requires the currents of $10^7$ A/cm$^2$, which seems to be quite achievable in experiment.
![\[tmeqlz\] The zero temperature plot of toroidal (anapole) moment projection onto the laboratory $Z$-axis vs the external current density for $\Delta = 0.003 j$: thicker curve (1) in equilibrium, the other curves in the case of the finite value of current sweeping rate $10^{13}$ A/(cm$^2\cdot$s), (2) from $\tau = + 3$ to $\tau = - 2.7$ and (3) from $\tau = - 3$ to $\tau = + 2.7$.](Graph02.jpg)
The tunneling processes when sweeping the current $j_z$ at a constant rate over an avoided energy level crossing can be treated in the frames of the Landau-Zener tunneling model. The probability $P$ to change the state characterized by the quantum number of the toroidal moment at the avoided level crossing is given by the expression $$\label{laze}
P = 1 - \exp \left( - \frac{\pi \delta^2}{(4\pi/c) \hbar \cdot T_0 \cdot |\tau_1 - \tau_2| \cdot |d j_z/d t|} \right),$$ where $\tau_1 = -3$ and $\tau_2 = +3$ are the toroidal moment quantum numbers of the avoided level crossing with the splitting $\delta = \frac{3}{2} j x^3$, and $d j_z/d t$ is the constant current sweeping rate. For the rate of $d j_z / d t \sim 10^{13}$ A/(cm$^2\cdot$s) and $x = 0.003$ we have $P \approx 0.95$.
With the Landau-Zener-Stückelberg model in mind, we can now start to understand qualitatively the hysteresis in the system considered (see fig. \[tmeqlz\]). Let us start at a sufficiently large negative current $j_z$. At very low temperature, all molecules are in the $| \tau = -3 \rangle$ ground state. When the current is come down to zero, all molecules will stay in the $| \tau = -3 \rangle$ ground state.
When passing the current over the avoided level crossing region at $j_z \approx 0$, there is Landau-Zener probability $P$ ($0 < P < 1$) to tunnel from the $| \tau = -3 \rangle$ to the $| \tau = +3 \rangle$ state. The dependence of the toroidal moment on the current then reads as follows $$\tau (x,j_z) = P \cdot \frac{\partial E_{+}(x,j_z)}{\partial j_z} + (1 - P) \cdot \frac{\partial E_{-}(x,j_z)}{\partial j_z}$$ shown in fig. \[tmeqlz\] (curve 3). The toroidal moment undergoes the jump of $\Delta \tau = 6 P$. So the average value of toroidal moment $\tau$ becomes less than 3, namely $\tau = 2.7$ for $P = 0.95$. The relevant mixed quantum state relaxes to the equilibrium $| \tau = +3 \rangle$ state due to interaction with environment (the $3 \rightarrow 1$ process in fig. \[tmeqlz\]).
Sweeping the current in the opposite direction from the $| \tau = +3 \rangle$ state, we arrive to the mixed state with toroidal moment average value of $-2.7$, see curve 2 in fig. \[tmeqlz\]. The state relaxes as above to the equilibrium $| \tau = -3 \rangle$ state (the $2 \rightarrow 1$ process in fig. \[tmeqlz\]). This way, we will come to the closed hysteresis loop.
Conclusion
==========
In the present work, it is shown for the first time that there exists a possibility of the macroscopic quantum tunneling of toroidal moment (spin chirality) in the molecular system based on Ising-type rare-earth ions. It is highly important that the ground state of such magnetically ordered systems has zero magnetic moment and should be characterized by toroidal moment. This is the reason why the slow relaxation observed experimentally in such systems cannot be caused by the MQT of the magnetic moment, but can possibly be accounted for the MQT of the toroidal moment instead.
We wish to acknowledge the financial support of the Russian Foundation for Basic Research (projects 08-02-01068, 09-02-11309, and 10-02-90475). One of the authors (A.K.Z.) thanks B. Barbara, A. Ceulemans, L.F. Chibotaru, and A. Soncini for discussions.
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---
abstract: 'By means of infrared spectroscopy we determine the temperature-doping phase diagram of the Fano effect for the in-plane Fe-As stretching mode in Ba$_{1-x}$K$_{x}$Fe$_{2}$As$_{2}$. The Fano parameter $1/q^2$, which is a measure of the phonon coupling to the electronic particle-hole continuum, shows a remarkable sensitivity to the magnetic/structural orderings at low temperatures. More strikingly, at elevated temperatures in the paramagnetic/tetragonal state we find a linear correlation between $1/q^2$ and the superconducting critical temperature $T_c$. Based on theoretical calculations and symmetry considerations, we identify the relevant interband transitions that are coupled to the Fe-As mode. In particular, we show that a sizable $xy$ orbital component at the Fermi level is fundamental for the Fano effect and possibly also for the superconducting pairing.'
author:
- 'B. Xu'
- 'E. Cappelluti'
- 'L. Benfatto'
- 'B. P. P. Mallett'
- 'P. Marsik'
- 'E. Sheveleva'
- 'F. Lyzwa'
- 'Th. Wolf'
- 'R. Yang'
- 'X. G. Qiu'
- 'Y. M. Dai'
- 'H. H. Wen'
- 'R. P. S. M. Lobo'
- 'C. Bernhard'
title: 'Scaling of the Fano effect of the in-plane Fe-As phonon and the superconducting critical temperature in Ba$_{1-x}$K$_{x}$Fe$_{2}$As$_{2}$'
---
The identification of the superconducting (SC) pairing mechanism of the iron arsenides is complicated by their multi-band and multi-gap structure and by the entanglement of the magnetic, orbital and structural degrees of freedom [@Paglione2010; @Stewart2011; @Mazin2010; @Wang2011; @Chubukov2012; @Medici2014; @Fernandes2014]. A prominent example is the stripe-like antiferromagnetic (AF) order in undoped and weakly doped [Ba$_{1-x}$K$_{x}$Fe$_{2}$As$_{2}$]{} (BKFA), described by a single $q$-vector along (0, $\pi$), that is accompanied by an orthorhombic distortion of the high temperature tetragonal structure with strong nematic fluctuations [@Fernandes2014]. This orthorhombic AF (o-AF) order persists well into the SC regime where it competes with SC at $0.15 \leq x \leq 0.3$. The strongest SC response, in terms of the $T_c$ value [@Avci2014; @Bohmer2015], condensate density [@Mallett2017PRB] and condensation energy [@Storey2013], occurs around optimum doping ($x$ = 0.3 – 0.35) just as the o-AF order vanishes. Shortly before this point, around $x$ = 0.24 – 0.26, a different AF order occurs that has the spins reoriented along the c-axis [@Waser2015] and a double-$q$ structure with a superposition of (0, $\pi$) and ($\pi$,0) that maintains the tetragonal symmetry [@Allred2016; @Fernandes2015; @Wang2015; @Lorenzana2008; @Gastiasoro2015; @Avci2014; @Mallett2015EPL]. This tetragonal AF (t-AF) order competes more strongly with SC than the o-AF one and leads to a $T_c$ reduction and a strong suppression of the SC condensate [@Bohmer2015; @Mallett2015PRL; @Mallett2017PRB].
The strong electronic correlations and, in particular, magnetic fluctuations have also pronounced effects on the band structure in the vicinity of the Fermi-surface (FS). In BKFA the FS is composed of several hole-pockets at the center of the Brillouin zone (BZ) ($\Gamma$-point) and electron pockets at the zone boundary (M-point) that have mainly Fe $xz$, $yz$, and $xy$ character. These bands are about two-times narrower than those predicted by DFT calculations [@Qazilbash2009; @Lu2008] and, in addition, are pushed toward the FS [@Charnukha2015]. The latter effect arises from the strong particle-hole asymmetry and a pronounced interband scattering between the M and $\Gamma$-points that most likely involves AF fluctuations [@Ortenzi2009; @Cappelluti2011; @Fanfarillo2016]. Especially near the M-point this yields very flat bands in the vicinity of the FS with a nearly singular behaviour that is very pronounced in the ARPES spectra of optimal doped Sm-1111 and BKFA [@Charnukha2015; @Evtushinsky2014].
Although the bare electron-phonon coupling is commonly believed to be very weak [@Boeri2008], it can be strongly increased by the spin-phonon interaction [@Boeri2010; @Egami2010; @Coh2016]. This could possibly explain why a strong enhancement of the Fano effect of the in-plane $E_u$ Fe-As mode was observed in the o-AF state of undoped BaFe$_2$As$_2$ [@Akrap2009; @Schafgans2011; @Xu2018]. On the other hand, a sizeable Fano effect was also reported for near optimally doped BKFA without AF order [@Xu2015; @Yang2017]. To the best of our knowledge, a systematic study of the evolution of this Fano effect as a function of doping and temperature, and an assignment of the underlying electronic and magnetic excitations responsible for it, are still lacking.
{width="2\columnwidth"}
Here we close this knowledge gap by showing the detailed temperature and doping dependence of the Fano effect of the $E_u$ mode in BKFA. At low temperatures the strength of the Fano coupling, expressed in terms of the asymmetry parameter $1/q^2$, appears to be very sensitive to the magnetic and structural transitions at $x < 0.3$. Most remarkably, at temperatures well above these magnetic and structural transitions (in the paramagnetic tetragonal state), we find that $1/q^2$ exhibits a linear scaling with the SC critical temperature $T_c$. This striking observation is suggestive of an intimate relationship between the electronic excitations coupled to the $E_u$ phonon mode and the SC pairing mechanism.
A series of BKFA single crystals with $0 \leq x \leq 0.6$ were grown with a flux method [@Karkin2014]. Their K-content was determined with X-ray diffraction and electron dispersive X-ray spectroscopy to an accuracy of ${\Delta}x \approx 0.02$ [@Mallett2015PRL; @Mallett2017PRB]. The AF and SC transition temperatures, $T_N$ and $T_c$ were derived from the anomalies in the dc resistivity curves. The *ab*-plane reflectivity $R(\omega)$ was measured at near-normal incidence with a Bruker VERTEX 70V spectrometer. An *in situ* gold overfilling technique [@Homes1993] was used to obtain the absolute reflectivity of the samples. The room temperature spectrum in the near-infrared to ultraviolet (4000–50000[$~\textrm{cm}^{-1}$]{}) was measured with a commercial ellipsometer (Woollam VASE). The optical conductivity was obtained from a Kramers-Kronig analysis of $R(\omega)$ [@Dressel2002]. For the extrapolation at low frequency, we used the function $R = 1 - A\sqrt{\omega}$ (Hagen-Rubens) in the normal state and $R = 1 - A\omega^4$ in the SC state. On the high-frequency side, we used the room temperature ellipsometry data and extended them by assuming a constant reflectivity up to 12.5 eV that is followed by a free-electron ($\omega^{-4}$) response.
The upper panels of Figure \[Fig1\] show the temperature dependent spectra of the real part of the optical conductivity, $\sigma_1(\omega)$, in the infrared range for selected doping levels of $x$ = 0, 0.08, 0.19 and 0.33. Their infrared conductivity is dominated by the strong electronic response that is composed of a Drude peak at the origin, due to the itinerant carriers, and a pronounced tail toward high frequency, that arises from inelastic scattering of the free carriers and/or low-lying interband transitions [@Benfatto2011; @Charnukha2014; @Marsik2013; @Calderon2014]. The spectra agree well with the previously reported ones [@Hu2008; @Wu2010; @Charnukha2013; @Dai2013PRL; @Xu2017; @Mallett2017PRB] and show the well-known changes due to the spin-density-wave (SDW) at $T_N$ = 138 K, 130 K and 90 K for $x$ = 0, 0.08 and 0.19, respectively, and the SC gap below $T_c$ = 18 K, and 38 K at $x$ = 0.19 and 0.33, respectively. The SDW and the SC gaps both reduce the spectral weight of the regular charge carrier response. For the former this spectral weight is shifted to higher energy, where it forms a so-called pair-breaking peak, whereas in the SC state it is transferred to a $\delta(\omega)$ function at the origin that accounts for the infinite dc conductivity.
{width="2\columnwidth"}
In addition to these broad electronic features, the infrared-active Fe-As stretching mode (with $E_u$ symmetry) [@Akrap2009] is clearly visible in all $\sigma_1(\omega)$ spectra as a weak but sharp mode around 255[$~\textrm{cm}^{-1}$]{}. Its remarkable temperature and doping dependence is detailed on the left hand side of the lower panels of Fig. \[Fig1\] which show a magnified view of the Fe-As phonon mode with the electronic background subtracted. A sketch of the eigenvectors of the $E_u$ Fe-As phonon is displayed in Fig. \[Fig2\]a. For a quantitative analysis we fitted (solid lines in Figs. \[Fig1\](a1-d1)) the optical properties of the $E_u$ mode with a Fano function [@Fano1961; @Cappelluti2010; @Cappelluti2012]: $$\label{Fano}
\sigma_{1}(\omega) = S\left[\frac{q^2 + 2qz -1}{q^2 (1 + z^2)}\right],$$ where $z = (\omega - \omega_0)/\Gamma$, and where $\omega_0$, $\Gamma$ and $S$ are the frequency, linewidth and strength of the phonon mode, respectively. An asymmetric profile of the phonon lineshape is ruled here by the Fano parameter $1/q^2$, which reflects the strength of the coupling between the phonon and the underlying electronic excitations. When the electronic excitations are lacking or they are not coupled with the phonon resonance we have $1/q^2 = 0$ and a symmetric Lorentzian lineshape is recovered.
The temperature dependence of the so-obtained phonon parameters $\omega_0$, $S$ and $1/q$, is shown in the lower right panels of Fig. \[Fig1\]. For all magnetic samples, the combined AF and structural transition into the o-AF state gives rise to clear anomalies in the $T$-dependence of $\omega_0$, $S$ and, especially, of $1/q$. In the following we focus on the evolution of the latter. At $x$ = 0, in agreement with previous reports [@Xu2018; @Chen2017; @Schafgans2011], $1/q$ has a very small negative value in the paramagnetic state that increases strongly in magnitude below $T_N$ (Fig. \[Fig1\]a4). For the doped samples with $x$ = 0.08 and 0.19 the o-AF transition gives rise to corresponding anomalies, except that $1/q$ increase from a negative value above $T_N$ (that is larger at $x$ = 0.19 than at 0.08) to a large [*positive*]{} value below $T_N$. The origin of the abrupt sign reversal of $1/q$ is further discussed in the Supplemental Material . Finally, for the optimally doped sample without any AF order ($x$ = 0.33, Fig. \[Fig1\]d4) $1/q$ has the largest negative value and it is only weakly temperature dependent. Quite remarkably, there is hardly a signature of the SC transition at $T_c$ in the temperature dependence of the phonon parameters. This is a clear indication that the Fano effect of the $E_u$ Fe-As mode does not arise from the coupling with the itinerant charge carriers, for which the spectral weight in the vicinity of the phonon mode decreases below $T_c$ due to the formation of the SC energy gap [@Li2008]. This implies that the Fano effect of this $E_u$ phonon mode is governed by the coupling to some interband transitions that are part of the electronic background at higher frequency (that is only weakly affected by the SC transition as shown in Fig. \[Fig1\]d).
The full doping and temperature dependence of $1/q$ is summarized as a color map in Fig. \[Fig2\]b. Also shown is the evolution of $T_N$ (solid black line), which is accompanied by abrupt changes of $1/q$, and of $T_c$ (solid white line) which hardly affects the $1/q$ value, as was already discussed above. In the following we focus on the doping dependence of $1/q$ without the influence of the AF and structural transitions as derived from the constant-temperature cut in the paramagnetic/tetragonal phase at $T$ = 150 K. Whereas the phonon frequency and intensity in Fig. \[Fig2\]c,d show only a weak doping dependence, the Fano parameter in Fig. \[Fig2\]e reveals a characteristic dome-like profile that closely resembles the one of the $T_c$ value and reproduces even the indentation at $x$ = 0.24 – 0.26 due to the strong competition of SC with the t-AF order. This striking dome-like doping dependence of the Fano parameter persists up to room temperature (as is further detailed in the Supplemental Material) .
The almost linear correlation in the doping range $x$ = 0 – 0.6 between the Fano parameter of the $E_u$ mode in the paramagnetic state and the SC transition temperature is further highlighted in Fig. \[Fig2\]f where we plot $1/q^2$ versus $T_c$. Clear deviations from a linear behavior occur only for the samples with $x$ = 0.24 and 0.26, for which the competition with the t-AF phase causes an anomalous suppression of superconductivity and where additional complex physics is probably at work (e.g. due to a residual o-AF phase the $T_c$ value might be overestimated).
To shed more light on the physical processes responsible for the Fano effect of the $E_u$ Fe-As mode in these materials, we analyzed the optical data within the charged-phonon scheme that was originally developed for carbon-based materials, [@Rice1992; @Kuzmenko2009; @Cappelluti2010; @Cappelluti2012]. In this context the Fano asymmetry parameter, $1/q$, is essentially ruled by the imaginary part of a complex function $\chi(\omega)$ that can be identified as a dynamical response function between the current operator and the electron-phonon operator relative to the $E_u$ phonon. In particular, $1/q$ scales as the imaginary part at the phonon frequency $\omega_0$, i.e. $1/q \propto \mbox{Im}\chi(\omega_0)$, which is different from zero whenever the $E_u$ phonon is coupled to an electronic interband transition at $\omega_0$. To identify the relevant processes responsible for the Fano effect, we analyze the paramagnetic/tetragonal state without any structural distortion. Using the Slater-Koster approach [@Daghofer2008; @Calderon2009], we consider a minimal tight-binding model containing the three orbitals ($xz$, $yz$, $xy$) relevant for the description of the low-energy band structure in the doping range under consideration. We compute the electron-phonon operator for the $E_u$ mode at linear order in the lattice displacement, and we analyze the properties of the corresponding charged-phonon response function $\chi(\omega)$. As detailed in the Supplemental Material we find that, when only the $xz/yz$ orbital subsector is considered, the charged-phonon response function $\chi(\omega)$ is vanishing at the leading order, implying a corresponding vanishing Fano effect. This property is a consequence of the underlying symmetries of the $xz/yz$ subsystem. On the other hand, a finite Fano asymmetry is possible when the $xy$ orbital component is taken into account, allowing for a finite contribution to $\chi(\omega)$ of interband particle-hole transitions between the $xy$ and $xz/yz$ components of the bands. Even though a quantitative estimate of $\chi(\omega_0)$ at the $E_u$ phonon frequency is beyond the scope of the present manuscript, we can nonetheless conclude that the contribution of the $xy$ orbital to the low-energy optical transitions plays a dominant role in the experimentally observed, dome-like doping dependence of the Fano asymmetry.
These conclusions hold true as long as the above-mentioned symmetries are preserved. In this respect, any breaking of the $xz/yz$ equivalence due e.g. to electronic or structural nematic phases can prompt additional low-energy transitions leading to an otherwise forbidden Fano-type coupling with the $E_u$ mode. We believe that this is the case in particular for the AF ordered states. Note, on the other hand, that the SC phase does [*not*]{} modify the symmetries ruling the charged-phonon response, explaining the observed lack of a significant change of the $E_u$ mode in the SC state.
The primary role of the $xy$ orbital character in understanding the phase diagram of the Fano effect of the $E_u$ mode, and the remarkable correlation between the Fano parameter $1/q^2$ and the SC critical temperature $T_c$ reported in Fig. \[Fig2\]f, shed new light on the current understanding of the SC phase diagram as well. Whereas it seems unlikely that the electron-phonon coupling can be the primary mechanism of the SC pairing, the observed correlation between $1/q^2$ and $T_c$ suggests that both the Fano effect of the $E_u$ mode and the SC pairing interaction require the presence of a sizeable $xy$ orbital component.
This calls for a closer look into the band structure and its doping evolution. As discussed above, in the paramagnetic/tetragonal state the Fano effect of the $E_u$ mode arises primarily from low-energy transitions between the $d_{xy}$ and the $d_{xz/yz}$ bands. Such optical transitions at low energy occur only close to the M points, where recent ARPES data [@Charnukha2015; @Evtushinsky2014] showed that as doping increases the Fermi level is pushed towards the bottom of the electron-like bands, so that the Fermi surface at the M points evolves from an elliptical shape to electron-like propellers. In this regime the bands close to M points are remarkably flat, leading to a substantial increase of the density of states. This can in turn affect both the strength of the AF spin-fluctuation exchange between the M and $\Gamma$ pockets, that is responsible for the pairing glue, and the joint density of states of the optical interband transitions near the M point, that govern the Fano effect. The flattening of the $xy$ bands at the M point can also explain a sizeable spin-fluctuation exchange between the M pockets [@Kreisel2017]. This can then provide a secondary $d$-wave pairing channel with respect to the dominant $s_\pm$ one, that manifests below $T_c$ with the appearance of excitonic-like Bardasis-Schrieffer resonances in the Raman spectra [@Maiti2016; @Bohm2018]. Moreover, it remains to be understood whether the $E_u$ phonon is just an “accidental witness” of the SC pairing interaction (via its sensitivity to the $xy$ orbital content at the Fermi-level) or whether it even plays a cooperative role and enhances $T_c$.
In summary, we performed a systematic study of the Fano effect of the in-plane FeAs stretching mode in Ba$_{1-x}$K$_{x}$Fe$_{2}$As$_{2}$. Firstly, we showed that the Fano effect of this phonon mode is strongly enhanced by the magnetic/structural transition into the orthorhombic AF state. Secondly, and most importantly, we observed a striking, linear relationship between the Fano parameter $1/q^2$, as measured at temperatures well above the magnetic/structural transitions, and the SC critical temperature, $T_c$. Theoretical calculations based on symmetry considerations show that the $xy$ orbital component of the low-energy bands near the M-point of the BZ plays a central role for the Fano effect of the $E_u$ phonon mode. This calls for a detailed investigation of the role played by the same orbital degrees of freedom on the orbital-selective pairing mechanism based on spin-fluctuations exchange, and their possibly cooperative interplay with electron-phonon coupling.
Work at the University of Fribourg was supported by the Schweizer Nationalfonds (SNF) by Grant No. 200020-172611. L.B. acknowledges financial support by the Italian MAECI under the Italian-India collaborative project SUPERTOP-PGR04879. Work at IOP was supported by MOST of China (Grant No. 2017YFA0302903) and NSFC of China (Grant No. 11774400).
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abstract: 'We use cyclotomy to design new classes of permutation polynomials over finite fields. This allows us to generate many classes of permutation polynomials in an algorithmic way. Many of them are permutation polynomials of large indices.'
address: |
School of Mathematics and Statistics\
Carleton University\
Ottawa, ON K1S 5B6\
Canada
author:
- Qiang Wang
title: Cyclotomy and permutation polynomials of large indices
---
[^1]
msbm10 at 12pt
Introduction
============
Let $p$ be prime and $q=p^m$. Let ${\mathbb{F}_q}$ be a finite field of $q$ elements and ${\mathbb{F}_q}^* = {\mathbb{F}_q}\setminus \{0\}$. A polynomial is a permutation polynomial (PP) of a finite field $\mathbb{F}_q$ if it induces a bijective map from ${\mathbb F}_q$ to itself. The study of permutation polynomials of a finite field goes back to 19-th century when Hermite and later Dickson pioneered this area of research. In recent years, interests in permutation polynomials have significantly increased because of their applications in coding theory and cryptography such as $S$-boxes. In some of these applications, the study of permutation polynomials over finite fields has also been extended to the study of permutation polynomials over finite rings and other algebraic structures. For more background material on permutation polynomials we refer to Chap. 7 of [@LN:97]. For a detailed survey of open questions and recent results see [@LM:88], [@LM:93], [@Mullen:93], and [@MullenWang:12].
In [@AGW:11], the authors provide a general theory which, in essence, reduces a problem of determining whether a given polynomial over a finite field ${\mathbb{F}_q}$ is a permutation polynomial to a problem of determining whether another polynomial permutes a smaller set. One of very useful smaller sets is the set of cyclotomic cosets. Earlier, Niederreiter and Winterhof [@NW:05] and Wang[@Wang] have studied so-called cyclotomic permutations. Namely, let $C_0$ be a subgroup of $\mathbb{F}_q^*$ with index $\ell \mid q-1$ and the factor group $\mathbb{F}_q^*/C_0$ consists of the [*cyclotomic cosets*]{} $$C_i := \gamma^i C_0, \ \ \ i = 0, 1, \cdots, \ell-1,$$ where $\gamma$ is a fixed primitive element of ${\mathbb{F}_q}$. For any $A_0, A_1, \cdots, A_{\ell-1} \in \mathbb{F}_q$ and positive integer $r$, the so-called [*$r$-th order cyclotomic mapping $f^{r}_{A_0, A_1, \cdots,
A_{\ell-1}}$ of index $\ell$* ]{} from $\mathbb{F}_q$ to itself is defined by $$f^{r}_{A_0, A_1, \cdots,
A_{\ell-1}} (x) =
\left\{
\begin{array}{ll}
0, & if ~ x=0; \\
A_0 x^{r}, & if~ x \in C_0; \\
\vdots & \vdots \\
A_i x^{r}, & if ~x \in C_i; \\
\vdots & \vdots \\
A_{\ell-1} x^{r}, & if~ x \in C_{\ell-1}.
\end{array}
\right.$$
It is shown that $r$-th order cyclotomic mappings produce the polynomials of the form $x^r f(x^s)$ where $s =\frac{q-1}{\ell}$. Furthermore, PPs of the form $x^r f(x^s)$ have been intensively studied in [@AAW:08; @AGW:09; @AW:05; @AW:06; @AW:07; @Chapuy:07; @WL:91; @Wang; @Zieve-2; @Zieve:09; @Zieve].
It is also well known that every polynomial $P(x)$ over $\mathbb{F}_q$ such that $P(0) =b$ has the form $ax^rf(x^s)+b$ with some positive integers $r, s$ such that $s\mid q-1$. Let $q-1 = \ell s$. More precisely, we observe that any polynomial $P(x)\in\mathbb{F}_q[x]$ can be written as $a(x^rf(x^{(q-1)/\ell}))+b$, for some $r \geq 1$ and $\ell\mid (q-1)$. To see this, without loss of generality, we can write $$P(x)=a(x^n+a_{n-i_1} x^{n-i_1}+\cdots+a_{n-i_k} x^{n-i_k})+b,$$ where $a,~a_{n-i_j}\neq 0$, $j=1, \cdots, k$. Here we suppose that $j\geq 1$ and $n-i_k=r$. Then $P(x)=a\left(x^r f(x^{(q-1)/\ell}) \right)+b,$ where $f(x)=
x^{e_0}+a_{n-i_1} x^{e_1}+\cdots+ a_{n-i_{k-1}}x^{e_{k-1}} + a_{r}
$, $$\ell=\frac{q-1}{\gcd(n-r,n-r-i_1,\cdots, n-r-i_{k-1}, q-1)},$$ and $\gcd(e_0, e_1, \cdots, e_{k-1}, \ell)=1 .$ The constant $\ell$ is called the [*index*]{} of polynomial $P(x)$ (see [@AGW:09]). The index of a polynomials is closely related to the concept of the least index of cyclotomic permutations. Many classes of PPs that are constructed recently have small indices $\ell$, see for example, [@AAW:08; @AW:05; @AW:06; @AW:07; @Chapuy:07; @Zieve-2; @Zieve:09].
In this paper, we extend the definition of cyclotomic mappings and study the permutation polynomials corresponding to these cyclotomic mappings. These polynomials have either the presentation given in terms of cyclotomic mappings of index $\ell$, $$f^{r_0(x), r_1(x), \ldots, r_{\ell-1}(x)}_{A_0, A_1, \cdots,
A_{\ell-1}} (x) =
\left\{
\begin{array}{ll}
0, & if ~ x=0; \\
A_0 r_0(x), & if~ x \in C_0; \\
\vdots & \vdots \\
A_i r_i(x), & if ~x \in C_i; \\
\vdots & \vdots \\
A_{\ell-1} r_{\ell -1}(x), & if~ x \in C_{\ell-1},
\end{array}
\right.$$ or the polynomial presentation $$\label{correspondence}
P(x) = \sum_{i=0}^{\ell-1} \frac{A_i}{\ell \zeta^{i(\ell-1)}} r_i(x) \left(x^{(\ell-1)s} + \zeta^i x^{(\ell-2)s} +\cdots + \zeta^{i(\ell-2)} x^s + \zeta^{i(\ell-1)} \right),$$ where $r_0(x), r_1(x), \ldots, r_{\ell-1}(x) \in {\mathbb{F}_q}[x]$ and $\zeta =\gamma^s$ be a fixed primitive $\ell$-th root of unity throughout this paper. Essentially, these polynomials are of the form $\sum_{i=0}^n x^{r_i} f_i(x^s)$. And indices of polynomials of this form are normally large, which are different from $\ell$ in general. After we study several cases when $r_i(x)$’s are general and $\ell$ is small, we study in detail the situation when $r_i(x)$’s are monomials $x^{r_i}$ for some positive integers $r_i$’s. We give some general criteria of determining these polynomials are permutation polynomials of finite fields (Theorems \[main\], \[construction\], \[main2\], \[specialMain\]). One of them can be written as follows:
\[main2\] Let $q-1 = \ell s$ and $A_0, \ldots, A_{\ell-1} \in {\mathbb{F}_q}$. Then $$P(x) =
\left\{
\begin{array}{ll}
0, & if ~ x=0; \\
A_0 x^{r_0}, & if~ x \in C_0; \\
A_1 x^{r_1}, & if~ x \in C_1; \\
\vdots & \vdots \\
A_{\ell-1} x^{r_{\ell-1}}, & if ~ x \in C_{\ell},
\end{array}
\right.$$ is a PP of ${\mathbb{F}_q}$ if and only if $(r_i, s) =1$ for any $i=0, 1, \ldots, \ell-1$ and $\mu_{\ell} = \{ A_i^s \zeta^{r_i i} \mid i=0, \ldots, \ell-1\}$, where $\mu_{\ell}$ is the set of all $\ell$-th roots of unity.
We note that each $A_i^s \zeta^{r_i i}$ is an $\ell$-th root of unity as long as $A_i$ is not zero. Hence what we really need is to check all $A_i^s \zeta^{r_i i}$ ($0\leq i \leq \ell-1$) are distinct in the above theorem. Using these criteria in different forms, we demonstrate our method by constructing many new classes of PPs (Theorems \[ZhaHuThm8Gen\], \[PPoverfthreepower1\], \[PPoverfthreepower2\], \[PPoverfthreepower3\], \[PPoverfthreepower4\], \[specialR\], \[allOneBranchesCongruenceMinusOne\], \[binomialBranches\], Corollaries \[rogersBranches\], \[exampleOfCongruenceOne\], \[exampleOfAllOneBranches\]). Here we only list very few particular examples of these results over fields with small characteristic.
The polynomial $P(x) = x^{\frac{2(2^n-1)}{3}+2^i} + x^{\frac{2(2^n-1)}{3}+2^j} + x^{\frac{2^n-1}{3}+2^i} + x^{\frac{2^n-1}{3}+2^j} + x^{2^i}$ is a PP of ${\mathbb{F}}_{2^n}$ for any even positive integer $n$ and non-negative integers $i, j$.
The polynomial $f(x) =x^{\frac{3^n-1}{2} +3^i}+ 2 x^{\frac{3^n-1}{2} +3}+ 2 x^{\frac{3^n-1}{2} +2} + 2x^{\frac{3^n-1}{2} +1} + x^{3^i} + x^3 + x^2 + x$ is a PP of ${\mathbb{F}}_{3^n}$ for any positive integer $n$ and non-negative integer $i$.
The polynomial $f(x) = x^{\frac{3^n-1}{2} +2^i}+ 2x^{\frac{3^n-1}{2} +3} + 2x^{\frac{3^n-1}{2} +2} + 2x^{\frac{3^n-1}{2} +1} + x^{2^i} + x^3 + x^2 + x$ is a PP of ${\mathbb{F}}_{3^n}$ for any odd positive integer $n$ and non-negative integer $i$.
\[PPoverfthreepower3\] The polynomial $f(x) = x^{\frac{3^n-1}{2} +3^i}+ 2x^{\frac{3^n-1}{2} +3} + x^{\frac{3^n-1}{2} +2} + 2x^{\frac{3^n-1}{2} +1} + 2 x^{2^i} + 2x^3 + x^2 + 2 x$ is a PP of ${\mathbb{F}}_{3^n}$ for any odd positive integer $n$ and non-negative integer $i$.
The polynomial $f(x) = x^{\frac{3^n-1}{2} +2^i}+ x^{\frac{3^n-1}{2} +3} + 2x^{\frac{3^n-1}{2} +2} + x^{\frac{3^n-1}{2} +1} + 2x^{2^i} + x^3 + 2x^2 + x$ is a PP of ${\mathbb{F}}_{3^n}$ for any odd positive integer $n$ and non-negative integer $i$.
\[PPoverfthreepower4\] Let $q=3^n$ and $\alpha, \beta, \gamma, \theta \in \mathbb{F}_{3^n}$. Let $f(x) = (\beta -\alpha) x^{(q-1)/2 +3} + (\beta \theta -\alpha \gamma) x^{(q-1)/2 +2} + (\beta \theta^2 -\alpha \gamma^2) x^{(q-1)/2 +1} - (\beta + \alpha) x^{3} - (\beta \theta + \alpha \gamma) x^{2} - (\beta \theta^2 +\alpha \gamma^2) x$. Then $f$ is a PP of $\mathbb{F}_{3^n}$ if and only if $\eta(\alpha) = \eta(\beta)$, $\eta(\gamma) =-1$, and $\eta(\theta) =1$.
One can easily see from the definition that these classes of PPs have indeed large indices because they contain terms with consecutive exponents. We also remark that our results not only generalize many previous results in [@AAW:08; @AW:06; @AW:07; @Zieve-2], but also generalize several more recent results including a class of PPs constructed by Hou in the study of reversed Dickson polynomials (Theorem 1.1 in [@Hou:11]) and several classes of PPs studied by Zha and Hu thereafter (Theorems 7-11 in [@ZhaHu]); see Theorems \[PPoverfthreepower1\], \[3branchesZha1\], \[3branchesZha2\], \[squaredivisor\]. Our method can also provide an algorithmic way to generate permutation polynomials over finite fields.
Cyclotomic mappings permutation polynomials
===========================================
Let $\gamma$ be a fixed primitive element of $\mathbb{F}_q$, $\ell \mid q-1$, and the set of all nonzero $\ell$-th powers be $C_0 = \{ \gamma^{\ell j}: j = 0, 1,
\cdots, s -1\}$. Then $C_0$ is a subgroup of $\mathbb{F}_q^*$ of index $\ell$. The elements of the factor group $\mathbb{F}_q^*/C_0$ are the [*cyclotomic cosets*]{} $$C_i := \gamma^i C_0, \ \ \ i = 0, 1, \cdots, \ell-1.$$ For any $A_0, A_1, \cdots, A_{\ell-1} \in \mathbb{F}_q$ and monic polynomials $r_0(x), \ldots, r_{\ell-1}(x) \in {\mathbb{F}_q}[x]$ we define a [*cyclotomic mapping $f^{r_0(x), r_1(x), \ldots, r_{\ell-1}(x)}_{A_0, A_1, \cdots,
A_{\ell-1}}$ of index $\ell$* ]{} from $\mathbb{F}_q$ to itself by $$f^{r_0(x), r_1(x), \ldots, r_{\ell-1}(x)}_{A_0, A_1, \cdots,
A_{\ell-1}} (x) =
\left\{
\begin{array}{ll}
0, & if ~ x=0; \\
A_0 r_0(x), & if~ x \in C_0; \\
\vdots & \vdots \\
A_i r_i(x), & if ~x \in C_i; \\
\vdots & \vdots \\
A_{\ell-1} r_{\ell -1}(x), & if~ x \in C_{\ell-1}.
\end{array}
\right.$$ Moreover, $f^{r_0(x), r_1(x), \ldots, r_{\ell-1}(x)}_{A_0, A_1, \cdots,
A_{\ell-1}}$ is called an [*cyclotomic mapping of the least index $\ell$*]{} if the mapping can not be written as a cyclotomic mapping of any smaller index. The polynomial of degree at most $q-1$ representing the cyclotomic mapping $f^{r_0(x), r_1(x), \ldots, r_{\ell-1}(x)}_{A_0, A_1, \cdots,
A_{\ell-1}} (x) $ is called an [*cyclotomic mapping polynomial*]{}. In particular, when $r_0(x) = \cdots = r_{\ell-1} (x) = x^r$ for a positive integer $r$, it is known as a [*$r$-th order cyclotomic mapping polynomial*]{}, denoted by $f^r_{A_0, A_1, \cdots, A_{\ell-1}}(x)$ (see [@NW:05] for $r=1$ or [@Wang]).
Let $s = \frac{q-1}{\ell}$ and $\zeta =\gamma^s$ be a primitive $\ell$-th root of unity. It is shown in [@AW:07] that polynomials of the form $x^rf(x^s)$ and the $r$-th order cyclotomic mapping polynomials $f^r_{A_0, A_1, \cdots, A_{\ell-1}}
(x)$ where $A_i = f(\zeta^i)$ for $0\leq i\leq \ell-1$ are the same. More generally, for any $P(x) = \sum_{i=0}^{\ell-1} r_i(x) f_i(x^s)$ with $P(0) =0$, we can also write $P(x)$ as a cyclotomic mapping as follows:
$$P(x) =
f^{P_0(x), P_1(x)\ldots, P_{\ell-1}(x)}_{A_0, A_1, \cdots, A_{\ell-1}} (x)
= \left\{
\begin{array}{ll}
0, & \mbox{if} ~ x=0; \\
A_0 P_0(x), & \mbox{if}~ x \in C_0; \\
\vdots & \vdots \\
A_i P_i(x), & \mbox{if} ~x \in C_i; \\
\vdots & \vdots \\
A_{\ell-1} P_{\ell-1} (x), & \mbox{if}~ x \in C_{\ell-1},
\end{array}
\right.$$ where $A_j P_j(x) = \sum_{i=0}^{\ell-1} r_i(x) f_i(\zeta^j)$ and $P_j(x)$ is the monic associated polynomial. Indeed, for any $x \in C_j$, $x=\gamma^{\ell i+ j}$ for some $ 0\leq i \leq s-1$ and thus $P(\gamma^{\ell i+ j}) = \sum_{i=0}^{\ell-1} r_i(\gamma^{\ell i+ j}) f_i(\zeta^j) = A_j P_j(\gamma^{\ell i+ j})$.
On the other hand, for any given cyclotomic mapping of index $\ell$, $$f^{r_0(x), r_1(x), \ldots, r_{\ell-1}(x)}_{A_0, A_1, \cdots,
A_{\ell-1}} (x) =
\left\{
\begin{array}{ll}
0, & if ~ x=0; \\
A_0 r_0(x), & if~ x \in C_0; \\
\vdots & \vdots \\
A_i r_i(x), & if ~x \in C_i; \\
\vdots & \vdots \\
A_{\ell-1} r_{\ell -1}(x), & if~ x \in C_{\ell-1},
\end{array}
\right.$$ we can find a unique polynomial $P(x)$ modulo $x^q -x$ corresponding to it. Namely, $$\label{correspondence}
P(x) = \sum_{i=0}^{\ell-1} \frac{A_i}{\ell \zeta^{i(\ell-1)}} r_j(x) \left(x^{(\ell-1)s} + \zeta^i x^{(\ell-2)s} +\cdots + \zeta^{i(\ell-2)} x^s + \zeta^{i(\ell-1)} \right).$$ Indeed, if each $x\in C_i$, we must have $x^s = \zeta^i$. So we have $$\frac{A_i}{\ell \zeta^{i(\ell-1)}} r_i(x) \left(x^{(\ell-1)s} + \zeta^i x^{(\ell-2)s} +\cdots + \zeta^{i(\ell-2)} x^s + \zeta^{i(\ell-1)} \right) = A_i r_i(x),$$ and for $j\neq i$, $$\frac{A_j}{\ell \zeta^{j(\ell-1)}} r_j(x) \left(\zeta^{i(\ell-1)} + \zeta^j\zeta^{i(\ell-2)} +\cdots + \zeta^{j(\ell-2)} \zeta^i + \zeta^{j(\ell-1)} \right) = \frac{A_j (\zeta^{i\ell} - \zeta^{j\ell})}{\ell \zeta^{j(\ell-1)} (\zeta^i - \zeta^j)} r_j(x) = 0.$$
The correspondence (\[correspondence\]) provides a way to construct permutation polynomials of finite fields. First of all, it is obvious to obtain the following result on cyclotomic mappings.
\[mainLemma\] Let $f^{r_0(x), r_1(x), \ldots, r_{\ell-1}(x)}_{A_0, A_1, \cdots,
A_{\ell-1}} (x)$ be a cyclotomic mapping of index $\ell$ over ${\mathbb{F}_q}$ given as $$f^{r_0(x), r_1(x), \ldots, r_{\ell-1}(x)}_{A_0, A_1, \cdots,
A_{\ell-1}} (x) =
\left\{
\begin{array}{ll}
0, & if ~ x=0; \\
A_0 r_0(x), & if~ x \in C_0; \\
\vdots & \vdots \\
A_i r_i(x), & if ~x \in C_i; \\
\vdots & \vdots \\
A_{\ell-1} r_{\ell -1}(x), & if~ x \in C_{\ell-1}.
\end{array}
\right.$$ Then $f^{r_0(x), r_1(x), \ldots, r_{\ell-1}(x)}_{A_0, A_1, \cdots,
A_{\ell-1}}$ induces a permutation of ${\mathbb{F}_q}$ if and only if $\cup_{i=0}^{\ell-1} A_i r_i(C) = {\mathbb{F}_q}^*$, where $A_i r_i(C_i) = \{ A_i r_i(x) \mid x \in C_i \}$ for $ 0\leq i \leq \ell-1$. In particular, if $A_0, \ldots, A_{\ell -1} \neq 0$ and each $A_i r_i(x)$ is a bijective map from $C_i$ to another coset $C_{j_i}$, then $f^{r_0(x), r_1(x), \ldots, r_{\ell-1}(x)}_{A_0, A_1, \cdots,
A_{\ell-1}}$ induces a permutation of ${\mathbb{F}_q}$ if and only if $$\{ A_0 r_0(C_0), \ldots, A_{\ell-1} r_{\ell-1} (C_{\ell-1}) \} = \{C_0, \ldots, C_{\ell -1} \}.$$
Lemma \[mainLemma\] and Equation (\[correspondence\]) provide us a general scheme to construct PPs of finite fields. We can design PPs of specific type in two steps. First, we choose an $\ell \mid q-1$ and set a pattern of permutation of cyclotomic cosets. For example, we may want to have a permutation which maps $C_0$ to $C_1$, $C_1$ to $C_2$, etc. Secondly, we choose different polynomials which maps one $C_i$ to another $C_j$ satisfying the previous requirements and use them to form a cyclotomic mapping. In the above example, we choose $\ell$ polynomials $A_i r_i(x)$ which map $C_0$ to $C_1$, $C_1$ to $C_2$, etc, respectively. The polynomial determined by Equation (\[correspondence\]) is the desired one. We note that $A_i r_i(x)$’s do not need to be PPs of ${\mathbb{F}_q}$, they only need to be bijective from one $C_i$ to another $C_j$ depending on the requirements. This gives a lot of flexibility and opens up a direction of studying polynomials that map one $C_i$ to another $C_j$ bijectively. In the rest of paper, we demonstrate our methodology and construct many new PPs of finite fields, many of them have large indices.
First of all, we obtain the following result for cyclotomic mappings polynomials of index $2$ which follows directly from Lemma \[mainLemma\].
Let $A_0, A_1 \in {\mathbb{F}_q}$ and $f_0(x), f_1 (x)$ be any two polynomials of ${\mathbb{F}_q}$ such that $$f(x) = f^{f_0(x), f_1(x)}_{A_0, A_1} (x) =
\left\{
\begin{array}{ll}
0, & if ~ x=0; \\
A_0 f_0(x), & if~ x \in C_0; \\
A_1 f_1(x), & if~ x \in C_1.
\end{array}
\right.$$ Then $f$ is a PP of ${\mathbb{F}_q}$ if either one of the following holds.
\(i) $A_0 f_0(C_0) = C_0$ and $A_1 f_1(C_1) = C_1$; or
\(ii) $A_0 f_0(C_0) = C_1$ and $A_1 f_1(C_1) = C_0$.
In particular, if we take $f_1(x), f_2 (x)$ as any two polynomials of ${\mathbb{F}_q}$ of indices at most $2$, then we have the following.
Let $q$ be odd and let $r_0, r_1$ be positive integers and $f_0(x), f_1(x) \in {\mathbb{F}_q}[x]$. Let $$f(x) =
\left\{
\begin{array}{ll}
0, & if ~ x=0; \\
x^r_0 f_0(x^{(q-1)/2}), & if~ x \in C_0; \\
x^r_1 f_1(x^{(q-1)/2}), & if~ x \in C_1.
\end{array}
\right.$$ Then $f$ is a PP of ${\mathbb{F}_q}$ if and only if $(r_0, (q-1)/2) = (r_1, (q-1)/2) =1$ and $\eta (f_0(1) f_1(-1) ) = (-1)^{r_1 +1}$, where $\eta$ is a quadratic character of ${\mathbb{F}_q}$.
Obviously $f_0(x^{(q-1)/2}) = f_0(1)$ for $x\in C_0$ and $f_1(x^{(q-1)/2}) = f_1(-1)$ for $x\in C_1$. If $f$ is a PP, we must have $(r_0, (q-1)/2) = (r_1, (q-1)/2) =1$. Moreover, $f_0(1) x^{r_0}$ maps $C_0$ onto $C_0$ if $\eta(f_0(1))=1$, and onto $C_1$ if $\eta(f_0(1))=-1$. Therefore $f_1(-1) x^{r_1}$ maps $C_1$ onto $C_1$ if $\eta (f_1(-1) ) = (-1)^{r_1 +1}$, and onto $C_0$ if $\eta (f_1(-1)) = (-1)^{r_1}$. In any case, $\eta (f_0(1) f_1(-1) ) = (-1)^{r_1 +1}$. The converse is obvious and we omit the proof.
The following result generalizes Theorem 8 [@ZhaHu] which only gives the sufficient part.
\[ZhaHuThm8Gen\] Let $p$ be an odd prime and $n$, $t$, $r$ be any positive integers. Then $f(x) = (1-x^t)x^{\frac{p^n-1}{2} + r} -x^r - x^{t+r}$ is a PP over $\mathbb{F}_{p^n}$ if and only if $(r, p^n-1)=1$ and $(t+r,\frac{p^n-1}{2} ) =1$.
We rewrite $f(x) = (1-x^t)x^{\frac{p^n-1}{2} + r} -x^r - x^{t+r} = x^r (x^{\frac{p^n-1}{2}} -1 ) - x^{r+t} (x^{\frac{p^n-1}{2}} + 1 )$. Obviously $s= \frac{p^n-1}{2}$, $\ell = 2$, $r_0=t+r$, $f_0(x) = -(x+1)$, $r_1=r$ and $f_1(x) =x-1$. So we write $f(x) = f_{-2, -2}^{x^{r+t}, x^r} (x)$. By the previous theorem, $f$ is a PP if and only if $(r, (p^n-1)/2)=1$, $(t+r,(p^n-1)/2)=1$, and $(r, 2) =1$.
If $(t+r, \frac{p^n-1}{2}) =1$ and $(t+r, p^n-1)=2$, then we must have $p^n \equiv 3 \pmod{4}$. Hence we have the following corollary.
Let $p$ be an odd prime and $n$, $t$, $r$ be any positive integers. Then $f(x) = (1-x^t)x^{\frac{p^n-1}{2} + r} -x^r - x^{t+r}$ is a PP over $\mathbb{F}_{p^n}$ provided
\(i) $(r, p^n-1)=1$ and $(t+r, p^n -1) =1$; or
\(ii) $(r, p^n-1)=1$, $(t+r, p^n-1) =2$ and $p^n \equiv 3 \pmod{4}$.
Next we obtain the following new classes of PPs over finite fields of characteristic $3$.
\[PPoverfthreepower1\] Let $q=3^n$ and $t$ be any positive integer. Let $\alpha, \beta, \theta \in \mathbb{F}_{q}^*$ and $$f(x) =
\left\{
\begin{array}{ll}
0, & if ~ x=0; \\
\alpha x^t , & if~ x \in C_0; \\
\beta ( x^3+\theta x^2+ \theta^2 x), & if~ x \in C_1.
\end{array}
\right.$$ Then $f$ is a PP of ${\mathbb{F}_q}$ if and only if $(t, \frac{q-1}{2}) =1$, $\eta(\theta) =1$, and $\eta (\alpha) = \eta(\beta)$, where $\eta$ is the quadratic character of ${\mathbb{F}_q}$. In this case, $f(x) = (\beta x^3 + \beta \theta x^2 + \beta \theta^2 x - \alpha x^t) x^{\frac{3^n-1}{2}} -
(\beta x^3 + \beta \theta x^2 + \beta \theta^2 x + \alpha x^t)$.
Assume $f$ is a PP. Because $x^t$ always map $C_0$ into $C_0$ and $\beta ( x^3+\theta x^2+ \theta^2 x) = \beta x (x-\theta)^2$, we must have $\eta (\alpha) = \eta(\beta)$. Indeed, we must have either $\eta (\alpha) = \eta(\beta) = 1$ so that $f$ maps $C_0$ into $C_0$ and maps $C_1$ into $C_1$, or $\eta (\alpha) = \eta(\beta) = -1$ so that $f$ maps $C_0$ into $C_1$ and maps $C_1$ into $C_0$. In either case, $(t, \frac{q-1}{2}) =1$ because $x^t$ permutes $C_0$. On the other hand, let $\beta ( x^3+\theta x^2+ \theta^2 x) = \beta ( y^3+\theta y^2+ \theta^2 y)$ for $x, y \in C_1$. Then we obtain $(x-y)(x^2 + (y-2\theta) x + (y-\theta)^2)=0$. It is obvious that $(x^2 + (y-2\theta) x + (y-\theta)^2)=0$ if and only if $\eta( (y-2\theta)^2 -4 (y-\theta)^2) = \eta(\theta y) =1$. Hence $\beta ( x^3+\theta x^2+ \theta^2 x)$ is one-to-one over $C_1$ if and only if $(x^2 + (y-2\theta) x + (y-\theta)^2) \neq 0$ over $C_1$. The latter is equivalent to $\eta(\theta y) \neq 1$ and thus $\eta(\theta) = 1$. The converse is similar and we omit the proof.
We note that in the case that $\theta =0$, $f$ is a PP of ${\mathbb{F}_q}$ if and only if $(t, \frac{q-1}{2})=1$ and $\eta(\alpha) = \eta(\beta)$.
In the study of permutation behaviour of the reversed Dickson polynomial, Hou [@Hou:11] proved that $D_{3^e+5}(1, x)$ is a PP over ${\mathbb{F}}_{3^e}$ when $e$ is positive even integer. Equivalently, Hou proved the following result which can also be put into the context of cyclotomic mappings. We observe that Hou’s result follows from Theorem \[PPoverfthreepower1\] for $t=3$, $\alpha = \theta =2$ and $\beta=1$ (with a linear shift by $-1$). We note that $\eta(2) = \eta(1) =1$ in $\mathbb{F}_{3^e}$ for any even positive $e$.
\[Theorem 1.1, [@Hou:11]\] Let $e$ be a positive even integer. Then $f(x) = (1-x-x^2)x^{\frac{3^e+1}{2}} -1 - x + x^2$ is a PP over ${\mathbb{F}}_{3^e}$.
Similarly, let $t=3^i$, $\alpha = \beta =2$ and $\theta=1$, we have the following result.
The polynomial $f(x) =x^{\frac{3^n-1}{2} +3^i}+ 2 x^{\frac{3^n-1}{2} +3}+ 2 x^{\frac{3^n-1}{2} +2} + 2x^{\frac{3^n-1}{2} +1} + x^{3^i} + x^3 + x^2 + x$ is a PP of ${\mathbb{F}}_{3^n}$ for any positive integer $n$ and non-negative integer $i$.
In particular, when $i=1$, we have
The polynomial $f(x) = x^{\frac{3^n-1}{2} +2} + x^{\frac{3^n-1}{2} +1} +x^3 +2 x^2 +2 x$ is a PP over ${\mathbb{F}}_{3^n}$ for any positive integer $n$.
The following result (Proposition 1 in [@ZhaHu]) follows also from Theorem \[PPoverfthreepower1\] for $\alpha =1$. We note $(t, 3^n-1) =1$ implies that $t$ is odd and thus $(t, (3^n-1)/2) =1$.
Let $t$ be a positive integer with $(t, 3^n-1) =1$. Assume $\theta, \beta \in {\mathbb{F}}_{3^n}^*$ with $\eta(\theta) = \eta(\beta) =1$. Then $f(x) = (\beta x^3 + \beta \theta x^2 + \beta \theta^2 x - x^t) x^{\frac{3^n-1}{2}} -
(\beta x^3 + \beta \theta x^2 + \beta \theta^2 x + x^t)$ is a PP over ${\mathbb{F}}_{3^n}$.
Theorem \[PPoverfthreepower1\] generalizes Proposition 1 in [@ZhaHu] in a few different ways. First, it gives a necessary and sufficient description. Secondly, a constant $\alpha$ could be interpreted as $f_i(x^{(q-1)/2})$ for any polynomial $f_i(x) \in {\mathbb{F}_q}[x]$. Thirdly, $t$ could be even if $n$ is odd because it is only required that $(t, (3^n-1)/2) =1$ in stead of $(t, 3^n-1)=1$. For example, plug $t=2^i$ and $\theta = \beta = \alpha =1$ in Theorem \[PPoverfthreepower1\] for ${\mathbb{F}}_{3^n}$ where $n$ is odd, we obtain
The polynomial $f(x) = x^{\frac{3^n-1}{2} +2^i}+ 2x^{\frac{3^n-1}{2} +3} + 2x^{\frac{3^n-1}{2} +2} + 2x^{\frac{3^n-1}{2} +1} + x^{2^i} + x^3 + x^2 + x$ is a PP of ${\mathbb{F}}_{3^n}$ for any odd positive integer $n$ and non-negative integer $i$.
In particular, when $i=1$, we have
The polynomial $f(x) = x^{\frac{3^n-1}{2} +3} + x^{\frac{3^n-1}{2} +1} + 2x^3 + x^2 + 2x$ is a PP of ${\mathbb{F}}_{3^n}$ for any odd positive integer $n$.
In a similar way, we obtain the following result which extends the previous results.
\[PPoverfthreepower2\] Let $q=3^n$ and $t$ be any positive integer. Let $\alpha, \beta, \theta \in \mathbb{F}_{q}^*$ and $$f(x) =
\left\{
\begin{array}{ll}
0, & if ~ x=0; \\
\beta ( x^3+\theta x^2+ \theta^2 x), & if~ x \in C_0;\\
\alpha x^t , & if~ x \in C_1.
\end{array}
\right.$$ Then $f$ is a PP of ${\mathbb{F}_q}$ if and only if $(t, \frac{q-1}{2} ) =1$, $\eta(\theta) = -1$, and any one of the following holds: (i) $t$ is odd and $\eta (\alpha) = \eta(\beta)$; (ii) $t$ is even and $\eta(\alpha) = -\eta(\beta)$. In this case, $f(x) = - (\beta x^3 + \beta \theta x^2 + \beta \theta^2 x - \alpha x^t) x^{\frac{3^n-1}{2}} - (\beta x^3 + \beta \theta x^2 + \beta \theta^2 x + \alpha x^t)$.
Assume $f$ is a PP. Obviously, $(t, \frac{q-1}{2} ) =1$ because $x^t$ maps $C_0$ onto either $C_0$ or $C_1$. Moreover, let $\beta ( x^3+\theta x^2+ \theta^2 x) = \beta ( y^3+\theta y^2+ \theta^2 y)$ for $x, y \in C_0$. Then we obtain $(x-y)(x^2 + (y-2\theta) x + (y-\theta)^2)=0$. It is obvious that $(x^2 + (y-2\theta) x + (y-\theta)^2)=0$ if and only if $\eta( (y-2\theta)^2 -4 (y-\theta)^2) = \eta(\theta y) =1$. Hence $\beta ( x^3+\theta x^2+ \theta^2 x)$ is one-to-one over $C_0$ if and only if $(x^2 + (y-2\theta) x + (y-\theta)^2) \neq 0$ over $C_0$. The latter is equivalent to $\eta(\theta y) \neq 1$ and thus $\eta(\theta) = - 1$. We now consider two cases of $t$. If $t$ is odd, then $x^t$ maps $C_1$ onto $C_1$. Because $\beta ( x^3+\theta x^2+ \theta^2 x) = \beta x (x-\theta)^2$, we must have $\eta (\alpha) = \eta(\beta)$. If $t$ is even, then $x^t$ maps $C_1$ onto $C_0$. Because $( x^3+\theta x^2+ \theta^2 x) = x (x-\theta)^2$ maps $C_0$ onto $C_0$, we must have $\eta(\alpha) = -\eta(\beta)$. The converse is similar and we omit the proof.
For $t=3^i$, $\theta =2$, and $\alpha = \beta =1$, we apply Theorem \[PPoverfthreepower2\] over $\mathbb{F}_{3^n}$ with odd $n$ to obtain the following result.
\[PPoverfthreepower3\] The polynomial $f(x) = x^{\frac{3^n-1}{2} +3^i}+ 2x^{\frac{3^n-1}{2} +3} + x^{\frac{3^n-1}{2} +2} + 2x^{\frac{3^n-1}{2} +1} + 2 x^{2^i} + 2x^3 + x^2 + 2 x$ is a PP of ${\mathbb{F}}_{3^n}$ for any odd positive integer $n$ and non-negative integer $i$.
For $t=2$, $\theta = \beta =2$, and $\alpha=1$, we apply Theorem \[PPoverfthreepower2\] over $\mathbb{F}_{3^n}$ with odd $n$ to obtain the following result.
The polynomial $f(x) = x^{\frac{3^n-1}{2} +2^i}+ x^{\frac{3^n-1}{2} +3} + 2x^{\frac{3^n-1}{2} +2} + x^{\frac{3^n-1}{2} +1} + 2x^{2^i} + x^3 + 2x^2 + x$ is a PP of ${\mathbb{F}}_{3^n}$ for any odd positive integer $n$ and non-negative integer $i$.
In particular, when $i=1$, we obtain
The polynomial $f(x) = x^{\frac{3^n-1}{2} +3} + x^{\frac{3^n-1}{2} +1} + x^3 + x^2 + x$ is a PP over ${\mathbb{F}}_{3^n}$ for any odd positive integer $n$.
We also obtain the following PPs such that both branches are cubic polynomials.
\[PPoverfthreepower4\] Let $q=3^n$ and $\alpha, \beta, \gamma, \theta \in \mathbb{F}_{3^n}$. Let $f(x) = (\beta -\alpha) x^{(q-1)/2 +3} + (\beta \theta -\alpha \gamma) x^{(q-1)/2 +2} + (\beta \theta^2 -\alpha \gamma^2) x^{(q-1)/2 +1} - (\beta + \alpha) x^{3} - (\beta \theta + \alpha \gamma) x^{2} - (\beta \theta^2 +\alpha \gamma^2) x$. Then $f$ is a PP of $\mathbb{F}_{3^n}$ if and only if $\eta(\alpha) = \eta(\beta)$, $\eta(\gamma) =-1$, and $\eta(\theta) =1$.
Obviously, we have $$f(x) =
\left\{
\begin{array}{ll}
0, & if ~ x=0; \\
\alpha ( x^3+\gamma x^2+ \gamma^2 x) , & if~ x \in C_0; \\
\beta ( x^3+\theta x^2+ \theta^2 x), & if~ x \in C_1.
\end{array}
\right.$$ Assume $f$ is a PP of ${\mathbb{F}_q}$. Because $\alpha ( x^3+\gamma x^2+ \gamma^2 x)= \alpha x (x-\gamma)^2$ and $\beta ( x^3+\theta x^2+ \theta^2 x) = \beta x (x-\theta)^2$, they map $C_0$ and $C_1$ into different cosets respectively, as long as $\eta(\alpha) = \eta (\beta)$. Moreover, let $\beta ( x^3+\theta x^2+ \theta^2 x) = \beta ( y^3+\theta y^2+ \theta^2 y)$ with $x, y \in C_1$. Then we obtain $(x-y)(x^2 + (y-2\theta) x + (y-\theta)^2)=0$. It is obvious that $(x^2 + (y-2\theta) x + (y-\theta)^2)=0$ if and only if $\eta( (y-2\theta)^2 -4 (y-\theta)^2) = \eta(\theta y) =1$. Hence $\beta ( x^3+\theta x^2+ \theta^2 x)$ is one-to-one over $C_1$ if and only if $\eta(\theta) = 1$. Similarly, $\alpha ( x^3+\gamma x^2+ \gamma^2 x)$ is one to one over $C_0$ if and only if $\eta(\gamma) = - 1$.
For the rest of paper, we concentrate on refinement of Lemma \[mainLemma\] with more branches. Obviously, $A_i \neq 0$ for all $i$’s if $f$ is a PP. Moreover, if $r_i(x)$’s are of certain special formats then we can simplify Lemma \[mainLemma\] significantly.
One of the most natural choice is that $r_i(x) = x^{r_i}$ for $i=0, \ldots, \ell-1$. In this case, we must have $(r_i, s) =1$ in order for $P(x)$ to be a PP; otherwise, $|C_i^{r_i} | \neq s$, a contradiction. Hence we have the following result.
\[main\] Let $\ell, s, r_0, \ldots, r_{\ell-1}$ be positive integers such that $s = (q-1)/\ell$ and $(r_i, s)=1$ for any $i=0, \ldots, \ell-1$. Let $q$ be prime power and $A_0, \ldots, A_{\ell-1} \in {\mathbb{F}_q}^*$. Let $$P(x) =
f^{x^{r_0}, x^{r_1}, \ldots, x^{r_{\ell-1}}}_{A_0, A_1, \cdots, A_{\ell-1}} (x)
= \left\{
\begin{array}{ll}
0, & if ~ x=0; \\
A_0 x^{r_0}, & if~ x \in C_0; \\
\vdots & \vdots \\
A_i x^{r_i}, & if ~x \in C_i; \\
\vdots & \vdots \\
A_{\ell-1} x^{r_{\ell -1}}, & if~ x \in C_{\ell-1}.
\end{array}
\right.$$ Then the following are equivalent.
1. $P(x)$ is a PP of ${\mathbb{F}_q}$;
2. $A_iC_{i r_i } \neq A_{i^\prime} C_{i^\prime r_{i^\prime}}$ for any $0 \leq i < i^\prime \leq \ell-1$, where the subscripts of $C_{i r_i}$ are taken modulo $\ell$.
3. $Ind_\gamma (\frac{A_i}{A_{i^\prime}}) \not\equiv r_{i^\prime} i^\prime
- r_i i \pmod{\ell}$ for any $0 \leq i < i^\prime \leq \ell-1$, where $ind_\gamma (a)$ is residue class $b \bmod q-1$ such that $a=\gamma^{b}$.
4. $\{ A_0, A_1\gamma^{r_i}, \cdots, A_{\ell-1} \gamma^{(\ell-1)r_i} \}$ is a system of distinct representatives of $\mathbb{F}_q^*/C_0$.
5. $\{A_i^s\zeta^{i r_i} \mid i = 0, \cdots, \ell -1 \}$ is the set $\mu_{\ell}$ of all distinct $\ell$-th roots of unity.
6. ${\displaystyle \sum_{i=0}^{\ell-1}
\zeta^{cr_i i} A_i^{cs} =0}$ for all $c=1, \cdots, \ell-1$.
The proof is similar to the proof of Theorem 1 in [@Wang] and we include it for the sake of completeness.
Since $ C_i = \{\gamma^{\ell j +i}: j = 0, 1, \cdots, s -1\}$, for any two elements $x \neq y \in C_i$, we have $x = \gamma^{\ell j+i}$ and $y =\gamma^{\ell j^\prime +i}$ for some $0\leq j \neq j^\prime
\leq s-1$ . Since $(r_i, s) =1$, we obtain $A_i x^{r_i} = A_i
\gamma^{\ell r_i j+ i r_i } \neq A_i y^{r_i} = A_i \gamma^{\ell r_i j^\prime +ir_i}$. Moreover, it is easy to prove that $C_0^{r_0} = C_0$ and more generally $C_i^{r_i} = C_{i r_i}$ for any $0\leq i \leq \ell-1$. Hence $(a)$ and $(b)$ are equivalent.
Because $A_i\gamma^{i r_i }$ is a coset representative of $A_iC_{ir_i }$, it is easy to see that $(c)$, $(d)$, and $(e)$ are equivalent. Finally, since all of $A_0^s, A_1^s\zeta^{r_1}, \cdots,
A_{\ell-1}^s \zeta^{(\ell-1)r_{\ell-1}}$ are $\ell$-th roots of unity, $(e)$ means that $A_0^s$, $A_1^s\zeta^{r_1}$, $\cdots$, $A_{\ell-1}^s
\zeta^{(\ell-1)r_{\ell-1}}$ are all distinct. By Lemma 2.1 in [@AW:07], $(e)$ is equivalent to $(f)$.
This result generalizes Theorem 2.2 [@AW:07], Theorem 1 [@Wang], and Lemma 2.1 [@Zieve-2], and all the consequences in these references. Furthermore, we obtain the following result in terms of the polynomial presentation.
\[construction\] Let $q$ be a prime power, $\ell \mid q-1$ and $s = (q-1)/\ell$. Let ${\mathbb{F}_q}$ be a finite field of $q$ elements and $\zeta \in {\mathbb{F}_q}$ be a primitive $\ell$-th root of unity. Let $A_0, \ldots, A_{\ell-1} \in {\mathbb{F}_q}^*$. Then $$P(x) = \sum_{i=0}^{\ell-1} \frac{A_i x^{r_i}}{\ell \zeta^{i(\ell-1)}} \left(x^{(\ell-1)s} + \zeta^i x^{(\ell-2)s} +\cdots + \zeta^{i(\ell-2)} x^s + \zeta^{i(\ell-1)} \right)$$ is a PP of ${\mathbb{F}_q}$ if and only if $(r_i, s) =1$ for all $i=0, \ldots, \ell-1$ and $\{A_i^s\zeta^{i r_i} \mid i = 0, \cdots, \ell -1 \} = \mu_{\ell}$, where $\mu_{\ell}$ is the set of all $\ell$-th roots of unity. The latter condition is equivalent to that $\{ t_i + ir_i \mid i=0, \ldots, \ell-1 \}$ is a complete set of residues modulo $\ell$, where $A_i^s = \zeta^{t_i}$ for $i=0, \ldots, \ell-1$. In particular, if $A_0^s = A_1^s = \ldots =A_{\ell-1}^s \neq 0$, then $P(x)$ is a PP of ${\mathbb{F}_q}$ if and only if $(r_i, s) =1$ for all $i=0, \ldots, \ell-1$ and $\{ ir_i \mid i=0, \ldots, \ell-1 \}$ is a complete set of residues modulo $\ell$.
By Theorem \[main\] and Equation (\[correspondence\]), $P(x)$ is a PP of ${\mathbb{F}_q}$ if and only if $(r_i, s) =1$ for all $i=0, \ldots, \ell-1$ and $\{A_i^s\zeta^{i r_i} \mid i = 0, \cdots, \ell -1 \}$ is the set $\mu_{\ell}$ of all distinct $\ell$-th roots of unity. Moreover, $\{A_i^s\zeta^{i r_i} = \zeta^{t_i + i r_i} \mid i = 0, \cdots, \ell -1 \}= \mu_{\ell}$ is equivalent to that $\{ t_i + ir_i \mid i=0, \ldots, \ell-1 \}$ is a complete set of residues modulo $\ell$.
Theorem \[construction\] provides a simple algorithmic way to construct PPs of ${\mathbb{F}_q}$ with large indices. First, take any factor $\ell$ of $q-1$ and let $s= \frac{q-1}{\ell}$. Then pick any $\ell$ positive integers $r_0, \ldots, r_{\ell-1}$ such that $(r_i, s) =1$ for $i=0, \ldots, \ell-1$ and any $\ell$ nonzero constants $A_0, \ldots, A_{\ell-1} \in {\mathbb{F}_q}^*$. As long as $A_i^s \zeta^{ir_i}$ ($0\leq i \leq \ell-1$) are all distinct (equivalently, $\{ t_i + ir_i \mid i=0, \ldots, \ell-1 \}$ is a complete set of residues modulo $\ell$), we obtain a PP of ${\mathbb{F}_q}$. In this way, one can construct a very large amount of classes of PPs of ${\mathbb{F}_q}$. Here we give a few more examples of PPs of finite fields produced by our construction method.
First we consider a few classes of PPs with three branches.
\[3branches\] Let $q=p^n$ such that $3\mid q-1$ and $s= \frac{q-1}{3}$. Let $\zeta$ be a primitive $3$-rd root of unity. Let $A_0, A_1, A_2 \in {\mathbb{F}_q}$. Then $P(x) = A_0 x^{r_0} \left(x^{2s} + x^{s} +1 \right) +
\zeta A_1 x^{r_1}$ $\left(x^{2s} + \zeta x^{s} +\zeta^{2} \right)$ $+ \zeta^{2} A_2 x^{r_2} \left(x^{2s} + \zeta^2 x^{s} +\zeta \right) $ is a PP of ${\mathbb{F}_q}$ if and only if $(r_i, s) =1$ for $i=0, 1, 2$ and $ \{ A_0^s, A_1^s \zeta^{r_1}, A_2^s \zeta^{2r_2} \}$ $= \{ 1, \zeta, \zeta^2\}$.
The following result generalizes Theorem 9 in [@ZhaHu]. Again, we show these conditions are both necessary and sufficient.
\[3branchesZha1\] Assume $p^n \equiv 1 \pmod{3}$. Let $s= \frac{p^n-1}{3}$ and $\zeta$ be an element of $\mathbb{F}_{p^n}$ of order $3$. Then $$f(x) = x (x^s - \zeta)(x^s-\zeta^2) + x^3 (x^s - 1)(x^s-\zeta^2) + \zeta x^p
(x^s - 1)(x^s-\zeta)$$ is a PP over ${\mathbb{F}_{p^n}}$ if and only if
\(a) $p \equiv 1 \pmod 3$ and $s \equiv 1 \pmod 3$; or
\(b) $p\equiv 2 \pmod 3$ and $s \equiv 2 \pmod 3$.
In this case, $\ell =3$ and $r_0 =1, r_1 =3, r_2=p$. Also $f_0(x) = (x- \zeta)(x-\zeta^2)$, $f_1(x) = (x-1)(x-\zeta^2)$, and $f_2(x) = \zeta (x-1)(x-\zeta)$. So $A_0 = (1-\zeta)(1-\zeta^2) = 1- \zeta - \zeta^2 + \zeta^3 = 2-\zeta -\zeta^2 = 3$, $A_1 = (\zeta-1)(\zeta-\zeta^2) = 3 \zeta^2$, and $A_2 =\zeta (\zeta^2 -1)(\zeta^2 -\zeta) = 3\zeta^2$. Hence $$f(x) = f_{A_0, A_1, A_2}^{x^1, x^{3}, x^p}(x) = \left\{
\begin{array}{ll}
0 & x =0;\\
3 x & x\in C_0; \\
3\zeta^2 x^{3} & x\in C_1; \\
3\zeta^2 x^p & x \in C_2.
\end{array} \right.$$
Obviously, we have $(r_i, s) =1$ for $i=0, 1, 2$. Moreover, $\{A_0^s, A_1^s \zeta^{3}, A_2^s \zeta^{2p} \} = \{3^s, 3^s \zeta^{2s+3}, 3^s \zeta^{2s+2p} \}$ is equal to $\{1, \zeta^{2s}, \zeta^{2s+2p} \}$ if and only if $p, s$ satisfy either $p \equiv 1 \pmod 3$ and $s \equiv 1 \pmod 3$, or $p\equiv 2 \pmod 3$ and $s \equiv 2 \pmod 3$. By Corollary \[3branches\], we complete our proof.
The following result also generalizes Theorem 10 in [@ZhaHu].
\[3branchesZha2\] Let $i$ be any positive integer and assume $p^n \equiv 1 \pmod{9}$. Let $s= \frac{p^n-1}{3}$ and $\zeta$ be an element of $\mathbb{F}_{p^n}$ of order $3$. Then $$f(x) = x (x^s - \zeta)(x^s-\zeta^2) + x^{p^i} (x^s - 1)(x^s-\zeta^2) + \zeta x^p
(x^s - 1)(x^s-\zeta)$$ is a PP over ${\mathbb{F}_{p^n}}$ if and only if
\(i) $p \equiv 1 \pmod 3$; or
\(ii) $i$ is odd and $p\equiv 2 \pmod 3$.
In this case, $\ell =3$ and $r_0 =1, r_1 =p^i, r_2=p$. Also $f_0(x) = (x- \zeta)(x-\zeta^2)$, $f_1(x) = (x-1)(x-\zeta^2)$, and $f_2(x) = \zeta (x-1)(x-\zeta)$. So $A_0 = (1-\zeta)(1-\zeta^2) = 1- \zeta - \zeta^2 + \zeta^3 = 2-\zeta -\zeta^2 = 3$, $A_1 = (\zeta-1)(\zeta-\zeta^2) = 3 \zeta^2$, and $A_2 = \zeta (\zeta^2 -1)(\zeta^2 -\zeta) = 3\zeta^2$. Hence $$f(x) = f_{A_0, A_1, A_2}^{x^1, x^{p^i}, x^p}(x) = \left\{
\begin{array}{ll}
0 & x =0;\\
3 x & x\in C_0; \\
3\zeta^2 x^{p^i} & x\in C_1; \\
3\zeta^2 x^p & x \in C_2.
\end{array} \right.$$
Obviously, we have $(r_j, s) =1$ for $j=0, 1, 2$. Therefore, by Corollary \[3branches\], $f(x)$ is a PP over ${\mathbb{F}_q}$ if and only if $\{A_j^{s} \zeta^{r_j j} \mid j=0, 1, 2 \} = \{ 1,\zeta, \zeta^2\}$. Indeed, $\{A_j^{s} \zeta^{r_j j} \mid j=0, 1, 2 \} = \{ 3^s, (3\zeta^2)^s\zeta^{p^i}, (3\zeta^2)^s \zeta^{2p} \}$. We only need to find conditions so that $1, \zeta^{2s+p^i}, \zeta^{2s+2p}$ are all distinct, equivalently, $2s+p^i \not \equiv 0 \pmod 3$, $s+p \not \equiv 0 \pmod{3}$ and $2s +p^i \not \equiv 2s+2p \pmod{3}$. Under the assumption of $p^n \equiv 1 \pmod{9}$, we have $s \equiv 0 \pmod{3}$. Hence we require $p^i \not \equiv 0 \pmod 3$, $p \not \equiv 0 \pmod{3}$ and $p^i \not \equiv 2p \pmod{3}$. Therefore either $p \equiv 1 \pmod{3}$, or $p \equiv 2 \pmod{3}$ and $i$ is odd.
In particular, if $A_0, A_1, A_2$ belong to the same cyclotomic coset, then the condition $ \{ A_0^s, A_1^s \zeta^{r_1}, A_2^s \zeta^{2r_2} \}= \{ 1, \zeta, \zeta^2\}$ reduces $\{1, \zeta^{r_1}, \zeta^{2r_2} \}= \{ 1, \zeta, \zeta^2\}$, which is equivalent to $r_1 \equiv r_2 \not\equiv 0 \pmod{3}$.
Let $q=p^n$ such that $3\mid q-1$ and $s= \frac{q-1}{3}$. Let $\zeta$ be a primitive $3$-rd root of unity. Let $A_0, A_1, A_2 \in {\mathbb{F}_q}$ such that $A_0^s = A_1^s = A_2^s$. Then $P(x) = A_0 x^{r_0} $$ \left(x^{2s} + x^{s}+ 1 \right) +
A_1 x^{r_1}$ $\left( \zeta x^{2s} + \zeta^2 x^{s} +1 \right)$ $+ A_2 x^{r_2} \left(\zeta^2 x^{2s} + \zeta x^{s} + 1\right) $ is a PP of ${\mathbb{F}_q}$ if and only if $(r_i, s) =1$ for $i=0, 1, 2$ and $r_1 \equiv r_2 \not\equiv 0 \pmod{3}$.
From this corollary, if we take $q=2^n$ with $n$ is even, $A_0=A_1= A_{2} = 1$, $r_0 = 2^i$ and $r_1=r_{2} = 2^j$ for some non negative integers $i, j$, we obtain the following classes of PPs with coefficients in $\mathbb{F}_2$.
The polynomial $P(x) = x^{\frac{2(2^n-1)}{3}+2^i} + x^{\frac{2(2^n-1)}{3}+2^j} + x^{\frac{2^n-1}{3}+2^i} + x^{\frac{2^n-1}{3}+2^j} + x^{2^i}$ is a PP over ${\mathbb{F}}_{2^n}$ for any even positive integer $n$ and non-negative integers $i, j$.
Similarly, we can construct PPs with coefficients in general base field $\mathbb{F}_p$.
Let $q=p^m$, $\ell$ be a prime factor of $q-1$ with $s= \frac{q-1}{\ell}$. Let $A_0, A_1\in {\mathbb{F}_q}^*$. Then $f(x) =A_0 x^{r_0} \left(x^{(\ell-1) s} + \cdots + x^{s} +1 \right)$ $-
A_1 x^{r_1} \left(x^{(\ell-1)s} + \cdots + x^{s} + \ell-1 \right)$ is a PP of ${\mathbb{F}_q}$ if and only if $(r_0, s) = (r_1, s) =1$ and $A_0^s = A_1^s$.
Let $P(x)$ be the cyclotomic mapping $f_{A_0, A_1, \ldots, A_1}^{r_0, r_1, \ldots, r_1}(x)$, we obtain $$\begin{aligned}
P(x) &=& \frac{A_0 x^{r_0}}{\ell} \left(x^{(\ell-1)s} + x^{(\ell-2)s} +\cdots + x^s + 1 \right) \\
& & + \sum_{i=1}^{\ell-1} \frac{A_i x^{r_i}}{\ell \zeta^{i(\ell-1)}} \left(x^{(\ell-1)s} + \zeta^i x^{(\ell-2)s} +\cdots + \zeta^{i(\ell-2)} x^s + \zeta^{i(\ell-1)} \right) \\
&=& \frac{A_0 x^{r_0}}{\ell} \left(x^{(\ell-1)s} + x^{(\ell-2)s} +\cdots + x^s + 1 \right) \\
& & + \frac{A_1 x^{r_1}}{\ell} \left( \sum_{i=1}^{\ell-1} \zeta^{-i(\ell-1)} x^{(\ell-1)s} + \sum_{i=1}^{\ell-1} \zeta^{-i(\ell-2)} x^{(\ell-2)s} +\cdots + \sum_{i=1}^{\ell-1} \zeta^{-i} x^s + \ell -1 \right)\\
&=& \frac{A_0 x^{r_0}}{\ell} \left(x^{(\ell-1)s} + x^{(\ell-2)s} +\cdots + x^s + 1 \right) - \frac{A_1 x^{r_1}}{\ell} \left(x^{(\ell-1)s} + \cdots + x^{s} + \ell -1 \right), \end{aligned}$$ where the last equality holds because $\sum_{i=1}^{\ell-1} \zeta^{-i(\ell-j)} = -1$ for all $j=1, \ldots, \ell-1$ when $\ell$ is prime. By Theorem \[construction\] and let $r_1=\cdots = r_{\ell-1}$ and $A_1 = \cdots = A_{\ell-1}$, $P(x)$ is a PP of ${\mathbb{F}_q}$ if and only if $(r_0, s) = (r_1, s) =1$ and $\{A_0^s, A_1^s \zeta^{r_1}, \ldots, A_1^s \zeta^{(\ell-1) r_1} \} = \mu_{\ell}$. The latter condition is equivalent to $A_0^s \neq A_1^s \zeta^{i r_1}$ for all $i=1, \ldots, \ell-1$, namely, $A_1^s = A_0^s$.
Taking $A_0 = A_1 =1$, we obtain the following PP with coefficients in the prime field $\mathbb{F}_p$.
Let $q=p^m$, $\ell$ be a prime factor of $q-1$ with $s= \frac{q-1}{\ell}$. Then $f(x) = x^{r_0} \left(x^{(\ell-1) s} + \cdots + x^{s} +1 \right)$ $-
x^{r_1} \left(x^{(\ell-1)s} + \cdots + x^{s} + \ell-1 \right)$ is a PP of ${\mathbb{F}_q}$ if and only if $(r_0, s) = (r_1, s) =1$.
Finally we give another application of Theorem \[construction\], which generalizes Theorem 11 in [@ZhaHu].
\[squaredivisor\] Assume $p^n \equiv 1 \pmod{\ell^2}$ and let $\theta$ be an element of ${\mathbb{F}_{p^n}}$ of order $\ell$. Then $$f(x) = \sum_{i=1}^t x^{p^i} \prod_{j=1, j\neq i}^{t} (x^{(p^n-1)/\ell} - \theta^j)$$ is a PP over ${\mathbb{F}_{p^n}}$ if and only if $\{ip^i \pmod{\ell} \mid i=0, \ldots, \ell-1\} = \mathbb{Z}_{\ell}$.
Let $s = \frac{p^n-1}{\ell}$. We note that $f$ is a cyclotomic mapping with $r_i = p^i$ for $i=0, \ldots, \ell-1$ and $ A_i = \prod_{j=1, j\neq i}^{t} (\theta^{i} - \theta^j) = \theta^{i(\ell-1)} (1-\theta^{-1})\cdots (1-\theta^{-(\ell-1)}) = \ell \theta^{i(\ell-1)} $. By Theorem \[construction\], $f$ is a PP of ${\mathbb{F}_q}$ if and only if $(p^i, q-1)=1$ for all $i=0, \ldots, \ell-1$ and $\{ \ell^s \theta^{i(\ell-1)s} \theta^{ip^i} \mid i=0, \ldots, \ell-1\} = \mu_{\ell}$. The condition $p^n \equiv 1 \pmod{\ell^2}$ implies $\ell \mid s$ and thus $\theta^{i(\ell-1)s} =1$. Hence $\{ \ell^s \theta^{ip^i} \mid i=0, \ldots, \ell-1\} = \mu_{\ell}$ if and only if $\{ip^i \pmod{\ell} \mid i=0, \ldots, \ell-1\} = \mathbb{Z}_{\ell}$.
\[ZhaHuTheorem11\] Assume $p \equiv 1 \pmod{\ell}$ and $p^n \equiv 1 \pmod{\ell^2}$ and let $\theta$ be an element of ${\mathbb{F}_{p^n}}$ of order $\ell$. Then $$f(x) = \sum_{i=1}^t x^{p^i} \prod_{j=1, j\neq i}^{t} (x^{(p^n-1)/\ell} - \theta^j)$$ is a PP over ${\mathbb{F}_{p^n}}$
Assume $p^n \equiv 1 \pmod{16}$ and let $\theta$ be an element of ${\mathbb{F}_{p^n}}$ of order $\ell$. Then $$f(x) = \sum_{i=1}^t x^{p^i} \prod_{j=1, j\neq i}^{t} (x^{(p^n-1)/\ell} - \theta^j)$$ is a PP over ${\mathbb{F}_{p^n}}$.
Obviously, $p$ must be odd. If $p \equiv 1 \pmod{4}$, then $f$ is PP over ${\mathbb{F}_{p^n}}$ by Corollary \[ZhaHuTheorem11\]. If $p \equiv 3 \pmod{4}$, then $\{ip^i \mid i=0, 1, 2, 3\} = \{0, p, 2p^2, 3p^3\}$ is indeed a complete set of residue modulo $4$. By Theorem \[squaredivisor\], $f$ is a PP over ${\mathbb{F}_{p^n}}$.
Similarly, we obtain the following corollary.
Assume $p^n \equiv 1 \pmod{25}$ and let $\theta$ be an element of ${\mathbb{F}_{p^n}}$ of order $\ell$. Then $$f(x) = \sum_{i=1}^t x^{p^i} \prod_{j=1, j\neq i}^{t} (x^{(p^n-1)/\ell} - \theta^j)$$ is a PP over ${\mathbb{F}_{p^n}}$ if and only if $p \equiv 1 \pmod{5}$.
Realization of constants by polynomials
=======================================
In this section, we give more applications of Theorem \[main\] (or another version as in Theorem \[main2\]) to construct many new classes of PPs which have large indices and simple descriptions. We mainly consider how to choose constants $A_0, \ldots, A_{\ell-1}$ in terms of polynomials of specific formats in the cyclotomic mappings constructions. This demonstrate that our results generalize these results in [@AAW:08; @AW:06; @AW:07; @Zieve-2].
First of all, another way to rewrite Theorem \[main\] is as follow:
\[main2\] Let $q-1 = \ell s$, $f_0(x), \ldots, f_{\ell-1} (x) \in {\mathbb{F}_q}[x]$ and $$P(x) =
\left\{
\begin{array}{ll}
0, & if ~ x=0; \\
x^{r_0} f_0(x^s), & if~ x \in C_0; \\
x^{r_1} f_1(x^s), & if~ x \in C_1; \\
\vdots & \vdots \\
x^{r_{\ell-1}} f_{\ell -1} (x^s), & if ~ x \in C_{\ell}.
\end{array}
\right.$$ Then $P(x)$ is a PP of ${\mathbb{F}_q}$ if and only if $(r_i, s) =1$ for any $i=0, 1, \ldots, \ell-1$ and $\mu_{\ell} = \{\zeta^{r_i i}f_i(\zeta^i)^s \mid i=0, \ldots, \ell-1\}$, where $\mu_{\ell}$ is the set of all $\ell$-th roots of unity.
Using Equation (\[correspondence\]) and Theorem \[main2\], we can construct many PPs in the following polynomial format with large indices.
$$\label{correspondence2}
P(x) = \sum_{i=0}^{\ell-1} \frac{x^{r_i} f_i(x^s)}{\ell \zeta^{i(\ell-1)}} \left(x^{(\ell-1)s} + \zeta^i x^{(\ell-2)s} +\cdots + \zeta^{i(\ell-2)} x^s + \zeta^{i(\ell-1)} \right).$$
As long as $f_i(\zeta^i) \neq 0$, we can rewrite Theorem \[main2\] as follow:
\[specialMain\] Let $q-1=\ell s$, $f_1(x), \ldots, f_{\ell-1}(x) \in {\mathbb{F}_q}[x]$, and $$P(x) =
\left\{
\begin{array}{ll}
0, & if ~ x=0; \\
x^{r_0} f_0(x^s), & if~ x \in C_0; \\
x^{r_1} f_1(x^s), & if~ x \in C_1; \\
\vdots & \vdots \\
x^{r_{\ell-1}} f_{\ell -1} (x^s), & if ~ x \in C_{\ell}.
\end{array}
\right.$$ Suppose $f_i(\zeta^i)^s = A \zeta^{n_i}$ for each $i=0, \ldots, \ell-1$ and a nonzero constant $A \in {\mathbb{F}_q}^*$. Then $P(x)$ is a PP of ${\mathbb{F}_q}$ if and only if
1. $(r_i, s) =1$ for any $i=0, \ldots, \ell-1$.
2. $\{ i r_i + n_i \mid i =0, \ldots, \ell-1\}$ is a complete set of residues modulo $\ell$.
In particular, if $f_i(\zeta^{i})^s = A$ for each $i=0,
\ldots, \ell-1$ and a nonzero constant $A\in {\mathbb{F}_q}^*$, then $P$ is a permutation polynomial of ${\mathbb{F}_q}$ if and only if $(r_i, s) =1$ for $i =0, \ldots, \ell -1$ and $\{r_i i \mid i =0, \ldots, \ell -1\}$ is a complete set of residues modulo $\ell$.
[ Because the constant $A$ appears in each branch of the definition of $P(x)$, we can assume $A=1$ without loss of generality. From Theorem \[main2\], we need to show that condition (ii) is equivalent to that $\mu_{\ell} = \{\zeta^{r_i i}f_i(\zeta^i)^s = \zeta^{ir_i + n_i}\mid i=0, \ldots, \ell-1\}$, which is obvious. ]{}
We note that if $r_0 = r_1 = \cdots = r_{\ell-1} :=r$, then $\{r_i i \mid i =0, \ldots, \ell -1\}$ is a complete set of residues modulo $\ell$ if and only if $(r, \ell) =1$. If $r_0 = \cdots = r_{\ell-1}$ and $n_0 = \cdots = n_{\ell-1}$, we obtain Theorem 4.1 in [@AW:07] as a corollary. As a special case of Theorem \[specialMain\], we also have the following result.
\[rogersBranches\] Let $q-1 = \ell s $, $g_1(x), \ldots, g_{\ell-1}(x)$ be any $\ell$ polynomials over ${\mathbb{F}_q}$, and $$P(x) =
\left\{
\begin{array}{ll}
0, & if ~ x=0; \\
x^{r_0} g_0(x^s)^\ell, & if~ x \in C_0; \\
x^{r_1} g_1(x^s)^\ell, & if~ x \in C_1; \\
\vdots & \vdots \\
x^{r_{\ell-1}} g_{\ell -1} (x^s)^\ell, & if ~ x \in C_{\ell}.
\end{array}
\right.$$ Then $P(x)$ is a permutation polynomial of ${\mathbb{F}_q}$ if and only if $\{r_i i \mid i =0, \ldots, \ell -1\}$ is a complete set of residues modulo $\ell$, $(r_i, s) =1$ and $g_i(\zeta^i) \neq 0$ for all $0\leq i \leq
\ell-1$.
[This is true since if we set $f_i(x)= g_i(x)^\ell$, then we have $f_i(\zeta^i)^s=g_i(\zeta^i)^{\ell s}=g_i(\zeta^i)^{q-1} =1$. The result follows from Theorem \[specialMain\].]{}
We note that earlier results of Wan and Lidl (see Corollary 1.4 in [@WL:91]), and Akbary and Wang (Theorem 3.1 in [@AW:07]) are also special cases of the above result.
We next construct cyclotomic permutations using classes of PPs with coefficients in some appropriate subfield which has been studied in [@AAW:08], [@AW:06], [@AW:07], [@Chapuy:07], and [@Zieve-2].
\[specialR\] Let $\ell, r_0, \ldots, r_{\ell-1}$ be a positive integer with $q-1 = \ell s$. Suppose $q = q_0^m$ where $q_0 \equiv 1 \pmod{\ell}$ and $\ell \mid m$. Let $f_1(x), \ldots, f_{\ell-1}(x)$ be polynomials in $\mathbb{F}_{q_0}[x]$ and $$P(x) =
\left\{
\begin{array}{ll}
0, & if ~ x=0; \\
x^{r_0} f_0(x^s), & if~ x \in C_0; \\
x^{r_1} f_1(x^s), & if~ x \in C_1; \\
\vdots & \vdots \\
x^{r_{\ell-1}} f_{\ell -1} (x^s), & if ~ x \in C_{\ell}.
\end{array}
\right.$$ Then the polynomial $P(x)$ is a permutation polynomial of ${\mathbb{F}}_q$ if and only if $\{r_i i \mid i =0, \ldots, \ell -1\}$ is a complete set of residues modulo $\ell$, $(r_i, s)=1$ and $f_i(\zeta^i) \neq 0$ for all $0\leq i \leq \ell-1$.
[ Let $m=\ell n$. The result is clear from Theorem \[specialMain\], since we have $$\begin{aligned}
f_i(\zeta^i)^{\frac{q-1}{\ell}}&=&f_i(\zeta^i)^{\frac{q_0^{\ell n}-1}{\ell}}\\
&=& f_i(\zeta^i)^{\frac{q_0^{n}-1}{\ell} \left((q_0^{n})^{\ell-1}+(q_0^{n})^{\ell-2}+\cdots+1 \right)}\\
&=& \left( \prod_{j=0}^{\ell-1} f_i(\zeta^i)^{q_0^{nj}} \right)^{\frac{q_0^{n}-1}{\ell}}\\
&=& \left( f_i(\zeta^i)^\ell \right)^{\frac{q_0^{n}-1}{\ell}}\\
&=& 1.\end{aligned}$$ ]{}
For example, let $v$ be the order of $p$ in $\mathbb{Z}/\ell\mathbb{Z}$. For any positive integer $n$, we can take $q=q_0^m = p^{\ell vn}$ in the above result. We note Theorem \[specialR\] generalizes Corollary 3.3 ([@AW:07] or Theorem 3.1 ([@Chapuy:07]) or Theorem 1.2 ([@Zieve-2]), which deal with the case $P(x) = x^rf(x^s)$.
Moreover, in [@AAW:08; @AW:07; @Zieve-2], classes of PPs of the form $x^r(1+x^{e_1 s} + \ldots +x^{e_k s})^t$ are studied. For $h(x) = 1 + x+ \cdots +x^k$, it is well known that $h(\zeta^0) = k+1 \neq 0$ if and only if $p \nmid (k+1)$ and $h(\zeta^i) = \frac{\zeta^{(k+1)i}-1}{\zeta^i -1} \neq 0$ if and only if $\ell \nmid (k+1)i$ for $i=1, \ldots, \ell-1$. Here we construct cyclotomic permutations from these classes which generalizes Theorem 5.2 ([@AAW:08]) and Corollary 2.3 ([@Zieve-2]).
\[exampleOfCongruenceOne\] Let $\ell$ be positive integer with $ q- 1 = \ell s$. For all $i=0, \ldots, \ell -1$, let $r_i, k_i, e_i, t_i$ be positive integers such that $(\ell, e_i) =1$ and $h_{k_i}(x) = 1 + x + \cdots + x^{k_i}$. Suppose $q = q_0^m \pmod{\ell}$ such that $q_0 \equiv 1 \pmod{\ell}$ and $\ell \mid m$. Then $$P(x) =
\left\{
\begin{array}{ll}
0, & if ~ x=0; \\
x^{r_0} h_{k_0}(x^{e_0 s})^{t_0}, & if~ x \in C_0; \\
x^{r_1} h_{k_1}(x^{e_1 s})^{t_1}, & if~ x \in C_1; \\
\vdots & \vdots \\
x^{r_{\ell-1}} h_{k_{\ell -1}} (x^{e_{\ell-1} s})^{t_{\ell-1}}, & if ~ x \in C_{\ell},
\end{array}
\right.$$ permutes ${\mathbb{F}_q}$ if and only if $\{r_i i \mid i =0, \ldots, \ell -1\}$ is a complete set of residues modulo $\ell$, $(r_i, s)=1$ for all $0\leq i \leq \ell-1$, $p \nmid k_0+1$, and $\ell \nmid i(k_i +1)$ for all $i=1, \ldots, \ell-1$.
Now we construct several classes of PPs obtained from Theorem \[specialMain\] such that $f_i(\zeta^i)^s$ ($i=0, \ldots, \ell-1$) are not necessarily the same. Again, the following result extends Theorem 4.4 in [@AW:07] and Theorem 1.3 in [@Zieve-2].
\[allOneBranchesCongruenceMinusOne\] Let $\ell$ be positive integer with $ q- 1 = \ell s$. For all $i=0, \ldots, \ell -1$, let $r_i, k_i, e_i, t_i, n_i$ be positive integers such that $(\ell, e_i) =1$. Put $h_{k^\prime_i}(x) = 1 + x + \cdots + x^{k^\prime_i}$ and $h_{k_i}(x) = 1 + x + \cdots + x^{k_i}$. Let $ \bar{k_i} = \ell/(\ell, k_i)$. Suppose $q = q_0^m$ such that $q_0 \equiv -1 \pmod{\ell}$ and $m$ is even. Pick $\hat{h}_i \in \mathbb{F}_{q_0} [x]$ and let $f_i (x) := h_{k^\prime_i}(x)^{t_i} \hat{h}_i (h_{k_i}(x)^{\bar{k_i}})$. Then $$P(x) =
\left\{
\begin{array}{ll}
0, & if ~ x=0; \\
x^{r_0} f_0(x^{e_0 s}), & if~ x \in C_0; \\
x^{r_1} f_1(x^{e_1 s}), & if~ x \in C_1; \\
\vdots & \vdots \\
x^{r_{\ell-1}} f_{\ell -1} (x^{e_{\ell-1} s}), & if ~ x \in C_{\ell},
\end{array}
\right.$$ permutes ${\mathbb{F}_q}$ if and only if $\{ (r_i+\frac{e_ik_i^\prime t_is}{2}) i \mid i =0, \ldots, \ell -1\}$ is a complete set of residues modulo $\ell$, $(r_i, s)=1$ and $f_i(\zeta^i) \neq 0$ for all $0\leq i \leq \ell-1$.
Let $m=2n$. We note that $\ell \mid q_0 +1$ implies that $\zeta^{q_0} = \zeta^{-1}$ and thus $h_{k_i^\prime}(\zeta^{ie_i})^{q_0-1} =$ $\left( \frac{ \zeta^{(k_i^\prime +1) i e_i q_0} - 1}{\zeta^{i e_i q_0} -1} \right)
\left (\frac{ \zeta^{i e_i} -1}{ \zeta^{(k_i^\prime +1) i e_i} - 1} \right)$ $= \zeta^{-k_i^\prime i e_i}$. Furthermore, $q_0 -1 \mid \frac{q_0^2-1}{q_0+1} \mid \frac{q-1}{q_0+1} \mid \frac{q-1}{\ell} =s$ implies that $h_{k_i^\prime}(\zeta^{ie_i})^{s} = \zeta^{-\frac{k_i^\prime i e_i s}{q_0-1}} = \zeta^{\frac{k_i^\prime i e_i s}{2}}$. Similarly, $h_{k_i}(\zeta^{ie_i})^{\bar{k_i} q_0} = \left( \frac{h_{k_i}(\zeta^{ie_i})}{\zeta^{k_i i e_i}} \right)^{\bar{k_i}} = h_{k_i}(\zeta^{ie_i})^{\bar{k_i}}$ implies that $h_{k_i}(\zeta^{ie_i})^{\bar{k_i}} \in \mathbb{F}_{q_0}$. Then the result follows from Theorem \[specialMain\], since we have $$\begin{aligned}
f_i(\zeta^{ie_i})^{\frac{q-1}{\ell}}&=& \left(h_{k^\prime_i}(\zeta^{ie_i})^{t_i} \right)^{\frac{q-1}{\ell}} \left( \hat{h}_i (h_{k_i}(\zeta^{ie_i})^{\bar{k_i}}) \right)^{\frac{q_0^{2n}-1}{\ell}}\\
&=& h_{k^\prime_i}(\zeta^{ie_i})^{t_i s} \left( \hat{h}_i (h_{k_i}(\zeta^{ie_i})^{\bar{k_i}} ) \right)^{\frac{q_0^{2}-1}{\ell} \left((q_0^{2})^{n-1}+(q_0^{2})^{n-2}+\cdots+1 \right)}\\
&=& \zeta^{\frac{i e_i k_i^\prime t_i s}{2}}\left( \prod_{j=0}^{n-1} \hat{h}_i (h_{k_i}(\zeta^{ie_i})^{\bar{k_i}} )^{q_0^{2j}} \right)^{\frac{q_0^{2}-1}{\ell}}\\
&=& \zeta^{\frac{i e_i k_i^\prime t_i s}{2}},\end{aligned}$$ as long as $f_i(\zeta^{ie_i}) \neq 0$.
Again, for $h(x) = 1 + x+ \cdots +x^k$, it is well known that $h(\zeta^0) = k+1 \neq 0$ if and only if $p \nmid (k+1)$ and that $h(\zeta^i) = \frac{\zeta^{(k+1)i}-1}{\zeta^i -1} \neq 0$ if and only if $\ell \nmid (k+1)i$ for $i=1, \ldots, \ell-1$. We therefore obtain a generalization of Theorem 4.4 ([@AW:07]) and Corollary 2.4 ([@Zieve-2]) as follows:
\[exampleOfAllOneBranches\] Let $\ell$ be positive integer with $ q- 1 = \ell s$. For all $i=0, \ldots, \ell -1$, let $r_i, k_i, e_i, t_i, n_i$ be positive integers such that $(\ell, e_i) =1$. Put $h_{k_i}(x) = 1 + x + \cdots + x^{k_i}$. Suppose $q = q_0^m$ such that $q_0 \equiv -1 \pmod{\ell}$ and $m$ is even. Then $$P(x) =
\left\{
\begin{array}{ll}
0, & if ~ x=0; \\
x^{r_0} h_{k_0}(x^{e_0 s})^{t_0}, & if~ x \in C_0; \\
x^{r_1} h_{k_1}(x^{e_1 s})^{t_1}, & if~ x \in C_1; \\
\vdots & \vdots \\
x^{r_{\ell-1}} h_{k_{\ell -1}} (x^{e_{\ell-1} s})^{t_{\ell-1}}, & if ~ x \in C_{\ell},
\end{array}
\right.$$ permutes ${\mathbb{F}_q}$ if and only if $\{ (r_i+\frac{e_ik_it_is}{2}) i \mid i =0, \ldots, \ell -1\}$ is a complete set of residue modulo $\ell$, $(r_i, s)=1$ for all $0\leq i \leq \ell-1$, $p \nmid k_0+1$, and $\ell \nmid i(k_i +1)$ for all $i=1, \ldots, \ell-1$.
Finally we take all branches as binomials and obtain a large class of PPs, which generalizes Theorem 3.1 [@AW:06] and Theorem 2.5 [@Zieve-2]. We note the necessary and sufficient description of a subclass of permutation binomials can be found in [@Wang; @Wang2].
\[binomialBranches\] Let $\ell$ be positive integer with $ q- 1 = \ell s$. Let $u_i > r_i >0$ and $a_i \in {\mathbb{F}_q}^*$ such that $\gcd(u_i-r_i, q-1) := s$ is a constant for all $i=0, \ldots, \ell-1$. Let $e_i := (u_i-r_i)/\ell$ and $\eta$ be a fixed primitive $2\ell$-th root of unity in the algebraic closure of ${\mathbb{F}_q}$ and $\zeta=\eta^2$. Suppose $(\eta^{i e_i} + a_i/\eta^{i e_i} )^s =1$ for each $i=0, \ldots, \ell-1$. Then $$P(x) =
\left\{
\begin{array}{ll}
0, & if ~ x=0; \\
x^{u_0} + a_0 x^{r_0}, & if~ x \in C_0; \\
x^{u_1} + a_1 x^{r_1}, & if~ x \in C_1; \\
\vdots & \vdots \\
x^{u_{\ell-1}} + a_{\ell-1} x^{r_{\ell-1}}, & if ~ x \in C_{\ell}.
\end{array}
\right.$$ permutes ${\mathbb{F}_q}$ if and only if $-a_i \neq \zeta^{i e_i}$ and $(r_i, s)=1$ for all $0\leq i \leq \ell-1$, $\{r_i i + \frac{e_i s i}{2} \mid i =0, \ldots, \ell -1\}$ is a complete set of residues modulo $\ell$.
Let $x^{u_i} + a_i x^{r_i} = x^{r_i} (x^{e_i s} + a)$. We have $$\begin{aligned}
(\zeta^{i e_i} + a)^{s}&=& (\eta^{2ie_i} + a)^s\\
&=& \eta^{ie_i s} \left( \eta^{ie_i} + a /\eta^{ie_i} \right)^s \\
&=& \eta^{ie_is} = \zeta^{ie_is/2}.\end{aligned}$$ The rest of proof follows easily from Theorem \[specialMain\].
conclusion
==========
In this paper we study permutation polynomials of finite fields in terms of cyclotomy. We provide both theoretical and algorithmic ways to generate permutation polynomials of finite fields. We have demonstrated how to construct concrete classes of PPs using our method. One can expect to generate more concrete classes of permutation polynomials by taking different polynomials as branches in our cyclotomic mapping construction. It is also expected to further extend our method to additive cyclotomy as studied in [@AGW:11; @YuanDing:12; @Zieve].
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School of Mathematics and Statistics, Carleton University, 1125 Colonel By Drive, Ottawa, Ontario, K1S 5B6, CANADA\
E–mail address: [wang@math.carleton.ca]{}
[^1]: Research of the authors was partially supported by NSERC of Canada
|
---
abstract: 'We consider Toeplitz operators defined on a concave corner-shaped subset of the square lattice. We obtain a necessary and sufficient condition for these operators to be Fredholm. We further construct a Fredholm concave corner Toeplitz operator of index one. By using this, a relation between Fredholm indices of quarter-plane and concave corner Toeplitz operators is clarified. As an application, topological invariants and corner states for some bulk-edges gapped Hamiltonians on two-dimensional (2-D) class AIII and 3-D class A systems with concave corners are studied. Explicit examples clarify that these topological invariants depend on the shape of the system. We discuss the Benalcazar–Bernevig–Hughes’ 2-D Hamiltonian and see that there still exists topologically protected corner states even if we break some symmetries as long as the chiral symmetry is preserved.'
address: 'Mathematics for Advanced Materials-OIL c/o AIMR Tohoku University, National Institute of Advanced Industrial Science and Technology, 2-1-1 Katahira, Aoba, Sendai 980-8577, Japan'
author:
- Shin Hayashi
title: Toeplitz operators on concave corners and topologically protected corner states
---
Introduction
============
Toeplitz operators and its index theory, which have been intensively studied in mathematics, are known to play an important role also in condensed matter physics. In this paper, we consider Toeplitz operators defined on a concave corner-shaped subset of the square lattice and study its index theory. We then apply these results to the study of topologically protected corner states on systems with codimension-two convex and concave corners. Benalcazar–Bernevig–Hughes’ 2-D model, which leads to the recent active study of higher-order topological insulators, is also studied from this viewpoint.
The topology of gapped Hamiltonians is known to be interesting from a physical point of view [@PS16]. One important aspect of [*topological insulators*]{} is the existence of topologically protected edge states while its bulk is gapped. For a quantum Hall system, its topological invariant, known as the TKNN number [@TKNN82], is defined as the first Chern number of the complex vector bundle (called the Bloch bundle) over the two-dimensional torus (called the Brillouin torus). Such edge states appear corresponding to this topology. This relation is proved by Hatsugai [@Hat93b] and is called the [*bulk-edge correspondence*]{}. Kellendonk–Richter–Schulz-Baldes explained this correspondence as an index theory for Toeplitz operators [@KRSB02; @KRSB00] and generalized it to disordered systems by using the noncommutative geometric technique developed by Connes and Bellissard [@BvES94; @Connes94]. Specifically, $K$-theory and index theory applied to the Toeplitz extension of the rotation $C^*$-algebra explains the bulk-edge correspondence for quantum Hall systems.
Apart from these studies, Toeplitz algebras associated with subsemigroups of abelian groups have been much studied [@BS06; @CD71; @Cu17; @Do73]. A cone of the square lattice is an example of such subsemigroups. Toeplitz operators defined on cones that appear as an intersection of two half-planes are called [*quarter-plane Toeplitz operators*]{} [@DH71; @Ji95; @Pa90; @Sim67]. Douglas–Howe studied these operators on a quarter-plane of a special shape by using the tensor product structure of the quarter-plane Toeplitz algebra [@DH71]. In this special case, Coburn–Douglas–Singer obtained an index formula to express a Fredholm index of a quarter-plane Toeplitz operator in a topological manner [@CDS72]. Park further developed Douglas–Howe’s technique to the case of general quarter-planes [@Pa90]. Combined with Jiang’s construction of Fredholm quarter-plane Toeplitz operators [@Ji95], boundary homomorphisms of $K$-theory for $C^*$-algebras associated with Park’s short exact sequence are computed. In this paper, we regard these cones (quarter-planes) as models of [*convex corners*]{}.
Since real materials have various shapes, to study the topology of Hamiltonians on systems of various shapes is a natural direction for further research. In [@Hayashi2], the index theory for quarter-plane Toeplitz operators is applied to the topological study of some gapped Hamiltonians on systems with codimension-two convex corners. It is shown that for gapped Hamiltonians that are gapped not just on the bulk but also on two edges, there exists a topological invariant that is related to corner states. In this paper, we refer this relation to the [*bulk-edge and corner correspondence*]{} [@Hayashi2]. These results are obtained by applying $K$-theory for $C^*$-algebras for the following quarter-plane Toeplitz extension obtained by Douglas–Howe and Park in [@DH71; @Pa90] (all symbols are defined in the main body of this paper): $$\label{seq1}
0 \to K({\hat{{\mathcal{H}}}^{\alpha,\beta}}) \to {\hat{{\mathcal{T}}}^{\alpha,\beta}}\overset{\hat{\gamma}}{\to} {\mathcal{S}^{\alpha, \beta}}\to 0.$$ The topological invariant for such a gapped bulk-edges Hamiltonian is defined as an element of some $K$-group of a $C^*$-algebra, and a boundary homomorphism of the six-term exact sequence associated with some short exact sequence of $C^*$-algebras relates these two. Moreover, in [@Hayashi2], a nontrivial example is obtained by using some tensor product construction.
Recently, topologically protected corner states are intensively studied in condensed matter physics [@BBH17a; @HWK17; @KPVW18] under the name of [*higher-order topological insulators*]{} [@Frank]. A trigger seems to be the Benalcazar–Bernevig–Hughes’ paper [@BBH17a]. They considered a specific 2-D (resp. 3-D) Hamiltonian on a square (resp. cube)-shaped domain. This system has four codimension-two (resp. eight codimension-three) convex corners of the special shape. It turns out that this system has corner states. In order to characterized these higher order phases, they proposed topological quantities named [*nested Wilson loops*]{}. On these studies, a role of some spatial symmetries is rather stressed [@BBH17a; @HF18; @Frank]
In this paper, we first study Toeplitz operators defined on a [*concave corner-shaped subset*]{} of the square lattice ${\mathbb{Z}}^2$. Such a concave corner appears as a union of two half-planes. We consider the $C^*$-algebra ${\check{{\mathcal{T}}}^{\alpha,\beta}}$ generated by the Toeplitz operators obtained by compressing the translation operators on ${\mathbb{Z}}^2$ onto the concave corner-shaped subset and show an extension of the following form (Theorem \[main\]): $$0 \to K({\check{{\mathcal{H}}}^{\alpha,\beta}}) \to {\check{{\mathcal{T}}}^{\alpha,\beta}}\overset{\check{\gamma}}{\to} {\mathcal{S}^{\alpha, \beta}}{\to} 0.$$ As a result, a necessary and sufficient condition for Fredholmness of concave corner Toeplitz operators is obtained (Theorem \[Fredholm\]). Further, we construct a nontrivial example of Fredholm concave corner Toeplitz operators of index one (Theorem \[construction\]). Comparing them with Jiang’s result [@Ji95], a relation between index theory for Toeplitz operators on convex corners (quarter-planes) and concave corners is clarified (Corollary \[relation\]). This result leads to a Coburn–Douglas–Singer-type index formula for Fredholm concave corner Toeplitz operators when the concave corner is of a special shape (Corollary \[concaveCDS\]). In the case of quarter-planes, a linear splitting of the sequence (\[seq1\]) is constructed by compressing half-plane Toeplitz operators onto quarter-planes [@Pa90]. However, when we study concave corners, they are a subset of neither half-planes nor subsemigroups of ${\mathbb{Z}}^2$, so compressions do not, at least directly, give a linear splitting. This is one technical difference between convex and concave cases, and so we adopt a slightly different approach although some discussions of previous results [@Ji95; @Pa90] still technically apply in concave cases. We first construct explicitly a rank-one projection as an element of the algebra ${\check{{\mathcal{T}}}^{\alpha,\beta}}$ and show that the compact operator algebra is contained in this algebra (Proposition \[contain\]). We then show that the quotient algebra ${\check{{\mathcal{T}}}^{\alpha,\beta}}/K({\check{{\mathcal{H}}}^{\alpha,\beta}})$ is isomorphic to the algebra ${\mathcal{S}^{\alpha, \beta}}$ (Proposition \[prop2\]). The surjectivity of the homomorphism $\check{\gamma}$ is proved by using the surjectivity of the homomorphism $\hat{\gamma}$ proved in [@Pa90] and specifying a dense subalgebra of ${\mathcal{S}^{\alpha, \beta}}$ (Lemma \[dense\]).
We next apply these results to the study of topologically protected corner states. In [@Hayashi2], only $3$-D class A systems with codimension-two convex corners are discussed, where the short exact sequence of Theorem \[main\] enables us to examine such corner states of systems with concave corners. In this paper, we mainly study $2$-D class AIII systems with codimension-two (convex and concave) corners. We consider Hamiltonians on the square lattice and assume that they are gapped at zero, not just on the bulk but also on two edges. For such gapped Hamiltonians, we define a topological invariant as an element of some $K$-group (Definition \[gappedinvAIII\]). We also define another topological invariant for a corner Hamiltonian that is related to corner states (Definition \[gaplessinvAIII\]) and show a relation between these two invariants (Theorem \[BECCAIII\]). Integer-valued numerical corner invariants are defined by using traces on $K({\hat{{\mathcal{H}}}^{\alpha,\beta}})$ and $K({\check{{\mathcal{H}}}^{\alpha,\beta}})$. When we consider two edges, we can associate convex and concave corners (see Fig. \[convconc\]). Correspondingly, we can define two numerical corner invariants under our assumption. We show that these two numerical corner invariants are different by the multiplication by $-1$ (Theorem \[minusAIII\]). Through this relation, the Coburn–Douglas–Singer index formula [@CDS72] and its concave corner analogue (Corollary \[concaveCDS\]) gives a topological method to compute numerical corner invariants from gapped bulk-edges Hamiltonians. We also see that if the rank of the space of the internal degree of freedom is two, then our corner topological invariants are necessarily zero (Proposition \[remrank\]). Thus, in order to find a nontrivial example, its rank must be greater than or equal to four.
We further give a construction of explicit examples by using tensor products, as in [@Hayashi2]. We construct some gapped Hamiltonians from two Hamiltonians of $1$-D class AIII (conventional) topological insulators, and the numerical convex corner invariant is given as a product of topological numbers of these two (Theorem \[prodthmAIII\]). By using this construction, we provide an explicit example of Hamiltonians with nontrivial convex and concave corner invariants (Sect. $5$). This example clarifies that these corner invariants may change depending on the shape of the system. Actually, the example discussed there corresponds to the 2-D Hamiltonian discussed by Benalcazar–Bernevig–Hughes in [@BBH17a] (Equation (6) of [@BBH17a]. We refer this model to the [*2-D BBH model*]{}) when we take parameters in some specific way. Based on the chiral symmetry, we define an integer-valued topological invariant for the 2-D BBH model and compute it. The bulk-edge and corner correspondence gives another explanation of the existence of topologically protected corner states for this model. While a role of spatial symmetries is much discussed in studies of higher-order topological insulators [@BBH17a], our method does not require any spatial symmetry. Through an example, we see that topologically protected corner states remain even if we break some symmetries which the 2-D BBH model originally have as long as the chiral symmetry is preserved. Some corresponding results in the case of 3-D class A systems are also collected in Sect. $4.2$.
This paper is organized as follows. In Sect. $2$, we define concave corner Toeplitz operators and introduce the $C^*$-algebra ${\check{{\mathcal{T}}}^{\alpha,\beta}}$ generated by these operators. In this section, we show a short exact sequence and obtain a necessary and sufficient condition for concave corner Toeplitz operators to be Fredholm. In Sect. $3$, we construct an explicit example of a concave corner Fredholm Toeplitz operator of index one and collects some of its consequences. In Sect. $4$, we apply these result to the study of topologically protected corner states. We mainly treat 2-D class AIII systems, though the results for 3-D class A systems are also collected. In Sect. $5$, we consider an explicit example of 2-D class AIII Hamiltonian whose corner invariant is nontrivial on a system with a codimension-two (convex and concave) corner. We also discuss the 2-D BBH model from our viewpoint there.
Concave corner Toeplitz algebras and their extension
====================================================
In this paper, we mainly consider concave corners, that is, corners whose angles are strictly greater than $\pi$. In particular, we study an index theory for Toeplitz operators defined on concave corners. In this section, we define such operators and study their properties. Specifically, we consider a $C^*$-algebra generated by concave corner Toeplitz operators and show a short exact sequence that clarifies a necessary and sufficient condition for these operators to be Fredholm. In this paper, we use only basics about $K$-theory for $C^*$-algebras. Details can be found in [@Bl98; @HR00; @Mur90; @RLL00], for example.
Setup
-----
Let ${\mathcal{H}}$ be the Hilbert space $l^2({\mathbb{Z}}^2)$. For a pair of integers $(m, n)$, let ${ e_{m,n}}$ be the element of ${\mathcal{H}}$ that is $1$ at $(m,n)$ and $0$ elsewhere. For $(m, n) \in {\mathbb{Z}}^2$, let $M_{m,n} \colon {\mathcal{H}}\to {\mathcal{H}}$ be the translation operator defined by $(M_{m,n}\varphi)(k,l) = \varphi(k-m, l-n).$[^1] We choose real numbers $\alpha < \beta$, and let ${{\mathcal{H}}^{\alpha}}$ and ${{\mathcal{H}}^{\beta}}$ be the closed subspaces of ${\mathcal{H}}$ spanned by $\{ { e_{m,n}} \mid -\alpha m + n \geq 0 \}$ and $\{ { e_{m,n}} \mid -\beta m + n \leq 0 \}$, respectively. ${{\mathcal{H}}^{\alpha}}$ and ${{\mathcal{H}}^{\beta}}$ model half-planes distinguished by lines $y = \alpha x$ and $y = \beta x$ (see the left-hand side of Fig. \[convconc\]). We here consider two models of spaces with codimension-two boundaries, which we call [*corners*]{}. One is an intersection of two half-planes, and the other is a union of these two. We refer to these two as a [*convex corner*]{} and a [*concave corner*]{}, respectively[^2] (see Fig. \[convconc\]). Specifically, let $\hat{\Sigma} := \{ (x,y) \in {\mathbb{Z}}^2 \mid -\alpha x + y \geq 0 \ \text{and} -\beta x + y \leq 0 \}$, and let ${\hat{{\mathcal{H}}}^{\alpha,\beta}}$ be the closed subspace of ${\mathcal{H}}$ spanned by elements in the set $\{ {{\bm e}}_{x,y} \mid (x,y) \in \hat{\Sigma} \}$. Note that the Hilbert space ${\hat{{\mathcal{H}}}^{\alpha,\beta}}$ is intersection ${{\mathcal{H}}^{\alpha}}\cap {{\mathcal{H}}^{\beta}}$ of ${{\mathcal{H}}^{\alpha}}$ and ${{\mathcal{H}}^{\beta}}$. We regard ${\hat{{\mathcal{H}}}^{\alpha,\beta}}$ as a model of a convex corner. Let ${\hat{P}^{\alpha,\beta}}$ be the orthogonal projection of ${\mathcal{H}}$ onto ${\hat{{\mathcal{H}}}^{\alpha,\beta}}$. Note that ${\hat{P}^{\alpha,\beta}}= {P^{\alpha}}{P^{\beta}}= {P^{\beta}}{P^{\alpha}}$. Let $\check{\Sigma} := \{ (x,y) \in {\mathbb{Z}}^2 \mid -\alpha x + y \geq 0 \ \text{or} -\beta x + y \leq 0 \}$, and let ${\check{{\mathcal{H}}}^{\alpha,\beta}}$ be the closed subspace of ${\mathcal{H}}$ spanned by elements in the set $\{ {{\bm e}}_{x,y} \mid (x,y) \in \check{\Sigma} \}$. We regard ${\check{{\mathcal{H}}}^{\alpha,\beta}}$ as a model of a concave corner. Let ${\check{P}^{\alpha,\beta}}$ be the orthogonal projection of ${\mathcal{H}}$ onto ${\check{{\mathcal{H}}}^{\alpha,\beta}}$. Note that ${\check{P}^{\alpha,\beta}}= {P^{\alpha}}+ {P^{\beta}}- {P^{\alpha}}{P^{\beta}}$. In what follows, we consider operators on these Hilbert spaces. The real numbers $\alpha$ and $\beta$ correspond to the slope of two edges (Fig. \[convconc\]). We can take $\alpha = - \infty$ or $\beta = +\infty$, but not both (if $\alpha = - \infty$ and $\beta = +\infty$, the “corner” will be the “edge”). If we fix $\alpha$ and $\beta$, we can consider two types of corners, that is, convex and concave corners. In this paper, we treat both of these cases[^3].
\[othercases\] In the main body of this paper, we just treat the case in which the corner (or edges) includes lattice points on lines $y = \alpha x$ and $y = \beta x$. We can consider variants that do not contain these points. For these cases, the results of this paper still hold. Some results in these cases are collected in the appendix of this paper.
![A convex corner (left) and a concave corner (right) correspond to shaded area[]{data-label="convconc"}](1.eps){width="110mm"}
The [*quarter-plane Toeplitz $C^*$-algebra*]{} [@DH71; @Pa90] is defined to be the $C^*$-subalgebra ${\hat{{\mathcal{T}}}^{\alpha,\beta}}$ of $B({\hat{{\mathcal{H}}}^{\alpha,\beta}})$ generated by $\{ {\hat{P}^{\alpha,\beta}}M_{m,n} {\hat{P}^{\alpha,\beta}}\mid (m,n) \in {\mathbb{Z}}^2 \}$. Similarly, we define the [*concave corner Toeplitz $C^*$-algebra*]{} to be the $C^*$-subalgebra ${\check{{\mathcal{T}}}^{\alpha,\beta}}$ of $B({\check{{\mathcal{H}}}^{\alpha,\beta}})$ generated by $\{ {\check{P}^{\alpha,\beta}}M_{m,n} {\check{P}^{\alpha,\beta}}\mid (m,n) \in {\mathbb{Z}}^2 \}$. We also define the [*half-plane Toeplitz $C^*$-algebras*]{} ${{\mathcal{T}}^{\alpha}}$ and ${{\mathcal{T}}^{\beta}}$ to be $C^*$-subalgebras of $B({{\mathcal{H}}^{\alpha}})$ and $B({{\mathcal{H}}^{\beta}})$ generated by $\{ P^\alpha M_{m,n} P^\alpha \mid (m,n) \in {\mathbb{Z}}^2 \}$ and $\{ P^\beta M_{m,n} P^\beta \mid (m,n) \in {\mathbb{Z}}^2 \}$, respectively. Let ${{\mathcal{C}}^\alpha}$, ${{\mathcal{C}}^\beta}$ and ${\check{{\mathcal{C}}}^{\alpha,\beta}}$ be the commutator ideals of ${{\mathcal{T}}^{\alpha}}$, ${{\mathcal{T}}^{\beta}}$ and ${\check{{\mathcal{T}}}^{\alpha,\beta}}$, respectively. As is shown in [@CD71], we have surjective $*$-homomorphisms ${\sigma^\alpha}\colon{{\mathcal{T}}^{\alpha}}\to C({\mathbb{T}}^2)$ and ${\sigma^\beta}\colon {{\mathcal{T}}^{\beta}}\to C({\mathbb{T}}^2)$ that map $P^\alpha M_{m,n} P^\alpha$ to $\chi_{m,n}$ and $P^\beta M_{m,n} P^\beta$ to $\chi_{m,n}$, respectively, where $\chi_{m,n}(\xi,\eta) = \xi^m \eta^n$. As in [@Pa90], we define a $C^*$-algebra $\mathcal{S}^{\alpha, \beta}$ to be the pullback of ${{\mathcal{T}}^{\alpha}}$ and ${{\mathcal{T}}^{\beta}}$ along $C({\mathbb{T}}^2)$, that is, ${\mathcal{S}^{\alpha, \beta}}:= \{ (T^\alpha, T^\beta) \in {{\mathcal{T}}^{\alpha}}\oplus {{\mathcal{T}}^{\beta}}\mid {\sigma^\alpha}(T^\alpha) = {\sigma^\beta}(T^\beta) \}$. As is shown in [@Pa90], we have surjective $*$-homomorphisms $\hat{\gamma}^\alpha \colon {\hat{{\mathcal{T}}}^{\alpha,\beta}}\to {{\mathcal{T}}^{\alpha}}$ and $\hat{\gamma}^\beta \colon {\hat{{\mathcal{T}}}^{\alpha,\beta}}\to {{\mathcal{T}}^{\beta}}$ that map ${\hat{P}^{\alpha,\beta}}M_{m,n} {\hat{P}^{\alpha,\beta}}$ to ${P^{\alpha}}M_{m,n} {P^{\alpha}}$ and ${\hat{P}^{\alpha,\beta}}M_{m,n} {\hat{P}^{\alpha,\beta}}$ to ${P^{\beta}}M_{m,n} {P^{\beta}}$, respectively. By using these two, we obtain surjective $*$-homomorphism $\hat{\gamma} \colon {\hat{{\mathcal{T}}}^{\alpha,\beta}}\to {\mathcal{S}^{\alpha, \beta}}$ given by $\hat{\gamma}(T) = (\hat{\gamma}^\alpha(T), \hat{\gamma}^\beta(T))$. We write $p^\alpha \colon {\mathcal{S}^{\alpha, \beta}}\to {{\mathcal{T}}^{\alpha}}$ and $p^\beta \colon {\mathcal{S}^{\alpha, \beta}}\to {{\mathcal{T}}^{\beta}}$ for the $*$-homomorphisms given by projections onto each component. $$\label{Sab}
\vcenter{
\xymatrix{
{\mathcal{S}^{\alpha, \beta}}\ar[r]^{p^\beta} \ar[d]_{p^\alpha}& {{\mathcal{T}}^{\beta}}\ar[d]^{{\sigma^\beta}}
\\
{{\mathcal{T}}^{\alpha}}\ar[r]^{{\sigma^\alpha}} & C({\mathbb{T}}^2).
}}\vspace{-1mm}$$ We write $\sigma$ for the composition ${\sigma^\alpha}\circ p^\alpha = {\sigma^\beta}\circ p^\beta$.
Note that the dense subalgebras of ${{\mathcal{T}}^{\alpha}}$, ${{\mathcal{T}}^{\beta}}$, ${\hat{{\mathcal{T}}}^{\alpha,\beta}}$ and ${\check{{\mathcal{T}}}^{\alpha,\beta}}$ consist of the following operators: $$\label{densea}
\text{For} \ \ {{\mathcal{T}}^{\alpha}}\ \colon \ \sum_{i=1}^l c_i {P^{\alpha}}M_{m_{i0}, n_{i0}} \biggl( \prod_{j=1}^{k_i} {P^{\alpha}}M_{m_{ij}, n_{ij}} \biggl) {P^{\alpha}},$$ $$\label{denseb}
\text{For} \ \ {{\mathcal{T}}^{\beta}}\ \colon \ \sum_{i=1}^l c_i {P^{\beta}}M_{m_{i0}, n_{i0}} \biggl( \prod_{j=1}^{k_i} {P^{\beta}}M_{m_{ij}, n_{ij}} \biggl) {P^{\beta}},$$ $$\label{denseab}
\text{For} \ \ {\hat{{\mathcal{T}}}^{\alpha,\beta}}\ \colon \ \sum_{i=1}^l c_i {\hat{P}^{\alpha,\beta}}M_{m_{i0}, n_{i0}} \biggl( \prod_{j=1}^{k_i} {\hat{P}^{\alpha,\beta}}M_{m_{ij}, n_{ij}} \biggl) {\hat{P}^{\alpha,\beta}},$$ $$\label{denses}
\text{For} \ \ {\check{{\mathcal{T}}}^{\alpha,\beta}}\ \colon \ \sum_{i=1}^l c_i {\check{P}^{\alpha,\beta}}M_{m_{i0}, n_{i0}} \biggl( \prod_{j=1}^{k_i} {\check{P}^{\alpha,\beta}}M_{m_{ij}, n_{ij}} \biggl) {\check{P}^{\alpha,\beta}},$$ where $c_i \in {\mathbb{C}}$.
\[dense\] A dense subalgebra of ${\mathcal{S}^{\alpha, \beta}}$ consists of the pairs of operators of the following form: $$\label{densess}
\biggl( \sum_{i=1}^l \hspace{-0.5mm} c_i {P^{\alpha}}\hspace{-0.5mm} M_{m_{i0}\hspace{-0.3mm}, n_{i0}} \hspace{-0.5mm} \biggl(\prod_{j=1}^{k_i} \hspace{-0.5mm} {P^{\alpha}}\hspace{-0.5mm} M_{m_{ij} \hspace{-0.3mm},n_{ij}} \hspace{-1mm} \biggl) \hspace{-0.5mm} {P^{\alpha}}\hspace{-0.5mm}, \hspace{-0.5mm} \sum_{i=1}^l \hspace{-0.5mm} c_i {P^{\beta}}\hspace{-0.5mm} M_{m_{i0}\hspace{-0.3mm}, n_{i0}} \hspace{-0.5mm} \biggl( \prod_{j=1}^{k_i} \hspace{-0.5mm} {P^{\beta}}\hspace{-0.5mm} M_{m_{ij}\hspace{-0.3mm}, n_{ij}} \biggl) {P^{\beta}}\hspace{-0.5mm} \biggl)$$ where $c_i \in {\mathbb{C}}$.
As is shown in [@Pa90], we have a surjective $*$-homomorphism $\hat{\gamma} \colon {\hat{{\mathcal{T}}}^{\alpha,\beta}}\rightarrow {\mathcal{S}^{\alpha, \beta}}$. An image of a dense subalgebra of the algebra ${\hat{{\mathcal{T}}}^{\alpha,\beta}}$ under the surjective $*$-homomorphism $\hat{\gamma}$ is a dense subalgebra of ${\mathcal{S}^{\alpha, \beta}}$. A dense subalgebra of ${\hat{{\mathcal{T}}}^{\alpha,\beta}}$ consists of operators of the form (\[denseab\]). For an operator $\hat{T}$ of the form (\[denseab\]), $$\begin{gathered}
\hat{\gamma}(\hat{T}) = (\hat{\gamma}^\alpha(\hat{T}), \hat{\gamma}^\beta(\hat{T}))=\\
\biggl( \sum_{i=1}^l \hspace{-0.5mm} c_i {P^{\alpha}}\hspace{-0.5mm} M_{m_{i0}\hspace{-0.3mm}, n_{i0}} \hspace{-0.5mm} \biggl(\prod_{j=1}^{k_i} \hspace{-0.5mm} {P^{\alpha}}\hspace{-0.5mm} M_{m_{ij} \hspace{-0.3mm},n_{ij}} \hspace{-1mm} \biggl) \hspace{-0.5mm} {P^{\alpha}}\hspace{-0.5mm}, \hspace{-0.5mm} \sum_{i=1}^l \hspace{-0.5mm} c_i {P^{\beta}}\hspace{-0.5mm} M_{m_{i0}\hspace{-0.3mm}, n_{i0}} \hspace{-0.5mm} \biggl( \prod_{j=1}^{k_i} \hspace{-0.5mm} {P^{\beta}}\hspace{-0.5mm} M_{m_{ij}\hspace{-0.3mm}, n_{ij}} \biggl) {P^{\beta}}\hspace{-0.5mm} \biggl).\vspace{-2mm}\end{gathered}$$ Thus, the pairs of operators of this form compose a dense subalgebra of ${\mathcal{S}^{\alpha, \beta}}$.
Surjective $*$-homomorphisms from ${\check{{\mathcal{T}}}^{\alpha,\beta}}$ to ${{\mathcal{T}}^{\alpha}}$ and ${{\mathcal{T}}^{\beta}}$
--------------------------------------------------------------------------------------------------------------------------------------
In this subsection, we construct $*$-homomorphisms from ${\check{{\mathcal{T}}}^{\alpha,\beta}}$ to ${{\mathcal{T}}^{\alpha}}$ and ${{\mathcal{T}}^{\beta}}$. We basically follow the proof of Proposition $1.2$ of [@Pa90], which treats convex corners, but some points should be modified in our concave case. We first prepare the following lemma.
\[lem2\] Let $\{ (m_i, n_i ) \}$ be a finite collection of pairs of integers. Then, there exists a pair of integers $(r,s)$ such that, for all $i$,
- $-\alpha (m_i - r) + (n_i - s) \geq 0$ if and only if $-\alpha m_i + n_i \geq 0$,
- $-\beta(m_i - r) + (n_i - s) > 0$.
We choose $\epsilon > 0$, $M > 0$ so that $\epsilon < \min \{ \alpha m_i - n_i \mid -\alpha m_i + n_i < 0 \}$ and $-M \leq \min \{ -\beta m_i + n_i \}$. Then, it suffices to show that there exist some integers $r$ and $s$ such that $$0 \leq \alpha r -s < \epsilon \qquad \text{and} \qquad -\beta r + s < -M.$$
As in [@Pa90], we here use the following result contained in [@HW08]: there exists a positive integer $r$ and an integer $s$ such that $$0 \leq \alpha - \frac{s}{r} < \frac{1}{r^2} \qquad \text{and} \qquad r > \max \biggl\{ \frac{1}{\epsilon}, \frac{M}{\beta - \alpha} \biggl\}.$$ For such $r$ and $s$, we have $$0 \leq \alpha r - s < \frac{1}{r} < \epsilon,
\vspace{-1mm}$$ and $$\vspace{-1mm}
-\beta r + s = -(\beta - \alpha)r + (-\alpha r + s) \leq -(\beta - \alpha) r \leq -M.$$ as desired.
\[prop1\] There exists surjective $*$-homomorphisms $${\check{\gamma}^\alpha}\colon {\check{{\mathcal{T}}}^{\alpha,\beta}}\to {{\mathcal{T}}^{\alpha}}, \ \ {\check{\gamma}^\beta}\colon {\check{{\mathcal{T}}}^{\alpha,\beta}}\to {{\mathcal{T}}^{\beta}}.$$
For $\check{T} = \sum_{i=1}^l c_i {\check{P}^{\alpha,\beta}}M_{m_{i0}, n_{i0}} \bigl( \prod_{j=1}^{k_i} {\check{P}^{\alpha,\beta}}M_{m_{ij}, n_{ij}} \bigl) {\check{P}^{\alpha,\beta}}$, we set $$\label{image}
{\check{\gamma}^\alpha}(\check{T}) := \sum_{i=1}^l c_i {P^{\alpha}}M_{m_{i0}, n_{i0}} \biggl( \prod_{j=1}^{k_i} {P^{\alpha}}M_{m_{ij}, n_{ij}} \biggl) {P^{\alpha}}.
\vspace{-2mm}$$ and $${\check{\gamma}^\beta}(\check{T}) := \sum_{i=1}^l c_i {P^{\beta}}M_{m_{i0}, n_{i0}} \biggl( \prod_{j=1}^{k_i} {P^{\beta}}M_{m_{ij}, n_{ij}} \biggl) {P^{\beta}}.$$ To show that ${\check{\gamma}^\alpha}$ and ${\check{\gamma}^\beta}$ are well-defined and extend to $*$-homomorphisms on ${\check{{\mathcal{T}}}^{\alpha,\beta}}$, it is sufficient to show $\| {\check{\gamma}^\alpha}(\check{T}) \| \leq \| \check{T} \|$ and $\| {\check{\gamma}^\beta}(\check{T}) \| \leq \| \check{T} \|$. We here discuss ${\check{\gamma}^\alpha}$ only. The result for ${\check{\gamma}^\beta}$ is proved in almost the same way.
Let $\epsilon > 0$. We take $f \in {{\mathcal{H}}^{\alpha}}$ such that $f$ has a finite support, $\| f \|= 1$ and $\| {\check{\gamma}^\alpha}(\check{T}) \| \leq \| {\check{\gamma}^\alpha}(\check{T}) f \| + \epsilon$. Let $S$ be the union of the set ${\mathrm{supp}}(f)$ and the following set $$\left\{ \biggl( m_0 + \sum_{j=N}^{k_i} m_{ij}, n_0 + \sum_{j=N}^{k_i} n_{ij} \biggl) \ \Biggl\vert
\begin{array}{ll}
(m_0, n_0) \in {\mathrm{supp}}(f),\\
\ 0 \leq i \leq l, 0 \leq N \leq k_i
\end{array}
\right\}.$$ The set $S$ is a finite subset of ${\mathbb{Z}}^2$. Applying Lemma \[lem2\] to the set $S$, we obtain a pair $(r, s)$ of integers such that for any $(m, n) \in S$, we have
- $-\alpha (m-r) + (n-s) \geq 0$ if and only if $-\alpha m + n \geq 0$,
- $-\beta(m-r) + (n - s) > 0$.
This leads to the following relation:
- ${\check{\gamma}^\alpha}(\check{T})M_{-r, -s} f = M_{-r, -s} {\check{\gamma}^\alpha}(\check{T}) f$,
- ${\check{\gamma}^\alpha}(\check{T})M_{-r, -s} f = \check{T} M_{-r, -s} f$.
By using this, we have $$\begin{aligned}
\| {\check{\gamma}^\alpha}(\check{T}) \| &\leq \| {\check{\gamma}^\alpha}(\check{T})f \| + \epsilon
= \| M_{-r,-s} {\check{\gamma}^\alpha}(\check{T}) f \| + \epsilon \\
&= \| {\check{\gamma}^\alpha}(\check{T}) M_{-r,-s} f \| + \epsilon
= \| \check{T} M_{-r,-s} f \| + \epsilon \\
&\leq \| \check{T} \| \| M_{-r,-s} \| \| f \| + \epsilon
= \| \check{T} \| + \epsilon.\end{aligned}$$ Thus, $\| {\check{\gamma}^\alpha}(\check{T}) \| \leq \| \check{T} \|$ holds.
Since ${\check{\gamma}^\alpha}$ is a $*$-homomorphism and operators of the form $(\ref{image})$ compose a dense subalgebra of ${{\mathcal{T}}^{\alpha}}$, the map ${\check{\gamma}^\alpha}$ is surjective.
Since ${\check{\gamma}^\alpha}\circ {\sigma^\alpha}= {\check{\gamma}^\beta}\circ {\sigma^\beta}$, we have a $*$-homomorphism $\check{\gamma} \colon {\check{{\mathcal{T}}}^{\alpha,\beta}}\to {\mathcal{S}^{\alpha, \beta}}$ given by $\check{\gamma}(\check{T}) = ({\check{\gamma}^\alpha}(\check{T}), {\check{\gamma}^\beta}(\check{T}))$. By Lemma \[dense\], the map $\check{\gamma}$ is surjective.
$K({\check{{\mathcal{H}}}^{\alpha,\beta}}) \subset {\check{{\mathcal{T}}}^{\alpha,\beta}}$
------------------------------------------------------------------------------------------
For $(x,y) \in \check{\Sigma}$, let $p_{x,y}$ be the orthogonal projection of ${\check{{\mathcal{H}}}^{\alpha,\beta}}$ onto ${\mathbb{C}}{{\bm e}}_{x,y}$. In this subsection, we show the following proposition by constructing explicit rank-one projections contained in the algebra ${\check{{\mathcal{T}}}^{\alpha,\beta}}$.
\[contain\] $K({\check{{\mathcal{H}}}^{\alpha,\beta}}) \subset {\check{{\mathcal{T}}}^{\alpha,\beta}}$. Moreover, $K({\check{{\mathcal{H}}}^{\alpha,\beta}})$ is contained in $\operatorname{\mathrm{Ker}}\check{\gamma}$.
To show this proposition, we employ a trick by Jiang [@Ji95]. We consider the action of $SL(2, {\mathbb{Z}})$ onto ${\mathbb{Z}}^2$. An action of $g \in SL(2,{\mathbb{Z}})$ maps a line through the origin whose slope is $s$ to the line through the origin of possibly different slope. We write $g(s)$ for its slope. It is shown in Sect. $1$ of [@Ji95] that there is a $g \in SL(2,{\mathbb{Z}})$ such that $0 < g(\alpha) \leq \frac{1}{2}$ and $1 \leq g(\beta) < + \infty$. The action of $g$ induces a unitary isomorphism between Hilbert spaces ${\check{{\mathcal{H}}}^{\alpha,\beta}}$ and $\check{{\mathcal{H}}}^{g(\alpha),g(\beta)}$ and thus induces an isomorphism between $C^*$-algebras ${\check{{\mathcal{T}}}^{\alpha,\beta}}$ and $\check{{\mathcal{T}}}^{g(\alpha),g(\beta)}$ without changing their Fredholm index theory. Thus, we assume the following condition without loss of generality: $$0 < \alpha \leq \frac{1}{2} , \quad 1 \leq \beta < + \infty. \tag{$\dagger$}$$
For $(m,n) \in {\mathbb{Z}}^2$, let $\check{{\mathcal{P}}}_{m,n} := {\check{P}^{\alpha,\beta}}M_{m,n} {\check{P}^{\alpha,\beta}}M_{-m,-n} {\check{P}^{\alpha,\beta}}$. The operator $\check{{\mathcal{P}}}_{m,n}$ is a projection contained in ${\check{{\mathcal{T}}}^{\alpha,\beta}}$ (e.g., the projection $1 - \check{{\mathcal{P}}}_{-1,0}$ is explained in Fig. \[P-10\]). For $k \in \{ 1, 2, \cdots \}$, let $${\tilde{\mathcal{P}}}_k := (1-\check{{\mathcal{P}}}_{-1,0})\check{{\mathcal{P}}}_{-k-1,0} - (1- \check{{\mathcal{P}}}_{-1,0})\check{{\mathcal{P}}}_{-k,0} \in {\check{{\mathcal{T}}}^{\alpha,\beta}},$$ and let $B_k := \{ (x, y) \in {\mathbb{Z}}^2 \mid 0 \leq -\alpha x + y < \alpha \ \text{and} \ k\beta < -\beta x + y \leq (k+1)\beta \}$. Then, ${\tilde{\mathcal{P}}}_k$ is the orthogonal projection of ${\check{{\mathcal{H}}}^{\alpha,\beta}}$ onto the closed subspace spanned by elements in the set $\{ {{\bm e}}_{x,y} \mid (x, y) \in B_k \}$ (the projection ${\tilde{\mathcal{P}}}_3$ is explained in Fig. \[N3\]).
![The case of $\frac{1}{4} < \alpha < \frac{1}{3}$ and $1 < \beta < \infty$. $1- \check{{\mathcal{P}}}_{-1,0}$ is the orthogonal projection onto closed subspace corresponding to lattice points contained in the shaded area[]{data-label="P-10"}](2.eps){width="80mm"}
![The case of $\frac{1}{4} < \alpha < \frac{1}{3}$ and $1 < \beta < \infty$. ${\tilde{\mathcal{P}}}_3$ is the orthogonal projection $p_{-4,-1}$ onto closed subspace ${\mathbb{C}}{{\bm e}}_{-4,-1}$. The set $B_3$ contains just one element, $(-4,-1)$, which corresponds to the lattice point contained in the shaded area[]{data-label="N3"}](3.eps){width="90mm"}
For $\alpha$ satisfying the condition ($\dagger$), there exists a unique $N \in \{ 2,3, \cdots \}$ such that $\frac{1}{N+1} < \alpha \leq \frac{1}{N}$. We show some ${\tilde{\mathcal{P}}}_k$ is a rank-one projection. The statement is divided into five cases corresponding to the values of $\alpha$ and $\beta$.
\[Ngeneral\] Let $N \geq 2$.
1. When $\alpha = \frac{1}{2}$ and $\beta = 1$, we have ${\tilde{\mathcal{P}}}_1 = p_{-4,-2}$.
2. When $N \geq 3$, $\alpha = \frac{1}{N}$ and $\beta = 1$, we have ${\tilde{\mathcal{P}}}_{N-2} = p_{-N,-1}$.
3. When $\alpha = \frac{1}{N}$ and $1 < \beta < \infty$, we have ${\tilde{\mathcal{P}}}_{N-1} = p_{-N,-1}$.
4. When $\frac{1}{N+1} < \alpha < \frac{1}{N}$ and $\beta = 1$, we have ${\tilde{\mathcal{P}}}_{N-1} = p_{-N-1,-1}$.
5. When $\frac{1}{N+1} < \alpha < \frac{1}{N}$ and $1 < \beta < \infty$, we have ${\tilde{\mathcal{P}}}_N = p_{-N-1,-1}$.
It is sufficient to show that, in each case, the set $B_k$ contains just one element $(x, y)$, where $k$ and $(x, y)$ correspond to subscripts of ${\tilde{\mathcal{P}}}_k$ and $p_{x,y}$ indicated above. The proof of 1) $\sim$ 5) goes almost in the same way, but note that it is convenient to distinguish the cases of $N = 2$ and $N \geq 3$ also in the case of 3) $\sim$ 5). We here present just the proof of 5) for the case of $N \geq 3$.
Let $N \geq 3$. We calculate the set $B_N$, i.e., all values of $(x, y) \in {\mathbb{Z}}^2$ satisfying inequalities $0 \leq -\alpha x + y < \alpha$ and $N\beta < -\beta x + y \leq (N+1)\beta$, and show that it is just one point, $(-N-1, -1)$. From these two inequalities, we obtain the following inequality: $$(-N-1)\frac{\alpha \beta}{\beta - \alpha} \leq y < (-N+1)\frac{\alpha \beta}{\beta - \alpha} < 0.
\vspace{-1mm}$$ Since $N \geq 3$, $\alpha < \frac{1}{N}$ and $1 < \beta$, the left-hand side is strictly greater than $-2$. Thus, an integer $y$ should be $-1$. When $y = -1$, we have $-1 - \frac{1}{\alpha} < x \leq - \frac{1}{\alpha}$. Since $\frac{1}{N+1} < \alpha < \frac{1}{N}$, $x$ should be $-N-1$. The point $(-N-1, -1)$ satisfies the desired inequality, and so $B_N = \{ (-N-1, -1) \}$.
[*Proof of Proposition \[contain\]*]{} By using Lemma \[Ngeneral\], we see that the algebra ${\check{{\mathcal{T}}}^{\alpha,\beta}}$ contains at least a rank-one projection $p_{x,y}$ for some $(x,y) \in \check{\Sigma}$. For any $(u,v) \in \check{\Sigma}$, we have $$p_{u,v} = ( {\check{P}^{\alpha,\beta}}M_{u-x,v-y} {\check{P}^{\alpha,\beta}}) p_{x,y} ({\check{P}^{\alpha,\beta}}M_{u-x,v-y} {\check{P}^{\alpha,\beta}})^* \in {\check{{\mathcal{T}}}^{\alpha,\beta}}.$$ Thus, ${\check{{\mathcal{T}}}^{\alpha,\beta}}$ contains rank-one projections $p_{u,v}$ for any $(u,v) \in \check{\Sigma}$ and thus contains operators of the form ${\check{P}^{\alpha,\beta}}M_{u-x,v-y} {\check{P}^{\alpha,\beta}}p_{x,y}$ for any $(u,v), (x,y) \in \check{\Sigma}$. By using this, we can see that every rank-one projection on ${\check{{\mathcal{H}}}^{\alpha,\beta}}$ is contained in ${\check{{\mathcal{T}}}^{\alpha,\beta}}$, and thus ${\check{{\mathcal{T}}}^{\alpha,\beta}}$ contains all finite-rank operators on ${\check{{\mathcal{H}}}^{\alpha,\beta}}$. Thus, the inclusion $K({\check{{\mathcal{H}}}^{\alpha,\beta}}) \subset {\check{{\mathcal{T}}}^{\alpha,\beta}}$ holds.
To further show that $K({\check{{\mathcal{H}}}^{\alpha,\beta}})$ is contained in $\operatorname{\mathrm{Ker}}\check{\gamma}$, it is sufficient to show that $\check{\gamma}({\tilde{\mathcal{P}}}_{k}) = ({\check{\gamma}^\alpha}({\tilde{\mathcal{P}}}_{k}), {\check{\gamma}^\beta}({\tilde{\mathcal{P}}}_{k})) = 0$ for $k \geq 1$. We have $$\begin{gathered}
{\check{\gamma}^\alpha}({\tilde{\mathcal{P}}}_{k}) = {\check{\gamma}^\alpha}(1-\check{{\mathcal{P}}}_{-1,0}) {\check{\gamma}^\alpha}(\check{{\mathcal{P}}}_{-k-1,0} - \check{{\mathcal{P}}}_{-k,0}) =\\
(1 - {P^{\alpha}}M_{-1,0} {P^{\alpha}}M_{1,0} {P^{\alpha}})({P^{\alpha}}M_{-k-1,0}{P^{\alpha}}M_{k+1,0}{P^{\alpha}}- {P^{\alpha}}M_{-k,0}{P^{\alpha}}M_{k,0}{P^{\alpha}}).\end{gathered}$$ $1 - {P^{\alpha}}M_{-1,0} {P^{\alpha}}M_{1,0} {P^{\alpha}}$ and ${P^{\alpha}}M_{-k-1,0}{P^{\alpha}}M_{k+1,0}{P^{\alpha}}- {P^{\alpha}}M_{-k,0}{P^{\alpha}}M_{k,0}{P^{\alpha}}$ are projections onto closed subspaces spanned by sets $\{ {{\bm e}}_{x,y} \mid 0 \leq -\alpha x +y < \alpha \}$ and $\{ {{\bm e}}_{x,y} \mid k \leq -\alpha x +y < (k+1)\alpha \}$, respectively. Thus, for $k \geq 1$, we have ${\check{\gamma}^\alpha}({\tilde{\mathcal{P}}}_{k}) = 0$. We also have ${\check{\gamma}^\beta}({\tilde{\mathcal{P}}}_{k}) = 0$ since ${\check{\gamma}^\beta}(1-\check{{\mathcal{P}}}_{-1,0}) = 0$.
Concave corner Toeplitz extension
---------------------------------
The following is the main theorem of this paper.
\[main\] There is the following short exact sequence of $C^*$-algebras: $$\label{exact}
0 \to K({\check{{\mathcal{H}}}^{\alpha,\beta}}) \to {\check{{\mathcal{T}}}^{\alpha,\beta}}\overset{\check{\gamma}}{\to} {\mathcal{S}^{\alpha, \beta}}\to 0,
\vspace{-1mm}$$ where $K({\check{{\mathcal{H}}}^{\alpha,\beta}})$ is the $C^*$-algebra of compact operators on ${\check{{\mathcal{H}}}^{\alpha,\beta}}$.
In this subsection, we give a proof of this theorem.
\[prop2\] There is a $*$-isomorphism $\theta \colon {\mathcal{S}^{\alpha, \beta}}\to {\check{{\mathcal{T}}}^{\alpha,\beta}}/K({\check{{\mathcal{H}}}^{\alpha,\beta}})$, that is, on the dense subalgebra of ${\mathcal{S}^{\alpha, \beta}}$ obtained from Lemma \[dense\], of the following form: $$\begin{gathered}
\theta \biggl( \sum_{i=1}^l \hspace{-0.5mm} c_i {P^{\alpha}}\hspace{-0.5mm} M_{m_{i0}\hspace{-0.3mm}, n_{i0}} \hspace{-0.5mm} \biggl(\prod_{j=1}^{k_i} \hspace{-0.5mm} {P^{\alpha}}\hspace{-0.5mm} M_{m_{ij} \hspace{-0.3mm},n_{ij}} \hspace{-1mm} \biggl) \hspace{-0.5mm} {P^{\alpha}}\hspace{-0.5mm}, \hspace{-0.5mm} \sum_{i=1}^l \hspace{-0.5mm} c_i {P^{\beta}}\hspace{-0.5mm} M_{m_{i0}\hspace{-0.3mm}, n_{i0}} \hspace{-0.5mm} \biggl( \prod_{j=1}^{k_i} \hspace{-0.5mm} {P^{\beta}}\hspace{-0.5mm} M_{m_{ij}\hspace{-0.3mm}, n_{ij}} \biggl) {P^{\beta}}\hspace{-0.5mm} \biggl)
\\
=
\biggl[ \sum_{i=1}^l c_i {\check{P}^{\alpha,\beta}}M_{m_{i0}, n_{i0}} \biggl( \prod_{j=1}^{k_i} {\check{P}^{\alpha,\beta}}M_{m_{ij}, n_{ij}} \biggl) {\check{P}^{\alpha,\beta}}\biggl].\end{gathered}$$ Its inverse is the $*$-homomorphism induced by $\check{\gamma}$.
Let $\check{T}$ be an element of ${\check{{\mathcal{T}}}^{\alpha,\beta}}$ of the form (\[denses\]) and $(T^\alpha, T^\beta)$ be an element of ${\mathcal{S}^{\alpha, \beta}}$ of the form (\[densess\]). By Lemma \[dense\], such operators form a dense subalgebra of ${\mathcal{S}^{\alpha, \beta}}$. We define $\theta(T^\alpha, T^\beta) := [\check{T}] \in {\check{{\mathcal{T}}}^{\alpha,\beta}}/K({\check{{\mathcal{H}}}^{\alpha,\beta}})$. To show the well-definedness of $\theta$ and that $\theta$ extends to a $*$-homomorphism on ${\mathcal{S}^{\alpha, \beta}}$, it suffices to show the following inequality: $$\| \theta(T^\alpha, T^\beta) \| \leq \| (T^\alpha, T^\beta) \|.$$
We relabel the set $\check{\Sigma}$ as in Fig. \[relabel\]. This gives an order on the set $\{ e_{x,y} \mid (x,y) \in \check{\Sigma} \}$. Let $P_n$ be the orthogonal projection onto the span of the first $n$ elements. Then, for $[\check{T}] \in {\check{{\mathcal{T}}}^{\alpha,\beta}}/K({\check{{\mathcal{H}}}^{\alpha,\beta}})$, we have $$\| [\check{T}] \| = \inf_{C \in K({\check{{\mathcal{H}}}^{\alpha,\beta}})} \| \check{T} + C \| = \lim_n \| \check{T}(1 - P_n) \|.$$ The last equality follows since $\{ P_n \}_{n=0}^{\infty}$ is an approximate unit for $K({\check{{\mathcal{H}}}^{\alpha,\beta}})$ (see Theorem $1.7.4$ of [@HR00], for example).
![Relabel lattice points as $0,1,2, \cdots$. Divide the set $\check{\Sigma}$ into two parts[]{data-label="relabel"}](4.eps){width="60mm"}
Further, we divide the set $\check{\Sigma}$ into two parts ${\mathcal{A}}$ and ${\mathcal{B}}$ by the line $y = \frac{\alpha + \beta}{2}x$, as in Fig. \[relabel\]. Specifically, let ${\mathcal{A}}:= \{ (x,y) \in \check{\Sigma} \mid -\frac{\alpha + \beta}{2}x + y \geq 0 \}$ and ${\mathcal{B}}:= \check{\Sigma} \setminus {\mathcal{A}}$. Let $M = \| (T^\alpha, T^\beta) \| = \max \{ \| T^\alpha \|, \| T^\beta \| \}$, and let $f \in l^2(\check{\Sigma})$, which has a finite support and satisfies $\| f \| = 1$. There exists $n_0 \in {\mathbb{N}}$ such that for any $n \geq n_0$, we have $\check{T}(1-P_n) f|_{\mathcal{A}}= T^\alpha (1-P_n) f|_{\mathcal{A}}$ and $\check{T}(1-P_n) f|_{\mathcal{B}}= T^\beta (1-P_n) f|_{\mathcal{B}}$. Since the operator $\check{T}$ is of the form (\[denses\]), we can take such $n_0$ uniformly with respect to $f$. Thus, for $n \geq n_0$, we have $$\begin{aligned}
\| \check{T}(1-P_n) f \| &\leq \| \check{T}(1-P_n) f|_{\mathcal{A}}\| + \| \check{T}(1-P_n) f|_{\mathcal{B}}\| \\
&= \|T^\alpha (1-P_n) f|_{\mathcal{A}}\| + \| T^\beta (1 - P_n) f|_{\mathcal{B}}\| \\
&\leq \|T^\alpha \| \| f|_{\mathcal{A}}\| + \| T^\beta \| \| f|_{\mathcal{B}}\| \\
&\leq M (\| f|_{\mathcal{A}}\| + \| f|_{\mathcal{B}}\|)
= M \| f \|
= M.\end{aligned}$$ Thus, we have $\| \check{T}(1-P_n) \| \leq M$. By taking $n \to \infty$, we have $\| \theta(T^\alpha, T^\beta) \| = \| [\check{T}] \| \leq M = \| (T^\alpha, T^\beta) \|$, as desired.
By Proposition \[contain\], $\check{\gamma}$ induces a $*$-homomorphism ${\check{{\mathcal{T}}}^{\alpha,\beta}}/K({\check{{\mathcal{H}}}^{\alpha,\beta}}) \to {\mathcal{S}^{\alpha, \beta}}$. By computing on dense subalgebras of ${\check{{\mathcal{T}}}^{\alpha,\beta}}/K({\check{{\mathcal{H}}}^{\alpha,\beta}})$ and ${\mathcal{S}^{\alpha, \beta}}$, we can check that this map is an inverse of $\theta$. Thus, $\theta$ is an isomorphism.
[*Proof of Theorem \[main\]*]{} By Proposition \[contain\], we have a short exact sequence $0 \to K({\check{{\mathcal{H}}}^{\alpha,\beta}}) \to {\check{{\mathcal{T}}}^{\alpha,\beta}}\to {\check{{\mathcal{T}}}^{\alpha,\beta}}/K({\check{{\mathcal{H}}}^{\alpha,\beta}}) \to 0$. By Proposition \[prop2\], we have the isomorphism $\theta \colon {\mathcal{S}^{\alpha, \beta}}\to {\check{{\mathcal{T}}}^{\alpha,\beta}}/K({\check{{\mathcal{H}}}^{\alpha,\beta}})$. Combined with these results, we obtain the desired result.
By Theorem \[main\], a necessary and sufficient condition for concave corner Toeplitz operators to be Fredholm is obtained.
\[Fredholm\] An operator $\check{T} \in {\check{{\mathcal{T}}}^{\alpha,\beta}}$ is Fredholm if and only if $\check{\gamma}(\check{T})$ is invertible in ${\mathcal{S}^{\alpha, \beta}}$ or, equivalently, if and only if ${\check{\gamma}^\alpha}(\check{T})$ and ${\check{\gamma}^\beta}(\check{T})$ are invertible in ${{\mathcal{T}}^{\alpha}}$ and ${{\mathcal{T}}^{\beta}}$, respectively.
A Fredholm operator of index one and an index formula
=====================================================
In this section, we study further concave corner Toeplitz operators from the viewpoint of index theory. We explicitly construct a Fredholm Toeplitz operator associated with a concave corner whose index is one. By using this result, we compute some $K$-groups associated with concave corners and boundary homomorphisms associated with the extension (\[exact\]) of Theorem \[main\]. Moreover, a relation with index theory for quarter-plane Toeplitz operators [@Ji95; @Pa90] is obtained. By using this relation, we show some corresponding results obtained previously for quarter-plane Toeplitz operators [@CDS72; @DH71]. Especially, a Coburn–Douglas–Singer-type index formula for Fredholm concave corner Toeplitz operators is obtained.
A Fredholm operator of index one
--------------------------------
We first construct a Fredholm concave corner Toeplitz operator of index one and compute $K$-groups of some $C^*$-algebras associated with concave corners.
As in [@Ji95], by using the action of $SL(2, {\mathbb{Z}})$ onto ${\mathbb{Z}}^2$, we assume the condition ($\dagger$) without loss of generality. In this section, we consider the following operator: $$\check{A} := \check{{\mathcal{P}}}_{0,1} + M_{1,1}(1 - \check{{\mathcal{P}}}_{-1,0}) + M_{1,0}(\check{{\mathcal{P}}}_{-1,0} - \check{{\mathcal{P}}}_{0,1}).$$ Since $\check{A} = \check{{\mathcal{P}}}_{0,1} + {\check{P}^{\alpha,\beta}}M_{1,1} {\check{P}^{\alpha,\beta}}(1 - \check{{\mathcal{P}}}_{0,-1}) + {\check{P}^{\alpha,\beta}}M_{1,0} {\check{P}^{\alpha,\beta}}(\check{{\mathcal{P}}}_{-1,0} - \check{{\mathcal{P}}}_{0,1})$, the operator $\check{A}$ is an element of the algebra ${\check{{\mathcal{T}}}^{\alpha,\beta}}$. The following theorem is the main theorem of this section.
\[construction\] $\check{A}$ is a surjective Fredholm operator whose Fredholm index is $1$. Its kernel is given as follows:
1. When $\alpha = \frac{1}{2}$ and $\beta = 1$, $\operatorname{\mathrm{Ker}}\check{A} = {\mathbb{C}}({{\bm e}}_{-3,-1} -{{\bm e}}_{-2, -1})$.
2. When $0 < \alpha < \frac{1}{2}$ and $\beta = 1$, $\operatorname{\mathrm{Ker}}\check{A} = {\mathbb{C}}({{\bm e}}_{-2, 0} - {{\bm e}}_{-1,0})$.
3. When $\alpha = \frac{1}{2}$ and $1 < \beta < \infty$, $\operatorname{\mathrm{Ker}}\check{A} = {\mathbb{C}}({{\bm e}}_{-1,0} - {{\bm e}}_{0,0})$.
4. When $0 < \alpha < \frac{1}{2}$ and $1 < \beta < \infty$, $\operatorname{\mathrm{Ker}}\check{A} = {\mathbb{C}}({{\bm e}}_{-1,0} - {{\bm e}}_{0,0})$.
Moreover, we have $\check{A} - 1 \in {\check{{\mathcal{C}}}^{\alpha,\beta}}$.
To examine the operator $\check{A}$, it is convenient to divide its domain and range as follows. We divide the set $\check{\Sigma}$ into three parts $\check{\Sigma} = {\mathcal{D}}_1 \sqcup {\mathcal{D}}_2 \sqcup {\mathcal{D}}_3$, where $${\mathcal{D}}_1 = \{ (x,y) \in \check{\Sigma} \mid 0 \leq -\alpha x + y < \alpha \ \text{and} \ 1 < - \beta x + y \},$$ $${\mathcal{D}}_2 = \{ (x,y) \in \check{\Sigma} \mid \alpha \leq -\alpha x + y < 1 \ \text{and} \ \beta < -\beta x + y \},$$ $${\mathcal{D}}_3 = \check{\Sigma} \setminus ({\mathcal{D}}_1 \sqcup {\mathcal{D}}_2).$$ Note that it can be checked that the set $\{ (x,y) \in \check{\Sigma} \mid 0 \leq -\alpha x + y < \alpha \ \text{and} \ 1 < -\beta x + y \leq \beta\}$ is empty under the assumption ($\dagger$).
We also divide $\check{\Sigma}$ in the following way, $\check{\Sigma} = {\mathcal{R}}_1 \sqcup {\mathcal{R}}_2 \sqcup {\mathcal{R}}_3$, where $${\mathcal{R}}_1 = \{ (x,y) \in \check{\Sigma} \mid 0 \leq -\alpha x + y < 1 - \alpha, \ \text{and} \ 1 < - \beta x + y \},$$ $${\mathcal{R}}_2 = \{ (x,y) \in \check{\Sigma} \mid 1 - \alpha \leq -\alpha x + y < 1 \ \text{and} \ 1 < -\beta x + y \},$$ $${\mathcal{R}}_3 = \check{\Sigma} \setminus ({\mathcal{R}}_1 \sqcup {\mathcal{R}}_2).$$ Note that ${\mathcal{D}}_3 = {\mathcal{R}}_3$ (see Fig. \[domain\] and Fig. \[range\]).
![${\mathcal{D}}_1$ and ${\mathcal{D}}_2$[]{data-label="domain"}](5.eps){width="110mm"}
![${\mathcal{R}}_1$ and ${\mathcal{R}}_2$[]{data-label="range"}](6.eps){width="110mm"}
First, we have $$\check{A} {{\bm e}}_{x,y} =
\left\{
\begin{aligned}
{{\bm e}}_{x+1, y+1} & \hspace{3mm} \text{if} \ (x, y) \in {\mathcal{D}}_1, \\
{{\bm e}}_{x+1,y} \ & \hspace{3mm} \text{if} \ (x, y) \in {\mathcal{D}}_2, \\
{{\bm e}}_{x,y} \ \ & \hspace{3mm} \text{if} \ (x, y) \in {\mathcal{D}}_3. \\
\end{aligned}
\right.$$ Actually, if $(x, y) \in {\mathcal{D}}_2$, we have $$\begin{aligned}
\check{A} {{\bm e}}_{x,y} &= \check{{\mathcal{P}}}_{0,1} {{\bm e}}_{x,y} + M_{1,1}(1 - \check{{\mathcal{P}}}_{0,-1}) {{\bm e}}_{x,y} + M_{1,0}(\check{{\mathcal{P}}}_{-1,0} - \check{{\mathcal{P}}}_{0,1}) {{\bm e}}_{x,y}\\
&= 0 + 0 + M_{1,0} {{\bm e}}_{x,y} = {{\bm e}}_{x+1, y},\end{aligned}$$ and the other cases follow via a similar computation. $\check{A}$ is surjective since $${{\bm e}}_{x,y} =
\left\{
\begin{aligned}
\check{A} {{\bm e}}_{x-1,y} \ & \hspace{3mm} \text{if} \ (x, y) \in {\mathcal{R}}_1,\\
\check{A} {{\bm e}}_{x-1, y-1} & \hspace{3mm} \text{if} \ (x, y) \in {\mathcal{R}}_2,\\
\check{A} {{\bm e}}_{x,y} \ \ & \hspace{3mm}\text{if} \ (x, y) \in {\mathcal{R}}_3. \\
\end{aligned}
\right.$$ Actually, if $(x, y) \in {\mathcal{R}}_2$, that is, $1 - \alpha \leq -\alpha x + y < 1$ and $1 < -\beta x + y$, then $0 \leq -\alpha(x-1) + (y-1) < \alpha$ and $\beta < -\beta (x-1) + (y-1)$. Thus, $(x-1, y-1) \in {\mathcal{D}}_1$, and we have $\check{A} {{\bm e}}_{x-1, y-1} = {{\bm e}}_{x,y}$; the other cases follow via similar computations.
We next show the following results:
- There exist two points $(x_0, y_0)$ and $(x_1, y_1)$ in the set $\check{\Sigma}$ such that $\check{A} {{\bm e}}_{x_0,y_0} = \check{A} {{\bm e}}_{x_1, y_1} = {{\bm e}}_{x_1, y_1}$.
- For any point $(u,v) \in \check{\Sigma} \setminus \{ (x_1,y_1) \}$, there exists unique point $(x, y) \in \check{\Sigma} \setminus \{ (x_0, y_0), (x_1,y_1) \}$ such that $\check{A} {{\bm e}}_{x,y} = {{\bm e}}_{u,v}$.
- These $(x_0, y_0)$ and $(x_1, y_1)$ take the following values:
1. When $\alpha \hspace{-0.5mm} = \hspace{-0.5mm} \frac{1}{2}$ and $\beta \hspace{-0.5mm} =\hspace{-0.5mm} 1$, $(x_0, y_0) \hspace{-0.5mm}=\hspace{-0.5mm} (-3, -1)$ and $(x_1, y_1) \hspace{-0.5mm}=\hspace{-0.5mm} (-2,-1)$,
2. When $0 \hspace{-0.5mm}<\hspace{-0.5mm} \alpha \hspace{-0.5mm}<\hspace{-0.5mm} \frac{1}{2}$ and $\beta \hspace{-0.5mm}=\hspace{-0.5mm} 1$, $(x_0, y_0) \hspace{-0.5mm}=\hspace{-0.5mm} (-2, 0)$ and $(x_1, y_1) \hspace{-0.5mm}=\hspace{-0.5mm} (-1,0)$,
3. When $\alpha \hspace{-0.5mm}=\hspace{-0.5mm} \frac{1}{2}$ and $1 \hspace{-0.5mm}<\hspace{-0.5mm} \beta \hspace{-0.5mm}<\hspace{-0.5mm} \infty$, $(x_0, y_0) \hspace{-0.5mm}=\hspace{-0.5mm} (-1, 0)$ and $(x_1, y_1) \hspace{-0.5mm}=\hspace{-0.5mm} (0, 0)$,
4. When $0 < \alpha < \frac{1}{2}$ and $1 < \beta < \infty$, $(x_0, y_0) = (-1, 0)$ and $(x_1, y_1) = (0, 0)$.
We here prove them only in the case 4), that is, when $0 < \alpha < \frac{1}{2}$ and $1 < \beta < \infty$. The other cases can be shown in the same way.
When $(x,y) \in {\mathcal{D}}_1$, that is, $0 \leq -\alpha x + y < \alpha$ and $\beta < -\beta x + y$, we have $$1 - \alpha \leq -\alpha (x+1) + (y+1) < 1 \ \text{and} \ \beta < -\beta (x+1) + (y+1),$$ and thus, the point $(x+1, y+1)$ is contained in ${\mathcal{R}}_2$. On the other hand, if $(x, y) \in {\mathcal{R}}_2$, then $(x-1, y-1) \in {\mathcal{D}}_2$. Thus, there is a bijection $$\{ {{\bm e}}_{x,y} \mid (x, y) \in {\mathcal{D}}_1 \} \overset{\check{A} = M_{1,1}}{\longrightarrow} \{ {{\bm e}}_{x,y} \mid (x, y) \in {\mathcal{R}}_2 \}.$$ We next compute points $(x, y)$ in ${\mathcal{R}}_1$ for which $(x+1, y)$ is not contained in ${\mathcal{R}}_1$. Such $(x,y) \in {\mathbb{Z}}^2$ satisfy $0 \leq -\alpha (x+1) + y < 1 - \alpha$ and $1 - \beta < -\beta(x+1) + y \leq 1$. There is just one point that satisfies these inequalities, and under the assumption of 4), this point is $(-1, 0)$. As in the case of ${\mathcal{D}}_1$, there is a bijection $$\{ {{\bm e}}_{x,y} \mid (x, y) \in {\mathcal{D}}_2 \setminus \{ (-1, 0) \} \} \overset{\check{A} = M_{1,0}}{\longrightarrow} \{ {{\bm e}}_{x,y} \mid (x, y) \in {\mathcal{R}}_2 \}.$$ The result follows since $$\{ {{\bm e}}_{x,y} \mid (x, y) \in {\mathcal{D}}_3 \} \overset{\check{A} = {\mathrm{id}}}{\longrightarrow} \{ {{\bm e}}_{x,y} \mid (x, y) \in {\mathcal{R}}_3 \},$$ is a bijection and $\check{A} {{\bm e}}_{-1,0} = \check{A} {{\bm e}}_{0,0} = {{\bm e}}_{0,0}$.
By applying the method in [@CD71] for our subset $\check{\Sigma}$, we obtain the following short exact sequence, $$0 \to {\check{{\mathcal{C}}}^{\alpha,\beta}}\to {\check{{\mathcal{T}}}^{\alpha,\beta}}\overset{\sigma \circ \check{\gamma}}{\to} C({\mathbb{T}}^2) \to 0.$$ Note that the set $\check{\Sigma}$ contains the subsemigroup $\hat{\Sigma}$ of the discrete abelian group ${\mathbb{Z}}^2$. Note also that $\hat{\Sigma}$ acts on the set $\check{\Sigma}$ and that $\hat{\Sigma}$ generates ${\mathbb{Z}}^2$. By using this sequence, to show that $\check{A} - 1 \in {\check{{\mathcal{C}}}^{\alpha,\beta}}= \operatorname{\mathrm{Ker}}(\sigma \circ \check{\gamma})$, it is sufficient to show $(\sigma \circ \check{\gamma}) (\check{A} - 1) = 0$. This holds since $(\sigma \circ \check{\gamma}) (\check{{\mathcal{P}}}_{m,n}) = 1$ for any $(m,n) \in {\mathbb{Z}}^2$.
By using Theorem \[construction\], we here compute $K$-groups of concave corner $C^*$-algebras ${\check{{\mathcal{T}}}^{\alpha,\beta}}$ and its commutator ideals ${\check{{\mathcal{C}}}^{\alpha,\beta}}$. Associated with the sequence (\[main\]), we have the following six-term exact sequence: $$\xymatrix{
K_1(K({\check{{\mathcal{H}}}^{\alpha,\beta}})) \ar[r]& K_1({\check{{\mathcal{T}}}^{\alpha,\beta}}) \ar[r] & K_1({\mathcal{S}^{\alpha, \beta}}) \ar[d]^{\check{\delta}_1}
\\
K_0({\mathcal{S}^{\alpha, \beta}}) \ar[u]^{\check{\delta}_0} & K_0({\check{{\mathcal{T}}}^{\alpha,\beta}}) \ar[l] & K_0(K({\check{{\mathcal{H}}}^{\alpha,\beta}})), \ar[l]
}$$ $K$-groups of ${\mathcal{S}^{\alpha, \beta}}$ is computed in [@Pa90]. The result is[^4], $$K_0({\mathcal{S}^{\alpha, \beta}}) \cong
\left\{
\begin{aligned}
{\mathbb{Z}}\ & \hspace{3mm} \text{if $\alpha$ and $\beta$ are both rational,}\\
{\mathbb{Z}}^2 & \hspace{3mm} \text{if either $\alpha$ or $\beta$ is rational and the other is irrational,}\\
{\mathbb{Z}}^3 & \hspace{3mm} \text{if $\alpha$ and $\beta$ are both irrational.}
\end{aligned}
\right.$$ and $K_1({\mathcal{S}^{\alpha, \beta}}) = {\mathbb{Z}}$. By Theorem \[construction\], we can see from the above six-term exact sequence that $\check{\delta}_1 \colon K_1({\mathcal{S}^{\alpha, \beta}}) \to K_0(K({\check{{\mathcal{H}}}^{\alpha,\beta}}))$ is an isomorphism. Now, we can compute $K$-groups of ${\check{{\mathcal{T}}}^{\alpha,\beta}}$, and the result is as follows: $$K_0({\check{{\mathcal{T}}}^{\alpha,\beta}}) \cong
\left\{
\begin{aligned}
{\mathbb{Z}}\ & \hspace{3mm} \text{if $\alpha$ and $\beta$ are both rational,}\\
{\mathbb{Z}}^2 & \hspace{3mm} \text{if either $\alpha$ or $\beta$ is rational and the other is irrational,}\\
{\mathbb{Z}}^3 & \hspace{3mm} \text{if $\alpha$ and $\beta$ are both irrational.}
\end{aligned}
\right.$$ and $K_1({\check{{\mathcal{T}}}^{\alpha,\beta}}) = 0$.
We next compute the $K$-group of the commutator algebra ${\check{{\mathcal{C}}}^{\alpha,\beta}}$. As in [@Ji95; @Pa90], if we restrict the sequence (\[exact\]) to ${\check{{\mathcal{C}}}^{\alpha,\beta}}\subset {\check{{\mathcal{T}}}^{\alpha,\beta}}$, we obtain the following short exact sequence: $$0 \to K({\check{{\mathcal{H}}}^{\alpha,\beta}}) \to {\check{{\mathcal{C}}}^{\alpha,\beta}}\overset{\check{\gamma}'}{\to} {{\mathcal{C}}^\alpha}\oplus {{\mathcal{C}}^\beta}\to 0,$$ where $\check{\gamma}'$ is the restriction of $\check{\gamma}$ onto ${\check{{\mathcal{C}}}^{\alpha,\beta}}$. Associated with this sequence, we have the following six-term exact sequence: $$\xymatrix{
K_1(K({\check{{\mathcal{H}}}^{\alpha,\beta}})) \ar[r]& K_1({\check{{\mathcal{C}}}^{\alpha,\beta}}) \ar[r] & K_1({{\mathcal{C}}^\alpha}) \oplus K_1({{\mathcal{C}}^\beta}) \ar[d]
\\
K_0({{\mathcal{C}}^\alpha}) \oplus K_0({{\mathcal{C}}^\beta}) \ar[u]& K_0({\check{{\mathcal{C}}}^{\alpha,\beta}}) \ar[l] & K_0(K({\check{{\mathcal{H}}}^{\alpha,\beta}})). \ar[l]
}$$ $K$-groups of ${{\mathcal{C}}^\alpha}$ and ${{\mathcal{C}}^\beta}$ are computed in [@JK88; @Xia88]. The result is $$K_0({{\mathcal{C}}^\alpha}) \cong K_0({{\mathcal{C}}^\beta}) \cong
\left\{
\begin{aligned}
{\mathbb{Z}}\ & \hspace{3mm} \text{if $\alpha$ (or $\beta$) is rational},\\
{\mathbb{Z}}^2 & \hspace{3mm} \text{if $\alpha$ (or $\beta$) is irrational}.
\end{aligned}
\right.$$ and $K_1({{\mathcal{C}}^\alpha}) \cong K_1({{\mathcal{C}}^\beta}) \cong {\mathbb{Z}}$. By Theorem \[construction\], the operator $\check{A} - 1$ is contained in the algebra ${\check{{\mathcal{C}}}^{\alpha,\beta}}$. Thus, the map $\check{\delta}_1 \colon K_1({{\mathcal{C}}^\alpha}) \oplus K_1({{\mathcal{C}}^\beta}) \to K_0(K({\check{{\mathcal{H}}}^{\alpha,\beta}}))$ is surjective. As in [@Ji95], we can compute $K$-groups of ${\check{{\mathcal{T}}}^{\alpha,\beta}}$ as follows: $$K_0({\check{{\mathcal{C}}}^{\alpha,\beta}}) \cong
\left\{
\begin{aligned}
{\mathbb{Z}}^2 & \hspace{3mm} \text{if $\alpha$ and $\beta$ are both rational},\\
{\mathbb{Z}}^3 & \hspace{3mm} \text{if either $\alpha$ or $\beta$ is rational, and the other is irrational},\\
{\mathbb{Z}}^4 & \hspace{3mm} \text{if $\alpha$ and $\beta$ are both irrational}.
\end{aligned}
\right.$$ and $K_1({\check{{\mathcal{C}}}^{\alpha,\beta}}) = {\mathbb{Z}}$.
A relation with the quarter-plane case and an index formula
-----------------------------------------------------------
We next compare index theory for quarter-plane (convex corner) Toeplitz operators [@DH71; @Ji95; @Pa90] and that for concave corner Toeplitz operators. There are group isomorphisms $K_0(\hat{\mathrm{Tr}}) \colon K_0(K({\hat{{\mathcal{H}}}^{\alpha,\beta}})) \to {\mathbb{Z}}$ and $K_0(\check{\mathrm{Tr}}) \colon K_0(K({\check{{\mathcal{H}}}^{\alpha,\beta}}))$ $\to {\mathbb{Z}}$ that map a class $[p]_0$ of a finite-rank projection $p$ to $\operatorname{\mathrm{rank}}(\mathrm{Image} (p))$. Associated with extensions (\[seq1\]) and (\[exact\]), there are the following two group isomorphisms from $K_1({\mathcal{S}^{\alpha, \beta}})$ to ${\mathbb{Z}}$: $$K_0(\hat{\mathrm{Tr}}) \circ \hat{\delta}_1 \colon K_1({\mathcal{S}^{\alpha, \beta}}) \to {\mathbb{Z}}\ \ \ \text{and} \ \ \ K_0(\check{\mathrm{Tr}}) \circ \check{\delta}_1 \colon K_1({\mathcal{S}^{\alpha, \beta}}) \to {\mathbb{Z}}.$$ where $\hat{\delta}_1$ is the boundary homomorphism of the six-term exact sequence of $K$-theory for $C^*$-algebras associated with the quarter-plane Toeplitz extension (\[seq1\]). Let $\hat{{\mathcal{P}}}_{m,n} := {\hat{P}^{\alpha,\beta}}M_{m,n} {\hat{P}^{\alpha,\beta}}M_{-m,-n} {\hat{P}^{\alpha,\beta}}$. Jiang considered in [@Ji95] the following operator: $$\hat{A} := \hat{{\mathcal{P}}}_{0,1} + M_{1,1}(1 - \hat{{\mathcal{P}}}_{-1,0}) + M_{1,0}(\hat{{\mathcal{P}}}_{-1,0} - \hat{{\mathcal{P}}}_{0,1}).$$ Note that $\hat{A} \in {\hat{{\mathcal{T}}}^{\alpha,\beta}}$. It is shown in [@Ji95] that $\hat{A}$ is the isometric Fredholm operator whose Fredholm index is $-1$. By comparing the Fredholm quarter-plane Toeplitz operator constructed in [@Ji95] and the Fredholm concave corner Toeplitz operator constructed in Theorem \[construction\], we obtain the following result:
\[relation\] $K_0(\check{\mathrm{Tr}}) \circ \check{\delta}_1= -K_0(\hat{\mathrm{Tr}}) \circ \hat{\delta}_1$.
By the map $\hat{\gamma}$ in the quarter-plane Toeplitz extension (\[seq1\]), we have $\hat{\gamma}(\hat{A}) \in {\mathcal{S}^{\alpha, \beta}}$. We can check the equality $\hat{\gamma}(\hat{A}) = \check{\gamma}(\check{A})$ and that this element gives a generator $[\hat{\gamma}(\hat{A})]_1$ of the $K$-group $K_1({\mathcal{S}^{\alpha, \beta}}) \cong {\mathbb{Z}}$. By Theorem $1$ of [@Ji95], Theorem \[construction\] and Proposition $9.4.2$ of [@RLL00], we have $(K_0(\check{\mathrm{Tr}}) \circ \check{\delta}_1)([\check{\gamma}(\check{A})]_1) = \operatorname{\mathrm{index}}(\check{A}) = 1 = - \operatorname{\mathrm{index}}(\hat{A}) = - (K_0(\hat{\mathrm{Tr}}) \circ \hat{\delta}_1)([\hat{\gamma}(\hat{A})]_1)$.
We now restrict our attention to the case of $\alpha = 0$ and $\beta = \infty$. Previous works studied quarter-plane Toeplitz operators in this case and obtained many results [@CDS72; @DH71]. Combined with Corollary \[relation\], we obtain corresponding results for concave corner Toeplitz operators, and we state it explicitly for the later use.
In [@CDS72], Coburn–Douglas–Singer obtained an index formula for Fredholm quarter-plane Toeplitz operators. A corresponding result for concave corner Toeplitz operators is as follows. Let $r$ be a positive integer. The map $\hat{\gamma}$ induces a surjective $*$-homomorphism $$1 \otimes \hat{\gamma} \colon M_r({\mathbb{C}}) {\otimes} \check{{\mathcal{T}}}^{0,\infty} \cong M_r(\check{{\mathcal{T}}}^{0,\infty}) \to M_r({\mathbb{C}}) {\otimes} \mathcal{S}^{0,\infty} \cong M_r(\mathcal{S}^{0,\infty})$$ which we denote $\hat{\gamma}$, for simplicity[^5]. The algebra $M_r(\mathcal{S}^{0,\infty})$ is a $C^*$-subalgebra of $M_r(\mathcal{T}^0) \oplus M_r(\mathcal{T}^\infty) \cong M_r(C({\mathbb{T}}) {\otimes} \mathcal{T}) \oplus M_r(\mathcal{T} {\otimes} C({\mathbb{T}}))$. We write $(\xi, \eta)$ for valuables in ${\mathbb{T}}^2$. $M_r(\mathcal{T}^0)$ and $M_r(\mathcal{T}^\infty)$ have valuables $\xi$ and $\eta$, respectively.
\[concaveCDS\] If $\check{T}$ is a Fredholm operator in $M_r(\check{{\mathcal{T}}}^{0,\infty})$ with symbol $\check{\gamma}(\check{T}) = (\check{\gamma}^0(\check{T}), \check{\gamma}^\infty(\check{T}))$ in $M_r(\mathcal{S}^{0,\infty})$. Then, there is a path $(F_t, G_t)$ in $M_r(\mathcal{S}^{0,\infty})$ such that $F_0 = \check{\gamma}^0(\check{T})$, $G_0 = \check{\gamma}^\infty(\check{T})$ and such that $$F_1(\xi) =
\begin{pmatrix}
\xi^m & & & \\
& 1 & & \\
& & \ddots & \\
& & & 1
\end{pmatrix}
\ \text{and} \ \
G_1(\eta) =
\begin{pmatrix}
\eta^n & & & \\
& 1 & & \\
& & \ddots & \\
& & & 1
\end{pmatrix}$$ for some $(m,n)$ in ${\mathbb{Z}}^2$. The Fredholm index of $\check{T}$ is given by $\operatorname{\mathrm{index}}(\check{T}) = m+n$.
Since $\hat{\gamma} \colon M_r(\hat{{\mathcal{T}}}^{0,\infty}) \to M_r(\mathcal{S}^{0,\infty})$ is surjective, there is $\hat{T} \in M_r(\hat{{\mathcal{T}}}^{0,\infty})$ satisfying $\hat{\gamma}(\hat{T}) = \check{\gamma}(\check{T})$. Since $\check{T}$ is Fredholm, $\check{\gamma}(\check{T})$ is invertible in $\mathcal{S}^{0,\infty}$, and thus, $\hat{T}$ is a Fredholm quarter-plane Toeplitz operator (see Theorem $2.6$ of [@Pa90]). By Theorem in p$589$ of [@CDS72], such a path $(F_t, G_t)$ exists, and we have $\operatorname{\mathrm{index}}(\hat{T}) = -(m+n)$. By Corollary \[relation\], we have $-(m+n) = \operatorname{\mathrm{index}}(\hat{T}) = (K_0(\hat{\mathrm{Tr}}) \circ \hat{\delta}_1) ([\hat{\gamma}(\hat{T})]_1) = -K_0(\hat{\mathrm{Tr}}) \circ \hat{\delta}_1 ([\check{\gamma}(\check{T})]_1) = - \operatorname{\mathrm{index}}(\check{T})$.
\[notunique\] According to [@CDS72], the path $(F_t, G_t)$ is not unique and each $m$ and $n$ are not uniquely determined in general.
We next see that when a Fredholm concave corner Toeplitz operators is of a special form, its Fredholm index is zero. The corresponding result for quarter-plane Toeplitz operators is obtained in [@DH71]. For a continuous function, $\varphi \colon {\mathbb{T}}^2 \to {\mathbb{C}}$, the multiplication operator generated by $\varphi$ defines a bounded linear operator on $L^2({\mathbb{T}}^2)$. Through the Fourier transform, this multiplication operator defines a bounded linear operator $M_\varphi$ on $l^2({\mathbb{Z}}^2)$. Then, $\check{P}^{0,\infty} M_\varphi \check{P}^{0,\infty}$ is a concave corner Toeplitz operator. For an operator of this form, we have the following result.
\[zero\] Let $\varphi \colon {\mathbb{T}}^2 \to {\mathbb{C}}$ be a continuous function. If the concave corner Toeplitz operator $\check{P}^{0,\infty} M_\varphi \check{P}^{0,\infty}$ is Fredholm, then its Fredholm index is zero.
By our assumption, $\check{\gamma}(\check{P}^{0,\infty} M_\varphi \check{P}^{0,\infty}) = (P^{0} M_\varphi P^{0}, P^{\infty} M_\varphi P^{\infty})$ is an invertible element in $\mathcal{S}^{0,\infty}$. Thus, $\hat{\gamma}(\hat{P}^{0,\infty} M_\varphi \hat{P}^{0,\infty}) = (P^{0} M_\varphi P^{0}, P^{\infty} M_\varphi P^{\infty})$ also is invertible, and the quarter-plane Toeplitz operator $\hat{P}^{0,\infty} M_\varphi \hat{P}^{0,\infty}$ is Fredholm. By Corollary in p$208$ of [@DH71], the Fredholm index of $\hat{P}^{0,\infty} M_\varphi \hat{P}^{0,\infty}$ is zero. By Corollary \[relation\], the Fredholm index of $\check{P}^{0,\infty} M_\varphi \check{P}^{0,\infty}$ is also zero.
Topological invariants and topologically protected corner states
================================================================
In [@Hayashi2], 3-D class A systems with codimension-two convex corners are studied, and a topological invariant is defined for a gapped bulk-edges Hamiltonian as an element of some $K$-group. Its relation with gapless corner states is also proved. Key ingredients are index theory for quarter-plane Toeplitz operators [@DH71; @Ji95; @Pa90]. A nontrivial example is obtained in [@Hayashi2] by constructing Hamiltonians from Hamiltonians of 2-D class A and 1-D class AIII (conventional) topological insulators. For such Hamiltonians, if we consider the convex corner of the special shape ($\alpha = 0$ and $\beta = +\infty$), corner topological invariants are defined, and the numerical corner invariant is equal to the product of two topological numbers of two (conventional) topological insulators (called the product formula). The study in [@Hayashi2] is based on previous works [@DH71; @Ji95; @Pa90] and is restricted to convex corners.
The results in Sect. $2$ and Sect. $3$ of this paper enable us to examine systems with concave corners. In this section, we define topological invariants for some Hamiltonians on 2-D class AIII systems (Sect. $4.1$). Moreover, we introduce the gapless corner topological invariant especially for concave corners and show the relation between gapped and gapless invariants. Correspondingly, for Hamiltonians that are gapped on two edges, we can define two corner invariants corresponding to these two corners. We show that there is a relation between these two corner invariants. We further formulate a product formula as in [@Hayashi2]. By using this, we obtain explicit examples of gapped bulk-edges Hamiltonians of nontrivial convex and concave corner invariants. They differ by the multiplication of $-1$, which clarifies that these corner invariants depend on the shape of the system. Since 3-D class A systems with convex corners are studied in [@Hayashi2], we mainly consider other cases.
We here collect the notations used in this subsection. Let $V$ be a finite rank Hermitian vector space and denote the complex dimension of $V$ by $N$. We write ${\mathcal{H}}_V$, ${{\mathcal{H}}^{\alpha}}_V$, ${{\mathcal{H}}^{\beta}}_V$, ${\hat{{\mathcal{H}}}^{\alpha,\beta}}_V$ and ${\check{{\mathcal{H}}}^{\alpha,\beta}}_V$ for ${\mathcal{H}}{\otimes} V$, ${{\mathcal{H}}^{\alpha}}{\otimes} V$, ${{\mathcal{H}}^{\beta}}{\otimes} V$, ${\hat{{\mathcal{H}}}^{\alpha,\beta}}{\otimes} V$ and ${\check{{\mathcal{H}}}^{\alpha,\beta}}{\otimes} V$, respectively. If there is an endomorphism $\Pi$ on $V$, we extend $\Pi$ onto ${\mathcal{H}}_V$, ${{\mathcal{H}}^{\alpha}}_V$, ${{\mathcal{H}}^{\beta}}_V$, ${\hat{{\mathcal{H}}}^{\alpha,\beta}}_V$ and ${\check{{\mathcal{H}}}^{\alpha,\beta}}_V$ by the pointwise operation, i.e., $1 {\otimes} \Pi$, and denote $\Pi$, also. Similarly, we write ${P^{\alpha}}$, ${P^{\beta}}$ ${\hat{P}^{\alpha,\beta}}$ and ${\check{P}^{\alpha,\beta}}$ for the orthogonal projections onto ${{\mathcal{H}}^{\alpha}}_V$, ${{\mathcal{H}}^{\beta}}_V$, ${\hat{{\mathcal{H}}}^{\alpha,\beta}}_V$ and ${\check{{\mathcal{H}}}^{\alpha,\beta}}_V$ defined by ${P^{\alpha}}{\otimes} 1$, ${P^{\beta}}{\otimes} 1$, ${\hat{P}^{\alpha,\beta}}{\otimes} 1$ and ${\check{P}^{\alpha,\beta}}{\otimes} 1$, for simplicity.
2-D class AIII system
---------------------
In this subsection, we consider 2-D class AIII systems with a codimension-two corner. We rather focus on the study of systems with concave corners, but we briefly study convex corners and show a relation between corner invariants defined on systems with these two types of corners.
In this subsection, we assume that the vector space $V$ has a ${\mathbb{Z}}_2$-grading given by $\Pi$. Specifically, $\Pi \colon V \to V$ is a complex linear map that satisfies $\Pi^2 = 1$. We consider a continuous map ${\mathbb{T}}^2 \to \operatorname{\mathrm{End}}_{\mathbb{C}}(V)$, $(\xi, \eta) \mapsto H(\xi, \eta)$, where, for each $(\xi,\eta) \in {\mathbb{T}}^2$, $H(\xi,\eta)$ is Hermitian. Moreover, we assume that $H(\xi, \eta)$ preserves chiral symmetry, that is, for any $(\xi,\eta) \in {\mathbb{T}}^2$, $H(\xi,\eta)$ anti-commutes with $\Pi$. Note that in this case, $N$ is necessarily an even number. Through the Fourier transform, the multiplication operator on $L^2({\mathbb{T}}^2; V)$ generated by $H(\xi, \eta)$ defines a bounded linear self-adjoint operator $H$ on the Hilbert space ${\mathcal{H}}_V$. We call $H$ the [*bulk Hamiltonian*]{}. Let $\alpha < \beta$ be real numbers (possibly $\alpha = -\infty$ or $\beta = +\infty$, but not both). By using them, we consider the following half-plane Toeplitz operators, $$H^\alpha := {P^{\alpha}}H {P^{\alpha}}\colon {{\mathcal{H}}^{\alpha}}_V \to {{\mathcal{H}}^{\alpha}}_V, \ \ H^\beta := {P^{\beta}}H {P^{\beta}}\colon {{\mathcal{H}}^{\beta}}_V \to {{\mathcal{H}}^{\beta}}_V.$$ and call them [*edge Hamiltonians*]{}. We also consider the following convex and concave corner Toeplitz operators: $$\hat{H}^{\alpha,\beta} := {\hat{P}^{\alpha,\beta}}H {\hat{P}^{\alpha,\beta}}\colon {\hat{{\mathcal{H}}}^{\alpha,\beta}}_V \to {\hat{{\mathcal{H}}}^{\alpha,\beta}}_V, \ \ \check{H}^{\alpha,\beta} := {\check{P}^{\alpha,\beta}}H {\check{P}^{\alpha,\beta}}\colon {\check{{\mathcal{H}}}^{\alpha,\beta}}_V \to {\check{{\mathcal{H}}}^{\alpha,\beta}}_V,$$ and call them [*corner Hamiltonians*]{}.
Note that $H^\alpha$, $H^\beta$, $\hat{H}^{\alpha,\beta}$ and $\check{H}^{\alpha,\beta}$ anti-commutes with $\Pi$. The following is our assumption in this subsection.
[**Assumption (Spectral gap condition)**]{} We assume that our edge Hamiltonians have a common spectral gap at the Fermi level $0$, i.e., $0$ is not contained in either $\mathrm{sp}(H^\alpha)$ or $\mathrm{sp}(H^\beta)$. We refer to this condition as the [*spectral gap condition*]{}.
Note that under this assumption, our bulk Hamiltonian is also gapped at zero [@Hayashi2]. By using chiral symmetry, we have following decomposition: $H =
\begin{pmatrix}
0 & h^*\\
h & 0
\end{pmatrix}$. We now fix an orthonormal basis of $V$ and identify it with ${\mathbb{C}}^N$. By our spectral gap condition, the operators ${P^{\alpha}}h {P^{\alpha}}$ and ${P^{\beta}}h {P^{\beta}}$ are both invertible. Let $u^\alpha := {P^{\alpha}}h {P^{\alpha}}/|{P^{\alpha}}h {P^{\alpha}}|$ and $u^\beta = {P^{\beta}}h {P^{\beta}}/|{P^{\beta}}h {P^{\beta}}|$.[^6] The pair $(u^\alpha, u^\beta)$ defines a unitary element in $M_{N/2}({\mathcal{S}^{\alpha, \beta}})$ and so defines an element of the $K_1$-group $K_1({\mathcal{S}^{\alpha, \beta}})$.[^7]
\[gappedinvAIII\] We define the gapped topological invariant as follows: $${\mathcal{I}}_{\mathrm{BE}}^{2d, {\mathrm{A\hspace{-.1em}I\hspace{-.1em}I\hspace{-.1em}I}}}(H) := [(u^\alpha, u^\beta)]_1 \in K_1({\mathcal{S}^{\alpha, \beta}}).$$
We next consider a system with a concave corner and introduce a corner topological invariant. By the spectral gap condition, $\check{\gamma}(\check{H}^{\alpha,\beta}) = (H^\alpha, H^\beta)$ and $\check{\gamma}({\check{P}^{\alpha,\beta}}h {\check{P}^{\alpha,\beta}}) = ({P^{\alpha}}h {P^{\alpha}}, {P^{\beta}}h {P^{\beta}})$ are invertible elements. Thus, by Theorem \[Fredholm\], the operators $\check{H}^{\alpha,\beta}$ and ${\check{P}^{\alpha,\beta}}h {\check{P}^{\alpha,\beta}}$ are Fredholm. By the polar decomposition, there is a unique partial isometry $v \in M_{N/2}({\check{{\mathcal{T}}}^{\alpha,\beta}})$ such that ${\check{P}^{\alpha,\beta}}h {\check{P}^{\alpha,\beta}}= v |{\check{P}^{\alpha,\beta}}h {\check{P}^{\alpha,\beta}}|$. By using this, we define the corner invariant.
\[gaplessinvAIII\] We define the [*gapless corner invariant*]{} of our system as follows: $$\check{{\mathcal{I}}}_{\mathrm{Corner}}^{2d, {\mathrm{A\hspace{-.1em}I\hspace{-.1em}I\hspace{-.1em}I}}}(H) := [1-v^*v]_0 - [1- vv^*]_0 \in K_0(K({\check{{\mathcal{H}}}^{\alpha,\beta}})).$$
The following is the bulk-edge and corner correspondence for our system.
\[BECCAIII\] The map $\check{\delta}_1 \colon K_1({\mathcal{S}^{\alpha, \beta}}) \to K_0(K({\check{{\mathcal{H}}}^{\alpha,\beta}}))$ maps the gapped topological invariant ${\mathcal{I}}_{\mathrm{BE}}^{2d, {\mathrm{A\hspace{-.1em}I\hspace{-.1em}I\hspace{-.1em}I}}}(H)$ to the gapless corner invariant $\check{{\mathcal{I}}}_{\mathrm{Corner}}^{2d, {\mathrm{A\hspace{-.1em}I\hspace{-.1em}I\hspace{-.1em}I}}}(H)$.
Since $\hat{\gamma}(v) = (u^\alpha, u^\beta)$, this follows from Proposition $9.2.4$ of [@RLL00].
By using the isomorphism $K_0(\check{\mathrm{Tr}}) \colon K_0(K({\check{{\mathcal{H}}}^{\alpha,\beta}})) \to {\mathbb{Z}}$, we obtain an integer, i.e., the [*numerical*]{} corner invariant. We here write it down explicitly. Since $\check{H}^{\alpha,\beta}$ is Fredholm, $\operatorname{\mathrm{Ker}}\check{H}^{\alpha,\beta}$ is of finite rank. Since $\Pi$ anti-commutes with $\check{H}^{\alpha,\beta}$, $\Pi$ acts on $\operatorname{\mathrm{Ker}}\check{H}^{\alpha,\beta}$. Moreover, since $\Pi^2 = 1$, the space $\operatorname{\mathrm{Ker}}\check{H}^{\alpha,\beta}$ decomposes into the direct sum of $+1$ eigenspace $W^+$ and $-1$ eigenspace $W^-$ of $\Pi |_{\operatorname{\mathrm{Ker}}\check{H}^{\alpha,\beta}}$. We define its [*signature*]{} $\mathrm{sign}( \Pi |_{\operatorname{\mathrm{Ker}}\check{H}^{\alpha,\beta}})$ as the difference of the rank of these spaces, that is, $$\mathrm{sign}(\Pi |_{\operatorname{\mathrm{Ker}}\check{H}^{\alpha,\beta}}) := \operatorname{\mathrm{rank}}W^+ - \operatorname{\mathrm{rank}}W^-.$$ Note that the signature is used to define edge indices of 1-D class AIII topological insulators (see [@PS16], for example). By using this, the [*numerical corner invariant*]{} of our 2-D class AIII system with a codimension-two concave corner is expressed as follows: $$K_0(\check{\mathrm{Tr}})(\check{{\mathcal{I}}}_{\mathrm{Corner}}^{2d, {\mathrm{A\hspace{-.1em}I\hspace{-.1em}I\hspace{-.1em}I}}}(H)) = \operatorname{\mathrm{index}}({\check{P}^{\alpha,\beta}}h {\check{P}^{\alpha,\beta}}) = \mathrm{sign}( \Pi |_{\operatorname{\mathrm{Ker}}\check{H}^{\alpha,\beta}}) \in {\mathbb{Z}}.$$
By using the extension (\[seq1\]) instead of (\[exact\]), we can also treat convex corners in the same way. By using the convex corner Hamiltonian $\hat{H}^{\alpha,\beta}$, the corner topological invariant $\hat{{\mathcal{I}}}_{\mathrm{Corner}}^{2d, {\mathrm{A\hspace{-.1em}I\hspace{-.1em}I\hspace{-.1em}I}}}(H)$ for a 2-D class AIII system with codimension-two convex corner is defined as an element of the $K$-group $K_0(K({\hat{{\mathcal{H}}}^{\alpha,\beta}}))$. Its numerical corner invariant is given by $K_0(\hat{\mathrm{Tr}})(\hat{{\mathcal{I}}}_{\mathrm{Corner}}^{2d, {\mathrm{A\hspace{-.1em}I\hspace{-.1em}I\hspace{-.1em}I}}}(H)) = \mathrm{sign}( \Pi |_{\operatorname{\mathrm{Ker}}\hat{H}^{\alpha,\beta}})$. Moreover, the bulk-edge and corner correspondence holds; that is, $$\label{BECCAIII2}
\hat{\delta}_1({\mathcal{I}}_{\mathrm{BE}}^{2d, {\mathrm{A\hspace{-.1em}I\hspace{-.1em}I\hspace{-.1em}I}}}(H)) = \hat{{\mathcal{I}}}_{\mathrm{Corner}}^{2d, {\mathrm{A\hspace{-.1em}I\hspace{-.1em}I\hspace{-.1em}I}}}(H).$$
The following is a relation between numerical corner invariants for convex and concave corners.
\[minusAIII\] $K_0(\check{\mathrm{Tr}})(\check{{\mathcal{I}}}_{\mathrm{Corner}}^{2d, {\mathrm{A\hspace{-.1em}I\hspace{-.1em}I\hspace{-.1em}I}}}(H)) = - K_0(\hat{\mathrm{Tr}})(\hat{{\mathcal{I}}}_{\mathrm{Corner}}^{2d, {\mathrm{A\hspace{-.1em}I\hspace{-.1em}I\hspace{-.1em}I}}}(H))$
This follows from Corollary \[relation\], Theorem \[BECCAIII\] and (\[BECCAIII2\]).
We now compare our gapped topological invariant ${\mathcal{I}}_{\mathrm{BE}}^{2d, {\mathrm{A\hspace{-.1em}I\hspace{-.1em}I\hspace{-.1em}I}}}(H)$ with [*bulk*]{} topological invariants for 2-D class AIII topological insulators. Let $u := h/|h|$. Through the Fourier transform, $u$ defines a unitary element in $M_{N/2}(C({\mathbb{T}}^2))$ and thus defines an element $[u]_1$ in $K_1(C({\mathbb{T}}^2))$. We have $K_1(C({\mathbb{T}}^2)) \cong {\mathbb{Z}}\oplus {\mathbb{Z}}$ and topological invariants for the bulk Hamiltonian corresponding to these two ${\mathbb{Z}}$ components are called [*weak invariants*]{}.
\[weakAIII\] For Hamiltonians satisfying our spectral gap condition, these two weak invariants are zero.
The algebra ${\mathcal{S}^{\alpha, \beta}}$ is defined as a pullback. By calculating the Mayer-Vietoris exact sequence for the pull-back diagram (\[Sab\]), we can check that $\sigma_* \colon K_1({\mathcal{S}^{\alpha, \beta}})$ $\to K_1(C({\mathbb{T}}^2))$ is the zero map. Since $\sigma_*([(u^\alpha, u^\beta)]_1) = [u]_1$, we have $[u]_1 = 0$, which means that these two weak invariants are both zero.
We next restrict our attention to the case of $\alpha = 0$ and $\beta = \infty$ and consider an explicit example. We first see the following constraint.
\[remrank\] When $N=\operatorname{\mathrm{rank}}V$ is $2$, the corner invariants $\hat{{\mathcal{I}}}_{\mathrm{Corner}}^{2d, {\mathrm{A\hspace{-.1em}I\hspace{-.1em}I\hspace{-.1em}I}}}(H)$ and $\check{{\mathcal{I}}}_{\mathrm{Corner}}^{2d, {\mathrm{A\hspace{-.1em}I\hspace{-.1em}I\hspace{-.1em}I}}}(H)$ for convex and concave corners are both zero.
We first consider the case of convex corners. Since $K_0(\hat{\mathrm{Tr}})$ is an isomorphism, it is sufficient to show that $K_0(\hat{\mathrm{Tr}})(\hat{{\mathcal{I}}}_{\mathrm{Corner}}^{2d, {\mathrm{A\hspace{-.1em}I\hspace{-.1em}I\hspace{-.1em}I}}}(H))$ is zero. Note that $K_0(\hat{\mathrm{Tr}})(\hat{{\mathcal{I}}}_{\mathrm{Corner}}^{2d, {\mathrm{A\hspace{-.1em}I\hspace{-.1em}I\hspace{-.1em}I}}}(H)) = \operatorname{\mathrm{index}}(\hat{P}^{0,\infty} h \hat{P}^{0,\infty})$. When $N = 2$, $h$ is a Fourier transform of a multiplication operator on $L^2({\mathbb{T}}^2)$ generated by a continuous function ${\mathbb{T}}^2 \to {\mathbb{C}}$. Then, the results follow from Corollary in p$208$ of [@DH71]. The result for concave cases follows from Corollary \[zero\].
Thus, to find 2-D class AIII Hamiltonians of nontrivial corner invariants, $N$ must be greater than or equal to $4$ since $N$ is an even integer.
We now give a construction of nontrivial examples. For $j=1,2$, let $V_j$ be ${\mathbb{Z}}_2$-graded finite rank Hermitian vector spaces whose ${\mathbb{Z}}_2$-gradings are given by complex linear maps $\Pi_j \colon V_j \to V_j$ that satisfy $\Pi_j^2 = 1$ $(j=1,2)$. Let $H_j$ be multiplication operators on $l^2({\mathbb{Z}}; V_j)$ generated by continuous maps ${\mathbb{T}}\to \operatorname{\mathrm{End}}(V_j)$, $t \mapsto H_j(t)$. We assume that $H_j$ is self-adjoint invertible and satisfies the relation $\Pi_j H_j \Pi_j^* = -H_j$ $(j=1,2)$. $H_1$ and $H_2$ are Hamiltonians of 1-D class AIII (conventional) topological insulators. Let ${\mathcal{I}}^{1d, {\mathrm{A\hspace{-.1em}I\hspace{-.1em}I\hspace{-.1em}I}}}(H_1)$ and ${\mathcal{I}}^{1d, {\mathrm{A\hspace{-.1em}I\hspace{-.1em}I\hspace{-.1em}I}}}(H_2)$ be their topological invariants, which are defined as follows[^8]. Let $\operatorname{\mathrm{Ker}}H_1 = W_1^+ \oplus W_1^-$ be the eigenspace decomposition with respect to $\Pi_1$, where the action of $\Pi_1$ on $W_1^\pm$ is $\pm 1$, respectively. Let $w_1^+ = \operatorname{\mathrm{rank}}W_1^+$ and $w_1^- = \operatorname{\mathrm{rank}}W_1^-$. Then, we have ${\mathcal{I}}^{1d, {\mathrm{A\hspace{-.1em}I\hspace{-.1em}I\hspace{-.1em}I}}}(H_1) = -w_1^+ + w_1^-$. We also take the eigenspace decomposition $\operatorname{\mathrm{Ker}}H_2 = W_2^+ \oplus W_2^-$ with respect to $\Pi_2$ and let $w_2^+ = \operatorname{\mathrm{rank}}W_2^+$ and $w_2^- = \operatorname{\mathrm{rank}}W_2^-$. Then, we have ${\mathcal{I}}^{1d, {\mathrm{A\hspace{-.1em}I\hspace{-.1em}I\hspace{-.1em}I}}}(H_2) = -w_2^+ + w_2^-$. We consider the following operator on $l^2({\mathbb{Z}}^2; V_1 {\otimes} V_2)$: $$H = H_1 {\otimes} \Pi_2 + 1 {\otimes} H_2.$$ which has a chiral symmetry given by $\Pi := \Pi_1 {\otimes} \Pi_2$. Then, the bulk and two edge Hamiltonians $H$, $H^0$ and $H^\infty$ are all invertible, i.e., gapped at zero (see Theorem $4$ (1) of [@Hayashi2]). Moreover, the following formulae hold:
\[prodthmAIII\]
1. $K_0(\hat{\mathrm{Tr}})(\hat{{\mathcal{I}}}_{\mathrm{Corner}}^{2d,{\mathrm{A\hspace{-.1em}I\hspace{-.1em}I\hspace{-.1em}I}}}(H)) = {\mathcal{I}}^{1d, {\mathrm{A\hspace{-.1em}I\hspace{-.1em}I\hspace{-.1em}I}}}(H_1) \cdot {\mathcal{I}}^{1d, {\mathrm{A\hspace{-.1em}I\hspace{-.1em}I\hspace{-.1em}I}}}(H_2)$,
2. $K_0(\check{\mathrm{Tr}})(\check{{\mathcal{I}}}_{\mathrm{Corner}}^{2d,{\mathrm{A\hspace{-.1em}I\hspace{-.1em}I\hspace{-.1em}I}}}(H)) = - {\mathcal{I}}^{1d, {\mathrm{A\hspace{-.1em}I\hspace{-.1em}I\hspace{-.1em}I}}}(H_1) \cdot {\mathcal{I}}^{1d, {\mathrm{A\hspace{-.1em}I\hspace{-.1em}I\hspace{-.1em}I}}}(H_2)$.
As in Theorem $4$ of [@Hayashi2], $\operatorname{\mathrm{Ker}}\hat{H}^{0,\infty} = \operatorname{\mathrm{Ker}}H_1 {\otimes} \operatorname{\mathrm{Ker}}H_2$ holds. We have $$\begin{aligned}
\operatorname{\mathrm{Ker}}\hat{H}^{0,\infty} &=& (W_1^+ \oplus W_1^-) {\otimes} (W_2^+ \oplus W_2^-)\\
&=& (W_1^+ {\otimes} W_2^+) \oplus (W_1^- {\otimes} W_2^+) \oplus (W_1^+ {\otimes} W_2^-) \oplus (W_1^- {\otimes} W_2^-).\end{aligned}$$ The operator $\Pi$ acts on this space, and we have $$\begin{aligned}
&K_0(\hat{\mathrm{Tr}})(\hat{{\mathcal{I}}}_{\mathrm{Corner}}^{2d,{\mathrm{A\hspace{-.1em}I\hspace{-.1em}I\hspace{-.1em}I}}}(H)) = \operatorname{\mathrm{sign}}\Pi|_{\operatorname{\mathrm{Ker}}\hat{H}^{0,\infty}}\\
&= \operatorname{\mathrm{rank}}(W_1^+ {\otimes} W_2^+) - \operatorname{\mathrm{rank}}(W_1^- {\otimes} W_2^+) - \operatorname{\mathrm{rank}}(W_1^+ {\otimes} W_2^-) + \operatorname{\mathrm{rank}}(W_1^- {\otimes} W_2^-)\\
&= w_1^+ w_2^+ - w_1^- w_2^+ - w_1^+ w_2^- + w_1^-w_2^-
= (w_1^+ - w_1^-)(w_2^+ - w_2^-)\\
&= {\mathcal{I}}^{1d, {\mathrm{A\hspace{-.1em}I\hspace{-.1em}I\hspace{-.1em}I}}}(H_1) \cdot {\mathcal{I}}^{1d, {\mathrm{A\hspace{-.1em}I\hspace{-.1em}I\hspace{-.1em}I}}}(H_2).\end{aligned}$$ This proves (1). (2) follows from (1) and Theorem \[minusAIII\].
Note that to find $H_1$ and $H_2$ of nontrivial topological invariants, the rank of $V_1$ and $V_2$ must be greater than or equal to $2$. Thus, to find an example of a nontrivial corner invariant in this way, the rank of $V_1 {\otimes} V_2$ must be greater than or equal to $4$. This is consistent with Proposition \[remrank\], and an example contained in Sect. \[sectexam\] provides an example of $N=4$.
\[CDScorner\] Numerical corner invariants for convex and concave corners are given by Fredholm indices of convex and concave corner Toeplitz operators, respectively. When $\alpha = 0$ and $\beta =\infty$, the Coburn–Douglas–Singer index formula [@CDS72] and its concave analog (Corollary \[concaveCDS\]) give a topological method to compute them by using gapped Hamiltonians. However, to find a necessary path in the algebra $M_{N/2}(\mathcal{S}^{0,\infty})$ is not necessarily easy in general [@CDS72; @Pa90].
Since we defined topological invariants (${\mathcal{I}}_{\mathrm{BE}}^{2d, {\mathrm{A\hspace{-.1em}I\hspace{-.1em}I\hspace{-.1em}I}}}(H)$ in Definition \[gappedinvAIII\] and $\check{{\mathcal{I}}}_{\mathrm{Corner}}^{2d, {\mathrm{A\hspace{-.1em}I\hspace{-.1em}I\hspace{-.1em}I}}}(H)$ in Definition \[gaplessinvAIII\]) and stated their relation (Theorem \[BECCAIII\]) in a $K$-theoretic way, a generalization to the higher-dimensional case is straightforward, as in Remark $5$ of [@Hayashi2]. For a $(n+2)$-D class AIII system with codimension-two concave corner, a topological invariant for gapped bulk-edges Hamiltonians is defined as an element of $K_1({\mathcal{S}^{\alpha, \beta}}{\otimes} {\mathbb{T}}^{n})$, and a gapless corner invariant is defined as that of $K_0(K({\check{{\mathcal{H}}}^{\alpha,\beta}}) {\otimes} {\mathbb{T}}^{n})$. Let $\check{\delta}_1 \colon K_1({\mathcal{S}^{\alpha, \beta}}{\otimes} {\mathbb{T}}^{n})$ $\to$ $K_0(K({\check{{\mathcal{H}}}^{\alpha,\beta}}) {\otimes} {\mathbb{T}}^{n})$ be a boundary homomorphism associated with a short exact sequence obtained by taking a tensor product of the sequence (\[exact\]) and $C({\mathbb{T}}^n)$. Then, $\check{\delta}_1$ maps the gapped topological invariant to the gapless corner invariant. Its definition and proof are parallel with the one in this subsection.
3-D class A system
------------------
In this subsection, we consider 3-D class A systems with codimension-two concave corners. The contents of this section are almost parallel with [@Hayashi2], but we here use the sequence (\[exact\]) instead of the quarter-plane Toeplitz extension (\[seq1\]) used in [@Hayashi2].
We consider a continuous map ${\mathbb{T}}^3 \to \operatorname{\mathrm{End}}_{\mathbb{C}}(V)$, $(\xi, \eta, t) \mapsto H(\xi, \eta, t)$, where, for each $(\xi,\eta,t) \in {\mathbb{T}}^3$, $H(\xi,\eta,t)$ is Hermitian. The multiplication operator generated by $H(\xi, \eta, t)$ defines a bounded linear operator on $L^2({\mathbb{T}}^3;V)$. Through the Fourier transform, we obtain a bounded linear self-adjoint operator $H$ on $l^2({\mathbb{Z}}^3;V)$ and We call $H$ the [*bulk Hamiltonian*]{}. By the Fourier transform in the last ${\mathbb{Z}}$ component, we obtain a continuous family of bounded linear self-adjoint operators $\{ H(t) \colon {\mathcal{H}}_V \to {\mathcal{H}}_V \}_{t \in {\mathbb{T}}}$. By taking their compressions onto ${{\mathcal{H}}^{\alpha}}_V$ and ${{\mathcal{H}}^{\beta}}_V$, we obtain one-parameter families of half-plane Toeplitz operators, $$\{ H^\alpha(t) := {P^{\alpha}}H(t) {P^{\alpha}}\}_{t \in {\mathbb{T}}}, \ \
\{ H^\beta(t) := {P^{\beta}}H(t) {P^{\beta}}\}_{t \in {\mathbb{T}}},$$ and we call them [*edge Hamiltonians*]{}. We also consider the compression onto ${\check{{\mathcal{H}}}^{\alpha,\beta}}_V$ and obtain the following family of concave corner Toeplitz operators: $$\{ \check{H}^{\alpha,\beta}(t) := {\check{P}^{\alpha,\beta}}H(t) {\check{P}^{\alpha,\beta}}\}_{t \in {\mathbb{T}}},$$ We call them the [*corner Hamiltonian*]{}. The following is our assumption in this subsection.
[**Assumption (Spectral gap condition)**]{} We assume that our edge Hamiltonians have a common spectral gap at the Fermi level $\mu \in {\mathbb{R}}$ for any $t$ in ${\mathbb{T}}$, i.e., $\mu$ is not contained in either $\mathrm{sp}(H^\alpha(t))$ or $\mathrm{sp}(H^\beta(t))$. We refer to this condition as a [*spectral gap condition*]{}.
In what follows, we assume $\mu = 0$ without loss of generality. Under the spectral gap condition, the gapped topological invariant is defined as an element of a $K$-group, that is, ${\mathcal{I}}_{\mathrm{BE}}^{3d, {\mathrm{A}}}(H) \in K_0(\mathcal{S}^{\alpha,\beta} {\otimes} C({\mathbb{T}}))$ (defined at Definition $1$ of [@Hayashi2] and denoted ${\mathcal{I}}_{\mathrm{BE}}(H)$ there). We here consider a concave corner that appears as a union of two half-planes and defines the corner invariant.
\[concavecornerinv\] By the spectral gap condition and Theorem \[main\], we have a continuous family $\{ \check{H}^{\alpha,\beta}(t) \}_{t \in {\mathbb{T}}}$ of bounded linear self-adjoint Fredholm operators. This family defines an element $\check{{\mathcal{I}}}_{\mathrm{Corner}}^{3d, {\mathrm{A}}}(H)$ of the $K$-group $K_1(C({\mathbb{T}}))$. We call $\check{{\mathcal{I}}}_{\mathrm{Corner}}^{3d, {\mathrm{A}}}(H)$ the [*gapless corner invariant*]{}.
Its numerical corner invariant is given by using spectral flow[^9] $\mathrm{sf} \colon$ $K_1(C({\mathbb{T}})) \to {\mathbb{Z}}$, that is, $\mathrm{sf}(\check{{\mathcal{I}}}_{\mathrm{Corner}}^{3d, {\mathrm{A}}}(H)) \in {\mathbb{Z}}$. The following is the bulk-edge and corner correspondence for our system.
\[BECCA\] The map $\check{\delta_0} \colon K_0({\mathcal{S}^{\alpha, \beta}}{\otimes} C({\mathbb{T}})) \to K_1(C({\mathbb{T}}))$ maps ${\mathcal{I}}_{\mathrm{BE}}^{3d, {\mathrm{A}}}(H)$ to the gapless corner invariant. That is, $\check{\delta}_0({\mathcal{I}}_{\mathrm{BE}}^{3d, {\mathrm{A}}}(H)) =\check{{\mathcal{I}}}_{\mathrm{Corner}}^{3d, {\mathrm{A}}}(H)$.
Definition \[concavecornerinv\] and Theorem \[BECCA\] are parallel with Definition $2$ and Theorem $3$ of [@Hayashi2], and we omit the detail. In our setting, we can define convex and concave corner invariants $\hat{{\mathcal{I}}}_{\mathrm{Corner}}^{3d, {\mathrm{A}}}(H)$ and $\check{{\mathcal{I}}}_{\mathrm{Corner}}^{3d, {\mathrm{A}}}(H)$ for convex and concave corners, respectively (the convex corner invariant $\hat{{\mathcal{I}}}_{\mathrm{Corner}}^{3d, {\mathrm{A}}}(H)$ is defined in Definition $2$ of [@Hayashi2] and denoted as ${\mathcal{I}}_{\mathrm{Corner}}(H)$). There is the following relation between these two.
\[minusA\] $\mathrm{sf}(\check{{\mathcal{I}}}_{\mathrm{Corner}}^{3d, {\mathrm{A}}}(H)) = - \mathrm{sf}(\hat{{\mathcal{I}}}_{\mathrm{Corner}}^{3d, {\mathrm{A}}}(H))$.
Let fix a base point of ${\mathbb{T}}$. We have the isomorphism $K_0({\mathcal{S}^{\alpha, \beta}}{\otimes} C({\mathbb{T}}))$ $\cong K_0({\mathcal{S}^{\alpha, \beta}}) \oplus K_0({\mathcal{S}^{\alpha, \beta}}{\otimes} C_0((0,1)))$. The projection onto the second component gives a homomorphism $p \colon K_0({\mathcal{S}^{\alpha, \beta}}{\otimes} C({\mathbb{T}})) \to K_0({\mathcal{S}^{\alpha, \beta}}{\otimes} C_0((0,1)))$. Let $\theta \colon K_1({\mathcal{S}^{\alpha, \beta}}) \to K_0({\mathcal{S}^{\alpha, \beta}}{\otimes} C_0((0,1)))$ be the suspension isomorphism, and let $\beta \colon K_0(K({\check{{\mathcal{H}}}^{\alpha,\beta}})) \to K_1(K({\check{{\mathcal{H}}}^{\alpha,\beta}}) {\otimes} C_0((0,1)))$ be the Bott isomorphism. Then, by Corollary \[relation\], we have $$\begin{gathered}
\mathrm{sf}(\check{{\mathcal{I}}}_{\mathrm{Corner}}^{3d, {\mathrm{A}}}(H)) = \mathrm{sf} \circ \check{\delta}_0 ({\mathcal{I}}_{\mathrm{BE}}^{3d, {\mathrm{A}}}(H))
= \mathrm{sf} \circ \check{\delta}_0 \circ p ({\mathcal{I}}_{\mathrm{BE}}^{3d, {\mathrm{A}}}(H))\\
= K_0(\check{\mathrm{Tr}}) \circ \beta^{-1} \circ \check{\delta}_0 \circ p ({\mathcal{I}}_{\mathrm{BE}}^{3d, {\mathrm{A}}}(H))
= K_0(\check{\mathrm{Tr}}) \circ \check{\delta}_1 \circ \theta^{-1} \circ p ({\mathcal{I}}_{\mathrm{BE}}^{3d, {\mathrm{A}}}(H))\\
= - K_0(\hat{\mathrm{Tr}}) \circ \hat{\delta}_1 \circ \theta^{-1} \circ p ({\mathcal{I}}_{\mathrm{BE}}^{3d, {\mathrm{A}}}(H))
= - \mathrm{sf}(\hat{{\mathcal{I}}}_{\mathrm{Corner}}^{3d, {\mathrm{A}}}(H)),\end{gathered}$$ where the last equality follows by the repetition of the previous equalities for convex corners.
We next consider the case of $\alpha = 0$ and $\beta = +\infty$ (we assume $\mu = 0$) and give a construction of an explicit example. Let $V_3$ be a finite-rank Hermitian vector space. Let $H_3$ be a multiplication operator on $l^2({\mathbb{Z}}^2; V_3)$ generated by a continuous map ${\mathbb{T}}^2 \to \operatorname{\mathrm{End}}(V_3)$. We assume that $H_3$ is self-adjoint and invertible (Hamiltonian of a 2-D class A (conventional) topological insulator). Let ${\mathcal{I}}^{2d, \mathrm{A}}(H_3)$ be the topological number of $H_3$. Let $H_2$ be a bounded linear operator $l^2({\mathbb{Z}}; V_2)$ introduced in Sect. $4.1$ (Hamiltonian of a 1-D class AIII (conventional) topological insulator whose chiral symmetry is implemented by $\Pi_2$). Using these operators, let us consider the following bounded linear self-adjoint operator $H$ on the Hilbert space $l^2({\mathbb{Z}}^3; V_3 {\otimes} V_2)$, $$\label{prodHam}
H = H_3 {\otimes} \Pi_2 + 1 {\otimes} H_2.$$ Its partial Fourier transform gives a family of bounded linear self-adjoint operators $\{ H(t) = H_3(t) {\otimes} \Pi_2 + 1 {\otimes} H_2 \}_{t \in {\mathbb{T}}}$ on the Hilbert space $l^2({\mathbb{Z}}^2; V_3 {\otimes} V_2)$.
\[productA\] We have $\mathrm{sf}(\check{{\mathcal{I}}}_{\mathrm{Corner}}^{3d, {\mathrm{A}}}(H)) = - {\mathcal{I}}^{2d, \mathrm{A}}(H_3) \cdot {\mathcal{I}}^{1d, {\mathrm{A\hspace{-.1em}I\hspace{-.1em}I\hspace{-.1em}I}}}(H_2)$, where the right-hand side is the product of two integers.
By Theorem $4$ (1) of [@Hayashi2], for our Hamiltonian $H$ of the form (\[prodHam\]), the edge Hamiltonians $H^0(t)$ and $H^\infty(t)$ are invertible, and thus, the corner invariant is defined. By Theorem $4$ (2) of [@Hayashi2], we have $\mathrm{sf}(\hat{{\mathcal{I}}}_{\mathrm{Corner}}^{3d, {\mathrm{A}}}(H)) = {\mathcal{I}}^{2d, \mathrm{A}}(H_3) \cdot {\mathcal{I}}^{1d, {\mathrm{A\hspace{-.1em}I\hspace{-.1em}I\hspace{-.1em}I}}}(H_2)$. Then, the results follow by Theorem \[minusA\].
By using these results, we provide an explicit example of a bulk Hamiltonian $H$ such that $H^0$ and $H^\infty$ are both gapped and its corner invariant for the concave corner is nontrivial.
\[examA\] Let $H'_3$ be the following bounded linear self-adjoint operator on $l^2({\mathbb{Z}}^2) {\otimes} {\mathbb{C}}^2 \cong l^2({\mathbb{Z}}^2; {\mathbb{C}}^2)$: $$H'_3 = \frac{1}{2 i} \sum_{j=1,2} (S_j - S_j^*) {\otimes} \sigma_j + \bigl(-1 + \frac{1}{2} \sum_{j=1,2}(S_j + S_j^*) \bigl) {\otimes} \sigma_3,$$ where $S_1 = M_{1,0}$ and $S_2 = M_{0,1}$ are translation operators. $H'_3$ is an example of a $2$-D type A (conventional) topological insulator. Its topological invariant is calculated in [@PS16] and is ${\mathcal{I}}^{2d, \mathrm{A}}(H'_1) = -1$. Let $H'_2$ and $\Pi'$ be following self-adjoint operators on the Hilbert space $l^2({\mathbb{Z}}) {\otimes} {\mathbb{C}}^2 \cong l^2({\mathbb{Z}}, {\mathbb{C}}^2)$: $$\tiny
H'_2 = \frac{1}{2} S {\otimes} (\sigma_1 + i \sigma_2) + \frac{1}{2} S^* {\otimes} (\sigma_1 - i \sigma_2), \
\Pi' = 1 {\otimes} \sigma_3.$$ where $\sigma_1$, $\sigma_2$ and $\sigma_3$ are Pauli matrices[^10] and $S$ is the translation operator given by $(S \varphi)(n) = \varphi(n-1)$. Then, we have $\Pi' H'_2 (\Pi')^* = - H'_2$. This is an example of 1-D class AIII (conventional) topological insulator. Its topological number is ${\mathcal{I}}^{1d, {\mathrm{A\hspace{-.1em}I\hspace{-.1em}I\hspace{-.1em}I}}}(H'_2) = -1$ (see [@PS16]). By using them, we consider the following bounded linear self-adjoint operator on $l^2({\mathbb{Z}}^3;{\mathbb{C}}^4)$: $$H = H'_3 {\otimes} \Pi' + 1 {\otimes} H'_2.$$ By Theorem \[productA\], its numerical corner invariant for the concave corner is computed as $\mathrm{sf}(\check{{\mathcal{I}}}_{\mathrm{Corner}}^{3d,{\mathrm{A}}}(H)) = - {\mathcal{I}}^{2d, \mathrm{A}}(H'_1) \cdot {\mathcal{I}}^{1d,{\mathrm{A\hspace{-.1em}I\hspace{-.1em}I\hspace{-.1em}I}}}(H'_2) = -(-1) \cdot (-1) = -1$. Note that by Theorem \[minusA\], the numerical corner invariant for convex corner is $\mathrm{sf}(\hat{{\mathcal{I}}}_{\mathrm{Corner}}^{3d,{\mathrm{A}}}(H)) = 1$ (see also Example $1$ of [@Hayashi2]).
Example and 2-D BBH model {#sectexam}
=========================
In this section, we introduce an explicit example of 2-D class AIII Hamiltonians whose corner invariant is nontrivial on a system with a codimension-two (convex and concave) corner. Comparing with this example, we discuss Benalcazar–Bernevig–Hughes’ 2-D Hamiltonian [@BBH17a] from our viewpoint.
We first study the following 1-D class AIII Hamiltonian; $$H_{{\mathrm{A\hspace{-.1em}I\hspace{-.1em}I\hspace{-.1em}I}}}(k; \gamma_1, \gamma_2, \lambda_1, \lambda_2) = H_{{\mathrm{A\hspace{-.1em}I\hspace{-.1em}I\hspace{-.1em}I}}}(k) := \gamma_1 \sigma_1 + \gamma_2 \sigma_2 + \lambda_1 \cos(k) \sigma_1 + \lambda_2 \sin(k) \sigma_2$$ where $k \in {\mathbb{R}}/2\pi{\mathbb{Z}}\cong {\mathbb{T}}$. Its chiral symmetry is given by $\sigma_3$. By the Fourier transform, we obtain a bounded linear self-adjoint operator $H_{{\mathrm{A\hspace{-.1em}I\hspace{-.1em}I\hspace{-.1em}I}}}$ on $l^2({\mathbb{Z}}, {\mathbb{C}}^2)$. For simplicity, we assume $\lambda_1 \neq 0$ and $\lambda_2 \neq 0$ . Since $$H_{{\mathrm{A\hspace{-.1em}I\hspace{-.1em}I\hspace{-.1em}I}}}(k) \hspace{-0.3mm} = \hspace{-0.3mm}
\begin{pmatrix}
0 & \gamma_1 \hspace{-0.3mm} - \hspace{-0.3mm} i \gamma_2 \hspace{-0.3mm} + \hspace{-0.3mm} \lambda_1 \hspace{-0.3mm} \cos(k) \hspace{-0.3mm} - \hspace{-0.3mm} i \lambda_2 \sin(k) \hspace{-0.3mm} \\
\gamma_1 \hspace{-0.3mm} + \hspace{-0.3mm} i \gamma_2 \hspace{-0.3mm} + \hspace{-0.3mm} \lambda_1 \hspace{-0.3mm} \cos(k) + i \lambda_2 \hspace{-0.3mm} \sin(k) \hspace{-3mm} & 0 \\
\end{pmatrix},$$ the (bulk) Hamiltonian is invertible (i.e. gapped at zero) when $\left| \gamma_1 / \lambda_1\right|^2 + \left| \gamma_2 / \lambda_2\right|^2 \neq 1$. This is a model of a 1-D class AIII (conventional) topological insulator, and its topological number, which is the winding number of $\gamma_1 + i \gamma_2 + \lambda_1 \cos(k) + i \lambda_2 \sin(k)$ around zero, is the following. $${\mathcal{I}}^{1d, {\mathrm{A\hspace{-.1em}I\hspace{-.1em}I\hspace{-.1em}I}}}(H_{{\mathrm{A\hspace{-.1em}I\hspace{-.1em}I\hspace{-.1em}I}}}) =
\left\{
\begin{aligned}
1, & \hspace{3mm} \text{if} \ \left| \gamma_1/\lambda_1 \right|^2 + \left| \gamma_2/\lambda_2 \right|^2 < 1,\\
0, & \hspace{3mm} \text{if} \ \left| \gamma_1/\lambda_1 \right|^2 + \left| \gamma_2/\lambda_2 \right|^2 > 1.
\end{aligned}
\right.$$
Let $\gamma_{x,1}$, $\gamma_{x,2}$, $\gamma_{y,1}$, $\gamma_{y,2}$, $\lambda_{x,1}$, $\lambda_{x,2}$, $\lambda_{y,1}$ and $\lambda_{y,2}$ be real numbers. By using $H_{\mathrm{A\hspace{-.1em}I\hspace{-.1em}I\hspace{-.1em}I}}$, we consider the following 2-D Hamiltonian, $$\begin{aligned}
&H(k_x, k_y; \gamma_{x,1}, \gamma_{x,2}, \lambda_{x,1}, \lambda_{x,2}, \gamma_{y,1}, \gamma_{y,2}, \lambda_{y,1}, \lambda_{y,2}) := \\
&H_{{\mathrm{A\hspace{-.1em}I\hspace{-.1em}I\hspace{-.1em}I}}}(k_x; \gamma_{x,1}, \gamma_{x,2}, \lambda_{x,1}, \lambda_{x,2}) \otimes 1
+ \sigma_3 \otimes H_{{\mathrm{A\hspace{-.1em}I\hspace{-.1em}I\hspace{-.1em}I}}}(k_y; \gamma_{y,1}, \gamma_{y,2}, \lambda_{y,1}, \lambda_{y,2})\\
&= \gamma_{x,1} \sigma_1 \otimes 1 + \gamma_{x,2} \sigma_2 \otimes 1 + \lambda_{x,1} \cos(k_x) \sigma_1 \otimes 1 + \lambda_{x,2} \sin(k_x) \sigma_2 \otimes 1\\
& + \gamma_{y,1} \sigma_3 \otimes \sigma_1 + \gamma_{y,2} \sigma_3 \otimes \sigma_2 + \lambda_{y,1} \cos(k_y) \sigma_3 \otimes \sigma_1 + \lambda_{y,2} \sin(k_y) \sigma_3 \otimes \sigma_2. \end{aligned}$$ where $k_x, k_y \in {\mathbb{R}}/2\pi{\mathbb{Z}}$. Just for simplicity, we assume that $\lambda_{x,1}$, $\lambda_{x,2}$, $\lambda_{y,1}$ and $\lambda_{y,2}$ are non-zero. This Hamiltonian preserves the chiral symmetry given by $\Pi = \sigma_3 \otimes \sigma_3$. Through the Fourier transform, we obtain a bounded linear self-adjoint operator $H$ on $l^2({\mathbb{Z}}^2, {\mathbb{C}}^4)$. We now take $\alpha = 0$ and $\beta = \infty$ and introduce two edge Hamiltonians $H^0$, $H^\infty$ and the corner Hamiltonian $H^{0,\infty}$. When $\left| \gamma_{x,1} / \lambda_{x,1}\right|^2 + \left| \gamma_{x,2} / \lambda_{x,2} \right|^2 \neq 1$ and $\left| \gamma_{y,1} / \lambda_{y,1} \right|^2 + \left| \gamma_{y,2} / \lambda_{y,2} \right|^2 \neq 1$, the (bulk) Hamiltonians $H_{{\mathrm{A\hspace{-.1em}I\hspace{-.1em}I\hspace{-.1em}I}}}(k_x)$ and $H_{{\mathrm{A\hspace{-.1em}I\hspace{-.1em}I\hspace{-.1em}I}}}(k_y)$ of 1-D class AIII (conventional) topological insulators are invertible. Thus, by Sect. $4.1$ (or Theorem $4$ (1) of [@Hayashi2]), the bulk and two edge Hamiltonians ($H$, $H^0$ and $H^\infty$) are invertible and the numerical corner invariant for the convex corner is defined. Moreover, by Theorem \[prodthmAIII\], its value is the product of topological numbers of two 1-D class AIII (conventional) topological insulators and is computed as follows. $$\begin{gathered}
K_0(\hat{\mathrm{Tr}})\hspace{-0.3mm}(\hat{{\mathcal{I}}}_{\mathrm{Corner}}^{2d,{\mathrm{A\hspace{-.1em}I\hspace{-.1em}I\hspace{-.1em}I}}}(H)\hspace{-0.3mm}) \hspace{-0.3mm}=\hspace{-0.3mm} K_0(\hat{\mathrm{Tr}})\hspace{-0.3mm}(\hat{{\mathcal{I}}}_{\mathrm{Corner}}^{2d,{\mathrm{A\hspace{-.1em}I\hspace{-.1em}I\hspace{-.1em}I}}}(H(\gamma_{x,\hspace{-0.3mm}1}\hspace{-0.3mm},\hspace{-0.3mm} \gamma_{x,\hspace{-0.3mm}2}\hspace{-0.3mm},\hspace{-0.3mm} \lambda_{x,\hspace{-0.3mm}1}\hspace{-0.3mm},\hspace{-0.3mm} \lambda_{x,\hspace{-0.3mm}2}\hspace{-0.3mm},\hspace{-0.3mm} \gamma_{y,\hspace{-0.3mm}1}\hspace{-0.3mm},\hspace{-0.3mm} \gamma_{y,\hspace{-0.3mm}2}\hspace{-0.3mm},\hspace{-0.3mm} \lambda_{y,\hspace{-0.3mm}1}\hspace{-0.3mm}, \hspace{-0.3mm} \lambda_{y,\hspace{-0.3mm}2})\hspace{-0.2mm})\hspace{-0.2mm})\\
= {\mathcal{I}}^{1d, {\mathrm{A\hspace{-.1em}I\hspace{-.1em}I\hspace{-.1em}I}}}(H_{{\mathrm{A\hspace{-.1em}I\hspace{-.1em}I\hspace{-.1em}I}}}(k_x; \gamma_{x,1},\hspace{-0.3mm} \gamma_{x,2},\hspace{-0.3mm} \lambda_{x,1},\hspace{-0.3mm} \lambda_{x,2})) \cdot {\mathcal{I}}^{1d, {\mathrm{A\hspace{-.1em}I\hspace{-.1em}I\hspace{-.1em}I}}}(H_{{\mathrm{A\hspace{-.1em}I\hspace{-.1em}I\hspace{-.1em}I}}}(k_y; \gamma_{y,1},\hspace{-0.3mm} \gamma_{y,2},\hspace{-0.3mm} \lambda_{y,1},\hspace{-0.3mm} \lambda_{y,2}))\\
= \left\{
\begin{aligned}
1, & \hspace{3mm} \text{if} \ \ \left| \frac{\gamma_{x,1}}{\lambda_{x,1}} \right|^2 + \left| \frac{\gamma_{x,2}}{\lambda_{x,2}} \right|^2 < 1, \ \text{and} \ \left| \frac{\gamma_{y,1}}{\lambda_{y,1}} \right|^2 + \left| \frac{\gamma_{y,2}}{\lambda_{y,2}} \right|^2 < 1\\
0, & \hspace{3mm} \text{otherwise}.
\end{aligned}
\right.\vspace{-1mm}\end{gathered}$$ By Theorem \[minusAIII\], the numerical corner invariant $K_0(\check{\mathrm{Tr}})(\check{{\mathcal{I}}}_{\mathrm{Corner}}^{2d,{\mathrm{A\hspace{-.1em}I\hspace{-.1em}I\hspace{-.1em}I}}}(H))$ for the concave corner is also (defined and) computed which is their negative. Thus, when parameters are taken as $\left| \gamma_{x,1} / \lambda_{x,1}\right|^2 + \left| \gamma_{x,2} / \lambda_{x,2} \right|^2 < 1$ and $\left| \gamma_{y,1} / \lambda_{y,1} \right|^2 + \left| \gamma_{y,2} / \lambda_{y,2} \right|^2 < 1$, there exist topologically protected corner states both for concave and concave corners associated with $\alpha = 0$ and $\beta = \infty$.
\[shape\] If we change $\alpha$ or $\beta$, the shape/angle of the corner changes. The previous results [@DH71; @Pa90; @Ji95] and results of Sect. $2$ and $3$ enables us to treat corners of angles less than $\pi$ and bigger than $\pi$, respectively. If we fix the bulk Hamiltonian and change $\alpha$ and $\beta$, a natural question is whether numerical corner invariants changes correspondingly. Example \[examA\] and the above one clarify that numerical corner invariants change depending on the shape of the corner. More precisely, as in Theorem \[minusAIII\] and Theorem \[minusA\], numerical corner invariants for concave and convex corners for fixed $\alpha$ and $\beta$ differ by the factor $-1$.
\[BBH\] Let $U$, $r_4$ and $\Theta$ be following transformations on ${\mathbb{C}}^4$ ; $$U :=
\begin{pmatrix}
0 & 0 & 0 & -1 \\
1 & 0 & 0 & 0 \\
0 & -1 & 0 & 0 \\
0 & 0 & 1 & 0
\end{pmatrix},
\ \
r_4 :=
\begin{pmatrix}
0 & 0 & 1 & 0 \\
1 & 0 & 0 & 0 \\
0 & 0 & 0 & -1 \\
0 & 1 & 0 & 0
\end{pmatrix}.
\ \
\Theta :=
\begin{pmatrix}
c & 0 & 0 & 0 \\
0 & c & 0 & 0 \\
0 & 0 & c & 0 \\
0 & 0 & 0 & c
\end{pmatrix}.$$ where[^11] $c$ is the complex conjugation on ${\mathbb{C}}$. Matrices $U$ and $r_4$ are unitary transformations and $\Theta$ is an anti-unitary transformation. If $\gamma_{x,2} = \gamma_{y,2} = 0$ is satisfied, our Hamiltonian preserves two anti-commuting reflection symmetries. Specifically, let $m_x := -\sigma_1 \otimes \sigma_3$ and $m_y := - 1 \otimes \sigma_1$, then we have, $$m_x H(k_x, k_y)m_x^* = H(-k_x, k_y), \ \ m_y H(k_x, k_y)m_y^* = H(k_x, -k_y).$$ Further, if $\gamma_x = \gamma_y$ and $\gamma_{x,1} = \gamma_{y,1}$ is satisfied, our Hamiltonian preserves time-reversal, particle-hole and $C_4$-symmetries $$\begin{gathered}
\Theta H(k_x, k_y) \Theta^* = H(-k_x, -k_y), \ \
\Xi H(k_x, k_y) \Xi^* = -H(-k_x, -k_y),\\
r_4 H(k_x, k_y) r_4^* = H(k_y, -k_x),\end{gathered}$$ where $\Xi = \Theta \circ \Pi$. In other words, we can see that if $\gamma_{x,2} \neq 0$ and $\gamma_{y,2} \neq 0$, two anti-commuting reflection symmetries, the time-reversal symmetry (TRS) and the particle-hole symmetry (PHS) are broken. If $\lambda_{x,1} \neq \lambda_{y,1}$ or $\lambda_{x,2} \neq \lambda_{y,2}$, the $C_4$-symmetry is broken[^12].
Let us consider the unitary transformation induced by $U$, specifically, consider the following 2-D Hamiltonian[^13]; $$\begin{aligned}
& U H(k_x, k_y; \gamma_{x,1}, \gamma_{x,2}, \lambda_{x,1}, \lambda_{x,2}, \gamma_{y,1}, \gamma_{y,2}, \lambda_{y,1}, \lambda_{y,2})U^* := \\
&= \gamma_{x,1} \sigma_1 \otimes 1 - \gamma_{x,2} \sigma_2 \otimes \sigma_3 + \lambda_{x,1} \cos(k_x) \sigma_1 \otimes 1 - \lambda_{x,2} \sin(k_x) \sigma_2 \otimes \sigma_3 \\
& - \gamma_{y,1} \sigma_2 \otimes \sigma_2 + \gamma_{y,2} \sigma_2 \otimes \sigma_1 + \lambda_{y,1} \cos(k_y) \sigma_2 \otimes \sigma_2 + \lambda_{y,2} \sin(k_y) \sigma_2 \otimes \sigma_1. \end{aligned}$$ When $\gamma_{x,1} = \gamma_{y,1}$, $\gamma_{x,2} = \gamma_{y,2} = 0$ and $\lambda_{x,1} = \lambda_{x,2} = \lambda_{y,1} = \lambda_{y,2}$, this 2-D model is discussed by Benalcazar–Bernevig–Hughes (Equation (6) of [@BBH17a]). In this case, this model preserves TRS, PHS, the chiral symmetry, two anti-commuting reflection symmetries and $C_4$-symmetry specified by the unitary transform of the above operators[^14] [@BBH17a]. For this model, they find the quadrupole phase which hosts topologically protected corner states where they stressed the role of reflection symmetries. Since the unitary transform does not change these topological invariants, as long as we keep track of its chiral symmetry, the above computation also computes the numerical corner invariant of 2-D BBH model both for convex and concave corners associated with $\alpha = 0$ and $\beta = \infty$. For such a special choice of parameters (as in [@BBH17a]), our result about the existence of topologically protected corner states is consistent with that of Benalcazar–Bernevig–Hughes’ and gives another explanation for that. Note that our results states that there exists topologically protected corner states even if we break TRS, PHS, two anti-commuting reflection symmetries and the $C_4$-symmetry.
\[experiment\] After the work of [@BBH17a], corner states are reported to have been observed experimentally in metamaterials [@PBHG18; @Gracia18].
Some variants
=============
As in Remark \[othercases\], most results in this paper also hold in the cases in which the corner (or edges) do not necessarily include lattice points on lines $y=\alpha x$ and $y=\beta x$. In this appendix, we make this statement precise by fixing the setups and clarifying the corresponding results. Although the proofs of the corresponding results are parallel with those contained in the main body of this paper, some parts of the discussions are based on the explicit construction of an example, especially the constructions of rank-one projections (Lemma \[contain\]) and that of the Fredholm concave corner Toeplitz operator of index one (Theorem \[construction\]). For these reasons, we collect the corresponding results in this appendix. The corresponding results for quarter-plane Toeplitz operators, briefly mentioned in [@Ji95], are also included for completeness.
Since we consider two edges, corresponding to whether the edge includes lattice points on boundaries, we can consider four cases. Each case corresponds to the case in which closed subspaces ${{\mathcal{H}}^{\alpha}}$ and ${{\mathcal{H}}^{\beta}}$ of ${\mathcal{H}}$ are spanned by the following sets:
Case $1$ : $\{ { e_{m,n}} \mid -\alpha m + n \geq 0 \}$ and $\{ { e_{m,n}} \mid -\beta m + n \leq 0 \}$, respectively.
Case $2$ : $\{ { e_{m,n}} \mid -\alpha m + n > 0 \}$ and $\{ { e_{m,n}} \mid -\beta m + n \leq 0 \}$, respectively.
Case $3$ : $\{ { e_{m,n}} \mid -\alpha m + n \geq 0 \}$ and $\{ { e_{m,n}} \mid -\beta m + n < 0 \}$, respectively.
Case $4$ : $\{ { e_{m,n}} \mid -\alpha m + n > 0 \}$ and $\{ { e_{m,n}} \mid -\beta m + n < 0 \}$, respectively.
For these cases, we associate concave corners and define concave corner $C^*$-algebras ${\check{{\mathcal{T}}}^{\alpha,\beta}}$ in the same way as in Sect. $2$. Note that Case $1$ is already treated in the main body of this paper. In the following, we assume the condition ($\dagger$) for $\alpha$ and $\beta$.
We first collect constructios of rank-one projections in Cases $2\sim4$. They correspond to Lemma \[contain\] in Case $1$. As in Lemma \[contain\], we take $N \in \{ 2,3, \cdots \}$ such that $\frac{1}{N+1} < \alpha \leq \frac{1}{N}$.
In Case $2 \sim 4$, some ${\tilde{\mathcal{P}}}_k$ is a rank-one projection. Explicitly, we have the following results.
In Case $2$, $\begin{cases}
\text{when} \ \frac{1}{N+1} < \alpha \leq \frac{1}{N} \ \text{and} \ \beta = 1, \text{we have} \ {\tilde{\mathcal{P}}}_{N-1} = p_{-N-1,-1}.\\
\text{when} \ \frac{1}{N+1} < \alpha \leq \frac{1}{N} \ \text{and} \ 1 < \beta < \infty, \text{we have} \ {\tilde{\mathcal{P}}}_N = p_{-N-1,-1}.
\end{cases}$
In Case $3$, $\begin{cases}
\text{when} \ \alpha = \frac{1}{N} \ \text{and} \ 1 < \beta \leq \infty, \ \text{we have} \ {\tilde{\mathcal{P}}}_{N-1} = p_{-N,-1}.\\
\text{when} \ \frac{1}{N+1} < \alpha < \frac{1}{N} \ \text{and} \ 1 < \beta \leq \infty, \text{we have} \ {\tilde{\mathcal{P}}}_N = p_{-N-1,-1}.
\end{cases}$
In Case $4$, $\begin{cases}
\text{when} \ \alpha = \frac{1}{N} \ \text{and} \ \beta = 1, \ \text{we have} \ {\tilde{\mathcal{P}}}_{1} = p_{-1,0}.\\
\text{in the other cases (under $(\dagger)$), we have} \ {\tilde{\mathcal{P}}}_{N} = p_{-N-1,-1}.
\end{cases}$
We here write down the result of computing the Fredholm index of the following operator in Cases $2\sim4$ which corresponds to Theorem \[construction\] in Case $1$. $$\check{A} := \check{{\mathcal{P}}}_{0,1} + M_{1,1}(1 - \check{{\mathcal{P}}}_{-1,0}) + M_{1,0}(\check{{\mathcal{P}}}_{-1,0} - \check{{\mathcal{P}}}_{0,1}).$$
In Cases $2\sim4$, $\check{A}$ is a surjective Fredholm operator whose Fredholm index is $1$. We also have $\check{A} - 1 \in {\check{{\mathcal{C}}}^{\alpha,\beta}}$. Its kernel is given as follows:
In Case $2$, $\begin{cases}
\text{when} \ 0 < \alpha \leq \frac{1}{2} \ \text{and} \ \beta = 1, \operatorname{\mathrm{Ker}}\check{A} = {\mathbb{C}}({{\bm e}}_{-2, 0} - {{\bm e}}_{-1,0}).\\
\text{when} \ 0 < \alpha \leq \frac{1}{2} \ \text{and} \ 1 < \beta < \infty, \operatorname{\mathrm{Ker}}\check{A} = {\mathbb{C}}({{\bm e}}_{-1,0} - {{\bm e}}_{0,0}).
\end{cases}$
In Case $3$, under the assumption $(\dagger)$, we have $\operatorname{\mathrm{Ker}}\check{A} = {\mathbb{C}}({{\bm e}}_{-1,0} - {{\bm e}}_{0,0})$.
In Case $4$, under the assumption $(\dagger)$, we have $\operatorname{\mathrm{Ker}}\check{A} = {\mathbb{C}}({{\bm e}}_{0,1} - {{\bm e}}_{1,1})$.
We next consider the following quarter-plane Toeplitz operator in Cases $1\sim4$. $$\hat{A} := \hat{{\mathcal{P}}}_{0,1} + M_{1,1}(1 - \hat{{\mathcal{P}}}_{-1,0}) + M_{1,0}(\hat{{\mathcal{P}}}_{-1,0} - \hat{{\mathcal{P}}}_{0,1}).$$ Note that $\hat{A} \in {\hat{{\mathcal{T}}}^{\alpha,\beta}}$. Jiang shows in [@Ji95] that, under the assumption ($\dagger$), this is an isometric Fredholm operator and compute its Fredholm index mainly in the Case $1$. The other cases are briefly mentioned (Remark (1) in p2828 of [@Ji95]), though their Fredholm indices are stated as $\pm 1$. We here need to fix its sign in order to obtain the corresponding result for Corollary \[relation\] especially in Cases $2\sim4$. For this reason, we (re)state necessary results in the following. Its proof is totally parallel with that of Jiang [@Ji95].
In Cases $1\sim4$, $\hat{A}$ is an isometric Fredholm operator whose Fredholm index is $-1$. Its cokernel is given as follows:
In Case $1$, under the assumption $(\dagger)$, we have $\operatorname{\mathrm{Coker}}\hat{A} = {\mathbb{C}}{{\bm e}}_{0,0}$.
In Case $2$, under the assumption $(\dagger)$, we have $\operatorname{\mathrm{Coker}}\hat{A} = {\mathbb{C}}{{\bm e}}_{1,1}$.
In Case $3$, $\begin{cases}
\text{when} \ 0 < \alpha \leq \frac{1}{2} \ \text{and} \ \beta = 1, \operatorname{\mathrm{Coker}}\hat{A} = {\mathbb{C}}{{\bm e}}_{2,1}.\\
\text{when} \ 0 < \alpha \leq \frac{1}{2} \ \text{and} \ 1 < \beta < \infty, \operatorname{\mathrm{Coker}}\hat{A} = {\mathbb{C}}{{\bm e}}_{1,1}.
\end{cases}$
In Case $4$, $\begin{cases}
\text{when} \ \alpha = \frac{1}{2} \ \text{and} \ \beta = 1, \operatorname{\mathrm{Coker}}\hat{A} = {\mathbb{C}}{{\bm e}}_{3,2}.\\
\text{when} \ 0 < \alpha < \frac{1}{2} \ \text{and} \ \beta = 1, \operatorname{\mathrm{Coker}}\hat{A} = {\mathbb{C}}{{\bm e}}_{2,1}.\\
\text{when} \ 0 < \alpha \leq \frac{1}{2} \ \text{and} \ 1 < \beta < \infty, \operatorname{\mathrm{Coker}}\hat{A} = {\mathbb{C}}{{\bm e}}_{1,1}.
\end{cases}$
Acknowledgments {#acknowledgments .unnumbered}
---------------
The author would like to thank Takeshi Nakanishi and Yukinori Yoshimura for showing him the result of a numerical calculation, which convinced him about the content of this paper. He also would like to thank Ken-Ichiro Imura and Ryo Okugawa for many discussions concerning [@BBH17a] and Max Lein for sharing the information regarding [@Gracia18]. The author acknowledge the support of the Erwin Schr[ö]{}dinger Institute where part of this work was conducted. He would like to thank organizers of the workshop “Bivariant K-theory in Geometry and Physics” for their hospitability. This work was supported by JSPS KAKENHI Grant Number JP17H06461 and JP19K14545.
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[^1]: Note that our choice of translation direction is the same as [@Ji95] and different from [@Pa90]. In our definition, $M_{m,n}{ e_{s,t}} = { e_{s+m, t+n}}$ holds.
[^2]: The square lattice ${\mathbb{Z}}^2$ is naturally embedded in the Euclidean space ${\mathbb{R}}^2$. As a subset of ${\mathbb{R}}^2$, what we called convex corners are [*not*]{} convex sets. We here use the words [*convex*]{} and [*concave*]{} just to distinguish the two models of corners indicated in Fig. \[convconc\].
[^3]: In order to distinguish these two cases, we use [*hat*]{} “$\wedge$” for objects associated with convex corners and [*check*]{} “$\vee$” for those with concave corners (e.g., ${\hat{{\mathcal{H}}}^{\alpha,\beta}}$ and ${\check{{\mathcal{H}}}^{\alpha,\beta}}$).
[^4]: In what follows, $K$-groups of $C^*$-algebras ${\check{{\mathcal{T}}}^{\alpha,\beta}}$ and ${\check{{\mathcal{C}}}^{\alpha,\beta}}$ are computed, and the result for ${\mathcal{S}^{\alpha, \beta}}$, ${{\mathcal{C}}^\alpha}$ and ${{\mathcal{C}}^\beta}$ are presented corresponding to the values of $\alpha$ and $\beta$. The case of $\alpha = -\infty$ or $\beta = + \infty$ is the same as that of rational $\alpha$ or rational $\beta$.
[^5]: If $A$ is an algebra, $M_r(A)$ denotes the algebra of all $r \times r$ matrices with entries in $A$.
[^6]: ${P^{\alpha}}h {P^{\alpha}}/|{P^{\alpha}}h {P^{\alpha}}|$ is defined by the continuous functional calculous by the continuous function ${\mathbb{C}}\setminus \{ 0\} \to {\mathbb{C}}$ given by $z \mapsto z/|z|$.
[^7]: This element does not depend on the choice of the identification $V \cong {\mathbb{C}}^N$.
[^8]: We here give the definition of edge topological invariants for 1-D class AIII topological insulators. By the bulk-edge correspondence, this coincides with the bulk topological invariant which is defined as the winding number of the determinant of its symbol, that is $\mathrm{Wind}(\{\det h_j(t)\}_{t \in {\mathbb{T}}})$ where $H_j =
\begin{pmatrix}
0 & h_j^*\\
h_j & 0
\end{pmatrix}$. (see [@PS16], for example).
[^9]: We here regard ${\mathbb{T}}$ as the unit circle in the complex plane and fix the counter-clockwise orientation.
[^10]: $
\sigma_1 =
\begin{pmatrix}
0 & 1\\
1 & 0
\end{pmatrix}, \
\sigma_2 =
\begin{pmatrix}
0 & -i\\
i & 0
\end{pmatrix}, \
\sigma_3 =
\begin{pmatrix}
1 & 0\\
0 & -1
\end{pmatrix}.
$
[^11]: We here employ the following identification: $
\begin{pmatrix}
a & b\\
c & d
\end{pmatrix}
\otimes A
=
\begin{pmatrix}
aA & bA\\
cA & dA
\end{pmatrix}.
$
[^12]: Note that $r_4 (\sigma_2 \otimes 1) r_4^* = - \sigma_3 \otimes \sigma_2$, $r_4(\sigma_1 \otimes 1)r_4^* = \sigma_3 \otimes \sigma_1$, $ r_4(\sigma_3 \otimes \sigma_1)r_4^* = \sigma_1 \otimes 1$ and $r_4 (\sigma_3 \otimes \sigma_2) r_4^* = \sigma_2 \otimes 1$ holds.
[^13]: Note that we have $U(\sigma_1 \otimes 1) U^* = \sigma_1 \otimes 1$, $U(\sigma_2 \otimes 1) U^* = - \sigma_2 \otimes \sigma_3$, $U(\sigma_3 \otimes \sigma_1) U^* = - \sigma_2 \otimes \sigma_2$ and $U(\sigma_3 \otimes \sigma_2) U^* = - \sigma_2 \otimes \sigma_1$
[^14]: Specifically, they are $U \Theta U^* = \Theta$, $U \Xi U^* = (\sigma_3 \otimes \sigma_1) \circ \Theta$, $U \Pi U^* = \sigma_3 \otimes \sigma_1$, $U m_x U^* = \sigma_1 \otimes \sigma_3$, $U m_y U^* = \sigma_1 \otimes \sigma_1$ and $U r_4 U^* =
\begin{pmatrix}
0 & 1 \\
-i\sigma_2 & 0
\end{pmatrix}$, respectively.
|
---
abstract: 'The 331 model offers an explanation of flavor by anomaly cancellation between three families. It predicts three exotic quarks, $Q=D$, $S$, $T$, and five extra gauge bosons comprising an additional neutral $Z_2$ and four charged dileptonic gauge bosons $(Y^{--},Y^-)$, $(Y^{++},Y^+)$. Production of $Q\overline{Q}$, $QY$, $YY$ and $Z_2$ at the SSC is calculated, and signatures are discussed.'
address:
- |
Institute of Field Physics, Department of Physics and Astronomy,\
University of North Carolina, Chapel Hill, NC 27599–3255, USA
- |
TRIUMF, 4004 Wesbrook Mall\
Vancouver, B.C., V6T 2A3, Canada
author:
- 'Paul H. Frampton, James T. Liu, B. Charles Rasco'
- Daniel Ng
title: |
SSC Phenomenology of the\
331 Model of Flavor
---
#### Introduction. {#introduction. .unnumbered}
The standard model (SM) of strong and electroweak interactions is extremely successful. All experimental data are consistent with the minimal version of the SM, and so theoretical extensions of the SM must be motivated not, alas, by experiment, but by attempting to understand features that are accommodated in the SM but not explained by it. Perhaps the most profound such feature is the replication of quark-lepton families where the lightest family $(u,d,e,\nu_e)$ is repeated twice more, $(c,s,\mu,\nu_\mu)$, $(t,b,\tau,\nu_\tau)$, with the only difference between the families lying in the particle masses.
For consistency of a gauge theory, chiral anomalies must cancel [@adler; @bell; @bouchiat], and it is a remarkable fact that this cancellation occurs between quarks and leptons in the SM within each family separately. For one family, this can be used with only a few simple assumptions to deduce the electric charges of the fundamental fermions including electric charge quantization without recourse to the assumption of any simple grand unifying group containing electric charge as a generator [@geng]. The presence of more than one family, however, is accommodated rather than explained in the SM. In the 331 model [@pleitez; @frampton], which is an extension of the SM, one obtains a theory with three extended families. While each extended family has a non-vanishing chiral anomaly, the three families taken together do not. This then offers a possible first step in understanding the flavor question.
#### The Model. {#the-model. .unnumbered}
The 331 model has gauge group ${\rm SU}(3)_c \times {\rm SU}(3)_L \times {\rm U}(1)_X$ (hence the name). There are five additional gauge bosons beyond the SM; a neutral $Z'$ and four dileptons, $(Y^{--},Y^-)$ with lepton number $L = +2$ and $(Y^{++},Y^+)$ with lepton number $L = -2$. Here $L = L_e + L_\mu + L_\tau$ is the total lepton number; the 331 model does not conserve the separate family lepton numbers $L_i$ ($i=e,\mu,\tau$). The new $Z'$ will mix with the $Z$ of the SM to give mass eigenstates $Z_2$ and $Z_1$, but the singly-charged dilepton will not mix with the $W^\pm$ in the minimal 331 model where total lepton number $L$ is conserved.
In the 331 model the leptons are the same in number as in the SM; however the usual doublet–singlet pattern per family is replaced by one antitriplet of electroweak ${\rm SU}(3)_L$. Namely the leptons in each family are in a $({\bf1},{\bf3}^*)_0$ under $(3_c,3_L)_X$. The quarks are in $({\bf3},{\bf3})_{-1/3}$ and $({\bf3}^*,{\bf1})_{-2/3,1/3,4/3}$ for the first and second families and in $({\bf3},{\bf3}^*)_{+2/3}$ and $({\bf3}^*,{\bf1})_{-5/3,-2/3,+1/3}$ for the third family [@twomodels]. The result is that there are three new quarks in the 331 model called $D$, $S$, and $T$ respectively with electric charges $-4/3$, $-4/3$ and $+5/3$, and these provide targets of discovery at the SSC. From the manner in which the dileptons couple to the heavy quarks, we see that the latter also carry lepton number. $D$ and $S$ have $L = +2$ and $T$ has $L = -2$.
The minimum Higgs structure necessary [@frampton] for symmetry breaking and giving quarks and leptons acceptable masses is three complex ${\rm SU}(3)_L$ triplets, $\Phi({\bf1},{\bf3})_1$, $\phi({\bf1},{\bf3})_0$, and $\phi'({\bf1},{\bf3})_{-1}$ and a complex sextet $H({\bf1},{\bf6})_0$. The breaking of 331 to the SM is accomplished by a vacuum expectation value (VEV) of the neutral component of the $\Phi({\bf1},{\bf3})_1$, and this gives the only tree-level contribution to the masses of the heavy quarks $D$, $S$, and $T$ and the principal such contribution to the masses of the new gauge bosons $Z'$ and $Y$. After ${\rm SU}(3)_L$ symmetry breaking we are left with a three-Higgs doublet SM with new heavy scalars, quarks $D$, $S$, and $T$ and gauge bosons $Z'$ and $(Y^{\pm\pm},Y^\pm)$. In this letter, we examine the production and signatures of these new quarks and gauge bosons.
#### Gauge Boson Masses and Mixing. {#gauge-boson-masses-and-mixing. .unnumbered}
For triplets, we define the ${\rm SU}(3)_L\times {\rm U}(1)$ couplings $g$ and $g_X$ according to the covariant derivative $$D_\mu=\partial_\mu-i g T^a W_\mu^a -i g_X XT^9 X_\mu\ ,$$ where $T^a=\lambda^a/2$ and $T^9=\rm diag(1,1,1)/\sqrt{6}$ are normalized according to ${\rm Tr}\, T^a T^b = {1\over2}\delta^{ab}$ and $\lambda^a$ are the usual Gell-Mann matrices. Note that $T^9=1/\sqrt{6}$ for ${\rm SU(3)}_L$ singlets. $X$ is the ${\rm U}(1)_X$ charge of the representations given above and is related to the usual hypercharge by $Y/2=\sqrt{3}T^8+\sqrt{6}XT^9$ where $Q=T^3+Y/2$.
When ${\rm SU}(3)_L$ is broken to ${\rm SU}(2)_L$ by the VEV $\langle\Phi^a\rangle = \delta^{a3} U/\sqrt{2}$, the new gauge bosons get masses at tree level, $M_Y^2 = (1/4)g^2U^2$ and $M_{Z'}^2 = (1/6)(2g^2 + g_X^2)U^2$, giving $$\label{eq:massrel}
{M_Y\over M_{Z'}}=\sqrt{3g^2\over4g^2+2g_X^2}\ ,$$ which is an ${\rm SU}(3)_L$ generalization of the $\rho$-parameter.
Using the matching conditions at the ${\rm SU}(3)_L$ breaking scale [@ng], $g_X$ is given by $$\label{eq:match}
{g_X^2\over g^2}={6\sin^2\theta_W(M_{Z'})\over1-4\sin^2\theta_W(M_{Z'})}\ ,$$ which allows us to rewrite $$\label{eq:mymzp}
{M_Y\over M_{Z'}}={\sqrt{3(1-4\sin^2\theta_W(M_{Z'}))}\over
2\cos\theta_W(M_{Z'})}\ ,$$ which gives a relationship between $M_Y$ and $M_{Z'}$ of $M_Y\le 0.26M_{Z'}$ for $\sin^2\theta_W(M_Z)=0.233$. This relationship is specific to the minimal Higgs structure where ${\rm SU}(3)_L$ breaking is accomplished only by triplet Higgs. Where possible we will discuss cross-sections for extended ranges of masses to allow for the possibility of a more general non-minimal 331 model, but will use Eq. (\[eq:mymzp\]) when a definite relationship between $M_Y$ and $M_{Z'}$ is required.
At the electroweak scale, Eq. (\[eq:massrel\]) picks up small corrections due to ${\rm SU}(2)_L$ breaking. This allows the dilepton doublet to be split in mass and also gives rise to $Z$–$Z'$ mixing. Here, $Z$ is the standard mixture of $W^3$ and $Y$; whereas $Z'$ is the gauge eigenstate orthogonal to $Z$ and $\gamma$. Because the quarks in the third family are in a different ${\rm SU}(3)_L$ representation, the $Z'$ coupling differentiates among families and hence there are tree-level flavor-changing neutral currents (FCNC) in the left-handed sector involving light quarks coupled to $Z'$ [@pleitez; @ng].
The $Z$–$Z'$ mixing arises from the mass matrix (in the $\{Z,Z'\}$ basis) $$\label{eq:mixmat}
{\cal M}^2=\pmatrix{M_Z^2&M_{ZZ'}^2\cr M_{ZZ'}^2&M_{Z'}^2}\ ,$$ where $M_Z^2=M_W^2/\cos^2\theta_W$ and $M_{ZZ'}^2$ are proportional to ${\rm SU}(2)_L$ breaking VEVs only. Diagonalizing the mass matrix gives the mass eigenstates $Z_1$ and $Z_2$ which can be taken as mixtures, $Z_1=Z\cos\phi-Z'\sin\phi$ and $Z_2=Z\sin\phi+Z'\cos\phi$. The mixing angle, $\phi$, is given by $$\tan^2\phi={M_Z^2-M_{Z_1}^2\over M_{Z_2}^2-M_Z^2}\ ,$$ where $M_{Z_1}=91.173$GeV and $M_{Z_2}$ are the physical mass eigenvalues.
Because the entries in Eq. (\[eq:mixmat\]) arise from the same Higgs VEVs, $M_{Z_2}$ and $\phi$ are related. The mixing angle is constrained to lie between the solid lines shown in Fig. \[figmix\]. Additional constraints on this mixing arise from analysis of weak neutral current and precision electroweak measurements [@ng; @langacker; @liu]. Shown on Fig. \[figmix\] is a lower bound on $M_{Z_2}$ from FCNC [@ng]. Since this bound comes from first–third family mixing, it is sensitive to the values of the CKM parameters as well as new mixing angles coming from the new quarks.
The matching condition, Eq. (\[eq:match\]), has an interesting consequence — namely $\sin^2\theta_W<1/4$ at the 331 breaking scale [@frampton]. Since $\sin^2\theta_W$ runs towards larger values at higher energies, this provides an upper bound on the new physics. Using the running of $\sin^2\theta_W(M_{Z_2})$ for a three-Higgs doublet SM and demanding that $\alpha_X(\mu)$ not be too strong puts an upper limit on $M_{Z_2}$. For $\alpha_X(\mu)$ we impose, [*faute de mieux*]{}, an upper bound $\alpha_X(\mu) < 2\pi$, which implies $M_{Z_2} < 2200$GeV as shown in Fig. \[figmix\].
An [*indirect*]{} lower limit on $M_{Z_2}$ follows from Eq. (\[eq:mymzp\]) and the empirical lower bound on the dilepton mass [@dng], particularly that coming from polarized muon decay [@carlson]. Measurement of the $e^+$ endpoint spectrum [@jodidio] gives the lower bound $M_Y>400$GeV while the muon spin rotation technique [@beltrami] gives $M_Y>300$GeV (both at 90% C.L.). Using the weakest lower bound, we find $M_{Z_2} > 1400$GeV, which is shown in Fig. \[figmix\]. In turn, this gives a strong limit on $Z$–$Z'$ mixing, $-.02<\phi<+.001$, assuming the minimal Higgs sector.
The minimal 331 model is thus very predictive, and equally very easy to rule out, giving a narrow window for the $Z_2$ mass between 1400GeV and 2200GeV and a corresponding window of between 300GeV and 430GeV for the dilepton mass.
We note that the ${\rm U(1)}_X$ coupling constant $g_X$ diverges at a Landau pole less than one order of magnitude above the $Z_2$ mass. Because of this divergence, we assume that $\alpha_X$ does not run above $M_{Z_2}$ (and hence $\alpha_X<2\pi$ is always satisfied) when evaluating the elementary cross-sections at SSC energies. We expect such a behavior to be imposed by new physics above the $Z_2$ mass in a more complete theory.
#### $Z_2$ Production. {#z_2-production. .unnumbered}
Production of $Z_2$, which, due to the negligible mixing, is the same as $Z'$, is dominantly by $q\overline{q}$ annihilation, and the resulting cross-section is given by the solid line in Fig. \[figZp\] (left-hand vertical axis). The branching ratios for $Z_2$ decay are also shown (right-hand vertical axis, dotted curves); here $M_Q=600$GeV has been assumed. The $\phi$ are the light physical scalars of the three-Higgs SM, assumed to have a common mass $M_\phi=200$GeV. Note from Fig. \[figZp\] that the branching ratio into leptons is extremely small — this is because leptons have $X=0$ and the $Z_2$ is dominantly in the ${\rm U}(1)_X$. Nevertheless, the cross-section for $Z_2$ production is so large that the charged lepton decay mode should be easily visible at the SSC. The decay into a pair of dileptons will provide healthy statistics for probing the $Z_2$. Both these features are different from usual $Z'$ phenomenologies [@robinett; @leung; @mahan; @glashow; @he]. Forward-backward asymmetry would also be important to distinguish this model from other $Z'$ models [@lanrobros].
#### Heavy Quark Production. {#heavy-quark-production. .unnumbered}
Production of $Q\overline{Q}$ in $pp$ collisions proceeds through both the strong and the electroweak interactions. The strong interaction is through gluon fusion $gg\to
Q\overline{Q}$ and quark-antiquark annihilation $q\overline{q}\to Q\overline{Q}$, and these are identical to top quark $t\overline{t}$ production. New production mechanisms in the 331 model, however, are available in the electroweak sector by $t$-channel dilepton exchange and, more importantly, by $s$-channel $Z_2$ exchange. The latter mechanism dominates production of $Q\overline{Q}$ for intermediate and large $M_Q$, being an order of magnitude larger in cross-section than the strong interactions for $M_Q\sim{1\over2}M_{Z_2}$ or above; this is because $\alpha_X(\mu)$ is much larger than $\alpha_3(\mu)$ at these energies.
The decay modes of $Q$ are dictated by the conservation of lepton number. Because, as already discussed, $D$ and $S$ have $L = +2$ they decay mainly by $Q\rightarrow qY$. This leads in $pp\rightarrow Q\overline{Q}$ to two or more jets from $q\overline{q}$ plus two lepton pairs from the $Y$s. When both $Y$s are doubly-charged, the leptons are like-sign, possibly resonant if $M_Q>M_Y$, and present a clear signature. For all $M_Q$ up to 1TeV, the $Q$ production cross-sections are in the multi-picobarn range and there are substantial event rates.
#### Dilepton Production. {#dilepton-production. .unnumbered}
By lepton number conservation, dileptons must be produced either in pairs, $YY$, or in association with a heavy quark $QY$. Pair production occurs from $q\overline{q}$ annihilation through $s$-channel $\gamma$, $Z_1$, $Z_2$ or $W$ exchange and $t$-channel heavy quark exchange. Depending on the flavors in the $q\overline{q}$ initial state, one can produce $Y^{++}Y^{--}$, $Y^{++}Y^-$, $Y^+Y^-$, or $Y^+Y^{--}$. We have computed the cross-sections using the leading log parametrization of the structure functions given by Morfin and Tung [@morfin] evaluated at $Q^2=\hat s$. The results are depicted by the solid lines in Fig. \[figYYQY\] for $M_Q=600$GeV. The pairs $Y^{++}Y^{--}$ and $Y^+Y^-$ are produced similarly: the cross-sections are nearly the same and are dominated by the $Z_2$ resonance below the threshold where $Z_2\to Q\overline{Q}$ becomes kinematically allowed.
Associated production of a dilepton with a heavy quark is by quark-gluon fusion, either by $s$-channel $q$ exchange or by $t$-channel $Q$ exchange diagrams. The results for the production cross-sections are depicted by the dotted lines in Fig. \[figYYQY\]. These are the cross-sections for production of a given dilepton in association with an arbitrary heavy quark $Q=D$ or $S$. In general, we find that associated production is larger in cross-section than pair production, and depends more strongly on the mass $M_Q$ of the heavy quark.
The decays of heavy quarks has already been discussed. The dilepton decays either into a lepton pair or, if kinematically allowed, into $qQ$ or scalar pairs. The doubly-charged dileptons $Y^{++}$ and $Y^{--}$ give the especially simple signature of like-sign di-lepton pairs and is the most striking effect predicted by the 331 model at the SSC. As discussed above, the minimal version of the theory demands that the dilepton mass is only just above the current empirical lower bound 300GeV $< M_Y < 430$GeV so the SSC production cross-sections are large, from one to several tens of picobarns. Aside from the $s$-channel production $e^-e^- \rightarrow Y^{--}$ [@paul], the creation in a $pp$ collider is the best discovery mode for dileptons.
Indirect searches for the existence of doubly-charged dileptons in $e^+e^-$ colliders such as LEP-II and NLC have been previously studied [@dng]. Deviations from the SM expectations could show up even at LEP-II. Direct searches have also been considered in $e^-p$ colliders [@agrawal]; although the cross-sections at HERA are undetectably small, dileptons would be readily observable at LEP-II–LHC.
#### Discussion. {#discussion. .unnumbered}
Because the dilepton mass is so restricted by the minimal 331 model, its striking decay signature into like-sign di-lepton pairs would be readily detectable at the SSC. Similarly the $Z_2$, despite its higher mass, is easily within reach of the SSC. These new gauge bosons would enrich the SM into a new extended standard model which has the advantage of requiring three families, rather than only accommodating them. Of course, we shall again be at merely another stepping-stone to the final theory. But unless we confirm what the stepping-stone is, it is not possible to guess with confidence what grand-unified theory or even superstring theory it signals at extreme high energy near or at the Planck scale. Once the families are dealt with, however, there may be a better chance for the bold extrapolation across an assumed desert to be successful.
Along these lines, the Landau pole in $\alpha_X(\mu)$ mentioned above makes clear that new physics beyond the minimal 331 model is crucial even before the desert begins. What this new physics is should be ascertainable or, at least, strongly restricted by the requirement of embedding the theory in a simple or quasi-simple unifying group and by the usual requirements about running couplings meeting together at the unification scale. The anomaly cancellation in the minimal 331 model already suggested a grand unified theory as the origin, and now the Landau pole shows why failure to find a simple unification for the minimal model was not surprising since something extra is needed to make the abelian ${\rm U}(1)_X$ more asymptotically free even before the long extrapolation towards the Planck mass. We expect that these considerations about even higher energies will not affect the predictions of the minimal 331 model we have presented at energies accessible to the SSC and hence await with interest what the SSC will reveal.
We thank Ernest Ma for useful discussions. This work was supported in part by the U.S. Department of Energy under Grant No. DE-FG-05-85ER-40219 and by the Natural Science and Engineering Research Council of Canada.
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|
---
abstract: 'Dilepton-jet final states are used to study physical phenomena not predicted by the standard model. The ATLAS discovery potential for leptoquarks and Majorana Neutrinos is presented using a full simulation of the ATLAS detector at the Large Hadron Collider. The study is motivated by the role of the leptoquark in the Grand Unification of fundamental forces and the see-saw mechanism that could explain the masses of the observed neutrinos. The analysis algorithms are presented, background sources are discussed and estimates of sensitivity and the discovery potential for these processes are reported.'
author:
- 'Vikas Bansal (ATLAS Collaboration)'
title: 'ATLAS Sensitivity to Leptoquarks, $W_{\bf R}$ and Heavy Majorana Neutrinos in Final States with High-${\bf {\ensuremath{p_{T}}}}$ Dileptons and Jets with Early LHC Data at 14 TeV proton-proton collisions'
---
Introduction
============
The Large Hadron Collider (LHC) will soon open up a new energy scale that will directly probe for physical phenomena outside the framework of the Standard Model (SM). Many SM extensions inspired by Grand Unification introduce new, very heavy particles such as leptoquarks. Extending the SM to a larger gauge group that includes, [*e.g.*]{} Left-Right Symmetry (LRS) [@Mohapatra:1975it], could also explain neutrino masses via the see-saw mechanism. The LRS-based Left-Right Symmetric Model (LRSM) [@K.-Huitu:1997ye] used as a guide for presented studies, extends the electroweak gauge group of the SM from SU(2)$_L$ $\times$ U(1)$_Y$ to SU(2)$_L$ $\times$ SU(2)$_R$ $\times$ U(1)$_{B-L}$ and thereby introduces $Z^\prime$ and right-handed $W$ bosons. If the LRS breaking in nature is such that all neutrinos become Majoranas, the LRSM predicts the see-saw mechanism [@Mohapatra:1981oz] that elegantly explains the masses of the three light neutrinos.
Search for scalar leptoquarks
=============================
Leptoquarks (LQ) are hypothetical bosons carrying both quark and lepton quantum numbers, as well as fractional electric charge [@Buchmuller:1986iq; @Georgi:1974sy]. Leptoquarks could, in principle, decay into any combination of any flavor lepton and any flavor quark. Experimental limits on lepton number violation, flavor-changing neutral currents, and proton decay favor three generations of leptoquarks. In this scenario, each leptoquark couples to a lepton and a quark from the same SM generation[@Leurer:1993em]. Leptoquarks can either be produced in pairs by the strong interaction or in association with a lepton via the leptoquark-quark-lepton coupling. Figure \[fig:LQ\_feynman\] shows Feynman diagrams for the pair production of leptoquarks at the LHC.
This contribution describes the search strategy for leptoquarks decaying to either an electron and a quark or a muon and a quark leading to final states with two leptons and at least two jets. The branching fraction of a leptoquark to a charged lepton and a quark is denoted as $\beta$[^1].
MC-simulated signal events have been studied[@ATLAS_CSC_PHYSICS_BOOK:2008] using Monte Carlo (MC) samples for first generation (1st gen.) and second generation (2nd gen.) scalar leptoquarks simulated at four masses of 300 GeV, 400 GeV, 600 GeV, and 800 GeV with the MC generator [Pythia]{} [@pythia] at 14 TeV $pp$ center-of-mass energy. The next to leading order (NLO) cross section [@Kramer:2004df] for the above simulated signal decreases with leptoquark mass from a few pb to a few fb with mass point of 400 GeV at (2.24$\pm$0.38) pb.
--------------------- ----------- ----------- --------------- --------------- ----------------------------------------- --
Physics Before Baseline $S_T$ $M_{ee}$ M$_{lj}^{1}$ - M$_{lj}^{2}$ mass window
sample selection selection $\ge 490$ GeV $\ge 120$ GeV \[ 320-480 \] - \[ 320-480 \]
LQ (400 GeV) 2.24 1.12 1.07 1.00 0.534
$Z$/DY $\ge$ 60 GeV 1808. 49.77 0.722 0.0664 0.0036
$t\bar{t}$ 450. 3.23 0.298 0.215 0.0144
VB pairs 60.94 0.583 0.0154 0.0036 0.00048
Multijet $10^{8}$ 20.51 0.229 0.184 0.0
--------------------- ----------- ----------- --------------- --------------- ----------------------------------------- --
\[cut\_flow\_LQ\_ee\]
------------------- ----------- ----------- -------------------------- -------------- -------------- ---------------------- --
Physics Before Baseline p$^{\mu}_{T}$$\ge$60 GeV S$_{T}$ $M(\mu\mu)$ M$_{lj}$ mass window
sample selection selection p$^{jet}_{T}$$\ge$25 GeV $\ge600$ GeV $\ge110$ GeV \[ 300 - 500 \]
LQ (400 GeV) 2.24 1.70 1.53 1.27 1.23 0.974
$Z$/DY$\ge$60 GeV 1808. 79.99 2.975 0.338 0.0611 0.021
$t\bar{t}$ 450. 4.17 0.698 0.0791 0.0758 0.0271
VB pairs 60.94 0.824 0.0628 0.00846 0.00308 0.00205
Multijet $10^{8}$ 0.0 0.0 0.0 0.0 0.0
------------------- ----------- ----------- -------------------------- -------------- -------------- ---------------------- --
\[cut\_flow\_LQ\_mm\]
Reconstruction and objects selection {#baseline_selection}
------------------------------------
Signal reconstruction requires selection of two high quality leptons and at least two jets. Each signal jet and lepton candidate is required to have transverse momentum (${\ensuremath{p_{T}}})>$ 20 GeV. This helps to suppress low ${\ensuremath{p_{T}}}$ background predicted by the SM. Leptons are required to have pseudorapidity $|\eta|$ below 2.5, which is the inner detector’s acceptance, whereas jets are restricted to $|\eta| < 4.5$ to suppress backgrounds from underlying event and minimum bias events that dominate in the forward region of the detector. In addition, leptons are required to pass identification criteria, which, in case of electrons, are based on electromagnetic-shower shape variables in the calorimeter and, in the case of muons, are based on finding a common track in the muon spectrometer and the inner detector together with a muon isolation[^2] requirement in the calorimeter. Electron candidates are also required to have a matching track in the inner detector. Furthermore, it is required that signal jet candidates are spatially separated from energy clusters in the electromagnetic calorimeter that satisfy electron identification criteria. Finally, a pair of leptoquark candidates are reconstructed from lepton-jet combinations. Given the fact that these four objects can be combined to give two pairs, the pair that has minimum mass difference between the two leptoquark candidates is assumed to be the signal.
Background Studies {#LQ_Bckg}
------------------
The main backgrounds to the signal come from $t\bar{t}$, and $Z /DY$+jets production processes. Multijet production where two jets are misidentified as electrons, represents another background to the dielectron(1st gen.) channel. In addition, minor contributions arise from diboson production. Other potential background sources, such as single-top production, were also studied and found to be insignificant.
The backgrounds are suppressed and the signal significance is improved by taking advantage of the fact that the final state particles in signal-like events have relatively large ${\ensuremath{p_{T}}}$. A scalar sum of transverse momenta of signal jets and lepton candidates, denoted by ${\ensuremath{S_{T}}}$, helps in reducing the backgrounds while retaining most of the signal. The other variable used to increase the signal significance is the invariant mass of the two leptons, $M_{ll}$. The distributions of these two variables for the first generation channel are shown in Fig. \[Lq\_ee\_ST\_Mee\_cuts\].
After applying optimized selection on these two variables, ${\ensuremath{S_{T}}}$ and $M_{ll}$, relative contributions from the background processes from $t\bar{t}$, $Z /DY$, diboson and multijet are 22%, 7%, 0.4% and 18%, respectively. Partial cross-section for the signal and the background processes passing the selection criteria are shown in tables \[cut\_flow\_LQ\_ee\] and \[cut\_flow\_LQ\_mm\] for the first and second generation channels, respectively. Figure \[Lq\_ee\_400\_2D\_proj\_before\_and\_after\_cuts\] shows the invariant masses[^3] of the reconstructed leptoquark candidates before and after background suppression criteria are applied to the MC data.
--------------------- ----------- ----------- --------------- ---------------- --------------- ---------------
Physics Before Baseline $M(ejj)$ $M(eejj)$ $M(ee)$ $S_T$
sample selection selection $\ge 100$ GeV $\ge 1000$ GeV $\ge 300$ GeV $\ge 700$ GeV
LRSM\_18\_3 0.248 0.0882 0.0882 0.0861 0.0828 0.0786
LRSM\_15\_5 0.470 0.220 0.220 0.215 0.196 0.184
$Z$/DY $\ge$ 60 GeV 1808. 49.77 43.36 0.801 0.0132 0.0064
$t\bar{t}$ 450. 3.23 3.13 0.215 0.0422 0.0165
VB pairs 60.94 0.583 0.522 0.0160 0.0016 0.0002
Multijet $10^{8}$ 20.51 19.67 0.0490 0.0444 0.0444
--------------------- ----------- ----------- --------------- ---------------- --------------- ---------------
\[lrsm\_ee\_table\_selection\_criteria\]
--------------------- ----------- ----------- --------------- ---------------- --------------- ---------------
Physics Before Baseline $M(\mu jj)$ $M(\mu\mu jj)$ $M(\mu\mu)$ $S_T$
sample selection selection $\ge 100$ GeV $\ge 1000$ GeV $\ge 300$ GeV $\ge 700$ GeV
LRSM\_18\_3 0.248 0.145 0.145 0.141 0.136 0.128
LRSM\_15\_5 0.470 0.328 0.328 0.319 0.295 0.274
$Z$/DY $\ge$ 60 GeV 1808. 79.99 69.13 1.46 0.0231 0.0127
$t\bar{t}$ 450. 4.17 4.11 0.275 0.0527 0.0161
VB pairs 60.94 0.824 0.775 0.0242 0.0044 0.0014
Multijet $10^{8}$ 0.0 0.0 0.0 0.0 0.0
--------------------- ----------- ----------- --------------- ---------------- --------------- ---------------
\[lrsm\_mm\_table\_selection\_criteria\]
Sensitivity and Discovery Potential
-----------------------------------
ATLAS’s sensitivity to leptoquark signal for a 400 GeV mass hypothesis and with an integrated $pp$ luminosity of 100 ${\ensuremath{\mathrm{pb^{-1}}}}$ is summarized in Fig. \[LQ\_sensitivity\]. The cross-sections include systematic uncertainties of 50%. Leptoquark-like events in the ATLAS detector are triggered by single leptons with an efficiency of 97%. ATLAS is sensitive to leptoquark masses of about 565 GeV and 575 GeV for 1st and 2nd generations, respectively, at the given luminosity of 100 ${\ensuremath{\mathrm{pb^{-1}}}}$ provided the predicted cross-sections for the pair production of leptoquarks are correct.
Search for $W_R$ bosons and heavy Majorana neutrinos
====================================================
$W_R$ bosons are the right-handed counterpart of the SM $W$ bosons. These right-handed intermediate vector bosons are predicted in LRSMs and can be produced at the LHC in the same processes as the SM’s $W$ and $Z$. They decay into heavy Majorana neutrinos. The Feynman diagram for $W_R$ production and subsequent decay to Majorana neutrino is shown in Fig. \[lrsm\_fig\_feynman\].
This section describes the analysis of $W_R$ production and its decays $W_R \to e N_e$ and $W_R \to \mu N_\mu$, followed by the decays $N_e \to e q^\prime \bar{q}$ and $N_\mu \to \mu q^\prime \bar{q}$, which are detected in final states with (at least) two leptons and two jets. The two leptons can be of either same-sign or opposite-sign charge due to the Majorana nature of neutrinos. This analysis in both the dielectron and the dimuon channels has been performed without separating dileptons into same-sign and opposite-sign samples.
Studies [@ATLAS_CSC_PHYSICS_BOOK:2008] of the discovery potential for $W_R$ and Majorana neutrinos N$_e$ and N$_{\mu}$ have been performed using MC samples where M(N$_l$) = 300 GeV; M(W$_R$)= 1800 GeV (referred to as LRSM\_18\_3) and M(N$_l$) = 500 GeV; M(W$_R$) = 1500 GeV (referred to as LRSM\_15\_5), simulated with PYTHIA according to a particular implementation [@A.-Ferrari:2000si] of LRSM [@K.-Huitu:1997ye]. The production cross-sections $\sigma(pp(14{\rm~TeV}) \rightarrow W_R X)$ times the branching fractions $(W_R \rightarrow l N_l \rightarrow l l j j)$ are 24.8 pb and 47 pb for LRSM\_18\_3 and LRSM\_15\_5, respectively.
Reconstruction and objects selection {#reconstruction-and-objects-selection}
------------------------------------
Signal event candidates are reconstructed using two electron or muon candidates and two jets that pass the standard selection criteria as discussed in section \[baseline\_selection\]. The two signal jet candidates are combined with each of the signal leptons and the combination that gives the smaller invariant mass is assumed to be the new heavy neutrino candidate. The other remaining lepton is assumed to come directly from the decay of the $W_R$ boson. If signal electrons and signal jets overlap in $\Delta R$ within 0.4 then, to avoid double counting, only the two signal jets are used to reconstruct the invariant masses of the heavy neutrino candidate and $W_R$.
Background Studies {#background-studies}
------------------
The main backgrounds to the LRSM analyses studied here are the same as mentioned in section \[LQ\_Bckg\]. The same background suppression criteria as in the leptoquark analyses are also effective here, namely S$_T$ and m$_{ll}$. The distributions of these two variables for the dimuon channel are shown in Fig. \[lrsm\_mm\_fig\_st\_mll\]. Partial cross-section for the signal and the background processes passing the selection criteria are shown in tables \[lrsm\_ee\_table\_selection\_criteria\] and \[lrsm\_mm\_table\_selection\_criteria\] for the dielectron and dimuon channels, respectively. Figure \[lrsm\_mm\_fig\_wr\_masses\] shows the invariant mass of the reconstructed $W_R$ candidates before and after background suppression criteria are applied to the MC data.
Sensitivity and Discovery Potential
-----------------------------------
Signal significance for $W_R$ analyses in the dielectron and dimuon channels as a function of integrated $pp$ luminosity at 14 TeV is summarized in Fig. \[lrsm\_fig\_discovery\]. The results include systematic uncertainty of 45% and 40% for dielectron and dimuon channel, respectively. The events in this analysis are also triggered by single leptons with an efficiency of 97%.
Conclusions
===========
Dilepton-jet based final states have been discussed in both electron and muon channels. Discovery potential for leptoquarks and LRSM with early LHC data have been investigated with the predicted cross-sections for these models. Assuming a $\beta$ = 1, both 1st and 2nd generations leptoquarks could be discovered with masses up to 550 GeV with 100 ${\rm pb^{-1}}$ of data. Two LRSM mass points LRSM\_18\_3 and LRSM\_15\_5 for the $W_R$ bosons and heavy Majorana neutrinos have been studied. The discovery of these new particles with such masses would require integrated luminosities of 150 ${\rm pb^{-1}}$and 40 ${\rm pb^{-1}}$, respectively.
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[^1]: $\beta = 1$ would mean that leptoquarks do not decay into quarks and neutrinos.
[^2]: $E_T^{iso}/p_T^\mu \le 0.3$, where $p_T$ is muon candidate’s transverse momentum and $E_T^{iso}$ is energy detected in the calorimeters in a cone of $\Delta$R=$\sqrt{(\Delta\eta^2 + \Delta\phi^2)}$=0.2 around muon candidate’s reconstructed trajectory.
[^3]: These distributions contain two entries per event corresponding to the two reconstructed leptoquark candidates.
|
---
abstract: 'Skin is a highly non-linear, anisotropic, rate dependent inelastic, and nearly incompressible material which exhibits substantial hysteresis even under very slow (quasistatic) loading conditions. In this paper, a series of uniaxial cyclic loading tests of porcine and rat skin at different strain rates and with samples oriented in different directions (with respect to the spine) were conducted to study the effect of strain rate and samples orientations with respect to spine on mullins effect and skin inelastic response. A noteworthy feature of skin is that, similar to certain filled rubbers, its mechanical response shifts after the first extension and exhibits softening and hysteresis when loaded under cyclic tension and mullins effect is observed. The results of these strain-controlled cyclic loading tests also indicated that the extent of softening is different for different strain rates and orientations. Also a substantial hysteresis persists even at very low strain rates indicating inelastic behavior beyond the rate sensitive viscoelastic response. Through this series of experiments, by investigating the effect of strain rate on pig skin and rat skin, we conclude that the skin response is rate dependent but inelastic and shows irreversible changes in fiber orientation which are observed in histology results. Also, skin shows persistent deformation that is only partially recovered even after a long period of unloading.'
author:
- 'N. Afsar-Kazerooni'
- 'A. R. Srinivasa'
- 'J.C. Criscione'
bibliography:
- 'reference.bib'
title: Experimental Investigation of the Inelastic Response of Pig and Rat Skin under Uniaxial Cyclic Mechanical Loading
---
Introduction
============
The largest organ in the body is skin. It is composed of several layers including a stratified, cellular epidermis and an underlying dermis of connective tissue [@McGrath10]. More recently a whole new layer of fluid vesicles (called the intersitium) has been discovered by Benias et al. and it exists right below skin surface. There is a layer of fat under the dermis which separates these layers of skin from the rest of the body [@Benias18]. The dermis contains collagen, elastin and extra fibrillar matrix materials. The collagen fibers are made of protein and provide the strength of skin and are necessary for the shape and structure of skin [@ricard11]. Elastin, which is another component of skin, has a role in mechanical properties of skin too. Oxlund et al. showed that after loading and deformation, elastin fibers play a role in the recoiling mechanism, but they are not primarily responsible for bearing load unlike collagen [@OXLUND88]. When these fibers are damaged or lost due to aging, skin loses its stiffness and elasticity and wrinkling is observed [@Naylor11].
Preliminary investigations have shown that these collagenous tissue found in skin, cartilage, blood vessels, and the cornea are tough, compliant, and capable of withstanding multiaxial cyclic loading with minimal damage. Experiments have shown that under loading [@LIAO99] and [@HILL12], the fibers are initially slack, then they are gradually “recruited” as deformation proceeds. Eventually the fibers become taut and the stress dramatically increases with very little further elongation.
Brown et al. showed that stress-strain curve of abdominal human skin under uniaxial tension test contains 3 stages on the curve [@Brown73]. Based on the results from scanning electron microscopy, Brown et al. observed that at the beginning of first stage, collagen fibers are wavy and not elongated [@Brown73]. At this point, elastin fibers play an important role in stretching and after a while, collagen fibers slowly elongate with increasing strain. This is now responsible for the extension of skin and at this stage the slope of stress strain curve is very low in comparison with the next stages. By further increasing the strain, collagen fibers get aligned. This stretching of collagen fibers at the second stage is known as the linear region [@Xu11]. In this stage, collagen fibers are much more aligned in the direction of loading and these aligned fibers increase the stiffness of skin. The stress as well as the stiffness at this stage is high in comparison with the previous stage. At the last stage, the stress reaches an ultimate tensile strength of collagen, the fibers start failing, and the slope decreases again until failure [@Brown73] and [@Xu11].
Haut et al. performed uniaxial tensile testing on rat skin until failure [@Haut89]. The results showed that the tensile strength is different when samples are cut in different directions with respect to spine, since they have different direction with respect to Langer lines. Langer lines are correlated to the natural orientation of collagen fibers in the dermis and they are mostly perpendicular to the spine in animals [@swaim90]. Also, the tensile strength are sensitive to strain rate and show different strength values at high and low strain rates. The tensile strength is highly correlated to the degree of collagen crosslinking [@Dombi93]. Annaidh et al. measured the slope of the stress-strain curve for human skin [@Annaidh12]. They carried out monotonic uniaxial tensile testing on human skin at different orientations with respect to the Langer lines and histology was done on three samples to find collagen fiber orientation. Slopes for with the angle of 45 degree with respect to Langer lines were between the parallel and perpendicular directions with respect to Langer lines. North et al. used a Schrader double acting hydraulic cylinder (which measures changes in volume by applying pressure), and showed that human abdominal skin can be considered as an incompressible material since changes in volume compressibility of skin is smaller than that of water for large changes in pressure [@North78].
In previous studies, most of the experiments on skin have been only for monotonic loading. While there has been quite a few researches on the inelasticity of soft tissue, there has been very little published on skin under cyclic loading. It has also long been known that when skin undergoes large deformation under cyclic loading, it exhibits strong hysteresis [@Fung13].
Kang et al. investigated the mechanical behavior of porcine skin under uniaxial cyclic loading and from the histological images, they observed irrecoverable micro-mechanisms such as sliding of collagen fibers and fibrils [@Kang11]. The experiments and histological studies have clearly demonstrated the inelastic behavior of skin and the mechanisms of damage, however, the modeling of this behavior has been quite challenging.
Under cyclic loading skin shows some similar response as filled rubbers does. Under these conditions filled rubbers, after loading in the first cycle, most of the softening happens. After the first cycle, the following cycles coincide with each other [@Mullins49]. This softening is called mullins effect in elastomers (e.g. [@Diani09]). The softening is higher at higher strains. Also, it was observed that at strains lower than the strain at peak stress, and at strains higher than the strain at peak, the softening will disappear and stress-strain response will go back to the first path and follow it. Lokshin et al. [@lokshin09] have studied the response of precodntioned skin by subjecting it to step strains and fully unloading the sample. They modeleed the skin as purely viscoelastic solid where the original shape and response is recovered with time upon unloading. On the other hand, mullins effect is, in essence, an irreversible shape and response change phenomenon. Purely viscoelastic response functions such as that developed by Lokshin et al. [@lokshin09] can not capture this effect and it is necessary to have an inelasticity model that can show this effect [@lokshin09]. Furthermore, Nava et al. [@Nava04] demonstrated that biological tissues during surgery behave more like tissue during the first cycle under cyclic loading rather than the tissue after preconditioning. They further go on to state that ”This analysis demonstrates that a quasi-linear viscoelastic model fails in describing the observed evolution from the “virgin” to the preconditioned state. Good agreement between simulation and measurement are obtained by introducing an internal variable changing according to an evolution equation”. Lanir et al. [@lokshin09]. Primarily focused on “precondioned” skin and therefore did not account for the inelastic changes. They also did not report on the measured response during preconditioning, focusing instead only on the stabilized response. In view of this, in this paper we present experimental results on the response of “virgin" skin by subjecting it strain cycling at different rates, with both full and partial unloading. It is hoped that the results will be of use to modelers of skin for use in surgical simulators or virtual reality simulators, diagnosis as well as simulation of the interaction of skin with wearable devices [@Avis00] and [@Satava99].
There are only a few studies on mullins effect on skin. In terms of the experimental evidence Munoz et al.[@Munoz08] performed uniaxial cyclic loading on abdominal region of mice skin . The results show softening after first cycle and they expressed it as mullins effect, similar to the case of the response of filled rubber. Several features of the response are similar to the response in this work which will be discussed in next sections. However, they have restricted their loading to only one direction with respect to the spine. Furthermore, since it is not clear from their paper what kind of stress and strain measures were used, it is difficult to have a detailed comparisons with models etc. More Recently, Zhu et al. [@Zhu14] performed monotonic and force controlled cyclic tension tests on the dorsal area of pig skin from two perpendicular direction . Since, they ran force controlled tests, it is difficult to see internal loops and measure the extent of softening. There are still many unanswered questions regarding the inelastic response of skin. For example, is the response elastic under quasistatic loading so that hysteresis is entirely due to viscoelasticty? Is there a directional dependence of the response? If so, are they qualitatively similar? What happens when skin is reloaded without being fully unloaded, does it follow the unloading curve or does it follow an entirely new path? The aim of this paper is to show that skin shows a strong mullins effect, i.e., irreversible change in response superimposed on a time dependent phenomenon and to quantify the effect in terms of both stress strain response as well as fiber reorientation (histological data) so that modelers can utilize the data. The fact that there is an irreversible change is demonstrated both via the stress strain response and histology.
For this purpose, the experimental data is provided on the response of pig and rat skin subjected to strain controlled partial cyclic loading at different strain rates and with three different directions compared to the spine axis followed by images from optical microscopy to show fiber alignment before and after cyclic tensile testing. The experiments demonstrate that both pig and rat skin have complex behavior (especially under partial unloading) but show strikingly similar features in common with the response of filled rubbers although the microstructural processes that give rise to this are different. The experiments also reveal that there is strong anisotropy of response (even though the response shows a similar nature in different directions). We show further that the internal hysteresis decreases with increasing strain rates and that hysteresis persists for quasistatic response (which is a mark of inelasticity) . We quantify the change in the hysteresis by measuring the amount of energy dissipated per unit strain for different strain rates.
Material and Method
===================
Sample Preparation and Set-up
-----------------------------
Large skins from the abdominal and back region of adult pigs were obtained from Veterinary School of Texas A$\&$M University. First, skins were shaved clean of hair using a razor blade and 12 samples with length of 20 mm, average thickness of 2 mm and average width of 7 mm were cut from the large skin with epidermis and dermis layers (the initial widths and the initial thicknesses were measured at four different location along the length and then their average were reported as an initial width and thickness). Samples were cut in three different directions with respect to the direction of spine, perpendicular, parallel and with the angle of 45 degrees with respect to spine to investigate the effect of anisotropy of skin (-(d)) on behavior of skin under uniaxial cyclic loading. In order to prevent slipping of samples from grips, sand paper was glued to a piece of rubber and placed between the skin sample and the wedge action grips of Instron 5567 testing machine with load cell capacity of $\pm$ 5 kN (-(a)). Also, in order to study the behavior of different kind of skin, 10 rat skin samples were harvested from adult rats from Veterinary School of Texas A$\&$M University with length of 20 mm, average thickness of 1.5 mm and average width of 5 mm. Samples preparation and gripping for rat skins were the same as pig skin. In order to find the relation between actual tissue deformation and grip displacement, after calculating and comparing the strain from grip displacement, and the strain from image correlation along the sample from different positions of the sample, the difference between both measurement was observed to be less than 5 percent. The details of the comparison between grip displacements and DIC is described in the Appendix.
Experiments
-----------
Pig skin samples were tested under cyclic uniaxial loading at strain rates of 0.011 $s^{-1}$ (displacement rate of 13 mm/min) and 0.22 $s^{-1}$ (displacement rate of 260 mm/min) at the room temperature. Similar tests were done on rat skin samples at the strain rate of 0.008 $s^{-1}$ (displacement rate of 10 mm/min) and 0.5 $s^{-1}$ (displacement rate of 500 mm/min) also at the room temperature. In order to keep the samples wet during the tension tests and prevent them from drying that it does not affect or change the behavior of skin [@Berardesca95], Veterinary 0.9 percent Sodium Chloride solution was sprayed on each sample during the test. The cyclic loading protocol was as follows: first, samples were stretched to a specific elongation (point B in -(a)) and then partially unloaded until reached a certain elongation (point C in -(a)). Then this strain controlled loading and unloading cycles were repeated for a few times and force was measured.
For each direction, two samples were tested to check the repeatability of the tests. In order to investigate the effect of time on recovery of skin, one sample from rat skin and one sample from pig skin were tested at day 0 and again after 3 days of keeping at refrigerator at $0^{\circ}C$ the same samples were tested. Also, the morphology of outer surface of skin (which is called stratum corneum of the same sample) was observed under the optical microscope to study changes that happened after 3 days. Finally, in order to find fiber orientations, after cyclic loading, pig skin samples were cut and after fixation, tissue processing and sectioning, they were stained. Then slides were observed under VHX-600 Digital Microscope. Fiber orientations were measured by ImageJ software.
Results
=======
Engineering stress and engineering strain were computed from force displacement data. Engineering stress was calculated by dividing the force over the reference area (initial thickness multiply by initial length). Engineering strain was computed by dividing displacement over initial length. -(b) shows the path of engineering stress and engineering strain versus time during partially loading and unloading.
\[ht\]
shows the stress strain response of pig skins. The cyclic uniaxial testing of pig skin dogbone samples which were cut in directions of perpendicular, parallel, and with an angel of 45 degrees with respect to the spine at the strain rate of 0.22 $s^{-1}$ are shown in (a)-(c). Tests were found to be repeatable and followed the same path and showed a repeatable response within the margin of less than 8 percent. Same experiments were carried out at the strain rate of 0.011 $s^{-1}$ and the results are shown in the (d)-(f), these figures show repeatability of the test.
\[ht\]
shows the results of uniaxial cyclic partially loading and unloading of rat skin at two different strain rates of 0.008 $s^{-1}$ and 0.5 $s^{-1}$. At the strain rate of 0.008 $s^{-1}$ samples were tested from three different orientations with respect to the spine: direction 1 which is perpendicular to the spine direction, direction 2 which is parallel to the spine direction and direction 3 which is cut by an angle of 45 degree with respect to spine. At the strain rate of 0.5 $s^{-1}$ samples were tested from two different orientations, direction 1 which is perpendicular to the spine direction and direction 2 which is parallel to the spine direction. The stress strain response under cyclic loading for both pig and rat skin shows the following features:
1. During the first loading (AB in -(a)) the initial modulus is very small but increases rapidly beyond a certain strain.
2. When the strain is reduced (BC in -(a)) the unloading curve shows evidence of hysteresis.
3. Upon reloading, the response approximates the unloading curve (but not exactly, i.e. it shows unloading reloading hysteresis) up to nearly the previous peak and then appears to continue along the original source this is usually referred to as softening.
4. The hysteresis between the first loading and unloading (i.e. path ABC) is much greater than the subsequent cycles (i.e. path CDEFG).
5. All of this clearly reveals mullins effect in skin.
6. If the unloading curve reaches zero load (i.e. full unloading) the response reveals a persistent strain (see point K in -(a)).
7. Comparing (a)-(c) and (d)-(f), we see that if the same experiment is carried out at a higher rate, the stresses are higher at a certain strain (i.e. there is a certain amount of strain rate dependent hardening).
8. However, the unloading, reloading hysteresis decreases with increasing rate ().
9. Also, at higher strain rates, the transition from lower modulus to higher modulus and hardening region happens at lower strain a confirming observations by Zhou et al. [@Zhou10].
10. Next, upon comparing the response between the 0 and 90 direction, the direction perpendicular with respect to spine has higher slope (especially at region 2 and 3) than the parallel direction ((a)-(b)).
11. The 45 degree experiment shows that it has the slope between the other two directions ((a)-(b)).
12. True stresses and logarithmic strains at each direction at different strain rate ((c)-(e) shows similar behavior as engineering stresses and strains.
13. The behavior of the rat skin (see ) also showed similar qualitative features although the stress amplitudes were quite different.
Skin shows anisotropic behavior [@Minns73]. This behavior of skin was studied under uniaxial cyclic loading on pig skin at different strain rates and different samples from 3 different directions with respect to spine. (a)-(b) show anisotropy in pig skin behavior. (c)-(e) shows that at the effect of strain rate is more dominant at higher strains. In order to investigate whether these are permanent or not, one sample of pig skin and one of rat skin were subject to uniaixal cycling upto a maximum strain of 60 $\%$ (see at a strain rate of 0.011 $s^{-1}$ on both rat skin and pig skin with the original orientation of perpendicular to the orientation of spine. Both samples were tested at day 0 and then kept in a refrigerator (at $0^{\circ}C$) for 3 days. After 3 days, uniaxial testing were carried out on them again. In order to investigate the effect of storing skin on mechanical behavior of skin, one previously untreated rat and one pig samples were tested after 3 days, as control samples.
shows images of the surface of outer layer of pig skin and rat skin which is called stratum corneum under optical microscopy. This layer is composed of dead and dying skin cells from the underlying epidermis. (A-a) shows the pig skin image before uniaxal cyclic testing at day 0 and (A-b) shows the pig skin image after uniaxial cyclic loading. (A-c) shows the image of the pig skin after cyclic loading and storing in a fridge for 3 days. (A-d) shows pig skin image after cyclic loading at day 3 and it is elongated. (B-a) is rat skin before uniaxial cyclic loading at day 0, as the figure shows, fibers have random orientation, and in (B-b), fibers are elongated. Then after storing in a fridge for 3 days, the sample was investigated under microscope. As (B-c) shows, fibers are still elongated in comparison with (B-a), and (B-d) shows the rat skin image after uniaxial cyclic testing after 3 days.
\[ht\] ![ Microstructure of samples after staining: before deformation (sample 0) and after deformation under uniaxial tensile testing: sample 1 was cut in the perpendicular direction with respect to spine and 2 was cut in the parallel direction with respect to spine.[]{data-label="fig:8"}](fig8 "fig:")
. \[fig:9\]
![a) Standard deviation over fiber distribution and b) Maximum probability of fiber distributions of each sample at different directions.[]{data-label="fig:10"}](fig10)
Fiber recruitments and realignment of collagen fibers plays an important role in mechanical behavior of biological tissues. Histology has been done to help modelers to have a more comprehensive data to model the behavior of skin. shows the histology results of pig skin samples before and after deformation. Sample 0 is an undeformed sample and it was cut in 3 different direction: A is a cut in a perpendicular direction with respect to spine, B is a cut in a parallel direction with respect to spine and C is a cut parallel to dermis. A and B are cuts through all layers. Sample 1 was cut in the perpendicular direction with respect to spine and 2 was cut in the parallel direction with respect to spine. For these two samples, A is a cut in a parallel direction with loading direction, B is a cut in the perpendicular direction with loading direction and C is a cut parallel to dermis. -a shows Standard deviation over fiber distribution and -b shows Maximum probability of fiber distributions of each sample at different directions. Quantitative experimental data was provided on the distribution of fibers before and after deformation. These fibers recruitment and reorientation were quantitatively measured in .
Discussion
==========
Usually experiments in the literature are about fully unloading conditions [@lokshin09]. In this paper, partially unloading was performed ( and ) which is not usually done, and it reveals information about structure of materials. In fully loading and unloading, the hysteresis is larger than partially unloading, so it is necessary to perform both types of unloading to find a more comprehensive constitutive equation that can capture the whole behavior of skin. One of the focuses of this study is to show the effect of strain rate on mullins effect (the softening that happens during the first unloading). As Eshel et al. showed, the relaxation time for the tissue is in the order of hundreds of seconds [@Eshel01]. When strain rate is fast, it means that strain rates are faster than relaxation time for the tissue. For example, a strain rate of 0.1 per second [@Eshel01] (to a strain of 0.6 implies that the experiment will be completed in 6 seconds is very fast as compared to the relaxation time of the tissue. On the other hand, 0.008 per second will be considered slow since the time taken to complete the loading to 0.6 will be about the relaxation time, ie the material will have time to relax before the experiment is complete so it is considered slow. Skin shows anisotropic behavior [@Minns73]. This behavior of skin was studied under uniaxial cyclic loading on pig skin at different strain rates and different samples from 3 different directions with respect to spine. compares skin responses at different direction and then compares different strain rates at each direction and it shows that the effect of strain rate is more dominant at higher strains. Orientation of samples with respect to Langer lines or spine affects the mechanical properties of skin and stress strain curves; especially modulus at the second stage will be different for each orientation. Stiffening is increased by increasing the angle between the sample and the orientation of spine. The maximum value is observed for the direction of perpendicular to the spine which is almost parallel to Langer lines of that region [@Ottenio15]. As it was expected, shows that the results of rat skin were similar to the results of the pig skins. In their stress strain curves, mullins effect were observed, they showed that softening occurs after the first cycle and the amount of softening is increased by increasing the strain [@Mullin48]. Also, they showed anisotropic behavior similar to pig skin and in direction perpendicular to the spine, rat skin is more stiff and by deceasing the angle between the direction of sample and the spine, stiffness decreases [@Ottenio15] but pig skin is more sensitive to direction with respect to spine than rat skin. In order to investigate the effect of strain rate and direction on the mullins effect and softening that happens to skin, maximum stresses at both first and second cycles are normalized to the maximum stress at first cycle. shows that stress at the second cycle approximately decreases by 11 percent for the strain rate of 0.011 $s^{-1}$. For the strain rate of 0.22 $s^{-1}$, softening happened by 15 percent. The results show that the amount of softening is more sensitive to strain rate than orientation of samples with respect to spine or Langer lines. The stress strain behavior dependency on strain rate is due to the movement of collagen fibers which are starting to be aligned and uncoiled [@lanir79]. expresses similar behaviors between pig skin and rat skin. It shows that rat skin is more sensitive to strain rate rather than orientation with respect to the spine orientation. At low strain rate, the amount of softening was approximately 13 percent and at higher strain rate (0.5 $s^{-1}$), it was approximately 20 percent.
\[tab:1\]
-------------- ------------ ------------ ------------ ------------ ------------ ------------
Direc. 1 Direc. 1 Direc. 2 Direc. 2 Direc. 3 Direc. 3
Normalized 0.011 0.22 0.011 0.22 0.011 0.22
stress ($s^{-1}$) ($s^{-1}$) ($s^{-1}$) ($s^{-1}$) ($s^{-1}$) ($s^{-1}$)
First cycle 1 1 1 1 1 1
Second cycle 0.89 0.86 0.89 0.85 0.88 0.85
-------------- ------------ ------------ ------------ ------------ ------------ ------------
: Normalized stresses at the end of each cycle at the strain of 0.6 at different strain rates on pig skins which were cut in direction 1 which is perpendicular to the spine direction, direction 2 which is parallel to the spine direction and direction 3 which is cut by the angle of 45 degree with respect to spine. This table shows the amount of reduction in stress after the first cycle.
\[tab:2\]
-------------- ------------ ---------------- ------------------ ----------------
Direc. 1 Direc. 1 Direc. 2 Direc. 2
Normalized 0.008 0.5 ($s^{-1}$) 0.008 ($s^{-1}$) 0.5 ($s^{-1}$)
stress ($s^{-1}$) ($s^{-1}$) ($s^{-1}$) ($s^{-1}$)
First cycle 1 1 1 1
Second cycle 0.86 0.80 0.87 0.82
-------------- ------------ ---------------- ------------------ ----------------
: Normalized stresses at the end of each cycle at the strain of 0.5 and 0.008 $s^{-1}$ at different strain rates on rat skins which were cut in direction 1 which is perpendicular to the spine direction, direction 2 which is parallel to the spine direction.
Persistent strain and original response (almost) recovers after unloading for a few days
----------------------------------------------------------------------------------------
As noted in the general observations, the stress strain cyclic loading graphs reveal that the samples do not fully recover to their original shapes upon unloading after cyclic loading. In this paper we do not investigate the effect of temperature on behavior of skin, however Xu et al. investigated the effect of temperature on behavior of skin and they showed that by increasing the temperature, skin is damaged [@XU08]. However, shows that after few days while the response was SIMILAR in shape, the actual stresses were much lower, i.e. there was substantial softening. One of the reason for this softening may be the permanent damage that may happen to collagen fibers during the cyclic loading and permanent stretches (3 mm) were observed on both rat and pig skin samples which did not recover after 3 days. Lokshin et al. in their work always did abrupt straining up to a particular level of strain, holding fixed and fully abrupt unloading[@lokshin09]. This kind of loading program mostly highlights viscoelasticity. They have actually modeled skin as a purely viscoelastic material and they have not considered weather or not there is a permanent deformation. The aim of our paper is to demonstrate with a combination of stress strain response as well as histology that there is an irreversible damage in the tissue. In their work[@lokshin09], the strain was up to 30 percent. At lower strains (such as 30 percent), mullins effect and permanent deformation are not easily identifiable. However, at a strain around 60 percent like what we did, mullins effect can be observed easily. shows that even after 3 days, the material did not recover. On the other hand, the viscoelastic model[@lokshin09] is recoverable and will be back to its original response. Residual strains were observed after unloading due to changes in tissues structure [@sellaro07]. Actually, the inelastic behavior during mechanical testing is due to the change in microstructure. Fiber recruitments and realignment of collagen fibers plays an important role in mechanical behavior of biological tissues. After uniaxil tensile testing, some of the fibers do not go back to their original configuration and they remain in a straight state which is their deformed position which leads to residual strains in the tissue after unloading. Also, by comparing the results between rat skin and pig skin, the amount of softening observed on pig skin after 3 days is approximately 30 percent (obtained by comparing the maximum stresses at the maximum strain at day 0 and day 3), which is more than the softening of rat skin which was approximately 20 percent. It must be added that control samples (previously untreated samples) show the same stress-strain curve with a negligible decrease in stress value, so it can be said that the softening is due to the cyclic loading, not storing for 3 days. By comparing pictures in , it is observed that after unixial loading, cells are elongated. Also, after cyclic loading and storing in a fridge for 3 days, fibers are still elongated with respect to undeformed state. These observations helped us to model the behavior of skin [@Afsar18]. Deformation mechanisms start with randomly orientated fibers with low stiffening, then fiber recruitment starts and stiffening increases, finally fiber slippage starts and competes with fiber recruitment until fiber tearing causes permanent damages. If unloaded before tearing, partial recovery is achieved [@Afsar18]. Histology results () show that microstructure of skin undergoes permanent changes. Compared to control samples which fibers are randomly oriented, after uniaxial testing, fibers were mostly oriented in one direction. As -a shows for the control sample in each cut orientation, standard deviation is the highest, which interprets the orientation distribution is wide and there is no preferred direction and it shows randomly distribution. However, the results show that after uniaxial tensile testing there is a preferred orientation in fiber distributions and it shows the alteration mechanisms, fiber recruitment and the inelastic fiber straightening.
Quantification of hysteresis at different strain rate
-----------------------------------------------------
The key difference between inelasticity and viscoelasticity has to do with the persistence of hystersis at quasi-static strain rates. In viscoelasticity, at very low strain rates, there is no hysteresis. However, here at very low strain rates, hysteresis still exists, which is a marker for inelasticity [@srinivasa09] and [@rajagopal16]. Also, after 3 days there is a permanent deformation which did not recover. In viscoelastic material, recovery occurs. To investigate the inelastic response of skin, it is necessary to measure the area of hystereses to find the amount of dissipated energy. Since the total work is $\int\sigma d\varepsilon$, it is possible to approximate this by Trapezoidal rule from the experimental data. The area of internal loops for pig skin samples (at strain rates of 0.22 $s^{-1}$ and 0.011 $s^{-1}$ ) an rat skin sample (at strain rate of 0.011 $s^{-1}$) which were cut in direction 1 (perpendicular to spine) were measured from A to B in -(c) at the maximum engineering strain of 40$\%$. The next two cycles were calculated from C to D in -(c) at the maximum engineering strain of 60$\%$. The calculated areas of internal loops which show the amount of dissipated energy are reported in . The amount of dissipated energy at low strains (A-B) for both strain rates are very close and they are lower than the amount of dissipated energy at high strains (C-D). Also, by increasing the strain (C-D), the amount of dissipated energy at lower strain rate (0.011 $s^{-1}$) is more than that at higher strain rate (0.22 $s^{-1}$) which indicates that by increasing the strain rate, there is a decrease in hystersis. At each strain rate, the hysteresis gets larger by increasing the maximum strain. For skin samples, the amount of dissipated energy at the third region of stress-strain curve is more than at the second region. The amount of dissipated energy for rat skin is less than the amount of dissipated energy for pig skin at the same strain rate. Also, in rat skin, the dissipated energy is less sensitive to the maximum strain compared to pig skin. For the rate skin, at very high strain rate (0.5/sec), since the next loading curve is lower than the previous unloading curve, the area between these two curves is negative although the magnitude is higher, indicating a stronger softening behavior during reloading at higher rates.
\[tab:3\]
--------- ------------- ----------------- --------- --------- ---------
Type of Maximum Strain
Skin Eng. Strain Rate $(s^{-1})$ Cycle 1 Cycle 2 Cycle 3
Rat 40 $\%$ 0.011 0.013 0.014 0.014
Pig 40 $\%$ 0.011 0.05 0.05 0.06
Rat 60 $\%$ 0.011 0.035 0.041 N/A
Pig 60 $\%$ 0.011 0.3 0.32 N/A
Pig 40 $\%$ 0.22 0.08 0.09 0.12
Pig 60 $\%$ 0.22 0.18 0.21 N/A
Rat 40 $\%$ 0.5 -0.05 -0.05 -0.01
Rat 60 $\%$ 0.5 -1.39 -1.31 N/A
--------- ------------- ----------------- --------- --------- ---------
: Comparison of the area of internal loops of pig skin samples and rat skin samples which were cut in direction 1 which is perpendicular to the spine direction at different strain rates at the maximum engineering strain of 40$\%$ and 60$\%$.
Conclusions
===========
We showed that under uniaxial cyclic loading test, skin shows hysteretic response, and mullins effect was observed. Upon investigating the effect of strain rate and comparing results of tests performed at different strain rates, softening was observed. The softening shows sensitivity to strain rate. At higher strain rates, mullins effect is more significant. Orientation of samples with respect to Langer lines or spine affects the mechanical properties of skin and stress strain curves: stiffening increases by increasing the angle between the sample and the orientation of spine. However, the extent of softening is more related to strain rate rather than sample orientation with respect to spine. Since at very low strain rates hystereses still exist, the skin behavior is more inelastic than viscoelastic. In addition to stress-strain curves, optical microscopic images helped us to a better understanding of skin behavior under cyclic loading. These images show fibers realignment which causes stiffening at higher strains. There is also fiber slippage at high stresses. These two phenomena are responsible for behavior of skin until rupture which is a permanent damage. Despite of all limitations during this study, like sample sizes or temperature effect on skin behavior, we characterized the skin behavior under cyclic loading. Also, we have done histologies and quantified density of fiber distributions which help modelers to have a comprehensive data to model the behavior of skin. Our study does not precondition the material before the experiments and the raw data was reported as is, so that comprehensive models can be developed. In our future work we propose a microstructurally motivated model to describe the inelastic behavior of skin and softening that happens to skin and we will investigate mullins effect on skin under biaxial testing.
The authors gratefully thank Dr. Terry Creasy from Texas A$\&$M University for allowing us to use his facilities.
Appendix
========
In order to compare the Instron cross-head strain and the strain measured by DIC, the strain from grip displacement and the strain from image correlation along the sample from different positions of the sample were calculated, it was seen that the difference between both measurement was less than 5 percent. $$\varepsilon_d\ = \frac{y_2-y_1}{u_2-u_1}$$
\[ht\] ![ a) Schematic calculation of Engineering Strain in Y direction from DIC. b) Displacement in Y direction measuring by DIC[]{data-label="fig:comp"}](Comparison "fig:")
\[ht\]
\[tab:4\]
Line $\varepsilon_d$ $\varepsilon_c$ Difference
--------- ----------------- ----------------- ------------
1 1.84 1.85 0.5
2 1.78 1.85 3.99
3 1.77 1.85 4.09
4 1.8 1.85 2.7
5 1.8 1.85 2.7
Average 1.8 1.85 2.7
: Both DIC and Instron cross-head engineering strain
Samples with width of 10.4 mm, thickness of 1.65 mm and initial length of 54 mm were tested under simple tension test. The measured engineering strain from crosshead displacement was 1.85 ($\varepsilon_c$). Also, engineering strain was measured by DIC, 5 different lines were chosen from different position of the sample . The engineering strain ($\varepsilon_d$) was calculated as the differences between displacements of the top and bottom point of the line ($y_1$ and $y_2$) and divided by the difference between positions of these points on the line ($u_1$ and $u_2$) . Table 1 shows strains measured by both DIC and Instron cross-head. The differences between $\varepsilon_d$ and $\varepsilon_c$ is less than 5 percent on each line and by getting the average of strain on all 5 lines, the difference is 2.7 percent which conforms to the strain measurement by the Instron cross-head.
|
---
abstract: 'This work addresses the question whether it is possible to design a computer-vision based automatic threat recognition (ATR) system so that it can adapt to changing specifications of a threat [*without having to create a new ATR each time*]{}. The changes in threat specifications, which may be warranted by intelligence reports and world events, are typically regarding the physical characteristics of what constitutes a threat: its material composition, its shape, its method of concealment, etc. Here we present our design of an AATR system (Adaptive ATR) that can adapt to changing specifications in materials characterization (meaning density, as measured by its x-ray attenuation coefficient), its mass, and its thickness. Our design uses a two-stage cascaded approach, in which the first stage is characterized by a high recall rate over the entire range of possibilities for the threat parameters that are allowed to change. The purpose of the second stage is to then fine-tune the performance of the overall system for the current threat specifications. The computational effort for this fine-tuning for achieving a desired PD/PFA rate is far less than what it would take to create a new classifier with the same overall performance for the new set of threat specifications.'
author:
- |
Ankit Manerikar Tanmay Prakash Avinash C. Kak\
School of Electrical and Computer Engineering, Purdue University\
West Lafayette, IN, USA\
[amanerik@purdue.edu]{}
title: 'Adaptive Target Recognition: A Case Study Involving Airport Baggage Screening '
---
Introduction
============
Automatic threat recognition (ATR) systems for applications such as airport passenger baggage screening are subject to expensive and time-consuming processes of certification before they can be deployed. As is to be expected, such systems are designed for a particular set of threat specifications. Unfortunately, the real world being what it is, the precise specifications of a threat do not remain constant with time and depend much on the world events and the evolving capabilities of the bad guys out there.
Since the cost of developing a totally new ATR for a new set of specifications can be expensive and time consuming, it is necessary to explore the possibilities related to the design of adaptive automatic threat recognition (AATR) systems that can be quickly adapted to changing threat specifications. [^1]
In the context of airport baggage screening using 3D imaging based on X-ray tomography, threats like home-made explosives (HMEs) [@wells2012review] and firearms are characterized by parameters such as materials and their composition, their shapes, the methods expected to be used for their concealment, and so on. When the specifications of such threats change, the modifications are generally with respect to these parameters. In this paper, we will focus exclusively on making an ATR system for HME detection adaptive with respect to changes in the materials density (as measured by its x-ray attenuation coefficient), its thickness, and its mass.
The adaptive framework we present in this paper employs a two-stage classifier cascade in which the first stage is designed to operate with a sufficiently high recall rate over the entire range of expected variability in the threat parameters that could change. The second stage classifier is then fine-tuned to the parameters of the current threat specification.
The two-stage classifier cascade is also provided with an [*adaptation protocol*]{} for each threat parameter with respect to which classifier adaptivity is desired. This protocol takes the classifier created for a given specification of the parameters and modifies it to suit another specification. Depending on the nature of the parameter involved, this protocol may entail a revisit to the training data. Even when the training data is revisited for adaptation, the amount of work involved in the adaptation process is far less than what it would take to create a brand new classifier for the new set of parameter specifications.
Our AATR system was tested on a dataset [@alertdataset] made available by the DHS sponsored ALERT center at Northeastern University specifically for the purpose of evaluating AATR algorithms for airport baggage screening. A unique feature of this dataset are its ORS (Object Requirements Specification) files. These files express the specifications of a threat in terms of its mass, density and thickness. We demonstrate the “adaptive power” of our approach by adapting our classifier to range of ORS files.
The organization of the paper is as follows: Section II provides a brief review of the existing work related to ATR systems and the response of prevalent ATR systems to varying parameter specifications. This is followed by a formal description of an Adaptive ATR system in Section III as well as the principle of operation behind our proposed approach. Section IV then introduces the Cascaded Classification Approach to AATR system design encompassing the two-stage classifier model, the adaptation protocols adopted for the threat parameters and the training methodology with the adaptation protocols using the technique of Dynamic Sample Weighting. The implemented AATR system is then described in detail in Section V explaining the overall system operation. Finally, the implementation results for testing adaptability on different threat parameters are tabulated and illustrated in Section VI.
Related Work
============
ATR based on CT imaging for airport baggage inspection is made challenging by the artifacts that result from metallic objects that can be in arbitrary locations in a bag [@karimi2015metal] [@xue2009metal]; by a lack of apriori structural information as compared to medical applications of CT [@mouton2013experimental]; and by large variability in the CT density range among the objects found in bags.
Much work on algorithms for automated baggage inspection has been carried out under the auspices of the Department of Homeland Security’s ALERT (Awareness and Localization of Explosive-Related Threats) initiatives on ATR segmentation and object classification [@alertdataset] [@alertto4initiative]. This has led to the development of a number of ATR segmenters and classifiers that could potentially be incorporated in the airport checkpoint security pipeline in the future. Amongst these, the Stratovan Tumbler proposed by Wiley et al. [@wiley2012automatic] makes use of a 3D flood-fill region-growing technique for segmenting out target blobs from the query image. The method proposed by Grady et al. [@grady2012automatic], on the other hand, employs isoperimetric graph partitioning [@grady2006isoperimetric] to perform segmentation for ATR. Song [@song2015vendor] proposed a sequential segmentation and carving method for ATR segmentation employing splitting and merging techniques for target extraction. Other proposed techniques use sieve decomposition algorithms and adaptive region growing for ATR.
More recent ALERT initiatives [@alertto4initiative] have focused on a contextual classification of the object blobs leading to the inception of more complex ATR systems using multiple stages of segmentation and classification. Several algorithms and ATR structures have been proposed in this direction that include graph-based segmenters, MRF-EM based image segmentation and decision-tree based ATR systems [@alertto4initiative]. Other methods also provide for an extension of ATR systems to dual-energy CT scans [@mouton2015materials] as well for joint metal artifact reduction and segmentation [@jin2015joint].
For all these cases, the performance of the ATR systems is evaluated on their ability to achieve desired values of precision and recall for specific threats and fixed values of threat parameters — little analysis is made on what the response of these systems would be if these threats and threat parameters were to vary during runtime. This would be especially difficult for ATR methods described in [@wiley2012automatic] and [@grady2012automatic] wherein the segmentation routines are pre-tuned for each threat specification. On the other hand, the ATR classifiers in [@song2015vendor][@alertto4initiative] [@jin2015joint] can be re-trained to adapt to a new threat but this involves retraining the ATR from a scratch.
This paper thus analyzes the problem of building an Adaptive ATR system, i.e., the problem of desensitizing an ATR system to variations in threat specifications. The two-stage classifier and the adaptation protocols proposed in the paper present a modular structure to carry out this desensitization with respect to specific threat parameters (density, mass and thickness) and without resorting to a complete rebuilding of the ATR. The detailed implementation of this model is elaborated in the following sections.
System Overview - Adaptive ATR
==============================
Problem Statement
-----------------
X-ray based threat recognition for airport baggage screening can be tricky in case of threats such as home-made explosives (HMEs) [@singh2003explosives] which do not conform to a distinct shape or form and can be easily concealed. Such threats are detected on the basis of a materials-based characterization that involves the Region-of-Responsibility (ROR) [@martz2009overview] or density range (in terms of its x-ray attenuation co-efficient), total object mass and thickness amongst other parameters. These specifications are, however, subject to frequent updates and modification with newer developments and alerts and therefore also require the deployed ATR systems to be adaptive to these updates. Adjusting and re-certifying the ATR system for newer updates, however, is time-consuming and may result in an undesirable operational downtime. This puts forth the notion of the design of an Adaptive ATR (AATR) system, which is characterized by the ability to handle modifications to threat specifications during runtime while taking minimal effort to retrain or reconstruct the ATR.
Principle of Operation:
-----------------------
In this paper, we present a two-stage classifier cascade architecture for the design of an AATR system - the focus for this ATR system design is exclusively with respect to the set of threat parameters that include the material density range, total object mass and object thickness. The two stage classifier structure makes use of the knowledge that while the precise specifications of these threat parameters may require modification during ATR operation, the range within which the parameters may vary can be pre-determined and remains fixed. Thus, the first stage of the cascade is designed to operate over this entire range with a high recall rate while the second stage fine-tunes the overall system performance by narrowing down the detection to the precise specification. Any modification to threat specifications therefore requires adjusting only the second stage of the cascade thus taking minimal effort to retrain the system.
Now, ATR system performance is generally evaluated on the basis of the following metrics described in [@wells2012review]: $$\begin{aligned}
& \textnormal{Probability of Detection, } PD
&=& & \frac{TP}{TP+FN} \\
& \textnormal{Probability of False Alarm, } PFA
&=& & \frac{FP}{FP+TN}\end{aligned}$$
where TP, FP, TN and FN are True Positive, False Positive, True Negative and False Negative values respectively for the detected and total number of target objects on the dataset used for training the ATR. Here, Probability of Detection (PD) denotes the probability that the detected object is a threat while the Probability of False Alarm (PFA) denotes the probability that the detected object is a false alarm. We use these same performance metrics to evaluate the AATR system response as well - the proposed AATR design aims at obtaining PD and PFA values that are as close to the target values as possible.
For each of the three threat parameters, namely, Density, Mass and Thickness, we employ an adaptation protocol specified by the Dynamic Sample Weighting technique explained in later sections. This technique allows tuning the system performance for the precise specification values of the threat parameters and also allows for adjusting system PD and PFA by moving along the ROC curve.
The dataset used for training and testing our architecture is the TO-4 dataset [@alertdataset] made available by ALERT center at Northeastern University - this dataset contains 188 instances of CT baggage scans with labeled ground truth images showing saline, rubber and clay objects. We used threat specifications expressed in terms of target ranges of density, mass and thickness to test the adaptive power of our system - system performance was evaluated by creating an AATR classifier for one specification set, creating new specification sets by modifying one or more threat parameter values and checking how well the new system adapts to these new specifications in terms of PD and PFA.
Proposed Methodology - Cascaded Classifier Architecture:
========================================================
Figure \[bd\_aatr\] shows the block diagram for the two-stage cascaded classifier AATR architecture along with an example illustration for the case of a varying density range specification for threat detection. (To characterize X-ray based detection, we express the density range in Modified Hounsfield Units or MHUs, i.e., Hounsfield Units offset by 1000 to yield a non-negative range of values). From the example illustration, we can see how the Stage I classifier which is designed to detect threats or targets over a large density range (380 - 2470 MHUs, encompassing all possible densities of the target objects) generates an output with a high recall rate. The second classifier in the cascade then fine-tunes this result by narrowing down on the precise density range for the particular target specification (for example, 1050-1215 MHUs for saline objects). Changes in target specifications therefore only warrant a change in the second classifier which avoids training the entire AATR system for the new modification (e.g., changing to a new density range of 1170-1290 MHUs for detecting rubber objects). A stage-wise description of the system is given ahead:
Stage-wise System Description:
------------------------------
### Stage I Classifier: {#stage-i-classifier .unnumbered}
The basis for operation of the cascaded classifier architecture lies in the notion that while the exact target specifications for ATR operation are subject to change, the overall range within which these values vary remains fixed. The Stage I classifier therefore is designed to identify any object whose threat specification falls within the acceptable global range for any of the threat parameters. Because this range remains fixed, the Stage I classifier does not need to be retrained or retuned for every new modification to the threat parameter values during the AATR operation. For the current implementation, the Stage I classifier is a Graph-based segmenter [@grady2012automatic] that segments out and generates a set of candidate target blobs from the input query image. Since all of these blobs do not satisfy the precise threat specification, the output of the Stage I classifier is riddled with a number of false alarms giving a high PFA rate - this is taken care of in the second stage of the cascade.
It is important to remember that the objective of an ATR system is not to partition the query image into threat and non-threat regions but to extract uniquely labeled segments from the image that correspond to different objects within the baggage. Thus, a challenging task for an ATR system in general is to be able to distinguish between two touching objects or one object contained within another - it is a non-trivial problem as one explosive concealed within a collection of similar but benign objects can be overlooked as a false alarm if not treated as a separate segment during the ATR operation. This problem is addressed in our implementation through the use of graph partitioning [@shi2000normalized].
### Stage II Classifier: {#stage-ii-classifier .unnumbered}
Once the set of candidate target blobs has been generated by the Stage I classifier, the Stage II classifier in the cascade fine-tunes the result by classifying these blobs as per the exact threat specifications. For the case in Figure \[bd\_aatr\], the Stage II classifier in the cascade filters only those blobs from the Stage I classifier output that match the density range provided in the threat specification. Depending on the threat parameter to be modified, the classifier may need to be retrained by revisiting the training data but since this constitutes only a part of the entire AATR cascade, it avoids retraining the entire system. We employ different adaptation protocols for the threat parameters under consideration, each of which is explained in the next section.
Adaptation Protocols:
---------------------
Diffferent threat parameters warrant different adaptation protocols to be adopted to adjust to a new modification. For parameters such as mass, this may be as simple as changing a threshold to prune lighter masses from the candidate blobs while for parameters like density, it may require retraining the entire Stage II classifier using histogram-based descriptors. The three different adaptation protocols for the three parameters under consideration, i.e., Mass, Density and Thickness are described as follows:
### Tuning for Target Mass Range: {#tuning-for-target-mass-range .unnumbered}
As mentioned above, the protocol for adapting to a modification in the target mass range simply involves changing the threshold for pruning light masses from the set of candidate blobs generated by the Stage I classifier. A very good approximation of the mass of the candidate blob can be obtained by integrating over the voxel-wise CT density values within the blob volume - this can be used to filter out those blobs which do not fit into the specified mass range.
### Tuning for Target Density Range: {#tuning-for-target-density-range .unnumbered}
Density range specifications for threat identification are the main parameters for the materials based characterization of the threat and are determined from the average density range of the constituent material of the target [@alertdataset]. However, since the material composition of any object is never completely homogeneous, it is difficult to determine the material composition by considering an absolute range for the density specification [@wiley2012automatic] - this is especially difficult for small and thin blobs wherein the density distribution is corrupted by even a small quantity of noise, artifacts or contamination.
To adapt the AATR system to varying densities, therefore, we make use of a random forest classifier that is trained over the ALERT TO-4 dataset to identify target blobs composed of the desired material of interest. This classifier makes use of Normalized Density Histograms as classification features and is trained and tested on the dataset using ten-fold cross-validation. Specification of a new density range for the threat, thus, requires re-training of this classifier for the new desired density window.
### Tuning for Target Thickness Range: {#tuning-for-target-thickness-range .unnumbered}
Thickness has become an important parameter for threat recognition since the occurrence of several recorded incidents of transporting HMEs concealed as thin plasticized sheets. Determining thickness of a sheet object can be difficult especially as the sheet within the baggage can be placed in a mangled or folded form. The adaptation protocol for the target thickness range involves construction of a 3D Thickness Vector for the candidate blob. This thickness vector is calculated for an object by calculating the median thickness of the object along each of its oriented principal axes and normalizing it over the largest thickness value (for the multiple folds of the sheet, the thickness vector is scaled by a suitable multiplier). A simple KNN Classifier based on this vector is then trained to identify objects within the target thickness range. In our implementation, the Thickness Vector is used for Dynamic Sample Weighting by concatenating it to the Density Histogram feature vector.
Dynamic Sample Weighting:
-------------------------
In our AATR implementation, the three adaptation protocols are implemented simultaneously using a single classifier through use of Dynamic Sample Weighting. This technique regulates the retraining of a classifier to adapt to the modification of one threat parameter by utilizing the knowledge of the values of all available parameters. By assigning a higher weight to those training samples whose parameter values are closer to the desired range and training the classifier accordingly, a better classification response can be obtained compared to the independent execution of the adaptation protocols.
In this method, a random forest classifier is constructed wherein the feature vector contains the Normalized Density Histogram and the Thickness Vector concatenated together. While training, each training sample in the dataset is assigned a weight that is determined by the vicinity of the sample parameter values to the target parameter range. To calculate the weights, we take into account the specified range for each of the three threat parameters and construct a Gaussian Weighing Function (See Figure \[gww\]). This can be exemplified for the case of Thickness as follows:
![ *Gaussian Weighting Function* for Target Thickness Range $[6.5, 10.0]$ mm with a standard deviation, $\sigma=1.0$ and binarizing threshold, $t=0.8$. The parameters $\sigma, t$ are optimized to obtain the desired PD-PFA performance while retraining.[]{data-label="gww"}](figures/gsw_aatr.png){width="\linewidth"}
Let us consider the thickness range specified by the minimum and maximum values of 6.5 mm and 10.0 mm respectively. Comparing the thickness of the current training sample with these limiting values, a sample weight is assigned to this sample using the Gaussian Weighting function illustrated in Figure \[gww\]. The expression for this Gaussian Weighting Window for minimum and maximum limiting values, $T_{l}$ and $T_{h}$ is given by: $$\begin{aligned}
w(t) &=& e^{\left[ \left( \frac{t - T_{l}-0.1T_d}{\sigma} \right)^2 \right]} & & t \leq T_{l}' \\
&=& 1.0 & & T_{l}' < t \leq T_{h}' \nonumber\\
&=& e^{\left[ \left( \frac{t - T_{h}+0.1T_d}{\sigma} \right)^2 \right]} & & T_{h}' \leq t \nonumber\end{aligned}$$
where $\sigma$ is the standard deviation, t is the thickness of the current sample and $T_d=|T_{h}-T_{l}|$ is the thickness range, $T_h' = T_{h}-0.1T_d$ and $T_l' = T_{l}+0.1T_d $.
The total sample weight is thus the product of the individual sample weights for the threat parameters, Mass, Thickness and Density. The classifier is retrained with these sample weights for any modification in the specified threat parameters. Dynamic Sample Weighting also allows tuning the classifier performance for a target PD and PFA - this is done by adjusting the standard deviation $\sigma$ of the Gaussian weighting function and the threshold $t$ on the sample weights that generates the positive/negative samples for the specific OOI. A grid-based search is used to find the optimum values of $\sigma, t$ that give the PD and PFA values closest to the target values during cross-validation.
Tuning for Desired PD and PFA Response:
---------------------------------------
![*Adjusting PD/PFA for the AATR system:* The two-stage cascade allows for a mechanism to adjust the target PD/PFA, $PD^{(T)}, PFA^{(T)}$ for the entire system by adjusting only the PD/PFA for $C_2$, i.e., $PD^{(2)}, PFA^{(2)}$. In the figure, $N^{(i)}$ denotes the set of filtered voxels at the $i^{th}$ stage; $PD^{(i)}, PFA^{(i)}$ denote the PD/PFA for the $i^{th}$ classifier.[]{data-label="pdpfa_tune"}](figures/pdpfa_tune_6.png){width="0.95\linewidth"}
To adapt to the desired PD/PFA requirements, the cascaded classification structure requires a mechanism which allows tuning the AATR classifier performance along the ROC curve without retraining the entire AATR system. For the cascaded architecture, this can be done by tuning the respective Stage II classifier as elaborated below:
Consider the Stage I classifier $C_1$ and the Stage II classifier $C_2$ connected in a cascade as shown in Figure \[pdpfa\_tune\] with the respective PD, PFA values $PD^{(1)}, PFA^{(1)}$ and $PD^{(2)}, PFA^{(2)}$ while the target PD, PFA for the total system are given by $PD^{(T)}, PFA^{(T)}$. As the output of $C_1$ is directly fed to $C_2$ as input, it is evident that the total PD of the cascaded system is equivalent to the product of the PD’s of the individual blocks and this holds true for PFA as well: $$\begin{aligned}
PD^{(T)} &=& PD^{(1)}\cdot PD^{(2)} \\
PFA^{(T)} &=& PFA^{(1)}\cdot PFA^{(2)} \end{aligned}$$
Now, because the Stage I classifier is only trained once, the values for $PD^{(1)}, PFA^{(1)}$ remain fixed but $PD^{(2)}, PFA^{(2)}$ can be adjusted for every new modification to the threat specifications as the Stage II classifier is retrained. The target PD, PFA can thus be obtained by tuning $C_2$ to attain the following PD, PFA values: $$PD^{(2)} = \frac{PD^{(T)}} {PD^{(1)}}; \>\>\>\>\>\>\>
PFA^{(2)} = \frac{PFA^{(T)}} {PFA^{(1)}}$$
By using Dynamic Sample Weighting in our implementation, we attempt to attain the desired PD, PFA values by adjusting the thresholds of the sample weights and the spread of the Gaussian weighting function.
System Implementation
=====================
We have implemented the proposed two-stage cascaded classification structure using a Graph-based ATR segmenter and a Random Forest classifier as the Stage I and Stage II classifiers in the cascade. The AATR system was tested for its adaptability by subjecting it to different sets of threat specifications grouped into three categories: *Varying Density (or constituent materials), Varying Mass* and *Varying Thickness* - the ALERT TO-4 dataset [@alertdataset] was used for the testing that included different target objects selected out of three Materials-of-Interest (MOIs), namely, *saline, rubber* and *clay*. The dataset contains 188 training samples consisting of CT scans of baggage that contained both target and non-target objects, in bulk as well as sheet form. The details for each stage of the structure are given ahead:
Stage I Classifier:
-------------------
{width="11cm"}
The Stage I Classifier is constructed based on the ATR described in [@wiley2012automatic] with its blocks explained below:
### Pre-Processing: {#pre-processing .unnumbered}
The CT query image is first pre-processed using Mumford-Shah energy minimization [@mumford1989optimal] via the Ambrosio-Tortorelli approximation [@ambrosio1990approximation] to reduce the noise and artifacts and generate smooth sections for further segmentation. The 3D energy minimization is carried out with a parameter set for the minimizer $(\alpha, \beta, \epsilon)$ [@ambrosio1990approximation] adjusted to $(1000,0.9,0.1)$.
### Supervoxel Segmentation: {#supervoxel-segmentation .unnumbered}
The pre-processed image is then subjected to Supervoxel segmentation using SLIC [@achanta2012slic] with an initial number of segments $n\_segments=1000$ and compactness, $c=40$. The supervoxels are then filtered depending on whether their mean density lies within the pre-determined global range for threat densities ($380-2470$ MHUs).
### Graph-Partitioning: {#graph-partitioning .unnumbered}
To address the problem of segmenting out individual objects uniquely, a Normalized Cuts algorithm [@shi2000normalized] is used to partition the set of filtered supervoxels into a set of candidate blobs on the basis of intensity difference and the presence of a distinct boundary between two supervoxels. To do so, a graph is constructed for the set of supervoxels with edge weights calculated as follows: $$w(i,j) = \eta_{e}\cdot\eta_{n}\cdot \exp \left[ \frac{(I_i-I_j)^2}{\sigma}\right]\exp \left[ \frac{S(i,j)^2}{0.25}\right]$$ where:
- $\footnotesize
S(i,j) =\frac{\textnormal{ No. of boundary voxels with a distinct edge} }{\textnormal{ Total No of boundary voxels between Nodes i, j}}$
- $\eta_{n}$ - Set to 1 if Node i and Node j are neighbors
- $\eta_{e}$ - Set to 1 if $S(i,j) > 0.25$.
- $I_i,I_j$ - Mean intensities of Nodes i and j.
- $\sigma$ - Global density range for all threats.
The Normalized Cuts algorithm is then applied with a threshold, $t=0.1$ to generate the set of candidate blobs.
Stage II Classifier - Random Forest:
------------------------------------
For the threat parameters specified for a known constituent material, i.e., with samples present in the TO-4 dataset (namely, saline, rubber or clay), the Stage II classifier used is a Random Forest classifier with 50 estimators and using the Gini impurity criterion. The Stage II classifier uses a feature vector consisting of a 100-bin Normalized Density histogram and a 3D Thickness vector concatenated together. The density histogram features are weighted with a value of $\frac{0.5}{100}=0.005$ while the thickness vector is weighted with $\frac{0.5}{3}=0.167$ to normalize the feature vector - if the target thickness does not change in the new modification, the respective weights are set to zero. The output blobs of the Stage II classifier are then filtered using the mass feature pruning lighter masses from the final output image.
Results
=======
[0.5]{} ![*PD/PFA Performance of AATRs with different systems for Stage I classification:* The figure shows the response of three different AATRs for the cases of Varying Density, Varying Mass and Varying Thickness. Each AATR is implemented with a different system as its Stage I classifier - these systems include a Graph-based ATR Segmenter (AATR 1), a Supervoxel-based ATR classifier (AATR 2) and a CCL-based ATR segmenter (AATR 3) respectively. The performance range of current ATRs for PD and PFA has been obtained from [@alertto4initiative] is denoted by a green band in the figure. \*AATR 1 gives the best adaptive response and has been explained in detail in this paper. ](./figures/graph_7.png "fig:"){width="94.00000%"}
[0.5]{} ![*PD/PFA Performance of AATRs with different systems for Stage I classification:* The figure shows the response of three different AATRs for the cases of Varying Density, Varying Mass and Varying Thickness. Each AATR is implemented with a different system as its Stage I classifier - these systems include a Graph-based ATR Segmenter (AATR 1), a Supervoxel-based ATR classifier (AATR 2) and a CCL-based ATR segmenter (AATR 3) respectively. The performance range of current ATRs for PD and PFA has been obtained from [@alertto4initiative] is denoted by a green band in the figure. \*AATR 1 gives the best adaptive response and has been explained in detail in this paper. ](./figures/supvox_7.png "fig:"){width="94.00000%"}
[0.5]{} ![*PD/PFA Performance of AATRs with different systems for Stage I classification:* The figure shows the response of three different AATRs for the cases of Varying Density, Varying Mass and Varying Thickness. Each AATR is implemented with a different system as its Stage I classifier - these systems include a Graph-based ATR Segmenter (AATR 1), a Supervoxel-based ATR classifier (AATR 2) and a CCL-based ATR segmenter (AATR 3) respectively. The performance range of current ATRs for PD and PFA has been obtained from [@alertto4initiative] is denoted by a green band in the figure. \*AATR 1 gives the best adaptive response and has been explained in detail in this paper. ](./figures/ccl_7.png "fig:"){width="94.00000%"}
----------------- --------------------- ---------------------
Material Stage I Classifier AATR
(Density Range) (PD $\%$, PFA $\%$) (PD $\%$, PFA $\%$)
saline$^*$ (91 $\%$, 53 $\%$) (90 $\%$, 12 $\%$)
rubber$^*$ (87 $\%$, 62 $\%$) (85 $\%$, 13 $\%$)
clay$^*$ (83 $\%$, 66 $\%$) (84 $\%$, 13 $\%$)
----------------- --------------------- ---------------------
: Adaptation to Varying Threat Parameters (AATR 1) - Varying Thickness[]{data-label="tb:am5"}
\
----------- --------------------- ---------------------
Mass Stage I Classifier AATR
Range (PD $\%$, PFA $\%$) (PD $\%$, PFA $\%$)
$ >400$ g (86 $\%$, 69 $\%$) (83 $\%$, 12 $\%$)
$ >300$ g (88 $\%$, 66 $\%$) (86 $\%$, 17 $\%$)
$ >100$ g (86 $\%$, 64 $\%$) (82 $\%$, 15 $\%$)
----------- --------------------- ---------------------
: Adaptation to Varying Threat Parameters (AATR 1) - Varying Thickness[]{data-label="tb:am5"}
\
-------------------- --------------------- ---------------------
Thickness Stage I Classifier AATR
Range (PD $\%$, PFA $\%$) (PD $\%$, PFA $\%$)
$> 10.0\>\>\>$ mm (89 $\%$, 65 $\%$) (85 $\%$, 7 $\%$)
$6.5-10.0$ mm (86 $\%$, 63 $\%$) (83 $\%$, 9 $\%$)
$0-6.5\>\>\>\>$ mm (86 $\%$, 62 $\%$) (83 $\%$, 9 $\%$)
-------------------- --------------------- ---------------------
: Adaptation to Varying Threat Parameters (AATR 1) - Varying Thickness[]{data-label="tb:am5"}
\
**Note:** The *saline, clay* and *rubber* objects correspond to the density ranges (1050-1215), (1170-1290) and (1530-1715) MHUs respectively. The difference between $PFA^{(T)}$ and $PFA^{(1)}$ for all cases shows the effect of fine-tuning the Stage I classifier output in the second stage of the cascade.
![*ROC Response Curve for the Implemented AATR*: (i) the implemented AATR system, (ii) a Supervoxel-based ATR classifier for reference [@alertto4initiative], (iii) the desired ROC curve for system operation. The systems were trained to detect saline objects with a fixed set of specifications to achieve the desired ROC curve. The ATR in this case was completely retrained for a new variation while the AATR adapts to the variations.[]{data-label="fig:results"}](figures/roc_curve_9.png){width="45.00000%"}
The implemented AATR system was tested for its adaptability on the ALERT TO-4 dataset [@alertdataset]. This dataset consists of 188 CT scans of bags that contain target objects made from *saline, rubber* and *clay* and non-target objects that can typically be found in passenger baggage at airports. The target objects are both in bulk and in sheet forms.
As previously mentioned in the Introduction, target specifications for testing for adaptability are in the form of ORS (Object Requirements Specification) files. Through a specification of the minimum and maximum value for the three parameters — mass, density, and range — an ORS file declares what it means for an object to be a target. Each ORS file also specifies the [*desired*]{} PD and PFA for the classifier performance for that target.
We tested our AATR system for adaptability by subjecting it to nine ORS files, each with a different threat specification for one of the three material specific parameters — mass, thickness, or density. In three of the ORS files, only the mass specification changed, in another grouping of three, only the density changed, and in yet another grouping of three, only the thickness changed. With regard to the desired AATR performance, it was the same for all ORS files: $90 \%$ for PD and $10 \%$ for PFA.
The PD/PFA performance as achieved by our AATR systems for the three ORS groupings is illustrated in Figure 5. The figure shows the PD/PFA values achieved for both the Stage I classifier (red) and the total AATR system (blue) after the Stage II classifier is added - this is shown for the case of three AATRs implemented with a different Stage I classifier: AATR 1 is implemented from a Connected Component Labeling (CCL)-based ATR segmenter [@alertto4initiative], AATR 2 is derived from a Supervoxel-based ATR classifier while AATR 3 is based on the Graph-based ATR segmenter [@grady2012automatic] that we have implemented and explained in detail in the previous sections. We see a marked drop in the PFA values as a result of the fine-tuning performed by the Stage II classifier for all three cases. For comparison, the green band in the figure is for the mainstream ATRs described in [@alertto4initiative]. This figure establishes that the overall performance of an AATR framework need be no worse than what it is for well-designed ATR algorithms out there. This is also exemplified by a comparison between the ROC curves of the AATR system with that of an ATR system trained on a fixed set of material-specific parameters, as shown in Figure 6 - our use of Dynamic Sample Weighting produces a response within an acceptable range from the ATR ROC curve.
The PD/PFA response for AATR 1 is also tabulated in Tables \[tb:am2\], \[tb:am4\] and \[tb:am5\] for a numerical demonstration. We see that the standard deviations for the PD and PFA do not exceed above $2.62 \%$ and $2.05 \%$ respectively in all cases showing stable behavior against varying threat specifications.
Conclusion
==========
The results obtained with our AATR architecture for designing adaptive classifiers for threat recognition show that the overall classification performance with an adaptive framework need be no worse than what can be achieved with a traditional approach that calls for creating a brand new classifier for each new definition of what constitutes a threat. Considering that the work required for the fine-tuning of the Stage II classifier in our AATR is much less than what it would take to create an ATR from ground zero, our work establishes the viability of AATR frameworks for automatic target recognition.
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[^1]: The need and technical requirements for AATR were developed by Carl Crawford (Csuptwo), Harry Martz (Lawrence Livermore National Laboratory) and Laura Parker (DHS Explosives Division Science and Technology Directorate) in collaboration with Northeastern University’s Awareness and Localization of Explosives-Related Threats (ALERT) Center, a DHS Center of Excellence. The datasets and scoring tools used in the paper were provided by ALERT, which was funded by DHS.
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The discovery of ’colossal ’ magnetoresistance (CMR) [@helm] together with its many unusual properties has received considerable attention lately.
Experiments have revealed very rich phase diagram interpreted in terms of magnetic ferro, antiferro, canted, polaronic, and non-saturated phases . Charge ordered phases also have been found[@Ramirez]. The phase diagram, as a function of concentration $x$, temperature, magnetic field, or magnitude of the superexchange interaction is not quite clear yet for the different compounds. The metallic phase can be reached by hole doping of the parent compound LaMnO$_{3}$, by substituting La for divalent alkalies, Pb, or by stoichiometry changes. Very recently, neutron scattering experiments have been interpreted in terms of polaronic droplets[@Hennion]. Much less is known about the electron doped compounds where doping does not seem to produce metallization.
From the theoretical point of view, the pioneering work of de Gennes [@degen] proposed a canted phase to resolve the competition between the ferromagnetic double interaction introduced by the presence of itinerant holes and the superexchange interaction. Recently, several contributions to this problem have been reported. Arovas and Guinea [@Arovas], studied this problem using a Schwinger boson formalism to obtain a phase diagram showing several homogeneous phases and pointing out that phase separation replaces the canted phase in a large region. Indeed phase separation appears in several numerical treatments of the problem [@Dagotto1]. In other analytical treatments, more adequate to treat local instabilities, non-saturated local magnetization states have appeared at zero temperature[@caty]. M.Yu.Kagan [*et. al.*]{} [@Khomsky] have studied the stability of the canted phases against the formation of large ferromagnetic ’droplets’ containing several particles and they conclude that the formation of droplets is favored in the ground state. The variety of results obtained from the different approaches points to the need of clarifying the picture and testing the results.
In this work, for the first time, we find the low energy quasiparticles and characterize their structure and dispersion relation in the low concentration limit. These quasiparticles correspond to the electron followed by a ferromagnetic local distortion (ferromagnetic polaron) in the antiferromagnetic (AF) background. The dispersion relation is dominated by $%
k\rightarrow k+\pi $ scattering due to the presence of AF order. In order to make a conection with transport properties, we also study the the tendency to localization of these polarons in the presence of impurities and magnetic field.
To render evident the nature of the ground state, we resort to the Lanczos method, which is free from approximations. The Hamiltonian is simplified to a single orbital per site, no lattice effects are considered, and we have to reduce to one dimensional chains. However our results provide a simple picture that, we presume, can be put to test of the dilute limit of electron doped systems. In these systems, the limitations of the model Hamiltonian may not be as stringent as in the hole doped systems for the following reasons: the lattice structure is more symmetric so Jahn Teller distortions should play a less important role, the large in-site Coulomb repulsion inhibits double occupation so that it may be possible to describe the physics by the use of a single effective orbital, and finally the antiferromagnetic structure of two interpenetrating lattices can be properly described in one dimension.
In order to describe the manganites we consider two degrees of freedom: localized spins that represent the $t_{2g}$ electrons at the Mn sites, and itinerant electrons that hop from $e_{2g}$ $Mn$ orbitals to nearest neighbor $e_{2g}$ orbitals. The model Hamiltonian includes exchange $(J)$ energies, an antiferromagnetic interaction between localized spins $(K)$ and a hopping term of strength $t$ which we will use as energy unit hereafter. It reads: $$\begin{aligned}
H &=&-J_{h}\sum_{i}{\bf S}_{i}\cdot {\bf \sigma }_{i}+K\sum_{<i,j>}{\bf S}%
_{i}\cdot {\bf S}_{j}+ \\
&&\sum_{<i,j>,\sigma }t_{ij}\left( c_{i\sigma }^{+}\cdot c_{j\sigma
}+h.c.\right) \,\end{aligned}$$
where $n_{i,\sigma }=$[** **]{}$c_{i\sigma }^{+}$ $c_{i\sigma }$ , and $c_{i\sigma }^{+},$ $c_{i\sigma }$ creates and destroys an itinerant electron with spin $\sigma $ at site $i,$ respectively. [** **]{}${\bf S}_{i}$ and ${\bf \sigma }_{i}$ are the localized and itinerant spin operators at site $i$, respectively. In what follows we take large values of J$_{h}$ which prevents double occupation and makes the on-site Coulomb interaction $%
U $ irrelevant. This model has been studied numerically for finite concentration in reference[@Moreo], and in the absence of AF coupling ($%
K=0$) in reference [@Dagotto1]. In this paper we focus on the dilute limit.
=3.5truein =2.5truein
We first investigate the homogeneity of the solutions for different values of the antiferromagnetic interaction $K$ (in units of $t$). To this end, we calculate the ground state with one electron added for chains of different sizes up to $N=20$. With the aim of looking for spin distortions around the charge, we calculate a correlation function which makes such a situation evident : $<n_{i}S_{j}S_{j+1}>$. Because of translational symmetry the results depend only on $\left| i-j\right| $. The results are shown in Fig. 1 where we plot $N<n_{0}S_{j}S_{j+1}>$ vs $j$, where $N$ is the number of sites. As it can be seen in Fig. 1, for large $j$, this correlation function takes a value very close to the one obtained from the Bethe ansatz solution for the Heisenberg chain, $<S_{j}S_{j+1}>\cong 0.443$. The extension and the magnitude of the spin distortion around the particle increases as K decreases. The oscillations observed in the curve corresponding to $K=1$ are also observed for larger values of $K$. They are a consequence of the weakening of the antiferromagnetic links around the charge position which produce a sort of local spin dimerization. To prove this point, we show in the inset the nearest neighbors spin-spin correlation functions for a Heisenberg chain of the same size where the link at site zero is a factor of two smaller than the rest.
However as K decreases, it is difficult to find an adequate approximation to describe the large polaronic distortion. In order to obtain the effective mass of these polarons, we investigate the dispersion relation for charge excitations. To this end we calculate the lowest energy state for different values of the momentum $k=2\pi n/N$ within the subspace where the total spin is that of the ground state. In Fig. 2 we show the dispersion relation scaled to the thermodynamic limit for $K=3$, $J_{h}=100$; $K=1$, $J_{h}=10$; and $K=0.3$, $J_{h}=10$.
=3.9truein =3.truein
We start analyzing the dynamics in the regime where $(J_{h}>>K>>t).$ In that case, the charge moves as a spin one $({\bf \Sigma })$. The effective hopping resulting from the projection of the hopping term onto the reduced $%
S=1$ Hilbert space is: $tP_{ij}({\bf \Sigma }_{i}{\bf S}_{j}+1/2)$, where $%
P_{ij}$ is the permutation operator between sites $i$ and $j$. We can picture the movement of the particle, in this limit, as going from a state $%
\downarrow $ $\uparrow $ $\downarrow $ $\Uparrow $ $\downarrow $ $\uparrow $ $\downarrow $ to an intermediate state $\downarrow $ $\uparrow $ $\downarrow
$ $\uparrow $ $0$ $\uparrow $ $\downarrow $, and finally to $\downarrow $ $%
\uparrow $ $\downarrow $ $\uparrow $ $\downarrow $ $\Uparrow $ $\downarrow ,$where $\Uparrow $ $(0)$ represents the $S_{z}=+1(0)$ components of the spin $%
S=1$. Thus, in order to move, the charge has to hop to the nearest neighbor, via a spin flip process, through states that differ in energy by $\Delta
\approx K/2.$ It can be easily verified that the efective hopping of this process is equal to $t_{ef}=t/\sqrt{2}.$ The dispersion relation given by this dynamics is: $\Delta /2\pm \sqrt{(\Delta /2)^{2}+4t_{ef}^{2}\cos ^{2}(k)%
\text{ }}$ . The expression corresponding to the lower band is plotted with full line in Fig. 2 and compared with the numerical result for $K=3$ and $%
J_{h}=100$. This expression is valid in general for a particle moving in an antiferromagnetic background where scattering between $k$ and $k+\pi $ states dominates the dynamics of the particle (dotted lines in Fig.2).
In the case where $K>>J_{h}>>t$ the spin distortion can be neglected and the particle propagates in an antiferromagnetic lattice. The Hund interaction alternates the site energy of the propagating particle so that the difference between the two sublattices is given by $\Delta =J_{h}(<\sigma
_{j}S_{j+1}>-<\sigma _{j}S_{j}>)\cong J_{h}(<S_{j}S_{j+1}>-<\sigma
_{j}S_{j}>)$ where we approximate $<\sigma _{j}S_{j}>\approx 1/4$ its value at the triplet state, and $<\sigma _{j}S_{j+1}>\approx $ $<S_{j}S_{j+1}>=\ln
2-1/4$ , the Bethe ansatz value. Using these values we find $\Delta
=0.19J_{h}.$ In this case $t_{ef}$ is equal to $t.$
=3.8truein =3.truein
When $J_{h}>>t\gtrsim K,$ the magnetic distortion around the charge is large and the effective hopping is dominated by the overlap between the magnetic distortions about the nearest neighbors sites. This last effect dominates the polaron effective mass. Therefore, the mass of polarons increases when $K
$ decreases, as obtained in Fig. 2, where $t_{ef}$ decreases from 0.75 for $%
K=1$ to 0.23 for $K=0.3$, in agreement with the above results showing that the spin distortion around the charge increases in magnitude and extension when $K$ decreases. In Fig. 3, we calculate the bandwidth for several values of $K$. One can distinguish clearly two regimes: $K<<t$ and $K>>t,$ the first corresponds to a large magnetic distortion and the second corresponds to a smaller one according to Fig. 1.
In order to test how robust is the polaronic description of the results, we pin the polaron to a site by lowering in $\epsilon _{0}$ the diagonal energy at site zero. This may be relevant to the real materials since the doping process necessarily introduces some disorder. Since we are treating a linear chain, this always localizes the particle, but the localization length should be very different for different effective masses, so that a small $%
\epsilon _{0}$ localizes much more the polaron for low values of $K$ than for larger ones. This is shown in Fig. 4 where we plot $<n_{i}>$ around site zero for different values of $\epsilon _{0\text{ }}$and $K$. For comparison we also show in full line the exponential fit of the different curves showing the change in localization length.
=3.5truein =3.truein
In Fig 5, we show the change in the values of $<n_{i}>$ for different magnetic fields It can be seen that the localization of the polaron decreases with magnetic field as a consequence of increased effective hopping between nearest neighbors. A fact that may be important for the transport properties of these systems since it implies a negative magnetoresistive behavior for conductivity due to hopping between localized states[@alalmex]
=3.8truein =2.5truein
Finally, in Fig. 6 we present some of the results for the two particles ground states. The charge-charge correlation function, normalized to the non correlated $(J_{h}=0)$ case, clearly shows repulsion between the particles. This long range repulsion increases with the magnetic distortion indicating its magnetic origin.
=3.5truein =2.5truein
The fact that polarons repel each other points to a picture of the electron doped systems similar to that proposed originally by de Gennes [@degen] of ’self-trapped electrons’. Further discussion on the non-diluted limit, will be postponed for a later publication where newer results will be shown.
In summary, we have investigated the possibility of a non uniform ground state in a model Hamiltonian using the Lanczos technique. The model describes chains of localized spins coupled antiferromagnetically on which electrons are added. These electrons suffer a strong ferromagnetic interaction with the local spins and can hop from site to site. Assuming the model adequately describes the physics of electron doped manganites, the results presented here point to a picture of these systems where heavy one electron polarons dominate the magnetic and transport properties. Their masses depend strongly on the relation between the hopping energy and the AF superexchange interaction. Clearly, the doping itself will localize the polarons so that transport will result from hopping between pinned sites. Negative magnetoresistance should appear as a consequence of the decrease of the pinning energy with magnetic field[@alalmex].
Adding two particles to the chains we find long range repulsion between them. This long range repulsion could give raise to charge ordering.
Finally, we would like to point out that the order of oxygen vacancies in $%
CaMnO_{3-\delta }$ makes real the possibility of one dimensional electron paths in these materials[@mate]. We hope that our results will stimulate more experimental and theoretical investigations on the electron doped manganites.
[*Acknowledgments*]{}
Two of us (C.D.B. and J.M.E.) are supported by the Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET). B. A. is partially supported by CONICET. M.A. is partially supported by the Centre National pour la Recherche Scientifique (CNRS). We would like to aknowledge support from the ’Fundacion Antorchas’ and the Program ECOS-SECyT A97E05.
Permanent address: Laboratoire d ’Etudes des Propriétés Electroniques des Solides (LEPES) - BP166, 38042 Grenoble Cedex 9, France.
R. von Helmholt, J. Wecker, B. Holzapfel, L. Schultz, and K. Samwer, Phys. Rev. Lett. [**71**]{}, 2331 (1993).
A. P. Ramirez, J. of Physics: Condens. Matter 8171 (97) and references therein.
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D. P. Arovas and F. Guinea, preprint.
E. Dagotto [*et. al.*]{}, cond-mat/9709029; J. Riera, K. Hallberg, and E. Dagotto, Phys. Rev. lett. [**79**]{}, 713 (1997); S. Yunoki [*et. al.*]{}, Phys. Rev. Lett. [**80** ]{}845 (1998).
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---
bibliography:
- 'neco\_2.bib'
---
1
\
[Towards a Kernel based Physical Interpretation of Model Uncertainty]{}
\
[**Rishabh Singh**]{}\
[**Jose C. Principe**]{}\
[Computational NeuroEngineering Laboratory, Department of Electrical and Computer Engineering, University of Florida, Gainesville, FL 32611, USA]{}\
[**Keywords:**]{} Model uncertainty, RKHS, point-prediction, neural networks, quantum physics, moment decomposition
\
[**Abstract**]{}
This paper introduces a new information theoretic framework that provides a sensitive multi-modal quantification of data uncertainty by leveraging a quantum physical description of its metric space. We specifically work with the kernel mean embedding metric which yields an intuitive physical interpretation of the signal as a *potential field*, resulting in its new energy based formulation. This enables one to extract multi-scale uncertainty features of data in the form of *information eigenmodes* by utilizing moment decomposition concepts of quantum physics. In essence, this approach decomposes local realizations of the signal’s PDF in terms of quantum uncertainty moments. We specifically present the application of this framework as a non-parametric and non-intrusive surrogate tool for predictive uncertainty quantification of point-prediction neural network models, overcoming various limitations of conventional Bayesian and ensemble based UQ methods. Experimental comparisons with some established uncertainty quantification methods illustrate performance advantages exhibited by our framework.
Introduction
============
Information Theory: Physics based Perspective
---------------------------------------------
The foundation of information theory lies in its ability to quantify uncertainty in random variables. This is primarily achieved by entropy, a metric having its origins in thermodynamics where it is used to describe micro-states of a system [@bol]. Shannon [@r1] first formulated entropy in the context of information theory which was later generalized by Renyi [@r2] among many others. It has, since then, become an indispensable tool in density estimation and other statistical evaluations that attempt to characterize the intrinsic generating functions of data [@r3; @r4; @r5]. However, since even before the introduction of Shannon’s work, Fisher information [@fish] has been well regarded as the cornerstone concept in measuring the gain of information from data and in quantifying the *order* of a system [@roy1], instead of disorder as is done by entropy. This presents a strong link between the role of information in data analysis and physical laws [@roy2]. Indeed, attempts to formulate quantum physical models of stock markets in the field of econophysics, for instance, have been quite prevalent in recent years [@q1; @q2].
Motivation and Contributions
----------------------------
### Data Uncertainty Viewpoint {#data-uncertainty-viewpoint .unnumbered}
In most real world scenarios, observed signals are generated by systems controlled by a multitude of source processes and noise resulting in very complicated data dynamics comprising of high uncertainty and stochasticity. Current machine learning models and information theoretic divergence measures fail to effectively characterize uncertainties associated with such data. Quantum based formulations in physics, on the other hand, have been well known for providing high resolution multi-scale characterizations of system dynamics. This is achieved through a stochastic description of the system in terms of energy modes in a Hilbert space of functions. The key qualities of such formulations is their non-parametric nature and a complete consideration for all internal associations of the system, leading to a description of the system at all points in space. We hypothesize that extending information theoretic measures in terms of such physics based formulations could yield similar advantages in the characterization of data and models, thereby providing us with a more enhanced view of information dynamics. This inspires us to develop data characterization tools that gather source information intrinsically from data while making minimal assumptions on their governing distributions, other than the fact that their generating processes follow laws of physics. Our conjecture is that frameworks that work with the intrinsic stochasticities associated with local *data-induced* metric spaces would be significantly more sensitive towards signal characterization. Perhaps the best contender for this metric space is the Gaussian reproducing kernel Hilbert space (RKHS) which has been well established to provide universal characterization of data [@parze; @gp; @g1; @g2]. Moreover, projecting data into an RKHS transforms them into Gaussian functions centered at the data coordinates that obey the properties of a potential field [@prin]. Hence the RKHS makes it possible to obtain quantum physical formulations of data properties with simplicity because of the uses of the Hilbert space.
Therefore, towards the goal of effectively quantifying data uncertainty, we introduce an RKHS based information theoretic framework that utilizes quantum physical interpretation of the data space to extract its various uncertainty moments. This methodology has solid foundations because it uses the kernel mean embedding (KME) metric [@emb] which embeds the data into a kernel feature space and non-parametrically characterizes its PDF as an element in the RKHS. We show that the information potential field is an empirical estimate of the KME which eventually leads to a quantum physical energy-based description of the data structure in terms of a Schrödinger’s equation. This allows us to extract the various modes of uncertainty related to the interaction of an upcoming data sample with the signal’s history by implementing a moment decomposition procedure based on orthogonal polynomial projections, similar to that used in quantum physics to extract the eigenmodes of a particle with respect to its neighboring force field. We also refer to extracted data uncertainty modes as information eigenmodes since they are essentially *energy-based* information features of data. Our uncertainty framework is depicted in fig. \[fr1\] and can be summarized in terms of the following key steps.
1. Definition of an RKHS based metric space using kernel mean embedding of data (information potential field).
2. Quantum physical energy-based formulation of the metric space in terms of the Schrödinger’s equation.
3. Moment decomposition procedure involving orthogonal polynomial projections of the wave-function to extract uncertainty eigenstates from the quantum formulation.
![Basic depiction of the proposed framework[]{data-label="fr1"}](frm.png)
Such a framework offers several key advantages. Firstly, owing to the use of a kernel based metric, our framework takes into account all intrinsic even order statistical moments (for the Gaussian kernel) and provides a universal characterization of data. Secondly, the quantum-physical formulation of the data space and its subsequent moment decomposition results in a high resolution (multi-scale) quantification of data uncertainties, which becomes increasingly sensitive at the tails of data distribution (where uncertainty is maximum). Lastly, the framework can be implemented in real time on a sample-by-sample basis. The kernel metric and its associated quantum formulations adapt to new data thereby making it an appropriate feature extraction framework for streaming data.
### Model Uncertainty Viewpoint {#model-uncertainty-viewpoint .unnumbered}
In the world of machine learning, feedforward point-prediction neural network models have made remarkable progress over the past two decades in a large variety of applications [@lec]. However, despite their success, such models do not provide information related to their prediction uncertainties. This information is crucial in sensitive application arenas such as personalized medicine and autonomous driving, especially given how prone neural networks are towards overfitting. Moreover, there have also been alarming revelations regarding the high susceptibility of such models towards adversarial attacks [@su; @adv]. All of this has led to a recent realization of the importance of uncertainty quantification of point-prediction models among the machine learning community thus making it an active area of research.
We are therefore motivated to explore the utility of our proposed framework for the predictive uncertainty quantification of point-prediction based learning models. The main idea here is to find the uncertain regions in the input-output mapping (instead of data space) learnt by the model. In this case, instead of working on data PDF, we propose a quantum based decomposition of the local realizations of distribution learnt by the model parameters based on the cross-entropy between the model output and its internal layers. This cross-entropy leads to the definition of the cross information potential. We hypothesize that such a decomposition of the model could yield useful uncertainty information related to its predictions. Before delving into more details, we first review some established methods of uncertainty quantification (UQ).
Existing UQ methods can be broadly classified into *forward UQ* and *inverse UQ* methods from an implementation point of view [@uncdef1; @uncdef2]. In uncertainty propagation (forward UQ), one attempts to directly characterize the model output uncertainty distribution from the implicit uncertainties present in the parameters. Inverse UQ, on the other hand, attempts to quantify uncertainty distributions over model parameters. In the context of machine learning, most of the focus has been on the latter category with a domination of Bayesian based inferencing methods which offer the most mathematically grounded approach to quantify model uncertainty by learning probability distributions of model weights [@mack; @neal; @bishop]. Early development of Bayesian based models revolved around Laplacian approximation [@mack], Hamiltonian Monte Carlo [@neal], Markov-chain Monte Carlo (MCMC) based Bayesian neural networks [@bishop]. Although such methods offer a principled approach of quantifying model uncertainty mainly by marginalizing over model parameters, they involve prohibitive computational costs and lack scalability towards large data and model architectures. Most of the recent work in this field has therefore been related to developing faster variational inference based approaches that offer more efficient ways of training Bayesian neural networks (BNNs) [@graves; @jord; @hof]. The high parameter dimensionality and the complexity of weight associations in modern neural networks still makes it very difficult for such variational inference approaches to adequately capture parameter dependencies [@projbnn]. Other methods involve surrogate modeling techniques that exploit the input-output mapping learnt by the model [@nag; @sur1; @sur2]. Here, computationally cheap approximations of models are used for easier extraction of the relevant information related to model uncertainty. Forward UQ methods include ensemble based methods where multiple instances of models with different initializations are trained on noisy data and the result is the aggregation of all model outputs [@tib; @osb; @pearce]. The variations of the results provide the necessary uncertainty information. A notable related work is that of Lakshminarayan [@laks] where authors use ensemble neural networks to implement forward UQ. Recent work of Gal and Ghahramani [@gal] has gained increased popularity due to its simplicity and effectiveness in quantifying predictive uncertainty. Here, authors propose Monte Carlo dropout where multiple instantiations of dropout are used during testing of models to obtain the uncertainty intervals associated with the model predictions.
We advocate an approach for predictive uncertainty quantification that is non-intrusive to the training process of a traditional point-prediction model and relies solely on extracting information from the internal dependencies of the *trained* model with respect to its output. In this regard, we hypothesize that the application of our framework as a forward UQ method could be advantageous. The idea is to create an alternate representation or an embedding of the learnt input-output mapping that makes it easier to quantify how uncertain (or probabilistically far) a prediction is with respect to it. Towards this end, we utilize our framework as a surrogate uncertainty quantifier of a trained neural network that constructs an RKHS embedding of the model’s discrete input-output mapping, realized at every test instance. This is done by projecting the model’s internal activation outputs (from one or more layers) into the RKHS and empirically evaluating it with respect to the corresponding model prediction (implemented during each test cycle). Multi-scale uncertainty modes can then be extracted by the subsequent quantum physical formulations and moment decompositions implemented in the same manner as before. In simpler terms, we aim to utilize our framework to provide an alternate richer representation of the model-output dynamics. The implementation is depicted in fig. \[fr2\]. From a physical perspective, *one can visualize the kernel embedding of the model’s internal activations as a drum membrane and the output as drumstick hitting the membrane during each test cycle.* The resultant modes of membrane vibration
![Implementation of proposed framework on a neural network.[]{data-label="fr2"}](nnfr_nnn.png)
(that depend on where exactly the membrane was hit) quantify the uncertainties of the model at various energy levels with respect to the output, which is what we are interested in extracting. We submit various advantages offered by our framework over traditional model uncertainty quantification paradigms.
- Our framework is non-parametric as well as non-intrusive to the model’s training process and hence can be implemented universally on trained machine learning models although we restrict ourselves to neural network models in this paper.
- Owing to the established statistical richness of the RKHS, the kernel mean embedding of the model utilizes all even order intrinsic associations of the model’s internal structure (valuable information that is usually ignored in conventional applications).
- We further posit that our framework, which yields high resolution multi-scale uncertainty features, exhibits increased sensitivity towards the characterization of the model’s internal realizations of data the and is able to better quantify regions of test data where the model hasn’t been trained, when compared to established methods.
Paper Structure
---------------
Further parts of the paper are organized as follows. We begin by introducing the kernel mean embedding (KME) theory in section 2, where we discuss some of its relevant properties. We also introduce the information potential field (IPF) metric as an empirical estimate of the KME and the primary metric used in this paper. We highlight its relevance with respect to Renyi’s entropy. In section 3, we introduce a quantum physical formulation of the data derived from the IPF and show in section 4 how this interpretation opens the doors to a novel uncertainty moment decomposition method, inspired from established methods in quantum physics. Section 5 presents a step-by-step summary of the entire proposed framework. We first analyze our framework from a completely data driven perspective (without involving a model) and provide pedagogical examples in section 6 using implementation of the framework on time-series signals, which provides an intuitive understanding of how it characterizes data dynamics. We subsequently present the application of the framework for quantifying predictive uncertainties of neural network models in section 7. Related experimental results are provided in section 8 which initially consist of examples illustrating advantages offered by our framework compared to established methods such as Monte Carlo dropout and GP regression. Further results on some benchmark datasets provide quantified evidence demonstrating the same.
Kernel Mean Embedding
=====================
General Definition and Properties
---------------------------------
Kernel methods have been very popular and well established in the field of machine learning [@smola]. The crux of their success is largely owed to a powerful property of the reproducing kernel Hilbert space (RKHS) associated with positive definite kernels called the “kernel trick" [@aron], which allows one to pose any problem in an input set $X$ as a linear-algebraic problem in its RKHS, $H$, with a non-linear transformation (embedding) of $X$ into $H$ induced by a kernel $k: X $ x $ X \rightarrow R$. In other words, the RKHS, constructed by an appropriate kernel, allows one to simplify any non-linear relationship in an input space as a linear expression in a higher dimensional space. This property has led to the advent of many popular kernel based algorithms in machine learning [@hoff; @jp]. Following similar intuition, another elegant property of the RKHS is the theory of kernel mean embedding (KME) which allows one to non-parametrically quantify a data distribution from the input space as an element of its associated RKHS [@embor]. The main idea stems from the probabilistic generalization of the measure-theoretic viewpoint of data. For a detailed explanation of the metric, we refer the reader to [@emb]. Its definition is summarized as follows.
Suppose that the space $\mathcal{Z(X)}$ consists of all probability measures $\mathbb{P}$ on a measurable space $(\mathcal{X},\Sigma)$. The kernel mean embedding of probability measures in $\mathcal{Z(X)}$ into an RKHS denoted by $\mathcal{H}$ and characterized with a reproducing kernel $k : X \times X \rightarrow \mathbb{R}$ is defined by a mapping $$\mu: \mathcal{Z(X)} \rightarrow \mathcal{H}, \ \ \mathbb{P} \mapsto \int k(x, .)d\mathbb{P}(x)
\label{5}$$
Hence the kernel mean embedding (KME) represents the probability distribution in terms of a mean function by utilizing the kernel feature map in the space of the distribution. In other words, $$\phi(\mathbb{P}) = \mu_{\mathbb{P}} = \int k(x, .)d\mathbb{P}(x)
\label{kl}$$ There are several useful properties associated with the KME. For instance, it is injective, meaning that $\mu_{\mathbb{P}} = \mu_{\mathbb{Q}}$ only when $\mathbb{P} = \mathbb{Q}$, thus allowing for unique characterizations of data distributions. It also makes minimal assumptions on the data generating process and enables extensions of most learning algorithms in the space of probability distributions.
In this paper, we propose a KME based feature decomposition of data wherein we extract the successive quantum *stochastic energy modes* associated with the data. This is accomplished by reinterpreting the data space as a physical system by utilizing the KME. We reformulate the empirical form of KME in terms of a probabilistic wave-function metric. In doing so, we treat the space of data as a quantum physical *force field* composed of inherent uncertainties wherein the interaction of an information particle (data sample) with the field associates an implicit potential energy with the information particle. Thereafter, we extract the various energy modes associated with the information particle from this formulation of data by mimicking established concepts of quantum physics used popularly in the Eigenstate decomposition of physical systems. In order to explain and motivate the physical characterization of the KME, we first describe its empirical estimate and the relevance of the estimate with respect to popular information theoretic measures in the next section before its physical interpretation and uncertainty extraction in sections 4 and 5 respectively.
Empirical Estimate
------------------
In most real world applications, there is no information available regarding the nature of $\mathbb{P}$. One must therefore resort to empirical estimation of the KME. The simplest method of empirically computing the KME is by computing its unbiased estimate given by $$\hat\mu = \frac{1}{n}\sum_{t=1}^{n}k(x_t,.).
\label{6}$$ Here $\hat\mu$ converges to $\mu$ for $n \rightarrow \infty$, in concordance with the law of large numbers. One can intuit that the empirical KME is also a result of the general Dirac formulation assigning a mass of $1/n$ to every data sample. This also gives rise to the interpretation of the empirical KME as an instance of a point process [@emb].
Relevance in Renyi’s Entropy Estimation
---------------------------------------
Renyi’s quadratic entropy [@r2] is given by $$H_2(X) = -log\int{p(x)^2 dx} = -logV(X).$$ The argument of the logarithm in Renyi’s entropy, $V(X)$, is an important quantity called the information potential (IP) of the data set [@ux], which is simply the mean value of the PDF. One can estimate this quantity by using the Parzen density estimator [@parz] for estimating $p(x)$. Hence, assuming a Gaussian kernel window of kernel width $\sigma$, one can readily estimate directly from experimental data ${x_i, i=1, ..., N}$ the information potential as $$\begin{aligned}
V(X) = \int p(x)^2 dx = \int\bigg(\frac{1}{N}\sum_{i=1}^{N}G_{\sigma}(x-x_i)\bigg)^2dx \\ = \frac{1}{N^2}\int
\bigg(\sum_{i=1}^{N}\sum_{j=1}^{N}G_\sigma(x-x_j).G_\sigma(x-x_i)\bigg)dx \\ = \frac{1}{N^2}
\sum_{i=1}^{N}\sum_{j=1}^{N}\int G_\sigma(x-x_j).G_\sigma(x-x_i)dx \\ = \frac{1}{N^2}
\sum_{i=1}^{N}\sum_{j=1}^{N} G_{\sigma/\sqrt{2}}(x_j - x_i)
\end{aligned}
\label{7}$$ Hence the IP is a number obtained by the double sum of the Gaussian functions centered at differences of samples with a larger kernel size. Exactly the same result is obtained using the empirical estimate of the KME in a RKHS defined by the Gaussian function. There is a physical interpretation of $V(X)$ if we think of the samples as particles in a potential field, hence the name information potential. It can also be interpreted as the total potential created by the data set in an RKHS, i.e. $$V(X) = \frac{1}{N}\sum\limits_{j=1}^{N}V(x_j),$$ where, $$V(x) = \frac{1}{N}\sum\limits_{i=1}^{N}G(x - x_i)
\label{cf}$$ represents the *field* due to each sample, which can be interpreted as an information particle. We refer to $V(x)$ as the *information potential field* (IPF) and it is basically a continuous function over the RKHS obtained by the sum of Gaussian bumps centered on the samples. We now delve into the quantum physical interpretation of the empirical KME which we now refer to as the IPF henceforth.
Quantum Interpretation of the IPF
=================================
From a physical perspective, systems are conceptualized as either *classical* or *quantum*. Classical systems are generally characterized by dynamic variables that are smoothly varying and can be modeled by deterministic parameters (Newtonian laws, for instance). Quantum physical systems, on the other hand, are characterized by jumps with increased stochasticity and uncertainty in the measurement of the their associated parameters. The composition of probabilistic wave-function modes determines system behavior in this case. The dynamics of a physical particle under the influence of a general quantum system can be described as follows [@qho].
The time-independent Schrödinger’s equation for a particle at position $x$ in a general quantum system is given by $$\hat{H}\psi = \bigg(-\frac{\hslash^2}{2m}\nabla^2 + V_r(x)\bigg)\psi(x) = E\psi
\label{ti}$$ where the notations have the following meaning,
- $\hat{H}$ denotes the Hamiltonian operator and is given by $\hat{H} = -\frac{\hslash^2}{2m}\nabla^2 + V(x)$, where
- $-\frac{\hslash}{2m}$ is the kinetic energy operator with $\hslash$ and $m$ being the Planck’s constant and particle mass respectively.
- $V_r(x)$ represents the potential energy of the particle at position $x$.
- $\nabla^2$ is the Laplacian operator.
- $\psi(x)$ is the wave-function value at position $x$ that also denotes the probability of finding the particle at that position given by $p(x) = {|\psi(x)|}^2$.
A similar interpretation can be extended to data systems as well [@prin]. We can infer that the IPF is always positive and regions of space with more samples will have a larger IP, while regions of the space with few samples will have a lower IP. Here, the shape of the kernel function will determine the “gravity”, instead of the inverse square law of physics. Following this intuition, the idea of a potential field over the space of the samples can be readily extended with quantum theoretical concepts [@prin]. A Schrödinger’s time-independent equation equivalent to (\[ti\]) was formulated to define a new potential energy function $V_s(x)$ (data-equivalent form of $V_r(X)$ in (\[ti\])) based on a wave-function defined by using the IPF as the probability measure $p(x)$. Since $p(x) = |\psi(x)|^2$, then for a set of information particles with a Gaussian kernel, the wave-function for one dimensional information particle becomes, $$\psi(x) = \sqrt{\frac{1}{N}\sum\limits_{i=1}^{N}G_{\sigma}(x-x_i)}
\label{ps}$$ Furthermore, we can assume $m = 1$ in (\[ti\]) since all information particles are assumed to have the same mass. We can also rescale $V_s(x)$ such that $\sigma$ (bandwidth of the kernel window) is the only free parameter that replaces all physical constants. This reformulates (\[ti\]) to yield the Schrödinger’s time-independent equation for information particles as $$H\psi(x) = \bigg(-\frac{\sigma^2}{2}\nabla^2 + V_s(x)\bigg)\psi(x) = E\psi(x)
\label{qsc}$$ where $H$ denoted the Hamiltonian. Solving for $V_s(x)$, we obtain: $$V_s(x) = E + \frac{\sigma^2/2\nabla^2\psi(x)}{\psi(x)}
\label{sf}$$ which was called the *quantum information potential field* (QIPF) denoted by $V_s(x)$. To determine the value of $V_s(x)$ uniquely, we require that $min(V_s(x)) = 0$, which makes $$E = -min\frac{\sigma^2/2\nabla^2\psi(x)}{\psi(x)}
\label{eig}$$ where $0 \leq E \leq 1/2$. Note that $\psi(x)$ is the eigenfunction of $H$ and $E$ is the lowest eigenvalue of the operator, which corresponds to the ground state. Given the data set, we expect $V_s(x)$ to increase quadratically outside the data region and to exhibit local minima associated with the locations of highest sample density (clusters). This can be interpreted as clustering since the potential function attracts the data distribution function $\psi(x)$ to its minima, while the Laplacian drives it away, producing a complicated potential function in the space. We should remark that, in this framework, $E$ sets the scale at which the minima are observed. This derivation can be easily extended to multidimensional data. We can see that $V_s(x)$ in (\[sf\]) is also a potential function that differs from $V(x)$ in (\[cf\]) because it is now an energy based formulation associated with the quantum description of the IPF.
Extraction of QIPF Energy Modes
===============================
Unlike the classical interpretation, the quantum interpretation provides a much more detailed decomposition of the system dynamics by assuming it to consist of a large (potentially infinite) number of stochastic *features*, given by the energy modes. Likewise, when applying this quantum field potential to data, the same interpretation holds. Because of the finite number of samples, the local structure of the PDF in the space of samples will be very difficult to quantify. In the input space, we normally use clustering or other techniques to achieve this goal, but we still have an enormous difficulty in characterizing the tails of distributions. Here it is relevant to remember the characteristic function of the PDF and the cumulants, which has been a work horse of statistics. The issue with the cumulants is the complexity of estimating the higher order moments in high dimensional data. Here, instead, we will follow the teachings of quantum theory and will employ a model decomposition of the wave function to subsequently extract uncertainty modes that focus on characterizing the tail regions of the data PDF. We first analyze the behavior of the quantum harmonic oscillator, a popular example of a quantum model that is pervasively used in many fields to describe system behavior (econometrics, for instance [@q1; @q2]). The following definition describes the extraction of the system’s wave-function modes [@qho].
The potential energy of a particle can be generalized using Hooke’s law as $V(X)=\frac{1}{2}m\omega^2x^2$. The Hamiltonian of the particle characterizes its dynamic parameters (position and momentum) and is formulated as $$\hat{H} = \frac{\hat{p}^2}{2m} + \frac{1}{2}m\omega^2x^2$$ where $\omega = \sqrt{\frac{k}{m}}$ is the angular frequency of the oscillator, $x$ is the position and $\hat{p} = -i\hslash\frac{d}{dx}$ represents the momentum operator. Given this Hamiltonian, the time-independent Schrödinger’s equation can be formulated as $$\hat{H}\psi(x) = \bigg[-\frac{\hslash^2}{2m}\frac{d^2}{dx^2} + \frac{1}{2}m\omega^2 x^2\bigg]\psi(x) = E\psi(x)
\label{qu}$$ This differential equation can be treated as an eigenvalue problem and solved using the spectral method to yield a family of wave-function modes, $\psi_n(x)$, that amount to successive Hermite polynomial moments. The solutions are given as:
$$\begin{aligned}
E_0 = \frac{{\hslash}w}{2}, &&& {\psi}_0 = {\alpha}_0e^{\frac{-y^2}{2}}\\
E_1 = \frac{3{\hslash}w}{2}, &&& {\psi}_1 = {\alpha}_0(2y)e^{\frac{-y^2}{2}}\\
E_2 = \frac{5{\hslash}w}{2}, &&& {\psi}_2 = {\alpha}_0(4y^2 - 2)e^{\frac{-y^2}{2}}\\
.
\label{sol}
\end{aligned}$$
Here, $y = \sqrt{\frac{mw}{\hslash}}x \ $, $ \ \psi_0, \psi_1, \psi_2...$ are the obtained wave-function modes and $E_0, E_1, E_2...$ are their corresponding eigenvalues. Therefore the solution to the Schrödinger equation for the harmonic oscillator yields infinite eigenfunctions successively associated with each other through Hermite polynomials.
Hence we see that the quantum interpretation enables one to extract the various intrinsic energy modes associated with the system, along with the corresponding eigenvalue of each mode. In the previous section, we have described the Schrödinger’s equation associated with the quantum IPF (QIPF) given by (\[qsc\]) which essentially provides a quantum interpretation of data dynamics similar to how (\[qu\]) does for the harmonic oscillator. Our goal is to now extract successive energy modes (analogical to those obtained in (\[sol\])) associated with the QIPF given by $V_s(x) = E + \frac{\sigma^2/2\nabla^2\psi(x)}{\psi(x)}$. The ground state of the wave-function, in the case of the QIPF formulation, has already been probabilistically defined as an expression of the empirical KME given by $\psi(t) = \sqrt{p(t)} = \sqrt{\frac{1}{n}\sum\limits_{i=1}^{n}k(x_i, t)}$. We use this information to summarize the QIPF state extraction procedure in the following conjecture.
Consider the QIPF of the data samples $x$ as $V_s(x) = E + \frac{\sigma^2/2\nabla^2\psi(x)}{\psi(x)}$ with the associated ground state wave-function given by $\psi(x) = \sqrt{\frac{1}{n}\sum\limits_{i=1}^{n}k(x_i, t)}$. The approximate higher order energy modes of $\psi(x)$ can be extracted by projecting the ground state wave-function into the corresponding order Hermite polynomial given by $\psi_k(x) = H^*_{k}(\psi(x))$, where $H^*_k$ denotes the normalized $k^{th}$ order Hermite polynomial, normalized so that $H^*_k = \int\limits_{x=-\infty}^{\infty}e^{-x^2}[H_k(x)]^2dx = 1$. This leads to the evaluation of the higher order QIPF states as $$\begin{aligned}
V_s^k(x) = E_k + \frac{\sigma^2/2\nabla^2H^*_k(\psi(x))}{H^*_k(\psi(x))}\\\\ = E_k + \frac{\sigma^2/2\nabla^2\psi_k(x)}{\psi_k(x)}
\label{vs}
\end{aligned}$$ where $k$ denotes the order number and $E_k$ denotes the corresponding eigenvalues of the various modes and is given by $$E_k = -min\frac{\sigma^2/2\nabla^2\psi_k(x)}{\psi_k(x)}
\label{ek}$$
The extracted modes of the data QIPF given by $V^k_s(x)$ are thus stochastic functionals depicting the different moments of *potential energy* of the data at any point $x$. This is different from the IPF formulation of (\[cf\]) because $V^k_s(x)$ is an energy based metric resembling the potential energy operator in a quantum harmonic oscillator at various energy levels (Eigenstates) depicted by $E_k$.
Summarizing the Framework
=========================
The key aspects of the framework are summarized as follows.
- **Metric Construction:** We construct an RKHS based metric space of the data using the kernel mean embedding function which provides a non-parametric characterization of the implicit PDF of the data without making any prior assumptions.
- **Quantum-Physical Interpretation:** We use the empirical KME (or the IPF) to define a probabilistic wave-function over the space of data thereby providing a quantum-physical interpretation of data spaces. Thereafter, we utilize the Schrödinger’s equation to provide an energy based interpretation of the data dynamics by imposing kinetic and potential energy operators that are ‘stochasticized’ by the wave-function over the data metric space, thereby defining the overall energy composition of the data.
- **Potential Energy Expression (QIPF):** We replace all physical constants related to the kinetic energy operator using the kernel bandwidth and thereafter express the potential energy of data, given by the QIPF, as a function of all other terms - eigenvalue, wave-function and the kernel bandwidth. The eigenvalue is simply determined to be a lower bound value which constraints the QIPF to always be positve.
- **Extraction of QIPF States:** We extract approximate energy moments (or modes) of the data QIPF by implementing orthogonal Hermite polynomial projections of the ground state wave-function and finding the corresponding QIPF state.
The implementation details of the framework are illustrated in figure \[frf\].
Mode Decomposition of Time Series
=================================
We begin our analysis of the proposed framework by studying how it characterizes time series signals. We used MATLAB R2019a to obtain the results shown in this section. For an intuitive understanding of how the different QIPF modes get configured in the space of data, we extracted the first 6 modes of a simple sine wave signal. We generated 3000 samples of a 50 Hz unweighted sine wave signal sampled at the rate of 6000 samples per second to mimic a continuous signal. The signal was also normalized to zero mean and unit standard deviation. We used all 3000 samples as centers to construct the wave-function given by (\[ps\]) and then evaluated it at each point in the data space range $x = (-6, 6)$ using a step size of 0.1. We then evaluated the Hermite projections of the wave-function value at each point to subsequently extract 6 QIPF modes using the formulation given by (\[vs\]). This was done for four different kernel widths whose corresponding QIPF plots (represented by solid color lines) are shown in fig. \[space\]. The dashed line represents the empirical KME estimate (or simply the IP) given by $p(x) = \psi^2(x) = \frac{1}{N}\sum\limits_{i=1}^{N}\kappa(x, x_i)$, which basically gives an estimate of the data PDF. All plots were normalized for easier visualization. Perhaps the most important property of the extracted QIPF modes that can be observed from the plots is that, for all kernel widths, they systematically signify the more uncertain
[.45]{} ![Analysis of mode locations in the data space using different kernel widths. Solid colored lines represent the different QIPF modes. Dashed line represents the empirical KME (IP).[]{data-label="space"}](0_6.png "fig:"){width="7.2cm" height="3.3cm"}
[.45]{} ![Analysis of mode locations in the data space using different kernel widths. Solid colored lines represent the different QIPF modes. Dashed line represents the empirical KME (IP).[]{data-label="space"}](1_2.png "fig:"){width="8cm" height="3.3cm"}
[.45]{} ![Analysis of mode locations in the data space using different kernel widths. Solid colored lines represent the different QIPF modes. Dashed line represents the empirical KME (IP).[]{data-label="space"}](1_8.png "fig:"){width="7.2cm" height="3.3cm"}
[.45]{}
regions of the data space closer to the tails of the data PDF. In fact, one can observe the significant increase in the density (or clustering) of the extracted QIPF modes as one moves farther away from the mean ($x=0$) and towards the PDF tails. Furthermore, we observe here that the modes appear sequentially based on their orders, with the lower order modes signifying regions closer to the mean and the higher order modes clustering together at the PDF tails. An interesting observation that must also be noted is that, for larger kernel widths (1.8 and 2.6), which exceed the dynamic range of the signal, we can see that some high order modes begin to emerge in the region around the mean. This behavior is remarkably similar to physical systems. If we consider the same drum membrane analogy we used in section I, one can visualize the space of the samples here as the membrane. If we increase the tension of the membrane and hit it, the drum will vibrate for a long time. In our potential field, the stiffness is controlled by the kernel size. If the kernel size is large, the QIPF becomes stiffer leading to the energy in the higher QIPF modes to increase. If one decreases the kernel size, the membrane becomes more elastic leading to many local modes that decay much faster.
As a pedagogical demonstration to understand how the framework characterizes different dynamical data structures, we implement it to compare the extracted uncertainty modes of a simple sine wave oscillator and a Lorenz series. The sine wave represents one of the simplest time series with a single generating function. The Lorenz series on the other hand is a chaotic deterministic dynamical system with complex state space defined by the following mutually coupled differential equations (with $\sigma$, $\rho$ and $\beta$ as system parameters) governing its dynamics: $$\frac{dx}{dy} = \sigma(y-x),\>
\frac{dy}{dt} = x(\rho - z) - y,\>
\frac{dz}{dt} = xy - \beta{z}.
\label{lorenz}$$ We generated 3000 samples of Lorenz series after setting the parameters as $\sigma = 10$, $\rho = 28$ and $\beta = 8/3$ and the initial conditions as $x_1 = 0$, $y_1 = 1$ and $z_1 = 1.05$. The signal was also normalized to zero mean and unit variance. We generated two sine wave signals with the first one having a fundamental frequency of 150 Hz and the second one with an added odd harmonic frequency component (150 Hz + 250 Hz). The signals were sampled at a rate of 6000 samples per second and were normalized to zero mean and unit variance. We extracted the first 10 QIPF modes using (\[vs\]) and (\[ek\]) to encode the different signals. The kernel width used for doing so was fixed to a moderate value of 1.2 for all signals. Fig. \[dom\] shows the signals (top row) and the corresponding histogram plots (bottom row) of the number of times the value of each QIPF mode dominated over the others throughout the durations of the signals. As can be seen in fig. \[dom\](a), there are only two dominant modes in case of the single frequency sine wave (modes 2 and 3). Addition of an odd harmonic leads to increase in the number of modes contributing to the signal dynamics to four (fig. \[dom\](b)). The dominant modes in the case of Lorenz series, on the other hand, are more spread out (across all 10 modes) thus indicating a more complex data dynamical structure. These trends are quite similar to what we would expect from a frequency decomposition of the signals, except that here we are able to perform this decomposition on a sample-by-sample basis.
[.3]{} ![*Top row:* Generated signals from different dynamical systems. *Bottom row:* Dominance frequencies of QIPF states corresponding to each signal.[]{data-label="dom"}](org.png "fig:")
[.3]{} ![*Top row:* Generated signals from different dynamical systems. *Bottom row:* Dominance frequencies of QIPF states corresponding to each signal.[]{data-label="dom"}](mix_n.png "fig:")
[.3]{}
[.3]{} ![*Top row:* Generated signals from different dynamical systems. *Bottom row:* Dominance frequencies of QIPF states corresponding to each signal.[]{data-label="dom"}](org_h.png "fig:")
[.3]{} ![*Top row:* Generated signals from different dynamical systems. *Bottom row:* Dominance frequencies of QIPF states corresponding to each signal.[]{data-label="dom"}](mix_h.png "fig:")
[.3]{} ![*Top row:* Generated signals from different dynamical systems. *Bottom row:* Dominance frequencies of QIPF states corresponding to each signal.[]{data-label="dom"}](lor_h.png "fig:")
Model Uncertainty Quantification
================================
The QIPF mode extraction framework can so be naturally extended for implementation on machine learning models. The fundamental idea here is to create a continuous surrogate embedding of the trained model that represents its intrinsic distribution. We can then extract the QIPF uncertainty modes associated with the interactions between model’s embedding and its output. We expect this to quantify the extent to which the prediction falls within the scope of the model’s intrinsic distribution. As evidenced in the previous section, we expect higher order QIPF modes to cluster in data regions where the model has not been trained, represented by the tails of the input-output mapping PDF learnt by the network, thereby providing a sensitive uncertainty characterization of data spaces unknown to the model. In this regard, we specifically focus on neural network models due to a recent surge in its research interest.
The basic implementation strategy of the QIPF decomposition framework on a trained neural network is the same as the data-based implementation, except for the way in which we construct the information potential field (IPF). In this case, we aim to decompose the interactions between the RKHS fields of different pairs of network layers (with one of them typically being the output layer), thereby obtaining a multi-scale uncertainty representation of the implicit mapping between them. Intuitively, this quantifies the *probabilistic discrepancies* between two layers of the network during each test cycle. The RKHS field of each layer is represented by the kernel feature map constructed by its corresponding node activation outputs. We evaluate the *cross information potential* (CIP) which measures interactions between two information potentials [@prin]. One can represent the cross information potential between two layers of the neural network using a generalized form of the kernel mean embedding formulation. Let us consider two layers of an ANN whose node outputs are represented by the random variables $L_1$ and $L_2$. The kernel feature map of $L_1$ can then be represented in the same form as (\[kl\]) as $$\mu_{\mathbb{P}_{L_1}} = \int \kappa(l_1, .)\mathbb{P}_{L_1}(l_1)dl_1.$$ The mean evaluation of $\mu_{L_1}$ at all points specified by $L_2$ is the cross information potential between $L_1$ and $L_2$ and can be represented as $$\mu_{\mathbb{P}_{L_1}}(\mathbb{P}_{L_2})=V_c({{\mathbb{P}_{L_1}},{\mathbb{P}_{L_2}}}) = \int\int \kappa(l_1, l_2)\mathbb{P}_{L_1}(l_1)\mathbb{P}_{L_2}(l_2)dl_1dl_2.
\label{km2}$$ The empirical evaluation of (\[km2\]) leads to $$\hat{V_c}({{\mathbb{P}_{L_1}},{\mathbb{P}_{L_2}}}) = \frac{1}{nm}\sum\limits_{i=1}^{n}\sum\limits_{j=1}^{m}\kappa\big(l_1(i)-l_2(j)\big)$$ The quantum decomposition can then be performed in the same way as before (summarized in fig. \[frann\]), thus leading to extraction of multi-scale uncertainty features associated with the layer-layer interactions of the neural network. We will typically consider $L_2$ to be the output layer in most empirical evaluations, thus measuring the probabilistic interactions between the model’s output and one or more of the hidden layers. Here we will show results with respect to the field created by the hidden layer output thereby finding out uncertainties in the output layer parameters. Since we expect the QIPF modes (resulting from the information potential interactions between the hidden layers and the output) to be more densely clustered in test data regions unknown to the model, we quantify uncertainty ranges associated with the model’s predictions by measuring the standard deviation of QIPF states extracted at each instance of testing.
Experiments and Analysis
========================
We present simulation results to illustrate and compare performance of the QIPF framework with respect to currently popular approaches for the problem of predictive uncertainty quantification. All simulations described in this section were performed using python 3.6. Being a kernel based approach, we compare the QIPF framework’s performance with that of Gaussian process regression (GPR) [@gp] which is a widely famed kernel method for machine learning that is known for providing reliable uncertainty estimates associated with its predictions. We also provide comparisons with Monte Carlo dropout [@gal] that has gained recent popularity as a an approximate variational inference based method for uncertainty quantification of neural networks.
Regression
----------
### Datasets {#datasets .unnumbered}
We generate two different regression datasets as didactic examples for experimental comparisons and analysis. The idea of such datasets is to simulate real world scenarios where tasks in machine learning encounter test data from outside the training domain or have to face external noise or outliers in their training set. Indeed these synthesized datasets have also been used in various uncertainty quantification literature for demonstration of methods [@osband; @gal]. The first dataset consists of 60 regression pairs $x_i$, $y_i$ from the following weighted sine signal: $$y_i = {x_i}sin(x_i).$$
[.32]{} ![Synthesized datasets for experimental evaluations. Blue circles depict the sampled training data and red dashed lines represent their associated generating functions. Pink bands represent regions with no training samples.[]{data-label="dats"}](syn2data.png "fig:")
[.32]{} ![Synthesized datasets for experimental evaluations. Blue circles depict the sampled training data and red dashed lines represent their associated generating functions. Pink bands represent regions with no training samples.[]{data-label="dats"}](syn2data_n.png "fig:")
[0.32]{} ![Synthesized datasets for experimental evaluations. Blue circles depict the sampled training data and red dashed lines represent their associated generating functions. Pink bands represent regions with no training samples.[]{data-label="dats"}](syn1data.png "fig:")
Here, the training inputs $x_i$ are drawn uniformly from $(-5, 5)$. The dataset is shown in fig. \[dats\]a as synthesized dataset I. The blue circles represent the training samples and the red dotted line represents its underlying governing function in the region $(-15, 15)$. Although the training pairs are sampled only from a specific region, testing (for all algorithms) is performed in the entire data region by sampling 120 test data pairs uniformly from the region $(-15, 15)$. The pink bands represent test data regions for which training data has not been provided. We therefore expect high predictive uncertainties in these regions. As part of the analysis, we also add 6 widely varying outlier samples (not lying on the governing function) to the training set of synthesized dataset I as shown in fig. \[dats\]b. Synthesized data II consists of noisy regression pairs $x_i$, $y_i$ sampled from the following signal: $$y_i = x_i + sin(\alpha(x_i + w_i)) + sin(\beta(x_i + w_i)) + w_i.$$ Here, we set $\alpha = 4$, $\beta = 13$ and $w_i\sim N(\mu=0, \sigma^2=0.03^2)$. We draw 40 input samples for training uniformly from $(-1, 0.2)$ and 10 from $(0.7, 1)$, leaving the region $(0.2, 0.7)$ as blank. For model testing, we draw 120 test sample pairs uniformly from $(-2, 2)$. The dataset is depicted in fig \[dats\]c. In addition to these datasets, we also perform model extrapolation experiments on the Mauna Loa CO2 dataset which consists of atmospheric CO2 concentrations measured from in situ air samples collected at the Mauna Loa Observatory, Hawaii [@keeling].
![Comparison of the different predictive UQ methods. Blue lines represent model predictions. Blue shaded areas represent their associated uncertainty ranges. Red dotted lines are test prediction errors. Pink bands represent untrained regions.[]{data-label="mainres"}](syn2gpr.png "fig:") ![Comparison of the different predictive UQ methods. Blue lines represent model predictions. Blue shaded areas represent their associated uncertainty ranges. Red dotted lines are test prediction errors. Pink bands represent untrained regions.[]{data-label="mainres"}](syn2mc.png "fig:") ![Comparison of the different predictive UQ methods. Blue lines represent model predictions. Blue shaded areas represent their associated uncertainty ranges. Red dotted lines are test prediction errors. Pink bands represent untrained regions.[]{data-label="mainres"}](syn2qp.png "fig:")
![Comparison of the different predictive UQ methods. Blue lines represent model predictions. Blue shaded areas represent their associated uncertainty ranges. Red dotted lines are test prediction errors. Pink bands represent untrained regions.[]{data-label="mainres"}](syn2ngpr.png "fig:") ![Comparison of the different predictive UQ methods. Blue lines represent model predictions. Blue shaded areas represent their associated uncertainty ranges. Red dotted lines are test prediction errors. Pink bands represent untrained regions.[]{data-label="mainres"}](syn2nmc.png "fig:") ![Comparison of the different predictive UQ methods. Blue lines represent model predictions. Blue shaded areas represent their associated uncertainty ranges. Red dotted lines are test prediction errors. Pink bands represent untrained regions.[]{data-label="mainres"}](syn2nqp.png "fig:")
![Comparison of the different predictive UQ methods. Blue lines represent model predictions. Blue shaded areas represent their associated uncertainty ranges. Red dotted lines are test prediction errors. Pink bands represent untrained regions.[]{data-label="mainres"}](syn1gpr.png "fig:") ![Comparison of the different predictive UQ methods. Blue lines represent model predictions. Blue shaded areas represent their associated uncertainty ranges. Red dotted lines are test prediction errors. Pink bands represent untrained regions.[]{data-label="mainres"}](syn1mc.png "fig:") ![Comparison of the different predictive UQ methods. Blue lines represent model predictions. Blue shaded areas represent their associated uncertainty ranges. Red dotted lines are test prediction errors. Pink bands represent untrained regions.[]{data-label="mainres"}](syn1qp.png "fig:")
![Comparison of the different predictive UQ methods. Blue lines represent model predictions. Blue shaded areas represent their associated uncertainty ranges. Red dotted lines are test prediction errors. Pink bands represent untrained regions.[]{data-label="mainres"}](gp_co2.png "fig:") ![Comparison of the different predictive UQ methods. Blue lines represent model predictions. Blue shaded areas represent their associated uncertainty ranges. Red dotted lines are test prediction errors. Pink bands represent untrained regions.[]{data-label="mainres"}](mc_co2.png "fig:") ![Comparison of the different predictive UQ methods. Blue lines represent model predictions. Blue shaded areas represent their associated uncertainty ranges. Red dotted lines are test prediction errors. Pink bands represent untrained regions.[]{data-label="mainres"}](qp_co2.png "fig:")
### Model Implementations {#model-implementations .unnumbered}
For implementing the MC dropout and QIPF framework on synthesized data I, we use a small fully connected and ReLu activated neural network with 3 hidden layers containing 20 neurons each. We train the network on the given training samples in the region $(-5, 5)$. Since the network is very small, the dropout rate for training was set to 0.05 (similar to that recommended in [@gal] for a similar network size) and we used 100 epochs with the batch size equal to the number of training samples. Thereafter we tested the network on 120 input points sampled uniformly from the entire data region $(-15, 15)$. For implementing MC dropout, the test dropout rate was set to 0.2 and we used 100 forward stochastic runs to quantify the uncertainty interval at each test point. We implemented the QIPF framework by extracting 5 cross-QIPF states of the prediction point with respect to the activation outputs of each hidden layer. We used a kernel width based on the values of hidden layer activation outputs and fixed it to 20 times the Silverman’s thumb rule for bandwidth estimation [@sil]. The criteria for kernel width depends on the range of data space on which the QIPF framework is to be implemented as well as the desired resolution of modes. Since we are operating on a relatively small neural network over a limited data span, we speculated that 5 QIPF modes would be enough for our implementation. Thereafter we quantified the uncertainty interval by measuring the standard deviation of the extracted cross-QIPF states at each point of prediction. For synthesized data II, we used a 2-layer network with 50 neurons each for training. The training dropout rate was set to 0.1 and we used 100 training epochs with the batch size equal to the total training data. We implemented MC dropout and QIPF frameworks in the same manner as before on the entire data region $(-2, 2)$ using 120 uniformly generated test samples. In the case of the CO2 data, we used a relatively larger fully connected ReLu network of 5 layers with 100 neurons each. Training dropout rate in this case was set as 0.2. We test the network over the entire training region as well as an extrapolated region outside of the it. We also fitted the Gaussian Process regression model on all datasets. In each case, the parameters associated with the covariance kernel function were chosen using grid search so as to maximize the log marginal likelihood of the data.
### Analysis of Results {#analysis-of-results .unnumbered}
The results of the different uncertainty quantifiers are shown in fig. \[mainres\]. The blue line represents the predictions and the blue shaded regions depict the uncertainty ranges (standard deviation) at the prediction points quantified by the different methods (GPR, MC dropout and QIPF). Red dotted line indicates the test set prediction errors with respect to the generating function values at those points and the pink bands represent regions in the input space where no training samples were generated. For synthesized data I in fig. \[mainres\]a, it can be observed that the GP regression model is able to better identify the generating signal dynamics than the neural network, producing low predictive errors for some distance outside of the training set. This is expected for small non-linear datasets where kernel methods outperform ANNs. It also produces well calibrated uncertainty ranges that are seen to be roughly proportional to the predictive errors. The QIPF framework can also be seen to produce uncertainty ranges that scale more proportionally with respect to the test error. MC dropout, on the other hand, produces comparatively less realistic uncertainties that soon converge to a constant level showing no change with respect to prediction errors thereafter. We observe wide disparities between the uncertainty quantification of the three algorithms in fig. \[mainres\]b when the outliers are added to the training data (synthesized data I). We observe here that all models converge to the center of the outlier data when making predictions at those points. This is expected since the mean of such widely varying data points would represent the lowest error region for most learning models. However, we also notice that MC dropout becomes very overconfident with low uncertainties associated with its predictions at the outlier regions, which is opposite of what we would ideally expect. This is also reported in [@hern]. GP regression model also shows unrealistically low uncertainties in its predictions at the particular outlier points though it still produces increased uncertainty range around them. Also, unlike before, the GP model can be observed to converge to an unrealistic constant level of uncertainty as one goes outside of the training region, regardless of the increase in predictive errors. The QIPF framework, on the other hand, shows a remarkable property of *increasing* its uncertainty range at the outlier regions, which is the ideal behavior. It is also seen to maintain its property of increasing proportionally with the predictive errors in all outside data regions. This indicates the sensitivity of the QIPF framework towards data variances and outliers in the training set. For synthesized data II (which consists of added normal noise), similar observations can be made related to the nature of uncertainties quantified by the different methods outside of the training domain. As before, the QIPF framework is seen to produce uncertainty estimates that increase more realistically outside the training domain and more proportional to the predictive errors when compared to MC dropout and GP regression. Both QIPF framework and MC dropout can be seen to be sensitive in their characterizations of uncertainty in the thin middle untrained band (given by the region (0.2, 0.7)) . However, we also observe here that, like before, MC dropout becomes unrealistically overconfident due to large variances in the training data pairs (on the left side of its corresponding graph in fig. \[mainres\]c). Similar analysis on the CO2 data (fig. \[mainres\]d) reveals GP regression to fit better than the other two models in the training region with very little error. However, it continues to be insensitive to predictive error outside the training domain by exhibiting a constant level of uncertainty when extrapolated during testing. The same trend of GP regression is reported in [@gal]. Both MC dropout and QIPF framework extrapolate more realistically in terms of uncertainty. We summarize the observations related to the QIPF framework from these results as follows.
- The framework is observed to be robust towards training set outliers and is able to effectively capture the model’s associated uncertainties with respect to them during testing. This can be attributed to the ability of the Gaussian RKHS in better capturing the true data distributions.
- For all datasets, the QIPF framework is observed to produce uncertainty estimates that are more consistent with predictive errors in all regions of the data domain consequently exhibiting realistic uncertainty ranges for both model interpolation and extrapolation applications.
- The framework also exhibits increased sensitivity towards inherent data variances.
Classification
--------------
We also demonstrate the ability of the QIPF framework quantify uncertainties related to classification problems. Towards this end, we train a ReLu activated and fully connected MLP with 3 hidden layers with 512 - 256 - 128 neurons respectively (from first to last hidden layers) on the MNIST dataset [@mnist]. We train the network without the implementation of dropout for 10 epochs using a batch size of 100. During testing, we rotated a single digit of 1 gradually (60 times uniformly) and fed each rotated version to the trained NN. During each test instance, we extracted the first 10 cross QIPF modes of the average node input value of the last layer (before thresholding) with respect to the activation outputs of hidden layer 1. In this application, we extracted the QIPF modes twice using different kernel widths (20x and 30x the Silverman bandwidth of the hidden layer 1 activation outputs) and considered the average of the two runs. This was done to ensure that the modes cover the range between the first and last hidden layer outputs. Fig. \[ro\] shows samples of the gradually rotated test sequence and fig. \[mn\] shows the graph of the standard deviation of the 10 average QIPF modes at each test input. The pink bands represent the rotations at which the network produced incorrect classification results. One can observe the sharp rise of the standard deviation of the uncertainty modes at the misclassified test regions thus indicating that the framework produces uncertainty results consistent with the model classification errors.
**N** **Q** **MC Dropout** **QIPF**
-- -- -- -- ------- -- ------- -- ---------------- -- -------------------- -- -- --
308 6 0.332 +- 0.051 **0.204 +- 0.050**
506 13 0.246 +- 0.038 **0.234 +- 0.042**
9568 4 0.170 +- 0.035 **0.124 +- 0.035**
1030 8 0.234 +- 0.035 **0.221 +- 0.044**
768 8 0.238 +- 0.028 0.268 +- 0.061
[c c c c c c c c c c c c c c]{} & & & & 308 & & 6 & & 0.294 +- 0.093 & & **0.169 +- 0.046**\
& & & & 506 & & 13 & & 0.222 +- 0.041 & & 0.234 +- 0.049\
& & & & 9568 & & 4 & & 0.151 +- 0.050 & & **0.150 +- 0.057**\
& & & & 1030 & & 8 & & 0.218 +- 0.036 & & **0.211 +- 0.043**\
& & & & 768 & & 8 & & 0.235 +- 0.032 & & 0.274 +- 0.046\
-- -- -- -- ------ -- ---- -- ---------------- -- -------------------- -- -- --
308 6 0.226 +- 0.063 **0.167 +- 0.038**
506 13 0.223 +- 0.023 0.234 +- 0.041
9568 4 0.146 +- 0.038 **0.144 +- 0.05**
1030 8 0.220 +- 0.030 **0.204 +- 0.035**
768 8 0.263 +- 0.034 **0.240 +- 0.048**
-- -- -- -- ------ -- ---- -- ---------------- -- -------------------- -- -- --
Benchmark Datasets
------------------
We quantify and compare the quality of uncertainty estimates of our method with MC dropout over UCI datasets that are typically used as benchmarking data in various uncertainty quantification literature. We measure the quality of uncertainty estimate by quantifying how calibrated the uncertainty estimates are with respect to prediction errors. We chose datasets with diverse numbers of samples in order to test the framework on different forms of non-linearities. We train neural networks with 50 neurons in each hidden layer on 20 randomly generated train-test splits of the normalized UCI datasets (similar to the experimental framework of [@gal; @hern]. A single kernel width of 1 was used in this case for extracting 10 QIPF states. We measured the RMSE of the uncertainty range (std. deviation of the QIPF modes at each test sample) with respect to the test error in each train-test split. This was done to measure how calibrated and scaled the estimated uncertainty range was with respect to the error. The average RMSE as well as its standard deviation for the 20 test splits are presented in table \[tab\] for the framework’s implementation on neural network architectures consisting of 1, 2 and 3 hidden layers respectively. It can be observed that the QIPF framework has lower RMSE values than MC dropout for most datasets in all network configurations thereby indicating that the estimated uncertainty using QIPF is more realistic.
Conclusion {#conclusion .unnumbered}
==========
In this paper, we introduced a new information theoretic approach for quantifying uncertainty that is inspired by quantum physical principles and concepts. We formulated a new moment decomposition framework for data by utilizing the RKHS based metric called the information potential field. The key advantages offered by our framework are its non-parametric nature, ability to be implemented on a sample-by-sample basis and its sensitivity towards unseen data or model regions. We gave pedagogical examples to show how our model provides a multi-scale characterization of the tails of the data PDF about which we generally have the least amount of information. We demonstrated how our framework can be utilized as a powerful tool for the predictive uncertainty quantification of models in both regression and classification tasks. Related results indicate that our framework is able to recognize and quantify training outlier regions of the data encountered during model testing, unlike MC dropout or GP regression. It is also able to more appropriately remain consistent with respect to the predictive errors.
In the future, we intend to conduct more data-based implementations of the framework in order to explore its utility in the signal processing domain.
Acknowledgments {#acknowledgments .unnumbered}
---------------
We would like to acknowledge the support provided for this work by DARPA under agreement number HR001116S0001.
|
---
abstract: 'This paper considers the problem of manipulating a uniformly rotating chain: the chain is rotated at a constant angular speed around a fixed axis using a robotic manipulator. Manipulation is quasi-static in the sense that transitions are slow enough for the chain to be always in “rotational equilibrium”. The curve traced by the chain in a rotating plane – its shape function – can be determined by a simple force analysis, yet it possesses a complex multi-solutions behavior typical of non-linear systems. We prove that the configuration space of the uniformly rotating chain is homeomorphic to a two-dimensional surface embedded in $\mathbb{R}^3$. Using that representation, we devise a manipulation strategy for transiting between different rotation modes in a stable and controlled manner. We demonstrate the strategy on a physical robotic arm manipulating a rotating chain. Finally, we discuss how the ideas developed here might find fruitful applications in the study of other flexible objects, such as circularly towed aerial systems, elastic rods or concentric tubes.'
author:
- |
Hung Pham Quang-Cuong Pham\
[^1]
bibliography:
- 'library.bib'
title: Robotic manipulation of a rotating chain
---
Introduction {#sec:intro}
============
An idle person with a chain in her hand will likely at some point starts rotating it around a vertical axis, as in Fig. \[fig:rotate\_by\_hand\]A. After a while, she might be able to produce another mode of rotation, whereby the chain would curve inwards, as in Fig. \[fig:rotate\_by\_hand\]B, instead of springing completely outwards. With sufficient dexterity, she might even reach more complex rotation modes, such as in Fig. \[fig:rotate\_by\_hand\]C. Transitions into such complex rotation modes are however difficult to reproduce reliably as instabilities can quickly lead to unsustainable rotations (Fig. \[fig:rotate\_by\_hand\]D). This paper investigates the mechanics of the transitions between different rotation modes, and proposes a strategy to perform those transitions in a stable and controlled manner.
Motivations {#motivations .unnumbered}
-----------
There are several reasons why this problem is hard to solve. First, there are multiple solutions for a given control input (distance $r$ between the attached end of the chain and the rotation axis, and angular speed $\omega$). This ambiguity makes it difficult to devise a manipulation strategy directly in the control space. Second, some control inputs can quickly lead to “uncontrollable” behaviors of the chain, as illustrated in in Fig. \[fig:rotate\_by\_hand\]D.
![Manual rotation of a chain around a vertical axis. **A, B, C**: Uniform rotation modes 0, 1, 2 respectively. **D**: Unstable behavior.[]{data-label="fig:rotate_by_hand"}](fig/RotateByHand.pdf){width="50.00000%"}
The theoretical study of the rotating chain and, in particular, of its rotation modes, has a long and rich history in the field of applied mathematics [@Kolodner1955; @caughey1958whirling; @Caughey1970; @Wu1972; @stuart1976steadily; @russell1977equilibrium; @Toland1979], which we review in Section \[sec:applied-mathematics\]. Here, by devising and implementing a *manipulation strategy* to stably transit between different rotation modes, we hope to provide a new, robotics-enabled, understanding of this problem. Indeed, at the core of our approach lie concepts specifically forged in the field of robotics, such as “configuration space”, “stable configurations”, “path-connectivity”, etc.
As opposed to rigid bodies, flexible objects are in general characterized by an infinite number of degrees of freedom, which entails significant challenges when it comes to manipulation. Specific approaches have therefore been developed in the field of robotics to study the manipulation of flexible objects, as reviewed in Section \[sec:robotic-manip\].
The above studies are motivated by a number of practical applications. For the rotating chain in particular, applications include aerial manipulation by Unmanned Air Vehicles (UAV), which has recently received some attention, as discussed in more details in Section \[sec:aerial-manip\].
Contribution and organization of the paper {#contribution-and-organization-of-the-paper .unnumbered}
------------------------------------------
Our contribution in this paper is threefold. First, we study the case of arbitrary non-zero attachment radii $r$ (Section \[sec:forward\_kinematics\]). This extends and generalizes existing works, which all focus on the case of zero attachment radius, and sets the stage for stable transitions between different rotation modes, which specifically require manipulating the attachment radius. In particular, we determine in this section the number of solutions to the shape equation for any given value $r$ and $\omega$.
Second, we show that the configuration space of the uniformly rotating chain with variable attachment radius is homeomorphic to a two-dimensional surface embedded in $\mathbb{R}^3$ (Section \[sec:configuration\_space\]). We study the subspace of stable configurations and establish that it is not possible to stably transit between rotation modes without going back to the low-amplitude regime.
Third, based on the above results, we propose a manipulation strategy for transiting between rotation modes in a stable and controlled manner (Section \[sec:manipulation\]). We show the strategy in action in a physical experiment where a robotic arm manipulates a rotating chain and makes it reliably transit between different rotation modes.
Before presenting our contribution, we review related works (Section \[sec:related-works\]) and recall Kolodner’s equations of motion of the rotating chain (Section \[sec:background\]). Finally, we discuss possible applications and extensions and sketch some perspectives for future work (Section \[sec:conclusion\]).
Related works {#sec:related-works}
=============
The manipulation of the rotating chain is relevant to a number of fields such as (i) applied mathematics, (ii) flexible object manipulation in robotics, and (iii) aerial manipulation. We now review the literature and describe the position of the current work with respect to each of these fields.
Theoretical studies of the rotating chain {#sec:applied-mathematics}
-----------------------------------------
In applied mathematics, the study of the rotating chain was initiated in 1955 by a remarkable paper by Kolodner [@Kolodner1955]. Kolodner established the existence of critical speeds $(\omega_i)_{i\in\mathbb{N}}$ such that there are no uniform rotations if the angular speed $\omega<\omega_1$, and there are exactly $n$ rotation modes for $\omega_n<\omega<\omega_{n+1}$. In [@caughey1958whirling], Caughey studied the rotating chain with small but non-zero attachment radii. The results obtained by Caughey extend Kolodner’s and agree with our study of the low-amplitude regime. In [@Caughey1970], Caughey investigated the rotating chain with both ends attached. In [@stuart1976steadily], Stuart considered the original rotating chain problem using bifurcation theory, and arrived at the same results as Kolodner. In [@Wu1972], Wu considered the large angular speeds regime. In [@Toland1979], Toland initiated a new approach based on the calculus of variation, but did not obtain new significant results, as compared to Kolodner.
The common point of all previous works is that the chain is attached to the rotation axis, or very close to it [@caughey1958whirling]. Yet, reliably observing and transiting between different rotation modes precisely require using arbitrary non-zero attachment radii $r$, the distance between the attached end and the rotation axis. The current paper extends previous studies by specifically considering arbitrary attachment radii.
Robotic manipulation of flexible objects {#sec:robotic-manip}
----------------------------------------
Within the field of robotics, the manipulation of flexible objects is studied along two main directions. A first direction is topological: one is mainly interested in the order and sequence of the manipulation rather than in the precise behavior of the flexible object. Examples include origami folding [@balkcom2008robotic], laundry folding [@miller2012geometric] or rope-knotting [@wakamatsu2006knotting; @Yamakawa2013a].
The second research direction is concerned with the precise shape and dynamics of the manipulated object. Within this research direction, one can distinguish two main approaches. The first approach discretizes the flexible object into a large number of small rigid elements, and subsequently carries out finite-element calculations, see [@Skop1971; @russell1977equilibrium; @murray1996trajectory] for inextensible cables or [@Gilbert2013a; @Rucker2010a] for concentric tube robots. This approach can be applied to any type of flexible objects as long as a dynamical model is available. However, it usually yields no *qualitative* understanding of the manipulation. For example, while finite-element calculations can compute the shape of the rotating chain for various control inputs, they can establish neither the existence of different rotation modes, nor the manipulation strategies to transit between different modes.
By contrast, the second approach considers the flexible object as the solution of a (partial) differential equation and tries to establish qualitative properties of this solution. While this approach is harder to put in place – usually because of the complex mathematical calculations and concepts involved – it can lead to stunning and insightful results. For example, Bretl and colleagues established that the configuration space of the Kirchhoff elastic rod is of dimension 6 [@bretl2014quasi] and that it is path-connected [@borum2015free]. Such results would be impossible to obtain via finite-element methods.
The present study of the rotating chain is inscribed within this analytical approach. From the dynamic model of the rotating chain, we investigate qualitative properties of its configuration space: dimension, connectivity, and stability. These properties are in turn crucial to devise a manipulation strategy to stably transit between different rotation modes.
Aerial manipulation {#sec:aerial-manip}
-------------------
Although the study of the rotating chain first stemmed out of scientific curiosity, it has recently found applications in aerial manipulation. In [@murray1996trajectory; @Williams2007], the authors considered a fixed-wing aircraft towing a long cable whose other end is free. The circular flying pattern imprints a pseudo-stationary shape to the cable, which in turn allows precisely controlling the position of the free end. Practical applications of this scheme include remote sensing in isolated areas [@Williams2007], payload delivery and pickup [@Skop1971; @merz2016feasibility; @Williams2007], or more recently, recovery of micro air vehicles [@Colton2011; @nichols2014aerial; @Sun2014]. In the latter application, the micro vehicles are able to attach themselves to the towed end, which moves at a relatively slower speed than that of the aircraft. The recent surge of interest in Unmanned Air Vehicles (UAVs) also offers many potential applications: [@merz2016feasibility] studies a single UAV flying circularly while towing a cable, [@sreenath2013trajectory] deals with general (non-circular) aerial manipulation, while [@michael2011cooperative] targets cooperative manipulation using a team of UAVs.
The above works are based on dynamic simulation [@Skop1971; @russell1977equilibrium; @Williams2008], or numerical optimal control [@Sun2014; @Williams2007a]. Physical experiments were found to agree with simulations [@Williams2007]. However, there are a number of questions these works are unable to address, for instance: (i) under which conditions are there multiple solutions to the same set of controls (fly radius and angular speed)? (ii) how to avoid or initiate “jumps” between different quasi-static rotational solutions [@russell1977equilibrium]? Here, we precisely answer these questions for the case of a simple rotating chain, without considering aerodynamic drag or end mass. We also discuss how the method can be extended to include these effects, offering thereby solid theoretical foundations for developing safe and stable applications in circular aerial manipulation.
Background and problem setting {#sec:background}
==============================
Equations of motion of the rotating chain
-----------------------------------------
Here we recall the main equations governing the motion of the rotating chain initially obtained by Kolodner [@Kolodner1955]. Fig. \[fig:schema\] depicts an inextensible and homogeneous chain of length $L$ and linear density $\mu$ that rotates around a vertical $Z$-axis. One end of the chain is maintained at the attachment radius $r$ from the rotation axis, while the other end is free. Note that the case of a chain with tip mass can be reduced to this case, see Appendix \[sec:tip-mass\].
![A chain rotating around a fixed vertical axis. At a time instant $t$, the chain describes a 3D curve parameterized by $s$: $s=0$ at the free end, $s=L$ at the attached end, where $L$ is the length of the chain.[]{data-label="fig:schema"}](fig/chain_sketch.pdf){width="30.00000%"}
Let ${\mathbf{x}}(s,t) := {[x(s,t),y(s,t),z(s,t)]}^\top\in {\mathbb{R}}^3$ denote a length-time parameterization of the chain where $s$ equals zero at the free end and equals $L$ at the attached end (Fig. \[fig:schema\]). Next, let $F(s,t)\geq 0$ be the tension of the chain. Neglecting aerodynamic effect, one writes the equation of motion for the chain as $$\label{eq:eom}
\mu \ddot{{\mathbf{x}}} = (F{{\mathbf{x}}}')' + \mu {\mathbf{g}},$$ where $\dot \Box$ and $\Box'$ denote differentiation with respect to $t$ and $s$ respectively; ${\mathbf{g}}:=[0, 0, -g]^{\top}$ is the gravitational acceleration vector. The inextensibility constraint can be written as $$\label{eq:inex}
\|{\mathbf{x}}(s, t)'\|_{2} = 1.$$
We seek solutions that are *uniform rotations*; those which have constant shape in a plane that rotates around the $Z$-axis. In this case, the motion of the chain becomes $$\begin{aligned}
x(s,t) & = \rho(s) \cos(\omega t),\\
y(s,t) & = \rho(s) \sin(\omega t),\\
z(s,t) & = z(s),
\end{aligned}$$ where the function $\rho(s)$ is called the *shape function* of the chain. Directly from inextensibility constraint , we have: $$\label{eq:11}
\|(\rho(s), z(s))\|_2 = \sqrt{\rho'(s)^2 + z'(s)^2} = 1.$$ Also, the tension of the chain $F(s, t)$ is time independent.
Substituting the above expressions into Eq. yields $$\begin{aligned}
\label{eq:x1z}
(F\rho')' + \mu\rho \omega^2 &= 0, \\
\label{eq:2}
(Fz')' - \mu g &= 0,\end{aligned}$$ where $F, \rho, z$ are functions of $s$. Integrating Eq. and noting that the tension at the free end vanishes ($F(0)=0$) yield $$\label{eq:24}
Fz' = \int_0^s\mu g \; {{\rm d}{\lambda}}= \mu g s.$$ Next, by the inextensibility constraint , we have $$\label{eq:F}
F =\frac{\mu g s}{z'} = \frac{\mu g s}{\sqrt{1-\rho'^2}}.$$ Substituting Eq. (\[eq:F\]) into Eq. (\[eq:x1z\]) yields the governing equation for the shape function $\rho(s)$ $$\label{eq:x1}
\frac{{{\rm d}{}}}{{{\rm d}{s}}} \left( \frac{\mu g s}{\sqrt{1-\rho'^2}}\rho' \right)
+ \mu\rho\omega^2
= 0$$ subject to the following boundary condition $$\label{eq:1}
\rho(L) = r.$$
Remark that we have applied two boundary conditions: (i) tension at the free end must be zero: $F(0, t)=0$ for any $t$; and (ii) ${\mathbf{x}}(L, t)$ equals the reference trajectory traced by the robotic manipulator (or the aircraft’s trajectory in the towing problem).
Problem formulation
-------------------
We can now define the *configurations* and the *control inputs* of a rotating chain.
\[def:conf\] (Configuration) A configuration of the rotating chain is a pair $q:=(\omega,\rho)$, where $\omega \geq 0$ is a rotation speed and $\rho$ is a shape function satisfying the governing equation (\[eq:x1\]) and that $\rho(0)\geq 0$. The set of all such configurations is called the configuration space of the rotating chain and denoted ${{\cal C}}$.
(Control input) A control input is a pair $(r, \omega)$, where $r\geq 0$ is an attachment radius and $\omega\geq 0$ is a rotation speed. The set of all inputs is called the control space and denoted ${{\cal V}}$. If equation (\[eq:x1\]) has non-trivial solutions with boundary conditions and parameters defined by the input ($r, \omega$) then the input is called *admissible*.
Note that the condition $\rho(0) \geq 0$ in Definition \[def:conf\] identifies duplicate solutions. Any configuration $(\omega,\rho)$ corresponds to two possible solutions: one has shape function $\rho$ and one has shape function $-\rho$, both rotate at angular speed $\omega$. The later solution can be obtained by rotating the former solution by $180$ degrees. A similar remark applied to the definition of the control space ${{\cal V}}$ where we require positive attachment radius.
We can formulate the chain manipulation problem as follows: given a pair of starting and goal configurations $(q_\mathrm{init}, q_\mathrm{goal})$ find a control trajectory $(0, 1) \rightarrow {{\cal V}}$ that brings the chain from $q_\mathrm{init}$ to $q_\mathrm{goal}$ without going through instabilities (instabilities will be discussed in Section \[sec:stability\_analysis\]).
Forward kinematics of the rotating chain with non-zero attachment radius {#sec:forward_kinematics}
========================================================================
Dimensionless shape equation {#sec:dimless}
----------------------------
Still following Kolodner, we convert Eq. into a dimensionless equation, more appropriate for subsequent analyses. Consider the changes of variable $$\label{eq:change_of_var}
u := \frac{\rho'}{\sqrt{1-{\rho'}^2}}\frac{s\omega^2}{g}, \quad \bar
s := \frac{s\omega^2}{g},$$ which by combining with Eq. leads to $$\label{eq:rho2}
\frac{{{\rm d}{u}}}{{{\rm d}{\bar s}}} + \rho \frac{\omega^2}{g} = 0.$$
One can now differentiate Eq. with respect to $\bar s$ to arrive at $$\frac{{{\rm d}{}}^2 }{{{\rm d}{\bar s}}^2 } u + \rho' = 0,$$ which is combined with the relation $$\label{eq:rho1}
\rho' =\frac{u }{\sqrt{{\bar s}^2+u^2}}$$ to yield the dimensionless differential equation $$\label{eq:new}
\frac{{{\rm d}{}}^2 }{{{\rm d}{\bar s}}^2 } u(\bar s) +
\frac{u(\bar s)} {\sqrt{\bar s^2+{u(\bar s)}^2}} = 0.$$
We first consider the boundary condition at $\bar s = 0$. By definition of $u$, one has $u(0)=0$. The end boundary condition $\rho(L)=r$ implies that $$\label{eq:boundary_2}
u'\left(L\frac{\omega^2}{g}\right)= -r\frac{\omega^2}{g},$$ where $\Box'$ denotes in this context differentiation with respect to $\bar s$.
We summarize the boundary conditions on $u$ as $$\label{eq:boundary}
u(0) = 0, \quad u'(\bar L) = \bar r,$$ where $$\label{eq:defa}
\bar L:=L\omega^2/g, \quad
\bar r:=-r\omega^2/g.$$ This is the standard form of a Boundary Value Problem (BVP).
**Remark** Denote by $\rho_{0}$ the distance from the free end to the $Z$-axis. Using Eq. , we have $$\label{eq:boundary_1}
u'(0)= a,$$ where $a = -\rho_0 \omega^2/ g$.
**Remark** Applying L’Hôpital rule twice, one finds that $$\lim_{\bar s\to 0}\frac{u(\bar s)} {\sqrt{\bar s^2+{u(\bar s)}^2}} = \frac{a}{\sqrt{1+a^2}}.$$ Thus, the differential equation (\[eq:new\]) is well-defined at $\bar s=0$.
Shooting method {#sec:shooting}
---------------
(image) at (0,0) [ ![**A**: Shooting from different initial guesses of $u'(0)=a$. There might be more than one initial value (green and red) that satisfy the end condition $u'(\bar L) = \bar r$. **B**: $a_1, a_2, ...$ are different initial values of $u'(0)$ that yield $u'(\bar L)=0$; $a_i$ denotes the initial guess such that the $i$-th zero of $u'$ coincides with $\bar L$[]{data-label="fig:2Dshooting"}](fig/variational_shooting.pdf "fig:"){width="30.00000%"} ]{};
at (0.53, -0.04) [$\bar s$]{}; at (-0.04, 0.29) [$u'$]{}; at (-0.04, 0.79) [$u'$]{};
(0.795, 0.135) circle (0.07cm) node \[anchor=south, color=black\] [$(\bar L, 0)$]{}; (0.67, 0.835) circle (0.07cm) node \[anchor=south east, color=black\] [$(\bar L, \bar r)$]{};
at (1.02, 0.96) [**A**]{}; at (1.02, 0.46) [**B**]{};
We numerically solve the BVP posed in the last section using the *simple shooting method* [@Stoer1982]. Given a control input $(r, \omega)$, the method finds resulting configurations as follows:
1. compute $(\bar r, \bar L)$ from $(r, \omega)$ using Eq. ;
2. repeat until convergence:
1. guess an initial value $a\in {\mathbb{R}}$ for $u'(0)$ or use the value from the last iteration;
2. integrate Eq. from the initial condition $(u(0),u'(0))=(0,a)$ at $\bar s = 0$ to $\bar s=\bar L$;
3. check whether $u'(\bar L)=\bar r$;
4. if not, refine the guess $a$ by Newton’s method;
3. recover $\rho(s)$ from ${u'}_{\rm{last\_iter}}(\bar s)$.
One can then recover $z(s)$ using $\rho(s)$, the inextensibility constraint , the boundary condition $z(L)=0$ and the fact that $z'(s)\geq 0$ (See Eq. ). Also, for a given tuple $(\bar r, \bar L)$, there might be multiple solutions to the BVP which translates to multiple configurations for a given control input (Fig. \[fig:2Dshooting\]**A**).
**Remark** It is straightforward to see that if $u(\bar s)_{\bar s\in [0, \bar L]}$ is a solution of Eq. , then $- u(\bar s)_{\bar s\in [0, \bar L]}$ is also a solution. Therefore there is no loss of generality to consider only non-negative values of $a$, as integrating from $-a$ leads to the same configuration.
Number of configuration {#sec:number-configuration}
-----------------------
(image) at (0,0) [ ![The graph of $|u'_a(\bar L)|$ versus $|a|$. The main text shows that if $\bar r_{i+1}<|\bar r|<\bar r_i$, then there are $2i+1$ non-trivial solutions. The green line illustrates the case $\bar r_3<|\bar r|<\bar r_2$ where there are 5 non-trivial solutions (green disks).[]{data-label="fig:kappa_i"}](fig/kappa_i_convex.pdf "fig:"){width="30.00000%"} ]{};
(0.155, 0.25) – (0.66, 0.25); (0.162,0.25) circle \[radius=0.055cm\]; (0.205,0.25) circle \[radius=0.055cm\]; (0.255,0.25) circle \[radius=0.055cm\]; (0.545,0.25) circle \[radius=0.055cm\]; (0.66,0.25) circle \[radius=0.055cm\];
at (-0.1, 0.5) [$|u'_a(\bar L)|$]{}; at (0.5, -0.05) [$|a|$]{}; (a) at (0.83, -0.035) [$7$ solutions]{}; (a) – (0.83, 0.1); at (0.83, 0.225) [$5$ solutions]{}; at (0.83, 0.715) [$3$ solutions]{}; at (0.83, 0.925) [$1$ solution]{};
at (0.11, 0.16) [$\bar r_3$]{}; at (0.19, 0.35) [$\bar r_2$]{}; at (0.35, 0.82) [$\bar r_1$]{};
at (0.14, 0.025) [$a_3$]{}; at (0.24, 0.025) [$a_2$]{}; at (0.6, 0.025) [$a_1$]{};
We now analyze the number of solutions for different parameters. Denote by $u'_a(\bar s)$ the function $u'(\bar s)$ obtained by integrating from $(u(0),u'(0))=(0,a)$. Following Kolodner, let $z_i(a)$ be the $i$-th zero of $u'_a(\bar s)$. The function $z_i(a)$ has the following properties (Theorem 2 [@Kolodner1955]):
- $z_i(a)$ is well-defined for all $i \in \mathbb{N}$ and is a strictly increasing function of $a$ over $(0, +\infty)$;
- $\lim_{a\rightarrow 0}z_i(a)=h_i^2 / 4=:\lambda_i$ where $h_i$ is the $i$-th zero of the Bessel function $J_0$ (Appendix \[sec:lowamp\]);
- $\lim_{a\rightarrow +\infty }z_i(a) = +\infty$.
Next, let us define $a_i$ as the absolute value of $a$ such that $z_i(a)=\bar L$, $$\label{eq:12}
a_i := |z_i^{-1}(\bar L)|.$$ By the properties of $z_i$, $a_i$ exists if and only if $\lambda_i \leq \bar L$ and when it exists, it is unique since $z_i(a)$ is a strictly increasing function of $a$.
Fig. \[fig:2Dshooting\]**B** shows the construction of $a_1$, $a_2$, $a_3$. One can also observe that the $a_i$’s form a decreasing sequence, $$a_1 > a_2 > a_3 > \dots > a_n,$$ where $n$ is the largest $i$ so that $ \lambda_i \leq \bar L$.
We now turn to the general case where $\bar r$ is not necessarily zero. Consider fixed parameters $(\bar r, \bar L)$, it can be seen that the number of configurations equals the number of intersections that the $u'_a(\bar L)$ versus $a$ graph makes with the horizontal lines $u'_a(\bar L) = \bar r$ and $u'_a(\bar L) = -\bar r$.
In fact, we can simplifies further. Since $\bar r$ and $- \bar r$ refer to the same radius and that $\rho(s)$ and $-\rho(s)$ refer to the same shape function, the number of intersections the $|\bar\rho_a(\bar L)|$ versus $|a|$ graph makes with the horizontal line $|u'_a(\bar L)|= |\bar r|$ equals the number of configurations (Fig. \[fig:kappa\_i\]).
By inspecting Fig. \[fig:kappa\_i\], $|u'_a(\bar L)|$ is zero at $a_i$ and $a_{i+1}$; note moreover that $|u'_a(\bar L)|$ increases as $a$ increases from $a_{i+1}$, reaches a maximum at some $a_i^*$, and then decreases as $a$ increases from $a_i^*$ to $a_i$ [^2]. Let us denote the maximum reached by $|u'_a(\bar L)|$ between $a_i$ and $a_{i+1}$ by $\bar r_i$, $$\begin{aligned}
\label{eq:13}
\bar r_i&:=|u'_{a_i^*}(\bar L)| = \max_{a_{i+1}<a<a_i} |u'_a(\bar L)|, \quad\text{for } i < n;\\
\bar r_n&:=\max_{0<a<a_n} |u'_a(\bar L)|.\end{aligned}$$ One can next observe that the $\bar r_i$’s form a decreasing sequence[^3], $$\bar r_1 > \bar r_2 > \bar r_3 > \dots > \bar r_n.$$
One can now state the following proposition, whose proof results directly from the examination of Fig. \[fig:kappa\_i\].
\[prop:nsol\] Let $n$ be the largest $i$ so that $\lambda_i \leq \bar L$. The number of non-trivial configurations of an uniformly rotating chain depends on $|\bar r|$ as follows:
1. if $|\bar r|=0$, there are $n$ non-trivial solutions;
2. if $0 < |\bar r|<\bar r_n$, there are $2n+1$ non-trivial solutions;
3. if $\bar r_{i+1}<|\bar r|<\bar r_i$ for $i\in[1, n-1]$, there are $2i+1$ non-trivial solutions;
4. if $|\bar r|=\bar r_i$ for $i=[1,n]$, there are $2i$ non trivial solutions;
5. if $|\bar r|>\bar r_1$, there is one non-trivial solution.
Rotation modes {#sec:modes}
--------------
By the change of variable , $u'=\rho\omega^2/g$, the number of zeros of $u'(\bar s)_{\bar
s\in(0,\bar L)}$ corresponds to the number of times the chain crosses the rotation axis. We can now give an operational definition of rotation modes.
A chain is said to be rotating in mode $i$ if its shape crosses the axis exactly $i$ times or, in other words, if the function $u'(\bar s)_{\bar s\in(0,\bar L)}$ has exactly $i$ zeros.
Let us re-interpret Prop. \[prop:nsol\] in terms of rotation modes. Consider a positive $\bar r$ verifying $\bar r_{i+1}<\bar r<\bar r_i$. In Fig. \[fig:kappa\_i\], the horizontal line $|u'(\bar L)| = \bar r$ intersects the graph of $|\bar \rho_a(\bar L)|$ versus $|a|$ at $2i+1$ points. Call the $X$-coordinates of these points $b_1>b_2>\dots>b_{2i+1}$. Remark that
- $b_1>a_1$, thus by definition of $a_1$, the function $u'_{b_1}(\bar s)$ has no zero in $(0,\bar L)$, the chain rotates in mode 0;
- $a_1>b_2>b_3>a_2$, thus by definition of $a_1,a_2$, the functions $u'_{b_2}(\bar s)$ and $u'_{b_3}(\bar s)$ have each one zero in $(0,\bar L)$, the chain rotates in mode 1;
- more generally, for any $k\in [1,i]$, $a_{k-1}>b_{2k}>b_{2k+1}>a_k$, thus by definition of $a_{k-1},a_k$, the functions $u'_{b_{2k}}(\bar s)$ and $u'_{b_{2k+1}}(\bar s)$ have each $k$ zeros in $(0,\bar L)$, the chain rotates in mode $k$.
Figure \[fig:multi\_solutions\] illustrates the above discussion for $i=2$.
{width="\textwidth"}
Analysis of the configuration space of the rotating chain {#sec:configuration_space}
=========================================================
In the previous section, we have established a relationship between the control inputs and the configurations. Here, we investigate the properties of the configuration space and of the subspaces of stable configurations. In particular, a crucial question for manipulation, which we address, is whether the stable subspace is *connected*, allowing for stable and controlled transitions between different modes.
Parameterization of the configuration space
-------------------------------------------
From now on, we make two technical assumptions: (i) the distance $\rho(0)$ from the free end of the chain to the rotation axis is upper-bounded by some $\rho_{\max}$; (ii) the rotation speed $\omega$ is upper-bounded by some $\omega_{\max}$. Note that these two assumptions do not reduce the generality of our formulation since they simply assert that there exist some finite bounds, which could be arbitrarily large. From Eq. (\[eq:defa\]) and , the two assumptions next imply that $a$ and $\bar L$ are upper-bounded by some constants $a_{\max}$ and $\bar L_{\max}$. We can now prove a first characterization of the configuration space.
\[prop:A\_to\_C\] Define the parameter space ${{\cal A}}$ by $${{\cal A}}:= (0, a_{\max}) \times (0, \bar L_{\max}).$$ There exists a homeomorphism $f:{{\cal A}}\to{{\cal C}}$.
This proposition implies that, despite (a) the potentially infinite dimension of the space of all shape functions $\rho$ and (b) the one-to-many mapping between control inputs and configurations, the configuration space of the rotating chain is actually of dimension 2 and has a very simple structure. Note that ${{\cal A}}$ is essentially a 2D box.
The first dimension, $a$, is proportional to the distance of the free end to the rotation axis. Thus, choosing the free end rather than the attached end as reference point allows finding a one-to-one mapping with the shape function. The second dimension, $\bar L$, is defined by $\bar L:=L\omega^2/g$. Since the length $L$ of the chain is fixed, $\bar L$ changes as a function of the angular speed $\omega$.
To simplify the notations, we define ${\mathbf{u}}:=(u,u')$ and rewrite Eq. as a dimensionless ODE $$\label{eq:X}
\frac{{{\rm d}{{\mathbf{u}}}}}{{{\rm d}{\bar s}}} = {\mathbf{X}}({\mathbf{u}},\bar s).$$ We can now give a proof for Proposition \[prop:A\_to\_C\].
The mapping $f$ is essentially the shooting method described in Section \[sec:shooting\]. Given a pair $(a,\bar L)\in {{\cal A}}$, we first obtain $\omega$ from $\bar L$ using the relationship $\bar L = L\omega ^2/g$. Next, we integrate the ODE from the initial condition $${\mathbf{u}}(0) = ( 0, a )$$ until $\bar s=\bar L$ to obtain $u'(\bar s)$ for $ \bar s \in (0, \bar L)$. Finally, we obtain $\rho$ from $u'$ using Eq. .
\(1) Surjectivity of $f$. Let $(\omega,\rho)\in {{\cal C}}$. Since $\rho$ verifies (\[eq:x1\]), one can perform the change of variables (\[eq:change\_of\_var\]) and obtain $u$ and $u'$. Next, consider $a=u'(0)$ and $\bar L=L\omega ^2/g$. One has clearly $a\in(0,a_{\max})$, $\bar L\in(0,\bar L_{\max})$, and $f((a,\bar L))=(\omega,\rho)$.
\(2) Injectivity of $f$. Assume that there are $(a_1, \bar L_1) \neq
(a_2, \bar L_2)$ such that $f(a_1,\bar L_1) = f(a_2,\bar L_2) =
(\omega,\rho)$. One has $a_1=a_2=-\rho(0)\omega^2/g$ and $\bar
L_1=\bar L_2=L\omega^2/g$, which implies the injectivity.
\(3) Continuity of $f$. We show in the Appendix \[sec:ODE\] that the ODE is Lipschitz. It follows that the function $u'(\bar s)$ for $0 \leq \bar s \leq \bar L$ depends continuously on its initial condition, which implies that $\rho(s)$ depends continuously on $a$.
\(4) Continuity of $f^{-1}$. It can be seen from the injectivity proof that $a$ and $\bar L$ depend continuously on $\omega$ and $\rho(0)$, and the latter depends in turn continuously on $\rho$.
Next, we establish a homeomorphism between the parameter space and a smooth surface in 3D, which allows an intuitive visualization of the configuration space.
\[prop:A\_to\_S\] For a given $a\in(0,a_{\max})$, integrate the differential equation from $(0,a)$ until $\bar s=\bar L_{\max}$. The set $(\bar s,u(\bar s),u'(\bar s))_{\bar s\in(0,\bar L_{\max})}$ is then a 1D curve in $\mathbb{R}^3$. The collection of those curves for $a$ varying in $(0,a_{\max})$ is a 2D surface in $\mathbb{R}^3$, which we denote by ${{\cal S}}$ (see Fig. \[fig:3Dshooting\]).
There exists a homeomorphism $l:{{\cal A}}\to{{\cal S}}$.
(image) at (0,0)[ ![The surface ${{\cal S}}$ that is homeomorphic to the configuration space ${{\cal C}}$. We depict two solution curves on the surface ${{\cal S}}$ (dashed lines), integrated from two different values of $a$ (large and medium). Red, blue, green, and purple lines represent respectively the first, second, third and fourth zero-radius loci (see Proposition \[prop:zero\]).[]{data-label="fig:3Dshooting"}](fig/3Dshooting.pdf "fig:"){width="50.00000%"}]{};
at (0.36, 0.78) [$u'$]{}; at (0.2, 0.54) [$u$]{}; at (0.75, 0.33) [$\bar s$]{}; (a) at (0.35, 0.3) [$\cal S$]{}; (a) – (0.4, 0.43);
The construction of $l$ follows from the definition: given a pair $(a,\bar L)\in {{\cal A}}$, integrate (\[eq:X\]) from $(0,a)$ until $\bar s=\bar L$. Then define $l(a,\bar L):=(\bar L,{\mathbf{u}}(\bar L))$.
\(1) Surjectivity of $l$. Consider a point $(\bar L,{\mathbf{u}})
\in{{\cal S}}$. By definition of $\cal S$, there exists $a\in(0,a_{\max})$ so that integrating (\[eq:X\]) from $(0,a)$ reaches ${\mathbf{u}}$ at $\bar s = \bar L$. Clearly, $l(a,\bar L)=(\bar L,{\mathbf{u}})$.
\(2) Injectivity of $l$. This results from the Uniqueness theorem for ODEs, see Appendix \[sec:ODE\].
\(3) Continuity of $l$. From the Continuity theorem for ODEs (Appendix \[sec:ODE\]), it is clear that the end point $(\bar
L,{\mathbf{u}}(\bar L)) \in {{\cal S}}$ depends continuously on the initial condition $a$.
\(4) Continuity of $l^{-1}$. Consider two points $(\bar L_1,{\mathbf{u}}^*_1),
(\bar L_2,{\mathbf{u}}^*_2)\in {{\cal S}}$ that are sufficiently close to each other, $$|\bar L_1-\bar L_2| \leq \delta,\quad \|{\mathbf{u}}^*_1-{\mathbf{u}}^*_2\| \leq \delta,$$ for some $\delta$ that we shall choose later. Consider the curves ${\mathbf{u}}_1,{\mathbf{u}}_2$ such that ${\mathbf{u}}_1(\bar L_1)={\mathbf{u}}_1^*$ and ${\mathbf{u}}_2(\bar L_2)={\mathbf{u}}_2^*$. By the Continuity theorem (Appendix \[sec:ODE\]) one has for some appropriate constant $K$, $$\|{\mathbf{u}}_1(0)-{\mathbf{u}}_2(0)\| \leq e^{M\bar L_1}\|{\mathbf{u}}_1(\bar
L_1)-{\mathbf{u}}_2(\bar L_1) \|$$ $$\leq e^{M\bar L_1} \left( \|{\mathbf{u}}_1(\bar L_1)-{\mathbf{u}}_2(\bar L_2))\| +
\|{\mathbf{u}}_2(\bar L_2)-{\mathbf{u}}_2(\bar L_1)\|\right)$$ $$\leq e^{M\bar L_1} (\delta + M|\bar L_1-\bar L_2|) = e^{M\bar L_1}(M+1)\delta,$$ where the last inequality come from the uniform boundedness of ${\mathbf{u}}$. For any $\epsilon$, it suffices therefore to choose $\delta:=\frac{\epsilon e^{-M\bar L_1}}{M+1}$ so that $|a_1-a_2|=\|{\mathbf{u}}_1(0)-{\mathbf{u}}_2(0)\|\leq \epsilon$, which proves the continuity of $l^{-1}$.
Combining Propositions \[prop:A\_to\_C\] and \[prop:A\_to\_S\], we obtain the following theorem.
\[theo:homeomorphism\] The configuration space ${{\cal C}}$ of the rotating chain is homeomorphic to the 2D surface ${{\cal S}}$ represented in Fig. \[fig:3Dshooting\].
Zero-radius loci and low-amplitude regime
-----------------------------------------
Before studying the stable subspaces, we need first to define the zero-radius loci and the low-amplitude regime in the configurations space.
\[prop:zero\] Zero-radius loci are configurations whose corresponding attachment radii verify $r=0$. Define $\bar L_i := L\omega_i^2/g$ where $\omega_i$ is the $i$-th discrete angular speed (Appendix \[sec:lowamp\]). We have the following properties on the surface ${{\cal S}}$
(i) The $i$-th zero-radius locus is an infinite curve that branches out from the $\bar s$-axis at $(\bar L_i,0,0)$, see Fig. \[fig:3Dshooting\];
(ii) The $i$-th zero-radius locus separates configurations in rotation mode $i-1$ from those in rotation mode $i$.
\(i) This property is implied by Kolodner’s results, see the first paragraph of Sec \[sec:number-configuration\] for more details.
\(ii) Consider a rotation in mode $i-1$ and the corresponding curve $(\bar s, u_1(\bar s), u_1'(\bar s))_{\bar s\in [0, \bar L_1]}$. By definition, $u_1'(\bar s)$ has $i-1$ zeros in the interval $[0, \bar L_{1}]$. Equivalently, we see that the 3D curve $(\bar s, u_1(\bar s), u_1'(\bar s))$ crosses the first, second…$i-1$-th zero-radius locus. Now, since the loci start infinitely near the $\bar s$-axis \[point (i)\] and extend to infinity, any curve deformed from $(\bar s, u_1(\bar s), u_1'(\bar s))_{\bar s\in [0, \bar L_1]}$ also crosses the same loci.
Consider now another rotation, which is in mode $i$, and the corresponding curve $(\bar s, u_2(\bar s), u_2'(\bar s))_{\bar s \in [0, \bar L_2]}$. By Theorem \[theo:homeomorphism\], one can associate the two rotations with their endpoints $(\bar L_1, u_1(\bar L_1), u_1'(\bar L_1))$ and $(\bar L_2, u_2(\bar L_2), u_2'(\bar L_2))$ on the surface ${{\cal S}}$. We will show that any continuous path that connect these two points necessarily crosses the $i$-th zero-radius locus. Indeed, assume the contradiction, it follows that there is a continuous curve ending at $(\bar L_2, u_2(\bar L_2), u_2'(\bar L_2))$ that does not cross the $i$-th locus. This is a contradiction to our assertion in the first paragraph of point (ii).
We have thus established that the $i$-th zero-radius locus separates configurations of rotation mode $i-1$ from those in rotation mode $i$.
\[prop:1\] The low-amplitude regime corresponds to configurations associated with infinitely small values of $u(\bar s)$ and $u'(\bar s)$, for all $\bar s\in(0,\bar L)$.
(i) The low-amplitude regime corresponds to points on the surface ${{\cal S}}$ that are infinitely close to the $\bar s$-axis (in Fig. \[fig:3Dshooting\]).
(ii) Moreover, this regime corresponds to points on the parameter space ${{\cal A}}$ that have small values of $a$.
\(i) It is clear that a low-amplitude rotation has $u(\bar L)$ and $u'(\bar L)$ infinitely small. Conversely, if $u(\bar L)$ and $u'(\bar L)$ are infinitely small, by the continuity of the mapping $l^{-1}$ in the proof of Proposition \[prop:A\_to\_S\], the initial condition $a$ is also infinitely small. Finally, integrating from an infinitely small $a$ will yield $u(\bar s)$ and $u'(\bar s)$ infinitely small for all $\bar s\in(0,\bar L)$.
\(ii) This is true from (i).
The low-amplitude rotations with zero attachment radius thus correspond to $(\bar L_i,\delta u,0)$, $i\in\mathbb N$ for small values of $|\delta u|$. In the sequel, we shall refer to the $i$-th small-amplitude rotation with zero radius as the point $(\bar L_i, 0, 0)$ instead of the more correct phase “$(\bar L_i,\delta u,0)$ for small values of $|\delta u|$”.
Stability analysis {#sec:stability_analysis}
------------------
So far we have considered the space of all configurations of the rotating chain, that is, all solutions to the equation of motion (\[eq:eom\]). However, not all configurations are *stable*; in fact, experiments show that many are not. This section investigates the structure of the stable subspace – the subset of stable configurations – and discuss stable manipulation strategies.
To analyze the stability of configurations, we model the chain by a series of lumped masses, connected by stiff links, see Fig. \[fig:discrete-chain\].
(image) at (0,0) [![\[fig:discrete-chain\] Discretized chain model with $N$ masses. ](fig/discrete-chain.pdf "fig:")]{};
at (0.06, 0.68) [$x$]{}; at (0.15, 0.95) [$z$]{}; at (0.32, 0.75) [$y$]{}; at (0.1, 0.825) [$\{\rm O\}$]{};
at (0.85, 0.02) [${\mathbf{x}}_0$]{}; at (0.93, 0.22) [${\mathbf{x}}_1$]{}; at (0.74, 0.5) [${\mathbf{x}}_{N-2}$]{}; at (0.75, 0.7) [${\mathbf{x}}_{N-1}$]{}; at (0.52, 0.84) [${\mathbf{x}}_{N}$]{}; at (0.75, 0.145) [${\mathbf{l}}_1$]{}; at (0.5, 0.6) [${\mathbf{l}}_{N-1}$]{}; at (0.5, 0.725) [${\mathbf{l}}_{N}$]{};
Denote the position of the $i$-th mass in the rotating frame $\{\rm O\}$ by ${\mathbf{x}}_i\in \mathbb{R}^{3}$. The attached end is fixed in $\{\rm{O}\}$ at ${\mathbf{x}}_N$. The state of the discretized chain is then given by a $6N$-dimensional vector consisting of the positions and velocities of the masses $$\label{eq:14}
{\mathbf{y}} := [{\mathbf{x}}_0, \dot {{\mathbf{x}}}_0, \dots, {\mathbf{x}}_{N-1}, \dot {{\mathbf{x}}}_{N-1}].$$ Applying Newton’s laws to the masses (see details in Appendix \[sec:discrete\]), one can obtain the dynamics equation $$\label{eq:discrete-dyn}
\dot{{\mathbf{y}}} = {\mathbf{f}}({\mathbf{y}}).$$
From Proposition \[prop:A\_to\_C\], the configurations of the rotating chain can be represented by a pair $(a,\bar L)$, which is associated with the position of the free end ${\mathbf{x}}_0$. Next, we discretize $(0,a_{\max})\times(0,\bar L_{\max})$ into a 2D grid. For each $(a,\bar L)$ in the grid, we integrate, from the free end ${\mathbf{x}}_0$, the shape function of the discretized chain (\[eq:14\]) at rotational equilibrium – in the same spirit as in Proposition \[prop:A\_to\_C\]. This discretized shape function corresponds to a state vector ${\mathbf{y}}^{\rm{eq}}:= [{\mathbf{x}}^{\rm{eq}}_0, {\mathbf{0}}, \dots, {\mathbf{x}}^{\rm{eq}}_{N-1}, {\mathbf{0}}]$. Finally, we assess the stability of ${\mathbf{y}}^{\rm{eq}}$ by looking at the Jacobian $${\mathbf{J}}({\mathbf{y}}^{\rm{eq}}):=\frac{\rm d{\mathbf{f}}}{\rm d {\mathbf{y}}}({\mathbf{y}}^{\rm{eq}}).$$ Specifically, if the largest real part $\lambda_{\max}:=\max_{i} \rm{Re}(\lambda_{i})$ of the eigenvalues of ${\mathbf{J}}({\mathbf{y}}^{\rm{eq}})$ is positive, then the system is unstable at ${\mathbf{y}}^{\rm{eq}}$; if it is negative, then the system is asymptotically stable at ${\mathbf{y}}^{\rm{eq}}$ [@khalil1996noninear Theorem 3.1].
Fig. \[fig:stability\_map\](A) depicts the values of $\lambda_{\max}$ for $(a,\bar L)\in(0,5)\times(0,40)$. One can observe an interesting distribution of these values; in particular, the sharp transitions around the zero-radius loci (black lines). However, even though $\lambda_{\max}$ gets very close to zero on the left side of the zero-radius loci or in the low-amplitude regime, it is never negative, hinting that the system is at best marginally stable. While this could be expected from our model, which does not include any energy dissipation, it is contrary to the experimental observation of stable rotation states.
We need therefore to take into account aerodynamic forces in the chain dynamics, see details in Appendix \[sec:discrete\]. Note that aerodynamic forces do not significantly affect the analysis of the previous sections, as their effect on the shape of the chain is negligible: for example, for a chain of length $\SI{0.76}{m}$ and parameters $(a, \bar L) = (2.0, 10.0)$, the changes in the equilibrium positions are less than $\SI{1}{mm}$, which is 0.14% of the chain length.
Fig. \[fig:stability\_map\](B) depicts the values of $\lambda_{\max}$ for the system with aerodynamic forces. One can note that the overall distribution of $\lambda_{\max}$ is very similar to that of the system *without* aerodynamic forces \[Fig. \[fig:stability\_map\](A)\], but with the key difference that the regions in Fig. \[fig:stability\_map\](A) with low but positive values now contain in Fig. \[fig:stability\_map\](B) *negative* values of $\lambda_{\max}$, which corresponds to asymptotically stable states.
(image) at (0,0) [![\[fig:stability\_map\] (Best viewed in color) Maps of $\lambda_{\max}$, the largest real part of the eigenvalues of the linearized dynamics of two 10-link lumped-mass models at equilibrium: (**A**) model without aerodynamic forces and (**B**) model with aerodynamic forces. Positive values (red color) indicate unstable behaviors while negative values (blue color) indicate asymptotically stable behaviors. Most configurations that are stable in the presence of aerodynamic forces can not be concluded to be stable when there is no aerodynamic forces. Black lines: zero-radius loci – configurations whose attachment radii are zeros. Green arrow: A path in the chain’s configuration space that contains only stable configurations. Black dashed arrow: A path that contains unstable configurations. ](fig/stability_map_nodrag.pdf "fig:")]{}; (imageright) at (0,-5) [![\[fig:stability\_map\] (Best viewed in color) Maps of $\lambda_{\max}$, the largest real part of the eigenvalues of the linearized dynamics of two 10-link lumped-mass models at equilibrium: (**A**) model without aerodynamic forces and (**B**) model with aerodynamic forces. Positive values (red color) indicate unstable behaviors while negative values (blue color) indicate asymptotically stable behaviors. Most configurations that are stable in the presence of aerodynamic forces can not be concluded to be stable when there is no aerodynamic forces. Black lines: zero-radius loci – configurations whose attachment radii are zeros. Green arrow: A path in the chain’s configuration space that contains only stable configurations. Black dashed arrow: A path that contains unstable configurations. ](fig/stability_map.pdf "fig:")]{};
at (0.12, 0.95) [(**B**)]{}; at (0.45, 0.02) [$\bar L$]{}; at (0.0, 0.535) [$a$]{};
(0.3, 0.61) circle \[x radius=0.018, y radius=0.03\] node\[anchor=south, very thick, color=green, xshift=-0.5cm\] (start) ;
(0.42, 0.33) circle \[x radius=0.018, y radius=0.03\] node\[anchor=south, very thick, color=orange, xshift=-0.5cm\] (start) ;
(0.29, 0.58) .. controls (0.23, 0.4) .. (0.19, 0.185) – (0.365, 0.185) – (0.4, 0.29);
(0.32, 0.59) .. controls (0.45, 0.5) .. (0.425, 0.38) ;
at (0.12, 0.95) [(**A**)]{}; at (0.45, 0.02) [$\bar L$]{}; at (0.0, 0.535) [$a$]{};
One can make three more specific observations:
1. Configurations that are immediately on the right-hand sides of the zero-radius loci and with $a$ relatively large are unstable (red color);
2. Configurations that are immediately on the left-hand sides of the zero-radius loci and with $a$ relatively large are stable (blue color);
3. Configurations with $a$ small (low-amplitude regime) are stable (light blue color).
Observation (1) hints that the upper portions of the zero-radius loci form “unstable barriers” in the configuration space. Therefore, it is *not* possible to stably transit between rotation modes $i-1$ and $i$ (which requires crossing the $i$-th zero-radius locus, see Proposition \[prop:zero\]) while staying in the upper portion of the configuration space \[dashed black arrow in Fig. \[fig:stability\_map\](B)\]. Observation (2) implies that transitions between configurations of the same mode can be stable. Observation (3) hints that a possible transition strategy might consist in (i) going down to the low-amplitude regime; (ii) traversing the $i$-th zero-radius locus while remaining in the low-amplitude regime; (iii) going up towards the desired end configuration \[green arrow in Fig. \[fig:stability\_map\](B)\]. This strategy thus traverses only regions with negative $\lambda_{\max}$ and can be expected to be stable. The next section experimentally assesses this strategy.
Manipulation of the rotating chain {#sec:manipulation}
==================================
Experiment
----------
We now experimentally test the manipulation strategy enunciated in the previous section. More precisely, to stably transit between two different rotation modes $i$ and $j$, we propose to \[see the green arrow in Fig. \[fig:stability\_map\](B)\]
1. Move from the rotation of mode $i$ towards $(\bar L_{i+1},0,0)$ while staying in the blue region of Fig. \[fig:stability\_map\](B);
2. Move along the $\bar L$-axis towards $(\bar L_{j+1},0,0)$;
3. Move from $(\bar L_{j+1},0,0)$ towards the rotation of mode $j$ while staying in the blue region of blue region of Fig. \[fig:stability\_map\](B).
In practice, the histories of the control inputs ($r$ and $\omega$) to achieve the transitions in steps 1 and 3 can be found by simple linear interpolation, see e.g. Fig. \[fig:exp\_control\](A).
**A**\
![**A**: Histories of the control inputs. Red: attachment radius $r$; blue: angular speed $\omega$. A: low-amplitude rotation at critical speed $\omega_1$. A $\to$ B: moving deep into rotation mode 0. B: stable rotation at mode 0. B $\to$ C: moving back to the low-amplitude regime with critical speed $\omega_1$ and subsequently increasing the speed to $\omega_2$ while staying in the low-amplitude regime. C: low-amplitude rotation at critical speed $\omega_2$. C $\to$ D: moving deep into rotation mode 1. D: stable rotation at mode 1. E: low-amplitude rotation at critical speed $\omega_3$. F: stable rotation at mode 2. Note that the attachment radius was not exactly zero in the low-amplitude regimes, but set to some small values. This was necessary to physically generate the desired rotation speeds. **B**: Snapshots of the chain at different time instants. The labels A–F refer to the same time instants as in the control inputs plot. A video of the experiment (including more types of transitions) is available at <https://youtu.be/EnJdn3XdxEE>.[]{data-label="fig:exp_control"}](fig/exp_control.pdf "fig:"){width="50.00000%"}\
**B** ![**A**: Histories of the control inputs. Red: attachment radius $r$; blue: angular speed $\omega$. A: low-amplitude rotation at critical speed $\omega_1$. A $\to$ B: moving deep into rotation mode 0. B: stable rotation at mode 0. B $\to$ C: moving back to the low-amplitude regime with critical speed $\omega_1$ and subsequently increasing the speed to $\omega_2$ while staying in the low-amplitude regime. C: low-amplitude rotation at critical speed $\omega_2$. C $\to$ D: moving deep into rotation mode 1. D: stable rotation at mode 1. E: low-amplitude rotation at critical speed $\omega_3$. F: stable rotation at mode 2. Note that the attachment radius was not exactly zero in the low-amplitude regimes, but set to some small values. This was necessary to physically generate the desired rotation speeds. **B**: Snapshots of the chain at different time instants. The labels A–F refer to the same time instants as in the control inputs plot. A video of the experiment (including more types of transitions) is available at <https://youtu.be/EnJdn3XdxEE>.[]{data-label="fig:exp_control"}](fig/states_transition.pdf "fig:"){width="50.00000%"}
We perform the following transitions $$\mathrm{Rest} \to \mathrm {Mode\ 0} \to \mathrm {Mode\ 1} \to \mathrm {Mode\ 2}$$ on a metallic chain of length $\SI{0.76}{\meter}$ (note that the weight of the chain is not involved in the calculations). The upper end of the chain was attached to the end-effector of a 6-DOF industrial manipulator (Denso VS-060). The critical speeds, calculated using equation (\[eq:critical\_velocities\]), are given in Table \[tab:critical\].
$i$ $1$ $2$ $3$
--------------- -------- -------- ---------
$\omega_i$ () $4.34$ $9.97$ $15.64$
: Critical speeds for a chain of length $\SI{0.76}{\meter}$
\[tab:critical\]
A video of the experiment (including more types of transitions) is available at <https://youtu.be/EnJdn3XdxEE>. Fig. \[fig:exp\_control\](A) shows the attachment radius and the angular speed as functions of time. Fig. \[fig:exp\_control\](B) shows snapshots of the chain at different rotation modes. As can be observed in the video, the chain could transit between different rotation modes in a stable and controlled manner.
As the final note, we observed that any manipulation sequence that traverses highly unstable regions (red regions in Fig. \[fig:stability\_map\]) definitely leads to unsustainable rotations, as illustrated by the last section of the video.
Implications for aerial manipulation
------------------------------------
For a circularly towing system, the ability to transit between rotation modes is desirable. Indeed, different modes have different functions. For instance, mode $0$ rotations are most suitable to initiate a rotation sequence from a straight flying trajectory. On the other hand, rotations at higher order modes such as $1$ and $2$ have more compact shapes, smaller tip radii and higher tip velocities, and are therefore more suitable to perform the actual deliveries or explorations.
It is furthermore desirable to switch modes in a quasi-static manner, as studied in this paper. Indeed, the *transient* dynamics of a heavily underactuated system such as the chain can be difficult to handle. The infinite dimensionality of the system, unavoidable modeling errors and aerodynamic effects make it challenging to design and reliably execute non-quasi-static mode switching trajectories.
Our result suggests however that it is *not* possible to realize quasi-static mode transitions with fixed-wing aircraft. Indeed, since the turning radii of such aircraft are lower-bounded, the resulting rotations cannot enter the low-amplitude regime, which is necessary for quasi-static mode transition, as shown in the above development. Therefore, although non-quasi-static mode transitions are more challenging to plan and execute, they must be studied in future works.
Conclusion {#sec:conclusion}
==========
The study of the rotating chain has a long and rich history. Starting from the 1950’s, a number of researchers have described its behavior, and identified the existence of rotation modes. In this paper, we have investigated for the first time the *manipulation* problem, how to stably transit between different rotation modes. For that, we developed a framework for understanding the kinematics of the rotating chain with non-zero attachment radii and its configuration space. Based on this understanding, we proposed a manipulation strategy for transiting between different rotation modes in a stable and controlled manner. In turn, on the practical side, this result has some implications for aerial manipulation.
It can be shown (see Appendix \[sec:tip-mass\]) that all the previous developments can be extended to the case of the chain with non-negligible tip mass. The key enabling notion here is that of *differential flatness* [@murray1996trajectory], with the flat output being the state of the free end. By differential flatness, given any trajectory of the free end, one can reversely compute the state trajectory and the control trajectory of the whole system. In fact, the property that we have “manually” discovered in this paper – the configuration space of a rotating chain is parameterized by the parameter space ${{\cal A}}$ – is related to the differential flatness of the rotating chain system. Indeed, each point $(a, \bar L)$ corresponds to a circular motion of the free end, which in turn, by differential flatness, corresponds to the state and control trajectory of the whole chain, which in turn defines the configuration. This observation suggests two possible extensions:
- the motion of the free end can be more general (an ellipse), and can thereby lead to more practical applications, such as swinging to hit some position with the tip mass;
- other differentially-flat systems, whose flat output can be parameterized.
Another idea developed here, namely the visualization of the configuration space based on forward integration of the shape function, might find fruitful applications in the study of other flexible objects with “mode transition”, such as elastic rods or concentric tubes subject to “snapping”. Our future work will explore these possible extensions.
Chain with non-negligible tip mass {#sec:tip-mass}
----------------------------------
Suppose that the free end of the chain carries a drogue of mass $M$. We show that all the previous development can be applied to this more general problem.
We first proceed similarly to Section \[sec:background\] and derive the dynamics equation of the rotating chain with tip mass. Writing the force equilibrium equation at the tip mass yields $$\begin{aligned}
\label{eq:17}
F(0) z'(0) &= Mg,\\
\label{eq:19}
F(0) \rho'(0) &= - M \rho(0) \omega^{2}.\end{aligned}$$ Next, integrate Eq. to obtain $$\label{eq:5}
F(s) z(s)' = g(\mu s + M),$$ where $\mu$ is again the linear density of the chain. This equation leads to $$\label{eq:6}
F(s) = g\frac{\mu s + M}{\sqrt{1 - \rho'^2}}.$$ One arrives at the governing equation $$\label{eq:7}
\frac{{{\rm d}{}}}{{{\rm d}{s}}} \left(\rho' \frac{\mu s + M}{\sqrt{1-\rho'^2}}
\right)
+ \rho \frac{\mu\omega^2}{g}
= 0,$$ with boundary condition $\rho(L)=r$ where $r$ is the attachment radius. One can now convert Eq. to a dimensionless equation $$\begin{aligned}
\label{eq:9}
\frac{{{\rm d}{}}^{2}u}{{{\rm d}{\bar s}}^{2}} + \frac{u}{\sqrt{(\bar s +
M\omega^2 / \mu g)^2 + u^2}} = 0\end{aligned}$$ by the following changes of variable $$\label{eq:8}
\begin{aligned}
u &:= \rho'\frac{\mu s + M}{\sqrt{1-{\rho'}^2}}\frac{\omega^2 }{\mu g}, \\ \quad \bar
s &:= \frac{s\omega^2}{g}.
\end{aligned}$$ The boundary conditions are $$\begin{aligned}
\label{eq:10}
u'(0) &= a,\\
u(0) &= a \frac{M\omega^2}{\mu g},\\
u'(\bar L) &= \bar r,\end{aligned}$$ where $a= - \rho(0) \omega ^2 / g$ and $\bar r = - r
\omega^2/g$.
Eq. is a BVP that can be solved using the shooting method as described in Section \[sec:shooting\]. Moreover, we see that $(a, \bar L)$ also parameterizes the solution space, which is the configuration space of the rotating chain with tip mass.
Low-amplitude regime {#sec:lowamp}
--------------------
Here we recall the results obtained by Kolodner [@Kolodner1955] for the low-amplitude regime. Low-amplitude rotations are defined by a zero attachment radius $r=0$ and infinitely small values for the shape function $\rho$. Linearizing equation (\[eq:x1\]) about $\rho=0$ yields
$$\label{eq:4}
\rho w^2/g + \rho' + s \rho'' = 0,$$
with the boundary condition $\rho(L) = 0$.
By a change of variable $v:=2\sqrt{s\omega^2/g}$, one can rewrite the above equation as $$\rho v + \rho_v + \rho_{vv} v =0,$$ which has solutions of the form $$\rho(v) = c J_0(v),\quad \mathrm{\ie}$$ $$\rho(s) = cJ_0(2\omega\sqrt{s/g}),$$ where $J_0$ is the zeroth-Bessel function. The boundary condition $\rho(L) = 0$ then implies that the angular speed can only take discrete values $(\omega_i)_{i\in\mathbb{N}}$ where $$\label{eq:critical_velocities}
\omega_i = \frac{h_i}{2} \sqrt{g/L}$$ where $h_i$ is the $i$-th zero of the Bessel function $J_0$.
Useful results from the theory of Ordinary Differential Equations {#sec:ODE}
-----------------------------------------------------------------
The ordinary differential equation (\[eq:X\]) satisfies Lipschitz condition in some convex bounded domain ${{\cal D}}$ that contains ${{\cal S}}$.
Note first that $|u''(u, \bar s)| < 1$ for all $u, \bar s \in {\mathbb{R}}$, which implies that ${{\cal S}}$ is bounded. Set now $${{\cal D}}:= (0, \bar L_{\max}) \times (u_{\inf}, u_{\sup}) \times (u'_{\inf}, u'_{\sup}),$$ where $u_{\inf}$, $u_{\sup}$, $u'_{\inf}$, $u'_{\sup}$ are bounds on ${{\cal S}}$. Clearly, ${{\cal D}}$ is bounded, convex and contains ${{\cal S}}$. Next, all partial derivatives $\frac{{\partial}{\mathbf{X}}_i}{{\partial}x_j}$ are continuous in ${{\cal D}}$ (with continuation at $\bar s=0$, see Remark in Section \[sec:dimless\]). This implies that ${\mathbf{X}}$ is Lipschitz in ${{\cal D}}$ [@HL2005elec].
We now recall two standard theorems in the theory of Ordinary Differential Equations, see [@HL2005elec].
If the vector field ${\mathbf{X}}({\mathbf{u}}, t)$ satisfies Lipschitz condition in a domain ${{\cal D}}$, then there is at most one solution ${\mathbf{u}}(t)$ of the differential equation $$\frac{{{\rm d}{{\mathbf{u}}}}}{{{\rm d}{t}}} = {\mathbf{X}}({\mathbf{u}}, t)$$ that satisfies a given initial condition ${\mathbf{u}}(a) = {\bm{c}}\in {{\cal D}}$.
Let ${\mathbf{u}}_1(t)$ and ${\mathbf{u}}_2(t)$ be any two solutions of the differential equation ${\mathbf{X}}({\mathbf{u}}, t)$ in $T_1 \leq t \leq T_2$, where ${\mathbf{X}}({\mathbf{u}}, t)$ is continuous and Lipschitz in some domain ${{\cal D}}$ that contains the region where ${\mathbf{u}}_1(t)$ and ${\mathbf{u}}_2(t)$ are defined. Then, there exists a constant $M$ such that $$\|{\mathbf{u}}_1(t) - {\mathbf{u}}_2(t)\| \leq e^{M|t-a|} \|{\mathbf{u}}(a) - {\bm{y}}(a)\|$$ for all $a, t \in [T_1, T_2]$.
The discretized chain model {#sec:discrete}
---------------------------
Here we describe the procedure to obtain Eq. , which is the dynamics equation of the discretized chain model employed in Section \[sec:stability\_analysis\], see also Fig. \[fig:discrete-chain\].
The net force ${\mathbf{F}}_{i}$ acting on the $i$-th mass is the sum of the following three components:
1. *fictitious forces*, which include the Coriolis force and centrifugal force associated with the rotating frame;
2. *constraint forces* generated by the $i$-th and $i+1$-th links;
3. *aerodynamic forces*, which include drag and lift.
Fictitious forces are computed using standard formulas, which can be found in any textbook on classical mechanics. To compute the constraint forces, we model the links as stiff linear springs whose stiffness approximates that of the chain used in the experiment of Section \[sec:manipulation\], which was $\simeq\SI{8e7}{N/m}$. Constraint forces are then computed using Hooke’s law.
Next, to compute aerodynamic forces, we follow the modelling choices of [@Williams2007], i.e. the aerodynamic forces acting on the $i$-th link is placed entirely on the $i$-th mass. Specifically, define the link length vector as ${\mathbf{l}}_{i}:= {\mathbf{x}}_{i} - {\mathbf{x}}_{i-1}$ and denote by ${\mathbf{v}}_i$ the actual air speed of the $i$-th mass, the angle of attack of the $i$-th link is given by $$\cos \xi_i = - \frac{{\mathbf{l}}_i \cdot {\mathbf{v}}_{i} } {\|{\mathbf{l}}_i\| \|{\mathbf{v}}_i\| }.$$ The drag and lift acting on the $i$-th link are then given by $$\begin{aligned}
{\mathbf{F}}_{i}^{D} &= 0.5 \rho_{a} C_{D} \|{\mathbf{l}}_i\| d \|{\mathbf{v}}_{i}\|^2{\mathbf{e}}_{D},\\
{\mathbf{F}}_{i}^{L} &= 0.5 \rho_{a} C_{L} \|{\mathbf{l}}_i\| d \|{\mathbf{v}}_{i}\|^2{\mathbf{e}}_{L},
\end{aligned}$$ where the directions and coefficents of drag and lift are $$\begin{aligned}
{\mathbf{e}}_{D} &= - \frac{{\mathbf{v}}_{i}}{\|{\mathbf{v}}_{i}\|}, &\quad
{\mathbf{e}}_{L} &= - \frac{({\mathbf{v}}_i\times {\mathbf{l}}_{i})\times {\mathbf{v}}_{i}}
{\|({\mathbf{v}}_i\times {\mathbf{l}}_{i})\times {\mathbf{v}}_{i}\|},\\
C_{D} &= C_f + C_{n}\sin^{3} (\xi_i), &\quad C_{L} &= C_{n}\sin^2\xi_{i} \cos \xi_{i}.
\end{aligned}$$ In the above equations, $d$ denotes the diameter of the chain, $C_{f}$ and $C_{n}$ are respectively the skin-fraction and crossflow drag coefficients, $\rho_{a}$ is the air density. These parameters have the following numerical values $$\begin{aligned}
d &= \SI{1}{mm},\; &\rho_a&= \SI{1.225}{kg/m^{3}}, \\
C_f &= 0.038, \; &C_n &= 1.17.
\end{aligned}$$
Summing the components we obtain the $i$-th net force ${\mathbf{F}}_i$, from which the acceleration of the $i$-th mass can be found as $$\ddot {{\mathbf{x}}}_i = {\mathbf{F}}_i / m_i,$$ where $m_i$ is the mass of the $i$-th mass. Rearranging the terms, one obtains the dynamics equation $$\dot{{\mathbf{y}}} = {\mathbf{f}} ({\mathbf{y}}).$$
[^1]: Hung Pham and Quang-Cuong Pham are with Air Traffic Management Research Institute (ATMRI) and Singapore Centre for 3D Printing (SC3DP), School of Mechanical and Aerospace Engineering, Nanyang Technological University, Singapore. This work was partially supported by grant ATMRI:2014-R6-PHAM (awarded by NTU and the Civil Aviation Authority of Singapore) and by the Medium-Sized Centre funding scheme (awarded by the National Research Foundation, Prime Minister’s Office, Singapore).
[^2]: This claim is based on numerical observations.
[^3]: We have not yet been able to prove rigorously that the sequence is indeed decreasing.\[fn:1\]
|
---
abstract: 'We consider the problem of design of the acoustic structure of arbitrary geometry with prescribed desired properties. We use optimization approach for the solution of this problem and minimize the Tikhonov functional on adaptively refined meshes. These meshes are refined locally only in places where the acoustic structure should be designed. Our special symmetric mesh refinement strategy together with interpolation procedure allows the construction of the symmetric acoustic material with prescribed properties. Efficiency of the presented adaptive optimization algorithm is illustrated on the construction of the symmetric acoustic material in two dimensions.'
author:
- 'L. Beilina [^1]'
- 'E. Smolkin [^2]'
title: Computational design of acoustic materials using an adaptive optimization algorithm
---
Introduction
============
In this work we present a new adaptive optimization algorithm which can construct acoustic materials with arbitrary geometry from desired scattering parameters. We formulate our problem as a Coefficient Inverse Problem (CIP), and our goal is to determine an unknown spatially distributed wave speed of the acoustic wave equation from boundary measurements on the adaptively refined meshes. To solve our CIP, we minimize the Tikhonov functional in order to find the wave speed distribution inside designed domain which satisfies prescribed scattering properties. In the case of numerical simulations of Section \[sec:numex\] we formulate these properties as obtaining as small as possible reflections from the designed structure. For minimization of the Tikhonov functional we use Lagrangian approach and search for a stationary point of it on the adaptively refined meshes. Compared with other works on this subject [@B; @BJ; @btkm14] we need to refine mesh locally only inside the known geometry. For construction of a new mesh we use symmetric mesh refinement strategy combined with the interpolation procedure over the neighboring vertices for every element in the mesh. This allows us finally to get acoustic material of the symmetric structure.
To construct the desired acoustic structure we formulate an adaptive optimization algorithm which includes solution of the forward and adjoint problems for the acoustic wave equation. The domain decomposition finite element/finite difference (FE/FD) method of [@hybrid] is used for the computational solution of these problems. This method is implemented efficiently using the software packages WavES [@waves] and PETSc [@petsc]. In the theoretical part of this work we present proof of the energy estimate for a hyperbolic equation with one unknown function - the wave speed- and different boundary conditions for the case of our domain decomposition. We illustrate efficiency of the proposed method in numerical examples on the construction of new acoustic material in two dimensions. The goal of our numerical simulations is to reconstruct the wave speed function of the hyperbolic equation from single observations of the solution of this equation in space and time which gives us as small reflections as possible. We note that the domain decomposition approach in this case is particularly feasible for implementing of absorbing boundary conditions [@EM].
Developed in this work adaptive optimization method can be used in construction and design of new materials including nano-materials with so-called cloaking properties, see [@cloak1; @cloak2; @cloak3]. To obtain cloaking structures in all these works are used methods of transformational optics which are based on the accordance between material parameters and coordinate transformations. In the current work we propose to use an adaptive optimization algorithm which is an alternative approach for the construction of an approximate cloaking. Depending on applications, this method can be used alone or as a compliment to the method of transformational optics. Advantage of a new technique compared to the transformational optics is fast construction of any material of arbitrary geometry with desired symmetric structure of any size. This structure is not dependent on the coordinate transformation and can be adapted to desired properties of the physical material. The mesh size of the symmetric structure can be defined as a parameter in the adaptive mesh refinement procedure used in the optimization algorithm. Thus, the new algorithm allows efficiently compute a new material of any symmetric structure with desired properties. A first version of a such algorithm was presented in [@AoA] for design of a nanophotonic structure.
The paper is organized as follows. In Section \[sec:modelhyb\] we present statements of the forward and inverse problems and in Section \[sec:opt\] we describe the Lagrangian approach for solution of our CIP. Stability estimates for the solution of forward and adjoint problems are given in Section \[sec:energyerror1\]. In Section \[sec:fem\] we present the domain decomposition FEM/FDM to solve the minimization problem of Section \[sec:opt\], and in Section \[sec:ad\_alg\] we present an adaptive conjugate gradient algorithm for the solution of our CIP. Finally, in our concluding Section \[sec:numex\] we demonstrate efficiency of the adaptive optimization algorithm identifying the wave speed function in two dimensions to construct material of symmetric structure which produce as small reflections as possible.
Statement of the forward and inverse problems {#sec:modelhyb}
=============================================
Let $x = (x_1, x_2)$ denote a point in $\mathbb{R}^2$ in an unbounded domain $D$. We model the wave propagation by the following Cauchy problem for the scalar wave equation: $$\label{modelhyb}
\begin{cases}
\tilde{c}(x) \frac{\partial^2 u}{\partial t^2} - \triangle u = 0 & ~ \mbox{in}~~\mathbb{R}^2 \times (0, \infty), \\
u(x,0) = f_0(x), ~~~u_t(x,0) = 0 &~ \mbox{in}~~ D.
\end{cases}
$$ Here, $u$ is the total wave pressure generated by the plane wave $p(t)$ which is incident at $x_1 = x_0$ and propagates along $x_2$ axis, $\tilde{c}(x)= \frac{1}{c(x)^2}$ is the isotropic function with the spatially distributed wave speed $c(x)$.
Let now $D \subset \mathbb{R}^{2}$ be a bounded domain with the boundary $\partial D$. We use the notation $D_T := D \times (0,T), \partial D_T := \partial D
\times (0,T), T > 0$ and assume that $$f_{0}\in H^{1}(D), \tilde{c}(x) \in C^2(D). \label{f1}$$ For computational solution of (\[modelhyb\]) we use the domain decomposition finite element/finite difference (FE/FD) method of [@hybrid] which was applied for the solution of different coefficient inverse problems for the acoustic wave equation in works [@B; @hybrid; @BJ; @BCL]. To apply method of [@hybrid] we decompose $D$ into two regions $D_{FEM}$ and $D_{FDM}$ such that the whole domain $D = D_{FEM} \cup D_{FDM}$, see Figure \[fig:0\_1\]. In $D_{FEM}$ we use the finite element method (FEM), and in $D_{FDM}$ we will use the Finite Difference Method (FDM), see details in [@hybrid]. Furthermore, we decompose the domain $D_{FEM}$ into three regions $G_0, G_1, G_2$ such that $D_{FEM} = G_0 \cup G_1 \cup G_2$, where $G_0$ is the innermost subdomain with the boundary $\partial G_0$, $G_1$ is the subdomain where we want to design the acoustic material, and $G_2$ is the outermost subdomain, see Figure \[fig:0\_1\]-b).
[c]{}\
a) $D$\
at (0,0) ; (7.5,-15pt) node\[anchor=north\] [$G_2$]{}; (7.5,-40pt) node\[anchor= north\] [$G_1$]{}; (7.5,-65pt) node\[anchor=north\] [$G_0$]{};
\
b) $D_{FEM}$
Let the boundary $\partial D$ be decomposed as $\partial D =\partial
_{1} D \cup \partial _{2} D \cup \partial _{3} D$ where $\partial
_{1} D$ and $\partial _{2} D$ are top and bottom sides of the domain $D$, respectively, and $\partial _{3} D$ is the union of left and right sides of this domain. At $S_T := (\partial_1 D \cup \partial_2
D) \times (0,T)$ we have time-dependent observations. We define $S_{1} = \partial_1 D \times (0,T)$, $S_{1,1} = \partial_1 D \times
(0,t_1]$, $S_{1,2} = \partial_1 D \times (t_1, T)$, $S_2 =
\partial_2 D \times (0, T)$ and $S_3 = \partial_3 D \times (0,
T)$, $S_4 = \partial G_0 \times (0, T)$.
We also introduce the following spaces of real valued functions $$\label{spaces}
\begin{split}
H_u^1(D_T) &:= \{ w \in H^1(D_T): w( \cdot , 0) = 0 \}, \\
H_{\lambda}^1(D_T) &:= \{ w \in H^1(D_T): w( \cdot , T) = 0\},\\
U^{1} &=H_{u}^{1}(D_T)\times H_{\lambda }^{1}(D_T)\times C\left( \overline{D}\right).
\end{split}$$
In our computations we have used the following model problem $$\label{model1}
\begin{cases}
\tilde{c} \frac{\partial^2 u}{\partial t^2} - \triangle u = 0&~ \mbox{in}~~ D_T, \\
u(x,0) = f_0(x), ~~~u_t(x,0) = 0 &~ \mbox{in}~~ D, \\
\partial _{n} u = p(x,t) & ~\mbox{on}~ S_{1,1}, \\
\partial _{n} u =-\partial _{t} u & ~\mbox{on}~ S_{1,2} \cup S_2, \\
\partial _{n} u =0 &~\mbox{on}~ S_3 \cup S_4.
\end{cases}
$$ In (\[model1\]) we use the first order absorbing boundary conditions [@EM] and $p(x,t) \in L_2(S_{1,1})$. We note that these conditions are exact in the case of computations of Section \[sec:numex\], since in our computations we initialize the plane wave orthogonal to the domain of propagation.
We choose the coefficient $\tilde{c}(x)$ in (\[model1\]) such that $$\label{coefic}
\begin{cases}
\tilde{c} \left( x\right) \in \left [ 1, M \right], M=const. > 0,
& \text{ for }x\in G_1, \\
\tilde{c}(x) =1
& \text{ for }x\in D_{FDM} \cup G_2.
\end{cases}$$
We consider the following inverse problem
**Inverse Problem (IP)**
*Let the coefficient* $\tilde{c} \left( x\right)$* in the problem (\[model1\]) satisfies conditions (\[coefic\]) and assume that* $ \tilde{c}\left( x\right) $* is unknown in the domain* $G_1$*. Determine the function* $ \tilde{c}\left(
x\right) $* in (\[model1\]) for* $x\in G_1$ * assuming that the following function* $\widetilde u\left(
x,t\right) $* is known* $$u\left( x,t\right) = \widetilde u \left( x,t\right), ~\forall \left( x,t\right) \in S_T. \label{2.5}$$
Optimization method {#sec:opt}
===================
In this section we present the reconstruction method to solve inverse problem **IP**. This method is based on the finding of the stationary point of the following Tikhonov functional $$F(u, \tilde{c}) = \frac{1}{2} \int_{S_T}(u - \widetilde{u})^2 z_{\delta }(t) dS dt +
\frac{1}{2} \gamma \int_{G_1}(\tilde{c}- \tilde{c}_0)^2~~ dx,
\label{functional}$$ where $u$ satisfies the equations (\[model1\]), $\tilde{c}_{0}$ is the initial guess for $\tilde{c}$ (see details about choice of this guess in Section \[sec:numex\] and [@btkm14; @BOOK]), $\widetilde{u}$ is the observed field at $S_T$, $\gamma > 0$ is the regularization parameter and $z_{\delta }(t)$ is the compatibility function in time and can be chosen as in [@btkm14].
To find minimum of (\[functional\]) we use the Lagrangian approach [@B; @BJ; @btkm14] and define the following Lagrangian in the week form $$\label{lagrangian1}
\begin{split}
L(v) &= F(u, \tilde{c})
- \int_{D_T} \tilde{c} \frac{\partial
\lambda }{\partial t} \frac{\partial u}{\partial t} ~dxdt
+ \int_{D_T}( \nabla u)( \nabla \lambda)~dxdt \\
& - \int_{S_{1,1}} \lambda p(x,t) ~dS dt + \int_{S_{1,2} \cup S_2} \lambda \partial_t u ~dS dt,
\end{split}$$ where $v=(u,\lambda, \tilde{c}) \in U^1$, and search for a stationary point with respect to $v$ satisfying for all $\bar{v}= (\bar{u}, \bar{\lambda}, \bar{\tilde{c}}) \in U^1$ $$L'(v; \bar{v}) = 0 , \label{scalar_lagr1}$$ where $ L^\prime (v;\cdot )$ is the Jacobian of $L$ at $v$.
In order to find the Fréchet derivative (\[scalar\_lagr1\]) of the Lagrangian (\[lagrangian1\]) we consider $L(v + \bar{v}) - L(v)~
\forall \bar{v} \in U^1$ and single out the linear part of the obtained expression with respect to $\bar{v}$. When we derive the Fréchet derivative we assume that in the Lagrangian (\[lagrangian1\]) functions in $v=(u,\lambda, \tilde{c}) \in U^1$ can be varied independent on each others. We note that by doing so we get the same Fréchet derivative of the Lagrangian (\[lagrangian1\]) as by assuming that functions $u$ and $\lambda$ are dependent on the coefficient $\tilde{c}$, see details in Chapter 4 of [@BOOK]. Similar to [@B; @hybrid; @BJ] we use conditions $\lambda \left(
x,T\right) =\partial _{t}\lambda \left( x,T\right) =0$ and imply such conditions on the function $\lambda $ to deduce that $ L\left( u,\lambda,
\tilde{c} \right) :=L\left( v\right) =F\left( u, \tilde{c}\right).$ We also use conditions (\[coefic\]) on $\partial D$, together with initial and boundary conditions of (\[model1\]) to get that for all $\bar{v} \in U^1$ we have $$L'(v; \bar{v}) = \frac{\partial L}{\partial \lambda}(v)(\bar{\lambda}) + \frac{\partial L}{\partial u}(v)(\bar{u}) + \frac{\partial L}{\partial \tilde{c}}(v)(\bar{\tilde{c}}) = 0, \label{scalar_lagrang1}$$ or $$\label{forward1}
\begin{split}
0 &= \frac{\partial L}{\partial \lambda}(v)(\bar{\lambda}) = \\
&- \int_{D_T} \tilde{c} \frac{\partial \bar{\lambda}}{\partial t} \frac{\partial u}{\partial t}~ dxdt
+ \int_{D_T} ( \nabla u) (\nabla \bar{\lambda}) ~ dxdt \\
&- \int_{S_{1,1}} \bar{\lambda} p(x,t)~dS dt \\
&+ \int_{S_{1,2} \cup S_2} \bar{\lambda} \partial_t u ~dS dt,~~\forall \bar{\lambda} \in H_{\lambda}^1(D_T),
\end{split}$$ $$\label{control1}
\begin{split}
0 &= \frac{\partial L}{\partial u}(v)(\bar{u}) = \\
&\int_{S_T}(u - \widetilde{u})~ \bar{u}~ z_{\delta}~ d S dt- \int_{D}
\tilde{c} \frac{\partial{\lambda}}{\partial t}(x,0) \bar{u}(x,0) ~dx\\
&- \int_{S_{1,2} \cup S_2}
\frac{\partial{\lambda}}{\partial t} \bar{u} ~dS dt
\\
&- \int_{D_T} \tilde{c} \frac{\partial \lambda}{\partial t} \frac{\partial \bar{u}}{\partial t}~ dxdt \\
&+ \int_{D_T} ( \nabla \lambda) (\nabla \bar{u}) ~ dxdt, ~\forall \bar{u} \in H_{u}^1(D_T),
\end{split}$$ $$\label{grad1new}
\begin{split}
0 &= \frac{\partial L}{\partial \tilde{c}}(v)(\bar{\tilde{c}})
= - \int_{D_T} \frac{\partial \lambda}{\partial t} \frac{\partial u}{\partial t} \bar{\tilde{c}}~dxdt \\
&+\gamma \int_{G_1} (\tilde{c} - \tilde{c}_0) \bar{\tilde{c}}~dx,~ x \in D.
\end{split}$$ We observe that (\[forward1\]) is the weak formulation of the state equation (\[model1\]) and (\[control1\]) is the weak formulation of the following adjoint problem $$\label{adjoint1}
\begin{cases}
\tilde{c} \frac{\partial^2 \lambda}{\partial t^2} -
\triangle \lambda = - (u - \widetilde{u}) z_{\delta} &~ x \in S_T, \\
\lambda(\cdot, T) = \frac{\partial \lambda}{\partial t}(\cdot, T) = 0, \\
\partial _{n} \lambda = \partial _{t} \lambda & ~\mbox{on}~ S_{1,2} \cup S_2,
\\
\partial _{n} \lambda =0 & ~\mbox{on}~ S_3 \cup S_4 \cup S_{1,1}.
\end{cases}$$
We define by $u(\tilde{c}), \lambda(\tilde{c})$ exact solutions of the forward and adjoint problems, respectively, for the known function $\tilde{c}$. Then using the fact that exact solutions $u(\tilde{c}), \lambda(\tilde{c})$ are sufficiently stable (see Chapter 5 of book [@lad] for details), we get from (\[lagrangian1\]) $$F( u(\tilde{c}), \tilde{c}) = L(v(\tilde{c})),$$ and the Fréchet derivative of the Tikhonov functional can be written as $$\label{derfunc}
\begin{split}
F'(\tilde{c}) := &F'(u(\tilde{c}), \tilde{c})= \frac{\partial F}{\partial \tilde{c}}(u(\tilde{c}), \tilde{c})
= \frac{\partial L}{\partial \tilde{c}}(v(\tilde{c}))
.
\end{split}$$ Inserting (\[grad1new\]) into (\[derfunc\]), we get the following space-dependent function: $$\label{derfunc2}
\begin{split}
F'(\tilde{c})(x) &:= F'(u(\tilde{c}),\tilde{c})(x) =\\
&- \int_0^T \frac{\partial \lambda(\tilde{c})}{\partial t} \frac{\partial u(\tilde{c})}{\partial t} (x,t)~dt
+\gamma (\tilde{c} - \tilde{c}_0)(x).
\end{split}$$
Stability estimates {#sec:energyerror1}
===================
The stability estimate for the forward problem (\[model1\]) follows from the stability estimate of [@hybrid] and can be derived using the technique of [@lad]. For analysis we first introduce the $L_2$ inner product and the norm over $D_T$ and $D$, correspondingly, as $$\begin{split}
((a,b))_{D_T} &= \int_{D} \int_0^T a b ~ dx dt,~ \|a \|_{L_2(D_T)}^2 = ((a,a))_{D_T}, \\
(a,b)_{D} &= \int_{D} a b ~dx,~ \| a \|_{L_2(D)}^2 = (a,a)_{D}.
\end{split}$$
**Theorem**
*Assume that the condition (\[coefic\]) for the function $\tilde{c}(x)$ holds. Let* $D \subset
\mathbb{R}^{n}, n=2,3,$* be a bounded domain with a piecewise smooth boundary* $\partial D$. * For any* $ t\in \left(
0,T\right) $* we define* $D_{t}= \partial_1 D\times \left( 0,t_1\right)
.$ * Assume that there exists a solution* $u$* of the problem (\[model1\]). Then $u \in
H^{1}(D_{T})$ is unique and there exists a positive constant* $A=A(\| \tilde{c} \|_{D}, t)$ * such that the following energy estimate is true for all $t \in (0,T ]$*
$$\label{estimate1}
\begin{split}
\left \Vert \sqrt{\tilde{c}}~ \partial _{t} u(x,t)
\right\Vert_{L_{2}\left( D \right) }^{2}
&+ \left \Vert \nabla u (x,t) \right\Vert _{L_{2}\left( D \right) }^{2} \\
& \leq A \left[ \left\Vert p(x,t) \right\Vert _{L_{2}\left( D _{t}\right)}^{2}+
\left\Vert \nabla f_{0}\right\Vert _{L_{2}\left( D\right) }^{2}\right] .
\end{split}$$
**Proof.**
A proof of this theorem follows from the stability estimate given in [@hybrid]. $\square$
The stability result for the adjoint problem is obtained similarly as for the forward problem, the only difference is in the integration in time $(t, T)$. **Theorem**
*Assume that the condition (\[coefic\]) for the function $\tilde{c}(x)$ holds. Let* $D \subset
\mathbb{R}^{n}, n=2,3$* be a bounded domain with a piecewise smooth boundary* $\partial D$. * For any* $ t\in \left(
0, T\right) $* we define by* $D_{t_a}= (\partial_1 D \cup \partial_2 D) \times \left( t, T\right)
.$ * Assume that there exists a solution* $ \lambda $* of the problem (\[adjoint1\]) and a solution* $u $* of the problem (\[model1\]). Then $\lambda \in
H^{1}(D_{T})$ is unique and there exists a positive constant* $B=B(\| \tilde{c} \|_{D}, t)$ * such that the following energy estimate is true for all $t \in (0,T ]$*
$$\label{estimate2}
\begin{split}
\left \Vert \sqrt{\tilde{c}}~ \partial _{t} \lambda(x,t)
\right\Vert_{L_{2}\left( D \right) }^{2}
+ \left \Vert \nabla \lambda (x,t) \right\Vert _{L_{2}\left( D \right) }^{2}
\leq B \left\Vert ( u - \tilde{u}) z_{\delta } \right\Vert _{L_{2}\left( D _{t_a}\right)}^{2}.
\end{split}$$
**Proof.**
We multiply the equation in (\[adjoint1\]) by $2 \partial_t \lambda$ and integrate over $ D\times \left( t, T\right)$ to get $$\label{eq1_mod4adj}
\begin{split}
&\int \limits_{t}^{T}\int\limits_{D} 2~\tilde{c}~
\partial_{tt} \lambda ~\partial_t \lambda ~ dxd\tau
- \int \limits_{t}^{T}\int\limits_{D} 2
\nabla \cdot ( \nabla \lambda)~\partial_t \lambda~ dxd\tau \\
&=
- 2\int\limits_{t}^{T}\int\limits_{\partial_1 D \cup \partial_2 D} (u - \tilde{u}) z_{\delta } ~\partial_t \lambda ~dSd\tau.
\end{split}$$
Next, we integrate by parts in time the first term of (\[eq1\_mod4adj\]) and noting zero initial condition in (\[adjoint1\]), we have $$\label{eq4_time1adj}
\begin{split}
\int\limits_{t}^{T} \int\limits_{D} \partial _{t}
\left( \tilde{c} \partial_t \lambda^{2} \right) dxd\tau
= -\int\limits_{D}\left( \tilde{c}\partial_t \lambda^{2} \right) \left( x,t\right) dx.
\end{split}$$
Next, we integrate by parts in space the second term of (\[eq1\_mod4adj\]). From (\[coefic\]) it follows that $\tilde{c}=1$ on $\partial D$. Thus, using (\[coefic\]) and absorbing boundary condition in (\[adjoint1\]), we get $$\label{grad1adj}
\begin{split}
&2\int\limits_{t}^{T}\int\limits_{D} \nabla \cdot \left(\nabla \lambda \right) \partial_t \lambda dx d\tau
=
2\int\limits_{t}^{T}\int\limits_{\partial D}\left( \partial_t \lambda
\right) \partial_n \lambda dS d\tau \\
&-
2 \int\limits_{t}^{T}\int\limits_{D}\left( \nabla \lambda \right) \left(
\nabla \partial_t \lambda \right) dxd\tau \\
&=
2 \int\limits_{t}^{T}\int\limits_{\partial_1 D \cup \partial_2 D}\left( \partial_t \lambda
\right)^2 ~dS d\tau - \int\limits_{t}^{T}\int\limits_{D} \partial _{t}|\nabla \lambda |^{2}dx d\tau.
\end{split}$$
Integrating last term of (\[grad1adj\]) in time and using initial conditions of the equation (\[adjoint1\]), we obtain $$\label{grad1_1adj}
\begin{split}
&\int\limits_{t}^{T}\int\limits_{D} \partial _{t} |\nabla \lambda| ^{2}dxd\tau
=\int\limits_{D} | \nabla \lambda| ^{2}\left( x,T\right) dx
- \int\limits_{D} |\nabla \lambda |^{2}\left( x, t\right) dx \\
&= -\int\limits_{D} | \nabla \lambda|^{2}\left( x,t\right) dx.
\end{split}$$
We insert (\[eq4\_time1adj\])-(\[grad1\_1adj\]) in (\[eq1\_mod4adj\]) to get $$\label{eq_mainadj1}
\begin{split}
&-\int\limits_{D}\left( \tilde{c} \partial_t \lambda^{2} \right) \left(
x,t\right) dx - \int\limits_{D} | \nabla \lambda |^{2}\left( x,t\right)
dx \\
&= 2 \left (\int\limits_{t}^{T}\int\limits_{\partial_1 D \cup \partial_2 D}\left(\partial_t
\lambda \right)^2 -
(u - \tilde{u} )z_{\delta } ~\partial_t \lambda \right)~~dS d\tau.
\end{split}$$ The equation above can be rewritten as $$\label{eq_mainadj2}
\begin{split}
&\int\limits_{D}\left( \tilde{c} \partial_t \lambda^{2} \right) \left(
x,t\right) dx + \int\limits_{D} | \nabla \lambda |^{2}\left( x,t\right)
dx \\
&= 2 \left (\int\limits_{t}^{T}\int\limits_{\partial_1 D \cup \partial_2 D}
( u -\tilde{u})z_{\delta } ~\partial_t \lambda
- \left(\partial_t
\lambda \right)^2\right )~~dS d\tau.
\end{split}$$
Young’s inequality applied to directly leads to $$\label{mod4_5adj}
\begin{split}
&\int\limits_{D}
\left(\tilde{c} \partial_t \lambda ^{2} + \left| \nabla \lambda \right|^{2} \right)(x,t)~ dx \\
&\leq
B \int\limits_{t}^{T}\int\limits_{\partial_1 D \cup \partial_2 D} |(\tilde{u} - u) z_{\delta }|^{2}(x, \tau)~dSd\tau,
\end{split}$$ with a constant $B=0.5$ which is the desired result.
$\square$
The finite element method in $D_{FEM}$ {#sec:fem}
======================================
As was mentioned above for the numerical solution of (\[model1\]) we use the domain decomposition FE/FD method of [@hybrid]. Similarly with this work, in our computations we decompose the finite difference domain $D_{FDM}$ into squares, and the finite element domain $D_{FEM}$ - into triangles. In $D_{FDM}$ we use the standard finite difference discretization of the equation (\[model1\]) and obtain an explicit scheme as in [@hybrid].
For the finite element discretization of $D_{FEM}$ we define a partition $K_{h}=\{K\}$ which consists of triangles. We define by $h$ the mesh function as $h|_{K}=h_{K}$, where $h_K$ is the local diameter of the element $K$, and assume the minimal angle condition on the $K_{h}$ [@Brenner]. Let $J_{\tau }=\left\{ J\right\} $ be a partition of the time interval $(0,\,T)$ into subintervals $J=(t_{k-1},\,t_{k}]$ of uniform length $\tau =t_{k}-t_{k-1}$.
To solve the state problem (\[model1\]) and the adjoint problem (\[adjoint1\]) we define the finite element spaces, $W_{h}^{u}\subset H_{u}^{1}\left( Q_{T}\right) $ and $W_{h}^{\lambda
}\subset H_{\lambda }^{1}\left( Q_{T}\right) $. First, we introduce the finite element trial space $W_{h}^{u}$ $$\begin{split}
W_{h}^{u} := &\{w\in H_{u}^{1}(Q_T):w|_{K\times J}\in P_{1}(K)\times
P_{1}(J), \\
&\forall K\in K_{h},~\forall J\in J_{\tau }\},
\end{split}$$ where $P_{1}(K)$ and $P_{1}(J)$ denote the set of linear functions on $K$ and $J$, respectively. We also introduce the finite element test space $W_{h}^{\lambda }$ as $$\begin{split}
W_{h}^{\lambda }:= &\{w\in H_{\lambda }^{1}(Q_T):w|_{K\times J}\in
P_{1}(K)\times P_{1}(J), \\
& \forall K\in K_{h},~\forall J\in J_{\tau }\}.
\end{split}$$ To approximate the function $\tilde{c}$, we use the space of piecewise constant functions $C_{h}\subset L_{2}\left( D \right) $, $$C_{h}:=\{u\in L_{2}(D ):u|_{K}\in P_{0}(K),~\forall K\in K_{h}\},
\notag$$where $P_{0}(K)$ is the set of constant functions on $K$.
Setting $V_{h}=W_{h}^{u}\times W_{h}^{\lambda }\times C_{h}$, the finite element method for (\[scalar\_lagr1\]) now reads: *Find* $v_{h}\in V_{h}$*, such that* $$L^{\prime }(v_{h})(\bar{v})=0, ~\forall \bar{v}\in V_{h}. \notag$$
To find approximate solution $v_h \in V_{h}$ we need to solve the forward problem , the adjoint problem and then find the discrete gradient $L^{\prime }_{\tilde{c}}(v_{h})$. For the fully discrete schemes of these equations we refer to [@hybrid].
Adaptive conjugate gradient algorithm {#sec:ad_alg}
=====================================
To compute minimum of the functional (\[functional\]) we use the adaptive conjugate gradient method (ACGM). The regularization parameter $\gamma$ in ACGM is computed iteratively via rules of [@BKS]. For the local mesh refinement we use a posteriori error estimate of [@B; @BJ] which means that the finite element mesh in $D_{FEM}$ should be locally refined where the maximum norm of the Fréchet derivative of the Lagrangian with respect to the coefficient is large. However, since our goal is to design material inside the known domain $G_1$, we refine mesh only inside this domain.
Now we define $$\label{Bhm}
\begin{split}
{g}^m(x) = - {\int_0}^T \frac{\partial \lambda_h^m}{\partial t}
\frac{\partial E_h^m}{\partial t}~ dt + \gamma^m (\tilde{c}_h^m - \tilde{c}_0),
\end{split}$$ where $\tilde{c}_{h}^{m}$ is approximation of the function $\tilde{c}_{h}$ on the iteration step $m$ in AGCM, $E_{h}\left(
x,t,\tilde{c}_{h}^{m}\right) ,\lambda _{h}\left( x,t,\tilde{c}_{h}^{m}
\right) $ are computed by solving the state problem (\[model1\]) and the adjoint problem (\[adjoint1\]), respectively, with $\tilde{c}:=\tilde{c}_{h}^{m}$. In our computations of section \[sec:numex\] we use the following algorithm.
**Algorithm (AGCM)**
- Step 0. Set number of mesh refinements $j:=0$. Choose initial mesh $K_{h}^j$ in $D_{FEM}$ and time partition $J_{\tau}^j$ of the time interval $\left( 0,T\right)$ as described in section \[sec:fem\]. Start with the initial approximation $\tilde{c}_{h}^{0}= \tilde{c}_0$ at $K_{h}^0$ and compute the sequences of $\tilde{c}_{h}^{m}$ via the following steps:
- Step 1. Compute solutions $E_{h}\left(
x,t,\tilde{c}_{h}^{m}\right) $ and $\lambda _{h}\left(
x,t,\tilde{c}_{h}^{m}\right) $ of state (\[model1\]) and adjoint (\[adjoint1\]) problems, respectively, on $K_{h}^j$ and $J_{\tau}^j$.
- Step 2. Update the coefficient $\tilde{c}_h:=\tilde{c}_{h}^{m+1}$ on $K_{h}^j$ (only inside the discretized domain $G_1$) and $J_{\tau}^j$ using the conjugate gradient method $$\label{cgm}
\begin{split}
\tilde{c}_h^{m+1} &= \tilde{c}_h^{m} + \alpha^m d^m(x),
\end{split}$$ where $$\begin{split}
d^m(x)&= -g^m(x) + \beta^m d^{m-1}(x),
\end{split}$$ with $$\begin{split}
\beta^m &= \frac{\| g^m(x)\|^2}{\| g^{m-1}(x)\|^2},
\end{split}$$ where $d^0(x)= -g^0(x)$. In (\[cgm\]) the step size $\alpha$ in the gradient update is computed as $$\alpha^m = -\frac{((g^m, d^m)) }{\gamma^m {{\left\Vertd^m\right\Vert}}^2},$$ and the regularization parameter $\gamma^m$ at iteration $m$ is computed iteratively accordingly to [@BKS] as $$\label{iterreg}
\gamma^m = \frac{\gamma_0 }{ (m+1)^p},~~ p \in (0,1).$$
- Step 3. Stop computing $\tilde{c}_{h}^{m}$ and obtain the function $\tilde{c}_h$ at $M=m$ if either $\| g^{m}\|_{L_{2}( D_{FEM})}\leq
\theta$ or norms $\|g^{m}\|_{L_{2}(D_{FEM})}$ are stabilized. Here $\theta$ is the tolerance in updates $m$ of gradient method. Otherwise set $m:=m+1$ and go to step 1.
- Step 4. Refine the mesh $K_h^j$ inside $G_1$ using symmetric mesh refinement procedure, for example, as shown in Figure \[fig:7\].
- Step 5. Set $j:=j+1$ and construct a new mesh $K_{h}^j$ in $D_{FEM}$ and a new partition $
J_{\tau}^j$ of the time interval $\left( 0,\,T\right)$ with the new time step $\tau $ which should be chosen correspondingly to the CFL condition of [@CFL67].
- Step 6. Interpolate the approximation $\tilde{c}_h$ computed on the step 3, from every element $K^{j-1}$ on the previous space mesh $K_h^{j-1}$ to the new elements $K^{j}$ in the mesh $K_h^{j}$, and obtain the initial guess $\tilde{c}_0$ on a new mesh. Set $m=1$ and return to step 1.
- Step 7. Stop refinements of $K_h^j$ and $J_{\tau}^j$ if norms defined in step 3 either increase or stabilize, compared to the previous space mesh.
Numerical Studies {#sec:numex}
=================
The goal of this section is to present possibility of the computational design of an acoustic structure with the property to generate as small reflections as possible. This problem is equivalent to **IP**. Thus, we will reconstruct a function $\tilde{c}(x)$ inside a domain $G_1$ using the ACGM algorithm of section \[sec:ad\_alg\]. We assume, that this function is known inside $D_{FDM} \cup G_2$ and is set to be $\tilde{c}(x)=1$.
Our computational geometry $D$ is split into two geometries $D_{FEM}$ and $D_{FDM}$ as described in section \[sec:modelhyb\], see Figure \[fig:0\_1\]. We denote by $\partial D_{FEM}$ the outer boundary of $D_{FEM}$ and by $\partial D_{FDM}$ the inner boundary of $D_{FDM}$. We set the dimensionless computational domain $D$ as $$D = \left\{ x= (x_1,x_2) \in (-1.1, 1.1) \times (-0.62,0.62)\right\},$$ and the domain $D_{FEM}$ as $$D_{FEM} = \left\{ x= (x_1,x_2) \in ((-1.0,1.0) \times (-0.52,0.52) \right\}.$$ The spatial mesh in $D_{FEM}$ and in $D_{FDM}$ consists of triangles and squares, respectively. We choose the initial mesh size $h=0.02$ in $D = D_{FEM} \cup D_{FDM}$, as well as in the contiguous regions between FE/FD domains. We also decompose the domain ${\mathrm{D}}_{FEM}$ into three different domains $G_0, G_1, G_2$ such that ${\mathrm{D}}_{FEM} = G_0 \cup G_1 \cup G_2$ which are intersecting only by their boundaries, see Figure \[fig:0\_1\]. The goal of our numerical tests is to reconstruct the function $\tilde{c}$ of the domain $G_{1}$ of Figure \[fig:0\_1\] which produces as small reflections as possible.
We initialize a plane wave in $D$ in time $T=[0,2.0]$ such that $$\label{f}
\begin{split}
p(t) =\left\{
\begin{array}{ll}
\sin \left( \omega t \right) ,\qquad &\text{ if }t\in \left( 0,\frac{2\pi }{\omega }
\right) , \\
0,&\text{ if } t>\frac{2\pi }{\omega }.
\end{array}
\right.
\end{split}$$
As for the forward problem in $D_{FDM}$ we solve the problem (\[model1\]) choosing $\tilde{c}=1$, and in $D_{FEM}$ we solve $$\label{3D_1}
\begin{split}
\tilde{c} \frac{\partial^2 u}{\partial t^2} - \triangle u &= 0~ \mbox{in}~~
D_{{FEM} \times (0,T)}, \\
u(x,0) = 0, ~~~u_t(x,0) &= 0~ \mbox{in}~~ D_{FEM}, \\
u(x,t)|_{\partial D_{FEM}} &= u(x,t)|_{\partial D_{{FDM}_I}},\\
\partial_n u &= 0~ \mbox{on}~~ \partial G_0.
\end{split}$$ Here, $\partial D_{{FDM}_I}$ denotes internal structured nodes of $D_{FDM}$ which have the same coordinates as structured nodes at the boundary $\partial D_{FEM}$, see details in [@hybrid]. We note, that we use the boundary condition $\partial_n u = 0$ on $\partial G_0$ which implies that waves are not penetrated into $G_0$.
We also note that in $D_{FDM}$ the adjoint problem will be the following wave equation with $\tilde{c}(x)=1$ for $ x \in D_{FDM}$: $$\label{adjwaveeq}
\begin{split}
\frac{\partial^2 \lambda}{\partial t^2} - \triangle \lambda &=
- (u- \tilde{u}) z_{\delta}~~\mbox{in}~~ D_{FDM} \times (0,T), \\
\lambda(x,T) = 0, ~~~\lambda_t(x,T) &=0~ \mbox{in}~~ D, \\
\lambda(x,t)|_{\partial D_{FDM}} &= \lambda(x,t)|_{\partial D_{{FEM}_{I}}}, \\
\partial _{n} \lambda(x,t)& =0~ \mbox{on}~S_3 \cup S_{1,1}, \\
\partial _{n} \lambda(x,t)& = \partial_t \lambda~ \mbox{on}~S_{1,2} \cup S_{2},
\end{split}$$ which we solve using finite difference method. In $D_{FEM}$ we solve the problem $$\label{adj3D_1}
\begin{split}
\tilde{c} \frac{\partial^2 \lambda}{\partial t^2} - \triangle \lambda &= 0~ \mbox{in}~~ D_{FEM} \times (0,T), \\
\lambda(x,T) = 0, ~~~\lambda_t(x,T) &= 0~ \mbox{in}~~ D_{FEM}, \\
\lambda(x,t)|_{\partial D_{FEM}} &= \lambda(x,t)|_{\partial D_{{FDM}_{I}}}, \\
\partial_n \lambda &= 0 ~ \mbox{on}~~ S_4, \\
\end{split}$$ using finite element method. Here, $\partial
D_{{FEM}_{I}}$ denotes internal structured nodes of $D_{FEM}$ lying on the inner boundary $\partial D_{FDM}$ of $D_{FDM}$, see details in [@hybrid] for the exchange procedure between FE/FD solutions.
As initial guess $\tilde{c}_0(x)$ we take different constant values of the function $\tilde{c}(x)$ inside domain of $G_1$ of Figure \[fig:0\_1\] on the coarse non-refined mesh, and we take $\tilde{c}(x)=1.0$ everywhere else in $D$. We choose three different constant values of $\tilde{c}_0(x)= \{1.5,2.0,2.5\}$ inside $G_1$. We define that the minimal and maximal values of the function $\tilde{c}(x)$ belong to the following set $M_{\tilde{c}} $ of admissible parameters $$\label{admpar}
\begin{split}
M_{\tilde{c}} := \left \{\tilde{c}\in C(\overline{D })| 1 \leq \tilde{c}(x)\leq \max_{G_1} \tilde{c}_0(x) \right \}.
\end{split}$$ The time step is chosen to be $\tau=0.002$ which satisfies the CFL condition [@CFL67].
Reconstructions
---------------
We generate data at the observation points at $S_{T}$ by solving the forward problem (\[model1\]) in the time interval $t=[0,2.0]$, with function $p(t)$ given by (\[f\]) and for different values of $\omega= \{40,60,80,100\}$. To generate non-reflected data $\tilde{u}$ at $S_{T}$ we take the function $\tilde{c}(x)=1$ for all $x$ in $D$ and solve the problem (\[model1\]) with a plane wave (\[f\]) and $\omega= \{40,60,80,100\}$. We regularize the solution of the inverse problem by starting computations with regularization parameter $\gamma=0.01$ in (\[functional\]) and then updating this parameter iteratively in ACGM by formula (\[iterreg\]). Computing the regularization parameter in this way is optimal for our problem. We refer to [@Engl] for different techniques for choice of a regularization parameter.
Figure \[fig:2\] shows real part of the Fourier transform of the time-dependent solution $u(x,t)$ of when the initial guess for $\tilde{c}$ was $\tilde{c}_0=1.5$ in all points of $G_1$ (left figures), and after application of the adaptive optimization algorithm on three times refined mesh in $G_1$ (right figures) for different values of $\omega$ in (\[f\]). All right figures in Figure \[fig:2\] show significant reduction of backscattered reflections for all tested frequencies compared with left figures.
Figures \[fig:4\], \[fig:5\] present reconstructions of $\tilde{c}$ which we have obtained on three time adaptively refined mesh inside the domain $G_1$ for different values of $\omega$ in (\[f\]). We note that different initial guesses $\tilde{c}_0$ in produce different symmetric structures inside $G_1$ with different values of the function $\tilde{c}(x)$, compare reconstructions presented on Figures \[fig:7\]. Left images of Figure \[fig:7\] present reconstructions obtained in ACGM when the optimized function $\tilde{c}$, obtained on a coarse mesh, is sequentially interpolated on the one, two and three times refined mesh. Then this interpolated function is taken as an initial guess $\tilde{c}_0$ in and optimized further to get reconstruction on the third refined mesh. Right images of Figure \[fig:7\] are obtained after direct application of the adaptive algorithm of Section \[sec:ad\_alg\]. Optimized values of $\tilde{c}(x)$ obtained on Figures \[fig:4\]–\[fig:7\] can be of physical interest since they present symmetric structured domains with almost the same material in every structured layer.
Acknowledgments {#acknowledgments .unnumbered}
===============
The research of L.B. is supported by the sabbatical programme at the Faculty of Science, University of Gothenburg. The research of E.S. is supported by the Ministry of Education and Science of the Russian Federation, Project No. 1.894.2017/$\Pi$.
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a\) $\omega= 40$ b\) $\omega= 40$
c\) $\omega= 60$ d\) $\omega= 60$
e\) $\omega= 80$ f\) $\omega= 80$
g\) $\omega= 100$ h\) $\omega= 100$
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[ a) $\omega = 40, j=3$]{} [zoomed]{}
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[^1]: Department of Mathematical Sciences, Chalmers University of Technology and Gothenburg University, SE-42196 Gothenburg, Sweden, e-mail: `larisa@chalmers.se`
[^2]: Department of Mathematics and Supercomputing, Penza State University, Penza, Russia, e-mail: ` smolkin@chalmers.se`
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---
abstract: 'Polarized neutron diffraction allows to determine the local susceptibility tensor on the magnetic site both in single crystals and powders. It is widely used in the studies of single crystals, but it is still hardly applicable to a number of highly interesting powder materials, like molecular magnets or nanoscale systems because of the low luminosity of existing instruments and the absence of an appropriate data analysis software. We show that these difficulties can be overcome by using a large area detector in combination with the two-dimensional Rietveld method and powder samples with magnetically induced preferred crystallite orientation. This is demonstrated by revisiting two test powder compounds, namely, low anisotropy (soft) ferrimagnetic compound Fe~3~O~4~ and spin-ice compound Ho~2~Ti~2~O~7~ with high local anisotropy. The values of magnetic moments in Fe~3~O~4~ and the susceptibility tensors of Ho~2~Ti~2~O~7~ at various temperatures and fields were found in perfect agreement with these found earlier in single crystal experiments. The magnetically induced preferred crystallite orientation was used to study the local susceptibility of a single-molecule magnet Co(\[(CH~3~)~2~N\]~2~CS)~2~Cl~2~. Hence, the studies of local magnetic anisotropy in powder systems might now become accessible.'
author:
- 'I.A.Kibalin$^{1, 2}$ and A.Gukasov$^{1}$'
bibliography:
- 'bibliography.bib'
title: 'Local magnetic anisotropy by polarized neutron powder diffraction: application of magnetically induced preferred crystallite orientation'
---
Introduction
============
Polarized neutron diffraction (PND), also called “flipping ratio method” is a powerful tool to investigate intra- or intermolecular magnetic interactions. It gives a direct access to the magnetization distribution in the unit cell[@Gillon_2012], permits separating the spin and orbital contributions[@Schweizer_2008] and allows determining the local susceptibility tensor on the magnetic sites[@Gukasov_2002_jpcm]. The magnetization distribution has contributed to the understanding of magnetic interactions by revealing the spin delocalization, the spin density distribution, and the wave functions of unpaired electrons[@Kibalin_2017]. In turn, the local susceptibility approach has been successfully used in recent studies of field induced magnetic order in R~2~Ti~2~O~7~ pyrochlore compounds with either uniaxial or planar anisotropy[@Cao_2008; @Cao_2009]. PND is becoming a reference in mapping the magnetic anisotropy at the atomic scale in molecular magnets[@Ridier_2015; @Klahn_2018]. Unfortunately PND currently applies only to single crystals, which makes it inadequate for a number of highly interesting topics due to the difficulties encountered in growing sufficiently large samples.
Motivated by challenging scientific subjects, several attempts have been performed to investigate magnetized powder samples with polarized neutrons[@LelievreBerna_2010; @Wills_2005; @Hiraka2014; @Rivin_2013]. This allowed to reveal magnetic moments of iron at different crystallographic sites in prussian blue[@Wills_2005] and in $\alpha$-Fe~16~N~2~ nanoparticles[@Hiraka2014], as well as, to amend the magnetic structure of highly anisotropic TbCo~2~Ni~3~[@Rivin_2013]. The validity of the method was illustrated by measurements of magnetic anisotropy in a polycrystalline sample of Tb~2~Sn~2~O~7~[@Gukasov_2010]. As a proof of concept the local susceptibility parameters of Tb were found by a two steps procedure. First, the integrated intensities of the spin-up and spin-down components $I_{+}(hkl)$ and $I_{-}(hkl)$ were obtained by profile matching from the corresponding powder patterns. Then, the program CHILSQ of the Cambridge Crystallography Subroutine Library[@Brown_1993] was used to fit the integrated intensities. It is clear that such a procedure of data treatment can be applied only to highly symmetric crystal structures with small unit cell. For more complex structures a Rietveld method needs to be developed.
Rietveld analysis has become mandatory in powder diffraction for nuclear and magnetic structure refinement[@Rodriguez-Carvajal1993; @Larson_1994; @Petricek_2014]. It refines various metrics, including lattice parameters, structure and magnetic parameters, a preferred orientation to derive a calculated diffraction pattern. Once the calculated pattern becomes nearly identical to an experimental one, various properties pertaining to that sample can be obtained. However, the Rietveld method for polarized neutron powder diffraction (PNPD) has not been implemented yet. In the above-mentioned polarized powder experiments special softwares (model dependent) were developed for the data treatment.
We note that at first PNPD measurements were performed on conventional powder diffractometers equipped by one-dimensional (1D) detectors while modern unpolarized neutron powder diffractometers (Super-D2B, D20, SPODI) at reactor sources are equipped with two-dimensional (2D) detectors. Area detectors increase the efficiency of the instrument by an order of magnitude, but the common approach at these instruments consists in reducing the accumulated 2D data from area detector into 1D diffraction pattern by “unbending” measured Debye cones. The resulting pattern is then treated using standard 1D Rietveld refinement[@Petricek_2014; @Rodriguez-Carvajal1993; @Larson_1994]. Most recent powder diffractometers at advanced neutron spallation sources (WISH, POWGEN) use very large area detectors and operate in TOF mode. This generates rather complex three-dimensional (3D) angular- and wavelength-dispersive data which are eventually transformed into one-dimensional diffraction patternI(2$\theta$) (or I($\lambda$))[@mantid] to allow standard Rietveld refinement. It has been noted that two-dimensional extension of Rietveld method for neutron TOF powder diffraction taking into account the variation of diffraction angle2$\theta$ and wavelength$\lambda$ decreases the number of data-reduction steps and avoids the loss of high-resolution information [@Jacobs_2015], but full scale multidimensional Rietveld software for neutron TOF powder diffraction still need to be developed.
Since area detectors increase considerably the efficiency of the instruments, we performed our PNPD measurements on diffractometers equipped with large 2D position sensitive detectors. When using area detectors, the polarized neutron scattering is a function of2$\theta$ and $\varphi$ but also of the angle between the magnetic field and the scattering vector. Moreover, neutrons are sensitive only to the magnetic moment perpendicular to the scattering vector. Therefore, the variation of intensity along the Debye cones can be used for the separation of nuclear and magnetic scattering contributions. For these reasons the transformation of angular-dispersive polarized neutron data from area detectors into one-dimensional 2$\theta$ pattern is not applicable. We note as well that the equation for powder averaging derived in the paper[@Gukasov_2010] is valid only for the vertical field and scattering in the horizontal plane. Here we give an expression for powder averaging valid for general scattering geometry, which allows an implementation of the full scale 2D Rietveld method in PNPD.
Another possibility of increasing the efficiency of PNPD consists in using magnetically induced preferred crystallite orientation. This technique can be applied to biaxial crystals in which the magnetic susceptibility tensor has different principal values (i. e. orthorhombic, monoclinic and triclinic systems)[@Kimura_2014; @Stekiel2015]. Under a strong magnetic field the crystallites overcome the steric hindrence of powder packing and align their easy magnetization axis parallel to the applied magnetic field, leading to crystallite preferred orientation. As a consequence, different reflections with similar Bragg angles $2\theta$ appear at different angles along the Debye cones. No overlapping of these reflections occurs, which allows to use diffractometers with low resolution (hence, high luminosity) for powder diffraction.
Here we show that the combination of a large area detector with 2D Rietveld analysis and magnetically induced preferred crystallite orientation enables PNPD in systems not available as single crystals. We illustrate this by the results of two test cases of magnetic materials; low anisotropy (soft) ferrimagnetic compound Fe~3~O~4~ and spin-ice compound Ho~2~Ti~2~O~7~ with high local anisotropy. We show that in both cases the combination of area detector with 2D Rietveld method shortens the acquisition time by an order of magnitude, without loosing the precision of parameters evaluation. Finally, we present the results of the local susceptibility studies on the single-molecule magnet Co(\[(CH~3~)~2~N\]~2~CS)~2~Cl~2~ with magnetically induced preferred orientation of crystallites, which shows that the PNPD now opens large opportunities in the local anisotropy quantification of complex structures.
Polarized neutron powder diffraction
====================================
It is well established that the flipping sum and difference of the integrated intensities ($I_{+}$ and $I_{-}$) of pollycrystalline samples are proportional to[@Gillon2007]
$$I_{+}+I_{-}\sim\left|N\right|^{2}+\left\langle \left|\vec{M}_{\perp}\right|^{2}\right\rangle ,\label{eq:flip_iint_sum}$$
$$I_{+}-I_{-}\sim N^{*}\left\langle\left( \vec{M}_{\perp}\cdot\vec{P}\right)\right\rangle +N\left\langle \left(\vec{M}_{\perp}^{*}\cdot\vec{P}\right)\right\rangle,\label{eq:flip_iint_diff}$$
where $N$ is the nuclear structure factor, $\vec{M}_{\perp}$ is the projection of the magnetic structure factor $\vec{M}(\vec{k})$ perpendicular to the scattering vector $\vec{k}$. $\vec{M}$ is induced by the magnetic field $\vec{H}$ applied in the vertical direction (figure\[fig:For-different-crystallites\]) and $\vec{P}$ is the neutron polarization vector parallel to $\vec{H}$. Angle brackets show the powder averaging over scattering crystallites.
![(Color online) The principal scheme of the scattering at the pollycrystalline sample. A notation is explained in the text. \[fig:For-different-crystallites\]](fig_1.pdf){width="7cm"}
In soft magnetic materials the atomic magnetic moments $\vec{M}_{a}$ are directed along the applied field $\vec{H}$. Thus, the powder averaging of $\left|\vec{M}_{\perp}\right|^{2}$, $\left( \vec{M}_{\perp}\cdot\vec{P}\right) $ can be written as:
$$\left\langle \left|\vec{M}_{\perp}\right|^{2}\right\rangle =\left|\sum_{a}M_{a}\sin\alpha f_{a}(\vec{k})\exp\left[2\pi i\vec{k}\cdot\vec{r}_{a}\right]\right|^{2}\label{eq:m_perp_sq_aver_smm}$$
and
$$\left\langle \left(\vec{M}_{\perp}\cdot\vec{P}\right)\right\rangle =\sum_{a} M_{a}P\sin^{2}\alpha f_{a}(\vec{k})\exp\left[2\pi i\vec{k}\cdot\vec{r}_{a}\right],\label{eq:m_perp_p_aver_smm}$$
where the sum over $a$ includes all atoms in the unit cell with radius vector $\vec{r}_{a}$, $f_{a}(\vec{k})$ is the magnetic form factor in spherical approximation[@IntTablesHB2004].
For paramagnets and diamagnets the structure factor $\vec{M}$ can be written as[@Gukasov_2010]:
$$\vec{M}(\vec{k})=\sum_{a}\frac{1}{N_{a}}f_{a}(\vec{k})\sum_{p}R_{p}\chi_{a}R_{p}^{-1}\vec{H}e^{2\pi i\vec{k}(R_{p}\vec{r}_{a}+\vec{t}_{p})},\label{eq:SFT-1}$$
where the sum over $a$ includes all independent atoms, the sum over $p$ includes those generated from atom $a$ by the $N_{g}$ symmetry operators $\left\lbrace R_{p}:\vec{t}_{p}\right\rbrace$ of the space group $\mathcal{G}$. $N_{a}$ is the number of operators $q$ in $\mathcal{G}$ for which $R_{q}\vec{r}_{a}+\vec{t}_{q}=\vec{r}_{a}$; $N_{g}/N_{a}$ is the multiplicity of the site $a$ which has point symmetry $Q_{a}$ generated by the rotational parts of the operators $q$. This implies that the local susceptibility tensor $\chi_{a}$ is invariant to the rotations in $Q_{a}$ so that $R_{q}\chi_{a}R_{q}^{-1}=\chi_{a}$ for all $R_{q}$ in $Q_{a}$. The number of independent components of the tensor $\chi_{a}$ varies from two for uniaxial site symmetries to six for triclinic ones.
The structure factor tensor $\chi(\vec{k})$, which is independent of the magnitude and direction of the applied field can be expressed as follows:
$$\chi(\vec{k})=\sum_{a}\frac{1}{N_{a}}f_{a}(\vec{k})\sum_{p}R_{p}\chi_{a}R_{p}^{-1}e^{2\pi i\vec{k}(R_{p}\vec{r}_{a}+\vec{t}_{p})}.\label{eq:SFT}$$
Expression for the powder averaging of $\left|\vec{M}_{\perp}\right|^{2}$ and $\left(\vec{M}_{\perp}\cdot\vec{P}\right)$ terms in the case of magnetic field applied vertically and the detector in the horizontal plane has been given in the Ref.[@Gukasov_2010]. It can be shown (see Supplemental Material) that this expression can be generalized for any scattering geometry. Namely, the structure factor tensor is to be transformed into a Cartesian coordinate system with the $z$ axis parallel to the scattering vector. If the transformation is expressed through the matrix$T$ (see Supplemental Material) the components of the tensor become $\Sigma=T\cdot\chi\cdot T^{-1}$ and the averaged terms above can be written as follows
$$\begin{array}{c}
\left\langle \left|\vec{M}_{\perp}\right|^{2}\right\rangle =\frac{1}{2}H^{2}\left[\left(\Sigma_{11}^{2}+2\Sigma_{12}^{2}+\Sigma_{22}^{2}\right)\sin^{2}\alpha+\right.\\
\left.+2\left(\Sigma_{13}^{2}+\Sigma_{23}^{2}\right)\cos^{2}\alpha\right]
\end{array}\label{eq:m_perp_sq_aver}$$
$$\left\langle \left( \vec{M}_{\perp}\cdot\vec{P}\right)\right\rangle =PH\left(\frac{\Sigma_{11}+\Sigma_{22}}{2}\right)\sin^{2}\alpha.\label{eq:m_perp_p_aver}$$
Here $\cos^{2}\alpha=\cos^{2}\theta\sin^{2}\phi$. These equations describe the scattering along the Debye cones in the 2D Rietveld refinement. We note that for the special case of scattering in the equatorial plane ($\phi=0$) the expressions (\[eq:m\_perp\_sq\_aver\], \[eq:m\_perp\_p\_aver\]) are in exact accordance with these given before[@Gukasov_2010].
2D-diffraction profile
----------------------
In the two-dimensional case the calculated intensity $y_{\pm}(2\theta,\phi)$ for a single-phase diffraction pattern can be expressed for every data point by
$$y_{\pm}(2\theta,\phi)=S\sum_{h}m_{h}L_{f}P_{h}I_{\pm}(\alpha)\psi_{h}(2\theta-2\theta_{h},\phi)+b(2\theta,\phi),
\label{eq:profile}$$
where $S$ is a scale factor, $m_{h}$ is the multiplicity of reflection, $L(\theta,\phi)$ is the Lorentz factor, $P_{h}$ is the density of (hkl) poles at the scattering vector (preferred orientation), $\psi_{h}(2\theta-2\theta_{h},\phi)$ is the peak profile function normalized to unit area, and $b(2\theta,\phi)$ is the background. The summation is done over all $h$ reflections for each data point. For a cylindrical detector[@Norby_1997] the Lorentz factor is $\sqrt{1-\sin^{2}2\theta\sin^{2}\phi}/\sin^{2}\theta\cos\theta$. In the case of one-dimensional Rietveld refinement the profile function is usually described by the pseudo-Voight function. For the two-dimensional description of the diffraction pattern, an appropriate profile function still needs to be found. Here we used the standard one-dimensional expression for the profile function $\psi_{h}(2\theta-2\theta_{h})$ neglecting the dependence of the peak profile from the polar angle $\phi$ . For these reason the part of the diffraction pattern with strong dependence of the peak profile from the polar angle $\phi$ was excluded from the refinement procedure.
It has been suggested[@Hiraka2014; @LelievreBerna_2010] that in the PNPD better quality information can be derived by using the flipping difference data, as contamination from the cryomagnet and sample is largely eliminated in the difference. However, we note that a simultaneous refinement of the sum and the difference patterns is mandatory for the scaling of magnetic moment values. Moreover, as has been noted in Ref.[@Gukasov_2010], in the cases of strong magnetic scatters with high anisotropy the sum patterns might contain a number of purely magnetic reflections which do not depend on neutron polarization.
Experiment and data treatment
-----------------------------
Neutron diffraction studies were performed at the Orphée 14MW reactor of the Laboratory Léon Brillouin, CEA Saclay. The diffraction patterns were collected on the diffractometer 5C1, equipped by position sensitive detector with cylindrical geometry covering $80^{\circ}$ and $25^{\circ}$ in horizontal and vertical directions, using neutrons of wavelength $\lambda=0.84$Å obtained with a Heusler alloy monochromator. The incident beam polarization $P$ is $0.91$.
PNPD data were collected on powder sample of Fe~3~O~4~ in an external field of 0T and 6T below the Verwey transition (at 10K) and above it (at 150K). The experiments with sintered powder sample of Ho~2~Ti~2~O~7~ were performed in the temperature range from 5K up to 50K in the magnetic field of 1T.
Measurements of the Co(II) complex with single-molecule magnet behavior have been carried out on the thermal polarized neutron lifting counter diffractometer 6T2 (LLB-Orphée, Saclay). Neutrons of wavelength 1.4Å were monochromated by a vertically focusing graphite crystal and polarized by a supermirror bender. The polarization factor of the beam was 0.95. The position sensitive detector has a flat geometry. Data treatment was performed using the newly developed 2D Rietveld software RhoChi[^1].
Soft ferrimagnetic Fe~3~O~4~
============================
Magnetite Fe~3~O~4~, as an original magnetic material with modern applications ranging from spintronics to MRI contrast agents was chosen as an example of soft (low anisotropy) ferrimagnetic for the software benchmarking. At ambient temperatures it orders in inverted cubic spinel ferrite with the tetrahedral ($A$) site occupied by Fe^3+^ ions and with Fe^2+^ and Fe^3+^ ions coexisting at the same octahedral ($B$) site[@Hamilton_1958]. Magnetite undergoes a first-order transition below 120K where the resistivity increases by two orders of magnitude and the structural distortions from cubic symmetry occur[@Okamura_1932; @Ellefson_1934]. It is suggested that this transition is driven by a charge ordering of Fe^2+^ and Fe^3+^ ions[@VERWEY_1939]. Polarized neutron diffraction measurements performed on the single crystal of magnetite earlier has shown the antiparallel orientation of the moments at the tetrahedral and octahedral sites but surprisingly no difference between the magnetic moments at the sites was found[@Rakhecha_1978].
![(Color online) Flipping sum diffraction patterns collected on Fe~3~O~4~ at $T=150$K, $H=0$T. The measured 2D pattern is shown on the top, the calculated is shown on the bottom (a), chi squares normalized per number of points is 5.87. Diffraction profile estimated near the equatorial plane (b). $\gamma$ is azimuthal angle and $\nu$ is elevation angle in the laboratory coodinate system $(xyz)$, where $\vec{x}||\vec{k}_{i}$, $\vec{z}||\vec{H}$\[fig:(Color-online)-Flipping\].](fig_2a.eps){width="8cm" height="4.36cm"}
![(Color online) Flipping sum diffraction patterns collected on Fe~3~O~4~ at $T=150$K, $H=0$T. The measured 2D pattern is shown on the top, the calculated is shown on the bottom (a), chi squares normalized per number of points is 5.87. Diffraction profile estimated near the equatorial plane (b). $\gamma$ is azimuthal angle and $\nu$ is elevation angle in the laboratory coodinate system $(xyz)$, where $\vec{x}||\vec{k}_{i}$, $\vec{z}||\vec{H}$\[fig:(Color-online)-Flipping\].](fig_2b.eps){width="8cm" height="4.36cm"}
In the absence of a magnetic field the flipping sum diffraction pattern corrected for background is presented in figure\[fig:(Color-online)-Flipping\] together with standard 1D diffraction pattern limited to the equatorial plane. One can see that the scattering intensity distribution along the Debye cones is rather homogeneous and the width of the cones increases with the $\phi$ angle.
Figure\[fig:BReflection111\] shows the $\phi$ dependence of the halfwidth $H_{pV}$ and of the integrated intensity for (111) ($2\theta_{h}=9.42^{\circ}$) reflection. As seen from the figure the width $H_{pV}$ remains approximately constant in the angular range from $0{^\circ}$ to $20^{\circ}$ and strongly increases at higher angles. Therefore the angular range from $0^{\circ}$ to $20^{\circ}$ was used in the refinement. In the meantime we note that the integrated intensity of the (111) reflection remains constant along the whole Debye cone, which is due to the fact that the magnetic moments are randomly oriented.
![(Color online) The distribution of $H_{pV}$ and the sum of the integrated intensities for the reflection $(111)$ measured with the magnetic field 6T (triangles) and without it (circles) at the diffractometer 5C1 at 150K over the Debye cone ($\phi=0{^\circ}$ correspond to the scattering in the equatorial plane). The model values after 2D Rietveld refinement are given by the dotted lines.\[fig:BReflection111\]](fig_3.eps){width="7cm" height="7cm"}
After refinement by the Rietveld method using equations (\[eq:m\_perp\_sq\_aver\_smm\], \[eq:m\_perp\_p\_aver\_smm\]) the magnetic moments of iron in tetrahedral and octahedral positions at 150K and 0T are found to be $-4.23(9)\mu_{B}$, $3.76(6)\mu_{B}$ for 1D data and $-4.09(2)\mu_{B}$, $3.94(2)\mu_{B}$ for 2D ones. Different signs of magnetic moments at two sites correspond to their antiparallel orientation to each other. One can see that the values of magnetic moments are in agreement for both refinements, while a significant decrease of error bars is observed for 2D data. The refined parameters are also in good agreement with literature: $-4.20(3)\mu_{B}$ and $3.97(3)\mu_{B}$[@Wright_2002].
Flipping sum and difference diffraction patterns measured at 150 K in magnetic field of 6 T are shown in figure\[fig:2d-Diffraction-profile-Fe3O4\_150K\_6T\]. The presence of magnetic scattering depending on the neutron spin orientation is clearly seen in the flipping difference pattern where reflections with significant magnetic contribution are easily recognizable by strong variation of their intensity along the Debye cone. The angular dependence of the integrated intensity of (111) reflection is shown in figure\[fig:BReflection111\]. It is in a good agreement with the model values calculated by using formula 4 (dotted lines on figure\[fig:BReflection111\]). Note, that strong dependence of magnetic scattering on polar angle $\phi$ rises a problem in the reduction of the two-dimensional diffraction pattern to the one-dimensional one.
![(Color online) Flipping sum and difference diffraction patterns collected on Fe~3~O~4~ at $T=150$K, $6$T. The measured 2D pattern is shown on the top, the calculated one is shown on the bottom (a), chi squares normalized per number of points is 3.93. Diffraction profiles estimated near the equatorial plane (b). The position of reflections is marked by “$|$”. Black lines show the differences between experimental points (blue) and model line (orange).\[fig:2d-Diffraction-profile-Fe3O4\_150K\_6T\]](fig_4a.eps){width="8cm" height="8cm"}
![(Color online) Flipping sum and difference diffraction patterns collected on Fe~3~O~4~ at $T=150$K, $6$T. The measured 2D pattern is shown on the top, the calculated one is shown on the bottom (a), chi squares normalized per number of points is 3.93. Diffraction profiles estimated near the equatorial plane (b). The position of reflections is marked by “$|$”. Black lines show the differences between experimental points (blue) and model line (orange).\[fig:2d-Diffraction-profile-Fe3O4\_150K\_6T\]](fig_4b.eps){width="8cm" height="8cm"}
Use of polarized neutron diffraction improve considerably the precision of the Fe magnetic moments determination. The values of magnetic moments obtained from the 2D refinement are $-4.03(1)\mu_{B}$ for the ion Fe^3+^ at the tetrahedral site and $3.95(1)\mu_{B}$ for that at the octahedral site. For the 1D data the moments are found to be $-4.05(7)\mu_{B}$ and $3.89(6)\mu_{B}$, respectively. Measurements performed below the Verwey transition did not show any evolution of the scattering signal (see Supplemental Material). No new magnetic reflections associated with the ordering of octahedral B irons (Fe^3+^, Fe^2+^) were observed, which is in agreement with previous polarized neutron single crystal diffraction measurements[@Rakhecha_1978].
Spin-ice compound Ho~2~Ti~2~O~7~
================================
Among rare earth pyrochlores titanates Ho~2~Ti~2~O~7~ is considered as canonical spin ice compound that shows various exotic magnetic states produced by the presence of geometric frustration[@Harris_1997; @Gardner_2010]. It shows Ising-like behavior, with the magnetic moments being constrained along the local $\left<111\right>$ axes. In the pyrochlore lattice, distinction between Ising, Heisenberg, or XY models cannot be based, as usual, on the analysis of the macroscopic properties of a single crystalline sample in a magnetic field because of the presence of four different anisotropy axes. The information about the local anisotropy of Ho~2~Ti~2~O~7~ has been first obtained by polarized neutron single crystal diffraction based on the so called “local susceptibility approach”[@Cao_2009]. The temperature behavior of the reported local susceptibility tensor has confirmed the Ising character of Ho local anisotropy and was in perfect agreement with that calculated from the rare earth crystal field parameters. Here we show that the same information about the local susceptibility tensor can be obtained by using 2D Rietveld refinement of polarized neutron powder diffraction patterns.
We collected a series of powder patterns from the Ho~2~Ti~2~O~7~ sintered powder sample in the temperature range 5–50K and in field of 1T. We found that applying a magnetic field to the sample led to dramatic changes in the diffraction pattern. As an example the flipping diffraction patterns measured at 5K in 1T are shown in figure\[fig:2dRietveld\]. As expected a strong variation of intensity along the Debye cone is observed. It can be seen as well that the intensities of reflections allowed by $Fd\overline{3}m$ symmetry (111, 220, 113, etc) are strongly polarization dependent (see the flipping difference pattern in figure\[fig:2dRietveld\](b)). We also found that the new reflections 200, 222, 240 appear which are forbidden by $Fd\overline{3}m$ symmetry. As seen from the difference plot the intensities of these reflections do not depend on neutron polarization but they are of purely magnetic origin. It has been shown that these reflections arise from the off-diagonal coefficient in the local susceptibility $\chi_{12}$ which becomes significant at low temperatures [@Gukasov_2010]. Note that the flipping difference pattern, proportional to$N\cdot M_{z, \perp}$, contains both positive and negative values, as its sign depends on the phase of the magnetic and nuclear structure factors.
For the space group $Fd\overline{3}m$, the symmetry constraints imply that local susceptibility tensor has only two independent matrix elements $\chi_{11}$and $\chi_{12}$ and the principal axes of Ho magnetization ellipsoids are oriented along the four local $\left<111\right>$ axes. Their lengths given by $\chi_{\parallel}=\chi_{11}+2\chi_{12}$ and $\chi_{\perp}=\chi_{11}-\chi_{12}$ were determined at each temperature. Thermal evolution of $\chi_{\parallel}$ and $\chi_{\perp}$ obtained by 2D Rietveld refinement on polycrystalline sample is shown in the figure\[fig:(color-online).-Susceptibility\] by closed symbols. Open symbols in the figure show the results of previous study performed using polarized neutron diffraction on single crystal[@Cao_2009]. One can see that the results of Rietveld refinement are in a good agreement with the single crystal ones and offer the same precision of the susceptibility parameters.
![(Color online) The measured and calculated flipping sum (top) and difference (bottom) diffraction patterns collected on Ho~2~Ti~2~O~7~ at diffractometer 5C1, $T=5$K, $H=1$T for 2D (a) and 1D (b) diffraction profiles. Chi squares normalized per number of points is 0.57 for 2D diffraction patterns\[fig:2dRietveld\].](fig_5a.eps){width="8cm" height="8cm"}
![(Color online) The measured and calculated flipping sum (top) and difference (bottom) diffraction patterns collected on Ho~2~Ti~2~O~7~ at diffractometer 5C1, $T=5$K, $H=1$T for 2D (a) and 1D (b) diffraction profiles. Chi squares normalized per number of points is 0.57 for 2D diffraction patterns\[fig:2dRietveld\].](fig_5b.eps){width="8cm" height="8cm"}
![(Color online) Temperature dependence of the susceptibility components $\chi_{\parallel}$ (circles) and $\chi_{\perp}$ (triangles) for single (open symbols, taken from[@Cao_2009]) and powder (full symbols) Ho~2~Ti~2~O~7~ at 1T. The insert shows Ho magnetization ellipsoids at 50K. \[fig:(color-online).-Susceptibility\]](fig_6.eps){width="7cm" height="7cm"}
Single-molecule magnet: Co(II) complex
======================================
The 2D Rietveld method is known to be a powerful tool allowing to study powder samples having preferred crystallite orientation[@Ferrari1994]. Application of magnetic field to anisotropic powder samples can induce the preferred crystallite orientation, as the net moment of the crystallites tends to align in the field direction. Since the resultant preferred orientation can be determined from the 2D patterns one cans use these “magnetically textured” samples in PNPD. We found that such an approach in combination with 2D Rietveld method have a number of advantages and we applied it to the studies of local susceptibility in the cobalt(II) complex with molecular formula Co(L~1~)~2~Cl~2~, where L~1~ is tetramethylthiourea \[(CH~3~)~2~N\]~2~CS[@Vaidya_2018]. The compound is a single-molecule magnet that shows superparamagnetic behavior below a certain blocking temperature and exhibits magnetic hysteresis of purely molecular origin.
The powder was filled in a vanadium container of 6mm diameter without compressing it. The sample was cooled to 2K and the diffraction patterns were measured as a function of magnetic field. The Debye rings in zero field were found homogeneous indicating the absence of preferred crystallite orientation. In magnetic fields above 1T the crystallite reorientation started to appear and at 5T the Debye rings were transformed in a series of well separated diffraction spots (figure\[fig:2dRietveld\_CoCl2\]). We note that the subsequent reduction of the magnetic field to 0T did not change the crystallite orientation back to a random one. The diffraction patterns measured at 2K and 5T in the “magnetically textured” sample were used to determine the local susceptibility of the cobalt ion.
![(Color online) The measured and calculated flipping sum (top) and difference (bottom) diffraction patterns collected on Co(L~1~)~2~Cl~2~ at diffractometer 6T2, $T=2$K, $H=5$T. Chi squares normalized per number of points is 6.71. The insert shows Co magnetization ellipsoids surrounded by S (yellow) and Cl (green).\[fig:2dRietveld\_CoCl2\]](fig_7.eps){width="8cm" height="8cm"}
To take into account the preferred orientation we used a modified March model[@Dollase_1986], which was developed to describe the mechanism of grain rotation that produces preferred orientation:
$$P_{h}=t+(1-t)\cdot\left[r^{2}\cos^{2}\alpha_{h}+\frac{\sin^{2}\alpha_{h}}{r}\right]^{-3/2},\label{eq:march_function}$$
where $t$ is the fraction of randomly oriented crystallites, $\alpha_{h}$ is the angle between the transfer momentum and the preferred orientation axis, and $r$ describes the anisotropic shape of crystallites. In the Debye-Scherrer geometry $r$ is more than one for platy crystallites and it is less than one for acicular crystallites. Although in our case the origin of preferred orientation is due to the application of magnetic field, using the March distribution allows estimation of an intuitively simple equivalent specimen compaction.
The studied single-molecule magnet has the monoclinic space group $P2_{1}/n$ with $a=9.88$Å, $b=12.69$Å, $c=14.13$Å, $\beta=92.99^{\circ}$. It is composed of 43 atoms in the asymmetric unit[@Vaidya_2018], including 24 hydrogen atoms. The flipping patterns measured at 2K and 5T were used to refine the crystalline texture parameters and the susceptibility tensor of cobalt. As seen from figure\[fig:2dRietveld\_CoCl2\] a very good agreement between patterns calculated after the refinement and the experimental ones is observed. Both the positions and the widths of the diffraction spots on the Debye cones are well reproduced in the model patterns. The refined March texture parameters $t=0.938(1)$ and $r=0.119(1)$ show that the magnetically induced preferred orientation is rather low at 5 T and only a small part of crystallites is aligned with their easy axes parallel to the field. Hence, stronger fields are needed to overcome the steric hindrence in the powder packing. It is clear, however, that the presence of preferred orientation gives a big advantage in the 2D Rietveld refinement when using area detectors. As seen from the figure different reflections with similar Bragg angles $2\theta$ appear at different $\varphi$ angles. In result no overlapping of these reflections occurs, which allows to use diffractometers with low resolution, like the single crystal diffractometer 6T2, for powder diffraction. We note as well that the conventional approach consisting in the projection of 2D data on the 1D one for Rietveld refinement would result in dramatic decrease of the resolution due to the reflection overlapping.
Finally, the refined magnetization tensor corresponding to an external field of 5T for cobalt atom in an asymmetric unit was found to be equal to
$$\left(\begin{array}{ccc}
1.9(3) & 0.0(3) & 0.1(1)\\
0.0(3) & 2.3(3) & -1.4(2)\\
0.1(1) & -1.4(2) & 2.7(3)
\end{array}\right)\mu_{B}.\label{eq:chi_co}$$
The corresponding magnetization ellipsoid is presented in the insert of figure\[fig:2dRietveld\_CoCl2\]. The averaged magnetization estimated as $2.3(2)$$\mu_{B}$ is close to magnetization per cobalt atom $2$$\mu_{B}$ taken from the magnetization measurements on polycrystalline sample at 5T[@Vaidya_2018]. A detailed analysis of the crystallite alignment, the evolution of the magnetization ellipsoids with temperature and field as well as the theoretical interpretation of the ellipsoid orientation are still in progress and will be published later[^2].
Conclusion
==========
Our results suggest that the combination of area detector, 2D Rietveld analysis and the technique of magnetically induced preferred crystallite orientation opens new route to the studies of local magnetic susceptibility in polycrystalline materials by polarized neutron diffraction. The results of 2D Rietveld analysis of diffraction patterns from soft (Fe~3~O~4~) and high (Ho~2~Ti~2~O~7~) magnetic compounds are in perfect agreement with the single crystal ones reported earlier. We demonstrate that using “magnetically textured” powder and the 2D Rietveld refinement allows to obtain the precision in the determination of the susceptibility parameters close to that obtained in the single crystal diffraction experiments. By applying this procedure to a single-molecule magnet in polycrystalline form we obtained the local susceptibility tensor for cobalt atom, which can be now confronted to the theory. More generally we suggest that the magnetic structure determination by applying 2D Rietveld method to the “magnetically textured” samples has significant perspectives, as it does not require high instrumental resolution due to the fact that different types of reflections with similar Bragg angles are spread over the Debye cones.
Acknowledgments {#acknowledgments .unnumbered}
===============
We are grateful to J.Overgaard for stimulating discussions and also providing us a single-molecule magnet with Co(II) complex. IK thanks CNRS for his postdoctoral position.
[^1]: The RhoChi source code is written in python3 using cryspy library. It is freely distributed through the GitHub service: https://github.com/ikibalin/cryspy, where a short guide of its application is given together with examples.
[^2]: J.Overgaard et.al. to be published
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abstract: 'The valley dependent optical selection rules in recently discovered monolayer group-VI transition metal dichalcogenides (TMDs) make possible optical control of valley polarization, a crucial step towards valleytronic applications. However, in the presence of Landaul level (LL) quantization such selection rules are taken over by selection rules between the LLs, which are not necessarily valley contrasting. Using MoS$_{2}$ as an example we show that the spatial inversion-symmetry breaking results in unusual valley dependent inter-LL selection rules, which is controlled by the sign of the magnetic field and directly locks polarization to valley. We find a systematic valley splitting for all LLs in the quantum Hall regime, whose magnitude is linearly proportional to the magnetic field and in comparable with the LL spacing. Consequently, unique plateau structures are found in the optical Hall conductivity, which can be measured by the magneto-optical Faraday rotations.'
author:
- 'Rui-Lin Chu$^1$'
- Xiao Li$^2$
- 'Sanfeng Wu$^{3}$'
- Qian Niu$^2$
- 'Wang Yao$^{4}$'
- 'Xiaodong Xu$^{3,5}$'
- 'Chuanwei Zhang$^{1}$'
title: 'Valley Splitting and Valley Dependent Inter-Landau-Level Optical Transitions in Monolayer MoS$_{2}$ Quantum Hall Systems'
---
Optical properties of two-dimensional (2D) charge carrier systems such as 2D electron gases (2DEG), graphene, and topological insulators are important for studying their underlying charge carrier properties and future applications in optoelectronics [Klem,IVK,Kerr,Ploog,Ploog2,deHeer,Carbotte,Jiangzg,Aguilar,Tse,Tokura,Abergel]{}. Generally, the charge carrier dynamics can be strikingly different with and without a magnetic field, as evidenced by the celebrated example of quantum Hall effect (QHE). In the quantum Hall regime, an external large magnetic field produces a series of Landau levels (LLs) with discrete energies and the optical transitions occur only between appropriate LLs following certain selection rules. In the past few decades, such selection rules and relevant optical phenomena have been intensively studied in various magneto-optical measurements [Klem,IVK,Kerr,Ploog,Ploog2,deHeer,Jiangzg,Aguilar,Tokura,Abergel,Iris,Shimano,Hofmann]{}.
Monolayers of MoS$_{2}$ and other group-VI transition-metal dichalcogenides (TMDs) represent a new family of 2D materials beyond graphene. Because of their coupled spin and valley physics and large band gap, monolayer TMDs have become exciting platforms for exploring novel valleytronic and optoelectronic applications [Mak-10prl,zhuzhiyong,AKis,xiaodi-mo,nat-com,Mak2, xiaodongcui]{}. Recently, the optical properties of TMDs in zero magnetic field have been widely studied in many experiments [@xiaodongcui; @Mak2; @xu1; @xu2]. However, the counterpart in a finite magnetic field has not been well explored.
In this work, we study the optical properties of monolayer MoS$_{2}$ quantum Hall systems with a large magnetic field. With a zero magnetic field, it is known that monolayer MoS$_{2}$ and graphene share similar valley contrasting physics, i.e., the circular polarizations ($\sigma _{+}$, $\sigma _{-}$) are locked with two inequivalent valleys K and K$^{\prime }$ due to the opposite orbital helicity of two valleys [@Geim; @yaowang; @beenakker; @white]. However, with a large magnetic field in graphene, the polarization is no longer associated with the valley degree of freedom because the transitions are between LLs, whose selection rules allow transitions in both polarizations ($\sigma _{+}$ and $\sigma _{-}$) at both valleys (K and K$%
^{\prime }$), as observed in many magneto-optical measurements [deHeer,Jiangzg,Iris,Shimano,Hofmann]{}. Furthermore, such inter-LL selection rules in graphene are the same as 2DEG with large band gaps (e.g., GaAs quantum wells), where valleys do not even exist [Klem,IVK,Kerr,Ploog,Ploog2]{}. These known results naturally indicate that the optical responses of monolayer MoS$_{2}$ in the quantum Hall region should also be valley independent.
Surprisingly, we find this is not the case for monolayer MoS$_{2}$, where the polarization selection rules for the inter-LL transitions are still valley dependent. More interestingly, the selection rules are controlled by the sign of the magnetic field, i.e., the valley index in the selection rules can be flipped by reversing the sign of the magnetic field, which is fundamentally different from the zero magnetic field case, where the polarization-valley locking is fixed. We also show that optical transitions in this system are made more unusual by a systematic valley splitting for all LLs, whose magnitude is linear against the magnetic field and is comparable with the LL spacing. The valley-polarization selection rules and valley splitting lead to a series of spin-valley polarized transitions between LLs as well as unique plateau structures in the optical Hall conductivity, which can be addressed in the circular dichroism, magnetoluminescence and Faraday rotations, showing distinguishable features from graphene and 2DEG. Our predictions also apply to other group-VI TMDs.
*LLs and valley splitting.* The monolayers of MoS$_{2}$ consist of a Mo layer sandwiched between two S layers in a trigonal prismatic arrangement. Although similar to graphene in many aspects, some of its properties are more favorable than graphene. It features a direct band gap $%
\Delta $ in the visible wavelength regime, which occurs at the two inequivalent valleys K and K$^{\prime }$ at the corners of the hexagonal Brillouin zone. The inversion symmetry is naturally broken in the monolayers, which induces both strong spin-orbit coupling and spin-valley coupling [@xiaodi-mo; @nat-com].
![ (color online) (a) schematic of the spin-valley coupled band structure of TMDs. Red(blue) represents spin up(down), respectively. (b) and (c): Conduction and valence band LLs for MoS$_{2}$ under $B_{\perp }$ = 20 T. K(K’) valley is on the left(right). The crossing-LL states in the conduction band are from the dangling bonds on the zigzag edges[Liuguibin]{}, which do not affect the LLs. Dash line is a guide to eye for valley spitting and also marks filling level $\protect\nu(K)=4$, $\protect\nu%
(K^{\prime })=2$. (d) $\Delta _{c}^{01}$($\Delta _{v}^{01})$ is the absolute energy difference between the LL 0(-1) in K valley and 1(0) in K’ valley in conduction(valence) band. $\hbar\protect\omega_0^c$($\hbar\protect\omega_0^v$) is the LL spacing in conduction(valence) band. (e) same as (d) calculated from orbital magnetic moment and effective mass approximation. $a$ is the lattice constant.](fig1.eps){width="48.00000%"}
The zero-field band structure for monolayer MoS$_{2}$ is schematically shown in Fig. 1(a), which is usually described by the effective Dirac model [xiaodi-mo,nat-com]{}. In each valley there are two sets of gapped Dirac spectra with red (blue) representing spin up (down) respectively. Because of the large effective mass at the band edges the LLs only scale as $n\hbar
eB_{\perp }/m^{\ast }$ at the low energy part, which resemble conventional 2D semiconductors more rather than Dirac fermions [Lixiao,haizhou,Asgari,Andor-prb,Zhouli]{}. Here $B_{\perp }$ is the perpendicular magnetic field and $n$ is the LL index. To obtain the LLs, we adopt a three-band atomic tight-binding model from Ref. [@Liuguibin] and apply $B_{\perp }$ via the Peierls substitution $t_{ij}=t_{1,2}e^{-ie/\hbar
\int \mathbf{A}\cdot d\mathbf{r}}$, where $\mathbf{A}=(-By,0,0)$ is the vector potential.
In Fig. 1(b-c) we present the low energy LLs, where a zigzag ribbon structure is used with a width $L_{y}=170$ nm, which is sufficiently larger than the magnetic length scale $l_{B}$. The set of LLs from the lower split-off valence bands are not shown since they are similar to Fig. 1(c). The Zeeman splitting is first neglected here and will be discussed later. We label the LLs and assign the $n=0$ LLs according to the analytic solutions from the effective two-band Dirac model [@Lixiao]. When $B_{\perp }>0$ they appear only in conduction band of K valley and valence band of K$%
^{\prime }$ valley. Therefore the valley degeneracy for them is already lifted. Here our focus is on the more general $n\neq 0$ LLs.
We notice a systematic valley splitting exists for all $n\neq 0$ LLs with the magnitude comparable to the LL spacing, which is not revealed by the effective model [@Lixiao]. A linear relation with $B_{\perp }$ is found. Here we let $B_{\perp }\geqslant 5$T to ensure $L_{y}>>l_{B}$. The linearity should extend to the low field situation in this single-particle calculation. The linear valley splitting and its discrepancy with the effective model can be intuitively understood from the orbital magnetic moment [@Chang; @prx-dot], which is of opposite sign at the two valleys ($\pm m$). Taking the conduction band as an example, in the presence of $B_{\perp }$ the valley energy difference is $\Delta _{c}^{01}=2m\cdot B_{\perp }$. In the effective model, this matches $\hbar \omega _{0}^{c}$, the LL spacing between 0 and 1, resulting in valley degeneracy in $n\neq 0$ LLs [@Cai]. Such matching is violated in the tight-binding model where $\Delta _{c}^{01}>\hbar \omega
_{0}^{c}$ giving rise to the valley splitting, as is shown in Fig. 1d and 1e. In the case of graphene, the valley degeneracy is known to be lifted in high magnetic fields via electron-electron or electron-phonon interactions [@Kim-prl; @Ong-prl; @Andrei-nat; @Kim-natphys]. Similar linear relations between the valley splitting and $B_{\perp }$ have also been experimentally observed in silicon and AlAs 2D electron systems [@Si-natphy; @AlAs-prl]. Their physical origin, however, remains controversial.
The valley splitting here has a few direct consequences: (i) The $n=0,1$ LLs in conduction band are always valley polarized and $n=0,-1$ in valence band are spin-valley polarized. (ii) The total filling factor follows a sequence $%
\nu =2,3,4\ldots $ in the electron doped regime and $\nu =-1,-2,-3\ldots $ in the hole doped regime. The lifting of valley degeneracy in $n=0$ LLs can be attributed to the broken spatial-inversion symmetry in the monolayer. Such symmetry is known to guarantee the valley degeneracy rigorously on graphene, regardless of the time-reversal symmetry [@Mccann]. However, it does not explain the splitting in $n\neq 0$ LLs [@Fuchs]. Instead, using graphene lattice as a toy model, we find the splitting in $n\neq 0$ LLs can be induced by the next-nearest-neighbour (NNN) electron hopping, which breaks the electron-hole symmetry. Therefore the valley-splitting in $%
n\neq 0$ LLs stems from spontaneous breaking of spatial-inversion, electron-hole and time-reversal symmetry. In fact, the low energy physics in MoS$_{2}$ is dominated by electron hopping between Mo atoms, which is indeed the NNN hopping on the honeycomb lattice [@xiaodi-mo; @nat-com].
{width="47.00000%"}{width="40.00000%"}
*Valley-dependent inter-LL selection rules:* We now turn to the optical properties of these valley-degeneracy-lifted LLs. Because $\hbar
\omega _{0}<<\Delta $, the intraband and interband optical transitions in this system belong to two completely different regimes: intraband in the microwave to terahertz and interband in the visible frequency range. We will set our focus on the latter because for MoS$_{2}$ the valley contrasting interband optical transitions have been the most intriguing property in experiments for valleytronics [xiaodi-mo,nat-com,xiaodongcui,Mak2,xu1,xu2]{}.
When considering the transitions between levels $n^{\prime }$ and $n$, the well-known selection rule for 2DEG and graphene requires $|n|=|n^{\prime
}|\pm 1$ [@Aoki-prl; @Carbotte; @Jiangzg; @Aoki-prb]. For MoS$_2$ such selection rule can also be obtained from the effective model[@Rose]. At first glance, since the LL spacing is comparable in the conduction and valence bands, four transitions would occur at very close but non-degenerate photon energies: $-n\leftrightarrow n+1$ and $-(n+1)\leftrightarrow n$ ($%
n\geqslant 1$) for both valleys. In Fig. 2 we calculate the optical absorption spectrum $$A(\omega )\propto \sum_{\epsilon _{n}<\mu ,\epsilon _{n^{\prime }}>\mu }%
\frac{|J_{x}^{nn^{\prime }}|^{2}}{i(\epsilon _{n^{\prime }}-\epsilon
_{n}-\omega -i\Gamma )},$$at three different filling levels corresponding to the LLs presented in Fig.1 through exact diagonalization, where $J_{x}$ is the current matrix and $\Gamma $ is the broadening parameter. We immediately notice several distinctive features. (i) The expected four-fold peaks only appear two-fold. Unlike in graphene, here the transitions $-n\leftrightarrow n+1$ in K valley and $-(n+1)\leftrightarrow n$ in K$^{\prime }$ valley are completely suppressed, which suggest highly valley dependent selection rules. Transitions from the spin-split lower valence bands at higher photon energies also follow the same rule except with opposite spins, as seen in Figs. 2(d-f). Such selection rules originate from the severely broken spatial-inversion symmetry in the monolayer. (ii) As the filling level goes up, the number of spin-valley polarized peaks has an alternating $2-1-2-1$ pattern, which can be attributed to the valley-imbalanced Pauli blocking caused by the $n=0$ LLs and the valley splitting as illustrated in Figs. 2(g-h).
{width="33.00000%"}
To further understand the role played by the valley degree of freedom in the optical Hall effect, we calculate the optical Hall conductivity using the Kubo formula [@Aoki-prl; @Aoki-prb; @Shimano], $$\begin{aligned}
\sigma _{ij}(\omega ) &=&\frac{i}{L_{x}L_{y}}\sum_{\epsilon _{n}<\mu
,\epsilon _{n^{\prime }}>\mu }\frac{1}{\epsilon _{n^{\prime }}-\epsilon _{n}}%
(\frac{J_{i}^{nn^{\prime }}J_{j}^{n^{\prime }n}}{\epsilon _{n^{\prime
}}-\epsilon _{n}-\omega -i\Gamma } \notag \\
&-&\frac{J_{j}^{nn^{\prime }}J_{i}^{n^{\prime }n}}{\epsilon _{n^{\prime
}}-\epsilon _{n}+\omega +i\Gamma }),\end{aligned}$$where $i,j=x,y$. Following Ref. [@Aoki-prl], here we retain 40 LLs and impose periodic boundary conditions along the $x$ and $y$ directions. $%
B_{\perp }$ is kept at $20.8$ T, as close as possible to that used Figs. 1 and 2, since in such calculations $B_{\perp }$ can only take discrete levels. The two valleys cannot be distinguished in the momentum space. However, since the valley index is associated with spin, we can distinguish valleys by spins. The result is shown in Fig. 3, where $\sigma _{\pm
}(\omega )=\sigma _{xx}(\omega )\pm i\sigma _{xy}(\omega )$ is the optical conductivity for the right and left circular polarized light. Spin-valley polarized resonance structures for $\sigma _{xy}$ is found, similar to Fig. 2. In this set-up, for the electron(hole) doped regime the spin-valley polarization is achieved for the spin up (down) and K$^{\prime }$ (K) valley, respectively. Upon switching the sign of $B_{\perp }$ the valley and spin polarization also flips. At resonance photon frequencies $\sigma _{xy}$ from the two valleys actually have the opposite signs. This is another distinctive feature from graphene, in which both valleys contribute equally to the total $\sigma _{xy}$ [@Iris; @Shimano]. But due to the difference in $\Delta _{c}^{01}$ and $\Delta _{v}^{01}$ (Fig. 1d), the resonance frequencies in the two valleys are slightly miss-matched, leading to spin-valley mixed resonance peaks in $\sigma _{xy}$ (starting from the third resonance in Fig. 3c) instead of cancelling out.
{width="40.00000%"}
A more important message from Fig. 3 when compared with Fig. 2 is that the allowed interband transitions in K and K$^{\prime }$ valleys are solely attributed to the left and right circular polarized light separately $$\begin{aligned}
&&K:~~~-(n+1)~\leftrightarrow ~n,~~~\sigma _{-} \notag \\
&&K^{\prime }:~~~-n~\leftrightarrow ~n+1,~~~\sigma _{+}\end{aligned}$$where $n\geqslant 0$. Consequently the circular polarization is directly locked with the valley degree of freedom in optical transitions in the quantum Hall regime. Upon flipping the direction of $B_\perp$, the valley index will switch, *i.e.* $K$($\sigma_+$) and $K^{\prime }$($\sigma
_- $).
*Optical Hall plateaus.* The optical conductivity as a function of the chemical potential $\mu $ is shown in Fig. 4, where $\omega $ is slightly away from resonance. The static quantum Hall conductivity is also presented, showing fully valley-degeneracy-lifted and well quantized plateaus. We notice that in the optical conductivity each spin or valley component also develops its own and contrasting plateaus, although like the net $\sigma _{xy}(\omega )$ they are not quantized either. The $n=0$ LLs and the valley splitting are manifested in the alternating sequence of the step structures in these two components, which also lead to a unique sequence of filling factors in the net $\sigma _{xy}(\omega )$ plateaus, as labelled in Fig. 4. To be specific, $2,3$ spans $n=0$ to $n=1$ in K valley and $4,5$ $%
n=1 $ in K to the $n=1$ in K$^{\prime }$ valley, and so on. Interestingly, the valley contrasting plateaus persist even when $\mu $ is in the band gap, as is seen for the $0$ plateau that extends all the way to the valence band top.
*Circular dichroism, magnetoluminescence and Faraday rotation.* Valley resolved interband optical transitions shown in Fig. 2 are readily detectable by the circular dichroism spectroscopy due to the polarization-valley locking. Given the already excellent photoluminescenence of monolayer TMDs in zero-field, magnetoluminescence would be an ideal test of the valley dependent selection rules, in which luminescence between individual LLs in a selected valley can be driven by resonant circular polarized excitations in the Faraday geometry [@Klem; @Ploog; @Ploog2]. The optical Hall conductivity $\sigma _{xy}(\omega )$ can be measured from the Faraday rotation angle $\theta (\omega )$ [@Iris; @Shimano; @Aoki-prl]. A spin-valley polarized excitation would be indicated by a single maximum absolute slope $|d\theta /d\omega |$ at resonant frequencies as shown in Fig. 3c.
*Interplay of valley and spin splitting.* The Zeeman splitting is estimated to be much smaller than the valley splitting when we assume an ordinary g-factor for electrons ($g=2$). The Zeeman spin splitting will slightly enlarge the valley spitting in both the valence and conduction bands with the same magnitude. However, The spin in optical transitions is conserved and the spin-Zeeman field does not change valley-dependent inter-LL transition frequency and selection rules. Additionally, the valence band tops at the K(K’) valleys are composed of $m=-2$($m=$2) $d$-orbitals from the Mo atoms[@xiaodi-mo; @Liuguibin], which induce an additional Zeeman-type splitting term in presence of $B_{\perp }$. The valley splitting in the valence band is further enlarged by this term. The conduction band bottoms are not affected by this term because they are composed of the $m=0$ $d$-orbitals of Mo atom[@xiaodi-mo; @Liuguibin]. Accordingly, this additional term induces valley-contrasting frequency shift for the inter-LL transitions at each valley, which will appear as a linear shift against $B_{\perp }$ for each valley in the spectrums of circular dichroism and magnetoluminescence. The valley-dependent inter-LL selection rules remain intact.
To conclude, we have shown that valley splitting exists for all the LLs in monolayer MoS$_{2}$ and other TMDs even without considering the interaction effects. Optical transitions in the quantum Hall regime follow valley dependent selection rules controlled by the sign of the magnetic field, which lock the circular polarization and valley degree of freedom together. Finally we propose circular dichroism spectroscopy, magnetoluminescence and Faraday rotation measurements as potential tests for the selection rules as well as the valley splitting. An interesting extension of current study would be the disorder and localization effect in the optical Hall effect, since in this system the mixing of valleys inevitably involves spin flipping, which is distinct from graphene [@haizhou; @Aoki-prl].
*Acknowledgements* R.C. and C.Z. are supported by ARO (W911NF-12-1-0334), AFOSR (FA9550-13-1-0045), and NSF-PHY (1249293). X.L. and Q.N. are supported by DOE-DMSE (DE-FG03-02ER45958), NBRPC (2012CB-921300), NSFC (91121004), and the Welch Foundation (F-1255). S. W. and X. X. are supported by US DoE, BES, Division of Materials Sciences and Engineering (DE-SC0008145). W. Y is supported by the Croucher Foundation(Croucher Innovation Award), and the RGC and UGC of Hong Kong(HKU705513P,HKU9/CRF/13G,AoE/P-04/08).
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abstract: |
We study the effects of mobility on the evolution of cooperation among mobile players, which imitate collective motion of biological flocks and interact with neighbors within a prescribed radius $R$. Adopting the prisoner’s dilemma game and the snowdrift game as metaphors, we find that cooperation can be maintained and even enhanced for low velocities and small payoff parameters, when compared with the case that all agents do not move. But such enhancement of cooperation is largely determined by the value of $R$, and for modest values of $R$, there is an optimal value of velocity to induce the maximum cooperation level. Besides, we find that intermediate values of $R$ or initial population densities are most favorable for cooperation, when the velocity is fixed. Depending on the payoff parameters, the system can reach an absorbing state of cooperation when the snowdrift game is played. Our findings may help understanding the relations between individual mobility and cooperative behavior in social systems.\
\
**Keywords**: cooperation, flocks, evolutionary games, prisoner’s dilemma, snowdrift game, mobility
author:
- |
Zhuo Chen$^{1*}$, Jianxi Gao$^1$, Yunze Cai$^2$ and Xiaoming Xu$^{1,2,3}$\
$^1$Shanghai Jiao Tong University, Shanghai, China\
$^2$University of Shanghai For Science and Technology, Shanghai, China\
$^3$Shanghai Academy of Systems Science, Shanghai, China\
$^*$jeffchen\_ch@yahoo.com.cn
title: Evolution of Cooperation among Mobile Agents
---
Introduction
============
Cooperation is commonly observed throughout biological systems, animal kingdoms and human societies. But from a Darwinian viewpoint, cooperators are at a disadvantage in natural selection, because they increase the fitness of others at the cost of their own survival and reproduction [@Okasha]. In a broad range of disciplines, understanding the emergence of cooperation is a fundamental problem, which is often studied within the framework of evolutionary game theory.
The prisoner’s dilemma (PD) game and the snowdrift game are commonly used two person games with two strategies, cooperation (C) and defection (D). Mutual cooperation pays each a reward $R$, while mutual defection brings each a punishment $P$. When one defector meets one cooperator, the former gains the temptation $T$ while the latter obtains the sucker’s payoff $S$. The PD is defined by the payoffs, if $T>R>P>S$ and $2R>S+T$. In a single round of the PD, though the individual interest can be maximized by defection, the collective payoff achieves the maximum only when both players cooperate. Hence the dilemma arises. As an alternative model to study cooperative behavior, the SD is produced when $T>R>S>P$. In contrast with the PD, the best strategy of the SD depends on the co-player: to defect if the opponent cooperates, but to cooperate if the opponent defects. Under replicator dynamics in well-mixed populations, defection is the only evolutionarily stable strategy in the PD, while cooperators may coexist with defectors in the SD. Note in the SD, the average population payoff at evolutionary equilibrium is smaller than that when everyone plays C [@Doebeli2005]. Thus SD is still a social dilemma.
One of possible mechanisms accounting for the establishment of cooperation is the so-called network reciprocity [@Nowak2006]. Discarding the well-mixed assumption for populations, this theory focuses on how spatial structure affects the evolution of cooperation. Axelrod first suggested to locate individuals on the two-dimensional array, where interactions only happened within local neighborhoods. Nowak and May developed this idea later, showing that unconditional cooperators could survive by forming clusters [@Nowak1993]. These pioneering studies have triggered an intensive investigation of spatial games, yielding enormous combinations of evolutionary rules, graphs and game models. In Ref. [@Szabo1998], the effect of noise is incorporated in the strategy adoption, and Darwinian selection of the noise level favors a specific parameter value that induces the highest level of cooperation [@Szolnoki2009a; @Szabo2009]. Diversity is another role facilitating cooperation, which takes various forms as heterogeneous graphs [@Santos2008], preferential imitations [@Yang2009], reproduction probabilities [@Wu2009], individual rationality [@Chen2009], fitness [@Perc2008] or behavioral preferences [@VanSegbroeck2009]. Since connectivity structures in the real world are far more than regular lattices, there are many interests in the impact of complex topologies on cooperative behavior [@Santos2005; @Gomez-Gardenes2007; @Ren2007; @Chen2008; @Du2009; @Pena2009]. The co-evolution of strategies and individual traits, such as teaching activities [@Szolnoki2008; @Szolnoki2009], learning rules [@Moyano2009; @Cardillo2010] and social ties [@Zimmermann2004; @Santos2006; @Szolnoki2008a; @Chen2009a; @Fu2008], constitutes a key mechanism for the sustainability of cooperation. Interestingly, cooperators can benefit from the continuous supply of new players [@Poncela2008; @Poncela2009], and the strategy-independent evolution of networks can evoke powerful mechanisms to promote cooperation [@Szolnoki2009c; @Szolnoki2009b]. More details about spatial evolutionary games can be found in Ref. [@Doebeli2005; @Nowak2006; @Szabo2007; @Perc2010] and references therein.
Mobility of individuals is responsible for various spatiotemporal dynamics on geographical scales, such as the spread of infectious diseases and wireless viruses [@Kleinberg2007]. And statistical properties of human motion have attracted much interest in recent years [@Brockmann2006; @Gonzalez2008; @Song2010]. Indeed, the motion of individuals is an important characteristic of social networks [@Gonzalez2006]. Though it is often neglected, the effects of mobility on the evolution of cooperation vary with movement forms and population structures. Vainstein et al. [@Vainstein2007] considered a random diffusive process in a population of agents with pure strategies, where each agent can jump to a nearest empty site with a certain probability. It was found that cooperation can be enhanced by the movement of players, provided that the mobility parameter is kept with a certain range. The weak form of the PD adopted in Ref. [@Vainstein2007] was later extended to other games [@Guan2007; @Sicardi2009],and it was found that cooperation in the SD is not so often inhibited as that reported in Ref. [@Hauert2004]. Besides, the movement of players may take an adaptive form for payoffs or neighbors, and contingent mobility is often expected to enhance cooperation. Aktipis [@Aktipis2004] proposed a walk-away strategy to avoid repeated interactions with defectors, which outperforms complex strategies under a number of conditions. Helbing and Yu introduced the success-driven migration, in which players determine destinations through fictitious play [@Helbing2009]. Besides, individuals can decide when to move based on the number of neighboring defectors [@Jiang2010].
The synchronised motion of animal groups, such as fish schools and bird flocks, is an intriguing phenomenon, which can be modeled by systems of self-driven agents [@Vicsek1995; @Nagy2007; @Dossetti2009]. Recently, the model by Vicsek et al. has gained much attention for minimalism styles and rich dynamics [@Vicsek1995]. Here we combine the Vicsek model with evolutionary games, focusing on the effect of mobility on the evolution of cooperation. We reserve well-known elements like direction alignment and circular neighborhoods, ignoring the influence of angular noise on the update of velocity. We also cancel the periodic boundary conditions for simplicity, which can strongly affect the system behavior at the large velocity regime [@Nagy2007]. Thus when players move, the system is split into some disconnected groups, within which agents move toward the same direction. Note in some social systems, individuals do divide into groups according to race, wealth, age, and so on. We think that the aggregation of individuals partly reflects the community structure in social networks. In Ref. [@Chen2011], we have investigated an evolutionary PD game in a Vicsek-like model, where each agent plays with constant number of neighbors. We have found that cooperation can be maintained and even enhanced by the motion of players, provided that certain conditions are fulfilled. In the current work, we will check the robustness of our conclusions, when each agent plays the PD game with those individuals within a certain distance. Besides, we will study how mobility affects the outcome of the SD game.
The Model
=========
We consider a system with $N$ autonomous agents, which have positions $x_i(t)$ and move synchronously with velocities $\overrightarrow{V_i}(t)$ in a two-dimensional plane. The velocity $\overrightarrow{V_i}(t)$ of the agent $i$ is characterized by a fixed absolute velocity $v$ and an angle $\theta_i(t)$ indicating the direction of motion. When $t=0$, all agents are randomly distributed in an $L\times L$ square without boundary restrictions. Rather than fixed within a periodic domain, individuals can cross the border of the square when $t>0$, and move in the whole plane. The square only represents the initial distribution of individuals with a density $\rho=N/L^2$. Besides, initial moving directions of agents, $\theta_i(0)$, are uniformly distributed in the interval $[0,2\pi)$. At each time step, the $i$th agent updates its position according to $$x_i(t+1)=x_i(t)+\overrightarrow{V_i}(t)\Delta t.$$ Here $\Delta t$ is set to 1 between two updates on the positions.
To simulate the process of direction alignment in flocks, the angle $\theta_i(t)$ of the agent $i$ is updated according to the average direction of nearby neighbors [@Vicsek1995]. Then we have $$\theta_i(t+1)=arctan\frac{sin\theta_i(t)+\sum_{j\in
W_i(t)}sin\theta_j(t)}{cos\theta_i(t)+\sum_{j\in
W_i(t)}cos\theta_j(t)},$$ where $W_i(t)$ denotes the neighbors set of the agent $i$ at time $t$. Here $W_i(t)$ is defined as agents in the spherical neighborhood of the radius $R$ centered on the agent $i$, $$W_i(t)=\{j|\mid x_j-x_i\mid<R,j\in N,j\neq i\},$$ where $|\bullet|$ denotes the Euclidean distance between j and i in two-dimensional space. And we assume that each agent has the same radius.
The equations given above characterize the motion of agents. When moving in the plane, the agents also play games in pairs. For the PD, we take a re-scaled form suggested by Nowak et al. [@Nowak1993] as $$\mathbf{A}=\left(
\begin{array}{ccc}
1 & 0 \\
b & 0
\end{array}
\right),$$ where $b$ denotes the temptation to defect, and $1<b<2$. And for the SD, we take a simplified form as $$\mathbf{A}=\left(
\begin{array}{ccc}
1 & 1-r \\
1+r & 0
\end{array}
\right),$$ where $r$ denotes the cost-to-benefit ratio of mutual cooperation, and $0<r<1$. The strategy $s_i$ of the agent $i$, cooperation or defection, can be denoted by a unit vector $(1,0)^T$ or $(0,1)^T$ respectively. During the evolution of strategies, an normalized payoff is calculated to exclude the effect coming from different degrees of players, $$P_i=\frac{\sum_{j\in W_i(t)}s_i^TAs_j}{\mid W_i(t) \mid},$$ where $|\bullet |$ represents the size of $W_i(t)$, and $A$ is the payoff matrix. Afterward, every agent compares its income with that of its neighbors, following the strategy which owns the highest payoff among its neighbors and itself [@Nowak1993].
The system begins with an equal percentage of cooperators and defectors. At each step, all agents collect payoffs and update strategies, and next, they modify positions and directions. The time scale that characterizes the evolution of strategies is the same as the time scale that represents the motion of players. This process is repeated until the system reaches equilibrium.
Fig. \[overview\] illustrates the segregation of players at equilibrium. And players in the same group can move coherently, as shown in Fig. \[detail\]. During the process of direction alignment, the movement of players may lead to time-variant neighborhoods. The total number of new neighbors appearing at time $t$ can be calculated as $$n(t)=\sum_{i=1}^{N}|W_i(t)-W_i(t)\bigcap W_i(t-1)|,$$ where $|\bullet |$ represents the set size. Fig. \[typicalevo\] shows typical evolution of $n(t)$, which is divided by $N$ for normalization. Besides, we also plot the evolution of the cooperator frequency $fc$ and the average normalized velocity $V_a$ [@Vicsek1995] for comparison. When $t>500$, one can see that $n(t)$ decreases to $0$, and $V_a$ reaches a steady value. These findings imply that a static interaction network has been constructed with fixed neighborhoods and velocities of players. When $t>1000$, $fc$ fluctuates stably. Then, the equilibrium frequency of cooperators can be obtained by averaging over a long period.
Simulations are carried out in a system with $N=1000$, $L=10$. To ensure fixed topology of the interaction network, the evolution of $n(t)$ is monitored after a suitable relaxation time, which is varied from $5000$ to $10^5$ time steps and dependent on the values of $R$, $v$ and $L$. If $n(t)\leq 1$, and this condition can hold for $q=1000$ time steps, the network will be treated as a static one. Then the equilibrium frequency of cooperators is evaluated by averaging over the last 1000 generations. All data points shown in each figure are acquired by averaging over 200 realizations of independent initial states.
Results and Discussions
=======================
Fig. \[fc-b:subfig\] shows the fraction of cooperators $fc$ as a function of the payoff parameter, $b$ for the PD and $r$ for the SD, under different values of $v$ when $R$ is fixed. Clearly, the cooperation level decreases with $b$ and $r$ in both games, no matter what $v$ is. Compared with the static case ($v=0$), it is worth noting that cooperation is greatly enhanced in a large region of $b$ ($r$), when players are allowed to move with a low velocity (for example, $v=0.01$). As shown in Fig. \[fc-b:subfig\], the proportion of cooperators for $v>0$ is higher than that for $v=0$,when $b<1.17$ or $r<0.6$, and $fc$ can even approach $1$ in the SD. But such an enhancement of cooperation can only be observed for small values of $b$ ($r$), as the velocity increases from $0.01$ to $0.15$. Indeed, a rapid decrease of the cooperator frequency can be observed in both games when $v=0.15$.
To clarify the effects of $v$ on the cooperator frequency, Fig. \[fcv\] presents the dependence of the cooperation level $fc$ on the absolute velocity $v$ for different values of $R$. It displays that the fraction of cooperators for $v=10^{-6}$ is very close to that for $v=0$, irrespective of the value of $R$. Meanwhile, it can be found that whether the movement of players promotes cooperation is largely determined by the value of $R$. As shown in Fig. \[fcv:a\] and Fig. \[fcv:d\], the movement of players fails to promote cooperation for small $R$. One can see that the maximum of $fc$ for $R=0.1$ appears at $v=10^{-6}$ in both games, and the cooperation level is lower than that for $v=0$ over the entire range of $v$. For large $R$, the enhancement of cooperation resulting from the movement of players is quite limited or even disappeared. As illustrated in Fig. \[fcv:c\] and Fig. \[fcv:f\], when $v\leq 10^{-2}$, the curve of $fc$ nearly coincides with the result for $v=0$ in the PD, and only a tiny increase of $fc$ can be observed in the SD. When $R=0.4$, however, the introduction of mobility can significantly improve cooperation in both games. As shown in Fig. \[fcv:b\] and Fig. \[fcv:e\], the cooperator frequency for $v>0$ is higher than that for $v=0$ in the whole region of $v$, and there is a maximum of $fc$ at $v=10^{-2}$. The resonance-like phenomenon also implies that for a fixed $b$ ($r$), decreasing the value of $v$ cannot always promote cooperation.
Before moving forward, we would like to add some remarks about the above result. Previous work has shown that cooperation is not only possible but may even be enhanced by the non-contingent movement of players when compared with the static case [@Vainstein2007; @Dai2007; @Meloni2009; @YangArx]. And our result provides another example that helps understand the universal role of mobility in the evolution of cooperation. Particularly, Meloni and his colleagues [@Meloni2009] investigated how cooperation emerges in a population of PD players, which move in a square plane with periodic boundary conditions. The final outcome of the system has only two possibilities, all-C or all-D, when the neighborhood of each player is defined by a fixed radius $R$. The authors claimed that the movement of players can promote cooperation when the temptation to defect and the velocity of players are small, and the probability of achieving an all-C state monotonically decreases with the velocity for a fixed payoff parameter. In our model, however, the maximum cooperation level does not occur at the limit $v\rightarrow 0$, when the movement of players promotes cooperation for modest values of $R$, and a non-monotonic dependence of the cooperator frequency on the velocity of players can be observed in Fig. \[fcv:b\] and Fig. \[fcv:e\]. Compared with the result in Ref. [@Meloni2009], this phenomenon can be explained by the difference between the rules of movement. In Ref. [@Meloni2009], the network of contacts is continuously changing, because individual directions are controlled by N-independent random variables. As a result, randomness among partnerships can be preserved all the time. But in the present work, a static network of interactions is gradually developed, when individuals successfully align themselves with neighbors. Such fixed partnerships are maintained by the cancellation of periodic boundary conditions, which allows the coexistence of cooperators and defectors. As shown in Fig. \[typicalevo\], the range of alignment interaction fluctuates only when $t<500$, while the variance of neighbors causes a sharp transition of the cooperator frequency. In the process of direction alignment, the larger the value of $v$, the higher the probability for each individual to encounter different neighbors. Different with the work in Ref. [@Meloni2009], the value of $v$ also determines how long random partnerships persist. In our work, for a fixed $R$, the larger the value of $v$, the sooner the system is expected to achieve static neighborhoods. It is not easy to describe how cooperation is promoted by small values of $v$, and a heuristic explanation is that a low degree of migration can trigger the expansion of cooperator clusters, as suggested in Ref. [@YangArx]. In our model, the cooperation level for $v=10^{-6}$ is near to that for $v=0$. And for large values of $v$, cooperative clusters may be destroyed by the movement of players, making cooperators vulnerable to defectors. Thus similar to the so called evolutionary coherence resonance [@Perc2006a; @Perc2006], the maximum level of cooperation can be induced by an optimal amount of randomness, which is determined by the absolute velocity $v$ of players. In addition, results in Fig. \[fcv\] have also shown that the movement of players can inhibit cooperation for a small (large) value of $R$. Next, we will make discussions about the role of $R$ in the evolution of cooperation.
In Fig. \[fc-R:subfig:c\] and Fig. \[fc-R:subfig:d\], we show that the cooperator frequency $fc$ varies with the radius $R$ for different values of $v$. It displays that the proportion of cooperators monotonously decreases with $R$, until the radius exceeds a certain value, and for $R<0.2$, the maximum of $fc$ appears at $v=0$. Note in the current work, interaction neighborhoods are determined by the radius $R$ at each time step. For near-zero values of $R$, there are few links among individuals in the instant network. As pairwise interactions increase with $R$, it is hard for isolated cooperators to resist the invasion of defectors. When players are allowed to move, defectors have more chances to exploit cooperators. Then for small $R$, defection becomes dominant in the population, and the movement of players inhibits the evolution of cooperation. But for larger values of $R$, cooperators are expected to get together, and the introduction of mobility causes a more rapid increase of $fc$. As shown in Fig. \[fc-R:subfig:c\] and Fig. \[fc-R:subfig:d\], the curves of $fc$ for $v>0$ depart around $R=0.09$, and then begin increasing with $R$ in both games. For each value of $v$, the increase of $R$ induces a resonance-like phenomena, and the cooperator frequency $fc$ reaches a maximum around $R=0.6$. This finding indicates that intermediate values of $R$ are most favorable for cooperation, since the system approaches a fully connected network for extremely large $R$ in the stationary state. It also helps explain why the movement of players fails to give evident enhancement to the cooperation level at large values of $R$. Indeed, for large $R$, the network of interactions is almost static, since individuals can quickly align themselves with neighbors. As a result, though the maximum of $fc$ decreases with $v$, the curves of $fc$ for different values of $v$ gradually merge at large values of $R$. In Fig. \[fc-R:subfig:a\] and Fig. \[fc-R:subfig:b\], we show the dependence of the cooperation level on the radius $R$ for different values of $b$ ($r$). It displays that the resonance-like phenomena is greatly influenced by the payoff parameter, and for a fixed $R$, the maximum of $fc$ decreases with $b$ ($r$). For large $b$ ($r$), the cooperator frequency $fc$ monotonously decreases with $R$, and the maximum level of cooperation appears at the limit $R\rightarrow 0$. But for $r=0.2$, the system can reach an absorbing state of all cooperators. This is because the SD is more favorable for cooperators than the PD.
In Fig. \[dens\], we plot the cooperator frequency $fc$ against the initial density $\rho$ for fixed payoff parameters when $R=0.5$, $v=0.05$. One can see that the behavior of $fc$ caused by the variance of $\rho$ is similar to that shown in Fig. \[fc-R:subfig\]. This is because both $R$ and $\rho$ can influence the size of neighborhood. For instance, when $t=0$, the average degree of the interaction network can be written as $<k>=NR^2\pi/L^2=\rho R^2\pi$. When the players are located on the vertices of a fixed network, previous results have shown the resonant behavior of the cooperator frequency around certain values of the average degree [@Tang2006]. And our work can be viewed as an extension to the dynamic interaction network that appears during the movement of players. For small $\rho$, all agents are widely dispersed in the plane, and cooperators cannot get enough support from cooperative neighbors. Though the chance of forming cooperative clusters increases with $\rho$, the proportion of cooperators monotonously decreases until $\rho>0.13$, as shown in the inset of Fig. \[dens\_pd\]. Large values of $\rho$ are also harmful to cooperators. This is because the mean field situation is nicely recovered for large $\rho$, in which interactions almost take place among each pair of players. Between these two limits, one can find that the cooperation level can reach the maximum for moderate values of $\rho$. It has been reported that the cooperation level can reach a peak at some values of $\rho$, when the players are running in a square with periodic boundary conditions [@Meloni2009]. And our results indicate the importance of the initial density $\rho$ to the evolution of cooperation, even when the boundary restrictions are removed. In the SD, one can find the similar phenomenon that observed in the PD: when $r=0.6$, the cooperator frequency $fc$ first decreases for small $\rho$, then increases with $\rho$ until reaching the maximum, and decreases for large $\rho$. Note when $r=0.2$, the cooperator frequency $fc$ monotonously increases with $\rho$, and the system can reach an absorbing state of full cooperators at last.
Conclusion
==========
To summarize, we have investigated the PD game and the SD game in a population of mobile players, which move at a constant speed $v$ while interacting with neighbors within a fixed radius $R$. Through alignment of traveling direction, individuals develop static interaction neighborhoods during movement. Numerical simulations show that cooperation can be maintained with simple strategies, provided the payoff parameter and the velocity are small. Compared with the case for $v=0$, cooperation can be enhanced by the movement of players, and there is an optimal value of $v$ to induce the maximum cooperation level. However, such enhancement of cooperation can only be observed for modest values of $R$. This is because cooperators have less chances to form clusters for small $R$, while interaction neighborhoods approach a static, full connected network for large $R$. Besides, when $v$ is fixed, our results suggest that intermediate values of $R$ and initial population densities $\rho$ are most favorable for cooperation, since both $R$ and $\rho$ can influence the size of neighborhood. The resonance-like phenomenon caused by $R$ and $\rho$ is also affected by the payoff parameter, and for a small cost-to-benefit ratio, the system can reach an absorbing state of full cooperators. Our results may be relevant for understanding the role of information flows in cooperative, multi-vehicle systems [@Olfati-Saber2007].
The work is jointly supported by the National Natural Science Foundation of China (61004088), the Key Foundation for Basic Research from Science and Technology Commission of Shanghai (09JC1408000) and the Aeronautic Science Foundation of China(20100157001).
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---
abstract: 'We discuss the applicability of classical control theory to problems in smart grids and smart cities. We use tools from iterated function systems to identify controllers with desirable properties. In particular, controllers are identified that can be used to design not only stable closed-loop systems, but also to regulate large-scale populations of agents in a predictable manner. We also illustrate by means of an example and associated theory that many classical controllers are not be suitable for deployment in these applications.'
author:
- |
André R. Fioravanti, Jakub Mareček, Robert N. Shorten,\
Matheus Souza, Fabian R. Wirth[^1]
bibliography:
- 'ref.bib'
- 'mdps.bib'
title: '**On Classical Control and Smart Cities**'
---
Introduction
============
Smart Grid and Smart City research, at a very high level, is about making the best use of existing resources, as we try to manage meet demand for them under application-specific constraints. This is a classical consideration of Control Theory, and while classical control has much to offer in such application areas, there are aspects of these contemporary applications that also offer the opportunity for practitioners to explore new boundaries in control [@7563957].
First, classical control is typically concerned with regulating a single system such that the system behaviour achieves a desired behaviour in an optimal way. In contrast, in Smart Grid and Smart City applications, the aggregate effect of the actions of an ensemble of (often human) agents is a variable of considerable interest. Further, classical control is concerned with the control of systems whose structure does not vary over time. On the other hand, in Smart Grid and Smart City applications, we typically wish to control and influence the behaviour of large-scale populations, where the number of agents in the ensemble may be uncertain and varying over time. Third, data sets involved are often obtained in a closed loop setting. That is, operators’ decisions are often reflected in the data sets. Finally, a fundamental difference between classical control and Smart Grid and Smart City control, is the need to study the effect of the control signals on the statistical properties of the populations that we wish to influence. Among all of these fundamental differences, it is this last issue, of the need of ergodic feedback systems, that is perhaps most alien to the classical control theorist, and yet the issue that is perhaps the most pressing in real-life applications since the predictability, at the level of individual agents, underpins operators’ ability to issue contracts.
In this paper, our objective is to discuss and explain the need for ergodic control design, present some initial results that identify controllers that give rise to ergodic feedback systems, and also identify some classical controllers, which may give rise to difficulties. Our starting point is the observation that many problems that are considered in Smart Grids and Smart Cities can be cast in a framework, where a large number of agents, such as people, cars, or machines, often with unknown objectives, compete for a limited resource. The challenge of allocating these resources in a manner that is not wasteful, which gives an optimal return on the use of these resources for society, and which, in addition, gives a guaranteed level of service to each of the agents competing for that resource, gives rise to a whole host of problems, which are best addressed in a control-theoretic manner. From a practical perspective, some of these problems may seem unrelated to traditional applications of control. For example, balancing supply and demand in an energy system may seem familiar to a control engineer, at a high level, there are many variants and contract types, which introduce much additional complexity. Allocating parking spaces, regulating cars competing for shared road space, allocating shared bikes, all at the same time guaranteeing fair and equal access to the participants, are examples of Smart City type applications that seem more removed from the traditional interests of control engineering. However, while each of these applications seem very different, certain key features remain the same. Resource utilisation should be maximised while delivering a certain quality of service to individual agents. From the perspective of a control engineer, this latter statement decompose into three objectives; two of which are familiar to the classical control engineer, one of which constitutes a relatively new consideration. Our first objective is to fully utilise the resource. From a control engineers’ perspective, this is a regulation problem. We would then like to make optimal use of the resource. While both of these objectives are concerned with affecting the aggregate behaviour of an agent population, they make no attempt to control the manner in which the agents orchestrate their behaviour to achieve this aggregate effect. Our final, and third, consideration thus focuses on the effects of the control on the microscopic properties of the agent population.
Ultimately, this third concern focuses on the stochastic process that governs the share of the resource that is allocated to an [*individual agent*]{}. For example, we may wish that each agent, on average, receives a fair share of the resource over time, or, at a much more fundamental level, we wish the average allocation of the resource to each agent over time to be a stable quantity that is entirely predictable and which does not depend on initial conditions, and which is not sensitive to noise entering the system. From the point of view of the design of the feedback system, these latter concerns are related to the existence of the [*unique invariant measure*]{} that governs the distribution of the resource amongst the agents in the long run. Thus, the design of feedback systems for deployment in cities must consider not only the traditional notions of regulation and optimisation, but also the guarantees concerning the existence of this unique invariant measure. As we shall see, this is not a trivial task and many familiar control strategies, in very simple situations, do not necessarily give rise to feedback systems, which posses all three of these features.
The Problem {#sec:problem}
===========
![Feedback model.[]{data-label="system"}](diagram/diagram-pi2){width="\columnwidth"}
\[c\][$-$]{} \[b\][[$r$]{}]{} \[b\][[$e(k)$]{}]{} \[b\][[$\pi(k)$]{}]{} \[bl\][[$\hat y(k)$]{}]{} \[c\][[$y(k)$]{}]{} \[c\][$\mathcal{C}$]{} \[c\][$\mathcal{F}$]{} \[c\][$\mathcal{S}_1$]{} \[c\][$\mathcal{S}_2$]{} \[c\][$\mathcal{S}_N$]{} ![Feedback model.[]{data-label="system"}](system.eps "fig:"){width="0.8\columnwidth"}
We consider a closed loop displayed in Figure \[system\], comprising a controller, a number of agents, and a filter, in discrete time. A controller $\mathcal{C}$ produces a signal $\pi(k)$ at time $k$. In response, $N \in {\mathbb{N}}$ agents modelled by systems $\mathcal{S}_1$, $\mathcal{S}_2$, …, $\mathcal{S}_N$ amend their use of the resource. We model the use $x_i(k)$ of agent $i$ at time $k$ as a random variable, where the randomness can be a result of the inherent randomness in the reaction of user $i$ to the control signal $\pi(k)$, or the response to a control signal that is intentionally randomized, or the receipt of the control signal subject to random perturbations. The aggregate resource utilisation $y(k) = \sum_{i = 1}^N x_i(k)$ at time $k$ is then also a random variable. The controller does not have access to either $x_i(k)$ or $y(k)$, but only to the error signal $e(k)$, which is the difference of $\hat y(k)$, the output of a filter $\mathcal{F}$, and $r$, the desired value of $y(k)$. Even though this setup is elementary, it represents many applications found in Smart Grid and Smart City applications.
Let us consider two simple examples of the structures ${\mathcal C} \, : \, (e, x_c) \mapsto \pi$ and ${\mathcal F} \, : \, (y, x_f) \mapsto \hat y$, where $x_c \in X_C \subseteq {\mathbb{R}}^{n_c}$ is the internal state of the controller in dimension $n_c$, $x_f \in X_F \subseteq {\mathbb{R}}^{n_f}$ is the internal state of the filter in dimension $n_f$, and $n_c, n_f$ are non-negative integers:
\[linear\] In the linear setting, the controller dynamics may be: $$\label{eq_c}
{\mathcal C} ~:~ \left\{ \begin{array}{rcl}
x_c(k+1) & = & A^c x_c(k) + B^c e(k), \vspace{0.1cm} \\
\pi(k) & = & C^c x_c(k) + D^c e(k),
\end{array} \right.$$ where $x_c \, : \, {\mathbb{N}}\to {\mathbb{R}}^{n_c}$ is the internal state of the controller in dimension $n_c$. One could adopt a linear model for the filter ${\mathcal F}$, based on the classic IIR/FIR structures [@Oppenheim]. Remembering that $y$ is a linear combination of the states, one has $$\label{eq_f}
{\mathcal F} ~:~ \left\{ \begin{array}{rcl}
x_f(k+1) & = & A^f x_f(k) + B^f y(k) + C^f \tilde y(k), \vspace{0.1cm} \\
\hat y(k) & = & D^f x_f(k),
\end{array} \right.$$ where $\tilde y$ stores the previous $M$ values of $y$; that is, $\tilde y$ evolves by $\tilde y(k+1) = J y(k) + L \tilde y(k)$ with $$\label{JL}
J = \left[ \begin{array}{c} 1 \\ 0 \\ \vspace{-0.12cm} 0 \\ \vdots \\ 0 \end{array} \right], \quad L = \left[ \begin{array}{ccccc} 0 & 0 & \cdots & 0 & 0 \\ 1 & 0 & \cdots & 0 & 0 \\\vspace{-0.12cm} 0 & 1 & \cdots & 0 & 0 \\ \vdots & \vdots & \ddots & \vdots & \vdots\\ 0 & 0 & \cdots & 1 & 0 \end{array} \right].$$
As a very special case, one could consider:
\[linear2\] A Proportional-Integral (PI) controller: $$\label{pid1}
{\mathcal C} ~:~ \pi(k) = \pi(k-1) + \kappa \big[ e(k) - \alpha e(k-1) \big],$$ with a state $x_c$ modelling one-element buffers for $e$ and $\pi$, and a moving-average filter: $$\label{fir}
{\mathcal F} ~:~
\hat y(k) = \frac{y(k) + y(k-1)}{2},$$ with state $x_f$ modelling one-element buffer for $y$.
Loosely speaking, our aim is for the controller ${\mathcal C}$ and filter ${\mathcal F}$ to make the long-run behaviour of the system acceptable, for all possible distributions of the initial state, both from the perspective of the individual agents, and from the perspective of the macroscopic behaviour of the network of agents. Somewhat more formally, we aim to find conditions such that for every stable linear controller $\mathcal{C}$ and every stable linear filter $\mathcal{F}$, the feedback loop regulates the aggregate resource use, and converges in distribution to a unique invariant measure. The existence of a unique invariant measure may mean many things in practice: for example that agents get a fair share of the resource, or even that their time-averaged share is predictable in some sense. For example within Smart Grids, many demand-response management (DRM) schemes [@Hiskens2011] consider interruptible loads (IL) [@192993], such as in heating, ventilation, and air conditioning (HVAC) systems [@Alagoz2013; @Salsbury2013]. Specifically, one aims to regulate the total active power demand $r$ of the ILs, while $x_i(k)$ determines the active power supplied to IL $i$ at time $k$. Although continuous changes to the demand are possible, in theory, many current schemes consider this binary notion of interruptibility [@192993], in practice. Then, it may desirable for an IL to have the same rate of interruptions as any other IL with the same costs, independent of whether the IL was on or off at the start. Many researchers, e.g. [@5560848; @Alagoz2013; @Salsbury2013], propose to use the classic Proportional-Integral-Derivative (PID) controllers in this setting. As we shall see, such controllers do not guarantee the existence of the unique invariant measure and it is not difficult to construct situations in which they fail to provide agent-specific outcomes independent of the agent’s initial state.
Notation and Preliminaries
==========================
Markov Chains
-------------
Let $\Sigma$ be a closed subset of ${\mathbb{R}}^n$ with the usual Borel $\sigma$-algebra ${\cal B}(\Sigma)$. We call the elements of ${\cal B}$ events. A Markov chain on $\Sigma$ is a sequence of ($\Sigma$-valued) random vectors $\{ X(k)\}_{k\in{\mathbb{N}}}$ with the Markov property, that is the probability of an event conditioned on past events is given by conditioning on the previous event, *i.e.*, we always have $$\begin{gathered}
{\mathbb{P}}(X(k+1) \in G \,|\, X(j)=x_{j}, \, j=0, 1, \dots, k) \\ = {\mathbb{P}}\big(X(k+1)\in G\,|\, X(k)=x_{k}\big),\end{gathered}$$ where $G$ is an event and $k \in {\mathbb{N}}$. We assume the Markov chain is time-homogeneous and the transition operator $P$ of the Markov chain is defined by $$P(x,G) := {\mathbb{P}}(X(k+1) \in G \vert X(k) = x).$$ If $X_0$ is distributed according to an initial distribution $\lambda$ we denote by ${\mathbb{P}}_\lambda$ the probability measure induced on the space of sequences with values in $\Sigma$. Conditioned on an initial distribution $\lambda$, the random variable $X(k)$ is distributed according to the measure $\lambda_k$ which is given by $$\label{eq:measureiteration}
\lambda_{k+1}(G) := \lambda_{k}P(G) := \int_{\Sigma} P(x, G) \, \lambda_k(d x), \quad G
\in {\cal B}.$$ A measure $\mu$ on $\Sigma$ is called invariant with respect to the Markov process $\{ X(k) \}$ if it is a fixed point for the iteration described by , *i.e.*, if $\mu P = \mu$. The existence of attractive invariant measures is intricately linked to ergodic properties of the system.
Invariant Measures and Ergodicity
---------------------------------
For the systems of interest to us a particular type of Markov chains are of interest: the so-called iterated function systems (IFS). In an iterated function system we are given a set of maps $\{ f_j : \Sigma \to \Sigma
\vert j \in {\cal J} \}$, where ${\cal J}$ is an index set. Associated to these maps there are probability functions $p_j: \Sigma \to [0,1]$ such that $$X(k+1) = f_j(X(k)) \quad \text {with probability } \quad p_j(X(k)).$$ It is, of course, required that $\sum_{j\in {\cal J}} p_j(x) = 1$ for all $x \in \Sigma$. Sufficient conditions for the existence of a unique attractive invariant measure can be given in terms of the central notion of “average contractivity”, which can be traced back to [@elton1987ergodic; @BarnsleyDemkoEltonEtAl1988; @barnsley1989recurrent]:
\[Barnsley\] Let $\Sigma \subset {\mathbb{R}}^n$ be closed. Consider an IFS with a finite index set ${\cal J}$, and Lipschitz maps $f_j:\Sigma \to \Sigma$, $j\in {\cal J}$. Assume that the probability functions $p_j$ are Lipschitz continuous and bounded below by $\eta > 0$. If there exists a $\delta>0$ such that for all $x, y \in \Sigma, x \not = y$ $$\begin{aligned}
\sum_{j\in {\cal J}} p_j(x) \log \left( \frac{ \|{f_j (x) - f_j (y)}\| }{ \|x-y\| } \right) < -\delta < 0,
\end{aligned}$$ then there exists an attractive (and hence unique) invariant measure $\mu$ for the IFS.
We can combine Theorem \[Barnsley\] with an ergodic theorem by Elton, [@elton1987ergodic], to obtain that for all (deterministic) initial conditions $x \in
\Sigma$ the limit $$\label{eq:ergodicprop}
\lim_{k \to\infty} \frac{1}{k+1} \sum_{\nu=0}^k X(\nu) = \mathbb{E}(\mu)$$ exists almost surely (${\mathbb{P}}_{x_0}$) and is independent of $x_0 \in \Sigma$. The limit is given by the expectation with respect to the invariant measure $\mu$. For more general theorems, the reader is referred to recent surveys [@iosifescu2009iterated; @stenflo2012survey]. For one of the first uses of IFS in the control community, see [@Branicky1994; @Branicky1998].
More generally, an invariant measure $\mu$ is called ergodic for a Markov process, if for $\mu$-almost all initial conditions holds almost surely.
From the point of view of applications in smart cities, such an ergodic property should be a minimum requirement. We want to avoid situations, where the average allocation of resources to agents depends on their initial conditions, on possible initial conditions of controllers and filters, etc. In addition, it is desirable to shape the expected value so that an overall optimum is obtained.
With this background, our general problem considered in this paper is modelled as a Markov chain on a state space representing all the system components. We thus let $X_S = \{ (x_i) \}$ be the set of vectors representing the possible values for the agents. The spaces $X_F,X_C$ contain the possible internal states for filter and central controller. Our system thus evolves on the state space $\Sigma :=
X_S \times X_C \times X_F$.
Ergodic Invariant Measures and Coupling
---------------------------------------
An important observation is that coupling arguments provide criteria for the non-existence of a unique invariant measure. Coupling arguments have been used since the theorem of Harris [@harris1960lower; @lindvall2012lectures], and are hence sometimes known as Harris-type theorems. Generally, they link the existence of a coupling with the forgetfulness of initial conditions.
To formalise these notions, let us denote the the space of trajectories of a $\Sigma$-valued Markov chain $\{ X(k)\}_{k\in{\mathbb{N}}}$, *i.e.*, the space of all sequences $(x(0),x(1),x(2),\ldots)$ with $x(k) \in \Sigma$, ${k\in{\mathbb{N}}}$, by $\Sigma^\infty$ (the “path space”). The measure space over $X^\infty$ is denoted by $M(\Sigma^\infty)$. Recall, for example, ${\mathbb{P}}_\lambda \in M(\Sigma^\infty)$, the probability measure induced on the path space by the initial distribution $\lambda$ of $X_0$.
A coupling of two measures $P_{\mu_1},P_{\mu_2} \in M(\Sigma^\infty)$ is a measure on $\Sigma^\infty\times \Sigma^\infty$ whose marginals coincide with $P_{\mu_1}, P_{\mu_2}$. To be precise, consider $\Gamma \in M(\Sigma^\infty\times \Sigma^\infty)$, i.e., a measure over the product of the two path spaces. Clearly, $\Gamma$ can be projected to the space of measures over one or the other path space $\Sigma^\infty$; we denote the projectors $\Pi^{(1)} \Gamma$ and $\Pi^{(2)} \Gamma$. The set $C(P_{\mu_1}, P_{\mu_1})$ of couplings of $P_{\mu_1}, P_{\mu_2} \in M(\Sigma^\infty)$ is then defined by $$\begin{aligned}
\{
\Gamma \in M(\Sigma^\infty \times \Sigma^\infty) \; : \;
\Pi^{(1)} \Gamma = P_{\mu_1}, \Pi^{(2)} \Gamma = P_{\mu_2}
\}.\end{aligned}$$ We say that a coupling $\Gamma$ is an asymptotic coupling if $\Gamma$ has full measure on the pairs of convergent sequences. To make this precise consider the following set denoted ${\cal D}$: $$\begin{aligned}
\left \{ (x_1, x_2) \in \Sigma^\infty \times \Sigma^\infty \; : \; \lim_{k\to \infty} \left\| x_1(k) - x_2(k) \right\| =0
\right \}\end{aligned}$$ $\Gamma$ is an asymptotic coupling if $\Gamma({\cal D}) = 1$. The following statement if a specialization of [@hairer2011asymptotic Theorem 1.1] to our situation:
\[coupling-argument\] Let $P$ be a Markov operator admitting two ergodic invariant measures $\mu_1$ and $\mu_2$. The following are equivalent:
1. $\mu_1 = \mu_2$.
2. There exists an asymptotic coupling of $P_{\mu_1}$ and $P_{\mu_2}$.
Consequently, if no asymptotic coupling of $P_{\mu_1}$ and $P_{\mu_2}$ exists, then $\mu_1$ and $\mu_2$ are distinct.
The Existence of a Unique Invariant Measure
===========================================
Let us consider the very simple setting of Example \[linear\] and show that the unique invariant measure exists:
\[thm01\] Consider the feedback system depicted in Figure \[system\], with ${\mathcal C}$ and ${\mathcal F}$ given in (\[eq\_c\]) and (\[eq\_f\]). Assume that each agent $i \in \{1,\cdots,N\}$ has state $x_i(k)$ governed by the following affine stochastic difference equation: $$x_i(k+1) = w_{ij}(x_i),$$ where the affine mapping $w_{ij}$ is chosen at each step of time according to a Dini-continuous probability function $p_{ij}(x_i, q(k))$, out of $w_{ij}(x_i) \defeq A_i x_i + b_{ij},$ where $A_i$ is a Schur matrix and for all $i$, $q(k)$, $\sum_j p_{ij}(x_i, q(k)) = 1$. In addition, suppose that there exist scalars $\delta_i > 0$ such that $p_{ij}(x_i,\pi)
\geq \delta_i > 0$; that is, the probabilities are bounded away from zero. Then, for every stable linear controller $\mathcal{C}$ and every stable linear filter $\mathcal{F}$, the feedback loop converges in distribution to a unique invariant measure.
Following [@BarnsleyDemkoEltonEtAl1988], the proof is centred at the construction of an iterated function system (IFS) with place-(state-)dependent probabilities that describes the feedback system. To this end, consider the augmented state $$\xi \defeq [x'~,~y~,~\tilde y~,~z_f'~,~\hat y~,~e~,~z_c'~,~q]'$$, whose dynamic behaviour is described by the difference equation $$\label{dynamical}
\xi(k+1) = w_\ell(x) \defeq \mathcal{A}\xi(k) + b_\ell,$$ where $\mathcal{A}$ is the matrix $$\notag
{\scalebox{0.8}{$
\begin{bmatrix}
\hat{A} & & & & & & & \\
\mathbf{1}'\hat A & 0 & & & & & & \\
0 & J & L & & & & & \\
0 & B^f & C^f & A^f & & & & \\
0 & D^f B^f & D^f C^f & D^f A^f & 0 & & & \\
0 & -D^f B^f & -D^f C^f & -D^f A^f & 0 & 0 & & \\
0 & 0 & 0 & 0 & 0 & B^c & A^c & & \\
0 & -D^c D^f B^f & - D^c D^f C^f & -D^c D^f A^f & 0 & C^c B^c & C^c A^c & 0
\end{bmatrix}$}},$$ where $J$ and $L$ are from , $\hat A \defeq \mathbf{diag}(A_i)$, and $b_\ell$ is built from all the combinations of the vectors $b_{ij}$ and other signals. To apply Corollary 2.3 from [@BarnsleyDemkoEltonEtAl1988], two observations must be made. First, note that each map $w_\ell$ is chosen with probability $p_\ell(\xi) \geq
\prod_{i=1}^N \delta_i > 0$ and, thus, they are bounded away from zero. Second, since $\sigma({\mathcal A}) = \sigma(\hat A) \cup \sigma(L)
\cup \sigma(A^f) \cup \sigma(A^c) \cup \{0\}$ and, by hypothesis, $A_i$, $A^f$ and $A^c$ are Schur matrices, then for any induced matrix norm $\|\cdot\|$ there exists an $m \in {\mathbb{N}}$ sufficiently large such that $\|{\mathcal A}^m\| < 1$. This provides the desired contraction on average (after a finite number of steps). The result then follows from a variant of Theorem \[Barnsley\], see [@BarnsleyDemkoEltonEtAl1988]. The proof is complete.
Some comments on the result of Theorem \[thm01\] are in order.
Dini’s condition on the probabilities may, obviously, be replaced by simpler, more conservative assumptions, such as Lipschitz or Hölder conditions [@BarnsleyDemkoEltonEtAl1988].
As we shall see, the requirement $p_{ij}(x_i,\pi) \geq \delta_i > 0$ in the theorem statement is not an artefact of our analysis, as $p_{ij}(x_i, \pi) = 0$ may lead to a non-ergodic behaviour.
We should also explicate:
Under the assumptions of Theorem \[thm01\]: The case of $x_i \in \{ 0, 1\}$ is obtained by setting the $A_i$ to zero and by introducing $b_{i0}=0$ and $b_{1i}=1$.
Finally, considering that Theorem \[Barnsley\] does not require linearity, it is clear that one can extend the results to non-linear systems, as we do in a follow-up of this paper.
Switched Controllers
====================
Next, let us consider negative results. Switched controllers are widely adopted by designers in practical applications that span several areas of engineering [@Shorten07]. The gain in flexibility provided by these controllers makes them suitable candidates for some smart cities problems. In this section, consider the following structure for the controller ${\mathcal C}$ $${\mathcal C} ~:~ \left\{ \begin{array}{rcl}
x_c(k+1) & = & A_{c\sigma(k)} x_c(k) + B_{c\sigma(k)} e(k), \vspace{0.1cm} \\
\pi(k) & = & C_{c\sigma(k)} x_c(k) + D_{c\sigma(k)} e(k),
\end{array} \right.$$ where, once again, $x_c\in {\mathbb{R}}^{n_c}$ is its internal state and $\sigma \,
: \, {\mathbb{N}}\to [n_s] \defeq \{1,\cdots,n_s\}$ is the [*switching signal*]{}, which, at each time step, assigns one of the $n_s$ constituent systems $(A_{ci}, B_{ci}, C_{ci}, D_{ci})$, $i \in [n_s]$. Two complementary behaviours for the switching function are usually considered whenever one is analysing switched systems: switching control and switching perturbation.
Let us consider a simple example consisting of two agents ${\mathcal S}_1$ and ${\mathcal S}_2$ with only one common resource that must be shared, which means $r = 1$. As before, let us denote by $x_1$ and $x_2$ the agents states, which indicate whether each agent has access to the common resource or not; that is, $x_i \in \{0,1\}$, for $i \in \{1,2\}$. For simplicity, consider ${\mathcal F} = 1$ and, thus, $\hat{y} = y = x_1 + x_2$. Once again, each agent’s behaviour is affected by the broadcast pricing signal $\pi$, which is computed by the controller ${\mathcal C}$. Let us assume each agent ${\mathcal S}_i$ switches its current state $x_i$ with probability $$\label{prb}
\mathbb{P}(x_i(k+1) = j ~|~ x_i(k) \neq j) = \pi,$$ for all $k \in {\mathbb{N}}$. Given the identity above, in an attempt to adequately control the agents stochastic behaviour, the switched controller $${\mathcal C} ~ : ~ \pi(k) = \frac{1}{2} |e(k)|, ~ \forall k \in {\mathbb{N}}.$$ seems to be a suitable candidate. Under the action of this controller, the closed-loop system depicted in Figure \[system\] corresponds to the Markovian process whose transitions are represented in Figure \[markov\_ne\], where the chain modes are $(x_1,x_2)$. As this Markov chain presents two different stationary states, the process is not ergodic; see [@Leon_Garcia]. Therefore, under mild conditions, it may happen that $x_1$ will never get access to the resource.
![Closed-loop transitions.[]{data-label="markov_ne"}](transitions){width="0.75\columnwidth"}
\[c\][$~(0,0)$]{} \[c\][$~(0,1)$]{} \[c\][$~(1,0)$]{} \[c\][$~(1,1)$]{} \[c\][$\frac{1}{4}$]{} \[c\][$1$]{} ![Closed-loop transitions.[]{data-label="markov_ne"}](markov_ne "fig:"){width="0.6\columnwidth"}
Inspired by the previous example, let us now consider a more comprehensive case. Consider the problem where $r = 50$ units of a certain resource must be shared among $N = 100$ agents. As before, each agent ${\mathcal S}_i$ has a state $x_i \in \{0,1\}$, with switching law given by (\[prb\]). The broadcast signal is $\pi$, computed by the controller ${\mathcal C}$. Once again we make use of an input-dependent switched controller without memory $${\mathcal C} ~:~ \pi(k) = \frac{1}{100}|e(k)|,\quad k \in {\mathbb{N}}.$$ For each initial number of active agents between $0$ and $N$, we performed Monte-Carlo experiments with $10^4$ realisations. In Figure \[markov\_ne\], the results for two complementary situations are shown. The upper sequence represents the expected value of resource consumption for agents whose initial state is $x_i(0) = 1$, whereas the lower series corresponds to the same value for agents with $x_i(0) =
0$. This plot shows that there exists a dependence of the expected value $\bar x_i$ on the initial condition of an agent, implying the process is non-ergodic.
From a control-theoretical viewpoint this result may seem adequate, since $\lim_{k
\to\infty} e(k) = 0$ almost surely. Thus, resources are totally distributed. However, as the obtained closed-loop system does not present ergodicity, the resources are not shared fairly amongst all agents. Hence, the controller fails to achieve predictability.
![Switched controller simulation results: average resource consumption against the number of initial active systems[]{data-label="nerg"}](fig_nerg){width="\columnwidth"}
\[t\][Number of initial active systems]{} \[b\][Average resource consumption]{} ![Switched controller simulation results.[]{data-label="nerg"}](fig_nerg "fig:"){width="\columnwidth"}
We now consider the switching control case, in which the switching signal $\sigma$ is a design variable, together with the controller matrices $(A_{ci}, B_{ci}, C_{ci}, D_{ci})$, $i \in [n_s]$, as in Example 1. A [*state dependent*]{} switching rule is a well-established structure for this control design problem [@Liberzon99]. In this setting, $\sigma$ is a composition of a function $u \, : \, {\mathbb{R}}^{n_c} \to [n_s]$ with the controller state $x_c \, : \, {\mathbb{N}}\to {\mathbb{R}}^{n_c}$, that is, $\sigma(k) = u(x_c(k))$ for all $k \in {\mathbb{N}}$. Let us illustrate an issue that could possibly arise in this formulation. Consider that $n_s = 2$ and that the function $u$ divides the state space in two disjoint sets $\mathbb{S}_1$ and $\mathbb{S}_2$ (e.g. two half-spaces), such that $u(x) = 1$ for all $x \in \mathbb{S}_1$ and $u(x) = 2$ for all $x \in \mathbb{S}_2$. Since the dynamic behaviour of (\[eq\_c\]) has to be considered, the mappings $w_\ell(\cdot)$ will also depend on the internal mode of the switched controller ${\mathcal C}$ and, specifically, on the switching function $\sigma$. Noting that, whenever $x_c(k)$ is in $\mathbb{S}_1$, the switching function is such that $\sigma(k) = 1$, there is a null probability of switching to the other controller mode, one concludes that the assumptions in [@BarnsleyDemkoEltonEtAl1988] are not satisfied by this controller. This issue can be circumvented if the designer allows a small probability of switching to every other mode at any point in the state space. In the switching perturbation case, the designer focuses on determining the controller matrices $(A_{ci}, B_{ci}, C_{ci}, D_{ci})$, $i
\in [n_s]$, aiming to provide a robust and stable closed-loop system with respect to any possible switching signal. It can be seen as a two-level design, where the controller realisation and the switching signal can be devised separately, targeting different objectives. For this particular case, we guarantee the existence of a unique invariant measure by ensuring the existence of a contraction property, and under some additional assumptions. For example, a contraction can be ensured using the following lemma, which is based on the classic stability result [@Daafouz02]. In the following statement, for symmetric matrices, $P,Q$ we write $P\prec Q$, if $Q-P$ is positive definite.
\[lem01\] Consider the switched linear system $$\label{eq:switchsys}
x(k+1) = A_{\sigma(k)} x(k),$$ where $x \in{\mathbb{R}}^{n}$ is the state and $\sigma \, : \, {\mathbb{N}}\to [n_s]$ is a switching sequence. If there exist positive definite symmetric matrices $P_1,\cdots,P_{n_s} \in \mathbb{S}_+^{n}$ satisfying the following linear matrix inequalities $$\label{jamal}
A_i' P_j A_i - P_i \prec 0,$$ for all $(i,j) \in [n_s]^2$, then $\eqref{eq:switchsys}$ is exponentially stable for arbitrary switching sequences $\sigma$ and there exists $m \in {\mathbb{N}}$ sufficiently large such that $$\label{norm}
\|A_{i_m} \cdots A_{i_1}\| < 1$$ holds for any sequence of indices $i_1,\cdots,i_m \in [n_s]$.
First, note that, if there exist matrices $P_1,\cdots,P_{n_s}$ satisfying (\[jamal\]), then there exists a sufficiently small scalar $\epsilon \in (0,1)$ such that $$\label{eps}
A_i' P_j A_i \prec (1 - \epsilon)^2 P_i$$ hold for all $(i,j) \in [n_s]^2$. Now define the quadratic, time-varying function $v \, : \, {\mathbb{N}}\times {\mathbb{R}}^n \to {\mathbb{R}}_+$, given by $$\label{lyap}
v(k,x) = x' P_{\sigma(k)} x.$$ It is clear [@Horn1] that there exist constants $\alpha$ and $\beta$ such that $$\label{bound}
\alpha \|x\|^2 \leq x' P_i x \leq \beta \|x\|^2 \quad \forall \, i \in [n_s]$$ hold for any $x \in {\mathbb{R}}^n$; here, $\|\cdot\|$ can be any vector norm.
Let us first prove that is exponentially stable for arbitrary switching signals $\sigma$. To this end, fix a switching sequence $\sigma$ and note that (\[eps\]) implies that $v$ satisfies along trajectories $\{ x(k) \}$ corresponding to this switching sequence that $$\label{dec}
v(k+1,x(k+1)) \leq (1 - \epsilon)^2 v(k,x(k))$$ for all $k \in {\mathbb{N}}$. Therefore, an inductive argument applied to (\[dec\]) yields $$v(k,x(k)) \leq (1 - \epsilon)^{2k} v(0,x(0)),$$ for any $k \in {\mathbb{N}}$ and any given initial condition $x(0)\in {\mathbb{R}}^n$. Using the bounds (\[bound\]), it follows that $$\label{exp}
\|x(k)\| \leq c (1 - \epsilon)^k \|x(0)\|$$ holds for all $k \in {\mathbb{N}}$, where $c \defeq \sqrt{\beta/\alpha}$. As $\sigma$ and $x(0)$ were arbitrary in this argument, we obtain exponential stability.
Finally, let us move our attention to (\[norm\]). Noting that $$\|A_{i_k} \cdots A_{i_1}\| = \max_{w \neq 0}\frac{ \|A_{i_k} \cdots A_{i_1} w\| } {\|w\|},$$ one can take $w$ as any initial condition $x(0)$ to $\Sigma$ and, since (\[exp\]) holds for any switching sequence $\sigma$ and any $x(0)$, it follows that $$\|A_{i_k} \cdots A_{i_1}\| = \max_{x(0) \neq 0} \frac{ \|x(k)\| }{\|x(0)\|} \leq c(1 - \epsilon)^k$$ holds for all $k \in {\mathbb{N}}$. Since there exists a sufficiently large $m \in
{\mathbb{N}}$ such that $(1 - \epsilon)^m < c^{-1}$, (\[norm\]) holds. This completes the proof.
The implication of Lemma \[lem01\] is that, provided Equation \[eq:switchsys\] represents the closed loop system, and some additional matrix inequality conditions hold, there always exists a sufficiently large number of jumps that ensures the contractivity of any possible switching chain. The existence of a unique invariant measure can be shown under assumptions similar to those of Theorems 1 and 3.
Control with poles on the unit circle {#comments-pi}
=====================================
We finally come to the most interesting observation of the paper. In many applications, one wishes the error $e = r - \hat y$ to be convergent, that is, $\lim_{k \to \infty} e(k) = 0$. Consequently, controllers with integral action, such as the Proportional-Integral (PI) controller, are widely adopted [@Franklin; @Franklin_dig]. See Example \[linear2\], which can be implemented as: $$\label{pid1}
\pi(k) = \pi(k-1) + \kappa \big[ e(k) - \alpha e(k-1) \big],$$ which means its transfer function from $e$ to $\pi$ is given by $$\label{pid2}
C(z) \defeq \frac{ \hat \pi (z) }{ \hat e(z) } = \kappa\frac{1 - \alpha z^{-1}}{1 - z^{-1}}.$$ Since this transfer function is not asymptotically stable, any associated realisation matrix $A_c$ will not be Schur. Note that this is the case for any controller with any sort of integral action, *i.e.*, pole at $z = 1$.
\[thm:pole\] Consider $N$ agents with states $x_i, i=1,\ldots,N$. Assume that there is an upper bound $M$ on the different values the agents can attain, *i.e.*, for each $i$ we have $x_i \in {\cal A}_i =\{
a_1,\ldots,a_M \}\subset {\mathbb{R}}$ for a given set ${\cal A}_i$.
Consider the feedback system in Figure \[system\], where ${\mathcal
F}\, : \, y \mapsto \hat y$ is a finite-memory moving-average (FIR) filter. Let ${\cal C}_L$ be a linear marginally stable single-input single-output (SISO) system with a pole $s_1 = e^{q i\pi}$ on the unit circle where $q$ is a rational number. Assume the controller ${\mathcal C} \, : \, e \mapsto \pi$ is the cascade of ${\cal C}_L$ and a continuous map ${\cal C}_p: {\mathbb{R}}\to [0,1]$, i.e. if $\hat \pi
(k)$ is the output of ${\cal C}_L$ at time $k$, then the signal from the controller is $\pi(k) = {\cal C}_p(\hat \pi(k))$. Then the following holds.
1. The set ${\cal O}_{\cal F}$ of possible output values of the filter ${\cal F}$ is finite.
2. If the real additive group ${\cal E}$ generated by $\{ r - \hat y \mid \hat y \in {\cal O}_{\cal F} \}$ is discrete, then the closed-loop system cannot be ergodic.
One implication of the theorem is to illustrate the separation of the classical performance of the closed loop, and ergodic behaviour. It is perfectly possible for the closed loop to perform its regulation function well, and an the same time destroy the ergodic properties of the closed loop.
\(i) By assumption, the states of the agents $x \in {\mathbb{R}}^N$ can only attain finitely many values. Consequently, the set of possible values of $y$ is finite and thus also the set of possible outputs of the filter is finite, as it is just the moving average over a history of finite length.
\(ii) We denote by ${\cal E}$ the additive subgroup of ${\mathbb{R}}$ generated by the filter outputs. By (i), the set of possible inputs to the linear part of the controller is finite at any time $k\in {\mathbb{N}}$. Let $(A,B,C)$ be a minimal realization of the linear controller with $A \in {\mathbb{R}}^{n_c \times n_c}$, $B,C^T\in {\mathbb{R}}^{n_c}$. Without any loss of generality, assume that $$A =
\begin{bmatrix}
Q & 0 \\ 0 & R
\end{bmatrix},\quad B =
\begin{bmatrix}
B_1 \\ B_2
\end{bmatrix} , \quad C =
\begin{bmatrix}
C_1 & C_2
\end{bmatrix}.$$ Here $Q$ is equal to $1,-1$ or a $2 \times 2$ orthogonal matrix with the eigenvalues $s_1$ and $\overline{s_1}$. The matrix $R$ is marginally Schur stable. We will concentrate on the first (or first two) component of the state of the controller, which we denote by $x^{(1)}$. Given an initial value $x_0^{(1)}$ these states are given by $$x^{(1)}(k) = Q^k x^{(1)}_0 + \sum_{\nu=0}^{k-1}Q^{k-\nu-1} B_1 e(\nu).$$ For some power $K$ we have by assumption that $Q^K = I_2$. Thus $x^{(1)}(k)$ is an element of the set ${\cal Z}(x_0)$ given by $$\left \{ Q^k x^{(1)}(0) + \sum_{\nu=0}^{K-1} J^\nu B_1 e_\nu \ \Big\vert
\ k =
0,\ldots,K-1 ,e_\nu \in {\cal E} \right\}.$$ By assumption, this set is discrete in ${\mathbb{R}}$ or ${\mathbb{R}}^2$, as the case may be. The state space of the controller may thus be partitioned into the uncountably many equivalence classes under the equivalence relation on ${\mathbb{R}}^{n_c}$ given by $x \sim y$, if $y^{(1)} \in {\cal Z}(x)$. These are invariant under the evolution of the Markov chain.
Ergodic invariant measures which are concentrated on different equivalence classes clearly cannot couple asymptotically as the respective trajectories remain a positive distance apart. By Theorem \[coupling-argument\] the Markov chain cannot be ergodic.
We note that the conditions for non-ergodicity are satisfied, in particular, if the resource consumption of agents is always a rational number and the coefficients of the FIR filter are also rational. For implementations on standard computers this will always be the case. Other numerical issues issues may arise, though.
Let us illustrate the undesirable behaviour that may arise whenever a PI controller is being used in the closed-loop system. In the first example, we point out that the integral action may be heavily dependent on the controller state initial condition. To this end, consider the feedback system in depicted in Figure \[system\] with $N = 4$ agents, whose states $x_i$ are in the set $\{0,1\}$; as before, if $x_i = 1$, we say that agent $i$ has taken the resource or is [*active*]{}.
Our main goal is to regulate the number of active agents around the reference $r = 2$. We assume that two agents, namely $x_1$ and $x_2$, have the following probabilities of being active $$p_{12}(x_i(k+1) = 1) = 0.02 + \frac{0.95}{ 1 + \exp(-100(\pi(k) - 5))},
\notag$$ whereas the remaining agents’ probability of consuming the resource is given by $$p_{34}(x_i(k+1) = 1) = 0.98 - \frac{0.95}{ 1 + \exp(-100(\pi(k) - 1))}.
\notag$$ Note that their behaviour is, thus, complementary; indeed, if the control signal $\pi(k) \gg 5$, then $x_1$ and $x_2$ tend to be active, and, on the other hand, if $\pi(k) \ll 1$, then $x_3$ and $x_4$ are more susceptible to take the resource.
In this design problem, we implement two types of controllers ${\mathcal C}$: a PI controller and its lag approximant. For this example, the filter ${\mathcal F}$ is the moving avergage (FIR) filter defined in (\[fir\]). The PI controller is the one implemented in (\[pid1\]) with $\kappa = 0.1$ and $\alpha = -4$; this controller is approximated by a lag controller with $\kappa = 0.1$, $\alpha = -4.01$ and $\beta = 0.99$. Figure \[sim1\] points out that the PI controller regulates the average number of active agents $\bar y$, whereas the lag controller presents a steady state error (as expected). However, Figure \[sim2\] shows different average trajectories for the first agent $\bar x_1$ for different initial conditions of the controller ${\mathcal C}$, namely $x_c(0) = 50$ and $x_c(0) = -50$. As the figure points out, this agent’s behaviour is completely dependent on the initial value of $x_c$ when ${\mathcal C}$ is the PI controller; this undesired behaviour vanishes on the long run when a lag controller is used – that is, the system becomes ergodic. Figure \[Xc\] illustrates the dynamic response of the controller state $x_c$ for both initial conditions and both controllers; both cases converge to the same value for the lag structure and this is not observed when the designer uses a PI.
\[c\][$\quad x_c(0) = 50$]{} \[l\][$x_c(0) = -50$]{} \[t\][$k$]{} \[b\][$\bar y(k)$]{} \[b\][$\bar x_1(k)$]{} ![Average number of active agents dynamics.[]{data-label="sim1"}](sim1 "fig:"){width=".7\columnwidth"}
\[c\][$\quad x_c(0) = 50$]{} \[l\][$x_c(0) = -50$]{} \[t\][$k$]{} \[b\][$\bar y(k)$]{} \[b\][$\bar x_1(k)$]{} ![Average trajectory of the first agent.[]{data-label="sim2"}](sim2 "fig:"){width=".7\columnwidth"}
\[l\][$x_c(0) = 50$]{} \[l\][$x_c(0) = -50$]{} \[t\][$k$]{} \[b\][$\bar x_c(k)$]{} ![Average controller state dynamics.[]{data-label="Xc"}](Xc "fig:"){width="0.9\columnwidth"}
Conclusions
===========
We have shown that across a number of problems studied under the banners of Smart Grid and Smart Cities, classical proportional-integral (PI) control cannot give rise to a feedback system with a unique invariant measure, which can lead to non-ergodic behaviour under benign conditions. The undesirable behaviour can be alleviated by the use of an IFS-based controller or by *ad hoc* modifications to the classical controllers. Notice, in particular, that the input-dependent switched controllers fail to ensure the probabilities are bounded away from zero. This can be solved by replacing $f(\cdot) = \alpha|\cdot|$ by $\hat f(\cdot) = \alpha|\cdot| + \beta$, where both $\alpha$ and $\beta$ are defined to ensure $\pi \in (\epsilon,1 - \epsilon)$ for some $\epsilon > 0$. Similarly, the PI control can be modified by introducing a lead/lag compensator [@Franklin; @Franklin_dig], with the effect of ensuring a contraction. The same effect can be achieved by adjusting the filter. Still, IFS-based controllers may present a principled approach to the problem.
[^1]: A. R. Fioravanti and M. Souza are at the University of Campinas, Brazil. J. Marecek is with IBM Research – Ireland, in Dublin, Ireland. R. N. Shorten is at the University College Dublin, Dublin, Ireland. F. Wirth is at the University of Passau, Germany. This work was in part supported by Science Foundation Ireland grant 11/PI/1177 and received funding from the European Union Horizon 2020 Programme (Horizon2020/2014-2020), under grant agreement no 68838.
|
---
abstract: 'We present an approach that combines the local density approximation (LDA) and the dynamical mean-field theory (DMFT) in the framework of the full-potential linear augmented plane waves (FLAPW) method. Wannier-like functions for the correlated shell are constructed by projecting local orbitals onto a set of Bloch eigenstates located within a certain energy window. The screened Coulomb interaction and Hund’s coupling are calculated from a first-principle constrained RPA scheme. We apply this LDA+DMFT implementation, in conjunction with a continuous-time quantum Monte-Carlo algorithm, to the study of electronic correlations in LaFeAsO. Our findings support the physical picture of a metal with intermediate correlations. The average value of the mass renormalization of the Fe 3$d$ bands is about $1.6$, in reasonable agreement with the picture inferred from photoemission experiments. The discrepancies between different LDA+DMFT calculations (all technically correct) which have been reported in the literature are shown to have two causes: i) the specific value of the interaction parameters used in these calculations and ii) the degree of localization of the Wannier orbitals chosen to represent the Fe 3$d$ states, to which many-body terms are applied. The latter is a fundamental issue in the application of many-body calculations, such as DMFT, in a realistic setting. We provide strong evidence that the DMFT approximation is more accurate and more straightforward to implement when well-localized orbitals are constructed from a large energy window encompassing Fe-3$d$, As-4$p$ and O-2$p$, and point out several difficulties associated with the use of extended Wannier functions associated with the low-energy iron bands. Some of these issues have important physical consequences, regarding in particular the sensitivity to the Hund’s coupling.'
author:
- Markus Aichhorn
- Leonid Pourovskii
- Veronica Vildosola
- Michel Ferrero
- Olivier Parcollet
- Takashi Miyake
- Antoine Georges
- Silke Biermann
title: |
Dynamical Mean-Field Theory within an Augmented Plane-Wave Framework:\
Assessing Electronic Correlations in the Iron Pnictide LaFeAsO
---
Introduction
============
This article has two purposes. The first one is to present a new implementation of dynamical mean-field theory (DMFT) within electronic structure calculation methods. This implementation is based on a highly precise full-potential linear augmented plane wave method (FLAPW), as implemented in the Wien2k electronic structure code [@Wien2k]. The second purpose of this article is to report on DMFT calculations for the iron oxypnictide LaFeAsO, the parent compound of the ‘$1111$’-family of recently discovered iron-based superconductors. The strength of electronic correlations in these materials is an important issue, which has been a subject of debate in the literature [@haule1; @haule2; @anisimov2; @shorikov1; @anisimov3].
The combination of dynamical mean-field theory with density-functional theory in the local density approximation (LDA+DMFT) provides a powerful framework for the quantitative description of electronic correlations in a realistic setting. A number of materials have been investigated in this framework over the past decade, such as transition metals and transition-metal oxides, rare-earth and actinide compounds, and organic conductors. These examples testify to the progress in our understanding of the key physical phenomena associated with the competition between the localized and itinerant characters of electrons belonging to different orbitals (see e.g. Refs. \[\] for reviews).
In the past few years, a new generation of LDA+DMFT implementations have been put forward [@pav04; @lechermann_wannier-bis; @ani05; @gavri05; @sol06; @ani06-bis; @amadon_pw_08], which emphasize the use of Wannier functions as a natural bridge between the band-structure and the real-space description of the solid in terms of orbitals. These functions span the subset of orbitals which are treated within the many-body DMFT framework. In this article, we present an implementation of LDA+DMFT within the full potential linear augmented plane wave (FLAPW) framework, using atomic orbitals that are promoted to Wannier functions by a truncated expansion over Bloch functions followed by an orthonormalization procedure. This is a simpler alternative to the previous implementation of DMFT within FLAPW [@lechermann_wannier-bis], which constructed the Wannier functions following the prescription of maximal localisation [@marzari_wannier_1997_prb; @sou01]. The choice of FLAPW is motivated by the high level of accuracy of this all-electron, full-potential method. In the present work, we use the Wien2k electronic structure package [@Wien2k], and we have constructed an interface to it that allows for the construction of Wannier-like functions used in DMFT. Our implementation is described in detail in Sec. \[sect:theory\]. As a benchmark, we perform calculations on a test material, SrVO$_3$, which are presented in Appendix \[sect:srvo3\] and compared to previously published results for this material [@lie03; @sek04; @pav04; @pav05; @nek05; @sol06; @nek06]. Throughout this article, many-body effects are treated in the DMFT framework using the recently developed continuous-time strong-coupling Quantum Monte Carlo algorithm of P.Werner and coworkers [@werner_ctqmc; @haule_ctqmc_prb_07]. Because very low temperatures can be reached, and very high accuracy can be obtained at low-frequency, this algorithm represents a major computational advance in the field.
In Sec. \[sect:results\], we address the issue of electronic correlations in LaFeAsO. The DMFT calculations which have been published soon after the experimental discovery of superconductivity in the iron oxypnictides have provided seemingly contradictory answers to this question. In Refs. \[\], K. Haule and G. Kotliar proposed that LaFeAsO is a strongly correlated metal, rather close to the Mott metal-insulator transition, and characterized by a reduced value of the quasiparticle coherence scale, resulting in bad metallic behavior. In contrast, in Refs. \[\], V.Anisimov and coworkers proposed that these materials are in a weak to intermediate regime of correlations.
Our LDA+DMFT calculations for LaFeAsO support the physical picture of a metal with intermediate correlations. The average value of the mass renormalization of the Fe 3$d$ bands is about $1.6$, in reasonable agreement with the picture inferred from photoemission experiments. We also find that there is no technical inconsistency between different DMFT results reported for LaFeAsO before. We show that the discrepancies in the literature are due to two causes: i) the specific value of the interaction parameters used in these calculations and ii) the degree of localization of the Wannier orbitals chosen to represent the Fe 3$d$ states, to which many-body terms are applied.
In Sec. \[sect:results\], we perform detailed comparisons between LDA+DMFT calculations performed with different degree of localization of the correlated orbitals, associated with different choices of energy windows for the Wannier construction (and accordingly, different degrees of screening of the interaction parameters). We point out several difficulties associated with the use of more extended Wannier functions associated with the low-energy iron bands only. Some of these issues have important physical consequences, in particular regarding the sensitivity to the Hund’s coupling.
This article ends with several appendices, reporting on more detailed aspects or technical issues. Appendix A is devoted to a benchmark of our implementation on a “classical” test compound, SrVO$_3$. Appendix B details some technical issues associated with the projection scheme used to display partial spectral functions with a given orbital character. Appendix C discusses the influence of spin-flip and pair-hopping terms on the degree of correlations, on the basis of model calculations. We conclude that while these terms are indeed important close to the Mott transition, they can safely be neglected in the regime of correlations relevant to LaFeAsO.
Theoretical framework {#sect:theory}
=====================
Implementation of LDA+DMFT in the APW framework
-----------------------------------------------
### LDA+DMFT in the basis of Bloch waves
To make this article self-contained and in order to define the main notations, this subsection begins by briefly reviewing some essential aspects of the LDA+DMFT framework. The presentation is close to that of Refs. \[\], where additional details can be found.
Dynamical mean-field theory is a quantitative method for handling electron correlations, which can be described as an “effective atom” approach. The self-energy in the solid is approximated by that of a local model, a generalized Anderson impurity model describing a specific set of atomic-like orbitals coupled to a self-consistent environment. The self-consistency requirement is that the local on-site Green’s function of the solid, calculated using this local self-energy, must coincide with the Green’s function of the effective impurity model.
In order to formulate the local effective atom problem, a set of (orthonormal) local orbitals $|\chi_m^{\alpha,\sigma}\rangle$, and corresponding Wannier-like functions $|w_{\mathbf{k}m}^{\alpha,\sigma}\rangle$, must be constructed. These Wannier functions span the “correlated” subspace $\mathcal C$ of the full Hilbert space, in which many-body correlations (beyond LDA) are taken into account. This set of orbitals spanning the correlated subspace must be clearly distinguished from the full basis set of the problem, in which the Green’s function of the solid can be expressed. Obviously, the basis set spans a much larger Hilbert space, involving all relevant electronic shells.
Below, we discuss in details how the $|w_{\mathbf{k}m}^{\alpha,\sigma}\rangle$ are constructed from the local orbitals $|\chi_m^{\alpha,\sigma}\rangle$. The index $m$ is an orbital index within the correlated subset, $\alpha$ denotes the atom in the unit cell, and $\sigma$ is the spin degree of freedom. Projections of quantities of interest on the subset $\mathcal C$ are done using the projection operator $$\hat P^{\alpha,\sigma}(\mathbf k) = \sum_{m\in \mathcal
C}|w_{\mathbf{k}m}^{\alpha,\sigma}\rangle\langle
w_{\mathbf{k}m}^{\alpha,\sigma}|.\label{eq:projop}$$
The effective impurity model is then constructed for the correlated subset $\mathcal C$. It is defined by the Green’s function of the effective environment, $\mathcal{G}^{0,\sigma}_{mm'}(i\omega_n)$ and Hubbard-Kanamori interaction parameters $U_{mm'm''m'''}$. By solving this model in a suitably chosen manner one obtains the impurity Green’s function $G_{mm'}^{\sigma,\rm{imp}}(i\omega_n)$ as well as the impurity self-energy $$\Sigma_{mm'}^{\sigma,\rm{imp}}(i\omega_n)=\left(\mathcal{G}^{\sigma,0}(i\omega_n)\right)^{-1}_{mm'}
- \left(G^{\sigma,\rm{imp}}(i\omega_n)\right)^{-1}_{mm'}.\label{eq:sigmaimp}$$
For the formulation of the self-consistency condition relating the lattice Green’s function of the solid to the impurity model, it is convenient to choose the Bloch basis $|\psi_{\mathbf{k}\nu}^\sigma\rangle $ as the complete basis set of the problem, since it is a natural output of any electronic structure calculation. The (inverse) Green’s function of the solid expressed in this basis set is given by: $$G^{\sigma}(\mathbf{k},i\omega_{n})^{-1}_{\nu\nu'} =
(i\omega_{n}+\mu-\epsilon_{\mathbf{k}\nu}^\sigma)\delta_{\nu\nu'}-
\Sigma_{\nu\nu'}^\sigma(\mathbf{k},i\omega_{n}),\label{eq:latt-G}$$ where $\epsilon_{\mathbf{k}\nu}^\sigma$ are the Kohn Sham eigenvalues and $\Sigma_{\nu\nu'}^\sigma(\mathbf{k},i\omega_{n})$ is the approximation to the self-energy obtained by the solution of the DMFT impurity problem. It is obtained by “upfolding” the impurity local self-energy as $$\Sigma_{\nu\nu'}^{\sigma}(\mathbf{k},i\omega_{n})
=\sum_{\alpha,mm'}P_{\nu m}^{\alpha,\sigma*}
(\mathbf{k})\Delta\Sigma_{mm'}^{\sigma,\rm{imp}}(i\omega_{n})
P_{m'\nu'}^{\alpha,\sigma}(\mathbf{k}),\label{eq:latt-Self}$$ where $P_{m\nu}^{\alpha,\sigma}(\mathbf{k})=\langle
w_{\mathbf{k}m}^{\alpha,\sigma}|\psi_{\mathbf{k},\nu}^{\sigma}\rangle$ are the matrix elements of the projection operator, Eq. (\[eq:projop\]) and $$\Delta\Sigma_{mm'}^{\sigma,\rm{imp}}(i\omega_{n})=\Sigma_{mm'}^{\sigma,\rm{imp}}(i\omega_{n})-\Sigma_{mm'}^{\rm{dc}}.
\label{sigma_mm}$$ Here, $\Sigma_{mm'}^{\sigma,\rm{imp}}$ is the impurity self-energy, Eq. (\[eq:sigmaimp\]), expressed in the local orbitals, and $\Sigma_{mm'}^{\rm{dc}}$ is a double-counting correction, which will be discussed in Sect. \[sect:impl\_compmeth\].
The local Green’s function is obtained by projecting the lattice Green’s function to the set of correlated orbitals $m$ of the correlated atom $\alpha$ and summing over the full Brillouin zone, $$G_{mm'}^{\sigma,\rm{loc}}(i\omega_{n})=
\sum_{\mathbf{k},\nu\nu'}P_{m\nu}^{\alpha,\sigma}(\mathbf{k})G_{\nu\nu'}^{\sigma}(\mathbf{k},i\omega_n)
P_{\nu'm'}^{\alpha,\sigma*}(\mathbf{k}).\label{eq:local-G}$$ Note that the local quantities $G_{mm'}^{\sigma,\rm{loc}}(i\omega_{n})$ and $\Delta\Sigma_{mm'}^{\sigma,\rm{imp}}(i\omega_{n})$ carry also an index $\alpha$, which we suppressed for better readability.
The self-consistency condition of DMFT imposes that the *local* Green’s function, Eq. (\[eq:local-G\]) must coincide with the one obtained from the effective impurity problem, $$\mathbf{G}^{\sigma,\rm{loc}}(i\omega_n)= \mathbf{G}^{\sigma,\rm{imp}}(i\omega_n).\label{eq:SC}$$ This equation implies that the Green’s function of the effective environment, $\mathcal{G}_0$, must be self-consistently related to the self-energy of the impurity model through: $$\label{eq:selfconsist-sigma}
\mathcal{G}_0^{-1} = \Sigma_{\rm{imp}} + G_{\rm{loc}}^{-1}$$ where the dependence of $G_{\rm{loc}}$ on $\Sigma_{\rm{imp}}$ is specified by Eqs. (\[eq:latt-G\],\[eq:local-G\]). In practice, the DMFT equations are solved iteratively: starting from an initial $\mathcal{G}_0$, the impurity model is solved for $\Sigma_{\rm{imp}}$, and a new $\mathcal{G}_0$ is constructed from (\[eq:selfconsist-sigma\]). The cycle is repeated until convergence is reached.
In order to construct the set of Wannier functions, we start from a set of local atomic-like orbitals $|\chi_m^{\alpha,\sigma}\rangle$ defined in the unit cell. These orbitals can be expanded over the full Bloch basis-set as: $$|\chi_{\mathbf{k}m}^{\alpha,\sigma}\rangle
=\sum_{\nu}\langle\psi_{\mathbf{k}\nu}^{\sigma}|\chi_{m}^{\alpha,\sigma}\rangle
|\psi_{\mathbf{k}\nu}^{\sigma}\rangle$$ This expansion is then truncated by choosing an energy window $\mathcal W$, and restricting the sum to those Bloch states with Kohn-Sham energies $\epsilon_{\mathbf{k}\nu}$ within $\mathcal W$. The number of bands included in $\mathcal W$ will in general depend on $\mathbf k$ and $\sigma$. We thus define the modified orbitals (which do not form an orthonormal set because of the truncation): $$|\tilde\chi_{\mathbf{k}m}^{\alpha,\sigma}\rangle
=\sum_{\nu\in \mathcal{W}}\langle\psi_{\mathbf{k}\nu}^{\sigma}|\chi_{m}^{\alpha,\sigma}\rangle
|\psi_{\mathbf{k}\nu}^{\sigma}\rangle.\label{eq:Correl-orb-1}$$ Let us denote the matrix elements of the projection operator for this subset as $$\widetilde P_{m\nu}^{\alpha,\sigma}(\mathbf{k})=\langle
\tilde\chi_{m}^{\alpha,\sigma}|\psi_{\mathbf{k}\nu}^{\sigma}\rangle,
\qquad \nu\in\mathcal{W}
\label{eq:proj-1}$$ The matrix $\widetilde P_{m\nu}^{\alpha,\sigma}(\mathbf{k})$ is not unitary, except when the sum in (\[eq:Correl-orb-1\]) is carried over all Bloch bands. It is also important to note that the matrices $\mathrm{\widetilde{P}}^{\alpha,\sigma}$ are in general non-square matrices. They reduce to square matrices only in the case when the number of Kohn-Sham bands contained in the chosen window equals at every $\mathbf{k}$-point the number of correlated local orbitals to be constructed.
The orbitals $\left|\tilde\chi_{\mathbf{k}m}^{\alpha,\sigma}
\right\rangle $ can be orthonormalized, giving a set of Wannier-like functions: $$\left|w_{\mathbf{k}m}^{\alpha,\sigma}\right\rangle
=\sum_{\alpha',m'}S_{m,m'}^{\alpha,\alpha'}
\left|\tilde \chi_{\mathbf{k}m'}^{\alpha',\sigma}\right\rangle ,
\label{eq:wannier}$$ where $S_{m,m'}^{\alpha,\alpha'}
=\left\{ O(\mathbf{k},\sigma)^{-1/2}\right\} _{m,m'}^{\alpha,\alpha'}$ and $O_{m,m'}^{\alpha,\alpha'}(\mathbf{k},\sigma)=\left\langle
\tilde \chi_{\mathbf{k}m}^{\alpha,\sigma}\right|\left.\tilde\chi_{\mathbf{k}m'}^{\alpha',\sigma}
\right\rangle $ the overlap matrix elements.
The overlap $O_{m,m'}^{\alpha,\alpha'}(\mathbf{k},\sigma)$ finally reads $$O_{m,m'}^{\alpha,\alpha'}(\mathbf{k},\sigma)=\sum_{\mathcal{W}}
\widetilde{P}_{m\nu}^{\alpha,\sigma}(\mathbf{k})
\widetilde{P}_{\nu m'}^{\alpha',\sigma*}(\mathbf{k}),\label{eq:over-wannier}$$ while the orthonormalized projectors are then written as $$P_{m\nu}^{\alpha,\sigma}
(\mathbf{k})=\underset{\alpha'm'}{\sum}\left
\{ \left[O(\mathbf{k},\sigma)
\right]^{-1/2}\right\}_{m,m'}^{\alpha,\alpha'}
\widetilde{P}_{m'\nu}^{\alpha',\sigma}(\mathbf{k}).
\label{eq:wannier-proj}$$
### Augmented plane waves
In this work, the Bloch basis $|\psi_{\mathbf{k}\nu}^\sigma\rangle $ are expanded in augmented plane waves (APW/LAPW), which will be briefly described in the following. As was first pointed out by Slater [@Slater], near atomic nuclei the crystalline potential in a solid is similar to that of a single atom, while in the region between nuclei (in the interstitial) the potential is rather smooth and weakly-varying. Hence, one may introduce a set of basis functions, [*augmented plane waves*]{}, adapted to this general shape of the potential. First the crystal space is divided into non-overlapping muffin-tin (MT) spheres centered at the atomic sites and the interstitial region in between. In the interstitial region($I$) the APW $\phi_{\mathbf{G}}^{\mathbf{k}}(\mathbf{r})$ is simply the corresponding plane wave for given reciprocal lattice vector $\mathbf{G}$ and crystal momentum $\mathbf{k}$: $$\phi_{\mathbf{G}}^{\mathbf{k}}(\mathbf{r})=\frac{1}{\sqrt{V}}e^{i(\mathbf{k}+\mathbf{G})\mathbf{r}} \quad \mathbf{r}\in I,$$ where $V$ is the unit cell volume. This plane wave is augmented inside each of the MT-spheres by a combination of the radial solutions of the Schr[ö]{}dinger equation in such way that the resulting APW is continuous at the sphere boundary. The APW are then employed to expand the Kohn-Sham (KS) eigenstates $\psi_{\mathbf{k}\nu}^{\sigma}(\mathbf{r})$ for the full Kohn-Sham (KS) potential, (without any shape approximation). In the original formulation of the APW method the radial solutions expanding a KS eigenstate inside MT-spheres had to be evaluated at the corresponding eigenenergy leading to an energy-dependent basis set and, hence, to a non-linear secular problem. In order to avoid this complication, linearized versions of the APW method have been proposed. There are two widely-used schemes for the APW linearization. In the first, the linear APW (LAPW) method [@LAPWSingh], the plane wave is augmented within MT-spheres by a combination of the radial solutions, evaluated at chosen linearization energies $E_{1l}$, and their energy derivatives. The resulting linear augmented plane wave then reads:
$$\phi_{\mathbf{G}}^{\mathbf{k}}(\mathbf{r})=\left\{ \begin{array}{ll}
\frac{1}{\sqrt{V}}e^{i(\mathbf{k}+\mathbf{G})\mathbf{r}} & \qquad\mathbf{r}\in I\\
\underset{lm}{\sum}\left[A_{lm}^{\alpha,\mathbf{k}+\mathbf{G}}u_{l}^{\alpha,\sigma}(r,E_{1l}^{\alpha})+
B_{lm}^{\alpha,\mathbf{k}+\mathbf{G}}\dot{u}_{l}^{\alpha,\sigma}(r,E_{1l}^{\alpha})\right]Y_{m}^{l}(\hat{r})
& \qquad\mathbf{r}\in
R_{MT}^{\alpha}\end{array}\right.\label{eq:LAPW}$$
where the index $\alpha=1,...,N_{at}$ runs over all $N_{\alpha}$ atomic sites in the unit cell, the coefficients $A_{lm}$ and $B_{lm}$ are determined from the requirement for the linear APW to be continuous and differentiable at the sphere boundary, $r$ and $\hat r$ are the radial and angular parts of the position vector, respectively. The energy-independent basis set (\[eq:LAPW\]) leads to a linear secular problem, however, compared to the energy-dependent APW, a larger number of the LAPW in the basis set is generally required to attain the same accuracy. In order to decrease the requirement for the number of APW another linearization scheme, APW+$lo$ [@APWlo] has been proposed. The APW+$lo$ basis set consists of the augmented plane waves evaluated at a fixed energy $E_{1l}$: $$\phi_{\mathbf{G}}^{\mathbf{k}}(\mathbf{r})=\left\{ \begin{array}{ll}
\frac{1}{\sqrt{V}}e^{i(\mathbf{k}+\mathbf{G})\mathbf{r}} & \qquad\mathbf{r}\in I\\
\underset{lm}{\sum}A_{lm}^{\alpha,\mathbf{k}+\mathbf{G}}u_{l}^{\alpha,\sigma}(r,E_{1l}^{\alpha})Y_{m}^{l}(\hat{r})
& \qquad\mathbf{r}\in
R_{MT}^{\alpha},\end{array}\right.\label{eq:APW}$$ where the coefficient $A_{lm}$ are determined from the requirement for $\phi_{\mathbf{G}}^{\mathbf{k}}(\mathbf{r})$ to be continuous at the sphere boundary. To increase the variational freedom of the APW+$lo$ basis set the fixed-energy APW (\[eq:APW\]) are supplemented for the physically important orbitals (with $l \leq 3$) by the local orbitals ($lo$) that are not matched to any plane wave in the interstitial and are defined only within the muffin tin spheres ($\mathbf{r}\in
R_{MT}^{\alpha}$) $$\phi_{lm,\alpha}^{lo}(\mathbf{r})=\left[A_{lm}^{\alpha,lo}u_{l}^{\alpha,\sigma}(r,E_{1l}^{\alpha})+
B_{lm}^{\alpha,lo}\dot{u}_{l}^{\alpha,\sigma}(r,E_{1l}^{\alpha})\right]Y_{m}^{l}(\hat{r})
\label{eq:lo-apw}$$ with the coefficients $A_{lm}$ and $B_{lm}$ chosen from the requirement of zero value and slope for the local orbital at the sphere boundary.
Additional local orbitals (usually abbreviated with the capital letters as $LO$, $\phi_{lm,\alpha}^{LO}$) can be introduced to account for semicore states. They have a similar form as Eq. (\[eq:lo-apw\]) with the redefined $A_{lm}^{\alpha,LO}$ and a second term with a coefficient $C_{lm}^{\alpha,LO}$ and the radial function evaluated at a corresponding energy $E_{2l}^{\alpha}$ for the semicore band. The coefficient $B_{lm}$ is set to $0$ in the APW+$lo$ framework.
Generally, in the full potential augmented plane wave method the LAPW, APW+$lo$ and $LO$ types of orbitals can be employed simultaneously. The Kohn-Sham eigenstate is expanded in this mixed basis as: $$\psi_{k\nu}^{\sigma}(\mathbf{r})=\overset{N_{b}}
{\underset{i=1}{\sum}}c_{i\nu}\phi_{i}^{\sigma}(\mathbf{r}),\label{eq:KS-eigen}$$ where $N_b$ is the number of the orbitals in the basis set. The LDA+DMFT framework introduced in the present work can also be used in conjunction with any mixed APW+$lo$/LAPW/$LO$ basis set.
### Local orbitals and Wannier functions in the APW basis {#sect:locorbsinAPW}
Having defined the basis set we may now write down the expression for the Bloch eigenstate expanded in the general APW basis (\[eq:KS-eigen\]). For $\mathbf{r}$ in the interstitial region, it reads: $$\psi_{\mathbf{k}\nu}^{\sigma}(\mathbf{r})=\frac{1}{\sqrt{V}}\overset{N_{PW}}
{\underset{\mathbf{G}}{\sum}}c_{\mathbf{G}}^{\nu,\sigma}(\mathbf{k})e^{i(\mathbf{k}
+\mathbf{G})\mathbf{r}},$$ while for the region within the MT-spheres $\mathbf{r}\in R_{MT}^{\alpha}(\alpha=1,...,N_{at})$, we have:
$$\begin{split}
\psi_{\mathbf{k}\nu}^{\sigma}(\mathbf{r})&=\overset{N_{PW}}{\underset{\mathbf{G}}{\sum}}
c_{\mathbf{G}}^{\nu,\sigma}(\mathbf{k})\underset{lm}{\sum}A_{lm}^{\alpha,\mathbf{k}+
\mathbf{G}}u_{l}^{\alpha,\sigma}(r,E_{1l}^{\alpha})Y_{m}^{l}(\hat{r})
+\overset{N_{lo}}{\underset{n_{lo}=1}{\sum}}c_{lo}^{\nu,\sigma}
\left[A_{lm}^{\alpha,
lo}u_{l}^{\alpha,\sigma}(r,E_{1l}^{\alpha})+B_{lm}^{\alpha,lo}
\dot{u}_{l}^{\alpha,\sigma}(r,E_{1l}^{\alpha})\right]Y_{m}^{l}(\hat{r})+\\
&+\overset{N_{LO}}{\underset{n_{LO}=1}{\sum}}c_{LO}^{\nu,\sigma}
\left[A_{lm}^{\alpha,
LO}u_{l}^{\alpha,\sigma}(r,E_{1l}^{\alpha})+C_{lm}^{\alpha,LO}u_{l}^{\alpha,\sigma}(r,E_{2l}^{\alpha})\right]Y_{m}^{l}(\hat{r}),
\end{split}
\label{eq:KS-eigen-2}$$
where $N_{PW}$ is the total number of plane waves considered in the interstitial which in turn is augmented inside each MT-sphere, $N_{lo}$ is the number of $\phi_{lm,\alpha}^{lo}(\mathbf{r})$ orbitals of Eq. (\[eq:lo-apw\]) and $N_{LO}$ the corresponding number of auxiliary orbitals for semicore states $\phi_{lm,\alpha}^{LO}(\mathbf{r})$.
In the framework of the APW method one has several choices for the ‘initial’ correlated orbitals $|\chi_m^{\alpha,\sigma}\rangle$. Any suitable combination of the radial solution of the Schrödinger equation and its energy derivative for a given correlated shell $\{\alpha,l\}$ can be employed, for example, the $lo$-orbital (\[eq:lo-apw\]). In the present paper we simply chose the $|\chi_m^{\alpha,\sigma}\rangle$’s as the solutions of the Schrödinger equation within the MT-sphere $\left|u_{l}^{\alpha,\sigma}(E_{1l})Y_{m}^{l}\right\rangle$ at the corresponding linearization energy $E_{1l}$.
Inserting $|\chi_m^{\alpha,\sigma}\rangle=\left|u_{l}^{\alpha,\sigma}(E_{1l})Y_{m}^{l}\right\rangle$ and the expansion (\[eq:KS-eigen-2\]) of the Bloch eigenstate in terms of APWs into Eqs. (\[eq:Correl-orb-1\],\[eq:proj-1\]), and making use of the orthonormality of the radial solutions and their energy derivatives: $$\begin{aligned}
\left\langle
u_{l}^{\alpha,\sigma}(E_{1l})Y_{m}^{l}\right.\left|u_{l'}^{\alpha,\sigma}(E_{1l})Y_{m'}^{l'}\right\rangle
&=\delta_{ll'mm'}
\label{eq:overlap-uu}\\
\left\langle u_{l}^{\alpha,\sigma}(E_{1l})Y_{m}^{l}\right.
\left|\dot{u}_{l'}^{\alpha,\sigma}(E_{1l})Y_{m'}^{l'}
\right\rangle &=0,
\label{eq:overlap-uudot}\end{aligned}$$ one obtains the following expression for the projection operator matrix element: $$\begin{split}
\widetilde P_{m\nu}^{\alpha,\sigma}(\mathbf{k})&=\left\langle u_{l}^{\alpha,\sigma}
(E_{1l})Y_{m}^{l}\right.\left|\psi_{\mathbf{k}\nu}^{\sigma}
\right\rangle\\
&=A_{lm}^{\nu,\alpha}(\mathbf{k},\sigma)
+\overset{N_{LO}}{\underset{n_{LO}=1}{\sum}}
C_{lm,LO}^{\nu,\alpha}(\mathbf{k},\sigma),
\end{split}
\label{eq:proj-2}$$ In this expression, the first term in the right hand side of (\[eq:proj-2\]) is due to the contribution from the LAPW and/or APW+$lo$ orbitals $$\begin{array}{ll}
A_{lm}^{\nu,\alpha}(\mathbf{k},\sigma)&=\overset{N_{PW}}
{\underset{\mathbf{G}}{\sum}}c_{\mathbf{G}}^{\nu,\sigma}(\mathbf{k})
A_{lm}^{\alpha,\mathbf{k}+\mathbf{G}}\\
&+\overset{N_{lo}}{\underset{n_{lo}=1}{\sum}}c_{lo}^{\nu,\sigma}
A_{lm}^{\alpha,lo}+\overset{N_{LO}}{\underset{n_{LO}=1}{\sum}}c_{LO}^{\nu,\sigma}
A_{lm}^{\alpha,LO}
\end{array}
\label{eq:coeff-proj-1}$$ and the contribution due to the $LO$ (semicore) orbitals that arises due to mutual non-orthogonality of the radial solutions of the Schrödinger equation for different energies $$C_{lm,LO}^{\nu,\alpha}(\mathbf{k},\sigma)=
c_{LO}^{\nu,\sigma}C_{lm}^{\alpha,LO}
\tilde{O}_{lm,l'm'}^{\alpha,\sigma},
\label{eq:coeff-proj-2}$$ where $\tilde{O}_{lm,l'm'}^{\alpha,\sigma}$ is the corresponding overlap: $$\tilde{O}_{lm,l'm'}^{\alpha,\sigma}=\left\langle u_{l}^{\alpha,\sigma}(E_{1l})Y_{m}^{l}\right.\left|u_{l'}^{\alpha,\sigma}(E_{l,LO})
Y_{m'}^{l'}\right\rangle \neq 0 \label{eq:overlap-u1u2}$$ Then we orthonormalize the obtained local orbitals to form a set of Wannier-like functions, Eq. (\[eq:wannier\]). The corresponding projection operator matrix elements (\[eq:proj-2\]) are orthonormalized accordingly using Eq. (\[eq:wannier-proj\]).
Implementation and computational methods {#sect:impl_compmeth}
----------------------------------------
### FLAPW code
For the electronic structure calculation we use the full potential APW+$lo$/LAPW code as implemented in the Wien2k package [@Wien2k]. We have built an interface that constructs the projectors to the correlated orbitals ($ P_{m\nu}^{\alpha,\sigma}(\mathbf{k})$) out of the eigenstates produced by the Wien2k code, as described in Sect. \[sect:locorbsinAPW\]. In order to obtain the local Green’s function, the summation over momenta, Eq. (\[eq:local-G\]), is done in the irreducible Brillouin zone (BZ) only, supplemented by a symmetrization procedure which is standard in electronic structure calculations, $$\sum_{\mathbf k}^{BZ} {\mathbf A}({\mathbf k}) =
\sum_{s=1}^{N_s}\sum_{\mathbf k}^{IBZ} {\mathcal O}_s{\mathbf
A}({\mathbf k}){\mathcal O}_s^{\dagger},$$ where ${\mathbf A}(\mathbf k)$ is any $\mathbf k$-dependent matrix, $N_s$ the number of symmetry operations and $\mathcal O_s$ the symmetrization matrices. Furthermore, we construct the local orbitals in the local coordinate system of the corresponding atom. This means that the equivalent atoms in the unit cell for which the DMFT should be applied, e.g. the two Fe atoms in the oxypnictides, are exactly the same and the impurity problem has to be solved only once. Afterward, the Green’s function and self-energies are put back to the global coordinate system of the crystal in which the Bloch Green’s function, Eq. (\[eq:latt-G\]) is formulated.
### Continuous-time quantum Monte-Carlo
For the solution of the impurity problem we use the strong-coupling version of the continuous-time quantum Monte Carlo method (CTQMC) [@werner_ctqmc; @haule_ctqmc_prb_07]. It is based on a hybridization expansion and has proved to be a very efficient solver for quantum impurity models in the weak and strong correlation regime. It allows us to address room temperature ($\beta\equiv 1/kT \approx 40$eV$^{-1}$) without problems. In our calculations, we used typically around $5\cdot 10^6$ Monte-Carlo sweeps and 1000 $\mathbf k$-points in the irreducible BZ. Since the CTQMC solver computes the Green’s function on the imaginary-time axis, an analytic continuation is needed in order to obtain results on the real-frequency axis. Here, we choose to perform a continuation of the impurity self-energy using a stochastic version of the Maximum Entropy method,[@beach_ME] yielding real an imaginary parts of the retarded self-energy $\rm{Re}\Sigma(\omega+i0^+),\rm{Im}\Sigma(\omega+i0^+)$ which can be inserted into Eq. (\[eq:latt-G\]) in order to obtain the lattice spectral function and density of states.
### Many-body interactions
The CTQMC strong-coupling algorithm can deal with the full rotationally invariant form of the interaction hamiltonian [@haule_ctqmc_prb_07]. However, most calculations presented in this article will consider only the Ising terms of the Hund’s coupling, yielding the interaction Hamiltonian: $$\begin{split}
H_{int}&=\frac{1}{2}\sum_{mm',\sigma}U_{mm'}^{\sigma\sigma}n_{m\sigma}n_{m'\sigma} \\
&+\frac{1}{2}\sum_{mm'}U_{mm'}^{\sigma\bar{\sigma}}\left(n_{m\uparrow}n_{m'\downarrow}
+ n_{m\downarrow}n_{m'\uparrow}\right),
\end{split}$$ with $U_{mm'}^{\sigma\sigma}$ and $U_{mm'}^{\sigma\bar{\sigma}}$ the reduced interaction matrices for equal and opposite spins, respectively. This enables us to take advantage of a maximal amount of conserved quantum numbers and, hence, perform the CTQMC calculation without any truncation of the local basis. The effects of spin-flip and ‘pair-hopping’ terms in the Hund’s interaction are discussed in Appendix \[sect:hundsrule\].
In our approach, the interaction matrices are expressed in terms of the Slater integrals $F^0$, $F^2$, and $F^4$, where for $d$-electrons these parameters are related to the Coulomb and Hund’s coupling via $U=F^0$, $J=(F^2+F^4)/14$, and $F^2/F^4=0.625$.[@anisimov_lda+u_review_1997_jpcm] Using standard techniques the four-index $U$-matrix is calculated, and the reduced interaction matrices are then given by $U_{mm'}^{\sigma\bar{\sigma}}=U_{mm'mm'}$ and $U_{mm'}^{\sigma\sigma}=U_{mm'mm'}-U_{mm'm'm}$. With the above definitions, the Coulomb parameters $U$ and $J$ are related to the matrices via $$\begin{aligned}
\label{eq:Uaverage}
U&=\frac{1}{N^2}\sum_{mm'}^N
U_{mm'}^{\sigma\bar{\sigma}}
\\
\label{eq:Javerage}
J&=U-\frac{1}{N(N-1)}\sum_{m\ne m'}^N U_{mm'}^{\sigma\sigma}. \end{aligned}$$
As mentioned above, the LDA+DMFT scheme (as the LDA+U one) involves a double-counting correction $\Sigma_{mm'}^{\rm{dc}}$ in Eqs. (\[eq:latt-Self\],\[sigma\_mm\]). Indeed, on-site Coulomb interactions are already treated on mean-field level in LDA. Several forms of the double-counting correction term have been proposed and investigated.[@anisimov_lda+u_review_1997_jpcm; @lichtenstein_magnetism_dmft_2001_prl; @Ylvisaker_LSDA+U_2009_prb] In this work we follow Ref. and use the following double-counting correction: $$\Sigma_{mm'}^{\sigma,\rm{dc}} =
\delta_{mm'}\left[ U\left(N_c-\frac{1}{2}\right)- J\left(N_c^\sigma
-\frac{1}{2}\right)\right],\label{eq:dc}$$ where $U$ is the average Coulomb interaction, $J$ the Hund’s rule coupling, $N_c^\sigma$ the spin-resolved occupancy of the correlated orbitals, and $N_c=N_c^\uparrow+N_c^\downarrow$. We compared the results obtained with Eq. (\[eq:dc\]) also with the double-counting correction given in Ref. which gave very similar results.
Choice of energy window, localization of Wannier functions, and screening {#sect:energyscales}
-------------------------------------------------------------------------
The Wannier functions defining the correlated subspace of orbitals for which a DMFT treatment is performed, are constructed by truncating the expansion of the initial atomic-like local orbitals to a restricted energy window $\mathcal{W}$, as described above. The choice of this energy window is an important issue, which deserves further discussion. Indeed, it will determine the shape and the degree of localization of the resulting Wannier-like functions.
Let us consider first the case of a rather small energy window, containing only those bands that have dominantly an orbital character which qualifies them as “correlated” (e.g., the Fe-$3d$ orbitals in LaFeAsO or the V-$3d$-t$_{2g}$ orbitals in SrVO$_3$). In that case, the dimension of the Kohn-Sham Hamiltonian used in Eq. (\[eq:latt-G\]) coincides with that of the correlated subspace $\mathcal{C}$ (i.e. with the number of orbitals involved in the effective impurity model, for a single correlated atom per cell). The Wannier-like functions are then quite extended in real space, and resemble strongly the Wannier orbitals constructed within other schemes, such as the maximally localized Wannier construction of Ref. , or the $N$-th order muffin-tin basis set downfolded to that set of bands.[@and00; @and00-2; @zur05] In such a situation, hybridization of the correlated orbitals with states that lie outside the energy window is neglected at the DMFT level. Some information about the hybridization, e.g., of the d-states with ligand orbitals is of course taken into account through the leakage of the Wannier orbitals on neighboring ligand sites (see, e.g., Ref. ).
In contrast, if a larger energy window is chosen, it will in general contain states treated as correlated as well as states on which no Hubbard interactions are imposed. In this case, the dimension of the Kohn-Sham Hamiltonian used in the LDA+DMFT calculation of the local Green’s function (\[eq:latt-G\],\[eq:local-G\]) exceeds the number of correlated orbitals involved in the effective impurity model. The Wannier functions are more localized in space, and the information about the hybridization of the correlated orbitals with other states within this larger window is carried by the off-diagonal blocks of the Hamiltonian between correlated and uncorrelated states.
An instructive case occurs when the correlated bands are well separated from the uncorrelated bands at all $\mathbf{k}$-points, but the bands overlap in energy. This situation is realized for instance for the t$_{2g}$ bands in SrVO$_3$ that extend into the energy region of the e$_g$ bands. In order to strictly pick the three correlated t$_{2g}$ bands at each $\mathbf{k}$-point, one would in that case have to introduce a $\mathbf{k}$-dependent energy window. For a $\mathbf{k}$-independent energy window, one will in general have more than three bands at some $\mathbf{k}$-points, corresponding to some $e_g$ contribution in the chosen window. This can then be expected to result in slightly more localized orbitals. An example is given in Appendix \[sect:benchmark\].
Local Coulomb interactions, screening and constrained RPA calculations {#sect:CRPA}
----------------------------------------------------------------------
The choice of the energy window influences the value of the interaction parameters $U_{mm'm''m'''}$ in a crucial manner, which can be traced back to two main reasons.
First, the interaction parameters are related to matrix elements of a screened interaction between the chosen Wannier functions. The more bands are included in their construction, the more localized they become and, hence, the matrix elements increase. Second, screening effects themselves affect the value of $U$. The more states are excluded from the screening process, the larger $U$ becomes. In what follows we distinguish carefully between these two effects.
In the present work, we apply the present LDA+DMFT implementation to one of the new high-T$_c$ superconductors, LaFeAsO. For constructing the Wannier functions, we focus on an energy window that contains the 10 bands around the Fermi level with dominantly Fe-$3d$ character and also the bands coming from the $p$-bands of O and As, which are mainly located in the energy region $[-6, -2]$eV, resulting in a “$dpp$-hamiltonian”. In addition, we performed calculations also for a smaller energy window containing only the Fe-$3d$ bands, yielding a “$d$-hamiltonian”, as well as for a very large window including around 60 Bloch bands.
The values of the Coulomb interactions $U$ and $J$ are calculated from the constrained random phase approximation (cRPA)[@ary04; @miyake2], using the recently developed scheme for entangled band structures[@miyake-disentanglement]. cRPA calculations for LaFeAsO have been performed before in Refs. . In Ref. the screened Coulomb parameters are obtained for three different situations: (i) by constructing Wannier functions from an energy window comprising the Fe-$d$ bands only, and screening calculated excluding the Fe-$d$ channels only, (ii) by considering a larger window which also includes the As and O-$p$ states, so that the screening processes, for instance, from the As-$p$ states to the Fe-$d$ ones are also excluded, (iii) and finally, a hybrid situation (dubbed ‘$d$-$dpp$’ in Ref. ), in which the Wannier functions are calculated from an energy window including Fe-$d$, O-$p$, and As-$p$, but only the Fe-$d$ states are excluded from the screening. In other words, the screening is calculated as in (i) and the Wannier functions as in (ii). As discussed in Ref. , this third option should be appropriate in a situation in which a full $dpp$-hamiltonian is used but Hubbard interactions are only applied to the $d$ states.
Here, we follow the same approach as in the “$d$-$dpp$” case of Ref. , but using the new disentanglement scheme of Ref. . First, a partially screened Coulomb interaction $W_r$ is constructed as follows: Wannier functions for the Fe-$d$ states are calculated, and a basis for the complementary subspace (containing in particular the ligands, but also higher lying f-states) is constructed. Based on the interpolating $d$-band structure, the $P_d$ polarization is computed, and the partially screened Coulomb interaction $W_r$ is obtained by screening the bare Coulomb interaction by all RPA screening processes [*except $P_d$*]{}. Finally, according to the cRPA procedure the Hubbard $U$ matrix is composed of the matrix elements of $W_r$ in the basis of $dpp$-Wannier functions. As argued in Ref. this procedure is suitable for calculations that deal with the full $dpp$-Hamiltonian in the many-body calculations while explicitly retaining Coulomb interactions on the $d$-submanifold only. In particular, we stress that to the extent that our projection method produces Wannier functions for the $dpp$-window, the Hubbard $U$ parameters are expressed in the same basis as the impurity quantities.
It is important to note that we keep the screening channels in cRPA, i.e. $P_d$, unchanged when the energy window in our calculation is varied. Thus, the different energy windows affect only the localization of the Wannier functions, but not the screening process of the bare Coulomb interaction, and therefore the effective interactions are increasing with increasing energy window. Keeping the screening channels fixed is fully consistent with the fact that correlations are only included for the $d$ electrons, but not for the ligand states.
For our purposes, we calculate the average Coulomb interaction $U$ and Hund parameter $J$ from the matrices calculated by cRPA. With this $U$ and $J$, the interaction matrices in the spherical symmetric approximation used in our calculation are obtained as discussed above. As we will discuss in more detail below, the comparison of the resulting $U_{mm'}^{\sigma\sigma}$ and $U^{\sigma\bar{\sigma}}_{mm'}$ with the cRPA matrices shows that for the $dpp$-hamiltonian, the approximation using atomic values for the ratios of Slater integrals $F^k$ is well justified, whereas for the $d$-hamiltonian the cRPA matrices show strong orbital anisotropies.
Results for the iron oxypnictide LaFeAsO {#sect:results}
========================================
Construction of the $dpp$-Hamiltonian
-------------------------------------
Let us start the discussion of correlation effects in LaFeAsO with our results for the $dpp$-hamiltonian, for which Wannier functions are constructed from the energy window $\mathcal{W}=[-5.5,2.5]$eV. These Wannier functions are quite well localised. The corresponding Kohn-Sham hamiltonian contains $22$ Bloch bands, corresponding to the $10$ Fe-$3d$ bands, the $6$ As-$p$ and the $6$ O-$p$ bands.
The local many-body interactions corresponding to this choice of Wannier functions are obtained from cRPA, as described in the previous section. They read: $$\begin{aligned}
U_{mm'}^{\sigma\sigma}|_{\rm{cRPA}}&=\left(\begin{array}{ccccc}
0.00 & 1.61 & 1.55 & 2.26 & 2.26 \\
1.61 & 0.00 & 2.50 & 1.82 & 1.82 \\
1.55 & 2.50 & 0.00 & 1.70 & 1.70 \\
2.26 & 1.82 & 1.70 & 0.00 & 1.74 \\
2.26 & 1.82 & 1.70 & 1.74 & 0.00\end{array}\right)\nonumber\\
U^{\sigma\bar{\sigma}}_{mm'}|_{\rm{cRPA}}&=\left(\begin{array}{ccccc}
3.77 & 2.35 & 2.21 & 2.71 & 2.71 \\
2.35 & 3.94 & 2.87 & 2.44 & 2.44 \\
2.21 & 2.87 & 3.31 & 2.29 & 2.29 \\
2.71 & 2.44 & 2.29 & 3.48 & 2.29 \\
2.71 & 2.44 & 2.29 & 2.29 & 3.48\end{array}\right)\nonumber\end{aligned}$$ The ordering of orbitals in those matrices is $d_{z^2}$, $d_{x^2-y^2}$, $d_{xy}$, $d_{xz}$, $d_{yz}$.
According to the conventions of the formulae Eqs. (\[eq:Uaverage\]), (\[eq:Javerage\]), these matrices correspond to the values: $U=2.69$eV and $J=0.79$eV. Using these values of $U$ and $J$, we construct the spherically symmetric interaction matrices, $$\begin{aligned}
U_{mm'}^{\sigma\sigma}&=\left(\begin{array}{ccccc}
0.00 & 1.49 & 1.49 & 2.30 & 2.30 \\
1.49 & 0.00 & 2.57 & 1.76 & 1.76 \\
1.49 & 2.57 & 0.00 & 1.76 & 1.76 \\
2.30 & 1.76 & 1.76 & 0.00 & 1.76 \\
2.30 & 1.76 & 1.76 & 1.76 & 0.00\end{array}\right)\nonumber\\
U_{mm'}^{\sigma\bar{\sigma}}&=\left(\begin{array}{ccccc}
3.59 & 2.19 & 2.19 & 2.73 & 2.73 \\
2.19 & 3.59 & 2.91 & 2.37 & 2.37 \\
2.19 & 2.91 & 3.59 & 2.37 & 2.37 \\
2.73 & 2.37 & 2.37 & 3.59 & 2.37 \\
2.73 & 2.37 & 2.37 & 2.37 & 3.59\end{array}\right)\nonumber\end{aligned}$$ It is obvious, that the approximation of the cRPA matrices by using spherical symmetrisation is well justified in this case, with the largest absolute deviation being $\Delta U\approx 0.35$eV, corresponding to a relative error of around 0.09. The reason for this good agreement is that in the present case the Wannier functions are already very close to atomic-like orbitals. We also checked that cRPA yields a significantly larger value of $U$ for iron-oxide (FeO), as expected physically.
LDA+DMFT Results ($dpp$ hamiltonian)
------------------------------------
![\[fig:laofeas\_dpp\_tot\] (Color online) Total DOS for LaOFeAs, $dpp$ Hamiltonian. Black line: LDA DOS. Red line: LDA+DMFT DOS.](fig1.eps){width="0.9\columnwidth"}
![\[fig:Sigma\] (Color online) Real (top) and imaginary (bottom) part of the orbital dependent self-energy in the $dpp$ Hamiltonian for $U=2.69$, $J=0.79$.](fig2.eps){width="0.9\columnwidth"}
![\[fig:laofeas\_dpp\_part\] (Color online) DOS of Fe orbitals in LaOFeAs, $dpp$ Hamiltonian. Color coding as in Fig. \[fig:laofeas\_dpp\_tot\].](fig3.eps){width="0.9\columnwidth"}
We carried out LDA+DMFT calculations for the $dpp$-hamiltonian using the above matrices at an inverse temperature $\beta=40$eV$^{-1}$ (room temperature $T=300$K), using the experimental crystal structure of LaFeAsO. In Fig. \[fig:laofeas\_dpp\_tot\] we display the resulting total densities of states (DOS) together with the corresponding LDA DOS. The total densities of states were computed from the lattice Green’s function, Eq. (\[eq:latt-G\]), traced over all $\nu \in \mathcal{W}$ and integrated over BZ. In order to obtain the corresponding LDA densities of states $\Sigma_{\nu\nu'}^{\sigma}(\mathbf{k},i\omega_{n})$ in Eq. (\[eq:latt-G\]) was set to zero.
One sees in Fig. \[fig:laofeas\_dpp\_tot\] that the LDA+DMFT DOS near the Fermi level displays characteristic features of a metal in an intermediate range of correlations. Both occupied and empty states are shifted towards the Fermi level due to the Fermi-liquid renormalizations. No high-energy features that would correspond to lower or upper Hubbard bands are present in the LDA+DMFT electronic structure. The Fermi-liquid behavior is clear from the self-energy on the real-frequency axis, which we plot in Fig. \[fig:Sigma\] for the $dpp$-hamiltonian. Although it shows a quite rich structure as a function of energy, the real part displays clear linear behavior at low-frequency. The imaginary part is small around $\omega=0$ and has a quadratic frequency dependence at low frequency. It does increase to rather large values at higher frequencies, however, especially for occupied states. Hence, our results are in general agreement with the previous calculations of Anisimov [*et al.*]{}[@anisimov2] and with the experimental photoemission (PES)[@malaeb1] and X-ray absorption (XAS)[@kurmaev] spectra of LaFeAsO, which report a moderately-correlated system with mass renormalisation around $1.8-2.0$.
In order to analyze the strength of correlations for different Fe 3$d$ orbitals we calculated the corresponding quasiparticle residues $Z_m=\left[1-{\rm Im}[\frac{d\Sigma_{mm}(\omega)}{d\omega}|_{\omega \to
0}]\right]^{-1}$ from the self-energy Eq. (\[sigma\_mm\]) on the Matsubara grid (hence, avoiding all uncertainties related to the analytical continuation). The values are 0.609, 0.663, 0.609, and 0.596 for the $d_{z^2}$, $d_{x^2-y^2}$, $d_{xy}$, and degenerate $d_{xz}$/$d_{yz}$ orbitals, resp. In this $dpp$ energy window the Wannier functions become quite localized and their spread is expected to be isotropic. Indeed, within the $dpp$-hamiltonian the difference in $Z_m$ between the orbitals is rather small. The resulting value for the average mass renormalization (between 1.5 and 1.7) is in reasonable agreement with the experimental estimate of 1.8 extracted in Ref. from experimental PES. The smaller mass renormalization found in our calculation compared to the experimental value can be attributed to the single-site approximation of DMFT. Spatial spin fluctuations, which are completely neglected in this approach, can eventually increase the effective mass of the quasiparticles.
The partial densities of states for all Fe 3$d$ orbitals computed within the $dpp$-model are displayed in Fig. \[fig:laofeas\_dpp\_part\]. The partial LDA+DMFT DOS for the $x^2-y^2$ and $yz,xz$ orbitals are shifted upwards relative to the $xy$. Indeed, we found that the crystal field (CF) splitting between the Fe 3$d$ orbitals is somewhat affected by correlations. The splitting between the lowest $xy$ and highest $z^2$ orbitals remains unchanged ($\approx 0.3$eV), while the $x^2-y^2$ and $yz,xz$ CF levels are shifted upwards by 0.15 and 0.08eV relative to their positions in LDA. In LDA+DMFT they are located at 0.25 and 0.18eV, respectively, above the $xy$ orbital.
![\[fig:laofeas\_dpp\_akw\] Momentum-resolved spectral function of LaOFeAs, $dpp$ Hamiltonian. Dark areas mark large spectral weight.](fig4.eps){width="0.9\columnwidth"}
![\[fig:laofeas\_dpp\_akwzoom\] Comparison of the momentum-resolved spectral function of LaOFeAs at low energies. Upper panel: LDA. Lower panel: LDA+DMFT $dpp$ hamiltonian.](fig5a.eps "fig:"){width="0.9\columnwidth"}\
![\[fig:laofeas\_dpp\_akwzoom\] Comparison of the momentum-resolved spectral function of LaOFeAs at low energies. Upper panel: LDA. Lower panel: LDA+DMFT $dpp$ hamiltonian.](fig5b.eps "fig:"){width="0.9\columnwidth"}
It is also instructive to look at the momentum-resolved spectral function $A({\mathbf k},\omega)$ of the crystal. It is obtained from the lattice Green’s function, Eq. (\[eq:latt-G\]), using the real-frequency self-energy and tracing over the orbital degrees of freedom. The result for the $dpp$-model is shown in Fig. \[fig:laofeas\_dpp\_akw\] for an energy range including Fe-$d$, As-$p$, and O-$p$ states. In agreement with the Fermi-liquid picture of moderately-correlated quasiparticles discussed above, one can see well-defined excitations around the Fermi level, which get more diffuse at higher binding energies. The bands above the Fermi level are less affected, since the self-energies are quite asymmetric and smaller for positive frequencies, see Fig. \[fig:Sigma\]. Additionally, it is easy to see that the As-$p$ states, dominantly in the energy range $[-3.5,-2]$eV, hybridize stronger with the Fe-$d$ states and get, thus, affected by correlations. This effect is almost absent for the O-$p$ states, since they hybridize much less with Fe-$d$.
In Fig. \[fig:laofeas\_dpp\_akwzoom\] we show a comparison between the LDA band structure and the LDA+DMFT ${\bf k}$-resolved electronic structure in a low-energy range around the Fermi level. This again reveals the coherent quasiparticles at the Fermi level, as well as more diffuse bands at higher energies. The crossover between long-lived quasiparticles and more diffuse states with a shorter lifetime is around $-0.4$eV, in qualitative agreement with existing ARPES data[@lu-d-h1]. A point to mention here is the effect of the CF splitting on the band structure. For example, a difference between the LDA and DMFT results can be seen for the excitation with predominantly $xy$ character. In LDA it forms a hole-pocket with an excitation energy of $+0.08$eV at the $\Gamma$ point. Due to correlations, however, this band is shifted down significantly to the Fermi level, and the third hole pocket stemming from the $d_{xy}$ orbital could eventually vanish upon electron doping.
One has to keep in mind that a direct comparison to experimental data is difficult for this compound, since (i) the experiments where done at low temperatures in the SDW phase, whereas our calculations are done at room-temperature using the tetragonal crystal structure, and (ii) ARPES experiments on the $1111$ family of pnictide superconductors are difficult to perform because of difficulties with single-crystal synthesis. Nevertheless, on a qualitative level, there is a satisfactory agreement between LDA+DMFT and experiments.
Interactions $z^2$ $x^2-y^2$ $xy$ $yz$,$zx$
-------------------------------- -------------- -------------- -------------- --------------
${\bf U=2.69}$, ${\bf J=0.79}$ [**0.61**]{} [**0.66**]{} [**0.61**]{} [**0.60**]{}
$U=2.69$, $J=0.60$ 0.72 0.76 0.73 0.71
$U=3.70$, $J=0.80$ 0.52 0.57 0.53 0.52
$U=5.00$, $J=0.80$ 0.41 0.45 0.43 0.42
: Quasiparticle weights for different interaction parameters, with Wannier orbitals constructed from $\mathcal{W}=[-5.5,2.5]$eV ($dpp$ hamiltonian). The values in boldface correspond to the interaction parameters obtained from cRPA.
\[tabl:Z\]
Interactions $z^2$ $x^2-y^2$ $xy$ $yz$,$zx$
-------------------- ------- ----------- ------ -----------
$U=3.00$, $J=0.80$ 0.62 0.66 0.58 0.58
$U=3.00$, $J=0.60$ 0.74 0.77 0.72 0.72
$U=3.70$, $J=0.80$ 0.58 0.61 0.52 0.56
: Quasiparticle weights for different interaction parameters, with the Wannier orbitals constructed for a very large window $\mathcal{W}=[-5.5,13.6]$eV
\[tabl:Z2\]
We also studied the dependence of the results on the values of the interaction parameters $U$ and $J$. The resulting quasi-particle renormalizations $Z_{m}$ are listed in Table \[tabl:Z\]. Comparing the first two rows, one can see that a smaller value of $J$ decreases the degree of correlations. The third line correspond to values similar to the ones used in Ref. , giving very similar results. We also increased $U$ to the (unphysically) large value of $U=5.0$eV, and the system still displays metallic behavior, although more correlated. Hence, our calculations strongly suggest that LaFeAsO is not close to a Mott metal-insulator transition.
In order to check the robustness of our results, we also investigated the effect of increasing even further the spatial localization of the Wannier functions, corresponding to a very large energy window $\mathcal{W}=[-5.5,13.6]$eV. We did several calculations for different parameter sets, and the resulting quasi-particle renormalizations $Z_{m}$ of all these calculations are listed in Table \[tabl:Z2\]. For this case, no cRPA calculations for the interaction matrices were performed, but it is expected that $U$ and $J$ will slightly increase with more localised Wannier orbitals. In that sense, the first row of Table \[tabl:Z2\] corresponds to interaction parameters that could be realised for these Wannier functions. It is very satisfying to see the calculations gave almost identical quasi-particle renormalizations. Also the dependence on $U$ and $J$ is very similar to the one we found for the $dpp$ hamiltonian. In that sense we consider our calculations to be converged in terms of the number of Bloch bands that are included for the construction of the Wannier functions and the local Hamiltonian.
Remarks on calculations using the $d$-Hamiltonian and extended Wannier functions
--------------------------------------------------------------------------------
In this section, we address the LDA+DMFT calculations performed with the so-called $d$-hamiltonian, where only the $10$ Fe-$d$ bands around the Fermi level are used for the construction of the Wannier orbitals. In doing so, we shall shed light on the discussion which has appeared in the literature[@haule1; @anisimov2; @haule2; @shorikov1; @anisimov3] regarding the results of LDA+DMFT calculations by different authors, and the degree of correlations of the $1111$ family of pnictide superconductors.
### Wannier functions and interaction matrices
![\[fig:laofeas\_d\] (Color online) Impurity spectral function within the $d$ hamiltonian using interaction parameters $U=4.0$eV and $J=0.7$eV as have been used in Ref. . For those values a very correlated metal is obtained.](fig6.eps){width="0.9\columnwidth"}
The first thing to note is that the Wannier functions constructed from a small energy window encompassing only the Fe-$d$ bands are quite extended and very anisotropic, as discussed in details in Ref. . This is directly reflected in the interaction matrices calculated by cRPA in this restricted energy window: $$\begin{aligned}
U_{mm'}^{\sigma\sigma}|_{\rm cRPA}&=\left(\begin{array}{ccccc}
0.00 & 1.41 & 1.26 & 1.87 & 1.87 \\
1.41 & 0.00 & 1.91 & 1.54 & 1.54 \\
1.26 & 1.91 & 0.00 & 1.33 & 1.33 \\
1.87 & 1.54 & 1.33 & 0.00 & 1.44 \\
1.87 & 1.54 & 1.33 & 1.44 & 0.00
\end{array}\right)\nonumber\\
U^{\sigma\bar{\sigma}}_{mm'}|_{\rm cRPA}&=\left(\begin{array}{ccccc}
3.17 & 2.02 & 1.72 & 2.22 & 2.22 \\
2.02 & 3.36 & 2.16 & 2.04 & 2.04 \\
1.72 & 2.16 & 2.17 & 1.73 & 1.73 \\
2.22 & 2.04 & 1.73 & 2.73 & 1.84 \\
2.22 & 2.04 & 1.73 & 1.84 & 2.73\end{array}\right)\nonumber\end{aligned}$$ which display a strong orbital dependence. For instance, the intraorbital (Hubbard) interaction spans from 2.17eV to 3.36eV. The interaction matrices in the spherical symmetric approximation using, the averages $U=2.14$ and $J=0.59$, are $$\begin{aligned}
U_{mm'}^{\sigma\sigma}&=\left(\begin{array}{ccccc}
0.00 & 1.25 & 1.25 & 1.85 & 1.85 \\
1.25 & 0.00 & 2.06 & 1.45 & 1.45 \\
1.25 & 2.06 & 0.00 & 1.45 & 1.45 \\
1.85 & 1.45 & 1.45 & 0.00 & 1.45 \\
1.85 & 1.45 & 1.45 & 1.45 & 0.00
\end{array}\right)\nonumber\\
U_{mm'}^{\sigma\bar{\sigma}}&=\left(\begin{array}{ccccc}
2.82 & 1.77 & 1.77 & 2.18 & 2.18 \\
1.77 & 2.82 & 2.31 & 1.91 & 1.91 \\
1.77 & 2.31 & 2.82 & 1.91 & 1.91 \\
2.18 & 1.91 & 1.91 & 2.82 & 1.91 \\
2.18 & 1.91 & 1.91 & 1.91 & 2.82\end{array}\right)\nonumber\end{aligned}$$ The largest deviation in this case is $\Delta
U=0.65$eV, corresponding to a relative error of about 0.26. This shows clearly that the spherical approximation is highly questionable when using only the $d$-bands for the Wannier construction. Of course, the full anisotropic interaction matrices can in principle be used in the LDA+DMFT calculation, but this raises the very delicate issue of a reliable [*orbital-dependent*]{} double-counting correction.
Another consequence of using delocalized Wannier functions is that they lead to significant non-local interactions $V_{dd}$, which we found to be (from cRPA) of order $0.23 U$ to $0.32 U$. These interactions are completely neglected in the single-site local DMFT approach, suggesting the need for a cluster extension in that case. For these various reasons, we have reservations against using a $d$-only Hamiltonian with extended Wannier functions for DMFT calculations on LaFeAsO, as also previously emphasized Ref. .
### Consistency with previous calculations
Nevertheless, in order to clarify apparent discrepancies between previously published LDA+DMFT results[@haule1; @anisimov2; @haule2; @shorikov1; @anisimov3], we performed calculations within the $d$-hamiltonian, for several interaction parameters reported in the literature. For the values $U=4.0$eV and $J=0.7$eV used in Ref. , we do confirm that the results then display very strong correlations with quasiparticle renormalizations ranging from $Z=0.11$ ($xy$ orbital) to $Z=0.34$ ($x^2-y^2$ orbital). One may note that within the $d$-model there is a substantial orbital dependence of $Z_m$, with a stronger renormalization predicted for the $xy$, $yz$ and $zx$ orbitals. This is a clear consequence of the Wannier functions being much more delocalized and anisotropic.
The scattering rate at this inverse temperature of $\beta=40$eV$^{-1}$ is quite sizable (${\rm Im}\Sigma(\omega^+=0)\approx -0.4\ldots-0.6$, depending on the orbital), showing that the system is on the verge of a coherence-incoherence crossover and a bad metal. The impurity spectral function is plotted in Fig. \[fig:laofeas\_d\]. It resembles very much the one shown in Fig. 3 of Ref. , showing clear signatures of lower and upper Hubbard bands. There are, though, some discrepancies with the total weight and the positions of the Hubbard bands, but given the differences in the calculation (underlying electronic structure method, temperature, interaction vertex which here is only density-density), this agreement with Ref. is quite satisfactory.
Furthermore, using the parameters $U=0.8$eV and $J=0.5$eV from Ref. , we find renormalizations in the range $Z\approx 0.7 - 0.8$. This is somewhat smaller than reported in Ref., although not in drastic disagreement.
Finally, we investigated the dependence on the Hund’s rule coupling of calculations performed with the $d$-only hamiltonian. Decreasing $J$ to the much lower value $J=0.2$eV but keeping $U=4$eV, we find the system to be much less correlated ($Z$ between 0.63 and 0.73). We thus confirm, for those calculations, the great sensitivity to the Hund’s coupling reported in Ref. . We note however that, although reducing $J$ does make the system somewhat less correlated in this case too, this sensitivity is much weaker when calculations are performed with the full $dpp$ hamiltonian, as reported above.
### Origin of the sensitivity to the Hund’s coupling: level crossings
In order to understand the origin of the remarkable sensitivity of the correlation strength to the value of $J$ observed with the $d$-Hamiltonian we have studied the evolution of the ground state of the Fe 3$d$ “atomic shell” as function of $J$. We obtained the 3$d$ level positions corresponding to two different choices of the energy window: the “small” one corresponding to the $d$-Hamiltonian and comprising 10 Fe 3$d$ bands and the “very large” one comprising all As 4$p$, O 2$p$ and Fe 3$d$ bands as well as all unoccupied bands up to 13 eV above $E_F$. The non-interacting level positions $\epsilon_{mm'}^{\alpha,\sigma}$ are then obtained as
$$\epsilon_{mm'}^{\alpha,\sigma}=\sum_{{\mathbf k},\nu\in \mathcal{W}} P_{m\nu}^{\alpha,\sigma}\epsilon_{\mathbf{k}\nu}^\sigma
P_{\nu m'}^{\alpha,\sigma *}-\tilde{\Sigma}_{mm'}^{\sigma,\rm{dc}},$$
where the double counting term $\tilde{\Sigma}_{mm'}^{\sigma,\rm{dc}}$ is calculated in accordance with Eq. (\[eq:dc\]) but with the “atomic” occupancy $N=6$ of the Fe 3$d$ shell. We used the same values of $U=2.14$ and 2.69eV for the “small” and “very large” window choices, respectively, while the value of $J$ was varied from 0.1 to 0.5eV. With $\epsilon_{mm'}^{\alpha,\sigma}$ corresponding to the $d$-Hamiltonian we observed a level crossing at $J \approx 0.2$ eV with the atomic ground state changing from the one with spin moment $S=1$ to the one with $S=2$. In the case of the “very large” window the ground state always corresponds to $S=2$, and the splitting between the ground state and first excited level is constant. It is obvious that a drastically different behavior of those two “atomic” models is related to the corresponding level positions $\epsilon_{mm'}$, which are computed using different choices for the Wannier orbitals. The observed change of the Fe 3$d$ atomic ground state, induced by increasing $J$, hints on a possible strong dependence of correlation strength on the Hund’s rule coupling for LaFeAsO, which is indeed observed in our LDA+DMFT calculations with the $d$-Hamiltonian. However, this sensitivity stems from a particular choice of delocalized and anisotropic Wannier functions and is much less pronounced when the energy window for the Wannier function construction is increased.
The bottom-line of this investigation is that all previously published calculations seem to be technically correct. However, as discussed above, one introduces several severe approximations when dealing with the $d$-hamiltonian only, and the justification of these approximations (restriction to local interactions, single-site DMFT, etc...) is questionable. This is especially true in this compound, due to the strong covalency between iron and arsenic states.
Conclusion and prospects
========================
In the first part of this work, we present an implementation of LDA+DMFT in the framework of the full-potential linearized augmented plane waves method. We formulate the DMFT local impurity problem in the basis of Wannier orbitals, while the full lattice Green’s function is written in the basis of Bloch eigenstates of the Kohn-Sham problem. In order to construct the Wannier orbitals for a given correlated shell we choose a set of local orbitals, which are then expanded onto the KS eigenstates lying within a certain energy window. In practice, we employ the radial solutions of the Schrödinger equation for a given shell evaluated at the corresponding linearization energy as local orbitals. By orthonormalizing the obtained set of basis functions we construct a set of true Wannier orbitals as well as projector operator matrices relating the Bloch and Wannier basis sets. We derive explicit formulas for the projected operator matrices in a general FLAPW framework, which may include different types of augmented plain waves, $lo$ and $LO$ orbitals. Our new implementation is benchmarked using the test case of SrVO$_3$, for which we have obtained spectral and electronic properties in very good agreement with results of previous LDA+DMFT calculations.
In the second part of this paper we apply this LDA+DMFT technique to LaFeAsO in order to assess the degree of electronic correlations in this compound and clarify the ongoing controversy about this issue in the literature. We solved the DMFT quantum impurity problem using a continuous-time quantum Monte Carlo approach. The Wannier functions are constructed using an energy window comprising Fe 3$d$, As 4$p$ and O 2$p$. The resulting Wannier orbitals are rather well localized and isotropic. We take the average values of $U=$2.69 eV and $J=$0.79 from constrained RPA calculations, where the Wannier functions and screening channels are consistent with our setting of the LDA+DMFT scheme. We have checked the robustness of these results by increasing the size of the energy window, what resulted in a very similar physical picture.
Our LDA+DMFT results indicate that LaFeAsO is a moderately correlated metal with an average value for the mass renormalization of the Fe 3$d$ bands about 1.6. This value is in reasonable agreement with estimates from photoemission experiments.
We also consider a smaller energy window that includes Fe-$d$ states only. The resulting Wannier functions in this case are quite extended, leading to anisotropic and non-local Coulomb interactions. We take different values for $U$ and $J$, including the ones used in previous theoretical LDA+DMFT approaches. We demonstrate that different physical pictures ranging from a strongly correlated compound on the verge of the metal insulator transition to a moderately to weakly correlated one can emerge depending, in particular, on the choice of the Hund’s rule coupling $J$ as observed in Ref. . However, there are conceptual difficulties when constructing a local Hamiltonian from rather delocalized Wannier orbitals. The interactions are very anisotropic and orbital dependent, and non-local interactions could also become important.
In summary, we demonstrate that the discrepancies in the results of several recent theoretical works employing the LDA+DMFT approach stem from two main causes: i) the choice of parameters of the local Coulomb interaction on the Fe 3$d$ shell and ii) the degree of localization of the Wannier orbitals chosen to represent the Fe 3$d$ states, to which many-body terms are applied. Regarding the first point, the calculated interaction parameters employed in the present work are significantly smaller than the values hypothesized in Refs. \[\]. Regarding the second point, we provide strong evidence that the DMFT approximation is more accurate and more straightforward to implement when well-localized orbitals are constructed from a large energy window encompassing Fe-3$d$, As-4$p$ and O-2$p$. This issue has fundamental implications for many-body calculations, such as DMFT, in a realistic setting.
Benchmark: SrVO$_3$ {#sect:benchmark}
===================
For benchmarking purposes, we present in this appendix LDA+DMFT results for an oxide that has become a classical test compound for correlated electronic structure calculations, namely the cubic perovskite 3. As a paramagnetic correlated metal with intermediate electron-electron interactions, it is in a regime that is neither well described by pure LDA calculations nor by approaches such as LDA+U that are geared at ordered insulating materials. From the experimental side, 3 has been characterized by different techniques (angle-resolved and angle-integrated photoemission spectroscopy, optics, transport, thermodynamical measurements etc)[@imada_mit_review; @fujimori_pes_oxides; @ono91; @maiti_2001; @maiti_phd; @inoue_casrvo3_1995_prl; @sek04; @yos05; @wad06-bis; @sol06; @egu06].
LDA+DMFT calculations have been performed both for an effective low energy model that comprises the three degenerate bands of mainly 2g character that are located around the Fermi energy – taking advantage of the cubic crystal field that singles out this group of bands – and for a bigger energy window comprising also the oxygen $p$-states. [@lie03; @sek04; @pav04; @nek05; @yos05; @wad06-bis; @sol06; @nek06; @amadon_pw_08].
In the low-energy effective 2g model a quasi-particle renormalization of $Z \sim 0.6$, compatible with experiments, is obtained for $U$ values around 4eV. The remaining spectral weight is shifted toward lower and upper Hubbard bands. The lower Hubbard band, located around $-1.5$eV binding energy has indeed been observed in photoemission; the high energy satellite of the 2g model is located around 2.5eV. [^1]
Concerning calculations taking into account also the ligand states, it should be noted that possible LDA errors on the separation of $p$- and $d$-states are not corrected by DMFT, since only the $d$-states are treated as correlated.
![\[fig:srvo3d\] (Colore online) DOS for 3, $d$-only model (small energy window). Top panel (a): Total DOS of LDA (black) and LDA+DMFT (red). Bottom panel (b): LDA local orbitals (black), impurity spectral function $A(\omega)$ (red), and vanadium 2g partial DOS (green). Coulomb parameters for these calculations are given as inset.](fig7.eps){width="0.9\columnwidth"}
![\[fig:srvo3dp\] (Color online) Total DOS for 3, $dp$ Hamiltonian (vanadium 2g and oxygen $p$).](fig8.eps){width="0.9\columnwidth"}
In the present work, we use 3 as a benchmark for our projector orbitals implementation of LDA+DMFT, with results very similar to previous theoretical studies. We performed two kinds of calculations: (i) We used as an energy window the range from $-1.35$eV to 1.90eV which comprises the 2g bands, and – at some k-points – one or both of the bands. This is closest in spirit to a 2g model within a Wannier function formalism, though not exactly the same due to the inclusion of some contribution. To recover a Wannier prescription one would in fact have to choose a k-dependent window, such as to include exactly three bands [*at each k-point*]{}, corresponding to the three-fold degenerate manifold of dominantly 2g bands. (ii) We used an energy window of $-8.10$eV to 1.90eV, spanning both, the bands used in (i) and the oxygen $p$ dominated bands located between $-8$ and $-2$eV.
Please note that, in order to be consistent with existing literature, we use a different parametrisation of the interaction matrix compared to Sect. \[sect:results\]. Here we define $U$ to be the onsite intraorbital Coulomb interaction, $U-2J$ to be the inter-orbital interaction for electrons with opposite spin, and $U-3J$ the inter-orbital interaction between electrons with equal spin.
The results for the first case are shown in Fig. \[fig:srvo3d\]. The upper panel, Fig. \[fig:srvo3d\] (a), displays the total spectral function of the thus defined model within LDA and LDA+DMFT. Our results recover previously published results, with a quasiparticle renormalisation of around $Z=0.60$ for a values $U=4.0$eV and $J=0.65$eV. The contribution of the bands to the total DOS can easily be identified from the LDA+DMFT spectra, where an additional hump between the quasiparticle peak and the upper Hubbard band appears.
Fig. \[fig:srvo3d\] (b) shows the local orbitals used for the DMFT calculations, together with the corresponding impurity spectral function $A(\omega)$ and the vanadium 2g partial DOS. The latter one is obtained bu projecting the lattice Green’s function to 2g character using the partial projectors to be introduced in App. \[sect:pc\_proj\]. The main difference to panel (a) is the absence of the additional character.
Finally, Fig. \[fig:srvo3dp\] shows the LDA+DMFT spectral function compared to the LDA density of states, as calculated within the larger energy defined in (ii) above. Since the Wannier function are more localised, as compared to case (i), the value for the Coulomb interactions has to be adjusted accordingly, and we chose a value of $U=6.0$eV. As can be seen in the figure, ligand states are barely modified by the correlations, and the results for the 2g-derived bands are very close to what is seen in the effective low energy model. The quasiparticle renormalisation is $Z=0.57$, in good agreement with the pure 2g treatment discussed before.
The results of these calculations correspond to what can be expected on the basis of previously published work, and thus validate our new implementation.
\[sect:srvo3\]
Projectors for partial DOS {#sect:pc_proj}
==========================
In order to calculate the partial density of states for a given atomic site and particular orbital character (correlated or not) we construct a different type of projectors, which we call $\hat{\bf{\Theta}}^{i,\sigma}$.
The Wannier operators of Eq. \[eq:wannier-proj\] project onto a given Wannier-like orbital. On the other hand, the new set $\hat{\bf{\Theta}}^{i,\sigma}$, as we will show, project onto a given orbital of certain character for which we do not apply any orthonormalization process as in the first. Unlike the Wannier projectors, the $\hat{\bf{\Theta}}^{i,\sigma}$’s can also project to other orbitals atoms apart from the correlated set.
A given orbital character contributes in the eigenstates through the solutions of the Schrödinger equation inside the sphere $u_{l}^{\sigma}(r,E_{l1})Y_{m}^{l}\chi_{\sigma}$, $\dot{u^{\sigma}}_{l}(r,E_{l1})Y_{m}^{l}\chi_{\sigma}$ and $u_{l}^{\sigma}(r,E_{l2})Y_{m}^{l}\chi_{\sigma}$ which do not form an orthonormalized basis set. It is more convenient to construct these projectors if the wave function is rewritten in an orthonormal basis set.
In a general form, inside a given sphere we can express $\psi_{\mathbf{k}\nu}^{\sigma}(\mathbf{r})$ as
$$\psi_{\mathbf{k}\nu}^{\sigma}(\mathbf{r})=\sum_{lm}A'_{lm}u_{l1}+\sum_{lm}B'_{lm}\dot{u}_{l}+\sum_{lm}C'_{lm}u_{l2},\label{eq:wave-func1}$$
where we simplify the notation by omitting the angular and spin parts and defining $A'_{lm}$, $B'_{lm}$ and $C'_{lm}$ as combined coefficients which are generally k-dependent and contain the sum over the plane waves and local orbitals. We also define $u_{l1}\equiv$$u_{l}(r,E_{l1})$, $\dot{u}_{l}\equiv\dot{u}_{l}(r,E_{l1})$ and $u_{l2}\equiv u_{l}(r,E_{l2})$.
We then rewrite $\psi_{\mathbf{k}\nu}^{\sigma}(\mathbf{r})$ as a function of a set of orthogonal orbitals $\phi_{j}(r)$, *j*=1,2,3 as follows
$$\psi_{\mathbf{k}\nu}^{\sigma}(\mathbf{r})=\sum_{lm}\sum_{j}\left(A'_{lm}c_{1j}^{lm}+B'_{lm}c_{2j}^{lm}+C'_{lm}c_{3j}^{lm}\right)\phi_{j}(r).\label{eq:wave-func2}$$
The coefficients $c_{ij}^{lm}$ are the matrix elements of the square root of the corresponding overlap matrix
$$\mathbf{C}=\left(\begin{array}{ccc}
1 & 0 & \left\langle u_{l1}\right|\left.u_{l2}\right\rangle \\
0 & \left\langle \dot{u}_{l}\right|\left.\dot{u}_{l}\right\rangle & \left\langle \dot{u}_{l}\right|\left.u_{l2}\right\rangle \\
\left\langle u_{l2}\right|\left.u_{l1}\right\rangle & \left\langle u_{l2}\right|\left.\dot{u}_{l}\right\rangle & \left\langle u_{l2}\right|\left.u_{l2}\right\rangle \end{array}\right)^{\frac{1}{2}}.\label{eq:overlap}$$
In this way, rewriting Eq. \[eq:wave-func2\] as
$$\psi_{\mathbf{k}\nu}^{\sigma}(\mathbf{r})=\sum_{lm}\sum_{j}\tilde{c}_{j}^{lm}\phi_{j}(r).\label{eq:wave-func3}$$
the matrix elements of the projector to a given atom with *lm* character finally reads,
$$\Theta_{m\nu j}^{i,\sigma}(\mathbf{k})=\tilde{c}_{j}^{lm}.\label{eq:theta-proj-2}$$
The spectral function of a given atom *i* with orbital character *m*, is obtained as
$$A_{m}^{i,\sigma}(\mathbf{k},\omega)=-\frac{1}{\pi}Im\left[\sum_{\nu\nu',j}\Theta_{m\nu j}^{i,\sigma}(\mathbf{k})G_{\nu\nu'}^{\sigma}(\mathbf{k},\omega^{+})\Theta_{\nu'm'j}^{i,\sigma*}(\mathbf{k})\right]$$
Influence of the rotational invariance of Hunds-rule coupling in multi-orbital systems {#sect:hundsrule}
======================================================================================
![\[figZ\] Quasi-particle renormalisation $Z$ in a 3-orbital Hubbard model. Calculations have been done using a semicircular DOS with bandwidth $W=4t$. Top panel: $n=3$ (half filling). Bottom panel: $n=2$.](fig9.eps){width="0.8\columnwidth"}
In our DMFT calculations using CTQMC as impurity solver, we restricted the Hund’s rule interaction to Ising-type interactions only, although there is no conceptual limitation of the algorithm to this type of interactions. The reason for doing this is of purely technical nature, since in this case one can diagonalize the local problem very efficiently, and furthermore, it enables us to use the so-called segment-picture update scheme,[@werner2] which increases the efficiency of the CTQMC method a lot.
One may now ask how results change if the fully rotational-invariant Hund’s rule exchange is taken into account. For this purpose, we study a multiband model Hamiltonian, assuming degenerate bands, no interband hybridizations, and a semicircular density of states. Applying the self-energy functional theory (SFT),[@sft_potthoff] we can study the quasiparticle renormalization $Z$ as function of interactions $U$ and $J$. In this study, we choose the convention of setting the intraorbital Coulomb repulsion to $U$ and the interorbital to $U^\prime=U-2J$, and give all energies in units of the single-particle hopping amplitude $t$, i.e., the band with of the DOS is $W=4t$.
In addition to the density-density interactions, we consider also the additional spin-flip and pair-hopping terms of the local Hamiltonian, $$\begin{aligned}
H_{\rm sf}&=-\frac{J}{2}\sum_{mm'}\left(c_{m\uparrow}^\dagger
c_{m\downarrow} c_{m'\downarrow}^\dagger c_{m'\uparrow} + {\rm
h.c.}\right) \\
H_{\rm ph}&=-\frac{J}{2}\sum_{mm'}\left(c_{m\uparrow}^\dagger
c_{m\downarrow}^\dagger c_{m'\uparrow} c_{m'\downarrow} + {\rm
h.c.}\right). \end{aligned}$$ We do calculations at $T=0$, and choose the reference system for the SFT framework to consist of one bath degree of freedom for each correlated orbital. Hence, going up to $M=5$ orbitals, we have to diagonalize a local problem consisting of at most 10 orbitals.
The upper panel of Fig. \[figZ\] shows $Z$ for a 3-orbital model at half-filling, $n=3$, for $J=0.1U$. A tremendous reduction of the critical $U_c$ of the metal-to-insulator transition (MIT) is observed, already for Ising-like interactions. This is a well known fact that for multi-orbital systems at or close to half-filling, the effect of $J$ should be strongest.[@ono03; @pruschke05; @in.ko.05] The inclusion of spin-flip and pair-hopping terms gives raise to two effects. (i) For moderate correlations, $Z\approx 0.6$, these terms lead to a slight reduction of $Z$, but (ii) the critical $U$ for the MIT is shifted upwards. This qualitatively holds also away from half-filling, which can be seen in the lower panel of Fig. \[figZ\], where we plotted $Z$ for $n=2$. Although the transition is not of first order any more, one can again identify two regimes. For moderate correlations, $Z$ [*decreases*]{}, whereas close to the transition the spin-flip and pair-hopping terms [*increase*]{} the renormalization $Z$ and the critical $U_c$ is pushed to higher values. This is consistent with a numerical renormalization group study for the 2-orbital Hubbard model.[@pruschke05]
We also considered the case relevant for pnicitde materials, i.e., $M=5$, $n=6$, and $Z$ around 0.5. This regime could be realize by setting (i) $U=3.5t$ and $J=0.35t$, which shows a reduction of $Z$ from 0.52 to 0.47 due to spin-flip and pair-hopping, or (ii) $U=2t$, $J=0.4t$, giving a reduction from 0.61 to 0.57. In conclusion, this analysis shows that the picture of a moderately-correlated metal as argued in Sect. \[sect:results\] holds also when a fully rotational-invariant Hund’s exchange is considered. For systems close to a MIT, this is no longer true and the spin-flip and pair-hopping terms become crucial.
We are grateful to Vladimir Anisimov, Ryotaro Arita, Gabriel Kotliar, Igor Mazin, Frank Lechermann, Alexander Lichtenstein and, especially, Kristjan Haule for useful discussions and correspondence. M.A. is grateful to F. Assaad for enlightening discussions on the stochastic Maximum Entropy method. We acknowledge the support of the Agence Nationale de la Recherche (under project CORRELMAT), and of GENCI and IDRIS (under project 091393) for supercomputer time. M.A. acknowledges financial support from the Austrian Science Fund (FWF), grant J2760-N16.
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[^1]: Even though comparisons with x-ray absorption spectroscopy have been attempted it is not clear that the low-energy description by a pure t2g model is still valid at these energies. Note in particular that in this energy region overlaps with the eg states, split off by the cubic crystal field, could come into play.
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