text stringlengths 4 2.78M |
|---|
---
abstract: 'The form of a sheath near a small electrode, with bias changing from below to above the plasma potential is studied using 2D particle-in-cell (PIC) simulations. Five cases are studied: (A) an electrode biased more than the electron temperature ($T_e/e$) below the plasma potential, (B) an electrode biased less than $T_e/2e$ below the plasma potential, (C) an electrode biased nearly at the plasma potential, (D) an electrode biased more than $T_i/2e$ but less than $T_e/2e$ above the plasma potential, and (E) an electrode biased much greater than $T_e/2e$ above the plasma potential. In case (A), the electron velocity distribution function (EVDF) is observed to be Maxwellian with a Boltzmann-type exponential density decay through the ion sheath and presheath. In cases (B) and (C), the EVDFs exhibit a loss-cone type truncation due to fast electrons overcoming the small potential difference between the electrode and plasma. No sheath is present in this regime, and the plasma remains quasineutral up to the electrode. The EVDF truncation leads to a presheath-like density and flow velocity gradient. In case (D) an electron sheath is present, and essentially all ions are repelled. Here the truncation driven behavior persists, but is accompanied by a shift in the maximum value of the EVDF that is not present in the negative bias cases. In case (E), the flow shift becomes greater and the loss-cone moves further into the tail of the EVDF. In this case the flow moment has significant contributions from both the flow shift of the EVDF maximum, and the loss-cone truncation.'
author:
- Brett Scheiner
- 'Scott D. Baalrud'
- 'Matthew M. Hopkins'
- 'Benjamin T. Yee'
- 'Edward V. Barnat'
bibliography:
- 'nosheath.bib'
title: 'Particle-in-cell study of the ion-to-electron sheath transition'
---
INTRODUCTION
============
Electron sheaths can form around plasma boundaries that are biased above the plasma potential. In the absence of secondary emission of electrons from the boundary, the electrode can only be biased above the plasma potential provided that the electron collecting surface is small enough that global current balance in the plasma can be maintained[@2007PhPl...14d2109B]. This requirement that the electron collecting area is small typically precludes this situation from happening near large boundaries, hence the typical electron sheath is found around small electrodes such as Langmuir probes collecting the electron saturation current[@1926PhRv...28..727M].
The conventional wisdom has been that electron sheaths interface with the bulk plasma without a presheath [@1926PhRv...28..727M; @2005PhPl...12e5502H], do not pose a significant perturbation to the bulk plasma, and do not require any significant acceleration region for electrons entering the sheath[@1991JPhD...24..493R; @2006PSST...15..773C]. However, recent experiments and simulations at low temperature plasma conditions, [@2015arXiv150805971Y] along with theoretical work [@2015PhPl...22l3520S], have shown that the electron sheath is accompanied by a presheath. In this presheath the electrons achieve a flow moment comparable to the electron thermal speed by the sheath edge. Unlike the ion presheath, which accelerates ions through a ballistic response to the electric field, the electron presheath accelerates electrons through a force due to a pressure gradient. Simulations of the electron presheath have demonstrated that this pressure gradient term in the electron momentum equation is in fact the dominant term[@2015PhPl...22l3520S].
The electron sheath and presheath are fundamental features of plasma boundary interactions. Although they have been rarely studied, there are potentially important applications of a deeper understanding of this fundamental phenomenon. One example is the low potential side of double layers, such as those found in the auroral upward current region[@2004JGRA..10912220E] and double layers in helicon plasma thrusters[@2007ApPhL..91t1505C]. Other areas where electron sheaths are known to be present are tethered space probes[@1996JGR...10117229S], near electrodes used to induce circulation in dusty plasmas[@1998PhRvL..80.4189L], as well as the sheath prior to the formation of anode spots on small and gridded electrodes [@1991JPhD...24.1789S; @2009PSST...18c5002B; @2008PSST...17c5006S].
In this paper, the pressure-gradient-driven flow that arises as the bias of a small electrode is swept from several volts below to several volts above the plasma potential is studied using 2D particle-in-cell (PIC) simulations. During the transition from ion to electron sheath the electron velocity distribution function (EVDF) develops a loss cone as the electrode bias nears the plasma potential. As the electrode bias begins to exceed the plasma potential, the EVDF acquires a significant flow shift in addition to the loss cone. The nature of the EVDF in this transition is quantified by focusing on five electrode biases. In case (A) the electrode is biased several times $T_e/2e$ below the plasma potential so that an ion sheath is present. Here $T_e/2e$ is the typical change in potential seen in ion presheaths, and is used as a measure of how weak the electrode can be biased below the plasma potential before a significant number of electrons overcome the potential barrier and are collected. In this case nearly all electrons are repelled and the EVDF remains Maxwellian. Case (B) consists of an electrode biased less than $T_e/2e$ below the plasma potential. In this case the plasma remains quasineutral up to the electrode surface. At this bias, not all electrons are repelled resulting in the formation of a loss cone, which leads to presheath-like density and flow velocity gradients that increase as the electrode is approached. In case (C), when the electrode is biased nearly at the plasma potential, the loss cone and presheath-like gradients persist. In case (D), an electron sheath is observed at biases more than $T_i/2e$ but less than $T_e/2e$ above the plasma potential. Here $T_i/2e$ is significant since below this bias a significant number of ions overcome the potential barrier and are collected by the electrode. In case (D) the EVDF continues to have a loss cone, but also has developed a significant flow shift. In case (E) the electrode is biased to a value much greater than the plasma potential. In this case the flow shift is more visible and the EVDF undergoes a distortion due to the acceleration in the strong electric field.
This paper is organized as follows. Sec. II describes the simulation setup and presents the key observations from simulations at different values of the electrode bias. Sec. III discusses the results of the simulations and explains the observations. Sec. IV gives concluding remarks.
SIMULATIONS
===========
Simulations were performed using Aleph, an electrostatic PIC code [@2012CoPP...52..295T]. For the present work, a 2D triangular mesh was used to discretize the 2D domain shown in Fig. 1a. The 15 cm by 5 cm domain had three boundaries with a Dirichlet $V=0$ boundary condition and one reflecting Neumann boundary condition so that the simulation represents a physical area of 15 cm by 10 cm. An electrode of length 0.1 cm and width 0.02 cm was placed perpendicular to the center of the reflecting boundary. The time step of 0.5$\times10^{-4} \ \mu s$ resolved the electron plasma frequency and ensured that the plasma particles satisfied the CFL condition[@courant1928partiellen]. Each simulation ran for 800,000 time steps totaling 40 $\mu s$ of physical time. The size of each mesh element was 0.02 cm such that the electron Debye length was resolved. Particles used included helium ions and electrons, each with a macroparticle weight of 2000. The helium plasma was sourced in the volume at a rate of $10^8 cm^{-3} \ \mu s^{-1}$ at temperatures of 0.08 eV for ions and 4 eV for electrons.
------------ -------------------- -------------------- ---------------- ----------------------------------------
Simulation $\ \ \phi_p \ \ $ $\ \ \phi_E \ \ $ $\ \ T_e \ \ $ Desired criteria
(V) (V) (eV) for $\phi_E-\phi_p$
A 2.2 -25 0.62 $\phi_E-\phi_p< -T_e/2e$
B 5.7 5.5 1.55 $\ 0>\phi_E-\phi_p> -T_e/2e$
C 6.5 6.5 1.88 $\phi_E-\phi_p\sim 0 \ \ \ \ \ \ \ \ $
D 7.6 8 2.29 $T_e/2e>\phi_E-\phi_p>T_i/2e \ \ \ $
E 18.5 25 5.17 $\phi_E-\phi_p>T_e/2e \ \ \ $
------------ -------------------- -------------------- ---------------- ----------------------------------------
: Summary of the electrode bias $\phi_E$, plasma potential $\phi_p$, electron temperature $T_e$, and the desired value for $\phi_E-\phi_p$ in each simulation.
Simulations for the five cases (A)-(E), mentioned previously in Sec. I, were carried out and results are summarized in Table I. The flow moments, densities for electrons and ions, and potential profiles are shown in Fig. 2. The figure displays values along a line perpendicular to the electrode, along the reflecting boundary. These values are from the average of the last 100,000 time steps of field data output by Aleph. Note that the non-smooth behavior in the velocity moment data is due to the velocity moment being a cell-based value, which is constant across each cell in the final output of the simulations. Since the simulations were 2D, the temperature values used here were computed from the x and y velocity components only, i.e. $T_e = n_e\int d^2v m_e(v_{r,x}^2+v_{r,y}^2)f_e/2$, where $f_e$ is the EVDF, $n_e$ is the electron density, and $v_{r,i}=({\mathbf}{v}-{\mathbf}{V_e})\cdot\hat{{\mathbf}{i}}$. Two-dimensional velocity moments have previously been used when comparing theory to 2D simulations[@2015PhPl...22l3520S; @2015PPCF...57d4003B]. The electron and ion flow velocities in Fig. 2 are normalized by $v_{T_e}=\sqrt{T_e/m_e}$ and $v_B=\sqrt{T_e/m_i}$ respectively. Here the 2D values of $T_e$ were computed approximately 0.5 cm from the electrode surface, and were 0.62 eV, 1.55 eV, 1.88 eV, 2.29 eV, and 5.17eV for cases (A)-(E) respectively.
Using data from simulations the terms of the momentum equation, $$m_en_e{\mathbf}{V}_e\cdot\nabla{\mathbf}{V}_e=-ne{\mathbf}{E}-\nabla\cdot\mathcal{P}_e-{\mathbf}{R}_e,$$ were evaluated. Here $m_e$, $n_e$, and ${\mathbf}{V_e}$ are the electron mass, density, and flow moment, $\mathcal{P}_e$ is the pressure tensor defined below, and ${\mathbf}{R}_e$ is a friction term. The $\hat{x}$ vector components $m_en_e{\mathbf}{V}_e\cdot\nabla{\mathbf}{V}_e\cdot\hat{x}$ and $\nabla\cdot\mathcal{P}_e\cdot\hat{x}$ of the terms of the momentum equation were computed from particle location and velocity data at 20 different time slices separated by $5\times10^{-3}\mu s$ each. The velocity moments of the electron velocity distribution function $f_e({\mathbf}{v})$ used above are $${\mathbf}{V}_e=\frac{1}{n_e}\int d^2v {\mathbf}{v} f_e({\mathbf}{v}),$$ $$n_e=\int d^2v f_e({\mathbf}{v}),$$ and $$\mathcal{P}_e=m_e\int d^2v ({\mathbf}{v}-{\mathbf}{V_e})({\mathbf}{v}-{\mathbf}{V_e})f_e({\mathbf}{v})=p_e\mathcal{I}+\Pi_e,$$ where the pressure tensor is decomposed into the scalar pressure $p_e$ multiplied by the identity tensor $\mathcal{I}$ and stress tensor $\Pi_e$. Based on observation, it was assumed that the dominant gradients in ${\mathbf}{V_e}$ and $\mathcal{P}_e$ were in the $\hat{x}$ direction. These moments along with the value of $-ne{\mathbf}{E}\cdot\hat{x}$ from the time averaged mesh based data are also shown in Fig. 2. Once again, all values were computed in 2D using only x and y velocity information.
Using particle location and velocity data from the same 20 time slices mentioned above, the EVDFs were obtained by plotting the 2D histogram of the x and y velocity components for particles. First, the distributions were obtained in three 0.1 by 0.1 cm boxes for all five simulations, the locations of which are indicated by the black boxes in Fig. 1b. EVDFs for all simulations are shown in Fig. 4. For the case of the electrode biased at 25V additional EVDFs, shown in Fig. 4, were obtained in the region indicated by the small red boxes in Fig. 1b to better resolve the EVDF within the sheath. Note that a slight asymmetry is present in the y velocity components in Fig. 3, this is due to the averaging box not being aligned with the symmetry axis as shown in Fig. 1b. This asymmetry is less visible in Fig. 4 due to the smaller averaging area. The interpretation of the data for these five simulations are discussed in the following section.
DISCUSSION
==========
In the previous section, simulations with five different values of electrode bias were presented: strong ion and electron sheath cases (A) and (E), and three cases (B)-(D) with the electrode biased near the plasma potential. In this section, flow moments and EVDFs calculated from these simulations are used to study the development of pressure gradient driven electron flow in the transition from ion to electron sheath.
First, the behavior of electrons near the strong ion sheath case (A) is examined. The fluid moments and fields for this case are shown in Fig. 2 and the EVDFs in Fig. 3. In the ion presheath and sheath the electrons have no flow, except for a slight flow shift directed out of the sheath within 0.2cm of the electrode, which is due to the expulsion of a small number of electrons sourced in the sheath. Although the flow is non-zero within 0.2 cm, the density is small enough that $n_e{\mathbf}{V_e}$ is negligible. The EVDFs show no truncation or flow shift, which corresponds to the expectation based on a Boltzmann density profile $n_e\sim \exp(-e\Delta\phi/T_e)$ in the presheath. This expectation is verified in Fig. 5 using the computed 2D electron temperature and simulated values of $\Delta\phi$. The EVDFs demonstrate that as the electrode is approached the density drops off rapidly, with only a few particles present within 0.1 cm of the electrode surface. A Boltzmann density profile is expected when the EVDF is Maxwellian, which can be seen from the electron momentum equation given in Eq. (1). In this cassette friction and flow terms can be neglected, the pressure tensor reduces to $n_eT_e\mathcal{I}$, and its gradient is balanced by the electric field, leading to the exponential relation mentioned above. While other distributions might satisfy the zero flow moment requirement, the Maxwellian supports an exponential decay for particles that follow characteristics defined by the electric field[@1873Natur...8..537C].
When the electrode was biased within $T_e/2e$ below the plasma potential, as it is in case (B), the assumptions of the Boltzmann relation for the density profile break down. At such a small bias a significant number of electrons have energy greater than $T_e/2$ and reach the electrode, resulting in a truncation of the EVDF. The only electrons directed away from the electrode near it’s surface are those sourced in this region. However, this population is small and essentially negligible. The effects of this truncation can be seen in Fig. 3. Near the surface of the electrode, the truncated part of the EVDF forms a loss-cone type distribution with an angle that narrows when moving further into the bulk plasma. As shown in Fig. 6, there is an area that is geometrically inaccessible to electrons moving from left to right, forming a shadow. In this region fast electrons directed towards the electrode are collected, and no electrons at high energy are available to fill in the part of the distribution directed away from the electrode. In the case of a strong ion sheath this does not occur, electrons directed towards the electrode are reflected and the distribution remains Maxwellian.
In the cases (B) and (C) the difference between the plasma potential and electrode are $0>\phi_E-\phi_p> -T_e/2e$ and $\phi_E-\phi_p\sim 0$, as can be seen in the potential plotted in Fig. 2. The density profiles for ions and electrons, along with the small potential gradients, indicate that in these cases the plasma remains quasineutral up to the electrode. Although there is no breakdown of quasineutrality, there are still significant flow moments and density gradients for electrons and ions. Unlike the typical ion presheath, the ions only achieve a flow velocity of $\sim0.4v_B$ by the electrode for the case slightly below the plasma potential, while the electrons have a flow of $\sim0.6v_{T_e}$, with both flows directed towards the electrode. The electric field term for case (B) is weaker than the pressure term, and for case (C) the field term is negligible, suggesting the flows in this regime are not related to the electric field. Gradients in the electron densities and flow velocities can be explained by the widening of the loss cone. As the electrode is approached the loss cone angle increases while there is little or no shift in velocity of the maximum value of the EVDF in Fig. 3. The increasing flow moment is a result of the increasing angle of the loss cone. The continuity equation $\nabla\cdot(n_e{\mathbf}{V_e})=0$ requires that an increase in flow moment must be accompanied by a decrease in density. The computed velocity moment gradients in Fig. 2 indicate that the pressure tensor gradient, due to the increasing truncation, plays an important role when the electrode is biased near the plasma potential.
In case (D) the behavior changes. The most notable difference is that the plasma no longer remains quasineutral up to the electrode, but a well defined electron sheath has formed. When $\phi_E-\phi_p\sim T_e/2e$ nearly all of the ions are repelled due to the fact that $T_e\gg T_i$. This can be seen in the ion density profile, which shows very few ions in the electron sheath. Here, the electrons achieve a flow velocity of $\sim0.6\sqrt{T_e/m_e}$ by the sheath edge, which is slightly slower than the previously predicted value of $\sqrt{T_e/m_e}$[@2015PhPl...22l3520S]. However, the exact sheath edge position is difficult to determine in time averaged data due to streaming instability driven fluctuations which have been discussed previously[@2015PhPl...22l3520S]. In addition, the form of the EVDF, including the extent of the loss cone, would need to be accounted for in the calculation of the sheath edge speed for electrons. Nevertheless, the flow of electrons into the sheath can be seen to be driven by pressure tensor gradients in Fig. 2, with some portion of the flow due to the shift in the maximum value of the EVDF, and the rest due to the truncation visible in Fig. 3.
The presheath behavior remains qualitatively similar as the electrode bias is further increased, as demonstrated by case (E). The presheath behavior is largely the same as that in case (D), a shift in maximum value of the EVDF and a loss cone is present near the sheath as shown in Fig. 3. The main difference between simulation (E) and simulation (D) is the deformation of the EVDF within the sheath, which is shown in detail in Fig. 4. Near the sheath edge, at x=0.165 cm, the EVDF resembles the loss cone shaped distributions of Fig. 3. Moving into the sheath, the EVDF becomes stretched out, with electrode-directed electrons becoming more and more depleted due to the increasing strength of the electric field. A similar deformation of the EVDF has been seen in Vlasov simulations of an electron sheath around a cylindrical probe[@2013PhPl...20a3504S].
The electron presheath behavior in simulations (D) and (E) were slightly different than that previously described in the 1D model of ref. \[7\], however, it also shares some of the same qualities. Previous analysis was based on the 1D projections of the EVDF which look largely Maxwellian in the direction perpendicular to the electrode surface. Based on the observation of 1D EVDFs the electron presheath was modeled describing the electrons with a flow-shifted Maxwellian and ions with a Boltzmann density profile. Using these assumptions the dominant term in the electron momentum equation, Eq. (1), is the pressure gradient, which was exactly $T_e/T_i$ larger than contribution of the presheath electric field. This model proposed a significant deviation from the conventional assumption of a half-Maxwellian at the sheath edge[@1961JAP....32.2512M; @1962JAP....33.3094M; @2005PhPl...12e5502H]. It also showed that the electron sheath has a presheath that causes a significant perturbation far into the plasma. The results in this paper present a refined picture compared to the previous 1D model. Like the 1D model, a long presheath was visible in the electron flow moments shown for simulations (D) and (E) in Fig. 2. The EVDFs discussed above demonstrate that, for the electron sheath cases, the flow velocity is due to both a loss-cone like truncation and a flow shift. The truncation due to the loss cone leads to a situation where the electron flow velocity is heavily influenced by gradients in the pressure tensor. These are treated using a scalar pressure gradient in the 1D model. When the electrode is biased at or slightly below the plasma potential the EVDFs have a loss-cone like truncation, but with no flow shift, demonstrating that the flow shift is present only when the electrode is biased sufficiently positive that an electron sheath forms.
CONCLUSION
==========
In this paper, the physics of the sheath and presheath was explored using electron velocity data from 2D PIC simulations. When the electrode was biased less than $T_e/2e$ below the plasma potential, as in case (B), the plasma remained quasineutral up to the electrode surface, but still had presheath-like density and flow velocity gradients for both electrons and ions. Using EVDFs obtained from individual electron velocity data, the flow and density gradients were shown to originate from a loss-cone-like truncation of the EVDF due to fast electrons overcoming the $T_e/2e$ potential barrier. This was described as an effect of the breakdown of the assumptions behind the Boltzmann density relation, that arises due to a gradient in the stress tensor.
In case (D), when the electrode was biased less than $T_e/2e$ and greater than $T_i/2e$ above the plasma potential, a well defined electron sheath was present. In this case, since $T_e\gg T_i$, ions were observed to be completely repelled from the electrode. When an electron sheath was present, the EVDFs had flow velocity moments with contributions from both a flow shift, and a loss-cone-like truncation. As the electrode was approached from the bulk plasma, the loss cone angle was observed to widen, increasing the flow moment and leading to presheath behavior driven by a pressure tensor gradient. The onset of this behavior occurs in the ion sheath case when the electrode bias is sufficiently close to the plasma potential that electrons are not completely repelled, leading to an addition to the pressure tensor gradient term in Eq. (1). This contribution is due to the increasing value of the stress tensor, in Eq. (4), which grows in magnitude as the electrode is approached and the EVDF truncation becomes more pronounced.
Finally, the electron presheath behavior was compared with the 1D model described previously. In both the present simulations, and the 1D model, a long electron presheath is present in which electrons are accelerated close to their thermal speed by forces associated with a gradient in the pressure. The 2D simulations reveal a loss-cone nature, in addition to the flow shift, that is absent in the 1D model. The 2D simulations observe contributions to the force driving flow from both the scalar pressure gradient and the stress tensor gradient, whereas a 1D model accounts only for the scalar pressure contribution.
Acknowledgments {#acknowledgments .unnumbered}
===============
The first author would like to thank James Franek and Andrew Fierro for reading the manuscript. This research was supported by the Office of Fusion Energy Science at the U.S. Department of Energy under contract DE-AC04-94SL85000. The first author was also supported by the U.S. Department of Energy, Office of Science, Office of Workforce Development for Teachers and Scientists, Office of Science Graduate Student Research (SCGSR) program. The SCGSR program is administered by the Oak Ridge Institute for Science and Education for the DOE under contract number DE-AC05-06OR23100.
|
---
abstract: |
The efficient numerical solution of many kinetic models in plasma physics is impeded by the stiffness of these systems. Exponential integrators are attractive in this context as they remove the CFL condition induced by the linear part of the system, which in practice is often the most stringent stability constraint. In the literature, these schemes have been found to perform well, [*e.g.*]{}, for drift-kinetic problems. Despite their overall efficiency and their many favorable properties, most of the commonly used exponential integrators behave rather erratically in terms of the allowed time step size in some situations. This severely limits their utility and robustness.
Our goal in this paper is to explain the observed behavior and suggest exponential methods that do not suffer from the stated deficiencies. To accomplish this we study the stability of exponential integrators for a linearized problem. This analysis shows that classic exponential integrators exhibit severe deficiencies in that regard. Based on the analysis conducted we propose to use Lawson methods, which can be shown not to suffer from the same stability issues. We confirm these results and demonstrate the efficiency of Lawson methods by performing numerical simulations for both the Vlasov–Poisson system and a drift-kinetic model of a ion temperature gradient instability.
author:
- Nicolas Crouseilles
- Lukas Einkemmer
- Josselin Massot
title: Exponential methods for solving hyperbolic problems with application to kinetic equations
---
[***Keywords—*** exponential integrators, Lawson schemes, kinetic equations, hyperbolic PDEs, numerical stability, drift-kinetic equations]{}
Introduction {#intro}
============
The goal of this work is to develop high order and efficient numerical methods for nonlinear kinetic models, such as the Vlasov-Poisson equations or drift-kinetic models. In most situations, the nonlinearity in the transport term originates from the coupling with a Poisson type problem that is used to compute the electric field.
Historically, particle in cell methods have been extensively used to treat kinetic problems. In this approach, the unknown is sampled by discrete particles which are advanced in time using an ODE solver, whereas the electric field is computed on a spatial grid. For some problems, these methods can tackle high dimensional kinetic problems with a relatively low computational cost. However, they also suffer from numerical noise which pollutes the accuracy in low density regions of phase space. Moreover, as the number of particles is increased the error only decreases as the inverse of the square of the number of particles. Thus, convergence is slow. For a review of particle methods we refer the reader to [@verboncoeur2005].
On the other hand, Eulerian methods ([*e.g.*]{} finite volumes or finite differences), which directly discretize the phase space, are able to reach high order accuracy in time, space, and velocity. However, in addition to the fact that they are costly, these methods usually suffer from stability constraints that force a relation between the time step and the phase space grid sizes, the so-called Courant–Friedrichs–Lewy (CFL) condition. Hence, a large number of time steps is required to reach the long times that are often required in plasma physics applications.
To overcome this CFL condition, semi-Lagrangian methods have been developed during the last decades. These methods realize a compromise between the Lagrangian ([*i.e.*]{} particle in cell) and Eulerian approaches by exploiting the characteristics equations to overcome the CFL condition, while still performing computations on a grid in both space and velocity [@cheng1976; @sonnendrucker1999; @filbet2003]. This approach is usually combined with splitting methods to avoid a costly multidimensional interpolation step. This allows for a separate treatment of the terms in the equation and the corresponding characteristic curves can then, at least in some situations, be computed analytically. For purely hyperbolic problems, the setting we consider here, it is also possible to construct high order splitting schemes [@casas2017].
Splitting results in a very accurate and efficient scheme for the Vlasov-Poisson equation. This is the case because the problem is only split into two parts, the characteristics of which can then be solved exactly in time. However, this is not necessarily true for more complicated equations such as gyrokinetic or drift-kinetic models. Indeed, for the drift-kinetic model, a three terms splitting has to be performed so that a relatively large number of stages are required to reach high order accuracy in time. In addition, some stages can not be solved exactly in time and thus require additional numerical work to approximate them.
In [@cep] an alternative approach based on exponential integrators was proposed. These schemes exploit the fact that in many applications where (gyro)kinetic models are used, the most stringent CFL condition is associated with the linear part of the model. This observation serves as the basis for the numerical methods we will consider in this work. Starting from the variation of constant formula, the linear part of the model will be solved exactly as part of an exponential integrator, whereas the nonlinear part, which is very often orders of magnitudes less stiff than the linear part, will be treated explicitly in time. In practice, the linear part can then be solved in phase space by using Fourier techniques or semi-Lagrangian schemes and the nonlinear part is approximated by standard finite difference/finite volume/discontinuous Galerkin techniques.
The numerical results presented in [@cep] were generally very favorable. The authors were able to take larger time steps compared to what has been reported for splitting methods in the literature and the computational cost was significantly reduced. In addition, since exponential integrators treat the nonlinear part explicitly, they can be adapted much more easily to different models. Despite these many favorable properties, the largest stable time step size was difficult to predict and varied significantly from method to method. Moreover, as we will see, many exponential integrators can behave rather erratically depending on the specific configuration of the simulation.
Thus, the main goal in this paper is to understand the stability of exponential integrators when applied to purely hyperbolic problems. While there is a large literature and well established theory for exponential integrators applied to parabolic problems (see [@ei] and references therein), we will see that for purely hyperbolic problems many surprises are encountered. Based on this analysis we will then propose to use a class of exponential methods, Lawson methods, that do not suffer from the described deficiency. Our analysis explains the efficient and robust behavior of Lawson integrators for this kind of problems. We will also present numerical results for both the Vlasov–Poisson equations and a drift-kinetic model that confirm the expected behavior and shows that using this approach significant performance improvements compared to the exponential integrators used in [@cep], and by extension compared to splitting methods, can be attained.
The paper is organized as follows. First, we offer a brief introduction to exponential methods (section \[sec:expint\]). Then, in section \[ode\] a linear stability analysis is performed for both the time and phase space discretization. For the explicit part, we consider both centered differences (such as Arakawa’s method) and weighted essentially non-oscillatory schemes (WENO) schemes. In sections \[sec:vp\] and \[sec:dk\] we investigate the performance of these methods for the Vlasov–Poisson equations and a four-dimensional drift-kinetic model, respectively.
Exponential integrators and Lawson methods \[sec:expint\]
=========================================================
Exponential methods are a class of time integration schemes that are applied to differential equations of the form $$\label{eq:expint-eq} \dot{u} = Au + F(u),$$ where $A$ is a matrix and $F$ is a, in general nonlinear, function of $u$. Usually, both $A$ and $F$ are the result of a spatial semi-discretization of a partial differential equation. Exponential methods are applied to problems where $A$ is stiff or otherwise poses numerical challenges, while $F$ can be treated explicitly. For the hyperbolic case, a prototypical example is the Vlasov equation . Exponential methods are advantageous if the largest velocity is large compared to the electric field. Then the linear part has a much more stringent CFL condition than the nonlinear part of the equation. We will consider this example in some detail later in the paper.
In this paper, we will consider two types of exponential methods. The idea of *exponential integrators* is to use the variation of constants formula to rewrite equation in the following form $$u(t_n+\Delta t) = \exp(\Delta t A) u(t_n) + \int_0^{\Delta t} \exp((\Delta t -s)A) F(u(t_n+s)) \,\mathrm{d}s,$$ where we denote the time step size by $\Delta t >0$ and $t_n = n\Delta t$ with $n\in\mathbb{N}$. This expression is still exact; [*i.e.*]{}no approximation has been made. Note, however, that this can not be used as a numerical method as evaluating the integral would require the knowledge of $u(t_n+s)$, which is not available. Approximating the integral in terms of available data results in an exponential integrator. In the simplest case we use the rectangular rule at the left endpoint to obtain $$u(t_n+\Delta t) \approx u^{n+1} = \exp(\Delta t A) u^n + \Delta t \varphi_1(\Delta t A) F(u^n),$$ where $\varphi_1(z)=(\mathrm{e}^z-1)/z$ is an entire function. This is the first order exponential Euler method. In a similar way exponential Runge–Kutta methods can be constructed. We refer to the literature, in particular the review article [@ei], for more details.
Another ansatz to remove the stiff linear term from equation is to introduce the change of variable $$v(t) := \exp(-t A) u(t).$$ Plugging this into equation yields $$\label{eq:lawson-eq} \dot{v}(t) = \exp(-t A) F(\exp(t A) v(t)).$$ Now we apply an explicit Runge–Kutta method to the transformed equation. In the simplest case, applying the explicit Euler scheme yields $$v(t_n+\Delta t) \approx v^{n+1} = v^{n} + \Delta t \exp(-t_n A) F(\exp(t_n A)v^{n}).$$ Reversing the change of variables yields $$u^{n+1} = \exp(\Delta t A) u^{n} + \Delta t \exp(\Delta t A) F(u^{n}).$$ This is the Lawson–Euler method, also a method of order one. Lawson methods are also commonly referred to as integrating factor methods. We immediately see that any explicit Runge–Kutta method applied to equation uniquely determines a Lawson scheme. We call the chosen Runge–Kutta method the *underlying Runge–Kutta method*. For more details we refer the reader to [@lawson1967; @canuto1988; @trefethen2000; @minchev2005].
The example of the Lawson–Euler method already shows the similarity between Lawson schemes and exponential integrators. In fact, Lawson methods can be considered a subclass of exponential integrators. That is, they are a type of exponential integrators that only involve the exponential, but no other matrix functions. For the purpose of this paper we keep the nomenclature distinct. A Lawson scheme is a numerical method obtained as described above, while an exponential integrator is a numerical scheme that, in addition to the matrix exponential, uses other matrix functions.
The efficiency of exponential methods crucially depends on a good method to evaluate the application of the required matrix functions to a vector. A range of methods has been developed to accomplish this. For example, Krylov methods or interpolation at Leja points can be used for a wide range of problems; see, for example, [@higham2008; @hochbruck1997; @al2011; @caliari2014; @ckor15]. However, often the most efficient approach is to exploit particular knowledge about the differential equation under consideration. For example, in the hyperbolic case $A$ might be a linear advection operator. In this case the application of $\exp(\Delta t A)$ can be computed by using Fourier techniques or semi-Lagrangian schemes. Much research effort has been dedicated towards improving spectral and semi-Lagrangian schemes for kinetic problems [@crouseilles2011; @einkemmer2014; @einkemmer2016; @filbet2003; @grandgirard; @klimas1994; @Morrison2017; @rossmanith2011; @sonnendrucker1999; @cheng1976; @sircombe2009valis; @crouseilles2015hamiltonian; @crouseilles2016asymptotic; @einkemmer2019comparison; @einkemmer2014convergence; @einkemmer2014dG] and obtaining good performance on state of the art HPC systems [@rozar2013; @einkemmer2015; @bigot2013; @latu2007gyrokinetic; @mehrenberger2013vlasov; @einkemmer2016mixed; @crouseilles2009parallel; @einkemmer2019gpu].
Before proceeding, let us note that for parabolic problems, [*i.e.*]{} where $A$ is an elliptic operator, a mature theory for exponential integrators is available. We again refer the reader to the review article [@ei]. In this setting there are relatively few surprises with respect to stability and even rigorous convergence results are available. In addition, exponential integrators have been considered for problems that include both hyperbolic and parabolic terms (see, for example [@martinez2009; @tambue2010; @einkemmer2016expintmhd; @einkemmer2013expintgpu]). An interesting point to make is that in this community Lawson methods have all but lost their appeal. In fact, there are many reasons why exponential integrators are to be preferred. For example, if a Krylov method is used to compute the matrix functions, the $\varphi_1$ function usually converges faster than the exponential. In addition, exponential integrators that retain their full order for non-homogeneous boundary conditions have been constructed [@hochbruck2005]. It has been shown that this property can not be achieved for Lawson methods [@hochbruck2017lawson]. However, the situation for purely hyperbolic problems is markedly different. Most of the theoretical results that have been obtained in an abstract framework do not apply and there is relatively little literature available. We will see that the stability for exponential integrators in the fully hyperbolic setting is full of surprises. Moreover, since for kinetic problems we usually have efficient methods to compute the matrix exponential and complicated boundary conditions are rather rare, Lawson methods are an attractive choice due to their improved stability, as we will see.
Linear analysis \[ode\]
=======================
Determining the stability of a numerical scheme by conducting an analysis of a linear and scalar test equation is very well established in the literature. Usually, the Dahlquist test equation $\dot{u} = \lambda u$ is considered. The justification for this is that a linear ODE can be written as $\dot{u} = A u$. Once the matrix $A$ is diagonalized we essentially obtain the test equation. For linear PDEs we first perform a space discretization. Then the same argument can be applied to the resulting differential equation and stability constraints, such as the famous CFL condition, can be deduced. In the nonlinear case this, of course, only gives an indication for stability. Nevertheless, in many practical problems the theory derived in this fashion agrees very well with what is observed in numerical experiments.
The situation for exponential integrators and Lawson methods is more complicated as we separate two parts of the differential equation. Thus, we will consider the following test equation $$\label{ode_linear}
\dot{u} = ia u + \lambda u, \qquad a\in\mathbb{R}, \lambda\in\mathbb{C}, \qquad u(0) = u_0\in\mathbb{C}.$$ We note that we are here exclusively interested in equations with two hyperbolic parts. The reason why we allow $\lambda$ to lie in the complex plane is that some space discretization schemes introduce numerical diffusion. Thus, the discretization moves the eigenvalues from the imaginary axis to the left half complex plane.
Although this test equation is used frequently in the literature, its use is also frequently criticized. The reason for this criticism is that in the linear case the equation $\dot{u} = Au + Bu$ can only be transformed to the form given in equation if $A$ and $B$ are simultaneously diagonalizable. This is a severe restriction which is usually not true in practice. Thus, the test equation, even in the linear case, gives only a necessary condition for stability. While this argument is certainly correct, we emphasize that if a numerical integrator does not work for the test equation there is not much hope that it would work for more complicated problems. Therefore, the test equation is still useful and in fact we will see that many of the deficiencies of exponential integrators observed in practice can be illustrated well using the test equation .
Lawson methods
--------------
Applying a Lawson method to the test equation proceeds as follows. First, we introduce the change of variable $$v(t)=e^{-iat} u(t).$$ which yields the equation $$\dot{v} = e^{-iat} \lambda (e^{iat }v) = \lambda v.$$ Thus, we precisely obtain the Dahlquist test equation. We now apply an explicit Runge–Kutta method to that equation. It is well known that this results in $$v^{n+1} = \phi(z) v^n, \qquad z=\lambda \Delta t,$$ where $\phi$ is the so-called stability function. Reversing the change of variable we obtain $$u^{n+1} = e^{ia\Delta t} \phi(z) u^{n}, \qquad z=\lambda \Delta t.$$ The condition for stability is $| e^{ia\Delta t} \phi(z) | = | \phi(z) | \leq 1$. Thus, the linear stability characteristics of a Lawson scheme is identical to that of its underlying Runge–Kutta method.
This makes the problem rather easy as the stability constraint for explicit Runge–Kutta methods has been extensively studied in the literature. In our present application we are primarily interested in obtaining numerical methods that maximize the part of the imaginary axis that is included in the domain of stability. It is well known that an $s$ stage method can include at most $i [-(s-1),s-1]$. This is, for example, stated as an exercise in [@sode Chapter. IV.2, exercise 3]. Thus, unfortunately, there is no analog to Runge–Kutta–Chebyshev methods for hyperbolic problems.
For the sake of completeness we plot in Figure \[fig:RK\_sd2\] the curve given by $|\phi(z)| = 1$ for different Runge-Kutta methods. The only non-standard method here is *RK(3,2) best* which is a three stage second order method that has been purposefully constructed to enhance stability on the imaginary axis (see Appendix \[butcher\] for its Butcher tableau). We also emphasize that the stability domain of the classic four stage fourth order Runge–Kutta method is quite close to the theoretical bound $i [-(s-1),s-1], s=4$.
![The domain of stability for some classic explicit Runge–Kutta methods is shown. The nomenclature *RK(s,p)* denotes a method with $s$ stages that is of order $p$. The Butcher tableaus of these methods are given in Appendix \[butcher\]. []{data-label="fig:RK_sd2"}](img/rk_sd.png){width="30.00000%"}
Exponential integrators
-----------------------
We now apply commonly used exponential integrators to the test equation . In this work we will consider the following methods: ExpRK22 (a classic two stage second order method), the method of Cox–Matthews [@cox], the method of Hochbruck–Ostermann [@hochbruck2005], and the method of Krogstad [@krogstad2005]. We refer to [@ei] for more details and to Appendix \[butcher\] for the Butcher tableaus of these methods.
For the sake of brevity we will only detail the calculation for the ExpRK22 scheme. Applying this method to the test equation we obtain $$\begin{aligned}
k_1&=&e^{ia\Delta t}u^n + \Delta t\varphi_1(ia\Delta t)\lambda u^n\nonumber\\
u^{n+1}&=& e^{ia\Delta t}u^n + \Delta t \Big[ (\varphi_1(ia\Delta t)-\varphi_2(ia\Delta t))\lambda u^n + \varphi_2(ia\Delta t)\lambda k_1\Big], \end{aligned}$$ where $\varphi_1(z)=(e^{z}-1)/z$ and $\varphi_2(z)=(e^{z}-1-z)/z^2$ are entire functions. This yields the stability function $$\phi(z) = e^{ia\Delta t} + \Big(\varphi_1(ia\Delta t)-\varphi_2(ia\Delta t)+e^{ia\Delta t}\varphi_2(ia\Delta t)\Big)z + \varphi_1(ia\Delta t)\varphi_2(ia\Delta t)z^2,$$ where, as before, we use $z=\lambda \Delta t$.
Our first observation is that, in contrast to Lawson methods, the behavior of this stability function can not be understood by the domain of stability of the underlying $RK(2, 2)$ method, [*i.e.*]{} the explicit method we obtain if we take $a \to 0$. In fact, as we vary $a$ the domain of stability changes drastically. The domain of stability for the four exponential integrators (ExpRK22, Cox–Matthews, Hochbruck–Ostermann, and Krogstad) is plotted in Figure \[fig:expRK\_sd\] for $a\Delta t=1.1$ and $a\Delta t=3.4$. It is most striking that for large $\vert a\Delta t \vert$ the domain of stability does not contain a symmetric interval of the imaginary axis. It should be evident that this has the potential to causes severe stability issues.
[0.3]{} ![Stability domain of exponential integrators for two different values of $a\Delta t\in\{1.1, 3.4\}$. From top left to bottom right: ExpRK22, Krogstad, Cox–Matthews and Hochbruck–Ostermann.[]{data-label="fig:expRK_sd"}](img/expRK22_sd.png "fig:"){width="\textwidth"}
[0.3]{} ![Stability domain of exponential integrators for two different values of $a\Delta t\in\{1.1, 3.4\}$. From top left to bottom right: ExpRK22, Krogstad, Cox–Matthews and Hochbruck–Ostermann.[]{data-label="fig:expRK_sd"}](img/K_sd.png "fig:"){width="\textwidth"}
[0.3]{} ![Stability domain of exponential integrators for two different values of $a\Delta t\in\{1.1, 3.4\}$. From top left to bottom right: ExpRK22, Krogstad, Cox–Matthews and Hochbruck–Ostermann.[]{data-label="fig:expRK_sd"}](img/CM_sd.png "fig:"){width="\textwidth"}
[0.3]{} ![Stability domain of exponential integrators for two different values of $a\Delta t\in\{1.1, 3.4\}$. From top left to bottom right: ExpRK22, Krogstad, Cox–Matthews and Hochbruck–Ostermann.[]{data-label="fig:expRK_sd"}](img/HO_sd.png "fig:"){width="\textwidth"}
Phase space discretization
--------------------------
We start from the two-dimensional linear transport equation $$\label{vp_linear}
\partial_t f + d\partial_x f + b\partial_v f = 0, \;\; d, b\in\mathbb{R}, \;\; x\in [0, 2 \pi], \;\; v\in [-v_{\max},v_{\max}],$$ where $v_{\max}>0$ refers to the truncated velocity domain. The sought-after distribution function is $f(t,x,v)$ and we impose periodic boundary conditions in the $x$-direction. We assume that $d$ and $b$ are constants and thus the corresponding operators commute. This is an idealization of the Vlasov equation we will consider in the next section. In preparation for that example it is most useful to think that $d$ is large and thus would induce a stringent CFL condition if discretized by an explicit scheme.
We now have to discretize this equation both in the $x$ and the $v$ direction. In the spatial direction $x$, we will consider a spectral approximation. Performing a Fourier transformation of equation yields $$\label{fourier_x_vlasov}
\partial_t \hat{f}_{k} + i d k\hat{f}_{k} +b \partial_v \hat{f}_{k} = 0,$$ where $\hat{f}_k(t,v)$ denotes the Fourier transform of $f(t,x,v)$ with respect to $x$. The corresponding frequency is denoted by $k$.
We now perform the discretization in the $v$ direction. The grid points are denoted by $v_j=-v_{\max}+j \Delta v$, with $\Delta v=2v_{\max}/N_v$, where $N_v$ is the number of points. We will consider two options here. Namely, either using a centered difference scheme or an upwind scheme.
#### Centered scheme in $v$.\
The classic centered scheme is obtained by approximating the velocity derivative in equation by $$(\partial_v \hat{f}_{k})(v_j) \approx \frac{\hat{f}_{k, j+1}-\hat{f}_{k, j-1}}{2\Delta v},$$ where $\hat{f}_{k,j}$ is an approximation of $\hat{f}_{k}(v_j)$. Inserting this centered approximation in yields $$\label{eq:half-fourier}
\partial_t \hat{f}_{k,j} + i d k\hat{f}_{k,j} +b \frac{\hat{f}_{k,j+1} -\hat{f}_{k,j-1} }{2\Delta v} = 0.$$ The system is already diagonal with respect to the index $k$. We now also diagonalize it with respect to the index $j$. To do that we express the function in terms of its Fourier modes with respect to $v$. That is, $$\hat{f}_{k,j} = \sum_m \bar{f}_{k, m}\exp\left(i \frac{2\pi m}{2v_{\max}} v_j\right),$$ where $\bar{f}_{k,m}$ denotes the (double) Fourier transform of $f$ with frequency in space $k$ and frequency in velocity $m$. Inserting this into equation yields $$\label{discrete_linear_transport}
\partial_t \bar{f}_{k,m} + i dk\bar{f}_{k,m} +b \frac{i\sin(2\pi m \Delta v/(2v_{\max})) }{\Delta v} \bar{f}_{k,m}= 0.$$ We immediately see that this equation is precisely in the form of equation as studied in the previous section. We also observe that $\lambda \in i\mathbb{R}$. That is, the eigenvalues for the centered difference approximation lie exclusively on the imaginary axis. One immediate consequence is that for Lawson methods the CFL condition is given by $b \Delta t < C \Delta v$, where $C$ is chosen such that $i [-C,C]$ lies in the domain of stability of the underlying Runge–Kutta method.
#### Linearized WENO approximation in $v$.\
A common technique to discretize hyperbolic partial differential equations is to use the so-called weighted essentially non-oscillatory schemes (WENO) schemes. These are nonlinear schemes that limit oscillations in regions where sharp gradients occur, but still yield high order accuracy in smooth regions of the phase space. In the linear case WENO schemes reduce to upwind discretizations. Here, we will consider the LW5 scheme (the linearized version of the WENO5 scheme as considered in [@baldauf; @lunet; @motamed; @wang]) that is given by (from now on we assume w.l.o.g. that $b>0$) $$\begin{aligned}
(\partial_v \hat{f}_k)(v_j) &\approx \frac{1}{\Delta v}\Big(-\frac{1}{30} \hat{f}_{k,j-3} +\frac{1}{4} \hat{f}_{k,j-2} -\hat{f}_{k,j-1} + \frac{1}{3} \hat{f}_{k,j} +\frac{1}{2} \hat{f}_{k,j+1} - \frac{1}{20} \hat{f}_{k,j+2}\Big). \end{aligned}$$ We now perform the same analysis as for the centered scheme (see [@baldauf; @fov]). This yields $$\begin{aligned}
(i m \bar{f}_{k, m} \approx ) \;\;\mu_m \bar{f}_{k, m} &:= \frac{\bar{f}_{k,m}}{\Delta v}\Big(-\frac{1}{30} e^{-\frac{3i m\pi \Delta v}{v_{\max}}} +\frac{1}{4} e^{-\frac{2i m \pi \Delta v}{v_{\max}}} -e^{-\frac{i m \pi \Delta v}{v_{\max}}} \nonumber\\
&\hspace{1cm}+ \frac{1}{3} +\frac{1}{2} e^{\frac{i m \pi \Delta v}{v_{\max}}} - \frac{1}{20} e^{\frac{2i m \pi \Delta v}{v_{\max}}}\Big).
\label{lw5symbol}\end{aligned}$$ We then obtain the equation $$\label{discrete_linear_transport_weno}
\partial_t \bar{f}_{k,m} + i dk\bar{f}_{k,m} +b \mu_m \bar{f}_{k,m}= 0.$$ Once again this is precisely the form of equation , where $a=dk\in\mathbb{R}$ and $\lambda=b\mu_m \in\mathbb{C}$. The main difference to the centered difference scheme is that $\lambda$ is not necessarily on the imaginary axis. In fact, the eigenvalues aquire a negative real part which stabilizes the scheme and avoids spurious oscillations, but also adds unphysical dissipation to the numerical method.
Computing the CFL condition
---------------------------
Equipped with the knowledge of the domain of stability for the time discretization and the eigenvalues of the space discretization, we are now in a position to determine the CFL condition for the linear transport equation . This task will be rather easy to accomplish for the Lawson schemes, where the stability does not depend on the advection speed for the transport in the $x$ direction. However, for exponential integrators even this linear analysis is rather complicated, as we will see.
### Centered scheme in $v$.
In the case of centered approximation of the velocity derivative, the Fourier multiplier is a pure imaginary complex number (see equation ). We thus look for $y_{\max}\in \mathbb{R}_+$ such that the interval $i(-y_{\max},y_{\max}) \subset {\cal D}$, where ${\cal D}$ is the domain of stability for the chosen time integrator.
#### Lawson integrators.\
We simply look for the largest value $y_{\max}$ such that $i(-y_{\max}, y_{\max}) \subset \mathcal{D}$. The corresponding values for a number of schemes are listed in Table \[tab:ymax\_Lawson\]. These values have to be understood in the following way: they induce the CFL condition $b \Delta t\leq y_{\max}\Delta v$ for the discretized equation , where $\Delta t$ denotes the time step size and $\Delta v$ is the velocity mesh size.
Methods Lawson($RK(3,2) \; best$) Lawson($RK(3,3)$) Lawson($RK(4,4)$)
------------ --------------------------- ------------------- -------------------
$y_{\max}$ $2$ $\sqrt{3}$ $2\sqrt{2}$
: CFL number for some Lawson schemes applied to . []{data-label="tab:ymax_Lawson"}
#### Exponential integrators.\
For the exponential integrators the domain of stability is very sensitive to the value of $(a\Delta t)$. To get an idea of what we can expect, we consider the quantity $y_{\max} = \min_{(a\Delta t)\in\mathbb{R}} \; y^{exp}_{\max}(a \Delta t)$. As before, $y^{exp}_{\max}(a\Delta t)$ is the largest value such that $i(-y^{exp}_{\max}(a\Delta t), y^{exp}_{\max}(a\Delta t))\subset {\cal D}$, where ${\cal D}$ is the domain of stability for the chosen exponential time integrator for a given $(a\Delta t)$. Even for relatively simple numerical methods it is not possible to compute this quantity analytically. Thus, we resort to numerical approximations. Unfortunately, it turns out that for most exponential integrators this value is zero. This can be appreciated by considering Figure \[fig:expRK\_sd\] once more. Clearly, there are values of $(a \Delta t)$ such that no relevant part of the imaginary axis (or only half the imaginary axis) is part of the domain of stability. Thus, most exponential integrators are unstable in the von Neumann sense. However, this is not what we observe in practice. In fact, already the results presented in [@cep] indicate that we can successfully run numerical simulations using, for example, the Cox–Matthews scheme. There are two major points to consider here
- The $y_{\max}$ obtained is a worst case estimate. In fact, we know that for $\Delta t \to 0$ we regain the stability of the underlying Runge–Kutta method. Thus, for small $(a \Delta t)$ the methods is expected to work well.
- As is usually done we have mandated that $\vert \phi(z) \vert \leq 1$. However, strictly speaking this is not necessary for practical simulation. If we assume that $\vert \phi(z) \vert \leq 1+\varepsilon$ and we take $n$ steps the amplification of the error is given by $(1+\varepsilon)^n$. In the limit $n \to +\infty$ this quantity diverges. However, since we usually do not take infinitely small time steps we still can hope to obtain a relatively accurate approximation, especially if $\varepsilon$ is small.
To investigate this further, we propose to relax the stability condition by introducing a threshold $\varepsilon>0$ in the definition of the stability domain $$\label{d_eps}
\mathcal{D}_\varepsilon = \{ z\in\mathbb{C} : |\phi(z)| \leq 1+\varepsilon \}.$$
In Figure \[ymax\_example\], we plot the domain of stability for the Cox–Matthews method and $a\Delta t = 3.4$ for $\varepsilon=0$ and $\varepsilon=10^{-2}$. One can observe that in the latter case a non-zero $y_{max}^{exp}(3.4)$ is obtained. We also call attention to the fact that the part of the imaginary axis included in this relaxed stability domain is not symmetric.
[0.33]{} ![Example of variation of $y_+$ and $y_-$ when we relax the stability condition for the Cox–Matthews scheme. We represent $\mathcal{D}_{0}$ on the left, and $\mathcal{D}_{10^{-2}}$ on the right with the values $y_+$ and $y_-$ such that $i(y_-, y_+)\subset \mathcal{D}_{\varepsilon}$.[]{data-label="ymax_example"}](img/CM_sd_ymax_e0p00.png "fig:"){width="\textwidth"}
[0.33]{} ![Example of variation of $y_+$ and $y_-$ when we relax the stability condition for the Cox–Matthews scheme. We represent $\mathcal{D}_{0}$ on the left, and $\mathcal{D}_{10^{-2}}$ on the right with the values $y_+$ and $y_-$ such that $i(y_-, y_+)\subset \mathcal{D}_{\varepsilon}$.[]{data-label="ymax_example"}](img/CM_sd_ymax_e0p01.png "fig:"){width="\textwidth"}
In Figure \[ymax\_expRK22\], we plot the dependence of $y^{exp}_{\max}$ as a function of $(a\Delta t)$ for $\varepsilon=10^{-2}$ and the ExpRK22 method. Let us recall that for $\varepsilon=0$, the method gives $y_{\max}=0$. One can observe that the domain of stability ${\cal D}_\varepsilon$ of this method is still symmetric with respect to the real axis. In addition, the method becomes more stable as $|a\Delta t|$ increases. Thus, the behavior of the method is completely different from the configuration with $\varepsilon=0$.
![$y^{exp}_{\max}$, $|y_+|$ and $|y_-|$ as a function of $a \Delta t$ for the ExpRK22 method with $\varepsilon=10^{-2}$. []{data-label="ymax_expRK22"}](img/ymax_expRK22_0p01)
In Figure \[ymax\_HO\], we plot $y^{exp}_{\max}$ as a function of $(a\Delta t)$ for the Hochbruck–Ostermann method (once again for $\varepsilon=10^{-2}$). This schemes also gives $y_{\max}=0$ for $\varepsilon=0$. In this case the domain of stability is not symmetric and the stability depends quite erratically on the value of $(a \Delta t)$.
![$y^{exp}_{\max}$, $|y_+|$ and $|y_-|$ as a function of $a\Delta t$ for the Hochbruck–Ostermann method with $\varepsilon=10^{-2}$.[]{data-label="ymax_HO"}](img/ymax_HO_0p01)
In Table \[tab:ymax\_expo\] we have summarized the values of $y_{\max}$ for the four exponential integrators considered in this paper.
Methods ExpRK22 Krogstad Cox–Matthews Hochbruck–Ostermann
---------------------------------------- --------- ---------- -------------- ---------------------
$y^{exp}_{\max} (\varepsilon=10^{-3})$ $0.300$ $0.100$ $0.150$ $0.250$
$y^{exp}_{\max} (\varepsilon=10^{-2})$ $0.551$ $0.200$ $0.450$ $0.501$
$y^{exp}_{\max} (\varepsilon=10^{-1})$ $1.001$ $0.601$ $1.351$ $1.702$
: CFL number, assuming the relaxed stability constraint, for some exponential integrators applied to .[]{data-label="tab:ymax_expo"}
### Linearized WENO5 (LW5) scheme.
In the case of a WENO5 approximation of the velocity derivative, we can not easily find a Fourier multiplier because of its nonlinearity. Recent studies about stability of WENO5 [@wang; @motamed; @lunet] considered the linearized version of WENO schemes by freezing the nonlinear weights, so that WENO5 reduces to a high order (linear) upwind scheme called LW5. For this LW5 scheme we can compute the eigenvalues, see equation , and different time stepping schemes can be studied to determine the stability limit. We consider only Lawson methods here since we found that the exponential schemes we considered ([*i.e.*]{} ExpRK22, Krogstad, Cox–Matthews, Hochbruck–Ostermann) lead to unstable results when they are combined with LW5 (even in the weak sense considered in the previous section).
The goal is then to determine the largest non-negative real number $\sigma>0$ such that the eigenvalues of the upwind scheme LW5 scaled by $\sigma$ are contained in the domain of stability for the time integrator. Since the eigenvalues of LW5 are not as simple as in the case of the centered scheme, we determine $\sigma$ numerically. The main idea of the algorithm to obtain an estimation of $\sigma$ is:
1. The argument $\varphi$ of the eigenvalues $\mu_m$ (normalized by $\Delta v$) given by are discretized using a fine angular grid $\{ \varphi_k\} \subset [-\pi/2, \pi/2]$, since the real part of $\mu_{m}$ is negative due to its diffusive character.
2. A discretized version of the boundary of the stability domain of the underlying Runge–Kutta method is computed.
3. For each discretized eigenvalue, we look for the closest boundary point of the Runge–Kutta stability domain. This enables us to compute the associated stretching factor $\sigma(\varphi_k)$.
4. Taking the minimum over all the discretized eigenvalues yields $\sigma:=\min_{k} \sigma(\varphi_k)$.
In Figure \[cfl\_rk44\_lw5\] (left), we plot the dependence of $\sigma$ with respect to the angle $\varphi\in [-\pi/2, \pi/2]$ for Lawson($RK(4, 4)$) coupled with LW5. We also plot (Figure \[cfl\_rk44\_lw5\] (right)) the stability domain of Lawson($RK(4, 4)$), the eigenvalues for LW5 (normalized by $\Delta v$) and the eigenvalues for LW5 scaled by $\sigma$. The CFL number for some Lawson schemes is shown in Table \[tab:ymax\_weno\_Lawson\]. It is interesting to note that the time step size is reduced compared to the centered scheme.
Methods Lawson($RK(3,2) \; best$) Lawson($RK(3,3)$) Lawson($RK(4,4)$)
----------- --------------------------- ------------------- -------------------
$\sigma $ $1.344$ $1.433$ $1.73$
: CFL number for some Lawson schemes applied to . []{data-label="tab:ymax_weno_Lawson"}
[0.4]{} ![Left: $\sigma$ as a function of the angle $\varphi$. Right: stability domain of Lawson($RK(4,4)$) (red), eigenvalues for LW5 normalized by $\Delta v$ (blue) and eigenvalues for LW5 normalized by $\Delta v$ stretched with factor $\sigma=1.73$ (dashed green). []{data-label="cfl_rk44_lw5"}](img/cfl_rk44_weno_phi.png "fig:"){width="\textwidth"}
[0.3]{} ![Left: $\sigma$ as a function of the angle $\varphi$. Right: stability domain of Lawson($RK(4,4)$) (red), eigenvalues for LW5 normalized by $\Delta v$ (blue) and eigenvalues for LW5 normalized by $\Delta v$ stretched with factor $\sigma=1.73$ (dashed green). []{data-label="cfl_rk44_lw5"}](img/cfl_rk44_weno.png "fig:"){width="\textwidth"}
Numerical simulation: Vlasov-Poisson equations \[sec:vp\]
=========================================================
In this section we apply Lawson methods and exponential integrators to the Vlasov-Poisson system. We will see that the linear theory developed in the last section gives a good indication of the stability even for this nonlinear problem. We consider a distribution function $f(t, x, v)$ depending on time $t\geq 0$, space $x$, with periodic boundary conditions, and velocity $v$, which satisfies the Vlasov equation $$\label{vlasov}
\partial_t f(t, x, v) + v\partial_x f(t, x, v) + E(f)(t, x)\partial_v f(t, x, v) = 0$$ coupled to a Poisson problem for the electric field $E(f)(t, x)$ $$\partial_x E(f)(t, x)= \int_{\mathbb{R}} f(t, x, v) \,dv -1.$$ We employ a Fourier approximation in space. In velocity we either use a centered discretization $$\partial_t \hat{f}_{k,j} + v_j i k \hat{f}_{k,j} + {\savestack{\tmpbox}{\stretchto{ \scaleto{ \scalerel*[\widthof{\ensuremath{\Big(E_{\cdot} \frac{f_{\cdot,j+1} -f_{\cdot,j-1}}{2\Delta v}\Big)}}]{\kern-.6pt\bigwedge\kern-.6pt} {\rule[-\textheight/2]{1ex}{1.2\textheight}} }{\textheight}}{0.6ex}}\stackon[1pt]{\Big(E_{\cdot} \frac{f_{\cdot,j+1} -f_{\cdot,j-1}}{2\Delta v}\Big)}{\tmpbox}}_k = 0$$ or the WENO5 discretization $$\displaystyle\partial_t \hat{f}_{k,j} + v_j ik \hat{f}_{k,j} + {\savestack{\tmpbox}{\stretchto{ \scaleto{ \scalerel*[\widthof{\ensuremath{\Big(E^+_{\cdot} \frac{f^+_{\cdot,j+1/2} - f^+_{\cdot,j-1/2}}{\Delta v} \Big)}}]{\kern-.6pt\bigwedge\kern-.6pt} {\rule[-\textheight/2]{1ex}{1.2\textheight}} }{\textheight}}{0.6ex}}\stackon[1pt]{\Big(E^+_{\cdot} \frac{f^+_{\cdot,j+1/2} - f^+_{\cdot,j-1/2}}{\Delta v} \Big)}{\tmpbox}}_k
+ {\savestack{\tmpbox}{\stretchto{ \scaleto{ \scalerel*[\widthof{\ensuremath{\Big(E^-_{\cdot} \frac{f^-_{\cdot,j+1/2} - f^-_{\cdot,j-1/2}}{\Delta v} \Big)}}]{\kern-.6pt\bigwedge\kern-.6pt} {\rule[-\textheight/2]{1ex}{1.2\textheight}} }{\textheight}}{0.6ex}}\stackon[1pt]{\Big(E^-_{\cdot} \frac{f^-_{\cdot,j+1/2} - f^-_{\cdot,j-1/2}}{\Delta v} \Big)}{\tmpbox}}_k = 0,
\label{vp_weno}$$ where $E^+=\max(E, 0)$, $E^-=\min(E, 0)$ and $f^\pm_{j+1/2}$ denote the numerical fluxes (see Appendix \[app\_weno\] for more details). Both of these phase space discretizations can be easily cast in the following form $$\partial_t \hat{f}_{k,j} = - v_j i k \hat{f}_{k,j} + F(f)_{k,j},$$ for an appropriately defined $F$. We can now apply an exponential integrator or a Lawson scheme. To illustrate this let us consider the exponential Euler method. This gives $$\hat{f}^{n+1}_{k,j} = \exp(-\Delta t v_j i k) \hat{f}^n_{k,j} + \Delta t \varphi_1(-\Delta t v_j i k) F(f^n)_{k,j}.$$ Since in Fourier space the exponential and $\varphi_1$ functions have only scalar arguments, their computation is easy and efficient ([*i.e.*]{} no matrix functions have to be computed). Due to the nonlinearity, it is favorable to compute $E \partial_v f$ in real space. This is done efficiently by using the fast Fourier transform. Generalizing this scheme to multiple dimensions in both space and velocity is straightforward.
To apply our theory from the linear analysis to the nonlinear Vlasov-Poisson case, we need a way to compute the CFL condition. Note that the coefficient of the $v$ advection depends on $E$ and thus implicitly on time. We choose the time step for the centered scheme as follows $$\label{cfl_vlasov_lc}
\Delta t_n = \frac{y_{\max} \Delta v}{\| E^n \|_{L^\infty}},$$ whereas for the WENO5 scheme we use the CFL condition computed from its linearized version (LW5) $$\label{cfl_vlasov_lw}
\Delta t_n = \frac{\sigma \Delta v}{\| E^n \|_{L^\infty}}.$$ The value $\| E^n \|_{L^\infty}$ is just the maximal value of the electric field at time $t_n$. The values for $y_{\max}$ and $\sigma$ are given in Tables \[tab:ymax\_Lawson\], \[tab:ymax\_expo\], and \[tab:ymax\_weno\_Lawson\] according to the chosen time integrator.
Landau damping test.
--------------------
We present numerical results for the standard Landau damping test case. The initial condition is given by $$f_0(x, v) = \frac{1}{\sqrt{2\pi}} e^{-\frac{v^2}{2}} (1+0.001 \cos(0.5 x)), \;\; x\in [0, 4\pi], v\in \mathbb{R}.$$ The numerical parameters are chosen as follows: the number of points in space is $N_x=81$ whereas the velocity domain is truncated to $[-v_{\max}, v_{\max}]$ with $v_{\max}=8$ and is discretized with $N_v=128$ grid points.
Let us remark that for the Landau damping test, the conditions and allow us to take very large time steps, since $\| E^n \|_{L^\infty} \leq \| E^0 \|_{L^\infty} = 2\cdot 10^{-3}$. Then, we get $\Delta t = C \Delta v \; 0.5\cdot 10^3 = 62.5 C$, where $C$ can be either $y_{\max}$ or $\sigma$ depending on the chosen time integrator. This means that in practice we can choose the time step $\Delta t$ independently from the mesh. This is clearly a desirable desirable feature of the time integrator.
In Figure \[ld\], the time history of the electric energy $\|E^n\|_{L^2}$ (in semi-log scale) with two different time steps ($\Delta t=1/8$ and $\Delta t=1$) using Lawson($RK(4, 4)$) for the time integration and a WENO5 scheme for the velocity direction is shown. One can observe that the expected damping rate ($\gamma=-0.153$) is recovered for both configurations. Although, the accuracy deteriorates for $\Delta t=1$, it is clear that the numerical scheme is stable.
![Landau damping test: time history of $\|E(t)\|_{L^2}$ (semi-log scale) obtained with Lawson($RK(4, 4)$) and WENO5 with $\Delta t=1/8$ and $\Delta t=1$.[]{data-label="ld"}](img/{Emax}.pdf){width="\textwidth"}
Bump on tail test.
------------------
Next, we consider the bump on tail test for which the initial condition is $$f_0(x, v) = \left[\frac{0.9}{\sqrt{2\pi}} e^{-\frac{v^2}{2}} + \frac{0.2}{\sqrt{2\pi}} e^{-2(v-4.5)^2} \right](1+0.04 \cos(0.3 x)), \;\; x\in [0, 20\pi], v\in \mathbb{R}.$$ The numerical parameters are chosen as follows: the number of points in space is $N_x=135$ whereas the velocity domain $[-v_{\max}, v_{\max}]$ (with $v_{\max}=8$) is discretized with $N_v=256$ grid points. Concerning the time step, as in the Landau damping example, the conditions and turn out to be very light for Lawson schemes. Indeed, we found $\max_n \| E^n \|_{L^\infty} \approx 0.6$ so that, with the considered velocity grid, the time step has to be smaller than $0.14$ in the worst case (Lawson($RK(3, 2) \; best)$ combined with WENO5). To capture correctly the phenomena involved in the bump on tail test, we take the following time step size $$\label{dtbot}
\Delta t_n = \min \Big( 0.1, \frac{C \Delta v}{\|E^n\|_{L^\infty}} \Big),$$ with $C=y_{\max}$ or $\sigma$ depending on the chosen scheme. Thus, also in this configuration we are mostly limited by the accuracy and not by the stability constraint.
In Figure \[space\], the full distribution function $f$ is plotted at time $t=40$ for Lawson($RK(4, 4)$) coupled with the WENO5 scheme, Lawson($RK(4, 4)$) coupled with the centered scheme and Hochbruck–Ostermann coupled with the centered scheme. In these figures, we look at the impact of the velocity approximation. One can observe spurious oscillations in the centered scheme case (middle figure) whereas the slope limiters of WENO5 are able to control this phenomena so that extremas are well preserved. This is consistent with what has been observed in the literature.
![Distribution function at time $t=40$ as a function of $x$ and $v$ for Lawson($RK(4, 4)$) + WENO5 (left), Lawson($RK(4, 4)$) + centered scheme (center), Hochbruck–Ostermann + centered scheme (right).[]{data-label="space"}](img/vp_cfl.png){width="\textwidth"}
In Figure \[total\_energy\], we plot the time evolution of $({\cal H}^n -{\cal H}(0))/{\cal H}(0)$, where ${\cal H}^n\approx {\cal H}(n\Delta t)$ and ${\cal H}(t)$ is the total energy defined by $${\cal H}(t) = \frac{1}{2}\int\int |v|^2 f(t, x, v) \,dxdv + \frac{1}{2}\int |E|^2(t, x) \,dx.$$ This quantity is known to be preserved with time and thus allows us to look at the accuracy of the different methods considered in this paper. We observe that Lawson/centered schemes (referred as ’CD2’ in the legend) preserve this quantity well. It is well known (see for example [@cf]) that centered schemes are better at preserving the total energy compared to upwind schemes. The reason is that upwind schemes introduce numerical diffusion. The exponential integrators that are considered show all very similar behavior with respect to energy conservation. They seem to include less drift than the Lawson methods, but for the time scales considered here their error is larger.
![Time evolution of the relative error of the total energy for the different methods. []{data-label="total_energy"}](img/H.png){width="90.00000%"}
Although being able to choose the time step size independently of the mesh is a desirable feature, it makes checking the sharpness of the CFL estimate derived in the previous section more difficult. To accomplish this, we consider the same parameters as before, except for the phase space mesh which now uses $N_x=81$ and $N_v=512$ grid points. Then the maximum time step becomes $\Delta t=\min_n C\Delta v/\|E^n\|_{L^\infty} \approx 0.052C$ (since $\max_n \|E^n\|_{L^\infty}\approx 0.6$ and $\Delta v=16/512=0.03125$). We consider two different time steps: $\Delta t=0.052C$ (which satisfies the linearized CFL condition) and $\Delta t = 1.4 \times 0.052C$ (which violates the linearized CFL condition). The results are shown in Figure \[unstable\]. There the Lawson($RK(4,4)$) method has been chosen for the time discretization whereas WENO5 and centered scheme are both considered for the velocity discretization. More specifically,
- for WENO5 we use $C=1.73$ (obtained from the linearized version LW5) and we compare the results obtained with $\Delta t=0.09$ (satisfies the CFL condition) and $\Delta t=0.13$ (does not satisfy the CFL condition).
- for the centered scheme, we use $C=2\sqrt{2}$ and we compare the results obtained with $\Delta t=0.14$ (satisfies the CFL condition) with $\Delta t=0.2$ (does not satisfy the CFL condition).
In Figure \[unstable\], the time evolution of the electric energy $\|E(t)\|^2_{L^2}$ is displayed for these two velocity discretizations. One can observe for the time step size that satisfies the CFL condition the simulation is stable and gives the expected results, whereas for the choice that violates the CFL condition the simulation blows up. Thus, the results confirm that the CFL condition obtained by the linear theory yields a good prediction for the nonlinear Vlasov–Poisson equation. On the right part of Figure \[unstable\], the time history of the quantity $C \Delta v/ \|E^n\|_{L^\infty}$ is shown (red) together with the time step size considered for the WENO velocity discretization. The choice $\Delta t=0.13$ (blue line) is larger than the allowed time step size around $t\approx 20$, which explains the numerical instability observed at that point in time.
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --
![Illustration of the accuracy of the CFL estimate obtained from the linear theory. History of electric energy with Lawson($RK(4,4)$) + WENO5 (left), Lawson($RK(4,4)$) + centered scheme (middle) and history of CFL condition for Lawson($RK(4,4)$) + WENO5 case (right)[]{data-label="unstable"}](img/ee_weno_rk44.png "fig:") ![Illustration of the accuracy of the CFL estimate obtained from the linear theory. History of electric energy with Lawson($RK(4,4)$) + WENO5 (left), Lawson($RK(4,4)$) + centered scheme (middle) and history of CFL condition for Lawson($RK(4,4)$) + WENO5 case (right)[]{data-label="unstable"}](img/ee_o2_rk44.png "fig:") ![Illustration of the accuracy of the CFL estimate obtained from the linear theory. History of electric energy with Lawson($RK(4,4)$) + WENO5 (left), Lawson($RK(4,4)$) + centered scheme (middle) and history of CFL condition for Lawson($RK(4,4)$) + WENO5 case (right)[]{data-label="unstable"}](img/bot_cfl_weno_rk44.png "fig:")
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --
Numerical simulation: drift-kinetic equations \[sec:dk\]
========================================================
In this section we will consider a model motivated by the simulation of strongly magnetized plasmas, such as those found in tokamaks. In this case the dynamics is governed by gyrokinetic equations. Gyrokinetics averages out the fast oscillatory motion of the charged particles around the magnetic field lines. In a simplified slab geometry, gyrokinetic models reduce to the drift-kinetic equation. In this case the unknown $f$ depends on three cylindrical spatial coordinates $(r,\theta,z)$ and one velocity direction $v$. This model is composed of a guiding-center dynamics in the plane orthogonal to the magnetic field lines and of a Vlasov type dynamics in the direction parallel (to the magnetic field lines). In addition to its relevance in physics, it is also a good test case for stressing exponential methods. The latter is due to the fact that after some time the nonlinearity can become strong enough such that the time step size is is dictated by stability constraints (especially for high order methods).
Our goal in this section is to find a numerical approximation of $f=f(t,r,\theta,z,v)$ satisfying the following $4D$ slab drift-kinetic equation (see [@grandgirard]) $$\label{dk}
\partial_tf-\frac{\partial_\theta \phi}{r}\partial_rf+\frac{\partial_r \phi}{r}\partial_\theta f
+v\partial_zf-\partial_z\phi\partial_{v}f=0,$$ for $(r,\theta,z,v)\in \Omega\times[0,L]\times \mathbb{R}$, $\Omega=[r_{\rm min},r_{\rm max}]\times [0, 2\pi]$. The self-consistent potential $\phi=\phi(r,\theta,z)$ is determined by solving the quasi neutrality equation $$\begin{aligned}
-\left[\partial_r^2\phi+\left(\frac{1}{r}+\frac{\partial_r n_0(r)}{n_0(r)}\right)\partial_r\phi \right.&+\left.\frac{1}{r^2}\partial_\theta^2\phi\right]+\frac{1}{T_e(r)}(\phi-\langle\phi\rangle)\nonumber\\
\label{qn}
&\hspace{-3cm}=\frac{1}{n_0(r)}\int_{\mathbb{R}} fdv-1,\end{aligned}$$ where $\langle\phi\rangle = \frac{1}{L}\int_0^L \phi(r,\theta,z)\,dz$ and the functions $n_0$ and $T_e$ depend only on $r$ and are given analytically.
In many situations the $v \partial_z f$ term yields the most restrictive CFL condition. In this setting exponential methods can be very successful as they remove the most stringent CFL condition, while still treating the remaining terms explicitly (which computationally is relatively cheap). The $\varphi$ functions can be computed in Fourier space (as has been discussed in some detail for the Vlasov–Poisson system in the previous section) or using a semi-Lagrangian approach. Exponential integrators for the drift-kinetic model have been proposed in [@cep]. They compare favorably to splitting schemes and have the advantage that they can be more easily adapted to different models. In [@cep] only a second order exponential integrator and the fourth order Cox–Matthews scheme have been considered. Due to the investigations in the present paper we now understand that this is not an ideal choice. Thus, the purpose of this section is to demonstrate that Lawson methods can be more efficient and to further corroborate the results obtained in the previous sections. The difference in stability for Lawson schemes and exponential integrators will be very evident in the numerical simulations that are presented.
Numerical discretization
------------------------
First, we remark that $z$ is a periodic variable which motivates us to consider the Fourier transform in this direction. The corresponding frequencies are denoted by $k$. Equation then becomes $$\partial_t \hat{f}_k -\partial_r \widehat{\left(\frac{\partial_\theta \phi}{r}f\right)}_k+\partial_\theta \widehat{\left(\frac{\partial_r \phi}{r} f\right)}_k +vik \hat{f}_k-\partial_v\widehat{\left( \partial_z\phi \, f\right)}_k=0,$$ Setting $F(t, f)= \partial_r \widehat{\left(\frac{\partial_\theta \phi}{r}f\right)}-\partial_\theta \widehat{\left(\frac{\partial_r \phi}{r} f\right)} +\partial_v\widehat{\left( \partial_z\phi \,f\right)}$, this equation can be written as $$\partial_t \hat{f} = - vik \hat{f} + F(t, f).$$ This is now precisely in the form to which we can apply an exponential method. In addition, computing the required matrix functions is very efficient as all the frequencies decouple (see the corresponding discussion in section \[sec:vp\]).
To complete the numerical scheme, one has to detail the phase space approximation. As in [@cep] we will use Arakawa’s method to approximate the derivatives needed to compute $F$. Arakawa’s method is a centered difference scheme that conserves three invariants. More details can be found in [@cep].
Numerical results \[subsec:driftkinetic-results\]
-------------------------------------------------
In this section, we detail the physical parameters of the considered test case. The set up is identical to [@cep] (see also [@BC2013; @vlasovia]). The initial value is given by $$\begin{aligned}
f(t=0,r,\theta,z,v) &=
f_{\rm eq}(r,v)\left[1+\epsilon \exp\left(-\frac{(r-r_p)^2}{\delta r}\right)\cos\left(\frac{2\pi n}{L}z+m\theta\right)\right],\end{aligned}$$ where the equilibrium distribution is given by $$\label{eq:equilibrium}
f_{\rm eq}(r,v)=\frac{n_0(r)\exp(-\frac{v^2}{2T_i(r)})}{(2\pi T_i(r))^{1/2}}.$$ The radial profiles $T_i$, $T_e$, and $n_0$ have the analytic expressions $$\mathcal{P}(r) = C_\mathcal{P}\exp\left(-\kappa_\mathcal{P}\delta r_{\mathcal{P}}\tanh(\frac{r-r_p}{\delta r_{\mathcal{P}}})\right), \; \mathcal{P}\in \{T_i,T_e,n_0\}$$ with the constants defined as follows $$\ C_{T_i}=C_{T_e}=1,\ C_{n_0}=\frac{r_{\rm max}-r_{\rm min}}{
\int_{r_{\rm min}}^{r_{\rm max}}\exp(-\kappa_{n_0}\delta r_{n_0}\tanh(\frac{r-r_p}{\delta r_{n_0}}))\,dr}.$$ Finally, we consider the parameters of [@BC2013] (MEDIUM case) $$\begin{aligned}
&&r_{\rm min} = 0.1,\ r_{\rm max} = 14.5,\\
&& \kappa_{n_0}= 0.055,\ \kappa_{T_i}=\kappa_{T_e}= 0.27586,\\
&&\delta r_{T_i}=\delta r_{T_e}=\frac{\delta r_{n_0}}{2}= 1.45,\ \epsilon=10^{-6},\ n=1,\ m=5,\\
&&L=1506.759067,\ r_p = \frac{r_{\rm min}+r_{\rm max}}{2},\delta r = \frac{4 \delta r_{n_0}}{\delta r_{T_i}}.\end{aligned}$$ and use a $v$-range of $v \in [-7.32,7.32]$.
We consider two configurations. A direct formulation, where the boundary conditions are given by $$f(r_{\rm min},\theta ,z,v)=f_{eq}(r_{\rm min},v) \qquad
f(r_{\rm max},\theta ,z,v)=f_{eq}(r_{\rm max},v).$$ Note that these are not homogeneous Dirichlet boundary conditions. It is well known (and supported by [@cep]) that the Arakawa scheme works better for homogeneous boundary conditions. In addition to the direct formulation, we therefore also introduce a so-called perturbation formulation (see also [@vlasovia; @latu2]). First, we note that the equilibrium function $f_{\rm eq}$ defined in (\[eq:equilibrium\]) is a steady state for our problem. We therefore divide $f$ into $$f(t,r,\theta ,v)=f_{eq}(r,v)+\delta f(t,r,\theta ,v).$$ With this formulation, our problem (\[dk\]) becomes $$\partial_t\delta f+\frac{E_\theta}{r}\partial_r (f_{eq} + \delta f)-\frac{E_r}{r}\partial_\theta \delta f+v\partial_z\delta f+E_z \partial_v (f_{eq} + \delta f)=0,$$ where $E_\theta=-\partial_\theta \phi$, $E_r=-\partial_r \phi$ and $E_z=-\partial_z \phi$. Expanding the various terms we obtain $$\partial_t\delta f+\frac{E_\theta}{r}\partial_r\delta f-\frac{E_r}{r}\partial_\theta \delta f+v\partial_z\delta f+ E_z \partial_v \delta f
+\frac{E_\theta}{r}\partial_r f_{eq} + E_z \partial _v f_{eq}=0$$ which can be written as $$\partial_t\delta f + v\partial_z \delta f -F(\delta f) +\frac{E_\theta}{r}\partial_r f_{eq} +E_z\partial _v f_{eq} = 0.$$ Note that the equation is very similar to equation (\[dk\]). We, however, have obtained two additional source terms, which depend on the equilibrium distribution $f_{eq}$ as well as on the electric field. Furthermore, the right hand side of the quasi-neutrality equation (\[qn\]) becomes $$\frac{1}{n_0}\int f_{eq} \,dv+\frac{1}{n_0}\int \delta f \,dv-1 = \frac{1}{n_0}\int \delta f \,dv.$$ Due to the similarity of the direct formulation and the perturbation formulation, the same code can be used for both by simply exchanging the right hand side of the quasi-neutrality equation, changing the boundary conditions, and adding the appropriate source terms. Thus, to implement the exponential integrator we consider the following equation $$\partial_t\delta f + v\partial_z \delta f = F_{pert}(\delta f),$$ with $$F_{pert} (\delta f)= F(\delta f) -\frac{E_\theta}{r}\partial_r f_{eq} - E_z\partial _v f_{eq},$$ and proceed as before (with $F$ replaced by $F_{pert}$). The space discretization of the source terms can be done either analytically or using a numerical approximation. In our implementation we have used standard centered differences. The Arakawa scheme that is used to discretize $F(\delta f)$ now employs homogeneous Dirichlet boundary conditions for $\delta f$ in the $r$-direction.
We have seen in section \[sec:vp\] that for the Vlasov–Poisson equation we can derive a constraint on the time step size which ensures stability. For Lawson methods this also gives a good estimate in practice. However, for exponential integrators the situation is far more complicated, see the discussion in section \[ode\]. Thus, a natural question that arises is how large time steps can we take in practice. To do that we will employ an adaptive step size controller that uses Richardson extrapolation to obtain an error estimate. By denoting a time step as follows $f^{n+1} = \varphi_{\Delta t_n}(f^n)$ and considering $\tilde{f}^{n+1} = \varphi_{\Delta t_n/2}\circ \varphi_{\Delta t_n/2}(f^n)$ we can construct the Richardson extrapolated numerical solution of a method of order $p$ as follows $f_R^{n+1} = (2^{p+1} \tilde{f}^{n+1} - f^{n+1})/(2^{p+1} -1)$, which turns out to be an approximation of order $(p+1)$ of the exact solution. Then, it is possible to determine an estimate of the local error $e_{n+1}$ of the time integrator through the following expression $$e_{n+1} = \left\Vert f_R^{n+1} - f^{n+1}\right\Vert_{L^{\infty}} + {\cal O}(\Delta t_n^{p+2}),
$$ where the $L^\infty$ norm is considered in the $r, \theta, z, v$ variables. If the estimate for the error $e_{n+1}$ is larger than a specified $\text{tol}$ we reject the step and start again from time $t_n$. Otherwise, the step is accepted and we proceed with the time integration. In either case we then determine the new step size $\Delta t_{new}$ such that the local error is smaller than the tolerance. That is, we choose $$\label{compute_dt}
\Delta t_{new}=s \Delta t_{n}\left(\frac{\text{tol}}{e_{n+1}}\right)^{1/(p+1)},$$ where $\text{tol}$ is the prescribed tolerance, $p$ is the order of the method, and $s=0.8$ is a safety factor. This process is very well established in the literature and we refer the interested reader to [@gustafsson1988; @gustafsson1994; @soderlind2002; @soderlind2006; @lukas]. Other strategies can also be considered such as embedded Lawson or exponential methods (see [@gni; @balac]). Such methods may be more efficient but we restrict ourselves here to the strategy based on Richardson extrapolation since it can be applied to any time integrator.
An interesting property of this adaptive step size controller is that it forces the time step size to satisfy the stability constraint of the numerical method. This is perhaps surprising at first sight since the scheme only controls the local error. However, numerical instability are characterized by error amplification as integration proceeds in time. Thus, a single step can violate the stability constraint, but later on the error amplification increases the local error in such a way that the adaptive step size controller is forced to reduce the time step size. Thus, the controller ensures that we obtain a stable numerical simulation for which the local error is below the specified tolerance.
This procedure allows us to perform a fair comparison between Lawson methods and exponential integrators. Since we are mainly interested in the stability of the methods and, particularly in the nonlinear regime, prescribing a stringent tolerance is infeasible in any case, we will choose a relatively large tolerance for our simulation ($\text{tol}=10^{-2}$ for the perturbation formulation). To avoid the problem of too large time steps at the beginning of the simulation, where accuracy and not stability dictates the time step, we limit the maximal step size to $\Delta t=11$ (coarse) and $\Delta t=10$ (fine) for second order methods, $\Delta t=30$ for third order methods and Lawson($RK(3, 2) \; best$), and $\Delta t=40$ for fourth order methods.
To evaluate the performances of the different time integrators, we consider the time evolution of the electric energy defined by $${\cal E}(t) = \left( \int_{0}^{L} \int_0^{2\pi} \phi^2(t, r_p, \theta, z ) \,d\theta dz \right)^{1/2}, \;\;\; \mbox{ with } r_p=\frac{r_{\min}+r_{\max}}{2},$$ as well as the time evolution of the total mass and the total energy $$\begin{aligned}
{\cal M}(t) &=& \int_{r_{\min}}^{r_{\max}} \int_{0}^{L} \int_0^{2\pi} \int_{\mathbb R} f(t, r, \theta, z, v ) \,dv d\theta dz dr, \nonumber\\
{\cal N}(t) &=&\int_{r_{\min}}^{r_{\max}} \int_{0}^{L} \int_0^{2\pi} \int_{\mathbb R} \frac{v^2}{2} f(t, r, \theta, z, v ) \,dv d\theta dz dr \nonumber\\
&& + \int_{r_{\min}}^{r_{\max}} \int_{0}^{L} \int_0^{2\pi} \int_{\mathbb R} f(t, r, \theta, z, v )\phi(t, r, \theta, z) \,dv d\theta dz dr. \end{aligned}$$ The numerical results for the perturbation formulation are given in Figure \[fig:driftkinetic-pert1\]. There the time history of the electric energy and the time step size as of function of time for two different discretizations in phase space, $32 \times 32 \times 32 \times 64$ and $64 \times 64 \times 64 \times 128$ grid points, are shown. We first see that all the time integrators agree very well; that is, we see an initial exponential growth (the rate is in good agreement with the linear theory; see, for example, [@BC2013]) in the electric energy. This phase is followed by saturation at very similar levels for all numerical methods used. We also observe that all exponential integrators, except the method of Krogstad, are forced to reduce their step size after time $t\approx 5000$ for the fine, [*i.e.*]{}$64\times 64 \times 64 \times 128$ grid points, case. This is particularly drastic for ExpRK33 and the Cox–Matthews method which suffer from stability issues even for the coarse discretization. In general, for the finer space discretization (see the right plot in Figure \[fig:driftkinetic-pert1\]), the problem becomes significantly more severe. It is also worth mentioning that ExpRK33 leads to unstable results in spite of the step size controller, which clearly highlights the unstable nature of that integrator. Neither of the Lawson schemes have similar issues and the size of the stability domain on the imaginary axis gives a good indication of the relative time steps this methods can take. We also note that at later times Lawson schemes are able to take significantly larger time steps compared to exponential integrators. The only exponential integrator that performs well in this regime is the method of Krogstad. Thus, the numerical results agree well with what we would expect based on the theoretical analysis.
[0.48]{} ![Numerical simulation for a number of Lawson methods and exponential integrators for the drift-kinetic model (perturbation formulation). The upper plots show the time step size as a function of time. The lower plots show the time evolution of the electric energy. The configuration on the left uses $32 \times 32 \times 32 \times 64$ grid points and the configuration on the right uses $64 \times 64 \times 64 \times 128$ grid points.[]{data-label="fig:driftkinetic-pert1"}](img/{driftkinetic-tol1.00e-02-32x32x32x64-pert1}.pdf "fig:"){width="\textwidth"}
[0.48]{} ![Numerical simulation for a number of Lawson methods and exponential integrators for the drift-kinetic model (perturbation formulation). The upper plots show the time step size as a function of time. The lower plots show the time evolution of the electric energy. The configuration on the left uses $32 \times 32 \times 32 \times 64$ grid points and the configuration on the right uses $64 \times 64 \times 64 \times 128$ grid points.[]{data-label="fig:driftkinetic-pert1"}](img/{driftkinetic-tol1.00e-02-64x64x64x128-pert1}.pdf "fig:"){width="\textwidth"}
The corresponding numerical results using the direct formulation are shown in Figure \[fig:driftkinetic-pert0\]. The situation for the direct formulation is very similar to the perturbation formulation, even if one can observe that the time steps are slightly larger than in the perturbation formulation. One explanation comes from the fact that the relative error computed in the perturbation case involves the norm of $\delta f$ which can be quite different from the norm of $f$ so that equation leads to different value of the time step even if the accuracy of the solution is the same.
[0.48]{} ![Numerical simulation for a number of Lawson methods and exponential integrators for the drift-kinetic model (direct formulation). The upper plots show the time step size as a function of time. The lower plots show the time evolution of the electric energy. The configuration on the left uses $32 \times 32 \times 32 \times 64$ grid points and the configuration on the right uses $64 \times 64 \times 64 \times 128$ grid points.[]{data-label="fig:driftkinetic-pert0"}](img/{driftkinetic-tol1.00e-02-32x32x32x64-pert0}.pdf "fig:"){width="\textwidth"}
[0.48]{} ![Numerical simulation for a number of Lawson methods and exponential integrators for the drift-kinetic model (direct formulation). The upper plots show the time step size as a function of time. The lower plots show the time evolution of the electric energy. The configuration on the left uses $32 \times 32 \times 32 \times 64$ grid points and the configuration on the right uses $64 \times 64 \times 64 \times 128$ grid points.[]{data-label="fig:driftkinetic-pert0"}](img/{driftkinetic-tol1.00e-02-64x64x64x128-pert0}.pdf "fig:"){width="\textwidth"}
In Figure \[fig:mass\_energy\], the time history of the relative error of the total mass and of the total energy are displayed with the phase space discretization $64\times 64\times 64\times 128$ and using Lawson($RK(4,4)$) and Cox–Matthews time integrators (the perturbation formulation is used here). Since mass is a linear invariant it is preserved, up to machine precision, by the exponential integrator, see [@le2015KP]. We also observe good conservation of energy even in the nonlinear phase, which confirms the excellent behavior of the methods.
![Numerical simulation for Lawson(RK(4,4)) and the Cox–Matthews method for the drift-kinetic model (perturbation formulation). Left: time history of the error in total mass. Right: time history of the error in total energy. The configuration uses $64 \times 64 \times 64 \times 128$ grid points.[]{data-label="fig:mass_energy"}](img/{diagnostics}.pdf){width="\textwidth"}
Finally, we show slices of the distribution function and the density at different times. The simulation in Figure \[fig:snapshots-lrk44\] is conducted with the Lawson($RK(4,4)$) scheme and the simulation in Figure \[fig:snapshots-cm\] with the Cox–Matthews scheme (both with the perturbation formulation). In both cases the configuration of Figure \[fig:driftkinetic-pert1\] and the fine space resolution has been employed. As comparison, a reference solution computed with the Lawson($RK(4,4)$) scheme and a step size controller that keeps the error below $10^{-5}$ per unit time step is shown in Figure \[fig:snapshots-ref\]. We remark that all simulations show good agreement: the $m=5$ modes in the $\theta$ direction are recovered and after initial growth of the unstable mode, we can observe a shearing of the structures and the appearance of small scale structures which are typical for the nonlinear phase.
![A slices at $(z,v)=(0,0)$ of the distribution function (on the left) and a slice at $z=0$ of the density (on the right) are shown for times $t=3000$, $4000$, and $5000$. The Lawson($RK(4,4)$) scheme, in the configuration described in section \[subsec:driftkinetic-results\], with $64\times64\times64\times128$ grid points is used. \[fig:snapshots-lrk44\]](img/{snapshots-lawson-rk44-tol1e-2}.pdf){width="\textwidth"}
![A slices at $(z,v)=(0,0)$ of the distribution function (on the left) and a slice at $z=0$ of the density (on the right) are shown for times $t=3000$, $4000$, and $5000$. The Cox–Matthews scheme, in the configuration described in section \[subsec:driftkinetic-results\], with $64\times64\times64\times128$ grid points is used. \[fig:snapshots-cm\]](img/{snapshots-coxmatthews-tol1e-2}.pdf){width="\textwidth"}
![A slices at $(z,v)=(0,0)$ of the distribution function (on the left) and a slice at $z=0$ of the density (on the right) are shown for times $t=3000$, $4000$, and $5000$. The Lawson($RK(4,4)$) scheme with a tolerance of $10^{-5}$ per unit step and $64\times64\times64\times128$ grid points is used. \[fig:snapshots-ref\]](img/{snapshots-reference-1e-5}.pdf){width="\textwidth"}
Acknowledgement {#acknowledgement .unnumbered}
===============
We would like to thank David C. Seal (U.S. Naval Academy) and Sigal Gottlieb (University of Massachusetts, Dartmouth) for the helpful discussion.
This work has been carried out within the framework of the EUROfusion Consortium and has received funding from the Euratom research and training programme 2014- 2018 and 2019-2020 under grant agreement No 633053. The views and opinions expressed herein do not necessarily reflect those of the European Commission. The work has been supported by the French Federation for Magnetic Fusion Studies (FR-FCM) and by the Austrian Science Fund (FWF): project number P 32143-N32.
A.H. Al-Mohy, N.J. Higham, *Computing the action of the matrix exponential, with an application to exponential integrators*, SIAM J. Sci. Comput. 33, (2011), pp. 488–511.
S. Balac, F. Mahé, *Embedded Runge-Kutta scheme for step-size control in the Interaction Picture method*, Comput. Phys. Commun. 184, (2013), pp. 1211-1219.
M. Baldauf, *Stability analysis for linear discretisations of the advection equation with Runge-Kutta time integration* J. Comput. Phys. 227, (2007), pp. 6638–6659.
J. Bigot, V. Grandgirard, G. Latu, C. Passeron, F. Rozar, O. Thomine, *Scaling GYSELA code beyond 32K-cores on Blue Gene/Q*, ESAIM: Proceedings 43, (2013), pp. 117–135.
M. Caliari, P. Kandolf, A. Ostermann, S. Rainer, *Comparison of software for computing the action of the matrix exponential*, BIT Numer. Math. 54, (2014), pp. 113–128.
M. Caliari, P. Kandolf, A. Ostermann, S. Rainer, *The Leja method revisited: backward error analysis for the matrix exponential*, SIAM J. Sci. Comput. 38, (2016), pp. 1639–1661.
C. Canuto, M.Y. Hussaini, A. Quateroni, T.A. Zang, *Spectral methods in fluid dynamics*. Springer Verlag, 1988.
F. Casas, N. Crouseilles, E. Faou, M. Mehrenberger, *High-order Hamiltonian splitting for Vlasov-Poisson equations*, Numer. Math. 135, (2017), pp. 769–801.
C. Cheng, G. Knorr, *The integration of the Vlasov equation in configuration space*. J. Comput. Phys. 22, (1976), pp. 330–351.
D. Coulette, N. Besse, *Numerical comparisons of gyrokinetic multi-water-bag models*, J. Comput. Phys. 248, (2013), pp. 1–32.
S.M. Cox, P.C. Matthews, *Exponential time differencing for stiff systems*, J. Comput. Phys. 176, (2002), pp. 430–455.
N. Crouseilles, L. Einkemmer, E. Faou, *Hamiltonian splitting for the Vlasov–Maxwell equations*, J. Comput. Phys. 283, (2015), pp. 224–240.
N. Crouseilles, L. Einkemmer, E. Faou, *An asymptotic preserving scheme for the relativistic Vlasov–Maxwell equations in the classical limit*. Comput. Phys. Commun. 209, (2016), pp. 13–26.
N. Crouseilles, L. Einkemmer, M. Prugger, *An exponential integrator for the drift-kinetic model*, Comput. Phys. Commun. 224, (2019), pp. 144–153.
N. Crouseilles, F. Filbet, *Numerical approximation of collisional plasmas by high order methods*, J. Comput. Phys., 201, (2004), pp. 546–572.
N. Crouseilles, P. Glanc, M. Mehrenberger, C. Steiner, *Finite volume schemes for Vlasov*, ESAIM Proceedings, (2013), pp. 275–297.
N. Crouseilles, P. Glanc, S. Hirstoaga, E. Madaule, M. Mehrenberger, J. Pétri, *A new fully two-dimensional conservative semi-Lagrangian method: applications on polar grids, from diocotron instability to ITG turbulence*, Eur. Phys. J. D 68, (2014), pp. 252–261.
N. Crouseilles, G. Latu, E. Sonnendr[ü]{}cker, *A parallel Vlasov solver based on local cubic spline interpolation on patches*. J. Comput. Phys. 228, (2009), pp. 1429–1446.
N. Crouseilles, M. Mehrenberger, F. Vecil, *Discontinuous Galerkin semi-Lagrangian method for Vlasov-Poisson*, ESAIM: Proceedings 32, (2011), pp. 211–230.
L. Einkemmer, *High performance computing aspects of a dimension independent semi-Lagrangian discontinuous Galerkin code*, Comput. Phys. Commun. 202, (2016), pp. 326–336.
L. Einkemmer, A. Ostermann, *A splitting approach for the Kadomtsev–Petviashvili equation* J. Comput. Phys. 299, (2015), pp. 716–730.
L. Einkemmer, *A mixed precision semi-Lagrangian algorithm and its performance on accelerators*, International Conference on High Performance Computing & Simulation (HPCS), 2016, pp. 74–80.
L. Einkemmer, *A study on conserving invariants of the Vlasov equation in semi-Lagrangian computer simulations*, J. Plasma Phys. 83, (2017).
L. Einkemmer, *An adaptive step size controller for iterative implicit methods*, Appl. Numer. Math. 132, (2018), pp. 182–204.
L. Einkemmer, *Semi-Lagrangian Vlasov simulation on GPUs*, Preprint, arXiv:1907.08316.
L. Einkemmer, *A performance comparison of semi-Lagrangian discontinuous Galerkin and spline based Vlasov solvers in four dimensions*, J. Comput. Phys. 376, (2019), pp. 937–951.
L. Einkemmer, A. Ostermann, *Convergence analysis of Strang splitting for Vlasov-type equations*, SIAM J. Numer. Anal. 52, (2014), pp. 140–155.
L. Einkemmer, A. Ostermann, *Convergence analysis of a discontinuous Galerkin/Strang splitting approximation for the Vlasov–Poisson equations*, SIAM J. Numer. Anal. 52, (2014), pp. 757–778.
L. Einkemmer, A. Ostermann, *Exponential integrators on graphic processing units*, High Performance Computing and Simulation (HPCS), International Conference On, 2013.
L. Einkemmer, A. Ostermann. *A strategy to suppress recurrence in grid-based Vlasov solvers*, Eur. Phys. J. D 68, (2014), pp. 197-203.
L. Einkemmer, M. Tokman, J. Loffeld, *On the performance of exponential integrators for problems in magnetohydrodynamics*, J. Comput. Phys. 330, (2016), pp. 550–565.
F. Filbet, E. Sonnendr[ü]{}cker, *Comparison of Eulerian Vlasov solvers*. Comput. Phys. Commun. 150, (2003), pp. 247–266.
V. Grandgirard, M. Brunetti, P. Bertrand, N. Besse, X. Garbet, P. Ghendrih, G. Manfredi, Y. Sarazin, O. Sauter, E. Sonnendr[ü]{}cker, J. Vaclavik, L. Villard, *A drift-kinetic Semi-Lagrangian 4D code for ion turbulence simulation*, J. Comput. Phys. 217, (2006), pp. 395–423.
K. Gustafsson, *Control-theoretic techniques for stepsize selection in implicit Runge-Kutta methods*, ACM Transactions on Mathematical Software 20, (1994), pp. 496–517.
K. Gustafsson, M. Lundh, G. S[ö]{}derlind, *A PI stepsize control for the numerical solution of ordinary differential equations*, BIT Numer. Math. 28, (1988), pp. 270–287.
E. Hairer, C. Lubich, G. Wanner, *Geometric Numerical Analysis*, Springer.
E. Hairer, G. Wanner, *Solving Ordinary Differential Equations II, Stiff and Differential-Algebraic Problems*, Springer.
N.J. Higham, *Functions of matrices: theory and computation*, SIAM, 2008.
M. Hochbruck, C. Lubich, *On Krylov subspace approximations to the matrix exponential operator*, SIAM J. Numer. Anal. 34, (1997), pp.1911–1925.
M. Hochbruck, A. Ostermann, *Explicit exponential Runge–Kutta methods for semilinear parabolic problems*, IMA Journal on Numerical Analysis 43, (2005), pp. 1069–1090.
M. Hochbruck, A. Osterman, *Exponential integrator*, Acta Numerica 19, (2010), pp. 209–286.
M. Hochbruck, A. Ostermann, *On the convergence of Lawson methods for semilinear stiff problems.* Preprint, arXiv:1709.01000.
A.J. Klimas, W.M. Farrell, *A splitting algorithm for Vlasov simulation with filamentation filtration*, J. Comput. Phys. 110, (1994), pp.150–163.
S. Krogstad, *Generalized integrating factor methods for stiff PDEs*, J. Comput. Phys. 203, (2005), pp. 72–88.
G. Latu, N. Crouseilles, V. Grandgirard, E. Sonnendr[ü]{}cker, *Gyrokinetic semi-lagrangian parallel simulation using a hybrid OpenMP/MPI programming*, 14th European PVM/MPI Userś Group Meeting, (2007), pp. 356–364. 2007.
G. Latu, V. Grandgirard, J. Abiteboul, N. Crouseilles, G. Dif-Pradalier, X. Garbet, P. Ghendrih, M. Mehrenberger, Y. Sarazin, E. Sonnendr[ü]{}cker, D. Zarzoso, *Improving conservation properties in a 5D gyrokinetic semi-Lagrangian code*, Eur. Phys. J. D 68, (2014), pp. 345–360.
J.D. Lawson, *Generalized Runge–Kutta processes for stable systems with large Lipschitz constants*. SIAM J. Numer. Anal. 4, (1967), pp. 372–380.
T. Lunet, C. Lac, F. Auguste, F. Visentin, V. Masson, J. Escobar, *Combination of WENO and Explicit Runge-Kutta Methods for Wind Transport in the Meso-NH Model*. Mon. Wea. Rev. 145, (2017), pp. 3817–3838.
A. Martínez, L. Bergamaschi, M. Caliari, M. Vianello, *A massively parallel exponential integrator for advection-diffusion models*, J. Comput. Appl. Math. 231, (2009), pp. 82–91.
M. Mehrenberger, C. Steiner, L. Marradi, N. Crouseilles, E. Sonnendrucker, B. Afeyan. *Vlasov on GPU*. ESAIM proc. 43, (2013), pp. 37–58.
B.V. Minchev, W. Wright, *A review of exponential integrators for first order semi-linear problems* Preprint, Numerics 2/2005.
P. J. Morrison, *Structure and structure-preserving algorithms for plasma physics*. Phys. Plasmas 24, (2017).
M. Motamed, C. B. Macdonald, S. J. Ruuth *On the linear stability of the fifth-order WENO discretization* J. Scientific Comput. 47, (2011), pp. 127–149.
J.A. Rossmanith, D.C. Seal, *A positivity-preserving high-order semi-Lagrangian discontinuous Galerkin scheme for the Vlasov–Poisson equations*, J. Comput. Phys. 230, (2011), pp. 6203–6232.
F. Rozar, G. Latu, J. Roman, *Achieving memory scalability in the GYSELA code to fit exascale constraints*, Parallel Processing and Applied Mathematics, (2013), pp. 185–195.
N. J. Sircombe, T. D. Arber, *VALIS: A split-conservative scheme for the relativistic 2D Vlasov–Maxwell system*, J. Comput. Phys. 228, (2009), pp. 4773–4788.
G. S[ö]{}derlind, *Automatic control and adaptive time-stepping*, Numerical Algorithms 31, (2002), pp. 281–310.
G. S[ö]{}derlind, *Time-step selection algorithms: Adaptivity, control, and signal processing*, Applied Numerical Mathematics 56, (2006), pp. 488–502.
E. Sonnendr[ü]{}cker, J. Roche, P. Bertrand, A. Ghizzo, *The semi-Lagrangian method for the numerical resolution of the Vlasov equation*. J. Comput. Phys. 149, (1999), pp. 201–220.
A. Tambue, G.J. Lord, S. Geiger, *An exponential integrator for advection-dominated reactive transport in heterogeneous porous media*, J. Comput. Phys. 229, (2010), pp. 3957–3969.
L.N. Trefethen, *Spectral methods in Matlab*. Soc. Industr. Appl. Math., 2000.
J. P. Verboncoeur. *Particle simulation of plasmas: review and advances*. Plasma Phys. Control. Fusion 47, (2005), pp. 231–260.
R. Wang, R. J. Spiteri *Linear instability of the fifth-order WENO method* SIAM J. on Numer. Anal. 45, (2007), pp 1871–1901.
Appendix
========
Butcher tableaus {#butcher}
----------------
In this section, we write down the different numerical methods used in this work. As in section \[sec:expint\], we consider the following equation $$\dot{u} = A u + F(u),$$ where $A$ is a matrix and $F$ a general nonlinear function of $u$. The Butcher tableaus for the Lawson integrators used in the main text are stated in this section. A Lawson method is uniquely determined by the underlying (explicit) Runge–Kutta methods and can be written as follows $$\begin{aligned}
u^{(\ell)} &= e^{c_\ell \Delta t A}u^n + \Delta t\sum_{j=1}^s a_{\ell, j} e^{-(c_j-c_\ell)\Delta t A} F(u^{(j)}), \\
u^{n+1} &= e^{\Delta t A}u^n + \Delta t\sum_{j=1}^s b_j e^{(1-c_j)\Delta t A} F(u^{(j)}),
\end{aligned}$$ where the coefficients $a_{\ell, j}$ and $b_j$ are given by the Butcher tableaus. The Butcher tableaus for the *RK(2,2) best*, *RK(3,3)* (the classic method of order 3), and *RK(4,4)* (the classic method of order 4) are shown in Table \[rks\].
[c|cccc]{} $0$\
$\frac{1}{2}$ & $\frac{1}{2}$\
$\frac{1}{2}$ &$0$ &$\frac{1}{2}$\
$1$& $0$& $0$& $1$\
& $\frac{1}{6}$ &$\frac{1}{3}$ &$\frac{1}{3}$ &$\frac{1}{6} $
[c|ccc]{} $0$\
$\frac{1}{2}$ & $\frac{1}{2}$\
$1 $ &$-1$ &$2$\
& $\frac{1}{6}$ &$\frac{2}{3}$ &$\frac{1}{6}$
[c|ccc]{} $0$\
$\frac{1}{2}$ & $\frac{1}{2}$\
$\frac{1}{2}$ &$0$ &$\frac{1}{2}$\
& $0$ &$0$ &$1$
A general exponential integrator can be written as $$\begin{aligned}
u^{(\ell)} &= u^n + \Delta t\sum_{j=1}^s a_{\ell, j}(\Delta t A)\left( F(u^{(j)}) + A u^n \right) \\
u^{n+1} &= u^n + \Delta t\sum_{j=1}^s b_j(\Delta t A)\left( F(u^{(j)}) + A u^n \right),
\end{aligned}$$ where the coefficients $a_{\ell, j}(\Delta t A)$ and $b_j(\Delta t A)$ can be written as a linear combination of $\varphi_\ell$ and $\varphi_{\ell, j}$ (see [@ei]) $$\varphi_\ell(z) = \frac{e^{z} - \sum_{k=0}^{\ell-1}\frac{1}{k!}z^k}{z^\ell},$$ and we use the notations $\varphi_\ell :=\varphi_\ell(\Delta t A)$ and $\varphi_{\ell,j} := \varphi_\ell(c_j \Delta t A)$. The coefficients are collected in tableau form, see Table \[tab:butcher\_expRK\].
---------- ------------ ---------- -------------- ------- --
$0$ $$&$$
$c_2$ $a_{2, 1}$ $$
$\vdots$ $\vdots$ $\ddots$ $$
$c_s$ $a_{s1}$ $\cdots$ $a_{s, s-1}$
$b_1$ $\cdots$ $b_{s-1}$ $b_s$
---------- ------------ ---------- -------------- ------- --
: Butcher tableau of a general exponential integrators[]{data-label="tab:butcher_expRK"}
The Butcher tableaus for the exponential integrators used in the main text are given in Tables \[butcherexprk22\], \[butcherK\], \[butcherHO\] and \[butcherCM\].
----- ------------------------- -------------
$0$
$1$ $\varphi_{1,2}$
$\varphi_1 - \varphi_2$ $\varphi_2$
----- ------------------------- -------------
: Butcher tableau of ExpRK22.[]{data-label="butcherexprk22"}
--------------- ------------------------------------------ ------------------------- ------------------------- -------------------------
$0$
$\frac{1}{2}$ $\frac{1}{2}\varphi_{1,2}$
$\frac{1}{2}$ $\frac{1}{2}\varphi_{1,3}-\varphi_{2,3}$ $\varphi_{2,3}$
$1$ $\varphi_{1,4}-2\varphi_{2,4}$ $0$ $2\varphi_{2,4}$
$\varphi_1-3\varphi_2+4\varphi_3$ $2\varphi_2-4\varphi_3$ $2\varphi_2-4\varphi_3$ $-\varphi_2+4\varphi_3$
--------------- ------------------------------------------ ------------------------- ------------------------- -------------------------
: Butcher tableau of the Krogstad method.[]{data-label="butcherK"}
--------------- --------------------------------------------- ----------------- ----------------- -------------------------------------- -------------------------
$0$
$\frac{1}{2}$ $\frac{1}{2}\varphi_{1,2}$
$\frac{1}{2}$ $\frac{1}{2}\varphi_{1,3}-\varphi_{2,3}$ $\varphi_{2,3}$
$1$ $\varphi_{1,4}-2\varphi_{2,4}$ $\varphi_{2,4}$ $\varphi_{2,4}$
$\frac{1}{2}$ $\frac{1}{2}\varphi_{1,5}-2a_{5,2}-a_{5,4}$ $a_{5,2}$ $a_{5,2}$ $\frac{1}{4}\varphi_{2,5} - a_{5,2}$
$\varphi_1-3\varphi_2+4\varphi_3$ $0$ $0$ $-\varphi_2+4\varphi_3$ $2\varphi_2-8\varphi_3$
--------------- --------------------------------------------- ----------------- ----------------- -------------------------------------- -------------------------
: Butcher tableau of the Hochbruck–Ostermann method.[]{data-label="butcherHO"}
$$\begin{aligned} a_{5,2} &= \frac{1}{2}\varphi_{2,5}-\varphi_{3,4}+\frac{1}{4}\varphi_{2,4}-\frac{1}{2}\varphi_{3,5} \\ a_{5,4} &= \frac{1}{4}\varphi_{2,5}-a_{5,2} \end{aligned}$$
--------------- --------------------------------------------- ---------------------------- ------------------------- ------------------------
$0$
$\frac{1}{2}$ $\frac{1}{2}\varphi_{1,2}$
$\frac{1}{2}$ $0$ $\frac{1}{2}\varphi_{1,3}$
$1$ $\frac{1}{2}\varphi_{1,3}(\varphi_{0,3}-1)$ $0$ $\varphi_{1,3}$
$\varphi_1-3\varphi_2+4\phi_3$ $2\varphi_2-4\varphi_3$ $2\varphi_2-4\varphi_3$ $4\varphi_3-\varphi_2$
--------------- --------------------------------------------- ---------------------------- ------------------------- ------------------------
: Butcher tableau of the Cox–Matthews method.[]{data-label="butcherCM"}
WENO5 scheme {#app_weno}
------------
The different ingredients of the WENO5 scheme used in are detailed here. First the fluxes are given by $$\begin{aligned}
{f}_{j+\frac{1}{2}}^+ =\ & w_0^+\left( \frac{2}{6}f_{j-2} - \frac{7}{6}f_{j-1} + \frac{11}{6}f_{j} \right)
+ w_1^+\left( -\frac{1}{6}f_{j-1} + \frac{5}{6}f_{j} + \frac{2}{6}f_{j+1} \right) \\
+ & w_2^+\left( \frac{2}{6}f_{j} + \frac{5}{6}f_{j+1} - \frac{1}{6}f_{j+2} \right)
\end{aligned}$$ and $$\begin{aligned}
{f}_{j+\frac{1}{2}}^- =\ & w_2^-\left( -\frac{1}{6}f_{j-1} + \frac{5}{6}f_{j} + \frac{2}{6}f_{j+1} \right)
+ w_1^-\left( \frac{2}{6}f_{j} + \frac{5}{6}f_{j+1} - \frac{1}{6}f_{j+2} \right) \\
+ & w_0^-\left( \frac{11}{6}f_{j+1} - \frac{7}{6}f_{j+2} + \frac{2}{6}f_{j+3} \right).
\end{aligned}$$ The weights are defined through the $\beta$ coefficients $$\begin{aligned}
\beta_0^+ &= \frac{13}{12}(\underbrace{f^+_{j-2} - 2f^+_{j-1} + f^+_{j} }_{\Delta x^2(f''_j + \mathcal{O}(\Delta x))}))^2 + \frac{1}{4}( \underbrace{f^+_{j-2} - 4f^+_{j-1} + 3f^+_{j}}_{2\Delta f'_j + \mathcal{O}(\Delta x^2))} )^2 \\
\beta_1^+ &= \frac{13}{12}( \underbrace{f^+_{j-1} - 2f^+_{j} + f^+_{j+1}}_{\Delta x^2(f''_j + \mathcal{O}(\Delta x^2))} )^2 + \frac{1}{4}( \underbrace{f^+_{j-1} - f^+_{j+1}}_{2\Delta x f'_j + \mathcal{O}(\Delta x^2))})^2 \\
\beta_2^+ &= \frac{13}{12}( \underbrace{f^+_{j} - 2f^+_{j+1} + f^+_{j+2}}_{\Delta x^2(f''_j + \mathcal{O}(\Delta x))} )^2 + \frac{1}{4}(\underbrace{3f^+_{j} - 4f^+_{j+1} + f^+_{j+2}}_{-2\Delta f'_j + \mathcal{O}(\Delta x^2))})^2 \\
\end{aligned}$$ with $$\begin{aligned}
\beta_0^- &= \frac{13}{12}(f^-_{j+1} - 2f^-_{j+2} + f^-_{j+3})^2 + \frac{1}{4}(3f^-_{j+1} - 4f^-_{j+2} + f^-_{j+3})^2 \\
\beta_1^- &= \frac{13}{12}(f^-_{j} - 2f^-_{j+1} + f^-_{j+2})^2 + \frac{1}{4}( f^-_{j} - f^-_{j+2})^2 \\
\beta_2^- &= \frac{13}{12}(f^-_{j-1} - 2f^-_{j} + f^-_{j+1})^2 + \frac{1}{4}( f^-_{j-1} - 4f^-_{j} + 3f^-_{j+1})^2 \\
\end{aligned}$$ Then, the normalized weights are $$\alpha_i^\pm = \frac{\gamma_i}{(\varepsilon + \beta_i^\pm)^2},\quad i=0,1,2,$$ where $\varepsilon$ is a numerical regularization parameter set to $10^{-6}$ and $\gamma_0=\frac{1}{10}$, $\gamma_1=\frac{6}{10}$ and $\gamma_2=\frac{3}{10}$. Finally the weights are given by $$w_i^\pm = \frac{\alpha_i^\pm}{\sum_m \alpha_m^\pm},\quad i=0,1,2.$$
|
---
abstract: 'We consider critical points $u:\Omega\to N$ of the bi-energy $$\int_\Omega |\Delta u|^2\,\d x,$$ where $\Omega\subset\R^m$ is a bounded smooth domain of dimension $m\ge 5$ and $N\subset\R^L$ a compact submanifold without boundary. More precisely, we consider variationally biharmonic maps $u\in W^{2,2}(\Omega,N)$, which are defined as critical points of the bi-energy that satisfy a certain stationarity condition up to the boundary. For weakly convergent sequences of variationally biharmonic maps, we demonstrate that the only obstruction that can prevent the strong compactness up to the boundary is the presence of certain non-constant biharmonic $4$-spheres or $4$-halfspheres in the target manifold. As an application, we deduce full boundary regularity of variationally biharmonic maps provided such spheres do not exist.'
address:
- |
Serdar Altuntas\
Fakultät für Mathematik, Universität Duisburg-Essen\
Thea-Leymann-Straße 9\
45127 Essen, Germany
- |
Christoph Scheven\
Fakultät für Mathematik, Universität Duisburg-Essen\
Thea-Leymann-Straße 9\
45127 Essen, Germany
author:
- Serdar Altuntas
- Christoph Scheven
title: 'Blow-up analysis and boundary regularity for variationally biharmonic maps'
---
Introduction and statement of the results
=========================================
Biharmonic maps are a higher order variant of harmonic maps $u\in C^\infty(\Omega,N)$ into a Riemannian manifold $N\subset\R^L$, which are defined as critical points of the Dirichlet energy $$E_1(u):=\int_\Omega |Du|^2\d x.$$ Analogously, we call a map $u\in C^\infty(\Omega,N)$ biharmonic if it is a critical point of the bi-energy $$E_2(u):=\int_\Omega |\Delta u|^2\d x.$$ More generally, maps $u\in W^{2,2}(\Omega,N)$ that satisfy the Euler-Lagrange equation of $E_2$ in the weak sense are called weakly biharmonic, and the class of weakly harmonic maps is defined accordingly. The analytical and geometric properties of harmonic maps have been extensively studied over the last decades and are quite well understood. We refer to [@TwoReports] for an overview over the classical theory. The theory of biharmonic maps, however, is not yet developed up to the same level as the one of harmonic maps. In the present article, we analyse the behaviour of biharmonic maps at the boundary and investigate the questions of compactness properties and regularity up to the boundary.
Before discussing the state of the regularity theory for biharmonic maps, let us briefly recall some of the main results on regularity of harmonic maps. For minimizing harmonic maps, i.e. minimizers of the Dirichlet energy in a given Dirichlet class, Schoen & Uhlenbeck proved that the singular set can have at most Hausdorff-dimension $m-3$, see [@Schoen-U]. An alternative proof was later given by Luckhaus [@Luckhaus]. Moreover, in [@Schoen-U2], Schoen & Uhlenbeck were even able to prove full regularity in a neighbourhood of the boundary. For harmonic maps that are not minimizing, only slightly weaker results are known. First of all, no regularity results can be derived in super-critical dimensions $m>2$ for weakly harmonic maps that do not satisfy a certain energy monotonicity formula, cf. [@Riviere-discont]. For the slightly smaller class of stationary harmonic maps, however, Bethuel [@Bethuel] established that the singular set has vanishing $(m-2)$-dimensional Hausdorff measure, see also [@Riviere; @Partial-Riviere] for an alternative proof. The reduction of the dimension to the upper bound $m-3$ as in the case of minimizers is not known in the general situation. On a technical level, the reason is that weakly convergent sequences of stationary harmonic maps may not have a strongly convergent subsequence [@Lin Example 1.1], differently from the case of minimizing harmonic maps [@Schoen-U; @Luckhaus]. Therefore, it is not possible to derive the dimension bound for the singular set by means of Federer’s dimension reduction principle. However, a deep result by Lin [@Lin] states that this lack of compactness can occur if and only if the target manifold contains a non-constant smooth harmonic $2$-sphere $v:S^2\to N$. As a consequence, under the assumption that the target manifold does not carry any non-trivial harmonic $2$-spheres, it is possible to prove full regularity in the neighbourhood of the boundary also for more general critical points and not only for minimizers [@Lin; @Scheven0].
The regularity of biharmonic maps was first investigated by Chang, Wang & Yang [@CWY], see also [@Wang_sphere; @Wang4d; @Wang], with the result that for any stationary biharmonic map, the set of interior singular points is negligible with respect to the $(m-4)$-dimensional Hausdorff measure. In the case of minimizing biharmonic maps, the dimension of the interior singular set was further reduced to at most $m-5$ by the second author [@Scheven1]. The latter article also contains results for stationary biharmonic maps under the assumption that the target manifold does not carry any non-constant Paneitz-biharmonic $4$-spheres (cf. Definition \[def:paneitz-biharmonic\]), which turns out to be the analogue of the condition found by Lin [@Lin] in the harmonic map case. As for harmonic maps, an indispensable tool for all mentioned partial regularity results in super-critical dimensions $m>4$ is an energy monotonicity formula. For biharmonic maps, this formula was derived in the interior case by Angelsberg [@Angelsberg]. In the boundary situation, however, the question for the corresponding monotonicity formula remained open for some time. In fact, since such a formula was unknown, the first results on partial boundary regularity [@GongLammWang] and full boundary regularity for minimizers [@Mazowiecka] had to impose this monotonicity property as an additional assumption. This gap in the theory has been closed by the first author [@Altuntas], who provided a suitable boundary monotonicity formula and thereby completed the mentioned results from [@GongLammWang; @Mazowiecka].
The present article now is concerned with the question whether full boundary regularity can also be derived for biharmonic maps that are not minimizing, but only critical points of the bi-energy. The suitable notion of critical point in the boundary situation is that of a variationally biharmonic map, see Definition \[def:variationally-bi\]. This notion, which is slightly stronger than that of a stationary biharmonic map, has been introduced in [@Scheven0] in the harmonic map case and allows in particular to use any variation of the domain that keeps the boundary values fixed. Our first main result is a compactness property for sequences of variationally biharmonic maps. For the proof, we adapt the strategy from [@Scheven1], which are in turn based on [@Lin], to the boundary case. The arguments consist in an intricate blow-up analysis of the defect measure, which detects a possible lack of strong convergence. We achieve basically the analogous result as in the harmonic map case, with the only exception that additionally to the non-existence of non-trivial Paneitz-biharmonic $4$-spheres, we also need to exclude the existence of non-constant Paneitz-biharmonic $4$-halfspheres with constant boundary values. The reason is that the non-existence proof of harmonic $2$-halfspheres from [@Lemaire] does not seem to carry over to the higher order case. However, it seems to be plausible that whenever it is possible to exclude nontrivial Paneitz-biharmonic $4$-spheres, the same arguments will also yield the nonexistence of the corresponding halfspheres. An example of this principle is given in Proposition \[prop:flat-torus\].
This compactness property is the prerequisite for our second main result, which ensures the full boundary regularity of variationally biharmonic maps under the assumption that no biharmonic $4$-spheres and $4$-halfspheres as above exist in the target manifold. For the proof, we follow Federer’s dimension reduction argument and analyse tangent maps of variationally biharmonic maps in singular boundary points. For the construction of the tangent maps, it is crucial to have the strong convergence properties from our first main result. Since it is possible to show that all tangent maps necessarily have to be constant, we can deduce that singular boundary points do not exist.
Next, we specify our assumptions and state our main results.
Variationally biharmonic maps
-----------------------------
Let $\Omega\subset\R^m$ be a bounded domain of dimension $m\ge5$. We prescribe Dirichlet boundary data on a boundary part $\Gamma\subset\partial\Omega$, where the boundary datum is given in form of a map $g\in C^3(\Gamma_\delta,N)$ defined on a neighborhood $\Gamma_\delta:=\{x\in\overline\Omega\,:\, \mathrm{dist}(x,\Gamma)<\delta\}$ of $\Gamma$. We consider critical points of the bi-energy $$E_2(u):=\int_\Omega|\Delta u|^2\d x
\qquad\mbox{for }u\in W^{2,2}(\Omega,N).$$
The following notion of weakly biharmonic maps can be derived by considering variations of the type $u_s(x):=\pi_N(u(x)+s V(x))$ with the nearest-point retraction $\pi_N$ onto $N$.
A map $u\in W^{2,2}(\Omega,N)$ is called weakly biharmonic iff $$\int_\Omega \Delta u\cdot \Delta V\,d x=0$$ holds true for every vector field $V\in W^{2,2}_0\cap L^\infty(\Omega,\R^L)$ that is tangential along $u$ in the sense $V(x)\in T_{u(x)}N$ for a.e. $x\in\Omega$.
In other words, a weakly biharmonic map is characterized by the fact that $\Delta^2u\perp T_uN$ holds in the weak sense. This is equivalent to the differential equation $$\label{weakly-bi}
\Delta^2u
=
\Delta \big(A(u)(Du\otimes Du)\big)
+2\mathrm{div}\,\big(D(\Pi(u))\cdot\Delta u\big)-\Delta(\Pi(u))\cdot\Delta u$$ in the distributional sense, where $A(u)$ denotes the second fundamental form of $N\subset\R^L$ and $\Pi(u(x)):\R^L\to T_{u(x)}N$ is the orthogonal projection onto the tangent space at $u(x)\in N$. For a detailed proof, we refer to [@Wang Prop. 2.2]. Classical solutions $u\in C^4(\Omega,N)$ of are called biharmonic maps.
As mentioned above, in super-critical dimensions $m\ge 5$ a monotonicity formula is crucial for the derivation of regularity results. Since such a formula can not be expected to hold for general weakly biharmonic maps, we have to consider a stronger notion of biharmonicity. Considering variations of the type $u_s(x)=u(x+s\xi(x))$ leads to the following notion of biharmonic maps.
\[def:variationally-bi\] A map $u\in W^{2,2}(\Omega,N)$ is called *stationary biharmonic* iff it is weakly biharmonic in $\Omega$ and satisfies the differential equation $$\begin{aligned}
\label{stationary-bi-interior}
&\int_\Omega \big(4\Delta u\cdot D^2uD\xi+2\Delta u\cdot
Du\,\Delta\xi-|\Delta u|^2\mathrm{div}\,\xi\big)\,\d x=0
\end{aligned}$$ for every $\xi\in C^\infty_0(\Omega,\R^m)$.
However, it turns out that this notion is still not sufficient for the treatment of the Dirichlet problem, since the differential equation only contains information on interior properties of solutions. Therefore, we rely on the following notion of biharmonic maps that is adapted to the Dirichlet boundary problem.
A map $u\in W^{2,2}(\Omega,N)$ is called *variationally biharmonic* with respect to the Dirichlet datum $g$ on $\Gamma$ if $$\label{Dirichlet-Gamma}
(u,Du)=(g,Dg)\qquad\mbox{on $\Gamma$ in the sense of traces,}$$ and if $$\frac{d}{ds}\bigg|_{s=0}E_2(u_s)=0$$ holds true for every variation $u_s\in W^{2,2}(\Omega,N)$, $s\in(-\eps,\eps)$, for which the above derivative exists, which satisfies $u_0=u$, the boundary condition with $u_s$ in place of $u$, and $u_s=u$ a.e. on $\Omega\setminus K$ for some compact set $K\subset\Omega\cup\Gamma$.
### Paneitz-biharmonic maps
For maps on the $k$-dimensional upper halfsphere $S^k_+$, there are corresponding notions of biharmonicity. For the purposes of the present article, it suffices to consider the case $k=4$. We prescribe boundary values in form of a map $g\in
C^3(U_\delta,N)$, where we abbreviated $U_\delta:=S^{4}_+\cap(\R^4\times[0,\delta))$. For maps defined on the $4$-sphere $S^4$, we define the Paneitz-bi-energy by $$P_{S^4}(u):=\int_{S^4}\big[|\Delta_S u|^2+2|Du|^2\big]\d x
\qquad\mbox{for }u\in W^{2,2}(S^k,N),$$ with the Laplace-Beltrami operator $\Delta_S$ on $S^4$. Analogously, we define $P_{S^4_+}(u)$ for any $u\in W^{2,2}(S^4_+,N)$. The Euler-Lagrange operator of this functional is given by the Paneitz-operator $\mathcal{P}u:=\Delta_S^2u-2\Delta_S u$ on $S^4$, which plays an important role in comformal geometry. In particular, well-known properties of the Paneitz operator imply that the Paneitz-bi-energy $P_{S^4}$ is conformally invariant, cf. [@Paneitz; @Chang]. Critical points of $P_{S^k}$ are called Paneitz-biharmonic maps in the following sense.
\[def:paneitz-biharmonic\] A map $u\in C^4(S^4,N)$ is called *Paneitz-biharmonic* if $$\label{paneitz-biharmonic}
\Delta_S^2u(x)-2\Delta_Su(x)\perp T_{u(x)}N$$ holds true for any $x\in S^4$. Analogously, a map $u\in C^4(S^4_+,N)$ is called *Paneitz-biharmonic* with Dirichlet datum $g$ if $(u,Du)=(g,Dg)$ on $\partial S^4_+$ and holds true for any $x\in S^4_+$.
We say that the Riemannian manifold $N$ does not carry any non-constant Paneitz-biharmonic $4$-spheres if every Paneitz-biharmonic map $u\in C^4(S^4,N)$ is constant.
Analogously, we say that $N$ does not carry any non-constant Paneitz-biharmonic $4$-halfspheres with constant boundary values if every Paneitz-biharmonic map $u\in C^4(S^4_+,N)$ with $(u,Du)=(c,0)$ on $\partial S^4_+$ for some constant $c\in N$ satisfies $u=c$ on $S^4_+$.
Statement of the results
------------------------
Now we are in a position to state our main results. In all statements, we restrict ourselves to the case of a flat boundary, i.e. to the case that $\Omega$ is a half ball $B_R^+$, with boundary values prescribed on the flat part of the boundary, which we denote by $T_R$. The general case of a smooth boundary can be reduced to this case by flattening the boundary. However, this procedure will change the Euclidean metric to a more general Riemannian one. Nevertheless, we decided to treat only the model case of the Euclidean metric in order not to overburden this work with additional technicalities.
Our first main result is the following compactness property for bounded sequences of variationally biharmonic maps.
\[thm:compact\] Let $N\subset\R^L$ be a compact, smooth Riemannian manifold that does neither carry non-constant Paneitz-biharmonic $4$-spheres nor non-constant Paneitz-biharmonic $4$-halfspheres with constant boundary values. Assume that $g_i\in C^\infty(B_4^+,N)$, $i\in\N$, is a sequence of boundary values and $u_i\in W^{2,2}(B_4^+,N)$ are variationally biharmonic maps with respect to the Dirichlet data $g_i$ on $T_4$, and that both sequences are bounded in the sense $$\sup_{i\in\N}\,\|u_i\|_{W^{2,2}(B_4^+)}<\infty
\qquad\mbox{and}\qquad
\sup_{i\in\N}\,\|g_i\|_{C^{4,\alpha}(B_4^+)}<\infty,$$ for some $\alpha\in(0,1)$. Then there is a map $u\in W^{2,2}(B_1^+,N)$ so that after passing to a subsequence, we have the convergence $$u_{i}\to u \qquad\mbox{in $W^{2,2}(B_{1}^+,\R^L)$, as $i\to\infty$.}$$
The preceding compactness result is the crucial step for the derivation of the full boundary regularity for variationally biharmonic maps.
\[thm:boundary-regularity\] Let $N\subset\R^L$ be a smooth compact Riemannian manifold that does neither carry a non-constant Paneitz-biharmonic $4$-sphere nor a non-constant Paneitz-biharmonic $4$-halfsphere with constant boundary values. Assume that the map $u\in W^{2,2}(B_1^+,N)$ is variationally biharmonic with $(u,Du)=(g,Dg)$ on $T_1$ in the sense of traces, for Dirichlet values $g\in
C^\infty(B_1^+,N)$. Then $u$ is smooth in a full neighbourhood of $T_1$.
Plan of the paper
-----------------
In Section \[sec:preliminaries\], we gather some technical tools that will be crucial for our arguments. In particular, we derive a Morrey space estimate that is a consequence of a boundary monotonicity formula, and we recall some partial regularity results for variationally biharmonic maps. Moreover, we prove a gradient estimate under a smallness assumption on the energy.
Section \[sec:compact\] is then devoted to the proof of our compactness result. We introduce the notion of defect measure for a sequence of variationally biharmonic maps and analyse in what sense this measure detects the lack of strong convergence. We show that if a nontrivial defect measure exists, then a blow-up procedure yields a flat defect measure that is supported on an $(m-4)$-dimensional plane. This means that we can find a sequence of variationally biharmonic maps that converge strongly away from this plane. In this more controlled situation, it is then possible to follow the ideas by Lin [@Lin] and to construct a suitable blow-up sequence around carefully chosen blow-up points. Depending on whether these points approach the boundary or not, we can show that the limit gives rise to a non-constant Paneitz-biharmonic $4$-halfsphere with constant boundary values or to a corresponding full $4$-sphere. Since the existence of such maps is excluded by assumption, we deduce the desired strong compactness for any bounded sequence of variationally biharmonic maps.
The next Section \[sec:liouville\] contains some Liouville type results for biharmonic maps on half-spaces that arise as tangent maps of biharmonic maps in boundary points. Since we can show that all possible tangent maps are constant, the implementation of Federer’s dimension reduction principle in Section \[sec:dim-red\] allows us to deduce our second main result on the full boundary regularity of variationally biharmonic maps.\
**Acknowledgments.** This work has been supported by the DFG-project SCHE 1949/1-1 “Randregularität biharmonischer Abbildungen zwischen Riemann’schen Mannigfaltigkeiten”.
Preliminaries {#sec:preliminaries}
=============
Notation
--------
We write $B_r(a)\subset\R^m$ for the open ball with radius $r>0$ and center $a\in\R^m$ and $S_r(a):=\partial B_r(a)$ for the corresponding sphere. For the upper halfspace, we use the abbreviation $\R^m_+:=\R^{m-1}\times[0,\infty)$, and write $B_r^+(a):=B_r(a)\cap\R^m_+$ for arbitrary centers $a\in\R^m_+$. Moreover, in the case of a center $a=(a',0)\in\partial\R^m_+$ we use the abbreviations $S_r^+(a):=S_r(a)\cap\R^m_+$ for the curved part and $T_r(a):=B_r(a')\times\{0\}$ for the flat part of the boundary of $B_r^+(a)$. If the center is clear from the context, we will often omit it in the notation and simply write $B_r$, $B_r^+$, $S_r$, $S_r^+$, $T_r$ instead of $B_r(a)$, $B_r^+(a)$, $S_r(a)$, $S_r^+(a)$, $T_r(a)$.
For the Lebesgue measure on $\R^m$ we write $\mathcal{L}^m$, and for $0\le k\le m$, the $k$-dimensional Hausdorff measure on $\R^m$ will be abbreviated by $\mathcal{H}^k$.
For $f\in L^2(B_2^+)$ and $\lambda>0$, we write $$\|f\|_{L^{2,\lambda}(B_2^+)}^2
:=
\sup_{B_\rho^+(y)\subset
B_2^+}\frac1{\rho^\lambda}\int_{B_\rho^+(y)}|f|^2\,\d x.$$
Finally, the singular set of a map $u:\Omega\to\R^L$ is defined by $$\operatorname{sing}(u):=\overline\Omega\setminus \left\{x\in\overline\Omega\,:\, u\in
C^\infty(B_\rho(x)\cap\overline\Omega,\R^L)\mbox{ for some }\rho>0 \right\}.$$
Monotonicity formula and consequences
-------------------------------------
The proof of the following lemma can be retrieved from [@Altuntas Lemma 2.1].
Let $u\in W^{2,2}(\Omega,N)$ be a variationally biharmonic map with respect to the Dirichlet datum $g$ on $\Gamma\subset\partial\Omega$. Then for every $\xi\in
C^\infty(\Omega\cup\Gamma,\R^m)$ with $\xi(x)\in T_x(\partial\Omega)$ for every $x\in\Gamma$ and $\operatorname{spt}\xi\Subset\Omega\cup\Gamma$, we have $$\begin{aligned}
\label{stationary-bi}
&\int_\Omega \big(4\Delta u\cdot D^2uD\xi+2\Delta u\cdot
Du\,\Delta\xi-|\Delta u|^2\mathrm{div}\,\xi\big)\,\d x\\\nonumber
&\quad=
\int_\Omega 2\Delta u\cdot \Delta \big[\Pi(u)(Dg\,\xi)\big]\,\d x,
\end{aligned}$$ with the orthogonal projection $\Pi(u(x)):\R^L\to T_{u(x)}N$.
The preceding lemma is the first step in the derivation of a boundary monotonicity formula, which has been proven in [@Altuntas] for the case of a flat boundary. More precisely, we specialize to the case $\Omega=B_R^+(a)$ and $\Gamma=T_R(a)$.
[([@Altuntas Theorem 1.2])]{}\[thm:monotonicity\] Let $m\ge 5$, $0<R\le1$ and assume that $u\in W^{2,2}(B_R^+(a),N)$ satisfies . Then, for a.e. radii $0<\rho<r<R$, we have the monotonicity formula $$\begin{aligned}
\label{bdry-monotonicity}
&\Phi_u(a;\rho)
+
4\int_{B_r^+(a)\setminus B_\rho^+(a)}e^{\chi|x-a|}\bigg(\frac{|D\partial_Xu|^2}{|x-a|^{m-2}}+(m-2)\frac{|\partial_Xu|^2}{|x-a|^m}\bigg)\,\d x\\\nonumber
&\quad\le
\Phi_u(a;r)
+
K\Psi_u(a;\rho,r),
\end{aligned}$$ where we abbreviated $$\begin{aligned}
\label{Def-Phi}
\Phi_u(a;r)&:=e^{\chi r}r^{4-m}\int_{B_r^+(a)}|\Delta u|^2\,\d x\\\nonumber
&\qquad+ e^{\chi
r}r^{3-m}\int_{S_r^+(a)}\big(\partial_X|Du|^2+4|Du|^2-4r^{-2}|\partial_Xu|^2\big)\,d\mathcal{H}^{m-1},
\end{aligned}$$ with the short-hand notation $\partial_X:=(x_i-a_i)\partial_i$, and $$\begin{aligned}
\label{Def-Psi}
\Psi_u(a;\rho,r)
&:=r+\int_{B_r^+(a)\setminus
B_\rho^+(a)}\bigg(\frac{|D^2u|^2}{|x-a|^{m-5}}+\frac{|Du|^2}{|x-a|^{m-3}}\bigg)\,\d x\\\nonumber
&\qquad+
\int_{S_r^+(a)\cup S_\rho^+(a)}\frac{|D^2u|^2}{|x-a|^{m-6}}\,d\mathcal{H}^{m-1}.
\end{aligned}$$ In the above formula, the constants $\chi,K\ge0$ depend only on the data $m,N$ and $\|Dg\|_{C^2}$. In particular, the constants $\chi$ and $K$ vanish in the limit $\|Dg\|_{C^2}\to0$.
An important consequence of the preceding monotonicity formula is the following Morrey space estimate.
\[lemma:morrey\] Let $m\ge 5$ and $g\in C^3(B_1^+,N)$ be given. Assume that $u\in W^{2,2}(B_1^+,N)$ satisfies and that $(u,Du)=(g,Dg)$ holds on $T_1$, in the sense of traces. Then for any ball $B_R^+(a)\subset B_1^+$ with radius $R\in(0,1]$ we have $$\begin{aligned}
\sup_{B_\rho^+(y)\subset B_{R/2}^+(a)}\,&
\rho^{4-m}\int_{B_\rho^+(y)}\big(|D^2u|^2+\rho^{-2}|Du|^2\big)\,\d x\\
&\le
c_1R^{4-m}\int_{B_R^+(a)}\big(|\Delta u|^2+R^{-2}|Du|^2\big)\,\d x
+c_2R,
\end{aligned}$$ with constants $c_1$, $c_2$ that depend at most on $m,N,$ and $\|Dg\|_{C^2}$. Moreover, we have $c_2\to0$ in the limit $\|Dg\|_{C^2}\to0$.
For the proof, we modify some ideas from [@Struwe] and [@Moser]. Throughout this proof, we write $c$ for a constant that may depend on $m$, $N$, and $\|Dg\|_{C^2}$. We first consider the case $a\in T_1$. For $0<s<\frac{R}{8}$, we consider radii $\rho\in[2s,4s]$ and $r\in[\frac R2,R]$ that will be chosen later. Our goal is to derive an estimate for the term $$\widetilde\Phi_u(a;\rho)
:=
e^{\chi\rho}\rho^{4-m}\int_{B_\rho^+(a)}|\Delta u|^2\,\d x
+
e^{\chi\rho}\rho^{3-m}\int_{S_\rho^+(a)}|Du|^2\,\d \HM^{m-1},$$ where $\chi\ge0$ is the constant from Theorem \[thm:monotonicity\], which depends only on $m,N,$ and $\|Dg\|_{C^2}$. For the difference of this term and the term $\Phi_u(a;\rho)$ defined in , we compute, using again the abbreviation $X(x):=x-a$, $$\begin{aligned}
&\widetilde\Phi_u(a;\rho)-\Phi_u(a;\rho)\\
&\qquad=
e^{\chi\rho}\rho^{3-m}
\int_{S_\rho^+(a)}\big(4\rho^{-2}|\partial_Xu|^2
-\partial_X|Du|^2-3|Du|^2\big)\d\mathcal{H}^{m-1} \\
&\qquad=
e^{\chi\rho}\rho^{3-m}
\int_{S_\rho^+(a)}\big(4\rho^{-2}|\partial_Xu|^2
-2D(\partial_Xu)\cdot Du-|Du|^2\big)\d\mathcal{H}^{m-1} \\
&\qquad\le
e^{\chi\rho}\rho^{3-m}
\int_{S_\rho^+(a)}\big(4\rho^{-2}|\partial_Xu|^2+|D\partial_Xu|^2\big)\d\mathcal{H}^{m-1},
\end{aligned}$$ where we applied Young’s inequality in the last step. Taking the mean integral over $\rho\in[2s,4s]$, we deduce $$\begin{aligned}
&{\mathchoice
{{{\setbox0=\hbox{$\displaystyle{\textstyle-}{\int}$}
\vcenter{\hbox{$\textstyle-$}}\kern-.5\wd0}}}
{{{\setbox0=\hbox{$\textstyle{\scriptstyle-}{\int}$}
\vcenter{\hbox{$\scriptstyle-$}}\kern-.5\wd0}}}
{{{\setbox0=\hbox{$\scriptstyle{\scriptscriptstyle-}{\int}$}
\vcenter{\hbox{$\scriptscriptstyle-$}}\kern-.5\wd0}}}
{{{\setbox0=\hbox{$\scriptscriptstyle{\scriptscriptstyle-}{\int}$}
\vcenter{\hbox{$\scriptscriptstyle-$}}\kern-.5\wd0}}}
\!\int}_{2s}^{4s}\widetilde\Phi_u(a;\rho)\d\rho\\
&\ \le
{\mathchoice
{{{\setbox0=\hbox{$\displaystyle{\textstyle-}{\int}$}
\vcenter{\hbox{$\textstyle-$}}\kern-.5\wd0}}}
{{{\setbox0=\hbox{$\textstyle{\scriptstyle-}{\int}$}
\vcenter{\hbox{$\scriptstyle-$}}\kern-.5\wd0}}}
{{{\setbox0=\hbox{$\scriptstyle{\scriptscriptstyle-}{\int}$}
\vcenter{\hbox{$\scriptscriptstyle-$}}\kern-.5\wd0}}}
{{{\setbox0=\hbox{$\scriptscriptstyle{\scriptscriptstyle-}{\int}$}
\vcenter{\hbox{$\scriptscriptstyle-$}}\kern-.5\wd0}}}
\!\int}_{2s}^{4s}\Phi_u(a;\rho)\d\rho\\
&\quad\ +
c\int_{B_{4s}^+(a)\setminus
B_{2s}^+(a)}e^{\chi|x-a|}\bigg(\frac{|D\partial_Xu|^2}{|x-a|^{m-2}}+(m-2)\frac{|\partial_Xu|^2}{|x-a|^m}\bigg)\d
x\\
&\ \le
{\mathchoice
{{{\setbox0=\hbox{$\displaystyle{\textstyle-}{\int}$}
\vcenter{\hbox{$\textstyle-$}}\kern-.5\wd0}}}
{{{\setbox0=\hbox{$\textstyle{\scriptstyle-}{\int}$}
\vcenter{\hbox{$\scriptstyle-$}}\kern-.5\wd0}}}
{{{\setbox0=\hbox{$\scriptstyle{\scriptscriptstyle-}{\int}$}
\vcenter{\hbox{$\scriptscriptstyle-$}}\kern-.5\wd0}}}
{{{\setbox0=\hbox{$\scriptscriptstyle{\scriptscriptstyle-}{\int}$}
\vcenter{\hbox{$\scriptscriptstyle-$}}\kern-.5\wd0}}}
\!\int}_{2s}^{4s}\Phi_u(a;\rho)\d\rho\\
&\quad\ +
c\,{\mathchoice
{{{\setbox0=\hbox{$\displaystyle{\textstyle-}{\int}$}
\vcenter{\hbox{$\textstyle-$}}\kern-.5\wd0}}}
{{{\setbox0=\hbox{$\textstyle{\scriptstyle-}{\int}$}
\vcenter{\hbox{$\scriptstyle-$}}\kern-.5\wd0}}}
{{{\setbox0=\hbox{$\scriptstyle{\scriptscriptstyle-}{\int}$}
\vcenter{\hbox{$\scriptscriptstyle-$}}\kern-.5\wd0}}}
{{{\setbox0=\hbox{$\scriptscriptstyle{\scriptscriptstyle-}{\int}$}
\vcenter{\hbox{$\scriptscriptstyle-$}}\kern-.5\wd0}}}
\!\int}_{2s}^{4s}\int_{B_{r}^+(a)\setminus
B_{\rho/2}^+(a)}e^{\chi|x-a|}\bigg(\frac{|D\partial_Xu|^2}{|x-a|^{m-2}}+(m-2)\frac{|\partial_Xu|^2}{|x-a|^m}\bigg)\d
x\d \rho,
\end{aligned}$$ where we used $\frac\rho2\le 2s$ and $4s\le\frac R2\le r$ in the last step. Both terms on the right-hand side can be estimated by an application of the monotonicity formula . This leads to the estimate $$\begin{aligned}
{\mathchoice
{{{\setbox0=\hbox{$\displaystyle{\textstyle-}{\int}$}
\vcenter{\hbox{$\textstyle-$}}\kern-.5\wd0}}}
{{{\setbox0=\hbox{$\textstyle{\scriptstyle-}{\int}$}
\vcenter{\hbox{$\scriptstyle-$}}\kern-.5\wd0}}}
{{{\setbox0=\hbox{$\scriptstyle{\scriptscriptstyle-}{\int}$}
\vcenter{\hbox{$\scriptscriptstyle-$}}\kern-.5\wd0}}}
{{{\setbox0=\hbox{$\scriptscriptstyle{\scriptscriptstyle-}{\int}$}
\vcenter{\hbox{$\scriptscriptstyle-$}}\kern-.5\wd0}}}
\!\int}_{2s}^{4s}\widetilde\Phi_u(a;\rho)\d\rho
&\le
c\,\Phi_u(a;r)\\
&\quad+
cK{\mathchoice
{{{\setbox0=\hbox{$\displaystyle{\textstyle-}{\int}$}
\vcenter{\hbox{$\textstyle-$}}\kern-.5\wd0}}}
{{{\setbox0=\hbox{$\textstyle{\scriptstyle-}{\int}$}
\vcenter{\hbox{$\scriptstyle-$}}\kern-.5\wd0}}}
{{{\setbox0=\hbox{$\scriptstyle{\scriptscriptstyle-}{\int}$}
\vcenter{\hbox{$\scriptscriptstyle-$}}\kern-.5\wd0}}}
{{{\setbox0=\hbox{$\scriptscriptstyle{\scriptscriptstyle-}{\int}$}
\vcenter{\hbox{$\scriptscriptstyle-$}}\kern-.5\wd0}}}
\!\int}_{2s}^{4s}\big(\Psi_u(a;\rho,r)+\Psi_u(a;\tfrac\rho2,r)\big)\d\rho,
\end{aligned}$$ for a.e. $r\in[\frac R2,R]$, with the constant $K=K(m,N,\|Dg\|_{C^2})$ from Theorem \[thm:monotonicity\]. We recall that $K\to0$ as $\|Dg\|_{C^2}\to0$. By definition of $\Psi_u$, the last integral can be estimated by $$\begin{aligned}
&{\mathchoice
{{{\setbox0=\hbox{$\displaystyle{\textstyle-}{\int}$}
\vcenter{\hbox{$\textstyle-$}}\kern-.5\wd0}}}
{{{\setbox0=\hbox{$\textstyle{\scriptstyle-}{\int}$}
\vcenter{\hbox{$\scriptstyle-$}}\kern-.5\wd0}}}
{{{\setbox0=\hbox{$\scriptstyle{\scriptscriptstyle-}{\int}$}
\vcenter{\hbox{$\scriptscriptstyle-$}}\kern-.5\wd0}}}
{{{\setbox0=\hbox{$\scriptscriptstyle{\scriptscriptstyle-}{\int}$}
\vcenter{\hbox{$\scriptscriptstyle-$}}\kern-.5\wd0}}}
\!\int}_{2s}^{4s}\big(\Psi_u(a;\rho,r)+\Psi_u(a;\tfrac\rho2,r)\big)\d\rho\\
&\qquad\le
2r+c\int_{B_r^+(a)\setminus
B_{s}^+(a)}\bigg(\frac{|D^2u|^2}{|x-a|^{m-5}}+\frac{|Du|^2}{|x-a|^{m-3}}\bigg)\,\d x\\\nonumber
&\qquad\quad+
2\int_{S_r^+(a)}\frac{|D^2u|^2}{|x-a|^{m-6}}\,d\mathcal{H}^{m-1}.
\end{aligned}$$ Integrating by parts twice, we estimate the second last integral as follows. $$\begin{aligned}
\label{169}
&\int_{B_r^+(a)\setminus B_s^+(a)}\left(\dfrac{\vert D^2u\vert^2}{\vert
x-a\vert^{m-5}}+\dfrac{\vert Du\vert^2}{\vert
x-a\vert^{m-3}}\right)\d x\\\nonumber
&\quad=\int_{s}^r
\left(\sigma^{5-m}\int_{S_{\sigma}^+(a)}\vert D^2u\vert^2\d \HM^{m-1}
+\sigma^{3-m}\int_{S_{\sigma}^+(a)}\vert Du\vert^2\d \HM^{m-1}\right)
\d \sigma\\\nonumber
&\quad\le
r^{5-m}\int_{B_{r}^+(a)}\big(\vert
D^2u\vert^2+r^{-2}|Du|^2\big)\d x\\\nonumber
&\quad\quad+c\int_{s}^r\sigma^{4-m}\int_{B_{\sigma}^+(a)}\big(\vert D^2u\vert^2+\sigma^{-2}|Du|^2\big)\d x\d\sigma\\\nonumber
&\quad\leq
cr\sup_{\sigma\in[s,r]}\E(\sigma),\end{aligned}$$ where we introduced the notation $$\begin{aligned}
\E(\sigma)
:=
\sigma^{4-m}\int_{B_{\sigma}^+(a)}\big(\vert
D^2u\vert^2+\sigma^{-2}\vert Du\vert^2\big)\d x.\end{aligned}$$ Combining the three preceding estimates and recalling the definition of $\Phi_u$, we deduce $$\begin{aligned}
&{\mathchoice
{{{\setbox0=\hbox{$\displaystyle{\textstyle-}{\int}$}
\vcenter{\hbox{$\textstyle-$}}\kern-.5\wd0}}}
{{{\setbox0=\hbox{$\textstyle{\scriptstyle-}{\int}$}
\vcenter{\hbox{$\scriptstyle-$}}\kern-.5\wd0}}}
{{{\setbox0=\hbox{$\scriptstyle{\scriptscriptstyle-}{\int}$}
\vcenter{\hbox{$\scriptscriptstyle-$}}\kern-.5\wd0}}}
{{{\setbox0=\hbox{$\scriptscriptstyle{\scriptscriptstyle-}{\int}$}
\vcenter{\hbox{$\scriptscriptstyle-$}}\kern-.5\wd0}}}
\!\int}_{2s}^{4s}\widetilde\Phi_u(a;\rho)\d\rho\\\nonumber
&\qquad\le
c\,\Phi_u(a;r) + cr^{6-m}\int_{S_r^+(a)}|D^2u|^2\,d\mathcal{H}^{m-1}
+cKR+cR\sup_{\sigma\in[s,r]}\E(\sigma)\\\nonumber
&\qquad\le
cr^{4-m}\int_{B_r^+(a)}|\Delta u|^2\d x+cr^{5-m}\int_{S_r^+(a)}\big(|D^2u|^2+r^{-2}|Du|^2\big)\,d\mathcal{H}^{m-1}\\\nonumber
&\quad\qquad
+cKR+
cR\sup_{\sigma\in[s,r]}\E(\sigma),\nonumber\end{aligned}$$ for a.e. $r\in[\frac R2,R]$. Now we choose a good radius $r\in[\frac R2,R]$ in the sense that $$\begin{aligned}
\nonumber
r^{5-m}\int_{S_{r}^+(a)}\big(\vert D^2u\vert^2+r^{-2}\vert
Du\vert^2\big)\d\mathcal{H}^{m-1}
\leq
c\E(R).\end{aligned}$$ Moreover, this radius can be chosen in such a way that the preceding estimate is valid for this choice of $r$, which implies $$\begin{aligned}
\label{estimate-of-the-mean}
{\mathchoice
{{{\setbox0=\hbox{$\displaystyle{\textstyle-}{\int}$}
\vcenter{\hbox{$\textstyle-$}}\kern-.5\wd0}}}
{{{\setbox0=\hbox{$\textstyle{\scriptstyle-}{\int}$}
\vcenter{\hbox{$\scriptstyle-$}}\kern-.5\wd0}}}
{{{\setbox0=\hbox{$\scriptstyle{\scriptscriptstyle-}{\int}$}
\vcenter{\hbox{$\scriptscriptstyle-$}}\kern-.5\wd0}}}
{{{\setbox0=\hbox{$\scriptscriptstyle{\scriptscriptstyle-}{\int}$}
\vcenter{\hbox{$\scriptscriptstyle-$}}\kern-.5\wd0}}}
\!\int}_{2s}^{4s}\widetilde\Phi_u(a;\rho)\d\rho
\le
c\E(R) +cKR
+
cR\sup_{\sigma\in[s,R]}\E(\sigma).\end{aligned}$$ Our next aim is to estimate $\widetilde\Phi_u(a;\rho)$ from below. First, we observe that with a standard cut-off function $\eta\in
C^\infty_0(B_\rho(a))$ with $\eta\equiv1$ in $B_{\rho/2}(a)$, two integrations by parts and Young’s inequality lead to the estimate $$\begin{aligned}
&\rho^{4-m}\int_{B_{\rho/2}^+(a)}|D^2(u-g)|^2\d x
\le
\rho^{4-m}\int_{B_{\rho}^+(a)}\eta^2|D^2(u-g)|^2\d x\\
&\qquad\le
c\rho^{4-m}\int_{B_\rho^+(a)}\big(|\Delta (u-g)|^2+\rho^{-2}|D(u-g)|^2\big)\d x,\end{aligned}$$ which implies $$\begin{aligned}
\label{L2-estimate}
&\rho^{4-m}\int_{B_{\rho/2}^+(a)}|D^2u|^2\d x\\\nonumber
&\qquad\le
c\rho^{4-m}\int_{B_\rho^+(a)}\big(|\Delta
u|^2+\rho^{-2}|Du|^2\big)\d x
+c\rho^2\|Dg\|_{C^1}^2,\end{aligned}$$ where $c=c(m)$. Two applications of Gauß’ theorem yield $$\begin{aligned}
&m\int_{B_\rho^+(a)}|Du|^2\d x\\
&\quad=
\int_{B_\rho^+(a)}\mathrm{div}\big((x-a)|Du|^2\big)\d x
-
2\int_{B_\rho^+(a)}\partial_XDu\cdot Du\,\d x\\
&\quad=
\rho\int_{S_\rho^+(a)}|Du|^2\d\mathcal{H}^{m-1}
+
2\int_{B_\rho^+(a)}\big(\partial_Xu\cdot \Delta
u+|Du|^2\big)\d x\\
&\quad\qquad
-2\rho^{-1}\int_{S_\rho^+(a)}|\partial_Xu|^2\,\d\mathcal{H}^{m-1}
+2\int_{T_\rho(a)}\partial_Xg\cdot\partial_mg\,\d x\\
&\quad\le
\rho\int_{S_\rho^+(a)}|Du|^2\d\mathcal{H}^{m-1}\\
&\quad\qquad+
\int_{B_\rho^+(a)}\big(\rho^2|\Delta u|^2+3|D u|^2\big)\d x
+c\rho^{m}\|Dg\|_{C^0}^2. \end{aligned}$$ Since $m>3$, we can re-absorb the integral of $|Du|^2$ into the left-hand side. Multiplying the resulting estimate by $\rho^{2-m}$, we infer $$\begin{aligned}
\label{Du-estimate}
&\rho^{2-m}\int_{B_\rho^+(a)}|Du|^2\d x\\\nonumber
&\quad\le
c\rho^{3-m}\int_{S_\rho^+(a)}|Du|^2\d\mathcal{H}^{m-1}
+
c\rho^{4-m}\int_{B_\rho^+(a)}|\Delta u|^2\d x
+c\rho^2\|Dg\|_{C^0}^2\\\nonumber
&\quad\le
c\,\widetilde\Phi_u(a;\rho) +c\rho^2\|Dg\|_{C^0}^2.\end{aligned}$$ Combining estimates and , we deduce $$\begin{aligned}
&\rho^{4-m}\int_{B_{\rho/2}^+(a)}\big(|D^2u|^2+\rho^{-2}|Du|^2\big)\d
x\\
&\qquad\le
c\rho^{4-m}\int_{B_\rho^+(a)}\big(|\Delta
u|^2+\rho^{-2}|Du|^2\big)\d x
+c\rho^2\|Dg\|_{C^1}^2\\
&\qquad\le
c\,\widetilde\Phi_u(a;\rho) +c\rho^2\|Dg\|_{C^1}^2\end{aligned}$$ for any $\rho\in[2s,4s]$, which implies in turn $$\begin{aligned}
E(s)
\le
c\,\widetilde\Phi_u(a;\rho)+cR^2\|Dg\|_{C^1}^2.\end{aligned}$$ We use this to estimate the left-hand side of from below, with the result $$\begin{aligned}
\E(s)&\le
c\E(R)+c\widetilde K R
+
cR\sup_{\sigma\in[s,R]}\E(\sigma),\nonumber\end{aligned}$$ for every $s\in (0,\frac R8)$, where we abbreviated $\widetilde K:=K+\|Dg\|_{C^1}^2$. We take the supremum over $s\in [\delta,\frac R8)$ on both sides, for some $\delta>0$, and infer $$\begin{aligned}
\sup_{s\in[\delta,R]}\E(s)
&\le
\sup_{s\in[\delta,R/8)}\E(s)
+c\E(R)\\
&\leq
c\E(R)+c\widetilde K R
+
cR \sup_{s\in[\delta,R]}\E(s).\end{aligned}$$ Now we choose a radius $R_0=R_0(m,N,\|Dg\|_{C^2})>0$ so small that $cR_0\le
\frac12$, which allows us to re-absorb the last term into the left-hand side, provided $R\le R_0$. Letting $\delta\downarrow0$, we deduce $$\begin{aligned}
\label{Morrey-boundary}
&s^{4-m}\int_{B_s^+(a)}\big(\vert
D^2u\vert^2+s^{-2}|Du|^2\big)\d x\\\nonumber
&\qquad\le
cR^{4-m}\int_{B_R^+(a)}\big(\vert
D^2u\vert^2+R^{-2}|Du|^2\big)\d x + c \widetilde KR\end{aligned}$$ for all $s\le R\le R_0$, in the case $a\in T_1$. For radii $s,R$ with $s\le R_0\le R\le1$, we obtain the same estimate by applying with $R_0$ in place of $R$ and then enlarging the domain of integration on the right-hand side. Finally, the estimate is immediate in the case $R_0\le s\le R$, since $R_0$ is a universal constant. Hence, we obtain for any $s\le
R\le 1$ and boundary points $a\in T_1$.
In the interior case $B_R(a)\subset B^+$, we can argue in the same way, starting from the interior version of the monotonicity formula from [@Angelsberg], to derive the estimate $$\begin{aligned}
\label{Morrey-interior}
&s^{4-m}\int_{B_s(a)}\big(\vert
D^2u\vert^2+s^{-2}|Du|^2\big)\d x\\\nonumber
&\qquad\le
cR^{4-m}\int_{B_R(a)}\big(\vert
D^2u\vert^2+R^{-2}|Du|^2\big)\d x\end{aligned}$$ for all $s\le R$. The two preceding estimates can be combined in a standard way to obtain the result. In fact, let $B_\rho^+(y)\subset B_{R/2}^+(a)$ be an arbitrary ball with $\rho\le \frac R4$ and $y_m\le\frac R4$. We use the notation $(y',y_m):=y\in\R^{m-1}\times\R$ and let $R_1:=\max\{y_m,\rho\}$. Then we use first the interior estimate and then the boundary version to deduce $$\begin{aligned}
&\rho^{4-m}\int_{B_\rho^+(y)}\big(\vert
D^2u\vert^2+\rho^{-2}|Du|^2\big)\d x\\
&\qquad\le
cR_1^{4-m}\int_{B_{R_1}^+(y)}\big(\vert
D^2u\vert^2+R_1^{-2}|Du|^2\big)\d x\\
&\qquad\le
cR_1^{4-m}\int_{B_{2R_1}^+(y',0)}\big(\vert
D^2u\vert^2+R_1^{-2}|Du|^2\big)\d x\\
&\qquad\le
cR^{4-m}\int_{B_{R/2}^+(y',0)}\big(\vert
D^2u\vert^2+R^{-2}|Du|^2\big)\d x +c\widetilde K R\\
&\qquad\le
cR^{4-m}\int_{B_{R}^+(a)}\big(\vert
D^2u\vert^2+R^{-2}|Du|^2\big)\d x+c\widetilde K R.\end{aligned}$$ In the remaining case $y_m>\frac R4$, the corresponding result follows from the interior estimate . Finally, for radii $\rho>\frac R4$, the above estimate is trivial. We note that $\widetilde K\to0$ in the limit $\|Dg\|_{C^2}\to0$. Hence, we have established the assertion in any case.
For later reference, we state another consequence of the monotonicity formula.
\[cor:density\] Assume that $u\in W^{2,2}(B_1^+,N)$ satisfies and $(u,Du)=(g,Dg)$ on $T_1$, in the sense of traces. Then the limit $$\lim_{\rho\downarrow0}{\mathchoice
{{{\setbox0=\hbox{$\displaystyle{\textstyle-}{\int}$}
\vcenter{\hbox{$\textstyle-$}}\kern-.5\wd0}}}
{{{\setbox0=\hbox{$\textstyle{\scriptstyle-}{\int}$}
\vcenter{\hbox{$\scriptstyle-$}}\kern-.5\wd0}}}
{{{\setbox0=\hbox{$\scriptstyle{\scriptscriptstyle-}{\int}$}
\vcenter{\hbox{$\scriptscriptstyle-$}}\kern-.5\wd0}}}
{{{\setbox0=\hbox{$\scriptscriptstyle{\scriptscriptstyle-}{\int}$}
\vcenter{\hbox{$\scriptscriptstyle-$}}\kern-.5\wd0}}}
\!\int}_{\rho/2}^\rho \Phi_u(a;\sigma)\d\sigma$$ exists for any $a\in T_1$.
For two radii $0<\rho<\frac R2<\frac14(1-|a|)$, the monotonicity formula from Theorem \[thm:monotonicity\] implies $$\label{averaged-monotonicity}
{\mathchoice
{{{\setbox0=\hbox{$\displaystyle{\textstyle-}{\int}$}
\vcenter{\hbox{$\textstyle-$}}\kern-.5\wd0}}}
{{{\setbox0=\hbox{$\textstyle{\scriptstyle-}{\int}$}
\vcenter{\hbox{$\scriptstyle-$}}\kern-.5\wd0}}}
{{{\setbox0=\hbox{$\scriptstyle{\scriptscriptstyle-}{\int}$}
\vcenter{\hbox{$\scriptscriptstyle-$}}\kern-.5\wd0}}}
{{{\setbox0=\hbox{$\scriptscriptstyle{\scriptscriptstyle-}{\int}$}
\vcenter{\hbox{$\scriptscriptstyle-$}}\kern-.5\wd0}}}
\!\int}_{\rho/2}^\rho \Phi_u(a;\sigma)\d\sigma
\le
{\mathchoice
{{{\setbox0=\hbox{$\displaystyle{\textstyle-}{\int}$}
\vcenter{\hbox{$\textstyle-$}}\kern-.5\wd0}}}
{{{\setbox0=\hbox{$\textstyle{\scriptstyle-}{\int}$}
\vcenter{\hbox{$\scriptstyle-$}}\kern-.5\wd0}}}
{{{\setbox0=\hbox{$\scriptstyle{\scriptscriptstyle-}{\int}$}
\vcenter{\hbox{$\scriptscriptstyle-$}}\kern-.5\wd0}}}
{{{\setbox0=\hbox{$\scriptscriptstyle{\scriptscriptstyle-}{\int}$}
\vcenter{\hbox{$\scriptscriptstyle-$}}\kern-.5\wd0}}}
\!\int}_{R/2}^{R} \Phi_u(a;s)\d s
+
K{\mathchoice
{{{\setbox0=\hbox{$\displaystyle{\textstyle-}{\int}$}
\vcenter{\hbox{$\textstyle-$}}\kern-.5\wd0}}}
{{{\setbox0=\hbox{$\textstyle{\scriptstyle-}{\int}$}
\vcenter{\hbox{$\scriptstyle-$}}\kern-.5\wd0}}}
{{{\setbox0=\hbox{$\scriptstyle{\scriptscriptstyle-}{\int}$}
\vcenter{\hbox{$\scriptscriptstyle-$}}\kern-.5\wd0}}}
{{{\setbox0=\hbox{$\scriptscriptstyle{\scriptscriptstyle-}{\int}$}
\vcenter{\hbox{$\scriptscriptstyle-$}}\kern-.5\wd0}}}
\!\int}_{\rho/2}^\rho{\mathchoice
{{{\setbox0=\hbox{$\displaystyle{\textstyle-}{\int}$}
\vcenter{\hbox{$\textstyle-$}}\kern-.5\wd0}}}
{{{\setbox0=\hbox{$\textstyle{\scriptstyle-}{\int}$}
\vcenter{\hbox{$\scriptstyle-$}}\kern-.5\wd0}}}
{{{\setbox0=\hbox{$\scriptstyle{\scriptscriptstyle-}{\int}$}
\vcenter{\hbox{$\scriptscriptstyle-$}}\kern-.5\wd0}}}
{{{\setbox0=\hbox{$\scriptscriptstyle{\scriptscriptstyle-}{\int}$}
\vcenter{\hbox{$\scriptscriptstyle-$}}\kern-.5\wd0}}}
\!\int}_{R/2}^{R} \Psi_u(a;\sigma,s)\d s\d\sigma.$$ Using an integration by parts argument similarly to and then applying Lemma \[lemma:morrey\], we can estimate $$\begin{aligned}
&{\mathchoice
{{{\setbox0=\hbox{$\displaystyle{\textstyle-}{\int}$}
\vcenter{\hbox{$\textstyle-$}}\kern-.5\wd0}}}
{{{\setbox0=\hbox{$\textstyle{\scriptstyle-}{\int}$}
\vcenter{\hbox{$\scriptstyle-$}}\kern-.5\wd0}}}
{{{\setbox0=\hbox{$\scriptstyle{\scriptscriptstyle-}{\int}$}
\vcenter{\hbox{$\scriptscriptstyle-$}}\kern-.5\wd0}}}
{{{\setbox0=\hbox{$\scriptscriptstyle{\scriptscriptstyle-}{\int}$}
\vcenter{\hbox{$\scriptscriptstyle-$}}\kern-.5\wd0}}}
\!\int}_{\rho/2}^\rho{\mathchoice
{{{\setbox0=\hbox{$\displaystyle{\textstyle-}{\int}$}
\vcenter{\hbox{$\textstyle-$}}\kern-.5\wd0}}}
{{{\setbox0=\hbox{$\textstyle{\scriptstyle-}{\int}$}
\vcenter{\hbox{$\scriptstyle-$}}\kern-.5\wd0}}}
{{{\setbox0=\hbox{$\scriptstyle{\scriptscriptstyle-}{\int}$}
\vcenter{\hbox{$\scriptscriptstyle-$}}\kern-.5\wd0}}}
{{{\setbox0=\hbox{$\scriptscriptstyle{\scriptscriptstyle-}{\int}$}
\vcenter{\hbox{$\scriptscriptstyle-$}}\kern-.5\wd0}}}
\!\int}_{R/2}^R \Psi_u(a;\sigma,s)\d s\d\sigma\\
&\qquad\le
cR\sup_{\sigma\in[\rho/2,R]}\sigma^{4-m}\int_{B_\sigma^+(a)}\big(|D
^2u|^2+\sigma^{-2}|Du|^2\big)\d x
\le
c(u) R.
\end{aligned}$$ Here, $c(u)$ denotes a constant that depends on $m,N,\|Dg\|_{C^2}$, and $\|u\|_{W^{2,2}}$. We use this to estimate the right-hand side of . Then, we first let $\rho\downarrow0$ and then $R\downarrow0$ in the resulting estimate, which implies $$\limsup_{\rho\downarrow0}{\mathchoice
{{{\setbox0=\hbox{$\displaystyle{\textstyle-}{\int}$}
\vcenter{\hbox{$\textstyle-$}}\kern-.5\wd0}}}
{{{\setbox0=\hbox{$\textstyle{\scriptstyle-}{\int}$}
\vcenter{\hbox{$\scriptstyle-$}}\kern-.5\wd0}}}
{{{\setbox0=\hbox{$\scriptstyle{\scriptscriptstyle-}{\int}$}
\vcenter{\hbox{$\scriptscriptstyle-$}}\kern-.5\wd0}}}
{{{\setbox0=\hbox{$\scriptscriptstyle{\scriptscriptstyle-}{\int}$}
\vcenter{\hbox{$\scriptscriptstyle-$}}\kern-.5\wd0}}}
\!\int}_{\rho/2}^\rho \Phi_u(a;\sigma)\,\d\sigma
\le
\liminf_{R\downarrow0}{\mathchoice
{{{\setbox0=\hbox{$\displaystyle{\textstyle-}{\int}$}
\vcenter{\hbox{$\textstyle-$}}\kern-.5\wd0}}}
{{{\setbox0=\hbox{$\textstyle{\scriptstyle-}{\int}$}
\vcenter{\hbox{$\scriptstyle-$}}\kern-.5\wd0}}}
{{{\setbox0=\hbox{$\scriptstyle{\scriptscriptstyle-}{\int}$}
\vcenter{\hbox{$\scriptscriptstyle-$}}\kern-.5\wd0}}}
{{{\setbox0=\hbox{$\scriptscriptstyle{\scriptscriptstyle-}{\int}$}
\vcenter{\hbox{$\scriptscriptstyle-$}}\kern-.5\wd0}}}
\!\int}_{R/2}^R \Phi_u(a;s)\,\d s.$$ This concludes the proof of the corollary.
Partial regularity for variationally biharmonic maps
----------------------------------------------------
The following $\eps$-regularity result was first established in [@GongLammWang Thm. 1.1] under the additional assumption that a boundary monotonicity inequality of the type is satisfied. The proof was later completed in [@Altuntas], where the boundary monotonicity formula was proved for arbitrary variationally biharmonic maps. The Morrey space estimate that results from the monotonicity formula is stated in Lemma \[lemma:morrey\]. Combining [@GongLammWang Lemma 3.1] with Lemma \[lemma:morrey\] leads to the following regularity result.
\[epsreg\] Let $m\ge 5$, $a\in\partial\R^m_+$, $\rho>0$, and $g\in
C^\infty(B_\rho^+(a),N)$ be given. There exists a constant $\eps_1>0$, depending only on $m,N$, and $\|Dg\|_{C^2}>0$ so that for every weakly biharmonic map $u\in W^{2,2}(B_\rho^+(a),N)$ with that attains the Dirichlet datum $g$ on $T_\rho(a)$ and fulfills the estimate $$\rho^{4-m}\int_{B_\rho^+(a)}\big(|\Delta u|^2+\rho^{-2}|Du|^2\big)\,\d x+\rho<\eps_1$$ we have $u\in C^\infty(B_{\rho/2}^+(a),N)$.
In a standard way, the preceding theorem and its interior counterpart from [@Wang] imply the following partial regularity result.
\[cor:partial\] Let $u\in W^{2,2}(B_4^+,N)$ be variationally biharmonic with respect to the Dirichlet datum $g\in C^\infty(B_4^+,N)$ on $T_4$. Then there is a subset $\Sigma\subset\Omega$ with $\HM^{m-4}(\Sigma)=0$ so that $u\in C^\infty(\Omega\setminus\Sigma,N)$.
Finally, we have the following quantitative estimate. In the case of harmonic maps, the corresponding result is due to Schoen [@Schoen]. For the higher order case considered here, we employ a technique that goes back to Moser [@Moser1 Lemma 5.3], see also [@Scheven2 Lemma 5.3].
\[lem:uniform\] For every $\delta>0$, $\Lambda>0$, and $\alpha\in(0,1)$, there is a constant $\eps=\eps(\delta,\Lambda,\alpha,m,N)>0$ so that the following holds. Assume that $u\in C^4(\overline B_1^+,N)$ is a biharmonic map with Dirichlet values $g\in C^{4,\alpha}(B_1^+,N)$ on $T_1$ for which $$\label{smalldelta}
\left\{\begin{array}{l}
\displaystyle\sup_{B_\rho^+(a)\subset B_1^+}\,
\rho^{2-m}\int_{B_\rho^+(a)}|Du|^2\,\d x<\eps\\[3.5ex]
\quad\|g\|_{C^{4,\alpha}(B_1^+)}\le \Lambda
\end{array}\right.$$ holds true, then we have the estimate $$\sum_{k=1}^4|D^ku(x)|^{1/k}\le \frac\delta{1-|x|}
\mbox{\qquad for all $x\in B_1^+$.}$$
We use the abbreviation $[u]_{C^4}(x):=\sum_{k=1}^4|D^ku(x)|^{1/k}$ for $x\in B_1^+$. If the assertion of the lemma was not true, we could find sequences of biharmonic maps $u_i\in C^4(\overline B_1^+,N)$ and boundary values $g_i\in C^{4,\alpha}(B_1^+,N)$ with $(u_i,Du_i)=(g_i,Dg_i)$ on $T_1$ for all $i\in\N$ so that $$\begin{aligned}
\label{smalldelta-sequence}
\left\{
\begin{array}{l}
\displaystyle\sup_{B_\rho^+(a)\subset
B_1^+}\,\rho^{2-m}\int_{B_\rho^+(a)}|Du_i|^2\,\d x\to0
\qquad\mbox{as }i\to\infty,\\[3.5ex]
\quad\;\displaystyle\sup_{i\in\N}\|g_i\|_{C^{4,\alpha}(B_1^+)}\le \Lambda,
\end{array}
\right.
\end{aligned}$$ but for all $i\in\N$, we have $$\begin{aligned}
\label{notsmall}
\max_{0\le r\le 1}\,(1-r)\max_{\overline B_r^+}\,[u_i]_{C^4}>\delta.
\end{aligned}$$
For every $i\in\N$, we choose a radius $r_i\in[0,1)$ with $$(1-r_i)\max_{\overline B_{r_i}^+}\,[u_i]_{C^4}
=
\max_{0\le r\le 1}(1-r)\max_{\overline B_r^+}\,[u_i]_{C^4}$$ and then a point $x_i\in\overline B_{r_i}^+$ with $$[u_i]_{C^4}(x_i)=\max_{\overline B_{r_i}^+}\,[u_i]_{C^4}.$$ With these choices, we define scaling factors by $$\lambda_i:=\frac\delta{2\,[u_i]_{C^4}(x_i)}
\qquad\mbox{for }i\in\N.$$ We observe that implies $\lambda_i<\frac{1-r_i}2<1$. With these factors we define rescaled maps $$v_i(x):=u_i(x_i+\lambda_ix)
\qquad\mbox{and}\qquad
h_i(x):=g_i(x_i+\lambda_ix)$$ for $x\in\Omega_i$, where $$\Omega_i:=\left\{(x^{(1)},\ldots,x^{(m)})\in
B_1\,:\,x^{(m)}\ge-\lambda_i^{-1}x_i^{(m)}\right\}\supset B_1^+.$$ The rescaled maps satisfy $$\label{delta2}
[v_i]_{C^4}(0)=\lambda_i[u_i]_{C^4}(x_i)= \frac\delta2$$ by definition of $\lambda_i$. Moreover, because of $B_{\lambda_i}(x_i)\subset B_{(1+r_i)/2}$ and the choice of $r_i$, $x_i$, and $\lambda_i$, we infer $$\label{delta3}
\max_{\overline \Omega_i}\,[v_i]_{C^4}
\le \lambda_i\max_{\overline B_{(1+r_i)/2}^+}\,[u_i]_{C^4}
\le
\lambda_i\,\frac{1-r_i}{1-\frac{1+r_i}2}\,[u_i]_{C^4}(x_i)
=
2\lambda_i[u_i]_{C^4}(x_i)
=\delta.$$ From $\lambda_i<1$ and $_2$, we obtain $$\label{G1}
\sup_{i\in\N}\|h_i\|_{C^{4,\alpha}(\Omega_i)}
<
\sup_{i\in\N}\|g_i\|_{C^{4,\alpha}(B_1^+)}
\le \Lambda.$$ By the scaling invariance of the biharmonic map equation, the maps $v_i$ are again biharmonic, which means by that they satisfy a boundary value problem of the form $$\label{boundary-problem}
\left\{
\begin{array}{ll}
\Delta^2v_i=\tilde f(v_i,Dv_i,D^2v_i,D^3v_i)=:f_i
&\mbox{in }\Omega_i,\\[1.5ex]
v_i=h_i,\ Dv_i=Dh_i&
\mbox{on $B_1\cap\big\{x^{(m)}=-\lambda_i^{-1}x_i^{(m)}\big\}$}.
\end{array}
\right.$$ Clearly, in the case that the last set is empty, the map $v_i$ satisfies the differential equation on the full ball $B_1$ and there is no boundary condition. From the form of the biharmonic map equation and , we infer $$\label{right-hand-side-bounded}
\sup_{i\in\N}\|f_i\|_{C^1(\Omega_i)}<\infty.$$ For a standard cut-off function $\zeta\in C^\infty_0(B_1,[0,1])$ with $\zeta\equiv 1$ in $B_{1/2}$, we use classical Schauder estimates for the maps $\zeta v_i$ on the halfspaces $\R^m\cap\{x^{(m)}=-\lambda_i^{-1} x_i^{(m)}\}$. In this way, we deduce that implies $$\|v_i\|_{C^{4,\alpha}(\Omega_i\cap B_{1/2})}
\le
c\big(\|h_i\|_{C^{4,\alpha}(\Omega_i)}+\|f_i\|_{C^{0,\alpha}(\Omega_i)}+\|v_i\|_{C^{3,\alpha}(\Omega_i)}\big).$$ In view of , , and , we infer that the restrictions $v_i|_{B_{1/2}^+}$ are bounded in $C^{4,\alpha}(B_{1/2}^+,N)$, independently of $i\in\N$. Therefore, after passing to a subsequence, the Arzelà-Ascoli theorem yields the convergence $$v_i\to v\qquad\mbox{in $C^4(B_{1/2}^+,N)$, as $i\to\infty$},$$ for some limit map $v\in C^4(B_{1/2}^+,N)$. Now on the one hand, the identity implies $$\label{notconstant}
[v]_{C^4}(0)=\lim_{i\to\infty}[v_i]_{C^4}(0)=\frac\delta2,$$ but on the other hand, the choice of $u_i$ according to $_1$ leads to the estimate $$\begin{aligned}
\int_{B_{1/2}^+}|Dv|^2\,\d x
&=
\lim_{i\to\infty}\int_{B_{1/2}^+}|Dv_i|^2\,\d x
\le
\lim_{i\to\infty}\int_{\Omega_i\cap B_{1/2}}|Dv_i|^2\,\d x\\
&=
\lim_{i\to\infty}\lambda_i^{2-m}\int_{B_{\lambda_i/2}^+(x_i)}|Du_i|^2\,\d
x=0,
\end{aligned}$$ which means that $v$ is constant on $B_{1/2}^+$. In view of , this yields the desired contradiction and completes the proof of the lemma.
By combining the preceding regularity results, we arrive at the following conclusion.
\[cor:uniform\] We consider a ball $B_r^+(x_0)=B_r(x_0)\cap\R^m_+$ for some $x_0\in\R^m_+$ and $r\in(0,1]$. There is a constant $\eps_0=\eps_0(\Lambda,\alpha,m,N)>0$, so that for every weakly biharmonic map $u\in W^{2,2}(B_r^+(x_0),N)$ that satisfies and attains the Dirichlet datum $g\in C^\infty(B_r^+(x_0),N)$ on $T_r(x_0)$, the estimates $$\label{small}
\left\{
\begin{array}{l}
\displaystyle r^{4-m}\int_{B_r^+(x_0)}\big(|\Delta u|^2
+r^{-2}|Du|^2\big)\,\d x<\eps_0,\\[3.5ex]
\displaystyle\|g\|_{C^{4,\alpha}(B_r^+(x_0))}\le \Lambda
\end{array}
\right.$$ imply $u\in C^4(B_{r/2}^+(x_0),N)$ with $\|Du\|_{C^3(B_{r/2}^+(x_0))}\le
c(\Lambda,\alpha,m,N)$.
By a scaling argument, it suffices to consider the case $r=1$. In the case that $B_{3/4}^+(x_0)$ intersects $\partial\R^m_+$, we apply Lemma \[lemma:morrey\] on balls $B_{1/4}^+(a)$ for any $a\in B_{3/4}^+(x_0)\cap \partial\R^m_+$. In view of this lemma and assumption $_1$, we infer $$\begin{aligned}
\sup_{B_\rho^+(y)\subset B_{R_0}^+(a)}\,
\rho^{4-m}\int_{B_\rho^+(y)}\big(|D^2u|^2+\rho^{-2}|Du|^2\big)\,\d x
\le
cR_0^{2-m}\eps_0+cR_0,
\end{aligned}$$ for any $R_0\in(0,\tfrac18)$. By choosing first $R_0$ and then $\eps_0$ small enough in dependence on $\Lambda,\alpha, m$, and $N$, we can ensure that the assumptions of Theorem \[epsreg\] and Lemma \[lem:uniform\] with $\delta=1$ are satisfied on the ball $B_{R_0}^+(a)$, which imply that $u\in C^4(B_{R_0/2}^+(a))$ with $\|Du\|_{C^3(B_{R_0/4}^+(a))}\le c$. We note that the application of Lemma \[lem:uniform\] is possible after a suitable rescaling. Since the last estimate holds for any $a\in B_{3/4}^+(x_0)\cap \partial\R^m_+$, we deduce the asserted gradient estimates in every point $y\in B_{1/2}^+(x_0)$ with $y_m<\frac14
R_0$. In the remaining case $y\in B_{1/2}^+(x_0)$ with $y_m\ge\tfrac14 R_0$, we apply the corresponding interior estimates on the ball $B_{R_0/4}(y)$, cf. [@Scheven1 Thm. 2.6]. In this way, we arrive at the desired bound $\|Du\|_{C^3(B_{1/2}^+(x_0))}\le c(\Lambda,\alpha,m,N)$.
Compactness for sequences of variationally biharmonic maps {#sec:compact}
==========================================================
The defect measure
------------------
Our goal is to prove compactness for a sequence of variationally biharmonic maps $u_i\in
W^{2,2}(B_4^+,N)$ with respect to Dirichlet values $g_i\in
C^{4,\alpha}(B_4^+,N)$ on $T_4$. We assume that the sequence is bounded in the sense that $$\begin{aligned}
\label{W22-bound}
\sup_{i\in\N}\,(\|D^2u_i\|_{L^2(B_4^+)}+\|Du_i\|_{L^2(B_4^+)})<\infty
\ \quad\mbox{and}\ \quad
\sup_{i\in\N}\|g_i\|_{C^{4,\alpha}(B_4^+)}<\infty.\end{aligned}$$ More precisely, we consider the slightly more general case of maps with and instead of variationally biharmonic maps, since the properties and are clearly preserved under strong convergence in $W^{2,2}$. In view of , Lemma \[lemma:morrey\] implies the Morrey space bound $$\label{bound:morrey}
\sup_{i\in\N}\Big(|D^2u_i\|^2_{L^{2,m-4}(B_2^+)}+\|Du_i\|^2_{L^{2,m-2}(B_2^+)}
+\|Dg_i\|_{C^{4,\alpha}(B_2^+)}^2\Big)\le\Lambda$$ for a constant $\Lambda\ge1$ that depends on $m,N$, and the sequences $(u_i)$ and $(g_i)$. Here we used the Morrey type norms that are defined by $$\label{def:morrey-norm}
\|f\|_{L^{2,\lambda}(B_2^+)}^2
:=
\sup_{B_\rho^+(y)\subset
B_2^+}\frac1{\rho^\lambda}\int_{B_\rho^+(y)}|f|^2\,\d x,$$ for $f\in L^2(B_2^+)$. From now on, we restrict ourselves to the ball $B_2^+$ and assume that a bound of the type is valid. This bound implies in particular that the sequence of Radon measures $\Lm\edge|\Delta u_i|^2$ is bounded on $\overline B_2^+$. Therefore, possibly after passing to a subsequence, we infer a map $u\in W^{2,2}(B_2^+,N)$ and a Radon measure $\nu$ on $\overline B_2^+$ with $u_i\wto u$ weakly in $W^{2,2}(B_2^+,\R^L)$, strongly in $W^{1,2}(B_2^+,\R^L)$ and a.e., and moreover $$\begin{aligned}
\Lm\edge|\Delta u_i|^2{\overset{\raisebox{-1ex}{\scriptsize $*$}}{\rightharpoondown}}\Lm\edge|\Delta u|^2+\nu\end{aligned}$$ weakly\* in the space of Radon measures, as $i\to\infty$. The lower semicontinuity of the $L^2$-norm with respect to weak convergence implies $\nu\ge 0$. We call the measure $\nu$ the *defect measure* of the sequence $u_i$, since it detects a possible lack of strong convergence in $W^{2,2}$, see Lemma \[defect\] below. The pair $(u,\nu)$ can be considered as the limit configuration of the sequence $u_i$. This motivates the following
\[def:BLambda\] For sequences of maps $u_i\in W^{2,2}(B_2^+,N)$ and nonnegative Radon measures $\nu_i$ on $\overline B_2^+$, where $i\in\N_0$, we write $(u_i,\nu_i)\Bto(u_0,\nu_0)$ as $i\to\infty$ if and only if convergence holds in the following sense. $$\begin{aligned}
\nonumber
\left\{
\begin{array}{cl}
u_i\wto u_0&\mbox{weakly in $W^{2,2}(B_2^+,\R^L)$}\\
u_i\to u_0&\mbox{strongly in $W^{1,2}(B_2^+,\R^L)$ and a.e.,}\\
\Lm\edge|\Delta u_i|^2+\nu_i{\overset{\raisebox{-1ex}{\scriptsize $*$}}{\rightharpoondown}}\Lm\edge|\Delta u_0|^2+\nu_0&
\mbox{weakly$^*$ as Radon measures.}
\end{array}\right.
\end{aligned}$$ For the set of all limit configurations of biharmonic maps, we write $$\B:=\left\{
(u,\nu)\left|
\begin{array}{l}
(u_i,0)\Bto (u,\nu)\mbox{ for maps $u_i\in
W^{2,2}(B_2^+,N)$ that satisfy}\\
\mbox{\eqref{weakly-bi} and \eqref{stationary-bi},
attain the boundary values $g_i\in C^\infty(B_2^+,N)$}\\
\mbox{in the sense
$(u_i,Du_i)=(g_i,Dg_i)$ on $T_2$, and that satisfy}\\
\mbox{$\|D^2u_i\|_{L^{2,m-4}(B_2^+)}^2+\|Du_i\|_{L^{2,m-2}(B_2^+)}^2+\|g_i\|_{C^{4,\alpha}(B_2^+)}^2\le\Lambda$}
\end{array}
\right.\right\}$$ for a given constant $\Lambda\ge1$ and $\alpha\in(0,1)$. Here, $"0"$ denotes the zero measure and we used the Morrey norms defined in . For a given pair $\mu=(u,\nu)\in\B$, we define the [*energy concentration set*]{} $\Sigma_\mu$ as the set of points $a\in \overline B_2^+$ with the property $$\liminf_{\rho\searrow 0}\left(\rho^{4-m}\int_{B_\rho^+(a)}(|\Delta
u|^2+\rho^{-2}|Du|^2)\,\d x+\rho^{4-m}\nu(B_\rho^+(a))\right)\ge \eps_0,$$ where the constant $\eps_0=\eps_0(\Lambda,\alpha,m,N)>0$ is chosen according to Corollary \[cor:uniform\].
The following lemma clarifies the meaning of the defect measure and the energy concentration set.
\[defect\] Assume that $u_i\in W^{2,2}(B_2^+,N)$, $i\in\N$, is a sequence of maps with , , boundary values $$(u_i,Du_i)=(g_i,Dg_i)\qquad\mbox{on $T_2$ in the sense of traces},$$ so that the bound $$\label{Lambda-bound}
\sup_{i\in\N}\big(\|D^2u_i\|_{L^{2,m-4}(B_2^+)}^2
+\|Du_i\|_{L^{2,m-2}(B_2^+)}^2+\|g_i\|_{C^{4,\alpha}(B_2^+)}^2\big)\le\Lambda$$ is satisfied. Moreover, we assume that $(u_i,0)\Bto(u,\nu)=:\mu$ as $i\to\infty$, for some $(u,\nu)\in\B$. Then there holds
1. $u_i\to u$ in $C^3_{\textup{loc}}(
B_2^+{\setminus}\Sigma_\mu,\R^L)$ as $i\to\infty$.
2. If the defect measure satisfies $\operatorname{spt}\nu\cap \overline B_1^+=\varnothing$, then we have strong convergence $u_i\to u$ in $W^{2,2}(B_{1}^+,\R^L)$, as $i\to\infty$.
In order to prove (i), we choose an arbitrary point $a\in B_2^+{\setminus}\Sigma_\mu$. By the definition of $\Sigma_\mu$, we may choose a $\rho\in(0,1)$ with $$\rho^{4-m}\int_{B_\rho^+(a)}(|\Delta u|^2
+\rho^{-2}|Du|^2)\,\d x+\rho^{4-m}\nu(B_\rho^+(a))<\eps_0.$$ By slightly diminishing the value of $\rho$ if necessary, we can additionally achieve that $\nu(\partial B_\rho^+(a))=0$, because $\nu(\partial B_\rho^+(a))>0$ can hold at most for countably many values of $\rho\in(0,1)$. Using the convergence $(u_i,0)\Bto(u,\nu)$, we conclude $$\begin{aligned}
&\lim_{i\to\infty}\rho^{4-m}\int_{B_\rho^+(a)}(|\Delta u_i|^2
+\rho^{-2}|Du_i|^2)\,\d x\\
&\qquad=
\rho^{4-m}\int_{B_\rho^+(a)}(|\Delta u|^2
+\rho^{-2}|Du|^2)\,\d x+\rho^{4-m}\nu(B_\rho^+(a))
<\eps_0.
\end{aligned}$$ Therefore, Corollary \[cor:uniform\] yields the uniform estimate $\|u_i\|_{C^4(B_{\rho/2}^+(a))}\le c(\Lambda,\alpha,m,N)$ for all sufficiently large $i\in\N$, from which we infer by Arzéla-Ascoli’s theorem that $u_i\to u$ holds in $C^3(B_{\rho/2}^+(a),N)$ as $i\to\infty$. Since $a\in B_2^+{\setminus}\Sigma_\mu$ was arbitrary, this implies (i).\
For the proof of (ii), we note that in the case of $\operatorname{spt}\nu\cap\overline B_1^+=\varnothing$, the set $\Sigma_\mu\cap \overline B_1^+$ is given by $$\Sigma_\mu\cap \overline B_1^+=
\Big\{y\in \overline B_1^+:\liminf_{\rho\searrow 0}
\rho^{4-m}\int_{B_\rho^+(y)}\big(|\Delta u|^2+\rho^{-2}|Du|^2\big)\,\d x\ge\eps_0\Big\}.$$ Since the values of $u$ are contained in the bounded manifold $N$, we have $u\in L^\infty\cap W^{2,2}(B_1^+,N)$, which implies by Gagliardo-Nirenberg embedding that $u\in W^{1,4}(B_1^+,N)$. Therefore, we infer from Hölder’s inequality and [@Ziemer Lemma 3.2.2] that $\HM^{m-4}(\Sigma_\mu\cap \overline B_1^+)=0$. This implies that for any given $\eps>0$, we can choose a cover $A:=\cup_{k\in\N}B_{\rho_k}(a_k)\supset\Sigma_\mu\cap \overline B_{1}^+$ of open balls with radii $\rho_k\in(0,1)$ and centers $a_k\in \overline B_{1}^+$ so that $\sum_{k\in\N}\rho_k^{m-4}<\eps$. Hence, the bound yields the following estimate for all $i\in\N$. $$\begin{aligned}
\int_{A\cap B_{1}^+}|D^2u_i-D^2u|^2\,\d x
&\le& 2\sum_{k\in\N}\int_{B_{\rho_k}^+(a_k)}\big(|D^2u_i|^2+|D^2u|^2\big)\,\d x\\
&\le& c\Lambda\sum_{k\in\N}\rho_k^{m-4}
\le c\Lambda\eps.
\end{aligned}$$ On the compact set $\overline B_{1}^+\setminus A\subset\overline
B_{1}^+{\setminus}\Sigma_\mu$, the conclusion (i) implies $u_i\to u$ in $C^3(\overline B_{1}^+\setminus A,\R^L)$. Hence, $$\limsup_{i\to\infty}
\int_{B_{1}^+}|D^2u_i-D^2u|^2\,\d x
\le c\Lambda\eps\,+\,\lim_{i\to\infty}\int_{B_{1}^+\setminus A}|D^2u_i-D^2u|^2\,\d x
= c\Lambda\eps.$$ Since $\eps>0$ was arbitrary, we conclude that $u_i\to u$ holds in $W^{2,2}(B_{1}^+,\R^L)$ in the limit $i\to\infty$, as claimed.
Next, we analyse the relation between the defect measure and the energy concentration set.
\[structure\] There are positive constants $c_1,c_2$, depending only on $m$, so that every pair $\mu=(u,\nu)\in\B$ satisfies $$\label{equiv}
c_1\eps_0\HM^{m-4}\edge(\Sigma_\mu\cap \overline B_1^+)
\le\nu\edge\overline B_1^+
\le c_2\Lambda\HM^{m-4}\edge(\Sigma_\mu\cap \overline B_1^+).$$ Furthermore, $\Sigma_\mu$ is a closed set and $\Sigma_\mu=\operatorname{sing}(u)\cup\operatorname{spt}(\nu)$.
The inclusion $\operatorname{sing}(u)\cup\operatorname{spt}(\nu)\subset\Sigma_\mu$ holds by Corollary \[cor:uniform\] and Lemma \[defect\](i). For the converse inclusion, we assume that there is some point $a\in\Sigma_\mu\setminus\operatorname{sing}(u)$. By the choice of $a$, the functions $|\Delta u|$ and $|Du|$ are bounded on a neighborhood of $a$. Therefore, the definition of $\Sigma_\mu$ implies $$\liminf_{\rho\searrow 0}\rho^{4-m}\nu(B_\rho^+(a))\ge\eps_0,$$ from which we infer $a\in\operatorname{spt}(\nu)$. We have thus proven $\Sigma_\mu=\operatorname{sing}(u)\cup\operatorname{spt}(\nu)$, which implies in particular that $\Sigma_\mu$ is a closed set.\
Now we turn our attention to the proof of . For a Borel set $A\subset\Sigma_\mu\cap\overline B_1^+$, we choose an arbitrary cover $\cup_{j\in\N}B_{\rho_j}(a_j)\supset A$ of balls with radii $\rho_j\in(0,1)$ and centers $a_j\in\overline B_1^+$. Since $(u,\nu)\in\B$, we infer that $$\nu(A)\le\sum_{j\in\N}\nu(B_{\rho_j}(a_j))
\le c_2\Lambda\sum_{j\in\N}\rho_j^{m-4}$$ holds true, for a constant $c_2=c_2(m)$. Since the cover of $A$ was arbitrary, we conclude $\nu(A)\le c_2\Lambda\HM^{m-4}(A)$. For the proof of the first estimate in , we choose a set $E\subset\Sigma_\mu$ with $\HM^{m-4}(E)=0$ and $$\lim_{\rho\searrow 0}\rho^{4-m}\int_{B_\rho^+(a)}\big(|\Delta u|^2+|Du|^4\big)\,\d x=0
\qquad\mbox{for all }a\in \Sigma_\mu\setminus E.$$ A set with this property exists by [@Ziemer Lemma 3.2.2]. From the choice of $E$ and the definition of $\Sigma_\mu$, we know $$\label{positive}
\liminf_{\rho\searrow 0}\rho^{4-m}\nu(B_\rho^+(a))\ge\eps_0
\qquad\mbox{for all }a\in\Sigma_\mu\setminus E.$$ Let $\eps>0$ be given. By the definition of the Hausdorff measure, we may choose $\delta>0$ small enough to ensure $$\HM^{m-4}(A\setminus E)
\le\eps+\inf\Big\{\alpha(m{-}4)\sum_{j\in\N}r_j^{m-4}\ \Big|\
A\setminus E\subset\bigcup_{j\in\N}B_{r_j}(a_j),\ 0<r_j\le\delta\,\Big\},$$ where $\alpha(m-4)$ denotes the volume of the $(m-4)$-dimensional unit ball. Since $\nu$ is a Radon measure, we may choose an open set $O_\eps\supset A$ with $\nu(O_\eps\cap\R^m_+)\le\nu(A)+\eps$. We consider the family of balls $B_{\rho_j}(a_j)\subset O_\eps$ with $0<\rho_j\le\delta/5$, centers $a_j\in A\setminus E$ and the property $$\label{positive2}
\rho_j^{4-m}\nu(B_{\rho_j}^+(a_j))\ge\frac{\eps_0}2.$$ By , the union of all balls with these properties covers the set $A\setminus E$. A Vitali type covering argument therefore yields the existence of a countable disjoint family $\{B_{\rho_j}(a_j)\}_{j\in\N}$ of balls with the property , $B_{\rho_j}(a_j)\subset O_\eps$ and $A\setminus E\subset\,\cup_jB_{5\rho_j}(a_j)$, where $5\rho_j\le\delta$ for all $j\in\N$. This implies $$\HM^{m-4}(A)=\HM^{m-4}(A\setminus E)\le \eps+\alpha(m{-}4)\sum_{j\in\N}
(5\rho_j)^{m-4}$$ by the choice of $\delta$. Since the balls $B_{\rho_j}(a_j)$ are pairwise disjoint and satisfy , we can further estimate $$\sum_{j\in\N}\rho_j^{m-4}
\le
\frac2{\eps_0}\sum_{j\in\N}\nu(B_{\rho_j}^+(a_j))
\le\frac2{\eps_0}\nu(O_\eps\cap\R^m_+)\le\frac2{\eps_0}\big(\nu(A)+\eps\big).$$ Combining the last two estimates, we arrive at $$\eps_0\HM^{m-4}(A)\le \eps_0\eps+c_1^{-1}(\nu(A)+\eps)$$ with a constant $c_1=c_1(m)>0$. Since $\eps>0$ was arbitrary, we can omit the terms involving $\eps$ in the preceding estimate. This completes the proof of the lemma.
For the blow-up analysis of the defect measure, the case of a constant limit map is of particular interest. In this case, the defect measure inherits a monotonicity property from the sequence of biharmonic maps.
\[monodefect\] We consider a pair $(c,\nu)\in\B$, where $c\in N$ denotes a constant map. More precisely, we assume that $(c,\nu)$ can be approximated by biharmonic maps $u_i$ in the sense $\B\ni (u_i,0)\Bto(c,\nu)$, where the approximating maps satisfy the boundary condition $(u_i,Du_i)=(g_i,Dg_i)$ on $T_2$, for boundary values $g_i\in C^\infty(B_2^+,N)$ with $$\label{C4-conv-bdry-values}
g_i\to c
\mbox {\qquad in $C^4(B_2^+,N)$, as $i\to\infty$}.$$ Then, the functions $$(0,1]\ni r\mapsto r^{4-m}\nu(B_r^+(a))$$ are monotonically nondecreasing for every $a\in \overline
T_1$. Moreover, there is a subsequence $\{i_j\}\subset\N$ so that $$\label{Phi-converge}
\left\{
\begin{array}{ll}
\displaystyle\lim_{j\to\infty}\Phi_{u_{i_j}}(a;r)=r^{4-m}\nu(B_r^+(a)),\\
\displaystyle\lim_{j\to\infty} K_{i_j}\Psi_{u_{i_j}}(a;\rho,r)\to 0
\end{array}
\right.$$ for a.e. $\rho,r\in(0,1]$, as $j\to\infty$, where the terms $\Phi_{u_i}$, $\Psi_{u_i}$ and $K_i$ are defined in Theorem \[thm:monotonicity\].
From Theorem \[thm:monotonicity\] we deduce that the approximating biharmonic maps $u_i$ satisfy the monotonicity formula with constants $\chi_i$, $K_i>0$ that satisfy $\chi_i\to0$ and $K_i\to0$ as a consequence of $\|Dg_i\|_{C^3(B_2^+)}\to0$ in the limit $i\to\infty$. Discarding the non-negative integral on the left-hand side of the monotonicity formula, we deduce $$\label{limit-Phi}
\Phi_{u_i}(a;\rho)
\le
\Phi_{u_i}(a;r)+K_i\Psi_{u_i}(a;\rho,r)$$ for all $i\in\N$ and a.e. radii $0<\rho<r<1$. Next, for a.e. radius $r\in(0,1)$ and $X(x):=x-a$, we may define $$f_i(r):=\int_{S_r^+(a)}\big(\partial_X|Du_i|^2+4|Du_i|^2-4r^{-2}|\partial_Xu_i|^2\big)d\Hm,$$ as well as $$g_i(r):= K_i r^{6-m}\int_{S_r^+(a)}|D^2u_i|^2\d \Hm.$$ Since $u_i\to c$ strongly in $W^{1,2}(B_2^+,\R^K)$, $\sup_i\|D^2u_i\|_{L^2(B_2^+)}^2<\infty$, and $K_i\to0$, we have $f_i\to 0$ and $g_i\to0$ in $L^1([0,1])$, as $i\to\infty$. Therefore, there is a subsequence $\{i_j\}\subset\N$ with $f_{i_j}\to 0$ a.e. and $g_{i_j}\to0$ a.e. in $[0,1]$, as $j\to\infty$. We conclude that for almost every $r\in[0,1]$, $$\begin{aligned}
\Phi_{u_{i_j}}(a;r)
&=e^{\chi_{i_j}r}r^{4-m}\int_{B_r^+(a)}|\Delta u_{i_j}|^2\,\d x+
e^{\chi_{i_j}r}r^{3-m}f_{i_j}(r)\\
&\to r^{4-m}\nu(B_r^+(a))
\end{aligned}$$ in the limit $j\to\infty$, if we choose in particular $r\in[0,1]$ in such a way that $\nu(\partial B_r(a))=0$. Similarly, we deduce $$\begin{aligned}
&K_{i_j}\Psi_{u_{i_j}}(a;\rho,r)\\
&\quad=K_{i_j}r+K_{i_j}\int_{B_r^+(a)\setminus
B_\rho^+(a)}\bigg(\frac{|D^2u_{i_j}|^2}{|x-a|^{m-5}}+\frac{|Du_{i_j}|^2}{|x-a|^{m-3}}\bigg)\,\d x+g_{i_j}(\rho)+g_{i_j}(r)\\\nonumber
&\quad\to0
\end{aligned}$$ as $j\to\infty$, for a.e. $0<\rho<r<1$. The two preceding convergences imply assertion . Using this result in , we infer $$\rho^{4-m}\nu(B_\rho^+(a))
\le
r^{4-m}\nu(B_r^+(a))$$ for a.e. $0<\rho<r\le 1$. Since the measures $\nu(B_\rho^+(a)),$ $\nu(B_r^+(a))$ depend left-continuously on $\rho,r\in[0,1]$, this implies the asserted monontonicity property.
Existence of flat tangent measures
----------------------------------
We consider tangent pairs of $\mu=(u,\nu)\in\B$ in a boundary point $a\in \overline T_1$ in the following sense. For scaling factors $r\in(0,1)$ we define rescaled versions $\mu_{a,r}=(u_{a,r},\nu_{a,r})$ by $$\begin{aligned}
u_{a,r}(x)&:=u(a+rx)&&\mbox{\ \ \ \ for $x\in B_{1/r}^+$,}&
\label{rescale}\\\nonumber
\nu_{a,r}(A)&:=r^{4-m}\nu(a+rA)&&\mbox{\ \ \ \
for every Borel set $A\subset B_{1/r}^+$}.&\end{aligned}$$ We note that the definition of $\B$ implies $$\label{bound-rescalings}
\int_{B_{2}^+}\big(|D^2u_{a,r}|^2+|Du_{a,r}|^2\big)\d x
+
\nu_{a,r}(\overline B_{2}^+)
\le c(m)\Lambda$$ for any $a\in\overline T_1$ and $r\in(0,\tfrac12)$. For a map $u_*\in W^{2,2}(B_2^+,N)$ and a Radon measure $\nu_*$ on $\overline B_2^+$, we call $\mu_*=(u_*,\nu_*)$ a *tangent pair* of $\mu$ in $a\in\overline
T_1$ if there exists a sequence $r_i\searrow 0$ so that $\mu_{a,r_i}\Bto\mu_*$. For the family of all tangent pairs of a given $\mu\in\B$ in a boundary point, we write $$\T(\mu):=\big\{\mu_*\,\big|\,\mu_{a,r_i}\Bto\mu_*\mbox{ for a sequence
$r_i\searrow 0$ and some $a\in \overline T_1$}\big\}.$$ A standard diagonal sequence argument yields $$\label{tangent}
\T(\mu)\subset\B,$$ cf. [@Scheven1 Lemma 3.3]. More precisely, assume that $\mu_{a,r_i}\Bto\mu_\ast$ as $i\to\infty$, and let $u_k\in
W^{2,2}(B_2^+,N)$ be biharmonic maps as in Definition \[def:BLambda\] with $(u_k,0)\Bto \mu$ in the limit $k\to\infty$. Then a diagonal sequence argument yields a sequence $(k_i)_{i\in\N}$ in $\N$ with $(\tilde u_i,0):=((u_{k_i})_{a,r_i},0)\Bto \mu_\ast$ as $i\to\infty$, which proves $\mu_\ast\in\B$, and thereby the claim . We note that the boundary values $\tilde g_i:=(g_{k_i})_{a,r_i}\in C^\infty(B_2^+,N)$ of $\tilde
u_i$ satisfy $$\|D\tilde g_i\|_{C^3}
\le
r_i\|Dg_{k_i}\|_{C^3}
\le
r_i\Lambda\to0
\qquad\mbox{in the limit }i\to\infty.$$ This means that every tangent pair $\mu_\ast\in\T(\mu)$ is the limit of biharmonic maps $\tilde u_i$ in the sense $(\tilde
u_i,0)\Bto\mu_\ast$, where the boundary values of $\tilde u_i$ converge to a constant $c_\ast\in N$ in the sense $$\label{eq:3}
\tilde g_i\to c_\ast
\quad\mbox{in $C^4(B_2^+,N)$, as $i\to\infty$.}$$ In the following lemma, we construct a flat defect measure by a double blow-up procedure.
\[blow-up\] Assume that $N$ does not carry a non-constant Paneitz-biharmonic $4$-sphere, and that there is a pair $\mu=(u,\nu)\in\B$ with $\operatorname{spt}\nu\cap\overline B_1^+\neq\varnothing$. Then there exists a pair $(c_\ast,\bar\nu)\in\T(\mu)\subset\B$, where $c_\ast\in N$ represents a constant map and $$\bar\nu=c_0\HM^{m-4}\edge\big(V\cap \overline B_2^+\big)\label{flat}$$ for an $(m-4)$-dimensional subspace $V\subset\partial\R^m_+$ and a constant $c_0>0$.
Since $u\in W^{2,2}\cap W^{1,4}(\overline B_2^+,N)$, we know that for $\HM^{m-4}$-a.e. $a\in \overline T_1$, $$\label{nodensity}
\lim_{\rho\searrow 0}\,\rho^{4-m}\int_{B_\rho^+(a)}\big(|D^2u|^2+|Du|^4)\,\d x=0,$$ cf. [@Ziemer Lemma 3.2.2]. Since $N$ does not carry a non-constant Paneitz-biharmonic $4$-sphere, we know from the analysis of the interior case in [@Scheven1 Thm. 1.6] that $\operatorname{spt}\nu\subset\partial\R^n_+$. On the other hand, we have $\operatorname{spt}\nu\cap\overline B_1^+\neq\varnothing$ by assumption, so that Lemma \[structure\] implies $\HM^{m-4}(\Sigma_\mu\cap\overline T_1)>0$. Therefore, we may choose a point $a\in\Sigma_\mu\cap\overline T_1 $ with the property . As a consequence of , there is a sequence $r_i\searrow 0$ so that the rescaled maps $u_{a,r_i}$ satisfy $$u_{a,r_i}\to c_\ast
\qquad\mbox{strongly in $W^{2,2}(B_2^+,\R^L)$ and in $W^{1,4}(B_2^+,\R^L)$,}$$ as $i\to\infty$, for some constant $c_\ast\in N$. In view of , by passing to a subsequence we can achieve the convergence $\mu_{a,r_i}\Bto (c_\ast,\nu_*)$ for some Radon measure $\nu_*$ on $\overline B_2^+$ as $i\to\infty$. The result of this first blow-up is a tangent pair $\mu_*:=(c_\ast,\nu_*)\in\T(\mu)\subset\B$ for which the terms $r^{4-m}\,\nu_*(B_r^+(a))$ depend monotonically nondecreasing on $r\in(0,1)$ for every $a\in\overline T_1$, see Lemma \[monodefect\]. We point out that assumption of this lemma is satisfied in view of . In particular, this monotonicity property implies that the $(m-4)$-dimensional density $$\Theta^{m-4}(\nu_*,a):=\lim_{r\searrow 0}\,r^{4-m}\nu_*(B_r^+(a))$$ exists for every $a\in\overline T_1$. Because of $(c_\ast,\nu_*)\in\B$ we have $\Theta^{m-4}(\nu_*,a)\le \Lambda$ for every $a\in\overline T_1$. As already noted above, the interior result [@Scheven1 Thm. 1.6] implies $\Sigma_{\mu_\ast}\subset\partial\R^m_+$. Since $\mu_*=(c_\ast,\nu_*)$ with a constant map $c_\ast$, we can characterize the set $\Sigma_{\mu_*}\cap\overline B_1^+$ by $$\Sigma_{\mu_*}\cap\overline B_1^+
=\big\{a\in \overline T_1\,:\,\Theta^{m-4}(\nu_*,a)\ge\eps_0\big\}.$$ Because Lemma \[structure\] implies $\nu_*(\overline
B_1^+\setminus\Sigma_{\mu_*})=0$, we deduce $$0<\eps_0\le \Theta^{m-4}(\nu_*,y)\le \Lambda\qquad\mbox{for
$\nu_*$-a.e. $y\in\overline B_1^+$.}$$ The above property implies the existence of an $(m-4)$-flat tangent measure of $\nu_*$, see [@Mattila Thm. 14.18]. More precisely, for $\nu_*$-a.e. $y\in\overline B_1^+$, there is a sequence $\rho_j\searrow 0$ with $(\nu_*)_{y,\rho_j}{\overset{\raisebox{-1ex}{\scriptsize $*$}}{\rightharpoondown}}\bar\nu$ weakly$^\ast$ in the space of Radon measures, where $\bar\nu=c_0\HM^{m-4}\edge V$ for an $(m-4)$-dimensional subspace $V\subset\R^m$ and a constant $c_0>0$. As above, we infer from the interior case in [@Scheven1 Thm. 1.6] that $\nu_*(\overline B_1^+\setminus
\overline T_1)=0$, so that we can find a boundary point $y\in\overline T_1$ with the above property and with $V\subset\partial\R^n_+$. Finally, we note that a diagonal sequence argument implies that tangent pairs of tangent pairs are again tangent pairs, which implies that $(c_\ast,\bar\nu)\in\T(\mu)$. This completes the proof.
Construction of a biharmonic $4$-sphere or $4$-halfsphere
---------------------------------------------------------
\[sphere\] Assume that there is a pair $\mu=(u,\nu)\in\B$ with $\operatorname{spt}\nu\cap\overline B_1^+\neq\varnothing$. Then there exists a non-constant Paneitz-biharmonic sphere $v\in
C^\infty(S^4,N)$ or a non-constant Paneitz-biharmonic half-sphere $C^\infty(S_+^4,N)$ with constant boundary values.
For the proof we will adapt techniques from [@Lin], see also [@Scheven1] for the higher order case. Assume for contradiction that $N$ does not contain any non-constant Paneitz-biharmonic $4$-spheres. Then, after a suitable rotation, Lemma \[blow-up\] yields the existence of a pair $\bar\mu=(c_\ast,\bar\nu)\in\T(\mu)\subset\B$, where $c_\ast\in N$ is a constant map and $\bar\nu$ is the measure on $\overline B_2^+$ given by $$\bar\nu
=c_0\HM^{m-4}\edge\big(\overline B_2^{m-4}\x\{0\}\big)$$ for a positive constant $c_0$. Clearly, we have $\Sigma_{\bar\mu}=\overline B_2^{m-4}\times\{0\}$. By definition of $\B$ there are maps $u_i\in W^{2,2}(B_2^+,N)$ and boundary values $g_i\in
C^{\infty}(B_2^+,N)$ with $(u_i,Du_i)=(g_i,Dg_i)$ on $T_2$ in the sense of traces, so that $$\|D^2u_i\|_{L^{2,m-4}(B_2^+)}^2+\|Du_i\|_{L^{2,m-2}(B_2^+)}^2+\|g_i\|_{C^{4,\alpha}(B_2^+)}^2
\le \Lambda\label{morrey-sequence}$$ for any $i\in\N$, and the following convergence holds as $i\to\infty$. $$\begin{aligned}
u_i\to&\ c_\ast&&\hspace{-8em}\mbox{in }W^{1,2}(B_2^+,N),&\label{W12}\\
u_i\to&\ c_\ast&&\hspace{-8em}\mbox{in }
C^3_{\mathrm{loc}}(B_2^+\setminus(B_2^{m-4}\times\{0\}),N),&\label{C2}\\
g_i\to &\ c_\ast&&\hspace{-8em}\mbox{in }C^4(B_2^+,N),&\label{gC4}\\
\Lm\edge|\Delta u|^2{\overset{\raisebox{-1ex}{\scriptsize $*$}}{\rightharpoondown}}&\ c_0\HM^{m-4}\edge\big(\overline
B_2^{m-4}\x\{0\}\big)&&
\mbox{as measures on $\overline B_2^+$}.&\label{Mass}
\end{aligned}$$ In particular, the convergence follows from Lemma \[defect\](i), and the convergence follows from .
In the sequel we will use the notation $\C_r(x)=\C_r(x',x''):=B_r^{m-4}(x')\x B^4_r(x'')$ for cylinders with barycenter $x=(x',x'')\in\R^{m-4}\x\R^4$, and $\C_r^+(x):=C_r(x)\cap\R^m_+$. The barycenter will be omitted in the notation if it is zero. For the proof of the theorem we will either construct a smooth biharmonic map $\R^4\to N$ or a biharmonic map $\R^4_+\to N$ with constant boundary values as the limit of a blow-up sequence $$\label{rescaled}
v_i(y):=u_i(p_i+\delta_i y)
\mbox{ with $p_i=(p_i',p_i'')\in B^{m-4}_{1/4}\x B^4_{1/8}\cap\R^m_+$ and
$\delta_i\searrow 0$,}$$ where the parameters $p_i$ and $\delta_i$ will be carefully chosen below. The above map is defined for $y\in\C_{R_i}$ with $y_m\ge
-\delta_i^{-1}p_{i,m}$, where $R_i:=1/8\delta_i\to\infty$, and we write $p_{i,m}$ for the $m$-th component of the vector $p_i\in\R^m$. The construction is carried out in several steps.\
[*Step 1.* ]{}We claim that after extracting a subsequence, there holds $$\label{claim1}
\sum_{k=1}^{m-4}\int_{B_{1/2}^+}|D\partial_k u_i|^2\,\d x\to 0,
\qquad\mbox{as }i\to\infty.$$ From the monotonicity formula stated in Theorem \[thm:monotonicity\] we infer $$\begin{aligned}
\label{bdry-mono}
&
4\int_{B_r^+(a)\setminus B_\rho^+(a)}e^{\chi_i|x-a|}\frac{|D\partial_Xu_i|^2}{|x-a|^{m-2}}\,\d x\\\nonumber
&\quad\le
\Phi_{u_i}(a;r)-\Phi_{u_i}(a;\rho)
+
K_i\Psi_{u_i}(a;\rho,r)
\end{aligned}$$ for a.e. $0<\rho<r<1$ and every $a\in B_1^{m-4}\times\{0\}$, where we used the abbreviations $\Phi_{u_i}$ and $\Psi_{u_i}$ introduced in and and $X(x):=x-a$. Moreover, since $g_i\to c_\ast$ in $C^4$, Theorem \[thm:monotonicity\] yields $K_i\to0$ and $\chi_i\to0$ in the limit $i\to\infty$. Lemma \[monodefect\] implies that after passing to a subsequence, we have the convergence $$\begin{aligned}
&\lim_{i\to\infty}\Big(\Phi_{u_{i}}(a;r)-\Phi_{u_{i}}(a;\rho)+K_i\Psi_{u_i}(a;\rho,r)\Big)\\
&\qquad=
r^{4-m}\bar\nu(B_r^+(a))-\rho^{4-m}\bar\nu(B_\rho^+(a))
=0
\end{aligned}$$ for a.e. $0<\rho<r<1$, where the last identity follows from the particular form of $\bar\nu$ and the fact $a\in B_1^{m-4}\times\{0\}$. Using the last formula to pass to the limit in , we deduce $$\begin{aligned}
&\limsup_{i\to\infty}r^{2-m}\int_{B_r^+(a)\setminus
B_\rho^+(a)}|D\partial_Xu_i|^2\,\d x\\
&\qquad\le
\limsup_{i\to\infty}\int_{B_r^+(a)\setminus
B_\rho^+(a)}e^{\chi_i|x-a|}\frac{|D\partial_Xu_i|^2}{|x-a|^{m-2}}\,\d x
=0.
\end{aligned}$$ Moreover, the Morrey bound implies $$\begin{aligned}
\limsup_{i\to\infty}r^{2-m}\int_{B_\rho^+(a)}|D\partial_Xu_i|^2\,\d x
\le
c\rho^{m-2}r^{2-m}\Lambda,
\end{aligned}$$ which can be made arbitrarily small by choosing $\rho>0$ small enough. Combining the last two formulae, we arrive at $$\begin{aligned}
&\lim_{i\to\infty}\int_{B_r^+(a)}|D\partial_Xu_i|^2\,\d x=0
\end{aligned}$$ for a.e. $r\in(0,1]$, where $X(x)=x-a$. We apply this identity once with $a_0:=0$ and once with $a_k:=\tfrac14 e_k\in B_1^{m-4}\times\{0\}$ for $k\in\{1,\ldots,m-4\}$. In this way, we deduce $$\begin{aligned}
&\tfrac1{16}\int_{B_{1/2}^+}|D\partial_ku_i|^2\,\d x\\
&\qquad\le
2\int_{B_{1/2}^+}|D\langle Du_i,x\rangle|^2\,\d x
+
2\int_{B_{1/2}^+}|D\langle Du_i,x-\tfrac14 e_k\rangle|^2\,\d x
\to0
\end{aligned}$$ in the limit $i\to\infty$. This yields the claim .\
[*Step 2 Choice of $p_i'$.*]{} We claim that the parameters $p_i'\in B_{1/4}^{m-4}$ can be chosen with the properties $$\begin{aligned}
\sup_{0<r\le1/8}\sum_{k=1}^{m-4}\,
r^{4-m}\int_{B_r^{m-4}(p_i')\x B_{1/4}^4\cap\R^m_+}|D\partial_k
u_i|^2\,\d x\to 0
\qquad\mbox{as $i\to\infty$}\label{step2}
\end{aligned}$$ and $$\begin{aligned}
\mbox{\ \ the maps $u_i$ are of class $C^\infty$
in a neighborhood of $\{p_i'\}\x\overline{B_{1/4}^4}\cap\R^m_+$.}
\label{step2-smooth}
\end{aligned}$$ For the proof of this claim, we consider the functions $f_i\in
L^1(\R^{m-4})$ defined by $$f_i(x'):=\sum_{k=1}^{m-4}\int_{B_{1/4}^4\cap\R^4_+}|D\partial_k
u_i(x',x'')|^2\,\d x''
\qquad\mbox{for }x'\in B_{3/8}^{m-4}$$ and $f_i(x')=0$ otherwise. Since $B_{3/8}^{m-4}\times B_{1/4}^4\subset
B_{1/2}$, the convergence implies $f_i\to 0$ in $L^1(\R^{m-4})$, as $i\to\infty$. The weak $L^1$-estimate for the Hardy-Littlewood maximal function implies $$\mathcal{L}^{m-4}(\{x'\in\R^{m-4}:\mathcal{M}f_i(x')>\eps_i\})\le\frac{c(m)}{\eps_i}\|f_i\|_{L^1}$$ for every $\eps_i>0$, where we used the abbreviation $$\mathcal{M}f_i(x'):= \sup_{r>0}\
r^{4-m}\int_{B_r^{m-4}(x')}f_i(y')\,\d y'\qquad\mbox{for }x'\in\R^{m-4}.$$ With the choices $\eps_i:=2c(m)\|f_i\|_{L^1}/\mathcal{L}^{m-4}(B_{1/4}^{m-4})\to0$, as $i\to\infty$, the above inequality implies $$\label{max_func}
\mathcal{L}^{m-4}(\{x'\in B_{1/4}^{m-4}:\mathcal{M}f_i(x')>\eps_i\})\le\frac12\mathcal{L}^{m-4}(B_{1/4}^{m-4}).$$ Furthermore, the partial regularity result from Corollary \[cor:partial\] implies $$\begin{aligned}
\nonumber
&\mathcal{L}^{m-4}\left(\left\{x'\in B_{1/4}^{m-4}:
\mbox{there is a $x''\in\overline{B_{1/4}^4}\cap\R^4_+$
with
$(x',x'')\in\operatorname{sing}(u_i)$}\right\}\right) \\
&\quad\le\mathcal{H}^{m-4}(\operatorname{sing}(u_i))=0.\label{reg}
\end{aligned}$$ Because of (\[max\_func\]) and (\[reg\]), there are points $p_i'\in
B_{1/4}^{m-4}$ satisfying and $\mathcal{M}f_i(p_i')\le\eps_i\to 0$ as $i\to\infty$. The latter property implies , so that the claim of Step 2 is verified.\
[*Step 3 Choice of the scaling factors $\delta_i$.*]{} The scaling factors have to be chosen carefully to make sure that the bi-energies of the rescaled maps neither tend to zero nor become unbounded, since we want to obtain a non-constant biharmonic map of finite bi-energy in the limit. In order to preserve a certain energy level during the blow-up, we consider the quantity $$\F_i(\delta):=
\max_{z\in \overline{B^4_{1/4}}\cap\R^m_+}\,\delta^{4-m}\int_{\C_{\delta}^+(p_i',z)}\,
\big(|\Delta u_i|^2+\delta^{-2}|Du_i|^2\big)\,\d x$$ for $\delta\in(0,1]$. We may assume that $p_i'\to:p_0'\in\overline B_{1/4}^{m-4}$ as $i\to\infty$. We will inductively construct a subsequence $i_j\in\N$, $j\in\N_0$, and a decreasing sequence $\delta_{i_j}\searrow 0$ with $$\label{energylevel}
\F_{i_j}(\delta_{i_j})=2^{1-m}\eps_0
\qquad\mbox{for any $j\in\N$}.$$ Note that we did not claim anything for the case $j=0$, so we can simply choose $i_0=1$ and $\delta_0=\frac12$. Now suppose that $i_{j-1}\in\N$ and $\delta_{i_{j-1}}>0$ have already been chosen for some $j\in\N$. For any $\delta_*\in(0,\frac12\delta_{i_{j-1}})$, we infer from and that $$\lim_{i\to\infty}\,\delta_*^{4-m}\int_{\C_{\delta_*}^+(p_i',0)}
\big(|\Delta u_i|^2+\delta_*^{-2}|Du_i|^2\big)\,\d x
=\delta_*^{4-m}\bar\nu(\C_{\delta_*}^+(p_0',0))\ge\eps_0,$$ where the last estimate holds because of $(p_0',0)\in\Sigma_{\bar\mu}$ and $\C_{\delta_*}(p_0',0)\supset B_{\delta_*}(p_0',0)$. Choosing $i_j\in\N$ large enough, depending on $\delta_*$, we can thus achieve $$\F_{i_j}(\delta_*)\ge 2^{1-m}\eps_0.$$ This fixes the index $i_j\in\N$. On the other hand, we know from the choice of $p_i'$ in Step 2 that the map $u_{i_j}$ is smooth on $A_{\delta}:=\overline{B_{\delta}^{m-4}(p_{i_j}')\x
B^4_{1/4+\delta}}\cap\R^m_+$ if $\delta>0$ is chosen small enough. Consequently, $$\F_{i_j}(\delta)\le \max_{A_\delta}\big(|\Delta u_{i_j}|^2+|Du_{i_j}|^2\big)\delta^{2-m}
\Lm(\C_{\delta}^+)\le 2^{-m}\eps_0$$ if $\delta\in(0,\delta_*]$ is chosen small enough in dependence on $i_j$. Combining the last two estimates and applying the intermediate value theorem, we deduce the existence of a number $\delta_{i_j}\in(0,\frac12\delta_{i_{j-1}})$ with the property . This construction yields the desired sequence $\delta_{i_j}\searrow0$ for $j\to\infty$ with , which concludes Step 3. In what follows, we will denote the subsequence $\{i_j\}$ again by $\{i\}$ for simplicity.\
[*Step 4 Choice of $p_i''$.*]{} We choose points $p_i''\in\overline{B_{1/4}^4}\cap\R^4_+$ for which $p_i:=(p_i',p_i'')$ satisfies $$\label{max0}
\delta_i^{4-m}\int_{\C_{\delta_i}^+(p_i)}
\big(|\Delta u_i|^2+\delta_i^{-2}|Du_i|^2\big)\,\d x
=\F_i(\delta_i)=2^{1-m}\eps_0$$ for all $i\in\N$, which is possible by and the definition of $\F_i$. We claim that for all but finitely many values of $i\in\N$, we have $p_i''\in B_{1/8}^4$. Indeed, if this was not the case, after passing to a subsequence we would have $$p_{i}\in\overline{B_{1/4}^{m-4}\x(B_{1/4}^4\setminus
B_{1/8}^4)}\qquad\mbox{for any }i\in\N.$$ We consider a radius $R_0\le\frac1{20}$ to be fixed later independently of $i\in\N$. We can assume $\sqrt2\delta_i\le\frac{R_0}4$ by choosing $i\in\N$ large enough. We write $p_i^0$ for the orthogonal projection of $p_i$ onto $\partial\R^m_+$ and distinguish between the cases $|p_i-p_i^0|<\frac{R_0}4$ and $|p_i-p_i^0|\ge\frac{R_0}4$. In the first case, we observe that $\C^+_{\delta_{i}}(p_i)\subset B^+_{\sqrt 2\delta_i}(p_i)
\subset B^+_{R_0/2}(p_i^0)$. Applying Lemma \[lemma:morrey\] with the center $a=p_i^0\in T_1$, we thus infer $$\begin{aligned}
\label{boundary-case}
&\delta_{i}^{4-m}\int_{\C^+_{\delta_{i}}(p_i)}
\big(|\Delta u_{i}|^2+\delta_{i}^{-2}|Du_{i}|^2\big)\,\d x\\\nonumber
&\qquad\le
cR_0^{4-m}\int_{B^+_{R_0}(p_i^0)}
\big(|\Delta u_{i}|^2+R_0^{-2}|Du_{i}|^2\big)\,\d x
+cR_0\\\nonumber
&\qquad\le
cR_0^{4-m}\int_{B^+_{5R_0/4}(p_i)}
\big(|\Delta u_{i}|^2+R_0^{-2}|Du_{i}|^2\big)\,\d x+cR_0\\
&\qquad\le
cR_0^{2-m}\int_{B_{1/2}^{m-4}\x(B^4_{1/2}\setminus B^4_{1/16})\cap\R^m_+}
\big(|\Delta u_i|^2+|Du_i|^2\big)\,\d x+cR_0,\nonumber
\end{aligned}$$ where we used the property $R_0\le\frac1{20}$ in the last step. In the remaining case $|p_i-p_i^0|\ge\frac{R_0}{4}$, we again assume $\sqrt2\delta_i\le\frac{R_0}4$, which implies $\C_{\delta_{i}}^+(p_i)\subset B_{\sqrt 2\delta_i}(p_i)
\subset B_{R_0/4}(p_i)$. Therefore, Lemma \[lemma:morrey\] implies $$\begin{aligned}
\label{interior-case}
&\delta_{i}^{4-m}\int_{\C^+_{\delta_{i}}(p_i)}
\big(|\Delta u_{i}|^2+\delta_{i}^{-2}|Du_{i}|^2\big)\,\d x\\\nonumber
&\qquad\le
cR_0^{2-m}\int_{B_{R_0/2}(p_i)}
\big(|\Delta u_{i}|^2+R_0^{-2}|Du_{i}|^2\big)\,\d x+cR_0\\\nonumber
&\qquad\le
cR_0^{2-m}\int_{B_{1/2}^{m-4}\x(B^4_{1/2}\setminus B^4_{1/16})\cap\R^m_+}
\big(|\Delta u_i|^2+|Du_i|^2\big)\,\d x+cR_0.
\end{aligned}$$ Now, we recall the choice of $p_i''$, use either or and then the convergences and . In this way, we deduce $$\begin{aligned}
2^{1-m}\eps_0
&=&\lim_{i\to\infty}\delta_{i}^{4-m}\int_{\C^+_{\delta_{i}}(p_i)}
\big(|\Delta u_{i}|^2+\delta_{i}^{-2}|Du_{i}|^2\big)\,\d x\\
&\le&cR_0^{2-m}\lim_{i\to\infty}\int_{B_{1/2}^{m-4}\x(B^4_{1/2}\setminus B^4_{1/16})\cap\R^m_+}
\big(|\Delta u_i|^2+|Du_i|^2\big)\,\d x+cR_0\\
&=& cR_0^{2-m}\,\bar\nu\big(B_{1/2}^{m-4}\x(B^4_{1/2}\setminus B^4_{1/16})\cap\R^m_+\big)+cR_0=cR_0.
\end{aligned}$$ By choosing $R_0$ so small that $cR_0<2^{1-m}\eps_0$, we arrive at the desired contradiction. This yields the claim $p_i''\in B_{1/8}^4$.\
[*Step 5 Blow-up.*]{} As before, we denote the $m$-th component of the vector $p_i\in\R^m$ by $p_{i,m}$. We distinguish between the case $\delta_i^{-1}p_{i,m}\to\infty$ and the case $\delta_i^{-1}p_{i,m}\to b$ for some $b\in[0,\infty)$ in the limit $i\to\infty$. By passing to a subsequence, we can ensure that one of these two alternatives is satisfied. In the first case, we define rescaled maps $v_i$ according to $$v_i(y):= u_i(p_i+\delta_iy),
\qquad\mbox{for $y\in \C_{R_i}$ with $y_m\ge -\delta_i^{-1}p_{i,m}$,}$$ where $R_i:=1/8\delta_i\to\infty$. In the case $\delta_i^{-1}p_{i,m}\to\infty$, the domain of definition of $v_i$ contains arbitrarily large balls centered in the origin. Therefore, this case can be treated analogously as in the interior situation, cf. the arguments following (3.24) in [@Scheven1]. In this way, it is possible to show that the limit map of the rescaled maps $v_i$ is of the form $\hat v(x',x'')=v(x'')$ for a non-constant biharmonic map $v\in C^\infty(\R^4,N)$ with $|D^2u|\in L^2(\R^4)$. Under stereographic projection, this map corresponds to a non-constant Paneitz-biharmonic map $v\in
C^\infty(S^4,N)$, which is a contradiction to our assumptions.
We omit the details, which have been carried out in [@Scheven1], and present only the corresponding arguments in the boundary situation, i.e. in the case $\delta_i^{-1}p_{i,m}\to b<\infty$, as $i\to\infty$. In this situation, we rescale around the orthogonal projections $p_i^0\in\partial\R^m_+$ of $p_i$, i.e. we define maps $v_i\in W^{2,2}(\C_{R_i}^+,N)$ by letting $$v_i(y):= u_i(p_i^0+\delta_iy),
\quad\mbox{for $y\in \C_{R_i}^+$ and $i\in\N$,}$$ where $R_i:=1/8\delta_i\to\infty$. For the corresponding rescaling of the boundary values, we write $$h_i(y):= g_i(p_i^0+\delta_iy),
\quad\mbox{for $y\in \C_{R_i}^+$ and $i\in\N$.}$$ We can assume in particular that $\delta_i^{-1}p_{i,m}<R_i$ for all $i\in\N$. As a consequence of our construction, the maps $v_i$ have the following properties. By and the fact $(p_i^0)''\in B_{1/8}^4$, there holds $$\label{con}
\sum_{k=1}^{m-4}R^{4-m}\int_{B_R^{m-4}\x
B_{R_i}^4\cap\R^m_+}|D\partial_kv_i|^2\,\d x\to 0$$ as $i\to\infty$, for every $R>0$. Moreover, and the definition of $\F_i$ implies $$\begin{aligned}
\label{max}
&\int_{\C_1^+(0,\delta_i^{-1}p_{i,m})}\big(|\Delta v_i|^2+|Dv_i|^2\big)\,\d y\\
&\qquad=\max_{z\in B^4_{R_i}\cap\R^4_+}\int_{\C_1^+(0,z)}\big(|\Delta v_i|^2+|Dv_i|^2\big)\,\d y
=2^{1-m}\eps_0\nonumber
\end{aligned}$$ for all $i\in\N$. Finally, from we infer $$\label{mor}
\sup_{B_\rho^+(a)\subset \C_{R_i}^+}\rho^{4-m}\int_{B_\rho^+(a)}
\big(|D^2v_i|^2+\rho^{-2}|Dv_i|^2\big)\,\d y\le \Lambda$$ for all $i\in\N$. From we deduce by Rellich’s theorem, combined with a diagonal sequence argument, that there is a limit map $\hat v\in W^{1,2}_{\mathrm{loc}}(\R^m_+,N)$ with $v_i\to\hat v$ strongly in $W^{1,2}(B_R^+,N)$ for every $R>0$ and almost everywhere, after passing to a subsequence. For a fixed $R>0$, we may additionally find a subsequence $\{i_j\}\subset\N$ with $v_{i_j}\wto \hat v$ weakly in $W^{2,2}(B_R^+,\R^L)$. Therefore, implies for every $1\le k\le m-4$ $$\label{conhat}
\int_{B_R}|D\partial_k\hat
v|^2\,\d y\le\lim_{j\to\infty}\int_{B_R}|D\partial_kv_{i_j}|^2\,\d y=0
\qquad\mbox{for any }R>0.$$ As a consequence, $\partial_k\hat v$ is constant, and since $N$ is compact, even $\partial_k\hat v\equiv 0$ for every $1\le k\le m-4$. This implies that there is a map $v\in W^{2,2}_{\textrm{loc}}(\R^4_+,N)$ with $\hat v(y',y'')=v(y'')$ for a.e. $(y',y'')\in\R^{m-4}\times\R^4_+$. Furthermore, we recall for further reference that the strong convergence $v_i\to\hat v$ in $W^{1,2}(B_R^+,N)$ implies $$\label{con1}
\lim_{i\to\infty}\int_{B_R^+}|\partial_k v_i|^2\,\d y=0
\qquad\mbox{for all $R>0$ and $1\le k\le m-4$.}$$ \
[*Step 6 ($C^3$-convergence).* ]{} In order to establish local $C^3$-convergence $v_i\to \hat v$, we will show that the bi-energy of the maps $v_i$ is small on every cylinder $\C_{1/2}(a)$ for $a\in B_R^+$, if $i>i_0(R)$ is sufficiently large. More precisely, we will show that the quantities $$\CF(a):=\int_{\C_1^+}\big(|\Delta v_i|^2+|Dv_i|^2\big)(y+a)\psi(y)\,\d y$$ are small, where $a\in\C^+_{R_i-1}$ and $\psi\in C^\infty_{\mathrm{cpt}}(\C_1,[0,1])$ is a cut-off function with $\psi\equiv 1$ on $\C_{1/2}$ and $|D^2\psi|+|D\psi|\le c$ for some constant $c=c(m)$. For $1\le k\le m-4$, we will use the test vector field $\xi:=\psi_ae_k$, where $\psi_a(y):=\psi(y-a)$, in the differential equation for the maps $v_i$. Note that the maps $v_i$ also satisfy by scaling invariance, and that the test vector field is admissible since $e_k$ is tangential to $\partial\R^m_+$. Before applying the differential equation, we calculate $$\begin{aligned}
\frac{\partial}{\partial a_k}\,\CF(a)
&=&-\int_{\R^m}\big(|\Delta v_i(y)|^2+|Dv_i(y)|^2\big)\partial_k\psi(y-a)\,\d y\\
&=&-\int_{\R^m}\big(|\Delta v_i|^2\operatorname{div}\xi-2Dv_i\cdot D\partial_kv_i\,\psi_a\big)\,\d y,
\end{aligned}$$ using integration by parts in the last step. We re-write the first term in the last integral by an application of the differential equation for the maps $v_i$, with the result $$\begin{aligned}
&\left|\frac{\partial}{\partial a_k}\,\CF(a)\right|^2\\
&\qquad\le
2\left|\int_{\R^m}
4\Delta v_i\cdot \partial_\ell\partial_k v_i\,\partial_\ell\psi_a
+2\Delta v_i\cdot\partial_kv_i\Delta\psi_a
-2(Dv_i\cdot D\partial_kv_i)\psi_a
\,\d y\right|^2\\
&\qquad\qquad+
2\left|\int_{\R^m}
2\Delta v_i\cdot \Delta \big[\Pi(v_i)(\psi_a\partial_kh_i)\big]
\,\d y\right|^2\\
&\qquad\le c\|v_i\|^2_{W^{2,2}(\C_1^+(a))}\int_{\C_1^+(a)}
\big(|D\partial_kv_i|^2+|\partial_kv_i|^2\big)\,\d y\\
&\qquad\qquad
+c\|\Delta
v_i\|_{L^2(\C_1^+(a))}^2(1+\|v_i\|^2_{W^{2,2}(\C_1^+(a))})
\|Dh_i\|^2_{C^2(\C_1^+(a))}.
\end{aligned}$$ The right-hand side vanishes in the limit $i\to\infty$ because the sequence $v_i$ is bounded in $W^{2,2}(\C_1^+(a))$ by and we have the convergences , , and $h_i\to c_\ast$ in $C^4(\C_1^+(a))$. Therefore, the above estimate implies $\frac{\partial}{\partial a_k}\,\CF\to 0$ uniformly on $B_R^+$ for every $R>0$ and every $1\le k\le m-4$, as $i\to\infty$. Since we know furthermore $\CF((0,a''))\le2^{1-m}\eps_0$ for all $a''\in B_{R_i}^4\cap \R^4_+$ by , we arrive at $$\int_{\C_{1/2}^+(a)}\big(|\Delta v_i|^2+|Dv_i|^2\big)\,\d y\le\CF(a)<2^{2-m}\eps_0$$ for all $a\in B_R^+$, if $i\ge i_0(R)$ is chosen sufficiently large. Applying Corollary \[cor:uniform\] on $B_{1/2}^+(a)\subset\C_{1/2}^+(a)$, we infer the bound $$\sup_{i\in\N}\|Dv_i\|_{C^3(B^+_{1/4}(a))}\le c(\Lambda,\alpha,m,N)
\qquad\mbox{for all }a\in B_R^+.$$ By Arzéla-Ascoli, this implies convergence $v_i\to\hat v$ in $C^3(B_R^+,N)$. Since we have already established almost everywhere convergence $v_i\to\hat v$, it is not necessary to pass to another subsequence. Therefore, the $C^3$-convergence holds on every ball $B_R^+$ with $R>0$.\
[*Step 7 (Conclusion).* ]{} Keeping in mind that $\delta_i^{-1}p_{i,m}\to b\in[0,\infty)$ as $i\to\infty$, we deduce from the $C^3$-convergence and the identity that $$\begin{aligned}
&\int_{\C_1^+(0,b)}\big(|\Delta\hat v|^2+|D\hat v|^2\big)\,\d y\\
&\qquad
=\lim_{i\to\infty}\int_{\C_1^+(0,\delta_i^{-1}p_{i,m})}\big(|\Delta v_i|^2+|Dv_i|^2\big)\,\d y=2^{1-m}\eps_0>0.
\end{aligned}$$ Consequently, the map $\hat v$, and thus also the restriction $v=\hat v|_{\{0\}\x\R^4_+}$, is not constant. On the other hand, implies for any $R>0$ $$\int_{B_R^4\cap\R^4_+}|D^2v|^2\,\d y
=c(m)R^{4-m}\int_{\C_R^+}|D^2\hat v|^2\,\d y\le c(m)\Lambda,$$ which yields $D^2v\in L^2(\R^4_+,N)$. Since the maps $v_i$ are weakly biharmonic and converge to $\hat v$ in $C^3_{\mathrm{loc}}(\R^m_+,N)$, the map $\hat v$ is biharmonic as well, and so is its restriction $v=\hat v|_{\{0\}\x\R^4_+}$. As a consequence, we even have $v\in C^\infty(\R^4_+,N)$, cf. [@LammWang]. Thus, $v\in C^\infty(\R^4_+,N)$ is a non-constant biharmonic map with finite bi-energy and constant boundary values on $\partial\R^4_+$. By stereographic projection and the conformal invariance of the Paneitz-bi-energy, this corresponds to a smooth, non-constant Paneitz-biharmonic $4$-halfsphere with constant boundary values, cf. Lemma \[lem:liouville-4\](ii). This completes the proof of the theorem.
Proof of Theorem \[thm:compact\]
--------------------------------
We consider a sequence $u_i\in W^{2,2}(B_4^+,N)$, $i\in\N$, of variationally biharmonic maps with respect to Dirichlet values $g_i\in C^\infty(B_4^+,N)$ on $T_4$ for which $$\sup_{i\in\N}\big(\|u_i\|_{W^{2,2}(B_4^+)}+\|g_i\|_{C^{4,\alpha}(B_4^+)}\big)<\infty$$ holds true, for some $\alpha\in(0,1)$. In view of Lemma \[lemma:morrey\], this estimate implies the uniform Morrey space bound $$\sup_{i\in\N}\big(\|D^2u_i\|_{L^{2,m-4}(B_2^+)}+\|Du_i\|_{L^{2,m-2}(B_2^+)}\big)<\infty.$$ This means that there exists a constant $\Lambda\ge1$ so that $(u_i,0)\in\B$ for all $i\in\N$. Therefore, after passing to a subsequence we have convergence $(u_i,0)\Bto(u,\nu)\in\B$ in the sense of Definition \[def:BLambda\], in the limit $i\to\infty$. Let us assume for contradiction that there is no strong subconvergence $u_i\to u$ in $W^{2,2}(B_{1}^+,\R^L)$. Then Lemma \[defect\](ii) implies $~\operatorname{spt}\nu\cap\overline B_1^+\neq\varnothing$. Therefore, Theorem \[sphere\] yields the existence of a non-constant Paneitz-biharmonic $4$-sphere or non-constant Paneitz-biharmonic $4$-halfsphere with constant boundary values. But the existence of such maps is excluded by the assumptions of Theorem \[thm:compact\], so that the claimed strong convergence holds true. $\square$\
Liouville type theorems for biharmonic maps on a half space {#sec:liouville}
===========================================================
The next theorem excludes the existence of certain non-constant biharmonic maps which might occur as tangent maps in singular boundary points. We remark that since these maps are homogeneous of degree zero, they can also be interpreted as maps $v:S^{m-1}_+\to N$ that are biharmonic with respect to a certain Paneitz-bi-energy, cf. [@Scheven1 Lemma 5.1]. We note that the case $m=5$, which corresponds to Paneitz-biharmonic $4$-halfspheres, can not be treated with the same methods and will be postponed to Lemma \[lem:liouville-4\].
\[thm:Liouville\] Let $m\ge 6$ and assume that $u\in W^{2,2}_{\mathrm{loc}}(\R^m_+,N)$ is a stationary biharmonic map that is homogeneous of degree zero and admits constant Dirichlet boundary values in the sense $(u,Du)=(c,0)$ on $\partial\R^m_+$. Moreover, we assume that $u$ is smooth in a neighbourhood of $\partial\R^m_+\setminus\{0\}$. Then $u$ is constant in $\R^m_+$.
We consider the vector field $\xi\in C^\infty(\R^m_+,\R^m)$ defined by $\xi(x):=\eta(|x|)e_m$ for a function $\eta\in
C^\infty_0((0,\infty),[0,\infty))$. Since this vector field does not vanish on $\partial\R^m_+$, it is not admissible in the differential equation . However, because $u$ is smooth in a neighbourhood of $\partial\R^m_+\setminus\{0\}$, we infer from Gauß’ theorem that $$\begin{aligned}
\nonumber\label{PDE-stationary}
\mathrm{I}:=&\int_{\R^m_+} \big(4\Delta
u\cdot \partial_i\partial_ju\,\partial_i\xi_j
+2\Delta u\cdot \partial_iu\,\Delta\xi_i
-|\Delta u|^2\partial_i\xi_i\big)\,\d x\\\nonumber
&\quad=
-\int_{\partial\R^m_+}\big(4\Delta
u\cdot \partial_m\partial_ju\,\xi_j
+2\Delta u\cdot \partial_iu\,\partial_m\xi_i\\
&\qquad\qquad\qquad\qquad
-2\partial_m(\Delta u\cdot\partial_iu)\xi_i
-|\Delta u|^2\xi_m\big)\,\d x=:\mathrm{II},
\end{aligned}$$ where we used the convention to sum all double indices from $1$ to $m$. Using the definition of $\xi$ and the fact $Du=0$ on $\partial\R^m_+$, we compute $$\begin{aligned}
\nonumber\label{calc-II}
\mathrm{II}
&=
-\int_{\partial\R^m_+}\eta(|x|)\big(4\Delta
u\cdot \partial_m^2u
-2\partial_m(\Delta u\cdot\partial_mu)
-|\Delta u|^2\big)\,\d x\\\nonumber
&=
-\int_{\partial\R^m_+}\eta(|x|)\big(2\Delta
u\cdot \partial_m^2u
-|\Delta u|^2\big)\,\d x\\
&=
-\int_{\partial\R^m_+}\eta(|x|)\,|\partial_m^2 u|^2\,\d x.
\end{aligned}$$ In the last step, we used the fact that $u$ is constant on $\partial\R^m_+$, which implies $\Delta u=\partial^2_mu$ on $\partial\R^m_+$. Moreover, with the abbreviation $r=|x|$, the definition of $\xi$ implies $$\begin{aligned}
\mathrm{I}&=
\int_{\R^m_+}\tfrac1r \eta'(r)4\Delta u\, x_i\partial_i\partial_mu\,\d x\\
&\qquad+\int_{\R^m_+} 2\Delta u\cdot \partial_m u
\,r^{1-m}\tfrac{\partial}{\partial r}(r^{m-1}\eta'(r))\,\d x\\
&\qquad-\int_{\R^m_+} \tfrac1r \eta'(r) |\Delta u|^2x_m\,\d x\\
&=: \mathrm{I}_1+\mathrm{I}_2+\mathrm{I}_3.
\end{aligned}$$ Since $u$ is homogeneous of degree zero, the derivative $\partial_mu$ is homogeneous of degree $-1$, which implies $x_i\partial_i\partial_mu=-\partial_mu$. By the homogeneity of $u$, we thus infer $$\begin{aligned}
\label{calc-I1}
\mathrm{I}_1
=
-4\int_0^\infty r^{m-5}\eta'(r) \,\d r \int_{S_1^+}\Delta u\cdot \partial_mu \,d\mathcal{H}^{m-1}.
\end{aligned}$$ Using integration by parts, we re-write the term $\mathrm{I}_2$ to $$\begin{aligned}
\label{calc-I2}
\mathrm{I}_2
&=
2\int_0^\infty r^{-3}\tfrac{\partial}{\partial r}(r^{m-1}\eta'(r))\,\d r
\int_{S_1^+} \Delta u\cdot \partial_m u\,d\mathcal{H}^{m-1}\\\nonumber
&=
6\int_0^\infty r^{m-5}\eta'(r)\,\d r
\int_{S_1^+} \Delta u\cdot \partial_m u\,d\mathcal{H}^{m-1}.
\end{aligned}$$ At this point, we used the assumption that $\eta$ has compact support in $(0,\infty)$, which implies that the boundary terms vanish. Finally, we compute $$\begin{aligned}
\label{calc-I3}
\mathrm{I}_3
=
-\int_0^\infty r^{m-5}\eta'(r)\,\d r
\int_{S_1^+} |\Delta u|^2x_m\,d\mathcal{H}^{m-1}.
\end{aligned}$$ Plugging , , , and into , we deduce $$\begin{aligned}
\label{conclusion-1}
&-\int_0^\infty r^{m-5}\eta'(r)\,\d r
\int_{S_1^+} \big[|\Delta u|^2x_m-2\Delta
u\cdot\partial_mu\big]\,d\mathcal{H}^{m-1}\\\nonumber
&\qquad=
-\int_{\partial\R^m_+}\eta(|x|)\,|\partial_m^2 u|^2\,\d x\le0.
\end{aligned}$$ Using the homogeneity of $u$ and Gauß’ theorem, we furthermore compute $$\begin{aligned}
-\int_{S_1^+}2\Delta u\cdot\partial_mu\,d\mathcal{H}^{m-1}
&=
-(m-3)\int_{B_1^+}2\Delta u\cdot\partial_mu\,\d x\\
&=
(m-3)\int_{B_1^+}2 Du\cdot\partial_m Du\,\d x,
\end{aligned}$$ where the boundary integrals vanish because of $\frac{\partial u}{\partial r}=0$ on $S_1^+$ and $\partial_mu=0$ on $\partial\R^m_+$. Another application of Gauß’ theorem implies $$\begin{aligned}
-\int_{S_1^+}2\Delta u\cdot\partial_mu\,d\mathcal{H}^{m-1}
&=
(m-3)\int_{B_1^+}\partial_m|Du|^2\,\d x\\
&=
(m-3)\int_{S_1^+}x_m|Du|^2 \,d\mathcal{H}^{m-1}.
\end{aligned}$$ We use this identity in and integrate by parts with respect to $r$. In this way, we arrive at $$\begin{aligned}
\label{final-liouville}
&(m-5)\int_0^\infty r^{m-6}\eta(r)\,\d r
\int_{S_1^+} x_m\big[|\Delta u|^2+(m-3)|Du|^2\big]
\,d\mathcal{H}^{m-1}\\\nonumber
&\qquad=
-\int_{\partial\R^m_+}\eta(|x|)\,|\partial_m^2 u|^2\,\d x\le0.
\end{aligned}$$ Since $m>5$ and $\eta\ge0$, we infer $|Du|=0$ on $S_1^+$. By the homogeneity of $u$, this implies that $u$ is constant on $\R^m_+$, as claimed.
\[rem:m=5\]
In the case $m=5$, we obtain the result corresponding to Theorem \[thm:Liouville\] under the additional assumption of non-existence of Paneitz-biharmonic $4$-halfspheres.
\[lem:liouville-4\] Assume that the manifold $N$ does not carry any non-constant Paneitz-biharmonic $4$-halfspheres with constant boundary values. Then the following two statements holds true.
1. Any biharmonic map $u\in C^\infty(\R^5_+\setminus\{0\},N)$ that is homogeneous of degree zero and attains constant boundary values $(u,Du)=(c,0)$ on $\partial\R^5_+$ is constant.
2. Any biharmonic map $w\in C^\infty(\R^4_+,N)$ with finite bi-energy and constant boundary values $(w,Dw)=(c,0)$ on $\partial\R^4_+$ is constant.
\(i) Since $u$ is homogeneous of degree zero, we have $$\label{Bi-Laplace}
\Delta^2u=\big(\tfrac{\partial^2}{\partial
r^2}+\tfrac{4}{r}\tfrac{\partial}{\partial
r}+\tfrac1{r^2}\Delta_S)\tfrac1{r^2}\Delta_Su
=\tfrac1{r^4}(\Delta_S^2u-2\Delta_Su)$$ in $\R^5_+\setminus\{0\}$, where $\Delta_S$ denotes the Laplace-Beltrami operator on $S^4_+$, and we abbreviated $r=|x|$. Since $u$ is biharmonic on $\R^5_+\setminus\{0\}$, we deduce that the restriction $v:=u|_{S^4_+}$ is Paneitz-biharmonic on $S^4_+$. Hence, the map $v$ is constant by assumption, and so is its homogeneous extension $u$.
\(ii) By stereographic projection, the biharmonic map $w\in
C^\infty(\R^4,N)$ gives rise to a map $v\in
C^\infty(S^4_+\setminus\{e_1\},N)$. The conformal invariance of the Paneitz-bi-energy, see [@Paneitz; @Chang], implies that $$P_{S^4_+}(v)=\int_{\R^4_+}|\Delta w|^2\,\d x<\infty$$ and that $v\in W^{2,2}(S^4_+,N)$ is a critical point of the Paneitz-bi-energy. More precisely, the map $v$ is Paneitz-biharmonic on $S^4_+\setminus\{e_1\}$. To conclude, it remains to show that the singularity in $e_1$ can be removed. To this end, we consider the homogeneous extension $u(x):=v(\frac{x}{|x|})$. The identity implies that $u\in W^{2,2}_{\mathrm{loc}}(\R^5_+,N)$ is biharmonic on $\R^5_+\setminus(\R_{\ge0}e_1)$. Since $\R_{\ge0}e_1$ is a one-dimensional set and is therefore negligible with respect to the $W^{2,2}$-capacity, we infer that $u$ is weakly biharmonic on $\R^5_+$. Moreover, since $u$ is the homogeneous extension of a map in $W^{2,2}\cap W^{1,4}(S^4_+,N)$, for any $\eps>0$ we can choose a radius $\delta>0$ sufficiently small to ensure $$\label{small-Morrey}
\sup_{B_\rho^+(y)\subset
B_\delta(e_1)}\frac1\rho\int_{B_\rho^+(y)}\big(|D^2u|^2+|Du|^4\big)\d
x\le\eps.$$ Therefore, the $\eps$-regularity result from [@GongLammWang Lemma 3.1] can be applied, which yields that $v$ is smooth in a neighbourhood of $e_1$. We point out that in view of , no monotonicity formula is required for this result. Consequently, we have shown that $v\in C^\infty(S^4_+,N)$ is a Paneitz-biharmonic map with constant Dirichlet boundary values. By assumption, the map $v$ is constant, and so is the map $w\in
C^\infty(\R^4_+,N)$. This completes the proof of the lemma.
We end this section with an example of a target manifold for which our standing assumptions on the non-existence of Paneitz-biharmonic spheres are satisfied.
\[prop:flat-torus\] Let $N=S_{r_1}^1\times\ldots\times S_{r_n}^1\subset\R^{2n}$ be a flat torus, where $n\ge1$ and $r_i>0$ for $i=1,\ldots,n$. Then $N$ does neither carry a non-constant Paneitz-biharmonic $4$-sphere nor a non-constant Paneitz-biharmonic $4$-halfsphere with constant boundary values.
The assertion concerning the full spheres has already been proved in [@Scheven1 Lemma 5.4]. Here, we demonstrate that the same argument can be applied to yield the constancy of the corresponding halfspheres with constant boundary values.
To this end, we consider a Paneitz-biharmonic map $u\in C^\infty(S^4_+,N)$ with constant boundary values in the sense $(u,Du)=(c,0)$ on $\partial
S^4_+$. We extend $u$ to a function on $\R^5_+\setminus\{0\}$ that is homogeneous of degree zero. Because of identity , the extended function, which we still denote by $u$, is biharmonic on $\R^5_+\setminus\{0\}$. Therefore, Remark \[rem:m=5\] implies $D^2u\equiv0$ on $\partial\R^5_+$. For every vector field $V\in C^\infty(S^4_+,\R^{2n})$ with $V(x)\in T_{u(x)}N$ for every $x\in S^4_+$, the fact $\Delta_S^2u(x)-2\Delta_S u(x)\perp T_{u(x)}N$ for all $x\in S^4_+$ and two integrations by parts imply $$\begin{aligned}
\label{weak-Euler}
0&=\int_{S^4_+}(\Delta_S^2u-2\Delta_Su)\cdot V\,\d\HM^4\\\nonumber
&=\int_{S^4_+}(\Delta_S u\cdot \Delta_S V-2\Delta_Su\cdot V)\,\d\HM^4
+\int_{\partial S^4_+}\big( \Delta_Su\cdot\partial_5V-\partial_5\Delta_S u\cdot V\big)\d\HM^3.
\end{aligned}$$ Since $\Delta_Su=0$ on $\partial S^4_+$, the boundary integral vanishes for every test vector field with $V\equiv0$ on $\partial S^4_+$. In particular, this holds true for the choice $V:=\Delta_S^\top u$, where $\Delta_S^\top$ denotes the tangential part of the Laplace-Beltrami operator on $S^4_+$. We assume that $V$ is extended to a function on $\R^5_+\setminus\{0\}$ that is homogeneous of degree zero. More precisely, this means that $V(x)=|x|^2\nabla^u_{e_i}\partial_iu(x)$ for any $x\in\R^5_+\setminus\{0\}$, where we used the notation $\nabla^u$ for the covariant derivative on the bundle $u^\ast\mathrm{T}N$. Since $u$ and $V$ are both homogeneous of degree zero, we have $$\Delta_S u
=
\partial_i\partial_i u
=
\Delta_S^\top u+A(u)(\partial_iu,\partial_iu)$$ and $$\Delta_SV
=
\partial_k\partial_kV
=
\partial_k\big[\nabla^u_{e_k}V+A(u)(\partial_ku,V)\big],$$ where $A$ denotes the second fundamental form of the submanifold $N\subset\R^{2n}$. Moreover, we employ the usual summation convention and sum all double indices from $1$ to $5$. Consequently, choosing $V:=\Delta_S^\top u$ in , we infer $$\begin{aligned}
\label{euler-lagrange}
0&=
\int_{S^4_+}(\Delta_S u\cdot\Delta_S V-2\Delta_Su\cdot
V)\,\d\HM^4\\
\nonumber
&=\int_{S^4_+}
\Big(\Delta_S^\top u\cdot \partial_k \nabla^u_{e_k}V
+
\Delta_S^\top u\cdot \partial_k[A(u)(\partial_ku,V)]\Big)
\d\HM^4\\\nonumber
&\qquad+
\int_{S^4_+}\Big(A(u)(\partial_iu,\partial_iu)\cdot \partial_k\nabla^u_{e_k}V
+
A(u)(\partial_iu,\partial_iu)\cdot\partial_k[A(u)(\partial_ku,V)\big]\Big)\,\d\HM^4\\\nonumber
&\qquad-\int_{S^4_+}2\Delta_S^\top u\cdot V\d\HM^4.
\end{aligned}$$ In the first and the third term appearing on the right-hand side, we use the identity $$\partial_k\nabla^u_{e_k}V
=
\nabla^u_{e_k}\nabla^u_{e_k}V+A(u)(\partial_ku,\nabla^u_{e_k}V),$$ which follows from the definition of the covariant derivative and the second fundamental form. Next, using the equations by Weingarten and Gauß and keeping in mind that the Riemannian curvature of $N$ vanishes, we deduce $$\begin{aligned}
\Delta_S^\top u\cdot \partial_k[A(u)(\partial_ku,V)]
&=
-A(u)(\partial_ku,\Delta_S^\top u)\cdot A(u)(\partial_ku,V)\\
&=
-A(u)(\partial_ku,\partial_ku)\cdot A(u)(\Delta_S^\top u,V).
\end{aligned}$$ Finally, because the flat torus has a parallel second fundamental form, i.e. $\nabla^\perp A\equiv 0$, we can compute $$\begin{aligned}
\partial_k\big[A(u)(\partial_ku,V)\big]
&=
A(u)(\Delta_S^\top u,V)+A(u)(\partial_ku,\nabla^u_{e_k}V).
\end{aligned}$$ Using the preceding observations in , we deduce $$\begin{aligned}
\label{flat-torus-1}
0&=
\int_{S^4_+}\big[\Delta_S^\top u\cdot
\nabla^u_{e_k}\nabla^u_{e_k}V
-2\Delta_S^\top u\cdot V\big]\d\HM^4\\\nonumber
&\quad+
2\int_{S^4_+}A(u)(\partial_iu,\partial_iu)\cdot
A(u)(\partial_ku,\nabla^u_{e_k}V)\,\d\HM^4\\\nonumber
&=:\mathrm{I}+\mathrm{II}.
\end{aligned}$$ Next, we use the identity $\nabla^u_{e_k}\nabla^u_{e_k}V=\operatorname{tr}_S(\nabla^u\nabla^uV)$ on $S^4_+$, where we abbreviated $\operatorname{tr}_S$ for the trace on $\mathrm{T}S^4_+$. Integrating by parts and recalling the definition of $V$, we then compute $$\label{eq:I}
\mathrm{I}
=
-\int_{S^4_+}\big[|\nabla^u\Delta_S^\top u|^2+2|\Delta_S^\top
u|^2\big]\d\HM^4\le
-2\int_{S^4_+}|\Delta_S^\top
u|^2\d\HM^4.$$ Since $N$ has vanishing Riemannian curvature, we have the following identity on $S^4_+$. $$\begin{aligned}
\nabla_{e_k}^uV
&=
\nabla_{e_k}\big(|x|^2\nabla^u_{e_\ell}\partial_\ell u\big)\\
&=
\nabla_{e_\ell}^u\nabla^u_{e_\ell}\partial_k u
+2x_k\nabla_{e_\ell}^u\partial_\ell u\\
&=
\operatorname{tr}_S(\nabla^u\nabla^u\partial_ku)
+\big(\tfrac{\nabla^2}{\partial r^2}+4\tfrac{\nabla}{\partial
r}\big)\partial_ku+2x_k\nabla_{e_\ell}^u\partial_\ell u\\
&=
\operatorname{tr}_S(\nabla^u\nabla^u\partial_ku)
-2\partial_ku
+2x_k\nabla_{e_\ell}^u\partial_\ell u,
\end{aligned}$$ where the last step follows from the fact that $\partial_ku$ is homogeneous of degree $-1$. We use this identity, recall the fact $x_k\partial_ku=0$ and integrate by parts. Since the second fundamental form is parallel, we infer $$\begin{aligned}
\label{flat-torus-2}
\mathrm{II}&=
2\int_{S^4_+}A(u)(\partial_iu,\partial_iu)\cdot
\operatorname{tr}_S \big[A(u)(\partial_ku,\nabla^u\nabla^u\partial_ku)\big]\,\d\HM^4\\\nonumber
&\qquad-4\int_{S^4_+}|A(u)(\partial_iu,\partial_iu)|^2\,\d\HM^4\\\nonumber
&=
-
2\int_{S^4_+}A(u)(\partial_iu,\partial_iu)\cdot
\operatorname{tr}_S\big[
A(u)(\nabla^u\partial_ku,\nabla^u\partial_ku)\big]\,\d\HM^4\\\nonumber
&\qquad-4\int_{S^4_+}\operatorname{tr}_S\big[A(u)(\partial_iu,\nabla^u\partial_iu)\cdot
A(u)(\partial_ku,\nabla^u\partial_ku)\big]\,\d\HM^4\\\nonumber
&\qquad
-4\int_{S^4_+}|A(u)(\partial_iu,\partial_iu)|^2\,\d\HM^4.
\end{aligned}$$ We note that the boundary term appearing in the integration by parts vanishes because of $Du\equiv0$ on $\partial\R^5_+$. The second integral on the right-hand side is non-positive, and so is the first one, because the Gauß equations on the flat manifold $N$ imply $$\begin{aligned}
&-2A(u)(\partial_iu,\partial_iu)\cdot
\operatorname{tr}_S\big[A(u)(\nabla^u\partial_ku,\nabla^u\partial_ku)\big]\\
&\qquad=
-2\operatorname{tr}_S\big[A(u)(\partial_iu,\nabla^u\partial_ku)\cdot
A(u)(\partial_iu,\nabla^u\partial_ku)\big]
\le0.
\end{aligned}$$ Consequently, equation implies $$\label{eq:II}
\mathrm{II}
\le
-4\int_{S^4_+}|A(u)(\partial_iu,\partial_iu)|^2\,\d\HM^4.$$ Using and in , we arrive at $$\int_{S^4_+}\big[2|\Delta_S^\top u|^2
+
4|A(u)(\partial_iu,\partial_iu)|^2\big]\,\d\HM^4
\le0.$$ This implies that both the tangential and the normal part of $\Delta u$ vanish on $S^4_+$, which means that $u:\R^5_+\setminus\{0\}\to\R^{2n}$ is a harmonic function. Classical regularity results for harmonic functions now yield $u\in C^\infty(\R^5_+,N)$. Since $u$ is homogeneous of degree zero, this implies that it is a constant map, as claimed. This completes the proof of the proposition.
Full boundary regularity {#sec:dim-red}
========================
Properties of tangent maps
--------------------------
The strategy for the proof of the full boundary regularity is to apply the dimension reduction argument by Federer in order to prove that the dimension of the singular set is zero. To this end, we first provide some properties of tangent maps of a variationally biharmonic map $u\in W^{2,2}(B^+_2,N)$ in a boundary point $a\in T_1$. For $r\in(0,1)$ we define the rescaled maps $$u_{a,r}\in W^{2,2}(B_{1/r}^+\,,N)\mbox{\ \ \ by\ \ \ }u_{a,r}(x):=u(a+rx).$$ A map $v\in W^{2,2}_{\mathrm{loc}}(\R^m_+,N)$ is called a [*tangent map*]{} of $u$ at the point $a\in T_1$ if there is a sequence $r_i\searrow 0$ with $u_{a,r_i}\to v$ strongly in $W^{2,2}_{\mathrm{loc}}(\R^m,N)$ as $i\to\infty$.\
The Compactness Theorem \[thm:compact\] is crucial at this point to ensure the strong convergence of suitable rescaled maps to a tangent map. Moreover, we obtain the following structure theorem for tangent maps of biharmonic maps.
\[theo:tangent\_maps\] Assume that $u\in W^{2,2}(B_2^+,N)$ is a weakly biharmonic map that satisfies the stationarity condition and satisfies $(u,Du)=(g,Dg)$ on $T_2$ in the sense of traces. Moreover, we assume that $N$ does not carry any nontrivial Paneitz-biharmonic $4$-spheres nor Paneitz-biharmonic $4$-halfspheres with constant boundary data. Then the following statements are true:
1. For any $a\in T_1$ and any sequence $r_i\searrow 0$ there is a tangent map $v\in W^{2,2}_{loc}(\R^m_+,N)$ so that $u_{a,r_i}$ subconverges to $v$ in $W^{2,2}_{loc}(\R^m_+,N)$ as $i\to\infty$. The tangent map is again weakly biharmonic, satisfies and attains constant Dirichlet boundary values on $\partial\R^m_+$.
2. Every tangent map of $u$ in a point $a\in T_1$ is homogeneous of degree zero.
3. Let $s\ge 0$. At $\mathcal{H}^{s}$-a.e. point $a\in\operatorname{sing}(u)\cap T_1$ there exists a tangent map $v$ of $u$ with $\mathcal{H}^{s}(\operatorname{sing}(v)\cap T_1)>0$.
The strong subconvergence $u_{a,r_i}\to v$ to some tangent map $v\in W^{2,2}_{\mathrm{loc}}(\R^m_+,N)$ is a consequence of the Compactness Theorem \[thm:compact\]. The differential equations and are preserved under strong convergence, which yields the statements on the biharmonicity of the tangent map. The boundary values of the rescaled maps $u_{a,r_i}$ are given by the maps $g_{a,r_i}$, which satisfy $$\sup_{B_{1/r}^+}\big(|D^2g_{a,r_i}|+|Dg_{a,r_i}|\big)
\le
r_i\|g\|_{C^2}\to0,
\qquad\mbox{as }i\to\infty.$$ This implies $g_{a,r_i}\to c$ for a constant $c\in N$ and proves that the tangent map attains constant boundary values. This completes the proof of (i).
Our next goal is to show (ii). To this end, let $v$ be a tangent map of $u$ in a point $a\in T_1$. Since $v$ possesses constant boundary values, the monotonicity formula from Theorem \[thm:monotonicity\] with center $a=0$ reads $$\label{v_monoton}
4\int_{B_r^+\setminus B_\rho^+}\left(\frac{|D\,\partial_X
v|^2}{|x|^{m-2}}+(m-2)\frac{|\partial_X v|^2}{|x|^m}\right)\d x
\le
\Phi_v(0,r)-\Phi_v(0,\rho)$$ for a.e. $0<\rho<r$, where $X(x)=x$. The strong convergence $u_{a,r_i}\to v$ implies that the functions $f_i(\rho):=\Phi_{u_{a,r_i}}(0,\rho)$ converge in $L^1_{\mathrm{loc}}([0,\infty))$ to $f(\rho):=\Phi_v(0,\rho)$, as $i\to\infty$. Therefore, for any $R>0$ we deduce $$\begin{aligned}
{\mathchoice
{{{\setbox0=\hbox{$\displaystyle{\textstyle-}{\int}$}
\vcenter{\hbox{$\textstyle-$}}\kern-.5\wd0}}}
{{{\setbox0=\hbox{$\textstyle{\scriptstyle-}{\int}$}
\vcenter{\hbox{$\scriptstyle-$}}\kern-.5\wd0}}}
{{{\setbox0=\hbox{$\scriptstyle{\scriptscriptstyle-}{\int}$}
\vcenter{\hbox{$\scriptscriptstyle-$}}\kern-.5\wd0}}}
{{{\setbox0=\hbox{$\scriptscriptstyle{\scriptscriptstyle-}{\int}$}
\vcenter{\hbox{$\scriptscriptstyle-$}}\kern-.5\wd0}}}
\!\int}_{R/2}^R \Phi_v(0,\rho)\,\d\rho
&=
\lim_{i\to\infty}\,{\mathchoice
{{{\setbox0=\hbox{$\displaystyle{\textstyle-}{\int}$}
\vcenter{\hbox{$\textstyle-$}}\kern-.5\wd0}}}
{{{\setbox0=\hbox{$\textstyle{\scriptstyle-}{\int}$}
\vcenter{\hbox{$\scriptstyle-$}}\kern-.5\wd0}}}
{{{\setbox0=\hbox{$\scriptstyle{\scriptscriptstyle-}{\int}$}
\vcenter{\hbox{$\scriptscriptstyle-$}}\kern-.5\wd0}}}
{{{\setbox0=\hbox{$\scriptscriptstyle{\scriptscriptstyle-}{\int}$}
\vcenter{\hbox{$\scriptscriptstyle-$}}\kern-.5\wd0}}}
\!\int}_{R/2}^R\Phi_{u_{a,r_i}}(0,\rho)\,\d\rho\\
&=
\lim_{i\to\infty}\,{\mathchoice
{{{\setbox0=\hbox{$\displaystyle{\textstyle-}{\int}$}
\vcenter{\hbox{$\textstyle-$}}\kern-.5\wd0}}}
{{{\setbox0=\hbox{$\textstyle{\scriptstyle-}{\int}$}
\vcenter{\hbox{$\scriptstyle-$}}\kern-.5\wd0}}}
{{{\setbox0=\hbox{$\scriptstyle{\scriptscriptstyle-}{\int}$}
\vcenter{\hbox{$\scriptscriptstyle-$}}\kern-.5\wd0}}}
{{{\setbox0=\hbox{$\scriptscriptstyle{\scriptscriptstyle-}{\int}$}
\vcenter{\hbox{$\scriptscriptstyle-$}}\kern-.5\wd0}}}
\!\int}_{R/2}^R\Phi_{u}(a,\rho r_i)\,\d\rho
=
\lim_{S\downarrow0}\,{\mathchoice
{{{\setbox0=\hbox{$\displaystyle{\textstyle-}{\int}$}
\vcenter{\hbox{$\textstyle-$}}\kern-.5\wd0}}}
{{{\setbox0=\hbox{$\textstyle{\scriptstyle-}{\int}$}
\vcenter{\hbox{$\scriptstyle-$}}\kern-.5\wd0}}}
{{{\setbox0=\hbox{$\scriptstyle{\scriptscriptstyle-}{\int}$}
\vcenter{\hbox{$\scriptscriptstyle-$}}\kern-.5\wd0}}}
{{{\setbox0=\hbox{$\scriptscriptstyle{\scriptscriptstyle-}{\int}$}
\vcenter{\hbox{$\scriptscriptstyle-$}}\kern-.5\wd0}}}
\!\int}_{S/2}^S\Phi_{u}(a,\sigma)\,\d\sigma,
\end{aligned}$$ since the last limit exists by Corollary \[cor:density\]. We deduce that the left-hand side does not depend on $R>0$. Therefore, inequality implies $$\begin{aligned}
\int_{B_{R/2}^+}\left(\frac{|D\,\partial_X
v|^2}{|x|^{m-2}}+(m-2)\frac{|\partial_X v|^2}{|x|^m}\right)\d x
=0
\end{aligned}$$ for any $R>0$, which is only possible if $\partial_Xv=0$. This yields the asserted homogeneity of the tangent map.
For the proof of the last statement (iii), we recall a standard result on densities of sets, which states that for $\mathcal{H}^s$-a.e. point $a\in\operatorname{sing}(u)\cap T_1$ there is a sequence $r_i\searrow 0$ with $$\label{pos_dens}
\lim_{i\to\infty}r_i^{-s}\mathcal{H}^s(\operatorname{sing}(u)\cap T_{r_i}(a))\ge 2^{-s}\alpha(s),$$ where $\alpha(s):=\Gamma(\frac12)^s/\Gamma(\frac s2+1)$, cf. Remark 3.7 in [@Simon]. As shown above, after passing to a subsequence, the rescaled maps $u_{a,r_i}$ converge to a tangent map $v\in W^{2,2}_{\mathrm{loc}}(\R^m_+,N)$ locally with respect to the $W^{2,2}$-norm. Let us assume for contradiction that $\mathcal{H}^s(\operatorname{sing}(v)\cap T_1)=0$. Then for any $\eps>0$ there is a cover $\cup_{j\in\N}B_j\supset\operatorname{sing}(v)\cap T_1$ of open balls $B_j$ with radii $\rho_j$ so that $$\label{eps_cover}
\sum_{j\in\N}\rho_j^s<\eps.$$ We claim that $\operatorname{sing}(u_{a,r_i})\cap T_1\subset U_\eps:=\cup_jB_j$ for all but finitely many values of $i\in\N$. If this was not the case, after passing to a subsequence we could choose singular boundary points $p_i\in\operatorname{sing}(u_{a,r_i})\cap T_1$ with $p_i\to p\not\in\operatorname{sing}(v)$ as $i\to\infty$. Since $v$ is smooth in a neighbourhood of $p$, we have $$r^{4-m}\int_{B_r^+(p)}\big(|\Delta v|^2+r^{-2}|Dv|^2\big)\d x+r<2^{2-m}\eps_1$$ for all sufficiently small radii $r\in(0,\delta)$, where $\eps_1>0$ denotes the constant from the Regularity Theorem \[epsreg\] for the case of constant boundary values. The preceding inequality implies for sufficiently large values of $i>i_0(r)$ that $$\begin{aligned}
\left(\tfrac
r2\right)^{4-m}\int_{B_{r/2}^+(p_i)}\big(|\Delta(u_{a,r_i})|^2+\left(\tfrac
r2\right)^{-2}|D(u_{a,r_i})|^2\big)\d x
+\tfrac r2<\eps_1
\end{aligned}$$ holds true, since we can in particular assume $B_{r/2}^+(p_i)\subset B_r^+(p)$. Theorem \[epsreg\] therefore implies $p_i\not\in\operatorname{sing}(u_{a,r_i})$, which is a contradiction. We conclude that $\operatorname{sing}(u_{a,r_i})\cap T_1\subset U_\eps$ holds true for all sufficiently large $i\in\N$. But this means $$r_i^{-s}\mathcal{H}^s(\operatorname{sing}(u)\cap T_{r_i}(a))=\mathcal{H}^s(\operatorname{sing}(u_{a,r_i})\cap T_1)
\le \alpha(s)\sum_{j\in\N}\rho_j^s<\alpha(s)\eps$$ for all sufficiently large $i\in\N$, where we used in the second last step. By choosing $\eps:=2^{-s-1}$, we achieve a contradiction to the density condition , which completes the proof of the theorem.
Proof of Theorem \[thm:boundary-regularity\]
--------------------------------------------
Having established the properties of the tangent maps gathered in the preceding theorem, the proof of Theorem \[thm:boundary-regularity\] follows in a standard way from the dimension reduction principle due to Federer, see for example [@Simon], Thm. A.4. We briefly sketch the remainder of the proof.
Let $d\in\N_0$ be the smallest natural number with the property that $$\label{def:d}
\Hdim(\operatorname{sing}u\cap T_1)\le d$$ holds true for every weakly biharmonic map $u\in W^{2,2}(B_1^+,N)$ with and smooth boundary values $g\in
C^\infty(B_1^+,N)$ on $T_1$. Here, $\Hdim$ denotes the Hausdorff dimension. We assume for contradiction that there are weakly biharmonic maps with and smooth boundary values that have singular boundary points. Then by definition of $d$, we can find such a weakly biharmonic map $u\in W^{2,2}(B_1^+,N)$ and some $s\in(d-1,d]$ with $\mathcal{H}^s(\operatorname{sing}u\cap T_1)>0$. By Theorem \[theo:tangent\_maps\] there exists a tangent map $v_0$ of $u$ in a singular boundary point $a\in \operatorname{sing}u\cap T_1$ that satisfies $\mathcal{H}^s(\operatorname{sing}v_0\cap T_1)>0$ as well. Moreover, $v_0$ possesses constant boundary values and is homogeneous of degree zero. Unfortunately, we do not know whether $v_0$ is smooth in a neighbourhood of $\partial\R^m_+\setminus\{0\}$, so that Theorem \[thm:Liouville\] is not applicable. Therefore, in the case $d>0$, we repeat the procedure and construct in turn a tangent map $v_1$ of $v_0$ in a point $b\in \operatorname{sing}v_0\cap T_1\setminus\{0\}$ that again satisfies $\mathcal{H}^s(\operatorname{sing}v_1\cap T_1)>0$. Moreover, this second tangent map is homogeneous of degree zero and satisfies moreover $\partial_bv_1\equiv0$. This implies in particular that the singular set contains the one-dimensional subspace $\R
b\subset\partial\R^m_+$. We repeat the last step $d$ times to successively construct tangent maps $v_1,v_2,\ldots,v_d$ whose singular sets contain subspaces of increasingly higher dimension. The result is a tangent map $v_d\in W^{2,2}(\R^m_+,N)$ whose singular set contains a $d$-dimensional linear subspace $V\subset\partial\R^m_+$. After a rotation, we may assume that $\operatorname{sing}v_d\supset \R^d\times\{0\}$. Moreover, by construction, the map $v_d$ is weakly biharmonic, satisfies , has constant boundary values on $T_1$, is homogeneous of degree zero and constant in direction of $\R^d\times\{0\}$. By the definition of $d$ according to , we know $$\operatorname{sing}v_d\cap \partial\R^m_+= \R^d\times\{0\}.$$ In view of the partial regularity result from Corollary \[cor:partial\], this is only possible if $d\le m-5$. We consider the restriction $\tilde
v:=v_d|_{\{0\}\times \R^{m-d}_+}\in W^{2,2}_{\mathrm{loc}}(\R^{m-d}_+,N)$. A standard cut-off argument implies that $\tilde
v$ is stationary biharmonic on $\R^{m-d}_+$, cf. [@Scheven1 Lemma 4.3]. Moreover, $\tilde v$ has constant boundary values in the sense of $(u,Du)=(c,0)$ on $\partial\R^{m-d}_+$, and $\tilde v$ is homogeneous of degree zero. Moreover, by construction we have $\operatorname{sing}\tilde v\cap \partial\R^{m-d}_+=\{0\}$, which means that $\tilde v$ is smooth on a neighbourhood of $\partial\R^{m-d}_+\setminus\{0\}$. In the case $d\le m-6$, we may therefore apply Theorem \[thm:Liouville\], which implies that $\tilde v$ is constant. If $d=m-5$, the partial regularity result from Corollary \[cor:partial\] and the homogeneity of $\tilde v$ imply that $\tilde v$ is smooth on $\R^{m-d}_+$. Therefore, our assumptions and Lemma \[lem:liouville-4\] again yield that $\tilde v$ is a constant map. In any case, we obtain a contradiction to $\operatorname{sing}(\tilde v)\cap\partial\R^{m-d}_+=\{0\}$. We conclude that our assumption must have been false, i.e. $\operatorname{sing}u\cap T_1=\varnothing$ holds for any weakly biharmonic map $u\in
W^{2,2}(B_1^+,N)$ with and smooth boundary values. This completes the proof of Theorem \[thm:boundary-regularity\].
[99]{} S. Altuntas, A boundary monotonicity inequality for variationally biharmonic maps and applications to regularity theory. *Ann. Global Anal. Geom.* 54(4):489–508, 2018.
G. Angelsberg. A monotonicity formula for stationary biharmonic maps. *Math. Z.* 252(2):287–293, 2006.
F. Bethuel, On the singular set of stationary harmonic maps. *Manuscripta Math.* 78:417–443, 1993.
A. Chang, P. Yang. *On a fourth order curvature invariant*, Spectral problems in geometry and arithmetic, Contemp. Math. 237, pp. 9-28, Amer. Math. Soc., Providence, 1999.
A. Chang, L. Wang, P. Yang, A regularity theory of biharmonic maps. *Comm. Pure Appl. Math.* 52(9):1113–1137, 1999.
J. Eells, L. Lemaire, Two reports on harmonic maps. World Scientific Publishing Co., Inc., River Edge, NJ, 1995.
H. Gong, T. Lamm, C. Wang, Boundary partial regularity for a class of biharmonic maps. *Calc. Var. Partial Differential Equations* 45(1-2):165–191, 2012.
T. Lamm, C. Wang, Boundary regularity for polyharmonic maps in the critical dimension. *Adv. Calc. Var.* 2(1):1–16, 2009.
L. Lemaire, Applications harmoniques de surfaces riemanniennes. *J. Differential Geom.* 13(1):51–78, 1978.
F.H. Lin, Gradient estimates and blow-up analysis for stationary harmonic maps. *Annals of Math.* 149:785–829, 1999.
S. Luckhaus, Partial Hölder continuity for minima of certain energies among maps into a Riemannian manifold. *Indiana Univ. Math. J.* 37(2):349–367, 1988.
P. Mattila, *Geometry of Sets and Measures in Euclidean Spaces - Fractals and rectifiability*. Cambridge Studies in advanced mathematics 44, Cambridge University Press, 1995.
K. Mazowiecka Boundary regularity for minimizing biharmonic maps. *Calc. Var. Partial Differential Equations* 57(6):Art. 143, 2018.
R. Moser, The blow-up behavior of the biharmonic map heat flow in four dimensions. *Int. Math. Res. Pap.* 2005:351–402, 2005.
R. Moser, A variational problem pertaining to biharmonic maps. *Comm. Partial Differential Equations* 33(7-9):1654–1689, 2008.
S. Paneitz, A quartic conformally covariant differential operator for arbitrary pseudo-Riemannian manifolds. *SIGMA Symmetry Integrability Geom. Methods Appl.* 4, paper 036, 2008.
T. Rivière, Everywhere discontinuous harmonic maps into spheres. *Acta Math.* 175(2):197–226, 1995.
T. Rivière, Conservation laws for conformally invariant variational problems. *Invent. Math.* 168(1):1–22, 2007.
T. Rivière, M. Struwe, Partial regularity for harmonic maps and related problems. *Comm. Pure Appl. Math.* 61(4):451–463, 2008.
C. Scheven, Variationally harmonic maps with general boundary conditions: boundary regularity. *Calc. Var. Partial Differential Equations* 25(4):409–429, 2006.
C. Scheven, Dimension reduction for the singular set of biharmonic maps. *Adv. Calc. Var.* 1:53–91, 2008.
C. Scheven, An optimal partial regularity result for minimizers of an intrinsically defined second-order functional. *Ann. Inst. Henri Poincaré, Anal. Non Linéaire.* 26(5):1585–1605, 2009.
R. Schoen, *Analytic aspects of the harmonic map problem*, in: S.S. Chern(ed.), Seminar on Nonlinear Partial Differential Equations, Springer, 1984, 321–358.
R. Schoen and K. Uhlenbeck, A regularity theory for harmonic maps. *J. Differential Geom.* 17(2):307–335, 1982.
R. Schoen and K. Uhlenbeck, Boundary regularity and the Dirichlet problem for harmonic maps. *J. Differential Geom.* 18(2):253–268, 1983.
L. Simon, [*Lectures on Geometric Measure Theory*]{}, Proc. of Centre for Math. Anal., Australian National University, Vol. 3, Canberra, 1983.
M. Struwe, Partial regularity for biharmonic maps, revisited. *Calc. Var. Partial Differential Equations* 33(2):249–262, 2008.
C. Wang, Remarks on biharmonic maps into spheres. *Calc. Var. Partial Differential Equations* 21(3):221–242, 2004.
C. Wang, Biharmonic maps from $\R^4$ into a Riemannian manifold. *Math. Z.* 247(1):65–87, 2004.
C. Wang, Stationary biharmonic maps from $\R^m$ into a Riemannian manifold. *Comm. Pure Appl. Math.* 57(4):419–444, 2004.
W. Ziemer, *Weakly Differentiable Functions. Sobolev Spaces and Functions of Bounded Variation*, Graduate Texts in Mathematics, 120, Springer, New York, 1989.
|
---
abstract: 'In the paper, a novel distributed stochastic approximation algorithm (DSAA) is proposed to seek roots of the sum of local functions, each of which is associated with an agent from the multiple agents connected in a network. At any time, each agent updates its estimate for the root utilizing the observation of its local function and the information derived from the neighboring agents. The key difference of the proposed algorithm from the existing ones consists in the expanding truncations (so it is called as DSAAWET), by which the boundedness of the estimates can be guaranteed without imposing the growth rate constraints on the local functions. The convergence of the estimates generated by DSAAWET to a consensus set belonging to the root set of the sum function is shown under weaker conditions on the local functions and on the observation noise in comparison with the existing results. We illustrate our approach by two applications, one from signal processing and the other one from distributed optimization. Numerical simulation results are also included.'
author:
- 'Jinlong Lei, and Han-Fu Chen, [^1][^2]'
title: ' Distributed Stochastic Approximation Algorithm With Expanding Truncations: Algorithm and Applications '
---
Distributed stochastic approximation, expanding truncation, multi-agent network, distributed optimization.
Introduction
=============
Distributed algorithms have been extensively investigated in connection with the problems arising from sensor networks and networked systems for recent years, for example, consensus problem [@ren1; @murray0; @huang1], distributed estimation [@Zhang1; @filter1], sensor localization [@localization1], distributed optimization [@optimization0; @nedic; @optimization1], distributed control [@control1; @control2] and so on. The distributed algorithms work in the situation, where the goal is cooperatively accomplished by a multi-agent network with computation and communication abilities allocated in a distributed environment. Their advantages over the centralized approaches for networked problems consist in enhancing the robustness of the networks, preserving privacy, and reducing the communication and computation costs.
Stochastic approximation (SA) was first considered in the work [@RM] of Robbins and Monro for finding roots of a function with noisy observations, now it is known as the RM algorithm. Then SA was used by Kiefer and Wolfowitz [@KW] to estimate the maximum of an expectation function only with noisy function observations. SA has found wide applications in signal processing, communications and adaptive control, see, e.g.,[@Kushner; @Chen_2002; @Chen_Zhao]. Recently, many distributed problems are solved by SA-based distributed algorithms, e.g., distributed parameter estimation [@Zhang1], distributed convex optimization over random networks [@optimization1], and searching local minima of a non-convex objective function [@Bianchi1]. As a result, investigating the distributed algorithm for SA is also of great importance.
Distributed stochastic approximation algorithms (DSAA) were proposed in [@DSA; @DSA2] to cooperatively find roots of a function, being a sum of local functions associated with agents in a multi-agent network. Each agent updates its estimate for the root based on its local information composed of: 1) the observation of its local function possibly corrupted by noise and 2) the information obtained from its neighbors. The weak convergence for DSAA with constant step-size is investigated in [@DSA2], while the almost sure convergence for DSAA with decreasing step-size is studied in [@DSA]. Performance gap between the distributed and the centralized stochastic approximation algorithms is investigated in [@Bianchi2]. However, DSAA discussed in [@Bokar0; @Bokar1; @Bokar2] is in a different setting in comparison with [@DSA; @DSA2]. In fact, all components of the root-vector are estimated by each agent in [@DSA; @DSA2], while in [@Bokar0; @Bokar1; @Bokar2] the components are separately estimated at different processors.
It is noticed that almost all aforementioned SA-based distributed algorithms require rather restrictive conditions to guarantee convergence For example, in [@DSA] it is required that each local function is globally Lipschitz continuous and the observation noise is a martingale difference sequence (mds). However, these conditions may not hold for some problems, e.g., distributed principle component analysis and distributed gradient-free optimization to be discussed in Section II.B. This paper aims at solving the distributed root-seeking problem under weaker conditions in comparison with those used in [@DSA].
Symbol Definition
-------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -- -- --
$\| v \| , \| A\|$ Euclidean ( $l^2$) norm of vector $v$, matrix $A$
$A \geq 0$ Each entry of matrix $A$ is nonnegative, and $A$ is called the nonnegative matrix.
$ \mathbf{I}_m $ $m \times m$ identity matrix
$ \mathbf{1} , \mathbf{0} $ Vector or matrix with all entries equal to 1, 0
$X^T$, $X^{-1}$ Transpose of matrix $X$, inverse of matrix $X$
$ col \{x_1, \cdots, x_m \} $ $ col \{x_1, \cdots, x_m \} \triangleq( x_1^T,\cdots, x_m^T)^T $ stacking the vectors or matrices $x_1,\cdots, x_m$
$I_{[\textrm{Inequality}]} $ Indicator function meaning that it equals 1 if the inequality indicated in
the bracket is fulfilled, and 0 if the inequality does not hold
$\otimes $ Kronecker product
$ a \wedge b $ min $\{ a, b \}$
$d(x, \Omega)$ $ \inf_y\{\parallel x-y\parallel: y\in \Omega \} $
$ E [\cdot]$ Expectation operator
$D_{\bot} $ $ D_{\bot} \triangleq ( \mathbf{I}_N -\frac{\mathbf{1}\mathbf{1}^T}{N}) \otimes \mathbf{I}_l$, $N=$ the number of agents in the network,
$l=$ the number of arguments of the local function
$m(k,T)$ $m(k,T) \triangleq \max \{ m: \sum_{i=k}^m \gamma_i \leq T \} $ is an integer valued function for $T>0$ and integer $k$
$\tau_{i,m}$ The smallest time when the truncation number of agent $i$ has reached $m$,
i.e., $\tau_{i,m}\deq\textrm{inf} \{k: \sigma_{i,k}=m \}$, where $\{ \sigma_{i,k}\}$ is generated by -.
$\tau_m$ $\tau_m\deq \min\limits_{i \in \mathcal{V}}\tau_{i,m},$ the smallest time when at least one of agents has its truncation number reached $ m$
$\tilde{\tau}_{j,m}$ $\tilde{\tau}_{j,m}\deq \tau_{j,m} \wedge \tau_{m+1}$
$\hat{\sigma}_{i,k}$ $ \hat{\sigma}_{i,k} \deq \max_{ j \in N_{i }(k)} \sigma_{j,k}, $ where $ N_{i}(k) $ is the set of neighboring agents of agent $i$ at time $k$.
$\sigma_{k}$ The largest truncation number among all agents at time $k$, i.e., $\sigma_{k} =\max\limits_{i \in \mathcal{V}}\sigma_{i,k}=\max\limits_{i \in \mathcal{V}} \hat{ \sigma}_{i,k} .$
$ \delta(t)$ $ \delta(t ) \rightarrow 0 \textrm{ as }t \rightarrow 0$.
Contributions of the paper are as follows. 1) We propose a novel DSAA with expanding truncations (DSAAWET) for networks with deterministic switching topologies by designing a network truncation mechanism. It is shown that estimates generated by DSAAWET at all agents approach with probability one to a consensus set, which is contained in the root set of the sum function. Compared with [@DSA], we neither assume the observation noise to be an mds nor impose any growth rate constraint and Lipschitz continuity on the local functions. 2) We apply the proposed algorithm to two application problems stated in Section II.B, and establish the corresponding theoretic results. In contrast, the algorithm proposed in [@DSA] is not applicable to these problems.
The rest of the paper is arranged as follows. The distributed root-seeking problem is formulated, and two motivation examples are given in Section II. DSAAWET is defined and the corresponding convergent results are presented in Section III. The proof of the main results is given in Section IV with some details placed in Appendices. The proposed algorithm is applied to solve the two application problems in Section V with numerical examples included. Some concluding remarks are given in Section VI.
Problem Formulation and Motivations
====================================
We first formulate the distributed root-seeking problem with the related communication model. Then we give two motivation problems that cannot be solved by the existing algorithms, but can be solved by DSAAWET to be proposed in the paper.
Problem Statement
------------------
Consider the case where all agents in a network collectively search the root of the sum function given by $$\label{summ1}
f(\cdot)= \frac{1}{N}\sum_{i=1}^N f_i(\cdot),$$ where $f_i(\cdot):\mathbb{R}^l \rightarrow \mathbb{R}^l$ is the local function assigned to agent $i$ and can only be observed by agent $i.$ Let $J \triangleq \{x\in \mathbb{R}^l : f(x)=0 \}$ denote the root set of $f(\cdot)$.
For any $ i \in \mathcal{V}$, denote by $x_{i,k} \in \mathbb{R}^l$ the estimate for the root of $f(\cdot) $ given by agent $i $ at time $k$. Agent $i$ at time $k+1$ has its local noisy observation $$\label{ob}
O_{i,k+1}= f_i(x_{i,k})+\varepsilon_{i,k+1},$$ where $\varepsilon_{i,k+1}$ is the observation noise. Agent $i$ is required to update its estimate $ x_{i,k}$ on the basis of its local observation and the information obtained from its neighbors.
The information exchange among the $N$ agents at time $k$ is described by a digraph $\mathcal{G }(k) = \{ \mathcal{V }, \mathcal{E }(k)\}$, where $\mathcal{V }=\{ 1,\cdots,N\}$ is the node set with node $i$ representing agent $i$; $\mathcal{E }(k) \subset \mathcal{V } \times \mathcal{V } $ is the edge set with $(j,i)\in\mathcal{E }(k)$ if agent $i$ can get information from agent $j$ at time $k$ by assuming $(i,i) \in \mathcal{E}(k) $. Let the associated adjacency matrix be denoted by $W(k) =[ \omega_{ij}(k)]_{i,j =1}^N$, where $ \omega_{ij}(k)>0$ if and only if $(j,i )\in \mathcal{E }(k)$, and $ \omega_{ij}(k)=0$, otherwise. Denote by $ N_{i}(k)=\{ j \in \mathcal{V}:(j,i )\in \mathcal{E }(k) \}$ the neighbors (neighboring agents) of agent $i$ at time $k$.
A time-independent digraph $\mathcal{G}=\{ \mathcal{V}, \mathcal{E}\}$ is called strongly connected if for any $ i,j\in \mathcal{V}$ there exists a directed path from $i$ to $j$. By this we mean a sequence of edges $ (i,i_1),(i_1,i_2),\cdots,(i_{p-1},j)$ in the digraph with distinct nodes $ i_m \in \mathcal{V}~~\forall m: 0 \leq m \leq p-1$, where $p$ is called the length of the directed path. A nonnegative square matrix $A $ is called doubly stochastic if $A\mathbf{1} =\mathbf{1}$ and $\mathbf{1}^T A =\mathbf{1}^T$, where $\mathbf{1}$ is the vector of compatible dimension with all entries equal to 1.
Motivation Examples
--------------------
We now give two motivation examples, and show that they cannot be solved by the existing distributed stochastic approximation algorithm [@DSA]. We will return to these examples in Section V to show that DSAAWET proposed in the paper can solve them.
### Distributed Principle Component Analysis
In signal processing and pattern recognition, effectively cluster large data sets is a major objective. Principle component analysis (PCA) is a powerful technique to process multivariate data by constructing a concise data representation through computing the dominant eigenvalues and the corresponding eigenvectors of the data covariance matrix [@DPCA1; @DPCA2]. It is delivered in [@DPCA1] that for recent applications, the large data sets are gathered by a group of spatially distributed sensors. So, the centralized PCA may no longer be applicable since these distributed data sets are often too large to send to a fusion center. Some techniques are developed in [@DPCA1] to compute the global PCA for distributed data sets with updating, but there is no convergence analysis. We aim to propose a distributed algorithm for the global PCA, and to establish its convergence.
The global data matrix $u_k$ at time $k$ is distributed among $N$ sensors that spatially distributed in the network: $$u_k= \begin{pmatrix}
& u_{1,k} \\
& \cdots \\
&u_{N,k}
\end{pmatrix} ,$$ where $u_{i,k} \in \mathbb{R}^{p_i\times d}$ is collected by sensor $i.$ The rows of $u_k$ denote the observations and columns denote the features. Assume that for each $i \in \mathcal{V}$, $\{u_{i,k}\}$ is an iid sequence with zero mean. Denote by $A = E[u_k^Tu_k] $ the covariance matrix of $u_k$. The primary objective is to estimate the largest eigenvalue and the corresponding eigenvector of $A$.
It has been shown in [@Oja] that finding the unit eigenvector corresponding to the largest eigenvalue of $A$ can be reduced to find the nonzero root of function $ f(x)=Ax-(x^T Ax)x.$ Then the distributed implementation of PCA is converted to finding the nonzero root of the function given by $$f(x)= \frac{1}{N}\sum_{i=1}^N f_i(x) ,$$ where $ f_i(x)=A_ix-(x^T A_ix)x$ with $ A_i=E[u_{i,k}^Tu_{i,k}]$.
Denote by $x_{i,k}$ the estimate given by agent $i$ at time $k$ for the unit eigenvector corresponding to the largest eigenvalue of matrix $A$. Since $A_i$ cannot be directly derived, by replacing $A_i$ with its sample $A_{i,k}=u_{i,k}^Tu_{i,k}$ at time $k$, the local observation of agent $i$ at time $k+1$ is as follows $$\label{pca}
O_{i,k+1}=A_{i,k}x_{i,k}-(x_{i,k}^T A_{i,k}x_{i,k})x_{i,k} .$$
Then, the distributed principle component analysis is in the distributed root-seeking form. Noting that $f_i(\cdot)$ contains a cubic term, the local function is not globally Lipschitz continuous. Consequently, DSAA proposed in [@DSA] cannot be directly applied to this example.
### Distributed Gradient-free Optimization
Consider a multi-agent network of $N$ agents, for which the objective is to cooperatively solve the following optimization problem $$\label{opt1}
\min_x \quad c(x)=\sum_{i=1}^N c_i(x),$$ where $c_i(\cdot) : \mathbb{R}^l \rightarrow \mathbb{R}$ is the local objective function of agent $i$, and $c_i(\cdot)$ is only known by $i$ itself. Assume the optimization problem has solutions. Consider the case where gradients of the cost functions are unavailable but the cost functions can be observed with noises. Then the finite time difference of the cost functions can be adopted to estimate the gradient, see [@KW] [@Chen_1999] [@gradienfree2]. This problem is referred to as the gradient-free optimization [@gradienfree1]. For the distributed optimization problem with a convex set constraint and with each cost function being convex, [@Yuan_NN_15][@Yuan_NN_16] have developed the distributed zeroth-order methods by only using functional evaluations, and proved that the expected function value sequence converges to the optimal value. In contrast, we aim at deriving the almost sure convergence without convex set constraint.
Assume that the cost functions $c_i(\cdot),i=1,\cdots N$ are differentiable and the global function $c(\cdot)$ is convex. Then solving the problem is equivalent to finding roots of the function $ f(x)= \sum_{i=1}^N f_i(x)/N$ with $f_i(\cdot) =- \nabla c_i(\cdot)$, where $\nabla c_i(\cdot)$ is the gradient of $c_i(\cdot)$. Here the randomized KW method proposed in [@Chen_1999] is adopted to estimate gradients of the cost functions. Denote by $x_{i,k}$ the estimate given by agent $i$ for the solution to problem at time $k$. Let $\triangle_{i,k} \in \mathbb{R}^l,k=1,2,\cdots$ be a sequence of mutually independent random vectors with each component independently taking values $\pm1$ with probability $\frac{1}{2}$. The observation of function $- \nabla c_i(\cdot)$ at point $x_{i,k}$ is constructed as follows $$\label{opt2}
O_{i,k+1}=-\frac{c_i(x_{i,k }^{+})+\xi_{i,k+1}^{+}-c_i(x_{i,k }^{-})-\xi_{i,k+1}^{-}}{2\alpha_k} \triangle_{i,k},$$ where $x_{i,k }^{+}=x_{i,k}+\alpha_k\triangle_{i,k}$, $x_{i,k }^{-}=x_{i,k}-\alpha_k\triangle_{i,k}$, $\xi_{i,k+1}^{+}$ and $\xi_{i,k+1}^{-}$ are the observation noises of the cost function $c_i(\cdot)$ at $x_{i,k }^{+}$ and $x_{i,k }^{-}$, respectively, and $\alpha_k >0 $ for any $k \geq 0$.
Thus, we have transformed the distributed gradient-free optimization to the distributed root-seeking form. It is shown in [@Chen_1999] that for any $i \in \mathcal{V}$, the observation noise $ \varepsilon_{i,k+1}=O_{i,k+1}+\nabla c_i(x_{i,k}) $ is not an mds, and hence DSAA proposed in [@DSA] cannot be directly applied to this problem.
The problems arising from these two examples motivate us to propose DSAAWET, by which the two motivation examples are well dealt with as to be shown in Section V.
$$\begin{aligned}
& \sigma_{i,0}=0,~~~ \hat{\sigma}_{i,k} \deq \max_{ j \in N_{i }(k)} \sigma_{j,k}, \label{comp1} \\
& x'_{i,k+1}= ( \sum_{ j \in N_{i }(k)} \omega_{ij}(k) ( x_{j,k} I_{[ \sigma_{j,k} = \hat{ \sigma}_{i,k}]}+
x^* I_{[ \sigma_{j,k} < \hat{\sigma}_{i,k}]}) + \gamma_{ k } O_{i,k+1} ) I_{[\sigma_{i,k}= \hat{ \sigma}_{i,k}]}
+ x^* I_{[\sigma_{i,k}< \hat{ \sigma}_{i,k}]}, \label{comp2} \\
& x_{i,k+1}=x^* I_{[ \parallel x'_{i,k+1} \parallel > M_{\hat{\sigma}_{i,k}} ]}
+x'_{i,k+1}I_{[ \parallel x'_{i,k+1} \parallel \leq M_{\hat{\sigma}_{i,k}}] }, \label{step1}\\
& \sigma_{i,k+1}= \hat{\sigma}_{i,k}+ I_{[ \parallel x'_{i,k+1} \parallel > M_{\hat{\sigma}_{i,k}}]}, \label{step2}\end{aligned}$$
DSAAWET and Its Convergence
===========================
In this section, we define DSAAWET and formulate the main results of the paper.
DSAAWET
--------
Let us first explain the idea of expanding truncations. In many important problems, the sequence of estimates generated by the RM algorithm may not be bounded, and it is hard to define in advance a region the sought-for parameter belongs to. This motives us to adaptively define truncation bounds as follows. When the estimate crosses the current truncation bound, the estimate is reinitialized to some pre-specified point $x^*$ and at the same time the truncation bound is enlarged. Due to the decreasing step-sizes, this operation makes as if we rerun the algorithm with initial value $x^*$, with smaller step-sizes and larger truncation bounds. The expanding truncation mechanism incorporating with some verifiable conditions makes the reinitialization cease in a finite number of steps, and hence makes the estimates bounded. As results, after a finite number of steps the algorithm runs as the RM algorithm. This is well explained in [@Chen_2002; @Chen_Zhao] and in the related references therein.
We now apply the idea of expanding truncation to distributed estimation, i.e., for the case $N>1$. This leads to DSAAWET - at the bottom of p.4 with initial values $x_{i,0}~,i=1,\cdots,N$, where $ O_{i,k+1}$ defined by is the local observation of agent $i$, $\{ \gamma_k\}_{k \geq 0}$ with $ \gamma_{ k} > 0$ are step-sizes used by all agents, $x^* \in \mathbb{R}^l$ is a fixed vector known to all agents, $\{M_k\}_{k \geq 0} $ is a sequence of positive numbers increasingly diverging to infinity with $M_0 \geq \| x^* \|$, $ \sigma_{i,k}$ is the number of truncations for agent $i$ up-to-time $k$, and $ M_{\hat{\sigma}_{i,k}}$ serves as the the truncation bound when the $(k+1)$th estimate for agent $i$ is generated.
The algorithm - is performed according to the following three steps.
i\) **Consensus for truncation numbers.** At time $k$, the algorithm for agent $i$ may or may not be truncated. This yields that at time $k,$ $\sigma_{i,k}$ may be different for $i=1,\cdots, N$. The truncation number for each agent is expected to achieve consensus, which can be set to the largest truncation number among the agents. Hence it is required to carry out a max consensus procedure [@maxc] on the truncation numbers. Thus, we set the truncation number $\hat{\sigma}_{i,k}$ to be the largest one among its neighbors $\{ \sigma_{j,k},~j \in \mathcal{N}_i(k) \} ,$ as indicated by .
ii\) **Average consensus + innovation update.** At time $k+1$, agent $i$ produces an intermediate value $x'_{i,k+1}$ as shown by . If $\sigma_{i,k}< \hat{ \sigma}_{i,k} $, then $x'_{i,k+1}=x^{*}$. Otherwise, $x'_{i,k+1}$ is a combination of the consensus part and the innovation part, where the consensus part is a weighted average of estimates derived at its neighbors, and the innovation part processes its local current observation.
iii\) **Local truncation judgement.** If $ x'_{i,k+1} $ remains inside its local truncation bound $M_{\hat{\sigma}_{i,k}}$, i.e., $\parallel x'_{i,k+1} \parallel \leq M_{\hat{\sigma}_{i,k}} $, then $ x_{i,k+1}= x'_{i,k+1}$ and $\sigma_{i,k+1}= \hat{\sigma}_{i,k}$. If $x'_{i,k+1}$ exits from the sphere with radius $M_{\hat{\sigma}_{i,k}}$, i.e., $\parallel x'_{i,k+1} \parallel >M_{\hat{\sigma}_{i,k}} $, then $ x_{i,k+1} $ is pulled back to the pre-specified point $ x^*$, and at the same time the truncation number $\sigma_{i,k+1}$ is increased to $ \hat{\sigma}_{i,k}+1$. This is described by , .
Denote by $$\label{largest}
\sigma_{k} \deq \max\limits_{i \in \mathcal{V}}\sigma_{i,k},$$ the largest truncation number among all agents at time $k$. If $N=1,$ then by denoting $$\hat{ \sigma}_{i,k} =\sigma_{i,k}\triangleq \sigma_k,~ x_{i,k} \triangleq x_k, ~x_{i,k}'\triangleq x_k', ~O_{i,k}\triangleq O_k$$ it can be easily seen that - becomes $$\begin{split}
& x'_{k+1}= x_k+ \gamma_{ k } O_{k+1} , \\
& x_{k+1}=x^* I_{[ \parallel x'_{k+1} \parallel > M_{ \sigma_k }]}
+x'_{k+1}I_{[ \parallel x'_{k+1} \parallel \leq M_{ \sigma_k}]}, \\
& \sigma_{k+1}= \sigma_k+ I_{[ \parallel x'_{k+1} \parallel > M_{ \sigma_k}].} \nonumber \end{split}$$ The equation is called the stochastic approximation algorithm with expanding truncations (SAAWET), which requires possibly the weakest conditions for its convergence among various modifications of the RM algorithm [@Kushner; @Chen_2002; @Ben; @Bokar]. Thus, the advantages of SAAWET over the RM algorithm might remain for the case $N>1.$
\[r1\] It is noticed that $\sigma_{i,k+1}\geq \hat{\sigma}_{i,k} \geq \sigma_{i,k} ~~\forall k \geq 0$ by and . Further, it is concluded that $$\label{sub1}
x_{i,k+1}= x^*\textrm{ if }\sigma_{i,k+1} > \sigma_{i,k}.$$ This can be seen from the following consideration: i) If $ \sigma_{j,k} \leq \sigma_{i,k} ~\forall j \in N_{i} (k)$, then from we derive $ \hat{\sigma}_{i,k} = \sigma_{i,k}$. Since $ \sigma_{i,k+1} > \sigma_{i,k}$, by it follows that $\parallel x'_{i,k+1} \parallel > M_{\hat{\sigma}_{i,k}}$, and hence from we derive $x_{i,k+1}=x^*.$ ii) If there exists $ j \in N_i(k)$ such that $ \sigma_{j,k} > \sigma_{i,k}$, then from we derive $ \hat{\sigma}_{i,k} = \max_{ j \in N_{i }(k)} \sigma_{j,k} > \sigma_{i,k},$ and from we have $x'_{i,k+1}=x^*$. Consequently, by we have $x_{i,k+1}=x^*$.
Assumptions
-----------
We list the assumptions to be used.
A1 $ \gamma_k>0, \gamma_k \xlongrightarrow [k \rightarrow \infty]{} 0 $, and $ \sum\limits_{k=1}^\infty \gamma_k=\infty$.
A2 There exists a continuously differentiable function $v(\cdot ) :
\mathbb{R}^l\rightarrow \mathbb{R}$ such that
& \_[[d(x,J )]{} ]{} f\^T(x)v\_x(x)<0& \[4.1\]
for any $ \Delta > \delta>0$, where $v_x(\cdot)$ denotes the gradient of $v(\cdot) $ and $d(x,J )=\inf_y\{\parallel x-y\parallel: y\in J \} $,
b\) $v(J )\triangleq \{v(x): x\in J \}$ is nowhere dense,
c\) $ \| x^* \| < c_0 \textrm{ and } v(x^*) < \textrm{inf }_{\| x \|=c_0} v(x) $ for some positive constant $c_0 $, where $x^*$ is used in .
A3 The local functions $f_i(\cdot)~~ \forall i \in \mathcal{V}$ are continuous.
A4 a) $W(k)~~\forall k \geq 0$ are doubly stochastic matrices;
b) There exists a constant $0< \eta <1$ such that $$\omega_{ij}(k) \geq \eta \quad \forall j \in \mathcal{N}_i(k) ~~\forall i \in \mathcal{V}~~\forall k \geq 0;$$
c) The digraph $\mathcal{G}_{\infty}=\{ \mathcal{V}, \mathcal{E}_{\infty}\}$ is strongly connected, where $$\mathcal{E}_{ \infty} =
\{ (j,i): (j,i) \in \mathcal{E}(k) \textrm{ for infinitely many indices } k \} ;$$
d) There exists a positive integer $B $ such that $$(j,i) \in \mathcal{E} (k) \cup \mathcal{E}(k+1) \cup \cdots \cup \mathcal{E}(k+B-1)$$ for all $(j,i) \in \mathcal{E}_{\infty}$ and any $k \geq 0$.
A5 For any $i \in \mathcal{V}$, the noise sequence $\{\varepsilon_{i,k+1}\}_{k\geq 0}$ satisfies
& \_k \_[i, k+1]{} 0,&\
& \_[T 0]{} \_[k ]{} \_[m=n\_k]{}\^[m(n\_k,t\_k)]{}\_[m ]{} \_[i,m+1]{} I\_[ \[x\_[i,m]{} K \]]{} =0\
& t\_k K &
along indices $\{n_k \}$ whenever $\{x_{i,n_k}\}$ converges, where $m(k,T) \triangleq \max
\{ m: \sum_{i=k}^m \gamma_i \leq T \} $.
Let us explain the conditions. A1 is a standard assumption for stochastic approximation, see [@Chen_2002; @Bokar]. From A2 it is seen that $v(\cdot)$ is not required to be positive. A2 a) means that $v(\cdot)$ serves as a Lyapunov function for the differential equation $\dot{x}=f(x).$ It is noticed that A2 b) holds if $J$ is finite, and A2 c) takes place if $v(\cdot)$ is radially unbounded.
Condition A4 describes the connectivity properties of the communication graphs. For detailed explanations we refer to [@optimization0]. Set $\Phi(k,k+1) =\mathbf{I}_N, ~\Phi(k,s)= W(k) \cdots W(s) ~ \forall k \geq s. $ By [@optimization0 Proposition 1] there exist constants $c>0$ and $0< \rho<1$ such that $$\label{graphmatrix}
\parallel \Phi(k,s) - \frac{1}{N} \mathbf{1} \mathbf{1}^T \parallel \leq c\rho^{k-s+1} \quad \forall k \geq s .$$
It is noticed that A5 b) is convenient for dealing with state-dependent noise. The indicator function $I_{ [\parallel x_{i,m} \parallel \leq K ]} $ in the condition will be casted away if the observation noise does not depend on the estimates. However, for the state-dependent noise, before establishing the boundedness of $\{ x_{i,k}\}$, the condition with an indicator function included is easier to be verified. It is worth noting that in A5 b) we do not assume existence of a convergent subsequence of $\{x_{i,k}\} $ for any $ i$, we only require A5 b) hold along indices of any convergent subsequence if exists. Verification of A5 b) along convergent subsequences is much easier than that along the whole sequence. If $\{ \varepsilon_{i,k}\}$ can be decomposed into two parts $\varepsilon_{i,k}=\varepsilon_{i,k}^{(1)}+\varepsilon_{i,k}^{(2)}$ such that $ \sum_{k=0}^{\infty } \gamma_{k } \varepsilon_{i,k+1}^{(1)} I_{ [\parallel x_{i,k} \parallel \leq K ]} < \infty
\textrm{ and }\varepsilon_{i,k}^{(2)} I_{ [\parallel x_{i,k} \parallel \leq K ]} \xlongrightarrow [k\rightarrow \infty] {} 0,$ then A5 b) holds. So, A5 holds when the observation noise is an iid sequence or an mds with bounded second moments if $ \sum\limits_{k=1}^\infty \gamma_k^2<\infty$.
Main Results
-------------
Define the vectors $ X_k \deq col\{ x_{1,k} ,\cdots, x_{N,k}\}, ~\varepsilon_k\deq col \{
\varepsilon_{1,k} , \cdots, \varepsilon_{N,k} \} ,
F(X_k)\deq col \{ f_1(x_{1,k}) , \cdots,f_N(x_{N,k})\} $. Denote by $ X_{\bot, k}\deq D_{\bot} X_k$ the disagreement vector of $X_k$ with $ D_{\bot} \triangleq ( \mathbf{I}_N -\frac{\mathbf{1}\mathbf{1}^T}{N}) \otimes \mathbf{I}_l$, and by $x_k
=\frac{1}{N} \sum_{i=1}^N x_{i,k}$ the average of the estimates derived at all agents at time $k$.
\[thm1\] Let $ \{ x_{i,k}\} $ be produced by - with an arbitrary initial value $ x_{i,0} $. Assume A1-A4 hold. Then for any sample path $\omega$ where A5 holds for all agents, the following assertions take place:
i\) $\{x_{i,k}\}$ is bounded and there exists a positive integer $k_0$ possibly depending on $\omega$ such that $$\label{notr}
x_{i,k+1}= \sum_{ j \in N_{i }(k)} \omega_{ij}(k) x_{j,k} + \gamma_{ k } O_{i,k+1} ~~ \forall k \geq k_0 ,$$ or in the compact form: $$\label{centalform}
X_{k+1}= ( W (k) \otimes \mathbf{I}_l) X_k+ \gamma_k (F(X_k)+\varepsilon_{k+1}) ~\forall k \geq k_0;$$
& X\_[, k]{} d( x\_k,J ) 0; &\[res1\]
iii\) there exists a connected subset $J^{*} \subset J$ such that $$\label{connected}
d( x_k,J^{*} ) \xlongrightarrow [k \rightarrow \infty] {} 0.$$
The proof of Theorem \[thm1\] is presented in Section IV. From the proof it is noticed that for deriving i) the condition A5 a) is not required. Theorem \[thm1\] establishes that the sequence $\{X_k\}$ is bounded; the algorithm - finally turns to be a RM-based DSAA without truncations; and the estimates given by all agents converge to a consensus set, which is contained in a connected subset of the root set $J $, with probability one when A5 holds for almost all sample paths for all agents. As a consequence, if $J$ is not dense in any connected set, then $x_k$ converges to a point in $J$. However, it is unclear how does $\{x_k\}$ behave when $J$ is dense in some connected set. This problem was investigated for the centralized algorithm in [@fang].
Compared with [@DSA], we impose weaker conditions on the local functions and on the observation noise. In fact, we only require the local functions be continuous, while conditions ST1 and ST2 in [@DSA Theorem 3] do not allow the functions to increase faster than linearly. In addition, we do not require the observation noise to be an mds as in [@DSA]. As shown in [@Chen_2002] [@Chen_Zhao], A5 is probably the weakest requirement for the noise since it is also necessary for convergence whenever the root $x^0$ of $f(\cdot)$ is a singleton and $f(\cdot)$ is continuous at $x^0 $. Different from the random communication graphs used in [@DSA], here we use the deterministic switching graphs to describe the communication relationships among agents.
Proof of Main Results
======================
Prior to analyzing $\{x_{i,k}\}$, let us recall the convergence analysis for SAAWET, i.e., DSAAWET with $N=1$. The key step in the analysis is to establish the boundedness of the estimates, or to show that truncations cease in a finite number of steps. If the number of truncations increases unboundedly, then SAAWET is pulled back to a fixed vector $x^{*}$ infinitely many times. This produces convergent subsequences from the estimation sequence. Then the condition A5 b) is applicable and it incorporating with A2 yields a contradiction. This proves the boundedness of the estimates.
Let us try to use this approach to prove the boundedness of $x_k=\frac{1}{N} \sum_{i=1}^{N} x_{i,k}$ with $\{x_{i,k}\}$ generated by -. In the case $ \sigma_k \xlongrightarrow [k \rightarrow \infty ]{} \infty$, we have $ \lim\limits_{k \rightarrow \infty } \sigma_{i,k} = \infty~~ \forall i \in \mathcal{V}$ by Corollary \[cor\] given below. Then from Remark \[r1\] it is known that the estimate $x_{i,k+1}$ given by agent $i$ is pulled back to $x^{*}$ when the truncation occurs at time $k+1$. This means that $\{x_{i,k}\}~\forall i \in \mathcal{V}$ contains convergent subsequences. However, $\{x_k\}$ may still not contain any convergent subsequence to make A5 b) applicable. This is because truncations may occur at different times for different $i \in \mathcal{V}.$ Therefore, the conventional approach used for convergence analysis of SAAWET cannot directly be applied to the algorithm -.
To overcome the difficulty, we first introduce auxiliary sequences $ \{\tilde{x}_{ i,k}\} $ and $ \{\tilde{\varepsilon}_{i,k+1} \} $ for any $i \in \mathcal{V}.$ It will be shown in Lemma \[lemau2\] that $ \{\tilde{x}_{i,k}\} $ satisfies the recursions -, for which the truncation bound at time $k$ is the same $M_{\sigma_k}$ for all agents and the estimates $\tilde{x}_{i,k+1} ~ \forall i \in \mathcal{V}$ are pulled back to $x^{*}$ when $\sigma_{k+1}>\sigma_k$. As a result, the auxiliary sequence $\{ \widetilde{X}_k\}$ has convergent subsequences, where $ \widetilde{X}_k\deq col \{ \tilde{x}_{1,k}, \cdots, \tilde{x}_{N,k} \}.$ Besides, it will be shown in Lemma \[lemma1\] that the noise $\{\tilde{\varepsilon}_{k} \}$ satisfies a condition similar to A5 b) along any convergent subsequence of $\{ \widetilde{X}_k\}$, where $\tilde{\varepsilon}_{k} \deq col \{ \tilde{\varepsilon}_{1,k}, \cdots, \tilde{\varepsilon}_{N,k} \}$. To borrow the analytical method from the centralized stochastic algorithm, we rewrite the algorithm - in the centralized form with observation noise $\{ \zeta_{m+1}\}$. By the results given in Lemma \[lemau2\] and Lemma \[lemma1\], it is shown in Lemma \[lemnoise\] that the noise sequence $\{ \zeta_{m+1}\}$ satisfies along convergent subsequences, which is similar to A5 b) when $N=1$. Then by algorithm and the noise property , we show that the number of truncations for all agents converge to the same finite value, and that $\{\tilde{x}_{i,k}\}_{i \in \mathcal{V}}$ reach a consensus to the root set. Thus, $\{x_{i,k}\}$ and $\{\tilde{x}_{i,k}\}$ coincide in a finite number of steps, and their convergence is equivalent. In the rest of this section, we will demonstrate Theorem \[thm1\] in details based upon the aforementioned ideas.
Auxiliary Sequences
-------------------
Denote by $\tau_{i,m}\deq\textrm{inf} \{k: \sigma_{i,k}=m \}$ the smallest time when the truncation number of agent $i$ has reached $m$, by $\tau_m\deq \min\limits_{i \in \mathcal{V}}\tau_{i,m} $ the smallest time when at least one agent has its truncation number reached $ m$. Set $\tilde{\tau}_{j,m}\deq \tau_{j,m} \wedge \tau_{m+1} $, where $a\wedge b =\min\{a,b\}. $
For any $i \in \mathcal{V}$, define the auxiliary sequences $ \{\tilde{x}_{i,k}\}_{k \geq 0}$ and $ \{\tilde{\varepsilon}_{i,k+1} \}_{k \geq 0}$ as follows: $$\begin{aligned}
& \tilde{x}_{i,k}\triangleq x^{*}, ~ \tilde{\varepsilon}_{i,k+1}\triangleq -f_i(x^{*})\quad \forall k: \tau_m \leq k < \tilde{\tau}_{i,m}, \label{aux0} \\
& \tilde{x}_{i,k}\triangleq x_{i,k}, ~ \tilde{\varepsilon}_{i,k+1}\triangleq \varepsilon_{i,k+1} \quad \forall k: \tilde{\tau}_{i,m} \leq k < \tau_{m+1} , \label{aux1}\end{aligned}$$ where $m$ is an integer.
Note that for the considered $\omega$ there exists a unique integer $m \geq 0$ corresponding to an integer $k \geq 0$ such that $\tau_m \leq k < \tau_{m+1}$. By definition $\tilde{\tau}_{i,m} \leq \tau_{m+1} ~ \forall i \in \mathcal{V}$. So, $ \{\tilde{x}_{i,k}\}_{k \geq 0}$ and $ \{\tilde{\varepsilon}_{i,k+1} \}_{k \geq 0}$ are uniquely determined by the sequences $ \{x_{i,k}\}_{k \geq 0}$ and $ \{ \varepsilon_{i,k+1} \}_{k \geq 0}$. Besides, for any $ k \in [ \tau_m , \tau_{m+1})$, the following assertions hold:
& \_[i,k]{}=x\^\* , \_[i,k+1]{}=-f\_i(x\^\*)\_[i,k]{}<m; & \[case1\]\
& \_[i,k]{}=x\_[i,k]{} , \_[i,k+1]{}=\_[i,k+1]{} \_[i,k]{}=m; & \[case2\]\
& \_[j,k]{}=x\^\* \_[j,k-1]{}<m; & \[case3\]\
& \_[j, k+1]{}= x\^[\*]{} j \_[k+1]{}=m+1. & \[sub2\]
[**Proof:**]{} i) From $\sigma_{i,k} <m$ we see that the truncation number of agent $i$ is smaller than $m$ at time $k$, then by the definition of $\tau_{i,m}$ we derive $\tau_{i,m} >k$. Thus, $ \tilde{\tau}_{i,m}= \tau_{i,m} \wedge \tau_{m+1} >k,$ and hence from we conclude .
ii\) From $\sigma_{i,k}=m$ by definition we have $\tau_{i,m} \leq k$, and hence $\tilde{\tau}_{i,m} = \tau_{m+1} \wedge \tau_{i,m } =\tau_{i,m} \leq k. $ Then by it is clear that holds.
iii\) By $\tau_m \leq k < \tau_{m+1}$ we see $\sigma_{j,k} \leq m ~ \forall j \in \mathcal{V}$. We show separately for the cases $\sigma_{j,k} = m$ and $\sigma_{j,k} < m$. 1) Let us first consider the case $\sigma_{j,k} = m$. Since $\sigma_{j,k-1} < m $ and $\sigma_{j,k} = m$, by we obtain $x_{j,k}=x^{*}$. Hence from we see $\tilde{x}_{j,p} =x_{j,p}=x^{*}$. 2) We now consider the case $\sigma_{j,k} < m$. By we see $\tilde{x}_{j,k}=x^{*}$, which is the assertion of .
iv\) From $ k \in [ \tau_m , \tau_{m+1})$ we see $\sigma_k=m$. Thus from $ \sigma_{k+1}= m+1$ by definition we derive $\tau_{m+1} =k+1$, and hence $ k+1 \in [ \tau_{m+1} , \tau_{m+2})$. By $\sigma_k=m$ we see $\sigma_{j,k} < m+1 ~ \forall j \in \mathcal{V}$, then we derive by . $\blacksquare$
\[lemau2\] The sequences $ \{\tilde{x}_{i,k}\}, \{ \tilde{\varepsilon}_{i,k+1}\} $ defined by satisfy the following recursions $$\begin{aligned}
\hat{x}_{i,k+1} & \deq\sum_{j \in N_i(k)} \omega_{ij} (k) \tilde{x}_{j,k} +
\gamma_{ k } ( f_i( \tilde{x}_{i,k})+\tilde{\varepsilon}_{i,k+1}), \label{algorithm01} \\
\tilde{x}_{i,k+1} & =\hat{x}_{i,k+1} I_{ [ \parallel \hat{x}_{j,k+1} \parallel \leq M_{\sigma_k}~\forall j \in \mathcal{V} ] } \nonumber
\\& +x^* I_{ [ \exists j \in \mathcal{V} ~ \parallel \hat{x}_{j,k+1} \parallel > M_{\sigma_k} ]}, \label{algorithm02} \\
\sigma_{k+1} &=\sigma_k+ I_{ [ \exists j \in \mathcal{V} ~\parallel \hat{x}_{j,k+1} \parallel > M_{\sigma_k} ]}, \quad \sigma_0=0. \label{algorithm03}\end{aligned}$$
The proof is given in Appendix \[PLA\].
Before clarifying the property of the noise sequence $ \{\tilde{\varepsilon}_{i,k+1} \}_{k \geq 0}$, we need the following lemma.
\[lem001\] Assume A4 holds. Then
& \_[j,k+B d\_[i,j]{}]{} \_[i,k]{} j k 0, & \[gap000\]
where $d_{i,j}$ is the length of the shortest directed path from $i$ to $j $ in $\mathcal{G}_{\infty}$, and $B $ is the positive integer given in A4 d),
& \_[j,m]{} \_m+BD j m 1, & \[gap11\]
where $D\deq\max\limits_{i,j \in \mathcal{V}}d_{i,j}$.
i\) Since $\mathcal{G}_{\infty}$ is strongly connected by A4 c), for any $j \in \mathcal{V}$ there exists a sequence of nodes $i_1, i_2, \cdots, i_{d_{i,j}-1}$ such that $(i, i_1) \in \mathcal{E}_{\infty},(i_1, i_2) \in \mathcal{E}_{\infty},
\cdots, (i_{d_{i,j}-1},j) \in \mathcal{E}_{\infty}$.
Noticing that $ ( i,i_1) \in \mathcal{E}_{\infty},$ by A4 d) we have $$(i,i_1) \in \mathcal{E}(k)\cup \mathcal{E}(k+1) \cup \cdots \cup \mathcal{E}(k+B-1). \nonumber$$ Therefore, there exists a positive integer $k' \in [k ,k+B-1]$ such that $(i,i_1) \in \mathcal{E}(k') .$ So, $i \in N_{i_1}(k')$, and hence by and we derive $$\label{ineq}
\begin{split}
\sigma_{i_1,k+B} & \geq \sigma_{i_1,k'+1} \geq \hat{\sigma}_{i_1,k'} \geq \sigma_{i,k'}\geq \sigma_{i,k } . \nonumber
\end{split}$$ Similarly, we have $\sigma_{i_2,k+2B} \geq \sigma_{i_1,k+B} \geq \sigma_{i,k } .$ Continuing this procedure, we finally reach the inequality .
ii\) Let $\tau_m=k_1 $ for some $m\geq1$. Then there is an $i$ such that $\tau_{i,m}=k_1.$ By we have $
\sigma_{j,k_1+B d_{i,j}} \geq \sigma_{i,k_1 }=m ~~\forall j \in \mathcal{V}. $
For the case where $\sigma_{j,k_1+ Bd_{i,j}} = m ~\forall j \in \mathcal{V},$ we have $\tau_{j,m}\leq k_1+Bd_{i,j} ~\forall j \in \mathcal{V}.$ By noticing $\tau_m=k_1,$ from here by the definition of $\tilde{\tau}_{j,m}$ we obtain : $$\begin{aligned}
\tilde{\tau}_{j,m}\leq \tau_{j,m}\leq \tau_m+Bd_{i,j}\leq \tau_m+BD ~~\forall j \in \mathcal{V}.\end{aligned}$$ For the case where $\sigma_{j,k_1+ Bd_{i,j}} > m$ for some $j \in \mathcal{V},$ we have $\tau_{m+1}\leq k_1+Bd_{i,j} \textrm{ for some }j \in \mathcal{V}, $ and hence $\tau_{m+1}\leq \tau_m+BD $. Again, by noticing $\tau_m=k_1 $ we obtain : $$\begin{aligned}
\tilde{\tau}_{j,m}\leq \tau_{m+1}\leq \tau_m+BD ~~\forall j \in \mathcal{V}.\end{aligned}$$
\[cor\] If $ \sigma_k \xlongrightarrow [k \rightarrow \infty ]{} \infty$, then $ \lim\limits_{k \rightarrow \infty } \sigma_{i,k} = \infty~ \forall i \in \mathcal{V}$. This is because there exists an $i_0 \in \mathcal{V}$ such that $ \sigma_{i_0,k} \xlongrightarrow [k \rightarrow \infty ]{} \infty$. Then from it follows that $\sigma_{j,k} \xlongrightarrow [k \rightarrow \infty ]{} \infty~\forall j \in \mathcal{V}$.
\[r3\] If $\{ \sigma_k\}$ is bounded, then $\{ \tilde{x}_{i,k}\}$ and $\{x_{i,k}\}$, $\{ \tilde{\varepsilon}_{i,k}\}$ and $\{\varepsilon_{i,k}\}$ coincide in a finite number of steps.
The result can be derived by and the following assertion $$\label{infinity}
\tau_{\sigma+1}=\infty\textrm{ when } \lim\limits_{k \rightarrow \infty} \sigma_{k} = \sigma .$$ We now verify . Since $\sigma_{k} $ is defined as the largest truncation number among all agents at time $k$, from $\lim\limits_{k \rightarrow \infty} \sigma_{k} = \sigma$ we have $
\sigma_{i, k} \leq \sigma ~~ \forall k \geq 0 ~~ \forall i \in \mathcal{V}.$ From here by the definition of $\tau_{i,m}$ it follows that $\tau_{i, \sigma+1} = \textrm{inf} \{k: \sigma_{i,k}= \sigma+1 \} = \infty ~~ \forall i \in \mathcal{V},$ and hence $\tau_{\sigma+1}=\infty$. Thus, holds.
\[lemma1\] Assume A5 b) holds at the sample path $\omega$ under consideration for all agents. Then for this $\omega$ $$\label{auxnoise}
\begin{split}
& \lim\limits_{T \rightarrow 0} \limsup\limits_{k \rightarrow \infty} \frac{1}{T}
\parallel \sum_{s=n_k}^{m(n_k,t_k)\wedge ( \tau_{\sigma_{n_k}+1}-1 )} \gamma_s \tilde{\varepsilon}_{ s+1}
I_{[ \parallel \tilde{X}_{ s}\parallel \leq K ]} \parallel
\\&~~~~~~~ =0 ~~\forall t_k \in[0,T] \textrm{ for sufficiently large } K >0
\end{split}$$ along indices $\{ n_k\}$ whenever $\{ \tilde{X}_{ n_k}\}$ converges at $\omega$.
The proof is shown in Appendix \[PL1\].
Local Properties Along Convergent Subsequences
----------------------------------------------
Set $\Psi(k,s)\triangleq [ D_{\bot} (W(k) \otimes \mathbf{I}_l)] [ D_{\bot} (W(k-1) \otimes \mathbf{I}_l)] \cdots$ $(W(s) \otimes \mathbf{I}_l)]~~\forall k\geq s, \quad \Psi(k-1,k)\triangleq \mathbf{ I}_{Nl} .$
Since the matrices $ W(k ) ~ \forall k \geq 1$ are doubly stochastic, by using the rule of Kronecker product $$\label{kr}
(A \otimes B)(C \otimes D)= (AC ) \otimes ( BD)$$ we conclude that for any $ k \geq s-1$ $$\begin{aligned}
& \Psi(k,s)= (\Phi(k,s) - \frac{1}{N} \mathbf{1} \mathbf{1}^T) \otimes \mathbf{I}_l ,\label{power23}\\
& \Psi(k,s) D_{\bot} = ( \Phi(k,s) - \frac{1}{N} \mathbf{1} \mathbf{1}^T) \otimes \mathbf{ I}_l . \label{power24}\end{aligned}$$
The following lemma measures the closeness of the auxiliary sequence $\{ \widetilde{X}_{ k}\}$ along its convergent subsequence $\{ \widetilde{X}_{n_k}\}$.
\[lemma2\] Assume A1, A3, A4 hold and that A5 b) holds for all agents at the sample path $\omega$ under consideration. Let $\{ \widetilde{X}_{n_k}\}$ be a convergent subsequence of $\{ \widetilde{X}_k\}:
\widetilde{X}_{n_k} \xlongrightarrow [k \rightarrow \infty] {} \bar{X}$ at the considered $\omega$. Then for this $\omega$ there is a $T>0$ such that for sufficiently large $k$ and any $T_k \in [0,T]$ $$\label{compactform}
\widetilde{X}_{m+1}= ( W(m) \otimes \mathbf{I}_l)
\widetilde{X}_m+ \gamma_{m } (F(\widetilde{X}_m)+\tilde{\varepsilon}_{m+1})$$ for any $m=n_k, \cdots, m(n_k, T_k)$, and $$\begin{aligned}
& \parallel \widetilde{X}_{m+1} -\widetilde{X}_{n_k} \parallel \leq c_1 T_k +M_0', \label{res21}\\
& \parallel \bar{x}_{m+1}-\bar{x}_{n_k} \parallel \leq c_2 T_k~ ~ \forall m: n_k \leq m \leq m(n_k,T_k) , \label{res22}\end{aligned}$$ where $ \bar{x}_k\triangleq \frac{1}{N} ( \mathbf{1}^T \otimes \mathbf{I}_l) \widetilde{X}_k =\frac{1}{N}\sum_{i=1} ^N \tilde{x}_{i,k} $, and $c_0,~c_1 ,~M_0' $ are positive constants which may depend on $\omega$.
The proof is given in Appendix \[PL2\].
By multiplying both sides of with $ \frac{1}{N} ( \mathbf{1}^T \otimes \mathbf{I}_l) $ from left, by $ \mathbf{1}^TW(m) =\mathbf{1}^T$ and it follows that $$\label{centrelized}
\begin{split}
\bar{x}_{m+1}=&\bar{x}_m + \gamma_m f(\bar{x}_m)+
(\mathbf{1}^T \otimes \mathbf{I}_l)\gamma_m \tilde{\varepsilon}_{m+1}/ N \\+&
\gamma_m \sum_{i=1}^N \big( f_i(\tilde{x}_{i,m})- f_i(\bar{x}_m) \big)/N. \end{split}$$ Setting $e_{i,m+1}\triangleq ( f_i(\tilde{x}_{i,m})- f_i(\bar{x}_m) \big)/N $, $e_{m+1}\triangleq \sum_{i=1}^N e_{i,m+1}$, and $ \zeta_{m+1}\triangleq (\mathbf{1}^T \otimes \mathbf{I}_l) \tilde{\varepsilon}_{m+1}/N+ e_{m+1},$ we rewrite in the centralized form as follows: $$\label{centrelized2}
\begin{split}
\bar{x}_{m+1}=\bar{x}_m + \gamma_m ( f(\bar{x}_m)+ \zeta_{m+1} ).
\end{split}$$
The following lemma gives the property of the noise sequence $\{ \zeta_{k+1}\}$. For the proof we refer to Appendix \[PLN\].
\[lemnoise\] Assume that all conditions used in Lemma \[lemma2\] are satisfied. Let $\{ \widetilde{X}_{n_k}\}$ be a convergent subsequence with limit $\bar{X}$ at the considered $\omega$. Then for this $\omega$ $$\label{err2}
\lim_{T \rightarrow 0} \limsup_{k \rightarrow \infty} \frac{1}{T}
\parallel \sum_{s=n_k}^{m(n_k,T_k) } \gamma_s \zeta_{s+1} \parallel =0 \quad \forall T_k \in[0,T] .$$
The following lemma gives the crossing behavior of $v(\cdot)$ at the trajectory $\bar{x}_k$ with respect to a non-empty interval that has no intersection with $v(J)$.
\[lemma3\] Assume A1-A4 hold and that A5 b) holds for all agents at a sample path $\omega$. Then any nonempty interval $[\delta_1, \delta_2 ]$ with $d([\delta_1, \delta_2 ], v(J))>0 $ cannot be crossed by $\{v(\bar{x}_{n_k}), \cdots, v(\bar{x}_{m_k})\}$ infinitely many times with $ \{ \| \widetilde{X}_{n_k} \| \}$ bounded, where by “ crossing $[\delta_1, \delta_2]$ by $\{v(\bar{x}_{n_k}), \cdots, v(\bar{x}_{m_k})\}$ " it is meant that $v(\bar{x}_{n_k}) \leq \delta_1, v(\bar{x}_{m_k}) \geq \delta_2 ,$ and $ \delta_1 < v(\bar{x}_s) < \delta_2 ~~\forall s: n_k< s<m_k$.
Assume the converse: for some nonempty interval $[\delta_1, \delta_2 ]$ with $d([\delta_1, \delta_2 ], v(J))>0 $, there are infinitely many crossings $\{ v(\bar{x}_{n_k}), \cdots, v(\bar{x}_{m_k})\}$ with $ \{ \|\widetilde{X}_{n_k}\| \} $ bounded.
By the boundedness of $ \{ \|\widetilde{X}_{n_k}\| \}$, we can extract a convergent subsequence still denoted by $\{ \widetilde{X}_{n_k}\}$ with $\lim\limits_{k \rightarrow \infty} \widetilde{X}_{n_k} = \bar{X}$. So, $\lim\limits_{k \rightarrow \infty} \bar{x}_{n_k}= \bar{x}$ with $\bar{x}\triangleq \frac{\mathbf{1}^T \otimes \mathbf{I}_l}{N} \bar{X}$. Setting $T_k=\gamma_{n_k}$ in , we derive $ \parallel \bar{x}_{n_k+1}-\bar{x}_{n_k} \parallel \leq c_2 \gamma_{n_k}
\xlongrightarrow [k \rightarrow \infty]{} 0 .$ By the definition of crossings $v(\bar{x}_{n_k}) \leq \delta_1 < v(\bar{x}_{n_k+1})$, we obtain $$\label{gap1}
v(\bar{x}_{n_k}) \xlongrightarrow [k \rightarrow \infty]{} \delta_1 =v(\bar{x}),
\quad d(\bar{x}, J) \triangleq \vartheta>0.$$ Then by it is seen that $$\label{gap2}
d(\bar{x}_s, J) > \frac{\vartheta}{2} \quad \forall s : n_k \leq s \leq m(n_k,t)+1$$ for sufficiently small $t>0$ and large $k$. From we obtain $$\label{vfunction}
\begin{array}{lll}
& v(\bar{x}_{m(n_k,t)+1})= v(\bar{x}_{n_k} + \sum_{s=n_k}^{m(n_k,t)} \gamma_s( f(\bar{x}_s)+ \zeta_{s+1})) \\
& = v(\bar{x}_{n_k})+ v_x(\xi_k)^T \sum_{s=n_k}^{m(n_k,t)} \gamma_s( f(\bar{x}_s)+ \zeta_{s+1}),
\end{array}$$ where $\xi_k$ is in-between $\bar{x}_{n_k}$ and $\bar{x}_{m(n_k,t)+1 } $. We then rewrite as follows: $$\label{vfunction2}
\begin{array}{lll}
& v(\bar{x}_{m(n_k,t)+1}) - v(\bar{x}_{n_k})= \sum_{s=n_k}^{m(n_k,t)}\gamma_s v_x( \bar{x}_s)^T f(\bar{x}_s)
\\
& + \sum_{s=n_k}^{m(n_k,t)} \gamma_s(v_x(\xi_k)- v_x( \bar{x}_s))^T f(\bar{x}_s) \\& +v_x(\xi_k)^T \sum_{s=n_k}^{m(n_k,t)} \gamma_s \zeta_{s+1}. \end{array}$$
By and there exists a constant $\alpha_1>0$ such that $$v_x( \bar{x}_s)^T f(\bar{x}_s) \leq - \alpha_1 \quad \forall s : n_k \leq s \leq m(n_k,t)$$ for sufficiently small $t>0$ and large $k$, and hence $$\label{estm41}
\begin{array}{lll}
\sum_{s=n_k}^{m(n_k,t)} \gamma_s v_x( \bar{x}_s)^T f(\bar{x}_s) \leq - \alpha_1 t.
\end{array}$$
Since $\{\bar{x}_{s } : n_k \leq s \leq m(n_k,t) \}$ are bounded, by continuity of $f(\cdot)$ there exists a constant $c_6>0$ such that $$\label{sum41}
\begin{array}{lll}
\sum_{s=n_k}^{m(n_k,t)} \gamma_s \parallel f(\bar{x}_s) \parallel \leq c_6t .
\end{array}$$
Since $\xi_k$ is in-between $\bar{x}_{n_k}$ and $\bar{x}_{m(n_k,t)+1 } $, by continuity of $v_x(\cdot)$ and we know that $$\label{D01}
v_x(\xi_k)
- v_x( \bar{x}_s) =\delta(t) \quad \forall s: n_k \leq s\leq m(n_k,t),$$ where $\delta(t) \rightarrow 0\textrm{ as }t \rightarrow 0$. Then by we derive $$\label{sum42}
\begin{array}{lll}
& \sum_{s=n_k}^{m(n_k,t)} \gamma_s (v_x(\xi_k)- v_x( \bar{x}_s))^T f(\bar{x}_s)
\\& \leq \delta(t) \sum\limits_{s=n_k}^{m(n_k,t)} \gamma_s \parallel f(\bar{x}_s) \parallel \leq \delta(t) t .
\end{array}$$
Since $ \bar{x}_{n_k} \xlongrightarrow [k \rightarrow \infty]{} \bar{x}$, by continuity of $v_x(\cdot)$ and it follows that for sufficiently small $t>0$ and large $k$ $$v(\bar{x}_{s})-v_x( \bar{x} )= o(1)+\delta(t) \quad \forall s: n_k \leq s \leq m(n_k,t) ,$$ where $o(1)\rightarrow 0\textrm{ as } k \rightarrow \infty.$ Then by we derive $$v_x(\xi_k)-v_x( \bar{x} )= o(1)+\delta(t) \quad \forall s: n_k \leq s \leq m(n_k,t).$$ Consequently, for sufficiently small $t>0$ and large $k$ $$\label{sum43}
\begin{array}{lll}
& v_x(\xi_k)^T \sum_{s=n_k}^{m(n_k,t)} \gamma_s \zeta_{s+1} \\&=[(v_x(\xi_k)-v_x( \bar{x} )) + v_x( \bar{x} ) ]^T \sum_{s=n_k}^{m(n_k,t)} \gamma_s \zeta_{s+1} \\
& \leq (o(1)+\delta(t)+\|v_x( \bar{x} )\|) \parallel \sum_{s=n_k}^{m(n_k,t)} \gamma_s \zeta_{s+1} \parallel.
\end{array}$$ Substituting into , we obtain $$\begin{array}{lll}
& v(\bar{x}_{m(n_k,t)+1}) - v(\bar{x}_{n_k}) \leq - \alpha_1 t + \delta(t) t
\\& + (o(1)+\delta(t)+ \| v_x( \bar{x} ) \|) \parallel \sum\limits_{s=n_k}^{m(n_k,t)} \gamma_s \zeta_{s+1} \parallel.
\end{array}$$ Then by it follows that $$\begin{array}{lll}
& \limsup\limits_{k \rightarrow \infty} v(\bar{x}_{m(n_k,t)+1}) \leq \delta_1 - \alpha_1 t+ \delta(t) t \\& +
( \delta(t) +\| v_x( \bar{x} ) \|) \limsup\limits_{k \rightarrow \infty} \parallel \sum_{s=n_k}^{m(n_k,t)} \gamma_s \zeta_{s+1}\parallel,
\end{array}$$ and hence from we have $$\label{vfunction4}
\begin{split}
& \limsup_{k \rightarrow \infty} v(\bar{x}_{m(n_k,t)+1}) \leq \delta_1- \frac{\alpha_1}{2} t \end{split}$$ for sufficiently small $t.$
However, by continuity of $v_x(\cdot)$ and we know that $$\lim_{t \rightarrow 0} \max_{n_k \leq m \leq m(n_k,t)}\parallel v(\bar{x}_{m+1})-
v(\bar{x}_{n_k}) \parallel=0,$$ which implies that $m(n_k,t) +1 < m_k$ for sufficiently small $t$. Therefore, $ v(\bar{x}_{m(n_k,t)+1}) \in (\delta_1, \delta_2 )$, which contradicts with . Consequently, the converse assumption is not true. The proof is completed.
Finiteness of Number of Truncations
-----------------------------------
\[lemma4\] Let $ \{ x_{i,k}\} $ be produced by - with an arbitrary initial value $ x_{i,0} $. Assume A1, A3, A4, and A5 b) hold.
i\) If $\lim\limits_{k \rightarrow \infty } \sigma_k = \infty, $ then there exists an integer sequence $\{n_k\} $ such that $\bar{x}_{n_k}= x^{*} $, and $\{\bar{x}_{n_k}\}$ starting from $ x^{*}$ crosses the sphere with $\| x \| =c_0 $ infinitely many times, where $\{\bar{x}_{n_k}\}$ is defined in Lemma 4.6 and $c_0$ is given in A2 c).
ii\) If, in addition, A2 also holds, then there exists a positive integer $\sigma$ possibly depending on $\omega$ such that $$\label{lem4}
\lim_{k \rightarrow \infty} \sigma_{ k}=\sigma .$$
The proof of the lemma is given in Appendix \[PL4\]. Lemma \[lemma4\] says that the largest truncation number among all agents converges, while the following lemma indicates that the truncation numbers at all agents converge to the same limit.
\[lem001\] Assume all conditions required by Lemma \[lemma4\] are satisfied. Then there exists a positive integer $\sigma$ such that $$\label{trb}
\lim_{k \rightarrow \infty} \sigma_{i, k}=\sigma ~~ \forall i\in \mathcal{V}.$$
Since all conditions required by Lemma \[lemma4\] hold, holds for some positive integer $\sigma$. Thus, $$\label{upper}
\sigma_{i, k} \leq \sigma ~~ \forall k \geq 0 ~~ \forall i \in \mathcal{V}.$$ From by we have $\tau_{\sigma+1}=\infty$, and hence $\tilde{\tau}_{i,\sigma} =\tau_{i,\sigma} \leq BD+ \tau_{\sigma} ~ \forall i \in \mathcal{V} $ by . This means that the smallest time when the truncation number of agent $i$ reaches $\sigma$ is not larger than $ BD+ \tau_{\sigma}$. So, the truncation number of agent $i$ after time $ BD+ \tau_{\sigma}$ is not smaller than $\sigma$, i.e., $ \sigma_{i, k} \geq \sigma ~ \forall k \geq BD+ \tau_{\sigma} ~ \forall i \in \mathcal{V},$ which incorporating with yields $$\label{consensus}
\sigma_{i, k}= \sigma ~~ \forall k \geq BD+ \tau_{\sigma} ~~ \forall i \in \mathcal{V}.$$ Consequently, holds.
Proof of Theorem \[thm1\]
--------------------------
[**Proof.**]{} i) By and there is a positive integer $\sigma$ possibly depending on $\omega$ such that $$\label{hat}
\hat{\sigma}_{i, k}=\sigma_{i, k}= \sigma ~~ \forall k \geq k_0 \deq BD+ \tau_{\sigma} ~~ \forall i \in \mathcal{V},$$ and hence by $$\label{prime}
x'_{i,k+1}= \sum_{ j \in N_{i }(k)} \omega_{ij}(k) x_{j,k} + \gamma_{ k } O_{i,k+1}
~~ \forall k \geq k_0~ ~ \forall i \in \mathcal{V}.$$ By we see $\sigma_{i,k+1}=\hat{\sigma}_{i,k} = \sigma ~ \forall k \geq k_0~\forall i \in \mathcal{V}$, and hence $ \|x_{i,k+1}' \| \leq M_{\sigma} $ by and $ x_{i,k+1}=x_{i,k+1}' $ by for any $ k \geq k_0$ and any $i \in \mathcal{V}$. So, we conclude that for any $i \in \mathcal{V}$, $\{x_{i,k}\}$ is bounded and follows from .
ii\) By multiplying both sides of with $ D_{\bot}$ from left we derive $$\begin{split}
& X_{\bot, k+1}= D_{\bot} (W(k) \otimes I_l) X_{ k }+ \gamma_{k } D_{\bot} (F( X_k)+\varepsilon_{k+1}) , \nonumber
\end{split}$$ and inductively $$\label{dis021}
\begin{array}{lll}
& X_{\bot, k+1}= \Psi(k, k_0) X_{k_0}+ \\
& \sum_{m=k_0} ^k
\gamma_{m } \Psi(k-1, m) D_{\bot} (F( X_m)+ \varepsilon_{m+1}) \quad \forall k \geq k_0. \nonumber
\end{array}$$ Then by we derive $$\label{disagree}
\begin{array}{lll}
& X_{\bot, k+1} = [(\Phi(k, k_0) - \frac{1}{N} \mathbf{1} \mathbf{1}^T) \otimes \mathbf{I}_l] X_{k_0}\\&+ \sum_{m=k_0} ^k
\gamma_{m } [(\Phi(k-1, m) - \frac{1}{N} \mathbf{1} \mathbf{1}^T) \otimes \mathbf{I}_l] F(X_m) \\
& +\sum_{m=k_0} ^k \gamma_{m } [( \Phi(k-1, m) - \frac{1}{N} \mathbf{1} \mathbf{1}^T) \otimes \mathbf{I}_l] \varepsilon_{m+1}. \nonumber
\end{array}$$ Therefore, from by continuity of $F(\cdot )$ and the boundedness of $\{ X_s\}$, we conclude that there exist positive constants $c_1 , c_2 ,c_3 $ possibly depending on $\omega$ such that $$\label{er2}
\begin{array}{lll}
& \parallel X_{\bot, k+1} \parallel \leq c_1 \rho^{k+1- k_0} + c_2 \sum_{m= k_0} ^k \gamma_m \rho^{k -m} \\
& + c_3\sum_{m=k_0} ^{k} \gamma_m \rho ^{k-m} \parallel \varepsilon _{m+1}\parallel
\quad \forall k \geq k_0 .
\end{array}$$
Noticing that for any given $\epsilon>0$ there exists a positive integer $k_1 $ such that $ \gamma_k \leq \epsilon~~ \forall k \geq k_1$, we then have $$\begin{array}{lll}
& \sum_{m=0}^k \gamma_m \rho^{k-m} = \sum_{m=0}^{k_1} \gamma_m \rho^{k-m}
+ \sum_{m=k_1+1}^{k} \gamma_m \rho^{k-m}
\\& \leq \rho^{k-k_1} \sum_{m=0}^{k_1 } \gamma_m + \epsilon \frac{1}{1-\rho} \xlongrightarrow [k \rightarrow \infty \atop \epsilon \rightarrow 0 ]{} 0 \nonumber.
\end{array}$$ Therefore, the second term at the right-hand side of tends to zero as $k \rightarrow \infty$. Similarly, the last term at the right-hand side of also tends to zero since $ \lim\limits_{k \rightarrow \infty} \gamma_k \varepsilon_{k+1}=0 .$ Therefore, by $0< \rho<1$ from we conclude that $$X_{\bot, k} \xlongrightarrow [k \rightarrow \infty]{} \mathbf{0}.$$
By i) and Corollary \[r3\] we see that $\{\tilde{x}_{i,k}~\forall i\in \mathcal{V}\} $ are bounded for this $\omega$, and hence $\{\bar{x}_k\}$ is bounded. The rest of the proof is similar to that given in [@Chen_2002].
We first show the convergence of $ v( \bar{x}_k).$ Since $$v_1 \triangleq \liminf_{k \rightarrow \infty } v( \bar{x}_k) \leq \limsup_{k \rightarrow \infty } v( \bar{x}_k)
\triangleq v_2 ,$$ we want to prove $v_1=v_2$. Assume the converse: $v_1< v_2$. Since $v(J)$ is nowhere dense, there exists a nonnegative interval $[\delta_1, \delta_2 ]
\in ( v_1, v_2)$ such that $d([\delta_1, \delta_2 ], v(J))>0 $. Then $v(\bar{x}_k)$ crosses the interval $[\delta_1, \delta_2 ]$ infinitely many times. This contradicts Lemma \[lemma3\]. Therefore, $v_1=v_2$, which implies the convergence of $v( \bar{x}_k)$.
We then prove $ d( \bar{x}_k ,J ) \xlongrightarrow [k \rightarrow \infty]{} 0$. Assume the converse. Then by the boundedness of $ \{\bar{x}_k\}$ there exists a convergent subsequence $ \bar{x}_{n_k} \xlongrightarrow [k \rightarrow \infty]{} \bar{x} $ with $ d(\bar{x}, J) \triangleq \vartheta>0$. From it follows that for sufficiently small $t>0$ and large $k$ $$d(\bar{x}_s, J) > \frac{\vartheta}{2} \quad \forall s : n_k \leq s \leq m(n_k,t),$$ and hence from there exists a constant $b>0$ such that $$v_x(\bar{x}_s)^T f(\bar{x}_s) < -b \quad \forall s: n_k \leq s \leq m(n_k,t).$$ Thus, similar to the proof for obtaining it is seen that for sufficiently small $t>0 $ $$\label{contradict1}
\limsup_{k \rightarrow \infty} v(\bar{x}_{m(n_k,t)+1}) \leq v(\bar{x})- \frac{b}{2} t \nonumber.$$ This contradicts with the convergence of $v( \bar{x}_k)$. Therefore, $ d( \bar{x}_k ,J ) \xlongrightarrow [k \rightarrow \infty]{} 0$, and hence $ d( x_k ,J ) \xlongrightarrow [k \rightarrow \infty]{} 0$.
iii\) Assume the converse: i.e., $J^{*}$ is disconnected. Then there exist closed sets $J_1^{*}$ and $J_2^{*}$ such that $J^{*}= J_1^{*} \cup J_2^{*}$ and $d(J_1^{*}, J_2^{*})>0$. Define $\rho=\frac{1}{3}d(J_1^{*}, J_2^{*}).$ Noticing $ d( \bar{x}_k ,J^{*} ) \xlongrightarrow [k \rightarrow \infty]{} 0$, we know there exists $k_0$ such that $$\label{E1}
\bar{x}_k \in B(J_1^{*}, \rho) \cup B(J_2^{*}, \rho) ~~ \forall k\geq k_0,$$ where $B(A, \rho)$ denotes the $\rho$-neighborhood of the set $A.$
Define $$\begin{aligned}
& n_0=\inf \{ k >k_0, d( \bar{x}_k ,J_1^{*} )<\rho \}, \nonumber\\
& m_p=\inf \{ k >n_p, d( \bar{x}_k ,J_2^{*} )<\rho \}, \nonumber\\
& n_p=\inf \{ k >m_p, d( \bar{x}_k ,J_1^{*} )<\rho \}, ~~ p\geq 0 .\nonumber\end{aligned}$$ By we have $ \bar{x}_{n_p} \in B(J_1^{*}, \rho), ~ \bar{x}_{n_p+1} \in B(J_2^{*}, \rho) $ for any $ p \geq 0$. Then by $d(J_1^{*}, J_2^{*})=3\rho$ we derive $$\label{E2}
\| \bar{x}_{n_p} - \bar{x}_{n_p+1} \| >\rho .$$
Since $\{\bar{x}_{n_k} \}$ is bounded, we can extract a convergent subsequence, still denoted by $\{\bar{x}_{n_k} \}$. By setting $T_k = \gamma_{n_k }$ in we derive $ \| \bar{x}_{n_k+1} - \bar{x}_{n_k } \| \leq c_2 \gamma_{n_k } \xlongrightarrow [k \rightarrow \infty]{} 0, $ which contradicts with . So, the converse assumption is not true. Hence the proof is completed. $\blacksquare$
Convergence for Application Problems
====================================
In this section, we establish convergence results for the two problems stated in Section II, and present the corresponding numerical simulations.
Distributed PCA
---------------
We now apply DSAAWET to the problem of distributed principle component analysis, and establish its convergence.
\[thm-PCA\]Let $\{x_{i,k}\}$ be produced by – with $O_{i,k+1}$ defined by and $x_{i,0}=\mathbf{1}/\sqrt{N},$ where $x^{*}=\mathbf{1}/\sqrt{N}$. Assume that A4 holds, and, in addition, that
B1 $ \gamma_k>0, ~\sum\limits_{k=1}^\infty \gamma_k=\infty,~ \textrm{ and } \sum\limits_{k=1}^\infty \gamma_k^2<\infty; $
B2 i) $\{u_{i,k}\}$ is an iid sequence with zero mean and with bounded fourth moment for any $i \in \mathcal{V}$;
ii\) the largest eigenvalue of $A$ is with unit multiplicity with the corresponding eigenvectors denoted by $u^{(0)}$ and $-u^{(0)}$.
Then $$\label{PCAresult} X_{\bot, k} \xlongrightarrow [k \rightarrow \infty]{} \mathbf{0}~
\textrm{ and }~ d(x_k, J) \xlongrightarrow [k \rightarrow \infty]{}0~~a.s.,$$ where $J\deq \{0, u^{(0)}, -u^{(0)}\}.$ Moreover, for any $ i,j \in \mathcal{V}$ $$\label{PCA_cov}
\begin{split}
\lim_{k \rightarrow \infty} x_{i,k}= \lim_{k \rightarrow \infty} x_{j,k} &=u^{(0)}\textrm{ or }-u^{(0)} \textrm{ or }\mathbf{0}~~ a.s.
\end{split}$$
We prove this result by adopting the similar procedures as that used in the proof of Theorem \[thm1\]. Firstly, we use the auxiliary sequences defined in to prove the finiteness of truncation numbers. Then we show that the estimates are bounded and finally reach consensus. At last, we show that the estimates either converge to the unit eigenvector corresponding to the largest eigenvalue or to zero.
By B1 the step size $\{\gamma_k\}$ satisfies A1. Since $f_i(x)=A_ix-(x^TA_ix)x$, we see A3 holds. By we know $$\begin{split} \varepsilon_{i,k+1}& = O_{i,k+1}-f_i(x_{i,k}) \\&=(A_{i,k}-A_i)x_{i,k}-(x_{i,k}^T ( A_{i,k} -A_i)x_{i,k})x_{i,k} . \nonumber
\end{split}$$ Then from B2 i) we conclude that $\{ \varepsilon_{i,k+1}I_{[ \| x_{i,k} \| \leq K]}\}$ is an mds with bounded second moments for any $K>0$, and hence by the convergence theorem for mds [@Yuan1997Probability] $$\label{filter5}
\begin{split}
& \sum_{k=0}^{\infty} \gamma_k \varepsilon_{i,k} I_{[ \| x_{i,k} \| \leq K]} < \infty ~a.s.
\end{split}$$ So, A5 b) holds almost surely for any agent $i$.
By B2 ii) we see that the nonzero roots of $f(x)=Ax-(x^TAx)x$ are $J^0=\{u^{(0)},-u^{(0)}\}$. Define $$v(x)=\frac{e^{\|x\|^2}}{ x^T Ax}~~\forall x \neq \mathbf{0}.$$ Then $$\label{ }
\begin{split}
v_x(x) & =2x v(x)-\frac{e^{\|x\|^2} 2Ax}{ (x^T Ax)^2}
= \frac{ 2v(x)}{x^TAx}\big( (x^TAx)x-Ax \big), \nonumber
\end{split}$$ and hence $$\label{lyap}
\begin{split}
f^T(x) v_x(x) = -\frac{ 2v(x)}{x^TAx} \| f(x)\|^2 <0~~ \forall x \notin J. \end{split}$$
We now prove that $ \lim\limits_{k \rightarrow \infty} \sigma_k=\sigma<\infty$.
Assume the converse $ \lim\limits_{k \rightarrow \infty} \sigma_k=\infty.$
By Lemma \[lemma4\] i) $\{\widetilde{X}_k\}$ has a convergent subsequence $\{\widetilde{X}_{n_k}\}$ with $\widetilde{X}_{n_k}=(\mathbf{1} \otimes \textrm{I}_m) x^*$. Since A1, A3, A4 and A5 b) hold, Lemma \[lemma2\] and \[lemnoise\] take place. By noticing $\bar{x}_{n_k}=\mathbf{1}/\sqrt{N}$, from we see that for sufficiently small $T$ and large $k$, all $\overline{x}_{m+1}~~\forall m: n_k \leq m \leq m(n_k,T)$ are uniformly above zero. This together with yields when we follow the proof procedure for Lemma \[lemma3\] with $v(J)$ replaced by $v(J^0)$. It is worth noting that $v(J)$ is not defined since $v(x)$ is not defined at $x=0,$ while $v(J^{(0)})$ is well defined. Then, any nonempty interval $[\delta_1, \delta_2 ]$ with $d([\delta_1, \delta_2 ], v(J^0))>0 $ cannot be crossed by $\{v(\bar{x}_{n_k}), \cdots, v(\bar{x}_{m_k})\}$ infinitely many times.
Since $v(x)$ is radially unbounded, there exists $c_0>1$ such that $v(x^*)<\textrm{inf }_{\parallel x \parallel =c_0} v(x)$. By Lemma \[lemma4\] i) $\{\bar{x}_{n_k}\}$ starting from $ x^{*}$ crosses the sphere with $\| x \| =c_0$ infinitely many times. Since $v(J^0)$ is nowhere dense, there exists a nonempty interval $[\delta_1, \delta_2 ]
\in (v(x^*),\textrm{inf }_{\parallel x \parallel =c_0} v(x))$ with $d([\delta_1, \delta_2 ], v(J^0))>0 $. Therefore, $[\delta_1, \delta_2 ]$ with $d([\delta_1, \delta_2 ], v(J^0))>0 $ is crossed by $\{v(\bar{x}_{n_k}), \cdots, v(\bar{x}_{m_k})\}$ infinitely many times. This yields a contradiction, hence the converse assumption is not true. Thus, $$\label{PCAT}
\lim\limits_{k \rightarrow \infty} \sigma_k=\sigma<\infty.$$
Since holds, the proof for Theorem \[thm1\] i) is still applicable. So, we conclude that $\{X_k\}$ is bounded, and there exists a positive integer $k_0$ such that holds. Then by $$\sum_{k=0}^{\infty} \gamma_k \varepsilon_{i,k} < \infty~~a.s.$$ and A5 holds. Since the proof for the first result in Theorem \[thm1\] ii) still holds, we derive $X_{\bot, k} \xlongrightarrow [k \rightarrow \infty]{} \mathbf{0}~a.s$.
We now prove $d(x_k, J) \xlongrightarrow [k \rightarrow \infty]{}0~a.s.$ and by considering the following two cases:
Case 1: If $X_{k}$ converges to zero a.s., then by the definition of $x_k$ it follows that $ x_k \xlongrightarrow [k \rightarrow \infty]{}0\in J~~a.s.$
Case 2: If $X_{k}$ does not converge to zero, then for sufficiently large $k_1>k_0$ $$\label{nonzero}
X_k\neq \mathbf{0}~~\forall k \geq k_1.$$ This is because the converse assumption that $X_{k_2}= \mathbf{0} $ for some $k_2>k_1 $ leads to $X_k= \mathbf{0}~~\forall k \geq k_2$ by and . This contradicts with the assumption that $X_{k}$ does not converge to zero. So, holds.
By and $X_{\bot, k} \xlongrightarrow [k \rightarrow \infty]{} \mathbf{0}$ we see that $x_k\neq \mathbf{0} $ for sufficiently large $ k$. Note that $\{x_{i,k}\}$ and $\{\tilde{x}_{i,k}\}$ coincide in a finite number of steps by Corollary \[r3\] and . Then $\overline{x}_k \neq \mathbf{0} $ for sufficiently large $ k$. Without loss of generality, we may assume $\overline{x}_k \neq \mathbf{0}~~\forall k \geq 0. $ Since A1, A3, A4, A5 and hold, by the similar proof as that for Theorem \[thm1\] ii) we derive $\lim\limits_{k \rightarrow \infty} d(x_k,J^0)=0.$ Since $J^0$ is composed of isolated points, we conclude that in this case $x_k$ converges to either $u^{(0)}$ or $-u^{(0)}.$
Thus, all assertions of the theorem have been proved.
![The estimation sequences and estimation error[]{data-label="Figure A1"}](PCA1.png){width="3.8in"}
Distributed Principle Componet Analysis
Let $N=1000$. The matrix $W(k)$ is as follows: $$\begin{split}
&W(3k-2)= \begin{pmatrix}
W_1 & \mathbf{0} \\
\mathbf{0} & \textrm{I}_{N/2}
\end{pmatrix},~ W(3k-1)= \begin{pmatrix}
\textrm{I}_{N/2} & \mathbf{0} \\
\mathbf{0} &W_2
\end{pmatrix}, \\
&W(3k)= \begin{pmatrix}
\frac{1}{2} \textrm{I}_{N/2} & \frac{1}{2} \textrm{I}_{N/2}\\
\frac{1}{2}\textrm{I}_{N/2}& \frac{1}{2}\textrm{I}_{N/2}
\end{pmatrix}, \nonumber
\end{split}$$ where matrices $W_1\in \mathbb{R}^{N/2\times N/2}$ and $W_2\in \mathbb{R}^{N/2\times N/2}$ are doubly stochastic. Further, they are the adjacency matrices of some strongly connected digraphs. Thus, A4 holds.
Each sensor $i=1,2, \cdots, N$ has access to an $9$-dimensional iid Gaussian sequence $\{u_{i,k}\}$ with zero mean. Set $\gamma_k=\frac{1}{k}$, $M_k=2^k$. Let the sequence $\{x_{i,k}\}$ be produced by – with $O_{i,k+1}$ defined by and with $x_{i,0}= x^{*}=\mathbf{1}/\sqrt{N}$. Denote by $x_k^i$ the $i$th component of $x_k=\frac{1}{N} \sum_{i=1}^N x_{i,k}$, and by $e(k)=\sum_{i=1}^N \| x_{i,k}-u^{(0)} \|_2/N$ the average of 2-norm errors for all agents at time $k$.
The estimates $x_k$ are demonstrated in Fig. \[Figure A1\] with the dashed lines denoting the true values and the real lines the corresponding estimates. The estimation errors $e(k)$ are demonstrated in Fig. \[Figure A1\]. From the figure it is seen that the estimates converge to the unit eigenvector corresponding to the largest eigenvalue of $A.$
For a comparison, we also present the simulation results computed by DSAA proposed in [@DSA] for the same example. The initial value for each agent, $ \alpha_k$ and $ \gamma_k $ are set to be the same as those used in DSAAWET. The simulation results are given in Table II, from which it is seen that the estimates are unbounded. When a smaller step size, say $\gamma_k=\frac{ 1}{ k+20 }$, is taken, then the estimates are bounded for some sample paths and are unbounded for others. It is hard to indicate how small the step-sizes should be for a given sample path in advance. As a matter of fact, DSAAWET is an adaptive method for choosing step-sizes for any sample path.
$k$ 0 1 2 3 4
--------- ----- ------- ------- ----------------- ----------------- --
$x_k^1$ 1/3 1.9 -587 $5.46*10^{11}$ $-1.49*10^{41}$
$x_k^2$ 1/3 3.2 -593 $3.64*10^{11}$ $-9.89*10^{40}$
$x_k^3$ 1/3 0.277 -45.4 $-9.9*10^{10}$ $2.93*10^{40}$
$x_k^4$ 1/3 -1.26 402 $-6*10^{11}$ $1.72*10^{41}$
$x_k^5$ 1/3 -3.26 736 $-2.23*10^{11}$ $4.76*10^{40}$
$x_k^6$ 1/3 0.25 -17.8 $7.86*10^{10}$ $-2.23*10^{40}$
$x_k^7$ 1/3 3.3 -711 $5.02*10^{11}$ $-1.34*10^{41}$
$x_k^8$ 1/3 0.19 2.48 $-1.76*10^{11}$ $5.54*10^{40}$
$x_k^9$ 1/3 -1.63 397 $-2.37*10^{11}$ $6.38^{40}$
: Estimates produced by DSAA in [@DSA]
Distributed Gradient-free Optimization
--------------------------------------
We now consider the convergence of DSAAWET applied to the distributed gradient-free optimization problem.
\[thmopt\] Let $ \{ x_{i,k}\} $ be produced by - with $O_{i,k+1}$ given by for any initial value $ x_{i,0} $. Assume that A4 holds, and, in addition,\
C1 i) $\gamma_k>0, \sum_{k=1}^{\infty } \gamma_k =\infty$, and $ \sum_{k=1}^{\infty } \gamma_k^p <\infty $ for some $p \in (1,2]$;
ii) $\alpha_k>0$, and $\alpha_k\rightarrow 0$ as $k \rightarrow \infty$;
iii) $ \sum_{k=1}^{\infty } \gamma_k^2 / \alpha_k^2 <\infty$;\
C2 i) $ c_i(\cdot), i=1,2,\cdots,N$ are continuously differentiable, and $\nabla c(\cdot)$ is locally Lipschitz continuous;
ii) $c(\cdot)$ is convex with a unique global minimum $x_{min}$;
iii) $c(x^{*}) < \sup_{ \| x \| =c_0} c(x) $ and $\| x^{* } \| <c_0$ for some positive constant $c_0$, where $x^{*}$ is used in ;\
C3 $\xi_{i,k+1}= \xi_{i,k+1}^{+}- \xi_{i,k+1}^{-} $ is independent of $ \{ \triangle_{i,s}, s=1,2,\cdots,k\}$ for any $k\geq 0$, and $\xi_{i,k+1}$ satisfies one of the following two conditions:
i) $\sup_k | \xi_{i,k } | \leq \xi_i ~ a.s.,$ where $\xi_i$ may be random;
ii) $\sup_k E [ \xi_{i,k+1}^2] < \infty$\
for any $i=1,2,\cdots N$. Then for any $i=1,2,\cdots, N$ $$\label{opt6} x_{i,k} \xlongrightarrow [k \rightarrow \infty] {} x_{min} ~a.s.$$
To apply Theorem \[thm1\] to this problem, we have to verify conditions A1-A5.
By C1 the step-size $\{ \gamma_k\}$ satisfies A1. Since $c(\cdot)$ is convex and differentiable, $x_{min}$ is the global minimum of $c(\cdot) $ if and only if $ \nabla c(x_{min})=0.$ So, the original problem is equivalent to finding the root $J=\{ x_{min}\}$ of $ f(\cdot)=\sum_{i=1}^N f_i(\cdot)/N$ with $f_i(\cdot) =- \nabla c_i(\cdot)$. By setting $v(\cdot)=c(\cdot)$, we derive $f^T(x)v_x(x)=- \| \nabla c(x) \|^2 /N $ and $v(J)=\{ c(x_{min})\}$. So, A2 a) and A2 b) hold. By C2 iii) we have A2 c). It is clear that C2 i) implies A3.
By we see $$\label{DO_noise}
\varepsilon_{i,k+1}=O_{i,k+1}+\nabla c_i(x_{i,k}) ,$$ where $O_{i,k+1}$ is given by . The analysis for the observation noise $\{\varepsilon_{i,k+1}\}$ is the same as that given in [@Chen_1999], so, is omitted here. It is shown in [@Chen_1999] that $\{ \varepsilon_{i,k+1} \}$ is not an mds for any $i \in \mathcal{V}$, but the following limit takes place: $$\label{opt5}
\begin{split}
& \lim_{T \rightarrow 0} \limsup_{k \rightarrow \infty }\frac{1}{T}
\parallel \sum_{n= k}^{m( k,t_k)}\gamma_{m } \varepsilon_{i,n+1} I_{ [\parallel x_{i,n} \parallel \leq K ]} \parallel =0
\\& ~~~ \forall t_k \in [0,T] \textrm{ for any positive integer }K , ~~a.s.
\end{split}$$ Therefore, A5 b) holds with probability one for any $i \in \mathcal{V}$. By Theorem \[thm1\] i) we see that $\{ x_{i,k}\}$ is bounded almost surely. So, A5 a) holds by taking $t_k= \gamma_k$ in .
In summary, we have shown that A1-A5 hold. Since $J=\{ x_{min}\}$, by Theorem \[thm1\] ii) we derive .
![ Trajectories of the estimates []{data-label="Figure A2"}](DO2.png){width="3.5in"}
If the convex function $c(\cdot)$ is allowed to have non-unique minima while other conditions in Theorem \[thmopt\] remain unchanged, then by Theorem \[thm1\] we know that $x_{i,k} ~\forall i=1,\cdots, N$ converge to a connected subset $J^{*} \subset J$, where $J=\{ x \in \mathbb{R}^l: \nabla c(x)=0\}$. Compared with [@Yuan_NN_15][@Yuan_NN_16], the almost sure convergence is established here without assuming that each cost function is convex. However, we require the cost functions to be differentiable and only consider the unconstrained optimization problem.
Distributed Gradient-free Optimization
Consider the network of three agents with local cost functions given by $$\label{sf}
\begin{split}
& L_1(x,y)=x^2+y^2+10\sin(x) ; \\
& L_2(x,y)=(x-4)^2+(y-1)^2-10\sin (x);\\
& L_3(x,y)=0.01(x-2)^4+(y-2)^2. \nonumber
\end{split}$$
Let the communication relationship among the agents be described by a strongly connected digraph with the adjacency matrix being doubly stochastic. The task of the network is to find the minimum $(x^0,y^0)=(2,1)$ of the cost function $ L(x,y) =\sum_{i=1}^3 L_i(x,y)$. Though each local cost function is non-convex, the global cost function $L(x,y)$ is convex.
Let the observation noise for the cost function of each agent be a sequence of iid random vectors $\in {\cal N}(0,I).$ The first and second component of the initial values for all agents are set to be mutually independent and uniformly distributed over the intervals $[-2,6]$ and $[-2,4]$, respectively. Set $x^*= (-1,4)^T$, $\gamma_k=\frac{2}{k}$, $\alpha_k= \frac{1}{k^{0.2}}$, and $M_k=2^k$. Let the estimates for the minimum be produced by – with $O_{i,k+1}$ defined by . The estimates of $x^0$ and $y^0$ produced by the three agents are demonstrated in Fig. \[Figure A2\], where $x_i(k) $ and $ y_i(k) $ denote agent $i$’s estimates for $x^0$ and $y^0$ at time $k$, respectively. From the figure it is seen that the estimates given by all agents converge to the minimum, which is consistent with the theoretic result.
Concluding Remarks
===================
In this paper, DSAAWET is defined to solve the formulated distributed root-seeking problem. The estimates are shown to converge to a consensus set belonging to a connected subset of the root set. Two problems as examples of those which can be solved by DSAAWET are demonstrated with numerical simulations provided.
For further research it is of interest to analyze the convergent rate of the proposed algorithm, and to consider the convergent properties of DSAAWET over random networks taking into account the possible packet loss in communication. It is also of interest to consider the possibility of removing the continuity assumption on $f_i(\cdot),$ since for the centralized SAAWET the function is only required be measurable and locally bounded.
Proof Lemma \[lemau2\] {#PLA}
======================
We prove the lemma by induction.
We first prove - for $k=0$. Since $0 \in [\tau_0, \tau_1)\textrm{ and } \sigma_{i,0}=0 ~ \forall i \in \mathcal{V}$, by we derive $ \tilde{x}_{i,0} = x_{i,0}, ~ \tilde{\varepsilon}_{i, 1} = \varepsilon_{i,1} ~ \forall i \in \mathcal{V}$. Then by noticing $\hat{\sigma}_{i,0}=\sigma_{i,0}=0 ~ \forall i \in \mathcal{V}$, from we see $$\label{equalityhat}
\hat{x}_{i, 1}= x'_{i,1}~~ \forall i \in \mathcal{V}.$$ We now show that $ \tilde{x}_{i,1}$ and $\sigma_1$ generated by - are consistent with their definitions by considering the following two cases:
i\) There is no truncation at $k=1$, i.e., $\sigma_{i,1}=0 ~\forall i \in \mathcal{V}$. In this case, from $\hat{\sigma}_{i,0} =0 ~ \forall i \in \mathcal{V}$ by we have $ \| x'_{i,1} \| \leq M_0 ~\forall i \in \mathcal{V}$, and hence $ \| \hat{x}_{i, 1}\| \leq M_0 ~\forall i \in \mathcal{V}$ by . Then $ x_{i, 1}= x'_{i,1} $ by , $ \tilde{x}_{i, 1} = \hat{x}_{i, 1}$ and $\sigma_1=0$ by and , respectively. These together with imply that $ \tilde{x}_{i,1}= x_{i, 1} ~ \forall i \in \mathcal{V},$ which is consistent with the definition of $\tilde{x}_{i,1} $ given by since $\tilde{\tau}_{i,0} \leq 1 < \tau_1$. By we see $\sigma_1=\max_{i \in \mathcal{V}} \sigma_{i,1}=0,$ which is consistent with that derived from -.
ii\) There is a truncation at $k=1$ for some agent $i_0$, i.e., $\sigma_{i_0,1}=1.$ In this case, by we derive $ x_{i_0,1}=x^{*}, ~ \| x'_{i_0,1} \|>M_0$, and hence $\| \hat{x}_{i_0, 1} \| > M_0$ by . Therefore, $ ~ \tilde{x}_{i, 1} = x^{*} ~\forall i \in \mathcal{V}
\textrm{ and }\sigma_1=1$ by . By from $\sigma_{i_0,1}=1 $ we derive $\sigma_1=\max_{i \in \mathcal{V}} \sigma_{i,1}=1$. Since $ 0 \in [ \tau_0 , \tau_{ 1})$ and $\sigma_1=1$, by we see $\tilde{x}_{i, 1} = x^{*} ~ \forall i \in \mathcal{V}$. Thus, $ \tilde{x}_{i,1}$ and $\sigma_1$ defined by and are consistent with those produced by -.
In summary, we have proved the lemma for $k=0$.
Inductively, we assume - hold for $k=0,1,\cdots,p$. At the fixed sample path $\omega$ for a given integer $ p $ there exists a unique integer $m $ such that $\tau_m \leq p < \tau_{m+1}$. We intend to show - also hold for $k=p+1$.
Before doing this, we first express $ \hat{x}_{i,p+1}~\forall i \in \mathcal{V}$ produced by for the following two cases:
Case 1: $\sigma_{i,p} <m$. Since $p \in [\tau_m, \tau_{m+1}), $ by we see $$\label{lem331}
\tilde{x}_{i,p}=x^* , \quad \tilde{\varepsilon}_{i,p+1}=-f_i(x^*).$$ From $\sigma_{i,p} <m$ it follows that $\sigma_{j,p-1} < m ~\forall j \in N_i(p)$, because otherwise, there would exist a $j_1 \in N_i(p)$ such that $\sigma_{j_1,p-1} \geq m$. Hence by we would derive $\sigma_{i,p} \geq \hat{\sigma}_{i,p-1} \geq \sigma_{j_1,p-1} \geq m $, yielding a contradiction.
From $\sigma_{j,p-1} < m ~\forall j \in N_i(p)$ by we have $\tilde{x}_{j,p}= x^*~ ~\forall j \in N_i(p) ,
$ which incorporating with yields $$\label{A11}
\hat{x} _{i,p+1}= x^* \quad \forall i: \sigma_{i,p}<m .$$
Case 2: $ \sigma_{i,p}= m$. By $\tau_m \leq p < \tau_{m+1}$ we see $\sigma_{j,p} \leq m ~ \forall j \in \mathcal{V}$, and hence by we obtain $$\label{hatd}
\hat{\sigma}_{i,p}=m~ ~ \forall i: \sigma_{i,p}=m .$$ Then by $$\label{aux2}
\begin{array}{lll}
x'_{i,p+1} &=& \sum_{ j \in N_{i }(p)} \omega_{ij}(p) ( x_{j,p} I_{[ \sigma_{j,p} = m]} +
x^* I_{[ \sigma_{j,p} < m]}) \\& +& \gamma_{ p } (f_i( x_{i,p})+\varepsilon_{i,p+1}).
\end{array}$$
From $\sigma_{i,p} =m$ and $p \in [\tau_m, \tau_{m+1}), $ by it is clear that $$\label{express1}
\tilde{x}_{i,p}=x_{i,p}, ~ ~ \tilde{\varepsilon}_{i,p+1}=\varepsilon_{i,p+1} .$$ By the first term in and , for any $j \in N_i(p)$ we have $$\label{express2}
\tilde{x}_{j,p}=x_{j,p}~\textrm{ if }\sigma_{j,p} = m, ~ ~ \tilde{x}_{j,p}=x^{*} ~\textrm{ if }\sigma_{j,p} < m.$$ Substituting into , from we derive $$\label{eq111}
\begin{split}
\hat{x}_{i,p+1}= x'_{i,p+1} \quad \forall i: \sigma_{i,p}=m .
\end{split}$$
Since $\tau_m \leq p < \tau_{m+1}$, from $p < \tau_{m+1}$ it follows that $\sigma_p < m+1$, and hence $\sigma_{p+1} \leq m+1$, while from $\tau_m \leq p $ it follows that $\sigma_p=m$, and hence $\sigma_{p+1}\geq m$. Thus, we have $m\leq \sigma_{p+1}\leq m+1$.
We now show that $ \tilde{x}_{i, p+1}$ and $\sigma_{p+1}$ generated by - are consistent with their definitions . We prove this separately for the cases $\sigma_{p+1}=m+1$ and $\sigma_{p+1}=m$.
Case 1: $\sigma_{p+1}=m+1$. We first show $$\label{if}
\sigma_{i,p+1} \leq m ~\textrm{ if }\sigma_{i,p} <m$$ separately for the following two cases 1) and 2): 1) $\sigma_{i,p} <m$ and $ \sigma_{j,p} <m ~\forall j \in N_i(p)$. For this case by we derive $\hat{\sigma}_{i,p} <m$, and hence $\sigma_{i,p+1} \leq \hat{\sigma}_{i,p}+1 \leq m$ by . 2) $\sigma_{i ,p} <m$ and $ \sigma_{j,p} =m ~\textrm{ for some }j \in N_i(p)$. For this case we obtain $\hat{\sigma}_{i,p} =m, ~x'_{i,p+1}=x^{*}$ by . Since $ \| x^{*}\| \leq M_0 \leq M_m $, then by $\sigma_{i,p+1} =\hat{\sigma}_{i,p} =m$. Thus, $\sigma_{i,p+1}\leq m $ when $\sigma_{i,p} <m$. Thereby holds. This means that $$\label{onlyif}
\sigma_{i,p+1} =m+1\textrm{ only if } \sigma_{i,p} =m.$$
By definition from $\sigma_{p+1}=m+1$ we know that there exists some $ i_0 \in\mathcal{V} $ such that $\sigma_{i_0, p+1}=m+1$. Then $\sigma_{i_0, p }=m$ by , and hence $\hat{\sigma}_{i_0, p }=m$ from . Then from $\sigma_{i_0, p+1}=m+1$ by we derive $\parallel x'_{i_0, p+1} \parallel > M_m$, and hence $ \parallel \hat{x}_{i_0, p+1} \parallel =\parallel x'_{i_0, p+ 1} \parallel >M_m $ by . Then from we derive $ \tilde{x} _{i, p+1}=x^*~~ \forall i \in \mathcal{V} $ and $\sigma_{p+1}=m+1 $, which is consistent with $\sigma_{p+1}$ defined by . Since $\sigma_{ p+1}=m+1$ and $p \in [\tau_m, \tau_{m+1}), $ by or from , we see $ \tilde{x} _{i, p+1}=x^*~ \forall i \in \mathcal{V} $. This is consistent with that produced by -.
Case 2: We now consider the case $\sigma_{p+1}=m$. In this case, $\sigma_{i , p+1} \leq m ~ \forall i \in \mathcal{V}.$ By , from we see $$\label{B20}
\parallel x'_{i , p+1} \parallel \leq M_m , ~x_{i,p+1}= x'_{i, p+1} ~~ \forall i :\sigma_{i, p }=m .$$ So, by we have $$\label{B23}
\| \hat{x}_{i,p+1} \| = \| x'_{i,p+1}\| \leq M_m \quad \forall i:\sigma_{i, p }=m .$$ From $\parallel x^* \parallel \leq M_0 \leq M_m$ and we derive $ \parallel \hat{x}_{i,p+1} \parallel \leq M_m ~ \forall i:\sigma_{i, p }<m ,$ which incorporating with yields $\parallel \hat{x}_{i ,p+ 1} \parallel \leq M_m
~ \forall i \in \mathcal{V}.$ Then from we have $$\label{B21}
\tilde{x} _{i,p+1}= \hat{x}_{i,p+1} \quad \forall i \in \mathcal{V}, ~~ \sigma_{p+1}=m.$$ Thus $\sigma_{p+1} $ is consistent with that defined by .
It remains to show that $ \tilde{x}_{i,p+1}$ generated by - is consistent with that defined by . We consider two cases: 1) $\sigma_{i , p } = m$. For this case, by we see $ \tilde{x}_{i,p+1}= x_{i,p+1} ~ \forall i:\sigma_{i, p }=m $. By $\sigma_{ p+1} = m$ we see $p+1\in [\tau_m, \tau_{m+1})$, and hence $ \tilde{x}_{i,p+1}= x_{i,p+1}$ by . So, the assertion holds for any $i$ with $ \sigma_{i , p } = m$. 2) $\sigma_{i , p} < m$. From $\sigma_{p+1}=m$ we see $p+1\in [\tau_m, \tau_{m+1})$, and hence by $\sigma_{i , p} < m$ from we see that $ \tilde{x} _{i, p+1} $ defined by equals $x^{*}$. By we derive $ \tilde{x} _{i, p+1}=x^{*} $, and hence the assertion holds for any $i$ with $ \sigma_{i , p } < m$.
In summary, we have shown that $ \tilde{x}_{i, p+1}$ and $\sigma_{p+1}$ generated by - are consistent with their definitions . This completes the proof. $\blacksquare$
Proof of Lemma \[lemma1\] {#PL1}
=========================
For the lemma it suffices to prove $$\label{lem1}
\begin{split}
& \lim\limits_{T \rightarrow 0} \limsup\limits_{k \rightarrow \infty} \frac{1}{T}
\parallel \sum_{s=n_k}^{m(n_k,t_k)\wedge ( \tau_{\sigma_{n_k}+1}-1 )} \gamma_s \tilde{\varepsilon}_{i,s+1}
I_{[ \parallel \tilde{x}_{i, s}\parallel \leq K ]} \parallel
\\&~~~~~~~ =0 ~~\forall t_k \in[0,T] \textrm{ for sufficiently large } K >0
\end{split}$$ along indices $\{ n_k\}$ whenever $\{ \tilde{x}_{i, n_k}\}$ converges for the sample path $\omega$ where A5 b) holds for agent $i$.
We consider the following two cases:
Case 1: $ \lim\limits_{k \rightarrow \infty} \sigma_{k} = \sigma < \infty $. We now show $$\label{infinity}
\tau_{\sigma+1}=\infty\textrm{ when } \lim\limits_{k \rightarrow \infty} \sigma_{k} = \sigma .$$ Recall that $\sigma_{k} $ is defined as the largest truncation number among all agents at time $k$, from $\lim\limits_{k \rightarrow \infty} \sigma_{k} = \sigma$ we have $
\sigma_{i, k} \leq \sigma ~~ \forall k \geq 0 ~~ \forall i \in \mathcal{V}.$ From here by the definition of $\tau_{i,m}$ it follows that $\tau_{i, \sigma+1} = \textrm{inf} \{k: \sigma_{i,k}= \sigma+1 \} = \infty ~~ \forall i \in \mathcal{V},$ and hence $\tau_{\sigma+1}=\infty$. Thus, holds.
From we have $\tilde{\tau}_{i,\sigma} \leq \tau_{\sigma}+BD $, and hence by $$\label{111}
\tilde{x}_{i,k} =x_{i,k}, ~ \tilde{\varepsilon}_{i,k+1}=\varepsilon_{i,k+1} ~~\forall k \geq \tau_{ \sigma} +BD.$$ So, $\parallel \sum_{s= k}^{m( k,t )\wedge ( \tau_{\sigma_{ k}+1}-1 )} \gamma_s \tilde{\varepsilon}_{i,s+1}
I_{[ \parallel \tilde{x}_{i, s}\parallel \leq K ]} \parallel \\ =\parallel \sum_{s=k}^{m( k,t ) } \gamma_s \varepsilon_{i,s+1} I_{[ \parallel x_{i, s}\parallel \leq K ]} \parallel$ for any $t>0$ and any sufficiently large $k$. Then we conclude by A5 b).
Case 2: $ \lim\limits_{k \rightarrow \infty} \sigma_{k} = \infty $. We prove separately for the following three cases:
i\) $\tilde{\tau}_{i, \sigma_{ n_p } } \leq n_{p } $. For this case, $[ n_p,\tau_{\sigma_{n_p }+1}) \subset [ \tilde{\tau}_{i, \sigma_{n_p }},\tau_{\sigma_{n_p }+1}) ,$ and hence from we derive $$\label{C1}
\tilde{x}_{i,s}= x_{i,s}, ~ \tilde{\varepsilon}_{i,s+1}= \varepsilon_{i,s+1} ~ \forall s: n_p \leq s < \tau_{\sigma_{n_p }+1}.$$ Thus, for sufficiently large $K$ and any $ t_p \in[0,T]$ $$\label{s1}
\begin{array}{lll}
& \parallel \sum_{s=n_p }^{m(n_p ,t_p)\wedge ( \tau_{\sigma_{n_p}+1}-1 )} \gamma_s \tilde{\varepsilon}_{i, s+1}
I_{[ \parallel \tilde{x}_{i, s}\parallel \leq K ]} \parallel \\& = \parallel
\sum_{s= n_p }^{m(n_p ,t_p)\wedge ( \tau_{\sigma_{n_p }+1}-1)}
\gamma_s \varepsilon_{i,s+1} I_{[ \parallel x_{i,s}\parallel \leq K ]} .
\end{array}$$ By we conclude that $ \{ x_{i,n_p } \} $ is a convergent subsequence. Noticing $\sum_{s= n_p }^{m(n_p ,t_p)\wedge ( \tau_{\sigma_{n_p }+1}-1)} \gamma_s \leq
\sum_{s= n_p }^{m(n_p ,t_p) } \gamma_s \leq t_p \leq T ,$ we then conclude by and A5 b).
ii\) $\tilde{\tau}_{i, \sigma_{ n_p } } > n_{p } $ and $\tilde{\tau}_{i, \sigma_{n_p }}=\tau_{\sigma_{n_p }+1}$. By the definitions of $\tau_k$ and $\sigma_k$ we derive $\tau_{\sigma_k} \leq k$, and hence $\tau_{\sigma_{n_p}} \leq n_p$. Then $[ n_p,\tau_{\sigma_{n_p }+1}) \subset [ \tau_{\sigma_{n_p}} , \tilde{\tau}_{i, \sigma_{n_p }}) $, and hence by we have $$\begin{array}{lll}
& \tilde{x}_{i,s}= x^*, \quad \tilde{\varepsilon}_{i,s+1}= -f_i(x^{*}) \quad
\forall s: n_p \leq s <\tau_{\sigma_{n_p }+1}. \nonumber
\end{array}$$ From $\tilde{\tau}_{i, \sigma_{n_p }}=\tau_{\sigma_{n_p }+1}$ by we see $ \tau_{\sigma_{n_p }+1} \leq \tau_{\sigma_{n_p }}+BD \leq n_p +BD. $ Then for sufficiently large $K$ and any $ t_p \in[0,T]$ $$\begin{array}{lll}
& \parallel \sum_{s=n_p }^{m(n_p ,t_p)\wedge ( \tau_{\sigma_{n_p}+1}-1 )} \gamma_s \tilde{\varepsilon}_{i, s+1}
I_{[ \parallel \tilde{x}_{i, s}\parallel \leq K ]} \parallel \\& \leq
\sum_{s= n_p }^{ n_p+BD}
\gamma_s \parallel f_i(x^*) \parallel \xlongrightarrow [p \rightarrow \infty]{} 0 , \nonumber
\end{array}$$ and hence holds.
iii\) $\tilde{\tau}_{i, \sigma_{ n_p } } > n_{p } $ and $\tilde{\tau}_{i, \sigma_{n_p }} < \tau_{\sigma_{n_p }+1}$. By the definition of $\tilde{\tau}_{i, \sigma_{ n_p } }$ from $\tilde{\tau}_{i, \sigma_{n_p }} < \tau_{\sigma_{n_p }+1}$ it follows that $\tilde{\tau}_{i, \sigma_{ n_p } }=\tau_{i, \sigma_{n_p }} $, and hence by $ \tau_{i, \sigma_{n_p }} \leq \tau_{\sigma_{n_p }}+BD$. By noticing $\tau_{\sigma_{n_p}} \leq n_p$ we conclude that $$\label{gap0}
\tau_{\sigma_{n_p}} \leq n_p < \tilde{\tau}_{i, \sigma_{ n_p } }= \tau_{i, \sigma_{n_p }} \leq n_p +BD .$$ So, $[ n_p,\tau_{i, \sigma_{n_p }}) \subset [ \tau_{\sigma_{n_p}} , \tilde{\tau}_{i, \sigma_{n_p }}) $. Then from here and $\tilde{\tau}_{i, \sigma_{ n_p } }= \tau_{i, \sigma_{n_p }} $ by we derive $$\label{s31}
\begin{array}{lll}
& \tilde{x}_{i,s}= x^*, \quad \tilde{\varepsilon}_{i,s+1}= -f_i(x^{*}) \quad
\forall s: n_p \leq s <\tau_{i, \sigma_{n_p }}, \\
& \tilde{x}_{i,s}= x_{i,s}, \quad \tilde{\varepsilon}_{i,s+1}= \varepsilon_{i,s+1} \quad
\forall s:\tau_{i, \sigma_{n_p }} \leq s < \tau_{\sigma_{n_p }+1}. \nonumber
\end{array}$$ Consequently, for sufficiently large $K$ and any $ t_p \in[0,T]$ $$\label{s3}
\begin{array}{lll}
& \parallel \sum_{s=n_p }^{m(n_p ,t_p)\wedge ( \tau_{\sigma_{n_p }+1}-1)}
\gamma _s \tilde{\varepsilon}_{i,s+1}
I_{[ \parallel \tilde{x}_{i,s}\parallel \leq K ]} \parallel
\\& \leq \parallel \sum_{s=n_p }^{m(n_p ,t_p)\wedge ( \tau_{\sigma_{n_p }+1}-1)} \gamma _s f_i(x^*)
I_{[ n_p \leq s <\tau_{i, \sigma_{n_p }}]} \parallel \\
& + \parallel \sum_{s= \tau_{i, \sigma_{n_p }} }^{m(n_p ,t_p)\wedge ( \tau_{\sigma_{n_p }+1}-1)}
\gamma_s \varepsilon_{i,s+1} I_{[ \parallel x_{i,s}\parallel \leq K ]} \| .
\end{array}$$
Note that the first term at the right hand of is smaller than $ \sum_{s=n_p }^{ \tau_{i, \sigma_{n_p }} } \gamma _s \| f_i(x^*)\|$, which tends to zero as $k \rightarrow \infty$ by the last inequality in and $ \lim\limits_{k \rightarrow \infty} \gamma_{k} = 0.$ The truncation number for agent $i$ at time $\tau_{i, \sigma_{n_p }}$ is $\sigma_{n_p }$, while it is smaller than $\sigma_{n_p }$ at time $\tau_{i, \sigma_{n_p }}-1$ since $\tau_{i, \sigma_{n_p }}$ is the smallest time when the truncation number of $i$ has reached $\sigma_{n_p }$. Consequently, by Remark \[r1\] we have $x_{i, \tau_{i, \sigma_{n_p }}}=x^*$, and hence $\{x_{i, \tau_{i, \sigma_{n_p }}} \}_{p \geq 1}$ is a convergent subsequence. Noticing $$\sum_{s= \tau_{i, \sigma_{n_p }} }^{m(n_p ,t_p)\wedge ( \tau_{\sigma_{n_p }+1}-1)}
\gamma_s \leq \sum_{s= n_p }^{m(n_p ,t_p) } \gamma_s \leq t_p ,$$ we conclude by and A5 b).
Since one of i), ii), iii) must take place for the case $ \lim\limits_{k \rightarrow \infty} \sigma_{k} = \infty $, we thus have proved in Case 2.
Combining Case 1 and Case 2 we derive . $\blacksquare$
Proof of Lemma \[lemma2\] {#PL2}
==========================
Let us consider a fixed $\omega $ where A5 b) holds.
Let $C > \| \bar{X}\| $. There exists an integer $k_C >0 $ such that $$\label{subsequence}
\| \widetilde{X}_{n_k} \| \leq C \quad \forall k \geq k_C.$$
By Lemma \[lemma1\] we know that there exist a constant $T_1>0$ and a positive integer $k_0 \geq k_C$ such that $$\label{lem23}
\begin{split}
& \| \sum_{s=n_k}^{m(n_k,t_k)\wedge ( \tau_{\sigma_{n_k}+1}-1)} \gamma_s \tilde{\varepsilon}_{ s+1}
I_{[\parallel \widetilde{X}_{ s}\parallel \leq K ]} \| \leq T_0
\\& \forall t_k \in[0,T_0] \quad \forall T_0 \in[0, T_1 ]~~\forall k \geq k_0
\end{split}$$ for sufficiently large $K >0$. Define $$\begin{aligned}
& M_0'\triangleq 1+C( c\rho +2), \label{def21} \\
&H_1\triangleq \max_{X} \{ \parallel F(X) \parallel : \parallel X \parallel \leq M_0'+1
+C \}, \label{def22} \\
&c_1\triangleq H_1+3 + \frac{c ( \rho+1) }{1-\rho},~~\mbox{and} \quad c_2\triangleq \frac{H_1+1}{\sqrt{N}}, \label{def23}\end{aligned}$$ where $c$ and $\rho$ are given by . Select $T >0$ such that $$\label{def24}
0< T \leq T_1\textrm{ and }c_1 T <1.$$
For any $ k \geq k_0$ and any $ T_k \in [0,T]$ define $$\label{lem24}
\begin{split}
& s_k\triangleq \sup \{ s \geq n_k: \parallel \widetilde{X}_j- \widetilde{X}_{n_k} \parallel\\&
\leq c_1T_k +M_0' \quad \forall j: n_k \leq j \leq s \}.
\end{split}$$ Then from and it follows that $$\label{lem22}
\begin{split}
\parallel \widetilde{X}_s \parallel
\leq M_0'+1+C\quad \forall s: n_k \leq s \leq s_k .
\end{split}$$
We intend to prove $ s_k > m(n_k, T_k).$ Assume the converse that for sufficiently large $ k \geq k_0$ and any $ T_k \in [0,T]$ $$\label{inverse2}
s_k \leq m(n_k, T_k).$$
We first show that there exists a positive integer $k_1 > k_0$ such that for sufficiently large $ k \geq k_1$ $$\label{notruncation}
s_k < \tau_{\sigma_{n_k}+1}~~ \forall k \geq k_1 ~~ \forall T_k \in [0,T].$$ We prove for the two alternative cases: $ \lim\limits_{k \rightarrow \infty } \sigma_k = \infty$ and $ \lim\limits_{k \rightarrow \infty } \sigma_k =\sigma <\infty $.
i\) $ \lim\limits_{k \rightarrow \infty } \sigma_k = \infty$. Since $\{M_k\}$ is a sequence of positive numbers increasingly diverging to infinity, there exists a positive integer $k_1 > k_0$ such that $M_{ \sigma_{n_k} } >M_0'+1+C$ for all $k \geq k_1$. Hence, from we know $ s_k < \tau_{\sigma_{n_k}+1} $.
ii\) $ \lim\limits_{k \rightarrow \infty } \sigma_k =\sigma <\infty $. For this case there exists a positive integer $k_1 > k_0$ such that $ \sigma_{n_k} =\sigma$ for all $k \geq k_1 $, and hence $\tau_{\sigma_{n_k}+1}=\tau_{\sigma+1}=\infty$ by . Then $m(n_k,T_k) < \tau_{\sigma_{n_k}+1}$, and hence by we derive .
By we see $ T_k \in [0,T] \subset [0, T_1]$, then from it follows that for sufficiently large $ k \geq k_1$ and any $ T_k \in [0,T]$ $$\label{lem208}
\begin{split}
&\| \sum_{s=n_k}^{m(n_k,t_k )\wedge ( \tau_{\sigma_{n_k}+1}-1)} \gamma_s \tilde{\varepsilon}_{ s+1}
I_{[\parallel \tilde{X}_{ s}\parallel \leq K ]} \| \leq T_k
~~ \forall t_k \in[0,T_k]
\end{split}$$ for sufficiently large $K >0$. By setting $t_k =\sum_{m=n_k}^s \gamma_m $ for some $s \in [ n_k,s_k],$ from we see $\sum_{m=n_k}^s \gamma_m \leq \sum_{m=n_k}^{s_k} \gamma_m \leq T_k $. Noticing $m(n_k,t_k) =s$, from we derive $m(n_k,t_k )\wedge ( \tau_{\sigma_{n_k}+1}-1) =s.$ Then by setting $ K \triangleq M_0'+1+C$, from it follows that $$\label{err}
\begin{split}
& \parallel \sum_{m=n_k}^{s} \gamma_m \tilde{\varepsilon}_{ m+1} \parallel \leq T_k \quad \forall s: n_k \leq s \leq s_k
\end{split}$$ for sufficiently large $ k \geq k_1$ and any $T_k \in [0,T]$.
Let us consider the following algorithm starting from $n_k$ without truncation $$\label{lem26}
Z_{m+1}= (W(m) \otimes \mathbf{I}_l) Z_m + \gamma_{m } (F(Z_m)+\tilde{\varepsilon}_{m+1}),
~ Z_{n_k}= \tilde{X}_{n_k}.$$ By we know that holds for $m=n_k, \cdots, s_k-1$ for $ \forall k \geq k_1 ~ \forall T_k \in [0,T]$. Then from here we derive $$\label{aux21}
Z_m = \widetilde{X}_m ~~ \forall m: n_k \leq m \leq s_k .$$ Hence by we know that for $ \forall k \geq k_1 ~ \forall T_k \in [0,T]$ $$\label{upper21}
\parallel F(Z_m) \parallel \leq H_1 ~~ \forall m: n_k \leq m \leq s_k .$$
Set $z_k = \frac{\mathbf{1}^T \otimes \mathbf{I}_l}{N} Z_k$. By multiplying both sides of from left with $ \frac{1}{N} ( \mathbf{1}^T \otimes \mathbf{I}_l) $, from $ \mathbf{1}^TW(m) =\mathbf{1}^T$ and we derive $$z_{s+1}=z_s + \frac{\mathbf{1}^T \otimes \mathbf{I}_l}{N } \gamma_s (F(Z_s)+\tilde{\varepsilon}_{s+1}),$$ and hence $$\label{lem27}
\begin{array}{lll}
& \parallel z_{s+1} - z_{n_k} \parallel = \parallel \frac{\mathbf{1}^T \otimes \mathbf{I}_l}{N } \sum_{m=n_k}^s \gamma_{m } (F(Z_m)+\tilde{\varepsilon}_{m+1}) \parallel \\
&\leq \frac{1}{ \sqrt{N}} \big( \sum_{m=n_k}^s \gamma_{m }\parallel F(Z_m) \parallel +
\parallel \sum_{m=n_k}^s \gamma_{m } \tilde{\varepsilon}_{m+1} \parallel \big). \nonumber
\end{array}$$ Then from here by we conclude that for sufficiently large $ k \geq k_1$ and any $ T_k \in [0,T]$ $$\label{lem28}
\begin{split}
\| z_{s+1} - z_{n_k} \| \leq \frac{H_1+ 1}{ \sqrt{N} } \sum_{m=n_k}^s \gamma_{m } =c_2 T_k
~~ \forall s: n_k \leq s \leq s_k.
\end{split}$$
Denote by $Z_{\bot, s}\triangleq D_{\bot} Z_{s}$ the disagreement vector of $Z_{s}$. By multiplying both sides of from left with $ D_{\bot}$ we derive $$\label{dis1}
\begin{split}
& Z_{\bot, s+1}= D_{\bot} (W(s) \otimes I_l)Z_{ s }+ \gamma_{s } D_{\bot} (F(Z_s)+\tilde{\varepsilon}_{s+1}),
\end{split}$$ and inductively $$\label{dis21}
\begin{array}{lll}
& Z_{\bot, s+1}= \Psi(s, n_k) Z_{n_k}+ \\
& \sum_{m=n_k} ^s
\gamma_{m } \Psi(s-1, m) D_{\bot} (F(Z_m)+\tilde{\varepsilon}_{m+1}) \quad \forall s \geq n_k. \nonumber
\end{array}$$ From here by we derive $$\label{dis22}
\begin{array}{lll}
& Z_{\bot, s+1} = [(\Phi(s, n_k) - \frac{1}{N} \mathbf{1} \mathbf{1}^T) \otimes I_l] Z_{n_k} \\&+ \sum_{m=n_k} ^s
\gamma_{m } [(\Phi(s-1, m) - \frac{1}{N} \mathbf{1} \mathbf{1}^T) \otimes I_l] F(Z_m) \\
& +\sum_{m=n_k} ^s \gamma_{m } [( \Phi(s-1, m) - \frac{1}{N} \mathbf{1} \mathbf{1}^T) \otimes I_l] \tilde{\varepsilon}_{m+1}.
\end{array}$$ From we have $\| Z_{n_k} \| \leq C$, and hence from it follows that $$\label{dis3}
\begin{array}{lll}
& \parallel Z_{\bot, s+1} \parallel \leq C c\rho^{s +1-n_k} + \sum_{m=n_k} ^s
\gamma_{m } H_1 c\rho^{s -m} + \\
& \parallel \sum_{m=n_k} ^s \gamma_{m } [( \Phi(s-1, m) - \frac{1}{N} \mathbf{1} \mathbf{1}^T) \otimes \mathbf{I}_l] \tilde{\varepsilon}_{m+1} \parallel .
\end{array}$$
By we derive $\parallel \Gamma_s- \Gamma_{n_k-1} \parallel \leq T_k \quad \forall s: n_k \leq s \leq s_k$, where $\Gamma_n \triangleq \sum_{m=1} ^n \gamma_m \tilde{\varepsilon}_{m+1}.$ Notice $$\label{errr3D}
\begin{array}{lll}
& \sum_{m=n_k} ^s
\gamma_{m } (\Phi(s-1, m) \otimes I_l) \tilde{\varepsilon}_{m+1} \\ &= \sum_{m=n_k} ^s (\Phi(s-1, m) \otimes I_l) (\Gamma_{m}-\Gamma_{m-1})
\\ &= \sum_{m=n_k} ^s (\Phi(s-1, m) \otimes I_l) (\Gamma_{m}-\Gamma_{n_k-1})
\\& -\sum_{m=n_k} ^s (\Phi(s-1, m) \otimes I_l) (\Gamma_{m-1}-\Gamma_{n_k-1}) . \nonumber
\end{array}$$ Summing by parts, by we have $$\label{errr3D}
\begin{array}{lll}
& \parallel \sum\limits_{m=n_k} ^s
\gamma_{m } (\Phi(s-1, m) \otimes I_l) \tilde{\varepsilon}_{m+1} \parallel \leq \parallel \Gamma_s- \Gamma_{n_k-1}\parallel + \\& \sum\limits_{m=n_k} ^{s-1} \parallel \Phi(s-1, m) - \Phi(s-1, m+1) \parallel
\parallel \Gamma_{m} - \Gamma_{n_k-1} \parallel \\
& \leq T_k + \sum\limits_{m=n_k} ^{s-1} ( c \rho^{s-m-1}+c \rho^{s-m}) T_k
\\& \leq T_k + \frac{c ( \rho+1) }{1-\rho} T_k ~~ \forall s: n_k \leq s \leq s_k, \nonumber
\end{array}$$ which incorporating with yields $$\label{D11f}
\begin{array}{lll}
& \| \sum_{m=n_k} ^s \gamma_{m } [( \Phi(s-1, m) - \frac{1}{N} \mathbf{1} \mathbf{1}^T) \otimes I_l] \tilde{\varepsilon}_{m+1} \|
\\& \leq (2+\frac{c ( \rho+1) }{1-\rho})T_k \quad \forall s: n_k \leq s \leq s_k
\end{array}$$ for sufficiently large $ k \geq k_1$ and any $ T_k \in [0,T]$.
By noticing $\sum_{m=n_k} ^s \gamma_{m } \rho^{s -m} \leq \frac{ 1 }{1-\rho} \sup_{ m \geq n_k } \gamma_{m }
,$ from $ \lim\limits_{k \rightarrow \infty} \gamma_{k} = 0 $ and it is concluded that $$\label{est4}
\begin{array}{lll}
& \parallel Z_{\bot, s+1} \parallel \leq Cc\rho + \frac{ c H_1 }{1-\rho} \sup_{ m \geq n_k } \gamma_{m }
+ (2 + \frac{c ( \rho+1) }{1-\rho} ) T_k
\\& \leq Cc\rho + 1
+ (2 + \frac{c ( \rho+1) }{1-\rho} ) T_k ~~ \forall s: n_k \leq s \leq s_k
\end{array}$$ for sufficiently large $ k \geq k_1$ and any $ T_k \in [0,T]$. By noticing $Z_s=Z_{\bot,s}+( \mathbf{1} \otimes I_l ) z_s ,$ we derive $$\label{D12}
\begin{array}{lll}
& \parallel Z_{s+1} - Z_{n_k} \parallel \\&=
\parallel ( \mathbf{1} \otimes \mathbf{I}_l)z _{s+1}+ Z_{\bot, s+1} - Z_{\bot, n_k} -
( \mathbf{1} \otimes \mathbf{I}_l ) z_{n_k} \parallel \\
& \leq \parallel Z_{\bot, s+1} \parallel + \parallel Z_{\bot, n_k} \parallel + \sqrt{N}\parallel
z_{s+1}- z_{n_k} \parallel.
\end{array}$$ Since $\parallel Z_{\bot, n_k} \parallel \leq 2\parallel Z_{ n_k} \parallel=2C $, from it follows that for sufficiently large $k \geq k_1$ and any $ T_k \in [0,T]$ $$\label{result1}
\begin{array}{lll}
& \parallel Z_{s+1} - Z_{n_k} \parallel \leq C c\rho + 1
+ (2+ \frac{c ( \rho+1) }{1-\rho} ) T_k +2C + \\& (H_1+ 1) T_k
\leq C( c\rho + 2 )+1 + (3+H_1+ \frac{c ( \rho+1) }{1-\rho} )
T_k \\& =M_0'+c_1 T_k \quad \forall s: n_k \leq s \leq s_k.
\end{array}$$ Therefore, from and $\| Z_{n_k} \| \leq C$ we know that for sufficiently large $k \geq k_1$ and any $ T_k \in [0,T]$ $$\label{D13}
\parallel Z_{s_k+1} \parallel \leq \parallel Z_{n_k} \parallel + M_0'+c_1 T_k \leq
M_0'+1+C. \nonumber$$ Rewrite in the compact form as follows $$\widehat{X}_{s_k+1}= [W(s_k) \otimes \mathbf{I}_l]
\widetilde{X}_{s_k} + \gamma_{s_k } (F( \widetilde{X}_{s_k})+\tilde{\varepsilon}_{s_k+1}) ,$$ where $ \widehat{X}_k \deq col \{ \hat{x}_{1,k}, \cdots, \hat{x}_{N,k} \}.$ Then by $\widehat{X}_{s_k+1}=Z_{s_k+1}$, and hence from it follows that $$\label{bound}
\parallel \widehat{X}_{s_k+1} \parallel \leq M_0'+1+C .$$
We now show $$\label{appentr}
\widetilde{X}_{s_k+1}= \widehat{X}_{s_k+1}\textrm{ and } s_k+1 < \tau_{\sigma_{n_k}+1}$$ for sufficiently large $k \geq k_1$ and any $ T_k \in [0,T]$. We consider the following two cases: $ \lim\limits_{k \rightarrow \infty } \sigma_k = \infty$ and $ \lim\limits_{k \rightarrow \infty } \sigma_k =\sigma <\infty $.
i\) $ \lim\limits_{k \rightarrow \infty } \sigma_k = \infty$. By noting $M_{ \sigma_{n_k} } >M_0'+1+C~~ \forall k \geq k_1$, from by we see $\widetilde{X}_{s_k+1}=\widehat{X}_{s_k+1}$ and $\sigma_{ s_k+1}=\sigma_{ s_k }.$ Hence $s_k+1 < \tau_{\sigma_{n_k}+1}$ by .
ii\) $ \lim\limits_{k \rightarrow \infty } \sigma_k =\sigma< \infty$. Since $ ~\tau_{\sigma_{n_k}+1}= \infty ~\forall k \geq k_1$, by we see $ s_k+1 < \tau_{\sigma_{n_k}+1} $ . Then by $\sigma_{n_k}=\sigma ~\forall k \geq k_1$ we conclude $\sigma_{s_k+1}=\sigma_{s_k}=\sigma $. Hence by we derive $\widetilde{X}_{s_k+1}= \widehat{X}_{s_k+1}$. Thus holds.
From we know that holds for $m=s_k$ for sufficiently large $k \geq k_1$ and any $ T_k \in [0,T]$. From $\widehat{X}_{s_k+1}=Z_{s_k+1}$ by we see $\widetilde{X}_{s_k+1}=Z_{s_k+1}$. Hence from and $Z_{n_k}= \widetilde{X}_{n_k}$ it follows that for sufficiently large $k \geq k_1$ and any $ T_k \in [0,T]$ $$\parallel \widetilde{X}_{s_k+1}
- \widetilde{X}_{n_k} \parallel \leq M_0'+c_1 T_k,$$ which contradicts with the definition of $s_k$ given by . Thus does not hold. So, $s_k > m(n_k,T_k)$ and hence by the definition of $s_k$ given in we derive .
Since $\{ \widetilde{X}_s: n_k \leq s \leq m(n_k,T_k)\}$ are bounded, similar to proving it can be shown that $m(n_k,T_k) +1 < \tau_{\sigma_{n_k}+1} $. Thus holds for $m=n_k,\cdots, m(n_k,T_k)$. Similar to we obtain $$\begin{split}
& \parallel \bar{x}_{m+1} - \bar{x}_{n_k} \parallel \leq c_2T_k
\quad \forall m: n_k \leq m \leq m(n_k,T_k) \nonumber
\end{split}$$ for sufficiently large $k $ and any $ T_k \in [0,T]$. Hence, holds.
The proof of Lemma \[lemma2\] is completed. $\blacksquare$
Proof of Lemma \[lemnoise\] {#PLN}
===========================
Since $ \lim\limits_{k \rightarrow \infty}\widetilde{X}_{n_k} = \bar{X}$, by setting $\bar{x}\triangleq \frac{\mathbf{1}^T \otimes \mathbf{I}_l}{N} \bar{X}$ we then have $\lim\limits_{k \rightarrow \infty} \bar{x}_{n_k} = \bar{x}$. Lemma \[lemma2\] ensures that there exists a $T\in (0,1) $ such that $m(n_k,T ) < \tau_{\sigma_{n_k}+1}$ and $\{\widetilde{X}_s: n_k \leq s \leq m(n_k,T ) +1\}$ are bounded for sufficiently large $k$. So, for any $ T_k \in[0,T]$ and any sufficiently large $K$ $$\parallel \sum_{s=n_k}^{m(n_k,T_k)\wedge ( \tau_{\sigma_{n_k}+1}-1 )} \gamma_s \tilde{\varepsilon}_{s+1}
I_{[ \parallel \tilde{X}_ s\parallel \leq K ]} \parallel \\ = \parallel \sum_{s=n_k}^{m(n_k,T_k) } \gamma_s \tilde{\varepsilon}_{ s+1} \parallel .$$ Therefore, by Lemma \[lemma1\] we derive $$\label{error20}
\lim_{T \rightarrow 0} \limsup_{k \rightarrow \infty} \frac{1}{T}
\parallel \sum_{s=n_k}^{m(n_k,T_k) } \gamma_s \tilde{\varepsilon}_{ s+1} \parallel =0 \quad \forall T_k \in[0,T]. \nonumber$$ So, for it suffices to show $$\label{error23}
\begin{split}
& \lim_{T \rightarrow 0} \limsup_{k \rightarrow \infty} \frac{1}{T}
\parallel \sum_{s=n_k}^{m(n_k,T_k) } \gamma_s e_{s+1} \parallel =0 \quad \forall T_k \in[ 0,T ].
\end{split}$$
Similar to , we can show that there exist positive constants $c_3 , c_4 ,c_5 $ such that for sufficiently large $k$ $$\label{lem31}
\begin{array}{lll}
\parallel \tilde{X}_{\bot, s+1} \parallel & \leq c_3 \rho^{s+1-n_k} +
c_4 \sup_{ m\geq n_k } \gamma_{m } \\& + c_5 T \quad \forall s: n_k \leq s \leq m(n_k,T ).
\end{array}$$ Since $ 0 < \rho<1$, there exists a positive integer $m' $ such that $\rho^{m'}<T.$ Then $\sum_{m=n_k}^{n_k+m'} \gamma_m \xlongrightarrow [k \rightarrow \infty]{} 0 $ by $\lim\limits_{ k \rightarrow \infty} \gamma_k =0 $. Thus, $ n_k+m' < m(n_k,T)$ for sufficiently large $k $. Therefore, from we know that for sufficiently large $k $ $$\label{bd1}
\parallel \tilde{X}_{\bot, s+1} \parallel \leq o(1) + (c_3+c_5) T
\quad \forall s: n_k+m' \leq s \leq m(n_k,T ),$$ where $o(1) \rightarrow 0$ as $k \rightarrow \infty$.
Since $ \bar{x}_{n_k} \xlongrightarrow [k \rightarrow \infty]{} \bar{x}$, from it follows that for sufficiently large $k $ and $\forall s: n_k+m' \leq s \leq m(n_k,T )$ $$\label{bound12}
\begin{split}
& \parallel \bar{x}_s -\bar{x} \parallel \leq \parallel \bar{x}_s - \bar{x}_{n_k} \parallel+ \parallel \bar{x}_{n_k} -\bar{x} \parallel=o(1) + \delta(T), \\
& \parallel \tilde{x}_{i,s} -\bar{x} \parallel \leq \parallel \tilde{x}_{i,s} - \bar{x}_s \parallel+
\parallel \bar{x}_s - \bar{x} \parallel =o(1) +\delta(T) , \nonumber
\end{split}$$ where $ \delta(T) \rightarrow 0$ as $T \rightarrow 0$. By continuity of $f_i(\cdot)$ we derive $ \parallel f_i(\tilde{x}_{i,s}) -f_i(\bar{x}_s) \parallel \leq \parallel f_i(\tilde{x}_{i,s}) -f_i(\bar{x} ) \parallel +\parallel f_i(\bar{x}_s)-f_i(\bar{x}) \parallel =o(1) + \delta(T).$ Consequently, $$\parallel e_{i,s+1} \parallel = \parallel f_i(\tilde{x}_{i,s}) -f_i(\bar{x}_s) \parallel /N =o(1) + \delta(T).$$ Then for sufficiently large $k $ $$\label{D00}
\parallel e_{s+1} \parallel =o(1) + \delta(T)
\quad \forall s: n_k+m' \leq s \leq m(n_k,T ).$$
By the boundedness of $\{\widetilde{X}_s: n_k \leq s \leq m(n_k,T )\}$ and by continuity of $f_i(\cdot)$ we know that there exists a constant $c_e >0$ such that $\parallel e_{s +1} \parallel \leq c_e $. Then from we derive $$\begin{split}
& \parallel \sum_{s=n_k}^{m(n_k,T_k) } \gamma_s e_{s+1} \parallel \\
& \leq \sum_{s=n_k}^{ n_k+m' }\gamma_s c_e +
( o(1) + \delta(T) ) \sum_{s= n_k+m'+1}^{m(n_k,T_k) } \gamma_s
\\& \leq c_e m' \sup_{s \geq n_k} \gamma_s + T( \delta(T) +o(1) ~~\forall T_k \in [0,T] \nonumber
\end{split}$$ for sufficiently large $k $. From here by $\lim\limits_{ k \rightarrow \infty} \gamma_k =0 $ we derive $$\label{error22}
\begin{split}
& \limsup_{k \rightarrow \infty } \frac{1}{T} \parallel \sum_{s=n_k}^{m(n_k,T_k) } \gamma_s e_{s+1} \parallel
= \delta(T) \quad \forall T_k \in[ 0,T ]. \nonumber
\end{split}$$ Letting $T\rightarrow 0$, we derive , and hence holds.
The proof of Lemma \[lemnoise\] is completed. $\blacksquare$
Proof of Lemma \[lemma4\] {#PL4}
==========================
i\) Assume $$\label{inverse}
\lim\limits_{k \rightarrow \infty } \sigma_k = \infty.$$ Then there exists a positive integer $ n_k $ such that $ \sigma_{n_k}=k $ and $\sigma_{n_k-1}=k-1$ for any $ k\geq 1$. Since $n_k-1 \in [\tau_{k-1},\tau_k)$ from $ \sigma_{n_k}=k $, by we see $\tilde{x}_{i,n_k} =x^*~~\forall i \in \mathcal{V}$. Consequently, $ \widetilde{X}_{n_k}= (\mathbf{1} \otimes I_l) x^{*}$ and hence $\{ \widetilde{X}_{n_k}\}$ is a convergent subsequence with $\bar{x}_{n_k}= x^{*}$.
Since $\{M_k \}$ is a sequence of positive numbers increasingly diverging to infinity, there exists a positive integer $k_0 $ such that $$\label{no2}
M_{ k } \geq 2\sqrt{N}c_0+ 2+M_1' \quad \forall k \geq k_0,$$ where $c_0$ is given in A2 c) and $$\label{F2}
M_1'=2+ ( 2\sqrt{N}c_0 +2)( c \rho +2).$$
In what follows, we show that under the converse assumption $\{\bar{x}_{n_k}\}$ starting from $ x^{*}$ crosses the sphere with $ \| x \|=c_0$ infinitely many times. Define $$\begin{aligned}
& m_k \triangleq \inf\{ s > n_k: \parallel \widetilde{X}_s \parallel \geq 2\sqrt{N}c_0+2+M_1' \}, \label{notion11} \\
& l_k \triangleq \sup\{ s < m_k: \parallel \widetilde{X}_s \parallel \leq 2\sqrt{N}c_0 +2 \}. \label{notion12}\end{aligned}$$
Noticing $ \| \widetilde{X}_{n_k} \| =\sqrt{N} \| x^{*}\|$ and $\| x^{*}\|< c_0$, we derive $ \| \widetilde{X}_{n_k} \| < \sqrt{N} c_0.$ Hence from it is seen that $n_k < l_k < m_k .$ By the definition of $l_k$ we know that $\{ \widetilde{X}_{l_k} \}$ is bounded, then there exists a convergent subsequence, denoted still by $\{ \widetilde{X}_{l_k}\}.$ By denoting $\bar{X} $ the limiting point of $\widetilde{X}_{l_k} $, from $\| \widetilde{X}_{l_k} \| \leq 2\sqrt{N}c_0 +2 $ it follows that $ \| \bar{X} \| \leq 2\sqrt{N}c_0+2$.
By Lemma \[lemma2\] there exist constants $M_0'>0$ defined by with $C= 2\sqrt{N}c_0+2 $, $c_1>0 $ and $c_2>0$ defined by , $ 0< T<1$ with $c_1 T \leq 1$ such that $$\parallel \widetilde{X}_{m+1} -\widetilde{X}_{l_k} \parallel \leq c_1 T +M_0'
\quad \forall m: l_k \leq m \leq m(l_k,T) \nonumber$$ for sufficiently large $k \geq k_0$. Then for sufficiently large $k \geq k_0$ $$\label{res71}
\begin{split}
& \parallel \widetilde{X}_{ m+1} \parallel \leq \parallel \widetilde{X}_{l_k} \parallel + c_1T +M_0' \\
& \leq 2\sqrt{N}c_0+ 2+1+1+( 2\sqrt{N}c_0+2) (c\rho+2)
\\& =2\sqrt{N}c_0+2+M_1' \quad \forall m: l_k \leq m \leq m(l_k,T).
\end{split}$$ Then $m(l_k,T) \leq n_{k+1}$ for sufficiently large $k \geq k_0 $ by .
From by the definition of $m_k$ defined in , we conclude $ m(l_k,T)+1 \leq m_k$ for sufficiently large $k \geq k_0.$ Then by we know that for sufficiently large $k \geq k_0$ $$\label{llm3}
\begin{split}
& 2\sqrt{N}c_0 +2 < \parallel \widetilde{X}_{m+1} \parallel \leq 2\sqrt{N}c_0+2+M_1'
\\&~~~~~~~~~~~~~~~~~~~~~~\quad \forall m: l_k \leq m \leq m(l_k,T) .
\end{split}$$
Since $0< \rho<1$, there exists a positive integer $m_0$ such that $4c\rho^{m_0} <1$. Then $\sum_{m=l_k}^{l_k+m_0} \gamma_m \xlongrightarrow [k \rightarrow \infty]{} 0 $ by $ \gamma_k\xlongrightarrow [k \rightarrow \infty]{} 0$, and hence $ l_k+m_0 < m(l_k,T) < n_{k+1} $ for sufficiently large $k \geq k_0$. So, from it is seen that for sufficiently large $k \geq k_0$ $$\label{F4}
\parallel \widetilde{X}_{ l_k+m_0} \parallel > 2\sqrt{N}c_0+2.$$
Noticing that $ \{\widetilde{X}_{m+1} : l_k \leq m \leq m(l_k,T)\}$ are bounded, similarly to we know that for sufficiently large $k \geq k_0$ $$\label{llem2}
\begin{split}
& \parallel \widetilde{X}_{\bot, m+1} \parallel \leq ( 2\sqrt{N}c_0+2)c\rho^{m+1-l_k} +
\frac{ c H_1 }{1-\rho} \sup_{ m \geq l_k } \gamma_{m } \\
& + 2 T + \frac{c ( \rho+1) }{1-\rho} T
\quad \forall m: l_k \leq m \leq m(l_k,T). \nonumber
\end{split}$$ From here, by $c_1T<1 $ and $\gamma_k \xlongrightarrow [k \rightarrow \infty]{} 0$ it follows that $$\label{llem3}
\begin{split}
& \parallel \widetilde{X}_{\bot, l_k+m_0} \parallel \leq 2 c\rho^{m_0}( \sqrt{N}c_0 +1) + \frac{1}{2}+ c_1
T \\ & \leq \frac{1}{2}( \sqrt{N}c_0 +1) + \frac{1}{2}+1 =\frac{\sqrt{N}c_0}{2}+2
\end{split}$$ for sufficiently large $k \geq k_0$. By noticing $ (\mathbf{1} \otimes \mathbf{I}_l )\bar{x}_{l_k+m_0}= \widetilde{X}_{ l_k+m_0}- \widetilde{X}_{\bot,l_k+m_0} ,$ from we conclude that $$\begin{split}
& \sqrt{N} \parallel \bar{x}_{l_k+m_0} \parallel= \parallel \widetilde{X}_{ l_k+m_0}- \widetilde{X}_{\bot, l_k+m_0} \parallel \\&
\geq \parallel \widetilde{X}_{ l_k+m_0} \parallel - \parallel\widetilde{X}_{\bot, l_k+m_0} \parallel > \frac{3}{2}\sqrt{N}c_0.
\end{split}$$ Therefore, $\parallel \bar{x}_{l_k+m_0} \parallel >c_0.$
Thus, we have shown that for sufficiently large $k \geq k_0$, starting from $ x^{*},$ $\{\bar{x}_{n_k}\}$ crosses the sphere with $ \| x \| =c_0$ before the time $n_{k+1}$. So, $\{\bar{x}_{n_k}\}$ starting from $ x^{*}$ crosses the sphere with $\| x \| =c_0$ infinitely many times.
ii\) Since $v(J)$ is nowhere dense, there exists a nonempty interval $[\delta_1, \delta_2 ]
\in (v(x^*),\textrm{inf }_{\parallel x \parallel =c_0} v(x))$ with $d([\delta_1, \delta_2 ], v(J))>0 $. Assume the converse $\lim\limits_{k \rightarrow \infty } \sigma_k = \infty.$ By i) $\{v(\bar{x}_k)\}$ crosses the interval $[\delta_1, \delta_2 ]$ infinitely many times while $ \{\widetilde{X}_{n_k}\}$ converges. This contradicts Lemma \[lemma3\]. Therefore, is impossible and holds. The proof is completed. $\blacksquare$
[1]{}
W. Ren, and R. W. Beard, “Consensus seeking in multiagent systems under dynamically changong interaction topologies," *IEEE Trans. Autom. Control*, vol. 50, no. 5, pp. 655–661, 2005.
R. Olfati-Saber, and R. M. Murray, “Consuensus problems in networks of agnets with switching topology ans time-delays," *IEEE Trans. Autom. Control*, vol.49, no. 9, pp. 1520–1533, 2004.
M. Y. Huang, “Stochastic approximation for consensus: a new approach via ergodic backward products," *IEEE Trans. Autom. Control*, vol. 57, no. 12, pp. 2994–3008, 2012.
Q. Zhang and J. F. Zhang, “Distributed parameter estimation over unreliable networks with Markovian switching topologies," *IEEE Trans. Autom. Control*, vol. 57, no. 10, pp. 2545–2560, 2012.
C. G. Lopes, and A. H. Sayed, “Diffusion least-mean square over adaptive networks: formulation and performance analysis," *IEEE Trans. Singnal Processing*, vol. 56, no. 7, pp. 3122–3136, 2008.
U. A. Khan, S. Kar, and José M. F. Moura, “Distributed Sensor Localization in Random Environments Using Minimal Number of Anchor Nodes," *IEEE Trans. Singnal Processing*, vol. 57, no. 5, pp. 2000–2016, 2009.
A. Nedić, and A. Ozdaglar, “Distributed subgradient methods for multi-agent optimization," *IEEE Trans. Autom. Control*, vol. 54, no. 1, pp. 48–61, 2009.
A. Nedić, A. Ozdaglar, and P. A. Parrilo, “Constrained consensus and optimization in multi-agent networks," *IEEE Trans. Autom. Control*, vol. 55, no. 4, pp. 922–938, 2010.
I. Lobel, and A. Ozdaglar, “Distributed Subgradient Methods for Convex Optimization Over Random Networks," *IEEE Trans. Autom. Control*, vol. 56, no. 6, pp. 1291–1306, 2011.
K. Y. You, Z. K. Li, and L. H. Xie, “Consensus Condition for Linear Multi-agent Systems over Randomly Switching Topologies ," *Automatica*, vol. 49, no. 10, pp. 3125–3132, 2013.
R. Olfati-Saber, J. A. Fax, and R. M. Murray, “Consensus and cooperation in networked nulti-agent systems," *Proc. IEEE*, vol. 95, no. 1, pp. 215–233, 2007.
H. J. Kushner, and G. Yin, “Asymptotic properties of distributed and communicating stochastic approximation algorithms,” *SIAM Journal on Control and Optimization*, vol. 25, no. 5, pp. 1266-1290, 1987.
P. Bianchi, G. Fort, and W. Hachem, “Performance of a distributed stochastic approximation algorithm," *IEEE Trans. Inf. Theory*, vol. 59, no. 11, pp. 7405–7418, 2013.
G. Morral, P. Bianchi, G. Fort, J. Jakubowicz, “Distributed stochastic approximation: The price of non-double stochasticity," *Proc. Asilomoar* 2012.
P. Bianchi, J. Jakubowitz, “Convergence of a multi-agent projected stochastic gradient algorithm for non-convex optimization," *IEEE Trans. Automa. Control,* vol. 58, no. 2, pp. 391–405, 2013.
L. M. Rios, and N. V. Sahinidis, “Derivative-free optimization: A review of algorithms and comparison of software implementations," *Journal of Global Optimization,* vol. 56, no.3, pp. 1247-1293, 2013.
D. Yuan and D. W. C. Ho, “Randomized gradient-free method for multiagent optimization over time-varying networks," *IEEE Trans. Neural Netw. Learn. Syst.,* vol. 26, no. 6, pp. 1342–1347, 2015.
D. Yuan, D. W. C. Ho, and S. Xu. “Zeroth-Order Method for Distributed Optimization With Approximate Projections," *IEEE Trans. Neural Netw. Learn. Syst.,* vol. 27, no. 2, pp. 284–294, 2016.
F. Iutzeler, P. Ciblat, J. Jakubowicz, “Analysis of max-consensus algorithms in wireless channels," *IEEE Transactions on Signal Processing,* vol. 60, no. 11, pp. 6103–6107, 2012.
H. Robbins, and S. Monro, “A stochastic approximation method,” *Annu. Math. Statist.*, vol. 22, no. 3, pp. 400–405, 1951.
J. Kiefer, and J. Wolfowitz, “Stochastic estimation of the maximum of a regression function," *Ann. Math. Statist.*, vol. 23, no.3, pp. 462–466, 1952.
H. J. Kushner, and G. Yin, *Stochastic Approximation Algorithms and Applications.* Springer-Verlag, New York, 1997.
A. Benveniste, M. Metivier, and P. Priouret, *Adaptive Algorithms and Stochastic Approximation*, Springer-Verlag, New York, 1990,
V. S. Bokar, *Stochastic approximation: a dynamical systems viewpoint.* Tata Institute of Fundamental Research, Mumbai, 2008.
H. F. Chen and Y. M. Zhu, “Stochastic approximation procedures with randomly varying truncations,” *Scientia Sinica (Series A),* vol. 29, no. 9, pp. 914–926, 1986.
H. F. Chen, “Stochastic approximation and its new applications,,” *Proceedings of 1994 Hong Kong International Workshop on New Directions of Control and Manufacturing,* pp. 2-12, 1994.
H. F. Chen, *Stochastic Approximation and Its Applications*. Dordrecht, The Netherlands: Kluwer, 2002.
H. F. Chen and W. X. Zhao, *Recursive Identification and Parameter Estimation.* CRC Press, 2014.
H. F. Chen, T. Ducan, and B. P. Ducan, “A Kiefer-Wolfowitz algorithm with randomized differences," *IEEE Trans. Autom. Control*, vol. 44, no. 3, pp. 442–453, 1999.
H. F. Chen, and G. Yin, “Asymptotic properties of sign algorithms for adaptive fitering," *IEEE Trans. Automatic Control*, vol. 48, no. 9, pp. 1545–1556, 2003.
J. C. Spall, “Multivariate stochastic approximation using a simultaneous perturbation gradient optimization ," *IEEE Trans. Automatic Control,* vol. 45, pp. 1839–1853, 1992.
V. S. Bokar, “Distributed computation of fixed points of $\infty$-nonexpansive maps," *Indian Acad. Sci. Math. Sci.,* vol. 106, no. 3, pp. 289–300, 1996.
V. S. Bokar, and V. V. Phansalkar, “Managing interprocessor delays in distributed recursive algorithms," *Sadhana,* vol. 19, pp. 995-1003, 1994.
V. S. Bokar, “Asynchronous stochastic approximation," *SIAM Journal on Control and Optimization,* vol. 36, no. 3, pp. 840-851, 1998.
E. Oja, and J. Karhunen, “On stochastic approximation of the eigenvectors and eigenvalues of the expectation of a random matrix," *Journal of Mathematical Analysis and Applications*, vol. 106, pp. 69–84, 19857.
Z. J. Bai, R. H. Chan, and F. T. Luk, “Principal component analysis for distributed data sets with updating", *International Workshop on Advanced Parallel Processing Technologies*, pp. 471-483, Springer Berlin Heidelberg, 2005.
Y. A. Borgne, S. Raybaud, and G. Bontempi, “Distributed principal component analysis for wireless sensor networks," *Sensors,* vol. 8, no. 8, pp. 4821–4850, 2008.
J. Lei, and H. F. Chen, “Distributed Stochastic Approximation Algorithm With Expanding Truncations: Algorithm and Applications ", *arXiv:1410.7180v3.*
S. C. Yuan and H. Teicher, [*Probability Theory*]{}. Springer, 1997.
H. T. Fang, and H. F. Chen, “Stability and instability of limit point of the stochastic approximation algorithms", *IEEE Trans. Automa. Control,* vol. 45, no. 3, pp. 413-420, 2000.
[Jinlong Lei]{} received the B.E. degree in Automation from the University of Science and Technology of China in 2011 and the Ph.D. degree in Operations Research and Cybernetics in 2016 from the Institute of Systems Science, Academy of Mathematics and Systems Science, Chinese Academy of Sciences. Her research interests are in recursive parameter estimation and optimization for networked systems. She is now with the Department of Industrial and Manufacturing Engineering, Pennsylvania State University as a postdoctoral fellow.
[Han-Fu Chen,]{} after his graduation from the Leningrad (St. Petersburg) State University, Russia, joined the Institute of Mathematics, Chinese Academy of Sciences (CAS). Since 1979 he has been with the Institute of Systems Science, which now is a part of the Academy of Mathematics and Systems Science, CAS. He is a Professor of the Key Laboratory of Systems and Control of CAS. His research interests are mainly in stochastic systems, including system identification, adaptive control, and stochastic approximation and its applications to systems, control, and signal processing. He authored and coauthored more than 210 journal papers and eight books.
He served as an IFAC Council Member (2002-2005), President of the Chinese Association of Automation (1993-2002), and a Permanent member of the Council of the Chinese Mathematics Society (1991-1999). He is an IFAC Fellow, a Member of TWAS, and a Member of Chinese Academy of Sciences.
[^1]: The authors are with the Key Laboratory of Systems and Control, Institute of Systems Science, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, P. R. China. ( e-mail: leijinlong11@mails.ucas.ac.cn (Jinlong Lei); hfchen@iss.ac.cn (Han-Fu Chen)).
[^2]: This work was supported by the NSFC under Grants 61273193, 61120106011, 61134013, the 973 program of China under grant No.2014CB845301, and the National Center for Mathematics and Interdisciplinary Sciences, Chinese Academy of Sciences.
|
---
abstract: |
-1.4truecm
We overview some attempts to find $S$-duality analogues of non-supersymmetric Yang-Mills theory, in the context of gravity theories. The case of MacDowell-Mansouri gauge theory of gravity is discussed. Three-dimensional dimensional reductions from the topological gravitational sector in four dimensions, enable to recuperate the $2+1$ Chern-Simons gravity and the corresponding $S$-dual theory, from the notion of self-duality in the four-dimensional theory.
address: |
$^{a}$ [*Departamento de Física, Centro de Investigación y de*]{}\
Estudios Avanzados del IPN\
P.O. Box 14-740, 07000, México D.F., México\
$^b$ [*Instituto de Física de la Universidad de Guanajuato*]{}\
P.O. Box E-143, 37150, León Gto., México\
$^c$ [*Facultad de Ciencias Físico Matemáticas, Universidad*]{}\
Autónoma de Puebla\
P.O. Box 1364, 72000, Puebla, México
author:
- 'H. García-Compeán$^{a}$[^1], O. Obregón$^{b}$[^2] and C. Ramírez$^c$[^3]'
title: 'Searching for $S$-duality in Gravitation[^4]'
---
-.5truecm
Introduction
============
Strong/weak coupling duality ($S$-duality) in superstring and supersymmetric gauge theories in various dimensions has been, in the last five years, the major tool to study the strong coupling dynamics of these theories. Much of these results require supersymmetry through the notion of BPS state. These states describe the physical spectrum and they are protected of quantum corrections leaving the strong/coupling duality under control to extract physical information. In the non-supersymmetric case there are no BPS states and the situation is much more involved. This latter case is an open question and it is still under current investigation.
In the specific case of non-supersymmetric gauge theories in four dimensions, the subject has been explored recently in the Abelian as well as in the non-Abelian cases [@witten; @oganor] (for a review see [@quevedo]). In the Abelian case, one considers $CP$ non-conserving Maxwell theory on a curved compact four-manifold $X$ with Euclidean signature or, in other words, U(1) gauge theory with a $\theta$ vacuum coupled to four-dimensional gravity. The manifold $X$ is basically described by its associated classical topological invariants: the Euler characteristic $\chi(X) = { 1
\over 16 \pi^2}\int_X {\rm tr} R \wedge \tilde R$ and the signature $\sigma(X) = - {1 \over 24 \pi^2}\int_X {\rm tr} R\wedge R$. In the Maxwell theory, the partition function $Z(\tau)$ transforms as a modular form under a finite index subgroup $\Gamma_0(2)$ of SL$(2,{\bf Z})$ [@witten], $Z(-1/\tau) =
\tau^u \bar{\tau}^v Z(\tau)$, with the modular weight $(u,v) =
({1 \over 4}(\chi + \sigma), {1\over 4}(\chi - \sigma))$. In the above formula $ \tau = {\theta \over 2 \pi} + {4 \pi i \over g^2}$, where $g$ is the U(1) electromagnetic coupling constant and $\theta$ is the usual theta angle.
In order to cancel the modular anomaly in Abelian theories, it is known that one has to choose certain holomorphic couplings $B(\tau)$ and $C(\tau)$ in the topological gravitational (non-dynamical) sector, through the action
$$I^{TOP} = \int_X \bigg( B(\tau) {\rm tr} R \wedge \tilde R + C(\tau)
{\rm tr} R \wedge R \bigg),
\label{1}$$
i.e., which is proportional to the appropriate sum of the Euler characteristic $\chi(X)$ and the signature $\sigma(X)$.
1truecm
$S$-Duality in MacDowell-Mansouri Gauge Theory of Gravity
=========================================================
Let us briefly review the MacDowell-Mansouri (MM) proposal [@mm]. The starting point for the construction of this theory is to consider an SO(3,2) gauge theory with a Lie algebra-valued gauge potential $A^{AB}_\mu$, where the indices $\mu = 0, 1, 2, 3$ are space-time indices and the indices $A, B= 0, 1, 2, 3, 4$. From the gauge potential $A^{~AB}_\mu$ we may introduce the corresponding field strength $F_{\mu\nu}^{~~AB} = \partial_\mu A_\nu^{~AB} - \partial_\nu A^{AB}_\mu
+ \frac{1}{2} f^{AB}_{CDEF} A^{CD}_\mu A^{EF}_\nu,$ where $f^{AB}_{CDEF}$ are the structure constants of SO(3,2). MM choose $F^{a4}_{\mu\nu} \equiv 0$ and as an action
$$S_{MM} = \int d^4 x \epsilon^{\mu\nu\alpha\beta} \epsilon_{abcd}
F^{~~ab}_{\mu\nu} F^{~~cd}_{\alpha\beta},$$
where $a,b,...{\rm etc.}=0,1,2,3.$
On the other hand, by considering the self-dual (or anti-self-dual) part of the connection, a generalization has been proposed [@nos]. The extension to the supergravity case is considered in [@prl].
One can then search whether the construction of a linear combination of the corresponding self-dual and anti-self-dual parts of the MacDowell-Mansouri action can be reduced to the standard MM action plus a kind of $\Theta$-term and, moreover, if by this means one can find the “dual-theory" associated with the MM theory. This was showed in [@pursuing] and the corresponding extension to supergravity is given at [@dual]. In what follows we follow Ref. [@pursuing]. Let us consider the action
$$S=\int d^4 x \epsilon^{\mu\nu\alpha\beta} \epsilon_{abcd} \bigg(
{^+} \tau {^+}F^{~~ab}_{\mu\nu} {^+}F^{~~cd}_{\alpha\beta} - {^-} \tau
{^-}F^{~~ab}_{\mu\nu} {^-}F^{~~cd}_{\alpha\beta} \bigg) ,$$
where ${^\pm}F^{~~ab}_{\mu\nu} = \frac{1}{2} \bigg( F^{~~ab}_{\mu\nu}
\pm \tilde F^{~~ad}_{\mu\nu} \bigg)$ and $\tilde F^{~~ab}_{\mu\nu} = - \frac{1}{2} i
\epsilon^{ab}_{~~cd}
F^{~~cd}_{\mu\nu}$. It can be easily shown [@pursuing], that this action can be rewritten as
$$S=\frac{1}{2} \int d^{4} x \epsilon^{\mu\nu\alpha\beta} \epsilon_{abcd}
\bigg[ ({^+}\tau - {^-}\tau) F^{~~ab}_{\mu\nu} F^{~~cd}_{\alpha\beta} + (
{^+}\tau + {^-}\tau) F^{~~ab}_{\mu\nu} \tilde F^{~~cd}_{\alpha\beta}
\bigg].$$
In their original paper, MM have shown [@mm] that the first term in this action reduces to the Euler topological term plus the Einstein-Hilbert action with a cosmological term. This was achieved after identifying the components of the gauge field $A^{~AB}_{\mu}$ with the Ricci rotation coefficients and the vierbein. Similarly, the second term can be shown to be equal to $i\theta P$, where $P$ is the Pontrjagin topological term [@nos]. Thus, it is a genuine $\theta$ term, with $\theta$ given by the sum ${^+}\tau + {^-}\tau$.
Our second task is to find the “dual theory", following the same scheme as for Yang-Mills theories [@oganor]. For that purpose we consider the parent action
$$I= \int d^{4}x \epsilon^{\mu\nu\alpha\beta} \epsilon_{abcd}
\bigg( c_1{^+} G^{~~ab}_{\mu\nu} {^+}G^{~~cd}_{\alpha\beta} + c_2
{^-} G^{~~ab}_{\mu\nu} {^-} G^{~~cd}_{\alpha\beta} + c_3 {^+}
F^{~~ab}_{\mu\nu} {^+}G^{~~cd}_{\alpha\beta} + c_4 {^-} F^{~~ab}_{\mu\nu}
{^-} G^{~~cd}_{\alpha\beta} \bigg) .$$
From which the action (3) can be recovered after integration on ${^+}G$ and ${^-}G$.
In order to get the “dual theory" one should start with the partition function
$$Z= \int {\cal D} {^+}G \, {\cal D} {^-}G \, {\cal D} A\,\,e^{-I}.$$
To proceed with the integration over the gauge fields we observe that $F^{~~ab}_{\mu\nu} = \partial_\mu A^{~ab}_{\nu} - \partial_\nu A^{~ab}_\mu
+ \frac{1}{2} f^{ab}_{CDEF} A^{~CD}_{\mu} A^{~EF}_{\nu}.$ Taking into account the explicit expression for the structure constants, the second term of $F^{~~ab}_{\mu\nu}$ will naturally split in four terms given by $A^{~ad}_{\mu} A^{~~b}
_{\nu d}$ $- A^{~ad}_\nu A^{~~b}_{\mu d}$ $ - \lambda^2 \bigg( A^{~a4}_\mu
A^{~b4}_\nu - A^{~a4}_\nu A^{~b4}_\mu \bigg).$ The integration over the components $A^{~a4}_\mu$ is given by a Gaussian integral, which turns out to be $det\, {\bf G}^{-1/2}$, where [**G**]{} is a matrix given by ${\bf G}^{~~\mu\nu}_{ab}$ $ = 8 i \lambda^2 \epsilon^{\mu\nu\alpha\beta}
\bigg( c_3 {^+}G_{\alpha\beta ab} - c_4 {^-}G_{\alpha\beta ab} \bigg).$
Thus, the partition function (6) can be written as
$$Z= \int {\cal D}{^+} G\, {\cal D} {^-}G \, {\cal D} A^{~ab}_\mu\,\, det\,
{\bf
G}^{-1/2}\,\, e^{- \, {I\!\!I}},$$
where
$${I\!\!I}=2 i \int d^4 x \epsilon^{\mu\nu\alpha\beta} \bigg[ c_1
{^+}G^{~~ab}_{\mu\nu} {^+}G_{\alpha\beta ab} - c_2 {^-} G^{~~ab}_{\mu\nu}
{^-}G_{\alpha\beta ab} + 2 H^{~~ab}_{\mu\nu} ( c_3
{^+} G_{\alpha\beta ab} - c_4 {^-}G_{\alpha\beta ab}) \bigg],$$ and $H^{~~ab}_{\mu\nu} = \partial_\mu A^{~ab}_\nu - \partial_\nu A^{~ab}_\mu
+ \frac{1}{2} f^{ab}_{cdef} A^{~cd}_\mu A^{~ef}_\nu$ is the SO(3,1) field strength.
Our last step to get the dual action is to integrate over $A^{~ab}_\mu$. This kind of integration is well known and has been performed in previous works [@oganor; @pursuing; @towards]. The result is
$$Z=\int {\cal D}{^+}G\, {\cal D} {^-}G \,\, det\, {\bf G}^{-1/2} \,\,
det ({^+}M)^{-1/2} det ({^-}M)^{-1/2} \,\, e^{- \int d^4x \tilde L},$$
with
$$\begin{array}{ll}
\tilde{L} &= \epsilon^{\mu\nu\rho\sigma} \bigg[ -{1 \over 4
{^+}\tau} {^+}G^{~~ab}_{\mu \nu} {^+}G_{\rho\sigma ab} + {1 \over 4
{^-}\tau} {^-}G^{~~ab}_{\mu \nu} {^-}G_{\rho \sigma ab} +
2 \partial_{\nu} {^+}G_{\rho \sigma ab} {({^+}M)}^{-1 \ abcd}_{\mu
\lambda} \epsilon^{\lambda \theta \alpha \beta}
\partial_{\theta} {^+}G_{\alpha \beta cd} \\
&- 2 \partial_{\nu} {^-}G_{\rho
\sigma
ab} {({^-}M)}^{-1 \ abcd}_{\mu
\lambda} \epsilon^{\lambda \theta \alpha \beta}
\partial_{\theta} G^-_{\alpha \beta cd}\bigg],
\end{array}
\label{tcinco}$$
where ${^\pm}M^{\mu\nu ~cd}_{~~ab} = \frac{1}{2} \epsilon^{\mu\nu\alpha\beta}
\bigg( - \delta^c_a {^\pm}G^{~~~d}_{\alpha\beta b} + \delta^c_b
{^\pm}G^{~~~ d}_{\alpha\beta a} + \delta^d_a {^\pm} G^{~~~ c}_{\alpha\beta
b}
- \delta^d_b {^\pm} G^{~~~ c}_{\alpha\beta a} \bigg)$ and ${^+} \tau = - \frac{1}{4c_1},$ ${^-}\tau = - \frac{1}{4c_2}$, $c_3 = c_4 = 1.$
The non-dynamical model considered in a previous work [@towards] results in a kind of non-linear sigma model [@oganor] of the type considered by Freedman and Townsend [@freed], as in the usual Yang-Mills dual models. The dual to the dynamical gravitational model (9) considered here, results in a Lagrangian of the same structure. However, it differs from the non-dynamical case by the features discussed above.
1truecm
(Anti)Self-duality of the Three-dimensional Chern-Simons Gravity
=================================================================
It is well known that the $2+1$ Einstein-Hilbert action with nonvanishing cosmological constant $\lambda$ is given by the “standard” and “exotic” Einstein actions [@wittenone]. It is well known that for $\lambda >0$ (and $\lambda <0$), these actions are equivalent to a Chern-Simons actions in $2+1$ dimensions with gauge group ${\cal G}$ to be SO(3,1) (and SO(2,2)).
In this section we will work out the Chern-Simons Lagrangian for (anti)self-dual gauge connection with respect to duality transformations of the internal indices of the gauge group ${\cal G}$, in the same philosophy of MM [@mm], and that of [@nos]
$$L{^{\pm}}_{CS} = \int_{\cal M} \varepsilon^{ijk} \bigg( {^{\pm}} A_i^{AB}
\partial_j {\ ^{\pm}} A_{kAB} + {\frac{2 }{3}} {^{\pm}} A_{iA}^B {^{\pm}}
A_{jB}^C {^{\pm}} A_{kC}^A \bigg), \label{csself}$$
where $A,B,C,D= 0,1,2,3,$ $\eta_{AB} = diag(-1,+1,+1,+1)$ and the complex (anti) self-dual connections are ${^{\pm}} A_{i}^{AB} = {\frac{1}{2}}(A_{i}^{AB} \mp {\frac{i}{2}}
\varepsilon^{AB}_{ \ \ CD} A_{i}^{CD})$, which satisfy the relation $\varepsilon^{AB}_{ \ \ CD}
{^{\pm}} A_i^{CD} = \pm i {^{\pm}} A^{AB}.$
Thus using the above equations, the action (10) can be rewritten as
$$L{^{\pm }}_{CS}=\int_{\cal M}{\frac{1}{2}}\varepsilon ^{ijk}\bigg(
A_{i}^{AB}\partial _{j}A_{kAB}+{\frac{2}{3}}A_{iA}^{\ \ B}A_{jB}^{\ \
C}A_{kC}^{\ \ A}\bigg)\mp {\frac{i}{4}}\varepsilon ^{ijk}\varepsilon ^{ABCD}
\bigg(A_{iAB}\partial _{j}A_{kCD}+{\frac{2}{3}}A_{i\text{ }A}^{\
E}A_{jEB}A_{kCD}\bigg). \label{eq13}$$
In this expression the first term is the Chern-Simons action for the gauge group ${\cal G}$, while the second term appears as its corresponding “$
\theta $-term”. The same result was obtained in $3+1$ dimensions when we considered the (anti)self-dual MM action [@pursuing], or the (anti)self-dual $3+1$ pure topological gravitational action [@towards].
One should remark that the two terms in the action (13) are the Chern-Simons and the corresponding “$\theta $-term” for the gauge group ${\cal G}$ under consideration. After imposing the particular identification $A_i^{AB}=(A_{i}^{ab},A_{i}^{3a})=(\omega_{i}^{ab},\sqrt{\lambda
}e_{i}^{a})$ and $\omega_{i}^{ab}=\varepsilon^{abc} \omega_{ic}$, the “exotic” and “standard” actions for the gauge group SO(3,1) are given respectively by
$$L_{CS}{^{\pm}} = \int_X {\frac{1}{2}} \varepsilon^{ijk} \bigg(
\omega^a_i(\partial_j \omega_{ka} - \partial_k \omega_{ja}) + {\frac{2 }{3}}
\varepsilon_{abc} \omega_i^a \omega_j^b \omega_k^c + \lambda
e^a_i(\partial_j e_{ka} - \partial_k e_{ia}) - 2 \lambda
\varepsilon_{abc}e_i^a e_j^b \omega_k^c\bigg)$$
$$\pm i \sqrt{\lambda}\varepsilon ^{ijk}\bigg(e_{i}^{a}(\partial _{j}\omega _{ka}-\partial
_{k}\omega _{ja})-\varepsilon _{abc}e_{i}^{a}\omega _{j}^{b}\omega _{k}^{c}+{
\frac{1}{3}}\lambda \varepsilon _{abc}e_{i}^{a}e_{j}^{b}e_{k}^{c}\bigg),
\label{eq14}$$
plus surface terms. It is interesting to note that the above action (12) can be obtained from action (1) (for a suitable choice of $B(\tau)$ and $C(\tau)$) by dimensional reduction from $X$ to its boundary ${\cal M} = \partial X$. Thus the “standard” action come from the Euler characteristic $\chi(X)$, while the “exotic” action come from the signature $\sigma(X).$
1truecm
Chern-Simons Gravity Dual Action in Three Dimensions
====================================================
This section is devoted to show that a “dual” action to the Chern-Simons gravity action can be constructed following [@sabido]. Essentially we will repeat the procedure to find the the “dual” action to MM gauge theory given in Sec. II.
We begin from the original non-Abelian Chern-Simons action given by
$$L=\int_{\cal M}d^{3}x{\frac{g}{4\pi }}\varepsilon ^{ijk}A_{i}^{AB}\bigg(\partial
_{j}A_{kAB}+{\frac{1}{3}}f_{ABCDEF}A_{j}^{CD}A_{k}^{EF}\bigg). \label{cs}$$
Now, as usual we propose a parent action in order to derive the dual action
$$L_D = \int_{\cal M} d^3 x \varepsilon^{ijk} \bigg( a B^{AB}_i H_{jkAB} + b A^{AB}_i
G_{jkAB} + c B^{AB}_i G_{jkAB} \bigg), \label{parent}$$
where $H_{jkAB}= \partial_j A_{kAB} + {\frac{1 }{3}} f_{ABCDEF}A^{CD}_jA^{EF}_k$ and $B^{AB}_i$ and $G^{AB}_{ij}$ are vector and tensor fields on ${\cal M}$. It is a very easy matter to show that the action (13) can be derived from this parent action after integration of $G$ fields
The “dual” action $L^*_D$ can be computed as usually in the Euclidean partition function, by integrating first out with respect to the physical degrees of freedom $A^{AB}_i.$ The resulting action is of the Gaussian type in the variable $A$ and thus, after some computations, it is easy to find the “dual” action
$$L_{D}^{\ast }=\int_{\cal M}d^{3}x\varepsilon ^{ijk}\bigg \{-{\frac{3}{4a}}
(a\partial _{i}B_{jAB}+bG_{ijAB})[{\bf R}^{-1}]_{kn}^{ABCD}\varepsilon
^{lmn}(a\partial _{l}B_{mCD}+bG_{lmCD})+c\alpha _{i}^{AB}G_{jkAB}\bigg\},
\label{bosonicdual}$$
where $[{\bf R}]$ is given by $[{\bf R}]_{ABCD}^{ij}=\varepsilon ^{ijk}f_{\
\ ABCD}^{EF}B_{kEF}$ whose inverse is defined by $[{\bf R}]_{ABCD}^{ij}[{\bf
R}^{-1}]_{jk}^{CDEF}=\delta _{k}^{i}\delta _{AB}^{EF}.$
The partition function is finally given by $$Z = \int {\cal D}G {\cal D}B \sqrt{det({\bf M}^{-1})} exp \big( - L^*_D
\big).$$ In this “dual action” the $G$ field is not dynamical and can be integrated out. The integration of this auxiliary field gives $$L^{**}_D=\int_{\cal M} d^3 x{\frac{4\pi }{g}}\varepsilon
^{lmn} \bigg(B_{l}^{AB}\partial _{m}B_{nAB}-{\frac{4\pi }{g}}
f_{ABCDEF}B_{l}^{AB}B_{m}^{CD}B_{n}^{EF}\bigg).$$ The fields $B$ cannot be rescaled if we impose “periodicity” conditions on them. Thus, this dual action has inverted coupling with respect to the original one (compare with [@bala] for the Abelian case).
2truecm
**Acknowledgments**
The results of sections 3 and 4 were obtained in collaboration with Miguel Sabido. We are very grateful to him for a critical reading of this manuscript.
1truecm
E. Witten, Selecta Mathematica, New Series, [**1**]{}, 383 (1995).
O. Ganor and J. Sonnenschein, Int. J. Mod. Phys. A [**11**]{} (1996) 5701; N. Mohammedi, hep-th/9507040; Y. Lozano, Phys. Lett. B [**364**]{} (1995) 19.
F. Quevedo, Nucl. Phys. Proc. Suppl. [**61**]{} A (1998) 23.
S.W. Mac Dowell and F. Mansouri, Phys. Rev. Lett. [**38**]{} (1977) 739; F. Mansouri, Phys. Rev. D [**16**]{} (1977) 2456.
J.A. Nieto, O. Obregón and J. Socorro, Phys. Rev. D [**50**]{} (1994) R3583.
J.A. Nieto, J. Socorro and O. Obregón, Phys. Rev. Lett. [**76**]{} (1996) 3482.
H. García-Compeán, O. Obregón and C. Ramírez, Phys. Rev. D [**58**]{}, 104012-1 (1998); Chaos, Solitons and Fractals [**10**]{} (1999) 373.
H. García-Compeán, J.A. Nieto, O. Obregón and C. Ramírez, Phys. Rev. D [**59**]{} (1999) 124003.
H. García-Compeán, O. Obregón, J.F. Plebański and C. Ramírez, Phys. Rev. D [**57**]{}, 7501 (1998).
D. Z. Freedman and P. K. Townsend, Nucl. Phys. B[** 177**]{}, 282 (1981); P. C. West, Phys. Lett. B [**76**]{}, 569 (1978); A. H. Chamseddine, Ann. Phys. (N.Y.) [**113**]{}, 219 (1978); S. Gotzes and A. C. Hirshfeld, [*ibid*]{}. [**203**]{}, 410 (1990).
E. Witten, Nucl. Phys. B [**311**]{} (1988) 46; Nucl. Phys. B [**323**]{} (1989) 113.
H. García-Compeán, O. Obregón, C. Ramírez and M. Sabido, “Remarks on $2+1$ Self-dual Chern-Simons Gravity”, hep-th/9906154, to appear in Phys. Rev. D.
A.P. Balachandran, L. Chandar and B. Sathiapalan, Int. J. Mod. Phys. A [**11**]{} (1996).
[^1]: E-mail: compean@fis.cinvestav.mx
[^2]: E-mail: octavio@ifug3.ugto.mx
[^3]: E-mail: cramirez@fcfm.buap.mx
[^4]: Invited talk at the [*Third Workshop on Gravitation and Mathematical-Physics,*]{} Nov. 28-Dec. 3 1999, León Gto. México.
|
---
author:
- 'Yang Zhang[^1]'
bibliography:
- 'AA2016.bib'
title: |
Lecture Notes on\
Multi-loop Integral Reduction\
and Applied Algebraic Geometry
---
These notes are for the author’s lectures, “Integral Reduction and Applied Algebraic Geometry Techniques” in School and Workshop on Amplitudes in Beijing 2016.
I would like to acknowledge Simon Badger, Niklas Beisert, Zvi Bern, Emil J. Bjerrum-Bohr, Jorrit Bosma, Poul Damgaard, Simon Caron-Huot, Lance Dixon, Claude Duhr, Bo Feng, Hjalte Frellesvig, Alessandro Georgoudis, Johannes Henn, Rijun Huang, Harald Ita, David Kosower, Kasper Larsen, Jianxin Lu, Qing Lu, Pierpaolo Mastrolia, Dhagash Mehta, Mads Sogaard, Martin Sprenger, Mingmin Shen, Henry Tye, Gang Yang, Huafeng Zhang and Huaxing Zhu for useful discussions in related directions. Especially I thank Jorrit Bosma for reading this manuscript carefully and correcting various typos. In particular, I express my gratitude to Michael Stillman for his excellent lectures on algebraic geometry at Cornell University.
I also acknowledge the support provided by Swiss National Science Foundation, via Ambizione grant (PZ00P2\_161341).
Introduction
============
From childhood, we know that integral calculus is more difficult than differential calculus, moreover a multiple integral can be a hard nut to crack. Multifold integrals appear ubiquitously in science and technology, for example, to understand high order quantum interactions, we have to deal with [*multi-loop Feynman integrals*]{}. For precision LHC physics, next-to-next-to-leading order (NNLO) and even next-to-next-to-leading order (N3LO) contribution should be calculated, to be compared with the experimental data. This implies we have to compute two-loop or three-loop, non-supersymmetric, frequently massive Feynman integrals. It is a tough task.
Recall that in college, when we get a complicated integral, usually we do not compute it directly by brute force. Instead, we may first:
- Reduce the integrand. For example, given a univariate rational function integral, we use partial fraction to split the integrand into a sum of fractions, each of which contains only one pole.
- Convert the integral to residue computations. For an analytic univariate integrand, sometimes we can deform the contour of integral and make it a residue computation. The latter is often much easier than the original integral.
- Rewrite the integral by integration-by-parts (IBP).
All these basic techniques are used every day in high energy physics, and in all other branches of physics. For example, Ossola, Papadopoulos and Pittau (OPP) [@Ossola:2006us; @Ossola:2007ax] developed a systematic one-loop integrand reduction method, in the fashion of partial fraction. This method reduces one-loop Feynman integrals to one-loop master integrals, whose coefficients can be automatically extracted from tree diagrams by unitarity analysis. Nowadays, OPP method becomes a standard programmable algorithm for computing next-to-leading order (NLO) contributions.
However, for two-loop and higher-loop Feynman integrals, these basic techniques for simplifying integrals often become insufficient. For example,
- For multi-loop orders, a Feynman integrand is still a rational function, however, in [*multiple variables*]{}. In this case, it is not easy to carry out partial fraction or general integrand reduction. This new issue is the [*monomial order*]{}, the order of variables. Naive reduction results may be too complicated for next steps, integral computation or unitarity analysis.
- For multi-loop generalized unitarity, sometimes we have residues not from one complex variable, but from multiple complex variables. It is well-known that the analysis of several complex variables is much harder than univariate complex analysis. For example,
> (Hartog) Let $f(z_1,\ldots, z_n)$ be an analytic function in $U \backslash \{P\}$, where $U$ is an open set of $\C^n$ ($n>1$) and $P$ is a point in $U$. Then $f(z_1,\ldots, z_n)$ is analytic in $U$.
Hartog’s theorem implies that any isolated singular point of a multivariate analytic function is removable. Hence, non-trivial singular points of multivariate analytic function have a much more complicated geometric structure than those in univariate cases. Besides, multivariate Cauchy’s theorem does not apply for the case when analytic functions have zero Jacobian at the pole. That makes residue computation difficult. For instance, $$\label{residue_question}
\oint \oint_{\text{around (0,0)}} \frac{dz_1 dz_2}{(a z_1^3+z_1^2+z_2^2)(z_1^3+z_1
z_2-z_2^2)} =~?$$
- For multi-loop integrals, the number of IBP relations becomes huge. We may need to list a large set of IBP relations, and then use linear algebra to eliminate unwanted terms to get [*useful*]{} IBPs. However, the linear system can be very large and Gauss elimination (especially in analytic computations) may exhaust computer RAM.
Is there a way to list only useful IBPs, by adding constraints on differential forms? The answer is “yes”, but these constraints are subtle. These are linear equations which only allow polynomial solutions [@Gluza:2010ws]. [^2] How do we solve them efficiently?
Most Feynman integral simplification procedures in multi-loop orders, suffer from the complicated structure of multiple variables. Note that, usually our targets are just polynomials or rational functions. However, multivariate polynomial problems can be extremely difficult. (One famous example is Jacobian conjecture, which stands unsolved today.)
The modern branch of mathematics dealing with multivariate polynomials and rational functions is [*algebraic geometry*]{}. Classically, algebraic geometry studies the geometric sets defined by zeros of polynomials. Polynomial problems are translated to geometry problems, and vice versa. Note that since only polynomials are allowed, algebraic geometry is more “rigid” than [*differential geometry*]{}. Classical algebraic geometry culminates at the classification theorem of algebraic surfaces by the [*Italian school*]{} in 19th century.
Modern algebraic geometry is rigorous, much more general and abstract. The classical geometric objects are replaced by the abstract concept [ *scheme*]{}, and powerful techniques like [*homological algebra*]{} and [ *cohomology*]{} are introduced in algebraic geometry thanks to Alexander Grothendieck and contemporary mathematicians [@MR3075000; @MR0163909; @MR0163911; @MR0173675; @MR0199181; @MR0217086; @MR0238860; @MR0463157]. Modern algebraic geometry shows its power in the proof of [*Fermat’s last theorem*]{} by Andrew Wiles. Now algebraic geometry applies on number theory, representation theory, complex geometry and theoretical physics.
Back to our cases, there are numerous polynomial/rational function problems. Clearly, they are not as sophisticated as [*Fermat’s last theorem*]{} or [*Riemann hypothesis*]{}. Apparently they resemble classical algebraic geometry problems. However, beyond the classification of curves or surfaces, we need computational power to solve polynomial-form equations, to compute multivariate residues in the real world. The computational aspect of algebraic geometry, was neglected for a long time.
When I was a graduate student, I was lucky taking a class by Professor Michael Stillman. One fascinating thing in the class was that many times after learning an important theorem, Michael turned on the computer and ran a program called “Macaulay2” [@M2]. He typed in number fields, polynomials, and geometric objects in the study. Then various commands in the program can automatically generate the dimension, the genus and various maps between objects. He taught us one essential tool behind the program was the so-called [ *Gröbner basis*]{}, which is the crucial concept in the new subject [*computational algebraic geometry*]{} (CAG) [@MR3330490; @opac-b1094391]. It was my first time hearing about CAG and soon found it useful.
CAG aims at multivariate polynomial and rational function problems in the real world. It began with [*Buchberger’s algorithm*]{} in 1970s, which obtained the Gröbner basis for a [*polynomial ideal*]{}. Buchberger’s algorithm for polynomials is similar to Gaussian Elimination for linear algebra: the latter finds a linear basis of a subspace while the former finds a “good” generating set for an ideal. With Gröbner basis, one can carry out multivariate polynomial division and simplify rational functions; one can eliminate variables from a polynomial system; one can apply polynomial constraints without solving them... Then CAG developed quickly and now it is so all-purpose that people use it outside mathematics, like in robotics, cryptography and game theory. I believe that CAG is crucial for the deep understanding of multi-loop scattering amplitudes.
Hence, the purpose of these lecture notes is to introduce a fast-developing research field: applied algebraic geometry in multi-loop scattering amplitudes. I would like to show CAG methods by examples,
- Multi-loop integrand reduction via Gröbner basis. This generalizes one-loop OPP integrand reduction method to all loop orders. In this section, I will introduce basic notations of polynomial ring, rudiments of algebraic geometry and the Gröbner basis method.
- Multivariate residue computation, in [*generalized*]{} unitarity analysis. A flavor of several complex variables will be provided in the section. Then I present the definition of multivariate residues and CAG based algorithms for computing multivariate residues. Finally I show that they are very useful in high-loop unitarity analysis.
- Multi-loop IBP with polynomial constraints. These constraints form a [*syzygy*]{} system, which can be solved by [@Gluza:2010ws] techniques. We show that we can combine this with unitarity cuts and the Baikov representation [@Baikov:1996rk] to further improve the efficiency.
I will illustrate mathematical concepts and methods by practical examples and exercises, even beyond mathematics/physics, like the game [ *Sudoku*]{}. The proof of many mathematical theorems will be skipped or just roughly sketched. I will not cover all the technical details of the research frontier from integral reduction, since I believe it is more important for readers to get the idea of basic algebraic geometry and find its applications in their own research fields.
Integrand reduction and Gröbner basis
=====================================
Basic physical objects
----------------------
In these notes we mainly focus on scattering amplitudes in perturbative quantum field theory and (super-)gravity. To make the reduction methods general, we aim at non-supersymmetric amplitudes. These methods definitely work with supersymmetric theories, however, it is more efficient to combine them with specific shortcuts in supersymmetric theories.
Referring to an $L$-loop Feynman diagram, we mean a [*connected*]{} diagram with $n$ external lines, $P$ propagators, and $L$ [*fundamental cycles*]{} [^3]. We further require that each external line is connected to some fundamental cycle. Define $V$ as the number of vertices in this diagram, then the graph theory relation holds, $$\label{Graph_theory}
L=P-V+1.$$ Note that this relation is not Euler’s famous formula, since this relation holds for both planar and nonplanar graphs in graph theory, but Euler characteristic does not enter this relation. (Of course, for a planar graph, by embedding fundamental cycles into a plane as face boundaries, it becomes Euler’s formula for planar graphs.)
For gauge theories, we have [*color-ordered*]{} Feynman diagrams such that the external color particles must be drawn from infinity in a given cyclic order, and the Feynman rules would differ from the unordered ones. Sometimes, with these constraints, we cannot draw a Feynman diagram on a plane without crossing lines. We call such a Feynman diagram a [*nonplanar diagram in the sense of color ordering.*]{} Note that this definition is different from [ *nonplanar diagram in the sense of graph theory*]{}, since by lifting the color order constraint, a colored-ordered nonplanar diagram may be embedded into a plane without crossing lines. See an example in Fig.\[xbox\_planar\].
Sometimes, for an $L$-loop diagram with $L>1$, two fundamental cycles do not share a common edge. In this case the diagram is [*factorable*]{}, i.e., factorized into two diagram. We consider a factorable diagram as two lower loop-order diagrams, instead of an “authentic” $L$-loop diagram. See an example in Figure \[reducible\_diagrams\]a. For a $n$-point $L$-loop diagram, if two external lines attach to one vertex, we consider this diagram as an $n-1$ point diagram. See an example in Figure \[reducible\_diagrams\]b.
A Feynman diagram has the associated Feynman integral, $$I=\int \frac{d^D l_1}{i \pi^{D/2}} \ldots \frac{d^D l_L}{i \pi^{D/2}}
\frac{N(l_1,\ldots l_L)}{D_1 \ldots D_P}\,.
\label{Feynman integral}$$ For each fundamental cycle, we assign an internal momenta $l_i$. Here the denominators of Feynman propagator have the form, $D_i=(\alpha_1 l_1 + \dots \alpha_l l_L+\beta_1 k_1 +\ldots
\beta_n k_n)^2-m_i^2$. $k_1 \ldots k_n$ are the external momenta. $\alpha$’s must be $\pm 1$. For fermion propagators, we complete the denominator squares to get this form. $N(l_1,\ldots l_L)$ is the numerator, which depends on Feynman rules and the symmetry factor. Here we hide the dependence of external momenta/polarizations in $N(l_1,\ldots l_L)$. The spacetime dimension $D$ may take the value $4-2\epsilon$ in the [*dimensional regularization scheme*]{} (DimReg). Sometimes we also discuss the case $D=4$ or some other fixed integer, for studying leading singularity and maximal unitarity cut.
Integrand reduction at one loop
-------------------------------
Consider the problem of reducing the integrand in (\[Feynman integral\]) before integration. Schematically integrand reduction, as a generalization of partial fractions, is to express the numerator $N$ as, $$N= \Delta + \sum_{j=1}^P h_j D_j,
\label{IntRed}$$ where $\Delta$ and $h_j$’s are polynomials in loop momenta components. The term $h_j D_j$ cancels a denominator $D_i$ and provides a Feynman integral with fewer propagators. Then this term merges with other Feynman integrals in the scattering amplitude. $\Delta$ remains for this diagram. If $\Delta$ is “significantly simpler” than $N$, this integrand reduction is useful.
### Box diagram
To make our discussion solid, we first introduce the classical OPP reduction method [@Ossola:2006us; @Ossola:2007ax] at one loop order. It is well known that if $D=4$, all one-loop Feynman integrals with more than $4$ distinct propagators can be reduced to Feynman integrals with at most $4$ distinct propagators, while if $D=4-2\epsilon$, one-loop Feynman integrals with more than $5$ distinct propagators are reduced to Feynman integrals with at most $5$ distinct propagators, at the integrand level. These statements can be proven by tensor calculations [@Melrose:1965kb]. Later in this section, we re-prove these by a straightforward algebraic geometry argument.
For simple $D=4$ cases, we only need to start from the box diagram. For instance, consider $D=4$ four-point massless box,
![One-loop massless box diagram[]{data-label="box"}](graphs/box.eps)
with denominators in propagators, $$\begin{gathered}
D_1=l^2,\quad D_2=(l-k_1)^2, \quad D_3=(l-k_1-k_2)^2,\quad D_4=(l+k_4)^2.\end{gathered}$$ The Mandelstam variables are $s=(k_1+k_2)^2$ and $t=(k_1+k_4)^2$. It is useful to re-parameterize the loop momentum $l$ instead of using its Lorentz components. There are several parametrization methods: (1) van Neerven-Vermaseren parameterization [@vanNeerven:1983vr] (2) spinor-helicity parameterization (3) Baikov parametrization. Here we use the straightforward van Neerven-Vermaseren parameterization, and postpone applications of other parameterizations later.
Note that by energy-momentum conservation, only external momenta $k_1$, $k_2$ and $k_4$ are independent. To make a $4D$ basis, we introduce an auxiliary vector $\omega_\mu\equiv\frac{2 i}{s}\epsilon_{\mu \nu \rho \sigma} k_1^\nu
k_2^\rho k_4^\sigma$ [^4] $$\label{eq:19}
\omega^2 = -\frac{t (s+t)}{s} \,.$$ Then the basis $\{e_1,e_2 \ldots e_4\}\equiv
\{k_1,k_2,k_4,\omega\}$. The [*Gram matrix*]{} of this basis is, $$\label{eq:3}
G=\left(
\begin{array}{cccc}
0 & \frac{s}{2} & \frac{t}{2} & 0 \\
\frac{s}{2} & 0 & \frac{-s-t}{2} & 0 \\
\frac{t}{2} & \frac{-s-t}{2} & 0 & 0 \\
0 & 0 & 0 & -\frac{t (s+t)}{s} \\
\end{array}
\right),\quad G_{ij}=e_i\cdot e_j\,.$$ Note that for any well-defined basis, Gram matrix should be non-degenerate. For any $4D$ momentum $p$, define van Neerven-Vermaseren variables as, $$\begin{gathered}
\label{box_vNV}
x_i(p)\equiv p \cdot e_i,\quad i=1,\ldots ,4\,.\end{gathered}$$ Then for any two $4D$ momenta, a scalar product translates to van Neerven-Vermaseren form, by linear algebra $$\label{eq:4}
p_1 \cdot p_2=\mathbf x(p_1)^T (G^{-1}) \mathbf x(p_2) \, ,$$ where the bold $\mathbf x(p)$ denotes the column $4$-vector, $(x_1,x_2,x_3,x_4)^T$. Back to our one-loop box, define $x_i\equiv
x_i(l)$. Hence a Lorentz-invariant numerator $N_\text{box}$ in has the form, $$\label{eq:5}
N_\text{box}=\sum_{m_1} \sum_{m_2} \sum_{m_3} \sum_{m_4} c_{m_1m_2m_3m_4}x_1^{m_1} x_2^{m_2} x_3^{m_3} x_4^{m_4}\,,$$ For a renormalizable theory, there is a bound on the sum, $m_1+m_2+m_3+m_4\leq 4$. The goal in integrand reduction is to expand $$N_\text{box}=\Delta_\text{box}+h_1 D_1 +\ldots h_4 D_4\,,
\label{box_IR}$$ such that the remainder polynomial $\Delta_\text{box}$ is as simple as possible.
Following [@Ossola:2006us], the simplest $\Delta_\text{box}$ can be obtained by a direct analysis. Note that $$\begin{aligned}
x_1 &= l \cdot k_1 =\half (D_1-D_2), \nn \\
x_2 &= l \cdot k_2 =\half (D_2-D_3)+\frac{s}{2}, \nn \\
x_3 &= l \cdot k_4 =\half (D_4-D_1),
\label{RSP_box}\end{aligned}$$ hence $x_1$ and $x_3$ can be written as combinations of $D_i$’s, while $x_2$ is equivalent to the constant $s/2$ up to combinations of $D_i$’s. A scalar product which equals combinations of denominators and constants is called a [*reducible scalar product*]{} (RSP). In this cases, $x_1,x_2,x_3$ are RSPs. The remainder $\Delta_\text{box}$ shall not depend on RSPs, hence, $$\label{eq:7}
\Delta_\text{box}= \sum_{m_4} c_{m_4} x_4^{m_4}.$$ $x_4$ is called a [*irreducible scalar product*]{} (ISP). Furthermore, using the expansion of $l^2$ and , $$\begin{aligned}
D_1=l_1^2&= \frac{1}{{4 s t (s+t)}}\big(-4 s^2 x_4^2 +s^2 t^2 +4 D_1 s^2 t-2 D_2
s^2 t-2 D_4 s^2 t+D_2^2 s^2+D_4^2 s^2\nn\\
&-2 D_2 D_4 s^2
+2 D_1 s t^2-2
D_3 s t^2+2 D_1 D_2 s t
-4 D_1 D_3 s t+2 D_2 D_3 s t+2 D_1 D_4 s t\nn\\
&-4 D_2 D_4 s t
+2 D_3
D_4 s t+D_1^2 t^2+D_3^2 t^2-2 D_1 D_3 t^2 \big),
\label{box_quadratic_eqn}\end{aligned}$$ which means $$\label{box_quadratic_eqn_cut}
x_4^2=\frac{t^2}{4} +\mathcal O (D_i).$$ Hence quadratic and higher-degree monomials in $x_4$ should be removed from the box integrand, and $$\Delta_\text{box}=c_0 + c_1 (l \cdot \omega).
\label{box_integrand_basis}$$ This is the [*integrand basis*]{} for the $4D$ box, which contains only $2$ terms. Note that by Lorentz symmetry, $$\label{eq:11}
\int d^D l \frac{l\cdot \omega}{D_1 D_2 D_3 D_4} =0,$$ for any value of $D$. So $c_1$ should not appear in the final expression of scattering amplitude. We call such a term a [*spurious term*]{}. But it is important for integrand reduction, as we will see soon.
There are two ways of using the integrand basis ,
1. Direct integrand reduction (IR-D). If the numerator $N$ is known, for instance from Feynman rules, we can use and to reduce $N$ explicitly to get $c_0$ and $c_1$. $h_1D_1+\ldots h_4D_4$ is kept for further triangle, bubble ... computations.
2. Integrand reduction with unitarity (IR-U). Sometimes, it is more efficient to fit the coefficients $c_0$ and $c_1$ from tree amplitudes, by unitarity. Here $c_0$ and $c_1$ correspond to the remaining information at the quadruple cut, $$\label{eq:12}
D_1 = D_2 =D_3 =D_4=0.$$ From and , there are two solutions for $l$, namely $l^{(1)}$ and $l^{(2)}$, characterized by, $$\begin{aligned}
\label{box_solution}
\text{(1)}\quad x_1&=0,\quad x_2=\frac{s}{2},\quad x_3=0,\quad x_4=\frac{t}{2}\,,\\
\text{(2)}\quad x_1&=0,\quad x_2=\frac{s}{2},\quad x_3=0,\quad
x_4=-\frac{t}{2}\,.\end{aligned}$$ On this cut, the box diagram becomes four tree diagrams, summed over different [*on-shell*]{} massless internal states. $$\begin{gathered}
\label{tree_product}
S^{(i)}_\text{box}=\sum_{h_1} \sum_{h_2} \sum_{h_3}\sum_{h_4} A(k_1,
l^{(i)}-k_1, -l^{(i)};s_1, h_2, -h_1)\times \nn\\
A(k_2, l^{(i)}-k_1-k_2,k_1-l^{(i)};s_2, h_3, -h_2) A(k_3,
l^{(i)}+k_4,k_1+k_2-l^{(i)};s_3, h_4, -h_3)\nn\\
\times A(k_4,l^{(i)},-k_4-l^{(i)};s_4,h_1,-h_4)\,,\end{gathered}$$ where $s_i$’s stand for external particles helicities, while $h_i$’s stand for internal particles helicities and should be summed. Unitarity implies that, $$\label{box_unitarity}\left\{
\begin{array}{cc}
c_0+\frac{t}{2}c_1=S^{(1)}_\text{box} & \\
c_0-\frac{t}{2}c_1=S^{(2)}_\text{box} & \\
\end{array}
\right. .$$ Generically, there is a unique solution for $(c_0,c_1)$. Here we see the importance of the box integrand basis . If there are fewer than $2$ terms in the basis (oversimplified), then the integrand cannot be fitted from unitarity. If there are more than $2$ terms in the basis (redundant), then the integrand will contain free parameters which mess up the amplitude computation for following steps.
### Triangle diagram
After the box integrand reduction is done, we proceed to the triangle cases. Note that there are more than one triangle diagrams, in a $4$-point scattering process, by pinching one internal line. Consider this one, $$\label{eq:16}
I=\int \frac{d^4l }{i \pi^2} \frac{N_\text{tri}}{D_1D_2D_3}\,,$$ where external lines $3$ and $4$ are combined.
![One-loop triangle diagram[]{data-label="box"}](graphs/tri.eps)
The kinematics is much simpler than that of the box case. Besides $\omega$, we introduce another imaginary auxiliary vector, $$\label{eq:17}
\tilde \omega=i\bigg(-\frac{s+t}{t} k_1+\frac{t}{s} k_2-k_4\bigg)\,.$$ Then, $$\begin{gathered}
\tilde \omega \cdot k_1=0,\quad \tilde \omega\cdot k_2=0,\quad
\omega\cdot \tilde \omega=0, \quad (\tilde \omega)^2=\omega^2=-\frac{t(s+t)}{s}.\end{gathered}$$ Note that the momentum $k_4$ does not appear in propagators of this triangle diagram, so we would better replace the variable $x_3=l\cdot k_4$ by a new variable $y_3\equiv l\cdot \tilde \omega$, $$\label{eq:20}
x_3=-\frac{s+t}{s} x_1+\frac{t}{s} x_2+i y_3 \,.$$
The integrand reduction for triangle reads $N_\text{tri}=\Delta_\text{tri}+h_1D_1+h_2D_2+h_3D_3$. Generically, $$\label{eq:21}
N_\text{tri} =\sum_{m_1} \sum_{m_2} \sum_{m_3} \sum_{m_4} d_{m_1m_2m_3m_4}x_1^{m_1} x_2^{m_2} y_3^{m_3} x_4^{m_4}\,,$$ with the renormalization constraint that $m_1+m_2+m_3+m_4 \leq
3$ [@Ossola:2006us]. Here we already replaced $x_3$. Again, $$\begin{aligned}
x_1 &= l \cdot p_1 =\half (D_1-D_2)\,, \nn \\
x_2 &= l \cdot p_2 =\half (D_2-D_3)+\frac{s}{2}\,.
\label{RSP_tri}\end{aligned}$$ we have $2$ RSPs, $x_1$, $x_2$ and $2$ ISPs, $y_3$, $x_4$. Again, from $D_1=l^2$, we have $$\label{eq:22}
y_3^2+x_4^2=\mathcal O (D_i) \,,$$ which means we can trade $y_3^2$ for $x_4^2$. Hence with the renormalization condition, $$\Delta_\text{tri}=d_0' +d_1' y_3+ d_2' x_4 +d_3' y_3 x_4 + d_4'
x_4^2+d_5' y_3 x_4^2+d_6' x_4^3 \,.$$ which contains $7$ terms. By Lorentz symmetry, $$\int d^Dl\frac{y_3^m x_4^n}{D_1D_2D_3}=0 \,,$$ as long as $m$ is odd or $n$ is odd. It seems that $x_4^2$ term survives the integration. To further simplify the integral, we redefine the integrand basis, $$\Delta_\text{tri}=d_0+d_1 y_3+ d_2 x_4 +d_3 y_3 x_4 + d_4
(x_4^2-y_3^2)+d_5 y_3 x_4^2+d_6 x_4^3 \,.
\label{integrand_basis_tri}$$ By the symmetry between $\tilde \omega$ and $\omega$, the term proportional to $d_4$ integrates to zero. Hence, the integrand basis of triangle contains $1$ scalar integral and $6$ spurious terms.[^5]
To use this basis, again, there are two manners as in the previous section.
1. (IR-D). Suppose that the box integrand reduction is finished and the triangle diagram integrand is obtained, say from Feynman rules. We combine the triangle integrand and the term proportional to $D_4$ in , and carry out the reduction process in this section explicitly. Finally, we get coefficients $d_0,\ldots,d_6$.
2. (IR-U). The goal is to determine $d_0,\ldots,d_6$ from unitarity. We need the triple cut, $$\label{eq:10}
D_0=D_1=D_3=0 \,,$$
There are two branches of solutions, $$\begin{aligned}
\label{tri_cut_solution}
\text{(1)}\quad x_1&=0,\quad x_2=\frac{s}{2},\quad y_3=i z,\quad x_4=z \,,\\
\text{(2)}\quad x_1&=0,\quad x_2=\frac{s}{2},\quad y_3=-i z,\quad x_4=z \,,\end{aligned}$$ where for each branch $z$ is a free parameter. On this cut, the numerator becomes a sum of products of tree amplitudes, $$\begin{gathered}
\label{eq:14}
S^{(i)}_\text{tri}(z)=\sum_{h_1} \sum_{h_2} \sum_{h_3} A(k_1,
l^{(i)}-k_1, \c -l^{(i)};s_1, h_2, \c -h_1)(z)\times \nn\\
A(k_2, l^{(i)}\c -k_1\c -k_2,k_1-l^{(i)};s_2, h_3, -h_2)(z) A(k_3,k_4,
l^{(i)},k_1+k_2\c -l^{(i)};s_3,s_4, h_1, \c -h_3)(z)\nn \,.\\\end{gathered}$$ for $i=1,2$. We try to fit coefficients in $\Delta_\text{tri}$ with $S^{(i)}_\text{tri}(z)$. However, the new issue is that $\Delta_\text{tri}$ on either branch, is a polynomial of $z$. $S^{(i)}_\text{tri}(z)$ in general is not a polynomial of $z$, since the last tree amplitude may have a pole when $(l+p_4)^2=0$. On the cut, $$\label{eq:1}
\frac{1}{(l+p_4)^2}=\frac{1}{t+2 i y_3}\,,$$ which becomes a fraction in $z$ for each branch. Note that this pole is from quadruple cut, hence we have to subtract the box integrand basis to avoid the double counting. The correct unitarity relation is, $$\Delta_\text{tri}\big(l^{(i)}(z)\big)=S^{(i)}_\text{tri}(z)
-\frac{c_0+c_1 \big(l^{(i)}(z)\cdot
\omega\big)}{\big(l^{(i)}(z)+p_4\big)^2}\,, \quad i=1,2 \,.
\label{OPP_subtraction}$$ If $c_0$ and $c_1$ are known from box integrand reduction, then both sides of the equation are polynomials in $z$ and Tylor expansions determine coefficients $d_0,\ldots,d_6$. [^6]
The further reduction for bubbles is similar.
### D-dimensional one-loop integrand reduction
Dimensional regularization is a standard way for QFT renormalization. Here we briefly introduce OPP integrand reduction [@Ossola:2007ax; @Giele:2008ve; @Ellis:2011cr] in D-dimension for one-loop diagrams.
Again, consider the four-point massless box integral in $D=4-2\epsilon$, $$\label{box_D}
I_\text{box}^D[N]=\int \frac{d^D l}{i \pi^{D/2}} \frac{N^D_\text{box}}{D_1
D_2 D_3 D_4}\,,$$ with the same definition of $D_i$’s. The loop momentum $l$ contains two parts $l=l^{[4]}+l^\perp$, where $l^{[4]}$ is the four-dimensional part and $l^\perp$ is the component in the extra dimension. $$\label{eq:24}
l^2=(l^{[4]})^2+(l^\perp)^2=(l^{[4]})^2 -\mu_{11}\,.$$ Here we introduce a variable $\mu_{11}=-(l^\perp)^2$. We use the scheme such that all external particles are in $4D$, hence, $$\label{eq:25}
(l^\perp) \cdot k_i=0,\quad i=1,\ldots, 4$$ and similar orthogonal conditions hold between $l^\perp$ and external polarization vectors hold. This implies $l^\perp$ appears in the integrand only in the form of $\mu_{11}$. $l^{[4]}$ is parameterized by the same van Neerven-Vermaseren variables $x_1,\ldots
x_4$, as before. Therefore, $$\label{eq:26}
N^D_\text{box}=\sum_{m_1} \sum_{m_2} \sum_{m_3} \sum_{m_4} \sum_m
c_{m_1 m_2 m_3 m_4 m}x_1^{m_1} x_2^{m_2} x_3^{m_3} x_4^{m_4} \mu_{11}^m\,,$$ with the renormalization condition $m_1+m_2+m_3+m_4+2m\leq
4$. ($\mu_{11}$ contains $2$ powers of $l$.) Again, as in the 4D case, $$\begin{aligned}
\label{box_D_RSP}
x_1 = \half (D_1-D_2),\quad x_2 =\half (D_2-D_3)+\frac{s}{2},\quad
x_3 = \half (D_4-D_1),\end{aligned}$$ so $x_1$, $x_2$ and $x_3$ are RSPs which do not appear in the integrand basis. The ISPs are $x_4$ and $\mu_{11}$. From the relation $D_1=(l^{[4]})^2 -\mu_{11}$, we get, $$\label{eq:27}
x_4^2=\frac{t^2}{4}-\frac{(s+t)t}{s} \mu_{11} +\mathcal O(D_i)\,,$$ Hence we can trade $x_4^2$ for $\mu_{11}$ in the integrand basis, $$\Delta_\text{box}^D=c_0+c_1 x_4+ c_2 \mu_{11}+ c_3 \mu_{11} x_4+c_4
\mu_{11}^2\,,
\label{box_integrand_basis_D}$$ which contains $5$ terms. The terms proportional to $x_4$ are again spurious, i.e., integrated to zero.
The coefficients $c_0,\ldots c_4$ can either be calculated from explicit reduction (IR-D) or unitarity (IR-U). For the latter, the quadruple cut $D_1=D_2=D_3=D_4=0$ is applied. There is one family of solutions which is one-dimensional, $$\label{eq:29}
x_1=0,\quad x_2=\frac{s}{2},\quad x_3=0,\quad x_4=z, \quad \mu_{11}=\frac{s(t^2-4z^2)}{4t(s+t)}.$$ Amazingly, the $4D$ quadruple cut contains two zero-dimensional solutions while $D$-dim quadruple cut has only one family of solution. The two roots in $4D$ are connected by a cut-solution curve, in DimReg. The Taylor series in $z$ fits coefficients $c_0,\ldots c_4$.
If only $\epsilon\to 0$ limit of the amplitudes is needed, can be further simplified by [*dimension shift*]{} identities, $$\begin{aligned}
\label{box_dimension_shift}
\int\frac{d^D}{i\pi^{D/2}}
\frac{\mu_{11}}{D_1D_2D_3D_4}&=\frac{D-4}{2} I_\text{box}^{D+2}[1]\\
\int\frac{d^D}{i\pi^{D/2}}
\frac{\mu_{11}^2}{D_1D_2D_3D_4}&=\frac{(D-4)(D-2)}{4} I_\text{box}^{D+4}[1]\end{aligned}$$ These identities can be proven via Baikov parameterization (Chapter \[cha:integr-parts-reduct\]) . It is well known that the $6D$ scalar box integral is finite and the $8D$ scalar box is UV divergent such that, $$\begin{aligned}
\label{eq:31}
\lim_{D\to 4}\frac{D-4}{2} I_\text{box}^{D+2}[1] &=0,\\
\lim_{D\to 4}\frac{(D-4)(D-2)}{4} I_\text{box}^{D+4}[1] &=-\frac{1}{3}.\end{aligned}$$ Hence the integrand basis after integration becomes, $$\begin{aligned}
\label{eq:32}
\lim_{D\to 4}\int\frac{d^Dl}{i\pi^{D/2}}
\frac{\Delta^D_\text{box}}{D_1D_2D_3D_4} &=& c_0
I_\text{box}^{D}[1]-\frac{1}{3} c_4\end{aligned}$$ in the $\epsilon\to 0$ limit. The second term is called a [*rational term*]{}, which cannot be obtained from the $4D$ quadruple cut.
It seems that $D$-dimensional integrand reduction is more complicated than the $4D$ case, with more variables and more integrals in the basis. However, it provides the complete amplitude for a general renormalizable QFT, and mathematically, its cut solution has simpler structure.
OPP method is programmable and highly efficient for automatic one-loop amplitude computation [@Ossola:2007ax; @Badger:2010nx; @Cullen:2011xs; @Hirschi:2011pa].
Issues at higher loop orders
----------------------------
Since OPP method is very convenient for one-loop cases, the natural question is: is it possible to generalize OPP method for higher loop orders?
Of course, higher loop diagrams contain more loop momenta and usually more propagators. Is it a straightforward generalization? The answer is “no”. For example, consider the $4D$ $4$-point massless double box diagram (see Fig. \[graph\_dbox\]),
![two-loop double box diagram[]{data-label="graph_dbox"}](graphs/dbox.eps)
associated with the integral, $$\begin{aligned}
\label{eq:33}
I_\text{dbox}[N]=\int \frac{d^4 l_1}{i \pi^2}\frac{d^4 l_2}{i \pi^2}\frac{N}{D_1 D_2 D_3 D_4 D_5 D_6 D_7}.\end{aligned}$$ The denominators of propagators are, $$\begin{gathered}
\label{dbox_propagators}
D_1=l_1^2,\quad D_2=(l_1-k_1)^2,\quad D_3=(l_1-k_1-k_2)^2,\quad D_4=(l_2+k_1+k_2)^2, \nn\\
D_5=(l_2-k_4)^2,\quad
D_6=l_2^2,\quad D_7=(l_1+l_2)^2\,.\end{gathered}$$ The goal of reduction is to express, $$\begin{gathered}
\label{dbox_IR}
N_\text{dbox}=\Delta_\text{dbox} +h_1 D_1 +\ldots +h_7 D_7\end{gathered}$$ such that $\Delta_\text{dbox}$ is the “simplest”. (In the sense that all its coefficients in $\Delta_\text{dbox}$ can be uniquely fixed from unitarity, as in the box case.)
We use van Neerven-Vermaseren basis as before, $\{e_1,e_2,e_3,e_4\}=\{k_1,k_2,k_4,\omega\}$. Define $$\label{dbox_vNV}
x_i=l_1\cdot e_i, \quad y_i=l_2\cdot e_i,\quad i=1,\ldots 4.$$ Then we try to determine $\Delta_\text{dbox}$ in these variables like one-loop OPP method. $$\begin{aligned}
x_1 &= \half (D_1-D_2)\,, \nn \\
x_2 &=\half (D_2-D_3)+\frac{s}{2}\,, \nn \\
y_2 &= \half (D_4-D_6)-y_1-\frac{s}{2}\,, \nn \\
y_3 &= \half (D_6-D_5) \,,
\label{dbox_RSP}\end{aligned}$$ Hence we can remove RSPs: $x_1$, $x_2$, $y_2$ and $y_3$ in $\Delta_\text{dbox}$. (We trade $y_2$ for $y_1$, by symmetry consideration: under the left-right flip symmetry of double box, $x_3 \leftrightarrow y_1$. ) There are $4$ ISPs, $x_3$, $y_1$, $x_4$ and $y_4$.
Then following the one-loop OPP approach, the quadratic terms in $(l_i
\cdot \omega)$ can be removed from the integrand basis, since, $$\begin{aligned}
x_4^2&=x_3^2-t x_3+\frac{t^2}{4} +\mathcal O(D_i)\,,\nn\\
y_4^2&=y_1^2-t y_1+\frac{t^2}{4}+\mathcal O(D_i) \,,\nn\\
x_4 y_4 &=\frac{s+2t}{s} x_3
y_1+\frac{t}{2}x_3+\frac{t}{2}y_1-\frac{t^2}{4}+\mathcal
O(D_i) \,.
\label{dbox_quadratic}\end{aligned}$$ Then the trial version of integrand basis has the form, $$\begin{aligned}
\label{eq:28}
\Delta_\text{dbox}=\sum_m \sum_n \sum_\alpha \sum_\beta
c_{m,n,\alpha,\beta} x_3^{m} y_1^{n} x_4^{\alpha} y_4^{\beta} \,,\end{aligned}$$ where $(\alpha,\beta)\in\{(0,0),(1,0),(0,1)\}$. The renormalization condition is, $$\begin{aligned}
\label{eq:34}
m+\alpha\leq 4,\quad n+\beta\leq 4,\quad m+n+\alpha+\beta\leq 6\, . \end{aligned}$$ By counting, there are $56$ terms in the basis. Is this basis correct?
Have a look at the unitarity solution. The heptacut $D_1=\ldots D_7=0$ has a complicated solution structure [@Kosower:2011ty]. (See table. \[dbox\_sol\]).
$x_1$ $x_2$ $x_3$ $x_4$ $y_1$ $y_2$ $y_3$ $y_4$
----- ------- --------------- ------------------- ---------------------- --------------------------- ------------------------ ------- ------------------------------
(1) $0$ $\frac{s}{2}$ $z_1$ $z_1-\frac{t}{2} $ $0$ $-\frac{s}{2}$ 0 $\frac{t}{2}$
(2) $0$ $\frac{s}{2}$ $z_2$ $-z_2+\frac{t}{2} $ $0$ $-\frac{s}{2}$ 0 $-\frac{t}{2}$
(3) $0$ $\frac{s}{2}$ $0$ $\frac{t}{2}$ $z_3$ $-z_3-\frac{s}{2}$ $0$ $z_3-\frac{t}{2}$
(4) $0$ $\frac{s}{2}$ $0$ -$\frac{t}{2}$ $z_4$ $-z_4-\frac{s}{2}$ $0$ $-z_4+\frac{t}{2}$
(5) $0$ $\frac{s}{2}$ $\frac{z_5-s}{2}$ $\frac{z_5-s-t}{2}$ $\frac{s(s+t-z_5)}{2z_5}$ $-\frac{s(s+t)}{2z_5}$ $0$ $\frac{(s+t)(s-z_5)}{2z_5}$
(6) $0$ $\frac{s}{2}$ $\frac{z_6-s}{2}$ $\frac{-z_6+s+t}{2}$ $\frac{s(s+t-z_6)}{2z_6}$ $-\frac{s(s+t)}{2z_6}$ $0$ $-\frac{(s+t)(s-z_6)}{2z_6}$
: solutions of the $4D$ double box heptacut.[]{data-label="dbox_sol"}
There are $6$ branches of solutions, each of which is parameterized by a free parameter $z_i$. Solutions (5) and (6) contain poles in $z_i$, hence we need Laurent series for tree products, $$\label{eq:35}
S^{(i)}=\sum_{k=-4}^4 d_k^{(i)} z_i^k,\quad i=5,6\,.$$ The bounds are from renormalization conditions, so there are $9$ nonzero coefficients for each case. Solutions (1), (2), (3), (4) are relatively simpler, $$\label{eq:35}
S^{(i)}=\sum_{k=0}^4 d_k^{(i)} z_i^k,\quad i=1,2,3,4\,.$$ So there are $5$ nonzero coefficients for each case. These solutions are not completely indenpendent, for example, solution (1) at $z_1=s$ and solution (6) at $z_6=t/2$ correspond to the same loop momenta. Therefore, $$\label{eq:36}
S^{(1)}(z_1\to s)=S^{(6)}(z_6 \to t/2)\,.$$ There are $6$ such intersections, namely between solutions (1) and (6), (1) and (4), (2) and (3), (2) and (5), (3) and (6), (4) and (5). Hence, there are $9\times 2+5\times 4 -6=32$ independent $d_k^{(i)}$’s.
Now the big problem emerges, $$\label{eq:38}
56>32\,.$$ There are more terms in the integrand basis than those determined from unitarity cut. That means this integrand basis is redundant. However, it seems that we already used all algebraic constraints in and . Which constraint is missing?
We need to reconsider , especially the meaning of “simplest” integrand basis. For simple example like massless double box diagram, it is possible to use the detailed structures like symmetries and Gram determinant constraints, to get a proper integrand basis [@Mastrolia:2011pr; @Badger:2012dp]. However, in general, we need an automatic reduction method, without looking at the details. So we refer to a new mathematical approach, [*computational algebraic geometry*]{}.
Elementary computational algebraic geometry methods
---------------------------------------------------
### Basic facts of algebraic geometry in affine space I
In order to apply the new method, we need to list some basic concepts and facts on algebraic geometry [@MR0463157].
We start from a polynomial ring $R=\F[z_1,\ldots z_n]$ which is the collection of all polynomials in $n$ variables $z_1,\ldots z_n$ with coefficients in the [*field*]{} $\F$. For example, $\F$ can be $\Q$, the rational numbers, $\C$, the complex numbers, $\Z/p\Z$, the [*finite field*]{} of integers modulo a prime number $p$, or $\C(c_1,c_2,\ldots
c_k)$, the complex rational functions of parameters $c_1,\ldots ,c_k$.
Recall that the right hand side of contains the sum $h_1 D_1 +\ldots + h_7 D_7$ where $D_i$’s are known polynomials and $h_i$’s are arbitrary polynomials. What are general properties of such a sum? That leads to the concept of [*ideal*]{}.
An ideal $I$ in the polynomial ring $R=\F[z_1,\ldots z_n]$ is a subset of $R$ such that,
- $0\in I$. For any two $f_1,f_2\in I$, $f_1+f_2 \in I$. For any $f\in I$, $-f\in I$.
- For $\forall f \in I$ and $\forall h \in R$, $h f\in I$.
The ideal in the polynomial ring $R=\F[z_1,\ldots z_n]$ generated by a subset $S$ of $R$ is the collection of all such polynomials, $$\sum_i h_i f_i, \quad h_i\in R, \quad f_i\in S.
\label{Ideal_generator}$$ This ideal is denoted as $\langle S \rangle$. In particular, $\langle
1 \rangle=R$, which is an ideal which contains all polynomials. Note that even if $S$ is an infinite set, the sum in is always restricted to a sum of a finite number of terms. $S$ is called the generating set of this ideal.
\[example\_ideal\] Let $I=\langle x^2+y^2+z^2-1, z\rangle$ in $\Q[x,y,z]$. By definition, $$I=\{h_1 (x^2+y^2+z^2-1)+h_2\cdot z,\ \forall h_1,h_2\in R\}\,,$$ Pick up $h_1=1$, $h_2=-z$, and we see $x^2+y^2-1\in I$. Furthermore, $$x^2+y^2+z^2-1 = (x^2+y^2-1)+z\cdot z\,.$$ Hence $I=\langle x^2+y^2-1, z\rangle$. We see that, in general, the generating set of an ideal is not unique.
Our integrand reduction problem can be rephrased as: given $N$ and the ideal $I=\langle D_1 ,\ldots, D_7 \rangle$, how many terms in $N$ are in $I$? To answer this, we need to study properties of ideals.
\[thm\_Noether\] The generating set of an ideal $I$ of $R=\F[z_1,\ldots z_n]$ can always be chosen to be finite.
See Zariski, Samuel [@MR0384768].
This theorem implies that we only need to consider ideals generated by finite sets in the polynomial ring $R$.
\[quotient\_ring\] Let $I$ be an ideal of $R$, we define an equivalence relation, $$\label{eq:59}
f\sim g,\quad \text{if and only if } f-g\in I\,.$$ We define an equivalence class, $[f]$ as the set of all $g\in R$ such that $g\sim f$. The [*quotient ring*]{} $R/I$ is set of equivalence classes, $$R/I=\{[f]| f\in R\}\,.$$ with multiplication $[f_1][f_2]\equiv [f_1f_2]$. (Check this multiplication is well-defined.)
To study the structure of an ideal, it is very useful to consider the algebra-geometry relation.
Let $\mathbb K$ be a field, $\F\subset \K$. The $n$-dimensional $\K$-affine space $\mathbf A^n_\K$ is the set of all $n$-tuple of $\K$. Given a subset $S$ of the polynomial ring $\F[z_1,\ldots,z_n]$, its [*algebraic set*]{} over $\K$ is, $$\label{eq:8}
\mathcal Z_\K(S)=\{p\in \mathbf A^n_\K | f(p)=0,\ \text{for every } f \in S\}.$$ If $\K=\F$, we drop the subscript $\K$ in $\mathbf
A^n_\K$ and $\mathcal Z_\K(S)$.
So the algebraic set $\mathcal Z(S)$ consists of all [*common solutions*]{} of polynomials in $S$. Note that to solve polynomials in $S$ is equivalent to solve all polynomials simultaneously in the ideal generated by $S$, $$\label{eq:18}
\mathcal Z(S)=\mathcal Z(\langle S \rangle ),$$ since if $p\in \mathcal Z(S)$, then $f(p)=0$, $\forall f\in S$. Hence, $$\label{eq:37}
h_1(p) f_1(p) + \ldots + h_k(p) f_k(p)=0,\quad \forall h_i \in R,\ \forall f_i \in S.$$ So we always consider the algebraic set of an ideal.
For example, $\mathcal Z(\langle 1 \rangle )=\emptyset$ (empty set) since $1\not =0$. For the ideal $I=\langle x^2+y^2+z^2-1, z\rangle$ in example \[example\_ideal\], $\mathcal Z(I)$ is the unit circle on the plane $z=0$.
We want to learn the structure of an ideal from its algebraic set. First, for the empty algebraic set,
\[weak\_Nullstellensatz\] Let $I$ be an ideal of $\Fpoly$ and $\K$ be an algebraically closed field [^7] , $\F\subset \K$. If $\mathcal Z_\K(I)=\emptyset$, then $I=\langle 1 \rangle$.
See Zariski and Samuel, [@MR0389876 Chapter 7].
The field extension $\K$ must be algebraically closed. Otherwise, say, $\K=\F=\Q$, the ideal $\la x^2-2 \ra$ has empty algebraic set in $\Q$. (The solutions are not rational). However, $\la x^2-2\ra\not=\la
1\ra$. On the other hand, $\F$ need not be algebraically closed. $I=\langle 1 \rangle$ means, $$1=h_1 f_1 +\ldots + h_k f_k,\quad f_i \in I, \ h_i\in \Fpoly\,.$$ where $h_i$’s coefficients are in $\F$, instead of an algebraic extension of $\F$.
We prove that, generally, the $4D$ pentagon diagrams are reduced to diagrams with fewer than $5$ propagators, $D$-dimensional hexagon diagram are reduced to diagrams with fewer than $6$ propagators, in the integrand level.
For the $4D$ pentagon case, there are $5$ denominators from propagators, namely $D_1,\ldots D_5$. There are $4$ Van Neerven-Vermaseren variables for the loop momenta, namely $x_1$, $x_2$, $x_3$ and $x_4$. So $D_i$’s are polynomials in $x_1,\ldots,x_4$ with coefficients in $\F=\Q(s_{12},s_{23},s_{34},s_{45},s_{15})$. Define $I=\la D_1,\ldots
D_5,\ra$. Generally $5$ equations in $4$ variables, $$\label{eq:40}
D_1=D_2=D_3=D_4=D_5=0\,,$$ have no solution (even with algebraic extensions). Hence by Hilbert’s weak Nullstellensatz, $I=\la 1\ra$. Explicitly, there exist $5$ polynomials $f_i$’s in $\F[x_1,x_2,x_3,x_4]$ such that $$\label{eq:41}
f_1D_1+f_2 D_2 +f_3 D_3 +f_4 D_4 +f_5 D_5=1\,.$$ Therefore, $$\begin{gathered}
\int d^4l \frac{1}{D_1 D_2 D_3 D_4 D_5}=\int d^4l \frac{f_1}{D_2 D_3
D_4 D_5} +\int d^4l\frac{f_2}{D_1 D_3 D_4 D_5}+\int d^4l\frac{f_3}{D_1 D_2 D_4 D_5}\nn\\
\int d^4l\frac{f_4} {D_1 D_2 D_3 D_5}+\int d^4l\frac{f_5} {D_1 D_2 D_3 D_4}\,,\end{gathered}$$ where each term in the r.h.s is a box integral (or simpler). Note that $f_i$’s are in $\F[x_1,x_2,x_3,x_4]$, so the coefficients of these polynomials are rational functions of Mandelstam variables $s_{12},s_{23},s_{34},s_{45},s_{15}$. Weak Nullstellensatz theorem does not provide an algorithm for finding such $f_i$’s. The algorithm will be given by the Gröbner basis method in next subsection, or by the resultant method [@opac-b1094391].
Notice that in the DimReg case, we have one more variable $\mu_{11}=-(l^\perp)^2$. The same argument using Weak Nullstellensatz leads to the result.
For a general algebraic set, we have the important theorem:
Let $\F$ be an algebraically closed field and $R=\F[z_1,\ldots
z_n]$. Let $I$ be an ideal of $R$. If $f\in R$ and, $$\label{eq:44}
f(p)=0,\quad \forall p\in \mathcal Z(I),$$ then there exists a positive integer $k$ such that $f^k\in I$.
See Zariski and Samuel, [@MR0389876 Chapter 7].
Hilbert’s Nullstellensatz characterizes all polynomials vanishing on $\mathcal Z(I)$, they are “not far away” from elements in $I$. For example, $I=\la
(x-1)^2 \ra$ and $\mathcal Z(I)=\{1\}$. The polynomial $f(x)=(x-1)$ does not belong to $I$ but $f^2\in I$.
Let $I$ be an ideal in $R$, define the [*radical ideal*]{} of $I$ as, $$\label{eq:45}
\sqrt I=\{f\in R| \exists k\in\Z^+, f^k\in I\}\,.$$ For any subset $V$ of $\mathbf A^n$, define the ideal of $V$ as $$\label{eq:46}
\mathcal I(V)=\{f\in R| f(p)=0, \ \forall p\in V\}\,.$$ Then Hilbert’s Nullstellensatz reads, over an algebraically closed field, $$\label{eq:47}
\mathcal I (\mathcal Z(I)) = \sqrt I\,.$$ An ideal $I$ is called [*radical*]{}, if $\sqrt I = I$.
If two ideals $I_1$ and $I_2$ have the same algebraic set $\mathcal
Z(I_1)=\mathcal Z(I_2)$, then they have the same radical ideals $\sqrt
I_1=\sqrt I_2$. On the other hand, if two sets in $\mathbb A^n$ have the same ideal, what could we say about them? To answer this question, we need to define topology of $\mathbb A^n$:
Define Zariski topology of $\mathbf A^n_\F$ by setting all algebraic set to be topologically closed. (Here $\F$ need not be algebraic closed.)
The intersection of any number of Zariski closed sets is closed since, $$\bigcap_i \mathcal Z(I_i) =\mathcal Z(\bigcup_i I_i ).
\label{algebraic_set_intersection}$$ The union of two closed sets is closed since, $$\mathcal Z(I_1)\bigcup \mathcal Z(I_2) =\mathcal Z( I_1 I_2
)=\mathcal Z( I_1 \cap I_2 ).
\label{algebraic_set_union}$$ $\mathbf A^n_\F$ and $\emptyset$ are both closed because $\mathbf
A^n_\F=\mathcal Z(\{0\})$, $\emptyset=\mathcal Z(\la 1\ra)$. That means Zariski topology is well-defined. We leave the proof of and as an exercise.
Note that Zariski topology is different from the usual topology defined by Euclidean distance, for $\F=\Q,\mathbb R,\C$. For example, over $\C$, the “open” unit disc defined by $D=\{z||z|<1\}$ is not Zariski open in $\mathbf A^1_\C$. The reason is that $\C-D=\{z||z|\geq 1\}$ is not Zariski closed, i.e. $\C-D$ cannot be the solution set of one or several complex polynomials in $z$.
Zariski topology is the foundation of affine algebraic geometry. With this topology, the dictionary between algebra and geometry can be established.
(Here $\F$ need not be algebraic closed.)
1. If $I_1 \subset I_2$ are ideals of $\Fpoly$, $\mathcal
Z(I_1) \supset \mathcal Z(I_2)$
2. If $V_1 \subset V_2$ are subsets of $\mathbf A^n_\F$, $\mathcal
I(V_1) \supset \mathcal I(V_2)$
3. For any subset $V$ in $\mathbf A^n_\F$, $\mathcal Z (\mathcal
I(V))=\overline V$, the Zariksi closure of $V$.
The first two statements follow directly from the definitions. For the third one, $V\subset \mathcal Z (\mathcal
I(V))$. Since the latter is Zariski closed, $\overline V \subset \mathcal Z (\mathcal
I(V))$. On the other hand, for any Zariski closed set $X$ containing $V$, $X=\mathcal Z(I)$. $I\subset \mathcal I(V)$. From statement 1, $X=\mathcal Z(I) \supset \mathcal Z (\mathcal
I(V))$. As a closed set, $\mathcal Z (\mathcal
I(V))$ is contained in any closed set which contains $V$, hence $\mathcal Z (\mathcal
I(V))=\overline V$.
In the case $\F$ is algebraic closed, the above proposition and Hilbert’s Nullstellensatz established the one-to-one correspondence between radical ideals in $\Fpoly$ and closed sets in $\mathbf
A^n_\F$. We will study geometric properties like reducibility, dimension, singularity later in these lecture notes. Before this, we turn to the computational aspect of affine algebraic geometry, to see how to explicitly compute objects like $I_1\cap I_2$ and $\mathcal Z(I)$.
### Gröbner basis
#### One-variable case
We see that ideal is the central concept for the algebraic side of classical algebraic geometry. An ideal can be generated by different generating sets, some may be redundant or complicated. In linear algebra, given a linear subspace $V=\sp\{v_1\ldots v_k\}$ we may use Gaussian elimination to find the linearly-independent basis of $V$ or Gram-Schmidt process to find an orthonormal basis. For ideals, a “good basis” can also dramatically simplify algebraic geometry problems.
\[GB\_one\_variable\] As a toy model, consider some univariate cases.
- For example, $I=\la x^3-x-1 \ra$ in $R=\Q[x]$. Clearly, $I$ consists of all polynomials in $x$ proportional to $x^3-x-1$, and every nonzero element in $I$ has the degree higher or equal than $3$. So we say $B(I)=\{x^3-x-1\}$ is a “good basis” for $I$. $B(I)$ is useful: for any polynomial $F(x)$ in $\Q[x]$, polynomial division determines, $$F(x) =q(x) (x^3-x-1)+r(x) , \quad q(x),r(x)\in \Q[x],\ \deg r(x)<3$$ Hence $F(x)$ is in $I$ if and only if the remainder $r$ is zero. It also implies that $R/I=\sp_\Q\{[1],[x],[x^2]\}$.
- Consider $J=\la x^3-x^2+3x-3,x^2-3x+2\ra$. Is the naive choice $B(J)=\{f_1,f_2\}=\{x^3-x^2+3x-3,x^2-3x+2\}$ a good basis? For instance, $f=f_1-x f_2=2x^2+x-3$ is in $I$ but it is proportional to neither $f_1$ nor $f_2$. Polynomial division over this basis is not useful, since $f$’s degree is lower than $f_1$, the only division reads, $$\label{eq:48}
f=2 f_2 + (7 x-7) \,.$$ The remainder does not tell us the membership of $f$ in $I$. Hence $B(J)$ does not characterize $I$ or $R/I$, and it is not “good”. Note that $\Q[x]$ is a principal ideal domain (PID), any ideal can be generated by one polynomial. Therefore, use Euclidean algorithm (Algorithm \[Euclid\]) to find the greatest common factor of $f_1$ and $f_2$, $$(x-1)=\frac{1}{7}f_1(x)-\frac{x+2}{7}f_2(x),\quad (x-1)|f_1(x),\ (x-1)|f_2(x)$$ Hence $J=\la x-1 \ra$. We can check that $\tilde B(J)=\{x-1\}$ is a “good” basis in the sense that Euclidean division over $\tilde B(J)$ solves membership questions of $J$ and determined $R/J=\sp_\Q\{[1]\}$.
[**Input:**]{} $f_1, f_2$, $\deg f_1\geq \deg f_2$ polynomial division $f_1=q f_2+r$ $f_1:=f_2$ $ f_2:=r$ $f_2$ (gcd)
Recall that in , given inverse propagators $D_1, \ldots, D_7$, we need to solve the membership problem of $I=\la D_1\ldots D_7\ra$ and compute $R/I$. However, in general, a set like $\{ D_1
\ldots D_7\}$ is not a “good basis”, in the sense that the polynomial division over this basis does not solve the membership problem or give a correct integrand basis (as we see previously). Since it is a multivariate problem, the polynomial ring $R$ is not a PID and we cannot use Euclidean algorithm to find a “good basis”.
Look at Example \[GB\_one\_variable\] again. For the univariate case, there is a natural monomial order $\prec$ from the degree, $$\label{eq:49}
1 \prec x \prec x^2 \prec x^3 \prec x^4 \prec \ldots\,,$$ and all monomials are sorted. For any polynomial $F$, define the [*leading term*]{}, $\LT(F)$ to be the highest monomial in $F$ by this order (with the coefficient). For multivariate cases, the degree criterion is not fine enough to sort all monomials, so we need more general monomial orders.
Let $M$ be the set of all monomials with coefficients $1$, in the ring $R=\Fpoly$. A monomial order $\prec$ of $R$ is an ordering on $M$ such that,
1. $\prec$ is a total ordering, which means any two different monomials are sorted by $\prec$.
2. $\prec$ respects monomial products, i.e., if $u\prec v$ then for any $w\in M$, $uw\prec vw$.
3. $1\prec u$, if $u\in M$ and $u$ is not constant.
There are several important monomial orders. For the ring $\Fpoly$, we use the convention $1\prec z_n\prec z_{n-1}\prec\ldots \prec z_1$ for all monomial orders. Given two monomials, $g_1=z_1^{\alpha_1}\ldots z_n^{\alpha_n}$ and $g_2=z_1^{\beta_1}\ldots z_n^{\beta_n}$, consider the following orders:
- Lexicographic order (). First compare $\alpha_1$ and $\beta_1$. If $\alpha_1<\beta_1$, then $g_1\prec g_2$. If $\alpha_1=\alpha_2$, we compare $\alpha_2$ and $\beta_2$. Repeat this process until for certain $\alpha_i$ and $\beta_i$ the tie is broken.
- Degree lexicographic order (). First compare the total degrees. If $\sum_{i=1}^n\alpha_i<\sum_{i=1}^n\beta_i$, then $g_1\prec g_2$. If total degrees are equal, we compare $(\alpha_1, \beta_1)$, $(\alpha_2, \beta_2)$ ... until the tie is broken, like .
- Degree reversed lexicographic order (). First compare the total degrees. If $\sum_{i=1}^n\alpha_i<\sum_{i=1}^n\beta_i$, then $g_1\prec g_2$. If total degrees are equal, we compare $\alpha_n$ and $\beta_n$. If $\alpha_n<\beta_n$, then $g_1 \succ g_2$ (reversed!). If $\alpha_n=\beta_n$, then we further compare $(\alpha_{n-1}$, $\beta_{n-1})$, $(\alpha_{n-2}$, $\beta_{n-2})$ ... until the tie is broken, and use the reversed result.
- Block order. This is the combination of and other orders. We separate the variables into $k$ blocks, say, $$\label{eq:51}
\{z_1,z_2,\ldots z_n\}=\{z_1,\ldots z_{s_1}\} \cup
\{z_{s_1+1},\ldots z_{s_2}\} \ldots \cup \{z_{s_{k-1}+1},\ldots z_n\}\,.$$ Furthermore, define the monomial order for variables in each block. To compare $g_1$ and $g_2$, first we compare the first block by the given monomial order. If it is a tie, we compare the second block... until the tie is broken.
Consider $\Q[x,y,z]$, $z\prec y \prec x$. We sort all monomials up to degree $2$ in , , and the block order $[x]\succ
[y,z]$ with in each block. This can be done be the following code:
$\pmb{F=1+x+x^2+y+x y+y^2+z+x z+y z+z^2;}\\
\pmb{\text{MonomialList}[F,\{x,y,z\},\text{Lexicographic}]}\\
\pmb{\text{MonomialList}[F,\{x,y,z\},\text{DegreeLexicographic}]}\\
\pmb{\text{MonomialList}[F,\{x,y,z\},\text{DegreeReverseLexicographic}]}\\
\pmb{\text{MonomialList}[F,\{x,y,z\},\{\{1,0,0\},\{0,1,1\},\{0,0,-1\}\}]\text{ }}$
and the output is,
$\left\{x^2,x y,x z,x,y^2,y z,y,z^2,z,1\right\}$
$\left\{x^2,x y,x z,y^2,y z,z^2,x,y,z,1\right\}$
$\left\{x^2,x y,y^2,x z,y z,z^2,x,y,z,1\right\}$
$\left\{x^2,x y,x z,x,y^2,y z,z^2,y,z,1\right\}$
Note that for , $x\succ y^2$, $y\succ z^2$ since we first compare the power of $x$ and the $y$. The total degree is not respected in this order. On the other hand, and both consider the total degree first. The difference between and is that, $ x z\succ_\text{\grlex} y^2$ while $ x z\prec_\text{\grevlex}
y^2$. So $\grevlex$ tends to set monomials with more variables, lower, in the list of monomials with a fixed degree. This property is useful for computational algebraic geometry. Finally, for this block order, $x\succ y^2$ since $x$’s degrees are compared first. But $y\prec z^2$, since $[y,z]$ block is in .
With a monomial order, we define the leading term as the highest monomial (with coefficient) of a polynomial in this order. Back to the second part of Example \[GB\_one\_variable\], $$\label{eq:50}
\LT(f_1)=x^3\quad \LT(f_2)=x^2, \quad \LT(x-1)=x$$ The key observation is that although $x-1\in J$, its leading term is not divisible by the leading term of either $f_1$ or $f_2$. This makes polynomial division unusable and $\{f_1,f_2\}$ is not a “ good basis”. This leads to the concept of .
####
For an ideal $I$ in $\Fpoly$ with a monomial order, a $G(I)=\{g_1,\ldots g_m\}$ is a generating set for $I$ such that for each $f\in I$, there always exists $g_i\in G(I)$ such that, $$\label{eq:52}
\LT(g_i) | \LT(f) \,.$$
We can check that for the ideal $J$ in Example \[GB\_one\_variable\], $\{f_1,f_2\}$ is not a Gröbner basis with respect to the natural order, while $\{x-1\}$ is.
#### Multivariate polynomial division
To harness the power of we need the multivariate division algorithm, which is a generalization of univariate Euclidean algorithm (Algorithm \[multivariate\_polynomial\_division\]). The basic procedure is that: given a polynomial $F$ and a list of $k$ polynomials $f_i$’s, if $\LT(F)$ is divisible by some $\LT(f_i)$, then remove $\LT(F)$ by subtracting a multiplier of $f_i$. Otherwise move $\LT(F)$ to the remainder $r$. The output will be $$\label{eq:53}
F=q_1 f_1 + \ldots q_k f_k +r\,,$$ where $r$ consists of monomials cannot be divided by any $LT(f_i)$. Let $B=\{f_1,\ldots f_k\}$, we denote $\overline{F}^B$ as the remainder $r$.
[**Input:**]{} $F$, $f_1\ldots f_k$, $\succ$ $q_1:=\ldots :=q_k=0$, $r:=0$ $reductionstatus:=0$ $q_i:=q_i+\frac{\LT(F)}{\LT(f_i)}$ $F:=F-\frac{\LT(F)}{\LT(f_i)} f_i$ $reductionstatus:=1$ [**break**]{} $r:=r+\LT(F)$ $F:=F-\LT(F)$
$q_1\ldots q_k$, $r$
Recall that the one-loop OPP integrand reduction and the naive trial of two-loop integrand reduction are very similar to this algorithm.
Note that for a general list of polynomials, the algorithm has two drawbacks: (1) the remainder $r$ depends on the order of the list, $\{f_1,\ldots f_n\}$ (2) if $F\in \la f_1\ldots f_n\ra$, the algorithm may not give a zero remainder $r$. These made the previous two-loop integrand reduction unsuccessful. eliminates these problems.
\[GB\_division\] Let $G=\{g_1,\ldots g_m\}$ be a in $\Fpoly$ with the monomial order $\succ$. Let $r$ be the remainder of the division of $F$ by $G$, from Algorithm \[multivariate\_polynomial\_division\].
1. $r$ does not depend on the order of $g_1,\ldots g_m$.
2. If $F\in I=\la g_1,\ldots g_m\ra$, then $r=0$.
If the division with different orders of $g_1,\ldots g_n$ provides two remainder $r_1$ and $r_2$. If $r_1\not =r_2$, then $r_1-r_2$ contains monomials which are not divisible by any $\LT(g_i)$. But $r_1-r_2\in I$, this is a contradiction to the definition of .
If $F\in I$, then $r\in I$. Again by the definition of , if $r\not=0$, $\LT(r)$ is divisible by some $\LT(g_i)$. This is a contradiction to multivariate division algorithm.
Then the question is: given an ideal $I=\la f_1\ldots f_k\ra$ in $\Fpoly$ and a monomial order $\succ$, does the exist and how do we find it? This is answered by , which was presented in 1970s and marked the beginning of computational algebraic geometry.
####
Recall that for one-variable case, Euclidean algorithm (Algorithm \[Euclid\]) computes the gcd of two polynomials hence the is given. The key step is to cancel leading terms of two polynomials. That inspires the concept of S-polynomial in multivariate cases.
\[S-polynomial\] Given a monomial order $\succ$ in $R=\Fpoly$, the S-polynomial of two polynomials $f_i$ and $f_j$ in $R$ is, $$S(f_i,f_j)=\frac{\LT(f_j)}{\gcd\big(\LT(f_i),\LT(f_j)\big)} f_i
-\frac{\LT(f_i)}{\gcd\big(\LT(f_i),\LT(f_j)\big)} f_j.$$
Note that the leading terms of the two terms on the r.h.s cancel.
Given a monomial order $\succ$ in $R=\Fpoly$, with respect to $\succ$ exists and can be found by (Algorithm \[Buchberger\]).
See Cox, Little, O’Shea [@MR3330490].
[**Input:**]{} $B=\{f_1\ldots f_n\}$ and a monomial order $\succ$ $queue:=\text{all subsets of B with exactly two elements}$
$\{f,g\}:=\text{head of } queue$ $r:=\overline{S(f,g)}^B$ $B:=B\cup{r}$ queue $<<$ $\{\{B_1,r\},\ldots \{{\text{last\ of}\ }B,r\}\}$ head of $queue$
$B$ ()
The uniqueness of is given via [*reduced* ]{}.
For $R=\Fpoly$ with a monomial order $\succ$, a reduced is a $G=\{g_1,\ldots g_k\}$ with respect to $\succ$, such that
1. Every $\LT(g_i)$ has the coefficient $1$, $i=1,\ldots,k$.
2. Every monomial in $g_i$ is not divisible by $\LT(g_j)$, if $j\not =i$.
For $R=\Fpoly$ with a monomial order $\succ$, $I$ is an ideal. The reduced of $I$ with respect to $\succ$, $G=\{g_1,\ldots g_m\}$, is unique up to the order of the list $\{g_1,\ldots g_m\}$. It is independent of the choice of the generating set of $I$.
See Cox, Little, O’Shea [@MR3330490 Chapter 2]. Note that given a $B=\{h_1\ldots h_m\}$, the reduced $G$ can be obtained as follows,
1. For any $h_i\in B$, if $\LT(h_j)|\LT(h_i)$, $j\not=i$, then remove $h_i$. Repeat this process, and finally we get the [*minimal basis*]{} $G'\subset B$.
2. For every $f\in G'$, divide $f$ towards $G'-\{f\}$. Then replace $f$ by the remainder of the division. Finally, normalize the resulting set such that every polynomial has leading coefficient $1$, and we get the reduced $G$.
Note that reduces only one polynomial pair every time, more recent algorithms attempt to (1) reduce many polynomial pairs at once (2) identify the “unless” polynomial pairs [*a priori*]{}. Currently, the most efficient algorithms are Faugere’s F4 and F5 algorithms [@Faugere199961; @Faugere:2002:NEA:780506.780516].
Usually we compute Gröbner basis by programs, for example,
- The embedded computes Gröbner basis by . The relation between and the original generating set is not given. Usually, computation in is not very fast.
- [Maple ]{} Maple computes by either or highly efficient F4 algorithm.
- [Singular ]{} is a powerful computer algebraic system [@DGPS] developed in University of Kaiserslautern. uses either or F4 algorithm to computer .
- [Macaulay2]{} is a sophisticated algebraic geometry program [@M2], which orients to research mathematical problems in algebraic geometry. It contains and experimental codes of F4 algorithm.
- Fgb package [@FGb]. This is a highly efficient package of F4 and F5 algorithms by Jean-Charles Faugére. It has both [Maple ]{} and [C++ ]{} interfaces. Usually, it is faster than the F4 implement in [Maple]{}. Currently, coefficients of polynomials are restricted to $\Q$ or $\Z/p$, in this package.
\[example\_Buchberger\] Consider $f_1=x^3 - 2 x y$, $f_2=x^2 y - 2 y^2 + x$. Compute the of $I=\la f_1,f_2\ra$ with and $x\succ y$ [@MR3330490].
We use .
1. In the beginning, the list is $B:=\{h_1,h_2\}$ and the pair set $P:=\{(h_1,h_2)\}$, where $h_1=f_1$, $h_2=f_2$, $$S(h_1,h_2)=-x^2,\quad h_3:=\overline{S(h_1,h_2)}^B=-x^2\,,$$ with the relation $h_3=y h_1-x h_2$.
2. Now $B:=\{h_1,h_2,h_3\}$ and $P:=\{(h_1,h_3),(h_2,h_3)\}$. Consider the pair $(h_1,h_3)$, $$S(h_1,h_3)=2xy,\quad h_4:=\overline{S(h_1,h_3)}^B=2 x y\,,$$ with the relation $h_4=- h_1-x h_3$.
3. $B:=\{h_1,h_2,h_3,h_4\}$ and $ P:=\{(h_2,h_3),(h_1,h_4),(h_2,h_4),(h_3,h_4)\}$. For the pair $(h_2,h_3)$, $$S(h_2,h_3)=-x+2y^2,\quad h_5:=\overline{S(h_2,h_3)}^B=-x+2y^2\,,$$ The new relation is $h_5=-h_2-y h_3$.
4. $B:=\{h_1,h_2,h_3,h_4,h_5\}$ and $$P:=\{(h_1,h_4),(h_2,h_4),(h_3,h_4),(h_1,h_5),(h_2,h_5),(h_3,h_5),(h_4,h_5)\}.$$ For the pair $(h_1,h_4)$, $$S(h_1,h_4)=-4 x y^2,\quad \overline{S(h_1,h_4)}^B=0$$ Hence this pair does not add information to . Similarly, all the rests pairs are useless.
Hence the Groebner basis is $$\label{eq:42}
B=\{h_1,\ldots h_5\}=\{x^3-2 x y,x^2 y+x-2 y^2,-x^2,2 x y,2 y^2-x\}.$$ Consider all the relations in intermediate steps, we determine the conversion between the old basis $\{f_1, f_2\}$ and $B$, $$\begin{gathered}
\label{eq:43}
h_1= f_1,\quad h_2= f_2,\quad h_3= f_1 y-f_2 x\nn\\
h_4= -f_1 (1+x y)+f_2 x^2,\quad h_5= -f_1 y^2 +(x y-1) f_2 \end{gathered}$$ Then we determine the reduced . Note that $\LT(h_3)|\LT(h_1)$, $\LT(h_4)|\LT(h_2)$, so $h_1$ and $h_2$ are removed. The minimal is $G'=\{h_3,h_4,h_5\}$. Furthermore, $$\label{eq:54}
\overline{h_3}^{\{h_4,h_5\}} =h_3,\quad \overline{h_4}^{\{h_3,h_5\}}
=h_4, \quad \overline{h_5}^{\{h_3,h_4\}} =h_5\quad$$ so $\{h_3,h_4,h_5\}$ cannot be reduced further. The reduced is $$\label{eq:56}
G=\{g_1,g_2,g_3\}=\{-h_3, \half h_4, \half h_5\}=\{x^2,x y,
y^2-\half x\}.$$ The conversion relation is, $$\label{GB_conversion}
g_1= -y f_1 +x f_2 ,\quad g_2= -\frac{ (1+x y)}{2}f_1+\half x^2 f_2
,\quad g_3= -\half y^2 f_1+\half (x y-1) f_2.$$ finds $G$ directly via $\pmb{
\text{GroebnerBasis}[\{x^3-2 x y,x^2 y-2 y^2+x\},\{x,y\},}$\
$\pmb{\text{MonomialOrder}\to \text{DegreeReverseLexicographic}]}
$. However, it does not provide the conversion . This can be found by [Maple]{} or .
As a first application of , we can see some fractions can be easily simplified (like integrand reduction), $$\begin{aligned}
\label{example_reduction}
\frac{x^2}{(x^3-2 x y)(x^2 y-2 y^2+x)}&=&\frac{-y f_1+x f_2}{f_1 f_2}
=-\frac{y}{f_2}+\frac{x}{f_1}\nn \\
\frac{x y}{(x^3-2 x y)(x^2 y-2 y^2+x)}&=&\frac{- (1+x y)f_1/2+ x^2 f_2/2}{f_1 f_2}
=-\frac{1+x y}{2f_2}+ \frac{x^2}{2f_1}\nn \\
\frac{y^2}{(x^3-2 x y)(x^2 y-2 y^2+x)}&=&\frac{h_5+x/2}{f_1f_2}
=\frac{x}{2f_1 f_2}-\frac{ y^2}{2f_2}+ \frac{x y-1}{2f_1}\end{aligned}$$ In first two lines, we reduce a fraction with two denominators to fractions with only one denominator. In the last line, a fraction with two denominators is reduced to a fraction with two denominators but lower numerator degree ($y^2\to x$). Higher-degree numerators can be reduced in the same way. Hence we conclude that all fractions $N(x,y)/(f_1 f_2)$ can be reduced to, $$\label{eq:55}
\frac{1}{f_1 f_2}, \quad \frac{x}{f_1 f_2},\quad \frac{y}{f_1 f_2}$$ and fractions with fewer denominators. Note that even with this simple example, one-variable partial fraction method does not help the reduction.
We have some comments on :
1. For $\Fpoly$, the computation of polynomial division and only used addition, multiplication and division in $\F$. No algebraic extension is needed. Let $\F\subset \K$ be a field extension. If $B=\{f_1,\ldots, f_k\}\subset \Fpoly$, then the computation of $B$ in $\K[x_1,\ldots,x_n]$ produces a which is still in $\Fpoly$, irrelevant of the algebraic extension.
2. The form of a and computation time dramatically depend on the monomial order. Usually, is the fastest choice while is the slowest. However, in some cases, with is preferred. In these cases, we may instead consider some “midway” monomial order the like block order, or convert a known basis to basis [@FAUGERE1993329].
3. If all input polynomials are linear, then the reduced is the [*echelon form*]{} in linear algebra.
### Application of
is such a powerful tool that once it is computed, most computational problems on ideals are solved.
#### Ideal membership and fraction reduction
A immediately solves the ideal membership problem. Given an $F\in
R=\Fpoly$, and $I=\la f_1,\ldots f_k\ra$. Let $G$ be a of $I$ with a monomial order $\succ$. $F\in I$ if and only if $\overline{F}^G=0$, i.e., the division of $F$ towards $G$ generates zero remainder (Proposition \[GB\_division\]).
$G$ also determined the structure of the quotient ring $R/I$ (Definition \[quotient\_ring\]). $f \sim g$ if and only if $f-g\in
I$. The division of $f_1-f_2$ towards $G$ detects equivalent relations. In particular,
Let $M$ be the set of all monic monomials in $R$ which are not divisible by any leading term in $G$. Then the set, $$V= \{[p]|p\in M\}$$ is an $\F$-linear basis of $R/I$. \[remainders\]
For any $F\in R$, $\overline{F}^G$ consists of monomials which are not divisible by any leading term in $G$. Hence $[F]$ is a linear combination of finite elements in $V$.
Suppose that $\sum_j c_j [p_j]=0$ and each $p_j$’s are monic monomials which are not divisible by leading terms of $G$ . Then $\sum_jc_jp_j\in I$, but by the Algorithm \[multivariate\_polynomial\_division\]. $\overline{\sum_jc_j
p_j}^G=\sum_jc_j
p_j$. So $\sum_jc_jp_j=0$ in $R$ and $c_j$’s are all zero.
As an application, consider fraction reduction for $N/(f_1 \ldots
f_k)$, where $N$ is polynomial in $R$, $$\label{eq:60}
\frac{N}{f_1 \ldots f_k}=\frac{r}{f_1 \ldots f_k} + \sum_{j=1}^k
\frac{s_i}{f_1 \ldots \hat{f_j}\ldots f_k}.$$ The goal is to make $r$ simplest, i.e., $r$ should not contain any term which belongs to $I=\la f_1,\ldots f_k \ra$. We compute the of $I$, $G=\{g_1,\ldots g_l\}$ and record the conversion relations $g_i=\sum_{j=1}^k f_j a_{ji}$ from the computation.
Polynomial division of $N$ towards $G$ gives, $$\label{eq:57}
N=r+\sum_{i=1}^l q_i g_i$$ where $r$ is the remainder. The result, $$\label{fraction_reduction}
\frac{N}{f_1 \ldots f_k}=\frac{r}{f_1 \ldots f_k} + \sum_{j=1}^k
\frac{\big(\sum_{i=1}^l a_{ji}q_i\big)}{f_1 \ldots \hat{f_j}\ldots f_k},$$ gives the complete reduction since by the properties of $G$, no term in $r$ belongs to $I$. solves integrand reduction problem for multi-loop diagrams. In practice, there are shortcuts to compute numerators like $\big(\sum_{i=1}^l a_{ji}q_i\big)$.
#### Solve polynomial equations with
In general, it is very difficult to solve multivariate polynomial equations since variables are entangled. characterizes the solution set and can also remove variable entanglements.
Let $f_1\ldots f_k$ be polynomials in $R=\F[x_1,\ldots x_n]$ and $I=\la f_1\ldots f_k\ra$. Let $\bar \F$ be the algebraic closure of $\F$. The solution set in $\bar \F$, $\mathcal Z_{\bar\F}(I)$ is finite, if and only if $R/I$ is a finite dimensional $\F$-linear space. In this case, the number of solutions in $\bar \F$, counted with multiplicity, equals $\dim_\F (R/I)$.
See Cox, Little, O’Shea [@opac-b1094391]. The rigorous definition of multiplicity is given in the next chapter, Definition \[local\_ring\].
Note that again, we distinguish $\F$ and its algebraic closure $\bar \F$, since we do not need computations in $\bar \F$ to count total number of solutions in $\bar \F$. $\dim_\F (R/I)$ can be obtained by counting all monomials not divisible by $\LT(G(I))$, leading terms of the . Explicitly, $\dim_\F (R/I)$ is computed by of .
\[eq:61\] Consider $f_1=-x^2+x+y^2+2, f_2=x^3-x y^2-1$. Determine the number of solutions $f_1=f_2=0$ in $\C^2$.
Compute the for $\{f_1,f_2\}$ in with $x\succ y$, we get, $$\label{eq:62}
G=\{y^2+3x+1,x^2+2x-1\}.$$ Then $\LT(G)$=$\{y^2,x^2\}$. Then $M$ in Proposition \[remainders\] is clearly $\{1, x ,y, x y\}$. The linear basis for $\Q[x,y]/\la f_1,
f_2\ra$ is $\{[1],[x],[y],[x y]\}$. Therefore there are $4$ solutions in $\C^2$. Note that Bézout’s theorem would give the number $2\times
3=6$. However, we are considering the solutions in affine space, so there are $6-4=2$ solutions at infinity. Another observation is that the second polynomial in $G$ contains only $x$, so the variable entanglement disappears and we can first solve for $x$ and then us $x$-solutions to solve $y$. This idea will be developed in the next topic, elimination theory.
Sudoku is a popular puzzle with $9\times 9$ spaces. The goal is to fill in digits from ${1,2,\ldots, 9}$, such that each row, each column and each $3\times 3$ sub-box contain digits $1$ to $9$. See two Sudoku problems in Figure \[sudoku\_1\].
Typically people solve Sudoku with backtracking algorithm: try to fill in as many digits as possible, and if there is no way to proceed then go one step back. It can be easily implemented in computer codes, and usually it is very efficient. Here we introduce solving Sudoku by . This method is not the most efficient way, however, besides finding a solution, it illustrates the global structure of solutions.
We convert this puzzle to an algebraic problem. Name the digit on $i$-th row and $j$-th column as $x_{ij}$. $x_{ij}$ must be in $\{1,\ldots 9\}$. Let, $$\label{eq:63}
F(x)=(x-1)(x-2) \ldots (x-9).$$ So there are $81$ equations, $F(x_{ij})=0$. Two spaces in the same row, or in the same column, or in the same sub-box, cannot contain the same digit. For example, $x_{11}\not=x_{12}$. Note this is not an equality, how do we write an algebraic equation to describe this constraint?
The standard trick to “differentiate” polynomials. Consider $F(y)-F(x)$, where $x$ and $y$ refer to two boxes that cannot contain the same digit. $F(y)-F(x)$ must be proportional to $y-x$. $$\label{eq:58}
\frac{F(y)-F(x)}{y-x}=g(x,y).$$ where $g(x,y)$ is a polynomial. It is clearly that when $y\not=x$, $g(x,y)=0$. On the other hand, from the Taylor series, $$\label{eq:64}
F(y)-F(x)=(y-x)\bigg(F'(x)+\half (y-x) F''(x) + \ldots
\bigg)=(y-x)g(x,y).$$ If $g(x,y)=0$ but $y=x$, then $F'(x)=0$. However $F(x)$ has no multiple root, that means $F(x)$ and $F'(x)$ cannot be both zero. So if $g(x,y)=0$ then $y\not=x$. There are $810$ such equations like $g(x_{11},x_{12})=0$. Then with the known input information in Sukodu, we have a polynomial equation system.
For the first Sudoku, there are $81+810+27=918$ equations. It is really a large system with high degree polynomials. Amazingly, we can still solve it by . Using command in , and the number field $Z/11$, this sudoku is solved on a laptop computer with in about $4.9$ seconds. The output is linear and gives the unique solution of the Sudoku (Figure \[sudoku\_solution\_1\]).
For Sudoku 2, there are $919$ equations. takes about $5.1$ seconds on a laptop to get , $$\begin{gathered}
G= \{x_{58}^2-4,x_{11}-5,x_{12}-3,x_{13}-4,x_{14}-6,x_{15}-7,x_{16}-8,x_{17}-9,x_{18}-1,x_{19}-2,\nn\\x_{21}-6,x_{22}-7,x_{23}-2,x_
{24}-1,x_{25}-9,x_{26}-5,x_{27}-3,x_{28}-4,x_{29}-8,x_{31}-1,x_{32}-9,\nn\\x_{33}-8,x_{34}-3,x_{35}-4,x_{36}-2,x_{37}-5,x_{38}-6,x
_{39}-7,x_{41}-8,x_{42}+9 x_{58}-9,\nn\\x_{43}+9 x_{58}-2,x_{44}-7,x_{45}+3
x_{58}-11,x_{46}-1,x_{47}-4,x_{48}+x_{58}-11,x_{49}-3,x_{51}-4,\nn\\
x_{52}+2 x_{58}-9,x_{53}+2 x_{58}-2,x_{54}-8,x_{55}+8
x_{58}-11,x_{56}-3,x_{57}-7,x_{59}-1,x_{61}-7,\nn\\x_{62}-1,x_{63}-3,x_{64}-9,x_{65}-2,x_{66}-4,x_{67}-8,x_{68}-5,x_{69}-6,x_{71}-
9,x_{72}-6,x_{73}-1,\nn\\x_{74}-5,x_{75}-3,x_{76}-7,x_{77}-2,x_{78}-8,x_{79}-4,x_{81}-2,x_{82}-8,x_{83}-7,x_{84}-4,x_{85}-1,\nn\\x_{86}
-9,x_{87}-6,x_{88}-3,x_{89}-5,x_{91}-3,x_{92}-4,x_{93}-5,x_{94}-2,\nn\\x_{95}-8,x_{96}-6,x_{97}-1,x_{98}-7,x_{99}-9\}\,.\end{gathered}$$ Note that the new feature is that $G$ contains a quadratic polynomial, which means the solution for this sudoku is not unique. From leading term counting, there are $2$ solutions. Explicitly, solve the first equation $$\begin{gathered}
\label{eq:67}
x_{58}^2=4 \mod 11\,,\end{gathered}$$ and we get two solutions, $x_{58}=2$ or $x_{58}=9$. Afterwards, we get two complete solutions (Figure \[sudoku\_solution\_2\]).
#### Elimination theory
We already see that can remove variable entanglement, here we study this property via elimination theory,
\[Elimination\] Let $R=\F[y_1,\ldots y_m,z_1,\ldots z_n]$ be a polynomial ring and $I$ be an ideal in $R$. Then $J=I\cap \Fpoly$, the [*elimination ideal*]{}, is an ideal of $\Fpoly$. $J$ is generated by $G(I)\cap
\Fpoly$, where $G(I)$ is the of $I$ in order with $y_1\succ
y_2\ldots \succ y_m\succ z_1\succ z_2\ldots \succ z_n$.
See Cox, Little and O’Shea [@MR3330490].
Note that elimination ideal $J$ tells the relations between $z_1\ldots
z_n$, without the interference with $y_i$’s. In this sense, $y_i$’s are “eliminated”. It is very useful for studying polynomial equation system. In practice, in may involve heavy computations. So frequently, we use block order instead, $[y_1,\ldots y_m]\succ [z_1,\ldots z_n]$ while in each block can be applied.
Eliminate theory applies in many scientific directions, for example, it transfers tree-level scattering equations (CHY formalism) [@Cachazo:2013iea; @Cachazo:2013hca; @Cachazo:2013gna; @Cachazo:2014nsa; @Cachazo:2014xea] with $n$ particles, in $(n-3)$ variables, to a univariate polynomial equation [@Dolan:2015iln]. Here we give a simple example in IMO,
[**Problem**]{} Solve the system of equations: $$\begin{aligned}
\label{eq:69}
x + y + z &=& a\nn \\
x^2 + y^2 + z^2 &=& b^2\nn \\
xy &=& z^2\end{aligned}$$ where $a$ and $b$ are constants. Give the conditions that $a$ and $b$ must satisfy so that $x, y, z$ (the solutions of the system) are distinct positive numbers.
[**Solution**]{} The tricky part is the condition for positive distinct $x,y,z$. Now with this problem can be solved automatically.
First, eliminate $x,y$ by in with $x\succ y \succ z$. For example, in $$\begin{gathered}
\pmb{ \text{GroebnerBasis}[\{-a+x+y+z,-b^2+x^2+y^2+z^2,x
y-z^2\},\{x,y,z\},}\\
\pmb{
\text{MonomialOrder}\to \text{Lexicographic},\text{CoefficientDomain}\to \text{RationalFunctions}]}\end{gathered}$$ and the resulting is, $$\label{eq:71}
G=\left\{a^2\c -2 a z\c -b^2,-a^4+y \left(2 a^3\c +2 a b^2\right)+2 a^2 b^2-4 a^2 y^2-b^4,a^2-2 a
x\c-2 a y+b^2\right\}.$$ The first element is in $\Q(a,b)[z]$, hence it generates the elimination ideal. Solve this equation, we get, $$\label{IMO_z}
z=\frac{a^2-b^2}{2 a}\,.$$ Then eliminate $y,z$ by in with $z\succ y \succ x$. We get the equation, $$\label{IMO_x}
a^4+x (-2 a^3-2 a b^2)-2 a^2 b^2+4 a^2 x^2+b^4=0\,.$$ To make sure $x$ is real we need the discriminant, $$\label{x_discriminant}
-4 a^2 (a^2-3 b^2) (3 a^2-b^2)\geq 0\,.$$ Similarly, to eliminate $x,z$, we use with $z\succ x \succ y$ and get $$\label{eq:75}
a^4+y (-2 a^3-2 a b^2)-2 a^2 b^2+4 a^2 y^2+b^4=0\,,$$ and the same real condition as . Note that $x$ and $y$ are both positive, if and only if $x,y$ are real, $x+y>0$ and $x y$. Hence positivity for $x,y,z$ means, $$\begin{aligned}
\label{eq:66}
z=\frac{a^2-b^2}{2 a}&>0\nn\\
x+y=a-z=a- \frac{a^2-b^2}{2a}&>0\\
-4 a^2 (a^2-3 b^2) (3 a^2-b^2)&\geq 0.\end{aligned}$$ which implies that, $$\label{admissible_region_pre}
a>0,\quad b^2<a^2\leq 3b^2.$$ To ensure that $x$, $y$ and $z$ are distinct, we consider the ideal in $\Q[a,b,x,y,z]$. $$\label{eq:70}
J=\{-a+x+y+z,-b^2+x^2+y^2+z^2,x
y-z^2,(x-y)(y-z)(z-x)\}.$$ Note that to study the $a$, $b$ dependence, we consider $a$ and $b$ as variables. Eliminate ${x,y,z}$, we have, $$\label{eq:74}
g(a,b)=(a-b) (a+b) (a^2-3 b^2)^2 (3 a^2-b^2) \in J.$$ If all the four generators in $J$ are zero for some value of $(a,b,x,y,z)$, then $g(a,b)=0$. Hence, if $g(a,b)\not=0$, $x$, $y$ and $z$ are distinct in the solution. So it is clear that inside the region defined by , the subset set $$\label{admissible_region}
a>0,\quad b^2<a^2<3b^2.$$ satisfies the requirement of the problem. On the other hand, if $a^2=3b^2$, explicitly we can check that $x$, $y$ and $z$ are not distinct in all solutions. Hence $x,y,z$ in a solution are positive and distinct, if and only if $a>0$ and $b^2<a^2<3b^2$. With and , it is trivial to obtain the solutions.
#### Intersection of ideals
In general, given two ideals $I_1$ and $I_2$ in $R=\Fpoly$, it is very easy to get the generating sets for $I_1+I_2$ and $I_1 I_2$. However, it is difficult to compute $I_1\cap I_2$. Hence again we refer to especially to elimination theory.
Let $I_1$ and $I_2$ be two ideals in $R=\Fpoly$. Define $J$ as the ideal generated by $\{t f|f\in I_1\}\cup \{(1-t) g|g\in
I_2\}$ in $\F[t,z_1,\ldots z_n]$. Then $I_1\cap I_2=J\cap R$, and the latter can be computed by elimination theory.
If $f\in I_1$ and $f\in I_2$, then $f=t f+(1-t)f \in J$. So $I_1\cap
I_2\in J\cap R$. On the other hand, if $F\in J\cap R$, then $$\label{eq:77}
F(t,z_1,\ldots,z_n)=a(t, z_1,\ldots,z_n) t f(z_1,\ldots,z_n)+b(t, z_1,\ldots,z_n) (1-t) g(z_1,\ldots,z_n)\,,$$ where $f\in I_1$, $g\in I_2$. Since $F\in R$, $F$ is $t$ independent. Plug in $t=1$ and $t=0$, we get, $$\label{eq:78}
F=a(1, z_1,\ldots,z_n) f(z_1,\ldots,z_n),\quad F=b(0, z_1,\ldots,z_n) g(z_1,\ldots,z_n)\,.$$ Hence $F\in I_1\cap I_2$, $ J\cap R\subset I_1\cap I_2$.
In practice, terms like $tf$ and $(1-t)g$ increase degrees by $1$, hence this elimination method may not be efficient. More efficient method is given by [*syzygy*]{} computation [@opac-b1094391 Chapter 5].
### Basic facts of algebraic geometry in affine space II
In this subsection, we look closer at properties of algebraic sets and ideals. Consider $I=\{x^2 - y^2, x^3 + y^3 - z^2\}$ in $\C[x,y,z]$. From naive counting, $\mathcal Z(I)$ is a curve since there are $2$ equations in $3$ variables. However, the plot of $\mathcal Z(I)$ (Figure \[fig:reducible\_curve\]) looks like a line and a cusp curve. So $\mathcal Z(I)$ is [*reducible*]{}, in the sense that it can be decomposed into smaller algebraic sets. So we need the concept of [*primary decomposition*]{}.
![A reducible algebraic set (in blue), defined by $\mathcal Z(\{x^2 - y^2, x^3 + y^3 - z^2\})$.[]{data-label="fig:reducible_curve"}](graphs/reducible_curve.eps)
An ideal $I$ in a ring $R$ is called prime, if $\forall a b\in I$ ($a$, $b\in R$) then $a\in I$ or $b\in I$. An ideal $I$ in $R$ is called primary is if $a b\in I$ ($a$, $b\in R$) then $a\in I$ or $b^n\in I$, for some positive integer $n$.
A prime ideal must be a primary ideal. On the other hand,
If $I$ is a primary ideal, then the radical of $I$, $\sqrt I$ is a prime ideal.
See Zariski and Samuel [@MR0384768 Chapter 3].
Note that $I=\{x^2 - y^2, x^3 + y^3 - z^2\}$ is not a prime ideal or primary ideal. Define $a=x-y$, $b=x+y$, clearly $ab\in I$, but $a\not
\in I$ and $b^n\not \in I$ for any positive integer $n$. (The point $P=(2,2,4)\in \mathcal Z(I)$. If $(x+y)^n\in I$ then $(x+y)^n|_P=0$. It is a contradiction.)
For another example, $J=\la (x-1)^2 \ra$ in $\C[x]$ is primary but not prime. $\mathcal Z(J)$ contains only one point $\{1\}$ with the multiplicity $2$. $(x-1)(x-1) \in J$ but $(x-1)\not \in J$. For there examples, we see primary condition implies that the corresponding algebraic set cannot be decomposed to smaller algebraic sets, while prime condition further requires that the multiplicity is $1$.
For an ideal $I$ in $\Fpoly$, $I$ has the primary decomposition, $$\label{primary_decomposition}
I=I_1 \cap \ldots \cap I_m\,,$$ such that,
- Each $I_i$ is a primary ideal in $\Fpoly$,
- $I_i\not \supset \cap_{j\not=i} I_j$,
- $\sqrt I_i \not = \sqrt I_j$, if $i\not =j$.
Although primary decomposition may not be unique, the radicals $\sqrt I_i$’s are uniquely determined by $I$ up to orders.
See Zariski, Samuel [@MR0384768 Chapter 4].
Note that unlike Gröbner basis, primary decomposition is very sensitive to the number field. For an ideal $I\subset \Fpoly$, $\F
\subset \mathbb K$, the primary decomposition results of $I$ in $\Fpoly$ and $\K[z_1,\ldots,z_n]$ can be different. Primary decomposition can be computed by or . However, the computation is heavy in general.
Primary decomposition was also used for studying string theory vacua [@Mehta:2012wk].
Consider $I=\{x^2 - y^2, x^3 + y^3 - z^2\}$. Use or , we find that, $I=I_1\cap I_2$, where, $$\label{eq:68}
I_1=\la z^2,x+y\ra,\quad I_2=\la 2y^3-z^2,x-y \ra$$ Then $\sqrt I_1=\la z,x+y \ra$ is a prime ideal, where $I_2$ itself is prime.
When $I\subset \Fpoly$ has a primary decomposition $I=I_1\cap \ldots
\cap I_m$, $m>1$, then $\mathcal Z_\F(I)=\mathcal Z_\F(I_1) \cup \ldots \cup
\mathcal Z_\F(I_m) $. Then algebraic set decomposed to the union of sub algebraic sets. We switch the study of reducibility to the geometric side.
Let $V$ be a nonempty closed set in $\mathbf A_\F$ in Zariski topology, $V$ is irreducible, if $V$ cannot be a union of two closed proper subsets of $V$.
Let $\mathbb K$ be an algebraic closed field. There is a one-to-one correspondence: $$\begin{array}{ccc}
\text{prime ideals in }\mathbb K[z_1,\ldots z_n]
& &\text{irreducible
algebraic sets in }\mathbf A_{\mathbb K}\\
I & \longrightarrow & \mathcal Z_\K(I)\\
\mathcal I(V) & \longleftarrow & V\\
\end{array}$$
(Sketch) This follows from Hilbert Nullstellensatz .
We call an irreducible Zariski closed set “affine variety”. Similar to primary decomposition of ideals, algebraic set has the following decomposition,
\[variety\_decomposition\] Let $V$ be an algebraic set. $V$ uniquely decomposes as the union of affine varieties, $V=V_1 \cup \ldots \cup V_m$, such that $V_i \not
\supset V_j$ if $i\not =j$.
Let $I=\mathcal I(V)$. The primary decomposition determines that $I=I_1 \cap\ldots \cap I_m$. Since $I$ is a radical ideal, all $I_i$’s are prime. Then $V=\mathcal
Z(I)=\cap_{i=1}^m \mathcal Z(I_i)$. Each $\mathcal Z(I_i)$ is an affine variety. If $\mathcal Z(I_i) \supset \mathcal Z(I_j)$, then $I_i \subset I_j$ which is a violation of radical uniqueness of Lasker-Noether theorem.
If there are two decompositions, $V=V_1 \cup \ldots \cup V_m=W_1\cup
\ldots \cup W_l$. $V_1=V_1\cap(W_1\cup
\ldots \cup W_l)=(V_1\cap W_1)\cup \ldots (V_1\cap W_l)$. Since $V_1$ is irreducible, $V_1$ equals some $V_1\cap W_j$, , say $j=1$. Then $V_1\subset W_1$. By the same analysis $W_1 \subset V_i$ for some $i$. Hence $V_1\subset V_i$ and so $i=1$. We proved $W_1=V_1$. Repeat this process, we see that the two decompositions are the same.
\[dbox\_primary\_decomposition\] As an application, we use primary decomposition to find cut solutions of $4D$ double box in Table \[dbox\_sol\]. It is quite messy to derive all unitarity solutions by brute force computation. In this situation, primary decomposition is very helpful.
Use van Neerven-Vermaseren variables, the ideal $I=\la D_1,\ldots D_7\ra$ decomposes as $I=I_1\cap I_2 \cap I_3 \cap I_4 \cap I_5 \cap I_6$. $$\begin{aligned}
\label{eq:76}
I_1&=& \{2 y_4-t,s+2 y_2,-t+2 x_3-2 x_4,y_3,\frac{s}{2}+y_1+y_2,x_2-\frac{s}{2},x_1\}\,,\nn\\
I_2&=& \{t+2 y_4,s+2 y_2,-t+2 x_3+2
x_4,y_3,\frac{s}{2}+y_1+y_2,x_2-\frac{s}{2},x_1\}\,,\nn\\
I_3&=& \{s+t+2 y_2+2 y_4,2
x_4-t,x_3,y_3,\frac{s}{2}+y_1+y_2,x_2-\frac{s}{2},x_1\}\,,\nn\\
I_4&=& \{s+t+2 y_2-2 y_4,t+2
x_4,x_3,y_3,\frac{s}{2}+y_1+y_2,x_2-\frac{s}{2},x_1\}\,,\nn\\
I_5&=& \{s+t+2 y_2+2 y_4,x_4 (2 s+2 t)+y_4 (2 s+2 t)+s t+t^2+4 x_4
y_4,\nn\\ &&-t+2 x_3-2
x_4,y_3,\frac{s}{2}+y_1+y_2,x_2-\frac{s}{2},x_1\}\,,\nn\\
I_6&=& \{s+t+2 y_2-2 y_4,x_4 (-2 s-2 t)+y_4 (-2 s-2 t)+s t+t^2+4 x_4
y_4,\nn\\ &&-t+2 x_3+2 x_4,y_3,
\frac{s}{2}+y_1+y_2,x_2-\frac{s}{2},x_1\}\,.
\end{aligned}$$ Each $I_i$ is prime and corresponds to a solution in Table \[dbox\_sol\]. computes this primary decomposition in about $3.6$ seconds on a laptop. In practice, the computation can be sped up if we first eliminate all RSPs.
Hence the unitarity solution set $\mathcal Z(I)$ consists of six irreducible solution sets $\mathcal Z(I_i)$, $i=1\ldots 6$. Each one can be parametrized by a free parameter.
For a variety $V$, we want to define its dimension. Intuitively, we may test if $V$ contains a point, a curve, a surface...? So the dimension of $V$ is defined as the length of variety sequence in $V$,
The dimension of a variety $V$, $\dim V$, is the largest number $n$ in all sequences $\emptyset \not=W_0\subset W_1 \ldots \subset W_n\subset V$, where $W_i$’s are distinct varieties.
On the algebraic side, let $V=\mathcal Z(I)$, where $I$ is an ideal in $R=\Fpoly$. Consider the quotient ring $R/I$. Roughly speaking, the remaining “degree of freedom” of $R/I$ should be the same as $\dim V$. Krull dimension counts “the degree of freedom”,
The Krull dimension of a ring $S$, is the largest number $n$ in all sequences $p_0\subset p_1 \ldots \subset p_n$, where $p_i$’s are distinct prime ideals in $S$.
If for a prime ideal $I$, $R/I$ is has Krull dimension zero then $I$ is a [*maximal ideal*]{}. A maximal ideal $I$ in $R$ is an ideal which such that for any proper ideal $J\supset I$, $J=I$. $I$ is a maximal idea, if and only if $R/I$ is a field. ($R$ itself is not a maximal idea of $R$). When $\F$ is algebraically closed, then any maximal ideal $I$ in $R=\Fpoly$ has the form [@MR3330490], $$\label{eq:79}
I=\la z_1-c_1,\ldots z_n-c_n\ra,\quad c_i\in \F.$$ Note that the point $(c_1,\ldots,c_n)$ is zero-dimensional, and $R/I=\F$ has Krull dimension $0$. More generally,
If $\F$ is algebraically closed and $I$ a prime proper ideal of $R=\Fpoly$. Then the Krull dimension of $R/I$ equals $\dim \mathcal
Z(I)$.
See Hartshorne [@MR0463157 Chapter 1]. Note that Krull dimension of $R/I$ is different from the linear dimension $\dim_\F R/I$.
In summary, we has the algebra-geometry dictionary (Table \[AG\_dictionary\]), where the last two rows hold if $\F$ is algebraic closed.
------------------------------------------------------------------------------------------------------------------
Algebra Geometry
------------------------------------ --------------- -------------------------------------------------------------
Ideal $I$ in $\Fpoly$ algebraic set $\mathcal Z(I)$
$I_1\cap I_2$ $\mathcal Z(I_1\cap I_2)=\mathcal Z(I_1)\cup
\mathcal Z(I_2)$
$I_1+ I_2$ $\mathcal Z(I_1+ I_2)=\mathcal Z(I_1)\cap
\mathcal Z(I_2)$
$I_1\subset I_2$ $\Rightarrow$ $\mathcal Z(I_1)\supset
\mathcal Z(I_2)$
prime ideal $I$ $\Rightarrow$ $\mathcal Z(I)$ (irreducible) variety
maximal ideal $I$ $\Rightarrow$ $\mathcal Z(I)$ is a point
Krull dimension of $\dim \Fpoly/I$ $=$ $\dim \mathcal Z(I) $
------------------------------------------------------------------------------------------------------------------
: algebraic geometry dictionary[]{data-label="AG_dictionary"}
We conclude this section by an example which applies Gröbner basis, primary decomposition and dimension theory.
\[Galois\_group\] (Galois theory) Galois theory studies the symmetry of a field extension, $\F \subset \K$ by the Galois group $\text{Aut}(\K/\F)$. Historically, Galois group of a polynomial is defined to be the permutation group of roots, such that algebraic relations are preserved. Galois completely determined if a polynomial equation can be solved by radicals. In practice, given a polynomial to find its Galois group may be difficult. Here we introduce an automatic method of computing Galois group.
For example, consider the polynomial $f(x)=x^4+3 x+3$ in $\Q[x]$. It is irreducible over $\Q[x]$ and contains no multiple root in $\mathbb C$. We denote the four distant roots as $x_1$, $x_2$, $x_3$, $x_4$. To ensure that these variables are distant, we use a classic trick in algebraic geometry: auxiliary variable. Introduce a new variable $w$, define that $$\label{Galois_ideal}
I=\la f(x_1), f(x_2), f(x_3), f(x_4),
w(x_1-x_2)(x_1-x_3)(x_1-x_4)(x_2-x_3)(x_2-x_4)(x_3-x_4)-1 \ra\,.$$ It is clear that in $\C[x_1,x_2,x_3,x_4,w]$, $\mathcal Z(I)$ is a finite set (for example via Gröbner basis computation.) The four variables must be distinct on the solution set, because of the last generator in . Back to $\Q[x_1,x_2,x_3,x_4,w]$, we want to find more algebraic relations over $\Q$ which are “consistent” with $I$. That is to find a maximal ideal $J$ in $\Q[x_1,x_2,x_3,x_4,w]$, $I\subset J$. In practice, we use primary decomposition and find that in $\Q[x_1,x_2,x_3,x_4,w]$, $$\label{eq:72}
I=I_1\cap I_2 \cap I_3\,,$$ where explicitly each $I_i$ is prime. Since $dim_\Q
(\Q[x_1,x_2,x_3,x_4,w]/I)$ is finite, $$\label{eq:73}
dim_\Q
(\Q[x_1,x_2,x_3,x_4,w]/I_1)<\infty\,.$$ $I_1$ is prime hence $\Q[x_1,x_2,x_3,x_4,w]/I_1$ has no zero divisor. A finite-dimensional $Q$-algebra with no zero divisor must be a field. Hence $I_1$ is a maximal ideal of $\Q[x_1,x_2,x_3,x_4,w]$.
Compute the Groebner basis of $I_1$ with the block order $[w]\succ
[x_1,x_2,x_3,x_4]$, we have $$\begin{gathered}
\label{eq:80}
G(I_1)=\{x_1+x_2+x_3+x_4,2 x_4^2+2 x_2 x_4+2 x_3 x_4+x_2+x_3-3,
2 x_3
x_2+x_2+x_3+3,\nn\\ x_2^2\c -x_2+x_3^2\c-x_3,4 x_4^3\c -2 x_4^2+6 x_4+5 x_2+5 x_3+9,2 x_4
x_3^2+x_3^2+2 x_4^2 x_3\c-3 x_3+x_4^2\c-3 x_4\c-3,\nn\\4 x_3^3\c-2 x_3^2+x_3\c-5 x_2+9,315 w\c-2
x_3^2-4 x_4 x_3-2 x_4^2+3\}\end{gathered}$$ Except the last one, polynomials in $G(I_1)$ provides all the algebraic relations over $\Q$ of the four roots. Note that some relations are trivial like $x_1+x_2+x_3+x_4=0$ which comes from coefficients of $f(x)$. Some relations like $2 x_3
x_2+x_2+x_3+3=0$, are nontrivial.
Consider all $24$ permutations of $(x_1, x_2, x_3, x_4)$, we find the $8$ of them preserves algebraic relations in $G(I_1)$, explicitly, $$\begin{aligned}
\label{eq:82}
&&(x_1,x_2,x_3,x_4),(x_1,x_3,x_2,x_4),(x_2,x_1,x_4,x_3),(x_3,x_1,x_4,x_2),\nn\\
&&(x_2,x_4,x_1,x_3),(x_3,x_4,x_1,x_2),(x_4,x_2,x_3,x_1),(x_4,x_3,x_2,x_1)\,.\end{aligned}$$ Hence Galois group of the $x^4+3 x+3$ is the dihedral group $D_4$. Clearly, this process applies to all irreducible polynomials without multiple root.
Note that $\Q[x_1,x_2,x_3,x_4,w]/I_1$ actually is the splitting field of this polynomial.
Multi-loop integrand reduction via Gröbner basis
------------------------------------------------
With the knowledge of basic algebraic geometry, now multi-loop integrand reduction is almost a piece of cake. We apply method [@Zhang:2012ce; @Mastrolia:2012an].
Consider the algorithm of direct integrand reduction (IR-D). Suppose that all terms with denominator set $\mathcal D$, $\{D_1,\ldots D_k\}\subsetneqq \mathcal D$ are already reduced, then,
1. Collect all integrand terms with inverse propagators $D_1,\ldots D_k$, which include terms from Feynman rules and also terms from the integrand reduction of parent diagrams. Denote the sum as $N/(D_1,\ldots D_k)$.
2. Define $I=\la D_1,\ldots ,D_k\ra$. Compute the of $I$ in , $G(I)=\{g_1,g_2,\ldots ,g_m\}$.
3. Polynomial division $N=a_1 g_1+\ldots a_m g_m+\Delta$. Use Gröbner basis convention relation, rewrite the division as $N=q_1
D_1+\ldots q_k D_k+\Delta$.
4. Add $\Delta/(D_1 \ldots D_k)$ to the final result. Keep terms $$\label{eq:86}
\frac{q_1}{\hat{D_1}D_2 \ldots D_k} + \frac{q_2}{D_1\hat{D_2}
\ldots D_k} +\ldots \frac{q_k}{D_1D_2\ldots \hat{D_k}}\,,$$ for child diagrams.
Repeat this process, until all terms left are integrated to zero (like massless tadpoles, integral without loop momenta dependences).
Integrand reduction (IR-U) is more subtle. Again, Suppose that all diagrams with denominator set $\mathcal D$, $\{D_1,\ldots D_k\}\subsetneqq \mathcal D$ are reduced, then,
1. Define $I=\la D_1,\ldots D_k \ra$. Compute the of $I$ in with numeric kinematics, $G(I)=\{g_1,g_2,\ldots ,g_m\}$.
2. Identify all degree-one polynomials in $G(I)$, and solve them linearly. The dependent variables are RSPs. Define $J$ as the ideal obtained by eliminate all RSPs in $I$.
3. Make a numerator ansatz $N$ in ISPs, with the power counting restriction from renormalization conditions. Divide $N$ toward $G(J)$, the remainder $\Delta$ is the integrand basis.
4. Cut all propagators by $D_1=\ldots =D_k=0$. Classify all solutions by the primary decomposition of $J$ and get $n$ irreducible solutions.
5. On the cut, compute the tree products summed over internal spins/helicities. Subtract all known parent diagrams on this cut. The result should be a list of $n$ functions $S_i$, defined on each cut solution.
6. Fit coefficients of $\Delta$ from $S_i$’s.
We have some comments here:
- To make an integrand basis with undetermined coefficients, we only need Gröbner basis with numeric kinematic conditions.
- RSPs can be automatically found, because any degree-one polynomial in $I$ should be a linear combination of degree-one polynomials in $G(I)$, via Algorithm \[multivariate\_polynomial\_division\]. Hence linear algebra computation determines RSPs.
- Integrand basis should not contain RSPs. Furthermore, it is helpful to eliminate RSPs before the primary decomposition.
- If the cut solution is complicated, primary decomposition helps finding all of solutions. And in general, solution sets cannot be parameterized rationally before primary decomposition.
The key idea of these algorithms is that polynomial division via provides the simplest integrand, in the sense that the resulting numerator does not contain any term which are divisible by denominators.
Back to our double box examples, we use algebraic geometry methods to automate most of the computations. Given $7$ propagators in Van Neerven-Vermaseren variables, we use number field $\F=\Q(s,t)$, define the ideal $I=\la D_1 , D_2, \ldots D_7\ra$.
First, we determine the RSPs. Compute $G(I)$ in , with numeric kinematics, $t\to -3, s\to 1$. We find that $G(I)$ contains $4$ linear polynomials, $$\begin{gathered}
\label{eq:83}
\{y_3,\frac{1}{2}+y_1+y_2,x_2-\frac{1}{2},x_1\}\subset G(I)\,.\end{gathered}$$ This allow us to define RSPs: we have $4$ linear polynomials and $5$ variables, pick up $y_1$ to be the free variable. And then we determined $x_1,x_2,y_2,y_3$ are RSPs. (If needed, the full RSP relations can be obtained from Groebner basis conversion.) $$\begin{gathered}
\label{dbox_RSP_2}
x_1=\frac{D_1-D_2}{2},\quad x_2=\frac{D_2-D_3}{2}+\frac{s}{2}\,,\nn\\
y_2=\frac{D_4-D_6}{2}-\frac{s}{2}-y_1,\quad y_3=\frac{-D_6+D_7}{2}\,.\end{gathered}$$
Then, we consider to eliminate RSPs. Define $J$ to be an ideal in $\F[x_3,y_1,x_4,y_4]$, which is the ideal after RSP elimination. With numeric kinematics, the of $J$ in and $y_4\succ
x_4\succ y_1\succ x_3$ is, $$\begin{gathered}
G(J)=\{-4 x_3^2-12 x_3+4 x_4^2-9,20 x_3 y_1+4 x_4 y_4+6 x_3+6 y_1+9,-4
y_1^2-12 y_1+4 y_4^2-9,\nn\\
4 x_3^2 y_4+20 x_4 x_3 y_1+12 x_3 y_4+6 x_4 y_1+6 x_4 x_3+9 x_4+9
y_4,\nn\\
4 x_4 y_1^2+12 x_4 y_1+20 x_3 y_4 y_1+6 x_3 y_4+9 x_4+6 y_4 y_1+9 y_4,
4 x_3^2 y_1^2+2 x_3^2 y_1+2 x_3 y_1^2+3 x_3 y_1,\nn\\
80 x_3^2 y_1 y_4+16 x_3^2 y_4+40 x_3 y_1 y_4+18 x_3 y_4-6 x_4 y_1+24
x_4 x_3-9 x_4-9 y_4\}.
\label{dbox_GB}\end{gathered}$$ Note that the first $3$ polynomials are just equations in . However, the rest algebra relations in are not obtained by the naive generalization of OPP method. So previously we got a redundant basis.
Consider the numerator in ISPs only, $$\begin{aligned}
\label{eq:28}
N_\text{dbox}=\sum_m \sum_n \sum_\alpha \sum_\beta
c_{mn\alpha\beta}' x_3^{m} y_1^{n} x_4^{\alpha} y_4^{\beta} ,\end{aligned}$$ where $c_{mn\alpha\beta}'$ are indeterminate coefficients. By renormalization condition, there $160$ such $c$’s. Divide $
N_\text{dbox}$ by $G(I)$, we get the remainder, $$\begin{aligned}
\label{eq:28}
\Delta_\text{dbox}=\sum_{(m,n,\alpha,\beta)\in S}
c_{mn\alpha\beta} x_3^{m} y_1^{n} x_4^{\alpha} y_4^{\beta} ,\end{aligned}$$ where the index set $S$ contains $32$ elements, $$\begin{gathered}
(0, 0, 0, 0), (1, 0, 0, 0), (2, 0, 0, 0), (3, 0, 0, 0), (4, 0, 0, 0),
(0, 1, 0, 0), (1, 1, 0, 0), (2, 1, 0, 0),\nn\\ (3, 1, 0, 0), (4, 1, 0, 0),
(0, 2, 0, 0), (1, 2, 0, 0), (0, 3, 0, 0), (1, 3, 0, 0), (0, 4, 0, 0),
(1, 4, 0, 0), \nn\\ (0, 0, 1, 0), (1, 0, 1, 0), (2, 0, 1, 0), (3, 0, 1, 0),
(0, 1, 1, 0), (0, 0, 0, 1), (1, 0, 0, 1), (2, 0, 0, 1),\nn\\ (3, 0, 0, 1),
(4, 0, 0, 1), (0, 1, 0, 1), (1, 1, 0, 1), (0, 2, 0, 1), (1, 2, 0, 1),
(0, 3, 0, 1), (1, 3, 0, 1).
\label{dbox_integrand_basis_GB}\end{gathered}$$ Note that the number of terms in $\Delta_\text{dbox}$ matches the number of independent relations from unitarity cuts. is the integrand basis of the $4D$ double box. Of these $32$ terms, the last $16$ terms integrated to zero by Lorentz symmetry, so they are spurious terms.
In Example \[dbox\_primary\_decomposition\], we already used primary decomposition to find all unitarity-cut solutions. Note that there is shortcut” it is enough to consider the primary decomposition of $J$. On a laptop computer, it takes only $0.22$ seconds to finish. Using and $4D$ tree amplitudes, we can easily determine the double box integrand for (super)-Yang-Mills theory [@Badger:2012dp; @Zhang:2012ce].
For $D=4-2\epsilon$, we need to introduce $\mu$ variables, $$\begin{gathered}
\label{eq:88}
l_i=l_i^{[4]}+l_i^\perp , \quad i=1,\ldots ,L,\nn \\
\mu_{ij}=-l_i ^\perp \cdot l_j ^\perp, \quad 1\leq i \leq j\leq L.\end{gathered}$$ In this case, we have further simplification: $I=\la D_1,\ldots D_k \ra$ must be a prime ideal, hence it is not necessary to consider the primary decomposition of $I$ [@Zhang:2012ce; @Badger:2013gxa].
Consider two-loop five-gluon pure Yang-Mills planar amplitude, with helicity $(+++++)$. Note that tree-level all-plus-helicity $5$-gluon amplitude in Yang-Mills theory is zero, while the one-loop-level is finite. The two-loop amplitude is much more challenging. We used algebraic geometry method to compute this amplitude.
For the integrand, we use both IR-D and IR-U methods [@Badger:2013gxa]. Note that this amplitude is well-define only with $D=4-2\epsilon$. Repeat the integrand reduction process, we get all the diagrams with non-vanishing integrands in Figure \[2l5g\_planar\] (and their permutations). For example, the box-pentagon diagram for this amplitude has a simple integrand, $$\begin{aligned}
&\Delta_{431}(1^+,2^+,3^+,4^+,5^+) =
\nonumber\\&
-\frac{i \, s_{12}s_{23}s_{45} \, F_1(D_s,\mu_{11},\mu_{22},\mu_{12})}{\A12\A23\A34\A45\A51
\trfive}
\left( \tr_+(1345) (l_1 + k_5)^2 + s_{15}s_{34}s_{45}
\right)
\label{delta431}\end{aligned}$$ where $$\begin{aligned}
F_1(D_s,\mu_{11},\mu_{22},\mu_{12})
= (D_s-2)\left( \mu_{11}\mu_{22} + \mu_{11}\mu_{33} + \mu_{22}\mu_{33} \right) + 16\left(
\mu_{12}^2 - \mu_{11}\mu_{22} \right),
\label{eq:F1def}\end{aligned}$$ and $\mu_{33}=\mu_{11}+\mu_{22}+2\mu_{12}$ and $D_s$ is the dimension for internal states. [@Badger:2013gxa]. $$\begin{aligned}
\trfive &= \tr \! \left( \gamma_5 {k\kern-0.45em/}_1 {k\kern-0.45em/}_2 {k\kern-0.45em/}_3
{k\kern-0.45em/}_4 \right) \; = \; \B12 \A23 \B34 \A41 - \A12 \B23
\A34 \B41 .\nn \\
\tr_{\pm}(abcd) &= \frac{1}{2} \tr \! \big( (1 \pm \gamma_5) {k\kern-0.45em/}_a {k\kern-0.45em/}_b {k\kern-0.45em/}_c {k\kern-0.45em/}_d \big),\end{aligned}$$ Results from IR-D and IR-U match each other. After getting these simple integrand, the complete integrals and final analytic result for this amplitude was obtained by differential equation method [@Gehrmann:2015bfy].
All-plus two-loop five-gluon non-planar integrand and all-plus two-loop six-gluon integrand were also obtained by integrand reduction method [@Badger:2015lda; @Badger:2016ozq].
\
See [@Feng:2012bm; @Mastrolia:2013kca; @Mastrolia:2016dhn] for more example of CAG based integrand reductions.
Exercises
---------
Derive the integrand basis of a box diagram with $k_1^2=k_2^2=k_3^2=k_4^2=m^2$ and inverse propagators $D_1=l_1^2$, $D_2=(l_1-k_1)^2$, $D_3=(l_1-k_1-k_2)^2$ and $D_4=(l_1+k_4)^2$, via OPP approach [@Ossola:2006us; @Ossola:2007ax].
$I$ and $J$ are two ideals in $\Fpoly$. Define $I+J=\{f+g|f\in I, \ g\in J\}$ and $IJ$ as the ideal generated by the set $\{f g|f\in I, \ g\in
J\}$.
1. Prove that $\sqrt I$, $I+ J$, $I\cap J$ are ideals.
2. Prove that $IJ \subset I\cap J$ and $\sqrt{I\cap J}=\sqrt{IJ}$.
3. Let $I=\la y(y-x^2)\ra$, $J=\la x y\ra$ in $\Q[x,y]$, determine $Z_\C(I+J)$, $Z_\C(I\cap J)$ and $Z_\C(IJ)$. Compute generating sets of $I+J$, $I\cap J$ and $IJ$. Is $IJ$ the same as $I\cap J$ in this case? Compute generating sets of $\sqrt{I\cap J}$ and $\sqrt{IJ}$.
Let $f_1(x)=2 x - 4 x^2 + x^3$ and $f_2(x)=x^2-1$. Prove that as an ideal in $\Q[x]$, $\la f_1,f_2\ra=\la 1 \ra$. Explicitly find two polynomials $h_1(x)$ and $h_2(x)$ in $Q[x]$ such that, $$\label{eq:39}
h_1(x) f_1(x) + h_2(x) f_2(x) =1\,.$$ (Hint: use Euclid’s algorithm, Algorithm \[Euclid\].)
Prove and . $$\begin{aligned}
\bigcap_i \mathcal Z(I_i) &=&\mathcal Z(\bigcup_i I_i )\,.
\nn \\
\mathcal Z(I_1)\bigcup \mathcal Z(I_2) &=&\mathcal Z( I_1 I_2
)=\mathcal Z( I_1 \cap I_2 )\,.
\end{aligned}$$
1. Use computer software like , [Maple]{}, or , to eliminate $y$ and $z$ from $$\label{eq:81}
I=\la -x^3-x z+y^2-1,x^2+x z+y^2,x y+x z+y\ra\,,$$ to get a equation in $x$ only. How many common zeros are there for the three polynomials over $\C$?
2. Use computer software to find the projection of the curve $\mathcal C$, $$\label{eq:84}
\mathcal C:\quad x^2 + x y + z^2= x^2 - z y - z^3+1=0\,,$$ on $x$-$y$ plane.
Let $f_1=y^2 - x^3 - 1$, $f_2=x y + y^2 + 1$ and $f_3=y^2 + x - y$. Use [Maple]{} or [Macaulay]{} to find the $G=\{g_1, \ldots
, g_m\}$ and the conversion, $$\label{eq:85}
g_j =\sum_{i=1}^3 f_i a_{ij} .$$ Reduce the fraction $1/(f_1 f_2 f_3)$ as, $$\label{eq:87}
\frac{1}{f_1 f_2 f_3}=\frac{q_1}{f_2 f_3}+\frac{q_2}{f_1 f_3}+\frac{q_3}{f_1 f_2}$$ where $q_1$, $q_2$ and $q_3$ are polynomials in $x$ and $y$.
Use [Macaulay]{} or to find the primary decomposition of $I=\la x z - y^2, x^3 - y z\ra$. Then parameterize each irreducible closed set.
Use the method in Example \[Galois\_group\], to determine the Galois group of $x^4-10 x^2+1$.
Massless crossed box diagram is the two-loop diagram with $k_1^2=k_2^2=k_3^2=k_4^2=0$ and inverse propagators $D_1=l_1^2$, $D_2=(l_1 - k_1)^2$, $D_3=(l_1 - k_1 - k_2)^2$, $D_4=l_2^2$, $D_5=(l_2 - k_4)^2$, $D_6=(l_1 + l_2 -
k_1 - k_2 - k_4)^2$, $D_7=(l_1 + l_2)^2$.
Find the $4D$ integrand basis via Gröbner basis.
Use the $4D$ double box integrand basis to determine the double box integrand form of the $4D$ $(--++)$ and $(-+-+)$ helicity color-ordered amplitude in pure-Yang-Mills theory. (Hint: see [@Badger:2012dp].)
Unitarity Cuts and Several Complex Variables
=============================================
Maximal unitarity
-----------------
Besides the integrand reduction method for loop amplitudes, we can also consider the (generalized) unitarity with residue approach [@Bern:1994zx; @Bern:1994cg; @Bern:1995db; @Britto:2004nc; @Britto:2005fq] $$A_n^\text{$L$-loop}=\sum_{i} c_i I_i + \text{rational terms}\,.
\label{MI}$$ The set $\{I_k\}$ is the master integral (MI) basis, i.e., minimal linear basis of Feynman integrals. For example, for one-loop order, we have [*scalar*]{} box, triangle, bubble (and tadpole) integrals . The MI basis is usually a proper subset of the integrand basis like , since spurious terms are removed and integration-by-parts (IBP) identities are used.
[*Maximal unitarity*]{} method gets coefficients $c_i$’s for a scattering process, from contour integrals. (Usually contour integrals are simpler than Euclidean Feynman integrals.) Let $k$ be the largest number of propagators for all integrals in MI basis. Suppose that there are $d(k)$ diagrams with exactly $k$ propagators in the master integral list, $\mathscr
D_1,\ldots, \mathscr D_{d(k)}$.
Maximal unitarity method first separate (\[MI\]) as, $$\begin{gathered}
A_n^\text{$L$-loop}=\sum_{\alpha=1}^{d(k)}\sum_j c_{\alpha,j} I_{\alpha,j} + \big(\text{simpler integrals}\big)+\text{rational terms}\,,
\label{maximal_unitarity}\end{gathered}$$ where for fixed $\alpha$, $I_{\alpha,j}$’s stand for all master integrals associated with the diagram $\mathscr D_\alpha$. “Simpler integrals” stands for integrals with fewer-than-$k$ propagators.
The coefficients $c_{\alpha,j}$’s can be obtained by maximal unitarity as follows: Let the propagators of $\mathscr
D_\alpha$ be $D_1,
\ldots, D_k$. For simplicity, we drop the index $\alpha$. The cut equation is $$\label{eq:90}
D_1=\ldots =D_k=0\,,$$ which has $m$ independent solutions. In algebraic geometry language, the ideal $I=\la D_1 \ldots D_k \ra$ has the primary decomposition, $$\label{eq:89}
I=I_1 \cap \ldots \cap I_m\,.$$ Each independent solution is an (irreducible) variety, $V_i=\mathcal
Z(I_i)$. For an [*integer value* ]{} of the spacetime dimension $D$, we replace a generic Feynman integral by a contour integral, $$\begin{aligned}
\label{contour_integral}
\int \frac{d^D l_1}{i\pi^{D/2}} \ldots \frac{d^D l_L}{i\pi^{D/2}}
\frac{N(l_1,\ldots l_L)}{D_1^{}\ldots
D_k^{}}
&\rightarrow &\oint \frac{d^D l_1}{(2\pi i)^{D}} \ldots \frac{d^D
l_L}{(2\pi i )^{D}}
\frac{N(l_1,\ldots l_L)}{D_1^{} \ldots
D_k^{}}\nonumber \\
&=& \sum_{i=1}^m \sum_b w_b^{(i)} \oint_{\mathcal C_b^{(i)}} \Omega^{(i)}(N)\,.\end{aligned}$$ In the first line, we have a $DL$-fold contour integral. Part of the contour integrals serve as “holomorphic” Dirac delta functions in $D_1,\ldots ,D_k$, and the original integral becomes $(\dim V_i)$-fold contour integrals on each $V_i$. The $\mathcal C_b^{[i]}$’s are non-trivial contours on $V_i$ for this integrand, which consists of poles in the integrand and fundamental cycles of $V_i$. On each cut solution, the original numerator $N(l_1,\ldots,l_L)$ becomes, $$\label{eq:91}
N(l_1,\ldots,l_L) \big|_{V_i}= S^{(i)}\,.$$ where $S^{(i)}$ is the sum of products of tree amplitudes obtained from the maximal cut. In general, there may be several nontrivial contours on $V_i$, so for each one we set up a weight $w_b^{(i)}$ to be determined later.
We demand that if the original integral is zero, or can be reduce to integrals with fewer propagators by IBPs, the corresponding contour integral is zero. If $$\begin{aligned}
\label{eq:92}
\int \frac{d^D l_1}{i\pi^{D/2}} \ldots \frac{d^D l_L}{i\pi^{D/2}}
\frac{F(l_1,\ldots l_L)}{D_1^{}\ldots
D_k^{}}=0\,,\quad \text{($F$ is spurious)},\end{aligned}$$ or $$\begin{aligned}
\int \frac{d^D l_1}{i\pi^{D/2}} \ldots \frac{d^D l_L}{i\pi^{D/2}}
\frac{F(l_1,\ldots l_L)}{D_1^{}\ldots
D_k^{}}= (\text{simpler integrals})\,,\quad \text{(IBP relation)},\end{aligned}$$ Then $$\label{eq:93}
\sum_{i=1}^m \sum_b w_b^{(i)} \oint_{\mathcal C_b^{(i)}}
\Omega^{(i)}(F)=0\,.$$
Spurious terms and IBPs fix $w_b^{(i)}$’s up to the normalization of master integrals. To extract the coefficients $c_i$’s in (\[MI\]), we can find a special set of weights $w_{b,j}^{(i)}$ such that, $$c_j = \sum_{i=1}^m \sum_b w_{b,j}^{(i)} \oint_{\mathcal C_b^{(i)}} \Omega^{(i)}(S^{(i)}).
\label{linear_fitting_c}$$ After getting all $k$-propagator master integrals’ coefficients, we repeat this process for $(k-1)$ propagator integrals. We need spurious terms and IBPs to fix the contour weights.
For example, consider the $4D$ massless four-point amplitude. reads. $$\begin{gathered}
A_4^\text{$1$-loop}=c_{box} I_{box} + \ldots \,.\end{gathered}$$ From , $D_1=D_2=D_3=D_4=0$ has two solutions. Change the original integral to contour integrals, $$\begin{aligned}
\label{eq:94}
\int \frac{d^4 l_1}{i \pi^2} \frac{1}{D_1 D_2 D_3 D_4} \to \frac{2}{t(s+t)}\oint
\frac{dx_1dx_2 dx_3 dx_4}{(2 \pi i)^4} \frac{1}{D_1
D_2 D_3 D_4}
=
\left\{\begin{array}{cc}
\frac{1}{4st} & \text{on }V_1\\
-\frac{1}{4st} & \text{on }V_2\\
\end{array}
\right.\end{aligned}$$ and $$\begin{aligned}
\label{spurious_contour}
\int \frac{d^4 l_1}{i \pi^2} \frac{l\cdot \omega}{D_1 D_2 D_3 D_4} \to\frac{2}{t(s+t)}\oint
\frac{dx_1dx_2 dx_3 dx_4}{(2 \pi i)^4} \frac{x_4}{D_1
D_2 D_3 D_4}
=
\left\{\begin{array}{cc}
\frac{1}{8s} & \text{on }V_1\\
\frac{1}{8s} & \text{on }V_2\\
\end{array}
\right.\end{aligned}$$ We have two weights $\omega^{(1)}$ and $\omega^{(2)}$. From the contour integral of spurious term , $$\label{eq:95}
\omega^{(1)} \frac{1}{8 s}+\omega^{(2)} \frac{1}{8 s} =0 \,.$$ Hence $\omega^{(2)}=-\omega^{(1)}$. Normalize the weights for the scalar box integral, $$\label{eq:13}
\omega^{(1)}=2 st,\quad \omega^{(2)}=-2 st\,.$$
Hence $$\begin{aligned}
\label{eq:96}
c_{box}&=2 st \cdot \frac{2}{t(s+t)} \big(\oint_{V_1} \frac{dx_1dx_2 dx_3
dx_4}{(2\pi i)^4} \frac{N}{D_1 D_2 D_3 D_4} -\oint_{V_2}\frac{dx_1dx_2 dx_3
dx_4}{(2\pi i)^4} \frac{N}{D_1 D_2 D_3 D_4} \big)\nn\\
&= \frac{1}{2} S^{(1)}+\frac{1}{2} S^{(2)}\,.\end{aligned}$$ which is the same as .
The two-loop maximal unitarity method was first invented in [@Kosower:2011ty] for the $4D$ massless double box diagram, in an elegant way of determining all contours and corresponding contour weights. Afterwards, this method was generalized for the double box diagram with external massive legs [@Johansson:2012sf; @Johansson:2012zv; @Johansson:2013sda; @CaronHuot:2012ab].
In general, for multi-loop cases, the contour integrals are multivariate, and can be complicated in some cases. There are complicated issues with :
1. The solution set $V_i$ is not a rational variety. For example, $V_i$ can be an elliptic curve or a hyper-elliptic curve. Then contour integrals are not only residue computations, but also integrals over the fundamental cycles. Some of these cases are treated by maximal unitarity with complete elliptic integrals or hyper-ellitpic integrals [@Sogaard:2014jla; @Georgoudis:2015hca]. There is a rich algebraic geometry structure in this direction and these integrals are important for the LHC physics. But we are not going to cover this direction in these notes, since the background knowledge of algebraic curves needs to be introduced.
2. The residue is multivariate and Cauchy’s formula does not work since the Jacobian at the pole is zero. For example, the $4D$ slashed box diagram and the $4D$ triple box diagram both have complicated multivariate residues. We discuss this direction in the rest of this chapter.
Note that, in a different context, [@Henn:2014qga; @Caron-Huot:2014lda; @Remiddi:2016gno; @Primo:2016ebd] contour integrals like from Feynman integrals are also important for determining the [*canonical*]{} MI basis [@Henn:2013pwa; @Lee:2014ioa; @Meyer:2016slj], for which the differential equation [@Kotikov:1990zk; @Kotikov:1990kg; @Kotikov:1991pm; @Bern:1993kr; @Remiddi:1997ny; @Gehrmann:1999as] has a simple $\epsilon$-form[^8].
### A multivariate residue example
Consider the $4D$ three-loop massless triple box diagram (Figure. \[tribox\]). There are $10$ inverse propagators, $$\begin{aligned}
D_1 = {} & l_1^2\;, &
D_2 = {} & l_2^2\;, &
D_3 = {} & l_3^2\;, &
D_4 = {} & (l_1+k_1)^2\;, \nn \\
D_5 = {} & (l_1-k_2)^2\;, &
D_6 = {} & (l_2+k_3)^2\;, &
D_7 = {} & (l_2-k_4)^2\;, &
D_8 = {} & (l_3+k_1+k_2)^2\;, \nn \\
D_9 = {} & (l_1-l_3-k_2)^2\;, &
D_{10} = {} & (l_3-l_2-k_3)^2\;,\end{aligned}$$ with $k_1^2=k_2^2=k_3^2=k_4^2=0$.
![Three loop triple box diagram[]{data-label="tribox"}](graphs/tribox.eps)
We parameterize loop momenta with the spinor helicity formalism [@Dixon:1996wi], $$\begin{aligned}
\ell_1(\alpha_1,\dots,\alpha_4) = {} &
\alpha_1 k_1+\alpha_2 k_2+
\alpha_3\frac{{\langle23\rangle}}{{\langle13\rangle}}1\tilde 2+
\alpha_4\frac{{\langle13\rangle}}{{\langle23\rangle}}2\tilde 1\;,
\nn \\[1mm]
\ell_2(\beta_1,\dots,\beta_4)\, = {} &
\beta_1 k_3+\beta_2 k_4+
\beta_3
\frac{{\langle14\rangle}}{{\langle13\rangle}} 3\tilde 4+
\beta_4
\frac{{\langle13\rangle}}{{\langle14\rangle}}
4\tilde 3\;.
\nn \\[1mm]
\ell_3(\gamma_1,\dots,\gamma_4)\, = {} &
\gamma_1 k_2+\gamma_2 k_3+
\gamma_3
\frac{{\langle34\rangle}}{{\langle24\rangle}}
2\tilde 3+
\gamma_4
\frac{{\langle24\rangle}}{{\langle34\rangle}}
3\tilde 2\;,\end{aligned}$$ The cut solution for $D_1=D_2=\ldots D_{10}=0$ can be found by primary decomposition [@Badger:2012dv; @Sogaard:2013fpa], $$I=I_1\cap \ldots \cap I_{14}.$$ There are $14$ independent solutions, each of which can be parameterized rationally. For example, on $V_1=\mathcal Z(I_1)$ the triple box Feynman integral with numerator $N(l_1,l_2,l_3)$ becomes a contour integral, $$\begin{aligned}
\frac{1}{(2\pi i)^2t^2s^{8}}\oint
\frac{dz_1\wedge dz_2 N(l_1,l_2,l_3)|_{V_1}}{(1+z_1)(1+z_2)(1+z_1-\chi z_2)}\;,\end{aligned}$$ where the denominators come from the Jacobian of evaluating holomorphic delta functions in $D_1,\ldots D_{10}$. $z_1$, $z_2$ are free variables parametrizing this solution. The difficulty is that on this cut solution, loop momenta $l_i$ are not polynomials in $z_1$ and $z_2$, but rational functions in $z_1$ and $z_2$ [@Sogaard:2013fpa]. Hence we get contour integrals like, $$\label{xbox_residue}
\frac{1}{(2\pi i)^2}\oint
\frac{dz_1\wedge dz_2 P(z_1,z_2)}{(1+z_1)(1+z_2)(1+z_1-\frac{t}{s} z_2)z_2}\;.$$ where $P(z_1,z_2)$ is a polynomial in $z_1$ and $z_2$. $(z_1,z_2)\to
(-1,0)$ is a multivariate residue. Note that at this point, $3$ factors in the denominator vanish, $$1+z_1,\quad z_2 \quad 1+z_1-\frac{t}{s} z_2\,.$$ Hence, the Jacobian of denominators must be vanishing at $(-1,0)$, so the residue cannot be calculated by inverse Jacobian (Cauchy’s theorem). Note that we cannot directly use polynomial division to simplify the integrand, since $I=\la 1+z_1 , z_2, 1+z_1-\frac{t}{s}
z_2\ra =\la 1+z_1,z_2\ra\not=\la 1\ra$. So Hilbert’s weak Nullstellensatz (Theorem \[weak\_Nullstellensatz\]) cannot be used here to reduce the number of denominators.
Difficult multivariate residues also arise from the maximal cut of integrals with doubled propagators from two-loop integrals. In the rest of this chapter, we use algebraic geometry techniques to compute these residues efficiently.
Basic facts of several complex variables
----------------------------------------
### Multivariate holomorphic functions
We first review some properties of several complex variables [@MR1288523; @MR1045639; @MR2176976].
Complex variables for $\C^n$ are $z_i=x_i+i y_i$ and the basis for the tangent space is $$\frac{\partial}{\partial
z_i}=\frac{1}{2}\big(\frac{\partial}{\partial x_i}-i
\frac{\partial}{\partial y_i}\big),\quad
\frac{\partial}{\partial
\bz_i}=\frac{1}{2}\big(\frac{\partial}{\partial x_i}+i \frac{\partial}{\partial y_i}\big).$$ For a point $\xi=\xis$ in $\C^n$, the (open) polydisc with radius $r$ is $$\Delta(\xi,r)=\{\zs\big||z_i-\xi_i|<r, \ i=1,\ldots n\},.$$
A differentiable function $f$ on $U$, an open set of $\C^n$, is [ *holomorphic* ]{} if, $$\frac{\partial f}{\partial
\bz_i}=0,\quad i=1,\ldots n.$$
Let $f$ a function holomorphic in $\Delta(\xi,r)$ and continuous on $\bar \Delta(\xi,r)$. Then for $z\in \Delta(\xi,r)$, $$f(z)=\frac{1}{(2\pi i)^n}\int_{|w_1-\xi_1|=r}\ldots \int_{|w_n-\xi_n|=r} \frac{ f(w_1,\ldots,w_n)dw_1
\ldots dw_n}{(w_1-z_1)\ldots (w_n-z_n)}.$$
Apply one-variable Cauchy’s formula $n$ times [@MR1045639].
From the Taylor expansion of $1/(w_i-z_i)$ in $(z_i-\xi_i)$, $f(z)$ has a multivariate Taylor expansion in $\bar
\Delta(\xi,r)$. Hence like in the univariate case, a holomorphic function is an analytic function. Similarly, for two holomorphic functions $f$ and $g$ on a connected open set $U\subset \C^n$, if $f=g$ on an open subset of $U$ then $f=g$ on $U$.
However, the pole structure of a multivariate function is very different from that in the univariate case.
\[Hartog\] Let $U$ be an open set of $\C^n$, $n>1$. Let $K$ be a compact subset of $U$ and $U-K$ be connected. Then any holomorphic function on $U-K$ extends to a holomorphic function of $U$.
See Hörmander [@MR1045639 Chapter 2].
Consider $n=2$, $U$ is the polydisc $\Delta(0,r)$ and $K=O=\{(0,0)\}$ in Theorem \[Hartog\]. Suppose that $f(z_1,z_2)$ is holomorphic in $U-K$. Define the function $$g(z_1,z_2)=\frac{1}{2 \pi i}\int_{|w_2|=r'} \frac{f(z_1,w_2)dw_2}{w_2-z_2}\,,$$ where $0<r'<r$. Clearly $g$ is well defined in the smaller polydisc $\Delta(0,r')$. $g$ is holomorphic in both $z_1$ and $z_2$. If $z_1\not =0$, then by the univariate Cauchy’s formula, $g(z_1,z_2)=f(z_1,z_2)$. $f=g$ in $\Delta(0,r')\cap
\{z|z_1\not=0\}$, hence $f=g$ in $\Delta(0,r')-O$. Define a new function $$F(z)=\left\{
\begin{array}{cc}
g(z) & z\in \Delta(0,r')\\
f(z) & z\not \in \Delta(0,r')\ \text{but } z \in \Delta(0,r)
\end{array}
\right. \,$$ Clearly $F$ is holomorphic in $U$ and $F=f$ in $U-K$, so $F$ is the extension.
Hartog’s extension means the pole of a multivariate holomorphic $f$ has complicated structure, say, cannot be a point. It also implies that we should not study the space of holomorphic functions on an open set like $U-K$ in Theorem \[Hartog\], since these functions can always be extended.
Laurent expansion of a multivariate holomorphic function is also subtle.
A subset $\Omega$ of $\C^n$ is called a Reinhardt domain, if $\Omega$ is open, connected and for any $\zs\in \Omega$, $(e^{i\theta_1}
z_1,\ldots , e^{i\theta_n}
z_n)\in \Omega$, $\forall \theta_1 \in \mathbb R,\ldots, \theta_n
\in \mathbb R$. This is a generalization of an annulus on the complex plane.
Let $f$ be a holomorphic function on a Reinhardt domain $\Omega$. Then there exists a Laurent series, $$\sum_{(\alpha_1,\ldots,\alpha_n) \in Z^n} c_{\alpha_1\ldots \alpha_n}z_1^{a_1} \ldots z_n^{a_n}\,,$$ which is uniformly convergent to $f$ on any compact subset of $\Omega$.
See Scheidemann [@MR2176976].
A multivariate function $f$ may be defined over a domain which is not a Reinhardt domain. For a simple example, the function $f=1/(z_1-z_2)$ is defined on $U=\{(z_1,z_2)|z_1\not =z_2,\ (z_1,z_2)\in
\C^2\}$, where $U$ is not a Reinhardt domain. It is hard to define the Laurent series for $f$ in $U$. Instead, consider $\Omega=\{(z_1,z_2)||z_1|>|z_2|,\ (z_1,z_2)\in
\C^2\}$. Then $\Omega$ is a Reinhardt domain, and on $\Omega$, $$\frac{1}{z_1-z_2}=\sum_{n=0}^\infty z_2^n z_1^{-n-1}, \quad
(z_1,z_2)\in \Omega$$ This Laurent series does not converge outside $\Omega$.
We turn to complex manifolds.
A complex manifold $M$ is a differentiable manifold, with an open cover $\{U_\alpha\}$ and coordinate maps $\phi_\alpha: U_\alpha \to
\C^n$, such that all $\phi_\alpha\phi_\beta^{-1}$’s components are holomorphic on $\phi_\beta(U_\alpha \cap U_\beta)$ for $U_\alpha \cap U_\beta \not = \emptyset$.
\[CPn\] Define $\CP^n$ as the quotient space $\C^{n+1}-\{0,\ldots 0\}$ over $$(Z_0,\ldots ,Z_{n})\sim (\lambda Z_0,\ldots \lambda Z_{n}),\quad
\lambda \in\C^*
\label{rescaling}$$ The equivalence class of $(Z_0,\ldots,Z_n)$ in $\CP^n$ is denoted as $[Z_0, \ldots, Z_n]$, which is the [*homogeneous coordinate*]{}. Define an open cover of $\CP^n$, $\{U_0,U_1,\ldots,U_n\}$, where $$U_i=\{[Z_0,\ldots,Z_n]|Z_i\not=0\},\quad i=0,\ldots, n\,.$$ For each $U_i$, the coordinate map $\phi_i: \ U_i \to \C^n$ is $$\phi_i([Z_0,\ldots,Z_n])=(\frac{Z_0}{Z_i},\ldots,
\widehat{\frac{Z_i}{Z_i}},\ldots, \frac{Z_n}{Z_i})\equiv
(z_0^{(i)},\ldots,\widehat{z_i^{(i)}}\ldots z_n^{(i)}).$$ Hence, for $i<j$, $$\label{projective_atlas}
\phi_i \phi_j^{-1}(z_0^{(j)},\ldots,\widehat{z_j^{(j)}}\ldots
z_n^{(j)})=(\frac{z_0^{(j)}}{z_i^{(j)}},\ldots,\widehat{\frac{z_i^{(j)}}{z_i^{(j)}}}\ldots,\frac{1}{z_i^{(j)}},\ldots
\frac{z_n^{(j)}}{z_i^{(j)}})\,.$$ Since on $\phi_j (U_i\cap U_j)$, $z_i^{(j)}\not =0$, the transformation is holomorphic. Hence $\CP^n$ is a compact complex space. In particular, we may identify $U_0$ as $\C^n$.
For a homogeneous polynomial $F(Z_0,\ldots Z_n)$, the equation $F(Z_0,\ldots
Z_n)=0$ is well defined, since the rescaling does not affect the value $0$.
Like the real manifold case, we can also study sub-manifolds of a manifold. In particular, the codimension-1 case is very important for our discussion of residues in this chapter.
An analytic hypersurface $V$ of a complex manifold $M$ is a subset of $M$ such that $\forall p\in V$, there exists a neighborhood of $p$ in $M$, such that locally $V$ is the set of zeros of a holomorphic function $f$, defined in this neighborhood.
Like in the algebraic variety case (Theorem \[variety\_decomposition\]), any analytic hypersurface uniquely decomposes as the union of irreducible analytic hypersurfaces. [@MR1288523].
For a complex manifold $M$, a divisor $D$ is a locally finite formal linear combination $$D=\sum_i c_i V_i\,,$$ where each $V_i$ is an irreducible analytic hypersurface in $M$.
### Multivariate residues
Recall that for in that univariate case, the residue of a meromorphic function $h(z)/f(z)$ at the point $\xi$, is defined as $$\Res{}_\xi\bigg(\frac{h(z)}{f(z)}dz\bigg)=\frac{1}{2\pi i}\oint_{|z-\xi|=\epsilon} \frac{h(z)dz}{f(z)}\,.$$ where $f$ and $h$ are holomorphic near $\xi$.
To define a multivariate residue in $\C^n$, we need $n$ vanishing denominators $f_1, \ldots f_n$ such that $f_1(z)=\ldots =f_n(z)=0$ defines isolated points.
\[Grothendieck\_residue\] Let $U$ be a ball in $\C^n$ centered at $\xi$, i.e. $||z-\xi||<\epsilon$ for $z\in U$. Assume that $f_1(z), \dots, f_n(z)$ are holomorphic function in $U$, and have only one isolated common zero, $\xi$ in $U$. Let $h(z)$ be a holomorphic function in a neighborhood of $\bar U$. Then for the differential form $$\label{omega}
\omega=\frac{h(z) dz_1 \wedge \cdots \wedge dz_n}{f_1(z) \cdots f_n(z)}\;,$$ the (Grothendieck) residue [@MR1288523] at $\xi$ is defined to be $$\label{local_residue}
\Res{}_{\{f_1,
\dots, f_n\},\xi}(\omega)=\bigg(\frac{1}{2\pi i}\bigg)^n\oint_{\Gamma}
\frac{h(z) dz_1 \wedge \cdots \wedge dz_n}{f_1(z) \cdots f_n(z)}\;,$$ where the contour $\Gamma$ is defined by the real $n$-cycle $\Gamma=\{z: z\in $U$, ~|f_i(z)|=\epsilon\}$ with the orientation specified by $d(\arg f_1) \wedge \cdots \wedge d(\arg f_n)$.
Note that $\Gamma$ in this definition ensures that $f_i$’s are nonzero for this contour integral. A naive contour choice $\Gamma'=\{z: z\in $U$, |z_i-\xi_i|=\epsilon,\ \forall i\}$ in general does not work. For instance, $$\frac{1}{(2\pi i)^2} \oint_{\Gamma'} \frac{dz_1\wedge dz_2}{(z_1+z_2) (z_1-z_2)},$$ with $\Gamma'=\{z: z\in $U$, |z_1|=\epsilon,|z_2|=\epsilon
\}$ is ill-defined. On this contour, both $(z_1+z_2)$ and $(z_1-z_2)$ have zeros.
Note that if we permute functions $\{f_1 ,\ldots , f_n\}$, the differential form is invariant but the contour orientation will be reversed if the permutation is odd. This is a new feature of multivariate residue, hence in Definition \[Grothendieck\_residue\], we keep $\{f_1,\ldots, f_n\}$ in the subscript.
Clearly, if $f_1(z)=f_1(z_1),\ldots, f_n(z)=f_n(z_n)$, then $$\label{factorable_residue}
\Res{}_{\{f_1,
\dots, f_n\},\xi}(\omega)=\bigg(\frac{1}{2\pi i}\bigg)^n\oint
\frac{dz_1}{f_{z_1}} \oint\frac{dz_2}{f_{z_2}} \ldots \oint \frac{h(z)dz_n}{f_{z_n}} \,,$$ i.e., the multivariate residue becomes iterated univariate residues.
In Definition \[Grothendieck\_residue\], if the Jacobian of $f_1,
\ldots, f_n$ in $z_1,\ldots z_n$ at $\xi$ is nonzero, we call this residue non-degenerate. Otherwise it is called degenerate.
If the residue in Definition \[Grothendieck\_residue\] is non-degenerate, then $$\label{non_degenerate_residue}
\Res{}_{\{f_1,
\dots, f_n\},\xi}(\omega)=\frac{h(\xi)}{J(\xi)}\;.$$ where $J(\xi)$ is the Jacobian of $f_1,
\ldots, f_n$ in $z_1,\ldots z_n$ at $\xi$ .
In this case, we can use implicit function theorem to treat $f_i$’s as coordinates and compute the residue directly [@MR1288523].
\[vanishing\_residue\] If $h$ in Definition \[Grothendieck\_residue\] satisfies, $$h(z)=q_1(z) f_1(z) + \ldots + q_n(z) f_n(z), \quad z\in U$$ where $q_i$’s are holomorphic functions in $U$. Then $$\Res{}_{\{f_1,
\dots, f_n\},\xi}(\omega)=0\;.$$
This is from Stokes’ theorem [@MR1288523].
In general a multivariate residue is not of a form like or non-degenerate. Unlike the univariate case, Laurent expansion, even if it is defined in a subset, in general does not help the evaluation of multivariate residues. Hence we need a sophisticated method to compute residues, like .
Let $M$ be a compact complex manifold, and $D_1 \ldots D_n$ be divisors of $M$, such that $D_1\cap \ldots \cap D_n=S$ is a finite set. If $\omega$ is a holomorphic $n$-form defined in $M-D_1\cup \ldots
\cup D_n$ whose polar divisor is $D=D_1+\ldots D_n$, then $$\sum_{\xi \in S}\Res{}_{\{D_1,
\dots, D_n\},\xi}(\omega) =0 .$$
This is from Stokes’ theorem for a complex manifold. See Griffiths and Harris [@MR1288523].
Note that to consider global residue theorem, we need a compact complex manifold but $\C^n$ is not. So residues on a complex manifold have to be defined. $\omega$ has the polar divisor $D=D_1+\ldots D_n$ means, near a point $\xi\in S$, locally each $D_i$ is a divisor of a holomorphic function $f_i$ and $\omega$ has the local form . Again, the subscript $\{D_1,
\dots, D_n\}$ indicates the ordering of denominators.
Consider the meromorphic differential form in $\C^n$, $$\omega=\frac{dz_1 \wedge dz_2}{(z_1+z_2)(z_1-z_2)}\,.$$ Extend $\omega$ to a meromorphic differential form in $\CP^2$ (Example \[CPn\]). Let $[Z_0,Z_1,Z_2]$ be the homogeneous coordinate. In the patch $U_0$, define $z_1=Z_1/Z_0$, $z_2=Z_2/Z_0$. For the patch $U_1$, let $u_0=Z_0/Z_1$, $u_2=Z_2/Z_1$. Then on $U_0\cap U_1$, $$z_1=\frac{1}{u_0},\quad z_2=\frac{u_2}{u_0}\,.$$ After a change of variables, on $U_0\cap U_1$, $$\omega=\frac{du_0 \wedge du_2}{u_0(u_2-1)(u_2+1)}\,.$$ Similarly, for the patch $U_2$, let $v_0=Z_0/Z_2$, $v_1=Z_1/Z_2$. On $U_0\cap U_2$, $$\omega=\frac{dv_0 \wedge dv_1}{v_0(v_1-1)(v_1+1)}\,.$$ Then in $\CP^2$, $\omega$ is defined except on $3$ irreducible hypersurfaces $V_1=\{Z_0=0\}$, $V_2=\{Z_1+Z_2=0\}$ and $V_3=\{Z_1-Z_2=0\}$. To apply the global residue theorem, consider $$D_1=V_0+V_1,\quad D_2=V_2 \,.$$ Then $D=D_1+D_2$ is the polar divisor of $\omega$. $D_1\cap
D_2=\{P_1,P_2\}$, where $P_1=[1,0,0]$ and $P_2=[0,1,1]$. The global residue theorem reads, $$\Res{}_{\{D_1, D_2\},P_1} (\omega)+\Res{}_{\{D_1, D_2\},P_2} (\omega)=0 \,.$$ Explicitly by , $$\Res{}_{\{D_1, D_2\},P_1} (\omega)=-\half,\quad \Res{}_{\{D_1, D_2\},P_2} (\omega)=\half \,.$$ Note that if we consider a different set of divisors, say, $D_1'=V_1$,$D_2'=V_0+V_2$, then $D_1'\cap
D_2'=\{P_1,P_3\}$, where $P_3=[0,1,-1]$. So there is another relation $\Res{}_{\{D_1', D_2'\},P_1}
(\omega)+\Res{}_{\{D_1', D_2'\},P_3} (\omega)=0$, and, $$\Res{}_{\{D_1', D_2'\},P_3} (\omega)=\half.$$ We see that for a multivariate case, there can be several global residues relations for one meromorphic form.
Multivariate residues via computational algebraic geometry
-----------------------------------------------------------
There are several algorithms for calculating multivariate residues in algebraic geometry. We mainly use two methods, the [*transformation law*]{} and the [*Bezoutian*]{}.
### Transformation law
For the residue in Definition \[Grothendieck\_residue\], and $g_i=\sum_j a_{ij} f_j$, where $a_{ij}$ are locally holomorphic functions near $\xi$. We have $$\label{transformation}
\Res{}_{\{f_1,
\dots, f_n\},\xi}(\omega)= \Res{}_{\{g_1,
\dots, g_n\},\xi}(\det A~\omega)$$ where $A$ is the matrix $(a_{ij})$.
See Griffiths and Harris [@MR1288523].
Note that this is a transformation of denominators, not the complex variables. In particular, if $f_1,\ldots f_n$ are polynomials, we can calculate the Gröbner basis for $I=\langle
f_1,\ldots f_n\rangle$ in to get a set of polynomial $g_i$’s, such that each $g_i$ is univariate. $g_i(z)=g_i(z_i)$ (Theorem \[Elimination\]). Then the r.h.s of can be calculated as univariate residues.
Consider the residue of $$\omega=\frac{dx \wedge dy }{f_1 f_2}$$ at $(0,0)$, where $f_1=a y^3 + x^2 + y^2$, $f_2=x^3 + x y - y^2$. This is a degenerate residue. By computations, $$A=\left(
\begin{array}{cc}
-\frac{2 a x^2+a x-a y x-a y+1}{a^2} & \frac{a x^4-a y x^2+a y^2 x-x+a y^2-y}{a} \\
\frac{a^2 y^5-2 a y^3-a x^2 y^2+x y+y+x^2}{a^3} & \frac{a x y^2-y-x}{a^3} \\
\end{array}
\right),$$ and, $$\{g_1,g_2\}=\big\{\frac{x^2 (a^2 x^5-3 a x^2-a x-1)}{a^2},\frac{y^3
(a^3 y^5-2 a^2 y^3+a y+1)}{a^3}\big\}. \,$$ Note that $g_1$, $g_2$ are univariate polynomials. Hence by , $$\Res{}_{\{f_1,f_2\},(0,0)}(\omega)=a(1-a)\,.$$
Consider the $4D$ triple box’s maximal cut , near $z_1=-1$ and $z_2=0$, $$\omega=\frac{dz_1\wedge dz_2 P(z_1,z_2)}{(1+z_1)(1+z_2)(1+z_1-\frac{t}{s}
z_2)z_2}\;.$$ Define $V_1=\{1+z_1=0\}$, $V_2=\{z_2=0\}$ and $V_3=\{1+z_1-\chi z_2\}$, which are irreducible hypersurfaces. So locally the polar divisor of $\omega$ is, $$D=V_1+V_2+V_3.$$ To define multivariate residues, we may consider two divisors $D_1=V_1+V_2$ and $D_2=V_3$. This corresponds to the denominator definitions, $f_1=(1+z_1)z_2$ and $f_2=(1+z_1-t/s z_2)$. Using to change denominators, we find that, for example if $P=1$, $$\Res{}_{\{f_1,f_2\},(0,0)}(\omega)=s/t.$$ Note that there are different ways to define the divisors for $\omega$, for instance, $D_1'=V_1+V_3$ and $D_2'=V_2$, i.e. $f_1'=(1+z_1)(1+z_1-\chi z_2)$ and $f_2'=z_2$. Multivariate residue dependence on the definition of divisors, for example if $P=1$, $$\Res{}_{\{f_1',f_2'\},(0,0)}(\omega)=0\not=\Res{}_{\{f_1,f_2\},(0,0)}(\omega).$$ Hence we need to consider all possible divisor definitions.
We calculated all $64$ residues from the maximal unitarity cut of a three-loop triple box diagram [@Sogaard:2013fpa], by Cauchy’s theorem and transformation law. Then the contours weights are determined by spurious integrals and IBPs. We used contour weights to derive the triple box master integrals part of $4$-gluon $3$-loop pure-Yang-Mills amplitude, which agrees with that from integrand reduction method [@Badger:2012dv].
For integrals with doubled propagators, we can also use the transformation law to compute residues for contour integrals [@Sogaard:2014ila; @Sogaard:2014oka].
1. Usually, computation in is heavy. It is better to first compute in order, $G(I)=\{F_1,\ldots
F_k\}$ and find the relations $F_i=b_{ij} f_j$. Then compute in a block order to get univariate polynomials $g_i(z_i)$. Divide $g_i(z_i)$ towards $G(I)$ and use $b_{ij}$’s, we get the matrix $A$.
2. This method also works if $f_1,\ldots f_n$ are holomorphic functions but not polynomials. Replace $f_i$’s by their Taylor series, we can apply the Gröbner basis computation.
### Bezoutian
Multivariate residue computation via , may be quite heavy since the transformation matrix $A$ may contain high-degree polynomials. The Bezoutian method provide a different approach.
With the convention of Definition \[Grothendieck\_residue\], for $\xi\in \C^n$, define the local symmetric form, for locally holomorphic functions $N_1$ and $N_2$, $$\begin{gathered}
\langle N_1, N_2\rangle_\xi\equiv\Res{}_{\{f_1,
\dots, f_n\},\xi}\bigg(\frac{N_1 N_2 \dnz}{f_1\ldots f_n}\bigg)\,.
\end{gathered}$$ If $f_1,\ldots f_n$, $N_1, N_2$ are globally holomorphic in $\C^n$ and $\mathcal
Z(\{f_1,\ldots f_n\})$ is a finite set, then the global symmetric form is $$\begin{gathered}
\langle N_1, N_2\rangle\equiv \sum_{\xi\in \mathcal
Z(\{f_1,\ldots f_n\})} \Res{}_{\{f_1,
\dots, f_n\},\xi}\bigg(\frac{N_1 N_2\dnz}{f_1\ldots f_n}\bigg)\,.
\end{gathered}$$
For the rest of the discussion, we assume $f_1,\ldots f_n$, $N_1, N_2$ are polynomials. In the previous Chapter, we used the ring $R=\mathbb F[x_1,\ldots,x_n]$ and ideals to study algebraic varieties. Here to discuss local properties of a variety, we need the concept of the local ring.
\[local\_ring\] Consider $R=\mathbb C[x_1,\ldots,x_n]$, for a point $\xi \in \C^n$, $R_\xi$ is the set of rational functions, $$R_\xi\equiv\bigg\{\frac{f(z)}{g(z)}\big|g(\xi)\not=0,\quad f,g\in R\bigg\}\,.$$ For an ideal $I$ in $R$, we denote $I_\xi$ as the ideal in $R_\xi$ generated by $I$. If $\xi\in \mathcal Z(I)$, and $\dim_{\C}R_\xi/I_\xi<\infty$, we define the multiplicity of $I$ at $\xi$ as $\dim_{\C}R_\xi/I_\xi$.
Let $I$ be $\la f_1,\ldots f_n\ra$. From Proposition \[vanishing\_residue\], it is clear that $\la ,\ra$ is defined in $R/I$ and $\la ,\ra_\xi$ is defined in $R_\xi/I_\xi$, because any polynomial in the ideal $I$ or the localized ideal $I_\xi$ must yield zero residue.
Let $I=\la f_1,\ldots, f_n\ra$ be an ideal in $\C[x_1,\ldots x_n]$, $\mathcal Z(I)$ is a finite set. Then $\la ,\ra$ is non-degenerate in $R/I$ and $\la ,\ra_\xi$ is non-degenerate in $R_\xi/I_\xi$.
See Griffiths and Harris [@MR1288523], Dickenstein et al. [@Dickenstein:2010:SPE:1965470].
Non-degeneracy of $\la, \ra$ implies that given a linear basis $\{e_1,
\ldots e_k\}$ of $R/I$, there is a dual basis $\{\Delta_1,
\ldots \Delta_n\}$, such that, $$\label{eq:5}
\langle e_i, \Delta_j\rangle =\delta_{ij}\,.$$ If these two bases are explicitly found, then we can compute any $\la
N_1, N_2\ra$. In particular, the sum of residues (in affine space) of $\omega=N\dnz /(f_1\ldots f_n)$ is obtained algebraically, $$\sum_{\xi\in \mathcal Z(I)}\Res{}_{\{f_1,\ldots f_n\},\xi}(\omega)=\la N, 1\ra=\la \sum_{i=1}^k c_i
e_i, \sum_{j=1}^k\mu_j \Delta_j\ra=\sum_{i=1}^k c_i \mu_i\,,$$ where in the second equality, we expand $N=\sum_i c_i e_i $ and $1=\sum_i \mu_i \Delta_i$. $c_i$’s and $\Delta_i$’s are complex numbers.
Explicitly, $\{e_i\}$’s are found by using of $I$ in , $G(I)$. They are monomials which are not divisible by any leading term in $G(I)$. The dual basis can be found via the Bezoutian matrix [@Dickenstein:2010:SPE:1965470]. First, calculate the Bezoutian matrix $B=(b_{ij})$, $$\begin{gathered}
\label{eq:4}
b_{ij}\equiv \frac{f_i(y_1,\ldots y_{j-1},z_j,\ldots,z_n)}{z_j-y_j}
-\frac{f_i(y_1,\ldots y_{j},z_{j+1},\ldots,z_n)}{z_j-y_j}\,,\end{gathered}$$ where $y_i$’s are auxiliary variables. Let $\tilde I$ be the ideal in $\C[y_1,\ldots y_n]$ which is obtained from $I$ after the replacement $z_1 \to
y_1,\ldots, z_n \to y_n$.
Then we divide the determinant $\det B$ over the double copy of the Gröbner bases, $G(I)\otimes G(\tilde I)$. The remainder can be expanded as, $$\label{eq:6}
\sum_{i=1}^k \Delta_i(y) e_i(z),$$ here the $\Delta_i(y)$’s, after the backwards replacement $y_1 \to
z_1,\ldots, y_n \to z_n$ become the elements of the dual basis [@Dickenstein:2010:SPE:1965470].
Let $f_1=z_1+9 z_2+14 z_3+6$, $f_2=11 z_2 z_1+12 z_3 z_1+3 z_1+4
z_2+16 z_2 z_3+14 z_3$ and $f_3=2 z_1 z_2+15 z_1 z_3 z_2+5 z_3 z_2+8
z_1 z_3$ be polynomials in $\C[z_1,z_2,z_3]$. Define $$\omega=\frac{z_1^3 dz_1\wedge dz_2 \wedge dz_3}{f_1 f_2 f_3}\,.$$ The Bezoutian determinant in $z_1,z_2,z_3$ and auxiliary variables $y_1, y_2, y_3$ is, $$\begin{gathered}
\det B=-180 y_1^2 z_3+2520 y_1 z_3^2-1485 y_2 y_1 z_2-576 y_1
z_2-1620 y_2 y_1 z_3\nn\\
+1620 y_1 z_2 z_3-408 y_1 z_3-207 y_2 z_2+612 y_2 z_3+2160 y_2 z_2 z_3+165 y_2 y_1^2+64 y_1^2-322 y_2 y_1\nn\\-128 y_1-115 y_2-3360 z_2 z_3^2-952 z_3^2+140 z_2+1372 z_2 z_3+700 z_3\,.\end{gathered}$$ Let $I=\la f_1,f_2,f_3\ra$. Divide $\det B$ towards $G(I)\otimes
G(\tilde I)$, and we get the basis $\{e_i\}$, $$e_1=z_3^3,\quad e_2=z_2 z_3, \quad e_3=z_3^2, \quad e_4=z_2, \quad e_5=z_3,\quad e_6=1\,,$$ and the dual basis $\{\Delta_i\}$, $$\begin{gathered}
\Delta_1=\frac{141120}{23},\quad \Delta_2= 2 (-12420 z_2-22680
z_3-\frac{203652}{23})\,,\nn\\\Delta_3=-22680 z_2-35280
z_3-\frac{335832}{23}\,,\nn\\ \Delta_4= 2 (-22680 z_3^2-12420 z_2 z_3-5436 z_3+1872
z_2+1278)\,,\nn\\ \Delta_5=-35280 z_3^2-22680 z_2 z_3-24528 z_3-5436
z_2-\frac{79884}{23}\,,
\nn\\ \Delta_6=\frac{141120 z_3^3}{23}-\frac{335832 z_3^2}{23}-\frac{203652 z_2 z_3}{23}-\frac{79884 z_3}{23}+1278 z_2+\frac{21282}{23}\,.\end{gathered}$$ From the dual basis, we find the linear relation, $$\begin{gathered}
1=\frac{23}{141120} \Delta_1\,.\end{gathered}$$ By polynomial division, we find $$\begin{gathered}
z_1^3=\frac{1568}{11} e_1 + c_2 e_2 +\ldots c_6 e_6 \mod I\,.\end{gathered}$$ Hence the sum of residues, $$\begin{aligned}
\sum_{\xi\in \mathcal Z(I)}\Res{}_{\{f_1,f_2,f_3\},\xi}(\omega)&=\la z_1^3,
1\ra\nn\\
&=\frac{23}{141120} \la\frac{1568}{11} e_1 + c_2 e_2 +\ldots + c_6
e_6, \Delta_1\ra
=\frac{23}{990}\,.\end{aligned}$$ Note that all points in $\mathcal Z(I)$ and all local residues are irrational, but the sum is rational.
This example is from the CHY formalism of scattering equation [@Cachazo:2013gna; @Cachazo:2013hca; @Cachazo:2013iea; @Cachazo:2014nsa; @Cachazo:2014xea] for $6$-point tree amplitudes. In the CHY formalism, scattering amplitudes are expressed as the sum of residues of CHY integrand. Here we calculate the amplitude without solving the scattering equations [@Sogaard:2015dba]. See alternative algebraic approaches in [@Baadsgaard:2015voa; @Baadsgaard:2015ifa; @Huang:2015yka].
1. Note that by this method, we get the sum of residues (in affine space) purely by and matrix determinant computations. It is not needed to consider algebraic extension or explicit solutions of $f_1=\ldots = f_n=0$.
2. The Bezoutian matrix is just an $n\times n$ matrix, i.e., the size of matrix is independent of the dimension $\dim_\C R/I$. Hence it is an efficient method for computing the sum of residues.
3. If $f_i$’s coefficients are parameters, this method proves that the sum of residues is a rational function of these parameters.
4. In some cases, the sum of residues can also be evaluated by the global residue theorem (GRT). However, in general, there are many poles at infinity so the GRT computation can be messy.
We can also use the Bezoutian matrix to find local residues. One approach is [*partition of unity* ]{} for an affine variety: For each $\xi\in\mathcal Z(I)$, we can find a polynomial $s_\xi$ [@opac-b1094391], such that, $$\begin{gathered}
\label{eq:7}
\sum_{ \xi\in \mathcal Z(I)} s_\xi=1\mod I, \quad s_\xi^2=s_\xi \mod I\nn\,,\\
s_{\xi_i} s_{\xi_j}=0 \mod I,\quad \text{if } i\not =j\,.\end{gathered}$$ Then the individual residue is extracted from the sum of residues, $$\label{eq:8}
\Res{}_{\{f_1,
\dots, f_n\},\xi}(\omega) =\sum_{ u\in \mathcal Z(I)} \Res{}_{\{f_1,
\dots, f_n\},u}(s_{\xi} \omega)\,,$$ where the r.h.s is again obtained by Bezoutian matrix computation [@Dickenstein:2010:SPE:1965470].
Exercises
---------
Consider the maximal unitarity cut of $D=2$ massless sunset diagram with $k_1^2=M^2$ and inverse propagators, $$D_1=l_1^2,\quad D_2=l_2^2,\quad D_3=(l_1+l_2-k_1)^2\,.$$
1. Define an auxiliary vector $\omega$, $k_1\cdot \omega=0$, $\omega^2=-M^2$. Let $e_1=(k_1+\omega)/2$ and $e_2=k_1-\omega$. Parameterize the loop momenta as, $$l_1 =a_1 e_1+ a_2 e_2,\quad l_2=b_1 e_1+b_2 e_2\,.$$ Rewrite the $D_i$’s as polynomials in $a_1,a_2,b_1,b_2$. Define $I=\la
D_1, D_2, D_3\ra$, use or to find independent solutions via primary decomposition, $$I=I_1 \cap \ldots \cap I_m\,.$$
2. Formally define, $$I[s_1,s_2,s_3;N]=\int \frac{d^2 l_1}{(2\pi)^2}\frac{d^2
l_2}{(2\pi)^2} \frac{N}{D_1^{s_1} D_2^{s_2} D_3^{s_3}}\,.$$ Consider the maximal cut of the scalar integral $I[1,1,1;1]$ on each of the cut solutions $\mathcal Z(I_i)$. From the resulting contour integrals, determine all the poles on maximal cut. How many of them are redundant?
3. Denote independent poles as $\{P_1,\ldots P_k\}$ and denote $I[s_1,s_2,s_3;N]_{P_i}$ as the residue of its corresponding contour integral at $P_i$. Compute $I[1,1,1;1]_{P_i}$ for all $P_i$.
4. Denote $$I[s_1,s_2,s_3;N]|_\text{cut}=\sum_i^k w_i I[s_1,s_2,s_3;N]|_{P_i}\,,$$ where the $w_i$’s are weights of contours. Require that $I[1,1,1;N]|_\text{cut}=0$ for spurious terms $N$, $$l_1\cdot \omega,\quad l_2\cdot \omega,\quad (l_1\cdot
\omega)(l_2\cdot k_1),\quad l_2\cdot k_1 -l_1\cdot k_1,\quad (l_2\cdot k_1)^2 -(l_1\cdot k_1)^2\,.$$ What are the linear constraints on the $w_j$’s?
5. Determine the ratio, $
I[2,1,1;1]|_\text{cut}/I[1,1,1;1]|_\text{cut}$. Derive the on-shell integral relation (by determining $c$) $$I[2,1,1;1]= c I[1,1,1;1] +(\text{simpler integrals})\,.$$ Similarly, determine $c'$ in $$\label{eq:9}
I[3,1,1;1]= c' I[1,1,1;1] +(\text{simpler integrals})\,.$$
Consider the meromorphic form, $$\omega=\frac{z_1 dz_1 \wedge dz_2}{(z_1+z_2)(z_1-z_2+z_1 z_2)}\,.$$ Extend $\omega$ on a meromorphic form in $\CP^2$. Find all residues of $\omega$ in $\CP^2$ and verify the global residue theorem explicitly.
Consider the meromorphic form, $$\omega=\frac{N(z_1,z_2)dz_1\wedge dz_2}{(z_1+a z_2)(z_1^3+z_2^2+b
z_1 z_2)}\,.$$
1. Use the transformation law and computation in [Maple]{} or , to compute the residue at $(0,0)$ with $N(z_1,z_2)=1$ and $N(z_1,z_2)=z_1$.
2. Without computation, argue that if $N(z_1,z_2)=z_1^2$ then the residue at $(0,0)$ is zero by Proposition \[vanishing\_residue\].
Consider the meromorphic form, $$\omega=\frac{N(z_1,z_2)dz_1\wedge dz_2}{(z_1+ z_2)(z_1-z_2)(z_1^2+z_2^2+z_1)}\,.$$ Define $f_1=(z_1+ z_2)$, $f_2=(z_1-z_2)$ and $f_3=(z_1^2+z_2^2+z_1)$. Use the transformation law to compute, $$\Res{}_{\{f_1, f_2 f_3\},(0,0)}(\omega),\quad \Res{}_{\{f_1 f_2, f_3\},(0,0)}(\omega)\,.$$
Consider $f_1=z_1^2+ z_1 z_2+a
z_2$, $f_2=z_1^3+z_2^2+b z_1 z_2$ and $I=\la f_1,f_2\ra$.
1. Using in , determine the basis $\{e_i\}$ for $\C[z_1,z_2]/I$.
2. Using Bezoutain matrix, find the dual basis $\{\Delta_i\}$.
3. Compute the sum of residues in $\C^n$ for $$\omega=\frac{z_1 z_2^2 dz_1\wedge dz_2}{f_1 f_2}\,.$$
4. Compute $\la e_i, e_j\ra$ for all elements in $\{e_i\}$. Define $s_{ij}=\la e_i, e_j\ra$ and check that $S=(s_{ij})$ is a symmetric non-degenerate matrix.
Integration-by-parts Reduction and Syzygies {#cha:integr-parts-reduct}
===========================================
Integration-by-parts (IBP) identities [@Tkachov:1981wb; @Chetyrkin:1981qh], arise from the vanishing integration of total derivatives. Combined with symmetry relations, IBPs reduce integrals to master integrals (MIs), i.e., the linearly independent integrals.
An $L$-loop $D$-dimensional [^9] IBP in general has the form, $$\int \frac{d^D l_1}{i\pi^{D/2}} \ldots \int \frac{d^D l_L}{i\pi^{D/2}}
\sum_{j=1}^L \frac{\partial}{\partial l_j^\mu}
\bigg(\frac{v_j^\mu \hspace{0.5mm} }{D_1^{a_1} \cdots D_k^{a_k}}\bigg)
\hspace{1mm}=\hspace{1mm} 0 \,, \label{eq:IBP_schematic}$$ where the vectors components $v_j^\mu$’s are polynomials in the internal and external momenta, the $D_k$’s denote inverse propagators, and the $a_i$’s are integers.
For many multi-loop scattering amplitudes, IBP reduction is a necessary step. After using unitarity and integrand reduction to obtain the integrand basis, we may carry out IBP reduction to get the minimal basis of integrals. For differential equations of Feynman integrals [@Kotikov:1990zk; @Kotikov:1990kg; @Kotikov:1991pm; @Bern:1993kr; @Remiddi:1997ny; @Gehrmann:1999as], after differentiating of the master integrals, we get a large number of integrals in general. Then IBP reduction is required to convert them to a linear combination of MIs, so that the differential equation system is closed.
Multi-loop IBP reduction in general is very difficult. The difficulty comes from the large number of choices of $v_i^\mu$ in : there are many IBP relations and integrals involved. After obtaining IBP relations, we need to apply linear reduction to find the independent set of IBPs. This process usually takes a lot of computing time and RAM. The current standard IBP generating algorithm is [Laporta]{} [@Laporta:2001dd; @Laporta:2000dc]. There are several publicly available implementations of automated IBP reduction: AIR [@Anastasiou:2004vj], FIRE [@Smirnov:2008iw; @Smirnov:2014hma], Reduze [@Studerus:2009ye; @vonManteuffel:2012np], LiteRed [@Lee:2012cn], along with private implementations. IBP computation can be sped up by using finite-field methods [@vonManteuffel:2014ixa; @vonManteuffel:2016xki].
One sophisticated way to improve the IBP generating efficiency is to pick up suitable $v_i^\mu$’s such that (\[eq:IBP\_schematic\]) contains no doubled propagator [@Gluza:2010ws]. Since from Feynman rules, usually we only have integrals without doubled propagators. Hence if we can work with integrals without doubled propagators during the whole IBP reduction procedure, the computation will be significantly simplified. Specifically, when $a_i= 1$, $\forall i=1,\ldots
,k$ in (\[eq:IBP\_schematic\]), if $$\label{no_double_propagator}
\sum_j \frac{\partial D_i}{\partial l_j^\mu} v_i^\mu =\beta_i
D_i,\quad i=1\ldots k\,,$$ where $\beta_i$ is a polynomial in loop momenta, then all double-propagator integrals are removed from the IBP relation .
Note that appears to be a linear equation system for $v_i^\mu$’s and $\beta_i$. However, $v_i^\mu$’s must be polynomials in loop momenta, otherwise the doubled propagators reappear. If we solve by standard linear algebra method, then the solutions are in general rational functions which do not help the IBP reduction. To distinguish with linear equations, is called a [*syzygy*]{} equation. It is not surprising that the form of is closely related to S-polynomials and polynomial division (Definition \[S-polynomial\] and Algorithm \[multivariate\_polynomial\_division\]), so the syzygy can be solved by .
It is convenient to consider IBPs in various integral representations, like Feynman parametric representation or Baikov representation [@Baikov:1996rk]. We believe that the syzygy approach maximaizes its power, when combined with Baikov representation and unitarity cuts. (See using syzygy approach in Feynman parametric representation [@Lee:2013hzt; @Lee:2014tja].) Baikov representation linearizes inverse propagators $D_i$’s so the syzygy equation becomes simpler. Furthermore, It is more efficient to compute IBPs with unitarity cuts, in a divide-and-conquer fashion, than to get complete IBPs at once.
In this chapter, we first introduce the Baikov representation and then review syzygy and the geometric meaning of . We will see that it defines [*polynomial tangent fields of a hypersurface*]{}, or formally [*derivations*]{} in algebraic geometry. Finally we sketch some recent IBP methods [@Ita:2015tya; @Larsen:2015ped].
Baikov representation
---------------------
The basic idea of the Baikov representation [@Baikov:1996rk] is to define inverse propagators and irreducible scalar products (ISP), except $\mu_{ij}$’s, as free variables.
For a simple example, consider the $D=4-2\epsilon$ one-loop box diagram . $$I_\text{box}^D[N]=\int \frac{d^D l}{i \pi^{D/2}} \frac{N}{D_1
D_2 D_3 D_4}\,.$$ Using van Neerven-Vermaseren variables, there are $5$ variables $x_1,x_2,x_3,x_4$ and $\mu_{11}$. Hence it is a $5$-variable system. The solid angel of $(-2\epsilon)$ directions in this integral is irrelevant, hence, $$\begin{aligned}
I_\text{box}^D[N]&=\frac{1}{i \pi^{D/2}} \int d^{-2\epsilon}l^\perp
\int d^4 l^{[4]}\frac{N}{D_1
D_2 D_3 D_4}\nn\\
&= \frac{1}{i \pi^{D/2}} \frac{ \pi ^{\frac{D-4}{2}}}{\Gamma
(\frac{D-4}{2})} \int_0^\infty \mu_{11}^{\frac{D-6}{2}} d\mu_{11} \int d^4 l^{[4]}\frac{N}{D_1
D_2 D_3 D_4}\nn\\
&= \frac{1}{i\pi^2 \Gamma
(\frac{D-4}{2})} \frac{2}{t(t+s)}\int_0^\infty \mu_{11}^{\frac{D-6}{2}} d\mu_{11}
\int dx_1 dx_2 dx_3 dx_4\frac{N}{D_1
D_2 D_3 D_4} \,,\end{aligned}$$ where the factor $2/(t(t+s))$ is the Jacobian of changing variables $l^{[4]} \to x_1,\ldots , x_4$. Note that in this form, the dimension shift identities are manifest.
Since in this case the ISPs are $x_4$ and $\mu_{11}$, we define Baikov variables $z_1,\ldots z_5$ as, $$\label{box_baikov_Direct}
z_1 \equiv D_1,\quad z_2 \equiv D_2,\quad z_3 \equiv D_3,\quad z_4 \equiv D_4,\quad
z_5 \equiv l_1 \cdot \omega\,,$$ Note that the Jacobian $$\frac{\partial(z_1,z_2,z_3,z_4,z_5)}{\partial(x_1,x_2,x_3,x_4,\mu_{11})}=-8$$ is a constant. This is not surprising since by , $$\begin{aligned}
x_1 = \half (z_1-z_2),\quad x_2 =\half (z_2-z_3)+\frac{s}{2},\quad
x_3 = \half (z_4-z_1),\end{aligned}$$ $z_5=x_4$ and $D_1$ is linear in $\mu_{11}$. The inverse map, $(z_1,z_2,z_3,z_4,z_5)\mapsto (x_1,x_2,x_3,x_4,\mu_{11})$ uniquely exists and has polynomial form, $$\begin{aligned}
\mu_{11}&=\frac{1}{4 s t (s+t)}\big( s^2 t^2-2 s^2 t z_2-2 s^2 t
z_4+s^2 z_2^2+s^2 z_4^2-4 s^2 z_5^2-2 s^2 z_2 z_4 \nn\\-2 s t^2 z_1
&-2 s t^2 z_3+2 s t z_1 z_2-4 s t z_1 z_3+2 s t z_2 z_3+2 s t z_1
z_4-4 s t z_2 z_4+2 s t z_3 z_4+t^2 z_1^2 \nn\\ & +t^2 z_3^2-2 t^2
z_1 z_3\big)
\equiv
F(z_1,z_2,z_3,z_4,z_5) \,.\end{aligned}$$ Then, we get the Baikov representation, $$I_\text{box}[N]=\frac{1}{i\pi^2 \Gamma
(\frac{D-4}{2})} \frac{1}{4t(t+s)}
\int_\Omega dz_1 dz_2 dz_3 dz_4 dz_5 F(z_1,z_2,z_3,z_4,z_5)^{\frac{D-6}{2}} \frac{N}{z_1
z_2 z_3 z_4}\,,$$ where $F(z_1,z_2,z_3,z_4,z_5)$ is called the Baikov polynomial. $N$ is a polynomial in $z_1,\ldots z_5$. The integral region $\Omega$ is defined by $F(z_1,z_2,z_3,z_4,z_5) \geq 0$. In general, the integral region of Baikov representation is complicated. However, for the purpose of deriving IBPs, the explicit region is not important [^10].
In practice, after OPP integrand reduction [@Ossola:2006us; @Ossola:2007ax], $N$ is a polynomial of $\mu_{11}$ and at most linear in $(l\cdot \omega)$ . The terms with $\mu_{11}$ lead to scalar integrals in higher dimension , while terms linear in $(l\cdot
\omega)$ are spurious. Hence we assume that $N$ is independent of $(l\cdot \omega)$ and $\mu_{11}$. That implies that we can [ *integrate out*]{} the $\omega$ direction.
Define $V=\sp\{k_1,k_2,k_4\}$ and $V^\sharp$ is the direct sum of $\sp\{\omega\}$ and $(-2\epsilon)$-dimensional spacetime. We decompose $l=l^{[3]}+l^\sharp$ according to $V\oplus V^\sharp$. Then $$(l^\sharp)^2=-\mu_{11}-\frac{s}{t(s+t)} x_4^2\equiv -\lambda_{11}.$$ It is clear that $D_1,\ldots, D_4$ are functions in $x_1,x_2,x_3$ and $\lambda_{11}$ only. We may redefine the Baikov variables, $$z_1 = D_1,\quad z_2 = D_2,\quad z_3 = D_3,\quad z_4 = D_4.$$ Only $4$ variables are needed. Repeat the previous process, $$I_\text{box}[N]=\frac{1}{i\pi^{3/2} \Gamma
(\frac{D-3}{2})} \frac{1}{4\sqrt{-st(t+s)}}
\int dz_1 dz_2 dz_3 dz_4 \tilde F(z_1,z_2,z_3,z_4)^{\frac{D-5}{2}} \frac{N}{z_1
z_2 z_3 z_4}\,.$$ if $N$ has no $l_1\cdot \omega$ dependence. $\tilde F(z_1,z_2,z_3,z_4)=F(z_1,z_2,z_3,z_4,0)$.
Baikov representation also works for higher-loop and both planar and nonplanar diagrams. For example, in a scheme within which all external particles are in $4D$, a two-loop integral with $n\geq 5$ points becomes $$I^{(2)}_{n\geq 5} [N]= \frac{2^{D-6}}{\pi^{5}\Gamma(D-5) J} \int \prod_{i=1}^{11} d z_i \hspace{0.6mm}
F(z)^{\frac{D-7}{2}} \frac{N}{z_1 \cdots z_k}\,,
\label{two-loop-integral-z}$$ where $J$ is a Jacobian without $D$ dependence. Here $F(z)$ is the determinant $\mu_{11}\mu_{22} -\mu_{12}^2$ in Baikov representation. In the same scheme, for a two-loop amplitude with $n< 5$ point, we can integrate out $5-n$ spurious directions and get, $$I^{(2)}_{n<5} [N]= \frac{2^{D-n-1}}{\pi^{n}\Gamma(D-n) J} \int \prod_{i=1}^{2n+1} d z_i \hspace{0.6mm}
F(z)^{\frac{D-n-2}{2}} \frac{N}{z_1 \cdots z_{k}} \,.
\label{two-loop-integral-4z}$$ We leave the Baikov representation of the massless double box diagram as an exercise (Exercise \[dbox\_Baikov\]).
For deriving IBP relations, the overall prefactors are irrelevant. In the rest of this chapter, we neglect these factors in Baikov representation. In general for an $L$-loop integral in a scheme of which external particles are in $4D$, $$\label{Baikov}
I^{(L)}_{n} [N]\propto \int \prod_{i=1}^{\phi(n) L+\frac{L(L-1)}{2}}
dz_i \ F(z)^\frac{D-L-\phi(n)}{2} \frac{N}{z_1 \ldots z_k}\,,$$ where $$\phi(n)=\left\{
\begin{array}{cc}
n, & n<5\\
5,& n\geq 5
\end{array}
\right . .$$ The Baikov polynomial $F(z)$ is the determinant $\det(\mu_{ij})$ if $n\geq 5$, or the determinant $\det(\lambda_{ij})$ is $n<5$.
### Unitarity cuts in Baikov representation
We see that in the Baikov representation, inverse propagators are simply linear monomials. Another feature is that the unitarity cut structure is clear.
Note that now all inverse propagators are linear, so a unitarity cut $D_i^{-1} \to \delta(D_i)$ just means to set certain $z_i$ as zero in . For a given $c$-fold cut ($0\leq c\leq k$), let $\mathcal{S}_\mathrm{cut}$, $\mathcal{S}_\mathrm{uncut}$ and $\mathcal{S}_\mathrm{ISP}$ be the sets of indices labelling cut propagators, uncut propagators and ISPs, respectively. $\mathcal{S}_\mathrm{cut}$ thus contains $c$ elements. Furthermore, we denote $m$ as the total number of $z_j$ variables, $$m=\phi(n) L+\frac{L(L-1)}{2}\,,$$ and set $\mathcal{S}_\mathrm{uncut}=\{r_1,\ldots, r_{k-c}\}$ and $\mathcal{S}_\mathrm{ISP}=\{r_{k-c+1},\ldots, r_{m-c}\}$. Then, by cutting the propagators, $z_i^{-1} \to \delta(z_i),
i \in \mathcal{S}_\mathrm{cut}$, the integrals (\[two-loop-integral-z\]) and (\[two-loop-integral-4z\]) reduce to $$I^{(L)}_\mathrm{cut} [N]= \int \frac{d z_{r_1} \cdots d z_{r_{m-c}} }
{z_{r_1}\cdots z_{r_{k-c}}} N F(z)^{\frac{D - L-\phi(n)}{2}} \bigg|_{z_i=0\,, \forall i\in \mathcal{S}_\mathrm{cut}} \,,
\label{cut-z}$$
\[dbox\_Baikov\_cut\] Consider the quintuple cut for the $D$-dimensional massless double box. (See Exercise \[dbox\_Baikov\]), $D_2=D_3=D_5=D_6=D_7=0$. In this cases, $m=9$. $\mathcal{S}_\mathrm{uncut}=\{1,4\}$, $\mathcal{S}_\mathrm{cut}=\{2,3,5,6,7\}$, $\mathcal{S}_\mathrm{ISP}=\{8,9\}$. The Baikov representation with this cut reads, $$I^{(2)}_\mathrm{penta-cut} [N]= \int \frac{d z_1 d z_4
d z_8 d z_9}{z_1 z_4 } F_{[5]}(z)^{\frac{D - 6}{2}}
N \bigg|_{z_2=z_3= z_5= z_6= z_7=0}\,,$$ where, $$\begin{gathered}
F_{[5]}(z)=\frac{(s t\c -s z_1\c -2 s z_8\c -2 s z_9\c -t z_1\c-t
z_4\c+2 z_4 z_8\c -4 z_8 z_9) (2 s z_1 z_9\c +4 s z_8 z_9\c +t z_1 z_4)}{4 s t (s+t)}.\end{gathered}$$
If we consider the maximal cut $D_1=D_2=\ldots =D_7=0$, then $\mathcal{S}_\mathrm{uncut}=\emptyset$, $\mathcal{S}_\mathrm{cut}=\{1,2,3,4,5,6,7\}$, $\mathcal{S}_\mathrm{ISP}=\{8,9\}$. The Baikov representation on this cut reads, $$I^{(2)}_\mathrm{hepta-cut} [N]= \int
d z_8 d z_9 F_{[7]}(z)^{\frac{D - 6}{2}}
N \bigg|_{z_i=0,\ 1\leq i\leq 7}\,,$$ and the Baikov polynomial on the maximal cut is simply, $$F_{[7]}(z)=\frac{z_8 z_9 (s t-2 s z_8-2 s z_9-4 z_8 z_9)}{t (s+t)}\,.$$
### IBPs in Baikov representation
Note that the higher the unitarity cut is, the simpler the Baikov polynomial becomes. So we try to use cuts as much as possible to reconstruct the full IBP, instead of finding the full set of IBPs at once. Suppose that we consider a $c$-fold cut and make an IBP ansatz as, $$\begin{aligned}
\label{ansatz}
0\hspace{-0.5mm}&=\hspace{-0.5mm}\int \hspace{-0.8mm}
d \bigg( \hspace{-0.4mm} \sum_{i=1}^{m-c} \hspace{-0.8mm}
\frac{(-1)^{i+1} a_{r_i} F(z)^{\frac{D-h}{2}}}{z_{r_1}\cdots z_{r_{k-c}}}
d z_{r_1} \hspace{-0.5mm} \wedge \hspace{-0.5mm} \cdots \widehat{d z_{r_i}} \cdots
\hspace{-0.5mm} \wedge \hspace{-0.5mm} d z_{r_{m-c}}
\hspace{-0.5mm} \bigg)\nn\\
\c&= \c \int \sum_{i=1}^{m-c} \c \bigg(\c \frac{\partial a_{r_i}}{\partial z_{r_i}}
\c \bigg)F(z)^{\frac{D-h}{2}} \omega\c +\c \frac{D\c -\c h}{2}\c \sum_{i=1}^{m-c}
\bigg(a_{r_i}\frac{\partial F}{\partial z_{r_i}}\bigg) F^\frac{D-h-2}{2}
\omega\c - \sum_{i=1}^{k-c} \frac{a_{r_i}}{z_{r_i}}F(z)^{\frac{D-h}{2}} \omega\,.\end{aligned}$$ where $\omega$ is the measure $dz_{r_1}\wedge \ldots
\wedge dz_{r_{m-c}}/(z_{r_1}\cdots z_{r_{k-c}})$ and $h=D-L-\phi(n)$. The second sum contains integrals in $D-2$ dimension while the third sum contains doubled propagators.
If it is required that resulting IBP has no dimensional shift or doubled poles [@Ita:2015tya; @Larsen:2015ped], we have the [*syzygy equations*]{}, $$\begin{aligned}
b F + \sum_{i=1}^{m-c} a_{r_i}\frac{\partial F}{\partial
z_{r_i}} &=0\,, \label{eq:syzygy_1} \\
a_{r_i} + b_{r_i} z_{r_i}&=0\,, \hspace{4mm} i=1,\ldots, k-c \,,
\label{eq:syzygy_2}\end{aligned}$$ where $a_{r_i}$, $b$ and $b_{r_i}$ must be polynomials in $z_j$. Note that the last $(k-c)$ equations in Eq. (\[eq:syzygy\_2\]) are trivial since they are solved as $a_{r_i}=-b_{r_i} z_{r_i}$. So alternatively, we have only one syzygy equation, $$b F -\sum_{i=1}^{k-c} b_{r_i}\bigg(z_{r_i}\frac{\partial F }{\partial z_{r_i}}\bigg) \hspace{0.8mm}+\hspace{0.8mm}
\sum_{j=k-c+1}^{m-c} a_{r_j}\frac{\partial F}{\partial z_{r_j}}
\hspace{0.8mm}=\hspace{0.8mm} 0
\label{syz}$$ for polynomials $b_{r_i}$, $a_{r_i}$ and $b$. These equations are similar to the tangent condition of a hypersurface in differential geometry, however, we require polynomial solutions. So we apply algebraic geometry to study these equations.
Syzygies
--------
Syzygy can be understood as relations between polynomials. Consider the ring $R=\Fpoly$ and $R^m$, the set of all $m$-tuple of $R$. $R^m$ in general is not a ring but $R\times R^m \to R$ is well-defined as, $$\begin{gathered}
f \cdot (f_1,\ldots ,f_m) \mapsto (f f_1, \ldots, f f_m).\end{gathered}$$ This leads to the definition of modules.
A module $M$ over a ring $R$ is an Abelian group ($+$) with a map $R\times M\to M$ such that,
1. $r\cdot (m_1+m_2)=r \cdot m_1+r \cdot m_2$, $\forall r\in R$, $m_1,m_2\in M$.
2. $(r_1+r_2)\cdot m=r_1 \cdot m+r_2 \cdot m$, $\forall
r_1,r_2\in R$, $m\in M$.
3. $(r_1 r_2) \cdot m=r_1 \cdot (r_2 \cdot m)$, $\forall
r_1,r_2\in R$, $m\in M$.
4. $1\cdot m=m$. $1\in R$, $\forall m\in M$.
For example $R^m$, $I$ and $R/I$ are all $R$-modules, where $I$ is an ideal of $R$. To simplify notations, we formally write an element $(f_1,\ldots f_m)\in R^m$ as $f_1\e_1+\ldots f_m \e_m$.
Any submodule of $R^m$ is finitely generated.
This is a generalization of Theorem \[thm\_Noether\]. See Cox, Little and O’Shea [@opac-b1094391].
Given an $R$ module $M$, the syzygy module of $m_1, \ldots m_k\in M$, $\syz(m_1\ldots m_k)$, is the submodule of $R^k$ which consists of all $(a_1, \ldots a_k)$ such that $$a_1 \cdot m_1 + a_2 \cdot m_2 + \ldots a_k \cdot m_k=0\,.$$
So defines a syzygy module with $M=R$, i.e., “relations” between polynomials. Naively, given $f_1,\ldots, f_k$, it is clear that $f_j \e_i -f_i \e_j\in R^k$, $i\not=j$ is a syzygy for $f_1,\ldots, f_k$. Such a syzygy is called a [*principal syzygy*]{} which is denoted as $P_{ij}$.
In some cases, principal syzygies generate the whole syzygy module of given polynomials. For example,
\[principle\_syzygy\] Given $f_1,\ldots, f_k$ in $R=\Fpoly$, if $\la f_1,\ldots,f_k\ra=\la
1\ra$, then $\syz(f_1,\ldots f_k)$ is generated by the principal syzygies $P_{ij}=f_j \e_i -f_i \e_j$, $1\leq i\not =j\leq k$.
We have $q_1 f_1 +\ldots q_k f_k=1$, where the $q_i$’s are in $R$. For any element in $\syz(f_1,\ldots f_k)$, $$a_1 f_1 +\ldots a_k f_k=0\,.$$ we can rewrite $a_i$ as, $$\begin{aligned}
a_i=\sum_{j=1}^k a_i q_j f_j=\big(\sum_{\substack{j=1\\j\not=i}}^ka_i q_j f_j
\big)+a_i q_i f_i \nn =\sum_{\substack{j=1\\j\not=i}}^ka_i q_j f_j -\sum_{\substack{j=1\\j\not=i}}^ka_j q_i f_j \equiv \sum_{\substack{j=1\\j\not=i}}^ks_{ij} f_j,\end{aligned}$$ where $s_{ij}=a_i q_j-a_j q_i$ is a polynomial and is antisymmetric in indices. Hence, this syzygy $$\begin{aligned}
\sum_{i=1}^k a_i \e_i= \sum_{i=1}^k\sum_{\substack{j=1\\j\not=i}}^k
s_{ij} f_j \e_i=\sum_{\substack{i=1,j=1\\j\not=i}}^k
\frac{s_{ij}}{2}(f_j \e_i -f_i \e_j)= \sum_{\substack{i=1,j=1\\j\not=i}}^k
\frac{s_{ij}}{2} P_{ij}\,,\end{aligned}$$ is generated by principal syzygies.
\[circle\_syzygy\] Consider the polynomial $F=x^2+y^2-1$ in $\Q[x,y]$. Define $f_1=\partial F/\partial x=2x$, $f_2=\partial F/\partial y=2y$ and $f_3=F$. It is clear that $\la 2x,2y,x^2+y^2-1\ra=\la 1\ra$. Hence $\syz(f_1,f_2,f_3)$ is generated by, $$\begin{gathered}
(y,-x,0),\quad (x^2+y^2-1,0,-2x),\quad (0,x^2+y^2-1,-2y)\,.\end{gathered}$$ Note that we see that $F=0$ defines the unit circle. The tangent vector at any point on the circle is, $$\begin{gathered}
y\frac{\partial }{\partial x}-x\frac{\partial }{\partial y}\,,\end{gathered}$$ which corresponds to the first generator in .
In general, syzygy module for given polynomials can be found by computation. For a $G=\{g_1,\ldots g_m\}$ in a certain monomial order, consider two elements $g_i$, $g_j$, $i<j$. Let $S(g_i,g_j)=a_i g_i
+a_j g_j$ be the S-polynomial (Definition \[S-polynomial\]). $S(g_i,g_j)$ must be divisible by $G$, hence, by polynomial division (Algorithm \[multivariate\_polynomial\_division\]), $$a_i g_i +a_j g_j = \sum_{l=1}^m q_l g_l\,.$$ Clearly, this is a syzygy of $g_1,\ldots g_m$, which explicitly reads $ q_1 \e_1 +\ldots
(q_i-a_i) \e_i +\ldots + (q_j -a_j) \e_j +\ldots q_m \e_m$. We call this syzygy, [*reduction of an S-polynomial*]{} and denote it as $s_{ij}\equiv \sum _{l=1}^m (s_{ij})_l \e_l$.
\[Schreyer\]
1. For a $G=\{g_1,\ldots g_m\}$ in $R=\Fpoly$, $\syz(g_1,\ldots g_m)$ is generated by reductions of S-polynomials, $s_{ij}$.
2. For generic polynomials $\{f_1,\ldots,f_k\}$ in $R=\Fpoly$, let $G=\{g_1,\ldots g_m\}$ be their in a certain monomial order. Suppose that the conversion relations are, $$g_i=\sum_{j=1}^k a_{ij} f_j\quad f_i=\sum_{j=1}^m b_{ij} g_j\,.$$ The $\syz(f_1,\ldots f_k)$ is generated by, $$\begin{gathered}
\sum_{i=1}^m\sum_{j=1}^k (s_{\alpha\beta})_i a_{ij} \e_j, \quad
1\leq \alpha<\beta \leq m\nn\\
\e_i-\sum_{l=1}^k \sum_{j=1}^m b_{ij} a_{jl} \e_l, \quad
1\leq i \leq k
\label{syzygy_GB}\end{gathered}$$
See Cox, Little and O’Shea [@opac-b1094391]. Note that in second line of , the relations are coming from the map from $f_i$’s to $G$ and the inverse map.
This theorem also generalizes to modules. Given several elements $m_1,\ldots m_k$ in $R^m$, we can define a module order which is an extension of monomial order. Then we can compute and the syzygy module of $m_1,\ldots
m_k$ [@opac-b1094391].
In practice, we may use in or in , to find the syzygy module of polynomials or elements in $R^m$. See alternative ways of finding syzygies with the linear algebra method [@Schabinger:2011dz] or by F5 algorithm [@MR2772175].
Polynomial tangent vector field
-------------------------------
In this section, we use the tool of syzygy to study [*polynomial tangent vector field*]{} [@Hauser1993].
Let $F(z)$ be a polynomial in $R=\mathbb
C[z_1,\ldots,z_n]$. $F=0$ defines a hypersurface (reducible or irreducible). The set of polynomial tangent fields, $\TF_F$, is the submodule in $R^n$ which consists of all $(a_1,\ldots, a_n)\in R^n$ such that, $$\label{polynomial_tangent_field}
\sum_{i=1}^n a_i \frac{\partial F}{\partial z_i} = b F\,,$$ for some polynomial $b$ in $z_i$’s. is a syzygy equation which can be solved by the algorithm in Theorem \[Schreyer\]. (We drop the factor $b$ in the definition of $\TF_F$, since this factor can be easily recovered later.) Mathematically, $\TF_F$ is called the set of [*derivations*]{}, from $R/\la F \ra$ to $R/\la F \ra$.
Geometrically, if a point $\xi=(\xi_1,\ldots ,\xi_n)\in \mathbb C^n$ is on the hypersurface $\mathcal Z(F)$, then $$\sum_{i=1}^n a_i (\xi_1,\ldots ,\xi_n) \frac{\partial F}{\partial z_i}(\xi_1,\ldots ,\xi_n) = 0\,,$$ and $(a_1(\xi),\ldots,a_n(\xi))$ is along the tangent direction of $\mathcal Z(F)$. This is the origin of the terminology, [*polynomial tangent vector field*]{}.
Although the syzygy computation by Theorem \[Schreyer\] can find $\mathbf T_F$ for any polynomial $F$, it is interesting to study the geometric properties of $F$ and $\mathbf T_F$.
\[singular\_ideal\] For a polynomial $F$ in $R=\mathbb
C[z_1,\ldots,z_n]$, the singular ideal $I_S$ for $F$ is defined to be, $$I_s=\la\frac{\partial F}{\partial z_1},\ldots, \frac{\partial
F}{\partial z_n}, F\ra \,.$$ If $I_s=\la 1 \ra$, then we call the hypersurface $\mathcal Z(F)$ smooth. Otherwise we call points in $\mathcal Z(I_s)$ singular points.
Intuitively, at a singular point $\xi\in \mathcal Z(I_s)$, $F$ and all its first derivates vanish. Hence near $\xi$, $F=0$ does not define a complex submanifold with codimension $1$.
If a hypersurface is smooth, then by Definition \[singular\_ideal\] and Proposition \[principle\_syzygy\], we have the following statement.
If $F$ in $R=\mathbb
C[z_1,\ldots,z_n]$ defines a smooth hypersurface, then $\TF_F$ is generated by principal syzygies of $\partial F/\partial z_1,\ldots, \partial
F/\partial z_n, F$.
For instance, in Example \[circle\_syzygy\], the unit circle is clearly smooth. Hence its polynomial tangent vector fields is generated by principal syzygies. This can be understood as an algebraic version of the implicit function theorem.
The singular cases are more interesting and subtle.
Let $F=y^2-x^3$. $F=0$ is not a smooth curve, since the singular variety is $I_s=\la -3x^2,2y,y^2-x^3\ra=\la x^2, y\ra \not=\la 1\ra$. So there is one singular point at $(0,0)$ which is a cusp point. We cannot just use principal syzygies to generate $\TF_F$, so we turn to Theorem \[Schreyer\].
Define $\{f_1,f_2,f_3\}=\{-3x^2,2y,y^2-x^3\}$. Note that this is a in , although it is not a reduced .
- $S(f_1,f_2)=(2y)f_1+(3x^2)f_2=0$ hence we get a syzygy generator $\mathcal S_1=(2y,-3x^2,0)$.
- $S(f_2,f_3)=(-x^3)f_2-(2y)f_3=-2y^3=-y^2 f_2$. $\mathcal S_2=(0,
-x^3+y^2 ,-2 y)$.
- $S(f_3,f_1)=-3f_3+x f_1=-3y^2=-\frac{3}{2}y f_2$. $\mathcal S_3=(x,
\frac{3}{2}y,-3)$.
$\mathcal S_3$ is not from principal syzygies. Locally it characterizes the scaling behavior of the curve $y^2-x^3=0$ near the cusp point $(0,0)$. It is a [*weighted Euler vector field*]{} [@Hauser1993].
Dropping the factor $b$ in , we find that $\TF_F$ is generated by, $$(2y, -3x^2),\quad (0,-x^3+y^2), \quad (x,\frac{3y}{2})\,.$$
Let $F\in R=\mathbb C[z_1,\ldots,z_n]$, $\TF_F$ is a Lie algebra with $[,]$ defined as that for vector fields.
Let $v_1=(a_1, \ldots ,a_n)$ and $v_2=(b_1,\ldots ,b_n)$ be two polynomial tangent vector fields, $$\sum_{i=1}^n a_i \frac{\partial F}{\partial z_i} = A F,\quad
\sum_{i=1}^n b_i \frac{\partial F}{\partial z_i} = B F\,,$$ where $A$ and $B$ are polynomials. $[v_1,v_2]$’s i-th component is, $$\begin{gathered}
\sum_{j=1}^n \bigg( a_j \frac{\partial
b_i}{\partial z_j}- b_j \frac{\partial a_i}{\partial
z_j}\bigg) \,,\end{gathered}$$ Hence $[v_1,v_2]$ acts on $F$ as, $$\begin{gathered}
\sum_{i=1}^n \sum_{j=1}^n \bigg( a_j \frac{\partial
b_i}{\partial z_j}- b_j \frac{\partial a_i}{\partial
z_j}\bigg) \frac{\partial F}{\partial z_i}=F\cdot \sum_{j=1}^n\bigg( a_j \frac{\partial B}{\partial z_i}-b_j \frac{\partial A}{\partial z_i}\bigg)\,.\end{gathered}$$ so $[v_1,v_2]$ is in $\TF_F$.
In general, $\TF_F$ is an infinite-dimensional Lie algebra over $\C$. We may call $\TF_F$ a [*tangent algebra*]{}.
If we require a polynomial vector field $(a_1, \ldots ,a_n)$ to be tangent to a list of hypersurfaces defined by $F_1,\ldots,F_k$, like the case of and , $$\begin{gathered}
\sum_{i=1}^n a_i \frac{\partial F_1}{\partial z_i} = A_1 F_1(z)\nn\\
\ldots \nn\\
\sum_{i=1}^n a_i \frac{\partial F_k}{\partial z_i} = A_k F_k(z)\,.
\label{vector_field_hypersurfaces}\end{gathered}$$ Then by definition, the solution set of such $(a_1, \ldots ,a_n)$’s is the intersection of modules $\TF_{F_1}\cap\ldots \cap\TF_{F_k}$, which is again a submodule of $R^n$. On the other hand,
\[tangent\_algebra\_component\] If a polynomial $F$ in $R=\C[z_1,\ldots,z_n]$ factorizes as, $$F=f_1^{s_1}\ldots f_k^{s_k} \,,$$ where $f_i$’s are irreducible polynomials in $R$ and $f_i \not | f_j$ if $i\not =j$. Here $s_i$’s are positive integers. Then $\TF_F=\TF_{f_1}\cap\ldots \cap\TF_{f_k}$.
It is clear that $\TF_F\supset \TF_{f_1}\cap\ldots
\cap\TF_{f_k}$. For $(a_1, \ldots ,a_n)\in \TF_F$, $$\sum_{l=1}^k s_l \bigg(\sum_{i=1}^n a_i \frac{\partial
f_l}{\partial z_i}\bigg) \frac{F}{f_l}=b F\,.$$ For a fixed index $t$, $1\leq t\leq k$, divide the above expression by $f_t^{s_t-1}$, $$s_t \bigg(\sum_{i=1}^n a_i \frac{\partial
f_t}{\partial z_i}\bigg) \frac{F}{f_t^{s_t}}+
\sum_{\substack{l=1\\ l\not= t}}^k s_l \big(\sum_{i=1}^n a_i \frac{\partial
f_l}{\partial z_i}\big) \frac{F}{f_l f_t^{s_t-1}}=b \frac{F}{ f_t^{s_t-1}}\,.$$ Note that the second term on the l.h.s and the r.h.s are polynomials proportional to $f_t$. Hence, $$s_t \bigg(\sum_{i=1}^n a_i \frac{\partial
f_t}{\partial z_i}\bigg) \frac{F}{f_t^{s_t}}\,,$$ is also proportional to $f_t$. However $f_t$ does not divide $F/f_t^{s_t}$, since $f_i$’s are distinct irreducible polynomials. So $f_t$ divides $\sum_{i=1}^n a_i \partial
f_t/\partial z_i$ and $(a_1, \ldots ,a_n)\in \TF_{f_t}$, and $\TF_F\subset \TF_{f_1}\cap\ldots
\cap\TF_{f_k}$.
It implies that for a reducible hypersurface, its tangent algebra is the intersection of tangent algebras of all its irreducible components [@Hauser1993].
In practice, give a syzygy equation system , we can first determine each $\TF_{F_i}$ and then calculate the intersection $\TF_{F_1}\cap\ldots \cap\TF_{F_k}$. (See [@opac-b1094391 Chapter 5] for the algorithm of computing intersection of submodules.) Furthermore, for each $\TF_{F_i}$, if $F_i$ is factorable, we can use Proposition \[tangent\_algebra\_component\] to further divide the problem. This divide-and-conquer approach is in general much more efficient than solving at once.
IBPs from syzygies and unitarity
--------------------------------
With the Baikov representation, unitarity cut and syzygy computations, we introduce some recent IBP generating algorithms [@Ita:2015tya; @Larsen:2015ped] with the two-loop double box as an example.
For the massless double box, define $$I[m_1,\ldots ,m_9]=\int \frac{d^D l_1}{i \pi^{D/2}} \frac{d^D l_2}{i
\pi^{D/2}} \frac{(l_1\cdot k_4)^{-m_8}(l_2\cdot k_1)^{-m_9}}{D_1^{m_1} \ldots D_7^{m_7}}\,.$$ Our target integral space is the set of all list $(m_1\ldots m_9)$ such that $m_i\leq 1$, $i=1,\ldots,7$, $m_j\leq 0$ , $j=8,\ldots,9$, since we try to find IBPs without doubled propagators.
Consider the massless double box with maximal cut. From Example \[dbox\_Baikov\_cut\], we see that with maximal cut the Baikov polynomial is $$\label{F_7cut}
F_{[7]}= \frac{z_8 z_9 (s t-2 s z_8-2 s z_9-4 z_8 z_9)}{t (s+t)}\,.$$ The syzygy equation is, $$a_8 \frac{\partial F_{[7]}}{\partial z_8}+a_9
\frac{\partial F_{[7]}}{\partial z_9}=\beta F_{[7]}\,.$$ Solutions of $(a_8,a_9)$ form $\TF_{F_{[7]}}$, the tangent algebra of $F_{[7]}$. We leave the computation of $\TF_{F_{[7]}}$ as an exercise. There are $3$ generators of $\TF_{F_{[7]}}$, $$\begin{gathered}
v_1=\big(-(t-2 z_8) z_8 , (t-2 z_9) z_9 \big),\quad
v_2=\big (2 (s+t) z_8 z_9, -(t-2 z_9) z_9 (s+2 z_9)\big),\nn\\
v_3=\big(0 , -z_9 (s t-2 s z_8-2 s z_9-4 z_8 z_9)\big)\,.\end{gathered}$$ Using these generators and the ansatz , we get IBPs without double propagators. For instance, from the first generator we have the IBP, $$I[1,1,1,1,1,1,1,-1,0]=I[1,1,1,1,1,1,1,0,-1] +
\ldots ,$$ and from the second generator, $$\begin{gathered}
4 (D-3) I[1,1,1,1,1,1,1,0,-2]+(3 D s-12 s-2 t)
I[1,1,1,1,1,1,1,0,-1]\nn\\
-\frac{1}{2} (D-4) s t I[1,1,1,1,1,1,1,0,0]
=0 + \dots \,.\end{gathered}$$ Note that with maximal cut, any integral with at least one $m_i<1$, $i=1,\ldots 7$, is neglected. “$\ldots$” stands for these integrals.
To get all IBPs with maximal cut, we need to consider vector fields $q_1 v_1+q_2 v_2+q_3 v_3$ where $q_1$, $q_2$ and $q_3$ are arbitrary polynomials in $z_8$, $z_9$ up to a given degree. When the smoke is clear, we find that all integrals with $m_i=1$, $i=1,\ldots 7$ and $m_j\leq 0$ , $j=8,\ldots,9$ are reduced to $I[1,1,1,1,1,1,1,0,0]$, $I[1,1,1,1,1,1,1,-1,0]$ and integrals with fewer-than-$7$ propagators.
\[slashed\_box\_cut\] Consider the quintuple cut of the massless double box, $D_2=D_3=D_5=D_6=D_7=0$. The goal is to study integrals $I_\text{dbox}[m_1,m_2,\ldots m_9]$ such that $m_2=m_3=m_5=m_6=m_7=1$, $m_1,m_4\leq 1$, $m_8,m_9$ non-positive. The syzygy equations read, $$\begin{aligned}
a_1 \frac{\partial F_{[5]}}{\partial z_1}+a_4
\frac{\partial F_{[5]}}{\partial z_4}+a_8 \frac{\partial F_{[5]}}{\partial z_8}+a_9
\frac{\partial F_{[5]}}{\partial z_9}&=&\beta F_{[5]} \\
a_1 &=& b_1 z_1\\
a_4 &=& b_4 z_4
\label{dbox_5cut}\end{aligned}$$ In the formal language, the solutions of the last two equations form a tangent algebra $\TF_{14}$ with generators, $$\begin{aligned}
(z_1,0,0,0),\quad (0,z_4,0,0),\quad (0,0,1,0),\quad (0,0,0,1).\end{aligned}$$ The first equation can be solved by in and , which leads to a tangent algebra $\TF_{F[5]}$. Then the solution set of of $\TF_{F[5]}\cap \TF_{14}$. This intersection of submodules can be calculated by in and .
Again we find IBPs with this tangent algebra. All integrals with $m_2=m_3=m_5=m_6=m_7=1$, $m_1,m_4\leq 1$, $m_8,m_9$ non-negative are reduced to $3$ master integrals $I[1,1,1,1,1,1,1,0,0]$, $I[1,1,1,1,1,1,1,-1,0]$, $I[0,1,1,0,1,1,1,0,0]$ and integrals with fewer-than-$5$ propagators.
In general it is easy to obtain IBPs with maximal cut, since the number of variable is small. We may use symmetries and IBPs with maximal cut, numerically, to find all MIs in our package [Azurite]{} [@Georgoudis:2016wff]. (Alternatively, the master integral list can be obtained by the critical point analysis in parametric representation [@Lee:2013hzt].) It takes only a few seconds to find all master integrals for the massless double box.
- double box, $I[1,1,1,1,1,1,1,-1,0]$, $I[1,1,1,1,1,1,1,0,0]$,
- slashed box, $I[0,1,1,0,1,1,1,0,0]$,
- box bubble, $I[0,1,0,1,1,1,1,0,0]$,
- double bubble, $I[1,0,1,1,0,1,0,0,0]$,
- bubble triangle, $I[0,1,0,1,0,1,1,0,0]$,
- $t$-channel sunset, $I[0,1,0,0,1,0,1,0,0]$,
- $s$-channel sunset, $I[0,0,1,0,0,1,1,0,0]$.
We define that $I[m_1,\ldots m_9]$ is lower than $I[n_1,\ldots n_9]$ if $m_i\leq n_i$, $i=1,\ldots 7$. For example, the $s$-channel sunset is lower than the slashed box. A triple cut $D_3=D_6=D_7=0$ contains all information of the quintuple cut in Example \[slashed\_box\_cut\]. Since here the lowest master integrals are double bubble, bubble triangle, $t$-channel sunset, $s$-channel sunset, we can see that the following four cuts, $$\begin{gathered}
D_1=D_3=D_4=D_5=0,\quad D_2=D_4=D_6=D_7=0\nn\\
D_2=D_5=D_7=0,\quad D_3=D_6=D_7=0\,,\end{gathered}$$ determine the complete IBPs without cut.
By this method [@Larsen:2015ped], a Mathematica code with communication with , analytically reduces all double box integrals with numerator rank $\leq 4$, to the $8$ master integrals in about $39$ seconds for the massless double box on a laptop. Similarly, it takes about $162$ seconds for the analytic IBP reduction of the one-massive double box. We expect that combined with sparse linear algebra and finite-field fitting techniques, it can solve some very difficult two-loop/three-loop IBP problems in the near future.
Exercise
--------
\[dbox\_Baikov\] Consider the two-loop massless double box diagram (Fig. \[graph\_dbox\]) with inverse propagators $D_1,\ldots, D_7$ defined in . Let $$I_\text{dbox}[N]=\int \frac{d^D l_1}{i \pi^{D/2}} \frac{d^D l_2}{i
\pi^{D/2}} \frac{N}{D_1 \ldots D_7}.$$ By integrand reduction, we see that $N$ can be a polynomial in $\mu_{11}$, $\mu_{22}$ and $\mu_{12}$, but at most linear in $(l_1\cdot \omega)$ and $(l_2 \cdot \omega)$. Terms linear in $(l_1\cdot \omega)$ and $(l_2 \cdot \omega)$ are spurious so dropped. Terms in $\mu$’s can be converted to integrals without $\mu$’s in higher dimension, via Schwinger parameterization [@Bern:2003ck]. Or alternatively, polynomials in $\mu$’s or $(l_i\cdot \omega)$ can be directly integrated out by [*adaptive integrand decomposition*]{} [@Mastrolia:2016dhn], using Gegenbauer polynomials techniques. Hence we assume $N$ contains no $\mu$’s, $(l_1\cdot \omega)$ or $(l_2 \cdot \omega)$.
1. The original Van Neerven-Vermaseren variables are defined in and $\mu_{ij}=-l_i^\perp \cdot l_j^\perp$. To integrate out the $\omega$ direction, define $V_1=\sp\{k_1,k_2,k_4\}$ and $V^\sharp$ as the direct sum of $\sp\{\omega\}$ and $(-2\epsilon)$ extra spacetime. Decompose $l_i=l_i^{[3]}+l_i^\sharp$ and denote $(l_i^\sharp\cdot
l_j^\sharp)=-\lambda_{ij}$. Prove that $$\lambda_{11}=\mu_{11}\c +\frac{s}{t(s+t)} x_4^2,\quad
\lambda_{22}=\mu_{22}\c +\frac{s}{t(s+t)} y_4^2,\quad
\lambda_{12}=\mu_{12}\c +\frac{s}{t(s+t)} x_4 y_4\,,$$ and the $D_1,\ldots ,D_7$ only depend on $x_1,x_2,x_3,y_1,y_2,y_3,\lambda_{11},\lambda_{22},\lambda_{12}$.
2. Integrate over the solid angle parts of $l_1^\sharp$ and $l_2^\sharp$ to get $$\begin{gathered}
I_\text{dbox}[N]=\frac{2^{D-5}}{\pi^4\Gamma(D-4)}\int_0^\infty d\lambda_{11}\int_0^\infty
d\lambda_{22} \int_{-\sqrt{\lambda_{11}
\lambda_{22}}}^{\sqrt{\lambda_{11} \lambda_{22}}} d\lambda_{12} (\lambda_{11}\lambda_{22}-\lambda_{12}^2)^{\frac{D-6}{2}}\times
\nn\\
\int d^3 l_1^{[3]} d^3 l_2^{[3]} \frac{N}{D_1 \ldots D_7}\,.
\end{gathered}$$
3. Define $9$ Baikov variables as $$\begin{gathered}
z_i=D_i,\quad 1\leq i \leq 7,\quad z_8=l_1\cdot k_4,\quad
z_9=l_2\cdot k_1\,.
\end{gathered}$$ Find the inverse map $(z_1,\ldots z_9)\mapsto
(x_1,x_2,x_3,y_1,y_2,y_3,\lambda_{11},\lambda_{22},\lambda_{12})$ and the Jacobian of the map.
4. Derive the Baikov form of integral, $$\begin{gathered}
I_\text{dbox}[N]=\frac{2^{D-5}}{\pi^4\Gamma(D-4) J}
\int \prod_{i=1}^9 dz_i F(z)^{\frac{D-6}{2}}\frac{N}{D_1 \ldots D_7}\,.
\end{gathered}$$ Calculate $J$ and $F(z)$ explicitly. Note that the Jacobian of the changing variables $l_i^{[3]}$ to $ (x_1,x_2,x_3,y_1,y_2,y_3)$ should be included.
Derive the Baikov representation for two-loop pentagon-box diagram, (Fig. \[graph\_pentabox\]).
![Pentagon box diagram[]{data-label="graph_pentabox"}](graphs/pentagon_box.eps)
with inverse propagators, $$\begin{gathered}
D_1=l_1^2, \quad D_2=(l_1 - k_1)^2, \quad D_3= (l_1 - k_1 - k_2)^2, \quad D_4= (l_1 - k_1 - k_2 - k_3)^2, \nn\\\quad D_5= (l_2 + k_1 +
k_2 + k_3)^2, \quad D_6= (l_2 + k_1 + k_2 + k_3 + k_4)^2, \quad
D_7=l_2^2, \quad D_8=(l_1 + l_2)^2.\end{gathered}$$ (Hint: define $z_i=D_i$, $i=1,\ldots,8$. $z_{9}=l_1 \cdot k_5$, $z_{10}=l_2 \cdot k_1$, $z_{11}=l_2 \cdot k_2$.)
Consider $f_1=x^3-2 x y$, $f_2=x^2 y-2 y^2 +x$ as the Example \[example\_Buchberger\]. We know that the in is $G=\{g_1,g_2,g_3\}=\{x^2, x y, y^2-\half x\}$. The conversion relations are, $$\begin{gathered}
g_1= -y f_1 +x f_2 ,\quad g_2= -\frac{ (1+x y)}{2}f_1+\half x^2 f_2
,\quad g_3= -\half y^2 f_1+\half (x y-1) f_2,\\
f_1=x g_1-2 g_2,\quad f_2=y g_1-2 g_3\,.
\end{gathered}$$ Find the generators of $\syz(f_1,f_2)$ by Theorem \[Schreyer\].
Let $F=\left(x^2+y^2\right)^2+3 x^2 y-y^3$, the plot of the curve $F=0$ is in Figure. \[graph\_triple\]. Determine the singular points of this curve and find the polynomial tangent vector fields $\TF_F$.
![A singular curve, $\left(x^2+y^2\right)^2+3 x^2 y-y^3=0$[]{data-label="graph_triple"}](graphs/triple_point.eps)
Compute $\TF_F$ for the double box on the maximal cut, where $F$ is the corresponding Baikov polynomial . (We drop the subscript “ ${[7]}$”.)
1. Use in or , to compute $\TF_F$ directly.
2. Note that $F$ has $3$ irreducible factors, $f_1=z_8$, $f_2=z_9$ and $f_3=(s t-2 s z_8-2 s z_9-4 z_8 z_9)$. $f_1$ is linear so $\TF_{f_1}$ is generated by, $$(z_8,0),\quad (0,1)\,.$$ Similarly, $\TF_{f_2}$ is generated by, $$(1,0),\quad (0,z_9)\,.$$ What is $\TF_{f_1}\cap \TF_{f_2}$? Note that $f_3=0$ is smooth. Use Proposition \[principle\_syzygy\] to find $\TF_{f_3}$.
3. Use in or , to compute $
\TF_{F} =\TF_{f_1}\cap \TF_{f_2}\cap \TF_{f_3}$. Compare the result with that from the direct computation.
Consider the three-loop massless triple box diagram (Figure. \[tribox\]).
1. Define $z_i=D_i$, $i=1,\ldots 10$, and $$\begin{gathered}
z_{11}=(l_1 + k_4)^2,\quad z_{12}=(l_2 + k_1)^2,\quad z_{13}=(l_3 + k_1)^2,\nn\\ z_{14}=(l_3 + k_4)^2, \quad l_{15}=(l_1 + l_2)^2.
\end{gathered}$$ Determine its Baikov representation.
2. Derive IBPs with the maximal cut $D_1=\ldots=D_{10}=0$, and determine the master integrals with $10$ propagators for this diagram.
[^1]: Department of physics, ETH Zürich, Wolfgang-Pauli-Strasse 27, 8093 Zürich, Switzerland, <yang.zhang@phys.ethz.ch>.
[^2]: As an analogy, consider the equation $6 x+ 9 y=15$ in $x$, $y$. If $x$, $y$ are allowed to be rational numbers, it is a simple linear equation. However, if only integer values for $x$, $y$ are allowed, it is a less-trivial Diophantine equation in number theory. Here we have polynomial-valued Diophantine equations.
[^3]: We need some graph theory concepts here: for a graph $G$, a [ *spanning tree*]{} $T$ is a tree subgraph which contains all vertices of $G$. Given any edge $e$ in $G$ which is not in $T$, we define a [*fundamental cycle*]{} $C_e$ as the simple cycle which consists of $e$ and a subset of $T$. The number of fundamental cycles is independent of the choice of $T$.
[^4]: The normalization is from the convention of spinor helicity formalism. So $\omega$ is a pure imaginary vector, and later on the unitarity solutions appear to be real in van Neerven-Vermaseren variables.
[^5]: We use the massless case as an illustrative example. Actually for a triangle diagram with two massless external lines, the scalar integral itself can be further reduced to bubble integrals, via IBPs.
[^6]: Note that in general, for a massive triangle diagram, the two cut branches may merge into one. In this case, a Laurent expansion over $z$ is needed and again remove the redundant pole.
[^7]: A field $\K$ is algebraically closed, if any non-constant polynomial in $\K[x]$ has a solution in $\K$. $\Q$ is not algebraically closed, the set of all algebraic numbers $\bar Q$ and $\C$ are algebraically closed.
[^8]: See [@Ablinger:2015tua] for the algorithm of solving differential equations without choosing a special basis, in the univariate case.
[^9]: In general, we need to consider IBP in $D$-dimension. Otherwise for a specific integer-valued $D$, IBP relations may contain non-vanishing boundary terms.
[^10]: Note that the boundary of the integration is defined by the hypersurface $F=0$, therefore the total derivatives vanish on the boundary and we do not need to worry about surface terms in IBPs.
|
---
abstract: 'We investigate the relation between circular velocity ${{v_{\text{c}}}}$ and bulge velocity dispersion $\sigma$ in spiral galaxies, based on literature data and new spectroscopic observations. We find a strong, nearly linear [${{v_{\text{c}}}}$–$\sigma$]{} correlation with a negligible intrinsic scatter, and a striking agreement with the corresponding relation for elliptical galaxies. The least massive galaxies ($\sigma<80~{{\text{km\,s$^{-1}$}}}$) significantly deviate from this relation. We combine this [${{v_{\text{c}}}}$–$\sigma$]{} correlation with the well-known [${M_{\text{BH}}}$–$\sigma$]{} relation to obtain a tight correlation between circular velocity and supermassive black hole mass, and interpret this as observational evidence for a close link between supermassive black holes and the dark matter haloes in which they presumably formed. Apart from being an important ingredient for theoretical models of galaxy formation and evolution, the relation between $M_{\text{BH}}$ and circular velocity has the potential to become an important practical tool in estimating supermassive black hole masses in spiral galaxies.'
author:
- 'Maarten Baes$^{1,2}$, Herwig Dejonghe$^1$, Pieter Buyle$^1$, Laura Ferrarese$^3$, Gianfranco Gentile$^{4,5}$'
date: '?? and in revised form ??'
---
Introduction
============
The existence of supermassive black holes (SMBHs) in the nuclei of galaxies has been suspected for almost half a decade, as accretion onto SMBHs seemed the only logical explanation for the existence of quasars. HST observations have provided evidence that SMBHs with masses ranging from $10^6$ to $10^9$ are present in the centre of a few dozens of nearby (quiescent) galaxies. Be this sufficient evidence for the existence of SMBHs, we can now tackle more fundamental questions concerning their formation and evolution. An obvious way to proceed is the study of the relation between SMBHs and the galaxies that host them. It was found that black hole masses are correlated with parameters of the hot stellar components of their host galaxies. The tight [${M_{\text{BH}}}$–$\sigma$]{} relation (Gebhardt et al. 2000; Ferrarese & Merritt 2000) is now the preferred paradigm to study SMBH demographics in galactic nuclei.
This apparently tight link between bulges and SMBHs reflects an important ingredient that should be reproduced (and thus hopefully explained) by theoretical models of galaxy formation. In fact, the tightness of the [${M_{\text{BH}}}$–$\sigma$]{} correlation is somewhat surprising. In most of the state-of-the-art models, the total galaxy mass (or dark matter mass ${M_{\text{DM}}}$), rather than the bulge mass, plays a fundamental role in shaping the SMBHs. A close correlation could therefore be expected between ${M_{\text{BH}}}$ and ${M_{\text{DM}}}$, rather than between ${M_{\text{BH}}}$ and the bulge properties. Establishing whether the [${M_{\text{BH}}}$–$\sigma$]{} or the [${M_{\text{BH}}}$–${M_{\text{DM}}}$]{}relation reflects the fundamental mode by which SMBHs form and evolve will ultimately rely on a comparison of the intrinsic scatter of the two correlations.
Unfortunately, a direct observational characterization of the [${M_{\text{BH}}}$–${M_{\text{DM}}}$]{}relation is currently impossible. Ferrarese (2002b) first argued that a correlation between ${M_{\text{BH}}}$ and ${M_{\text{DM}}}$ should be reflected in an [${M_{\text{BH}}}$–${{v_{\text{c}}}}$]{} correlation, where ${{v_{\text{c}}}}$ is the circular velocity in the flat part of the rotation curve of spiral galaxies. Indeed, in most of the state-of-the-art galaxy formation models, there is a one-to-one correspondence between the circular velocity and the mass of the dark matter halo. Unfortunately, there are (presently) only a handful of spiral galaxies with secure SMBH masses, and only two of them have a well-measured extended rotation curve. A way to avoid this problem is to adopt the tight [${M_{\text{BH}}}$–$\sigma$]{} correlation in order to estimate black hole masses in a larger sample of galaxies. A tight correlation between SMBH mass and dark matter halo mass should thus appear in the form of a correlation between central velocity dispersion and circular velocity. Ferrarese (2002b) presented a first attempt at establishing such a correlation. Baes et al. (2003) significantly improved on these results by almost doubling the sample size. The present contribution is focused on the latter results.
Sample selection
================
A simple measure for the circular velocity of galaxies is half of the integrated line width from spatially unresolved H[i]{} 21cm measurements, corrected for inclination. Various authors have recovered a nearly linear correlation between integrated line width and bulge velocity dispersion (e.g. Whitmore & Kirshner 1981; Whittle 1992; Franx 1993). This correlation has significant scatter and galaxy type could act as a third parameter in this correlation. This can be due to the fact that the integrated line width is not an accurate measure for the flat part of the rotation curve, and hence of the total dark matter content. Therefore, we chose to consider only those galaxies with an extended rotation curve measured well beyond the optical radius, in order to reliably trace the circular velocity in the flat part of the rotation curve. In order to see the benefits of this approach, it is interesting to consider the Tully-Fisher (TF) study of the Ursa Major cluster spiral galaxies by Verheijen (2001): the scatter in the TF relation strongly decreases when he considers the flat part in the rotation curve instead of the integrated line width.
We constructed a data set of 28 spiral galaxies with central velocity dispersion data and a rotation curve measured beyond the optical radius. For 16 spirals, the data could be retrieved from the literature (see references in Ferrarese 2002b). For the remaining 12 galaxies, rotation curve data were available in the literature (Palunas & Williams 2000), and the velocity dispersions were measured with the EFOSC2 instrument on the ESO 3.6m telescope. The total sample of 28 spiral galaxies can be found in Baes et al. (2003).
{width="90.00000%"}
The [${{v_{\text{c}}}}$–$\sigma$]{} correlation
===============================================
In the left panel of figure [\[vcsigma.eps\]]{} we plot the circular velocity versus the velocity dispersion for the 28 spiral galaxies in our sample. For the 24 galaxies with a velocity dispersion greater than about 80 , there is a very tight correlation between ${{v_{\text{c}}}}$ and $\sigma$. We fitted a straight line to these data, taking into account the errors on both quantities and obtained $$\log\left(\frac{{{v_{\text{c}}}}}{{v_0}}\right)
=
(0.96 \pm 0.11) \log\left(\frac{\sigma}{{v_0}}\right)
+
(0.21 \pm 0.023),
\label{vcsigma2}$$ where ${v_0}=200\,{{\text{km\,s$^{-1}$}}}$. Two issues concerning this [${{v_{\text{c}}}}$–$\sigma$]{}correlation deserve some special attention.
Firstly, the tightness of the correlation is astonishing: we find $\chi_{\text{red}}^2 = 0.281$, corresponding to a goodness-of-fit of 99.9 per cent. The [${{v_{\text{c}}}}$–$\sigma$]{} relation can hence be regarded as having a negligible intrinsic scatter. Moreover, this correlation appears to be robust: there are no significant outliers in the range $\sigma>80~{{\text{km\,s$^{-1}$}}}$. The correlation appears to break down for galaxies with dispersions below about 80 however: all four galaxies with $\sigma<80~{{\text{km\,s$^{-1}$}}}$ lie significantly above the correlation defined by the more massive spirals. Interestingly, this is also the mass range in which nearly all bulgeless spiral galaxies are located. New observations are indispensable to understand the behaviour of the [${{v_{\text{c}}}}$–$\sigma$]{} correlation in the low mass regime. Moreover, all galaxies in our sample are high surface brightness galaxies, and it presently unclear how the [${{v_{\text{c}}}}$–$\sigma$]{} relation behaves in diverse environments (see also Pizzella et al. 2004).
Secondly, it is interesting to compare this correlation to a similar one recently found for elliptical galaxies. Based on stellar dynamical models for 20 round ellipticals constructed by Kronawitter et al. (2000), Gerhard et al. (2001) discovered a very tight relation between the central dispersion and the circular velocity (the circular velocity curves of ellipticals were found to be flat to within 10 per cent). Both the slope and zero-point of this correlation agree amazingly well with the [${{v_{\text{c}}}}$–$\sigma$]{} correlation of our spiral galaxy sample (see right panel of figure 1). Gerhard et al. (2001) argue that a proportionality between $\sigma$ and ${{v_{\text{c}}}}$ can be expected for ellipticals on the basis of their dynamical homology. For spiral galaxies this proportionality cannot be explained by simple dynamical arguments, as convincingly argued by Ferrarese (2002b). Moreover, the fact that both spiral and elliptical galaxies seem to obey exactly the same correlation is absolutely striking.
The correlation between ${M_{\text{BH}}}$ and ${{v_{\text{c}}}}$
================================================================
As both the [${{v_{\text{c}}}}$–$\sigma$]{} and [${M_{\text{BH}}}$–$\sigma$]{} correlations seem to hold over the entire Hubble range, we can combine them to derive a correlation between the circular velocity and SMBH mass. Although the slope of the [${M_{\text{BH}}}$–$\sigma$]{} relation is not well established (Tremaine et al. 2002; Ferrarese 2002a), this little affects the conclusion that the [${{v_{\text{c}}}}$–$\sigma$]{} relation entails a connection between SMBH mass and the large scale velocity of the host galaxy (and henceforth the mass of the surrounding dark matter halo). For instance, using the characterization of the [${M_{\text{BH}}}$–$\sigma$]{} relation from Tremaine et al. (2002), we obtain $$\log\left(\frac{{M_{\text{BH}}}}{{{\text{M$_\odot$}}}}\right)
=
(4.21 \pm 0.60) \log\left(\frac{{{v_{\text{c}}}}}{{v_0}}\right)
+
(7.24 \pm 0.17).
\label{MBHvc}$$ The correlation between SMBH mass and circular velocity is useful for two different goals. Firstly, combined with other tight relations such as the [${M_{\text{BH}}}$–$\sigma$]{} relation and the TF relation, it clearly points at an intimate interplay between the various galactic components (dark matter, discs, bulges and SMBHs) and forms a strong test for galaxy formation and evolution models. In particular, the [${{v_{\text{c}}}}$–$\sigma$]{} relation can be used to discriminate between various theoretical models of galaxy formation. For example, if SMBHs form mainly through coalescence of smaller black holes during galaxy mergers, a relation ${M_{\text{BH}}}\propto v_{\text{c}}^3$ is expected, whereas theories in which accretion and feedback are the main ingredients for black hole growth prefer a ${M_{\text{BH}}}\propto v_{\text{c}}^5$ relation (e.g. Silk & Rees 1998; Haiman & Loeb 1998; Haehnelt, Natarajan & Rees 1998; Kauffman & Haehnelt 2000; Wyithe & Loeb 2003; Di Matteo et al. 2003).
Apart from being an ingredient in theoretical galaxy formation models, the derived [${M_{\text{BH}}}$–${{v_{\text{c}}}}$]{} relation can also serve as a practical tool to estimate the black hole masses in spiral galaxies. The most preferred means of estimating ${M_{\text{BH}}}$ in galaxies is the [${M_{\text{BH}}}$–$\sigma$]{}relation. Unfortunately, the number of spiral galaxies with reliable velocity dispersion measurements is relatively small. Since extended rotation curves have been measured for large samples of spiral galaxies (mainly for use in TF studies), the [${M_{\text{BH}}}$–${{v_{\text{c}}}}$]{} relation has the potential to become an important practical tool in estimating supermassive black hole masses in spiral galaxies.
Baes M., Buyle P., Hau G. K. T., Dejonghe H., 2003, MNRAS, 341, L44
Di Matteo T., Croft R. A. C., Springel V., Hernquist L., 2003, ApJ, 593, 56
Ferrarese L., 2002a, in “Current high-energy emission around black holes”, C. Lee and H. Chang eds., p. 3
Ferrarese L., 2002b, ApJ, 578, 90
Ferrarese L., Merritt D., 2000, ApJ, 539, L9
Franx M., 1993, IAUS, 153, 243
Gebhardt K., et al., 2000, ApJ, 539, L13
Gerhard O., Kronawitter A., Saglia R. P., Bender R., 2001, AJ, 121, 1936
Haehnelt M. G., Natarajan P., Rees M. J., 1998, MNRAS, 300, 817
Haiman Z., Loeb A., 1998, ApJ, 503, 505
Kauffmann G., Haehnelt M., 2000, MNRAS, 311, 576
Kormendy J., Richstone D., 1995, ARA&A, 33, 581
Kronawitter A., Saglia R. P., Gerhard O., Bender R., 2000, A&AS, 144, 53
Palunas P., Williams T. B., 2000, AJ, 120, 2884
Pizzella A., Dalla Bontà E., Corsini E. M., Coccato L., Sarzi M., Bertola F., 2004, this volume
Silk J., Rees M. J., 1998, A&A, 331, L1
Tremaine S., et al., 2002, ApJ, 574, 740
Verheijen M. A. W., 2001, ApJ, 563, 694
Whitmore B. C., Kirshner R. P., 1981, ApJ, 250, 43
Whittle M., 1992, ApJ, 387, 109
Wyithe J. S. B., Loeb A., 2003, ApJ, 595, 614
|
---
abstract: 'We discuss Beauville groups whose corresponding Beauville surfaces are either always strongly real or never strongly real producing several infinite families of examples.'
address: 'Ben Fairbairn, Department of Economics, Mathematics and Statistics, Birkbeck, University of London, Malet Street, London WC1E 7HX, United Kingdom'
author:
- Ben Fairbairn
title: 'Purely (Non-)Strongly Real Beauville Groups'
---
Introduction
============
A surface $\mathcal{S}$ is a **Beauville surface** if
- the surface $\mathcal{S}$ is isogenous to a higher product, that is, $\mathcal{S}\cong(\mathcal{C}_1\times\mathcal{C}_2)/G$ where $\mathcal{C}_1$ and $\mathcal{C}_2$ are algebraic curves of genus at least 2 and $G$ is a finite group acting faithfully on $\mathcal{C}_1$ and $\mathcal{C}_2$ by holomorphic transformations in such a way that it acts freely on the product $\mathcal{C}_1\times\mathcal{C}_2$, and
- each $\mathcal{C}_i/G$ is isomorphic to the projective line $\mathbb{P}_1(\mathbb{C})$ and the covering map $\mathcal{C}_i\rightarrow\mathcal{C}_i/G$ is ramified over three points.
These surfaces were first defined by Catanese in [@C] and the first significant investigation of them was conducted by Bauer, Catanese and Grunewald in [@BCG]. They have numerous nice properties and are relatively easy to construct making them useful for producing counterexamples and testing conjectures. The following condition is also investigated in [@BCG].
Let $\mathcal{S}$ be a complex surface. We say that $\mathcal{S}$ is **strongly real** if there exists a biholomorphism $\sigma\colon\mathcal{S}\rightarrow\overline{\mathcal{S}}$ such that $\sigma\circ\overline{\sigma}$ is the identity map.
What makes these surfaces particularly easy to work with is that all of the above can be easily translated into group theoretic terms.
\[MainDef\] Let $G$ be a finite group. Let $x,y\in G$ and let $$\Sigma(x, y) :=\bigcup_{i=1}^{|G|}\bigcup_{g\in G}\{(x^i)^g,(y^i)^g,((xy)^i)^g\}.$$
A **Beauville structure** for the group $G$ is a set of pairs of elements $\{(x_1, y_1),(x_2,y_2)\}\subset G\times G$ with the property that $\langle x_1, y_1\rangle = \langle x_2, y_2\rangle=G$ such that $$\Sigma(x_1, y_1)\cap \Sigma(x_2, y_2)=\{e\}.$$ If $G$ has a Beauville structure we say that $G$ is a **Beauville group**.
A group defines a Beauville surface if and only if it has a Beauville structure. Furthermore, the Beauville surface defined by a particular Beauville structure is strongly real if and only if the corresponding Beauville structure has the property of being strongly real that we define as follows.
\[SRDef\] Let $G$ be a Beauville group and let $X =\{(x_1, y_1),(x_2, y_2)\}$ be a Beauville structure for $G$. We say that $G$ and $X$ are **strongly real** if there exists an automorphism $\phi\in\mbox{Aut}(G)$ and elements $g_i\in G$ for $i = 1, 2$ such that $$g_i\phi(x_i)g_i^{-1}=x_i^{-1}\mbox{ and }g_i\phi(y_i)g_i^{-1}=y_i^{-1}.$$
Most of the Beauville structures appearing in the literature are either explicitly shown to be strongly real or the question of reality is never pursued. In many groups there are elements that can never be inverted by automorphisms meaning any Beauville structure defined using such elements cannot be strongly real, indeed groups typically have numerous Beauville structures some of which are strongly real, some of which are not. Here we are interested in the extreme cases and thus make the following definition.
\[PureDef\] A finite group $G$ is a **purely strongly real Beauville group** if $G$ is a Beauville group such that every Beauville structure of $G$ is strongly real. A finite group $G$ is a **purely non-strongly real Beauville group** if $G$ is a Beauville group such that none of its Beauville structures are strongly real.
Throughout we shall follow the conventions that in a group $G$ and elements $g,h\in G$ we have that $g^h=hgh^{-1}$ and $[g,h]=ghg^{-1}h^{-1}$.
This paper is organised as follows. In Section 2 we will give infinitely many examples of Beauville groups that have strongly real Beauville structures as well as Beauville structures that are not suggesting that groups typically lie in neither of the categories in Definition \[PureDef\]. Despite this, we go on in Section 3 to give infinitely many examples of purely strongly real Beauville groups before in the final section giving infinitely many examples of purely non-strongly real Beauville groups.
Neither Case
============
We first prove results showing that Beauville groups typically fall into neither case with several different examples.
If $n>5$, then the alternating group A$_n$ is neither a Purely strongly real Beauville group nor a non-strongly real Beauville group.
Strongly real Beauville structures for these groups were constructed by Fuertes and González-Diez in [@FG] so it is sufficient to construct non-strongly real Beauville structures for these groups. We consider the cases $n$ odd and $n$ even separately. Recall that in either case, if two permutations have different cycle type, then they cannot be conjugate.
First suppose that $n\geq7$ is odd. Let $$x_1:=(1,2,4)\mbox{ and }y_1:=(1,2,3,4,\ldots,n).$$ We have that their product is the $n$-cycle $x_1y_1=(1,3,4,2,5,\ldots,n)$ by direct calculation. By considering the subgroup generated by the elements $x_1^{y_1^i}$ we have that the subgroup these elements generate is 2-transitive and therefore primitive. Since this subgroup contains the 3-cycle $x_1$ it follows that these elements generate the whole of the alternating group. It is easy to see that no automorphism simultaneously inverts both of these elements so any permutations that can be used to extend this to a Beauville structure gives a non-strongly real Beauville structure. Consider the permutations $$x_2:=(5,4,3,2,1)\mbox{ and }y_2:=(3,4,5,6,\ldots,n).$$ We have that their product is the $(n-2)$-cycle $x_2y_2=(1,6,\ldots,n,3,2)$ by direct calculation. By considering the subgroup generated by the elements $x_2^{y_2^i}$ we have that the subgroup these elements generate is 2-transitive and therefore primitive. Since this subgroup contains the double-transposition $[x_2,y_2]=(1,5)(3,n)$ it follows that these elements generate the whole of the alternating group. We therefore have a non-strongly real Beauville group.
Next, suppose $n\geq8$ is even. Let $$x_1:=(1,2)(3,4)\mbox{ and }y_1:=(2,3,4,\ldots,n).$$ We have that their product is the $(n-1)$-cycle $x_1y_1=(1,3,5,\ldots,n,2)$ by direct calculation. By considering the subgroup generated by the elements $x_1^{y_1^i}$ we have that the subgroup these elements generate is 2-transitive and therefore primitive. Since this subgroup contains the double-transposition $x_1$ it follows that these elements generate the whole of the alternating group. It is easy to see that no automorphism simultaneously inverts both of these elements so any permutations that can be used to extend this to a Beauville structure gives a non-strongly real Beauville structure. Finally, let $$x_2:=(1,2,3)(4,5,\ldots,n)\mbox{ and }y_2:=(5,4,3,2,1).$$ We have that their product is the $(n-3)$-cycle $x_2y_2=(3,5,\ldots,n)$. By considering the subgroup generated by the elements $y_2^{x_2^i}$ we have that the subgroup these elements generate is 2-transitive and therefore primitive. Since this subgroup contains the double transposition $[x_2,y_2^2]=(2,5)(3,n)$ it follows that these elements generate the whole of the alternating group. We therefore have a non-strongly real Beauville group.
The case of $A_6$ is complicated by the existence of exceptional outer automorphisms but despite this can easily be handled separately.
Further examples are given by the following.
None of the Suzuki groups $^2B_2(2^{2n+1})$ are purely (non-)strongly real Beauville groups.
The Beauville structures constructed by Fuertes and Jones in [@FJ Theorem 6.2] are non-strongly real since they take $y_1$ as having order 4 and no automorphism maps such elements to their inverses in these groups. Strongly real Beauville structures for these groups were constructed by the author in [@F2].
Aside from the Mathieu groups M$_{11}$ and M$_{23}$ none of the sporadic simple groups are purely (non-)strongly real Beauville groups.
Aside from M$_{11}$ and M$_{23}$, strongly real Beauville structures for these groups were constructed by the author in [@F1]. Non-strongly real Beauville structures are easily obtained computationally and with character theory, the more difficult cases being easily dealt with thanks to the existence of elements that cannot be sent to their inverses by any automorphism at all such as elements of order 71 in the monster group $\mathbb{M}$ and elements of order 47 in the baby monster group $\mathbb{B}$.
We will return to the cases of M$_{11}$ and M$_{23}$ in Section \[NonSec\] since they are genuine exceptions to the above.
It is clear that numerous further examples can be constructed from the above using direct products.
Purely Strongly Real Beauville Groups
=====================================
Firstly, the following observation has been made elsewhere in the literature many times and provides infinitely many purely strongly real Beauville groups.
\[AbelianProof\] A finite abelian group is a strongly real Beauville group if and only if it is a purely strongly real Beaville group.
In any abelian group the homomorphism $x\mapsto-x$ inverts every element.
Abelian Beauville groups were first constructed by Catanese in [@C] and classified by Bauer, Catanese and Grunewald in [@BCG]. For non-soluble examples, we have the following infinite supply.
\[PruityL2(q)\] Let $q>2$ be a power of 2 and let $k$ be a positive integer. If $(q,k)\not=(4,1)$, then the characteristically simple group $L_2(q)^k$ is a purely strongly real Beaville group whenever it is 2-generated.
In [@MacB] MacBeath observed that any generating pair for the groups $L_2(q)$ can be inverted by an inner automorphism when $q$ is even and since these groups have only one class of involutions, the only elements of even order.
We remark that the exception $(q,k)=(4,1)$ is a genuine exception thanks to the isomorphism A$_5\cong\mbox{L}_2(4)$ and this group is well known to not be a Beauville group.
For each prime $p\geq5$ we can also construct infinitely many (new) non-abelian nilpotent examples as follows.
\[ExtraSpecProp\] If $p\geq5$ is a prime and $n\geq r\geq1$ are integers, then the group $$G:=\langle x,y,z\,|\,x^{p^n},\,y^{p^n},z^{p^r}\,[x,y]=z,\,[x,z],\,[y,z]\rangle$$ is a purely strongly real Beauville group.
Let $\{(x_1,y_1),(x_2,y_2)\}$ be a Beauville structure of $G$. We first note that Aut($G$) acts transitively on the non-central elements of $G$ of a given order and a generating pairs must consist of two element of order $p^n$, so without loss we may assume that $x_1=x$. Similarly $C_{\mbox{Aut}(G)}(x)$ acts transitively the elements $x$ can generate with so without loss we may also assume that $y_1=y$ and that $\phi\in$Aut($G$) acts by $x^\phi=x^{-1}$ and $y^{\phi}=y^{-1}$ (so we can take the element $g_1$ of Definition \[SRDef\] to be trivial).
Observe the following. For any element $g\in G\setminus\Phi(G)$ we have that its conjugates are $g^G=\{gz^i\,|\,i=0,\ldots,p^r-1\}$. Every element of this group can be written in the form $x^iy^jz^k$ for some $0\leq i,j\leq p^n-1$ and $0\leq k\leq p^r-1$, so we have that $x_2=x^{i_1}y^{j_1}z^{k_1}$ and $y_2=x^{i_2}y^{j_2}z^{k_2}$ for some $1\leq i_1,i_2,j_1,j_2\leq p^n-1$ and $0\leq k_1,k_2\leq p^r-1$. Notice that if these are chosen so that $i_1\not=j_1$, $i_2\not= j_2$, $i_1+i_2\not=j_1+j_2$ and $o(x_2)=o(y_2)=p^n$, then these elements provide a Beauville structure and since $p\geq5$ finding such integers is straightforward. Moreover $$(x^iy^jz^k)^{\phi}=x^{-i}y^{-j}z^k\mbox{ and }(x^iy^jz^k)^{-1}=x^{-i}y^{-j}z^{-k-ij}$$ We also have that for any $0\leq a,b\leq p^n-1$ $$(x^{-i}y^{-j}z^k)^{x^ay^b}=x^{-i}y^{-j}z^{k+bi-aj}.$$ It follows that to have $(x_2^{\phi})^{x^ay^b}=x_2^{-1}$ and $(y_2^{\phi})^{x^ay^b}=y_2^{-1}$ we must have that $$bi_1-aj_1\equiv-2k_1-ij\mbox{ (mod }p^r)\mbox{ and }bi_2-aj_2\equiv-2k_2-ij\mbox{ (mod }p^r).$$ If the values of $i_1, i_2, j_1, j_2, k_1$ and $k_2$ are chosen so $x_2$ and $y_2$ generate the group and the conjugacy condition is satisfied, then values of $a$ and $b$ satisfying these equations can be found if $j_2i_1\not\equiv j_1i_2$ (mod $p$). We claim that if $j_2i_1\equiv j_1i_2$ (mod $p$), then $x_2$ and $y_2$ do not generate the group. Note that under this condition we have that $$(x^{i_1}y^{j_1})^{i_2}=x^{i_1i_2}y^{j_1i_2}z^k=x^{i_1i_2}y^{j_2i_1}z^k$$ for some $k$ but we also have that $$(x^{i_2}y^{j_2})^{i_1}=x^{i_1i_2}y^{j_2i_1}z^{k'}$$ for some $k'$ in other words $y_2\in\langle x_2,z\rangle$ which is a proper subgroup.
We remark that strongly real Beauville $p$-groups have proved somewhat difficult to construct the only previously known examples being given in [@F3; @F4; @Gul1; @Gul2]. The above provides infinitely many further new examples and does so for each prime $p\geq5$.
Clearly Proposition \[PruityL2(q)\], Lemma \[AbelianProof\] and Proposition \[ExtraSpecProp\] can be combined to produce infinitely many other examples like L$_2(8)\times C_5^2$ but it is not clear what other examples can arise.
Find other examples of purely strongly real Beauville groups.
In particular, we have the following question.
Do there exist purely strongly real Beauville 2-groups and 3-groups.
In the opinion of the author it seems likely that 2-generated 2-groups are more likely to be purely strongly real Beauville groups: there is a general philosophy in the study of $p$-groups that ‘the automorphism group of a $p$-group is typically a $p$-group’ thanks to the results of Helleloid and Martin [@HM]. In particular, if $p$ is odd, then typically no automorphism like the $\phi$ in of Definition \[SRDef\] exists since such an automorphism must necessarily have even order. On the other side of the coin however, most of their results give examples with large numbers of generators and examples that are 2-generated seem difficult to construct.
Purely non-strongly Beauville groups {#NonSec}
====================================
What about purely non-strongly real Beauville groups? In [@F1 Lemma 2.2] the author shows, in the terminology defined here, that the Mathieu groups M$_{11}$ and M$_{23}$ are purely non-strongly real Beauville groups, indeed the real content of [@F2 Conjecture 1] is that among the non-abelian finite simple groups these are really the only ones. For an infinite supply of examples we have the following.
\[nonSR\] If $G$ and $H$ are Beauville groups of coprime order, such that $G$ is a purely non-strongly real Beauville group, then $G\times H$ is a purely non-strongly real Beauville group.
Let $\{(x_1,y_1),(x_2,y_2)\}$ be a Beauville structure for $G$ and let $\{(u_1,v_1),(u_2,v_2)\}$ be a Beauville structure for $H$. Since $|G|$ and $|H|$ are coprime it follows that $$\{((x_1,u_1),(y_1,v_1)),((x_2,u_2),(y_2,v_2))\}$$ is a Beauville structure for $G\times H$. Since $|G|$ is coprime to $|H|$ it follows that $G$ and $H$ are not isomorphic and so Aut($G\times H$)=Aut($G)\times$Aut($H$). Since elements of Aut($G$) cannot be used to provide a strongly real Beauville structure, it follows that none of the Beauville structures of $G\times H$ are strongly real.
There exist infinitely many purely non-strongly real Beauville groups.
As noted above, $M_{11}$ is a purely non-strongly real Beauville group. In Proposition \[nonSR\] we can therefore take $G$ to be M$_{11}$. Since $|\mbox{M}_{11}|=11\times5\times3^2\times2^4$ we can take $H$ to be any Beauville group has order coprime to 11, 5, 3 and 2. Infinitely many examples of such groups are constructed in [@BBF; @F4; @Gul1; @Gul2; @SV] as well as in Proposition \[ExtraSpecProp\].
The next example shows that Proposition \[nonSR\] is far from being the best possible.
The group $M_{11}\times A_5$ is easily seen to be a Beauville group as the elements $$x_1:=(1,2,3,4,5,6,7,8,9,10,11)(12,13,14,15,16),$$ $$y_1:=(1,5,3,4,10,2,8,9,11,6,7)(12,14,15,13,16)$$ along with $$x_2:=(1,2,9,10,6)(3,11,5,4,7)(12,13,14,15,16),$$ $$y_2:=(1,4,8,11,3)(2,9,7,5,6)(12,14,15,13,16)$$ provide a Beauville structure of type ((55,55,55),(5,5,5)). The lack of automorphisms that make $M_{11}$ a purely non-strongly real Beauville group clearly also make $M_{11}\times A_5$ a purely non-strongly real Beauville group. Note that every prime dividing the order $A_5$ also divides the order of $M_{11}$ and that $A_5$ not even a Beauville group.
What is the most general form of Proposition \[nonSR\]?
The author is not aware of any nilpotent or even soluble examples. As previously mentioned ‘the automorphism group of a $p$-group is typically a $p$-group’. Among the groups of order $p^n$ for small $n$ few examples of 2-generated groups with an automorphism group of odd order exist and the few that do appear to not be Beauville groups, suggesting that such examples are actually quite difficult to construct.
Do there exist any nilpotent or soluble purely non-strongly real Beauville groups?
This would be immediately answered by combining Proposition \[nonSR\] with an answer to the following.
Do there exist Beauville $p$-groups whose automorphism groups have odd order?
[99]{}
N.W. Barker, N. Bosten and B.T. Fairbairn “A note on Beauville $p$-groups" Exp. Math., 21(3): 298–306 (2012) [`arXiv:1111.3522v2`](http://arxiv.org/abs/1111.3522) doi:10.1080/10586458.2012.669267
I. Bauer, F. Catanese and F. Grunewald, Beauville surfaces without real structures, in *Geometric methods in algebra and number theory* pp. 1–42, *Progr. Math.*, 235, Birkhäuser Boston, Boston, MA, 2005.
F. Catanese, Fibered surfaces, varieties isogenous to a product and related moduli spaces, *Amer. J. Math.* 122 (2000), no. 1, pp. 1–44
B.T. Fairbairn “Some Exceptional Beauville Structures" J. Group Theory, 15(5), pp. 631–639 (2012) [`arXiv:1007.5050`](http://arxiv.org/abs/1007.5050) `DOI: 10.1515/jgt-2012-0018`
B.T. Fairbairn, “Strongly Real Beauville Groups" in *Beauville Surfaces and Groups, Springer Proceedings in Mathematics & Statistics, Vol. 123* (eds I. Bauer, S. Garion and A. Vdovina), Springer-Verlag (2015) pp. 41–61
B.T. Fairbairn, More on Strongly Real Beauville Groups, in *Symmetries in Graphs Maps and Polytopes 5th SIGMAP Workshop, West Malvern, UK, July 2014* (eds. J. Širáň and R. Jajcay) Springer Proceedings in Mathematics & Statistics 159 (2016) pp. 129–146
B.T. Fairbairn, A new infinite family of non-abelian strongly real Beauville $p$-groups for every odd prime $p$, *Bull. Lond. Math. Soc.* 49(4) (2017) doi:10.1112/blms.12060 [`arXiv:1608.00774`](http://arxiv.org/abs/1608.00774)
Y. Fuertes and G. González-Diez “On Beauville structures on the groups $S_n$ and $A_n$" Math. Z. 264 (2010), no. 4, 959–968
Y. Fuertes and G. Jones “Beauville surfaces and finite groups" J. Algebra 340 (2011) 13–27
Ş. Gül, A note on strongly real Beauville $p$-groups, *Monatsh. Math.* (2017)\
doi:10.1007/s00605-017-1034-1 `arXiv:1607.08907`
Ş. Gül, An infinite family of strongly real Beauville $p$-groups, preprint 2016\
[`arXiv:1610.06080`](https://arxiv.org/pdf/1610.06080.pdf)
G.T. Helleloid and U. Martin, The automorphism group of a finite $p$-group is almost always a $p$-group, *J. Algebra* 312 (2007) pp. 284–329
A. M. MacBeath “Generators of the linear fractional groups" Number Theory (Proc. Sympos. Pure Math., Vol. XII, Houston, Tex., 1967), Amer. Math. Soc., Providence, R.I., 1969, pp. 14–32
J. Stix and A. Vdovina, Series of $p$-groups with Beauville structure, *Monatsh. Math.* 181 (2016), no. 1, pp. 177–186 doi:10.1007/s00605-015-0805-9 `arXiv:1405.3872`
|
---
abstract: 'A recently developed theory for eliminating decoherence and design constraints in quantum computers, “encoded recoupling and decoupling”, is shown to be fully compatible with a promising proposal for an architecture enabling scalable ion-trap quantum computation \[D. Kielpinski *et al.*, Nature **417**, 709 (2002)\]. Logical qubits are encoded into pairs of ions. Logic gates are implemented using the S$\o$rensen-M$\o$lmer (SM) scheme applied to pairs of ions at a time. The encoding offers continuous protection against collective dephasing. Decoupling pulses, that are also implemented using the SM scheme directly to the encoded qubits, are capable of further reducing various other sources of qubit decoherence, such as due to differential dephasing and due to decohered vibrational modes. The feasibility of using the relatively slow SM pulses in a decoupling scheme quenching the latter source of decoherence follows from the observed $1/f$ spectrum of the vibrational bath.'
author:
- 'D.A. Lidar and L.-A. Wu'
title: 'Encoded Recoupling and Decoupling: An Alternative to Quantum Error Correcting Codes, Applied to Trapped Ion Quantum Computation'
---
Introduction
============
In the quest to construct a scalable quantum computer, a recent proposal [@Kielpinski:02] advocating an array-based approach to quantum computing (QC) with trapped ions appears particularly promising. Ions are stored for later processing in a memory region, then transported to an interaction region, where pairs are coupled in order to enact quantum logic gates. This proposal overcomes some of the design constraints that plagued the original Cirac-Zoller (CZ) ion-trap-QC proposal [@Cirac:95], which prevented the latter from becoming fully scalable. In the new proposal an encoding of a single logical qubit into the states $\{|0_{L}\rangle =|\!\!\downarrow \uparrow
\rangle ,|1_{L}\rangle =|\!\!\uparrow \downarrow \rangle \}$ of two trapped-ion (physical) qubits is used. Quantum logic gates are implemented using the S$\o$rensen-M$\o$lmer (SM) scheme [@Sorensen:99; @Sorensen:00] (see also related schemes by Milburn [*et al.*]{} [@Milburn:99; @Milburn:00]), which has the advantage of reduced sensitivity to motional state heating compared to the CZ proposal. The encoding into $\{|\!\!\downarrow \uparrow \rangle
,|\!\!\uparrow \downarrow \rangle \}$ is useful because these states form a decoherence-free subspace (DFS) [@Zanardi:97c; @Zanardi:97a; @Duan:98; @Lidar:PRL98] with respect to collective dephasing, a process whereby the environment introduces identical random phase modulations on groups of physical qubits [@ERD:comment1]. In the context of the quantum charge coupled device (QCCD) proposed in [@Kielpinski:02], such a process is one of two dominant sources of decoherence. The DFS encoding reduces the collective dephasing problem by several orders of magnitude [@Kielpinski:comment]. A method to perform universal QC using the SM scheme on these DFS qubits was proposed in [@Kielpinski:02], and independently in [@Bacon:thesis].
The DFS encoding $\{|\!\!\downarrow \uparrow \rangle ,|\!\!\uparrow \downarrow
\rangle \}$ is well known [@Palma:96; @Duan:98; @Lidar:PRL99; @Kempe:00], and its utility against collective dephasing was demonstrated experimentally using photons [@Kwiat:00] and trapped ions [@Kielpinski:01]. The notion of universal QC using a DFS has been explored in general in [@Bacon:99a; @Kempe:00; @Zanardi:99a; @Viola:99a] for Hamiltonians that always preserve the DFS, in [@Lidar:00b; @KhodjastehLidar:02; @Alber:02] using a combination of DFS and active quantum error correction methods [@Shor:95; @Steane:96a; @Aharonov:96; @Laflamme:96; @Preskill:97a; @Knill:98; @Steane:02], and in [@Beige:00], using an approach wherein transitions out of the code space are suppressed by continuous observation. Still more generally, the notion of *universal QC while overcoming decoherence as well as design constraints* has been explored by us and coworkers at a theoretical level in a series of recent papers [@WuLidar:01; @LidarWu:01; @WuLidar:01a; @WuLidar:01b; @ByrdLidar:01a; @WuLidar:02; @WuByrdLidar:02; @LidarWuBlais:02; @WuLidar:02a]. This theory uses a combination of qubit encoding into a DFS with selective recoupling [@LidarWu:01] and dynamical decoupling [@Viola:98; @Viola:98a; @Duan:98e; @Viola:99; @Zanardi:98b; @Vitali:99; @Vitali:01; @ByrdLidar:01; @Uchiyama:02] operations, so we refer to it as encoded recoupling and decoupling (ERD). The utility of ERD as a general method for quantum simulation, universal QC, and decoherence suppression has also been stressed and explored by Viola [@Viola:01a].
The main focus of ERD so far was on solid state [@WuLidar:01; @LidarWu:01; @WuLidar:01a; @WuLidar:01b; @ByrdLidar:01a; @WuLidar:02; @WuByrdLidar:02; @LidarWuBlais:02; @WuLidar:02a] and NMR [@Viola:01a; @Fortunato:01] QC proposals. Here we present a *unified treatment* of the ERD ideas, and show that they apply also in an atomic physics setting, namely the QCCD ion-trap proposal [@Kielpinski:02]. Specifically, we show here how to perform universal QC on DFS qubits comprised of pairs of trapped ions, by using the SM scheme for quantum logic, in a manner that involves manipulating only pairs of ions at any given time, while always perfectly preserving the DFS encoding (see Brown *et al*. [@Brown:02] for an interesting alternative set of ion pairs-only logic gates, which, however, does not preserve the DFS at all times). By applying strong and fast dynamical decoupling (bang-bang [@Viola:98; @Viola:98a]) SM pulses we show how to further drastically reduce sources of decoherence beyond collective dephasing. While a qubit is being *stored* an example of such a source of decoherence is deviations from the collective dephasing approximation. While a qubit is being *manipulated* for the purposes of QC, coupling to vibrational modes is necessary [@Cirac:95; @Sorensen:99; @Sorensen:00; @Milburn:99; @Milburn:00], and decoherence of these vibrational modes due to patch-potential noise is the second dominant source of qubit decoherence [@Wineland:98; @Myatt:00; @Kielpinski:01; @Turchette:00]. A method to suppress vibrational mode decoherence (as well as heating, which is not as serious a problem when the SM scheme is used), employing a version of the dynamical decoupling method known as parity kicks, was proposed and discussed in detail by Vitali & Tombesi (VT) [@Vitali:99; @Vitali:01]. This method uses a fast+strong modulation of the trapping potential. We present here an alternative decoupling method for suppressing decoherence of ion-trap qubits due to their coupling to decohered vibrational modes, that operates directly on the qubit (spin-) states. The feasibility of this scheme, in spite of the relative slowness of the SM pulses, follows from the observed $1/f$ spectrum of the vibrational bath [@Turchette:00; @Wineland:comment]. The concentration of most of the bath spectral density in the vicinity of the low, rather than the high-frequency cutoff, implies much relaxed constraints on the decoupling pulses compared to those usually assumed [@Viola:98; @Viola:98a; @Vitali:99; @Vitali:01]. A full analysis of this result will be presented in a forthcoming publication [@ShiokawaLidar:tbp].
More generally, we show here how *all* sources of decoherence beyond collective dephasing can in principle be suppressed using sufficiently strong+fast SM pulses. This includes so-called bath-induced leakage errors, wherein the system-bath coupling induces transitions into or out of the qubit subspace [@Zanardi:98b; @ByrdLidar:01a; @WuByrdLidar:02]. We provide feasibility estimates for the decoupling pulses and find that they are within current experimental reach. The overall picture emerging from this work is that ERD provides a means for a robust, decoherence-resistant implementation of universal QC with trapped ions. Experimental implementation of the ERD method should be possible with current ion-trap technology and we suggest a few experiments.
The structure of the paper is as follows. In section \[logic\] we review the DFS encoding into two spins and the associated logic gates. We show how our previous formulation thereof can be reinterpreted in the context of acting on pairs of trapped ions within the SM scheme. We also present a method for coupling pairs of encoded qubits using pulses that involve controlling only pairs of ions at a time, while always preserving the DFS encoding. In the subsequent sections we discuss how to reduce decoherence. In section \[decoupling\] we review the decoupling method, emphasizing its application to trapped ion arrays. We then proceed to apply the ERD method:in section \[createDFS\] we show how to eliminate the residual differential dephasing contribution to decoherence using SM pulses; and, in section \[leakage-elim\] we discuss how to reduce all further sources of decoherence, including the component that arises due to coupling to decohered motional states. Then, in section \[all\] we show how to fully implement ERD, i.e., we show how to combine universal QC via recoupling over DFS-encoded qubits with decoherence suppression via encoded decoupling. To make ERD fully effective for trapped ions we suggest to combine it with the VT potential-modulation method. Concluding remarks are presented in section \[conclusions\].
Encoded universal logic gates in ion traps {#logic}
==========================================
To fix terminology we first connect the methods developed in [@WuLidar:01; @LidarWu:01; @WuLidar:01a] to the gates proposed for trapped ions in [@Kielpinski:02]. Let $X_{i},Y_{i},Z_{i}$ denote the standard Pauli matrices $\sigma _{i}^{x},\sigma _{i}^{y},\sigma _{i}^{z}$, acting on the $i$th physical qubit (we will use both notations interchangeably). In [@WuLidar:01] it was shown that for the code $\{|0_{L}\rangle =|\!\!\downarrow \uparrow \rangle ,|1_{L}\rangle =|\!\!\uparrow \downarrow \rangle \}$ the encoded logical operations (involving the first two physical qubits) are
$$\begin{aligned}
\overline{X}_{12} &=&\frac{1}{2}(X_{1}X_{2}+Y_{1}Y_{2}),\quad \notag \\
\overline{Y}_{12} &=&\frac{1}{2}(Y_{1}X_{2}-X_{1}Y_{2}),\quad \notag \\
\overline{Z}_{12} &=&\frac{1}{2}(Z_{1}-Z_{2}). \label{eq:bars}\end{aligned}$$
These operations form an $su(2)$ algebra (i.e., we think of them as Hamiltonians rather than unitary operators). We use a bar to denote logical operations on the encoded qubits. In [@WuLidar:01; @LidarWu:01; @WuLidar:01a] these logical operations were denoted by $T_{12}^{\alpha }$, $\alpha \in
\{x,y,z\}$, and a detailed analysis was given on how to use typical solid-state Hamiltonians (Heisenberg, XXZ, and XY models) to implement quantum logic operations using this DFS encoding. E.g., the term $
X_{1}X_{2}+Y_{1}Y_{2}$ is the spin-spin interaction in the XY model, and $
Z_{1}-Z_{2}$ represents a Zeeman splitting. A static Zeeman splitting and a controllable XY interaction can be used to generate a universal set of logic gates, a result that has very recently been applied in the context of spin-based QC using semiconductor quantum dots and cavity QED [@Feng:02]. Similar conclusions hold when the XY interaction is replaced by a Heisenberg [@LidarWu:01; @Levy:01; @Benjamin:01] or XXZ interaction [@WuLidar:01a], or even by a Heisenberg interaction that includes an anistropic spin-orbit term [@WuLidar:02]. We remark that, as first shown in [@Bacon:99a; @Kempe:00], the various types of exchange interactions can be made universal also without any single-qubit terms (such as a Zeeman splitting), by encoding into three or more qubits [@Kempe:00; @DiVincenzo:00a; @Kempe:01; @Kempe:01a; @Vala:02], a result that has been termed encoded universality [@Bacon:Sydney].
Logic gates on two ions encoding a single logical qubit {#SMgates}
-------------------------------------------------------
S$\o$rensen and M$\o$lmer proposed a quantum logic gate that couples two ions via a two photon process that virtually populates the excited motional states of the ions [@Sorensen:00]. The SM scheme works well even for ions in thermal motion, while the CZ scheme requires cooling the ions to their motional ground state. The SM scheme involves applying two lasers of opposite detuning $\delta $ to the two ions. Ideally the Lamb-Dicke limit should be satisfied:
$$(n+1)\eta ^{2}\ll 1, \label{eq:LD}$$
where $\eta $ is the Lamb-Dicke parameter and $n$ is the mean vibrational number. Deviations from the Lamb-Dicke limit lead to fidelity reduction that is proportional to $\eta ^{4}$ [@Sorensen:00]. The time required to prepare a maximally entangled state using the SM scheme is
$$\tau_{\mathrm{SM}}=\frac{\pi }{\eta \Omega }\sqrt{K} \label{eq:tauSM}$$
where $\Omega $ is the Rabi frequency and $K$ is an integer [@Sorensen:00]. For realistic parameters, in the strong field limit ($K=1$ in Eq. (12) of [@Sorensen:00]), $\tau _{\mathrm{SM}}$ can be made as short as $1\mu $sec.
In [@Kielpinski:02] it was shown that the SM two-ion gate can be expressed as follows. The unitary gate $U_{2}(\theta ,\phi _{1},\phi _{2})$ was introduced, which we here rename $U_{ij}(\theta ,\phi _{i},\phi _{j})$: $$\begin{aligned}
U_{ij}(\theta ,\phi _{i},\phi _{j}) &\equiv &\exp (i\theta X_{\phi
_{i}} X_{\phi _{j}}) \notag \\
&=&\cos \theta I_{i} I_{j}+i\sin \theta X_{\phi _{i}} X_{\phi
_{j}}, \label{eq:Uij}\end{aligned}$$ where $$X_{\phi }\equiv X\cos \phi +Y\sin \phi .$$ The phase $\phi _{i}$ is the phase of the driving laser at the $i$th ion, while $\theta \propto \Omega $ and can be set over a wide range of values [@Sorensen:00; @Kielpinski:thesis]. Introducing the operators $$\tilde{X}_{ij}\equiv \frac{1}{2}(X_{i}X_{j}-Y_{i}Y_{j}),\quad \tilde{Y}
_{ij}\equiv \frac{1}{2}(Y_{i}X_{j}+X_{i}Y_{j})$$ (denoted $R_{ij}^{x}$, $R_{ij}^{y}$ respectively in [@WuLidar:01; @LidarWu:01; @WuLidar:01a]) we can express
$$\begin{aligned}
U_{ij}(\theta ,\phi _{i},\phi _{j}) &=&\cos \theta \bar{I}+i\sin \theta
(\cos \Delta \phi _{ij}\overline{X}_{ij} \notag \\
&&+\sin \Delta \phi _{ij}\overline{Y}_{ij} +\cos \Phi _{ij}\tilde{X}_{ij}+\sin \Phi _{ij}\tilde{Y}_{ij}), \notag \\
\label{eq:Uij2}\end{aligned}$$
where $\Phi _{ij}=$ $\phi _{i}+\phi _{j}$. It is simple to check that $
\tilde{X}_{ij}$ and $\tilde{Y}_{ij}$ annihilate the code subspace $
\{|0_{L}\rangle =|\!\!\downarrow \uparrow \rangle ,|1_{L}\rangle =|\!\!\uparrow
\downarrow \rangle \}$ and have non-trivial action (as encoded $X$ and $Y$) on the orthogonal subspace $\{|\!\!\downarrow \downarrow \rangle ,|\!\!\uparrow
\uparrow \rangle \}\!$. Therefore, as also observed in [@Kielpinski:02] and [@Bacon:thesis], upon restriction to the DFS we can write: $$\begin{aligned}
U_{ij}(\theta ,\phi _{i},\phi _{j}) &\overset{\mathrm{DFS}}{\mapsto }&\bar{U}
_{ij}(\theta ,\Delta \phi _{ij}) \notag \\
&=&\exp (i\theta \overline{X}_{\Delta \phi _{ij}})=\cos \theta \bar{I}+i\sin
\theta \overline{X}_{\Delta \phi _{ij}}. \notag \\
&& \label{eq:Ubar}\end{aligned}$$ The fact that $\bar{U}_{ij}$ depends only on the relative phase $\Delta \phi
_{ij}$ is crucial: this quantity can be controlled by adjusting the angle between the driving laser and the interatomic axis, as well as by small adjustments of the trap voltages (which, in turn, control the trap oscillation frequency, and hence the ion spacing), whereas it is much harder to control the absolute phase $\phi _{i}$ [@Sackett:00; @Kielpinski:thesis; @Kielpinski:02], and hence also $\Phi _{ij}$. This is why the code subspace $\{|\!\!\downarrow \uparrow \rangle,|\!\!\uparrow\downarrow \rangle \}$ enjoys a preferred status over the subspace $\{|\!\!\downarrow \downarrow \rangle ,|\!\!\uparrow
\uparrow \rangle \}$. A thorough theoretical analysis of the approximations leading to the gate $
U_{ij}(\theta ,\phi _{i},\phi _{j})$ is given in [@Sorensen:00] (see also [@Bacon:thesis] for an abbreviated exposition that emphasizes the connection to computation in a DFS).
Let us establish the connection between the seemingly distinct sets of logic operations in Eqs. (\[eq:bars\]),(\[eq:Uij\]). To do so, we only need to use Eqs. (\[eq:Uij2\]),(\[eq:Ubar\]) while ignoring the component that annihilates the DFS ($\tilde{X},\tilde{Y}$). Then: $$\begin{aligned}
\exp (i\theta \overline{X}_{12}) &=&U_{12}(\theta ,\phi ,\phi )=\bar{U}
_{12}(\theta ,0) \notag \\
\exp (i\theta \overline{Y}_{12}) &=&U_{12}(\theta ,\phi ,\phi +\frac{\pi }{2}
)=\bar{U}_{12}(\theta ,\frac{\pi }{2}) \notag \\
\exp (i\theta \overline{Z}_{12}) &=&\exp (i\frac{\pi }{4}\overline{Y}
_{12})\exp (i\theta \overline{X}_{12})\exp (-i\frac{\pi }{4}\overline{Y}
_{12}) \notag \\
&=&\bar{U}_{12}(\frac{\pi }{4},\pi /2)\bar{U}_{12}(\theta ,0)\bar{U}_{12}(-
\frac{\pi }{4},\pi /2). \label{eq:XYZ}\end{aligned}$$ The third line follows from the elementary operator identity $$X_{\phi }=X\cos \phi +Y\sin \phi =e^{-i\phi Z/2}Xe^{i\phi Z/2}
\label{eq:identity}$$ which holds for any $su(2)$ angular momentum set $\{X,Y,Z\}$, i.e., operators that satisfy the commutation relation $[X,Y]=2iZ$ (and cyclic permutations thereof), in particular also the encoded operators $\{\overline{
X},\overline{Y},\overline{Z}\}$.
This proves the equivalence of the two sets of operators. Using these results and Eq. (\[eq:Uij\]), a more direct connection can be written in terms of the Hamiltonians: $$\begin{aligned}
\overline{X}_{12}\quad &\Longleftrightarrow &\quad X_{\phi } X_{\phi }
\label{eq:xbar} \\
\overline{Y}_{12}\quad &\Longleftrightarrow &\quad X_{\phi } X_{\phi
+\pi /2}, \label{eq:ybar}\end{aligned}$$ where the equivalence is meant in terms of a projection of the RHS Hamiltonians to the DFS. In the context of ion-trap QC the logic gate $\bar{
U
}(\theta ,\Delta \phi )$ can be performed directly, so it may be more intuitively useful than the $\{\overline{X},\overline{Y},\overline{Z}\}$ set. Eq. (\[eq:XYZ\]) shows that by properly choosing $\theta $ and $
\Delta \phi _{ij}$ all single DFS-encoded qubit gates can be performed.
Entangling gate between pairs of encoded qubits involving four ions {#4ions}
-------------------------------------------------------------------
In [@Kielpinski:02] the following unitary gate was introduced $$\begin{aligned}
U_{4} &=&\exp (-i\frac{\pi }{4}X_{\phi _{_{1}}} X_{\phi
_{_{2}}} X_{\phi _{_{3}}} X_{\phi _{_{4}}}) \notag \\
&=&\frac{1}{\sqrt{2}}\left( I_{1} I_{2} I_{3}
I_{4}-iX_{\phi _{_{1}}} X_{\phi _{_{2}}} X_{\phi
_{_{3}}} X_{\phi _{_{4}}}\right) \notag \\
&&\overset{\mathrm{DFS}}{\mapsto }\frac{1}{\sqrt{2}}\left( \bar{I}
_{12} \bar{I}_{34}-i\overline{X}_{\Delta \phi _{_{12}}}
\overline{X}_{\Delta \phi _{_{34}}}\right) \notag \\
&=&\exp (-i\frac{\pi }{4}\overline{X}_{\Delta \phi _{_{12}}}
\overline{X}_{\Delta \phi _{_{34}}}).
\label{eq:U4}\end{aligned}$$ This gate, also considered in slightly less general form in [@Bacon:thesis], can be used to entangle two DFS-qubits. It involves simultaneous control over two phase differences $\Delta \phi _{_{12}},\Delta
\phi _{34}$, and thus control over the motion of two pairs of ions. The case $\Delta \phi _{_{12}}=\Delta \phi _{34}=0$ was used in [@Sackett:00] to demonstrate entanglement of four trapped-ion qubits, but this choice is not unique.
We now come to an important point that was not addressed in [@Kielpinski:02]: in order for the DFS encoding to offer protection against collective dephasing during the exection of the entangling gate, *collective dephasing conditions must prevail over all four ions*. To see this, note that a differential dephasing term such as $(Z_{1}-Z_{3})\otimes
B $ (where $B$ is a bath operator) does not commute with $U_{4}$, so that if such a term exists during gate execution then the DFS will not be preserved, according to a theorem in [@Bacon:99a]. On the other hand, collective dephasing over all four ions, expressed by a system-bath coupling of the form $(\sum_{i=1}^{4}Z_{i})\otimes B$, does commute with $U_{4}$, so that in this case the DFS is preserved [@Bacon:99a]. While deviations from collective dephasing over pairs of ions have been shown experimentally to be small [@Kielpinski:01], this may not be the case over the length scales involving four ions [@Kielpinski:comment]. We discuss in section \[createDFS\] how to create such extended collective dephasing conditions.
Taken together, the results in this section show how universal QC can be implemented using trapped ions by applying the SM scheme to pairs of ions at a time, each encoding a DFS qubit. The DFS encoding takes care of protecting the encoded information against collective dephasing. We now move on to a discussion of how to reduce additional source of decoherence.
Dynamical decoupling pulses and their application to trapped ions {#decoupling}
=================================================================
Let us briefly review the decoupling technique, as it pertains to our problem (for an overview see, e.g., [@Viola:01a]). Decoupling, as proposed by Viola and Lloyd [@Viola:98; @Viola:98a], relies on the ability to apply *strong and fast* pulses, in a manner which effectively averages the system-bath interaction Hamiltonian $H_{SB}$ to zero. A quantitative analysis was first performed in [@Viola:98; @Viola:98a] for pure dephasing in the linear spin-boson model (which corresponds to the ohmic case of the Caldeira-Leggett model [@Leggett:87]): $H_{SB}=\gamma \sigma
^{z}\otimes B$, where $B$ is a Hermitian boson operator. The analysis was recently extended to the non-linear spin-boson model, with similar conclusions about performance [@Uchiyama:02]. Imperfections in the pulses were considered in [@Duan:98e], and it was shown that an optimal value for the pulse period can be found. Since the decoupling pulses are *strong* one ignores the evolution under $H_{SB}$ while the pulses are on, and since the pulses are *fast* one ignores the evolution of the bath under its free Hamiltonian $H_B$ during the pulse cycle. The simplest example of eliminating an undesired unitary evolution $U=\exp [-it(H_{SB}+H_{B})]$, is the “*parity-kick sequence*” [@Viola:98; @Viola:98a; @Vitali:99]. Suppose we have at our disposal a fully controllable interaction generating a gate $R$ such that $R$ *conjugates* $U$: $R^{\dagger
}UR=U^{\dagger }$. Then the sequence $UR^{\dagger }UR=I$ serves to eliminate $U$. A simple example of a parity kick sequence is the following. Assume we can turn on the single-qubit Hamiltonian $\Omega X_{j}$ for a time $\pi
/2\Omega $. This generates the single-qubit gate $X_{j}=i\exp (-i\frac{\pi }{
2}X_{j})$. Suppose that $H_{SB}=\sum_{i=1}^{N}\sum_{\alpha \in
\{x,y,z\}}\gamma _{i}^{\alpha }\sigma _{i}^{\alpha }\otimes B_{i}^{\alpha }$. Each term in $H_{SB}$ either commutes or anti-commutes with $X_{j}$. If a term $A$ in $H_{SB}$ anticommutes with $X_{j}$ then the evolution under it will be conjugated by the gate $X_{j}$: $X_{j}\exp (-iA\Delta t)X_{j}=\exp
(-iX_{j}AX_{j}\Delta t)=\exp (iA\Delta t)$. This allows for selectively removing this term using the parity-kick cycle, which we write as: $[\Delta
t,X_{j},\Delta t,X_{j}]$. Reading from right to left, this notation means: apply $X_{j}$ pulse, free evolution for time $\Delta t$, repeat. Suppose that we can also apply the single qubit gate $Y_{j}$. Then, since every system factor in the above $H_{SB}$ contains a single-qubit operator, it follows that we can selectively keep or remove each term in $H_{SB}$ by using the parity-kick cycle. Note, however, that without additional symmetry assumptions, this procedure, if used to eliminate *all errors*, requires a number of pulses that is exponential in the number of qubits $N$ [@Duan:98e; @Viola:99]. The reason is that without symmetry assumptions we will need at least two non-commuting single-qubit operators per qubit (e.g., $X$, $Y$), and we will need to concatenate decoupling pulse sequences. Below we show how to dynamically generate such symmetries, in a way that avoids this exponential scaling (for a discussion of this point see the Conclusions section).
Note that in the analysis of the parity kick cycle we ignored $H_{SB}$ and $
H_{B}$ while $R$ was operating; this is justified by the *strength* assumption. The bath Hamiltonian $H_{B}$ commutes with the applied pulses, but its effect is very important since $[H_B,H_{SB}]\neq 0$ in general. Therefore if the bath has spectral components at frequencies higher than the inverse of the interval between decoupling pulses, then the bath density matrix will pick up phases that are essentially random, and this effect will show up as decoherence (for a quantitative analysis see [@Viola:98; @Viola:98a; @Duan:98e; @Vitali:01; @Uchiyama:02]. Hence it is commonly assumed that the pulse interval, $\Delta t$, should be small compared to the inverse of the high-frequency cutoff $\omega _{c}$ of the bath spectral density $I(\omega)$ [@Viola:98; @Viola:98a], which also sets the scale of the bath-induced noise correlation time $t_c$ (the *speed* assumption). It can be shown that the overall system-bath coupling strength $\gamma _{SB}$ is then renormalized by a factor $\Delta t\omega _{c}$ after a cycle of decoupling pulses [@Viola:99], or that the bath-induced error rate is reduced by a factor proportional to $(\Delta t/t_{c})^2$ [@Duan:98e]. Using a Magnus expansion [@Ernst:book], it can be shown that there is a hierarchy of decoupling schemes, whereby $\gamma _{SB}$ is renormalized by a factor $(\Delta t\omega _{c})^{k}$, where $k\geq 1$ is the order of the decoupling scheme [@Viola:99]. The implication for single-qubit dephasing, $H_{SB}=\frac{1}{2}\gamma _{SB}Z\otimes B$ ($B$ is a dimensionless bath operator), is that the dephasing time $T_{2}$ is increased by a factor $1/(\Delta t\omega _{c})^{2k}$ [@ERD:comment2]. Thus it seems crucial to be able to apply pulses at intervals $\Delta t\ll 1/\omega _{c}$. However, as shown first by Viola & Lloyd [@Viola:98], and then by VT in their quantitative analysis of a vibrational mode linearly coupled to a boson bath, a finite-temperature bath sets another, thermal timescale that must be beat in order for the decoupling method to work [@Vitali:01]. Let the system-bath coupling be $$H_{SB}^{\mathrm{vib}}=\gamma \sum_{k}(ab_{k}^{\dagger }+a^{\dagger }b_{k}),
\label{eq:vibHSB}$$ where $a$ ($b)$ is an annihilation operator for the system ($k$th bath) vibrational mode, and $\gamma $ is the (for simplicity uniform) energy damping rate. In the context of trapped ions the bath is provided by fluctuating patch-potentials (due, e.g., to randomly oriented domains at the surface of the electrodes or adsorbed materials on the electrodes) [@Turchette:00]. Then VT showed that the decoupling pulse interval (in fact, the entire cycle time) must be shorter also than the thermal decoherence time $$t_{\mathrm{dec}}(T)=\{\gamma \lbrack 1+2n(T)]\}^{-1},$$ where $n(T)=[e^{\hbar \omega _{0}/k_{B}T}-1]^{-1}$ is the mean vibrational number of the system oscillator at thermal equilibrium with temperature $T$, and $\omega _{0}$ is the frequency of the oscillator, i.e., the system is described by the harmonic oscillator Hamiltonian $H_{S}=\hbar \omega
_{0}a^{\dagger }a$. Thus the timescale condition for successful decoupling is $$\Delta t\ll \min \{1/\omega _{c},t_{\mathrm{dec}}(T)\}.$$ As shown in the VT analysis, the timescale $t_{\mathrm{dec}}(T)$ is especially relevant for vibrational mode decoherence in ion traps, which as already mentioned above, is responsible for qubit decoherence during quantum logic gate operations.
However, for trapped ions experimental evidence so far points to a $1/f^{\alpha }$ spectrum for the vibrational bath over a range $1-100$MHz [@Turchette:00 p.5], implying that there is no clear high-frequency cutoff $\omega _{c}$. Encouragingly, in a recent experiment involving a charge qubit in a small superconducting electrode (Cooper-pair box), a version of parity-kick decoupling was successfully used to suppress low-frequency energy-level fluctuations (causing dephasing) due to $1/f$ charge noise [@Nakamura:02]. This suggests that decoupling can help even in the absence of a clear cutoff frequency. Recent theoretical results support this observation [@ShiokawaLidar:tbp]: for $1/f$ noise most of the bath spectral density $I(\omega)$ is concentrated in the low, rather than the high-end of the frequency range.
In spite of the apparent $1/f^{\alpha }$ spectrum in trapped ions, VT used a cutoff estimate of $\omega _{c}\leq 100$ MHz [@Vitali:01], and showed that suppression of vibrational decoherence can be accomplished by *pulsing the oscillation frequency* $\omega _{0}$ *of the ion chain* (i.e., by pulsing the trapping potential), provided $\Delta t<1/\omega
_{c}\sim 1$nsec, *and* $T\leq 10$mK.
Given the estimate in Section \[SMgates\] of $\tau _{\mathrm{SM}}\gtrsim 1\mu $s for the SM gate, it is clear that we cannot hope to satisfy the strict $\Delta t<1$nsec timescale requirement which would be needed in order to use decoupling directly on the qubit, rather than the vibrational modes, assuming the VT estimate of $\omega _{c}$. However, the theoretical analysis [@ShiokawaLidar:tbp] and the success of parity-kick decoupling in the presence of $1/f$ noise in the charge qubit case [@Nakamura:02] suggests that it may well be worthwhile to apply decoupling pulses on the qubit *in addition* to, or perhaps instead of, the VT trapping-potential-modulation scheme.
Now let us comment on the strength assumption. Here we must make sure that the amplitude of the decoupling pulses, $\Omega $, can be made much stronger than the system-bath interaction $\gamma $ in Eq. (\[eq:vibHSB\]), i.e., the heating rate from the vibrational ground state of the ion chain. Experimental measurements of $\gamma $ are very sensitive to trap geometry, secular frequency, and size [@Turchette:00], and range from a few Hz to a few tens of KHz [@Turchette:00; @Roos:99]. On the other hand, one can have $\Omega $ as high as $1$MHz [@Steane:00], so the strength assumption can be comfortably satisfied. This does come at a price, however, since in the strong field limit the SM gate is perturbed by a term that yields direct, off-resonant coupling of the qubit $|\uparrow \rangle $ and $
|\downarrow \rangle $ states without changes in the vibrational motion [@Sorensen:00]. This is a *unitary* gate error that decreases the gate fidelity by $(N/2)(\Omega /\delta )^{2}$, where $N$ is the number of ions participating in the gate [@Sorensen:00 Table II]. This forces us to be in a parameter regime where $\Omega \ll \delta $. In principle it is possible to exactly cancel this effect if the duration of the laser pulses is chosen so that both Eq. (\[eq:tauSM\]) and the condition $\tau _{
\mathrm{SM}}=K^{\prime }\pi /\delta $ are satisfied, where $K^{\prime }$ is an integer and $\delta $ is the detuning. However, in the context of decoupling we will also need to satisfy conditions such as $\Omega \tau _{
\mathrm{SM}}=\pi /m$ where $m$ is an integer. Putting these conditions together yields $$\begin{aligned}
\Omega \frac{K^{\prime }\pi }{\delta } &=&\pi /m\quad \Rightarrow \quad
\delta =mK^{\prime }\Omega \\
\Omega \frac{\pi }{\eta \Omega }\sqrt{K} &=&\pi /m\quad \Rightarrow \quad
\eta =m\sqrt{K}\end{aligned}$$ While there is no problem with the first of these, the second condition implies that we cannot be in the Lamb-Dicke limit, Eq. (\[eq:LD\]). Therefore exact cancellation is not possible in our case, and we must resort to $\Omega \ll \delta $ in order to keep the fidelity reduction to a minimum. On the other hand, the kind of unitary error that is caused by off-resonant coupling can be corrected by optimal control pulse shaping methods [@Palao:02].
Finally, we note that fluctuations of various experimental parameters, such as intensity and phase fluctuations of the exciting lasers, can cause pure dephasing of the vibrational modes, in addition to the dissipative coupling described above [@Schneider:98]. Clearly, the success of decoupling strategies hinges on strong suppression of such fluctuations, as in the threshold theorem of fault tolerant quantum error correction [@Aharonov:96; @Preskill:97a; @Knill:98; @Steane:02].
To conclude, the discussion in this section indicates that the experimental viability of decoupling schemes in ion traps is rather promising, although it is hard to estimate their success at this point. In the following sections the analysis will be carried out at a more abstract level, emphasizing the algebraic conditions for a successful implementation of ERD. In the end it will be up to an experiment to decide the usefulness of the proposed schemes.
Creating collective dephasing conditions using decoupling pulses:reducing decoherence during storage {#createDFS}
====================================================================================================
One of the important advantages of the DFS encoding $\{|\!\!\downarrow \uparrow
\rangle ,|\!\!\uparrow \downarrow \rangle \}$ is that it is immune to collective dephasing. However, other sources of decoherence inevitably remain. In this and the following section, we algebraically classify all additional decoherence effects and show how they can be eliminated, in particular in the context of trapped ions.
Creating collective dephasing on a pair of ions
-----------------------------------------------
First, let us analyze the effect of breaking the collective dephasing symmetry, by considering a system-bath interaction of the form $$H_{SB}^{\mathrm{deph}(2)}=Z_{1}\otimes B_{1}^{z}+Z_{2}\otimes B_{2}^{z}$$ where $B_{1}^{z},B_{2}^{z}$ are arbitrary bath operators. This describes a general dephasing interaction on two qubits, and we can expect this to be the case during *storage* of trapped ion qubits in the QCCD proposal. The source of such dephasing during storage is long wavelength, randomly fluctuating ambient magnetic fields [@Kielpinski:01], that randomly shift the relative phase between the qubit $|\uparrow \rangle $ and $
|\downarrow \rangle $ states through the Zeeman effect. The interaction can be rewritten as a sum over a collective dephasing term $Z_{1}+Z_{2}$ and another, differential dephasing term $Z_{1}-Z_{2}$, that is responsible for errors on the DFS: $$H_{SB}^{\mathrm{deph}(2)}=\left( Z_{1}+Z_{2}\right) \otimes B_{\mathrm{col}
}^{z}+\left( Z_{1}-Z_{2}\right) \otimes B_{\mathrm{dif}}^{z}.$$ Here $B_{\mathrm{col}}^{z}=\left( B_{1}^{z}+B_{2}^{z}\right) /2$ and $B_{
\mathrm{dif}}^{z}=\left( B_{1}^{z}-B_{2}^{z}\right) /2$. If $B_{\mathrm{dif}
}^{z}$ were zero then there would only be collective dephasing and the DFS encoding would offer perfect protection [@ERD:comment3]. However, in general $B_{\mathrm{dif}}^{z}\neq 0$, and the DFS encoding will not suffice to offer complete protection.
The crucial observation is that, since $Z_{1}-Z_{2}\propto \overline{Z}_{12}$ \[recall Eq. (\[eq:bars\])\], the offending term causes *logical* errors on the DFS [@ByrdLidar:01a]. Then the problem of $B_{\mathrm{dif}
}^{z}\neq 0$ can be solved using a series of pulses that symmetrize $H_{SB}^{
\mathrm{deph}(2)}$ such that only the collective term remains, as shown in [@WuLidar:01b; @Viola:01a] [@ERD:comment4]. To do so note that since the offending term $\propto \overline{
Z}_{12}$, it anticommutes with $\overline{X}_{12}=\frac{1}{2}(X_1 X_2
+ Y_1 Y_2)$. At the same time $
\overline{X}_{12}$ commutes with $Z_{1}+Z_{2}$. This allows us to flip the sign of the offending term by using a pair of $\pm \pi /2$ pulses in $
\overline{X}_{12}$, while leaving only the collective term. Evolution with the flipped sign followed by unaltered evolution leads to cancellation of the offending term. Specifically [@WuLidar:01b]:
$$e^{-iH_{SB}\tau }e^{-i\frac{\pi }{2}\overline{X}_{12}}e^{-iH_{SB}\tau }e^{i
\frac{\pi }{2}\overline{X}_{12}}=e^{-i(Z_{1}+Z_{2})\otimes B_{\mathrm{col}
}^{z}2\tau },$$
or, in ion-trap terms: $$e^{-iH_{SB}\tau }\bar{U}_{12}(-\frac{\pi }{2},0)e^{-iH_{SB}\tau }\bar{U}
_{12}(\frac{\pi }{2},0)=e^{-i(Z_{1}+Z_{2})\otimes B_{\mathrm{col}}^{z}2\tau
}, \label{eq:sym}$$ where $\bar{U}_{ij}(\theta ,\Delta \phi _{ij})$ was defined in Eq. (\[eq:Ubar\]), and we used the identification found in Eq. (\[eq:XYZ\]). This equation means that the system-bath coupling effectively looks like collective dephasing at the end of the pulse sequence. Thus, the system is periodically (every $2\tau $) projected into the DFS.
In order for the the procedure described in Eq. (\[eq:sym\]) to work, the SM gate $\bar{U}_{12}(\pm \frac{\pi }{2},0)$ must be executed at a timescale faster than the cutoff frequency associated with the fluctuating magnetic fields causing the differential dephasing term in $H_{SB}^{\mathrm{
\ deph}(2)}$. This cutoff has not yet been characterized experimentally, but the decay rate of the DFS-encoded state of two ions has been measured to be $
2.2$KHz [@Kielpinski:01]. Using this as a rough estimate for the cutoff frequency, we see that the procedure of Eq. (\[eq:sym\]) is likely to be attainable with fast ($\tau _{\mathrm{SM}}\approx 1\mu $s) SM pulses.
Creating collective dephasing on a block of four ions
-----------------------------------------------------
So far we have discussed creation of collective dephasing conditions on a single DFS qubit. However, as mentioned in Section \[4ions\], it is essential for the reliable execution of an entangling logic gate to have collective dephasing over all four ions participating in the gate, even if only two are coupled at a time. A procedure for creating collective *decoherence* conditions over blocks of $3,4,6$ and $8$ qubits was given in [@WuLidar:01b]. Here we show how to do the same for a block of 4 qubits with collective dephasing.
Let us start with a general dephasing Hamiltonian on $N$ ions, and rewrite it in terms of nearest-neighbor sums and differences: $$\begin{aligned}
H_{SB}^{\mathrm{deph}} &=&\sum_{i=1}^{N}Z_{i}\otimes B_{i} \\
&=&\sum_{j=1}^{N/2}\left( Z_{2j}+Z_{2j-1}\right) \otimes B_{2j}^{+}+\left(
Z_{2j}-Z_{2j-1}\right) \otimes B_{2j}^{-},\end{aligned}$$ where $B_{2j}^{\pm }\equiv (B_{2j}\pm B_{2j-1})/2$. As noted above, $
Z_{2j}-Z_{2j-1}\propto \overline{Z}_{2j-1,2j}$, so that to eliminate all nearest-neighbor differences of the form $\left(
Z_{2j}-Z_{2j-1}\right) $ we can use the collective decoupling pulse $X_{nn}=\bigotimes_{j=1}^{N/2}e^{i\frac{\pi }{2} \overline{X}_{2j-1,2j}}$:
$$e^{-iH_{SB}\tau }X_{nn}e^{-iH_{SB}\tau }X_{nn}^{\dagger }=e^{-i2\tau
\sum_{j=1}^{N/2}\left( Z_{2j}+Z_{2j-1}\right) \otimes B_{2j}^{+}},$$
or, in ion-trap terms:
$$\begin{aligned}
e^{-iH_{SB}\tau }\left[ \bigotimes_{j=1}^{N/2}\bar{U}_{2j-1,2j}(-\frac{\pi }{
2},0)\right] e^{-iH_{SB}\tau }\left[ \bigotimes_{j=1}^{N/2}\bar{U}_{2j-1,2j}(
\frac{\pi }{2},0)\right]
=e^{-i2\tau \sum_{j=1}^{N/2}\left(
Z_{2j}+Z_{2j-1}\right) \otimes B_{2j}^{+}}.\end{aligned}$$
The next step is to eliminate next-nearest neighbor differential terms. To this end let us rewrite the outcome of the $X_{nn}$ pulse in terms of sums and differences over blocks of four ions: $$\begin{aligned}
\sum_{j=1}^{N/2}\left( Z_{2j}+Z_{2j-1}\right) \otimes B_{2j}^{+}
&=& \sum_{j=1}^{N/2}\left[ Z_{2j+2}+Z_{2j+1}+Z_{2j}+Z_{2j-1}\right] \otimes
B_{2j}^{+,+} \notag \\
&+& \sum_{j=1}^{N/2} \left[ (Z_{2j+2}-Z_{2j})+(Z_{2j+1}-Z_{2j-1})\right]
\otimes B_{2j}^{+,-},\end{aligned}$$ where $B_{2j}^{+,\pm }\equiv (B_{2j+2}^{+}\pm B_{2j}^{+})/2$. The term in the first line contains only the desired block-collective dephasing over $4$ ions. The term in the second line contains undesired differential dephasing terms that we wish to eliminate. But these terms once again have the appearance of encoded $Z$ operators, between next-nearest neighbor ion pairs. Therefore we need to apply a second collective pulse $
X_{nnn}=\bigotimes_{j=1}^{N/2}e^{i\frac{\pi }{2}\overline{X}_{2j-1,2j+1}}e^{i
\frac{\pi }{2}\overline{X}_{2j,2j+2}}$, that applies encoded $X$ operators on these ion pairs. At this point we are left just with collective dephasing terms on blocks of $4$ ions, as required: $$\begin{aligned}
e^{-i2\tau \sum_{j=1}^{N/2}\left( Z_{2j}+Z_{2j-1}\right) \otimes
B_{2j}^{+}}\left[ \bigotimes_{j=1}^{N/2}\bar{U}_{2j-1,2j+1}(-\frac{\pi }{2}
,0)\bar{U}_{2j,2j+2}(-\frac{\pi }{2},0)\right] &\times& \notag \\
e^{-i2\tau \sum_{j=1}^{N/2}\left( Z_{2j}+Z_{2j-1}\right) \otimes B_{2j}^{+}}
\left[ \bigotimes_{j=1}^{N/2}\bar{U}_{2j-1,2j+1}(\frac{\pi }{2},0)\bar{U}
_{2j,2j+2}(\frac{\pi }{2},0)\right] &=&
e^{-i4\tau \sum_{j=1}^{N/2}\left(
Z_{2j+2}+Z_{2j+1}+Z_{2j}+Z_{2j-1}\right) \otimes B_{2j}^{+,+}}.\notag \\
\label{eq:create4}\end{aligned}$$
This pulse sequence is important to ensure that collective dephasing conditions will prevail during the execution of logic gates between DFS qubits, as emphasized in Section \[4ions\].
To conclude, the procedures discussed in this section provide a means for *engineering collective dephasing conditions in an ion trap experiment*. We propose here to implement these symmetrization schemes experimentally. Moreover, we propose to combine them with the logic gates described in Sec. \[logic\]. How to do this efficiently is discussed in Sec. \[all\] below.
Reduction of all remaining decoherence on a single DFS qubit during logic gate execution {#leakage-elim}
========================================================================================
The reduction of differential dephasing errors, as in the previous subsection, is particularly relevant for storage errors. However, this is only the first step. Additional sources of decoherence may take place during storage, and in particular during the execution of logic gates, the most dominant of which is qubit decoherence due to coupling to decohered vibrational modes, as discussed above. It is useful to provide a complete algebraic classification of the possible decoherence processes. This will allow us to see what can be done using SM-decoupling pulses. To this end let us now write the system-bath Hamiltonian on two physical qubits in the general form $$H_{SB}=H_{\mathrm{Leak}}+H_{\mathrm{Logi}}+H_{\mathrm{DFS}}$$ where $$\begin{aligned}
H_{\mathrm{DFS}} &=&\mathrm{Span}\{\frac{ZI+IZ}{2},\frac{XY+YX}{2},\frac{
XX-YY}{2}, \notag \\
&& ZZ,II\} \notag \\
H_{\mathrm{Leak}} &=&\mathrm{Span}\{XI,IX,YI,IY,XZ,ZX,YZ,ZY\} \notag \\
H_{\mathrm{Logi}} &=&\mathrm{Span}\{\bar{X}=\frac{XX+YY}{2},\bar{Y}=\frac{
YX-XY}{2}, \notag \\
\bar{Z} &=&\frac{ZI-IZ}{2}\} \label{eq:class}\end{aligned}$$ where $I$ is the identity operator, $XZ\equiv X_{1}Z_{2}$ (etc.), and where $
\mathrm{Span}$ means a linear combination of these operators tensored with bath operators. The $16$ operators in Eq. (\[eq:class\]) form a complete basis for all $2$-qubit operators. This classification, first introduced in [@ByrdLidar:01a], has the following significance. The operators in $H_{
\mathrm{DFS}}$ either vanish on the DFS, or are proportional to identity on it. In either case their effect is to generate an overall phase on the DFS, so they can be safely ignored from now on. The operators in $H_{\mathrm{Leak}
}$ are the *leakage errors*: terms that cause transitions between states inside and outside of the DFS. A universal and efficient decoupling method for eliminating such errors, for arbitrary numbers of (encoded)qubits was given in [@WuByrdLidar:02]. Finally, the operators in $H_{
\mathrm{Logi}}$ have the form of logic gates on the DFS. However, these are undesired logic operations, since they are coupled to the bath, and thus cause decoherence.
It is worthwhile to already emphasize that the operator $YI+IY\in H_{\mathrm{
Leak}}$ is of particular importance: As shown in [@Sorensen:00 Eq.43], this is the operator that describes qubit decoherence due to motional decoherence during application of the SM gate.
In the previous subsection we showed how to eliminate the logical error $
\bar{Z}$, but we see now that this was only one error in a much larger set. To deal with the additional errors it is useful at this point to introduce a more compact notation for the pulse sequences. We denote by $[\tau ]$ a period of evolution under the free Hamiltonian, i.e., $U(\tau )\equiv \exp
(-iH_{SB}\tau )\equiv \lbrack \tau ]$, and further denote $$P\equiv \bar{U}_{12}(-\frac{\pi }{2},0)=\exp (-i\frac{\pi }{2}\overline{X}
_{12}).$$ Thus Eq. (\[eq:sym\]) can be written as: $$\exp [-i(B_{1}^{z}+B_{2}^{z})(Z_{1}+Z_{2})\tau ]=[\tau ,P,\tau ,P^{\dagger
}].$$ Notice that this is an example of a parity-kick pulse sequence.
As a first step in dealing with the additional errors, note that the symmetrization procedure $[\tau ,P,\tau ,P^{\dagger }]$ can in fact achieve more than just the elimination of the differential dephasing $Z_{1}-Z_{2}$ term. Since $\overline{X}_{12}$ also anticommutes with $\overline{Y}_{12}=
\frac{1}{2}(Y_{1}X_{2}-X_{1}Y_{2})\in H_{\mathrm{Logi}}$, if such a term appears in the system-bath interaction it too will be eliminated using the same procedure.
So far we have used a $\frac{\pi }{2}\overline{X}_{12}$ pulse. Interestingly, the Hamiltonian $\overline{X}_{12}$ can also be used to eliminate all leakage errors [@ByrdLidar:01a]. To see this, note that $
\bar{U}_{12}(\pm \pi ,0)=\exp (\pm i\pi \overline{X}_{12})=Z_{1}Z_{2}$. This operator anticommutes with *all* terms in $H_{\mathrm{Leak}}$. Hence it too can be used in a parity-kick pulse sequence, that will eliminate all the leakage errors. In particular, this pulse sequence will eliminate qubit decoherence due to motional decoherence, i.e., the error $YI+IY\in H_{
\mathrm{Leak}}$.
At this point we are left with just a single error: $\overline{X}
_{12}\otimes B$ itself, in $H_{\mathrm{Logi}}$. Clearly, we cannot use a pulse generated by $\overline{X}_{12}$ to eliminate this error. Instead, to deal with this error we need to introduce one more pulse pair that anticommutes with $\overline{X}_{12}$, e.g., $\exp (\pm i\frac{\pi }{2}
\overline{Y}_{12})=\bar{U}_{12}(\pm \frac{\pi }{2},\frac{\pi }{2})$.
Let us now see how to combine all the decoherence elimination pulses into one efficient sequence. First we introduce the abbreviations $$\begin{aligned}
\Pi &\equiv &\bar{U}_{12}(\pm \pi ,0)=\exp (\pm i\pi \overline{X}_{12})=\Pi
^{\dagger }=PP \notag \\
Q &\equiv &\bar{U}_{12}(-\frac{\pi }{2},\frac{\pi }{2})=\exp (-i\frac{\pi }{
2 }\overline{Y}_{12}) \notag \\
\Lambda &\equiv &\bar{U}_{12}(\pm \pi ,\frac{\pi }{2})=\exp (\pm i\pi
\overline{Y}_{12})=\Lambda ^{\dagger }=QQ \label{eq:PQetc}\end{aligned}$$ As argued above, the $\pi $ pulse $\Pi $ eliminates $H_{\mathrm{Leak}}$: $$\exp [-i(H_{\mathrm{Logi}}+H_{\mathrm{DFS}})2\tau ]=[\tau ,\Pi ,\tau ,\Pi ].$$ *This may be sufficient in practice*, since as argued above this pulse sequence eliminates the $YI+IY$ term, and the DFS encoding takes care of collective dephasing. Thus we expect that using cycles of two pulses we can almost entirely eliminate the two most important sources of decoherence. This expectation of course depends on the time scale requirement for decoupling being satisfied, as discussed in detail in Section \[decoupling\] above. In practice it may well be advantageous to combine the DFS encoding and $[\tau ,\Pi ,\tau
,\Pi]$ pulse sequence with the VT method of pulsing the trapping potential [@Vitali:99; @Vitali:01].
Now let us discuss adding the extra pulses needed to achieve full decoherence elimination. The $\pi /2$ pulse $P$ eliminates $\bar{Y}$ and $
\bar{Z}$ in $H_{\mathrm{Logi}}$. Combining this with the sequence for leakage elimination we have the sequence of $4$ pulses: $$\begin{aligned}
e^{-i(H_{\mathrm{DFS}}+\bar{X}\otimes B_{\bar{X}})4\tau } &=&[U(\tau )\Pi
U(\tau )\Pi ]P^{\dagger }[U(\tau )\Pi U(\tau )\Pi ]P \notag \\
&=&[\tau ,\Pi ,\tau ,P,\tau ,\Pi ,\tau ,P^{\dagger }], \label{eq:4pulses}\end{aligned}$$ (where we have used $\Pi P^{\dagger }=P$, $\Pi P=P^{\dagger }$).
If we wish to entirely eliminate decoherence then we are left just with getting rid of the logical error due to $\overline{X}$. To eliminate it we now combine with the $\bar{Y}$-direction, $\pi /2$ pulse, $Q$:
$$\begin{aligned}
e^{-iH_{\mathrm{DFS}}8\tau } &=&[U(\tau )\Pi U(\tau )PU(\tau )\Pi U(\tau
)P^{\dagger }]Q^{\dagger }
[U(\tau )\Pi U(\tau )PU(\tau )\Pi U(\tau )P^{\dagger }]Q \notag \\
&=&[\tau ,\Pi ,\tau ,P,\tau ,\Pi ,\tau ,P^{\dagger },Q^{\dagger },
\tau ,\Pi ,\tau ,P,\tau ,\Pi ,\tau ,P^{\dagger },Q]\end{aligned}$$
\[eq:10pul\]
which takes ten pulses. Unfortunately it is not possible to compress this further, since $P^{\dagger }Q=(i\overline{X})(-i\overline{Y})=i\overline{Z}$ and $P^{\dagger }Q^{\dagger }=-i\overline{Z}$, neither of which cannot be generated directly (in one step) from the available gate $\bar{U}
_{ij}(\theta ,\Delta \phi _{ij})=\cos \theta \bar{I}+i\sin \theta \overline{
X }_{\Delta \phi _{ij}}$. Finally, note that in principle the last pulse sequence is applicable also to other QC proposals, such as NMR and quantum dots.
One important caveat (mentioned in Section \[decoupling\] above) is that, because we need very strong and fast pulses, our gate operation may become imperfect. Specifically, off-resonant coupling and deviations from the Lamb-Dicke approximation may become important. The former introduces a term $
XI+IX$ into the Hamiltonian generating the $U_{ij}(\theta ,\phi _{i},\phi
_{j})$ gate [@Sorensen:00 Sec. IIIA], which can cause *unitary* leakage errors from the DFS. These can in turn be reduced using the methods in [@Tian:00; @Palao:02]. Whether the decoupling method we have proposed offers an improvement will have to be put to an experimental test.
Combining logic gates with decoupling pulses {#all}
============================================
So far we have discussed computation using the encoded recoupling method (Section \[logic\]), and encoded decoupling (Sections \[createDFS\],\[leakage-elim\]). We now put the two together in order to obtain the full ERD picture. At least two methods are available for combining quantum computing operations with the sequences of decoupling pulses we have presented above. For a general analysis of this issue see [@Viola:99a].
Fast + Strong Gates Method
--------------------------
The decoupling pulse sequences given in Sec. \[leakage-elim\] stroboscopically create collective dephasing conditions at the conclusion of each cycle. As noted above, this is equivalent to a periodic projection into the DFS. This property allows for stroboscopic quantum computation at the corresponding projection times [@Viola:99a]. Here the computation pulses need to be synchronized with the decoupling pulses, and inserted at the end of each cycle. The amount of time available for implementation of a logic gate is no more than the bath correlation time $\tau _{c}=2\pi
/\omega_{c}$. Assuming the dominant decoherence contributions not accounted for by the DFS encoding to come from differential dephasing (setting the $\tau_c$ time-scale) and $1/f$ noise, and that we already assumed that we can use pulses with interval $\Delta t\ll $ $\tau _{c}$, it is consistent to assume that we can then also perform logic gates on the same time scale.
Fast + Weak Gates Method
------------------------
There may be an advantage to using fast but weak pulses for the logic gates, while preserving the fast + strong property of the decoupling pulses. To see how to combine logic gates with decoupling in this case, let us denote by $
H_{S}=X_{\phi _{i}} X_{\phi _{j}}$ the controllable system Hamiltonian that generates the entangling gate $U_{ij}(\theta ,\phi
_{i},\phi _{j})$ \[recall Eq. (\[eq:Uij\])\]. Suppose first that we turn on this logic-gate generating Hamiltonian in a manner that is neither very strong nor very fast, so that the system-bath interaction is not negligible while $H_{S}$ is on (this obviously puts less severe demands on experimental implementation). Then the corresponding unitary operator describing the dynamics of system plus bath is: $$\tilde{U}(t)=\exp [-it(H_{S}+H_{SB}+H_{B})].$$ Now, *if we choose* $H_{S}$* so that it commutes with the decoupling pulses*, then we can show that after decoupling $$\tilde{U}(t)\mapsto \exp [-i2t(H_{S}+H_{B})], \label{eq:Ucomp}$$ provided $t$ is sufficiently small. Tracing out the bath then leaves a purely unitary, decoherence-free evolution on the system. To prove this, assume we have chosen $t^{\prime }$ and the decoupling Hamiltonian $
H_{S}^{\prime }$ so that (i) $\exp (-it^{\prime }H_{S}^{\prime })H_{SB}\exp
(it^{\prime }H_{S}^{\prime })=-H_{SB}$ (the parity kick transformation), and (ii) $[H_{S}^{\prime },H_{S}]=0$. Then
$$\begin{aligned}
\tilde{U}(t)e^{-it^{\prime }H_{S}^{\prime }}\tilde{U}(t)e^{-it^{\prime
}H_{S}^{\prime }} &=&\tilde{U}(t)e^{-it[H_{S}+e^{-it^{\prime }H_{S}^{\prime
}}H_{SB}e^{it^{\prime }H_{S}^{\prime }}+H_{B}]} \\
&=&e^{-it(H_{S}+H_{SB}+H_{B})}e^{-it(H_{S}-H_{SB}+H_{B})} \\
&=&e^{-\{2it(H_{S}+H_{B})+t^{2}([H_{SB},H_{S}]+[H_{SB},H_{B}])+O(t^{3})\}},\end{aligned}$$
where we have used the Baker-Campbell-Hausdorff formula, $\exp (\alpha
A)\exp (\alpha B)=\exp \{\alpha (A+B)+\frac{\alpha ^{2}}{2}[A,B]+O(\alpha
^{3})\}$.
Setting $H_{S}=\Omega S$ and $H_{SB}=\gamma _{SB}S^{\prime }\otimes B$ we have the condition $t\ll 1/\sqrt{\Omega \gamma _{SB}}$, in order to be able to neglect the $O(t^{2})$ term $[H_{SB},H_{S}]$. Using $\Omega \sim 1$MHz, $
\gamma _{SB}\sim 10$KHz as in Section \[decoupling\], we find $t\ll 10\mu $sec. However, the more stringent constraint comes from the $[H_{SB},H_{B}]$ term, since $H_{B}$ is not bounded for a harmonic oscillator. A more careful analysis then shows the familiar conclusion, that the bath should not be allowed to evolve for longer than its correlation time [@Viola:98; @Viola:98a; @Vitali:01]. Hence the actual requirement may still the far more stringent condition $t\ll 1/\omega _{c}\leq 1$nsec for the decoupling pulse interval; see Sec. \[decoupling\]. This cannot be satisfied with SM pulses, but in this case we can resort to the VT potential modulation method. When we do this in conjunction with SM decoupling pulses we can be sure that Eq. (\[eq:Ucomp\]) is an excellent approximation. On the other hand, the requirements for a $1/f$ bath spectral density are far less stringent and may be satisfied even with SM pulses alone [@ShiokawaLidar:tbp]. Furthermore, for the rotation angle $\theta =\Omega t$ describing the computation we have $\theta \ll \sqrt{\Omega /\gamma _{SB}}\leq 10$, which means that there is no restriction on applying large rotations.
Let us now show how to efficiently combine logic operations and decoupling pulses. For simplicity consider only the case where we can neglect the $\bar{
X}$ error, i.e., our decoupling sequence is the 4-pulse one given in Eq. (\[eq:4pulses\]). Suppose we wish to implement a logical $X$ operation, i.e., $\exp (-i\theta \overline{X}_{12})$. Recall \[Eq. (\[eq:xbar\])\] that this involves turning on the Hamiltonian $H_{S}^{X}=\Omega _{X}X_{\phi
} X_{\phi }\overset{\mathrm{DFS}}{\mapsto }\Omega _{X}\overline{X}
_{12}$ between two physical qubits. Because the decoupling pulses $P=\exp (-i
\frac{\pi }{2}\overline{X}_{12})$ and $\Pi =\exp (\pm i\pi \overline{X}
_{12}) $ are generated in terms of the same Hamiltonian, they commute with $
H_{S}^{X}$ while eliminating $H_{SB}$ (except for the terms in $H_{SB}$ that have trivial action on the DFS). Thus the conditions under which Eq. (\[eq:Ucomp\]) were shown to hold are satisfied. This allows us to insert the logic gates into the four free evolution periods involved in the pulse sequence of Eq. (\[eq:4pulses\]). Thus, the full pulse sequence that combines creation of collective dephasing conditions with execution of the logic gate is: $$e^{-it(\Omega _{X}\overline{X}_{12}+H_{\mathrm{DFS}})}=\tilde{U}(t/4)\Pi
\tilde{U}(t/4)P\tilde{U}(t/4)\Pi \tilde{U}(t/4)P^{\dagger }, \label{eq:X}$$ with $\tilde{U}(t)=\exp [-it(H_{S}^{X}+H_{SB}+H_{B})]$, and which, using the DFS encoding, is equivalent to the desired $\exp (-i\theta \overline{X}
_{12}) $. This involves 8 control pulses, 4 of which are of the fast+strong type (those involving $P$ and $\Pi $), and 4 of which must be fast, but need not be so strong that we can neglect $H_{SB}$.
If we wish to implement logical $Y$ operation, i.e., $\exp (-i\theta
\overline{Y}_{12})$, then we cannot now use $P$ and $\Pi $, since they anticommute with $\overline{Y}_{12}$ and will eliminate it. Instead we should use decoupling pulses generated in terms of $\overline{Y}_{12}$, which will also have the desired effect of eliminating $H_{\mathrm{Leak}}$, as well as $\bar{X}$ and $\bar{Z}$ logical errors, while commuting with the $
\overline{Y}$ logic operations (and for this reason can of course not eliminate $\bar{Y}$ errors). These are just the $Q$ and $\Lambda $ pulses defined in Eq. (\[eq:PQetc\]). In ion trap terms this implies \[recall Eq. (\[eq:ybar\])\] turning on the Hamiltonian $H_{S}^{Y}=\Omega _{Y}X_{\phi
} X_{\phi +\pi /2}\overset{\mathrm{DFS}}{\mapsto }\Omega _{Y}
\overline{Y}_{12}$ between two physical qubits. Thus: $$e^{-it(\Omega _{Y}\overline{Y}_{12}+H_{\mathrm{DFS}})}=\tilde{U}(t/4)\Lambda
\tilde{U}(t/4)Q\tilde{U}(t/4)\Lambda \tilde{U}(t/4)Q^{\dagger },
\label{eq:Y}$$ with $\tilde{U}(t)=\exp [-it(H_{S}^{Y}+H_{SB}+H_{B})]$, and which, using the DFS encoding, is equivalent to the desired $\exp (-i\theta \overline{Y}
_{12}) $.
Finally, to generate single DFS-qubit rotations about an arbitrary axis we can combine Eqs. (\[eq:X\]),(\[eq:Y\]) according to the Euler angles construction. Given that Eqs. (\[eq:X\]),(\[eq:Y\]) each take 8 pulses, the Euler angle method will generate an arbitrary DFS-qubit rotation in at most 24 pulses.
Concerning gates that entangle two DFS-qubits, the situation is more involved, since now the next-nearest neighbor pulses in Eq. (\[eq:create4\]), that create the collective dephasing conditions on four ions, do not all commute with the $U_4$ gate of Eq. (\[eq:U4\]). Therefore here we must resort to the strong + fast method of the previous subsection, i.e., we need to synchronize the $U_4$ pulses with the end of the decoupling pulse sequence.
Taken together, the methods described in this section provide an explicit way to implement universal QC using trapped ions in a manner that offers protection against all sources of qubit decoherence, using a fast + strong (or fast + weak) version of the SM scheme, possibly in combination with the VT potential modulation method.
Discussion and Conclusions {#conclusions}
==========================
We have proposed a method of encoded recoupling and decoupling (ERD) for performing decoherence-protected quantum computation in ion traps. Our method combines the S$\o$rensen-M$\o$lmer (SM) scheme for quantum logic gates with an encoding into ion-pair decoherence-free subspaces (each pair yielding one encoded qubit), and sequences of recoupling and decoupling pulses. The qubit encoding protects against collective dephasing processes, while the decoupling pulses symmetrize all other sources of decoherence into a collective dephasing interaction. The recoupling pulses are used to implement encoded quantum logic gates, either during or in between the decoupling pulses. All pulses are generated directly using the SM scheme. We have provided numerical estimates of the feasibility of our scheme, which seem quite favorable. In order to achieve full protection against all decoherence it may be necessary to supplement ERD with the potential modulation method due to Vitali & Tombesi, in order to reduce vibrational mode decoherence. However, it may be worthwhile to test ERD without potential modulation first, as a significant reduction in decoherence can already be expected according to the results presented here. This is so because the vibrational bath has been found experimentally to have a $1/f^\alpha$ spectral density [@Wineland:comment], and there exists evidence that in such a case decoupling may be possible under moderate timing contraints [@ShiokawaLidar:tbp].
As mentioned in Section \[decoupling\], the dynamical decoupling method requires an exponential number of pulses if the most general form of decoherence is to be suppressed, that can couple arbitrary numbers of qubits to the environment (total decoherence [@Lidar:PRL98]). This exponential scaling is avoided here by focusing on decoherence elimination inside blocks of *finite* size (e.g., at most four ions) where arbitrary decoherence is allowed. However, we have implicitly assumed that there are no decoherence processes coupling different blocks. This is a reasonable assumption for trapped ions, where the different blocks can be kept sufficiently far apart until they need to be brought together in order to execute inter-block logic gates. When this happens, ERD can still be efficiently applied on the temporarily larger block.
It may be questioned whether there is any advantage in using ERD compared to methods of active quantum error correcting codes (QECC). Both ERD and QECC are capable of dealing with arbitrary decoherence processes, and are fully compatible with universal quantum computation. There are two main advantages to ERD: First, we need only two ions per qubit, compared to a redundancy of five ions per qubit to handle all single-qubit errors in QECC [@Laflamme:96]. So far experiments involving trapped ions have used up to four ions [@Sackett:00], so that this encoding economy is a distinct advantage for near-term experiments. Second, our method is directly compatible with the SM scheme for logic gates in ion traps. On the other hand it is not clear how to directly use SM gates for QECC. These are general features of ERD: economy of encoding redundancy and use of only the most easily controllable interactions. On the other, the disadvantage of ERD compared to QECC is that there does not exist, at this point, a result analogous to the threshold theorem of fault tolerant quantum error correction. This means that we cannot yet guarantee full scalability of ERD as a stand-alone method. However, in principle it is always possible to concatenate ERD with QECC, as done, e.g., for DFS with QECC in [@Lidar:PRL99; @Lidar:00b; @KhodjastehLidar:02; @Alber:02], and then the standard fault tolerance results apply.
Finally, we note that ERD is a general method, that is not limited to trapped ions. We hope that the methods proposed here will inspire experimentalists to implement encoded recoupling and decoupling in the lab, thus demonstrating the possibility of fully decoherence-protected quantum computation, in particular using trapped ions.
This material is based on research sponsored by the Defense Advanced Research Projects Agency under the QuIST program and managed by the Air Force Research Laboratory (AFOSR), under agreement F49620-01-1-0468. The U.S. Government is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright notation thereon. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of the Air Force Research Laboratory or the U.S. Government. D.A.L. further gratefully acknowledges financial support from PRO, NSERC, and the Connaught Fund. We thank Prof. C. Monroe and Dr. S. Schneider for very useful discussions.
[78]{} natexlab\#1[\#1]{}bibnamefont \#1[\#1]{}bibfnamefont \#1[\#1]{}citenamefont \#1[\#1]{}url \#1[`#1`]{}urlprefix\[2\][\#2]{} \[2\]\[\][[\#2](#2)]{}
, ****, ().
, ****, ().
, ****, ().
, ****, ().
,
, ****, ().
, ****, ().
, ****, ().
, ****, ().
, ****, ().
, Ph.D. thesis, (), .
, ****, ().
, ****, ().
, ****, ().
, ****, ().
, ****, ().
, ****, ().
, ****, ().
, ****, ().
, , ****, ().
, ****, ().
,
, in ** (, , ), p. , .
, ****, ().
, ****, ().
, ****, ().
, ****, ().
, ****, ().
, .
, ****, ().
, ****, ().
, ****, ().
, ****, ().
, ****, ().
, ****, ().
, ,
, ****, ().
, ****, ().
,
, ****, ().
, .
, ****, ().
, ****, ().
, ****, ().
, ****, ().
, ****, ().
, ****, ().
, ****, ().
, ****, ().
, ****, ().
,
, ****, (), .
, ****, ().
, ****, ().
,
,
, ****, ().
, ****, ().
, ****, ().
, ****, ().
, ****, ().
, .
, in **, edited by (, , ), p. , .
, Ph.D. thesis, (), .
, ****, ().
, ****, ().
, **, no. in (, , ).
, ****, ().
, ****, ().
, ****, ().
, ****, ().
, ****, ().
, ****, ().
|
---
abstract: 'The spin state transition in LaCoO$_3$ has eluded description for decades despite concerted theoretical and experimental effort. In this study, we approach this problem using fully charge self-consistent Density Functional Theory + Embedded Dynamical Mean Field Theory (DFT+DMFT). We show from first principles that LaCoO$_3$ cannot be described by a single, pure spin state at any temperature. Instead, we observe a gradual change in the population of higher spin multiplets with increasing temperature, with the high spin multiplets being excited at the onset of the spin state transition followed by the intermediate spin multiplets being excited at the metal insulator transition temperature. We explicitly elucidate the critical role of lattice expansion and oxygen octahedral rotations in the spin state transition. We also reproduce, from first principles, that the spin state transition and the metal-insulator transition in LaCoO$_3$ occur at different temperature scales. In addition, our results shed light on the importance of electronic entropy in driving the spin state transition, which has so far been ignored in all first principles studies of this material.'
author:
- Bismayan Chakrabarti
- Turan Birol
- Kristjan Haule
title: 'Role of Entropy and Structural Parameters in the Spin State Transition of LaCoO$_3$'
---
Introduction
============
The spin state transition in LaCoO$_3$ has been the subject of intensive investigation for decades.[@Heikes; @Naiman; @Goodenough] This compound is established to be a narrow bandgap insulator at low temperature with Pauli-like magnetic susceptibility. However between 90-150 K, it transitions to a local moment phase with a Curie-Weiss like susceptibility which reaches its peak around 150K. It also undergoes a gradual closing of the insulating gap and is known to be metallic above 600K [@BhideMoss; @English; @Saitoh].
There is considerable debate regarding the mechanism of this transition, mainly due to the uncertainty regarding the multiplet of the $Co^{3+}$ ion which characterizes the excited state of the compound. The cobalt ion in LaCoO$_3$ is commonly assumed to be in the $d^6$ state with a formal valence of $3+$. Due to the fact that the scale of the crystal field splitting is comparable to the Hunds coupling energy scale, one would expect that as temperature is increased, there would be an entropy-driven transition from the low spin (LS) $S=0$ state with a fully filled $t_{2g}$ shell($t^6e^0$) to an $S=2$ high spin (HS) state ($t^4e^2$)[@Goodenough]. Indeed there is considerable experimental evidence to support such a scenerio. Electron spin resonance[@Zopka], neutron scattering [@Podlesnyak], X-ray absorbtion spectroscopy and magnetic circular dichroism experiments[@Haverfort] all point towards a transition to an HS state. In addition, no inequivalent Co-O bond is found in EXAFS experiements, which also supports the formation of an HS state due to the HS state not being strongly Jahn-Teller active [@Sundaram]. However, it has been noted that in order to explain the XAS experimental data, one would have to assume that the crystal field grows with temperature, which is counter-intuitive.[@Eder; @Haverfort] This led to some authors suggesting that there is an LS-HS alternating structure caused by breathing distortions[@Bari] [@Goodenough] and interatomic repulsion between the HS atoms.[@Asaka; @Eder]
A competing explanation, whereby the excited state is the $S=1$ intermediate spin (IS) state ($t^5e^1$), has also become popular[@Heikes; @Radaelli], mainly because of LDA+U results which show that the IS state is lower in energy compared to the HS state.[@Korotin; @Pandey; @Anisimov] The stability of the IS state has been justified by the large hybridization of the Co 3d electrons with neighboring O 2p electrons. This causes charge transfer between the ions resulting in the Co ion having a $d^7$ structure according to the Zaanen-Sawatzky-Allen scheme,[@Zaanen] which in turn would cause stabilization of the IS state. The intermediate spin state hypothesis also seems to explain experimental findings such as Raman Spectroscopy, X-Ray photoemission, XAS, EELS, as well as susceptibility and thermal expansion measurements. [@Saitoh; @Abbate; @Masuda; @Klie; @Zobel; @Gne; @Maris; @Vogt].
To summarize, there has been significant debate regarding the true nature of the spin state transition in LaCoO$_3$. Interest in this compound has also been enhanced in light of recent discoveries of ferromagnetism induced by Sr (hole) doping [@Kriener; @Masayuki; @Kunes_doped; @nemeth], and by experiments reporting strain induced magnetism in epitaxially grown thin films.[@Fuchs1; @Fuchs2; @Rondinelli; @Freeland; @Fuchs3; @Herklotz; @Hsu] Additionally, there have been reports of the emergence of a striped phase in thin films (with alternating LS and HS/IS regions)[@Striped], as well as the presence of low temperature ferromagnetism in experiments performed on LaCoO$_3$ nanoparticles[@Belanger1]. Furthermore, there have also been experiments conducted on single crystals of LaCoO$_3$ in the presence of a strong magnetic field[@highfield1; @highfield2; @highfield3; @highfield4] which have reported the presence of multiple metamagnetic transitions, which have been attempted to be explained by spin-state superlattices as well as excitonic condensation.[@kunes_highfield] Hence, we can see that there is great interest in understanding the complex physical nature of this material.
There have been multiple previous DMFT studies that have approached this material and the transitions therein. For example, in Ref. , Augustinski et al. studied the effect of carrier doping on the spin state of LaCoO$_3$. Zhang et al, in Ref. , considered hydrostatic pressure as well as substitution of La with other lanthanides, and provided an explicit picture of the effect of the crystal field splitting on the Co ion, while Krapek et al. (Ref. ) showed that the charge fluctuations, in addition to the spin state fluctuations, are important in understanding the physics of this compound. In this paper, we use Density Functional Theory + embedded Dynamical Mean Field Theory (DFT+DMFT)[@dmft_website] to analyze the spin state transition in bulk LaCoO$_3$. Unlike earlier studies, our implementation is fully charge self-consistent and extremizes the DFT+DMFT functional in real space (for details the reader is referred to the appendix and the references contained therein), thereby avoiding the downfolding approximation and uses the numerically exact CTQMC impurity solver[@Dmft3; @haule]. In addition, to our knowledge none of the earlier studies provide a comprehensive analysis of all of the factors governing the transition such as octahedral rotations and electronic entropy. Our main findings can be summarized as- i) We show that LaCoO$_3$ has large charge fluctuations and it is not possible to explain the spin state with a single multiplet at any temperature. However at the onset of the spin state transition, the HS multiplets are excited, with the IS multiplets being excited later around the onset of the metal-insulator transition. ii) We illustrate that the crystal field splitting is very sensitive to the crystal structure, and taking into account not only the thermal expansion but also the oxygen octahedral rotations are very important for understanding the behavior of the material. We repeat our calculations for four different crystal structures that correspond to two different temperatures and two different oxygen octahedral rotation angles. This way, for the first time, we isolate the effect of the oxygen octahedral rotations and thermal expansion on the electronic and spin structure of LaCoO$_3$. This should be an important factor in determining the effect of Ln substitution, as discussed in Ref. and Ref. , but this connection has not been studied explicitly in LaCoO$_3$ yet. iii) We demonstrate conclusively that it is possible to stabilize (without orbital order) an insulating phase at intermediate temperatures where local moments are present, thereby showing that the metal-insulator transition is distinct from the spin state transition in this compound.We provide a detailed picture of the evolution of the spectral function and the spin state probabilities with temperature and conclusively show from first principles that the spin state and metal insulator transitions occur at very different temperatures. iv)We also show that electronic entropy difference between the high and low temperature states is necessary for the stabilization of the excited spin states, which is a fact that has, though expected to be important, been overlooked in various first principle studies so far primarily because accurate first principles methods for calculation of the electronic entropy have not been available.
Crystal Structure
=================
LaCoO$_3$ is a perovskite, which has the rare earth element La on the A-site and Co on the B-site at the center of an oxygen octahedron (see Fig \[Perov\_struct\]). Like most perovskites, LaCoO$_3$ has oxygen octahedral rotations (Fig. \[structure\]) which result in the oxygen octahedra rotating out-of-phase around the \[111\] axes of the undistorted cubic highsymmetry structure. This rotation pattern is denoted by $a^-a^-a^-$ in the Glazer notation, and corresponds to the space group R$\bar 3$c (\#167).
![Unit cell for a perovskite with no octahedral rotations ($a^0a^0a^0$ structure). In LaCoO$_3$, the green atoms would be La, the red atoms O and the blue atom Co. The figure also shows the octahedra formed by the oxygen atoms around the Co atom.[]{data-label="Perov_struct"}](figure1.png){height="2in"}
As noted by Thornton et. al.[@Thornton], LaCoO$_3$ shows large thermal expansion as well as variation in the octahedral rotation angle with increase in temperature. In our study, we use four different crystal structures to isolate and study the effect of different lattice parameters on the spin state transition. We use two different experimental structures observed at 1143K and 4K, which we denote by HTa$^-$ and LTa$^-$. Comparing the electronic structure for these two crystal structures provides a means to study the temperature evolution of the electronic structure. In addition to these two, we also built two crystal structures with the same strain state (unit cell vectors) as them, but with no octahedral rotations. These structures, denoted by HTa$^0$ and LTa$^0$, enable us to isolate the effect of oxygen octahedral rotations on the spin state of LaCoO$_3$.
![A depiction of the pattern of octahedral rotations that is present in LaCoO$_3$. Each of the oxygen octadra rotates in opposite direction to all nearest neighbour octahedra by the same amount relative to all three cartesian axes ($a^-a^-a^-$ structure).[]{data-label="structure"}](figure2.png){height="2in"}
Density of States
=================
In Fig \[dos\_fig\] we show the density of states for all 4 structures, calculated at both low temperature and high temperature (116K and 1160K) using DFT+DMFT. In our DMFT calculations, we use $U=6.0$ eV and $J=0.7$ eV. Further details of our calculations and methodology are presented in the appendix. Unlike DFT, which always predicts a metallic state, our calculations correctly reproduce an insulating ground state at low temperature for all the structures. The $t_{2g}$ orbitals are below the fermi level whereas the $e_g$ orbitals are above the fermi level. For the two experimental structures (LTa$^-$ and HTa$^-$), this low temperature charge gap closes continuously with increasing temperature and there is a large overlap in energy between the $t_{2g}$ and $e_g$ orbitals at high temperatures. This overlap, however, is much smaller if the structures without rotations are used. (See Fig. \[fig4\]b).[^1] The HTa$^0$ structure shows some overlap at high temperatures, while the LTa$^0$ structure almost remains an insulator for the entire range of temperatures studied, with a small overlap developing above 900K. This shows clearly that octahedral rotations play a large role in decreasing the strength of the crystal field splitting. This can be explained by the fact that the rotation of the oxygen octahedra causes misalignment of the crystal field of the O atoms with that of the La atoms, which normally reinforce each other if the perovskite has no octahedral rotations. This leads to an overall reduction of the effective crystal field which reduces the charge gap between the $t_{2g}$ and $e_g$ orbitals. [^2]. This effect seems to overcome the expected decrease in the bandwith of the $e_g$ orbitals caused by the octahedral rotations. Finally, note that there is a considerable overlap in energy of the O 2p orbitals with the Co 3d orbitals, which is very important in producing charge fluctuations on the Co ion, making it highly mixed-valent.
{width="\textwidth" height="3.3in"}
![(a) Evolution of $|S_z|$ with temperature for all four structures. (b) Evolution of Density of states at fermi level with temperature for all four structures.[]{data-label="fig4"}](figure4.png){width="\columnwidth"}
The spin state transition
=========================
In order to focus on the spin state of the Co ion, we calculate the expectation value of the magnitude of z-component of the spin $\langle|S_z|\rangle$. Note that all our calculations are in the paramagnetic state and hence the value of $\langle S_z \rangle=0$. The results are presented in Fig. \[fig4\]a as a function of temperature.[^3] The largest value of $|S_z|$ at 1160K is seen for the HTa$^-$ structure, followed by the LTa$^-$ structure. This is in line with the stronger crystal field in the LTa$^-$ structure due to the smaller lattice constant. We also observe that the spin state transition starts at a higher temperature for the LTa$^-$ structure ($\sim$580K) compared to the the HTa$^-$ ($\sim$380K). This is also consistent with the the low temperature structure having higher stability for the LS state. The structures without rotations consistently show lower buildup of higher spin states than the ones with rotations. The HTa$^0$ structure displays a spin state transition, but with an eventual high temperature value of $|S_z|$ that is lower than both the structures with octahedral rotations (LTa$^-$ and HTa$^-$). On the other hand, the LTa$^0$ structure shows almost no transition. This shows that the role that the octahedral rotations play in the reduction of the crystal field is essential for the spin state transition.
Figures \[fig4\]a and \[fig4\]b also show that the spin state transition and the charge gap closing occur at different temperatures, which is a trend that has been observed in experiment but has not been captured in earlier DMFT simulations. For example, Fig \[fig4\]b shows that both the HTa$^-$ and the LTa$^-$ structures show a complete closure of the charge gap at $\sim$ 600K whereas Fig 4a shows that the spin state transition in the two structures occurs at very different temperatures.
Nature of the excited spin state
================================
Because of the large hybridization between Co and O ions, the Co-d orbitals have large charge fluctuations and all the four structures have an effective d-shell occupation of $n_d \sim 6.6$. As a result, any analysis of the spin states in terms of the LS, IS and HS states of the $d^6$ configuration of the Co ion is necessarily inadequate. In fact, our calculations show that the $d^7$ configuration has a higher occupation probability than $d^6$, and there are also significant probabilities for $d^5$ and $d^8$. (Importance of large charge fluctuations in LaCoO$_3$ has been discussed before, for example by Abbate et al.[@Abbate] who emphasized the role of covalency between the Co d orbitals and the O anions in order to explain their XAS data. Our results agree with their observation of highly covalent states and gradual transitions, and show that both the $3d^6$ and $3d^7$ occupancies are fundamental to understanding the behavior of the material.)
Fig. \[spin\_prob\] shows the evolution of the occupation probabilities for the different values of $|S_z|$ with temperature. Even at high temperatures, $|S_z|=0$ and $|S_z|=0.5$ (the LS states for the even and odd occupancy sectors of the d orbital) remain the states with the highest probability. However, with the increase of temperature, the weight of the higher spin states increases. At the onset of the transition, the initial change in the value of the spin state is predominantly caused by the excitation of the $|S_z|=2$ and the $|S_z|=1.5$ multiplets. For example, from Fig \[fig4\]a, we can see that for the HTa$^-$ structure the $|S_z|$ value first starts rising at around 300K. Looking at the relevant portion of Fig 5, we see that the only spin states which show an increase in probability are the $|S_z|=2$ and the $|S_z|=1.5$ multiplets. The $|S_z|=1$ multiplet sees an increase in probability at higher temperatures (above 600K) which is the temperature of the charge gap closing, as evidenced by Fig \[fig4\]b. A similar trend is seen for the other structures as well. Therefore the initial signature of the transition is best seen in the behavior of the $|S_z|=2.0$ and $|S_z|=1.5$ multiplets, which can be said to be the HS multiplets for the $d^6$ and $d^7$ occupancies respectively, whereas the $|S_z|=1$ state sees in increase in occupation at the temperature scale of the metal-insulator transition in each of the structures and not at the spin state transition temperature. (Note that these effects are not seen in the LTa$^0$ structure where no significant transition occurs.)
![Evolution of occupation probabilities for all the spin states for the four structures with temperature. []{data-label="spin_prob"}](figure5.png){width="\columnwidth"}
{width="\textwidth" height="3.3in"}
In Fig. \[hist\], we show the occupancy histograms below and above the transition (at 116K and 1160K) (CTQMC gives us access to the state space probability for each of the 1024 states of the d orbital. However, in order to aid visualization, we only show states which have an occupation probability above 0.001 in any of the structures at any temperature). This figure displays clearly how the transition is marked by the excitation of states in the higher spin multiplets. We see that the low temperature state for all of the structures is marked by the presence of a few states with large probability (mainly corresponding to the $|S_z|=0$ and $|S_z|=0.5$ states). As the spin state transition sets in, a large number of higher spin states get excited and the LS spin states lose weight. Note that the high spin states are highly degenerate so there is no one large peak for the high spin states, but instead there is a multitude of lower peaks. This supports the idea that the transition is primarily an entropy driven transition. We can also get a good idea of the relative strengths of the transition for the different structures: The largest change occurs in the HTa$^-$ structure, and the smallest one happens in the LTa$^0$.
In passing, we would like to emphasize that even though the occupancy histograms show that LaCoO$_3$ is in a mixed spin state, this is a spatially homogenous mixed state (such as those discussed in references \[\]) and not an inhomogeneous spin state, such as those discussed in various theoretical studies [@Zhang; @Knizek; @Zhuang; @Kunes2011] and also discussed in relation to the Magnetic Circular Dichroism results in Ref. \[\]. All of our calculations are performed using structures with only one crystallographically inequivalent Co ion. Thus, a spin state superstructure is beyond what our current calculations can reproduce. However, a very important conclusion of our DFT+eDMFT study is that a *spatially inhomogenous mixed state is not necessary to reproduce the spin state and metal-insulator transitions in LaCoO$_3$*.
In this context, we would also like to point out that the possibility of a spatially inhomogenous spin state in which low- and intermediate- or high-spin Co$^{3+}$ ions form a spin state crystal is considered as a possible scenerio to explain the metamagnetic phase transitions observed under high magnetic fields.[@highfield1; @highfield2; @highfield3] A recent two-orbital DMFT study, on the other hand, favors an excitonic condensation scenerio.[@kunes_highfield] Our study does not address the effect of a magnetic field, nor does it cover the low-temperature range where the field induced transitions are observed. Nevertheless, our results underline the importance of both the intermediate spin and high spin states coexisting on the same ion, as well as the importance of charge fluctuations on the Co ion, which supports some of the findings of Ref. along with some of the explanations considered for the original high field experiments [@highfield4].
Contribution of Electronic Entropy
==================================
According to the entropy driven transition scenerio, which is supported by calorimetric measurements[@Stolen], LaCoO$_3$ favors higher spin multiplets at elevated temperatures because of the associated gain in electronic entropy as a result of the high degeneracy of these high spin states - a point missed by first principles calculations at the level of DFT. Access to higher spin states is also made easier by a larger lattice constant due to the reduced crystal field splitting, so the gain in electronic entropy could also be a driving factor for the large thermal expansion seen in this material.
We calculated the contribution of the electronic entropy to the free energy using our state of the art DFT+DMFT implementation[@Birol]. In particular, we evaluated the Free Energy and the Electronic Entropy for both the 4K and 1143K structures (LTa$^-$ and HTa$^-$) at 1160K to predict if the structural changes make a considerable difference.[^4] The HTa$^-$ structure is indeed much higher in electronic entropy compared to the LTa$^-$ structure at 1160K and we observe the difference in $T\cdot S$ between these two structures to be $\sim$ 110 meV per formula unit. This unusually large difference emphasises the importance of electronic entropy to the transition.We also calculate the energy difference between the HTa$^-$ and LTa$^-$ structures to be $\sim$ 70 meV at 1160K with the LTa$^-$ being lower in energy. Thus we see that when the entropy is taken into account and the Free Energy (F=E-TS) is calculated, the high temperature structure HTa$^-$ becomes more stable purely due to the contribution of electronic entropy. This result confirms the structural phase transition that is observed as a function of temperature. So, we can conclude that electronic entropy, which has been ignored in many first-principles studies of this material, is a leading factor in creating an anomalously large thermal expansion and driving the material to a high spin state.
Summary
=======
We studied the spin state transition of LaCoO$_3$ using state of the art fully charge self consistent DFT+ Embedded DMFT. By using different experimental and hypothetical crystal structures, we disentangled the effect of different components of the crystal structure and showed that both the thermal expansion and the presence of oxygen octahedral rotations have tremendous effect on the spin state transition of LaCoO$_3$. Our single site DMFT approach reproduced not only the spin state transition but also the intermediate phase which has nonzero magnetic moment but is insulating. This shows that the spin state and the metal-insulator transitions occur at different temperature scales and that the insulating phase with local magnetic moments can be reproduced without necessarily involving cell doubling via mechanisms such as breathing distortions of spatially inhomogenous mixed spin states. Our results emphasize the importance of charge fluctuations on the Co ion due to hybridization with the O anions, and thus point to the inadequacy of a simple spin state picture with only one formal valence. We find that while the spin state transition is concurrent with a sudden change in occupation in the high spin multiplets and the metal-insulator transition with a jump in the intermediate spin probability, both low and intermediate spin states also have significant occupation in the whole temperature range. Finally, our work is the first calculation of the electronic entropy of LaCoO$_3$ and it points to the fact that the change in electronic entropy with temperature is significant and is large enough to drive the spin state transition in this material.
Acknowledgements
================
TB was supported by the Rutgers Center for Materials Theory. KH was supported by the NSF-DMR 1405303.
Details of the DFT+DMFT Calculations
====================================
We performed fully charge self-consistent DFT+ Embedded DMFT calculations[@Dmft1][@Dmft3][@Dmft2] on LaCoO$_3$. Our implementation is based on the Wien2K all-electron DFT package.[@wien2k] For the DFT functional we use the GGA-PBE functional. We employed a 10 atom unit cell for the 2 structures with rotations and 5 atom unit cells for the two structures without rotations and used 512 k points in the first Brillouin zone. Our DMFT implementation uses state-of-the-art CTQMC[@haule] impurity solver based on the hybridization expansion (CTQMC-HYB) and all simulations are iterated to self-consistency. It is to be noted that our implementation makes use of projectors to embed/project out the impurity self-energy onto the lattice degrees of freedom. This prevents errors associated with downfolding using Wannier orbitals and allows us to achieve highly accurate charge self-consistency.[@Dmft3] We also account for octahedral rotations, wherever present, by applying local rotations so as to align our correlated orbitals with the local crystal field set up by the neighboring atoms. In our simulations, we use a Hubbard U of 6.0 eV and Hund’s coupling (J) of 0.7eV. We have also investigated the effects of varying the value of J, and found that this does not change the temperature at which the transitions occur, but merely changes the value of the observed $|S_z|$ by a small amount, with higher J values resulting in slightly larger values. We ignored spin-orbit interaction, as it is expected to be small in this 3d transition metal oxide.
In our calculations, we included all d electrons of the Co ion in the impurity, performing the CTQMC calculation with all 5 orbitals. We used the Slater parametrization of the Coulomb U.[@dmft_coulombU] The typical self-consistent calculation requires 30 self consistency cycles each with around 10 DFT iterations and one CTQMC iteration per cycle. Our LDA+DMFT calculations gave us the Green’s function ($G(i\omega)$) on the imaginary (Matsubara) axis which we analytically continued using the maximum entropy method to get the density of states on the real axis. The electronic entropy we calculate is the entropy of the impurity part, which includes all the d-shell electrons and is expected to be the dominant contribution.
Our implementation of DMFT is based on mapping the original lattice problem to an auxilliary impurity problem. We solve numerically exactly the impurity problem defined by the action:
$$\begin{gathered}
\label{equ:action}
\mathcal{S}=\int_0^\beta d\tau\psi^\dagger_{L\sigma}(\tau)\frac{\partial}{\partial\tau} \psi_{L\sigma}(\tau) +\int_0^\beta d\tau \int_0^\beta d\tau'\psi^{\dagger}_{L_1 \sigma}(\tau')\Delta_{L_1\sigma,L_2\sigma'} (\tau - \tau') \psi_{L_2 \sigma'}(\tau)\\
%
+\frac{1}{2}\int_0^\beta d\tau \sum_{L_1,L_2,L_3,L_4,\sigma,\sigma'} U_{L_1,L_2,L_3,L_4}\psi^\dagger_{L_1 \sigma}(\tau) \psi^\dagger_{L_2 \sigma'}(\tau) \psi_{L_3 \sigma'}(\tau) \psi_{L_4 \sigma}(\tau) \end{gathered}$$
where $\tau$ is imaginary time, $L$ and $\sigma$ denote angular momentum and spin labels,$\psi$ and $\psi^\dagger$ are the annihilation and creation operators for impurity electrons and $\Delta$ is the hybridization function between our impurity and the bath, which encodes most of the lattice information. The on-site Coulomb repulsion between the Co d-electrons is given by $$\hat{U}=\frac{1}{2}\sum_{L_1,L_2,L_3,L_4,\sigma,\sigma'} U_{L_1,L_2,L_3,L_4}c^\dagger_{L_1 \sigma} c^\dagger_{L_2 \sigma'} c_{L_3 \sigma'} c_{L_4 \sigma}$$ where $$U_{L_1,L_2,L_3,L_4}=\sum_k \frac{4 \pi}{2k+1}\langle Y_{L_1}|Y_{km}|Y_{L_4}\rangle\langle Y_{L_2}|Y^*_{km}|Y_{L_3}\rangle F^k_{l_1,l_2,l_3,l_4}$$
where $Y$ denotes spherical harmonics and $F^k$ denote Slater integrals. The CTQMC impurity solver gives us the impurity self-energy $\Sigma$. Since correlations are very local in real space, we embed this self-energy by expanding it in terms of quasi-localized atomic orbitals,$\ket{\phi^{m}_{n}}$;
$$\Sigma_{i\omega}(r,r')=\sum_{mm',nn'}\braket{r|\phi^{m}_{n}}\braket{\phi^{m}_{n}|\Sigma|\phi^{m'}_{n'}}\braket{\phi^{m'}_{n'}|r'}$$
where $m,m'$ denote the sites of Co atoms in the most general cluster-DMFT implementation and $n,n'$ denote the atomic degrees of freedom for each site. Since, we perform single site DMFT, $\sum_{mm'}$ becomes $\delta_{m,m'}$. We then solve the Dyson equation in real space (or an equivalent complete basis such as the Kohn-Sham basis) according to the equation:
$$G_{i\omega}(r,r')=\left( \left(i \omega +\mu +\triangledown^{2} +V_{KS}(r) \right)\delta(r-r') - \Sigma_{i \omega}(r,r') \right)$$
This procedure is what we define as embedded DMFT, because the self energy is embedded into a large Hilbert Space instead of constructing a Hubbard-like model by downfolding to a few bands using Wannier Orbitals. This method has the advantage that the correlations are much more localized in real space compared to the Wannier representation, which makes DMFT a much better approximation. In addition, this formulation of DFT+DMFT can be shown to be derivable from the Luttinger Ward Functional which makes the formulation stationary and conserving [@Birol].
[61]{} natexlab\#1[\#1]{}bibnamefont \#1[\#1]{}bibfnamefont \#1[\#1]{}citenamefont \#1[\#1]{}url \#1[`#1`]{}urlprefix\[2\][\#2]{} \[2\]\[\][[\#2](#2)]{}
, , , ****, (), ISSN , <http://www.sciencedirect.com/science/article/pii/003189146490182X>.
, , , , , ****, (), <http://scitation.aip.org/content/aip/journal/jap/36/3/10.1063/1.1714092>.
, ****, (), <http://link.aps.org/doi/10.1103/PhysRev.155.932>.
, , , , ****, (), <http://link.aps.org/doi/10.1103/PhysRevB.6.1021>.
, , , ****, (), <http://link.aps.org/doi/10.1103/PhysRevB.65.220407>.
, , , , , , ****, (), <http://link.aps.org/doi/10.1103/PhysRevB.55.4257>.
, ****, (), <http://link.aps.org/doi/10.1103/PhysRevB.67.172401>.
, , , , , , , , , ****, (), <http://link.aps.org/doi/10.1103/PhysRevLett.97.247208>.
, , , , , , , , , , , ****, (), <http://link.aps.org/doi/10.1103/PhysRevLett.97.176405>.
, , , , , , , , , ****, (), <http://link.aps.org/doi/10.1103/PhysRevLett.102.026401>.
, ****, (), <http://link.aps.org/doi/10.1103/PhysRevB.81.035101>.
, ****, (), <http://link.aps.org/doi/10.1103/PhysRevB.5.4466>.
, , , ****, (), <http://link.aps.org/doi/10.1103/PhysRevB.71.024418>.
, ****, (), <http://link.aps.org/doi/10.1103/PhysRevB.66.094408>.
, , , , , , ****, (), <http://link.aps.org/doi/10.1103/PhysRevB.54.5309>.
, , , , , , , , , ****, (), <http://link.aps.org/doi/10.1103/PhysRevB.77.045123>.
, , , , ****, (), <http://link.aps.org/doi/10.1103/PhysRevB.68.235113>.
, , , ****, (), <http://link.aps.org/doi/10.1103/PhysRevLett.55.418>.
, , , , , , , , , ****, (), <http://link.aps.org/doi/10.1103/PhysRevB.47.16124>.
, , , , , , , ****, (), <http://link.aps.org/doi/10.1103/PhysRevLett.71.4214>.
, , , , , , ****, (), <http://link.aps.org/doi/10.1103/PhysRevLett.99.047203>.
, , , , , , , , ****, (), <http://link.aps.org/doi/10.1103/PhysRevB.66.020402>.
, , , , , , , , , ****, (), <http://scitation.aip.org/content/aip/journal/ltp/32/2/10.1063/1.2171521>.
, , , , , , ****, (), <http://link.aps.org/doi/10.1103/PhysRevB.67.224423>.
, , , , ****, (), <http://link.aps.org/doi/10.1103/PhysRevB.67.140401>.
, , , , , , , , , , ****, (), <http://link.aps.org/doi/10.1103/PhysRevB.69.094417>.
, , , , ****, (), , <http://dx.doi.org/10.1143/JPSJ.63.1486>.
, , , ****, (), <http://link.aps.org/doi/10.1103/PhysRevLett.110.267204>.
, , , , , , , , , ****, (), <http://link.aps.org/doi/10.1103/PhysRevB.88.035125>.
, , , , , , , , , , ****, (), <http://link.aps.org/doi/10.1103/PhysRevB.75.144402>.
, , , , , , ****, (), <http://link.aps.org/doi/10.1103/PhysRevB.77.014434>.
, ****, (), <http://link.aps.org/doi/10.1103/PhysRevB.79.054409>.
, , , ****, (), <http://scitation.aip.org/content/aip/journal/apl/94/12/10.1063/1.3097551>.
, , , , , , , , , , ****, (), <http://link.aps.org/doi/10.1103/PhysRevB.79.024424>.
, , , , ****, (), <http://link.aps.org/doi/10.1103/PhysRevB.79.092409>.
, , , , ****, (), <http://link.aps.org/doi/10.1103/PhysRevB.82.100406>.
, , , , , , , , , , , ****, (), , , <http://dx.doi.org/10.1021/nl302562>.
, , , , , , , , , ****, (), <http://stacks.iop.org/0953-8984/27/i=17/a=176003>.
, , , , , , ****, (), <https://link.aps.org/doi/10.1103/PhysRevB.93.220401>.
, , , , , , , , , , , ****, (), <https://link.aps.org/doi/10.1103/PhysRevLett.109.037201>.
, , , , , , **** ().
, , , , ****, ().
, ****, ().
, , , , ****, (), <http://link.aps.org/doi/10.1103/PhysRevB.86.184413>.
, , , , , , ****, (), <http://link.aps.org/doi/10.1103/PhysRevB.86.195104>.
**, .
, , , ****, (), <http://link.aps.org/doi/10.1103/PhysRevB.81.195107>.
, ****, (), <http://link.aps.org/doi/10.1103/PhysRevB.75.155113>.
, , , ****, ().
, , , ****, ().
, , , , , , ****, ().
, , , , , ****, (), <http://link.aps.org/doi/10.1103/PhysRevB.79.014430>.
, , , ****, (), <http://link.aps.org/doi/10.1103/PhysRevB.57.10705>.
, ****, (), <http://link.aps.org/doi/10.1103/PhysRevLett.106.256401>.
, , , , , ****, (), <http://link.aps.org/doi/10.1103/PhysRevB.55.14103>.
, ****, (), <http://link.aps.org/doi/10.1103/PhysRevLett.115.256402>.
, , , , ****, (), <http://link.aps.org/doi/10.1103/RevModPhys.68.13>.
, , , , , , ****, (), <http://link.aps.org/doi/10.1103/RevModPhys.78.865>.
, , , , , ** (, ).
**, .
, ** (, ).
[^1]: While calculating the density of states (DOS) at the Fermi level for the different structures at different temperatures, we ensured that their Fermi energies were adjusted such that the energy levels for the oxygen densities of states lay at the same energy values. This was required because there was an ambiguity in the value of the chemical potential at temperatures where the structure gave rise to an insulating band-gap and we believe an accurate comparison can only be made if some features of the DOS are held fixed. This procedure required a shift in the chemical potential of some of the simulations of the order of 0.1 eV. The results we plot in Fig. \[fig4\]b are obtained after these shifts are put in. Fig 3 on the other hand plots the densities of states before any such post-processing has been done. This leads to small differences between the two figures. Instead of fixing the Oxygen levels, we also tried fixing the Lanthanum f levels and this gave rise to very similar results. We firmly believe that our results displayed in Fig. 4b are robust and it is merely the relevant magnitudes of the y-axis values at high temperatures that fluctuate by a small amount (depending on which features are held fixed) and not the actual temperature at which the charge-gap closure takes place. We also do not plot the DOS at the Fermi level but the average of the DOS at five points around $E=E_F$ as this takes care of some of the numerical noise that creeps into our calculation due to both Monte Carlo noise and the errors in analytic continuation. We also tested our results by averaging over different number of points and no significant changes take place that would affect our claims.
[^2]: In addition, this trigonal distortion also leads to a splitting of the $t_{2g}$ orbitals into 2+1 orbitals, thereby again reducing the gap with the $e_g$ orbitals. However this effect is very small and hence is not shown in our plots for the sake of simplicity
[^3]: Note that the quantitative value of the transition temperature is overestimated in our calculations. This can be explained by the fact that DMFT does not take into account finite wavelength fluctuations, and as a result, has a tendency to overestimate order like many other mean field methods.
[^4]: In our calculations, the phononic contribution to the entropy is not included since we employ the Born Oppenheimer approximation and the ion cores are considered to have well defined, stationary positions. While a first principles calculation of the phononic entropy is in principle possible, for example using the quasiharmonic approximation[@Dove_Book], such an endeavour is very computationally demanding, and, to the best of our knowledge, has never been performed using an advanced first principles method such as DFT+DMFT. In this particular system, there is no indication of a particularly interesting soft phonon mode, and so ignoring the phononic entropy probably does not change any results in a significant way.
|
---
address: |
Computer Science Division\
UC Berkeley\
Berkeley, CA\
\
\
author:
-
-
-
title: Variational Consensus Monte Carlo
---
,
,
|
---
abstract: 'Time projection chambers (TPCs) have found a wide range of applications in particle physics, nuclear physics, and homeland security. For TPCs with high-resolution readout, the readout electronics often dominate the price of the final detector. We have developed a novel method which could be used to build large-scale detectors while limiting the necessary readout area. By focusing the drift charge with static electric fields, we would allow a small area of electronics to be sensitive to particle detection for a much larger detector volume. The resulting cost reduction could be important in areas of research which demand large-scale detectors, including dark matter searches and detection of special nuclear material. We present simulations made using the software package Garfield of a focusing structure to be used with a prototype TPC with pixel readout. This design should enable significant focusing while retaining directional sensitivity to incoming particles. We also present first experimental results and compare them with simulation.'
author:
-
title: 'Charge-Focusing Readout of Time Projection Chambers'
---
IEEE Copyright Notice
=====================
Copyright 2012 IEEE. Published in the conference record of the 2012 IEEE Nuclear Science Symposium and Medical Imaging Conference, October 29-November 3, 2012, Anaheim, CA, USA. Personal use of the material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works, must be obtained from the IEEE.
Introduction
============
We are working to apply charge focusing to an existing TPC which uses Gas Electron Multipliers (GEMs) to amplify the drift charge and the ATLAS pixel chip to detect it. The idea for charge focusing was originally proposed by Sven Vahsen (University of Hawaii) and John Kadyk (Lawrence Berkeley National Laboratory). Pixel electronics have the advantage of drastically reducing detector noise, which scales with the capacitance of each detector cell and thus also scales with the cell area. Pixels also have excellent timing performance (sampling at 40 MHz), which determines the resolution in the drift direction. The combination of pixel readout and charge focusing allows us to retain the advantages of pixels while instrumenting a large detector. Additionally the increased charge density due to focusing gives higher readout efficiency at constant threshold.
In developing the charge focusing idea, there were two main questions we needed to address. First, could we make this focusing homogeneous? This is desirable as it makes the detector response uniform across the pixel chip. Second, could we limit charge diffusion during focusing? Low levels of diffusion are necessary for retaining directional sensitivity when reconstructing the short tracks expected from WIMP and neutron recoils. Since the focused track will be even shorter, we must keep the diffusion below that of the rest of the detector divided by the focusing factor.
Initial Simulation
==================
Simulations were performed using the Garfield drift program developed at CERN. We have assumed 5 cm$^2$ GEMs and a 1 cm$^2$ pixel chip. The proposed focusing geometry is simulated as one GEM and one pixel chip, each represented as a .005 cm thick box held at a set potential, as well as a series of square rings arranged around the pixel chip. These rings act as electrodes, each held at a set voltage. In order to produce our desired electric field, we have placed the rings in a pattern somewhat reminiscent of a waveguide and used a 5 cm spacing and 1.4 kV voltage differential between GEM and pixel chip (see Figure \[waveguidegeometry\]).
![Proposed focusing geometry[]{data-label="waveguidegeometry"}](waveguidegeometry.pdf){width="\linewidth"}
To simulate the paths of drifting electrons, we used Garfield’s built-in Runge-Kutta-Fehlberg (RKF) solver. The RKF method gives the average expected path for a single electron drifting from a certain point in the detector, and allows a relatively quick assessment of the general drift characteristics of a particular geometry as compared to performing a full Monte Carlo simulation. We simulated electrons drifting from 26 different starting points arranged in a line running along the x-axis from the center of the GEM to one of its edges. Comparing the initial and final x-positions of the electrons allowed us to calculate an approximate “focusing factor” for each starting point (i.e. an initial track which is contracted to 1/5 its length at readout will have a focusing factor of 5).
![Electron drift lines determined with RKF method[]{data-label="driftlines"}](waveguidedriftlines.pdf){width="\linewidth"}
![Linear focusing factor along x-direction[]{data-label="focusingfactor"}](waveguidefocusingfactor.pdf){width="\linewidth"}
![RMS values of the transverse diffusion (z-direction)[]{data-label="RMS"}](waveguideRMS.pdf){width="\linewidth"}
According to this simulation, our design should give us a fairly consistent focusing factor throughout a large part of the drift region (see Figures \[driftlines\] and \[focusingfactor\]).
We also wanted to ensure that we could limit diffusion of the electrons during focusing. To measure the transverse diffusion, we performed Monte Carlo drift simulations of groups of electrons from the same starting positions as those used in our previous RKF drift simulation. The results of this simulation are shown in Figure \[RMS\]. We found that, for the majority of the chip, the diffusion was on the order of 200 microns.
Experimental Verification
=========================
Our initial simulations were promising, but we wanted to make sure they were accurate. One concern was the fact that Garfield does not account for inter-electron Coulomb repulsion.
![2D cross-sectional view of test setup[]{data-label="labtestgeometry"}](labtestgeometry.pdf){width="\linewidth"}
We decided to create a simplified test setup to determine the validity of our simulations. This design consists of a single, square ring placed between the GEMs and pixel chip, as shown in Figure \[labtestgeometry\]. By adjusting the voltage on the ring, we can switch between focusing and non-focusing modes.
![COMSOL simulation of non-focusing test setup[]{data-label="comsolnonfocusing"}](comsolnonfocusing.pdf){width="\linewidth"}
![COMSOL simulation of focusing test setup[]{data-label="comsolfocusing"}](comsolfocusing.png){width="\linewidth"}
These two modes, simulated with COMSOL Multiphysics, are pictured in Figures \[comsolnonfocusing\] and \[comsolfocusing\].
Drift Simulation
----------------
To simulate electron drift with this simplified geometry, we again built a 3D simulation of the geometry using Garfield. As before, we used the Runge-Kutta-Fehlberg technique to determine the drift lines for electrons with starting positions along the x-axis. For this simulation, we used 250 evenly-spaced starting points and drifted in a 70% Argon, 30% CO$_2$ gas mixture at 1 atm.
![Linear focusing factor along x-direction[]{data-label="labtestfocusingfactor"}](labtestfocusingfactor.pdf){width="\linewidth"}
The focusing factor values across the extent of the pixel chip are shown in Figure \[labtestfocusingfactor\] This focusing is clearly non-homogeneous, but it should give us a measurable difference between the focusing and non-focusing modes.
In order to predict the behavior of this experimental setup, we simulated the difference in the electron hit rate that should exist between the focusing and non-focusing modes.
![RKF simulation of drift lines originating across one quadrant of GEM. Here focusing is turned off.[]{data-label="nonfocusingrateRKF"}](nonfocusingrateRKF.pdf){width="\linewidth"}
![RKF simulation of drift lines originating across one quadrant of GEM. Here focusing is turned on.[]{data-label="focusingrateRKF"}](focusingrateRKF.pdf){width="\linewidth"}
To do so, we simulated RKF drift lines originating at evenly-spaced intervals across one quadrant of the GEM, as shown in Figures \[nonfocusingrateRKF\] and \[focusingrateRKF\]. We then counted the number of drift lines ending on the pixel chip. This should be equivalent to the hit rate assuming the ionization is produced by radiation that is homogeneously distributed throughout the detector volume. We found that 83 drift lines terminated on the chip with focusing turned on, while only 37 did so with the focusing off. This is equivalent to a 2.24$\times$ rate increase from non-focusing to focusing.
Lab Installation
----------------
![CAD drawing of focusing ring and pixel installation[]{data-label="focusingsetup"}](FocusingSetup.png){width="\linewidth"}
To implement this design in our detector, we cut the focusing ring out of copper sheet and installed it in our TPC as depicted in Figure \[focusingsetup\].
![Simplified drawing of lab test setup with alpha particle source[]{data-label="labtestsetup"}](labtestsetup.pdf){width="\linewidth"}
We used a Polonium-210 alpha particle source to produce ionization tracks in our detector. This setup is shown in Figure \[labtestsetup\].
Hit Rate Analysis
-----------------
To determine the actual hit rate for our experimental setup, we replaced the pixel chip with a solid copper pad and connected it to a pulse height analyzer. We then measured the number of hits in a given time period with the focusing turned on and off. By fitting the resulting alpha energy peak with a Gaussian, we determined the total hit rate due to alpha particles.
![Electron hit count due to alpha particles in detector with focusing off. Hits were recorded over 299 seconds.[]{data-label="nonfocusingrate"}](nonfocusingrate.pdf){width="\linewidth"}
![Electron hit count due to alpha particles in detector with focusing on. Hits were recorded over 299 seconds.[]{data-label="focusingrate"}](focusingrate.pdf){width="\linewidth"}
These results are shown in Figures \[nonfocusingrate\] and \[focusingrate\].
In order to find accurate values for these measured hit rates, we needed to account for the detector dead time, which differs between the focusing and non-focusing runs and affects the amount of active time for which the detector is actually recording data. To find the actual rate, we used the equation $$R=\frac{R_m}{1-R_m\tau}\cite{deadtime}$$ where R is the actual rate, R$_m$ is the measured rate, and $\tau$ is the total dead time.
Applying the dead time correction to our measured data, we found a hit rate of 6.04$\times 10^4$hits/s with focusing turned off, and $1.52\times 10^5$hits/s with focusing on, giving a 2.51$\times$ rate increase from non-focusing to focusing. This value is not in perfect agreement with the 2.24$\times$ increase our simulation predicted, but it does support the fact that our design is focusing the ionization tracks of the alpha particles.
Preliminary Pixel Data
----------------------
After performing the rate analysis with the pulse height analyzer, we installed the pixel chip in the detector and again recorded data for both focusing and non-focusing modes. More work still needs to be done on this data to correctly separate alpha tracks from noise events, but we do have some encouraging preliminary analysis.
![Number of pixel hits recorded per event read out by pixel chip with detector in non-focusing mode[]{data-label="nonfocusingpixel"}](nonfocusingpixel.pdf){width="\linewidth"}
![Number of pixel hits recorded per event read out by pixel chip with detector in focusing mode[]{data-label="focusingpixel"}](focusingpixel.pdf){width="\linewidth"}
Figures \[nonfocusingpixel\] and \[focusingpixel\] show the number of pixel hits recorded for each event that was read out by the pixel chip for the non-focusing and focusing modes, respectively. This is essentially equivalent to a measurement of each event’s size. As can be seen from these figures, there is an approximately 3$\times$ event size decrease from non-focusing to focusing. This suggests that with the focusing turned on the electrons from each alpha track are focused onto fewer pixels.
Conclusion
==========
We have created simulations using Garfield which demonstrate that it should be possible to implement charge focusing while minimizing diffusion and keeping focusing relatively homogeneous across the readout area.
We have also created a simplified experimental setup in order to verify the accuracy of our simulations. Both electron hit rate and preliminary event size analysis show evidence of focusing in our lab setup; we have rough but imperfect agreement with the predictions of our simulation. We will continue to analyze the alpha track data recorded by the pixel chip.
![Proposed layout for next-generation TPC[]{data-label="nextgen"}](nextgen.pdf){width="0.8\linewidth"}
We plan to further develop charge focusing with the next-generation D$^3$ detector prototype. This detector (as shown in Figure \[nextgen\]) will incorporate four readout chips. We believe the adjacent cell placement should improve edge effects seen in our simulations.
Acknowledgment {#acknowledgment .unnumbered}
==============
We acknowledge fruitful discussions with John Kadyk and Maurice Garcia-Sciveres of Lawrence Berkeley National Laboratory, and thank them for providing the ATLAS FE-I3 pixel chip and associated readout electronics used in this work. We acknowledge support from the U.S. Department of Homeland Security under Award Number 2011-DN-077-ARI050-03 and the U.S. Department of Energy under Award Number DE-SC0007852. The views and conclusions contained in this document are those of the authors and should not be interpreted as necessarily representing the official policies, either expressed or implied, of the United States Government or any agency thereof.
[1]{}
M. L. Larsen and A. B. Kostinski, “Simple dead-time corrections for discrete series of non-Poisson data,” *Meas. Sci. Technol.*, vol. 20, no. 9, Sept. 2009
|
---
author:
- 'F. Ratnikov'
title: 'Using Machine Learning to Speed Up and Improve Calorimeter R&D'
---
Introduction
============
Design of new experiments, as well as upgrade of ongoing experiments, is a continuous process in experimental high energy physics.
This is a many-fold process: global optimisation requires different steps to be coordinated. For example, when varying the material of the calorimeter absorber, the reconstruction algorithm should be re-tuned to accommodate a new Molière radius for the detector. If done manually, this is a time-consuming procedure, which significantly slows down the optimisation loop turnover.
The ultimate goal of the detector construction is to establish the necessary performance of physics measurements. However, the propagation of particular future calorimeter technologies to the metrics quantifying the ultimate physics performance of the detector is not immediate, and requires several steps in between.
![\[fig:pipeline\] General pipeline for the calorimeter optimisation includes several steps. Blue blocks indicate data processing pipeline steps; pink bubbles represent configurations and conditions for pipeline steps; yellow blocks close down the optimisation loop.](figures/pipeline.png){width="80.00000%"}
The typical workflow for optimisation of the calorimeter components is sketched in Fig. \[fig:pipeline\].
- Selected event samples, both signal and background, are used to initiate an optimisation cycle for comparing performance for signal recovery and background suppression.
- Calorimeter is usually installed downstream of the detector, so propagation of events from the origin point to the calorimeter is necessary. This step is dependent on the properties of the rest of the detector in front of the calorimeter. Also, if the calorimeter detector has an inhomogenous configuration, details of the global geometry are to be accounted for in this step. The latter allows optimisation of the global geometry based on the physics quality metrics.
- The technology for the individual calorimeter modules is a central point for the detector R&D. To evaluate technology, we need to simulate its response to the event. This is done using response simulation models, e.g. Geant4 [@geant4]. The calorimeter technology details drive such simulation.
- Behaviours of the front end electronics are another important contribution into physics quality of the detector. Although such properties are hard to simulate, good data samples may be obtained from beam or bench tests.
- The reconstruction algorithm is absolutely necessary to evaluate the quality of converting the detector response into the physics objects.
- The physics quality metrics may be calculated using reconstructed objects. This metric can be used as a target function for the optimisation procedure.
- All aspects of the calorimeter may be optimised: the details of the calorimeter technology, the geometrical layout, and possible reconstruction algorithms.
Optimisation cycle built on top of event processing pipeline allows to derive physics motivated optimal parameters for the detector.
Machine Learning Based Approach
===============================
To evaluate the physics performance of a particular configuration of the possible future calorimeter detector, one needs to run the optimisation cycle described above. A good fine-tuning of individual blocks is important to properly propagate properties of the configuration under study to the ultimate physics performance. For regular stable detector operation these blocks are carefully tuned for the actual detector configuration. In contrast, for the detector R&D process, many different possible configurations are studied simultaneously. Nevertheless, the reasonable representations of simulation and reconstruction steps, which are tuned for every studied configuration, are necessary for inferring consistent conclusions about physics performance of these configurations. Being made manually, this is a time consuming work. Fortunately, these studies use well labelled datasets either from MC simulation or from test beam measurements. Thus surrogate models may be built and trained on labelled data using regular Machine Learning (ML) approaches. This allows to speed up building models for different pipeline steps. Importantly, such training may be automated and requires minor expert supervision. In the following sections we demonstrate the possibility of applying ML based solutions to different steps of the pipeline.
Generating Detector Response
============================
<span style="font-variant:small-caps;">Geant4</span> simulation of the calorimeter response is computationally intensive. This is primarily because shower development and collecting responses are done on the micro-level of the individual shower particles. At the same time, calorimeters usually have granularity much coarser than that of the simulation micro-level. As a result, detailed shower information is aggregated into relatively few responses of the calorimeter. This means that the transfer function converting impact particle parameters into the calorimeter responses is relatively simple and maybe substituted by the surrogate generative model trained by standard means of ML.
![\[fig:calogan\] Using generative models may significantly speed up simulation of detector response. Top row: Geant4 simulated showers; bottom row: generated calorimeter responses for impact particle identical to those producing showers in the top row [@lhcbcalogan].](figures/calogan.png){width="80.00000%"}
Fig. \[fig:calogan\] illustrates application of the surrogate generative model to the LHCb electromagnetic calorimeter [@lhcbcalogan]. While the calorimeter has modules with three different granularities of 12, 6, and 4 cm in size, the generative model based on the Wasserstein flavor [@wsgan] of Generative Adversarial Network approach [@gan] converts kinematics and position of the impact particle into virtual calorimeter response on the 30x30 matrix of cells of 2 cm in size. This allows to aggregate the obtained response into every actual calorimeter granularity around the impact point. Details in Ref. [@lhcbcalogan] demonstrate that such a model easily learns main properties of the signal thus allowing 3 orders of magnitude faster simulation of the calorimeter response for the R&D optimisation cycle.
ML-based Reconstruction
=======================
The big slow down factor for running optimisation cycle is a necessity to fine tune reconstruction algorithm for every new calorimeter technology and geometry configuration. ML may help to tune the reconstruction in an automatic way. Indeed, as soon as a sample of calorimeter responses is available, the corresponding regressor may be trained to extract physics information from the raw response.
![\[fig:scurve\] ML-based reconstruction of the calorimeter cluster position provides spatial resolution similar to the customised reconstruction procedure, but without [*a priori*]{} knowledge about the particular spatial properties of the calorimeter under study. Left - correlation between cluster centre and the true track position; middle - correlation corrected using parametrised correction; right - correlation using ML trained regressor. ](figures/scurve.png){width="80.00000%"}
Fig. \[fig:scurve\] demonstrates the quality of the spatial reconstruction of the calorimeter cluster for the case of the LHCb 4 cm and 12 cm modules. Agnostic to particular calorimeter details, automatically trained ML model (based on xgboost [@xgboost] in this case) produces a slightly better performance than the manually selected parametric model. Importantly, the automatically trained generic regressor provides performance comparable with the manually tuned one. This justifies the use of this regressor in the optimisation cycle in place of a well tuned reconstruction algorithm for extracting physics observables from the calorimeter response.
Pileup Mitigation with Timing
=============================
A significant contribution from pileup events is expected after LHC and detectors upgrade during long shutdown in 2031 [@lhcls3]. The effective way of reducing this contribution is using signal time, thus separating contributions from distinct primary interactions. To evaluate this approach, consistent estimation of timing resolution for different technologies and configurations of the upgraded calorimeter are required. Using ML for this task allows making this estimation automatic, consistent, and agnostic to details of the system.
For this analysis we use responses from 30 GeV electrons collected at DESY test beam facility [@testbeamdata]. Signals are read out at a sampling rate of 5 GHz, which allows re-sampling to lower the effective sampling rates. Details of data and its pre-processing are described in Ref. [@chepposter].
ML regressor may be trained on test beam data to convert a set of signal sampling series into the reference time of the signal. To test stability of the ML approach, several different ML techniques are used to train the regressor. Fig. \[fig:timeresolution\] demonstrates similar best results for different sampling rates obtained from several different approaches. This consistency confirms that different flexible enough surrogate ML models reproduce time measurement behaviours with reasonable accuracy, and it is good enough for approximate R&D studies. The result illustrates that for these kinds of signals using sampling rates above 500 MHz insignificantly improves timing resolution. The xgboost ML implementation is used in following examples as the most common, flexible and stable approach.
[0.58]{} {width="1.\textwidth"}
[0.38]{} {width="100.00000%"}
At high pileup conditions, there is a possibility that the same channel will contain smaller background contribution from another signal in addition to the primary signal. That background contribution will have a different time, and thus may disturb the timing measurement for the primary signal. To evaluate the efficiency of the pileup mitigation, it is important to estimate this effect. This requires a customised reconstruction algorithm, that accounts for details of signals behaviours. The ML approach to the same problem is generic: it does not require [*a priori*]{} knowledge details of signal properties. All the necessary information is extracted from the train data sample. Following this approach, the dataset was prepared by constructing composite signals as an artificial mixture of the primary and the background signals with known amplitude ratio and time offset. To evaluate the ability to detect the presence of another, background contribution, the ML classifier is trained on this dataset. The ML classification quality for identifying the presence of background contribution for different relative signal strengths and time offsets is presented in Fig. \[fig:onetwodisc\].
[0.44]{} {width="1.\textwidth"}
[0.52]{} {width="1.\textwidth"}
The next problem is to quantify the effect of the background contribution on the time resolution of the primary signal. The regressor, which converts composed signal time series to the reference time of the primary signal, was trained using a set of composite signals. Fig. \[fig:timeresolution2\] illustrates time resolution degradation for different relative amplitudes and time offsets of the two contributions. This distribution is necessary to plug time resolution effects into the signal reconstruction step of the optimisation loop. Fig. \[fig:timeresolution1\] illustrates time resolution dependency on the sampling rate. The difference between performances for one signal and for two signals including background contribution $\alpha$=0 is driven by insensitive region in Fig. \[fig:onetwodisc\] where the regressor can not reliably identify the presence of the second contribution. The [*a priori*]{} knowledge that it is a single contribution thus improves the timing resolution for the signal.
Global Optimisation
===================
The pipeline in Fig. \[fig:pipeline\] with plugged in ML driven steps allows automation of the optimisation cycle. Although the automatic training of ML components is much faster than manual tuning, it still takes significant time. Therefore, the global optimisation procedure requires minimising the total number of optimisation iterations . ML suggests effective approaches to this kind of optimisation problems, like Bayesian optimisation and others [@mloptimisation]. ML based multi-parameter optimisation with a massive Geant4 simulation step included into the optimisation cycle is illustrated in Ref. [@shipopt].
Conclusions
===========
The calorimeter R&D process requires time consuming computation steps to evaluate physics performance for different detector techniques and configurations. Surrogate ML models may be used for most steps that are necessary for evaluating quality of different solutions. Such models are automatically trained on available datasets and provide possibility to consistently estimate the resulting physics performance. Using automatic training speeds up the turnover for the performance studies and ensures consistency and uniformity of obtained results.
The research leading to these results has received funding from the Russian Science Foundation under agreement No 19-71-30020.
[99]{}
J. Allison et al., *Recent developments in Geant4*, *Nuclear Instruments and Methods in Physics Research A* [**835**]{} (2016) pp.186-225.
V. Chekalina et al., *Generative Models for Fast Calorimeter Simulation: the LHCb case*, *EPJ WOK* [**v214**]{} (2019) pg. 2034.
M. Arjovsky, S. Chintala, L. Bottou, *Wasserstein GAN*, arXiv:1701.07875.
I. Goodfellow et al., *Generative adversarial nets*, in *Advances in neural information processing systems* (2014), pp. 2672–2680.
XGBoost: *A Scalable Tree Boosting System*, arXiv:1603.02754.
LHCb Collaboration, *Expression of Interest for a Phase-II LHCb Upgrade: Opportunities in flavour physics, and beyond, in the HL-LHC era*, *CERN report CERN-LHCC-2017-003* (2017).
*The Phase 2 Upgrade of the LHCb Calorimeter system*, *Calorimetry for High Energy Frontier 2019*, in these proseedings.
F. Ratnikov et al., *Using ML to Speed Up New and Upgrade Detector Studies*, *24th International Conference on Computing in High Energy and Nuclear Physics*, will appear in proceedings.
Mehdad, Ehsan and Kleijnen, Jack, *Stochastic intrinsic Kriging for simulation metamodeling*, *Applied Stochastic Models in Business and Industry* (2018).
A. Baranov et al., *Optimising the Active Muon Shield for the SHiP Experiment at CERN*, *Journal of Physics: Conference Series* [**934**]{} (2017) pg.12050
|
---
abstract: 'We study the holographic hydrodynamics in the Einstein-Gauss-Bonnet(EGB) gravity in the framework of the large $D$ expansion. We find that the large $D$ EGB equations can be interpreted as the hydrodynamic equations describing the conformal fluid. These fluid equations are truncated at the second order of the derivative expansion, similar to the Einstein gravity at large $D$. From the analysis of the fluid flows, we find that the fluid equations can be taken as a variant of the compressible version of the non-relativistic Navier-Stokes equations. Particularly, in the limit of small Mach number, these equations could be cast into the form of the incompressible Navier-Stokes equations with redefined Reynolds number and Mach number. By using numerical simulation, we find that the EGB holographic turbulence shares similar qualitative feature as the turbulence from the Einstein gravity, despite the presence of two extra terms in the equations of motion. We analyze the effect of the GB term on the holographic turbulence in detail.'
author:
- 'Bin Chen$^{1,2,3}$, Peng-Cheng Li$^{1}$, Yu Tian$^{4,5,6}$ and Cheng-Yong Zhang$^{3}$[^1]'
title: '**Holographic Turbulence in Einstein-Gauss-Bonnet Gravity at Large $D$**'
---
*$^1$Department of Physics and State Key Laboratory of Nuclear Physics and Technology,\
Peking University, 5 Yiheyuan Road, Beijing 100871, China\
*
$^2$Collaborative Innovation Center of Quantum Matter, 5 Yiheyuan Road, Beijing 100871, China\
$^3$Center for High Energy Physics, Peking University, 5 Yiheyuan Road, Beijing 100871, China\
$^4$School of Physics, University of Chinese Academy of Sciences, Beijing 100049, China\
$^5$Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China\
$^6$ Center for Gravitation and Cosmology, College of Physical Science and Technology, Yangzhou University, Yangzhou 225009, China
Introduction
============
In recent years, it has been found that black hole physics simplify significantly in the limit that the number of spacetime dimensions $D$ is very large [@Emparan1302; @Emparan1402; @Emparan1406; @Emparan1502]. The key feature in the large $D$ limit is that the gravitational field of a black hole is strongly localized near its horizon due to the very large radial gradient of the gravitational potential. By performing the $1/D$ expansion in the near region of the black hole, the Einstein equations can be reduced to several effective equations which capture the dynamics of the black hole. As a consequence one can solve these equations to construct various black hole solutions, including black string and black ring with different asymptotic behaviors and topologies. Furthermore one can study the linear stability of these solutions perturbatively to obtain the quasinormal modes, and even study the non-linear evolutions of the solutions numerically [@Emparan1504; @Suzuki1505; @Suzuki1506; @Emparan1506; @Tanabe1510; @Chen1702; @Tanabe1511; @Rozali1607; @Miyamoto1705; @Emparan:2018bmi].
The fact that the dynamics of the black holes are captured by the effective equations of the horizons is reminiscent of the membrane paradigm[@Bhattacharyya1504; @Bhattacharyya1511; @Dandekar1607; @Dandekar1609; @Bhattacharyya1704; @Dandekar:2017aiv], initialized in [@Damour:1978cg][^2]. One essential feature of the membrane paradigm is that the evolution of the horizon is like a viscous fluid. In [@Emparan1602] it was found that the large $D$ effective equations for (AdS or asymptotically flat) black branes can be interpreted as the equations for dynamical fluid. For AdS black branes the transport coefficients were found to match well with the result obtained from the holographic study. More importantly, the study in [@Emparan1602] indicates that the hydrodynamical gradient expansion is truncated at a finite order and the higher order transport coefficients are vanishing in the $D \to \infty$ limit.
On the other hand, according to the AdS/CFT correspondence, the geometry of dynamical black holes in asymptotically anti-de Sitter spaces (AAdS) can be described in terms of a conformal fluid living on the conformal boundary of the spaces [@Bhattacharyya0712]. This fluid/gravity duality not only provides a geometric way to study fluid dynamics but also help to reveal new phenomena that never occurs in gravity. For instance, it is well-known that the turbulence is a ubiquitous property of the fluid if the Reynolds number is sufficient large. The presence of the turbulence in hydrodynamics indicates a similar turbulent behavior should appear for the AdS black holes. As expected, it was found that by numerical simulation a perturbed black brane in AAdS can exhibit turbulent behavior when the Reynolds number of the fluid counterparts is sufficiently large [@Adams1307; @Green1309; @Carrasco1210]. Within the fluid/gravity duality it is possible to use the geometric tool to study the turbulence, such as the geometric interpretation of turbulence [@Adams1307]. Therefore, the natural questions are what is the relation between the large $D$ effective equations for the AdS black branes and the large $D$ limit of the conformal fluid living on the AdS boundary, and can the fluid equations exhibit turbulent behavior in a certain regime?
Indeed, as demonstrated by a recent work [@Rozali1707], the large $D$ effective equations for the AdS black branes are equivalent to the large $D$ limit of relativistic hydrodynamics which is naturally truncated at the second order in derivatives. Actually, in deriving the large $D$ limit of the relativistic hydrodynamics, one need to decompose $D=n+q+1$ and consider the dynamical variables depending only on $q+1$ coordinates. Moreover one has to rescale the time and space coordinates appropriately such that the hydrodynamics on $q+1$ dimensional system is simplified significantly in the large $n$ limit. Such scaling laws for the coordinates are precisely the ones used in studying the large $D$ limit of the gravity. More interestingly the effective equations for the black brane in the large $D$ limit are exactly the same as the hydrodynamic equations of motions in a fluid frame. These equations are a variant of the compressible version of the non-relativistic Navier-Stokes equations. Then by using analytic and numerical techniques the authors in [@Rozali1707] analyzed two and three-dimensional turbulent flow of the fluid in various regime and the relation with the geometry of the black branes. In this paper we would like to extend the study to the holographic hydrodynamics in the framework of Einstein-Gauss-Bonnet (EGB) gravity at large $D$.
The Einstein-Gauss-Bonnet gravity provides a nice platform in studying the holographic hydrodynamics. According to the holographic dictionary, the higher-derivative terms in gravity may come from the stringy correction or the string interaction. The Gauss-Bonnet term, including the quadratic terms of the curvature tensors, appears as the leading order correction in the low energy effective action of the heterotic string theory [@Zwiebach1985; @Boulware1985]. Another appealing feature of the EGB gravity is that its equations of motion remain second order such that the fluctuations around the vacuum do not have ghost-like mode. The EGB gravity has been well studied in the holographic hydrodynamics, in particular on the Kovtun-Son-Starinets(KSS) viscosity bound [@Kovtun:2004de] . In [@Brigante:2007nu; @Brigante:2008gz], it was shown that the KSS bound was violated in the EGB gravity =(1-4ł\^[D=5]{}\_[GB]{}), where $\l_{GB}$ is the Gauss-Bonnet coefficient in five dimensions. The causality constraints require that in five dimensions [@Hofman:2008ar; @Buchel0906; @Hofman:2009ug] -ł\^[D=5]{}\_[GB]{}. In general dimensions $D$, the viscosity satisfies \[etaovers\] =(1-ł\^[D]{}\_[GB]{}), and the causality constraints require that [@Buchel0911] - ł\^[D]{}\_[GB]{} .\[causal\]
The large $D$ study of the EGB gravity [@Chen1511; @Chen1703; @Chen1707] shares many similar features with that of the Einstein gravity. For example, there are also two classes of quasinormal modes for the EGB black holes: one consists of the decoupled modes, which characterize the dynamics of the black hole, and the other one consists of featureless non-decoupling modes [@Chen1511]. For the EGB black strings (branes), the final states of the non-linear evolution are non-uniform black strings if the strings are thin enough. This is qualitatively the same as the behavior of black strings in the Einstein gravity. Besides, the large $D$ effective equations for the (asymptotically flat) EGB black branes can be interpreted as the fluid equations as well[@Chen1707]. So for the AdS black branes in the EGB gravity one may expect that the large $D$ effective equations are also of the equations for a dynamical fluid, which allows us to study the holographic turbulence in the EGB gravity.
In the paper, based the work of [@Rozali1707] we will study the holographic turbulence in the EGB gravity by using the large $D$ expansion method. Unlike the case in the Einstein gravity, up to second order the hydrodynamics dual to the EGB gravity is unknown. In this work, we focus on the holographic EGB hydrodynamics. As we show in section \[section:equationofmotion\], if we take the same scalings for the coordinates as those used in constructing the EGB black strings [@Chen1707] at large $D$, then we obtain the effective equations which have the form of the hydrodynamical equations naturally truncated at second order in derivatives. Moreover, the transport coefficients are the same as those obtained from AdS/CFT [@Buchel0911].
In section \[section:analyticalanalysis\] we give an analytical discussion on the the large $D$ EGB fluid flows. We show that in the small Mach number limit, the EGB equations could be reduced to the incompressible Navier-Stokes equations with modified Reynolds and Mach number. However, if the Mach number is not small, the extra terms coming from the GB term may play an important role, especially in 2D flow. Then in section \[section:numericalstudy\] we numerically solve the equations of motion and analyze the effect of the GB term on the turbulence. We surprisingly find that the extra terms play negligible role in the evolution of the turbulence. We find that the 2D EGB turbulence has qualitatively similar behavior as the one from the Einstein gravity. For example, we notice an inverse and direct cascade in the turbulence, with the energy spectrum of the direct cascade obeying a $k^{-4}$ power law. In section \[section:geometricinter\] we verify that the relation between the horizon curvature power spectrum and the hydrodynamic energy power spectrum proposed by [@Adams1307] holds for the EGB gravity. We end with a summary and some discussions in section \[section:summary\].
Equations of motion of relativistic hydrodynamics {#section:equationofmotion}
=================================================
$1/n$ expansion of the EGB equations
-------------------------------------
The action of the $D$ dimensional EGB gravity with a negative cosmological constant $\Lambda=-(D-1)(D-2)/2$ (we have set the radius of AdS space to unity) is given by I=d\^Dx(R-2+L\_[GB]{}), with L\_[GB]{}=R\_R\^-4R\_R\^+R\^2, where $\alpha$ is the GB coefficient. Here we follow the conventions in [@Chen1511; @Chen1703; @Chen1707]. The GB coefficient $\a$ is related to the one used in [@Buchel0911] by =ł\^D\_[GB]{}. As the GB term appear in the low energy effective action in the heterotic string theory [@Boulware1985], the GB coefficient is usually taken to be positive. However, just taking the EGB theory as a gravity theory, there is no reason to restrict the GB coefficient to be positive. Actually from the causal constraints [@Buchel0911] it can be negative as well. In this work, we do not make any constraints on the GB coefficient.
From the action, we obtain the equations of motion for the metric \[EGBequations\] R\_-g\_R+g\_+H\_=0, where H\_=-g\_L\_[GB]{}+2(RR\_-2R\_R\^\_+2R\^R\_ +R\_R\_\^). The black brane solution in EGB gravity can be written as [@Cai0109] ds\^2=-r\^2 h(r)dt\^2++(\_[ij]{}dx\^idx\^j),i,j=1,, D-2, where h(r)&=&(1-),\[blackbrane\]\
&=&(D-3)(D-4)=ł\^D\_[GB]{},\[blackbrane:alpha\] $r_+$ is the horizon radius, and $\tilde{\alpha}$ is exactly the Gauss-Bonnet coefficient used in [@Buchel0911]. Note that in the above expression $L_c$ is introduced as the effective radius of the AdS space in EGB gravity, since as $r\to \infty$, $h(r)\to1/L_c^2$, with L\_c\^2=. From above we can find that there exists an upper bound for the GB coefficient $\tilde{\alpha}$, i.e. \[upperboundforalpha\] , beyond which the theory is not well-defined. On the other hand, the causality constraint (\[causal\]) requires that in the large D limit, - 1/4. It turns out that the upper bound is the same as (\[upperboundforalpha\]).
Using the Eddington-Finkelstin coordinates the solution (\[blackbrane\]) can be written as ds\^2=-r\^2 h(r)dv\^2+2dvdr+(\_[ij]{}dx\^idx\^j). $L_c$ can be absorbed into the redefinition of $x^i$, i.e. $x^i\to x^i/L_c$, such that $L_c$ does not appear in the metric explicitly. This form of the metric leads to the following metric ansatz \[metricansatz\] ds\^2&=&-r\^2A(v,r,z\^c) dv\^2-2u\_v(v,r,z\^c) dvdr-2u\_a(v,r,z\^c) dz\^a dr\
&&+r\^2(-2C\_a(v,r,z\^c) dz\^a dv+G\_[ab]{}(v,r,z\^c)dz\^adz\^b +d\^2), where $a, b, c=1, \cdots q$, $\vec{x}$ is an $n$ dimensional vector and $D=n+q+2$. Note that as in [@Emparan1506; @Rozali1707] we keep $q$ finite and let $n\to\infty$. In this case most of the spatial directions remain translational invariance, so that we only study the deformations of the black brane that depend only on a finite number of coordinates $z^a$. Moreover, in the following we use $1/n$ rather than $1/D$ as the expansion parameter.
In order to perform the $1/n$ expansion properly we need to specify the large $D$ scalings of the metric functions. According to the properties of the boundary hydrodynamics [@Buchel0911], the speed of sound of long-wavelength perturbations is of $\mc O(1/\sqrt{n})$, this indicates that to capture the physics of the long-wavelength perturbations, we should rescale $z^a\to z^a/\sqrt{n}$, $\vec{x}\to\vec{x}/\sqrt{n}$. In addition, we consider small velocities $\mc O(1/\sqrt{n})$ along the brane directions. Therefore, the large $n$ scalings of the metric functions are respectively A=O(1), u\_v=O(1),u\_a=O(n\^[-1]{}),C\_a=O(n\^[-1]{}),G\_[ab]{}=(\_[ab]{}+O(n\^[-1]{})). These scalings are exactly the same as the one in discussing the black string in the large $D$ EGB theory[@Chen1707]. We can rewrite the metric ansatz (\[metricansatz\]) as ds\^2=-r\^2A dv\^2-2u\_v(v,r,z\^c) dvdr+r\^2(-\_a dz\^a dv+\_[ab]{}dz\^adz\^b +d\^2), where by gauge choice we have set $u_a=0$ and all the functions in the above expression are of $\mc O(1)$ now.
At large $D$ the radial gradient becomes dominant, i.e. $\partial_r=\mc O(n)$, $\partial_v=\mc O(1)$, $\partial_a=\mc O(1)$, so in the near region of the black brane it is better to introduce a new radial coordinate =()\^n, such that the derivative with respect to ${\mathsf{R}}$ is finite in the large $D$ limit, where $r_0$ is a horizon length scale which can be set to be unity $r_0=1$. To solve the EGB equations we need to specify the boundary conditions at large ${\mathsf{R}}$, which are A=L\_c\^[-2]{}+O(\^[-1]{}),\_a=O(\^[-1]{}),\_[ab]{}=\_[ab]{}+O(\^[-1]{}). On the other hand, the solutions have to be regular at the horizon.
At the leading order of the $1/n$ expansion, the EGB equations only contain ${\mathsf{R}}$-derivatives so they can be solved by performing ${\mathsf{R}}$-integrations. After imposing the boundary conditions the leading order solutions are obtained as A=(1-),u\_v=-1, \_a=(-),\_[ab]{}=\_[ab]{}+, where \[G0\] G\_[0]{}&=&\
&&+, and to simplify the expressions we have introduced the quantities ${\mathbf { b}}$ and $\beta$ \[bandbeta\] &=&,\
&=&, 02. As shown in [@Emparan1602] and [@Chen1707], the $1/n$ terms in $\tilde{G}_{ab}$ are obtained at the next-to-leading order in the $1/n$ expansion of the EGB equations. It must be included since it also appears in the EGB equations at the leading order of the $1/n$ expansion. $m(v,z^a)$ and $p_a(v,z^a)$ are the integration functions of ${\mathsf{R}}$-integrations of the EGB equations. At the next-to-leading order of the $1/n$ expansion, $m$ and $p_a$ must satisfy the effective equations \[effeq1\] \_v m-\_b\^b m=-\_b p\^b, \[effeq2\] &&\_v p\_a-\_b\^b p\_a -\_a\^b p\_b+p\_a\_b\^b m +p\_b\^b\_a m\
&&+\_b() -(p\_b\^bm\_a m+p\_a \^bm\_b m)\
&&+(\^bm\_bp\_a+\_am\^bp\_b)+=0, In the limit $\tilde{\alpha}\to0$, i.e. $\beta\to1$, the above equations reproduce the ones in the Einstein gravity [@Emparan1602].
Linear analysis {#subsection:linearanalysis}
---------------
Considering a small perturbation around the static uniform black brane solution, i.e. $m=m_0=$ const., $p_a=0$, m&=&m\_0+m e\^[-iv+ i k\_b z\^b]{},\
p\_a&=&p\_a e\^[-iv+ i k\_b z\^b]{}, and plugging these into the effective equations (\[effeq1\]) and (\[effeq2\]), we can read the quasinormal mode frequencies of the shear mode and the sound mode.
#### Shear mode.
The frequency is \[QNMfrequency\] =-i k\^2. As we will see in the next subsection, this dispersion relation is related to the transport coefficient of the viscous hydrodynamics.
#### Sound modes.
The frequencies are \_=-i k\^2. It is easy to see that the perturbation is always stable. Up to the leading order of $k$, one finds $\omega_{\pm}=\pm \sqrt{\frac{2}{\beta+1}}k$. As we will see in the next subsection, this coefficient corresponds to the speed of sound of the long-wavelength perturbation of the fluid.
Dynamical fluid
---------------
In [@Emparan1602], it was shown that the effective equations for the black branes can be interpreted as the equations for a dynamical fluid. This turns out to be true for the asymptotically flat EGB black branes as well[@Chen1707]. Here we show that the effective equations (\[effeq1\]) and (\[effeq2\]) can be actually transformed into the fluid equations. Firstly, we introduce ${\mathrm{v}}_a$ by p\_a=(m \_a+\_a m), then we find that the first effective equation (\[effeq1\]) becomes the continuity equation for the energy density \[fluideq1\] \_v m+\^a( m \_a)=0, with $m$ and ${\mathrm{v}}_a$ being taken as the energy density and the velocity of the fluid flows.
Furthermore, we notice that (\[effeq2\]) can be written in terms of ${\mathrm{v}}_a$ as
\[fluideq2\] \_v(m \^a)+\_b(m \^a \^b+\^[ab]{})=0 where \[stresstensor\] \_[ab]{}=\_[ab]{} m- m\_[(a]{} \_[b)]{}-m\_a\_bm. This equation is for the momentum conservation, with $\tau_{ab}$ being the stress tensor. Therefore, we may interpret the effective equations as the equations of motion for non-relativistic[^3], compressible fluid naturally truncated at second order in derivatives.
As shown in [@Rozali1707], the Einstein equations at large $D$ are equivalent to the large $D$ hydrodynamics within the fluid/gravity duality. This statement holds for the EGB gravity as well: the EGB equations at large $D$ are equivalent to the large $D$ hydrodynamics describing the conformal fluid living on the conformal boundary of the AdS space. Although the relativistic hydrodynamics that dual to AdS black branes in the EGB gravity up to second order is unknown at present, we can still find some hints by comparing the properties of the fluid flows (\[fluideq1\]) and (\[fluideq2\]) with the results obtained from AdS/CFT [@Buchel0911]. From (\[stresstensor\]) we find the pressure P=m, which gives the equation of state of the black string. The speed of sound of the long-wavelength perturbations is then \[soundspeed\] c\_s==. Due to the scaling relation we used in the metric ansatz, the physical speed of sound is $\mathbf{c_s}=c_s/\sqrt{n}$, which is small in the large $n$ limit. Moreover, the shear viscosity is \[viscosity\] = m, then the ratio of the shear viscosity to the entropy density is given by =, with the entropy density $s=4\pi m$. Both this result and the speed of sound are in accord with the results found previously in [@Buchel0911] by taking the large $D$ limit and using our convention[^4].
General analysis of the large $D$ EGB fluid flows {#section:analyticalanalysis}
=================================================
In this section we give a general discussion on the holographic EGB fluid flows. We will relate the equations of motion (\[fluideq1\]) and (\[fluideq2\]) to the compressible Navier-Stokes equations. In terms of f\^a=, the equations of motion (\[fluideq1\]) and (\[fluideq2\]) can be rewritten as[^5] \[feq1\] \_v m+m+m =\^2 m, \[feq2\] &&\_[v]{}+()-\^[2]{}-()\
&&+-()+(-)=0. It is convenient to introduce the following dimensionless variables, \[dimensionlessvars\] =,=,=,=L\_0 , where $L_0$ is a characteristic length scale of the system, $U$ is a characteristic velocity and $E$ is a characteristic energy density. In terms of these dimensionless variables, (\[feq1\]) and (\[feq2\]) become of the forms \[Eq:dimenless1\] \_ + +=\^2 , \[Eq:dimenless2\] &&\_+()- \^[2]{}- ()\
&&+- ( )+( - )=0, where R\_e=L\_0 U,M=U, are the Reynolds number and the Mach number. Although the above equations are slightly more complicated than the ones appearing in the Einstein gravity, they can actually be taken as a variant of the compressible version of the non-relativistic Navier-Stokes equations. This can be seen by taking the incompressible limit and redefining the Reynolds number and the Mach number.
Let us first consider the simple case when the Mach number is small, and the fluid flow is nearly incompressible. We can expand $\vec{u}$ and $\epsilon$ in series of $M$, i.e. =\_[i=0]{}M\^i \_[(i)]{},=\_[i=0]{}M\^i\_[(i)]{}. Then from the above equations we find that $\epsilon_{(0)}$ and $\epsilon_{(1)}$ are constant, and $\vec{u}_{(0)}$ and $\epsilon_{(2)}$ satisfy the following equations, \[incomNS1\] \_[(0)]{}=0,\_\_[(0)]{}+(\_[(0)]{})\_[(0)]{}- \^[2]{}\_[(0)]{} +=0, with $\epsilon_{(0)}$ being the density and $\epsilon_{(2)}$ being the pressure. If we redefine the Reynolds number and the Mach number as \[ReandM\] = L\_0 U,=U, then instead of (\[incomNS1\]), $\vec{u}_{(0)}$ and $\epsilon_{(2)}$ satisfy \_[(0)]{}=0,\_\_[(0)]{}+(\_[(0)]{})\_[(0)]{} += \^[2]{}\_[(0)]{},\[NS\] which are exactly the same as the incompressible Navier-Stokes equations. In this case, it should be appropriate to view ${\mathbf {R_e}}$ and ${\mathbf { M}}$ as the Reynolds number and the Mach number of our large $D$ fluid flow. In fact, the Reynolds number and the Mach number in the viscous fluid are defined as =,=, where $\eta$ and $c_s$ are the shear viscosity and the sound velocity. From the relations (\[viscosity\]) and (\[soundspeed\]), it is easy to see that they are exactly the same as (\[ReandM\]).
Up to now, we have found that in the small Mach number limit, the equations of motion for the holographic fluid flow in the EGB theory take exactly the form of the Navier-Stokes equations for an incompressible fluid. The impact of the GB term on the fluid flow is totally encoded in the redefinition of the Reynolds number and the Mach number. The incompressible flow has been well studied (see for example in the textbook [@Davidson2015]), and here we just give a brief review on the main results for the freely decaying fluid turbulence.
One may define the kinetic energy and the enstrophy by E\_I=|u\_[(0)]{}|\^2d\^qx, \_I=\_[(0)ij]{}\_[(0)]{}\^[ij]{}d\^qx, where \_[(0)ij]{}=\_iu\_[(0)j]{}-\_ju\_[(0)i]{} is the vorticity 2-form. It is illuminating to write the kinetic energy and the enstrophy in terms of the energy spectrum $E(k)$, E\_I&=&\_0\^E(k)dk\
\_I&=&\_0\^k\^2E(k)dk, The energy spectrum $E(k)$ provides a measure of how the energy is distributed across the various eddy size $\sim 1/k$. From the equations (\[NS\]), one can get the evolution equations for $E_I$ and $\Omega_I$ \_E\_I&=&-\_I,\
\_\_I&=&\^[(0)ij]{}\_[(0)jk]{}\_[(0)]{}\^[ki]{}d\^qx-P\_I, \[Omevolution\] where $\s_{(0)}$ and $P_I$ are the non-relativistic shear tensor and the palinstrophy, defined by \_[(0)ij]{}=\_iu\_[(0)j]{}+\_ju\_[(0)i]{},P\_I=\_k\_[(0)ij]{}\^k\_[(0)]{}\^[ij]{}d\^qx As $\Om_I \geq 0$, the energy $E_I$ is always decreasing. For the enstrophy equation (\[Omevolution\]), the two terms on the right-hand side correspond to the the generation of the enstrophy via the vortex line stretching and the destruction of the enstrophy by the viscous forces. In the large Reynolds number limit, there is an approximate balance between the production and the viscous dissipation of the enstrophy. Thus, to keep $\Om_I$ a constant, the energy spectrum has to be distributed in such a way that the energy will flow from the lower momentum modes to the higher momentum modes. Under Kolmogorov’s Similarity Hypothesis, the energy spectrum behaves as E(k)\~\^[2/3]{} k\^[-5/3]{}, which is known as the Kolmogorov’s five-thirds law, where $\varepsilon$ denotes the rate of energy dissipation at small scales. The turbulence in two spatial dimensions is a little bit special. In this case, the vortex stretching term vanishes for an incompressible flow, and the enstrophy declines monotonically as the flows evolves so that it is bounded from above by its initial value. In the large Reynolds number limit, the energy is conserved. It turns out that there could be another kind of energy cascade, the so-called inverse energy cascade, in which the energy flow to the lower momentum modes. Besides, before the turbulence adjusts to its fully developed state, there is a direct cascade of the enstrophy from the large scales down to the small scales due to the continual filamentation of the vorticity. In the inertial range, according to Batchelor [@Batchelor1969] , the energy spectrum for the direct cascade could be \[directcascadelaw\] E(k)\~\^[2/3]{}k\^[-3]{}, where $\delta$ is the rate of dissipation of the enstrophy. This is the two-dimensional analogue of the Kolmogorov’s five-third law. However, unlike Kolmogorov’s five-thirds law, the direct cascade law (\[directcascadelaw\]) is far from being well accepted. In fact, the numerical simulations suggest that $E\sim k^{-n}$, where $n$ is typically a little larger than 3. For more details, one can see [@Davidson2015] and the references therein. On the other hand, according to Kraichnan [@Kraichnan1967], the possibility of an inverse cascade depends on the relative strengths of the non-linear interactions between scales. If the input of the energy is not sustained for sufficiently long time, the inverse cascade cannot develop and there appears only the direct cascade in the flow. Thus, unlike the case of the forced two-dimensional flows, the inverse cascade with a Kolomogrov-like energy scaling $\sim k^{-5/3}$ is puzzling in the decaying case [@Mininni2013].
It is illuminating to study the equations (\[Eq:dimenless1\]) and (\[Eq:dimenless2\]) beyond the small Mach number limit. We may define the energy and enstrophy via E\_C&=&|u|\^2d\^qx\
\_C&=& d\^qx. Here and in the following we need the vorticity and the shear tensor \_[ij]{}=\_iu\_j-\_ju\_i,\_[ij]{}=\_iu\_j+\_ju\_i. From the equations (\[Eq:dimenless1\]) and (\[Eq:dimenless2\]), we have \_E\_C&=&(u)d\^qx-((\_[ij]{}\^[ij]{}+\_[ij]{}\^[ij]{})+)\
\_\_C&=&\_[ij]{}\^[jk]{}(\^i\_k-\^i\_ku)d\^qx+()\[OmC\] There are two remarkable points on these two equations. The first is that the effective Reynolds number ${\mathbf {R_e}}$ and Mach number ${\mathbf { M}}$ appear in the equations. This suggest that the effect of the GB term could be largely encoded by these two parameters. The second is that the first term on the right hand side in the equation (\[OmC\]) could be taken as a vortex stretching term, by which the energy is passed down the cascade to the small scales. The other terms are proportional to $1/{\mathbf {R_e}}$, and play the similar role as the palinstrophy that characterizes the viscous dissipation at small scales.
If the Mach number is not small, the equations (\[Eq:dimenless1\]) and (\[Eq:dimenless2\]) are more complicated than the ones appearing in the Einstein gravity. Actually there are two more terms proportional to $(\beta-1)$ in (\[Eq:dimenless2\]), whose physical implication is not clear. However, both terms are inversely proportional to $1/{\mathbf {R_e}}$, and consequently when the Reynolds number is very large, they may not play significant role on the dynamics of the flow except the effect on the viscous dissipation. In three spatial dimensions, if the Reynolds number ${\mathbf {R_e}}$ is large, the Kolmogorov cascade should appear in the turbulence as the vortex stretching term dominates. In two spatial dimensions, the vortex stretching term is vanishing. Therefore one may expect the inverse energy cascade in this case. However, due to the presence of two extra terms induced by the GB term, it is not clear if the inverse energy cascade do appear or present different feature. In order to understand the effects of the GB term, we need to do numerical simulation.
Numerical Study of Turbulent Flows {#section:numericalstudy}
==================================
In this section by numerically solving the equations of motion (\[effeq1\]) and (\[effeq2\]) we will give a detailed analysis for the turbulent flows in the EGB theory. In particular, we focus on the distinctions between the Einstein gravity and the EGB theory.
As in [@Rozali1707], we consider the flows in a toroidal domain of size $L$. To solve the equations of motion (\[effeq1\]) and (\[effeq2\]) we use the Fourier spectral method in the spatial directions and the fourth order Runge-Kutta method for time evolution. The initial conditions are taken to be shear flows with small random perturbations, since in the inviscid case, i.e. $\eta\to0$, they are stationary solutions of the fluid equations. Explicitly, the initial conditions are p\_x=p\_x(),p\_y=(),m=m\_0=, for $q=2$, and p\_x&=&()+p\_x(),p\_y=()+p\_y(),\
p\_z&=&()+p\_z(),m=m\_0=, for $q=3$, where $\delta p_i$ denote the perturbations p\_i=\_ A\_[i,]{}(\_[i,]{}+ ). In the above expressions, $ A_{i,\vec{m}}$ denotes the amplitude of the perturbation and is chosen from a uniform distribution ranging from $0$ to a small value. The phase $\delta\phi_{i,\vec{m}}$ is chosen from a uniform distribution ranging from $0$ to $2\pi$. $\vec{m}$ is the wavenumber in unit of $2\pi/L$. For more details about the initial conditions, one can see [@Rozali1707; @Green1309; @Carrasco1210].
Due to the decay caused by the shear viscosity, the shear flow we study is not steady. The Reynolds number, as a useful dimensionless quantity to predict the stability of the steady flow, may still be useful in our case. Thus the stability of the flow depends only on the instantaneous value of the Reynolds number [@Green1309].
For our flow, we follow the definition in [@Rozali1707] where the characteristic velocity appearing in (\[ReandM\]) is given by[^6] \[characteristicvelocity\] U=(f)-f. $U$ characterizes the velocity fluctuation. For the initial conditions of the flow, it is easy to obtain the Reynolds number and the Mach number \[initialRe\] \_0=,\_0=, where we have used $L_0=\frac{L}{n_I}$. From the above expressions we can see the effect of the GB term on the initial conditions. On the one hand, the presence of the GB term always lowers the initial Reynolds number. On the other hand, for a positive GB coefficient $\tilde{\alpha}$ (i.e. $0\leq\beta<1$), the presence of the GB term always lowers the Mach number, but for a negative $\tilde{\alpha}$ (i.e. $1<\beta\leq2$) it would always lead to a larger Mach number. Similar to the steady flow case, we expect that if the initial Reynolds number is sufficiently small the flow is stable against small perturbations, and the final state is laminar. While for a sufficiently large initial Reynolds number, the flow is unstable and we should expect the emergence of the turbulent flow. Between these two extremes, for an intermediate initial Reynolds number, there exists a critical Reynolds number [@Green1309] beyond which the transition from the laminar flow to the turbulent flow may occur, as we will see in the following.
As a preview, we show the typical simulations of two dimensional fluid flow in Figs. \[fig:2Dvorticitybeta=1\], Fig. \[fig:2Dvorticitybeta=05\] and Fig. \[fig:2Dvorticitybeta=2\] and the simulations of the three dimensional fluid flow in Fig. \[fig:vorticity3D\] with large Reynolds numbers for different values of $\beta$. Qualitatively, the behavior of the fluid flows in the EGB theory is very similar to the one in the Einstein gravity. It can be described by three phases: the initial growth of the instabilities, the turbulent regime in which the inverse/direct energy cascade is observed, and the late time decay into an equilibrium. An obvious distinction between these two theories is that, for a positive GB coefficient the time for the turbulence to emerge is later, but for a negative GB coefficient the time is earlier, under the same initial conditions ($L$, $m_0$ and $n_I$). This is largely because the presence of the positive GB coefficient lowers both the initial values of the Reynolds number and the Mach number as can be see from (\[initialRe\]), thus it costs more time for the flow to reach the turbulent phase. In contrast, the presence of the negative GB coefficient lowers the initial value of the Reynolds number but increase the the initial value of the Mach number. As a consequence, it costs less time for the flow to reach the turbulence phase.
On the other hand, if we keep the initial Reynolds number and Mach number fixed, then the behavior of the fluid for various $\beta$’s should not be the same, because their dynamical equations are different. As shown in Fig. \[fig:2DRe500\], we can see that in comparison with the case of the Einstein gravity, a positive GB coefficient has a larger evolution rate while a negative GB coefficient has a smaller evolution rate. This may be ascribed to the distinction of the dynamical equation. After we adopt the definition (\[ReandM\]), the dynamical equation has two extra terms proportional to the inverse of the Reynolds number. These two terms induce viscous forces and affect the evolution of the fluid flow. As we will see in the next subsection, from the linear analysis, the positive GB coefficient has the smaller viscous effect and the larger evolution rate, while the negative GB coefficient has the larger viscous effect and the smaller evolution rate. In the following we will pay our attention to the effects of the GB term on the fluid flows in these different stages.
![The vorticity field $\omega = \partial_x p_y - \partial_y p_x$ for a two dimensional fluid flow in the Einstein gravity ($\beta=1$), with initial Reynolds number ${\mathbf {R_e}}_0 = 937.5$, $m_0=2$, initial mode $n_I=8$, and $A=10^{-5}$ at various times. We can see the initial growth of the instability, and later on the formation of filaments which indicates the direct cascade, and then the turbulent regime at which a large scale structure is formed and the inverse energy cascade is apparent. Finally, we can see the slow decay of two counter-rotating vortices. Note that here and in the following we employ $\tau$ as the units of time with $\tau=L_0 m_0$.[]{data-label="fig:2Dvorticitybeta=1"}](2Dvorticity1beta=1.png "fig:")![The vorticity field $\omega = \partial_x p_y - \partial_y p_x$ for a two dimensional fluid flow in the Einstein gravity ($\beta=1$), with initial Reynolds number ${\mathbf {R_e}}_0 = 937.5$, $m_0=2$, initial mode $n_I=8$, and $A=10^{-5}$ at various times. We can see the initial growth of the instability, and later on the formation of filaments which indicates the direct cascade, and then the turbulent regime at which a large scale structure is formed and the inverse energy cascade is apparent. Finally, we can see the slow decay of two counter-rotating vortices. Note that here and in the following we employ $\tau$ as the units of time with $\tau=L_0 m_0$.[]{data-label="fig:2Dvorticitybeta=1"}](2Dvorticity2beta=1.png "fig:")![The vorticity field $\omega = \partial_x p_y - \partial_y p_x$ for a two dimensional fluid flow in the Einstein gravity ($\beta=1$), with initial Reynolds number ${\mathbf {R_e}}_0 = 937.5$, $m_0=2$, initial mode $n_I=8$, and $A=10^{-5}$ at various times. We can see the initial growth of the instability, and later on the formation of filaments which indicates the direct cascade, and then the turbulent regime at which a large scale structure is formed and the inverse energy cascade is apparent. Finally, we can see the slow decay of two counter-rotating vortices. Note that here and in the following we employ $\tau$ as the units of time with $\tau=L_0 m_0$.[]{data-label="fig:2Dvorticitybeta=1"}](2Dvorticity3beta=1.png "fig:") ![The vorticity field $\omega = \partial_x p_y - \partial_y p_x$ for a two dimensional fluid flow in the Einstein gravity ($\beta=1$), with initial Reynolds number ${\mathbf {R_e}}_0 = 937.5$, $m_0=2$, initial mode $n_I=8$, and $A=10^{-5}$ at various times. We can see the initial growth of the instability, and later on the formation of filaments which indicates the direct cascade, and then the turbulent regime at which a large scale structure is formed and the inverse energy cascade is apparent. Finally, we can see the slow decay of two counter-rotating vortices. Note that here and in the following we employ $\tau$ as the units of time with $\tau=L_0 m_0$.[]{data-label="fig:2Dvorticitybeta=1"}](2Dvorticity4beta=1.png "fig:") ![The vorticity field $\omega = \partial_x p_y - \partial_y p_x$ for a two dimensional fluid flow in the Einstein gravity ($\beta=1$), with initial Reynolds number ${\mathbf {R_e}}_0 = 937.5$, $m_0=2$, initial mode $n_I=8$, and $A=10^{-5}$ at various times. We can see the initial growth of the instability, and later on the formation of filaments which indicates the direct cascade, and then the turbulent regime at which a large scale structure is formed and the inverse energy cascade is apparent. Finally, we can see the slow decay of two counter-rotating vortices. Note that here and in the following we employ $\tau$ as the units of time with $\tau=L_0 m_0$.[]{data-label="fig:2Dvorticitybeta=1"}](2Dvorticity5beta=1.png "fig:")![The vorticity field $\omega = \partial_x p_y - \partial_y p_x$ for a two dimensional fluid flow in the Einstein gravity ($\beta=1$), with initial Reynolds number ${\mathbf {R_e}}_0 = 937.5$, $m_0=2$, initial mode $n_I=8$, and $A=10^{-5}$ at various times. We can see the initial growth of the instability, and later on the formation of filaments which indicates the direct cascade, and then the turbulent regime at which a large scale structure is formed and the inverse energy cascade is apparent. Finally, we can see the slow decay of two counter-rotating vortices. Note that here and in the following we employ $\tau$ as the units of time with $\tau=L_0 m_0$.[]{data-label="fig:2Dvorticitybeta=1"}](2Dvorticity7beta=1.png "fig:") ![The vorticity field $\omega = \partial_x p_y - \partial_y p_x$ for a two dimensional fluid flow in the Einstein gravity ($\beta=1$), with initial Reynolds number ${\mathbf {R_e}}_0 = 937.5$, $m_0=2$, initial mode $n_I=8$, and $A=10^{-5}$ at various times. We can see the initial growth of the instability, and later on the formation of filaments which indicates the direct cascade, and then the turbulent regime at which a large scale structure is formed and the inverse energy cascade is apparent. Finally, we can see the slow decay of two counter-rotating vortices. Note that here and in the following we employ $\tau$ as the units of time with $\tau=L_0 m_0$.[]{data-label="fig:2Dvorticitybeta=1"}](2Dvorticity8beta=1.png "fig:")![The vorticity field $\omega = \partial_x p_y - \partial_y p_x$ for a two dimensional fluid flow in the Einstein gravity ($\beta=1$), with initial Reynolds number ${\mathbf {R_e}}_0 = 937.5$, $m_0=2$, initial mode $n_I=8$, and $A=10^{-5}$ at various times. We can see the initial growth of the instability, and later on the formation of filaments which indicates the direct cascade, and then the turbulent regime at which a large scale structure is formed and the inverse energy cascade is apparent. Finally, we can see the slow decay of two counter-rotating vortices. Note that here and in the following we employ $\tau$ as the units of time with $\tau=L_0 m_0$.[]{data-label="fig:2Dvorticitybeta=1"}](2Dvorticity9beta=1.png "fig:")
![The vorticity field of a two dimensional fluid flow in the EGB theory with $\beta=0.5$ and the other parameters are the same as those in Fig. \[fig:2Dvorticitybeta=1\] at various times.[]{data-label="fig:2Dvorticitybeta=05"}](2Dvorticity1beta=05.png "fig:")![The vorticity field of a two dimensional fluid flow in the EGB theory with $\beta=0.5$ and the other parameters are the same as those in Fig. \[fig:2Dvorticitybeta=1\] at various times.[]{data-label="fig:2Dvorticitybeta=05"}](2Dvorticity2beta=05.png "fig:")![The vorticity field of a two dimensional fluid flow in the EGB theory with $\beta=0.5$ and the other parameters are the same as those in Fig. \[fig:2Dvorticitybeta=1\] at various times.[]{data-label="fig:2Dvorticitybeta=05"}](2Dvorticity3beta=05.png "fig:") ![The vorticity field of a two dimensional fluid flow in the EGB theory with $\beta=0.5$ and the other parameters are the same as those in Fig. \[fig:2Dvorticitybeta=1\] at various times.[]{data-label="fig:2Dvorticitybeta=05"}](2Dvorticity4beta=05.png "fig:") ![The vorticity field of a two dimensional fluid flow in the EGB theory with $\beta=0.5$ and the other parameters are the same as those in Fig. \[fig:2Dvorticitybeta=1\] at various times.[]{data-label="fig:2Dvorticitybeta=05"}](2Dvorticity5beta=05.png "fig:") ![The vorticity field of a two dimensional fluid flow in the EGB theory with $\beta=0.5$ and the other parameters are the same as those in Fig. \[fig:2Dvorticitybeta=1\] at various times.[]{data-label="fig:2Dvorticitybeta=05"}](2Dvorticity7beta=05.png "fig:")![The vorticity field of a two dimensional fluid flow in the EGB theory with $\beta=0.5$ and the other parameters are the same as those in Fig. \[fig:2Dvorticitybeta=1\] at various times.[]{data-label="fig:2Dvorticitybeta=05"}](2Dvorticity8beta=05.png "fig:")![The vorticity field of a two dimensional fluid flow in the EGB theory with $\beta=0.5$ and the other parameters are the same as those in Fig. \[fig:2Dvorticitybeta=1\] at various times.[]{data-label="fig:2Dvorticitybeta=05"}](2Dvorticity9beta=05.png "fig:")
![The vorticity field of a two dimensional fluid flow in the EGB theory with $\beta=2$ and the other parameters are the same as those in Fig. \[fig:2Dvorticitybeta=1\] at various times.[]{data-label="fig:2Dvorticitybeta=2"}](2Dvorticitybeta=2f1.pdf "fig:")![The vorticity field of a two dimensional fluid flow in the EGB theory with $\beta=2$ and the other parameters are the same as those in Fig. \[fig:2Dvorticitybeta=1\] at various times.[]{data-label="fig:2Dvorticitybeta=2"}](2Dvorticitybeta=2f2.pdf "fig:")![The vorticity field of a two dimensional fluid flow in the EGB theory with $\beta=2$ and the other parameters are the same as those in Fig. \[fig:2Dvorticitybeta=1\] at various times.[]{data-label="fig:2Dvorticitybeta=2"}](2Dvorticitybeta=2f3.pdf "fig:") ![The vorticity field of a two dimensional fluid flow in the EGB theory with $\beta=2$ and the other parameters are the same as those in Fig. \[fig:2Dvorticitybeta=1\] at various times.[]{data-label="fig:2Dvorticitybeta=2"}](2Dvorticitybeta=2f4.pdf "fig:") ![The vorticity field of a two dimensional fluid flow in the EGB theory with $\beta=2$ and the other parameters are the same as those in Fig. \[fig:2Dvorticitybeta=1\] at various times.[]{data-label="fig:2Dvorticitybeta=2"}](2Dvorticitybeta=2f5.pdf "fig:") ![The vorticity field of a two dimensional fluid flow in the EGB theory with $\beta=2$ and the other parameters are the same as those in Fig. \[fig:2Dvorticitybeta=1\] at various times.[]{data-label="fig:2Dvorticitybeta=2"}](2Dvorticitybeta=2f7.pdf "fig:")![The vorticity field of a two dimensional fluid flow in the EGB theory with $\beta=2$ and the other parameters are the same as those in Fig. \[fig:2Dvorticitybeta=1\] at various times.[]{data-label="fig:2Dvorticitybeta=2"}](2Dvorticitybeta=2f8.pdf "fig:")![The vorticity field of a two dimensional fluid flow in the EGB theory with $\beta=2$ and the other parameters are the same as those in Fig. \[fig:2Dvorticitybeta=1\] at various times.[]{data-label="fig:2Dvorticitybeta=2"}](2Dvorticitybeta=2f9.pdf "fig:")
-- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![The vorticity field $\vec{\omega} = \nabla\times\vec{p}$ of three dimensional fluid flow with initial parameters $L=250$, $m_0=1$, $n_I=1$ and $A=10^{-2}$ at different times. For simplicity here we only show $\omega_z$, the other components are similar. The upper row is for the case of the Einstein gravity (with $\beta=1$) and the lower row is for the case of the EGB theory (with $\beta=0.5$). Different from the two dimensional flow, at the turbulent regime the formation of small structure is observed which indicates the direct energy cascade.[]{data-label="fig:vorticity3D"}](3Dvorticity1beta=1.png "fig:") ![The vorticity field $\vec{\omega} = \nabla\times\vec{p}$ of three dimensional fluid flow with initial parameters $L=250$, $m_0=1$, $n_I=1$ and $A=10^{-2}$ at different times. For simplicity here we only show $\omega_z$, the other components are similar. The upper row is for the case of the Einstein gravity (with $\beta=1$) and the lower row is for the case of the EGB theory (with $\beta=0.5$). Different from the two dimensional flow, at the turbulent regime the formation of small structure is observed which indicates the direct energy cascade.[]{data-label="fig:vorticity3D"}](3Dvorticity2beta=1.png "fig:") ![The vorticity field $\vec{\omega} = \nabla\times\vec{p}$ of three dimensional fluid flow with initial parameters $L=250$, $m_0=1$, $n_I=1$ and $A=10^{-2}$ at different times. For simplicity here we only show $\omega_z$, the other components are similar. The upper row is for the case of the Einstein gravity (with $\beta=1$) and the lower row is for the case of the EGB theory (with $\beta=0.5$). Different from the two dimensional flow, at the turbulent regime the formation of small structure is observed which indicates the direct energy cascade.[]{data-label="fig:vorticity3D"}](3Dvorticity3beta=1.png "fig:")
![The vorticity field $\vec{\omega} = \nabla\times\vec{p}$ of three dimensional fluid flow with initial parameters $L=250$, $m_0=1$, $n_I=1$ and $A=10^{-2}$ at different times. For simplicity here we only show $\omega_z$, the other components are similar. The upper row is for the case of the Einstein gravity (with $\beta=1$) and the lower row is for the case of the EGB theory (with $\beta=0.5$). Different from the two dimensional flow, at the turbulent regime the formation of small structure is observed which indicates the direct energy cascade.[]{data-label="fig:vorticity3D"}](3Dvorticity1beta=05.png "fig:") ![The vorticity field $\vec{\omega} = \nabla\times\vec{p}$ of three dimensional fluid flow with initial parameters $L=250$, $m_0=1$, $n_I=1$ and $A=10^{-2}$ at different times. For simplicity here we only show $\omega_z$, the other components are similar. The upper row is for the case of the Einstein gravity (with $\beta=1$) and the lower row is for the case of the EGB theory (with $\beta=0.5$). Different from the two dimensional flow, at the turbulent regime the formation of small structure is observed which indicates the direct energy cascade.[]{data-label="fig:vorticity3D"}](3Dvorticity2beta=05.png "fig:") ![The vorticity field $\vec{\omega} = \nabla\times\vec{p}$ of three dimensional fluid flow with initial parameters $L=250$, $m_0=1$, $n_I=1$ and $A=10^{-2}$ at different times. For simplicity here we only show $\omega_z$, the other components are similar. The upper row is for the case of the Einstein gravity (with $\beta=1$) and the lower row is for the case of the EGB theory (with $\beta=0.5$). Different from the two dimensional flow, at the turbulent regime the formation of small structure is observed which indicates the direct energy cascade.[]{data-label="fig:vorticity3D"}](3Dvorticity3beta=05.png "fig:")
-- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![The vorticity field of two dimensional fluid flow in the EGB theory with various values of $\beta$. Here by adjusting $L$ and $m_0$ we keep the initial Reynolds number and Mach number invariant with $\beta$.[]{data-label="fig:2DRe500"}](2DRe500beta1.png "fig:") ![The vorticity field of two dimensional fluid flow in the EGB theory with various values of $\beta$. Here by adjusting $L$ and $m_0$ we keep the initial Reynolds number and Mach number invariant with $\beta$.[]{data-label="fig:2DRe500"}](2DRe500beta05.png "fig:") ![The vorticity field of two dimensional fluid flow in the EGB theory with various values of $\beta$. Here by adjusting $L$ and $m_0$ we keep the initial Reynolds number and Mach number invariant with $\beta$.[]{data-label="fig:2DRe500"}](2DRe500beta2.png "fig:")
Stability of shear flow
-----------------------
In this subsection we analyze the early stage of the flow. Due to the restriction of our computational resource, in the following we mainly show our results of the two dimensional flows. As we mentioned before, depending on the initial Reynolds number, the small random perturbations of the shear flow grow or decay with time. The growth of the perturbations can be quantified by the growth of $p_x$ field [@Green1309] or as in [@Rozali1707] by the energy spectrum $E_C$ whose definition is [^7] \[energyspectrum\] E\_C(v,k) = \_[|’|k]{}|\_q(v,’)|\^[2]{}, with \_q(v,) = d\^[q]{}x(v,)e\^[-i]{}. In what follows we will take either one of the two quantities when necessary.
At low values of ${\mathbf {R_e}}_0$ the flow is laminar, as shown in Fig. \[fig:2DinitialstageRe10\]. We can see that the $L_2$ norms of the components of $p_i$, L\_2(p\_i)=, decay exponentially both for the Einstein gravity and the EGB theory. From the linear analysis in section \[subsection:linearanalysis\], we know that the presence of the positive GB coefficient lowers the quasinormal mode frequency (\[QNMfrequency\]) and so the viscosity (\[viscosity\]), thus the $L_2$ norms of $p_i$ decay slower than the ones in the Einstein gravity. In contrast, the presence of the negative GB coefficient increases the decay rate so that we have steeper lines in the right panel of Fig. \[fig:2DinitialstageRe10\]. Moreover, we find that for larger Mach numbers the results are essentially the same as the ones of small Mach number case.
![$L_2$ norms of $p_i$ as a function of time for a laminar flow, with $L=50$, $n_I=5$ and $m_0=2$. []{data-label="fig:2DinitialstageRe10"}](2DinitialstageRe10beta1.pdf "fig:") ![$L_2$ norms of $p_i$ as a function of time for a laminar flow, with $L=50$, $n_I=5$ and $m_0=2$. []{data-label="fig:2DinitialstageRe10"}](2DinitialstageRe10beta05.pdf "fig:") ![$L_2$ norms of $p_i$ as a function of time for a laminar flow, with $L=50$, $n_I=5$ and $m_0=2$. []{data-label="fig:2DinitialstageRe10"}](2DinitialstageRe10beta2.pdf "fig:")
![The energy spectrum $E_C$ for two dimensional fluid flow, with $L=15000$, $n_I=8$ and $m_0=2$. The left panel corresponds to $\beta=1$ and the right panel corresponds to $\beta=0.5$.[]{data-label="fig:2DinitialstageRe937"}](2DinitialstageRe=937beta=1.png "fig:") ![The energy spectrum $E_C$ for two dimensional fluid flow, with $L=15000$, $n_I=8$ and $m_0=2$. The left panel corresponds to $\beta=1$ and the right panel corresponds to $\beta=0.5$.[]{data-label="fig:2DinitialstageRe937"}](2DinitialstageRe=937beta=05.png "fig:")
![Intermediate flow: $L_2(p_x)$ and the characteristic velocity $U$ as the functions of time for a laminar flow, with $L=2000$, $n_I=5$ and $m_0=2$. For the positive GB coefficient ($\beta=0.5$), the decay rate of $\log U$ is smaller and the maximum of $\log L_2(p_x)$ appears at later time. In contrast, for the negative GB coefficient ($\beta=2$), the decay rate is larger and it takes less time for $\log L_2(p_x)$ to reach the peak. []{data-label="fig:2DcriticalRe"}](2DcriticalRebeta=1.pdf "fig:") ![Intermediate flow: $L_2(p_x)$ and the characteristic velocity $U$ as the functions of time for a laminar flow, with $L=2000$, $n_I=5$ and $m_0=2$. For the positive GB coefficient ($\beta=0.5$), the decay rate of $\log U$ is smaller and the maximum of $\log L_2(p_x)$ appears at later time. In contrast, for the negative GB coefficient ($\beta=2$), the decay rate is larger and it takes less time for $\log L_2(p_x)$ to reach the peak. []{data-label="fig:2DcriticalRe"}](2DcriticalRebeta=05.pdf "fig:") ![Intermediate flow: $L_2(p_x)$ and the characteristic velocity $U$ as the functions of time for a laminar flow, with $L=2000$, $n_I=5$ and $m_0=2$. For the positive GB coefficient ($\beta=0.5$), the decay rate of $\log U$ is smaller and the maximum of $\log L_2(p_x)$ appears at later time. In contrast, for the negative GB coefficient ($\beta=2$), the decay rate is larger and it takes less time for $\log L_2(p_x)$ to reach the peak. []{data-label="fig:2DcriticalRe"}](2DcriticalRebeta=2.pdf "fig:")
At a large enough ${\mathbf {R_e}}_0$ we observe the turbulent instabilities, as shown in Fig. \[fig:2DinitialstageRe937\]. At the first glance the energy spectrums in the Einstein gravity and the EGB theory are very similar. In both cases we find the decay of the initial disturbance at $k=8\times\frac{2\pi}{L}$ and then an unstable mode emerging at a lower wavenumber $k\sim \frac{3}{2}\times\frac{2\pi}{L}$, which may indicate the appearance of the inverse cascade. However, the direction cascade is not apparent, this may attribute the insufficient precision of our numerical simulation. As we mentioned before since the flow in the EGB theory has a lower Reynolds number, the initial disturbance decays slower and the new unstable mode grows slower, comparing with the Einstein gravity.
For an intermediate value of the initial Reynolds number, as shown in Fig. \[fig:2DcriticalRe\], the initial small perturbations characterized by $L_2(p_x)$ grow exponentially to reach a maximum that is smaller than $L_2(p_y)$, and then decay exponentially. From Fig. \[fig:2DcriticalRe\] we can find the exponential decay of the characteristic velocity $U$ defined in (\[characteristicvelocity\]). From the definition (\[ReandM\]) the Reynolds number varies with time. In this case we can define a critical Reynolds number $R_c$ as in [@Green1309]. Initially, we have ${\mathbf {R_e}}_0>R_c$, thus the unstable mode $L_2(p_x)$ grows exponentially. As time progresses, ${\mathbf {R_e}}$ decreases, then when ${\mathbf {R_e}}< R_c$, $L_2(p_x)$ is not growing any more, instead it decays exponentially. Therefore, at the peak of $L_2(p_x)$ we may identify ${\mathbf {R_e}}=R_c$.
${\mathbf {R_e}}_0$ $150$ $160$ $170$ $180$
--------------------- --------- --------- --------- ---------
$R_c$ $17.45$ $17.46$ $18.82$ $20.68$
: The critical Reynolds numbers of the fluid flow in the Einstein gravity with the initial Mach number ${\mathbf { M}}_0=0.5$.[]{data-label="Rc:fixedMach"}
${\mathbf { M}}_0$ $0.25$ $0.5$ $1$ $2$
-------------------- --------- --------- --------- ---------
$R_c$ $18.11$ $17.46$ $16.30$ $15.16$
: The critical Reynolds numbers of the fluid flow in the Einstein gravity with the initial Reynolds number ${\mathbf {R_e}}_0=160$.[]{data-label="Rc:fixedReynolds"}
${\mathbf { M}}_0=1$
$\beta$ $1/3$ $0.5$ $1$ $2$
--------- --------- --------- --------- ---------
$R_c$ $16.42$ $16.24$ $16.30$ $16.29$
: The critical Reynolds numbers of the fluid flow in the EGB gravity with different GB coefficients, here the initial Reynolds number is fixed ${\mathbf {R_e}}_0=160$.[]{data-label="Rc:differentGBcoe"}
\
${\mathbf { M}}_0=2$
$\beta$ $1/3$ $0.5$ $1$ $2$
--------- --------- --------- --------- ---------
$R_c$ $15.00$ $15.05$ $15.16$ $15.13$
: The critical Reynolds numbers of the fluid flow in the EGB gravity with different GB coefficients, here the initial Reynolds number is fixed ${\mathbf {R_e}}_0=160$.[]{data-label="Rc:differentGBcoe"}
We have identified the critical Reynolds number ${\mathbf {R_e}}_c$ for different initial Reynolds numbers, different initial Mach numbers and different $\beta$’s. Firstly, for the Einstein gravity $\beta=1$, we find that for a given Mach number the larger is initial Reynolds number, the larger is the critical Reynolds number, as shown in Table \[Rc:fixedMach\]. Secondly, for a given initial Reynolds number, we find that the larger is the Mach number, the smaller is the critical Reynolds number, as shown in Table \[Rc:fixedReynolds\]. Finally, we can see from Table \[Rc:differentGBcoe\] that for the fixed initial Reynolds number and Mach number, the critical Reynolds numbers of different GB coefficients are pretty close.
Turbulent regime
----------------
![The $L_2$ norm of $p_i$ as a function of time for the turbulent flow, with $L=15000$, $n_I=8$ and $m_0=2$. We can see that $p_x$ grows exponentially fast until it is of similar amplitude to $p_y$.[]{data-label="fig:2DpxpyRe937"}](2DpxpyRe937beta1.pdf "fig:") $\qquad$ ![The $L_2$ norm of $p_i$ as a function of time for the turbulent flow, with $L=15000$, $n_I=8$ and $m_0=2$. We can see that $p_x$ grows exponentially fast until it is of similar amplitude to $p_y$.[]{data-label="fig:2DpxpyRe937"}](2DpxpyRe937beta05.pdf "fig:")
![The log-log plots of the energy spectrum for two dimensional flow at various times, with $L=15000$, $n_I=8$ and $m_0=2$. There appears an power law $E_C\sim k^{-4}$ in the inertial range. []{data-label="fig:2Dpowerlawbeta"}](2Dpowerlawbeta1.pdf "fig:") ![The log-log plots of the energy spectrum for two dimensional flow at various times, with $L=15000$, $n_I=8$ and $m_0=2$. There appears an power law $E_C\sim k^{-4}$ in the inertial range. []{data-label="fig:2Dpowerlawbeta"}](2Dpowerlawbeta05.pdf "fig:")
If the initial Reynolds number is sufficiently large, the initial instability grows until it reach the same amplitude as that of the initial shear mode, then the fluid goes into the turbulent regime, as shown in Fig. \[fig:2DpxpyRe937\]. In this stage, the small eddies merge into vortices, which continue to merge into increasingly large vortices until two counter-rotating vortices are formed, as shown in Figs. \[fig:2Dvorticitybeta=1\] and \[fig:2Dvorticitybeta=05\]. This process is the so-call inverse energy cascade. Besides, there is a direct cascade caused by the formation of the filaments of the vorticity. These two cascades can be characterized by the shape of the energy spectrum $E_C$ in the inertial range. The detailed analysis of the energy spectrum during the turbulent phase can be found in [@Rozali1707]. Here we only want to emphasize that for the two dimensional fluid flow in the EGB theory the energy spectrum of the direct cascade satisfies a $k^{-4}$ power law for a small Mach number and a large Reynolds number, as shown in Fig. \[fig:2Dpowerlawbeta\]. Similar to the Einstein gravity, the $k^{-5/3}$ law for the inverse cascade is absent from the energy spectrum analysis in the case of the EGB theory. From the analysis in [@Mininni2013], it was found that the range at which the $k^{-5/3}$ law applies is very short and the distinction from the $k^{-3}$ law demands high numerical precision. For three dimensional fluid flow, the celebrated $k^{-5/3}$ power law has been found in [@Rozali1707], even the Mach number is large. Due to the limited computational power, we are not able to carry out the numerical analysis on three dimensional fluid flow. Nevertheless, we expect the same energy cascade happens in the EGB turbulence as well.
Late time decay
---------------
![The energy density and the velocity field at the final stage of the two dimensional turbulent flow. []{data-label="fig:2Dlatetimedecaykspace"}](2Dlatetimedecaym.png "fig:")![The energy density and the velocity field at the final stage of the two dimensional turbulent flow. []{data-label="fig:2Dlatetimedecaykspace"}](2Dlatetimedecaypx.png "fig:")![The energy density and the velocity field at the final stage of the two dimensional turbulent flow. []{data-label="fig:2Dlatetimedecaykspace"}](2Dlatetimedecaypy.png "fig:")
The final phase of the turbulent flow is the formation of two counter-rotating vortices, as shown in Figs. \[fig:2Dvorticitybeta=1\], \[fig:2Dvorticitybeta=05\] and \[fig:2Dvorticitybeta=2\]. We find that the late time decay takes the following form m(x,y)&=&m\_0,\
p\_x(x,y)&=& p\_[x0]{} e\^[- ()\^2v]{} (y+\_1),\
p\_y(x,y)&=& p\_[y0]{} e\^[- ()\^2 v]{} (x+\_1), where $m_0$, $p_{x0}$ and $p_{y0}$ are constant. The above behavior can be understood as follows. Since in this stage the velocity field becomes small, thus the linear analysis for the fluid flows is applicable. The decay rate is just the damping part of the quasinormal mode of the black brane. On the other hand, due to the inverse cascade at late time the lowest mode dominates the flow, as shown in Fig. \[fig:2Dlatetimedecaykspace\].
Geometric Interpretation of Turbulence {#section:geometricinter}
======================================
In this section, we would like to see if the horizon power spectrum defined in [@Adams1307] is applicable to the turbulence of the holographic EGB fluid.
First of all, let us write down the leading order metric of the EGB-AdS black brane \[leadingordermetric\] ds\^2=- dv\^2+2dvdr- dz\^a dv+(\_[ab]{}+)dz\^adz\^b +d\^2, where ${\mathbf { b}}$ and $\beta$ are introduced in (\[bandbeta\]), and $G_0$ is given in (\[G0\]). Then the event horizon is at \[horizonposition\] =m(v,z\^a). The extrinsic curvature on the event horizon is given by \_\^\_ \^\_\_ n\_,\^\_\^\_+\^n\_, where $n_\mu$ is the null normal to the horizon and $\ell_\mu$ ia an auxiliary null vector whose normalization is conveniently chosen to satisfy $\ell_\mu n^\mu=-1$. Next, we define the rescaled traceless horizon curvature \^i\_[ j]{}\^i\_[ j]{}, where $\sqrt{\gamma}$ is the horizon area element, $\Sigma^i_{\; j}\equiv \Theta^i_{\; j}-\frac{1}{D-2}\Theta^l_{\; l} \delta^i_{\; j}$ is the traceless part of the extrinsic curvature and $\kappa$ is defined by the geodesic equation $n^\mu \nabla_\mu n_\nu=\kappa n_\nu$. Then the horizon curvature power spectrum is defined as \[horizoncurpow\] (v,k)\_[||k]{} \^[\*i]{}\_[ j]{}(v,) \^[j]{}\_[ i]{}(v,), with $\tilde{\theta}^i_{\; j}\equiv \int d^q x \,\theta^i_{\; j} e^{-i \vec{k}\cdot \vec{x}}$.
From (\[leadingordermetric\]) and (\[horizonposition\]), we obtain n\_dx\^=dr-( dv+ dz\^a ),\_dx\^=-dt, and =,=, from which we have \^i\_[j]{}=. Let us take the incompressible limit in which $m$ is treated as a constant, then \^i\_[j]{}(\^ip\_j+\_j p\^i), and \^i\_[ j]{}(i k\^i \_j+i k\_j \^i), where $\tilde{p}_i$ is the Fourier transformation of $p_i$. Plugging this into (\[horizoncurpow\]) and comparing with (\[energyspectrum\]) we find ()\^2 k\^2. Thus in the low Mach number case the emergent $k^2$ relation holds for EGB gravity as well, with the coefficient embodying the effect of the GB term. As in the case of the Einstein gravity[@Rozali1707], if the Mach number is not small the simple relation $\mathcal{A}\sim k^2 E_C$ is expected to be broken.
Summary {#section:summary}
=======
In this paper we studied the holographic hydrodynamics of the fluid flow in the Einstein-Gauss-Bonnet gravity by using the large $D$ effective theory. After performing the $1/D$ expansion and integrating out the radial direction of the EGB equations, we obtained the effective equations for the EGB-AdS black branes. We found that the effective equations can be easily turned into the form of the equations for a dynamical fluid, the properties of which are consistent with the results obtained from the other studies in AdS/CFT. In the small Mach number limit, we found that the holographic hydrodynamic fluid equations in the EGB gravity could reduce to the Navier-Stokes equations for the incompressible fluid, with the modified Reynolds number and the modified Mach number \[redfReM\] = L\_0 U,=U. Therefore, as in the case of the Einstein gravity, the Kolmogorov scaling laws are expected to emerge for three dimensional fluid flow, and an inverse/direct cascade should exhibit for two dimensional turbulent fluid flow.
Compared to the hydrodynamical equations derived in the Einstein gravity, there are two extra terms in the equations of motion (\[Eq:dimenless2\]), being proportional to $1/{\mathbf {R_e}}$. In three dimensions, the vortex stretching term has the dominant influence on the kinetic energy transfer between different scales, and it leads to the Kolmogorov scaling. In this case, the two extra terms play minor role. In two dimensions, as the vortex stretching term is vanishing, and the evolution of the enstrophy is determined by the palinstrophy-like terms, which include the contribution from the extra terms. One may expect that the 2D turbulence may present different feature due to the extra terms.
Surprisingly, we found quite similar behaviors for the 2D holographic turbulence in the EGB theory as the one in the Einstein gravity. We generated the turbulent flow in a toroidal domain by numerically solving the equations of motion. The evolution of the flow can be divided into three stages: the initial growth of the instabilities, the turbulent regime and the late time decay. Qualitatively, in all these stages the behavior of the EGB fluid flow is similar to the one in the Einstein gravity: the initial instability grows accompanied with the formation of filaments, then the flow enters the turbulence regime in which the large scale structure is formed, and finally the formed counter-rotating vortices decay slowly. The formation of the large scale structure indicates an inverse cascade and the formation of filaments indicates the direct cascade. We found that the energy spectrum of the direct cascade obeys a $k^{-4}$ power law but the $k^{-5/3}$ law for the inverse cascade is absent. A possible reason for this could be the insufficient numerical precision [@Mininni2013].
Despite the qualitative similarity, the EGB turbulence presents some different features. The effect of the GB term is twofold. On the one hand, the GB term affects the initial Reynolds number and Mach number. If the initial conditions are fixed, for a positive GB coefficient, both the Reynolds number and Mach number are smaller than those of the fluid in the Einstein gravity, such that the EGB fluid needs more time to reach the turbulent phase. However, for a negative GB coefficient, the Reynolds number is smaller but the Mach number is larger, so that the fluid needs less time to reach the turbulent phase. On the other hand, if we keep the initial Reynolds number and Mach number fixed, the extra terms in the dynamical equation affect the dissipation rate and so the evolution rate of the fluid flow. Moreover, we found that the critical Reynolds number is not sensitive to the value of the Gauss-Bonnet coefficient.
Due to the simplification at large $D$, we showed analytically that in the regime of small Mach number the proposed relation between the horizon curvature power spectrum and the hydrodynamic energy power spectrum $\mathcal{A}/E_C\propto k^2 $ [@Adams1307] still holds in the EGB gravity.
In this work, we only did spectrum analysis for the 2D holographic EGB turbulence. It would be interesting to consider the 3D case. Even though a Kolmogorov cascade is expected in 3D case, it would be nice to show it explicitly.
Acknowledgments {#acknowledgments .unnumbered}
===============
BC would like to thank the participants in “Gravity - New perspectives from strings and higher dimensions" at Centro de Ciencias de Benasque Pedro Pascual for stimulated discussions. In particular, he thanks R. Emparan, V. Hubeny, M. Rangamani and A. Yarom for the discussion which initiated this project. We are grateful to M. Rozali, E. Sabat and A. Yarom for the correspondence and the clarification on the numerical simulation. We thank Shan-Quan Lan for the discussion on the energy spectrum analysis. B. Chen and P.-C. Li were supported in part by NSFC Grant No. 11275010, No. 11325522, No. 11335012 and No. 11735001. Y. Tian is partly supported by the National Natural Science Foundation of China (Grant Nos. 11475179 and 11675015) and also supported by the “Strategic Priority Research Program of the Chinese Academy of Sciences”, grant No. XDB23030000. C.-Y. Zhang is supported by National Postdoctoral Program for Innovative Talents BX201600005.
R. Emparan, R. Suzuki, and K. Tanabe, “The large $D$ limit of General Relativity," JHEP [**1306**]{} (2013) 009 \[arXiv:1302.6382\].
R. Emparan, R. Suzuki and K. Tanabe, “Instability of rotating black holes: large $D$ analysis," JHEP [**06**]{} (2014) 106, \[arXiv:1402.6215\].
R. Emparan, R. Suzuki and K. Tanabe, “ Decoupling and non-decoupling dynamics of large $D$ black holes," JHEP **1407**, 113 (2014) \[arXiv:1406.1258\].
R. Emparan, R. Suzuki and K. Tanabe, “Quasinormal modes of (Anti-)de Sitter black holes in the $1/D$ expansion," JHEP [**04**]{} (2015) 085, \[arXiv:1502.02820\].
R. Emparan, T. Shiromizu, R. Suzuki, K. Tanabe, and T. Tanaka, “ Effective theory of Black Holes in the $1/D$ expansion," JHEP **06** (2015) 159, \[arXiv:1504.06489\].
R. Suzuki and K. Tanabe, “Stationary black holes: Large $D$ analysis," JHEP **1509**, 193(2015) \[arXiv:1505.01282\].
R. Suzuki and K. Tanabe, “ Non-uniform black strings and the critical dimension in the $1/D$ expansion," JHEP [**10**]{} (2015) 107, \[arXiv:1506.01890\].
R. Emparan, R. Suzuki and K. Tanabe, “Evolution and End Point of the Black String Instability: Large $D$ Solution," Phys. Rev. Lett. [**115**]{}, no. 9, 091102 (2015) \[arXiv:1506.06772\].
K. Tanabe, “Black rings at large $D$," JHEP [**1602**]{}, 151 (2016) \[arXiv:1510.02200\].
B. Chen, P.-C. Li and Z. z. Wang, “Charged Black Rings at large D," JHEP [**1704**]{} (2017) 167, \[arXiv:1702.00886\].
K. Tanabe, “Instability of de Sitter Reissner-Nordstrom black hole in the $1/D$ expansion," Class. Quant. Grav. **33** no. 12, 125016 (2016) \[arXiv:1511.06059\].
M. Rozali and A. Vincart-Emard, “On Brane Instabilities in the Large $D$ Limit," doi:10.1007/JHEP08(2016)166 \[arXiv:1607.01747\].
U. Miyamoto, “Non-linear perturbation of black branes at large $D$," JHEP [**06**]{} (2017) 033, \[arXiv:1705.00486\].
R. Emparan, R. Luna, M. Martinez, R. Suzuki and K. Tanabe, “Phases and Stability of Non-Uniform Black Strings,” arXiv:1802.08191 \[hep-th\].
S. Bhattacharyya, A. De, S. Minwalla, R. Mohan and A. Saha, “ A membrane paradigm at large $D$," JHEP **1604**, 076 (2016) \[arXiv:1504.06613\].
S. Bhattacharyya, M. Mandlik, S. Minwalla and S. Thakur, “A Charged Membrane Paradigm at Large $D$," JHEP **1604**, 128 (2016) \[arXiv:1511.03432\].
Y. Dandekar, A. De, S. Mazumdar, S. Minwalla and A. Saha, “The large $D$ black hole Membrane Paradigm at first subleading order,” JHEP [**1612**]{}, 113 (2016)\[arXiv:1607.06475\].
Y. Dandekar, S. Mazumdar, S. Minwalla and A. Saha, “Unstable ‘black branes’ from scaled membranes at large $D$,” JHEP [**1612**]{}, 140 (2016) \[arXiv:1609.02912\].
S. Bhattacharyya, P. Biswas, B. Chakrabarty, Y. Dandekar and A. Dinda, “The large $D$ black hole dynamics in AdS/dS backgrounds," \[1704.06076\].
Y. Dandekar, S. Kundu, S. Mazumdar, S. Minwalla, A. Mishra and A. Saha, “An Action for and Hydrodynamics from the improved Large D membrane,” arXiv:1712.09400 \[hep-th\].
T. Damour, “Black Hole Eddy Currents,” Phys. Rev. D [**18**]{}, 3598 (1978).
K. S. Thorne, R. H. Price and D. A. Macdonald, “Black Holes: The Membrane Paradigm,” NEW HAVEN, USA: YALE UNIV. PR. (1986) 367p.
R. Emparan, K. Izumi, R. Luna, R. Suzuki and K. Tanabe, “ Hydro-elastic complementarity in black branes at large $D$," JHEP **06** (2016) 117, \[arXiv:1602.05752\].
S. Bhattacharyya, V. E. Hubeny, S. Minwalla and M. Rangamani, “Nonlinear Fluid Dynamics from Gravity," JHEP [**02**]{} (2008) 045, \[arXiv:0712.2456\].
A. Adams, P. M. Chesler, and H. Liu, “Holographic turbulence," Phys. Rev. Lett. [**112**]{}, 151602 (2014), \[arXiv:1307.7267\].
F. Carrasco, L. Lehner, R. C. Myers, O. Reula, and A. Singh, “Turbulent flows for relativistic conformal fluids in 2+1 dimensions," Phys. Rev. [**D86**]{}, 126006 (2012), \[arXiv:1210.6702\].
S. R. Green, F. Carrasco and L. Lehner, “A Holographic path to the turbulent side of gravity," Phys. Rev. X [**4**]{}, 011001 (2014) \[arXiv:1309.7940\].
M. Rozali, E. Sabat and A. Yarom, “Holographic Turbulence in a Large Number of Dimensions," \[arXiv:1707.08973\].
B. Zwiebach, “Curvature Squared Terms and String Theories," Phys. Lett. B **156** (1985) 315.
D. G. Boulware and S. Deser, “String-Generated Gravity Models," Phys. Rev. Lett. **55** (1985) 2656.
P. Kovtun, D. T. Son and A. O. Starinets, “Viscosity in strongly interacting quantum field theories from black hole physics,” Phys. Rev. Lett. [**94**]{}, 111601 (2005) doi:10.1103/PhysRevLett.94.111601 \[hep-th/0405231\].
M. Brigante, H. Liu, R. C. Myers, S. Shenker and S. Yaida, “Viscosity Bound Violation in Higher Derivative Gravity,” Phys. Rev. D [**77**]{}, 126006 (2008) doi:10.1103/PhysRevD.77.126006 \[arXiv:0712.0805\].
M. Brigante, H. Liu, R. C. Myers, S. Shenker and S. Yaida, “The Viscosity Bound and Causality Violation,” Phys. Rev. Lett. [**100**]{}, 191601 (2008) doi:10.1103/PhysRevLett.100.191601 \[arXiv:0802.3318\].
D. M. Hofman and J. Maldacena, “Conformal collider physics: Energy and charge correlations,” JHEP [**0805**]{}, 012 (2008) doi:10.1088/1126-6708/2008/05/012 \[arXiv:0803.1467\].
A. Buchel and R. C. Myers, “Causality of Holographic Hydrodynamics," JHEP [**08**]{} (2009) 016 \[arXiv: 0906.2922\].
D. M. Hofman, “Higher Derivative Gravity, Causality and Positivity of Energy in a UV complete QFT,” Nucl. Phys. B [**823**]{}, 174 (2009) doi:10.1016/j.nuclphysb.2009.08.001 \[arXiv:0907.1625\].
A. Buchel, J. Escobedo, R. C. Myers, M. F. Paulos, A. Sinha and M. Smolkin, “Holographic GB gravity in arbitrary dimensions," JHEP [**[03]{}**]{} (2010) 111\[arXiv:0911.4257\].
B. Chen, Z.-Y. Fan, P. Li and W. Ye, “Quasinormal modes of Gauss-Bonnet black holes at large $D$," JHEP [**01**]{} (2016) 085 \[arXiv:1511.08706\].
B. Chen and P.-C. Li, “Static Gauss-Bonnet black holes at large $D$," JHEP [**1705**]{} (2017) 025 \[arXiv:1703.06381\].
B. Chen, P.-C. Li and C.-Y. Zhang, “Einstein-Gauss-Bonnet Black Strings at Large $D$," JHEP [**10**]{} (2017) 123 \[arXiv:1707.09766\].
R.-G. Cai,“Gauss-Bonnet Black Holes in AdS Spaces," Phys. Rev. D [**65**]{}, 084014 (2002) \[arXiv:hep-th/0109133\].
P. Davidson, “Turbulence: an introduction for scientists and engineers", Oxford University Press (2015).
G. K. Batchelor, “Computation of the energy spectrum in homogeneous two-dimensional turbulence," The Physics of Fluids [**12**]{} (1969) II-233.
R. H. Kraichnan, “Inertial ranges in two-dimensional turbulence," The Physics of Fluids [**10**]{} (1967) 1417-1423.
P. D. Mininni and A. Pouquet, “Inverse cascade behavior in freely decaying two-dimensional fluid turbulence," Phys. Rev. E [**87**]{} (Mar, 2013) 033002.
[^1]: bchen01@pku.edu.cn, wlpch@pku.edu.cn, ytian@ucas.ac.cn, zhangcy0710@pku.edu.cn,
[^2]: For the subsequent developments, see [@Thorne:1986iy].
[^3]: In fact, since the physical velocity for the fluid flow is $\mathbf{v}^i=\mathrm{v}^i/\sqrt{n}$, the fluid flow is non-relativistic in the large $n$ limit. In [@Rozali1707], due to the extra factor $1/\sqrt{n}$ carried by the velocity filed, the large $D$ hydrodynamics is shown to be non-relativistic.
[^4]: In fact, if we use the convention in [@Buchel0911] and for simplicity only consider one momentum along the brane direction, then we find the effective equations \_v m-\_z\^2 m+\_z p\_z=0, \_v p\_z+\_z\^2 p\_z+\_z\^2 m +\_z p\_z \_z m +(-)\_z m+=0. In terms of $p_z=\frac{1}{\beta}m v_z+\frac{1+\beta}{2\beta}\partial_z m$, the effective equations have the similar form as (\[fluideq1\]) and (\[fluideq2\]). In this case the ratio of the shear viscosity to the entropy density is found to be $\frac{\eta}{s}=\frac{1}{4\pi}\frac{\beta^2+1}{2}$ which is exactly the same as the large $D$ limit of (\[etaovers\]).
[^5]: Note that $\nabla\vec{f}\cdot\nabla m$ is different from $\nabla m\cdot\nabla \vec{f}$: the former one is written as $\partial_a f^b\partial_bm$ but the latter one is $\partial_bm \partial^bf_a$.
[^6]: Note that this definition is slightly different from the one given in [@Green1309], however it is easy to see that for $q=2$, they are pretty close.
[^7]: The definition of the energy spectrum is a little different from the one in [@Rozali1707]: here we use $\vec{p}$ rather than $\vec{u}$ in $E_C$. We find that this would be easier for numerical analysis. It works pretty well.
|
---
abstract: 'We study the Berezinskii-Kosterlitz-Thouless mechanism for vortex-antivortex pair formation in two-dimensional superfluids for nonequilibrium condensates. Our numerical study is based on a classical field model for driven-dissipative quantum fluids that is applicable to polariton condensates. We investigate the critical noise needed to create vortex-antivortex pairs in the systems, starting from a state with uniform phase. The dependence of the critical noise on the nonequilibrium and energy relaxation parameters is analyzed in detail.'
author:
- 'Vladimir N. Gladilin'
- Michiel Wouters
title: 'Noise-induced transition from superfluid to vortex state in two-dimensional nonequilibrium polariton condensates'
---
Introduction
============
With the advent of synthetic quantum systems, the interest in driven-dissipative many-body systems has grown substantially in the last decade. Where particles in ultracold atomic gases can to a very good approximation be conserved, losses can be engineered [@ott16] or are unavoidable in strongly interacting Bose gases [@bose-unit]. In systems based on electromagnetic degrees of freedom, both in the microwave [@fitzpatrick17] and optical domain [@carusotto-ciuti; @brenecke13], cavity losses are often not negligible. In order to reach a steady state, some driving of the system is then necessary to compensate for the losses. This raises the question on the modifications of the steady state with respect to the thermal equilibrium state in conservative systems.
Here, we will consider the case of two-dimensional weakly interacting bosons that are subject to single particle losses, which are compensated by a nonresonant drive. These systems are realized by microcavity polariton condensates [@Kasprzak06], but there may be also the possibility to construct them with ultracold atoms [@ott16]. Microcavity polaritons are hybrid light-matter quasiparticles resulting from the strong coupling between an excitonic transition in a quantum well and a photonic mode in an enveloping microcavity. From their photonic component, they inherit a light effective mass, enhancing the spatial coherence, whereas from their excitonic component, they inherit interactions (see Refs. [@rodriguez17; @delteil19] for the experimental determination of the interaction constant). Under nonresonant excitation, a large density of excitons is created, which subsequently relax to the lower polariton region. At equilibrium, the two-dimensional bose gas features a Berezinskii-Kosterlitz-Thouless (BKT) phase transition: for increasing temperature, thermally excited vortex-antivortex pairs become unbound, resulting in the loss of superfluidity. A natural question is then how this transition will be affected by losses and driving.
From the experimental side, this physics was addressed in Ref. [@caputo2016]. In their system with a long polariton life time, they did find a phase transition that was interpreted as the binding to unbinding transition of vortex-antivortex pairs, reminiscent of the equilibrium situation.
Already at the level of small phase fluctuations, nonequilibrium systems behave differently from their equilibrium counterparts. Where for the latter, the phase dynamics is for small fluctuations to a good approximation described by a linear equation, in the nonequilibrium case, a nonlinear term appears, which brings them in the Kardar-Parisi-Zhang universality class [@wachtel; @sieberer; @he15; @altman15; @sieberer16; @squizzato18; @gladilin14].
Some first theoretical insight in the modification of the BKT transition due to the nonequilibrium condition can be gained by considering the modification of vortices when going away from equilibrium. It turns out that the gain/losses introduce an additional current with the vortex core as its source. The phase profile is consequently deformed, resulting in a spiral wave [@aranson2002]. This modification in the vortex flow field subsequently affects the interactions between vortices and antivortices: when the outward flows are more important than the usual azimuthal flows, the interaction between vortex and antivortex becomes repulsive. These repulsive interactions hamper the vortex-antivortex recombination, enabling the formation of vortex-antivortex clusters with a very long life time [@gladilin18]. A renormalization group based approach has shown that these repulsions are fatal for the superfluid phase. The renormalization flow always goes toward the normal phase, even though this physics may manifest itself only at very large distances [@wachtel].
In order to shed further light on the phase diagram of 2D nonequilibrium polaritons, we resort here to numerical simulations of the noisy generalized Gross-Pitaevskii equation (ngGPE). This equation can be derived within the truncated Wigner approximation [@wouters09] or with the Keldysh field theory formalism [@szymanska07; @sieberer14] and has been widely used to model nonresonantly pumped polariton condensates [@16; @kalinin18; @comaron18; @bobrovska17; @estrecho18; @ohadi18]. At equilibrium, when the Bose gas is fully characterized by its density, temperature and interaction constant, the critical temperature is found to equal $T_c \approx 2\pi \hbar^2 n/[m \ln(380 \hbar^2/mg)]$, where $n$ is the density, $m$ the mass and $g$ the interaction strength [@prokofev01]. Away from equilibrium however, there are more microscopic parameters that enter the theoretical description. We investigate here how they affect the critical noise strength (the analog of the temperature out of equilibrium).
In a previous theoretical study [@gladilin18], based on a noise free generalized GPE, we found that starting from an initial state with a large number of vortices, several can survive in the steady state, because of the repulsive interactions between vortices and antivortices. We even found that these can form quite regular structures. With this physics understood, we will start here from the opposite initial condition with a homogeneous phase. In polariton condensates, such an initial condition can be achieved by sending a resonant pulse with a flat phase profile. As expected, we find that only when sufficiently strong noise is present, vortex-antivortex pairs can be formed in the subsequent time evolution. We will show that the pair production shows a well defined noise threshold, allowing us to draw a phase diagram for the system.
For systems that are far from equilibrium, we have shown [@30; @gladilin18] that the gGPE predicts a self-acceleration of vortices and production of new pairs in vortex-antivortex collisions, leading to chaotic dynamics. In this parameter regime, we find that a moderate noise suppresses this mechanism, leading to a stabilization of the system.
The structure of the paper is as follows. In Sec. \[sec:model\], our model for nonequilibrium condensation is recapitulated. The phase diagram is discussed in Sec. \[sec:results\] and conclusions are drawn in Sec. \[sec:concl\].
Model \[sec:model\]
===================
We consider nonresonantly excited two-dimensional polariton condensates. In the case of sufficiently fast relaxation in the exciton reservoir, this reservoir can be adiabatically eliminated and the condensate is described by the noisy generalized Gross-Pitaevskii equation [@wouters09; @szymanska07; @sieberer14; @15; @16; @a31] $$\begin{aligned}
({\rm i}-\kappa)\hbar \frac{\partial \psi}{\partial t} =&&
\left[-\frac{\hbar^2\nabla^2}{2m} +g |\psi|^2 \right. \nonumber
\\
&&\left.+\frac{{\rm i}}{2} \left(\frac{P}{1+|\psi|^2/n_s}-\gamma
\right) \right] \psi+\sqrt{D} \xi . \label{ggpe}\end{aligned}$$ Here $m$ is the effective mass and the contact interaction between polaritons is characterized by the strength $g$. The imaginary term in the square brackets on the right hand side describes the saturable pumping (with strength $P$ and saturation density $n_s$) that compensates for the losses ($\gamma$). We take into account the energy relaxation $\kappa$ in the condensate [@38; @39]. The complex stochastic increments have the correlation function $\langle
\xi^*(x,t) \xi(x',t') \rangle=2 \delta({\bf r}-{\bf r}')
\delta(t-t') $.
For polariton condensates, the validity of Eq. is not always straightfoward to justify. The repulsive interactions between the condensate and the exciton reservoir may lead to an effective attraction between the polaritons, leading to instability for a positive polariton mass [@baboux18; @bobrovska14; @bobrovska17]. This unstable state can be stabilized by a negative effective mass, that can be obtained in a polariton microcavity lattice [@baboux18]. We will not consider the instability physics in this work and assume positive mass and positive interactions (but the physics remains unaltered when the signs of the mass, interaction strength and energy relaxation parameter ($\kappa$) are simultaneously changed).
It is then convenient to rewrite Eq. (\[ggpe\]) in a dimensionless form, by expressing the particle density $|\psi|^2$ in units of $n_0\equiv n_s(P/\gamma-1 )$, time in units of $\hbar/(gn_0)$, length in units of $\hbar/\sqrt{2mgn_0}$, and noise intensity in units of $\hbar^3 n_0/(2m)$: $$\begin{aligned}
({\rm i}-\kappa)\frac{\partial\psi}{\partial t}=&& \left[-\nabla^2
+|\psi|^2 \phantom{\frac{\psi^2}{\psi^2}}\right. \nonumber
\\
&&\left.+{\rm i}c\frac{1-|\psi|^2}{1+\nu |\psi|^2} \right] \psi
+\sqrt{D} \xi . \label{ggpe2}\end{aligned}$$ Besides the noise intensity $D$, equation (\[ggpe2\]) contains three other dimensionless scalar parameters: $\kappa$ characterizes, as described above, damping in the system dynamics, $c=\gamma/(2gn_s)$ is a measure of the deviation from equilibrium, and $\nu=n_0/n_s$ is proportional to the relative excess of the pumping intensity $P$ over the threshold intensity. In the absence of noise, Eq. has the steady state solution $\psi = \sqrt{n} e^{- i n t}$, where the density $n$ satisfies $$\kappa n = \frac{c(1-n)}{1+\nu n}, \label{eq:dens_eq}$$ so that $n$ is a decreasing function of $\kappa$.
For weak noise, the fluctuations on top of this homogeneous solution can in first approximation be analyzed within the Bogoliubov approach [@chiocchetta13], but a more refined analysis has revealed that the dynamics of the phase fluctuations is in the KPZ universality class [@gladilin14; @ji15; @altman15; @he15; @squizzato18; @he17], where the phase nonlinearity cannot be neglected.
Here, we will study the critical noise needed to create vortices. In order to address this problem, Eq. (\[ggpe2\]) is solved numerically using the same finite-difference scheme as in Ref. [@30]. Specifically, we use periodic boundary conditions for a square of size $L_x=L_y=40$ with grid step equal to 0.2 corresponding to a cutoff in momentum space equal to $K_c=5 \pi$. The model is a classical field model, that suffers from the usual ultraviolet catastrophe [@prokofev01]. This implies that our results will be cutoff dependent. Below, we will discuss this dependence and how it will affect the comparison of our theoretical results with experiments.
When analyzing the noise-induced BKT transition, each run starts from a uniform condensate distribution at $D=0$. Then we apply noise with a fixed nonzero intensity $D$ during a time interval $t_D$. The detection of vortex pairs in the presence of a strong noise is somewhat tricky, but fortunately in the absence of noise their annihilation time is known to be rather long even at very weak nonequilibrium [@kulczy; @Comaron17] and this time strongly increases with increasing $c$ [@gladilin18]. For these reasons, it is more convenient to check the presence of vortex pairs sometime later after switching off the noise. The used time delay (typically few our units of time) is sufficient for significant relaxation of the noisy component in the density and phase distributions of the condensate and, at the same time, is too short for vortex pair annihilation. To determine the critical noise for the BKT transition, $D_{\rm BKT}$, we use the following criterion. If for a noise intensity $D$ vortex pairs are present after a noise exposure time $t_D$ (and hence $D>D_{\rm BKT}$), while for a certain noise intensity $D^\prime<D$ no vortex pairs appear even at noise exposures few times longer then $t_D$, then $D^\prime$ lies either below $D_{\rm BKT}$ or above $D_{\rm BKT}$ and closer to $D_{\rm
BKT}$ then to $D$. Therefore, the critical noise intensity can be estimated as $D_{\rm BKT}=D^\prime \pm (D-D^\prime)$. An illustration is displayed in Fig. \[example\]. For $D=0.015$ and noise exposure time $t_D=100$, a noisy density distribution shown in Fig. \[example\](a) evolves after switching off the noise into a pattern with clearly seen vortices and antivortices, which persist during a relatively long time \[see Figs. \[example\](b) and (c)\]. For a slightly lower noise intensity, $D=0.014$, and significantly longer noise exposure time, $t_D=600$ , after switching off the noise the corresponding noisy distribution \[Fig. \[example\](d)\] rapidly relaxes towards a uniform vortex-free state \[Figs. \[example\](e) and (f)\]. According to our approach, the estimate for the critical noise in this case is $D_{\rm
BKT}=0.014\pm 0.001$.
![Distributions of the particle density $n$ for $c=1.2$, $\nu=1$, $\kappa=0.01$, $D=0.015$ and $t_D=100$ at the time moment $t=0$ when the noise is switched off (a) and at $t=50$ (b). Distribution of the phase $\theta$ of the order parameter at $t=5$ is shown in panel (c). Panels (d), (e) and (f): same as in panels (a), (b) and (c), respectively, but for $D=0.014$, $t_D=600$ and a shorter time delay after switching off the noise ($t=5$). \[example\]](f0.eps){width="0.9\linewidth"}
Results and Discussion \[sec:results\]
=======================================
In Fig. \[dkap\](a), the critical noise intensity corresponding to the BKT transition, $D_{\rm BKT}$, is shown as a function of the damping parameter $\kappa$ for the cases of moderate ($c=0.3$) and strong ($c=4$) deviations from equilibrium. In both cases the critical noise $D_{\rm BKT}$ is seen to considerably increase with $\kappa$, despite the fact that the average density of the condensate at $D=D_{\rm BKT}$ is a monotonously decreasing function of $\kappa$ \[see also Eq. \], this decrease being especially pronounced at smaller $c$ \[see Fig. \[dkap\](b)\].
![Noise intensity $D_{\rm BKT}$ corresponding to the BKT transition \[panel (a)\] and average density of the condensate $n_{\rm
BKT}$ at $D=D_{\rm BKT}$ \[panel (b)\] as a function of damping at $\nu=1$ and two different values of the nonequilibrium parameter $c$. The error bars are smaller than the symbol size. Inset of panel (b): ratio $d_{\rm BKT}=D_{\rm BKT}/n_{\rm BKT}$ for $c=4$ (stars) and $c=0.3$ (diamonds). \[dkap\]](f1n.eps){width="0.9\linewidth"}
The inset of Fig. \[dkap\](b) shows the ratio $d_{\rm BKT}\equiv
D_{\rm BKT}/n_{\rm BKT}$ as a function of $\kappa$. For the two very different values of the nonequilibrium parameter $c$, the curves $d_{\rm BKT}(\kappa)$ appear to lie relatively close to each other. One can also notice that at $c=0.3$ the dependence of $d_{\rm BKT}$ on $\kappa$ is nearly linear (except for the smallest values of $\kappa$).
At $\kappa \gg 1$ linear scaling of the noise intensity with $\kappa$ directly follows from an additional rescaling of time \[$t\rightarrow t / \kappa$; see Eq. (\[ggpe2\])\]. Our numerical results show that this linear dependence of $d_{\rm BKT}$ on $\kappa$ remains to good approximation unaltered even when $\kappa
<1$.
At equilibrium ($c=0$), the increase of the critical noise with increasing $\kappa$ can be understood by making the connection with the thermal equilibrium case. For vanishing nonequilibrium and $\kappa \gg 1$, our equation reduces to model A dynamics [@hohenberg-halperin], which has a Boltzmann-Gibbs steady state distribution at temperature $T = D/2\kappa$. In this limit, the transition occurs at the equilibrium BKT temperature [@prokofev01], where $T_{\rm BKT} = \eta n$, with $\eta$ a numerical cutoff dependent constant. For the critical noise, one then obtains $D_{\rm BKT} =2 \eta \kappa n$.
Numerically, we have observed that for $c\rightarrow 0$, the critical noise obeys this relation, even for $\kappa < 1$. The fact that the dissipative part of the dynamics does not alter the steady state can be understood from the following argument. It is well established that pure Gross-Pitaevskii dynamics ($D=\kappa=c=0$) samples the thermal equilibrium state in the microcanonical ensemble, at an energy determined by the initial state. Similarly, the Langevin dynamics \[when omitting the $i$ in the l.h.s. of Eq. and taking $c=0$\] samples the phase space according to the canonical ensemble at a temperature determined by the balance between noise and dissipation. Since the thermal state is the steady state of both the GP and Langevin dynamics, it is natural that the steady state of the combined dynamics is also at thermal equilibrium, with the temperature determined by the Langevin part.
As implied by the results displayed in Fig. \[dkap\](a), the critical noise intensity $D_{\rm BKT}$ increases when moving away from equilibrium. The dependence of $D_{\rm BKT}$ on the nonequlibrium parameter $c$ is further illustrated in Fig. \[dc\](a) for the cases of weak ($\kappa=0.01$ and 0.1) and zero damping. At nonzero damping the increase of $D_{\rm BKT}$ with $c$ is partly related to the simultaneous increase of the average condensate density $n_{\rm BKT}(c)$ shown in Fig. \[dc\](b). The latter originates from the growing contribution of the pumping-loss term in Eq. (\[ggpe2\]) \[the last term in the square brackets in Eq. \]. Indeed, this term tends to keep the condensate density as close to 1 as possible. As a result, the degrading effect of damping or noise on the condensate density weakens with increasing $c$. This can be seen, e.g., from Fig. \[dkap\](b) by comparing to each other the values of $n_{\rm BKT}$ at different $c$: for each fixed, not too small $\kappa$, the value of $n_{\rm
BKT}|_{c=4}$ is significantly larger than $n_{\rm BKT}|_{c=0.3}$, even though the density $n_{\rm BKT}|_{c=4}$ corresponds to a considerably higher noise intensity than that for $n_{\rm
BKT}|_{c=0.3}$ \[see Fig. \[dkap\](a)\].
![Noise intensity $D_{\rm BKT}$ corresponding to the BKT transition \[panel (a)\] and average density of the condensate $n_{\rm
BKT}$ at $D=D_{\rm BKT}$ \[panel (b)\] as a function of the nonequilibrium parameter $c$ at $\nu=1$ and different values of the damping parameter $\kappa$. The error bars are smaller than the symbol size. Inset of panel (a): ratio $d_{\rm BKT}=D_{\rm
BKT}/n_{\rm BKT}$ for $\kappa=0.1$ (triangles) and $\kappa=0.01$ (circles). Inset of panel (b): ratio $d_{\rm BKT}=D_{\rm BKT}/n_{\rm
BKT}$ for $\kappa=0$ (squares). The black line corresponds to the linear fitting $d_{\rm BKT}=a c$, with $a=0.003$.\[dc\]](f2.eps){width="0.9\linewidth"}
At the same time, as follows from the behavior of the ratio $d_{\rm
BKT}\equiv D_{\rm BKT}/n_{\rm BKT}$ \[see the inset of Fig. \[dc\](a)\], the increase of $D_{\rm BKT}$ with $c$ is considerably faster as compared to that of $n_{\rm BKT}$. This means that the effect of pumping-loss processes on $D_{\rm BKT}$ cannot be explained solely by the corresponding stabilization of the average condensate density against noise and damping. This becomes even more evident when looking at the results for $\kappa=0$. Indeed, while the critical noise $D_{\rm BKT}$ demonstrates a clear rise with increasing $c$ \[see Fig. \[dc\](a)\], the average density $n_{\rm
BKT}(c)$ remains almost constant \[see Fig. \[dc\](b)\]. At small $c$, the values of $n_{\rm BKT}$ even slightly decrease with $c$ suggesting that the stabilizing effects of the pumping-loss processes do not completely compensate the average-density reduction caused by the increase of the noise intensity at the BKT transition. However, an important aspect of the the pumping-loss processes, which is not directly reflected in the [*average*]{} density, is that they impede formation of deep [*local*]{} suppressions of the condensate density. Those deep density suppressions, together with the appropriate phase gradient, are necessary prerequisites for vortex pair formation and hence for the BKT transition. The discussed numerical results imply that just the stabilizing effect of the pumping-loss term on the local density of the condensate governs the increase of $D_{\rm BKT}$ with $c$ at $\kappa
\rightarrow 0$. At $\kappa = 0$ the dependence of $D_{\rm BKT}$ on $c$ becomes close to linear \[see Fig. \[dc\](a)\], while for $d_{\rm BKT}(c)$ a linear function $d=ac$ with $a=0.003$ provides a nearly perfect approximation \[see Fig. \[dc\](b)\].
This interpretation of our numerical results can be elucidated by a linear analysis of the fluctuations. Writing the field as $\psi(x,t)=\sqrt{1+\delta n(x,t)}e^{i \theta(x,t)}$, one obtains in linearized approximation for the Fourier components $$\begin{aligned}
\frac{\partial}{\partial t} \left(
\begin{array}{c}
\theta_k \\
n_k
\end{array}
\right) =&& \left(
\begin{array}{cc}
-\kappa \epsilon_k & -\frac{\epsilon_k}{2}-1 \\
2 \epsilon_k & - \frac{2c}{1+\nu}+4 \kappa
\end{array}
\right) \left(
\begin{array}{c}
\theta_k \\
n_k
\end{array}
\right) \nonumber \\ &&+\left(
\begin{array}{c}
\sqrt{D} \xi^{(\theta)}_k \\
\sqrt{4 D} \xi^{(n)}_k
\end{array}
\right),\end{aligned}$$ where $\epsilon_k=k^2$. The white noise in Fourrier space has the correlation function $\langle \xi^{(\theta)}(k,t)
\xi^{(\theta)}(k',t) \rangle = 4\pi \delta(k+k') \delta(t-t') $, analogous for $\xi^{(n)}$, and $\langle \xi^{(\theta)} \xi^{(n)}
\rangle=0$.
The steady state density fluctuations can be obtained in closed form. In the limit of large $k$, they simplify to
[n\_k n\_[k’]{} = ]{} 2 (k+k’) & $\kappa \neq 0$\
2 (k+k’) & $\kappa = 0$
For the density fluctuations in real space, we then obtain for a sufficiently large momentum cutoff $K_c$
[n\^2(x) ]{} (2+\^2) K\_c & $\kappa \neq 0$ \[eq:n2knz\]\
K\_c\^2 & $\kappa = 0$ \[eq:n2kz\]
The local density fluctuations obtained here in the linear approximation show behavior that is in line with what was observed numerically for the BKT transition. Density fluctuations are first suppressed by the damping $\kappa$ and for $\kappa=0$ by the nonequilibrium parameter $c$.
Note the very different dependence of both results on the momentum cutoff. In the presence of a nonzero damping, the cutoff dependence is a very weak logarithm, where in its absence, it becomes quadratic. This difference is due to the fact that the momentum space density tends to a constant in the $\kappa=0$ case, where it decays as $k^{-2}$ for $\kappa \neq 0$. Even the latter is not fast enough to ensure convergence in two dimensions, hence the logarithmic divergence of the density fluctuations.
The values of $D_{\rm BKT}$, obtained from the numerical simulations, correlate with the above analysis. At $c=1.2$, $\nu=1$, and $\kappa=0$, an increase of the grid step from 0.2 to 0.4 leads to a change of the numerically determined $D_{\rm BKT}$ from $0.00355\pm 0.0001$ to $0.0139 \pm 0.0001$ that is in line with the quadratic dependence of $\langle n^2(x) \rangle$ on $K_c$ in the absence of energy relaxation \[see Eq. \]. At the same time, for nonzero damping ($\kappa \geq 0.01$) the relative difference between the values of $D_{\rm BKT}$ corresponding to the grid step 0.4 and 0.2 is about 40% for $\kappa=0.01$ and does not exceed 15% for $\kappa \geq 0.1$. The critical density increases somewhat when increasing the grid step, such that $d_{\rm BKT}=
D_{\rm BKT}/n_{\rm BKT}$ increases only by about 10% for $\kappa
\geq 0.1$. Our analytical arguments and those of Ref. [@prokofev01] suggest a logarithmic dependence on the cutoff, but we were not able to test this scaling numerically due to a lack of accessible range in the grid spacing. [Choosing the grid spacing much larger than 0.4, it becomes too coarse to give a good description of the vortex cores, while choosing a grid spacing smaller than 0.2 slows down the calculations too much (the maximal kinetic energy increases quadratically with momentum cutoff). Moreover, decreasing the grid spacing below 0.2 would give a kinetic energy that is typically larger than what is required to justify the lower polariton approximation. ]{}
As seen from Eq. (\[ggpe2\]), the pumping-loss term increases in magnitude when decreasing the parameter $\nu$. One can expect, therefore, that a decrease of $\nu$ leads to an increase of the critical noise $D_{\rm BKT}$. Our simulations confirm this expectation but, at the same time, show that the influence of $\nu$ on $D_{\rm BKT}$ is relatively weak for nonvanishing damping. At $\kappa=0.01$ a decrease of $\nu$ by one order of magnitude (from 1 down to $0.1$) results in an increase of $D_{\rm BKT}$ approximately by 4% at $c=0.3$ and by 12% at $c=4$. The corresponding increase of $d_{\rm BKT}$ is about 20% for both $c=0.3$ and 4. In other words, the effect of $\nu$ on the critical noise is rather minor as compared to the much stronger impact of $\kappa$ and $c$. Let us now look in some more detail at the simultaneous dependence of $d_{\rm BKT}$ on $\kappa$ and $c$. Taking into account, on the one hand, the nearly linear dependence $d_{\rm BKT}$ on $\kappa$ at large $\kappa$ \[see the inset in Fig. \[dkap\](b)\] and, on the other hand, the linear dependence on $c$, $d_{\rm BKT}\approx ac$ at $\kappa \rightarrow 0$, it seems [reasonable]{} to consider the renormalized quantity $d_{\rm BKT}/(\kappa+ac)$. As seen from Fig. \[dnorm\], all the results of our simulations lie in a rather narrow interval around 1: $0.6<d_{\rm BKT}/(\kappa+ac)<1.3$. This suggests that the simple expression $\kappa + 0.003c$ already can serve as a crude but quite reasonable estimate for $d_{\rm BKT}$ at any $\kappa$ and $c$. A better approximation of the numerical results is obtained (see Fig. \[dnorm\]) by using the fitting function, [that summarizes our numerical calculations rather accurately:]{} $$\begin{aligned}
d^*=&& 0.609 \kappa+0.003c \nonumber \\ &&+\frac{ \kappa^{1.411}
c^{0.411} }{2.646\kappa^{0.610} + 0.0706 c^{0.889} + 1.622
\kappa^{1.514} c^{1.308} }\end{aligned}$$ Since the numerical results for the dependence of $d_{\rm BKT}$ on both $\kappa$ and $c$ are seen to be fitted quite well by $d^*(c,\kappa)$, we may expect that this function can provide also meaningful results for inter- and extrapolation [\[see the inset in Fig. \[dnorm\](b)\]]{}.
![Renormalized noise intensity $d_{\rm BKT}/(\kappa+ac)$ (symbols) and its fitting by $d^*/(\kappa+ac)$ (lines) as a function of the damping parameter $\kappa$ \[panel (a)\] and nonequilibrium parameter $c$ \[panel (b)\].The inset shows the dependence of $d^*/(\kappa+ac)$ on both $c$ and $\kappa$. Straight dashed lines correspond to the parameter values covered by the curves in panels (a) and (b). \[dnorm\]](f3n.eps){width="0.9\linewidth"}
Finally, we address the question of whether the BKT transition under consideration can be influenced by the effect of vortex pair generation by moving vortices [@gladilin18]. This effect has been predicted to occur in strongly nonequilibrium condensates when self-accelerated vortices acquire sufficiently high velocities with respect to the surrounding condensate. The results of [@gladilin18] were obtained in the absence of noise. In our numerics containing the noise term, we have observed that fluctuations of the condensate density and currents tend to impede the acceleration of vortices. Consequently, the generation of vortex pairs in vortex collisions becomes impossible at high noise intensities. This is illustrated in Fig. \[dm\], which shows the upper boundary $D_{\rm m}$ for the noise intensity range, where generation of new vortex pairs by self-accelerated vortices is possible, as a function of $c$ at $\nu=1$ and $\kappa=0.01$. The corresponding calculations are performed, like in [@gladilin18], starting with one pinned vortex pair in the region under consideration and simulating the dynamics during a time interval $\Delta t \sim 1000$ after depinning of vortices, now in the presence of noise.
![Noise intensity $D_{\rm BKT}(c)$ corresponding to the BKT transition in comparison with $D_{\rm m}(c)$, the maximum noise intensity, which still allows for vortex pair generation by moving vortices, at $\nu=1$ and $\kappa=0.01$. \[dm\]](f4.eps){width="0.9\linewidth"}
In Fig. \[dm\], the dependence of $D_{\rm m}$ on $c$ is given in comparison with the critical noise $D_{\rm BKT}$ at the same $\kappa$ and $\nu$. As seen from Fig. \[dm\], the region of $c$ and $D$, where pair generation by vortices is possible, lies at large $c$ ($c > 3.3$) and has no overlap with the curve $D_{\rm BKT}(c)$. Of course, the latter remains true for any $\kappa
\geq 0.01$. Indeed, the critical noise $D_{\rm BKT}$ increases with $\kappa$. At the same time, damping slows down vortex motion and thus impedes pair generation by moving vortices. As a result, the aforementioned region will shrink with increasing $\kappa$. Our simulations show that already at $\kappa$ as small as 0.1 the velocities of vortices are insufficient for pair generation. So, we can conclude that at $\kappa \geq 0.01$ the BKT transition is not affected by the processes of pair generation by moving vortices. As concerns the limit of very weak damping $\kappa< 0.01$, the nearly linear behavior of $D_{\rm BKT}(c)$, obtained for $\kappa=0$, manifests no visible peculiarities in the region of large $c$, where the processes of pair generation by vortices become possible. This implies that, also in the limit $\kappa \to 0$, the BKT transition is not considerably affected by these processes.
Conclusions \[sec:concl\]
=========================
We have investigated the critical noise for the spontaneous formation of unbound vortex-antivortex pairs in a driven-dissipative bosonic system where particle losses are compensated by nonresonant pumping. In this work, we have focused on the noise needed to form vortex-antivortex pairs starting from a uniform phase. In our model, the critical noise strength depends – with a suitable choice of units – on three dimensionless parameters: the energy relaxation rate ($\kappa$), the nonequilibrium parameter ($c$) and the gain saturation parameter ($\nu$).
In the absence of energy relaxation, the nonequilibrium parameter determines the critical noise strength, but this critical value depends quadratically on the momentum space cutoff, giving our numerical results limited predictivity for specific experiments. In the presence of sufficiently strong energy relaxation, the cutoff dependence is much weaker and experimentally relevant results can be extracted from the numerics in this case. In this regime, the critical noise strength increases both with energy relaxation (as expected from equilibrium calculations) and with nonequilibrium parameter $c$. The latter dependence could seem counterintuitive, because with increasing $c$, vortices and antivortices repel each other at large distances, which could favor their unbinding. We interpret the impeding of vortex-antivortex formation further away from nonequilibrium as a consequence of the reduction of the density fluctuations. The effect of nonequilibrium on the *formation* of vortices is therefore opposite to its effect on the *annihilation*: the life time of existing vortices is dramatically enhanced by nonequilibrium [@gladilin18].
Quantitatively, we have found that the effect of the nonequilibrium parameter is actually small for $\kappa \gg 0.003 c$, which is satisfied when damping is not too small and the system is not too far from equilibrium. Very far from equilibrium, self acceleration of vortices can lead to the production of new vortex antivortex pairs. We have shown here that this pair production ceases for increasing noise.
Our numerical simulations were performed for systems with periodic boundary conditions whose size is comparable to the systems employed in current experiments. The study of the thermodynamical limit of infinite system size remains a challenge for both theoretical analysis and experimental investigation.
Acknowledgements {#acknowledgements .unnumbered}
================
We thank Sebastian Diehl for a stimulating discussion. This work was financially suppported by the FWO Odysseus program.
R. Labouvie, B. Santra, S. Heun, and H. Ott, Phys. Rev. Lett. **116**, 235302 (2016).
P. Makotyn, C. E. Klauss, D. L. Goldberger, E. A. Cornell and D. S. Jin, Nat. Phys. **10**, 116 (2014); C. Eigen, J. A. P. Glidden, R. Lopes, E. A. Cornell, R. P. Smith and Z. Hadzibabic Nature **563**, 221 (2018).
M. Fitzpatrick, N. M. Sundaresan, A. C. Y. Li, J. Koch, and A. A. Houck, Phys. Rev. X **7**, 011016 (2017).
I. Carusotto and C. Ciuti, Rev. Mod. Phys. [**85**]{}, 299 (2013).
F. Brennecke, R. Mottl, K. Baumann, R. Landig, T. Donner, and T. Esslinger, Proceedings of the National Academy of Sciences [**110**]{}, 11763 (2013).
J. Kasprzak, M. Richard, S. Kundermann, A. Baas, P. Jeambrun, J. Keeling, F. Marchetti, M. Szyma[ń]{}ska, R. Andre, J. Staehli, V. Savona, P. B. Littlewood, B. Deveaud and Le Si Dang, Nature [**443**]{}, 409 (2006).
A. Delteil, T. Fink, A. Schade, S. Höfling, C. Schneider, A. Imamoğlu, Nat. Materials [**18**]{}, 219 (2019).
S. R. K. Rodriguez, W. Casteels, F. Storme, N. Carlon Zambon, I. Sagnes, L. Le Gratiet, E. Galopin, A. Lemaître, A. Amo, C. Ciuti, and J. Bloch, Phys. Rev. Lett. [**118**]{}, 247402 (2017).
D. Caputo, D. Ballarini, G. Dagvadorj, C. S. Mu[ñ]{}oz, M. De Giorgi,L. Dominici, K. West, L. N. Pfeiffer, G. Gigli, F. P. Laussy, M. H. Szymańska and D. Sanvitto, Nat. Mat. [**17**]{},145 (2018).
G. Wachtel, L. M. Sieberer, S. Diehl and E. Altman, Phys. Rev. B [**94**]{}, 104520 (2016).
L. M. Sieberer, G. Wachtel, E. Altman and S. Diehl, Phys. Rev. B [**94**]{}, 104521 (2016).
D. Squizzato, L. Canet and A. Minguzzi, Phys. Rev. B **97**, 195453 (2018).
V. N. Gladilin, K. Ji, and M. Wouters, Phys. Rev. A **90**, 023615 (2014).
L. He, L. M. Sieberer, E. Altman, and S. Diehl, Phys. Rev. B 92, 155307 (2015).
E. Altman, L. M. Sieberer, L. Chen, S. Diehl, and J. Toner, Phys. Rev. X 5, 011017 (2015).
L. M. Sieberer, M. Buchhold, and S. Diehl, Reports on Progress in Physics 79, 096001 (2016).
I. S. Aranson and L. Kramer, Rev. Mod. Phys. **74**, 99 (2002).
V. N. Gladilin and M. Wouters, arXiv:1810.06365.
M. Wouters and V. Savona, Phys. Rev. B **79**, 165302 (2009).
M. H. Szymanśka, J. Keeling, and P. B. Littlewood, Phys. Rev. B **75**, 195331 (2007).
L. M. Sieberer, S. D. Huber, E. Altman, S. Diehl, Phys. Rev. B **89**, 134310 (2014).
K. Kalinin, P. G. Lagoudakis, N. G. Berloff, Phys. Rev. B [**97**]{}, 094512 (2018).
P. Comaron, G. Dagvadorj, A. Zamora, I. Carusotto, N. P. Proukakis, M. H. Szymańska, Phys. Rev. Lett. [**121**]{}, 095302 (2018).
N. Bobrovska, M. Matuszewski, K. S. Daskalakis, S. A. Maier, S. Kéna-Cohen, ACS Photonics [**5**]{}, 111 (2017).
E. Estrecho, T. Gao, N. Bobrovska, M. D. Fraser, M. Steger, L. Pfeiffer, K. West, T. C. H. Liew, M. Matuszewski, D. W. Snoke, A. G. Truscott and E. A. Ostrovskaya, Nat. Comm. [**9**]{}, 2944 (2018).
H. Ohadi, Y. del Valle-Inclan Redondo, A. J. Ramsay, Z. Hatzopoulos, T. C. H. Liew, P. R. Eastham, P. G. Savvidis, J. J. Baumberg, Phys. Rev. B [**97**]{}, 195109 (2018).
J. Keeling and N. G. Berloff, Phys. Rev. Lett. [**100**]{}, 250401 (2008).
N. Prokof’ev, O. Ruebenacker, and B. Svistunov Phys. Rev. Lett. 87, 270402 (2001).
V. N. Gladilin and M. Wouters, New J. Phys. [**19**]{}, 105005 (2017).
M. Wouters and I. Carusotto, Phys. Rev. Lett. [**99**]{}, 140402 (2007).
N. Bobrovska and M. Matuszewski, Phys. Rev. B [**92**]{}, 035311 (2015).
L.P. Pitaevskii, Zh. Eksp. Teor. Fiz. [**35**]{}, 408 (1958) \[Sov. Phys. JETP [**35**]{}, 282 (1959)\].
M. Wouters, New J. Phys. [**14**]{}, 075020 (2012).
F. Baboux, D. De Bernardis, V. Goblot, V.N. Gladilin, C. Gomez, E. Galopin, L. Le Gratiet, A. Lemaître, I. Sagnes, I. Carusotto, M. Wouters, A. Amo, J. Bloch, Optica [**5**]{}, 1163 (2018).
N. Bobrovska, E. A. Ostrovskaya and Michal Matuszewski, Phys. Rev. B [**90**]{}, 205304 (2014).
A. Chiocchetta and I. Carusotto, EPL, **102**, 67007 (2013).
K. Ji, V. N. Gladilin, and M. Wouters, Phys. Rev. B [**91**]{}, 045301 (2015).
L. He and S. Diehl, New J. Phys. 19, 115012 (2017).
M. Kulczykowski and M. Matuszewski, Phys. Rev. B [**95**]{}, 075306 (2017).
P. Comaron, G. Dagvadorj, A. Zamora, I. Carusotto, N. P. Proukakis, M. H. Szymanska, Phys. Rev. Lett. [**121**]{}, 095302 (2018).
P. C. Hohenberg and B. I. Halperin, Rev. Mod. Phys. [**49**]{},435 (1977).
|
---
abstract: 'A star coloring of a graph $G$ is a proper vertex coloring such that the subgraph induced by any pair of color classes is a star forest. The star chromatic number of $G$ is the minimum number of colors needed to star color $G$. In this paper we determine the star-chromatic number of the splitting graphs of cycles of length $n$ with $n \equiv 1 \pmod 3$ and $n=5$, resolving an open question of Furnma[ń]{}czyk, Kowsalya, and Vernold Vivin.'
address: $^1$Williams College
author:
- Sumun Iyer$^1$
title: 'Star-coloring Splitting Graphs of Cycles'
---
2010 [*Mathematics Subject Classification*]{}: 05C15; 05C75.
*Keywords: Star coloring; splitting graph; cycle.*
Introduction
============
Let $G=(V,E)$ be a simple, undirected graph. A proper vertex $n$-coloring of $G$ is a surjective mapping $\phi: V \to \{1,2, \dots, n\}$ such that if $u$ is adjacent to $v$, then $\phi(u) \neq \phi(v)$. An *n-star-coloring* of $G$ is a proper vertex $n$-coloring with one additional condition: each path on four vertices in $G$ is colored by at least three distinct colors. Alternatively, a star coloring of $G$ is a coloring such that the subgraph induced by any pair of color classes of $G$ is a star forest. Star coloring strengthens the notion of *acyclic coloring* in which the subgraph induced by any pair of color classes is a forest. The *star chromatic number* of $G$, denoted $\chi_s(G)$, is the minimum number of colors needed to star color $G$.
Star coloring was first introduced by Branko Gr[ü]{}nbaum in 1973 in the context of strengthening acyclic colorings of planar graphs [@grunbaum]. Star coloring also arises naturally in combinatorial computing. As one would imagine, finding an optimal star coloring of a general graph is NP-hard. Coleman and Mor[é]{} showed that star-coloring remains an NP-hard problem even on bipartite graphs [@coleman]. Coloring variants (like acyclic or star coloring) have been used to compute sparse Hessian and Jacobian matrices with techniques like finite differences and automatic differentiation. Gebremedhin, Tarafdar, Manne, and Pothen provided algorithms for finding heuristic solutions to star coloring and acyclic coloring problems [@gebre]. Their techniques utilize the structure of subgraphs induced by color classes and their findings have applications to efficient computation of Hessian matrices. Because the problems of computing these matrices can be recast as graph coloring problems, employing graph coloring as a model for computation can yield particularly effective algorithms. See [@colorsurvey] for a detailed survey of using graph coloring to compute derivatives.
In 2004 Fertin, Raspaud, and Reed determined the star chromatic number of trees, cycles, complete bipartite graphs, and other families of graphs [@fertin]. Star chromatic numbers of other types of graphs–including sparse graphs, bipartite planar graphs, and planar graphs with high girth–are studied in [@bu], [@kierstead], [@mohan], and [@timmons].
For a vertex $v$ of a graph $G = (V,E)$ let $N(v)$ denote the open neighborhood of $v$. The *splitting graph*, $S(G)$, is obtained by adding a new vertex $v'$ corresponding to each $v$ in $V$ and edges from $v'$ such that $N(v)=N(v')$ (see Figure \[Fig1\]). The splitting graph construction plays an important role in the theory of graph labeling. For a comprehensive survey of results in graph labeling and more references on splitting graphs see Gallian’s “Dynamic survey of graph labeling" [@gallian]. In 2017 Furma[ń]{}czyk, Kowsalya, and Vernold Vivin determined the star chromatic number of splitting graphs of complete and complete bipartite graphs, paths, and some cycles [@furman]. They posed as an open question the problem of determining the star chromatic number of splitting graphs of cycles on $n$ vertices where $n =5$ or $n \equiv 1 \pmod 3$. In this paper we provide a construction that shows that $\chi_S(S(C_n)) = 4$ for all $n \equiv 1 \pmod 3$, $n \geq 10$ and prove that $\chi_S(S(C_4))=\chi_S(S(C_5))=\chi_S(S(C_7)) = 5$.
Splitting graphs of cycles
==========================
We include the following two results for completeness. The first is due to Fertin, Raspaud, and Reed [@fertin] and the second is due to Furma[ń]{}czyk, Kowsalya, and Vernold Vivin [@furman].
\[cycle\]*(Fertin, Raspaud, Reed)* Let $C_n$ be a cycle on $n \geq 3$ vertices. Then, $$\chi_S(C_n)=
\begin{cases}
4 & \textrm{when} \ n=5 \\
3 & \textrm{otherwise.}
\end{cases}$$
\[furmanthm\] *(Furma[ń]{}czyk, Kowsalya, Vernold Vivin)* Let $C_n$ be a cycle on $n \geq 3$ vertices. Then $$\chi_S(S(C_n))
\begin{cases}
=4 & \textrm{if} \ n \not\equiv 1 \mod 3 \ \textrm{and} \ n \neq 5 \\
\leq 5 & \textrm{otherwise.}
\end{cases}$$
To resolve the case of splitting graphs of cycles $C_n$ with $n \equiv 1 \pmod 3$, we first present a construction that shows that $\chi_S(S(C_n))$ is 4 for $n \geq 10$.
\[construction\] If $n \equiv 1 \mod 3$ and $n \geq 10$, then $\chi_S(S(C_n)) = 4$.
Let $n \in {\mathbb{N}}$ with $n \equiv 1 \pmod 3$ and $n \geq 10$. By [@furman], we know that for all $n$, $\chi_S(S(C_n)) \geq 4$. We now provide a construction to star color $S(C_n)$ with four colors. Label the vertices of the copy of $C_n$ in $S(C_n)$ with $v_0, v_1, \ldots , v_{n-1}$ clockwise. Label the vertex corresponding to $v_i$ in the splitting graph construction with $v_i'$ for $0 \leq i \leq n-1$.
Define $\phi : V(S(C_n)) \to \{1,2,3,4\}$ as follows. For $0 \leq i \leq n-8$: $$\phi(v_i) =
\begin{cases}
1 & \textrm{if} \ n \equiv 0 \pmod 3;\\
2 & \textrm{if} \ n \equiv 1 \pmod 3;\\
3 & \textrm{if} \ n \equiv 2 \pmod 3.
\end{cases}$$
We color the remaining seven vertices of $C_n$ as follows. Let $\phi(v_{n-1})=\phi(v_{n-4})=4$, $\phi(v_{n-2})=\phi(v_{n-5})= 3$, $\phi(v_{n-3})=\phi(v_{n-7})=1$, and $\phi(v_{n-6})=2$.
Now, we color the splitting vertices. Let $\phi(v_i')=4$ for $1 \leq i \leq n-6$. Let $\phi(v_j')=2$ for $n-4 \leq j \leq n-1$. Let $\phi(v_0')=3$ and $\phi(v_{n-5}')=1$.
We claim that $\phi$ is a star coloring of $S(C_n)$. The proof follows from Figure \[Fig1\] and Figure \[Fig2\]. Figure \[Fig1\] shows the four star coloring of $S(C_{10})$. If $n=10+3k$, then our construction for $\phi$ essentially glues a copy of Figure \[Fig2\] with $3k$ nodes at the appropriate spot (marked by dotted lines) in Figure \[Fig1\]. It is easy to check that this does not create any new 2-colored $P_4$’s and so $\phi$ is a four star coloring of $S(C_n)$.
![The four star coloring of $S(C_{10})$ constructed in Theorem \[construction\].[]{data-label="Fig1"}](coloringofsc10.png){height="3cm"}
![The four star coloring of the “insertion” piece from Theorem \[construction\].[]{data-label="Fig2"}](coloringinsertion.png){height="3cm"}
We will now argue that the star-chromatic numbers of $S(C_4)$, $S(C_5)$, and $S(C_7)$ are five. All three proofs have essentially the same flavor with some additional technical detail for $S(C_7)$. The idea of two colored graphs being “the same" will be useful and so we provide a formal definition:
Let $G_1$ and $G_2$ be two graphs with vertex sets $V(G_1)$ and $V(G_2)$ respectively and vertex colorings $\phi_1$ and $\phi_2$ respectively. Then, $G_1$ and $G_2$ are *isomorphic as vertex-colored graphs* if there are bijective functions $\pi: V(G_1) \to V(G_2)$ and $\theta: \{\phi_1(v) \ : \ v \in V(G_1)\} \to \{\phi_2(v) \ : \ v \in V(G_2)\}$ such that $u$ is adjacent to $v$ in $G_1$ if and only if $\pi(u)$ is adjacent to $\pi(v)$ in $G_2$ and for all $v \in V(G_1)$ we have $\theta(\phi_1(v))=\phi_2(\pi(v))$.
The star chromatic number of the splitting graph of $C_4$ is 5.
Label the vertices of $C_5$ clockwise with $v_0,v_1,v_2,v_3$ and the vertex corresponding to $v_i$ under the splitting graph construction with $v_i'$. Suppose for sake of contradiction that $\phi$ is a four star-coloring of $S(C_4)$. By Theorem \[cycle\], it suffices to consider the following two cases.
**Case 1**: Suppose that $\phi$ uses all four colors to color the copy of $C_4$ in $S(C_4)$. Since $\phi$ is a proper vertex coloring, either $\phi(v_0') = \phi(v_0)$ or $\phi(v_0')=\phi(v_2)$. If $\phi(v_0')=\phi(v_0)$, then it follows that $\phi(v_1')=\phi(v_3)$. Then, $v_1' \to v_0 \to v_3 \to v_0'$ is a 2-colored $P_4$, a contradiction. On the other hand, if $\phi(v_0')=\phi(v_2)$, then it follows from considering the path $v_0' \to v_1 \to v_2 \to v_3'$ that $\phi(v_3')=\phi(v_3)$. Now, $v_0' \to v_3 \to v_2 \to v_3'$ is a 2-colored $P_4$, a contradiction.
**Case 2**: Suppose $\phi$ uses three colors to color the copy of $C_4$ in $S(C_4)$. Without loss of generality, we can assume $\phi(v_0)=\phi(v_2)$ (the other cases are isomorphic as vertex-colored graphs). Since $\phi$ is a proper vertex coloring, either $\phi(v_3') = \phi(v_3)$ or $\phi(v_3')=\phi(v_1)$. In the former case, $v_3' \to v_2 \to v_3 \to v_0$ is a 2-colored $P_4$ and in the latter case, $v_3' \to v_0 \to v_1 \to v_2$ is a 2-colored $P_4$, a contradiction.
The star chromatic number of the splitting graph of $C_5$ is 5.
Label the vertices of $C_5$ clockwise with $v_0, \ldots, v_4$ and label the vertex corresponding to $v_i$ under the splitting graph construction with $v_i'$.
Suppose for the sake of contradiction that $\phi$ is a four star coloring of $S(C_5))$. By Theorem \[cycle\], $\phi$ uses all four colors to color $C_5$. Since $\phi$ is a proper star coloring, $\phi$ must use one of the four colors to color two distinct vertices and the other three colors to color the remaining three vertices. Without loss of generality, we can assume $\phi(v_0)$ is used twice. We now consider two cases depending on which other vertex of $C_5$ has the same color as $v_0$.
**Case 1**: Suppose $\phi(v_0) = \phi(v_2)$. This implies $\phi(v_4) = \phi(v_4')$ and therefore that $\phi(v_0')=\phi(v_3)$. Then, $v_0' \to v_4 \to v_3 \to v_4'$ is a 2-colored $P_4$, a contradiction.
**Case 2**: Suppose $\phi(v_0) = \phi(v_3)$. This implies $\phi(v_1)=\phi(v_1')$ and therefore that $\phi(v_0') = \phi(v_2)$. Then, $v_0' \to v_1 \to v_2 \to v_1'$ is a 2-colored $P_4$, a contradiction.
Thus, $\chi_S(S(C_5)) = 5$.
To prove that the star chromatic number of $S(C_7)$ is 5, we first give two helpful lemmas.
\[bicolp3\] Suppose $C_7$ is three star colored. Then, some $P_3$ in $C_7$ is 2-colored.
Suppose for the sake of contradiction that $\phi: V(C_7) \to \{1,2,3\}$ is a three star coloring of $C_7$ with no bi-colored $P_3$. Label the vertices of $C_7$ clockwise with $v_0, \ldots , v_6$. Since $\phi$ is a star coloring, for some $i$ we know $\phi(v_i)=1$. Since $\phi$ has no bi-colored $P_3$, we know $\phi(v_{i-1}) \neq \phi(v_{i+1})$. Suppose without loss of generality (the other case is symmetric) that $\phi(v_{i-1})=2$ and $\phi(v_{i+1})=3$. The fact that $\phi$ has no 2-colored $P_3$ completely determines the colors of the remaining vertices and it follows that $v_{i-2} \to v_{i-3} \to v_{i+3}$ is a 2-colored $P_3$, a contradiction.
\[babcb\] Label the vertices of $C_n$ clockwise with $v_0, \ldots, v_{n-1}$ and label the vertex corresponding to $v_i$ under the splitting graph construction with $v_i'$. Suppose $\phi$ is a $k$-star coloring of $S(C_n)$ and there exists $i$ such that $\phi(v_i)=\phi(v_{i+2})=\phi(v_{i+4})$. Then $k \geq 5$.
Since $\phi$ is a star coloring, $\phi(v_{i+1}) \neq \phi(v_{i+3})$. It follows that $\phi$ assigns $v_{i+1}'$ a different color from the three distinct colors used to color $v_i$, $v_{i+1}$, and $v_{i+3}$ (if the color of $v_{i+1}$ matches any of the three colors used to color $v_i$, $v_{i+1}$, or $v_{i+3}$, then it is easy to check that $\phi$ is not a star coloring). Considering the path $v_i \to v_{i+1}' \to v_{i+2} \to v_{i+3}'$, we see that $\phi(v_{i+1}') \neq \phi(v_{i+3}')$. Thus, $k \geq 5$.
![The four star coloring of $C_7$ used in Case 2b of the proof of Theorem \[c7\][]{data-label="Fig3"}](4coloringc7.png){height="2cm"}
\[c7\] The star chromatic number of $S(C_7)$ is 5.
Suppose for the sake of contradiction that $\phi$ is a four star-coloring of $S(C_7)$. We then consider two cases depending on whether $\phi$ uses three or four colors to color $C_7$.
**Case 1**: Suppose $\phi$ uses three colors to color $C_7$. We know by Lemma \[bicolp3\] that, colored by $\phi$, $C_7$ has some 2-colored $P_3$. That is, for some $i$, $\phi(v_{i-1})=\phi(v_{i+1})$. Since $\phi$ is a proper vertex coloring, $\phi(v_{i+2}) \neq v_{i-1})$ and similarly, $\phi(v_{i-2}) \neq \phi(v_{i-1})$. Since $\phi$ is a star coloring, we have that $\phi(v_{i_2}) \neq \phi(v_i)$ and $\phi(v_{i+2}) \neq \phi(v_i)$. This implies that either $\phi(v_{i-3})=\phi(v_{i+1})$ or $\phi(v_{i+3})=\phi(v_{i+1})$. In either case, Lemma \[babcb\] gives a contradiction.
**Case 2**: Suppose $\phi$ uses four colors to color $C_7$. It is easy to check that $\phi$ cannot assign any one color to three distinct vertices of $C_7$. The proof now proceeds through two subcases.
**Case 2a**: Suppose $\phi$ has a bi-colored $P_3$. That is, there exists $i$ such that $\phi(v_{i-1})=\phi(v_{i+1})$. Since $\phi$ is a star-coloring, $\phi(v_{i-2}) \neq \phi(v_i)$ and $\phi(v_{i+2}) \neq \phi(v_i)$. It is easy to see that $\phi(v_{i-2}) = \phi (v_{i+2})$ (otherwise, $v_i'$ requires a fifth color to be properly star colored). Since $\phi$ is a star coloring that uses four colors to star color $C_7$, one of $\phi(v_{i-3})$ or $\phi(v_{i+3})$ is distinct from $\phi(v_i)$, $\phi(v_{i+1})$, and $\phi(v_{i+2})$. Because the two cases are symmetric, we may suppose $\phi(v_{i-3})$ is the distinct color. Lemma \[babcb\] and the fact that $\phi$ is a proper vertex coloring imply that $\phi(v_{i+3})=\phi(v_i)$. Now, it follows that $\phi(v_i') = \phi(v_{i-3})$. This implies $\phi(v_{i+2}') = \phi(v_{i+2})$ and thus $\phi(v_{i+3}') = \phi(v_{i+1})$. Then, $v_{i+3}' \to v_{i+2} \to v_{i+1} \to v_{i+2}'$ is a 2-colored $P_3$, a contradiction.
**Case 2b**: Assume next that $\phi$ has no bi-colored $P_3$. One can check that any four star coloring of $C_7$ without any 2-colored $P_3$ is isomorphic as a vertex coloring to the coloring presented in Figure \[Fig3\].
First, we claim that for some $i$, $\phi(v_i) = \phi(v_i')$. Suppose for the sake of contradiction that for all $i$, $\phi(v_i) \neq \phi(v_i')$. Because $\phi$ has no 2-colored path with three vertices in $C_7$, this fully determines the color of each of the vertices added in the splitting construction. If we label the vertices of $C_7$ (clockwise by $v_0, \dots , v_6$) such that $\phi(v_i)=4$, then we can check that this leads to the 2-colored $P_4$ $v_{i-2}' \to v_{i-1} \to v_i \to v_{i+1}'$, a contradiction.
Thus, there exists an $i$, $0 \leq i \leq 6$, such that $\phi(v_i)=\phi(v_i')$. Since $C_7$ has no 2-colored path with three vertices, the condition $\phi(v_i)=\phi(v_i')$ for any $i$ fully determines $\phi$. It is easy to check that for a fixed $i$ with $0 \leq i \leq 6$, the coloring $\phi$ determined by setting $\phi(v_i)=\phi(v_i')$ contains a 2-colored $P_3$.
Conclusion
==========
Here we have computed the star chromatic number of splitting graphs of cycles. It would be interesting to consider star colorings of splitting graphs of other families–including complete multipartite graphs and direct products (alternatively called tensor products or Kronecker products) of cycles and paths. The shadow graph is another common construction in graph labeling (see [@gallian]). One could also bound the star chromatic number of the shadow graphs of various basic families.
Acknowledgments
===============
This research was conducted at the 2017 REU at the University of Minnesota Duluth, supported by NSF/DMS-1659047. We would like to thank Joe Gallian for his incredible support at the REU as well as for reading through this paper.
[1]{}
M. O. Albertson, G. G. Chappell, H. A. Kierstead, A. K[ü]{}ndgen, and R. Ramamurthi, Coloring with no 2-colored $P_4$’s, *Electron. J. Combin.* 11 (2004), \#R26.
Y. Bu, D. W. Cranston, M. Montassier, A. Raspaud, and W. Wang, Star coloring of sparse graphs, *J. Graph Theory* 62(3)(2009), 201-219.
T. F. Coleman and J. J. Mor[é]{}, Estimation of spare Hessian matrices and graph coloring problems, *Math. Program.* 28(1984), 243-270.
G. Fertin, A. Raspaud, and B. Reed, On star coloring of graphs, *J. Graph Theory* 47(3)(2004), 163-182.
H. Furma[ń]{}czyk, Kowsalya V., and Vernold Vivin J., On Star Coloring of Splitting Graphs, arXiv:1705.09357 \[math.CO\] May 2017.
J. Gallian, A dynamic survey of graph labeling, *Electron. J. Combin.* (2016) \# DS6.
A. H. Gebremedhin, F. Manne, and A. Pothen, What color is your Jacobian? Graph coloring for computing derivatives, *SIAM Review* 47(4) (2005), 629-705.
A. H. Gebremedhin, A. Tarafdar, F. Manne, and A. Pothen, New acyclic and star coloring algorithms with application to computing Hessians, *SIAM J. Sci. Comput.* 29(3)(2007), 1042-1072.
B. Gr[ü]{}nbaum, Acyclic colorings of planar graphs, *Israel J. Math.* 14(4)(1973), 390-408.
H. A. Kierstead, A. K[ü]{}ndgen, and C. Timmons, Star coloring bipartite planar graphs, *J. Graph Theory* 60(1)(2009), 1-10.
B. Mohar and S. [Š]{}pacapan, Degenerate and star colorings of graphs on surfaces, *European J. Combin.* 33(3)(2012), 340-349.
C. Timmons, Star coloring high girth planar graphs, *Electron. J. Combin.* (2008) \# R124.
|
---
abstract: |
The goal of this paper is to introduce [*Hodge 1-motives*]{} of algebraic varieties and to state a corresponding cohomological Grothendieck-Hodge conjecture, generalizing the classical Hodge conjecture to arbitrarily singular proper schemes.
We also construct generalized cycle class maps from the ${{\cal K}}$-cohomology groups $H^{p+i}({{\cal K}}_p)$ to the sub-quotiens $W_{2p}H^{2p+i}/W_{2p-2}$ given by the weight filtration. However, in general, the image of this cycle map (as well as the image of the canonical map from motivic cohomology) is strictly smaller than the rational part of the Hodge filtration $F^p$ on $H^{2p+i}$.
author:
- 'by Luca [Barbieri-Viale]{}'
title: 'On algebraic 1-motives related to Hodge cycles'
---
Introduction
============
Let $X$ be an algebraic ${\mbox{$\mathbb C$}}$-scheme. The singular cohomology groups $H^*(X,{\mbox{$\mathbb Z$}}(\cdot))$ carry a mixed Hodge structure, see [@D III]. Deligne theory of 1-motives (see [@D III]) is an algebraic framework in order to deal with [*some*]{} mixed Hodge structures extracted from $H^*(X,{\mbox{$\mathbb Z$}}(\cdot))$, [[*i.e.*]{}, ]{}those having non-zero Hodge numbers in the set $\{(0,0), (0,-1), (-1,0),
(-1,-1)\}$. Therefore, these cohomological invariants of algebraic varieties would be algebraically defined as 1-motives over arbitrary base fields or schemes. Note that a general theory of mixed motives can be regarded as an algebraic framework in order to deal with [*all*]{} mixed Hodge structures $H^*(X,{\mbox{$\mathbb Z$}}(\cdot))$.
A 1-motive $M$ over a scheme $S$ is given by an $S$-homomorphism of group schemes $$M = [L{\stackrel{u}{\rightarrow}} G]$$ where $G$ is an extension of an abelian scheme $A$ by a torus $T$ over $S$, and the group scheme $L$ is, locally for the étale topology on $S$, isomorphic to a given finitely-generated free abelian group. There are Hodge, De Rham and $\ell$-adic realizations (see [@D III] and [@DM]).
If $X$ is a smooth proper ${\mbox{$\mathbb C$}}$-scheme then $H^i(X,{\mbox{$\mathbb Z$}}(j))$ is pure of weight $i-2j$. If $i=2p$ is even a natural 1-motive would be given by the lattice of Hodge cycles in $H^{2p}(X,{\mbox{$\mathbb Z$}}(p))$, [[*i.e.*]{}, ]{}of those integral cohomology classes (modulo torsion) which are of type $(0,0)$. Classical Hodge conjecture claims that (over ${\mbox{$\mathbb Q$}}$) such a 1-motive would be obtained from classes of algebraic cycles on $X$ only. For $i=2p+1$ odd the 1-motive corresponding to $H^{2p+1}(X,{\mbox{$\mathbb Z$}}(p+1))$ is given by the abelian variety associated to the largest sub-Hodge structure whose types are $(-1,0)$ or $(0,-1)$. Grothendieck-Hodge conjecture characterize (over ${\mbox{$\mathbb Q$}}$) this sub-Hodge structure as the coniveau $\geq p$ sub-space, [[*i.e.*]{}, ]{}the abelian variety as the algebraic part of the intermediate jacobian.
Grothendieck-Hodge conjectures are concerned with the quest of an algebraic definition for the named 1-motives. In fact, the usual Hodge conjecture can be reformulated by saying that the Hodge realization of the algebraically defined ${\mbox{$\mathbb Q$}}$-vector space of codimension $p$ algebraic cycles modulo numerical (or homological) equivalence is the 1-motivic part of $H^{2p}(X,{\mbox{$\mathbb Q$}}(p))$. Moreover, the 1-motivic part of $H^{2p+1}(X,{\mbox{$\mathbb Q$}}(p+1))$ would be the Hodge realization of the isogeny class of the universal regular quotient.
The main task of this paper is to define [*Hodge 1-motives*]{} of singular varieties and to state a corresponding cohomological Grothendieck-Hodge conjecture, by dealing with their Hodge realizations.
A short survey of the subject {#a-short-survey-of-the-subject .unnumbered}
-----------------------------
The classical Hodge conjecture along with a tantalizing overview can be found in [@DH]. Recall that Grothendieck corrected the general Hodge conjecture in [@GH]. The book of Lewis [@LW] is a very good compendium of methods and results.
Recall that Jannsen [@JA] formulated an homological version of the Hodge conjecture for singular varieties. Moreover, Bloch in a letter to Jannsen (see the Appendix A in [@JA] [[*cf.*]{} ]{}Section 5), gave a counterexample to a naive cohomological Hodge conjecture for curves on a singular 3-fold. However, in the same letter, Bloch was guessing that the Hodge conjecture for divisors, [[*i.e.*]{}, ]{}$F^1\cap H^2(X,{\mbox{$\mathbb Z$}})$ is generated by $c_1$ of line bundles on $X$, holds true in the singular setting “because one has the exponential”. Anyways, jointly with V. Srinivas, we gave a counterexample to this claim and questioned a reformulation of the Hodge conjecture for divisors in [@BS] by restricting to Zariski locally trivial cohomology classes, [[*i.e.*]{}, ]{}let $L^pH^*(X, {\mbox{$\mathbb Z$}})$ be the filtration induced by the Leray spectral sequence along the canonical continuous map $X_{\rm an}\to X_{\rm Zar}$, is $F^1\cap L^1H^2(X, {\mbox{$\mathbb Z$}})$ given by $c_1$ of line bundles on $X$ ? Still, this reformulation doesn’t hold in general, [[*e.g.*]{}, ]{}see [@BiS] where it is also proved for $X$ normal.
From the work of Carlson (see [@CA] and [@C]) and the theory of Albanese and Picard 1-motives [@BSAP] it now appears that the theory of 1-motives is a natural setting for a formulation of a cohomological version of the Hodge conjectures for singular varieties. For example, $F^1\cap
H^2(X,{\mbox{$\mathbb Z$}})$ is simply given by $c_1$ of simplicial line bundles on a smooth proper hypercovering $\pi : X_{{\mbox{\LARGE $\cdot $}}}\to X$ via universal cohomological descent $\pi^* :H^2(X,{\mbox{$\mathbb Z$}})\cong H^2(X_{{\mbox{\LARGE $\cdot $}}},{\mbox{$\mathbb Z$}})$. This Néron-Severi group ${{\rm NS}\,}({X_{{\mbox{\LARGE $\cdot $}}}}) \cong F^1\cap H^2(X,{\mbox{$\mathbb Z$}})$ is actually independent of the choice of the smooth simplicial scheme. Furthermore, ${{\rm NS}\,}({X_{{\mbox{\LARGE $\cdot $}}}})$ admits an algebraic definition as the quotient of the simplicial Picard group scheme ${\mbox{$\bf Pic$}}_{{X_{{\mbox{\LARGE $\cdot $}}}}}$ by its connected component of the identity ([[*cf.*]{} ]{}[@BSAP]). However, the 1-motivic part of $H^2(X,{\mbox{$\mathbb Z$}})$ is still larger than ${{\rm NS}\,}({X_{{\mbox{\LARGE $\cdot $}}}})$. Therefore the largest algebraic part of $H^2(X,{\mbox{$\mathbb Z$}})$ will be detected from a honest 1-motive only (see [@BRS]).
Note that the rank of the usual ${{\rm NS}\,}(X)$ (= the image of ${{\rm Pic}\,}(X)$ in $H^2(X,{\mbox{$\mathbb Z$}})$) is actually smaller than ${{\rm NS}\,}({X_{{\mbox{\LARGE $\cdot $}}}})$, in general. Moreover $F^1\cap W_0H^2(X,{\mbox{$\mathbb Q$}}) =0$ thus ${{\rm NS}\,}({X_{{\mbox{\LARGE $\cdot $}}}})_{{\mbox{\scriptsize{${\mbox{$\mathbb Q$}}$}}}}$ is naturally a subspace of $H^2(X,{\mbox{$\mathbb Q$}})/W_0$.
Considering the Leray filtration $L^pH^{2p}(X, {\mbox{$\mathbb Q$}})$ a natural question formulated in [@BiS] is if $F^p\cap L^pH^{2p}(X, {\mbox{$\mathbb Q$}})$ will be given by higher Chern classes. However, this would not be true without some extra hypothesis on $X$ and does not tell enough about the algebraic part of all $H^{2p}(X, {\mbox{$\mathbb Q$}})$.
An outline of the conjectural picture {#an-outline-of-the-conjectural-picture .unnumbered}
-------------------------------------
Let $X$ be a proper integral ${\mbox{$\mathbb C$}}$-scheme. Let $H {\mbox{\,$\stackrel{\rm def}{=}$}\,}H^{2p+i}(X)$ be our mixed Hodge structure on $H^{2p+i}(X, {\mbox{$\mathbb Z$}})/ {\rm (torsion)}$ for a fixed pair of integers $p\geq 0$ and $i\in{\mbox{$\mathbb Z$}}$. First remark that we always have an extension $$0\to{{\rm gr}\,}^W_{2p-1}H \to W_{2p}H/W_{2p-2}H\to {{\rm gr}\,}^W_{2p}H\to 0.$$ An extension always defines an extension class map $$e^p: H^{p,p}_{{\mbox{\scriptsize{${\mbox{$\mathbb Z$}}$}}}}
\to J^p(H)$$ which is not, in general, a 1-motive. In fact, $J^p(H)$ is a complex torus which is not an abelian variety, in general. Recall that Carlson [@CA] studied abstract extensions of Hodge structures showing their geometric content for low-dimensional varieties. For higher dimensional schemes consider the largest abelian subvariety $A^p(H)$ of the torus $J^p(H)$. Denote $H^p(H)$ the group of Hodge cycles, that is the preimage in $H^{p,p}_{{\mbox{\scriptsize{${\mbox{$\mathbb Z$}}$}}}}$ of $A^p(H)$ under the extension class map. Note that an example due to Srinivas shows that the group $H^p(H)$ of Hodge cycles in this sense can be strictly smaller than $H^{p,p}_{{\mbox{\scriptsize{${\mbox{$\mathbb Z$}}$}}}}$ (see Section 5.2).
Define the [*Hodge 1-motive*]{} of the mixed Hodge structure $H$ the so obtained 1-motive $$e^p: H^{p}(H)\to A^p(H).$$ Conversely, Deligne’s theory of 1-motives [@D] grant us of a mixed Hodge structure $H^h$ corresponding to this 1-motive (see Section 2.2 for details). According to Deligne’s philosophy of 1-motives there should be an algebraically defined 1-motive whose Hodge realization is $H^h$. The algebraic definition (see Section 2.1) is predictable [*via*]{} Grothendieck-Hodge conjectures and Bloch-Beilinson motivic world as follows.
Assume $X$ smooth. Consider the filtration $F^*_a$ on the Chow group $CH^p(X)$ given by $F^0_a = CH^p(X)$, $F^1_a = CH^p(X)_{\rm alg}$ the sub-group of cycles algebraically equivalent to zero and $F^2_a=$ the kernel of the Abel-Jacobi map. Thus the graded pieces are ${{\rm gr}\,}^0_{F_a} =
NS^p(X)$, the Néron-Severi group, and ${{\rm gr}\,}^1_{F_a} = J^p_a(X)=$ the group of ${\mbox{$\mathbb C$}}$-points of an abelian subvariety of the intermediate jacobian. We then get an extension $$0 \to J^p_a(X) \to CH^p(X)/F^2_a \to NS^p(X)\to
0.$$ Note that Grothendieck-Hodge conjecture claims that $J^p_a(X)=A^p(H^{2p-1}(X))$ up to isogeny, [[*i.e.*]{}, ]{}$J^p_a(X)$ is the largest abelian subvariety of the intermediate jacobian.
If $X$ is not smooth then let ${X_{{\mbox{\LARGE $\cdot $}}}}$ be a smooth proper simplicial scheme along with $\pi : {X_{{\mbox{\LARGE $\cdot $}}}}\to X$, a universal cohomological descent morphism ([[*cf.*]{} ]{}[@GRO]). In zero characteristic, such ${X_{{\mbox{\LARGE $\cdot $}}}}$ was firstly provided by the construction of hypercoverings in [@D], then by that of cubical hyperresolutions in [@GN] where the dimensions of the components are bounded or by the method of hyperenvelopes given in [@GS]. In [@DJ] such a simplicial scheme is provided in positive characteristics.
The above extension, given by the filtration $F^*_a$ on the Chow groups of each component of the so obtained simplicial scheme ${X_{{\mbox{\LARGE $\cdot $}}}}$, yields a short exact sequence of complexes. Let $(NS^p)^{\bullet}$ and $(J^p_a)^{\bullet}$ denote such complexes. By taking homology groups we then get boundary maps $$\lambda^i_a : H^i((NS^p)^{\bullet}) \to
H^{i+1}((J^p_a)^{\bullet}).$$ We conjecture that the boundary map $\lambda^i_a$ behave well with respect to the extension class map $e^p$ yielding a motivic cycle class map, [[*i.e.*]{}, ]{}the following diagram $$\begin{array}{ccc}
H^{i}((NS^p)^{\bullet})&{\stackrel{\lambda^i_a}{\rightarrow}}& H^{i+1}((J^p_a)^{\bullet})\\
\downarrow & &\downarrow\\
H^{2p+i}(X)^{p,p} & {\stackrel{e^p}{\rightarrow}} & J^p(H^{2p+i}(X))
\end{array}$$ commutes. Note that all maps in the square are canonically defined. We guess that the image 1-motive (up to isogeny!) is the above Hodge 1-motive of the mixed Hodge structure (see Conjecture \[MHC\] for a full statement). In fact, we may expect $J^p_a$ would be obtained as the universal regular quotient of $CH^p(X)_{\rm alg}$ and that the filtration $F^*_a$ would be induced by the motivic filtration conjectured by Bloch, Murre and Beilinson. Accordingly we can sketch an algebraic definition of such Hodge 1-motives (see Section 2.1).
If $X$ is singular one is then puzzled by the role of $${{\rm Hom}\,}_{\rm MHS}
({\mbox{$\mathbb Z$}}(-p), H^{2p+i}(X))$$ where the ${{\rm Hom}\,}$ is taken in the abelian category of mixed Hodge structures. That is the integral part of $F^p$ (the Hodge filtration on $H^{2p+i}(X,{\mbox{$\mathbb C$}})$) which is contained in the kernel of the extension class map $e^p$ above. Note that $F^p\cap W_{2p-2}H^{2p+i}(X,{\mbox{$\mathbb Q$}})=
0$ here. In the smooth case, such a target is usually reached by algebraic cycles. In order to obtain cycle class maps we may use local higher Chern classes and edge maps in coniveau spectral sequences (see [@BO] and [@BV2]).
In the singular case, we show that such edge maps can be recovered by weight arguments. In order to do this we define [*Zariski sheaves*]{} of mixed Hodge structures, obtaining [*infinite dimensional*]{} mixed Hodge structures on their cohomology (see Section 3). The main example is given by the Zariski sheaf ${{\cal H}}^*_X$ associated to the presheaf $U\subset X
\mapsto H^*(U)$ of mixed Hodge structures. On the smooth simplicial scheme ${X_{{\mbox{\LARGE $\cdot $}}}}$ we also have a simplicial sheaf ${{\cal H}}_{{X_{{\mbox{\LARGE $\cdot $}}}}}^*$ of mixed Hodge structures. Since $\pi : {X_{{\mbox{\LARGE $\cdot $}}}}\to X$ yields $H^*(X)\cong {\mbox{$\mathbb H$}}^{*}(X_{{\mbox{\LARGE $\cdot $}}})$ we then obtain a local-to-global spectral sequence $$L^{p,q}_2 =
{\mbox{$\mathbb H$}}^p(X_{{\mbox{\LARGE $\cdot $}}}, {{\cal H}}_{{X_{{\mbox{\LARGE $\cdot $}}}}}^q){\mbox{$\Rightarrow$}}H^{p+q}(X)$$ in the category of infinite dimensional mixed Hodge structures. The sheaf ${{\cal H}}_{{X_{{\mbox{\LARGE $\cdot $}}}}}^q$ has weights $\leq 2q$ and the same holds for its cohomology. There is an edge map (see Section 4) $$s\ell^{p+i}: {\mbox{$\mathbb H$}}^{p+i}(X_{{\mbox{\LARGE $\cdot $}}}, {{\cal H}}_{{X_{{\mbox{\LARGE $\cdot $}}}}}^p)/W_{2p-2} \to
W_{2p}H^{2p+i}(X)/W_{2p-2}.$$ We expect that the image of $s\ell^{p+i}$ is the mixed Hodge structure $H^{2p+i}(X)^h$ corresponding to the Hodge 1-motive.
Moreover, consider ${{\cal K}}$-cohomology groups ${\mbox{$\mathbb H$}}^*(X_{{\mbox{\LARGE $\cdot $}}},
{{\cal K}}_p)$ where ${{\cal K}}_p$ are the simplicial sheaves associated to Quillen’s higher $K$-theory. Recall that there are local higher Chern classes $$c_p : {{\cal K}}_p \to {{\cal H}}_{{X_{{\mbox{\LARGE $\cdot $}}}}}^p(p)$$ for each $p\geq 0$. We thus obtain a generalized cycle class map $$c\ell^{p+i} : {\mbox{$\mathbb H$}}^{p+i}(X_{{\mbox{\LARGE $\cdot $}}}, {{\cal K}}_p)_{{\mbox{\scriptsize{${\mbox{$\mathbb Q$}}$}}}} \to
W_{2p}H^{2p+i}(X, {\mbox{$\mathbb Q$}})/W_{2p-2}.$$ However the image of $c\ell^{p+i}$ is not $F^p\cap H^{2p+i}(X, {\mbox{$\mathbb Q$}})$, [[*i.e.*]{}, ]{}the rational part of the Hodge filtration can be larger (see Section 5.1 where Bloch’s counterexample is explained). The same applies to the canonical map $H^{2p+i}_m(X, {\mbox{$\mathbb Q$}}(p)) \to H^{2p+i}(X, {\mbox{$\mathbb Q$}}(p))$ from motivic cohomology.
Towards Hodge mixed motives {#towards-hodge-mixed-motives .unnumbered}
---------------------------
Any reasonable theory of mixed motives would include the theory of 1-motives, [[*i.e.*]{}, ]{}it would be a fully faithful functor from the ${\mbox{$\mathbb Q$}}$-linear category of 1-motives to that of mixed motives. This is the case of the triangulated category of geometrical motives introduced by Voevodsky (see [@V 3.4], [[*cf.*]{} ]{}[@LM] and [@LI]). Hanamura’s construction (see [@HA] and [@HAM]) doesn’t apparently provide such a property as yet.
As remarked by Grothendieck [@GH §2] and Deligne [@DH §5] the Hodge conjecture yields nice properties of the Hodge realization of pure motives, [[*i.e.*]{}, ]{}the usual Hodge conjecture means that the Hodge realization functor is fully faithful. It would be interesting to investigate such a property in the mixed case, [[*e.g.*]{}, ]{}if this formulation of the Hodge conjecture provide such a property of mixed motives.
We remark that M. Saito recently observed (see [@MS 2.5 (ii)] and [@MSC]) that the canonical functor from arithmetic Hodge structures to mixed Hodge structures is not full. Even if the Hodge realization factors through arithmetic Hodge structures, this non-fullness doesn’t imply the non-fullness of the Hodge realization of mixed motives (as noticed by M. Saito as well).
However, the first natural attempt to go further with Hodge mixed motives is to provide an intrinsic definition of such objects internally. In fact, since 1-motives provide mixed motives we may claim that such Hodge mixed motives exist and would be naturally defined over any field or base scheme.
Acknowledgements {#acknowledgements .unnumbered}
----------------
Papers happen because many people co-operate with one another. Often the author is just one of a whole group of people who pitch in, and that was the case here.
Gratitude is due therefore, firstly, to V. Srinivas: he has been exceedingly generous to me with his time, shared enthusiasms over the years and told quite a few tricks.
A huge intellectual debt is due to P. Deligne who also interceded with advice that helped on several portions of the manuscript.
I am grateful also to S. Bloch, O. Gabber, H. Gillet, M. Hanamura, U. Jannsen, M. Levine, J. D. Lewis, J. Murre, A. Rosenschon, M. Saito, C. Soulé and V. Voevodsky for discussions on some matters treated herein.
This research was carried out with smooth efficiency thanks to several foundations. I like to mention Tata Institute of Fundamental Research and Institut Henri Poincaré for their support and hospitality.
Filtrations on Chow groups
==========================
Following the general framework of mixed motives ([[*e.g.*]{}, ]{}see [@DM], [@LM] and [@JM] for a full overview) we may expect the following picture for non-singular algebraic varieties over a field $k$ (algebraically closed of characteristic zero for simplicity).
Let $X$ be a smooth proper $k$-scheme. Bloch, Beilinson and Murre ([[*cf.*]{} ]{}[@JM], [@JF] and [@MU3]) conjectured the existence of a finite filtration $F_m^*$ on Chow groups $CH^p(X){\mbox{\,$\stackrel{\rm def}{=}$}\,}{{\cal Z}}^p(X)/\equiv_{\rm rat}$ of codimension $p$ cycles modulo rational equivalence such that
- $F_m^0CH^p(X)=CH^p(X)$,
- $F_m^1CH^p(X)$ is given by $CH^p(X)_{\rm hom}$, [[*i.e.*]{}, ]{}by the sub-group of those codimension $p$ cycles which are homologically equivalent to zero for some Weil cohomology theory,
- $F_m^*CH^p(X)$ should be functorial and compatible with the intersection pairing, and
- this filtration should be motivic, [[*e.g.*]{}, ]{}${{\rm gr}\,}_{F_m}^iCH^p(X)_{{\mbox{\scriptsize{${\mbox{$\mathbb Q$}}$}}}}$ depends only on the Grothendieck motive $h^{2p-i}(X).$
Regular homomorphisms
---------------------
Consider the sub-group $CH^p(X)_{\rm alg}$ of those cycles in $CH^p(X)$ which are algebraically equivalent to zero, [[*i.e.*]{}, ]{}$CH^p(X)_{\rm alg}{\mbox{\,$\stackrel{\rm def}{=}$}\,}\ker
(CH^p(X)\to NS^p(X)).$ Denote $CH^p(X)_{\rm ab}$ the sub-group of $CH^p(X)_{\rm alg}$ of those cycles which are abelian equivalent to zero, [[*i.e.*]{}, ]{}$CH^p(X)_{\rm ab}$ is the intersection of all kernels of regular homomorphisms from $CH^p(X)_{\rm alg}$ to abelian varieties.
Assume the existence of a universal regular homomorphism $\rho^p
:CH^p(X)_{\rm alg}\to A^p_{X/k}(k)$ to (the group of $k$-points) of an abelian variety $A^p_{X/k}$ defined over the base field $k$ ([[*cf.*]{} ]{}[@LB]). This is actually proved for $p=1, 2, \dim(X)$ (see [@MU1]).
Thus $CH^p(X)_{\rm alg}\subseteq CH^p(X)_{\rm hom}$ and there would be an induced functorial filtration $F_a^*$ on $CH^p(X)$ such that
- $F_a^0CH^p(X) = CH^p(X)$,
- $F_a^1CH^p(X) = CH^p(X)_{\rm alg}$ is the sub-group of cycles algebraically equivalent to zero, and
- $F_a^2CH^p(X) = CH^p(X)_{\rm ab}$, [[*i.e.*]{}, ]{}is the kernel of the universal regular homomorphism $\rho^p$ defined above.
Note that the existence of the abelian variety $A^p_{X/k}$ is not explicitly mentioned in the context of mixed motives but is a rather natural property after the case $k={\mbox{$\mathbb C$}}$.
For $X$ smooth and proper over ${\mbox{$\mathbb C$}}$ one obtains that the motivic filtration is such that [*(i)*]{} $F_m^1CH^p(X)=CH^p(X)_{\rm hom}$ is the sub-group of cycles whose cycle class in $H^{2p}(X,{\mbox{$\mathbb Z$}}(p))$ is zero, and [*(ii)*]{} $F_m^2CH^p(X)$ is contained in the kernel of the Abel-Jacobi map $CH^p(X)_{\rm hom}\to J^p(X)$ and $F_m^2CH^p(X)\cap CH^p(X)_{\rm alg}$ is the kernel of the Abel-Jacobi map $CH^p(X)_{\rm alg}\to J^p(X)$. In this case, $CH^p(X)_{\rm ab}$ will be the kernel of the Abel-Jacobi map $CH^p(X)_{\rm alg}\to J^p(X)$, [[*i.e.*]{}, ]{}[*(iii)*]{} $A^p_{X/{\mbox{\scriptsize{${\mbox{$\mathbb C$}}$}}}}$ is the algebraic part of the intermediate jacobian. Is well known that the image $J^p_a(X)$ of $CH^p(X)_{\rm alg}$ into $J^p(X)$ yields a sub-torus of $J^p(X)$ which is an abelian variety: moreover, is known to be universal for $p=1, 2,
\dim(X)$ (see [@MU1] and [@MU2]).
In the following, for the sake of simplicity, the reader can indeed assume that $k ={\mbox{$\mathbb C$}}$ and [*(i)–(iii)*]{} are satisfied by the first two steps of the filtration $F^i_m$. In fact, for $k={\mbox{$\mathbb C$}}$, S. Saito has obtained (up to torsion!) such a result (see [@SA Prop. 2.1], [[*cf.*]{} ]{}[@LF] and [@JF]). Moreover, in the following, the reader could also avoid reference to the motivic filtration by dealing with the first two steps of the “algebraic” filtration $F^i_a$ defined above.
Extensions
----------
Let $X$ be smooth and proper over $k$. For our purposes just consider the following extension $$\label{cyclext}
0\to {{\rm gr}\,}^1_{F_m}CH^p(X) \to CH^p(X)/F_m^2 \to {{\rm gr}\,}^0_{F_m}CH^p(X) \to 0$$ Note that $A^p_{X/k}(k)$ is contained in ${{\rm gr}\,}^1_{F_m}CH^p(X)$ (since $F^2_a= F^2_m\cap CH^p(X)_{\rm alg}$) and ${{\rm gr}\,}^0_{F_m}CH^p(X)$ has finite rank.
For $p=1$ this extension is the usual extension associated to the connected component of the identity of the Picard functor, [[*i.e.*]{}, ]{}$A^1_{X/k}={{\rm Pic}\,}_{X/k}^0$ and ${{\rm gr}\,}^0_{F_m}$ is the Néron-Severi of $X$. If $p=\dim X$ then $F_m^1$ will be the kernel of the degree map and $F_m^2$ is the Albanese kernel, [[*i.e.*]{}, ]{}$A^{\dim X}_{X/k}$ is the Albanese variety and ${{\rm gr}\,}^0_{F_m}= {\mbox{$\mathbb Z$}}^{\oplus c}$ where $c$ is the number of components of $X$.
However, if $1<p<\dim X$ then $CH^p(X)_{\rm alg}\neq CH^p(X)_{\rm hom}$ in general. Let $Grif^p(X)$ denote the quotient group, [[*i.e.*]{}, ]{}the Griffiths group of $X$. Since ${{\rm gr}\,}^1_{F_a}CH^p(X) = A^p_{X/k}(k)$ and ${{\rm gr}\,}^0_{F_a}CH^p(X)=NS^p(X)$ we then have a diagram with exact rows and columns $$\label{abelext}
\begin{array}{ccccccc}
&& &0&&0&\\
&& &\downarrow&&\downarrow&\\
&0&\to &F^2_m/F^2_a&\to&Grif^p(X)&\\
&\downarrow& &\downarrow&&\downarrow&\\
0\to &A^p_{X/k}(k)&\to& CH^p(X)/F_a^2& \to & NS^p(X)& \to 0\\
&\downarrow& &\downarrow&&\downarrow&\\
0\to &{{\rm gr}\,}^1_{F_m}CH^p(X)& \to& CH^p(X)/F_m^2 &\to & {{\rm gr}\,}^0_{F_m}CH^p(X)&
\to 0\\
&& &\downarrow&&\downarrow&\\
&& &0&&0&
\end{array}$$ Note that these extensions still fail to be of the same kind of the Pic extension. However, considering the extension $$0\to CH^p(X)_{\rm alg} \to
CH^p(X) \to NS^p(X) \to 0$$ we may hope for a natural regular homomorphism to $k$-points of an abstract extension in the category of group schemes (locally of finite type over $k$) $$0\to A \to G \to N \to
0$$ where $G$ is a commutative group scheme which is an extension of a discrete group of finite rank $N$, associated to the abelian group of codimension p cycles modulo numerical equivalence, by an abelian variety $A$, isogenous to the universal regular quotient ([[*cf.*]{} ]{}[@GR]). If $k={\mbox{$\mathbb C$}}$ it is easy to see that such extension $G({\mbox{$\mathbb C$}})$ exists trascendentally.
Hodge 1-motives
===============
Let $k$ be a field, for simplicity, algebraically closed of characteristic zero. Consider the ${\mbox{$\mathbb Q$}}$-linear abelian category ${\rm 1-Mot}_k$ of 1-motives over $k$ with rational coefficients (see [@D] and [@BSAP]). Denote $M_{{\mbox{\scriptsize{${\mbox{$\mathbb Q$}}$}}}}$ the isogeny class of a 1-motive $M=[L\to G]$. The category ${\rm 1-Mot}_k$ contains (as fully faithful abelian sub-categories) the tensor category of finite dimensional ${\mbox{$\mathbb Q$}}$-vector spaces as well as the semi-simple abelian category of isogeny classes of abelian varieties. The Hodge realization (see [@D] and [@BSAP]) is a fully faithful functor $$T_{\rm Hodge}: {\rm 1-Mot}_{{\mbox{\scriptsize{${\mbox{$\mathbb C$}}$}}}}{\hookrightarrow}{\rm MHS}\hspace*{1cm}
M_{{\mbox{\scriptsize{${\mbox{$\mathbb Q$}}$}}}}\mapsto T_{\rm Hodge}(M_{{\mbox{\scriptsize{${\mbox{$\mathbb Q$}}$}}}})$$ defining an equivalence of categories between ${\rm 1-Mot}_{{\mbox{\scriptsize{${\mbox{$\mathbb C$}}$}}}}$ and the abelian sub-category of mixed ${\mbox{$\mathbb Q$}}$-Hodge structures of type $\{(0,0), (0,-1), (-1,0), (-1,-1)\}$ such that ${{\rm gr}\,}_{-1}^W$ is polarizable. Under this equivalence a ${\mbox{$\mathbb Q$}}$-vector space corresponds to a ${\mbox{$\mathbb Q$}}$-Hodge structure purely of type $(0,0)$ and an isogeny class of an abelian variety corresponds to a polarizable ${\mbox{$\mathbb Q$}}$-Hodge structure purely of type $\{(0,-1), (-1,0)\}$.
Algebraic construction
----------------------
Let $X$ be a proper scheme over $k$. We perform such a construction for simplicial schemes ${X_{{\mbox{\LARGE $\cdot $}}}}$ coming from universal cohomological descent morphisms $\pi :
{X_{{\mbox{\LARGE $\cdot $}}}}\to X$ ([[*cf.*]{} ]{}[@D], [@GN], [@GS], [@HA] and [@DJ]).
Let ${X_{{\mbox{\LARGE $\cdot $}}}}$ be such a proper smooth simplicial scheme over the base field $k$. By functoriality, the filtration $F_m^jCH^p$ on each component $X_i$ of ${X_{{\mbox{\LARGE $\cdot $}}}}$ yields a complex $$(F_m^jCH^p)^{ \bullet}: \cdots\to
F_m^jCH^p(X_{i-1}){\stackrel{\delta^{*}_{i-1}}{\rightarrow}}
F_m^jCH^p(X_{i}){\stackrel{\delta^{*}_{i}}{\rightarrow}}F_m^jCH^p(X_{i+1})\to\cdots$$ where $\delta^{*}_{i}$ is the alternating sum of the pullback along the face maps $\partial^{k}_{i}: X_{i+1}\to X_i$ for $0\leq k \leq i+1$.
The complex of Chow groups $(CH^p)^{ \bullet}$, induced from the simplicial structure as above, is filtered by sub-complexes: $$0\subseteq (F_m^pCH^p)^{ \bullet}\subseteq \cdots \subseteq (F_m^1CH^p)^{
\bullet} \subseteq (F_m^0CH^p)^{ \bullet}= (CH^p)^{ \bullet}.$$ Define $F_m^*(CH^p)^{ \bullet}{\mbox{\,$\stackrel{\rm def}{=}$}\,}(F_m^*CH^p)^{ \bullet}.$
The extension (\[cyclext\]) given by the filtration $F_m^*CH^p(X_i)$ on each component $X_i$ of the simplicial scheme ${X_{{\mbox{\LARGE $\cdot $}}}}$, for a fixed $p\geq
0$, yields the following short exact sequence of complexes $$\label{simpext}
0\to {{\rm gr}\,}^1_{F_m}(CH^p)^{ \bullet} \to (CH^p)^{ \bullet}/F_m^2 \to
{{\rm gr}\,}^0_{F_m}(CH^p)^{ \bullet} \to 0$$ Note that $ {{\rm gr}\,}^1_{F_m}(CH^p)^{i}$ contains the group of $k$-points of the abelian variety $A_{X_i/k}^p$ and, moreover ${{\rm gr}\,}^0_{F_m}(CH^p)^{i}_{{\mbox{\scriptsize{${\mbox{$\mathbb Q$}}$}}}}$ is the finite dimensional vector space of codimension p cycles on $X_i$ modulo homological (or numerical) equivalence.
From (\[simpext\]) we then get a long exact sequence of homology groups and, in particular, we obtain boundary maps $$\lambda^i_m
:H^{i}({{\rm gr}\,}^0_{F_m}(CH^p)^{\bullet}) \to
H^{i+1}({{\rm gr}\,}^1_{F_m}(CH^p)^{\bullet}).$$ Denote $A_{{X_{{\mbox{\LARGE $\cdot $}}}}/k}^p$ the complex of abelian varieties $A_{X_i/k}^p$. Since $A_{{X_{{\mbox{\LARGE $\cdot $}}}}/k}^p(k)$ is a sub-complex of ${{\rm gr}\,}^1_{F_m}(CH^p)^{\bullet}$ we then get induced (functorial) maps on homology groups $$\theta^i: H^{i+1}(A_{{X_{{\mbox{\LARGE $\cdot $}}}}/k}^p(k))\to
H^{i+1}({{\rm gr}\,}^1_{F_m}(CH^p)^{\bullet}).$$ Note that (\[simpext\]) is involved in a functorial diagram (\[abelext\]). The corresponding complex of Néron-Severi groups $(NS^p)^{\bullet}$ yield boundary maps $$\label{bound}
\lambda^i_a :H^{i}((NS^p)^{\bullet}) \to H^{i+1}(A_{{X_{{\mbox{\LARGE $\cdot $}}}}/k}^p(k)).$$ These maps fit into the following commutative square $$\begin{array}{ccc} H^{i}({{\rm gr}\,}^0_{F_m}(CH^p)^{\bullet})&
{\stackrel{\lambda^i_m}{\longrightarrow}}& H^{i+1}({{\rm gr}\,}^1_{F_m}(CH^p)^{\bullet})\\
\uparrow& &\uparrow{\ \scriptsize \theta^i}\\
H^{i}((NS^p)^{\bullet})&{\stackrel{\lambda^i_a}{\longrightarrow}}&H^{i+1}(A_{{X_{{\mbox{\LARGE $\cdot $}}}}/k}^p(k)).
\end{array}$$ Moreover, the kernel of $\theta^i$ is clearly equal to the image of the boundary map $$\tau^i :H^{i}({{\rm gr}\,}^1_{F_m}(CH^p)^{\bullet}/A_{{X_{{\mbox{\LARGE $\cdot $}}}}/k}^p(k))\to
H^{i+1}(A_{{X_{{\mbox{\LARGE $\cdot $}}}}/k}^p(k)).$$
Is the image of the connected component of the identity of $H^{i+1}(A_{{X_{{\mbox{\LARGE $\cdot $}}}}/k}^p)$ under $\theta^i$ an abelian variety, [[*e.g.*]{}, ]{}is $\tau^i = 0$ up to a finite group ?
This is clearly the case if $p=1$ (see below for the case $k={\mbox{$\mathbb C$}}$). For $k={\mbox{$\mathbb C$}}$ this question is related to Griffiths Problem E in [@GR] asking a description of the “invertible points” of the intermediate jacobians (also [[*cf.*]{} ]{}Mumford-Griffiths Problem F in [@GR]).
Assume that the above question has a positive answer and denote $H^{i+1}(A_{{X_{{\mbox{\LARGE $\cdot $}}}}/k}^p)^{\dag}$ the so obtained abelian variety. We then obtain an algebraically defined 1-motive as follows.
Let $H^{i}({{\rm gr}\,}^0_{F_m}(CH^p)^{\bullet})^{\dag}$ be the sub-group of those elements in $H^{i}({{\rm gr}\,}^0_{F_m}(CH^p)^{\bullet})$ which are mapped to $H^{i+1}(A_{{X_{{\mbox{\LARGE $\cdot $}}}}/k}^p)^{\dag}$ under the boundary map $\lambda^i_m$ above.
[Let ${X_{{\mbox{\LARGE $\cdot $}}}}$ be such a smooth proper simplicial scheme over $k$. Denote $$\Xi^{i,p}{\mbox{\,$\stackrel{\rm def}{=}$}\,}[ H^{i}({{\rm gr}\,}^0_{F_m}(CH^p)^{\bullet})^{\dag}
{\stackrel{\lambda^i_m}{\longrightarrow}}
H^{i+1}(A_{{X_{{\mbox{\LARGE $\cdot $}}}}/k}^p)^{\dag}]_{{\mbox{\scriptsize{${\mbox{$\mathbb Q$}}$}}}}$$ the isogeny 1-motive obtained from the construction above. Call $ \Xi^{i,p}$ the [*Hodge 1-motive*]{} of the simplicial scheme.]{}
We expect that $\Xi^{i,p}$ is independent of the choice of $\pi : {X_{{\mbox{\LARGE $\cdot $}}}}\to
X$. The main motivation for questioning the existence of such a purely algebraic construction is given by the following analytic counterpart.
Analytic construction
---------------------
Let ${\rm MHS}$ be the abelian category of usual Deligne’s mixed Hodge structures [@D]. An object $H$ of ${\rm MHS}$ is defined as a triple $H = (H_{{\mbox{\scriptsize{${\mbox{$\mathbb Z$}}$}}}}, W, F)$ where $H_{{\mbox{\scriptsize{${\mbox{$\mathbb Z$}}$}}}}$ is a finitely generated ${\mbox{$\mathbb Z$}}$-module, $W$ is a finite increasing filtration on $H_{{\mbox{\scriptsize{${\mbox{$\mathbb Z$}}$}}}}\otimes{\mbox{$\mathbb Q$}}$ and $F$ is a finite decreasing filtration on $H_{{\mbox{\scriptsize{${\mbox{$\mathbb Z$}}$}}}}\otimes{\mbox{$\mathbb C$}}$ such that $W, F$ and ${\overline}F$ is a system of opposed filtrations.
Let $H\in {\rm MHS}$ be a torsion free mixed Hodge structure with positive weights. Let $W_*H$ denote the sub-structures defined by the intersections of the weight filtration and $H_{{\mbox{\scriptsize{${\mbox{$\mathbb Z$}}$}}}}$. Let $p$ be a fixed positive integer and assume that ${{\rm gr}\,}^W_{2p-1}H$ is polarizable.
Consider the following extension in the abelian category ${\rm MHS}$ $$\label{ext} 0\to {{\rm gr}\,}^W_{2p-1}H
\to \frac{W_{2p}H}{W_{2p-2}H}\to {{\rm gr}\,}^W_{2p}H \to 0$$ Taking ${{\rm Hom}\,}({\mbox{$\mathbb Z$}}(-p), -)$ we get the extension class map $$e^p: {{\rm Hom}\,}({\mbox{$\mathbb Z$}}(-p),{{\rm gr}\,}^W_{2p}H)\to {{\rm Ext}\,}({\mbox{$\mathbb Z$}}(-p), {{\rm gr}\,}^W_{2p-1}H)$$ where ${{\rm Hom}\,}({\mbox{$\mathbb Z$}}(-p),{{\rm gr}\,}^W_{2p}H)=H_{{\mbox{\scriptsize{${\mbox{$\mathbb Z$}}$}}}}^{p,p}$ is the sub-structure of $(p,p)$-classes in ${{\rm gr}\,}^W_{2p}H$ and $${{\rm Ext}\,}({\mbox{$\mathbb Z$}}(-p), {{\rm gr}\,}^W_{2p-1}H)\cong J^p(H){\mbox{\,$\stackrel{\rm def}{=}$}\,}\frac{{{\rm gr}\,}^W_{2p-1}H_{{\mbox{\scriptsize{${\mbox{$\mathbb C$}}$}}}}}{F^p +{{\rm gr}\,}^W_{2p-1}H_{{\mbox{\scriptsize{${\mbox{$\mathbb Z$}}$}}}}}$$ is a compact complex torus. Note that ([[*cf.*]{} ]{}[@CA]) $${{\rm Ext}\,}(H_{{\mbox{\scriptsize{${\mbox{$\mathbb Z$}}$}}}}^{p,p},
{{\rm gr}\,}^W_{2p-1}H)\cong {{\rm Hom}\,}(H_{{\mbox{\scriptsize{${\mbox{$\mathbb Z$}}$}}}}^{p,p},J^p(H)).$$ Thus $e^p\in {{\rm Hom}\,}(H_{{\mbox{\scriptsize{${\mbox{$\mathbb Z$}}$}}}}^{p,p}, J^p(H))$ corresponds to a unique extension class $$\label{he} 0\to
{{\rm gr}\,}^W_{2p-1}H \to H^e \to H_{{\mbox{\scriptsize{${\mbox{$\mathbb Z$}}$}}}}^{p,p}\to 0$$ which is the pull-back extension associated to $H_{{\mbox{\scriptsize{${\mbox{$\mathbb Z$}}$}}}}^{p,p}{\hookrightarrow}{{\rm gr}\,}^W_{2p}H$ and (\[ext\]). Moreover, since we always have ${{\rm gr}\,}^W_{2p-1}H\cap F^p =0$ then $$F^p\cap H^e_{{\mbox{\scriptsize{${\mbox{$\mathbb Z$}}$}}}} = \ker (H_{{\mbox{\scriptsize{${\mbox{$\mathbb Z$}}$}}}}^{p,p}{\stackrel{e^p}{\rightarrow}}J^p(H)).$$ Now, if ${{\rm gr}\,}^W_{2p-1}H$ is (polarizable) of level $1$ then the torus $J^p(H)$ is an abelian variety and $H^e$ is the Hodge realization of the 1-motive over ${\mbox{$\mathbb C$}}$ defined by the extension class map $e^p$ above.
In general, let $H^{\prime}$ be the largest sub-structure of $W_{2p-1}H$ which is polarizable and purely of type $\{(p-1,p), (p,p-1)\}$ modulo $W_{2p-2}H$, [[*i.e.*]{}, ]{}if $$H^{2p-1}_a {\mbox{\,$\stackrel{\rm def}{=}$}\,}(H^{p-1,p}+ H^{p,p-1})_{{\mbox{\scriptsize{${\mbox{$\mathbb Z$}}$}}}}$$ is the polarizable sub-structure of ${{\rm gr}\,}^W_{2p-1}H$ of those elements which are purely of the above type, then $H^{\prime}$ is defined by the following pull-back extension $$0\to W_{2p-2}H \to H^{\prime}\to H^{2p-1}_a\to 0,$$ along the canonical projection $W_{2p-1}H{\mbox{$\to\!\!\!\!\to$}}{{\rm gr}\,}^W_{2p-1}H.$
Let $H^{\prime\prime}\subseteq W_{2p}H$ be defined by the following pull-back extension $$0\to W_{2p-1}H \to H^{\prime\prime}\to H_{{\mbox{\scriptsize{${\mbox{$\mathbb Z$}}$}}}}^{p,p}
\to 0,$$ along the canonical projection $W_{2p}H{\mbox{$\to\!\!\!\!\to$}}{{\rm gr}\,}^W_{2p}H.$ Thus, the extension (\[he\]) can be regarded as the push-out of such extension involving $H^{\prime\prime}$ along $W_{2p-1}H{\mbox{$\to\!\!\!\!\to$}}{{\rm gr}\,}^W_{2p-1}H.$ Namely, we obtain that $$\frac{H^{\prime\prime}}{W_{2p-2}H}=H^e$$ fitting in the extension $$\label{heprime} 0\to \frac{H^{\prime}}{W_{2p-2}H} \to H^e
\to \frac{H^{\prime\prime}}{H^{\prime}}\to 0.$$ Let $$h^p: {{\rm Hom}\,}({\mbox{$\mathbb Z$}}(-p),\frac{H^{\prime\prime}}{H^{\prime}})\to {{\rm Ext}\,}({\mbox{$\mathbb Z$}}(-p),
\frac{H^{\prime}}{W_{2p-2}H})$$ be the extension class map obtained from (\[heprime\]).
The map $h^p$ above yields a 1-motive over ${\mbox{$\mathbb C$}}$ which is just the restriction of $e^p:H_{{\mbox{\scriptsize{${\mbox{$\mathbb Z$}}$}}}}^{p,p}\to J^p(H)$ to the largest abelian subvariety in $J^p(H)$. In particular: $\ker (h^p) =
\ker (e^p).$
Since $H^{\prime}/W_{2p-2}H=
H_a^{2p-1}$ by construction we have that $ {{\rm Ext}\,}({\mbox{$\mathbb Z$}}(-p),H_a^{2p-1})$ is the largest abelian sub-variety of $ {{\rm Ext}\,}({\mbox{$\mathbb Z$}}(-p),{{\rm gr}\,}^W_{2p-1}H )=J^p(H)$. Moreover, note that we also have induced extensions $$0\to \frac{H^{\prime}}{W_{2p-2}H} \to
{{\rm gr}\,}^W_{2p-1}H \to \frac{W_{2p-1}H}{H^{\prime}}\to 0$$ and $$0\to
\frac{W_{2p-1}H}{H^{\prime}} \to \frac{H^{\prime\prime}}{H^{\prime}}\to
H_{{\mbox{\scriptsize{${\mbox{$\mathbb Z$}}$}}}}^{p,p}\to 0$$ yielding, together with (\[he\]) and (\[heprime\]), the following commutative diagram with exact rows $$\begin{array}{c} {{\rm Hom}\,}({\mbox{$\mathbb Z$}}(-p),\frac{W_{2p-1}H}{H^{\prime}})\to {{\rm Hom}\,}({\mbox{$\mathbb Z$}}(-p),\frac{H^{\prime\prime}}{H^{\prime}})\to {{\rm Hom}\,}({\mbox{$\mathbb Z$}}(-p),H_{{\mbox{\scriptsize{${\mbox{$\mathbb Z$}}$}}}}^{p,p})\to {{\rm Ext}\,}({\mbox{$\mathbb Z$}}(-p),\frac{W_{2p-1}H}{H^{\prime}})\\ {\mbox{\large $\parallel$}}\hspace*{3cm}h^p\downarrow \hspace*{3cm}\downarrow e^p\hspace*{3cm} {\mbox{\large $\parallel$}}\\
{{\rm Hom}\,}({\mbox{$\mathbb Z$}}(-p),\frac{W_{2p-1}H}{H^{\prime}})\to {{\rm Ext}\,}({\mbox{$\mathbb Z$}}(-p),
\frac{H^{\prime}}{W_{2p-2}H})\to{{\rm Ext}\,}({\mbox{$\mathbb Z$}}(-p), {{\rm gr}\,}^W_{2p-1}H)\to{{\rm Ext}\,}({\mbox{$\mathbb Z$}}(-p),\frac{W_{2p-1}H}{H^{\prime}}) \end{array}$$ Since ${{\rm Hom}\,}({\mbox{$\mathbb Z$}}(-p),\frac{W_{2p-1}H}{H^{\prime}}) = 0 = {{\rm Hom}\,}({\mbox{$\mathbb Z$}}(-p),H_a^{2p-1})$ everything follows from a diagram chase.
[Let $A^p(H){\mbox{\,$\stackrel{\rm def}{=}$}\,}{{\rm Ext}\,}({\mbox{$\mathbb Z$}}(-p),H_a^{2p-1})$ denote the abelian part of the compact complex torus $J^p(H)$. Denote $H^p(H){\mbox{\,$\stackrel{\rm def}{=}$}\,}{{\rm Hom}\,}({\mbox{$\mathbb Z$}}(-p),H^{\prime\prime}/H^{\prime})$ the group of Hodge cycles, [[*i.e.*]{}, ]{}the sub-group of $H_{{\mbox{\scriptsize{${\mbox{$\mathbb Z$}}$}}}}^{p,p}$ mapped to $A^p(H)$ under the extension class map $e^p$. Define $$H^h{\mbox{\,$\stackrel{\rm def}{=}$}\,}T_{\rm Hodge}([H^p(H){\stackrel{h^p}{\rightarrow}}A^p(H)])$$ the mixed Hodge structure corresponding to the 1-motive defined from (\[heprime\]) above. Call this 1-motive the [*Hodge 1-motive*]{} of $H$.]{}
We remark that the mixed Hodge structure $H^h$ clearly corresponds to the following extension $$0\to H_a^{2p-1}\to H^h\to H^p(H)\to 0,$$ obtained by pulling back $H^p(H) = F^p\cap
(H^{\prime\prime}/H^{\prime})_{{\mbox{\scriptsize{${\mbox{$\mathbb Z$}}$}}}}$ along the projection $H^e{\mbox{$\to\!\!\!\!\to$}}H^{\prime\prime}/H^{\prime}$ in (\[heprime\]). In particular $H^h\subseteq H^e$ and $F^p\cap H_{{\mbox{\scriptsize{${\mbox{$\mathbb Z$}}$}}}}\subseteq F^p\cap H^h_{{\mbox{\scriptsize{${\mbox{$\mathbb Z$}}$}}}} = F^p\cap H^e_{{\mbox{\scriptsize{${\mbox{$\mathbb Z$}}$}}}}.$
Hodge conjecture for singular varieties
---------------------------------------
Let $X$ be a proper smooth ${\mbox{$\mathbb C$}}$-scheme. The coniveau or arithmetic filtration ([[*cf.*]{} ]{}[@GH]) $$N^iH^j(X) {\mbox{\,$\stackrel{\rm def}{=}$}\,}\ker (H^j(X) \to {\mathop{\rm
lim}_{\buildrel\longrightarrow\over{{\rm codim}_XZ\geq i}}} H^j
(X-Z))$$ yields a filtration by Hodge sub-structures of $H^j(X)$. We have that $N^iH^j(X)$ is of level $j-2i$ and $$\label{coin}
N^iH^j(X)_{{\mbox{\scriptsize{${\mbox{$\mathbb Q$}}$}}}}\subseteq H^j(X, {\mbox{$\mathbb Q$}})\cap F^iH^j(X).$$
[*(Grothendieck-Hodge conjecture [@GH])*]{} The left hand side of (\[coin\]) is the largest sub-space of the right hand side, generating a sub-space of $H^j(X, {\mbox{$\mathbb C$}})$ which is a sub-Hodge structure.
Let $X$ be a proper (integral) ${\mbox{$\mathbb C$}}$-scheme. Recall that the weight filtration on $H^{*}(X, {\mbox{$\mathbb Q$}})$ is given by the canonical spectral sequence of mixed ${\mbox{$\mathbb Q$}}$-Hodge structures $$E_1^{s, t}=H^{t}(X_s){\mbox{$\Rightarrow$}}{\mbox{$\mathbb H$}}^{s+t}({X_{{\mbox{\LARGE $\cdot $}}}})$$ for any smooth and proper hypercovering $\pi : {X_{{\mbox{\LARGE $\cdot $}}}}\to X$ and universal cohomological descent $H^{*}(X)\cong {\mbox{$\mathbb H$}}^{*}({X_{{\mbox{\LARGE $\cdot $}}}})$ (see [@D]). In fact, the spectral sequence degenerates at $E_2$. Denote $(H^{t})^{\bullet}$ the complexes $E_1^{\cdot, t}$ of $E_1$-terms. We clearly have $$H^i((H^{t})^{\bullet})={{\rm gr}\,}^W_{t}H^{t+i}(X).$$ Consider the complexes $(N^lH^{t})^{\bullet}$ induced from the coniveau filtration $N^lH^{t}(X_{i})$ on the components $X_i$ of the simplicial scheme ${X_{{\mbox{\LARGE $\cdot $}}}}$. We then have a natural map of Hodge structures $$\nu^{i,l}
:H^i((N^lH^{t})^{\bullet}) \to {{\rm gr}\,}^W_{t}H^{t+i}(X).$$ Note that the image of $\nu^{i,l}$ is contained in the sub-space ${{\rm gr}\,}^W_{t}H^{t+i}(X, {\mbox{$\mathbb Q$}})\cap F^l$.
\[GHC\] The image of $\nu^{i,l}$ is the largest sub-space of ${{\rm gr}\,}^W_{t}H^{t+i}(X,{\mbox{$\mathbb Q$}})\cap F^l$ which is a sub-Hodge structure of ${{\rm gr}\,}^W_{t}H^{t+i}(X)$.
It is reasonable to expect that such a statement will follow from the original Grothendieck-Hodge conjecture and abstract Hodge theory.
Grothendieck-Hodge conjecture (for coniveau $p$ and degrees $2p, 2p+1$) can be reformulated as follows ([[*cf.*]{} ]{}Grothendieck’s remark on motives in [@GH]). Consider ${{\rm gr}\,}^0_{F_m}CH^{p}(X)$ and $A^{p+1}_{X/k}\subseteq {{\rm gr}\,}^1_{F_m}CH^{p+1}(X)$ (for $k={\mbox{$\mathbb C$}}$ this is the algebraic part of $J^{p+1}(X)$) as 1-motives with rational coefficients. The Hodge realization of these algebraically defined 1-motives is $N^pH^{2p}(X)$ and $N^pH^{2p+1}(X)$ respectively.
Let $X$ be smooth and proper over ${\mbox{$\mathbb C$}}$. Then $$T_{\rm Hodge}([{{\rm gr}\,}^0_{F_m}CH^{p}(X)\to 0]_{{\mbox{\scriptsize{${\mbox{$\mathbb Q$}}$}}}}) = H^{p,p}_{{\mbox{\scriptsize{${\mbox{$\mathbb Q$}}$}}}}$$ and $$T_{\rm Hodge}([0\to
A^{p+1}_{X/k}]_{{\mbox{\scriptsize{${\mbox{$\mathbb Q$}}$}}}})=(H^{p,p+1}+H^{p+1,p})_{{\mbox{\scriptsize{${\mbox{$\mathbb Q$}}$}}}}.$$
Note that ${{\rm gr}\,}^0_{F_m}CH^{p}(X)$ would be better defined as ${{\cal Z}}^p(X)/\equiv_{\rm num}$, up to torsion.
Now apply to the mixed ${\mbox{$\mathbb Q$}}$-Hodge structure $H=H^{2p+i}(X)$ the construction performed in the previous section. For a fixed pair $(i,p)$ of integers recall that (\[ext\]) is an extension of ${{\rm gr}\,}^W_{2p}H^{2p+i}(X)$ by ${{\rm gr}\,}^W_{2p-1}H^{2p+i}(X)$, where: $$H^{i+1}((H^{2p-1})^{\bullet})={{\rm gr}\,}^W_{2p-1}H^{2p+i}(X)=
\frac{\ker (H^{2p-1}(X_{i+1}) \rightarrow H^{2p-1}(X_{i+2}))}{{{\rm im}\,}(H^{2p-1}(X_{i}) \rightarrow H^{2p-1}(X_{i+1}))}$$ and $$H^{i}((H^{2p})^{\bullet})={{\rm gr}\,}^W_{2p}H^{2p+i}(X)=
\frac{\ker(H^{2p}(X_{i}) \rightarrow H^{2p}(X_{i+1}))}{{{\rm im}\,}(H^{2p}(X_{i-1}) \rightarrow H^{2p}(X_{i}))}.$$ We then have that $J^p(H)= J^p(H^{i+1}((H^{2p-1})^{\bullet})))$ is isogenous to $H^{i+1}((J^p)^{\bullet}))$ where $(J^p)^{\bullet}$ is the complex of jacobians $J^p(X_i)$ of the components $X_i$.
Consider the complex $A_{{X_{{\mbox{\LARGE $\cdot $}}}}/{\mbox{\scriptsize{${\mbox{$\mathbb C$}}$}}}}^p$ of abelian sub-varieties given by the algebraic part of intermediate jacobians. The complex $A_{{X_{{\mbox{\LARGE $\cdot $}}}}/{\mbox{\scriptsize{${\mbox{$\mathbb C$}}$}}}}^p({\mbox{$\mathbb C$}})$ is a sub-complex of the complex of compact tori $(J^p)^{\bullet}$. Therefore, the induced maps $$H^{i+1}(A_{{X_{{\mbox{\LARGE $\cdot $}}}}/{\mbox{\scriptsize{${\mbox{$\mathbb C$}}$}}}}^p({\mbox{$\mathbb C$}})) \to J^p(H)$$ are holomorphic mappings (which factor through $\theta^i$) and whose image is isogenous to an abelian sub-variety of the maximal abelian sub-variety $A^p(H)$ of $J^p(H)$.
Moreover, the homology of the complex of $(p,p)$-classes is mapped to $H^{p,p}_{{\mbox{\scriptsize{${\mbox{$\mathbb Q$}}$}}}} = F^p\cap H^{i}((H^{2p})^{\bullet})$. Thus, there are canonical maps $$H^{i}((NS^p)^{\bullet}) \to H^{p,p}_{{\mbox{\scriptsize{${\mbox{$\mathbb Q$}}$}}}}$$ which factor through $H^{i}({{\rm gr}\,}_{F_m}^0(CH^p)^{\bullet})$.
We expect that the Hodge 1-motive of the simplicial scheme ${X_{{\mbox{\LARGE $\cdot $}}}}$ would be canonically isomorphic to the Hodge 1-motive of $H^{2p+i}(X)$.
\[MHC\] Let $X$ be a proper ${\mbox{$\mathbb C$}}$-scheme and let $\pi : {X_{{\mbox{\LARGE $\cdot $}}}}\to X$ be a smooth and proper hypercovering. Let $H^{2p+i}(X)$ denote Deligne’s mixed ${\mbox{$\mathbb Q$}}$-Hodge structure on $H^{2p+i}(X, {\mbox{$\mathbb Q$}})$, [[*i.e.*]{}, ]{}obtained by the universal cohomological descent isomorphism $H^{2p+i}(X, {\mbox{$\mathbb Q$}})\cong {\mbox{$\mathbb H$}}^{2p+i}({X_{{\mbox{\LARGE $\cdot $}}}}, {\mbox{$\mathbb Q$}})$.
1. The following square $$\begin{array}{ccc}
H^{i}((NS^p)^{\bullet})&{\stackrel{\lambda^i_a}{\rightarrow}}& H^{i+1}(A_{{X_{{\mbox{\LARGE $\cdot $}}}}/{\mbox{\scriptsize{${\mbox{$\mathbb C$}}$}}}}^p({\mbox{$\mathbb C$}}))\\
\downarrow & &\downarrow\\
H^{2p+i}(X)^{p,p} & {\stackrel{e^p}{\rightarrow}} & J^p(H^{2p+i}(X))
\end{array}$$ commutes, yielding a motivic “cycle class map”.
2. The image of the motivic “cycle class map” is the Hodge 1-motive of the ${\mbox{$\mathbb Q$}}$-Hodge structure $H^{2p+i}(X)$.
3. We have that $$T_{\rm Hodge}(\Xi^{i,p}) \cong H^{2p+i}(X)^h.$$
[*Note that if $X$ is smooth and proper then $H^{2p+i}(X)$ is pure and $H^{2p+i}(X)^h\neq 0$ if and only if $i= 0, -1$ ($p$ fixed). In this case, the above conjecture follows from the reformulation of Grothendieck-Hodge conjecture for $H^{2p}(X)$ and $H^{2p-1}(X)$.*]{}
Local Hodge theory
==================
See [@D] for notations, definitions and properties of mixed Hodge structures.
Infinite dimensional mixed Hodge structures
-------------------------------------------
Let ${\rm MHS}$ denote the abelian category of usual Deligne’s $A$-mixed Hodge structures [@D], [[*i.e.*]{}, ]{}for $A$ a noetherian subring of ${\mbox{$\mathbb R$}}$ such that $A\otimes{\mbox{$\mathbb Q$}}$ is a field, an object $H$ of ${\rm MHS}$ is defined as a triple $H = (H_A, W, F)$ where $H_A$ is a finitely generated $A$-module, $W$ is a finite increasing filtration on $H_A\otimes{\mbox{$\mathbb Q$}}$ and $F$ is a finite decreasing filtration on $H_A\otimes{\mbox{$\mathbb C$}}$ such that $W, F$ and ${\overline}F$ is a system of opposed filtrations.
*An $\infty$-mixed Hodge structure $H$ is a triple $(H_A,
W, F)$ where $H_A$ is any $A$-module, $W$ is a finite increasing filtration on $H_A\otimes{\mbox{$\mathbb Q$}}$ and $F$ is a finite decreasing filtration on $H_A\otimes{\mbox{$\mathbb C$}}$ such that $W, F$ and ${\overline}F$ is a system of opposed filtrations.*
Denote ${\rm MHS}^{\infty}$ the category of $\infty$-mixed Hodge structures or “infinite dimensional” mixed Hodge structures, [[*i.e.*]{}, ]{}where the morphisms are those which are compatible with the filtrations.
The category ${\rm MHS}^{\infty}$ is abelian and ${\rm MHS}$ is a fully faithful abelian subcategory of ${\rm MHS}^{\infty}$. Note that similar categories of infinite dimensional mixed Hodge structures already appeared in the literature, see Hain [@HN] and Morgan [@MO]. For example the category of limit mixed Hodge structures ${\rm
MHS}^{\lim}$, [[*i.e.*]{}, ]{}whose objects and morphisms are obtained by formally add to ${\rm MHS}$ (small) filtered colimits of objects in ${\rm MHS}$ with colimit morphisms.
Consider the case $A={\mbox{$\mathbb Q$}}$. In this case, in the category ${\rm
MHS}^{\infty}$ we have infinite products of those families of objects $\{H_i\}_{i\in I}$ such that the induced families of filtrations $\{W_{i}\}_{i\in I}$ and $\{F_{i}\}_{i\in I}$ are finite. Moreover such a (small) product of epimorphisms is an epimorphism.
For the sake of exposition we often call mixed Hodge structures the objects of ${\rm MHS}$ as well as those of ${\rm MHS}^{\infty}$ (or ${\rm
MHS}^{\lim}$).
Zariski sheaves of mixed Hodge structures
-----------------------------------------
Let $X$ denote the (big or small) Zariski site on an algebraic ${\mbox{$\mathbb C$}}$-scheme. However, most of the results in this section are available for any topological space or Grothendieck site.
Denote $X_{{\mbox{\LARGE $\cdot $}}}$ a simplicial object of the category of algebraic ${\mbox{$\mathbb C$}}$-schemes over $X$: recall that (see [@D 5.1.8]) simplicial sheaves on $X_{{\mbox{\LARGE $\cdot $}}}$ can be regarded as objects of a Grothendieck topos with enough points.
Consider presheaves of mixed Hodge structures. Note that a presheaf of usual Deligne’s $A$-mixed Hodge structures will have its stalks in ${\rm
MHS}^{\lim}$.
Consider those presheaves (resp. simplicial presheaves) of ${\mbox{$\mathbb Q$}}$-mixed Hodge structures on $X$ (resp. on $X_{{\mbox{\LARGE $\cdot $}}}$) such that the filtrations are finite as filtrations of sub-presheaves on $X$. These presheaves can be sheafified to sheaves having finite filtrations and preserving the above conditions on the stalks.
Make the following working definition of sheaves (or simplicial sheaves) of mixed Hodge structures. Let $A, {\mbox{$\mathbb Q$}}$ and ${\mbox{$\mathbb C$}}$ denote as well the constant sheaves on $X$ (or $X_{{\mbox{\LARGE $\cdot $}}}$) associated to the ring $A$, the rationals and the complex numbers.
A (simplicial) sheaf ${{\cal H}}$ of $A$-mixed Hodge structures, or “$A$-mixed sheaf” for short, is given by the following set of datas:
[*i)*]{}
: a (simplicial) sheaf ${{\cal H}}_A$ of $A$-modules,
[*ii)*]{}
: a finite (exhaustive) increasing filtration ${{\cal W}}$ by ${\mbox{$\mathbb Q$}}$-subsheaves of ${{\cal H}}_{{\mbox{\scriptsize{${\mbox{$\mathbb Q$}}$}}}}{\mbox{\,$\stackrel{\rm def}{=}$}\,}{{\cal H}}_A\otimes{\mbox{$\mathbb Q$}}$,
[*iii)*]{}
: a finite (exhaustive) decreasing filtration ${{\cal F}}$ by ${\mbox{$\mathbb C$}}$-subsheaves of ${{\cal H}}_{{\mbox{\scriptsize{${\mbox{$\mathbb C$}}$}}}}{\mbox{\,$\stackrel{\rm def}{=}$}\,}{{\cal H}}_A\otimes{\mbox{$\mathbb C$}}$;
satisfying the condition that ${{\cal W}},{{\cal F}}$ and ${\overline}{{\cal F}}$ is a system of opposed filtrations, [[*i.e.*]{}, ]{}we have that $$gr^p_{{{\cal F}}}gr^q_{{\overline}{{\cal F}}}gr_{n}^{{{\cal W}}}({{\cal H}}) = 0$$ for $p+q\neq n$.
There is a canonical decomposition $$gr_n^{{{\cal W}}}({{\cal H}}) =
\bigoplus_{p+q=n}^{}
{{\cal A}}^{p,q}$$ where ${{\cal A}}^{p,q} {\mbox{\,$\stackrel{\rm def}{=}$}\,}{{\cal F}}^p\cap {\overline}{{\cal F}}^q$ and conversely.
In the case of a simplicial sheaf assume that the filtrations are given by simplicial subsheaves, [[*i.e.*]{}, ]{}the simplicial structure should be compatible with the filtrations on the components. A simplicial $A$-mixed sheaf ${{\cal H}}_{{X_{{\mbox{\LARGE $\cdot $}}}}}$ on the simplicial space ${X_{{\mbox{\LARGE $\cdot $}}}}$ can be regarded ([[*cf.*]{} ]{}[@D 5.1.6]) as a family of $A$-mixed sheaves ${{\cal H}}_{X_i}$ (on the components $X_i$) such that the simplicial structure is also compatible with the filtrations ${{\cal W}}_{X_i}$ and ${{\cal F}}_{X_i}$ of ${{\cal H}}_{X_i}$.
A morphism of $A$-mixed sheaves is a morphism of (simplicial) sheaves of $A$-modules which is compatible with the filtrations. Denote ${\cal MHS}_X$ and ${\cal MHS}_{X_{{\mbox{\LARGE $\cdot $}}}}$ the corresponding categories.
In order to show the following Lemma one can just use Deligne’s Theorem [@D 1.2.10].
\[abel\] The categories ${\cal MHS}_X$ and ${\cal
MHS}_{X_{{\mbox{\LARGE $\cdot $}}}}$ of ${\mbox{$\mathbb Q$}}$-mixed sheaves are abelian categories. The kernel (resp. the cokernel) of a morphism $\varphi : {{\cal H}}\to {{\cal H}}'$ has underlying ${\mbox{$\mathbb Q$}}$ and ${\mbox{$\mathbb C$}}$-vector spaces the kernels (resp. the cokernels) of $\varphi_{{\mbox{\scriptsize{${\mbox{$\mathbb Q$}}$}}}}$ and $\varphi_{{\mbox{\scriptsize{${\mbox{$\mathbb C$}}$}}}}$ with induced filtrations. Any morphism is strictly compatible with the filtrations. The functors $gr_{{{\cal W}}}$ and $gr_{{{\cal F}}}$ are exacts.
Note that if $X=\{\infty\}$ is the singleton then ${\cal
MHS}_{\infty}$ is equal to ${\rm MHS}^{\infty}$. Examples of ${\mbox{$\mathbb Q$}}$-mixed sheaves are clearly given by constant sheaves associated to ${\mbox{$\mathbb Q$}}$-mixed Hodge structures, yielding a canonical fully faithful functor $${\rm MHS}^{\infty}\to {\cal MHS}_X.$$
Stalks of a ${\mbox{$\mathbb Q$}}$-mixed sheaf ${{\cal H}}$ are in ${\rm MHS}^{\infty}$, the filtrations being induced stalkwise. In fact, the condition on the filtrations given with any ${\mbox{$\mathbb Q$}}$-mixed sheaf is local, at any point of $X$. Skyscraper sheaves $x_*(H)$ associated to an object $H\in{\rm
MHS}^{\infty}$ and a point $x$ of $X$ are in ${\cal MHS}_X$. There is a natural isomorphism $${{\rm Hom}\,}_{{\rm MHS}^{\infty}}({{\cal H}}_x, H) \cong {{\rm Hom}\,}_{{\cal MHS}_X}({{\cal H}},
x_*(H)).$$
More generally, a presheaf in ${\rm MHS}^{\infty}$, with finite filtrations presheaves, can be sheafified to an $A$-mixed sheaf, in a canonical way, by applying the usual sheafification process to the filtrations together with the presheaf.
[Say that an $A$-mixed sheaf ${{\cal H}}$ is flasque if ${{\cal H}}_A$ is a flasque sheaf.]{}
For a given ${\mbox{$\mathbb Q$}}$-mixed sheaf ${{\cal H}}$ we then dispose of a canonical flasque ${\mbox{$\mathbb Q$}}$-mixed sheaf $$\prod_{x\in X}x_*({{\cal H}}_x)$$ where the product is taken over a set of points of $X$.
Hodge structures and Zariski cohomology
---------------------------------------
We show that, if ${{\cal H}}$ is a ${\mbox{$\mathbb Q$}}$-mixed sheaf then there is a unique ${\mbox{$\mathbb Q$}}$-mixed Hodge structure on the sections such that $\Gamma (-, gr(\dag)) = {{\rm gr}\,}\Gamma
(-,\dag)$. In fact, the mixed Hodge structure on the ${\mbox{$\mathbb Q$}}$-vector space of (global) sections is such that the following $$\Gamma (X,{{\cal H}}_{{\mbox{\scriptsize{${\mbox{$\mathbb Q$}}$}}}}) \subset
\prod_{x \in
X}{{\cal H}}_x$$ is strictly compatible with the filtrations; in the same way, for a simplicial sheaf, the following $$\Gamma (X_{{\mbox{\LARGE $\cdot $}}},{{\cal H}}_{{\mbox{\scriptsize{${\mbox{$\mathbb Q$}}$}}},_{{\mbox{\LARGE $\cdot $}}}}) \subset \ker \prod_{x \in X_0}{{\cal H}}_x\to
\prod_{x \in X_1}{{\cal H}}_x$$ is strictly compatible with the filtrations.
Let ${{\cal H}}_{X}\in{\cal MHS}_X$ and ${{\cal H}}_{X_{{\mbox{\LARGE $\cdot $}}}}\in{\cal
MHS}_{X_{{\mbox{\LARGE $\cdot $}}}}$ as above. There are left exact functors $${{\cal H}}_{X}\mapsto \Gamma (X,{{\cal H}}_X) \mbox{\hspace*{1cm}} {\cal MHS}_X \to
{\rm MHS}^{\infty}$$ and $${{\cal H}}_{X_{{\mbox{\LARGE $\cdot $}}}}\mapsto\Gamma (X_{{\mbox{\LARGE $\cdot $}}},{{\cal H}}_{X_{{\mbox{\LARGE $\cdot $}}}}) \mbox{\hspace*{1cm}}
{\cal MHS}_{X_{{\mbox{\LARGE $\cdot $}}}} \to {\rm MHS}^{\infty}$$ These functors yield ${\mbox{$\mathbb Q$}}$-mixed Hodge structures on the usual cohomology, which we denote by $H^*(X,{{\cal H}}_X)$ and $H^*(X_{{\mbox{\LARGE $\cdot $}}},{{\cal H}}_{X_{{\mbox{\LARGE $\cdot $}}}})$ respectively, such that if $$0\to {{\cal H}}' \to {{\cal H}}\to {{\cal H}}'' \to 0$$ is exact in ${\cal MHS}_X$, respectively in ${\cal MHS}_{X_{{\mbox{\LARGE $\cdot $}}}}$, then $$\cdots\to H^i(X,{{\cal H}}_X)\to H^i(X,{{\cal H}}''_X)\to H^{i+1}(X,{{\cal H}}'_X)\to \cdots$$ is exact in ${\rm MHS}^{\infty}$, and respectively for the cohomology of $X_{{\mbox{\LARGE $\cdot $}}}$: moreover, in the latter case we have a spectral sequence $$E^{p,q}_1 = H^q(X_p,{{\cal H}}_{X_p}) {\mbox{$\Rightarrow$}}{\mbox{$\mathbb H$}}^{p+q}(X_{{\mbox{\LARGE $\cdot $}}},{{\cal H}}_{X_{{\mbox{\LARGE $\cdot $}}}})$$ in the category ${\rm MHS}^{\infty}$.
In fact, there is an extension in ${\cal MHS}$ $$0\to {{\cal H}}\to \prod_{x\in X}x_*({{\cal H}}_x)\to {{\cal Z}}^1\to 0$$ where ${{\cal Z}}^1$ has the quotient ${\mbox{$\mathbb Q$}}$-mixed structure; as usual, we then get another extension $$0\to {{\cal Z}}^1\to \prod_{x\in X}x_*({{\cal Z}}^1_x)\to {{\cal Z}}^{2}\to 0$$ and so on. We therefore get a flasque resolution $$\prod_{x\in X}x_*({{\cal H}}_x)\to \prod_{x\in X}x_*({{\cal Z}}^1_x)\to
\prod_{x\in X}x_*({{\cal Z}}^2_x)\to \cdots$$ in ${\cal MHS}$: the canonical ${\mbox{$\mathbb Q$}}$-mixed flasque resolution.
If ${{\cal H}}_X$ is flasque then $\Gamma (X,{{\cal H}}_X)$ has a canonical ${\mbox{$\mathbb Q$}}$-mixed Hodge structure as claimed above; note that the filtrations would be given by flasque sub-sheaves.
In general, by construction, the cohomology is the homology of the complex of sections in ${\rm MHS}^{\infty}$. Thus $H^*(X,{{\cal H}}_X)$ has a canonical ${\mbox{$\mathbb Q$}}$-mixed Hodge structure. The same argument applies to the total complex of the double complex of flasque ${\mbox{$\mathbb Q$}}$-mixed sheaves given by the canonical resolutions on each component of a simplicial sheaf.
Refer to \[SGA4\] and [@LM Chapter IV] for a construction of canonical Godement resolutions available on any site and compare [@D 1.4.11] for the existence of bifiltered resolutions.
In particular, the mixed Hodge structure $H^* (X,{{\cal H}}_X)$ is such that $H^*
(X,{{\cal H}}_X)_{{\mbox{\scriptsize{${\mbox{$\mathbb Q$}}$}}}} = H^* (X,{{\cal H}}_{{\mbox{\scriptsize{${\mbox{$\mathbb Q$}}$}}}})$, $WH^* (X,{{\cal H}}_X)_{{\mbox{\scriptsize{${\mbox{$\mathbb Q$}}$}}}} = H^*
(X,{{\cal W}}{{\cal H}}_{{\mbox{\scriptsize{${\mbox{$\mathbb Q$}}$}}}})$ and $FH^* (X,{{\cal H}}_X)_{{\mbox{\scriptsize{${\mbox{$\mathbb C$}}$}}}} = H^* (X,{{\cal F}}{{\cal H}}_{{\mbox{\scriptsize{${\mbox{$\mathbb C$}}$}}}})$. There is a decomposition $${{\rm gr}\,}_n^{W}H^* (X, {{\cal H}}_X) = \bigoplus_{p+q=n}^{} H^* (X,
{{\cal A}}^{p,q}_X).$$
[*Note that any (non-canonical) ${\mbox{$\mathbb Q$}}$-mixed flasque resolution in ${\cal MHS}$ yields a bifiltered complex and a bifiltered quasi-isomorphism with the canonical resolution. Therefore, the so obtained ${\infty}$-mixed Hodge structure on the cohomology is unique up to isomorphism.*]{}
Considering complexes in ${\cal MHS}_X$ and ${\cal
MHS}_{X_{{\mbox{\LARGE $\cdot $}}}}$ we construct the derived categories of ${\mbox{$\mathbb Q$}}$-mixed sheaves ${{\cal D}}^*({\cal MHS}_X)$ and ${{\cal D}}^*({\cal MHS}_{X_{{\mbox{\LARGE $\cdot $}}}})$ as usual, as well as ${{\cal D}}^*({\rm MHS}^{\infty})$. We have a total derived functor $$R\Gamma (X, -): {{\cal D}}^*({\cal MHS}_X)\to {{\cal D}}^*({\rm MHS}^{\infty})$$ sending a complex of ${\mbox{$\mathbb Q$}}$-mixed sheaves to the total complex of sections of its canonical resolution. Moreover, if $f:X\to Y$ is a continuous map, we have a higher direct image ${\mbox{$\mathbb Q$}}$-mixed sheaf $R^qf_*({{\cal H}}_X)$ on $Y$, which corresponds as well to an exact functor $$Rf_*:{{\cal D}}^*({\cal MHS}_X)\to {{\cal D}}^*({\cal MHS}_Y).$$ There is an inverse image exact functor $f^*:{{\cal D}}^*({\cal MHS}_Y)\to {{\cal D}}^*({\cal MHS}_X).$ Moreover, Grothendieck six standard operations can be otained in the derived category of ${\mbox{$\mathbb Q$}}$-mixed sheaves.
Local-to-global properties
--------------------------
Let $X$ be an algebraic ${\mbox{$\mathbb C$}}$-scheme and let $X_{\rm an}$ be the associated analytic space. For any Zariski open subset $U\subseteq X$ the corresponding integral cohomology $H^r(U_{\rm an},{\mbox{$\mathbb Z$}}(t))$ carry a mixed Hodge structure (see [@D 8.2]) such that the restriction maps $H^r(U_{\rm
an},{\mbox{$\mathbb Z$}}(t))\to H^r(V_{\rm an},{\mbox{$\mathbb Z$}}(t))$ for $V\subseteq U$ are maps of mixed Hodge structures. Thus the presheaf of mixed Hodge structures $$\label{sheaf}
U \mapsto H^r(U_{\rm an},{\mbox{$\mathbb Q$}}(t))$$ can be sheafified to a Zariski ${\mbox{$\mathbb Q$}}$-mixed sheaf. In fact, for a fixed $r$, the resulting non-zero Hodge numbers of $H^r(U_{\rm
an},{\mbox{$\mathbb Q$}})$, for any $U$, are in the finite set $[0,r]\times [0,r]$ (see [@D 8.2.4]).
[*Denote ${{\cal H}}^r_X({\mbox{$\mathbb Q$}}(t))$ the ${\mbox{$\mathbb Q$}}$-mixed sheaf obtained hereabove. For ${X_{{\mbox{\LARGE $\cdot $}}}}$ a simplicial ${\mbox{$\mathbb C$}}$-scheme denote ${{\cal H}}^r_{{X_{{\mbox{\LARGE $\cdot $}}}}}$ the simplicial ${\mbox{$\mathbb Q$}}$-mixed sheaf given by ${{\cal H}}^r_{X_p}$ on the component $X_p$.* ]{}
If $X$ has algebraic dimension $n$ then all its Zariski open affines $U$ do have dimension $\leq n$ thus ${{\cal H}}^r_X=0$ for $r>n$.
\[zarhodge\] The Zariski cohomology groups $H^*(X,{{\cal H}}^r_X)$ carry $\infty$-mixed Hodge structures. Possibly non-zero Hodge numbers of $H^*(X,{{\cal H}}^r_X)$ are in the finite set $[0,r]\times [0,r]$. The Zariski cohomology ${\mbox{$\mathbb H$}}^{*}(X_{{\mbox{\LARGE $\cdot $}}},{{\cal H}}_{X_{{\mbox{\LARGE $\cdot $}}}}^r)$ carry $\infty$-mixed Hodge structure and the canonical spectral sequence $$E^{p,q}_1 = H^q(X_p,{{\cal H}}_{X_p}^r) {\mbox{$\Rightarrow$}}{\mbox{$\mathbb H$}}^{p+q}(X_{{\mbox{\LARGE $\cdot $}}},{{\cal H}}_{X_{{\mbox{\LARGE $\cdot $}}}}^r)$$ is in the category ${\rm MHS}^{\infty}$.
Let $\omega : X_{\rm an} \to X_{\rm Zar}$ be the continuous map of sites induced by the identity mapping. We then have a Leray spectral sequence $$L^{q,r}_2 = H^q(X_{\rm Zar},R^r\omega_*({\mbox{$\mathbb Z$}})){\mbox{$\Rightarrow$}}H^{q+r}(X_{\rm
an},{\mbox{$\mathbb Z$}})$$ of abelian groups. Since $R^r\omega_*({\mbox{$\mathbb Q$}})\cong {{\cal H}}_{X}^r$ these sheaves can be regarded as ${\mbox{$\mathbb Q$}}$-mixed sheaves and its Zariski cohomology carry $\infty$-mixed Hodge structures as above.
For ${X_{{\mbox{\LARGE $\cdot $}}}}$ a simplicial scheme we thus have $\omega_{{\mbox{\LARGE $\cdot $}}} : (X_{{\mbox{\LARGE $\cdot $}}})_{\rm
an} \to
(X_{{\mbox{\LARGE $\cdot $}}})_{\rm Zar}$ as above and a Leray spectral sequence $$L^{q,r}_2 = {\mbox{$\mathbb H$}}^q(X_{{\mbox{\LARGE $\cdot $}}}, R^r(\omega_{{\mbox{\LARGE $\cdot $}}})_*({\mbox{$\mathbb Z$}})){\mbox{$\Rightarrow$}}{\mbox{$\mathbb H$}}^{q+r}(X_{{\mbox{\LARGE $\cdot $}}},{\mbox{$\mathbb Z$}})$$ where $R^r(\omega_{{\mbox{\LARGE $\cdot $}}})_*({\mbox{$\mathbb Z$}})\cong {{\cal H}}_{{X_{{\mbox{\LARGE $\cdot $}}}}}^r$.
\[l2g\] [*(Local-to-global)*]{} There are spectral sequences $$L^{q,r}_2 = H^q(X,{{\cal H}}_{X}^r){\mbox{$\Rightarrow$}}H^{q+r}(X_{\rm an},{\mbox{$\mathbb Q$}})$$ and $$L^{q,r}_2 = {\mbox{$\mathbb H$}}^q(X_{{\mbox{\LARGE $\cdot $}}}, {{\cal H}}_{{X_{{\mbox{\LARGE $\cdot $}}}}}^r){\mbox{$\Rightarrow$}}{\mbox{$\mathbb H$}}^{q+r}(X_{{\mbox{\LARGE $\cdot $}}},{\mbox{$\mathbb Q$}})$$ in the category of $\infty$-mixed Hodge structures.
The proof of this compatibility result will appear elsewhere; however, for smooth ${\mbox{$\mathbb C$}}$-schemes and using (\[ares\]) below, the compatibility follows from [@KP Corollary 4.4].
Edge maps
=========
Recall that the classical cycle class maps can be obtained [*via*]{} edge homomorphisms in the coniveau spectral sequence. This is a consequences of Bloch’s formula [@BO]. Working simplicially we then construct certain cycle class maps for singular varieties [*via*]{} edge maps in the local-to-global spectral sequence. We first show that the results of [@BO] hold in the category of $\infty$-mixed Hodge structures.
Bloch-Ogus theory
-----------------
From Deligne [@D 8.2.2 and 8.3.8] the cohomology groups $H^*_Z(X)$ (= $H^*(X {\rm mod} X-Z,{\mbox{$\mathbb Z$}})$ in Deligne’s notation) carry a mixed Hodge structure fitting into long exact sequences $$\label{loc}
\cdots \to H_Z^j(X)\to H_T^j(X)\to
H_{T-Z}^j(X-Z)\to H_Z^{j+1}(X)\to \cdots$$ for any pair $Z\subset T$ of closed subschemes of $X$.
Since classical Poincaré duality is compatible with the mixed Hodge structures involved, then the functors $$Z\subseteq X \mapsto (H_Z^*(X),H_*(Z))$$ yield a Poincaré duality theory with supports (see [@BO] and [@JA]) in the abelian tensor category of mixed Hodge structures. Furthermore we have that the above theory is appropriate for algebraic cycles in the sense of [@BV2].
Let $X^p$ be the set of codimension $p$ points in $X$. For $x\in X^p$ let $$H^{*}_x(X) {\mbox{\,$\stackrel{\rm def}{=}$}\,}{\mathop{\rm
lim}_{\buildrel\longrightarrow\over{U\subset X}}}
H^*_{\overline{\{x\}}\cap U}(U).$$ Taking direct limits of (\[loc\]) over pairs $Z\subset T$ filtered by codimension and applying the exact couple method to the resulting long exact sequence we obtain the coniveau spectral sequence $$C^{p,q}_1 =\coprod_{x\in X^p}^{} H^{q+p}_x(X) {\mbox{$\Rightarrow$}}H^{p+q}(X)$$ in the abelian category ${\rm MHS}^{\infty}$ ([[*cf.*]{} ]{}[@BV2]).
Consider $X$ smooth over ${\mbox{$\mathbb C$}}$. By local purity, we have that $H^{q+p}_x(X,{\mbox{$\mathbb Z$}}(r))\cong H^{q-p}(x,{\mbox{$\mathbb Z$}}(r-p))$ if $x$ is a codimension $p$ point in $X$, [[*i.e.*]{}, ]{}here we have set $$H^*(x) {\mbox{\,$\stackrel{\rm def}{=}$}\,}{\mathop{\rm
lim}_{\buildrel\longrightarrow\over{V {\rm\ open\ } \subset \overline{\{ x\}}}}} H^*(V).$$ Sheafifying the (limit) sequences (\[loc\]), we obtain the following exact sequences of ${\mbox{$\mathbb Q$}}$-mixed sheaves on $X$: $$\label{shortloc} 0\to
{{\cal H}}^r_{Z^{p}}\to \coprod_{x\in X^p}^{} x_*
(H^{r-2p}(x)) \to {{\cal H}}^{r+1}_{Z^{p+1}}\to 0$$ where ${{\cal H}}^r_{Z^{p}}$ is the ${\mbox{$\mathbb Q$}}$-mixed sheaf associated to the presheaf $$U\mapsto {\mathop{\rm
lim}_{\buildrel\longrightarrow\over{{\rm codim}_XZ\geq p}}} H_{Z\cap U}^r(U).$$ In fact, the claimed short exact sequencs (\[shortloc\]) are obtained [*via*]{} the “locally homologically effaceable” property (see [@BO Claim p. 191]), [[*i.e.*]{}, ]{}the following map of sheaves on $X$ $${{\cal H}}^*_{Z^{p+1}} {\stackrel{\rm zero}{\rightarrow}}
{{\cal H}}^*_{Z^p}$$ vanishes for all $p \geq 0$.
[*(Arithmetic resolution)*]{} Let ${{\cal H}}^q_X({\mbox{$\mathbb Q$}}(t))$ be the ${\mbox{$\mathbb Q$}}$-mixed sheaf defined in (\[sheaf\]). Assuming $X$ smooth over ${\mbox{$\mathbb C$}}$ then $$\label{ares}
0\to{{\cal H}}^q({\mbox{$\mathbb Q$}}(t))\to \coprod_{x\in X^0}^{}x_*( H^{q}(x)(t))\to
\coprod_{x\in X^1}^{} x_*( H^{q-1}(x)(t-1))\to \cdots\to
\coprod_{x\in X^q}^{}x_*({\mbox{$\mathbb Q$}}(t-q))\to 0$$ is a flasque resolution in the category ${\cal MHS}_X$. Therefore, the coniveau spectral sequence $$\label{coniveau}
C^{p,q}_2 = H^p(X,{{\cal H}}^q({\mbox{$\mathbb Q$}}(t))) {\mbox{$\Rightarrow$}}H^{p+q}(X, {\mbox{$\mathbb Q$}}(t))$$ is in the category ${\rm MHS}^{\infty}$.
Follows from construction as sketched above. In fact, all axioms stated in [@BO Section 1] are verified in ${\rm MHS}$ and the results in [@BO Sections 3-4] can be obtained in ${\rm MHS}^{\infty}$.
In particular, consider the presheaf of vector spaces $$U\mapsto F^iH^*(U)
\mbox{\hspace*{1cm} (resp.\ } U\mapsto W_iH^*(U))$$ and the associated Zariski sheaves ${{\cal F}}^i{{\cal H}}^*$ (resp. ${{\cal W}}_i{{\cal H}}^*$) on $X$ filtering the sheaves ${{\cal H}}^*({\mbox{$\mathbb C$}})$ (resp. ${{\cal H}}^*({\mbox{$\mathbb Q$}})$). These filtrations are defining the sheaf of mixed Hodge structures ${{\cal H}}^*_X$ above according to (\[sheaf\]). From Lemma \[abel\] ([[*cf.*]{} ]{}[@D Theor.1.2.10 and 2.3.5]) the functors $F^i$, $W_i$, $gr_i^W$ and $gr^i_F$ (any $i\in{\mbox{$\mathbb Z$}}$) from the category of ${\mbox{$\mathbb Q$}}$-mixed sheaves to that of ordinary sheaves are exact. Applying these functors to the arithmetic resolution (\[ares\]) we obtain resolutions of ${{\cal F}}^i{{\cal H}}^*$ (resp. ${{\cal W}}_i{{\cal H}}^*$) as follows.
\[arifilt\] The arithmetic resolution (\[ares\]) yields a bifiltered quasi-isomorphism $$({{\cal H}}^*,{{\cal F}}^{\dag},{{\cal W}}_{\sharp}){\stackrel{\simeq}{\rightarrow}}
(\coprod_{x\in X^{\odot}}^{}
x_*H^{*-\odot}(x),\coprod_{x\in X^{\odot}}^{} x_*F^{\dag-\odot},
\coprod_{x\in
X^{\odot}}^{} x_*W_{\sharp-2\odot}),$$ [[*i.e.*]{}, ]{}there are flasque resolutions: $$0\to {{\cal F}}^i{{\cal H}}^q\to
\coprod_{x\in X^0}^{} x_*(F^iH^{q}(x))\to \coprod_{x\in X^1}^{} x_*(
F^{i-1}H^{q-1}(x)) \to \cdots$$ and $$0\to {{\cal W}}_j{{\cal H}}^q\to \coprod_{x\in X^0}^{} x_*( W_jH^{q}(x))\to
\coprod_{x\in X^1}^{} x_*(W_{j-2}H^{q-1}(x)) \to \cdots$$ as well as $$0\to gr^i_{{{\cal F}}}gr_j^{{{\cal W}}}{{\cal H}}^q({\mbox{$\mathbb C$}})\to\coprod_{x\in
X^{0}}^{} x_*(gr^i_{F}gr_j^{W}H^{q}(x))\to\cdots\to
\coprod_{x\in
X^{q}}^{} x_*(gr^{i-q}_{F}gr_{j-2q}^{W}H^{0}(x))\to 0.$$
Consider the twisted Poincaré duality theory $(F^nH^{*},F^{-m}H_*)$ where the integers $n$ and $m$ play the role of twisting, [[*i.e.*]{}, ]{}we have $$F^{d-n}H^{2d-k}_Z(X)\cong F^{-n}H_k(Z)$$ for $X$ smooth of dimension $d$. From the arithmetic resolution of ${{\cal F}}^i{{\cal H}}^q$ in Scholium \[arifilt\] we obtain the following:
Assume $X$ smooth and let $i$ be a fixed integer. We then have a coniveau spectral sequence $$\label{conifilt} F^iC^{p,q}_2 =H^p(X,{{\cal F}}^i{{\cal H}}^q)
{\mbox{$\Rightarrow$}}F^iH^{p+q}(X)$$ where $H^p(X,{{\cal F}}^i{{\cal H}}^q)=0$ if $q<\mbox{min}(i,p)$.
Concerning the Zariski sheaves $gr^i_{{{\cal F}}}{{\cal H}}^q$ and ${\overline}{{\cal F}}^i$ we indeed obtain corresponding coniveau spectral sequences as above.
Note that the spectral sequence $F^iC_2$ can be obtained applying $F^i$ to the coniveau spectral sequence (\[coniveau\]).
**
Remark that applying $F^i$ (resp. $W_i$) to the long exact sequences (\[loc\]), taking direct limits over pairs $Z\subset T$ filtered by codimension and sheafifying, we do obtain the claimed flasque resolutions of ${{\cal F}}^i{{\cal H}}^*$ and ${{\cal W}}_i{{\cal H}}^*$ without reference to the category of ${\mbox{$\mathbb Q$}}$-mixed sheaves.
Note that for $X$ of dimension $d$, the fundamental class $\eta_X$ belongs to $W_{-2d}H_{2d}(X)\cap
F^{-d}H_{2d}(X)$ so that “local purity” yields the shift by two for the weight filtration and the shift by one for the Hodge filtration. Therefore one has to keep care of Tate twists when dealing with arithmetic resolutions.
Note that for $x\in X^0$ we have $H^{q}(x) = {{\cal H}}_x^q$ and there is a natural projection in ${\cal MHS_X}$ $$\prod_{x\in X}^{}
x_*({{\cal H}}_x^q)\to
\coprod_{x\in X^0}^{}x_*(H^{q}(x))$$ which is the identity on ${{\cal H}}^q$. The mixed Hodge structure induced by the arithmetic resolution on $H^*(X,
{{\cal H}}^q)$ is not the canonical one (which is the one induced by the canonical ${\mbox{$\mathbb Q$}}$-mixed flasque resolution) but yields a mixed Hodge structure which is naturally isomorphic to the canonical one (being induced by a natural isomorphism in the derived category ${{\cal D}}^*({\cal MHS}_X)$).
Coniveau filtration
-------------------
Let $X$ be a smooth ${\mbox{$\mathbb C$}}$-scheme. The coniveau filtration ([[*cf.*]{} ]{}[@GH]) $N^iH^j(X)$ is a filtration by (mixed) sub-structures of $H^j(X)$. This filtration is clearly induced from the coniveau spectral sequence (\[coniveau\]) [*via*]{} (\[loc\]). Remark that from the coniveau spectral sequence (\[coniveau\]) $${{\rm gr}\,}_N^{i-1}H^j(X) =C^{i-1, j-i+1}_{\infty}$$ which is a substructure of $H^{i-1}(X,{{\cal H}}^{j-i+1})$ for $i\leq 2$. In fact, from the arithmetic resolution we have that $C^{p,q}_2=H^p(X,{{\cal H}}^q(t))=0$ for $p>q$.
### Case $i=1$ {#case-i1 .unnumbered}
Let $X$ be a proper smooth ${\mbox{$\mathbb C$}}$-scheme. Note that $N^1H^j(X) = \ker (H^j(X)\to H^0(X,{{\cal H}}^j))=\{\mbox{\ Zariski
locally trivial classes in\ } H^j(X)\}.$ Thus $$\frac{H^j(X, {\mbox{$\mathbb Q$}})\cap
F^1H^j(X)}{N^1H^j(X)} = {{\rm gr}\,}_N^{0}H^j(X)\cap F^1\subseteq H^0(X,{{\cal H}}^j)\cap
F^1.$$ We remark that ${{\cal H}}^j/{{\cal F}}^1$ is the constant sheaf associated to $H^j(X,{{\cal O}}_X)$. Thus $$F^1\cap H^0(X,{{\cal H}}^j) \cong \ker (H^0(X,{{\cal H}}^j)\to
H^j(X,{{\cal O}}_X)).$$ If $j=1$ then $H^1(X)= H^0(X,{{\cal H}}^1)$ from (\[coniveau\]) and (\[coin\]) is trivially an equality. If $j=2$ then $F^1\cap
H^0(X,{{\cal H}}^2)=0$ from the exponential sequence. But for $j=3$ and $X$ the threefold product of an elliptic curve with itself Grothendieck’s argument in [@GH] yields a non-trivial element in $F^1\cap H^0(X,{{\cal H}}^3)$.
### Case $i=p$ and $j=2p$ {#case-ip-and-j2p .unnumbered}
Let $X$ be a smooth ${\mbox{$\mathbb C$}}$-scheme. If $j=2p$ then $N^iH^{2p}(X)(t)=0$ for $i>p$ and $N^pH^{2p}(X)(t)=C^{p,p}_{\infty}$. Moreover, from (\[coniveau\]) there is an induced edge map $$s\ell^{p}_0: H^p(X,{{\cal H}}^p_X(p))\to H^{2p}(X)(p)$$ which is a map of $\infty$-mixed Hodge structures and whose image is $N^pH^{2p}(X)(p)$. This equal the image of the classical cycle class map $c\ell^{p}: CH^p(X)\to H^{2p}(X)(p)$. In fact, by [@BO 7.6], the cohomology group $$H^p(X,{{\cal H}}^p_X(p))\cong {\rm coker} (\coprod_{x\in
X^{p-1}}^{}H^{1}(x) \to \coprod_{x\in X^p}^{} {\mbox{$\mathbb Z$}})$$ coincide with $NS^p(X)$, the group of algebraic cycles of codimension $p$ in $X$ modulo algebraic equivalence. Thus $c\ell^{p}$ factors through $s\ell^{p}_0$ and the canonical projection (see [@BV2]).
Recall that $F^iH^p(X,{{\cal H}}^q){\mbox{\,$\stackrel{\rm def}{=}$}\,}H^p(X,{{\cal F}}^i{{\cal H}}^q){\hookrightarrow}H^p(X,{{\cal H}}^q({\mbox{$\mathbb C$}}))$ is injective and $$H^p(X,{{\cal F}}^p{{\cal H}}^p)\cong {\rm coker} (\coprod_{x\in
X^{p-1}}^{}F^1H^{1}(x) \to \coprod_{x\in X^p}^{} {\mbox{$\mathbb C$}})$$ whence the canonical map $H^p(X,{{\cal F}}^p{{\cal H}}^p)\to NS^p(X)\otimes{\mbox{$\mathbb C$}}$ is also surjective. As an immediate consequence of this fact, [[*e.g.*]{}, ]{}from the coniveau spectral sequence (\[conifilt\]), we get the following.
\[Nero\] Let $X$ be a proper smooth ${\mbox{$\mathbb C$}}$-scheme. Then $$F^0H^p(X,{{\cal H}}^p(p)) {\mbox{\,$\stackrel{\rm def}{=}$}\,}H^p(X,{{\cal F}}^0{{\cal H}}^p(p))\cong NS^p(X)\otimes{\mbox{$\mathbb C$}}$$ and the image of the cycle map is in $H^{2p}(X,{\mbox{$\mathbb Q$}}(p))\cap
F^0H^{2p}(X, {\mbox{$\mathbb C$}}(p))$.
Now ${{\rm im}\,}c\ell^p_{{\mbox{\scriptsize{${\mbox{$\mathbb Q$}}$}}}}=N^pH^{2p}(X, {\mbox{$\mathbb Q$}}(p))$ and $H^{2p}(X, {\mbox{$\mathbb Q$}}(p))\cap
F^0H^{2p}(X, {\mbox{$\mathbb C$}}(p))$ is equal to $H^{p,p}_{{\mbox{\scriptsize{${\mbox{$\mathbb Q$}}$}}}}$, [[*i.e.*]{}, ]{}the sub-structure of rational $(p,p)$-classes in $H^{2p}(X)$. Note that $H^{p,p}_{{\mbox{\scriptsize{${\mbox{$\mathbb Q$}}$}}}}$ corresponds to the 1-motivic part of $H^{2p}(X)(p)$. For $X$ a smooth proper ${\mbox{$\mathbb C$}}$-scheme, the Hodge conjecture then claims that ${{\rm im}\,}c\ell^p_{{\mbox{\scriptsize{${\mbox{$\mathbb Q$}}$}}}} = H^{p,p}_{{\mbox{\scriptsize{${\mbox{$\mathbb Q$}}$}}}}$.
In this case ${{\rm gr}\,}_N^{p-1}H^{2p}(X)(p)=C^{p-1,p+1}_{\infty}$ is a quotient of $H^{p-1}(X,{{\cal H}}^{p+1}_X(p))$. For example: ${{\rm gr}\,}_N^{1}H^{4}(X)(2)=
H^{1}(X,{{\cal H}}^{3}_X(2)).$
Is $F^2\cap H^{1}(X,{{\cal H}}^{3}_X) =0$ ?
### Case $i=p$ and $j=2p+1$ {#case-ip-and-j2p1 .unnumbered}
Let $X$ be a smooth ${\mbox{$\mathbb C$}}$-scheme. If $j=2p+1$ then $N^iH^{2p+1}(X)=0$ for $i>p$ and $N^pH^{2p+1}(X)(t)=C^{p,p+1}_{\infty}$ which is a quotient of $H^{p}(X,{{\cal H}}^{p+1}_X)$, [[*i.e.*]{}, ]{}there is an edge map $$s\ell^{p+1}_{-1}: H^{p}(X,{{\cal H}}^{p+1}_X(p+1))\to H^{2p+1}(X)(p+1)$$ with image $N^pH^{2p+1}(X)(p+1)$. In this case the Grothendieck-Hodge conjecture characterize $N^pH^{2p+1}(X)$ as the largest sub-Hodge structure of type $\{(p,p+1), (p+1,p)\}$. This is the same as the 1-motivic part of $H^{2p+1}(X)(p+1)$.
This 1-motivic part yields an abelian variety which is the maximal abelian subvariety of the intermediate jacobian $J^{p+1}(X)$. On the other hand, it is easy to see that $N^pH^{2p+1}(X)(p+1)$ yields the algebraic part of $J^{p+1}(X)$, [[*i.e.*]{}, ]{}defined by the images of codimension $p+1$ cycles on $X$ which are algebraically equivalent to zero modulo rational equivalence ([[*cf.*]{} ]{}[@GH] and [@MU2]).
Exotic $(1,1)$-classes
----------------------
Consider $X$ singular. We briefly explain the Conjecture \[MHC\] for $p=1$. Moreover we show that there are edge maps generalizing the cycle class maps constructed in the previous section.
For $X$ a proper irreducible ${\mbox{$\mathbb C$}}$-scheme, consider the mixed Hodge structure on $H^{2 + i}(X,{\mbox{$\mathbb Z$}})$ modulo torsion. The extension (\[ext\]) is the following $$\label{ext1}
0\to H^{1+ i}((H^1)^{\bullet})\to W_2H^{2 + i}(X)/W_0 \to
H^{i}((H^2)^{\bullet})\to 0.$$ Since the complex $(H^1)^{\bullet}$ is made of level $1$ mixed Hodge structures then $H^{2 + i}(X)^h = H^{2 + i}(X)^e$ in our notation.
If $X$ is nonsingular then $H^{2 + i}(X)$ is pure and there are only two cases where this extension is non-trivial. In the case $i= -1$ the above conjecture corresponds to the well known fact that $H_1({{\rm Pic}\,}^0(X))= H^1(X,
{\mbox{$\mathbb Z$}})$. The case $i = 0$ corresponds to the celebrated theorem by Lefschetz showing that the subgroup $H^{1,1}_{{\mbox{\scriptsize{${\mbox{$\mathbb Z$}}$}}}}$ of $H^2(X,{\mbox{$\mathbb Z$}})$ of cohomology classes of type $(1,1)$ is generated by $c_1$ of line bundles on $X$. Since homological and algebraic equivalences coincide for divisors, the Néron-Severi group ${{\rm NS}\,}^1 (X)$ coincide with $H^{1,1}_{{\mbox{\scriptsize{${\mbox{$\mathbb Z$}}$}}}}$. For such a nonsingular variety $X$ we then have $${{\rm NS}\,}^1 (X) = F^1\cap
H^2(X,{\mbox{$\mathbb Z$}}) =
H^{1,1}_{{\mbox{\scriptsize{${\mbox{$\mathbb Z$}}$}}}} = H^1(X,{{\cal H}}^1_X) = N^1H^2(X,{\mbox{$\mathbb Z$}}).$$
For $i= -1$ and $X$ possibly singular, the conjecture corresponds to the fact (proved in [@BSAP]) that the abelian variety corresponding to ${{\rm gr}\,}_1^WH^1$ is $\ker^0 ({{\rm Pic}\,}^0(X_0) \to {{\rm Pic}\,}^0(X_1))$.
For $i = 0$ the Conjecture \[MHC\] is quite easily verified by checking the claimed compatibility of the extension class map. Such a statement then corresponds to a Lefschetz $(1,1)$-theorem for complete varieties with arbitrary singularities.
For $i\geq 1$ we may get [*exotic $(1, 1)$-classes*]{} in the higher cohomology groups $H^{2 + i}(X)$ of an higher dimensional singular variety $X$. We ignore the geometrical meaning of these exotic $(1, 1)$-classes. It will be interesting to produce concrete examples. The conjectural picture is as follows.
Let $\pi : X_{{\mbox{\LARGE $\cdot $}}}\to X$ be an hypercovering. Let $(H^q({{\cal H}}^1))^{\bullet}$ be the complex of $E^{\bullet ,q}_1$-terms of the spectral sequence in Corollary \[zarhodge\] for $r=1$. Now $E^{i ,q}_1 =
H^q(X_i,{{\cal H}}_{X_i}^1)=0$ for $q\geq 2$ (where $X_i$ are the smooth components of the hypercovering $X_{{\mbox{\LARGE $\cdot $}}}$ of $X$) and all non-zero terms are pure Hodge structures: therefore the spectral sequence degenerates at $E_2$. Thus, from Corollary \[zarhodge\], we get an extension $$\label{zarext1}
0\to H^{1+ i}((H^0({{\cal H}}^1))^{\bullet})\to {\mbox{$\mathbb H$}}^{1 +i}({X_{{\mbox{\LARGE $\cdot $}}}}, {{\cal H}}^{1}_{{X_{{\mbox{\LARGE $\cdot $}}}}})
\to H^{i}((H^1({{\cal H}}^1))^{\bullet})\to 0$$ in the category of mixed ${\mbox{$\mathbb Q$}}$-Hodge structures. We have $H^{i+1}((H^0({{\cal H}}^1))^{\bullet})=
H^{i+1}((H^1)^{\bullet})= {{\rm gr}\,}_1^WH^{2 + i}$ and $H^{i}((H^1({{\cal H}}^1))^{\bullet})= H^{i}((NS)^{\bullet}) =
H^{i}((N^1H^{2})^{\bullet})$.
Moreover, from the local-to-global spectral sequence in Claim \[l2g\] and cohomological descent we get the following edge map $$s\ell^{1+i}:{\mbox{$\mathbb H$}}^{1 +i}({X_{{\mbox{\LARGE $\cdot $}}}}, {{\cal H}}^{1}_{{X_{{\mbox{\LARGE $\cdot $}}}}})\to W_2H^{2 + i}(X)/W_0.$$ In fact, first observe that $W_0H^{2 + i}(X) = {\mbox{$\mathbb H$}}^{2 +i}({X_{{\mbox{\LARGE $\cdot $}}}},
{{\cal H}}^{0}_{{X_{{\mbox{\LARGE $\cdot $}}}}}).$ From (\[zarext1\]) above we then see that $W_0{\mbox{$\mathbb H$}}^{1
+i}({X_{{\mbox{\LARGE $\cdot $}}}}, {{\cal H}}^{1}_{{X_{{\mbox{\LARGE $\cdot $}}}}})=0.$ The map $s\ell^{1+i}$ is then easily obtained as an edge homomorphism of the cited local-to-global spectral sequence and weight arguments. This cycle map will fit in a diagram $$\begin{array}{ccccccc}
0\to & H^{1+ i}((H^1)^{\bullet})&\to & W_2H^{2 + i}(X)/W_0 &\to
&H^{i}((H^2)^{\bullet})&\to 0\\
&\uparrow{\mbox{\large $\parallel$}}& &\uparrow {\scriptsize s\ell^{1+i}}& &\uparrow &\\
0\to& H^{1+ i}((H^0({{\cal H}}^1))^{\bullet})&\to &{\mbox{$\mathbb H$}}^{1 +i}({X_{{\mbox{\LARGE $\cdot $}}}},
{{\cal H}}^{1}_{{X_{{\mbox{\LARGE $\cdot $}}}}})& \to & H^{i}((H^1({{\cal H}}^1))^{\bullet})&\to 0
\end{array}$$ mapping the extension (\[zarext1\]) to (\[ext1\]).
The image of the map $s\ell^{1+i}$ is $H^{2 + i}(X)^e$.
Following [@BSAP] consider the simplicial sheaf ${{\cal O}}_{{X_{{\mbox{\LARGE $\cdot $}}}}}^*$ and the corresponding Zariski cohomology groups ${\mbox{$\mathbb H$}}^{1 +i}({X_{{\mbox{\LARGE $\cdot $}}}},{{\cal O}}_{{X_{{\mbox{\LARGE $\cdot $}}}}}^*)$. Since the components of ${X_{{\mbox{\LARGE $\cdot $}}}}$ are smooth, the canonical spectral sequence $$E^{p ,q}_1 = H^q(X_p, {{\cal O}}^*_{X_p}) {\mbox{$\Rightarrow$}}{\mbox{$\mathbb H$}}^{p+ q}({X_{{\mbox{\LARGE $\cdot $}}}},{{\cal O}}_{{X_{{\mbox{\LARGE $\cdot $}}}}}^*)$$ yields a long exact sequence $$\cdots \to H^{1+ i}((H^0({{\cal O}}^*))^{\bullet})\to {\mbox{$\mathbb H$}}^{1
+i}({X_{{\mbox{\LARGE $\cdot $}}}},{{\cal O}}_{{X_{{\mbox{\LARGE $\cdot $}}}}}^*) \to H^{i}(({{\rm Pic}\,}))^{\bullet}){\stackrel{d^{i}}{\longrightarrow}} H^{2+
i}((H^0({{\cal O}}^*))^{\bullet}) \to \cdots$$ According to [@BSAP] (see the construction in [@BRS]) we may regard ${\mbox{$\mathbb H$}}^{1+i}({X_{{\mbox{\LARGE $\cdot $}}}},{{\cal O}}_{{X_{{\mbox{\LARGE $\cdot $}}}}}^*)$ as the group of $k$-points of a group scheme whose connected component of the identity yields a semi-abelian variety $$0 \to H^{1+ i}((H^0({{\cal O}}^*))^{\bullet})/\sigma\to {\mbox{$\mathbb H$}}^{1
+i}({X_{{\mbox{\LARGE $\cdot $}}}},{{\cal O}}_{{X_{{\mbox{\LARGE $\cdot $}}}}}^*)^0 \to H^{i}(({{\rm Pic}\,}^0))^{\bullet})^0\to 0$$ where $\sigma$ is a finite group. The Hodge realization of the so obtained isogeny 1-motive is $$T_{\rm Hodge} ([0\to {\mbox{$\mathbb H$}}^{1+i}({X_{{\mbox{\LARGE $\cdot $}}}},{{\cal O}}_{{X_{{\mbox{\LARGE $\cdot $}}}}}^*)^0]_{{\mbox{\scriptsize{${\mbox{$\mathbb Q$}}$}}}}) =
W_1H^{1+i}(X,{\mbox{$\mathbb Q$}})(1).$$ This last claim is clearly related to Deligne’s conjecture [@D 10.4.1]. For $i=-1, 0$ this is actually proven in [@BSAP] and for all $i$ in [@BRS].
Recall the existence of a canonical map of sheaves $c_1: {{\cal O}}_{{X_{{\mbox{\LARGE $\cdot $}}}}}^*\to
{{\cal H}}^{1}_{{X_{{\mbox{\LARGE $\cdot $}}}}}$ yielding a map $$c_1: {\mbox{$\mathbb H$}}^{1 +i}({X_{{\mbox{\LARGE $\cdot $}}}},{{\cal O}}_{{X_{{\mbox{\LARGE $\cdot $}}}}}^*)\to {\mbox{$\mathbb H$}}^{1
+i}({X_{{\mbox{\LARGE $\cdot $}}}}, {{\cal H}}^{1}_{{X_{{\mbox{\LARGE $\cdot $}}}}}).$$ By composing $s\ell^{1+i}$ and $c_1$ we then obtain a cycle map $${\mbox{$\mathbb H$}}^{1 +i}({X_{{\mbox{\LARGE $\cdot $}}}},{{\cal O}}_{{X_{{\mbox{\LARGE $\cdot $}}}}}^*)\to W_2H^{2 + i}(X)/W_0.$$ We may regard the image of this cycle map as the discrete part of ${\mbox{$\mathbb H$}}^{1
+i}({X_{{\mbox{\LARGE $\cdot $}}}},{{\cal O}}_{{X_{{\mbox{\LARGE $\cdot $}}}}}^*)$. Over ${\mbox{$\mathbb Q$}}$, it is clearly equal to $F^1\cap H^{2 +
i}(X,{\mbox{$\mathbb Q$}})$. The reader can easily check that this is the case, [[*e.g.*]{}, ]{}${\mbox{$\mathbb H$}}^{2
+ i}({X_{{\mbox{\LARGE $\cdot $}}}},{{\cal O}}_{{X_{{\mbox{\LARGE $\cdot $}}}}}^*[-1])$ coincides with Deligne-Beilinson cohomology (see [@BE 5.4]). In general, we may expect the following picture for cycle maps.
${{\cal K}}$-cohomology and motivic cohomology
----------------------------------------------
Let ${X_{{\mbox{\LARGE $\cdot $}}}}$ be a smooth simplicial scheme. Consider the local-to-global spectral sequence in Claim \[l2g\]. For a fixed $p$ we then obtain a spectral sequence $$W_{2p}{\mbox{$\mathbb H$}}^q(X_{{\mbox{\LARGE $\cdot $}}}, {{\cal H}}_{{X_{{\mbox{\LARGE $\cdot $}}}}}^r)/W_{2p-2}{\mbox{$\Rightarrow$}}W_{2p}{\mbox{$\mathbb H$}}^{q+r}(X_{{\mbox{\LARGE $\cdot $}}},{\mbox{$\mathbb Q$}})/W_{2p-2}.$$ The sheaf ${{\cal H}}_{{X_{{\mbox{\LARGE $\cdot $}}}}}^r$ has weights $\leq 2r$ and so the mixed Hodge structure on its cohomology has weights $\leq 2r$. Thus $W_{2p}{\mbox{$\mathbb H$}}^q(X_{{\mbox{\LARGE $\cdot $}}}, {{\cal H}}_{{X_{{\mbox{\LARGE $\cdot $}}}}}^p) = {\mbox{$\mathbb H$}}^q(X_{{\mbox{\LARGE $\cdot $}}}, {{\cal H}}_{{X_{{\mbox{\LARGE $\cdot $}}}}}^p)$ and $W_{2p}{\mbox{$\mathbb H$}}^q(X_{{\mbox{\LARGE $\cdot $}}}, {{\cal H}}_{{X_{{\mbox{\LARGE $\cdot $}}}}}^r)/W_{2p-2}=0$ if $r<p$. Thus, there is an edge map $$\label{sedge}
s\ell^{p+i}:{\mbox{$\mathbb H$}}^{p+i}(X_{{\mbox{\LARGE $\cdot $}}}, {{\cal H}}_{{X_{{\mbox{\LARGE $\cdot $}}}}}^p)/W_{2p-2}\to
W_{2p}{\mbox{$\mathbb H$}}^{2p+i}({X_{{\mbox{\LARGE $\cdot $}}}})/W_{2p-2}.$$ Note that if ${X_{{\mbox{\LARGE $\cdot $}}}}= X$ is constant then $s\ell^{p+0}=s\ell^{p}_0$ and $s\ell^{p-1}=s\ell^{p}_{-1}$ in the notation of Section 4.2, modulo $W_{2p-2}$.
The image of the edge map $s\ell^{p+i}$ is $H^{2p
+ i}(X)^h$.
Consider Quillen’s higher $K-$theory. Consider Zariski sheaves associated to Quillen’s $K$-functors. The ${{\cal K}}$-cohomology groups are ${\mbox{$\mathbb H$}}^*(X_{{\mbox{\LARGE $\cdot $}}},
{{\cal K}}_p)$ (as usual we consider Zariski simplicial sheaves ${{\cal K}}_p$). Local higher Chern classes give us maps of simplicial sheaves $c_p : {{\cal K}}_p \to
{{\cal H}}_{{X_{{\mbox{\LARGE $\cdot $}}}}}^p(p)$ for each $p\geq 0$ ([[*cf.*]{} ]{}[@BV2]). We thus obtain a map $$c_p : {\mbox{$\mathbb H$}}^{p+i}(X_{{\mbox{\LARGE $\cdot $}}}, {{\cal K}}_p) \to {\mbox{$\mathbb H$}}^{p+i}(X_{{\mbox{\LARGE $\cdot $}}}, {{\cal H}}_{{X_{{\mbox{\LARGE $\cdot $}}}}}^p(p)).$$ Note that in the canonical spectral sequence $$E^{s ,t}_1 = H^t(X_s,
{{\cal K}}_{p}) {\mbox{$\Rightarrow$}}{\mbox{$\mathbb H$}}^{s+t}({X_{{\mbox{\LARGE $\cdot $}}}},{{\cal K}}_{p})$$ we have $ H^t(X_s, {{\cal K}}_{p}) =
0$ if $t >p$. The same hold for the sheaf ${{\cal H}}_{{X_{{\mbox{\LARGE $\cdot $}}}}}^p$. We then have a commutative square $$\begin{array}{ccc}
{\mbox{$\mathbb H$}}^{p+i}(X_{{\mbox{\LARGE $\cdot $}}}, {{\cal K}}_p)& {\stackrel{c_p}{\rightarrow}} &{\mbox{$\mathbb H$}}^{p+i}(X_{{\mbox{\LARGE $\cdot $}}}, {{\cal H}}_{{X_{{\mbox{\LARGE $\cdot $}}}}}^p(p))\\
\downarrow & &\downarrow\\
H^{i}((CH^p)^{\bullet})&\to & H^{i}((NS^p)^{\bullet}).
\end{array}$$ Thus the image of $c_p$ in $H^{i}((NS^p)^{\bullet})$ is clearly contained in the kernel of the map $\lambda_a^i$ defined in (\[bound\]). We also have the following commutative square $$\begin{array}{ccc}
{\mbox{$\mathbb H$}}^{p+i}(X_{{\mbox{\LARGE $\cdot $}}}, {{\cal H}}_{{X_{{\mbox{\LARGE $\cdot $}}}}}^p)/W_{2p-2}& {\stackrel{s\ell^{p+i}}{\rightarrow}} &
W_{2p}{\mbox{$\mathbb H$}}^{2p+i}({X_{{\mbox{\LARGE $\cdot $}}}})/W_{2p-2}\\
\downarrow & &\downarrow\\
H^{i}((NS^p)^{\bullet})&\to & {{\rm gr}\,}_{2p}{\mbox{$\mathbb H$}}^{2p+i}({X_{{\mbox{\LARGE $\cdot $}}}}).
\end{array}$$ Composing $s\ell^{p+i}$ and $c_p$ above we then obtain a simplicial cycle map $$\label{scycle}
c\ell^{p+i}: {\mbox{$\mathbb H$}}^{p+i}(X_{{\mbox{\LARGE $\cdot $}}}, {{\cal K}}_p) \to W_{2p}H^{2p+i}({X_{{\mbox{\LARGE $\cdot $}}}})/W_{2p-2}.$$ Let $X$ be a proper ${\mbox{$\mathbb C$}}$-scheme and let ${X_{{\mbox{\LARGE $\cdot $}}}}\to X$ be a universal cohomological descent morphism. By descent, ${\mbox{$\mathbb H$}}^{*}({X_{{\mbox{\LARGE $\cdot $}}}})\cong H^{*}(X)$ as mixed Hodge structures. Let $H$ denote the mixed Hodge structure on $H^{2p+i}(X, {\mbox{$\mathbb Z$}})/({\rm torsion})$. Let $F^p$ denote the Hodge filtration. Note that $$F^p\cap H_{{\mbox{\scriptsize{${\mbox{$\mathbb Z$}}$}}}} =
{{\rm Hom}\,}_{\rm MHS} ({\mbox{$\mathbb Z$}}(-p), H) =
{{\rm Hom}\,}_{\rm MHS} ({\mbox{$\mathbb Z$}}(-p), W_{2p}H).$$ Moreover $${{\rm Hom}\,}_{\rm MHS} ({\mbox{$\mathbb Z$}}(-p), W_{2p}H) \subseteq
{{\rm Hom}\,}_{\rm MHS} ({\mbox{$\mathbb Z$}}(-p), W_{2p}H/W_{2p-2}H) = F^p \cap
H^e_{{\mbox{\scriptsize{${\mbox{$\mathbb Z$}}$}}}}.$$ Thus $$F^p\cap H_{{\mbox{\scriptsize{${\mbox{$\mathbb Z$}}$}}}}\subseteq F^p \cap H^e_{{\mbox{\scriptsize{${\mbox{$\mathbb Z$}}$}}}}= F^p \cap H^h_{{\mbox{\scriptsize{${\mbox{$\mathbb Z$}}$}}}} =
\ker (H^{p,p}_{{\mbox{\scriptsize{${\mbox{$\mathbb Z$}}$}}}}{\stackrel{e^p}{\rightarrow}} J^p(H)).$$ Therefore we have the following natural question.
\[cycleimage\] Let $X$ be a proper ${\mbox{$\mathbb C$}}$-scheme and let $\pi :{X_{{\mbox{\LARGE $\cdot $}}}}\to X$ be a proper smooth hypercovering. Is $F^p \cap
H^{2p+i}(X, {\mbox{$\mathbb Q$}})$ the image of the cycle class map $c\ell^{p+i}$ in (\[scycle\]) ?
Bloch’s counterexample answer this question in the negative (see Section 5.1 below). Note that here we actually deal with the simplicial scheme ${X_{{\mbox{\LARGE $\cdot $}}}}$ (not just $X$) as, [[*e.g.*]{}, ]{}in [@BS] it is shown that $F^1 \cap H^{2}(X, {\mbox{$\mathbb Z$}})$ can be larger than the image of ${{\rm Pic}\,}(X)$, if $X$ is singular. However, $F^1 \cap
H^{2}(X, {\mbox{$\mathbb Z$}})$ is the image of the ${{\rm Pic}\,}$ of any hypercovering of $X$. However, $F^2 \cap H^{4}(X, {\mbox{$\mathbb Q$}})$ is larger than the image of ${\mbox{$\mathbb H$}}^2({X_{{\mbox{\LARGE $\cdot $}}}},{{\cal K}}_2)$ if $X$ is the singular 3-fold in Section 5.1 below.
Let’s then consider the case $p =2$ in the above. In this case we have that $H^q(X_i,{{\cal H}}^2_{X_i})$ is purely of weight $q+2$ by the coniveau spectral sequence (\[conifilt\]). Thus the canonical spectral sequence in Corollary \[zarhodge\] degenerates yielding the following extension $$\label{zarext2}
0\to H^{1+ i}((H^1({{\cal H}}^2))^{\bullet})\to {\mbox{$\mathbb H$}}^{2+i}({X_{{\mbox{\LARGE $\cdot $}}}}, {{\cal H}}^{2}_{{X_{{\mbox{\LARGE $\cdot $}}}}})/W_2
\to H^{i}((NS^2)^{\bullet})\to 0.$$ Note that $H^1(X_i,{{\cal H}}^2_{X_i})= N^1H^3(X_i)$ and $W_2{\mbox{$\mathbb H$}}^{j}({X_{{\mbox{\LARGE $\cdot $}}}},
{{\cal H}}^{2}_{{X_{{\mbox{\LARGE $\cdot $}}}}})= H^{j}((H^0({{\cal H}}^2))^{\bullet})$. The map in (\[sedge\]) is mapping the extension (\[zarext2\]) to the following canonical extension $$0\to H^{1+ i}((H^3)^{\bullet})\to W_4H^{4 + i}(X)/W_2 \to
H^{i}((H^4)^{\bullet})\to 0.$$ According to Conjectures \[GHC\]–\[MHC\] we may expect that the image of ${\mbox{$\mathbb H$}}^{2+i}({X_{{\mbox{\LARGE $\cdot $}}}}, {{\cal H}}^{2}_{{X_{{\mbox{\LARGE $\cdot $}}}}})/W_2$ under this map is $H^{4 + i}(X)^h$.
Finally, making use of the triangulated category of motives (see [@V] and [@LM]) let $H^*_m(X,{\mbox{$\mathbb Q$}}(\cdot))$ denote the motivic cohomology of the proper ${\mbox{$\mathbb C$}}$-scheme $X$. Since motivic cohomology is universal we may get a canonical map $H^{2p+i}_m(X,{\mbox{$\mathbb Q$}}(p))\to H^{2p+i}(X,{\mbox{$\mathbb Q$}}(p))$ compatibly with the weight filtrations. This map will factors trhough Beilinson’s absolute Hodge cohomology [@BE]. However, in general, its image will not be larger than $c\ell^{p+i}$ in (\[scycle\]), [[*i.e.*]{}, ]{}smaller than the rational part of $F^pH^{2p+i}(X,{\mbox{$\mathbb C$}})$. In fact, we can see that only Beilinson’s absolute Hodge cohomology (or Deligne-Beilinson cohomology) would have image equal to $F^2\cap
H^{4}(X,{\mbox{$\mathbb Q$}})$ if $X$ is the singular 3-fold in Bloch’s counterexample below.
Examples
========
We finally discuss a couple of examples where one can test the conjectures.
Bloch’s example
---------------
We now consider Bloch’s example explained in a letter to U. Jannsen, reproduced in the Appendix A of [@JA] (see also Appendix A.I in [@LW]). This example, originally requested by Mumford, is a counterexample to a naive extension of the cohomological Hodge conjecture to the singular case. Moreover (as indicated by Bloch’s Remark 1 in [@JA Appendix A]) it shows that no cohomological invariants of algebraic varieties, that agree with Chow groups of non-singular varieties, can provide all Hodge cycles for singular varieties.
Let $P$ be the blow-up of ${\mbox{$\mathbb P$}}^3$ at a point $x$ in $S_0\subset {\mbox{$\mathbb P$}}^3$ a smooth hypersurface of degree $\geq 4$ over ${\overline}{{\mbox{$\mathbb Q$}}}$. The point $x$ is assumed ${\overline}{{\mbox{$\mathbb Q$}}}$-generic. Let $S$ be the blow-up of $S_0$ at $x$ over ${\mbox{$\mathbb C$}}$. Thus $S\subset P$ and $H^3(S,{\mbox{$\mathbb Q$}})=0$.
Let $X$ be the gluing of two copies of $P$ along $S$, [[*i.e.*]{}, ]{}the singular projective variety defined as the pushout $$\begin{array}{ccc}
S\coprod S&{\stackrel{i\coprod i}{\rightarrow}}& P\coprod P\\
c\downarrow\quad&&\quad\downarrow f\\
S&{\stackrel{j}{\rightarrow}}&X
\end{array}$$ Such a Mayer-Vietoris diagram always defines a cohomological descent morphism ${X_{{\mbox{\LARGE $\cdot $}}}}\to X$ ([[*e.g.*]{}, ]{}a distinguished (semi)simplicial resolution in the sense of Carlson [@CA §3 and §13]).
Thus we obtain a short exact sequence $$0 \to H^4(X,{\mbox{$\mathbb Q$}}(2)) \to H^4(P,{\mbox{$\mathbb Q$}}(2))^{\oplus 2} \to H^4(S,{\mbox{$\mathbb Q$}}(2)) \to 0$$ where $CH^2(P)_{{\mbox{\scriptsize{${\mbox{$\mathbb Q$}}$}}}} \cong H^4(P,{\mbox{$\mathbb Q$}}(2))$. Thus $H^4(X, {\mbox{$\mathbb Z$}})$ has rank 3, is purely of type $(2,2)$ and the Hodge 1-motive is $[H^4(X,{\mbox{$\mathbb Q$}}(2)) \to 0 ]$. From (\[zarext2\]) we obtain ${\mbox{$\mathbb H$}}^{2}({X_{{\mbox{\LARGE $\cdot $}}}}, {{\cal H}}^{2}_{{X_{{\mbox{\LARGE $\cdot $}}}}})/W_2 = H^0(({{\rm NS}\,}^2)^{\bullet})) =H^4(X,{\mbox{$\mathbb Q$}}(2)).$
Since the Albanese of $S$ vanishes, the algebraically defined Hodge 1-motive is given by $H^0(({{\rm NS}\,}^2)^{\bullet})) =\ker (NS^2(P)^{\oplus 2} \to NS^2(S))$ and we clearly have that $$[H^0(({{\rm NS}\,}^2)^{\bullet})) \to 0]\cong [H^4(X,{\mbox{$\mathbb Q$}}(2)) \to 0 ]$$ as predicted by Conjecture \[MHC\]. However $H^0((CH^2)^{\bullet})) = \ker (CH^2(P)^{\oplus 2} \to CH^2(S))$ has rank 2, and it is strictly smaller than $H^4(X,{\mbox{$\mathbb Q$}}(2))$, as Bloch’s observed. Moreover, from the above we may regard ${\mbox{$\mathbb H$}}^2({X_{{\mbox{\LARGE $\cdot $}}}},{{\cal K}}_2)_{{\mbox{\scriptsize{${\mbox{$\mathbb Q$}}$}}}}$ mapping to both $H^0((CH^2)^{\bullet}))$ and $H^4(X,{\mbox{$\mathbb Q$}}(2))$. Then $c\ell^{2}$ in (\[scycle\]) is not surjective because $H^0((CH^2)^{\bullet}))\neq H^4(X,{\mbox{$\mathbb Q$}}(2))$. The same argument applies to motivic cohomology $H^4_m(X,{\mbox{$\mathbb Q$}}(2))$. In fact, if $Y$ is a smooth variety $H^4_m(Y,{\mbox{$\mathbb Q$}}(2))\cong CH^2(Y)_{{\mbox{\scriptsize{${\mbox{$\mathbb Q$}}$}}}}$. However, Beilinson’s absolute Hodge cohomology is $H^4_{{{\cal D}}}(X,{\mbox{$\mathbb Z$}}(2))\otimes {\mbox{$\mathbb Q$}}\cong H^4(X,{\mbox{$\mathbb Q$}}(2))$. Thus, Beilinson’s formulation of the Hodge conjecture in [@BE §6] doesn’t hold in the singular case.
Note that in this example, all Hodge classes are involved, as the Hodge structure is pure.
Srinivas example
----------------
The following example has been produced by Srinivas upon author’s request. It is similar to Bloch’s example however, in this example, the space of “Hodge cycles" is strictly smaller than $H^{2,2}_{{\mbox{\scriptsize{${\mbox{$\mathbb Q$}}$}}}}$.
Let $Y$ be a smooth projective complex 4-fold with $H^1(Y,{\mbox{$\mathbb Z$}})=H^3(Y,{\mbox{$\mathbb Z$}})=0$, and with an algebraic cycle $\alpha\in CH^2(Y)$ whose singular cohomology class ${\overline}{\alpha}\in H^4(Y,{\mbox{$\mathbb Q$}})$ is a non-zero primitive class. For example, $Y$ could be a smooth quadric hypersurface in ${\mbox{$\mathbb P$}}^5$, and $\alpha\in CH^2(Y)$ the difference of the classes of two planes, taken from the two distinct connected families of planes in $Y$. Let $Z$ be a general hypersurface section of $Y$ of any fixed degree $d$ such that $H^{3,0}(Z)\neq 0$ (this holds for any large enough degree $d$; for example, if $Y$ is a quadric then we may take $Z=Y\cap H$ to be the intersection with a general hypersurface $H$ of any degree $\geq 3$).
Then $Z$ is a smooth projective 3-fold, and by the theorem of Griffiths, if $i:Z\to Y$ is the inclusion, then $i^*\alpha\in CH^2(Z)$ is homologically trivial, but no non-zero multiple of $i^*\alpha$ is algebraically equivalent to 0. In fact, $H^3(Z,{\mbox{$\mathbb Q$}})$ has no proper Hodge substructures, and so (because $H^{3,0}(Z)\neq 0$) the Abel-Jacobi map vanishes on the group $CH^2(Z)_{\rm alg}$ of cycle classes algebraically equivalent to 0; on the other hand, the Abel-Jacobi image of $i^*\alpha$ is non-torsion.
Now let $X$ be the singular projective variety defined as a push-out $$\begin{array}{ccc}
Z\coprod Z&{\stackrel{i\coprod i}{\rightarrow}}& Y\coprod Y\\
c\downarrow\quad&&\quad\downarrow f\\
Z&{\stackrel{j}{\rightarrow}}&X
\end{array}$$ so that $X$ is obtained by gluing two copies of $Y$ along $Z$.
Consider the simplicial scheme $X_{{\mbox{\LARGE $\cdot $}}}$ obtained as above ([[*e.g.*]{}, ]{}the Čech hypercovering of $X$, with $X_0\to X$ taken to be the quotient map $f:Y\coprod Y\to X$). Then $H^*(X_{{\mbox{\LARGE $\cdot $}}},{\mbox{$\mathbb Q$}})\cong H^*(X,{\mbox{$\mathbb Q$}})$ as mixed Hodge structures, and we have an exact sequence of mixed Hodge structures (of which all terms except $H^4(X,{\mbox{$\mathbb Z$}})$ are in fact pure) $$0\to H^3(Z,{\mbox{$\mathbb Z$}})\to H^4(X,{\mbox{$\mathbb Z$}})\to H^4(Y,{\mbox{$\mathbb Z$}})^{\oplus 2}{\stackrel{s}{\rightarrow}} H^4(Z,{\mbox{$\mathbb Z$}})$$ where $s(a,b)=i^*a-i^*b$. Then $({\overline}{\alpha},0)$ and $(0,{\overline}{\alpha})$ are linearly independent elements of $\ker s$, since $i^*{\overline}{\alpha}=0$ in $H^4(Z,{\mbox{$\mathbb Q$}})$ (this is essentially the definition of ${\overline}{\alpha}$ being a primitive cohomology class).
In this situation, the group of Hodge classes in $H^4(X,{\mbox{$\mathbb Q$}})/W_3$ is non-trivial, but since $H^3(Z,{\mbox{$\mathbb Q$}})$ has no non-trivial sub-Hodge structures, the intermediate Jacobian $J^2(Z)$ has no non-trivial abelian subvariety. The extension of Hodge structures determined by the Hodge classes is not split; for example the extension class of the pullback of $$0\to H^3(Z,{\mbox{$\mathbb Z$}})\to H^4(X,{\mbox{$\mathbb Z$}})\to \ker s\to 0$$ under ${\mbox{$\mathbb Z$}}(-2)\to \ker s$ determined by $({\overline}{\alpha},0)$ is (up to sign) the Abel-Jacobi image of $i^*\alpha$, which is non-torsion. Here ${{\rm rank}\,}(\ker s)=3$, so we get an extension class map ${\mbox{$\mathbb Z$}}^3\to J^2(Z)$; one checks that the image has rank 1, generated by the image of $({\overline}{\alpha},0)$ (or equivalently by the image of $(0,{\overline}{\alpha})$).
So the lattice for the corresponding Hodge 1-motive is, by definition $$\ker ({\mbox{$\mathbb Z$}}^3\to J^2(Z))=F^2\cap H^4(X,{\mbox{$\mathbb Z$}}),$$ which is strictly smaller than the lattice of all Hodge classes in $H^4(X,{\mbox{$\mathbb Z$}})$. Moreover, since $H^1(Z,{{\cal H}}^2)= N^1H^3(Z)=0$, from the extension (\[zarext2\]) we obtain ${\mbox{$\mathbb H$}}^{2}({X_{{\mbox{\LARGE $\cdot $}}}}, {{\cal H}}^{2}_{{X_{{\mbox{\LARGE $\cdot $}}}}})/W_2 \cong H^{0}((NS^2)^{\bullet})$.
Finally, the cycle $\alpha\in CH^2(Y)$ projects to a cycle in ${{\rm NS}\,}^2(Y)$ which restricts to a non-zero class $i^*\alpha\in NS^2(Z)_{{\mbox{\scriptsize{${\mbox{$\mathbb Q$}}$}}}}$ by construction. Since $CH^2(Z)_{\rm ab}=CH^2(Z)_{\rm alg}$ then the algebraically defined 1-motive is given by the image of $H^{0}((NS^2)^{\bullet})=\ker (NS^2(Y)^{\oplus
2} \to NS^2(Z))$ in $H^4(X,{\mbox{$\mathbb Z$}})$, providing generators for $\ker ({\mbox{$\mathbb Z$}}^3\to J^2(Z))$ as claimed in Conjecture \[MHC\].
: ${{\cal H}}$-cohomologies versus algebraic cycles, [*Math. Nachr.*]{} [**184**]{} (1997) 5-57.
: Deligne’s conjecture on 1-motives, preprint math.AG/0102150, 2001.
: The Néron-Severi group and the mixed Hodge structure on $H^2$, [*J. Reine Ang. Math.*]{} [**450**]{} (1994) 37-42.
: Albanese and Picard 1-motives, [*C. R. Acad. Sci.*]{} Paris, t. 326, Série I (1998) 1397–1401, and forthcoming Mémoire SMF.
: Notes on absolute Hodge cohomology, Contemporary Mathematics [**55**]{}, Part I, (1986) 35-68.
: A Lefschetz (1,1)-Theorem for normal projective varieties, [*Duke Math. J.*]{} [**101**]{} (2000) 427–458.
and[ A. Ogus]{}: Gersten’s conjecture and the homology of schemes, [*Ann.Sci.Ecole Norm.Sup.*]{} [**7**]{} (1974) 181–202.
: The obstruction to splitting a mixed Hodge structure over the integers, I, University of Utah, Salt Lake City, 1979.
: The one-motif of an algebraic surface, [*Compositio Math.*]{} [**56**]{} (1985) 271–314.
: Théorie de Hodge II, III [*Publ. Math.*]{} IHES [**40**]{} (1972) 5–57 and [*Publ. Math.*]{} IHES [**44**]{} (1974) 5–78.
: A quoi servent les motifs ?, in “Motives” [*Proc. Symp. Pure Math*]{} AMS [**55**]{}, Part 1, (1994) 143–161.
: The Hodge conjecture, Millennium Prize Problems, available browsing from [http://www.claymath.org/]{}
: Smoothness, semistability and alterations [*Publ. Math.*]{} IHES [**83**]{} (1996) 51–93.
and [C. Soulé]{}: Descent, motives and $K$-theory, [*J. Reine Ang. Math.*]{} [**478**]{} (1996) 127-176.
and [V. Navarro-Aznar]{}: Un critère d’extension d’un foncteur défini sur les schémas lisses, preprint, 1995.
: Some trascendental methods in the study of algebraic cycles, Springer LNM [**185**]{}, Heidelberg, 1971, 1-46.
: Technique de descente et théorèmes d’existence en géométrie algébrique I–VI, “Fondements de la Géométrie Algébrique” Extraits du [*Séminaire*]{} [Bourbaki]{} 1957/62.
: Hodge’s general conjecture is false for trivial reasons, [*Topology*]{} [**8**]{} (1969) 299–303.
: The de Rham homotopy theory of complex algebraic varieties, [*K-theory*]{} [**1**]{} (1987) I: 271–324, II: 481–497.
: Mixed motives and algebraic cycles, I II III, preprints.
: The mixed motive of a projective variety, in “Banff Proceedings” CRM Proc. Lecture Notes [**24**]{}, 2000.
: Mixed motives and algebraic $K$-theory, [*Springer*]{} LNM [**1400**]{}, 1990.
: Motivic sheaves and filtrations on Chow groups, in “Motives” AMS [*Proc. of Symp. in Pure Math*]{} [**55**]{} Part 1, 1994.
: Equivalence relations on algebraic cycles, in “Banff Proceedings” CRM Proc. Lecture Notes [**24**]{}, 2000.
: Mixed Motives, AMS Math. Surveys and Monographs Vol. 57, 1998.
: A survey of the Hodge conjecture, second edition, appendix B by B. Brent Gordon. AMS-CRM Monograph Series Vol. 10, Providence, RI, 1999.
: A filtration on the Chow groups of a complex projective variety, to appear in [*Compositio Mathematica*]{}.
: Suslin homology and Deligne 1-motives, in “Algebraic $K$-theory and algebraic topology” NATO ASI Series, 1993, 189–197.
: Intermediate jacobians, in “Algebraic Geometry” Oslo, Wolters-Nordhoff, 1972.
: The algebraic topology of smooth algebraic varieties, [*Publ. Math.*]{} IHES [**48**]{} (1978) 137–204 (Errata Corrige in [*Publ. Math.*]{} IHES [**64**]{}).
: Applications of algebraic $K$-theory to the theory of algebraic cycles, in “Proc. Algebraic Geometry”, Sitges, Springer LNM [**1124**]{}, 1985.
: Abel-Jacobi equivalence versus incidence equivalence for algebraic cycles of codimension two, [*Topology*]{} [**24**]{} (1985) 361–367.
: On a conjectural filtration on the Chow groups of an algebraic variety, I, II, [*Indag. Math.*]{} New Series [**4**]{} (1993) 177–201.
: Some spectral sequences for filtered complexes and applications, [*Jour. of Algebra*]{} [**186**]{} (1996) 793–806.
: Arithmetic mixed sheaves, preprint math.AG/9907189.
: Refined cycle maps, preprint math.AG/0103116.
: Motives, algebraic cycles and Hodge theory in “Banff Proceedings” CRM Proc. Lecture Notes [**24**]{}, 2000.
: Triangulated categories of motives over a field, in “Cycles, Transfers, and Motivic Cohomology Theories” Princeton Univ. Press, Annals of Math. Studies [**143**]{} 2000.
\
[*Dipartimento di Metodi e Modelli Matematici\
Università degli Studi di Roma “La Sapienza”\
Via A. Scarpa, 16\
I-00161 — Roma (Italy)*]{}
|
---
address:
- |
Niels Bohr Institute\
Blegdamsvej 17\
Copenhagen, 2100 Denmark\
ambjorn@nbi.dk
- |
Institute of Theoretical and Experimental Physics\
B. Cheremushkinskaya 25\
Moscow, 117218 Russian Federation\
makeenko@itep.ru
- |
Department of Physics and Astronomy\
University of British Columbia\
6224 Agricultural Road\
Vancouver, British Columbia V6T 1Z1, Canada\
semenoff@physics.ubc.ca
author:
- 'J. Ambj[ø]{}rn'
- 'Y. M. Makeenko'
- 'G. W. Semenoff'
title: 'Thermodynamics of D0-branes in matrix theory [^1]'
---
\#1\#2\#3\#4[[\#1]{} [**\#2**]{}, \#3 (\#4)]{}
‘=11
mycitex\[\#1\]\#2[@fileswauxout citeamycite[forciteb:=\#2]{}[\#1]{}]{}
mycite\#1[\[[\#1]{}\]]{}
‘=12
-1.4cm -0.8cm -0.8cm
v
NBI–HE–98–27\
MPS–RR–1998-25\
ITEP–TH–50/98
Introduction
============
Dirichlet p-branes are p+1-dimensional hypersurfaces on which superstrings can begin and end (see [@Pol96; @wati] for a review). The low energy dynamics of an ensemble of N parallel Dp-branes can be described by the U(N) supersymmetric gauge theory obtained by dimensional reduction of ten dimensional supersymmetric Yang-Mills theory to the p+1-dimensional world-volume of the brane.[@witt2] The Yang-Mills theory gives an accurate perturbative representation of the Dp-brane dynamics when the separations between the branes is large.[@gab; @kp; @dkps] It represents a truncation of the full string spectrum to the lowest energy modes. The full string theoretical interactions between a pair of Dp-branes is computed by considering the annulus diagram shown in fig. \[f:annulus\]. The short distance asymptotics of this diagram are dominated by the open string sector whose lowest modes are the fields of ten dimensional supersymmetric Yang-Mills theory. On the other hand, long distance asymptotics are most conveniently described by the dual description of this diagram as a closed string exchange and the relevant field theoretical modes are those of ten dimensional supergravity. That these are also represented by the dimensionally reduced super Yang-Mills theory is a result of supersymmetry and the fact that, for fixed Dp-brane positions, the ground state is a BPS state. At zero temperature, because of supersymmetry, the interaction potential between a pair of static D0-branes vanishes independently of their separation. Their effective action has been computed in an expansion in their velocities, divided by powers of the separation and is known to be [@b; @dkps] $$S_{\rm eff}(T=0)= \int dt \left(
\frac{1}{2g_s\sqrt{\alpha'}} \sum_{\alpha=1}^2 ( {\dot {\vec q}}\,{}^\alpha
) ^2 -
\frac{15}{16}\left( \alpha'\right)^3
\frac{\vert {\dot{\vec q}}\,{}^1-{\dot{\vec q}}\,{}^2
\vert^4}{\vert {\vec q}\,{}^1-{\vec q}\,{}^2 \vert^7}+\ldots \right)
\label{T=0}$$ This result agrees with the effective potential for the interaction of D0-branes in ten dimensional supergravity. Note that, for weak string coupling, the D0-brane is very heavy.
In this paper, we shall consider the description of D0-brane interactions in type IIA superstring theory using matrices. Even at very low temperatures, non-BPS states are important to the leading temperature dependence. We perform 1-loop computation of the effective interaction between static D0-branes in the matrix theory at finite temperature and compare with the known superstring computations. We show that the results of the two computations are similar in the low temperature limit but an extra integration over the temporal component of the gauge field, is present in the matrix theory. At finite temperature, because the Euclidean time is compact, the temporal gauge field can not be removed by a gauge transformation. [*This integration is needed in order to describe correctly thermodynamics of $D0$-branes both in the matrix and superstring theories*]{}.
The paper is organized as follows. In section 2 we discuss the formulation of the matrix theory at finite temperature. In section 3 we perform one loop computation of the effective interaction between static D0-branes at finite temperature and show that it is attractive, and short-ranged. In section 4 we compare this result with the superstring computations and discuss the conditions under which the two computations agree. Section 5 is devoted to the discussion of our results and, in particular, the origin of the divergence of the classical thermal partition function of D0-branes which is cured by quantum statistics.
Matrix theory at finite temperature
===================================
We shall consider the matrix theory description [@bfss] of the effective dynamics of D0-branes in a type-IIA superstring theory which is derived by the reduction of ten dimensional supersymmertic Yang-Mills theory which has the action $$S_{\rm YM}[A,\theta]=\frac{1}{g_{YM}^2}\int d\tau {\rm TR}\left(
\frac{1}{4}F_{\mu\nu}^2+\frac{i}{2}\theta\gamma_\mu D_\mu\theta\right)$$ to zero spatial dimension: $A_\mu=A_\mu(\tau)$, $\theta=\theta(\tau)$.
The thermal partition function of this theory is given by $$Z_{\rm YM}=\int[dA(\tau)][d\theta(\tau)]\exp\left( -S_{\rm YM}[A,\theta]\right)
\label{thermal}$$ where $S_{\rm YM}$ is the Euclidean action and the time coordinate is periodic. The bosonic and fermionic coordinates have periodic and anti-periodic boundary conditions, $$\begin{aligned}
A_\mu(\tau+\beta)= A_\mu(\tau), \\
\theta(\tau+\beta)=-\theta(\tau),\\
\beta=1/k_BT ,
\label{b.c}\end{aligned}$$ where $T$ is the temperature and $k_B$ is Boltzmann’s constant. Gauge fixing will be necessary and will involve introducing ghost fields which will have periodic boundary conditions.
The representation (\[thermal\]) of the thermal partition function can be derived in the standard way starting from the known Hamiltonian of the matrix theory [@bfss] and representing the thermal partition function $$Z_{\rm YM}= {\rm tr}\, e^{-\beta H}$$ via the path integral. The trace is calculated here over all states obeying Gauss’s law which is taken care by the integration over $A_0$ in (\[thermal\]). This representation of the matrix theory at finite temperature have been already discussed [@OZ98; @MOP98; @Sat98], but the temperature induced interaction between $D0$-branes described below was never identified.
In matrix theory, the diagonal components of the gauge fields, $\vec{a}^\alpha\equiv \vec{A}^{\alpha\alpha}$, are interpreted as the position coordinates of the $\alpha$-th D0-brane and they should be treated as collective variables. Static configurations play a special role since they satisfy classical equations of motion with the periodic boundary conditions and dominate the path integral as $g^2_{\rm YM}\rightarrow 0$. Notice that there are no such static zero modes for fermionic components since they would not satisfy the antiperiodic boundary conditions. [^2] This is an important difference from the zero temperature case and a manifestation of the fact that supersymmetry is explicitly broken at non-zero temperature.
In the following, we will construct an effective action for these coordinates by integrating the off-diagonal gauge fields, the fermionic variables and the ghosts, $$S_{\rm eff}[ \vec a^\alpha]\equiv -\ln \int[d a^\alpha_0]
\prod_{\alpha\neq\beta}[dA^{\alpha\beta}_\mu][d\theta][d{\rm ghost}]\exp\left(
-S_{\rm YM}-S_{\rm gf}-S_{\rm gh}\right).$$ Generally, this integration can only be done in the a simultaneous loop expansion and expansion in the number of derivatives of the coordinates $\vec a^\alpha$. Such an expansion is accurate in the limit where $\left| \vec a^\alpha-\vec a^\beta\right|$ are large for each pair of D0-branes and where the velocities are small. Since these variables are periodic in Euclidean time, small velocities are only possible at low temperatures. The remaining dynamical problem then defines the statistical mechanics of a gas of D0-branes, $$Z_{\rm YM}=\int\prod_{\tau,\alpha}
[d \vec a^\alpha(\tau)]\exp\left( -S_{\rm eff}[\vec a^\alpha]
\right).
\label{effec}$$ We expect that the zero temperature limit of $S_{\rm eff}$ reduces to (\[T=0\]).
We shall find several subtleties with this formulation. If the effective D0-brane action is to reproduce the results of a string theoretical computation, the integration over $a_0^\alpha$ must be performed in both cases.
The effective action is a symmetric functional of the position variables $\vec a^\alpha(\tau)$. Only the configuration of these coordinates needs to be periodic. Therefore the individual position should be periodic up to a permutation. The variables in the path integral (\[effec\]) should therefore be periodic up to a permutation and the integral should be summed over the permutations.
One loop computation
====================
We will compute the effective action $S_{\rm eff}$ in a simultaneous expansion in the number of loops and in powers of time derivatives of the D0-brane positions.
We decompose the gauge field into diagonal and off-diagonal parts, $$A_\mu^{\alpha\beta}=a_\mu^{\alpha}\delta^{\alpha\beta}+g_{\rm YM}
\bar A^{\alpha\beta}_\mu$$ where $\bar A_{\mu}^{\alpha\alpha}=0$ so that the curvature is $$F_{\mu\nu}^{\alpha\beta}=\delta^{\alpha\beta}f^\alpha_{\mu\nu}+
g_{\rm YM}D_\mu^{\alpha\beta}
\bar A^{\alpha\beta}_\nu-g_{\rm YM}D_\nu^{\alpha\beta}
\bar A_\mu^{\alpha\beta}- ig^2_{\rm YM}
\left[ \bar A_\mu, \bar A_\nu \right]^{\alpha\beta}$$ where $$f_{\mu\nu}^\alpha= \partial_\mu a^\alpha_\nu-\partial_\nu a^\alpha_\mu$$ and $$D_\mu^{\alpha\beta}=\partial_\mu-i\left(a^\alpha_\mu-a^\beta_\mu\right).$$ In the Yang-Mills term in the action, we keep all orders of the diagonal parts of the gauge field and expand up to second order in the off-diagonal components, $$\frac{1}{g_{\rm YM}^2}{\rm TR}\left( F_{\mu\nu}^2
\right)=\sum_\alpha \frac{1}{g_{\rm YM}^2}\left(f^\alpha_{\mu\nu}\right)^2
+2\sum_{\alpha\beta}\bar A_{\mu}^{\beta\alpha}\left(
\delta_{\mu\nu}{\buildrel \leftarrow\over D}_\lambda^{\beta\alpha}
\vec D_\lambda^{\alpha\beta}-{\buildrel \leftarrow\over
D}_\mu^{\beta\alpha}\vec D_\nu^{\alpha\beta}+2i\left(
f^\alpha_{\mu\nu}- f^\beta_{\mu\nu} \right)\right)\bar
A_\nu^{\alpha\beta}+\ldots$$ We will fix the gauge $$D_\mu^{\alpha\beta}\bar A_\mu^{\alpha\beta}=0.$$ This entails adding the Fadeev-Popov ghost term to the action $$S_{\rm gh}=\int\sum_{\alpha\beta} \left\{\bar c^{\alpha\beta}\left(
-D^{\alpha\beta}_\mu\right)^2c^{\beta\alpha}+i
g_{\rm YM}\bar c^{\beta\alpha}D_\mu^{\alpha\beta}
\left[\bar A_\mu,c\right]\right\}$$ There is a residual gauge invariance under the abelian transformation, $$\begin{aligned}
\bar A^{\alpha\beta}_\mu\rightarrow \bar A^{\alpha\beta}_\mu
e^{i(\chi^\alpha -\chi^\beta)},
\nonumber \\
a_\mu^\alpha\rightarrow a_\mu^\alpha+\partial_\mu\chi^\alpha.\end{aligned}$$ We shall use this gauge freedom to set the additional condition $$\partial_0 a^{\alpha}_0=0$$ and to fix the constant[^3] $$-\pi/\beta < a_0^\alpha\leq\pi/\beta.$$
The ghost for this gauge fixing condition decouples.
Keeping terms up to quadratic order in $\bar A, c,\bar c, \theta$, the action is $$\begin{aligned}
S=\int\left\{\frac{1}{4g_{\rm YM}^2}\sum_\alpha \left(
f^\alpha_{\mu\nu}\right)^2 +\frac{1}{2}
\sum_{\alpha\beta}\bar A_{\mu}^{\beta\alpha}
\left( -\delta_{\mu\nu} D_\lambda^{\beta\alpha}
D_\lambda^{\alpha\beta}+D_\mu^{\beta\alpha}
D_\nu^{\alpha\beta}+2i\left( f^\alpha_{\mu\nu}- f^\beta_{\mu\nu}
\right)\right)\bar A_\nu^{\alpha\beta}
\right.\nonumber\\
\left.
+\sum_{\alpha\beta}\bar
c^{\beta\alpha}(D_\mu^{\alpha\beta})^2c^{\alpha\beta}
+\frac{i}{2}\theta^{\beta\alpha}\gamma_\mu D_\mu^{\alpha\beta}
\theta^{\alpha\beta}
\right\}.\end{aligned}$$ The effective action obtained by integrating over $\bar A,\bar
c,c,\theta$ is $$\begin{aligned}
S_{\rm eff}=\int\sum_\alpha\frac{1}{4g_{\rm YM}^2}(f^\alpha_{\mu\nu})^2
+\sum_{\alpha\neq\beta}\left\{ \frac{1}{2}{\rm TR}\ln\left( -\delta_{\mu\nu}(D^{\alpha\beta}_\mu)^2 +2i(f_{\mu\nu}^\alpha-f_{\mu\nu}^\beta)\right)
\right. \nonumber \\ \left.
-{\rm TR}\ln\left( -(D_\mu^{\alpha\beta})^2\right)-\frac{1}{2}{\rm TR}\ln
\left( i\gamma_\mu D^{\alpha\beta}_\mu\right) \right\}.
\label{seff}\end{aligned}$$
Leading order in time derivatives
---------------------------------
We will evaluate the determinants on the right-hand-side of (\[seff\]) in an expansion in powers of the derivatives of $\vec
a(\tau)$. The leading order term can be found by setting $\vec a={\rm
const.}$. In this case, $f^\alpha_{\mu\nu}=0$ and (here we retain the tree-level term with time derivatives) $$S_{\rm eff}=8\sum_{\alpha <\beta}\left\{{\rm
TR}_B\ln\left( -(D^{\alpha\beta}_\mu)^2 \right) -{\rm TR}_F\ln\left(
-(D^{\alpha\beta}_\mu)^2 \right)\right\}$$ where the subscript $B$ denotes contributions from the gauge fields and ghosts, whereas $F$ denotes those from the adjoint fermions. The determinants should be evaluated with periodic boundary conditions for bosons and anti-periodic boundary conditions for fermions. (Note that, because of supersymmetry, if both bosons and fermions had identical boundary conditions the determinants would cancel. This would give the well-known result that the lowest energy state is a BPS state whose energy does not depend on the relative separation of the D0-branes.) The boundary conditions are taken into account by introducing Matsubara frequencies, so that $$e^{-S_{\rm eff}}=e^{-S_{0}} \beta^{\rm N}
\int \frac{da^\alpha_0}{2\pi} \prod_{\alpha<\beta}
\prod_{n=-\infty}^\infty
\left( \frac{ \left(\frac{2\pi n}{\beta}
+\frac{\pi}{\beta}+a_0^\alpha-a_0^\beta\right)^2+\vert\vec
a_\alpha-\vec a_\beta\vert^2}{\left(
\frac{2\pi n}{\beta}+a_0^\alpha-a_0^\beta\right)^2 +\vert\vec
a_\alpha-\vec a_\beta\vert^2}
\right)^8$$ Using the formula $$\prod_{n=-\infty}^\infty
\left( \frac{2\pi n}{\beta}+\omega\right)=\sin\left( \frac{\beta\omega}
{2}\right)
\label{21}$$ we obtain the result $$e^{-S_{\rm eff}}=e^{-S_{0}} \beta^{\rm N}
\int \frac{da^\alpha_0}{2\pi} \prod_{\alpha<\beta}
\left( \frac{ \cosh \beta\vert\vec a^\alpha-\vec a^\beta\vert+
\cos \beta \left(a_0^\alpha- a_0^\beta\right)}
{\cosh\beta\vert\vec a^\alpha-\vec a^\beta\vert -
\cos\beta \left( a_0^\alpha-a_0^\beta\right) }\right)^8
\label{23}$$ In order to find the effective action for $\vec a^\alpha$, it is now necessary to integrate the temporal gauge fields $a_0^\alpha$ over the domain $(-\pi/\beta,\pi/\beta]$. This integration implements the projection onto the gauge invariant eigenstates of the matrix theory Hamiltonian.
In the case where there is a single pair of D0-branes, N=2, the integration over $a_0^\alpha$ in (\[23\]) can be done explicitly to obtain the effective action $$S_{\rm eff}=\int_0^\beta d\tau\left\{
\sum_1^2\frac{(\dot{\vec a}{}^\alpha)^2}{2g_{\rm YM}^2}
-\frac{1}{\beta}\ln\left(
\frac{ P(z) }{ (1-z^2)^{15} }
\right)\right\}
\label{effac}$$ where $$\begin{aligned}
P(z)= 1+241z^2+12649z^4+254009z^6+2434901z^8+12456773z^{10}
+36119181z^{12}\nonumber \\*+61178589z^{14}+6191459z^{16}
+36109171z^{18}
+12462779z^{20}+2432171z^{22}\nonumber\\*
+254919z^{24}+12439z^{26}+271z^{28}
-z^{30},\end{aligned}$$ $z=\exp(-\beta\vert {\vec a}^1-{\vec a}^2\vert)$ and we have included the tree level term, which gives the non-relativistic kinetic energies of the D0-branes.
The effective action has the low temperature expansion $$\begin{aligned}
S_{\rm eff}=\int_0^\beta d\tau\left( \frac{1}{2 g_{\rm YM}^2}\sum_\alpha
\left(\dot{\vec a}^\alpha\right)^2-\frac{1}{\beta}\left( 256 e^{-2\beta\vert
\vec a^1-\vec a^2\vert}-16384 e^{-4\beta\vert \vec a^1-a^2
\vert} \right.\right. \nonumber \\* \left.\left.
+\frac{5614336}{3}e^{-6\beta\vert \vec a^1-\vec a^2\vert}+
\ldots\right)\right)
\label{finalSeff}\end{aligned}$$ We shall compare in the following section this result with the superstring computation of the effective interaction between D0-branes.
String theoretical interactions
===============================
The effective interactions of D0-branes in superstring theory is given by computing the annulus diagram shown in fig. \[f:annulus\]. This was done in ref. [@green] (and in [@VM96] for Dp-branes). The result of summing over physical (GSO projected) superstring states gives the free energy $$F[L,\beta,\nu]=\frac{8}{\sqrt{\pi \alpha'}}\int_0^\infty \frac{dl}{l^{3/2}}\,
e^{ -L^2l/4\pi^2\alpha'}
\Theta_2 \left(\nu \left \vert \frac{i\beta^2}{\pi \alpha' l} \right. \right)
\prod_{n=1}^\infty\left( \frac{1+e^{-nl}}{1-e^{-nl}}\right)^8
\label{GreenF}$$ where $$\Theta_2 \left(\nu \left \vert iz \right. \right)
=\sum_{q=-\infty}^\infty
e^{ -\pi z(2q+1)^2/4+i\pi(2q+1)\nu},$$ $L$ is the brane separation and $\nu$ is a parameter which weights the winding numbers of strings around the periodic time direction. An extra factor of 2 accounting for the exchange of the two ends of the superstring [@Pol96] ending on each of the two D0-branes is inserted in (\[GreenF\]).
The product in the integrand represents the sum over string states, with requisite degeneracies, $$8\prod_{n=1}^\infty\left( \frac{1+e^{-nl}}{1-e^{-nl}}\right)^8
=\sum_{N=0}^{\infty} d_N e^{-Nl}
\label{degeneracy}$$ where $d_N$ is the degeneracy of the either superstring state at level $N$. For the lowest few levels, $d_0=8$ and $E_0=L/2\pi\alpha'$.
Inserting (\[degeneracy\]) in (\[GreenF\]) and integrating over $l$, the free energy has the form $$F(\beta,L,\nu)=\frac{2}{\beta}\sum_{N=0}^\infty d_N~
\ln\left| \frac{ 1-e^{-\beta E_N+i\pi\nu}}{ 1+e^{-\beta E_N+i\pi\nu} }
\right|
\label{Fsuperstring}$$ where the string energies are given by the formula $$E_N=\frac{L}{2\pi\alpha'}\sqrt{ 1+\frac{4\pi^2\alpha'N}{L^2}}.
\label{Esuperstring}$$ This results in the partition function $$Z_{\rm str}(\beta,L,\nu)\equiv e^{-\beta F}= \prod_{N=0}^\infty \left|
\frac{1+e^{-\beta E_N+i\pi \nu}}{1-e^{-\beta E_N+i\pi \nu}}
\right|^{2d_N}.
\label{Zsuperstring}$$ The physical meaning of the last formula is obvious: the partition function equals the ratio of the Fermi and Bose distributions with the power being twice the degeneracy of states and $i\nu$ playing the role of a chemical potential. The factor of 2 in the exponent $2d_N$ in (\[Zsuperstring\]) is due to the interchange of the superstring ends as is already mentioned. It will provide the agreement with the matrix theory computation.
In order to compare with the Yang-Mills computation, we should first re-scale the coordinates so that the mass of the D0-brane appears in the kinetic term as in (\[T=0\]). The mass is given by the formula $$M=\frac{1}{g_s\sqrt{\alpha'}}$$ and the Yang-Mills coupling $g_{\rm YM}$ is related to the string coupling $g_s$ by the equation $$g_{\rm YM}^2=\frac{g_s}{4\pi^2(\alpha')^{3/2}}.$$ The physical coordinate of the $\alpha$-th D0-brane is identified with $$\vec q^\alpha= 2\pi\alpha'\vec a^\alpha .
\label{qvsa}$$ Taking N=2 in (\[23\]) and identifying $L=2\pi\alpha' \vert \vec a^1-\vec a^2 \vert$, we see that the integrand in (\[23\]) coincides with (\[Zsuperstring\]) truncated to the massless modes ($N=0$) provided $\nu=\beta(a_0^1-a_0^2)$.
It is clear that the integral over $a_0^\alpha$ is responsible for the “mismatch” of the effective actions between the string theory computation and matrix theory computation of the free energy. In the string theory done in the spirit of ref. [@green], the parameter $\nu$ appears in the same place as $\beta(a_0^1-a_0^2)/\pi$ but is not integrated. It is associated with the interaction of the ends of the open string with an Abelian gauge field background ${A}_{\mu}(\tau,\vec x)$: $$S_{\rm int}=\int dx^\mu {A}_\mu.$$ If the ends are separated by the distance $L$, e.g. along the first spatial axis, then $$\nu=\int_0^\beta d\tau \left( {A}_0(\tau,0,\ldots)-
{A}_0(\tau,L,\ldots)\right)$$ since $\dot x_\mu (\tau)=(1,\vec 0)$ on the boundaries. The matrix theory automatically takes into account the integration over the background field while in the string theory calculation of ref. [@green] the background field is fixed. This is just a reflection of the fact that matrix theory is an effective low-energy theory of $D$-branes, while the older string theory did not treat the boundaries as dynamical objects. However, it is interesting to notice how close some of the earlier string papers came to such a description simply by the requirement of consistency [@green1]. Further, in the context of matrix theory it is natural to take the exponential of the free energy (\[Fsuperstring\]) as in eq. (\[Zsuperstring\]), and only integrete over $\nu$ afterwards, a procedure not entirely obvious in a string theory where the boundaries are not dynamical objects. This will be discussed further in the next section.
An exact coincidence between the matrix theory and superstring results is possible only when the higher stringy modes are suppressed. Usually, the truncation of the string spectrum to get Yang-Mills theory is valid for small $\alpha'$, that is when we are interested in temperatures which are much smaller than $1/\sqrt{\alpha'}$. In fact, the condition in our case is a little different than this once the length $L$ appears as a parameter in the spectrum (\[Esuperstring\]). Then, the spectrum can be truncated at the first level only when $$\frac{1}{\beta}\equiv k_BT\ll
\sqrt{ \left( \frac{L}{2\pi\alpha'}\right)^2+\frac{1}{\alpha'} }-\frac{L}
{2\pi\alpha'}
\label{trunk}$$ which is the energy gap between the first two levels. If the temperature is small, this condition is always satisfied unless the length $L$ is not too large. In other words the truncation of the spectrum to the lightest modes is valid for $\beta \gg L$ (or $TL\ll 1$).
It is also interesting to discuss what happens in the opposite limit $L \gg \beta$ where the interaction between D0-branes is mediated by the lightest closed string modes. The superstring free energy can be evaluated in this limit by the standard modular transformation which relates the annulus diagram for an open string with a cylinder diagram for a closed string. Introducing the new integration variable $s=2\pi^2/l$, we rewrite (\[GreenF\]) as [@green] $$F[L,\beta,\nu]= \frac{8 \pi^4}{\sqrt{2 \pi \alpha'}}
\int_0^\infty \frac{ds}{s^{9/2}}\;
e^{s}\,e^{-L^2/2s\alpha'}
\Theta_2 \left(\nu \left \vert \frac{i\beta^2 s}{2\pi^3 \alpha' }
\right. \right)
\prod_{n=1}^\infty\left( \frac{1-e^{-(2n+1)s}}{1-e^{-2ns}}\right)^8.
\label{viathetas}$$ In the limit where the brane separation is large the integration over $s$ is concentrated in the region of large $s\sim L^{2}$. Substituting the large-$z$ asymptotics $$\Theta_2 \left(\nu \left \vert i z \right. \right)
\rightarrow
2\cos{(\pi\nu)}e^{-\pi z/4}$$ and evaluating the saddle-point integral, we get $$F[L,\beta,\nu] \propto \cos{(\pi\nu)}
\frac{\left(\beta^2-8\pi^2\alpha'\right)^{3/2}}{L^4}
e^{-L\sqrt{\beta^2-8\pi^2\alpha'}/2\pi \alpha'}.$$
Exponentiating and integrating over $\nu$, we have $$\int_{-1}^{1} {d\nu} Z_{\rm str}(\beta,L,\nu)
\propto \frac{\beta^2 \left(\beta^2-8\pi^2\alpha'\right)^{3}}{L^8}
e^{-L\sqrt{\beta^2-8\pi^2\alpha'}/\pi \alpha'}.$$ Taking into account (\[qvsa\]) the exponent at the low temperatures is the same as in (\[finalSeff\]) but the pre-exponential differs. The dependence of the pre-exponential on $L$ in the superstring case emerges because the splitting between energy states in (\[Esuperstring\]) is of order $1/L$ and the truncation condition (\[trunk\]) is no longer satisfied when $L$ is large. Higher stringy modes are then not separated by a gap and the continuum spectrum results in the $L$-dependence of the preexponential. As usual, the limits of $L\rightarrow\infty$ and $T\rightarrow 0$ are not interchangeable in the superstring theory.
Discussion
==========
Our main results concern D0-brane dynamics at finite temperatures. We have computed the 1-loop effective action for the interaction of static D0-branes in the matrix theory at finite temperature and compared it with the analogous superstring computation. We have seen that an extra integration over the eigenvalues of the holonomy along the compactified Euclidean time is present in the matrix theory. The two computations agrees in the low temperature limit provided the superstring thermal partition function is integrated over the Abelian gauge fields $a_0$’s living on D0-branes.
The integration over $a_0$’s is of course natural in the context of the Yang-Mills theory, where it expresses that only gauge-invariant states should contribute to ${\rm TR}\,e^{-\beta H}$. But it is also natural from the point of view of the D0-brane physics. It can be seen as follows. Suppose we make a T-duality transformation, which interchanges the Neumann and Dirichlet boundary conditions, along the compactified Euclidean time direction. Then $a_0$’s become the coordinates of D-instantons on the dual circle. The integration over $a_0$’s becomes now the integration over the positions of D-instantons. The partition function should involve such an integration over the collective coordinates and since they are collective coordinates the integration appears in front of the exponential of the effective action, not in the action itself. Viewed in terms of D0-branes and open strings, we have a gas of D0-branes with open strings between them. The individual strings might have a winding number $q$ (more precisely $2q+1$ in the case of superstrings), describing the winding around the finite-temperature space-time cylinder. The energy of such states are $\propto \beta q/2\pi \alpha'$. However, the $q$’s satisfy $\sum q=0$ as a result of the integration over $a_0$. Physically this constraint is most easily understood by going to the closed string channel where we have closed string boundary state on the dual circle with radius $\tilde{\beta} = 4\pi^2 \alpha'/\beta$ localized at the point $(\nu \tilde \beta, \vec q ) $. Passing to the momentum representation, we write $$\Big\vert B,\vec q, \nu \Big\rangle =
\sum_{q=-\infty}^\infty e^{-2i\pi\nu q} \, \Big\vert B,\vec q,
p_0=2\pi q /\tilde{\beta} \Big\rangle .$$ Here the temporal momentum is quantized as $p_0=2\pi q /\tilde{\beta}=q\beta/2\pi\alpha'$, which lead to the same energy as the above mentioned open string states. In this representation $\sum q=0$ simply expresses momentum conservation in the thermal direction.
The effective static potential between two D0-branes emerges because supersymmetry is broken by finite temperature. This effect of breaking supersymmetry is somewhat analogous to the velocity effects at zero temperature where the matrix theory and superstring computations agree to the leading order of the velocity expansion [@dkps]. We have thus shown that the leading term in a low temperature expansion is correctly reproduced by the matrix theory. The discrepancy between the matrix theory and superstring computations, which we have observed in the limit of large distances $L T\gg 1$, does not contradict this statement since temperature the limits of large distances and small temperatures are not interchangeable.
An interesting feature of the effective static potential between D0-branes is that it is logarithmic and attractive at short distances. In the matrix theory, the singularity of the computed 1-loop potential occurs when the distance between the D0-branes vanishes and the SU(N) symmetry which is broken by finite distances is restored. The integration over the off-diagonal components can no longer be treated in the 1-loop approximation! In the superstring theory, the singularity is exactly the same as in the matrix theory since it is determined only by the massless bosonic modes (the NS sector in the superstring theory). Its origin is [*not*]{} due to the presence of massless photon states in the spectrum. Putting $\nu_1=\nu_2$ in the above D-instanton picture on the dual temporal circle, we see that the mass of the lowest states, associated with the winding numbers $2q+1=\pm 1$ is $\tilde \beta{} /2\pi\alpha'$. The divergence at $L=0$ emerges, in this picture, after summing over all the open string states since no single state has such a divergence. It shows up only at finite temperature where the winding number $q$ exists.
It is important to notice that the computed partition functions take into account only thermal fluctuations of superstring stretched between D0-branes but not the fluctuations of D0-branes themselves. To calculate the thermal partition function of D0-branes, a further path integration over their periodic trajectories $\vec a(\tau)$ is to be performed as in (\[effec\]). One might think that classical statistics is applicable to this problem since the D0-branes are very heavy as $g^2_{\rm YM}\rightarrow0$ so that one could restrict himself by the static approximation. This is however not the case due to the singularity of the effective static potential at small distances. The integral over the D0-brane positions $\vec{a}$’s is divergent when the two positions coincide.
However, this singularity is only in the classical partition function. The path integral over the periodic trajectories $\vec a(\tau)$ that we actually have to do can not diverge since the 2-body quantum mechanical problem has a well-defined spectrum. The path integral can then be evaluated as $\sum_n \exp(-\beta E_n)$ where $E_n$ are in the spectrum of the operator $ H=P^2/M+V_{\rm eff}$. There certainly should not be the bound state energy eigenvalue at negative infinity for this quantum mechanical problem which implies the convergence of the path integral. These issues which are related to thermodynamics of D0-branes will be considered in a separate publication.
Let us finally discuss when the 1-loop appoximation that we have done is applicable. The loop expansion in Yang-Mills theory computation is valid only in the limit where $$g_{\rm YM}^2/\vert\vec a\vert^3\sim
g_s \left(\frac{ \sqrt{\alpha'}}{L}\right)^3$$ is small. This is due to the fact that the distance $L$ plays the role of a Higgs mass which cuts off the infrared divergences of the loop expansion in the 0+1 -dimensional gauge theory. Thus, the perturbative Yang-Mills theory computation is good when $$g_s^{1/3}\sqrt{\alpha'} \ll L.$$ This can be satisfied if either the string coupling is small or if the D0-brane separation is large compared with the string length scale. In the latter case, the truncation of the spectrum to the lightest modes is still valid when $$k_BT\ll \frac{1}{L} \ll\frac{1}{g_s^{1/3}\sqrt{\alpha'}}$$ Note that the first inequality which is independent of both the string scale and the string coupling is the one already discussed in the previous section. In this case, the temperature must be less than the inverse distance between D0-branes. In the case where the string coupling $g_s$ is small, the criterion for validity of truncation of the spectrum becomes $$k_BT<1/\sqrt{\alpha'}$$ that is the usual one.
Acknowlegments {#acknowlegments .unnumbered}
==============
We are gratefull to P. Di Vecchia, G. Ferretti, A. Gorsky, M. Green, M. Krogh, N. Nekrasov, N. Ohta, P. Olesen, L. Thorlacius and K. Zarembo for very useful discussions. J.A. and Y.M. acknowledge the support from MaPhySto financed by the Danish National Research Foundation and from INTAS under the grant 96–0524. The work by Y.M. is supported in part by the grants CRDF 96–RP1–253 and RFFI 97–02–17927.
[99]{} J. Polchinski, hep-th/9611050. W. Taylor, hep-th/9801182. E. Witten, Nucl. Phys. B460 (1996) 335, hep-th/9510135. U. Danielsson, G. Ferretti and B. Sundborg, Int. J. Mod. Phys. A11 (1996) 5463, hep-th/9603081. D. Kabat and P. Pouliot, Phys. Rev. Lett. 77 (1996) 1004, hep-th/9603127. M. R. Douglas, D. Kabat, P. Pouliot and S. Shenker, Nucl. Phys.. B485 (1997) 85, hep-th/9608024. C. Bachas, Phys. Lett. B374 (1996) 37, hep-th/9511043. T. Banks, W. Fischler, S. Shenker and L. Susskind, Phys. Rev. D55 (1997) 5112, hep-th/9610043. N. Ohta and J.-G. Zhou, Nucl. Phys. B522 (1998) 125, hep-th/9801023. M. L. Meana, M. A. R. Osorio and J. P. Peñalba, hep-th/9803058. B. Sathiapalan, Mod. Phys. Lett. A13 (1998) 2085, hep-th/9805126. M. B. Green, Nucl. Phys. B381 (1992) 201. M. M. Vázquez-Mozo, Phys. Lett. B388 (1996) 494, hep-th/9607052. M. B. Green, Phys. Lett. B266 (1991) 325.
[^1]: This work is supported in part by NATO Grant CRG 970561, by NSERC and by MaPhySto – Centre for Mathematical Physics and Stochastics.
[^2]: This is a difference between our computation at finite temperature and computations of the Witten index for the matrix theory where fermions obey periodic boundary conditions.
[^3]: These constants are related to the eigenvalues of the holonomy $${\rm P} \exp{\left(i \int_0^\beta d\tau A_0(\tau)\right)}=
\Omega^\dagger\;{\rm diag}\; \left(
e^{i\beta a_0^1},\ldots,e^{i\beta a_0^{\rm N}}\right) \Omega$$ known as the Polyakov loop winding along the compact Euclidean time. It can not be made trivial by the gauge transformation if $T\neq 0$.
|
---
abstract: 'We construct a quotient ring of the ring of diagonal coinvariants of the complex reflection group $W=G(m,p,n)$ and determine its graded character. This generalises a result of Gordon for Coxeter groups. The proof uses a study of category ${\mathcal{O}}$ for the rational Cherednik algebra of $W$.'
address: 'Department of Mathematics, University of Glasgow, Glasgow, G12 8QW, U.K.'
author:
- Richard Vale
bibliography:
- 'penguin.bib'
title: 'Rational Cherednik algebras and diagonal coinvariants of $G(m,p,n)$'
---
Introduction
============
{#gordon}
Let $\mathfrak{h}$ be a finite–dimensional complex vector space. An element $s \in {\mathrm{End}}({\mathfrak{h}})$ is called a complex reflection if ${\mathrm{rank}}_{\mathfrak{h}}(1-s) =1$ and $s$ has finite order. A finite group generated by complex reflections is called a complex reflection group. If $W$ is a complex reflection group then the ring of invariants ${\mathbb{C}}[ {\mathfrak{h}}]^W$ is a polynomial ring by the Shepherd-Todd theorem [@benson Theorem 7.2.1] and if ${\mathbb{C}}[{\mathfrak{h}}]^{W}_{+}$ denotes the elements with zero constant term then it is well-known that the ring of coinvariants $$\frac{{\mathbb{C}}[{\mathfrak{h}}]}{\langle {\mathbb{C}}[{\mathfrak{h}}]^{W}_{+} \rangle}$$ is a finite–dimensional algebra isomorphic to ${\mathbb{C}}W$ as a $W$–module. There is interest in analogues of this construction with the representation ${\mathfrak{h}}\oplus {\mathfrak{h}}^*$ in place of ${\mathfrak{h}}$, see for example [@Hai]. The ring $$D_W := \frac{{\mathbb{C}}[{\mathfrak{h}}\oplus {\mathfrak{h}}^*]}{\langle {\mathbb{C}}[{\mathfrak{h}}\oplus {\mathfrak{h}}^*]^{W}_{+} \rangle}$$ is called the ring of diagonal coinvariants of $W$. The ring $D_W$ has a natural grading with $\deg({\mathfrak{h}}^*) =1$ and $\deg({\mathfrak{h}}) = -1$. The following result was conjectured by Haiman and proved in Gordon [@Go]:
\[Gordon\][@Go] Let $W$ be a finite Coxeter group of rank $n$ with Coxeter number $h$ and sign representation ${\varepsilon}$. Then there exists a $W$–stable quotient ring $R_W$ of $D_W$ with the properties:
1. $\dim(R_W) = (h+1)^n$.
2. $R_W$ is graded with Hilbert series $t^{-hn/2}(1+t + \cdots + t^h)^n$.
3. The image of ${\mathbb{C}}[{\mathfrak{h}}]$ in $R_W$ is ${\mathbb{C}}[{\mathfrak{h}}] / \langle {\mathbb{C}}[{\mathfrak{h}}]^{W}_{+} \rangle$.
4. The character $\chi$ of the $W$–module $R_W \otimes {\varepsilon}$ satisfies $\chi(w) = (h+1)^{\dim \ker (1-w)} \quad \forall w \in
W$.
{#mainthm}
In [@vale], this result was generalised to the complex reflection groups $G(m,1,n)$. The aim of this paper is to obtain a further generalisation to the groups $G(m,p,n)$, with some mild restrictions on $m,p,n$. The following result will be proved:
\[mainthm\] Let $W = G(m,p,n)$ where $m \neq p$ and let ${\mathfrak{h}}$ be the reflection representation of $W$. Let $d=m/p$. Then there exists a $W$–stable quotient ring $S_W$ of $D_W$ with the properties:
1. $\dim(S_W) = (m(n-1)+d+1)^n$.
2. $S_W$ is graded with Hilbert series $t^{-n-m \binom{n}{2} }(1 + t + \cdots +
t^{m(n-1)+d})^n$.
3. The image of ${\mathbb{C}}[{\mathfrak{h}}]$ in $S_W$ is ${\mathbb{C}}[{\mathfrak{h}}] / \langle {\mathbb{C}}[{\mathfrak{h}}]^{W}_{+} \rangle$.
4. The character $\chi$ of $S_W \otimes \wedge^n {\mathfrak{h}}^*$ as a $W$–module satisfies $\chi(w) = (m(n-1)+d+1)^{\dim \ker (1-w)} \quad \forall w \in W$.
{#section}
Theorem \[gordon\] is proved by obtaining $R_W$ as the associated graded module of a finite–dimensional module over the rational Cherednik algebra of $W$. The properties of this module are derived by studying the category $\mathcal{O}$ for the rational Cherednik algebra. This is also the method that will be used to prove Theorem \[mainthm\].
{#section-1}
The structure of the paper is as follows. In Section \[cherednik\], the rational Cherednik algebra $H_\kappa$ is introduced for $W=G(m,p,n)$. This is a certain deformation of the skew group algebra ${\mathbb{C}}[{\mathfrak{h}}\oplus {\mathfrak{h}}^*]*W$ which depends on parameters $\kappa \in {\mathbb{C}}^{m/p}$. Next, in Section \[cato\], we recall some important properties of category ${\mathcal{O}}$ for $H_\kappa$, including the Knizhnik-Zamolodchikov functor ${\texttt{KZ}}$, which we use to relate category ${\mathcal{O}}$ to the category of modules over a Hecke algebra $\mathcal{H}$. A parametrisation of the simple $\mathcal{H}$–modules given by Genet and Jaçon [@GJ] enables us to prove that, for suitable choices of the parameters, there is only one finite-dimensional simple object $L(\mathsf{triv})$ in category ${\mathcal{O}}$ (Theorem \[infdim\]). We then define, in Section \[shifting\], a one-dimensional $H_{\kappa^{'}}$–module $\Lambda^\psi$ for “shifted" values of the parameters $\kappa^{'}$, and, using a shift isomorphism due to Berest and Chalykh [@BC], we construct the shifted module $$L:= H_{\kappa} e_{\varepsilon}\otimes_{e_{\varepsilon}H_{\kappa} e_{\varepsilon}} \Lambda^\psi$$ where $e_{\varepsilon}$ is a certain idempotent in ${\mathbb{C}}W$. The associated graded module $\mathrm{gr} L$ is, up to tensoring by a one-dimensional $W$–module, naturally a quotient of the ring of diagonal coinvariants. But $L$ is also a finite-dimensional object of category ${\mathcal{O}}$, and we are able to use our results on category ${\mathcal{O}}$ to show that it is isomorphic to $L(\mathsf{triv})$. By results of Chmutova and Etingof [@CE], $L(\mathsf{triv})$ is well-understood, and this enables us to compute the Hilbert series and character of $L$ and hence of $\mathrm{gr} L$, proving Theorem \[mainthm\]. These calculations are given in Section \[mainproof\].
{#section-2}
The main difference between our proof and the proof of Theorem \[Gordon\] is that for some of the Coxeter groups considered in [@Go], the rational Cherednik algebra depends on only one parameter, and there is only one choice of parameter for which the proof will work. In the $G(m,p,n)$ case with $m /neq p$, there is greater freedom to choose the parameters, and hence it is possible to have a lot of control over the simple modules in category ${\mathcal{O}}$, by considering what happens when the parameters are chosen generically. We note here that in the $m=p$ case the rational Cherednik algebra usually depends on only one parameter, and hence the proof of Theorem \[mainthm\] will not work for the groups $G(m,m,n)$. However, it is likely that an analogue of Theorem \[mainthm\] can be proved for these groups, following the arguments of [@Go]. We hope to return to this in future work.
{#section-3}
It may appear that, in the case $m=1$, the Hilbert series of $S_W$ should be $t^{-n-{n \choose 2}} (1 + t + \cdots + t^n )^n$, which does not generalise Theorem \[Gordon\] in type $A$. However, in order to make Theorem \[mainthm\] and Theorem \[Gordon\] agree in this case, we should write the Hilbert series of $S_W$ as $t^{-\dim {\mathfrak{h}}- m {\dim {\mathfrak{h}}\choose 2}} (1 + t+ \cdots +
t^{m(n-1)+d})^{\dim {\mathfrak{h}}}$. Note that the type $A$ case of Theorem \[Gordon\] does *not* follow from Theorem \[mainthm\] because we will assume throughout that $m>1$.
Acknowledgements
----------------
The research described here will form part of the author’s PhD thesis at the University of Glasgow. The author thanks K. A. Brown and I. Gordon for suggesting this problem and for their advice and encouragement. The author also wishes to thank Y. Berest and O. Chalykh for allowing us to look at a preliminary version of their paper [@BC], and O. Chalykh for explaining in detail to us the proof of Theorem \[shift\].
The group $G(m,p,n)$
====================
{#section-4}
Let ${\mathfrak{h}}$ be an $n$–dimensional complex vector space equipped with a sesquilinear form $\langle -,- \rangle$. Let $m \ge 1$ and let $p$ be a natural number such that $p|m$. We fix the notation $d=m/p$ and ${\varepsilon}= e^{\frac{2 \pi i}{m}}$ throughout. The complex reflection group $G(m,p,n)$ is defined to be the subgroup of $GL({\mathfrak{h}})$ consisting of those matrices with exactly one nonzero entry in each row and column, such that the nonzero entries are powers of ${\varepsilon}$ and such that the $d^{\mathrm{th}}$ power of the product of the nonzero entries is $1$.
Complex reflections {#def}
-------------------
We wish to identify the complex reflections in $G(m,p,n)$. It turns out that the set of complex reflections depends on $(m,p,n)$, so we will make the following assumption
From now on, assume $m>p$. Equivalently, $d>1$.
Under this assumption, the complex reflections in $G(m,p,n)$ are as follows. If $\{y_1 , \ldots , y_n \}$ is an orthonormal basis of ${\mathfrak{h}}$ then the set $\mathcal{S}$ of complex reflections in $W$ consists of the elements $s_i^{qp}$ for $1 \le i \le n$ and $1 \le
q \le d-1$, and $\sigma_{ij}^{(\ell)}$ for $1 \le i < j \le n$ and $0 \le \ell \le m-1$, defined by: $$\begin{aligned}
s_{i}^{qp} ( y_i) = {\varepsilon}^{qp} y_i & s_{i}^{qp} (y_j) = y_j & j \neq i \\
\sigma_{ij}^{(\ell)} (y_i) = {\varepsilon}^{-\ell} y_j & \sigma_{ij}^{(\ell)}
(y_j) =
{\varepsilon}^\ell y_i & \sigma_{ij}^{(\ell)} (y_k) = y_k , \: k \neq i,j .\\\end{aligned}$$
{#def2}
We now list the $W$–conjugacy classes in $\mathcal{S}$. For each $q$, $\{ s_i^{qp} | 1 \le i \le n\}$ form a single conjugacy class in $\mathcal{S}$. If $n \ge 3$ or $n=2$ and $p$ is odd, then $\{
\sigma_{ij}^{(\ell)} | i < j, \: 0 \le \ell \le m-1 \}$ also form a single conjugacy class in $\mathcal{S}$. For convenience, we now make the following assumption.
Either $n \ge 3$, or $n=2$ and $p$ is odd.
Under assumptions \[def\] and \[def2\], we see that there are exactly $d$ $W$–conjugacy classes of complex reflections. Furthermore, it follows from [@BMR Section 3] that the defining representation ${\mathfrak{h}}$ is irreducible when these assumptions hold. We are now in a position to construct the rational Cherednik algebra of $W$.
Theorem \[mainthm\] still holds in the case where $n=2$ and $p$ is even. To avoid clutter, the modifications necessary to prove this case are explained in Section \[fudge\].
The rational Cherednik algebra {#cherednik}
==============================
In [@DO] and [@GGOR] we have the following definition. Let $\mathcal{A}$ be the set of reflection hyperplanes of $W$ and for $H \in \mathcal{A}$ let $W_H = \mathrm{stab}_W(H)$, a cyclic group of order $e_H$. For $1 \le i \le e_H-1$, let ${\varepsilon}_{H,i} =
\frac{1}{e_H} \sum_{w \in W_H} \det (w)^i w$. For $H \in
\mathcal{A}$, let $\{k_{H, i} \}_{i=0}^{e_H}$ be a family of scalars such that $k_{H,i} = k_{H',i}$ whenever $H, H'$ are in the same $W$–orbit, and $k_{H,0} = k_{H, e_H} = 0$ for all $H$. For each $H \in \mathcal{A}$, pick a linear form $\alpha_H \in {\mathfrak{h}}^*$ with kernel $H$, and choose $\alpha_H^{\vee} \in {\mathfrak{h}}$ such that ${\mathbb{C}}\alpha_H^{\vee}$ is a $W_H$–stable complement to $H$ and $\alpha_H (\alpha_H^{\vee}) = 2$.
The rational Cherednik algebra is defined to be the quotient of $T({\mathfrak{h}}\oplus {\mathfrak{h}}^*) \ast W$, the skew product of $W$ with the tensor algebra on ${\mathfrak{h}}\oplus {\mathfrak{h}}^*$, by the relations $[y_1,
y_2]=0$ for all $y_1, y_2 \in {\mathfrak{h}}$, $[x_1, x_2]=0$ for all $x_1,
x_2 \in {\mathfrak{h}}^*$, and $$\label{commute} [y,x]= \langle
y, x \rangle + \sum_{H \in \mathcal{A}} \frac{\langle \alpha_H, y
\rangle \langle \alpha_H^{\vee}, x \rangle}{2} e_H
\sum_{j=0}^{e_H-1} (k_{H, j+1} - k_{H,j} ) {\varepsilon}_{H,j}$$ for all $y \in {\mathfrak{h}}$ and all $x \in {\mathfrak{h}}^*$, where $\langle -,-
\rangle$ here denotes the evaluation pairing between ${\mathfrak{h}}$ and ${\mathfrak{h}}^*$.
In this paper, the signs in the commutation relation have been chosen so that our parameters $k_{H,i}$ are the same as those of the paper [@GGOR]. However, we will also use results from the paper [@BC], in which the parameters denoted $k_{H,i}$ are the negatives of those given here. See Section \[shift\].
{#commutationrel}
In the case $W=G(m,p,n)$, we may write out the commutation relation more explicitly. Let $H_i$ be the reflection hyperplane of $s_i^{qp}$ and $H_{ij \ell}$ be the reflection hyperplane of $\sigma_{ij}^{(\ell)}$. Let $\{ y_1, \ldots , y_n\}$ be the standard basis of ${\mathfrak{h}}$ and $\{x_1, \ldots , x_n\}$ the dual basis of ${\mathfrak{h}}^*$. Then we may choose $\alpha_{H_i} = x_i$, $\alpha_{H_i}^{\vee} = 2y_i$, $\alpha_{H_{ij \ell}} = x_i -
{\varepsilon}^{\ell} x_j$ and $\alpha_{H_{ij \ell}}^{\vee} = y_i - {\varepsilon}^{-
\ell} y_j$. We have $e_{H_i} =d$ and ${\varepsilon}_{H_i, j} = \frac{1}{d}
\sum_{r=0}^{d-1} {\varepsilon}^{prj} s_i^{pr}$, and $e_{H_{ij \ell}} =2$ and $ {\varepsilon}_{H_{ij \ell}} = \frac{1}{2} (1+ (-1)^j
\sigma_{ij}^{(\ell)})$. The commutation relation (\[commute\]) becomes $$\begin{gathered}
\label{commute2} [y_a, x_b] = \delta_{ab} +
\sum_{i=1}^n \delta_{ia}\delta_{ib} \left[ \sum_{j=0}^{d-1}
(\kappa_{j+1} - \kappa_j) \sum_{r=0}^{d-1} {\varepsilon}^{prj} s_i^{pr}
\right] \\ + \sum_{1 \le i < j \le n } \sum_{\ell =0}^{m-1}
\frac{1}{2} (\delta_{ia} - {\varepsilon}^{\ell} \delta_{ja} )(\delta_{ib} -
{\varepsilon}^{-\ell} \delta_{jb}) 2 \kappa_{00} \sigma_{ij}^{\ell}\end{gathered}$$ where $k_{H_{ij \ell}} = \kappa_{00}$ for all $i,j
, \ell$ and $k_{H_i, j} = \kappa_j$. We will denote the rational Cherednik algebra with these parameters by $H_\kappa$.
{#pbw}
It was proved in [@EtGi Theorem 1.3] that $H_\kappa$ satisfies a PBW–property, that is, it is isomorphic as a vector space to ${\mathbb{C}}[{\mathfrak{h}}] \otimes {\mathbb{C}}W \otimes {\mathbb{C}}[{\mathfrak{h}}^*]$ via the multiplication map.
{#section-5}
Let $H_\kappa (G(m,p,n))$ denote the rational Cherednik algebra of $G(m,p,n)$ with parameters $\kappa$. We will make considerable use of the fact that there is an embedding $$H_\kappa (G(m,p,n))
\hookrightarrow H_\mu (G(m,1,n))$$ for an appropriate choice of $\mu$. This observation is essentially due to Dunkl and Opdam [@DO].
\[inclusion\] Given $\kappa = (\kappa_{00}, \kappa_1, \ldots, \kappa_{d-1})$, define $\mu= (\mu_{00}, \mu_1, \ldots, \mu_{d-1}, \mu_d , \ldots,
\mu_{m-1})$ by $\mu_{00}= \kappa_{00}$, $\mu_0=0$, $\mu_i =
\kappa_i/p$, $1 \le i \le d-1$, and $\mu_{sd+t} = \mu_t$ for $1
\le s \le p-1$ and $1 \le t \le d-1$. Then $H_\kappa(G(m,p,n))$ is the subalgebra of $H_\mu (G(m,1,n))$ generated by ${\mathfrak{h}}$, ${\mathfrak{h}}^*$, $\sigma_{ij}^{(\ell)}$ for all $i,j,\ell$, and $s_i^p$, $1 \le i
\le n$.
Write $H_\kappa:= H_\kappa(G(m,p,n))$ and $H_\mu :=
H_\mu(G(m,1,n))$. We need to check that the copies of ${\mathfrak{h}}$, ${\mathfrak{h}}^*$ in $H_\mu$ obey the commutation relations for $H_\kappa$. It is simply a question of substituting the $\mu$ values into (\[commute2\]) above, with $p=1$. We obtain, in $H_\mu$, $$\begin{gathered}
[y_a, x_b] = \delta_{ab} + \sum_{i=1}^n
\delta_{ia}\delta_{ib} \left[ \sum_{j=0}^{m-1} (\mu_{j+1} - \mu_j)
\sum_{r=0}^{m-1}
{\varepsilon}^{rj} s_i^{r} \right] \\
+ \sum_{1 \le i < j \le n } \sum_{\ell =0}^{m-1} (\delta_{ia} -
{\varepsilon}^{\ell} \delta_{ja} )(\delta_{ib} - {\varepsilon}^{-\ell} \delta_{jb})
\mu_{00} \sigma_{ij}^{\ell} \end{gathered}$$ where ${\varepsilon}:= e^\frac{2
\pi i}{m}$. This may be rewritten as $$\begin{gathered}
[y_a, x_b] = \delta_{ab} + \sum_{1 \le i < j
\le n } \sum_{\ell =0}^{m-1} (\delta_{ia} -
{\varepsilon}^{\ell} \delta_{ja} )(\delta_{ib} - {\varepsilon}^{-\ell} \delta_{jb})
\kappa_{00} \sigma_{ij}^{\ell} + \sum_{i=1}^n
\delta_{ia}\delta_{ib} \frac{1}{p} \sum_{q=0}^{m-1} x_q s_i^q\end{gathered}$$ where $$x_q = \sum_{j=0}^{d-1} {\varepsilon}^{qj} (\kappa_{j+1}- \kappa_j) +
\sum_{j=d}^{2d-1} {\varepsilon}^{qj} (\kappa_{[j+ 1]}-\kappa_{[j]}) + \cdots
+ \sum_{j= (p-1)d}^{pd-1} {\varepsilon}^{qj} (\kappa_{[j+1]} - \kappa_{[j]}
).$$ where $[j]$ denotes the remainder modulo $d$. Write $q =
ap+b$, $0 \le a \le d-1$, $0 \le b \le p-1$. Then $$\begin{aligned}
x_q &= \sum_{j=0}^{d-1} ({\varepsilon}^{qj}+ {\varepsilon}^{q(j+d)} + \cdots +
{\varepsilon}^{q(j+(p-1)d)}) (\kappa_{j+1} - \kappa_j ) \\
&= \sum_{j=0}^{d-1} \sum_{r=0}^{p-1} {\varepsilon}^{q(j+rd)}( \kappa_{j+1} -
\kappa_j ) \\ &= \sum_{j=0}^{d-1} \alpha_j (\kappa_{j+1}-\kappa_j)\end{aligned}$$ where $\alpha_j = {\varepsilon}^{qj} \sum_{r=0}^{p-1} {\varepsilon}^{qrd} = {\varepsilon}^{qj}
\sum_{r=0}^{p-1} (e^\frac{2 \pi i}{p} )^{br}$. So $$\alpha_j = \begin{cases}
{\varepsilon}^{apj} & \text{if $q=ap$,} \\
0 & \text{if not.} \end{cases}$$ And so $$x_q = \sum_{j=0}^{d-1} \alpha_j (\kappa_{j+1}-\kappa_j) =
\begin{cases} 0 & \text{if $q \neq ap$,} \\ p \sum_{j=0}^{d-1}
{\varepsilon}^{apj} (\kappa_{j+1} - \kappa_j) & \text{$q= ap$.} \end{cases}$$ So in $H_\mu$ we have $$\begin{gathered}
[y_a, x_b] = \delta_{ab} + \sum_{i=1}^n
\delta_{ia}\delta_{ib}\sum_{\stackrel{q=0}{q=ap}}^{m-1} \sum_{j=0}^{d-1} {\varepsilon}^{apj}
(\kappa_{j+1} - \kappa_j) s_i^q \\
+ \sum_{1 \le i < j \le n } \sum_{\ell =0}^{m-1} (\delta_{ia} -
{\varepsilon}^{\ell} \delta_{ja} )(\delta_{ib} - {\varepsilon}^{-\ell} \delta_{jb})
\kappa_{00} \sigma_{ij}^{\ell}. \end{gathered}$$ The first term on the right hand side is $\sum_{i=1}^n \delta_{ia}\delta_{ib}
\sum_{r=0}^{d-1}\sum_{j=0}^{d-1} {\varepsilon}^{prj} (\kappa_{j+1}- \kappa_j)
s_i^{rp}$. So $y_a, x_b$ obey the commutation relation for $H_\kappa$.
To finish the proof of the theorem, we may define a map $$T({\mathfrak{h}}\oplus {\mathfrak{h}}^*) \ast G(m,p,n) \rightarrow H_\mu$$ in the obvious way. We have checked above that the commutation relations for $H_\kappa$ are in the kernel, and it is easily seen that the other relations for $H_\kappa$ are in the kernel as well. Thus, there is a well-defined map $$H_\kappa \rightarrow H_\mu.$$ To check that this is injective, consider an element of $H_\kappa$ which is mapped to zero. Write it in terms of a PBW-basis of $H_\kappa$, and observe that all the coefficients must therefore be 0, since a PBW-basis of $H_\kappa$ is mapped into a subset of a PBW-basis of $H_\mu$.
The Dunkl representation {#dunkl}
------------------------
It is well–known (see for instance, [@DO], [@EtGi Proposition 4.5]) that $H_\kappa$ acts on ${\mathbb{C}}[{\mathfrak{h}}] \otimes
\textsf{triv}$ where $\textsf{triv}$ denotes the trivial representation of $W$. Furthermore, this action is faithful, and if ${\mathbb{C}}[{\mathfrak{h}}] \otimes \textsf{triv}$ is identified with ${\mathbb{C}}[{\mathfrak{h}}]$, then the action of $y \in {\mathfrak{h}}$ is given by a differential–difference operator called a Dunkl operator: $$T_y = \partial_y + \sum_{H \in \mathcal{A}}
\frac{\langle \alpha_H , y \rangle }{\alpha_H} \sum_{i=1}^{e_H -1} e_H k_{H, i} {\varepsilon}_{H, i}$$ If ${\mathfrak{h}}^{\mathrm{reg}} = {\mathfrak{h}}\setminus \cup_{H \in \mathcal{A}} H$ then the Dunkl representation defines an injective homomorphism $$H_\kappa \hookrightarrow \mathcal{D}({\mathfrak{h}}^{\mathrm{reg}})*W$$ called the Dunkl representation. If $\delta = \prod_{H \in
\mathcal{A}} \alpha_H \in {\mathbb{C}}[{\mathfrak{h}}]$, then ${\mathbb{C}}[{\mathfrak{h}}^{\mathrm{reg}}] =
{\mathbb{C}}[{\mathfrak{h}}]_\delta$ and the induced map $$H_\kappa|_{{\mathfrak{h}}^{\mathrm{reg}}}:=H_\kappa \otimes_{{\mathbb{C}}[{\mathfrak{h}}]} {\mathbb{C}}[{\mathfrak{h}}^{\mathrm{reg}}]
\rightarrow \mathcal{D}({\mathfrak{h}}^{\mathrm{reg}})*W$$ is an isomorphism ([@GGOR Theorem 5.6]).
{#section-6}
Let $\theta_\mu : H_\mu \rightarrow
\mathcal{D}({\mathfrak{h}}^{\mathrm{reg}}) \ast G(m,1,n)$ be the Dunkl representation of $H_\mu$ and let $\theta_\kappa: H_\kappa
\rightarrow \mathcal{D}({\mathfrak{h}}^{\mathrm{reg}}) \ast G(m,p,n)$ be the Dunkl representation of $H_\kappa$. Regarding $H_\kappa$ as a subalgebra of $H_\mu$, we wish to show that $\theta_\mu
|_{H_\kappa} = \theta_\kappa$. For this, it suffices to check that $\theta_\mu (y) = \theta_\kappa(y)$ for all $y \in {\mathfrak{h}}$. But $\theta_\mu(y)$ and $\theta_\kappa (y)$ may be regarded as differential-difference operators acting on the polynomial ring ${\mathbb{C}}[{\mathfrak{h}}]$, so it suffices to check that their values on polynomials are the same. If $p \in {\mathbb{C}}[{\mathfrak{h}}]$ then $\theta_\mu(y) (p) = y \cdot
p \otimes 1$ where we identify $p \in {\mathbb{C}}[{\mathfrak{h}}]$ with $p \otimes 1
\in {\mathbb{C}}[{\mathfrak{h}}] \otimes \mathsf{triv}$. But $y \cdot p \otimes 1 =
[y,p]_\mu \otimes 1$ where $[y,p]_\mu$ denotes the commutator in $H_\mu$. This may be written in terms of commutators $[y,x]_\mu$ for $x \in {\mathfrak{h}}^*$. But $[y,x]_\mu = [y,x]_\kappa$ for all $y \in
{\mathfrak{h}}$, $x \in {\mathfrak{h}}^*$, where $[y,x]_\kappa$ denotes the commutator in $H_\kappa$. So $[y,p]_\mu \otimes 1 = [y,p]_\kappa \otimes 1 =
\theta_\kappa(y) (p)$. So $\theta_\kappa(y) = \theta_\mu (y)$ as required. We have proved the following lemma.
\[dunklrep\] If $H_\kappa$ is the rational Cherednik algebra of $G(m,p,n)$, and we consider $H_\kappa$ as a subalgebra of $H_\mu$ as in Theorem \[inclusion\], then the Dunkl representation ${\mathbb{C}}[{\mathfrak{h}}] \otimes
{\mathsf{triv}}$ of $H_\mu$ restricts to the Dunkl representation of $H_\kappa$.
Category $\mathcal{O}$ {#cato}
======================
{#section-7}
In this section, we will review the theory for a general complex reflection group $W$ and its rational Cherednik algebra $H_\kappa$ depending on some collection of complex parameters $\kappa =
(k_{H,i})_{H \in \mathcal{A}, \: 0 \le i \le e_H-1}$.
{#section-8}
Following [@BEG1], let $\mathcal{O}$ be the abelian category of finitely-generated $H_\kappa$–modules $M$ such that for $P \in {\mathbb{C}}[{\mathfrak{h}}^*]^W$, the action of $P - P(0)$ is locally nilpotent. Let $\textsf{Irrep}(W)$ denote the set of isoclasses of simple $W$–modules. Given $\tau \in {\textsf{Irrep}}(W)$, define the standard module $M(\tau)$ by: $$M(\tau) = H_{\kappa} \otimes_{{\mathbb{C}}[{\mathfrak{h}}^*]*W} \tau$$ where for $p \in {\mathbb{C}}[{\mathfrak{h}}^*], w \in W$ and $v \in \tau$, $pw \cdot v
:= p(0) wv$.
{#section-9}
In [@DO], it is proved that $M(\tau)$ has a unique simple quotient $L(\tau)$, and [@GGOR] prove that $\{ L(\tau) | \tau
\in {\textsf{Irrep}}(W) \}$ is a complete set of nonisomorphic simple objects of ${\mathcal{O}}$, and that every object of ${\mathcal{O}}$ has finite length.
{#zdef}
By [@GGOR], if $z := \sum_{H \in \mathcal{A}}
\sum_{i=1}^{e_H -1} e_H k_{H,i} {\varepsilon}_{H, i} \in Z({\mathbb{C}}W)$, and $\mathfrak{d} := \sum_i x_i y_i \in H_\kappa$, then $\textsf{eu}_\kappa = \textsf{eu}:= \mathfrak{d} -z$ has the property that $[\textsf{eu},x]=x$ for all $x \in {\mathfrak{h}}^*$ and $[\textsf{eu},y] = -y$ for all $y \in {\mathfrak{h}}$ and $[\textsf{eu},w]=0$ for all $w \in W$. The action of $\textsf{eu}$ on $M(\tau)$ is diagonalisable and the eigenspaces are ${\mathbb{C}}[{\mathfrak{h}}]_d \otimes \tau$, $d \ge 0$, where ${\mathbb{C}}[{\mathfrak{h}}]_d$ denotes the homogeneous polynomials in ${\mathbb{C}}[{\mathfrak{h}}]$ of degree $d$. The eigenvalue of $\mathsf{eu}$ on ${\mathbb{C}}[{\mathfrak{h}}]_d \otimes \tau$ is $d- \theta(z)$ where $\theta(z)$ is the eigenvalue of $z$ on $\tau$. In particular, the lowest eigenvalue of $\mathsf{eu}$ on $M(\tau)$ is $-\theta(z)$.
{#catoaltdef}
A useful alternative definition of category ${\mathcal{O}}$ is quoted in [@CE Section 2.1]. Category ${\mathcal{O}}$ may be defined as the category of $H_\kappa$–modules $V$ such that $V$ is a direct sum of generalised $\mathsf{eu}_\kappa$ eigenspaces, and such that the real part of the spectrum of $\mathsf{eu}_\kappa$ is bounded below. It is clear from this definition that every finite-dimensional $H_\kappa$–module belongs to ${\mathcal{O}}$.
{#section-10}
The group $B_W:= \pi_1 ({\mathfrak{h}}^{\mathrm{reg}}/W)$ is called the braid group of $W$. In [@GGOR], a functor $$\texttt{KZ} : {\mathcal{O}}\rightarrow {\mathbb{C}}B_W-\mathrm{mod}$$ is constructed as follows: If $M \in {\mathcal{O}}$ then $M|_{{\mathfrak{h}}^{\mathrm{reg}}}:= {\mathbb{C}}[{\mathfrak{h}}^{\mathrm{reg}}] \otimes_{{\mathbb{C}}[{\mathfrak{h}}]}
M$ is a finitely-generated module over ${\mathbb{C}}[{\mathfrak{h}}^{\mathrm{reg}}]
\otimes_{{\mathbb{C}}[{\mathfrak{h}}]} H_\kappa \cong \mathcal{D}({\mathfrak{h}}^{\mathrm{reg}})*W$. In particular, $M$ is a $W$–equivariant $\mathcal{D}$–module on ${\mathfrak{h}}^{\mathrm{reg}}$ and hence corresponds to a $W$–equivariant vector bundle on ${\mathfrak{h}}^{\mathrm{reg}}$ with a flat connection $\nabla$. The horizontal sections of $\nabla$ define a system of differential equations on ${\mathfrak{h}}^{\mathrm{reg}}$ which, by a process described in [@BMR] and [@rouquiersurvey], give a monodromy representation of $\pi_1 ({\mathfrak{h}}^{\mathrm{reg}}/W)$. By definition, $\texttt{KZ}(M)$ is the monodromy representation of $\pi_1 ({\mathfrak{h}}^{\mathrm{reg}}/W)$ associated to $M$.
{#kznumbers}
By [@BMR 4.12] and [@GGOR Section 5.25], the monodromy representation factors through the Hecke algebra $\mathcal{H}$ of $W$. This is the quotient of ${\mathbb{C}}B_W$ by the relations: $$(T-1) \prod_{j=1}^{e_H-1} (T- \det(s)^{-j} e^{-2 \pi i k_{H,j}})$$ for $H \in \mathcal{A}$, $s \in W$ the reflection around $H$ with nontrivial eigenvalue $e^{2 \pi i /e_H}$, and $T$ an $s$–generator of the monodromy around $H$. The parameters differ from those given in [@GGOR] because the idempotent ${\varepsilon}_j(H)$ of [@BMR] is the ${\varepsilon}_{-j , H}$ of [@GGOR].
{#otor}
Therefore, $\texttt{KZ}$ gives a functor $\texttt{KZ}: {\mathcal{O}}\rightarrow \mathcal{H}-\mathrm{mod}$. By [@GGOR Section 5.3], $\texttt{KZ}$ is exact, and if ${\mathcal{O}}_{\mathrm{tor}}$ is the full subcategory of those $M$ in ${\mathcal{O}}$ such that $M|_{{\mathfrak{h}}^{\mathrm{reg}}} =0$ then $\texttt{KZ}$ gives an equivalence ${\mathcal{O}}/{\mathcal{O}}_{\mathrm{tor}} \tilde{\rightarrow}
\mathcal{H}-\mathrm{mod}$ [@GGOR Theorem 5.14].
The Hecke algebra of $G(m,p,n)$ {#hecke}
===============================
We now identify the algebra $\mathcal{H} =
\mathcal{H}(\kappa_{00}, \kappa_1, \ldots, \kappa_{d-1})$ through which $\texttt{KZ}$ factors, in the case of $W= G(m,p,n)$. By [@BMR Prop 4.22], this algebra is generated by $(T_s)_{s \in
\mathcal{N}(D)}$ where $\mathcal{N}(D)$ is the set of nodes of the braid diagram $D$ of $W$. The generators $T_s$ are subject to the braid relations defined by $D$, together with the relations of \[kznumbers\]. From the braid diagram in [@BMR Table 1] we see that if $p>2$ then $\mathcal{H}$ is generated by $T_s,
T_{t_2}, T_{t^{'}_2}, T_{t_3} , \ldots, T_{t_n}$ subject to the following relations $$\begin{aligned}
T_s T_{t^{'}_2} T_{t_2} - T_{t^{'}_2} T_{t_2} T_s &=0 \\
T_{t^{'}_2} T_{t_3} T_{t^{'}_2} - T_{t_3} T_{t^{'}_2} T_{t_3} &=0
\\
T_{t_2} T_{t_3} T_{t_2} - T_{t_3} T_{t_2} T_{t_3} &=0 \\
T_{t_3} T_{t^{'}_2} T_{t_2} T_{t_3} T_{t^{'}_2} T_{t_2} -
T_{t^{'}_2} T_{t_2} T_{t_3} T_{t^{'}_2} T_{t_2} T_{t_3} &=0 \\
T_{t_2} T_s (T_{t^{'}_2} T_{t_2})_{p-1} - T_s (T_{t^{'}_2}
T_{t_2})_p &=0 \\
[T_{t_i}, T_s] &=0 & i \ge 3\\
[T_{t_2}, T_{t_i}] &=0 & i \ge 4\\
[T_{t^{'}_2}, T_{t_i}] &=0 & i \ge 4\\
[T_{t_i}, T_{t_j}] &=0 & i, j \ge 3, &\: |i-j| \ge 2\\
T_{t_i} T_{t_{i+1}} T_{t_i} - T_{t_{i+1}} T_{t_i} T_{t_{i+1}} &=0
& i \ge 3 \\
(T_s -1) \prod_{j=1}^{d-1} (T_s - {\varepsilon}^{-pj} e^{2 \pi i \kappa_j} )
&= 0 \\
(T_{t_i} - 1)(T_{t_i} + e^{2 \pi i \kappa_{00}} ) &=0 & \forall i
\\ (T_{t_2^{'}} -1)(T_{t_2^{'}} + e^{2 \pi i \kappa_{00}}) &= 0 \\\end{aligned}$$ where if $x, y$ are generators then $(xy)_r$ denotes the word $(xy)^{r/2}$ if $r$ is even or $(xy)^{(r-1)/2}x$ if $r$ is odd. If $p=2$ then we see from [@BMR Table 2] that $\mathcal{H}$ is the algebra described above, except that the relation $T_{t_3}
T_{t^{'}_2} T_{t_2} T_{t_3} T_{t^{'}_2} T_{t_2} - T_{t^{'}_2}
T_{t_2} T_{t_3} T_{t^{'}_2} T_{t_2} T_{t_3} =0$ is omitted.
We wish to verify that the algebra presented by these generators and relations is the same as the Hecke algebra of $G(m,p,n)$ as defined in [@GJ 2.A]. We set $a_0 = T_s$, $a_1=
-T_{t^{'}_2}$, $a_2 = -T_{t_2}$ and $a_i = -T_{t_i}$ for $i \ge
3$. In the $p=2$ case, we see that we have exactly the relations of [@GJ 2.A], except that [@GJ] have the additional relation $(a_1 a_2 a_3)^2 = (a_3 a_1 a_2)^2$. However, this additional relation follows from the other relations, since $(T_s
-1) \prod_{j=1}^{d-1} (T_s - {\varepsilon}^{-pj} e^{2 \pi i \kappa_j} ) = 0$ implies that $a_0 = T_s$ is invertible, and we can then check that $(a_1 a_2 a_3)^2 a_0 = (a_3 a_1 a_2)^2 a_0$, using the relations listed above.
If $p>2$, we again have the same relations as [@GJ 2.A], except that the relation $$\label{rel1}a_0 a_1 a_2 = (q^{-1} a_1 a_2)^{2-p} a_2
a_0 a_1 + (q-1) \sum_{k=1}^{p-2} (q^{-1} a_1 a_2)^{1-k} a_0
a_1$$ of [@GJ] has been replaced by the relation $$\label{rel2}a_2 a_0 (a_1 a_2)_{p-1} = a_0 (a_1
a_2)_p.$$ An explicit calculation shows that in the presence of the other relations, (\[rel1\]) and (\[rel2\]) are equivalent, where $q= e^{2 \pi i \kappa_{00}}$. We therefore obtain the following lemma.
The Hecke algebra $\mathcal{H}$ through which the functor $\texttt{KZ}$ factors is isomorphic to the Hecke algebra denoted $\mathfrak{H}^{q,x}_{m,p,n} (\mathbb{C})$ in [@GJ], with parameters $q= e^{2 \pi i \kappa_{00}}$ and $x_1 = 1$, $x_j =
{\varepsilon}^{-p(j-1)} e^{-2 \pi i \kappa_{j-1}}$ for $j >1$.
We will need some facts about the representation theory of this algebra. Specifically, we will need to use the parametrisation of the simple modules that is the main result of [@GJ Section 3].
{#section-11}
As in [@GJ 2.B], we make the following definitions. For $1\le
i \le m$, write $i=sp+t$, with $0 \le s \le d-1$ and $1 \le t \le
p$. Let $\eta_p = e^{\frac{2 \pi i}{p}} = {\varepsilon}^d$ and let $Q_i =
\eta_p^{t-1}y_{s+1}$ where $y_{s+1}$ is chosen so that $y_{s+1}^p
= x_{s+1}$. In this way, we get a new sequence of complex numbers $Q:= (Q_1, \ldots , Q_m)$. Now we follow [@GJ 2.C]. Let $\Pi^m_n$ denote the set of multipartitions $\lambda =
(\lambda^{(i)})_{1 \le i \le m}$ of $n$ with $m$ parts. Then there is a permutation $\varpi$ that acts on $\Pi^m_n$ as follows. The permutation $\varpi$ may be expressed in cycle notation as $$\varpi = (1,2, \ldots, p)(p+1, p+2, \ldots 2p) \cdots ((d-1)p
+1, \ldots dp).$$ The action of $\varpi$ on a multipartition $\lambda = (\lambda^{(i)})$ is defined by $\varpi (\lambda)^{(i)}
= \lambda^{\varpi^{-1} (i)}$. Let $\mathcal{L}$ be a set of representatives of the orbits of this action of $\varpi$ on $\Pi^m_n$ and for $\lambda \in \Pi^m_n$, let $o_\lambda =
\mathrm{min} \{ k \in \mathbb{N}_{>0} | \varpi^k (\lambda) =
\lambda \}$.
{#section-12}
In Theorem \[hsimples\] below, the set of Kleshchev multipartitions in $\Pi^m_n$ is defined with respect to the parameters $q$ and $Q= (Q_1, \ldots, Q_m)$. To define the set of Kleshchev multipartitions, we first need the definition of the *residue* of a node in a multipartition $(\lambda^{(1)},
\ldots \lambda^{(m)} )$. If $x$ is a node in column $j(x)$ and row $i(x)$ of $\lambda^{(k)}$, we define the residue ${\mathrm{res}}(x) = Q_k
q^{j(x) -i(x)}$. We say $y \notin \lambda$ is an *addable* $a$–node if $\lambda \cup \{ y\}$ is a multipartition and ${\mathrm{res}}(y)= a$. We say $y \in \lambda$ is a *removable* $a$–node if $\lambda \setminus \{ y\}$ is a multipartition and ${\mathrm{res}}(y)=a$. A node $x \in \lambda^{(i)}$ is said to be *below* a node $y \in \lambda^{(j)}$ if either $i>j$ or else $i=j$ and $x$ is in a lower row than $y$. A removable node $x$ is called a *normal* $a$–mode if whenever $y$ is an addable $a$–node which is below $x$ then there are more removable $a$–nodes between $x$ and $y$ than there are addable $a$–nodes. A removable $a$–node is called *good* if it is the highest normal $a$–node of $\lambda$. The set of Kleshchev multipartitions is defined inductively by declaring that the empty multipartition is Kleshchev, and that a multipartition $\lambda$ is Kleshchev if and only if there is a node $y \in \lambda$ which is a good $a$–node, for some $a$, such that $\lambda \setminus \{
y\}$ is Kleshchev. A more detailed exposition, including examples, may be found in the introduction to the paper [@arikimathas].
By the remark following [@GJ Theorem 3.1], we have the following theorem.
[@GJ]\[hsimples\] Suppose $q$ is not a root of unity. Then the set of simple $\mathfrak{H}^{q,x}_{m,p,n}$–modules is in bijection with the set $\{ (\lambda, i) | \lambda \in \Lambda^{0} \cap \mathcal{L} \:
\mathrm{and} \: i \in [0, \frac{p}{o_\lambda} -1]\}$ where $\Lambda^0$ denotes the set of *Kleshchev* multipartitions in $\Pi^m_n$, and $\mathcal{L}$ is defined as above.
{#section-13}
In this paper, we will be primarily interested in values of the parameters such that the equation $$\label{kapparels} d \kappa_1 + m (n-1)\kappa_{00}
= -1-m(n-1)-d$$ holds. If this is the case, then $$q^{p(n-1)} = {\varepsilon}^{-p} e^{-2 \pi i \kappa_1} = x_2.$$ We will use the parametrisation given by Theorem \[hsimples\] to work out how many simple modules $\mathcal{H}$ has for generic choices of $\kappa_{00}, \kappa_1, \ldots , \kappa_{d-1}$ satisfying (\[kapparels\]). This will be used in the next section to give an upper bound on the number of finite-dimensional simple objects in category ${\mathcal{O}}$.
Simple modules for the Hecke algebra
====================================
{#section-14}
In this section, we have the standing assumptions that the parameters are $q= e^{2 \pi i \kappa_{00}}$ and $Q= (Q_1, \ldots,
Q_m)$ where the $Q_i$ are defined as above. In particular, $Q_{sp+t} = \eta_p^{t-1} x_{s+1}^{\frac{1}{p}}$. But $x_{s+1} =
{\varepsilon}^{-ps} e^{-2 \pi i \kappa_{s}}$ so $Q_{sp+t} = \eta_p^{t-1}
{\varepsilon}^{-s} e^{-2 \pi i \kappa_s /p} = {\varepsilon}^{d(t-1) -s} e^{-2 \pi i
\kappa_s /p}$. In particular, $Q_1 = 1$, $Q_2 = \eta_p$, $\ldots$, $Q_p = \eta_p^{p-1}$ and $Q_{p+1} = y_2 = q^{n-1}$, $Q_{p+2} =
\eta_p q^{n-1}$, $\ldots$, $Q_{2p} = \eta_p^{p-1} q^{n-1}$. We claim that, for these values of $Q_i$ and $q$, and for a generic choice of the $\kappa_{00}$ and $\kappa_i$, there are exactly $p$ multipartitions in $\Pi^m_n$ which are not Kleshchev and they can be described as follows.
Let $\rho = \begin{matrix} {\tiny\yng(3)} & \cdots & \tiny\yng(1)
\end{matrix}$, a partition of $n$, and for $1 \le i \le m$, define $\rho_i
\in \Pi^m_n$ to be the multipartition with $\rho$ in the $i^{\mathrm{th}}$ place and $\O$ everywhere else. Then we have the following lemma.
\[kleshchev\] With the above choices of $Q_i$ and $q$, the non-Kleshchev multipartitions in $\Pi^m_n$ are prescisely the $\rho_i$ where $1
\le i \le p$.
First, note that for a generic choice of the parameters, $\kappa_{00} \notin \mathbb{Q}$ and so we may assume that $q$ is not a root of unity.
Let $\lambda = (\lambda^{(1)}, \ldots,
\lambda^{(m)}) \in \Pi^m_n$. Suppose that $\lambda \neq \rho_i$ for $1 \le i \le p$. We must show that $\lambda$ is Kleshchev. We will show that we can repeatedly remove good nodes from $\lambda$ until we reach the empty partition. First, let $i
>2p$. We will show that we can reduce to the case $\lambda^{(i)}
=\O$. Recall that $Q_i$ is of the form $y_a \eta_p^b$ for some $a$ and $b$. Thus, the residue of a node in $\lambda^{(i)}$ is of the form $q^c y_a \eta_p^b$ for some $a,b,c$. If this is equal to the residue of a node in some $\lambda^{(j)}$ where $j \neq i$, then we must have $q^c y_a \eta_p^b = q^{c^{'}} y_{a^{'}}
\eta_p^{b^{'}}$ for some $a^{'}, b^{'}, c^{'}$. So $q^{c- c^{'}} =
y_{a^{'}}y_a^{-1} \eta_p^{b^{'}-b}$. Hence, $e^{2 \pi i
\kappa_{00} (c-c^{'})} = {\varepsilon}^{-(a^{'}-1)} e^{-2 \pi i
\kappa_{a^{'}} /p} {\varepsilon}^{(a-1)} e^{2 \pi i \kappa_a /p}
{\varepsilon}^{d(b^{'}-b)}$. So $$\mathrm{exp} (2 \pi i \kappa_{00} (c-c^{'}) + 2 \pi i \frac{a^{'}-1}{m}
+2 \pi i \frac{\kappa_{a^{'}}}{p} - 2 \pi i \frac{a-1}{m} -2 \pi i
\frac{\kappa_a}{p} - 2 \pi i \frac{d(b^{'}-b)}{m} ) =1.$$ For a generic choice of the $\kappa_i$ and $\kappa_{00}$, this can only happen if $c=c^{'}$ and $a=a^{'}$. But then this forces $i=j$. Hence, the only nodes in $\lambda$ which can have the same residue as a node in $\lambda^{(i)}$ are the other nodes in $\lambda^{(i)}$. Let $x$ be the rightmost node in the bottom row of $\lambda^{(i)}$. Any node with the same residue as $x$ has residue $q^{j(x)-i(x)} Q_i$ and so must lie on the same diagonal as $x$. But, by the choice of $x$, there can be no addable or removable nodes on the same diagonal as $x$. So $x$ is a good node, and we may remove $x$. Continuing inductively, the nodes of $\lambda^{(i)}$ may be removed one at a time, and we conclude that the multipartition with $\lambda^{(i)}$ replaced by $\O$ is Kleshchev. Note that this is not necessarily a multipartition of $n$; it is a multipartition of the integer $\sum_{j \neq i}
|\lambda^{(j)} | \le n$.
We are now reduced to checking that all the multipartitions of the form $$\lambda = (\lambda^{(1)} , \lambda^{(2)}, \ldots ,
\lambda^{(p)} , \ldots ,\lambda^{(2p)}, \O, \O , \ldots, \O)$$ with $\sum_i |\lambda^{(i)}| \le n$ and $\lambda \neq \rho_i$, $1
\le i \le p$, are Kleshchev. We now show that we can reduce to the case $\lambda^{(2p)} = \lambda^{(2p-1)} = \cdots = \lambda^{(p+1)}
= \O$. First, consider $\lambda^{(2p)}$. Suppose $\lambda^{(2p)}$ has $b$ rows and that the lowest row has length $a$. Let $x$ be the rightmost node of the bottom row of $\lambda^{(2p)}$. Then the residue of $x$ is $q^{a-b} \cdot q^{n-1} \eta_p^{p-1} =: r$. Then $x$ is a normal $r$–node because $\lambda^{(2p)}$ has no addable $r$–nodes which are below $x$, since such a node would have to lie on the same diagonal as $x$, which is impossible by choice of $x$. And if $i>2p$ then $\lambda^{(i)} =\O$ has no addable $r$–node, since the only node that can be added to $\lambda^{(i)}$ has residue $Q_i=Q_{sp+t}$ for some $s \ge 2$ and some $t$. This equals ${\varepsilon}^{d(t-1)-s}e^{-2 \pi i \kappa_s /p}$ which cannot equal $r$ for a generic choice of $\kappa_s$. Hence, there are no addable $r$–nodes of $\lambda$ which are below $x$, so $x$ is a normal $r$–node. We must show that $\lambda$ contains no higher normal $r$–nodes. Certainly $\lambda^{(2p)}$ contains no higher normal $r$–node, since any $r$–node must lie on the same diagonal as $x$ and so cannot be removable. Any normal $r$–node not in $\lambda^{(2p)}$ must lie in $\lambda^{(p)}$, because no power of $q$ can be equal to a power of $\eta_p$ since $q$ is not a root of unity. Suppose then that the node $y \in
\lambda^{(p)}$ has residue $r$. Say $y$ lies in row $i(y)$ and column $j(y)$ of $\lambda^{(p)}$. Then $q^{j(y)-i(y)} \eta_p^{p-1}
= q^{a-b+n-1} \eta_p^{p-1}$. Hence, $j(y)-i(y) = a-b+n-1$. Suppose $\lambda^{(p)}$ has $d$ columns. Then $a-b+n-1 =j(y)-i(y) \le
d-1$. So $n+1 \le n+a \le b+d$. But $b+d \le |\lambda^{(2p)}|
+|\lambda^{(p)}| \le n$, a contradiction. Hence, no such $y$ exists, and $x$ is the highest normal $r$–node, and so is good. Removing $x$ and continuing inductively, we may take $\lambda^{(2p)} = \O$. The same argument shows that we may take $\lambda^{(p+i)} = \O$, $1 \le i \le p$, as claimed.
We are now reduced to showing that those $\lambda$ of the form $\lambda = (\lambda^{(1)} , \ldots, \lambda^{(p)} , \O , \ldots
,\O)$ with $\lambda \neq \rho_i$, $1 \le i \le p$, are Kleshchev. First, consider $\lambda^{(p)}$. Let $b$ be the number of columns of $\lambda^{(p)}$. Then $b<n$ since $\lambda \neq \rho_p$. If $x
\in \lambda^{(p)}$ then $\mathrm{res}(x) = q^{j(x)-i(x)}
\eta_p^{p-1} $ where $j(x)-i(x) \le b -1$. Let $x$ be the rightmost node in the bottom row of $\lambda^{(p)}$ and let $r=
\mathrm{res}(x)$. Then $x$ is the only removable $r$–node in $\lambda$ since there can be no $r$–nodes in $\lambda^{(i)}$ with $i \neq p$, again because $q$ is not a root of unity. Furthermore, the only way there can be an addable $r$–node below $x$ is if such a node can be added to the empty diagram $\lambda^{(2p)}$. Such a node would have residue $q^{n-1} \eta_p^{p-1}$ and so we would have to have $j(x)-i(x) = n-1 \le b-1$. This contradicts $b<n$. Hence, $x$ is a good node and may be removed. Continuing inductively, we may remove all the nodes in $\lambda^{(p)}$. The same argument works for all the $\lambda^{(i)}$, $1 \le i \le p$, and so we can get to the empty partition from $\lambda$ by successively removing good nodes. Hence, $\lambda$ is Kleshchev.
To complete the proof, we observe that $\rho_i$ is not Kleshchev for $1 \le i \le p$, since the only removable node in $\rho_i$ is the node at the end of the row $\rho_i^{(i)}$. This node has residue $q^{n-1} \eta_p^{i-1}$. It is not normal as there is an addable node in $\rho_i$ which is the unique node that may be added to the empty diagram $\rho_i^{(p+i)}$, and this node also has residue $q^{n-1} \eta_p^{i-1}$.
\[numbersimples\] For generic values of the parameters satisfying $d \kappa_1 +
m(n-1) \kappa_{00} = -1-m(n-1)-d$, the algebra $\mathcal{H}$ has $|{\textsf{Irrep}}(W)|-1$ nonisomorphic simple modules.
By Theorem \[hsimples\], the simples are indexed by pairs $
(\lambda,i)$ such that $\lambda \in \Lambda^0 \cap \mathcal{L}$ and $0 \le i \le \frac{p}{o_\lambda}-1$. Let $T$ be the set of all such pairs. Note that $\{ \rho_1, \rho_2, \ldots, \rho_p\}$ is an orbit of $\varpi$ on $\Pi^m_n$. Let $S$ be the set of all pairs $(\lambda,i)$ with $\lambda \in \mathcal{L}$ and $0 \le i \le
\frac{p}{o_\lambda} -1$. Choose $\rho_1$ to be the representative of the orbit of $\rho_1$ in $\mathcal{L}$. Then $(\rho_1,0) \in S$ and $(\rho_1,0) \notin T$. Furthermore, if $\lambda \in
\mathcal{L}$ and $\lambda \neq \rho_1$ then $\lambda$ is Kleshchev by \[kleshchev\] and so $(\lambda,i) \in T$ for all $0 \le i \le
\frac{p}{o_\lambda}-1$. Hence, $T = S \setminus \{(\rho_1, 0)\}$ and so $|T| = |S|-1$. It remains to show that $|S|= |{\textsf{Irrep}}(W)|$.
By [@arikigrpn Theorem 2.6], $|S|$ is the number of simple modules for the generic version $K \mathcal{H}$ of the Hecke algebra $\mathcal{H}$, defined over the field $K=\mathbb{C}(q,x_1,
\ldots, x_d)$ with $q$ and the $x_i$ being indeterminates. But by the proof of [@BMR Theorem 4.24], $K \mathcal{H}$ is isomorphic to the group algebra $KW$ and so $|S|=|{\textsf{Irrep}}(W)|$.
Application to category ${\mathcal{O}}$
---------------------------------------
We may use Corollary \[numbersimples\], together with the functor ${\texttt{KZ}}$, to get some information about the category ${\mathcal{O}}$ of $H_\kappa$–modules.
\[infdim\] Let $\kappa_i, \kappa_{00}$ be chosen generically so that $d
\kappa_1 +m(n-1) \kappa_{00} = -1-m(n-1)-d$. Then there is exactly one finite-dimensional simple module in category ${\mathcal{O}}$, namely $L(\mathsf{triv})$.
First, recall from Section \[otor\] that ${\texttt{KZ}}$ is an exact functor, and that every $\mathcal{H}$–module is the image of some object of ${\mathcal{O}}$ under ${\texttt{KZ}}$. Suppose $X$ is a simple $\mathcal{H}$–module. Then, using exactness of ${\texttt{KZ}}$, we can find a simple object $L(\tau) \in {\mathcal{O}}$ such that ${\texttt{KZ}}(L(\tau)) = X \neq
0$. Since ${\texttt{KZ}}(L(\tau)) \neq 0$, we get $L(\tau)|_{{\mathfrak{h}}^{\mathrm{reg}}} \neq 0$ and hence $\dim (L(\tau)) =
\infty$. Since each of the $|{\textsf{Irrep}}(W)|-1$ simple $\mathcal{H}$–modules is then the image of some infinite-dimensional $L(\tau)$, we conclude that at least $|{\textsf{Irrep}}(W)|-1$ of the $L(\tau)$ are infinite-dimensional. (This argument is based on [@BEG2 Lemma 3.11]).
It remains to show that $L(\mathsf{triv})$ is finite-dimensional. It was shown in Lemma \[dunklrep\] that the Dunkl representation ${\mathbb{C}}[{\mathfrak{h}}] = M(\mathsf{triv})$ of $H_\kappa$ is the restriction to $H_\kappa$ of the Dunkl representation of the larger algebra $H_\mu$, where $H_\mu$ is the rational Cherednik algebra for $G(m,1,n)$ with parameters $\mu_{00} = \kappa_{00}$ and $\mu_i =
\kappa_i/p$. It follows from the hypothesis on $\kappa$ that $$m \mu_1 + m(n-1) \mu_{00}= -1-m(n-1)-d.$$ In the terminology of [@CE], this says that $(\mu_{00}, \mu_1,
\ldots, \mu_{m-1})$ belongs to the set $E_r$ where $r= m(n-1) +d+1
= m(n-1) +q$ where $q=d+1$ and $1 \le q \le m-1$. We are now in a position to apply [@CE Proposition 4.1]. Let ${\mathfrak{h}}_q$ be the representation of $G(m,1,n)$ with ${\mathfrak{h}}_q = {\mathbb{C}}^n$ as a vector space, and on which $S_n$ acts by permuting the coordinates, and $s_i$ acts by multiplying the $i^{\mathrm{th}}$ coordinate by ${\varepsilon}^{-d-1}$. Then [@CE Proposition 4.1] states that the polynomial representation $M({\mathsf{triv}})$ of $H_\mu$ contains a copy of ${\mathfrak{h}}_q$ in degree $m(n-1) +d+1$ consisting of singular vectors (vectors which are killed by ${\mathfrak{h}}\subset H_\mu$). Furthermore, if we set $\tilde{Y}_c$ to be the quotient of $M(\mathsf{triv})$ by the ideal generated by this copy of ${\mathfrak{h}}_q$, then [@CE Theorem 4.3(i)] states that $\tilde{Y}_c$ is finite-dimensional provided $\kappa_{00} \notin \mathbb{Q}$. Hence, there is an exact sequence of $H_\mu$–modules $$M({\mathsf{triv}}) \rightarrow \tilde{Y}_c \rightarrow 0.$$ Since $H_\kappa \subset H_\mu$, these are also $H_\kappa$–module maps, and so the Dunkl representation $M({\mathsf{triv}})$ of $H_\kappa$ has a finite-dimensional quotient. It follows that the unique smallest quotient $L({\mathsf{triv}})$ of the $H_\kappa$–module $M({\mathsf{triv}})$ is finite-dimensional, as required.
{#section-15}
This ends our study of the Hecke algebra and category ${\mathcal{O}}$. We now wish to apply Theorem \[infdim\] to study a special object in ${\mathcal{O}}$ whose associated graded module will yield the desired quotient of the ring of diagonal coinvariants.
Shifting
========
{#section-16}
In this section we will construct a one-dimensional $H_\kappa$–module $\Lambda$ for particular values of $\kappa$, and then construct a shifted version of $\Lambda$ which will be a finite-dimensional $H_\kappa$–module $L$. In the next section we will show how $L$ is related to the diagonal coinvariants of $W$.
A one-dimensional module
------------------------
\[onedim\] Suppose we choose the parameters $\kappa_{00}, \kappa_i$ such that $d \kappa_1 + m(n-1) \kappa_{00} = -1$. Then $H_\kappa$ has a one-dimensional module $\Lambda$ on which ${\mathfrak{h}}$ and ${\mathfrak{h}}^*$ act by zero and $W=G(m,p,n)$ acts by the trivial representation.
Let $\Lambda$ be the trivial $W$–module. We can make $\Lambda$ into a $T({\mathfrak{h}}\oplus {\mathfrak{h}}^*) \ast W$–module by making ${\mathfrak{h}}$, ${\mathfrak{h}}^*$ act by 0. So we are reduced to showing that the defining relations of $H_\kappa$ act by $0$ on $\Lambda$. The only relation that may cause difficulty is the commutation relation of section \[commutationrel\]. We need to check that the commutation relation between $y_a$ and $x_b$ acts by $0$ on $\Lambda$ for all $a,b$. Recall that $$\begin{gathered}
\label{commute3} [y_a, x_b] = \delta_{ab} +
\sum_{i=1}^n \delta_{ia}\delta_{ib} \left[ \sum_{j=0}^{d-1}
(\kappa_{j+1} - \kappa_j) \sum_{r=0}^{d-1} {\varepsilon}^{prj} s_i^{pr}
\right] \\ + \sum_{1 \le i < j \le n } \sum_{\ell =0}^{m-1}
\frac{1}{2} (\delta_{ia} - {\varepsilon}^{\ell} \delta_{ja} )(\delta_{ib} -
{\varepsilon}^{-\ell} \delta_{jb}) 2 \kappa_{00} \sigma_{ij}^{\ell}\end{gathered}$$ First, if $a \neq b$ then the right hand side of (\[commute3\]) becomes $$\sum_{1 \le i < j \le n } \sum_{\ell =0}^{m-1}
(\delta_{ia} - {\varepsilon}^{\ell} \delta_{ja} )(\delta_{ib} - {\varepsilon}^{-\ell}
\delta_{jb}) \kappa_{00} \sigma_{ij}^{\ell}$$ which acts on $\Lambda$ by the scalar $$\sum_{1 \le i < j \le n } \sum_{\ell
=0}^{m-1}
(\delta_{ia} - {\varepsilon}^{\ell} \delta_{ja} )(\delta_{ib} - {\varepsilon}^{-\ell}
\delta_{jb}) \kappa_{00} = \kappa_{00} \sum_{1 \le i < j \le n }
\sum_{\ell =0}^{m-1} (-{\varepsilon}^{\ell} \delta_{ja}\delta_{ib} -
{\varepsilon}^{-\ell} \delta_{ia} \delta_{jb})$$ which vanishes since $\sum_{\ell=0}^{m-1} {\varepsilon}^{\ell} =0$.
Second, if $a=b$ then we must show that $$1+
\left[ \sum_{j=0}^{d-1}
(\kappa_{j+1} - \kappa_j) \sum_{r=0}^{d-1} {\varepsilon}^{prj} \right] +
\sum_{1 \le i < j \le n } \sum_{\ell =0}^{m-1} (\delta_{ia} -
{\varepsilon}^{\ell} \delta_{ja} )(\delta_{ia} - {\varepsilon}^{-\ell} \delta_{ja})
\kappa_{00}$$ vanishes. So we must show that $1+ d \kappa_1 +
\kappa_{00} (m(n-a) + (a-1)m) =0$, which holds by the hypothesis.
A shift isomorphism
-------------------
We require the following theorem from [@BC].
\[shift\] Let $(\kappa_{00}, \kappa_1, \ldots , \kappa_{d-1}) \in {\mathbb{C}}^d$ and define $\kappa_{00}^{'} = \kappa_{00} +1 , \kappa_1^{'} = \kappa_1
+1$ and $\kappa_i^{'} = \kappa_i$ for all other $i$. Then there is an isomorphism $$\psi: e H_{\kappa^{'}} e \rightarrow e_{\varepsilon}H_\kappa e_{\varepsilon}$$ where $e$ is the symmetrising idempotent $e= \frac{1}{|W|} \sum_{w
\in W} w \in {\mathbb{C}}W$ and $e_{\varepsilon}= \frac{1}{|W|} \sum_{w \in W}
\det(w) w$.
This is a minor modification of [@BC Theorem 5.8(2)].
{#section-17}
Given $\kappa$ with $d \kappa_1 + m(n-1) \kappa_{00} =
-1-m(n-1)-d$, define $\kappa^{'}$ as in the statement of theorem \[shift\]. Then $d \kappa^{'}_1 + m(n-1) \kappa_{00}^{'} = -1$ and hence by theorem \[onedim\], there is a one-dimensional module $\Lambda$ for $H_{\kappa^{'}}$ on which $W$ acts trivially. Hence, $e \Lambda = \Lambda$ becomes a $e H_{\kappa^{'}}
e$–module. Via the isomorphism of \[shift\], we may define a $e_{\varepsilon}H_{\kappa} e_{\varepsilon}$–module $\Lambda^{\psi}$. Finally, we set $$L = H_\kappa e_{\varepsilon}\otimes_{e_{\varepsilon}H_\kappa e_{\varepsilon}} \Lambda^{\psi},$$ an $H_\kappa$–module.
Then, because $H_{\kappa} e_{\varepsilon}$ is a finite $e_{\varepsilon}H_{\kappa}
e_{\varepsilon}$–module (this follows by considering the associated graded modules, for instance), $L$ is finite-dimensional, and it then follows from Section \[catoaltdef\] that $L$ belongs to the category ${\mathcal{O}}$ of $H_{\kappa}$–modules.
Suppose we have chosen $\kappa$ generically. Then we may apply the results of the previous section. In particular, Theorem \[infdim\] says that the only finite-dimensional simple object in category ${\mathcal{O}}$ is $L({\mathsf{triv}})$. Since $L$ has a composition series with composition factors $L(\tau)$, $\tau \in {\textsf{Irrep}}(W)$, we see that every composition factor of $L$ must be $L({\mathsf{triv}})$. There is a functor $F: H_\kappa-\mathrm{mod} \rightarrow e_{\varepsilon}H_\kappa e_{\varepsilon}-\mathrm{mod}$ defined by $FM = e_{\varepsilon}M$. This is an exact functor and it takes a composition series of $L$ to a composition series of $FL \cong \Lambda^{\psi}$. Hence, $e_{\varepsilon}L({\mathsf{triv}}) \neq 0$ and $L \cong L({\mathsf{triv}})$. This proves the following lemma.
\[lequalsltriv\] If $\kappa_i$, $\kappa_{00}$ are chosen generically then the $H_\kappa$–module $L$ is isomorphic to $L({\mathsf{triv}})$.
A quotient ring of the diagonal coinvariants {#mainproof}
============================================
{#section-18}
We follow the proof of [@Go Section 5] to obtain the desired ring $S_W$ of Theorem \[mainthm\]. Choose generic $\kappa^{'}$ with $d \kappa_1^{'} + m(n-1)\kappa_{00}^{'}= -1$, let $\kappa$ be defined as above, and define $L = H_{\kappa} e_{\varepsilon}\otimes_{e_{\varepsilon}H_{\kappa} e_{\varepsilon}} \Lambda^\psi$ as above. Consider the filtration on $H_{\kappa}$ with $\deg({\mathfrak{h}}) = \deg({\mathfrak{h}}^*) =1$ and $\deg(W)=0$, and the associated graded module $\mathrm{gr} L$. As in [@Go], one obtains a surjection of ${\mathbb{C}}[{\mathfrak{h}}\oplus
{\mathfrak{h}}^*]*W$–modules: $${\mathbb{C}}[{\mathfrak{h}}\oplus {\mathfrak{h}}^*]*W e_{\varepsilon}\otimes_{{\mathbb{C}}[{\mathfrak{h}}\oplus {\mathfrak{h}}^*]^W}
\mathrm{gr} \Lambda^\psi \rightarrow \mathrm{gr} L.$$ By definition , ${\mathbb{C}}[{\mathfrak{h}}\oplus {\mathfrak{h}}^*]^W_+$ acts on $\mathrm{gr} \Lambda^\psi$ by 0, and hence $S_W:= \mathrm{gr} L \otimes \wedge^n {\mathfrak{h}}$ is a quotient ring of $D_W = {\mathbb{C}}[{\mathfrak{h}}\oplus {\mathfrak{h}}^*]/\langle {\mathbb{C}}[{\mathfrak{h}}\oplus
{\mathfrak{h}}^*]^W_+ \rangle$. We wish to determine the graded character of $S_W$. To do so, we will determine the graded character of $L
\cong L({\mathsf{triv}})$. This requires a more delicate study of the results of [@CE] that were used above.
{#section-19}
Recall from the proof of \[infdim\] that we have the $H_\kappa$–module $\tilde{Y}_c$ which is a finite-dimensional quotient of the Dunkl representation $M({\mathsf{triv}})$ obtained by factoring out an ideal $J$ of the polynomial ring ${\mathbb{C}}[{\mathfrak{h}}] =
M({\mathsf{triv}})$, where $J$ is generated by a copy of ${\mathfrak{h}}_q$ in degree $m(n-1)+d+1$ consisting of singular vectors. Call this copy $U$. Note that $U$ is an irreducible representation of $G(m,p,n)$ since it follows from the definition of ${\mathfrak{h}}_q$ (quoted in the proof of \[infdim\]) that $U \cong {\mathfrak{h}}^*$ as $G(m,p,n)$–modules. Now we are in a position to apply [@CE Theorem 2.3] with $W=G(m,p,n)$. This says that we have a BGG-resolution $$0 \leftarrow \tilde{Y}_c \leftarrow M({\mathsf{triv}}) \leftarrow M(U)
\leftarrow \cdots \leftarrow M(\wedge^n U) \leftarrow 0.$$ Hence in the Grothendieck group $K_0 ({\mathcal{O}})$, we have an equality $$[\tilde{Y}_c] = \sum_{i=0}^n (-1)^i [M(\wedge^i {\mathfrak{h}}^*)].$$ Now, by theorem \[infdim\], there is only one finite-dimensional simple module in ${\mathcal{O}}$, namely $L({\mathsf{triv}})$. Hence, $[\tilde{Y}_c] =
a \cdot [L({\mathsf{triv}})]$ for some $a \ge 1$. By [@DO Section 2.5 (32)], there is an ordering on ${\textsf{Irrep}}(W)$ such that the matrix with entries $[M(\tau): L(\sigma)]$ is unipotent upper triangular. Therefore the classes $[M(\tau)]$ give a $\mathbb{Z}$–basis for $K_0 ({\mathcal{O}})$ and there is a unique expression $$[L({\mathsf{triv}})] = \sum_\tau c_\tau [M(\tau)]$$ with $c_\tau \in \mathbb{Z}$. Hence, $a \cdot c_{\mathsf{triv}}= 1$ and so $a=1$. It follows that $[\tilde{Y}_c] = [L({\mathsf{triv}})]$ and hence $\tilde{Y}_c \cong L({\mathsf{triv}})$. Furthermore, $$[L({\mathsf{triv}})] = \sum_{i=0}^n (-1)^i [M(\wedge^i {\mathfrak{h}}^*)]$$ in $K_0 ({\mathcal{O}})$.
{#section-20}
To prove Theorem \[mainthm\], we will require the following lemma. Recall from Section \[zdef\] that the element $z \in {\mathbb{C}}W \subset
H_{\kappa}$ is defined by $z = \sum_{H \in
\mathcal{A}}\sum_{i=1}^{e_H-1} e_H k_{H,i} {\varepsilon}_{H,i}$. When $W=G(m,p,n)$ we have $$z = \sum_{i=1}^n \sum_{t=1}^{d-1} \kappa_t \sum_{j=0}^{d-1}
{\varepsilon}^{ptj} s_i^{pj} + \kappa_{00} \sum_{i<j} \sum_{0 \le r \le m-1}
(1 - \sigma_{ij}^{(r)} ).$$
\[zscalar\] $z$ acts on $\wedge^i {\mathfrak{h}}^*$ by the scalar $i(d \kappa_1 + m(n-1)
\kappa_{00}).$
Since $z$ is central, it acts on $\wedge^i {\mathfrak{h}}^*$ by the scalar $\chi_{\wedge^i {\mathfrak{h}}^*}(z)/{n \choose i}$. Choosing a basis for $
{\mathfrak{h}}^*$ for which $s_i^j$ is diagonal, we see that $\chi_{\wedge^i
{\mathfrak{h}}^*}(s_i^j) = {\varepsilon}^j {n-1 \choose i-1} + {n-1 \choose i}$. Similarly, $\chi_{\wedge^i {\mathfrak{h}}^*}(\sigma_{ij}^{(r)}) = {n-1
\choose i} - {n-1 \choose i-1}$. Substituting these values into the expression for $z$ gives the result.
{#section-21}
All of the properties of $S_W$ listed in Theorem \[mainthm\] apart from Theorem \[mainthm\](3) are immediate consequences of the following theorem. Recall that $D_W$ is graded with $\deg({\mathfrak{h}})=-1$ and $\deg({\mathfrak{h}}^*)=1$ and that this grading is $W$–stable. In general, if $M$ is a $\mathbb{Z}$–graded module and $\chi_k$ is the character of the $k^{\mathrm{th}}$ graded piece then the *graded character* of $M$ is defined to be the formal power series $\sum_i \chi_i t^i$.
\[gradedchar\] The graded character of $\mathrm{gr} L = S_W \otimes \wedge^n
{\mathfrak{h}}^*$ is $$w \mapsto t^{-n-m{n \choose 2}} \frac{ \det|_{{\mathfrak{h}}^*}(1-t^{m(n-1)+d+1} w)}
{\det|_{{\mathfrak{h}}^*}(1-tw)}$$
Recall the element $\mathsf{eu}_{\kappa} \in H_{\kappa}$ from Section \[zdef\]. By Lemma \[zscalar\], $z$ acts by 0 on the trivial representation $\mathsf{triv}= \wedge^0 {\mathfrak{h}}^*$ of $W$ and hence $\mathsf{eu}_{\kappa}$ also acts by 0 on the trivial representation $\mathsf{triv}$. Hence, the eigenvalue of $\mathsf{eu}_{\kappa}$ on the subspace ${\mathbb{C}}[{\mathfrak{h}}]_d \otimes
\mathsf{triv} \subset M(\mathsf{triv})$ is $d$.
By [@CE Theorem 4.2], the representation $\tilde{Y}_c=L(\mathsf{triv})$ of $H_{\kappa}$ is isomorphic as a graded $G(m,1,n)$–module to $U_{m(n-1)+d+1}^{\otimes n}$ where $U_{m(n-1)+d+1} = {\mathbb{C}}[u]/(u^{m(n-1)+d+1})$, regarded as a representation of $\mathbb{Z}_m = \langle s_1 \rangle$ via $s_1
(u) = {\varepsilon}^{-1} u$, and where $S_n$ acts by permuting the factors of the tensor product. Since $\{u^{i}| 0 \le i \le m(n-1)+d \}$ is a basis of $U_{m(n-1)+d+1}$, we may define distinct basis elements $a_i$ by $a_i := u^{m(i-1) +1}$, $1 \le i \le n$. Then the element $v := \sum_{\sigma \in S_n} \mathrm{sgn}(\sigma) a_{\sigma(1)}
\otimes a_{\sigma(2)} \otimes \cdots \otimes a_{\sigma(n)}$ affords the representation $\wedge^n {\mathfrak{h}}^*$ of $G(m,1,n)$, and lies in degree $n+ \sum_{i=1}^n m(i-1) = n +m{n \choose 2}$. Hence, $\mathsf{eu}_{\kappa} v =( n + m {n \choose 2}) v$. Note that $v$ also affords the representation $\wedge^n {\mathfrak{h}}^*$ of $G(m,p,n)$ when we consider $\tilde{Y}_c$ as a $W=G(m,p,n)$–module.
We define ${\bf h} = \mathsf{eu}_{\kappa} - n -
m{n \choose 2} \in H_{\kappa}$. We now calculate the graded character of $L =
L(\mathsf{triv})$ with respect to the ${\bf h}$–eigenspaces. We have shown above that in the Grothendieck group of ${\mathcal{O}}$, $$[L(\mathsf{triv})] = \sum_{i=0}^n (-1)^i [M(\wedge^i {\mathfrak{h}}^*)].$$
Now, by Lemma \[zscalar\], $z$ acts by $-i(m(n-1)+d+1)$ on $\wedge^i {\mathfrak{h}}^*$, and hence, by Section \[zdef\], the lowest eigenvalue of ${\bf h}$ on $M(\wedge^i {\mathfrak{h}}^*)$ is $i(m(n-1)+d+1) -
n -m{n \choose 2}$. Therefore, the graded character of $M(\wedge^i
{\mathfrak{h}}^*)$ is $$t^{-n-m{n \choose 2}}\frac{ \chi_{\wedge^i {\mathfrak{h}}^*}(w)
t^{i(m(n-1)+d+1)}}{\det|_{{\mathfrak{h}}^*} (1-tw)}.$$ But $\det|_{{\mathfrak{h}}^*} (1-
t^{m(n-1)+d+1}w) = \sum_i (-1)^i \chi_{\wedge^i {\mathfrak{h}}^*} (w)
t^{i(m(n-1)+d+1)}$ (this follows readily from diagonalising $w$) which gives the graded character of $L(\mathsf{triv})$ with respect to the ${\bf h}$–eigenspaces as $$w \mapsto t^{-n-m{n \choose 2}} \frac{ \det|_{{\mathfrak{h}}^*}(1-t^{m(n-1)+d+1} w)}
{\det|_{{\mathfrak{h}}^*}(1-tw)}$$ By the definition of the diagonal coinvariant ring $D_W$, there is a unique copy of the trivial representation in $D_W$, which lies in degree 0, and hence a unique copy of $\wedge^n {\mathfrak{h}}^*$ in $S_W \otimes \wedge^n {\mathfrak{h}}^*$, and hence a unique copy of $\wedge^n {\mathfrak{h}}^*$ in $L(\mathsf{triv})$ (since $S_W \otimes \wedge^n {\mathfrak{h}}^* = \mathrm{gr} L$, which is isomorphic to $L$ as a $W$–module). The unique copy of $\wedge^n
{\mathfrak{h}}^*$ in $L(\mathsf{triv})$ must be spanned by $v$. But ${\bf h} v
= 0$ and hence, by Lemma \[lequalsltriv\], ${\bf h}$ must act by 0 on the element $e_{\varepsilon}\otimes 1 \in L$ which affords the unique copy of $\wedge^n {\mathfrak{h}}^*$ in $L$. But because the grading induced by ${\bf h}$ on $L$ gives $e_{\varepsilon}\otimes 1$ degree 0 and $x \in {\mathfrak{h}}^*$ degree 1, and $y \in {\mathfrak{h}}$ degree $-1$, we see that $\mathrm{gr} L$ has the same graded character as $L(\mathsf{triv})$, which proves the theorem.
Proof of Theorem \[mainthm\] (3)
--------------------------------
This is similar to a proof in [@Go Section 5]. It is well-known (see for example [@kane]) that the ring of coinvariants $A={\mathbb{C}}[{\mathfrak{h}}]/{\mathbb{C}}[{\mathfrak{h}}]^W_+$ satisfies Poincaré duality. Therefore the highest degree graded component of $A$, which lies in degree $\sum_i (d_i-1)$ where the $d_i$ are the degrees of the fundamental invariants of $W$, is an ideal of $A$ which is contained in every nonzero ideal. This ideal is called the socle of $A$. In the case of $W=G(m,p,n)$ the degrees are $m, 2m, \ldots (n-1)m
, nd$ by [@BMR Table 1, Table 2], so the socle lies in degree $m {n \choose 2} +nd -n$. The image of $\mathbb{C}[{\mathfrak{h}}]$ in $S_W$ corresponds to the subspace $\mathbb{C}[{\mathfrak{h}}] e_{{\varepsilon}}\otimes
\Lambda^\psi$ of $L$. If $p\in \mathbb{C}[{\mathfrak{h}}]^W_+e_{{\varepsilon}}$ then by the definition of the shift isomorphism $\psi$ given in [@BC], we see that $\psi(epe)= e_{{\varepsilon}}pe_{{\varepsilon}}$. It follows that in $L$ we have: $$p\otimes \Lambda^\psi = e_{{\varepsilon}}pe_{{\varepsilon}}\otimes \Lambda^\psi =
e_{{\varepsilon}}\otimes e_{\varepsilon}p e_{\varepsilon}\Lambda^\psi = e_{\varepsilon}\otimes epe \cdot
\Lambda = 0.$$ Thus the ideal generated by $\mathbb{C}[{\mathfrak{h}}]_+^W$ annihilates $e_{{\varepsilon}}\otimes \Lambda^\psi$. On the other hand, the quotient $\mathbb{C}[{\mathfrak{h}}]/{\mathbb{C}}[{\mathfrak{h}}]^W_+$ contains a unique (up to scalar) element of maximal degree $m {n \choose 2} +nd-n$, say $q$. The space $\mathbb{C}q$ is the socle of $\mathbb{C}[{\mathfrak{h}}]/{\mathbb{C}}[{\mathfrak{h}}]^W_+$. We claim $qe_{{\varepsilon}}\otimes \Lambda^\psi \neq 0$. By the PBW theorem, any element of $H_\kappa$ can be written as a sum of terms of the form $p_-wp_+$ where $p_-\in\mathbb{C}[{\mathfrak{h}}^*]$, $p_+\in\mathbb{C}[{\mathfrak{h}}]$ and $w\in W$. Since $p_-$ and $w$ do not increase degree, it would follow if $qe_{{\varepsilon}}\otimes \Lambda^\psi$ were zero, then $L$ could have no subspace in degree $m {n \choose
2}+nd-n$. But the Hilbert series of $L$ has highest order term $t^{-n-m{n \choose 2} + mn(n-1) +nd} = t^{m{n \choose 2} +nd-n}$. Thus $qe_{{\varepsilon}}\otimes \Lambda^\psi$ is non–zero and $\mathbb{C}[{\mathfrak{h}}]e_{{\varepsilon}}\otimes \Lambda^\psi$ is isomorphic to $(\mathbb{C}[{\mathfrak{h}}]/{\mathbb{C}}[{\mathfrak{h}}]^W_+) e_{\varepsilon}\otimes \Lambda^\psi$. This proves Theorem \[mainthm\] (3). $\Box$
Appendix: the case $W=G(m,p,2)$ with $p$ even {#fudge}
=============================================
In this case, we have $d+1$ conjugacy classes of complex reflections in $W$. In the notation of Section \[def\] they are $C_q:=\{s_i^{qp}|1 \le i \le n\}$, (for $1 \le q \le d-1$), $C_{odd}:=\{ \sigma_{12}^{(\ell)} | \text{$\ell$ odd}\}$ and $C_{even}:=\{\sigma_{12}^{(\ell)}| \text{$\ell$ even} \}$. Thus, the rational Cherednik algebra depends on $d+1$ complex parameters: $\kappa_1, \kappa_2, \ldots, \kappa_{d-1}$ corresponding to $C_1$, $\kappa_{00}^{\rm{odd}}$ corresponding to $C_{odd}$ and $\kappa_{00}^{\rm{even}}$ corresponding to $C_{even}$. We choose these parameters so that $\kappa_{00}^{\rm{odd}} = \kappa_{00}^{\rm{even}}=: \kappa_{00}$, so our parameters become $(\kappa_{00}, \kappa_1, \ldots,
\kappa_{d-1})$. We must now verify that the rest of the proof of Theorem \[mainthm\] still works.
Using [@BMR Table 1, Table 2], we see that in the definition of the Hecke algebra $\mathcal{H}$ of Section \[hecke\], we get the same braid relations and the same relation for $T_s$ as in Section \[hecke\] for the $p=2$ case, but the relations for the $T_i$ are: $$\begin{aligned}
(T_{t_2^{'}}-1)(T_{t_2^{'}}+ e^{2 \pi i \kappa_{00}^{\rm even}})
&=0 \\
(T_{t_2} -1)(T_{t_2}+e^{2 \pi i \kappa_{00}^{\rm odd}}) &=0\end{aligned}$$ Thus, when $\kappa_{00}^{\rm odd}=\kappa_{00}^{\rm even}$, we get the Hecke algebra of [@GJ 2.A]. Furthermore, in [@CE Section 4.1], it is assumed that $\kappa_{00}^{\rm{odd}}=\kappa_{00}^{\rm{even}}$, so the constructions of [@CE] are still valid. So the proofs of Sections \[hecke\] and \[hsimples\] go through in the present case. Hence, Theorem \[infdim\] holds.
The only potential obstacle to completing the proof is the shift isomorphism of Theorem \[shift\]. But, under the shift isomorphism $e H_{\kappa^{'}} e \rightarrow e_{\varepsilon}H_\kappa e_{\varepsilon}$ from [@BC], the parameters $\kappa_{00}^{\rm{odd}}$ and $\kappa_{00}^{\rm{even}}$ are both shifted by $1$, so we can regard $\kappa_{00}$ as also being shifted by 1. The rest of the proof of Theorem \[mainthm\] now goes through.
|
---
abstract: |
Starting from the hypothesis of scaling solutions, the general exact form of the scalar field potential is found. In the case of two fluids, it turns out to be a negative power of hyperbolic sine. In the case of three fluids the analytic form is not found, but is obtained by quadratures.
PACS: 98.80.Cq, 98.80.Hw, 04.20.Jb
address: |
$^{1}$ Dept. of Phys. Sciences - Univ. Federico II and INFN Sez. di Napoli\
Complesso Universitario di Monte S. Angelo,\
via Cintia, Ed. G, I-80126 Napoli - Italy\
$^{2}$ DAMTP, Centre for Mathematical Sciences, Cambridge University,\
WilberforceRd., Cambridge CB3 0WA, U.K.
author:
- 'Claudio Rubano$^{1}$ and John D. Barrow$^{2}$'
title: Scaling Solutions and Reconstruction of Scalar Field Potentials
---
\[theorem\][Acknowledgement]{} \[theorem\][Algorithm]{} \[theorem\][Axiom]{} \[theorem\][Claim]{} \[theorem\][Conclusion]{} \[theorem\][Condition]{} \[theorem\][Conjecture]{} \[theorem\][Corollary]{} \[theorem\][Criterion]{} \[theorem\][Definition]{} \[theorem\][Example]{} \[theorem\][Exercise]{} \[theorem\][Lemma]{} \[theorem\][Notation]{} \[theorem\][Problem]{} \[theorem\][Proposition]{} \[theorem\][Remark]{} \[theorem\][Solution]{} \[theorem\][Summary]{}
Introduction {#introduction .unnumbered}
------------
In cosmological theories containing scalar fields with a self-interaction potential $V(\varphi),$ it is sometimes possible to reconstruct the required scalar field potential for a simple cosmological solution. In the context of inflationary theory, this approach was used by various authors [@ellis], [@barr1], [@mang], [@deritis], [@BP],[@jantzen1] [@jantzen2] [@jantzen3] [@capozziello1], [@capozziello2], and was primarily interested in the behaviour of solutions containing only scalar fields undergoing inflation. The more recent invocation of a scalar field as a dark matter source responsible for accelerating the universe today, under the pseudonym of “quintessence”, [@ostriker], [@caldwell], [@zlatev], [@steinhardt], [@barr2], is mathematically almost identical, but places different requirements on the solutions. In particular, it is of interest to find solutions which contain both perfect fluids and scalar fields. In this new context, interesting applications of the reconstruction approach were made by Chiba and Nakamura [@chiba] and Saini et al. [@saini].
In this paper, we seek exact cosmological solutions for a universe containing a perfect fluid and a scalar field. We start from the assumption that the energy density of the scalar field scales as an exact power of the scale factor: $\rho_{\varphi}=K\,a^{-n}$ , which is equivalent to imposing an equation of state linking the pressure and density, of the form $p_{\varphi }=w\rho_{\varphi}$ with constant $w=n/3-1$. For a flat Friedmann universe, it is then possible to find an explicit exact form of the potential in terms of $n$, $H_{0}$ , and $\Omega_{m0}$. The form of the resulting general solution has instructive features which will be discussed below.
Derivation of the Potential {#derivation-of-the-potential .unnumbered}
---------------------------
Consider a cosmological fluid with two non-interacting components: perfect-fluid matter and a scalar field $\varphi$ with potential $V(\varphi)$. In the flat universe case, we have the equations $$\begin{gathered}
3H^{2}={\cal G(\rho}_{m}+{\cal \rho}_{\varphi}) \\
\ddot{\varphi}+3H\dot{\varphi}+V^{\prime}(\varphi)=0 \\
{\cal \rho}_{\varphi}=\frac{1}{2}\dot{\varphi}^{2}+V(\varphi) \\
{\cal \rho}_{m}=Da^{-m}\ ,\end{gathered}$$ where $H=\dot{a}/a$, where $a(t)$ is the expansion scale factor is the Hubble expansion rate, and overdot denotes differentiation with respect to the comoving proper time $t$; ${\cal G}=8\pi G/c^{2}$ and $V^{\prime
}=dV/d\varphi$. The constant $m$ depends on the type of perfect fluid present.
We normalize the present value of the scale factor to $a_{0}=1$, without loss of generality, and for brevity denote the present matter-density parameter, $\Omega_{m0},$ by $\Omega_{0}$, and define $$D=3H_{0}^{2}\Omega_{0}/{\cal G\ }.$$
If we make the assumption that $${\cal \rho}_{\varphi}=Ka^{-n},$$ with $n<m$, so the scalar field can dominate at late times, and define $$K=3H_{0}^{2}(1-\Omega_{0})/{\cal G\ },$$ then from Eqs. (2), (3) and (6) we obtain $$\dot{\varphi}^{2}=\frac{Kn}{3}a^{-n}.$$ Since $$\frac{d\varphi}{dt}=Ha\frac{d\varphi}{da}\ ,$$ we have $$\left( \frac{d\varphi}{da}\right) ^{2}=\frac{K}{3H_{0}^{2}}\frac{\sqrt{n}}{%
\Omega_{0}a^{n-m+2}+(1-\Omega_{0})a^{2}}\ ,$$ which gives $$\begin{gathered}
\varphi(a)=\int\sqrt{\frac{1-\Omega_{0}}{{\cal G}}}\frac{\sqrt{n}\,da}{\sqrt{%
\Omega_{0}a^{n-m+2}+(1-\Omega_{0})a^{2}}} \\
=\frac{2\sqrt{n}}{\sqrt{{\cal G}}(m-n)}{\rm arc}\sinh\left( \sqrt {\frac{%
1-\Omega_{0}}{\Omega_{0}}}a^{\frac{m-n}{2}}\right) +\varphi_{in}\ .\end{gathered}$$
Returning to the potential, we get from Eqs. (3) and (9) $$V=Ka^{-n}-\frac{1}{2}\dot{\varphi}^{2}=\frac{3H_{0}^{2}}{{\cal G}}\left(
1-\Omega_{0}\right) \left( 1-\frac{n}{6}\right) a^{-n}\ .$$
The scale factor can be easily eliminated, giving eventually $$\begin{gathered}
V(\varphi)=\frac{3H_{0}^{2}}{{\cal G}}\left( 1-\Omega_{0}\right) \left( 1-%
\frac{n}{6}\right) \left( \frac{1-\Omega_{0}}{\Omega_{0}}\right) ^{\frac{n}{%
m-n}} \\
{\times}\left( \sinh\left( \sqrt{{\cal G}}\frac{m-n}{\sqrt{n}}%
(\varphi-\varphi_{in})\right) \right) ^{-\frac{2n}{m-n}}. \label{pot}\end{gathered}$$
This expression, in the case of dust ($m=3$), coincides with the one presented by [@urena] (which is in turn a particular case of the treatment of [@chimento])as well as with the one by [@sahni]. A general discussion about exact solutions for Friedmann equations, which includes ours as a particular case, can be found also in [@jantzen2], [@jantzen3], [@jantzen1]. In [@urena] and [@chimento], one can find explicit solutions for $a(t)$ and $\varphi(t)$, as well as an extensive discussion. We remark only that this form of potential is good for a tracker solution [@zlatev], [@steinhardt]. Indeed, straightforward computation of the function $\Gamma = V^{\prime\prime}V/(V^{\prime})^2 $, introduced and discussed in these papers, gives $$\Gamma = 1+ \frac{m-n}{2 n} \left ( {\rm sech\/} \left ( \frac{m-n}{\sqrt n}
\varphi \right ) \right)^2 > 1$$ as required by the tracking condition.
Our derivation differs from [@urena], [@chimento] and [@sahni] because it is simpler and is generalised to include all perfect fluid equations of state (other than the $n=6$ case which would correspond to a pure scalar field with no potential). Moreover, it proves that this form of $V(\varphi)$ is the [*unique*]{} solution, if condition (6) is imposed.
Discussion {#discussion .unnumbered}
----------
The first interesting feature of Eq. (\[pot\]) is that the slope (as well as the amplitude) of $V(\varphi)$ depends on $n$. This means that it is impossible to obtain a scaling solution, with the same potential slope, when passing from a radiation-dominated $(m=4)$ epoch to a matter-dominated $(m=3)
$ epoch. Even if we assume that $n$ changes in such a way that the slope remains constant, the coefficient $(1-n/6)$ changes. Moreover, the effective equation of state of the scalar field also changes and there is no physical mechanism for it to be influenced in this way by the dominating type of matter. Although this situation seems to be unphysical, it is merely an artifact of having sought a solution containing a single perfect fluid. The full solution must be found by including dust, radiation and scalar field from the outset, not by joining the radiation + scalar solution to the dust + scalar solution. If this is done for the case of dust + radiation + scalar field, with the same arguments as before, it is easy to derive $$\begin{aligned}
\varphi & =\int\sqrt{\frac{\Omega_{\varphi}}{{\cal G}}}\frac{\sqrt{n}\,da}{%
\sqrt{\Omega_{r}a^{n-2}+\Omega_{d}a^{n-1}+\Omega_{\varphi}a^{2}}}%
+\varphi_{in} \\
V & =\frac{3H_{0}^{2}}{{\cal G}}\Omega_{\varphi}\left( 1-\frac{n}{6}\right)
a^{-n}\ ,\end{aligned}$$ where $\Omega_{r}$, $\Omega_{d}$ and $\Omega_{\varphi}$ are the present values of the radiation, dust and scalar field density parameters, respectively, so $\Omega_{r}+\Omega_{d}+\Omega_{\varphi}=1$. These equations give a parametric representation of $V(\varphi)$, which cannot be solved analytically with simple functions, but is otherwise perfectly well defined and interpolates between the dust and radiation solutions of Eq. (\[pot\]). In our universe today, $\Omega_{r}<<\Omega_{d}$, but at early times the radiation term $\Omega_{r}a^{n-2}$ dominates the dust and cannot be dropped.
A numerical example illustrates the situation: let us set $n=1$, $%
\Omega_{\varphi}=0.7$, $\Omega_{r}=0.0001$, $\Omega_{d}=0.3-\Omega_{r}$, $%
3H_{0}^{2}=1$, ${\cal G}=1$. Fig. 1 shows that on the first part the “true” potential fits well with Eq. (\[pot\]) and $m=4$, while Fig. 2 shows that in the late regime the fit should be done with $m=3$.
The simple and attractive form of Eq. (\[pot\]) is lost, although it might be recovered by a suitable choice of the exponent: via some weighted mean with of $\Omega_{r}$ and $\Omega_{d}$. But there is no way of doing this other than a fit of the numerical values. Moreover, there is no reason why the exponent should depend on the particular values of $\Omega_{r}$ and $%
\Omega_{d}$. Also, since in this case the scaling feature of the solution is only approximate, the tracker behavior could be affected.
Simple numerical evaluations (with the parameters within the allowed range) show that the “true” potential is very well approximated by $$V=\frac{\alpha}{\varphi^{\beta}}-\gamma\ ,$$ with $\alpha$, $\beta$, $\gamma$ [*positive*]{} constants depending on the parameters; in particular, $\beta$ $\approx$ $2n/(m-n)$. The negative additive term, $\gamma$, is an artifact of the approximation over a finite range of $\varphi$. This result, already found by [@urena] for the dust case, shows that the inverse-power potential is effectively equivalent to Eq. (\[pot\]), so that it is possible to apply to this situation all the known results about tracker solutions. On this point, it is interesting to note that this approximation is very good over the whole range of $a$ from zero to the present-day value ($a_{0}=1$ according to our normalization). The asymptotic exponential behavior of the hyperbolic potential is therefore important only in the very far future and does not affect the dominance of scalar field now or its behaviour in the recent past.
As a final remark, we should stress that our arguments are based on the arbitrary assumption of Eq. (6). We have shown elsewhere [@rubano] that other forms of exponential potential are perfectly able to reproduce observational data, but of course in these cases $w$ is not constant. It is interesting also to note that, in one of the cases treated in that paper, $w$ is almost perfectly constant, and yet the form of the potential is substantially different from that of Eq. (\[pot\]).
G.F.R. Ellis and M.S. Madsen, [*Class. Quantum Grav.* ]{}[**8**]{}, 667 (1991)
J.D. Barrow, [*Phys. Rev*]{}[*.* ]{}D [**48**]{}, 1585 (1993)
G. Mangano, G. Miele, C. Stornaiolo, Mod. Phys. Lett. A10 1977 (1995)
R. de Ritis et al., [*Il Nuovo Cimento*]{} [**109**]{}, 403 (1994)
J.D. Barrow and P. Parsons, [* Phys. Rev.*]{} [**D 52**]{}, 5576 (1995)
R.T. Jantzen and C. Uggla [*Gen. Rel. Grav.*]{} [**24**]{}, 59 (1991)
R.T. Jantzen, C. Uggla and K. Rosquist, [*Gen. Rel. Grav.*]{} [**24**]{}, 409 (1992)
C. Uggla, R.T. Jantzen and K. Rosquist, [*Phys. Rev.*]{}, [**D 51**]{}, 5522 (1995)
S. Capozziello et al., [*Il Nuovo Cimento*]{} [**111**]{}, 623 (1996)
S. Capozziello et al., In “[*Proc. XI It. Conf. on Gen. Rel. and Grav. Phys.*]{}”, Trieste (Italy), Sept. 1994, p. 291, World Scientific, Singapore,1996
J.P. Ostriker and P.J. Steinhardt, [*Nature*]{} [**377**]{}, 600 (1995)
R.R. Caldwell, R. Dave, and P.J. Steinhardt, [*Phys. Rev. Lett.*]{} [**80**]{}, 1582 (1998)
I. Zlatev, L. Wang, and P.J. Steinhardt, [*Phys. Rev. Lett.*]{} [**82**]{}, 896 (1999)
P. J. Steinhardt, L. Wang, and I. Zlatev, [*Phys. Rev.* ]{}[**D 59**]{}, 123504 (1999)
J.D. Barrow, R. Bean & J. Magueijo, [*Mon. Not. Roy. Astron. Soc*]{}. [**316**]{}, L41 (2000)
T. Chiba and T. Nakamura,[* Phys. Rev.* ]{}[**D 62**]{}, 121301 (2000)
T.D. Saini et al., [*Phys. Rev. Lett.*]{} [**85**]{}, 1162 (2000)
L.A. Ureña-López and T. Matos, [*Phys. Rev.* ]{}[**D 62**]{}, 081302 (2000); T. Matos and L.A. Ureña-López, Class.Quant.Grav. 17 (2000) L75-L81
L.P. Chimento and A.S. Jakubi, [*Int. J. Mod. Phys.*]{} [**D 5**]{}, 71 (1995)
V. Sahni and A. Starobinsky, [*Int. J. Mod. Phys.*]{} [**D 9**]{}, 373 (2000)
C. Rubano and P. Scudellaro, astro-ph/0103335
[**Figure Captions**]{}
Figure 1: Early-time regime: the dashed curve is the n = 1 scalar field potential with dust (m = 3); the solid curve is the n = 1 scalar field potential with radiation (m = 4) and lies closer to the points plotting the full numerical solution for dust plus radiation and n = 1 scalar field potential.
Figure 2: Late-time regime: the dashed curve is the n = 1 scalar field potential with radiation (m = 4); the solid curve is the n = 1 scalar field potential with dust (m = 3) and lies closer to the points plotting the full solution for dust plus radiation and n = 1 scalar field potential.
|
---
author:
- 'Maurizio Salaris, Thomas de Boer, Eline Tolstoy, Giuliana Fiorentino'
- Santi Cassisi
title: The Horizontal Branch of the Sculptor Dwarf galaxy
---
Introduction
============
The Horizontal Branch (HB) is a luminous feature seen in the Colour-Magnitude Diagrams (CMDs) of all galaxies and globular clusters, corresponding to the central He-burning phase of low mass stars. Its shape – in a given bandpass combination – varies significantly, with an obvious dependence upon the age and metallicity range of the stellar populations of the system. However, several detailed investigations of the HB morphology in globular clusters have since long demonstrated that age and metallicity alone do not allow a unique interpretation of the HB [see, e.g., @dotterHB for a recent study, and references therein].
The most critical problem is that it is not possible to predict from first principles the mass loss during the Red Giant Branch (RGB) phase, which determines the location along the HB for a star with a fixed initial mass and chemical composition. Other factors, such as the initial helium abundance, also appear to play a crucial role in determining the form of the observed HB in Galactic globulars [see, e.g., @HBsp; @danto; @ema and references therein]. The role played by He appears to be related to the recently emerged new picture of the globular cluster origin and evolution, whereby each cluster hosts first generation (FG) stars with [*normal*]{} He (mass fraction roughly equal to the cosmological He abundance) and $\alpha$-enhanced metal mixtures, and second generation objects with C-N, O-Na (and sometimes Mg-Al) anticorrelations and enhanced He [see, e.g. @gratton and references therein for a review].
So far both these critical parameters – RGB mass loss and initial He of individual stars – have proven hard to measure [see, i.e., @origlia; @villanova; @villanova_b; @ema_b for some determinations on a small sample of globular clusters], and thus the HB is usually carefully avoided in the interpretation of either the age or the metallicity properties of a resolved stellar population.
This is particularly unfortunate for the study of the ancient ($>$10 Gyr old) stellar populations in nearby galaxies. These populations are the fossil record of the early stages of galaxy evolution, and provide crucial information about the epoch of formation of cosmic structures. The standard age diagnostics for these stars are to be found at faint magnitudes in a CMD, compressed into the oldest main sequence (MS) turn off region. This region of the CMD, apart from being small (in terms of magnitude and colour extension) and thus very sensitive to photometric errors, may also contain overlapping younger populations. Also, the age-metallicity degeneracy, although much better behaved than on the RGB, may still create problems in determining a unique solution for the ages and metallicities of the ancient stars.
Stellar evolution suggests that, at least in principle, the detailed properties of stars older than $\sim$10 Gyr could be recovered by analyzing the much more luminous and extended HB, if only its morphology could be sorted out in terms useful for understanding the relation to the star formation history and metallicity evolution of the galaxy.
The purpose of this work is to see whether it is possible to accurately model the resolved HB of the Sculptor dwarf spheroidal galaxy taking into account its past star formation history (SFH – Here we denote with SFH the star formation rate as a function of age and metal content). Sculptor has an exceptionally detailed SFH determination – derived without considering the HB – obtained by combining CMD analysis with detailed spectroscopic metallicities along the RGB [@deboer12]. We also use the well measured RR Lyrae properties and the optical CMD of non variable objects, as constraints to determine the best synthetic HB populations that match the observed CMD, using BaSTI stellar models @basti [@basti2]. The only free parameter in our analysis is the total amount of mass lost along the RGB, given that age and metal abundance distributions are fixed by the SFH. To the best of our knowledge this is the first time that a complete and detailed modelling of the HB of a resolved galaxy has been performed; our investigation is therefore a first step to assess the potential of synthetic HB modelling to add further constraints on a galaxy SFH, an issue particularly important when studying galaxies so distant that only RGB and HB stars can be resolved.
Any mismatch between the observed and predicted magnitude distribution along the HB may be a signature of the presence of enhanced-He populations – the counterpart of second generation stars in Galactic globulars – not included in the SFH determinations. This will provide an important additional piece of information regarding the comparison of photometric and chemical properties of dwarf galaxies and globular clusters. To date, comparisons of chemical abundance patterns between Galactic globulars and dwarf galaxies reveal that these latter lack the abundance anticorrelations (e.g. O-Na) typical of second generation stars in individual globular clusters [see, i.e., @gei], that are associated to varying degrees of He enhancement [see, i.e., @pasquini]. By [*enhanced-He*]{} populations we mean stars born with initial He mass fractions (Y) larger than the cosmological He abundance Y=0.246 [see @coc for a recent reevaluation]. The BaSTI models employ initial Y values that scale with the metal mass fraction Z as dY/dZ$\sim$1.4, derived by considering a cosmological Y$=$0.245 and the initial solar Z and Y as derived from a calibration of the standard solar model [see, e.g. @basti]. In the metallicity regime of Sculptor the resulting Y values are practically constant – ranging between 0.245 and 0.248 – and essentially equal to the cosmological He. It is important to notice that the determination of the SFH – see next section – employed the same dY/dZ scaling, hence approximately the cosmological He, across the whole Z range covered by the galaxy population.
The RGB mass loss law derived from the HB modelling – eventually a function of initial metallicity and/or age – will also provide baseline values to be compared with analogous determinations on globular clusters, and to be tested on the HB of other dwarf galaxies. As a byproduct of our investigation we will compare in the appendix the observed difference between RGB bump and HB magnitudes, with the theoretical prediction based on the galaxy SFH, and verify whether a discrepancy exists, as found in Galactic globulars [see, e.g., @dicecco].
The paper is structured as follows. Section 2 describes briefly the data and the SFH used in this investigation, whilst the next two sections present our synthetic HB analysis and results, followed by our conclusions.
Data
====
The Sculptor dwarf galaxy is well known to contain a predominately old stellar population [see, e.g., @dacosta], so the contamination of populations $<$10 Gyr old along the MS and overlapping the HB is minimal. Early work on the Sculptor HB by @bumpSc was mainly qualitative and made use of simple ZAHB fitting, to assess the galaxy metallicity distribution. It was later confirmed that the red and blue HBs in Sculptor exhibit differences in spatial distribution that correspond to differences in ages and metallicities within the oldest population [@Tolstoy04; @deboer11]. These previous works simply highlighted that the HB is complex, and age and metallicity spreads play a role in creating the complexity.
A healthy HB extended in colour from the red to the blue indicates that a significant number of RR Lyrae variable stars must populate the instability strip, as is observed. The most complete survey to date comes from @kal:95, where 226 RR Lyrae stars were identified and their light curves classified. Their properties are consistent with a spread in metallicities, and \[Fe/H\]$< -1.7$. This was confirmed by @clementini low resolution spectroscopy of 107 variables, which showed the metallicity to peak at \[Fe/H\]$\sim -1.8$, with a range covering $-2.40 <$ \[Fe/H\] $< -0.8$.
For the accurate modelling of the resolved HB of the Sculptor dwarf spheroidal, we make use of a deep optical B,V CMD presented in @deboer11. In the V-(B-V) CMD the HB is the most horizontal, hence most sensitive to potential He variations. The observed HB extends to (B-V)$\sim$0 in the blue, and even at these colours a change of Y (at fixed Z) affects both the luminosity of the zero age HB (ZAHB) and the luminosity of the end of the HB phase, thus enabling to test the presence of enhanced-He stars also along the bluest part of the galaxy HB.
The photometry was obtained using the CTIO 4-m MOSAIC II camera, and carefully calibrated using observations of Landolt standard fields [@Landolt07; @Landolt92]. This resulted in an accurate photometric catalogue, covering a region ranging from the Sculptor centre out to an elliptical radius r$_{ell}$$\le$1 deg. The HB of Sculptor can be seen not have an extended blue tail in the optical, and the hottest ${\rm T_{eff}}$ is lower than the observed limit ($\sim$ 12000 K) for the onset of radiative levitation.
The synthetic HB calculations also make use of the detailed SFH of Sculptor determined by @deboer12. This SFH is obtained using the same optical photometry presented in @deboer11, combined with detailed spectroscopic metallicities along the RGB, not using the observed HB stars. The distance modulus ${\rm (m-M)_V}$=19.72 and reddening E(B-V)=0.018 used in the SFH determination are also employed in the HB modelling. This SFH determination uses the DSEP stellar evolution models [@dsep] – that employ the same value as BaSTI for the cosmological He, and a very similar dY/dZ$\sim$1.5, that produces essentially the same initial Y for Sculptor metallicity range – and there is good agreement between the Sculptor SFH determined with DSEP or BaSTI models, as also shown by [@deboer12].
The SFH is provided for 5 annuli, extending from the centre out to a radius ${\rm r_{ell}}$=1 deg. The region of interest for our simulation comprises the innermost two annuli, with ${\rm r_{ell}} \le$0.183 deg, that include the survey of RR Lyrae variable stars by @kal:95. The adopted star formation rate within ${\rm r_{ell}} \le$0.183 deg is displayed in Figs. 2 and 13 of @deboer12; it is provided in terms of solar masses per year in 1 Gyr age bins, between 5 and 14 Gyr. Broadly speaking, the star formation rate peaks in the oldest age bin, and then declines slowly with time; it reaches half the peak value between 9 and 10 Gyr ago, and decreases down to zero between 5 and 6 Gyr ago. The low star formation rate means that the age bins between 5 and 8 Gyr have a negligible impact on the resulting synthetic HB.
For each age bin the SFH is further subdivided into 8 \[Fe/H\] bins 0.2 dex wide, covering the range between \[Fe/H\]=$-$2.5 and $-$0.9. The mean value of \[Fe/H\] increases slowly with the mean age of the population. A range of \[$\alpha$/Fe\] values (0.2 or 0.3 dex wide) is associated to each \[Fe/H\] bins [see Fig. 2 of @deboer12]. The values of \[$\alpha$/Fe\] decrease slowly with increasing \[Fe/H\] and are generally positive, but for the more metal rich bins, where \[$\alpha$/Fe\] also reaches negative values down to $-$0.2 dex for the higher \[Fe/H\] bin.
The properties of RR Lyrae stars are taken into account using @kal:95 survey in the centre of Sculptor. The survey employed many short expsoure images covering a wide range in time to identify the RR Lyrae stars in the central region of Sculptor and determine the light curve profiles. Their sample can be considered complete in terms of photometric and temporal sampling of the variable objects on the HB. Intensity-weighted magnitudes are provided in the V band, averaged over the pulsation period of each pulsator. Furthermore, to prevent contamination of non-variable HB features by the presence of variable stars at random phase, we have identified and removed the confirmed RR Lyrae variable stars from @deboer11 data – being taken at random phase, these objects were scattered to the blue, red and within the instability strip – based on spatial position. Due to the square field of view of the @kal:95 fields, compared to the elliptical region considered here, the area covered by the SFH is $\sim$30% larger than the area for which RR Lyrae measurements are available. Therefore, we will apply a 30% reduction of the number of synthetic stars in the instability strip, in comparisons with the observed RR Lyrae populations, to account for the different size of the area covered by @kal:95 observations. This also implies that there could be a relatively low number of undetected RR Lyrae variables taken at random phase (about 70 at most) scattered along and around the HB, that however do not affect appreciably the number distributions used for our analysis. As suggested by the referee, we have performed the full analysis described in the next sections also for the objects with ${\rm r_{ell}} \le$0.116 deg, e.g., within the innermost region of @deboer12 SFH determinations. This reduces by a factor $\approx$2 the observed number of HB objects used in the analysis. In this case @kal:95 field covers the whole region, implying that the RR Lyrae sample is essentially complete and also all variables can be removed from @deboer11 photometry. The results we obtain are completely consistent with what found for the case of ${\rm r_{ell}} \le$0.183 deg, that will be detailed in the next sections.
Methods
=======
We describe in this section the methods employed in our theoretical modelling of Sculptor HB.
Synthetic CMD generation
------------------------
In a complex old stellar population, the morphology of the HB will depend on both the chosen RGB mass loss and the input SFH. For this reason we have completely rewritten the BaSTI synthetic CMD generator [SYNTHETIC MAN – @basti] to account for both a generic SFH and an arbitrary RGB total mass loss. The code now produces a full synthetic CMD that includes also – for old populations – a HB of arbitrary morphology.
Briefly, the code reads first the full grid of models – tracks from the MS to the tip of the RGB and HB tracks – from the BaSTI database for varying Z and a scaled solar metal distribution [@basti], and the SFH of the population – in our case the SFH by @deboer12. For each age and \[Fe/H\] bin the synthetic CMD generator first draws randomly a stellar mass $m_i$, within the range 0.1 and 120 ${\rm M_\odot}$ [for consistency with @deboer12], according to a @kroupa initial mass function. A value of the age $t_i$ and ${\rm [Fe/H]_i}$ are then drawn randomly according to uniform probability distributions [in agreement with the derivation of the SFH by @deboer12], within the ranges associated to the selected age and \[Fe/H\] bin. In addition, a value of ${\rm [\alpha/Fe]_i}$ is also drawn randomly (uniform probability distribution) within the range prescribed by the SFH for the selected age and \[Fe/H\] bin.
If the model corresponding to $m_i$ and the selected metal composition has a lifetime at the first thermal pulse or at carbon ignition – the relevant age range for Sculptor SFH is fully covered by the BaSTI models – shorter than $t_i$, the star won’t appear in the synthetic CMD, and a new set of $m_i$, ${\rm [Fe/H]_i}$, ${\rm [\alpha/Fe]_i}$ and $t_i$ is drawn. We remark that for a given ${\rm [Fe/H]_i}$ and ${\rm [\alpha/Fe]_i}$ we consider scaled solar models with total metallicity ${\rm [M/H]\sim[Fe/H]+log(0.638 \ 10^{[\alpha/Fe]}+0.362)}$, that closely mimic models with the same \[M/H\] and varying \[$\alpha$/Fe\] in the low mass, low metallicity regime covered by the stars currently evolving in this galaxy [@scs:93].
If the lifetime of the model with mass $m_i$ is larger than $t_i$, and $t_i$ is smaller than the model age at He-ignition, interpolation in age and Z amongst the BaSTI tracks will determine the CMD location of this synthetic star. If $t_i$ is larger than the lifetime at He-ignition, but smaller than the lifetime at the first thermal pulse, the code subtracts from $m_i$ a specified value of the total mass loss along the RGB $\Delta M_{RGB}$, to provide a value for $m_{i, HB}$, e.g., the HB mass of the synthetic star with initial mass $m_i$. Interpolations in mass, Z and age amongst the BaSTI HB tracks provide the CMD location of the synthetic star.
Finally, the synthetic magnitudes determined with the described procedures, are perturbed by a mean Gaussian photometric error to approximate the observational error of @deboer11 data. For the magnitude range of the HB stars, the typical 1$\sigma$ photometric uncertainty of non-variable stars is $\sim$0.007 mag in B and V.
The values of $m_i$ are added, until the total mass of stars formed in each SFH bin satisfies the derived star formation rates. We have often arbitrarily multiplied the star formation rates in each bin by constant factors, to have a larger population of HB stars and minimize the Poisson noise in the simulations.
It is clear from this brief sketch of the synthetic CMD calculation, that the only free parameter entering our analysis is $\Delta M_{RGB}$.
![Synthetic (top) vs observed (bottom) CMD for the region in Sculptor dwarf spheoridal within ${\rm r_{ell} < 0.183}$ deg. The synthetic CMD has a number of HB stars close to the observed number (see text for details of the simulation). The three boxes (red, intermediate and blue) mark the three areas where star counts are compared with the simulations. The empty (blue) circles with dot represent the RR Lyrae stars found in the simulations. The observed RR Lyrae stars (as they are taken at random phase) are excluded from the bottom panel CMD. There are some non-pulsating stars in both synthetic and observed samples, located between the [*red*]{} and [*intermediate*]{} blue boxes (see text for details).[]{data-label="CMD"}](figpapCMD1.eps){width="\columnwidth"}
Modelling RR Lyrae stars
------------------------
For the RR Lyrae stars observed by @kal:95 only the mean V magnitudes of individual pulsators are available. Therefore, we had to employ theoretical determinations of the instability strip (IS) colour boundaries in our synthetic CMDs, together with the pulsational equation [@dcr:04] that provides the fundamental period (P) of a synthetic object as a function of its mass, metallicity, bolometric luminosity and effective temperature. First overtone periods (${\rm P_{FO}}$) are related to the fundamental ones by the relation P=${\rm P_{FO}}$+0.13 [@dcr:04]. The boundaries of the fundamental (F) and first overtone (FO) strips were taken from @dcr:04, with small adjustments – within the quoted theoretical uncertainties – to account for the empirical constraints given by the period distribution determined by @kal:95, that poses strong constraints on the width of the IS. As an example, if the theoretical red boundary is too red, the synthetic objects reach too long periods. We have therefore [*adjusted*]{} the boundaries of the F and FO regions in order to match as well as possible the period range covered by the observed F and FO pulsators. When using the [*standard*]{} IS boundaries provided by @dcr:04 from calculations with mixing length [*ml*]{}=1.5${\rm H_p}$, the synthetic objects reached too long periods. This constraint forced us to consider a larger value of [*ml*]{} in the pulsational results, and that shifts the red edge of the FF region to hotter temperatures, and lowers the upper boundary of the F periods. Thus we have employed the derivatives provided by @dcr:04, to determine the IS edges at varying [*ml*]{}. However, an increased [*ml*]{} tends to move the blue edge of the FO region to the red, narrowing down too much the portion of the IS populated by FO pulsators. We therefore further adjusted the F and FO boundaries at fixed [*ml*]{}, within their nominal uncertainties of the order of $\pm$50-100 K [@mm:03].
The final V magnitudes of the synthetic stars that lie within the IS will be compared to the RR Lyrae observations (their mean V magnitudes) by @kal:95 paper. The photometric uncertainty on the V magnitude of the observed variables is greater then the photometric uncertainty of @deboer11 data, due to the shorter exposure times employed by @kal:95. In the comparison with the RR Lyrae V distribution, to match the observational conditions of these data, the V magnitudes of the synthetic stars that lie within the IS have been perturbed by a Gaussian photometric error with $\sigma$=0.05 mag, as determined from Fig. 1 of @kal:95.
Comparing models and observations
---------------------------------
The bottom panel of Fig. \[CMD\] displays the observed HB, divided into three regions, delimited by rectangular boxes, that we denote as [*red*]{}, [*intermediate*]{} and [*blue*]{}. These boxes contain, respectively, 457, 194, and 401 stars. The bluer boundary of the [*red*]{} box and the red boundary of the [*intermediate*]{} box have been drawn at the approximate colours of the boundaries of the instability strips in Galactic globular cluster CMDs. The region between the [*red*]{} and [*intermediate*]{} box contains mainly RR Lyrae variables, but also a few non variable stars displayed in the CMD[^1]. We considered a synthetic HB model for a given $\Delta M_{RGB}$ choice to be satisfactory match to the observations when:
1. [the relative number of stars in the three boxes and within the region containing the RR Lyrae IS is reproduced within the Poisson uncertainty;]{}
2. [the observed mean V-magnitude of the non variable stars in each of the three boxes is matched within 0.01 mag;]{}
3. [the total (B-V) extension of the observed HB is well reproduced.]{}
When these conditions were satisfied, we also checked that there was general agreement with the overall shapes of the histograms of observed star counts as a function of both V and (B-V) for the non variable stars, and with the mean level of the RR Lyrae brightness (averaged over a pulsational cycle).
Two points must be mentioned regarding these additional constraints.
First, we could in principle have enforced the constraint of perfect statistical agreement between the theoretical and observed star counts as a function of both V magnitude and colour for the non variable objects. However, a perfect fit depends on a precise knowledge of the functional dependence of $\Delta M_{RGB}$ (and eventually the initial Y) on one or more stellar parameters. Given the current lack of solid theoretical and empirical guidance, this dependence may be extremely complicated, discontinuous or involving additional physical processes not included in stellar models. We have therefore used simple parametrizations that, as we will see in the next section, produce synthetic CMDs that are able to satisfy our three main criteria listed above. In this way we put strong constraints on $\Delta M_{RGB}$ and Y.
Regarding the RR Lyrae mean magnitudes, they were determined from independent observational data. The calibrated V magnitudes have a maximum zero point systematic uncertainty of $\sim$0.03 mag, according to @kal:95, whilst this uncertainty is much smaller for the non variable star photometry [@deboer11]. Given a possible small zero-point mismatch between the two photometries, we have simply checked that, when the main criteria were satisfied, the mean level of the synthetic RR Lyrae sample matched the observed one within the uncertainty on the relative zero-point of the two photometries. As a final, albeit weaker consistency check, we also compared the predicted and observed period distributions, and the predicted RR Lyrae \[Fe/H\] distribution, with the spectroscopic observations of @clementini.
Results
=======
We have calculated several synthetic CMDs, each one with typically much larger numbers of stars than observed – by scaling appropriately the SFH – to minimize the Poisson error on the synthetic star counts. For each test we fixed $\Delta M_{RGB}$, and checked the agreement between synthetic and observed HB morphology. As a zero order approximation, we tried with a standard Gaussian $\Delta M_{RGB}$ and mean value (and 1$\sigma$ spread) independent of \[M/H\] and age. This mass loss prescription did not match the observations, for any choices of the mean value and $\sigma$ spread around the mean. Also the use of a uniform probability with varying mean values and $\sigma$ spread, both independent of \[M/H\] and age did not help. The analysis of these failed attempts demonstrated that $\Delta M_{RGB}$ must be made dependent on \[M/H\], and also that only a small dispersion around these metallicity dependent mean values is consistent with the observations.
![\[Fe/H\] as a function of (B-V) for the best fit synthetic HB sample. Dots (filled circles) denote stars with age equal or larger (smaller) than 10 Gyr. (see text for details).[]{data-label="fehBV"}](figpapHBfeht.eps){width="\columnwidth"}
The simulations that matched the observed HB according to the criteria described in the previous section had a Gaussian distribution of $\Delta M_{RGB}$ with mean values:
1. [$<\Delta M_{RGB}>$=0.10${\rm M_{\odot}}$, for \[M/H\] $< -$1.8]{}
2. [$<\Delta M_{RGB}>$=0.14${\rm M_{\odot}}$ , for $-1.8\le$ \[M/H\]$\le -$1.6]{}
3. [$<\Delta M_{RGB}>$=0.14 - 0.15${\rm M_{\odot}}$ , for $-1.6 <$ \[M/H\]$\le -$1.4]{}
4. [$<\Delta M_{RGB}>$=0.14 - 0.16${\rm M_{\odot}}$ , for $-1.4 <$ \[M/H\]$\le -$1.3]{}
5. [$<\Delta M_{RGB}>$=0.16${\rm M_{\odot}}$, for \[M/H\]$> -$1.3]{}
and a very small dispersions, also metallicity independent, $\sigma$=0.005 $M_{\odot}$. We determined $<\Delta M_{RGB}>$ taking into account the error bars on the best-fit star formation rates – as provided by @deboer12 – hence the range of values for $-1.8\le$ \[M/H\]$\le -$1.3. In the other metallicity ranges the uncertainty on the star formation rates causes $<\Delta M_{RGB}>$ variations below 0.01${\rm M_{\odot}}$. All discussions and figures that follow display results obtained from simulations with the best-fit value of the star formation rate.
In place of step functions for $<\Delta M_{RGB}>$ vs \[M/H\] we tried also linear or quadratic analytical expressions as a function of only \[M/H\] or both \[M/H\] and age, but the match to the observations got generally worse.
Although the Reimers law [@reimers] – still widely employed to calculate RGB mass loss rates – predicts slightly increasing values of $<\Delta M_{RGB}>$ at increasing metallicity, when the free parameter $\eta$ that enters Reimers formula is kept fixed, the variations necessary to model Sculptor HB are larger. To this purpose, using the BaSTI models, we checked how our derived $<\Delta M_{RGB}>$ values can be transposed into values of the parameter $\eta$. For \[M/H\] $< -$1.8 $<\Delta M_{RGB}>$ is slightly higher than what predicted by $\eta$=0.2 (that provides integrated RGB mass loss values between 0.06 and 0.07$M_{\odot}$, depending on the age), whilst for \[M/H\]$> -$1.8 $<\Delta M_{RGB}>$ is close to what is predicted by $\eta$=0.4.
With this mass loss calibration the mean V magnitudes in the [*red*]{}, [*intermediate*]{} and [*blue*]{} boxes are equal to, respectively, 20.20, 20.25 and 20.33, identical within 0.01 mag to the observed values. The mean brightness of the RR Lyrae IS is within 0.03 mag (the simulation being fainter) of the observed value, equal to 20.14 mag. The top panel of Fig. \[CMD\] displays one realization of the HB, without scaling the galaxy SFH. The total absolute number of HB objects and the absolute number counts (not just the relative ones) in the three boxes and the region of the IS turn out to be equal – within the Poisson errors on the star counts – to the observations (again, considering the scaling factor for the limited area of the observed RR Lyrae population). This is an extremely important independent check of the derived star formation rates.
Figure \[fehBV\] displays \[Fe/H\] (more directly linked to observations than \[M/H\]) as a function of (B-V), for the HB stars in the simulation of Fig. \[CMD\]. There is a general trend of decreasing \[Fe/H\] with decreasing colour, in agreement with the early analysis by @bumpSc that made use of ZAHB fitting to the observed HB. However, the \[Fe/H\] dispersion at a given (B-V) is very large. If we split the synthetic sample into objects older than 10 Gyr and younger ones, we obtain ${\rm [Fe/H]=1.46 \ (B-V) - 2.29 }$ for the old sample, and a much steeper dependence ${\rm [Fe/H]=2.80 \ (B-V) - 3.10 }$ for the young objects, as is also clear from Fig. \[fehBV\]. The dispersion of \[Fe/H\] around these mean relationships is equal to 0.31 dex for the old sample, and 0.18 dex for the young one.
Figures \[Vhist\] and \[BVhist\] compare observed and synthetic star counts as a function of V and (B-V), respectively. Here we used simulations with a much larger number of objects than observed, and rescaled the star counts appropriately, to compare with observations.
![Observed (solid black line and filled circles) vs synthetic star counts (red line) as a function of the V magnitude in the (from left to right) [*blue*]{}, [*intermediate*]{} and [*red*]{} boxes, respectively (bin size of 0.05 mag for stars in the [*red*]{} box and 0.03 mag for the other two samples). Poisson errors on the observed star counts are also displayed.[]{data-label="Vhist"}](figpapV.eps){width="\columnwidth"}
![As Fig. \[Vhist\], but for the (B-V) colour (bin size of 0.015 mag).[]{data-label="BVhist"}](figpapBV.eps){width="\columnwidth"}
The overall shape of the theoretical histograms reproduces very closely the observed one. The comparison is worse for the star counts as a function of colour in the [*red*]{} box. It is remarkable that the magnitude distributions, very sensitive to the initial chemical composition, are well matched by the simulations, not just the mean values of V in the three boxes. There is no significant excess of stars brighter than what is predicted by the simulations, hence there is no clear evidence of He-rich stars.
![As Fig. \[Vhist\], but for the RR Lyrae population (bin size of 0.04 mag).[]{data-label="RRV"}](figpapRR.eps){width="\columnwidth"}
We now consider comparisons with the RR Lyrae IS, for which not only mean V magnitudes, but also \[Fe/H\] estimates from @clementini are available, to provide an additional consistency check of the HB simulations. These authors determined \[Fe/H\] for about half of the variables observed by @kal:95, using a revised version of the ${\rm \Delta S}$ method [@ds], calibrated on both the @zw:84 and @cg:97 (hereafter ZW and CG, respectively) globular cluster \[Fe/H\] scales. Typical errors on individual \[Fe/H\] estimates are equal to $\sim$0.15 dex.
Figure \[RRV\] compares observed and synthetic RR Lyrae star counts as a function of V. The overall shapes of the histograms are very similar, with just a slight offset of the synthetic sample towards fainter magnitudes, as indicated by the mean brightness, fainter by 0.03 mag. This offset is equal to the estimated maximum zero point error on the calibration of the observations [@kal:95].
Period [data from @kal:95] and \[Fe/H\] distributions for the sample of variables are displayed in Fig. \[RRPFEH\]. The theoretical star counts have been rescaled to account for the smaller number of objects with \[Fe/H\] determinations, compared to the photometric sample of RR Lyare stars. The fit to the period distribution is not perfect even after tuning – within the constraints imposed by pulsational models – the boundaries of the IS to match as well as possible the range covered by the observations. Only an [*ad hoc*]{} tuning of the F and FO boundaries would provide a better match. As noticed by @RRproc, the IS boundaries would be probably better determined with a lower [*ml*]{} for FO pulsators compared, to F ones. This would certainly improve the fit of the P histogram, that displays a substantial discrepancy for the ratio of FO to F pulsators, such that it is too low in the simulation. In this respect one also has to consider that we could not take into account the objects that are in the so called OR zone of the IS, where stars pulsate FO or F depending on where they evolved from (FO or F region). This adds an additional uncertainty when trying to match precisely the observed period distribution.
![Comparison with observations (displayed as histograms and filled circles with error bars) of star counts in the IS, as a function of P (bin size of 0.025 days) and \[Fe/H\] (bin size of 0.15 dex). Results for both the @zw:84 and [@cg:97] \[Fe/H\] scales are displayed in the middle and top panel, respectively.[]{data-label="RRPFEH"}](figpapRR2.eps){width="\columnwidth"}
Regarding the metallicity distribution, we display the results for both the ZW and CG \[Fe/H\] scales. On the whole, the synthetic sample matches the observed values on the ZW scale fairly well. The average \[Fe/H\] of the synthetic sample is ${\rm <[Fe/H]>}=-$1.88, very close to the observed one on the ZW scale, equal to ${\rm <[Fe/H]_{ZW}>}=-$1.84. For comparison, the observed mean value on the CG scale is ${\rm <[Fe/H]_{CG}>}=-$1.65
All these results are based on CMD simulations of the central regions of the galaxy, covered by the RR Lyrae photometry. It is impossible to constrain completely the synthetic HB models without information about the variable star population within the IS. We can however check whether our mass loss calibration based on the inner regions, is able to match the mean magnitudes and observed number of HB stars in the [*blue*]{}, [*intermediate*]{} and [*red*]{} boxes of Fig. \[CMD\], when the whole observed area from the centre out to an elliptical radius ${\rm r_{ell}}=$1 deg is considered[^2]. To this purpose we calculated synthetic CMDs as described, considering the appropriate SFH for the whole observed area, using the same integrated RGB mass loss prescription of our best fit to the HB of the inner region. As a result, we can match within 0.01 mag the observed mean V magnitudes of stars in the three boxes, and the empirical star counts (within the associated Poisson error) in these three regions of the CMD.
Conclusions
===========
We have performed the first detailed simulation of the HB of a resolved dwarf galaxy, taking consistently into account the SFH determined from MS and RGB photometric and spectroscopic observations, and using synthetic HB techniques usually applied to study the HB in globular clusters. The number of HB stars predicted by our simulations is consistent with observations, within the Poisson error on the star counts. The colour and V-magnitude distribution of all non-variable HB stars in Sculptor is matched well by the synthetic model, for simple choices of the integrated RGB mass loss. This latter needs to be metallicity dependent – with a stronger dependence than predicted by the Reimers law at fixed $\eta$ – to satisfy the observational constraints, and also must have a very small dispersion at fixed metallicity. The magnitude, metallicity (on the ZW scale) and period distribution of the RR Lyrae stars are also satisfactorily reproduced, when taking into account the current uncertainties on the IS boundaries.
There is no indication of enhanced-He subpopulations along the HB from the V-magnitude distribution of the non-variable and variable stars – within the uncertainty on the relative photometric zero-point. The metallicity range covered by the SFH, as constrained by spectroscopy of RGB stars, plus a simple RGB mass loss law, enable to cover both the full magnitude and colour range of HB stars. There is no excess of bright objects to be matched with enhanced-He populations. The agreement of the synthetic model with observations dictates that any enhancement of He – if present – has to be lower than $\Delta$Y=0.01. The good agreement of the synthetic sample with the \[Fe/H\] distribution of the RR Lyrae stars (on the ZW scale) lend additional support to the results of our simulations.
The lack of signatures of enhanced-He stars along the HB is consistent with the lack of the O-Na anticorrelation observed in Sculptor and other dwarf galaxies, and confirms the intrinsic difference between Local Group dwarf galaxies and globular cluster populations.
Regarding the RGB integrated mass loss $<\Delta M_{RGB}>$, our simulations suggest a very simple prescription for the case of Sculptor, with $<\Delta M_{RGB}>$ slowly increasing with increasing \[M/H\]. We also find an extremely small spread around these mean values. It is natural to try and compare $<\Delta M_{RGB}>$ determined from our analysis, with similar estimates in globular clusters to assess whether, at least in case of FG stars, the RGB mass loss is approximately the same as in this galaxy. We compare here with the results by @ema [@ema_b], who used synthetic HB models (using BaSTI models) to determine initial Y and mass distribution of HB stars in a few globular clusters. For NGC 2808, a cluster with \[Fe/H\]$\sim -$1.2 (\[M/H\]$\sim -$1.0, when considering a standard value \[$\alpha$/Fe\]=0.3-0.4), $<\Delta M_{RGB}>$=0.15${\rm M_{\odot}}$ for FG stars, that compares well with the value obtained for Sculptor at this metallicity. In case of M3 and M13, both with \[Fe/H\]$\sim -$1.6, the HB analysis provides $<\Delta M_{RGB}>$=0.12${\rm M_{\odot}}$ and 0.21${\rm M_{\odot}}$, respectively. Whereas the value for M3 is reasonably close to what we derive for Sculptor at the appropriate metallicity ($<\Delta M_{RGB}>\sim$0.14${\rm M_{\odot}}$), the result for M13 is obviously very discrepant. On the other hand, even when $<\Delta M_{RGB}>$ values are similar to Sculptor, the dispersion around these mean values is usually larger in the globular clusters. Before drawing any strong conclusions about the similarity (or lack) of the integrated RGB mass loss in globular cluster FG stars and dwarf galaxies, it is clearly necessary to extend analyses like ours and @ema [@ema_b] to much larger samples of objects.
In summary, our results show that in case of Sculptor a simple mass loss law is able to explain the observed detailed HB morphology. The next obvious step is to verify whether synthetic HB models with the same mass loss law can reproduce the HB of other resolved dwarf galaxies, with well established and diverse SFHs. If this is the case, the combined analysis of the RGB (strongly affected by the initial metallicity) and the HB in distant galaxies where only these phases can be resolved, will be able to provide constraints on the age and metallicity distributions of their oldest populations.
Finally, as a byproduct of our simulations, we have been able to compare the observed magnitude of the RGB bump with theory. As detailed in the appendix, we find a discrepancy between observed and theoretical RGB bump, consistent with results for Galactic globulars, that point towards a too bright bump in stellar models, at least for intermediate metallicity and metal poor clusters. This is at odds with recent results by @monelli10 for Sculptor, that were however based on older photometric data. We are able to explain the difference with our results for this galaxy, but the analysis of the RGB bump brightness in dwarf galaxies clearly deserves further detailed investigations.
We thank the anonymous referee for several comments that improved the presentation of our results. MS wishes to dedicate this paper to the memory of his father, who sadly passed away a few months ago. He also thanks the Kapteyn Astronomical Institute for their hospitality and support of a visit, during which an important point of this analysis was clarified. SC is grateful for financial support from PRIN-INAF 2011 “Multiple Populations in Globular Clusters: their role in the Galaxy assembly” (PI: E. Carretta), and from PRIN MIUR 2010-2011, project , prot. 2010LY5N2T (PI: F. Matteucci).
The RGB bump of Sculptor
========================
The RGB bump is produced when the advancing H-burning shell encounters the H-abundance discontinuity left over by the outer convection at its maximum depth reached during the first dredge-up. The consequent sudden increase of the H-abundance in the shell alters the efficiency of the H-burning shell, and causes a temporary drop in the surface luminosity. After the shell has crossed the H-abundance discontinuity, the luminosity starts to increase again. As a consequence, a low-mass RGB star crosses the same luminosity interval three times, and a bump (a local maximum) appears in the star counts per magnitude bin [see, i.e., @iben; @salarisbook].
The analysis of the photometry by @bumpSc disclosed the presence of two distinct RGB bumps; a bright blue bump at V$\sim$19.3-19.4, and a faint red bump one at V$\sim$20.0-20.1. Figure \[RGBlf\] displays our differential and cumulative RGB luminosity functions for Sculptor RGB and AGB stars, with ${\rm r_{ell}}<$0.183 deg. The break in the slope of the cumulative luminosity function displayed by dashed lines, points to the location of the RGB bump [@fp:90]. This is strongly corroborated by the CMD of Fig. \[RGB\], that displays along the RGB a tilted strip with a strong concentration of stars, approximately centred around the magnitude of the break in the slope of the luminosity function. We thus find a single, very extended – in both V and (B-V) – and continuous bump region whose ridge line is marked in Fig. \[RGB\]. The magnitude range agrees approximately with the brightness of the fainter bump claimed by @bumpSc. The bump gets fainter towards red colours as expected, given that more metal rich populations are redder along the RGB and have a fainter bump (at constant age). The spread of the bump over a large range of magnitudes makes the feature somewhat not well defined in the differential luminosity function.
We do not find any signature of an additional bump at V$\sim$19.3-19.4, as it is quite clear by just examining the CMD, although we find a discontinuity in the slope of the cumulative luminosity function at V$\sim$19.4 – approximately the magnitude of the bright bump claimed by @bumpSc – that we are going to discuss later.
Figure \[RGB\] also displays the ridge line of the bump region from our simulations calculated with the galaxy SFH. There is a clear offset of $\sim$0.35 mag in V, where the theoretical is brighter. Notice that there is no freedom to shift the simulated CMD to fainter magnitudes, because it would destroy the agreement of the synthetic HB with observations. Using the simulated CMD as a guide, the colour range of the observed bump corresponds to \[Fe/H\] values ranging between $\sim-$1.5 and $\sim -$2.0. This disagreement is consistent with the discrepancy found in a sample of Galactic globular clusters, in approximately the same metallicity interval, by @dicecco [@bumpGC][^3].
![Differential (lower panel) and cumulative (upper panel) RGB luminosity functions for Sculptor RGB and AGB stars, with ${\rm r_{ell}}<$0.183 deg. The break in the slope of the cumulative luminosity function (dashed lines) points to the location of RGB bump, whose average magnitude is marked in both panels by vertical dotted lines. The thick solid mark denotes a discontinuity in the slope of the cumulative luminosity function, the we ascribe to the appearance of the AGB clump (see text for details).[]{data-label="RGBlf"}](figpapLFbump.eps){width="\columnwidth"}
![Sculptor CMD for RGB stars around the bump (for ${\rm r_{ell}}<$0.183 deg). The solid line marks the ridge line of the observed bump region; the dashed line denotes the ridge line of the bump for the synthetic sample of RGB stars.[]{data-label="RGB"}](figpapRGBbump.eps){width="\columnwidth"}
A previous analysis by @monelli10 compared the difference between the V magnitude of the RGB bump and the HB at the level of the RR Lyrae instability strip (taken at the ZAHB) ${\rm \Delta V_{HB}^{bump}}$ observed in Sculptor, with results from the BaSTI models. They used data from the literature that were available at that time, and considered two values of ${\rm \Delta V_{HB}^{bump}}$, associated to the two bumps found by @bumpSc. As mentioned before, we do not find any trace of the bright bump in our adopted photometry (and there is no double RGB bump in the synthetic CMDs either), and we speculate that this may correspond to the asymptotic giant branch (AGB) clump, that marks the first ignition of the He burning shell around the inert CO core. Essentially AGB stars begin to contribute more substantially to the population of red giants from the level of the clump, when the evolutionary speed tends to (relatively) slow down.
Figure \[AGB\] displays a synthetic CMD for the population with ${\rm r_{ell}}$=0.183 deg – and approximately the same number of stars as observed – compared to the observations. The dotted horizontal line marks the magnitude of the discontinuity of the RGB+AGB cumulative luminosity function, brighter than the RGB bump, disclosed by Fig. \[RGBlf\]. The AGB clump in the synthetic diagram appears in the region (B-V)$\sim$0.70-0.75, on the blue side of the RGB, and V$\sim$19.1-19.4. [^4] The dotted line essentially marks the lower envelope of AGB clump stars and agrees with the magnitude of the brighter and bluer bump claimed by @bumpSc.
Let’s consider the case of the fainter bump, roughly consistent with the bump region found in our data, that gives ${\rm \Delta V_{HB}^{bump}}$=$-$0.35$\pm$0.21 This was derived from @bumpSc data, by considering the ZAHB level of the red HB – that agrees with our CMD – and the faint bump, and an average \[M/H\]$-$1.30$\pm$0.15 from @kir:09. With these values and associated error bars, the observed ${\rm \Delta V_{HB}^{bump}}$ appears broadly consistent with theoretical models, for ages between 10 and 14 Gyr [see Fig.5 of @monelli10]. Our analysis suggests instead that the theoretical ${\rm \Delta V_{HB}^{bump}}$ is smaller than observations (the theoretical bump is too bright with respect to the HB level, taken as reference).
![Synthetic CMD for stars around the bump and AGB clump (left panel – SFH for ${\rm r_{ell}}<$0.183 deg), compared with observations (right panel). The ridge lines of both observed and synthetic RGB bumps are also marked (solid and dashed lines, respectively). The AGB clump is the feature on the blue side of the RGB, in the range V$\sim$19.1-19.4 and (B-V)$\sim$0.70-0.75. The dotted line marks the level of the discontinuity of the slope of the RGB+AGB cumulative luminosity function (see text for details).[]{data-label="AGB"}](figpapAGBbump.eps){width="\columnwidth"}
The difference with our result may be traced back to a combination of factors. First of all, \[Fe/H\] for the stars that define the bump region, turns out to lie in the range ${\rm -2.0 < [Fe/H] < -1.5}$. The \[$\alpha$/Fe\] values from the SFH, combined with these \[Fe/H\], give a range of \[M/H\] values lower than the \[M/H\]$-$1.30$\pm$0.15 assumed by @monelli10, and hence a brighter theoretical bump. In second instance, our HB simulations show that for the red HB (stars within the [*red*]{} box in our analysis) the ZAHB level, taken as lower envelope of the observed stellar distribution, is determined by the more metal rich population, at variance with the metallicity range that dominates the bump region. Both these factors go in the direction of reducing the theoretical ${\rm \Delta V_{HB}^{bump}}$ to be used against the observed value, compared to the theoretical values in @monelli10 study.
[40]{} natexlab\#1[\#1]{}
, E. & [Gratton]{}, R. G. 1997, , 121, 95
, S., [Mar[í]{}n-Franch]{}, A., [Salaris]{}, M., [et al.]{} 2011, , 527, A59
, G., [Ripepi]{}, V., [Bragaglia]{}, A., [et al.]{} 2005, , 363, 734
, A., [Uzan]{}, J.-P., & [Vangioni]{}, E. 2013, ArXiv e-prints
, G. S. 1984, , 285, 483
, E., [Salaris]{}, M., [Ferraro]{}, F. R., [et al.]{} 2011, , 410, 694
, E., [Salaris]{}, M., [Ferraro]{}, F. R., [Mucciarelli]{}, A., & [Cassisi]{}, S. 2013, , 430, 459
, F., [Caloi]{}, V., [Montalb[á]{}n]{}, J., [Ventura]{}, P., & [Gratton]{}, R. 2002, , 395, 69
, T. J. L., [Tolstoy]{}, E., [Hill]{}, V., [et al.]{} 2012, , 539, A103
, T. J. L., [Tolstoy]{}, E., [Saha]{}, A., [et al.]{} 2011, , 528, A119
, A., [Bono]{}, G., [Stetson]{}, P. B., [et al.]{} 2010, , 712, 527
, M., [Marconi]{}, M., & [Caputo]{}, F. 2004, , 75, 190
, M., [Marconi]{}, M., & [Caputo]{}, F. 2004, , 612, 1092
, A., [Chaboyer]{}, B., [Jevremovi[ć]{}]{}, D., [et al.]{} 2008, , 178, 89
, A., [Sarajedini]{}, A., [Anderson]{}, J., [et al.]{} 2010, , 708, 698
, F., [Ferraro]{}, F. R., [Crocker]{}, D. A., [Rood]{}, R. T., & [Buonanno]{}, R. 1990, , 238, 95
, D., [Wallerstein]{}, G., [Smith]{}, V. V., & [Casetti-Dinescu]{}, D. I. 2007, , 119, 939
, R. G., [Carretta]{}, E., & [Bragaglia]{}, A. 2012, , 20, 50
, Jr., I. 1968, , 154, 581
, J., [Kubiak]{}, M., [Szymanski]{}, M., [et al.]{} 1995, , 112, 407
, E. N., [Guhathakurta]{}, P., [Bolte]{}, M., [Sneden]{}, C., & [Geha]{}, M. C. 2009, , 705, 328
, P. 2001, , 322, 231
, A. U. 1992, , 104, 340
, A. U. 2007, , 133, 2502
, S. R., [Siegel]{}, M. H., [Patterson]{}, R. J., & [Rood]{}, R. T. 1999, , 520, L33
, M., [Caputo]{}, F., [Di Criscienzo]{}, M., & [Castellani]{}, M. 2003, , 596, 299
, M., [Cassisi]{}, S., [Bernard]{}, E. J., [et al.]{} 2010, , 718, 707
, L., [Ferraro]{}, F. R., [Fusi Pecci]{}, F., & [Rood]{}, R. T. 2002, , 571, 458
, L., [Mauas]{}, P., [K[ä]{}ufl]{}, H. U., & [Cacciari]{}, C. 2011, , 531, A35
, A., [Cassisi]{}, S., [Salaris]{}, M., & [Castelli]{}, F. 2004, , 612, 168
, A., [Cassisi]{}, S., [Salaris]{}, M., & [Castelli]{}, F. 2006, , 642, 797
, G. W. 1959, , 130, 507
, D. 1975, Memoires of the Societe Royale des Sciences de Liege, 8, 369
, R. T. 1973, , 184, 815
, M. & [Cassisi]{}, S. 2005, [Evolution of Stars and Stellar Populations]{}, Wiley
, M., [Chieffi]{}, A., & [Straniero]{}, O. 1993, , 414, 580
, E., [Irwin]{}, M. J., [Helmi]{}, A., [et al.]{} 2004, , 617, L119
, S., [Geisler]{}, D., [Piotto]{}, G., & [Gratton]{}, R. G. 2012, , 748, 62
, S., [Piotto]{}, G., & [Gratton]{}, R. G. 2009, , 499, 755
, R. & [West]{}, M. J. 1984, , 55, 45
[^1]: In case of our simulations for the innermost region with ${\rm r_{ell}} \le$0.116 deg, that is fully covered by @kal:95 RR Lyrae observations, we find also some stars in the CMD region between [*red*]{} and [*intermediate*]{} boxes. This strenghtens the case for the small number of objects with ${\rm r_{ell}} \le$0.183 deg lying in the same CMD region, to be mostly non-variable stars, rather then – another viable possibility – RR Lyrae objects taken at random phase and not covered by @kal:95 field of view
[^2]: We cannot eliminate from the observed CMD the RR Lyrae stars taken at random phase, located at ${\rm r_{ell}}>$0.183 deg, because we cannot identify them
[^3]: Synthetic CMDs considering the appropriate SFH for the whole observed area within ${\rm r_{ell}}=$1 deg, confirm that the theoretical RGB bump is systematically brighter than the observed one, and the position of both bumps is substantially the same as for the case of ${\rm r_{ell}}<$0.183 deg
[^4]: The clump appears slightly better defined, at the position predicted by the simulations, in the observed CMD of the whole region with ${\rm r_{ell}}<$1 deg.
|
---
abstract: 'In this note we present three representations of a 248-dimensional Lie algebra, namely the algebra of Lie point symmetries admitted by a system of five trivial ordinary differential equations each of order forty-four, that admitted by a system of seven trivial ordinary differential equations each of order twenty-eight and that admitted by one trivial ordinary differential equation of order two hundred and forty-four.'
author:
- 'M.C. Nucci and P.G.L. Leach[^1]'
date: 'Dipartimento di Matematica e Informatica, Università di Perugia, 06123 Perugia, Italy'
title: Much ado about 248
---
Introduction
============
A system of $n$ ordinary differential equations each of order $M>1$, $$u_k^{(M)}=f_k(u_j^{(s)},t),\;\;\; j,k=1,n,\;\;\; s=0,M-1, \label{1.1}$$ has a variable number of Lie point symmetries depending upon the structure of the functions $f_k$. The maximal dimension $D$ of the algebra of admitted Lie point symmetries can be obtained by the formulæ [@Gonsalez-Gascon; @84] $$\begin{aligned}
M=2 &\Longrightarrow& D=n^2+4n+3 \label{1.2} \\
M>2 &\Longrightarrow & D=n^2+M n+3. \label{1.3}\end{aligned}$$ Some explicit numbers are given in Table \[tab 1\].
$_n{\rm \backslash}^{{\mbox{\tiny M}}}$ 2 3 4 5 6 7 8 9 10
----------------------------------------- ----- ----- ----- ----- ----- ----- ----- ----- -----
1 8 7 8 9 10 11 12 13 14
2 15 13 15 17 19 21 23 25 27
3 24 21 24 27 30 33 36 39 42
4 35 31 35 39 43 47 51 55 59
5 48 43 48 53 58 63 68 73 78
6 63 57 63 69 75 81 87 93 99
7 80 73 80 87 94 101 108 115 122
8 99 91 99 107 115 123 131 139 147
9 120 111 120 129 138 147 156 165 174
10 143 133 143 153 163 173 183 193 203
: \[tab 1\]: The maximal dimension of the algebra of admitted Lie point symmetries for systems of equations of varying order (horizontal) and number (vertical).
Recently the elaboration of the elements of the Lie algebra, $E8$, of order 248 has been variously announced [@BBC; @Derspiegel; @Lemonde; @TimesLon; @TimesNY] in the serious popular media. The authoritative source is the Atlas of Lie Groups and Representations [@atlas] which is funded by the National Science Foundation through the American Institute of Mathematics [@institute]. The results of the E8 computation were announced in a talk at MIT by David Vogan on Monday, March 19, 2007, and the details may be found at [@technical]. The Atlas of Lie Groups and Representations is a project to make available information about representations of semisimple Lie groups over real and $p$-adic fields. Of particular importance is the problem of the unitary dual, [*ie*]{} the classification of all of the irreducible unitary representations of a given Lie group. The goal of the Atlas of Lie Groups and Representations is to classify the unitary dual of a real Lie group, $G$, by computer. A step in this direction is to compute the admissible representations of $G$ including their Kazhdan-Lusztig-Vogan polynomials. The computation for $E8$ was an important test of the technology. While the computation is an impressive achievement, it is only a small step towards the unitary dual and should not be ranked as important as the original work of Kazhdan, Lusztig, Vogan, Beilinson, Bernstein [*et al*]{}. (See for example [@Beilinson; @83; @a; @Beilinson; @81; @a; @Bernstein; @86; @a; @Kazhdan; @79; @a; @Kazhdan; @80; @a; @Lusztig; @83; @a; @Vogan; @83; @a; @Gelfand; @82; @a].) Nevertheless the result was regarded as being suitable for a concerted campaign of publicity to heighten awareness of Mathematics in the community at large:\
“Symmetrie ist möglicherweise das erfolgreichste Prinzip der Physik überhaupt" [@Derspiegel].\
“Un groupe de chercheurs américains et européens, parmi lesquels on trouve deux Français, est parvenu à décoder une des structures les plus vastes de l’histoire des mathématiques" [@Lemonde].\
“It may be that some day this calculation can help physicists to understand the universe" [@TimesLon].\
“Eighteen mathematicians spent four years and 77 hours of supercomputer computation to describe this structure" [@TimesNY].\
In this note we demonstrate three representations of a Lie algebra of dimension 248. The two of us spent four hours and 77 seconds of pocket-calculator computation to describe these three structures.
Three simple systems
====================
For $D=248$ formula (\[1.2\]) does not have integral solutions and so there is no system of second-order ordinary differential equations of maximal symmetry possessing a 248-dimensional algebra of its Lie point symmetries[^2]. About formula (\[1.3\]) the factors of 248-3=245 are 1, 5 and 7 (49 is out of question because $49^2>245$). Consequently possible values of $n$ are 1, 5 and 7. The corresponding values of $M$ are 244, 44 and 28, respectively. The systems of maximal symmetry are easily obtained as one simply puts $f_k = 0$ $\forall$ $k$. Thus the systems we construct are the simplest representations of the equivalence class under point transformation of systems of equations of maximal symmetry.
Firstly we consider the following system: $$u_k^{(44)}=0,\quad \quad k=1,5.
\label{sys1}$$ It is easy to show that this simple system admits a 248-dimensional algebra of its Lie point symmetries since $5^2+5\cdot
44+3=248$. The algebra is generated by the operators $$\begin{array}{lcl}
\Gamma_1&=&t^2\partial_t+43 t\sum_{i=1}^{5} u_i\partial_{u_i},\\
[0.2cm] \Gamma_2&=&t\partial_t, \\ [0.2cm]\Gamma_3&=&\partial_t,\\
[0.2cm] \Gamma_{i,k}&=&u_{k}\partial_{u_i},\quad k=1,5,\;
i=1,5\\ [0.2cm] \Gamma_{i+5,s}&=&t^s\partial_{u_i},\quad
s=0,43,\; i=1,5.
\end{array}$$ Secondly we consider the system $$u_r^{(28)}=0,\quad \quad r=1,7.
\label{sys2}$$ This equally simple system admits a 248-dimensional algebra ($7^2+7\cdot
28+3=248$) of its Lie point symmetries generated by $$\begin{array}{lcl}
\Gamma_1&=&t^2\partial_t+27 t\sum_{j=1}^{7} u_j\partial_{u_j},\\
[0.2cm] \Gamma_2&=&t\partial_t, \\
[0.2cm]\Gamma_3&=&\partial_t,\\
[0.2cm] \Gamma_{j,r}&=&u_{r}\partial_{u_j},\quad r=1,7,\;
j=1,7\\
[0.2cm] \Gamma_{j+7,n}&=&t^n\partial_{u_j},\quad
n=0,27,\; j=1,7.
\end{array}$$ Thirdly and finally the scalar equation, $$u^{(244)}=0,
\label{sys3}$$ admits a 248-dimensional Lie algebra ($1^2+1\cdot
244+3=248$) of its point symmetries generated by the operators $$\begin{array}{lcl}
\Gamma_1&=&t^2\partial_t+243 t u\partial_{u},\\
[0.2cm] \Gamma_2&=&t\partial_t, \\ [0.2cm]\Gamma_3&=&\partial_t,\\
[0.2cm] \Gamma_4&=&u\partial_{u},\\
[0.2cm] \Gamma_{n+5}&=&t^n\partial_{u},\quad n=0,243.
\end{array}$$
Conclusion
==========
We have demonstrated three representations of Lie algebras of dimension 248 which is the dimension of $E8$. Although the algebras we present are not simple, their method of construction is. The reason for this simplicity is that we used representations for systems of equations of maximal symmetry. We do not deny that larger systems, be that in order or number, of less than maximal symmetry could possibly have an algebra of dimension 248, but even on the assumption that such systems be linear the complexity of the calculation becomes immense [@Gorringe; @88; @a] and defeats the purpose of the present note.
Note that we have used the simplest forms for the generators of the algebras of the three systems, (\[sys1\]), (\[sys2\]) and (\[sys3\]), for our primary interest is the demonstration of the existence of the algebras. Normally one would use combinations which reflect subalgebraic structures. For example in the case of (\[sys3\]) for which the algebra is obviously $sl(250,I\!\!R)$ one would replace $\Gamma_2$ with ${\tilde\Gamma}_2 = 2t\partial_t
+ 243u\partial_u$ to underline the subalgebraic structure $\{sl(2,I\!\!R)\oplus A_1\}\oplus_s 244 A_1$, where $\Gamma_1$, ${\tilde\Gamma}_2$ and $\Gamma_3$ constitute a representation of $sl(2,I\!\!R)$, $\Gamma_4$ reflects the homogeneity of the equation in the dependent variable and the 244-element abelian subalgebra is composed of the solution symmetries, so called because the coefficient functions are solutions of (\[sys3\]).
Acknowledgements {#acknowledgements .unnumbered}
================
PGLL thanks the University of Kwazulu-Natal for its continued support.
[99]{}
American Institute of Mathematics.\
http://aimath.org/E8/
Atlas of Lie Groups and Representations.\
http://www.liegroups.org/
BBC Monday, 19 March 2007, 12:28 GMT.\
http://news.bbc.co.uk/2/hi/science/nature/6466129.stm
Beilinson A (1983) Localization of representations of reductive Lie algebras [*Proceedings of the International Congress of Mathematicians, Warsaw*]{} 699-710
Beilinson A & Bernstein J (1981) Localisation de g-modules [*Comptes Rendus de l’Académie des Sciences de Paris Séries I Mathématiques*]{} [**292**]{} 15-18
Bernstein J (1986) On the Kazhdan-Lusztig conjectures [*AMS Summer Research Conference*]{} (University of California, Santa Cruz, July 1986)
Der Spiegel, 19 März 2007.\
http://www.spiegel.de/wissenschaft/mensch/0,1518,472569,00.html
Gelfand S & MacPherson R (1982) Verma modules and Schubert cells: a dictionary in [*Seminaire d’algebre Paul Dubriel et MP Malliavin*]{} (Lecture Notes in Mathematics [**925**]{}, Springer Verlag, Berlin–New York) 1–50
González-Gascón F & González-López A (1983) Symmetries of differential equations IV [*Journal of Mathematical Physics*]{} [**24**]{} 2006-2021
Gorringe VM & Leach PGL (1988) Lie point symmetries for systems of second order linear ordinary differential equations [*Quæstiones Mathematicæ*]{} [**11**]{} 95-117
Kazhdan D & Lusztig G (1979) Representations of Coxeter groups and Hecke algebras [*Inventiones Mathematicæ *]{} [**53**]{} 165–184
Kazhdan D & Lusztig G (1980) Schubert varieties and Poincaré duality in [*Geometry of the Laplace Operator*]{}, (Proceedings of Symposium on Pure Mathematics [**36**]{}, American Mathematical Society) 185–203
LEMONDE.FR avec AFP 19.03.07\
http://www.lemonde.fr/web/article/0,1-0@2-3244,36-884723@51-884724,0.html
Lusztig G & Vogan D (1983) Singularities of closures of K-orbits on flag manifold [*Inventiones Mathematicæ*]{} [**71**]{} 365–370
http://www.liegroups.org/AIM$_{}$E8/technicaldetails.html
NEW YORK TIMES 2007/03/20.\
http://select.nytimes.com/gst/abstract.html?res=F40613FE3C540C738EDDAA0894DF404482
The Times March 19, 2007.\
http://www.timesonline.co.uk/tol/news/uk/science/article1533648.ece
Vogan D (1983) Irreducible characters of semisimple Lie groups III: Proof of the Kazhdan-Lusztig conjecture in the integral case [*Inventiones Mathematicæ*]{} [**71**]{} 381–417
[^1]: permanent address: School of Mathematical Sciences, Westville Campus, University of KwaZulu-Natal, Durban 4000, Republic of South Africa
[^2]: Is this another instance of the intrinsically uniqueness of Classical Mechanics?
|
---
abstract: 'In this talk, I review the status of the calculation of the photon production rate in a hot quark-gluon plasma. Particular emphasis is given to the discussion of the various length scales of the problem.'
address: 'Brookhaven National Laboratory, Nuclear Theory, Bldg 510A, Upton, NY-11973, USA'
author:
- François Gelis
title: 'Photon production by a quark-gluon plasma[^1]'
---
Introduction - Model
====================
Photon or dilepton production by a hot plasma is expected to reflect rather cleanly the state of the system. Indeed, in the collision of two heavy nuclei, the typical size of the system is much smaller than the mean free path of particles that feel only the electromagnetic interactions, which enables photons and leptons to escape freely. However, it has been realized recently that those rates are non-perturbative quantities. In other words, even for an idealized situation where the QCD coupling constant is small, the rate at leading order receives contributions from an infinite set of diagrams.
To keep the model as simple as possible, I assume in this talk that the temperature is much larger than all the quark masses ($T\gg m_{\rm
q}$), that the strong coupling constant at this temperature scale is extremely small ($g\ll 1$), and that the system is in both thermal and chemical equilibrium. Under these conditions, thermal field theory seems to be the tool of choice to calculate the photon production rate, which is obtained by calculating the imaginary part of the retarded photon polarization tensor [@rate]: $${{dN^\gamma}\over{dtdV\,d\omega d^3{\mathbf q}}}\propto
{\rm Im}\,\Pi^\mu_\mu{}_{\rm ret}(\omega,{\mathbf q})\; .$$ In turn, the discontinuity of the retarded self-energy can be obtained directly by using the cutting-rules of the retarded/advanced formalism [@Gelis3]. This formula automatically sums over all the possible processes and interferences thereof at a given order in $g$ (and takes care of the statistical weights).
Lowest order
============
At lowest non trivial order in the loop expansion for ${\rm
Im}\,\Pi^\mu_\mu{}_{\rm ret}$, the relevant processes for the production of hard photons are the ones depicted on the following figure: 1=to $$\raise -11mm\box1$$ Ignoring prefactors, their contribution to ${\rm
Im}\,\Pi^\mu_\mu{}_{\rm ret}$ behave like [@1-loop] $${\rm Im}\,\Pi^\mu_\mu{}_{\rm ret}(\omega,{\mathbf q})\propto e^2 g^2 T^2
\ln\Big({{\omega T}\over{m^2_{\rm th}}}\Big)\; ,$$ where $m_{\rm th}\sim gT$ is a thermal mass of order $gT$. Such a regulator is needed here, because the production rate of [*massless*]{} photons exhibit a logarithmic collinear singularity if evaluated with massless quarks and gluons. The resummation of hard thermal loops [@HTL] naturally provides this thermal mass.
Bremsstrahlung-like processes
=============================
One can note that a process like bremsstrahlung never appears in the lowest order diagram. It shows up only in the next order (i.e. at two-loop in the effective theory resumming the HTLs), together with a process that differs from bremsstrahlung by crossing symmetry: 1=to $$\raise -11mm\box1$$ The contribution of these processes to ${\rm Im}\,\Pi^\mu_\mu{}_{\rm
ret}$ can be written in the following form [@2-loop]: $${\rm Im}\,\Pi^\mu_\mu{}_{\rm ret}(\omega,{\mathbf q})\propto
e^2 g^4 {{T^2}\over{m^2_{\rm th}}}
\Big[{{T^3}\over{\omega}}\oplus T\omega\Big]\; ,$$ where the first term dominates the low energy part of the spectrum, while the second term dominates its high energy part. Naively, one would expect those processes to come with four powers of $g$. However, this is altered by very strong collinear singularities, that bring the factor[^2] $T^2/m^2_{\rm th}\sim g^{-2}$ after regularization by a thermal mass. Therefore, it turns out that these contributions are as large as the 1-loop ones (or even larger for soft photons). The natural question one should ask after observing this collinear enhancement at two loops is whether it happens also in higher order diagrams, and whether the perturbative expansion can be kept under control.
Photon formation time and other scales
======================================
The collinear singularities in photon production diagrams are controlled by the virtuality of the quarks. It is particularly instructive to calculate the virtuality of an off-shell quark of momentum $R\equiv P+Q$ splitting into a photon of momentum $Q$ (and invariant mass $Q^2$) and an on-shell quark of momentum $P$ [@Gelis11]: $$R^2-M^2\approx {\omega\over p}\left[ {\mathbf p}_\perp^2+M^2_{\rm eff}\right]\;{\rm with\ }
M^2_{\rm eff}\equiv M^2 + {{Q^2}\over{\omega^2}}p_0 r_0\; .$$ This virtuality can be used to write down the expression of the more intuitive “photon formation time” $t_F$. Using the uncertainty principle, we have indeed: $$t_F^{-1}\sim \Delta E\sim {{R^2-M^2}\over{2r_0}}\sim {\omega\over{p_0 r_0}}
\left[ {\mathbf p}_\perp^2+M^2_{\rm eff}\right] \sim {{\omega M^2_{\rm eff}}\over{p_0 r_0}}$$ We therefore arrive at an important observation: the collinear enhancement is due to very small quark virtualities, which corresponds to very long photon formation times. One can easily check that the photon formation time increases if the photon becomes soft, or if the invariant mass of the photon becomes very small.
There are a priori three other length scales that may also play a role in this problem:
- The mean free path of the quarks in the plasma $\lambda_{\rm
mfp}\sim (g^2T\ln(1/g))^{-1}$.
- The range of the electric interactions $\lambda_{\rm el}\sim
(gT)^{-1}$.
- The range of magnetic interactions $\lambda_{\rm mag}\sim
(g^2T)^{-1}$.
In the limit where $g\ll 1$, those scales satisfy $\lambda_{\rm el}\ll
\lambda_{\rm mfp} \ll \lambda_{\rm mag}$.
From there, one can check that the condition for having higher order diagrams at the same order in $g$ as the bremsstrahlung is $\lambda_{\rm mfp} \le t_F$, and that the relevant topologies correspond to multiple scatterings of the quark in the medium [@Gelis11; @3-loop]. In other words, if producing the photon lasts more than the typical time between two scatterings of the quarks, then multiple scattering diagrams are also important. This effect has already been studied in a slightly different context, where a very fast fermion is going through some cold medium. In that case, it is known as the “Landau-Pomeranchuk-Migdal” effect. This effect reduces the rate of radiative energy loss in the low energy end of the spectrum. For photon production by a plasma, a preliminary (but incomplete) study indicates that the LPM effect reduces the photon rate both in the low end and in the high end of the spectrum [@3-loop].
Nature of the relevant diagrams
===============================
The last question we have to address is the nature of the multiple scattering topologies that are dominant, and the short answer is that it depends on the range of the interactions.
If the relevant interactions are short ranged, like the Debye-screened electric interactions for which $\lambda_{\rm el}\ll \lambda_{\rm
mfp}$, then on can check that only independent scatterings can occur, as illustrated on the following figure: 1=to $$\raise -11mm\box1$$ In that case, only ladder topologies are important for the calculation of the photon self-energy (together with the appropriate modification of the quark propagators, in order to preserve gauge invariance). Indeed, configurations like the diagram on the right in the previous figure require an interaction range at least comparable to the mean free path.
The situation can however become much more complicated if the interactions are long ranged, like for static magnetic interactions, since for them $\lambda_{\rm mfp}\ll \lambda_{\rm mag}$. In that case, the above argument does not apply to restrict the set of relevant topologies: successive scatterings are not independent from one another and any topology can a priori contribute, unless some unexpected cancellations occur. Those cancellations occur when the process under study can only happen with hard scatterings, which means that the relevant mean free path is not $\sim (g^2 T\ln(1/g))^{-1}$ (this is the average distance between two soft scatterings) but rather $(g^4 T)^{-1}$ (which is the average distance between two hard scatterings). When the relevant mean free path if larger than all the interaction scales, then only ladder diagrams contribute.
Even if no extra cancellation occurs, a naive power counting seems to indicate that sensitivity to the magnetic mass is suppressed by $1/\ln(1/g)$ compared to the sensitivity on the mean free path. Therefore, even in the worst scenario, one could calculate the photon production rate at leading logarithmic accuracy by only summing ladder diagrams. If no cancellation occurs, then terms beyond the leading logarithm are truly non-perturbative.
Conclusions
===========
The problem of collinear singularities in the photon rate by a plasma is closely related to the interplay between the various distance scales in the problem. Higher order topologies become important if the photon formation time is larger that the quark mean free path, and the nature and complexity of those topologies is controlled by the range of the interactions compared to the mean free path. Two questions can probably be addressed analytically: the production of massive hard photons is dominated by a single scattering if the invariant mass of the lepton pair is large enough and one can probably perform the resummation of ladder topologies, which can at least give the leading log rate.
[**Acknowledgments:**]{} My work is supported by DOE under grant DE-AC02-98CH10886.
[10]{}
, Phys. Rev. [**D**]{} [**28**]{}, 2007 ([1983]{}); [C. Gale, J.I. Kapusta]{}, Nucl. Phys. [**B**]{} [**357**]{}, 65 ([1991]{}).
, Nucl. Phys. [**B**]{} [**508**]{}, 483 ([1997]{}).
, Z. Phys. [**C**]{} [**53**]{}, 433 ([1992]{}); [J.I. Kapusta, P. Lichard, D. Seibert]{}, Phys. Rev. [**D**]{} [**44**]{}, 2774 ([1991]{}).
, Phys. Rev. [**D**]{} [**54**]{}, 5274 ([1996]{}); [*ibid.*]{}, Z. Phys. [**C**]{} [**75**]{}, 315 ([1997]{}); [P. Aurenche, F. Gelis, R. Kobes, H. Zaraket]{}, Phys. Rev [**D**]{} [**58**]{}, 085003 ([1998]{}).
, Phys. Rev. [**D**]{} [**60**]{}, 076002 ([1999]{}).
, Phys. Lett. [**B**]{} [**493**]{}, 182 ([2000]{}).
, Nucl. Phys. [**B**]{} [**337**]{}, 569 ([1990]{}).
, Phys. Rev. [**D**]{} [**61**]{}, 116001 ([2000]{}); [*ibid*]{}, [**D**]{} [**62**]{}, 096012 ([2000]{}).
[^1]: Work done in collaboration with P. Aurenche and H. Zaraket.
[^2]: This collinear factor decreases very fast if the photon has a non-zero invariant mass. When the invariant mass is maximal for a given energy, this factor is of order $1$ [@AurenGKZ2].
|
---
author:
- 'Volodymyr Maistrenko [^1]'
- Oleksandr Sudakov
- Yuri Maistrenko
title: Spiral wave chimeras for coupled oscillators with inertia
---
Introduction
============
Spiral wave chimeras are fascinating two-dimensional (2D) patterns first reported by Kuramoto & Shima in [@ks2003; @sk2004]. Their striking name was proposed seven years later in [@mls2010]. Manifesting the regular 2D spiraling, they possess, nevertheless, finite-sized incoherent cores, where the behavior is characterized a bell-shaped average frequency distribution of individual oscillators. In the last decade, spiral wave chimeras have been intensively studied both analytically and numerically [@OWYYS2012; @pa2013; @XKK2015; @L2017; @OWK2017; @TRT2018; @OK2019], and, very recently, they have got an experimental confirmation [@T2019]. In this paper with the use of bulky numerical simulations, we will demonstrate the appearance of spiral wave chimeras for the Kuramoto model with inertia and will study their bifurcations with increasing the coupling strength $\mu$. The transition starts with a standard chimeric pattern with an incoherent bell-shaped core and goes to the situation where the core is coherent. The latter means that all in-core oscillators are rotating with the same average frequency different from those for the spiraling oscillators outside the core. The transition including also intermediate partially synchronized states ends, eventually, in a spatiotemporal chaos. It is illustrated for chimeras with 4 cores in detail. We believe that it is characteristic also of more complex spiral chimera states with a larger number of cores.
Depending on the parameters, all coherent cores of a spiral chimera state can have the same or, on the contrary, different core frequencies. After entering the so-called “solitary region,”’ spiral cores are normally surrounded by solitary oscillators [@JMK2015; @JBLDKM2018] which follow the average core frequency.
Our model is a two-dimensional array of $N{\times}N$ identical phase oscillators of the Kuramoto–Sakaguchi type with inertia, where phases $\varphi_{i,j}$ evolve according to the equation
$$m\ddot{\varphi}_{ij} + \epsilon \dot{\varphi}_{ij} = \frac{\mu}{B_{P}(i,j)} \sum\limits_{(i^{\prime},j^{\prime})\in B_{P}(i,j) }\sin(\varphi_{i^{\prime}j^{\prime}} - \varphi_{ij}- \alpha),$$
where indices $i,j=1, ... , N$ are periodic modulo $N$, $m$ is the mass, $\epsilon$ is the damping coefficient, and $\alpha$ is the phase lag. The coupling is assumed to be non-local and isotropic: each oscillator $\varphi_{ij}$ is coupled with equal strength $\mu$ to all its nearest neighbors $\varphi_{i^{\prime}j^{\prime}}$ within a range $P$, i.e., to the oscillators falling in the 2D circle-like neighborhood
$$B_{P}(i,j):=\{ (i^{\prime},j^{\prime}){:} (i^{\prime}-i)^{2}+(j^{\prime}-j)^{2}\le P^{2}\}.$$ Without loss of generality, we put $m =1$ and $\epsilon=0.1$ in system (1).
Numerical simulations were performed on the basis of the Runge–Kutta solver DOPRI5 on the computer cluster “CHIMERA”, http://nll.biomed.kiev.ua/cluster. The Ukrainian Grid Infrastructure kindly provided us the distributed cluster resources and the parallel software with graphics processing units [@sls2011; @scm2017].
Spiral wave chimeras with 2 and 4 cores
=======================================
![Snapshots of phase distributions $\varphi_{i,j}$ for spiral wave chimera states with incoherent core: (a) - 2-core spiral wave chimera state ($\alpha=0.82, \mu=0.005, \epsilon=0.1, P=14, N=200$), (b) - 4-core spiral wave chimera state ($\alpha=0.77, \mu=0.012, \epsilon=0.1 , P=42, N=600$).[]{data-label="f1"}](Fig1a.png "fig:"){width="0.42\linewidth"} ![Snapshots of phase distributions $\varphi_{i,j}$ for spiral wave chimera states with incoherent core: (a) - 2-core spiral wave chimera state ($\alpha=0.82, \mu=0.005, \epsilon=0.1, P=14, N=200$), (b) - 4-core spiral wave chimera state ($\alpha=0.77, \mu=0.012, \epsilon=0.1 , P=42, N=600$).[]{data-label="f1"}](Fig1b.png "fig:"){width="0.42\linewidth"} ![Snapshots of phase distributions $\varphi_{i,j}$ for spiral wave chimera states with incoherent core: (a) - 2-core spiral wave chimera state ($\alpha=0.82, \mu=0.005, \epsilon=0.1, P=14, N=200$), (b) - 4-core spiral wave chimera state ($\alpha=0.77, \mu=0.012, \epsilon=0.1 , P=42, N=600$).[]{data-label="f1"}](Faza-insert-2.PNG "fig:"){width="0.07\linewidth"}
Figure 1 illustrates two examples of spiral wave chimera states with 2 and 4 incoherent cores, which typically exist in model (1) at small enough values of the coupling parameter $\mu$ and at intermediate $\alpha$ (obtained at the parameter points $A$ and $B$ shown in Fig. 2(a)). What is the fate of these states with an increase in the coupling strength $\mu$? We will demonstrate this later on by the example of the 4-core chimera shown to the right. First, we present the main parameter diagram to give a general comprehension of the model.
![\[fig:epsart\] Stability regions in the $(\alpha, \mu)$ parameter plane for 2- (blue) and 4-core (green) spiral wave chimera states and the region for spiral waves with coherent cores (red); stability region for solitary states (gray): (a) - $r=0.07$, (b) - $r=0.16$. $\varepsilon=0.1$. Snapshots of the states are shown in inserts.[]{data-label="f2"}](Fig2a-small.png "fig:"){width="0.8\linewidth"} ![\[fig:epsart\] Stability regions in the $(\alpha, \mu)$ parameter plane for 2- (blue) and 4-core (green) spiral wave chimera states and the region for spiral waves with coherent cores (red); stability region for solitary states (gray): (a) - $r=0.07$, (b) - $r=0.16$. $\varepsilon=0.1$. Snapshots of the states are shown in inserts.[]{data-label="f2"}](Fig2b-small.png "fig:"){width="0.8\linewidth"}
The results of direct numerical simulations of model (1) with $N=100$ in the two-parameter plane of the phase lag $\alpha$ and the coupling strength $\mu$ are presented in Fig. 2 for the coupling radii $r=0.07$ (a) and $0.16$ (b). This figure reveals the appearance of regions of 2- and 4-core spiral wave chimera states, shown in green and blue, at intermediate values of the phase lag $\alpha$. Alternatively, if the oscillator interactions are not phase-lagged ($\alpha=0$) or if $\alpha$ is small, the network displays the full synchronization.
This synchronous behavior is Lyapunov-stable for all $\alpha<\pi/2$ and for any $\mu>0$. However, its basin of attraction shrinks with an increase in $\alpha,$ since many other stable states arises. All together they capture the major part of the phase space. The “synch basin" size is an essential characteristic of multistable systems [@wsg2006] which is actual for our model (1) as well.
In the opposite situation where the phase lag $\alpha$ is close to $\pi/2$, the so-called [*solitary states*]{} obey the major part of the parameter space: A number of the network oscillators start to behave themselves differently, as compared to all frequency-synchronized other ones \[10\]. The solitary state region (shown in gray) persists at large $\alpha$ and at $\mu$ detached from $0$ up to the inverse homoclinic bifurcation curve HB, where the very last limit cycle (image of a single-solitary state) sticks on the homoclinic contour and disappears.
The regions of spiral wave chimeras are shown in Fig. 2 (a-b) as shaded (color) tongues. The tongues are issued from the $\mu=0$ level at some $\alpha_0<\pi2$, each is characterized by the number of cores, as it is illustrated by insets. Only the regions for 2- and 4-core chimeras are shown. Of a special interest is the region, where chimeras are characterized by the coherent core dynamics (shown in red). There is also a blank strip between the chimera and solitary regions, where the dynamics is represented by a developing spiral chaos. The pink dotted line in Fig 2(a) indicates the upper border for all spiral regions including the spiral chaos.
[r]{}[0.58]{}
{width="1.0\linewidth"}
![Spiral wave chimera transformation. Snapshots of the phase distributions $\varphi_{i,j}$ (left column), average frequencies $\bar{\omega}_{i,j}$ (middle column), and their cross-sections (right column) for a single core of spiral wave chimeras: (a) - incoherent core ($\mu=0.01$), (b) - partially coherent core ($\mu=0.06$), (c) - coherent core ($\mu=0.07)$, (d) - coherent core with solitary cloud ($\mu=0.11)$, $P=56, N=800$; (e) - spatiotemporal chaos $ (\mu=0.55, N=100 )$. $\alpha=0.42, \epsilon=0.1$. Interval of average frequencies $\Delta T = 1000$.[]{data-label="f4"}](Fig4a-f.png "fig:"){width="0.27\linewidth"} ![Spiral wave chimera transformation. Snapshots of the phase distributions $\varphi_{i,j}$ (left column), average frequencies $\bar{\omega}_{i,j}$ (middle column), and their cross-sections (right column) for a single core of spiral wave chimeras: (a) - incoherent core ($\mu=0.01$), (b) - partially coherent core ($\mu=0.06$), (c) - coherent core ($\mu=0.07)$, (d) - coherent core with solitary cloud ($\mu=0.11)$, $P=56, N=800$; (e) - spatiotemporal chaos $ (\mu=0.55, N=100 )$. $\alpha=0.42, \epsilon=0.1$. Interval of average frequencies $\Delta T = 1000$.[]{data-label="f4"}](Faza-AW-insert-a.PNG "fig:"){width="0.06\linewidth"} ![Spiral wave chimera transformation. Snapshots of the phase distributions $\varphi_{i,j}$ (left column), average frequencies $\bar{\omega}_{i,j}$ (middle column), and their cross-sections (right column) for a single core of spiral wave chimeras: (a) - incoherent core ($\mu=0.01$), (b) - partially coherent core ($\mu=0.06$), (c) - coherent core ($\mu=0.07)$, (d) - coherent core with solitary cloud ($\mu=0.11)$, $P=56, N=800$; (e) - spatiotemporal chaos $ (\mu=0.55, N=100 )$. $\alpha=0.42, \epsilon=0.1$. Interval of average frequencies $\Delta T = 1000$.[]{data-label="f4"}](Fig4a-AV.PNG "fig:"){width="0.27\linewidth"} ![Spiral wave chimera transformation. Snapshots of the phase distributions $\varphi_{i,j}$ (left column), average frequencies $\bar{\omega}_{i,j}$ (middle column), and their cross-sections (right column) for a single core of spiral wave chimeras: (a) - incoherent core ($\mu=0.01$), (b) - partially coherent core ($\mu=0.06$), (c) - coherent core ($\mu=0.07)$, (d) - coherent core with solitary cloud ($\mu=0.11)$, $P=56, N=800$; (e) - spatiotemporal chaos $ (\mu=0.55, N=100 )$. $\alpha=0.42, \epsilon=0.1$. Interval of average frequencies $\Delta T = 1000$.[]{data-label="f4"}](Fig4a-AV-cross-j.PNG "fig:"){width="0.28\linewidth"}\
![Spiral wave chimera transformation. Snapshots of the phase distributions $\varphi_{i,j}$ (left column), average frequencies $\bar{\omega}_{i,j}$ (middle column), and their cross-sections (right column) for a single core of spiral wave chimeras: (a) - incoherent core ($\mu=0.01$), (b) - partially coherent core ($\mu=0.06$), (c) - coherent core ($\mu=0.07)$, (d) - coherent core with solitary cloud ($\mu=0.11)$, $P=56, N=800$; (e) - spatiotemporal chaos $ (\mu=0.55, N=100 )$. $\alpha=0.42, \epsilon=0.1$. Interval of average frequencies $\Delta T = 1000$.[]{data-label="f4"}](Fig4b-f.PNG "fig:"){width="0.27\linewidth"} ![Spiral wave chimera transformation. Snapshots of the phase distributions $\varphi_{i,j}$ (left column), average frequencies $\bar{\omega}_{i,j}$ (middle column), and their cross-sections (right column) for a single core of spiral wave chimeras: (a) - incoherent core ($\mu=0.01$), (b) - partially coherent core ($\mu=0.06$), (c) - coherent core ($\mu=0.07)$, (d) - coherent core with solitary cloud ($\mu=0.11)$, $P=56, N=800$; (e) - spatiotemporal chaos $ (\mu=0.55, N=100 )$. $\alpha=0.42, \epsilon=0.1$. Interval of average frequencies $\Delta T = 1000$.[]{data-label="f4"}](Faza-AW-insert-b.PNG "fig:"){width="0.06\linewidth"} ![Spiral wave chimera transformation. Snapshots of the phase distributions $\varphi_{i,j}$ (left column), average frequencies $\bar{\omega}_{i,j}$ (middle column), and their cross-sections (right column) for a single core of spiral wave chimeras: (a) - incoherent core ($\mu=0.01$), (b) - partially coherent core ($\mu=0.06$), (c) - coherent core ($\mu=0.07)$, (d) - coherent core with solitary cloud ($\mu=0.11)$, $P=56, N=800$; (e) - spatiotemporal chaos $ (\mu=0.55, N=100 )$. $\alpha=0.42, \epsilon=0.1$. Interval of average frequencies $\Delta T = 1000$.[]{data-label="f4"}](Fig4b-AV.PNG "fig:"){width="0.27\linewidth"} ![Spiral wave chimera transformation. Snapshots of the phase distributions $\varphi_{i,j}$ (left column), average frequencies $\bar{\omega}_{i,j}$ (middle column), and their cross-sections (right column) for a single core of spiral wave chimeras: (a) - incoherent core ($\mu=0.01$), (b) - partially coherent core ($\mu=0.06$), (c) - coherent core ($\mu=0.07)$, (d) - coherent core with solitary cloud ($\mu=0.11)$, $P=56, N=800$; (e) - spatiotemporal chaos $ (\mu=0.55, N=100 )$. $\alpha=0.42, \epsilon=0.1$. Interval of average frequencies $\Delta T = 1000$.[]{data-label="f4"}](Fig4b-AV-cross.PNG "fig:"){width="0.27\linewidth"}\
![Spiral wave chimera transformation. Snapshots of the phase distributions $\varphi_{i,j}$ (left column), average frequencies $\bar{\omega}_{i,j}$ (middle column), and their cross-sections (right column) for a single core of spiral wave chimeras: (a) - incoherent core ($\mu=0.01$), (b) - partially coherent core ($\mu=0.06$), (c) - coherent core ($\mu=0.07)$, (d) - coherent core with solitary cloud ($\mu=0.11)$, $P=56, N=800$; (e) - spatiotemporal chaos $ (\mu=0.55, N=100 )$. $\alpha=0.42, \epsilon=0.1$. Interval of average frequencies $\Delta T = 1000$.[]{data-label="f4"}](Fig4c-f.PNG "fig:"){width="0.27\linewidth"} ![Spiral wave chimera transformation. Snapshots of the phase distributions $\varphi_{i,j}$ (left column), average frequencies $\bar{\omega}_{i,j}$ (middle column), and their cross-sections (right column) for a single core of spiral wave chimeras: (a) - incoherent core ($\mu=0.01$), (b) - partially coherent core ($\mu=0.06$), (c) - coherent core ($\mu=0.07)$, (d) - coherent core with solitary cloud ($\mu=0.11)$, $P=56, N=800$; (e) - spatiotemporal chaos $ (\mu=0.55, N=100 )$. $\alpha=0.42, \epsilon=0.1$. Interval of average frequencies $\Delta T = 1000$.[]{data-label="f4"}](Faza-AW-insert-c.PNG "fig:"){width="0.06\linewidth"} ![Spiral wave chimera transformation. Snapshots of the phase distributions $\varphi_{i,j}$ (left column), average frequencies $\bar{\omega}_{i,j}$ (middle column), and their cross-sections (right column) for a single core of spiral wave chimeras: (a) - incoherent core ($\mu=0.01$), (b) - partially coherent core ($\mu=0.06$), (c) - coherent core ($\mu=0.07)$, (d) - coherent core with solitary cloud ($\mu=0.11)$, $P=56, N=800$; (e) - spatiotemporal chaos $ (\mu=0.55, N=100 )$. $\alpha=0.42, \epsilon=0.1$. Interval of average frequencies $\Delta T = 1000$.[]{data-label="f4"}](Fig4c-AV.PNG "fig:"){width="0.27\linewidth"} ![Spiral wave chimera transformation. Snapshots of the phase distributions $\varphi_{i,j}$ (left column), average frequencies $\bar{\omega}_{i,j}$ (middle column), and their cross-sections (right column) for a single core of spiral wave chimeras: (a) - incoherent core ($\mu=0.01$), (b) - partially coherent core ($\mu=0.06$), (c) - coherent core ($\mu=0.07)$, (d) - coherent core with solitary cloud ($\mu=0.11)$, $P=56, N=800$; (e) - spatiotemporal chaos $ (\mu=0.55, N=100 )$. $\alpha=0.42, \epsilon=0.1$. Interval of average frequencies $\Delta T = 1000$.[]{data-label="f4"}](Fig4c-AV-cross-j.PNG "fig:"){width="0.28\linewidth"}\
![Spiral wave chimera transformation. Snapshots of the phase distributions $\varphi_{i,j}$ (left column), average frequencies $\bar{\omega}_{i,j}$ (middle column), and their cross-sections (right column) for a single core of spiral wave chimeras: (a) - incoherent core ($\mu=0.01$), (b) - partially coherent core ($\mu=0.06$), (c) - coherent core ($\mu=0.07)$, (d) - coherent core with solitary cloud ($\mu=0.11)$, $P=56, N=800$; (e) - spatiotemporal chaos $ (\mu=0.55, N=100 )$. $\alpha=0.42, \epsilon=0.1$. Interval of average frequencies $\Delta T = 1000$.[]{data-label="f4"}](Fig4d-f.PNG "fig:"){width="0.27\linewidth"} ![Spiral wave chimera transformation. Snapshots of the phase distributions $\varphi_{i,j}$ (left column), average frequencies $\bar{\omega}_{i,j}$ (middle column), and their cross-sections (right column) for a single core of spiral wave chimeras: (a) - incoherent core ($\mu=0.01$), (b) - partially coherent core ($\mu=0.06$), (c) - coherent core ($\mu=0.07)$, (d) - coherent core with solitary cloud ($\mu=0.11)$, $P=56, N=800$; (e) - spatiotemporal chaos $ (\mu=0.55, N=100 )$. $\alpha=0.42, \epsilon=0.1$. Interval of average frequencies $\Delta T = 1000$.[]{data-label="f4"}](Faza-AW-insert-d.PNG "fig:"){width="0.06\linewidth"} ![Spiral wave chimera transformation. Snapshots of the phase distributions $\varphi_{i,j}$ (left column), average frequencies $\bar{\omega}_{i,j}$ (middle column), and their cross-sections (right column) for a single core of spiral wave chimeras: (a) - incoherent core ($\mu=0.01$), (b) - partially coherent core ($\mu=0.06$), (c) - coherent core ($\mu=0.07)$, (d) - coherent core with solitary cloud ($\mu=0.11)$, $P=56, N=800$; (e) - spatiotemporal chaos $ (\mu=0.55, N=100 )$. $\alpha=0.42, \epsilon=0.1$. Interval of average frequencies $\Delta T = 1000$.[]{data-label="f4"}](Fig4d-AV.PNG "fig:"){width="0.27\linewidth"} ![Spiral wave chimera transformation. Snapshots of the phase distributions $\varphi_{i,j}$ (left column), average frequencies $\bar{\omega}_{i,j}$ (middle column), and their cross-sections (right column) for a single core of spiral wave chimeras: (a) - incoherent core ($\mu=0.01$), (b) - partially coherent core ($\mu=0.06$), (c) - coherent core ($\mu=0.07)$, (d) - coherent core with solitary cloud ($\mu=0.11)$, $P=56, N=800$; (e) - spatiotemporal chaos $ (\mu=0.55, N=100 )$. $\alpha=0.42, \epsilon=0.1$. Interval of average frequencies $\Delta T = 1000$.[]{data-label="f4"}](Fig4d-AV-cross-j.PNG "fig:"){width="0.28\linewidth"}
![Spiral wave chimera transformation. Snapshots of the phase distributions $\varphi_{i,j}$ (left column), average frequencies $\bar{\omega}_{i,j}$ (middle column), and their cross-sections (right column) for a single core of spiral wave chimeras: (a) - incoherent core ($\mu=0.01$), (b) - partially coherent core ($\mu=0.06$), (c) - coherent core ($\mu=0.07)$, (d) - coherent core with solitary cloud ($\mu=0.11)$, $P=56, N=800$; (e) - spatiotemporal chaos $ (\mu=0.55, N=100 )$. $\alpha=0.42, \epsilon=0.1$. Interval of average frequencies $\Delta T = 1000$.[]{data-label="f4"}](Fig4e-f.png "fig:"){width="0.27\linewidth"} ![Spiral wave chimera transformation. Snapshots of the phase distributions $\varphi_{i,j}$ (left column), average frequencies $\bar{\omega}_{i,j}$ (middle column), and their cross-sections (right column) for a single core of spiral wave chimeras: (a) - incoherent core ($\mu=0.01$), (b) - partially coherent core ($\mu=0.06$), (c) - coherent core ($\mu=0.07)$, (d) - coherent core with solitary cloud ($\mu=0.11)$, $P=56, N=800$; (e) - spatiotemporal chaos $ (\mu=0.55, N=100 )$. $\alpha=0.42, \epsilon=0.1$. Interval of average frequencies $\Delta T = 1000$.[]{data-label="f4"}](Faza-AW-insert-e.png "fig:"){width="0.06\linewidth"} ![Spiral wave chimera transformation. Snapshots of the phase distributions $\varphi_{i,j}$ (left column), average frequencies $\bar{\omega}_{i,j}$ (middle column), and their cross-sections (right column) for a single core of spiral wave chimeras: (a) - incoherent core ($\mu=0.01$), (b) - partially coherent core ($\mu=0.06$), (c) - coherent core ($\mu=0.07)$, (d) - coherent core with solitary cloud ($\mu=0.11)$, $P=56, N=800$; (e) - spatiotemporal chaos $ (\mu=0.55, N=100 )$. $\alpha=0.42, \epsilon=0.1$. Interval of average frequencies $\Delta T = 1000$.[]{data-label="f4"}](Fig4e-AV.png "fig:"){width="0.27\linewidth"} ![Spiral wave chimera transformation. Snapshots of the phase distributions $\varphi_{i,j}$ (left column), average frequencies $\bar{\omega}_{i,j}$ (middle column), and their cross-sections (right column) for a single core of spiral wave chimeras: (a) - incoherent core ($\mu=0.01$), (b) - partially coherent core ($\mu=0.06$), (c) - coherent core ($\mu=0.07)$, (d) - coherent core with solitary cloud ($\mu=0.11)$, $P=56, N=800$; (e) - spatiotemporal chaos $ (\mu=0.55, N=100 )$. $\alpha=0.42, \epsilon=0.1$. Interval of average frequencies $\Delta T = 1000$.[]{data-label="f4"}](Fig4e-AV-cross-j.png "fig:"){width="0.28\linewidth"}
Figure 3 shows regions for the spiral wave chimera states with 2 and 4 cores in the $(\alpha, P)$-parameter plane at fixed $\mu=0.015$. As it can be observed at $r=0.016,$ the $\alpha$-interval for the 4-core chimera is much wider than that for the 2-core one. Hence, the stability region for the 2-core chimera in Fig. 2(b) is smaller with respect to the case of $r=0.07$ (Fig. 2(a)). In addition, the stability region for the chimeras with coherent cores (red) lies inside the solitary region (gray) for $r=0.016$. It is in contrary to the $r=0.07$ case where the chimeras with coherent cores can exist also outside the solitary region.
A scenario typical of the transition of the 4-core incoherent chimera state to a coherent one is illustrated in Fig. 4, where we fix the phase lag $\alpha=0.42$ and increase the coupling strength $\mu$ along the vertical line with small circles in Fig. 2(a) from bottom to top. Only one of four cores is shown, the others are qualitatively similar.
First, in Fig. 4(a) which is obtained for $\mu=0.01$, the solution profile represents a classical chimera with an incoherent bell-shaped core. For larger $\mu,$ the frequency profile sharpens up and, at some value $\mu\cong0.05$, loses its smoothness. An additional concentric ring arises, as shown in Fig 4(b), and the chimera becomes quasiperiodic. With a further increase in $\mu,$ the new concentric rings with constant average frequency appear, and the quasiperiodic dynamics becomes more involved (not shown in the figure, see the figures of the next chapter).
Eventually, after a sequence of metamorphoses, only one region of this kind remains, i.e., all oscillators inside the core start to rotate with the same average frequency. Thus, the chimera state with a [*coherent core*]{} is born, as illustrated in Fig. 4(c) for $\mu=0.07$. We find that such striking behavior with the bi-stable frequency characteristic, one per core, persists at a further increase of $\mu$. However, after entering the solitary region at $\mu\approx0.081$, the isolated oscillators outside the cores can arise, and they begin to follow not the spiraling, but the core dynamics.
Spiral wave chimera of this kind with isolated oscillators is illustrated in Fig. 4(d) for $\mu=0.11$. The pattern is obtained from random initial conditions. It is characterized by a cloud of randomly allocated solitary oscillators around the core oscillating with its own frequency. The number of solitary oscillators in the cloud can be any, by depending on the initial conditions. In particular, the cloud can include only one or a few solitary oscillators or even no ones at all (cloud disappears); all these states can be obtained only by choosing the specially prepared initial conditions.
{width="0.8\linewidth"}
With a further increase in $\mu$, the shape of a coherent core undergoes dramatic deformations. Moreover, new frequencies can appear in the solitary cloud. This is illustrated in Fig. 4(d). The scenario ends eventually, when the spiral dynamics in Eq.(1) ceases to exist with a further increase in $\mu$. After that, only solitary states are left in system’s dynamics: a huge variety of multifrequency states can be generated due to specially prepared initial conditions; one striking example is illustrated in Fig. 4(e). The number of different stable solitary states grows exponentially with system’s size $N$. Then the network dynamics is identified as a [*spatial chaos*]{} (see [@omps2011]).
{width="0.232\linewidth"} {width="0.232\linewidth"} {width="0.232\linewidth"}
{width="0.232\linewidth"} {width="0.232\linewidth"} {width="0.232\linewidth"}
{width="0.232\linewidth"} {width="0.232\linewidth"} {width="0.232\linewidth"}
{width="0.232\linewidth"} {width="0.228\linewidth"} {width="0.225\linewidth"} {width="0.236\linewidth"}
In Fig. 4, the typical transformations of only one core of a 4-core chimera are shown. A question arises about the behavior in other three cores: Are the frequencies of oscillators the same or can be different? In Fig. 5, we present the evolution of all four frequencies $\bar{\omega}_{1}$ $\bar{\omega}_{2}$, $\bar{\omega}_{3}$, and $\bar{\omega}_{4},$ as well as the basic spiral frequency $\bar{\omega}_{0}$. Fixing the parameters $\alpha=0.42$, $r=0.07$, $P=7$ and varying $\mu$ from 0.01 to 0.1, we observe that different situations can occur. First, at small $\mu$, the core frequencies are essentially different. Further, with increase in $\mu$ beyond some $\mu\approx 0.071$, they become practically identical. The more precise inspection approves, however, that only three such frequencies coincide, the fourth one is slightly deviating. This situation still valid, if we remain inside of the coherent oscillator region (colored by red in Fig. 2(a)).
Examples of different core frequencies will be shown in the next section.
In Fig. 6, four more examples of the striking 4-core dynamics are presented to illustrate the variability of the behavior.
Conclusion: Emerging network dynamics
=====================================
As we observed above, the network dynamics becomes much richer, if the inertia is added into the standard Kuramoto model. Numerous new states that are not possible without inertia arise. Spiral wave chimeras are influenced by solitary states, and the [*spatial chaos*]{} behavior become a universal phenomenon in a wide parameter region.
As one can observed, isolated oscillators are always present in the phase space, as soon as the simulation is performed in the solitary parameter region with random initial conditions (although the solitaries can always be remove by slight corrections of the initial states of oscillators).
Interestingly, the number of different frequencies can be quite large. At least, we observed ten, as illustrated in Fig. 7(b). The number of incoherent cores can also be quite large, e.g., 10 and 26 in Fig. 7(a) and (b), respectively. Then the behavior develops in the form of a classical spiral chaos, but with incoherent core dynamics.
![Multiple spiral wave chimeras. Snapshots of the phase distributions $\varphi_{i,j}$: (a) - 10-core spiral wave chimera state ($\alpha=0.76, \mu=0.017, \epsilon=0.1, P=56, N=800$), (b) - 26-spiral wave chimera with solitary oscillators ($\alpha=0.88, \mu=0.025, \epsilon=0.1 , P=7, N=100$).[]{data-label="f7"}](Fig7a.PNG "fig:"){width="0.4\linewidth"} ![Multiple spiral wave chimeras. Snapshots of the phase distributions $\varphi_{i,j}$: (a) - 10-core spiral wave chimera state ($\alpha=0.76, \mu=0.017, \epsilon=0.1, P=56, N=800$), (b) - 26-spiral wave chimera with solitary oscillators ($\alpha=0.88, \mu=0.025, \epsilon=0.1 , P=7, N=100$).[]{data-label="f7"}](Fig7b.PNG "fig:"){width="0.4\linewidth"} ![Multiple spiral wave chimeras. Snapshots of the phase distributions $\varphi_{i,j}$: (a) - 10-core spiral wave chimera state ($\alpha=0.76, \mu=0.017, \epsilon=0.1, P=56, N=800$), (b) - 26-spiral wave chimera with solitary oscillators ($\alpha=0.88, \mu=0.025, \epsilon=0.1 , P=7, N=100$).[]{data-label="f7"}](Faza-insert-2.PNG "fig:"){width="0.06\linewidth"}
Figures 8 and 9 illustrate the examples of spiral wave chimeras typically observed in model (1), when varying the parameters.
In Fig. 8, the phase distributions $\varphi_{i,j}$ of spiral waves, spiral wave average frequencies $\bar{\omega}_{i,j}$, cross-sections along a dashed white line (third column), and ordered oscillator index for oscillators with average frequencies $\bar{\omega}_{i,j} > \bar{\omega}_{0}$ are shown. These patterns were obtained from random initial conditions. Here, for the clearly seen in color differences of average frequencies $\bar{\omega}_{i,j},$ their interval in Fig. 8 was taken from $-0.16$ to $-0.07$, although the average frequency of the main cluster $\bar{\omega}_{0}\approx-0.44...$.
{width="0.31\linewidth"} {width="0.07\linewidth"} {width="0.31\linewidth"}
{width="0.31\linewidth"} {width="0.31\linewidth"}
{width="0.31\linewidth"} {width="0.07\linewidth"} {width="0.31\linewidth"}
{width="0.31\linewidth"} {width="0.31\linewidth"}
{width="0.31\linewidth"} {width="0.07\linewidth"} {width="0.31\linewidth"}
{width="0.31\linewidth"} {width="0.31\linewidth"}
Figure 8 shows that the coherent spiral cores can have individual average frequencies. Moreover, solitary oscillators are located on core’s border or outside the spiral cores and can have average frequencies distinguished from every other spiral core. The number of the clusters of average frequencies does not depend on the number of spiral wave cores. For 2 (Fig. 8(a)), 4 (Fig. 8(b)) and 6 (Fig. 8(c)) spiral waves, the number of the of clusters is the same and is more than 10 (see the ordered oscillator indices). This number is determined by parameters of system (1).
{width="0.45\linewidth"} {width="0.1\linewidth"} {width="0.45\linewidth"} {width="1.02\linewidth"} {width="0.28\linewidth"} {width="0.06\linewidth"} {width="0.28\linewidth"} {width="0.34\linewidth"}
Finally, we present the 2-core spiral wave chimeras with solitary clouds in Fig. 9 for the high dimension $N=800$ at the parameter values $\alpha=0.45, \mu=0.15, \epsilon=0.1$, and $P=56$ obtained from random initial conditions. While the average frequencies $\bar{\omega}_{i,j}$ of the spiral cores are identical (green color in Fig. 9(b)), the solitary clouds can have different average frequencies $\bar{\omega}_{i,j}$ (yellow and red colors in Fig. 9(b)), as clearly seen in the cross-section at $j=419$ in Fig. 9(c). Note that the both spiral cores have incoherence at core’s borders in this example. The structure of the left spiral core is presented in Fig. 9(d-f) in detail on an enlarged scale at the cross-section of average frequencies $\bar{\omega}_{i,j}$ at $j=413$.
In conclusion, we have identified a novel scenario for the spiral wave transition in networks of coupled oscillators with inertia. It consists in the appearance of quasiperiodic chimera states with quasiperiodic, multifrequency cores ending eventually in chimeras with pure coherent core with a solitary cloud of isolated oscillators in the spiraling part of the phase space. We believe that this kind of behavior indicates a common probably universal inertia-induced phenomenon in Newtonian networks of very different nature.
Acknowledgments {#acknowledgments .unnumbered}
===============
We thank E. Knobloch and A. Pikovsky for the fruitful illumunating discussions. We also thank the Ukrainian Grid Infrastructure for providing the computing cluster resources and the parallel and distributed software used during this work.
[100]{} Y. Kuramoto, S.I. Shima. Prog. Theor. Phys. Supp. [**150**]{}, 115-125. (2003).
S.I. Shima, Y. Kuramoto. Phys. Rev. E. [**69**]{}, 036213. (2004).
E. Martens, C. Laing, S. Strogatz. Phys. Rev. Lett. [**104**]{}, 044101. (2010).
O. Omel’chenko, M. Wolfrum, S. Yanchuk, Yu. Maistrenko, O. Sudakov. Phys. Rev. E [**85 (3)**]{}, 036210. (2012).
M. J. Panaggio, D.M. Abrams. Phys. Rev. Lett. [**110**]{}, 094102. (2013).
J. Xie, E. Knobloch, H .Kao. Phys. Rev. E. [**92**]{}, 042921. (2015).
C.R. Laing. Proc. SIAM J. Appl. Dyn. Syst., [**16(2)**]{}, 974–1014. (2017).
O. Omel’chenko, M. Wolfrum, E. Knobloch. Proc. SIAM J. Appl. Dyn. Syst., [**17(1)**]{}, 97–127. (2017).
J.F. Totz, J. Rode, M.R. Tinsley, et al. Nature Phys [**14**]{}. 282–285 (2018).
O. Omel’chenko, E. Knobloch. New Journal of Physics, [**21**]{}. (2019).
J.F. Totz. Communications in Computer and Information Science, 55–97. (2019).
P. Jaros, Yu. Maistrenko, T. Kapitaniak. Phys. Rev. E [**91**]{}, 022907. (2015).
P. Jaros, S. Brezetsky, R. Levchenko, D. Dudkowski, T. Kapitaniak, Yu. Maistrenko. Chaos [**28**]{}, 011103. (2018).
A. Salnikov, R. Levchenko, O. Sudakov. Proc. 6th IEEE International Conference (IDAACS) . (2011).
O. Sudakov, A. Cherederchuk, V. Maistrenko. Proc. 9th IEEE International Conference (IDAACS). (2017).
D. Wiley, S. Strogatz, M. Girvan. Chaos [**16**]{}, 015103. (2006).
I. Omelchenko, Yu. Maistrenko, P. Hovel, E. Scholl. Phys. Rev. Lett. [**106**]{}, 234102. (2011).
[^1]:
|
---
abstract: 'We construct quantum mechanical observables and unitary operators which, if implemented in physical systems as measurements and dynamical evolutions, would contradict the Church-Turing thesis which lies at the foundation of computer science. We conclude that either the Church-Turing thesis needs revision, or that only restricted classes of observables may be realized, in principle, as measurements, and that only restricted classes of unitary operators may be realized, in principle, as dynamics.'
address: |
Center for Advanced Studies, Department of Physics and Astronomy, University of New Mexico, Albuquerque NM 87131-1156\
[*and*]{}\
Norman Bridge Laboratory of Physics 12-33, California Institute of Technology, Pasadena, CA 91125
author:
- 'M. A. Nielsen [^1]'
title: 'Computable functions, quantum measurements, and quantum dynamics'
---
Quantum mechanical measurements on a physical system are represented by [*observables*]{} - Hermitian operators on the state space of the observed system. It is an important question whether all observables may be realized, in principle, as measurements on a physical system. Dirac’s influential text ([@Dirac58a], page 37) makes the following assertion on the question:
[*The question now presents itself – Can every observable be measured? The answer theoretically is yes. In practice it may be very awkward, or perhaps even beyond the ingenuity of the experimenter, to devise an apparatus which could measure some particular observable, but the theory always allows one to imagine that the measurement can be made.*]{}
This Letter re-examines the question of whether it is possible, even in principle, to measure every quantum mechanical observable. Unexpectedly, ideas from computer science are crucial to the analysis. We also investigate a related question, namely, whether it is possible to realize, in principle, the dynamics corresponding to an arbitrary unitary operator on the state space of a quantum system. Of course, for specific systems particular measurements and unitary dynamics may be forbidden by system specific features, such as superselection rules. We will not be concerned with such specific features, but rather with general considerations.
In the remarkable paper which founded modern computer science, Turing [@Turing36a] defined a class of functions which are now known as the [*recursive*]{} or [*computable*]{} functions. The [*Church-Turing thesis*]{} [@Church36a; @Turing36a] of computer science states that this class of functions corresponds precisely to the class of functions which may be computed via what humans intuitively call an [*algorithm*]{} or [*procedure*]{}. More formally,
[*Every function which can be computed by what we would naturally regard as an algorithm is a computable function, and vice versa.*]{}
The Church-Turing thesis is fundamental to theoretical computer science, since it asserts that the mathematical class of functions studied by computer scientists, the computable functions, is the most general class of functions which may be calculated using a computer. It is an empirical statement, not a theorem of mathematics, and has been verified through over sixty years of testing [@Hofstadter79a; @Penrose89a]. For a review of different formulations of the Church-Turing thesis, see [@Hofstadter79a].
For convenience we refer to computable partial functions of a single non-negative integer as [*programs*]{}. That programs are only partial functions means that there may be some values of the input for which no output is defined. An example of a program is the function $f(x) = x^2$, which may be computed using a suitable Turing machine. Using Turing’s results it is possible to show [@Davis83a] that the set of programs may be numbered $0,1,2,3,\ldots$. Not all programs need to terminate or [*halt*]{} for all possible inputs. A simple example is the program $f$ which on input $x$ loops forever if $x$ is not a perfect square, or prints $\sqrt x$ if $x$ is a perfect square. This is an example of a partial function.
The question of whether or not a given program, numbered $x$, halts on the input of $y$ is obviously a question of considerable practical importance: we would like to know whether a given algorithm will terminate or not. To understand this question better, Turing defined a function, the [*halting function*]{}, $h$, by $$\begin{aligned}
h(x) \equiv \left\{ \begin{array}{ll} 1 & \mbox{if program} \, x \,
\mbox{halts on input} \, x; \\
0 & \mbox{if program} \, x\, \mbox{does not halt on input} \, x.
\end{array} \right.\end{aligned}$$ Turing [@Turing36a] demonstrated that the halting function is not a computable function. That is, there exists no algorithmic means for computing the value of the halting function for all values $x = 0,1,\ldots$. Thus there is no algorithm for determining whether a given program terminates or not.
Many non-computable functions other than the halting function are now known, and the reasoning which follows applies to any such function $h$. For concreteness we continue to imagine that $h$ is the halting function.
We define the [*halting observable*]{}, $\hat h$, by $$\begin{aligned}
\hat h \equiv \sum_{x = 0}^{\infty} h(x) |x\rangle \langle x|, \end{aligned}$$ where $|x\rangle$ is an orthonormal basis for the state space of some physical system with a countably infinite dimensional state space. We will suppose that the system is one such that all the states $|x\rangle$ may be prepared, in principle. For example, the states $|x\rangle$ might be number states of a single mode of the electromagnetic field. The halting observable is clearly an observable in the usual quantum mechanical sense: it is a Hermitian operator on the state space of the system of interest.
Logically, one of two possibilities must hold:
1. It is possible, in principle, to construct a measuring device capable of performing a measurement of the observable $\hat h$.
2. It is not possible, in principle, to construct a measuring device capable of performing a measurement of the observable $\hat h$.
Suppose the first possibility is true. Then in order to compute the value of $h(x)$ one performs the following procedure: Construct the measuring apparatus to measure $\hat h$, and prepare the system to be measured in the state $|x\rangle$. Now perform the measurement. With probability one the result of the measurement will be $h(x)$. This gives a procedure for computing the halting function. If one accepts the Church-Turing thesis this is not an acceptable conclusion, since the halting function is not computable.
Acceptance of the Church-Turing thesis therefore forces us to conclude that the second option is true, namely, that it is not possible, in principle, to construct a measuring device capable of performing a measurement of the observable $\hat h$. That is, only a limited class of observables correspond to measurements which may be performed, in principle, on quantum mechanical systems. An important question arises: to determine the precise class of observables which may be realized as measurements.
Might it be possible to perform an approximate measurement of $\hat h$? Suppose it is possible to measure an observable $\hat h'$ which is close to $\hat h$. Preparing the system in the state $|x\rangle$ and measuring $\hat h'$, a result in the range $h(x) \pm \delta$ is obtained with probability at least $1-\epsilon$, for some small $\epsilon$ and $\delta$. Clearly, by performing repeated measurements of this type it is possible to determine $h(x)$ with arbitrarily high confidence. Thus, approximate measurements of $\hat h$ give an algorithmic means for computing $h(x)$. Once again, if we accept the Church-Turing thesis then we are forced to conclude that it is not possible to perform such an approximate measurement. Note, however, that we are implicitly using a stronger version of the Church-Turing thesis than hitherto, since now we are regarding as an algorithm a procedure which outputs $h(x)$ with [*arbitrarily high confidence*]{}, rather than a purely deterministic procedure.
This second conclusion should be compared to work by Wigner [@Wigner52a], and Araki and Yanase [@Araki60a] on the WAY theorem [@Peres93a]. The WAY theorem shows that if we require certain conservation laws to be respected during the measurement process, then there are restrictions on the class of observables which may be realized using measuring devices. This is fundamentally different to the conclusion we have obtained, which does not depend on the imposition of externally imposed conservation requirements. Another difference is that it was shown in [@Araki60a] and [@Ghirardi81a] that it is possible to perform approximate measurements of the observables forbidden realization as measurements by the WAY theorem. As we have already seen, approximate measurements of $\hat h$ are not allowed by the Church-Turing thesis.
Up to this point we have considered the physical realization of measurements corresponding to quantum mechanical observables. Similar arguments apply also to the physical realization of unitary operators as dynamical evolutions. Define a function $g$ as follows: $$\begin{aligned}
g(x) & \equiv & \left\{ \begin{array}{ll} 2m-2 & \mbox{if}\,x\,\mbox{is the}\,
m\mbox{th smallest non-negative} \\
& \mbox{ integer such that}\, h(x)=0 \\
2m-1 & \mbox{if}\,x\,\mbox{is the}\,
m\mbox{th smallest non-negative} \\
& \mbox{integer such that}\, h(x)=1. \end{array}
\right. \nonumber \\
& & \end{aligned}$$ It is easy to verify that the operator $$\begin{aligned}
U & \equiv & \sum_{x=0}^{\infty} |g(x)\rangle \langle x| \end{aligned}$$ is unitary. Suppose we prepare the system in the state $|x\rangle$, perform the unitary dynamics $U$, and then do a measurement in the $|x\rangle$ basis with outcome $x'$ (note that there are systems where such a measurement can certainly be done, in principle, such as a single mode of the electromagnetic field). Note that $x'$ is even if and only if $h(x)=0$ and $x'$ is odd if and only if $h(x)=1$, so this gives a procedure for computing the halting function. Once again, logically, one of two possibilities must hold:
1. It is possible, in principle, to construct a system whose dynamics is described by the unitary operator $U$.
2. It is not possible, in principle, to construct a system whose dynamics is described by the unitary operator $U$.
Once again, if we accept the Church-Turing thesis then we are forced to the second conclusion: there are unitary operators which do not describe the dynamics of any system which can, even in principle, be constructed. By arguments similar to those used for observables, it is easy to see that an approximate dynamical realization of $U$ can also be used as part of a procedure for evaluating the halting function, so acceptance of the Church-Turing thesis implies that approximate realizations of $U$ are not possible, either.
The examples we have discussed take place in infinite dimensional state spaces. A similar construction for a spin $\frac 12$ system starts by defining (see chapter seven of [@Cover91a] for a review of definitions along these lines, and references) $$\begin{aligned}
\Omega \equiv \sum_{x: h(x)=1} \frac{1}{2^x}. \end{aligned}$$ Note that $0 < \Omega < 1$, and that the $x$th bit in the binary expansion of $\Omega$ is one if and only if $h(x) = 1$, so knowing the binary expansion of $\Omega$ is equivalent to knowing $h(x)$ for all $x$. Define $U \equiv \exp(-i \Omega \sigma_y)$. Starting the system in the $|\frac 12,\frac 12\rangle$ state (spin up in the $z$ direction) and applying the dynamics $U$ we see that the state after the dynamics is $$\begin{aligned}
\cos (\Omega) |\frac 12,\frac 12\rangle + \sin(\Omega) |\frac12,-\frac 12\rangle.\end{aligned}$$ By repeatedly performing this procedure and making measurements of $\sigma_z$ we may determine $\cos (\Omega)$ and thus $\Omega$ to any desired accuracy, with arbitrarily high confidence. It follows that we can determine the value of $h(x)$ for all $x$. Once again, if we accept the Church-Turing thesis, then we are forced to conclude that $U$ cannot be realized. Notice that, unlike the earlier examples, this procedure is not stable under perturbations of $U$. A slight change in $U$ can result in an incorrect evaluation of $h(x)$. Physically, uncontrolled interactions with the environment will necessarily mean that $U$ is not implemented exactly, and thus it is not possible to evaluate the halting function using a dynamical realization of $U$. Based on similar arguments it seems likely, though I know of no rigorous general proof, that any [*finite dimensional*]{} construction which allows evaluation of a non-computable function is unstable against perturbations, and therefore is not physically interesting.
Returning to the two physically interesting infinite dimensional examples, what conclusions can be drawn? There are two programs one might pursue.
The first program is to modify the Church-Turing thesis. Perhaps there exist in nature quantum processes which can be used to compute functions which are classically non-computable. It is far-fetched, but not logically inconsistent, to imagine some type of experiment - perhaps a scattering experiment - which can be used to evaluate the halting function.
Recognizing such a process poses some problems. How could we verify that a process computes the halting function (or any other non-computable function)? Because of the unsolvability of the halting problem, it is not possible to verify directly that the candidate “halting process” is, in fact, computing the halting function. Nevertheless, one can imagine inductively verifying that the process computes the halting function. One would do this by running a large number of programs on a computer for a long time, and checking that all the programs which halt are predicted to halt by the candidate halting process, and that programs predicted not to halt by the candidate halting process have not halted. Given sufficient empirical evidence of this sort, one could then [*postulate*]{} as a new physical law that the process computes the halting function.
What types of modification of the Church-Turing thesis might be considered in this program? One approach is to exclude quantum phenomena [*by fiat*]{} from the area of application of the thesis. Approaches of this type have numerous problems. First, the boundary between quantum and classical phenomena is rather fuzzy; where precisely does one draw the line? Second, the approach is [*ad hoc*]{}; what motivates the rejection of quantum phenomena from the area of application of the Church-Turing thesis? Many other modifications of the Church-Turing might be attempted, however we will not discuss them here, as no fully satisfactory modification has been found by the author.
It is the author’s conjecture that the Church-Turing thesis is essentially correct, and that a more satisfactory program is to address the problem of achieving a sharp characterization of the class of observables and unitary dynamics which may be realized in physical systems. At least two properties must be satisfied by such a characterization:
1. It should be consistent with a (possibly sharpened) form of the Church-Turing thesis.
2. It should be clear that the measurements and dynamics contained in that class are, in principle, realizable, and that all other measurements and dynamics could never be realized, even in principle, in the laboratory.
How might one achieve such a characterization? Deutsch [@Deutsch85a] has proposed what he calls the Church-Turing [*principle*]{}, to distinguish it from the less well formulated Church-Turing [*thesis*]{}. The statement of this principle reads:
[*Every finitely realizable physical system can be perfectly simulated by a universal model computing machine operating by finite means.*]{}
Note that [*finite means*]{} here has the meaning that on any [*given*]{} computation finite computational resources are used. As for classical computers, unbounded resources are in principle available, provided only finite resources are used on any given computation. [*Finitely realizable*]{} is being used in the same sense as we have used [*realizable*]{}: in principle, the system can be constructed in a laboratory.
Given such a universal machine, the problem of studying what classes of measurements and dynamics are realizable is reduced to the study of properties of a single physical system, the universal computing machine, since that system can be used to simulate all other finitely realizable physical systems. This reduction to studying the properties of a single system may make it considerably easier to study the classes of measurements and dynamics which are realizable.
For example, Deutsch showed that his proposed universal computing machine can not compute any function which is not computable on a classical computer. Thus, his machine can not compute the halting function. But, if we assume that the halting observable can be measured using some physical system, then by the Church-Turing principle the measurement could be simulated on a universal computing machine, and the result of the measurement determined. This contradicts Deutsch’s assertion that such a machine can not compute any function which is not computable classically, and we conclude that measurements of the halting observable are not possible, in principle. Thus, if we accept that Deutsch’s proposed machine satisfies the Church-Turing principle then it follows that the halting observable can not be measured.
What is gained by using arguments based on the Church-Turing principle instead of arguments based on the Church-Turing thesis, is that it may be possible to prove the Church-Turing principle within known physical theory, for a suitable universal model computing machine. Unfortunately, it is not clear to this author that the theory of quantum computation (see [@Ekert96a; @DiVincenzo95a; @Bennett95a] for a review), which has developed from Deutsch’s original proposal, provides a candidate universal model computing machine. In particular, it is not clear that the finite dimensional state spaces accessed by quantum computers are sufficient to simulate, with arbitrary accuracy, all the processes one finds in nature. Natural processes may take place in infinite dimensional state spaces, and it is has not been demonstrated that all such processes can be well simulated using a system with only a finite number of state space dimensions. Regardless of whether quantum computers satisfy the Church-Turing principle, it is certainly the case that the specification of a universal model computing machine satisfying the Church-Turing principle, besides being important in its own right, would also greatly simplify the question of characterizing the classes of measurements and dynamics which are realizable in physical systems.
This Letter has discussed two questions: what class of observables may be realized as quantum measurements; and what unitary operators may be realized as quantum dynamics. Using concepts from computer science we have constructed observables and unitary operators whose physical implementation would contradict the fundamental Church-Turing thesis of computer science. We conclude that the introduction of new concepts into computer science, physics, or both, is necessary to resolve this contradiction.
I thank Tony Bracken, Ike Chuang, Phil Diamond, Hoi-Kwong Lo, Gerard Milburn, John Preskill and Howard Wiseman for discussions about this paper. This work began with the support of a 1993 summer vacation scholarship in the University of Queensland Mathematics Department, and was continued with the support of the Office of Naval Research (Grant No. N00014-93-1-0116), the support of DARPA through the Quantum Information and Computing (QUIC) institute administered by the Army Research Office, and the Australian-American Educational Foundation (Fulbright Commission).
[10]{}
P. A. M. Dirac, [*The principles of quantum mechanics 4th ed.*]{} (Oxford University Press, Oxford, 1958).
A. M. Turing, Proc. Lond. Math. Soc. 2 (reprinted in [@Davis65a]) [**42**]{}, 230 (1936).
A. Church, Am. J. Math. (reprinted in [@Davis65a]) [**58**]{}, 345 (1936).
D. R. Hofstadter, [*Gödel, Escher, Bach: an eternal golden braid*]{} (Basic Books, New York, 1979).
R. Penrose, [*The emperor’s new mind*]{} (Oxford University Press, Oxford, 1989).
M. D. Davis and E. J. Weyuker, [*Computability, Complexity, and Languages*]{} (Academic Press, New York, 1983).
E. P. Wigner, Z. Phys. [**133**]{}, 101 (1952).
H. Araki and M. M. Yanase, Phys. Rev. [**120**]{}, 622 (1960).
A. Peres, [*Quantum Theory: Concepts and Methods*]{} (Kluwer Academic, Dordrecht, 1993).
G. C. Ghirardi, F. Miglietta, A. Rimini, and T. Weber, Phys. Rev. D [**24**]{}, 347 (1981).
T. M. Cover and J. A. Thomas, [*Elements of Information Theory*]{} (John Wiley and Sons, New York, 1991).
D. Deutsch, Proc. R. Soc. Lond. A [**400**]{}, 97 (1985).
A. Ekert and R. Jozsa, Rev. Mod. Phys. [**68**]{}, 1 (1996).
D. P. DiVincenzo, Science [**270**]{}, 255 (1995).
C. H. Bennett, Physics Today 255 (1995).
M. D. Davis, [*The Undecidable*]{} (Raven Press, Hewlett, New York, 1965).
[^1]: Electronic address: mnielsen@tangelo.phys.unm.edu
|
---
bibliography:
- 'library.bib'
---
=1
\[section\] \[thm\][Remark]{}
References {#references .unnumbered}
==========
|
---
abstract: 'We will show that every element of a finitely generated abelian group is automorphically equivalent what we will define to be a [*representative element*]{} in a [*repeat-free subgroup*]{}, and for finite abelian groups we can count the number of automorphism classes of elements.'
author:
- 'Charles F. Rocca Jr.'
title: 'Automorphism Classes of Elements in Finitely Generated Abelian Groups.'
---
Representative Elements
=======================
A finite abelian $p$-group $G_p$ can be written $$G_p=\mathbb{Z}_{p^{r_1}}^{k_1}\oplus\cdots\oplus\mathbb{Z}_{p^{r_n}}^{k_n},\: r_i<r_{i+1}\ for\ all\ i.$$ However for this paper we will write such a group as follows $$G_p=[\mathbb{Z}_{p^{r_1}}\oplus \cdots \oplus \mathbb{Z}_{p^{r_n}}]
\oplus[\mathbb{Z}_{p^{r_1}}^{k_1-1}\oplus\cdots\oplus\mathbb{Z}_{p^{r_n}}^{k_n-1}],\:
r_i<r_{i+1}.$$ This splits the group into two subgroups, a [*repeat-free subgroup*]{}, $$G_p^{rf}=\mathbb{Z}_{p^{r_1}}\oplus \cdots \oplus \mathbb{Z}_{p^{r_n}},$$ which contains one copy of each of the factors $\mathbb{Z}_{p^{r_i}}$ and a [*remainder subgroup*]{}, $$G_p^{rm}=\mathbb{Z}_{p^{r_i}}^{k_1-1}\oplus\cdots\oplus\mathbb{Z}_{p^{r_n}}^{k_n-1},$$ that contains the remaining factors of the group.
For the remainder of this article let $G$ be a finitely generated abelian group of the form $$G=G_{p_1}\oplus\cdots\oplus G_{p_m} \oplus \mathbb{Z}^l,$$ where $l\mbox{ and }m \geq 0$ and the $p_i$ are distinct primes with $p_i<p_{i+1}$ for all $i$. The above definition of a repeat free subgroup can be extended to any finitely generated abelian group $G$ as follows.
Given a finitely generated abelian group $G$ a [*repeat-free subgroup*]{} has one of the following forms
1. $G^{rf}=G_{p_1}^{rf}\oplus\cdots\oplus G_{p_m}^{rf} \oplus
\mathbb{Z}$ if $l$ and $m\geq 1$, or
2. $G^{rf}=G_{p_1}^{rf}\oplus\cdots\oplus G_{p_m}^{rf}$ if $l=0$ and $m\geq 1$, or
3. $G^{rf}=\mathbb{Z}$ if $l\geq 1$ and $m=0$.
The next definition defines our set of [*representative elements*]{} in a repeat-free finite abelian $p$-group.
An element $g=(g_1,\ldots,g_n)$ of a repeat-free finite abelian $G_p^{rf}=\mathbb{Z}_{p^{r_1}}\oplus \cdots
\oplus \mathbb{Z}_{p^{r_n}}$ $r_i<r_{i+1}$ for all $i$ is a [*representative element*]{} if
1. for all $i$ $g_i$ is either $0$ or a power of $p$,
2. for all $i,j$ if $i<j$, then $g_i<g_j$, and
3. for all $i,j$ if $i<j$, then the order of $g_i$ in $\mathbb{Z}_{p^{r_i}}$ is less than the order of $g_j$ in $\mathbb{Z}_{p^{r_j}}$.
We will show that in a finite abelian $p$-group every element is automorphically equivalent to a representative element of a repeat-free subgroup. Therefore we let a [*representative element*]{} of a finite abelian $p$-group be an element of a repeat-free subgroup which is a representative element in that subgroup. Extending this definition to all finitely generated abelian groups we get.
Let $G$ ba a finitely generated abelian group, then we say $g=(g_1,g_2,\ldots,g_m,Z)\in G$ is a [*representative element*]{} if
1. for all $j$, $g_j$ is a representative element of $G_{p_j}$,
2. the element $Z\in \mathbb{Z}^l$ is of the form $(z,0,\ldots,0)$ and
3. for all $j$, each term of $g_j$ is relatively prime to $z$.
So, the primary goal of this paper is the following theorem.
\[thm:main\]Every element of a finitely generated abelian group is automorphically equivalent to an unique representative element.
The proof of this result has two main steps:
1. Every element is automorphically equivalent to an element in a repeat-free subgroup.
2. Every element of a repeat-free subgroup is automorphically equivalent to a representative element.
The following lemma, in which we think to endomorphisms of finite abelian groups as matrices, is a major tool in proving these results.
[*[@RT]*]{}\[lemma:autolemma\] An endomorphism $\varphi$ of $G_p=\mathbb{Z}_{p^{r_1}} \oplus
\mathbb{Z}_{p^{r_2}} \oplus \cdots \oplus \mathbb{Z}_{p^{r_m}}$, $p$ prime and $r_i \leq r_{i+1}$ for all $i$, is an automorphism if and only if the reduction of $\varphi$ modulo $p$, written $\varphi_p$, is an automorphism of $\mathbb{Z}_p^m$.
The following proposition establishes the first step in the proof of the theorem.
\[prop:prop1\] Every element in a finitely generated abelian group $G$ is automorphically equivalent to an element in the repeat-free subgroup. In particular:
1. Each element $g=(g_1,\ldots,g_n)\in\mathbb{Z}_{p^r}^n$, $n \geq 1$ is automorphically equivalent to an element of the form $(p^l,0,\cdots,0)$, where the order of $g$ is $p^{r-l}$.
2. Each element $g=(g_1,\ldots,g_k)$ of $\mathbb{Z}^k$ is automorphically equivalent to one of the form $(d,0,\cdots,0)$, where $d=\gcd(g_1,...,g_k)$.
[*Proof:*]{}
Let $g\in \mathbb{Z}_{p^r}^n$; each term in $g$ is of the form $g_i=a_ip^{l_i}$, where $a_i$ is relatively prime to p. Therefore $g$ is mapped automorphically to the element $g'=(p^{l_1},\cdots,p^{l_n})$ by multiplying each $g_i$ by the inverse of $a_i$ modulo ${p^r}$.
The order of $g'$ which is the same as the order of $g$, is $p^{r-l}$. Therefore at least one term of $g'$ is equal to $p^l$, without loss of generality let $g'_1=p^l$. Also, every term of $g'$ will be divisible by $p^l$, that is $l_i \geq l$ for all $i$. Therefore $$\left[
\begin{array}{cccc}
1 & 0 & \cdots & 0\\
-p^{l_2-l}&1&\cdots &0\\
\vdots & \; &\ddots &\vdots\\
-p^{l_n-l}&0&\cdots &1
\end{array}
\right]
\left[
\begin{array}{c}
p^{l}\\
p^{l_2}\\
\vdots\\
p^{l_n}\\
\end{array}
\right]=
\left[
\begin{array}{c}
p^{l}\\
0\\
\vdots\\
0\\
\end{array}
\right],$$ where it is clear from the lemma \[lemma:autolemma\] that this matrix is an automorphism, thus proving part (a).
The element $g=(g_1,\ldots,g_k)$ of $\mathbb{Z}^k$ can be reduced using the Euclidian Algorithm by the repeated application of elementary row operations and this establishes (b).
To show that every element of a repeat-free subgroup is automorphically equivalent to a representative element we first restrict ourselves to a finite repeat-free $p$-group and define a [*basic reduction*]{} of an element.
Given a repeat-free $p$-group $$G_p^{rf}=\mathbb{Z}_{p^{r_1}}\oplus\cdots\oplus\mathbb{Z}_{p^{r_n}},\ r_i<r_{i+1}\ for\ all\ i$$ and an element $g=(p^{l_1},\ldots,p^{l_n})$, a [*Basic Reduction about position i of $g$*]{} is the following:
i.e. if the $j^{th}$ term has a greater value but lesser order than the $i^{th}$ term replace it by 0.
If the number of non-zero terms decreases, then the basic reduction is a [*non-trivial basic reduction*]{}, otherwise it is a [*trivial reduction*]{}. We will say that an element is [*reduced*]{} if there are no non-trivial basic reductions.
The matrix for the reduction transformation has elements $$a_{lk}=\left\{
\begin{array}{cl}
1 &if\; l=k\\
-p^{l_j-l_i} & if\; l=j\ and\ k=i\\
0 & otherwise
\end{array} \right.$$ which is similar to the matrix in Proposition \[prop:prop1\] and is again by by Lemma \[lemma:autolemma\] is an automorphism.
Let $$G_p^{rf}=\mathbb{Z}_{p^{r_1}}\oplus\cdots\oplus\mathbb{Z}_{p^{r_n}},\ r_i<r_{i+1}\ for\ all\ i$$ be a repeat-free $p$-group. Then an element $g \in G_p^{rf}$ is reduced if and only if $g$ is a representative element.
[*Proof:*]{}
Let $g=(p^{l_1},\ldots,p^{l_n})$ be a reduced element of $G_p^{rf}$ and let $1 \leq i < j \leq n$.
[**Case 1:**]{} If $p^{l_i} \geq p^{l_j}$, then the order of $p^{l_i}$ in $\mathbb{Z}_{p^{r_i}}$ is strictly less than the order of $p^{l_j}$ in $\mathbb{Z}_{p^{r_j}}$. Therefore, we can perform a basic reduction about position $j$, contradicting the assumption that $g$ is reduced.
[**Case 2:**]{} If $p^{l_i} < p^{l_j}$ and the order of $p^{l_i}$ in $\mathbb{Z}_{p^{r_i}}$ is greater than or equal to the order of $p^{l_j}$ in $\mathbb{Z}_{p^{r_j}}$, then we can perform a basic reduction about position $i$ again contradicting the assumption that $g$ was reduced. Therefore, if $g$ in $G_p^{rf}$ is reduced, then it is a representative element.
Now suppose that $g$ is a representative element. Let $1 \leq i <
j \leq n$ so that $p^{l_i} < p^{l_j}$ and the order of $p^{l_i}$ in $\mathbb{Z}_{p^{r_i}}$ is strictly less than the order of $p^{l_j}$ in $\mathbb{Z}_{p^{r_j}}$, when $p^{l_i}$ and $p^{l_j}$ are non-zero. If we performed a basic reduction about position $i$, then $p^{l_j}$ would be unchanged since the order of $p^{l_i}$ is less than that of $p^{l_j}$. Similarly, if we performed a basic reduction about position $j$, $p^{l_i}$ would remain unchanged since $p^{l_i} < p^{l_j}$. Since $i$ and $j$ are arbitrary if $g$ is a representative element, then $g$ is reduced.
We are now in a position to prove the main result for the case of repeat-free $p$-groups.
[*Proof:*]{}
(Theorem \[thm:main\] for repeat-free $p$-groups) Let $g=(g_1,\ldots,g_n)$ be an element in a finite repeat-free $p$-group $G_p^{rf}$. Applying Proposition \[prop:prop1\] we know that $g$ is automorphically equivalent to some $g'$ in which each term is a power of $p$. Now, if $g'$ is not a reduced element, then we can perform a non-trivial basic reduction and the the resulting element, which is automorphically equivalent to $g'$, will have strictly fewer non-zero terms. Since the rank of $G_p^{rf}$ is finite the process of making non-trivial reductions will terminate, and the resulting element will be a reduced element and therefore a representative element.
Finally we show that representative elements of automorphism classes are unique.
If $g$ and $h$ are automorphically equivalent representative elements in $G_p$, then $g=h$.
[*Proof:*]{}
(of claim) Let $g=(g_1,\ldots,g_n)$, $h=(h_1,\ldots,h_n)$ and let $A$ be the matrix representing the automorphism taking $h$ to $g$. Suppose $h_i \neq 0$ and let $p^{l_i}$, $0 \leq l_i \leq r_i$ be the order of $h_i$ in $\mathbb{Z}_{p^{r_i}}$ so that we may write $h_i=p^{r_i-l_i}$. By Lemma \[lemma:autolemma\] the automorphism $A$ must have the form $$A=
\left[
\begin{array}{cccc}
a_{11} & a_{12} & \cdots & a_{1n} \\
p^{r_2-r_1} a_{21} & a_{22} & \cdots & a_{2n}\\
\vdots & & & \vdots \\
p^{r_n-r_1} a_{n1} &p^{r_n-r_2} a_{n2} & \cdots & a_{nn}\\
\end{array}
\right]$$
where $a_{ii}$ is relatively prime to $p$. Therefore, assuming that $h_i\neq 0$, $$g_i=p^{r_i-l_1} a_{i1}+\cdots+p^{r_i-l_{i}}a_{ii}+\cdots+p^{r_n-l_{n}}a_{in}.$$
Since $h$ is a representative element, for any non-zero term $h_j$ if $j<i$, then the order of $h_j$ in $\mathbb{Z}_{p^{r_j}}$ is less than the order of $h_i$ in $\mathbb{Z}_{p^{r_i}}$; i.e. $l_j<l_i$. Hence, $r_i-l_j>r_i-l_i$ and $p^{r_i-l_i}$ divides $p^{r_i-l_j}$. If $j>i$, then $h_i<h_j$ and $p^{r_i-l_i}$ divides $p^{r_j-l_j}$. Thus $$g_i=p^{r_i-l_i}(p^{l_i-l_1} a_{i1}+\cdots+a_{ii}+\cdots+p^{r_n-l_{n}-(r_i-l_i)}a_{ni}).$$
Since $a_{ii}$ is relatively prime to $p$ $$b_i=(p^{l_i-l_1} a_{i1}+\cdots+a_{ii}+\cdots+p^{r_n-l_{n}-(r_i-l_i)}a_{ni}).$$
is also relatively prime to $p$. Thus, when $h_i$ is non-zero, $g_i=b_ih_i$, $b_i$ relatively prime to $p$. However $g$ is a representative element therefore $g_i$ is a power of $p$, $b_i=1$ and $g_i=h_i$.
An identical argument shows that $h_i=g_i$ when $g_i$ is non-zero. Therefore for all $i$, $g_i=h_i$ and so $g=h$.
If $G_p$ is any finite abelian $p$-group, then every element of $G_p$ is automorphically equivalent to a unique representative element in $G_p^{rf}$, which is also a representative element for $G_p$. Finally, if $G$ is any finitely generated abelian group, then we have established that any element of $G$ is automorphically equivalent to element satisfying the first two conditions of the definition of representative element. It can be shown that any element satisfying the first two conditions is automorphically equivalent to one that satisfies the third condition and has the same number or fewer nonzero terms. As an illustration consider the following example.
Let $G^{rf}=\mathbb{Z}_{p^{r_1}}\oplus\mathbb{Z}_{p^{r_2}}\oplus\mathbb{Z}$ and let $g=[p^{l_1},p^{l_2},kp]$ be any element of $G^{rf}$ satisfying the first two conditions of the definition of a representative element but not the third. Then $$\left[\begin{array}{ccc}
1 & 0 & p^{r_1-1}-p^{l_1-1} \\
0 & 1 & p^{r_2-1}-p^{l_2-1} \\
0 & 0 & 1
\end{array}\right]
\left[\begin{array}{c}p^{l_1} \\
p^{l_2} \\
kp \\
\end{array}\right]=
\left[\begin{array}{c}0 \\
0 \\
kp \\
\end{array}\right]$$ and so $g$ is automorphically equivalent to an element satisfying the third condition of our definition and has strictly fewer nonzero terms.
Counting Automorphism Classes
=============================
Having completed the main result we will now show how to count the number of representative elements in a given finite abelian group.
Let $G_p^{rf}=\mathbb{Z}_{p^{r_1}}\oplus\mathbb{Z}_{p^{r_2}}\oplus\cdots\oplus\mathbb{Z}_{p^{r_k}}$ where $r_i<r_{i+1}$ for all $i$ be a repeat-free $p$-group and $g=(p^{l_1},p^{l_2},\ldots,p^{l_k})$ be an automorphism class representative in $G_p^{rf}$. The non-zero terms of $g$ are both increasing and order increasing, thus for $0\leq i<j\leq k$ if $p^{l_i}$ and $p^{l_j}$ are non-zero terms of $g$ we know that $$p^{l_i} < p^{l_j}$$ and $$p^{r_i-l_i} < p^{r_j-l_j}.$$ Hence, $$l_i < l_j < l_i+(r_j-r_i).$$ We shall refer to the value $r_j-r_i$ as the [*gap*]{} between the terms $\mathbb{Z}_{p^{r_i}}$ and $\mathbb{Z}_{p^{r_j}}$ of $G_p^{rf}$, and in particular we will denote $r_{i+1}-r_i$ by $n_i$. As an immediate consequence of the above conclusion we know that if $p^{l_i}$ is non-zero, then either $p^{l_j}$ is zero or we have at most $r_j-r_i-1$ choices for $l_j$. Further, if $j=i+1$ and $n_i=r_{i+1}-r_i=1$, then one or the other of these two terms must be zero. Therefore, if the gap between two terms of a group is one, then in every automorphism class representative at least one of the terms will be zero.
For motivation let us count the number of automorphism class representatives in $G_p^{rf}=\mathbb{Z}_{p^{r_1}}\oplus\mathbb{Z}_{p^{r_2}}\oplus\mathbb{Z}_{p^{r_3}}$ where $r_1<r_2<r_3$. We begin by counting the number of automorphism classes with a given number of non-zero terms. $$\begin{array}{|c|l|}\hline
\mbox{\# of Non-Zero Terms} & \mbox{\# of Automorphism Classes}\\
\hline
0 & 1\\
1 & r_1+r_2+r_3\\
2 & r_1(n_{1}-1)+r_1(r_3-r_1-1)+r_2(n_{2}-1)\\
3 & r_1(n_{1}-1)(n_{2}-1)\\\hline
\end{array}$$
However $r_2=r_1+n_{1}$ and $r_3=r_1+n_{1}+n_{2}$ therefore the above equations can be rewritten as,
$$\begin{array}{|c|l|}\hline
\mbox{\# of Non-Zero Terms} & \mbox{\# of Automorphism Classes}\\
\hline
0 & 1\\
1 & r_1+(r_1+n_{1})+(r_1+n_{1}+n_{2})\\
2 & r_1(n_{1}-1)+r_1(n_{1}+n_{2}-1)+(r_1+n_{1})(n_{2}-1)\\
3 & r_1(n_{1}-1)(n_{2}-1)\\\hline
\end{array}$$
which gives us,
$$\begin{array}{|c|l|}\hline
\mbox{Non-Zero Terms} & \mbox{\# of Automorphism Classes}\\ \hline
0 & 1\\
1 & 3r_1+2n_{1}+n_{2}\\
2 & r_1(2n_{1}+2n_{2}-3)+n_{1}n_{2}-n_{1}\\
3 & r_1(n_{1}n_{2}-n_{1}-n_{2}+1)\\\hline
\end{array}$$
and the number of automorphism classes of elements equals
$$r_1(1+n_{1}+n_{2}+n_{1}n_{2})+1+n_{1}+n_{2}+n_{1}n_{2}=(r_1+1)(n_{1}+1)(n_{2}+1).$$
The number of automorphism classes of elements in a repeat-free finite abelian $p$-group $$G_p^{rf}=\mathbb{Z}_{p^{r_1}}\oplus\mathbb{Z}_{p^{r_2}}\oplus\cdots\oplus\mathbb{Z}_{p^{r_k}},\: r_i<r_{i+1}\ for\ all\ i$$ is equal to $$(r_1+1)(n_{1}+1)(n_{2}+1)\cdots(n_{k-1}+1),$$ where $$n_{i}=r_{i+1}-r_i.$$
[*Proof:*]{}
We have shown above that this is true for a group with three terms. We proceed by induction on the number of terms in $G_p^{rf}$. By induction the group $$H_p^{rf}=\mathbb{Z}_{p^{r_1}}\oplus\mathbb{Z}_{p^{r_2}}\oplus\cdots\oplus\mathbb{Z}_{p^{r_{k-1}}}$$ has $(r_1+1)(n_{1}+1)(n_{2}+1)\cdots(n_{k-2}+1)$ automorphism classes of elements. This is also the number of automorphism class representatives in $G_p^{rf}$ in which the $k^{th}$ term is 0. Therefore we complete the proposition with the following claim.
If $1<j\leq k$, then in $$G_p^{rf}=\mathbb{Z}_{p^{r_1}}\oplus\mathbb{Z}_{p^{r_2}}\oplus\cdots\oplus\mathbb{Z}_{p^{r_k}}$$ the number of automorphism class representatives in which the $j^{th}$ term is the last non-zero term is $$(r_1+1)(n_{1}+1)(n_{2}+1)\cdots(n_{j-2}+1)n_{j-1}.$$
[*Proof:*]{}
(of claim) If $j=2$, then there are $r_2$ automorphism class representatives in which the second term is the only non-zero term and $r_1(n_{1}-1)$ in which the first and second terms are the only non-zero terms. Therefore the number of automorphism class representatives in which the second term is the last non-zero term is
$$\begin{aligned}
r_1(n_{1}-1)+r_2&=&r_1n_{1}-r_1+r_1+n_{1}\\
&=&(r_1+1)n_{1}.\end{aligned}$$
Now let $2< j \leq k$, if $2\leq i <j$ and the $i^{th}$ term is the last non-zero term of an element prior to the $j^{th}$ term, then there are $(r_j-r_i-1)$ non-zero choices for the value of the $j^{th}$ term and by induction $$(r_1+1)(n_{1}+1)(n_{2}+1)\cdots(n_{i-2}+1)n_{i-1}(r_j-r_i-1)$$automorphism class representatives in which the $j^{th}$ term is the last non-zero term and the $i^{th}$ term is the second to last non-zero term. So letting $$A_i=(r_1+1)(n_{1}+1)(n_{2}+1)\cdots(n_{i-2}+1)n_{i-1}$$ and summing over all $i$ between $2$ and $j$, the total number of automorphism classes in which the $j^{th}$ term is the last non-zero term is $$A_j=r_j+r_1(r_j-r_1-1)+\sum_{i=2}^{j-1}A_i(r_j-r_i-1).$$ However for all $1\leq i < j \leq k$: $$r_j=r_{j-1}+n_{i-1}$$ and $$(r_j-r_i-1)=(r_{j-1}-r_i-1+n_{j-1})$$ so we may rewrite the previous equation as $$(r_{j-1}+n_{j-1})+(r_1(r_{j-1}-r_1-1)+r_1n_{j-1})+\sum_{i=2}^{j-1}(A_i(r_{j-1}-r_i-1)+A_in_{j-1})$$ And since $$A_{j-i}=r_{j-1}+r_1(r_{j-1}-r_1-1)+\sum_{i=2}^{j-2}A_i(r_{j-1}-r_i-1)$$ we get
$$\begin{aligned}
A_{j}&=&A_{j-1}+n_{j-1}+r_1n_{j-1}+A_{j-1}(r_{j-1}-r_{j-1}-1)+\sum_{i=2}^{j-1}A_in_{j-1}\\
&=&A_{j-1}-A_{j-1}+\left(r_1n_{j-1}+n_{j-1}+\sum_{i=2}^{j-1}A_in_{j-1}\right)\\
&=&r_1n_{j-1}+n_{j-1}+\sum_{i=2}^{j-1}A_in_{j-1}\\
&=&n_{j-1}\left((r_1+1)+\sum_{i=2}^{j-1}A_i\right)\\
&=&(r_1+1)(n_{1}+1)(n_{2}+1)\cdots(n_{j-2}+1)n_{j-1},\end{aligned}$$
thus proving the claim. In order to finish the proposition we observe that, by induction, the number of automorphism classes in which the $k^{th}$ term is zero is $$(r_1+1)(n_{1}+1)(n_{2}+1)\cdots(n_{k-2}+1)$$ and from the claim the number of automorphism classes in which the $k^{th}$ term is non-zero is $$(r_1+1)(n_{1}+1)(n_{2}+1)\cdots(n_{k-2}+1)n_{k-1},$$ and the sum of these two gives the desired result.
Since automorphisms respect the prime decomposition of finite abelian groups we get this final general result.
The number of automorphism classes of elements in a finite abelian group $$G=G_{p_1}\oplus\cdots\oplus G_{p_m}$$ where $$G_{p_j}=\mathbb{Z}_{p_j^{r_1}}^{k_1}\oplus\cdots\oplus\mathbb{Z}_{p_j^{r_n}}^{k_n},\: r_i<r_{i+1}$$ is equal to the product of the number of automorphism classes in each individual $G_{p_j}^{rf}$.
[9999]{}
C.F. Rocca Jr. and E.C. Turner, *Test Elements in Finitely Generated Abelian Groups*, Int. J. Algebra and Computation, Vol. 12, No. 4 (2002) pp.569-573.
J.C. O’Neill and E. C. Turner, *Test elements in direct products*, Int. J. Algebra and Comput, Vol. 10, No. 6 (2000), pp.751-756.
|
---
abstract: 'Many important forms of data are stored digitally in XML format. Errors can occur in the textual content of the data in the fields of the XML. Fixing these errors manually is time-consuming and expensive, especially for large amounts of data. There is increasing interest in the research, development, and use of automated techniques for assisting with data cleaning. Electronic dictionaries are an important form of data frequently stored in XML format that frequently have errors introduced through a mixture of manual typographical entry errors and optical character recognition errors. In this paper we describe methods for flagging statistical anomalies as likely errors in electronic dictionaries stored in XML format. We describe six systems based on different sources of information. The systems detect errors using various signals in the data including uncommon characters, text length, character-based language models, word-based language models, tied-field length ratios, and tied-field transliteration models. Four of the systems detect errors based on expectations automatically inferred from content within elements of a single field type. We call these [*single-field*]{} systems. Two of the systems detect errors based on correspondence expectations automatically inferred from content within elements of multiple related field types. We call these [*tied-field*]{} systems. For each system, we provide an intuitive analysis of the type of error that it is successful at detecting. Finally, we describe two larger-scale evaluations using crowdsourcing with Amazon’s Mechanical Turk platform and using the annotations of a domain expert. The evaluations consistently show that the systems are useful for improving the efficiency with which errors in XML electronic dictionaries can be detected.'
author:
-
-
bibliography:
- 'paper.bib'
title: Data Cleaning for XML Electronic Dictionaries via Statistical Anomaly Detection
---
=1
Introduction
============
There is increasing interest in the research, development, and use of automated techniques for assisting with [*data cleaning*]{}, also called [*data cleansing*]{} or [*scrubbing*]{}, which deals with detecting and removing errors and inconsistencies from data in order to improve the quality of data [@rahm2000]. In this paper we deal with data cleaning of electronic dictionaries stored in Extensible Markup Language (XML) format. XML is a markup language that defines a set of rules for encoding documents in a format that is both human-readable and machine-readable. Defined by free open standards, XML is a textual data format with strong support via Unicode for different human languages. It is widely used for the representation of electronic dictionaries and other forms of structured data [@ide1995; @francopoulo2006].
Electronic dictionaries are a fundamentally important semantic computing resource. They are a core resource consumed by downstream processes in the provision of various human language technologies as well as consumed directly by human language learners and as reference materials more generally. When dictionaries are digitized, whether via manual entry, Optical Character Recognition (OCR), or a mixture of these methods, it is inevitable that errors are introduced into the digitized version that is produced.
Prior work has focused on providing editing tools to assist with manual curation of the data and on providing tools for automatically detecting structural errors using only structural information. Although these systems have had success in finding dictionary errors, there are many errors that cannot be detected without analyzing the text content of the dictionary. For example, suppose that in some dictionary a lexical entry requires a headword, a part of speech, and a definition. Suppose there is an error in which the definition for some entry appears in the part of speech field, and the definition field contains the headword of the next entry. These errors will not be found by examining only the structure. Unlike previous work, in this paper we present methods for flagging statistical anomalies as likely errors in the textual content itself in electronic dictionaries in XML format using information from the textual content of the data.
We present six systems that detect errors using various signals in the data. The types of data quality problems that our systems are designed to detect are single-source instance level data quality problems [@rahm2000]. Four of our systems detect errors based on information automatically inferred from content within elements of a single field type. We call these systems [*single-field*]{} systems. Two of the systems detect errors based on correspondence information automatically inferred from content within elements of multiple related field types. We call these latter systems [*tied-field*]{} systems. For each system, we provide an intuitive analysis of the type of error that it is successful at detecting. Finally, we describe two larger-scale evaluations using crowdsourcing with Amazon’s Mechanical Turk platform and using domain expert annotations. The evaluations consistently show that the systems are useful for improving the efficiency with which errors in XML electronic dictionaries can be detected.
In the next section we situate our work with respect to previous related work. In section \[methods\] we describe the error detection systems in detail and provide intuitive examples of the sorts of errors that each error detection system finds. In section \[evaluation\] we describe our experimental tests and provide experimental results and discussion of results. Finally, in section \[conclusion\] we conclude.
Related Work {#relatedWork}
============
A categorization of data quality problems addressed by data cleaning and an overview of data cleaning methods is provided in [@rahm2000]. Many data cleaning methods are based on identifying discrepancies for user auditing, e.g., [@raman2001; @guyon1996]. The system in [@raman2001] is highly interactive; discrepancy detection is not their sole main focus. The discrepancy detection approach they outline is for users to define domains and then for discrepancies to be located through checking against the user-defined domains for constraint violation. In contrast, the methods in the current paper do not require user specification of domains and the methods in the current paper operate on the basis of different sources and indicators of discrepancies. In particular, the methods in the current paper are fundamentally different in that they operate on the basis of statistically anomalous events instead of constraint violations. In [@guyon1996], a similar overall workflow is presented whereby the most suspicious examples are flagged for a human operator to inspect and annotate as clean or “garbage." In contrast to the current paper, there is only one method in [@guyon1996], which is to flag the examples with the highest information gain for inspection by a human operator. The work in [@guyon1996] assumes the context of construction of an automated classifier in the computation of the information gain. The method in [@guyon1996] is evaluated on the task of handwritten digit recognition by seeing how well classifier performance improves with the addition of the data cleaning approach. In contrast, the current paper uses different methods for detecting suspicious examples and does not assume the context of construction of automated classifiers. Also in contrast, the methods in the current paper are evaluated on XML electronic dictionaries by measuring how many and what percentages of the detected anomalies are annotated as data errors by domain experts. The methods in the current paper may be able to be used in a complementary fashion with the methods from [@raman2001] and [@guyon1996] in future work.
Past work presented a method for repairing a digital dictionary in an XML format using a dictionary markup language called DML [@zajic2011]. It remains time-consuming and error-prone however to have a human exhaustively read through and manually correct a digital version of a dictionary, even with languages such as DML available for making corrections once errors are detected. The methods we present in the current paper automatically scan through and detect errors in dictionaries. The methods in the current paper can be used in concert with error correction techniques such as dictionary markup languages.
Previous approaches have been presented for detecting structural errors in digitized dictionaries [@bloodgood2012; @rodrigues2011]. The method in [@rodrigues2011] works by linearizing the lexicon structure, converting the opening tags in XML into tokens and then considering the likelihoods of various strings of tokens using a language modeling approach. Anomalous branches of XML tags are flagged as structural errors. The method ignores the underlying text data within the dictionary and only detects structural errors. The methods in [@bloodgood2012] outperform the method from [@rodrigues2011]. The methods in [@bloodgood2012] use a mixture of unsupervised methods, supervised machine learning methods, and system combination approaches. The highest-performing method uses a random forest system combination approach. The methods in [@bloodgood2012] only detect structural errors. Errors in the textual data content within the XML elements are not detected. In contrast, the current paper presents methods that detect errors in the text (data) content of XML elements. The methods in the current paper also use different approaches and different sources of information than were used in [@bloodgood2012; @rodrigues2011] and do not require training data, which is often not available. The error detection methods in the current paper can be used in concert with structural error detection methods.
Methods
=======
This section describes how our methods work. We categorize our methods into two types: [*single-field*]{} methods and [*tied-field*]{} methods. Single-field methods work by utilizing information within the data content of a single XML field type in order to detect errors. Tied-field methods work by utilizing information within the data content of multiple XML field types, exploiting various relationships between the data in the different fields, in order to detect errors.
For each candidate error, all of our methods return a numeric score indicating the system’s confidence that the candidate is an error. A threshold can be set for each method to control which candidates are detected as errors. The threshold for each method can be adjusted according to recall-precision[^1] preferences and dictionary characteristics. In general, higher thresholds will return results with higher precision and lower recall whereas lower thresholds will return results with lower precision and higher recall. The optimal threshold for each method depends on data characteristics and user preferences. We are not aware of a method for determining optimal thresholds. We set our thresholds to a level that yielded a reasonable number of error candidates for human review. The exact thresholds for each experiment are given in the following subsections.
Subsection \[single-FieldMethods\] describes how the single-field methods work and subsection \[tied-FieldMethods\] describes how the tied-field methods work. Subsection \[examples\] provides examples of anomalies detected by the various methods.
Single-Field Methods {#single-FieldMethods}
--------------------
The single-field error detection systems do not require any advance knowledge about the structure of the electronic dictionary. The only structural context information they use is the tag name of the elements containing the text data to be checked. Single-field methods can be used to check for errors in elements of any individual tag name. All single-field methods work according to the following high-level description: all entries of an individual tag name are processed and then any entries that are anomalous are flagged as errors. Each single-field method processes the entries and flags anomalies in different ways based on different aspects of the data. The rest of this subsection describes four single-field methods in detail.
### Uncommon Characters Method {#uncommonCharactersMethod}
Uncommon characters can be a frequent source of errors in electronic dictionaries. They can arise due to OCR errors, author typographical errors, and mislabeled and/or incorrectly merged fields. Texts that contain uncommon characters are reported as potential errors. For each element in the dictionary with a particular tag, we consider the texts inside those elements to be a collection of documents $D$, and the characters in the texts as the tokens. We calculate the inverse document frequency of each character $c$ observed in $D$ as follows: $$\label{idf}
idf(c,D) = \log_{10} \frac{N}{|\{ d \in D : c \in d \}|},$$ where $N$ is the number of documents in the collection.
When $idf(c,D) > threshold$, we consider $c$ to be an uncommon character. The threshold is configurable; we use a default value of four. Users can adjust the threshold according to their recall-precision preferences and according to dictionary characteristics. The elements containing uncommon characters are flagged by the system as potential errors.
### Text Length Method {#textLengthMethod}
When two text fields are inappropriately combined into one field, the result can be text that is unusually long. When a single text unit is inappropriately truncated, or split across two fields, the result can be text that is unusually short. Texts with unusually long or unusually short length are reported as potential errors. For each element in the dictionary with a given tag, we treat the lengths of the texts inside those elements as a sample of a normally distributed population. We calculate the mean and standard deviation of the sample, and for each value, we calculate the z-score, i.e., the signed number of standard deviations the value is above or below the mean. If the absolute value of the z-score of the length of a text is above a threshold, we flag the text as a possible error. The threshold is configurable; we use a default value of four. As the threshold is raised, only the most unusually long, or short, fields will be returned as errors.
### Word Sequence Method {#wordSequenceMethod}
Language modeling can capture the probability of a sequence of words occurring in textual content. This gives us the capability to flag unlikely sequences of words. These unlikely sequences can often be indicative of typographical errors, OCR errors, incorrect field joining and splitting, etc. Texts that are unlikely given a word-level language model of texts in the same context are reported as potential errors. For each element in the dictionary with a given tag, we build a language model of the text content of all the elements using n-grams of words. We calculate the entropy of the model with respect to each individual element’s text, and treat the entropies as a sample of a normally distributed population. We calculate the z-score of each entropy score. High z-scores indicate texts that are unlikely in the language model. The texts with entropy z-scores above a threshold are flagged as errors. The size of the n-grams in the language model and the threshold are configurable. We use 4-grams and threshold five by default. Using a larger n-gram size in the language model allows one to potentially capture more nuanced sequence characteristics, however, it would require a much larger amount of data to estimate properly and avoid introducing spurious estimates due to data sparsity. Also, larger n-gram sizes are more computationally intensive. The entropy z-score thresholds can be adjusted according to recall-precision preferences and dictionary characteristics.
### Character Sequence Method {#characterSequenceMethod}
This method is similar to the Word Sequence Method, except that instead of n-grams of words, we build language models using n-grams of characters. The size of the n-grams in the language model and the threshold are configurable. We use 4-grams and a threshold of five by default. Larger n-gram sizes could potentially capture more nuanced character sequence models, however, they would require a much larger amount of data to estimate properly and avoid introducing spurious estimates due to data sparsity. The n-gram size can also be adjusted based on the language of the field’s textual content. The entropy z-score thresholds can be adjusted according to recall-precision preferences and dictionary characteristics. Note that the error detection system based on character sequences will in some cases find errors that the Uncommon Character system also detects, but the Character Sequence Method is also capable of finding some errors that the Uncommon Character system is not able to find. This is because the Character Sequence Method can find errors in which none of the characters in the textual content is particularly uncommon, but in which the ordering of those characters is incorrect.
Tied-Field Methods {#tied-FieldMethods}
------------------
In many structured data sets, there are pairs of fields that are related to each other in predictable ways. For example, in dictionaries a word in a language’s native orthography and a phonetic transcription of the word are related because in many languages there are predictable relationships between spelling and pronunciation. Another related pair of fields in bilingual dictionaries is an example sentence demonstrating usage of a word and its translation. We call these sorts of related fields [*tied fields*]{} and we call error detection methods that exploit relationships between content in different field types [*tied-field*]{} methods. It is possible for there to be errors in a single field where the data value in that field is not anomalous in the context of only other values in that field type. However, the value might be anomalous in the context of data values in related fields. The single-field methods presented in subsection \[single-FieldMethods\] will be unable to detect these sorts of errors. The rest of this subsection describes two tied-field methods that can detect these sorts of errors.
### Tied-Field Length Ratio Method {#tied-FieldLengthRatioMethod}
This method determines the ratio of length in characters of tied fields; pairs of data values with unusual length ratios are then reported as potential errors. We treat the ratio of the length in characters of the tied-field data values to be a sample of a normally distributed population. We calculate the absolute value of the z-score of each length ratio. High values indicate tied-field instances where the data values have an unusual length ratio. Tied-field instances with scores above a threshold are flagged as potential errors. The threshold is configurable; we use a threshold of two by default. The threshold can be adjusted according to recall-precision preferences and dictionary characteristics.
To handle situations in which the distribution of length ratios is significantly different for short and long strings, we have an option to partition the tied-field instances by the length of the data in the first field, and treat each partition as its own population. Table \[t:shortAndLongStrings\] illustrates why it could be beneficial to partition tied-field instances by length. In these examples of correct pairs of data, the length ratios of the short tied-field pairs of data can be seen to vary more than the length ratios of the long tied-field pairs of data.
------ ------ ------- ------------ ----------------------------------------------------------- -------
Orth Pron Ratio Orth Pron Ratio
t tē 0.50 groundwork ground"wûrk‘ & 0.83\
ease & ēz & 2.00 & lithargyrum & lĭ\*thär“jĭ\*rŭm & 0.79\
v & vē & 0.50 & haidingerite & hī”dĭng\*er\*īt & 0.92\
------ ------ ------- ------------ ----------------------------------------------------------- -------
: Examples of orthography-pronunciation pairs. Short strings have a different distribution of length ratios than long strings.[]{data-label="t:shortAndLongStrings"}
### Tied-Field Transliteration Method {#tied-FieldTransliterationMethod}
For some tied fields, the data in the two fields have a more specific relationship than a length relationship. For example, in dictionaries some fields are transliterations of each other. Such transliterations will usually have character-level correspondences (not necessarily a one-to-one correspondence). If the correspondence can be modeled, then pairs of texts that do not correspond well can be reported as errors. We use Phonetisaurus[^2] to learn transliteration models across tied-fields. The Phonetisaurus system has been described in detail and has been shown to perform well in [@novak2012]. The resulting transliteration model represents how the first field can be transliterated into the second field, and can be used to generate scored transliteration candidates of the first field. For each pair of tied-fields, we use the transliteration model to transliterate the first field into the n-best candidates for the second field. Each candidate is given a transliteration cost by Phonetisaurus. By taking the inverse of the cost, we obtain a score indicating the model’s confidence in that transliteration candidate. We calculate the normalized edit distance (NED) of each candidate to the observed data that actually is present in the second field, and calculate the mean NED weighted by transliteration score. We treat the weighted means as a sample of a normally distributed population. We calculate the z-score of each weighted mean. High values indicate that the data occurring in the second field is an unusually large NED away from what our learned model would have expected based on the data that occurred in the first field. Instances with weighted mean z-scores above a threshold are flagged as errors. The threshold is configurable; we use a default of two. The weighted mean z-score threshold can be adjusted according to recall-precision preferences and dictionary characteristics.[^3]
Examples
--------
In this subsection we provide examples of anomalies detected by the various systems. To obtain these examples, we ran the systems over a digitized sample of an Urdu-English dictionary [@qureshi1991] and selected illustrative examples that can be understood by most readers without having to know too many details about specific dictionary representations in XML. Table \[t:examplesSingleField\] shows examples detected by the various single-field systems and Table \[t:examplesTiedField\] shows an example detected by the tied-field systems.
ExampleID FieldName Value
----------- ---------------- ----------
Example 1 GENDER /F.
Example 2 NUMBER PLU.
Example 3 PART-OF-SPEECH PARTICLE
: Examples of anomalies detected by single-field systems.[]{data-label="t:examplesSingleField"}
ExampleID FieldName Value
----------- --------------- ---------------- -- --
Example 4 ORTHOGRAPHY رجعت قہقری>
PRONUNCIATION rā
: Example anomaly detected by tied-field systems.[]{data-label="t:examplesTiedField"}
Example 1 in Table \[t:examplesSingleField\] was detected by all four of our single-field systems. This is an error in the data since the value should have been just “F.” without the “/”. Since the GENDER field almost exclusively contains values of “M.” or “F.”, the Word Sequence Method found any other words such as “/F.” to have unusually high entropy. The character-based language model detected this error similarly. The Text Length Method detected this error since it is three characters long, which is unusually long given the predominance of two-character-long values in this field. The Uncommon Characters Method detected this error since the “/” character is uncommon in this field. Example 1 shows how the different methods can sometimes all detect the same error albeit through different views of the data.
Example 2 in Table \[t:examplesSingleField\] shows an example of an error that was detected by some of the systems and not others. This is an error in the data since the value should have been just “PL.” without the “U”. The Word Sequence Method, the Character Sequence Method, and the Uncommon Characters Method all found this error. The Uncommon Characters Method detected this error since “U” is an uncommon character for values in the NUMBER field. The Text Length Method did not detect this error. This is because the length of four characters is not unusually long or short for this field - the NUMBER field often contains “PL.” with length three characters and “SING.” with length five characters.
Example 3 in Table \[t:examplesSingleField\] shows an example of an anomaly that was detected by all four single-field methods that was not erroneous. The Word Sequence Method and the Character Sequence Method detected it since it had unusually high entropy. The Text Length Method detected it since it was unusually long. The Uncommon Characters Method detected it since the characters “C", “E", “L", “P", and “R" are uncommon for this field. For reference, some of the most common values for the part of speech field include “V.", “N.", “ADJ.", “ADV.", etc. Although “PARTICLE" is an uncommon value for this field, it is not an error.
Example 4 in Table \[t:examplesTiedField\] shows an example detected by both the Tied-Field Text Length Ratio Method and the Tied-Field Transliteration Method. The ORTHOGRAPHY field had the value “رجعت قہقری>" and the PRONUNCIATION field had the value “rā". This is an error resulting from incorrect merging and splitting of fields. The Text Length Ratio Method detected this as an error since the ratio was unusually high for values of these two fields. The Transliteration Method detected this as an error since the weighted mean Normalized Edit Distance from the generated pronunciation candidates to the observed pronunciation “rā" was unusually high. Note that the single-field methods did not detect this error since “رجعت قہقری>" is not an anomalous value for ORTHOGRAPHY in isolation and neither is “rā" an anomalous value for PRONUNCIATION in isolation. It is only when they are tied to each other that an anomaly is detected.
The examples help to illustrate on a small scale what types of errors the various methods can detect. Also, the examples show how sometimes the methods have overlapping behavior and sometimes the methods have complementary behavior. The examples also illustrate how sometimes the methods detect errors in the data that need to be corrected and sometimes the methods detect anomalies that, while rare, are not errors that need to be corrected. In the next section we provide larger-scale evaluations of the error detection methods.
Evaluation
==========
We used Amazon’s Mechanical Turk crowdsourcing platform to evaluate the Tied-Field Length Ratio Method, the Tied-Field Transliteration Method, and a random sample of data. Mechanical Turk is an online crowdsourcing platform where workers, also called Turkers, complete simple tasks called Human Intelligence Tasks (HITs). Crowdsourcing can allow inexpensive and rapid data collection for various Natural Language Processing (NLP) tasks [@snow2008; @bloodgood2010a; @negri2011], including human evaluations of NLP systems [@callison-burch2009; @bloodgood2010b; @amigo2010; @chen2011; @bloodgood2014].
The tied fields that we used in our evaluation were the orthography and the corresponding pronunciation fields from the GNU Collaborative International Dictionary of English (GCIDE). GCIDE is a freely available dictionary of English based on Webster’s 1913 Revised Unabridged Dictionary and supplemented with entries from WordNet [@miller1990; @miller1995; @fellbaum1998] and additional submissions from users. GCIDE is formatted in XML and is available for download from [www.ibiblio.org/webster/](www.ibiblio.org/webster/). Out of the 16704 pairs of orthography-pronunciation values in the dictionary, our tied-field error detection systems identified 2797 of the pairs as possible errors. From this set of detected candidate errors, we randomly selected 1000 pairs detected by the tied-field length ratio system and 1000 pairs detected by the tied-field transliteration system for evaluation by Turkers.
For each candidate error, we asked five Turkers if the orthography-pronunciation pair was correct. Figure \[f:screenshot\] shows a screenshot of the Mechanical Turk interface we used. If a Turker judged that a pair was not correct, i.e., that the pair was truly an error, then the Turker was required to provide an explanation. By requiring an explanation when pairs are incorrect, we are, if anything, creating a bias where workers will tend towards saying pairs are correct since that is easier for them. This will cause, if anything, the efficacy of our error detection systems to be understated.
![Screenshot of interface for evaluating tied-field error detection systems.[]{data-label="f:screenshot"}](figures/screenshot){width="2.5in"}
Figure \[f:counts\] displays counts of proposed orthography-to-pronunciation errors judged to be real errors by the Turkers for the tied-field transliteration and the tied-field length ratio error detection systems as well as for randomly selected examples. The x-axis shows the number of Turkers that agreed a proposed error was a real error. Recall that for each of the proposed errors we had asked five Turkers to judge whether it was a real error or not. The y-axis shows the number of proposed errors that were judged to be real errors. The main observation is that both the tied-field transliteration system and the tied-field length ratio system find many more errors than the random selection system. In particular, observe that for the tied-field transliteration system three or more Turkers agree its proposed errors are really errors more than 60% of the time; for the tied-field length ratio system three or more Turkers agree its proposed errors are really errors more than 76% of the time. In contrast, for randomly selected proposed errors, three or more Turkers agree they are really errors only about 45.5% of the time. These results are evidence that the error detection systems could substantially increase the efficiency with which domain experts can clean XML data.
![Counts of proposed errors judged to be real errors by at least 5, 4, 3, 2, 1, and 0 Turkers.[]{data-label="f:counts"}](figures/counts){width="2.5in"}
Diving a little deeper, Figure \[f:lengthRatioScoreCutoff\] and Figure \[f:transliterationScoreCutoff\] show the average number of Turkers that agree a proposed error is really an error for proposed errors at varying score cutoffs. Figure \[f:lengthRatioScoreCutoff\] shows the information for the tied-field length ratio error detection system. In Figure \[f:lengthRatioScoreCutoff\], the score cutoff on the x-axis is the absolute value of the z-score. The absolute value is used because this error detection system finds errors with both unusually high and unusually low z-scores. The y-axis is the average number of Turkers (out of 5) who marked the proposed errors with scores above the corresponding cutoff as real errors. Figure \[f:transliterationScoreCutoff\] shows similar information for the tied-field transliteration error detection system. In Figure \[f:transliterationScoreCutoff\], the score cutoff on the x-axis is the z-score. The z-score is used because this error detection system finds errors with unusually high z-scores. The y-axis is the average number of Turkers (out of 5) who marked the proposed errors with scores above the corresponding score cutoff as real errors.
For systems that will be used for purposes of ranking proposed errors in order from most likely to least likely, it is desirable that they predict errors with higher accuracy above a particular score threshold. This has positive implications for application settings where users will go through errors in a sorted order from most likely to least likely. Figure \[f:lengthRatioScoreCutoff\] and Figure \[f:transliterationScoreCutoff\] show that from this perspective the length ratio system produces better results. The length ratio system correctly predicts consistently with a cutoff over 5.5 standard deviations from the average. In contrast, the transliteration system does not perform as well because it is unable to predict errors with as high a degree of precision as the length ratio system at any score cutoff. A perhaps surprising result in Figure \[f:transliterationScoreCutoff\] is that the transliteration system decreases in precision when the score cutoff increases past about 2.5 standard deviations from the average. This result can be explained by the low number of proposed errors with very high scores with the transliteration system. This can create the situation where there are too few data points to compute a precision score reliably.
![The average number of Turkers (out of 5) who believe the orthography-pronunciation pair is an error for varying score cutoffs for the length ratio system. The score is the absolute value of the z-score.[]{data-label="f:lengthRatioScoreCutoff"}](figures/lengthRatioScoreCutoff){width="2.5in"}
![The average number of Turkers (out of 5) who believe the orthography-pronunciation pair is an error for varying score cutoffs for the transliteration system. The score is the z-score.[]{data-label="f:transliterationScoreCutoff"}](figures/transliterationScoreCutoff){width="2.5in"}
Table \[t:overlap\] displays the overlap of the errors most strongly proposed by the length ratio error detection system and the transliteration error detection system for varying cutoff sizes. This table allows us to evaluate the similarity between the two systems. The two systems do not have a high degree of similarity; for example, only 4 of the top 100 anomalies detected by the length ratio system also appear in the top 100 anomalies detected by the transliteration system. The low levels of overlap indicate that the systems have complementary behavior. This complementary behavior can be leveraged to build improved hybrid systems.
The four single-field methods can also be combined with each other and with the tied-field methods. By combining methods a hybrid system could be built that takes into account different perspectives on the data. There are many possible ways of combining methods to build a hybrid system, e.g., see [@bloodgood2012]. In future work, it would be worthwhile to explore how to optimally combine the error detection methods to create an improved system.
Number of Proposed Errors Number of Common Results Percent
--------------------------- -------------------------- ---------
10 0 0%
25 1 4%
50 3 6%
100 4 4%
200 16 8%
300 31 10%
400 52 13%
500 81 16%
600 110 18%
700 143 20%
800 173 22%
900 186 21%
1000 227 23%
1500 396 26%
2000 563 28%
: Overlap of the anomalies most strongly proposed by the length ratio and the transliteration systems.[]{data-label="t:overlap"}
We conducted an additional evaluation of our systems using the annotations of a language expert in the process of correcting errors in a Tamasheq dictionary containing 5968 lexical entries. The language expert evaluated 175 anomalies proposed by the single-field text length system and the single-field uncommon character system for data in fields POS (part of speech) and MAIN (headword). The language expert marked each proposed anomaly as either a real error or not a real error. The results are in Table \[t:languageExpertResults\]. These results are further evidence that the error detection systems could substantially increase the efficiency with which domain experts can clean XML data.
System Field Real Error No Error Total
------------------ ------- ------------ ---------- -------
Uncommon POS 16 1 17
Character System MAIN 4 10 14
Text Length POS 110 3 113
System MAIN 30 1 31
: Language expert annotations on 175 anomalies proposed by the text length system and the uncommon character system for data in fields POS and MAIN.[]{data-label="t:languageExpertResults"}
Conclusion
==========
There is increasing interest in methods for computer-aided rapid data cleaning. We presented multiple methods for data cleaning of XML electronic dictionaries. The methods detect errors in the data content of the XML, unlike previous work that detected errors in the structure of the XML electronic dictionaries and ignored the content. The methods are based on different underlying sources and indicators of errors and have complementary behavior with each other and with previously developed methods. The methods can be classified into single-field methods and tied-field methods. Single-field methods detect anomalies on the basis of expectations inferred from content in the same single field type. Tied-field methods detect anomalies on the basis of expectations of content correspondences inferred from content in multiple related fields. Four single-field methods for error detection were presented that work by using expectations of word sequence information, expectations of character sequence information, expectations of length, and expectations of individual characters in particular fields. Two tied-field methods for error detection were presented that work by using expectations of length ratios and expectations of string correspondences via transliteration models.
The precision of the error detection systems tends to correlate with the internal scores of the systems. This desirable behavior supports the scenario where a domain expert would go down a sorted list of proposed errors from most likely to least likely. The performance of the different error detection systems varies based on the XML fields and the dictionaries on which they are applied. Domain experts can choose to invoke error detection system-field combinations with score cutoffs to suit their needs.
We evaluated these methods using the crowdsourcing platform Mechanical Turk and using expert annotations on multiple datasets. Sometimes the systems have overlapping behavior, detecting the same errors albeit through different views of the data. Often the systems have complementary behavior, which is promising for hybrid system construction in the future. We provided intuitive examples of the sorts of errors each of the systems can detect. In the evaluations, the systems are consistently helpful in increasing the efficiency with which data errors can be identified.
Acknowledgment {#acknowledgment .unnumbered}
==============
This material is based upon work supported, in whole or in part, with funding from the United States Government. Any opinions, findings and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the University of Maryland, College Park and/or any agency or entity of the United States Government.
[^1]: Recall and precision are standard measures for systems that perform search. For the case of detecting dictionary errors, recall is the percentage of true errors that are found by the system. Precision is the percentage of system-proposed errors that are in fact true errors in the dictionary.
[^2]: <https://code.google.com/p/phonetisaurus/>
[^3]: We also offer an option to normalize the tied-field text data to all lowercase letters.
|
---
abstract: 'We survey research relating algebraic properties of powers of squarefree monomial ideals to combinatorial structures. In particular, we describe how to detect important properties of (hyper)graphs by solving ideal membership problems and computing associated primes. This work leads to algebraic characterizations of perfect graphs independent of the Strong Perfect Graph Theorem. In addition, we discuss the equivalence between the Conforti-Cornuéjols conjecture from linear programming and the question of when symbolic and ordinary powers of squarefree monomial ideals coincide.'
address:
- 'Department of Mathematics, Oklahoma State University, 401 Mathematical Sciences, Stillwater, OK 74078'
- |
Tulane University\
Department of Mathematics\
6823 St. Charles Ave.\
New Orleans, LA 70118, USA
- 'Department of Mathematics, Oklahoma State University, 401 Mathematical Sciences, Stillwater, OK 74078'
author:
- 'Christopher A. Francisco'
- Huy Tài Hà
- Jeffrey Mermin
title: Powers of squarefree monomial ideals and combinatorics
---
[^1]
Introduction {#intro}
============
Powers of ideals are instrumental objects in commutative algebra. In addition, squarefree monomial ideals are intimately connected to combinatorics. In this paper, we survey work on secant, symbolic and ordinary powers of squarefree monomial ideals, and their combinatorial consequences in (hyper)graph theory and linear integer programming.
There are two well-studied basic correspondences between squarefree monomial ideals and combinatorics. Each arises from the identification of squarefree monomials with sets of vertices of either a simplicial complex or a hypergraph. The Stanley-Reisner correspondence associates to the nonfaces of a simplicial complex $\Delta$ the generators of a squarefree monomial ideal, and vice-versa. This framework leads to many important results relating (mostly homological) ideal-theoretic properties of the ideal to properties of the simplicial complex; see [@Bruns-Herzog Chapter 5] and [@Peeva Sections 61-64].
The edge and cover ideal constructions identify the minimal generators of a squarefree monomial ideal with the edges (covers) of a simple hypergraph. The edge ideal correspondence is more naïvely obvious but less natural than the Stanley-Reisner correspondence, because the existence of a monomial in this ideal does not translate easily to its presence as an edge of the (hyper)graph. Nevertheless, this correspondence has proven effective at understanding properties of (hyper)graphs via algebra. We focus on powers of squarefree monomial ideals when they are viewed as edge (or cover) ideals of hypergraphs. To the best of our knowledge, there has been little systematic study of the powers of squarefree ideals from the Stanley-Reisner perspective.
The general theme of this paper is the relationship between symbolic and ordinary powers of ideals. This topic has been investigated extensively in the literature (cf. [@BH; @ELS; @HoHu; @HKV]). Research along these lines has revealed rich and deep interactions between the two types of powers of ideals, and often their equality leads to interesting algebraic and geometric consequences (cf. [@HHT; @MT; @TTerai; @TT; @Varbaro]). We shall see that examining symbolic and ordinary powers of squarefree monomial ideals also leads to exciting and important combinatorial applications.
The paper is organized as follows. In the next section, we collect notation and terminology. In Section \[s:properties\], we survey algebraic techniques for detecting important invariants and properties of (hyper)graphs. We consider three problems:
1. computing the chromatic number of a hypergraph,
2. detecting the existence of odd cycles and odd holes in a graph, and
3. finding algebraic characterizations of bipartite and perfect graphs.
We begin by describing two methods for determining the chromatic number of a hypergraph via an ideal-membership problem, one using secant ideals, and the other involving powers of the cover ideal. Additionally, we illustrate how the associated primes of the square of the cover ideal of a graph detect its odd induced cycles.
The results in Section \[s:properties\] lead naturally to the investigation of associated primes of higher powers of the cover ideal. This is the subject of Section \[s:higherpowers\]. We explain how to interpret the associated primes of the $s^{\text{th}}$ power of the cover ideal of a hypergraph in terms of coloring properties of its $s^{\text{th}}$ expansion hypergraph. Specializing to the case of graphs yields two algebraic characterizations of perfect graphs that are independent of the Strong Perfect Graph Theorem.
Section \[s:packing\] is devoted to the study of when a squarefree monomial ideal has the property that its symbolic and ordinary powers are equal. Our focus is the connection between this property and the Conforti-Cornuéjols conjecture in linear integer programming. We state the conjecture in its original form and discuss an algebraic reformulation. This provides an algebraic approach for tackling this long-standing conjecture.
*We congratulate David Eisenbud on his $65^{\text{th}}$ birthday, and this paper is written in his honor.*
Preliminaries {#s:prel}
=============
We begin by defining the central combinatorial object of the paper.
A *hypergraph* is a pair $G=(V,E)$ where $V$ is a set, called the *vertices* of $G$, and $E$ is a subset of $2^{V}$, called the *edges* of $G$. A hypergraph is *simple* if no edge contains another; we allow the edges of a simple hypergraph to contain only one vertex (i.e., isolated *loops*). Simple hypergraphs have also been studied under other names, including *clutters* and *Sperner systems*. All hypergraphs in this paper will be simple.
A *graph* is a hypergraph in which every edge has cardinality exactly two. We specialize to graphs to examine special classes, such as cycles and perfect graphs.
If $W$ is a subset of $V$, the *induced sub-hypergraph* of $G$ on $W$ is the pair $(W,E_{W})$ where $E_{W}=E\cap 2^{W}$ is the set of edges of $G$ containing only vertices in $W$.
Throughout the paper, let $V=\{x_{1},\dots,x_{n}\}$ be a set of vertices. Set $S=K[V]=K[x_{1},\dots,x_{n}]$, where $K$ is a field. We will abuse notation by identifying the squarefree monomial $x_{i_{1}}\dots x_{i_{s}}$ with the set $\{x_{i_{1}},\dots, x_{i_{s}}\}$ of vertices. If the monomial $m$ corresponds to an edge of $G$ in this way, we will denote the edge by $m$ as well.
The *edge ideal* of a hypergraph $G=(V,E)$ is $$I(G)=(m:m\in E)\subset S.$$ On the other hand, given a squarefree monomial ideal $I\subset S$, we let $G(I)=(V,\operatorname{gens}(I))$ be the hypergraph associated to $I$, where $\operatorname{gens}(I)$ is the unique set of minimal monomial generators of $I$.
A *vertex cover* for a hypergraph $G$ is a set of vertices $w$ such that every edge hits some vertex of $w$, i.e., $w\cap e\neq \varnothing$ for all edges $e$ of $G$.
Observe that, if $w$ is a vertex cover, then appending a variable to $w$ results in another vertex cover. In particular, abusing language slightly, the vertex covers form an ideal of $S$.
The *cover ideal* of a hypergraph $G$ is $$J(G)=(w:w \text{ is a vertex cover of }G).$$
In practice, we compute cover ideals by taking advantage of duality.
\[d:alexdual\] Given a squarefree monomial ideal $I\subset S$, the *Alexander dual* of $I$ is $$I^{\vee}=\bigcap_{m\in \operatorname{gens}(I)} \mathfrak{p}_{m},$$ where $\mathfrak{p}_{m}=(x_{i}:x_{i}\in m)$ is the prime ideal generated by the variables of $m$.
Observe that if $I=I(G)$ is a squarefree monomial ideal, its Alexander dual $I^\vee$ is also squarefree. We shall denote by $G^*$ the hypergraph corresponding to $I^\vee$, and call $G^*$ the *dual hypergraph* of $G$. That is, $I^\vee = I(G^*)$. The edge ideal and cover ideal of a hypergraph are related by the following result.
\[alexduality\] The edge ideal and cover ideal of a hypergraph are dual to each other: $J(G)=I(G)^{\vee} = I(G^*)$ (and $I(G)=J(G)^{\vee}$). Moreover, minimal generators of $J(G)$ correspond to minimal vertex covers of $G$, covers such that no proper subset is also a cover.
Suppose $w$ is a cover. Then for every edge $e$, $w\cap e\neq \varnothing$, so $w\in \mathfrak{p}_{e}$. Conversely, suppose $w\in I(G)^{\vee}$. Then, given any edge $e$, we have $w\in \mathfrak{p}_{e}$, i.e., $w\cap e\neq \varnothing$. In particular, $w$ is a cover.
We shall also need generalized Alexander duality for arbitrary monomial ideals. We follow Miller and Sturmfels’s book [@MS], which is a good reference for this topic. Let $\mathbf{a}$ and $\mathbf{b}$ be vectors in $\mathbb{N}^n$ such that $b_i \le a_i$ for each $i$. As in [@MS Definition 5.20], we define the vector $\mathbf{a \setminus b}$ to be the vector whose $i^{\text{th}}$ entry is given by $$a_i \setminus b_i = \left\{
\begin{array}{l l}
a_i+1-b_i & \text{if } b_i \ge 1 \\
0 & \text{if } b_i=0. \\
\end{array}
\right.$$
\[def.generalalexanderdual\] Let $\mathbf{a} \in \mathbb{N}^n$, and let $I$ be a monomial ideal such that all the minimal generators of $I$ divide $\mathbf{x}^\mathbf{a}$. The *Alexander dual* of $I$ *with respect to* $\mathbf{a}$ is the ideal $$I^{[\mathbf{a}]} = \bigcap_{\mathbf{x}^\mathbf{b} \in \operatorname{gens}(I)} \,
(x_1^{a_1 \setminus b_1}, \dots, x_n^{a_n \setminus b_n}).$$
For squarefree monomial ideals, one obtains the usual Alexander dual by taking $\mathbf{a}$ equal to $\mathbf{1}$, the vector with all entries 1, in Definition \[def.generalalexanderdual\]. By Definition \[d:alexdual\], Alexander duality identifies the minimal generators of a squarefree ideal with the primes associated to its dual. The analogy for generalized Alexander duality identifies the minimal generators of a monomial ideal with the *irreducible components* of its dual.
A monomial ideal $I$ is *irreducible* if it has the form $I=(x_{1}^{e_{1}},\dots, x_{n}^{e_{n}})$ for $e_{i}\in {{\mathbb Z}}_{>0}\cup\{\infty\}$. (We use the convention that $x_{i}^{\infty}=0$.) Observe that the irreducible ideal $I$ is $\mathfrak{p}$-primary, where $\mathfrak{p}=(x_{i}:e_{i}\neq \infty)$.
Let $I$ be a monomial ideal. An *irreducible decomposition* of $I$ is an irredundant decomposition $$I=\bigcap Q_{j}$$ with the $Q_{j}$ irreducible ideals. We call these $Q_{j}$ *irreducible components* of $I$. By Corollary \[c:irreddecomp\] below, there is no choice of decomposition, so the irreducible components are an invariant of the ideal.
Let $I$ be a monomial ideal, and $\mathbf{a}$ be a vector with entries large enough that all the minimal generators of $I$ divide $\mathbf{x}^{\mathbf{a}}$. Then $(I^{[\mathbf{a}]})^{[\mathbf{a}]}=I$.
\[c:irreddecomp\] Every monomial ideal has a unique irreducible decomposition.
A recurring idea in our paper is the difference between the powers and symbolic powers of squarefree ideals. We recall the definition of the symbolic power.
For a squarefree monomial ideal $I$, the *$s^{\text{th}}$ symbolic power* of $I$ is $$I^{(s)}=\bigcap_{\mathfrak{p}\in\operatorname{Ass}(S/I)}\mathfrak{p}^{s}.$$ (This definition works because squarefree monomial ideals are the intersection of prime ideals. For general ideals (even general monomial ideals) the definition is more complicated.) In general we have $I^{s} \subseteq I^{(s)}$, but the precise nature of the relationship between the symbolic and ordinary powers of an ideal is a very active area of research.
In commutative algebra, symbolic and ordinary powers of an ideal are encoded in the symbolic Rees algebra and the ordinary Rees algebra. More specifically, for any ideal $I \subseteq S = K[x_1, \dots, x_n]$, the *Rees algebra* and the *symbolic Rees algebra* of $I$ are $${{\mathcal R}}(I) = \bigoplus_{q \ge 0}I^q t^q \subseteq S[t] \text{ and } {{\mathcal R}}_s(I) = \bigoplus_{q \ge 0} I^{(q)} t^q \subseteq S[t].$$ The symbolic Rees algebra is closely related to the Rees algebra, but often is richer and more subtle to understand. For instance, while the Rees algebra of a homogeneous ideal is always Noetherian and finitely generated, the symbolic Rees algebra is not necessarily Noetherian. In fact, non-Noetherian symbolic Rees algebras were used to provide counterexamples to Hilbert’s Fourteenth Problem (cf. [@Nagata; @Roberts]).
Chromatic number and odd cycles in graphs {#s:properties}
=========================================
In this section, we examine how to detect simple graph-theoretic properties of a hypergraph $G$ from (powers of) its edge and cover ideals. Since the results in this section involving chromatic number are the same for graphs as for hypergraphs, modulo some essentially content-free extra notation, we encourage novice readers to ignore the hypergraph case and think of $G$ as a graph.
Let $k$ be a positive integer. A *$k$-coloring* of $G$ is an assignment of colors $c_{1},\dots, c_{k}$ to the vertices of $G$ in such a way that every edge of cardinality at least 2 contains vertices with different colors. We say that $G$ is *$k$-colorable* if a $k$-coloring of $G$ exists, and that the *chromatic number* $\chi(G)$ of $G$ is the least $k$ such that $G$ is $k$-colorable.
Since loops do not contain two vertices, they cannot contain two vertices of different colors. Thus the definition above considers only edges with cardinality at least two. Furthermore, since the presence or absence of loops has no effect on the chromatic number of the graph, we will assume throughout this section that all edges have cardinality at least two.
For hypergraphs, some texts instead define a coloring of $G$ to be an assignment of colors to the vertices such that no edge contains two vertices of the same color. However, this is equivalent to a coloring of the one-skeleton of $G$, so the definition above allows us to address a broader class of problems.
\[running\] Let $G$ be the graph obtained by gluing a pentagon to a square along one edge, shown in Figure \[f:examplefigure\]. The edge ideal of $G$ is $I(G)=(ab,bc,cd,de,ae,ef,fg,dg)$. The chromatic number of $G$ is 3: for example, we may color vertices $a$, $c$, and $g$ red, vertices $b$, $d$, and $f$ yellow, and vertex $e$ blue.
![The graph $G$ in the running example[]{data-label="f:examplefigure"}](example){height="1.25in"}
The chromatic number of $G$ can be determined from the solutions to either of two different ideal membership problems.
Observe that a graph fails to be $k$-colorable if and only if every assignment of colors to its vertices yields at least one single-colored edge. Thus, it suffices to test every color-assignment simultaneously. To that end, let $Y_{1},\dots, Y_{k}$ be distinct copies of the vertices: $Y_{i}=\{y_{i,1},\dots, y_{i,n}\}$. We think of $Y_{i}$ as the $i^{\text{th}}$ color, and the vertices of $Y_{i}$ as being colored with this color. Now let $I(Y_{i})$ be the edge ideal $I = I(G)$, but in the variables $Y_{i}$ instead of $V$. Now an assignment of colors to $G$ corresponds to a choice, for each vertex $x_{j}$, of a colored vertex $y_{i,j}$; or, equivalently, a monomial of the form $y_{i_{1},1}y_{i_{2},2}\dots y_{i_{n},n}$. This monomial is a coloring if and only if it is not contained in the the monomial ideal $\widetilde{I}=I(Y_{1})+\dots + I(Y_{k})$. In particular, $G$ is $k$-colorable if and only if the sum of all such monomials is not contained in $\widetilde{I}$.
We need some more notation to make the preceding discussion into a clean statement. Let $\mathbf{m}=x_{1}\dots x_{n}$, let $T_{k}=K[Y_{1},\dots,Y_{k}]$, and let $\phi_{k}:S\to T_{k}$ be the homomorphism sending $x_{i}$ to $y_{1,i}+\dots + y_{k,i}$. Then $\phi_{k}(\mathbf{m})$ is the sum of all color-assignments, and we have shown the following:
\[l:secantcoloringlemma\] With notation as above, $G$ is $k$-colorable if and only if $\phi_{k}(\mathbf{m})\not\in\widetilde{I}$.
We recall the definition of the $k^{\text{th}}$ secant ideal. Secant varieties are common in algebraic geometry, including in many recent papers of Catalisano, Geramita, and Gimigliano (e.g., [@CGG]), and, as Sturmfels and Sullivant note in [@SS], are playing an important role in algebraic statistics.
Let $I\subset S$ be any ideal, and continue to use all the notation above. Put $T=K[V,Y_{1},\dots, Y_{k}]$ and regard $S$ and $T_{k}$ as subrings of $T$. Then the *$k^{\text{th}}$ secant power* of $I$ is $$I^{\{k\}}=S\cap\left(\widetilde{I}+\left(\{x_{i}-\phi_{k}(x_{i})\}\right)\right).$$
Lemma \[l:secantcoloringlemma\] becomes the following theorem of Sturmfels and Sullivant [@SS]:
\[t:secantcoloring\] $G$ is $k$-colorable if and only if $\mathbf{m}\not\in I(G)^{\{k\}}$. In particular, $$\chi(G) = \min \{ k ~|~ \mathbf{m} \not\in I(G)^{\{k\}} \}.$$
Let $G$ and $I$ be as in Example \[running\]. Then $I^{\{1\}}=I$ and $I^{\{2\}}=(abcde)$ both contain the monomial $abcdefg$. However, $I^{\{3\}}=0$. Thus $G$ is 3-colorable but not 2-colorable.
Alternatively, we can characterize chromatic number by looking directly at powers of the cover ideal.
Observe that, given a $k$-coloring of $G$, the set of vertices which are not colored with any one fixed color forms a vertex cover of $G$. In particular, a $k$-coloring yields $k$ different vertex covers, with each vertex missing from exactly one. That is, if we denote these vertex covers $w_{1},\dots, w_{k}$, we have $w_{1}\dots
w_{k}=\mathbf{m}^{k-1}$. In particular, we have the following result of Francisco, Hà, and Van Tuyl [@FHVTperfect].
\[t:covercoloring\] $G$ is $k$-colorable if and only if $\mathbf{m}^{k-1}\in J(G)^{k}$. In particular, $$\chi(G) = \min \{k ~|~ \mathbf{m}^{k-1} \in J(G)^k \}.$$
Let $J = J(G)$. Given a $k$-coloring, let $w_{i}$ be the set of vertices assigned a color other than $i$. Then $\mathbf{m}^{k-1}=w_{1}\dots w_{k}\in J^{k}$. Conversely, if $\mathbf{m}^{k-1}\in J^{k}$, we may write $\mathbf{m}^{k-1}=w_{1}\dots w_{k}$ with each $w_{i}$ a squarefree monomial in $J$. Assigning the color $i$ to the complement of $w_{i}$ yields a $k$-coloring: indeed, we have $\prod \frac{\mathbf{m}}{w_i} = \frac{\mathbf{m}^k}{\mathbf{m}^{k-1}}=\mathbf{m}$, so the $\frac{\mathbf{m}}{w_i}$ partition $V$.
In Example \[running\], let $\mathbf{m}=abcdefg$. The cover ideal $J(G)$ is $(abdf,acdf,bdef,aceg,bceg,bdeg)$. Because $J$ does not contain $\mathbf{m}^{0}=1$, $G$ is not $1$-colorable. All 21 generators of $J^{2}$ are divisible by the square of a variable, so $G$ is not $2$-colorable. Thus $\mathbf{m}\not\in J^{2}$, so $J$ is not $2$-colorable. However, $J^{3}$ contains $\mathbf{m}^{2}$, so $G$ is $3$-colorable.
\[r.bfold\] One can adapt the proof of Theorem \[t:covercoloring\] to determine the $b$-fold chromatic number of a graph, the minimum number of colors required when each vertex is assigned $b$ colors, and adjacent vertices must have disjoint color sets. See [@FHVTperfect Theorem 3.6].
The ideal membership problems in Theorems \[t:secantcoloring\] and \[t:covercoloring\] are for monomial ideals, and so they are computationally simple. On the other hand, computing the chromatic number is an **NP**-complete problem. The bottleneck in the algebraic algorithms derived from Theorems \[t:secantcoloring\] and \[t:covercoloring\] is the computation of the secant ideal $I(G)^{\{k\}}$ or the cover ideal $J(G)$ given $G$; these problems are both **NP**-complete.
It is naturally interesting to investigate the following problem.
Find algebraic algorithms to compute the chromatic number $\chi(G)$ based on algebraic invariants and properties of the edge ideal $I(G)$.
For the rest of this section, we shall restrict our attention to the case when $G$ is a graph (i.e., not a hypergraph), and consider the problem of identifying odd cycles and odd holes in $G$. As before, let $I = I(G)$ and $J = J(G)$.
Recall that a *bipartite graph* is a two-colorable graph, or, equivalently, a graph with no odd circuits. This yields two corollaries to Theorem \[t:covercoloring\]:
\[c:bipartite\] $G$ is a bipartite graph if and only if $\mathbf{m}\in J^{2}$.
\[c:oddcycle\] If $G$ is a graph, then $G$ contains an odd circuit if and only if $\mathbf{m}\not\in J^{2}$.
It is natural to ask if we can locate the offending odd circuits. In fact, we can identify the *induced odd cycles* from the associated primes of $J^{2}$.
Let $C=(x_{i_{1}},\dots, x_{i_{s}}, x_{i_{1}})$ be a circuit in $G$. We say that $C$ is an *induced cycle* if the induced subgraph of $G$ on $W=\{x_{i_{1}},\dots, x_{i_{s}}\}$ has no edges except those connecting consecutive vertices of $C$. Equivalently, $C$ is an induced cycle if it has no chords.
$G$ has induced cycles $abcde$ and $defg$. The circuit $abcdgfe$ isn’t an induced cycle, since it has the chord $de$.
Simis and Ulrich prove that the odd induced cycles are the generators of the second secant ideal of $I$ [@SU].
\[simisulrich\] Let $G$ be a graph with edge ideal $I$. Then a squarefree monomial $m$ is a generator of $I^{\{2\}}$ if and only if $G_{m}$ is an odd induced cycle.
If $G_{m}$ is an odd induced cycle, then $G_{m}$ and hence $G$ are not $2$-colorable. On the other hand, if $m\in I^{\{2\}}$, then $G_{m}$ is not $2$-colorable and so has an odd induced cycle.
Now suppose that $G$ is a cycle on $(2\ell -1)$ vertices, so without loss of generality $I=(x_{1}x_{2}, x_{2}x_{3}, \dots,
x_{2\ell-1}x_{1})$. Then the generators of $J$ include the $(2\ell -1)$ vertex covers $w_{i}=x_{i}x_{i+2}x_{i+4}\dots x_{i+2\ell -2}$ obtained by starting anywhere in the cycle and taking every second vertex until we wrap around to an adjacent vertex. (Here we have taken the subscripts mod ($2\ell -1$) for notational sanity.) All other generators have higher degree. In particular, the generators of $J$ all have degree at least $\ell$, so the generators of $J^{2}$ have degree at least $2\ell$. Thus $\mathbf{m}\not\in
J^{2}$, since $\deg(\mathbf{m})=2\ell -1$. However, we have $\mathbf{m}x_{i}=w_{i}w_{i+1}\in J^{2}$ for all $x_{i}$. Thus $\mathbf{m}$ is in the socle of $S/J^{2}$, and in particular this socle is nonempty, so $\mathfrak{p}_{\mathbf{m}}=(x_{1},\dots, x_{2\ell -1})$ is associated to $J^{2}$. In fact, it is a moderately difficult computation to find an irredundant primary decomposition:
\[p:oddcycledecomp\] Let $G$ be the odd cycle on $x_{1},\dots,x_{2\ell -1}$. Then $$J^{2}=\left[\bigcap_{i=1}^{2\ell -1} (x_{i},x_{i+1})^{2}\right] \cap
(x_{1}^{2},\dots, x_{2\ell -1}^{2}).$$
Proposition \[p:oddcycledecomp\] picks out the difference between $J^{2}$ and the symbolic square $J^{(2)}$ when $G$ is an odd cycle. The product of the variables $\mathbf{m}$ appears in $\mathfrak{p}^{2}$ for all $\mathfrak{p}\in \operatorname{Ass}(S/J)$, but is missing from $J^{2}$. (Combinatorially, this corresponds to $\mathbf{m}$ being a double cover of $G$ that cannot be partitioned into two single covers.) Thus $\mathbf{m}\in J^{(2)}\smallsetminus J^{2}$.
We can attempt a similar analysis on an even cycle, but we find only two smallest vertex covers, $w_{\text{odd}}=x_{1}\dots x_{2\ell-1}$ and $w_{\text{even}}=x_{2}\dots x_{2\ell}$. Then $\mathbf{m}=w_{\text{odd}}w_{\text{even}}\in J^{2}$ is not a socle element. In this case Theorem \[t:oddcycledecomp\] will tell us that $J^{2}$ has primary decomposition $\bigcap
(x_{i},x_{i+1})^{2}$, i.e., $J^{(2)}=J^{2}$.
In fact, Francisco, Hà, and Van Tuyl show that, for an arbitrary graph $G$, the odd cycles can be read off from the associated primes of $J^{2}$ [@FHVT]. Given a set $W\subset V$, put $\mathfrak{p}_{W}^{\langle 2\rangle }=(x_{i}^{2}:x_{i}\in W)$. Then we have:
\[t:oddcycledecomp\] Let $G$ be a graph. Then $J^{2}$ has irredundant primary decomposition $$J^{2}=\left[\bigcap_{e\in E(G)}\mathfrak{p}_{e}^{2}\right] \cap
\left[\bigcap_{G_{W}\text{ is an induced odd
cycle}}\mathfrak{p}_{W}^{\langle 2\rangle}\right].$$
\[t:oddcycleass\] Let $G$ be a graph. Then we have $$\operatorname{Ass}(S/J^{2})=\left\{\mathfrak{p}_{e}:e\in E(G)\right\} \cup
\left\{\mathfrak{p}_{W}:G_{W} \text{ is an induced odd cycle}\right\}.$$
Corollary \[t:oddcycleass\] and Theorem \[simisulrich\] are also connected via work of Sturmfels and Sullivant [@SS], who show that generalized Alexander duality connects the secant powers of an ideal with the powers of its dual.
We have $\operatorname{Ass}(S/J^{2})=E(G)\cup\{(a,b,c,d,e)\}$. The prime $(a,b,c,d,e)$ appears here because $abcde$ is an odd induced cycle of $G$. The even induced cycle $defg$ does not appear in $\operatorname{Ass}(S/J^{2})$, nor does the odd circuit $abcdgfe$, which is not induced. Furthermore, per Theorem \[simisulrich\], $I^{\{2\}}$ is generated by the odd cycle $abcde$.
Theorem \[t:oddcycledecomp\] and Corollary \[t:oddcycleass\] tell us that the odd cycles of a graph $G$ exactly describe the difference between the symbolic square and ordinary square of its cover ideal $J(G)$. It is natural to ask about hypergraph-theoretic interpretations of the differences between higher symbolic and ordinary powers of $J(G)$, and of the differences between these powers for the edge ideal $I(G)$. The answer to the former question involves *critical hypergraphs*, discussed in §\[s:higherpowers\]. The latter question is closely related to a problem in combinatorial optimization theory. We describe this relationship in §\[s:packing\].
The importance of detecting odd induced cycles in a graph is apparent in the Strong Perfect Graph Theorem, proven by Chudnovsky, Robertson, Seymour, and Thomas in [@CRST] after the conjecture had been open for over 40 years. A graph $G$ is *perfect* if for each induced subgraph $H$ of $G$, the chromatic number $\chi(H)$ equals the clique number $\omega(H)$, where $\omega(H)$ is the number of vertices in the largest clique (i.e., complete subgraph) appearing in $H$. Perfect graphs are an especially important class of graphs, and they have a relatively simple characterization. Call any odd cycle of at least five vertices an *odd hole*, and define an *odd antihole* to be the complement of an odd hole.
\[SPGT\] A graph is perfect if and only if it contains no odd holes or odd antiholes.
Let $G$ be a graph with complementary graph $G^c$ (that is, $G^c$ has the same vertex set as $G$ but the complementary set of edges). Let $J(G)$ be the cover ideal of $G$ and $J(G^c)$ be the cover ideal of $G^c$. Using the Strong Perfect Graph Theorem along with Corollary \[t:oddcycleass\], we conclude that a graph $G$ is perfect if and only if neither $S/J(G)^2$ nor $S/J(G^c)^2$ has an associated prime of height larger than three. It is clear from the induced pentagon that the graph from Running Example \[running\] is imperfect; this is apparent algebraically from the fact that $(a,b,c,d,e)$ is associated to $R/J(G)^2$.
Associated primes and perfect graphs {#s:higherpowers}
====================================
Theorem \[t:oddcycledecomp\] and Corollary \[t:oddcycleass\] exhibit a strong interplay between coloring properties of a graph and associated primes of the square of its cover ideal. In this section, we explore the connection between coloring properties of hypergraphs in general and associated primes of higher powers of their cover ideals. We also specialize back to graphs and give algebraic characterizations of perfect graphs.
A *critically $d$-chromatic hypergraph* is a hypergraph $G$ with $\chi(G) =d$ whose proper induced subgraphs all have smaller chromatic number; $G$ is also called a *critical hypergraph*.
The connection between critical hypergraphs and associated primes begins with a theorem of Sturmfels and Sullivant on graphs that generalizes naturally to hypergraphs.
\[ss-secant\] Let $G$ be a hypergraph with edge ideal $I$. Then the squarefree minimal generators of $I^{\{s\}}$ are the monomials $W$ such that $G_W$ is critically $(s+1)$-chromatic.
Higher powers of the cover ideal $J = J(G)$ of a hypergraph have more complicated structure than the square. It is known that the primes associated to $S/J^2$ persist as associated primes of all $S/J^s$ for $s \ge 2$ [@FHVTperfect Corollary 4.7]. As one might expect from the case of $J^{2}$, if $H$ is a critically $(d+1)$-chromatic induced subhypergraph of $G$, then $\mathfrak{p}_H \in \operatorname{Ass}(S/J^{d})$ but $\mathfrak{p}_H \notin \operatorname{Ass}(S/J^{e})$ for any $e < d$. However, the following example from [@FHVTperfect] illustrates that other associated primes may arise as well.
\[newkindofprimes\] Let $G$ be the graph with vertices $\{x_1,\dots,x_6\}$ and edges $$x_1x_2,x_2x_3,x_3x_4,x_4x_5,x_5x_1,x_3x_6,x_4x_6,x_5x_6,$$ where we have abused notation by writing edges as monomials. Thus $G$ is a five-cycle on $\{x_1,\dots,x_5\}$ with an extra vertex $x_6$ joined to $x_3$, $x_4$, and $x_5$. Let $J$ be the cover ideal of $G$. The maximal ideal $\mathfrak{m}=(x_1,\dots,x_6)$ is associated to $S/J^3$ but to neither $S/J$ nor $S/J^2$. However, $G$ is not a critically $4$-chromatic graph; instead, $\chi(G)=3$.
Consequently, the critical induced subhypergraphs of a hypergraph $G$ may not detect all associated primes of $S/J^s$. Fortunately, there is a related hypergraph whose critical induced subhypergraphs do yield a complete list of associated primes. We define the expansion of a hypergraph, the crucial tool.
\[d.expansion\] Let $G$ be a hypergraph with vertices $V=\{x_1,\dots,x_n\}$ and edges $E$, and let $s$ be a positive integer. We create a new hypergraph $G^s$, called *the $s^{\text{th}}$ expansion of $G$*, as follows. We create vertex sets $V_{1}=\{x_{1,1},\dots, x_{n,1}\}$, …, $V_{s}=\{x_{1,s},\dots, x_{n,s}\}$. (We think of these vertex sets as having distinct flavors. In the literature, the different flavors $x_{i,j}$ of a vertex $x_{i}$ are sometimes referred to as its *shadows*.) The edges of $G^{s}$ consist of all edges $x_{i,j}x_{i,k}$ connecting all differently flavored versions of the same vertex, and all edges arising from possible assignments of flavors to the vertices in an edge of $G$.
We refer to the map sending all flavors $x_{i,j}$ of a vertex $x_{i}$ back to $x_{i}$ as *depolarization*, by analogy with the algebraic process of polarization.
\[e.five\] Consider a five-cycle $G$ with vertices $x_1,\dots,x_5$. Then $G^2$ has vertex set $\{x_{1,1}, x_{1,2}, \dots, x_{5,1}, x_{5,2}\}$. Its edge set consists of edges $x_{1,1}x_{1,2}, \dots, x_{5,1}x_{5,2}$ as well as all edges $x_{i,j}x_{i+1,j'}$, where $1 \le j \le j' \le 2$, and the first index is taken modulo 5. Thus, for example, the edge $x_1x_2$ of $G$ yields the four edges $x_{1,1}x_{2,1}$, $x_{1,1}x_{2,2}$, $x_{1,2}x_{2,1}$, and $x_{1,2}x_{2,2}$ in $G^2$.
![The second expansion graph of a 5-cycle[]{data-label="f:expansion"}](expansion){height="1.5in"}
Our goal is to understand the minimal monomial generators of the generalized Alexander dual $(J(G)^{s})^{[\mathbf{s}]}$, where $\mathbf{s}$ is the vector $(s,\dots,s)$, one entry for each vertex of $G$. Under generalized Alexander duality, these correspond to the ideals in an irredundant irreducible decomposition of $J(G)^s$, yielding the associated primes of $S/J(G)^s$.
By generalized Alexander duality, Theorem \[ss-secant\] identifies the squarefree minimal monomial generators of $(J(G)^s)^{[\mathbf{s}]}$. Understanding the remaining monomial generators requires the following theorem [@FHVTperfect Theorem 4.4]. For a set of vertices $T$, write $\mathbf{m}_T$ to denote the product of the corresponding variables.
\[t.expansion\] Let $G$ be a hypergraph with cover ideal $J = J(G)$, and let $s$ be a positive integer. Then $$(J^s)^{[\mathbf{s}]} = (\overline{\mathbf{m}_T} ~\big|~ \chi(G^s_T) > s )$$ where $\overline{{{\bf m}}_T}$ is the depolarization of $\mathbf{m}_T$.
The proof relies on a (hyper)graph-theoretic characterization of the generators of $I(G^s)^{\{s\}}$ from Theorem \[ss-secant\]. One then needs to prove that $(J^s)^{[\mathbf{s}]}$ is the depolarization of $I(G^s)^{\{s\}}$, which requires some effort; see [@FHVTperfect].
Using Theorem \[t.expansion\], we can identify all associated primes of $S/J(G)^s$ in terms of the expansion graph of $G$.
\[c.allprimes\] Let $G$ be a hypergraph with cover ideal $J = J(G)$. Then $P=(x_{i_1},\dots,x_{i_r}) \in \operatorname{Ass}(S/J^s)$ if and only if there is a subset $T$ of the vertices of $G^s$ such that $G^s_T$ is critically $(s+1)$-chromatic, and $T$ contains at least one flavor of each variable in $P$ but no flavors of other variables.
We outline the rough idea of the proof. If $P \in \operatorname{Ass}(S/J^s)$, then $(x_{i_1}^{e_{i_1}},\dots,x_{i_r}^{e_{i_r}})$ is an irreducible component of $J^s$, for some $e_{i_j} > 0$. This yields a corresponding minimal generator of $(J^s)^{[\mathbf{s}]}$, which gives a subset $W$ of the vertices of $G^s$ such that $G^s_W$ is critically $(s+1)$-chromatic, and $W$ depolarizes to $x_{i_1}^{e_{i_{1}}} \dots x_{i_r}^{e_{i_{r}}}$. Conversely, given a critically $(s+1)$-chromatic expansion hypergraph $G^s_T$, we get a minimal generator of $(J^s)^{[\mathbf{s}]}$ of the form $x_{i_1}^{e_{i_1}} \cdots x_{i_r}^{e_{i_r}}$, where $1 \le e_{i_j} \le s$ for all $i_j$. Duality produces an irreducible component of $J^s$ with radical $P$.
Corollary \[c.allprimes\] explains why $\mathfrak{m} \in \operatorname{Ass}(S/J^3)$ in Example \[newkindofprimes\]. Let $T$ be the set of vertices $$T = \{x_{1,1},x_{2,1},x_{2,2},x_{3,1},x_{4,1},x_{5,1},x_{6,1}\},$$ a subset of the vertices of $G^3$. Then $G^3_T$ is critically 4-chromatic.
As a consequence of this work, after specializing to graphs, we get two algebraic characterizations of perfect graphs that are independent of the Strong Perfect Graph Theorem. First, we define a property that few ideals satisfy (see, e.g., [@HostenThomas]).
\[d:saturated\] An ideal $I \subset S$ has the *saturated chain property for associated primes* if given any associated prime $P$ of $S/I$ that is not minimal, there exists an associated prime $Q \subsetneq P$ with height$(Q)= $ height$(P)-1$.
We can now characterize perfect graphs algebraically in two different ways [@FHVTperfect Theorem 5.9]. The key point is that for perfect graphs, the associated primes of powers of the cover ideal correspond exactly to the cliques in the graph.
\[perfectgraphs\] Let $G$ be a simple graph with cover ideal $J$. Then the following are equivalent:
1. $G$ is perfect.
2. For all $s$ with $1 \le s < \chi(G)$, $P=(x_{i_1}, \dots, x_{i_r}) \in \operatorname{Ass}(R/J^s)$ if and only if the induced graph on $\{x_{i_1},\dots,x_{i_r}\}$ is a clique of size $1 < r \le s+1$ in $G$.
3. For all $s \geq 1$, $J^s$ has the saturated chain property for associated primes.
We sketch (1) implies (2) to give an idea of how expansion is used. Suppose $G$ is a perfect graph. A standard result in graph theory shows that $G^s$ is also perfect. Let $P \in \operatorname{Ass}(S/J^s)$, so $P$ corresponds to some subset $T$ of the vertices of $G^s$ such that $G^s_T$ is critically $(s+1)$-chromatic. Because $G^s$ is perfect, the clique number of $G^s_T$ is also $s+1$, meaning there exists a subset $T'$ of $T$ such that $G^s_{T'}$ is a clique with $s+1$ vertices. Thus $G^s_{T'}$ is also a critically $(s+1)$-chromatic graph contained inside $G^s_T$, forcing $T=T'$. Hence $G^s_T$ is a clique, and the support of the depolarization of $\bar{\mathbf{m}_T}$ is a clique with at most $s+1$ vertices. Therefore $G_P$ is a clique.
If $J$ is the cover ideal of a perfect graph, its powers satisfy a condition stronger than that of Definition \[d:saturated\]. If $P \in \operatorname{Ass}(S/J^s)$, and $Q$ is any monomial prime of height at least two contained in $P$, then $Q \in \operatorname{Ass}(S/J^s)$. This follows from the fact that $P$ corresponds to a clique in the graph.
Theorem \[perfectgraphs\] provides information about two classical issues surrounding associated primes of powers of ideals. Brodmann proved that for any ideal $J$, the set of associated primes of $S/J^s$ stabilizes [@Brodmann]. However, there are few good bounds in the literature for the power at which this stabilization occurs. When $J$ is the cover ideal of a perfect graph, Theorem \[perfectgraphs\] demonstrates that stabilization occurs at $\chi(G)-1$. Moreover, though in general associated primes may disappear and reappear as the power on $J$ increases (see, e.g., [@BHH; @HH] and also [@MV Example 4.18]), when $J$ is the cover ideal of a perfect graph, we have $\operatorname{Ass}(S/J^s) \subseteq \operatorname{Ass}(S/J^{s+1})$ for all $s \ge 1$. In this case, we say that $J$ has the *persistence property for associated primes*, or simply the *persistence property*. Morey and Villarreal give an alternate proof of the persistence property for cover ideals of perfect graphs in [@MV Example 4.21].
While there are examples of arbitrary monomial ideals for which persistence fails, we know of no such examples of *squarefree* monomial ideals. Francisco, Hà, and Van Tuyl (see [@FHVT; @FHVT2]) have asked:
\[sfpersistence\] Suppose $J$ is a squarefree monomial ideal. Is $\operatorname{Ass}(S/J^s) \subseteq \operatorname{Ass}(S/J^{s+1})$ for all $s \ge 1$?
While Question \[sfpersistence\] has a positive answer when $J$ is the cover ideal of a perfect graph, little is known for cover ideals of imperfect graphs. Francisco, Hà, and Van Tuyl answer Question \[sfpersistence\] affirmatively for odd holes and odd antiholes in [@FHVT2], but we are not aware of any other imperfect graphs whose cover ideals are known to have this persistence property. One possible approach is to exploit the machinery of expansion again. Let $G$ be a graph, and let $x_i$ be a vertex of $G$. Form the expansion of $G$ at $\{x_i\}$ by replacing $x_i$ with two vertices $x_{i,1}$ and $x_{i,2}$, joining them with an edge. For each edge $\{v,x_i\}$ of $G$, create edges $\{v,x_{i,1}\}$ and $\{v,x_{i,2}\}$. If $W$ is any subset of the vertices of $G$, form $G[W]$ by expanding all the vertices of $W$. Francisco, Hà, and Van Tuyl conjecture:
\[c.expansion\] Let $G$ be a graph that is critically $s$-chromatic. Then there exists a subset $W$ of the vertices of $G$ such that $G[W]$ is critically $(s+1)$-chromatic.
In [@FHVT2], Francisco, Hà, and Van Tuyl prove that if Conjecture \[c.expansion\] is true for all $s \ge 1$, then all cover ideals of graphs have the persistence property. One can also state a hypergraph version of Conjecture \[c.expansion\]; if true, it would imply persistence of associated primes for all squarefree monomial ideals.
Finally, in [@MV], Morey and Villarreal prove persistence for edge ideals $I$ of any graphs containing a leaf (a vertex of degree 1). Their proof passes to the associated graded ring, and the vital step is identifying a regular element of the associated graded ring in $I/I^2$. Morey and Villarreal remark that attempts to prove persistence results for more general squarefree monomial ideals lead naturally to questions related to the Conforti-Cornuéjols conjecture, discussed in the following section.
Equality of symbolic and ordinary powers and linear programming {#s:packing}
===============================================================
We have seen in the last section that comparing symbolic and ordinary powers of the cover ideal of a hypergraph allows us to study structures and coloring properties of the hypergraph. In this section, we address the question of when symbolic and ordinary powers of a squarefree monomial ideal are the same, and explore an algebraic approach to a long-standing conjecture in linear integer programming, the Conforti-Cornuéjols conjecture. In what follows, we state the Conforti-Cornuéjols conjecture in its original form, describe how to translate the conjecture into algebraic language, and discuss its algebraic reformulation and related problems.
The Conforti-Cornuéjols conjecture states the equivalence between the packing and the max-flow-min-cut properties for *clutters* which, as noted before, are essentially simple hypergraphs.
As before, $G = (V,E)$ denotes a hypergraph with $n$ vertices $V =
\{x_1, \dots, x_n\}$ and $m$ edges $E = \{e_1, \dots, e_m\}$. Let $A$ be the *incidence matrix* of $G$, i.e., the $(i,j)$-entry of $A$ is 1 if the vertex $x_i$ belongs to the edge $e_j$, and 0 otherwise. For a nonnegative integral vector $\c \in {{\mathbb Z}}_{\ge 0}^n$, consider the following dual linear programming system $$\begin{aligned}
\max \{\langle \1, \y\rangle ~|~ \y \in {{\mathbb R}}^m_{\ge 0}, A\y \le \c\} = \min \{\langle \c, \z\rangle ~|~ \z \in {{\mathbb R}}^n_{\ge 0}, A^\text{T}\z \ge \1\}. \label{eq.CCdualsystem}\end{aligned}$$
\[def.properties\] Let $G$ be a simple hypergraph.
1. The hypergraph $G$ is said to *pack* if the dual system (\[eq.CCdualsystem\]) has integral optimal solutions $\y$ and $\z$ when $\c = \1$.
2. The hypergraph $G$ is said to have the *packing property* if the dual system (\[eq.CCdualsystem\]) has integral optimal solutions $\y$ and $\z$ for all vectors $\c$ with components equal to 0, 1 and $+\infty$.
3. The hypergraph $G$ is said to have the *max-flow-min-cut (MFMC) property* or to be *Mengerian* if the dual system (\[eq.CCdualsystem\]) has integral optimal solutions $\y$ and $\z$ for all nonnegative integral vectors $\c \in {{\mathbb Z}}_{\ge 0}^n$.
In Definition \[def.properties\], setting an entry of $\c$ to $+\infty$ means that this entry is sufficiently large, so the corresponding inequality in the system $A\y \le \c$ can be omitted. It is clear that if $G$ satisfies the MFMC property, then it has the packing property.
The following conjecture was stated in [@Cornuejols Conjecture 1.6] with a reward prize of \$5,000 for the solution.
\[conj.CC\] A hypergraph has the packing property if and only if it has the max-flow-min-cut property.
As we have remarked, the main point of Conjecture \[conj.CC\] is to show that if a hypergraph has the packing property then it also has the MFMC property.
The packing property can be understood via more familiar concepts in (hyper)graph theory, namely *vertex covers* (also referred to as *transversals*), which we recall from Section \[intro\], and *matchings*.
A *matching* (or *independent set*) of a hypergraph $G$ is a set of pairwise disjoint edges.
Let $\alpha_0(G)$ and $\beta_1(G)$ denote the minimum cardinality of a vertex cover and the maximum cardinality of a matching in $G$, respectively. We have $\alpha_0(G) \ge \beta_1(G)$ since every edge in any matching must hit at least one vertex from every cover.
The hypergraph $G$ is said to be *König* if $\alpha_0(G)=\beta_1(G)$. Observe that giving a vertex cover and a matching of equal size for $G$ can be viewed as giving integral solutions to the dual system (\[eq.CCdualsystem\]) when $\c = \1$. Thus, $G$ is König if and only if $G$ packs.
There are two operations commonly used on a hypergraph $G$ to produce new, related hypergraphs on smaller vertex sets. Let $x \in V$ be a vertex in $G$. The *deletion* $G \setminus x$ is formed by removing $x$ from the vertex set and deleting any edge in $G$ that contains $x$. The *contraction* $G / x$ is obtained by removing $x$ from the vertex set and removing $x$ from any edge of $G$ that contains $x$. Any hypergraph obtained from $G$ by a sequence of deletions and contractions is called a *minor* of $G$. Observe that the deletion and contraction of a vertex $x$ in $G$ has the same effect as setting the corresponding component in $\c$ to $+\infty$ and 0, respectively, in the dual system (\[eq.CCdualsystem\]). Hence,
*$G$ satisfies the packing property if and only if $G$ and all of its minors are König.*
Let $G$ be a 5-cycle. Then $G$ itself is not König ($\alpha_0(G) = 3$ and $\beta_1(G) = 2$). Thus, $G$ is does not satisfy the packing property.
Any bipartite graph is König. Therefore, if $G$ is a bipartite graph then (since all its minors are also bipartite) $G$ satisfies the packing property.
We shall now explore how Conjecture \[conj.CC\] can be understood via commutative algebra, and more specifically, via algebraic properties of edge ideals.
As noted in Section \[s:prel\], symbolic Rees algebras are more complicated than the ordinary Rees algebras, and could be non-Noetherian. Fortunately, in our situation, the symbolic Rees algebra of a squarefree monomial ideal is always Noetherian and finitely generated (cf. [@HHT Theorem 3.2]). Moreover, the symbolic Rees algebra of the edge ideal of a hypergraph $G$ can also be viewed as the *vertex cover algebra* of the dual hypergraph $G^*$.
Let $G = (V,E)$ be a simple hypergraph over the vertex set $V = \{x_1, \dots, x_n\}$.
1. We call a nonnegative integral vector $\c = (c_1, \dots, c_n)$ a *$k$-cover* of $G$ if $\sum_{x_i \in e} c_i \ge k$ for any edge $e$ in $G$.
2. The *vertex cover algebra* of $G$, denoted by ${\mathcal{A}}(G)$, is defined to be $${\mathcal{A}}(G) = \bigoplus_{k \ge 0} {\mathcal{A}}_k(G),$$ where ${\mathcal{A}}_k(G)$ is the $k$-vector space generated by all monomials $x_1^{c_1} \dots x_n^{c_n}t^k$ such that $(c_1, \dots, c_n) \in {{\mathbb Z}}^n_{\ge 0}$ is a $k$-cover of $G$.
\[lem.vertexcoveralgebra\] Let $G$ be a simple hypergraph with edge ideal $I = I(G)$, and let $G^*$ be its dual hypergraph. Then $${{\mathcal R}}_s(I) = {\mathcal{A}}(G^*).$$
We are now ready to give an algebraic interpretation of the MFMC property.
\[lem.sigmagamma\] Let $G = (V,E)$ be a simple hypergraph with $n$ vertices and $m$ edges. Let $A$ be its incidence matrix. For a nonnegative integral vector $\c \in {{\mathbb Z}}^n_{\ge 0}$, define\
$\sigma(\c) = \max \{ \langle \1, \y \rangle ~|~ \y \in {{\mathbb Z}}^m_{\ge 0}, A\y \le \c \}$, and\
$\gamma(\c) = \min \{ \langle \c, \z \rangle ~|~ \z \in {{\mathbb Z}}^n_{\ge 0}, A^{\text{T}}\z \ge \1 \}.$\
Then
1. $\c$ is a $k$-cover of $G^*$ if and only if $k \le \gamma(\c)$.
2. $\c$ can be written as a sum of $k$ vertex covers of $G^*$ if and only if $k \le \sigma(\c)$.
By definition, a nonnegative integral vector $\c = (c_1, \dots, c_n) \in {{\mathbb Z}}^n_{\ge 0}$ is a $k$-cover of $G^*$ if and only if $$\begin{aligned}
k \le \min \{ \sum_{x_i \in e} c_i ~|~ e \text{ is any edge of } G^* \}. \label{eq.k-cover}\end{aligned}$$ Let $\z$ be the $(0,1)$-vector representing $e$. Observe that $e$ is an edge of $G^*$ if and only if $e$ is a minimal vertex cover of $G$, and this is the case if and only if $A^{\text{T}}\z \ge \1$. Therefore, the condition in (\[eq.k-cover\]) can be translated to $$\begin{aligned}
k & \le \min \{ \langle \c, \z \rangle ~|~ \z \in \{0,1\}^n, A^{\text{T}}\z \ge \1 \} \\
& = \min \{ \langle \c, \z \rangle ~|~ \z \in {{\mathbb Z}}^n_{\ge 0}, A^{\text{T}}\z \ge \1 \} = \gamma(\c).\end{aligned}$$
To prove (2), let $\mba_1, \dots, \mba_m$ be representing vectors of the edges in $G$ (i.e., the columns of the incidence matrix $A$ of $G$). By Proposition \[alexduality\], $\mba_1, \dots, \mba_m$ represent the minimal vertex cover of the dual hypergraph $G^*$. One can show that a nonnegative integral vector $\c \in {{\mathbb Z}}^n$ can be written as the sum of $k$ vertex covers (not necessarily minimal) of $G^*$ if and only if there exist integers $y_1, \dots, y_m \ge 0$ such that $k = y_1 + \dots + y_m$ and $y_1\mba_1 + \dots + y_m\mba_m \le \c$. Let $\y = (y_1, \dots, y_m)$. Then $$\langle \1, \y \rangle = y_1 + \dots + y_m \text{ and } A\y = y_1\mba_1 + \dots + y_m\mba_m.$$ Thus, $$\sigma(\c) = \max \{ k ~|~ \c \text{ can be written as a sum of } k \text{ vertex covers of } G^* \}.$$
\[thm.MFMCreduce\] Let $G$ be a simple hypergraph with dual hypergraph $G^*$. Then the dual linear programming system (\[eq.CCdualsystem\]) has integral optimal solutions $\y$ and $\z$ for all nonnegative integral vectors $\c$ if and only if ${{\mathcal R}}_s(I(G)) = {\mathcal{A}}(G^*)$ is a standard graded algebra; or equivalently, if and only if $I(G)^{(q)} = I(G)^q$ for all $q \ge 0$.
Given integral optimal solutions $\y$ and $\z$ of the dual system (\[eq.CCdualsystem\]) for a nonnegative integral vector $\c$, we get $$\sigma(\c) = \gamma(\c).$$ The conclusion then follows from Lemmas \[lem.vertexcoveralgebra\] and \[lem.sigmagamma\].
The following result (see [@HHTZ Corollary 1.6] and [@GVV Corollary 3.14]) gives an algebraic approach to Conjecture \[conj.CC\].
\[thm.33\] Let $G$ be a simple hypergraph with edge ideal $I = I(G)$. The following conditions are equivalent:
1. $G$ satisfies the MFMC property,
2. $I^{(q)} = I^q$ for all $q \ge 0$,
3. The associated graded ring $\gr_{I} := \bigoplus_{q \ge 0} I^q/I^{q+1}$ is reduced,
4. $I$ is *normally torsion-free*, i.e., all powers of $I$ have the same associated primes.
The equivalence between (1) and (2) is the content of Theorem \[thm.MFMCreduce\]. The equivalences of (2), (3) and (4) are well known results in commutative algebra (cf. [@HSV]).
The Conforti-Cornuéjols conjecture now can be restated as follows.
\[conj.CCalgebraic\] Let $G$ be a simple hypergraph with edge ideal $I = I(G)$. If $G$ has packing property then the associated graded ring $\gr_I$ is reduced. Equivalently, if $G$ and all its minors are König, then the associated graded ring $\gr_I$ is reduced.
It remains to give an algebraic characterization for the packing property. To achieve this, we shall need to interpret minors and the König property. Observe that the deletion $G \setminus x$ at a vertex $x \in {\mathcal{X}}$ has the effect of setting $x = 0$ in $I(G)$ (or equivalently, of passing to the ideal $(I(G),x)/(x)$ in the quotient ring $S/(x)$), and the contraction $G / x$ has the effect of setting $x=1$ in $I(G)$ (or equivalently, of passing to the ideal $I(G)_x$ in the localization $S_x$). Thus, we call an ideal $I'$ a *minor* of a squarefree monomial ideal $I$ if $I'$ can be obtained from $I$ by a sequence of taking quotients and localizations at the variables. Observe further that $\alpha_0(G) = \operatorname{ht}I(G)$, and if we let $\operatorname{m-grade} I$ denote the maximum length of a regular sequence of monomials in $I$ then $\beta_1(G) = \operatorname{m-grade} I(G)$. Hence, a simple hypergraph with edge ideal $I$ is König if $\operatorname{ht}I = \operatorname{m-grade} I$. This leads us to a complete algebraic reformulation of the Conforti-Cornuéjols conjecture:
\[conj.trans\] Let $I$ be a squarefree monomial ideal such that $I$ and all of its minors satisfy the property that their heights are the same as their m-grades. Then $\gr_I$ is reduced; or equivalently, $I$ is normally torsion-free.
The algebraic consequence of the conclusion of Conjecture \[conj.trans\] (and equivalently, Conjecture \[conj.CC\]) is the equality $I^{(q)} =
I^q$ for all $q \ge 0$ or, equivalently, the normally torsion-freeness of $I$. If one is to consider the equality $I^{(q)} = I^q$, then it is natural to look for an integer $l$ such that $I^{(q)} = I^q$ for $0 \le q \le
l$ implies $I^{(q)} = I^q$ for all $q \ge 0$, or to examine squarefree monomial ideals with the property that $I^{(q)} = I^q$ for all $q \ge q_0$. On the other hand, if one is to investigate the normally torsion-freeness then it is natural to study properties of minimally not normally torsion-free ideals. The following problem is naturally connected to Conjectures \[conj.CC\] and \[conj.trans\], and part of it has been the subject of work in commutative algebra (cf. [@HS]).
\[prob.algebraic\] Let $I$ be a squarefree monomial ideal in $S = K[x_1, \dots, x_n]$.
1. Find the least integer $l$ (may depend on $I$) such that if $I^{(q)} =
I^q$ for $0 \le q \le l$ then $I^{(q)} = I^q$ for all $q \ge 0$.
2. Suppose that there exists a positive integer $q_0$ such that $I^{(q)} =
I^q$ for all $q \ge q_0$. Study algebraic and combinatorial properties of $I$.
3. Suppose $I$ is minimally not normally torsion-free (i.e., $I$ is not normally torsion-free but all its minors are). Find the least power $q$ such that $\operatorname{Ass}(S/I^q) \not= \operatorname{Ass}(S/I)$.
[9999999]{}
S. Bandari, J. Herzog, and T. Hibi, Monomial ideals whose depth function has any given number of strict local maxima. Preprint, 2012. [arXiv:1205.1348]{}
C. Bocci and B. Harbourne, Comparing powers and symbolic powers of ideals. *J. Algebraic Geom.* [**19**]{} (2010), no. 3, 399–417.
M. Brodmann, Asymptotic stability of ${\rm Ass}(M/I\sp{n}M)$. *Proc. Amer. Math. Soc.* **74** (1979), 16–18.
W. Bruns and J. Herzog, Cohen-Macaulay rings. Cambridge Studies in Advanced Mathematics, [**39**]{}. *Cambridge University Press, Cambridge*, 1993.
M. V. Catalisano, A. V. Geramita, and A. Gimigliano, On the ideals of secant varieties to certain rational varieties. *J. Algebra* [**319**]{} (2008), no. 5, 1913–-1931.
M. Chudnovsky, N. Robertson, P. Seymour, and R. Thomas, The strong perfect graph theorem. *Ann. of Math.* (2) [**164**]{} (2006), 51–229.
G. Cornuéjols, Combinatorial optimization: Packing and covering. CBMS-NSF Regional Conference Series in Applied Mathematics, [**74**]{}, *SIAM, Philadelphia*, 2001.
L. Ein, R. Lazarsfeld, and K.E. Smith, Uniform bounds and symbolic powers on smooth varieties. *Invent. Math.* [**144**]{} (2001), no. 2, 241–252.
C. A. Francisco, H. T. Hà, and A. Van Tuyl, Associated primes of monomial ideals and odd holes in graphs. *J. Algebraic Combin.* [**32**]{} (2010), no. 2, 287–301.
C. A. Francisco, H. T. Hà, and A. Van Tuyl, A conjecture on critical graphs and connections to the persistence of associated primes. *Discrete Math.* [**310**]{} (2010), no. 15–16, 2176–2182.
C. A. Francisco, H. T. Hà, and A. Van Tuyl, Colorings of hypergraphs, perfect graphs, and associated primes of powers of monomial ideals. *J. Algebra* [**331**]{} (2011), 224–-242.
I. Gitler, C.E. Valencia, and R. Villarreal, A note on Rees algebras and the MFMC property. *Beiträge Algebra Geom.* [**48**]{} (2007), no. 1, 141–150.
H.T. Hà, S. Morey, Embedded associated primes of powers of square-free monomial ideals. *J. Pure Appl. Algebra* [**214**]{} (2010), no. 4, 301–308.
J. Herzog, T. Hibi, The depth of powers of an ideal. *J. Algebra* [**291**]{} (2005), 534–550.
J. Herzog, T. Hibi, and N.V. Trung, Symbolic powers of monomial ideals and vertex cover algebras. *Adv. Math.* [**210**]{} (2007), no. 1, 304–322.
J. Herzog, T. Hibi, N.V. Trung, and X. Zheng, Standard graded vertex cover algebras, cycles and leaves. *Trans. Amer. Math. Soc.* [**360**]{} (2008), no. 12, 6231–6249.
M. Hochster and C. Huneke, Comparison of symbolic and ordinary powers of ideals. *Invent. Math.* [**147**]{} (2002), no. 2, 349–369.
S. Hosten and R. Thomas, The associated primes of initial ideals of lattice ideals. *Math. Res. Lett.* **6** (1999), no. 1, 83–97.
C. Huneke, D. Katz, and J. Validashti, Uniform equivalence of symbolic and adic topologies. *Illinois J. Math.* [**53**]{} (2009), no. 1, 325–338.
C. Huneke, A. Simis, and W. Vasconcelos, Reduced normal cones are domains. Invariant theory (Denton, TX, 1986), 95–101, Contemp. Math. [**88**]{}. *Amer. Math. Soc., Providence, RI*, 1989.
E. Miller and B. Sturmfels, Combinatorial Commutative Algebra. GTM [**227**]{}, *Springer-Verlag*, 2004.
N.C. Minh and N.V. Trung, Cohen-Macaulayness of monomial ideals and symbolic powers of Stanley-Reisner ideals. *Adv. Math.* [**226**]{} (2011), no. 2, 1285–1306.
S. Morey and R.H. Villarreal, Edge ideals: algebraic and combinatorial properties. Progress in Commutative Algebra: Ring Theory, Homology, and Decomposition. *de Gruyter, Berlin*, 2012, 85–126.
M. Nagata, On the $14^\text{th}$ problem of Hilbert. *Amer. J. Math.* [**81**]{} (1959), 766–772.
I. Peeva, Graded syzygies. Algebra and Applications, [**14**]{}. *Springer-Verlag London, Ltd., London*, 2011.
P. Roberts, A prime ideal in a polynomial ring whose symbolic blow-up is not Noetherian. *Proc. Amer. Math. Soc.* [**94**]{} (1985), 589–592.
A. Simis and B. Ulrich, On the ideal of an embedded join. *J. Algebra* [**226**]{} (2000), no. 1, 1–14.
B. Sturmfels and S. Sullivant, Combinatorial secant varieties. *Pure Appl. Math. Q.* [**2**]{} (2006), no. 3, part 1, 867–891.
N. Terai and N.V. Trung, Cohen-Macaulayness of large powers of Stanley-Reisner ideals. *Adv. Math.* [**229**]{} (2012), no. 2, 711–730.
N.V. Trung and T.M. Tuan, Equality of ordinary and symbolic powers of Stanley-Reisner ideals. *J. Algebra* [**328**]{} (2011), 77–93.
M. Varbaro, Symbolic powers and matroids. *Proc. Amer. Math. Soc.* [**139**]{} (2011), no. 7, 2357–2366.
[^1]: This work was partially supported by grants from the Simons Foundation (\#199124 to Francisco and \#202115 to Mermin). Hà is partially supported by NSA grant H98230-11-1-0165.
|
---
abstract: 'Rheological phase transitions open the door to the less explored realm of non-equilibrium phase transitions. The main mechanism driving these transitions is usually mechanical perturbation by shear— an unjamming mechanism. Investigating discontinuous shear thickening (DST) is challenging because the shear counterintuitively acts as a jamming mechanism. Moreover, at the brink of this transition, a thickening material exhibits fluctuations that extend both spatially and temporally. Despite recent extensive research, the origins of such spatiotemporal fluctuations remain unidentified. Here, we investigate large fluctuations in DST by using versatile tools of stochastic thermodynamics. We discover a non-equilibrium dichotomy in the underlying mechanisms that give rise to large fluctuations and demonstrate that this dichotomy is a manifestation of novel collective behaviors across the transition. We then reveal the origin of spatiotemporal fluctuations in the shear thickening transition. Our study emphasizes the roles of stochastic thermodynamics tools in investigating non-equilibrium phase transitions, and demonstrates that these transitions are accompanied by simple dichotomies. We expect that our general approach will pave the way to unmasking the nature of non-equilibrium phase transitions.'
author:
- 'S. H. E. Rahbari$^1$'
title: Origin of Spatiotemporal Fluctuations in Discontinuous Shear Thickening
---
[**Introduction.**]{} Discontinuous shear thickening (DST), an abrupt shear-driven fluid-solid transition, occurs in a wide range of soft material systems such as Brownian and non-Brownian suspensions as well as granular materials [@brown_2014]. In suspensions, the transition is the result of an interplay between a stabilizing mechanism and frictional contacts due to the roughness of particles. At low flow speeds, the stabilizing mechanism keeps particles apart. This mechanism is usually a repulsive interaction such as steric forces and/or electrostatic repulsion of double layers that introduces a threshold stress. At high flow speeds, when the threshold stress is overcome by the shear forces, a proliferation of frictional contacts results in an abrupt increase of more than an order of magnitude of viscosity— the hallmark of a DST [@morris_2009]. The stabilizing mechanism is system dependent; however, the frictional contacts are essential for DST [@seto_2013]. This behavior is well captured by the Wyart-Cates model, which describes DST as a transition from frictionless to frictional rheologies, due to the proliferation of frictional contacts [@wyart_2014]. In granular materials, DST is believed to be a result of frustration of the tendency of granular systems to dilate under shear [@otsuki_2011; @otsuki_2018]. We stress that the canonical mechanism responsible for DST in all soft material systems undergoing DST is inter-particle friction. Notably, the friction is a (non-Brownian) granular mechanism [@kawasaki_2018].\
Despite different mechanisms underlying DST in all the aforementioned soft material systems, DST has one consistent aspect: near thickening transition a system becomes unstable with spatiotemporal fluctuations. In suspensions, temporal fluctuations appear as oscillations and chaotic time series akin to turbulence [@saint_2018], an effect dubbed rheochaos [@cates_2002]. Moreover, spatial fluctuations result in intermittent stress heterogeneities. These stress anomalies propagate along the vorticity direction, and they are referred to as vorticity bands [@olmsted_2008; @chacko_2018]. The strength of these fluctuations is enhanced with increasing stress, as confirmed by recent experiments using advanced techniques for the measurement of local rheology [@rathee_2017; @saint_2018]. Similar spatiotemporal fluctuations have also been reported in extensive detail for frictional granular materials undergoing DST, in a series of publications by Heussinger, Zippelius and co-workers [@grob_2014; @grob_2016; @saw_2019].\
These spatiotemporal fluctuations are believed to be the precursors of the shear thickening transition, yet the origin of these fluctuations remains a mystery. One reason for this lack of understanding is that, similarly to the glass transition [@holmes_2005], usual measures, such as pair correlation functions, show no signature of any dramatic change in micro-physics across DST [@thomas_2018]. Here, we reveal the origin of the spatiotemporal fluctuations by using a stochastic thermodynamic approach [@seifert_2012; @sekimoto_2010] that has been specifically adopted for rheological phase transitions [@rahbari_2017]. Stochastic thermodynamics is a powerful tool enabling thermodynamic quantities such as work, heat and power to be defined at the mesoscopic scale.\
[**Results.**]{} Here, we perform two dimensional molecular dynamics simulations of bidisperse frictional disks in a simple shear flow. We neglect thermal and hydrodynamic forces to focus on the central role of frictional forces in DST. However, a generalization of our approach to include hydrodynamic and thermal effects would be straightforward. Details of the simulations are given in the Methods of Supplementary Information (SI). A typical flow curve of the system is given in Supplementary Fig. 1-a, where the stress abruptly changes by more than one order of magnitude at a critical shear rate ${\ensuremath{\dot\gamma}}_c \simeq 10^{-5}$. The critical shear rate separates two distinct fluid and solid states. We demonstrate that for ${\ensuremath{\dot\gamma}}<{\ensuremath{\dot\gamma}}_c$, the normal component of the shear stress is dominant; however, above ${\ensuremath{\dot\gamma}}_c$, the tangential component predominates in the momentum transfer. This result once again emphasizes the fundamental role played by friction in DST. Most of physical measures such as pressure (Supplementary Fig. 2), coordination number (Supplementary Fig. 3) faithfully present the inherent state of the system consistent with the rheology. However, the kinetic temperature of the system (Supplementary Fig. 1-b), known as the granular temperature $T_G$, has misleading behavior for ${\ensuremath{\dot\gamma}}> {\ensuremath{\dot\gamma}}_c$. Therefore, the kinetic temperature cannot be used as a real measure of fluctuations in this system, and an alternative measure is required. To resolve this discrepancy, an effective temperature that complies with the rheology is required. To this end, we investigate large fluctuations in injected power $p$ by using the tools of stochastic thermodynamics to derive the effective temperature.\
The injected power due to simple shear is consumed at both translational and rotational degrees of freedom. Thus, we define the local injected power as: $$p = {\ensuremath{\sigma_{xy,n}}}{\ensuremath{\dot\gamma}}+ \delta{\ensuremath{\sigma_{xy}}}{\ensuremath{\dot\gamma}},
\label{eq:power}$$ where ${\ensuremath{\dot\gamma}}$ is the local shear rate, ${\ensuremath{\sigma_{xy,n}}}$ is the normal component of the local shear stress, and $\delta {\ensuremath{\sigma_{xy}}}={\ensuremath{\sigma_{xy,t}}}-{\ensuremath{\sigma_{yx,t}}}$ is the couple-stress, which is equal to difference in the off-diagonal components of the tangential part of the shear stress (the normal component cancels out, because ${\ensuremath{\sigma_{xy,n}}}= \sigma_{yx, n}$). The power is computed locally in the rectangle of the length of the system size and width of $w = 2R$, with $R = 0.7$ radius of larger particles, along the shearing direction. We cross-checked our results for a wider bin of width $w = 4R$ and obtained similar results. A typical probability distribution function (PDF) of power is shown in the inset in Fig.\[fig:tot\_power\_FT\]-a. The distribution is a double-exponential Boltzmann-type distribution. Notably, such a Boltzmann-type PDF of stochastic thermodynamics quantities has been reported for various far-from equilibrium systems, for example, the PDF of work done on particles in a frictional granular system under pure shear [@zheng_2018_2], injected power in turbulence [@bos_2019], power in frictionless disks under shear [@rahbari_2017], and some cases reported by Gerloff and Klapp on the confined colloidal suspensions in shear flow [@gerloff_2018]. Thus, the PDF of stochastic thermodynamic quantities has some common features across various non-equilibrium systems that have been overlooked to date. A power equal to $p$ given by Eq. \[eq:power\] is dissipated in a sub-system. This is akin to entropy production in a thermodynamic system. However, owing to a large fluctuation, the sub-system can give up the power, akin to entropy consumption [@seifert_2012]. As shown in the inset, there exists a substantial part for large fluctuations in power given by $p<0$. We examine an instantaneous detailed fluctuation relation comparing the ratio of the PDF of the entropy production rate $\mathcal{P}(p)$ to that of the entropy consumption rate $\mathcal{P}(-p)$ via: $$\ln \frac{\mathcal{P}(p)}{\mathcal{P}(-p)} = \beta p,
\label{eq:ft}$$ in which $1/\beta = T_e / \tau_e$ is the ratio of an effective temperature to a time-scale. In the main panel of Fig.\[fig:tot\_power\_FT\]-a, we plot this ratio for various shear rates. A linear dependence is clearly recovered, and thus the fluctuation relation is verified by our data. For ${\ensuremath{\dot\gamma}}< {\ensuremath{\dot\gamma}}_c$ (the blue data), we observe a large slope reflecting a very small effective temperature in the fluid phase. Although not visible in this regime, the slope slightly changes with shear rate. For ${\ensuremath{\dot\gamma}}>{\ensuremath{\dot\gamma}}_c$, the data show a substantially smaller slope, thus implying a considerably larger $T_e$. Moreover, in the solid-branch, data of various shear rates superimpose, and the slope becomes independent of the shear rate, in agreement with the rheology. To provide a clear demonstration, in Fig.\[fig:tot\_power\_FT\]-b we display the effective temperature $T_e$ versus the shear rate. The time scale $\tau_e$ is related to repulsive forces, because dissipative forces are negligible at very small shear rates. The effective temperature is computed via a direct linear regression by using Eq. \[eq:ft\]. Whereas a Bagnold dependence is obtained in the fluid-branch, $T_e$ resembles the behavior of the shear stress in the solid branch. Therefore, our proposed fluctuation relation gives rise to an effective temperature that behaves consistently with the rheology. Whereas the kinetic temperature shows an order of magnitude increase in fluctuations after thickening, the effective temperature increases more than two orders of magnitude. An effective temperature is a thermodynamic tool that enables mapping of a non-equilibrium state to an equilibrium thermodynamic state [@makse_2002]. Therefore, our measured effective temperature $T_e$ may be interpreted as follows: DST may be mapped to an equilibrium transition from a finite temperature to an infinite temperature with large fluctuations. We will show later that this case is true for DST. Because the contribution of frictional forces predominates the rheology, we now focus on large fluctuations due to frictional forces.\
{width="54.00000%"} {width="44.00000%"}\
[**Large fluctuations due to frictional forces.**]{} A large fluctuation caused by frictional forces occurs when the second term of the right-hand-side (rhs) of Eq. \[eq:power\] becomes negative, [[ *i.e., *]{}]{}$p_t = \delta{\ensuremath{\sigma_{xy}}}\times {\ensuremath{\dot\gamma}}<0 $. In the Fig. \[fig:neg\_power\_t\]-a inset, the probability for such a large fluctuation $P(\delta{\ensuremath{\sigma_{xy}}}{\ensuremath{\dot\gamma}}< 0)$ is plotted versus the shear rate. $P(\delta{\ensuremath{\sigma_{xy}}}{\ensuremath{\dot\gamma}}< 0)$ is almost constant (within errorbars) in the fluid branch, and it decreases monotonically in the solid branch. Because $p_t $ is a product of two terms, the couple-stress $\delta{\ensuremath{\sigma_{xy}}}$ and the shear rate ${\ensuremath{\dot\gamma}}$, a negative $p_t$ can be due to either $\delta{\ensuremath{\sigma_{xy}}}< 0$ with ${\ensuremath{\dot\gamma}}>0$ or ${\ensuremath{\dot\gamma}}<0$ with $\delta{\ensuremath{\sigma_{xy}}}> 0$. This can be mathematically expressed as a decomposition relation $$P(\delta{\ensuremath{\sigma_{xy}}}{\ensuremath{\dot\gamma}}< 0) = {\ensuremath{P(\delta\sigma_{xy}^-,{\ensuremath{\dot\gamma}}^+)}}+ {\ensuremath{P({\ensuremath{\dot\gamma}}^-, \delta\sigma_{xy}^+)}},
\label{eq:joint_p_t}$$ in which $P( , )$ is a joint probability. In the main panel of Fig. \[fig:neg\_power\_t\], we display $P(\delta{\ensuremath{\sigma_{xy}}}^-, {\ensuremath{\dot\gamma}}^+)$ (left Y-axis) and $P({\ensuremath{\dot\gamma}}^-, \delta{\ensuremath{\sigma_{xy}}}^+)$ (right Y-axis) with blue circles and red squares, respectively. A dazzling pattern emerges, indicating a decomposition of $P(\delta{\ensuremath{\sigma_{xy}}}{\ensuremath{\dot\gamma}}< 0)$ into two distinct branches for ${\ensuremath{P(\delta\sigma_{xy}^-,{\ensuremath{\dot\gamma}}^+)}}$ and ${\ensuremath{P({\ensuremath{\dot\gamma}}^-, \delta\sigma_{xy}^+)}}$. Moreover, this decomposition has two unique features. First, ${\ensuremath{P(\delta\sigma_{xy}^-,{\ensuremath{\dot\gamma}}^+)}}$ and ${\ensuremath{P({\ensuremath{\dot\gamma}}^-, \delta\sigma_{xy}^+)}}$ are approximately mirror images of one another, and second, in both solid and fluid states, their dependence on ${\ensuremath{\dot\gamma}}$ is dichotomous, meaning that when ${\ensuremath{P(\delta\sigma_{xy}^-,{\ensuremath{\dot\gamma}}^+)}}$ increases by ${\ensuremath{\dot\gamma}}$, ${\ensuremath{P({\ensuremath{\dot\gamma}}^-, \delta\sigma_{xy}^+)}}$ inversely decreases and vice versa. This finding provides evidence that large fluctuations in DST are governed by a simple dichotomy. We previously discovered another dichotomy for large fluctuations in frictionless particulate matter [@rahbari_2017]. We showed that such dichotomies reveal unprecedented information about the collective behavior in rheological phase transitions, thus once again emphasizing that non-equilibrium phase transitions are described by simple dichotomies whose importance has been overlooked to date. We now focus on this novel dichotomy to determine what lessons might be learned.\
We start with the interpretation of a large fluctuation due to ${\ensuremath{\dot\gamma}}<0$. In a simple shear flow, each layer along the shearing direction has a larger drift velocity with respect to the layer beneath it, thus resulting in a positive local shear rate. However, when ${\ensuremath{\dot\gamma}}<0$, a given layer is slower than the one below it. Therefore, a large fluctuation of ${\ensuremath{\dot\gamma}}<0$ corresponds to a non-monotonic change in local drift velocity due to a retarded layer, thus resulting in a local negative power $p_t<0$ providing $\delta{\ensuremath{\sigma_{xy}}}>0$. In Fig. \[fig:neg\_power\_t\] for ${\ensuremath{\dot\gamma}}< {\ensuremath{\dot\gamma}}_c$, ${\ensuremath{P({\ensuremath{\dot\gamma}}^-, \delta\sigma_{xy}^+)}}$ increases as ${\ensuremath{\dot\gamma}}_c$ is approached. This means that the flow becomes non-monotonic as the transition point is approached from below. This increasing non-monotonicity can be rationalized by the well-known instability near ${\ensuremath{\dot\gamma}}_c$ [@grob_2014; @grob_2016; @saw_2019]. In the solid-branch for ${\ensuremath{\dot\gamma}}>{\ensuremath{\dot\gamma}}_c$, the instabilities are washed out, and ${\ensuremath{P({\ensuremath{\dot\gamma}}^-, \delta\sigma_{xy}^+)}}$ decreases by increasing the shear rate. This finding is consistent with conventional wisdom, because the shear is a bias, and it removes all the retarded layers at large ${\ensuremath{\dot\gamma}}$, thus explaining the decreasing trend in ${\ensuremath{P({\ensuremath{\dot\gamma}}^-, \delta\sigma_{xy}^+)}}$ in the solid branch.\
Because of the dichotomy, ${\ensuremath{P(\delta\sigma_{xy}^-,{\ensuremath{\dot\gamma}}^+)}}$ has opposite behavior with respect to ${\ensuremath{P({\ensuremath{\dot\gamma}}^-, \delta\sigma_{xy}^+)}}$ across the transition region. To interpret the behavior of ${\ensuremath{P(\delta\sigma_{xy}^-,{\ensuremath{\dot\gamma}}^+)}}$, we first describe the relation of the couple-stress to the micro-physics. The couple-stress is related to the micro-physics via the total torque $\tau$ acting on particles according to $${\ensuremath{\sigma_{xy,t}}}- \sigma_{yx,t} \propto \sum_{i}\tau_{i},
\label{eq:coupled_stress}$$ in which $i$ runs through all particles in the system. The relation between the couple-stress and torque has been well documented in the context of coarse-graining by Goldhirsch [@goldhirsch_2010] and micropolar fluids by Mitarai, Hayakawa and Nakanishi [@mitarai_2002]. Importantly, the left-hand side (lfh) of Eq. \[eq:coupled\_stress\] is a coarse-grained field quantity, and the right-hand side is a particle property. Consequently, the results must be interpreted with caution. An alternative interpretation of the local shear rate is given by vorticity $\omega$ as $$\omega = \frac{\partial u_x}{\partial y} - \frac{\partial u_y}{\partial x},
\label{eq:virticity}$$ where $u_x$ and $u_y$ are the drift velocity along the x- and y-directions, respectively. In a simple shear flow along the x-direction $\partial_x u_y= 0$; therefore, $\omega = {\ensuremath{\dot\gamma}}$. As a result, $ p_t \propto \tau\times \omega$ and ${\ensuremath{P(\delta\sigma_{xy}^-,{\ensuremath{\dot\gamma}}^+)}}$ correspond to a large fluctuation due to negative total torque of a sub-system whose vorticity is positive $\omega>0$. Re-inspection of Fig. \[fig:neg\_power\_t\] reveals that ${\ensuremath{P(\delta\sigma_{xy}^-,{\ensuremath{\dot\gamma}}^+)}}$, or equivalently $P( \tau^-, \omega^+)$, decreases as the transition point in the fluid branch is approached. This behavior is non-trivial, because it indicates that the instability enhances the uniformity of the rotational degrees of particles (it reduces negative events due to inverse torque).\
![[**Decomposition of probabilities.**]{} The inset shows $P(\delta{\ensuremath{\sigma_{xy}}}{\ensuremath{\dot\gamma}}< 0)$ as a function of the shear rate. In the solid phase, $P(\delta{\ensuremath{\sigma_{xy}}}{\ensuremath{\dot\gamma}}< 0)$ decreases monotonically with increasing shear rate. No discontinuity is observed at the critical shear rate. Main figure: joint probabilities ${\ensuremath{P(\delta\sigma_{xy}^-,{\ensuremath{\dot\gamma}}^+)}}$ and ${\ensuremath{P({\ensuremath{\dot\gamma}}^-, \delta\sigma_{xy}^+)}}$ are displayed by the blue circles and red squares, respectively. Notably, a decomposition of joint probabilities into a mirror-image dichotomy has been discovered. Whereas $ {\ensuremath{P(\delta\sigma_{xy}^-,{\ensuremath{\dot\gamma}}^+)}}$ is reduced almost twice at ${\ensuremath{\dot\gamma}}_c$ and then increases as a function of the shear rate, ${\ensuremath{P({\ensuremath{\dot\gamma}}^-, \delta\sigma_{xy}^+)}}$ is enhanced by the same rate, and it decreases with the shear rate in the fluid phase. Interestingly at ${\ensuremath{\dot\gamma}}= {\ensuremath{\dot\gamma}}_c$, ${\ensuremath{P(\delta\sigma_{xy}^-,{\ensuremath{\dot\gamma}}^+)}}= {\ensuremath{P({\ensuremath{\dot\gamma}}^-, \delta\sigma_{xy}^+)}}$. The packing fraction is $\phi = 0.81$, and the number of particles is $N = 16384$. \[fig:neg\_power\_t\]](fig3 "fig:"){width=".5\textwidth"}\
To gain a better understanding of the enhancement of the uniformity of rotational degrees of freedom, we display subsequent snapshots as the system is sheared from left to right in Fig. \[fig:snapshot1\]-a to -d. The color coding corresponds to the total torque of each particle in which the blue particles have $\tau \leq -5\times 10^{-5}$, the red particles have $\tau \geq 5\times 10^{-5}$, and the green particles have nearly zero torque. The shear rate is ${\ensuremath{\dot\gamma}}= 4.467\times 10^{-6}$, which is below ${\ensuremath{\dot\gamma}}_c$. In snapshot-a, the system is homogeneous except for anomalies appearing as tiny clusters of large negative and positive like-torque particles. These domains of large torque particles show that large torque is spatially localized. These clusters grow spatially in snapshot-b, with more red (positive) clusters. In contrast, in snapshot-c, the blue (negative) clusters appear, and finally in snapshot-d, the red clusters nearly percolate in the system. These clusters of like-torque particles are analogous to domains of like-spin sites in an Ising model at finite temperature. To show how the total torque of the system changes in these snapshots, we display the mean torque in the system as a function of strain in panel-e. In this figure, the corresponding mean torque of the snapshots is marked by red letters/arrows. At $\tau = 1420$, the mean torque is zero; however, it begins to exhibit an oscillatory behavior whose amplitude is first enhanced, then decays and finally fades to zero at $\gamma=1455$. Moreover, the torque subsequently undergoes another oscillatory behavior for ${\ensuremath{\dot\gamma}}>1470$. This is a typical pattern that repeats throughout the simulations for ${\ensuremath{\dot\gamma}}< {\ensuremath{\dot\gamma}}_c$. Indeed, we see that the mean torque is positive at snapshot-b in which the red (positive) clusters predominate the system; in contrast, in snapshot-c, where the blue (negative) clusters predominate the system, the mean torque is negative. The mean torque reaches its maximum in snapshot-d, where the red (positive) clusters nearly percolate in the system. Here, we reach an important conclusion: the oscillatory behavior of the torque, which is reminiscent of rheochaos [@cates_2002], originates from the collective behavior of clusters of like-torque particles. Moreover, from a critical phenomena viewpoint, these spatially extended clusters resemble domains of like-spin sites in the Ising model at finite temperature.\
In snapshot-f, a typical configuration of the system is displayed after thickening. The shear rate is ${\ensuremath{\dot\gamma}}= 1.122 \times 10^{-5}$. In this snapshot, the system is homogeneous except for a few anomalous domains of opposite-torque particles. Interestingly, in these clusters, particles with anomalously large positive torque co-exist with those of anomalously large negative torque. A closeup view of such a cluster is given in panel-g. These regions of very large positive and negative torque particles may be compared to the Ising model at infinite temperature. Now we are ready to explain the behavior of the effective temperature $T_e$ in Fig. \[fig:tot\_power\_FT\]-b. As we discussed earlier, the effective temperature increases more than two orders of magnitude, and this increase might imply that the thickening transition is equivalent to an equilibrium phase transition from a finite temperature to the infinite temperature— an order-disorder-type transition. Indeed, this argument is supported by the snapshots across the thickening that imply a transition from a state of like-torque particles to that of opposite-torque particles. To characterize the oscillation of torque, we compute the auto-correlation function of torque $\left< \tau(0) \tau(\gamma)\right>$ in panel-h for two shear rates below thickening, which, as might be expected, is a damped oscillation of the form $$\left< \tau(0) \tau(\gamma)\right> = e^{-\gamma/\gamma_{rel.}} \cos(\pi\gamma),
\label{eq:auto_torque}$$ where $\gamma_{rel.}$ is a relaxation strain. Solid symbols indicate our data from simulations, and solid lines correspond to fits via Eq. \[eq:auto\_torque\]. The $\gamma_{rel.}$ increases as the transition point at ${\ensuremath{\dot\gamma}}= {\ensuremath{\dot\gamma}}_c$ is approached. In panel-i $\gamma_{rel.}\propto {\ensuremath{\dot\gamma}}^{1.59}$ for ${\ensuremath{\dot\gamma}}<{\ensuremath{\dot\gamma}}_c$ in the fluid phase. This damped oscillation is a direct consequence of the instability, and its increasing relation with ${\ensuremath{\dot\gamma}}$ is consistent with the findings from recent experiments reporting increased instability as the transition point is approached [@rathee_2017; @saint_2018]. In the solid phase, the auto-correlation function becomes a step function reminiscent of a Markov-process, thus indicating that the instability vanishes in the solid-phase after thickening.\
![[**Torque across thickening**]{} Panels-a to -d show subsequent snapshots of the system as it undergoes instability at ${\ensuremath{\dot\gamma}}= 4.467\times 10^{-6}$. The color coding corresponds to the total torque of each particle. As the system is sheared, domains of like-torque particles nucleate and grow, thus resulting in an enhancement of the rotational degrees of freedom, which we show to underlie the well-known instability near the thickening transition. The mean torque as a function of strain is shown in panel-e. In panel-f, a snapshot of the system is shown after thickening for ${\ensuremath{\dot\gamma}}= 1.122 \times 10^{-5}$. The system is homogeneous except for localized clusters of opposite-torque particles. A closeup view of such a cluster is given in panel-g, which shows that the instability diminishes in the solid branch. The auto-correlation function of the total torque of the system shows a damped-oscillatory behavior in the fluid phase (panel-h) whose relaxation strain scales with ${\ensuremath{\dot\gamma}}$ with an exponent of $1.59$. \[fig:snapshot1\]](snapshot_jpg "fig:"){width=".5\textwidth"}\
We now demonstrate the interplay between the above-mentioned collective behavior of the rotational degrees of freedom of particles and the rheology of the system. In Fig. \[fig:snapshot2\], we display snapshots of the system with color coding corresponding to the total torque (first row) and shear stress (second row) of each particle . We display the shear stress per particle, which is different from the coarse-grained shear stress over a domain. Panels-a to -d row $1$ show the total torque per particle as the system enters the instability region, where the total torque in the system oscillates ( according to Fig. \[fig:snapshot1\]-e). Each snapshot shows a configuration of the system after $\delta \gamma = 1$ for ${\ensuremath{\dot\gamma}}= 4.467\times 10^{-6}$. Remarkably, in panel-a row $2$, where the same configuration is colored by the total shear stress per particle, particles with large shear stress form stress-bearing chains along the compression direction (yellow chains). In snapshot-b, clusters of like-torque particles become larger, and as a result, the stress-bearing chains become more pronounced, with a color shift to larger stresses (red) along the compression direction. Notably, as a result of larger negative like-torque clusters in panel-b, negative stress-bearing chains in blue (negative stress) form along the dilation direction. These red and blue stress-bearing structures become more pronounced in panels-c and -d, where clusters of like-torque particles become larger. In addition, the configuration of the system changes dramatically from one snapshot to the other below thickening. In panels-e to -h, we display similar snapshots after thickening for ${\ensuremath{\dot\gamma}}= 1.122 \times 10^{-5}$. Each snapshot shows a configuration of the system after a strain difference of $\delta \gamma = 2$. There are strong stress-bearing structures along the compression direction (red chains), and particles with negative stress also form chains along the dilation direction (blue chains). Moreover, chains of positive stress percolate through the system, whereas chains of negative stress do not. Interestingly, even though the strain difference between each snapshot here is twice that in panels-a to -d, the structure of stress-bearing chains does not change dramatically from one snapshot to the next, because the system is in solid state. To quantitatively determine the correspondence between like-torque clusters and stress-bearing structures in the fluid phase, we plot the second moment of the torque $\left<{\ensuremath{\sigma_{xy}}}^2\right>$ and stress $\left<\tau^2\right>$ per particle versus the strain in Supplementary Fig. 4. It can be seen that $\left<{\ensuremath{\sigma_{xy}}}^2\right>$ and $\left<\tau^2\right>$ change proportionately as a function of the strain, thus indicating that fluctuations in torque directly influence the rheology, and the oscillation of torque results in spatiotemporal fluctuations in stress. However, in the solid phase, Supplementary Fig. 5, these quantities are independent of one another, and we do not observe an appreciable correlation between the torque and stress.\
[**Conclusion.**]{} We investigated fluctuations of power in a model system undergoing DST. We showed that large fluctuations caused by frictional forces are governed by a simple dichotomy that underlies the novel collective behaviors across the thickening transition. The joint probability of large fluctuations due to frictional forces ${\ensuremath{P(\delta\sigma_{xy}^-,{\ensuremath{\dot\gamma}}^+)}}$ decreases as the thickening transition is approached, thus indicating an enhanced uniformity of rotational degrees of freedom. Consequently, we discovered clusters of like-torque particles akin to an Ising model in finite temperature. We showed that (1) the growth of these clusters directly correlates with the rheology (2) the formation of the clusters results in spatially heterogeneous structures of stress-bearing chains below the thickening and (3) a competition between the opposite like-torque clusters underlies the origin of temporal fluctuations. Accordingly, we have identified the origin of spatiotemporal fluctuations near the thickening transition. After thickening in the solid state, we observe clusters of opposite-torque particles, analogously to an Ising model at infinite temperature. Moreover, in this regime, particles grind against each other, as they would be obliged to do by frictional forces that glue particles to one another. This opposite motion is similar to that of [*gear wheels*]{} in a mechanical watch. The increase of the effective temperature by more than two orders of magnitude at the transition point can thus be explained as the system transitions from the fluid state, in which large clumps of particles rotate rigidly in one or the other direction, to a solid state, in which particles grind against each other.\
We now conclude with some points discussing the possible implications of our results for a broader audience:\
(i) [ [**Order-disorder scenario.**]{} Besides the Wyart-Cates picture, several scenarios have been proposed to describe the mechanisms underlying shear thickening. Two well-known mechanisms are (1) cluster formation due to hydrodynamic interactions [@cheng_2011] and (2) an order-disorder transition [@hoffman_1974; @hoffman_1998]. In these scenarios, particles form ordered structures at small flow velocities, and these structures become unstable at large flow velocities. Although the formation of such ordered structures cannot occur in a bidisperse system, the similarity of these scenarios to our proposed order-disorder transition is notable. However, our order-disorder scenario involves only a transition in rotational degrees of freedom, in contrast to the above-mentioned mechanisms involving steric preferences of translational degrees of freedom.\
]{}
![[**Torque-stress snapshots .**]{} Snapshots of the system both below and above thickening. The color code in row number $1$ in each set corresponds to the total torque of each particle, and row number $2$ corresponds to the total shear stress per particle. In panels-a to -d, the shear rate is ${\ensuremath{\dot\gamma}}= 4.467\times 10^{-6}$, and the difference between each snapshot from left to right is a strain difference of unit length $\delta\gamma = 1$. The total torque of the system is given in panel-e of Fig. \[fig:snapshot1\], which corresponds to the nucleation of the instability. Clusters of like-torque grow larger as the instability sets in. In panel-b, larger clusters than those in panel-a result in stress-bearing structures. Remarkably, very large positive stress-bearing chains form along the compression direction (red chains), and shorter chains of very large negative stress form along the dilation direction (blue chains). In panel-c, the clusters of very large negative like-torque particles have become larger, thus resulting in an appreciable enhancement in very large negative stress-bearing structures along the dilation direction (blue chains). The second set at the bottom shows similar snapshots for ${\ensuremath{\dot\gamma}}= 1.122 \times 10^{-5}$ above thickening, and the difference between each snapshot from left to right is $\delta\gamma = 2$. Locally, like-torque clusters are absent, and instead we see domains of particles with very large positive and negative torques. In contrast to the stress-bearing chains below thickening, those structures here seem to have no correlation with the torque, and their configurations persist for a long time. \[fig:snapshot2\]](snapshot_3_jpg "fig:"){width=".5\textwidth"}\
(ii) [**Yielding of frictional systems.**]{} We note that Chattoraj [[ *et al. *]{}]{}have recently demonstrated that a deformed very dense frictional system exhibits oscillatory instability near yielding, as a result of a pair of complex eigenvalues of the Hessain matrix [@chattoraj_2019; @chattoraj_2019_2]; this is in contrast to a frictionless system whose Hessian has only real eigenvalues, and a failure occurs when one of those eigenvalues becomes zero— a saddle node bifurcation [@maloney_2004]. Chattoraj [[ *et al. *]{}]{} have also discussed a possible relation between the oscillatory amplifications in a frictional system with a long-standing problem in earthquake physics, remote triggering [@felzer_2006]. We expect that our discovery of the collective behavior of the rotational degrees of freedom of frictional particles might also shed light on the micro-physics of the oscillatory instability near the yielding transition.
(iii) [ [**Order parameter of non-equilibrium.**]{} Collective behaviors in equilibrium phase transitions are described by order parameters [@kardar_2007]. Non-equilibrium phase transitions have a much richer phenomenology, and as a result of that their collective behavior, cannot be explored by a single order parameter. Instead, as we demonstrated here, distributions of some order-like parameters must be examined. We showed that such examination can be performed well by using the stochastic thermodynamics of some stochastic energetics parameters. The investigation of large fluctuations in such stochastic energetics parameters, such as power, led us to the discovery of dichotomies describing the underlying collective modes. The scheme developed here can be easily applied to a much wider class of non-equilibrium phase transitions.\
]{}
(iv) [ [**Effective temperatures.**]{} The 21st century is the era of non-equilibrium statistical physics, which has a wide focus on many real-life, everyday phenomena. Yet, equilibrium thermodynamics is key to understanding the nature of non-equilibrium phenomena. A quantitative means of establishing such a connection is provided by the so-called effective temperatures that bridge equilibrium and non-equilibrium worlds [@makse_2002]. The first such attempt may be the work by Edwards and Oakeshott in an acclaimed paper that proposed that the principles of equilibrium statistical mechanics can be applied to granular materials [@edwards_1989]. The authors suggested that the packing fraction in granular materials may play the same role as energy in equilibrium statistical mechanics and defined the analogous density of states $\Omega(\phi) = \sum_{\nu} \delta(\phi - \phi_v)$, where the subscript $\nu$ corresponds to jammed states with the condition of force and torque balance being satisfied [@bi_2015_2]. Edwards’ entropy can then be defined as $S = \ln \Omega(\phi)$, which results in a temperature-like quantity $X$ via $1/X = \partial
S(\phi) / \partial \phi$. Later, because of the importance of normal and tangential forces in configurations of jammed states, stress was considered as an additional state variable. The resulting temperature, known as angoricity, corresponds to a jammed canonical ensemble. Several methods have been developed to test the ideas of Edwards. Probably the most renowned method is the overlapping histogram method by Dean and Leferevre [@dean_2003], which has been used in many recent studies [@zhao_2012; @mcnamara_2009; @bililign_2019]. Although Edwards’ original intention was to apply ensemble theory to describe the dynamics of slowly driven granular materials, in which the system moves from one jammed state into another, most recent literature has focused only on static jammed states. Furthermore, some recent work has suggested that angoricity is analogous to an effective temperature that describes the strength of the mechanical noise in driven granular materials [@behringer_2008; @zheng_2018_2]. However, this approach is in early stages of development. One large obstacle to applying Edwards’ theory is that the density of states $\Omega(\phi)$ and the corresponding partition function may be unknown for a given granular ensemble [@bi_2015_2]. Our stochastic thermodynamic approach provides an alternative way to compute an effective stress-temperature. We showed here and in a previous report [@rahbari_2017] that the resulting effective temperature is stress-like, [[ *i.e., *]{}]{}, it remains constant at the vanishing limit of the shear rate in the jammed configurations. However, it is Bagnoldian in the fluid phase. This method provides an alternative approach to investigate fluctuations in a wide range of driven non-equilibrium systems. The main advantage of this approach is that it does not require strictly jammed states.\
]{}
[**Acknowledgments**]{} We acknowledge fruitful discussions with Hyunggyu Park, Mike Cates, Hisao Hayakawa, Ludovic Berthier, Michio Otsuki, Takeshi Kawasaki, Abbas Ali Saberi, and Ji Woong Yu.
**Supplemental Materials: Origin of spatiotemporal fluctuations in Discontinuous Shear Thickening**
Methods {#sec:methods}
=======
We use a linear dashpot-spring to model both normal and tangential forces [@poeschel2005]. Two particles at positions ${\bf r_i}$ and ${\bf r_j}$ with radii $a_i$ and $a_j$, respectively, interact when they overlap, $\delta = |{\bf r_i} - {\bf r_j} | - (a_i + a_j) <0 $. A spring whose force is proportional to the overlap $\delta$ acts as a repulsive mechanism between two colliding particles. The interaction force along the normal direction is then given by $$f_{ij,n} = k_n \delta - \eta_n ({\bf v_i} - {\bf v_j}) \cdot {\bf n}_{ij},$$ where $k_n$ and $\eta_n$ are the elastic, and damping constants, respectively, and ${\bf n}_{ij}$ is a unit vector along the line connecting the centers of two particles ${\bf n}_{ij} = ({\bf r_i} - {\bf r_j}) / |{\bf r_i} - {\bf r_j} |$.
With ${\bf \omega_i}$ and ${\bf \omega_j}$, the angular velocities, the total tangential velocity at the contact point can be written as $${\bf v}_{ij,t} = ({\bf I} - {\bf n}_{ij}{\bf n}_{ij} ) \cdot[ {\bf v_i - v_j} - ( a_i{\bf \omega_i} + a_j{\bf \omega_j}) \times {\bf n}_{ij} ].$$
Integrating the tangential velocity ${\bf v}_{ij,t}$ from the initiation of contact to the current time gives the tangential overlap as $\xi = \int_{0}^{t_{coll}} | {\bf v}_{ij,t}| dt^\prime$. A spring proportional to $\xi$ acts in the tangential direction along the contact plane to model the static friction $$f_{ij,t} = k_t \xi - \eta_t {\bf v}_{ij,t}\cdot {\bf t}_{ij},$$ where $k_t$ and $\eta_t$ are the spring and damping coefficients, and ${\bf t}_{ij}$ is a unit vector along the contact plane, ${\bf t}_{ij} \cdot {\bf n}_{ij} = 0$. A torque proportional to the tangential force acts on each particle. Accordingly, the total force is equal to $${\bf f}_{ij} = f_{ij,n} {\bf n}_{ij} + f_{ij,t} {\bf t}_{ij},$$ from which translational and rotational degrees of freedom are coupled in this model.\
We use a $50:50$ bidisperse mixture of particles whose ratio of radii is $1.4$. The diameter of small particles is chosen to be the unit of length. The mass is equal to the area of each particle. The spring constants are $k_n = 1$ and $k_t = 0.5k_n$ and the damping coefficients are $\eta_n = \eta_t = 1$. The magnitude of the tangential force is bound by Coulomb’s frictional law $|f_{ij,t}|\le \mu |f_{ij,n}|$ where $\mu = 1$ is the friction coefficient.
We use Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) to integrate equations of motion of particles. This process is done by using pair style $gran/hooke/history$ to model the interactions plus Lees-Edwards boundary conditions by using $deform$.
Complementary results
=====================
A typical flow curve of the system is given in Fig. \[fig:mean\_Sxy\]-a. The simulations start at a relatively high shear rate ${\ensuremath{\dot\gamma}}= 10^{-4}$, and the shear rate is decreased stepwise such that the steps are equidistant on a logarithmic scale. Each simulation runs for a total strain of $\gamma_{tot}=20L$, and we store a snapshot of the system at each strain increment equal to the unit of length $\delta\gamma = 1$. There exists a critical shear rate ${\ensuremath{\dot\gamma}}_c \simeq 10^{-5}$ at which the stress abruptly changes by more than one order of magnitude. This finding is consistent with those reported by Otsuki and Hayakawa [@otsuki_2011]. The packing fraction is $\phi=0.81$ and the system shows DST for densities above a fictitious critical density of $\phi_c=0.795$, again in agreement with [@otsuki_2011]. The critical shear rate separates two distinct fluid and solid states. For ${\ensuremath{\dot\gamma}}< {\ensuremath{\dot\gamma}}_c$, a Bagnoldian rheology $\sigma_{xy}, P \propto {\ensuremath{\dot\gamma}}^2$ is observed, thus indicating a fluid-like behavior. For ${\ensuremath{\dot\gamma}}> {\ensuremath{\dot\gamma}}_c$, the system is solid-like, and the flow curves are generally described by Herschel-Bulkley rheology $\sigma_{xy}, P = \sigma_y + k{\ensuremath{\dot\gamma}}^y$ with a distinct offset $\sigma_y$ that indicates the appearance of shear-driven yield stress. We separately measure the contributions of tangential and repulsive interactions to the stress tensor. Green squares and blue circles correspond to repulsive, ${\ensuremath{\sigma_{xy,n}}}$, and tangential, $({\ensuremath{\sigma_{xy,t}}}+ {\ensuremath{\sigma_{yx,t}}})/2$, shear stress, respectively, and the inset shows the ratio of the two. For ${\ensuremath{\dot\gamma}}<{\ensuremath{\dot\gamma}}_c$, the normal component is dominant; however, above ${\ensuremath{\dot\gamma}}_c$, the tangential component predominates in the momentum transfer. This finding is in contrast to the pressure in which the normal component is dominant both above and below the thickening (Appendix Fig. \[fig:P\_t\_n\]). These results once again emphasize the canonical role played by friction in DST.\
Jamming of frictional disks occurs at a coordination number of $z_J = d+1 = 3$ [@vanhecke_2010]. Across the transition region in DST, the coordination number also increases from $z = z_f({\ensuremath{\dot\gamma}})<z_{J}$ in the fluid state to $z = z_s({\ensuremath{\dot\gamma}})>z_{J}$ in the solid branch (see Appendix Fig. \[fig:coordination\]), thus demonstrating that shear thickening is a consequence of a shear driven jamming transition [@bi_2011], which is generally accompanied by proliferation of the contacts [@seto_2013]. All aforementioned physical measures faithfully present the inherent state of the system as it transitions from the fluid to the solid states. However, the kinetic temperature of the system in Fig. \[fig:mean\_Sxy\]-b, known as the granular temperature $T_G$, has misleading behavior for ${\ensuremath{\dot\gamma}}> {\ensuremath{\dot\gamma}}_c$. Whereas it shows a quadratic dependence below ${\ensuremath{\dot\gamma}}_c$, consistently with the rheology, above ${\ensuremath{\dot\gamma}}_c$, $T_G$ has an algebraic dependence on ${\ensuremath{\dot\gamma}}$ with an exponent of $1.16$, which is reminiscent of the behavior of a dense fluid. This finding is in contrast to the jammed nature of the system above ${\ensuremath{\dot\gamma}}_c$. Therefore, the kinetic temperature cannot be used as a real measure of fluctuations in this system, and an alternative measure is required. To resolve this discrepancy, an effective temperature that complies with the rheology is required. To this end, we investigate large fluctuations in injected power $p$ by using the tools of stochastic thermodynamics to derive the effective temperature.\
![[**Rheology across thickening.**]{} (a) Shear stress as a function of the shear rate for normal (squares) and tangential (circles) components. For ${\ensuremath{\dot\gamma}}< {\ensuremath{\dot\gamma}}_c$, the behavior is Bagnoldian (${\ensuremath{\dot\gamma}}^2$): a typical fluid-like behavior. For ${\ensuremath{\dot\gamma}}> {\ensuremath{\dot\gamma}}_c$, the system is solid-like and exhibits a finite yield stress. The inset shows that below thickening, the normal component is dominant, whereas above the transition, the tangential part dominates. (b) Kinetic temperature as a function of shear rate. Whereas for ${\ensuremath{\dot\gamma}}< {\ensuremath{\dot\gamma}}_c$, the behavior is Bagnoldian, in agreement with the rheology, above ${\ensuremath{\dot\gamma}}_c$, it shows a dense fluid behavior not consistent with the rheology. The packing fraction is $\phi = 0.81$, and the number of particles is $N = 16384$. \[fig:mean\_Sxy\]](fig1-1 "fig:"){width="47.50000%"} ![[**Rheology across thickening.**]{} (a) Shear stress as a function of the shear rate for normal (squares) and tangential (circles) components. For ${\ensuremath{\dot\gamma}}< {\ensuremath{\dot\gamma}}_c$, the behavior is Bagnoldian (${\ensuremath{\dot\gamma}}^2$): a typical fluid-like behavior. For ${\ensuremath{\dot\gamma}}> {\ensuremath{\dot\gamma}}_c$, the system is solid-like and exhibits a finite yield stress. The inset shows that below thickening, the normal component is dominant, whereas above the transition, the tangential part dominates. (b) Kinetic temperature as a function of shear rate. Whereas for ${\ensuremath{\dot\gamma}}< {\ensuremath{\dot\gamma}}_c$, the behavior is Bagnoldian, in agreement with the rheology, above ${\ensuremath{\dot\gamma}}_c$, it shows a dense fluid behavior not consistent with the rheology. The packing fraction is $\phi = 0.81$, and the number of particles is $N = 16384$. \[fig:mean\_Sxy\]](fig1-2 "fig:"){width="47.50000%"}\
In this appendix, we display some auxiliary figures to complement the descriptions of the main figures in this manuscript.
![[**Rheology across thickening.**]{} Pressure defined as $(\sigma_{xx} + \sigma_{yy} )/2$ for both normal and tangential components given by circles and squares, respectively. Below thickening, Bagnold behavior (${\ensuremath{\dot\gamma}}^2$) is seen; above the transition, the pressure shows a behavior of Herschel-Bulkley form. In contrast to the shear stress, the normal part of the pressure is dominant both above and below the transition. \[fig:P\_t\_n\]](fig2 "fig:"){width="45.00000%"}\
![[**Coordination number.**]{} The mean coordination numbers as a function of the shear rate. The jamming coordination number for this system is $z_J = 3$. Below thickening, $z<z_J$. At the transition point, the coordination number increases to $z>z_J$. \[fig:coordination\]](fig3_si "fig:"){width="45.00000%"}\
![[**Correlated behavior of stress and torque in the fluid phase.**]{} To quantitatively show that the formation of like-torque clusters results in stress heterogeneities, we display the second-moment of shear stress per particle (circles) and total torque of each particle (squares). These two quantities are inter-correlated, and any fluctuation in the total torque of particles accordingly results in that in the shear stress per particle. This figure is a quantitative demonstration of the correlation of stress and torque snapshots in Fig. \[fig:snapshot2\]. The shear rate is ${\ensuremath{\dot\gamma}}= 4.467\times 10^{-6}$. \[fig:mean\_Sxy\_per\_part\_fluid\]](fig_mean_Sxy_per_part_fluid "fig:"){width="50.00000%"}\
![[**Un-correlated behavior of stress and torque in the solid phase.**]{} Similarly to the previous figure, we display the second-moment of shear stress per particle (circles) and total torque of each particle (squares). The shear rate is ${\ensuremath{\dot\gamma}}= 1.122 \times 10^{-5}$ above thickening. Fluctuations in the shear stress per particle and torque can be seen to be un-correlated. \[fig:mean\_Sxy\_per\_part\_solid\]](fig_mean_Sxy_per_part_solid "fig:"){width="50.00000%"}\
|
---
abstract: 'The number of $n$-edge embedded graphs (rooted maps) on the $g$-torus is known to grow as $t_gn^{5(g-1)/2}12^n$ when $g$ is fixed and $n$ tends to infinity. The constants $t_g$ can be computed thanks to the non-linear “$t_g$-recurrence”, strongly related to the KP hierarchy and the double scaling limit of the one-matrix model. The combinatorial meaning of this simple recurrence is still mysterious, and the purpose of this note is to point out an interpretation, via a connection with random (Brownian) maps on surfaces. Namely, we show that the $t_g$-recurrence is equivalent, via known combinatorial bijections, to the fact that $\mathbf{E}X_g^2=\frac{1}{3}$ for any $g\geq 0$, where $X_g,1-X_g$ are the masses of the nearest-neighbour cells surrounding two randomly chosen points in a Brownian map of genus $g$. This raises the question (that we leave open) of giving an independent probabilistic or combinatorial derivation of this second moment, which would then lead to a concrete (combinatorial or probabilistic) interpretation of the $t_g$-recurrence. We also compute a similar moment in the case of three marked points, for which a similar phenomenon occurs. In fact, we conjecture that for any $g\geq 0$ and any $k\geq 2$, the masses of the $k$ nearest-neighbour cells induced by $k$ uniform points in the genus $g$ Brownian map have the same law as a uniform $k$-division of the unit interval. We leave this question open even for $(g,k)=(0,2)$.'
author:
- 'Guillaume Chapuy[^1]'
bibliography:
- 'biblio.bib'
title: 'On tessellations of random maps and the $t_g$-recurrence'
---
Introduction and results
========================
In this note a *map* is a graph embedded without edge crossings on a closed oriented surface, in such a way that the connected components of the complement of the graph, called *faces*, are each homeomorphic to a disk. Loops and multiple edges are allowed, and maps are considered up to oriented homeomorphisms. A map is *rooted* if an edge is distinguished and oriented. The number $m_g(n)$ of rooted maps with $n$ edges on the surface of genus $g$ satisfies, for fixed $g\geq 0$ and $n\rightarrow \infty$: $$\begin{aligned}
\label{eq:mgn}
m_g(n) \sim t_g n^{\frac{5(g-1)}{2}} 12^n, \mbox{ for }t_g >0.\end{aligned}$$ In genus $0$, this result follows from the exact formula $m_0(n)=\frac{2\cdot 3^n}{(n+2)(n+1)}{2n \choose n}$ due to Tutte [@Tutte:census]. In higher genus, it was proved in [@BC0] using generating functions. A direct combinatorial interpretation of Tutte’s formula for maps of genus $0$ was given by Cori and Vauquelin [@CV] and much simplified by Schaeffer [@Schaeffer:phd; @ChassaingSchaeffer]. A combinatorial interpretation of was given in [@CMS] using the Marcus-Schaeffer bijection [@MS] and further developped in [@Chapuy:trisections].
None of the methods just mentioned enable to say much about the sequence of constants $(t_g)_{g\geq 0}$ that appear in , and indeed these references give explicit values only for very small values of $g$. There is however a remarkable recurrence formula to compute these numbers, that we call the *$t_g$-recurrence*. It is better expressed in terms of the numbers $\tau_g=2^{5g-2}\Gamma\left(\frac{5g-1}{2}\right)t_g$ and is given by: $$\begin{aligned}
\label{eq:tg}
\tau_{g+1} = \frac{(5g+1)(5g-1)}{3}\tau_{g}+\tfrac{1}{2}\sum_{g_1=1}^{g}\tau_{g1}\tau_{g+1-g1}, \ \ g \geq 0,\end{aligned}$$ which enables to compute these numbers easily starting from $\tau_0=-1$. This result was first stated in mathematical physics in relation with the *double scaling limit* of the one-matrix model, and obtained via a non-rigorous scaling of expressions involving orthogonal polynomials (we refer to [@LZ p201] for historical references). A more algebraic approach is based on the fact that the partition function of maps on surfaces, with infinitely many parameters marking vertex degrees, is a tau-function of the KP hierarchy. Going from the KP hierarchy to the recurrence (or to an equivalent Painlevé-I ODE for an associated generating function) relies on a trick of elimination of variables that can be performed in different ways and whose generality is, as far as we know, yet to be fully understood (for the case of triangulations see [@mcfly Appendix B.] or [@GJ; @tg] and for general maps see[@CC]).
The main observation of this note is to relate the recurrence to another side of the story, namely the study of random maps and their scaling limits. We refer to [@miermont:survey] for an introduction to this topic. To state our main observation we first need a few more definitions. A *quadrangulation* is a map in which each face contains exactly four corners, [*i.e.*]{} is bordered by exactly four edge-sides. It is *bipartite* if its vertices can be colored in black and white in such a way that there is no monochromatic edge. For each $n,g\geq0$, there is a classical bijection, due to Tutte, between rooted maps of genus $g$ with $n$ edges and rooted bipartite quadrangulations of genus $g$ with $n$ faces. For $n,g\geq 0$, we let ${\mathcal{Q}_n^{(g)}}$ be the set of rooted bipartite quadrangulations of genus $g$ with $n$ faces (with the convention that there is a single quadrangulation with $0$ face, which has genus $0$, no edge, and two vertices). We let ${{\mathbf{q}}_{n}^{(g)}}\in_u {\mathcal{Q}_n^{(g)}}$ be a bipartite quadrangulation of genus $g$ with $n$ faces chosen uniformly at random (the notation $\in_u$ to denote a uniform random element of a set will be used throughout this note). We equip the vertex set of ${{\mathbf{q}}_{n}^{(g)}}$ with the graph distance, noted $\mathbf{d}_n$, and with the uniform measure, noted $\mathbf{\mu}_n$. This makes ${{\mathbf{q}}_{n}^{(g)}}\equiv ({{\mathbf{q}}_{n}^{(g)}},\mathbf{d}_n,\mu_n)$ into a compact measured metric space. The set of (isometry classes of) such spaces is equipped with the Gromov-Hausdorff-Prokhorov (GHP) topology as in [@Miermont:tessellations Sec. 6]. A *Brownian map of genus $g$* is a random compact measured metric space $({\mathbf{q}}_\infty, d_\infty, \mu_\infty)$ that is such that: $$({{\mathbf{q}}_{n}^{(g)}},\tfrac{1}{n^{1/4}}\mathbf{d}_n,\mu_n)
\longrightarrow
({{\mathbf{q}}_\infty^{(g)}}, d_\infty, \mu_\infty),$$ in distribution along some subsequence for the GHP topology. The existence of Brownian maps of genus $g$ was proved in [@Miermont:tessellations], and their unicity for each $g \geq 1$ has been announced by Bettinelli and Miermont [@BettinelliMiermont] (in genus $0$ the unicity is an important result proved independently by Miermont [@Miermont:GH] and Le Gall [@LG:GH]). However the unicity of the limit is not needed for our discussion since we will prove the convergence of all the observables we are interested in. Also note that some authors prefer to introduce an additional scaling factor $(8/9)^{1/4}$ to the distance, but this is irrelevant to our discussion so we prefer avoid it.
\[thm:obs1\] For $g\geq 0$, let $({{\mathbf{q}}_\infty^{(g)}}, d_\infty, \mu_\infty)$ be a Brownian map of genus $g$. Let ${\mathbf{v}}_1,{\mathbf{v}}_2 \in {\mathbf{q}}^{(g)}_\infty$ be chosen independently according to the probability measure $\mu_\infty$, and let ${\mathbf{X}}_g, 1-{\mathbf{X}}_g$ be the masses of the corresponding cells in the nearest neighbour tessellation of ${{\mathbf{q}}_\infty^{(g)}}$ induced by ${\mathbf{v}}_1$ and ${\mathbf{v}}_2$, that is to say: $${\mathbf{X}}_g := \mu_\infty\Big(\big\{x\in {{\mathbf{q}}_\infty^{(g)}},\ d_\infty(x,{\mathbf{v}}_1)<\mathbf{d}_\infty(x,{\mathbf{v}}_2) \big\}\Big).$$ Then the sequence of numbers $\tau_g=2^{5g-2}\Gamma\left(\frac{5g-1}{2}\right)t_g$ satisfies: $$\tau_{g+1} = 2(5g+1)(5g-1)\tau_{g}\cdot \mathbf{E}[{\mathbf{X}}_{g}(1-{\mathbf{X}}_g)]
+\frac{1}{2}\sum_{g_1=1}^{g}\tau_{g1}\tau_{g+1-g1}, \ \ \ g\geq 0.$$
From we immediately deduce:
\[cor:main\] For any $g\geq 0$, the random variable ${\mathbf{X}}_g$ satisfies $$\mathbb{E}[{\mathbf{X}}_g(1-{\mathbf{X}}_g)]=\frac{1}{6},$$ or equivalently $\mathbf{E}{\mathbf{X}}_g^2=\frac{1}{3}.$
The reader may find surprising that $\mathbf{E}{\mathbf{X}}_g^2$ does not depend on $g\geq0$: indeed, although it is natural to expect that *local* statistics of Brownian maps do not depend on the genus, the nearest-neighbour tessellation depends *globally* of the metric space ${{\mathbf{q}}_\infty^{(g)}}$, that *is* genus dependent. In fact, this unexpected property is the main reason why this note is written. It suggests that there exists a simple probabilistic or combinatorial interpretation of this mysterious fact, based on a symmetry of the Brownian map, but we have not been able to find it. We emphasize that, via Theorem \[thm:obs1\], such an interpretation would provide a proof of the $t_g$-recurrence independent of orthogonal polynomials, matrix models or integrable hierarchies.
It is natural to ask if other moments of the variables ${\mathbf{X}}_g$ or related random variables are computable, and in which way they depend on the genus. Unfortunately we won’t go very far in this direction. Let ${\mathbf{v}}_1,{\mathbf{v}}_2,\dots,{\mathbf{v}}_k$ be $k\geq 2$ points in ${{\mathbf{q}}_\infty^{(g)}}$ chosen independently at random according to the Lebesgue measure $\mu_\infty$. Let $({\mathbf{Y}}_g^{(i:k)})_{1\leq i \leq k}$ be the masses of the $k$-nearest-neighbour cells induced by the ${\mathbf{v}}_i$’s, *i.e.* for $i\in[1..k]$ let $${\mathbf{Y}}_g^{(i:k)}:=\mu_\infty \big\{ x\in {{\mathbf{q}}_\infty^{(g)}}, \ \forall j\in [1..k]\setminus\{i\}, \ d_\infty(x,{\mathbf{v}}_i)<d_\infty(x,{\mathbf{v}}_j)\big\}.$$ We note that ${\mathbf{X}}_g={\mathbf{Y}}_g^{(1:2)}$, so we could have used a single notation, but we prefer to keep the lighter notation ${\mathbf{X}}_g$ for ${\mathbf{Y}}_g^{(1:2)}$ throughout this note. The following result is similar to, and as mysterious as Theorem \[cor:main\]:
\[thm:3points\] For $g\geq 0$, the masses ${\mathbf{Y}}_g^{(1:3)}, {\mathbf{Y}}_g^{(2:3)}, {\mathbf{Y}}_g^{(3:3)}$ of the Voronoï cells induced by three independent Lebesgue distributed points in the Brownian map of genus $g$ satisfy, for $g\geq 0$: $$\mathbf{E}[{\mathbf{Y}}_g^{(1:3)}{\mathbf{Y}}_g^{(2:3)}{\mathbf{Y}}_g^{(3:3)}]
=\frac{1}{60}.$$
As we will see, the fact that this moment is computable reflects the existence of a combinatorial device known as the “trisection lemma” [@Chapuy:trisections]. The fact that it does not depend on the genus, and that it coincides[^2] with the corresponding moment for a uniform three-division of the interval $[0,1]$, is as mysterious as for the previous result (or even more, since as we will see the computations leading to Theorem \[thm:3points\] are quite delicate and involve intermediate expressions that are complicated and magically become simpler at the last minute).
We won’t prove anything on higher moments or other values of $k$ since we lack the tools to study them. However, numerical simulations suggest that the first joint moments of the random variables $({\mathbf{Y}}_g^{(i:k)})_{1\leq i \leq k}$, for small values of $g$ and $k$, are close to what they are for a uniform partition of $[0,1]$ into $k$ intervals. Of course one has to be prudent with simulations, given that the metric observables in random discrete maps of size $n$ typically converge to their Brownian map analogue at speed $O(n^{-1/4})$, and that our numerical simulations are performed only for $n\approx 10^6$ to $10^7$. However Theorems \[thm:obs1\] and \[thm:3points\] support this conjecture, so we dare to state it explicitly:
\[conjecture\] For $k\geq 2, g\geq 0$, let ${{\mathbf{q}}_\infty^{(g)}}\equiv ({{\mathbf{q}}_\infty^{(g)}}, d_\infty, \mu_\infty)$ be a genus $g$ Brownian map and let ${\mathbf{v}}_1,\dots,{\mathbf{v}}_k$ be chosen according to $\mu_\infty^{\otimes k}$. Then the random vector $({\mathbf{Y}}_g^{(1:k)},{\mathbf{Y}}_g^{(2:k)},\dots, {\mathbf{Y}}_g^{(k:k)})$ has the same law as the subdivision of the unit interval induced by $k-1$ independent uniform variables. In particular, for any $g\geq 0$, ${\mathbf{X}}_g={\mathbf{Y}}_g^{(1:2)}$ is uniform on $[0,1]$.
To conclude this introduction, we emphasize that our main observation relates the moment $\mathbf{E}{\mathbf{X}}_g^2$ to the $g$-th step of the $t_g$-recurrence. In particular, the fact that $\mathbf{E}{\mathbf{X}}_0^2=1/3$ for the *genus $0$* Brownian map is only “equivalent” to the computation of the genus $1$ constant $t_1$, that can be performed by hand in several ways (and similarly, our proof of Theorem \[thm:3points\] for $g=0$ relies only on the value of the constants $t_1$ and $t_2$). However, proving Conjecture \[conjecture\] even for $(g,k)=(0,2)$ would be interesting in itself. Readers familiar with Miermont’s bijection [@Miermont:tessellations] may try to approach this problem by exact counting of well-labelled 2-face maps (we have failed trying to do so). One could also hope that in the future purely probabilistic methods (for example using the QLE viewpoint on the Brownian map [@MillerSheffield]) will enable to determine the full law of ${\mathbf{X}}_0$ or even the law of the vector $({\mathbf{Y}}_0^{(i:k)})_{1\leq i\leq k}$ for each $k$. In an opposite direction, we recall that the $t_g$-recurrence is only a “shadow” of the fact that the generating functions of maps satisfy a set of infinitely many partial differential equations called the KP hierarchy. It is natural to expect that other joint moments of the variables ${\mathbf{Y}}_g^{(i:k)}$, apart from the two cases we have been able to track, are related to these equations. This may lead to a way, based on integrable hierarchies, of approaching Conjecture \[conjecture\].
Proof of our main observation (Theorem \[thm:obs1\])
====================================================
Preliminaries
-------------
For $g\geq 0$ we let $Q_g(z)$ be the generating function of rooted bipartite quadrangulations of genus $g$ by the number of faces, and we let $Q_g^\bullet(z)$ be the g.f. of the same objects where an additional vertex is pointed. We let $m_g(n) = [z^n] Q_g(z)$ and we use the same notation with $~^\bullet$. In what follows the notation $a(n)\sim b(n)$ means (classically) that $a(n)/b(n)\rightarrow 1$ when $n$ tends to infinity, while the notation $F(z)\sim G(z) $ means that both $F$ and $G$ are algebraic functions of radius of convergence $\tfrac{1}{12}$, both have a unique dominant singularity at $z=\tfrac{1}{12}$, and we have $F(z)= G(z) (1+o(1))$ when $z\rightarrow \tfrac{1}{12}$ uniformly in a neighbourhood of $z=\tfrac{1}{12}$ slit along the line $[\tfrac{1}{12}, \infty)$.
From [@BC0] (see also [@CMS] for purely combinatorial proofs) we have for fixed $g\geq 0$: $$m_g(n) \sim t_g n^{\frac{5g-5}{2}} 12^n
\ \ , \ \
m^\bullet_g(n)
= (n+2-2g) m_g(n) \sim t_g n^{\frac{5g-3}{2}} 12^n$$ $$\begin{aligned}
\label{eq:singQg}
Q^\bullet_g(z) \sim \Gamma(\tfrac{5g-1}{2}) t_g (1-12z)^{\frac{1-5g}{2}}
=
2^{2-5g}\tau_{g}
(1-12z)^{\frac{1-5g}{2}}
.
$$
A *labelled map* of genus $g$ is a rooted map $M$ of genus $g$ equipped with a fonction $\ell: V(M)\rightarrow \mathbb{Z}$ such that for any edge $(u,v)$ of $M$ one has $\ell(u)-\ell(v)\in \{-1,0,1\}$. We consider these objects up to global translation of the labels (we will often fix a translation class by fixing the label of a particular vertex, often the root, to $0$). A *labelled one-face map (l.1.f.m.)* is a labelled map having only one face. We let $\mathcal{L}_n^{(g)}$ be the set of all (rooted) l.1.f.m. of genus $g$ with $n$ edges.
The Marcus-Schaeffer bijection ([@MS], see also [@CMS] for the version needed here) is an explicit bijection: $$\mathcal{Q}_n^{(g)\bullet} \longrightarrow \{\uparrow,\downarrow\} \times \mathcal{L}_n^{(g)},$$ where $\mathcal{Q}_n^{(g)\bullet}$ is the set of rooted bipartite quadrangulations of genus $g$ and $n$ faces equipped with a pointed vertex. It follows that $Q^\bullet_g(z)=2L_g(z)$ where $L_g(z)$ is the generating function of rooted l.1.f.m. of genus $g$ by the number of edges. Moreover, in genus $0$, rooted one-face maps are nothing but rooted plane trees, and a standard root-edge decomposition leads to the quadratic equation $
L_0(z)= 1+3zL_0(z)^2,
$ from which we get the explicit formula: $$\begin{aligned}
\label{eq:valKernel}
1-6zL_0(z) = \sqrt{1-12z}.\end{aligned}$$
The decomposition equation, Miermont’s bijection, and proof of Theorem \[thm:obs1\]
-----------------------------------------------------------------------------------
![Illustration of the decomposition leading to Equation \[eq:Tutte\][]{data-label="fig:Tutte"}](decomp2.pdf){width="\linewidth"}
We now come to the substance of this note, which is simply to try to write an equation for the generating function of l.1.f.m. by root-edge decomposition, and see what happens. We fix $g\geq 0$, and we consider a l.1.f.m. $M$ of genus $g+1$. If we remove the root edge of this map, two things can happen (see Figure \[fig:Tutte\]):
- we disconnect the map into two l.1.f.m. $M_1$ and $M_2$ whose genera sum up to $g+1$;
- we do not disconnect the map; in this case we are left with a map $M'$ of genus $g$ with two faces. Each face of $M'$ carries a distinguished corner, and the labels of these two corners differ by $-1$, $0$, or $1$.
Translating this operation into an equation for generating functions we obtain $$\begin{aligned}
\label{eq:Tutte}
L_{g+1}(z)
=
3z\sum_{g_1+g_2=g+1\atop g_1,g_2\geq 0}
L_{g_1}(z)
L_{g_2}(z)
+
z A_g(z)\end{aligned}$$ where:
- in the first term the factor $3z$ takes into account the choice of the increment of label along the root-edge in $\{-1,0,1\}$;
- $A_g(z)$ is the generating function by the number of edges, of unrooted labelled two-face maps, with faces numbered $F_1,F_2$, such that the face $F_i$ contains a marked corner $c_i$ for $i=1..2$, and that $|\ell(c_i)-\ell(c_2)|\leq 1$.
Objects counted by $A_g(z)$ can be related to quadrangulations thanks to Miermont’s bijection [@Miermont:tessellations]. This bijection is a generalization of the Marcus-Schaeffer bijection where the l.1.f.m is replaced by a labelled map having an arbitrary number, say $K$, of faces. We will apply it for $K=2$. In the following discussion, where we assume some familiarity with Miermont’s bijection, we will show how to arrive informally at Lemma \[lemma:convergenceCell\] below, and why this implies Theorem \[thm:obs1\]. Details of the proof of Lemma \[lemma:convergenceCell\] are postponed to the next sections.
Let us consider an object counted by $[z^n] A_g(z)$. Let us fix the translation class of the labels by saying that the minimum label in face $F_1$ is zero, and let us call $\delta$ the minimum label in face $F_2$. Let $i_1\geq 0$ and $i_2\geq \delta$ be the labels of the two marked corners $c_1$ and $c_2$, respectively, and recall that $i_1-i_2\in \{-1,0,1\}$. Applying Miermont’s bijection [@Miermont:tessellations] to this object, we construct a bipartite quadrangulation $Q$ of genus $g$ by adding a new vertex $s_1, s_2$ inside each face $F_1, F_2$, and applying a certain closure operation. At the end of the construction, we obtain a quadrangulation such that $d(s_1,s_2)+\delta$ is even. Moreover, the two corners $c_1$ and $c_2$ of the original two-face map are naturally associated to two edges $e_1$ and $e_2$ of the quadrangulation, and the construction is such that if $m_i$ is the endpoint of $e_i$ closer from $s_i$ in $Q$, for $i\in \{1,2\}$ one has: $$d(s_1,m_1)=i_1 \ \ ,\ \ d(s_2,m_2)=i_2-\delta \ \ , \ \
d(s_2,m_1)\geq i_1-\delta \ \ ,\ \ d(s_1,m_2)\geq i_2.$$ These constraints can simply be rewritten as: $$\begin{aligned}
\label{eq:crossed}
d(s_1,m_1) \leq d(s_1,m_2) -\epsilon\ \ , \ \
d(s_2,m_2) \leq d(s_2,m_1) -\epsilon,\end{aligned}$$ where $\epsilon = i_2-i_1$ is such that $|\epsilon|\leq 1$. Loosely speaking, the properties in say that, *up to an error at most 1*, $s_i$ is (weakly) closer to $m_i$ than to $m_{3-i}$ for $i=1..2$. Unfortunately these constraints do not entirely characterize these objects (see next section) but they do, in some sense, asymptotically. Thinking heuristically for a moment, we can expect that the analogue in the continuum limit of these discrete configurations is a Brownian map with four marked points $(m_1^\infty, m_2^\infty, s_1^\infty, s_2^\infty)$ such that if we subdivide the space in two nearest-neighbour cells induced by $m_1^\infty$ and $m_2^\infty$, the point $s_i^\infty$ belongs the nearest-neighbour cell induced by $m_i^\infty$ for each $i=1..2$. Up to technical details that we will carry out in the next section, this leads us quite naturally to the following conclusion:
\[lemma:convergenceCell\] The coefficient $[z^n]A_g(z)$ is such that, as $n$ goes to $\infty$: $$\begin{aligned}
\label{eq:convergenceCell}
\frac{[z^n]A_g(z)}{3/2 \cdot n^3m_{g}(n)} \longrightarrow \mathbb{E}[{\mathbf{X}}_g(1-{\mathbf{X}}_g)]\end{aligned}$$ with the notation of Theorem \[thm:obs1\].
\[rem:1\] The reader can understand heuristically the meaning of the denominator $3/2 \cdot n^3 m_{g}(n)$ as follows. The tuple $(Q,s_1,s_2,e_1,e_2)$ is a quadrangulation with two vertices and two marked edges. We can use $e_1$ as the root-edge of $Q$, and orient it by deciding that its source is at even distance from $s_1$. We can choose the “error” $\epsilon$ freely in $\{-1,0,1\}$ (since asymptotically we do not expect this error to play any role), and set $i_1:=d(s_1,m_1)$ and $\delta:=i_1+\epsilon - d(s_2,m_2)$. Since Miermont’s bijection requires that $d(s_1,s_2)+\delta$ is even, we are left with a rooted quadrangulation with one marked edge $e_2$, and two marked vertices $(s_1,s_2)$ subject to *two* parity constraints (that $s_1$ is at even distance from the root, and that $d(s_1,s_2)+\delta$ is even). Since a quadrangulation with $n$ faces has $2n$ edges and $n+2-2g$ vertices, and since it is natural to expect each parity constraint to contribute an asymptotic factor $\frac{1}{2}$, the total number of “base configurations” we obtain is $\sim 3\times (2n) n^2/4 \cdot m_g(n)$, hence the denominator in .
We can now conclude the proof of Theorem \[thm:obs1\]. First, we can rewrite the decomposition equation as: $$\begin{aligned}
\label{eq:TutteKernel}
(1-6zL_0(z)) L_{g+1}(z)
-
3z\sum_{g_1+g_2=g+1,\atop g_1,g_2>0}
L_{g_1}(z)
L_{g_2}(z)
=
z A_g(z),\end{aligned}$$ which expresses the generating function $L_{g+1}(z)$ in terms of the lower genus functions $L_i(z)$ for $i=1..g$, and of the “unknown” quantity $A_g(z)$. We recall that $Q^\bullet_g(z)=2L_g(z)$ and , from which we observe that each term in the L.H.S. of has a dominant singularity at $z=\tfrac{1}{12}$ with the same order of magnitude. More precisely, for the first term, using , we obtain $(1-6zL_0(z))L_{g+1}(z)\sim 2^{1-5(g+1)}\tau_{g+1}(1-12z)^{1-\frac{5}{2}(g+1)}$. For product terms we have $L_{g_1}(z) L_{g_2}(z)\sim
2^{2-5(g+1)}\tau_{g_1}\tau_{g_2}(1-12z)^{1-\frac{5}{2}(g_1+g_2)}$. It follows, using standard transfer theorems for algebraic functions [@Flajolet] that when $n$ goes to infinity: $$\begin{aligned}
[z^{n-1}] A_g(z) \sim 12^n n^{\frac{5g+1}{2}}
2^{1-5(g+1)}\Gamma\left(\tfrac{5g+3}{2}\right)^{-1}
\left(\tau_{g+1} - \tfrac{1}{2}\sum_{g_1+g_2=g+1\atop g_1,g_2>0} \tau_{g_1}\tau_{g_2} \right).\end{aligned}$$ But from Lemma \[lemma:convergenceCell\], we have another expansion of the “unknown” coefficient $[z^{n-1}] A_g(z)$, namely: $$[z^{n-1}] A_g(z) =\mathbb{E}{\mathbf{X}}_g(1-{\mathbf{X}}_g) \cdot 3/2\cdot n^3 m_{g}(n-1)
\sim\mathbb{E}{\mathbf{X}}_g(1-{\mathbf{X}}_g)\cdot 3\cdot 2^{1-5g}\Gamma(\tfrac{5g-1}{2})^{-1} \tau_g n^{\frac{5g+1}{2}} 12^{n-1}.$$ Theorem \[thm:obs1\] follows by comparing the last two expansions of the “unknown” quantity $[z^{n-1}]A_g(z)$ (we recall that $\Gamma(\frac{5g+3}{2})/\Gamma(\tfrac{5g-1}{2})=\frac{(5g+1)(5g-1)}{4}$).
Remaining proofs, I: general properties
---------------------------------------
Because we will need to pick both edges and vertices at random, we first need a lemma that compares both:
\[lemma:vertexvsedge\] Given a quadrangulation $Q$ of genus $g$ with $n$ faces, there exists a probability measure $\mu_n^E$ on edges of $Q$ and a mapping $\phi:E(Q)\rightarrow V(Q)$ that associates to each edge of $E$ a vertex at distance at most one of one of its endpoints, such that the distance in total variation between $\mu_n^{E}$ and the uniform measure on edges, and between $\phi \circ \mu_n^{E}$ and the uniform measure on vertices, are both $O(\tfrac{1}{n})$.
Let $L$ be the l.1.f.m associated to $Q$ via the Marcus-Schaeffer bijection and let $v_0\in V(L)$. The map $L$ has $n+1-2g$ vertices, so it is possible to choose a set $E'_L$ of $n-2g$ edges of $L$ and an orientation of edges of $E'_L$ such that each vertex of $V\setminus\{v_0\}$ has exactly one outgoing edge from $E'_L$ (to see this, take a spanning tree of $L$ and orient edges towards $v_0$). If $e \in E'_L$, we let $v(e)$ be its source, which is an element of $V\setminus\{v_0\}$. The edge $e$ of $E'_L$ is associated, via the Marcus-Schaeffer bijection, to a face $f(e)$ of the quadrangulation $Q$ that is incident to the vertex $v(e)$. We let $E'_Q$ be the subset of edges of $Q$ that border a face of the form $f(e)$ for some $e\in E_L$ and that are oriented from white to black when going clockwise around $f(e)$ (in some fixed bicoloration of $Q$). If $\tilde{e}\in E'_Q$, corresponding to the face $f(e)$, we define $\phi(\tilde{e}):=v(e)$. Since $\tilde{e}$ and $\psi(\tilde{e})$ both border the face $f(e)$, they are at distance at most one from each other. Moreover, if we choose $\tilde{e}$ uniformly at random from $E'_Q$, then by construction $\psi(\tilde{e})$ is uniform in $V\setminus \{v_0\}$. Since $E'_Q$ contains $2(n-2g)$ edges of $Q$ (among $2n$) and $V\setminus\{v_0\}$ contains $n-2g$ vertices of $Q$ (among $n+2-2g$), we are done.
In the following discussion we will implicitly restrict ourselves to a subsequence along which we have the GHP distributional convergence: $$({{\mathbf{q}}_{n}^{(g)}},\tfrac{1}{n^{1/4}}\mathbf{d}_n,\mu_n)
\longrightarrow
({{\mathbf{q}}_\infty^{(g)}}, d_\infty, \mu_\infty).$$ We will need the following direct consequence of [@Chapuy:trisections Thm. 4]. We state separately a discrete and a continuous statement, although they are intimately related:
\[lemma:zeroProba\]
\(i) Let $({{\mathbf{q}}_\infty^{(g)}},d_\infty, \mu_\infty)$ be a Brownian map of genus $g$ and let $({\mathbf{v}}_1^\infty, {\mathbf{v}}_2^\infty, {\mathbf{v}}_3^\infty)$ chosen at random according to $\mu_\infty^{\otimes 3}$. Then almost surely we have $d_\infty ({\mathbf{v}}_1^\infty,{\mathbf{v}}_3^\infty) \neq
d_\infty ({\mathbf{v}}_2^\infty,{\mathbf{v}}_3^\infty)$.
\(ii) Fix $K\geq 0$, and pick three uniform random vertices ${\mathbf{v}}_1^n$, ${\mathbf{v}}_2^n$, ${\mathbf{v}}_3^n$ in ${{\mathbf{q}}_{n}^{(g)}}$. Then the probability that $|d({\mathbf{v}}_1^n,{\mathbf{v}}_3^n)-d({\mathbf{v}}_2^n,{\mathbf{v}}_3^n)|\leq K$ goes to zero when $n$ goes to infinity.
It is proved in [@Chapuy:trisections] that if ${\mathbf{v}}_1\in_u V({{\mathbf{q}}_{n}^{(g)}})$, the random measure $$\eta_n^{(g)}:=\displaystyle \tfrac{1}{|V({{\mathbf{q}}_{n}^{(g)}})|}\sum_{v \in V({{\mathbf{q}}_{n}^{(g)}})} \delta_{d_n(v,{\mathbf{v}}_1)/n^{1/4}}$$ converges in distribution to a random measure $\eta_g$ that, almost surely, has no atoms (this last fact following from the fact that it is true for the ISE measure, see e.g. [@MBMJanson], and from the relation between $\eta_g$ and ISE given in [@Chapuy:trisections]). Now for $\alpha>0$, let ${p_{n,\alpha}}:=\mathbf{P}\big\{|d({\mathbf{v}}_1^n,{\mathbf{v}}_3^n)-d({\mathbf{v}}_2^n,{\mathbf{v}}_3^n)|\leq \alpha n^{1/4}\big\} = \mathbf{E}\langle ({\eta_n^{(g)}})^{\otimes 2} | h_\alpha\rangle$ where $h_\alpha(x,y):=\mathbf{1}_{|y-x|\leq \alpha}$ (here for a measure $\nu$ and a function $h$ we note $\langle\nu|h\rangle:=\int h(x)d\nu(x)$). Convergence in law implies that $\lim_n \mathbf{E}\langle(\eta_n^{(g)})^{\otimes 2} | f \rangle = \mathbf{E} \langle \eta_g^{\otimes 2}| f\rangle$ for any bounded and continuous function $f$, so choosing $f=f_\alpha$ continuous such that $h_\alpha \leq f_\alpha \leq h_{2\alpha}$ we get $\limsup_n p^n_\alpha \leq \mathbf{E}\langle ({\eta_g})^{\otimes 2} | h_{2\alpha}\rangle$, and since $\eta_g$ has no atoms we get: $$\begin{aligned}
\label{eq:limsup}
\lim_{\alpha\rightarrow 0} \limsup_n {p_{n,\alpha}}=0,\end{aligned}$$ from which (ii) follows (in fact, in a much stronger form that allows $K$ to be as large as $o(n^{1/4})$).
Now, by GHP convergence and [@Miermont:tessellations Prop. 6] one can define on the same probability space $({{\mathbf{q}}_{n}^{(g)}},\tilde{\mathbf{v}}_1^n,\tilde{\mathbf{v}}_2^n,\tilde{\mathbf{v}}_3^n)$ and $({{\mathbf{q}}_\infty^{(g)}},\tilde{\mathbf{v}}_1^\infty, \tilde{\mathbf{v}}_2^\infty, \tilde{\mathbf{v}}_3^\infty)$ such that $\tilde{\mathbf{v}}_i^{M}$ is $o(1)$-close to ${\mathbf{v}}_i^{M}$ in total variation distance for each $i\in\{1,2,3\}$ and $M\in\{n,\infty\}$, and such that almost surely $|d_n(\tilde{\mathbf{v}}_i^n,\tilde{\mathbf{v}}_j^n)/n^{1/4}-d_\infty(\tilde{\mathbf{v}}_i^\infty, \tilde{\mathbf{v}}_j^\infty)|=o(1)$ for each $i$, $j$. Letting $q_\alpha:=\mathbf{P}\{|d_\infty ({\mathbf{v}}_1^\infty,{\mathbf{v}}_3^\infty) -
d_\infty ({\mathbf{v}}_2^\infty,{\mathbf{v}}_3^\infty)|\leq \alpha\}$, it easily follows that $$q_{\alpha/2} \leq \limsup_n {p_{n,\alpha}}.$$ From this implies that $\limsup_{\alpha\rightarrow 0} q_\alpha =0$, which implies (i).
Remaining proofs, II: Lemma \[lemma:convergenceCell\] {#sec:technical}
-----------------------------------------------------
Before proving Lemma \[lemma:convergenceCell\], we need to describe more precisely the objects Miermont’s bijection leaves us with. We use the same notation as in the previous section for objects counted by $A_g(z)$ (marked faces $F_1, F_2$, minimum label in each face $0,\delta$, marked corners $c_1,c_2$). We will introduce the refinement $$A_g(z) = \sum_{\epsilon\in\{-1,0,1\}} A_g^\epsilon(z)$$ where $A_g^\epsilon(z)$ counts the same objects as $A_g(z)$ but with the restriction that $\ell(c_2)-\ell(c_1)=\epsilon$. Unpacking the definitions in [@Miermont:tessellations] we have:
\[lemma:MiermontConstraints\] For each $\epsilon\in\{-1,0,1\}$, the labelled two-face maps counted by $[z^{n}]A^{\epsilon}_g(z)$ are in bijection with tuples $(Q,s_1,s_2,e_1,e_2)$ such that:
- $Q$ is a bipartite quadrangulation of genus $g$ with $n$ faces (unrooted);
- $s_1,s_2$ are two vertices of $Q$ and $e_1,e_2$ are two marked edges of $Q$;
- for $j=1..2$ let $m_j$ be the endpoint of $e_j$ closer from $s_j$, and let $\delta := d(s_1,m_1)+\epsilon-d(s_2,m_2).$ Then the quantity $d(s_1,s_2)+\delta$ is even.
- Label each vertex $v$ of the quadrangulation $Q$ by $\ell(v):=\min(d(v,s_1),d(v,s_2)+\delta)$, and orient each edge towards its vertex of minimum label [^3]. From an edge $e$, define the *leftmost geodesic oriented path* as the oriented path starting from $e$ and continuing after each edge with the first oriented edge encountered in counterclockwise order around its endpoint, stopped when it reaches $s_1$ or $s_2$. Then for $i=1..2$, the leftmost geodesic oriented path starting at $e_i$ ends at $s_i$.
Note that, since $Q$ is bipartite, the property $(M2)$ is equivalent to the following:
- $d(m_1,m_2) \equiv \epsilon \mod 2.$
As for the complicated property $(M3)$, up to subdominating cases, it can be rephrased in simpler terms closely related to nearest neighbours tessellations. Indeed, we have:
\[lemma:GeoToVor\] Let $b_g^\epsilon(n)$ be the number of tuples $(Q,s_1,s_2,e_1,e_2)$ satisfying (M0), (M1), (M2) of the last lemma, and such that moreover we have:
- $d(s_1,e_1) < d(s_1,e_2) -4$ and $
d(s_2,e_2) < d(s_2,e_1) -4 .$
Then for each $\epsilon\in \{-1,0,1\}$ we have $[z^n]A_g^{\epsilon}(z) = b^\epsilon_g(n) + o(n^3m_g(n))$.
Note that $(M'3)$ implies that $d(s_1,m_1) < d(s_1,m_2) -2$ and $
d(s_2,m_2) < d(s_2,m_1) -2.$
Let $(Q,s_1,s_2,e_1,e_2)$ satisfying the hypotheses of Lemma \[lemma:GeoToVor\]. Then we claim that it also satisfies the hypotheses of Lemma \[lemma:MiermontConstraints\]. Indeed define $\ell(v)=\min(d(s_1,v),d(s_2,v)+\delta)$ as in Lemma \[lemma:MiermontConstraints\]. We observe that (M’3) implies that $$d(s_1,m_1) < d(s_1,m_1)+(d(s_2,m_1)-d(s_2,m_2)+\epsilon) = d(s_2,m_1)+\delta,$$ which shows that the minimum in the definition of $\ell(v)$ for $v=m_1$ is reached only by its *first* argument. Similarly, for $v=m_2$ we have by (M’3) that: $$d(s_2,m_2) < d(s_2,m_2)+(d(s_1,m_2)-d(s_1,m_1)-\epsilon)
=d(s_1,m_2)-\delta,$$ which shows that the minimum in the definition of $\ell(v)$ for $v=m_2$ is reached only by its *second* argument. Thus the discussion in [@Miermont:tessellations Sec 2.2] (more precisely the first inclusion sign $\subsetneq$ in the last displayed equation of that section) precisely says that (M3) is satisfied.
Conversely assume the hypotheses of Lemma \[lemma:MiermontConstraints\]. By general properties of Miermont’s labelling (namely, the fact that leftmost oriented geodesic paths as defined in the lemma are, in particular, geodesic paths), property $(M3)$ ensures that the minima defining $\ell(m_1)=\min(d(m_1,s_1),d(m_1,s_2)+\delta)$ and $\ell(m_2)=\min(d(m_2,s_1),d(m_2,s_2)+\delta)=\min(d(m_2,s_1),d(m_1,s_1)+\epsilon)$ are reached respectively by their first and second argument (and possibly reached twice). This implies: $$d(s_1,m_1)+\epsilon \leq d(s_1,m_2), \ d(s_2,m_2)\leq d(m_1,s_2)+\epsilon.$$ Thus, if hypothesis (M’3) is *not* satisfied, it must hold that either $|d(s_1,m_1)-d(s_1,m_2)|\leq 2$ or $|d(s_2,m_2)-d(s_2,m_1)|\leq 2$. It thus suffices to show that there are at most $o(n^3m_g(n))$ tuples $(Q,s_1,s_2,e_1,e_2)$ such that one of these two properties holds. For this it suffices to show that if $({\mathbf{q}},{\mathbf{e}}_1) \in_u {\mathcal{Q}_n^{(g)}}$ is a random rooted quadrangulation and $({\mathbf{s}}_1,{\mathbf{s}}_2,{\mathbf{e}}_2)$ are two vertices and an edge chosen independently uniformly at random in ${\mathbf{q}}$, the probability that $|d({\mathbf{s}}_1,{\mathbf{e}}_2)-d({\mathbf{s}}_1,{\mathbf{e}}_2)|\leq 2$ or $|d({\mathbf{s}}_2,{\mathbf{e}}_2)-d({\mathbf{s}}_2,{\mathbf{e}}_1)|\leq 2$ goes to zero as $n$ goes to infinity. This directly follows from Lemmas \[lemma:vertexvsedge\] and \[lemma:zeroProba\](ii).
In view of getting rid of the constraint $(M'2)$, we state the following lemma:
\[lemma:equalContrib\] For any $\epsilon\in \{-1,0,1\}$ we have as $n$ goes to infinity: $$[z^n] A^\epsilon_g (z) \sim \frac{1}{3} [z^n]A_g(z).$$
This can be proved by asymptotic analysis of generating functions using a simple adaptation of the method developed in [@CMS] for the enumeration of labelled one-face maps by *scheme decomposition*: one can enumerate objects counted by $A^\epsilon_g(z)$ with this approach and realize that changing the parameter $\epsilon$ only affects the principal singularity by a factor $1-O((1-12z)^{1/4})$, from which the result follows. We leave details to the reader.
We are now ready to conclude the proof.
First, we remark that from the last lemma: $$\begin{aligned}
\label{eq:linearSmart}
[z^n] \big(A_g^0(z) + A_g^1(z)\big) \sim \frac{2}{3} [z^n] A_g(z),\end{aligned}$$ while from Lemma \[lemma:GeoToVor\] and the remark preceeding it, $[z^n] A_g^0(z) + A_g^1(z)$ is equivalent to the number of tuples $(Q,s_1,s_2,e_1,e_2)$ satisfying properties $(M0),(M1)$, and $(M'3)$ (note that property $(M'2)$ disappears since we sum over both parities $\epsilon=0,1$). We will thus focus on such objects in the rest of the proof.
We note that a bipartite quadrangulation with a marked edge $e_1$ and a marked vertex $v_1$ can be canonically rooted by orienting $e_1$ towards its unique endpoint at even distance from $v_1$. This gives a one-to-two correspondence between elements of $\mathcal{Q}^{g}_n$ with a marked vertex and (unrooted) bipartite quadrangulations of genus $g$ with a marked vertex and a marked unoriented edge. We thus have: $$\begin{aligned}
\label{eq:proba}
\frac{[z^n]\big(A_g^0(z) + A_g^1(z)\big)}
{(2n) \cdot n^2 \cdot m_g(n)}
\sim \frac{1}{2}\cdot \mathbf{P}\Big\{d({\mathbf{s}}_1,{\mathbf{e}}_1) < d({\mathbf{s}}_1,{\mathbf{e}}_2) -4, \
d({\mathbf{s}}_2,{\mathbf{e}}_2) < d({\mathbf{s}}_2,{\mathbf{e}}_1) -4 \Big\},\end{aligned}$$ where the probability is taken over ${{\mathbf{q}}_{n}^{(g)}}\in_u \mathcal{Q}^g_n$ with two uniform marked vertices ${\mathbf{s}}_1,{\mathbf{s}}_2$, a uniform marked edge ${\mathbf{e}}_2$ and ${\mathbf{e}}_1$ is the root edge (in the denominator, the factor $n^2$ corresponds to the choice of the two vertices, while the factor $(2n)$ corresponds to the choice of the edge ${\mathbf{e}}_2$).
We recall that we implicitly restrict ourselves to a subsequence along which the GHP distributional convergence $$({{\mathbf{q}}_{n}^{(g)}},\tfrac{1}{n^{1/4}}\mathbf{d}_n,\mu_n)
\longrightarrow
({{\mathbf{q}}_\infty^{(g)}}, d_\infty, \mu_\infty)$$ holds. We will make use of this convergence using a coupling between ${{\mathbf{q}}_{n}^{(g)}}$ and ${{\mathbf{q}}_\infty^{(g)}}$. More precisely, according to [@Miermont:tessellations Proposition 6], we can build ${{\mathbf{q}}_{n}^{(g)}}$ and ${{\mathbf{q}}_\infty^{(g)}}$ on the same probability space, and define a measure $\nu$ on ${{\mathbf{q}}_{n}^{(g)}}\times {{\mathbf{q}}_\infty^{(g)}}$ such that for each $k\geq 1$, if $(\tilde{\mathbf{w}}^i_n, \tilde{\mathbf{w}}^i_\infty)_{1\leq i \leq k} \sim \nu^{\otimes k}$ we have almost surely $|d_n(\tilde{\mathbf{w}}^i_n,\tilde{\mathbf{w}}^j_n)n^{-1/4}-d_\infty(\tilde{\mathbf{w}}_\infty^i, \tilde{\mathbf{w}}_\infty^j)| \leq \epsilon_n$ for any $i,j$, and moreover the law of $\tilde{\mathbf{w}}^i_n$ (resp. $\tilde{\mathbf{w}}^i_\infty)$ differs from $\mu_n$ (resp. $\mu_\infty$) by at most $\epsilon_n$ in total variation distance, where $\epsilon_n$ is a nonnegative real sequence going to zero when $n$ goes to infinity. We will apply this with $k=4$. Using Lemma \[lemma:vertexvsedge\], we can moreover assume the vertices $\tilde{\mathbf{w}}_3^n$ and $\tilde{\mathbf{w}}_4^n$ are at distance at most $2$ of two random edges $\tilde{\mathbf{e}}_1^n$ and $\tilde{\mathbf{e}}_2^n$ respectively, and that the law of $\tilde{\mathbf{e}}_1^n$ and $\tilde{\mathbf{e}}_2^n$ is $\epsilon_n$-close in total variation to that of two uniform random edges (if necessary, we modify the sequence $\epsilon_n$ for this to be true, still asking that $\epsilon_n\rightarrow 0$).
If $v_1,v_2,v_3,v_4$ are points (or subsets) in some metric space of underlying distance $d$, and $K\in\mathbb{R}$ let us define the events: $$V_K(v_1,v_2,v_3,v_4) := \{ d(v_1,v_3)<d(v_1,v_4)-K, d(v_2,v_4)<d(v_2,v_3)-K\}$$ $$W_K(v_1,v_2,v_3):=\{|d(v_1,v_3)-d(v_2,v_3)|\leq K\}.$$ By the assumptions made on the coupling between ${{\mathbf{q}}_{n}^{(g)}}$ and ${{\mathbf{q}}_\infty^{(g)}}$ and from the triangle inequality we have, denoting $\Delta$ the symmetric difference: $$\begin{aligned}
V_{4}(\tilde{\mathbf{w}}_1^n, \tilde{\mathbf{w}}_2^n, \tilde{\mathbf{e}}_1^n,\tilde{\mathbf{e}}_2^n)
\Delta
V_0^\infty(\tilde{\mathbf{w}}_1^\infty, \tilde{\mathbf{w}}_2^\infty, \tilde{\mathbf{w}}_3^\infty,\tilde{\mathbf{w}}_4^\infty)
\subset W_{\delta_n}^\infty(\tilde{\mathbf{w}}_1^\infty, \tilde{\mathbf{w}}_2^\infty, \tilde{\mathbf{w}}_3^\infty)
\cup W_{\delta_n}^\infty(\tilde{\mathbf{w}}_1^\infty, \tilde{\mathbf{w}}_2^\infty, \tilde{\mathbf{w}}_4^\infty)\end{aligned}$$ with $\delta_n = O(\epsilon_n+n^{-1/4})$. We thus have: $$\begin{aligned}
\limsup_n \big|\mathbf{P}
V_{4}(\tilde{\mathbf{w}}_1^n, \tilde{\mathbf{w}}_2^n, \tilde{\mathbf{e}}_1^n,\tilde{\mathbf{e}}_2^n)
-\mathbf{P}
V_0(\tilde{\mathbf{w}}_1^\infty, \tilde{\mathbf{w}}_2^\infty, \tilde{\mathbf{w}}_3^\infty,\tilde{\mathbf{w}}_4^\infty)
\big|
&\leq
\limsup_n
2 \mathbf{P}W_{\delta_n}^\infty(\tilde{\mathbf{w}}_1^\infty, \tilde{\mathbf{w}}_2^\infty, \tilde{\mathbf{w}}_3^\infty) \\
&\leq \limsup_n \left(\epsilon_n+
2 \mathbf{P}W_{\delta_n}^\infty({\mathbf{w}}_1^\infty, {\mathbf{w}}_2^\infty, {\mathbf{w}}_3^\infty)\right)\end{aligned}$$ where $({\mathbf{w}}_i^\infty)_{1\leq i\leq 3} \sim \mu_\infty^{\otimes 3}$ are three random vertices in ${{\mathbf{q}}_\infty^{(g)}}$ chosen according to $\mu_\infty$, and where we just used the definition of total variation distance. From Lemma \[lemma:zeroProba\](i), the last $\limsup$ is equal to zero, which implies: $$\begin{aligned}
\limsup_n \big|\mathbf{P}
V_{4}({\mathbf{w}}_1^n, {\mathbf{w}}_2^n, {\mathbf{e}}_1^n,{\mathbf{e}}_2^n)
-\mathbf{P}
V_0({\mathbf{w}}_1^\infty, {\mathbf{w}}_2^\infty, {\mathbf{w}}_3^\infty,{\mathbf{w}}_4^\infty)
\big|
&=0\end{aligned}$$ where ${\mathbf{w}}_1^n, {\mathbf{w}}_2^n, {\mathbf{e}}_1^n,{\mathbf{e}}_2^n$ are two vertices and two edges of ${{\mathbf{q}}_{n}^{(g)}}$ chosen independently uniformly at random, and where $({\mathbf{w}}_i^\infty)_{1\leq i\leq 4} \sim \mu_\infty^{\otimes 4}$ are uniform in ${{\mathbf{q}}_\infty^{(g)}}$.
Now by rerooting invariance of random quadrangulations $\mathbf{P}
V_{4}({\mathbf{w}}_1^n, {\mathbf{w}}_2^n, {\mathbf{e}}_1^n,{\mathbf{e}}_2^n)$ is equal to the probability appearing in the R.H.S. of , while it follows directly from Lemma \[lemma:zeroProba\](i) and the Fubini theorem that the quantity $
\mathbf{P}
V_0({\mathbf{w}}_1^\infty, {\mathbf{w}}_2^\infty, {\mathbf{w}}_3^\infty,{\mathbf{w}}_4^\infty)$ is equal to $\mathbf{E}{\mathbf{X}}_g(1-{\mathbf{X}}_g)$. This concludes the proof.
To be fully complete we also state the:
The only thing to prove is that $\mathbf{E}\mathbf{X}_g = \tfrac{1}{2}$, which is a direct consequence of Lemma \[lemma:zeroProba\].
Three marked points (proof of Theorem \[thm:3points\])
======================================================
In this section we sketch the proof of Theorem \[thm:3points\]. We will insist on the combinatorial decompositions and the computation, since the details of the convergence results are very similar to what we did in the previous section.
We first need some definitions from [@CMS; @Chapuy:trisections]. If $L$ is a one-face map, its *skeleton* is the map obtained by removing all vertices of degree $1$ in $L$, and continuing to do so recursively until only vertices of degree at least $2$ remain. Vertices of a one-face map that are vertices of degree at least $3$ of its skeleton are called *nodes*. A node $v$ that has degree $k$ in the skeleton is called a $k$-node (note that its degree as a vertex in the one-face map can be larger than $k$). A one-face map is *dominant* if all vertices of its skeleton have degree at most $3$, *i.e.* if all its nodes are $3$-nodes. It is proved in [@CMS] that for fixed $g$, as $n$ goes to infinity, a proportion at least $1-O(n^{-1/4})$ of l.1.f.m. of genus $g$ with $n$ edges are dominant. By Euler’s formula, a dominant one-face map has $4g-2$ nodes.
Following [@Chapuy:trisections], we introduce the operation of *opening*. If $L$ is a one-face map and $v$ is a $3$-node of $L$, the *opening* of $v$ is the operation that consists in replacing $v$ by three new vertices, each linked to one edge of the skeleton, and distributing the three (possibly empty) subtrees attached to $v$ among these new vertices as on the following figure:
{width="0.4\linewidth"}
Following [@Chapuy:trisections][^4], we distinguish two types of $3$-nodes in a one-face map: *intertwined nodes*, that are such that their opening results in a one-face map of genus $g-1$ with three marked vertices; and *non-intertwined nodes*, that are such that their opening results in a map of genus $g-2$ with three faces, and one marked vertex inside each face (here the map can be disconnected, and its genus and number of faces are defined additively on connected components). The *trisection lemma* [@Chapuy:trisections Lemma 5], which is the key result underlying this section, asserts that any dominant map of genus $g\geq 1$ has exactly $2g$ intertwined nodes, hence $2g-2$ non-intertwined ones.
It follows that the number $K_{g+2}(n)$ of l.1.f.m. of genus $g+2$ with $n$ edges whose root edge is a skeleton-edge leaving a *non-intertwined* $3$-node satisfies: $$\begin{aligned}
\label{eq:doubleRooting}
K_{g+2}(n)\sim\frac{6(g+1)}{2n} [z^n] L_{g+2}(z).\end{aligned}$$ Indeed, the first-order contribution is given by dominant l.1.f.m., and in a dominant l.1.f.m. of genus $g+2$ we can choose $3(2(g+2)-2)=6(g+1)$ edges outgoing from a non-intertwined node as a new root edge, but we obtain each map $2n$ times in this way (since maps counted by $L_{g+2}(z)$ are already rooted at one of their $2n$ oriented edges).
We are now going to obtain another expression for the number $K_{g+2}(n)$ by performing a combinatorial decomposition. Comparing the two expressions will, in the end, lead us to Theorem \[thm:3points\].
Let $L$ be a dominant l.1.f.m of genus $g+2$ whose root edge is a skeleton-edge leaving a non-intertwined $3$-node $v$. We distinguish three cases, according to what happens when we perform the opening of the node $v$ (see Figure \[fig:Tutte3\]):
- we disconnect the map into three components;
- we disconnect the map into two components;
- we do not disconnect the map.
We let $C_{g+2}^{(i)}(z)$, $C_{g+2}^{(ii)}(z)$, $C_{g+2}^{(iii)}(z)$ be the generating function for these three cases, respectively.
![The three cases for a one-face map of genus $g+2$ rooted at skeleton edge leaving a non-intertwined $3$-node $v$.[]{data-label="fig:Tutte3"}](decomp3.pdf){width="\linewidth"}
Configurations corresponding to (i) can be reconstructed by starting with three rooted l.1.f.m. of positive genera summing up to $g+2$, and joining the three root vertices by new edges to a new vertex $v$. The generating function for the contribution of this case is thus: $$\begin{aligned}
\label{eq:case1}
C_{g+2}^{(i)}(z) = (3zL_0(z))^3
\sum_{g_1+g_2+g_3=g+2\atop g_1,g_2,g_3>0} L_{g_1}(z)L_{g_2}(z)L_{g_3}(z)\end{aligned}$$ where for each new edge a factor $3zL_0(z)$ takes into account the increment of this edge, and the attachment of a rooted tree (possibly empty) in the newly created corner (see Figure \[fig:Tutte3\]–Left).
Configurations corresponding to (ii) can be reconstructed by joining with a new edge the root vertex of a l.1.f.m. to the root vertex of another one which is rooted at a *non-isthmic* edge of its skeleton (see Figure \[fig:Tutte3\]–Center). Now, arguing as in the previous section, for each $h\geq 1$, the generating function $S_{h}(z)$ of l.1.f.m. of genus $h$ rooted at a non-isthmic edge of their skeleton satisfies: $$L_h(z) = S_h(z) + 3z\sum_{g_1+g_2=h\atop g_1,g_2\geq 0} L_{g_1}(z)L_{g_2}(z),$$ from which we get: $$S_h(z)=(1-6zL_0(z))L_h - 3z \sum_{g_1+g_2=h\atop g_1,g_2>0} L_{g_1}(z)L_{g_2}(z).$$ It follows that the contribution for case (ii) is given by: $$\begin{aligned}
\label{eq:case2}
C_{g+2}^{(ii)} =
3\cdot (3z L_0(z)) \sum_{h+h'=g+2\atop h,h'>0} L_{h'}(z) \left(
(1-6zL_0(z))L_h - 3z \sum_{g_1+g_2+g_3=h\atop g_1,g_2>0} L_{g_1}(z)L_{g_2}(z)\right),\end{aligned}$$ where as before the factor $3zL_0(z)$ takes into account the increment of the newly created edge, and the (possibly empty) rooted tree to attach in the newly created corner, and where the global factor of $3$ takes into account the choice of the root edge among the three skeleton-edges incident to the newly created vertex.
Summing up and we obtain the leading-order contribution for the sum of the first two cases: $$C_{g+2}^{(i)}(z) + C_{g+2}^{(ii)} (z) \sim C\cdot (1-12z)^{\frac{3}{2}-\frac{5}{2}(g+2)}$$ with $$\begin{aligned}
C&=& 2^{-5(g+2)} \sum_{\scriptscriptstyle g_1+g_2+g_3=g+2} \tau_{g_1}\tau_{g_2}\tau_{g_3}
+
\frac{3\cdot 4}{2} 2^{-5(g+2)}
\sum_{\scriptscriptstyle g_1+g_2=g+2} \tau_{g_1}\tau_{g_2}
-
3\cdot 2^{-5(g+2)}
\sum_{\scriptscriptstyle g_1+g_2+g_3=g+2} \tau_{g_1}\tau_{g_2} \tau_{g_3}\\
&=&
2^{-5(g+2)} \left(
6\cdot
\sum_{\scriptscriptstyle g_1+g_2=g+2} \tau_{g_1}\tau_{g_2}
-2\cdot
\sum_{\scriptscriptstyle g_1+g_2+g_3=g+2} \tau_{g_1}\tau_{g_2} \tau_{g_3}
\right)\end{aligned}$$ where we have used that $L_0(\frac{1}{12})=2$, and all sums are taken over positive indices ([*i.e.*]{} $g_1,g_2,g_3 >0$).
Now, the leading-order contribution for the sum of three cases (i), (ii), (iii), which from corresponds to the dominant singularity of the generating function $3(g+1) \int L_{g+2}(z) \frac{dz}{z}$ is given by (since all series are algebraic we can integrate expansions with no fear): $$3(g+1) \frac{1}{\frac{3}{2}-\frac{5}{2}(g+2)} 2^{1-5(g+2)} \tau_{g+2} (1-12z)^{\frac{3}{2}-\frac{5}{2}(g+2)}.$$ Taking the difference with the previous expression, we obtain that the leading order contribution corresponding to case (iii) is given by $$\begin{aligned}
\label{eq:case3}
C_{g+2}^{(iii)}(z) \sim C' 2^{-5(g+2)}(1-12)^{\frac{3}{2}-\frac{5}{2}(g+2)}\end{aligned}$$ where: $$C'= \frac{12(g+1)}{5g+7} \tau_{g+2}-\left(
6\cdot
\sum_{\scriptscriptstyle g_1+g_2=g+2} \tau_{g_1}\tau_{g_2}
-2\cdot
\sum_{\scriptscriptstyle g_1+g_2+g_3=g+2} \tau_{g_1}\tau_{g_2} \tau_{g_3}
\right)$$ This expression can be considerably simplified. To this end, define the formal power series $U(s)=\sum_{g\geq 1} \tau_g s^g$. Then the $t_g$-recurrence is equivalent to the equation: $$\begin{aligned}
\label{eq:tgdiff}
U(s)=\tfrac{s}{3} + \tfrac{1}{2}U(s)^2 + \frac{s}{3}
(5(\tfrac{sd}{ds})+1)
(5(\tfrac{sd}{ds})-1)
U(s).\end{aligned}$$ In view of the bi- and tri-linear sums appearing in the definition of $C'$, we would like to find an equation involving the series $6U(s)^2-2U(s)^3$. Luckily, we have:
\[lemma:eliminate\] The following differential equation holds: $$\begin{aligned}
\label{eq:tgdiff2}
\tfrac{4}{15}
(5\tfrac{sd}{ds}-3 )_{((5))} \big(s^2U\big)
=
-(5(\tfrac{sd}{ds})-3) \big(6U^2-2U^3\big)
+ 12 (\tfrac{sd}{ds}-1)\big(U-\tfrac{s}{3}\big)
- 28 s^2 \end{aligned}$$ where we use the notation $(5\tfrac{sd}{ds}-3 )_{((5))} =
(5\tfrac{sd}{ds}-3 )(5\tfrac{sd}{ds}-5 )(5\tfrac{sd}{ds}-7 )(5\tfrac{sd}{ds}-9 )(5\tfrac{sd}{ds}-11)$.
Extracting the coefficient of $s^{g+2}$ in , we directly obtain that the constant $C'$ can be rewritten in the much simpler form: $$C' = \frac{4}{15} (5g+5)(5g+3)(5g+1)(5g-1) \tau_{g}.$$
To sum up the present discussion, we have determined the first order asymptotic of the generating function $C_{g+2}^{(iii)}(z)$ of rooted maps counted by case (iii). Applying standard transfer theorems, the corresponding coefficient $c_{g+2}^{(iii)}(n):=[z^n]C_{g+2}^{(iii)}(z)$ satisfies: $$\begin{aligned}
\label{eq:cg3}
c_{g+2}^{(iii)}(n)\sim \frac{1}{15} 2^{6-5(g+2)} / \Gamma(\tfrac{5}{2}g-\tfrac{1}{2}) \tau_g \cdot n^{\frac{5}{2}(g+1)} 12^n,\end{aligned}$$ where we have used that $(5g+5)(5g+3)(5g+1)(5g-1) \Gamma(\tfrac{5}{2}g-\tfrac{1}{2})=2^4\cdot \Gamma(\tfrac{5(g+2)}{2}-\tfrac{3}{2})$
It is now time to apply Miermont’s bijection. If we disconnect the three endpoints belonging to the skeleton and the root vertex in a map from case (iii), we obtain a labelled map of genus $g$ with three faces, with one marked vertex inside each face, subject to the constraint that those three vertices have the same label (see Figure \[fig:Tutte3\]-Right). Miermont’s bijection transforms this object into a bipartite quadrangulation of genus $g$ with *six* marked vertices $(s_1,s_2,s_3,v_1,v_2,v_3)$, such that for $i=1..3$ the source $v_i$ is closer from the vertex $s_i$ than from the two other vertices $s_j$ (to see this, write precisely the inequalities analogue to as in the previous section). Arguing as in the previous section (see the sketch of proof below), up to subdominating cases, this property asymptotically characterizes those configurations, and we get:
\[combiToMoment3points\] The number $c_{g+2}^{(iii)}(n)$ of configurations in case (iii) satisfies: $$\frac{c_{g+2}^{(iii)}(n)}{n^5/4 \cdot 2^{-2} m_g(n)} \sim \mathbf{E}[{\mathbf{Y}}_g^{(1:3)}{\mathbf{Y}}_g^{(2:3)}{\mathbf{Y}}_g^{(3:3)}].$$
The reader can understand heuristically the denominator in the previous expression as follows. The factor $n^5/4$ comes from the fact that we have $\sim n^6$ ways to mark 6 vertices (among $n+2-2g$) but that the quadrangulation is unrooted so we divide by $4n$. The factor $2^{-2}$ corresponds to the fact that we have two parity constraints relating the distances of the six points together (these constraints enable us to choose the delays in such a way that the target vertices get the same label while respecting the parity constraints on delays required by Miermont’s bijection). We only sketch the proof of the lemma, since it is similar to what we did in the previous section.
First, $A(n):=2n \cdot c_{g+2}^{(iii)}(n)$ counts *rooted* labelled three-face maps of genus $g$ with $n$ edges, with faces numbered $F_1,F_2,F_3$, with three marked vertices $v_1,v_2,v_3$ such that $v_i$ is incident to the face $F_i$ only, and such that $\ell(v_1)=\ell(v_2)=\ell(v_3)$. For $(\epsilon_2,\epsilon_3)\in\{0,1\}^2$, we introduce a variant $A^{\epsilon_2,\epsilon_3}(n)$ of this number, counting the same objects but where the last property is replaced by $\ell(v_1)=\ell(v_2)-\epsilon_2=\ell(v_3)-\epsilon_3$. For such an object we let $\delta_i$ be the minimum label in face $F_i$ for $i=1,2,3$, and we fix a translation class of labels by assuming that $\delta_1=0$. We also note $\epsilon_1:=0$.
Let $L$ be a three-face map counted by $A^{\epsilon_2,\epsilon_3}(n)$ and $Q$ be its associated quadrangulation by Miermont’s bijection. Then $Q$ is a bipartite quadrangulation of genus $g$ with $n$ faces, carrying three source vertices $s_1,s_2,s_3$ and the three marked vertices $v_1,v_2,v_3$, and is such that $$\begin{aligned}
\label{eq:constraints3}
\ell(v_i) = d(v_i,s_i)+\delta_i \leq d(v_i,s_j)+\delta_j, i\neq j.\end{aligned}$$ Indeed this equation says that the minimum defining the label $\ell(v_i):=\min_{1\leq j\leq 3} d(v_i,s_j)+\delta_j$ in the Miermont labelling of the delayed quadrangulation $Q$ is reached by its $i$-th argument, which corresponds to the fact that vertex $v_i$ is incident to the face $F_i$ in $L$. Writing that $\ell(v_i)-\epsilon_i =\ell(v_j)-\epsilon_j$ and applying , we find that : $$\begin{aligned}
\label{eq:voro3}
d(v_i,s_i)-\epsilon_i \leq d(v_j,s_i)-\epsilon_j.\end{aligned}$$ Since $|\epsilon_i-\epsilon_j|\leq 2$ we can say, loosely speaking, that up to an error at most $2$, $s_i$ is closer from $v_i$ than from other $v_j$’s in $Q$. Note that another constraint from Miermont’s bijection is that, in $Q$, we have that $d(s_i,s_j)\equiv \delta_i+\delta_j\mod 2$ for all $i,j$, or equivalently, from , that $d(v_i,v_j)\equiv \epsilon_i-\epsilon_j \mod 2$.
Conversely, let $B^{\epsilon_2,\epsilon_3}(n)$ be the number of rooted bipartite quadrangulations of genus $g$ with $n$ faces and six marked vertices $s_1,s_2,s_3,v_1,v_2,v_3$ such that we have, for all $i\neq j$: $$\begin{aligned}
\label{eq:constraints3Strong}
d(v_i,s_i) < d(v_j,s_i)-2\end{aligned}$$ and such that $d(v_i,v_j)\equiv \epsilon_i-\epsilon_j\mod 2$. Given such an object, defining for each vertex $\ell(v)=\min_{1\leq i \leq 3}d(s_i,v)+\delta_i$, where $\delta_i:= d(v_i,s_i)-\epsilon_i+\epsilon_1-d(v_1,s_1)$, we see from that the minimum defining $\ell(v_i)$ is reached only by its $i$-th argument, and that $\ell(v_1)=\ell(v_2)-\epsilon_2=\ell(v_3)-\epsilon_3$. This ensures that the three-face map associated to such a quadrangulation by the (reverse) Miermont bijection is one of the objects counted by $A^{\epsilon_2,\epsilon_3}(n)$. In fact, the converse is true up to asymptotically negligible terms. Indeed, configurations counted by $A^{\epsilon_1,\epsilon_2}(n)-B^{\epsilon_2,\epsilon_3}(n)$ correspond to cases where for at least one $i\neq j$, holds but does not: such configurations are few by Lemma \[lemma:zeroProba\](ii). Details are similar to the proof of Lemma \[lemma:GeoToVor\] and we obtain that $$B^{\epsilon_2,\epsilon_3}(n) = 2 A^{\epsilon_1,\epsilon_2}(n) + o(n^6 m_g(n)),$$ where the factor of $2$ comes from the 2-to-1 nature of Miermont’s bijection.
Now, similarly as in Lemma \[lemma:equalContrib\], it is easy to see that for different $\epsilon_1,\epsilon_2$ the numbers $A^{\epsilon_1,\epsilon_2}(n)$ have the same first order contribution and: $$\sum_{(\epsilon_1,\epsilon_2)\in\{0,1\}^2} A^{\epsilon_1,\epsilon_2}(n) \sim 4 \cdot A^{0,0}(n).$$ Now $\sum_{(\epsilon_1,\epsilon_2)\in\{0,1\}^2} B^{\epsilon_2,\epsilon_3}(n) $ counts rooted bipartite quadrangulations with marked vertices $v_1,v_2,v_3,$ $s_1,s_2,s_3$ such that holds, and in which no parity constraints remain. Recalling that $c_{g+2}^{(iii)}(n)=\tfrac{1}{2n}A^{0,0}(n)$ we finally get $$c_{g+2}^{(iii)}(n) \sim \frac{1}{8\cdot 2n} n^6 m_g(n)
\cdot \mathbf{P}\big\{\forall\, 1\leq i\neq j \leq 3:\, d({\mathbf{v}}_i,{\mathbf{s}}_i) < d({\mathbf{v}}_j,{\mathbf{s}}_i)-2\big\}$$ where the probability is over ${{\mathbf{q}}_{n}^{(g)}}\in_u{\mathcal{Q}_n^{(g)}}$ and six uniform independent vertices $({\mathbf{v}}_1,{\mathbf{v}}_2,{\mathbf{v}}_3,{\mathbf{s}}_1,{\mathbf{s}}_2,{\mathbf{s}}_3)$ in ${{\mathbf{q}}_{n}^{(g)}}$. Finally, arguing exactly as in the proof of Lemma \[lemma:convergenceCell\], the last probability converges to $\mathbf{E}[{\mathbf{Y}}_g^{(1:3)}{\mathbf{Y}}_g^{(2:3)}{\mathbf{Y}}_g^{(3:3)}]$ and we are done.
From the last lemma and the expansion of $m_g(n)$ it follows that $$c_{g+2}^{(iii)}(n) \sim 2^{-2-5g}/\Gamma(\tfrac{5}{2}g-\tfrac{1}{2})\tau_g
\cdot \mathbf{E}[{\mathbf{Y}}_g^{(1:3)}{\mathbf{Y}}_g^{(2:3)}{\mathbf{Y}}_g^{(3:3)}]
\cdot n^{5+\frac{5}{2}(g-1)} 12^n.$$ Comparing with the previously obtained expansion of $c_{g+2}^{(iii)}(n)$ we find: $$\mathbf{E}[{\mathbf{Y}}_g^{(1:3)}{\mathbf{Y}}_g^{(2:3)}{\mathbf{Y}}_g^{(3:3)}] \sim \frac{1}{15 \cdot 2^2} =\frac{1}{60}$$ as claimed!
It only remains to prove Lemma \[lemma:eliminate\]:
We give a simple proof based on linear algebra, relatively brutal and (therefore) easily computerized. Let $E_0$ denote Equation , and consider its derivatives $E_i:=(\tfrac{d}{ds})^i E_0$ for $i\in[1..3]$. We thus obtain a polynomial system of four equations $\{E_i,i\in[0..3]\}$, involving the quantities $U_i=(\tfrac{d}{ds})^i U(s)$ for $i=0..5$. This system is *linear* and *triangular* in $U_2,U_3,U_4,U_5$ (note that it is *not* linear in $U_0$ and $U_1$). We can then solve for these four quantities and we obtain an expression of each $U_i$ for $i\in[2..5]$ as a (nonlinear) polynomial of $U_0=U(s)$ and $U_1=\tfrac{d}{ds}U(s)$ (with coefficients that are Laurent polynomials of $s$). Now expand the quantity $(5\tfrac{sd}{ds}-3 )_{((5))} \big(s^2U\big)$ as a linear combination of the $U_i$’s, and substitute the expressions just obtained of $U_i$ for $i=2..5$ in it. We obtain an equation of the form: $$(5\tfrac{sd}{ds}-3 )_{((5))} \big(s^2U\big) = \mbox{Polynomial}(s,s^{-1},U(s),\tfrac{d}{ds}U(s))$$ which, computations made, is .
### Acknowledgements {#acknowledgements .unnumbered}
I thank Jean-François Marckert and Grégory Miermont for interesting discussions related to the content of this note.
[^1]: Support from *Agence Nationale de la Recherche*, grant number ANR 12-JS02-001-01 “Cartaplus”, and from the City of Paris, grant “Émergences Paris 2013, Combinatoire à Paris”. Email: [guillaume.chapuy@liafa.univ-paris-diderot.fr]{}.
[^2]: if $U_1,U_2$ are two independent uniforms on $[0,1]$ and $I_1, I_2, I_3$ are the lengths of the three intervals they define, then $\mathbf{E} (I_1I_2I_3)=2\int_0^1dx\int_0^x dy \cdot y (y\!-\!x)(1\!-\!x\!-\!y) = \frac{1}{60}$. For a combinatorial calculation, notice that $\mathbf{E} (I_1I_2I_3)$ is the probability that five independent uniforms $U_1,U_2,V_1,V_2,V_3$ are ordered as $V_1<U_{1}\wedge U_2<V_2<U_1\vee U_{2}<V_3$, which is clearly equal to $\frac{2}{5!}=\tfrac{1}{60}$.
[^3]: As observed in [@Miermont:tessellations] under assumption (M2) the label varies by $\pm1$ along each edge so this is well defined.
[^4]: Strictly speaking, these notions are defined only for dominant maps in that reference. Here it will be convenient for the presentation of the decompositions to extend them to general $3$-nodes – but this is not a fundamental need, since in all quantities involved in our discussion will be led by dominant maps at the first order.
|
---
abstract: 'Critical probes of dark matter come from tests of its elastic scattering with nuclei. The results are typically assumed to be model-independent, meaning that the form of the potential need not be specified and that the cross sections on different nuclear targets can be simply related to the cross section on nucleons. For point-like spin-independent scattering, the assumed scaling relation is $\sigma_{\chi A} \propto A^2 \mu_A^2 \sigma_{\chi N}\propto A^4 \sigma_{\chi N}$, where the $A^2$ comes from coherence and the $\mu_A^2\simeq A^2 m_N^2$ from kinematics for $m_\chi\gg m_A$. Here we calculate where model independence ends, i.e., where the cross section becomes so large that it violates its defining assumptions. We show that the assumed scaling relations generically fail for dark matter-nucleus cross sections $\sigma_{\chi A} \sim 10^{-32}-10^{-27}\;\text{cm}^2$, significantly below the geometric sizes of nuclei, and well within the regime probed by underground detectors. Last, we show on theoretical grounds, and in light of existing limits on light mediators, that point-like dark matter cannot have $\sigma_{\chi N}\gtrsim10^{-25}\;\text{cm}^2$, above which many claimed constraints originate from cosmology and astrophysics. The most viable way to have such large cross sections is composite dark matter, which introduces significant additional model dependence through the choice of form factor. All prior limits on dark matter with cross sections $\sigma_{\chi N}>10^{-32}\;\text{cm}^2$ with $m_\chi\gtrsim 1\;\text{GeV}$ must therefore be re-evaluated and reinterpreted.'
author:
- 'Matthew C. Digman'
- 'Christopher V. Cappiello'
- 'John F. Beacom'
- 'Christopher M. Hirata'
- 'Annika H. G. Peter'
bibliography:
- 'main.bib'
title: '(Not as) Big as a Barn: Upper Bounds on Dark Matter-Nucleus Cross Sections'
---
Introduction
============
The nature of dark matter is one of the most pressing problems in both fundamental physics and cosmology. Decades of observations indicate that dark matter makes up the vast majority of matter in our universe, yet increasingly advanced experiments have yet to determine its physical nature. Once discovered, the particle properties of dark matter will be a guidepost to physics beyond the Standard Model as well as to an improved understanding of galaxies and cosmic structure [@Bertrev04; @Pet12; @bau12; @Bertone:2016nfn; @Plehn:2017fdg; @Buckley:2017ijx; @Baudis:2018bvr].
Progress depends on accurately assessing the regions of dark matter parameter space that remain viable. One of the best motivated dark-matter candidates is a single weakly interacting massive particle: a WIMP. There are several ways to search for WIMPs. First, through missing transverse momentum searches at colliders [@Baltz:2006fm; @Goodman:2010yf; @Goodman:2010ku; @Buchmueller:2014yoa; @Abd15; @ATLAS17; @CMS18; @Boveia:2018yeb]. Second, through searches for WIMP self-annihilation products and decay [@Delahaye:2007fr; @Meade:2009iu; @Hooper:2010mq; @Steigman:2012nb; @Albuquerque:2013xna; @Bulbul:2014sua; @Jeltema:2014qfa; @Ackermann:2015zua; @Leane:2018kjk; @Abdalla:2018mve; @Queiroz:2019acr; @Cholis:2019ejx; @Smirnov:2019ngs]. Third, by energy transfer through elastic scattering with nuclei and electrons. Laboratory direct-detection experiments [@Armengaud:2016cvl; @damic_result; @Akerib:2016vxi; @Amole:2017dex; @Akerib:2017kat; @xenon_1t; @pandax_result; @Agnes:2018fwg; @supercdms_result1; @dd9; @Abdelhameed:2019hmk] provide the tightest bounds on dark matter-nucleus elastic scattering cross sections, with other constraints provided by cosmology and astrophysics [@Cyb02; @Gor10; @Pro13; @Dvorkin:2013cea; @Kouvaris:2014lpa; @Ali-Haimoud:2015pwa; @Gluscevic:2017ywp; @Bod18; @Xu:2018efh; @Slatyer:2018aqg; @Bhoonah:2018wmw; @Cappiello:2018hsu; @Bringmann:2018cvk; @Ema:2018bih; @Gluscevic:2019yal; @Wadekar:2019xnf; @Nadler:2019zrb; @Alvey:2019zaa; @Cappiello:2019qsw]. While there are no robust signals yet, progress is rapid.
![Claimed constraints on the spin-independent dark matter-nucleon cross section [@Wandelt:2000ad; @Erickcek:2007jv; @Kavanagh:2017cru; @Mahdawi:2018euy; @Nadler:2019zrb]. Those from cosmology directly probe scattering with protons, but all others here are based on scattering with nuclei, and thus require the use of ‘model-independent’ scaling relations. Below, we show that assumptions used to derive these results are invalid over most of the plane.[]{data-label="fig:prexist_limit_simpl"}](figures/limitsformat.pdf){width="\columnwidth"}
For these searches, the two most common benchmarks for the performance of dark matter detection experiments are the dark matter self-annihilation cross section and the spin-independent dark matter-nucleon elastic scattering cross section, the simplest case. These benchmarks allow constraints set by different experiments to be scaled to each other. Here, we focus on spin-independent elastic dark matter-nucleus scattering for dark matter with $m_{\chi}\gtrsim 1$ GeV. For generality, we do not require that dark matter be a thermal relic.
Most direct-detection searches focus on pushing sensitivity to small cross sections, but it is also important to consider constraints on large cross sections [@Starkman:1990nj; @Wandelt:2000ad; @Albuquerque:2003ei; @Erickcek:2007jv; @Mack:2007xj; @Albuquerque:2010bt; @Mahdawi:2017cxz; @Mahdawi:2017utm; @Kavanagh:2017cru; @Hooper:2018bfw; @Emken:2018run; @Bramante:2018qbc; @Mahdawi:2018euy; @Bhoonah:2018gjb]. Direct-detection experiments are typically located beneath the atmosphere, rock, and detector shielding, such that dark matter with too large of a cross section loses too much energy above the detector. Energy loss in the detector overburden may open a window where strongly interacting dark matter is allowed [@Starkman:1990nj].
Figure \[fig:prexist\_limit\_simpl\] summarizes prior claimed constraints. The ‘IMP+IMAX+SKYLAB’ region is based on atmospheric and space-based detectors and is dashed because the results are commonly cited but are not based on detailed analyses in peer-reviewed papers [@Wandelt:2000ad]. There are several similar proposed XQC regions [@Wandelt:2000ad; @Erickcek:2007jv; @Mahdawi:2018euy]; we adopt that of Ref. [@Mahdawi:2018euy]. The ‘Underground Detectors’ region is taken directly from the summary plot in Ref. [@Kavanagh:2017cru]. For the ‘Cosmology’ region, we plot the strongest constraint that depends only on dark matter-proton scattering [@Nadler:2019zrb] (including helium would make the constraints somewhat stronger [@Xu:2018efh]). The details of which constraints are plotted do not affect our conclusions.
Direct-detection searches for spin-independent interactions benefit from an essentially model-independent $A^2$ coherent enhancement, as well as a kinematic factor of $\mu_A^2$, such that $\sigma_{\chi A}$ is related to the dark matter-nucleon elastic scattering cross section $\sigma_{\chi N}$ by $\sigma_{\chi A}\propto A^2\mu_A^2\sigma_{\chi N}$. For $m_\chi\gg m_A$, the dark matter-nucleus reduced mass $\mu_A\simeq A m_N$, such that the scaling becomes $\sigma_{\chi A}\propto A^4\sigma_{\chi N}$. This straightforward scaling allows constraints on dark matter-nucleus scattering to be related to each other and to the cross section on nucleons. This scaling is ‘model-independent’ in the sense that it is independent of the detailed shape of the potential. In Fig. \[fig:prexist\_limit\_simpl\], all constraints except the one labeled ‘Cosmology’ deal with nuclear targets with $A>1$, and hence assume this scaling relation.
How large of cross sections are allowed before the defining assumptions are violated? Here we systematically calculate the theoretical upper limits on dark matter-nucleon cross sections. We show that most of the parameter space of Fig. \[fig:prexist\_limit\_simpl\] is beyond the point where the simple scaling relations above are valid, or where point-like dark matter is even allowed. Our results are based first on generic considerations of theoretically allowed cross sections for short-range interactions with nuclei, and second on classes of models where we consider light mediators as a mechanism to obtain large cross sections. As far as we are aware, this is the first systematic exploration of these issues for dark matter-nucleus scattering (for related considerations in strongly self-interacting dark matter sectors, see, e.g., Ref. [@Tulin:2013teo]). Our results will require the reinterpretation of a large and varied body of work, e.g. Refs. [@Goodman:1984dc; @PhysRevLett.61.510; @Rich:1987st; @Starkman:1990nj; @McGuire:1994pq; @Bernabei:1999ui; @Derbin:1999uh; @Wandelt:2000ad; @Chen:2002yh; @Albuquerque:2003ei; @Zaharijas:2004jv; @Mack:2007xj; @Erickcek:2007jv; @Albuquerque:2010bt; @Dvorkin:2013cea; @Jacobs:2014yca; @Mahdawi:2017cxz; @Mahdawi:2017utm; @Kavanagh:2017cru; @Gluscevic:2017ywp; @Hooper:2018bfw; @Emken:2018run; @Xu:2018efh; @Bramante:2018qbc; @Slatyer:2018aqg; @Mahdawi:2018euy; @Bhoonah:2018gjb; @Bramante:2018tos; @Janish:2019nkk].
In Sec. \[sec:scattering\_theory\], we review the nonrelativistic scattering theory used to obtain the model-independent scaling relations. In Sec. \[sec:contact\], we examine the various ways that scaling relations can break down for contact interactions. In Sec. \[sec:light\_mediator\], we examine the possibility of achieving a larger cross section with a light mediator in light of present constraints on light mediators. In Sec. \[sec:composite\_dm\], we briefly discuss the possibility that dark matter itself could have a nonzero physical extent. In Sec. \[sec:implications\], we discuss the implications for existing constraints and future experiments. Finally, we summarize our results and the outlook for future work in Sec. \[sec:disc\_conc\].
Dark Matter Scattering Theory {#sec:scattering_theory}
=============================
We briefly review the basic nonrelativistic scattering theory required to derive the ‘model-independent’ scaling relation for the spin-independent elastic scattering cross section. We also discuss how some of the key assumptions may break down. Throughout, we set $\hbar=c=1$.
Overview of basic assumptions {#ssec:basic_assumptions}
-----------------------------
1. [**Single particle:** ]{} Dark matter is primarily a single unknown particle. The number density of dark matter is then determined only by its mass and the local dark matter density.
2. [**Point-like:**]{} Dark matter is a point-like particle with no excitation spectrum.
3. [**Electrically neutral:**]{} It is typically assumed that dark matter is electrically neutral. Millicharged dark matter has different dynamics and is too strongly constrained to produce large cross sections.
4. [**Equal coupling to all nucleons:**]{} For simplicity, we assume that dark matter has equal coupling to both protons and neutrons, although this assumption is not essential to any of our conclusions.
5. [**Local:**]{} The interaction is assumed to be local, $\left<{\bf x'}\left|\widehat{V}\right|{\bf x}\right>=V({\bf x'})\delta^3({\bf x'}-{\bf x})$.
6. [**Energy-independent potential:**]{} The potential for the interaction is assumed to be energy-independent, such that the cross section for the interaction is also energy-independent up to a form factor. For a spin-independent interaction, the potential must also be independent of the incident angular momentum $l$.
7. [**Elastic:**]{} For laboratory experiments, dark matter-nucleus scattering is assumed to occur at typical Milky Way virial velocities, $v\sim 10^{-3}\,c$. Typical recoil energies of $\mathcal{O}(1\;\text{keV})$ are not sufficient to produce Standard-Model particles, or to excite internal degrees of freedom of nuclei. Therefore, elastic scattering is the dominant interaction channel. In any case, all physical scattering processes have at least some elastic component [@joachain75_collision].
8. [**Coherence:**]{} Closely related to the assumption of purely elastic scattering is the assumption of coherence. For coherence to hold, it must be a good approximation to treat the dark matter as interacting with the nucleus as a whole, rather than with individual nucleons. If the momentum transfer $q$ is insufficient to excite internal degrees of freedom in a nucleus, which is true for $q r_N\ll1$, coherence is typically a good approximation [@bransden1983physics]. The breakdown of coherence can be parametrized by including a momentum-dependent form factor in the differential cross section.
9. [**No bulk effects:**]{} The scattering should be well approximated as being with a single nucleus, such that initial and final state effects in the bulk medium can be ignored. This approximation is good as long as the characteristic momentum transfer $q$ is large compared to the characteristic inter-atomic spacing, which is typical.
The rest of this paper deals with the failure of the following additional assumptions:
1. [**S-wave scattering**]{} For s-wave ($l=0$) scattering, the scattering is isotropic in the center of momentum frame. As shown in Sec. \[sssec:partial\_waves\], assuming $l=0$ is required to derive the model-independent $A^4$ scaling relation. However, real interactions may deviate significantly from isotropic scattering and we do not require $l=0$ in this analysis.
2. [**Weak Interaction:**]{} For $A^4$ scaling to hold, the interaction must be weak enough for the Born approximation to hold. We discuss this assumption in Sec. \[sssec:born\_approx\].
Basic scattering theory
-----------------------
Here we provide a brief review of the scattering theory formalism [@newton1964complex; @regge65_scattering; @joachain75_collision; @goldberger1975collision; @bransden1983physics; @sakurai; @khare2012introduction; @burke_joachain] used in later sections.
To be detectable, dark matter must have some kind of interaction with ordinary matter in a detector, written here as a potential $V({\bf r})$. We specialize to spin-independent interactions, and restrict our analysis to spherically symmetric potentials $V({\bf r}) = V(r)$ that fall off faster than $r^{-1}$ as $r\rightarrow\infty$. In the center of momentum frame, the time-independent Schrödinger equation giving the evolution of a nonrelativistic two-particle system with wavefunction $\psi({\bf r})$ and reduced mass $\mu$ is given $$\label{tise}
\left(-\frac{1}{2\mu}\nabla^2_{\bf r}+V({\bf r})\right)\psi({\bf r}) = E\psi({\bf r}).$$ As shown in Appendix \[app:lippmann\_schwinger\], far from the potential the solution of Eq. may be written: $$\begin{aligned}
\psi({\bf r})\xrightarrow[]{r\rightarrow\infty}&\psi_0({\bf r})-\frac{\mu e^{ikr}}{2\pi r} \int V({\bf r'})\psi({\bf r'})e^{-i{\bf k}_f\cdot{\bf r'}}d^3{\bf r'} \nonumber\\
=&\psi_0({\bf r})+(2\pi)^{-3/2}\frac{e^{ikr}}{r}f\left({\bf k}_i,{\bf k}_f\right),\label{scattering_amp_lipp}\end{aligned}$$ where $f\left({\bf k}_i,{\bf k}_f\right)=f(k,\theta)$ is the scattering amplitude, $\psi_0({\bf r})\equiv(2\pi)^{-3/2}e^{i {\bf k}_i\cdot{\bf r}}$, and ${\bf k}_i\equiv k{\bf \hat{z}}$ and ${\bf k}_f$ are the initial and final dark matter momenta, respectively. From the scattering amplitude, we obtain the differential cross section: $$\label{diff_scattering}
\frac{d\sigma}{d\Omega} =|f(k,\theta)|^2,$$ and the total elastic scattering cross section: $$\label{tot_scattering}
\sigma_{\chi A}=\int \frac{d\sigma}{d\Omega} d\Omega.$$
If the scattering is isotropic, $f(k,\theta)=f(k)$, Eq. is proportional to the rate of detectable scattering events in a detector. However, to be detectable, a collision must deposit sufficient energy into the detector. If the scattering angle is peaked close to $\theta=0$, very little momentum is transferred, and hence insufficient energy deposited in the detector. Therefore, it is sometimes more useful to weight the integral in Eq. by the momentum transfer to obtain the momentum-transfer cross section, $$\label{mt_scattering}
\sigma^{\text{mt}}_{\chi A}=\int \frac{d\sigma}{d\Omega}\left(1-\cos\theta\right) d\Omega.$$
For isotropic scattering, $\sigma^{\text{mt}}_{\chi A}=\sigma_{\chi A}$. For a potential with characteristic radius $r_A$, isotropic scattering is generically a good approximation at low energies, $k r_A\ll 1$, as discussed further in Sec. \[sssec:partial\_waves\]. Forward scattering is a major concern for light mediators (Sec. \[sec:light\_mediator\]); in the Coulomb scattering limit where the mediator mass $m_\phi\rightarrow0$, then $\sigma_{\chi A}\rightarrow\infty$, while $\sigma^{\text{mt}}_{\chi A}$ remains finite.
Derivation of ‘model-independent’ scaling
-----------------------------------------
Now we discuss approximation methods for $f(k,\theta)$. The two approaches we consider here are the Born approximation and the partial wave expansion. Both allow us to derive the $\sigma_{\chi A}=A^2\mu_A^2/\mu_N^2\sigma_{\chi N}$ scaling with nuclear mass number $A$. We begin with the Born approximation because it is the simple and familiar derivation. Because the partial wave expansion is valid even when the Born approximation fails, it allows us to more concretely show the behavior at large scattering cross sections.
### Born approximation {#sssec:born_approx}
Inspecting Eq. , a natural first approach to obtaining $f(k,\theta)$ is to solve for $\psi({\bf r})$ by iteration, which is the Born approximation, as demonstrated in Appendix \[app:born\_der\]. The first Born approximation to $f(k,\theta)$ is simply the Fourier transform of the potential: $$\label{first_born_fourier_body}
f^{(1)}\left(k,\theta\right)=f^{(1)}\left(q\right)=-\frac{2\mu_A}{q}\int_0^\infty V(r')\sin(qr')r' dr',$$ where $q=|{\bf q}|=2k\sin\theta$ is the momentum transfer.
Now, assume that the potential has some maximum radius $r_A$, and we have low energy scattering, $k r_A\ll 1$. Then we can approximate $\sin(qr')\approx q r'$ and integrate only up to the maximum radius $r_A$: $$\label{potential_ind_approx}
f^{(1)}\left(k,\theta\right)\approx-2\mu_A\int_0^{r_A} V(r')r'^2 dr'.$$
Eq. is a remarkable result. Provided the required approximations are valid, $f^{(1)}\left(k,\theta\right)$ depends only on the volume integral of the potential; it contains no information at all about the shape. Provided the volume integral of the potential is proportional to the nuclear mass number $A$, we have the scaling: $$\label{model_ind_scaling_amp}
f^{(1)}\left(k,\theta\right)\propto A\mu_A.$$ Plugging into Eq. : $$\label{model_ind_scaling_prop}
\sigma^{(1)}_{\chi A}\propto A^2\mu_A^2,$$ which can be recast more precisely in terms of the dark matter-nucleon reduced mass $\mu_N$ and scattering cross section $\sigma_{\chi N}^{(1)}$: $$\label{model_ind_scaling_tot}
\sigma^{(1)}_{\chi A}= A^2\frac{\mu_A^2}{\mu_N^2}\sigma_{\chi N}^{(1)}.$$
Eq. is the famous ‘model-independent’ scaling relation for the spin-independent elastic scattering cross section. Provided the potential falls off faster than $1/r$, this scaling relation is generally a good approximation at sufficiently low energies, so long as the first Born approximation reasonably approximates $f(k,\theta)$. However, we must examine when the first Born approximation fails.
We discuss the validity of the Born approximation in Appendix \[app:born\_der\]. A useful condition for the validity of the first Born approximation is [@khare2012introduction]: $$\label{final_cond_born_use}
\frac{\mu_A}{k}\left|\int_0^\infty V(r')\left(e^{2ikr'}-1\right)dr'\right|\ll 1.$$
We can simplify Eq. using our assumption of a maximum range $r_A$ and $k r_A\ll 1$: $$\label{born_cond_approx}
2\mu_A\left|\int_0^{r_A} V(r')r'dr'\right|\ll 1.$$ Eq. is equivalent to the statement that the potential is much too weak to form a bound state even if $V(r)$ was purely attractive [@levinson1949uniqueness]. While Eq. is a volume integral, Eq. is an area integral of the potential. Therefore, the first Born approximation is valid when some potential-weighted effective area is small. The effective area in question is, in fact, the elastic scattering cross section, as shown for a contact interaction in Sec. \[sec:contact\].
### Partial wave expansion {#sssec:partial_waves}
To investigate what happens when the Born approximation fails, the first step is to expand the scattered wave function in terms of Legendre polynomials and calculate the phase shift of each contribution. The phase shifts may be found by numerically integrating the Schrödinger equation, as described in Appendix \[app:partial\_wave\]. The elastic scattering cross section may be written in terms of the phase shifts $\delta_l(k)$:
$$\label{partial_wave_cross_sec}
\sigma_{\chi A}=\frac{4\pi}{k^2}\sum_{l=0}^\infty{(2l+1)\sin^2\left(\delta_l\right)}.$$
The momentum-transfer cross section in Eq. may also be written in terms of partial wave phase shifts:
$$\label{partial_mt_cross_section}
\sigma^{\text{mt}}_{\chi A}=\frac{4\pi}{k^2}\sum_{l}(l+1)\sin^2(\delta_{l+1}-\delta_l).$$
The mathematical decomposition in terms of partial waves is valid even beyond interactions that can be described in nonrelativistic potential scattering theory. However, when the number of partial waves becomes too large, it may be impractical to compute the phase shifts individually, and semiclassical approximations become useful [@joachain75_collision; @sakurai].
Physically, the sum over partial waves in Eq. is equivalent to the classical operation of averaging over all possible impact factors $b=\sqrt{l(l+1)}/k$ [@burke_joachain; @joachain75_collision]. Classically, for a potential with maximum range $r_A$, there would be no collisions for $b>r_A$. Therefore, a useful approximate upper limit on the highest partial wave that can meaningfully contribute to the sum in nonrelativistic quantum scattering is $l_{\text{max}}\approx k r_A$, and contributions from higher $l>l_{\text{max}}$ fall off quickly [@joachain75_collision]. Our derivation of the ‘model-independent’ scaling of Eq. in the Born approximation assumes $k r_A\ll 1$, which is equivalent to saying only the $l=0$ (s-wave) term contributes a nonvanishing phase shift.
Using the same iterative procedure as for the Born approximation, we can obtain the ‘model-independent’ form of the s-wave phase shift [@burke_joachain]: $$\label{swave_model_ind}
\delta_0(k)\approx -2\mu_A k\int_0^{r_A} V(r')r'^2 dr',$$ where the required approximation is $\delta_0(k)\ll 1$. Plugging Eq. into Eq. and again using $\delta_0(k)\ll 1$, we obtain precisely the same expression for the scattering amplitude we obtained for the Born approximation in Eq. , as expected. Again, the requirement $\delta_0(k)\ll 1$ places an upper bound on the maximum $\sigma_{\chi A}$ where the relation can apply.
If the cross section were instead the maximum allowed by unitarity, $\delta_0(k)=\pi/2$, we would obtain $$\label{swave_max_cross}
\sigma_{\chi A}=\frac{4\pi}{k^2},$$ which *decreases* as $1/A^2$ with increasing $A$, assuming $k\propto A$, rather than increasing as $A^4$.
Higher partial waves necessarily scale differently with $k$ [@joachain75_collision], $\delta_l(k)\propto k^{2l+1}$ for $k r_A\ll 1$. Higher $\delta_l(k)$ also contain information about the shape of the potential. Therefore, we do not expect any special ‘model-independent’ scaling when higher partial waves contribute.
Contact interactions {#sec:contact}
====================
In this section, we consider the limits on cross sections that can be obtained through a contact interaction, and how the scaling relations break down. A contact interaction is useful as an illustrative case because we do not need to consider the specific mechanism that produces the interaction.
Contact interaction with Born approximation {#ssec:contact_born}
-------------------------------------------
As a simple case, we consider a contact interaction with a nucleus, as could be produced by a heavy mediator. We roughly approximate the nuclear charge density as having a top hat shape with radius $r_A$: $$\label{potential_well}
V(r)=
\begin{cases}
V_0 &r<r_A\\
0 &\text{otherwise}.
\end{cases}$$ We assume the maximum charge density is roughly independent of atomic mass number $A$, such that $r_A\approx A^{1/3}r_N$, where $r_N\simeq 1.2\;\text{fm}$.
We use this toy model with a sharp cutoff because both the Born approximation and the partial wave phase shift $\delta_l(k)$ can be found analytically. The effect of using a more realistic charge distribution is discussed in Sec. \[sssec:realistic\_charge\].
Fourier transforming Eq. using Eq. gives: $$f^{(1)}(q)=\frac{2\mu_A V_0}{q^3}\left[qr_A\cos(q r_A)-\sin(q r_A)\right].$$
The total elastic scattering cross section in the first Born approximation is then: $$\begin{gathered}
\label{tophat_1born}
\sigma^{(1)}_{\chi N}=\frac{\pi\mu_A^2 V_0^2}{16k^6}\big[4kr_A\sin(4kr_A)+\cos(4kr_A)\\
+32k^4r_A^4-8k^2r_A^2-1\big].\end{gathered}$$ In the limit $kr_A\ll 1$, Eq. becomes: $$\label{tophat_1born_lowk}
\sigma^{(1)}_{\chi A}\approx\frac{16\pi}{9}\mu_A^2r_A^6V_0^2.$$ For scattering with a nucleon, Eq. would become $\sigma^{(1)}_{\chi N}\approx\frac{16\pi}{9}\mu_N^2r_N^6V_0^2$. Substituting $r_A\approx A^{1/3}r_N$, we recover the required scaling relation of Eq. : $$\label{tophat_scaling}
\sigma^{(1)}_{\chi A}\approx A^2\frac{\mu_A^2}{\mu_N^2}\sigma^{(1)}_{\chi N}.$$
In the $k r_A\ll 1$ limit, the condition for validity of the first Born approximation in Eq. is simply $$\label{tophat_1born_lowk_cond}
\mu_A r_A^2 V_0\ll 1.$$ Comparing to Eq. , we can rewrite the condition Eq. as: $$\label{tophat_1born_lowk_crosscond}
\sigma^{(1)}_{\chi A}\ll \frac{16}{9}\pi r_A^2.$$ Eq. has a simple physical interpretation. The first Born approximation is only applicable for elastic scattering cross sections much smaller than the geometric cross section of the nucleus. Using $r_N\approx 1.2\;\text{fm}$ in Eq. , the Born approximation result only applies for $\sigma^{(1)}_{\chi N}\ll 10^{-25}\;\text{cm}^2$.
Going to higher orders in the Born approximation does not unlock cross sections significantly exceeding the geometric limit of the potential. For $\sigma^{(1)}_{\chi A}>\frac{16}{9}\pi r_A^2$, the Born series is not even guaranteed to converge for all energies [@joachain75_collision].
However, it may still be possible to obtain a meaningful cross section in regimes where the Born approximation fails using partial wave analysis. We explore this technique below.
Contact interaction with partial waves {#ssec:tophat_swave}
--------------------------------------
For $r<r_A$ in Eq. , the radial wave function decomposed in partial waves has an analytic solution in terms of partial waves, $u_l(r)=C_l r j_l(k' r)$, where $k'\equiv\left(k^2-2\mu_A V_0\right)^{1/2}$ could be either pure real or pure imaginary. First, we consider the s-wave cross section, with $l=0$. Expanding in the limit where $V_0$ and $k$ are small, $(kr_A)^2\ll 1$, $|V_0|\ll 1/(2\mu_A r_A^2)$, we recover (see Appendix \[app:partial\_wave\]):
$$\label{delta_l_tophat_order2}
\delta_0(k)\approx-\frac{2\mu_A k r_A^3V_0}{3}+\frac{8\mu_A^2kr_A^5V_0^2}{15}+\mathcal{O}\left(|V_0|^3\right).$$
The corresponding s-wave cross section is $$\label{s_wave_sigma_order2_tophat}
\sigma^{l=0}_{\chi A}\approx\frac{16\pi}{9} \mu_A^2 r_A^6 V_0^2-\frac{128\pi}{45}\mu_A^3r_A^8V_0^3+\mathcal{O}\left(|V_0|^4\right),$$ which is identical to Eq. to lowest order in $|V_0|$, as anticipated in Sec. \[sssec:partial\_waves\].
Now we can see how the ‘model-independent’ $A$ scaling fails as the coupling strength gets stronger. The second-order term in Eq. scales $\propto\mu_A^3/\mu_N^3 A^{8/3}$, and either reduces the cross section for a repulsive potential $V_0>0$ or increases it for an attractive one. For $V_0$ large enough for the second-order and higher corrections to matter, there is, therefore, no model-independent scaling with $A$, because $\sigma^{l=0}_{\chi A}$ depends on the details of the potential. The breakdown of the scaling as a function of $A$ is shown in Fig. \[fig:scale\_saturation\]. Once details of the potential begin to matter, corrections from a more realistic charge distribution would also become important, as discussed in Sec. \[sssec:realistic\_charge\].
To illustrate further, if we instead considered the strong coupling limit, $|V_0|\gg1/{2\mu_A r_A^2}$, for $V_0>0$ we would obtain:
$$\label{large_V0_delta}
\delta_0(k)=-k r_A,$$
and $$\begin{aligned}
\label{sigma_s_wave_large_V0}
\sigma^{l=0}_{\chi A}&=\frac{4\pi}{k^2}\sin^2\left(-k r_A\right)\\
&=4\pi r_A^2,\label{tophat_geo_hard_sphere}\end{aligned}$$ so the repulsive cross section completely saturates at four times the geometric cross section. The saturation is plotted as a function of $|V_0|$ in Fig. \[fig:v0\_scale\_figure\]. Physically, the well has simply become an impenetrable hard sphere with a fixed radius. Therefore, we have obtained a physical maximum cross section for the repulsive contact interaction at small $k$ which is only $9/4$ times larger than the maximum we obtained using the first Born approximation.
### Higher partial waves {#sssec:higher_partial}
While the s-wave cross section is limited to the geometric cross section, it is natural to wonder if the contributions from higher partial waves could allow a larger total cross section. For $k r_A\gg1$, we can approximate semi-classically $l_\text{max}\approx k r_A$, as described in Sec. \[sssec:partial\_waves\]. Of course, in the quantum case, it is possible for higher partial waves to contribute, but their contributions fall off rapidly for $l>l_\text{max}$. Assuming a sharp cutoff is adequate for deriving an approximate maximum physically achievable cross section. The maximum possible cross section is achieved by saturating partial wave unitarity, i.e., taking $\delta_{l\leq l_\text{max}}(k)=\pi/2$: $$\begin{aligned}
\label{partial_wave_max_tophat}
\sigma_{\chi A}&=\frac{4\pi}{k^2}\sum_{l=0}^{l_\text{max}}(2l+1)\\
&=\frac{4\pi}{k^2}(1+l_\text{max})^2\\
&\approx 4\pi r_A^2.\label{partial_wave_max_tophat_fin}\end{aligned}$$ Now, we see that the saturation at approximately the geometric cross section, found for $k r_A\ll 1$ in Eq. , also holds for $k r_A\gg1$.
In fact, for a very strong repulsive contact interaction, the phase shifts for $l\leq l_\text{max}$ approach $\delta_{l\leq l_\text{max}}(k)\approx l\pi/2-ka$ [@joachain75_collision], such that $\sigma_{\chi A}\approx 2\pi r_A^2$. Therefore, a repulsive hard sphere almost saturates the unitarity limit of Eq. .
Including higher partial waves is therefore not a useful way of increasing the cross section, because the potential remains limited by the characteristic radius $r_A$.
Figure \[fig:sigma\_sigma\] shows the breakdown of the $A^4$ scaling for several example nuclei, fully taking into account contributions from higher partial waves. Direct detection constraints for underground detectors are affected by the breakdown of scaling at the $\mathcal{O}(1)$ level for $\sigma_{\chi N}\simeq10^{-32}\;\text{cm}^2$.
Attractive resonances {#ssec:attract_res}
---------------------
A final possible approach would be to saturate unitarity at $\delta_{l}=\pi/2$ while $k r_A \ll 1$, such that $$\label{swave_max}
\sigma_{\chi A}^{\text{max}}=\frac{4\pi}{k^2}(2l+1)$$ can become large. The limit $\delta_{l}=\pi/2$ at $k r_A\lesssim 1$ can be achieved through resonances, which occur when an attractive potential becomes strong enough to support a bound state.
In reality, the resonant scattering cross section would achieve large values only for a narrow range of $k$ relative to the incident dark matter velocity distribution [@breit_wigner; @joachain75_collision]. Because $k=\mu_A v$, the resonances are generically at different incident dark matter velocities for different elements, which guarantees that there are not any useful model-independent scaling relations relating the observed cross sections for strongly attractive potentials between different target materials.
In Fig. \[fig:scale\_saturation\], we show the behavior as a function of $A$ for two different values of $V_0$. When resonances are possible, the scaling with $A$ need not be monotonic. The behavior is fairly complex for even the simple rectangular well of Eq. . Realistic scaling is likely to be even more complicated, because the nuclear charge distribution changes as a function of $A$. Even two different nuclei with the same $A$ but different atomic numbers could have different charge distributions, and hence different resonant cross sections. Scaling with $A$ for strong attractive couplings is therefore highly model-dependent. In Fig. \[fig:v0\_scale\_figure\], we show the saturation of the s-wave cross section as a function of the coupling strength, as well as the resonant behavior which occurs once the potential becomes strong enough to support a quasi-bound state. The resonances for $A=4$ are fairly narrow, as for a nucleon the scattering is still well approximated in the low-$k$ limit. However, for $A=131$, the low-$k$ limit is no longer a good approximation, and resonances are broadened to the point that they do not significantly increase the scattering cross section. Additionally, there are many more resonances, from multiple partial waves. Note we have not implemented any velocity dispersion for this plot; the spreading of resonances is entirely due to broadening of peaks and overlapping contributions from multiple partial waves at finite $k$. Applying a realistic dark matter velocity distribution would smooth the peaks. For heavy nuclei with multiple naturally occurring isotopes (e.g. xenon), averaging over a distribution of isotopes would smooth the peaks even further.
![[**Top:**]{} Scaling with $A$ for the contact interaction in Sec. \[sec:contact\] with $|V_0|=1.18\times10^{-5}\;\text{GeV}$, computed using $k=0.005 A\;\text{fm}^{-1}$, $R_A=1.2 A^{1/3}\;\text{fm}$. We include partial waves up to $l_\text{max}=8$, which is sufficient to converge $\sigma_{\chi A}$ to $\sim 10^{-16}$ precision. Attractive and repulsive interactions scale similarly, although the scaling deviates from $A^4$ at high $A$ due to form-factor suppression, accounted for here by including the contributions from higher partial waves.\
[**Bottom:**]{} Same as above, but with $|V_0|=1.18\times10^{-3}\;\text{GeV}$, which corresponds to the ‘scaling relations unreliable for $A>12$’ line in Fig. \[fig:scattering\_scaling\]. Repulsive and attractive interactions no longer scale the same way, and both saturate close to $4\pi R_A^2$. The attractive potential shows resonances with $A$, which are sensitive to the specific choice of potential. For cross sections approaching the geometric cross section, any scaling with $A$ is highly model-dependent.[]{data-label="fig:scale_saturation"}](figures/scale_saturate_figure.pdf){width="\columnwidth"}
Overall, even if a carefully tuned resonance could achieve a large cross section for a single light nucleus, other nuclei would not necessarily have correspondingly large cross sections. Scaling relations between specific nuclear cross sections would also be highly model-dependent, such that constraints from different types of nuclei would be difficult to compare because the full resonance structure would not be known. For example, using more realistic charge distributions, such as an exponential potential for $A=4$ and a Woods-Saxon potential for $A=131$ [@charge_dist_1987] would shift the positions of the resonances somewhat.
![[**Top**]{}: Scaling of cross section with $|V_0|$ for $A=4$ (helium), calculated using the contact interaction in Sec. \[sec:contact\]. The cross sections are computed using the analytic partial wave results. For attractive potentials, once $|V_0|$ becomes large enough to support quasi-bound states, resonances can increase the cross section by several orders of magnitude, but only in a narrow range.\
[**Bottom**]{}: Same as above, but with $A=131$ (xenon). A larger number of partial waves contribute due to the larger $k\propto A$. There are many resonances, but they are not large enough to meaningfully increase the cross section above the geometric limit. Additionally, the resonances are not at the same values of $|V_0|$, which prevents resonances from achieving a large cross section which scales predictably with $A$, as shown in Fig. \[fig:scale\_saturation\].[]{data-label="fig:v0_scale_figure"}](figures/v0_scale_figure.pdf){width="\columnwidth"}
![Scaling of the nuclear cross section with nucleon cross section for the repulsive contact interaction of Sec. \[sec:contact\] at fixed $k_N=0.005\;\text{fm}^{-1}$. The contact interaction cannot achieve nucleon cross sections larger than the geometric cross section, denoted by the vertical red line. The cross section visibly deviates from $A^4$ scaling at the $\mathcal{O}(1)$ level for heavy nuclei even for $\sigma_{\chi N}\simeq10^{-32}\;\text{cm}^2$, and by the time scaling fails at the $\mathcal{O}(1)$ level for $^4$He at $\sigma_{\chi N}\simeq4\times10^{-28}\;\text{cm}^2$, the cross sections for heavy nuclei have completely saturated. The scaling could break down in different ways in other models.[]{data-label="fig:sigma_sigma"}](figures/sigma_sigma_figure.pdf "fig:"){width="\columnwidth"}\
### More realistic charge distributions {#sssec:realistic_charge}
The rectangular barrier potential in Eq. is a toy model. Realistic nuclear charge distributions have a smooth cutoff, and an exponential tail to larger radii [@charge_dist_1987], as in a Woods-Saxon potential. Because there is not a sharp cutoff, allowing the interaction strength to be arbitrarily large would cause the potential to grow logarithmically with $|V_0|$. However, there are limits to how strong $|V_0|$ can be. For $|V_0|\gtrsim 10^{-1}\;\text{GeV}$, QCD corrections break the simple nonrelativistic contact interaction picture. For $|V_0|\gtrsim 2\;\text{GeV}$, the interaction may be strong enough to pull proton-antiproton pairs out of the vacuum.
We have verified by numerically computing partial-wave amplitudes that for $|V_0|\lesssim 10^{-1}\;\text{GeV}$ using a Woods-Saxon potential increases the maximum $\sigma_{\chi A}$ by a factor of $\lesssim 10$. By definition, an increase in the computed cross section due to a different potential can only appear for $|V_0|$ strong enough that the ‘model-independent’ form of the cross section has already significantly broken down. Therefore, using a more realistic charge distribution cannot significantly change our conclusion that the cross section for a contact interaction cannot be much larger than the geometric cross section of the nucleus.
Realistic potentials are also not perfectly spherically symmetric, but the same basic picture of geometrical limitations still applies, and resonances are still possible.
Beyond contact interactions {#ssec:beyond_contact}
---------------------------
We have established that a contact interaction with a nucleus cannot achieve cross sections much larger than the geometrical cross section of the nucleus. The case where large cross sections might be achieved, a strongly attractive potential, produces resonances that are sensitive to the detailed structure of the potential and is far too model-dependent to possess any simple scaling relation relating the cross sections at different $A$. To circumvent these problems, we need an interaction with a larger characteristic range. One possible way to achieve a larger characteristic range is to insert a light mediator for the interaction, as discussed in Sec. \[sec:light\_mediator\]. Another possibility is composite dark matter with an intrinsic radius, discussed in Sec. \[sec:composite\_dm\].
Light Mediator {#sec:light_mediator}
==============
A simple approach to achieving a larger characteristic radius is to insert a light mediator, of mass $m_\phi=1/r_\phi$, which generically results in a potential of the form $$\label{yukawa_potential}
V(r) = \frac{\lambda_A\lambda_\chi}{4\pi}\frac{e^{-r/r_\phi}}{r},$$ where $\lambda_\chi$ and $\lambda_A=A\lambda_N$ are the coupling strengths of the particle $\phi$ to the dark matter and nucleus respectively. To achieve cross sections much larger than a nucleus, we should have $r_\phi\gg1\;\text{fm}$. The dark matter and target nucleus are distinguishable particles, so the Yukawa potential can be either attractive or repulsive. We assume the mediator is a scalar, although the general form of the potential would be similar for other light mediator candidates. The scattering amplitude of Eq. is easily calculated using Eq. : $$\label{yukawa_fourier}
f^{(1)}\left(q\right)=-\frac{\mu_A\lambda_\chi\lambda_A}{2\pi(q^2+1/r_\phi^2)},$$ which gives the total elastic scattering cross section: $$\label{tot_cross_yukawa}
\sigma^{(1)}_{\chi A}=\frac{\mu_A^2\lambda_\chi^2\lambda_A^2r_\phi^4}{\pi(1+4k^2r_\phi^2)}.$$ Because the characteristic radius is larger than the geometric radius of a nucleon, Eq. can in principle achieve larger cross sections within the domain of validity of the Born approximation than a contact interaction could. Because $k\propto \mu_A$, the scaling with $A$ is now more complicated due to the $k^2 r_\phi^2$ term in the denominator. Assuming $m_\chi\gg m_A$ such that $\mu_A\approx A m_N$, we have two limits:
$$\label{sigma_yukawa_limits}
\sigma^{(1)}_{\chi A}\approx \begin{cases}
A^4\sigma^{(1)}_{\chi N} & k_A r_\phi\ll 1\\
A^2\sigma^{(1)}_{\chi N} & k_A r_\phi\gg1.
\end{cases}$$
For direct-detection, $k_N\sim0.005\;\text{fm}^{-1}$ is set by Milky Way halo velocities $v\simeq10^{-3}\,c$ and the mass of a single nucleon, $k_A\simeq A k_N$. For $^{131}$Xe, $k_A\approx0.7\;\text{fm}^{-1}$, such that $k_A r_\phi>1$ occurs for $r_\phi\gtrsim1.4\;\text{fm}^{-1}$. Therefore, $A^4$ scaling is at best marginal for heavy nuclei for any $r_\phi$ that could conceivably produce a cross section $\sigma_{\chi A}\gtrsim 10^{-25}\;\text{cm}^2$. Additionally, as established for a general potential with a characteristic radius $r_\phi$ in Sec. \[sssec:born\_approx\], increasing the coupling strengths $\lambda_\chi$ or $\lambda_N$ at fixed $r_\phi$ causes the Born approximation, and therefore the $A^4$ scaling, to fail before cross sections larger than the geometric cross section are achieved. Therefore, for a light mediator, the $A^4$ scaling is only possible if $\sigma^{(1)}_{\chi A}\ll 10^{-25}\;\text{cm}^2$. The failure of $A^4$ scaling occurs without even considering the constraints on the existence of light mediators discussed in Subsection \[ssec:yukawa\_med\_lim\]. The $A^4$ scaling is preserved to somewhat larger cross sections than for the contact interaction shown in Fig. \[fig:scale\_saturation\].
![[**Top:**]{} Elastic scattering cross section contours as a function of mediator mass and coupling strength for the repulsive Yukawa potential in Eq. . We also show various constraints on the existence of such mediators from Ref. [@light_mediator]. The largest cross sections achieved in unconstrained regions are $\sigma_{\chi N}\lesssim10^{-27}\;\text{cm}^2$. For $m_\phi<10^{-9}\;\text{GeV}$, fifth-force constraints become many orders of magnitude stronger and dominate other constraints [@Murata:2014nra].\
[**Bottom:**]{} Same as above, but for the momentum-transfer cross section. The largest cross sections achieved in unconstrained regions are $\sigma^{\text{mt}}_{\chi N}\lesssim10^{-32}\;\text{cm}^2$.[]{data-label="fig:combine_cross"}](figures/sigma_combine_paper.pdf){width="\columnwidth"}
Momentum-transfer cross section
-------------------------------
In fact, even the $A^2$ scaling is too optimistic for the detectable momentum transfer in a detector with high $A$. Inspection of Eq. shows that for $k r_\phi\gg1$, the scattering becomes strongly peaked at $\theta=0$. Therefore, it is more useful to consider the momentum-transfer cross section, Eq. . Using the Born approximation, we can calculate Eq. analytically for the Yukawa potential: $$\begin{gathered}
\label{mt_yukawa}
\sigma^{\text{mt},(1)}_{\chi A}=\frac{\mu_A^2\lambda_\chi^2\lambda_A^2}{8\pi k^4(1+4k^2r_\phi^2)}\\
\times\big[(1+4 k^2r_\phi^2)\log(1+4 k^2r_\phi^2)
-4k^2r_\phi^2\big].\end{gathered}$$ For $k r_\phi\ll 1$, Eq. simplifies to $\sigma^{\text{mt},(1)}_{\chi A}\approx\sigma^{(1)}_{\chi A}$ as expected for isotropic scattering. However, for $k r_\phi\gg1$, we have: $$\label{mt_yukawa_highk}
\sigma^{\text{mt},(1)}_{\chi A}\approx\frac{\mu_A^2\lambda_\chi^2\lambda_A^2}{8\pi k^4}\left(\log(4k^2r_\phi^2)-1\right).$$ Eq. grows only $\propto \log(A)$, such that, for a fixed total detector mass, the total energy deposited in the detector would be larger for nuclei with *smaller* $A$. Direct detection experiments that focus on protons and other light nuclei, such as Refs. [@Behnke:2016lsk; @Amole:2017dex; @Lippincott:2017yst; @Collar:2018ydf], may therefore be effective ways of constraining the landscape for model-dependent direct detection.
Existing limits on light mediators {#ssec:yukawa_med_lim}
----------------------------------
If there were no other constraints on $r_\phi$ or $\lambda_A$, Eq. would allow $\sigma^{\text{mt},(1)}_{\chi A}\gg10^{-25}\;\text{cm}^2$, albeit with a less useful scaling relation between different nuclei. However, because the light mediator couples to the Standard Model directly, other experiments already place constraints on such a particle. Figure \[fig:combine\_cross\] shows the maximum achievable $\sigma_{\chi N}$ and $\sigma^{\text{mt}}_{\chi N}$ for a repulsive Yukawa potential, conservatively using the perturbativity limit $\lambda_\chi=4\pi$, $\mu_N\approx m_p$, and $k=0.005\;\text{fm}^{-1}$. When Eq. is $>10^{-4}$, Fig. \[fig:combine\_cross\] uses the results from a numerical partial wave expansion with adaptive $l_\text{max}$, rather than the Born approximation. In practice, the Born approximation is adequate in the entire unconstrained region.
Including all such constraints, we have $\sigma_{\chi N}\lesssim10^{-27}\;\text{cm}^2$ and $\sigma^{\text{mt}}_{\chi N}\lesssim10^{-32}\;\text{cm}^2$. Constraints that rely on lower relative velocities, such as the cosmological constraints discussed in Sec. \[ssec:cosmology\], could achieve larger cross sections, but their constraints would need to be be scaled correctly to compare them to direct detection constraints. The momentum-transfer cross section is restricted to be $\sigma^{\text{mt}}_{\chi N}\lesssim10^{-25}\;\text{cm}^2$ even for velocities as low as $10^{-6}\,c$.
It is also possible to produce the light mediator in a collision [@Feng:2017uoz]. Particle production is an inelastic scattering process and beyond the scope of this paper, but it could be another avenue to transfer momentum between dark matter and a detector.
The detailed constraints in Fig. \[fig:combine\_cross\] could be different for different types of mediators. For example, for a vector mediator, the BBN constraints would be stronger [@light_mediator]. Other constraints might be weaker. However, $\sigma_{\chi N}\lesssim10^{-27}\;\text{cm}^2$ is already smaller than the geometric cross section of the nucleus, and circumventing individual constraints is unlikely to drastically change the overall conclusion that light mediators do not appear to be a promising approach to achieving large cross sections.
Composite Dark Matter {#sec:composite_dm}
=====================
Another mechanism for achieving a larger characteristic interaction radius is dark matter that is not a point particle, but instead has a finite physical extent [@Kusenko:1997si; @Khlopov:2005ew; @Khlopov:2008ty; @Krnjaic:2014xza; @Detmold:2014qqa; @Hochberg:2014kqa; @Jacobs:2014yca; @Hardy:2015boa; @Garcia:2015loa; @Hochberg:2015vrg; @Farrar:2017ysn; @DeLuca:2018mzn; @Graham:2018efk; @Hochberg:2018vdo; @Grabowska:2018lnd; @Coskuner:2018are]. Such dark matter could take the form of a composite particle. Because such dark matter would likely require an entire dark sector, any conclusions about the largest possible cross section with composite dark matter would be intrinsically model-dependent. Because the largest physical scale in the problem is no longer related to a property of the target nucleus, the cross section need not scale with $A$ at all.
The actual scaling with $A$ could only be determined by examining the particular model of composite dark matter. Additionally, achieving cross sections significantly larger than a nucleus with composite dark matter will always require $k r_{\text{dm}}\gtrsim 1$ for typical Milky Way virial velocities, so constraints on composite dark matter will need to be computed with a specific dark matter form factor in mind. See Sec. \[ssec:cosmology\] for discussion of limits at the lower velocities relevant to cosmological limits. Analyses setting constraints on specific form factors at large cross sections should consider whether their specific choice of form factor can be achieved at the cross sections they are constraining in a physically realistic model.
Therefore, limits on composite dark matter need to be calculated in specific models. Calculation of constraints on specific models of composite dark matter is left to future work.
Implications for Existing Constraints {#sec:implications}
=====================================
Figure \[fig:scattering\_scaling\] summarizes the approximate limits for the repulsive contact-interaction cross sections discussed in Sec. \[sec:contact\]. In the colored regions, the Born approximation begins to break down when the proton cross section is scaled to heavier nuclei, ultimately failing even for light nuclei. For point-like dark matter with a contact interaction, cross sections much larger than the geometric cross section are completely forbidden. As discussed in Sec. \[sec:light\_mediator\], the limits for a light mediator are similarly below the geometric cross section. For $m_\chi\gtrsim 10^{16}\;\text{GeV}$, the entire (small) exclusion region for underground detectors is affected by the failure of scaling relations. Future improvements to constraints could change the region where the entire exclusion region would fail. Additionally, all detectors’ computed ceilings are affected by the breakdown of scaling relations.
Scaling constraints
-------------------
In the regime where scaling relations are unreliable, it becomes more difficult to compare constraints between experiments. When the scaling relations fail, scaling constraints from different nuclei to the dark matter-nucleon cross section using the $A^4$ is no longer meaningful.
For both contact interactions and light mediators, as the cross section begins to saturate, the momentum-transfer cross section scales less than linearly with $A$. Therefore, at fixed total detector mass, there is more detectable momentum transfer into the detector for *lighter* target nuclei. The failure of the scaling relations also occurs at larger cross sections for smaller $A$. For example, a $^{12}$C-based detector would be able to use the Born approximation, and therefore the scaling relations, up to about 3000 times larger dark matter-nucleon cross section than a $^{131}$Xe-based detector. Therefore, robustly covering the large cross section regime may be best accomplished by detectors using light nuclei, e.g. [@Behnke:2016lsk; @Amole:2017dex; @Lippincott:2017yst; @Collar:2018ydf].
One option is to simply not scale constraints at large cross sections. While with resonances it could be possible for heavy nuclei to have smaller cross sections than a single nucleon, broadening by the dark matter velocity dispersion may limit the effect of narrow resonances on the overall detectable signature. Therefore, a relatively conservative approach could be to plot the actual momentum-transfer cross section constraints obtained from different nuclei on the same scale. In fact, if composite dark matter as discussed in Sec. \[sec:composite\_dm\] is indeed the most plausible strongly interacting dark matter candidate, disregarding scaling with $A$ may be the most correct way of plotting constraints.
![Summary of theoretically allowed regions for dark matter candidates. For a contact interaction, $A^4$ scaling breaks down for heavy nuclei for $\sigma_{\chi N}\gtrsim10^{-32}\;\text{cm}^2$, and by $\sigma_{\chi N}\gtrsim4\times10^{-28}$ any scaling between different nuclei is model dependent. Here we define the failure of scaling as setting the LHS of Eq. equal to $0.5$. This choice approximately agrees with where scaling obviously fails in Fig. \[fig:sigma\_sigma\]. The breakdown is purely on theoretical grounds. Also shown is the maximum allowed momentum-transfer cross section for a $m_\phi=10^{-4}\;\text{GeV}$ light mediator using the constraints shown in Fig. \[fig:combine\_cross\], coincidentally at a comparable scale. For $m_\chi\lesssim10^4\;\text{GeV}$ we have applied a conservative self interaction constraint $\sigma_{\chi\chi}/m_{\chi}<10\;\text{cm}^2/\text{g}$ [@Tulin:2017ara]. For $\sigma_{\chi N}\gtrsim10^{-25}\;\text{cm}^2$, no viable point-like dark matter candidates exist.[]{data-label="fig:scattering_scaling"}](figures/scattering_scaling.pdf "fig:"){width="\columnwidth"}\
Detection ceilings
------------------
Now we briefly consider if the detection ceilings (i.e., the largest cross sections that can be probed by a given detector based on the detector’s overburden) shown in Fig. \[fig:prexist\_limit\_simpl\] are preserved. In our simple model in Sec. \[sec:contact\], cross sections simply saturate at four times the geometric cross section for heavier nuclei. Even if all nuclei in the detector overburden have an elastic scattering cross section equal to their geometric cross section, dark matter cannot be stopped by the overburden above some $m_\chi$ [@Goodman:1984dc].
Because all currently computed detector ceilings exist at cross sections where the breakdown of the $A^4$ scaling is severe, correctly calculated detector ceilings must be specialized to a specific model. For basic energy-independent cross section scaling, the weakened ceilings likely lead to stronger direct detection constraints for $m_\chi\lesssim 10^{16}\;\text{GeV}$. For such models, direct detection may even have exhausted the parameter space for cross sections up to the largest cross sections achievable with point-like dark matter. For other dark matter form factors, the behavior around the ceiling could be more complicated.
Further work is required to make detailed adjustments to existing constraint contours to determine what dark matter parameter space has been constrained at large cross sections.
![Claimed constraints from Fig. \[fig:prexist\_limit\_simpl\], with the problematic regions identified in Fig. \[fig:scattering\_scaling\] highlighted. All existing detector ceiling calculations are deeply in the model-dependent regime, or entirely excluded for point-like dark matter. To the right of the dashed vertical line, the entire (small) direct-detection region must be reanalyzed.[]{data-label="fig:limit_summary"}](figures/limits_format_orange.pdf "fig:"){width="\columnwidth"}\
Dark matter-proton scattering constraints {#ssec:cosmology}
-----------------------------------------
Constraints that rely only on dark matter scattering directly with protons are not directly affected by the breakdown of scaling relations with $A$. These are primarily constraints from cosmology and astrophysics, although at least one laboratory experiment uses proton targets [@Collar:2018ydf]. Astrophysics constraints (e.g. disk stability, stars, cosmic ray interactions, gas clouds, etc) are typically assumed to occur at galactic virial velocities, as for direct detection. Cosmology constraints, such as CMB and structure formation constraints, typically assume collisions occur at smaller relative velocities.
As shown in Fig. \[fig:scattering\_scaling\], the cross sections of interest for cosmological/astrophysical constraints are too large to be point-like dark matter. Therefore, they should be reinterpreted as constraints on specific models of composite dark matter with a specified form factor, as discussed in Sec. \[sec:composite\_dm\].
For cosmology constraints set at lower relative velocities, the suppression of the cross section by the form factor of dark matter is not as severe. One consequence is that it is possible to achieve somewhat larger cross sections for point-like dark matter with a light mediator than those shown in Fig. \[fig:combine\_cross\], although even for velocities as low as $v\simeq 0.3\;\text{km/s}$, existing constraints would still require $\sigma^{\text{mt}}_{\chi N}\lesssim 10^{-25}\;\text{cm}^2$. However, invoking such a model would require additional caution, as direct detection constraints would not be scaled correctly relative to the cosmology constraints, such that it would no longer be appropriate to plot cosmology and direct detection constraints on the same axes, as done in Fig. \[fig:prexist\_limit\_simpl\].
Cosmological and astrophysical constraints set at masses $m_\chi<1\;\text{GeV}$, discussed in Sec. \[ssec:low\_mass\_dm\], are at lower cross sections, and may still be meaningful constraints on point-like dark matter. However, analyses at lower masses should either directly investigate how high their limits can be extrapolated, or make it much clearer that there are caveats in extrapolating their results to much larger masses.
Low-mass dark matter {#ssec:low_mass_dm}
--------------------
Because for $m_\chi\ll1\;\text{GeV}$, $\mu_A\simeq m_\chi$, low-mass dark matter constraints benefit only from a single factor of $A^2$ from coherence. Therefore, the loss of the $A^2$ scaling at large cross sections will be orders of magnitude less severe than the impact from the loss of $A^4$ scaling at larger masses. The momentum transfer is also smaller, so the loss of coherence due to an assumed form factor for the dark matter would be less severe. Contact interactions are still limited by the geometric size of the nucleus, but constraints on light mediators will become a function of $m_\chi$ [@light_mediator].
We leave a detailed assessment of the impact of our considerations at low mass to future work. However, we reiterate our caution that constraints set at low masses should carefully state the limitations on extrapolating their constraints to $m_\chi\gtrsim1\;\text{GeV}$.
Conclusions {#sec:disc_conc}
===========
How do dark matter particles interact with matter? One of the most commonly considered cases to probe is the spin-independent interactions of $m_\chi > 1\;\text{GeV}$ point-like dark matter with nuclei. In the literature, a vast array of constraints — based on astrophysical and cosmological tests, as well as direct-detection searches with a wide range of nuclei and overburdens — are all compared to each other in simple plots of the dark matter-nucleon cross section and dark-matter mass. Comparing searches in this way requires the assumption of scaling relations, e.g., $\sigma_{\chi A} \propto A^4 \sigma_{\chi N}$ for $m_\chi \gg m_A$, that are widely assumed to be model-independent.
We systematically examine the validity of the assumptions used to derive these relations, calculating where model independence ends. Figure \[fig:limit\_summary\] summarizes our results. We find:
1. For small cross sections, $\sigma_{\chi N}\ll10^{-32}\;\text{cm}^2$, the usual scaling relations are valid, and multiple reasonable models can produce the same scaling relation.
2. For $10^{-32}\;\text{cm}^2\lesssim\sigma_{\chi N}\lesssim10^{-25}\;\text{cm}^2$, the assumed $A^4$ scaling for a contact interaction progressively fails for all nuclear targets as cross sections for heavier nuclei begin to saturate at their geometric cross sections. Experimental constraints on the existence of light mediators prevent simple light mediator models from achieving cross sections in this range at all, such that constraints set in this range of cross sections should be specialized to a model.
3. For $\sigma_{\chi N}>10^{-25}\;\text{cm}^2$, dark matter cannot be point-like. Contact interactions cannot achieve cross sections larger than the geometric cross section $\sigma_{\chi A}\simeq 4\pi r_A^2$, and simple light mediators are strongly ruled out. Dark matter with cross sections in this range must be composite.
The failure of the scaling relations should influence the design of future dark matter searches. For interactions with cross sections that scale less than linearly with $A$, such as some models of composite dark matter, dark matter detectors with lighter nuclei are more efficient per unit detector mass. As a result, future direct-detection searches for strongly interacting dark matter may benefit from constructing detectors with light nuclei.
Constraints on dark matter parameter space are most useful if they can be compared between different experiments. Where the $A^4$ scaling is not reliable, results need to be recast in terms of specific models. A comprehensive analysis should include clear statements about the mass ranges their results can reasonably be extrapolated to. Because constraints will not be the same for different models, plots including cross sections $\sigma_{\chi N}\gtrsim10^{-32}\;\text{cm}^2$ must specify a model, whether it involves a contact interaction, light mediator, composite dark matter, or something else.
We are grateful for useful discussions with Laura Baudis, Juan Collar, Adrienne Erickcek, Vera Gluscevic, Rafael Lang, Hitoshi Murayama, Ethan Nadler, and Juri Smirnov.
MCD and CMH were supported by the Simons Foundation award 60052667, NASA award 15-WFIRST15-0008, and the US Department of Energy award DE-SC0019083. CVC and JFB are supported by NSF grant PHY-1714479 to JFB. AHGP is supported by NASA grant ATP 80NSSC18K1014 and NSF grant AST-1615838.
Lippmann-Schwinger Equation {#app:lippmann_schwinger}
===========================
It is useful to write $E=\frac{k^2}{2\mu}$, $U({\bf r})\equiv2\mu V(r)$ and rearrange Eq. $$\label{tise_green}
\left(\nabla^2_{\bf r}+k^2\right)\psi({\bf r}) =U({\bf r})\psi({\bf r}).$$ Recognizing Eq. as an inhomegenous Helmholtz equation, we can write the general solution in integral form [@arfken; @burke_joachain]:
$$\label{tise_integral}
\psi({\bf r})=\phi({\bf r})+\int G_0\left({\bf r},{\bf r'}\right)U({\bf r'})\psi({\bf r'})d{\bf r'},$$
where $G_0\left({\bf r},{\bf r'}\right)$ is the Green’s function for an outgoing wave in the Helmholtz equation: $$\label{helmholtz_green_def}
\left(\nabla^2_{\bf r}+k^2\right)G_0\left({\bf r},{\bf r'}\right)=\delta({\bf r}-{\bf r'})$$ and $\left(\nabla^2_{\bf r}+k^2\right)\phi({\bf r})=0$ is a homogeneous solution. The Green’s function is given [@arfken]: $$\label{helmholtz_green_sol}
G_0\left({\bf r},{\bf r'}\right)=-\frac{1}{4\pi|{\bf r}-{\bf r'}|}e^{ik|{\bf r}-{\bf r'}|}.$$ Plugging in an incident plane wave for the homogeneous solution, $\phi({\bf r})= (2\pi)^{-3/2}e^{i {\bf k}_i\cdot{\bf r}}$, where ${\bf k}_i\equiv k{\bf\hat{z}}$, we arrive at the Lippmann-Schwinger equation, $$\label{lippmann_schwinger}
\psi({\bf r})=(2\pi)^{-3/2}e^{i {\bf k}_i\cdot{\bf r}}-\int U({\bf r'})\psi({\bf r'})\frac{e^{ik|{\bf r}-{\bf r'}|}}{4\pi|{\bf r}-{\bf r'}|}d^3{\bf r'}.$$
Now, the goal here is to discover what measurable effect the potential has on the scattered wave. Physically, any measurement we make of the scattered wave must occur long after particle has finished interacting with the potential. Therefore, we may safely assume $|{\bf r}|\gg|{\bf r'}|$, such that $|{\bf r}-{\bf r'}|\xrightarrow[]{r\rightarrow\infty}r-{\bf\hat{r}}\cdot{\bf r'}+\mathcal{O}(r^{-1})$, such that, defining ${\bf k}_f\equiv k\cdot{\bf\hat{r}}$, Eq. becomes $$\begin{aligned}
\psi({\bf r})\xrightarrow[]{r\rightarrow\infty}&\psi_0({\bf r})-\frac{e^{ikr}}{4\pi r} \int U({\bf r'})\psi({\bf r'})e^{-i{\bf k}_f\cdot{\bf r'}}d^3{\bf r'} \nonumber\\
\equiv&\psi_0({\bf r})+(2\pi)^{-3/2}\frac{e^{ikr}}{r}f\left({\bf k}_i,{\bf k}_f\right),\label{lippmann_schwinger_larger}\end{aligned}$$ where $\psi_0({\bf r})\equiv(2\pi)^{-3/2}e^{i {\bf k}_i\cdot{\bf r}}$. Physically, this equation represents an incoming plane wave and a radially outgoing spherical wave with scattering amplitude $f\left({\bf k}_i,{\bf k}_f\right)=f\left(k,\theta\right)$.
Born Approximation {#app:born_der}
==================
Now we want to calculate an approximation to the scattering amplitude for a given potential. If we assume the potential is a perturbation to the incident wavefunction, we can attempt to solve Eq. by iteration:
$$\begin{aligned}
\psi({\bf r})\xrightarrow[]{r\rightarrow\infty}&\psi_0({\bf r})-\frac{e^{ikr}}{4\pi r} \int U({\bf r'})\psi({\bf r'})e^{-i{\bf k}_f\cdot{\bf r'}}d^3{\bf r'}\nonumber\\
=&\psi_0({\bf r})-\frac{e^{ikr}}{4\pi r} \int U({\bf r'})\left[\psi_0({\bf r'})-...\right]e^{-i{\bf k}_f\cdot{\bf r'}}d^3{\bf r'}\nonumber\\
=&\psi_0({\bf r})+(2\pi)^{-3/2}\frac{e^{ikr}}{r}(f^{(1)}\left({\bf k}_i,{\bf k}_f\right)+...),\label{lippmann_schwinger_iteration}\end{aligned}$$
where we have assumed the correction is small, such that higher-order corrections can be ignored. Then we can read off our approximation to $f\left({\bf k}_i,{\bf k}_f\right)$ from Eq. : $$\label{first_born_scattering}
f^{(1)}\left({\bf k}_i,{\bf k}_f\right)=-\frac{1}{4\pi}\int U({\bf r'})e^{i ({\bf k}_i-{\bf k}_f)\cdot{\bf r'}}d^3{\bf r'}.$$ $f^{(1)}\left({\bf k}_i,{\bf k}_f\right)$ is the first Born approximation to $f\left({\bf k}_i,{\bf k}_f\right)$.
Inspecting Eq. , we recognize that the first Born approximation of $f\left({\bf k}_i,{\bf k}_f\right)$ is nothing more than the Fourier transform of the potential. Defining ${\bf q}\equiv{\bf k}_i-{\bf k}_f$ such that $q=|{\bf q}|=2k\sin\left(\frac{\theta}{2}\right)$, and assuming the potential to be spherically symmetric $U({\bf r'})=U({r'})$, we can perform the angular integration to obtain: $$\label{first_born_fourier}
f^{(1)}\left({\bf k}_i,{\bf k}_f\right)=f\left(q\right)=-\frac{1}{q}\int_0^\infty U(r')\sin(qr')r' dr'.$$ Eq. is a useful starting point for analysis. However, before we begin using the result, we should clarify when the approximation breaks down. It can be shown robustly [@joachain75_collision] that a sufficient condition for the Born series to converge for all $k$ is that the magnitude of the potential would not be strong enough to support a bound state if it were purely attractive [@levinson1949uniqueness], which is to say $$\label{born_convergence_guarantee}
\int_0^\infty r|U(r)|dr<1.$$ A useful heuristic condition for the validity of the first Born approximation at a given $k$ can be obtained by simply assuming the first order correction term in Eq. must be small in the scattering region, such that $\psi({\bf r})\approx\psi_0({\bf r})$ near ${\bf r}=0$ [@sakurai; @khare2012introduction]. Therefore, we require: $$\frac{1}{4\pi}\left|\int U({\bf r'})e^{i {\bf k}_i\cdot{\bf r'}}\frac{e^{ik|{\bf r}-{\bf r'}|}}{|{\bf r}-{\bf r'}|}d^3{\bf r'}\right|\ll 1.$$ Taking ${\bf r}=0$, replacing ${\bf k}_i\cdot{\bf r'}=kr'\cos\theta'$, and performing the angular integration, we have
$$\label{final_cond_born}
\frac{1}{2k}\left|\int_0^\infty U(r')\left(e^{2ikr'}-1\right)dr'\right|\ll 1.$$
Once we have the scattering amplitude, we can calculate the total cross section as in Eq. .
Partial wave analysis {#app:partial_wave}
=====================
The general scattering amplitude for a spherically symmetric potential in Eq. can be written as an arbitrary expansion in of Legendre polynomials $P_l\left(\cos\theta\right)$ [@sakurai; @joachain75_collision; @burke_joachain]: $$\label{amplitude_partial}
f(k,\theta)=\frac{1}{k}\sum_{l=0}^\infty(2l+1)e^{i\delta_l}\sin\left(\delta_l\right) P_l\left(\cos\theta\right).$$ Given the phase shifts $\delta_l(k)$, the total elastic scattering cross section can be readily evaluated: $$\label{partial_wave_cross_section}
\sigma^{\text{tot}}=\frac{4\pi}{k^2}\sum_{l=0}^\infty{(2l+1)\sin^2\left(\delta_l\right)}.$$
A spherically symmetric wave function can be written as a linear combination of Bessel functions of the first and second kind, $j_l(k r)$ and $n_l(k r)$. The scattered part of the wave function for $r>R$, where $R$ is some arbitrarily large cutoff radius for the potential, can then also be expanded in terms of Legendre polynomials: $$\label{scattered_wave}
\psi_{\text{scattered}}(r,\theta)=\sum_{l=0}^\infty{A_l(r)P_l(\cos\theta)},$$ where $$A_l(r)=e^{i\delta_l}\left[\cos\left(\delta_l\right)j_l(k r)-\sin\left(\delta_l\right) n_l(k r)\right]$$ is the radial wave function for the $l$th partial wave. We can obtain $\delta_l(k)$ by enforcing continuity of the logarithmic derivative of the wave function, $$\label{wf_derivative}
\beta_l=\frac{r}{A_l}\frac{d A_l}{d r},$$ at $r=R$. We can obtain the wave function for $r<R$ by directly integrating the one-dimensional Schrödinger equation, $$\label{1d_schro}
\frac{d^2u_l}{d r^2}+\left(k^2-2\mu V(r)-\frac{l(l+1)}{r^2}\right)u_l(r)=0,$$ where we have defined $u_l(r)\equiv r A_l(r)$. Note that because we are matching the phase shift with the logarithmic derivative of the wave function, the overall normalization of $A_l(r)$ is irrelevant for our purposes and can be chosen arbitrarily. We can then obtain $A_l(R)$ for any arbitrary potential $V(r)$ by analytically or numerically evaluating Eq. from $r=0$ to $r=R$.
Once we have $A_l(R)$, we can obtain $\delta_l$ using $$\label{delta_l}
\tan(\delta_l)=\frac{k R A_l j'_l(k R)-(d A_l/d \ln{r})|_{r=R}j_l(k R)}{k R A_l n'_l(k R)-(d A_l/d \ln{r})|_{r=R}n_l(k R)}.$$ We have avoided canceling $A_l(r)$ in order to preserve signs, to ensure we obtain the correct quadrant for $\delta_l(k)$.
The boundary condition at $r=0$ should properly be $u_l(0)=0$, but for numerical solutions taking the boundary to be at $r=0$ is inconvenient because of the $1/r^2$ centrifugal term. Instead we can take advantage of the arbitrary normalization of $A_l(r)$ and fix the boundary conditions $u_l(r_{\text{min}})=1$, $d u_l/dr(r_{\text{min}})=(l+1)/r_{\text{min}}$, where $r_{\text{min}}$ is some small minimum radius.
As $k\rightarrow0$, it can be shown [@joachain75_collision] that generically $\delta_l(k)\propto k^{2l+1}$, except in special cases where it is possible to achieve $\delta_l(k)\propto k^{2l-1}$ for a specific value of $l$. Inspecting Eq. , we can see the contributions from $l=0$ and $l=0$ are the only values of $l$ which can be nonvanishing as $k\rightarrow0$. The $l=0$ cross section is called the s-wave cross section.
|
---
abstract: |
*Dedicated to our friends Sylvain Gallot and Albert Schwarz*
We analyze an upper bound on the curvature of a Riemannian manifold, using “${\operatorname{\sqrt{R}ic}}$" curvature, which is in between a sectional curvature bound and a Ricci curvature bound. (A special case of ${\operatorname{\sqrt{R}ic}}$ curvature was previously discovered by Osserman and Sarnak for a different but related purpose.) We prove that our ${\operatorname{\sqrt{R}ic}}$ bound implies Günther’s inequality on the candle function of a manifold, thus bringing that inequality closer in form to the complementary inequality due to Bishop.
author:
- 'Benoît R. Kloeckner'
- Greg Kuperberg
bibliography:
- 'dg.bib'
title: 'A refinement of Günther’s candle inequality'
---
[^1]
Introduction
============
Two important relations between curvature and volume in differential geometry are Bishop’s inequality [@BC:book §11.10], which is an upper bound on the volume of a ball from a lower bound on Ricci curvature, and Günther’s inequality [@Gunther:volume], which is a lower bound on volume from an upper bound on *sectional* curvature. Bishop’s inequality has a weaker hypothesis then Günther’s inequality and can be interpreted as a stronger result. The asymmetry between these inequalities is a counterintuitive fact of Riemannian geometry.
In this article, we will partially remedy this asymmetry. We will define another curvature statistic, the root-Ricci function, denoted ${\operatorname{\sqrt{R}ic}}$, and we will establish a comparison theorem that is stronger than Günther’s inequality[^2]. ${\operatorname{\sqrt{R}ic}}$ is not a tensor because it involves square roots of sectional curvatures, but it shares other properties with Ricci curvature.
After the first version of this article was written, we learned that a special case of ${\operatorname{\sqrt{R}ic}}$ was previously defined by Osserman and Sarnak [@OS:entropy], for the different but related purpose of estimating the entropy of geodesic flow on a closed manifold. (See [Section \[s:exp\]]{}.) Although their specific results are different, there is a common motivation arising from volume growth in a symmetric space.
Growth of the complex hyperbolic plane {#s:complex}
--------------------------------------
Consider the geometry of the complex hyperbolic plane ${\mathbb{C}\mathrm{H}}^2$. In this 4-manifold, the volume of a ball of radius $r$ is $${\operatorname{Vol}}(B(r)) = \frac{\pi^2}{2} \sinh(r)^4 \sim \frac{\pi^2}{32}\exp(4r).$$ The corresponding sphere surface volume has a factor of $\sinh(2r)$ from the unique complex line containing a given geodesic $\gamma$, which has curvature $-4$, and two factors of $\sinh(r)$ from the totally real planes that contain $\gamma$, which have curvature $-1$. Günther’s inequality and Bishop’s inequality yield the estimates $$\frac{\pi^2}{48}\exp(3\sqrt{2} r) \gtrsim
{\operatorname{Vol}}(B(r)) \gtrsim \frac{\pi^2}{12}\exp(3r).$$ The true volume growth of balls in ${\mathbb{C}\mathrm{H}}^2$ (and in some other cases, see [Section \[s:exp\]]{}) is governed by the average of the square roots of the negatives of the sectional curvatures. This is how we define the ${\operatorname{\sqrt{R}ic}}$ function, for each tangent direction $u$ at each point $p$ in $M$.
Root-Ricci curvature
--------------------
Let $M$ be a Riemannian $n$-manifold with sectional curvature $K \le
\rho$ for some constant $\rho \ge 0$; we will implicitly assume that $\rho\ge\kappa$. For any unit tangent vector $u \in UT_pM$ with $p \in M$, we define $${\operatorname{\sqrt{R}ic}}(\rho,u) {\stackrel{\mathrm{def}}{=}}{\operatorname{Tr}}(\sqrt{\rho-R(\cdot,u,\cdot,u)}).$$ Here $R(u,v,w,x)$ is the Riemann curvature tensor expressed as a tetralinear form, and the square root is the positive square root of a positive semidefinite matrix or operator.
The formula for ${\operatorname{\sqrt{R}ic}}(\rho,u)$ might seem arcane at first glance. Regardless of its precise form, the formula is both local ([i.e.]{}, a function of the Riemannian curvature) and also optimal in certain regimes. Any such formula is potentially interesting. One important, simpler case is $\rho=0$, which applies only to non-positively curved manifolds: $${\operatorname{\sqrt{R}ic}}(0,u) = {\operatorname{Tr}}(\sqrt{-R(\cdot,u,\cdot,u)}).$$ In other words, ${\operatorname{\sqrt{R}ic}}(0,u)$ is the sum of the square roots of the sectional curvatures $-K(u,e_i)$, where $(e_i)$ is a basis of $u^\perp$ that diagonalizes the Riemann curvature tensor. This special case was defined previously by Osserman and Sarnak [@OS:entropy] ([Section \[s:exp\]]{}), which in their notation would be written $-\sigma(u)$.
For example, when $M={\mathbb{C}\mathrm{H}}^2$, one sectional curvature $K(u,e_i)$ is $-4$ and the other two are $-1$, so $${\operatorname{\sqrt{R}ic}}(0,u)= \sqrt{4}+\sqrt{1}+\sqrt{1} = 4,$$ which matches the asymptotics in [Section \[s:complex\]]{}.
In the general formula ${\operatorname{\sqrt{R}ic}}(\rho,u)$, the parameter $\rho$ is important because it yields sharper bounds at shorter length scales. In particular, in the limit $\rho\to\infty$, ${\operatorname{\sqrt{R}ic}}(\rho,u)$ becomes equivalent to Ricci curvature. [Section \[s:relations\]]{} discusses other ways in which ${\operatorname{\sqrt{R}ic}}$ fits the framework of classical Riemannian geometry. Our definition for general $\rho$ was motivated by our proof of the refined Günther inequality, more precisely by equation . The energy of a curve in a manifold can be viewed as linear in the curvature $R(\cdot,u,\cdot,u)$. We make a quadratic change of variables to another matrix $A$, to express the optimization problem as quadratic minimization with linear constraints; and we noticed an allowable extra parameter $\rho$ in the quadratic change of variables.
Another way to look at root-Ricci curvature is that it is equivalent to an average curvature, like the normalized Ricci curvature ${\operatorname{Ric}}/(n-1)$, but after a reparameterization. By analogy, the $L^p$ norm of a function, or the root-mean-square concept in statistics, is also an average of quantities that are modified by the function $f(x) = x^p$. In our case, we can obtain a type of average curvature which is equivalent to ${\operatorname{\sqrt{R}ic}}$ if we conjugate by $f(x) = \sqrt{\rho - x}$. Taking this viewpoint, we say that the manifold $M$ is of *${\operatorname{\sqrt{R}ic}}$ class $(\rho,\kappa)$* if $K \le \rho$, and if also $$\frac{{\operatorname{\sqrt{R}ic}}(\rho,u)}{n-1} \ge \sqrt{\rho-\kappa}$$ for all $u \in UTM$. This is the ${\operatorname{\sqrt{R}ic}}$ curvature analogue of the sectional curvature condition $K \le \kappa$.
A general candle inequality
---------------------------
The best version of either Günther’s or Bishop’s inequality is not directly a bound on the volume of balls in $M$, but rather a bound on the logarithmic derivative of the *candle function* of $M$. Let $\gamma=\gamma_u$ be a geodesic curve in $M$ that begins at $p = \gamma(0)$ with initial velocity $u\in UT_pM$. Then the candle function $s(\gamma,r)$ is by definition the Jacobian of the map $u\mapsto \gamma_u(r)$. In other words, it is defined by the equations $${\mathrm{d}}q = s(\gamma_u,r)\, {\mathrm{d}}u \,{\mathrm{d}}r \qquad q = \gamma_u(r) = \exp_p(ru),$$ where ${\mathrm{d}}q$ is Riemannian measure on $M$, ${\mathrm{d}}r$ is Lebesgue measure on ${\mathbb{R}}$, and ${\mathrm{d}}u$ is Riemannian measure on the sphere $UT_pM$. This terminology has the physical interpretation that if an observer is at the point $q$ in $M$, and if a unit candle is at the point $p$, then $1/s(\gamma,r)$ is its apparent brightness[^3].
The candle function $s_\kappa(r)$ of a geometry of constant curvature $\kappa$ is given by $$s_{\kappa}(r) = \begin{cases} \left(\frac{\sin(\sqrt{\kappa}r)}
{\sqrt{\kappa}}\right)^{n-1} & \kappa > 0 \\
r^{n-1} & \kappa = 0 \\ \left(\frac{\sinh(\sqrt{-\kappa}r)}
{\sqrt{-\kappa}}\right)^{n-1} & \kappa < 0
\end{cases}.$$
Let $M$ be a Riemannian $n$-manifold is of ${\operatorname{\sqrt{R}ic}}$ class $(\rho,\kappa)$ for some $\kappa \le \rho \ge 0$. Then $$(\log s(\gamma,r))' \ge (\log s_{\kappa}(r))'$$ for every geodesic $\gamma$ in $M$, when $2r\sqrt{\rho} \le \pi$. \[th:main\]
The prime denotes the derivative with respect to $r$.
When $\rho = 0$, the conclusion of [Theorem \[th:main\]]{} is identical to Günther’s inequality for manifolds with $K \le \kappa$, but the hypothesis is strictly weaker. When $\rho > 0$, the curvature hypothesis is weaker still, but the length restriction is stronger. The usual version of the inequality holds up to a distance of $\pi/\sqrt{\kappa}$. For our distance restriction, we replace $\kappa$ with $\rho$ and divide by 2.
The rest of this article is organized as follows. In [Section \[s:relations\]]{} we give several relations between curvature bounds and volume comparisons. In [Section \[s:applications\]]{} we list applications of [Theorem \[th:main\]]{}, and we prove [Theorem \[th:main\]]{} in [Section \[s:proof\]]{}.
The authors would like to thank Sylvain Gallot and John Hunter for especially helpful conversations.
Relations between conditions {#s:relations}
============================
Candle conditions
-----------------
We first mention two interesting properties of the candle function $s(\gamma,r)$:
1. $s(\gamma,r)$ vanishes when $\gamma(0)$ and $\gamma(r)$ are conjugate points.
2. The candle function is symmetric: If $\bar{\gamma}(t) =
\gamma(r-t)$, then $s(\bar{\gamma},r) = s(\gamma,r).$
The second property is not trivial to prove, but it is a folklore fact in differential geometry [@Yau:isoperimetric]\[Lem. 5\] (and a standard principle in optics).
Say that a manifold $M$ is ${\operatorname{Candle}}(\kappa)$ if the inequality $$s(\gamma,r) \ge s_\kappa(r)$$ holds for all $\gamma,r$; or ${\operatorname{LCD}}(\kappa)$, for *logarithmic candle derivative* [^4], if the logarithmic condition $$(\log s(\gamma,r))' \ge (\log s_\kappa(r))'$$ holds for all $\gamma,r$; or ${\operatorname{Ball}}(\kappa)$ if the volume inequality $${\operatorname{Vol}}(B(p,r)) \ge {\operatorname{Vol}}(B_\kappa(r))$$ holds for all $p$ and $r$; here $B_\kappa$ denotes a ball in the simply connected space of constant curvature $\kappa$. (If $\kappa
> 0$, then the first two conditions are only meaningful up to the distance $\pi/\sqrt{\kappa}$ between conjugate points in the comparison geometry.) We also write ${\operatorname{Candle}}(\kappa,\ell)$, ${\operatorname{LCD}}(\kappa,\ell)$, and ${\operatorname{Ball}}(\kappa,\ell)$ if the same conditions hold up to a distance of $r = \ell$.
The logarithmic derivative $(\log s(\gamma,r))'$ of the candle function has its own important geometric interpretation: it is the mean curvature of the geodesic sphere with radius $r$ and center $p = \gamma(0)$ at the point $\gamma(r)$. So it also equals $\Delta r$, where $\Delta$ is the Laplace Beltrami operator, and $r$ is the distance from any point to $p$. So if $M$ is ${\operatorname{LCD}}(\kappa)$, then we obtain the comparison $\Delta r \ge
\Delta_\kappa r_\kappa$, and the statement that spheres in $M$ are more extrinsically curved than spheres in a space of constant curvature $\kappa$.
Curvature and volume comparisons
--------------------------------
If $\kappa \le \rho = 0$, then we can organize the comparison properties of an $n$-manifold $M$ that we have mentioned as follows: $$\begin{gathered}
K \le \kappa \implies \text{${\operatorname{\sqrt{R}ic}}$ class $(0,\kappa)$} \implies
{\operatorname{LCD}}(\kappa) \\ \implies {\operatorname{Candle}}(\kappa) \implies {\operatorname{Ball}}(\kappa,{\operatorname{inj}}(M)),
\label{e:implies}\end{gathered}$$ where ${\operatorname{inj}}(M)$ is the injectivity radius of $M$. The first implication is elementary, while the second one is [Theorem \[th:main\]]{}. The third and fourth implications are also elementary, given by integrating with respect to length $r$.
If $\kappa \le \rho > 0$, then $$\begin{gathered}
K \le \kappa \implies \text{${\operatorname{\sqrt{R}ic}}$ class $(\rho,\kappa)$} \implies
{\operatorname{LCD}}(\kappa,\frac{\pi}{2\sqrt{\rho}}) \\
\implies {\operatorname{Candle}}(\kappa,\frac{\pi}{2\sqrt{\rho}})
\implies {\operatorname{Ball}}(\kappa,\ell),\end{gathered}$$ where $$\ell=\min({\operatorname{inj}}(M),\frac{\pi}{2\sqrt{\rho}}).$$
Finally, for all $\ell>0$, $${\operatorname{Candle}}(\kappa,\ell) \implies {\operatorname{Ric}}\le (n-1)\kappa g,$$ where $g$ is the metric on $M$, because [$$\label{e:deriv}s(\gamma,r) = r^{n-1} - {\operatorname{Ric}}(\gamma'(0))r^n + O(r^{n+1}).$$]{} In particular, in two dimensions, all of the implications in are equivalences.
Curvature bounds
----------------
The function ${\operatorname{\sqrt{R}ic}}(\rho)$ increases with $\rho$ faster than $$(n-1)\sqrt{\rho-\kappa}$$ in the sense that for all $\kappa\le \rho \le \rho'$, $$\text{${\operatorname{\sqrt{R}ic}}$ class $(\rho,\kappa)$} \implies \text{${\operatorname{\sqrt{R}ic}}$ class $(\rho',\kappa)$}.$$ In addition, the conjugate version of root-Ricci curvature converges to normalized Ricci curvature for large $\rho$: $$\lim_{\rho \to \infty} \rho-\left(\frac{{\operatorname{\sqrt{R}ic}}(\rho,u)}{n-1}\right)^2
= \frac{{\operatorname{Ric}}(u,u)}{n-1} \quad \forall u\in UTM.$$ The corresponding limit $\rho \to \infty$ in [Theorem \[th:main\]]{} has the interpretation that the upper bound looks more and more like a bound based on Ricci curvature at short distances. This is an optimal limit in the sense that Ricci curvature is the first non-trivial derivative of $s(\gamma,r)$ at $r=0$ by . On the other hand, without the length restriction, the limit $\rho \to \infty$ is impossible. That limit would be exactly Günther’s inequality with Ricci curvature, but such an inequality is not generally true.
Finally we can deduce a root-Ricci upper bound from a combination of sectional curvature and Ricci bounds. The concavity of the square root function implies that given the value of ${\operatorname{Ric}}(u,u)$, the weakest possible value of ${\operatorname{\sqrt{R}ic}}(\rho,u)$ is achieved when $R(\cdot,u,\cdot,u)$ has one small eigenvalue and all other eigenvalues equal. For all $\kappa \le
\alpha \le \rho$, we then get a number $\beta=\beta(\kappa,\alpha,\rho)$, decreasing in $\alpha$, such that [$$\label{e:mixed}K \le \alpha \text{ and } {\operatorname{Ric}}\le \beta g
\implies \text{${\operatorname{\sqrt{R}ic}}$ class $(\rho,\kappa)$}.$$]{} An explicit computation yields the optimal value $$\beta = \rho+(n-2)\alpha - \left((n-1)\sqrt{\rho-\kappa}
- (n-2)\sqrt{\rho-\alpha}\right)^2.$$ In particular, $$\begin{aligned}
\beta(\kappa,\rho,\rho) &= (n-1)^2\kappa-n(n-1)\rho\\
\beta(\kappa,\kappa,\rho) &= (n-1)\kappa.\end{aligned}$$ In order to deduce ${\operatorname{\sqrt{R}ic}}$ class $(\rho,\kappa)$ from classical curvature upper bounds, we can therefore ask for the strong condition $K \le \kappa$ (which implies ${\operatorname{Ric}}\le (n-1)\kappa g$), or ask for the weaker $K \le
\rho$ together with ${\operatorname{Ric}}\le \beta(\kappa,\rho,\rho) g$, or choose from a continuum of combined bounds on $K$ and ${\operatorname{Ric}}$. Moreover, the above calculation holds pointwise, so that in , $\alpha$ can be a function on $UTM$ instead of a constant.
Applications {#s:applications}
============
Most of the established applications of Günther’s inequality are also applications of [Theorem \[th:main\]]{}. The subtlety is that different applications use different criteria in the chain of implications . We give some examples. In general, let ${\tilde{M}}$ denote the universal cover of $M$.
Exponential growth of balls {#s:exp}
---------------------------
One evident application of our result is to estimate the rate of growth of balls, as already given by . This is related to the volume entropy of a closed Riemannian manifold $M$, which is by definition $${\operatorname{h_{\mathrm{vol}}}}(M) {\stackrel{\mathrm{def}}{=}}\lim_{r \to +\infty} \frac{\log {\operatorname{Vol}}B_{{\tilde{M}}}(p,r)}{r}.$$ By abuse of notation, we will use this same volume entropy expression when $M = {\tilde{M}}$ is simply connected rather than closed. Since a hyperbolic space of curvature $\kappa<0$ and dimension $n$ has volume entropy $(n-1)\sqrt{-\kappa}$, [Theorem \[th:main\]]{} implies that when $K \le 0$, [$$\label{e:vol}{\operatorname{h_{\mathrm{vol}}}}(M) \ge \alpha {\stackrel{\mathrm{def}}{=}}\inf_u {\operatorname{\sqrt{R}ic}}(0,u).$$]{} The estimate is sharp for every rank one symmetric space. (Recall that the rank one symmetric spaces are the generalized hyperbolic spaces ${{\mathbb{R}}\mathrm{H}}^n$, ${\mathbb{C}\mathrm{H}}^n$, ${\mathbb{H}\mathrm{H}}^n$, and ${\mathbb{O}\mathrm{H}}^2$.) The reason is that the operator $R(\cdot,\gamma',\cdot,\gamma')$ is constant along any geodesic $\gamma$. So by the Jacobi field equation ([Section \[s:proof\]]{}), the volume of $B(p,r)$ has factors of $\sinh \sqrt{\lambda_k} r$ for each eigenvalue $\lambda_k$ of $R(\cdot,\gamma',\cdot,\gamma')$. So we obtain the estimate $${\operatorname{Vol}}B(p,r) \propto \prod_k (\sinh \sqrt{\lambda_k} r)
\sim \exp(\alpha r).$$
However, although is a good estimate, it is superseded by the previous discovery of ${\operatorname{\sqrt{R}ic}}(0,u)$, for the specific purpose of estimating entropies. In addition to the volume entropy of $M$, the geodesic flow on $M$ has a topological entropy ${\operatorname{h_{\mathrm{top}}}}(M)$ and a measure-theoretic entropy ${\operatorname{h_\mu}}(M)$ with respect to any invariant measure $\mu$. Manning [@Manning:entropy] showed that ${\operatorname{h_{\mathrm{top}}}}(M) \ge {\operatorname{h_{\mathrm{vol}}}}(M)$ for any closed $M$, with equality when $M$ is nonpositively curved. Goodwyn [@Goodwyn:compare] showed that ${\operatorname{h_{\mathrm{top}}}}(M) \ge {\operatorname{h_\mu}}(M)$ for any $\mu$, with equality for the optimal choice of $\mu$. (In fact he showed this for any dynamical system.)
With these background facts, Osserman and Sarnak [@OS:entropy] defined ${\operatorname{\sqrt{R}ic}}(0,u)$ and established that [$$\label{e:OSBW}{\operatorname{h_\mu}}(M) \ge \int_{UTM} {\operatorname{\sqrt{R}ic}}(0,u) d\mu(u)$$]{} when $M$ is negatively curved, [i.e.]{}, $K \le \kappa < 0$, and $\mu$ is normalized Riemannian measure on $UTM$. This result was generalized to non-positive curvature by Ballmann and Wojtkowski [@BW:entropy].
This use of ${\operatorname{\sqrt{R}ic}}$ curvature concludes a topic that began with the Schwarz-Milnor theorem [@Milnor:curvature; @Schwarz:volume] that if $M$ is negatively curved, then $\pi_1(M)$ has exponential growth. Part of their result is that if $M$ is compact, then $\pi_1(M)$ has exponential growth if and only if ${\operatorname{h_{\mathrm{vol}}}}(M) > 0$. So equation , together with Manning’s theorem, shows that if $M$ is compact and nonpositively curved, then either $M$ is flat, or the growth of $\pi_1(M)$ is bounded below by .
Ballmann [@Ballmann:lectures] also showed that a non-positively curved manifold $M$ of finite volume satisfies the weak Tits alternative: either $M$ is flat, or its fundamental group contains a non-abelian free group. This is qualitatively a much stronger version of the Schwarz-Milnor theorem, and even its extension due to Manning, Osserman, Sarnak, Ballmann, and Wojtkowski.
Isoperimetric inequalities
--------------------------
Yau [@Yau:isoperimetric; @BZ:inequalities] established that if $M$ is complete, simply connected, and has $K \le \kappa < 0$, and $D \subseteq M$ is a domain, then $D$ satisfies a linear isoperimetric inequality: $${\operatorname{Vol}}(\partial D) \ge (n-1)\sqrt{-\kappa}{\operatorname{Vol}}(D).$$ His proof only uses a weakening of condition ${\operatorname{LCD}}(\kappa)$, namely that $$(\log s(\gamma,r))' \ge (n-1)\sqrt{-\kappa}.$$ So [Theorem \[th:main\]]{} yields Yau’s inequality when $M$ is of ${\operatorname{\sqrt{R}ic}}$ class $(0,\kappa)$.
McKean [@McKean:spectrum] showed that the same weak ${\operatorname{LCD}}(\kappa)$ condition also implies a spectral gap $$\lambda_0(\tilde M) \ge \frac{-\kappa n^2}{4}$$ for the first eigenvalue of the positive Laplace-Beltrami operator acting on $L^2(M)$. This spectral gap follows from a Poincaré inequality that is independently interesting: $$\int_M f^2 \le \frac{4}{-\kappa n^2} \int_M |\nabla f|^2$$ for all smooth, compactly supported functions $f$. McKean stated his result under the hypothesis $K \le \kappa$; it has been generalized by Setti [@Setti:ricci] and Borbély [@Borbely:spectrum] to mixed sectional and Ricci bounds; [Theorem \[th:main\]]{} provides a further generalization. Note in particular that Borbély’s result is optimal for complex hyperbolic spaces (and we get the same bound in this case), but we get better bounds for quaternionic and octonionic hyperbolic spaces.
Croke [@Croke:sharp] establishes the isoperimetric inequality for a compact non-positively curved 4-manifold $M$ with unique geodesics. In other words, if $B$ is a Euclidean 4-ball with $${\operatorname{Vol}}(M) = {\operatorname{Vol}}(B),$$ then $${\operatorname{Vol}}(\partial M) \ge {\operatorname{Vol}}(\partial B).$$ His proof only uses the condition ${\operatorname{Candle}}(0)$, in fact only for maximal geodesics between boundary points[^5]. So, Croke’s theorem also holds if $M$ is of ${\operatorname{\sqrt{R}ic}}$ class $((\frac{\pi}{2 L})^2,0)$, where $L$ is the maximal length of a geodesic; for any given $L$, this curvature bound is weaker than $K \le 0$. It is a well-known conjecture that if $M$ is $n$-dimensional and non-positively curved, then the isoperimetric inequality holds. The conjecture can be attributed to Weil [@Weil:negative], because his proof in dimension $n=2$ initiated the subject. More recently, Kleiner [@Kleiner:isoperimetric] established the case $n=3$. We are led to ask whether Weil’s isoperimetric conjecture still holds for ${\operatorname{Candle}}(0)$ or ${\operatorname{LCD}}(0)$ manifolds.
In a forthcoming paper, we will partially generalize Croke’s result to signed curvature bounds. In these generalizations, the main direct hypotheses are the ${\operatorname{Candle}}(\kappa)$ and ${\operatorname{LCD}}(\kappa)$ conditions, which are natural but not local. [Theorem \[th:main\]]{} provides important local conditions under which these hypotheses hold.
Almost non-positive curvature
-----------------------------
As mentioned above, one strength of root-Ricci curvature estimates is that we can adjust the parameter $\rho$; however, most of the applications mentioned so far are in the non-positively curved case $\rho = 0$. It is therefore natural to ask to which extent manifolds with almost non-positive sectional curvature and negative root-Ricci curvature behave like negatively curved manifolds.
More precisely, suppose that $M$ is compact, has diameter $\delta$ and satisfies both curvature bounds $$K\le \rho \quad \mbox{and} \quad {\operatorname{\sqrt{R}ic}}(\rho) \le \kappa.$$ Say that $M$ is *almost non-positively curved* if $0<\rho\ll
\delta^{-2}$, and that $M$ is *strongly negatively root-Ricci curved* if $\kappa\ll -\delta^{-2}$. Under these assumptions, [Theorem \[th:main\]]{} shows that the balls in ${\tilde{M}}$ grow exponentially up to a large multiple of the diameter $\delta$. We conjecture that if $M$ is also compact, then $\pi_1(M)$ has exponential growth or equivalently that $M$ has positive volume entropy.
In light of Ballmann’s result that a non-positively curved manifold $M$ of finite volume satisfies the weak Tits alternative, we ask whether a compact, almost-non-positively curved, strongly negatively root-Ricci curved manifold must contain a non-abelian free group in its fundamental group. We conjecture at the very least that an almost non-positively curved manifold with strongly negative root-Ricci curvature cannot be a torus. This would be an interesting complement to the result of Lohkamp [@Lohkamp:metrics] that every closed manifold of dimension $n \ge 3$ has a Ricci-negative metric.
The proof {#s:proof}
=========
In this section, we will prove [Theorem \[th:main\]]{}. The basic idea is to analyze the energy functional that arises in a standard proof of Günther’s inequality, with the aid of the change of variables $R = A^2 - \rho I$.
Using the Jacobi field model, [Theorem \[th:main\]]{} is really a result about linear ordinary differential equations. The normal bundle to the geodesic $\gamma(t)$ can be identified with ${\mathbb{R}}^{n-1}$ using parallel transport. Then an orthogonal vector field $y(t)$ along $\gamma$ is a Jacobi field if it satisfies the differential equation [$$\label{e:vector}y'' = -R(t)y,$$]{} where $$R(t) = R(\cdot,u(t),\cdot,u(t))$$ is the sectional curvature matrix and $u(t) = \gamma'(t)$ is the unit tangent to $\gamma$ at time $t$. By the first Bianchi identity, $R(t)$ is a symmetric matrix. The candle function $s(r) = s(\gamma,r)$ is determined by a matrix solution [$$\label{e:matrix}Y'' = -R(t)Y \qquad Y(0) = 0$$]{} by the formula $$s(r) = \frac{\det Y(r)}{\det Y'(0)}.$$ Its logarithmic derivative is given by $$(\log s(r))' = \frac{s'(r)}{s(r)}
= \frac{(\det Y)'(r)}{\det Y(r)}.$$ All invertible solutions $Y(r)$ to are equivalent by right multiplication by a constant matrix, and yield the same value for $s(r)$ and its derivative. In particular, if we let $Y(r) = I$, then the logarithmic derivative simplifies to $$(\log s(r))' = {\operatorname{Tr}}(Y'(r)).$$
Following a standard proof of Günther’s inequality [@GHL:riemannian]\[Thm. 3.101\], we define an energy functional whose minimum, remarkably, both enforces and minimizes the objective $(\log s(r))'$. Namely, we assume Dirichlet boundary conditions $$y(0) = 0 \qquad y(r) = v,$$ and we let [$$\label{e:energy}E(R,y) = \int_0^r \left({\langle {y',y'} \rangle}
- {\langle {y,Ry} \rangle}\right) \,{\mathrm{d}}t.$$]{} By a standard argument from calculus of variations, the critical points of $E(R,y)$ are exactly the solutions to with the given boundary conditions.
We can repeat the same calculation with the matrix solution $$Y(0) = 0 \qquad Y(r) = I,$$ with the analogous energy $$E(R,Y) = \int_0^r \left({\langle {Y',Y'} \rangle} - {\langle {Y,RY} \rangle}\right) \,{\mathrm{d}}t.$$ Here the inner product of two matrices is the Hilbert-Schmidt inner product $${\langle {A,B} \rangle} = {\operatorname{Tr}}(A^TB).$$ Moreover, if $Y$ is a solution to , then $E(R,Y)$ simplifies to $(\log s(r))'$ by integration by parts: $$\begin{aligned}
E(R,Y) &= \int_0^r \left({\langle {Y',Y'} \rangle} - {\langle {Y,RY} \rangle}\right) \,{\mathrm{d}}t \\
&= {\langle {Y(r),Y'(r)} \rangle} - {\langle {Y(0),Y'(0)} \rangle}
- \int_0^r {\langle {Y,Y''+RY} \rangle} \,{\mathrm{d}}t \\
&= {\langle {I,Y'(r)} \rangle} - 0 - 0 = {\operatorname{Tr}}(Y'(r)) = (\log s(r))'.\end{aligned}$$ Thus, our goal is to minimize $E(R,Y)$ with respect to both $Y$ and $R$. We want to minimize with respect to $Y$ in order to solve . Then for that $Y$, we want to minimize with respect to $R$ to prove [Theorem \[th:main\]]{}.
The following proposition tells us that or has a unique solution with Dirichlet boundary conditions, and that it is an energy minimum. Here and below, recall the matrix notation $A \le B$ (which was already used for Ricci curvature in the introduction) to express the statement that $B-A$ is positive semidefinite.
If $R \le \rho I$, and if $y$ is continuous with an $L^2$ derivative, then $E(R,y)$ is a positive definite quadratic function of $y$ when $\sqrt{\rho}r < \pi$, with the Dirichlet boundary conditions $y(0)
= y(r) = 0$. \[p:posdef\]
Let $$E(\rho,y) = E(\rho I,y) = \int_0^r \left({\langle {y',y'} \rangle}
- \rho{\langle {y,y} \rangle}\right)\,{\mathrm{d}}t$$ be the corresponding energy of the comparison case with constant curvature $\rho$. (Recall that the ultimate comparison is with constant curvature $\kappa$, but to get started we use $\rho$ instead.) Then $$E(\rho,y) \le E(R,y),$$ so it suffices to show that $E(\rho,y)$ is positive definite. When $\rho
= 0$, $E(\rho,y)$ is manifestly positive definite. Otherwise $E(\rho,y)$ is diagonalized in the basis of functions $$y_k(t) = \sin(\frac{\pi k t}{r})$$ with $k \ge 1$. A direct calculation yields $$E(\rho,y_k) = \frac{\pi^2 k^2-r^2\rho}{r} > 0,$$ as desired.
There is also a geometric reason that the comparison case $E(\rho,y)$ is positive definite: When $\rho = 0$, a straight line segment in Euclidean space is a minimizing geodesic; when $\rho > 0$, the same is true of a geodesic arc of length $r < \pi/\sqrt{\rho}$ on a sphere with curvature $\sqrt{\rho}$. We give a direct calculation to stay in the spirit of ODEs.
Let $\rho$ and $r < \pi/\sqrt{\rho}$ be fixed and suppose that $R \le \rho I$. Then $s(r)$ and $(\log s(r))'$ are both bounded below. \[p:bound1\]
We will simply prove the usual Günther inequality. As in the proof of [Proposition \[p:posdef\]]{}, $$E(R,Y) \ge E(\rho,Y)$$ for all $R$ and $Y$ with $Y(0) = 0$ and $Y(r) = I$. For each fixed $R$, the minimum of the left side is $(\log s(r))'$. The minimum of the right side (which may occur for a different $Y$, but no matter) is $(\log s_{\rho}(r))'$, which is a positive number. We obtain the same conclusion for $s(r)$ by integration.
Assume the hypotheses of [Proposition \[p:bound1\]]{}. If $R$ is $L^\infty$, then the solution $Y$ to is bounded uniformly, [i.e.]{}, with a bound that depends only on $||R||$ (and $r$ and $\rho$). Also $Y'$ is uniformly bounded and Lipschitz, and $Y''$ is uniformly bounded and $L^\infty$. \[p:bound2\]
In this proposition and nowhere else, it is more convenient to assume the initial conditions $${\hat{Y}}(0) = 0 \qquad {\hat{Y}}'(0) = I$$ rather than Dirichlet boundary conditions. The fact that ${\hat{Y}}$ and its derivatives are uniformly bounded, with these initial conditions, is exactly Grönwall’s inequality. To convert back to Dirichlet boundary conditions, we want to instead bound $$Y(t) = {\hat{Y}}(t){\hat{Y}}(r)^{-1}.$$ This follows from [Proposition \[p:bound1\]]{} by the formula $${\hat{Y}}(r)^{-1} = {\operatorname{adj}}({\hat{Y}}(r))\det({\hat{Y}}(r))^{-1},$$ where ${\operatorname{adj}}$ denotes the adjugate of a matrix.
Finally, $Y''(t)$ is $L^\infty$ and uniformly bounded because $Y(t)$ satisfies . Also $Y'(0) = {\hat{Y}}(r)^{-1}$ is uniformly bounded, so we can integrate to conclude that $Y'(t)$ is uniformly bounded and Lipschitz.
To prove [Theorem \[th:main\]]{}, we want to minimize $(\log s(r))'$ or $E(R,Y)$ over all $R$ such that [$$\label{e:Rcond}R \le \rho I \qquad {\operatorname{Tr}}(\sqrt{\rho I-R})
\ge \alpha {\stackrel{\mathrm{def}}{=}}\sqrt{\rho-\kappa}.$$]{} To better understand this minimization problem, we make a change of variables. Let $A(t)$ be a symmetric matrix such that [$$\label{e:A}R(t) = \rho I - A(t)^2 \qquad {\operatorname{Tr}}(A(t)) \ge \alpha.$$]{} In order to know that every $R(t)$ is realized, we can let $$A = \sqrt{\rho I-R}$$ be the positive square root of $\rho I-R$. Even if $A$ is not positive semidefinite, $R(t)$ still satisfies . This simplifies the optimization problem: in the new variable $A$, the semidefinite hypothesis can be waived.
Now the energy function becomes: $$\begin{aligned}
E(A,Y) &= \int_0^r \left({\langle {Y',Y'} \rangle}
- {\langle {Y,(\rho-A^2)Y} \rangle}\right)\,{\mathrm{d}}t \\
&= \int_0^r \left( {\operatorname{Tr}}((Y')^TY') + {\operatorname{Tr}}(Y^TA^2Y)
- \rho{\operatorname{Tr}}(Y^TY)\right)\,{\mathrm{d}}t.\end{aligned}$$ For the moment, fix $Y$ and let $Z = YY^T$. Then as a function of $A$, $$E(A) = \int_0^r {\operatorname{Tr}}(A^2YY^T)\, {\mathrm{d}}t + \text{constant}.$$ Since $YY^T$ is symmetric and strictly positive definite, $E$ is a positive-definite quadratic function of $A$, and we can directly solve for the minimum as [$$\label{e:AYY}A = \frac{\alpha (YY^T)^{-1}}{{\operatorname{Tr}}((YY^T)^{-1})}.$$]{} Even though we waived the assumption that $A$ is positive semidefinite, minimization restores it as a conclusion. Moreover, [$$\label{e:trap}{\operatorname{Tr}}(A) = {\operatorname{Tr}}(\sqrt{\rho I-R}) = \alpha.$$]{}
With the hypotheses , and if $r <
\pi/\sqrt{\rho}$, a minimum of $(\log s(r))'$ exists. Equivalently, a joint minimum of $E(A,Y)$ or $E(R,Y)$ exists. \[p:exists\]
The above calculation lets us assume , which means that $R$ is uniformly bounded. By [Proposition \[p:bound2\]]{}, so is $Y''$. We can restrict to a set of pairs $(R,Y'')$ of class $L^\infty$, which is compact in the weak-\* topology by the Banach-Alaoglou theorem. Equivalently, we can restrict to a uniformly bounded, uniformly Lipschitz set of pairs $(\smallint R,Y')$, which is compact in the uniform topology by the Arzela-Ascoli theorem. By integration by parts, we can write $$\begin{aligned}
E(R,Y) &= \int_0^r \left({\langle {Y',Y'} \rangle} - {\langle {Y,RY} \rangle}\right) \,{\mathrm{d}}t. \\
&= \left[{\langle {Y,(\smallint R) Y} \rangle} \right]_0^r +
\int_0^r \left({\langle {Y',Y'} \rangle} \,{\mathrm{d}}t + 2{\langle {Y',(\smallint R) Y} \rangle} \right) \,{\mathrm{d}}t.\end{aligned}$$ Thus the energy is continuous as a function of $\smallint R$ and $Y'$ and has a minimum on a compact family.
[Proposition \[p:exists\]]{} reduces [Theorem \[th:main\]]{} to solving the following non-linear matrix ODE, which is obtained by combining and : $$\begin{aligned}
Y'' &= (A^2-\rho)Y & A &= \frac{\alpha (YY^T)^{-1}}{{\operatorname{Tr}}((YY^T)^{-1})} \\
Y(0) &= 0 & Y(r) &= I.\end{aligned}$$ [Proposition \[p:exists\]]{} tells us that this ODE has at least one solution; we will proceed by finding all solutions with the given boundary conditions. First, if we suppress the boundary condition $Y(r) = I$, the solutions $Y(t)$ are invariant under both left and right multiplication by $O(n-1)$. So we can write $$Y(t) = U{\hat{Y}}(t)V,$$ where ${\hat{Y}}'(0)$ is diagonal with positive entries. In this case ${\hat{A}}(0)$ is also diagonal, and we obtain that ${\hat{Y}}(t)$ is diagonal for all $t$, and with positive entries because the entries cannot cross 0. Therefore $UV
= I$, because the identity is the only diagonal orthogonal matrix with positive entries.
So we can assume that $Y = {\hat{Y}}$, with diagonal entries $$\lambda_1(t), \lambda_2(t), \ldots, \lambda_{n-1}(t) > 0.$$ Each of these entries satisfies the same scalar ODE, [$$\label{e:scalar}w'' = \beta(t) w^{-1} - \rho w \qquad w(0) = 0 \qquad w(r) = 1,$$]{} where $$\beta(t) = \frac{\alpha}{{\operatorname{Tr}}((Y(t)Y(t)^T)^{-1})^2}.$$ We claim that if $w > 0$, then $w' > 0$ as well. If $\rho = 0$, then this is immediate. Otherwise, a positive solution $w(t)$ satisfies $$w(t) > \frac{\sin(\sqrt{\rho}t)}{\sin(\sqrt{\rho}r)} \qquad
w'(t) > \frac{\sqrt{\rho}\cos(\sqrt{\rho}t)}{\sin(\sqrt{\rho}r)},$$ because the right side is the solution to $w'' = - \rho w$ with the same boundary conditions. So we obtain that $w' > 0$ provided that $$r < \frac{\pi}{2\sqrt{\rho}}.$$ (This is where we need half of the distance allowed in the usual form of Günther’s inequality.)
To complete the proof, consider the phase diagram in the strip $[0,1] \times
(0,\infty)$ of the positive solutions $(w(t),w'(t))$ to . If we let $x = w(t)$, then the total elapsed time to reach $x = 1$ is $$r = \int_0^1 \frac{{\mathrm{d}}t}{{\mathrm{d}}x}{\mathrm{d}}x
= \int_0^1 \frac{{\mathrm{d}}x}{w'(w^{-1}(x))},$$ which is a positive integral. On the other hand, if $w_1$ and $w_2$ are two distinct solutions with $$w_1(0) = w_2(0) = 0 \qquad w'_1(0) > w'_2(0),$$ then the solutions cannot intersect in the phase diagram; we must have $$w'_1(w_1^{-1}(x)) > w'_2(w_2^{-1}(x)) > 0.$$ So two distinct, positive solutions to cannot reach $w(t)
= 1$ at the same time, which means with given the boundary conditions that there is only one solution. Thus, the diagonal entries $\lambda_k(t)$ of $Y(t)$ are all equal. In conclusion, $Y$, $A$, and $R$ all are isotropic at the minimum of the logarithmic candle derivative $(\log s(r))'$. This additional property implies the estimate for $(\log s(r))'$ immediately. (Note that when $R$ is isotropic, the hypothesis becomes equivalent to $K\le \kappa$, the usual assumption of Günther’s inequality.)
[^1]: Supported by NSF grant CCF \#1013079.
[^2]: We take the “ic" in the Ricci tensor ${\operatorname{Ric}}$ to mean taking a partial trace of the Riemann tensor $R$, but we take a square root first.
[^3]: Certain distant objects in astronomy with known luminosity are called *standard candles* and are used to estimate astronomical distances.
[^4]: And not to be confused with liquid crystal displays.
[^5]: We credit [@Croke:sharp] as our original motivation for this article.
|
---
abstract: 'We give a formula for the modular operator and modular conjugation in terms of matrix coefficients of corepresentations of a quantum group in the sense of Kustermans and Vaes. As a consequence, the modular autmorphism group of a unimodular quantum group can be expressed in terms of matrix coefficients. As an application, we determine the Duflo-Moore operators for the quantum group analogue of the normaliser of $SU(1,1)$ in $SL(2,\mathbb{C}$).'
address: 'Radboud Universiteit Nijmegen, IMAPP, FNWI, Heyendaalseweg 135, 6525 AJ Nijmegen, the Netherlands'
author:
- 'Martijn Caspers, Erik Koelink'
date: |
April 11, 2011\
[*Keywords:*]{} Locally compact quantum groups, Orthogonality relations, Duflo-Moore operators, Modular automorphism group, Plancherel measure.\
[*2000 Mathematics Subject Classification numbers:*]{} 20G42, 47D03, 47A67.
title: Modular properties of matrix coefficients of corepresentations of a locally compact quantum group
---
Introduction {#SectIntroduction}
============
The definition of locally compact quantum groups has been given by Kustermans and Vaes [@KusV], [@KusVII] at the turn of the millenium, and we use their definition of locally compact quantum groups in this paper. We stick mainly to the von Neumann algebraic setting [@KusVII]. Since the introduction of quantum groups in the 1980ies and their theoretical development, many results known in the theory of groups have been generalised to quantum groups in some setting. In particular, the theory of compact quantum groups has been settled satisfactorily by Woronowicz establishing analogues of the Haar measure and the Schur orthogonality relations for matrix elements of corepresentations analogous to the group case, see [@Tim] and references given there. In particular, in the Kustermans-Vaes approach to locally quantum groups there is a well-defined notion of dual locally compact quantum group. Moreover, the double dual gives back the original locally compact quantum group.
In his thesis [@Des §3.2] Desmedt generalises the Plancherel theorem for locally compact groups to the setting of quantum groups. Imposing sufficient conditions on a quantum group reminiscent of the conditions of the classical Plancherel theorem, he proves a decomposition of the biregular corepresentation in terms of tensor products of irreducible corepresentations. The intertwining operator, also called the Plancherel transformation, is given in terms of fields of positive self-adjoint operators which correspond to classical Duflo-Moore operators. One consequence of the quantum Plancherel theorem is the existence of orthogonality relations of matrix coefficients in terms of these operators.
The present paper focusses on the modular properties of matrix coefficients of a locally compact quantum group that satisfies the assumptions of Desmedt’s Plancherel theorem. The orthogonality relations suggest that modular properties of integrals of the matrix coefficients of corepresentations of a locally compact quantum can be expressed in terms of the corresponding operators of Duflo-Moore type. Here, we give the polar decomposition of the second operator as in the Tomita-Takesaki theorem for a general locally compact quantum group satisfying the conditions of the Plancherel theorem, see Theorem \[ThmPlancherelLeft\]. In the case of a unimodular locally compact quantum group, we obtain an explicit expression for the action of the modular automorphism group on matrix elements of corepresentations. This result is presented in Theorem \[ThmModularExpression\].
In the second part of this paper, we determine the modular conjugation and the modular automorphism group for the case of the locally compact quantum group associated with the normaliser of $SU(1,1)$ in $SL(2,\mathbb{C})$. This quantum group was introduced in [@KoeKus] and further studied in [@GrKoeKus], where the explicit decomposition of the left regular corepresentation is presented. We calculate the Duflo-Moore operators for almost all corepresentations in the decomposition of the left regular corepresentation. This extends Desmedt’s result in [@Des §3.5], where he determines Duflo-Moore operators for the discrete series corepresentations using summation formulas for basic hypergeometric series instead of the modular formula obtained in the present paper.
This paper is structured as follows. After introducing the notational conventions, we recall Desmedt’s Plancherel theorem in Section \[SectPlancherel\]. We indicate how his theorem implies orthogonality relations between matrix coefficients and prove a result about integrals of matrix coefficients that are square integrable, see Theorem \[ThmDomain\]. Next, in Section \[SectModular\] we give a formula of the modular automorphism group of a unimodular quantum group in terms of matrix coefficients. In Section \[SectExample\] we apply the theory of Sections \[SectPlancherel\] and \[SectModular\] to determine the Duflo-Moore operators of the normaliser of $SU(1,1)$ in $SL(2, \mathbb{C})$. Appendix \[AppendixA\] contains a technical result on direct integration and Appendix \[AppendixB\] proves that the example of Section \[SectExample\] satisfies the assumptions of the Plancherel theorem.
Conventions and notation
========================
For results on weight theory on von Neumann algebras our main reference is [@TakII]. If $\varphi$ is a weight on a von Neumann algebra $M$, we use the notation ${\mathcal{N}_{\varphi}}= \left\{ x \in M \mid \varphi(x^\ast x) < \infty \right\}$ and ${\mathcal{M}_{\varphi}}= {\mathcal{N}_{\varphi}}^\ast {\mathcal{N}_{\varphi}}$, ${\mathcal{M}_{\varphi}}^+ = {\mathcal{M}_{\varphi}}\cap M^+$. $\sigma_t^\varphi$ denotes the modular automorphism group of $\varphi$.
The definition of a locally compact quantum group we use is the one by Kustermans and Vaes [@KusV], [@KusVII]. We briefly recall their notational conventions, see also [@KusLec], [@Tim]. Let $(M, \Delta)$ be a locally compact quantum group, where $M$ denotes the von Neumann algebra and $\Delta$ the comultiplication. So $\Delta$ is normal $\ast$-homomorphism $\Delta\colon M\to M\otimes M$ satisfying $(\Delta{\otimes}\iota)\Delta=(\iota{\otimes}\Delta)\Delta$, where $\iota$ denotes the identity. Moreover, there exist two normal semi-finite faithful weights ${\varphi}$, $\psi$ on $M$ so that $$\begin{split}
{\varphi}\bigl((\om{\otimes}\iota )\Delta(x)\bigr)\, &=\, {\varphi}(x)\om(1), \qquad
\forall \ \om\in M^+_*,\, \forall\ x\in \cM^+_{\varphi}\qquad \text{(left invariance),}\\
\psi\bigl((\iota {\otimes}\om)\Delta(x)\bigr)\, &=\, \psi(x)\om(1), \qquad
\forall \ \om\in M^+_*,\, \forall\ x\in \cM^+_\psi
\qquad \text{(right invariance)}.
\end{split}$$ ${\varphi}$ is the left Haar weight and $\psi$ the right Haar weight. $({H_{\varphi}}, \Lambda, {\pi_{\varphi}})$ and $({H_{\psi}}, \Gamma, {\pi_{\psi}})$ denote the GNS-constructions with respect to the left Haar weight $\varphi$ and the right Haar weight $\psi$ respectively. Without loss of generality we may assume that ${H_{\varphi}}={H_{\psi}}$ and $M\subset B({H_{\varphi}})$. The operator $W\in B({H_{\varphi}}{\otimes}{H_{\varphi}})$ defined by $W^\ast \bigl( \La(a){\otimes}\La(b) \bigr) =
\bigl( \La{\otimes}\La\bigr) \bigl( \De(b)(a{\otimes}1)\bigr)$ is a unitary operator known as the multiplicative unitary. It implements the comultiplication $\De(x)= W^\ast(1{\otimes}x)W$ for all $x\in M$ and satisfies the pentagonal equation $W_{12}W_{13}W_{23}=W_{23}W_{12}$ in $B({H_{\varphi}}{\otimes}{H_{\varphi}}{\otimes}{H_{\varphi}})$. In [@KusV], [@KusVII], see also [@KusLec], [@Tim], it is proved that there exists a dual locally compact quantum group $(\hat{M},\hat{\Delta})$, so that $(\hat{\hat{M}},\hat{\hat{\Delta}}) = (M,\Delta)$.
A unitary corepresentation $U$ of a von Neumann algebraic quantum group on a Hilbert space $H$ is a unitary element $U\in M{\otimes}B(H)$ such that $(\De{\otimes}\iota)(U)=U_{13}U_{23}\in M{\otimes}M{\otimes}B(H)$, where the standard leg-numbering is used in the right hand side. A closed subspace $L\subseteq H$ is an invariant subspace for the unitary corepresentation $U$ if $(\om{\otimes}\iota)(U)$ preserves $L$ for all $\om\in M_\ast$. A unitary corepresentation $U$ in the Hilbert space $H$ is irreducible if there are only trivial (i.e. equal to $\{0\}$ or the whole Hilbert space $H$) invariant subspaces. If $U_1$ is a corepresentation on a Hilbert space $H_1$ and $U_2$ is a corepresentation on a Hilbert space $H_2$, then $U_1$ is equivalent to $U_2$ if there is a unitary map $\Upsilon: H_1 \rightarrow H_2$, such that $(\iota \otimes \Upsilon) U_1 = U_2 (\iota \otimes \Upsilon)$. We use the notation ${{\rm IC}}(M)$ for the equivalence classes of irreducible, unitary corepresentations of $(M,\Delta)$.
$(\hat{M}_u, \hat{\Delta}_u)$ denotes the universal dual and $(\hat{M}_c, \hat{\Delta}_c)$ denotes the reduced dual C$^\ast$-algebraic quantum groups [@Kus]. The dual weights are denoted by $\hat{\varphi}_u$ and $\hat{\psi}_u$ for $(\hat{M}_u, \hat{\Delta}_u)$ and $\hat{\varphi}_c$ and $\hat{\psi}_c$ for $(\hat{M}_c, \hat{\Delta}_c)$. Similarly, we have GNS-constructions $({H_{\varphi}}, \hat{\Lambda}_{\hat{\varphi}_u}, \pi_{\hat{\varphi}_u})$ and $({H_{\psi}}, \hat{\Gamma}_{\hat{\psi}_u}, \pi_{\hat{\psi}_u})$ for $(\hat{M}_u, \hat{\Delta}_u)$ and $({H_{\varphi}}, \hat{\Lambda}_{\hat{\varphi}_c}, \pi_{\hat{\varphi}_c})$ and $({H_{\psi}}, \hat{\Gamma}_{\hat{\psi}_c}, \pi_{\hat{\psi}_c})$ for $(\hat{M}_c, \hat{\Delta}_c)$. Recall that without loss of generality we may assume that ${H_{\varphi}}$ equals ${H_{\psi}}$.
By ${\textrm{IR}}(\hat{M}_u)$ and ${\textrm{IR}}(\hat{M}_c)$ we denote the equivalence classes of irreducible, unitary representations of $\hat{M}_u$ and $\hat{M}_c$ respectively. We recall from [@Kus] that there is a bijective correspondence between ${\textrm{IR}}(\hat{M}_u)$ and ${{\rm IC}}(M)$ and that ${\textrm{IR}}(\hat{M}_c)$ is contained in ${\textrm{IR}}(\hat{M}_u)$.
$W$ denotes the multiplicative unitary associated with $M$. For $\omega \in M_\ast$ we define $\lambda(\omega) = (\omega\otimes
\iota)(W) \in \hat{M}$. We set $$\mathcal{I} = \left\{\omega \in M_\ast \mid \Lambda(x) \mapsto \omega(x^\ast), x \in {\mathcal{N}_{\varphi}}\textrm{ is a continuous functional on } {H_{\varphi}}\right\}.$$ $\mathcal{I}$ is dense in $M_\ast$ [@KusV Lemma 8.5]. By the Riesz representation theorem, for every $\omega \in \mathcal{I}$ one can associate a unique vector denoted by $\xi(\omega)$ such that $\langle \xi(\omega), \Lambda(x) \rangle = \omega(x^\ast)$. The set $\xi(\omega)$, $\omega \in \mathcal{I}$, is dense in ${H_{\varphi}}$ [@KusV Lemma 8.5]. Then the dual weight $\hat{\varphi}$ on $\hat{M}$ is the weight defined by the GNS-construction $\lambda(\omega) \mapsto \xi(\omega)$. This GNS-construction of $\hat{M}$ is denoted by $\hat{\Lambda}$. All these definitions have right analogues. $$\mathcal{I}_R = \left\{\omega \in M_\ast \mid \Gamma(x) \mapsto \omega(x^\ast), x \in {\mathcal{N}_{\psi}}\textrm{ is a continuous functional on } {H_{\psi}}\right\}.$$ For $\omega \in \mathcal{I}_R$, there is a vector $\xi_R(\omega)$ such that $\langle \xi_R(\omega), \Gamma(x) \rangle = \omega(x^\ast)$. The set $\xi_R(\omega)$, $\omega \in \mathcal{I}_R$, is dense in ${H_{\varphi}}$.
For $\alpha \in M_\ast$, we denote $\overline{\alpha} \in M_\ast$ for the functional defined by $\overline{\alpha}(x) = \overline{\alpha(x^\ast)}$. Define $M_\ast^\sharp = \left\{ \alpha \in M_\ast \mid \exists \theta \in M_\ast: (\theta \otimes \iota)(W) = (\alpha \otimes \iota)(W)^\ast \right\}.$ It can be shown [@Kus] that for every $\alpha \in M_\ast^\sharp$ there is a unique $\theta \in M_\ast$ such that $(\theta \otimes \iota)(W) = (\alpha \otimes \iota)(W)^\ast$ and $\theta$ is determined by $\theta(x) = \overline{\alpha}(\mathcal{S}(x)), x \in {\mathcal{D}}(\mathcal{S})$, where $\mathcal{S}$ is the unbounded antipode of $(M, \Delta)$. We will write $\alpha^\ast$ for this $\theta$.
Basic results on direct integration can be found in [@Dix]. For direct integrals of unbounded operators we refer to [@Lan], [@Nus] and [@Schm Chapter 12]. If $X$ is a standard measure space with measure $\mu$, we use the notation $(H_U)_{U\in X}$ or simply $(H_U)_U$ for a field of of Hilbert spaces $H_U$ over $X$. If $(H_U)_U$ is a measurable field of Hilbert spaces we denote its direct integral by $\int^\oplus_{X} H_U d\mu(U)$. Similarly we add subscripts to denote fields of vectors, operators and representations.
Let $H$ be a Hilbert space. We define the inner product to be linear in the first entry and anti-linear in the second entry. We denote the Hilbert-Schmidt operators on $H$ by $B_2(H)$. Recall that $B_2(H)$ is a Hilbert space itself, which is isomorphic to $H \otimes \overline{H}$, the isomorphism being given by $\xi \otimes \overline{\eta}: h \mapsto \langle h, \eta \rangle \xi$. Here $\overline{H}$ denotes the conjugate Hilbert space. We denote vectors in $\overline{H}$ and operators acting on $\overline{H}$ with a bar. For $\xi, \eta \in H$ the normal functional $\omega_{\xi, \eta}$ on $B(H)$ is defined as $\omega_{\xi,\eta}(A) = \langle A \xi, \eta\rangle$. The domain of an (unbounded) operator $A$ on $H$ is denoted by ${\mathcal{D}}(A)$. The symbol $\otimes$ denotes either the tensor product of Hilbert spaces, the tensor product of operators or the von Neumann algebraic tensor product. It will always be clear from the context which tensor product is meant.
Plancherel Theorems {#SectPlancherel}
===================
The classical Plancherel theorem for locally compact groups [@DixC Theorem 18.8.1] has a quantum group analogue, which has been proved by Desmedt in [@Des]. This section recalls part of Desmedt’s Plancherel theorem and elaborates on minor modifications and implications of this theorem which turn out to be useful for explicit computations in Section \[SectExample\].
For two unbounded operators $A$ and $B$, we denote $A \cdot B$ for the closure of their product.
\[ThmPlancherelLeft\] Let $(M, \Delta)$ be a locally compact quantum group such that $\hat{M}$ is a type I von Neumann algebra and such that $\hat{M}_u$ is a separable C$^\ast$-algebra. There exist a standard measure $\mu$ on ${{\rm IC}(M)}$, a measurable field $(H_U)_U$ of Hilbert spaces, a measurable field $(D_U)_U$ of self-adjoint, strictly positive operators and an isomorphism ${\mathcal{Q}}_L$ of ${H_{\varphi}}$ onto $\int^\oplus {\textrm{B}_2}(H_U) d\mu(U)$ with the following properties:
1. For all $\alpha \in {\mathcal{I}}$ and $\mu$-almost all $U \in {{\rm IC}(M)}$, the operator $(\alpha \otimes \iota)(U)D_U^{-1}$ is bounded and $(\alpha \otimes \iota)(U) \cdot D_U^{-1}$ is a Hilbert-Schmidt operator on $H_U$.
2. For all $\alpha, \beta \in {\mathcal{I}}$ one has the Parseval formula $$\langle \xi(\alpha), \xi(\beta)\rangle = \int_{{{\rm IC}(M)}} \!\!{\rm Tr}\left(\left((\beta \otimes \iota)(U) \cdot D_U^{-1}\right)^\ast \left((\alpha \otimes \iota)(U) \cdot D_U^{-1}\right)\right) d\mu (U),$$ and ${\mathcal{Q}}_L$ is the isometric extension of $$\hat{\Lambda}(\lambda({\mathcal{I}})) \rightarrow \int^{\oplus}_{{{\rm IC}(M)}} {\textrm{B}_2}(H_U) d\mu(U): \:\:\xi(\alpha) \mapsto \int^\oplus_{{{\rm IC}(M)}} (\alpha \otimes \iota)(U) \cdot D_U^{-1}d\mu(U).$$
Here $\mu$ is called the left Plancherel measure and ${\mathcal{Q}}_L$ is called the left Plancherel transform. We will be dealing with a right analogue of the Plancherel theorem as well, see [@Des Remark 3.4.11]. Here we explicitly state the part of this theorem that is relevant for the present paper.
\[ThmPlancherelRight\] Let $(M, \Delta)$ be a locally compact quantum group such that $\hat{M}$ is a type-I von Neumann algebra and such that $\hat{M}_u$ is a separable C$^\ast$-algebra. There exist a standard measure $\mu_R$ on ${{\rm IC}(M)}$, a measurable field $(K_U)_U$ of Hilbert spaces, a measurable field $(E_U)_U$ of self-adjoint, strictly positive operators and an isomorphism ${\mathcal{Q}}_R$ of ${H_{\psi}}$ onto $\int^\oplus
{\textrm{B}_2}(K_U) d\nu(U)$ with the following properties:
1. \[ThmPlancherelRightI\] For all $\alpha \in {\mathcal{I}}_R$ and $\mu_R$-almost all $U \in {{\rm IC}(M)}$, the operator $(\alpha \otimes \iota)(U)E_U^{-1}$ is bounded and $(\alpha \otimes \iota)(U)\cdot E_U^{-1}$ is a Hilbert-Schmidt operator on $K_U$.
2. \[ThmPlancherelRightII\] For all $\alpha, \beta \in {\mathcal{I}}_R$ one has the Parseval formula $$\langle \xi_R(\overline{\alpha^\ast}), \xi_R(\overline{\beta^\ast})\rangle = \int_{{{\rm IC}(M)}} \!\!\!\!{\rm Tr}\left(\left((\beta \otimes \iota)(U) \cdot E_U^{-1}\right)^\ast \left((\alpha \otimes \iota)(U) \cdot E_U^{-1}\right)\right) d\mu_R(U),$$ and ${\mathcal{Q}}_R$ is the isometric extension of $$\xi_R(\overline{\mathcal{I}_R^\ast}) \rightarrow \int^{\oplus}_{{{\rm IC}(M)}} {\textrm{B}_2}(H_U) d\mu_R(U): \:\:\xi_R(\overline{\alpha^\ast}) \mapsto \int^\oplus_{{{\rm IC}(M)}} (\alpha \otimes \iota)(U) \cdot E_U^{-1}d\mu_R(U).$$
3. The measure $\mu_R$ can be choosen equal to the measure $\mu$ of Theorem \[ThmPlancherelLeft\] and the measurable field of Hilbert spaces $(K_U)_U$ can be choosen equal to $( H_U)_U$, the measurable field of Hilbert spaces of Theorem \[ThmPlancherelLeft\].
Parts (\[ThmPlancherelRightI\]) and (\[ThmPlancherelRightII\]) of this theorem can be obtained from [@Des Theorem 3.4.5] using the relations between the right Haar weight $\psi$ and the right Haar weight $\hat{\psi}_u$ on the universal dual quantum group. The prove is similar to how Theorem \[ThmPlancherelLeft\] is obtained from [@Des Theorem 3.4.5]. We elaborate a bit on the third statement. Since $\hat{\varphi}_u$ and $\hat{\psi}_u$ are both approximately KMS-weights on the universal dual $\hat{M}_u$, their W\*-lifts are n.s.f. weights so that [@TakII Theorem VIII.3.2] implies that the representations ${\pi_{\varphi}}$ and ${\pi_{\psi}}$ are equivalent. Hence $$\label{EqnEquivReps}
\pi_{\hat{\varphi}_u}(\hat{M}_u)'' = \pi_{\hat{\varphi}}(\hat{M}) \simeq \pi_{\hat{\psi}}(\hat{M}) = \pi_{\hat{\psi}_u}(\hat{M}_u)''.$$ The proofs of Theorems \[ThmPlancherelLeft\] and \[ThmPlancherelRight\] show that the measures $\mu$ and $\nu$ together with the measurable fields of Hilbert spaces $(H_U)_U$ and $(K_U)_U$ in Theorems \[ThmPlancherelLeft\] and \[ThmPlancherelRight\] arise from the direct integral decompositions of $\pi_{\hat{\varphi}_u}(\hat{M}_u)''$ and $\pi_{\hat{\psi}_u}(\hat{M}_u)''$, respectively. That is: $$\pi_{\hat{\varphi}_u}(\hat{M}_u)'' = \int^\oplus_{X} B(H_\sigma) d\mu(\sigma), \qquad \pi_{\hat{\psi}_u}(\hat{M}_u)'' = \int^\oplus_{Y} B(K_\sigma) d\mu_R(\sigma).$$ By (\[EqnEquivReps\]) we may assume that $\mu = \mu_R$, $X=Y$ and $(H_U)_U = (K_U)_U$. Furthermore, by [@Des Eqn. (3.4.2)], $\pi_{\hat{\varphi}}(y) = y = \pi_{\hat{\psi}}(y), \forall y \in \hat{M}$, which shows that the correspondence between $X$ and the measurable subspace ${\textrm{IR}}(\hat{M}_u)$ is the same for $\pi_{\hat{\varphi}_u}$ and $\pi_{\hat{\psi}_u}$. This proves the third statement of Theorem \[ThmPlancherelRight\].
${\mathcal{Q}}_R$ is called the right Plancherel transform. In the rest of this paper we will assume that $\mu = \mu_R$ and we simply call $\mu$ the Plancherel measure. Similarly, we identify $(K_U)_U$ with $(H_U)_U$.
\[RmkReducedSetting\] Theorems \[ThmPlancherelLeft\] and \[ThmPlancherelRight\] remain valid when the assumption that $\hat{M}_u$ is separable (universal norm) is replaced by the assumption that $\hat{M}_c$ is separable (reduced norm) and the measure space ${{\rm IC}}(M)$ is replaced by the measure space ${\textrm{IR}}(\hat{M}_c)$. The proof is a minor modification of the proof of [@Des Theorem 3.4.1]. Here, ${\textrm{IR}}(\hat{M}_u)$ can be replaced by ${\textrm{IR}}(\hat{M}_c)$ and $\hat{\mathcal{V}}$ should be read as the multiplicative unitary $W$, see [@Kus] for the definition of $\hat{\mathcal{V}}$. The proof of this modification can be obtained by using the following relations instead of [@Des p. 118-119]: $$\begin{split}
\pi_\sigma\left((\omega \otimes \iota)(W)\right)& = (\omega \otimes \iota)(U_\sigma) \textrm{ where } \sigma\in {\textrm{IR}}(\hat{M}_c) \textrm{ corresponds to } U_\sigma \in {{\rm IC}}(M), \\
\xi(\omega) &= \hat{\Lambda}\left( (\omega \otimes \iota)(W) \right) = \hat{\Lambda}_{\hat{\varphi}_c} \left( (\omega \otimes \iota) (W) \right).
\end{split}$$ In [@Des Theorem 3.4.8] Desmedt proves that the support of the left and right Plancherel measures equal ${\textrm{IR}}(\hat{M}_c)$, which is in agreement with this observation.
\[RmkSqIntCoreps\] The corepresentations that appear as discrete mass points in the Plancherel measure correspond to the square integrable correpresentations in the sense of [@BusMey Definition 3.2] or the equivalent definition of left square integrable corepresentations as in [@Des Definition 3.2.29]. A proof of this can be found in [@Des Theorem 3.4.10].
\[NtNotationDrieEnVier\] In the rest of this section as well as in Section \[SectModular\] we adopt the following notational conventions. $(M,\Delta)$ is a fixed locally compact quantum group satisfying the conditions of Theorems \[ThmPlancherelLeft\] and \[ThmPlancherelRight\]. We set $D = \int^\oplus_{{{\rm IC}}(M)} D_U d\mu(U)$, $E = \int^\oplus_{{{\rm IC}}(M)} E_U d\mu(U)$ and $H = \int^\oplus_{{{\rm IC}}(M)} H_U d\mu(U)$, where $\mu$ is the Plancherel measure. All (direct) integrals are taken over ${{\rm IC}}(M)$. In the proofs we omit this in the notation.
In the remainder of this Section, we express the Plancherel transformation in terms of matrix coefficients to arrive at Theorems \[ThmOrthogonalitRel\] and \[ThmDomain\]. These theorems can be considered as direct implications of the Plancherel theorems. We will need them in Section \[SectExample\].
\[LemGNS\] We have the following:
1. Let $x\in M$, such that the linear map $f: \hat{\Lambda}(\lambda(\mathcal{I})) \rightarrow \mathbb{C}: \xi(\alpha) \mapsto \alpha(x^\ast)$ is bounded. Then $x \in
{\mathcal{D}}(\Lambda)$ and $f(\xi(\alpha)) = \langle \xi(\alpha), \Lambda(x)\rangle$.
2. Let $x\in M$, such that the linear map $f: \hat{\Gamma}(\lambda(\mathcal{I}_R)) \rightarrow \mathbb{C}: \xi_R(\alpha) \mapsto \alpha(x^\ast)$ is bounded. Then $x \in
{\mathcal{D}}(\Gamma)$ and $f(\xi_R(\alpha)) = \langle \xi_R(\alpha), \Gamma(x)\rangle$.
We prove the first statement, the second being analogous. The claim is true for $x \in {\mathcal{N}_{\varphi}}$, since ${\mathcal{D}}(\Lambda) = {\mathcal{N}_{\varphi}}$ and by definition $\langle \xi(\alpha), \Lambda(x) \rangle =
\alpha(x^\ast)$, for all $\alpha \in \mathcal{I}$. Now, let $x \in M$ be arbitrary. The set $\{ \xi(\alpha) \mid \alpha \in \mathcal{I}
\}$ is dense in ${H_{\varphi}}$ by [@KusV Lemma 8.5] and its subsequent remark. Hence, by the Riesz theorem, there is a $v \in
{H_{\varphi}}$ such that for every $\alpha \in \mathcal{I}$ $
\alpha(x^\ast) = \langle \xi(\alpha), v\rangle.
$ Let $(e_j)_{j \in J}$ be a bounded net in the Tomita algebra $$\mathcal{T}_\varphi = \left\{ x \in {\mathcal{N}_{\varphi}}\cap {\mathcal{N}_{\varphi}}^\ast \mid x \textrm{ is analytic w.r.t. } \sigma^\varphi \textrm{ and } \sigma_z^\varphi(x) \in {\mathcal{N}_{\varphi}}\cap {\mathcal{N}_{\varphi}}^\ast, \forall z \in \mathbb{C} \right\},$$ converging $\sigma$-weakly to 1 and such that $\sigma^\varphi_{i/2}(e_j)$ converges $\sigma$-weakly to 1 (using the residue formula for meromorphic functions, one can see that the net $(e_j)_{j \in J}$ defined in [@TerpII Lemma 9] satisfies these properties). Let $a, b \in \mathcal{T}_\varphi$ and fix the normal functional $\alpha$ by $\alpha(x) = \varphi(axb), x \in M$. Using [@KusV Lemma 8.5] we find $$\langle \xi(\alpha), \Lambda(x e_j) \rangle = \varphi(a e_j^\ast x^\ast b) = \langle \Lambda(b \sigma_{-i}^\varphi(a e_j^\ast) ), v \rangle = \langle \Lambda(b \sigma_{-i}^\varphi(a) ), J {\pi_{\varphi}}(\sigma_{i/2}^\varphi(e_j)^\ast) J v \rangle.$$ Hence, $\Lambda(x e_j) = J {\pi_{\varphi}}(\sigma_{i/2}^\varphi(e_j)^\ast) J v $, so that $\Lambda(x e_j)$ converges weakly to $v$. Since $x e_j \rightarrow x$ $\sigma$-weakly and $\Lambda$ is $\sigma$-weak/weakly closed, this implies that $x \in {\mathcal{D}}(\Lambda) = {\mathcal{N}_{\varphi}}$ and $v = \Lambda(x)$.
Recall that $\int^\oplus B_2(H_U) d\mu(U) \simeq \int^\oplus H_U \otimes \overline{H_U} d\mu(U)$. For $\eta = \int^\oplus \eta_U d\mu(U), \xi =
\int^\oplus \xi_U d\mu(U) \in H$ the mesurable field of vectors $(\xi_U \otimes \overline{\eta_U})_U$ is not necessarily square integrable. If it is square integrable, $\int^\oplus \xi_U \otimes \overline{\eta_U} d\mu(U) \in \int^\oplus B_2(H_U) d\mu(U)$.
We obtain the following expression for the left Plancherel transformation.
\[LemCompI\] Let $\eta = \int^\oplus_{{{\rm IC}(M)}} \eta_U d\mu(U) \in H$ and $\xi =
\int^\oplus_{{{\rm IC}(M)}} \xi_U d\mu(U) \in H$ be such that $\eta \in {\mathcal{D}}(D^{-1})$ and $(\xi_U \otimes \overline{\eta_U})_U$ is square integrable. Then ${{\rm IC}(M)}\ni U \mapsto (\iota \otimes
\omega_{\xi_U, D_U^{-1} \eta_U})(U^\ast) \in M$ is $\sigma$-weakly integrable with respect to $\mu$ and $\int_{{{\rm IC}}(M)} (\iota
\otimes \omega_{\xi_U, D_U^{-1} \eta_U})(U^\ast) d\mu(U) \in
{\mathcal{N}_{\varphi}}$, and $$\label{EqnPlancherelMatrix}
{\mathcal{Q}}_L^{-1}(\int^{\oplus}_{{{\rm IC}(M)}} \xi_U \otimes \overline{\eta_U} d\mu(U)) = \Lambda\left( \int_{{{\rm IC}(M)}} (\iota \otimes \omega_{\xi_U, D_U^{-1} \eta_U})(U^\ast) d\mu(U)\right).$$
For $\alpha \in \mathcal{I}$, Theorem \[ThmPlancherelLeft\] implies that $$\begin{split}
& \langle \xi(\alpha), {\mathcal{Q}}_L^{-1}(\int^{\oplus} \xi_U \otimes \overline{\eta_U} d\mu(U)) \rangle = \int \langle(\alpha \otimes \iota)(U)D_U^{-1}, \xi_U \otimes \overline{\eta_U} \rangle_{{\rm HS}} d\mu(U) \\
= & \int (\alpha \otimes \omega_{D_U^{-1}\eta_U, \xi_U})(U) d\mu(U) = \alpha (\int (\iota \otimes \omega_{\xi_U,
D_U^{-1}\eta_U})(U^\ast) d\mu(U)^\ast) ,
\end{split}$$ where the last integral exists in the $\sigma$-weak sense. We see by Lemma \[LemGNS\] that, $\int (\iota \otimes \omega_{\xi_U,
D_U^{-1} \eta_U})(U^\ast) d\mu(U) \in {\mathcal{D}}(\Lambda) = {\mathcal{N}_{\varphi}}$, and (\[EqnPlancherelMatrix\]) follows.
\[RmkIntExists\] As in the proof of Lemma \[LemCompI\] we see that for $\xi = \int^\oplus \xi_U d\mu(U) \in H$, $\eta = \int^\oplus \eta_U d\mu(U) \in H$, the $\sigma$-weak integral $\int (\iota \otimes \omega_{\xi_U, \eta_U})(U^\ast) d\mu(U)\in M$ exists, and for $\alpha \in M_\ast$, $\vert \int (\alpha \otimes \omega_{\xi_U, \eta_U})(U) d\mu(U) \vert \leq \Vert \alpha \Vert \Vert \xi \Vert \Vert \eta \Vert$.
The previous lemma shows that ${\mathcal{Q}}_L^{-1}$ is an analogue of what Desmedt calls the (left) Wigner map [@Des Section 3.3.1]. This map is defined as $$B_2(H_U) \rightarrow H: \xi \otimes \overline{\eta} \mapsto \Lambda\left( (\iota \otimes \omega_{\xi, D_U^{-1} \eta})(U^\ast) \right),$$ where $U$ is a corepresentation on a Hilbert space $H_U$ that appears as a discrete mass point in the Plancherel measure, cf. the remarks about square integrable corepresentations at the end of Section \[SectPlancherel\]. This map is also considered in [@BusMey Page 203], where it is denoted by $\Phi$.
The next Lemma is the right analogue of Lemma \[LemCompI\], the proof being similar.
\[LemCompII\] Let $\eta = \int^\oplus_{{{\rm IC}(M)}} \eta_U d\mu(U) \in H$ and $\xi =
\int^\oplus_{{{\rm IC}(M)}} \xi_U d\mu(U) \in H$ be such that $\eta \in {\mathcal{D}}(E^{-1})$ and $( \xi_U \otimes \overline{\eta_U})_U $ is square integrable. Then ${{\rm IC}(M)}\ni U \mapsto (\iota \otimes
\omega_{\xi_U, E_U^{-1} \eta_U})(U) \in M$ is $\sigma$-weakly integrable with respect to $\mu$. Furthermore, $\int_{{{\rm IC}(M)}} (\iota \otimes
\omega_{\xi_U, E_U^{-1} \eta_U})(U) d\mu(U) \in {\mathcal{N}_{\psi}}$ and $${\mathcal{Q}}_R^{-1}(\int^{\oplus}_{{{\rm IC}(M)}} \xi_U \otimes \overline{\eta_U} d\mu(U)) = \Gamma\left( \int_{{{\rm IC}(M)}} (\iota \otimes \omega_{\xi_U, E_U^{-1} \eta_U})(U) d\mu(U)\right).$$
The Plancherel theorems imply the following orthogonality relations. The theorem follows immediately from the expressions for the Plancherel transformations given in Lemmas \[LemCompI\] and \[LemCompII\]. The orthogonality relations will be used in Section \[SectExample\] where we give a method to determine the Duflo-Moore operators of a locally compact quantum group that satisfies the assumptions of the Plancherel theorem.
\[ThmOrthogonalitRel\] Let $(M, \Delta)$ be a locally compact quantum group, such that $\hat{M}_u$ is separable and $\hat{M}$ is a type I von Neumann algebra. Let $\eta = \int^\oplus \eta_U d\mu(U) \in H$, $\xi = \int^\oplus
\xi_U d\mu(U) \in H$, $\eta' = \int^\oplus \eta_U' d\mu(U) \in H$ and $\xi' = \int^\oplus \xi_U' d\mu(U) \in H$. We have the following orthogonality relations:
1. Suppose that $\eta, \eta' \in {\mathcal{D}}(D)$ and that $ (\xi_U \otimes \overline{D_U \eta_U})_U ,
(\xi_U ' \otimes \overline{D_U \eta_U '} )_U$ are square integrable fields of vectors, then $$\label{EqnOrthI}
\begin{split}
&\varphi\left( \left(\int (\iota \otimes \omega_{\xi_U, \eta_U})(U^\ast) d\mu(U)\right) ^\ast \int (\iota \otimes \omega_{\xi_U', \eta_U'})(U^\ast) d\mu(U) \right) = \\
& \int \langle D_U \eta_U, D_U \eta'_U \rangle \langle \xi'_U, \xi_U
\rangle d\mu(U).
\end{split}$$
2. Suppose that $\eta, \eta' \in {\mathcal{D}}(E)$ and that $ (\xi_U \otimes \overline{E_U \eta_U})_U ,
(\xi_U ' \otimes \overline{E_U \eta_U '})_U $ are square integrable fields of vectors, then: $$\label{EqnOrthII}
\begin{split}
&\psi\left(\left( \int (\iota \otimes \omega_{\xi_U, \eta_U})(U) d\mu(U)\right) ^\ast\int (\iota \otimes \omega_{\xi_U', \eta_U'})(U) d\mu(U) \right) = \\
& \int \langle E_U \eta_U, E_U \eta'_U \rangle \langle \xi'_U, \xi_U
\rangle d\mu(U).
\end{split}$$
Here $\int (\iota \otimes \omega_{\xi_U, \eta_U})(U) d\mu(U), \int (\iota \otimes \omega_{\xi_U, \eta_U})(U^\ast) d\mu(U), \int (\iota \otimes \omega_{\xi_U', \eta_U'})(U) d\mu(U)$,\
$\int (\iota \otimes \omega_{\xi_U', \eta_U'})(U^\ast) d\mu(U)$ are defined in Lemma \[LemCompI\] and \[LemCompII\]. The integrals are taken over ${{\rm IC}(M)}$.
As observed in Remark \[RmkIntExists\] the element $\int (\iota \otimes \omega_{\xi_U,\eta_U})(U^\ast)d\mu(U) \in M$ exists for $\eta = \int^\oplus \eta_U d\mu(U) \in H$, $\xi = \int^\oplus
\xi_U d\mu(U) \in H$ and the next theorem investigates the consequences of $\int (\iota \otimes \omega_{\xi_U, \eta_U})(U^\ast)d\mu(U) \in {\mathcal{N}_{\varphi}}$.
\[ThmDomain\] Let $\xi = \int^\oplus_{{{\rm IC}(M)}} \xi_U d\mu(U) \in H$ be an essentially bounded field of vectors.
1. Let $\eta = \int^\oplus_{{{\rm IC}(M)}} \eta_U d\mu(U) \in H$ be such that $\int_{{{\rm IC}(M)}} (\iota \otimes \omega_{\xi_U, \eta_U})(U^\ast) d\mu(U) \in {\mathcal{N}_{\varphi}}$. Then, for almost every $U$ in the support of $(\xi_U)_U$, we have $\eta_U \in {\mathcal{D}}(D_U)$.
2. Let $\eta = \int^\oplus_{{{\rm IC}(M)}} \eta_U d\mu(U) \in H$ be such that $\int_{{{\rm IC}(M)}} (\iota \otimes \omega_{\xi_U, \eta_U})(U) d\mu(U) \in {\mathcal{N}_{\psi}}$. Then, for almost every $U$ in the support of $(\xi_U)_U$, we have $\eta_U \in {\mathcal{D}}(E_U)$.
We only give a proof of the first statement. Consider the sesquilinear form $$q(\eta, \eta') = \varphi\left( \int (\iota \otimes \omega_{\xi_U, \eta_U})(U^\ast) d\mu(U) ^\ast \int (\iota \otimes \omega_{\xi_U, \eta_U'})(U^\ast) d\mu(U) \right),$$ with $$q(\eta) = q(\eta,\eta), \quad {\mathcal{D}}(q) = \left\{ \eta = \int^\oplus \eta_U d\mu(U) \mid \int (\iota \otimes \omega_{\xi_U, \eta_U})(U^\ast) d\mu(U) \in {\mathcal{N}_{\varphi}}\right\}.$$ $q$ is a closed form on $H$. Indeed, assume that $\eta_n\in {\mathcal{D}}(q)$ converges in norm to $\eta \in H$ and that $q(\eta_n - \eta_m) \rightarrow 0$. Then $\int (\iota \otimes \omega_{\xi_U, \eta_{n,U}})(U^\ast) d\mu(U)$ converges to $\int (\iota \otimes \omega_{\xi_U, \eta_{U}})(U^\ast) d\mu(U)$ $\sigma$-weakly. By assumption $\Lambda(\int (\iota \otimes \omega_{\xi_U, \eta_{n,U}})(U^\ast) d\mu(U))$ is a Cauchy sequence in norm. The $\sigma$-weak-weak closedness of $\Lambda$ implies that $\int (\iota \otimes \omega_{\xi_U, \eta_{U}})(U^\ast) d\mu(U) \in {\mathcal{D}}(\Lambda) = {\mathcal{N}_{\varphi}}$, so $\eta \in {\mathcal{D}}(q)$ and $\Lambda(\int (\iota \otimes \omega_{\xi_U, \eta_{n,U}})(U^\ast) d\mu(U))$ converges to $\Lambda(\int (\iota \otimes \omega_{\xi_U, \eta_{U}})(U^\ast) d\mu(U))$ weakly. Since we know that $\Lambda(\int (\iota \otimes \omega_{\xi_U, \eta_{n,U}})(U^\ast) d\mu(U))$ is a actually a Cauchy sequence in the norm topology it is norm convergent to $\Lambda(\int (\iota \otimes \omega_{\xi_U, \eta_{U}})(U^\ast) d\mu(U))$. This proves that $q(\eta - \eta_n) \rightarrow 0$.
Since $(\xi_U)_U$ is a square integrable, essentially bounded field of vectors, $\int^\oplus \xi_U \otimes \overline{ \eta_U}
d\mu(U) \in B_2(H)$. By Lemma \[LemCompI\], ${\mathcal{D}}(D) \subseteq
{\mathcal{D}}(q)$, so that $q$ is densely defined. $q$ is symmetric and positive by its definition. By [@Kat Theorem VI.2.23], there is a unique positive, self-adjoint, possibly unbounded operator $A$ on $H$ such that $q(\eta, \eta') = \langle A \eta, A \eta' \rangle$ and ${\mathcal{D}}(A) = {\mathcal{D}}(q)$. By Theorem \[ThmOrthogonalitRel\] we see that for $\eta, \eta'\in {\mathcal{D}}(D)$ we have $\int \langle D_U
\eta_U, D_U \eta_U' \rangle \Vert \xi_U \Vert^2 d\mu(U) = \langle
A \eta, A \eta' \rangle$. Since both $A$ and $\int^\oplus \Vert
\xi_U \Vert d\mu(U)$ are positive, self-adjoint operators this yields $A = \int^\oplus \Vert \xi_U \Vert D_U d\mu(U)$. In particular $\eta_U \in {\mathcal{D}}(D_U)$ for almost every $U\in {\rm
supp}\left((\xi_U)_U \right) = \overline{\left\{U \in {{\rm IC}}(M) \mid
\Vert \xi_U \Vert \not = 0 \right\}}$.
Modular properties of matrix coefficients {#SectModular}
=========================================
In this section we work towards expressions for the modular automorphism group of the left and right Haar weight in terms of matrix elements of corepresentations, culminating in Theorem \[ThmModularExpression\]. The matrix coefficients of corepresentations are preserved under the modular automorphism group. The idea of proving this formula is to describe the polar decomposition of the conjugation operator $\Gamma(x) \mapsto \Lambda(x^\ast), x \in {\mathcal{N}_{\psi}}\cap {\mathcal{N}_{\varphi}}^\ast$ explicitly in terms of corepresentations. Then, for a unimodular quantum group, where $\Gamma = \Lambda$, the modular automorphism group is implemented by the absolute value of this operator.
Recall that in this section we use the notational conventions of Notation \[NtNotationDrieEnVier\].
At this point we recall the relevant results from the theory of normal, semi-finite, faithful (n.s.f.) weights and their modular automorphism groups. This is contained in [@TakII Chapters VI, VII, VIII]. We emphasize that the notation sometimes differs from [@TakII].
Consider the following two operators [@TakII Section VIII.3] $$\label{EqnConjugationI}
\begin{split}
{S}_{\psi,0}: {H_{\psi}}\rightarrow {H_{\psi}}: \Gamma(x) &\mapsto \Gamma(x^\ast), \quad x \in {\mathcal{N}_{\psi}}\cap {\mathcal{N}_{\psi}}^\ast, \\
{S}_0: {H_{\psi}}\rightarrow {H_{\varphi}}: \Gamma(x) &\mapsto \Lambda(x^\ast), \quad x \in {\mathcal{N}_{\psi}}\cap {\mathcal{N}_{\varphi}}^\ast.
\end{split}$$ Both operators are densely defined and preclosed. We denote their closures by ${S}_\psi$ and ${S}$, respectively. ${S}_\psi$ and ${S}$ correspond to $S_\psi$ and $S_{\varphi, \psi}$ in [@TakII Section VIII.3, (13)]. We denote their polar decompositions by ${S}_\psi = J_\psi {\nabla^{\frac{1}{2}}}_\psi, {S}= J {\nabla^{\frac{1}{2}}}$. By construction, $J_\psi$ and $\nabla_\psi$ are the modular conjugation and the modular operator appearing in the Tomita-Takesaki theorem. In particular, $\nabla_\psi$ implements the modular automorphism group $\sigma_t^\psi$, i.e. $$\label{EqnModularAutomorphism}
\sigma_t^\psi({\pi_{\psi}}(x)) = \nabla_\psi^{it} {\pi_{\psi}}(x) \nabla_\psi^{-it}, \qquad x \in M,\: t \in \mathbb{R}.$$ Furthermore, $\nabla_\psi^{it}, t \in \mathbb{R}$, is a homomorphism of the left Hilbert algebra $\Gamma({\mathcal{N}_{\psi}}\cap {\mathcal{N}_{\psi}}^\ast)$, i.e. $$\label{EqnHilbertProduct}
\nabla_\psi^{it} {\pi_{\psi}}(x) \nabla_\psi^{-it} \Gamma(y) = \pi_l(\nabla_\psi^{it} \Gamma(x)) \Gamma(y),$$ where $\pi_l(a)b = ab$ for $a, b \in \Gamma({\mathcal{N}_{\psi}}\cap {\mathcal{N}_{\psi}}^\ast)$. By [@TakII Section VIII.3, (11) and (29)] the modular automorphism group $\sigma_t^\varphi$ is implemented by $\nabla$, i.e. $$\label{EqnImplementation}
\sigma_t^\varphi(x) = \nabla^{it} x \nabla^{-it}, \qquad x \in M,\: t \in \mathbb{R}.$$ We emphasize that in general $\nabla^{it}, t \in \mathbb{R}$, fails to be a Hilbert algebra homomorphism of the left Hilbert algebra $\Gamma({\mathcal{N}_{\psi}}\cap {\mathcal{N}_{\psi}}^\ast)$. In case $(M, \Delta)$ is unimodular, we find that $\nabla = \nabla_\psi$ and $\nabla^{it}, t \in \mathbb{R}$, satisfies the relation (\[EqnHilbertProduct\]). This fact will eventually lead to Theorem \[ThmModularExpression\]. However, we present the theory more general and do not suppose that $(M, \Delta)$ is unimodular until this theorem.
It turns out that the polar decomposition of ${S}$ can be expressed in terms of corepresentations by means of the Plancherel theorems. The polar decomposition of ${\mathcal{Q}}_L \circ {S}\circ {\mathcal{Q}}_R^{-1}$ and the morphisms ${\mathcal{Q}}_L$ and ${\mathcal{Q}}_R$ give the polar decomposition of ${S}$. Eventually this yields Theorems \[ThmDecompositionI\] and \[ThmDecompositionII\].
\[RmkNotationDom\] For $\xi = \int ^\oplus\xi_U
d\mu(U)\in H$ and $\eta = \int^\oplus \eta_U d\mu(U)\in H$, and $A
= \int^\oplus A_U d\mu(U)$, $B=\int^\oplus B_U d\mu(U)$ decomposable operators on $H$, we will use $(\xi, \overline{\eta})\in {\mathcal{D}^\otimes}(A,\overline{B})$ to mean $\xi \in {\mathcal{D}}(A), \overline{\eta} \in
{\mathcal{D}}(\overline{B})$, $(\xi_U \otimes \overline{\eta_U})_U$ is square integrable and $\int^\oplus\xi_U\otimes \overline{\eta_U}
d\mu(U) \in {\mathcal{D}}(\int ^\oplus(A_U \otimes
\overline{B_U})d\mu(U))$. For closed opeators $A$ and $B$ the set of $\int^\oplus \xi_U \otimes \overline{\eta_U} d\mu(U)$ with $(\xi_U, \overline{\eta_U})
\in {\mathcal{D}^\otimes}(A,\overline{B})$ is a core for $\int^\oplus (A_U \otimes \overline{B_U}) d\mu(U)$ by Lemma \[LemTensorCore\]. In particular this set is dense in $\int^\oplus H_U \otimes
\overline{H_U} d\mu(U)$.
Let $\Sigma$ be the anti-linear flip $
\Sigma: \int ^\oplus H_U \otimes \overline{H_U} d\mu(U) \rightarrow \int ^\oplus H_U \otimes \overline{H_U} d\mu(U):\:\:
\int ^\oplus \xi_U \otimes \overline{\eta_U} d\mu(U) \mapsto
\int ^\oplus \eta_U \otimes \overline{\xi_U} d\mu(U).
$ $\Sigma$ is an anti-linear isometry of $\int ^\oplus H_U \otimes \overline{H_U} d\mu(U)$.
\[LemCompIII\] For $\eta = \int ^\oplus_{{{\rm IC}(M)}} \eta_U d\mu(U), \xi =
\int^\oplus_{{{\rm IC}(M)}} \xi_U d\mu(U) \in H$, with $(\eta, \overline{\xi}) \in {\mathcal{D}^\otimes}(E^{-1}, \overline{D}) )$, we have ${\mathcal{Q}}_R^{-1}\left(
\int^\oplus_{{{\rm IC}(M)}} \xi_U \otimes \overline{ \eta_U} d\mu(U)\right) \in {\mathcal{D}}(S)$ and: $$\label{EqnConjPlancherel}
{\mathcal{Q}}_L \circ {S}\circ {\mathcal{Q}}_R^{-1}\left( \int^\oplus_{{{\rm IC}(M)}} \xi_U \otimes \overline{ \eta_U} d\mu(U)\right) = \left( \int^\oplus_{{{\rm IC}(M)}} E_U^{-1}\eta_U \otimes \overline{ D_U \xi_U} d\mu(U)\right).$$
By Lemma \[LemCompII\]: $$\label{EqnLemCompIII}
{\mathcal{Q}}_R^{-1}\left( \int^\oplus \xi_U \otimes \overline{ \eta_U}
d\mu(U)\right) = \Gamma\left( \int (\iota \otimes \omega_{\xi_U,
E_U^{-1} \eta_U})(U) d\mu(U)\right).$$ By Lemmas \[LemCompI\] and \[LemCompII\] we obtain $$\int (\iota \otimes
\omega_{\xi_U, E_U^{-1} \eta_U})(U) d\mu(U) = \left(\int (\iota \otimes
\omega_{E_U^{-1} \eta_U, \xi_U})(U^\ast) d\mu(U)\right)^\ast \in {\mathcal{N}_{\psi}}\cap{\mathcal{N}_{\varphi}}^\ast.$$ Hence, by (\[EqnConjugationI\]), (\[EqnLemCompIII\]) and Lemma \[LemCompI\] $$\begin{split}
{\mathcal{Q}}_L \circ {S}\circ {\mathcal{Q}}_R^{-1}\left( \int^\oplus \xi_U \otimes \overline{ \eta_U} d\mu(U)\right) & = {\mathcal{Q}}_L \left(\Lambda\left( \int (\iota \otimes \omega_{\xi_U,E_U^{-1} \eta_U})(U) d\mu(U)^\ast\right) \right)\\
& = \left( \int^\oplus E_U^{-1}\eta_U \otimes \overline{ D_U \xi_U}
d\mu(U)\right),
\end{split}$$ from which the lemma follows.
We are now able to give the polar decomposition of ${\mathcal{Q}}_L \circ {S}\circ {\mathcal{Q}}_R^{-1}$.
\[ThmPolarDecI\] Consider ${S}_{\mathcal{Q}}:= {\mathcal{Q}}_L \circ {S}\circ {\mathcal{Q}}_R^{-1}$ as an operator on $\int^{\oplus}_{{{\rm IC}(M)}} H_U \otimes \overline{H_U} d\mu(U)$. Then the polar decomposition of ${S}_{\mathcal{Q}}$ is given by the self-adjoint, strictly positive operator $\int^{\oplus}_{{{\rm IC}(M)}} D_U
\otimes \overline{E_U^{-1}} d\mu(U)$ and the anti-linear isometry $\Sigma$.
Throughout this proof, let $\eta = \int^\oplus
\eta_U d\mu(U) \in H$, $\xi = \int^\oplus \xi_U d\mu(U) \in H$, $\eta' = \int^\oplus \eta_U' d\mu(U) \in H$ and $\xi' =
\int^\oplus \xi_U' d\mu(U) \in H$ be such that $(\eta_U \otimes
\overline{\xi_U})_U$ and $(\eta_U' \otimes \overline{\xi_U'})_U$ are square integrable.
Assume $(\eta, \overline{\xi}) \in {\mathcal{D}^\otimes}(D,
\overline{E^{-1}})$, $(\xi', \overline{\eta'}) \in {\mathcal{D}^\otimes}(D,
\overline{E^{-1}})$, so that by (\[EqnConjPlancherel\]), $$\begin{split}
&\langle \int^\oplus (\xi_U \otimes \overline{\eta_U}) d\mu(U), {S}_{\mathcal{Q}}\int^\oplus (\xi_U' \otimes \overline{\eta_U'}) d\mu(U) \rangle = \\
& \langle
\int^\oplus (\xi_U \otimes \overline{\eta_U}) d\mu(U), \int^\oplus (E_U^{-1}\eta_U' \otimes \overline{D_U \xi_U'}) d\mu(U) \rangle = \\ &
\int \langle \xi_U, E_U^{-1} \eta_U' \rangle \langle D_U \xi_U', \eta_U \rangle d\mu(U) = \\
& \int \langle E_U^{-1} \xi_U, \eta_U' \rangle \langle \xi_U', D_U\eta_U \rangle d\mu(U) = \\ &
\langle \int^\oplus (\xi_U' \otimes \overline{\eta_U'})
d\mu(U),\int^\oplus (D_U \eta_U \otimes \overline{E_U^{-1}\xi_U})
d\mu(U) \rangle.
\end{split}$$ So ${S}_{\mathcal{Q}}^\ast \left( \int^\oplus (\xi_U \otimes
\overline{\eta_U}) d\mu(U) \right) = \int^\oplus (D_U \eta_U
\otimes \overline{E_U^{-1}\xi_U}) d\mu(U)$.
Assuming $(\xi, \overline{\eta}) \in {\mathcal{D}^\otimes}(D^2, \overline{E^{-2}})$, it follows $${S}_{\mathcal{Q}}^\ast {S}_{\mathcal{Q}}\left( \int^\oplus (\xi_U \otimes \overline{\eta_U}) d\mu(U) \right) = \int^\oplus (D_U^2\xi_U \otimes \overline{E_U^{-2}\eta_U})
d\mu(U).$$ $\int ^\oplus D_U^2 \otimes \overline{E_U^{-2}} d\mu(U)$ is a positive, self-adjoint operator for which the set $$C := \textrm{span}_\mathbb{C} \left\{ \int ^\oplus (\xi_U
\otimes \overline{\eta_U}) d\mu(U) \mid (\xi, \eta) \in
{\mathcal{D}^\otimes}(D^2, E^{-2}) \right\},$$ forms a core by Lemma \[LemTensorCore\]. Since ${S}_{\mathcal{Q}}^\ast {S}_{\mathcal{Q}}$ is self-adjoint and agrees with the self-adjoint operator $\int
^\oplus D_U^2 \otimes \overline{E_U^{-2}} d\mu(U)$ on $C$ we find ${S}_{\mathcal{Q}}^\ast {S}_{\mathcal{Q}}= \int ^\oplus D_U^2 \otimes
\overline{E_U^{-2}} d\mu(U)$.
Assuming that $(\xi, \overline{\eta}) \in {\mathcal{D}^\otimes}(D,
\overline{E^{-1}})$, $$\begin{split}
&\Sigma \circ \left(\int ^\oplus D_U \otimes \overline{E_U^{-1}}
d\mu(U)\right) \left( \int^\oplus (\xi_U \otimes \overline{\eta_U}) d\mu(U) \right) =\\ &
\Sigma \left( \int^\oplus (D_U \xi_U \otimes \overline{E_U^{-1} \eta_U}) d\mu(U) \right) =
\int^\oplus (E_U^{-1} \eta_U \otimes \overline{D_U \xi_U})
d\mu(U) ,
\end{split}$$ so that ${S}_{\mathcal{Q}}$ and $\Sigma \circ \left(\int^\oplus
D_U \otimes \overline{E_U^{-1}} d\mu(U)\right)$ agree on a core, cf. Remark \[RmkNotationDom\].
Finally we translate everything back to the level of the GNS-representations ${H_{\varphi}}$ and ${H_{\psi}}$.
\[PropNabla\] Let $$\begin{split}
{D_{\nabla^\frac{1}{2}_0}}= & {\rm span}_\mathbb{C} \{ \int_{{{\rm IC}(M)}}
{(\iota \otimes \omega_{\xi_U, \eta_U})(U)} d\mu(U) \mid {\rm where } \\ & \qquad \qquad \eta \in {\mathcal{D}}(E) \cap
{\mathcal{D}}(E^{-1}), (\xi, E \eta) \in {\mathcal{D}^\otimes}(D,E^{-1}) \},
\end{split}$$ and define ${\nabla^{\frac{1}{2}}}_0 :\: \Gamma({D_{\nabla^\frac{1}{2}_0}}) \rightarrow {H_{\psi}}$ by $$\Gamma(\int_{{{\rm IC}(M)}} {(\iota \otimes \omega_{\xi_U, \eta_U})(U)} d\mu(U)) \mapsto \Gamma(\int_{{{\rm IC}(M)}} {(\iota \otimes \omega_{D_U \xi_U, E_U^{-1}\eta_U})(U)} d\mu(U)).$$ Then ${\nabla^{\frac{1}{2}}}_0$ is a densely defined, preclosed operator and its closure ${\nabla^{\frac{1}{2}}}$, is a self-adjoint, strictly positive operator satisfying ${\mathcal{Q}}_R \circ {\nabla^{\frac{1}{2}}}\circ
{\mathcal{Q}}_R^{-1} = \int^{\oplus}_{{{\rm IC}(M)}} D_U \otimes \overline{E_U^{-1}} d\mu(U)
$.
Let $C := \textrm{span}_\mathbb{C}\left\{ \int^\oplus \xi_U
\otimes \overline{\eta_U} d\mu(U) \mid (\xi, \eta)\in {\mathcal{D}^\otimes}(D_U^2,
E_U^{-2}) \right\}$. Then $C$ is a core for $\int^{\oplus} D_U
\otimes \overline{E_U^{-1}} d\mu(U)$. Indeed, $C$ is a core for $\int^{\oplus} D_U^2 \otimes \overline{E_U^{-2}} d\mu(U)$ by Lemma \[LemTensorCore\], and hence this is a core for $\int^{\oplus} D_U \otimes \overline{E_U^{-1}} d\mu(U)$.
Now, let $\eta = \int^\oplus \eta_U d\mu(U)\in H$ and $\xi = \int^\oplus \xi_U d\mu(U)\in H$ be such that $$\eta \in {\mathcal{D}}(E) \cap {\mathcal{D}}(E^{-1}), \qquad (\xi, E \eta) \in {\mathcal{D}^\otimes}(D,E^{-1}).$$ So $\eta \in {\mathcal{D}}(E)$ and $(\xi_U \otimes E_U \eta_U)_U$ is square integrable, so that $\int {(\iota \otimes \omega_{\xi_U, \eta_U})(U)} d\mu(U) \in {\mathcal{N}_{\psi}}$ by Lemma \[LemCompII\]. Similarly, since $E^{-1}\eta \in {\mathcal{D}}(E)$ and $(D_U \xi_U \otimes \eta_U)_U$ is square integrable, $\int {(\iota \otimes \omega_{D_U
\xi_U, E_U^{-1}\eta_U})(U)} d\mu(U)) \in {\mathcal{N}_{\psi}}$. Furthermore, we have the following inclusions: $$C \subseteq {\mathcal{Q}}_R( \Gamma({D_{\nabla^\frac{1}{2}_0}})) \subseteq
{\mathcal{D}}(\int^\oplus D_U \otimes \overline{E_U^{-1}}d\mu(U) )$$ and for $x \in \Gamma({D_{\nabla^\frac{1}{2}_0}})$ we have, $${\nabla^{\frac{1}{2}}}_0 (x) = {\mathcal{Q}}_R^{-1}\left(
\int^{\oplus} D_U \otimes \overline{E_U^{-1}} d\mu(U)\right)
{\mathcal{Q}}_R(x) .$$ Since ${\mathcal{Q}}_R$ is an isometric isomorphism, the claims follow from the fact that $\int^{\oplus} D_U \otimes
\overline{E_U^{-1}} d\mu(U)$ is a self-adjoint, strictly positive operator for which $C$ is a core.
\[PropJee\] Let ${D_{J_0}}$ be the linear space $$\begin{split}
& {\rm span}_\mathbb{C} \{ \int_{{{\rm IC}(M)}}^\oplus
{(\iota \otimes \omega_{\xi_U, \eta_U})(U)} d\mu(U) \mid {\rm where } \\ & \qquad \qquad \xi \in {\mathcal{D}}(D^{-1}), \eta \in
{\mathcal{D}}(E), (\xi_U \otimes \overline{ E_U \eta_U})_U \:\: {\rm is\:\:square\:\: integrable} \},
\end{split}$$ and define $J_0 : \Gamma({D_{J_0}}) \rightarrow {H_{\varphi}}:$ $$\Gamma(\int_{{{\rm IC}(M)}} {(\iota \otimes \omega_{\xi_U, \eta_U})(U)} d\mu(U)) \mapsto \Lambda(\int_{{{\rm IC}(M)}} {(\iota \otimes \omega_{D_U^{-1} \xi_U, E_U\eta_U})(U)} d\mu(U)^\ast).$$ Then $J_0$ is a densely defined anti-linear isometry, and its closure, denoted by $J$, is a surjective anti-linear isometry satisfying ${\mathcal{Q}}_L \circ J \circ {\mathcal{Q}}_R^{-1} = \Sigma$.
Let $C := \textrm{span}_\mathbb{C}\left\{ \int
\xi_U \otimes \overline{\eta_U} d\mu(U) \mid (\xi, \eta) \in
{\mathcal{D}^\otimes}(D^{-1}, \overline{E}) \right\}$. $C$ is dense in $\int^\oplus H_U
\otimes \overline{H_U} d\mu(U)$, c.f. Remark \[RmkNotationDom\].
For $\eta = \int^\oplus \eta_U d\mu(U)\in H$ and $\xi = \int^\oplus
\xi_U d\mu(U)\in H$ so that $\xi \in {\mathcal{D}}(D^{-1})$, $\eta \in
{\mathcal{D}}(E)$ and $(\xi_U \otimes \overline{ E_U \eta_U})_U$ is square integrable, we find $\int {(\iota \otimes \omega_{\xi_U, \eta_U})(U)} d\mu(U) \in {\mathcal{N}_{\psi}}$ and $\int
{(\iota \otimes \omega_{D_U^{-1} \xi_U, E_U\eta_U})(U)} d\mu(U)^\ast \in {\mathcal{N}_{\varphi}}$ by Lemmas \[LemCompI\] and \[LemCompII\]. So $C \subseteq {\mathcal{Q}}_R(\Gamma({D_{J_0}}))$, and for $x \in
\Gamma({D_{J_0}})$, $J_0(x) = {\mathcal{Q}}_L^{-1} \circ \Sigma \circ {\mathcal{Q}}_R(x).$ Then, since ${\mathcal{Q}}_L$ and ${\mathcal{Q}}_R$ are isomorphisms, the claim follows from $\Sigma$ being a surjective anti-linear isometry.
Note that the previous proposition is an analogy of the classical situation. Suppose that $G$ is a locally compact group for which the classical Plancherel theorem [@DixC Theorem 18.8.1] holds. The anti-linear operator $f \mapsto f^\ast$ acting on $L^2(G)$ is transformed into the anti-linear flip acting on $\int^\oplus K(\zeta) \otimes \overline{K}(\zeta) d\mu(\zeta)$ by the Plancherel transform. Here $f^\ast(x) = \overline{f(x^{-1})} \delta_G(x^{-1})$ and $\delta_G$ is the modular function on $G$.
From Theorem \[ThmPolarDecI\] and Propositions \[PropNabla\] and \[PropJee\] we obtain the following result.
\[ThmDecompositionI\] The polar decomposition of ${S}$ is given by $S = J {\nabla^{\frac{1}{2}}}$.
The roles of $\varphi$ and $\psi$ can be interchanged. Consider the operator: $$\label{EqnConjugationIII}
{S}_0': {H_{\varphi}}\rightarrow {H_{\psi}}: \Lambda(x) \mapsto \Gamma(x^\ast), \quad x \in {\mathcal{N}_{\varphi}}\cap {\mathcal{N}_{\psi}}^\ast.$$ This operator is densely defined and preclosed. We denote its closure by ${S}'$. The polar decomposition of ${S}'$ can be expressed in terms of corepresentations in a similar way.
\[ThmDecompositionII\] Consider ${S}': {H_{\varphi}}\rightarrow {H_{\psi}}$. Let ${D_{J_0}}'$ be the linear space $$\begin{split}
& {\rm span}_\mathbb{C} \{ \int^\oplus_{{{\rm IC}(M)}}
{(\iota \otimes \omega_{\xi_U, \eta_U})(U)}^\ast d\mu(U) \mid {\rm where }\\ & \qquad \qquad \xi \in {\mathcal{D}}(D), \eta \in
{\mathcal{D}}(E^{-1}), (D_U \xi_U \otimes \overline{ \eta_U} )_U \:\: {\rm is\:\: sq. \:\: int.} \},
\end{split}$$ and define $J_0' : \Lambda({D_{J_0}}') \rightarrow {H_{\psi}}$: $$\Lambda(\int_{{{\rm IC}(M)}} {(\iota \otimes \omega_{\xi_U, \eta_U})(U)}^\ast d\mu(U)) \mapsto \Gamma(\int_{{{\rm IC}(M)}} {(\iota \otimes \omega_{D_U \xi_U, E_U^{-1}\eta_U})(U)} d\mu(U)).$$ Then $J_0'$ is densely defined and isometric, and its closure, denoted by $J'$, is a surjective anti-linear isometry. Let $$\begin{split}
& {D_{\nabla^\frac{1}{2}_0}}' = \textrm{span}_\mathbb{C} \{ \int_{{{\rm IC}(M)}}
{(\iota \otimes \omega_{\xi_U, \eta_U})(U)} d\mu(U) \mid {\rm where } \\ & \qquad \qquad \xi \in {\mathcal{D}}(D) \cap
{\mathcal{D}}(D^{-1}), (D \xi, \overline{\eta}) \in {\mathcal{D}^\otimes}(D^{-1},\overline{E}) \},
\end{split}$$ and define ${\nabla'^{\frac{1}{2}}}_0 : \Lambda({D_{\nabla^\frac{1}{2}_0}}) \rightarrow {H_{\psi}}$: $$\Lambda(\int_{{{\rm IC}(M)}} {(\iota \otimes \omega_{\xi_U, \eta_U})(U)} d\mu(U)^\ast) \mapsto \Lambda(\int_{{{\rm IC}(M)}} {(\iota \otimes \omega_{D_U^{-1} \xi_U, E_U\eta_U})(U)} d\mu(U)^\ast).$$ Then ${\nabla'^{\frac{1}{2}}}_0$ is a densely defined, preclosed operator and its closure, denoted by ${\nabla'^{\frac{1}{2}}}$, is a self-adjoint, strictly positive operator.
Moreover, the polar decomposition of ${S}'$ is given by ${S}'
= J' {\nabla'^{\frac{1}{2}}}$.
We now assume that $(M, \Delta)$ is unimodular, so that $S = S'= S_\psi$ and Theorem \[ThmDecompositionI\] give an explicit expression for the modular operator and modular conjugation. This leads to the following expression for the modular automorphism group. In this case we write $\sigma_t$ for $\sigma_t^\varphi = \sigma_t^\psi$.
\[ThmModularExpression\] Suppose that $(M, \Delta)$ is unimodular. Let $(\xi_U)_U, (\eta_U)_U$ be square integrable vector fields. The modular automorphism group $\sigma_t$ of the Haar weight $\psi$ can be expressed as:
$$\label{EqnModularExpression}
\sigma_t \left( \int_{{{\rm IC}(M)}} {(\iota \otimes \omega_{\xi_U, \eta_U})(U)}d\mu(U) \right) = \int_{{{\rm IC}(M)}} {(\iota \otimes \omega_{D_U^{2it}\xi_U, E_U^{2it}\eta_U})(U)}d\mu(U).$$
For $\eta = \int^\oplus \eta_U d\mu(U) \in H$, $\xi =
\int^\oplus \xi_U d\mu(U) \in H$, such that $( \xi_U \otimes \overline{\eta_U})_U$ is a square integrable field of vectors and $\eta \in {\mathcal{D}}(E)$, we find $$\label{EqnModularII}
\nabla^{it} \Gamma\left(\int {(\iota \otimes \omega_{\xi_U, \eta_U})(U)} d\mu(U)\right) = \Gamma\left(\int {(\iota \otimes \omega_{D_U^{2it}\xi_U, E_U^{2it}\eta_U})(U)} d\mu(U)\right).$$ Indeed, $\left( \int^\oplus (D_U\otimes \overline{E_U^{-1}})d\mu(U)\right)^{2it} (\xi \otimes \overline{\eta}) = \int^\oplus (D_U^{2it}\xi_U\otimes
\overline{E_U^{2it}\eta_U})d\mu(U)$ by [@Lan Theorem 1.10], so (\[EqnModularII\]) follows from Lemma \[LemCompII\] and Proposition \[PropNabla\]. Since $\sigma_t({\pi_{\psi}}(x)) = \nabla^{it} {\pi_{\psi}}(x) \nabla^{-it}, x \in M$, (\[EqnHilbertProduct\]) implies $$\label{EqnModExpression}
\sigma_t \left({\pi_{\psi}}\left( \int {(\iota \otimes \omega_{\xi_U, \eta_U})(U)} d\mu(U) \right)\right) = {\pi_{\psi}}\left( \int {(\iota \otimes \omega_{D_U^{2it}\xi_U, E_U^{2it}\eta_U})(U)} d\mu(U) \right),$$ so the theorem follows from the identification of $M$ with ${\pi_{\psi}}(M)$, in this case.
Now let $\eta = \int^\oplus \eta_U d\mu(U) \in H$ and $\xi =
\int^\oplus \xi_U d\mu(U) \in H$ be arbitrary. We take sequences of square integrable vector fields $\xi_n = \int^\oplus \xi_{U,n} d\mu(U)$, $\eta_n = \int^\oplus \eta_{U,n} d\mu(U)$ such that $(\xi_{U,n} \otimes \overline{\eta_{U,n}})_U$ is a square integrable field of vectors, $\eta_n \in {\mathcal{D}}(E)$ and such that $\xi_n$ converges to $\xi$ and $\eta_n$ converges to $\eta$. Then $\int {(\iota \otimes \omega_{\xi_{U,n}, \eta_{U,n}})(U)} d\mu(U)$ is $\sigma$-weakly convergent to $\int {(\iota \otimes \omega_{\xi_{U}, \eta_{U}})(U)} d\mu(U)$ and hence $$\begin{split}
& \sigma_t\left(\int {(\iota \otimes \omega_{\xi_{U}, \eta_{U}})(U)} d\mu(U) \right) = \lim_{n\rightarrow\infty} \sigma_t\left(\int {(\iota \otimes \omega_{\xi_{U,n}, \eta_{U,n}})(U)} d\mu(U)
\right) = \\
& \lim_{n\rightarrow\infty} \left(\int {(\iota \otimes \omega_{D_U^{2it}\xi_{U,n}, E_U^{2it}\eta_{U,n}})(U)} d\mu(U) \right)
= \int {(\iota \otimes \omega_{D_U^{2it}\xi_{U}, E_U^{2it}\eta_{U}})(U)} d\mu(U) ,
\end{split}$$ which yields (\[EqnModularExpression\]).
We used (\[EqnHilbertProduct\]) to obtain (\[EqnModExpression\]). The unimodularity assumption is essential for Theorem \[ThmModularExpression\].
Let $(M, \Delta)$ be unimodular. Let $\eta = \int^\oplus
\eta_U d\mu(U) \in H$, $\xi = \int^\oplus \xi_U d\mu(U) \in H$, $r \in \mathbb{R}$ be such that $\eta \in {\mathcal{D}}(E^{2r})$ and $\xi \in {\mathcal{D}}(D^{2r})$, then: $$\int_{{{\rm IC}(M)}} {(\iota \otimes \omega_{\xi_U, \eta_U})(U)}d\mu(U) \in {\mathcal{D}}(\sigma_z),$$ for all $z$ in the strip $S(r) := \left\{ z \in \mathbb{C} \mid 0 \leq \textrm{Im}(z) \leq r, \textrm{ or } r \leq \textrm{Im}(z) \leq 0 \right\}$. In particular, if $\eta$ is analytic for $E$ and if $\xi$ is analytic for $D$, then $\int_{{{\rm IC}(M)}} {(\iota \otimes \omega_{\xi_U, \eta_U})(U)}d\mu(U)$ is analytic for the one-parameter group $\sigma_t$.
For $\alpha \in M_\ast$, define $$\begin{split}
&F_\alpha (z) = \alpha \left(\int {(\iota \otimes \omega_{D_U^{2iz} \xi_U, E_U^{2i\overline{z}}\eta_U})(U)}d\mu(U) \right) = \\
&\langle \int^\oplus \!\!\!\! (\alpha \otimes \iota)(U) d\mu(U) (\int^\oplus \!\!\!\! D_U d\mu(U) )^{2iz} \int^\oplus \!\!\!\! \xi_U d\mu(U), (\int^\oplus \!\!\!\! E_U d\mu(U) ) ^{2i\overline{z}} \int^\oplus \!\!\!\!\eta_U d\mu(U)\rangle.
\end{split}$$ Here the last equality follows from [@Lan Theorem 1.10]. By [@TakII Lemma VI.2.3], $F_\alpha (z)$ is an analytic continuation of $\alpha\left(\sigma_t^\varphi\left(\int {(\iota \otimes \omega_{\xi_U, \eta_U})(U)}d\mu(U)\right)\right)$ to the strip $S(r)$ such that $F_\alpha(z)$ is bounded by a constant $C \Vert \alpha \Vert$ where $C$ is independent of $\alpha$. Moreover, $F_\alpha(z)$ is continuous on $S(r)$ and analytic on the interior $S(r)^\circ$. Therefore $F(z) = \int {(\iota \otimes \omega_{D_U^{2iz} \xi_U, E_U^{2i\overline{z}}\eta_U})(U)}d\mu(U) $ is a continuation of $\sigma_t \left(\int {(\iota \otimes \omega_{\xi_U, \eta_U})(U)}d\mu(U)\right)$ to the strip $S(r)$ such that $F(z)$ is bounded and $\sigma$-weakly continuous on $S(r)$ and analytic on the interior $S(r)^\circ$ [@KusOneParam Result 1.2].
Example {#SectExample}
=======
Using the theory of square integrable corepresentions, Desmedt [@Des] determined the operators $D_U$ and $E_U$ for the corepresentions that appear as discrete mass points of the Plancherel measure, see also Remark \[RmkSqIntCoreps\]. In particular, his theory applies to compact quantum groups, for which every corepresentation is square integrable. As a non-compact example, Desmedt was able to determine the operators $D_U$ for the discrete series corepresentations of the quantum group analogue of the normalizer of $SU(1,1)$ in $SL(2,\mathbb{C})$, which we denote by $(M, \Delta)$ from now on, see [@KoeKus] and [@GrKoeKus]. Having the theory of Sections \[SectPlancherel\] and \[SectModular\] at hand we determine the operators $D_U$ and $E_U$ for the principal series corepresentations of $(M, \Delta)$.
We refer to [@KoeKus] and [@GrKoeKus] for the relevant properties of $(M, \Delta)$ and use the same notational conventions. In [@GrKoeKus Theorem 5.7] a decomposition of the multiplicative unitary in terms of irreducible corepresentations is given: $$\label{EqnWDecomposition}
W = \bigoplus_{p \in q^\mathbb{Z}} \left( \int_{[-1,1]}^\oplus
W_{p,x} dx \oplus \bigoplus_{x\in \sigma_d(\Omega_p)} W_{p,x}
\right).$$ Here $\sigma_d(\Omega_p)$ is the discrete spectrum of the Casimir operator [@GrKoeKus Definition 4.5, Theorem 4.6] restricted to the subspace given in [@GrKoeKus Theorem 5.7]. $W_{p,x}$ is a corepresention that is a direct sum of at most 4 irreducible corepresentations [@GrKoeKus Propositions 5.3 and 5.4]. An orthonormal basis for the corepresentation Hilbert space $\mathcal{L}_{p,x}$ of $W_{p,x}$ is given the vectors $e^{{\varepsilon}, \eta}_m (p,x), {\varepsilon}, \eta \in \{ -, + \}, m \in
\mathbb{Z}$. The corepresentations $W_{p,x}, p \in q^\mathbb{Z}, x
\in \sigma(\Omega_p)$ are called the discrete series corepresentations and the corepresentations $W_{p,x}, p \in
q^\mathbb{Z}, x \in [-1,1]$ are called the principal series corepresentations. We denote $D_{p,x}$ and $E_{p,x}$ for $D_{W_{p,x}}$ and $E_{W_{p,x}}$. The operators $D_{p,x}$ have been computed by Desmedt [@Des] for the discrete series. Hence we focus on the principal series. In Appendix \[AppendixB\] we verify that $(M, \Delta)$ satisfies the conditions of the Plancherel theorem, so that the theory of Sections \[SectPlancherel\] and \[SectModular\] applies. Furthermore, $(M, \Delta)$ is unimodular [@KoeKus]. We denote the modular automorphism group of the Haar weight by $\sigma_t$.
By [@GrKoeKus Lemmas 10.9] the action of the matrix elements in the GNS-space can be calculated explicitly: $$\begin{split}
&(\iota \otimes \omega_{e^{{\varepsilon},\eta}_m, e^{{\varepsilon}',\eta'}_{m'}})\left(W_{p,x} \right) f_{m_0, p_0, t_0} = \\
& C(\eta{\varepsilon}x;m',{\varepsilon}',\eta';{\varepsilon}{\varepsilon}'\vert p_0\vert
p^{-1}q^{-m-m'},p_0,m-m') \delta_{sgn(p_0),\eta\eta'} f_{m_0 - m +
m',{\varepsilon}{\varepsilon}' \vert p_0\vert p^{-1} q^{-m-m'} , t_0}.
\end{split}$$ Fix $p \in q^\mathbb{Z}$. Let ${\varepsilon}, \eta, m, {\varepsilon}', \eta', m'$ be $\mu$-measurable functions of $x \in [-1,1]$, thus ${\varepsilon}= {\varepsilon}(x),
\eta = \eta(x), \ldots$. Let $f, f'$ be $\mu$-square integrable complex functions on $[-1,1]$. Then $f(x) e^{{\varepsilon}, \eta}_{m} = f(x) e^{{\varepsilon},
\eta}_{m}(p,x)$ and $ f'(x) e^{{\varepsilon}', \eta'}_{m'} = f'(x) e^{{\varepsilon}',
\eta'}_{m'}(p,x)$ are $\mu$-square integrable fields of vectors. Since the modular automorphism group $\sigma_t$ is implemented by $\gamma^\ast \gamma$ [@KoeKus Section 4], Theorem \[ThmModularExpression\] yields $$\label{EqnDubbelModular}
\begin{split}
& \left( \int_{[-1,1]} (\iota \otimes \omega_{f(x)D_{p,x}^{2it} e^{{\varepsilon},\eta}_m, f'(x)E_{p,x}^{2it}e^{{\varepsilon}',\eta'}_{m'}})\left(W_{p,x} \right) d\mu(x) \right) f_{m_0, p_0, t_0} \\
= & \sigma_t\left( \int_{[-1,1]} (\iota \otimes \omega_{f(x)e^{{\varepsilon},\eta}_m,
f'(x)e^{{\varepsilon}',\eta'}_{m'}})\left(W_{p,x} \right) d\mu(x) \right) f_{m_0,
p_0, t_0 } \\
= &\vert \gamma \vert^{2it} \left( \int_{[-1,1]} (\iota \otimes \omega_{f(x)e^{{\varepsilon},\eta}_m, f'(x)e^{{\varepsilon}',\eta'}_{m'}})\left(W_{p,x} \right) d\mu(x) \right) \vert \gamma \vert^{-2it} f_{m_0, p_0, t_0} \\
= & \left(\frac{p_0^2}{p_0^2p^{-2}q^{-2m-2m'}}\right)^{it} \int_{[-1,1]}
\!\!\!\! \!\!\!\! f(x)\overline{f'(x)}C(\eta{\varepsilon}x;m',{\varepsilon}',\eta';{\varepsilon}{\varepsilon}'\vert p_0\vert p^{-1}q^{-m-m'},p_0,m-m') \\
&\qquad\qquad\times \delta_{sgn(p_0),\eta\eta'} f_{m_0 - m + m',{\varepsilon}{\varepsilon}' \vert p_0\vert p^{-1} q^{-m-m'} , t_0} d\mu(x) \\
= & (p^{2}q^{2m+2m'})^{it} \int_{[-1,1]} (\iota \otimes
\omega_{f(x)e^{{\varepsilon},\eta}_m, f'(x)e^{{\varepsilon}',\eta'}_{m'}})\left(W_{p,x} \right)
d\mu(x) f_{m_0, p_0, t_0}.
\end{split}$$ Define $A$ and $B$ as the unbounded self-adjoint operators on $\int^\oplus_{[-1,1]} \mathcal{L}_{p,x} d\mu(x)$ determined by $A
= \int^\oplus_{[-1,1]} A_{p,x} d\mu(x)$, $A_{p,x} e_m^{{\varepsilon},
\eta}(p,x) = p^2q^{2m} e_m^{{\varepsilon}, \eta}(p,x)$. $B =
\int^\oplus_{[-1,1]} B_{p,x} d\mu(x)$, $B_{p,x} e_m^{{\varepsilon},
\eta}(p,x) = q^{-2m} e_m^{{\varepsilon}, \eta}(p,x)$. So (\[EqnDubbelModular\]) yields $$\label{EqnVoorbeeldGelijk}
\begin{split}
& \int (\iota \otimes \omega_{f(x)D_{p,x}^{2it} e^{{\varepsilon},\eta}_m,
f'(x)E_{p,x}^{2it}e^{{\varepsilon}',\eta'}_{m'}})\left(W_{p,x} \right) d\mu(x)
\\ = & \int (\iota \otimes \omega_{f(x)A_{p,x}^{it}e^{{\varepsilon},\eta}_m,
f'(x)B_{p,x}^{it}e^{{\varepsilon}',\eta'}_{m'}})\left(W_{p,x} \right) d\mu(x),
\end{split}$$ where the integrals are taken over $[-1,1]$. For any two bounded operators $F = \int^\oplus_{[-1,1]} F_{p,x} d\mu(x)$, $G
= \int^\oplus_{[-1,1]} G_{p,x} d\mu(x)$ on $\int^\oplus_{[-1,1]}
\mathcal{L}_{p,x} d\mu(x)$, the map $\left( \int^\oplus_{[-1,1]}
\mathcal{L}_{p,x} d\mu(x)\right) \otimes
\overline{\left(\int^\oplus_{[-1,1]} \mathcal{L}_{p,x}
d\mu(x)\right)} \rightarrow M$ given by $$v \otimes \overline{w} = \int^\oplus_{[-1,1]}
v_x d\mu(x) \otimes \overline{\int^\oplus_{[-1,1]} w_x d\mu(x)}
\mapsto \int_{[-1,1]} (\iota \otimes \omega_{F_{p,x} v_x, G_{p,x}
w_x})\left(W_{p,x} \right) d\mu(x)$$ is norm-$\sigma$-weakly continuous since $$\vert \int_{[-1,1]}
\alpha \otimes \omega_{v_x, w_x}(W_{p,x}) d\mu(x) \vert \leq \Vert
\alpha \Vert \Vert F \Vert \Vert G \Vert \Vert v \Vert \Vert w
\Vert, \alpha \in M_\ast.$$ Therefore, for $v =
\int^\oplus_{[-1,1]} v_x d\mu(x), w = \int^\oplus_{[-1,1]} w_x
d\mu(x) \in \int^\oplus_{[-1,1]} \mathcal{L}_{p,x} d\mu(x)$, using [@Dix II.1.6, Proposition 7] and (\[EqnVoorbeeldGelijk\]), $$\label{EqnEqualMatrixElements}
\int_{[-1,1]} (\iota \otimes \omega_{D_{p,x}^{2it} v_x,
E_{p,x}^{2it}w_x })\left(W_{p,x} \right) d\mu(x)
= \int_{[-1,1]} (\iota \otimes \omega_{A_{p,x}^{it}v_x,
B_{p,x}^{it} w_x})\left(W_{p,x} \right) d\mu(x).$$ For $v = \int^\oplus_{[-1,1]} v_x d\mu(x), w =
\int^\oplus_{[-1,1]} w_x d\mu(x) \in \int^\oplus_{[-1,1]}
\mathcal{L}_{p,x} d\mu(x)$, with $(v_x)_x $ essentially bounded, $w \in {\mathcal{D}}\left(
\int^\oplus_{[-1,1]} E_{p,x} d\mu(x) \right)$, Theorem \[ThmOrthogonalitRel\] implies that $ \int_{[-1,1]} (\iota \otimes
\omega_{D_{p,x}^{2it} v_x, E_{p,x}^{2it}w_x })\left(W_{p,x}
\right) d\mu(x) \in {\mathcal{N}_{\psi}}. $ By (\[EqnEqualMatrixElements\]) and Theorem \[ThmDomain\], $B_{p,x}^{it} w_x \in {\mathcal{D}}(E_{p,x})$ almost everywhere in the support of $(v_x)_x$. Theorem \[ThmOrthogonalitRel\] implies that for $v' = \int^\oplus_{[-1,1]} v_x' d\mu(x), w' =
\int^\oplus_{[-1,1]} w_x' d\mu(x) \in \int^\oplus_{[-1,1]}
\mathcal{L}_{p,x} d\mu(x)$ with the extra assumptions $w' \in {\mathcal{D}}\left(
\int^\oplus_{[-1,1]} E_{p,x}^2 d\mu(x) \right)$ and $(v'_x \otimes
E_{p,x} w'_x)_x$ is square integrable, $$\begin{split}
&\int_{[-1,1]} \langle B^{it}_{p,x} w_x, E_{p,x}^2 w_x' \rangle \langle v'_x,
A^{it}_{p,x} v_x \rangle d\mu(x) \\=&
\psi\left(\left( \int_{[-1,1]} (\iota \otimes \omega_{A^{it}_{p,x} v_x, B^{it}_{p,x} w_x})(W_{p,x}) d\mu(x)\right) ^\ast\int_{[-1,1]} (\iota \otimes \omega_{v_x', w_x'})(W_{p,x}) d\mu(x) \right) \\
=& \psi\left(\left( \int_{[-1,1]} (\iota \otimes \omega_{D_{p,x}^{2it} v_x, E_{p,x}^{2it}w_x})(W_{p,x}) d\mu(x)\right) ^\ast\int_{[-1,1]} (\iota \otimes \omega_{v_x', w_x'})(W_{p,x}) d\mu(x) \right)\\
=& \int_{[-1,1]} \langle E_{p,x}^{2it} w_x, E_{p,x}^2 w_x' \rangle
\langle v_x', D_{p,x}^{2it} v_x \rangle d\mu(x).
\end{split}$$ $E_{p,x}$ is strictly positive by the Plancherel theorem. The elements $\int^\oplus_{[-1,1]} v'_x \otimes \overline{ E_{p,x}^2 w'_x}
d\mu(x)$ are dense in $ \int^\oplus_{[-1,1]} \mathcal{L}_{p,x}
d\mu(x) \otimes \overline{ \int^\oplus_{[-1,1]} \mathcal{L}_{p,x}
d\mu(x)}$, so $ \int^\oplus_{[-1,1]} D_{p,x}^{2it} \otimes
\overline{E_{p,x}^{2it}} d\mu(x) = \int^\oplus_{[-1,1]}
A_{p,x}^{it} \otimes \overline{B_{p,x}^{it}} d\mu(x)$. By Stone’s theorem and [@Lan Theorem 1.10] $ \int^\oplus_{[-1,1]}
D_{p,x} \otimes \overline{E_{p,x}} d\mu(x) = \int^\oplus_{[-1,1]}
A_{p,x}^{\frac{1}{2}} \otimes \overline{B_{p,x}}^{\frac{1}{2}} d\mu(x)$. Hence we see that there is a positive function $c(p,x)$, such that $$\begin{split}
D_{p,x} e^{{\varepsilon}, \eta}_m &= p q^m c(p,x) e^{{\varepsilon}, \eta}_m,\\
E_{p,x} e^{{\varepsilon}, \eta}_m &= q^{-m} c(p,x) e^{{\varepsilon}, \eta}_m.
\end{split}$$ The function $c(p,x)$ depends on the choice of the Plancherel measure $\mu$, see [@Des Theorem 3.4.1, part 6].
Desmedt [@Des §3.5] obtains a similar result using summation formulas for basic hypergeometric series, a method different from the one presented here. Note that the present method also applies to discrete series corepresentations and avoids caclulations involving special functions.
Appendix {#AppendixA}
========
For the theory of direct integrals of bounded operators we refer to [@Dix]. For the theory of direct integrals of unbounded closed operators we refer to [@Lan], [@Nus] and [@Schm Chapter 12].
\[LemTensorCore\] Let $(X, \mu)$ be a standard measure space. Let $(H_p)_p$ and $(K_p)_p$ be measurable fields of Hilbert spaces. Let $(A_p)_p$ and $(B_p)_p$ be measurable fields of closed operators on $(H_p)_p$ and $(K_p)_p$ respectively. Let $(e^n_p)_p, n\in
\mathbb{N}$ be a fundamental sequence for $(A_p)_p$ and let $(f^n_p)_p, n\in \mathbb{N}$ be a fundamental sequence for $(B_p)_p$. Set $A = \int^\oplus_X A_p d\mu(p)$, $B = \int^\oplus_X
B_p d\mu(p)$, $H = \int^\oplus_X H_p d\mu(p)$ and $K =
\int^\oplus_X K_p d\mu(p)$.
1. $(A_p \otimes B_p)_p$ is a measurable field of closed operators.
2. The countable set $$R = \left\{ (e^n_p \otimes f^m_p)_p
\mid n,m \in \mathbb{N} \right\},$$ is a fundamental sequence for $(A_p \otimes B_p)_p$.
3. The set $$T = {\rm span}_{\mathbb{C}}\left\{ \int^\oplus_X \!\!\!\! \xi_p \otimes \eta_p d\mu(p) \mid \!\!\!\! \begin{array}{l} \xi = \int^\oplus_X \xi_p d\mu(p) \in {\mathcal{D}}(A), \\ \eta = \int^\oplus_X \eta_p d\mu(p) \in {\mathcal{D}}(B),\\ \int^\oplus_X(\xi_p\otimes \eta_p) d\mu(p) \in {\mathcal{D}}(\int ^\oplus(A_p
\otimes B_p)d\mu(p)) \end{array} \right\},$$ is a core for $\int^\oplus_X (A_p \otimes B_p) d\mu(p)$.
We first prove (a) and (b). By [@Dix II.1.8, Proposition 10], for $(\xi_p)_p$, $(\eta_p)_p$ measurable fields of vectors, there is a unique measurable structure so that $(\xi_p \otimes \eta_p)_p$ is a measurable field of vectors. We check (1) - (3) of [@Lan Remark 1.5, (1) - (3)].\
(1) $(e_p^n \otimes f_p^m)_p$ is a $\mu$-measurable field of vectors and $e^n_p \otimes f_p^m \in
{\mathcal{D}}(A_p \otimes B_p)$ for all $p$. The function $$p \mapsto \langle (A_p \otimes B_p) (e_p^n \otimes f_p^m)_p, (e_p^{n'} \otimes f_p^{m'})_p\rangle = \langle A_p e_p^n, e_p^{n'}\rangle \langle B_p f_p^m, f_p^{m'}\rangle,$$ is $\mu$-measurable, so (2) follows. For (3) fix a $p \in X$. By definition $\{ e_p^n \mid n\in \mathbb{N} \}$ is a core for $A_p$ and $\{ f_p^n \mid n\in \mathbb{N} \}$ is a core for $B_p$. Then it follows from [@Kad Lemma 11.2.29] that $\textrm{span}_{\mathbb{C}}\left\{ e^n_p \otimes f^m_p \mid n,m
\in \mathbb{N} \right\}$ is a core for $A_p \otimes B_p$, so that $R$ is total in ${\mathcal{D}}(A_p \otimes B_p)$ with respect to the graph norm. In all, we have proved (a) and (b).
Using [@Dix II.1.3, Remarque 1], we may assume that $(e^n_p)_p$ (resp. $(f^n_p)_p$) satisfies $p \mapsto
\Vert(e^n_p)_p\Vert$ (resp. $p \mapsto \Vert(f^n_p)_p\Vert$) is bounded and vanishes outside a set of finite measure. Let $$\lambda^{n,m}_p = \left( \textrm{max}(1, \Vert (A_p \otimes B_p) (e_p^n \otimes f_p^m) \Vert, \Vert A_p e_p^n \Vert, \Vert, \Vert B_p f_p^m \Vert)\right)^{-1},$$ so $\lambda^{n,m}_p$ is measurable and $0 < \lambda^{n,m}_p \leq 1
$. Using the assumption $\lambda^{n,m}_p (e_p^n \otimes f_p^m)\in
T$. Moreover, $p \mapsto \Vert \lambda_p^{n,m} (e_p^n \otimes
f_p^m)\Vert_{\textrm{Graph}(A_p \otimes B_p)}^2$ is bounded. Let $S = \{ (\lambda_p^{n,m} (e_p^n \otimes f_p^m))_p \mid n,m \in
\mathbb{N} \} \subseteq T$. Now define $$M = \bigcup_{f \in \mathcal{C}} m_f S,$$ where $\mathcal{C}$ is the set of bounded measurable scalar-valued functions vanishing outside a set of finite measure and $m_f$ is multiplication by $f$. Then $M \subseteq T \subseteq
{\mathcal{D}}(\int^{\oplus}_X (A_p \otimes B_p) d\mu(p))$ and by [@Dix II.1.6, Proposition 7], $M$ is total in ${\mathcal{D}}(\int^{\oplus}_X (A_p \otimes B_p) d\mu(p))$ equipped with the graph norm. Hence $T$ is a core for $\int^{\oplus}_X (A_p \otimes
B_p) d\mu(p)$.
Appendix {#AppendixB}
========
$(M,\Delta)$ denotes the quantum group analogue of the normalizer of $SU(1,1)$ in $SL(2, \mathbb{C})$. We use the same notation as in [@KoeKus] and [@GrKoeKus]. The Casimir operator $\Omega$ is defined in [@GrKoeKus Definition 4.5]. $\sigma(\Omega)$ and $\sigma_d(\Omega)$ denote the spectrum and the discrete spectrum of $\Omega$ respectively.
\[PropDiscContCoreps\] Let $x \in [-1,1]$ and $x' \in \sigma_d(\Omega)$, so in particular $x \not= x'$. Then the irreducible summands of $W_{p,x}$ are all inequivalent from $W_{p,x'}$.
This follows from [@GrKoeKus], since the eigenvalues of $\Omega$ when restricted to $W_{p,x} '$ are contained in $\mathbb{R} \backslash [-1,1]$, whereas for $W_{p,x}$ the eigenvalues of $\Omega$ are in $[-1,1]$.
The next propositions show that $(M, \Delta)$ satisfies the conditions of the Plancherel theorem, cf. Remark \[RmkReducedSetting\].
$\hat{M}$ is a type I von Neumann algebra.
We start with some preliminary remarks. The projections in $\hat{M}'$ correspond to the invariant subspaces of $W$ and the minimal projections in $\hat{M}'$ correspond to the irreducible subspaces of $W$. The partial isometries in $\hat{M}'$ correspond to intertwiners of closed subcorepresentations of $W$.
Let $P \in \hat{M}'$ be the projection on $\bigoplus_{p \in
q^\mathbb{Z}} \int_{[-1,1]}^\oplus \mathcal{L}_{p,x}$. There are no intertwiners between closed subcorepresentations of $\bigoplus_{p \in q^\mathbb{Z}} \int_{[-1,1]}^\oplus W_{p,x} dx $ and the direct sum $\bigoplus_{p \in q^\mathbb{Z}} \int_{x \in
\sigma_d(\Omega)}^\oplus W_{p,x} $, see Proposition \[PropDiscContCoreps\]. Therefore, $P$ commutes with every partial isometry in $\hat{M}'$ so that $P$ is central. We have $\hat{M}' = P\hat{M}'P \oplus (1-P)\hat{M}'(1-P)$. The von Neumann algebra $(1-P)\hat{M}'(1-P)$ is of type I since the direct sum decomposition $\bigoplus_{p \in q^\mathbb{Z}} \int_{x \in
\sigma_d(\Omega)}^\oplus W_{p,x}$ together with the preliminary remarks yield that every projection majorizes a minimal projection.
Now we prove that $P\hat{M}'P$ is a type I von Neumann algebra. Define the Hilbert spaces $$\mathcal{L}_x = \left( \bigoplus_{p \in q^{\mathbb{Z}}}
\mathcal{L}_{p,x} \right) \oplus \left( \bigoplus_{p \in
q^{\mathbb{Z}}} \mathcal{L}_{p,-x}\right), \quad x \in
(0,1);\qquad \mathcal{L}_0 = \oplus_{p \in q^{\mathbb{Z}}}
\mathcal{L}_{p,0}.$$ Then, $$\label{EqnDirectIntDec}
P\mathcal{K} = \int^\oplus_{[0,1]} \mathcal{L}_x dx,$$ and we let $\mathcal{Z}$ denote the diagonizable operators with respect to this direct integral decomposition.
We claim that $\mathcal{Z} \subseteq \hat{M}' \subseteq
\mathcal{Z}'$. For the former inclusion, note that the stepfunctions in $\mathcal{Z}$ are linear combinations of projections onto invariant subspaces for $\hat{M}$. By the preliminary remarks we find $\mathcal{Z} \subseteq \hat{M}'$. To prove that $\hat{M}' \subseteq \mathcal{Z}'$, note that by [@GrKoeKus Corollary 4.11], $\hat{M}'$ is the [$\sigma$-strong-$\ast$]{} closure of the linear span of elements $\hat{J} Q(p_1, p_2, n) \hat{J}$, $p_1, p_2 \in q^\mathbb{Z}$, $n
\in \mathbb{Z}$. The operators $Q(p_1, p_2, n)$ are decomposable with respect to the direct integral decomposition (\[EqnDirectIntDec\]) as was proved in [@GrKoeKus]; combine [@GrKoeKus Proposition 10.5] together with the direct integral decomposition [@GrKoeKus Theorem 5.7] and the definition of $Q(p_1, p_2, n)$ [@GrKoeKus Equation (20)]. We prove that $\hat{J}$ is a decomposable operator with respect to (\[EqnDirectIntDec\]). It suffices to show that $\hat{J}
\subseteq \mathcal{Z}'$ [@Dix Theorem II.2.1].
Let $B \subseteq [0,1]$ be a Borel set and let $P_B \in
\mathcal{Z}$ be the operator $P_B = \int^\oplus_{[0,1]} \chi_B(x)
1_{\mathcal{L}_x} dx$, where $\chi_B$ is the indicator function on $B$. $P_B$ is a projection and we have $$\label{EqnProjectionPB}
\begin{split}
& \chi_{B\cup -B }(\Omega) \mathcal{K} = \chi_{B\cup -B }(\Omega) \bigoplus_{p,m,\epsilon, \eta} \mathcal{K}(p,m,\epsilon, \eta) =\\
& \bigoplus_p \left( \bigoplus_{m,\epsilon\eta = 1} \int^\oplus_{ x \in B \cup -B} \mathbb{C} dx \oplus \bigoplus_{m,\epsilon\eta = -1} \int^\oplus_{-x \in B \cup -B} \mathbb{C} dx \right) =\\
& \bigoplus_p \bigoplus_{m,\epsilon,\eta} \int^\oplus_{ x \in B \cup -B} \mathbb{C} dx
= \bigoplus_p \int^\oplus_{ x \in B \cup -B} \mathcal{L}_{p,x} dx
= \int^\oplus_{ x \in B} \mathcal{L}_{x} dx
= P_B \mathcal{K},
\end{split}$$ where the second equation uses [@Lan Theorem 1.10] and the fact that there is a direct integral decomposition $\mathcal{K}(p,m,\epsilon, \eta) = \int^\oplus_{\sigma(\Omega)}
\mathbb{C} dx$ such that $\chi_B(\Omega) \mathcal{K}(p,m,\epsilon,
\eta) = \int^\oplus_{\varepsilon \eta x \in B} \mathbb{C} dx$, see [@GrKoeKus Theorem 8.13]. Other equations are a matter of changing the order and combining direct integrals.
Note that $\Omega$ leaves the spaces $\mathcal{K}^+$ and $\mathcal{K}^-$ invariant. Let $P^+$ and $P^-$ be the projections onto respectively $\mathcal{K}^+$ and $\mathcal{K}^-$. Write, again using the notation of [@GrKoeKus] $$\Omega = \left(
\begin{array}{ll}
\Omega^+ & 0 \\
0 & \Omega^-
\end{array}
\right), \qquad \Omega_0 = \left(
\begin{array}{ll}
\Omega^+_0 & 0 \\
0 & \Omega^-_0
\end{array}
\right),$$ where $\Omega^\pm = \Omega P^\pm$ and $\Omega_0^\pm =
\Omega_0 P^\pm$. Note that $\Omega^\pm$ is a self-adjoint extension of $\Omega_0^\pm$. By [@GrKoeKus Equation (11)] we see that $\hat{J}$ leaves the spaces $\mathcal{K}^+$ and $\mathcal{K}^-$ invariant. We claim that $$\label{EqnOperatorOmega}
\hat{J}\vert_{\mathcal{K}^+} \Omega^+ \hat{J}\vert_{\mathcal{K}^+} = \Omega^+, \qquad \hat{J}\vert_{\mathcal{K}^-} \Omega^- \hat{J}\vert_{\mathcal{K}^-} = -\Omega^-.$$ By [@GrKoeKus Equations (11) and (19)] we find that $\hat{J} \Omega_0 \hat{J} f_{m,p,t} = {\textrm{sgn}}(pt) \Omega_0
f_{m,p,t}$, so that $\hat{J} \Omega_0^+ \hat{J} = \Omega_0^+$ and $\hat{J} \Omega_0^- \hat{J} = -\Omega_0^-$. Hence $\hat{J}
\Omega^+ \hat{J} \supseteq \Omega_0^+$, and $\hat{J} \Omega^-
\hat{J} \supseteq -\Omega_0^-$. Let $x \in \hat{M}'$, and write: $$\begin{split}
\hat{J} x \hat{J} = y^+ \oplus y^-, &\:\:\qquad y^+ = \left(
\begin{array}{ll}
y^{+}_1 & 0 \\
0 & y^{+}_2
\end{array}
\right) \in M_+, \:\: y^- = \left(
\begin{array}{ll}
0 & y^{-}_2 \\
y^{-}_1 & 0
\end{array}
\right) \in M_-,
\end{split}$$ where the decomposition is as in [@GrKoeKus Proposition 4.8]. By that same proposition, we find that $y^-_1
\Omega^+ \subseteq - \Omega^- y^-_1$, $y^-_2 \Omega^- \subseteq -
\Omega^+ y^-_2$, $y^+_1 \Omega^+ \subseteq \Omega^+ y^+_1$ and $y^+_2 \Omega^- \subseteq \Omega^- y^+_2$. This implies the inclusion in the following computation: $$\begin{split}
& x \hat{J}\left(
\begin{array}{ll}
\Omega^+ & 0 \\
0 & -\Omega^-
\end{array}
\right)\hat{J} = \hat{J} \hat{J} x \hat{J} \left(
\begin{array}{ll}
\Omega^+ & 0 \\
0 & -\Omega^-
\end{array}
\right) \hat{J} = \\
& \hat{J} \left(
\begin{array}{ll}
y^{+}_1 & 0 \\
0 & y^{+}_2
\end{array}
\right)\left(
\begin{array}{ll}
\Omega^+ & 0 \\
0 & -\Omega^-
\end{array}
\right) \hat{J} \oplus \hat{J} \left(
\begin{array}{ll}
0 & y^{-}_2 \\
y^{-}_1 & 0
\end{array}
\right)\left(
\begin{array}{ll}
\Omega^+ & 0 \\
0 & -\Omega^-
\end{array}
\right) \hat{J} \subseteq \\
& \hat{J} \left(
\begin{array}{ll}
\Omega^+ & 0 \\
0 & -\Omega^-
\end{array}
\right) (y^+ \oplus y^-) \hat{J}
= \hat{J} \left(
\begin{array}{ll}
\Omega^+ & 0 \\
0 & -\Omega^-
\end{array}
\right) \hat{J} x.
\end{split}$$ So $\hat{J} \Omega^+ \hat{J} \oplus - \hat{J} \Omega^-
\hat{J}$ is a self-adjoint operator affiliated to $\hat{M}$ extending $\Omega_0$. So [@GrKoeKus Theorem 4.6] implies that $\left(\hat{J}\vert_{\mathcal{K}^+} \Omega^+ \hat{J}\vert_{\mathcal{K}^+} \oplus - \hat{J}\vert_{\mathcal{K}^-} \Omega^-
\hat{J}\vert_{\mathcal{K}^-} \right) = \Omega$, which results in (\[EqnOperatorOmega\]).
To prove that $\hat{J} \subseteq \mathcal{Z}'$, it suffices to prove that for all Borel sets $B \subseteq [0,1]$, $\hat{J} P_B \hat{J} = P_B$. Indeed we have $$\begin{split}
\hat{J} P_B \hat{J} = \hat{J} \chi_{B\cup -B}(\Omega) \hat{J} &= \hat{J}\vert_{\mathcal{K}^+} \chi_{B \cup -B}(\Omega^+) \hat{J}\vert_{\mathcal{K}^+} \oplus \hat{J}\vert_{\mathcal{K}^-} \chi_{B\cup -B}(\Omega^-) \hat{J}\vert_{\mathcal{K}^-} \\&= \chi_{B\cup-B}(\Omega^+) \oplus \chi_{B\cup -B}(\Omega^-) = \chi_{B\cup -B}(\Omega) =
P_B.
\end{split}$$ The first and last equality are due to (\[EqnProjectionPB\]); the third equality is due to (\[EqnOperatorOmega\]). In all, we have proved that $\mathcal{Z}
\subseteq \hat{M} \subseteq \mathcal{Z}'$.
Let $W_x
= \left( \bigoplus_{p \in q^\mathbb{Z}} W_{p,x}\right) \oplus
\left( \bigoplus_{p \in q^\mathbb{Z}} W_{p,-x}\right)$ for $x \in
(0,1]$ and $W_0 = \bigoplus_{p \in q^\mathbb{Z}} W_{p,0}$. The operators $Q(p_1,p_2,n)$ form a countable family that generates $\hat{M}$ [@GrKoeKus Proposition 4.9]. We apply [@Dix Theorem II.3.2] and its subsequent remark, together with [@Dix Theorem II.3.1] to conclude that
$$\label{EqnMDecomposition}
P\hat{M}P = \int^\oplus_{x \in [0,1]} \hat{M}_x dx,$$
where $\hat{M}_x$ is generated by $\left\{
(\omega\otimes \iota)(W_x) \mid \omega \in M_\ast \right\}$ almost everywhere. The projections in $\hat{M}_x'$ correspond to irreducible subspaces of $W_x$. Since $W_x$ decomposes as a direct sum of irreducible corepresentations [@GrKoeKus Proposition 5.4], every projection in $\hat{M}_x'$ majorizes a minimal projection. We find that $\hat{M}_x'$ is type I and by [@Kad Theorem 14.1.21], [@TakI Corollary V.2.24] and (\[EqnMDecomposition\]) we conclude that $P\hat{M}P$ is type I .
$\hat{M}_c$ is separable.
Note that if $\omega_n \in M_\ast$ is sequence that converges in norm to $\omega \in M_\ast$, then $\Vert
\lambda(\omega_n) - \lambda(\omega) \Vert \leq \Vert \omega_n -
\omega \Vert$ so that $\lambda(\omega_n)$ converges in norm to $\lambda(\omega)$. Since the norm on $\hat{M}_c$ is the operator norm on the GNS-space and $\hat{M}_c$ is the C\*-algebra obtained as the closure of $\{ \lambda(\omega) \mid \omega \in M_\ast \}$. It suffices to check that $M_\ast$ is separable. The $\mathbb{Q}$-linear span of $\{ \omega_{f_{m_0,p_0,t_0},
f_{m_1,p_1,t_1}} \mid m_i \in \mathbb{Z}, p_i, t_i \in I_q, i
=0,1\}$ is weakly dense, hence norm dense in $M_\ast$.
[99]{}
Buss, A., and R. Meyer, *Square-integrable coactions of locally compact quantum groups*, Rep. Math. Phys. [**63**]{} (2009), 191-224.
Desmedt, P., “Aspects of the theory of locally compact quantum groups: Amenability - Plancherel measure”, PhD-Thesis, Katholieke Universiteit Leuven 2003.
Diestel, J., and J.J. Uhl Jr., “Vector measures”, Math. Surv. 15., AMS, 1977.
Dixmier, J., “Les algèbres d’opérateurs dans l’espace hilbertien”, Gauthiers-Villars, 1957.
Dixmier, J., “Les C\*-algèbres et leurs représentations”, Gauthiers-Villars, 1969.
Groenevelt, W., E. Koelink and J. Kustermans, *The dual quantum group for the quantum group analog of the normalizer of $SU(1,1)$ in $SL(2, \mathbb{C})$*, IMRN [**7**]{} (2010), 1167-1314.
Kadison, R.V., and J.R. Ringrose, “Fundamentals of the theory of operator algebras I and II”, AMS, 1997.
Kato, T., “Perturbation theory for linear operators”, Springer, 1976.
Koelink, E., and J. Kustermans, *A locally compact quantum group analogue of the normalizer of $\rm SU(1,1)$ in ${\rm SL}(2,\mathbf{C})$*, Comm. Math. Phys. [**233**]{} (2003), 231–296.
Kustermans, J., *Locally compact quantum groups in the universal setting*, Internat. J. Math. [**12**]{} (2001), 289–338.
Kustermans, J., “Locally compact quantum groups”, LNM 1865, Springer 2005, 99-180.
Kustermans, J., and S. Vaes, *Locally compact quantum groups*, Ann. Scient. Éc. Norm. Sup. [**33**]{} (2000), 837-934.
Kustermans, J., and S. Vaes, *Locally compact quantum groups in the von Neumann algebraic setting*, Math. Scand. [**92**]{} (2003), 68-92.
Kustermans, J., *One-parameter representations of C$^\ast$-algebras*, arXiv:funct-an/9707009, 1997.
Lance, C., *Direct integrals of left Hilbert algebras*, Math. Ann. [**216**]{} (1975), 11–28.
Nussbaum, A.E., *Reduction theory for unbounded closed operators in Hilbert space*, Duke Math. J. [**31**]{} (1964), 33-44.
Takesaki, M., “Theory of operator algebras I”, Springer, 1979.
Takesaki, M., “Theory of operator algebras II”, Springer, 2000.
Terp, M., *Interpolation spaces between a von Neumann algebra and its predual*, J. Operator Theory [**8**]{} (1982), 327-360.
Schmüdgen, K., “Unbounded operator algebras and representation theory”, Birkhäuser Verlag, 1990.
Timmermann, T., “An invitation to quantum groups and duality”, EMS, 2008.
Vaes, S., *A Radon-Nikodym theorem for von Neumann algebras.* J. Operator Theory [**46**]{} (2001), 477-489.
Vaes, S., and L. Vainerman, *Extensions of locally compact quantum groups and the bicrossed product construction.* Adv. in Math. [**175**]{} (2003), 1 - 101.
Woronowicz, S.L., *Compact matrix pseudogroups*, Commun. Math. Phys. [**111**]{} (1987), 613-665.
|
---
abstract: |
In a previous paper we showed that for every polarization on an abelian variety there is a dual polarization on the dual abelian variety. In this note we extend this notion of duality to families of polarized abelian varieties. As a main consequence we obtain an involution on the set of moduli spaces of polarized abelian varieties of dimension $g$. In particular, the moduli spaces $\mathcal
A_{(d_1,\ldots,d_g)}$ and $\mathcal
A_{(d_1,\frac{d_1d_g}{d_{g-1}},\ldots,\frac{d_1d_g}{d_2},d_g)}$ are canonically isomorphic.
address:
- |
Ch. Birkenhake\
Universität Mainz\
Fachbereich Mathematik\
Staudingerweg $9$\
D-$55099$ Mainz
- |
H. Lange\
Mathematisches Institut\
Universität Erlangen-Nürnberg\
Bismarckstraße $1\frac{ 1}{2}$\
D-$91054$ Erlangen\
Germany
author:
- Christina Birkenhake
- Herbert Lange
title: An Isomorphism between Moduli Spaces of Abelian Varieties
---
[^1]
Introduction
============
Let $k$ be an algebraically closed field and $(d_1,\ldots,d_g)$ a vector of positive integers such that $d_i|d_{i+1}$ for all $i=1,\ldots,g-1$ and $\mathrm{char}\,k\!\not\!|\,\,d_g$. Then the coarse moduli space $\mathcal A_{(d_1,\ldots,d_g)}$ of polarized abelian varieties of type $(d_1,\ldots,d_g)$ over $k$ exists and is a quasi-projective variety of dimension $\frac{1}{2}g(g+1)$ over $k$. The main result of this paper is the following theorem.
There is a canonical isomorphism of coarse moduli spaces $$\mathcal
A_{(d_1,\ldots,d_g)}\stackrel{\sim}{\longrightarrow} \mathcal
A_{(d_1,\frac{d_1d_g}{d_{g-1}},\ldots,\frac{d_1d_g}{d_2},d_g)}.$$
For the proof we show that there is a canonical isomorphism of the corresponding moduli functors. Namely, for every polarized projective abelian scheme $(A\longrightarrow
S,\lambda)$ we define a dual abelian scheme $({\Hat{A}}\longrightarrow
S,\lambda^\delta)$. Here ${\Hat{A}}\longrightarrow S$ is the usual dual abelian scheme and $\lambda^{\delta}$ is a polarization on ${\Hat{A}}/S$ with the property $(\lambda^{\delta})^{\delta}=\lambda$. Hence it makes sense to call $\lambda^\delta$ the *dual polarization* of ${\Hat{A}}/S$.
There are several possibilities to define the polarization $\lambda^\delta$. In section 2 we define a polarization $\lambda^D$ by inverting the isogeny $\lambda:A\longrightarrow{\Hat{A}}$. If $\lambda$ is given by a line bundle $\mathcal L$ we define in section 4, using Mukai’s Fourier functor, a relatively ample line bundle ${\Hat{\mathcal L}}$ on ${\Hat{A}}$ inducing a polarization $\phi_{{\Hat{\mathcal L}}}$ of ${\Hat{A}}$. Both polarizations are defined even in the non separable case. We show that they are multiples of each other. If ${\lambda}$ is a separable polarization of type $(d_1,\ldots,d_g)$, then $\lambda^{\delta}=d_1{\lambda}^D$, and if moreover ${\lambda}=\phi_{\mathcal L}$ with a relatively ample line bundle ${\mathcal L}$ on $A/S$, then $\phi_{{\Hat{\mathcal L}}}=d_1\cdots d_{g-1}\phi_{\mathcal L}^D=d_2\cdots d_{g-1}{\lambda}^{\delta}$.
The Dual Polarization $\lambda^\delta$
======================================
Let $\pi:A\longrightarrow S$ be an abelian scheme over a connected Noetherian scheme $S$. Grothendieck showed in [@Groth] that if $A/S$ is projective, the relative Picard functor $\mathcal{P}ic^0_{A/S}$ is represented by a projective abelian scheme ${\Hat{\pi}}:{\Hat{A}}\longrightarrow S$, called the dual abelian scheme. According to a theorem of Cartier and Nishi there is a canonical isomorphism $A\stackrel{\sim}{\longrightarrow}{\Hat{{\Hat{A}}}}$ over $S$. We always identify $A$ with its bidual ${\Hat{{\Hat{A}}}}$.\
For any $a\in A(S)$ let $t_a:A\longrightarrow A$ denote the translation by $a$ over $S$. Any line bundle $\mathcal L$ on $A$ defines a homomorphism $$\phi_{\mathcal L}:A\longrightarrow{\Hat{A}}$$ over $S$ characterized by $\phi_{\mathcal L}(a):=t_a^*{\mathcal L}\otimes{\mathcal L}^{-1}$ for all $a\in
A(S)$. The biduality implies that $\phi_{\mathcal L}$ is symmetric: ${\Hat{\phi}}_{\mathcal L}=\phi_{\mathcal L}$. Clearly $\phi_{{\mathcal L}\otimes\pi^*\mathcal M}=\phi_{\mathcal L}$ for any line bundle $\mathcal M$ on $S$. Moreover $\phi_{\mathcal L}$ is an isogeny if ${\mathcal L}$ is relatively ample.\
A *polarization* of $\pi:A\longrightarrow S$ is a homomorphism $\lambda:A\longrightarrow{\Hat{A}}$ over $S$ such that for every geometric point $s$ of $S$ the induced map $\lambda_s:A_s\longrightarrow{\Hat{A}}_s$ is of the form $\lambda_s=\phi_L$ for some ample line bundle $L$ of $A_s$. (Here ${\Hat{A}}_s:=({\Hat{A}})_s=(A_s){\Hat{}}$ )\
Obviously for any relatively ample line bundle ${\mathcal L}$ on $A/S$ the homomorphism $\phi_{\mathcal L}$ is a polarization. Conversely not every polarization of $A$ is of the form $\phi_{\mathcal L}$. However, if $\lambda:A\longrightarrow{\Hat{A}}$ is a polarization, then $2\lambda=\phi_{\mathcal L}$ for some relatively ample line bundle ${\mathcal L}$ on $A/S$ (see [@GIT] Prop 6.10, p.121).\
Let $\lambda:A\longrightarrow{\Hat{A}}$ be a polarization of $A$. Its kernel $K(\lambda)$ is a commutative group scheme, finite and flat over $S$. According to a Lemma of Deligne (see [@ot] p. 4) there is a positive integer $n$ such that $K(\lambda)$ is contained in the kernel $A_n:=\mathrm{ker}\{n_A:A\longrightarrow A\}$ where $n_A$ denotes multiplication by $n$. The *exponent* $e=e(\lambda)$ of the polarization $\lambda$ is by definition the smallest such positive integer.
\[prop2.1\] There is a polarization $\lambda^D:{\Hat{A}}\longrightarrow{\Hat{{\Hat{A}}}}=A$ of ${\Hat{A}}$, uniquely determined by $\lambda$, such that $\lambda^D\circ\lambda=e_A$ and $\lambda\circ\lambda^D=e_{{\Hat{A}}}$.
Since $K(\lambda)\subset A_e$ by definition of the exponent, there is a uniquely determined homomorphism $\lambda^D:{\Hat{A}}\longrightarrow A$ such that $\lambda^D\circ\lambda=e_A$. Hence $$(\lambda\circ{\lambda}^D)\circ{\lambda}={\lambda}\circ({\lambda}^D\circ{\lambda})={\lambda}\circ
e_A=e_{{\Hat{A}}}\circ{\lambda}.$$ So ${\lambda}\circ{\lambda}^D=e_{{\Hat{A}}}$, the homomorphism ${\lambda}$ being surjective. It remains to show that ${\lambda}^D$ is a polarization. For this suppose $s$ is a geometric point of $S$. So $A_s$ and ${\Hat{A}}_s$ are abelian varieties over an algebraically closed field. It is well known that a homomorphism $\varphi:{\Hat{A}}_s\longrightarrow{\Hat{{\Hat{A}}}}_s=A_s$ is of the form $\phi_M$ for some $M\in\mathrm{Pic}({\Hat{A}}_s)$ if and only if $\varphi$ is symmetric: ${\Hat{\varphi}}=\varphi$. So using ${\Hat{\lambda_s}}=\lambda_s$ and dualizing equation ${\lambda}^D_s\circ{\lambda}_s=e_{A_s}$ give ${\lambda}_s\circ{\Hat{{\lambda}^D_s}}={\Hat{{\lambda}}}_s\circ{\Hat{{\lambda}^D_s}}=e_{{\Hat{A}}_s}$ Comparing this with the equation ${\lambda}_s\circ{\lambda}^D_s=e_{{\Hat{A}}_s}$ and using the fact that ${\lambda}_s$ and $e_{{\Hat{A}}_s}$ are isogenies, implies ${\Hat{{\lambda}^D_s}}={\lambda}^D_s$. Hence there is a line bundle $M$ on ${\Hat{A}}_s$ such that ${\lambda}^D_s=\phi_M$, and it remains to show that $M$ is ample. For this note that $\phi_{{\lambda}_s^*M}={\Hat{{\lambda}}}_s\circ\phi_M\circ{\lambda}_s={\lambda}_s\circ{\lambda}_s^D\circ{\lambda}_s=e{\lambda}_s.$ Now $e{\lambda}_s$ being a polarization implies that ${\lambda}_s^*M$, and thus also $M$ is ample.
A polarization $\lambda:A\longrightarrow {\Hat{A}}$ is called *separable* if for every geometric point $s:\mathrm{Spec}(k)\longrightarrow S$ the characteristic of $k$ does not divide the exponent $e(\lambda_s)$ of $\lambda_s$. In this case $$K(\lambda_s)=\mathrm{ker}(\lambda_s)\simeq({\mathbb Z}/d_1{\mathbb Z}\times\cdots{\mathbb Z}/d_{g}{\mathbb Z})^2$$ with positive integers $d_1,\ldots,d_g$ such that $d_i|d_{i+1}$ and $d_g=e(\lambda_s)$ (see [@M] p. 294). Since the base scheme $S$ is connected and $K(\lambda)$ is an algebraic group scheme over $S$ with $K(\lambda)_s=K(\lambda_s)$ the integers $d_1,\ldots,d_g$ do not depend on $s$. The vector $(d_1,\ldots,d_g)$ is called the *type of the polarization* $\lambda$.
\[prop2.2\] Suppose $\lambda$ is a separable polarization of the abelian scheme $A/S$ of type $(d_1,\ldots,d_g)$. Then the polarization $\lambda^D$ of ${\Hat{A}}/S$ is separable of type $(1,\frac{d_g}{d_{g-1}},\ldots,\frac{d_g}{d_1})$.
Let $s$ be a geometric point of $S$. Equation $\lambda^D_s\circ\lambda_s=e(\lambda_s)_{A_s}$ implies that the exponent $e(\lambda_s^D)$ of $\lambda_s^D$ divides $e(\lambda_s)=d_g$, hence $\lambda^D$ is separable. Moreover $$\begin{aligned}
K(\lambda_s^D)=&\mathrm{ker}(\lambda^D_s)=\mathrm{ker}(e(\lambda_s)_{A_s})/\mathrm{ker}(\lambda_s)\\
\simeq&
({\mathbb Z}/d_g{\mathbb Z})^{2g}/({\mathbb Z}/d_1{\mathbb Z}\times\cdots\times{\mathbb Z}/d_g{\mathbb Z})^2\\
\simeq&\textstyle({\mathbb Z}/\frac{d_g}{d_{g-1}}{\mathbb Z}\times\cdots\times{\mathbb Z}/\frac{d_g}{d_1}{\mathbb Z})^2.\end{aligned}$$
If $\lambda$ is a separable polarization of type $(d_1,\ldots,d_g)$, then Proposition \[prop2.2\] implies that $(\lambda^D)^D$ is of type $(1,\frac{d_2}{d_1},\ldots,\frac{d_g}{d_1})$ and hence does not coincide with $\lambda$ if $d_1\not= 1$. However for $$\lambda^\delta:=d_1\lambda^D$$ we have
\[prop2.3\] $(\lambda^\delta)^\delta=\lambda$
By definition ${\lambda}^\delta$ is of type $(d_1,\frac{d_1d_g}{d_{g-1}},\ldots,\frac{d_1d_g}{d_2},d_g)$. So both ${\lambda}$ and ${\lambda}^\delta$ have exponent $d_g$. Applying Theorem \[prop2.1\] to $\lambda$ and $\lambda^\delta$ we get $\lambda^\delta\circ\lambda=d_1({\lambda}^D\circ{\lambda})=(d_1d_g)_A=d_1({\lambda}^\delta\circ({\lambda}^\delta)^D)=
\lambda^\delta\circ(\lambda^\delta)^\delta$. This implies the assertion.
Applications to Moduli Spaces
=============================
Let $k$ be an algebraically closed field and $\mathcal S$ the category of schemes of finite type over $k$. Fix a vector $(d_1,\ldots,d_g)$ of positive integers such that $d_i|d_{i+1}$ and $\mathrm{char}\,k\!\!\not|\,d_g$. Consider the functor $\underline{\mathcal A}_{(d_1,\ldots,d_g)}:\mathcal S
\rightarrow \{sets\}$ defined by $$S\mapsto\{\text{isomorphism classes of pairs }(A\rightarrow S,{\lambda})\}$$ where $A\rightarrow S$ is a projective abelian scheme of relative dimension $g$ over $S$, and ${\lambda}:A\rightarrow{\Hat{A}}$ is a polarization of type $(d_1,\ldots,d_g)$. Note that any such polarization ${\lambda}$ of $A$ is separable by the assumption on the characteristic $k$ of $S$.
Recall that a *coarse moduli scheme* for the functor $\underline{\mathcal A}_{(d_1,\ldots,d_g)}$ is a scheme $\mathcal A_{(d_1,\ldots,d_g)}$ of finite type over $k$ admitting a morphism of functors $${\alpha}:\underline{\mathcal A}_{(d_1,\ldots,d_g)}
(\,\cdot\,)\rightarrow\mathrm{Hom}(\,\cdot\,,\mathcal A_{(d_1,\ldots,d_g)})$$ such that
1. ${\alpha}(\mathrm{Spec}\,k):\underline{\mathcal A}_{(d_1,\ldots,d_g)}(\mathrm{Spec}\,k)
\rightarrow
\mathbf{Hom}(\mathrm{Spec}\,k,\mathcal A_{(d_1,\ldots,d_g)})$ is a bijection, and
2. for any morphism of functors $\beta:\underline{\mathcal A}_{(d_1,\ldots,d_g)}
(\,\cdot\,)\rightarrow\mathrm{Hom}(\,\cdot\,, B)$ there is a morphism of schemes $f:\mathcal A_{(d_1,\ldots,d_g)}\rightarrow B$ such that $\beta=\mathrm{Hom}(\,\cdot\,,f)\circ{\alpha}$.
It is well known that a uniquely determined coarse moduli scheme $\mathcal A_{(d_1,\ldots,d_g)}$ exists (see [@GIT]). Moreover it is clear from the definition that an isomorphism of functors induces an isomorphism of the corresponding coarse moduli schemes. This will be used to prove the following
\[thm4.1\] There is a canonical isomorphism of coarse moduli schemes $$\mathcal A_{(d_1,\ldots,d_g)}\longrightarrow
\mathcal A_{(d_1,\frac{d_1d_g}{d_{g-1}},\ldots,\frac{d_1d_g}{d_2},d_g)}.$$
By what we have said above it suffices to show that there is a canonical isomorphism of functors $\underline{\mathcal A}_{(d_1,\ldots,d_g)}\rightarrow
\underline{\mathcal A}_{(d_1,\frac{d_1d_g}{d_{g-1}},
\ldots,\frac{d_1d_g}{d_2},d_g)}$. But for any $S\in\mathcal S$ the canonical map $\underline{\mathcal A}_{(d_1,\ldots,d_g)}(S)\rightarrow
\underline{\mathcal A}_{(d_1,\frac{d_1d_g}{d_{g-1}},
\ldots,\frac{d_1d_g}{d_2},d_g)}(S)$, defined by $$(A\rightarrow S,{\lambda})\mapsto({\Hat{A}}\rightarrow S,{\lambda}^\delta)$$ has an inverse, since ${\Hat{{\Hat{A}}}}=A$ by the biduality theorem and $({\lambda}^\delta)^\delta={\lambda}$ by Proposition \[prop2.3\]. Moreover this is an isomorphism of functors, since $({\lambda}_T)^\delta=({\lambda}^\delta)_T$ for any morphism $T\rightarrow S$ in $\mathcal S$.
The Dual Polarization via The Fourier Transform
===============================================
In Section 2 we defined for every polarization $\lambda$ of a projective abelian scheme $A/S$ a polarization $\lambda^D$ of the dual abelian scheme ${\Hat{A}}/S$. Now suppose $\lambda=\phi_{\mathcal L}$ for a relatively ample line bundle ${\mathcal L}$ on $A/S$. In this section we apply Mukai’s Fourier transform to define a relatively ample line bundle ${\Hat{{\mathcal L}}}$ on ${\Hat{A}}/S$ which induces a multiple of $\lambda^D$.\
Let $\pi:A\longrightarrow S$ be a projective abelian scheme over a connected Noetherian scheme $S$ of relative dimension $g$. Let $\mathcal P=\mathcal P_A$ denote the normalized Poincaré bundle on $A\times_S{\Hat{A}}/S$. Here normalized means that both $(\epsilon\times1_{{\Hat{A}}})^*\mathcal P$ and $(1_A\times{\Hat{\epsilon}})^*\mathcal P$ are trivial, where $\epsilon:S\longrightarrow A$ and ${\Hat{\epsilon}}:S\longrightarrow {\Hat{A}}$ are the zero sections. Denote by $p_1$ and $p_2$ the projections of $A\times_S{\Hat{A}}$. A coherent sheaf $\mathcal M$ on $A$ is called -*sheaf of index* $i$ on $A$ if $R^j{p_2}_*(\mathcal
P\otimes p_1^*\mathcal M)=0$ for $j\not =i$. In this case $$F(\mathcal M):=R^i{p_2}_*(\mathcal P\otimes p_1^*\mathcal M)$$ is called the *Fourier transform* of $\mathcal M$. Let ${\mathcal L}$ be a relatively ample line bundle on $A/S$. The scheme $S$ being connected, the degree $d:=h^0({\mathcal L}_s)$ is constant for all geometric points $s$ of $S$, it is called the *degree* of ${\mathcal L}$. By the base change theorem ${\mathcal L}$ is a -sheaf of index $0$ and its Fourier transform $F({\mathcal L})$ is a vector bundle of rank $d$. Define $${\Hat{{\mathcal L}}}:=\Bigl(\det{F({\mathcal L})}\Bigr)^{-1}.$$ Let $e=e(\phi_{\mathcal L})$ be the exponent of the polarization $\phi_{\mathcal L}$. The main result of this section is the following
\[thm3.1\] ${\Hat{{\mathcal L}}}$ is a relatively ample line bundle on ${\Hat{A}}/S$ such that $$e\phi_{{\Hat{{\mathcal L}}}}=d\phi_{\mathcal L}^D.$$
If $\phi_{\mathcal L}$ is a separable polarization of type $(d_1,\ldots,d_g)$ then $e=d_g$ and $d=d_1\cdot\ldots\cdot d_g$. So $\phi_{{\Hat{{\mathcal L}}}}$ is a multiple of $\phi_{\mathcal L}^D$: $$\phi_{{\Hat{{\mathcal L}}}}=d_1\cdot\ldots\cdot d_{g-1}\phi_{\mathcal L}^D.$$ In the non separable case $e|d^2$ by Deligne’s Lemma but it is not clear to us whether $e|d$.
For the proof we need some preliminaries.
\[lem3.2\] Let $\varphi:A\longrightarrow B$ be an isogeny of abelian schemes and $\mathcal
M$ a -sheaf on $A$, then $\varphi_*\mathcal M$ is a -sheaf of the same index on $B$ and $$F_B(\varphi_*\mathcal M)={\Hat{\varphi}}\,^*F_A(\mathcal M).$$ Here $F_A$ and $F_B$ denote the Fourier transforms of $A$ and $B$ respectively.
The proof is the same as in the absolute case (see [@M1] (3.4)).
\[lem3.3\] Let $\mathcal M$ be a -sheaf of index $i$ on $A$ and $\mathcal F$ a locally free sheaf on $S$. Then $\pi^*\mathcal F\otimes \mathcal M$ is a -sheaf of index $i$ on $A$ and $$F(\pi^*\mathcal F\otimes \mathcal M)={\Hat{\pi}}^*\mathcal F\otimes F(\mathcal M).$$
Using $\pi\circ
p_1={\Hat{\pi}}\circ p_2$ and the projection formula we get $$\begin{aligned}
R^j{p_2}_*(\mathcal P\otimes p_1^*\pi^*\mathcal F\otimes p_1^*\mathcal M)
=& R^j{p_2}_*(p_2^*{\Hat{\pi}}^*\mathcal F\otimes\mathcal P\otimes p_1^*\mathcal M)\\
=&{\Hat{\pi}}^*\mathcal F\otimes R^j{p_2}_*(\mathcal P\otimes p_1^*\mathcal M)\\
=&\begin{cases}
{\Hat{\pi}}^*\mathcal F\otimes F(\mathcal M) & \text{if $j=i$}\\
0 & \text{otherwise}.
\end{cases}\end{aligned}$$ This implies the assertion.
\[prop3.4\] Let ${\mathcal L}$ be a relatively ample line bundle on $A/S$. Then ${\mathcal L}^{-1}$ is a -sheaf of index $g$ satisfying $\phi_{\mathcal L}^*F({\mathcal L})=\pi^*\pi_*{\mathcal L}\otimes{\mathcal L}^{-1}$.
The first assertion follows from Serre duality. Denote by $q_1,q_2:A\times_SA\rightarrow A$ the projections. Then $$\begin{aligned}
\pi^*\pi_*{\mathcal L}\otimes{\mathcal L}^{-1}&={q_2}_*(q_2-q_1)^*{\mathcal L}\otimes{\mathcal L}^{-1}\\
&\hspace{1,2cm}\Bigl(\text{(using flat base change with}\quad\pi\circ
q_2=\pi\circ(q_2-q_1)\Bigr)\\
&={q_2}_*\Bigl((q_2-q_1)^*{\mathcal L}\otimes q_2^*{\mathcal L}^{-1}\Bigr)\\
&={q_2}_*\Bigl(((-1)_A\times\phi_{\mathcal L})^*\mathcal P\otimes q_1^*(-1)_A^*{\mathcal L}\Bigr)\\
\intertext{\hspace{1cm}$ \Bigl($applying the formula $((-1)_A\times\phi_{\mathcal L})^*\mathcal P=(q_2-q_1)^*{\mathcal L}\otimes
q_1^*(-1)_A^*{\mathcal L}^{-1}\otimes q_2^*{\mathcal L}^{-1}\Bigr)$}
&= {q_2 }_*\Bigl((-1)_A\times\phi_{\mathcal L})^*(\mathcal P\otimes p_1^*{\mathcal L})\Bigr)\\
&=\phi_{\mathcal L}^*{p_2 }_*(\mathcal P\otimes p_1^*{\mathcal L})\quad\text{ (by flat base change)}\\
&=\phi_{\mathcal L}^*F({\mathcal L}).
\end{aligned}$$
By Proposition \[prop3.4\] we have $$\phi_{\mathcal L}^*({\Hat{{\mathcal L}}})=\Bigl(\det\phi_{\mathcal L}^*F({\mathcal L})\Bigr)^{-1}=
\Bigl(\det(\pi^*\pi_*{\mathcal L}\otimes{\mathcal L}^{-1})\Bigr)^{-1}=\pi^*(\det\pi_*{\mathcal L})^{-1}\otimes{\mathcal L}^d,$$ since $\pi^*\pi_*{\mathcal L}$ is a vector bundle of rank $d$. This implies that ${\Hat{{\mathcal L}}}$ is relatively ample. Moreover $$\phi_{\mathcal L}\circ\phi_{{\Hat{{\mathcal L}}}}\circ\phi_{\mathcal L}=\phi_{\phi_{\mathcal L}^*{\Hat{{\mathcal L}}}}
=\phi_{\pi^*(\det\pi_*{\mathcal L})^{-1}\otimes{\mathcal L}^d}=\phi_{{\mathcal L}^d}=d\phi_{\mathcal L}$$ implies that $$\phi_{{\Hat{{\mathcal L}}}}\circ\phi_{\mathcal L}=d_A\quad\text{and}\quad\phi_{\mathcal L}\circ\phi_{{\Hat{{\mathcal L}}}}=d_{{\Hat{A}}}.$$ Comparing this with Theorem \[prop2.1\] gives the assertion.
[CAV]{} Birkenhake, Ch., Lange, H., The Dual Polarization of an Abelian Variety, Arch. Math. 73, 380-389 (1999)
Grothendieck, A., Fondements de la Geométrie Algébrique, Sem. Bourbaki 1957-1962, Secrétariat Math., Paris (1962)
Mumford, D., Geometric Invariant Theory, Springer (1965)
Mumford, D., On the equations defining abelian varieties I, Inv. Math. 1, 287-354 (1966)
Mukai, S., Duality Between $D(X)$ and $D({\Hat{X}})$ with its Application to Picard Sheaves, Nagoya Math. J., Vol. 81, 153-175 (1981)
Mukai, S., Fourier Functor and its Application to the Moduli of Bundles on an Abelian Variety, Adv. Studies in Pure Math. 10, Algebraic Geometry, Sendai 1985, 515-550 (1987)
Oort, F., Tate, Group scheme of Prime Order, Ann. Scient. Ec. Norm. Sup. 3 (1970)
[^1]: Supported by DFG Contracts Ba 423/8-1 and HU 337/5-1
|
---
abstract: 'We use all-electrical methods to inject, transport, and detect spin-polarized electrons vertically through a 350-micron-thick undoped single-crystal silicon wafer. Spin precession measurements in a perpendicular magnetic field at different accelerating electric fields reveal high spin coherence with at least 13$\pi$ precession angles. The magnetic-field spacing of precession extrema are used to determine the injector-to-detector electron transit time. These transit time values are associated with output magnetocurrent changes (from in-plane spin-valve measurements), which are proportional to final spin polarization. Fitting the results to a simple exponential spin-decay model yields a conduction electron spin lifetime ($T_1$) lower bound in silicon of over 500ns at 60K.'
author:
- Biqin Huang
- 'Douwe J. Monsma'
- Ian Appelbaum
title: ' Coherent spin transport through a 350-micron-thick Silicon wafer '
---
Silicon (Si) has been broadly viewed as the ideal material for spintronics due to its low atomic weight, lattice inversion symmetry, and low isotopic abundance of species having nuclear spin.[@ZUTICRMP; @ZUTICPRL; @LYON] These qualities are in contrast to the high atomic weight, inversion-asymmetric zinceblende lattice, and high nuclear spin of the well-studied semiconductor GaAs,[@KIKKAWA1; @KIKKAWA2; @CROWELL1; @CROWELL2; @JIANG1; @JIANG2; @BHATTACHARYA] which consequently has a relatively large spin-orbit and hyperfine interaction.[@ZUTICRMP] The resulting long spin lifetime and spin coherence lengths in Si may therefore enable spin-based Si integrated circuits.[@ZUTICNV; @QC]
Despite this appeal, however, the experimental difficulties of achieving coherent spin transport in silicon were first overcome only recently, by using unique spin-polarized hot-electron injection and detection techniques.[@APPELBAUMNATURE; @BIQINJAP; @35percentAPL; @SPINFETEXPT] (Subsequently, tunnel spin injection was demonstrated using optical detection with circular polarization analysis of weak indirect-bandgap electroluminescence.[@JONKERNATPHYS]) In Refs. [@APPELBAUMNATURE] and [@BIQINJAP], spin transport through 10 $\mu$m of silicon was demonstrated and a spin lifetime lower bound of $\approx$1 ns at 85K was estimated. Using a new type of hot-electron spin injector that gives higher spin polarization and output current, we now show that (like in GaAs)[@KIKKAWA2] coherent spin transport can be observed over much longer lengthscales: we demonstrate transport vertically through a 350 $\mu$m-thick silicon wafer, and derive a spin lifetime of at least 500 ns at 60K (two orders of magnitude higher than metals or other semiconductors such as GaAs at similar temperature[@KIKKAWA1; @AWSCHALOMLOSSSAMARTH]).
{width="7.5cm" height="7cm"}
As in Refs. [@APPELBAUMNATURE; @BIQINJAP; @35percentAPL; @SPINFETEXPT], we use ultra-high vacuum metal-film wafer bonding[@SVTSCIENCE] to build a semiconductor-metal-semiconductor hot-electron spin detection structure. A 2" diameter double-side polished 350-micron-thick undoped (resistivity $>$ 13 k$\Omega\cdot$cm @ room-temperature) single-crystal Si(100) wafer is bonded to a 1-10 $\Omega\cdot$cm n-type Si(100) wafer with a 4nm Ni$_{80}$Fe$_{20}$/ 4nm Cu bilayer. [@SVTSCIENCE; @JANSEN] This process began with wafer cleaning in buffered HF solution and immediate loading into our wafer-bonding chamber. After pump-down and bakeout to the base pressure of 1E-8 Torr, 4nm of Cu was thermally evaporated onto the n-Si wafer only. (This layer is necessary to reduce the hot-electron collector Schottky barrier height there.)[@SVTPRL] During subsequent thermal evaporation of 2nm Ni$_{80}$Fe$_{20}$ on both wafers, the ultra-clean surfaces of the deposited metal films were pressed together [*in-situ*]{} with nominal force, forming a cohesive bond with a re-crystallized 4nm-thick Ni$_{80}$Fe$_{20}$ layer. [@SVTSCIENCE]
Although these bonding steps are identical to our previous reports with 10 $\mu m$-thick transport layer devices, the subsequent procedure used to fabricate 350 $\mu m$-thick transport layer devices differs significantly. In the present work, the outside polished surface of the undoped Si wafer in the bonded pair was covered by a protective 1 $\mu$m-thick SiO$_2$ layer deposited by an electron-beam source. A wafer saw was used to first cut through the undoped Si wafer and buried metal bonding layer, partially through the n-Si wafer to define individual device mesas. Then, the saw was used to cut trenches in the undoped Si wafer close to, but not through, the buried metal bilayer. Wet chemical etching with tetramethyl ammonium hydroxide (TMAH) removed the remaining Si and exposed the buried Ni$_{80}$Fe$_{20}$ for electrical contact.[@SVPT] After protective SiO$_2$ removal with buffered HF, a 40nm Al/10nm Co$_{84}$Fe$_{16}$/Al$_2$O$_3$/5nm Al/5nm Cu tunnel junction hot-electron spin injector was deposited using electron-beam evaporation through shadow masks for lateral patterning.[@35percentAPL]
Figure 1(a-c) illustrates the geometry of one of our completed four-terminal silicon spin-transport devices. The optical image in Fig. 1(a) shows a device (before contacting with wire-bonds) having a lateral size of approximately 1$\times$1.4mm. The schematic side-view and associated conduction band diagram in Fig. 1(b) and (c), respectively, shows the vertical geometry and can be used to elucidate the means of spin injection and detection. When a voltage bias $V_E$ is applied across the emitter tunnel junction, electrons that are spin polarized at the cathode Co$_{84}$Fe$_{16}$/Al$_2$O$_3$ interface tunnel through the oxide barrier and some travel ballistically through the nonmagnetic Al/Cu anode bilayer. Those electrons with energy above the Cu/Si Schottky barrier ($\approx$0.6eV)[@SZEBOOK] can couple with Si conduction band states and then quickly thermalize to the conduction band minimum.[@SIBEEM] These spin-polarized electrons are then accelerated in an applied electric field vertically through the 350 micron-thick wafer and toward the opposite side of the undoped Si, where they are ejected from the conduction band into the buried metal layer. Because the ferromagnetic Ni$_{80}$Fe$_{20}$ layer has a spin-dependent bandstructure, the inelastic scattering rates of these hot electrons to the Fermi energy is also spin-dependent. Therefore, the number of ballistic electrons that can couple with conduction band states in the n-Si collector on the other side (forming the “second collector current” $I_{C2}$) is dependent on the relative orientation of final spin direction and ferromagnet (FM) magnetization.
The spin-polarized electron injector we use here is notably different from the design in previous studies, where spin-dependent scattering in the base anode (ballistic spin filtering) was the operating mechanism.[@APPELBAUMNATURE; @BIQINJAP; @SPINFETEXPT] In the devices used in the present work, initial spin polarization is obtained by direct tunneling from the cathode FM (Co$_{84}$Fe$_{16}$) through the Al$_2$O$_3$ tunnel junction oxide. This design gives several advantages: 1. the FM is removed from the Si surface, preventing the formation of a non-magnetic silicide having strong, randomly-oriented magnetic moments. The elimination of this “magnetically-dead” region (which could cause significant spin scattering) maintains a high initial spin polarization.;[@35percentAPL] 2. Ballistic hot-electron transport before injection into the Si conduction band is through non-magnetic Al and Cu, which have much larger ballistic mean-free-paths than typical FMs, resulting in higher injected current ($I_{C1}$) and the spin-signal output current ($I_{C2}$) it drives; and 3. The Cu/Si Schottky barrier height is relatively low,[@SZEBOOK] further increasing $I_{C1}$.
{width="6.5cm" height="5.75cm"}
If spin “up” is both injected and detected with parallel FM magnetizations (and no spin flipping or rotating process occurs in the Si bulk) a relatively high $I_{C2}$ should be measured. On the other hand, if spin “up” is injected, but spin “down” is detected (with anti-parallel FM magnetizations), $I_{C2}$ will be relatively lower, again assuming no spin flips or rotations. The ferromagnetic layers chosen for the injector (Co$_{84}$Fe$_{16}$) and detector (Ni$_{80}$Fe$_{20}$) have different coercive (or switching) fields, which enables external control over the relative orientation of spin injection and detection axes with an in-plane magnetic field. At 150K, clean spin-valve signals at constant emitter bias $V_E=-1.3$V and accelerating voltage $V_{C1}=20$V (resulting in $\approx$580 V/cm electric field)[@BIQINJAP] indicate a $\approx$18% change in $I_{C2}$ when the magnetizations of injector and detector are switched from a parallel (P) to anti-parallel (AP) configuration by an externally-applied in-plane magnetic field, according to our expectations (as shown in Fig. 2). This magnetocurrent ratio ($MC=(I_{C2}^P-I_{C2}^{AP})/I_{C2}^{AP}$) corresponds to an electron current spin polarization of approximately $\mathcal{P}=MC/(MC+2)\approx$ 8%.[@SPINFETEXPT] However, this evidence for spin transport is not conclusive without observation of spin precession and dephasing (Hanle effect[@JOHNSON1985; @JOHNSON1988]) in a perpendicular magnetic field. [@MONZON]
![(a) Spin precession and dephasing (Hanle effect) of Si conduction-band electrons in a perpendicular magnetic field at 150K using the same voltage bias conditions as in Fig. 2, showing up to 13$\pi$ rad precession angles. The “FM switch” is caused by a residual in-plane magnetic field component switching the in-plane magnetization of the Ni$_{80}$Fe$_{20}$ detector at $\approx$+38 Oe, which inverts the maxima and minima at higher positive field values. (b) Simulation of the measurement in (a), using the drift-diffusion spin precession model given by Eq. \[IC2EQN\].](clip003nnn.eps){width="6.5cm" height="10cm"}
A perpendicular magnetic field $\vec{B}$ exerts a torque $(g \mu_B/\hbar)\vec{S} \times \vec{B}$ on the electron spin $\vec{S}$, causing spin rotation (precession) about $\vec{B}$. Here, $g$ is the electron spin g-factor, $\mu_B$ is the Bohr magneton, and $\hbar$ is the reduced Planck constant. Our spin detector measures the projection of final spin angle on an axis determined by the Ni$_{80}$Fe$_{20}$ magnetization, so we observe oscillations in $I_{C2}$ as the precession frequency $\omega=g\mu_B B/\hbar$ is varied.
Fig. 3(a) shows our measurement of $I_{C2}$ in varying perpendicular magnetic field with the same temperature and bias conditions as in Fig. 2. The measurement begins at negative field values when the injector/detector magnetizations are in a parallel orientation. As the field is increased, we see multiple oscillations due to spin precession. However, when the field reaches $\approx$+38 Oe, a small in-plane component of the applied field switches the magnetization of the magnetically softer Ni$_{80}$Fe$_{20}$, resulting in an antiparallel injector/detector orientation that inverts the magnitudes of maxima and minima.
The final precession angle $\theta$ at the detector is simply the product of transit time from injector to detector, $\tau$, and spin precession frequency $\omega$. Since our measurement is an average of the precession angles over all electrons arriving at the detector regardless of transit time $\tau$, the magnitudes of higher-order extrema (labeled in Fig 3(a)) are reduced by the dephasing associated with a distribution in transit times $\Delta \tau$ caused by random diffusion.
We can simulate our measurement in the presence of both drift and diffusion by integrating the contributions to our signal from an ensemble of precessing spins with a diffusion-controlled distribution of transit times using a simple model[@CROWELL2; @SPINFETTHEORY]:
$$\label{IC2EQN}
\Delta I_{C2} \sim \int_0^{\infty}\frac{1}{2\sqrt{\pi D t}}e^{-\frac{(x-vt)^2}{4Dt}}\cdot \cos(\omega t)\cdot e^{-t/\tau_{sf}}dt,$$
where $D$ is the diffusion constant, $v$ is drift velocity, and $\tau_{sf}$ is effective spin lifetime. The integrand is simply the product of the effects of drift and diffusion, precession, and finite spin lifetime. Using $x=L=350\mu$m, $D=200$ cm$^2$/s, $v=2.9\times 10^6$ cm/s,[@CANALI] and $\tau_{sf}=73$ ns (see below), we find excellent agreement between experiment and model in Fig. 3(b). (In this simulation, the sign is inverted for magnetic field values $>$38 Oe to match the experimental results.)
Despite transport through 350 microns of undoped Si, high spin coherence with at least $13\pi$ spin precession angle (more than six full rotations) is evident in Fig. 3(a), which is even greater than what was previously demonstrated using a much shorter 10 $\mu m$-thick transport layer.[@APPELBAUMNATURE] Because the transit time is therefore much longer in the thicker devices, it could be argued that diffusion should play a larger role and dephasing should suppress multiple oscillations in precession measurements. The results of the experiment and consistent model simulation clearly conflict with this reasoning.
The somewhat counterintuitive result can be explained with a simple argument: If transport is dominated by drift in the applied electric field[@BIQINJAP], the transit time is given by $\tau=L/v=L^2/(\mu V_{C1})$, where $\mu$ is the electron mobility, $L$ is the transport length, and $v$ is drift velocity.[@SPINFETTHEORY] The width $d$ of an initially injected infinitesimally-narrow gaussian spin distribution will increase by diffusion during this transit time to $d=\sqrt{D\tau}=L\sqrt{D/(\mu V_{C1})}$. Since the width of the distribution of transit times $\Delta \tau$ is $d/v$, the relative uncertainty in the distribution of final precession angle $\theta$ at the detector is $\Delta\theta/\theta=\frac{\omega \cdot \Delta \tau}{\omega\tau}= \sqrt{D/(\mu V_{C1})}$. This result is independent of the transit length $L$, so we can expect the same amount of dephasing regardless of the distance from injector to detector for any fixed precession angle (assuming ohmic behavior, $v=\mu E$, where $E$ is internal electric field).
From the oscillation period of spin precession measurements ($2B_\pi $, as shown in Fig 3(a)), we can determine the average spin transit time in any given accelerating electric drift field conditions (induced by $V_{C1}$) through $\tau=h/(2g\mu_B B_\pi)$. The normalized magnetocurrent $\Delta I_{C2}/I_{C1}$ determined by spin-valve measurements like those in Fig. 2 gives a quantity that is proportional to conduction electron current spin polarization, $\mathcal{P}$.[@APPELBAUMNATURE; @BIQINJAP] Associating this value with the transit times given by precession measurements (see above) gives data which can be fit with a simple exponential decay model, where
$$\label{EXPEQN}
\mathcal{P} \propto exp(-\tau/T_1).$$
The timescale $T_1$ is the longitudinal spin lifetime, since our spin-polarization data is derived from spin-valve measurements with in-plane magnetic fields colinear to the spin direction.
![(a) Fitting the normalized magnetocurrent ($\Delta I_{C2}/I_{C1}$) to an exponential decay model (Eq. \[EXPEQN\]) using transit times derived from spin precession measurements (like those in Fig. 3(a)) at variable internal electric field yields measurement of longitudinal spin lifetimes ($T_1$) in undoped bulk Si. (b) The experimental $T_1$ values obtained as a function of temperature are compared to Yafet’s $T^{-5/2}$ power law for indirect-bandgap semiconductors[@YAFET].](clip004nn.eps){width="6.25cm" height="10.5cm"}
The best-fits to our data at 60K, 85K, 100K, 125K and 150K using the expression in Eq. \[EXPEQN\] are 520ns, 269ns, 201ns, 96ns, and 73ns, respectively, as shown in Fig. 4(a) and (b). These lifetimes are much greater than the $\approx$1 ns lifetime lower bound estimated in previous work, because with the much longer transit lengths here, the applied accelerating voltage $V_{C1}$ varies the transit time over a range of $\approx$200 ns; previously the range was only several hundred ps, and parasitic electronic effects suppressed our estimate.[@APPELBAUMNATURE; @BIQINJAP] The temperature dependence of these spin lifetimes fit well to the expected behavior in an indirect-bandgap semiconductor predicted by Yafet ($\propto T^{-5/2}$), as shown in Fig. 4(b).[@YAFET; @LEPINE; @JAROSLOVACA] The relative absence of other relaxation mechanisms in Si is responsible for the long spin lifetimes.
Certainly, higher temperature operation is desirable. However, thermionic leakage at the second collector Schottky barrier and the difficulties of reliably operating our tunnel junction spin injector at high voltages necessary are the present limitation to increasing this temperature. Although observation of spin precession at high electric fields are possible at lower temperatures, measurements of spin lifetime below 60K are currently prevented by carrier freeze-out effects.
The long lifetimes measured here are lower bounds, with the possibility that parasitic electronic effects artificially suppress the values obtained.[@BIQINJAP] Hence, spin lifetimes could be higher with associated longer transport lengths. Due to the thickness limitations of Si wafers, we will explore these longer distances with lateral transport devices. This achievement should enable true spintronic circuits intimately compatible with existing Si CMOS logic, and potentially extend the performance trend of Si devices beyond its limits set by conventional approaches.
We gratefully acknowledge J. Fabian for helpful discussions. This work was supported by DARPA/MTO.
[10]{}
I. Žutić, J. Fabian and S. Das Sarma, Rev. Mod. Phys. [**[76]{}**]{}, 323 (2004).
I. Žutić, J. Fabian, and S. C. Erwin, Phys. Rev. Lett. [**[97]{}**]{}, 026602 (2006)
A. M. Tyryshkin, S. A. Lyon, A. V. Astashkin, and A. M. Raitsimring, Phys. Rev. B [**[68]{}**]{}, 193207 (2003).
J.M. Kikkawa and D.D. Awschalom, Phys. Rev. Lett. [**[80]{}**]{}, 4313 (1998).
J.M. Kikkawa and D.D. Awschalom, Nature [**[397]{}**]{}, 139 (1999).
X. Lou et al., Phys. Rev. Lett. [**[96]{}**]{}, 176603 (2006).
X. Lou et al., Nature Physics [**[3]{}**]{}, 197 (2007).
X. Jiang et al., Phys. Rev. Lett. [**[90]{}**]{}, 256603 (2003).
X. Jiang et al., Phys. Rev. Lett. [**[94]{}**]{}, 056601 (2005).
M. Holub, J. Shin, D. Saha, and P. Bhattacharya, Phys. Rev. Lett. [**[98]{}**]{}, 146603 (2007).
I. Žutić and J. Fabian, Nature [**[447]{}**]{}, 269 (2007).
M. A. Nielsen and I. L. Chuang, [*[Quantum Computation and Quantum Information]{}*]{} (Cambridge University Press, 2000).
I. Appelbaum, B. Huang, and D. Monsma, Nature [**[447]{}**]{}, 295 (2007).
B. Huang, D. Monsma and I. Appelbaum, J. Appl. Phys. [**[102]{}**]{}, 013901 (2007).
B. Huang, D. Monsma and I. Appelbaum, Appl. Phys. Lett. [**91**]{}, 072501 (2007).
B. Huang, L. Zhao, D. Monsma and I. Appelbaum, Appl. Phys. Lett. [**[91]{}**]{}, 052501 (2007).
B.T. Jonker, G. Kioseoglou, A.T. Hanbicki, C.H. Li, and P.E. Thompson, Nature. Physics [**3**]{}, 542 (2007).
D. D. Awschalom, D. Loss, and N. Samarth, [*Semiconductor Spintronics and Quantum Computation*]{}, (Springer, New York, 2004).
D.J. Monsma, R. Vlutters, and J.C. Lodder, Science [**[281]{}**]{}, 407 (1998).
R. Jansen, J. Phys. D [**[36]{}**]{}, R289 (2003).
D.J. Monsma, J.C. Lodder, Th. J.A. Popma, B. Dieny, Phys. Rev. Lett. [**[74]{}**]{}, 5260 (1995).
B. Huang, I. Altfeder, and I. Appelbaum, Appl. Phys. Lett. [**[90]{}**]{}, 052503 (2007).
S. Sze, [*Physics of Semiconductor Devices*]{}, 2nd edition, (Wiley, New York, 1981). L.D. Bell, S.J. Manion, M.H. Hecht, W.J. Kaiser, R.W. Fathauer, and A.M. Milliken, Phys. Rev. B [**[48]{}**]{}, 5712 (1993).
M. Johnson and R.H. Silsbee, Phys. Rev. B [**[37]{}**]{}, 5326 (1988).
M. Johnson and R.H. Silsbee, Phys. Rev. Lett. [**[55]{}**]{}, 1790 (1985).
F.G. Monzon, H.X. Tang, and M.L. Roukes, Phys. Rev. Lett. [**[84]{}**]{}, 5022 (2000).
I. Appelbaum and D.J. Monsma, Appl. Phys. Lett. [**[90]{}**]{}, 262501 (2007).
C. Canali, C. Jacoboni, F. Nava, G. Ottaviani, and A. Alberigi-Quaranta, Phys. Rev. B [**12**]{}, 2265 (1975).
Y. Yafet, in [*Solid State Physics, Vol. 14*]{}, ed. Seitz and Turnbull, (Academic, New York, 1963).
D. J. Lepine, Phys. Rev. B [**[2]{}**]{}, 2429 (1970).
J. Fabian, A. Matos-Abiague, C. Ertler, P. Stano, and I. Žutić, Acta Physica Slovaca [**57**]{}, 565 (2007).
|
---
address: 'Service de Physique Théorique, Université Libre de Bruxelles, B-1050 Brussels, Belgium'
author:
- Julian Heeck
title: 'Lepton Number Violation with *and without* Majorana Neutrinos [^1]'
---
Introduction
============
The observation of a Brout–Englert–Higgs-like scalar boson at the LHC completes the Standard Model (SM). It is however evident that the SM can not be the final description of nature, with
- neutrino masses and mixing,
- dark matter (DM),
- and the baryon asymmetry of the Universe (BAU)
among the most pertinent observations that require new physics. All three problems can be solved by relatively simple SM extensions, but there is no unique or (arguable) even simplest solution, so experimental input and theoretical motivation are required to lead the way. In this talk we will consider baryon ($B$) and lepton number ($L$) as guiding principles towards a solution to the three problems above. DM is not the focus here, but we will remark on it parenthetically.
It is well known that the *classical* SM Lagrangian has the accidental global symmetry $U(1)_B\times U(1)_{L_e}\times U(1)_{L_\mu}\times U(1)_{L_\tau}$ due to its particle content/gauge group representations and the requirement for renormalizability. Non-perturbative quantum effects – instantons at zero temperature, sphalerons at $T\neq 0$ – break both $B$ and $L$ by three units each, so $\Delta (B+L) = 6$, while $B-L$ remains conserved: $\Delta (B-L) =0$. The global symmetry of the *quantum* SM Lagrangian is hence only [@Araki:2012ip] $$\mathcal{G}_\mathrm{sym} = U(1)_{B-L}\times U(1)_{L_e-L_\mu}\times U(1)_{L_\mu-L_\tau} \,,$$ picking a convenient basis in generator space. We know from neutrino oscillations that $U(1)_{L_e-L_\mu}\times U(1)_{L_\mu-L_\tau}$ is a broken symmetry, whereas we have yet to observe a process violating $B-L$. (In fact, no process violating $B$ or $L$ has ever been observed, but we are very confident in the unobservable $\Delta (B+L) = 6$ breaking predicted by theory.)
$\mathcal{G}_\mathrm{sym}$ is the anomaly-free *global* symmetry of the SM Lagrangian at quantum level, and it is tempting to promote it to a *local* symmetry, i.e. a gauge symmetry alongside $SU(3)_C\times SU(2)_L \times U(1)_Y$. This only requires the introduction of three right-handed neutrinos $\nu_R$, uncharged under the SM gauge group, to cancel anomalies – a cheap price to pay for such an enlarged gauge group. Furthermore, the quantum numbers of the $\nu_R$ allow us to write down additional couplings $$\Delta {\mathcal{L}} = -\overline{\nu}_{R} y_\nu H^\dagger L + \text{h.c.} ,$$ which automatically give rise to a (Dirac) neutrino mass matrix $m_D = y_\nu \langle H\rangle$ after electroweak symmetry breaking. Promoting the global symmetry of the SM to a local symmetry thus requires neutrino masses *for consistency*, which can be taken as a motivation for this approach.
Flavored subgroups of $\mathcal{G}_\mathrm{sym}$, such as $U(1)_{B+3 (L_e-L_\mu - L_\tau)}$ or $U(1)_{L_\mu-L_\tau}$, make for simple flavor symmetries that can shed light on the leptonic mixing pattern [@Araki:2012ip] and neutrino hierarchies [@Heeck:2011wj; @Heeck:2012cd] (see also Ref. [@Heeck:2013bla]). Gauged $L_\mu-L_\tau$ in particular has recently received attention as an explanation for some tantalizing hints in $h\to \mu\tau$ [@Heeck:2014qea] and lepton-nonuniversal $B$-meson decays [@Altmannshofer:2014cfa; @Crivellin:2015mga; @Crivellin:2015lwa] (see contribution by A. Crivellin in these proceedings).
For simplicity, we will here focus on the unflavored part of $\mathcal{G}_\mathrm{sym}$, i.e. consider a gauged $B-L$ symmetry. This still requires three right-handed neutrinos, so the argument regarding automatically massive neutrinos from above applies. In the next sections we will explore the different realizations of a gauged $U(1)_{B-L}$ and their different phenomenology, in particular their very different solutions to the problems of neutrinos mass, the BAU, and DM (parenthetically).
Majorana B-L {#sec:majorana}
============
We start with the most popular realization of gauged $U(1)_{B-L}$, in which the symmetry is broken spontaneously by two units, i.e. $\Delta (B-L) = 2$. For this, a new SM-singlet scalar $\phi_{B-L=2}$ is introduced which carries $B-L = 2 \ (= -L)$ and can hence couple to the $\nu_R$ via $${\mathcal{L}} \ \supset \ -\overline{\nu}_{R} y_\nu H^\dagger L + \tfrac12 \overline{\nu}_R K \nu^c_R \ \phi_{B-L=2}^*+ \text{h.c.} ,$$ which gives rise to a right-handed Majorana mass matrix $\mathcal{M}_R = K \langle \phi_{B-L=2}\rangle$ after $B-L$ breaking and ultimately light Majorana neutrino masses via seesaw: $$\mathcal{M}_\nu \simeq - m_D^T \mathcal{M}_R^{-1} m_D \sim y_\nu^T K^{-1} y_\nu \left( \frac{10^{14}\,\mathrm{GeV}}{\langle \phi_{B-L = 2}\rangle}\right) \,\mathrm{eV} .$$ For Yukawa couplings of order one, the $B-L$ breaking scale is untestably high and the only signature of “Majorana $B-L$” is neutrinoless double beta decay ($0\nu 2\beta$), mediated by the light Majorana neutrinos. While the $0\nu 2\beta$ rate is definitely non-zero in this scenario, it could still be unobservably small for normal-hierarchy neutrinos if $(\mathcal{M}_\nu)_{ee} \simeq 0$. Additional signatures arise if the Yukawa couplings are chosen to be small, lowering the right-handed masses below the electroweak scale. In particular, choosing the flavor structure in such a way that one of the right-handed neutrinos, say $\nu_{R,1}$ barely couples to the left-handed neutrinos and has a mass around keV, it can be sufficiently stable to form (warm) DM.[^2] The small mixing of $\nu_{R,1}$ then effectively decouples it from the seesaw mechanism, so one of the active neutrinos remains massless.
“Majorana $B-L$” can also explain the BAU by means of leptogenesis, i.e. the out-of-equilibrium decay of the heavy right-handed neutrinos $\nu_R\to L H^* , \overline{L} H$ in the early Universe. $CP$-violation arises via loops and results in a lepton asymmetry $\Delta_L$, i.e. a different number of leptons and antileptons. Since the sphaleron processes ($\Delta B = \Delta L = 3$) are in equilibrium with the rest of the SM plasma at temperatures $T \gtrsim 100\,\mathrm{GeV}$, the lepton asymmetry will partly be converted to a baryon asymmetry $\Delta_B$.
Breaking $B-L$ by two units can hence solve the three main problems of the SM: neutrinos obtain Majorana masses via seesaw, the BAU is explained by leptogenesis, and one can even make one of the right-handed neutrinos stable enough to form DM. This is however not the only viable realization of a gauged $U(1)_{B-L}$, and we will cover two very different scenarios in the next sections.
Unbroken B-L {#sec:unbroken}
============
As already stated in the introduction, we have yet to observe a process that violates $B-L$. It is hence tempting to keep $U(1)_{B-L}$ as an unbroken gauge symmetry [@Heeck:2014zfa], making $B-L$ a properly conserved quantum number alongside electric charge and color. Neutrinos are then Dirac particles, and one either has to chose the Yukawa couplings very small to obtain the sub-eV required masses, $y_\nu = m_\nu/\langle H\rangle\lesssim 10^{-11}$, or introduce additional new physics that gives a more natural solution.
Surprisingly, even the BAU can be explained in this framework, with a mechanism dubbed neutrinogenesis [@Dick:1999je]. For this, new heavy doublet scalars $\Psi_j$ are introduced which decay out of equilibrium in the early Universe. $CP$ violation via loops can give rise to lepton asymmetries in the decays $\Psi_j\to \overline{L} \nu_R, L\overline{\nu}_R$, which take the form $\Delta_{\nu_L} = - \Delta_{\nu_R} \neq 0$. Lepton number is hence not broken in the decays, but merely distributed among left- and right-handed leptons. The crucial observation is now that the Yukawa couplings $y_\nu = m_\nu/\langle H\rangle$ are *too small* to put the $\nu_R$ in equilibrium with the rest of the SM plasma, and in particular with the sphalerons. These will therefore only see $\Delta_{\nu_L}$, and process it into a baryon asymmetry $\Delta_B$ via the usual $\Delta (B+L ) = 6$ processes, even though the total $B-L$ number of the Universe is zero at all times.
With the BAU and neutrino masses resolved, let us discuss the gauge boson $Z'$ coupled to $B-L$. If massless, the gauge coupling $g'$ is required to be tiny ($g' \lesssim 10^{-24}$) in order to be compatible with tests of the weak equivalence principle. However, since $U(1)_{B-L}$ is abelian, we can actually introduce a $Z'$ mass with the Stückelberg mechanism in a gauge-invariant way without breaking the symmetry. This makes the phenomenology of the $Z'$ much more interesting, because the mass is not coupled to neutrino masses, leptogenesis or the weak scale, and can hence sit at any scale. For low masses, constraints in the $M_{Z'}$–$g'$ plane arise from cosmology, astrophysics (stellar evolution), Big Bang nucleosynthesis and colliders [@Heeck:2014zfa]. Unavoidable kinetic mixing results in a $Z'$ coupling to hypercharge and gives rise to additional effects.
As far as DM is concerned, the $Z'$ can be long-lived if the gauge coupling and/or mass are small. The correct abundance can then be obtained by a misalignment mechanism analogous to axions/hidden photons [@Nelson:2011sf]. An alternative way to solve the DM problem in unbroken $B-L$ would be to introduce a new fermion (boson) with even (odd) $B-L$ charge; seeing as all SM fermions (bosons) are odd (even) under $B-L$, the new particle would be stable due to its $U(1)_{B-L}$ charge (similar to the stability of the electron due to the $U(1)_\mathrm{EM}$). A simple freeze-out mechanism using the $Z'$ interactions is then sufficient to obtain the desired DM abundance.
An unbroken gauged $U(1)_{B-L}$ can hence solve all of the three major problems of the SM: neutrinos obtain simple Dirac masses, the BAU can be obtained by neutrinogenesis, and DM can be obtained either with the $Z'$, or with new particles stabilized by the unbroken $U(1)$.
Dirac B-L {#sec:dirac}
=========
Let us turn to the third possibility regarding the fate of gauged $U(1)_{B-L}$, where the symmetry is broken – but not by two units. Breaking $B-L$ by any number $\Delta (B-L)\neq 2$ gives Dirac neutrinos, but since $B-L$ is still broken, this framework still allows for lepton number violation [@Heeck:2013rpa; @Heeck:2013vha].
Seeing as all SM fermions are odd under $B-L$, only $B-L$ breaking by even numbers can be observable (otherwise spin would be violated), so we focus on the simplest $\Delta (B-L)\neq 2$ case: $\Delta (B-L)=4$. Effective operators can be written down without effort: $$\begin{aligned}
\mathcal{O}^{d=6} &= \overline{\nu}^c_R \nu_R \ \overline{\nu}^c_R \nu_R\,,\label{eq:effective_ops} &
\mathcal{O}^{d=8}_1 &= |H|^2\ \overline{\nu}^c_R \nu_R \ \overline{\nu}^c_R \nu_R \,, \\
\mathcal{O}^{d=8}_2 &= (\overline{L}^c \tilde{H}) (H^\dagger L)\ \overline{\nu}^c_R \nu_R \,, &
\mathcal{O}^{d=8}_3 &= F_Y^{\mu\nu} \overline{\nu}^c_R \sigma_{\mu\nu} \nu_R \ \overline{\nu}^c_R \nu_R \,.\end{aligned}$$ At $d=10$, we only give a selection: $$\begin{aligned}
\mathcal{O}^{d=10}_1 &= (\overline{L}^c \tilde{H}) (H^\dagger L) \ (\overline{L}^c \tilde{H}) (H^\dagger L) \,, &
\mathcal{O}^{d=10}_2 &= F_Y^{\mu\nu} (\overline{L}^c \tilde{H}) (H^\dagger L) \ \overline{\nu}^c_R \sigma_{\mu\nu} \nu_R \vphantom{\sum} \,, \\
\mathcal{O}^{d=10}_3 &= W_a^{\mu\nu} (\overline{L}^c \tilde{H}) (H^\dagger \tau^a L) \ \overline{\nu}^c_R \sigma_{\mu\nu} \nu_R \,, &
\mathcal{O}^{d=10}_4 &= (\overline{u}_R d_R^c) ( \overline{d}_R H^\dagger L) ( \overline{\nu}^c_R \nu_R) \,,\end{aligned}$$ and operators without neutrinos arise at higher dimension still, e.g. $$\mathcal{O}^{d=20} = \left[(\overline{(D_\mu L)}^c \tilde{H}) (H^\dagger D_\nu L) \right]^2 \ \supset\ (\overline{e}_L^c W_\mu^+ W_\nu^+ e_L)^2 \,.
\label{eq:charged_leptons}$$
Let us present a simple model to show how these effective operators can be obtained and that they are indeed the lowest lepton-number-violating operators, i.e. $\Delta (B-L) = 2$ processes do not arise. We introduce a scalar $\phi$ with $B-L$ charge $4$ to break the $U(1)_{B-L}$ spontaneously by four units; in order to connect the symmetry breaking to the fermion sector, a second scalar $\chi$ with $B-L$ charge $-2$ is introduced which serves as a mediator and does not acquire a vacuum expectation value (VEV). The important parts of the Lagrangian are $$- {\mathcal{L}} \ \supset \ \overline{\nu}_{R} y_\nu H^\dagger L + \tfrac{1}{2} \overline{\nu}_{R} K \nu_{R}^c \, \chi + \mu\, \chi^2 \phi + \text{h.c.}
\label{eq:model}$$ One can easily realize a scalar potential with minimum at $\langle \chi\rangle =0$, $\langle H\rangle\neq 0 \neq \langle \phi \rangle$, which breaks $SU(2)_L\times U(1)_Y \times U(1)_{B-L}$ to $U(1)_\mathrm{EM}\times \mathbb{Z}_4^L$. An exact $\mathbb{Z}_4^L$ symmetry remains, under which leptons transform as $\ell \to -i \,\ell$ and $\chi \to - \chi$, making the neutrinos Dirac particles but still allowing for $\Delta L = 4$ processes.[^3] (The remaining $\mathbb{Z}_4$ symmetry could naturally be used as a stabilizing symmetry for a new DM particle, interacting with the SM via the $Z'$ and the scalars.)
Since $\chi$ does not acquire a VEV, the neutrinos will be Dirac particles $\nu = \nu_L + \nu_R$ with mass matrix $m_D = y_\nu \langle H \rangle$, just like in the unbroken $B-L$ case of Sec. \[sec:unbroken\]. The VEV of $\phi$ splits the masses of the real and imaginary part of $\chi$ due to the coupling $ \mu \chi^2 \phi$, so we end up with two scalars $\chi_{r,i}$ that couple to $\overline{\nu}_R \nu_R^c$. If these scalars are heavy, we can integrate them out to obtain the $\Delta (B-L) = 4$ operator $(\overline{\nu}_R \nu_R^c)^2$ of Eq. (see Fig. \[fig:uv\_model\]). Other $\Delta (B-L) = 4$ operators can be obtained by attaching SM interactions, or by going to a left–right extension of this simple model (see below). We stress again that neutrinos are Dirac particles here, and that there are no $\Delta (B-L) = 2$ processes allowed by the symmetry (such as $0\nu 2\beta$).
The $\Delta (B-L) = 4$ interactions can give rise to a new leptogenesis mechanism with Dirac neutrinos that differs qualitatively from the neutrinogenesis mechanism described in Sec. \[sec:unbroken\]. For this, we assume several heavy mediator scalars $\chi_j$, which decay out-of-equilibrium in the early Universe. Due to the couplings of Eq. , the scalars decay either into $\nu_R\nu_R$ or $\nu_R^c\nu_R^c$, and loop corrections induce a different rate for both channels [@Heeck:2013vha]. After all the scalars have decayed, we thus end up with an asymmetry in the right-handed neutrinos $\Delta_{\nu_R}$. This in itself is not helpful, because the right-handed neutrinos are decoupled from the rest of the SM plasma (which was the main trick in neutrinogenesis). In our case, we *need* them to be in equilibrium, so we have to introduce a second scalar doublet to the SM that has stronger couplings to the $\nu_R$ than the doublet that generates the neutrino mass. Such a model has already been proposed independently of Dirac $B-L$ in order to explain the smallness of Dirac neutrino masses [@Davidson:2009ha]. In this neutrinophilic two-Higgs-doublet model an additional global symmetry ensures that the second doublet couples only to $\overline{L}\nu_R$, and that it only acquires a tiny VEV (say eV). Because of this, the neutrino masses are small even if the Yukawa couplings to the second scalar doublet are large, solving the issue of small Dirac neutrino masses. Even better, the large Yukawa couplings imply that in our leptogenesis scenario the $\Delta_{\nu_R}$ asymmetry is transferred to an asymmetry in the left-handed leptons by the second doublet, and consequently converted to a baryon asymmetry by the sphalerons. As a consequence of the required thermalization of the $\nu_R$, we expect a contribution to the effective number of neutrinos in the early Universe, namely $N_\mathrm{eff} \gtrsim 3.14$, to be tested with future Planck-like experiments.
![$\Delta (B-L) = 4$ operator $(\overline{\nu}_R^c \nu_R)^2$ realized by exchange of scalars. Arrows show flow of lepton number.[]{data-label="fig:uv_model"}](uv_model){width="85.00000%"}
Above we have seen that $\Delta (B-L) = 4$ processes can be the lowest-order lepton-number-violating effect if neutrinos are Dirac particles, and also that it can lead to a new kind of Dirac leptogenesis mechanism. Compared to the (already hard to measure) $\Delta L = 2$ processes searched for in $0\nu 2\beta$ experiments, it is even harder to probe $\Delta L = 4$ processes directly, because of the high dimensionality of the underlying effective operators. Sensitive nuclear probes analogous to $0\nu 2\beta$ exist – namely the $0\nu 4\beta$ decay ${}^{150}_{\phantom{1}60}\mathrm{Nd} \to {}^{150}_{\phantom{1}64}\mathrm{Gd} + 4 e^-$ with energy release $Q_{0\nu 4\beta} \simeq 2.08\,\mathrm{MeV}$ testable with existing data from NEMO – but the expected rates in the toy model from above are unmeasurable small [@Heeck:2013rpa]. It is hence desirable to construct $\Delta (B-L) = 4$ models that can lead to stronger effects, which can be achieved in left–right extensions.
Let us embed the electroweak gauge group $SU(2)_L \times U(1)_Y$ into the left–right symmetry group $SU(2)_L\times SU(2)_R \times U(1)_{B-L}$. Consistency again requires the introduction of right-handed neutrinos (similar to just gauged $B-L$) to complete the right-handed lepton doublet $\Psi_R = (\nu_R, \, e_R)^T \sim (\boldsymbol{1},\boldsymbol{2},-1)$, while the scalar $H$ is promoted to a bi-doublet $H \sim (\boldsymbol{2},\overline{\boldsymbol{2}}, 0)$. The most common left–right model corresponds to an extension of “Majorana $B-L$”, i.e. features Majorana neutrinos. It is however not difficult to also extend “Dirac $B-L$” to a left–right model, simply by promoting the scalars $\chi$ and $\phi$ from above to $$\begin{aligned}
\chi_R &=& \frac{1}{\sqrt{2}}
\begin{pmatrix}
\chi^{-}_R & \chi^{0}_R & 0 \\
\chi^{--}_R & 0 & \chi^{0}_R \\
0 & \chi^{--}_R & -\chi^{-}_R
\end{pmatrix} \sim (\boldsymbol{1},\boldsymbol{3}, -2)\,, \\
\phi_R &=& \frac{1}{\sqrt{6}}
\begin{pmatrix}
\phi^{++}_R & \sqrt{3} \phi^{+++}_R & \sqrt{6} \phi^{++++}_R \\
\sqrt{3} \phi^{+}_R & -2 \phi^{++}_R & -\sqrt{3} \phi^{+++}_R \\
\sqrt{6} \phi^{0}_R & -\sqrt{3} \phi^{+}_R & \phi^{++}_R
\end{pmatrix} \sim (\boldsymbol{1},\boldsymbol{5},4)\,.\end{aligned}$$ The couplings analogous of Eq. then take the form (add $\chi_L$ and $\phi_L$ for LR parity) $${\mathcal{L}} \ \supset \ y \overline{\Psi}_L H \Psi_R + \kappa \, \overline{\Psi}_{R} \chi_{R} \Psi_{R}^c +\mu\, \text{tr}\left[\chi_{R} \phi_{R} \chi_{R}\right] + \text{h.c.} ,$$ so $\phi^0_R$ and $\chi^0_R$ play the same role as $\phi$ and $\chi$ from above. Note that the triplet $\chi$ does not acquire a VEV in this model, so the neutrinos are Dirac. $SU(2)_R$ is nevertheless broken above the electroweak scale via $\langle \phi_R\rangle\gg \langle H\rangle$: $$M_{W_R^{\pm}}^2 \simeq 2 g_R^2 \langle \phi^0_R\rangle^2 \,, \quad
M_{Z_R}^2 \simeq 8 (g_R^2 + 4 g_{B-L}^2) \langle \phi^0_R\rangle^2\,.$$ Compared to the toy model from above, it is now possible to consider processes that do not involve neutrinos, and are in particular not suppressed by small neutrino masses (see Fig. \[fig:left-right-operator\]). This opens the way towards collider searches for $\Delta L = 4$ processes such as $pp\to 4 \ell^- + 4 W^+$ at the LHC or $e^- e^- \to \ell^+ \ell^+ + 4 W^-$ at a future like-sign lepton collider [@preparation].
![$\Delta (B-L) = 4$ operator $e_R^4 W_R^4$ by exchange of scalars in a left–right model.[]{data-label="fig:left-right-operator"}](left-right-operator){width="60.00000%"}
Conclusion {#sec:conclusion}
==========
The incredible success of the Standard Model just deepens the mystery of its anomaly-free global symmetry $U(1)_{B-L}$. Consistently promoting this global symmetry to a local one automatically results in massive neutrinos, amending a major problem of the SM. The matter–antimatter asymmetry of our Universe is also deeply connected to the quantum number $B-L$, and the new particles in the wake of the $U(1)_{B-L}$ are potential candidates for DM. We presented an overview of the three phenomenologically distinct realizations of a gauged $U(1)_{B-L}$: 1) as an unbroken symmetry with a Stückelberg $Z'$, Dirac neutrinos and neutrinogenesis; 2) broken by two units with Majorana neutrinos, seesaw, and leptogenesis; 3) broken by $n\neq 2$ units, e.g. $n=4$, leading to lepton-number-violating Dirac neutrinos and Dirac leptogenesis. Experiments will have to decide the fate of $B-L$ and resolve the mystery surrounding it.
Acknowledgments {#acknowledgments .unnumbered}
===============
I thank Werner Rodejohann for collaboration on some of the work presented here, and the organizers of the *Moriond EW 2015* for financial support and the opportunity to present my results. This work is funded in part by IISN and by Belgian Science Policy (IAP VII/37).
[99]{}
T. Araki, J. Heeck and J. Kubo, JHEP [**1207**]{} (2012) 083 \[arXiv:1203.4951 \[hep-ph\]\]. J. Heeck and W. Rodejohann, Phys. Rev. D [**84**]{} (2011) 075007 \[arXiv:1107.5238 \[hep-ph\]\]. J. Heeck and W. Rodejohann, Phys. Rev. D [**85**]{} (2012) 113017 \[arXiv:1203.3117 \[hep-ph\]\]. J. Heeck, Nucl. Phys. Proc. Suppl. [**237-238**]{} (2013) 336. J. Heeck, M. Holthausen, W. Rodejohann and Y. Shimizu, arXiv:1412.3671 \[hep-ph\]. W. Altmannshofer, S. Gori, M. Pospelov and I. Yavin, Phys. Rev. D [**89**]{} (2014), 095033. A. Crivellin, G. D’Ambrosio and J. Heeck, arXiv:1501.00993 \[hep-ph\]. A. Crivellin, G. D’Ambrosio and J. Heeck, arXiv:1503.03477 \[hep-ph\]. J. Heeck, Phys. Lett. B [**739**]{} (2014) 256 \[arXiv:1408.6845 \[hep-ph\]\]. K. Dick, M. Lindner, M. Ratz and D. Wright, Phys. Rev. Lett. [**84**]{} (2000) 4039. A. E. Nelson and J. Scholtz, Phys. Rev. D [**84**]{} (2011) 103501 \[arXiv:1105.2812 \[hep-ph\]\]. J. Heeck and W. Rodejohann, Europhys. Lett. [**103**]{} (2013) 32001 \[arXiv:1306.0580\]. J. Heeck, Phys. Rev. D [**88**]{} (2013) 076004 \[arXiv:1307.2241 \[hep-ph\]\]. E. Witten, Nucl. Phys. Proc. Suppl. [**91**]{} (2001) 3 \[hep-ph/0006332\]. S. M. Davidson and H. E. Logan, Phys. Rev. D [**80**]{} (2009) 095008 \[arXiv:0906.3335\]. J. Heeck, in preparation.
[^1]: Talk presented at the 50th Rencontres de Moriond (EW Session), La Thuile, 15 March 2015.
[^2]: An alternative approach would be to use the remaining $\mathbb{Z}_2^{L}$ symmetry to stabilize a newly introduced particle with appropriate $B-L$ charge.
[^3]: Conservation of lepton number modulo $n > 2$ to forbid Majorana masses was also mentioned in Ref. [@Witten:2000dt].
|
---
bibliography:
- 'refs.bib'
title: 'GamePad: A Learning Environment for Theorem Proving'
---
|
---
abstract: |
We report spectroscopic and interferometric observations of the moderately metal-poor double-lined binary system HD 195987, with an orbital period of 57.3 days. By combining our radial-velocity and visibility measurements we determine the orbital elements and derive absolute masses for the components of $M_A = 0.844 \pm
0.018$ M$_{\sun}$ and $M_B = 0.6650 \pm 0.0079$ M$_{\sun}$, with relative errors of 2% and 1%, respectively. We also determine the orbital parallax, $\pi_{\rm orb} = 46.08 \pm 0.27$ mas, corresponding to a distance of $21.70 \pm 0.13$ pc. The parallax and the measured brightness difference between the stars in $V$, $H$, and $K$ yield the component absolute magnitudes in those bands. We also estimate the effective temperatures of the stars as $T_{\rm eff}^A = 5200 \pm
100$ K and $T_{\rm eff}^B = 4200 \pm 200$ K. Together with detailed chemical abundance analyses from the literature giving \[Fe/H\]$\approx
-0.5$ (corrected for binarity) and \[$\alpha$/Fe\]$=+0.36$, we use these physical properties to test current models of stellar evolution for metal-poor stars. Among the four that we considered, we find that no single model fits all observed properties at the measured composition, although we identify the assumptions in each one that account for the discrepancy and we conclude that a model with the proper combination of assumptions should be able to reproduce all the radiative properties. The indications from the isochrone fits and the pattern of enhancement of the metals in HD 195987 are consistent with this being a thick disk object, with an age of 10–12 Gyr.
author:
- 'Guillermo Torres, Andrew F. Boden, David W. Latham, Margaret Pan, Robert P. Stefanik'
title: 'An interferometric-spectroscopic orbit for the binary HD 195987: Testing models of stellar evolution for metal-poor stars'
---
Introduction {#sec:introduction}
============
Accurate determinations of the physical properties of stars in binary systems (mass, radius, temperature, etc.) provide for fundamental tests of models of stellar structure and stellar evolution. The most basic of those stellar properties is the mass. Several dozen eclipsing binary systems have component mass and radius determinations that are good to 1–2% [e.g., @Andersen1991], and show that main-sequence models for stars with masses in the range from about 1 M$_{\sun}$ to 10 M$_{\sun}$ and heavy element abundances near solar are in fairly good agreement with the observations. However, models for stars with masses that are significantly higher or lower, or that are in very early (pre-main sequence) or very advanced (post-main sequence) stages of evolution, or models for chemical compositions that are much different from solar, are largely untested by observations due to a lack of suitable systems or a lack of accuracy.
Stellar evolution theory has a wide range of applications in modern astrophysics, some of which have profound cosmological implications. One such application is the estimate of the ages of globular clusters, which represent the oldest of the stellar populations in our Galaxy. Because the metallicities of globular clusters are typically much lower than solar, model fits to color-magnitude diagrams can be viewed as useful tests of theory at these metal-poor compositions, at least regarding the general shape of the isochrones. However, the predicted mass at any given point along a fitted isochrone cannot be tested directly against accurate observations because such a constraint is unavailable for low metallicities. While it is true that there is no immediate cause for concern regarding the accuracy of current metal-poor models, the use of such calculations for deriving the age and other properties of globular clusters remains an extrapolation to some degree, and it would be reassuring to have detailed observational support.
Double-lined spectroscopic binary systems with known compositions below \[Fe/H\]$=-0.5$ that are also eclipsing and are therefore particularly suitable for accurate determinations of the masses and radii are virtually non-existent, both in clusters and in the general field[^1]. But the absolute masses can still be determined even when the binary is not eclipsing, if the pair can be spatially resolved so that the inclination angle of the orbit can be measured. With the highly precise astrometry now achievable using modern interferometers, this can significantly extend the pool of objects available for testing models of stellar evolution [e.g., @Armstrong1992; @Hummel1994; @Boden1999a; @Boden2000; @Hummel2001].
HD 195987 (HIP 101382, G209-35, Gliese 793.1, Groombridge 3215, $\alpha = 20^h 32^m 51\fs6$, $\delta = +41\arcdeg 53\arcmin
55\arcsec$, J2000, $V=7.08$) is a binary with a period of 57.3 days [@Imbert1980] reported to have a metallicity as low as \[m/H\]$=-0.83$ [@Laird1988], although more recent estimates are not as low. Originally listed as a single-lined spectroscopic binary, further observations of HD 195987 have revealed the weak secondary spectrum so that the system is now double-lined. The possibility of eclipses has been mentioned occasionally in the literature, and would obviously make the system even more interesting. However, the astrometric observations described in this paper clearly rule that out. The object was marginally resolved on one occasion by the speckle technique [@Blazit1987], but subsequent attempts have failed to separate the components [@Balega1999; @Mason2001a]. Measurements by the HIPPARCOS mission [@Perryman1997] detected the photocentric motion of the system with a semimajor axis of about 5 mas, and allowed the inclination angle and therefore the absolute masses to be measured for the first time, albeit with relatively large uncertainties [see @Osborn1999]. More recently the components have been clearly resolved by the Palomar Testbed Interferometer (PTI), opening the possibility of establishing the inclination angle of the orbit with much higher accuracy. This makes it a potentially important system in which the absolute masses can be determined and used to compare with stellar evolution theory for metal-poor stars, and is the motivation for this paper.
We report here our spectroscopic and interferometric observations of HD 195987, the analysis of which has provided accurate masses (good to 1–2%) and luminosities for the two components. We compare these measurements with recent models for the appropriate metallicity.
Observations {#sec:observations}
============
Spectroscopy {#subsec:spectroscopy}
------------
HD 195987 was originally placed on the observing program at the Harvard-Smithsonian Center for Astrophysics (CfA) as part of a project to monitor the radial velocities of a sample of high proper motion objects selected from the Lowell Proper Motion Survey [@Giclas1971; @Giclas1978]. The goal of this project is to investigate a variety of issues related to Galactic structure [see @Carney1987]. Observations were obtained from 1983 July to 2001 January mostly with an echelle spectrograph mounted on the 1.5-m Wyeth reflector at the Oak Ridge Observatory (Harvard, Massachusetts), and occasionally also with nearly identical instruments on the 1.5-m Tillinghast reflector at the F. L. Whipple Observatory (Mt. Hopkins, Arizona) and the Multiple Mirror Telescope (also on Mt. Hopkins, Arizona) prior to its conversion to a monolithic 6.5-m mirror. A single echelle order was recorded with photon-counting intensified Reticon detectors at a central wavelength of 5187 Å, with a spectral coverage of 45 Å. The strongest features present in this window are the lines of the b triplet. The resolving power is $\lambda/\Delta\lambda\approx
35,\!000$, and the signal-to-noise (S/N) ratios achieved range from about 13 to 50 per resolution element of 8.5 . A total of 73 usable spectra were obtained over a period of more than 17 years. The stability of the zero-point of our velocity system was monitored by means of exposures of the dusk and dawn sky, and small systematic run-to-run corrections were applied in the manner described by @Latham1992.
The observations for the first 3 years were analyzed by @Latham1988 with the techniques available at the time, but the lines of the secondary are so faint that they escaped detection and the object was treated as single-lined. With the introduction of more sophisticated analysis methods it was later discovered that the lines from the secondary could in fact be seen [@Carney1994]. A preliminary analysis has been presented by @Goldberg2002 on the basis of the same spectra as the earlier study. For the present investigation many more observations were obtained at higher S/N ratios and at phases that contribute more information to the mass determinations. Radial velocities from all of the spectra were derived by cross-correlation using the two-dimensional algorithm TODCOR [@ZM94]. The templates used for the primary and secondary of HD 195987 were taken from an extensive library of synthetic spectra based on model atmospheres by R. L. Kurucz[^2], computed by Jon Morse specifically for the wavelength window and resolution of our spectra. They are available for a range of effective temperatures ($T_{\rm eff}$), projected rotational velocities ($v \sin i$), surface gravities ($\log
g$), and metallicities (\[m/H\]). Grids of correlations were run to establish the template parameters giving the best match to each component, based on the peak correlation value averaged over all exposures and weighted by the S/N ratio of each spectrum. The stars present no measurable rotational broadening in our spectra, so we adopted $v \sin i$ values of 0 , along with $\log g$ values of 4.5, appropriate for dwarfs. The temperature and metallicity determinations are complicated by the fact that the secondary component is extremely faint (only 10% of the light of the primary; see below), and in addition those two quantities are very strongly correlated. We fixed \[m/H\] at values ranging from $-1.0$ to $+0.5$ (in steps of 0.5 dex) and determined the effective temperatures in each case. Lower metallicities lead to cooler temperatures because the changes in the strength of the lines from these two effects tend to compensate each other. Formally, the best match to the observed spectra was found for a metallicity around \[m/H\]$= -0.2$ and effective temperatures of $T_{\rm eff}^A = 5350$ K and $T_{\rm eff}^B = 4550$ K for the primary and secondary, respectively. However, because of the tradeoff mentioned above, different combinations of \[m/H\] and $T_{\rm
eff}$ give nearly equally good fits without changing the velocities very much, and an external constraint is needed on one of these parameters to break the degeneracy. For this we chose the metallicity, since several determinations are available and appear to agree that the system is metal-deficient compared to the Sun (§\[sec:physics\]). We adopted the value \[m/H\]$ = -0.5$, which leads to effective temperatures of $T_{\rm eff}^A = 5200 \pm 150$ K and $T_{\rm eff}^B = 4450 \pm 250$ K. The light ratio in our spectral window was found to be $(l_B/l_A)_{5187} = 0.09 \pm 0.01$.
Systematic errors in the radial velocities resulting from the narrow spectral window were investigated by means of numerical simulations, as discussed in detail by @Latham1996. Briefly, we generated a set of artificial binary spectra by combining the primary and secondary templates in the appropriate ratio and applying velocity shifts for both components as computed from a preliminary orbital solution at the actual times of observation of each of our spectra. These artificial spectra were then processed with TODCOR in exactly the same way as the real observations, and the resulting velocities were compared with the input (synthetic) values. The differences were found to be smaller than 0.4 , but were nevertheless applied to the raw velocities as corrections. The effect on the minimum masses derived from the spectroscopic orbit, which depend on the velocity semi-amplitudes $K_A$ and $K_B$, is at the level of 0.5%. The final velocities, including corrections, are listed in Table \[tab:rvs\].
The spectroscopic orbital elements resulting from these velocities are given in column 2 of Table \[tab:sb2\]. They are the period ($P$, in days), the center-of-mass velocity ($\gamma$, in ), the radial velocity semi-amplitudes of the primary and secondary ($K_A$ and $K_B$, in ), the orbital eccentricity ($e$), the longitude of periastron of the primary ($\omega_A$, degrees), and the time of periastron passage ($T$, Julian days). Earlier single-lined orbital solutions for HD 195987 were published by @Imbert1980, @Latham1988, and @Duquennoy1991, and @Goldberg2002 recently reported a double-lined solution. They are included in Table \[tab:sb2\] for comparison with ours. The orbit published by @Latham1988, which is based on a small subset of the same spectra used in the present paper, was superseded by that by @Goldberg2002 that used the same material, which in turn is superseded by our new definitive results. The solution by @Duquennoy1991 is not independent of that by @Imbert1980, but represents an update using additional observations with the same instrument. As seen in the table, all these solutions are fairly similar.
The uncertainties given in Table \[tab:sb2\] for the velocity amplitudes of our new solution, upon which the masses of the components depend critically, are strictly internal errors. In addition to the biases described earlier, further systematic errors in the velocities can occur because of uncertainties in the template parameters, particularly the temperatures and the metallicity. Extensive grids of correlations showed that the sensitivity of the velocity amplitudes to the secondary temperature is minimal because that star is so faint. They also showed that the combined effects of errors in the primary temperature and the metallicity contribute an additional $0.04~\kms$ uncertainty to $K_A$ and $0.3~\kms$ to $K_B$. These have been combined with the internal errors in the amplitudes, and propagated through to the final masses that we report later in §\[sec:physics\].
Interferometry {#sec:interferometry}
--------------
Near-infrared, long-baseline interferometric measurements of HD 195987 were conducted with the Palomar Testbed Interferometer (PTI), which is a 110-m baseline $H$- and $K$-band ($\lambda\sim$1.6 $\mu$m and $\sim$2.2 $\mu$m) interferometer located at Palomar Observatory. It is described in full detail elsewhere [@Colavita1999a]. The instrument gives a minimum fringe spacing of about 4 mas at the sky position of our target, making the HD 195987 binary system readily resolvable.
The interferometric observable used for these measurements is the normalized fringe contrast or [*visibility*]{} (squared), $V^2$, of an observed brightness distribution on the sky. The analysis of such data in the context of a binary system is discussed in detail by @Hummel1998, @Boden1999a, @Boden1999b, @Boden2000, and @Hummel2001, and will not be repeated here.
HD 195987 was observed with PTI in conjunction with objects in our calibrator list in the $K$ band ($\lambda\sim$2.2 $\mu$m) on 32 nights between 1999 June 24 and 2001 September 28, covering roughly 14 periods of the system. Additionally, the target was observed in the $H$ band ($\lambda\sim$1.6 $\mu$m) on five nights between 2000 July 20 and 2001 September 27. HD 195987, along with the calibration objects, was observed multiple times during each of these nights, and each observation, or scan, was approximately 130 seconds long. For each scan we computed a mean $V^2$ value from the scan data, and the error in the $V^2$ estimate from the rms internal scatter [@Colavita1999b]. HD 195987 was always observed in combination with one or more calibration sources within $\sim$20$\arcdeg$ on the sky. For our study we used four stars as calibrators: HD 195194, HD 200031, HD 177196, and HD 185395. Table \[tab:calibrators\] lists the relevant physical parameters for these objects. The last two calibration objects are known to have at least one visual companion [Washington Double Star Catalog; @Mason2001b] that could conceivable affect our visibility measurements. The companion of HD 177196 is so distant (44 at last measurement in 1925) that it has no effect because of the effective 1 field stop of the PTI fringe camera focal plane [@Colavita1999a]. Of the three recorded companions of HD 185395 two are more than 40 away, and the other has an angular separation of 29. This close companion is more than 5 magnitudes fainter than the primary in the $K$ band, and can therefore also be safely ignored.
The calibration of the HD 195987 $V^2$ data was performed by estimating the interferometer system visibility ($V^{2}_{\rm sys}$) using calibration sources with model angular diameters, and then normalizing the raw HD 195987 visibility by $V^{2}_{\rm sys}$ to estimate the $V^2$ measured by an ideal interferometer at that epoch [@Mozurkewich1991; @Boden1998]. Calibrating our HD 195987 dataset with respect to the four calibration objects listed in Table \[tab:calibrators\] results in a total of 171 calibrated scans (134 in $K$, 37 in $H$) on HD 195987 on 37 different nights. Our calibrated synthetic wide-band visibility measurements in the $H$ and $K$ bands are summarized in Table \[tab:visibH\] and Table \[tab:visibK\], which include the time of observation, the calibrated $V^2$ measurement and its associated error, the residual from the final fit (see below), the ($u$,$v$) coordinates in units of the wavelength $\lambda$ (weighted by the S/N ratio), and the orbital phase for each of our observations.
Determination of the orbit {#sec:orbit}
==========================
The radial velocities and interferometric visibilities for HD 195987 contain complementary information on the orbit of the system, which is described by the 7 elements mentioned in §\[subsec:spectroscopy\] and the additional elements $a$ (relative semimajor axis, expressed here in mas), $i$ (inclination angle), and $\Omega$ (position angle of the ascending node). Only by combining data from both techniques can all 10 elements of the three-dimensional orbit be determined. While the center-of-mass velocity and the velocity amplitudes depend only on the spectroscopy, the information on $a$, $i$, and $\Omega$ is contained solely in the interferometric visibilities. However, the $V^2$ values are invariant to a $180\arcdeg$ change in the position angle of the binary, so distinguishing between the descending node and the ascending node (where, according to convention, the secondary is receding) relies on our knowledge of the radial velocities.
As in previous analyses using PTI data [@Boden1999a; @Boden1999b; @Boden2000; @BL2001], the orbit of HD 195987 is solved by fitting a Keplerian orbit model directly to the calibrated $V^2$ and radial velocity data simultaneously. This allows the most complete and efficient use of the observations[^3], so long as the two types of measurements are free from systematic errors (see below). It also takes advantage of the redundancy between spectroscopy and interferometry for the four elements $P$, $e$, $\omega_A$, and $T$. Relative weights for the velocities (which are different for the primary and the secondary) and the visibilities were applied based on the internal errors of each type of observation.
Formally the interferometric visibility observables have the potential for resolving not only the HD 195987 relative orbit, but the binary components themselves, and these two effects must be considered simultaneously [e.g., @Hummel1994]. However, in the case of HD 195987 at a distance of approximately 22 pc the dwarf components have typical apparent sizes of less than 0.5 mas, which are highly unresolved by the 3–4 mas PTI fringe spacing. We have therefore estimated the apparent sizes of the components using the bolometric flux and effective temperatures [see @Blackwell94 and references therein], and constrained the orbital solutions to these model values (§\[sec:physics\]). We have adopted apparent component diameters of $0.419 \pm 0.018$ mas and $0.314 \pm 0.049$ mas for the primary and secondary components, respectively. These values are much smaller than the PTI $H$ and $K$ fringe spacings and have a negligible effect on the parameters of the orbit.
The results of this joint fit are listed in Table \[tab:combsol\] (“Full-fit" solution), where we have used the $H$-band and $K$-band visibilities simultaneously, weighted appropriately by the corresponding errors. In addition to the orbital elements, the visibility measurements provide the intensity ratio in $H$ and $K$, which are critical for the model comparisons described later.
The uncertainties listed for the elements in Table \[tab:combsol\] include both a statistical component (measurement error) and our best estimate of the contribution from systematic errors, of which the main components are: (1) uncertainties in the calibrator angular diameters (Table \[tab:calibrators\]); (2) the uncertainty in the center-band operating wavelength ($\lambda_0 \approx$ 1.6 $\mu$m and 2.2 $\mu$m), taken to be 20 nm ($\sim$1%); (3) the geometrical uncertainty in our interferometric baseline ($< 0.01$%); and (4) listed uncertainties in the angular diameters assumed for the stars in HD 195987, that were held constant in the fitting procedure.
Figure \[fig:hd195987\_orbit\] depicts the relative visual orbit of the HD 195987 system on the plane of the sky, with the primary component rendered at the origin and the secondary component shown at periastron. We have indicated the phase coverage of our $V^2$ data on the relative orbit with heavy line segments. Our interferometric data sample essentially all phases of the orbit (see also Figure \[fig:hd195987\_fitResiduals\]), leading to a reliable determination of the elements. The fit to these data is illustrated in Figure \[fig:hd195987\_V2trace\], which shows four consecutive nights of PTI visibility data on HD 195987 (24–27 June 1999), and $V^2$ predictions based on the “Full Fit” model for the system (Table \[tab:combsol\]). The fit to the radial velocity measurements is shown in Figure \[fig:hd195987\_RVfit\]. Phase plots of the $V^2$ and velocity residuals are shown in Figure \[fig:hd195987\_fitResiduals\], along with the corresponding histograms. We note here in passing that the agreement between the speckle measurement by @Blazit1987 (which by chance was obtained very near periastron passage, at phase 0.981) and our interferometric-spectroscopic orbit is very poor. However, this is not surprising given the limited resolution of the speckle observation (32 mas in the visible) compared to the predicted separation at the time of the observation (10.5 mas).
For comparison, we list also in Table \[tab:combsol\] the solutions we obtain for HD 195987 using only the interferometric visibilities (“$V^2$-only"), and only the radial velocities (“RV-only", repeated from Table \[tab:sb2\]). These two separate fits give rather similar results, indicating no significant systematic differences between the spectroscopic and interferometric data sets. Also included in the table is the solution reported in the HIPPARCOS Catalogue, where the semimajor axis refers to the photocentric motion of the pair rather than the relative motion. In the HIPPARCOS solution the elements $P$, $e$, $\omega_A$, and $T$ were adopted from the work by @Duquennoy1991, and held fixed. Given the much larger formal errors, the resulting elements are as consistent with our solution as can be expected.
The light ratio in the optical {#sec:lightratio}
==============================
The interferometric measurements of HD 195987 with PTI provide the intensity ratio between the components in the $H$ and $K$ bands, and allow the individual luminosities to be determined in the infrared. A similar determination in the optical based on our spectroscopy was given in §\[subsec:spectroscopy\]. A small correction from the 5187 Å region to the visual band yields $(l_B/l_A)_V = 0.10 \pm
0.02$. Because the secondary is so faint, this estimate may be subject to systematic errors that are difficult to quantify.
The detection of the photocentric motion of the binary by HIPPARCOS, along with the measurement of the apparent separation with PTI, allows an independent estimate of the light ratio to be made in the optical. The relative semimajor axis ($a$) and the photocentric semimajor axis ($\alpha$) are related by the classical expression $\alpha = a (B -
\beta)$, where $B = M_B/(M_A+M_B)$ is the mass fraction (also expressed as $B=K_A/(K_A+K_B)$ in terms of the observables) and $\beta=l_B/(l_A+l_B)$ is the fractional luminosity. The ratio $l_B/l_A$ at the effective wavelength of the HIPPARCOS observations ($H_p$ passband) can therefore be determined.
In order to take advantage of the much improved orbital elements compared to those available to the HIPPARCOS team, we have re-analyzed the HIPPARCOS intermediate astrometric data (abscissae residuals) as described in the original Catalogue [@HIP1997] [see also @Pourbaix2000]. The orbital period, the eccentricity, the longitude of periastron, and the time of periastron passage were adopted from our combined visibility-radial velocity fit as listed in Table \[tab:combsol\], and held fixed. In addition, the inclination angle and the position angle of the node are now known much better than can be determined from the HIPPARCOS data, so we adopted our own values here as well. The only parameters left to determine are then the semimajor axis of the photocenter ($\alpha$), the corrections to the position and proper motion of the photocenter, and the correction to the HIPPARCOS parallax.
The relevant results from this analysis are $\alpha = 5.04 \pm
0.47$ mas and $\pi_{\rm HIP} = 45.30 \pm 0.46$ mas. The uncertainties may be somewhat underestimated because they do not account for correlations between the elements (since we have fixed several of them in the new solution), although we do not expect the effect to be large in this case. As it turns out, the new results are not very different from the original determinations by the HIPPARCOS team ($\alpha = 5.24
\pm 0.66$ mas and $\pi_{\rm HIP} = 44.99 \pm 0.64$ mas). Forcing the HIPPARCOS parallax to come out identical to our orbital parallax makes a negligible change in the semimajor axis of the photocenter.
The luminosity ratio we determine from this is $(l_B/l_A)_V = 0.13 \pm
0.04$, which includes a very small correction from the $H_p$ band to the Johnson $V$ band. This is in good agreement with our spectroscopic determination, and we adopt the weighted average of the two results, $(l_B/l_A)_V = 0.11 \pm 0.02$, which corresponds to $\Delta V = 2.4 \pm 0.2$ mag.
Physical properties of HD 195987 {#sec:physics}
================================
The absolute masses of the components follow from the results of our combined interferometric-spectroscopic solution summarized in Table \[tab:combsol\], and are $M_A = 0.844 \pm 0.018$ M$_{\sun}$ and $M_B = 0.6650 \pm 0.0079$ M$_{\sun}$[^4]. The high accuracy achieved (2.1% error for the primary and 1.2% error for the secondary) is the result of the high quality of the observations as well as the favorable geometry of the system ($\sin i \approx 0.99$). The limiting factor is the spectroscopy, and in particular the velocities for the faint secondary.
The combination of the angular separation from interferometry and the linear separation from spectroscopy yields the orbital parallax of the system independently of any assumptions beyond Newtonian physics. The result, $\pi_{\rm orb} = 46.08 \pm 0.27$ mas (distance$= 21.70 \pm
0.13$ pc) is nearly a factor of two more precise than our revised HIPPARCOS value of $\pi_{\rm HIP} = 45.30 \pm 0.46$ mas, and the two determinations differ by about 1.5$\sigma$ (1.7%). The corresponding difference in the distance modulus, and consequently in the absolute magnitudes derived below, is $\Delta(m-M) = 0.037$ mag. The ground-based trigonometric parallax of HD 195987 from the weighted average of 5 determinations is $\pi = 49.1 \pm 5.1$ mas [@vanaltena1995].
The component luminosities in the $V$, $H$, and $K$ passbands depend on the flux ratios that we have measured, on the orbital parallax, and on the combined-light photometry. For the $V$ magnitude of the system we adopt the average value given by @Mermilliod1994, which is the result of 10 individual measurements from 7 different sources: $V
= 7.080 \pm 0.016$. Two measurements of $K$ are available from @Voelcker1975 and from the 2MASS Catalog [@2MASS]. Conversion of both to the CIT system [@Elias1983], using the transformations by @Bessell1988 and @Carpenter2001, gives $K=4.98 \pm 0.03$. The $H$ magnitude of the system measured by @Voelcker1975 is $H = 5.23 \pm 0.05$. A 2MASS measurement in $H$ is not available because the object is saturated in this passband, although not in $J$ and $K$. This is somewhat unexpected if we assume normal colors for stars of this mass, and it casts some doubt on the reliability of the $H$-band measurement. As a check we obtained a new measurement using an infrared camera on the 1.2-m telescope at the F.L. Whipple Observatory equipped with a $256\times 256$ InSb detector array, with standard stars adopted from @Elias1982 and 2MASS. The result, $H = 5.04 \pm 0.03$ (on the CIT system), is significantly different from the previous estimate, confirming our suspicions, and it is the value we adopt for the remainder of the paper. Any extinction and reddening corrections are negligible at a distance of only 22 pc. The individual absolute magnitudes and colors are listed in Table \[tab:physics\] along with the other physical properties.
The $V\!-\!K$ color indices for each component allow for an independent estimate of the effective temperatures of the stars. Based on the color-temperature calibrations by @Martinez1992, @Alonso1996, and @Carney1994 (the later being for the CIT system), we obtain average values of $T_{\rm eff}^A = 5200 \pm
100$ K and $T_{\rm eff}^B = 4100 \pm 200$ K that are nearly independent of the adopted metal abundance (for which these calibrations include a corrective term). The primary estimate is identical to our spectroscopic value in §\[subsec:spectroscopy\], while the secondary estimate is somewhat lower than the spectroscopic value and its uncertainty has a substantial contribution from the photometric errors.
An additional estimate of the effective temperature may be obtained from photometry in other bands, via deconvolution of the combined light. For this we used the indices for HD 195987 in the Johnson and Strömgren systems as listed by @Mermilliod1994 ($B\!-\!V =
0.800 \pm 0.014$; $U\!-\!B = 0.370 \pm 0.026$) and by @Olsen1983 [@Olsen1993] ($b\!-\!y = 0.480 \pm 0.002$, $m_1 = 0.296
\pm 0.005$, $c_1 = 0.271\pm 0.005$). The deconvolution was performed using tables of standard colors for normal stars by @Lejeune1998 (for a metallicity of \[Fe/H\]$= -0.5$) and by @Olsen1984, and adopting a magnitude difference between the primary and secondary in the visual band of $\Delta V = 2.4$ mag (§\[sec:lightratio\]). In addition to the calibrations mentioned above we used those by @Carney1983 and @Olsen1984. The results based on the deconvolved $B\!-\!V$, $b\!-\!y$, and also the $V\!-\!K$ indices for the system (accounting for the small difference between the different photometric systems in the infrared) give values of $5200 \pm 100$ K for the primary and $4100 \pm 200$ K for the secondary, identical to the estimates from the observed $V\!-\!K$ colors.
We adopt for the stars in HD 195987 the weighted average of our three determinations of the effective temperature (one spectroscopic and two photometric): $T_{\rm eff}^A = 5200 \pm 100$ K and $T_{\rm eff}^B =
4200 \pm 200$ K. From these effective temperatures, an estimated system bolometric flux of ($5.308\pm 0.062$) $\times$ 10$^{-8}$ erg cm$^{-2}$ s$^{-1}$ based on archival broadband photometry, and our observed 2.2 $\mu$m intensity ratio we estimate component bolometric fluxes of ($4.28\pm 0.19$) and ($1.02\pm 0.28$) $\times$ 10$^{-8}$ erg cm$^{-2}$ s$^{-1}$. These combined again with the adopted component temperatures yield estimated apparent diameters of $0.419\pm0.018$ mas and $0.314\pm0.049$ mas for the primary and secondary components, respectively, which are the values adopted for the orbital fit (§\[sec:orbit\]). At our estimated system distance to HD 195987 these angular diameters imply physical radii of $0.979\pm0.043$ R$_{\sun}$ and $0.73\pm0.11$ R$_{\sun}$ for the primary and secondary components, respectively.
As mentioned in §\[sec:introduction\], early high-resolution spectroscopic estimates of the metal abundance of HD 195987 placed the system in an interesting range for absolute mass determinations, at a metallicity of \[m/H\]$ = -0.83 \pm 0.15$ [@Laird1988]. Subsequently the same authors revised their estimate upward to \[m/H\]$
= -0.60 \pm 0.15$ using the same techniques and more spectra [@Carney1994]. Other determinations in the literature have given values that are similar or somewhat closer to the solar abundance. @Fulbright2000 used high-resolution spectroscopy to derive an estimate of \[Fe/H\]$ = -0.66 \pm 0.13$, while @Beers1999 obtained \[Fe/H\]$ = -0.52 \pm 0.20$ with a combination of different spectroscopic methods based on lower-resolution spectra. @Marsakov1988 inferred \[Fe/H\]$ = -0.34$ based on the ultraviolet excess of the system, and @Wyse1995 used Strömgren photometry to estimate the metallicity at \[Fe/H\]$ =
-0.31$. Despite the range of values, these sources do seem to point toward a heavy element abundance moderately lower than the Sun, by perhaps a factor of 3 or so. We discuss possible biases in the metallicity estimates in §\[sec:iron\].
Discussion {#sec:discussion}
==========
With many of the fundamental physical properties of HD 195987 now known to high accuracy, we proceed in this section with a detailed comparison between the observations and recent models of stellar evolution. In addition to the prospect of an interesting comparison for a chemical composition lower than solar, both components of our system have masses below 1 M$_{\sun}$, a regime in which relatively few stars have absolute mass determinations good to 1–2% [see, e.g., @Andersen1991; @Clausen1999; @Delfosse2000].
Comparison with stellar evolution models {#sec:modelcomp}
----------------------------------------
In order for a given model to be successful, we require that it agree with all the measurements for HD 195987 *simultaneously*, and *for both components* at the same time since they are presumably coeval. The measurements are the absolute masses, the absolute magnitudes in three different passbands ($M_V$, $M_H$, $M_K$), the effective temperatures, and the metal abundance.
In Figure \[fig:yale\] we show the absolute magnitudes and temperatures of the primary and secondary of HD 195987 as a function of mass, compared with isochrones from the Yale-Yonsei models by @Yi2001[^5]. Changes in the isochrones with age for a fixed metallicity equal to solar are illustrated on the left, and consist not only of a vertical displacement but also of a change in slope, becoming steeper for older ages. A similar effect is seen in the diagrams on the right-hand side, which show changes in the models for a fixed age (12 Gyr) and a range of metallicities bracketing the values reported for HD 195987. The isochrones shown with a solid line, which correspond to solar metallicity and an age of 12 Gyr, provide a fairly good fit to the absolute magnitudes in the $H$ and $K$ bands, as well as to the $M_V$ of the primary star. However, the secondary component appears underluminous in $V$, or else the slope of the models in the mass-$M_V$ plane is too shallow. In addition, the temperatures predicted by the models may be somewhat too hot, particularly for the secondary. Setting these differences aside for the moment, the @Yi2001 models would appear to point toward a metallicity for HD 195987 that is not far from solar, along with a fairly old age. This metallicity seems to conflict with the observational evidence indicated earlier. Formally the best fit to the magnitudes in $H$, $K$, and $V$ (primary only) gives a metallicity of \[Fe/H\]$ = -0.07$ and an age of 11.5 Gyr. Lowering the age of the isochrones to 10 Gyr and at the same time decreasing the metallicity to \[Fe/H\]$ = -0.15$ still produces tolerably good fits, but metallicities much lower than this are inconsistent with the measured magnitudes for any age. The disagreement with the $M_V$ of the secondary and with the temperatures remains.
In order to investigate the discrepancy in the absolute visual magnitude for the secondary, we focus in Figure \[fig:mvplane\] on the mass-$M_V$ plane and add observations for other binary components that have accurately determined masses and luminosities: FL Lyr B, HS Aur A, and HS Aur B [@Andersen1991]; Gliese 570 B and Gliese 702 B [@Delfosse2000]; and V818 Tau B and YY Gem AB [@Torres2002]. Figure \[fig:mvplane\]a shows that it is not only the secondary of HD 195987 (filled circles) that appears underluminous compared to the @Yi2001 isochrones, but four other stars with similar masses suggest the same trend (triangles). In Figure \[fig:mvplane\]b we show the same observations compared to the Lyon models by @Baraffe1998, for the same ages and heavy-element composition as in Figure \[fig:mvplane\]a. The agreement for the secondary is now much better, and is presumably due to the use by @Baraffe1998 of sophisticated model atmospheres as boundary conditions to the interior equations, whereas the @Yi2001 models use a gray approximation. The latter has been shown to be inadequate for low mass stars, where molecular opacity becomes important [@Chabrier1997]. Figure \[fig:mvplane\] suggests that the discrepancy begins at a mass intermediate between that of the primary and the secondary in HD 195987. One other consequence of the use of the gray approximation is that the effective temperature is typically overestimated in those models, just as hinted by Figure \[fig:yale\].
In Figure \[fig:baraffesiess\] we illustrate the agreement between the observations for HD 195987 and the predictions from the models by @Baraffe1998 and also by @Siess1997 [see also @Siess2000]. The latter isochrones also use a non-gray approximation (from fits to a different set of model atmospheres than those used by the Lyon group), and should therefore be fairly realistic as well. Once again the @Baraffe1998 models (left-hand side) show reasonably good agreement in the mass-luminosity planes for solar composition and an old age (12 Gyr), but do not quite reproduce the estimated effective temperature of the primary. The @Siess1997 models are similar, although the agreement in $M_V$ is not as good and the fit to the primary temperature is somewhat better.
In addition to the boundary conditions, another difference between the @Baraffe1998 models and the @Yi2001 models is the extent of mixing allowed, as described in the standard convection prescription by the mixing-length parameter $\alpha_{\rm ML}$. The models by @Yi2001 adopt a value of $\alpha_{\rm ML}=1.74$ that best fits the observed properties of the Sun. A much lower value of $\alpha_{\rm ML}=1.0$ is used by @Baraffe1998, whereas the best fit to the Sun in those models requires $\alpha_{\rm ML}=1.9$. This has significant consequences for the temperature profile and other properties such as the radius. In particular, less mixing leads to a lower effective temperature. The effect is illustrated in the diagrams on the left side of Figure \[fig:baraffesiess\], where the dot-dashed lines gives the predictions from the @Baraffe1998 models for \[Fe/H\]$= 0.0$ and an age of 12 Gyr if the solar value of $\alpha_{\rm ML}$ is used. Changes in $M_K$, $M_H$, and $M_V$ compared to the isochrones for the same age and metallicity but with $\alpha_{\rm ML}=1.0$ (solid lines) are relatively small and in fact tend to improve the agreement in $M_V$, while the temperatures predicted with $\alpha_{\rm ML}=1.9$ are several hundred degrees hotter than with $\alpha_{\rm ML}=1.0$ and show better agreement for the primary (the slope is also a better match to the observations). Presumably an intermediate value for the mixing-length parameter used in the @Baraffe1998 models would provide an optimal fit to HD 195987 for the composition and age indicated.
Other recent models give similar fits to the observations for HD 195987. For example, the recent series of isochrones from the Padova group [@Girardi2000] give fits that are also good for a metallicity near solar, and a slightly older age of 14 Gyr.
A shortcoming of the comparisons above is that a number of additional parameters in the isochrones are fixed, and they are somewhat different for each series of models. One of such parameters is the helium abundance, $Y$. The calculations by @Yi2001 use $Y =
0.266$ for solar metallicity (Figure \[fig:yale\]), while @Baraffe1998 adopt $Y = 0.275$ for their models with $\alpha_{\rm ML}=1.0$, but use a higher value of $Y = 0.282$ for the more realistic models that fit the Sun with $\alpha_{\rm ML}=1.9$ (Figure \[fig:baraffesiess\]). @Siess1997 adopted $Y =
0.277$, and @Girardi2000 used $Y = 0.273$. While the differences in these adopted helium abundances for solar metallicity are not large, they do affect the comparisons to some degree. An increase in $Y$ will shift the isochrones upwards in the mass-luminosity diagrams (opposite effect as an increase in metallicity, $Z$), and will yield slightly higher effective temperatures. Changes in $Y$ for other metallicities are governed by the enrichment law adopted in each series of models.
The effect of the treatment of convective overshooting is illustrated on the right-hand side of Figure \[fig:baraffesiess\] for the @Siess1997 models. The dot-dashed line corresponds to the calculations for an age of 12 Gyr and an overshooting parameter of $\alpha_{\rm ov} = 0.2 H_{\rm p}$ (where $H_{\rm p}$ is the pressure scale height), while the other isochrones assume no overshooting. The change compared to the 12 Gyr isochrone with no overshooting (solid line) is hardly noticeable.
The iron abundance of HD 195987 {#sec:iron}
-------------------------------
The model comparisons above seem to point toward a metallicity for the system near solar that is at odds with the observations, along with a fairly old age ($\sim$10–12 Gyr), perhaps a somewhat unusual combination. Because of the possible implications of this disagreement for our confidence in the models, in this section we examine each metallicity determination more closely in an effort to understand these discrepancies, and we attempt to quantify possible systematic errors.
The presence of the secondary, even though it is faint, affects the total light of the system at some level introducing subtle biases in the photometric estimates of the metallicity of HD 195987, which are derived from the combined colors assuming that they correspond to a single star. The photometric estimates of the temperature based on the same assumption are also affected, and this may propagate through and affect some of the spectroscopic abundance determinations as well.
In order to quantify these effects we have simulated binary systems by combining the single-star photometry of a primary and a secondary, each with normal colors, based on the same tabulations used in §\[sec:physics\]. We computed various photometric indices from the combined light, and used them to estimate the metallicity of HD 195987 following the same procedures employed by @Marsakov1988 (who relied on the ultraviolet excess in the Johnson system) and @Wyse1995 (based on Strömgren photometry). We then compared these results with those obtained for the primary alone, for a range of magnitude differences $\Delta V$ between the primary and secondary. In all cases we selected the primary so as to reproduce the actual observed colors of HD 195987 in each photometric system at $\Delta V =
2.4$ mag.
Figure \[fig:johnson\]a shows how the presence of the secondary affects the $U\!-\!B$ and $B\!-\!V$ colors, as well the ultraviolet excess $\delta(U\!-\!B)$ used by @Marsakov1988, as a function of the magnitude difference. The normalized ultraviolet excess $\delta(U\!-\!B)_{0.6}$ [corrected for the guillotine; see @Sandage1969], also frequently used for metallicity estimates, has a similar behavior. Figure \[fig:johnson\]b illustrates the effect on the derived metallicity using the calibration by @Marsakov1988, and also two calibrations by @Carney1979 based on $\delta(U\!-\!B)_{0.6}$. The \[Fe/H\] estimates are biased toward *lower* values[^6], and the maximum effect is for a secondary about 2 magnitudes fainter than the primary. If the original determination by @Marsakov1988 (\[Fe/H\]$=-0.34$) is corrected for this effect ($\sim$0.2 dex at $\Delta V=2.4$ mag), the result is much closer to the solar abundance.
The normalized ultraviolet excess observed for HD 195987 is $\delta(U\!-\!B)_{0.6} = 0.061$. This leads to \[Fe/H\]$=-0.13$ or \[Fe/H\]$=-0.18$ using the linear or quadratic calibration formulae by @Carney1979. From Figure \[fig:johnson\]b it is seen that in this case the corrected values would be \[Fe/H\]$\sim +0.1$.
Similar effects occur in the Strömgren system. The changes in the photometric indices are shown in Figure \[fig:stromgren\]a. The estimate of \[Fe/H\]$ = -0.31$ by @Wyse1995 is based on the quantities {$b\!-\!y$, $m_1$, $c_1$} and the calibration by @Schuster1989. A different calibration by @Olsen1984 involving {$\delta m_1$, $\delta c_1$} yields a nearly identical value of \[Fe/H\]$=-0.32$. As seen in Figure \[fig:stromgren\]b the bias toward lower metallicities due to the secondary happens to reach a maximum quite near the $\Delta V$ of our system. If accounted for, the resulting metal abundance from the Strömgren indices would be essentially solar.
Though it may seem that these corrections bring the photometric estimates of the metallicity of HD 195987 in line with the indications from the model comparisons described earlier, the photometric determinations of \[Fe/H\] are as a rule less trustworthy than those based on high-resolution spectroscopy [see, e.g., @Gehren1988].
The spectroscopic determinations in turn are not, however, completely insensitive to the presence of the secondary either. Two different effects must be considered. On the one hand the continuum from the secondary typically tends to fill in the spectral lines of the primary, which then appear weaker as if the star were more metal-poor. On the other hand the combined-light photometry is reddened. Therefore, the temperature estimates that are used to begin the spectroscopic analysis are biased toward lower values.
The metallicity determination by @Carney1994, \[m/H\]$= -0.60$, which supersedes that by @Laird1988, is based on a subset of the same spectra that we have used in this paper, which are of high resolution but low S/N. To derive the metal abundance they used a $\chi^2$ technique to compare the observations against a grid of synthetic spectra for a range of metallicities, adopting a fixed temperature determined from photometric indices. As they discuss, the \[m/H\] values derived for double-lined binaries with this method can be quite different depending on whether the spectra that are used have the lines of the two components well separated or exactly aligned. In the first case the continuum from the secondary tends to fill in the lines of the primary, as mentioned earlier, and the metallicity is biased toward lower values. In the second case, when the velocities of the stars are similar (close to the center-of-mass velocity $\gamma$ of the binary), the effect is the opposite because the average line strength is slightly increased due to the contribution of the cooler secondary. @Carney1994 used only the few spectra near the $\gamma$ velocity available to them, because then the bias toward higher metallicities is balanced to some degree by the bias toward lower metallicities resulting from the lower temperature inferred from the combined light. The latter is the result of the strong correlation between metallicity and temperature, which is essentially the same as that mentioned in §\[subsec:spectroscopy\].
In Figure \[fig:teff\]a we illustrate the magnitude of the effect of the secondary on the broadband and intermediate-band color indices $B\!-\!V$, $b\!-\!y$, and also $V\!-\!K$, from simulations analogous to those performed earlier. The contamination is much more noticeable in $V\!-\!K$, of course, because the secondary becomes comparatively brighter in $K$. Changes in the effective temperature inferred from the combined light compared to those for the primary alone are shown in Figure \[fig:teff\]b, based on the same calibrations used by @Carney1994.
Simulations performed by @Carney1994 to estimate the residual bias on the metallicity for double-lined systems indicate that in general the two effects do not quite cancel each other out. We estimate from their experiments that their metal abundance \[m/H\] for HD 195987 may still be too low by 0.15–0.20 dex, accounting for the fact that this system is a bit more extreme than the simulated binaries they considered. Note also that this metallicity estimate is labeled \[m/H\] instead of \[Fe/H\] because it includes all metals with lines present in the spectral window. The distinction could in principle be an important one due to the presence of the strong b lines in the CfA spectra, and the typically enhanced abundance of this element relative to iron in metal-poor stars (see next section). Tests described by @Carneyetal1987 indicate that the correction required for this effect is not likely to be more than about $-0.05$ dex for this system, especially given that it may be partially masked by other corrections they applied to the abundances based on comparisons with metallicity standards that may also have enhanced magnesium (J. Laird & B. Carney 2002, priv.comm.). The contribution of the two effects described above leads to an adjusted @Carney1994 estimate of \[Fe/H\] between $-0.50$ and $-0.45$.
The metallicity determination by @Beers1999, \[Fe/H\]$= -0.52$, is based on lower-resolution spectra (1–2 Å) and relies on both the strength of a K-line index and on the height of the peak of the Fourier autocorrelation function of the spectrum, which is a line strength indicator. Both indices are calibrated against the observed $B\!-\!V$ so as to remove the temperature dependence. The unrecognized presence of the secondary is unlikely to weaken the spectral lines of the primary very much in this case since the observations cover only the blue portion of the spectrum, where the secondary is fainter. However the reddening in the $B\!-\!V$ color, estimated to be $\sim$0.04 mag from Figure \[fig:johnson\]a, may have some effect on \[Fe/H\]. It is difficult to quantify its effect due to the complexity of the technique used by @Beers1999, but on the basis of their Figure 6 and Figure 9 the bias will be toward lower abundances for both of their metallicity indices, and a correction of $+0.1$ dex or more may be appropriate. With this adjustment the value for HD 195987 would be \[Fe/H\]$\approx -0.45$ to $-0.40$.
Finally, the detailed high-resolution ($\lambda/\Delta\lambda \sim
50,\!000$) abundance analysis by @Fulbright2000 gives \[Fe/H\]$=-0.66$, but also does not account for the presence of the secondary star. The measured equivalent widths and consequently the metal abundances for iron and other elements are thus expected to be slightly underestimated due to light of the secondary filling in the lines of the primary, given that their spectrum was obtained at an orbital phase when the lines were not exactly aligned. The magnitude of the effect may depend on the strength of the lines used in the analysis, and could be as large as 0.1 dex. In addition, @Fulbright2000 adopted an initial estimate of the effective temperature based on the $V\!-\!K$ index, which according to Figure \[fig:teff\]b leads to an underestimate of $\Delta T_{\rm
eff}\sim$180 K due to the infrared excess. Although their use of iron lines of different strength (to constrain the microturbulence), different excitation potentials (to constrain $T_{\rm eff}$), and different ionization stages (to contrain $\log g$) should tend to reduce the sensitivity of the results to this problem, the parameter sensitivity experiments in Table 8 by @Fulbright2000 indicate that such an error in $T_{\rm eff}$ can still produce changes in the resulting abundance of 0.07–0.16 dex (depending on whether other parameters are held fixed or allowed to vary to compensate for $\Delta
T_{\rm eff}$). The changes will be in the direction of lower abundances for lower temperatures. The combined effect of these corrections results in a metallicity of \[Fe/H\]$\approx -0.50$ to $-0.40$.
It may seem that many of these adjustments to account for the secondary are rather small and perhaps even within the uncertainties of some of the determinations. Indeed they are, but they are also *systematic* in nature and *known to be present*. And because we are trying to understand a systematic difference between the estimates of \[Fe/H\] and indications from all the models, these biases cannot be ignored, particularly since they often go in the same direction. To summarize the state of the (corrected) metallicity estimates for HD 195987, it appears that both of the photometric determinations result in abundances very near solar, while the three spectroscopic estimates, which should be more reliable, yield \[Fe/H\] values between $-0.50$ and $-0.40$. These are still decidedly lower than the solar value, and so the apparent disagreement with the evolutionary models persists if we give preference to the spectroscopic abundances.
Detailed abundances for other elements; further model comparisons {#sec:alpha}
-----------------------------------------------------------------
The detailed chemical analysis by @Fulbright2000 has revealed certain patterns in HD 195987 that are quite typical in metal-poor stars, and that turn out to be key to understanding the comparison with theoretical isochrones. In particular, it is well known that the abundance of the so-called $\alpha$-elements (O, Ne, Mg, Si, S, Ar, Ca, and Ti) in metal-poor stars is usually enhanced relative to iron when compared to the same ratios in Sun [@Conti1967; @Greenstein1970]. The enhancement, \[$\alpha$/Fe\], is actually observed to depend on the metallicity as a result of enrichment from Type II and Type Ia supernovae on different timescales throughout the history of the Galaxy [see, e.g., @Wheeler1989]. The average enhancements for the different $\alpha$-elements increase from the solar values at \[Fe/H\]$= 0.0$ to about $+0.3$ to $+0.5$ dex at \[Fe/H\]$\sim -1$ or so, and then they remain approximately constant for more metal-deficient stars. There is even evidence that different stellar populations in the Galaxy are chemically distinct [see, e.g., @Edvardsson1993; @Mashonkina2000; @Prochaska2000]. Four of the $\alpha$-elements in HD 195987 have been measured by @Fulbright2000, and have been found to be enhanced. The ratios measured are \[Mg/Fe\]$=+0.44 \pm 0.07$, \[Si/Fe\]$=+0.41 \pm 0.07$, \[Ca/Fe\]$=+0.26 \pm 0.07$, and \[Ti/Fe\]$=+0.31 \pm 0.09$ in the standard logarithmic notation.
A stellar mixture rich in these elements produces an increase in the opacity that affects the structure of a star and changes both the temperature and the luminosity predicted by the models. Compared to model calculations with solar ratios for all elements, $\alpha$-enhancement leads to cooler temperatures and lower luminosities. Therefore, these anomalies must be accounted for if a proper interpretation is to be made of the observations. A number of authors have noted, however, that these effects can be mimicked in models that assume solar ratios by simply increasing the overall metallicity used to compute the tracks [see, e.g., @Chieffi1991; @Chaboyer1992; @Baraffe1997; @VandenBerg2000]. Prescriptions for how to do this as a function of the $\alpha$-enhancement have been presented by @Salaris1993 and @VandenBerg2000, among others. In the absence of a detailed measurement of the abundance of the $\alpha$-elements for a particular object, average values of \[$\alpha$/Fe\] have often been adopted in different ranges of \[Fe/H\]. This carries some risk, however, given that there is considerable scatter in the measured enhancements for different stars at any given \[Fe/H\] value [@Edvardsson1993; @Carney1996; @Prochaska2000].
In the case of HD 195987 we have a direct measurement of \[$\alpha$/Fe\] if we assume that all the $\alpha$-elements follow the trend of Mg, Si, Ca, and Ti, which were the ones actually measured. This is generally observed to be true to first order in metal-poor stars. The average overabundance for the four measured elements is \[$\alpha$/Fe\]$
= +0.36 \pm 0.12$[^7]. Following @VandenBerg2000, the overall metallicity adjustment required for this level of \[$\alpha$/Fe\] is $+0.27$ dex. Given the estimates of \[Fe/H\] in the range $-0.50$ to $-0.40$ (§\[sec:iron\]), one would then expect that model isochrones computed for a metallicity between $-0.23$ and $-0.13$ should provide reasonably good fits to the observations. This is in fact quite close to what we find, as shown by the comparisons in §\[sec:modelcomp\], which suggested an overall composition between solar and $-0.15$.
Though adjusting the metallicity of models with scaled-solar mixtures may mimic those with $\alpha$-enhancements to a good approximation, the match is not perfect, as shown by @VandenBerg2000. Isochrones incorporating $\alpha$-enrichment for a range of values of \[$\alpha$/Fe\] between 0.0 to $+0.6$ have recently been published by @Bergbusch2001. We compare them with the observations for HD 195987 in Figure \[fig:vanden\], in the $M_V$ vs. mass and $T_{\rm eff}$ vs. mass diagrams ($H$- and $K$-magnitude predictions are not available in this series of calculations). Because these models use a gray approximation for the boundary conditions between the photosphere and the interior, we do not expect them to reproduce the brightness of the secondary or the temperature of either star very well (see §\[sec:modelcomp\]). Thus we must rely only on the absolute visual magnitude of the primary component for this test.
For a fixed value of \[$\alpha$/Fe\]$= +0.36$ the iron abundance that provides the best match is \[Fe/H\]$ = -0.45$ for an age of 12 Gyr (Figure \[fig:vanden\]), and \[Fe/H\]$ = -0.52$ for 10 Gyr. These values are in excellent agreement with the measured abundances by @Carney1994, @Beers1999, and @Fulbright2000, after correcting for the biases due to the secondary as described before. The effect of the $\alpha$-enhancement is illustrated in the figure by the shift between the solid line (\[$\alpha$/Fe\]$= +0.36$) and the dashed line (\[$\alpha$/Fe\]$ = 0.0$) for the same metallicity. The uncertainty in the adopted value of \[$\alpha$/Fe\] (0.12 dex) translates into the same uncertainty for the fitted value of \[Fe/H\]. The discrepancy for the $M_V$ of the secondary and for the temperatures is in the same direction as shown earlier, i.e., the models are overluminous for a star with the mass of the secondary, and both temperatures are overestimated.
In addition to the $\alpha$-elements, abundance ratios relative to iron were measured by @Fulbright2000 for the light elements Na and Al, the iron-peak elements V, Cr, and Ni, and the heavy elements Ba, Y, and Eu. A comparison with similar measurements for other stars reported by @Prochaska2000 reveals that the pattern of enhancements seen in HD 195987 is very similar to that exhibited by thick disk stars, and is clearly distinct from that of the thin disk stars [see also @Fuhrmann1998]. On that basis one might conclude that HD 195987 belongs to the thick disk population of our Galaxy. This would typically imply a rather old age for the binary, in qualitative agreement with the indications from the models in §\[sec:modelcomp\]. The kinematics of the system computed from the proper motion, the systemic radial velocity, and our orbital parallax (Galactic velocity components $U = 18~\kms$, $V = 13~\kms$, $W =
48~\kms$, relative to the Local Standard of Rest) lend some support to this idea: the $W$ velocity is quite consistent with thick disk membership [see @Carney1989], although the $V$ component is not as extreme.
The abundances of Eu and Ba are of particular astrophysical interest. Europium in the solar system is produced mostly by rapid neutron capture (“*r*-process"), for which the principal formation site is believed to be Type II supernovae. It is one of the few *r*-process elements with relatively unblended atomic lines present in the visible part of the spectrum, which makes it very useful as an indicator for the *r*-process history of stellar material. Barium, on the other hand, is predominantly a slow neutron capture (“*s*-process") element that is thought to be produced mainly during the thermal pulses of asymptotic giant branch stars. The abundance ratio between Eu and Ba is therefore particularly sensitive to whether nucleosynthesis of heavy elements occured by the *s-* or *r*-process, and as such it is a useful diagnostic for studying the chemical evolution of the Galaxy. \[Eu/Ba\] is observed to increase with metal deficiency, and has also been shown to correlate quite well with age [@Woolf1995]. For HD 195987 @Fulbright2000 measured \[Eu/Fe\]$ = +0.29 \pm 0.10$ and \[Ba/Fe\]$
= -0.10 \pm 0.12$, so that \[Eu/Ba\]$ = +0.39 \pm 0.14$. Once again this is quite typical of the values for thick disk stars [see @Mashonkina2000]. Furthermore, according to the correlation reported in Figure 14 by @Woolf1995 this ratio suggests an age for HD 195987 that is at least 10–12 Gyr, in agreement with indications from the stellar evolution models used here and also consistent with the old age of the thick disk [@Carney1989].
The overall agreement with the models {#sec:overall}
-------------------------------------
The comparisons in §\[sec:modelcomp\] and §\[sec:alpha\] indicate that current stellar evolution theory can successfully predict all of the observed characteristics of HD 195987 to high precision for a chemical composition matching what is measured. Unfortunately, however, none of the published models is capable of giving a good fit to all properties of both stars simultaneously. The various isochrones used in this paper do at least agree in predicting an old age for the system, which is also consistent with the pattern of enhancement of the heavy elements, as described above.
The low luminosity of the secondary star in the visual band lends strong support to models that use sophisticated boundary conditions to the interior equations, as in @Baraffe1998, particularly for the lower main sequence. This also appears to be required in order to reproduce the effective temperatures of these stars, but not with a mixing-length parameter as low as in those models. A value of $\alpha_{\rm ML}$ much closer to the one for the Sun [as used in most models *except* those by @Baraffe1998] seems indicated.
The enhanced levels of the $\alpha$-elements in HD 195987 and in other metal-poor stars cannot be ignored in the models, although their effect can be approximated in standard calculations by increasing the overall metallicity. Note, however, that this approach defeats the true purpose of the model comparisons, which is to test whether theory can reproduce all observed properties at the measured value of \[Fe/H\]. In the absence of a detailed chemical analysis for metal-poor stars, the constraint on models is considerably weakened because a satisfactory fit can usually be found by leaving the metallicity as a free parameter. The models by @Bergbusch2001 incorporate the effect of $\alpha$-enhancements, but regretably not the more refined boundary conditions used by @Baraffe1998, which probably explains their failure to reproduce the $M_V$ of the secondary and the effective temperatures of both stars.
The example of HD 195987 emphasizes the importance of detailed spectroscopic analyses as a crucial ingredient for proper comparisons of metal-poor stars with stellar evolution theory. Not only is an accurate determination of \[Fe/H\] needed, but also abundances for other elements, particularly the $\alpha$-elements that contribute the most to the opacities of these objects. To our knowledge the binary studied in this paper represents the most metal-poor object with such an analysis that has absolute masses determined to 2% or better. HD 195987 is perhaps a favorable case for abundance determinations, however, because the secondary is faint enough that it does not interfere significantly with that analysis. This is seldom the case in double-lined eclipsing binaries that have accurately determined physical properties.
While the tests described in previous sections appear to validate stellar evolution theory, the constraint is actually not as strong as it could be because the absolute radii for the components are not directly measured (but see below). The stellar radius is a very sensitive measure of evolution (age), and the fact that it can be derived by purely geometrical means in binaries that are eclipsing is what makes such systems so valuable. From accurate observations of a number of systems that do eclipse, there is now clear evidence that while current models are quite successful in predicting the radiative properties of stars, they fail to reproduce the radii of stars in the lower main-sequence [see @Popper1997; @Lastennet1999; @Clausen1999], predicting sizes that are too small. The discrepancy can be as large as 10–20%, as demonstrated recently for YY Gem AB and V818 Tau B by @Torres2002. This may have a significant impact on ages inferred from these models.
Incidentally, we note that the absolute radii that we derive in §\[sec:physics\] for the components of HD 195987 based on the radiative properties of the system are also somewhat larger than predicted for stars of this mass by all of the models considered here, by roughly 10%. Though more uncertain and possibly not as reliable as the geometric determinations that might be obtained in eclipsing systems, our estimates are consistent with the trend described above.
Final remarks {#sec:finalremarks}
=============
New spectroscopic observations of the nearby moderately metal-poor double-lined spectroscopic binary system HD 195987 (\[Fe/H\]$\sim -0.5$, corrected for binarity), along with interferometric observations that clearly resolve the components for the first time, have allowed us to obtain accurate masses ($\sigma\leq 2$%), the orbital parallax, and component luminosities in $V$, $H$, and $K$. Both the detailed chemical composition (including the pattern of enhancement of the $\alpha$-elements and heavier elements such as Ba and Eu) and the kinematics suggest that the system is a member of the thick disk population of our Galaxy, and an age of 10–12 Gyr or more (consistent with that conclusion) is inferred based on isochrone fits.
The determination of these stellar properties places useful constraints on stellar evolution theory for the lower main sequence. We have shown that none of the models considered here fits all the properties for both components simultaneously, although the discrepancies in each case can be understood in terms of the physical assumptions (convection prescriptions, boundary conditions, detailed chemical composition). Indications are that a model incorporating all the proper assumptions *together* (which is not done in any available series of calculations) would allow for a good match to the observations of HD 195987. While this may not be straightforward to do in practice due to the complexity of the problem, it would seem to be an obvious action item for theorists, which would enable observational astronomers to perform more useful tests for stars under 1 M$_{\sun}$. The ingredients identified here as making the most significant difference in the fit for the mass regime of this binary are: (a) the use of non-gray boundary conditions between the photosphere and the interior, based on modern model atmospheres that incorporate molecular opacity sources; (b) a mixing-length parameter close to that required for the Sun; and (c) the inclusion of the effect of enhanced abundances of the $\alpha$-elements.
Though it seems that all the radiative properties of HD 195987 can in principle be predicted accurately from current theory at the observed metallicity and $\alpha$-enhancement, there is no constraint in this particular case on the radius, a key diagnostic of evolution. Evidence cited earlier from other studies indicates that some adjustments are still needed in the models, which tend to underestimate the sizes of low-mass stars.
Further progress on the observational side in testing models for metal-poor stars requires additional candidates suitable for accurate determinations of their physical properties. The recent paper by @Goldberg2002 presents a list of nearly 3 dozen double-lined spectroscopic binaries from a proper motion sample, many of which are metal-deficient. Improvements in the sensitivity of ground-based interferometers in the coming years should allow several of them to be spatially resolved. Eclipsing binary candidates found in globular clusters in a number of recent studies are also valuable, but are fainter and the spectroscopy will be more challenging. In both cases detailed chemical analyses will be very important for a meaningful test of theory, as shown by the system reported here.
We are grateful to Joe Caruso, Bob Davis, and Joe Zajac for obtaining many of the spectroscopic observations, and to Bob Davis for also maintaining the CfA echelle database. We thank Lucas Macri for assistance in obtaining the $H$ magnitude of HD 195987, and Bruce Carney and John Laird for a careful reading of the manuscript and for very helpful comments on abundance issues. The anonymous referee is also thanked for a number of useful suggestions. Some of the observations reported here were obtained with the Multiple Mirror Telescope, a joint facility of the Smithsonian Institution and the University of Arizona. Interferometer data were obtained at Palomar Observatory using the NASA Palomar Testbed Interferometer (PTI), supported by NASA contracts to the Jet Propulsion Laboratory. Science operations with PTI are conducted through the efforts of the PTI Collaboration ([http://huey.jpl.nasa.gov/palomar/ptimembers.html]{}), and we acknowledge the invaluable contributions of our PTI colleagues and the PTI professional observer Kevin Rykoski. MP acknowledges support from the National Science Foundation Research Experiences for Undergraduates program at SAO. This research has made use of software produced by the Interferometry Science Center at the California Institute of Technology. This research has also made use of the SIMBAD database, operated at CDS, Strasbourg, France, of NASA’s Astrophysics Data System Abstract Service, of the Washington Double Star Catalog maintained at the U.S. Naval Observatory, and of data products from the Two Micron All Sky Survey, which is a joint project of the University of Massachusetts and the Infrared Processing and Analysis Center, funded by the National Aeronautics and Space Administration and the National Science Foundation.
Alonso, A., Arribas, S., & Martínez-Roger, C. 1996, , 313, 873
Andersen, J. 1991, , 3, 91
Armstrong, J. T., Hummel, C. A., Quirrenbach, A., Buscher, D. F., Mozurkewich, D., Vivekanand, M., Simon, R. S., Denison, C. S., Johnston, K. J., Pan, X.-P., Shao, M., & Colavita, M. M. 1992, , 104, 2217
Balega, I. I., Balega, Y. Y., Maksimov, A. F., Pluzhnik, E. A., Shkhagosheva, Z. U., & Vasyuk, V. A. 1999, , 140, 287
Baraffe, I., Chabrier, G., Allard, F., & Hauschildt, P. H. 1997, , 327, 1054
Baraffe, I., Chabrier, G., Allard, F., & Hauschildt, P. H. 1998, , 337, 403
Beers, T. C., Rossi, S., Norris, J. E., Ryan, S. G., & Shefler, T. 1999, , 117, 981
Bergbusch, P. A., & VandenBerg, D. A. 2001, , 556, 322
Bessell, M., & Brett, J. 1988, , 100, 1134
Blackwell, D., & Lynas-Gray, A. 1994, , 282, 899
Blazit, A., Bonneau, D., & Foy, R. 1987, , 71, 57
Boden, A. F., Colavita, M. M., van Belle, G. T., & Shao, M. 1998, , 3350, 872
Boden, A. F., Creech-Eakman, M., & Queloz, D. 2000, , 536, 880
Boden, A. F., Koresko, C. D., van Belle, G. T., Colavita, M. M., Dumont, P. J., Gubler, J., Kulkarni, S. R., Lane, B. F., Mobley, D., Shao, M., Wallace, J. K., The PTI Collaboration, & Henry, G. W. 1999a, , 515, 356
Boden, A. F., & Lane, B. F. 2001, , 547, 1071
Boden, A. F., Lane, B. F., Creech-Eakman, M. J., Colavita, M. M., Dumont, P. J., Gubler, J., Koresko, C. D., Kuchner, M. J., Kulkarni, S. R., Mobley, D. W., Pan, X.-P., Shao, M., van Belle, G.T., Wallace, J. K., & Oppenheimer, B. R. 1999b, , 527, 360
Carney, B. W. 1979, , 233, 211
Carney, B. W. 1983, , 88, 623
Carney, B. W. 1996, , 108, 900
Carney, B. W., Laird, J. B., Latham, D. W., & Kurucz, R. L. 1987, , 94, 1066
Carney, B. W., & Latham, D. W. 1987, , 92, 116
Carney, B. W., Latham, D. W., & Laird, J. B. 1989 , 97, 423
Carney, B. W., Latham, D. W., Laird, J. B., & Aguilar, L. A. 1994, , 107, 2240
Carpenter, J. M. 2001, , 121, 2851
Chaboyer, B., Sarajedini, A., & Demarque, P. 1992, , 394, 515
Chabrier, G., & Baraffe, I. 1997, , 327, 1039
Chieffi, A., Straniero, O., & Salaris, M. 1991, in ASP Conf. Ser.13, The Formation and Evolution of Star Clusters, ed. K. Janes (San Francisco: ASP). 219
Clausen, J. V., Baraffe, I., Claret, A., & VandenBerg, D. B. 1999, in Theory and Tests of Convection in Stellar Structure, ASP Conf.Ser. 173, eds. A. Giménez, E. F. Guinan & B. Montesinos (ASP: San Francisco), 265
Colavita, M. M. 1999, , 111, 111
Colavita, M. M., Wallace, J. K., Hines, B. E., Gursel, Y., Malbet, F., Palmer, D. L., Pan, X.-P., Shao, M., Yu, J. W., Boden, A. F., Dumont, P. J., Gubler, J., Koresko, C. D., Kulkarni, S. R., Lane, B. F., Mobley, D. W., & van Belle, G. T. 1999, , 510, 505
Conti, P. S., Greenstein, J. L., Spinrad, H., Wallerstein, G., & Vardya, M. S., 1967, , 148, 105
Delfosse, X., Forveille, T., Ségransan, D., Beuzit, J.-L., Udry, S., Perrier, C., & Mayor, M. 2000, , 364, 217
Duquennoy, A., & Mayor, M. 1991, , 248, 485
Edvardsson, B., Andersen, J., Gustafsson, B., Lambert, D. L., Nissen, P. E., & Tomkin, J. 1993, , 275, 101
Elias, J., Frogel, J., Hyland, A., & Jones, T. 1983, , 88, 1027
Elias, J., Frogel, J., Matthews, K., & Neugebauer, G. 1982, , 87, 1029
ESA 1997, The Hipparcos and Tycho Catalogues, ESA SP-1200
Fulbright, J. P. 2000, , 120, 1841
Fuhrmann, K. 1998, , 338, 161
Gehren, T. 1988, Reviews in Modern Astronomy 1, ed. G. Klare (Heidelberg: Springer), 52
Giclas, H. L., Burnham, R. Jr., & Thomas, N. G. 1971, Lowell Proper Motion Survey, Northern Hemisphere (Flagstaff: Lowell Observatory)
Giclas, H. L., Burnham, R. Jr., & Thomas, N. G. 1978, Lowell Obs. Bull. No. 164
Girardi, L., Bressan, A., Bertelli, G., & Chiosi, C. 2000, , 141, 371
Goldberg, D., Mazeh, T., Latham, D. W., Stefanik, R. P., Carney, B. W., & Laird, J. B. 2002, , submitted
Greenstein, J. L. 1970, Comm. Astrophys. Space Sci., 2, 85
Hummel, C. A., Armstrong, J. T., Quirrenbach, A., Buscher, D. F., Mozurkewich, D., Elias, N. S. II, & Wilson, R. E. 1994, , 107, 1859
Hummel, C. A., Carquillat, J.-M., Ginestet, N., Griffin, R. F., Boden, A. F., Hajian, A. R., Mozurkewich, D., & Nordgren, T. E. 2001, , 121, 1623
Hummel, C. A., Mozurkewich, D., Armstrong, J. T., Hajian, A. R., Elias, N. M. II, & Hutter, D. J. 1998, , 116, 2536
Imbert, M. 1980, , 42, 331
Kaluzny, J., Thompson, I. B., Krzeminski, W., Olech, A., Pych, W., & Mochejska, B. 2001, astro-ph/0111089
Laird, J. B., Carney, B. W., & Latham, D. W. 1988, , 95, 1843
Lastennet, E., Valls-Gabaud, D., Lejeune, Th., & Oblak, E. 1999, , 349, 485
Latham, D. W. 1992, in IAU Coll. 135, Complementary Approaches to Double and Multiple Star Research, ASP Conf. Ser. 32, eds. H. A.McAlister & W. I. Hartkopf (San Francisco: ASP), 110
Latham, D. W., Mazeh, T., Carney, B. W., McCrosky, R. E., Stefanik, R. P., & Davis, R. J. 1988, , 96, 567
Latham, D. W., Nordström, B., Andersen, J., Torres, G., Stefanik, R. P., Thaller, M., & Bester, M. 1996, , 314, 864
Lejeune, Th., Cuisinier, F., & Buser, R. 1998, , 130, 65
Marsakov, V. A., & Shevelev, Yu. G. 1988, Bull. Inf. CDS, 35, 129
Martínez-Roger, C., Arribas, S., & Alonso, A. 1992, Mem. Soc.Astr. Italiana, 63, 263
Mashonkina, L., & Gehren, T. 2000, , 364, 249
Mason, B. D., Hartkopf, W. I., Holdenried, E. R., & Rafferty, T.J. 2001a, , 121, 3224
Mason, B. D., Wycoff, G. L., Hartkopf, W. I., Douglass, G. G., & Worley, C. E. 2001b, , 122, 3466
Mermilliod, J.-C., & Mermilliod, M. 1994, Catalogue of Mean UBV Data on Stars, (New York: Springer)
Mozurkewich, D., Johnston, K. J., Simon, R. S., Bowers, P. F., Gaume, R., Hutter, D. J., Colavita, M. M., Shao, M., & Pan, X.-P. 1991, , 101, 2207
Olsen, E. H. 1983, , 54, 55
Olsen, E. H. 1984, , 57, 443
Olsen, E. H. 1993, , 102, 89
Osborn, W., & Hershey, J. L. 1999, , 111, 566
Perryman, M. A. C., Lindegren, L., Kovalevsky, J., Hoeg, E., Bastian, U., Bernacca, P. L., Crézé, M., Donati, F., Grenon, M., van Leeuwen, F., van der Marel, H., Mignard, F., Murray, C. A., Le Poole, R. S., Schrijver, H., Turon, C., Arenou, F., Froeschl, M., & Petersen, C. S. 1997, , 323, L49
Popper, D. M. 1997, , 114, 1195
Pourbaix, D., & Jorissen, A. 2000, , 145, 161
Prochaska, J. X., Naumov, S. O., Carney, B. W., McWilliam, A., & Wolfe, A. M. 2000, , 120, 2513
Salaris, M., Chieffi, A., & Straniero, O. 1993, , 414, 580
Sandage, A. 1969, , 158, 1115
Siess, L., Dufour, E., & Forestini, M. 2000, , 358, 593
Siess, L., Forestini, M., & Dougados, C. 1997, , 324, 556
Schuster, W. J., & Nissen, P. E. 1989, , 221, 65
Skrutskie M. F., Schneider S. E., Stiening R., Strom S. E., Weinberg M. D., Beichman C., Chester T., Cutri R., Lonsdale C., Elias J., Elston R., Capps R., Carpenter J., Huchra J., Liebert J., Monet D., Price S., & Seitzer P. 1997, in The Impact of Large Scale Near-IR Sky Surveys, eds. F. Garzon et al. (Dordrecht: Kluwer), 25
Thompson, I. B., Kaluzny, J., Pych, W., Burley, G., Krzeminski, W., Paczyński, B., Persson, S. E., & Preston, G. W. 2001, , 121, 3089
Torres, G., & Ribas, I. 2002, , 567, 1140
Udalski A., Szymanski M., Kaluzny J., Kubiak M., Krzeminski W., Mateo M., Preston, G. W., & Paczyński B. 1993, Acta Astr. 43, 289
van Altena, W. F., Lee, J. T., & Hoffleit, E. D. 1995, The General Catalogue of Trigonometric Stellar Parallaxes, (4th ed.; New Haven: Yale Univ. Obs.)
VandenBerg, D. A., Swenson, F. J., Forrest, J. R., Iglesias, C.A., & Alexander, D. R. 2000, , 532, 430
Voelcker, K. 1975, , 22, 1
Wheeler, J. C., Sneden, C., & Truran, J. W. 1989, , 27, 279
Woolf, V. M., Tomkin, J., & Lambert, D. L. 1995, , 453, 660
Wyse, R. F. G., & Gilmore, G. 1995, , 110, 2771
Yi, S., Demarque, P., Kim, Y.-C., Lee, Y.-W., Ree, C. H., Lejeune, T., & Barnes, S. 2001, , 136, 417
Zucker, S., & Mazeh, T. 1994, , 420, 806
-1.0in -0.8in
[crrrrc]{} 45546.727& $-$19.53 & $+$14.51 & $-$0.08 & $+$3.08 & 0.6932\
45563.682& $+$32.17 & $-$51.84 & $+$0.71 & $+$1.35 & 0.9890\
45623.550& $+$30.44 & $-$52.12 & $+$0.40 & $-$0.74 & 0.0334\
45688.410& $+$2.07 & $-$14.98 & $-$0.02 & $+$0.93 & 0.1649\
45689.413& $-$1.49 & $-$10.75 & $-$0.21 & $+$0.88 & 0.1824\
45694.419& $-$13.96 & $+$5.34 & $+$0.08 & $+$0.78 & 0.2697\
45695.426& $-$15.67 & $+$8.57 & $+$0.22 & $+$1.66 & 0.2873\
45721.434& $-$15.66 & $+$8.75 & $-$0.74 & $+$3.07 & 0.7410\
45742.957& $+$12.13 & $-$32.08 & $-$0.63 & $-$2.63 & 0.1165\
45757.909& $-$22.44 & $+$16.21 & $+$0.11 & $+$0.84 & 0.3774\
45887.911& $-$22.70 & $+$14.83 & $-$0.11 & $-$0.59 & 0.6453\
45896.729& $-$7.20 & $-$4.13 & $-$0.17 & $+$0.21 & 0.7991\
45917.758& $+$1.72 & $-$14.66 & $-$0.15 & $+$0.97 & 0.1660\
45935.641& $-$25.91 & $+$19.89 & $-$0.21 & $+$0.52 & 0.4780\
45945.660& $-$22.10 & $+$13.39 & $+$0.08 & $-$1.51 & 0.6527\
45949.647& $-$16.94 & $+$11.16 & $-$0.06 & $+$2.99 & 0.7223\
45962.577& $+$25.87 & $-$46.00 & $+$0.25 & $-$0.22 & 0.9479\
45981.527& $-$14.93 & $+$5.53 & $+$0.05 & $-$0.23 & 0.2785\
46005.529& $-$19.54 & $+$11.15 & $-$0.41 & $+$0.12 & 0.6972\
46008.517& $-$14.41 & $+$1.93 & $-$0.44 & $-$2.55 & 0.7493\
46215.852& $-$22.06 & $+$13.87 & $-$0.10 & $-$0.74 & 0.3663\
46246.856& $+$16.19 & $-$32.33 & $+$0.08 & $+$1.38 & 0.9072\
46274.870& $-$24.09 & $+$14.82 & $-$0.65 & $-$1.67 & 0.3959\
46310.591& $+$31.08 & $-$53.54 & $-$0.33 & $-$0.41 & 0.0191\
46489.937& $+$6.06 & $-$20.05 & $+$0.42 & $+$0.37 & 0.1479\
46493.936& $-$7.15 & $-$1.58 & $+$0.05 & $+$2.53 & 0.2176\
48199.555& $+$29.89 & $-$50.92 & $+$0.02 & $+$0.25 & 0.9728\
51257.899& $-$19.04 & $+$10.90 & $+$0.30 & $-$0.39 & 0.3268\
51258.890& $-$20.53 & $+$12.46 & $+$0.05 & $-$0.40 & 0.3440\
51260.898& $-$22.53 & $+$13.91 & $+$0.11 & $-$1.57 & 0.3791\
51261.884& $-$23.54 & $+$14.28 & $-$0.09 & $-$2.23 & 0.3963\
51263.902& $-$24.36 & $+$18.06 & $+$0.37 & $-$0.08 & 0.4315\
51264.901& $-$25.27 & $+$18.55 & $-$0.08 & $-$0.17 & 0.4489\
51267.884& $-$25.59 & $+$19.63 & $+$0.30 & $+$0.02 & 0.5009\
51268.880& $-$25.97 & $+$19.16 & $-$0.06 & $-$0.48 & 0.5183\
51269.877& $-$25.62 & $+$18.30 & $+$0.21 & $-$1.23 & 0.5357\
51270.870& $-$25.37 & $+$17.62 & $+$0.26 & $-$1.66 & 0.5530\
51273.886& $-$24.28 & $+$16.68 & $+$0.05 & $-$0.95 & 0.6057\
51274.884& $-$23.33 & $+$16.42 & $+$0.33 & $-$0.35 & 0.6231\
51277.857& $-$20.64 & $+$13.07 & $+$0.15 & $-$0.07 & 0.6749\
51279.868& $-$17.78 & $+$11.49 & $+$0.25 & $+$1.86 & 0.7100\
51293.838& $+$26.71 & $-$47.09 & $-$0.07 & $+$0.16 & 0.9537\
51295.873& $+$31.48 & $-$52.45 & $+$0.00 & $+$0.76 & 0.9892\
51296.859& $+$32.24 & $-$52.68 & $+$0.32 & $+$1.09 & 0.0064\
51297.814& $+$31.41 & $-$51.70 & $+$0.30 & $+$1.05 & 0.0231\
51298.869& $+$28.23 & $-$50.08 & $-$0.70 & $-$0.10 & 0.0415\
51299.858& $+$26.18 & $-$45.17 & $+$0.26 & $+$0.99 & 0.0587\
51300.861& $+$22.61 & $-$40.39 & $+$0.39 & $+$1.07 & 0.0762\
51380.600& $-$25.37 & $+$20.08 & $+$0.18 & $+$0.91 & 0.4673\
51413.642& $+$28.58 & $-$49.05 & $+$0.00 & $+$0.49 & 0.0437\
51424.724& $-$10.20 & $-$0.90 & $-$0.19 & $-$0.34 & 0.2371\
51445.726& $-$24.67 & $+$18.91 & $-$0.26 & $+$1.18 & 0.6035\
51464.487& $+$21.74 & $-$41.97 & $-$0.10 & $-$0.99 & 0.9308\
51466.675& $+$29.24 & $-$51.29 & $-$0.10 & $-$0.79 & 0.9689\
51468.667& $+$32.45 & $-$52.45 & $+$0.51 & $+$1.35 & 0.0037\
51476.569& $+$7.09 & $-$21.63 & $+$0.06 & $+$0.54 & 0.1415\
51521.513& $+$20.77 & $-$42.93 & $+$0.15 & $-$3.50 & 0.9256\
51525.510& $+$32.26 & $-$53.03 & $+$0.48 & $+$0.57 & 0.9953\
51530.440& $+$20.79 & $-$42.20 & $-$0.28 & $-$2.20 & 0.0813\
51618.884& $-$23.54 & $+$18.30 & $+$0.06 & $+$1.60 & 0.6243\
51640.847& $+$31.85 & $-$53.88 & $-$0.06 & $-$0.12 & 0.0074\
51667.839& $-$25.55 & $+$18.42 & $+$0.16 & $-$0.95 & 0.4783\
51682.802& $-$14.67 & $+$10.21 & $+$0.44 & $+$4.29 & 0.7393\
51699.727& $+$29.26 & $-$51.44 & $-$0.63 & $-$0.24 & 0.0346\
51712.696& $-$12.89 & $+$4.55 & $+$0.13 & $+$1.28 & 0.2609\
51788.531& $-$24.41 & $+$18.05 & $+$0.59 & $-$0.43 & 0.5838\
51808.604& $+$22.50 & $-$42.91 & $-$0.10 & $-$0.97 & 0.9340\
51815.609& $+$26.25 & $-$47.50 & $-$0.16 & $-$0.73 & 0.0562\
51842.553& $-$26.18 & $+$17.75 & $-$0.29 & $-$1.85 & 0.5263\
51866.503& $+$24.55 & $-$46.12 & $-$0.28 & $-$1.34 & 0.9441\
51874.448& $+$20.28 & $-$39.25 & $-$0.47 & $+$0.35 & 0.0827\
51921.441& $+$14.54 & $-$33.52 & $-$0.39 & $-$1.31 & 0.9025\
52004.892& $-$21.96 & $+$13.64 & $-$0.47 & $-$0.38 & 0.3583\
[lccc@cc]{}\
$P$ (days)& 57.32161 $\pm$ 0.00034& 57.3210 $\pm$ 0.0014& 57.325 $\pm$ 0.009& 57.3240 $\pm$ 0.0013& 57.3199 $\pm$ 0.0065\
$\gamma$ ()& $-5.867$ $\pm$ 0.038& $-5.630$ $\pm$ 0.075& $-5.56$ $\pm$ 0.12& $-6.13$ $\pm$ 0.07& $-$5.59 $\pm$ 0.12\
$K_{\rm A}$ ()& 28.944 $\pm$ 0.046& 28.80 $\pm$ 0.12& 28.89 $\pm$ 0.20& 28.73 $\pm$ 0.10& 29.16 $\pm$ 0.19\
$K_{\rm B}$ ()& 36.73 $\pm$ 0.21& & & & 34.6 $\pm$ 1.2\
$e$& 0.3103 $\pm$ 0.0018 & 0.305 $\pm$ 0.003 & 0.316 $\pm$ 0.006 & 0.306 $\pm$ 0.003 & 0.3083 $\pm$ 0.0060\
$\omega_{\rm A}$ (deg)& 357.03 $\pm$ 0.35& 356.8 $\pm$ 0.7& 355.9 $\pm$ 0.8& 356.8 $\pm$ 0.7& 357.1 $\pm$ 1.1\
$T$ (HJD$-$2,400,000)& 49404.825 $\pm$ 0.045 & 49404.6 $\pm$ 0.1 & 49404.8 $\pm$ 0.3 & 49404.933 $\pm$ 0.097 & 49404.78 $\pm$ 0.16\
\
$f(M)$ (M$_{\sun}$)& & 0.122 $\pm$ 0.002 & 0.1225 $\pm$ 0.0026 & 0.1219 $\pm$ 0.0017 &\
$M_{\rm A}\sin^3 i$ (M$_{\sun}$)& 0.808 $\pm$ 0.010 & & & & 0.721 $\pm$ 0.052\
$M_{\rm B}\sin^3 i$ (M$_{\sun}$)& 0.6369 $\pm$ 0.0046 & & & & 0.607 $\pm$ 0.024\
$q\equiv M_{\rm B}/M_{\rm A}$& 0.7881 $\pm$ 0.0047 & & & & 0.84 $\pm$ 0.03\
$a_{\rm A}\sin i$ (10$^6$ km)& 21.689 $\pm$ 0.036& 21.6 $\pm$ 0.1& 21.65 $\pm$ 0.16& 21.57 $\pm$ 0.10&\
$a_{\rm B}\sin i$ (10$^6$ km)& 27.52 $\pm$ 0.16& & & &\
$a \sin i$ (R$_{\sun}$)& 70.70 $\pm$ 0.24& & & & 68.7 $\pm$ 1.3\
\
$N_{\rm obs}$& 73$+$73 & 60 & 31 & 64 & 28$+$28\
Time span (days)& 6458 & 758 & 1003 & 4365 & 2653\
$\sigma_{\rm A}$ ()& 0.30 & 0.54 & 0.68 & 0.54 & 0.63\
$\sigma_{\rm B}$ ()& 1.38 & & & & 4.51\
[cccccc]{}
HD 195194& & 7.0 & 4.8 & 2.7 & 0.67 $\pm$ 0.10\
HD 200031& & 6.8 & 4.7 & 6.0 & 0.56 $\pm$ 0.05\
HD 177196& & 5.0 & 4.5 & 17 & 0.62 $\pm$ 0.10\
HD 185395& & 4.5 & 3.5 & 17 & 0.84 $\pm$ 0.08\
\[tab:calibrators\]
[ccccrrc]{} 51745.813& 0.322 & 0.135 & $-$0.123 & $-$30.683 & $-$57.824 & 0.8386\
51745.815& 0.407 & 0.127 & $-$0.050 & $-$30.198 & $-$58.251 & 0.8386\
51745.827& 0.639 & 0.259 & $+$0.123 & $-$27.951 & $-$59.560 & 0.8388\
51745.828& 0.608 & 0.174 & $+$0.083 & $-$27.624 & $-$59.703 & 0.8389\
51745.894& 1.051 & 0.339 & $+$0.166 & $-$12.316 & $-$65.245 & 0.8400\
51745.901& 0.920 & 0.272 & $+$0.006 & $-$10.291 & $-$65.561 & 0.8401\
51745.941& 1.025 & 0.414 & $+$0.067 & $+$0.135 & $-$66.600 & 0.8408\
52103.920& 0.743 & 0.231 & $+$0.015 & $-$44.026 & $-$26.884 & 0.0859\
52103.922& 0.740 & 0.031 & $-$0.019 & $-$43.768 & $-$27.217 & 0.0859\
52103.946& 1.054 & 0.341 & $+$0.080 & $-$39.373 & $-$31.461 & 0.0864\
52175.716& 0.584 & 0.037 & $+$0.001 & $-$45.253 & $-$25.448 & 0.3384\
52175.719& 0.638 & 0.025 & $-$0.048 & $-$44.871 & $-$25.984 & 0.3385\
52175.729& 0.911 & 0.084 & $-$0.039 & $-$43.101 & $-$27.937 & 0.3386\
52175.740& 0.956 & 0.050 & $+$0.123 & $-$41.183 & $-$29.784 & 0.3388\
52175.742& 0.846 & 0.047 & $+$0.073 & $-$40.779 & $-$30.131 & 0.3389\
52175.775& 0.481 & 0.038 & $-$0.014 & $-$33.720 & $-$35.322 & 0.3394\
52175.777& 0.499 & 0.033 & $-$0.077 & $-$33.055 & $-$35.692 & 0.3395\
52175.788& 0.854 & 0.101 & $+$0.004 & $-$30.364 & $-$37.135 & 0.3397\
52175.799& 0.905 & 0.094 & $+$0.029 & $-$27.430 & $-$38.497 & 0.3399\
52177.609& 0.980 & 0.063 & $+$0.049 & $-$50.012 & $-$4.245 & 0.3714\
52177.616& 0.977 & 0.088 & $+$0.032 & $-$50.413 & $-$5.798 & 0.3716\
52177.662& 0.789 & 0.066 & $+$0.034 & $-$49.975 & $-$15.690 & 0.3724\
52177.671& 0.977 & 0.132 & $+$0.008 & $-$49.397 & $-$17.432 & 0.3725\
52177.710& 0.534 & 0.106 & $+$0.072 & $-$45.059 & $-$25.216 & 0.3732\
52177.712& 0.662 & 0.086 & $+$0.109 & $-$44.757 & $-$25.646 & 0.3732\
52177.730& 0.944 & 0.194 & $+$0.012 & $-$41.940 & $-$28.921 & 0.3735\
52177.738& 0.651 & 0.139 & $-$0.025 & $-$40.397 & $-$30.397 & 0.3737\
52177.740& 0.643 & 0.089 & $+$0.045 & $-$39.995 & $-$30.719 & 0.3737\
52179.617& 0.972 & 0.059 & $+$0.028 & $-$50.720 & $-$7.199 & 0.4065\
52179.625& 0.978 & 0.061 & $+$0.075 & $-$50.815 & $-$8.877 & 0.4066\
52179.662& 0.692 & 0.072 & $+$0.045 & $-$49.836 & $-$16.701 & 0.4072\
52179.669& 0.942 & 0.082 & $+$0.019 & $-$49.322 & $-$18.258 & 0.4074\
52179.704& 0.373 & 0.059 & $+$0.027 & $-$45.322 & $-$25.308 & 0.4080\
52179.706& 0.446 & 0.027 & $+$0.036 & $-$45.064 & $-$25.641 & 0.4080\
52179.713& 0.802 & 0.077 & $+$0.061 & $-$43.940 & $-$27.019 & 0.4081\
52179.731& 0.736 & 0.068 & $-$0.029 & $-$40.884 & $-$30.179 & 0.4085\
52179.739& 0.415 & 0.058 & $-$0.026 & $-$39.393 & $-$31.485 & 0.4086\
[ccccrrc]{} 51353.925& 0.749 & 0.042 & $+$0.021 & $-$16.722 & $-$46.365 & 0.0020\
51353.949& 0.945 & 0.044 & $+$0.015 & $-$12.573 & $-$47.823 & 0.0024\
51353.974& 0.792 & 0.040 & $-$0.034 & $-$7.839 & $-$48.933 & 0.0028\
51353.996& 0.572 & 0.036 & $-$0.032 & $-$3.398 & $-$49.354 & 0.0032\
51354.912& 0.360 & 0.035 & $-$0.048 & $-$18.491 & $-$45.828 & 0.0192\
51354.935& 0.723 & 0.070 & $-$0.023 & $-$14.629 & $-$47.393 & 0.0196\
51354.958& 0.892 & 0.100 & $-$0.039 & $-$10.290 & $-$48.625 & 0.0200\
51354.999& 0.667 & 0.069 & $-$0.016 & $-$2.352 & $-$49.666 & 0.0207\
51355.888& 0.331 & 0.020 & $+$0.017 & $-$22.144 & $-$44.552 & 0.0362\
51355.935& 0.531 & 0.027 & $+$0.015 & $-$14.339 & $-$48.212 & 0.0370\
51356.918& 0.290 & 0.027 & $-$0.005 & $-$16.691 & $-$46.600 & 0.0542\
51356.962& 0.400 & 0.040 & $-$0.006 & $-$8.602 & $-$48.910 & 0.0549\
51356.985& 0.674 & 0.053 & $-$0.009 & $-$3.942 & $-$49.566 & 0.0553\
51360.887& 0.693 & 0.044 & $+$0.010 & $-$19.901 & $-$45.257 & 0.1234\
51360.937& 0.331 & 0.021 & $+$0.016 & $-$11.361 & $-$48.578 & 0.1243\
51360.938& 0.312 & 0.010 & $+$0.005 & $-$11.031 & $-$48.563 & 0.1243\
51375.901& 0.369 & 0.059 & $+$0.031 & $-$10.375 & $-$48.550 & 0.3853\
51385.791& 0.340 & 0.025 & $-$0.067 & $-$24.229 & $-$43.148 & 0.5579\
51385.833& 0.628 & 0.021 & $+$0.025 & $-$17.894 & $-$46.942 & 0.5586\
51386.788& 0.265 & 0.015 & $-$0.035 & $-$24.378 & $-$43.274 & 0.5753\
51386.815& 0.860 & 0.031 & $+$0.000 & $-$20.403 & $-$45.756 & 0.5757\
51386.848& 0.420 & 0.021 & $+$0.029 & $-$14.703 & $-$48.179 & 0.5763\
51399.766& 0.427 & 0.016 & $-$0.021 & $-$22.243 & $-$43.927 & 0.8017\
51399.802& 0.181 & 0.022 & $-$0.026 & $-$16.355 & $-$46.763 & 0.8023\
51399.843& 0.350 & 0.022 & $-$0.012 & $-$8.860 & $-$49.032 & 0.8030\
51401.762& 0.689 & 0.040 & $-$0.042 & $-$21.834 & $-$43.935 & 0.8365\
51401.803& 0.914 & 0.037 & $+$0.020 & $-$15.229 & $-$47.152 & 0.8372\
51401.841& 0.980 & 0.043 & $+$0.004 & $-$8.129 & $-$49.167 & 0.8379\
51415.769& 0.873 & 0.037 & $-$0.044 & $-$14.464 & $-$47.350 & 0.0809\
51415.789& 0.766 & 0.051 & $+$0.025 & $-$10.851 & $-$48.303 & 0.0812\
51416.744& 0.511 & 0.032 & $+$0.095 & $-$18.247 & $-$45.574 & 0.0979\
51416.772& 0.795 & 0.052 & $+$0.068 & $-$13.292 & $-$47.424 & 0.0984\
51472.637& 0.386 & 0.088 & $+$0.024 & $-$10.119 & $-$48.863 & 0.0729\
51694.947& 0.269 & 0.024 & $+$0.023 & $-$23.303 & $-$42.350 & 0.9512\
51694.962& 0.246 & 0.026 & $+$0.015 & $-$21.226 & $-$43.806 & 0.9515\
51694.973& 0.299 & 0.032 & $+$0.016 & $-$19.566 & $-$44.658 & 0.9517\
51694.992& 0.456 & 0.068 & $-$0.011 & $-$16.527 & $-$46.106 & 0.9520\
51695.002& 0.766 & 0.073 & $+$0.182 & $-$14.756 & $-$46.764 & 0.9522\
51707.915& 0.581 & 0.074 & $+$0.134 & $-$22.996 & $-$42.913 & 0.1774\
51707.929& 0.469 & 0.066 & $+$0.009 & $-$20.956 & $-$44.202 & 0.1777\
51707.956& 0.632 & 0.080 & $+$0.162 & $-$16.666 & $-$46.282 & 0.1782\
51707.971& 0.415 & 0.082 & $-$0.052 & $-$14.123 & $-$47.192 & 0.1784\
51707.975& 0.483 & 0.143 & $+$0.015 & $-$13.324 & $-$47.500 & 0.1785\
51707.989& 0.430 & 0.142 & $-$0.030 & $-$10.823 & $-$48.181 & 0.1787\
51707.992& 0.493 & 0.118 & $+$0.039 & $-$10.127 & $-$48.263 & 0.1788\
51708.003& 0.525 & 0.136 & $+$0.080 & $-$8.120 & $-$48.712 & 0.1790\
51716.888& 0.276 & 0.045 & $-$0.046 & $-$23.348 & $-$42.595 & 0.3340\
51716.898& 0.286 & 0.029 & $+$0.020 & $-$21.859 & $-$43.446 & 0.3342\
51716.953& 1.015 & 0.075 & $+$0.135 & $-$12.895 & $-$47.520 & 0.3351\
51716.963& 0.935 & 0.101 & $+$0.078 & $-$11.014 & $-$48.043 & 0.3353\
51716.990& 0.565 & 0.059 & $-$0.052 & $-$5.850 & $-$48.819 & 0.3358\
51717.001& 0.570 & 0.065 & $+$0.065 & $-$3.743 & $-$49.051 & 0.3359\
51723.876& 0.874 & 0.067 & $+$0.025 & $-$22.044 & $-$42.796 & 0.4559\
51723.890& 0.873 & 0.075 & $+$0.044 & $-$20.086 & $-$43.970 & 0.4561\
51723.919& 0.284 & 0.046 & $-$0.044 & $-$15.387 & $-$46.184 & 0.4566\
51723.933& 0.426 & 0.061 & $+$0.024 & $-$12.929 & $-$46.946 & 0.4569\
51723.949& 0.715 & 0.066 & $+$0.045 & $-$9.883 & $-$47.728 & 0.4572\
51723.963& 0.866 & 0.080 & $+$0.055 & $-$7.236 & $-$48.294 & 0.4574\
51723.974& 0.822 & 0.224 & $+$0.014 & $-$5.106 & $-$48.406 & 0.4576\
51723.988& 0.686 & 0.089 & $+$0.001 & $-$2.333 & $-$48.729 & 0.4579\
51723.996& 0.710 & 0.227 & $+$0.105 & $-$0.918 & $-$48.809 & 0.4580\
51741.797& 0.976 & 0.055 & $+$0.048 & $-$36.915 & $-$2.364 & 0.7685\
51741.804& 0.735 & 0.090 & $-$0.153 & $-$37.236 & $-$3.540 & 0.7687\
51741.902& 0.397 & 0.019 & $-$0.024 & $-$33.607 & $-$18.647 & 0.7704\
51741.909& 0.559 & 0.036 & $+$0.019 & $-$32.877 & $-$19.660 & 0.7705\
51741.944& 1.082 & 0.038 & $+$0.094 & $-$28.023 & $-$24.126 & 0.7711\
51741.951& 1.042 & 0.041 & $+$0.066 & $-$26.960 & $-$24.913 & 0.7712\
51744.796& 0.254 & 0.024 & $-$0.011 & $-$37.043 & $-$3.596 & 0.8209\
51744.803& 0.235 & 0.032 & $-$0.021 & $-$37.251 & $-$4.670 & 0.8210\
51744.851& 0.200 & 0.035 & $-$0.003 & $-$36.705 & $-$12.146 & 0.8218\
51744.858& 0.191 & 0.020 & $-$0.013 & $-$36.328 & $-$13.234 & 0.8219\
51744.928& 0.484 & 0.047 & $-$0.024 & $-$29.065 & $-$23.065 & 0.8232\
51744.935& 0.542 & 0.077 & $-$0.023 & $-$28.034 & $-$23.957 & 0.8233\
51746.822& 0.206 & 0.045 & $+$0.000 & $-$21.122 & $-$44.293 & 0.8562\
51746.824& 0.226 & 0.049 & $+$0.019 & $-$20.782 & $-$44.390 & 0.8562\
51746.836& 0.253 & 0.044 & $+$0.042 & $-$19.017 & $-$45.391 & 0.8564\
51746.839& 0.287 & 0.086 & $+$0.076 & $-$18.522 & $-$45.707 & 0.8565\
51746.893& 0.301 & 0.023 & $+$0.047 & $-$8.869 & $-$48.761 & 0.8574\
51746.895& 0.300 & 0.099 & $+$0.043 & $-$8.427 & $-$48.856 & 0.8575\
51752.793& 0.460 & 0.032 & $-$0.008 & $-$37.534 & $-$6.427 & 0.9604\
51752.799& 0.596 & 0.040 & $+$0.017 & $-$37.575 & $-$7.511 & 0.9605\
51752.806& 0.701 & 0.058 & $+$0.007 & $-$37.545 & $-$8.587 & 0.9606\
51752.829& 0.955 & 0.075 & $-$0.015 & $-$36.897 & $-$12.119 & 0.9610\
51752.835& 0.999 & 0.082 & $+$0.011 & $-$36.571 & $-$13.156 & 0.9611\
51752.842& 0.944 & 0.066 & $-$0.024 & $-$36.237 & $-$14.146 & 0.9612\
51752.872& 0.468 & 0.048 & $-$0.040 & $-$33.585 & $-$18.607 & 0.9617\
51791.675& 0.927 & 0.084 & $-$0.007 & $-$24.303 & $-$41.631 & 0.6387\
51791.683& 0.925 & 0.068 & $+$0.082 & $-$23.256 & $-$42.516 & 0.6388\
51794.671& 0.282 & 0.050 & $+$0.019 & $-$23.622 & $-$41.847 & 0.6909\
51794.714& 0.995 & 0.096 & $+$0.064 & $-$17.173 & $-$45.574 & 0.6917\
51794.780& 0.420 & 0.051 & $+$0.047 & $-$5.164 & $-$48.662 & 0.6928\
51831.634& 0.981 & 0.282 & $+$0.117 & $-$13.669 & $-$47.076 & 0.3358\
51831.644& 0.957 & 0.364 & $+$0.077 & $-$11.832 & $-$47.527 & 0.3359\
51832.645& 0.618 & 0.038 & $-$0.016 & $-$11.295 & $-$48.013 & 0.3534\
51832.655& 0.513 & 0.057 & $+$0.026 & $-$9.402 & $-$48.524 & 0.3536\
51832.687& 0.327 & 0.032 & $-$0.006 & $-$3.219 & $-$49.407 & 0.3541\
51832.703& 0.388 & 0.051 & $-$0.017 & $+$0.028 & $-$49.456 & 0.3544\
51832.741& 0.613 & 0.079 & $-$0.006 & $+$7.522 & $-$48.791 & 0.3551\
51832.749& 0.747 & 0.185 & $+$0.080 & $+$9.092 & $-$48.683 & 0.3552\
51833.587& 0.779 & 0.073 & $+$0.054 & $-$20.747 & $-$44.399 & 0.3698\
51833.594& 0.797 & 0.076 & $-$0.030 & $-$19.686 & $-$44.961 & 0.3700\
51833.628& 0.542 & 0.064 & $-$0.062 & $-$13.791 & $-$47.277 & 0.3706\
51833.636& 0.370 & 0.090 & $-$0.100 & $-$12.457 & $-$47.827 & 0.3707\
51833.671& 0.404 & 0.065 & $+$0.020 & $-$5.797 & $-$49.071 & 0.3713\
51833.678& 0.456 & 0.055 & $+$0.005 & $-$4.458 & $-$49.294 & 0.3714\
51833.713& 0.875 & 0.229 & $+$0.136 & $+$2.534 & $-$49.312 & 0.3720\
51833.720& 0.959 & 0.244 & $+$0.184 & $+$3.949 & $-$49.290 & 0.3722\
51850.576& 0.299 & 0.206 & $+$0.092 & $-$35.890 & $-$14.261 & 0.6662\
51850.587& 0.333 & 0.143 & $-$0.066 & $-$35.123 & $-$16.014 & 0.6664\
51850.589& 0.774 & 0.457 & $+$0.331 & $-$34.953 & $-$16.216 & 0.6664\
51850.627& 0.842 & 0.286 & $+$0.130 & $-$30.666 & $-$21.528 & 0.6671\
51850.636& 0.425 & 0.144 & $+$0.020 & $-$29.413 & $-$22.756 & 0.6673\
51851.630& 0.586 & 0.128 & $-$0.095 & $-$29.867 & $-$22.233 & 0.6846\
51851.632& 0.588 & 0.039 & $-$0.025 & $-$29.662 & $-$22.568 & 0.6846\
52120.799& 0.244 & 0.015 & $-$0.020 & $-$37.790 & $-$8.720 & 0.3804\
52120.806& 0.435 & 0.028 & $+$0.017 & $-$37.598 & $-$9.746 & 0.3805\
52120.821& 0.933 & 0.039 & $+$0.063 & $-$37.127 & $-$12.097 & 0.3807\
52120.827& 1.037 & 0.034 & $+$0.058 & $-$36.800 & $-$13.153 & 0.3809\
52120.850& 0.517 & 0.030 & $-$0.009 & $-$35.175 & $-$16.534 & 0.3812\
52154.730& 0.700 & 0.112 & $+$0.016 & $-$17.204 & $-$46.179 & 0.9723\
52154.753& 1.056 & 0.075 & $+$0.153 & $-$13.188 & $-$47.569 & 0.9727\
52154.776& 1.025 & 0.099 & $+$0.114 & $-$8.897 & $-$48.621 & 0.9731\
52156.711& 0.447 & 0.017 & $+$0.023 & $-$19.236 & $-$44.927 & 0.0069\
52156.741& 0.924 & 0.032 & $+$0.086 & $-$14.316 & $-$46.944 & 0.0074\
52156.767& 0.949 & 0.025 & $+$0.030 & $-$9.421 & $-$48.279 & 0.0078\
52156.788& 0.715 & 0.025 & $-$0.046 & $-$5.397 & $-$48.945 & 0.0082\
52180.651& 0.373 & 0.026 & $-$0.017 & $-$37.374 & $-$11.148 & 0.4245\
52180.658& 0.657 & 0.027 & $+$0.021 & $-$37.059 & $-$12.252 & 0.4246\
52180.671& 1.034 & 0.036 & $+$0.052 & $-$36.298 & $-$14.354 & 0.4249\
52180.678& 0.955 & 0.031 & $+$0.013 & $-$35.826 & $-$15.419 & 0.4250\
52180.692& 0.458 & 0.034 & $-$0.028 & $-$34.620 & $-$17.493 & 0.4252\
52180.699& 0.280 & 0.018 & $-$0.007 & $-$33.960 & $-$18.397 & 0.4253\
52180.713& 0.310 & 0.017 & $+$0.004 & $-$32.398 & $-$20.408 & 0.4256\
52180.720& 0.543 & 0.021 & $+$0.018 & $-$31.499 & $-$21.269 & 0.4257\
[lcccc]{}\
$P$ (days)& 57.3240 & 57.3298 $\pm$ 0.0035& 57.32161 $\pm$ 0.00034& 57.32178 $\pm$ 0.00029\
$a$ (mas)& 5.24 $\pm$ 0.66 & 15.368 $\pm$ 0.028& & 15.378 $\pm$ 0.027\
$\gamma$ ($\kms$)& & & $-$5.867 $\pm$ 0.038 & $-$5.841 $\pm$ 0.037\
$K_A$ ($\kms$)& & & 28.944 $\pm$ 0.046& 28.929 $\pm$ 0.046\
$K_B$ ($\kms$)& & & 36.73 $\pm$ 0.21& 36.72 $\pm$ 0.21\
$e$& 0.3060 & 0.30740 $\pm$ 0.00067 & 0.3103 $\pm$ 0.0018 & 0.30626 $\pm$ 0.00057\
$i$ (deg)& 89.50 $\pm$ 8.43& 99.379 $\pm$ 0.088& & 99.364 $\pm$ 0.080\
$\omega_A$ (deg)& 356.8 & 358.89 $\pm$ 0.53& 357.03 $\pm$ 0.35& 357.40 $\pm$ 0.29\
$\Omega$ (deg)& 327.66 $\pm$ 7.56& 335.061 $\pm$ 0.082& & 334.960 $\pm$ 0.070\
$T$ (HJD$-$2,400,000)& 43327.589 & 51354.000 $\pm$ 0.069 & 49404.825 $\pm$ 0.045 & 51353.813 $\pm$ 0.038\
$\Delta K_{\rm CIT}$ (mag)& & 1.063 $\pm$ 0.031 & & 1.056 $\pm$ 0.013\
$\Delta H_{\rm CIT}$ (mag)& & 1.18 $\pm$ 0.16 & & 1.154 $\pm$ 0.065\
\
$\pi$ (mas)& 45.30 $\pm$ 0.46& & & 46.08 $\pm$ 0.27\
\
$N_{\rm obs}$ (RV)& & & 73$+$73 & 73$+$73\
$N_{\rm obs}$ ($V^2$)& & 37$+$134 & & 37$+$134\
$\sigma_{V^2}$ ($H$/$K$)& & 0.0599 / 0.0613 & & 0.0592 / 0.0617\
$\sigma_{\rm RV}$ (A/B, )& & & 0.30 / 1.38 & 0.31 / 1.41\
[lcc]{}
Mass (M$_{\sun}$)& 0.844 $\pm$ 0.018 & 0.6650 $\pm$ 0.0079\
$T_{\rm eff}$ (K)& 5200 $\pm$ 100& 4200 $\pm$ 200\
$\pi_{\rm orb}$ (mas)&\
Dist (pc)&\
$M_V$ (mag)& 5.511 $\pm$ 0.028 & 7.91 $\pm$ 0.19\
$M_H$ (mag)& 3.679 $\pm$ 0.037 & 4.835 $\pm$ 0.059\
$M_K$ (mag)& 3.646 $\pm$ 0.033 & 4.702 $\pm$ 0.034\
$V\!-\!K$ (mag)& 1.865 $\pm$ 0.039 & 3.21 $\pm$ 0.19\
[^1]: A recent exception is the binary system OGLEGC-17 in the globular cluster $\omega$ Cen, discovered by the OGLE microlensing group [@Udalski1993]. A preliminary study by @Thompson2001 yielded encouraging results with errors in the masses and radii of $\sim$7% and $\sim$3%, respectively, and a more detailed analysis using additional observations can be expected in the near future [see @Kaluzny2001]. At an estimated metallicity of \[Fe/H\]$=
-2.29$ this is an extremely important system.
[^2]: Available at [http://cfaku5.harvard.edu]{}.
[^3]: In principle it would be possible to derive the position angle ($\theta$) and the angular separation ($\rho$) of the binary from the visibility measurements for each night. These intermediate data could then be used instead of the visibilities in what might be considered a more conventional astrometric-spectroscopic solution. However, the result would be inferior to the fit we describe here for several reasons [see, e.g., @Hummel2001], not the least of which is the fact that, due to the pattern of our observations, the number of scans on many nights is insufficient to solve for the separation vectors, which would result in a rather serious loss of information (phase coverage). Therefore, we refrain from even listing values of {$\rho$, $\theta$} in Table \[tab:visibH\] and Table \[tab:visibK\].
[^4]: These uncertainties include the contribution due to the template parameters discussed in §\[subsec:spectroscopy\].
[^5]: The infrared magnitudes in these isochrones are based on the color tables by @Lejeune1998, which adopt filter transmission functions from @Bessell1988 for the Johnson-Glass system. For the comparison with our observations we transformed the isochrones to the CIT system using the corrections by @Bessell1988.
[^6]: The sign of these changes may seem counterintuitive. The secondary star tends to redden both the $U\!-\!B$ and the $B\!-\!V$ indices, which ought to make the combined light of the system appear more metal-rich. As it turns out, the slope of this “reddening" vector in the $U\!-\!B$ vs. $B\!-\!V$ plane is smaller than that of the blanketing vector, so that the net effect is a bias toward *larger* ultraviolet excesses, and therefore lower metallicity estimates. We find, however, that in general this depends both on the color of the primary and on the magnitude difference $\Delta V$. For bluer primaries than that of HD 195987 the sign of the change is reversed in certain ranges of $\Delta V$.
[^7]: Because we are dealing in this case with *ratios* of element abundances, the effect that the secondary component might have on those determinations is not expected to be as significant as in the case of \[Fe/H\]. Table 8 by @Fulbright2000 indicates that the bias due to temperature errors averaged over Mg, Si, Ca, and Ti (which have different signs for the correlation with $T_{\rm eff}$) is 0.03 dex at most, with the sign depending on whether only the temperature is changed or whether other fitted parameters are allowed to vary simultaneously to compensate. Similarly, the continuum from the secondary should have a minimal effect on the abundance ratios.
|
On higher order Codazzi tensors\
on complete Riemannian manifolds {#on-higher-order-codazzi-tensors-on-complete-riemannian-manifolds .unnumbered}
================================
**I. G. Shandra**${}^{1 }$**S. E. Stepanov**${}^{1,2}$
${}^{1 }$Dept. of Data Analysis and Financial Technology, Finance University,
49-55, Leningradsky Prospect, 125468 Moscow, Russia
e-mail: ma-tematika@yandex.ru
${}^{2}$ Dept. of Mathematics, Russian Institute for Scientific and
Technical Information of the Russian Academy of Sciences,
20, Usievicha street, 125190 Moscow, Russia
e-mail: s.e.stepanov@mail.ru
****
**Abstract.** [We prove several Liouville-type non-existence theorems for higher order Codazzi tensors and classical Codazzi tensors on complete and compact Riemannian manifolds, in particular. These results will be obtained by using theorems of the connections between the geometry of a complete smooth manifold and the global behavior of its subharmonic functions. In conclusion, we show applications of this method for global geometry of a complete locally conformally flat Riemannian manifold with constant scalar curvature because its Ricci tensor is a Codazzi tensor and for global geometry of a complete hypersurface in a standard sphere because its second fundamental form is also a Codazzi tensor.]{}
**Keywords**: **** complete Riemannian manifold, higher order Codazzi tensor, subharmonic function.
**Mathematical Subject Classification:** 53C20; 53C25; 53C40
**1. Introduction**
Let $\left({\rm M,g}\right)$ be a Riemannian manifold of dimension *n* $\geq$ 2 with the Levi-Civita connection $\nabla $. Everywhere in the following we denote by $\Lambda ^{{\rm q}} M$ and $S^{p} M$ the vector bundles of exterior differential $q$-forms $\left(1\le q\le n-1\right)$ and symmetric differential $p$-forms $\left(p\ge 2\right)$ on ${\rm M}$. Throughout this paper we will consider the vector spaces of their *C*${}^{\infty}$-sections denoted by *${\rm C}^{\infty } \Lambda ^{{\rm p}} M$* and $C^{\infty } S^{p} M$, respectively. The Riemannian metric $g$ induces a point-wise inner product on the fibres of each of these spaces.
A symmetric bilinear form ${\rm T}\in {\rm C}^{\infty } S^{2} M$on a Riemannian manifold $\left(M,\, g\right)$ is said to be a *Codazzi tensor* if it satisfies the equation \[1\]; \[2, p. 435\]; \[3, pp. 24; 56; 68\] $$\label{GrindEQ__1_1_}
\left(\nabla _{X} T\right)\, \left(Y,\, Z\right)=\left(\nabla _{Y} T\right)\, \left(X,\, Z\right)$$ for any tangent vector fields $X,Y,Z$ on ${\rm M}$. The Codazzi tensor ${\rm T}\in {\rm C}^{\infty } S^{2} M$ is called *non-trivial* if it is not a constant multiple of metric \[1, p. 15\]. Alongside with it we know from \[1, p. 17\] that every smooth manifold ${\rm M}$ carries a $C^{\infty } $-metric $g$ such that $\left(M,\, g\right)$ admits a nontrivial Codazzi tensor ${\rm T}\in {\rm C}^{\infty } S^{2} M$. We remark here that this result is essentially local.
Codazzi tensors appear in a natural way in many geometric situations. Therefore, the research on Codazzi tensors is vast and has found many applications, and it would require a very long read to cover all aspects, even superficially. Some of these results can be found in the monograph \[2\] which was published in 1987. On the other hand, there are many papers on the geometry of Codazzi tensors \[5\] - \[10\] which were published in subsequent years.
In turn, we introduced in \[11\] the notion of Codazzi ${\rm p}$-tensors $\left({\rm p}\ge 2\right)$ which extends the well known concept for $p=2$. Let us recall that a *Codazzi ${\rm p}$-tensor* or, in other words, a *higher order Codazzi tensor* is a $C^{\infty } $-section *T* of the vector bundle ${\rm S}^{{\rm p}} M$on *${\rm M}$* satisfying the following equation: $$\label{GrindEQ__1_2_}
\left(\nabla _{X_{0} } T\right)\, \left(X_{1} ,\, X_{2} ,\, ...,\, X_{p} \right)=\left(\nabla _{X_{1} } T\right)\, \left(X_{0} ,\, X_{2} ,\, ...,\, X_{p} \right)$$ for any tangent vector fields $X_{0} ,\, X_{1} ,\, ...,\, X_{p} $ on *$M$*. The theory of higher order Codazzi tensors was developed in the papers from the following list \[12\] - \[15\].
In the present paper we study the question of nonexistence of higher order Codazzi tensors and classical Codazzi tensors on complete and compact Riemannian manifolds, in particular. The classical *Bochner technique* \[16\]; \[17\]; \[18, pp. 333-363\]; \[19\] and its generalized version \[20\] will help us to accomplish this task. We must recall here that the classic Bochner technique is an analytical method to obtain vanishing theorems for some topological or geometrical invariants on a compact (without boundary) Riemannian manifold, under some curvature assumption. The proofs of such theorems apply the *Bochner maximum principle* and the *theorem of Green* \[16, pp. 30-31\]. We will also use a generalized version of the Bochner technique in the present paper. Therefore, we will use the *maximum principle of Hopf* \[21, p. 47\], Yau, Li and Schoen results on the connections between the geometry of a complete smooth manifold and the global behavior of its subharmonic functions \[22\]; \[23\]; \[24\]. We have demonstrated in our papers \[25\]; \[26\] and \[27\] that this generalized version of the Bochner technique is effective for the differential geometry “in the large”. In addition, the theorems and corollaries of the present paper will illustrate the effectiveness of this method for studying the global geometry of higher order Codazzi tensors and, in particular, Codazzi tensor of order 2 on complete and compact Riemannian manifolds (see our Theorem 1, Corollary 2 and Theorem 3).
It is well known that the Ricci tensor $Ric$ is a Codazzi 2-tensor on an *n*-dimensional $\left(n\ge 3\right)$ locally conformally flat manifold *$\left({\rm M,g}\right)$* with constant (not necessarily zero) scalar curvature ${\rm s}={\rm trace}_{{\rm g}} Ric$ \[28\]. By using this fact, we will give an application of the Bochner technique to the global geometry of complete locally conformally flat Riemannian manifolds (see our Corollary 5 and Theorem 4). On the other hand, if $\left({\rm M,}\, {\rm g}\right)$ is a minimal hypersurface in the standard sphere ${\rm S}^{{\rm n}+1} $, then its second fundamental form ${\rm S}$ is a traceless Codazzi 2-tensor \[29, p. 388\]. By using this theorem, we will give an application of the Bochner technique to the global geometry of complete minimal hypersurfaces in a sphere (see Theorem 5).
The theorems and corollaries of this paper supplement our results from \[1\] and the results of other authors from \[2, Theorem 16.9\]; \[13\]; \[28\] and \[30\].
**2. Main results**
Everywhere in this paper we consider a higher order Codazzi tensor $T$ as a smooth section of the subbundle $S_{0}^{p} M$ of the bundle $S^{{\rm p}} M$ on a Riemannian manifold *$\left({\rm M,g}\right)$* defined by the condition $${\rm trace}_{{\rm g}} \, T=\sum _{i=1,...,n}T\, \left(\, e_{i} ,\, e_{i} ,\, X_{3} ,\, ...,\, X_{p} \right) =0$$ for $T\in C^{\infty } S_{0}^{p} M$ and orthonormal basis $\left\{\, e_{i} \, \right\}$ of $T_{x}M $ at an arbitrary point $x\in M$. In this case, ${\rm T}$ is called *traceless*. It can be proved that an arbitrary traceless Codazzi ${\rm p}$-tensor *T* is a *divergence-free* tensor field, i.e. ** $\delta \, T=0$ for the formal adjoint operator $\delta {\rm :}C^{\infty } \left(\otimes ^{p+1} {\rm T}^{*} M\right)\to C^{\infty } \left(\otimes ^{p} {\rm T}^{*} M\right)$ of the covariant derivative $\nabla {\rm :}C^{\infty } \left(\otimes ^{p} {\rm T}^{*} M\right)\to C^{\infty } \left(\otimes ^{p+1} {\rm T}^{*} M\right)$ where ${\rm T}^{*} M$ is the cotangent bundle on ${\rm M}$(see \[2, p. 54\]; \[29\]).
**Remark 1**. In \[12\] the dimension of the vector space of traceless Codazzi ${\rm p}$-tensors $\left({\rm p}\ge 2\right)$ on compact Riemannian surfaces of genus $\gamma $ was determined. It depends only on ${\rm p}$ and $\gamma $. Additionally the result was extended to genus zero.
The following *Bochner-Weitzenböck formula* for an arbitrary Codazzi ${\rm p}$-tensor ${\rm T}\in {\rm C}^{\infty } S_{0}^{{\rm p}} M$ holds $$\label{GrindEQ__2_1_}
\frac{1}{2} \, \Delta _{{\rm B}} \, \left\| \, {\rm T}\, \right\| ^{2} =Q_{P} \, \left(T{\rm ,}\, {\rm T}\right)+\left\| \, \nabla \, T\, \right\| ^{2} ,$$ where $\Delta _{B} ={\rm div}\circ \, {\rm grad}$ is the *Laplace-Beltrami operator*, $\left\| \, {\rm T}\, \right\| ^{2} =g\, \left(\, T,\, T\right)$, $\left\| \, \nabla {\rm T}\, \right\| ^{2} =g\, \left(\, \nabla T,\, \nabla T\right)$ and ${\rm Q}_{{\rm P}} $ is a quadratic form ${\rm Q}_{{\rm p}} {\rm :}\; S_{0}^{p} M\otimes S_{0}^{p} M\to $$\mathbb{R}$ which can be algebraically expressed through the curvature tensor $R$ and the Ricci tensor $Ric$ of *$\left({\rm M,g}\right)$.*
The curvature tensor ${\rm R}$ of *$\left({\rm M,g}\right)$* induces an algebraic *curvature operator* $\mathop{R}\limits^{\circ } :S_{0}^{2} M\to S_{0}^{2} M$. The symmetries of the curvature tensor $R$ imply that $\mathop{R}\limits^{\circ } $ is a selfadjoint operator, with respect to the point-wise inner product on $S_{0}^{2} M$. That is why, the eigenvalues of $\mathop{R}\limits^{\circ } $ are all real numbers at each point $x\in M$. Thus, we say $\mathop{R}\limits^{\circ } $ is *positive semidefinite* (resp. *strictly positive*), or simply $\mathop{R}\limits^{\circ } \ge 0$ (resp. $\mathop{R}\limits^{\circ } >0$), if all the eigenvalues of $\mathop{R}\limits^{\circ } $ are nonnegative (resp. strictly positive). In the next paragraph we will prove that $Q_{P} \left(T,\, T\right)\ge 0$ for an arbitrary ${\rm T}\in {\rm C}^{\infty } S_{0}^{p} M$ if the curvature operator $\mathop{R}\limits^{\circ } $ is positive semidefinite on $S_{0}^{2} M$ and $Q_{P} \left(T,\, T\right)>0$ for an arbitrary nonzero ${\rm T}\in {\rm C}^{\infty } S_{0}^{p} M$ if the curvature operator $\mathop{R}\limits^{\circ } $ is positive definite on $S_{0}^{2} M$.
**Remark 2.** The curvature operator $\mathop{R}\limits^{\circ } $ has been studied in many papers and monographs. It is famous for its numerous applications \[2, pp. 51-52; 346-347\]; \[31\]; \[32\]; \[34\]; \[35\]. Beside the curvature operator $\mathop{R}\limits^{\circ } $ there is a curvature operator $\bar{{\rm R}}{\rm :}\Lambda ^{2} M\to \Lambda ^{2} M$ (see \[18, pp. 83\]). It is also widely used in Riemannian geometry. Examples are given by the well known Gallot-Meyer theorem \[18, p. 351\] on harmonic $p$-forms $\omega \in {\rm C}^{\infty } \Lambda ^{p} M$ on a compact manifold *$\left({\rm M,g}\right)$* and its $p$-th Betti number $\beta _{{\rm p}} \left(M\right)$, and theorems from \[25\] and \[26\]. Therefore $\mathop{R}\limits^{\circ } $ is also referred to as the *curvature operator of the second kind* \[34\].
Taking into account and applying the “Hopf maximum principle” \[9, p. 47\] or, in other words, “strong maximal principal of Hopf” we will prove the following statement.
**LEMMA 1**. *Let the curvature operator* $\mathop{R}\limits^{\circ } \ge 0$ *at* *any point of a connected open domain $U$ of $\left({\rm M,g}\right)$ and ${\rm T}$ be a Codazzi* $p$-*tensor* ($p\ge 2$) *defined at any point of ${\rm U}$. If* *the scalar function $\left\| \, {\rm T}\, \right\| ^{2} $ has a local maximum at some point of* ${\rm U}$*, then $\left\| \, {\rm T}\, \right\| ^{2} $ is a constant function and ${\rm T}$ is invariant under parallel translations* *in* ${\rm U}$*. Moreover, if* $\mathop{R}\limits^{\circ } >0$ *at some point* $x\in U$*, then $T\equiv 0$.*
Let *$\left({\rm M,g}\right)$* be a compact Riemannian manifold with the curvature operator $\mathop{R}\limits^{\circ } \ge 0$. Then there exists a point ${\rm x}\in {\rm M}$ at which the function *$\left\| \, {\rm T}\, \right\| ^{2} $* attains the maximum. At the same time, *$\left\| \, {\rm T}\, \right\| ^{2} $* satisfies the condition $\Delta \, \left\| \, {\rm T}\, \right\| ^{2} \ge 0$ everywhere in *$\left({\rm M,g}\right)$*. If, moreover, $\mathop{R}\limits^{\circ } >0$ at some point $x\in {\rm M}$, then $T\equiv 0$ everywhere on *$\left({\rm M,g}\right)$*. As a result, we obtain the corollary that was proved in our paper \[11\]. Moreover, it is a generalization of the Berger-Ebin theorem on an arbitrary Codazzi 2-tensor with constant trace on a compact Riemannian manifold \[2, p. 436\]; \[29\].
**COROLLARY 1.** *Every traceless Codazzi $p$-tensor* (*$p\ge 2$*) *on a compact Riemannian manifold $\left({\rm M,g}\right)$ with nonnegative curvature operator is invariant under parallel translations. Moreover, if* $\mathop{R}\limits^{\circ } >0$ *at some point* $x\in {\rm M}$*, then there is no non-zero traceless Codazzi $p$-tensor* (*$p\ge 2$*)*.*
**Remark 3**. In \[37\] was proved that *g* ($\mathop{R}\limits^{\circ } \, \left({\rm T}\right),\, T$) $\geq$ 0 for any $T\in C^{\infty } S_{0}^{p} M$ and ${\rm p}\ge 2$ if ${\rm sec}\ge 0$ for the sectional curvature ${\rm sec}$ of $\left(M,g\right)$. Therefore, we can reformulate our Lemma 1 and Corollary 1 by using this statement. In particular, we can state that every traceless higher order Codazzi tensor on a compact Riemannian manifold with nonnegative sectional curvature is invariant under parallel translations.
Let us formulate a theorem that supplements Theorem 2 from our paper \[11\] which was proved for a compact Riemannian manifold.
**THEOREM** ** **1***. Let $\left({\rm M,g}\right)$ be a complete noncompact Riemannian manifold with nonnegative curvature operator* $\mathop{R}\limits^{\circ } $*. Then there is no non-zero traceless Codazzi $p$-tensor* ($p\ge 2$) *on $\left({\rm M,g}\right)$ such that* $\int _{{\rm M}}\left\| \, {\rm T}\, \right\| ^{{\rm q}} dVol_{g} <+\infty $ *for some $q\ge 1$.*
A Riemannian manifold *$\left({\rm M,g}\right)$* is *locally conformally flat* if, for any ${\rm x}\in {\rm M}$, there exists a neighborhood $U$of $x$ and $C^{\infty } $-function $f$ on $U$ such that $\left(\, U,\, e^{2f} g\right)$ is flat \[2, p. 60\]. We remind that a manifold *$\left({\rm M,g}\right)$* of dimension ${\rm n}\; \; \left({\rm n}\ge 4\right)$ is locally conformally flat if and only if its Weyl tensor ${\rm W}$ is identically zero \[2, p. 60\]. The Schouten tensor ${\rm Sch}$ plays an important role in the description of such manifolds \[38\]. This tensor has the form ${\rm S}ch=\left(n-2\right)^{-1} \, \left(\, R{\rm ic}-s\, \left(2n-2\right)^{-1} g\right)$ for the Ricci tensor ${\rm Ric}$ and the scalar curvature ${\rm s}={\rm trace}_{{\rm g}} \, {\rm Ric}$ of *$\left({\rm M,g}\right)$*. Let us formulate the following statement.
**COROLLARY** **2**. *Let $\left({\rm M,g}\right)$ be a complete noncompact locally conformally flat Riemannian manifold of dimension ${\rm n}\; \; \left({\rm n}\ge 4\right)$ with the nonnegative definite Schouten tensor*. *Then there is no non-zero traceless Codazzi $p$-tensor* ($p\ge 2$) *on $\left({\rm M,g}\right)$ such that* $\int _{{\rm M}}\left\| \, {\rm T}\, \right\| ^{{\rm q}} dVol_{g} <+\infty $ *for some $q\ge 1$.*
**Remark 4**. Corollary 2 supplements to some results on compact locally conformally flat Riemannian manifolds from \[13\]. We note that the condition of the nonnegative definiteness of the Schouten tensor in the case of constant positive scalar curvature means for a complete manifold that it is compact \[18, p. 251\].
Let us now consider a non-zero Codazzi 2-tensor on a Riemannian manifold, i.e. a symmetric bilinear form ${\rm T}\in C^{\infty } S^{2} M$ satisfying the *Codazzi equation* .
First, we consider a symmetric bilinear form ${\rm T}\in C^{\infty } S^{2} M$ as a 1-form with values in the cotangent bundle $T^{*} M$. This bundle comes equipped with the Levi-Civita covariant derivative $\nabla $, thus there is an induced exterior differential $d^{\nabla } {\rm :C}^{\infty } S^{2} M\to C^{\infty } \left(\Lambda ^{2} M\otimes T^{*} M\right)$ on $T^{*} M$-valued differential one-forms such as ${\rm d}^{\nabla } T\, \left(X,\, Y,\, Z\right)=\left(\nabla _{X} {\rm T}\right)\, \left(Y,\, Z\right)-\left(\nabla _{Y} T\right)\, \left(X,\, Z\right)$ for any tangent vector fields ${\rm X,}\, {\rm Y,}\, {\rm Z}$ on ${\rm M}$ and an arbitrary ${\rm T}\in C^{\infty } S^{2} M$ \[2, p. 355\]; \[18, pp. 349-350\]; \[39\]. In this case, a symmetric bilinear form ${\rm T}\in C^{\infty } S^{2} M$ is a Codazzi 2-tensor if and only if $d^{\nabla } {\rm T}$ vanishes \[18, p. 350\]; \[39\].
On the other hand, P. Peterson called \[18, p. 350\] a symmetric bilinear form ${\rm T}\in C^{\infty } S^{2} M$ *harmonic* if $T\in {\rm Ker}\, d^{\nabla } \bigcap {\rm Ker}\, \delta $. In this case, ${\rm T}$ is a divergence free Codazzi 2-tensor. In addition, from we obtain the equation $\delta \, T=-\, \; d\left(\, {\rm trace}_{g} T\right)$ for an arbitrary Codazzi 2-tensor ${\rm T}\in {\rm C}^{\infty } S^{2} M$. Therefore, we can conclude that a bilinear form ${\rm T}\in C^{\infty } S^{2} M$ is harmonic if and only if it is a Codazzi 2-tensor with constant trace \[18, p. 350\]. Moreover, the following statement holds.
**THEOREM 2.** *The vector space of harmonic symmetric bilinear forms* *on a compact Riemannian manifold is finite dimensional.*
**Remark 5.** J. P. **** Bourguignon proved \[40, p. 281\] that a compact orientable Riemannian four-manifold admitting a non-trivial Codazzi tensor of order 2 with constant trace (harmonic symmetric bilinear form) must have signature zero. ****
Everywhere in the following we will consider a harmonic symmetric bilinear form or, in other words, we will consider a Codazzi 2-tensor with constant trace.
We have the Bochner-Weitzenböck formula $$\label{GrindEQ__2_2_}
\frac{1}{2} \, \Delta _{B} \, \left\| \, {\rm T}\, \right\| ^{2} =Q_{2} \, \left(T,\, T\right)+\left\| \, \nabla \, T\, \right\| ^{2} ,$$ for an arbitrary harmonic symmetric bilinear form. Here the sign of the quadratic form $Q_{2} \left(T,\, T\right)$ depends on the sign of the sectional curvature ${\rm sec}$ of $\left({\rm M,g}\right)$ \[29\]. We remind here that an arbitrary Codazzi 2-tensor *T* on *$\left({\rm M,g}\right)$* commutes with the Ricci tensor $Ric$ at each point ${\rm x}\in {\rm M}$ \[1\]; \[2, p. 439\]. Therefore, the eigenvectors of an arbitrary Codazzi tensor $T$ determine the principal directions of the Ricci tensor at each point ${\rm x}\in {\rm M}$ \[41, pp.113-114\]. The converse is also true. Then taking into account of and using the “Hopf maximum principle”, we will prove in the next paragraph that the following lemma holds.
**LEMMA 2**. *Let $U$be a connected open domain $U$ of a Riemannian manifold $\left({\rm M,g}\right)$ and ${\rm T}$ be a harmonic symmetric bilinear form defined at any point of ${\rm U}$. If the sectional curvature ${\rm sec}\, \left(\, {\rm e}_{{\rm i}} \wedge \, e_{j} \right)\ge 0$ for all vectors of the orthonormal basis $\left\{\, e_{i} \right\}$ of $T_{x} M$ which is determined by the principal directions of the Ricci tensor ${\rm Ric}$ at an arbitrary point $x\in U$ and $\left\| \, {\rm T}\, \right\| ^{2} $ has a local maximum in the domain U, then $\left\| \, {\rm T}\, \right\| ^{2} $ is a constant function and ${\rm T}$ is invariant under parallel translations* *in* ${\rm U}$*. Moreover, if ${\rm sec}\, \left(\, {\rm e}_{{\rm i}} \wedge \, e_{j} \right)>0$ at some point* $x\in U$*, then ${\rm T}$ is trivial.*
If *$\left({\rm M,g}\right)$* is a compact manifold and a harmonic symmetric bilinear form *T* is given in a global way on *$\left({\rm M,g}\right)$* then due to the “Bochner maximum principle” for compact manifold it follows the classical Berger-Ebin theorem \[2, p. 436\] and \[29\] which is a corollary of our Lemma 2 (and see also Remark 3).
**COROLLARY 3**. *Every harmonic symmetric bilinear form* ${\rm T}\in {\rm C}^{\infty } S^{2} M$ *on a compact Riemannian manifold $\left({\rm M,g}\right)$ with nonnegative sectional curvature is invariant under parallel translations. Moreover, if ${\rm sec}>0$ at some point, then* ${\rm T}\in {\rm C}^{\infty } S^{2} M$ *is trivial.*
**Remark 6**. It is well known that every parallel symmetric tensor field $T\in C^{\infty } S^{2} M$ on a connected locally irreducible Riemannian manifold *$\left({\rm M,g}\right)$* is proportional to $g$, i.e. *$T=\lambda \, g$* for some constant $\lambda $. Due to this the second parts of Corollary ** 3 can be reformulated in the following form: Moreover, if *$\left({\rm M,g}\right)$* a connected locally irreducible Riemannian then $T$ is trivial.
For example, let $\left({\rm M,}\; {\rm g}\right)$ be a *Riemannian symmetric space of compact type* that is a compact Riemannian manifold with non-negative sectional curvature and positive-definite Ricci tensor (see \[42, p. 256\]). Moreover, if a Riemann symmetric space of compact type is a locally irreducibility Riemannian manifold *$\left({\rm M,g}\right)$* then it is a compact Riemannian manifold with positive sectional curvature \[43\]. Therefore, we can formulate the following corollary.
**COROLLARY ** 4***.* *Every harmonic symmetric bilinear form* *on a Riemannian symmetric manifold of compact type is invariant under parallel translations. If in addition to the above mentioned the manifold is locally irreducible then harmonic symmetric bilinear forms are trivial.*
The following theorem supplements the Berger-Ebin theorem \[2, p. 436\] and \[29\] for the case of a complete noncompact Riemannian manifold.
**THEOREM** **3**. *Let $\left({\rm M,g}\right)$ be a connected complete noncompact Riemannian manifold with nonnegative sectional curvature. Then there is no a non-zero harmonic symmetric bilinear form ${\rm T}$* *which satisfies the condition* $\int _{{\rm M}}\left\| \, T\, \right\| ^{{\rm q}} \, dVol_{g} <+\infty $ *at least for one* $q\ge 1$.
**Remark 7**. In the case of a locally conformally flat *n*-dimensional $\left(\, n\ge 4\, \right)$ Riemannian manifold *$\left({\rm M,g}\right)$* the following equalities hold \[28\]; \[44\] $$\sec \, \left(\, e_{i} \wedge \, e_{j} \right)=\frac{1}{n-2} \, \left(r_{i} +r_{j} -\frac{1}{n-1} \left(\, r_{\, 1} +...+r_{\, n} \right)\right); \sec \, \left(e_{i} \wedge \, e_{j} \right)=\frac{1}{n-2} \, \left(\, \lambda _{i} +\lambda _{j} \right)$$ where *$\left\{\, e_{i} \right\}$* is a orthonormal basis of $T_{x} M$ at an arbitrary point $x\in M$ such that *$Ric\, \left(e_{i} ,\, e_{j} \right)=r_{i} \, \delta _{ij} $*, *$Sch\, \left(e_{i} ,\, e_{j} \right)=\lambda _{i} \, \delta _{ij} $* and $\lambda _{{\rm i}} =\left(n-2\right)^{-1} \left(\, r_{i} -\left(2n-2\right)^{-1} s\, \right)$. Due to these equations we can formulate analogues of the Lemma 2 and the Corollary ** 3 for a locally conformally flat Riemannian manifold where the inequality *$\sec \, \left(\, e_{i} \wedge \, e_{j} \right)\ge 0$* can be replaced by *$\lambda _{i} +\lambda _{j} \ge 0$* or *$r_{i} +r_{j} \ge \left({\rm n}-1\right)^{-1} \left(\, r_{\, 1} +...+r_{\, n} \right)$* for any ${\rm i}\ne {\rm j}$.
For an *n*-dimensional $\left(n\ge 3\right)$ locally conformally flat manifold *$\left({\rm M,g}\right)$* with constant (not necessarily zero) scalar curvature its Ricci tensor ${\rm Ric}$ is a Codazzi 2-tensor with constant trace or, in other words, a harmonic symmetric bilinear form \[2, p. 444\]; \[28\]. Therefore, from Theorem 3 we conclude that the following corollary holds.
**COROLLARY 5*.*** *Let $\left({\rm M,g}\right)$ be an $n$-dimensional $\left(\, n\ge 3\, \right)$ connected complete noncompact locally conformally flat Riemannian manifold with nonnegative sectional curvature and constant scalar curvature. If $\left({\rm M,g}\right)$ is not locally flat then* $\int _{{\rm M}}\left\| \, {\rm Ric}\, \right\| ^{\, q} \, dVol_{g} =+\infty $ *for the Ricci tensor ${\rm Ric}$ and an arbitrary* $q\ge 1$*.*
**Remark 8**. Our Corollary 5 ** is a supplement to the theorem of M. Tani \[28\].
We strengthen the Corollary ** 5 by proving the validity of the following theorem.
**THEOREM 4**. *Let $\left({\rm M,g}\right)$ be an $n$-dimensional $\left(\, n\ge 3\, \right)$ connected complete locally conformally flat Riemannian manifold such that* $\, \left\| \, {\rm Ric}\, \right\| ^{2} <\left(n-1\right)^{-1} s^{2} $ *for* *the Ricci tensor ${\rm Ric}$ and the positive constant scalar curvature ${\rm s}$*. *If one of the following conditions is satisfied*: **
\(i) *$\, \left\| \, {\rm Ric}\, \right\| ^{2} $ has a global maximum point*;
\(ii) $\int _{{\rm M}}\left\| \, {\rm Ric}\, \right\| ^{{\rm q}} \, dVol_{g} <+\infty $ *at least for one* $q\ge 2$;
\(iii) *$\left({\rm M,g}\right)$ is a parabolic manifold,*
*then $\left({\rm M,g}\right)$ is a spherical space form.*
**Remark 9.** Theorem 4 is a supplement to the following S. Goldberg theorem \[30\].
Let *$\left({\rm M,}\, {\rm g}\right)$* be an ${\rm n}$-dimensional connected complete minimal hypersurface in the standard sphere $\left(\, {\rm S}^{{\rm n}+1} ,\, g_{0} \right)$ where ${\rm S}^{{\rm n}+1} \subset $ $\mathbb{R}$${}^{B}$ for ${\rm m}>{\rm n}+1$. In this case, we know that the second fundamental form ${\it S}$ of the hypersurface *$\left({\rm M,}\, {\rm g}\right)$* is a traceless Codazzi tensor \[29, p. 388\]. Therefore, can be rewritten in the form \[45\] $$\label{GrindEQ__2_3_}
\frac{1}{2} \, \Delta _{B} \, \left\| \, {\it S}\, \right\| ^{2} =\left\| \, {\rm S}\, \right\| ^{2} \left(\, n-\left\| \, {\rm S}\, \right\| ^{2} \right)+\left\| \, \nabla \, S\, \right\| ^{2}$$ The following theorem holds.
**Theorem 5.** *Let $\left({\rm M,}\, {\rm g}\right)$ be a connected complete minimal hypersurface in the standard sphere* $\left(\, {\rm S}^{{\rm n}+1} ,\, g_{0} \right)$ *such that its second fundamental form ${\rm S}$ satisfies the inequality $\left\| \, {\rm S}\, \right\| ^{2} \le {\rm n}$.* *If one of the following conditions is satisfied*: **
\(i) *$\, \left\| \, {\rm S}\, \right\| ^{2} $ has a global maximum point*;
\(ii) *$\int _{{\rm M}}\left\| \, {\rm S}\, \right\| ^{{\rm q}} \, dVol_{g} <+\infty $ at least for one $q\ge 2$;*
\(iii) *$\left({\rm M,g}\right)$ is a parabolic manifold*
*then one of the following occurs*: **
\(i) *$\left({\rm M,}\, {\rm g}\right)$ is an equator of* $\left(\, {\rm S}^{{\rm n}+1} ,\, g_{0} \right)$;
\(ii) *$\left({\rm M,}\, {\rm g}\right)$is locally isometric to a generalized Clifford torus.*
**Remark 10.** If *$\left({\rm M,}\, {\rm g}\right)$* is a complete Riemannian manifold with finite volume then the conditions (ii) of Theorem 4 and Theorem 5 are satisfied. For example, from *$\left\| \, {\rm S}\, \right\| ^{2} \le {\rm n}$* we obtain *$\int _{{\rm M}}\left\| \, S\, \right\| ^{\, 2} dVol_{g} \le n\, \int _{{\rm M}}\, dVol_{g} \le n\, Vol_{g} \left(M\right)<+\infty $.*
**3. Proofs of the statements**
Let us deduce the Weitzenböck formula for a traceless Codazzi ${\rm p}$-tensor *T* for $p\ge 2$. For this purpose, we remind that the local components of the Ricci tensor $Ric$ of the manifold $\left(M, g\right)$ are calculated from the identity $Ric\, \left(\partial _{i} ,\partial _{j} \right)=R_{ij} =R_{ikj}^{k} $ for the local components $R_{ikj}^{l} $ of the curvature tensor *R* determined from the equality $R_{lij}^{k} X^{l} =\nabla _{i} \nabla _{j} X^{k} -\nabla _{j} \nabla _{i} X^{k} $ where $\nabla _{i} =\nabla _{\partial _{{\rm i}} } $ and $X=X^{i} \partial _{i} $. Let us denote by $s:=g^{ij} R_{ij} $ the scalar curvature of the metric *g* for $\left(g^{ij} \right)=\left(g_{ij} \right)\, ^{-1} $. Then direct calculations give $$\frac{1}{2} \, \Delta _{B} \, \left\| \, T\, \right\| ^{2} {\rm :}=\frac{1}{2} g^{kl} \, \nabla _{k} \, \nabla _{l} \, \left(T^{{\rm k}_{1} k_{2} ...k_{p} } T_{k_{1} k_{2} ...k_{p} }^{} \right)=$$ $$=\left(g^{kl} \, \nabla _{k} \, \nabla _{l} \, T_{k_{1} k_{2} ...k_{p} }^{} \right)\, \, T^{{\rm k}_{1} k_{2} ...k_{p} } +g^{kl} \left(\nabla _{k} \, T_{k_{1} k_{2} ...k_{p} }^{} \right)\, \left(\nabla _{l} \, \, T^{{\rm k}_{1} k_{2} ...k_{p} } \right)=$$ $$=R_{ij} \, T^{ik_{2} ...k_{p} } T_{k_{2} ...k_{p} }^{j} -\left({\rm p}-1\right)\, R_{ijkl} T^{ikk_{3} ...k_{p} } T^{jl} _{k_{3} ...k_{p} } +g^{kl} \left(\nabla _{k} \, T_{k_{1} k_{2} ...k_{p} }^{} \right)\, \left(\nabla _{l} \, \, T^{{\rm k}_{1} k_{2} ...k_{p} } \right)=$$ $$=Q_{p} \, \left(T{\rm ,}\, {\rm T}\right)+\left\| \, \nabla \, T\, \right\| ^{2}$$ where the quadratic form $Q_{p} \, \left(T{\rm ,}\, {\rm T}\right)$ has the form $$\label{GrindEQ__3_1_}
{\rm Q}_{p} \left(\, T\, ,T\right)=R_{ij} \, T^{ik_{2} ...k_{p} } T_{k_{2} ...k_{p} }^{j} -\left({\rm p}-1\right)\, R_{ijkl} T^{ikk_{3} ...k_{p} } T^{jl} _{k_{3} ...k_{p} }$$ for components $T_{k_{1} ...k_{p} } $ of an arbitrary Codazzi ${\rm p}$-tensor ${\rm T}\in {\rm C}^{\infty } S_{0}^{p} M$with respect to a local coordinate system $x^{1} ,\, ...,\, x^{n} $. In deriving the formula we have used the condition of divergence-free of the Codazzi ${\rm p}$-tensor ${\rm T}$ and well-known Ricci identity (11.16) from \[41\].
The curvature operator $\mathop{R}\limits^{\circ } :S_{0}^{2} M\to S_{0}^{2} M$ is defined by equations \[2, pp. 51-52\] $$\label{GrindEQ__3_2_}
\mathop{R}\limits^{\circ } \left(\varphi _{il} \right)=R_{ijkl} \, {\rm T}^{jk} =g^{km} g^{jp} R_{ijkl} T_{mp}$$ for local components $T_{ij} \; $ of arbitrary $T\in $$S_{0}^{2} M$. Everywhere else we assume that the curvature operator $\mathop{R}\limits^{\circ } $ is nonnegative definite on an arbitrary section of the bundle $S_{0}^{2} M$, i.e. the inequality $R_{ijkl} \, T^{il} T_{\quad }^{jk} \ge 0$ is true for an arbitrary ${\rm T}\in {\rm S}_{0}^{2} M $and then $R_{ijkl} \, T^{il\, k_{3} ...k_{p} } T_{\quad k_{3} ...k_{p} }^{jk} \ge 0$ for any ${\rm T}\in {\rm S}_{0}^{p} M$. As a result, the second term in will be nonpositive, i.e. $\left({\rm p}-1\right)\, R_{ijkl} T^{ikk_{3} ...k_{p} } T^{jl} _{k_{3} ...k_{p} } \le 0$.
At an arbitrary point $ x\in M$ we choose orthogonal unit vectors $X,Y\in T_{x} M$ and define the tensor $\theta \in S_{0}^{2} M$ by the equality $$\theta =2^{-1} \left(X\otimes Y+Y\otimes X\right)$$ then $$g\left(\mathop{R}\limits^{\circ } \left(\theta \right),\, \theta \right)=2g\left(R\left(X,Y\right)\, Y,\, X\right) = 2 \sec \, \left(X\wedge Y\right).$$ That is why the sectional curvature of a manifold $\left(M, g\right)$ is everywhere nonnegative if the operator $\mathop{R}\limits^{\circ } $ is nonnegative definite on any section of the bundle $S_{0}^{2} M$. Let $X\in T_{x} M$ is a unit vector and we complete it to an orthonormal basis ${\rm X,}\, e_{2} ,\, ...\, ,e_{n} $ for $T_{x}M $ at an arbitrary point $x\in M$, then \[18, p. 86\] $$Ric\left(X,X\right)=\sum _{a=2}^{n}sec\left(X\wedge \, e_{a} \right)\, .$$ Therefore, the Ricci curvature is also nonnegative definite. Thus, if the operator $\mathop{R}\limits^{\circ } $ is nonnegative definite on sections of the bundle $S_{0}^{2} M$, then $Q_{p} \left(T,\, T\right)\ge 0$. As a result of it follows that $\Delta _{{\rm B}} \, \left\| \, {\rm T}\, \right\| ^{2} \ge 0$, i.e. $\left\| \, {\rm T}\, \right\| ^{2} $ is a nonnegative subharmonic function. Moreover, we note that $Q_{p} \left(T_{x} ,\, T_{x} \right)>0$ for a nonzero ${\rm T}_{{\rm x}} \in S_{0}^{p} \left(T_{x} M\right)$ at some point $x\in M$ if $\mathop{R}\limits^{\circ } >0$ for all non-zero $\theta _{{\rm x}} \in S_{0}^{2} \left(T_{x} M\right)$ at this point.
Let us prove our Lemma 1. Consider a traceless Codazzi $p$-tensor $(p\ge 2)$ in the connected open domain $U$ of $\left({\rm M,g}\right)$ where the curvature operator $\mathop{R}\limits^{\circ } \ge 0$ then $Q_{p} \left(T,\, T\right)\ge 0$. And according to we conclude that $\Delta _{{\rm B}} \, \left\| \, {\rm T}\, \right\| ^{2} \ge 0$, i.e. $\left\| \, {\rm T}\, \right\| ^{2} $ is a subharmonic function in the domain $U$. Suppose also that $\left\| \, {\rm T}\, \right\| ^{2} $ has a local maximum at some point ${\rm C}\in {\rm U}$ then according to the “Hopf’s maximum principle” \[9, p. 47\] we have that $\left\| \, {\rm T}\, \right\| ^{2} $ is a constant function in the domain ${\rm U}\subset {\rm M}$. In this case, $\Delta \, \left\| \, T\, \right\| ^{2} =0$ and as a consequence of we obtain $\left\| \, \nabla \, T\, \right\| ^{2} =0$ which means that the Codazzi $p$-tensor *T* is parallel.
Let $\left\| \, T\, \right\| ^{2} ={\rm C}$ (where *C* is a constant), then $T$ does not become zero anywhere in the domain $U$ and at the same time $\Delta _{{\rm B}} \, \left\| \, {\rm T}\, \right\| ^{2} =0$. If there is a point ${\rm x}\in {\rm U}$ where $\mathop{R}\limits^{\circ } >0$ then, as we stated above, $Q_{p} \left(T_{x} ,\, T_{x} \right)>0$. In this case, the left site of is equal to zero and its right side is greater than zero, so it follows that ${\rm T}=0$. The Lemma 1 is proved.
Corollary 1 of our Lemma 1 does not require any proof.
For the proof of our Theorem 1 in the case ${\rm q}=1$ we use the Theorem 1 from \[23\] according to which any nonnegative subharmonic function ${\rm f}\in {\rm C}^{\infty } {\rm M}$ on a connected complete noncompact Riemannian manifold with nonnegative sectional curvature must satisfy the condition $\int _{{\rm M}}f\, dVol_{g} =+\infty $.
Let us suppose that $\left({\rm M,g}\right)$ is a complete noncompact Riemannian manifold with nonnegative curvature operator $\mathop{{\rm R}}\limits^{\circ } $ (and that is why with nonnegative sectional curvature) and with a globally defined non-zero Codazzi ${\rm p}$-tensor ${\rm T}\in {\rm C}^{\infty } S_{0}^{p} M$ for an arbitrary $p\ge 2$.
By direct calculation we find the following $$\frac{1}{2} \, \Delta _{B} \, \left\| \, {\rm T}\, \right\| ^{2} =\left\| \, {\rm T}\, \right\| \, \, \Delta _{B} \, \left\| \, {\rm T}\, \right\| +\left\| \, {\rm d}\, \left\| \, {\rm T}\, \right\| \, \right\| ^{2} .$$ Then the equation can be rewritten in the form $$\left\| \, {\rm T}\, \right\| \, \, \Delta _{B} \, \left\| \, {\rm T}\, \right\| =Q_{p} \left(T,\, T\right)+\left\| \, \nabla \, T\, \right\| ^{2} -\left\| \, d\, \left\| \, {\rm T}\, \right\| \, \right\| ^{2} .$$ By using the first Kato inequality $\left\| \, \nabla \, T\, \right\| ^{2} \ge \left\| \, d\, \left\| \, {\rm T}\, \right\| \, \right\| ^{2} $(see \[46\]), we can write $$\left\| \, {\rm T}\, \right\| \, \, \Delta _{B} \, \left\| \, {\rm T}\, \right\| \ge Q_{p} \left(T,\, T\right).$$ Therefore, if ${\rm Q}_{{\rm p}} \left(T,\, T\right)\ge 0$ then we have $\Delta \, \left\| \, {\rm T}\, \right\| \ge 0$ and as a result of this $\, \left\| \, {\rm T}\, \right\| $ is a nonnegative subharmonic function. Now according to Theorem 1 from \[23\] we come to the conclusion that $\int _{{\rm M}}\left\| \, T\, \right\| \, dVol_{g} =+\infty $ for an arbitrary non-zero Codazzi $p$-tensor ${\rm T}\in {\rm C}^{\infty } S_{0}^{p} M$. Thus, on a complete noncompact Riemannian manifold with nonnegative curvature operator (and that is why with nonnegative sectional curvature) there exists no non-zero Codazzi tensor ${\rm T}\in {\rm C}^{\infty } S_{0}^{p} M$ such that $\int _{{\rm M}}\left\| \, T\, \right\| \, dVol_{g} <+\infty $.
For the proof of our Theorem 1 in the case ${\rm q}>1$ we use the Theorem 7 from \[22\]. According to this theorem any nonnegative subharmonic function on a connected complete noncompact Riemannian manifold satisfies the condition $\int _{{\rm M}}f^{{\rm q}} dVol_{g} =+\infty $ for ${\rm q}>1$ or ${\rm f}={\rm const}$.
Let us suppose that a non-zero traceless Codazzi $p$-tensor ${\rm T}$ is globally defined on a complete noncompact Riemannian manifold *$\left({\rm M,g}\right)$* with nonnegative curvature operator $\mathop{{\rm R}}\limits^{\circ } $. In this case$\, \left\| \, {\rm T}\, \right\| $ is a nonnegative subharmonic function. If in addition to the above mentioned ${\rm T}$ satisfies the condition $\int _{{\rm M}}\left\| \, {\rm T}\, \right\| ^{q} dVol_{g} <+\infty $ for some ${\rm q}>1$, then due to the Yau theorem we conclude that $\left\| \, {\rm T}\, \right\| =C$ for some nonnegative constant ${\rm C}$. Therefore, the condition $\int _{{\rm M}}\left\| \, {\rm T}\, \right\| ^{q} dVol_{g} <+\infty $ can be rewritten in the form $C^{q} \int _{{\rm M}}dVol_{g} <+\infty $. On the other hand, every complete noncompact Riemannian manifold with nonnegative Ricci curvature or with nonnegative sectional curvature has infinite volume \[22\]; \[23\]. Therefore, from the inequality $C^{q} \int _{{\rm M}}dVol_{g} <+\infty $ we conclude that $\left\| \, {\rm T}\, \right\| =C=0$. This completes our proof.
Let us prove Corollary 2. Suppose that *$\left({\rm M,g}\right)$* is a ${\rm n}$-dimensional $\left(\, n\ge 4\, \right)$ complete noncompact locally conformally flat Riemannian manifold. In this case, the curvature tensor ${\rm R}$ has the following local components \[2, pp. 60-61\]; \[28\]: $$\label{GrindEQ__3_3_}
{\rm R}_{{\rm ijkl}} =\frac{1}{n\, -\, 2} \, \, \left(R_{jl} g_{ik} -R_{jk} g_{il} +R_{ik} g_{jl} -R_{il} g_{jk} \right)\, \, -\frac{s}{\left(n-1\right)\left(n-2\right)} \, \left(g_{jl} g_{ik} -g_{jk} g_{il} \right).$$ Then from we obtain the following equalities $${\rm R}_{{\rm ijkl}} \, \theta ^{jk} \theta ^{il} =\frac{2}{n-2} \, \left(R_{ij} \, \theta ^{ik} \theta _{k}^{j} -\frac{s}{2\, \left(n-1\right)} \, \left\| \, \theta \, \right\| ^{2} \right)=$$ $$=\frac{1}{n\, -\, 2} \, \left(\left(R_{jl} -\frac{s}{2\, \left(n-1\right)} \, g_{j{\rm l}} \right)\, g_{ik} +\left(R_{{\rm ik}} -\frac{s}{2\, \left(n-1\right)} \, g_{ik} \right)\, g_{jl} \right)\, \theta ^{il} \theta ^{jk} =$$ $$=\, \left(S_{jl} \, g_{ik} +S_{ik} \, g_{jl} \right)\, \theta ^{il} \theta ^{jk} =2\, {\rm S}_{{\rm ij}} \theta ^{ik} \theta _{k}^{j} .$$ for the local component ${\rm S}_{{\rm ik}} $ of the Schouten tensor $SA{\rm h}\in C^{\infty } S^{2} M$ and the local components $\theta _{ik} $ of an arbitrary traceless tensor $\theta \in S_{0}^{2} M$. Then from the condition of nonnegative definiteness of the Schouten tensor $Sch \in C^{\infty } S^{2} M$ we obtain the inequality $\mathop{{\rm R}}\limits^{\circ } \ge 0$. After that we should repeat the proof of Theorem 1.
Let us prove Theorem 2. We define the following differential operator: $\delta ^{*} {\rm :}C^{\infty } T^{*} M\to C^{\infty } S^{2} M$ by the equality $\left(\delta ^{*} \xi \right)\, \left(\, {\rm X,}\, {\rm Y}\right)=2^{-1} \left(\, \left(\nabla _{X} \xi \right)\, Y+\left(\nabla _{Y} \xi \right)\, X\right)$ for any tangent vector fields ${\rm X,}\, {\rm Y,}\, {\rm Z}$ on ${\rm M}$ and an arbitrary one-form $\xi \in {\rm C}^{\infty } T^{*} M$. Moreover, we denote by $\delta ^{\nabla } $ the adjoint operator of ${\rm d}^{\nabla } $ \[29, p. 380; 388\]; \[2, p. 355\]. If $\left({\rm M,}\, {\rm g}\right)$ is a compact manifold then the condition $T\in {\rm Ker}\, d^{\nabla } \bigcap {\rm Ker}\, \delta $ is equivalent to $T\in {\rm Ker}\, \Psi $ for $\Psi =\delta ^{\nabla } d^{\nabla } +2\, \delta ^{*} \delta $ \[29, p. 388\]. The differential operator $\Psi $ is an elliptic operator with injective symbol and ${\rm Ker}\, \Psi ={\rm Ker}\, d^{\nabla } \bigcap {\rm Ker}\, \delta $ \[29, p. 388\]. Therefore, ${\rm Ker}\, \Psi $ is a finite-dimensional vector space of harmonic symmetric bilinear form on a compact manifold $\left({\rm M,}\, {\rm g}\right)$ \[2, p. 464\]. This completes our proof.
Let us prove Lemma 2. First, we rewrite the Bochner-Weitzenböck formula in the form $$\label{GrindEQ__3_4_}
\frac{1}{2} \, \Delta _{B} \, \left\| \, \bar{{\rm T}}\, \right\| ^{2} =Q_{2} \left(\bar{T},\, \bar{T}\right)+\left\| \, \nabla \, \bar{T}\, \right\| ^{2}$$ where ${\rm Q}_{2} \left(\, \bar{T},\, \bar{T}\right)=R_{ij} \bar{T}^{ik} \bar{T}_{k}^{j} -R_{ijkl} \bar{T}^{ik} \bar{T}^{jl} $ for a traceless Codazzi 2-tensor $\bar{T}$.
Second, if $T$ is a Codazzi tensor with constant trace, then $\bar{T}=T-{1 \mathord{\left/{\vphantom{1 n}}\right.\kern-\nulldelimiterspace} n} \, \, \left(trace_{g} \, T\right)\, g$ is a traceless Codazzi tensor such that $\Delta _{B} \, \left\| \, \overline{{\rm T}}\, \right\| ^{2} =\Delta _{B} \, \left\| \, T\, \right\| ^{2} $ and $\nabla \bar{{\rm T}}=\nabla T$. In addition, one can prove that $Q_{2} \left(\, \bar{T},\, \bar{T}\right)=Q_{2} \left(\, T,\, T\right)=R_{ij} T^{ik} T_{k}^{j} -R_{ijkl} T^{ik} T^{jl} $. Then can be rewritten in the form $$\label{GrindEQ__3_5_}
\frac{1}{2} \, \Delta _{B} \, \left\| \, {\rm T}\, \right\| ^{2} =Q_{2} \left(T,\, T\right)+\left\| \, \nabla \, T\, \right\| ^{2}$$ According to \[2, p. 436\] and \[29, p. 388\] we conclude that $$\label{GrindEQ__3_6_}
{\rm Q}_{2} \left(\, T,\, T\right)=\sum _{{\rm i}\, <{\rm j}}{\rm sec} \left(e_{i} \wedge \, e_{j} \right)\, \left(\lambda _{i} -\lambda _{j} \right)\, ^{2} ,$$ where *$\left\{\, e_{i} \right\}$* is a such orthonormal basis of the tangent space $T_{x} M$ at an arbitrary point $x\in M$ that the Codazzi tensor $T_{x} \left(\, e_{i} ,\, e_{j} \right)=\lambda _{i} \left(x\right)\, \delta _{ij} $ where $\delta _{ij} $ is the Kronecker symbol and ${\rm sec}\, \left(\, {\rm e}_{{\rm i}} \wedge \, e_{j} \right)$ is the sectional curvature in the direction of subspace $\pi \left(x\right)\subset T_{x} M$ such that $\pi \left(x\right)=span\, \left\{\, e_{i} ,\, e_{j} \right\}$. It is known from \[2, p. 439\] and \[1\] that an arbitrary Codazzi tensor *T* on a manifold *$\left({\rm M,g}\right)$* commutes with its Ricci tensor $Ric$, and therefore the eigenvectors *$\left\{\, e_{i} \right\}$* of the Codazzi tensor $T$ determine the *principal directions of the Ricci tensor* at each point ${\rm x}\in {\rm U}$\[41, pp.113-114\]. The converse is also true. Next, let us suppose that ${\rm sec}\, \left(\, {\rm e}_{{\rm i}} \wedge \, e_{j} \right)\ge 0$ in some connected open domain ${\rm U}\subset {\rm M}$ then ${\rm Q}\left(\, T\, \right)\ge 0$. Moreover, if there is a non-zero Codazzi tensor ${\rm T}$given in ${\rm U}\subset {\rm M}$ then from we conclude that $\Delta _{B} \, \left\| \, T\, \right\| ^{2} \ge 0$, i.e. $\left\| \, T\right\| ^{2} $ is a nonnegative subharmonic function in $U$. Let us suppose $\left\| \, T\right\| ^{2} $ has a local maximum at some point ${\rm x}\in {\rm U}$ then $\left\| \, T\, \right\| ^{2} $ is a constant function in ${\rm U}\subset {\rm M}$ according to the “Hopf’s maximum principle” \[21, p. 47\]. In this case, $\Delta _{B} \, \left\| \, T\, \right\| ^{2} =0$ and $\left\| \, \nabla \, T\, \right\| ^{2} =0$. Then the latter equation means that the Codazzi tensor ${\rm T}$ is a parallel tensor field.
Let $\left\| \, T\, \right\| ^{2} ={\rm C}$ (*C* is a constant) then from we obtain that $Q_{2} \left(T,\, T\right)+\left\| \, \nabla \, T\, \right\| ^{2} =0$. Since ${\rm sec}\, \left(\, {\rm e}_{{\rm i}} \wedge \, e_{j} \right)\ge 0$ it means that ${\rm Q}_{2} \left(\, T,\, T\right)=0$ and $\nabla \, T=0$. If there is a point ${\rm x}\in {\rm U}$ such that ${\rm sec}\, \left(\, {\rm e}_{{\rm i}} \wedge \, e_{j} \right)>0$ then from we come to the conclusion that $\lambda _{1} ={\rm ...}=\lambda _{{\rm n}} =\lambda $ which is equivalent to $T={1 \mathord{\left/{\vphantom{1 n}}\right.\kern-\nulldelimiterspace} n} \, \, \left(trace\, T\right)\, g$ (see \[2, p. 436\]). Lemma 2 is proved.
Assume that the manifold *$\left({\rm M,g}\right)$* is compact and the Codazzi tensor is globally defined on *$\left({\rm M,g}\right)$* then due to the “Bochner maximum principle” comes into force \[16, p. 30\] according to which a subharmonic function on a compact manifold is a constant. As a result, from our Lemma 2 we obtain Corollary 3 which is essentially the Berger-Ebin theorem \[2, p. 436\]; \[29\].
Let us prove our Theorem 3. Let *$\left({\rm M,g}\right)$* be a complete noncompact Riemannian manifold with nonnegative sectional curvature and ${\rm T}$ be a globally defined non-zero Codazzi 2-tensor with a constant trace. In this case ${\rm Q}_{2} \left(T,\, T\right)\ge 0$ and according to the Theorem 1 the norm $\, \left\| \, {\rm T}\, \right\| $ is a subharmonic function. Therefore, due to the Theorem 1 from \[23\] we come to the conclusion that there is no non-zero Codazzi tensor ${\rm T}$ on *$\left({\rm M,g}\right)$* such that $\int _{{\rm M}}\left\| \, T\, \right\| \, dVol_{g} <+\infty $.
Let us turn to the Yau theorem \[22, p. 663\]. Yau’s theorem states the following: If $\int _{{\rm M}}\left\| \, {\rm T}\, \right\| ^{{\rm q}} dVol_{g} <+\infty $ ** for some $q>1$ on a complete *$\left(M,g\right)$* then $\left\| \, {\rm T}\, \right\| =C$ ($C$ is a constant). In this case, the condition $\int _{{\rm M}}\left\| \, {\rm T}\, \right\| ^{{\rm q}} dVol_{g} <+\infty $ is equivalent to $C^{q} \int _{{\rm M}}dVol_{g} <+\infty $. The latter inequality is not feasible on a complete noncompact Riemannian manifold *$\left({\rm M,g}\right)$* with nonnegative sectional curvature \[23\]. So $\, \left\| \, T\, \right\| ={\rm C}=0$ that contradicts the existence of a non-zero Codazzi tensor. The Theorem 2 is proved.
The Corollary 5 does not require any proof.
And now we prove Theorem 4. Let, as before, *$\left({\rm M,g}\right)$* be an ${\rm n}$-dimensional $(n\ge 3)$ locally conformally flat Riemannian manifold with positive constant scalar curvature $s={\rm trace}_{{\rm g}} {\rm Ric}$. In this case, for the traceless Ricci tensor $\overline{Ric}=Ric-n^{-1} s\, g$ we have $$Q_{2} \left(\, Ric,\, Ric\right)=Q_{2} \left(\, \overline{Ric,}\, \overline{Ric}\, \right)=$$ $$=R_{ij} \bar{R}^{ik} \bar{R}_{k}^{j} -R_{ijkl} \bar{R}^{ik} \bar{R}^{jl} =\frac{1}{n-1} \, s\, \left\| \, \overline{Ric}\, \right\| ^{2} +\frac{n}{n-2} \bar{R}_{ij} \bar{R}^{ik} \bar{R}_{k}^{j} \ge$$ $$\label{GrindEQ__3_7_}
\ge \frac{1}{n-1} \, \left\| \, \overline{Ric}\, \right\| ^{2} \left(s-\sqrt{n\left(n-1\right)} \, \left\| \, \overline{Ric}\, \right\| \, \right)$$ where, due to Lemma 2.1 from \[47\], we used the inequality $$\bar{R}_{ij} \bar{R}^{ik} \bar{R}_{k}^{j} \ge -\frac{n-2}{\sqrt{n\left(n-1\right)} } \, \left\| \, \overline{Ric}\, \right\| ^{3} .$$ Then from the formula we obtain the inequality $$\label{GrindEQ__3_8_}
\frac{1}{2} \, \Delta _{{\rm B}} \, \left\| \, \overline{Ric}\, \right\| ^{2} \ge \frac{1}{n-1} \, \left\| \, \overline{Ric}\, \right\| ^{2} \left(s-\sqrt{n\left(n-1\right)} \, \left\| \, \overline{Ric}\, \right\| \, \right)+\left\| \, \nabla \, \overline{Ric}\, \right\| ^{2} .$$ One can prove that $\Delta _{B} \, \left\| \, Ric\, \right\| ^{2} =\Delta _{{\rm B}} \, \left\| \, \overline{Ric}\, \right\| ^{2} $. Therefore, we can rewrite in the following form $$\label{GrindEQ__3_9_}
\frac{1}{2} \, \Delta _{B} \, \left\| \, {\rm Ric}\, \right\| ^{2} \ge \frac{1}{n-1} \, \left\| \, \overline{Ric}\, \right\| ^{2} \left(s-\sqrt{n\left(n-1\right)} \, \left\| \, \overline{Ric}\, \right\| \, \right)$$ If we suppose that $\, \left\| \, {\rm Ric}\, \right\| ^{2} <\left(n-1\right)^{-1} s^{2} $ then from we conclude that $\Delta _{B} \, \left\| \, Ric\, \right\| ^{2} \ge 0$, i.e.$\, \left\| \, Ric\, \right\| ^{2} $ is a nonnegative subharmonic function.
In the first case, if *$\left({\rm M,g}\right)$* is a connected complete manifold and*$\, \left\| \, {\rm Ric}\, \right\| ^{2} $* has a global maximum point, then due the “Hoph maximum principle” we obtain $\, \left\| \, Ric\, \right\| ^{2} ={\rm C}$ where *C* is a constant.
In the second case, we rewrite in the following form $$\left\| \, {\rm Ric}\, \right\| \, \Delta _{B} \, \left\| \, {\it Ric}\, \right\| =\frac{1}{n-1} \, \left\| \, \overline{Ric}\, \right\| ^{2} \left(s-\sqrt{n\left(n-1\right)} \, \left\| \, \overline{Ric}\, \right\| \, \right)+\left\| \, \nabla \, Ric\, \right\| ^{2} -\left\| \, {\rm d}\, \left\| \, {\it Ric}\, \right\| \, \right\| ^{2} \ge$$ $$\label{GrindEQ__3_10_}
\ge \frac{1}{n-1} \, \left\| \, \overline{Ric}\, \right\| ^{2} \left(s-\sqrt{n\left(n-1\right)} \, \left\| \, \overline{Ric}\, \right\| \, \right)$$ where we used the first Kato inequality $\left\| \, \nabla \, {\rm Ric}\, \right\| ^{2} \ge \left\| \, d\, \left\| \, {\rm Ric}\, \right\| \, \right\| ^{2} $(see \[46\]). Let $\, \left\| \, {\rm Ric}\, \right\| ^{2} <\left(n-1\right)^{-1} s^{2} $ then from we conclude that $\left\| \, {\rm Ric}\, \right\| $ is a nonnegative subharmonic function. Then for an arbitrary ${\rm q}\ge 2$, either $\int _{{\rm M}}\left\| \, {\rm Ric}\, \right\| ^{{\rm q}} \, dVol_{g} =+\infty $ or $\left\| \, {\rm Ric}\, \right\| =C$ \[22, p. 664\]. Therefore, if we suppose that $\int _{{\rm M}}\left\| \, {\rm Ric}\, \right\| ^{{\rm q}} \, dVol_{g} <+\infty $ at least for one $q\ge 2$, then $\left\| \, {\rm Ric}\, \right\| ={\rm C}$.
In the third case, we remind that a complete manifold *$\left({\rm M,g}\right)$* is said to be *parabolic* if it does not admit a positive *Green’s function*. In addition, if *$\left({\rm M,g}\right)$* is a parabolic manifold then every subharmonic and bounded-above function on $M$ is constant \[20, p. 147\]. In our case, $\, \left\| \, Ric\, \right\| ^{2} $ is a subharmonic function on *$\left({\rm M,g}\right)$* such as $\left\| \, {\rm Ric}\, \right\| ^{2} <\left(n-1\right)^{-1} s^{2} $ for the constant scalar curvature$s>0$. Therefore, for the case of parabolic manifold *$\left({\rm M,g}\right)$* we have $\, \left\| \, Ric\, \right\| ^{2} ={\rm C}$.
Finally, if $\, \left\| \, Ric\, \right\| ^{2} ={\rm C}$ and $\left\| \, {\rm Ric}\, \right\| ^{2} <\left(n-1\right)^{-1} s^{2} $ then from we obtain $\, \left\| \, \overline{{\rm Ric}}\, \right\| ^{2} =0$, that is why ${\rm g}$ is an Einstein metric. Therefore, *$\left({\rm M,g}\right)$* becomes a complete Riemannian manifold with positive constant curvature, that means *$\left({\rm M,g}\right)$* is a *spherical space form* \[48, p. 69\].
Let us prove our Theorem 5. In the first case, if *$\left\| \, {\rm S}\, \right\| ^{2} \le {\rm n}$* then from we conclude that $\left\| \, {\rm S}\, \right\| ^{2} $ is a nonnegative subharmonic function. If, moreover, *$\left({\rm M,g}\right)$* is a connected complete manifold and*$\, \left\| \, {\rm S}\, \right\| ^{2} $* has a global maximum point, then due to the “Hoph maximum principle” we obtain $\, \left\| \, Ric\, \right\| ^{2} ={\rm C}$ where *C* is a constant.
In the second case, we rewrite in the following form $$\label{GrindEQ__3_11_}
\left\| \, {\it S}\, \right\| \, \Delta _{B} \, \left\| \, {\it S}\, \right\| \ge \left\| \, {\rm S}\, \right\| ^{2} \left(\, n-\left\| \, {\rm S}\, \right\| ^{2} \right)$$ Let *$\left\| \, {\rm S}\, \right\| ^{2} \le {\rm n}$* then from we conclude that $\left\| \, {\rm S}\, \right\| $ is a nonnegative subharmonic function. Then for an arbitrary ${\rm q}\ge 1$, either $\int _{{\rm M}}\left\| \, {\rm S}\, \right\| ^{{\rm q}} \, dVol_{g} =+\infty $ or $\left\| \, {\rm S}\, \right\| =C$ \[22, p. 664\]. Therefore, if we supouse $\int _{{\rm M}}\left\| \, {\rm S}\, \right\| ^{{\rm q}} \, dVol_{g} <+\infty $ for some $q\ge 2$, then $\left\| \, {\rm S}\, \right\| =C$.
In the third case, if *$\left({\rm M,g}\right)$* is a parabolic manifold then *$\left\| \, {\rm S}\, \right\| ^{2} ={\rm C}$* because *$\, \left\| \, {\rm S}\, \right\| ^{2} $* is a subharmonic function on *$\left({\rm M,g}\right)$* such that *$\left\| \, {\rm S}\, \right\| ^{2} \le {\rm n}$*.
Finally, if *$\left\| \, {\rm S}\, \right\| ^{2} ={\rm C}$* and *$\left\| \, {\rm S}\, \right\| ^{2} \le {\rm n}$* then from we obtain either $\left\| \, {\rm S}\, \right\| ^{2} \equiv 0$ or $\left\| \, {\rm S}\, \right\| ^{2} \equiv n$. ** In the first case, $\left({\rm M,}\, {\rm g}\right)$ must be totally geodesic. At the same time, we know that an arbitrary ${\rm n}$-dimensional complete totally geodesic submanifold of the sphere ${\rm S}^{n+p} $ is a sphere ${\rm S}^{n} $ \[49\]. Therefore, $\left({\rm M,}\, {\rm g}\right)$ has to be an equator ${\rm S}^{n} \subset {\rm S}^{n+1} $. In the second case, $\left({\rm M,}\, {\rm g}\right)$ is locally isometric to a generalized Clifford torus ${\rm S}^{{\rm k}} \left(r_{1} \right)\times S^{n-k} \left(r_{2} \right)$, which is the standard product embedding of the product of two spheres of radius ${\rm r}_{1} =\sqrt{k\, n^{-1} } $ and $r_{2} =\sqrt{\left(n-k\right)\, n^{-1} } $, respectively \[50\].
[**Acknowledgments.** Our work was supported by the Russian Foundation for Basic Research of the Russian Academy of Science (projects Nos. 16-01-00053 and 16-01-00756). The authors would like to express their thanks to Professor Nelli Makievskaya for her editorial suggestions, that contributed to the improvement of this paper as well as to the referee for his careful reading of the previous version of the manuscript and suggestions about this paper which led to various improvements.]{}
**References**
\[1\] Derdzinski A., Shen C., “Codazzi tensor fields, curvature and Pontryagin forms“, Proc. London Math. Soc., **47**:3 (1983), 15-26.
\[2\] Besse, A. Einstein Manifolds, Springer-Verlag, Berlin-Heidelberg (1987).
\[3\] Simon U., Schwenk-Schellschmidt A., Viesel H., Introduction to the affine differential geometry of hypersurfaces, Science University Tokyo Press, Tokyo (1991).
\[4\] Catino G., Mantegazza C., Mazzieri L., “A note on Codazzi tensors”, *Mathematische Annalen*, **362**: 1-2 (2015), 629-638.
\[5\] Merton G., “Codazzi tensors with two eigenvalue functions”, *Proc. Amer. Math. Soc.*, **141** (2013), 3265-3273.
\[6\] Aledo J.A., Espinar J.M., Galvez J.A., “The Codazzi equation for surfaces”, *Advances in Math*., **224** (2010), 2511-2530.
\[7\] Bonsante F., Seppi A., “On Codazzi Tensors on a hyperbolic surface and flat Lorentzian geometry”, *International Mathematics Research Notesá* **2016**: 2 (2016), 343-417.
\[8\] Catino G., Mantegazza C., Mazzieri L., “A note on Codazzi tensors”, *Mathematische Annalen*, **362**:1-2 (2015), 629-638.
\[9\] Baum H., “Holonomy group of Lorentzian manifolds: A status report”, *Global differential geometry*, Springer-Verlag, Berlin-Heidelberg (2012), 163-200.
\[10\] Catino C., “On conformally flat manifolds with constant positive scalar curvature”, *Proc. Amer. Math. Soc*., **144** (2016), 2627-2634.
\[11\] Stepanov S. E., "Fields of symmetric tensors on a compact Riemannian manifold”, *Math. Notes*, **52**:4 (1992), 1048-1050.
\[12\] Liu H.L., Simon U., Wang C.P., “Codazzi tensor and the topology of surfaces”, *Annals of Global Analysis and Geometry*, **16** (1998), 189-202.
\[13\] Liu H.L., Simon U., Wang C.P., “Higher order Codazzi tensors on conformally flat spaces”, *Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry*, **39**:2 (1998) 329-348.
\[14\] Leder J., Schwenk-Schellschmidt A., Simon U., Wiehe M., “Generating higher order Codazzi tensors by functions”*, Geometry and Topology of Submanifolds* IX”, London and Singapore, Word Scientific Publishing Co. Pte. Ltd. (1999), 174-191.
\[15\] Huang G., “Rigidity of Riemannian manifolds with positive scalar curvature”, Annals of Global Analysis and Geometry, 54:2 (2018), 257-272.
\[16\] Yano K., Bochner S., Curvature and Betti numbers, Princeton, Princeton Univ. Press (1953).
\[17\] Wu H., The Bochner technique in differential geometry, New York, Harwood Academic Publishers (1988).
\[18\] Peterson P., Riemannian geometry, Springer Int. Publ. AG, Switzerland (2016).
\[19\] Bérard P.H., “From vanishing theorems to estimating theorems: the Bochner technique revisited”, *Bulletin of the American Math. Soc*., **19**:2 (1988), 371-406.
\[20\] Pigola S., Rigoli M., Setti A.G., Vanishing and Finiteness Results in Geometric Analysis. A Generalization of the Bochner Technique, Birkhäuser Verlag AG, Berlin (2008).
\[21\] Calabi E., “An extension of E. Hopf’s maximum principle with an application to Riemannian geometry”, *Duke Math. J.*, **25** (1958), 45-56.
\[22\] Yau S.T., “Some function-theoretic properties of complete Riemannian manifold and their applications to geometry”, *Indiana Univ. Math. J*., **25**:7 **** (1976), 659-679.
\[23\] Greene R.E., Wu H., “Integrals of subharmonic functions on manifolds of nonnegative curvatures”, *Inventions Math*., **27** (1974), 265-298.
\[24\] Li P., Schoen R., “${\rm L}^{{\rm p}} $and mean value properties of subharmonic functions on Riemannian manifolds”, *Acta Mathematica*, **153** (1984), 279-301.
\[25\] Stepanov S.E., Tsyganok I.I., “Conformal Killing *L*${}^{2}$-forms on complete Riemannian manifolds with nonpositive curvature operator”, *Journal of Mathematical Analysis and Applications*, **458**:1 (2018), 1-8.
\[26\] Stepanov S.E., Tsyganok I.I., “Theorems on conformal mappings of complete Riemannian manifolds and their applications”, *Balkan Journal of Geometry and its Applications*, **22**:1 (2017), 81-86.
\[27\] Stepanov S.E., Mikeš J., “Liouville-type theorems for some classes of Riemannian almost product manifolds and for special mappings of Riemannian manifolds”, Differential Geometry and its Applications, 54 (2017), Part A, 111-121.
\[28\] Tani M., “On a conformally flat Riemannian space with positive Ricci curvature”, *Tohoku Math. Journal*, **19**:2 (1967), 227-231.
\[29\] Berger M., Ebin D., “Some decomposition of the space of symmetric tensors on a Riemannian manifold”, *Journal of Differential Geometry*, **3** (1969), 379-392.
\[30\] Goldberg S.I., “An application of Yau’s maximum principle to conformally flat spaces”, *Proceeding of the American Mathematical Society*, **79**:2 (1980), 260-270.
\[31\] Bouguignon J.-P., H. Karcher, “Curvature operators: pinching estimates and geometric examples”, *Ann. Sc. Éc. Norm. Sup.*, **11** (1978), 71-92.
\[32\] Tachibana S., Ogiue K., “Les variétés riemanniennes dont l’opérateur de coubure restreint est positif sont des sphères d’homologie réelle”, *C. R. Acad. Sc. Paris*, **289** (1979), 29-30.
\[33\] Stepanov S. E., Mikes. J, "Hodge-de Rham Laplacian and Tachibana operator on a compact Riemannian manifold with a curvature operator of fixed sign”, *Izvestiya*: *Mathematics*, **79**:2 (2015) 375-387.
\[34\] Kashiwada T., “On the curvature operator of the second kind”, *Natural Science Report* *Ochanomizu University*, **44**:2 (1993), 69-73.
\[35\] Stepanov S.E., Tsyganok I.I., “Theorems of existence and of vanishing of conformally Killing forms”, *Russian Mathematics,* **58**:10 (2014), 46-51.
\[36\] Stepanov S., Tsyganok I.I., “Vanishing theorems for harmonic mappings into non-negatively curved manifolds and their applications”, *Manuscripta Mathematica*, 154:1-2 (2017), 79-90.
\[37\] Bettiol R.G., Mendes R.A.E., “Sectional curvature and Weitzenböck formulae”, *Preprint*, arXiv:1708.09033v1 \[math.DG\] 29 Aug 2017, 24 pp.
\[38\] Guan P., Viaclovsky J., Wang G., “Some properties of the Schouten tensor and applications to conformal geometry”, *Transactions of the American Mathematical Society*, **355** (2002), no. 3, 925-933.
\[39\] Bourguignon J.P., “Formules de Weitzenbök en dimension 4”, Seminare A. Besse sur la geometrie Riemannienne dimension 4, Cedic. Ferman, Paris, (1981), 156-177.
\[40\] Bourguignon J.P., “Les variétés de dimension 4 á signature non nulle dont la courbure est harmonique sont d’Einstein”, *Invent. Math.,* 63 (1981), 263-286.
\[41\] Eisenhart L. P., Riemannian geometry, Princeton Univ. Press (1926).
\[42\] Kobayashi Sh., Nomizu K., Foundations of differential geometry, vol. 2, Interscience Publishers, New York-London-Sydney (1969).
\[43\] Donaldson S.K., “Symmetric spaces, Kähler geometry and Hamiltonian dynamics”, *Amer. Math. Soc. Transl*. **196** (1999), 13-33.
\[44\] Wegner B., “Kennzeichnungen von Räumen konstanter Krümmung unter lokal konform Euklidischen Riemannschen Mannigfaltigkeiten“, *Geom. Dedicata*, **2** (1973) 269–281.
\[45\] Peng C.-K., Terng C.-L., “The scalar curvature of minimal hypersurfaces in spheres”, *Math. Ann*., **266** (1983), 105-113.
\[46\] Calderbank D.M.J., Gauduchon P., Herzlich M., “Refined Kato inequalities and conformal weights in Riemannian geometry”, *Journal of Functional Analysis*, **173** (2000), 214-255.
\[47\] Okumura M., “Hypersurfaces and a pinching problem on the second fundamental tensor”, *Amer. J. Math.*, **96** (1974), 207–213.
\[48\] Wolf G., Spaces of constant curvature, California Univ. Press, Berkley (1972).
\[49\] Morvan J.-M., “Differential geometry of Riemannian submanifolds: recent results”, Proceedings of the Conference on Algebra and Geometry ( Kuwait, 1981), Kuwait Univ. Press, Kuwait (1981), 145-151.
\[50\] Lawson B., “Local rigidity theorems for minimal hypersurfaces”, *Ann. of Math*., **89**:2 (1969), 187-197.
|
---
abstract: 'We give an explicit and purely combinatorial description of the Duskin nerve of any $(r+1)$-point suspension $2$-category, and in particular of any $2$-category belonging to Joyal’s disk category $\Theta_2$.'
address:
- 'Fakultät für Mathematik, Ruhr-Universität Bochum, Bochum, Germany'
- 'Mathematical Sciences Institute, The Australian National University, Canberra, Australia '
author:
- Viktoriya Ozornova
- Martina Rovelli
bibliography:
- 'ref.bib'
title: 'The Duskin nerve of $2$-categories in Joyal’s disk category $\Theta_2$'
---
Overview of results {#overview-of-results .unnumbered}
===================
The $2$-categorical analog of the nerve of ordinary categories goes by the name of *Duskin nerve*. It was implicitly defined by Street [@StreetOrientedSimplexes] and further studied by Duskin [@duskin]. Roughly speaking, the objects, $1$-morphisms and $2$-morphisms of the given $2$-category are incorporated suitably in the $0$-, $1$- and $2$-simplices of the Duskin nerve, which is $3$-coskeletal.
The Duskin nerve is a classical construction, and many of its homotopical properties have been established. For instance, Duskin [@duskin] showed that the Duskin nerve of a $(2,0)$-category is always a Kan complex and that the Duskin nerve of a $(2,1)$-category is always a quasi-category. Bullejos, Carrasco, Cegarra, and Garzón showed in different combinations that analogs of Quillen’s Theorems A and B hold for the Duskin nerve of $2$-categories [@BullejosCegarra; @cegarra], and that the Duskin nerve is homotopically equivalent to other nerve constructions for $2$-categories [@CCG]. To mention one application, Nanda [@nanda] then built on their work showing that the Duskin nerve of the discrete flow $2$-category associated to a simplicial complex (with extra structure) has the same homotopy type as that of the simplicial complex.
The machinery developed by Steiner [@SteinerSimpleOmega] implicitly provides methods to study the Duskin nerve of $2$-categories. However, we are not aware of many explicit computations and descriptions of the Duskin nerve, even for small and rather simple $2$-categories.
As a first analysis in this direction, one could observe that in the nerve of finite $1$-categories there are only finitely many non-degenerate simplices, and imagine that the Duskin nerve of finite $2$-categories would enjoy the same property. Somewhat surprisingly, we discovered that the Duskin nerve of $2$-categories is much more complex than expected. For instance, we show in \[section2\] that the Duskin nerve of the free $2$-cell $$\begin{tikzcd}[row sep=3cm, column sep=2cm]
x \arrow[r, bend left, "f_0", ""{name=U,inner sep=2pt,below}]
\arrow[r, bend right, "f_1"{below}, ""{name=D,inner sep=2pt}]
& y,
\arrow[Rightarrow, from=U, to=D, "\alpha"]
\end{tikzcd}$$ which is a very simple $2$-category that does not contain any non-trivial composition, has non-degenerate simplices in each dimension.
The Duskin nerve of the free $2$-cell has precisely two non-degenerate simplices in each positive dimension.
This result was unexpected to us, and we were able to conjecture it in the first place only after having a computer produce all $n$-simplices of the Duskin nerve of the free $2$-cell for $n\le 6$. In order to prove the proposition, we developed a more general study of the Duskin nerve of $2$-categories of the form $\Sigma{\mathcal{D}}$, sometimes referred to as a *suspension $2$-categories*, of which the free $2$-cell is an example for ${\mathcal{D}}=[1]$.
The suspension $2$-category $\Sigma{\mathcal{D}}$ of a category ${\mathcal{D}}$, which can be pictured as $$\Sigma{\mathcal{D}}\ :=\ \ \ \
\begin{tikzpicture}[baseline=-5pt]
\draw (0,0) node[inner sep=0.2cm](a){} node(x){$x$};
\draw (2,0) node[inner sep=0.2cm](b){} node(y){$y$};
\draw[->] (a) edge[bend right] node[below](xy){${\mathcal{D}}$}(b);
\draw[->] (b) edge[bend right] node[above](xy){$\varnothing$}(a);
\draw[->] (a.180) arc (30:330:2mm)node[xshift=-0.8cm](xx){$[0]$};
\draw[->] (b.1) arc (-210:-510:2mm)node[xshift=0.8cm](yy){$[0],$};
\end{tikzpicture}$$ appears often in the literature as a special case of a simplicial suspension. For instance, the homwise nerve $N_*(\Sigma{\mathcal{D}})$ of the suspension $\Sigma{\mathcal{D}}$ is a simplicial category that agrees with what would be denoted as $U(N{\mathcal{D}})$ in [@bergner], as $S(N{\mathcal{D}})$ in [@Joyal2007], as $[1]_{N{\mathcal{D}}}$ in [@htt], and as ${\mathbbm{2}}[N{\mathcal{D}}]$ in [@RiehlVerityNcoh].
In \[section1\] we prove the following description for the Duskin nerve of suspension $2$-categories. Our methods were inspired by those used in [@BuckleyGarnerLackStreet], where Buckley, Garner, Lack and Street face a similar situation, showing that the monoidal nerve of a rather simple monoidal category is the highly non-trivial “Catalan simplicial set”.
An $n$-simplex of the Duskin nerve of the suspension $\Sigma{{\mathcal{D}}}$ can be uniquely described as a grid valued in ${\mathcal{D}}$ of the form $${
\small \begin{tikzcd} d_{0l} \arrow{d} \arrow{r} & d_{0 (l-1)} \arrow{d} \arrow{r} & \cdots \arrow{r} & d_{0 0} \arrow{d}\\
d_{1 l} \arrow{d}\arrow{r} & d_{1 (l-1)} \arrow{d} \arrow{r} & \cdots \arrow{r} & d_{1 0 } \arrow{d}\\
\vdots \arrow{d}& \vdots \arrow{d} & \ddots & \vdots \arrow{d} \\
d_{k l} \arrow{r} & d_{k (l-1)} \arrow{r} & \cdots \arrow{r} & d_{k 0},
\end{tikzcd}
}$$ for $k,l\ge-1$ and $k+l=n-1$, and the simplicial structure is understood as suitably removing or doubling rows or columns.
The proof of the theorem relies on a coskeletality argument, which hides the meaning of this correspondence for simplices in dimension higher than $4$. To address this, we devote \[section2\] to explaining how to convert a simplex of $N(\Sigma{\mathcal{D}})$ to a ${\mathcal{D}}$-valued grid and vice versa.
After having understood the Duskin nerve of suspension $2$-categories, we then study the Duskin nerve of $(r+1)$-point suspension $2$-categories $\Sigma[{\mathcal{D}}_1,\dots,{\mathcal{D}}_r]$, which are $2$-categories obtained by pasting together suspension $2$-categories of categories ${\mathcal{D}}_1\dots,{\mathcal{D}}_r$ along objects as in the following picture: $$\Sigma[{\mathcal{D}}_1,\dots,{\mathcal{D}}_{r}]\ :=
\begin{tikzpicture}[baseline=0pt]
\draw (0,0) node[inner sep=0.2cm](a){} node(x0){$x_0$};
\draw (2,0) node[inner sep=0.2cm](b){} node(x1){$x_1$};
\draw (4,0) node[inner sep=0.2cm](c){} node(x2){$x_2$};
\draw (6,0) node[inner sep=0.2cm](d){} node(dots){$\ldots$};
\draw (8,0) node[inner sep=0.2cm](e){} node(x_r){$x_r$.};
\draw[->] (a) to[bend right] node[below](xy1){${\mathcal{D}}_1$}(b);
\draw[->] (b) to[bend right] node[above](xy2){$\varnothing$}(a);
\draw[->] (b) to[bend right] node[below](xy3){${\mathcal{D}}_2$}(c);
\draw[->] (c) to[bend right] node[above](xy4){$\varnothing$}(b);
\draw[->] (c) to[bend right] node[below](xy5){${\mathcal{D}}_3$}($(d)+(-0.4, -0.1)$);
\draw[->] ($(d)+(-0.4, 0.1)$) to[bend right] node[above](xy6){$\varnothing$}(c);
\draw[->] ($(d)+(0.4, -0.1)$) to[bend right] node[below](xy7){${\mathcal{D}}_r$}(e);
\draw[->] (e) to[bend right] node[above](xy8){$\varnothing$}($(d)+(0.4, 0.1)$);
\end{tikzpicture}$$
This type of horizontal gluing of suspension $2$-categories appears e.g. in [@VerityComplicialAMS 11.5]. Motivating examples of $(r+1)$-point suspension $2$-categories are the $2$-categories that belong to Joyal’s disk category $\Theta_2$ from [@JoyalDisks], which are all $(r+1)$-point suspension $2$-categories of the form $\Sigma[[n_1],\dots,[n_r]]$ for $n_1,\dots,n_r\ge0$. An example would be the $2$-category $\Sigma[[2],[0],[1]]$, which is generated by the following data: $$\begin{tikzcd}[row sep=3.2cm, column sep=2.2cm]
x \arrow[r, bend left=50, "f", ""{name=U,inner sep=2pt,below}]
\arrow[r, "g"{near end, xshift=0.2cm}, ""{name=D,inner sep=2pt},""{name=M,inner sep=2pt, below}]
\arrow[r, bend right=50, "h"{below}, ""{name=DD,inner sep=2pt, xshift=0.05cm}]
& y\arrow[r, "l"]
& [-0.9cm]
z\arrow[r, bend left=50, "m", ""{name=U1,inner sep=2pt,below}]
\arrow[r, "k"{near end, xshift=0.2cm}, ""{name=D1,inner sep=2pt, xshift=0.05cm},""{name=M1,inner sep=2pt, below}]
& w.
\arrow[Rightarrow, from=U, to=D, "\alpha"]
\arrow[Rightarrow, from=M, to=DD, "\beta"{near start}]
\arrow[Rightarrow, from=U1, to=D1, "\gamma"]
\end{tikzcd}$$
We are able to describe the Duskin nerve of $(r+1)$-point suspension $2$-categories in \[section3\].
Let ${\mathcal{D}}_1,\dots,{\mathcal{D}}_r$ be given $1$-categories. An $n$-simplex of the Duskin nerve of the $(r+1)$-point suspension $\Sigma[{\mathcal{D}}_1,\dots,{\mathcal{D}}_r]$ can be uniquely described as a list of grids valued in ${\mathcal{D}}_i$ whose numbers of rows are suitably increasing.
The explicit description of the Duskin nerve of $2$-categories from this paper can then also be used to prove finer homotopical properties. For instance, in ongoing work, we use these results to show that the canonical inclusion $$N(\Sigma[1])\underset{N(\Sigma[0])}{\amalg}N(\Sigma[1]) \hookrightarrow N(\Sigma[2])$$ is a categorical equivalence.
Acknowledgements {#acknowledgements .unnumbered}
----------------
We would like to thank Clark Barwick, Andrea Gagna, Lennart Meier and Emily Riehl for helpful conversations.
The Duskin nerve of suspension $2$-categories - the theory {#section1}
==========================================================
\[descriptionDuskinnerve\]
We start by recalling the definition of the Duskin nerve of $2$-categories[^1].
The *Duskin nerve* $N({\mathcal{C}})$ of a $2$-category ${\mathcal{C}}$ is a $3$-coskeletal simplicial set in which
1. a $0$-simplex consists of an object of ${\mathcal{C}}$: $$x;$$
2. a $1$-simplex consists of a $1$-morphism of ${\mathcal{C}}$: $$\begin{tikzcd}
x \arrow[rr, "a"{below}]&& y;
\end{tikzcd}$$
3. a $2$-simplex consists of a $2$-cell of ${\mathcal{C}}$ of the form $c\Rightarrow b\circ a$: $$\begin{tikzcd}[baseline=(current bounding box.center)]
& y \arrow[rd, "b"] & \\ x \arrow[ru, "{a}"]
\arrow[rr, "c"{below}, ""{name=D,inner sep=1pt}]
&& z;
\arrow[Rightarrow, from=D,
to=1-2, shorten >= 0.1cm, shorten <= 0.1cm, ""]
\end{tikzcd}$$
4. a $3$-simplex consists of four $2$-cells of ${\mathcal{C}}$ that satisfy the following relation. $${ \def\tempa{d} \def\tempb{c} \def\tempc{e} \def\tempd{a} \def\tempe{b} \def\tempf{f} \def\tempg{} \def\temph{} \def\tempi{}}{ \begin{tikzcd}[column sep=1.3cm, row sep=0.4cm, baseline=(current bounding box.center), ampersand replacement=\&]
w \& z \arrow[l, "\tempc" swap] \&[-7mm] \&[-7mm] w \& z \arrow[l, "\tempc" swap]\\
{} \& \&[-7mm] {=} \&[-7mm] \& \\
x \arrow[r, "\tempa" swap]\arrow[uu, "\tempd", ""{name=a031,inner sep=2pt, swap} ] \arrow[uur, "\tempe"{near start, xshift=0.15cm}, ""{name=a02,inner sep=2pt, swap}]\& y \arrow[uu, "\tempb" swap]\&[-7mm] \&[-7mm] x \arrow[uu, "\tempd", ""{name=a032,inner sep=2pt, swap}]\arrow[r, "\tempa" swap]\& y\arrow[uu, "\tempb" swap] \arrow[uul, "\tempf"{near end, xshift=0.25cm}, ""{name=a13,inner sep=2pt, swap}]
\arrow[Rightarrow, from=a031, to=1-2, shorten >= 0.3cm, "\tempg" ]
\arrow[Rightarrow, from=a02, to=3-2, shorten >= 0.3cm, "\temph" swap ]
\arrow[Rightarrow, from=a032, to=3-5, shorten >= 0.3cm, "\tempi" swap ]
\arrow[Rightarrow, from=a13, to=1-5, shorten >= 0.3cm, "" ]
\end{tikzcd}
}$$
The following type of $2$-category is of interest in this paper.
\[suspension\] The *suspension* of a $1$-category ${\mathcal{D}}$ is the $2$-category $\Sigma{\mathcal{D}}$ with two objects $x,y$ and hom categories given by $$\operatorname{Map}_{\Sigma{\mathcal{D}}}(x,y)={\mathcal{D}},\ \operatorname{Map}_{\Sigma{\mathcal{D}}}(y,x)=[-1],\ \operatorname{Map}_{\Sigma{\mathcal{D}}}(x,x)=\operatorname{Map}_{\Sigma{\mathcal{D}}}(y,y)=[0].$$
We want to identify each $n$-simplex of the Duskin nerve of the suspension $2$-category $\Sigma{\mathcal{D}}$ with a functor $\sigma\colon [k]\times [l]^{\operatorname{op}} \to{\mathcal{D}}$, which is in turn completely described by $(k+1)\times(l+1)$ objects of ${\mathcal{D}}$ connected horizontally and vertically by morphisms of ${\mathcal{D}}$ $${
\scriptsize \begin{tikzcd} d_{0l} \arrow{d} \arrow{r} & d_{0 (l-1)} \arrow{d} \arrow{r} & \cdots \arrow{r} & d_{0 0} \arrow{d}\\
d_{1 l} \arrow{d}\arrow{r} & d_{1 (l-1)} \arrow{d} \arrow{r} & \cdots \arrow{r} & d_{1 0 } \arrow{d}\\
\vdots \arrow{d}& \vdots \arrow{d} & \ddots & \vdots \arrow{d} \\
d_{k l} \arrow{r} & d_{k (l-1)} \arrow{r} & \cdots \arrow{r} & d_{k 0},
\end{tikzcd}
}$$ such that all the resulting squares are all commutative. We call such a diagram a *matrix*[^2] valued in ${\mathcal{D}}$. We follow the convention that $[-1]=\varnothing$ is the empty category.
To this end, we first discuss how the collections of such morphisms assemble into a simplicial set.
\[matrixcoskeletal\] Let ${\mathcal{D}}$ be a category. The collection of ${\mathcal{D}}$-matrices $${\operatorname{Mat}_{n}({\mathcal{D}})}:=\{\sigma\colon [k]\times [l]^{\operatorname{op}} \to{\mathcal{D}}\ |\ k,l\ge-1,\ k+l=n-1\}$$ for $n\ge0$ defines a $3$-coskeletal simplicial set ${\operatorname{Mat}({\mathcal{D}})}$ with respect to the following simplicial structure. The faces and degeneracies of a ${\mathcal{D}}$-valued matrix $\sigma\colon [k]\times [l]^{\operatorname{op}} \to{\mathcal{D}}$ are given by $$d_i\sigma=
\left\{\begin{array}{lr}
\sigma (d^i\times\operatorname{id}_{[l]^{\operatorname{op}}}) \colon [k-1]\times [l]^{\operatorname{op}} \to {\mathcal{D}}&\mbox{ for } 0\leq i\leq k,\\
\sigma (\operatorname{id}_{[k]}\times d^{i-(k+1)})\colon [k]\times [l-1]^{\operatorname{op}} \to {\mathcal{D}}&\mbox{ for } k+1\leq i\leq n;
\end{array}\right.$$ $$s_i\sigma=
\left\{\begin{array}{lr}
\sigma (s^i\times\operatorname{id}_{[l]^{\operatorname{op}}}) \colon [k+1]\times [l]^{\operatorname{op}} \to {\mathcal{D}}&\mbox{ for } 0\leq i\leq k,\\
\sigma (\operatorname{id}_{[k]}\times s^{i-(k+1)})\colon [k]\times [l+1]^{\operatorname{op}} \to {\mathcal{D}}& \mbox{ for } k+1\leq i\leq n.
\end{array}\right.$$
Roughly speaking, in the simplicial set ${\operatorname{Mat}({\mathcal{D}})}$:
1. faces are given by removing precisely one row or one column;
2. degeneracies are given by doubling precisely one row or one column;
3. the non-degenerate simplices are the ones where no two rows and no two columns coincide.
The fact that ${\operatorname{Mat}({\mathcal{D}})}$ is indeed a simplicial set can be verified by means of a straightforward computation. For simplicity of exposition, we show $3$-coskeletality of ${\operatorname{Mat}({\mathcal{D}})}$ for the case of ${\mathcal{D}}$ being a poset; the general case only requires the treatement of a slightly larger distinction of cases.
Suppose we are given a collection of ${\mathcal{D}}$-valued matrices $\tau_i \colon [k-1]\times [n-1-k]^{\operatorname{op}}\to{\mathcal{D}}$ for $0\leq i \leq k$ and $\tau_i \colon [k]\times [(n-1-k)-1]^{\operatorname{op}} \to{\mathcal{D}}$ for $k+1\leq i\leq n$ satisfying the relation $d_i\tau_j=d_{j-1}\tau_i$ for all $0\leq i<j \leq n$ with $n\geq 4$; we then need to define a functor $\tau\colon [k]\times [n-1-k]^{\operatorname{op}}\to {\mathcal{D}}$ so that $d_i\tau=\tau_i$, and show its uniqueness. If $k=-1$ or $k=n$, then the uniqueness and existence are immediate; assume thus $0\le k\le n-1$. Since $k+(n-1-k)=n-1\geq 3$, either $k\geq 2$ or $(n-1-k) \geq 2$ or both. If $k\geq 2$, a map $[k]\times [n-1-k]^{\operatorname{op}} \to {\mathcal{D}}$ is the same data as a $k$-simplex in $N({\mathcal{D}}^{[n-1-k]^{\operatorname{op}}})$, which is a $1$-coskeletal simplicial set being the nerve of a poset, so it is completely determined by its boundary. A dual argument applies to the case $n-1-k\geq 2$.
As announced in Theorem A, we now identify the Duskin nerve of the suspension $2$-category $\Sigma{\mathcal{D}}$ with the simplicial set of ${\mathcal{D}}$-valued matrices.
\[simplicesasmatrices\] Let ${\mathcal{D}}$ be a $1$-category. There is an isomorphism of simplicial sets $$N(\Sigma{{\mathcal{D}}})\cong{\operatorname{Mat}({\mathcal{D}})}.$$
In particular, an $n$-simplex of the Duskin nerve of the suspension $\Sigma{{\mathcal{D}}}$ can be described uniquely as a functor $[k]\times [l]^{\operatorname{op}} \to {\mathcal{D}}$, with $k+l=n-1$ and $k,l\ge-1$.
We recall that the Duskin nerve of $\Sigma{{\mathcal{D}}}$ is $3$-coskeletal, and we showed in \[matrixcoskeletal\] that the set of ${\mathcal{D}}$-valued matrices also assembles into a $3$-coskeletal simplicial set. Therefore, to prove the theorem it is enough to identify the simplices of these two simplicial sets up to dimension $3$ compatibly with the simplicial structure.
We identify all simplices in dimension up to $3$ with ${\mathcal{D}}$-valued matrices as follows.
1. Any of the two objects $$x\quad\text{ and }\quad y$$ of $\Sigma{{\mathcal{D}}}$ defines a $0$-simplex of the Duskin nerve of $\Sigma{{\mathcal{D}}}$; we identify them with the unique functor $[0] \times [-1]^{\operatorname{op}}\to{\mathcal{D}}$ and the unique functor $[-1]\times [0]^{\operatorname{op}}\to{\mathcal{D}}$, respectively. Similarly, for all $n=1,2,3$ any of the two objects $x$ and $y$ of $\Sigma{{\mathcal{D}}}$ defines a unique degenerate $n$-simplex of the Duskin nerve; we identify them with the unique functor $[n]\times [-1]^{\operatorname{op}}\to{\mathcal{D}}$ and the unique functor $[-1]\times [n]^{\operatorname{op}}\to{\mathcal{D}}$, respectively.
2. Any object $a$ in ${\mathcal{D}}$ gives rise to a $1$-simplex in the Duskin nerve: $$\begin{tikzcd}
x \arrow[rr, "a"{below}]&& y
\end{tikzcd}$$ and all $1$-simplices of the Duskin nerve of $\Sigma{{\mathcal{D}}}$ that are not degeneracies of a $0$-simplex can uniquely be written in this form for some $a$ in ${\mathcal{D}}$. These $1$-simplices can be identified with the functor $[0]\times [0]^{\operatorname{op}}\to {\mathcal{D}}$ with image $$a.$$
3. Any morphism $\varphi\colon a\to b$ in ${\mathcal{D}}$ gives rise to two $2$-simplices in the Duskin nerve of $\Sigma{{\mathcal{D}}}$: $$\begin{tikzcd}[baseline=(current bounding box.center)]
& x \arrow[rd, "b"] & \\ x \arrow[ru, "{s_0x}"]
\arrow[rr, "a"{below}, ""{name=D,inner sep=1pt}]
&& y
\arrow[Rightarrow, from=D, to=1-2, shorten >= 0.1cm, shorten <= 0.1cm, "\varphi"]
\end{tikzcd}
\mbox{ and }
\begin{tikzcd}[baseline=(current bounding box.center)]
& y\arrow[rd, "{s_0y}"] & \\ x \arrow[ru, "b"]
\arrow[rr, "a"{below}, ""{name=T,inner sep=1pt}]
&& y.
\arrow[Rightarrow, from=T, to=1-2, shorten >= 0.1cm, shorten <= 0.1cm, "\varphi"]
\end{tikzcd}$$ Moreover, all $2$-simplices of the Duskin nerve of $\Sigma{{\mathcal{D}}}$ that are not degeneracies of a $0$-simplex can uniquely be written in one of these two forms for some $\varphi\colon a\to b$ in ${\mathcal{D}}$. These $2$-simplices can be identified with the functors $[1]\times [0]^{\operatorname{op}}\to{\mathcal{D}}$ and $[0]\times [1]^{\operatorname{op}}\to{\mathcal{D}}$ with image $$\begin{tikzcd}a \arrow[d,"\varphi"] \\
b
\end{tikzcd}
\quad\quad
\text{and}
\quad\quad
\begin{tikzcd}a\arrow[r,"\varphi"] & b.
\end{tikzcd}$$
4. Any commutative square $$\begin{tikzcd}a \arrow[d,"\psi" swap]\arrow[r,"\varphi"]&b\arrow[d,"\gamma"] \\
f \arrow[r,"\theta" swap]&c\\
\end{tikzcd}$$ in ${\mathcal{D}}$ gives rise to three $3$-simplices in the Duskin nerve of $\Sigma{\mathcal{D}}$: $${ \def\tempa{\operatorname{id}_x} \def\tempb{\operatorname{id}_x} \def\tempc{c} \def\tempd{a} \def\tempe{\operatorname{id}_x} \def\tempf{f} \def\tempg{\theta\psi} \def\temph{\operatorname{id}_x} \def\tempi{\psi}}{ \begin{tikzcd}[column sep=1.3cm, row sep=0.4cm, baseline=(current bounding box.center), ampersand replacement=\&]
y \& x \arrow[l, "\tempc" swap] \&[-7mm] \&[-7mm] y \& x \arrow[l, "\tempc" swap]\\
{} \& \&[-7mm] {=} \&[-7mm] \& \\
x \arrow[r, "\tempa" swap]\arrow[uu, "\tempd", ""{name=a031,inner sep=2pt, swap} ] \arrow[uur, "\tempe"{near start, xshift=0.15cm}, ""{name=a02,inner sep=2pt, swap}]\& x \arrow[uu, "\tempb" swap]\&[-7mm] \&[-7mm] x \arrow[uu, "\tempd", ""{name=a032,inner sep=2pt, swap}]\arrow[r, "\tempa" swap]\& x\arrow[uu, "\tempb" swap] \arrow[uul, "\tempf"{near end, xshift=0.25cm}, ""{name=a13,inner sep=2pt, swap}]
\arrow[Rightarrow, from=a031, to=1-2, shorten >= 0.3cm, "\tempg" ]
\arrow[Rightarrow, from=a02, to=3-2, shorten >= 0.3cm, "\temph" swap ]
\arrow[Rightarrow, from=a032, to=3-5, shorten >= 0.3cm, "\tempi" swap ]
\arrow[Rightarrow, from=a13, to=1-5, shorten >= 0.3cm, "\theta" ]
\end{tikzcd}
},$$ $${ \def\tempa{\operatorname{id}_x} \def\tempb{c} \def\tempc{\operatorname{id}_y} \def\tempd{a} \def\tempe{b} \def\tempf{f} \def\tempg{\varphi} \def\temph{\gamma} \def\tempi{\psi}}{ \begin{tikzcd}[column sep=1.3cm, row sep=0.4cm, baseline=(current bounding box.center), ampersand replacement=\&]
y \& y \arrow[l, "\tempc" swap] \&[-7mm] \&[-7mm] y \& y \arrow[l, "\tempc" swap]\\
{} \& \&[-7mm] {=} \&[-7mm] \& \\
x \arrow[r, "\tempa" swap]\arrow[uu, "\tempd", ""{name=a031,inner sep=2pt, swap} ] \arrow[uur, "\tempe"{near start, xshift=0.15cm}, ""{name=a02,inner sep=2pt, swap}]\& x \arrow[uu, "\tempb" swap]\&[-7mm] \&[-7mm] x \arrow[uu, "\tempd", ""{name=a032,inner sep=2pt, swap}]\arrow[r, "\tempa" swap]\& x\arrow[uu, "\tempb" swap] \arrow[uul, "\tempf"{near end, xshift=0.25cm}, ""{name=a13,inner sep=2pt, swap}]
\arrow[Rightarrow, from=a031, to=1-2, shorten >= 0.3cm, "\tempg" ]
\arrow[Rightarrow, from=a02, to=3-2, shorten >= 0.3cm, "\temph" swap ]
\arrow[Rightarrow, from=a032, to=3-5, shorten >= 0.3cm, "\tempi" swap ]
\arrow[Rightarrow, from=a13, to=1-5, shorten >= 0.3cm, "\theta" ]
\end{tikzcd}
},$$ $${ \def\tempa{c} \def\tempb{\operatorname{id}_y} \def\tempc{\operatorname{id}_y} \def\tempd{a} \def\tempe{b} \def\tempf{\operatorname{id}_y} \def\tempg{\varphi} \def\temph{\gamma} \def\tempi{\gamma\varphi}}{ \begin{tikzcd}[column sep=1.3cm, row sep=0.4cm, baseline=(current bounding box.center), ampersand replacement=\&]
y \& y \arrow[l, "\tempc" swap] \&[-7mm] \&[-7mm] y \& y \arrow[l, "\tempc" swap]\\
{} \& \&[-7mm] {=} \&[-7mm] \& \\
x \arrow[r, "\tempa" swap]\arrow[uu, "\tempd", ""{name=a031,inner sep=2pt, swap} ] \arrow[uur, "\tempe"{near start, xshift=0.15cm}, ""{name=a02,inner sep=2pt, swap}]\& y \arrow[uu, "\tempb" swap]\&[-7mm] \&[-7mm] x \arrow[uu, "\tempd", ""{name=a032,inner sep=2pt, swap}]\arrow[r, "\tempa" swap]\& y\arrow[uu, "\tempb" swap] \arrow[uul, "\tempf"{near end, xshift=0.25cm}, ""{name=a13,inner sep=2pt, swap}]
\arrow[Rightarrow, from=a031, to=1-2, shorten >= 0.3cm, "\tempg" ]
\arrow[Rightarrow, from=a02, to=3-2, shorten >= 0.3cm, "\temph" swap ]
\arrow[Rightarrow, from=a032, to=3-5, shorten >= 0.3cm, "\tempi" swap ]
\arrow[Rightarrow, from=a13, to=1-5, shorten >= 0.3cm, "\operatorname{id}_y" ]
\end{tikzcd}
}.$$ Moreover, it can be seen by direct inspection that all $3$-simplices of the Duskin nerve that are not degeneracies of a $0$-simplex can uniquely be written in one of these three forms for some commutative square in ${\mathcal{D}}$ as above. These $3$-simplices can be identified with the functors $[2]\times [0]^{\operatorname{op}} \to {\mathcal{D}}$, $[1]\times [1]^{\operatorname{op}} \to {\mathcal{D}}$ and $[0]\times [2]^{\operatorname{op}} \to {\mathcal{D}}$ displayed as $$\begin{tikzcd}a \arrow[d,"\psi"] \\
f \arrow[d,"\theta"] \\
c
\end{tikzcd}
\quad\quad
\text{and}
\quad\quad
\begin{tikzcd}a \arrow[d,"\psi" swap]\arrow[r,"\varphi"]&b\arrow[d,"\gamma"] \\
f \arrow[r,"\theta" swap]&c\\
\end{tikzcd}
\quad\quad
\text{and}
\quad\quad
\begin{tikzcd}a\arrow[r,"\varphi"] &b\arrow[r,"\gamma"]& c.
\end{tikzcd}$$
The given identification between simplices of the Duskin nerve in dimension up to $3$ and ${\mathcal{D}}$-valued matrices can be checked to be compatible with the simplicial structure, using the explicit formulas from \[matrixcoskeletal\].
The Duskin nerve of suspension $2$-categories - the practice
============================================================
The proof of \[simplicesasmatrices\] relies on the coskeletality of the simplicial sets $N(\Sigma{\mathcal{D}})$ and ${\operatorname{Mat}({\mathcal{D}})}$, and does not enlighten how the correspondence between ${\mathcal{D}}$-valued matrices $[k]\times[n-1-k]^{\operatorname{op}}\to{\mathcal{D}}$ and $n$-simplices in the Duskin nerve of $\Sigma{\mathcal{D}}$ really works for $n\ge4$. In this section we collect a few useful observations in this direction, and illustrate with an example how one can reconstruct a matrix from a simplex and vice versa.
Given the fact that $\Sigma{\mathcal{D}}$ has only two objects, for $n\ge2$ any $n$-simplex $\sigma$ in $N(\Sigma{\mathcal{D}})$ has exactly zero or one non-degenerate edges of the form $(k,k+1)$. More precisely,
- if the simplex $\sigma$ is the degeneracy of one of the $0$-simplices $x$ or $y$, each edge of $\sigma$ is degenerate at the same vertex $x$ or $y$. In this case, we read from the proof of \[simplicesasmatrices\] that the matrix corresponding to $\sigma$ is the unique functor $[n]\times [-1]^{\operatorname{op}}\to{\mathcal{D}}$ or the unique functor $[-1]\times [n]^{\operatorname{op}}\to{\mathcal{D}}$.
- if the simplex is not the degeneracy of a $0$-simplex, it has precisely one non-degenerate edge of the form $(k,k+1)$ for some $0\le k\le n-1$.
The fact that these two very different behaviours partition the simplices of $N(\Sigma{\mathcal{D}})$ is fundamental, and we therefore make the following definition.
For $n\ge1$ we say that an $n$-simplex $\sigma$ in $N(\Sigma{\mathcal{D}})$ is
- *maximally degenerate* if it is the degeneracy of one of the $0$-simplices $x$ or $y$.
- *of type $k$* for $0\le k\le n-1$ if it has one non-degenerate edge of the form $(k,k+1)$.
The $3$-simplex of $N(\Sigma{\mathcal{D}})$ $${ \def\tempa{\operatorname{id}_x} \def\tempb{c} \def\tempc{\operatorname{id}_y} \def\tempd{a} \def\tempe{b} \def\tempf{f} \def\tempg{\varphi} \def\temph{\gamma} \def\tempi{\psi}}{ \begin{tikzcd}[column sep=1.3cm, row sep=0.4cm, baseline=(current bounding box.center), ampersand replacement=\&]
y \& y \arrow[l, "\tempc" swap] \&[-7mm] \&[-7mm] y \& y \arrow[l, "\tempc" swap]\\
{} \& \&[-7mm] {=} \&[-7mm] \& \\
x \arrow[r, "\tempa" swap]\arrow[uu, "\tempd", ""{name=a031,inner sep=2pt, swap} ] \arrow[uur, "\tempe"{near start, xshift=0.15cm}, ""{name=a02,inner sep=2pt, swap}]\& x \arrow[uu, "\tempb" swap]\&[-7mm] \&[-7mm] x \arrow[uu, "\tempd", ""{name=a032,inner sep=2pt, swap}]\arrow[r, "\tempa" swap]\& x\arrow[uu, "\tempb" swap] \arrow[uul, "\tempf"{near end, xshift=0.25cm}, ""{name=a13,inner sep=2pt, swap}]
\arrow[Rightarrow, from=a031, to=1-2, shorten >= 0.3cm, "\tempg" ]
\arrow[Rightarrow, from=a02, to=3-2, shorten >= 0.3cm, "\temph" swap ]
\arrow[Rightarrow, from=a032, to=3-5, shorten >= 0.3cm, "\tempi" swap ]
\arrow[Rightarrow, from=a13, to=1-5, shorten >= 0.3cm, "\theta" ]
\end{tikzcd}
},$$ is of type $1$, whereas the $3$-simplex $${ \def\tempa{\operatorname{id}_x} \def\tempb{\operatorname{id}_x} \def\tempc{c} \def\tempd{a} \def\tempe{\operatorname{id}_x} \def\tempf{f} \def\tempg{\theta\psi} \def\temph{\operatorname{id}_x} \def\tempi{\psi}}{ \begin{tikzcd}[column sep=1.3cm, row sep=0.4cm, baseline=(current bounding box.center), ampersand replacement=\&]
y \& x \arrow[l, "\tempc" swap] \&[-7mm] \&[-7mm] y \& x \arrow[l, "\tempc" swap]\\
{} \& \&[-7mm] {=} \&[-7mm] \& \\
x \arrow[r, "\tempa" swap]\arrow[uu, "\tempd", ""{name=a031,inner sep=2pt, swap} ] \arrow[uur, "\tempe"{near start, xshift=0.15cm}, ""{name=a02,inner sep=2pt, swap}]\& x \arrow[uu, "\tempb" swap]\&[-7mm] \&[-7mm] x \arrow[uu, "\tempd", ""{name=a032,inner sep=2pt, swap}]\arrow[r, "\tempa" swap]\& x\arrow[uu, "\tempb" swap] \arrow[uul, "\tempf"{near end, xshift=0.25cm}, ""{name=a13,inner sep=2pt, swap}]
\arrow[Rightarrow, from=a031, to=1-2, shorten >= 0.3cm, "\tempg" ]
\arrow[Rightarrow, from=a02, to=3-2, shorten >= 0.3cm, "\temph" swap ]
\arrow[Rightarrow, from=a032, to=3-5, shorten >= 0.3cm, "\tempi" swap ]
\arrow[Rightarrow, from=a13, to=1-5, shorten >= 0.3cm, "\theta" ]
\end{tikzcd}
},$$ is of type $2$.
In particular, it is consistent to think of the maximally degenerate $n$-simplex of the $0$-simplex $y$ as the (unique) $n$-simplex *of type $-1$* and to the maximally degenerate $n$-simplex of the $0$-simplex $x$ as the (unique) $n$-simplex *of type $n$*. The following corollary relates the type $k$ of an $n$-simplex $\sigma$ to the size of the matrix corresponding to $\sigma$.
\[sizematrix\] Let ${\mathcal{D}}$ be a $1$-category, $n\ge1$ and $-1\le k\le n$. There is a bijective correspondence between the $n$-simplices of $N(\Sigma{\mathcal{D}})$ of type $k$, and the functors $[k]\times[n-1-k]^{\operatorname{op}}\to{\mathcal{D}}$.
The corollary is proven by induction on $n\ge1$. For $n=1,2$, the list of simplices of $n$-simplices of $N(\Sigma{\mathcal{D}})$ has been matched explicitly to a ${\mathcal{D}}$-valued matrix in the proof of \[simplicesasmatrices\], and the reader can see by direct inspection that the statement holds. Suppose now that we are given an $n$-simplex $\sigma$ of $N(\Sigma{\mathcal{D}})$ for $n>2$ and that $\sigma$ is of type $k$ for $-1\le k\le n$. We know that the $i$-th $(n-1)$-face of the $n$-simplex $\sigma$ is of type $k-1$ if $0\leq i \leq k$ and of type $k$ if $k+1 \leq i \leq n$. By induction hypothesis, the face $d_i\sigma$ therefore corresponds to a functor of the form $\tau_i \colon [k-1]\times [n-1-k]^{\operatorname{op}}\to{\mathcal{D}}$ for $0\leq i \leq k$ and $\tau_i \colon [k]\times [(n-1-k)-1]^{\operatorname{op}} \to{\mathcal{D}}$ for $k+1\le i\le n$. Recall that, for $n>2$, any $n$-simplex in $N(\Sigma{\mathcal{D}})$ or ${\operatorname{Mat}({\mathcal{D}})}$ is determined by its boundary. The only elements in ${\operatorname{Mat}({\mathcal{D}})}$ having such boundary are functors of the form $[k]\times[n-1-k]^{\operatorname{op}}\to{\mathcal{D}}$, as desired. A similar argument shows that the $n$-simplex corresponding to a functor $M\colon [k]\times [n-1-k]^{\operatorname{op}} \to{\mathcal{D}}$ must be of type $k$.
The next corollary describes a correspondence between *triangulations* labeled in the $2$-faces of a given simplex of the Duskin nerve of $\Sigma{\mathcal{D}}$, and *monotone paths* inside the corresponding ${\mathcal{D}}$-valued matrix.
For triangulations, we make use of the formalism from [@DK Ex. 2.2.15]. A *triangulation* ${\mathcal{T}}$ of a convex $(n+1)$-gon with cyclically numbered vertices only contains triangles with vertices being vertices of the original polygon. To any such triangulation ${\mathcal{T}}$, we can associate a simplicial subset $\Delta[{\mathcal{T}}] \subset \operatorname{sk}_2\Delta[n]\subset \Delta[n]$ by choosing the $2$-faces corresponding to the triangles in the triangulation.
Let $n\ge2$. Given an $n$-simplex $\sigma$ of $N(\Sigma{\mathcal{D}})$ of type $k$ for $0\le k\le n-1$, a *$\sigma$-labeled triangulation* consists of a triangulation ${\mathcal{T}}$ of an $(n+1)$-gon that does not have any triangle completely contained neither in $\{0,\ldots, k\}$ nor in $\{k+1,\ldots, n\}$, together with the composite $$\Delta[{\mathcal{T}}] \hookrightarrow \Delta[n] \xrightarrow{\sigma} N(\Sigma{\mathcal{D}}).$$
In particular, the definition requires a compatibility between the triangulation ${\mathcal{T}}$ and the simplex $\sigma$, namely that the $2$-simplices in the image of the composite $\Delta[{\mathcal{T}}]\to N(\Sigma{\mathcal{D}})$ above are not degeneracies of a $0$-simplex.
Given $\sigma$ a $3$-simplex of $N(\Sigma{\mathcal{D}})$ of type $1$ given by $${ \def\tempa{\operatorname{id}_x} \def\tempb{c} \def\tempc{\operatorname{id}_y} \def\tempd{a} \def\tempe{b} \def\tempf{f} \def\tempg{\varphi} \def\temph{\gamma} \def\tempi{\psi}}{ \begin{tikzcd}[column sep=1.3cm, row sep=0.4cm, baseline=(current bounding box.center), ampersand replacement=\&]
y \& y \arrow[l, "\tempc" swap] \&[-7mm] \&[-7mm] y \& y \arrow[l, "\tempc" swap]\\
{} \& \&[-7mm] {=} \&[-7mm] \& \\
x \arrow[r, "\tempa" swap]\arrow[uu, "\tempd", ""{name=a031,inner sep=2pt, swap} ] \arrow[uur, "\tempe"{near start, xshift=0.15cm}, ""{name=a02,inner sep=2pt, swap}]\& x \arrow[uu, "\tempb" swap]\&[-7mm] \&[-7mm] x \arrow[uu, "\tempd", ""{name=a032,inner sep=2pt, swap}]\arrow[r, "\tempa" swap]\& x\arrow[uu, "\tempb" swap] \arrow[uul, "\tempf"{near end, xshift=0.25cm}, ""{name=a13,inner sep=2pt, swap}]
\arrow[Rightarrow, from=a031, to=1-2, shorten >= 0.3cm, "\tempg" ]
\arrow[Rightarrow, from=a02, to=3-2, shorten >= 0.3cm, "\temph" swap ]
\arrow[Rightarrow, from=a032, to=3-5, shorten >= 0.3cm, "\tempi" swap ]
\arrow[Rightarrow, from=a13, to=1-5, shorten >= 0.3cm, "\theta" ]
\end{tikzcd}
},$$ an example of a $\sigma$-labeled triangulation is $$\begin{tikzcd}[column sep=1.3cm, row sep=0.4cm, baseline=(current bounding box.center), ampersand replacement=\&]
y \& y \arrow[l, "\operatorname{id}_y" swap] \\
{} \& \\
x \arrow[r, "\operatorname{id}_x" swap]\arrow[uu, "a", ""{name=a031,inner sep=2pt, swap} ] \arrow[uur, "b"{near start, xshift=0.15cm}, ""{name=a02,inner sep=2pt, swap}]\& x \arrow[uu, "c" swap]
\arrow[Rightarrow, from=a031, to=1-2, shorten >= 0.3cm, "\varphi" ]
\arrow[Rightarrow, from=a02, to=3-2, shorten >= 0.3cm, "\gamma" swap ]
\end{tikzcd}$$
Recall e.g. from [@VerityComplicialI Def. 65] that a *shuffle* of $\Delta[k]\times\Delta[l]$ for $k,l\ge0$ is a non-degenerate $(k+l)$-simplex of $\Delta[k]\times\Delta[l]$. An easy and useful characterization of these is that they are precisely the functors[^3] $$S:=(\alpha,\beta)\colon[k+l]\to[k]\times[l]^{\operatorname{op}}$$ that satisfy the *ordinate summation property*: for all $0\le i\le k+l$ $$\alpha(i) + l-\beta(i) = i .$$
Given a ${\mathcal{D}}$-valued matrix $M\colon[k]\times[n-1-k]^{\operatorname{op}}\to{\mathcal{D}}$ with $0\le k\le n-1$, a *monotone path inside the matrix $M$* consists of a shuffle $S\colon[n-1]\to[k]\times[n-1-k]^{\operatorname{op}}$, together with the restriction $$[n-1]\stackrel{S}{\longrightarrow}[k]\times[n-1-k]^{\operatorname{op}}\stackrel{M}{\longrightarrow}{\mathcal{D}}$$ of $M$ along $S$.
If $M\colon[1]\times[1]^{\operatorname{op}}\to{\mathcal{D}}$ is a functor given by $$\begin{tikzcd}a \arrow[d,"\psi" swap]\arrow[r,"\varphi"]&b\arrow[d,"\gamma"] \\
f \arrow[r,"\theta" swap]&c
\end{tikzcd}$$ a monotone path inside $M$ is for instance $$\begin{tikzcd}a\arrow[r,"\varphi"]&b\arrow[d,"\gamma"] \\
&c.\\
\end{tikzcd}$$
Let ${\mathcal{D}}$ be a $1$-category, $n\ge2$, $\sigma$ an $n$-simplex of $N(\Sigma{\mathcal{D}})$ of type $k$ for $0\le k\le n-1$, and $M_{\sigma}\colon[k]\times[n-1-k]^{\operatorname{op}}\to{\mathcal{D}}$ the corresponding ${\mathcal{D}}$-valued matrix according to \[simplicesasmatrices\]. There is a bijective correspondence between $\sigma$-labeled triangulations $\Delta[{\mathcal{T}}]\to N(\Sigma{\mathcal{D}})$ and monotone paths $P\colon[n-1]\to{\mathcal{D}}$ inside $M_{\sigma}\colon[k]\times[n-1-k]^{\operatorname{op}}\to{\mathcal{D}}$.
The corollary is a direct consequence of \[simplicesasmatrices\] along with the following combinatorial fact[^4].
Let $n\ge2$ and let $0\leq k \le n-1$. Then there is a bijective correspondence between triangulations $\Delta[{\mathcal{T}}]\to\Delta[n]$ of an $(n+1)$-gon which do not have a triangle completely contained neither in $\{0,\ldots, k\}$ nor in $\{k+1,\ldots, n\}$ and shuffles $S\colon [n-1]\to [k]\times[n-1-k]^{\operatorname{op}}$.
The lemma is proven by induction on $n\ge2$. If $n=2$ there is a unique triangulation of the $(2+1)$-gon given by $$\begin{tikzcd}[baseline=(current bounding box.center)]
& 1 \arrow[rd] & \\
0 \arrow[ru]
\arrow[rr]
&& 2
\end{tikzcd}$$ both when $k=0$ or $k=1$. On the other side, there is a unique shuffle given by $[1]\to [0]\times [1]^{\operatorname{op}}$ for $k=0$ and a unique shuffle $[1]\to [1]\times [0]^{\operatorname{op}}$ for $k=1$.
If $n>2$, we first show that for a given triangulation as above the edge $(0,n)$ is contained exactly in one triangle that is of the form $(0,n-1,n)$ or $(0,1,n)$ . To see this, assume otherwise that $(0,n)$ is contained in the triangle $(0,p,n)$ for some $1<p<n-1$. We only consider $p\leq k$, the other case being symmetric. Then the triangulation of the $(n+1)$-gon we started with induces a triangulation of the $(p+1)$-gon with vertices $0, 1, \ldots, p$, since $(0,p)$ is one of the edges in the triangulation. But we assumed that no triangles include only vertices in $0, 1, \ldots, k$, leading to a contradiction.
It then follows that the given triangulation of the $(n+1)$-gon includes exactly one of the triangles $(0,n-1,n)$ and $(0,1,n)$, and is completely and uniquely described by such a triangle and the triangulation of the remaining $n$-gon. By induction hypothesis, this corresponds to a shuffle of the form $[n-1-1]\to [k]\times [n-2-k]^{\operatorname{op}}$ in the first case and of the form $[n-1-1]\to [k-1]\times [n-1-k]^{\operatorname{op}}$ in the second case, together with an extra arrow that can be connected to this shuffle (horizontally in the first case and vertically in the second case). By connecting these two pieces together we obtain a shuffle of the form $[n-1]\to [k]\times[n-1-k]^{\operatorname{op}}$, and all such shuffles arise in this way.
We illustrate with an example how the proposition can be used to write down the matrix associated to a simplex and vice versa. The idea is that, given a triangulation labeled in a simplex, each simplex with a degenerate $2$-nd face contributes as a vertical step in the corresponding path and each $2$-simplex with a degenerate $0$-th face contributes as a horizontal step.
Let ${\mathcal{P}}$ be a poset. Consider the $4$-simplex $\sigma$ of type $1$ of the Duskin nerve of $\Sigma{\mathcal{P}}$ determined by the following $2$-skeleton $$\begin{tikzpicture}[scale=.9, font=\scriptsize]
\def\l{1.8cm}
\def\vertexa{x}\def\vertexb{x}\def\vertexc{y}\def\vertexd{y}\def\vertexe{y}\def\edgeab{}\def\edgebc{ p_{10}}\def\edgecd{}\def\edgede{}\def\edgeae{p_{02}}\def\edgebe{p_{12}}\def\edgebd{p_{11}}\def\edgead{p_{01}}\def\edgece{}\def\edgeac{p_{00}}\def\triangleabe{}\def\trianglebde{}\def\trianglebcd{}\def\triangleade{}\def\triangleabd{}\def\triangleacd{}\def\triangleabc{}\def\triangleace{}\def\trianglecde{}\def\trianglebce{}
\begin{scope} \draw[fill] (0,0) node (b0){$\vertexa$};
\draw[fill] (\l,0) node (b1){$\vertexb$};
\draw[fill] (1.5*\l,0.75*\l) node (b2){$\vertexc$};
\draw[fill] (0.5*\l,1.5*\l) node (b3){$\vertexd$};
\draw[fill] (-0.5*\l,0.75*\l) node (b4){$\vertexe$};
\draw[<-] (b1)--node[below]{$\edgeab$}(b0);
\draw[->] (b1)--node[below,xshift=0.1cm]{$\edgebc$}(b2);
\draw[->] (b2)--node[above]{$\edgecd$}(b3);
\draw[->] (b3)--node[above]{$\edgede$}(b4);
\draw[<-] (b4)--node[left](A2){}node[left](A3){$\edgeae$}(b0);
\draw[->] (b1)--node[right](A4){}node[below,xshift=-0.06cm]{$\edgebd$}(b3);
\draw[<-] (b3)--node[left](A1){}node[right, yshift=0.1cm]{$\edgead$}(b0);
\draw[twoarrowlonger] (A1)--node[below]{$\triangleabd$}(b1);
\draw[twoarrowlonger] (A2)--node[below, yshift=-0.2cm]{$\triangleade$}(b3);
\draw[twoarrowlonger] (A4)--node[below]{$\trianglebcd$}(b2);
\end{scope}
\begin{scope}[xshift=2.5*\l] \draw[fill] (0,0) node (b0){$\vertexa$};
\draw[fill] (\l,0) node (b1){$\vertexb$};
\draw[fill] (1.5*\l,0.75*\l) node (b2){$\vertexc$};
\draw[fill] (0.5*\l,1.5*\l) node (b3){$\vertexd$};
\draw[fill] (-0.5*\l,0.75*\l) node (b4){$\vertexe$};
\draw[<-] (b1)--node[below]{$\edgeab$}(b0);
\draw[->] (b1)--node[below,xshift=0.1cm]{$\edgebc$}(b2);
\draw[->] (b2)--node[below](B2){}node[above]{$\edgecd$}(b3);
\draw[->] (b3)--node[above]{$\edgede$}(b4);
\draw[<-] (b4)--node[left](B4){}node[left](A3){$\edgeae$}(b0);
\draw[->] (b0)--node[left, xshift=0.2cm, yshift=-0.1cm](B1){}node[above, near start]{$\edgeac$}(b2);
\draw[<-] (b3)--node[below](B3){}node[right, yshift=0.1cm]{$\edgead$}(b0);
\draw[twoarrowlonger] ($(B3)+(0,-0.2cm)$)--node[above]{$\triangleacd$}(b2);
\draw[twoarrowlonger] ($(B1)+(-0.1cm, 0.1cm)$)--node[above, xshift=0.2cm]{$\triangleabc$}(b1);
\draw[twoarrowlonger] (B4)--node[below, yshift=-0.2cm]{$\triangleade$}(b3);
\end{scope}
\begin{scope}[xshift=4*\l, yshift=-1.5*\l]
\draw[fill] (0,0) node (b0){$\vertexa$};
\draw[fill] (\l,0) node (b1){$\vertexb$};
\draw[fill] (1.5*\l,0.75*\l) node (b2){$\vertexc$};
\draw[fill] (0.5*\l,1.5*\l) node (b3){$\vertexd$};
\draw[fill] (-0.5*\l,0.75*\l) node (b4){$\vertexe$};
\draw[<-] (b1)--node[below]{$\edgeab$}(b0);
\draw[->] (b1)--node[below,xshift=0.1cm]{$\edgebc$}(b2);
\draw[->] (b2)--node[above]{$\edgecd$}(b3);
\draw[->] (b3)--node[above]{$\edgede$}(b4);
\draw[<-] (b4)--node[left](C2){}node[left](C3){$\edgeae$}(b0);
\draw[->] (b0)--node[left, xshift=0.2cm, yshift=-0.1cm](C1){}node[above, near start]{$\edgeac$}(b2);
\draw[->] (b2)--node[below]{$\edgece$}node[above](C4){}(b4);
\draw[twoarrowlonger] ($(C1)+(-0.1cm, 0.1cm)$)--node[above, xshift=0.2cm]{$\triangleabc$}(b1);
\draw[twoarrowlonger] (C3)--node[below, near start]{$\triangleace$}($(b2)+(-0.5cm,-0.15cm)$);
\draw[twoarrowlonger] (C4)--node[left]{$\trianglecde$}(b3);
\end{scope}
\begin{scope}[xshift=1.25*\l, yshift=-2*\l]
\draw[fill] (0,0) node (b0){$\vertexa$};
\draw[fill] (\l,0) node (b1){$\vertexb$};
\draw[fill] (1.5*\l,0.75*\l) node (b2){$\vertexc$};
\draw[fill] (0.5*\l,1.5*\l) node (b3){$\vertexd$};
\draw[fill] (-0.5*\l,0.75*\l) node (b4){$\vertexe$};
\draw[<-] (b1)--node[below]{$\edgeab$}(b0);
\draw[->] (b1)--node[below,xshift=0.1cm]{$\edgebc$}(b2);
\draw[->] (b2)--node[above]{$\edgecd$}(b3);
\draw[->] (b3)--node[above]{$\edgede$}(b4);
\draw[<-] (b4)--node[left](D2){}node[left](D3){$\edgeae$}(b0);
\draw[->] (b2)--node[below]{$\edgece$}node[above](D4){}(b4);
\draw[->] (b1)--node[above]{$\edgebe$}node[above](D1){}(b4);
\draw[twoarrowlonger] (D3)--node[below, xshift=-0.2cm]{$\triangleabe$}($(b1)+(-0.5cm,0.15cm)$);
\draw[twoarrowlonger] (D4)--node[left]{$\trianglecde$}(b3);
\draw[twoarrowlonger] (D1)--node[below]{$\trianglebce$}($(b2)+(-0.5cm,-0.15cm)$);
\end{scope}
\begin{scope}[xshift=-1.75*\l, yshift=-1.5*\l]
\draw[fill] (0,0) node (b0){$\vertexa$};
\draw[fill] (\l,0) node (b1){$\vertexb$};
\draw[fill] (1.5*\l,0.75*\l) node (b2){$\vertexc$};
\draw[fill] (0.5*\l,1.5*\l) node (b3){$\vertexd$};
\draw[fill] (-0.5*\l,0.75*\l) node (b4){$\vertexe$};
\draw[<-] (b1)--node[below]{$\edgeab$}(b0);
\draw[->] (b1)--node[below,xshift=0.1cm]{$\edgebc$}(b2);
\draw[->] (b2)--node[above]{$\edgecd$}(b3);
\draw[->] (b3)--node[above]{$\edgede$}(b4);
\draw[<-] (b4)--node[left](E2){}node[left](E3){$\edgeae$}(b0);
\draw[->] (b1)--node[right](E4){}node[below,xshift=-0.06cm]{$\edgebd$}(b3);
\draw[->] (b1)--node[above](E1){$\edgebe$}(b4);
\draw[twoarrowlonger] (E3)--node[below, xshift=-0.2cm]{$\triangleabe$}($(b1)+(-0.5cm,0.15cm)$);
\draw[twoarrowlonger] (E4)--node[below]{$\trianglebcd$}(b2);
\draw[twoarrowlonger] (E1)--node[left]{$\trianglebde$}(b3);
\end{scope}
\end{tikzpicture}$$ where $p_{ij}$ belongs to ${\mathcal{P}}$ and let’s determine the ${\mathcal{P}}$-valued matrix $M_{\sigma}$ that corresponds to it according to \[simplicesasmatrices\].
Given that the edge $(1,2)$ of $\sigma$ is non-degenerate, the given $4$-simplex is of type $1$ and we can use \[sizematrix\] to assert that the matrix $M_{\sigma}$ has to be of the form $$M_{\sigma}\colon[1]\times[4-1-1]^{\operatorname{op}}=[1]\times[2]^{\operatorname{op}}\to{\mathcal{P}}.$$ The $\sigma$-triangulation $$\begin{tikzpicture}[scale=1.2, font=\scriptsize]
\def\l{1.8cm}
\def\vertexa{x}\def\vertexb{x}\def\vertexc{y}\def\vertexd{y}\def\vertexe{y}\def\edgeab{}\def\edgebc{p_{10}}\def\edgecd{}\def\edgede{}\def\edgeae{p_{02}}\def\edgebe{p_{12}}\def\edgebd{p_{11}}\def\edgead{p_{01}}\def\edgece{}\def\edgeac{p_{00}}\def\triangleabe{}\def\trianglebde{}\def\trianglebcd{}\def\triangleade{}\def\triangleabd{}\def\triangleacd{}\def\triangleabc{}\def\triangleace{}\def\trianglecde{}\def\trianglebce{} \begin{scope}[xshift=-1.75*\l, yshift=-1.5*\l]
\draw[fill] (0,0) node (b0){$\vertexa$};
\draw[fill] (\l,0) node (b1){$\vertexb$};
\draw[fill] (1.5*\l,0.75*\l) node (b2){$\vertexc$};
\draw[fill] (0.5*\l,1.5*\l) node (b3){$\vertexd$};
\draw[fill] (-0.5*\l,0.75*\l) node (b4){$\vertexe$};
\draw[<-] (b1)--node[below]{$\edgeab$}(b0);
\draw[->] (b1)--node[below, xshift=0.1cm]{$\edgebc$}(b2);
\draw[->] (b2)--node[above]{$\edgecd$}(b3);
\draw[->] (b3)--node[above]{$\edgede$}(b4);
\draw[<-] (b4)--node[left](E2){}node[left](E3){$\edgeae$}(b0);
\draw[->] (b1)--node[right](E4){}node[below,xshift=-0.06cm]{$\edgebd$}(b3);
\draw[->] (b1)--node[above](E1){$\edgebe$}(b4);
\draw[twoarrowlonger] (E3)--node[below, xshift=-0.2cm]{$\triangleabe$}($(b1)+(-0.5cm,0.15cm)$);
\draw[twoarrowlonger] (E4)--node[below]{$\trianglebcd$}(b2);
\draw[twoarrowlonger] (E1)--node[left]{$\trianglebde$}(b3);
\end{scope}
\end{tikzpicture}$$ corresponds to the monotone path in $M_{\sigma}$ that covers fully the left column and the bottom row and the $1$-st row, and is as follows: $${
\scriptsize \begin{tikzcd} p_{02} \arrow{d} & \\
p_{1 2} \arrow{r} & p_{11}\arrow{r} & p_{10}.
\end{tikzcd}
}$$ The $\sigma$-triangulation $$\begin{tikzpicture}[scale=1.2, font=\scriptsize]
\def\l{1.8cm}
\def\vertexa{x}\def\vertexb{x}\def\vertexc{y}\def\vertexd{y}\def\vertexe{y}\def\edgeab{}\def\edgebc{p_{10}}\def\edgecd{}\def\edgede{}\def\edgeae{p_{02}}\def\edgebe{p_{12}}\def\edgebd{p_{11}}\def\edgead{p_{01}}\def\edgece{}\def\edgeac{p_{00}}\def\triangleabe{}\def\trianglebde{}\def\trianglebcd{}\def\triangleade{}\def\triangleabd{}\def\triangleacd{}\def\triangleabc{}\def\triangleace{}\def\trianglecde{}\def\trianglebce{} \begin{scope} \draw[fill] (0,0) node (b0){$\vertexa$};
\draw[fill] (\l,0) node (b1){$\vertexb$};
\draw[fill] (1.5*\l,0.75*\l) node (b2){$\vertexc$};
\draw[fill] (0.5*\l,1.5*\l) node (b3){$\vertexd$};
\draw[fill] (-0.5*\l,0.75*\l) node (b4){$\vertexe$};
\draw[<-] (b1)--node[below]{$\edgeab$}(b0);
\draw[->] (b1)--node[below, xshift=0.1cm]{$\edgebc$}(b2);
\draw[->] (b2)--node[above]{$\edgecd$}(b3);
\draw[->] (b3)--node[above]{$\edgede$}(b4);
\draw[<-] (b4)--node[left](A2){}node[left](A3){$\edgeae$}(b0);
\draw[->] (b1)--node[right](A4){}node[below, xshift=-0.06cm]{$\edgebd$}(b3);
\draw[<-] (b3)--node[left](A1){}node[right, yshift=0.1cm]{$\edgead$}(b0);
\draw[twoarrowlonger] (A1)--node[below]{$\triangleabd$}(b1);
\draw[twoarrowlonger] (A2)--node[below, yshift=-0.2cm]{$\triangleade$}(b3);
\draw[twoarrowlonger] (A4)--node[below]{$\trianglebcd$}(b2);
\end{scope}
\end{tikzpicture}$$ corresponds to the monotone path in $M_{\sigma}$ that goes through the $1$-st column, and is as follows: $${
\scriptsize \begin{tikzcd} p_{02} \arrow{r}&p_{01}\arrow{d}& \\
& p_{11}\arrow{r} & p_{10}.
\end{tikzcd}
}$$ The $\sigma$-triangulation $$\begin{tikzpicture}[scale=1.2, font=\scriptsize]
\def\l{1.8cm}
\def\vertexa{x}\def\vertexb{x}\def\vertexc{y}\def\vertexd{y}\def\vertexe{y}\def\edgeab{}\def\edgebc{p_{10}}\def\edgecd{}\def\edgede{}\def\edgeae{p_{02}}\def\edgebe{p_{12}}\def\edgebd{p_{11}}\def\edgead{p_{01}}\def\edgece{}\def\edgeac{p_{00}}\def\triangleabe{}\def\trianglebde{}\def\trianglebcd{}\def\triangleade{}\def\triangleabd{}\def\triangleacd{}\def\triangleabc{}\def\triangleace{}\def\trianglecde{}\def\trianglebce{} \begin{scope}[xshift=2.5*\l] \draw[fill] (0,0) node (b0){$\vertexa$};
\draw[fill] (\l,0) node (b1){$\vertexb$};
\draw[fill] (1.5*\l,0.75*\l) node (b2){$\vertexc$};
\draw[fill] (0.5*\l,1.5*\l) node (b3){$\vertexd$};
\draw[fill] (-0.5*\l,0.75*\l) node (b4){$\vertexe$};
\draw[<-] (b1)--node[below]{$\edgeab$}(b0);
\draw[->] (b1)--node[below, xshift=0.1cm]{$\edgebc$}(b2);
\draw[->] (b2)--node[below](B2){}node[above]{$\edgecd$}(b3);
\draw[->] (b3)--node[above]{$\edgede$}(b4);
\draw[<-] (b4)--node[left](B4){}node[left](A3){$\edgeae$}(b0);
\draw[->] (b0)--node[left, xshift=0.2cm, yshift=-0.1cm](B1){}node[above, near start]{$\edgeac$}(b2);
\draw[<-] (b3)--node[below](B3){}node[right, yshift=0.1cm]{$\edgead$}(b0);
\draw[twoarrowlonger] ($(B3)+(0,-0.2cm)$)--node[above]{$\triangleacd$}(b2);
\draw[twoarrowlonger] ($(B1)+(-0.1cm, 0.1cm)$)--node[above, xshift=0.2cm]{$\triangleabc$}(b1);
\draw[twoarrowlonger] (B4)--node[below, yshift=-0.2cm]{$\triangleade$}(b3);
\end{scope}
\end{tikzpicture}$$ corresponds to the monotone path in $M_{\sigma}$ that covers fully the $0$-th row and the last column, and is as follows: $${
\scriptsize \begin{tikzcd} p_{02} \arrow[r] & p_{01} \arrow[r]& p_{00}\arrow[d]\\
& & p_{10}.
\end{tikzcd}
}$$
We conclude that $M_{\sigma}$ is the functor $[1]\times [2]^{\operatorname{op}}\to{\mathcal{P}}$ given by $${
\scriptsize \begin{tikzcd} p_{02} \arrow{d} \arrow{r}&p_{01}\arrow{r}\arrow{d}&p_{00}\arrow{d} \\
p_{1 2} \arrow{r} & p_{11}\arrow{r} & p_{10}.
\end{tikzcd}
}$$
The Duskin nerve of the free $2$-cell {#section2}
=====================================
As an instance of Theorem A, we obtain a full description of the non-degenerate simplices of the Duskin nerve of the free $2$-cell $$\begin{tikzcd}[row sep=3cm, column sep=2cm]
x \arrow[r, bend left, "f_0", ""{name=U,inner sep=2pt,below}]
\arrow[r, bend right, "f_1"{below}, ""{name=D,inner sep=2pt}]
& y,
\arrow[Rightarrow, from=U, to=D, "\alpha"]
\end{tikzcd}$$ it being the suspension of the $1$-category $[1]$.
\[intrinsicdescription\] In dimension $n$, the Duskin nerve of the free $2$-cell has precisely two non-degenerate simplices $\sigma_{n}$ and $\sigma_n'$. More precisely, $\sigma_0:=y$ and $\sigma_0':=x$ are the two $0$-simplices of the Duskin nerve, $\sigma_1:=1\colon x\to y$ and $\sigma_1':=0\colon x\to y$ are the two $1$-simplices of the Duskin nerve, and for $n>1$ the $n$-simplices $\sigma_n$ and $\sigma_n'$ are described as follows.
- if $n=2m$, the non-degenerate $2m$-simplices $\sigma_{2m}$ and $\sigma_{2m}'$ are uniquely determined by the relations $$d_i\sigma_{2m}=\left\{
\begin{array}{lr}s_{m-1+i}\sigma_{2m-2}&\mbox{ for }0 \leq i \leq m-1 \\
\sigma'_{2m-1}&\mbox{ for }i=m\\
s_{i-m-1}\sigma_{2m-2}&\mbox{ for }m+1\leq i\leq 2m-1\\
\sigma_{2m-1}&\mbox{ for }i=2m
\end{array}\right.$$ $$d_i\sigma_{2m}'=\left\{
\begin{array}{lr}
\sigma_{2m-1}&\mbox{ for }i=0\\
s_{m-1+i}\sigma_{2m-2}'&\mbox{ for }1 \leq i \leq m-1 \\
\sigma_{2m-1}'&\mbox{ for }i=m\\
s_{i-m-1}\sigma_{2m-2}'&\mbox{ for }m+1\leq i\leq 2m;
\end{array}
\right.$$
- if $n=2m+1$, the non-degenerate $(2m+1)$-simplices $\sigma_{2m+1}$ and $\sigma_{2m+1}'$ are uniquely determined by the relations $$d_i\sigma_{2m+1}=\left\{\begin{array}{lr} s_{m+i}\sigma_{2m-1},&\mbox{ for }0 \leq i \leq m-1\\
\sigma_{2m},&\mbox{ for }i=m\\
\sigma_{2m}',&\mbox{ for }i=m+1\\
s_{i-m-2}\sigma_{2m-1},&\mbox{ for }m+2\leq i\leq 2m+1
\end{array}
\right.$$ $$d_i\sigma_{2m+1}'=\left\{\begin{array}{lr} \sigma_{2m},&\mbox{ for }i=0\\
s_{m-1+i}\sigma_{2m-1}',&\mbox{ for }1 \leq i \leq m \\
s_{i-m-1}\sigma_{2m-1}',&\mbox{ for }m+1\leq i\leq 2m\\
\sigma_{2m}',&\mbox{ for }i=2m+1.
\end{array}
\right.$$
By \[simplicesasmatrices\] we know that simplices of the Duskin nerve of the free $2$-cell can be enumerated by means of functors $[k]\times [l]^{\operatorname{op}}\to[1]$, with $k,l\ge-1$ and $k+l=n-1$. Moreover, an $n$-simplex is non-degenerate if and only if all rows are different and all columns are different, meaning that $k+1\leq l+2$ and $l+1\leq k+2$. Since we have $k+l=n-1$, we obtain that $n-2\leq 2k \leq n$.
According to this analysis, in dimension $n$ the Duskin nerve of the free $2$-cells has precisely two non-degenerate simplices $\sigma_{n}$ and $\sigma_n'$.
- If $n=2m$, the non-degenerate simplices $\sigma_{2m}$ and $\sigma'_{2m}$correspond to the functors $$M_{2m}\colon[m-1]\times [m]^{\operatorname{op}}\to[1]\text{ and }M_{2m}'\colon[m]\times [m-1]^{\operatorname{op}}\to[1]$$ given on objects by $$M_{2m}(i,j)=\begin{cases}
0, \mbox{ if }i<j,\\
1, \mbox{ else.}
\end{cases}
\text{ and }\quad
M_{2m}'(i,j)=\begin{cases}
0, \mbox{ if }i\leq j,\\
1, \mbox{ else.}
\end{cases}$$
- If $n=2m+1$, the non-degenerate simplices $\sigma_{2m+1}$ and $\sigma'_{2m+1}$ correspond to the functors $$M_{2m+1}\colon[m]\times [m]^{\operatorname{op}}\to[1]\text{ and }M'_{2m+1}\colon[m]\times [m]^{\operatorname{op}}\to[1]$$ given on objects by $$M_{2m+1}(i,j)=\begin{cases}
0, \mbox{ if }i<j,\\
1, \mbox{ else.}
\end{cases}
\text{ and }\quad
M_{2m+1}'(i,j)=\begin{cases}
0, \mbox{ if }i\leq j,\\
1, \mbox{ else.}
\end{cases}$$
In particular, in both $M$ and $M'$ no two rows or columns are equal, and each row and column is increasing. They can be depicted as follows. $${
\small \begin{tikzcd}[row sep=small, column sep=small]
0 \arrow{d} \arrow{r} & 0 \arrow{d} \arrow{r} & \cdots \arrow{r}&0\arrow{d}\arrow{r}& 0\arrow{d}\arrow{r} & 1 \arrow{d}\\
0 \arrow{d}\arrow{r} & 0 \arrow{d} \arrow{r} & \cdots \arrow{r}&0\arrow{r} \arrow{d}& 1 \arrow{r} \arrow{d} & 1 \arrow{d}\\
0 \arrow{d}\arrow{r} & 0 \arrow{d} \arrow{r} & \cdots \arrow{r}& 1\arrow{r}\arrow{d}& 1\arrow{r}\arrow{d} & 1 \arrow{d}\\
\vdots \arrow{d}& \vdots \arrow{d} & \ddots & \vdots\arrow{d} &\vdots\arrow{d}&\vdots\arrow{d} \\
0\arrow{r} \arrow{d} &1 \arrow{d}\arrow{r} & \cdots \arrow{r} &1\arrow{r}\arrow{d}&1\arrow{r}\arrow{d}& 1\arrow{d}\\
1\arrow{r} &1 \arrow{r} & \cdots \arrow{r} &1\arrow{r}&1\arrow{r}& 1
\end{tikzcd}
}$$ An induction argument shows that these simplices satisfy the desired relations. Uniqueness can be checked directly for simplices in dimension $1,2,3$ and follows from $3$-coskeletality of $N(\Sigma[1])$ for simplices in dimension at least $4$.
The Duskin nerve of $(r+1)$-point suspension $2$-categories {#section3}
===========================================================
As announced informally in Theorem B, we now describe the Duskin nerve of $(r+1)$-point suspension $2$-categories $\Sigma[{\mathcal{D}}_1,\dots,{\mathcal{D}}_r]$, of which a large class of examples is given by all elements of Joyal’s disk category $\Theta_2$. This description was inspired by the argument used in [@rezkTheta Prop.4.9].
\[iteratedsuspension\] The *$(r+1)$-point suspension* of given $1$-categories ${\mathcal{D}}_1,\dots{\mathcal{D}}_r$ is the $2$-category $\Sigma[{\mathcal{D}}_1,\dots,{\mathcal{D}}_r]$ constructed inductively as the pushout of $2$-categories $$\begin{tikzcd}
{[0]} \arrow[r, "\text{first object}"] \arrow[d, "\text{last object}" swap] & \Sigma{\mathcal{D}}_r \arrow[d, ""]\\
\Sigma[{\mathcal{D}}_1,\dots,{\mathcal{D}}_{r-1}]\arrow[r, "" swap] & \Sigma[{\mathcal{D}}_1,\dots,{\mathcal{D}}_{r-1},{\mathcal{D}}_r].
\end{tikzcd}$$
Alternatively, it can be seen using [@AraMaltsiniotisVers 7.2] that the $(r+1)$-point suspension $\Sigma[{\mathcal{D}}_1,\dots,{\mathcal{D}}_r]$ can be described as the $2$-category with $r+1$ objects $x_0,\dots,x_r$ and hom categories given by $$\operatorname{Map}_{\Sigma[ {\mathcal{D}}_{1},\dots, {\mathcal{D}}_{r}]}(x_{i},x_{j})\cong\left\{
\begin{array}{ccl}
{{\mathcal{D}}_{i+1}} \times \ldots \times{{\mathcal{D}}_{j}}&\text{ if }i<j\\
{[0]}&\text{ if }i=j\\
{[-1]}&\text{ if }i>j.
\end{array}
\right.$$
\[rmk2\] Given $r\ge1$, there is a map of categories $$s_r\colon[r] \hookrightarrow [1]^r$$ given on objects by $$s_r\colon i\mapsto(\underbrace{1,\dots,1}_{\text{$i$ times}},\underbrace{0,\dots,0}_{\text{$(r-i)$ times}}\!\!).$$ This map is injective on objects and fully faithful. When taking nerves, the induced simplicial map $$N(s_r)\colon\Delta[r] \hookrightarrow \Delta[1]^r$$ is a monomorphism, and the image is described as follows.
Any $n$-simplex of $\Delta[1]$ is of the form $f_{k}\colon[n]\to[1]$ for $k=-1,\dots,n$, with $f_k$ defined on objects by $$f_k\colon i\mapsto\left\{
\begin{array}{rcl}
0&i\le k\\
1&i>k.
\end{array}\right.$$ In particular, $s_r=(f_0, \ldots, f_{r-1})$. According to this notation an $n$-simplex $(f_{k_1},\dots,f_{k_r})\colon\Delta[n]\to\Delta[1]^r$ of $\Delta[1]^r$ is in the image of $N(s_r)$ if and only if $k_1\le\dots\le k_r$.
\[rmk1\] Given categories ${\mathcal{D}}_1,\dots,{\mathcal{D}}_r$, there are canonical maps of $2$-categories defined on $\Sigma[{\mathcal{D}}_1,\dots,{\mathcal{D}}_r]$.
1. For any $1\le i\le r$ there are canonical maps of $2$-categories $$\Sigma[{\mathcal{D}}_1,\dots,{\mathcal{D}}_r]\to\Sigma[{\mathcal{D}}_i],$$ which are induced by collapsing all $2$-categories $\Sigma{\mathcal{D}}_j$ for $j<i$ to the point $x_{i-1}$ and all $2$-categories $\Sigma{\mathcal{D}}_j$ for $j> i$ to the point $x_{i}$,
2. There is a canonical map of $2$-categories $$\Sigma[{\mathcal{D}}_1,\dots,{\mathcal{D}}_r]\to\Sigma[[0],\dots,[0]]\cong[r],$$ which is induced by collapsing each generating $2$-cell to a $1$-cell.
3. As a special case of the previous one, for any $1\le i\le r$ there are canonical maps of $2$-categories $$\Sigma[{\mathcal{D}}_i] \to \Sigma[0].$$ When taking nerves, the induced map $$N(\Sigma[{\mathcal{D}}_i]) \to N(\Sigma[0])=N([1])=\Delta[1],$$ sends an $n$-simplex $\sigma$ of $N(\Sigma[{\mathcal{D}}_i])$
- to the $n$-simplex $f_{-1}$ of $\Delta[1]$ if $\sigma$ is maximally degenerate at $y$;
- to the $n$-simplex $f_{k}$ of $\Delta[1]$ if $\sigma$ is of type $k$;
- to the $n$-simplex $f_{n}$ of $\Delta[1]$ if $\sigma$ is maximally degenerate at $x$.
Similarly, using the identification from \[simplicesasmatrices\] the induced map $${\operatorname{Mat}({\mathcal{D}}_i)} \to\Delta[1],$$ sends a matrix $M\colon[k]\times[n-1-k]^{\operatorname{op}}\to{\mathcal{D}}_i$ to $f_k$.
Given categories ${\mathcal{D}}_1,\dots,{\mathcal{D}}_r$, there is a pullback square of simplicial sets $$\begin{tikzcd}
N(\Sigma[{\mathcal{D}}_1,\dots,{\mathcal{D}}_r]) \arrow[r]\arrow[d] & N(\Sigma[{\mathcal{D}}_1]) \times \ldots \times N(\Sigma[{\mathcal{D}}_r])\arrow[d]\\
{\Delta[r]} \arrow[r, ""] & {\Delta[1]}\times\dots\times {\Delta[1]},
\end{tikzcd}$$ where
- the top horizontal map is induced by the canonical maps $\Sigma[{\mathcal{D}}_1,\dots,{\mathcal{D}}_r]\to\Sigma[{\mathcal{D}}_i]$;
- the left vertical map is induced by the canonical map $\Sigma[ {\mathcal{D}}_1,\dots, {\mathcal{D}}_r]\to[r]$;
- the right vertical arrow is induced by the canonical maps $\Sigma[{\mathcal{D}}_i]\to[1]$;
- the bottom horizontal arrow is induced by the canonical map $s_r\colon[r] \hookrightarrow [1]^r$.
Before proving the theorem, we observe that combining the result with \[simplicesasmatrices\], we obtain that there is also a pullback of simplicial sets $$\begin{tikzcd}
N(\Sigma[{\mathcal{D}}_1,\dots,{\mathcal{D}}_r]) \arrow[r]\arrow[d] & {\operatorname{Mat}({\mathcal{D}}_1)} \times \ldots \times {\operatorname{Mat}({\mathcal{D}}_r)}\arrow[d]\\
{\Delta[r]} \arrow[r, ""] & {\Delta[1]}\times\dots\times {\Delta[1]}.
\end{tikzcd}$$ In particular, it follows from \[rmk1,rmk2\] that an $n$-simplex of the Duskin nerve of the $(r+1)$-point suspension $\Sigma[{\mathcal{D}}_1,\dots,{\mathcal{D}}_r]$ can be uniquely described as a $r$-uple of functors $[k_i]\times [l_i]^{\operatorname{op}} \to {\mathcal{D}}_i$ for $i=1,\dots,r$, with $k_i,l_i\ge-1$, $k_i+l_i=n-1$ and subject to the condition that $k_i\le k_j$ for $0\le i\le j\le r$.
We argue that there is a pullback square of $2$-categories $$\begin{tikzcd}
\Sigma[{\mathcal{D}}_1,\dots,{\mathcal{D}}_r]\arrow[r]\arrow[d] &\Sigma[{\mathcal{D}}_1] \times \ldots \times \Sigma[{\mathcal{D}}_r]\arrow[d]\\
{[r]} \arrow[r, "s_r"] & {[1]}\times\dots\times {[1]},
\end{tikzcd}$$ From there we can then conclude, given that the Duskin nerve respects pullbacks and products, being a right adjoint.
The square of $2$-categories above commutes by direct inspection. In order to prove that the square is a pullback of $2$-categories, we check that it is a pullback at the level of objects, and that it is a locally a pullback at the level of hom-categories of any pair of objects in $\Sigma[{\mathcal{D}}_1,\dots,{\mathcal{D}}_r]$.
At the level of objects, we ought to look at the commutative square of sets $$\begin{tikzcd}
\operatorname{Ob}(\Sigma[{\mathcal{D}}_1,\dots,{\mathcal{D}}_r])\arrow[r]\arrow[d] &\operatorname{Ob}(\Sigma[{\mathcal{D}}_1] \times \ldots \times \Sigma[{\mathcal{D}}_r])\arrow[d]\\
\operatorname{Ob}( {[r]})\arrow[r, "s_r"] &\operatorname{Ob}( {[1]}\times\dots\times {[1]}).
\end{tikzcd}$$ This square is expressed as the following square, where both vertical maps are bijections, $$\begin{tikzcd}
\{x_0,\dots,x_r\}\arrow[r ]\arrow[d,"\cong"] &\{x,y\} \times \ldots \times \{x,y\}\arrow[d,"\cong"]\\
\{0,\dots,r\} \arrow[r, "s_r"] & \{0,1\}\times\dots\times \{0,1\}.
\end{tikzcd}$$ The square is therefore a pullback of sets.
At the level of hom-categories, given any two objects $x_i$ and $x_j$ of $\Sigma[{\mathcal{D}}_1,\dots,{\mathcal{D}}_r]$, we ought to look at the commutative square of categories $$\begin{tikzcd}
\operatorname{Map}_{\Sigma[{\mathcal{D}}_1,\dots,{\mathcal{D}}_r]}(x_{i},x_{j})\arrow[r]\arrow[d]& \operatorname{Map}_{\Sigma[{\mathcal{D}}_1] \times \ldots \times \Sigma[{\mathcal{D}}_r]}(\vec x_{s_r(i)},\vec x_{s_r(j)})\arrow[d]\\
\operatorname{Map}_{[r]}(i,j) \arrow[r] & \operatorname{Map}_{{[1]}\times\dots\times {[1]}}(s_r(i),s_r(j)),
\end{tikzcd}$$ where $\vec x_{s_r(i)}$ and $\vec x_{s_r(j)}$ denote the images of $x_i$ and $x_j$ in $\Sigma[{\mathcal{D}}_1] \times \ldots \times \Sigma[{\mathcal{D}}_r]$. If $i>j$, this square is easily expressed as the following square, where the left vertical map is an isomorphism of empty categories $[-1]$, $$\begin{tikzcd}
{[-1]}\arrow[r]\arrow[d,"\cong"] &\operatorname{Map}_{\Sigma[{\mathcal{D}}_1] \times \ldots \times \Sigma[{\mathcal{D}}_r]}(\vec x_{s_r(i)},\vec x_{s_r(j)})\arrow[d]\\
{[-1]} \arrow[r] & \operatorname{Map}_{{[1]}\times\dots\times {[1]}}(s_r(i),s_r(j)).
\end{tikzcd}$$ If instead $i\le j$, this square is easily expressed as the following square, where both horizontal arrows are isomorphisms of categories, $$\begin{tikzcd}
\operatorname{Map}_{\Sigma[{\mathcal{D}}_{1},\dots,{\mathcal{D}}_{r}]}(x_i,x_{j})\arrow[r,"\cong"]\arrow[d] &{{\mathcal{D}}_{i+1}} \times \ldots \times{{\mathcal{D}}_{j}}\arrow[d]\\
{[0]} \arrow[r, "\cong"] & {[0]}\times\dots\times {[0]}.
\end{tikzcd}$$ The square is therefore a pullback of categories in both cases.
[^1]: In this paper we are only concerned with *strict* $2$-categories.
[^2]: We warn the reader that the use of matrices from this paper is not directly related with the matrices used by Duskin in [@duskin].
[^3]: We choose to have $[l]^{\operatorname{op}}$ rather than $[l]$ in the second factor. This convention is more convenient to the setup of the paper and not restrictive, modulo consistent adaptation of the relation between the indices $\alpha(i)$ and $\beta(i)$ in the next formula.
[^4]: The lemma appears to be a variant of the classical fact that the Catalan number $C_{n-1}$ can be expressed in two equivalent ways: as the number of triangulations of an $(n+1)$-gon, or as the number of monotone lattice paths along the edges of a grid with $(n-1)\times (n-1)$ square cells which do not pass above the diagonal. However, we are not aware of a direct comparison with the statement of our lemma.
|
---
abstract: 'In this work we investigate the effect of local dissipation on the presence of density-wave ordering in spinful fermions with both local and nearest-neighbor interactions as described by the extended Hubbard model. We find density-wave order to be robust against decoherence effects up to a critical point where the system becomes homogeneous with no spatial ordering. Our results will be relevant for future cold-atom experiments using fermions with non-local interactions arising from the dressing by highly-excited Rydberg states, which have finite lifetimes due to spontaneous emission processes.'
author:
- Jaromir Panas
- Michael Pasek
- Arya Dhar
- Tao Qin
- 'Andreas Gei[ß]{}ler'
- 'Mohsen Hafez-Torbati'
- 'Max E. Sorantin'
- Irakli Titvinidze
- Walter Hofstetter
bibliography:
- 'main\_text\_file.bib'
title: 'Density-wave steady-state phase of dissipative ultracold fermions with nearest-neighbor interactions'
---
Introduction {#sec:intro}
============
Coupling to the environment is expected to change the properties of a quantum system. In experiments it usually leads to linewidth broadening [@goldschmidt2016; @aman2016], decoherence and finite lifetime of states [@zeiher2015; @zeiher2016]. These dissipative effects are usually limiting experiments. However, coupling between a system and the environment can also lead to exciting new phenomena, such as the quantum Zeno effect [@vidanovic2014; @bernier2014; @sarkar2014]. Dissipation might also be seen as a tool, allowing to drive the system towards a desired state. Recent proposals include engineering dissipative dynamics to create entangled states [@kraus2008] or to drive the system to Bose-Einstein condensation [@diehl2008]. The study of open quantum systems is also useful for investigating transport properties of quantum dots [@dzhioev2011; @dorda2014; @schwarz2016; @fugger2018] or correlated structures [@ajisaka2012; @knap2013; @titvinidze2015; @titvinidze2016].
An approach often followed in investigations of open quantum systems involves using well-established methods in quantum optics, e.g., the master equation [@breuer2002][@carmichael2002], to describe dissipation in lattice models, e.g., the Hubbard model [@hubbard1963; @gutzwiller1963; @konamori1963], or the Bose-Hubbard model [@gersch1963]. These can be experimentally realized with optical lattices [@jaksch1998; @greiner2002; @joerdens2008; @schneider2008], which allow for a close comparison between theory and experiment.
The fermionic Hubbard model was originally proposed to study magnetic properties of materials with strong electronic correlations [@varney2009]. Its extended version, including non-local interactions, has been extensively studied due to its relevance for understanding strongly-correlated electronic materials [@micnas1988; @dagotto1994; @chattopadhyay1997; @aichhorn2004; @kapcia2017; @terletska2017; @terletska2018; @mckenzie2001; @merino2001; @kobayashi2004; @calandra2002]. In these theoretical investigations phases such as spin density-wave (SDW), charge density-wave (CDW) and charge ordered metals (COM) were observed. Some evidence suggests also that a “half-metallic” phase could be found [@garg2014].
In experiment, long-range interactions between ultracold atoms can be realized in several ways by using dipolar quantum gases [@baranov2012]. Recently, the extended Hubbard model has been realized in experiments with polar molecules [@yan2013] and magnetic atoms [@baier2016]. Another promising approach is one, in which Rydberg excitations [@gallagher1988] are used. Due to the extreme properties of atoms excited to Rydberg state the van der Waals interaction between them can become the dominant energy scale in the system. Loading Rydberg atoms into deep optical lattices has been achieved recently [@schauss2012; @schauss2015; @zeiher2016; @zeiher2017; @schauss2018], allowing the realization of spin-lattice models and observation of spatial ordering due to long-range interaction. Corresponding theoretical investigations for Rydberg atoms in optical lattices have been recently performed predicting crystallization in the frozen limit [@pohl2010; @schachenmayer2010; @weimer2010; @vermersch2015]. Beyond the frozen limit, melting of crystalline structure and formation of supersolid due to kinetic energy has been observed [@lauer2012; @geissler2017; @li2018]. Effects of dissipation have also been investigated in the frozen limit with a variational principle [@weimer2015].
However, to our knowledge there has been so far no thorough investigation of the competition between all of the relevant energy scales set by (i) local interaction between atoms, (ii) kinetic energy due to their itinerant nature, (iii) non-local interaction, and (iv) dissipation. The last process is particularly relevant both for experimental realizations of the extended Hubbard model with Rydberg atoms, which are inherently dissipative, and for a better understanding of the possible ordered phases which can appear in open quantum many-body systems.
To investigate this problem we employ the recently developed Lindblad dynamical mean-field theory (L-DMFT) [@knap2013; @titvinidze2015; @titvinidze2016]. Although DMFT has some limitations due to its local self-energy – without non-local extensions it cannot describe, e.g., *d*-wave superconductivity [@lichtenstein2000] – it has proven highly successful in the study of correlated lattice problems [@georges1996]. Several approaches have been previously proposed to extend the method to the non-equilibrium regime [@aron2013; @aoki2014; @li2015]. However, none of these allowed to introduce dissipation on the level of the master equation. The L-DMFT method, on the other hand, treats out-of-equilibrium lattice problems by using an appropriately chosen type of impurity solver for the corresponding Anderson impurity model, called the auxiliary master equation approach (AMEA) [@knap2013; @dorda2014]. In contrast to the previous works using this method, which considered a closed quantum system of infinite size, we study a model of an open quantum system. We use the L-DMFT method to investigate the effect of physical dissipation processes on strongly-correlated many-body phases.
The outline of the article is as follows. In Sec. \[sec:ham\] we describe in more details the problem which we investigate. In Sec. \[sec:model\] we introduce our model Hamiltonian, the extended Hubbard model. Possible experimental realization of this model is then discussed in Sec. \[sec:rydberg\]. The experimentally-relevant dissipative processes that are included in our calculations are introduced in Sec. \[sec:dis\]. A short overview of the L-DMFT technique and its adaptation to dissipative systems is given in Sec. \[sec:ldmft\]. We present our results in Sec. \[sec:results\] on the competition between density-wave ordering and decoherence. Finally, in Sec. \[sec:end\], results are summarized.
Description of the system {#sec:ham}
=========================
Model Hamiltonian {#sec:model}
-----------------
We describe the coherent part of the dynamics by the 2-dimensional (2D) spin-$\frac{1}{2}$ extended Hubbard model (EHM), as illustrated in Fig. \[fig:model\], with the following Hamiltonian $$\label{eq:ehm}
\begin{split}
\hat{H} & = - J\sum_{\langle i,j\rangle, \sigma} \hat{c}^\dag_{i,\sigma} \hat{c}^{\phantom\dag}_{j,\sigma}
+ \epsilon\sum_{i,\sigma} \hat{c}^\dag_{i,\sigma} \hat{c}^{\phantom\dag}_{i,\sigma} \\
& + U\sum_{i} \hat{n}_{i,\downarrow} \hat{n}_{i,\uparrow}
+ \frac{V}{2}\sum_{\langle i,j\rangle,\sigma,\sigma'} \hat{n}_{i,\sigma} \hat{n}_{j,\sigma'}
\end{split}$$ where index $i$ runs over all lattice sites, $\langle i, j\rangle$ indicates a sum over nearest-neighbor (NN) sites independently, $\hat{c}^{\phantom\dag}_{i,\sigma}$ ($\hat{c}^\dag_{i,\sigma}$) is the fermionic annihilation (creation) operator for a particle on site $i$ with spin $\sigma\in\{\uparrow,\downarrow\}$, $\epsilon$ is the on-site energy, $U$ is the local interaction strength, $V$ is the NN interaction strength, $J$ is the hopping amplitude between NN and $\hat{n}_{i,\sigma}=\hat{c}^\dag_{i,\sigma}\hat{c}^{\phantom\dag}_{i,\sigma}$. In the following we set $\hbar=1$ and $J=1$ as the unit of energy, unless stated otherwise. At zero temperature and with a Fermi energy $\epsilon_F=0$, the on-site energy $\epsilon$ determines the filling of the system. For example, if it is equal to $\epsilon=-U / 2 - 4 V$ then the system is at half-filling, i.e., at total density $\sum_i \langle \hat{n}_i \rangle= \sum_i ( \langle \hat{n}_{i,\uparrow}\rangle + \langle \hat{n}_{i,\downarrow}\rangle) = N$, with $N$ being the number of sites in the lattice.
The 2D EHM for spin-$\frac{1}{2}$ fermions has been extensively studied in condensed-matter physics, due to its relevance for understanding $d$-wave pairing and the spatial charge ordering in a wealth of materials. On the square lattice at half-filling, seminal mean-field calculations [@micnas1988; @dagotto1994; @chattopadhyay1997] showed that there exist only two stable equilibrium phases for the EHM with repulsive interactions, namely the spin density wave (SDW) and charge density wave (CDW) phases, separated by a phase boundary at $V_c = U/4$. More recent beyond-mean-field calculations were also performed with the variational cluster approach [@aichhorn2004], single-site DMFT [@kapcia2017], cluster DMFT [@terletska2017], and GW+EDMFT [@ayral2017]. These studies found metallic, Mott-insulating, and charge-ordered phases at half-filling, but long-range antiferromagnetic order was suppressed at the outset. At quarter-filling $\sum_i \langle \hat{n}_i \rangle=0.5N$, for both non-zero local and nearest-neighbor repulsion, previous studies have shown that a checkerboard CDW phase appears at large $V$ [@mckenzie2001] and a $d_{xy}$-wave superconducting phase at intermediate values of $V$ [@merino2001; @kobayashi2004]. In the limit $U,V\gg t$, the ground state was found to be insulating with checkerboard CDW ordering and long-range antiferromagnetism [@mckenzie2001; @calandra2002], based on slave boson calculations and exact diagonalization. The extended Hubbard model has also been recently investigated beyond half- and quarter-fillings [@kapcia2017; @terletska2018]. These works reported the observation of charge-ordered insulator (COI), charge-ordered metal (COM) and Fermi liquid (FL) phases. It was shown that the phase transition from FL phase to quarter-filled COI is discontinuous while the transitions from quarter-filled COI to COM phase and from COM to FL phase are continuous. However, in these studies only non-magnetically ordered phases were considered. In the following we work mainly away from half-filling, and focus our study on the above-mentioned charge-ordered phase and its robustness with respect to dephasing processes.
Relevance to experiments with Rydberg-dressed fermions in optical lattices {#sec:rydberg}
--------------------------------------------------------------------------
The fermionic Hubbard model with only local interactions can be experimentally simulated by ultracold alkali atoms, e.g., $^{40}$K potassium atoms, loaded into optical lattices [@joerdens2008; @schneider2008]. Such an experimental set-up is indeed very flexible, and allows for the fine-tuning of physical parameters and realization of lattices with different geometries. This allows to explore different parts of the phase diagram of the Hubbard model, including the observation of the Mott insulator and metallic phases [@joerdens2008; @schneider2008].
A non-local interaction of the type present in the EHM can be achieved by coupling fermionic atoms in their ground state to highly-excited Rydberg states [@gallagher1988]. These Rydberg states have a large principal quantum number $n$ which results in exaggerated properties, such as a large radius of the valence electron orbital [@gallagher1988] which scales as $n^2$ and can reach distances on the order of $\sim \mu m$. The large spatial extension of the Rydberg atom, in extreme cases comparable to the typical spacing between lattice sites in an optical lattice, results in significant non-local inter-atom interactions with the van der Waals profile [@singer2005; @saffman2010] $$\label{eq:VvdW}
V_{vdW;ij} = \frac{C_6}{a^6|\textbf{i}-\textbf{j}|^6}$$ where $a$ is the lattice spacing, $|\textbf{i}-\textbf{j}|$ the distance between lattice sites $i$ and $j$, and $C_6$ determines the strength of the interaction. This coupling to a Rydberg state can be realized through interaction of atoms with coherent light of appropriately-chosen frequency and intensity. With each atom represented as a two-level system, the coupling is described in the rotating wave approximation [@cohen1998] by an effective Rabi frequency $\Omega_\mathrm{eff}$ and detuning $\delta_\mathrm{eff}$. In order to simulate experimentally the EHM for spin-$\frac{1}{2}$, one could focus on the so-called dressing regime, where the effective detuning is much larger than the Rabi frequency $\delta_\mathrm{eff}\gg \Omega_\mathrm{eff}$ [@balewski2014]. In this regime a new eigenstate emerges, with properties arising dominantly from the ground state of the atom with a small admixture of properties from the excited Rydberg state. The strength and shape of the interaction potential can then be tuned through the amount of admixture between the two states. The effective potential in this Rydberg-dressing regime is then [@henkel2010] $$\label{eq:VvdWef}
V_{\mathrm{eff};ij} = \frac{\tilde{C}_6}{a^6|\textbf{i}-\textbf{j}|^6+R_c^6},$$ with effective strength $\tilde{C}_6$ and soft-core radius $R_c$. With appropriate choice of experimental parameters, one can reach a regime in which only nearest-neighbor and local interaction processes are relevant for the dynamics. More detailed discussion of the dressed regime can be found in Appendix \[appsec:dress\]. In particular, the EHM can be realized with $^{40}$K potassium atoms in the dressing regime, with spin obtained by using two different hyperfine states which in turn are coupled with large detuning to the excited Rydberg states (see Fig. \[fig:model\]) to realize non-local interaction.
Recently, several experiments with bosonic Rydberg atoms loaded into optical lattice have been performed [@viteau2011; @zeiher2016; @schauss2012; @schauss2015; @zeiher2017; @schauss2018]. In order to reach longer time scales in the experiments the regime of vanishingly small hopping between lattice sites has been used [@zeiher2016; @schauss2012; @schauss2015; @zeiher2017; @schauss2018]. This allowed observation of an emerging ordering in the lattice due to the long-range interaction [@schauss2012; @schauss2015; @schauss2018]. However, even in the frozen limit, where the Hamiltonian of Rydberg-dressed atoms can be rewritten as an Ising quantum spin model [@schauss2018], dissipation was already seen as a major obstacle causing, e.g., avalanche loss of particles from the system [@zeiher2016]. In the following, we will go beyond the frozen limit and investigate possible steady-state phases of itinerant atoms which emerge from the full competition between kinetic processes and both short- and long-range interaction.
Dissipative processes {#sec:dis}
---------------------
### Model
In the above discussion we have focused on *coherent* processes that are present in experiments. However, Rydberg excited states have a relatively short lifetime due to spontaneous emission and black-body radiation [@loew2012; @desalvo2016; @goldschmidt2016]. To take these into account one must include the coupling of the system to its environment, and treat it as a many-body open quantum system. In this paper, we aim in particular at studying the effects of dephasing that occur due to the Rydberg-dressing.
As is often the case in the theory of open quantum systems, we will use the Born-Markov approximation to describe the evolution of the system. As a result, we can use the Lindblad master equation [@breuer2002; @carmichael2002] $$\label{eq:liouvillian}
\frac{d \hat{\rho}}{d t} = -i\left[\hat{H}, \hat{\rho} \right] + { \settoheight{\dhatheight}{\ensuremath{\hat{\mathcal{L}}}} \addtolength{\dhatheight}{-0.35ex} \hat{\vphantom{\rule{1pt}{\dhatheight}} \smash{\hat{\mathcal{L}}}}}[\hat{\rho}]$$ where $\hat{\rho}$ is the density matrix operator, and ${ \settoheight{\dhatheight}{\ensuremath{\hat{\mathcal{L}}}} \addtolength{\dhatheight}{-0.35ex} \hat{\vphantom{\rule{1pt}{\dhatheight}} \smash{\hat{\mathcal{L}}}}}$ is the superoperator which describes dissipation. In the Lindblad equation it is defined according to $$\label{eq:liouvillian2}
{ \settoheight{\dhatheight}{\ensuremath{\hat{\mathcal{L}}}} \addtolength{\dhatheight}{-0.35ex} \hat{\vphantom{\rule{1pt}{\dhatheight}} \smash{\hat{\mathcal{L}}}}}[\hat{\rho}] = \frac{1}{2} \sum_{\mu\nu} \Gamma_{\mu,\nu} \left( 2 \hat{L}_\nu \hat{\rho} \hat{L}^\dagger_\mu - \left\{ \hat{L}^\dagger_\mu \hat{L}_\nu, \hat{\rho} \right\} \right),$$ where $\hat{L}_\mu$ are jump operators, $\Gamma_{\mu\nu}$ are dissipation coefficients and $\mu$, $\nu$ iterate over relevant quantum numbers. Regarding the relevant jump operators to include in our description, we assume here that the dominant dissipative effects for Rydberg atoms are spontaneous emission processes with rate $\Gamma_\mathrm{se}$ (see Fig. \[fig:model\]). This can be mapped within the dressing regime to a dephasing process and described by the effective dephasing rate $\Gamma_\mathrm{dp}$ and the following jump operator (see App. \[appsec:dress\]) $$\label{eq:Ldp}
\hat{L}_{{dp},i,\sigma} = \hat{c}_{i,\sigma}^\dag\hat{c}^{\phantom\dag}_{i,\sigma}.$$ Note that with this type of jump operator the time evolution given by Eq. is effectively quartic in terms of creation and annihilation operators. Therefore, effects of dissipation are incorporated into the self-energy, together with the effects of interaction.
Such dephasing terms conserve the local particle number and hence cannot change the local occupation. One of the effects of this type of dissipation on the many-body state is, however, to cause the decay of the off-diagonal elements of the density matrix and drive the system towards the infinite-temperature state in the absence of an external thermal bath (cf. Appendix \[appsec:disheat\] and [@sarkar2014; @bernier2014]).
Dephasing terms of the form in Eq. correspond, in the theory of open quantum systems, to a continuous measurement process of the site occupation variable (see [@breuer2002], Ch. 3.5). Such terms are also useful to describe the coupling of the local fermion density to the environment through non-local interactions [@sarkar2014; @bernier2014].
Non-Equilibrium DMFT: auxiliary master equation approach {#sec:ldmft}
========================================================
The auxiliary master equation approach (AMEA) to non-equilibrium DMFT, here referred to as L-DMFT, has been proposed and developed to study transport properties of a correlated electronic layer coupled to non-interacting leads [@knap2013; @titvinidze2015; @titvinidze2016]. However, we show here that the method has potential for applications to other types of out-of-equilibrium problems, e.g., a lattice system with local dissipation. We describe how to adapt the method to such a problem and apply it to the extended Hubbard model with local dephasing. As described in the preceding section, this model can be simulated with Rydberg atoms loaded into optical lattices, where the dissipative processes are naturally present.
The L-DMFT method allows to find the steady-state of a system far-from-equilibrium in a self-consistent way and then to calculate static and dynamic local quantities. Due to the self-consistent approach it is better to have a unique steady-state. If we were to consider a closed system described by the Hamiltonian of Eq. , we would face the problem of non-unique steady-states. Indeed, for a closed system there are as many steady-states as there are eigenstates of the Hamiltonian. As we are interested in an open quantum system, where the lattice is subject to dephasing, this issue should not occur. While the dissipation indeed renders the steady-state unique, it also generically heats the system and drives it towards an uninteresting, infinite-temperature steady-state [@bernier2014] (App. \[appsec:disheat\]).
In order to have a non-trivial steady-state we assume that each site of the extended Hubbard model is coupled to a separate heat and particle reservoir that is always in thermal equilibrium, as depicted in Fig. \[fig:model\]. The local reservoirs act as a heat drain and allow us to study the limit of vanishing dissipation strength by lifting the degeneracy of the steady-states. We will use an exactly solvable model for the bath known as Davies’ model of heat conduction [@davies1978] or Büttiker’s heat-bath model [@buttiker1985; @buttiker1986]. This heat-bath model was recently employed to study the effect of dissipation on interacting many-body systems, using a variant of non-equilibrium DMFT [@tsuji2010; @aron2013; @aoki2014; @li2015; @qin2018] and quantum Monte Carlo path integrals [@yan2018]. To put the additional thermal baths into context we note that they are commonly used in context of the Floquet-DMFT to achieve *inhomogeneous* equations with a unique, non-trivial solution of the equations for the steady-state [@aoki2014; @qin2018]. Without heat-baths a periodically driven system approaches an infinite temperature state in the long timescale, but in the intermediate timescale it can be found in a quasistationary, Floquet prethermalized state [@peronaci2018]. Overall, heat-bath can be treated there as a theoretical “trick” for numerical methods, which allows to study the intermediate timescale state as a steady-state. We also note that recently an experiment was performed, in which an optical lattice was coupled to a thermal reservoir of atoms captured in a magneto-optical trap [@chong2018].
The local thermal baths are assumed to be one-dimensional semi-infinite chains of non-interacting fermions with hopping $J_b$ between neighboring sites and a retarded Green function given by $$\label{eq:leadGf}
g_b^R(\omega) = \frac{\omega}{2 J_b^2}-\rmi\frac{\sqrt{4J_b^2-\omega^2}}{2J_b^2}.$$ As the bath is assumed to be always in thermal equilibrium, its Keldysh Green function is given by the fluctuation-dissipation theorem $$\label{eq:fl_dis_rel}
g_b^K(\omega) = 2\rmi\left[ 1 -2f_b(\omega) \right]\mathrm{Im}\{g_b^R(\omega)\},$$ where $f_b(\omega)$ is the Fermi-Dirac distribution. We set the Fermi energy (chemical potential) $\epsilon_F=0$ and the temperature $T=0$. The value of hopping in the thermal baths is set to $J_b=7.5$ which gives a half-bandwidth ($2J_b$) on the order of magnitude of the maximal considered value of $U$. This allows for thermalization in a broad energy spectrum even in the presence of the Hubbard band splitting due to the local interaction. The coupling of the local thermal baths to the system is realized via exchange of particles with hopping amplitude $\nu$. The particular form of this coupling will be introduced in the next section.
DMFT self-consistency {#subsec:dmftsc}
---------------------
In DMFT a single approximation is made that the self-energy is a purely local quantity [@georges1996], such that $$\Sigma_{ij,\sigma}(\omega) = \delta_{ij} \Sigma_{i,\sigma}(\omega),$$ where $i$ and $j$ are lattice indices. As a consequence one is able to map a full lattice problem onto a set of local effective quantum impurity models, which significantly reduces the size of the many-body problem while fully preserving the nature of local quantum correlations. These impurity problems are coupled in the self-consistent approach via a Dyson equation given further in text.
Due to the local character of DMFT, we must however treat the non-local nearest-neighbor interaction term in Eq. within a Hartree mean-field approximation, where the interaction operator is mapped onto $$\frac{V}{2}\sum_{\langle i,j\rangle,\sigma} \hat{n}_{i,\sigma} \hat{n}_{j,\sigma} \to
V\sum_{\langle i,j\rangle,\sigma} \hat{n}_{i,\sigma} \langle \hat{n}_{j,\sigma} \rangle$$ with $\langle \hat{n}_{j,\sigma}\rangle$ determined self-consistently.
The model which we consider is translationally invariant, which allows to find a symmetry between lattice sites and reduce the size to a small number of inequivalent impurity problems. However, the symmetry of the ground state on the lattice can be spontaneously reduced in certain ordered phases such as the CDW phase, which we want to investigate. We assume throughout this work that the system has two translationally invariant sublattices, $A$ and $B$, which results in two different impurity problems to solve. The sublattices are defined such that each site from sublattice $A$ is neighboring only with sites belonging to sublattice $B$ and vice versa, see Fig. \[fig:model\].
The derivation of the DMFT equations in the non-equilibrium Keldysh formalism is well established in the literature [@knap2013; @aoki2014; @titvinidze2015]. We refer the reader in particular to Ref. , while here we present the differences with respect to this reference, which arise from the two-sublattice structure and geometry of the system that we consider here. We use a notation for the Green function in which $$\mathbf{G} = \begin{pmatrix} G_{AA} & G_{AB} \\ G_{BA} & G_{BB}\end{pmatrix},$$ where the $A$ and $B$ indices mark each sublattice. On top of that, to introduce retarded ($R$), advanced ($A$) and Keldysh ($K$) components of the Green functions, we use the notation $${ \underline{\mathbf{G}}} = \begin{pmatrix} \mathbf{G}^R & \mathbf{G}^K \\ \mathbf{0} & \mathbf{G}^A \end{pmatrix}.$$ Due to the translational invariance of the sublattices $A$ and $B$ one can perform a Fourier transform with a reduced Brillouin zone ($BZ'$). In analogy to Ref. [@titvinidze2015] we express the Green functions in momentum space. The Green functions of the non-interacting model decoupled from the thermal bath is given by $$\mathbf{g}_0^R(\vec{k},\omega) = \begin{pmatrix} \omega + \rmi 0^+ -\epsilon & -E_c(\vec{k}) \\ -E_c(\vec{k}) & \omega + \rmi 0^+ - \epsilon
\end{pmatrix}^{-1},$$ where $E_c(\vec{k})=-2J(\cos(k_x)+\cos(k_y))$, and with wave-vectors $\vec{k}$ from the reduced Brillouin zone. Correspondingly, as the local baths are decoupled from one another, their Green functions after Fourier transform have the form $$\label{eq:ABnot}
\begin{split}
\mathbf{g}_b^{R/K/A} (\vec{k},\omega) & = \mathbf{g}_b^{R/K/A} (\omega)\\
& = \begin{pmatrix}
g_b^{R/K/A}(\omega) & 0 \\ 0 & g_b^{R/K/A}(\omega)
\end{pmatrix}.
\end{split}$$ We note that the Keldysh part $\mathbf{g}^K_0(\vec{k},\omega)$ is state dependent and not uniquely defined because it corresponds to a system decoupled from any thermal bath. Nevertheless, we will only need to use the inverse of this Green function, for which the Keldysh part $\left[\mathbf{g}^{-1}_0(\vec{k},
\omega)\right]^K$ is infinitesimally small [@knap2013][^1].
Based on the above Green functions, we can determine the Green functions of a two-dimensional non-interacting system coupled to the thermal bath. We get $$\label{eq:cpl2res}
{ \underline{\mathbf{G}}}_0^{-1} (\vec{k},\omega) = { \underline{\mathbf{g}}}_0^{-1}(\vec{k},\omega) - \nu^2 { \underline{\mathbf{g}}}_b (\omega).$$ This equation determines the form of the coupling between the lattice sites and the thermal reservoir with $\nu^2$ defining the coupling strength. The Green function of the interacting model coupled to thermal baths is then given by the Dyson equation [@georges1996; @knap2013; @titvinidze2015] $${ \underline{\mathbf{G}}}^{-1} (\vec{k},\omega) = { \underline{\mathbf{G}}}_0^{-1}(\vec{k},\omega) - { \underline{\mathbf{\Sigma}}} (\omega).$$ Here ${ \underline{\mathbf{G}}}^{-1} (\vec{k},\omega)$ is the Green function of the full system in momentum space. The self-energy ${ \underline{\mathbf{\Sigma}}} (\omega)$ here describes effects of local and non-local interaction (the latter on the mean-field level) as well as the effects of dephasing. As it is assumed to be local, it is also momentum independent in the reduced Brillouin zone, but it might be different for sublattices $A$ and $B$ $$\mathbf{\Sigma}^{R/K/A}(\omega) = \begin{pmatrix} \Sigma^{R/K/A}_A(\omega) & 0 \\ 0 & \Sigma^{R/K/A}_B(\omega).
\end{pmatrix}$$ To close the self-consistency equations we extract the local part of the lattice Green function, i.e., for the sublattice $A$ ($B$) we have $$\label{eq:loc_Green_f}
{ \underline{G}}_{A(B)}(\omega) = \int_{BZ'} \frac{\d \vec{k}}{(2\pi)^2}{ \underline{G}}_{A(B)}(\vec{k},\omega).$$ Note that while here we perform the operation for sublattice $A$ and $B$ separately, we still have a matrix equation with retarded, advanced and Keldysh parts. We use the above result in the local Dyson equation, which reads $$\label{eq:locDyson}
{ \underline{\Delta}}_{A(B)}(\omega) = { \underline{G}}_{0,A(B)}^{-1}(\omega) - { \underline{\Sigma}}_{A(B)}(\omega) - { \underline{G}}^{-1}_{A(B)}(\omega),$$ with ${ \underline{\Delta}}_{A(B)}$ being the hybridization function which describes the effect of coupling the non-interacting impurity to both the thermal bath and the interacting lattice that surrounds it, and with $$G^R_{0,A}(\omega) = G^R_{0,B}(\omega) = \left(\omega + \rmi 0^+ - \epsilon\right)^{-1} .$$ representing the local Green function of the single, non-interacting lattice site decoupled from both the thermal bath and the surrounding lattice. The Keldysh part of the inverse Green function is again negligible.
The remaining problem is to solve the two emerging impurity problems for sublattice $A$ and $B$, given the hybridization functions ${ \underline{\Delta}}_{A(B)}$. Once one can do this, one can solve the full problem self-consistently.
Impurity solver: auxiliary master equation approach {#subsec:amea}
---------------------------------------------------
Solving the impurity problems is usually the bottleneck of the DMFT method. Here we deal with two independent problems, one for each of the sublattices. In the following the sublattice index $\alpha\in\{A,B\}$ denotes which impurity problem we consider. What significantly adds to the complexity of the task is the fact that the system is not in thermal equilibrium (at least in the general case) but rather in a steady-state of some non-trivial dissipative dynamics.
A method well-suited to solve such an impurity problem is the auxiliary master equation approach (AMEA) [@knap2013; @dorda2014; @titvinidze2015]. In our implementation of non-equilibrium DMFT, we adapt it to a problem with physical local dissipation. Below we briefly list the main points of this method, focusing on what is most relevant to our problem.
The foundation of this method is laid by the exact diagonalization approach to the impurity problem [@georges1996], in which it is mapped onto an effective finite size problem. A single impurity is coupled to a finite number $N_b$ of non-interacting bath sites, which imitate the surrounding of the impurity as closely as possible. These bath sites are completely auxiliary and should not be confused with the thermal bath introduced at the beginning of this section. The effective impurity Hamiltonian reads $$\label{eq:aux_ham}
H_{\alpha,aux} = \sum_{i,j=0;\sigma}^{N_b} E_{\alpha,ij} \hat{d}^\dag_{i,\sigma} \hat{d}_{j,\sigma} + U\hat{d}^\dag_{0,\downarrow} \hat{d}^\dag_{0,\uparrow}\hat{d}_{0,\uparrow} \hat{d}_{0,\downarrow},$$ where $\hat{d}_{i,\sigma}$ is the annihilation operator on site $i$ with spin $\sigma$, the $i$ and $j$ indices run over all possible sites in auxiliary impurity problem with $i=0$ referring to the impurity. $E_{\alpha,ij}$ are arbitrary parameters subject to the constraint that they should form a hermitian matrix and with the value of $E_{\alpha,00}$ fixed by the original lattice problem [^2]. We also choose to work with a star geometry of the bath, without loss of generality, cf. Fig. \[fig:amea\]. However, this Hamiltonian is by itself not sufficient to describe the time evolution of an open quantum system.
To circumvent this issue, in the AMEA approach the bath sites are coupled to a Markovian auxiliary reservoir, see Fig. \[fig:amea\]. This allows to describe the time evolution with the Lindblad master equation Eq. in which the Hamiltonian is given by Eq. . The dissipative part of the master equation is determined by two terms. The first term is the local, physical dissipation given in Eq. which acts only on the impurity state with index 0. The second term is determined by $$\label{eq:aux_dis}
\begin{split}
{ \settoheight{\dhatheight}{\ensuremath{\hat{\L}}} \addtolength{\dhatheight}{-0.35ex} \hat{\vphantom{\rule{1pt}{\dhatheight}} \smash{\hat{\L}}}}_{\alpha,aux}\left[\hat{\rho}\right] & = \sum_{\sigma;i,j=1}^{N_b} 2 \\
& \times\left[ \Gamma^{(1)}_{\alpha,ij}\left( \hat{d}_{i,\sigma}^{\phantom\dag} \hat{\rho} \hat{d}_{j,\sigma}^\dag - \frac{1}{2}\{\hat{\rho},\hat{d}^\dag_{j,\sigma} \hat{d}_{i,\sigma}^{\phantom\dag} \} \right)\right.\\
& +\left.\Gamma^{(2)}_{\alpha,ij}\left( \hat{d}^\dag_{j,\sigma} \hat{\rho} \hat{d}_{i,\sigma}^{\phantom\dag} - \frac{1}{2}\{\hat{\rho},\hat{d}_{i,\sigma}^{\phantom\dag} \hat{d}_{j,\sigma}^\dag \} \right) \right]
\end{split}$$ and describes the coupling of the bath sites to the Markovian reservoir. Here $\Gamma_{\alpha,ij}^{(1)}$ and $\Gamma_{\alpha,ij}^{(2)}$ are arbitrary parameters subject to the constraint that they should form a hermitian, positive-definite matrix.
This type of non-equilibrium impurity model has been extensively studied in the literature [@dzhioev2011; @ajisaka2012; @dorda2014; @schwarz2016]. Using exact diagonalization of the Liouvillian in the super-fermionic representation (which doubles the Hilbert space limiting achievable $N_b$) one can solve this impurity problem [@knap2013; @dzhioev2011super]. Note also that if we switch off the local interaction and dissipation on the impurity, the model becomes quadratic and therefore analytically solvable. Consequently, one can calculate the effective hybridization function ${ \underline{\Delta}}_{aux}$ with little computational effort [@knap2013].
The free parameters $E_{\alpha,ij}$ and $\Gamma_{\alpha,ij}^{(1/2)}$ of the impurity model form a set of variables $\{x_\alpha\} = \bigcup_{ij}\{E_{\alpha,ij}, \Gamma^{(1)}_{\alpha,ij}, \Gamma^{(2)}_{\alpha,ij}\}$. This set can be further reduced, cf. Ref [@knap2013], using symmetries of $\Gamma$ matrix and an appropriate geometry of the impurity model, e.g., the star geometry from Fig. \[fig:amea\]. The values of the variables are chosen in such a way that the cost function $$\label{eq:cost_fun}
\begin{split}
& \chi_\alpha\left(\{x_\alpha\} \right) \\
& = \int_{-\infty}^\infty \d \omega \left[ \chi_\alpha^R(\omega,\{x_\alpha\}) + \chi_\alpha^K(\omega,\{x_\alpha\})+ \chi_\alpha^f(\omega,\{x_\alpha\}) \right]
\end{split}$$ is minimized. Different contributions to $\chi$ are defined as $$\label{eq:cost_fun_el}
\begin{split}
\chi_\alpha^R\left(\omega,\{x_\alpha\} \right) & = \mathrm{Im}\left[\Delta_\alpha^R(\omega) - \Delta^R_{\alpha,aux}\left(\omega,\{x_\alpha\}\right)\right]^2, \\
\chi_\alpha^K\left(\omega,\{x_\alpha\} \right) & = \mathrm{Im}\left[\Delta_\alpha^K(\omega) - \Delta^K_{\alpha,aux}\left(\omega,\{x_\alpha\}\right)\right]^2, \\
\chi_\alpha^f\left(\omega,\{x_\alpha\} \right) & = \left|f_\alpha(\omega) - f_{\alpha,aux}\left(\omega,\{x_\alpha\}\right)\right|^2 |\mathrm{Im}[\Delta^R_{\bar{\alpha}}(\omega)]|.
\end{split}$$ The bar in $\bar{\alpha}$ denotes the complement of $\alpha$ in the set $\{A,B\}$, $\Delta_\alpha(\omega)$ is the physical hybridization function for sublattice $\alpha$ obtained from the Dyson equation , $\Delta_{\alpha,aux}(\omega,\{x_\alpha\})$ is the auxiliary hybridization function for sublattice $\alpha$ in the impurity model [@knap2013; @titvinidze2015], $f_\alpha(\omega)$ is the distribution function calculated using the fluctuation-dissipation relation and reads $$\label{eq:delta2distr}
f_\alpha(\omega) = \frac{1}{2} - \frac{1}{4}\frac{\mathrm{Im}[\Delta^K_\alpha(\omega)]}{\mathrm{Im}[\Delta^R_\alpha(\omega)]}.$$ An analogous formula is used for $f_{\alpha,aux}(\omega)$.
The terms $\chi_\alpha^R$ and $\chi_\alpha^K$ are responsible for obtaining the best fit of the retarded and Keldysh parts of the hybridization function, respectively. However, in the case of a small number of bath sites, the accuracy of the fit for these terms might come at the cost of a less accurate reproduction of the distribution function. This is compensated by our inclusion of the last term, $\chi^f_\alpha$. Obtaining an accurate fit of the distribution function is necessary only in the region of significant spectral weight, hence the factor $|\mathrm{Im}[\Delta_{\bar{\alpha}}(\omega)]|$.
Limitations {#subsec:limit}
-----------
It is clear that the approximation made in the course of the AMEA gets better with increasing number of bath sites $N_b$. However, due to the exponential scaling of the size of the problem with the number of bath sites and doubling of the Hilbert space one cannot reach large values of $N_b$ when using an exact diagonalization based solver. Within our implementation we are able to set this parameter up to $N_b=5$. In other works a number of bath sites up to $N_b=6$ has been reported [@dorda2014; @titvinidze2015; @titvinidze2016]. However, higher number of baths sites in the L-DMFT has not been reached yet. An alternative to the exact diagonalization based solver in the AMEA impurity problem is to use the matrix product states approach [@dorda2014; @schwarz2016; @fugger2018]. With this method values of up to $N_b=20$ have been reached [@fugger2018]. However, currently combining this method and the DMFT self-consistency is not practical, as the computational effort to solve a single impurity problem is too large to be used in a self-consistent approach.
With a limited number of bath sites some physical quantities might not be recovered accurately. For example, upon investigation of the occupation of different energy states, the Fermi distribution in thermal equilibrium might not be reproduced precisely. This leads to deviations between the results obtained with a standard, equilibrium DMFT solver, which can reach higher accuracy, and the L-DMFT solver used here.
One of the features that are difficult to capture using a small number of bath sites is the magnetic response of the system. E.g., with $N_b=5$ we could not reproduce the anti-ferromagnetic phase of the standard Hubbard model at zero temperature. Nevertheless, using the AMEA within stochastic wave function approach [@sorantin2018] we checked that the discrepancy between the results of the equilibrium and AMEA impurity solvers is decreasing with increasing number of bath sites. Also for other types of impurity magnetic response the value of $N_b>10$, which might be reached with the matrix product states method, was enough to get accurate results [@fugger2018].
Another relevant effect occurs if one of the impurity energy levels lies outside of the band specified by the heat-bath. In such case a localized state appears, whose evolution is not captured within the Markov approximation of the auxiliary reservoir [@zhang2012]. This issue is here amended by the choice of the local thermal baths with a broad energy spectrum.
Results {#sec:results}
=======
Equilibrium {#sec:eq}
-----------
We begin with the investigation of a system in thermal equilibrium, i.e., without dephasing process. As in this regime one can employ alternative methods to study the model described by Eq. , this serves both as a reference for the calculation with dephasing and as a benchmark of the L-DMFT method. We will compare it to the equilibrium DMFT and Hartree-Fock methods. Note that we allow only for two phases: charge density wave and normal phase with homogeneous particle density and neglect any magnetic ordering possibly emerging in the system, such that $\langle\hat{n}_{i,\uparrow}\rangle = \langle \hat{n}_{i,\downarrow}\rangle$.
In our equilibrium DMFT and Hartree-Fock calculations we set the chemical potential to $\mu=0$, temperature to $T=0$ and switch off the coupling to external thermal baths. For the L-DMFT calculations the chemical potential and the temperature are set to the same values indirectly, through coupling to the thermal reservoir, see Eq. and . We set the strength of this coupling to $\nu^2=0.5$. To obtain both CDW and normal phases we set the NN-interaction strength to $V=2$. We perform calculations with local-interaction strength ranging from $U=1$ to $U=16$. On-site energy is set to $\epsilon=-\frac{U}{2}$. This would correspond to half-filling in the absence of non-local interaction. However, with $V=2$ the filling is lower and thus results are away from half-filling.
In Fig. \[fig:occ\_eq\] we present the comparison of local occupations for a single spin species, $\langle\hat{n}_{A(B)}\rangle=\langle\hat{n}_{A(B),\downarrow}+\hat{n}_{A(B),\uparrow}\rangle$, obtained within equilibrium DMFT and Hartree-Fock mean-field. The equilibrium DMFT results were obtained with the exact diagonalization impurity solver [@hafez-torbati2018]. We observe that for intermediate values of the interaction strength a checkerboard CDW phase emerges, resulting in spontaneous symmetry breaking with non-zero value of the checkerboard order parameter $\Delta n=|\langle \hat{n}_A-\hat{n}_B\rangle|$.
The phase transition at high $U$ (and as a result high $|\epsilon|$) occurs due to the competition between the on-site energy $\epsilon$ and NN interaction $V$. Approximately, the energy cost of adding a particle at (almost empty) sublattice $B$ is given by $4V=8J$ (due to four singly occupied neighbors) and the energy gain is given by $\epsilon=-U/2$. Therefore, in the atomic limit one can expect a phase transition around $U=16J$. The hopping processes lead to hybridization of the two sublattices. This results in decreasing value of the checkerboard order parameter $\Delta n$ as we approach the phase transition point, and a shift of the critical interaction strength $U_c$ to lower values of around $U_c=14.5\pm0.5$. At low values of $U$ the phase transition occurs around $U_c=2.5\pm0.5$ with a jump in the total filling. As all four energy scales, namely $U$, $J$, $V$ and $\epsilon$, are comparable and we have not been able to find a simple explanation for the nature of this phase transition.
To benchmark the L-DMFT technique we performed a series of tests for an arbitrarily chosen value of on-site interaction $U=8$. Firstly we investigated how the accuracy of the method depends on the number of bath sites $N_b$ used in the impurity solver. We checked how the spectral function and the filling of sublattices changes for $1\leqslant N_b \leqslant 5$ (results not shown in here). While we observe significant deviations for $N_b\leqslant 2$, the values $N_b=3,\ 4$ and $5$ give comparable results.
Next we focused on the performance of the method with $N_b=4$. In Fig. \[fig:hyb\] we present the comparison of the hybridization functions at $U=8$ for sublattice $B$ obtained from the Dyson equation and after mapping onto the impurity AMEA model. We observe that while some finer details are lost in the mapping procedure, the main features are properly reproduced. Upon closer investigation of the distribution functions $f_{B,aux}(\omega)$ describing the environment of a site from sublattice $B$ (displayed in Fig. \[fig:hyb\]), we notice that: (i) the auxiliary hybridization function does not reproduce perfectly the Fermi-Dirac distribution function, (ii) there are small discrepancies between the physical and auxiliary distribution functions in the region where the spectral weight $\mathrm{Im}[\Delta^R(\omega)]$ is large. Discrepancies in the remaining regions do not lead to significant issues – as the spectral weight is small in these regions it does not contribute strongly to the dynamics or the total occupation of the system.
The discrepancies in the reproduced distribution function might lead to different values of occupation and double occupancy between the equilibrium and non-equilibrium solvers. This issue is indeed observed when comparing the DMFT and L-DMFT results (cf. Fig. \[fig:occ\_eq\]). Nevertheless, it is minimized by an appropriate choice of the cost function, which minimizes the error in the relevant regions through $\chi^f_\alpha$.
The discrepancies in the hybridization functions in the AMEA have yet another consequence. As not all features are perfectly reproduced, there might be local minima of the cost function , in which features which are captured more accurately appear in different parts of the spectrum. As a result, one might expect more than one self-consistently converged solution of the full L-DMFT approach. We checked that this leads to at most small quantitative differences in the converged solution. Qualitative features of the results remain unchanged.
Having established that the method gives a good qualitative description of the system we compared the results of equilibrium DMFT with those of L-DMFT for a wide range of values of $U$, Fig. \[fig:occ\_eq\]. The non-smoothness of the L-DMFT results originates from the emergence of multiple self-consistent solutions discussed above and from discrepancies in the effective distribution function $f(\omega)$. We observe that the methods yield similar results, but one can observe some quantitative differences. In all cases the system does not exhibit a CDW phase for weak local interaction. As $U$ is increased the system undergoes a phase transition, which occurs at a critical value around $U_c\approx 2.5\pm 0.5$ with a sharp change of the order parameter $\Delta n$. Investigating the type of this phase transition goes beyond the scope of this paper. As the value of $U$ is further increased the order parameter decreases until it vanishes completely at around $U_c\approx 14.5\pm0.5$. Overall, the comparison of DMFT and L-DMFT shows that the latter agrees qualitatively with an equilibrium method which is well established in the literature and which captures effects of strong local correlations. Quantitative differences can serve as a measure of the accuracy for our method.
To check the effect of exchange (Fock) terms due to the nearest-neighbor interaction, which are absent in DMFT, we additionally perform a mean-field study of the ground state long-range order of the Hamiltonian in Eq. , by following the self-consistent mean-field method that includes both Hartree- and Fock-decoupling of local and non-local interaction terms [@blaizot1986] . This self-consistent mean-field method was used previously for two-dimensional dipolar fermions [@bhongale2012; @bhongale2013]. Although anomalous mean-field terms that allow for a description of pairing with arbitrary spatial symmetry can in principle be included in the self-consistent method, we set them to zero in this work. As can be seen in Fig. \[fig:occ\_eq\], both L-DMFT and DMFT give qualitatively similar results to the self-consistent mean-field method, which shows that terms beyond Hartree approximation in the nearest-neighbor interaction do not modify qualitatively the density-wave ordering in this system. The discrepancies at larger values of $U$ are expected to be an effect of mean-field treatment of local interactions rather than neglecting exchange terms in DMFT or L-DMFT.
Non-equilibrium results {#sec:noneq}
-----------------------
We now turn to the non-equilibrium case. In Fig. \[fig:gm0\_vs\_gm05\] we compare the occupations of sublattices $A$ and $B$ for different interaction strengths with and without the dephasing $\Gamma_\mathrm{dp}=0.05$. We observe that the dephasing has the effect of reducing the differences between the occupation of sublattice $A$ ($\langle\hat{n}_A\rangle$) and sublattice $B$ ($\langle\hat{n}_B\rangle$). We also observe a change of the critical value of the local interaction, at which the system undergoes a phase transition between the homogeneous and CDW phases. The range of values of $U$ for which the CDW phase is present thus becomes smaller due to dephasing.
In order to further estimate the destructive effect of dephasing we investigate the $U-\Gamma_\mathrm{dp}$ phase diagram of our system, Fig. \[fig:phase\_diag\]. It is evident that with increasing strength of dephasing the range in $U$ for which one obtains a CDW phase shrinks, until it vanishes completely at around $\Gamma_\mathrm{dp}\approx 0.085\pm0.005$. Note that this value is at least one order of magnitude smaller than the other energy scales of the system, $\epsilon$, $U$, $V$ and $J$. We also note that the $U$-dependence of the critical value of $\Gamma_\mathrm{dp}$ is approximately symmetric with respect to the maximum at around $U=8$.
Next, we investigate how the coupling strength to the local thermal baths $\nu^2$ affects the results. In the inset of Fig. \[fig:phase\_diag\] we present the dependence of the critical dephasing strength $\Gamma^c_\mathrm{dp}$ on this quantity for $U=8$. We observe that for $\nu^2\lessapprox0.5$ the two quantities are proportional to each other $\Gamma^c_\mathrm{dp}\sim\nu^2$, whereas at large $\nu^2$ the critical value of $\Gamma^c_\mathrm{dp}$ seems to be shifted away from proportionality to lower values. One possible mechanism to explain the proportionality for small $\nu^2$ is the presence of heat exchange between the lattice and the baths, as we expect the heat current in lattice systems to be proportional to $\nu^2$ [@titvinidze2017]. In this regime the critical value of the dephasing would then be determined by the rate at which the heat generated by the dephasing is taken out of the system.
At high $\nu^2$ we expect this behavior to change because of the increasing hybridization between the system and the local thermal baths. In this regime, the baths have a stronger effect on the system. Although the rate of cooling is faster, at the same time they do not favor CDW ordering since they are identical for both sublattices. However, whether this is the only mechanism affecting the behavior at high $\nu^2$ cannot be concluded.
In the cold-atom experiments the cooling rate is not easy to control (though not impossible [@chong2018]). In the absence of controllable coupling to a thermal bath, however, one could still observe experimentally the CDW phase if the time scale at which the system is heated by dissipation is much longer than the time scale at which the CDW ordering emerges.
To check whether we work with experimentally realistic physical parameters, we consider now a possible experimental realization with fermionic $^{40}$K atoms loaded into an optical lattice [@joerdens2008; @schneider2008]. As discussed in Sec. \[sec:ham\] such a system can be well described by the Hubbard model and in order to introduce non-local interactions one can couple the two spin states to highly excited Rydberg states in the weak dressing regime, see also App. \[appsec:dress\]. For sufficiently large detuning compared to the Rabi frequency $\delta\gg\Omega$ one obtains effectively dressed ground states with Eq. describing effective interaction potential of two atoms [@henkel2010]. Here the effective coefficient is given by $\tilde{C}_6=(\Omega/2\delta)^4 C_6$, with $C_6$ determining the strength of the van der Waals interactions between two Rydberg states. The soft-core radius is given by $R_c=(C_6/2|\delta|)^{1/6}$. Finally, the dephasing strength is determined via the spontaneous emission rate $\Gamma_\mathrm{se}$ of the excited state via $\Gamma_\mathrm{dp} = (\Omega/2\delta)^2\Gamma_\mathrm{se}$, see App. \[appsec:dress\].
Let us now consider a particular choice of the Rydberg state for the $^{40}$K atoms, namely the $|26S\rangle$ state. For this choice we obtain $C_6\approx 27\ \hbar\ \mathrm{MHz}\ \mu\mathrm{m}^6$ [@singer2005], and $\Gamma_\mathrm{se}\approx 60\ \hbar\ $kHz [@beterov2009]. A typical value of the hopping amplitude in optical lattice is on the order of $J\approx 0.5\ \hbar\ $kHz and the lattice spacing is on the order of $a\approx 0.5\mu$m.
We aim at realizing a model in which only nearest-neighbor interaction is relevant. Therefore, we set the parameters to $R_c=0.5\mu$m and $\tilde{C}_6= 31\ \hbar\ \mathrm{Hz}\ \mu\mathrm{m}^6$. In this case we would obtain a nearest-neighbor interaction strength $V\approx 2J$, the same as in our calculations, and a next-nearest-neighbor interaction which is at least one order of magnitude weaker. To obtain the required value of $R_c$ one needs to set the detuning on the order of $\delta\approx860\ \hbar\ $MHz. With this value of the detuning and in order to get the appropriate value of $\tilde{C}_6$, we need to set the Rabi frequency to $\Omega=56\ \hbar\ $MHz (suggesting the need for further development of current experimental capabilities). Finally, we use the values of $\Omega$ and $\delta$ to estimate the effective dephasing strength, which is approximately given by $\Gamma_\mathrm{dp}\approx 64 \ \hbar\ $Hz $=0.128J$ – on the order of magnitude of the maximal $\Gamma_\mathrm{dp}$ considered here. The time scale at which dephasing heats the system is approximately given by $\hbar/\Gamma_\mathrm{dp}\approx 16 ms$.
We note that there are several ways in which one can decrease the dephasing strength in experiment, e.g., (i) with higher values of the Rabi frequency one can target higher excited states, which have a longer lifetime, (ii) one can use a lattice with a larger lattice constant, which also allows to use higher excited states without increasing the Rabi frequency. Note, however, that in our analysis we have neglected effects of black body radiation, which are present in a typical experiment and can lead to both stronger dephasing and avalanche loss of particles from the system [@goldschmidt2016; @zeiher2016].
Conclusions {#sec:end}
===========
In this work we have studied the effect of dissipation on charge ordered density-wave phases in a strongly-correlated many-body quantum system with local and non-local interaction, as encompassed by the fermionic extended Hubbard model, with dissipation effects treated at the level of the quantum master equation. This model was solved using a recent variant of non-equilibrium dynamical mean-field theory, the Lindblad-DMFT, that allows to include local dissipation effects non-perturbatively.
By studying the behavior of the checkerboard CDW order parameter, we have demonstrated that a CDW phase, similar to the one present in the zero-temperature equilibrium model, survives the introduction of a dephasing process up to a critical strength, where the density ordering is destroyed and the system becomes homogeneous. We studied the steady-state phase diagram of the model as a function of the local interaction $U$ and dissipation strength $\Gamma_\mathrm{dp}$ and found that a broad region of density-ordered steady-states exists at relatively weak and moderate dephasing strengths. We observed that the critical value of local interaction $U$, where the phase transition between the homogeneous and CDW phases occurs, depends on the dephasing strength, with the CDW phase shrinking as the dephasing strength is increased. Importantly, we observed that to a certain extent the effect of dephasing on the CDW order seems to be due dominantly to heating, as we have observed that the critical value of $\Gamma^c_\mathrm{dp}$ is proportional to the coupling strength $\nu^2$ to the bath.
We expect that using cold atomic fermionic gases dressed with a Rydberg state – thus acquiring long-range interactions – and loaded into optical lattices could present an experimental realization of the extended Hubbard model. We showed that the parameters considered in our work are experimentally realistic. The remaining issue is to estimate the effect of other types of dissipation, and estimate the time scale at which CDW order emerges in such a system and make sure that it is much shorter then the time scale at which the system is heated by dephasing.
Support by the Deutsche Forschungsgemeinschaft via DFG SPP 1929 GiRyd, SFB/TR 49 and the high-performance computing center LOEWE-CSC is gratefully acknowledged. I.T. and M. S. acknowledge funding from the Austrian Science Fund (FWF) within Projects P26508 and F41 (SFB ViCoM). The authors also acknowledge useful discussions with K. Byczuk, C. Groß, H. Weimer, S. Whitlock and J. Zeiher.
Dressed regime {#appsec:dress}
==============
In order to simulate a spin-$\frac{1}{2}$ fermionic Hubbard model one can use two hyperfine states of $^{40}$K potassium atoms [@joerdens2008; @schneider2008]. To introduce the long range interaction we need to couple these states to high lying Rydberg excited states. Because we want the non-local interaction to be isotropic we either need to use $|nS\rangle$ Rydberg states with a three level excitation scheme [@goldschmidt2016; @aman2016] or $|nP\rangle$ Rydberg states with a two level excitation scheme [@zeiher2016], but then one needs to appropriately arrange the orientation of Rydberg states with respect to the 2D lattice. In both cases it is enough to work with a single effective Rabi frequency $\Omega_{eff}$ and detuning $\delta_{eff}$.
Because we need to couple two hyperfine states to excited Rydberg states one can choose either to couple both hyperfine states to the same Rydberg state, or to couple each hyperfine state to a different Rydberg state, as depicted in Fig. \[fig:model\]. The first approach gives the same inter- and intra-species non-local interaction, but introduces small coherent and incoherent spin flip processes. The second approach does not introduce spin-flip terms, but results in small differences in the non-local interaction strength of different species. In the following we assume that these differences are negligible.
The full model for the corresponding experimental set-up, in the rotating wave approximation, has the following Hamiltonian [@saha2014] $$\label{apeq:mod}
\begin{split}
\hat{H}_1=&\sum_{\sigma\in\{\uparrow,\downarrow\}} \sum_{\langle i, j \rangle} \left(-J\hat{f}^\dagger_{i,\sigma}\hat{f}^{\phantom\dag}_{j,\sigma} -\tilde{J}\hat{f}^\dagger_{i,R_\sigma}\hat{f}^{\phantom\dag}_{j,R_\sigma}\right) \\
& +\frac{\Omega_{eff}}{2} \sum_{i,\sigma\in\{\uparrow,\downarrow\}} \left(\hat{f}^\dagger_{i,\sigma}\hat{f}^{\phantom\dag}_{i,R_\sigma}+\mathrm{h.c.}\right) \\
&-\delta_{eff} \sum_{i,\sigma} \hat{n}_{i,R_\sigma} + U \sum_i \hat{n}_{i, \uparrow}\hat{n}_{i, \downarrow} \\
& + \sum_{i, j, \sigma, \sigma'}\frac{V_{RR}(\textbf{r}_i,\textbf{r}_j)}{2} \hat{n}_{i,R_\sigma} \hat{n}_{j,R_{\sigma'}} \\
& + \sum_{i, j,\sigma,\sigma'}V_{gR}(\textbf{r}_i,\textbf{r}_j) \hat{n}_{i,\sigma} \hat{n}_{j,R_{\sigma'}}.
\end{split}$$ Here, apart from terms appearing in the Eq. , we have the hopping amplitude $\tilde{J}$ of the excited states, the effective Rabi frequency $\Omega_{eff}$ and detuning $\delta_{eff}$, and non-local interaction strengths $V_{RR}(\textbf{r}_i,\textbf{r}_j)$, $V_{gR}(\textbf{r}_i,\textbf{r}_j)$. $\hat{f}_{i,\sigma}$ annihilates a ground state atom in a hyperfine state $\sigma$ on site $i$. We use the notation $\sigma\in\{\uparrow,\downarrow\}$ for the two hyperfine states as they are later interpreted as two spin states of the Hubbard model. $\hat{f}_{i,R_\sigma}$ annihilates on site $i$ an atom in an excited Rydberg state $R_\sigma$, to which the hyperfine state $\sigma$ is coupled, see Fig. \[fig:model\].
As the system is subject to dissipative processes, it is not enough to determine the Hamiltonian, but we also need to determine the Lindblad operators. Here we will consider only spontaneous emission. The Lindblad operator for the spontaneous emission is $$\label{apeq:spem}
\hat{L}^{se}_{i,\sigma} = \hat{f}^{\dag}_{i,\sigma}\hat{f}^{\phantom\dag}_{i,R_\sigma}$$ with the strength of dissipation given by the constant $\Gamma_\mathrm{se}$ independent of spin and position. When coupling to the $|nS\rangle$ state via a 3-level scheme [@goldschmidt2016; @aman2016], this form of the Lindblad operator is approximate, assuming that the decay of the atom to any intermediate state is immediately followed by decay to the ground state.
For the moment we consider a single atom without dephasing. Due to the Rabi driving its ground and excited states are no longer eigenstates of the full Hamiltonian. E.g., the lowest energy eigenstate has the form $|\tilde{\sigma}\rangle = \alpha|\sigma\rangle + \beta |R_{\sigma}\rangle$. Assuming that we are in the regime where $\delta_{eff}\gg\Omega_{eff}$, we have that $\alpha\approx1$ and $\beta\approx \Omega_{eff}/(2\delta_{eff})\ll 1$ [@zeiher2016]. The admixture of the excited state to the new eigenstate is small. The state $|\tilde{\sigma}\rangle$ is called the dressed ground state. Similarly, the dressed high energy eigenstate $|\tilde{R}_\sigma\rangle$ will be predominantly a Rydberg state, with a small admixture of the ground state. It is safe to assume that due to its high energy, the $|\tilde{R}_\sigma\rangle$ is empty, and only the dressed ground state is occupied.
Next we consider two atoms without dephasing. Due to the small admixture of Rydberg excitation to the dressed ground state, and due to the very strong non-local interaction between two Rydberg states, the dressed ground state will effectively be subject to non-local interaction. The strength of this interaction will be approximately proportional to $V_{eff}\approx\beta^4 V_{RR}$ [@zeiher2016]. This interaction shifts the atom even further from resonance and as a results the value of $\beta$ will depend on the distance separating two atoms, therefore the effective interaction will also have a renormalized shape, with a soft core cut-off for small atom-atom separations. In this way we get to the Eq. for the effective potential [@henkel2010].
Note that through appropriate choice of $\Omega_{eff}$ and $\delta_{eff}$ we can control the strength and shape of the interaction potential. Thanks to this flexibility we can set the parameters such that only nearest-neighbor interaction is relevant in our model. Using the notation in which $\hat{c}_{i,\sigma}$ and $\hat{c}_{i,R_\sigma}$ annihilate the dressed ground and excited states, respectively, corresponding to spin $\sigma$ on site $i$, we obtain the Hamiltonian .
Finally, we consider the effect of dissipation in the dressed regime. The operators corresponding to the dressed states can be written as $\hat{f}_{i,\sigma} = \alpha^\ast \hat{c}_{i,\sigma} - \beta \hat{c}_{i,R_\sigma}$ and $\hat{f}_{i,R_\sigma} = \beta^{\ast} \hat{c}_{i,\sigma} + \alpha \hat{c}_{i,R_\sigma}$ with $|\alpha|^2+|\beta|^2=1$. In this representation the Lindblad operator becomes $$\hat{L}^{se}_{i,\sigma} = \left(\alpha^\ast \hat{c}^\dag_{i,\sigma} - \beta \hat{c}^\dag_{i,R_\sigma}\right) \left(\beta^\ast \hat{c}^{\phantom\dag}_{i,\sigma} + \alpha \hat{c}^{\phantom\dag}_{i,R_\sigma}\right).$$ Under conditions in which $\alpha\approx 1$ and $\beta\ll1$ the term with pre-factor $\beta^2$ will vanish. As we also mentioned the dressed excited state is empty, as it is a high-energy eigenstate. Therefore terms with $\hat{c}^{\phantom\dag}_{i,R_\sigma}$ can also be neglected. One finally obtains $$\hat{L}^{se}_{i,\sigma} \approx \beta^\ast \hat{c}^\dag_{i,\sigma} \hat{c}^{\phantom\dag}_{i,\sigma}.$$ We obtain effectively a new type of dissipation, namely dephasing, with Lindblad operator given by and dissipation strength given by $\Gamma_\mathrm{dp}=|\beta|^2\Gamma_\mathrm{se}$. In this picture the dressed excited states corresponding to $\hat{c}_{i,R_\sigma}$ drop out completely.
Dissipative heating {#appsec:disheat}
===================
To show that the dephasing considered in section \[sec:ham\] indeed heats the system, we consider a case in which our model is decoupled from thermal baths and the dissipation strength is finite. We notice that the Liouvillian with Lindblad operator conserves the number of particles in the system for both spin species. Therefore, we will consider states with fixed number of particles at half-filling. In other cases the Fock space of the system can be split into subspaces with different occupations and analyzed separately.
The maximally mixed state of a system corresponds to an infinite temperature. We will show that such a state is indeed a steady-state of the system. The uniqueness of the steady-state [@kraus2008; @spohn1976; @spohn1977] has been studied for certain specific classes of Liouvillians, but we have not found a proof applicable in our case. Nevertheless, we will assume that within the subspace of fixed particle number the steady-state is unique. With this one can conclude that dephasing heats up the system.
The maximally mixed state within the Fock space of dimension $M$ is proportional to the identity operator $$\hat{\rho}_{MM}=\frac{1}{M} \hat{\mathbb{1}}.$$ The condition for it to be a steady-state reads $$\frac{d \hat{\rho}_{MM}}{d t} = -i\left[\hat{H}, \hat{\rho}_{MM} \right] + { \settoheight{\dhatheight}{\ensuremath{\hat{\mathcal{L}}}} \addtolength{\dhatheight}{-0.35ex} \hat{\vphantom{\rule{1pt}{\dhatheight}} \smash{\hat{\mathcal{L}}}}}[\hat{\rho}_{MM}] = 0$$ As any operator commutes with the identity operator, the first term on the right-hand side vanishes. Now we consider the dissipative part given by with the Lindblad jump operator and $\Gamma_{\mu,\nu}=\delta_{\mu,\nu} \Gamma_\mathrm{dp}$, where $\mu$ iterates over lattice $i$ and spin $\sigma$ indices. Notice that as the jump operator is hermitian we get $$2 \hat{L}^{\phantom\dag}_{dp,i,\sigma} \hat{\mathbb{1}} \hat{L}^\dag_{dp,i,\sigma} - \left\{\hat{L}^\dag_{dp,i,\sigma}\hat{L}^{\phantom\dag}_{dp,i,\sigma},\hat{\mathbb{1}} \right\} =0.$$ As a result, the time derivative of $\hat{\rho}_{MM}$ vanishes, which means that it is indeed a steady-state.
[^1]: If we neglect the Keldysh part in the decoupled system it is essential that the full system can thermalize either through interaction or through coupling to thermal bath.
[^2]: It includes effects of non-local interaction treated on the mean-field level
|
---
abstract: 'Based on the mathematical similarity between the Friedman open metric and Gödel’s one in the case of nearby distances, we investigate a new scenario for universe’s evolution, where the present Friedman universe is originated from a primordial Gödel’s one by a phase transition during which the cosmological constant vanishes. Using Hubble’s constant and the present matter density as input, we show that the radius and density of the primordial Gödel’s universe are close, in order of magnitude, to the present values, and that the time of expansion coincides with the age of universe in the standard Friedman model. Besides, the conservation of angular momentum provides, in this context, a possible origin for the rotation of galaxies, leading to a relation between masses and spins corroborated by observational data.'
---
[Saulo Carneiro]{}[^1]
[*Instituto de Física, Universidade Federal da Bahia\
40210-340, Salvador, BA, Brasil*]{}
Introduction
============
${ }$
In spite of the paradigmatic role that standard Friedman model has played in cosmology, astronomical observation is not sufficiently precise and rich to close definitely the question of what metric describes universe evolution [@Burbidge]. In fact, since the original Einstein static universe, many cosmological models have been proposed along the years [@Weinberg], each one of them in accordance with some particular aspect of observation.
An important aspect of universe evolution, from both theoretical and observational points of view, is the global rotation, present in the literature since the pioneer works of Gamow [@Gamow] and Gödel [@Godel], the latter presenting a stationary solution for Einstein equations of gravitation. Evidences of cosmological rotation have been reported [@Birch]-[@Kuhne], but always being object of controversy. Another difficulty is related to the fact “that it is impossible to combine pure rotation and expansion in a solution of the general relativity field equations for a simple physical matter source" [@KO].
In this paper we discuss a new scenario for universe evolution, in which the universe observed nowadays, described by a Friedman open metric, is originated from a primordial Gödel’s universe, through a phase transition during which the negative cosmological constant, characteristic of Gödel’s solution, vanishes. The possibility of such a scenario is suggested by a mathematical similarity between the Friedman open metric and Gödel’s one, when we consider nearby distances.
This scenario is in accordance with our paradigm of an expanding homogeneous universe and, in addition, it solves some problems of the standard model, as, for example, the presence of the initial singularity. In fact, using as input the Hubble constant and the observed matter density, we show that the radius and density of the primordial Gödel’s universe are close to the present values, while the time of expansion coincides with the age of universe in the standard Friedman model.
As a consequence of the conservation of angular momentum during the Gödel-Friedman phase transition, we show that the primordial global rotation can naturally explain the origin of rotation of galaxies and clusters, a hypothesis corroborated by the observed dipole anisotropy in the distribution of the rotation axes of galaxies and, even quantitatively, by the empirical relation between masses and spins of multi-stellar objects.
Of course, this model should be seen only as a [*possible*]{} scenario, among many others, for universe evolution. Within the spirit referred in the first paragraph, our goal is only to provide an additional theoretical alternative to understand the observational data we have in hand.
The Gödel-Friedman universe
===========================
${ }$
In spherical coordinates ($r=sh\chi$, $\theta$, $\phi$, $\eta$), the open metric of Friedman is given by [@Landau]
$$\label{1}
ds_F^2=a^2(\eta)[d\eta^2-d\chi^2-sh^2\chi(d\theta^2+sin^2\theta\;d\phi^2)]$$
while the Gödel one, written in cylindrical coordinates ($\rho=sh\xi$, $y$, $\phi$, $\eta$), has the form [@Godel]
$$\label{2}
ds_G^2=a^2[d\eta^2-d\xi^2-dy^2+(sh^4\xi-sh^2\xi)d\phi^2+2\sqrt{2}\;sh^2\xi
d\phi d\eta]$$
For nearby distances, $\xi\ll1$, and we have
$$\label{3}
ds_G^2 \approx a^2[d\eta^2-d\xi^2-dy^2-sh^2\xi d\phi^2+2\sqrt{2}\;sh^2\xi
d\phi d\eta]$$
From the transformations
$$\rho=r\;sin\theta$$
$$y=r\;cos\theta$$
relating spherical and cylindrical coordinates, it follows
$$\label{4}
sh\xi=sh\chi\;sin\theta$$
$$y=sh\chi\;cos\theta$$
By differentiation we have
$$d\xi = \frac{1}{ch\xi} (sh\chi\;cos\theta d\theta+ch\chi\;sin\theta d\chi)$$
$$dy=-sh\chi\;sin\theta d\theta+ch\chi\;cos\theta d\chi$$
So, by using
$$\frac{1}{ch^2\xi} = \frac{1}{1 + sh^2\xi} \approx 1 - sh^2\xi = 1 - sh^2\chi\;sin^2\theta$$
we obtain
$$\label{5}
dy^2+d\xi^2\approx\;sh^2\chi d\theta^2+d\chi^2$$
Substituting (\[4\]) and (\[5\]) into (\[3\]), one obtains
$$\label{6}
ds_G^2\approx a^2[d\eta^2-d\chi^2-sh^2\chi(d\theta^2+sin^2\theta
d\phi^2)+2\sqrt2\;sh^2\chi\;sin^2\theta d\phi d\eta]$$
Comparing (\[1\]) and (\[6\]), we see that, unless for the non-diagonal term, they are formally equal. This identification leads us to conjecture the following possible model for universe’s evolution: $ds^2=ds_G^2$ for $\eta
<\eta _0$; $ds^2=ds_F^2$ for $\eta \geq \eta _0$ and $\xi \ll 1$.
Initially, we have a Gödel universe, described by metric (\[2\]) or, for nearby distances, by metric (\[6\]). At $\eta=\eta_0$ a phase transition occurs, and the cosmological constant vanishes. Now, the metric is non-stationary and, for nearby distances, it coincides with the open metric of Friedman. In the next section, we shall see that this match can be performed in such a way that the scale parameter $a(\eta)$ changes continuously.
Though in principle we could find different ways to explain the phase transition from Gödel’s universe to the Friedman one, a plausible mechanism is provided if the cosmological constant is identified with the energy density of a scalar field $\phi$, with a self-interaction potential given, for instance, by [@FL]
$$\label{FL}
V(\phi) = a\phi^2 - b\phi^3 + c\phi^4$$
with $a,b,c>0$.
It is easy to verify that, for $b>2\sqrt{ac}$, this potential presents a local minimum at $\phi=0$, for which $V(\phi)=0$, and an absolute minimum at
$$\phi = \frac{3b}{8c}\left[1 + \left(1-\frac{32ac}{9b^2}\right)^{1/2}\right]$$
for which $V(\phi)$ has a negative value.
Therefore, the desired phase transition can be achieved, for instance, by a tunneling of the scalar field from the absolute minimum, for which the potential (\[FL\]) (and then the cosmological constant) has a negative value – as should be in Gödel’s model – to the local minimum, for which the potential vanishes, leading to a null cosmological constant – as expected in the standard Friedman model. In section 6 we will return to this point, particularly in what concerns the probability of such a transition.
Numerical predictions
=====================
${ }$
For $\eta\geq\eta_0$, the open model predicts [@Landau]
$$\label{8}
a(\eta)=a_0(ch\eta-1)$$
$$\label{9}
t(\eta)=\frac{a_0}{c}(sh\eta-\eta)$$
$$\label{10}
\rho a^3(\eta)=\frac{3c^2}{4\pi k}a_0$$
where $a_0$ is a constant, $\rho$ is the matter density, $k$ is the gravitational constant and we have introduced the light velocity $c$.
From (\[8\]) and (\[10\]), it follows
$$\label{11}
a=a_0(ch\eta_0-1)=\frac{4\pi k\rho_G}{3c^2}a^3(ch\eta_0-1)$$
which, by using the Gödel radius [@Godel] $a^2=c^2(2\pi k\rho_G)^{-1}$, leads to $%
ch\eta_0=5/2$, or $\eta_0=1.6$, whatever the values of $a$ and $\rho_G$, the matter density in the Gödel phase.
Equations (\[8\])-(\[10\]) give
$$\label{12}
ch\left(\frac{\eta}{2}\right)=H_0\left(\frac{3}{8\pi k\rho}\right)^{\frac{1}{%
2}}$$
$$\label{14}
a(\eta)=\frac{c}{H_0} coth\left(\frac{\eta}{2}\right)$$
$$\label{15}
t(\eta)=\frac{1}{%
H_0}\left[\frac{coth(\eta/2)(sh\eta-\eta)}{ch\eta-1}\right]$$
where $H_0 \equiv \dot{a}(\eta)/a^2(\eta) = h \times 100$ km/(sMpc) is the Hubble constant. Using $h = 0.75$, $k=6.7 \times
10^{-11}$ m$^3$/kg.s$^2$ and $\rho=1.0\times10^{-27}$ kg/m$^3$ ($\Omega = 0.1$), we have, for the present values of $\eta$, $a(\eta)$ and $t(\eta)$, the values $\eta=3.7$, $%
a(\eta)=1.3 \times 10^{26}$m and $t(\eta)=3.8
\times 10^{17}$s.
Now, from (\[8\]) we obtain
$$\label{13}
a(\eta)=a\left(\frac{ch\eta-1}{ch\eta_0-1}\right)$$
which gives $a= 1.1
\times 10^{25}$m.
Moreover, (\[9\]) leads to
$$\label{16}
t(\eta_0)=t(\eta)\left(\frac{sh\eta_0-\eta_0}{sh\eta-\eta}\right)$$
i.e., to a time of expansion given by $\Delta t \equiv t(\eta) - t(\eta_0) = 3.6 \times
10^{17}$s.
For the open model, (\[10\]) gives $\rho a^3(\eta)=\mbox{constant}$. Therefore, $\rho a^3(\eta)=\rho_G a^3$, which leads to $\rho_G= 1.7 \times
10^{-24}$kg/m$^3$.
Finally, this last result leads us to an angular velocity for Gödel’s primordial universe given by [@Godel] $\omega_G = 2(\pi k \rho_G)^{1/2}
= 3.8 \times 10^{-17}$s$^{-1}$.
Therefore, by using only two observed parameters as input, namely the values for the Hubble constant and for the present matter density, we conclude that the values for Gödel’s radius and density are close to the present values, and that the time of expansion coincides with the age of universe in the standard model.
It is important to verify how sensitive these results are respect to the used value for the matter density of the present universe. If we use, instead of $\Omega = 0.1$, the value $\rho=1.0%
\times10^{-28}$kg/m$^3$ ($\Omega = 0.01$), it is easy to obtain from the above formulae the results $\eta=6.0$, $a(\eta)=1.3\times10^{26}$m, $a=1.0\times10^{24}$m, $%
t(\eta)=3.2\times10^{17}$s, $\Delta t=3.2\times10^{17}$s, $%
\rho_G=2.2\times10^{-22}$kg/m$^3$ and $\omega_G=4.3\times10^{-16}$s$^{-1}$.
A possible origin for galaxies rotation
=======================================
${ }$
An important problem related to the Gödel-Friedman phase transition is the conservation of angular momentum: the Friedman model is isotropic, while in Gödel’s one matter everywhere rotates relative to local gyroscopes. Now, we will see that the conservation of angular momentum in the context of the Gödel-Friedman phase transition leads naturally to a possible explanation for the origin of rotation of galaxies and clusters. For this purpose, we shall adapt to our context a reasoning originally made by L. Li [@Li] to describe galaxies formation in a rotating and expanding universe.
Let us consider, in the primordial Gödel’s universe, a proto-galaxy with mass $M$, radius $r$ and density $\rho_G$, rotating with angular velocity $%
\omega_G$ with respect to a local inertial frame (a gyroscope frame). Assuming for this proto-galaxy a spherically symmetric distribution of matter, its angular momentum relative to the gyroscope frame will be given by $J=2Mr^2\omega_G/5$, which can be rewritten, using $M=4\pi r^3\rho_G/3$, as
$$\label{Li}
J=\frac{2}{5}\left(\frac{3}{4\pi\rho_G}\right)^{2/3}\omega_GM^{5/3}$$
On the other hand, with respect to a galaxy frame whose origin is fixed at the proto-galaxy center, the proto-galaxy angular momentum is zero because, by definition, the galaxy frame co-rotates with the global rotation.
After the phase transition from Gödel’s universe to the Friedman one, the galaxy frame does not rotate anymore with respect to gyroscope ones, and the galaxy angular momentum relative to it will be given, due to the conservation of angular momentum, by (\[Li\]).
In this way, besides providing a possible origin for galaxies rotation, the Gödel-Friedman phase transition leads to a relation between angular momentum and mass of multi-stellar objects corroborated by observational data. In fact, the masses and spins of spiral galaxies, globular clusters and clusters of galaxies (including the Local Supercluster)[^2] can be fitted by the relation $J=\kappa
M^{5/3}$, with $\kappa\approx6.2\times10^{-2}$ (SI units)[^3].
A theoretical value for $\kappa$ can be predicted from (\[Li\]), but it will depend on $\rho_G$ and $\omega_G$, i.e., on the value of the present matter density used as input. For $\Omega = 0.1$ we have $%
\kappa\approx4.1\times10^{-2}$ and for $\Omega = 0.01$ we obtain $\kappa\approx1.8\times10^{-2}$ (SI units).
If this picture for the origin of galaxies angular momentum is correct, an alignment of galaxies angular momenta along the direction of global rotation of the primordial Gödel’s universe would be expected. This prediction was already supported in the literature by the discovery of a dipole anisotropy in the distribution of the rotation axes of galaxies [@Li], but, as pointed out by Li, an irregular shape of the proto-galaxies can lead, after the phase transition, to an almost random distribution [@Li].
The global metric
=================
${ }$
What happens for large distances after the phase transition? To answer this question, let us write the Gödel metric in the form
$$\label{global}
ds_G^2=a^2\{[d\eta + \Lambda sh^2(\xi/2)d\phi]^2
-(d\xi^2+dy^2+sh^2\xi d\phi^2)\}$$
with $\Lambda = 2\sqrt{2}$. The $\Lambda$-term originates the non-diagonal one in the nearby approximation (\[6\]), what suggests that the global metric arising from the phase transition should be given by the non-stationary, anisotropic metric
$$\label{anisotropic}
ds^2=a^2(\eta)[d\eta^2-(d\xi^2+dy^2+sh^2\xi d\phi^2)]$$
This metric (which reduces to the Friedman open metric for nearby distances) is solution of Einstein equations of gravitation, with $\rho<\rho_c$, where $\rho_c = 3H_0^2/8\pi k$ is the critical density at the moment of observation. From it, instead of relations (\[8\])-(\[10\]), we can obtain the following ones
$$\label{8'}
a(\eta)= a_0 \left[ch\left(\frac{\eta}{\sqrt{3}}\right)-1\right]$$
$$\label{9'}
t(\eta)=\frac{\sqrt{3}a_0}{c}\left[sh\left(\frac{\eta}{\sqrt{3}}\right)
-\frac{\eta}{\sqrt{3}}\right]$$
$$\label{10'}
\rho a^3(\eta)=\frac{c^2}{4\pi k}a_0$$
From them, it is easy to re-obtain the numerical predictions of section $3$. For $\Omega=0.1$, we obtain $\eta_0=3.1$ (actually this value does not depends on $\rho$), $\eta=6.6$, $a(\eta)=0.7\times10^{26}$m, $a=6.2\times10^{24}$m, $%
t(\eta)=3.5\times10^{17}$s, $\Delta t=3.3\times10^{17}$s, $%
\rho_G=1.4\times10^{-24}$kg/m$^3$ and $\omega_G=3.4\times10^{-17}$s$^{-1}$. Besides, for the parameter $\kappa$ introduced in section $4$, we obtain the value $\kappa=4.2\times10^{-2}$ (SI units). On the other hand, for $\Omega=0.01$, we have $\eta=10.5$, $a(\eta)=0.7\times10^{26}$m, $a=6.5\times10^{23}$m, $%
t(\eta)=4.0\times10^{17}$s, $\Delta t=4.0\times10^{17}$s, $%
\rho_G=1.3\times10^{-22}$kg/m$^3$, $\omega_G=3.3\times10^{-16}$s$^{-1}$ and $\kappa=2.0\times10^{-2}$ (SI units).
Concluding remarks
==================
${ }$
We have seen that the mathematical similarity between the Friedman open metric and Gödel’s one in the limit of nearby distances suggests conjecturing about a new cosmological scenario for universe’s evolution, where the nowadays expanding universe is originated from a primordial Gödel’s one. Besides providing a possible origin for the rotation of galaxies, such a cosmology solves some problems of the standard Friedman model, as the presence of an initial singularity and problems related to the age of universe: before the Gödel-Friedman phase transition, universe could have existed for a long time...
Of course, there remain several open questions. How to explain hot universe phenomena, e.g. the cosmic microwave background radiation or the nucleosynthesis, in the context of a cosmological scenario where no dense phase exists? Despite the possibility of explaining those phenomena without reference to a hot phase [@Burbidge], this question should be elucidated in order to put this model in a more physical basis.
Another point which deserves better understanding is the mechanism of phase transition. What is the probability of tunneling for the scalar field? Is there inflation or the production of bubbles and topological defects? Must the cosmological constant be exactly zero after the phase transition?
Actually, it is possible that all these problems are related, and that a more detailed study of the phase transition mechanism could shed light on problems like the origin of the cosmic microwave background.
As discussed in section 2, a possible scenario for the phase transition is provided by a tunneling of the scalar field from the absolute minimum of potential (\[FL\]) to the local one. This kind of transition may be very improbable, but it is precisely this fact that turns this scenario an interesting one. Before the phase transition, we have a Gödel, stationary universe, where no cosmological time can be defined. During the phase transition, we have a huge fluctuation with decrease of entropy of the matter content, after which the universe starts expanding and the cosmological time arrow coincides with the thermodynamic one. Despite to be much improbable, this scenario is not forbidden by any physical law, and the anthropic principle could be used to justify it.
Another possibility is to consider (\[FL\]) as an effective, temperature dependent potential. In this case, the phase transition can be caused by a change in the potential itself, due to possible thermal processes in the primordial phase. Such kind of transition of potential (\[FL\]) was already investigated in the scenario of QCD deconfinement [@QCD].
In any case, a detailed study of the phase transition should provide us a continuous match between the Gödel metric and the Friedman one, instead of the discontinuous match considered in this paper, that could turn the model richer in several aspects. A continuous transition from the absolute minimum of (\[FL\]) to its local minimum necessarily crosses its local, positive maximum, from which the potential rolls down to zero. This means that in the latest phase of the transition a huge amount of energy should be released, perhaps followed by an inflation process, and that a reminiscent, positive cosmological constant could exist nowadays, as suggested by recent observations [@Supernova]. Besides, the necessity of a finite interval of time for the transition to occur can lead to a primordial phase more dense than the one obtained in this work.
Its also important to investigate the role of closed time-like curves, characteristic of the Gödel universe, in the process of phase transition. Though we are considering, in the Gödel phase, only gravitational interaction between dust matter, the appearance of CTC´s can occur during the phase transition itself, where a self-interacting scalar field enters into play. In any case, note that the eventual appearance of CTCs does not affect the match considered in section 2, where only nearby distances are considered. However, their presence can be important when considering large ones, as done in section 5.
Finally, we would like to refer to the relation between this model and the ideas concerning the similarity between the universe and elementary particles [@SC]. If such similarity has a true physical significance, we could say that the Gödel-Friedman model describes a [*decaying*]{} universe. In this context, the Dirac hypothesis [@Dirac] about the dependence of physical quantities, as the gravitational constant, on the cosmological time is not necessary anymore: the similarity should be found between particles (the electron, in the Dirac original conjecture, or hadrons, in more recent ones) and the primordial Gödel’s universe. In fact, using the values obtained for $a$, $\rho_G$ and $\omega_G$ and typical values for the radius $r$, mass $m$ and spin $\hbar$ of a particle, one can check the relations $a/r \sim
(M/m)^{1/2} \sim (J/\hbar)^{1/3} \sim 10^{40}$, where $M \sim \rho_G a^3$ and $J \sim M \omega_G a^2$ are, respectively, the mass and angular momentum of the primordial universe.
Acknowledgements {#acknowledgements .unnumbered}
================
${ }$
I would like to thank P.F. González-Díaz and A. Saa for the encouragement, R. Muradian, E. Radu and A.E. Santana for useful discussions, and N. Andion for the reading of the manuscript.
G. Burbidge, IAU Symposium No. 183, Kyoto, Japan, 1997 (Kluwer Academic Publishers) \[astro-ph/9711033\].
S. Weinberg, 1972, [*Gravitation and Cosmology*]{} (John Wiley & Sons), chapter 16.
G. Gamow, Nature 158(1946)549.
K. Gödel, Rev.Mod.Phys. 21(1949)447.
P. Birch, Nature 298(1982)451; 301(1983)736.
B. Nodland and J.P. Ralston, Phys.Rev.Lett. 78(1997)3043.
Y.N. Obukhov, V.A. Korotky and F.W. Hehl, astro-ph/9705243.
R.W. Kühne, Mod.Phys.Lett.A 12(1997)2473.
V.A. Korotky and Y.N. Obukhov, [*Gravity, Particles and Space-Time*]{}, eds. P. Pronin and G. Sardanashvily (World Scientific: Singapore, 1996), pp. 421-439 \[gr-qc/9604049\].
L.D. Landau and E.M. Lifshitz, 1983, [*The Classical Theory of Fields*]{} (Oxford: Pergamon), §§111-114.
R. Friedberg and T.D. Lee, Phys.Rev.D 15(1977)1694.
L. Li, Gen.Rel.Grav. 30(1998)497.
R. Muradian, Astrophys. Space Sci. 69(1980)339.
I. Bednarek, M. Biesiada and R. Manka, astro-ph/9608053.
P.M. Garnavich [*et al.*]{}, Astrophys.J. 509(1998)74.
S. Carneiro, Found.Phys.Lett. 11(1998)95, and references therein.
P.A.M. Dirac, Nature 139(1937)323.
[^1]: E-mail: saulo@fis.ufba.br
[^2]: See Table I of [@Muradian].
[^3]: If we use the relation $J=\kappa M^n$, with $n$ free, the fitting gives $n
\approx 1.69$ (while $5/3 \approx 1.67$) and $\kappa\approx0.48\times10^{-2}$ (SI units), in accordance with the value referred by Li [@Li].
|
---
abstract: 'The rapid advances in sensors and ultra-low power wireless communication has enabled a new generation of wireless sensor networks: Wireless Body Area Networks (WBAN). To the best of our knowledge the current paper is the first to address broadcast in WBAN. We first analyze several broadcast strategies inspired from the area of Delay Tolerant Networks (DTN). The proposed strategies are evaluated via the OMNET++ simulator that we enriched with realistic human body mobility models and channel models issued from the recent research on biomedical and health informatics. Contrary to the common expectation, our results show that existing research in DTN cannot be transposed without significant modifications in WBANs area. That is, existing broadcast strategies for DTNs do not perform well with human body mobility. However, our extensive simulations give valuable insights and directions for designing efficient broadcast in WBAN. Furthermore, we propose a novel broadcast strategy that outperforms the existing ones in terms of end-to-end delay, network coverage and energy consumption. Additionally, we performed investigations of independent interest related to the ability of all the studied strategies to ensure the total order delivery property when stressed with various packet rates. These investigations open new and challenging research directions.'
author:
- '[^1]'
bibliography:
- 'biblioJournal.bib'
title: 'Broadcast Strategies and Performance Evaluation of IEEE 802.15.4 in Wireless Body Area Networks WBAN'
---
Omnet++, WBAN, Broadcast, mobility model.
Simulation settings and results {#sec:simulation}
===============================
In order to test the algorithms described above (Section \[BS\]) in a specific WBAN scenario, we implemented them under the Omnet++ simulator. Omnet++ includes a set of modules that specifically model the lower network layers of WSN and WBAN through the Mixim project [@mixim]. It includes a set of propagation models, electronics and power consumption models and medium access control protocols. The MoBAN framework for Omnet++ adds mobility models for WBANs of $12$ nodes in $4$ different postures. Unfortunately studying in deep the MoBAN code, we discovered that it mainly focuses on the mobility resulting from the change of position, rather than describing coherent and continuous movements. Besides, it models the movement of each node with respect to the centroid of the body and the signal attenuation between couples of nodes is approximated with a simple propagation formula that is not accurate enough to model low-power on-body transmission. It does not model absorption and reflection effects due to the body, alterations due to the presence of clothes and eventually interference from other technologies at the same frequency since 2.4GHz is a crowded band.
Channel model
-------------
We therefore decided to implement a realistic channel model published in [@channel] over the physical layer implementation provided by the Mixim framework. This channel model of an on-body $2.45\,GHz$ channel between $7$ nodes, that belong to the same WBAN, using small directional antennas modeled as if they were $1.5cm$ away from the body. Nodes are assumed to be attached to the human body on the head, chest, upper arm, wrist, navel, thigh, and ankle.
The nodes positions are calculated in $7$ postures: walking (walk), walking weakly (weak), running (run), sitting down (sit), wearing a jacket (wear), sleeping (sleep), and lying down (lie). these postures are represented on Fig. \[fig:postures\]. Walk, weak, and run are variations of walking motions. Sit and lie are variations of up-and-down movement. Wear and sleep are relatively irregular postures and movements. Channel attenuation is calculated between each couple of nodes for each of these positions as the average attenuation (in dB) and the standard deviation (in dBm). The model takes into account: the shadowing, reflection, diffraction, and scattering by body parts.
[0.9]{} ![Postures used in [@channel] to model the WBAN channel (Pictures source: [@channel])[]{data-label="fig:postures"}](Pictures/MobilityModel/pos-walk.png "fig:"){width="\textwidth" height="2.7cm"}
[0.9]{} ![Postures used in [@channel] to model the WBAN channel (Pictures source: [@channel])[]{data-label="fig:postures"}](Pictures/MobilityModel/pos-weak.png "fig:"){width="\textwidth" height="2.5cm"}
[0.9]{} ![Postures used in [@channel] to model the WBAN channel (Pictures source: [@channel])[]{data-label="fig:postures"}](Pictures/MobilityModel/pos-run.png "fig:"){width="\textwidth" height="2.5cm"}
[0.9]{} ![Postures used in [@channel] to model the WBAN channel (Pictures source: [@channel])[]{data-label="fig:postures"}](Pictures/MobilityModel/pos-sit.png "fig:"){width="\textwidth" height="2.5cm"}
[0.9]{} ![Postures used in [@channel] to model the WBAN channel (Pictures source: [@channel])[]{data-label="fig:postures"}](Pictures/MobilityModel/pos-lie.png "fig:"){width="\textwidth" height="2.5cm"}
[0.9]{} ![Postures used in [@channel] to model the WBAN channel (Pictures source: [@channel])[]{data-label="fig:postures"}](Pictures/MobilityModel/pos-sleep.png "fig:"){width="\textwidth" height="2.5cm"}
[0.9]{} ![Postures used in [@channel] to model the WBAN channel (Pictures source: [@channel])[]{data-label="fig:postures"}](Pictures/MobilityModel/pos-jacket.png "fig:"){width="\textwidth" height="2.5cm"}
Simulation Settings
-------------------
Above the channel model described in the previous section, we used standard protocol implementations provided by the Mixim framework. In particular, we used, for the medium access control layer, the IEEE 802.15.4 implementation. The sensitivity levels, header length of the packets and other basic information and parameters are taken from the 802.15.4 standards.
Each data point is the average of 50 simulations run with different seeds. The simulations are performed with only one packet transmission from the gateway (i.e the sink). The transmission power is set at the minimum limit level $-60\,dBm$ that allows an intermittent communication given the channel attenuation and the receiver sensitivity $-100\,dBm$. With this value of transmission power, we guarantee that at each time t of the simulation, we have a connected network and at the same time we ensure a limited energy consumption.
Simulation results
------------------
To compare between the different broadcast strategies presented in the previous section \[BS\] ([^2]), we studied for each one three basic parameters:
- **Percentage of covered nodes:** Since our unique source is the sink which broadcasts a single message in the network, we therefore calculate the percentage of nodes that have received this message.
- **End-to-end delay**:The average End-to-end delay is the time a message takes to reach the destination(i.e. every node except the sink).
- **Number of transmissions and receptions**:This is a key parameter, it helps us to have an estimation of the energy consumption at each node and for each posture for all the strategies.
### Percentage of covered nodes
We first compare the percentage of covered nodes after the broadcast of the message from the sink. Figures \[Fig2\] and \[Fig3\] show this percentage variations for each algorithm in function of the *TTL* “Time-To-Live” set for the packets. The choice of the *TTL* is justified by the fact that it is the only parameter in common between all the considered algorithms. The first observation is about the value of the percentage of covered nodes when *TTL* is equal to one. As shown in figures \[Fig2\] and \[Fig3\], it is almost the same for all strategies and is equal to 51% except for *EBP*. This value shows that half of the network is covered after the first transmission by the sink. For these simulations, human body posture is *walk posture*. In this posture, the sink has at least three neighbors that can reach at the first time which explain the percentage of covered nodes with $TTL=1$ (the other postures are shown in the figure \[Fig4\]). The slight decrease seen with *EBP* strategy is due to packet losses caused by collisions with control packets.
![Percentage of covered nodes (first group)[]{data-label="Fig2"}](Pictures/OnePacketPerSecond/Fig2.jpg){width="0.9\columnwidth"}
The strategy *Flooding* represents the reference of flooding-based strategies. With this strategy nodes continue to retransmit the packet while its *TTL* is greater than $1$, regardless of the past. For this reason, and in absence of conditions on retransmissions, strategy *Flooding* shows the highest percentage of covered nodes. For *EBP*, $I$ represents the interval between two successive control message *Hello message*. Each node send periodically a *Hello message* to discover the neighborhood since the decision to rebroadcast or not the message depends on the neighborhood state. *EBP* shows good results with a slight increase when $I=0.25$. Indeed, with $I=0.5$, the node has a less realistic view of the neighborhood which could force it to decide to not rebroadcast the message. A notable observation is that EBP stops automatically the broadcast when the each node’s neighbors are covered, while the classical *Flooding* algorithm (Figure \[Fig2\]) has to wait until the *TTL* of all packets reaches $1$. *Plain Flooding* and *Tabu Flooding* coverage starts to increase with the *TTL* and stabilize rapidly. Both algorithms stop even when the network is not fully covered. In case of *Plain Flooding* a few number of transmissions is allowed, only once, which corresponds to the first reception of the message. For *Tabu Flooding*, nodes only rebroadcast messages, if it is intended for them, and only for uncovered node. So in case the destination is not in the sender’s range, the message will be dropped by the other nodes. As it is very hard for these two algorithms to reach a full coverage, the notions of completion delay and the required amount of transmissions to cover the whole network are meaningless. Therefore these algorithms will be left out of the subsequent evaluations. The *Probabilistic Flooding* with decreasing probability ($P=P/2$) achieves a better coverage than the constant probability version with $P=0.5$. In fact, with *Probabilistic Flooding*, $P$ initially equal to $1$, which means that the probability to rebroadcast a message for the first time is equal to $1$ too, since the chosen random variable is necessarily less than $P$.
![Percentage of covered nodes (Second group)[]{data-label="Fig3"}](Pictures/OnePacketPerSecond/Fig3.jpg){width="0.9\columnwidth"}
For *Pruned Flooding* strategy , $K$ represents the number of nodes, a node chooses randomly to forwards the message to. In order to study the influence of this parameter on the behavior of the strategy, we varied the value of $K$ from $2$ to $5$. Figure\[Fig3\] shows that the number of nodes to choose randomly has a direct effect on the coverage percentage. With $K=2 and K=3$ it is almost impossible to cover all the nodes even with higher values of *TTL*. When $K=5$, we have better results but still the lowest percentage comparing to the other strategies. For *MBP*, $NH$ represents the threshold on the total number of hops the message passed through. We therefore studied the influence of this parameter on *MBP* results. Figure\[Fig3\] shows a slight variation depending on $NH$ parameter especially between $NH=2$ and $NH=3$. In general, *MBP* shows a good percentage of covered nodes,specially with low values of *TTL*, for example, with $TTL=3$, the strategy covers between $80$% and $85$% (with different values of $NH$) of the nodes. The novel strategy *Optimized Flooding* has good results. For example, with $TTL=4$, $90$% of the network is covered. As this strategy is an adaptation of the strategy *Flooding*, the percentage of covered nodes values and even the variation in function of the TTL are almost the same.
![Percentage of covered nodes per posture[]{data-label="Fig4"}](Pictures/OnePacketPerSecond/Fig4.jpg){width="0.9\columnwidth"}
Figure \[Fig4\] represents the network coverage percentage in different postures. The lowest percentage, independently of the posture, is with *Pruned Flooding* strategy, with $K=3$ a low number of nodes are designated to rebroadcast the message, knowing that a node doesn’t rebroadcast a message destined to another node. *Probabilistic Flooding* shows a better percentage than *Pruned Flooding* strategy but still the lowest. Both strategies have higher variations among the postures due to their random nature. Strategy *Flooding* shows the highest percentage in almost all postures. However, *MBP*, *EBP* and *Optimized Flooding* show good percentage of covered nodes between $70$% and $96$%. For these strategies and for the percentage of covered nodes parameter, even if a variation in function of the postures is observed, it still less significant than for the two other studied parameters: End-To-End delay (\[EED\]) and number of transmissions and receptions (\[TXRX\]). It is important to point that the sleep posture is a specific case that represents a static posture where some nodes are hidden and remain so all along the simulation. Even so, *Optimized Flooding* reaches $70$% of covered nodes.
### Average End-To-End Delay {#EED}
![Average End-to-End delay comparison between strategies[]{data-label="Fig55"}](Pictures/OnePacketPerSecond/Fig55.jpg){width="0.9\columnwidth"}
Figure \[Fig55\] represents the average End-to-End delay, i.e. the average time required for a message sent from the sink to reach a node. As expected, *Flooding* algorithm has the best performance due to the huge amount of message copies in the network. The strategy *Probabilistic Flooding (P=P/2)* shows a higher end-to-end delay due to the decreasing amount of retransmissions with respect to the flooding. The end-to-end delay achieved by *EBP* is strictly dependent from the *Hello message* interval ($I$). When $I$ is short, for instance $I=0.25s$, each time the connections in the model change, it is soon considered by nodes, resulting in a precise acquisition of the network state and a quick decision making. *Pruned Flooding* end-to-end delay is also directly related to the value of $K$ parameter: the higher is the value of $K$, the lower is the end-to-end delay. In particular in cases when $K=2$ and $K=3$, the algorithm spends a lot of time trying to cover the whole network. This is due to the low number of nodes that receive the packet and are designated to retransmit. *MBP* presents better performances in term of End-to-End delay than *EBP* and *Pruned Flooding*, even if, with this strategy, a node delays packet retransmission after T timer expiration. *MBP* End-to-End delay depends also on $NH$ value. With $NH=3$, *MBP* attempts to cover the nodes more quickly than for lower values, because, in this case, nodes use to rebroadcast the message faster based on simple flooding without delaying the retransmission. Finally, *Optimized Flooding* has an end-to-end delay close to *Flooding* strategy and it is also the best End-To-End delay after *Flooding*. Our first goal is reached since the idea is to maintain the low End-To-End delay given by *Flooding*.
![Average End-to-End Delay per node[]{data-label="Fig6"}](Pictures/OnePacketPerSecond/Fig6.jpg){width="0.9\columnwidth"}
Figure \[Fig6\] details the average End-to-End delay per node. We can notice that all algorithms behave similarly: they spend more time to cover the peripheral nodes in the network (example: node at the ankle). Indeed, peripheral nodes are highly mobile and located at a greater distance from the sink than the other ones. As a result, even if an algorithm is able to cover the central part of the network in a very short time, it then has to wait for a connection opportunity with the last nodes. *EBP* and *pruned Flooding* present the worst End-To-End delay. In case of *EBP*, nodes wait before retransmitting the message until all conditions are satisfied, i.e, the number of nodes in the neighborhood. Both of *Flooding* and *Optimized Flooding* strategies have the lowest End-To-End delay for all the nodes. These results are especially important regarding the end-to-end delay of the peripheral nodes. *Optimized Flooding* comes to deal moderately with this limit comparing to the other strategies including *MBP*.
![Average End-to-End delay per posture[]{data-label="Fig7"}](Pictures/OnePacketPerSecond/Fig7.jpg){width="0.9\columnwidth"}
Figure \[Fig7\] shows the average End-To-End delay per posture. *EBP* seems to perform better in the high mobility postures. Indeed, in *EBP*, because the number of neighbors is important, a peak is noticed in sleep position where some nodes are hidden and remain so all along the simulation, for that, nodes decide to not rebroadcast the message and keep it until all conditions are satisfied. In other hand, *Flooding*, *Optimized Flooding* and *MBP* are the less affected by human body postures than *EBP* and *Pruned Flooding*. However, we notice better results (closest to *Flooding*) and less variations in function of the postures with *Optimized Flooding* than with *MBP*.
### Number of transmissions and receptions {#TXRX}
The number of transmissions and receptions is a key parameter for WBAN network. The number of transmissions reflects the channel load and also gives an indication about the amount of electro-magnetic energy that will be absorbed by the body. The number of receptions gives an indication on the energy consumption of the different nodes in the network and hence on the devices autonomy, their capability to rely on a reduced size battery or even to harvest energy.
![Number of transmissions and receptions[]{data-label="Fig8"}](Pictures/OnePacketPerSecond/Fig88.jpg){width="0.9\columnwidth"}
Figure \[Fig8\] compares the total number of transmissions and receptions for all the studied algorithms. *Flooding* pays its good End-To-End delay and coverage performance as it exhibits the highest number which indicates a high energy consumption. It is the only algorithm that has no limitation on the number of transmissions besides the *TTL*. The number of transmissions of *EBP* varies with $I$. In fact, *EBP* strongly relies on the transmission of *hello messages* that cannot be neglected, this causes collisions with data packets (especially with shorter interval $I$) leading to lots of dropped packets and to unnecessary retransmissions. However it is important to note here that these results only account for data packets transmissions and receptions. The adaptive *Probabilistic Flooding (P=P/2)* has a better performance than the classical *Flooding*, because the probability decrease limits the number of retransmissions as the packet travels in the network. The energy consumed by *Pruned Flooding* depends on the parameter $K$: while for the end-to-end delay and coverage, the algorithm performs well with high values of $K$, in this case a visible reduction in number of transmissions and receptions is observed with lower values of $K$. For each retransmission, the node duplicates the packet as much as the number of nodes to choose randomly i.e as much as $K$ value. *MBP* does not suffer from the same issue, even if control messages such as acknowledgments are also necessary. In fact, acknowledgments are only sent when a packet is effectively forwarded, the control traffic is directly related to the network activity. *MBP* performance therefore mostly depends on the $NH$ parameter and lowering this value results in a better energy efficiency. Finally, with the strategy *Optimized Flooding*, a significant decrease in the number of transmissions and receptions is observed: $50$% compared to *Flooding*. It’s an important result. Remember that the unique limitation of *Flooding* strategy is related to the number of transmissions and receptions in other words the energy consumptions.
![Number of transmissions and receptions per node[]{data-label="Fig9"}](Pictures/OnePacketPerSecond/Fig9.jpg){width="0.9\columnwidth"}
Figure \[Fig9\] shows the total number of transmissions and receptions for each node. We observe that nodes at the center of the network (navel and chest) have the highest number of transmissions and receptions, their position allows them to communicate with more nodes. On the contrary the node on the ankle is the one with the lowest number of transmissions and receptions because standing in the periphery has few occasions to communicate with the rest of the network which explain again the highest End-To-End delay for this node. Besides, when the message reaches this node (node at the ankle), its *TTL* has already been decremented multiple times and it generally has no uncovered neighbor as far as knowledge-based algorithms are concerned. *Flooding* and *Pruned Flooding* present the highest number for all nodes. Nodes continue to rebroadcast messages while $TTL$ is greater than $1$. It is worst with *Pruned Flooding* where each message is duplicated three times (these results are for $K=3$). *Probabilistic Flooding (P=P/2)* performs better than *Flooding* and *Pruned Flooding*. Indeed, the central nodes happen to transmit more than the others, but their neighbors will limit their transmission probability immediately, resulting in a lower number of receptions. This algorithm therefore better distributes the consumption across the network. One of the main weaknesses of the flooding algorithms that their performance strongly depends on the *TTL*. Increasing the *TTL* lets the total amount of transmissions and receptions rise very quickly. This issue is partially solved in the adaptive *Probabilistic Flooding (P=P/2)*: the probability decrease works as an automatic brake, drastically reducing the transmission probability. For *MBP* and *EBP* the total number of transmissions and receptions is $50$% less than *Flooding* and *Pruned Flooding*. Indeed, *MBP* uses the information from the acknowledgement to control and stop the transmissions, while *EBP* uses information included in the control messages (*Hello message*). Even if, for our strategy *Optimized Flooding*, no control message or acknowledgement are exchanged between nodes, we observe a significant decrease on the total number of transmissions and receptions per node. The decision to stop the retransmission is related to the validity of the information contained in each copy of the message. This information is used to stop sending the message if the network is fully covered as well as eliminate the back and forth of the same copy between two nodes, especially between highly connected nodes.
![Number of transmissions and receptions per posture[]{data-label="Fig11"}](Pictures/OnePacketPerSecond/Fig11.jpg){width="0.9\columnwidth"}
Figure \[Fig11\] represents the number of transmissions and receptions for each posture. It’s important to study each strategy behavior in function of the wearer posture. We can notice, on this graph, for all strategies, the highest number is for *Run and Sit* postures. In fact, *Run* posture represents the highest mobility, even with frequent disconnections, nodes are able to meet more frequent to exchange messages. In *Sit* posture, nodes are closer to each other so they are able to exchange packets with a minimum loss. *Sleep* posture presents the lowest number of transmissions and receptions for all strategies. At a first glance, it seems an advantage for this posture but reconsidering the observations related to the percentage of covered nodes \[Fig4\] and to the end-to-end delay \[Fig7\], this is rather related to the high disconnections between nodes. The two strategies *MBP* and *Optimized Flooding* are less independent from the postures (ie less variations). Still, *Optimized Flooding* presents a lower number of transmissions and receptions than *MBP* except for some postures which could be explained by the fact that *Optimized Flooding* covers more nodes as shown in figure \[Fig4\].
Total order resilience
======================
In this section we push further our study in order to detect the capacity of the strategies described to be reliable and to ensure the total order message delivery. That is, messages sent by the sink in a specific order should be received by each node in the system in that specific order.
We stress strategies with transmission rate ranging from $1$ to $1000$ packets per second broadcasted from the sink to the other nodes. We fixed the MAC buffer capacity to 100[^3].
We study two parameters:
1. Percentage of covered nodes: we calculate the number of messages received at each node than we deduce the percentage.
2. Percentage of desequencing: the percentage of messages received in the wrong order.
#### Percentage of covered Nodes
[0.9]{} ![Percentage of covered nodes per posture[]{data-label="ReceivedMsg1"}](Pictures/DefaultQueueLength/FigPaperPosture1.png "fig:"){width="\textwidth" height="3.25cm"}
[0.9]{} ![Percentage of covered nodes per posture[]{data-label="ReceivedMsg1"}](Pictures/DefaultQueueLength/FigPaperPosture2.png "fig:"){width="\textwidth" height="3.25cm"}
[0.9]{} ![Percentage of covered nodes per posture[]{data-label="ReceivedMsg1"}](Pictures/DefaultQueueLength/FigPaperPosture3.png "fig:"){width="\textwidth" height="3.25cm"}
[0.9]{} ![Percentage of covered nodes per posture[]{data-label="ReceivedMsg1"}](Pictures/DefaultQueueLength/FigPaperPosture4.png "fig:"){width="\textwidth" height="3.25cm"}
[0.9]{} ![Percentage of covered nodes per posture[]{data-label="ReceivedMsg1"}](Pictures/DefaultQueueLength/FigPaperPosture5.png "fig:"){width="\textwidth" height="3.25cm"}
[0.9]{} ![Percentage of covered nodes per posture[]{data-label="ReceivedMsg1"}](Pictures/DefaultQueueLength/FigPaperPosture6.png "fig:"){width="\textwidth" height="3.25cm"}
[0.9]{} ![Percentage of covered nodes per posture[]{data-label="ReceivedMsg1"}](Pictures/DefaultQueueLength/FigPaperPosture7.png "fig:"){width="\textwidth" height="4.75cm"}
Figure \[ReceivedMsg1\] presents the percentage of covered nodes in function of packets number per second. Contrary to the expectations, all studied strategies behave similarly: when the packets rate goes to $1000$ packets per second the percentage of covered nodes almost linearly decreases to $10$%. At $100$ packets per second, percentage of covered nodes barely exceeds $50$%.
For *Flooding* and *MBP* percentage stagnates until $10$ packets per second than the curve starts decrease. However, For *Optimized Flooding* and *Probabilistic Flooding*, percentage remains constant until $20$ packets/s
*PrunedFlooding* presents the lowest percentage. Other strategies results are quite close at the beginning and starting from certain point curves overlap and converge to the same point.
#### Percentage of Desequencing
[0.9]{} ![Percentage of Desequencing per posture[]{data-label="Deseq1"}](Pictures/DefaultQueueLength/FigPaperDeseqPosture1.png "fig:"){width="\textwidth" height="3.25cm"}
[0.9]{} ![Percentage of Desequencing per posture[]{data-label="Deseq1"}](Pictures/DefaultQueueLength/FigPaperDeseqPosture2.png "fig:"){width="\textwidth" height="3.25cm"}
[0.9]{} ![Percentage of Desequencing per posture[]{data-label="Deseq1"}](Pictures/DefaultQueueLength/FigPaperDeseqPosture3.png "fig:"){width="\textwidth" height="3.25cm"}
[0.9]{} ![Percentage of Desequencing per posture[]{data-label="Deseq1"}](Pictures/DefaultQueueLength/FigPaperDeseqPosture4.png "fig:"){width="\textwidth" height="3.25cm"}
[0.9]{} ![Percentage of Desequencing per posture[]{data-label="Deseq1"}](Pictures/DefaultQueueLength/FigPaperDeseqPosture5.png "fig:"){width="\textwidth" height="3.25cm"}
[0.9]{} ![Percentage of Desequencing per posture[]{data-label="Deseq1"}](Pictures/DefaultQueueLength/FigPaperDeseqPosture6.png "fig:"){width="\textwidth" height="3.25cm"}
[0.9]{} ![Percentage of Desequencing per posture[]{data-label="Deseq1"}](Pictures/DefaultQueueLength/FigPaperDeseqPosture7.png "fig:"){width="\textwidth" height="4.75cm"}
Figure \[Deseq1\] presents the percentage of desequencing.
Contrary to results about the percentage of covered nodes, from curves, three phases can be seen:
- At the beginning, all strategies present $0$% of desequencing. At this point, strategies are able to handle more than one packet in the network.
- Then, from certain point (depending on the strategy), this percentage increases linearly. Here, referring to Figure \[ReceivedMsg1\], percentage of covered nodes decreases due to collisions and packets loss. Sequencing is no longer ensured.
- Finally, from certain point, percentage decreases to converge to 0% again. Also, referring to figure \[ReceivedMsg1\], this is due to the fact that few packets are received.
*MBP* strategy presents the highest percentage of desequencing starting from $2$ packets/s.
Percentage of desequencing increases starting from $10$ packets/s for *Flooding* and *Pruned Flooding*, and from $20$ packets/s for *Optimized Flooding* and *Probabilistic Flooding*.
*PrunedFlooding* low percentage of desequencing is due to a low percentage of covered nodes.
*Optimized Flooding* strategy presents the lowest percentage of desequencing compared to *Flooding* and *MBP* strategies.
Conclusion and future works
===========================
In this paper we evaluated through simulation the performance of several DTN-inspired broadcast strategies in a WBAN context. Our simulations, realized with the Omnet++ simulator, the Mixim framework and a WBAN channel model proposed in the literature and issued from realistic data-sets, allowed us to compare flooding-like strategies that forward packets blindly and differ mostly by how their stopping criterion with a representative knowledge-based algorithm, EBP, which relies on the knowledge of the neighborhood of each node and its evolutions. The simulations realized over a 7 nodes network in 7 types of movements allow a fine characterization of the compromise that exists between the capacity to flood the whole network quickly and the cost induced by this performance. Simulations also allowed us to identify some less intuitive behaviors: for all strategies, most of the time is spent trying to reach leaf nodes, which makes us think that the key lies in adaptive algorithms that are able to mix different strategies.
We described a novel protocols that relies on such an adaptive approach: *MBP*, for Mixed Broadcast Protocol that applies a more aggressive strategy in the center of the network, where connections are more stable, and becomes more cautious at the border of the network, where a blind transmission has a good chance of success. This protocol has been reported in the extended abstract of this work [@BCPP15]. Additionally we proposed the *Optimized Flooding* protocol that improves the performances of *MBP*. Furthermore, we propose a preliminary study related to the total order reliability of the existing strategies. This study advocates for further investigations on this specific direction.
As future works, we will adapt this work to the context of IR-UWB transmissions, which is a promising technology for WBAN. Another future work would be a detailed study of existing channel models [@CHModel1] and a comparison with the channel model. Furthermore, another interesting future direction would be to consider collisions among multiple WBANs [@WBAN3; @WBAN2].
Appendix: Additional simulations
================================
Impact of MAC queue length {#impact-mac}
--------------------------
![Percentage of covered nodes for different queue length []{data-label="ReceivedMsgMAC1"}](Pictures/QueueLengthVariation/FigPaperQueueVarPosture100pckt1.jpg){width="0.9\columnwidth"}
![Percentage of Desequencing for different queue length[]{data-label="DeseqMAC1"}](Pictures/QueueLengthVariation/FigPaperDeseqQueueVar100pcktPosture1.jpg){width="0.9\columnwidth"}
We run simulations with $100$ packets per second and we vary MAC buffer capacity from $1$ to $10000$ packets. Figures \[ReceivedMsgMAC1\] and \[DeseqMAC1\] show respectively the percentage of covered nodes and the percentage of desequencing function of the MAC queue length. *Flooding* shows less resilience to high transmission rate. *Probabilistic Flooding* is less affected by queue length in MAC layer. Our novel strategy stabilizes to the highest highest percentage of covered nodes. *MBP* shows again a high percentage of desequencing.
We can conclude that MAC queue length influences our strategies performance while its value is lower than the transmission rate. For a queue length superior than the transmission rate, no variation is observed in the percentage of covered nodes nor in the percentage of desequencing.
In the following we fixe the transmission rate at 10 packets per second (the rate usually used in the biomedical applications) and a variation of the queue length from 1 to 10. The simulations results are proposed in Figures \[DeseqMAC2\] and \[ReceivedMsgMAC2\].
![Percentage of covered nodes for different queue length []{data-label="ReceivedMsgMAC2"}](Pictures/QueueLengthVariation/10paquetsParSeconde/FigPaperQueueVarPosture10pckt1.jpg){width="0.9\columnwidth"}
![Percentage of Desequencing for different queue length[]{data-label="DeseqMAC2"}](Pictures/QueueLengthVariation/10paquetsParSeconde/FigPaperDeseqQueueVar10pcktPosture1.jpg){width="0.9\columnwidth"}
MBP: Timer study {#MBP-timer}
----------------
In the following we focus on the timer used for *MBP* strategy. The objective of this study is to have a closer inside on the *MBP* parameters (i.e. timer) impact on the overall performances of this strategy.Recall that in *MBP* strategy, starting from a certain threshold on the number of hops ($NH$), a node delays message broadcasting after a timer expiration. Thus, we will examine the influence of this timer on *MBP* strategy performances. For simulations, we considered $NH$ equal to $2$. We also varied timer’s value to obtain $15$ different values from $0.005$ to $1.000$ second.
Figure \[CoveredNodesMBPTimer\] shows the percentage of covered nodes for each timer value per posture.
For all postures, the obtained percentages for each timer value are extremely close: the average difference is around $2.5$%.
We also noted that with the increase of the timer value, the simulation results are closer to each other.
The first four timer values (the smallest ones) show better percentage for covered nodes. Short timer pushes nodes to broadcast more often the message in the network which increases the probability of receiving the message by the neighboring nodes.
[0.8]{} ![Percentage of covered nodes per posture for each timer[]{data-label="CoveredNodesMBPTimer"}](Pictures/MBPTimer/CoveredNodes/FigP1.jpg "fig:"){width="\textwidth" height="3.7cm"}
[0.8]{} ![Percentage of covered nodes per posture for each timer[]{data-label="CoveredNodesMBPTimer"}](Pictures/MBPTimer/CoveredNodes/FigP2.jpg "fig:"){width="\textwidth" height="3cm"}
[0.8]{} ![Percentage of covered nodes per posture for each timer[]{data-label="CoveredNodesMBPTimer"}](Pictures/MBPTimer/CoveredNodes/FigP3.jpg "fig:"){width="\textwidth" height="3cm"}
[0.8]{} ![Percentage of covered nodes per posture for each timer[]{data-label="CoveredNodesMBPTimer"}](Pictures/MBPTimer/CoveredNodes/FigP4.jpg "fig:"){width="\textwidth" height="3cm"}
[0.8]{} ![Percentage of covered nodes per posture for each timer[]{data-label="CoveredNodesMBPTimer"}](Pictures/MBPTimer/CoveredNodes/FigP5.jpg "fig:"){width="\textwidth" height="3cm"}
[0.8]{} ![Percentage of covered nodes per posture for each timer[]{data-label="CoveredNodesMBPTimer"}](Pictures/MBPTimer/CoveredNodes/FigP6.jpg "fig:"){width="\textwidth" height="3cm"}
[0.8]{} ![Percentage of covered nodes per posture for each timer[]{data-label="CoveredNodesMBPTimer"}](Pictures/MBPTimer/CoveredNodes/FigP7.jpg "fig:"){width="\textwidth" height="3cm"}
Figure \[EndToEndDelayMBPTimer\] shows the average end-to-end delay for each timer value per posture. It should be noted the performance of MBP strategy. With the increase of the timer value, the MBP strategy spends more time to cover all nodes in the network. The best end-to-end delay is obtained with timer’s value equal to $0.005$ second.
[0.8]{} ![Average End-To-End Delay per posture for each timer[]{data-label="EndToEndDelayMBPTimer"}](Pictures/MBPTimer/EndToEndDelay/FigP1.jpg "fig:"){width="\textwidth" height="3.7cm"}
[0.8]{} ![Average End-To-End Delay per posture for each timer[]{data-label="EndToEndDelayMBPTimer"}](Pictures/MBPTimer/EndToEndDelay/FigP2.jpg "fig:"){width="\textwidth" height="3cm"}
[0.8]{} ![Average End-To-End Delay per posture for each timer[]{data-label="EndToEndDelayMBPTimer"}](Pictures/MBPTimer/EndToEndDelay/FigP3.jpg "fig:"){width="\textwidth" height="3cm"}
[0.8]{} ![Average End-To-End Delay per posture for each timer[]{data-label="EndToEndDelayMBPTimer"}](Pictures/MBPTimer/EndToEndDelay/FigP4.jpg "fig:"){width="\textwidth" height="3cm"}
[0.8]{} ![Average End-To-End Delay per posture for each timer[]{data-label="EndToEndDelayMBPTimer"}](Pictures/MBPTimer/EndToEndDelay/FigP5.jpg "fig:"){width="\textwidth" height="3cm"}
[0.8]{} ![Average End-To-End Delay per posture for each timer[]{data-label="EndToEndDelayMBPTimer"}](Pictures/MBPTimer/EndToEndDelay/FigP6.jpg "fig:"){width="\textwidth" height="3cm"}
[0.8]{} ![Average End-To-End Delay per posture for each timer[]{data-label="EndToEndDelayMBPTimer"}](Pictures/MBPTimer/EndToEndDelay/FigP7.jpg "fig:"){width="\textwidth" height="3cm"}
Figure \[nbreTxRxMBPTimer\] shows the number of transmissions and receptions for each timer per posture. Contrary to the end-to-end delay parameter, we obtain a descending curve. This is due to the fact that when the timer is longer, nodes spend more time waiting before rebroadcasting the message. Obviously, this has a direct impact on the average end-to-end delay. However, short timers impact the energy consumption (see Figure \[nbreTxRxMBPTimer\]).
[0.8]{} ![Number of transmissions and receptions per posture for each timer[]{data-label="nbreTxRxMBPTimer"}](Pictures/MBPTimer/nbreTxRx/FigP1.jpg "fig:"){width="\textwidth" height="3.7cm"}
[0.8]{} ![Number of transmissions and receptions per posture for each timer[]{data-label="nbreTxRxMBPTimer"}](Pictures/MBPTimer/nbreTxRx/FigP2.jpg "fig:"){width="\textwidth" height="3cm"}
[0.8]{} ![Number of transmissions and receptions per posture for each timer[]{data-label="nbreTxRxMBPTimer"}](Pictures/MBPTimer/nbreTxRx/FigP3.jpg "fig:"){width="\textwidth" height="3cm"}
[0.8]{} ![Number of transmissions and receptions per posture for each timer[]{data-label="nbreTxRxMBPTimer"}](Pictures/MBPTimer/nbreTxRx/FigP4.jpg "fig:"){width="\textwidth" height="3cm"}
[0.8]{} ![Number of transmissions and receptions per posture for each timer[]{data-label="nbreTxRxMBPTimer"}](Pictures/MBPTimer/nbreTxRx/FigP5.jpg "fig:"){width="\textwidth" height="3cm"}
[0.8]{} ![Number of transmissions and receptions per posture for each timer[]{data-label="nbreTxRxMBPTimer"}](Pictures/MBPTimer/nbreTxRx/FigP6.jpg "fig:"){width="\textwidth" height="3cm"}
[0.8]{} ![Number of transmissions and receptions per posture for each timer[]{data-label="nbreTxRxMBPTimer"}](Pictures/MBPTimer/nbreTxRx/FigP7.jpg "fig:"){width="\textwidth" height="3cm"}
[^1]: This work was funded by SMART-BAN project (Labex SMART) http://www.smart-labex.fr. En extended abstract of this work [@BCPP15] has been published in ACM MSWIM 2015.
[^2]: Note that we performed an additional study on the timer used in the MBP strategy in order to choose the one offering the best compromise. The results of this study are reported in \[MBP-timer\]
[^3]: Additional simulations for various MAC buffer capacities are presented in Section \[impact-mac\].
|
---
author:
- 'Jean-Marie Lescure'
bibliography:
- 'biblio.bib'
title: 'Elliptic symbols, elliptic operators and Poincaré duality on conical pseudomanifolds'
---
[Abstract: In [@DL], a notion of noncommutative tangent space is associated with a conical pseudomanifold and the Poincaré duality in $K$-theory is proved between this space and the pseudomanifold. The present paper continues this work. We show that an appropriate presentation of the notion of symbols on a manifold generalizes right away to conical pseudomanifolds and that it enables us to interpret the Poincaré duality in the singular setting as a noncommutative symbol map.]{}
Introduction
============
In this paper we give a concrete description of the Poincaré duality in $K$-theory for a conical pseudomanifold as stated and proved in [@DL]. This duality holds between the algebra $C(X)$ of continuous fonctions on a (compact) pseudomanifold $X$ and the $C^*$-algebra $C^*(T^cX)$ of a suitable [*tangent space*]{} of this pseudomanifold.
The tangent space $T^cX$ introduced in [@DL] is a smooth groupoid. It is no more commutative, but it restricts to the usual tangent space of a manifold outside the singularity and the singular contribution is quite simple.
The duality between $C(X)$ and $C^*(T^cX)$ is defined in terms of bivariant $K$-theory but it is important to recall that it implies the existence of an isomorphism: $$\label{PD-map-conic-case}
\Sigma^c\ : \ K_0(X)\overset{\simeq}{\longrightarrow} K_0(C^*(T^cX))$$
The main purpose of this paper is to identify this isomorphism with a noncommutative symbol map, as one does in the smooth case with the usual symbol map. Indeed, the Poincaré duality in the case of a smooth closed manifold $V$ induces an isomorphism between $K_0(V)$ and $K_0(C^*(TV))\simeq K^0(T^*V)$ which is nothing else but the principal symbol map: $$\label{PD-map-smooth-case}
\begin{matrix}
K_0(V) & \longrightarrow & K^0(T^*V) \\
[P] & \longmapsto & [\sigma(P)]
\end{matrix}$$ sending classes of elliptic pseudodifferential operators (the basic cycles of the $K$-homology of $V$) to classes of their principal symbols (the basic cycles of the $K$-theory with compact supports of $T^*V$).
The interpretation of (\[PD-map-conic-case\]) as a noncommutative symbol map is really important for two reasons. Firstly, it validates the choice of a tangent space which is $K$-dual to the singular manifold and thus motivates further investigations toward analysis or differential geometry of singular spaces by using this noncommutative object as well as other tools of noncommutative geometry. Secondly, this approach can be of interest for people looking for Fredholmness conditions in elliptic equations in singular situations like stratified spaces. Indeed, the notion of tangent space of a stratified space is very intuitive as soon as one understands the conical case, and the notion of elliptic noncommutative symbols appears directly. The case of general stratifications will be treated in forthcoming articles.
In [@DL], we propose two $KK$–equivalent definitions of the tangent space of a pseudomanifold $X$ and the main results were stated for the first one, noted $T^cX$ in the present article. To explain in what sense cycles of the $K$-theory of the tangent space of a pseudomanifold are [*noncommutative symbols*]{} and cycles of its $K$-homology are [*pseudodifferential operators*]{}, we will use here the second definition given in [@DL], noted in the sequel $T^qX$. The equivalence in $K$-theory of both tangent spaces allows us to state all the results of [@DL] for $T^qX$ and in particular the isomorphism (\[PD-map-conic-case\]). Even if this equivalence is obvious for people familiar with groupoids, one will give full details about it in section \[section2\].
Now, surprisingly, one can define [*noncommutative symbols*]{} on a pseudomanifold exactly as one defines symbols on a smooth manifold. More precisely, symbols on a smooth manifold $V$ are functions on the cotangent space $T^*V$ with adequate behavior in the fibers. They can be considered as pointwise multiplication operators on, for instance, $C^\infty_c(T^*V)$. Under a Fourier transform in the fibers, they can also be viewed as families parametrized by $V$ of convolutions operators in the fibers of $TV$. Thus:
Symbols on $V$ are pseudodifferential operators on the tangent space $TV$,
where $TV$ is considered as a [**]{} groupoid and we talk about pseudodifferential calculus for groupoids [@NWX; @MP; @Vas; @Vas2].
This simple observation is already important to understand that the tangent groupoid defined by A. Connes in [@co0] gives the analytic index of elliptic pseudodifferential operators. Next, it suggests the following definition of noncommutative symbols on the pseudomanifold $X$.
Noncommutative symbols on $X$ are pseudodifferential operators on the tangent space $T^qX$.
We will see that, after some technical precautions on the Schwartz kernels and on the behavior near the “end” of $T^qX$ of these pseudodifferential operators, this apparently naive idea works. For instance, one can recover in a single object the notions of interior and conormal symbols arising in boundary problems and the notion of full ellipticity is quite immediate here.
Concerning the operators involved in the description of the Poincaré duality, some freedom is allowed: basically, all calculi based on the work of R. Melrose [@Mel] ($b$ or $c$ calculi for instance) as well as on the work of B.W. Schulze [@schulze1; @schulze2] can be used indifferently and lead to various representants of the same $K$-homology class (that is, to the Poincaré dual of a given elliptic noncommutative symbol).
The main tools used in this paper are Lie groupoids (see [@DL] and the corresponding bibliography), pseudodifferential calculus (see [@Shub; @Mel; @MP; @NWX; @Vas2]) bivariant $K$-theory (see [@Ka1; @Ka2; @Ska1; @CoS; @Bla; @WO; @BJ]).
The author mentions that different techniques have been precedently used to produce results close from the present work by A. Savin ([@Sav1], see also joint works by V. Nazaikinskii, A. Savin and B. Sternin [@NSS1; @NSS2]).
Reviews and Notations {#notations}
---------------------
The range and source maps of groupoids are noted $r$ and $s$. If $A$ is a subset of the space of units $G^{(0)}$ of a groupoid $G$ then $G|_A$ denotes the subgroupoid $G|_A=r^{-1}(A)\cap s^{-1}(A)$. All groupoids in the sequel are smooth (Lie groupoids), endowed with Haar systems in order to define their $C^*$-algebras. Moreover, they are amenable (as continuous fields of amenable groupoids [@ARe]). In particular, there is no ambiguity about their $C^*$-algebras and notations for their $K$-theory will be shortened: $$K^i(G):= K_i(C^*(G)) \text{ and }
KK(G_1,G_2):= KK(C^*(G_1),C^*(G_2))$$ If $f$ is a homomorphism between two $C^{*}$-algebras $A, B$, the corresponding class in $KK(A, B)$ will be denoted by $[f]$.
When a vector bundle $E\to G^{(0)}$ is given, we define a $C^*(G)$-Hilbert module noted $C^*(G,E)$ by taking the completion of $C^\infty_c(G,r^*E)$ for the norm associated with the $C^*(G)$-valued product : $$<f , g> (\gamma) = \int_{\eta\in G^{r(\gamma)}}
<f(\eta^{-1}),g(\eta^{-1}\gamma)>_{s(\eta)}.$$ We shall use various deformation groupoids $G=G_1\times\{t=0\}\cup G_2\times]0,1]_{t}$. The restriction morphism ${\mbox{ev}_{t=0}}: G\to G_1$ at $t=0$ gives an exact sequence: $$\label{exact-seq-def-gpd}
0{\rightarrow}C^*(G_2\times]0,1]) {\rightarrow}C^*(G)\overset{{\mbox{ev}_{t=0}}}{{\rightarrow}} C^*(G_1){\rightarrow}0$$ whose ideal is contractible in $KK$-theory. If $G_1$ is amenable (which will always be the case in this paper), one gets that $[{\mbox{ev}_{t=0}}]\in KK(G,G_2)$ is invertible. The [*deformation element*]{} associated with the deformation groupoid is the Kasparov element defined by $$\label{def-elt}
\partial_G = [{\mbox{ev}_{t=0}}]^{-1}\otimes[{\mbox{ev}_{t=1}}]\in KK(G_1,G_2)$$ For convenience, the pair groupoid on a set $E$ will be denoted by $\cC_E$ .
The (open) cone over a space $L$ is the quotient space $cL = (L\times [0,+\infty[)/ L\times\{0\}$.A conical pseudomanifold is a compact metrisable space $X$ equipped with the following data. There is one singular point (but everything in the sequel can be written for a finite number) which means that a point $c\in X$ is given and that $X^{o}:=X\setminus\{c\}$ is a manifold. Moreover there is an open neighborhood $\cN$ of $c$, a smooth manifold $L$, continuous maps $h : \cN\to [0,+\infty[$ and $\varphi_c:\cN\to cL$ satisfying the following:
- $h$ is surjective, $h^{-1}\{0\}=\{c\}$ and $h : \cN\setminus\{c\}\to
]0,+\infty[$ is a smooth submersion,
- $\varphi_c:\cN\to cL$ is a homeomorphism, smooth outside $c$, such that: $$\begin{CD}
\cN @>\varphi_c>> cL \\
@VhVV @V\overline{p_2}VV \\
[0,+\infty[ @>=>>[0,+\infty[
\end{CD}$$ commutes. Here $\overline{p_2}$ denotes the quotient map of the second projection $L\times\RR_+\to \RR_+$.
Conical pseudomanifolds are the simplest examples of a stratified space [@BHS]. We distinguish two parts in the regular stratum $X^{o}$: $$X^{o}= X_-\cup X_+$$ where $X_-=h^{-1}]0,1[$, and $X_+=X\setminus X_-$ is a smooth compact manifold with boundary, the latter being identified with $L$. The identification $X_-\simeq]0,1[\times L$ given by $\varphi_c$ will be often used without mention. The compactification $M=\overline{X^{o}}$ of $X^{o}$ into a manifold with boundary $L$ will be sometimes useful. The following picture illustrates the notations just defined:
A riemannian metric $g$ on $X^{o}$ satisfying: $$\label{product-metric}
(\varphi_c)_*g(h,y)=dh^2+g_L(y), \ \ (h,y)\in]0,+\infty[\times L$$ on $\cN\setminus\{c\}$ is chosen, where $g_L$ is a riemannian metric on $L$. The corresponding exponential maps for $X^{o}$ and $L$ are denoted $e$ and $e_L$, and the injectivity radius is assumed to be greater than $1$ in both cases. The geodesic distances are denoted $\dist,\dist_L$. The associated riemannian measure on $X^{o}$ and $L$ will be noted $d\mu$ and $d\mu_L$ and the associated Lebesgue measures on $T_x X^{o}$ and $T^*_xX^{o}$ for $x\in
X^{o}$ will be noted $dX$ and $d\xi$. We will note $d\mu^\RR$ the Lebesgue measure on $\RR$.
We shall assume that $X^{o}$ is oriented, but all constructions below can be done in the general case with half densities bundles.
The tangent space of a conical pseudomanifold $X$ was defined in [@DL] by: $$\label{TcX}
T^cX=\cC_{X_-}\cup TX_+\rightrightarrows X^{o}$$ The unit space is $X^{o}$. This is a disjoint union where $\cC_{X_-}$ is the pair groupoid of $X_-$ and $TX_+$ has groupoid structure equal to its vector bundle structure. We will mainly use in this paper a slightly different but equivalent (at the level of $K$-theory) definition of the tangent space which was also given in [@DL]: $$\label{TqX}
T^qX=T]0,1[_h\times \cC_{L}\cup TX_+\rightrightarrows X^{o}$$ We will refer to (\[TqX\]) as the ’q’ version and (\[TcX\]) as the ’c’ version of the tangent space of $X$.
The [*tangent groupoid*]{} is defined for the ’c’ and the ’q’ version by: $$\label{cGc}
\cG^c = T^cX\times\{0\}\cup\cC_{X^{o}} \times ]0,1]_t.$$ $$\label{cGq}
\cG^q = T^qX\times\{0\} \cup \cC_{X^{o}}\times ]0,1]_t$$ In order to write down on $T^qX$ some constructions made in [@DL] for $T^cX$, one needs the following deformation groupoids: $$\label{H}
H= T^qX\times\{u=0\} \cup T^c X\times ]0,1]_u,$$ $$\label{cH}
\cH = \cG^q\times\{u=0\} \cup \cG^c\times]0,1]_u$$ Let us recall that $\cH$ has three deformation parameters noted $h,t,u$ and contains all previous groupoids: $$\label{faces-cH}
\cH|_{t=0}=H,\ \cH|_{u=0}=\cG^q,\ \cH|_{u=1}=\cG^c,\
\cH|_{t=0,u=0}=T^qX,\ \cH|_{t=0,u=1}=T^cX.$$
{width="7cm"}
We describe now a differentiable structure for $\cH$. This is rather technical, and the unfamiliar reader can skip this construction up to the remark \[exp-cGq\] which will be reused later.
The unit space of $\cH$ is $\cH^0=X^{o}\times
[0,1]^2_{t,u}$. We cover $\cH^0$ by four open subsets : $\cH^0=A\cup B\cup C\cup D$ with: $$\label{cover-units-of-cH}
\begin{matrix}
A = {\mbox{int}}(X_+)\times[0,1]^2_{t,u} & B =
X^{o}\times]0,1]_t\times[0,1]_u \\
C = X_-\times[0,1]^2_{t,u} & D=
h^{-1}(]1-\varepsilon,1+\varepsilon[) \times[0,\varepsilon[_t\times[0,1]_u
\end{matrix}$$ Here $\varepsilon$ is an arbitrary small number and ${\mbox{int}}(X_+)=X_+\setminus\partial X_+$. We get the following cover of $\cH$: $$\label{cover-cH}
\cH = \cH|_A\cup\cH|_B \cup\cH|_C \cup\cH|_D$$ We have $\cH|_A=\cG_{{\mbox{int}}(X_+)}\times[0,1]_u$ where $\cG_{{\mbox{int}}(X_+)}$ is the tangent groupoid [@co0] of ${\mbox{int}}(X_+)$. We provide it with its usual smooth structure and $\cH|_A$ inherits the obvious product smooth structure. Next, the smooth structure of $\cH|_B=\cC_{X^{o}}\times]0,1]_t\times[0,1]_u$ is the product one. For the two remaining subgroupoids, we need to specify some [*gluing*]{} functions in the deformation parameters. We choose once for all a smooth decreasing function: $$\label{gluingfunction-tau}
\tau : [0,+\infty[\to [0,1]$$ satisfying $\tau(h)=1$ on $[0,1/2]$ and $\tau^{-1}\{0\}=[1,+\infty[$. Let: $$\label{gluingfunction-kappa}
\kappa : [0,+\infty[\times [0,1]\to [0,1]$$ be a smooth function satisfying $\min\big(1,\tau(h)+t\big)\le\kappa(h,t)\le 1$ and $\kappa(h,t)=\tau(h)+t$ if $\tau(h)+t\le 3/4$. Let: $$\label{gluingfunction-mu}
\mu : [0,+\infty[\times [0,1]\times[0,1]\to [0,1]$$ be a smooth function satisfying $\min\big(1,u\tau(h)+t\big)\le\mu(h,t,u)\le 1$ and $\mu(h,t,u)=u\tau(h)+t$ if $u\tau(h)+t\le3/4$. Let: $$\label{gluingfunction-l}
l : \RR_+=]0,+\infty[\to \RR$$ be a smooth bijective function satisfying $\frac{d}{d h}l>0$ and $l={\mbox{Id}}$ on a neighborhood of $[1,+\infty[$ in $]0,+\infty[$.
Coming back to the subgroupoid $\cH|_C$, observe that: $$\label{subgrpd-cHc}
\cH|_C = \left(T]0,1[_h\times\{(0,0)\}\cup
\cC_{]0,1[_h}\times[0,1]^2_{t,u}\setminus\{(0,0)\}\right)\times\cC_L$$ We note shortly $G_I$ the first factor in $\cH|_C$. Let $\cV$ be the open subset of $T]0,1[_h\times[0,1]^2_{t,u}$ given by: $$\cV = \{ (h,\lambda,t,u) \ | \ l^{-1}(l(h)+\mu(h,t,u)\lambda)<1\}$$ Then we provide $G_I$ with the smooth structure such that the bijective map $E_{G_I} : \cV\to G_I$ given by: $$\label{exp-GI}
E_{G_I}(h,\lambda,t,u)=
\left\lbrace\begin{matrix} \big(h,\lambda,0,0\big) &
\text{ if } (t,u)=(0,0) \\
\big(h,l^{-1}(l(h)+\mu(h,t,u)\lambda),t,u\big)&
\text{ if } (t,u)\not=(0,0)
\end{matrix}\right.$$ is a diffeomorphism. Thus, $\cH|_C$ inherits the product structure of $G_I\times\cC_L$.
It is tedious but not difficult to check that the smooth structures given to $\cH|_A,\cH|_B,\cH|_C$ are compatible on their common domain and it remains to give $\cH|_D$ with a compatible smooth structure.
Remember that $X^{o}$ and $L$ are riemmannian with exponential maps denoted by $e$ and $e^{L}$ (see paragraph \[notations\]) and consider now the open subset $\cU$ of $TX^{o}\times[0,1]^2_{t,u}$ given by the set of $(x,V,t,u)\in TX^{o}\times[0,1]^2$ satisfying:
- if $h(x)\ge1$ then $tV\in{\mbox{dom}}(e_{x})$,
- if $h(x)<1$ then writing $x=(h,y)\in]0,1[\times L$, $V=(\lambda,W)\in\RR\times T_yL$ under the identification $X_-=]0,1[_h\times L$, we have $\kappa(h,t)W\in{\mbox{dom}}(e^L_{y})$ and $(h,\lambda,t,u)\in\cV$.
We define a injective map $E_{\cH} : \cU\to \cH$ by setting:
- for $(x,V)\in TX^{o}$ with $h(x)\ge 1$: $$\label{exp-cH-h-biggerthan1}
E_{\cH}(x,V,t,u)=
\left\lbrace\begin{matrix} \big(x,e_x(tV),t,u\big) &
\text{ if } t>0 \\
\big(x,W,0,u\big)&
\text{ if } t=0
\end{matrix}\right.$$
- for $(x,V)\in TX^{o}$ with $h(x)< 1$ and $x=(h,y)$, $V=(\lambda,W)$ as above: $$\label{exp-cH-h-lessthan1}
E_{\cH}(h,y,\lambda,W,t,u)=
\left\lbrace\begin{matrix} \big(h,y,\lambda,e^L_y(\tau(h)W),0,0\big) &
\text{ if } (t,u)=(0,0) \\
\big(h,y,l^{-1}(l(h)+\mu(h,t,u)\lambda),e^L_y(\kappa(h,t)W),t,u\big)&
\text{ if } (t,u)\not=(0,0)
\end{matrix}\right.$$
One can check that $\cU_D=E_{\cH}^{-1}(\cH|_D)$ is an open subset of $\cU$. We provide $\cH|_D'=\cH|_D\cap E_{\cH}(\cU)$ with the smooth structure such that the map : $E_{\cH}:\cU_D\to \cH|_D'$ is a diffeomorphism. On the other hand, $\cH|_D''=\cH|_D\cap(\cH|_A\cup\cH|_B\cup\cH|_C)$ is open in $\cH|_A\cup\cH|_B\cup\cH|_C$ so it inherits a smooth structure as a submanifold of $\cH|_A\cup\cH|_B\cup\cH|_C$. The smooth structure given to $\cH|_D'$ and $\cH|_D''$ are compatible and cover $\cH|_D$. The resulting smooth structure of $\cH|_D$ is compatible with the ones given to the three other subgroupoids, so we have given to $\cH$ a smooth structure for which $E_{\cH}$ is an exponential map.
\[exp-cGq\]
1. The gluing function $l$ is important in the description of the Poincaré duals of elliptic noncommutative symbols. We will see that choosing $l=\log$ near $h=0$ leads to this description with the help of $b$-calculus, while choosing $l(h)=-1/h$ near $h=0$ would lead to $c$-calculus. Different choices of $l$ produce different but diffeomorphic smooth structures on $\cH$. Indeed, if $l,m$ are two such choices, the map $\cH\to \cH$ equal to identity if $(t,u)\not=(0,0)$ or $h\ge 1$ and sending $(h,\lambda,x,y,0,0)$ to $(h,\lambda.\frac{m'(h)}{l'(h)},x,y,0)$ is a smooth isomorphism between $\cH$ with the smooth structure given by $l$ and $\cH$ with the smooth structure given by $m$. This follows from a simple but tedious calculation and the fact that for any smooth function $f:\RR \to \RR$, the map: $$\phi : \RR^{3}\to \RR; \ (x, \mu,\lambda)\mapsto
\begin{cases}
\frac{f(x)-f(x-\mu\lambda)}{\mu} & \hbox{ if } \mu\not=0 \\
\lambda f'(x) & \hbox{ if } \mu=0
\end{cases}$$ is smooth on $\RR^{3}$.
2. All other gluing functions are technical ingredients and their choice has no incidences on the desired description.
3. All subgroupoids listed in (\[faces-cH\]) inherit smooth structures and exponential maps from those of $\cH$.
We will often use $\cG^q=\cH|_{u=0}$ in the sequel. Using $E_{\cH}$, we get an exponential map for this groupoid:
- for $(x,V,t)\in \cU|_{u=0}$ with $h(x)\ge 1$: $$\label{exp-cGq-h-biggerthan1}
E_{\cG^q}(x,V,t)=
\left\lbrace\begin{matrix} \big(x,e_x(tV),t,u\big) &
\text{ if } t>0 \\
\big(x,W,0,u\big)&
\text{ if } t=0
\end{matrix}\right.$$
- for $(x,V,t)\in \cU|_{u=0}$ with $h(x)< 1$ and $x=(h,y)$, $V=(\lambda,W)$: $$\label{exp-cGq-h-lessthan1}
E_{\cG^q}(h,y,\lambda,W,t)=
\left\lbrace\begin{matrix} \big(h,y,\lambda,e^L_y(\tau(h)W),0,0\big) &
\text{ if } t=0 \\
\big(h,y,l^{-1}(l(h)+t\lambda),e^L_y(\kappa(h,t)W),t,u\big)&
\text{ if } t>0
\end{matrix}\right.$$ where we have replaced $\mu(h,t,0)$ by $t$ to simplify.
The inverse of the exponential map $E_{\cG^q}$ will be noted shortly $\Theta$.
The following define a Haar system for $\cG^q$ which is necessary to define in a convenient way its $C^*$-algebra: $$\label{Haar-system-cGq}
\begin{array}{ccc}
t>0, & \cG^q_{(x,t)}=X^{o}\times\{t\}, &
d\lambda^{(x,t)}(x')=\frac{l'(h')}{t\tau(t,h')^n}d\mu_{x'}
=d\lambda^t \ (h'=h(x'))\\
t=0 \text{ and } h<1, & \cG^q_{(h,y,0)}=\RR\times L\times\{0\}, &
d\lambda^{(h,y,0)}=\frac{1}{\tau(h)^n}d\mu^\RR d\mu^L =d\lambda^{h,0}\\
t=0 \text{ and } h\ge1, & \cG^q_{(x,0)}=T_xX^{o}, & d\lambda^{(x,0)}=d\nu_{x}=d\lambda^x
\end{array}$$ Remark that $d\lambda^1$ is equal to $\frac{1}{h}d\mu$ near $h=0$, in other words it coincides with the density coming from a $b$-metric like: $$\label{bmetric}
g_b(h,y) = \frac{dh^2}{h^2} + g_L(y)$$
Equivalences of tangent spaces and Dirac elements {#section2}
=================================================
Two equivalent tangent spaces
-----------------------------
The main results of [@DL] are the construction of a Dirac element $D^c\in KK(T^cX\times X,\cdot)$, a dual Dirac element $\lambda^c\in KK(\cdot,T^cX\times X)$, where $\cdot$ stands for a point space, and the computations in bivariant $K$-theory: $$\lambda^c\underset{T^cX}{\otimes} D^c = 1\in KK(X,X)
\text{ and }
\lambda^c\underset{X}{\otimes} D^c = 1\in KK(T^cX,T^cX)$$ which give in particular the isomorphism $$\Sigma^c=(\lambda^c\underset{X}{\otimes}\cdot)=(.\underset{T^cX}{\otimes} D^c)^{-1}$$ in (\[PD-map-conic-case\]). In this paper, we will prefer to work with $T^qX$ rather than $T^cX$, because the analog for $T^qX$ of $\Sigma^c$: $$\label{PD-map-conic-q-case}
\Sigma^q=(\lambda^q\underset{X}{\otimes}\cdot)=(.\underset{T^qX}{\otimes} D^q)^{-1}: \
K_0(X)\overset{\simeq}{\longrightarrow} K^0(T^qX)$$ has a nice description. We are going to describe the $KK$-equivalence between $T^qX$ and $T^cX$ in order to have a correct representant of $D^q$.
\[KK-equiv-qc\] The deformation element $\partial_H\in
KK(T^qX,T^cX)$ associated with $H$ in (\[H\]) is a $KK$-equivalence.
Let ${\mbox{ev}_{+}} : H\to TX_+\times[0,1]$ be the restriction morphism and consider the commutative diagram: $$\label{CD-H}
\begin{CD}
0 & & 0 & & 0 \\
@AAA @AAA @AAA \\
C^*(TX_+) @<{\mbox{ev}_{u=1}}<< C^*([0,1]_u\times TX_+)
@>{\mbox{ev}_{u=0}}>> C^*(TX_+) \\
@A{\mbox{ev}_{+}}AA @A{\mbox{ev}_{+}}AA @A{\mbox{ev}_{+}}AA \\
C^*(T^cX) @<{\mbox{ev}_{u=1}}<< C^*(H) @>{\mbox{ev}_{u=0}}>> C^*(T^qX) \\
@AAA @AAA @AAA \\
C^*(\cC_{]0,1[\times L}) @<{\mbox{ev}_{u=1}}<< C^*(H|_{h<1}) @>{\mbox{ev}_{u=0}}>> C^*(T]0,1[\times \cC_L) \\
@AAA @AAA @AAA \\
0 & & 0 & & 0
\end{CD}$$ The columns are exact. On the top line the induced maps in $K$-theory provide $$\label{topinvertibility}
[{\mbox{ev}_{u=0}}]^{-1}{\otimes}[{\mbox{ev}_{u=1}}]=1\in KK(TX_+,TX_+)$$ Observe that $$\label{Hmoins}
H|_{h<1}= T]0,1[\times\cC_L \times\{0\} \cup \cC_{]0,1[\times L}\times
]0,1] \simeq \left(T]0,1[\times\{0\} \cup \cC_{]0,1[}\times]0,1]_u\right)\times\cC_L,$$ and that $\cG_{]0,1[}=T]0,1[\times\{0\} \cup \cC_{]0,1[}\times]0,1]$ is the usual tangent groupoid of the manifold $]0,1[$. The associated Kasparov element $$\partial_{\cG_{]0,1[}}\in KK(T]0,1[,\cC_{]0,1[})\simeq
KK(C_0(\RR^2),\CC)\simeq \ZZ$$ is invertible with inverse given by the Bott generator of $KK(\CC,C_0(\RR^2))$. It follows that in the bottom line $$\label{bottominvertibilty}
[{\mbox{ev}_{u=0}}]^{-1}{\otimes}[{\mbox{ev}_{u=1}}]=\mathbf{s}_{C^*(\cC_{L})}\left(\partial_{\cG_{]0,1[}}\right)\in
KK(T]0,1[\times\cC_L,\cC_{]0,1[}\times\cC_L)$$ is invertible. Here $\mathbf{s}_A:KK(B,C)\to KK(B{\otimes}A,C{\otimes}A)$ is the usual tensorisation operation in Kasparov theory.
In particular the Kasparov elements $[{\mbox{ev}_{u=1}}]$ in (\[topinvertibility\]) and (\[bottominvertibilty\]) are invertible. Hence, applying the five lemma to the long exact sequences in $KK$-theory associated with the first two columns of (\[CD-H\]) give the invertibility of the element $[{\mbox{ev}_{u=1}}]$ in the middle line of (\[CD-H\]). This yields that $\partial_H$, equal to $[{\mbox{ev}_{u=0}}]^{-1}{\otimes}[{\mbox{ev}_{u=1}}]$ in the middle line, is invertible.
The Dirac element for the ’q’ version
-------------------------------------
Let us turn to the description of $D^q$. We define: $$\label{df-Dq}
D^q:=\partial_H\underset{T^cX}{{\otimes}} D^c \in KK(T^qX\times X, \cdot)$$ We recall the construction of $D^c$. We set: $$\partial^c:= \partial_{\cG^c}{\otimes}\nu \in KK(T^cX,\cdot)$$ where $\nu$ is the Morita equivalence $\nu\in KK(\cC_{X^{o}},\cdot)$ given by: $$\nu = \left(L^2(X^{o}),m,0\right)\in KK(\cC_{X^{o}},\cdot)$$ $\partial^c$ is called the pre-Dirac element and the Dirac element is $$D^c:= \Phi^c\otimes \partial^c \in KK(T^cX\times X,\cdot)$$ where $\Phi^c$ is the $KK$-element associated to the map defined by $$\label{Psic}
\begin{matrix}
\Phi^c\ : & C^*(T^cX){\otimes}C(X) & \longrightarrow & C^*(T^cX) \\
& a{\otimes}f & \longmapsto & a.f\circ\pi^c
\end{matrix}$$ and $\pi^c$ is the composition of the range map of $T^cX$ with the projection $X^{o}\to X^{o}/\overline{X_-}\simeq X$. In the ’q’ version we set: $$\partial^q=\partial_{\cG^q}\otimes\nu$$ and $$\label{Psiq}
\begin{matrix}
\Phi^q\ : & C^*(T^qX){\otimes}C(X) & \longrightarrow & C^*(T^qX) \\
& a{\otimes}f & \longmapsto & a.f\circ\pi^q
\end{matrix}$$ where $\pi^q$ projects $T^qX$ onto $X$ like $\pi^c$ does $T^cX$ on $X$. We check that :
The following equality holds : $$D^q= \Phi^q\otimes \partial^q \in KK(T^qX\times X,\cdot)$$ where $D^q$ is defined by (\[df-Dq\]).
Let us consider the commutative diagram: $$\label{CD-cqH}
\begin{CD}
C^*(\cC_{X^{o}}) @<{\mbox{ev}_{t=1}}<< C^*(\cG^q) @>{\mbox{ev}_{t=0}}>> C^*(T^qX) \\
@A{\mbox{ev}_{u=0}}AA @A{\mbox{ev}_{u=0}}AA @A{\mbox{ev}_{u=0}}AA \\
C^*(\cC_{X^{o}}\times\lbrack 0,1\rbrack) @<{\mbox{ev}_{t=1}}<< C^*(\cH)
@>{\mbox{ev}_{t=0}}>> C^*(H) \\
@V{\mbox{ev}_{u=1}}VV @V{\mbox{ev}_{u=1}}VV @V{\mbox{ev}_{u=1}}VV \\
C^*(\cC_{X^{o}}) @<{\mbox{ev}_{t=1}}<< C^*(\cG^c) @>{\mbox{ev}_{t=0}}>> C^*(T^cX)
\end{CD}$$ At the level of $KK$-theory, the bottom line of the diagram gives $\partial^q$, up to the Morita equivalence $\nu$, while the top line gives $\partial^c$ (up to $\nu$). The right column gives the $KK$-equivalence $\partial_H$ while the product ${\mbox{ev}_{u=0}}^{-1}{\otimes}{\mbox{ev}_{u=1}}$ in the left column is obviously $1$ in $KK(\cC_{X^{o}},\cC_{X^{o}})$. This gives: $$\partial_H{\otimes}\partial^c=\partial^q.$$ To finish, let us introduce the multiplication morphism: $$\Phi^H : C^*(H){\otimes}C(X){\rightarrow}C^*(H)$$ given by $\Phi^H(a,f)(\gamma)=a(\gamma)f(\pi^H(\gamma))$, where the projection map $\pi^H : H{\rightarrow}X$ extends $\pi^q$ and $\pi^c$ in the obvious way. Denoting in the same way the restriction morphisms for the product groupoid $H\times X$, we get the formulas: $$\mathbf{s}_{C(X)}\partial_H= \mathbf{s}_{C(X)} ([{\mbox{ev}_{u=0}}]^{-1}{\otimes}[{\mbox{ev}_{u=1}}])=[{\mbox{ev}_{u=0}}]^{-1}{\otimes}[{\mbox{ev}_{u=1}}],$$ $$[{\mbox{ev}_{u=1}}]{\otimes}\Phi^c = \Phi^H {\otimes}[{\mbox{ev}_{u=1}}]\quad ; \quad
[{\mbox{ev}_{u=0}}]^{-1} {\otimes}\Phi^H = \Phi^q {\otimes}[{\mbox{ev}_{u=0}}]^{-1}.$$ Hence: $$\begin{aligned}
D^q=\partial_H\underset{T^cX}{{\otimes}} D^c &=& \mathbf{s}_{C(X)}(\partial_H) {\otimes}D^c
= [{\mbox{ev}_{u=0}}]^{-1}{\otimes}[{\mbox{ev}_{u=1}}]{\otimes}\Phi^c{\otimes}\partial^c_X\\
\nonumber
&=& \Phi^q {\otimes}[{\mbox{ev}_{u=0}}]^{-1}{\otimes}[{\mbox{ev}_{u=1}}] {\otimes}\partial^c
= \Phi^q{\otimes}\partial^q\end{aligned}$$
Cycles of the $K$-theory of the tangent space
=============================================
Symbols on a manifold as operators on the tangent space {#symbols-manifolds}
-------------------------------------------------------
To proceed, we need some definitions about pseudodifferential calculus on groupoids. The notions summed up below can be found with full details in the litterature: [@Vas; @MP; @NWX; @CoS].
Let $G$ be a smooth groupoid (the space of units is allowed to be a manifold with boundary, but the fibers are manifolds without boundary). Let $U_{\gamma} : C^{\infty}(G_{s(\gamma)}){\rightarrow}C^{\infty}(G_{r(\gamma)})$ be the isomorphism induced by right multiplication: $U_{\gamma}f(\gamma')=f(\gamma'\gamma)$. A linear operator $P : C^{\infty}_{c}(G) \to C^{\infty}(G)$ is a [*$G$-operator*]{} if there exists a family $P_{x} : C^{\infty}_{c}(G_{x}) \to C^{\infty}(G_{x}) $ such that $P(f)(\gamma)=P_{s(\gamma)}(f\vert_{G_{s(\gamma)}})(\gamma)$ and $U_{\gamma}P_{s(\gamma)}=P_{r(\gamma)}U_{\gamma}$.
A $G$-operator $P$ is a [*pseudodifferential operator on $G$*]{} (resp. of order $m$) if for any open local chart $\Phi :\Omega {\rightarrow}s(\Omega)\times W$ of $G$ such that $s=pr_1\circ \Phi$ (that is, for any distinguished chart) and any cut-off function $\chi \in C^\infty_c(\Omega)$, we have $(\Phi^{*})^{-1}(\chi P \chi)_{x}\Phi^{*}= a(x,w,D_w)$ where $a(x,w,\xi)\in S^*(s(\Omega)\times T^*W)$ is a classical symbol (resp. of order $m$).
One says that $P$ has [*support*]{} in $K\subset G$ if $\mbox{supp}(Pf)\subset K.\mbox{supp}(f)$ for all $f\in
C^{\infty}_{c}(G)$.
These definitions extend immediately to the case of operators acting between sections of bundles on $G^{(0)}$ pulled back to $G$ with the range map $r$. The space of compactly supported pseudodifferential operators on $G$ acting on sections of $r^*E$ and taking values in sections of $r^*F$ will be noted $\Psi_c^*(G,E, F)$. If $F=E$ we get an algebra denoted by $\Psi_c^*(G,E)$.
Basic examples of the usefulness of these operators are the case of foliations [@CoS; @Vas] and manifolds with corners [@Mo2]. This calculus is also used in [@DL] to define $KK$-theory classes and to compute some Kasparov products. Here, to motivate our definition of [*noncommutative symbols*]{} on a singular manifold, we explain in more details what is suggested in the introduction.
Let $V$ be a smooth compact riemannian manifold, $E$ a smooth vector bundle over $V$ and consider the tangent space $TV$ as a smooth groupoid (thus $r$ and $s$ are equal to the canonical projection map $TV\to V$).
Let $a\in\Psi_c^*(TV,E)$. By definition, $a$ is a smooth family $(a_{x})_{x\in V}$ where $a_{x}$ is a translation invariant pseudodifferential operators on $T_xV$ (with coefficients in $\End E_x$) and thus can be regarded as a distribution $a_{x}(X)$ on $T_{x}V$ acting by convolution on $C^{\infty}_{c}(T_{x}V, E_{x})$, so: $$u\in C^{\infty}_{c}(TV, E), \quad a(u)(x, X)=a*u(x, X)=\int_{Y\in
T_{x}V} a(x, X-Y)u(Y)dY$$ where the last integral is understood in the distributional sense. The distribution $a_{x}(X)$ being compactly supported, it has a Fourier transform $\widehat{a_{x}}(\xi)$ which is just its symbol. The whole family $(\widehat{a_{x}})_{x}$ identifies with a classical symbol on $V$ taking values in $\End E)$, that is, $\widehat{a}\in
S^*(V,\End E)$ and since the Fourier transform exchanges convolution with pointwise multiplication, we get an algebra homomorphism: $$\label{symbolsV}
\begin{array}{cccc}
\cF : & \Psi_c^*(TV,E) &{\rightarrow}& S^*(V,\End E) \\
& a &\longmapsto & \widehat{a}(x,\xi)
\end{array}$$ which is obviously injective. Conversely, the inverse Fourier transform associates to any symbol $b(x, \xi)\in S^*(V,\End E) $ a distribution $\overset{\vee}{b}(x, X)$ which, as a convolution operator, is a $TV$-pseudodifferential operator by the formula: $$u\in C^{\infty}_{c}(TV, E), \quad \overset{\vee}{b}*u(x, X)=
\int_{T_{x}V\times T^{*}_{x}V}e^{i(X-Y). \xi} b(x,
\xi)u(x,Y)dYd\xi$$ Moreover, introducing a smooth function $\phi(x, X)$ on $TV$ equal to $1$ if $X=0$ and equal to $0$ if $|X|>1$, we get: $$\overset{\vee}{a} = \phi. \overset{\vee}{a}+(1-\phi)\overset{\vee}{a}\in \Psi_c^*(TV,E) + \cS(TV,\End E)$$ where $\cS(TV, \End E)$ stands for the (Schwartz) space of smooth sections whose partial derivatives are rapidly decaying in the fibers of $TV$.
Thus, enlarging $\Psi_c^*(TV,E)$ as follows: $$\Psi^*(TV,E) := \Psi_c^*(TV,E) + \cS(TV,\End E),$$ the algebra monomorphism (\[symbolsV\]) extends to an algebra isomorphism $$\label{symbolsV+}
\cF \ :\ \Psi^*(TV,E)\longrightarrow S^*(V,\End E)$$ which preserves the filtrations. For a general discussion about the enlargement of spaces of compactly supported pseudodifferential operators by adding regularizing ones, see [@Vas; @LMN2005].
One can then reformulate the classical description of the $K$-theory with compact supports of $T^*V$:
\[KTVbySymbols\] Every element in $K^0(T^*V)\simeq K^0(TV)$ has a representant of the following form: $$[a] = \left( C^*(TV,E\oplus F), \begin{pmatrix}0 & a \\ b & 0
\end{pmatrix}\right)\in KK(\cdot,TV)$$ where $E,F$ are smooth vector bundles over $V$ and $a\in\Psi^0(TV,E,F)$, $b\in\Psi^0(TV,F,E)$ satisfy $ba-1\in\Psi^{-1}(TV,E)$, $ab-1\in\Psi^{-1}(TV,F)$.
Noncommutative symbols and their ellipticity on a conical pseudomanifold {#symbols-pseudomanifolds}
------------------------------------------------------------------------
Motivated by the previous approach, we enlarge the space of compactly supported pseudodifferential operators on $T^qX$ and define them as noncommutative symbols on the pseudomanifold $X$. Definitions are given in the scalar case since the presence of vector bundles bring no issues. We introduce: $$\label{TqXbar}
{\overline{T^qX}}= \{0\}\times\RR\times\cC_L \cup T^qX =T[0,1[_h\times\cC_L\cup
TX_+ \rightrightarrows M=\overline{X^o}$$
\[def.schwartz.TqXbar\] Let $\tau$ be the function choosed in (\[gluingfunction-tau\]) and define the function $|.| : T^qX\to \RR_+$ by: $$|\gamma| =\begin{cases}
\sqrt{\left({\displaystyle{\frac{\mbox{dist}_L(x,y)}{\tau(h)}}}\right)^2+\lambda^2}& \hbox{ if }
\gamma=(h,\lambda,x,y)\in T]0,1[\times\cC_L \ (ie, \ h<1) \\
\sqrt{g_x(X,X)} & \hbox{ if } \gamma=(x,X)\in TX_+ \ (ie, \ h\ge 1)
\end{cases}$$
The restriction at $t=0$ of the Haar system $\cG^{q}$ defined in (\[Haar-system-cGq\]) provides a Haar system for $T^{q}X$, and extended at $h=0$ in the obvious way, we get a Haar system for ${\overline{T^qX}}$. It is then easy to check that $|. |$ is a length function with polynomial growth on ${\overline{T^qX}}$ and the corresponding Shwartz algebra is denoted by $\cS({\overline{T^qX}})$ ([@LMN2005]). Using the seminorms: $$p_{D,N}(f) = \sup_{\gamma\in T^qX}(1+|\gamma|)^{N}|Df(\gamma)|,$$ where $N$ is a positive integer and $D\in\hbox{Diff}({\overline{T^qX}})$ is a differential operator on ${\overline{T^qX}}$, this Schwartz algebra can be presented as follows: $$\cS({\overline{T^qX}}) = \{ f\in C^\infty({\overline{T^qX}}) \ | \ p_{D,N}(f)<+\infty\ \forall
N\in \NN, \ \forall D\in \hbox{Diff}({\overline{T^qX}})\}$$
By restriction at $h=0$ of these functions, we get the Schwartz algebra $\cS(\RR\times\cC_L)$ of the groupoid $\RR\times\cC_{L}$ endowed with the restricted Haar system.
\[symbol\] The algebra of noncommutative symbols on $X$ is defined by: $$\label{def-symb-alg}
S^*(X) = \Psi_c^*({\overline{T^qX}}) + \cS({\overline{T^qX}}) \subset \Psi^*({\overline{T^qX}})$$ We also define $S^*_0(X)$ as the kernel of the restriction homomorphism at $h=0$: $$\label{restrictions-symb}
\begin{matrix}
\rho \ : & S^*(X) & {\rightarrow}& \Psi^*_c(\RR\times\cC_L)+\cS(\RR\times\cC_L)\\
& a & \longmapsto \ a|_{h=0}
\end{matrix}$$
The image of $\rho$ is exactly the algebra $\cP^*_{inv}(\RR\times L)$ of translation invariant pseudodifferential operators on $\RR\times L$ defined by R. Melrose in [@Mel2].
The smoothness of noncommutative symbols up to $h=0$ can be relaxed and singular behaviors can be of interest, see paragraph \[FuchsSymbols\].
The following observations will lead to the notion of ellipticity for these noncommutative symbols.
\[symbols-as-multipliers\] The following inclusion holds $$\label{symbols-are-multipliers}
S^0(X)\subset \cM(C^*(T^qX)).$$ Moreover, we have $S^{-1}(X)\subset C^*({\overline{T^qX}})$ and $$\label{symb-1-0-are-compact}
S^{-1}_0(X) = S^{-1}(X)\cap C^*(T^qX).$$
It is known from [@MP; @Vas] that $\Psi^0({\overline{T^qX}})\subset\cM(C^*({\overline{T^qX}}))$ and $\Psi^{-1}({\overline{T^qX}})\subset C^*({\overline{T^qX}})$. Since $C^*(T^qX)$ is an ideal of $C^*({\overline{T^qX}})$ and since any $a\in \Psi^0({\overline{T^qX}})$ maps $C^*(T^qX)$ to itself, (\[symbols-are-multipliers\]) is true. Since $C^*(T^qX)$ is the kernel of the restriction morphism $C^*({\overline{T^qX}})\to C^*(\RR\times\cC_L)$ at $h=0$, (\[symb-1-0-are-compact\]) is obvious.
In the sequel, the algebra of small $b$-calculus [@Mel] will be denoted by $\cP^*_b(M)$ and its ideal consisting of operators with vanishing indicial families will be denoted by $\cP^*_{b,0}(M)$. Given a $\cG^{q}$-pseudodifferential operator $P$, its restriction $P|_{t}$ at any $t>0$ is a $\cC_{X^{o}}$-pseudodifferential operator, that is, an ordinary pseudodifferential operator on the (open) manifold $X^{o}$. In fact, we will denote by $\Psi^*_b(\cG^q)$ the algebra of $\cG^{q}$-pseudodifferential operators whose restrictions $P|_{t}$ at any $t>0$ are in the $b$-calculus of $M=\overline{X^{o}}$, that is, such that for all $t>0$, $P|_{t}\in\cP^*_b(M)$. The ideal of operators $P\in\Psi^*_{b}(\cG^q)$ such that $P|_{t}\in\cP^*_{b,0}(M)$ for all $t>0$ will be denoted by $\Psi^*_{b,0}(\cG^q)$. The previous proposition extends to these spaces:
\[cGq-operators-as-multipliers\] The following inclusions hold: $$\label{cGqpsib-are-multipliers}
\Psi^0_{b}(\cG^q)\subset\cM(C^*(\cG^q))$$ $$\label{cGqpsib0-are-compact}
\Psi^{-1}_{b,0}(\cG^q)\subset C^*(\cG^q)$$
This follows from properties of $b$-calculus and the proposition \[symbols-as-multipliers\].
A noncommutative symbol $a\in S^*(X)$ is elliptic if it is invertible in $S^*(X)$ modulo $S^{-1}_0(X)$.\
A noncommutative symbol $a\in S^*(X)$ is relatively elliptic if it is invertible in $S^*(X)$ modulo $S^{-1}(X)$.
The relative ellipticity of $a\in S^*(X)$ is exactly its ellipticity as a pseudodifferential operator on ${\overline{T^qX}}$. The notion of ellipticity for our noncommutative symbols is stronger and is similar to the notion of full ellipticity [@Mel; @Mo2003].
Indeed, let $\sigma(a)\in C^\infty(S^*M)$ be the principal symbol of $a\in S^*(X)$ viewed as a pseudodifferential operator on ${\overline{T^qX}}$. We call $(\sigma(a),\rho(a))\in C^\infty(S^*M)\times\cP^*_{inv}(\RR\times L)$ the [*leading part of*]{} $a$.
The following assertions are equivalent:
1. The noncommutative symbol $a$ is elliptic on $X$.
2. The leading part of $a$ is invertible.
(i)$\Rightarrow$ (ii) is obvious. Conversely, let $a\in \Psi^d({\overline{T^qX}})$ be a noncommutative symbol whose principal part is invertible. Since $a$ is an elliptic ${\overline{T^qX}}$-pseudodifferential operator, we can choose $\widetilde{b}\in
\Psi^{-d}({\overline{T^qX}})$ inverting $a$ modulo $\Psi^{-1}({\overline{T^qX}})$. From the smoothness of the family $a$ we get a continuous map $h\in[0,1]\mapsto a|_h\in \cP^*_{inv}(\RR\times L)$ and the invertibility of $\rho(a)$ implies the invertibility of $a|_h$ if $h<\alpha$ for some $\alpha>0$. We pick a cut-off function $\omega \in C^\infty_c[0,\alpha[$ such that $\omega(0)=1$ and we set : $$b = \omega (a|_{h})^{-1}+(1-\omega)\widetilde{b}.$$ Then $$ab = \omega + (1-\omega)a\widetilde{b} =
\omega +(1-\omega)(1+q) = 1 +(1-\omega)q,$$ where $q\in\Psi^{-1}({\overline{T^qX}})$, and $ab-1\in S^{-1}_0(X)$ is proved. Things are similar for $ba-1$.
$K$-theory of the tangent space
-------------------------------
We prove in this paragraph that elliptic noncommutative symbols, when vector bundles are allowed, are the cycles of $K^0(T^qX)$. As already quoted, definitions \[def.schwartz.TqXbar\] and \[symbol\] extend immediately to the case of vector bundles and we note $S^*(X,E,F)$ the space of noncommutative symbols on $X$ acting between sections of bundles $E,F$ over $M$. With the convention $S^*(X,E):=S^*(X,E,E)$, the proposition \[symbols-as-multipliers\] becomes: $$S^0(X,E)\subset\cL(C^*(T^qX,E)),\qquad
S^{-1}_0(X,E)\subset\cK(C^*(T^qX,E))$$ Therefore, we can associate to each elliptic noncommutative symbol $a\in S^0(X;E,F)$ on $X$ of order $0$, an element in the $K$-theory of $T^qX$:
\[bounded-symbol\] Let $a\in S^0(X;E,F)$ be an elliptic noncommutative symbol on $X$. We set: $$[a]:=[C^*(T^qX,E\oplus F),\mathbf{a}]\in KK(\cdot,T^qX)\simeq K^0(T^qX)$$ where: $$\mathbf{a}=\begin{pmatrix} 0 & b\\ a &
0\end{pmatrix}$$ and $b$ is any noncommutative symbol inverting $a$ modulo $S^{-1}_0$.
It is straightforward that $[a]$ does not depend on the choice of the quasi-inverse $b$. The main result of this section is that proposition \[KTVbySymbols\] holds in this new framework:
\[symbgenerate\] Every element of $K^0(T^qX)$ has a representant among elliptic noncommutative symbols. More precisely: $$K^0(T^qX) = \{ [a] \ | \ a \text{ is an elliptic noncommutative symbol on } X
\text{of order } 0\},$$
1. Considering the Kasparov ungraded modules given by $ (C^*(T^qX,E),a)$ where $a\in S^0(X,E)$ and $a^2-1\in S^{-1}_0(X,E)$, the conclusion is the same for $K^1(T^qX)$.
2. In the same way, relative elliptic noncommutative symbols span the $K$-theory of ${\overline{T^qX}}$. Observe that ${\overline{T^qX}}$ is $KK$-equivalent to $TX^{o}$ which is $KK$-dual to $M=\overline{X^{o}}$.
[**Proof of the theorem:** ]{} Let us denote by $\Delta$ the following subset of $K^0(T^qX)$: $$\{ [a] \ | \ a \text{ is an elliptic noncommutative symbol on } X
\text{of order } 0\}.$$ From the exact sequence of $C^*$-algebras: $$\label{ES-TqX-TX_+}
0\to C^*(\cC_L\times TI)\overset{i}{\to}
C^*(T^qX)\overset{{\mbox{ev}_{+}}}{\to}
C^*(TX_+)\to 0,$$ we get the exactness of $$K^0(\cC_L\times TI)\overset{i}{{\rightarrow}}
K^0(T^qX)\overset{{\mbox{ev}_{+}}}{\to} K^0(TX_+).$$ If $i(K^0(\cC_L\times TI))\subset\Delta$ and ${\mbox{ev}_{+}}(K^0(T^qX))\subset{\mbox{ev}_{+}}(\Delta)$ then the theorem is true. These inclusions are checked in the following lemmas.
\[im-i-ev+\] The inclusion $i(K^0(\cC_L\times TI))\subset\Delta$ holds.
[**Proof of the lemma:** ]{} It is sufficient to find a generator $e$ of $K^0(C^*(\cC_L\times
TI))\simeq\ZZ$ such that $i_*(e)\in\Delta$. We will define first an appropriate generator of $K_0(C_0(\RR^2){\otimes}\cK(L^2(L)))$ and then we will use an isomorphism $C^*(\cC_L\times TI)\simeq C_0(\RR^2){\otimes}\cK(L^2(L))$,
Let us choose the following generator of $K_0(C_0(\RR^2))$: $$x = [\cE, 1, F] \hbox{ where: } \cE = C_0(\RR^2)\oplus C_0(\RR^2),
\ F := \frac{d}{\sqrt{1+d^2}} \hbox{ and } d :=
\begin{pmatrix} 0 & h-i\lambda \\ h+i\lambda & 0 \end{pmatrix}.$$ On the other hand, let $B_+$ be an elliptic pseudodifferential operator on $L$ with index $1$. Without loss of generality, we may assume that $B_+$ is of order $1$, is almost unitary (ie, unitary modulo $0$ order operators) and acts between sections of a trivial bundle $L\times \CC^k$. Let $b_+\in C^\infty(S^*L, U_k(\CC))$ be its principal symbol. Then the following represents $1\in K^0(\cC_L)$: $$x' = [\cE', 1, F'],\hbox{ where: } \cE' =
\cK((L^2(L;\CC^k))^2) , \
F' := \frac{B}{\sqrt{1+B^2}} \hbox{ and } B :=
\begin{pmatrix} 0 & B_+^* \\B_+ & 0 \end{pmatrix};$$ Now the Kasparov product $x''=x\underset{\CC}{\otimes}x'$ is a generator of $K_0(C_0(\RR^2){\otimes}\cK(L^2(L)))$ and is represented by $(\cE",F")$ where: $$\cE''= \cE\underset{C_0(\RR^2)}{{\hat{\otimes}}}(C_0(\RR^2){\otimes}\cE') \simeq
\cE\underset{\CC}{\otimes}\cE', \qquad F" = \frac{D}{\sqrt{1+D^2}}
\quad \text{ and } D = d{\hat{\otimes}}I_{2k}+I_2{\hat{\otimes}}B.$$ Here $I_n$ denotes the identity matrix of rank $n$ and ${\hat{\otimes}}$ is the graded tensor product. Recall the matricial expression of $D$: $$D= d{\hat{\otimes}}I_{2k}+I_2{\hat{\otimes}}B=
\begin{pmatrix} 0 & 0 & 1{\otimes}B_+^{*} & d_-{\otimes}1 \\
0 & 0 & d_+{\otimes}1 & -1{\otimes}B_+ \\
1{\otimes}B_+ & d_-{\otimes}1 & 0 &0 \\
d_+{\otimes}1 & -1{\otimes}B_+^{*} & 0 & 0
\end{pmatrix}$$ It is clear that $D$ is a pseudodifferential operator on $L$ with parameters $(h,\lambda)\in \RR^2$ (of order $1$) in the sense of [@Shub], acting on the sections of the product bundle $L\times\CC^{4k}$. Following the construction of complex powers given in [@Shub], we see that $F"$ remains in the same space of operators with parameters (but of course, it is of order $0$). Let us find a better representant of $x''$ by trivializing $F''$ at $+\infty$. Let us introduce the matrix: $$J=\begin{pmatrix} 0 & K \\
K & 0 \end{pmatrix}\in M_{4k}(\CC) \text{ where }
K = \begin{pmatrix} 0 & I_k \\
I_k & 0
\end{pmatrix}\in M_{2k}(\CC).$$ We choose a smooth decreasing function $M$ equal to $1$ on $]-\infty,0]$ and vanishing near $h=+\infty$. We set: $$\label{triv-infty-F}
C = M^{1/2} F''+(1-M)^{1/2}J.$$ We are going to check that $C^2-1\in\cK(\cE'')$. Observe that: $$C^2 = M (F'')^2 + 1-M + [M(1-M)]^{1/2}[F'',J] = 1+
[M(1-M)]^{1/2}[F'',J] \mod \cK(\cE'')$$ where the bracket is $\ZZ_2$-graded. To compute $[F'',J]$, let us proove that $(1+D^2)^{-1/2}$ commutes with $J$. Setting : $$\Delta = 1+ D^2 =
\begin{pmatrix}
\Delta_+ & 0 \\
0 & \Delta_-
\end{pmatrix}
= \begin{pmatrix}
H_+ & 0 & 0 & 0 \\
0 & H_- & 0 & 0 \\
0 & 0 & H_- & 0 \\
0 & 0 & 0 & H_+
\end{pmatrix},$$ one gets: $$\Delta J = \begin{pmatrix} 0 & \Delta_+K \\
\Delta_- K & 0 \end{pmatrix}, \qquad
J \Delta = \begin{pmatrix} 0 & K\Delta_- \\
K\Delta_+ & 0 \end{pmatrix},$$ and: $$\Delta_+ K = \begin{pmatrix} 0 & H_+ \\
H_- & 0 \end{pmatrix}=K\Delta_-, \qquad
\Delta_- K = \begin{pmatrix} 0 & H_- \\
H_+ & 0 \end{pmatrix}=K\Delta_+,$$ hence $J\Delta=\Delta J$ which implies, using functional calculus, that $J$ commutes with $\Delta^{-1/2}$, hence: $$[F'',J]=\Delta^{-1/2}(DJ+JD)=\Delta^{-1/2}(2h) =
(2h)(1+h^2+\lambda^2+ I_2{\otimes}B^2)^{-1/2}.$$ Since $h\mapsto M(h)(1-M(h))$ has compact support, we conclude that $[M(1-M)]^{1/2}[F'',J]\in \cK(\cE'')$, hence $C^2=1 \mod \cK(\cE'')$.
We thus get $[\cE'',C]\in K_0(C_0(\RR^2){\otimes}\cK(L^2(L)))$ and $C_t = M_t^{1/2} F''+(1-M_t)^{1/2}J$ with $M_t(h)=M(th)$ provides an operatorial homotopy between $x"=(\cE'',F'')$ and $(\cE'',C)$.
Using a Fourier transform with respect to the variable $\lambda$ and a reparametrization $\RR\simeq]0,1[$ on $h$, we get an isomorphism $\phi: C_0(\RR^2){\otimes}\cK(L^2(L))\overset{\simeq}{\to} C^*(\cC_L\times TI)$ and $C$ gives rises to an element still noted $C$ and belonging to $\Psi^0(\cC_L\times TI,\CC^{4k})$. We now set $$e=\phi_*(x'')=\phi_*(\cE'',C)=[C^*(\cC_L\times TI,\CC^{4k}),C]\in KK(\cdot, \cC_L\times TI)$$ Finally we extend $C$ to $T^qX$ by setting $C= J$ on $TX_+$ thanks to the formula (\[triv-infty-F\]). Hence: $$i_*(e) = [C^*(T^qX,\CC^{4k}),C]\in KK(\cdot,T^qX) \text{ and }
C\in\Psi^0({\overline{T^qX}},\CC^{4k}),\text{ hence } i_*(e)\in\Delta.$$
\[delta-restricts-onto\] The equality ${\mbox{ev}_{+}}([\Delta])= K^{0}(TX_+)$ holds.
[**Proof of the lemma:** ]{} To each $K$-theory class $\sigma\in K^0(T^*X_+)$ we shall associate $a_\sigma\in \Delta$ with $(a_\sigma)|_{TX_+}=\sigma$.
Each element of $\sigma\in K^0(T^*X_+)$ can be represented by a continuous section $f$ over $T^*X_+ \setminus X_+$ of the bundle ${\text{Iso}}(\pi^*E,\pi^*F)$ for some complex vector bundles $E,F$ over $X_+$ pulled-back by $\pi : T^*X_+ \to X_+$. One can assume that $f$ is homogeneous of degre $0$ in the fibers of $T^*X_+ $ and independent of $h$ near $\{h= 1\}=\partial X_+$.
One sets $E=F=X_+ \times \CC$ since the general case is identical. Using the Stone-Weierstrass theorem, we can find $g\in C^\infty(T^*X_+)$ polynomial in $\xi$, independent of $h$ near $h=1$ and approximating uniformily $f$ in the corona $\{\xi\in T^*X_+ \ | \ 1/2\le |\xi|\le 2\}$ up to an arbitrary small $\varepsilon>0$. Thus, modifying $f$ by: $$\label{analytic-symbol}
f (x,\xi) =
g(x,\frac{\xi}{\sqrt{1+|\xi|^2}})$$ one gets another representant of $\sigma\in K^0(T^*X_+)$.
Choosing an exponential map $\theta$ for ${\overline{T^qX}}$, a cut-off function $\phi\in
C^\infty_c({\overline{T^qX}})$ equal to $1$ on units and extending $f$ to $T^*X^{o}$ in the obvious way, one can define the following noncommutative symbol on $X$: $$\label{af}
a_f(u)(\gamma)\!=\!\mbox{Op}_{\theta,\phi}(f)(u)(\gamma)\! :=\!\int_{ \xi\in
T^*_{x}X^{o},\ s(\gamma')=x'}\hspace{-1cm}
e^{i<\theta^{-1}(\gamma'\gamma^{-1}),\xi>}f(x,\xi)
\phi(\gamma'\gamma^{-1})u(\gamma')d\gamma'd\xi,$$ where $x=r(\gamma)$, $x'=s(\gamma)$ and $u\in C^\infty_c({\overline{T^qX}})$. This noncommutative symbol is relatively elliptic on $X$, which is not sufficient here. We then consider the restriction $a_0$ of $a_f$ at $h=0$. If we note $\theta_0,\phi_0$ the corresponding restrictions of $\theta,\phi$, we have, using the same formula as (\[af\]): $$a_0=\mbox{Op}_{\theta_0,\phi_0}(f_0)\in \cP^0_{inv}(\RR\times L).$$ Taking the Fourier transform with respect to the real variable in the above operator, we get a pseudodifferential operator $\widehat{a_0}(\lambda)$ on $L$ with parameter $\lambda\in\RR$ which satisfies the condition of ellipticity with parameters, hence, by a classical result on operators with parameters, $\widehat{a_0}(\lambda)$ is invertible for large $|\lambda|$. Note that $\theta,\phi$ can be chosen so that: $$\theta_0 : \RR\!\times\! TL\!\ni\!(\lambda,y,V)\!\mapsto\!
(\lambda,\exp^L_y(V))\!\in \!\RR\!\times\!\cC_L \text{ for small } |V|
\text{ and }
\phi_0(\lambda,y,y')=\phi_L(y,y'),$$ where ${\mbox{exp}}^L$ is for instance the exponential map associated with the metric (\[product-metric\]) and $\phi_L$ is compactly supported in the range of $\exp_L$ and satisfies $\phi_L(y,y)=1$. It follows that, writing $x=(h,y)\in [0,1]\times L$; $\xi=(\lambda,\eta)\in T_x^*X_+\simeq \RR\times T^*_yL$ and $f_0(x,\xi)=f_0(y,\lambda,\eta)$: $$\label{indicial-family}
\widehat{a_0}(\lambda)(u)(y)=\int_{y'\in L,\eta\in T^*_yL}
e^{i<(\exp_y^L)^{-1}(y'),\eta>}f_0(y,\lambda,\eta)\phi_L(y,y')u(y')dy'd\eta$$ where $u\in C^\infty(L)$ and $f_0$ denotes the restriction of $f$ at $h=0$.
Observe that $f_0$ has a holomorphic extension with respect to the cotangent variable $\lambda\in\RR$ in the strip $$\label{holomorphic-strip}
\cB=\{ z=\lambda+iu \in\CC\ | \ -1/2<u <1/2\}.$$ Indeed, the following function: $$f_0(y,z,\eta)=g(x,\frac{(z,\eta)}{\sqrt{1+z^2+|\eta|^2}})$$ makes sense as a holomorphic function in $z=\lambda+iu\in\cB$ taking values in the space $C^\infty(T^*L)$ and is equal to $f_0$ when $u=0$. Moreover, for fixed $u\in]-1/2,1/2[$, the function: $$(y,\lambda,\eta)\mapsto f_0(y,\lambda+iu,\eta)$$ is a symbol of order $0$ on $L$ with parameter $\lambda\in\RR$ and one can find a constant $C$ independent of $y,\lambda,\eta$ such that: $$|f_0(y,\lambda+iu,\eta)- f_0(y,\lambda,\eta)|\le
C.u$$ Since $f_0(y,\lambda,\eta)$ satisfies by construction the condition of ellipticity for symbols on $L$ with parameters $\lambda\in\RR$, the previous estimate ensures that the same is true for $f_0(y,\lambda+iu,\eta)$ assuming that $|u|<\alpha$ for some $\alpha>0$ small enough. In the sequel we restrict the strip $\cB$ according to this ellipticity condition.
It follows that (\[indicial-family\]) gives rise to a holomorphic family $z\mapsto \widehat{a_0}(z)$ taking values in elliptic pseudodifferential operators on $L$ of order $0$. We have noted earlier that there exists $z\in\cB$ such that $\widehat{a_0}(z)$ is invertible, so by a classical result on homolorphic families of Fredholm operators, the sets: $$\{ z=\lambda+i\rho\ | \ |\rho|\le\alpha' \text{ and } p_z \text{ is
not invertible}\}$$ are finite for all $\alpha'<\alpha$. Hence, there exists $\beta$ such that $\widehat{a_0}(\lambda+i\beta)$ is invertible for all $\lambda\in\RR$.
Observe also that each $\widehat{a_0}(z)$ restricted to horizontal lines $\mbox{im}(z)=u$ in $\cB$ is a pseudodifferential operator on $L$ with parameter $\lambda=\mbox{re}(z)$, which allows to define $a_u\in\cP^*_{inv}(\RR\times L)$ by: $$\widehat{a_u}(\lambda)=\widehat{a_0}(\lambda+iu)$$ Choosing a smooth function $u(h)$ such that $u(0)=\beta$ and $u(1)=0$, we can define the required elliptic noncommutative symbol $a_\sigma$ on $X$ by: $$a_\sigma|_{X_+}=a_f \text{ and }
a_\sigma|_h=a_{u(h)}\text{ for all } 0\le h\le 1$$
Unbounded noncommutative symbols, Fuchs type noncommutative symbols {#FuchsSymbols}
-------------------------------------------------------------------
We can also associate $K$-theory classes to elliptic noncommutative symbols on $X$ of positive order. To do that, we state:
\[regular\] Let $a\in S^m(X,E)$ be an elliptic noncommutative symbol on $X$ with $m>0$. Let us consider $a$ as an unbounded operator on $C^*(T^qX,E)$ with domain $C^\infty_c(T^qX,E)$. Then its closure $\overline{a}$ is regular ([@BJ]).
The closure of $a$ with domain $C^\infty_c({\overline{T^qX}},E)$ is regular as an unbounded operator on $C^*({\overline{T^qX}},E)$ and following the proof of this result in [@Vas], we see that everything remains true if we consider $a$ as an unbounded operator on $C^*(T^qX,E)$ with domain $C^\infty_c(T^qX,E)$. Note that the result still holds if $a$ is only relatively elliptic on $X$.
As a consequence, to each elliptic noncommutative symbol $a$ on $X$ of order $m>0$ corresponds a morphism $q(a)=a(1+a^*a)^{-1/2}\in \cL(C^*(T^qX,E))$ and using the construction of complex powers given in [@Vas2], we get:
Let $a\in S^m(X,E)$ be an elliptic noncommutative symbol of order $m>0$.\
1) $(1+a^*a)^{-1/2}$ belongs to $S^{-m}(X,E)$.\
2) $q(a)$ is an elliptic noncommutative symbol on $X$.
1\) Done in [@Vas2].\
2) Let $b$ be a parametrix for $a$, that is $ab=1+r$, $ba=1+s$ with $r,s$ regularizing operators vanishing at $h=0$. Then $(1+a^*a)^{1/2}b$ is a parametrix for $q(a)$.
Now we can associate to each elliptic $a\in S^m(X,E,F)$ the following $K$-theory class : $[q(a)]$. Note that these noncommutative symbols do not produce directly [*unbounded*]{} $KK$-theoritic elements ([@BJ]) since $(1+a^*a)^{-1/2}$ is not a compact operator on the $C^*(T^qX)$-Hilbert module $C^*(T^qX,E)$. This defect leads us to consider [*Fuchs type noncommutative symbols*]{}. Let $\varphi$ be a positive smooth increasing function of $h$, equal to $1$ if $h\ge 1$ and satisfying $\varphi(h)=h$ near $h=0$.
An element $p\in \Psi^*(T^qX,E)$ is a Fuchs type noncommutative symbol on $X$ if $\varphi^lp$ belongs to $S^m(X,E)$ for some $l\in\RR_+$. The infimum of such $l$ is then called the fuchs type order of $p$. A Fuchs type noncommutative symbol $p$ with Fuchs type order $l$ strictly positive is elliptic if the noncommutative symbol $\varphi^lp$ is elliptic on $X$.
For an elliptic Fuchs type noncommutative symbol $p$, we can define as before $(1+p^*p)^{-1/2}\in \cL(C^*(T^qX,E))$. Thanks to the unbounded behavior of $p$ with respect to $h$ at $h=0$, the operator $(1+p^*p)^{-1/2}$ is actually compact so $(C^*(T^qX,E),p)$ provides an unbounded $KK$-theoritic element in the sense of [@BJ].
Examples of such symbols come from Dirac type operators on $X$, where the latter is provided with a conical metric $g=dh^2+h^2g_L$, and their typical expression near $h=0$ is: $$p = h^{-1} \begin{pmatrix} 0 & -\partial_\lambda +S \\
\partial_\lambda +S & 0\end{pmatrix}, $$ where $S$ is a Dirac type operator on $L$. See [@DLN2006] for a developpement of this example.
Poincaré dual of elliptic noncommutative symbols
================================================
Construction of a noncommutative symbol map
-------------------------------------------
We are going to define a [*noncommutative symbol map*]{} for $b$-operators using a deformation process encoded by $\cG^{q}$. We then get a generalization of the complete symbol map for manifolds [@Get; @Wi1980], and like the notion that it generalizes, the noncommutative symbol is not canonical and depends on several choices: exponential maps, cut-off functions, connections on vector bundles. The idea of the construction is very close to [@Get; @ENN1996].
We motivate the forthcoming constructions by recalling the case of differential operators on a smooth manifold $V$. Let $Q$ be a differential operator on $V$ and: $$Q(x,D_x)=\sum_\alpha a_\alpha(x)D_x^\alpha$$ its expression in a given local chart. For each $t\in ]0,1]$, the differential operator $P_t$ on $V$ defined locally by: $$\label{diff-tang-manifold-tpositive}
P_t(x,D_x)=Q(x,tD_x)$$ is well defined and setting: $$\label{diff-tang-manifold-tzero}
P_0(x,D_X)=\sum_\alpha a_\alpha(x)D_X^\alpha\in \mbox{Diff}(T_xV),$$ we get a differential operator $P=(P_t)_{t\in [0,1]}$ on the tangent groupoid $\cG_V=TV\times\{0\}\cup
\cC_V\times]0,1]$ of $V$. As explained in paragraph \[symbols-manifolds\], $P_0$ represents exactly the (total) symbol of $Q$.
Let us do the same thing for $b$-differential operators on $M=\overline{X^{o}}$ with the tangent groupoid $\cG^q$ (\[cGq\]) of the pseudomanifold $X$.
From now on, the gluing function $l$ (\[gluingfunction-l\]) is equal to logarithm function $l=\log$ near $h=0$. Let $Q$ be a $b$-differential operator on $M$ [@Mel]. That means that near $\{h=0\}=\partial M$, one has, writing $x=(h,y)\in]0,1[\times L$: $$Q = \sum_{k} a_k(h,y,D_y)(h\partial_h)^k$$ where $a_k$ are differential operators on $L$, depending smoothly in $h$. We define the family $(P_t)_{t\in[0,1]}$ as in (\[diff-tang-manifold-tpositive\]), (\[diff-tang-manifold-tzero\]) on $X_+$, while we set on $X_-$, writing $X=(\lambda,V)\in\RR\times T_yL$: $$\label{diff-tpositive}
P_t= \sum_k a_k(h,y,\kappa(t,h)D_y)\left(\frac{t}{l'(h)}\partial_h\right)^k \text{ if }
t>0,$$ $$\label{diff-tzero}
P_0= \sum_k a_k(h,y,\tau(h)D_y)(D_\lambda)^k \text{ if }
h<1 \text{ if } t=0.$$ The functions $\kappa$ and $\tau$ are those chosen in (\[gluingfunction-kappa\]) and (\[gluingfunction-tau\]). Observe that for $h$ close enough to $0$, (\[diff-tpositive\]) and (\[diff-tzero\]) give: $$P_t= \sum_k a_k(h,y,D_y)(th\partial_h)^k \text{ and } P_0= \sum_k a_k(h,y,D_y)(D_\lambda)^k.$$ Note that $P_0$ is a noncommutative symbol on $X$ and that $P_0|_{h=0}=p(0,y,D_y,D_\lambda)$ is exactly the [ *indicial operator*]{} of $Q$ [@Mel]. Moreover, the full ellipticity of $Q$ as a $b$-operator (that is its interior ellipticity and the invertibility of the indicial family) is the same as the ellipticity of $P_0$ as a noncommutative symbol on $X$. Hence, we have defined a map $$\sigma : Q\mapsto P_0$$ defined on $b$-differential operators and taking values in noncommutative symbols on $X$. It remains to extend this map to the pseudodifferential case: the idea is basically the same but things are more technical.
We will use a cover of $M\times M$ by three open subsets $R_1,R_2,R_3$ as shown below and a partition of unity $\omega_1,\omega_2,\omega_3$ subordinated to this cover. For instance, $R_1=([0,1/2[\times L)^2$ while $$R_2 = \{ (x,x')\in M^2\ | \dist(x,y)<1,\ h(x)+h(x')>3/2 \}$$ and $R_3$ is some open neighborhood of the complement of $R_1\cup R_2$ into $M^2$.
Let $Q\in\cP^*_b(M)$ with Schwartz kernel $\kappa$. Let $Q_i$ $i=1,2,3$ be the operators with Schwartz kernel $\kappa_i=\omega_i\kappa$ so that: $Q=Q_1+Q_2+Q_3$.
Let us focus on $Q_1$. Applying a Mellin transfom on $\kappa_1$, we get: $$a_1(h,\eta,y,y') =
\int_{\RR_+^*}\left(\frac{h}{h'}\right)^{i\eta}
\kappa_1(h,y,h',y')\frac{dh'}{h'}$$ where $a_1$ is a smooth function of $h\in[0,1/2[$ taking values in the space of pseudodifferential operators on $L$ with one parameter $\eta\in\RR$. Using a cut-off function $\phi_1$ such that $\omega_1\phi_1=\omega_1$, one recovers the action of $Q_1$ on functions as follows: $$u\in C^\infty_c(X^{o}),\quad Q_1 u (h,y) = \int_{\RR_+^*\times\RR} \left(\frac{h}{h'}\right)^{i\eta}
\left(a_1(h,\eta)\cdot\left(\phi_1(h,h')u(h',.)\right)\right)(y)
\frac{dh'}{h'}d\eta$$ where $\left(a_1(h,\eta)\cdot u(h',.)\right)(y)$ is the action of the operator $a_1(h,\eta)$ on $u(h',.)\in C^\infty(L)$ evaluated at $y\in L$. Since $\kappa(t,h)=1$ and $l(h)=\log(h)$ when $h\in[0,1/2]$, setting: $$\label{boundary-term-positive-t}
u\in C^\infty_c(X^{o}),\quad
P_{1,t} u (h,y) = \int_{\RR_+^*\times\RR} \left(\frac{h}{h'}\right)^{i\eta/t}
\left(a_1(h,\eta).\phi_1(h,h')u(h',.)\right)(y)\frac{dh'}{th'}d\eta$$ for $t>0$ and defining $P_{1,0}\in\Psi^*({\overline{T^qX}})$ by: $$\label{boundary-term-zero-t}
u\in C^\infty_c(]0,1/2[\times L\times\RR),\quad
P_{1,0} u (h,x,\lambda) = \int_{\RR^2} e^{i(\lambda-\lambda').\eta}
\left(a_1(h,\eta).u(h,.,\lambda')\right)(x)d\lambda'd\eta$$ we get a pseudodifferential operator on $\cG^q$ given by $P_1=(P_{1,t})_{t\in[0,1]}$ and such that $P_1|_{t=1}=Q_1$.
Since $\kappa_2$ is supported in $R_2$ which is included both in a compact subset of $X^{o}\times X^{o}$ and in the range of $E_{\cG^q}=\Theta^{-1}$, we can set: $$\widehat{a_2}(x,V)=\kappa_2(E_{\cG^q}(x,V,1)) \text{ and }
a_2(x,\xi)=\int_{M} e^{i\Theta(x,x',1).\xi}
\widehat{a_2}(x,\Theta(x,x',1)) d\lambda^1(x').$$ Then, choosing any function $\phi_2$ compactly supported in a neighborhood of $R_{2}$ and satisfying $\omega_2\phi_2=\omega_2$, we have for all functions $u\in C^\infty_c(X^{o})$: $$Q_2u(x) = \int_{M\times T^*_xM} e^{i\Theta(x,x',1).\xi}a_2(x,\xi)\phi_2(x,x')u(x')
d\lambda^1(x')d\xi.$$ To extend $Q_2$ as we did for $Q_1$, we set: $$\label{interior-term-positive-t}
P_{2,t}u(x) =\int_{M\times T^*_xM} e^{i\Theta(x,x',t).\xi}
a_2(x,\xi)\phi_2(x,x')u(x')d\lambda^t(x')d\xi .$$ This is for $t>0$, and we define $P_{2,0}\in\Psi({\overline{T^qX}})$ by: $$\label{interior-term-zero-t-small-h}
h<1,\, u\in C^\infty_c(\RR\times L),\, P_{2,0}|_h u(\lambda,y)
\!\! =\!\!\int_{\RR\times L\times T^*_{(h,y)}M}\hspace{-1.5cm}
e^{i\Theta(h,\lambda-\lambda',y,y',0).\xi}
a_2(h,y,\xi)u(\lambda',y')d\lambda^{h,0}(\lambda',y')d\xi$$ and $$\label{interior-term-zero-t-large-h}
u\in C^\infty_c(TX_+),\quad P_{2,0}|_{X_+}u(x,X)=
\int_{T_xM\times T^*_xM}\!\!\!\!\!\!
e^{i\Theta(x,X-X',0).\xi}a_2(x,\xi) u(x,X')d\lambda^x(X')d\xi.$$ The last piece $Q_3$ is smoothing and its Schwartz kernel $\kappa_3(x,x')$ vanishes both on a neighborhood of the diagonal and on a neighborhood of $\partial M\times \partial M$ in $M^2$. This implies that $\widetilde{\kappa_3}$ defined by $\widetilde{\kappa_3}(x,x',t)=\kappa_3(x,x')$ if $t>0$ and $\widetilde{\kappa_3}|_{t=0}=0$, belongs to $C^\infty(\cG^q)$ and the behavior of $\kappa_3(x,x')$ near $h(x)=h(x')=0$, resulting from the assumption that $Q$ is in the small calculus, yields also $\kappa_3(x,x',t)\in C^*(\cG^q)$. Thus setting for $t>0$ and $u\in C^\infty_c(X^{o})$: $$P_{3,t}u(x)=\int_M \kappa_3(x,x')u(x')d\lambda^t(x')$$ and for $t=0$: $P_{3,0}=0$, we have extended $Q_3$ in $P_3\in\Psi^{-\infty}(\cG^q)\cap C^*(\cG^q)$. We get a linear map: $$\begin{matrix}
\cP_b^*(M) & \longrightarrow & \Psi^*(\cG^q) \\
Q & \longmapsto & P_Q:=P_1+P_2+P_3 .
\end{matrix}$$ Restricting $P_Q$ at $t=0$ gives the desired noncommutative symbol map:
Let $Q\in\cP^*_b(M)$. With the notations above, we define the noncommutative symbol of $Q$ by : $$\sigma(Q) = P|_{t=0}=P_{1,0}+P_{2,0} \in S^*(X).$$
The following facts are obvious:
1. $Q$ is fully elliptic as a $b$-operator if and only if $\sigma(Q)$ is elliptic as a noncommutative symbol on $X$.
2. If $P\in\cP^p_b(M)$ and $Q\in\cP^q_b(M)$ then $$\sigma(PQ)=\sigma(P)\sigma(Q) \text{ modulo } S^{p+q-1}(X).$$
3. Everything above can be written in the same way for operators acting on sections of a vector bundle.
The noncommutative symbol map depends on the choices of the cover of $M^2$, of the partition of unity and of the exponential map of $\cG^q$, but the $K$-theory class of the noncommutative symbols of fully elliptic $b$-operators does not depend on these choices, as we will see in the next paragraph.
Conversely, one can define a [*quantification*]{} map ${\text{op}}_b$ which is a quasi inverse of $\sigma$. We describe it now. Let us choose $\omega\in C^\infty_c([0,1/2[)$ such that $\omega(h)=1$ near $h=0$. Let $a$ be a noncommutative symbol on $X$ and write $$a=\omega a + (1-\omega)a = a_1+a_2.$$ We extend $a_1$ as a $\cG^q$-pseudodifferential operator $\widetilde{a_1}$ by reverting the process used in (\[boundary-term-positive-t\],\[boundary-term-zero-t\]). Let $f_2$ be a symbol of $a_2$ viewed as a pseudodifferential operator on $T^qX$. That means that $f_2$ is an ordinary symbol on $A^*(T^qX)=T^*X^{o}$ such that: $$a_2 = {\text{op}}_{T^qX}(f_2) \text{ modulo } S^{-\infty}_0(X,E)$$ where ${\text{op}}_{T^qX}$ is given by: $$u\in C^\infty_c(T^qX), \ {\text{op}}_{T^qX}(f_2)(u)(\gamma)=
\int_{(T^qX)_{s(\gamma)}\times T^*_{r(\gamma)}X^{o}}
\hspace{-1.5cm}e^{i<\Theta^q(\gamma'\gamma^{-1}),\xi>}
f_2(r(\gamma),\xi)\phi(\gamma'\gamma^{-1})u(\gamma')d\lambda^{s(\gamma)}d\xi .$$ Here $\Theta^q=(E_{T^qX})^{-1}$ is the inverse of the exponential map of $T^qX$ given by restriction of $E_\cH$, and $\phi$ is a cut-off function equal to $1$ on units and supported in the range of $\Theta^q$. We can use the formulae (\[interior-term-positive-t\],\[interior-term-zero-t-small-h\], \[interior-term-zero-t-large-h\]) to build from $f_2$ a $\cG^q$-pseudodifferential operator $\widetilde{a_2}$ with the property : $$\widetilde{a_2}|_{t=0} = a_2 \text{ modulo } S^{-\infty}_0(X).$$ Thus we get an approximate lifting of noncommutative symbols: $$\label{liftsymbolstocGq}
\widetilde{a}:=\widetilde{a_1}+\widetilde{a_2}\in\Psi^*_b(\cG_q)$$ satisfying: $$\widetilde{a}|_{t=0}=a\text{ modulo } S^{-\infty}_0(X),$$ $$\label{opb}
{\text{op}}_b(a):= \widetilde{a}|_{t=1}\in\cP^*_b(M).$$ By construction: $$\label{rightinverse-symbol}
\sigma({\text{op}}_b(a))=a \text{ modulo } S^{-\infty}_0(X)$$ In the same way, if $P\in\cP_b^*(M)$ then ${\text{op}}_b(\sigma(P))-P$ is a smoothing operator with vanishing indicial operator.
The Poincaré duality as a noncommutative symbol map
---------------------------------------------------
All ingredients are now at hands to finish. Observe that to each fully elliptic $b$-operator $P: C^\infty(M,E)\to C^\infty(M,F)$ acting on sections of complex vectors bundles $E,F$, corresponds a $K$-homology class $[P]=[(L^2(M;E\oplus F;d\lambda^1),\rho_1,\mathbf{P})]$ on $X$ where:
- $L^2(M;E\oplus F;d\lambda^1)$ is the $\ZZ_2$-graded Hilbert space modeled on the measure $d\lambda^1$ and on product type hermitian structures on $E$ and $F$,
- $\rho_1$ is the action of $C(X)$ onto $L^2(M;E\oplus F;d\lambda^1)$ in the natural way through the quotient map $M\to X=M/\overline{X_-}$,
- $\mathbf{P}=\begin{pmatrix} 0& Q\\ P & 0 \end{pmatrix}$ where $Q$ is a full parametrix of $P$.
With the previous notations and those of the definition \[bounded-symbol\], the isomorphism $\Sigma^q:K_0(X)\to
K^0(T^qX)$ defined in (\[PD-map-conic-q-case\]) is given by: $$\label{interpretationPD}
[P] \longmapsto [\sigma(P)]$$
Recall that from theorem \[symbgenerate\], we know that every $K$-theory class $[a]\in K^0(T^qX)$ has a representant $a$ among elliptic noncommutative symbols on $X$. From (\[rightinverse-symbol\]), we know that $a$ is in the same $K$-theory class than the noncommutative symbol $\sigma(P)$ of a fully elliptic $b$-operator $P$. Eventually, since $\Sigma^q$ is an isomorphism, we get that each $K$-homology class of $X$ is represented by a fully elliptic $b$-operator.
Using the deformation of $T^{q}X$ into $T^{c}X$, leading to the $KK$-equivalence $T^qX\sim T^cX$, one could also get a concrete interpretation for $\Sigma^c$. However, the adequate adaptation of the notion of noncommutative symbols is more difficult to relate directly to what is done in boundary values problems or former studies about pseudodifferential calculus for groupoids.
Let $P:C^\infty(M,E_0)\to C^\infty(M,E_1)$ be a fully elliptic $0$-order b-operator. Let $E=E_0\oplus E_1$ and $a=\sigma(P)\in S^0(X,E)$. We need to prove that $\Sigma^q[P]=[a]$ or equivalently that $[P]=[a]\underset{T^qX}{\otimes} D^q$ (cf. section \[section2\]). Recall that: $$\label{recallPDformula}
[a]\underset{T^qX}{\otimes} D^q=\mathbf{s}_X([a]){\otimes}\Phi^q {\otimes}\partial^q.$$ Firstly, $\mathbf{s}_X([a]){\otimes}\Phi^q \in KK(X,T^qX)$ is represented by: $$\label{firststep}
(C^*(T^qX,E), \rho, \mathbf{a})$$ where $\rho : C(X){\rightarrow}\cL(C^*(T^qX,E))$ is given by $\rho(f)(\xi)(\gamma)=\xi(\gamma)f(\pi^q(\gamma))$.
The next step is to find $$\label{secondstep}
[\widetilde{\cE},\widetilde{\rho},\widetilde{\mathbf{a}}]\in KK(X,\cG^q)$$ such that $$\label{pb-lift-sigma}
(e^q_0)_*[\widetilde{\cE},\widetilde{\rho},\widetilde{\mathbf{a}}]=[C^*(T^qX,E), \rho, \mathbf{a}].$$ The desired lifting is made as follows. Let us note again $E$ the pull back of the original bundle $E$ to $X^{o}\times [0,1]$ with the range map of $\cG^q$. Let $\widetilde{\pi^q}$ be the composite map of the range map $\cG^q\to X^{o}\times[0,1]$ with the projection maps $X^{o}\times[0,1]\to X^{o}$ and $X^{o}\to X=X^{o}/\overline{X_-}$.
We set $\widetilde{\cE}=C^*(\cG^q,E)$, we define $\widetilde{\rho}$ by $\widetilde{\rho}(f)(\xi)(\gamma)=f(\widetilde{\pi^q}(\gamma))\xi(\gamma)$, and $ \widetilde{\mathbf{a}}$ is defined from $\mathbf{a}$ using (\[liftsymbolstocGq\]). By construction $\widetilde{{\mathbf{a}}}\in\Psi^0_b(\cG^q,E)\subset \cL(C^*(\cG^q,E))$ and $\widetilde{{\mathbf{a}}}^2=1$ modulo $\Psi^{-1}_{b,0}(\cG^q,E)\subset \cK(C^*(\cG^q,E))$. It follows that the triple $(\widetilde{\cE},\widetilde{\rho},\widetilde{\mathbf{a}})$ defined above satisfies (\[secondstep\]) and (\[pb-lift-sigma\]). Evaluating this element at $t=1$ gives: $$(e^q_1)_*[\widetilde{\cE},\widetilde{\rho},\widetilde{a}]
=[\widetilde{\cE}|_{t=1},\widetilde{\rho}_{t=1},{\text{op}}_b(a)] \in
KK(X,\cC_{X^{o}})$$ and applying the Morita equivalence $C^*(\cC_{X^{o}})\overset{\nu}{\sim} \CC$ produces the final result: $$[a]\underset{T^qX}{\otimes} D^q= [\widetilde{\cE}|_{t=1},\widetilde{\rho}_{t=1},{\text{op}}_b(a)]
\otimes \nu =
[L^2(M;E;d\lambda^1),\rho_1,P]=[P]\in
KK(X,\cdot)=K_0(X)$$
Index map
---------
Since $X$ is a compact Hausdorff space the map $p$ sending $X$ to a point gives rise to a morphism: $$p_*\ : K_0(X) {\rightarrow}K_0(\cdot)=\ZZ$$ called, for obvious reasons, the index map. We can capture $p_*$ with the pre-Dirac element and Poincaré duality:
\[analytic-index\] Let us denote by $\hbox{Ind}^q$ the map: $$\hbox{Ind}^q : [a]\in K(T^qX) \mapsto [a]\otimes \partial^q\in\ZZ,$$ then the following holds: $$\forall [a]\in K(T^qX),\qquad p_*((\Sigma^q)^{-1}[a]) =\hbox{Ind}^q[a] .$$ In other words, the index of a fully elliptic $0$-order $b$-operator $P: C^\infty(M,E)\to C^\infty(M,F)$ viewed as a Fredholm operator between $L^2(M,E,d\lambda^1)$ and $L^2(M,F,d\lambda^1)$ is equal to ${\mbox{Ind}^{q}(\sigma(P))}$.
The homomorphism $\CC\to C(X)$ corresponding to $p:X\to\cdot$ is denoted by $\widetilde{p}$. We have $$p_*((\Sigma^q)^{-1}[a]) = [\widetilde{p}]{\otimes}\left([a]\underset{T^qX}{\otimes}([\Phi^q]\otimes
\partial^q)\right)= [\widetilde{p}]{\otimes}\mathbf{s}_{X}[a]{\otimes}[\Phi^q]{\otimes}\partial^q.$$ Observe that $$[\widetilde{p}]{\otimes}\mathbf{s}_{X}[a]=[\widetilde{p}]\underset{\CC}{{\otimes}}[a]$$ so by the commutativity of the Kasparov product over $\CC$: $$[\widetilde{p}]{\otimes}\mathbf{s}_{X}[a]= {\otimes}[a]\underset{\CC}{{\otimes}}
[\widetilde{p}]= a{\otimes}\mathbf{s}_{T^{q}X}[\widetilde{p}].$$ But $\mathbf{s}_{T^{q}X}[\widetilde{p}]{\otimes}[\Phi^q]$ is equal to the class of the identity homomorphism of $C^{*}(T^{q}X)$, hence: $$\begin{aligned}
p_*((\Sigma^q)^{-1}[a])&=& ([\widetilde{p}]{\otimes}\mathbf{s}_{X}[a]){\otimes}[\Phi^q]{\otimes}\partial^q \\
&=& (a{\otimes}\mathbf{s}_{T^{q}X}[\widetilde{p}]){\otimes}[\Phi^q]{\otimes}\partial^q \\
&=& a{\otimes}(\mathbf{s}_{T^{q}X}[\widetilde{p}]{\otimes}[\Phi^q]){\otimes}\partial^q \\
&=& a{\otimes}\partial^q \end{aligned}$$
|
---
abstract: 'Within the field of computational materials discovery, the calculation of phase diagrams plays a key role. Uncertainty quantification for these phase diagram predictions enables a quantitative metric of confidence for guiding design in computational materials engineering. In this work, an assessment of the CALPHAD method trained on only density functional theory (DFT) data is performed for the Li-Si binary system as a case study. with applications to the modeling of Si as an anode for Li-ion batteries. Using a parameter sampling approach based on the Bayesian Error Estimation Functional (BEEF-vdW) exchange-correlation. By using built-in ensemble of functionals from BEEF-vdW, the uncertainties of the Gibbs Free Energy fitting parameters are obtained and can be propagated to the resulting phase diagram. To find the best fitting form of the CALPHAD model, we implement a model selection step using the Bayesian Information Criterion (BIC) applied to a specific phase and specific temperature range. Applying the best selected CALPHAD model from the DFT calculation, to other sampled BEEF functionals, an ensemble of CALPHAD models is generated leading to an ensemble of phase diagram predictions. The resulting phase diagrams are then compiled into a single-phase diagram representing the most probable phase predicted as well as a quantitative metric of confidence for the prediction. This treatment of uncertainty resulting from DFT provides a rigorous way to ensure the correlated errors of DFT is accounted for in the estimation of uncertainty. For the uncertainty analysis of the single-phase diagram of the Li-Si system, we explore three different methods using BEEF as three kinds of samplers with various assumptions of statistical independence: independent points of phases, independent pairs of phases, and independent convex hulls of phases. We find that each method of propagating the uncertainty can lead to different phases being identified as stable on the phase diagram. For example, the phase Li$_{4.11}$Si at 300K is predicted to be stable by all functionals using the second and third method, but only 15% of functionals predict it to be stable using the first method. From the phase diagram, we have also determined intercalation voltages for lithiated silicon. We see that the same phase can have a distribution of predicted voltages depending on the functional. In combination, we can generate a better understanding of the phase transitions and voltage profile to make a more analysis-informed prediction for experiments and the performance of Si-anodes within batteries.'
author:
- Ying Yuan
- Gregory Houchins
- 'Pin-Wen Guan'
- Venkatasubramanian Viswanathan
bibliography:
- 'cite.bib'
title: 'Uncertainty Quantification of First Principles Computational Phase Diagram Predictions of Li-Si System Via Bayesian Sampling '
---
Introduction {#introduction .unnumbered}
============
The prediction of phase diagrams and phase transformations are important for many energy applications, especially in Li-ion batteries.[@persson2010thermodynamic] Computational prediction of equilibrium phase diagrams using density functional theory has been used successfully to predict the thermodynamics of intercalation electrodes.[@persson2010thermodynamic; @pande2018robust] However, the prediction of equilibrium phase diagrams involves various sources of uncertainty: uncertainty associated with numerical predictions of density functional theory, choice of the exchange correlation functional,[@wellendorff2012density] uncertainty associated with the choice of a model for describing the thermodynamics of the system such as cluster expansion [@ruban2008configurational] and uncertainty associated with fitting the parameters of the chosen model.[@choi2008inductive]
Quantifying the uncertainty is important as this could lead to vastly different conclusions on the identified stable phases and the associated thermodynamics.[@houchins2020towards] The challenge associated with systematic uncertainty quantification and propagation through a model has limited the application of these methods to calculation of phase diagrams (CALPHAD). There have been approaches proposed for uncertainty quantification within CALPHAD.[@stan2003bayesian; @otis2017high] Stan et al., proposed a weighted genetic algorithm sampling tool to estimate the posterior probability of a free energy model parameters. Otis and Liu performed model selection of a CALPHAD model using both Akaike Information Criterion (AIC) and F-test. Using this in conjunction with a Monte-Carlo sampling scheme determined the posterior probability distribution. In a recent work, Honarmandi et al. performed a thorough evaluation of uncertainty of CALPHAD model parameters and the resulting phase diagrams of Hf-Si binary system.[@honarmandi2019bayesian] A large source of uncertainty within these predictions stems from the choice of the exchange correlation functional for DFT calculated data.[@wellendorff2012density; @decolvenaere2015testing; @kitchaev2016energetics; @lejaeghere2014error] Bayesian error estimation capabilities of the BEEF-vdW exchange correlation function has been used to quantify uncertainty associated with a variety of DFT-predicted material properties.[@PhysRevB.96.134426; @PhysRevB.94.064105; @Parks2019Uncertainty; @guan2019uncertainty]
In this work, we implement model-parameter selection through the use of Bayesian Information Criterion (BIC) applied to specific phases and specific temperature range to find the optimal number of parameters and parameter types in the CALPHAD models.
\[tab:1\]
------------------- ---------------------------- -----------
Phasename Pearsonsymbol/Space Reference
group/Latticeparameter(pm)
Li$_{17}$Si$_4$ cF420 \[\]
F$\bar{4}$3m
a = 1872.59(1)
Li$_{22}$Si${5}$ cF432 \[\]
F23
a = 1875.0
Li$_{21}$Si$_{5}$ cF416 \[\]
F$\bar{4}$3m
a = 1871.0
Li$_{4.11}$Si orthorhombic \[\]
Cmcm
a = 452.46(2)
b = 2194.4(1)
c = 1320.01(6)
Li$_{15}$Si$_4$ cI76 \[\]
I$\bar{4}$3d
a = 1063.22(9)
Li$_{13}$Si$_4$ oP34 \[\]
Pbam
a = 794.88(4)
b = 1512.48(8)
c = 446.61(2)
Li$_7$Si$_2$ oP34 \[\]
Pbam
a = 799
b = 1521
c = 443
Li$_7$Si$_3$ hR7 \[\]
R$\bar{3}$m
a = 443.5(1)
c = 1813.4(3)
Li$_2$Si mS12 \[\]
C2/m-$C^3_{2h}$
a = 770
b = 441
c = 656
Li$_{12}$Si$_7$ oP152 \[\]
Pnma
a = 860.0
b = 1975.5
c = 1433.6
LiSi tI32 \[\]
I 4$_1$/a($n^o88$)
a = 935.45(5)
c = 573.74(5)
------------------- ---------------------------- -----------
: Phases in the Li-Si binary system: Phase names and crystallographic data accepted in this work and experimental data reported in literature.
An implementation of Bayesian Error Estimation with van der Waals correction, the BEEF-vdW [@wellendorff2012density] was trained on a set of benchmark data spanning solid-state properties, covalent organic systems, noncovalent and van der Waals interactions, and chemisorption on solid metallic surfaces. A functional form for the exchange-correlation potential is fit through the least-squares fitting of the error in predictions for this training set. The parameters space can then be perturbed slightly to sample a collection of models that are marginally above the minimum of the least-squares fit and therefore should provide a nearly equally good fit. Within this work, we use the ensemble of functionals from BEEF as a sampler for the generation of CALPHAD models in order to propagate the uncertainty from DFT to the final prediction of phase diagrams. We assume that every phase diagram of different functionals from BEEF can provide potentially useful information as it samples the predictions of a point in exchange-correlation space. The BIC model selection method is applied for each chosen solid phase in our Li-Si binary system and a specific temperature range of 200K-450K. We implement a Bayesian Information Criterion (BIC) model selection step to determine best model from a 10-parameter CALPHAD model space. Using this approach, we not only explore the method of using only DFT to make a phase prediction but also give the quantitative range of uncertainty using DFT data at the GGA-level.
Li-Si Binary System {#li-si-binary-system .unnumbered}
===================
Li-ion batteries have played an important role in electrifying transporation.[@goodenough2013li; @thackeray2012electrical] Next-generation batteries are required for electrifying trucking[@sripad2017performance; @Sripad2017EvaluationOpportunities] and aviation.[@fredericks2018performance; @Viswanathan2019] Silicon anodes are a promising candidate among next-generation chemistries to improve the energy density of batteries.[@mcdowell201325th] Hence, a detailed and consistent thermodynamic description of the Li-Si binary system is critical for silicon anode development.
Uncertainty quantification of first principles computational phase diagram predictions via Bayesian sampling can help achieve reasonable phase predictions of Li-Si system and a full understanding of phase transitions during charging and discharging. In this work, we used the set of phases in the Li-Si binary system as in Table I \[tab:1\], which compiles the crystallographic data information of the solid phases including Li$_{17}$Si$_4$[@zeilinger2013single], Li$_{22}$Si$_5$[@axel1966kenntnis], Li$_{21}$Si$_{5}$[@nesper1987li21si5], Li$_{4.11}$Si[@zeilinger2013revision], Li$_{15}$Si$_4$[@zeilinger2013stabilizing], Li$_{13}$Si$_4$[@zeilinger2013], Li$_7$Si$_2$[@schafer1965kristallstruktur], Li$_7$Si$_3$[@von1980struktur], Li$_2$Si[@axel1965kristallstruktur], Li$_{12}$Si$_7$[@nesper1986li12si7], LiSi[@tang2013synthesis], as well as solid-phase BCC Li, solid-phase Diamond Si as previously used in work by Lang *et. al.* 2017.[@Liang2017thermodynamics]
Methods {#methods .unnumbered}
=======
To compute the energy of each structure, the Projector Augmented Wave method of Density functional theory, as implemented in GPAW [@enkovaara2010electronic], was used. The exchange correlation potential was treated at the genearalized gradient approximation level using he Bayesian Error Estimation Functional with van der Waals (BEEF-vdW)[@wellendorff2012density]. For each structure, the atomic positions were first relaxed to a maximum force of 0.03 eV/Å, and the computed energy at various volumes was fit to a vinet equation of state[@vinet1987compressibility] using either 7 or 9 different volumes depending on the difficulty of fitting the equation of state for the ensemble of functionals.
\[tab:2\]
phasename BICmodelselectionresult numberofparameters
------------------- ---------------------------------------------- --------------------
Li a+bT+d$T^2$+e$T^{-1}$+g$T^7$+i$T^4$+jlnT 7
Si a+bT+cTlnT+e$T^{-1}$+jlnT 5
Li$_{15}$Si$_4$ a+bT+cTlnT+e$T^{-1}$+i$T^4$+jlnT 6
Li$_{22}$Si$_{5}$ a+bT+cTlnT+d$T^2$+f$T^3$+jlnT 6
Li$_{21}$Si$_5$ a+bT+cTlnT+d$T^2$+f$T^3$+jlnT 6
Li$_{13}$Si$_4$ a+bT+cTlnT+e$T^{-1}$+g$T^7$+jlnT 6
LiSi a+bT+cTlnT+d$T^2$+f$T^3$+g$T^7$+i$T^4$ 7
Li$_7$Si$_2$ bT+cTlnT+d$T^2$+e$T^{-1}$+f$T^3$+jlnT 6
Li$_2$Si a+bT+cTlnT+d$T^2$+jlnT 5
Li$_{12}$Si$_7$ bT+cTlnT+d$T^2$+d$T^{-1}$+f$T^3$+i$T^4$+jlnT 7
Li$_{17}$Si$_4$ a+bT+cTlnT+d$T^2$+f$T^3$+jlnT 6
Li$_{4.11}$Si a+cTlnT+d$T^2$+f$T^3$+g$T^7$+i$T^4$ 6
Li$_7$Si$_3$ a+bT+cTlnT+d$T^2$+i$T^4$+jlnT 6
Using the fitted properties of the equation of state, a Debye-Grunessen theory analysis[@moruzzi1988calculated] was used to incorporate vibrational properties and predict the Gibbs free energy as a function of temperature as the Debye model is a reasonable approximation that yields finite temperature thermodynamics of sufficient accuracy[@guan2019uncertainty]. This process was repeated for the ensemble of 2000 non-self consistent exchange-correlation functionals within the BEEF-vdW model space framework, but not every functional can lead to successful result (see computational details of Supplimentary Information).
Next, to best fit the predicted Gibbs energy curve for each phase, we employ model selection through the use of BIC. The BIC was chosen as it tends to penalize complex models more heavily, giving preference to simpler models in selection than the Akaike information criterion (AIC)[@hastie2005elements] especially for a small number of training points. The Gibbs energy of the pure element $i$ in phase $\Phi$, namely the Gibbs energy function used within CALPHAD modeling, is adopted in this work as the following equation [@wang2013thermodynamic]: $$\begin{gathered}
G^{0,\Phi}_i(T)=G^\Phi_i(T)-H^{SER}_i=a+bT+cTlnT+dT^2\\
+eT^{-1}+fT^3+gT^7+hT^{-9}+iT^4+jlnT\end{gathered}$$ In this equation, the Gibbs energies of the pure solid (Li)-bcc, (Si)-diamond and all stoichiometric compounds are described with absolute reference state. $H^{SER}_i$ is the molar enthalpy of the element (i) at 298.15 K and 1 bar in its standard element reference (SER) state. T is the absolute temperature. There are 10 possible parameters, corresponding to different functional dependencies on T. While the physical meaning of each parameter is largely empirical, here, parameter a relates to the static energy, parameter b describes the entropy, and parameters c, d, e, f, g, h, i and j are determined from the temperature dependence of the heat capacity, $C_p$, for each compound: $$\begin{gathered}
C_p=-c-2dT-6eT^2-2fT^{-2}-42gT^6-90hT^{-10}\\-12iT^3+jT^{-1}\end{gathered}$$ Although each parameter has its relation to a specific physical meaning, for fitting part of the Gibbs energy result, it is not necessary to choose all of these parameters to get the best fitting equation[@lukas2007computational]. For the full 10-parameter model space, we iterate through all unique combinations of these 10 parameters for a total of $2^{10}-1=1023$ different G-T fitting models tested for each phase. BIC is defined as [@Schwarz1978estimating] $$\begin{aligned}
BIC(M)&=k\log(n)-2\log(\bar{L})\\
&=k\log(n)+n*\log(\frac{\sum(Y_i-f_i(X))^2}{n})\end{aligned}$$ where k is the number of parameters of each model, $\bar{L}$ is the maximized likelihood function, n is the number of training points of each model, here we take n as 26 for each model and each phase in temperature range 200K-450K. $Y_i$ is the true value of one training point, $f_i$(X) is the value of prediction after G-T fitting by using the model. For the results of BIC values of different models for each phase, choosing the model with minimum BIC is equivalent to choosing the model with the largest (approximate) posterior probability[@hastie2005elements], the model with the lowest BIC value is predicted to be the most ideal model, namely the model that best trades off accuracy and model complexity.
![A collection of 50 predictions from the full ensemble of convex hulls is shown is shown in blue. The BEEF optimal function is also shown as a black solid line. The stable phases appear in each convex hull are labeled with red cross. The lowest energy phase of each convex hull is labeled with green x, which represents the maximum stable lithiated silicon phase predicted by each functional.[]{data-label="fig:2"}](convexhull.png){width="48.00000%"}
After determining the best fitting model for each phase by applying BIC criterion to the data from the optimal BEEF functional, we fit the Gibbs energy curve for each phase and the ensemble of functionals. The results are then stored in a thermodynamic database (TDB) file as is conventional in CALPHAD. These TDB files are then read into the pycalphad software [@otis2017pycalphad] to get phase diagram data of each functional and replotted to produce the uncertainty phase diagram for the Li-Si system automatically, which is a new method to achieve automating first-principles phase diagram calculations different from A. van de Walle and G. Ceder’s former work[@van2002automating], another automated algorithm.
In CALPHAD, the uncertainty can be propagated from the model parameters to Gibbs free energy, then to the phase diagram. In previous work incorporating uncertainty within phase diagram predictions, as discussed above, Markov Chain Monte Carlo (MCMC) sampling approach has conventionally been used to obtain plausible optimum values and uncertainties of the parameters [@honarmandi2019uncertainty; @honarmandi2019bayesian]. This propagation of uncertainty assumes that the errors in the prediction of two materials from DFT are independent of one another. It is well known, however, that the errors of DFT predictions contain correlated, systematic prediction errors. Thus, within this work, we utilize BEEF as a sampler of GGA-level exchange-correlation potential space to understand the correlated uncertainty of the underlying ab initio thermodynamics data and its effect on the final prediction. Different functionals of BEEF-vdW lead to different predictions of both enthalpy from DFT and vibrational properties from the Debye analysis, and thus different value of G-T fitting parameters. This will then result in various predictions of stable phases. By propagating this uncertainty from the different functionals to the model parameters, then to the Gibbs free energy and ultimately the phase diagram, we can then assign a quantitative prediction confidence to the result.
In this work, we also considered the computation of equilibrium cell voltages based on the thermodynamics[@urban2016computational; @pande2018robust]. The electrochemical lithium-coupled ion transfer reaction with silicon is given by: $$\begin{gathered}
x(Li^++e^-)+Si \rightleftharpoons Li_xSi\end{gathered}$$ The Gibbs free energy change associated with this reaction is then given by: $$\begin{gathered}
\Delta{G} = G_{Li_xSi}-G_{Si}-xG_{Li^+}-xG_{e^-}\end{gathered}$$ where G$_{Li_xSi}$ is the free energy of the given Li-Si phase, G$_{Si}$ is the free energy of the solid pure silicon phase, G$_{Li^+}$ is the free energy of the Li-ion including the energy of solvation with the electrolyte and G$_{e^-}$ is the free energy of the electron at the potential of Si electrode. For ease, we can relate the sum of the free energy of the Li-ion and the electron to the free energy of bulk lithium given by the reaction: $$\begin{gathered}
Li^++e^- \rightleftharpoons Li_{(s)}\end{gathered}$$ and therefore $G_{Li^+} + G_{e^-_{U=0V}} = G_{Li_{(s)}}$. This sets the zero potential to the $Li/Li^+$ redox potential within the given electrolyte so $G_{e^-} = G_{e^-_{U=0V}} -eU_{Li/Li^+}$. We finally can write the change in Gibbs energy as: $$\begin{gathered}
\Delta{G} = G_{Li_xSi}-G_{Si}-xG_{Li_{(s)}}+x(eU_{Li/Li^+})\end{gathered}$$ and compute the the intercalation potential of a particular Li-Si phase. Additionally we can derive the intercalation potential by considering the phase transformation from a phase Li$_{x_1}$Si to another phase Li$_{x_2}$Si given by: $$\begin{gathered}
U_{Li/Li^+} = \dfrac{-1}{e(x_2-x_1)}(G_{Li_{x_2}Si}-G_{Li_{x_1}Si}-(x_2-x_1)G_{Li_{(s)}})\end{gathered}$$ Thus the important quantity needed is the free energies of the stable phases and from this, the itercalation potential can also be determined. This free energy not only contains the enthalpy which can be estimated as the internal energy given from DFT, but also contains vibrational contributions in the was of entropy and zero point energy. $\Delta{G} = \Delta{H}-T\Delta{S}+\Delta{ZPE}$.
To estimate the vibrational properties of the Gibbs energy relating to the zero-point energy and entropy, a Debye-Grunessen theory analysis [@moruzzi1988calculated] was performed using the DePye software [@guan2019uncertainty] which enables the efficient processing and vibrational predictions of the ensemble of functionals.
[points.png]{} (-0.15,5)
[points\_phaseD1.png]{} (-0.15,5)
[points\_phaseD2.png]{} (-0.15,5)
Results and Discussion {#results-and-discussion .unnumbered}
======================
A sample BIC model selection for Li$_{15}$Si$_4$ within the 10-parameter model space for the temperature range 200K-450K is shown in Figure \[fig:1\], with the BIC model selection of all other materials available in the Supplementary Information. In this case, the lowest value of the BIC corresponds to a 6 parameters model, while for other materials the number of parameters varied from 5 to 7 as can be seen in Table II\[tab:2\] along with the specific model chosen. Although the number of parameters chosen is consistent with the number commonly used in the literature about CALPHAD, different phases have different specific combinations of parameters rather than a fixed set of parameters as is conventionally used. The results show that although a higher number of parameters is not necessarily useful in practice, the specific terms used are important for the goodness of fit and thus will provide a more accurate and reasonable phase diagram prediction. The inset picture shows the results of root mean squared error (RMSE) of the prediction of training data corresponding to different models of solid-phase Li$_{15}$Si$_4$. For this plot and all corresponding plots for RMSE in the SI, we see that the error continues to decrease as the number of parameters increases while there is a distinct minimum for the BIC as it is expected that as the number of model parameters increases, over-fitting will occur. Thus BIC allows for the selection of a model with both a small error and a small number of parameters.
[pairs.png]{} (-0.15,5)
[pairs\_phaseD.png]{} (-0.15,5)
[combs.png]{} (-0.15,5)
[combs\_phaseD.png]{} (-0.15,5)
The phase diagram is determined by the structures on the convex hull of the free energy vs x in Li$_x$Si diagram. We show a sampling of the convex hull identified by 50 exchange correlation functionals within the BEEF-vdW model space at 300 K in Figure \[fig:2\]. We find that many of the GGA functionals predict similar convex hulls, while some predict vastly different stable phases and formation energies. Further, even for functionals having the same convex hull, the energy differences are vastly different, which will influence the intercalation voltages. The features are similar to that identified in an earlier work of the Li-C phase diagram by Pande et al.[@pande2018robust] This shows the importance of the exchange correlation functional in determining the phase diagram.[@lenchuk2019comparative]
As mentioned previously, we apply the results of this BIC model selection from the 10-parameter model space from optimal BEEF functional into other functionals of the ensemble for the temperature range is between 200K-450K. For the uncertainty analysis in phase diagram, we explore three different methods using BEEF as three kinds of samplers with various assumptions of statistical independence: independent points of phases, independent pairs of phases, and independent convex hulls of phases.
For the first case in which we assume the probability of prediction of a phase is independent of the prediction of any other phase, we define the c-value as the normalized number of times a functional predicts a phase to occur at a given temperature. That is the number of times a particular phase appears in the ensemble of phase diagrams. The specific c-value of each phase for each temperature is shown in Figure \[fig:3\](a). Figure \[fig:3\](b) and (c), shows this c-value plotted as a function of composition and temperature where the larger the c-value, the darker the color is. It can be seen that at a fixed temperature of 300K in Figure \[fig:3\](b), not all phases appear with high c-value of near 1. Rather some phases are predicted to be stable with higher confidence than others. Here we regard the confidence value beyond 0.05 as a good prediction value. In Figure \[fig:3\](a), phases Li$_{12}$Si$_7$, Li$_2$Si, and Li$_{21}$Si$_5$ are predicted to be unstable with confidence close to 0. In comparison with the phase diagrams from literature built on experimental data[@Liang2017thermodynamics] where phases only appear or do not appear, this incorporation of uncertainty allows for a more reliable prediction given the uncertainties present and the sensitivity of those uncertainties on the final result. There are many influencing factors and errors in experimental data. However, in the uncertainty phase diagram, we consider all of the different prediction results from the BEEF ensemble so that we can predict the occurrence probability of one phase point. At one phase with increasing temperature, if c-value also increases, the phase is stable at high temperatures, otherwise, the phase is stable at low temperatures. For example, we can say that solid Li is stable at low temperatures and Li$_{12}$Si$_7$ is very unlikely to be stable according to the c-value of the UQ phase diagram. In the previous uncertainty phase diagram found in literature,[@honarmandi2019uncertainty; @honarmandi2019bayesian] only the uncertainty range of a specific phase is given. Therefore, in this uncertainty phase diagram, it can not only give the uncertainty range but also give the occurrence probability compared with the uncertainty phase diagrams in P. Honarmandi et al’s work [@honarmandi2019bayesian]. Hence, this kind of uncertainty phase diagram can help determine a better phase prediction and understand the process of phase transition of a system better. When we consider the composition of mixing phases or the correlation between two adjacent predicted stable phases even all predicted stable phases, the former method is not enough, especially when trying to predict the exact phase mixture at compositions that are not a distinct phase. To do this, we utilize two additional methods that consider the occurrence of pairs together, as well as the occurrence of a the full set of phases on a hull. For the pairs of phases case in Li-Si system, similarly, we define the c-value as the normalized number of times a functional predicts a pair of phases to occur at a given temperature. Though there are 13 phases in Li-Si system, there are just 22 kinds of valid pairs collected form each convex hull, after removing the pairs with small probability(less than 0.05), there are just 8 kinds of valid pairs to be considered. Among them, there are just 2 kinds of pairs show with max probability when x in range 0 to 1 and T in range 200 to 450K. The specific c-value of each pair of phases for each temperature is shown in Figure \[fig:4\](a). Figure \[fig:4\](b) shows the corresponding phase diagram. Blue lines represents the phase boundaries of pairs of phases. For the phases between the phase boundaries, the deeper the color, the bigger the c-value. We can see when temperature is below 300K, phase with x in range between 0 and 0.196 will be predicted as the mixing of solid Li phase and solid Li$_{4.11}$Si phase with the maximum probability 0.75. With temperature increasing, this prediction will not change but the maximum c-value will decrease, whose physical meaning is that the solid Li phase may disappear because of the melting point. As for a phase with x in the range between 0.196 and 1.0, it is predicted to be the mixing of solid Li$_{4.11}$Si phase and solid Si phase with the maximum c-value 0.6 or a bit higher during the temperature range from 200 to 450K.
For the case of comparing final predicted convex hulls in Li-Si system, we define the c-value as the normalized number of times a functional predicts a specific convex hull of phases to occur at a given temperature. That is the number of appearance times of a convex hull of phases in the ensemble of phase diagrams divided by the number of functionals of an ensemble. Though there are 13 phases in Li-Si system, there are just 27 kinds of valid convex hulls predicted by the functionals of the ensemble, after removing convex hulls with a small probability(less than 0.05), there are just 6 kinds of the valid convex hull to be considered. Among them, there are just 2 convex hulls shown with max probability when x in range 0 to 1 and T in range 200 to 450K. The specific c-value of each convex hull of phases for each temperature is shown in Figure \[fig:5\](a). Figure \[fig:5\](b) shows the corresponding phase diagram. The crossing points of each horizontal T line and blue lines represent the predicted stable phases belong to the convex hull appearing with maximum probability at that temperature. For the phases between blue lines, the deeper the color, the bigger the c-value. We can see when the temperature is below 400K, the convex hull including stable solid phase Li, solid Li$_{4.11}$Si, and solid Si will be predicted appearing with the maximum c-value. Specifically, this prediction stays as 0.58 when T is below 300K, but with temperature increasing, the c-value will decrease, as the solid Li phase is predicted to disappear because of melting. When T increases from 400K to 450K, the convex hull including stable solid phase Li$_{4.11}$Si and solid Si will be predicted appearing with the maximum c-value but the value decreases. As for phase with x in range between 0.196 and 1.0, it is predicted to be the mixing of solid Li$_{4.11}$Si phase and solid Si phase with equal c-value of the convex hull it belongs to. The physical meaning of the white area is that the liquid phase Li may appear with the same c-value.
From the previous convex hull, we then evaluate the intercalation potentials, using the predicted energy of stable phases. The ensemble of energy predictions then generates and ensemble of intercalation potentials. In order to visualize these various predictions, the results at each composition were binned into a histogram with 0.066 V bin width. This then generates a probability distribution function at every composition with is then plotted as the contour map in Figure \[fig:6\]. From this figure, we see the maximum intercalation potential predicted with maximum probability varies slightly from the the BEEF optimal but give the range that includes the experimental data well. We can see here, different functionals can not only predict different phase transition but also predict different energy difference, both of them will influence the prediction of voltage. The same phase transition (namely the same pair of phases) may have huge voltage prediction difference because of the huge difference of energy difference between two phases from different functionals. And different phase transition (namely different pairs of phases) may have very close voltage prediction. If we just consider the c-value of phase, we may get better phases transition prediction; if we just consider the probability density of voltage, we may get better voltage profile prediction. By using both in combination, we can have a better understanding of the phase transitions and voltage profile to make a better prediction for experiments and the performance of the Li-Si system within batteries.
Conclusion {#conclusion .unnumbered}
==========
In this work, we employed model selection using the BIC to determine the best model from a list of independently generated models for a specific phase and specific temperature range using DFT-data and the Debye-Grunessen model. A sampling of GGA-space within the DFT data is carried out using the built-in error estimation capabilities of the BEEF-vdW exchange correlation function. Using this, we determine the uncertainty associated with Li-Si binary system, an important candidate Li-ion battery anode. We carry out three different approaches to uncertainty quantification for the Li-Si phase diagram. These three methods include various levels of statistical correlation between the prediction of phases. This analysis provides a basis to further extend uncertainty quantification of first principles data into the phase diagram predictions. We believe that quantifying the uncertainty will provide a more detailed assessment of the possible phase diagram and one particular use is to identify the regions of largest uncertainty to guide the most useful experiment to be done for the most information gain related to the phase diagram.
Y.Y would like to thank Dilip Krishnamurthy and Olga Vinogradova for their insightful input. G.H gratefully acknowledges funding support from the National Science Foundation under award CBET-1604898. Acknowledgment is also made to the Extreme Science and Engineering Discovery Environment (XSEDE) for providing computational resources through award number TG-CTS180061.
|
---
author:
-
title: '**High Resolution Spectroscopy and Spectropolarimetry of some late F-/early G-type sun-like stars as targets for Zeeman Doppler imaging.**'
---
Introduction
============
The study of young sun-like stars provides a window onto the Sun’s intensely active past and an understanding of early solar evolution. In particular, the observations of starspots and associated magnetic activity gives clues to the underlying dynamo processes operating in young sun-like stars. Thus, the study of magnetic fields helps us understand the solar interior as well as its atmosphere. A key question in this respect is: how does the young Sun differ in its internal structure and energy transport systems when compared with the modern-day Sun? The technique of Zeeman Doppler imaging (ZDI) [@Semel89; @Semel93; @Donati03] can be used to map the magnetic topologies of rapidly rotating young sun-like stars to help address this question.
In the Sun today, strong shear forces are formed at the interface between the solid-body rotation of the radiative zone and the differentially rotating convective layer in a region called the tachocline. In the tachocline differential rotation wraps north-south magnetic field lines around the Sun in the direction of rotation and convective motions act to raise the magnetic fields through the convection zone to emerge at the surface. This interaction converts the global poloidal magnetic field to a toroidal field. This effect is known as the “$\Omega$-effect". The $\alpha$-effect is the reverse process, converting the global toroidal field to the poloidal field. In contrast, studies by @Donati03 of K-dwarf stars and by @Marsden11a [@Marsden11b] and @Waite11 of pre-main-sequence G-type stars show regions of azimuthal fields. @Donati03 interpret this in terms of an $\alpha^{2}\Omega$ dynamo process [@Brandenburg89; @Moss95] being distributed throughout the entire convection zone and close to the surface of the star itself.
The Sun today undergoes activity cycles in the form of magnetic reversals, but at what stage do these cycles begin during the early evolution of the star? Recent theoretical work by @Brown10 has suggested that young stars undergo “attempted" field reversals, where the magnetic field begins to break-up, a signature of an impending reversal, only to reinforce again in the original direction. Resolving the origin of the solar dynamo will help us address the more general question of how stellar magnetic cycles develop in young stars, and affect any attendant emerging planetary systems. The search for ZDI targets is thus motivated by the need to study a sample of young Suns to test recent dynamo theory for these stars.
The initial search for potential ZDI targets by @Waite05 found two pre-main-sequence stars: HD 106506 and HD 141943. ZDI has been used to map the magnetic topologies of HD 106506 [@Waite11] and evolution of the magnetic topologies and variations in the surface differential rotation of HD 141943 [@Marsden11a; @Marsden11b]. This paper is a follow-on from this initial search for late F-/ early G-type stars. Our search specifically aims to measure the projected rotational velocity, [$\!$[*v*]{}sin[*i*]{}]{}, radial velocity, the level of magnetic and chromospheric activity, and confirm the expected youthful evolutionary status of these stars for studying the origins of the magnetic dynamo in young sun-like stars.
Observations at the Anglo-Australian Telescope
==============================================
Selection criteria
------------------
The Hipparcos space mission [@Perryman97] has provided a wealth of stellar astrometry, and revealed many previously unknown variable stars. A large number of these unresolved variables [@Koen02] are likely to be eclipsing binaries, but some are expected to be active stars with starspot modulation. From the Hipparcos database, late F-/early G-type unresolved variable stars were extracted from the original list of @Koen02. To reduce the sample to a manageable number of stars, only those sun-like stars with a variability between $\sim$ 0.04 and $\sim$ 0.1 magnitude were selected. If the variability was less than 0.04, the spot activity (if the variability was due to starspots) on the star would unlikely be sufficient to deform the stellar profiles sufficiently for any spatial information to be recovered using the technique of Doppler imaging (DI). Any variation above $\sim$ 0.1 would most likely be a result of a companion star. A final list of 38 stars was compiled for follow-up high-resolution spectroscopy and spectropolarimetry at the Anglo-Australian Telescope (AAT).
Spectroscopy
------------
High-resolution spectra of 38 late F-/early G- type stars were observed over two nights of Service observations on the 14th of April and 7th of September, 2008 using the using the University College of London Échelle Spectrograph (UCLES) at the AAT. The EEV2 chip was used with the central wavelength set to 526.8 nm. The 31 lines per mm grating was used with a slit width of 0.73 mm and slit length of 3.18 mm for observations on the 14th April while the slit width was set to 0.74mm and slit length of 3.17mm was set for observations on the 7th September, 2008. This gave an approximate resolution of 50500, extending from order \#84 to order \#129. A journal of the observations is shown in Table \[journal\_ucles\].
------------- -------------- ---------- ---------- ----------- ----------------- --------------------------------- --------------------- --------------
HIP Spectral UTSTART Exp Time S/N$^{a}$ v$_{rad}$$^{b}$ [$\!$[*v*]{}sin[*i*]{}]{}$^{b}$ EEW$^{c}$ H$\alpha$ EqW$^{d}$ Li
Type (sec) (mÅ) (mÅ)
UTDATE 2008, APR 14
23316 G5V$^{1}$ 10:13:49 600 77 22.8 $\sim$ 6 330$\pm$6 198$\pm$7
27518 G3$^{2}$ 10:26:08 400 72 5.2 $<$5 -46$\pm$19 $<$5
31021${^e}$ G3V$^{3}$ 10:48:24 400 73 – – – –
33111${^f}$ G5V$^{1}$ 10:57:02 400 79 – – – –
33699 F8V$^{4}$ 11:05:36 400 59 29.8 $<$5 56$\pm$10 41$\pm$3
41688 G6IV/V$^{6}$ 11:15:18 60 46 -20.4 $<$5 62$\pm$10 66$\pm$10
43720 G1V$^{1}$ 11:18:17 400 62 2.2 38 400$\pm$28 $<$5
46949 G2/3V$^{5}$ 11:26:48 600 64 26.8 $<$5 23$\pm$11 68$\pm$3
48146 G6IV/V$^{6}$ 11:39:30 600 49 0.5 $<$5 43$\pm$9 68$\pm$1
48770 G7V$^{1}$ 11:52:21 1200 67 19.6 35 990$\pm$90 234$\pm$5
60894 G0/1V$^{7}$ 12:14:57 600 29 29.4 $<$5 -36$\pm$30 $<$5
62517 G0$^{8}$ 12:27:56 400 68 -25.5 52 265$\pm$25 47$\pm$24
63734 F7/8V$^{9}$ 12:36:18 200 97 1.3 $\sim$ 6 121$\pm$11 141$\pm$6
63936 F8$^{8}$ 12:42:02 600 70 -10.0 $<$5 80$\pm$8 $<$5
64732${^e}$ F5V$^{6}$ 12:54:22 150 84 – – – –
66387 G0$^{8}$ 12:59:05 600 55 -24.1 $<$5 48$\pm$6 $<$5
67651${^g}$ F8$^{10}$ 13:10:25 400 57 – – – –
68328 G0$^{11}$ 13:20:13 600 76 8.8 $\sim$6 850$\pm$45 263$\pm$4
69338 G1V$^{6}$ 13:32:39 300 81 -8.6 $<$5 9$\pm$2 61$\pm$3
70053 G0$^{8}$ 13:39:00 450 68 7.8 $<$5 17$\pm$7 45$\pm$4
71933 F8V$^{1}$ 13:57:17 150 84 6.0 75 274$\pm$25 139$\pm$7
15:20:59 150 82 -1.9 75 $''$ $''$
71966 F7V$^{3}$ 14:01:16 450 75 9.3 $<$5 90$\pm$10 30$\pm$3
73780${^e}$ G0IV/V$^{7}$ 14:10:35 300 77 – – – –
75636${^g}$ G9V$^{1}$ 14:17:19 900 88 43.7 50 820$\pm$90 $<$5
77144 G1V$^{11}$ 14:33:28 450 93 -1.6 65 466$\pm$24 207$\pm$7
79090 F8$^{8}$ 14:43:39 600 60 6.1 $<$5 56$\pm$12 46$\pm$4
79688 G1V$^{5}$ 14:56:07 200 86 12.6 11 172$\pm$12 13$\pm$5
89829 G5V$^{1}$ 15:01:17 200 74 -10.6 114 280$\pm$52 211$\pm$13
15:25:42 200 75 1.2 114 $''$ $''$
90899 G1V$^{12}$ 15:06:00 450 82 -2.7 19 408$\pm$14 176$\pm$6
93378 G5V$^{1}$ 15:16:07 150 76 – 225 0 322$\pm$56
15:30:48 150 68 – 229 $''$ $''$
16:11:01 200 62 – 226 $''$ $''$
105388 G7V$^{1}$ 16:04:01 300 58 -1.8 17 520$\pm$50 216$\pm$5
UTDATE 2008, SEPT 7
5617 G2/3$^{7}$ 13:49:05 400 70 56.5 $<$5 – $<$5
10699 G7IV$^{1}$ 13:58:52 600 82 39.1 7 220$\pm$20 32$\pm$3
11241 F8V$^{9}$ 14:13:43 400 83 -3.5 $<$5 164$\pm$8 98$\pm$3
19072$^{f}$ F8$^{8}$ 18:19:56 240 79 – – – –
18:56:21 240 71 – – – –
20994 G0$^{8}$ 18:25:13 900 74 57.8 $<$5 85$\pm$31 44$\pm$2
21632${^h}$ G3V$^{1}$ 14:23:12 400 81 19.5 18 385$\pm$21 190$\pm$2
19:04:37 400 97 19.3 18 517$\pm$32 $''$
25848 G0$^{13}$ 18:42:35 600 68 27.3 69 668$\pm$60 250$\pm$13
19:13:43 600 65 28.7 69 $''$ $''$
------------- -------------- ---------- ---------- ----------- ----------------- --------------------------------- --------------------- --------------
$^{a}$S/N: Mean Signal-to-noise at Order 107, which was the centre of the spectrum. $^{b}$The radial velocity (v$_{rad}$) and projected rotational velocity ([$\!$[*v*]{}sin[*i*]{}]{}). The errors are estimated to be $\pm$1 , although for rapidly rotating stars with substantial deformation of the line profiles due to spot features, the errors could increase to $\pm$3 [km s$^{-1}$ ]{}or higher. $^{c}$EEW H$\alpha$: Emission equivalent width of the H$\alpha$ line, see Section 3.3. $^{d}$EqW Li: Equivalent width for the Li-670.78 nm spectral line. ${^e}$Binary system. ${^f}$Possible triple system. $^{g}$Possible binary system. $^{h}$ This star has shown variation in the H$\alpha$ profile hence both measurements for the emission equivalent width have been given.\
$^{1}$@Torres06, $^{2}$@Jackson54, $^{3}$@Houk75, $^{4}$@Rousseau96, $^{5}$@Houk82, $^{6}$@Houk99, $^{7}$@Houk78, $^{8}$@SAO66, $^{9}$@Houk88, $^{10}$@Dieckvoss75, $^{11}$@Sartori03, $^{12}$@Turon93, $^{13}$@Li98\
\
Spectropolarimetry {#sempol_obs}
------------------
Follow-up spectropolarimetric observations of stars that exhibited rapid rotation were undertaken on a number of Director’s nights at the AAT using the Semel Polarimeter (SEMPOL) [@Semel89; @Semel93; @Donati03]. Again the EEV2 chip was used, with a central wavelength of 526.8nm and coverage from 437.6497 nm to 681.8754 nm. The dispersion of $\sim$ 0.002481 nm at order \# 129, with an average resolution across the chip of approximately $\sim$ 70 000. Observations in circular polarisation (Stokes [*V*]{}) consists of a sequence of four exposures. After each of the exposures, the half-wave Fresnel Rhomb of the SEMPOL polarimeter was rotated between +45$^o$ and -45$^o$ so as to remove instrumental polarisation signals from the telescope and the polarimeter. Section \[spectro\_sect\] gives more details regarding spectropolarimetric observations while more details on the operation of SEMPOL is given in @Semel93, @Donati97 and @Donati03. A journal of the observations is shown in Table \[journal\_sempol\].
-------- ------------ ------------- ---------------- ------------------- -------------------- --------------------
HIP UTDATE UT Time$^a$ Exposure Time Mean S/N$^c$ Magnetic FAP$^d$
number (seconds) $^b$ (Stokes $\it{V}$) Detection?
21632 2008 Dec10 12:39:15 4 x 900 5898 No Detection 2.245 x 10$^{-01}$
21632 2008 Dec13 14:27:53 2 x 900 $^e$ 2710 Definite 4.761 x 10$^{-06}$
43720 2008 Dec09 16: 4:50 4 x 900 5304 Definite 0.000
43720 2008 Dec14 14:12:40 4 x 900 2626 No Detection 1.667 x 10$^{-02}$
43720 2009 Apr09 11: 8:13 4 x 800 3800 Definite 5.194 x 10$^{-10}$
43720 2009 Apr09 12:58:33 4 x 800 3759 Definite 6.871 x 10$^{-12}$
48770 2009 Dec03 16:42:47 4 x 750 1734 No Detection 1.528 x 10$^{-02}$
48770 2010 Apr01 10:30:40 4 x 800 2530 Definite 7.011 x 10$^{-10}$
62517 2009 Apr09 15: 7:23 4 x 800 1559 No Detection 8.504 x 10$^{-01}$
62517 2010 Apr02 12: 0:30 4 x 800 6685 Marginal Detection 1.786 x 10$^{-03}$
71933 2008 Dec18 17:44:15 4 x 900 1748 No Detection 7.146 x 10$^{-01}$
71933 2010 Apr01 15:37:08 4 x 800 8843 Definite 6.457 x 10$^{-11}$
77144 2010 Mar31 18:02:03 4 x 800 4315 Definite 0.000
77144 2010 Apr03 17:54:31 4 x 800 3783 No Detection 1.362 x 10$^{-01}$
89829 2009 Apr13 17:10:25 4 x 600 2544 No Detection 1.432 x 10$^{-01}$
89829 2010 Apr01 16:41:14 4 x 800 7274 Definite 4.663 x 10$^{-15}$
90899 2010 Mar28 17:50:20 2 x 800$^e$ 669 No Detection 6.837 x 10$^{-01}$
90899 2010 Apr02 17:42:47 4 x 800 3663 Marginal Detection 3.647 x 10$^{-3}$
93378 2010 Apr01 17:43:42 4 x 800 9727 No Detection 7.102 x 10$^{-2}$
105388 2008 Dec09 10:17:50 4 x 900 3822 Definite 1.732 x 10$^{-14}$
105388 2008 Dec10 10:23:32 4 x 900 3388 No Detection 5.084 x 10$^{-01}$
-------- ------------ ------------- ---------------- ------------------- -------------------- --------------------
\
$^a$ Mid-observing time\
$^b$ Generally a cycle consists of four sub-exposures.\
$^c$ Mean Signal-to-noise in the Stokes $\it{V}$ LSD profile, see Section \[Data\_Analysis\].\
$^d$ FAP: False Alarm Probability. See Section \[spectro\_sect\] for more details.\
$^e$ Due to cloud, this cycle was reduced to two sub-exposures.\
Data Analysis {#Data_Analysis}
-------------
The aim of this project is to determine the projected rotation velocity ([$\!$[*v*]{}sin[*i*]{}]{}), radial velocity, level of magnetic and chromospheric activity and estimate the age of each of the targets. The chromospheric activity indicators included the H$\alpha$, magnesium triplet and sodium doublet spectral lines. The Li[i]{} 670.78 nm spectral line was used as an age indicator. The initial data reduction was completed using the ES[p]{}RIT software package (Échelle Spectra Reduction: An Interactive Tool, [@Donati97]). The technique of Least Squares Deconvolution (LSD) [@Donati97] was applied to each spectra. LSD assumes that each spectral line in the spectrum from a star can be approximated by the same line shape. LSD combines several thousand weak absorption lines to create a high signal-to-noise (S/N) single-line profile. Whereas the average S/N of a typical profile was $\sim$ 50-100, the resulting LSD profile has a combined S/N of the order of 1000 or higher. This substantial multiplex gain has the advantage of removing the noise inherent in each line profile while preserving Stokes $\it{I}$ and $\it{V}$ signatures. The line masks that were used to produce the LSD profile were created from the Kurucz atomic database and ATLAS9 atmospheric models [@Kurucz93] and were closely matched to the spectral type of each individual star.
Results and Analysis
====================
Projected Rotational Velocity
-----------------------------
The projected rotational velocity, [$\!$[*v*]{}sin[*i*]{}]{}, was measured by rotationally broadening a solar LSD profile to match the LSD profile of the star. This method was shown to be reliable, particularly with rapidly rotating stars, by @Waite05 when they compared this technique with the Fast Fourier Transform technique of @Gray92. Table \[journal\_ucles\] shows the projected rotational velocities for the target stars. The error bars on each measurement are usually $\pm$1 , however, for rapidly rotating stars with substantial deformation of the line profiles due to spot features, the errors could increase to $\pm$3 [km s$^{-1}$ ]{}or higher. Many of these [$\!$[*v*]{}sin[*i*]{} ]{}values have not been previously determined.
The term Ultrafast Rotator (UFR) has been used extensively in the literature but without an explicit definition being applied. Hence the need to refine this terminology, particularly for solar-type stars. Stars that have projected rotational velocities less than 5 [km s$^{-1}$ ]{}will be considered as Slow Rotators (SR) as this is the lower limit at which we can accurately measure the [$\!$[*v*]{}sin[*i*]{}]{} of the star in this dataset. Those stars with [$\!$[*v*]{}sin[*i*]{} ]{}between 5 [km s$^{-1}$ ]{}and 20 [km s$^{-1}$ ]{}will be considered as Moderate Rotators (MR). This upper limit is considered a critical velocity where dynamo saturation has been theorised to slow the angular momentum loss of rapidly rotating stars (e.g. @Iwrin07 [@Krishnamurthi97; @Barnes96]). Below 20 , the strength of the star’s magnetic dynamo is related to the star’s rotation rate but above this speed, it is believed that dynamo saturation is occurring where the strength of the star’s magnetic dynamo is no longer dependent upon stellar rotation. One empirical measure of this saturation in young solar-type stars is coronal X-ray emission. This emission, defined as the ratio of the star’s X-ray luminosity to that of the star’s bolometric luminosity [@Vilhu84], increases as rotation increases when it plateau’s at the [$\!$[*v*]{}sin[*i*]{} ]{}of 20 . @Stauffer97 theorised that this is consistent with dynamo saturation. Those stars with [$\!$[*v*]{}sin[*i*]{} ]{}greater than 20 [km s$^{-1}$ ]{}to 100 [km s$^{-1}$ ]{}will be referred to as Rapid Rotators (RR). The upper limit of $\sim$ 100[km s$^{-1}$ ]{}was selected as at this rotational speed, the X-ray luminosity decreases below the saturated level [@Randich98], an effect that @Prosser96 called supersaturation. These definitions are consistent with those used by @Marsden09, @Marino03 and @Prosser96. Those stars between 100 [km s$^{-1}$ ]{}to 200 [km s$^{-1}$ ]{}will be referred to as Ultra-Rapid Rotators (URR). Stars that exceed 200 [km s$^{-1}$ ]{}will be referred to as Hyper-Rapid Rotators (HRR) and are likely to be very oblate. However many of the stars in this sample have not had their inclination determined, thus the [$\!$[*v*]{}sin[*i*]{} ]{}is likely to be an underestimate of the true equatorial rotation of the star, and thus depending on the inclination, the star may be more rapidly rotating than indicated by the [$\!$[*v*]{}sin[*i*]{}]{}. In addition, we limit these definitions to solar-type stars. Table \[vsini\_def\] gives a summary of the definitions used in this paper.
--------------------------- ----------------------------------
Classification [$\!$[*v*]{}sin[*i*]{}]{} range
()
Slow Rotator (SR) 0 - $<$ 5
Moderate Rotator (MR) 5 - $<$ 20
Rapid Rotator (RR) 20 - $<$ 100
Ultra-Rapid Rotator (URR) 100 - $<$ 200
Hyper-Rapid Rotator (HRR) 200+
--------------------------- ----------------------------------
: Classification of Solar-type stars based on projected rotational velocities.[]{data-label="vsini_def"}
\
Heliocentric Radial Velocity {#vrad}
----------------------------
Each spectrum, when extracted, was shifted to account for two effects. Initially, small instrumental shifts in the spectrograph were corrected for by using the positions of the telluric lines embedded in each spectrum. A telluric line mask was used to produce an LSD profile and the exact position of this profile was used to determine these small corrections [@Donati03]. Secondly, the heliocentric velocity of the Earth towards the star was determined and corrected for.
The LSD profile of the star was used to measure the radial velocity of the star by first re-normalising the profile, then fitting a gaussian curve to the profile and measuring the location of the minimum. The radial velocities are listed in Table \[journal\_ucles\]. The error in the radial velocity was estimated to be $\pm$1 [km s$^{-1}$ ]{}although the presence of spots on the surface did have some effect on the location of the minimum, especially on stars with rotational speeds in excess of 100 [km s$^{-1}$ ]{}such as HIP 89829 where the fitting of the gaussian profile was problematic given the rapid rotation. The large variation seen in the radial velocity measurements for HIP 89829 could be a result of this star being a binary star. However, we have searched for evidence of a companion star deforming the LSD profile of the spectropolarimetry data and conclude that this star is probably single. This still does not preclude the existence of a secondary component as a very low mass companion such as an M-dwarf is likely to modify the radial velocity of the primary without generating a detectable line in the LSD profile, because the line-mask employed here is optimised for the primary, not for the companion. It was impossible to accurately measure the radial velocity of the HRR HIP 93378 due to its extreme deformation of the LSD profiles.
Chromospheric Indicators: Hydrogen $\alpha$, Magnesium-I triplet and Sodium-I doublet.
---------------------------------------------------------------------------------------
The H$\alpha$, magnesium-I triplet (516.733, 517.270 and 518.362 nm) and sodium D$_{1}$ and D$_{2}$ doublet (588.995 and 589.592 nm) lines are often used as a proxies for stellar activity and in particular chromospheric activity. The H$\alpha$ line is formed in the middle of the chromosphere [@Montes04] and is often associated with plages and prominences. The magnesium I triplet and sodium D$_{1}$ and D$_{2}$ doublet lines are collisionally dominated and are formed in the lower chromosphere and upper photosphere. This makes them good indicators of changes in that part of the atmosphere of stars [@Montes04]. Many authors (e.g. @Zarro83, @Young89, @Soderblom93a, @Montes04, @Waite05) determine the emission component of the H$\alpha$ line by subtracting the stellar spectrum from a radial velocity-corrected, inactive star that has been rotationally broadened to match the [$\!$[*v*]{}sin[*i*]{} ]{}of the target. This technique is temperature dependent, but since all of our targets have similar spectral types, a solar spectrum was used as the inactive standard star in this survey. The resulting emission equivalent width (EEW) is a measure of the active chromospheric component of the these spectral lines, including the H$\alpha$ line. Figure \[activity\_indicators\] shows the core emission of the magnesium triplet and H$\alpha$ line for the more active stars in this sample. Many of the likely targets for ZDI exhibit core emissions; however, the HRR star HIP 93378 exhibits no activity in either the magnesium triplet or the H$\alpha$ line. This may be due to the extreme broadening of the spectral lines “washing out" the emission component.
![The core emission in the magnesium triplet lines, left panel, and the H$\alpha$ spectral line, right panel, are shown for rapid rotators in this sample. This core emission was obtained by subtracting a radial velocity corrected, rotationally-broadened solar spectrum. []{data-label="activity_indicators"}](Activity_Indicators.eps)
Lithium-I 670.78-nm: An Age Indicator {#lithium_paragraph}
-------------------------------------
In the absence of a companion star, an enhanced Li[i]{} 670.78 nm line can be used as an indicator of youth [@Martin96] for stars that are cooler than mid-G type (0.6 $\le$ B-V $\le$ 1.3) although this is not as useful for F-type stars where there appears to be a plateau in the depletion of the lithium due to age [@Guillout09]. However, @doNascimento10 point out that there is a large range in lithium depletion for solar-type stars which may reflect different rotational histories or as a result of different mixing mechanisms such as shear mixing caused by differential rotation [@Bouvier08]. In analysing the spectra, the equivalent width EqW$_{Li}$ was measured using the [SPLOT]{} task in [IRAF]{}. This was done so as to allow comparison with those measured by @Torres06. The error in the measured values of EqW$_{Li}$ is primarily due to uncertainties in the continuum location. When the rotational velocity, [$\!$[*v*]{}sin[*i*]{}]{}, exceeded 8 , the Li[i]{} spectral line is blended with the nearby 670.744 nm Fe[i]{} line. This was corrected using the same correction factor developed by @Soderblom93a [@Soderblom93b]. The correction used is shown in equation \[lithium\]. $$\label{lithium}
EW_{Li}\it{corr} = EW_{Li} - 20 (B-V) - 3$$
Some of the results appear slightly discrepant when compared with @Torres06. It is unclear whether @Torres06 corrected for the Fe[i]{} line in their measurements but such a difference in processing may possibly explain the discrepancies. Figure \[age\_indicator\] shows the strength of the Li[i]{} 670.78 nm line, compared with the nearby Ca[i]{} 671.80 nm line, for a number of stars that are likely to be future targets for the ZDI programme.
![The lithium-I 670.78 nm line, compared with the calcium-I 671.77 nm line, for a number of stars in this survey.[]{data-label="age_indicator"}](Age_Indicator.eps)
Spectropolarimetry {#spectro_sect}
------------------
The magnetic signatures embedded in starlight are extremely difficult to detect. The typical Zeeman signature is very small, with a circular polarisation signature (Stokes [*V*]{}) of $\sim$0.1% of the continuum level for active stars [@Donati97]. As discussed in Section \[sempol\_obs\], observations in Stokes [*V*]{} consists of a sequence of four sub-exposures with the half-wave Fresnel Rhomb being rotated between +45$^o$ and -45$^o$. To detect these signatures, LSD is applied to increase the signal-to-noise of the signature when creating the Stokes $\it{V}$ profile. The Stokes $\it{V}$ profile is the result of constructively adding the individual spectra from the four exposures by “pair-processing" sub-exposures corresponding to the opposite orientations of the half-wave Fresnel Rhomb. To determine the reliability of the process, a “null" profile is produced as a measure of the noise within the LSD process. This null profile is found by “pair-processing" sub-exposures corresponding to the identical positions of the half-wave Fresnel Rhomb of the SEMPOL polarimeter during each sequence of 4 sub-exposures.
When producing figures, such as Figure \[hip21632\_lsd\] for example, the Stokes $\it{V}$ (upper) and null (middle) profiles have been multiplied by 25 so as to show the variation within each profile. Both profiles have been shifted vertically for clarity. The dots on the Stokes $\it{V}$ and null profiles are the actual data while the smooth curve is a 3-point moving average. The deformation in the Stokes $\it{V}$ profile is a direct result of the magnetic field observed on the star while deformation in the Stokes $\it{I}$ (intensity) profile (lower) is a result of starspots on the surface of the star. For more information on ZDI see @Carter96 and @Donati97.
For each observation a false-alarm probability (FAP) of magnetic field detection was determined. FAP is a measure of the chance of the signal found in the Stokes $\it{V}$ profile being a result of noise fluctuations rather than a real magnetic detection. The FAP is based on a $\chi^2$ probability function [@Donati92] and is estimated by considering the reduced $\chi^{2}$ statistics both inside and outside the spectral lines, as defined by the position of the unpolarised LSD profiles, for both the Stokes $\it{V}$ and the null profiles [@Donati97]. The FAP for each observation is listed in Table \[journal\_sempol\]. A definite magnetic detection in the Stokes $\it{V}$ was considered if the associated FAP was smaller than $10^{-5}$ (i.e. $\chi^2$ probability was larger than 99.999 %) while a marginal detection was observed if the FAP was less than $10^{-3}$ but greater than $10^{-5}$. In addition to this, the signal must only have been detected in the Stokes $\it{V}$ profile and not within the null profile, and be within the line profile velocity interval, from $v_{rad}$-[$\!$[*v*]{}sin[*i*]{} ]{}to $v_{rad}$+[$\!$[*v*]{}sin[*i*]{}]{}. This criteria is consistent with the limits used by @Donati97.
Some comments on individual stars
=================================
Many of the stars in this survey exhibited some indication of activity either due to the presence of a companion star, or in the case of single stars, activity due to youth and/or rapid rotation. We will consider each likely ZDI target in more detail in this section.
Moderate and rapid rotators suitable for ZDI studies
----------------------------------------------------
### HIP 21632
HIP 21632 is a G3V star [@Torres06]. The Hipparcos space mission measured a parallax of 18.27$\pm$1.02 milli-arcseconds (mas) [@vanLeeuwen07], giving a distance of $178_{-9}^{+11}$ light-years (ly). Using the bolometric corrections of @Bessell98 and the formulations within that paper, the effective temperature of this star was determined to be 5825$\pm$45 K and the radius was estimated to be 0.96$\pm$0.04 R$_\odot$. The luminosity was subsequently estimated to be $0.93_{-0.11}^{+0.12}$ L$_\odot$. @Zuckerman04 proposed that HIP 21632 was a member of the Tucana/Horologium Association indicating an age of $\sim$30 Myr. This star has an emission equivalent width for the H$\alpha$ line in the range from $\sim$ 385 mÅ to $\sim$ 517 mÅ demonstrating the presence of a very active, and variable, chromosphere. This variation could be due to prominences occuring on this star. Two exposures, separated by 4 hours 41.41 minutes, were taken on April 14, 2008 demonstrated noticeable variation in the core emission of the H$\alpha$ line, as shown in Figure \[hip21632\_comparisonHA\]. Yet there was no variation in the magnesium triplet and indeed, there was very little core emission in the three lines, with some filling-in in the wings of the lines. However, there were some minor changes in the sodium D$_{1}$ line but the variation is no where near as pronounced as in the mid-level chromospheric level. There is no evidence of the presence of a companion star in the LSD profile. @Torres06 measured a radial velocity of 18.8 [km s$^{-1}$ ]{}(using cross correlation methods) and an equivalent width for the Li[i]{} line of 200 mÅ. These values are consistent with the values obtained from this survey of an average radial velocity of 19.4$\pm$1 [km s$^{-1}$ ]{}and a equivalent width for the Li[i]{} line of 190$\pm$2 nm (see Section \[lithium\_paragraph\] on a possible reason for slightly discrepant values).
This star was observed spectropolarimetrically on two occasions using SEMPOL. On the first occasion, a no magnetic signal was detected with a mean S/N of 5898 with 4 sub-exposures, yet achieved a definite magnetic detection with only 2 sub-exposures with a mean S/N of 2710 on the second occasion. The resulting LSD profile for this cycle is shown in Figure \[hip21632\_lsd\]. Whereas this star is only a moderate rotator, it is a worthwhile target for more detailed spectropolarimetric studies.
![The magnetic detection of young G3V star HIP 21632. The lower profile is the Stokes $\it{I}$ LSD profile, the middle profile is the null profile and the upper profile is the Stokes $\it{V}$ profile. The dots are the actual data while the smooth line is a moving 3-point average of the data. The Stokes $\it{V}$ and Null profiles have been vertically shifted for clarity. In addition to this, the Stokes $\it{V}$ and Null profiles each were multiplied by 25 in order to highlight the actual signatures. The Stokes $\it{V}$ profile clearly shows a strong magnetic detection. This was achieved using only 2 sub-exposures with a mean S/N of 2560.[]{data-label="hip21632_lsd"}](hip21632.eps)
![The variation of the H$\alpha$ profile of HIP 21632. Two exposures were taken, separated by 4 hours, 41.41 minutes. The variation in the H$\alpha$ profile is shown as a shaded region in the difference spectrum. The sharp absorption line at $\sim$656.4 nm in the 12:23:12UT spectra is most likely due to telluric lines at 656.4049 and 656.4200 nm. []{data-label="hip21632_comparisonHA"}](hip21632_comparing_ha.eps)
### HIP 25848
HIP 25848 is a G0 weak-lined T Tau-type star [@Li98]. It has a trigonometric parallax of 7.95$\pm$1.29 mas [@vanLeeuwen07] giving a distance of $410_{-57}^{+79}$ ly. Using the bolometric corrections by @Bessell98, the effective temperature was estimated to be 5700$\pm130$ K. Using the formulation contained in @Bessell98 the star is estimated to be $\sim$ $1.67_{-0.17}^{+0.23}$ R$_\odot$. Placing this star on the theoretical isochrones of @Siess00, it is estimated to be 1.3$\pm$0.1 M$_\odot$ with an age between 10Myr to 20Myr. This is shown in Figure \[evolution\]. This is consistent with the age found by @Tetzlaff11, however slightly less massive than that quoted in that paper. @Norton07 used SuperWASP to measure a period of 0.9426 d. This survey measured a [$\!$[*v*]{}sin[*i*]{} ]{}of 69 . This star has an emission equivalent width for the H$\alpha$ line of 668$\pm$60 mÅ, meaning that it is very active and one of the most active stars from this survey. The sodium doublet lines were filled in, almost to the continuum. It has a very deep Li[i]{} line suggesting, in the absence of a companion star, a youthful star. No spectropolarimetric observations were obtained were obtained. With a declination of +23$^{o}$, it would be a difficult target for ZDI at the AAT. However, this star would be an ideal target for ESPaDOnS at the CFHT (Canada-France-Hawaii Telescope, Hawaii) or NARVAL at the TBL (Télescope Bernard Lyot, Pic du Midi, France).
### HIP 43720
HIP 43720 is a particularly active, G1V star [@Torres06]. It has a trigonometric parallax of 5.38$\pm$0.94 mas [@vanLeeuwen07] giving a distance of $606_{-90}^{+128}$ ly. Using the star’s V-I value and the formulation in @Bessell98, the star’s temperature was estimated to be $5700_{-45}^{+40}$ K while its radius was estimated to be $2.6_{-0.4}^{+0.7}$ R$_\odot$. Placing HIP 43720 onto the theoretical isochrones of @Siess00, as shown in Figure \[evolution\], suggests that this star’s age is $\le$ 10 Myr years and has a mass of between 1.6 and 1.8 M$_\odot$. However, this age estimate is not supported by the depth of the Li[i]{} line, with an equivalent width of $<$5 mÅ. One can speculate that the lithium has already been depleted. @Guillout09 suggest that stars with deep convective envelopes, such as M-dwarfs, are very efficient at depleting the lithium concentration. Being a pre-main-sequence star, HIP 43720 may also possess a very deep convective zone. Alternatively, @Bouvier08 suggest that this depletion may be due to large velocity shear at the base of the convective zone as a result of star-disk interaction.
There is evidence of an active chromosphere with an emission equivalent width for the H$\alpha$ line of $\sim$400$\pm$28 mÅ and strong emission in the magnesium triplet lines, as shown in Figure \[activity\_indicators\]. Definite magnetic fields were observed on three occasions, however no magnetic field was detected on a fourth observation. On that occasion, the mean S/N was $\sim$ 2626. Figure \[hip43720\_lsd\] shows the Stokes $\it{V}$, null and Stokes $\it{I}$ profile. This is an interesting target for follow-up spectropolarimetric studies at the AAT and as a result, is the subject of a forthcoming intense ZDI study (Waite et al., in preparation).
![The various LSD profiles, as explained in Figure \[hip21632\_lsd\], for HIP 43720. The Stokes $\it{V}$ profile (upper profile) shows evidence of a strong magnetic feature on the young star HIP 43720. The LSD profile (lower profile) has a flat bottom, possibly indicating the presence of a polar spot feature.[]{data-label="hip43720_lsd"}](hip43720.eps)
### HIP 48770
HIP 48770 is a G7V pre-main-sequence star [@Torres06]. It has a trigonometric parallax of 5.83$\pm$1.55 mas [@vanLeeuwen07] giving a distance of $554_{-115}^{+199}$ ly. The [$\!$[*v*]{}sin[*i*]{} ]{}was measured to be 35 , which is consistent with that found by @Torres06. The radial velocity was determined to be 19.6 , which is different from the 22.6 [km s$^{-1}$ ]{}observed by @Torres06. There appears to be no evidence of a secondary component in the spectra; but the presence of a companion in a large orbit cannot be ruled out. HIP 48770 is very young with a predominate lithium feature, as shown in Figure \[age\_indicator\]. @Ammons06 calculated an effective temperature of 5539 K. However, using the bolometric corrections of @Bessell98, our estimate is higher at 6000$\pm$300 K. It is also very active with the H$\alpha$ spectral line being almost entirely filled in; and the magnesium triplet is also very strong. A visual magnitude of 10.5 would normally make it a challenging target for SEMPOL at the AAT. However, a magnetic detection was observed at the AAT demonstrating its highly active nature. The Stokes $\it{I}$ LSD profile, along with the Stokes $\it{V}$ profile is shown in Figure \[hip48770\_lsd\].
![The various LSD profiles, as explained in Figure \[hip21632\_lsd\], for HIP 48770.[]{data-label="hip48770_lsd"}](hip48770.eps)
### HIP 62517
HIP 62517 is an active G0 star [@SAO66]. It has a parallax of 2.38$\pm$1.6 mas [@vanLeeuwen07] giving a distance of $420_{-169}^{+862}$ ly. This star has a [$\!$[*v*]{}sin[*i*]{} ]{}of 52 . This particular star has a strong H$\alpha$ emission of 265 mÅ coupled with some filling in of the core of the magnesium triplet. However, its lithium line is rather weak, indicating that it may not be as young as some of the other stars in the sample. @Ammons06 calculated an effective temperature of 5336K, which is slightly higher than our estimate of 5250$\pm$65 K found using the bolometric corrections of @Bessell98. A marginal detection of a magnetic field was recorded on 2010, April 2, as shown in Figure \[hip62517\_lsd\]. The S/N was 6685. Although only two snapshots were taken several months apart, with the indications that this star is single, the global magnetic field may be relatively weak and would be difficult to recover any magnetic features if observed over several epochs. This makes this star a difficult target for ZDI at the AAT.
![The marginal magnetic detection in the Stokes $\it{V}$ profile (upper profile) for HIP 62517. The profiles are as explained in Figure \[hip21632\_lsd\].[]{data-label="hip62517_lsd"}](hip62517.eps)
### HIP 71933
HIP 71933 is a pre-main-sequence F8V star [@Torres06]. It has a parallax of 11.91$\pm$0.99 mas [@vanLeeuwen07] giving a distance of $274_{-21}^{+25}$ ly. It has a projected rotational velocity of 75 . The radial velocity was measured to be $\sim$4 . As mentioned previously, the large spots on the surface of this star makes accurate radial velocity measurements difficult. However, @Torres06 measured 8.7 [km s$^{-1}$ ]{}while @Gontcharov06 measured 12.3$\pm$0.4 [km s$^{-1}$ ]{}and @Kharchenko09 measured 12.1$\pm$0.4 . Perhaps this star is part of a wide binary system. @Torres06 flagged that this star might be a spectroscopic binary star. If the star was a binary, and the companion’s profile is overlapping the primary’s profile, it could be mistaken for spots. Alternatively, the companion may be a faint M-dwarf star thereby not deforming the profile at all. Careful consideration of the LSD profiles produced from the high-resolution spectra obtained using the normal UCLES setup (R$\sim$50000) and spectropolarimetry (R$\sim$70000), suggest that the deformations are due to spots rather than a companion. However, the presence of a secondary component cannot be ruled out by this survey.
@Holmberg09 estimated the effective temperature to be 5900K while @Ammons06 estimated the temperature to be 5938K. The equivalent width of the Li[i]{} was measured to be 139$\pm$7 mÅ, after accounting for the Fe[i]{} blended line using equation \[lithium\]. When using the theoretical isochrones of @Siess00, as shown in Figure \[evolution\], this star’s age is estimated to be $\sim$20 Myr years and has a mass of $\sim$ 1.2 M$_\odot$.
This is a particularly active star, as shown in Figure \[activity\_indicators\], with emission in the magnesium triplet lines and the H$\alpha$ line. However, there was some core emission in the Na[i]{} D$_{2}$, but not in the D$_{1}$ line. Spectropolarimetry was conducted on two occasions, once when the seeing was very poor ($\sim$ 2.5 to 3.5 arcsec) and only a S/N of 1748 but on the other occasion, reasonable seeing ($\sim$1.5 arcsec) resulted in a definite detection of a magnetic field. This detection is shown in Figure \[hip71933\_lsd\].
![The various LSD profiles, as explained in Figure \[hip21632\_lsd\], for HIP 71933. The S/N for this was 8843.[]{data-label="hip71933_lsd"}](hip71933.eps)
### HIP 77144
HIP 77144 is a post T-Tauri G1V star @Sartori03 in the Scorpius-Centaurus group [@Mamajek02]. It has a parallax of 7.12$\pm$1.28 mas [@vanLeeuwen07] giving a distance of $458_{-70}^{+100}$ ly. It has a [$\!$[*v*]{}sin[*i*]{} ]{}of 65 [km s$^{-1}$ ]{}with a particularly strong H$\alpha$ (EEW = 466$\pm$24 mÅ) and Li [I]{} lines (EqW = 207$\pm$7 mÅ). The temperature is estimated to be $\sim$6000$\pm$130 K. It is approximately $1.45_{-0.18}^{+0.25}$ R$_\odot$ when using the bolometric corrections of @Bessell98, giving a luminosity of $2.4_{-0.7}^{+1.2}$ L$_\odot$. This young star, when placed on the theoretical isochrones of @Siess00 gives an age of this star of $\sim$ 20 Myr and is approximately 1.3$\pm$0.1 M$_\odot$. This is shown in Figure \[evolution\]. The radial velocity was measured to be -1.6 , which is consistent with that found by @Madsen02 and @Kharchenko07 to within the respective errors of each measurement. A definite magnetic field was detected on 2010, March 31, with a mean S/N in the Stokes $\it{V}$ profile of 4315. This magnetic detection is shown in Figure \[hip77144\_lsd\].
![The various LSD profiles, as explained in Figure \[hip21632\_lsd\], for HIP 77144.[]{data-label="hip77144_lsd"}](hip77144.eps)
### HIP 90899
HIP 90899 is a G1V star [@Turon93]. According to Hipparcos database, this star has a parallax of 10.87$\pm$1.34 mas [@vanLeeuwen07], giving a distance of $300_{-33}^{+42}$ ly. Using the V-I determined by the Hipparcos Star Mapper Photometry, a value of 0.62$\pm$0.03 gives a temperature of $6090_{-140}^{+130}$ K using the formulation given in @Bessell98. This is consistent with other authors such as @Wright03 (6030 K) and @Ammons06 (5988 K). Using the bolometric corrections of @Bessell98 gives a radius of $0.97_{-0.08}^{+0.1}$R$_\odot$ and a luminosity of $1.13_{-0.25}^{+0.37}$ L$_\odot$. This star has an emission equivalent width for the H$\alpha$ line of 408$\pm$14 mÅ, with some filling of the core of the magnesium triplet lines, meaning that it is a very active star. It has a very deep Li[i]{} line with an equivalent width of 176$\pm$6 mÅ, suggesting that, in the absence of a companion star, is youthful in nature. The magnetic detection, as shown in Figure \[hip90899\_lsd\] was only marginal but still, at a [$\!$[*v*]{}sin[*i*]{} ]{}of 19 , the magnetic topologies should be recoverable at the AAT.
![The various LSD profiles, as explained in Figure \[hip21632\_lsd\], for HIP 90899.[]{data-label="hip90899_lsd"}](hip90899.eps)
### HIP 105388
HIP 105388 is a G7V pre-main-sequence star [@Torres06]. This star has a [$\!$[*v*]{}sin[*i*]{} ]{}of 17 . This compares with the measurement of @Torres06 of 15.4 . The value of this star’s radial velocity was determined to be -1.8$\pm$1.0 . This value is consistent with those measured by @Torres06 of -0.9 [km s$^{-1}$ ]{}and @Bobylev06 -1.6$\pm$0.2 , to within the respective errors. This star has an emission equivalent width for the H$\alpha$ line of 520 mÅ, however, there is little core emission in either the magnesium triplet or sodium doublet. @Zuckerman04 proposed that HIP 105388 was a member of the Tucana/Horologium Association and an age estimate for this moving group, hence this star, is 30 Myr. Further indication of the youthful nature of this star is the very strong Li[i]{} line, as shown in Figure \[lithium\]. @Tetzlaff11 estimate that this is a 1.0$\pm$0.1 M$_\odot$ star. Whereas the [$\!$[*v*]{}sin[*i*]{} ]{}is at the lower limit for ZDI at the AAT, a magnetic detection on this star was secured with an S/N of only 3822. There is limited spot activity as evidenced by the smooth Stokes $\it{I}$ profile (lower LSD profile) in Figure \[hip105388\_lsd\]. Recovering magnetic features from slow to moderate rotators is possible, as shown by @Petit05 and @Petit08.
![The various LSD profiles, as explained in Figure \[hip21632\_lsd\], for HIP 105388. The Stokes $\it{I}$ LSD profile shows limited spot activity on the star yet there is a definite magnetic detection in the Stokes $\it{V}$ profile. This is supported by the absence of signal in the null profile.[]{data-label="hip105388_lsd"}](hip105388.eps)
Ultra-Rapid Rotator: HIP 89829
------------------------------
HIP 89829 is a G5V star [@Torres06]. With a [$\!$[*v*]{}sin[*i*]{} ]{}of 114 , this star has been classified as an URR. This measurement is consistent with @Torres06. @Pojmanski02 quote a rotational period of 0.570751d with a photometric amplitude of $\delta$V = 0.07. This star is very active and has an emission equivalent width for the H$\alpha$ line of 280$\pm$52 mÅ yet little if any emission in the magnesium triplet or the sodium doublet is seen. It has a very deep, albeit broadened, lithium line with an equivalent width of 211$\pm$13 mÅ. This indicates, in the absence of a companion star, a youthful star. This is consistent with that found by @Torres06. When placed on the theoretical isochrones of @Siess00, this star is approximately 25-30 Myr and has a mass of between 1.0 and 1.2 M$_\odot$. The magnetic detection is shown in Figure \[hip89829\_lsd\]. This is one of the most rapidly rotating stars that has had its magnetic field detected at the AAT.
As mentioned in Section \[vrad\], the large variation seen in the radial velocity measurements for HIP 89829 could be a result of this star being a binary star. However, after carefully considering the resulting LSD profiles from both the normal UCLES setup (R$\sim$50000) and SEMPOL setup (R$\sim$70000), we feel that this star is [**probably**]{} single.
![The various LSD profiles, as explained in Figure \[hip21632\_lsd\], for HIP 89829.[]{data-label="hip89829_lsd"}](hip89829.eps)
Hyper-Rapid Rotator: HIP 93378
------------------------------
HIP 93378 is a pre-main-sequence, G5V star [@Torres06]. According to Hipparcos database, this star has a parallax of 9.14$\pm$0.92 mas [@vanLeeuwen07], giving a distance of $357_{-33}^{+40}$ ly. It is a HRR with a [$\!$[*v*]{}sin[*i*]{} ]{}of 226 , which is consistent with the 230 [km s$^{-1}$ ]{}value measured by @Torres06 within the large uncertainty created by such a rapid rotation. This star’s H$\alpha$ profile, when matched against a rotationally broadened solar profile, exhibitesd no emission in the core. Thus it appears that there is limited chromospheric activity occurring on this star. However, as mentioned previously, this lack of chromospheric emission may be due to the extreme broadening of the spectral lines thereby “washing out" the emission component. Another reason for this decreased H$\alpha$ emission may be due to supersatuation or even a modification of the chromospheric structure by the extremely strongs shear forces as a result of such rapid rotation.
No magnetic detection was observed on this star, even when the data were binned to increase the relative signal-to-noise in excess of 9000. Again this extreme rotation may have simply washed out any magnetic signature. The resulting LSD profile is shown in Figure \[hip93378\_lsd\]. Although this star is an extremely difficult target for ZDI at the AAT, its extreme rotation makes it an interesting target for Doppler imaging.
![The various LSD profiles, as explained in Figure \[hip21632\_lsd\], for HIP 93378. []{data-label="hip93378_lsd"}](hip93378.eps)
Active, young, slowly rotating stars.
-------------------------------------
This survey also found a number of slower rotating stars that are very young. Due to time constraints we did not take spectropolarimetric observations of these stars. However, the CFHT and TBL have been able to recover magnetic fields on slow rotators [@Petit08]. One such star is the early G0 [@Sartori03] star HIP 68328. It has a relatively slow rotation of $\sim$ 6 . It has a parallax of 8.34$\pm$1.56 mas [@vanLeeuwen07] giving a distance of $391_{-61}^{+90}$ ly. Using the bolometric corrections of @Bessell98, the temperature of HIP 68328 was estimated to be $5420_{-107}^{+114}$ K. This is consistent with the temperature estimated by @Lafrasse10 even though it is lower than the most recent estimate by @BailerJones11 of 5871 K. This project estimates that this star has a radius of $1.4_{-0.17}^{+0.26}$ R$_\odot$, giving a luminosity of $1.49_{-0.44}^{+0.78}$ L$_\odot$. This value is similar, within the relative error bars, to that estimated by @Sartori03 of 1.66 L$_\odot$. @DeZeeuw99 identified this star as a possible member of the Scorpius-Centaurus OB association. This is a young star-forming region with stars less than 20 Myr old. When placing this star on the theoretical isochrones of @Siess00, as shown in Figure \[evolution\], the star has an age of $\sim$20$\pm$10 Myr and a mass of $\sim$1.2$\pm$0.1 M$_\odot$. This age is consistent with the observation of a very deep Li[i]{} line with an equivalent width of 263$\pm$4 mÅ. It has a emission equivalent width for the H$\alpha$ of 850$\pm$45 mÅ, the most active star in this sample by this measure. These observations of youth and activity are shown in Figure \[hip68328\_activity\]. Other slow rotators with substantial lithium lines include HIP 23316, HIP 41688, HIP 63734 and HIP 11241. All have indicators of having an active chromosphere with core emission in the H$\alpha$ line. Another interesting target is the slow rotator is HIP 5617. It has very prominent emission in the wings of the H$\alpha$ line, extending above the continuum, indicating the presence of circumstellar material. However, the lithium line is very weak. This is perhaps a T-Tauri star.
![The activity indicators for HIP 68328. This star has a very strong emission component of the H$\alpha$ line with some moderate filling in of the sodium doublet and magnesium triplet. Also shown in this figure is the strong Li[i]{} 670.78 nm line.[]{data-label="hip68328_activity"}](hip68328_activity.eps)
Binary and Multiple Stellar Systems
-----------------------------------
The LSD profile is an excellent way of identifying a binary star [@Waite05], as the LSD profiles often show both stars, except if the star is undergoing an eclipse or is a faint star such as an M-dwarf. Where stars that appear to be rapidly rotating, more than one spectra (often two, three or more) were taken to make sure that the star was indeed single. HIP 31021, HIP 64732 and HIP 73780 were identified as binary stars based on their individual LSD profiles. HIP 64732 may have a giant polar spot on one of the components as one of the LSD profiles exhibited a “flat bottom", indicating the likely presence of a giant polar spot or high latitude features. This is shown in top left panel of Figure \[binary\]. HIP 19072, HIP 67651 and HIP 75636 are likely to be spectroscopic binary stars while HIP 33111 could be a triple system. The associated LSD profiles are shown in Figure \[binary\]. There is a slight deformation of the LSD profile of HIP 75636 on the blue side that could be due to a companion star just moving into an eclipse of the second star. Also, the radial velocity of this star was measured to be 43.7$\pm$1 [km s$^{-1}$ ]{}whereas @Torres06 measured the radial velocity to be 5.9 . While some of the stars in this survey exhibit slight shifts in the radial velocity when compared with the work of @Torres06, this difference is far too great to conclude anything else except that it is a binary star.
![LSD profiles of some likely binary stars.[]{data-label="binary"}](binary_stars.eps)
Table \[summary\] gives a summary of the likely targets for follow-up magnetic studies.
Confirmed targets HIP number
-------------------------- -----------------------------------
MR 21632, 90899, 105388
RR 43720, 48770, 62517, 71933, 77144
URR 89829
Potential targets ${^1}$
SR & MR 5617, 10699, 11241, 23316, 25848,
63734, 68328
HRR 93378 (DI target)
Binary Stars
Binary Stars 31021, 64732, 73780
Probable Binary Stars 19072, 33111${^2}$, 67651, 75636
\
See Table \[vsini\_def\] for definitions used in this table.\
${^1}$ Based solely on activity (if SR or MR) or rapid rotation.\
${^2}$ This star may even well be a triple system.\
Conclusion
==========
This survey aimed to determine the nature of a number of some unresolved variable stars from the Hipparcos database and to identify a number of targets for follow-up spectropolarimetric studies at the AAT. Of the 38 stars observed, three stars (HIP 31021, HIP 64732, HIP 73780) were spectroscopic binary stars while further three stars, (HIP 19072, HIP 67651 and HIP 75636) are likely to be a spectroscopic binary stars while HIP 33111 could be a triple system. Two stars rotate with speeds in excess of 100 : HIP 93378 ([$\!$[*v*]{}sin[*i*]{} ]{}$\sim$ 226 ) and HIP 89829 ([$\!$[*v*]{}sin[*i*]{} ]{}$\sim$ 114 ). Magnetic fields were detected on a number of the survey stars: HIP 21632, HIP 43720, HIP 48770, HIP 62517, HIP 71933, HIP 77144, HIP 89829, HIP 90899 and HIP 105388. All of these stars would be suitable for follow-up spectropolarimetric studies using SEMPOL at the AAT.
Acknowledgments {#acknowledgments .unnumbered}
===============
The authors thanks the Director of the Australian Astronomical Observatory for allowing us some of his observing time to observe potential targets. This time will permit the expansion of our study into activity cycles on solar-type stars. We thank the referee of this paper, Pascal Petit, for his diligence and insightful comments that has made this a better paper. We thank J.-F Donati on supplying ES[p]{}RIT. This project is supported by the Commonwealth of Australia under the International Science Linkages program. This project used the facilities of SIMBAD, HIPPARCOS and IRAF. This research has made use of NASA’s Astrophysics Data System.
Allende Prieto, C & Lambert, D.L., 1999, A&A, 352, 555 Ammons, S.M., Robinson, S.E., Strader, J., Laughlin, G., Fischer, D., & Wolf, A., 2006, ApJ, 638, 1004 Bailer-Jones, C.A.L., 2011, MNRAS, 411, 435 Barnes S. A., Sofia S., Prosser C. F., Stauffer J. R., 1999, ApJ, 516, 263 Bessell, M.S., Castelli, F., & Plez, B. 1998, A&A, 333, 231 Bobylev, V.V., Goncharov, G.A., & Bajkova, A.T., 2006, VizieR On-line Data Catalog: J/AZh/83/821 \[Date accessed December 27, 2010\] Bouvier, J., 2008, A&A, 489, 53L Brandenburg, A., Krause, F., Meinel, R., Moss, D. & Tuominen, I., 1989 A&A, 213, 411 Brown, B.P., Browning, M.K., Brun, A.S., Miesch, M.S., & Toomre, J., 2010, ApJ, 711, 424 Bruntt, H., De Cat, P., & Aerts, 2008, A&A, 478, 487 Carter, B.D., Brown, S., Donati, J.-F., Rees, D., & Semel, M., 1996, PASA, 13, 150 de Zeeuw, P.T., Hoogerwerf, R., de Bruijne, Brown, A.G.A., & Blaauw, A., 1999, AJ, 117, 354 Dieckvoss W., & Heckmann O., 1975 AGK3 Catalogue, Hamburg-Bergedorf Donati, J.-F., Semel, M., & Rees, D.E., 1992, A&A, 265, 669 Donati, J.-F., Semel, M., Carter, B.D., Rees, D.E., & Cameron, A.C., 1997, MNRAS, 291, 658 Donati, J.-F., Collier Cameron, A., Semel, M., Hussain, G.A.J., Petit, P., Carter, B.D., Marsden, S.C., Mengel, M., López Ariste, A., Jeffers, S.V., & Rees, D.E,. 2003, MNRAS, 345, 1145 Donati, J.-F., & Landstreet, J.D., 2009, ARA&A, 47, 333 do Nascimento Jr, J.D., da Costa, J.S., & De Medeiros, J.R., 2010, A&A, 529, 101 Gontcharov, G.A., 2006, AstL, 32, 759 Gray, D.F., 1992, The Observation and Analhysis of Stellar Photospheres, 2nd edn (Cambridge: Cambridge University Press), 392 Guillout, P., Klutsch, A., Frasca, A., Freire Ferrero, R., Marilli, E., Mignemi, G., Biazzo, K., Bouvier, J., Monier, R., Motch, G, Sterzik, M., 2009. A&A, 504, 829 Holmberg, J., Nordström, B., & Andersen, J., 2009, A&A, 501, 941 Houk, N., & Cowley, A.P., 1975, Michigan catalogue for the HD stars, vol. 1, Ann Arbor, Univ. of Michigan Houk, N., 1978, Michigan catalogue for the HD stars, vol. 2, Ann Arbor, Univ. of Michigan Houk, N., 1982, Michigan catalogue for the HD stars, vol. 3, Ann Arbor, Univ. of Michigan Houk, N., & Smith-Moore, M., 1988, Michigan catalogue for the HD stars, vol. 4, Ann Arbor, Univ. of Michigan Houk, N., & Swift, C., 1999, Michigan catalogue for the HD stars, vol. 5, Ann Arbor, Univ. of Michigan Irwin J., Hodgkin S., Aigrain S., Hebb L., Bouvier J., Clarke C., Moraux E., Bramich D. M., 2007, MNRAS, 377, 741 Jackson J. & Stoy R.H., 1954, Cape Photographic Catalogue for 1950.0 zone -30 to -64, Ann. Cape Obs. Kharchenko, N.V., Scholz, R.-D., Piskunov, A.E., Röser, S., & Schilbach, E., 2007, AN, 328, 889 Kharchenko, N.V., & Roeser, S., 2009, VizieR On-line Data Catalog: I/280B/ascc \[Date accessed December 27, 2010\] Koen, C., & Eyer, L., 2002, MNRAS, 331, 45 Krishnamurthi A., Pinsonneault M. H., Barnes S., Sofia S., 1997, ApJ, 480, 303 Kurucz, R.L., 1993, CDROM \#13 (ATLAS9 Atmospheric Models) and \#18 (ATLAS9 and SYNTHE routines) spectral line databases Lafrasse, S., Mella, G., Bonneau, D., Duvert, G., Delfosse, Z., & Chelli, A., 2010, VizieR On-Line Data Catalog: II/300 \[Date accessed February 19, 2011\] Li, J.Z., Hu, J.Y., 1998, A&AS, 132, 173 Madsen, S., Dravins, D., & Lindegren, L., 2002, A&A, 381, 446 Mamajek, E.E., Meyer, M.R., & Liebert, J., 2002, AJ, 124, 1670 Marino. A., Micela, G., Peres, G., & Sciortino, S., 2003, A&A, 407, L63 Marsden, S.C., Carter, B.D., & Donati, J.-F., 2009, MNRAS, 399, 888 Marsden, S.C., Jardine, M.M., Ramírez Vélez, E. Alecian, J.C., Brown, C.J., Carter, B.D., Donati, J.-F., Dunstone, N., Hart, R., Semel, M., & Waite, I.A., 2011a, MNRAS, 413, 1922 Marsden, S.C., Jardine, M.M., Ramírez Vélez, E. Alecian, J.C., Brown, C.J., Carter, B.D., Donati, J.-F., Dunstone, N., Hart, R., Semel, M., & Waite, I.A., 2011b, MNRAS, 413, 1939 Martín, E.L., & Claret, A., 1996, A&A, 306, 408 Masana, E., Jordi, C., & Ribas, I., 2006, A&A, 450, 735 Montes, D, Crespo-Chacón, I., Gálvez, M.C., Fernández-Figueroa, M.J., López-Santiago, J., de Castro, El., Cornide, J.m Hernán-Obispo, J., 2004, in “Lecture Notes and Essays in Astrophysics I" Proceedings of the Astrophysics Symposium held during the First Centennial of the Royal Spanish Physical Society, ed. A. Ulla and M. Manteiga, Madrid, 119 Moss, D., Barker, D.M., Brandenburg, A. & Tuominen, I., 1995, A&A, 294, 155 Norton A.J., Wheatley P.J., West R.G., Haswell C.A., Street R.A., Collier Cameron A., Christian D.J., Clarkson W.I., Enoch B., Gallaway M., Hellier C., Horne K., Irwin J., Kane S.R., Lister T.A., Nicholas J.P., Parley N., Pollacco D., Ryans R., Skillen I. & Wilson D.M., 2007, A&A, 467, 785 Perryman, M.A.C., Lindegren, L., Kovalevsky, J., H[ø]{}g, E., Bastian, U., Bernacca, P.L., Crézé, M., Donati, F., Grenon, M., Grewing, M., van Leeuwen, F., van der Marel, H., Mignard, F., Murray, C.A., Le Poole, R.S., Schrijver, H., Turon, C., Arenou, F., Froeschlé, M., & Petersen, C.S., 1997, A&A, 323, L49 Petit, P., Donati, J.-F., & Collier Cameron, A., 2002, MNRAS, 334, 374 Petit, P. & Donati, J.-F., 2005, Proceedings of the 13th Cambridge Workshop on Cool Stars, Stellar Systems and the Sun. ed. F. Favata, G.A.J. Hussain, and B. Battrick. ESA 560, 875 Petit, P., Dintrans, B., Solanki, S.K., Donati, J.-F., Aurière, M., Lignières, F., Morin, F., Paletou,F., Ramirez, J., Catala, C. & Fares, R., 2008, MNRAS, 388, 80 Pojmański, G., 2002, Acta Astronomica, 52, 397 Prosser, C.F., Randich, S., Stauffer, J.R., Schmitt, J.H.M.M., & Simon, T., 1996, AJ, 112, 1570 Randich, S., 1998, in Cool Stars, Stellar Systems, and the Sun, ASP Conf. Ser., ed. R. A. Donahue and J. A. Bookbinder, 154, 501 Rousseau J.M., Perie J.P., & Gachard M.T., 1996, A&AS, 116, 301 SAO Star Catalog J2000 \[Date accessed December, 27, 2010\] Sartori, M.J., Lépine, J.R.D., & Dias, W.S., 2003, A&A, 404, 913 Semel, M., 1989, A&A, 225, 456 Semel, M., Donati, J.-F., & Rees, D.E., 1993, A&A, 278, 231 Siess, L., Dufour E., & Forestini, M. 2000 A&A, 358, 593 Soderblom, D.R., Stauffer, J.R,. Hudon, J.R., & Jones, B.F., 1993a, ApJSS, 85, 315 Soderblom, D.R., Pilachowski, C.A., Fedele, S.B., & Jones, B.F., 1993b, AJ, 105, 2299 Stauffer J. R., Hartmann L. W., Prosser C. F., Randich S., Balachandran S., Patten B. M., Simon T., Giampapa M., 1997, ApJ, 479, 776 Tetzlaff, N., Neuhäuser, R., & Hohle, M.M., 2011, MNRAS, 410, 190 Thatcher, J.D., & Robinson, R.D., 1993, MNRAS, 262, 1 Torres, C.A.O., Quast, G.R., da Silva, L., de la Reza, R., Melo, C.H.F., & Sterzik, M., 2006, A&A, 460, 695 Turon C., Egret D., Gomez A., Grenon M., Jahreiss H., Requieme Y., Argue A.N., Bec-Borsenberger A., Dommanget J., Mennessier M.O., Arenou F., Chareton M., Crifo F., Mermilliod J.C., Morin D., Nicolet B., Nys O., Prevot L., Rousseau M., Perryman M.A.C 1993, BICDS, 43, 5 van Leeuwen, F., 2007, A&A, 474, 653 Vilhu, O., 1984, A&A, 133, 117 Waite, I.A., Carter, B.D., Marsden, S.C., & Mengel, M.W., 2005, PASA, 22, 29 Waite, I.A., Marsden, S.C., Carter, B.D., Hart, R., Donati, J.-F., Ramírez Vélez, J.C., Semel, M., & Dunstone, N., 2011, MNRAS, 413, 1949 Wright, C.O., Egan, M.P., Kraemer, K.E., & Price, S.D., 2003, AJ, 125, 359 Young, A., Skumanich, A., Stauffer, J.R., Bopp, B.W., Harlan, E., 1989, ApJ, 344, 427 Zarro, D.M., & Rodgers, A.W., 1983, ApJSS, 53, 815 Zuckerman, B., & Song, I., 2004, ArA&A, 42, 685
|
---
author:
- 'A. A. Abdo$^{(1,2)}$'
- 'M. Ackermann$^{(3)}$'
- 'M. Ajello$^{(3)}$'
- 'A. Allafort$^{(3)}$'
- 'W. B. Atwood$^{(4)}$'
- 'L. Baldini$^{(5)}$'
- 'J. Ballet$^{(6)}$'
- 'G. Barbiellini$^{(7,8)}$'
- 'D. Bastieri$^{(9,10)}$'
- 'K. Bechtol$^{(3)}$'
- 'R. Bellazzini$^{(5)}$'
- 'B. Berenji$^{(3)}$'
- 'R. D. Blandford$^{(3)}$'
- 'E. D. Bloom$^{(3)}$'
- 'E. Bonamente$^{(11,12)}$'
- 'A. W. Borgland$^{(3)}$'
- 'A. Bouvier$^{(3)}$'
- 'T. J. Brandt$^{(13,14)}$'
- 'J. Bregeon$^{(5)}$'
- 'M. Brigida$^{(15,16)}$'
- 'P. Bruel$^{(17)}$'
- 'R. Buehler$^{(3)}$'
- 'T. H. Burnett$^{(18)}$'
- 'S. Buson$^{(9,10)}$'
- 'G. A. Caliandro$^{(19)}$'
- 'R. A. Cameron$^{(3)}$'
- 'A. Cannon$^{(20,21)}$'
- 'P. A. Caraveo$^{(22)}$'
- 'J. M. Casandjian$^{(6)}$'
- 'C. Cecchi$^{(11,12)}$'
- 'Ö. Çelik$^{(20,23,24)}$'
- 'E. Charles$^{(3)}$'
- 'A. Chekhtman$^{(1,25)}$'
- 'J. Chiang$^{(3)}$'
- 'S. Ciprini$^{(12)}$'
- 'R. Claus$^{(3)}$'
- 'J. Cohen-Tanugi$^{(26)}$'
- 'J. Conrad$^{(27,28,29)}$'
- 'C. D. Dermer$^{(1)}$'
- 'A. de Angelis$^{(30)}$'
- 'F. de Palma$^{(15,16)}$'
- 'S. W. Digel$^{(3)}$'
- 'E. do Couto e Silva$^{(3)}$'
- 'P. S. Drell$^{(3)}$'
- 'A. Drlica-Wagner$^{(3)}$'
- 'R. Dubois$^{(3)}$'
- 'C. Favuzzi$^{(15,16)}$'
- 'S. J. Fegan$^{(17)}$'
- 'P. Fortin$^{(17)}$'
- 'M. Frailis$^{(30,31)}$'
- 'Y. Fukazawa$^{(32)}$'
- 'S. Funk$^{(3)}$'
- 'P. Fusco$^{(15,16)}$'
- 'F. Gargano$^{(16)}$'
- 'S. Germani$^{(11,12)}$'
- 'N. Giglietto$^{(15,16)}$'
- 'F. Giordano$^{(15,16)}$'
- 'M. Giroletti$^{(33)}$'
- 'T. Glanzman$^{(3)}$'
- 'G. Godfrey$^{(3)}$'
- 'I. A. Grenier$^{(6)}$'
- 'M.-H. Grondin$^{(34)}$'
- 'S. Guiriec$^{(35)}$'
- 'M. Gustafsson$^{(9)}$'
- 'D. Hadasch$^{(19)}$'
- 'A. K. Harding$^{(20)}$'
- 'K. Hayashi$^{(32)}$'
- 'M. Hayashida$^{(3)}$'
- 'E. Hays$^{(20)}$'
- 'S. E. Healey$^{(3)}$'
- 'P. Jean$^{(13)}$'
- 'G. Jóhannesson$^{(36)}$'
- 'A. S. Johnson$^{(3)}$'
- 'R. P. Johnson$^{(4)}$'
- 'T. J. Johnson$^{(20,37)}$'
- 'T. Kamae$^{(3)}$'
- 'H. Katagiri$^{(32)}$'
- 'J. Kataoka$^{(38)}$'
- 'M. Kerr$^{(18)}$'
- 'J. Knödlseder$^{(13)}$'
- 'M. Kuss$^{(5)}$'
- 'J. Lande$^{(3)}$'
- 'L. Latronico$^{(5)}$'
- 'S.-H. Lee$^{(3)}$'
- 'M. Lemoine-Goumard$^{(34)}$'
- 'F. Longo$^{(7,8)}$'
- 'F. Loparco$^{(15,16)}$'
- 'B. Lott$^{(34)}$'
- 'M. N. Lovellette$^{(1)}$'
- 'P. Lubrano$^{(11,12)}$'
- 'G. M. Madejski$^{(3)}$'
- 'A. Makeev$^{(1,25)}$'
- 'P. Martin$^{(39)}$'
- 'M. N. Mazziotta$^{(16)}$'
- 'J. Mehault$^{(26)}$'
- 'P. F. Michelson$^{(3)}$'
- 'W. Mitthumsiri$^{(3)}$'
- 'T. Mizuno$^{(32)}$'
- 'A. A. Moiseev$^{(23,37)}$'
- 'C. Monte$^{(15,16)}$'
- 'M. E. Monzani$^{(3)}$'
- 'A. Morselli$^{(40)}$'
- 'I. V. Moskalenko$^{(3)}$'
- 'S. Murgia$^{(3)}$'
- 'M. Naumann-Godo$^{(6)}$'
- 'P. L. Nolan$^{(3)}$'
- 'J. P. Norris$^{(41)}$'
- 'E. Nuss$^{(26)}$'
- 'T. Ohsugi$^{(42)}$'
- 'A. Okumura$^{(43)}$'
- 'N. Omodei$^{(3)}$'
- 'E. Orlando$^{(39)}$'
- 'J. F. Ormes$^{(41)}$'
- 'M. Ozaki$^{(43)}$'
- 'D. Paneque$^{(3)}$'
- 'J. H. Panetta$^{(3)}$'
- 'D. Parent$^{(1,25)}$'
- 'M. Pepe$^{(11,12)}$'
- 'M. Persic$^{(7,31)}$'
- 'M. Pesce-Rollins$^{(5)}$'
- 'F. Piron$^{(26)}$'
- 'T. A. Porter$^{(3)}$'
- 'S. Rainò$^{(15,16)}$'
- 'R. Rando$^{(9,10)}$'
- 'M. Razzano$^{(5)}$'
- 'A. Reimer$^{(44,3)}$'
- 'O. Reimer$^{(44,3)}$'
- 'S. Ritz$^{(4)}$'
- 'R. W. Romani$^{(3)}$'
- 'H. F.-W. Sadrozinski$^{(4)}$'
- 'P. M. Saz Parkinson$^{(4)}$'
- 'C. Sgrò$^{(5)}$'
- 'E. J. Siskind$^{(45)}$'
- 'D. A. Smith$^{(34)}$'
- 'P. D. Smith$^{(14)}$'
- 'G. Spandre$^{(5)}$'
- 'P. Spinelli$^{(15,16)}$'
- 'M. S. Strickman$^{(1)}$'
- 'L. Strigari$^{(3)}$'
- 'A. W. Strong$^{(39)}$'
- 'D. J. Suson$^{(46)}$'
- 'H. Takahashi$^{(42)}$'
- 'T. Takahashi$^{(43)}$'
- 'T. Tanaka$^{(3)}$'
- 'J. B. Thayer$^{(3)}$'
- 'D. J. Thompson$^{(20)}$'
- 'L. Tibaldo$^{(9,10,6,47)}$'
- 'D. F. Torres$^{(19,48)}$'
- 'G. Tosti$^{(11,12)}$'
- 'A. Tramacere$^{(3,49,50)}$'
- 'Y. Uchiyama$^{(3)}$'
- 'T. L. Usher$^{(3)}$'
- 'J. Vandenbroucke$^{(3)}$'
- 'G. Vianello$^{(3,49)}$'
- 'N. Vilchez$^{(13)}$'
- 'V. Vitale$^{(40,51)}$'
- 'A. P. Waite$^{(3)}$'
- 'P. Wang$^{(3)}$'
- 'B. L. Winer$^{(14)}$'
- 'K. S. Wood$^{(1)}$'
- 'Z. Yang$^{(27,28)}$'
- 'M. Ziegler$^{(4)}$'
date: 'Received 15 September 2010 / Accepted 19 October 2010'
title: '[*Fermi*]{} Large Area Telescope observations of Local Group galaxies: Detection of M31 and search for M33'
---
[Cosmic rays (CRs) can be studied through the galaxy-wide gamma-ray emission that they generate when propagating in the interstellar medium. The comparison of the diffuse signals from different systems may inform us about the key parameters in CR acceleration and transport. ]{} [We aim to determine and compare the properties of the cosmic-ray-induced gamma-ray emission of several Local Group galaxies. ]{} [We use 2 years of nearly continuous sky-survey observations obtained with the Large Area Telescope aboard the *Fermi* Gamma-ray Space Telescope to search for gamma-ray emission from and . We compare the results with those for the Large Magellanic Cloud, the Small Magellanic Cloud, the Milky Way, and the starburst galaxies and . ]{} [We detect a gamma-ray signal at 5$\sigma$ significance in the energy range 200 MeV – 20 GeV that is consistent with originating from . The integral photon flux above 100 MeV amounts to $(9.1 \pm 1.9_{\rm stat} \pm 1.0_{\rm sys}) \times 10^{-9}$ . We find no evidence for emission from and derive an upper limit on the photon flux $>100$ MeV of $5.1 \times 10^{-9}$ ($2\sigma$). Comparing these results to the properties of other Local Group galaxies, we find indications of a correlation between star formation rate and gamma-ray luminosity that also holds for the starburst galaxies.]{} [The gamma-ray luminosity of is about half that of the Milky Way, which implies that the ratio between the average CR densities in and the Milky Way amounts to $\xi = 0.35\pm0.25$. The observed correlation between gamma-ray luminosity and star formation rate suggests that the flux of is not far below the current upper limit from the LAT observations. ]{}
Introduction {#sec:intro}
============
Cosmic rays (CRs) produce high-energy gamma rays through interactions with interstellar matter and radiation fields. The resulting diffuse emissions directly probe CR spectra and intensities in galactic environments [e.g. @strong07]. The detection of the Small Magellanic Cloud [SMC; @abdo10a] and detailed studies of the Large Magellanic Cloud [LMC; @abdo10b] and the Milky Way [MW; e.g. @abdo09a] with the data collected by the Large Area Telescope (LAT) onboard the [*Fermi*]{} Gamma-ray Space Telescope enable comparative studies of cosmic rays in environments that differ in star formation rate (SFR), gas content, radiation fields, size, and metallicities.
Other galaxies in the Local Group that have been predicted to be detectable high-energy gamma-ray emitters are (Andromeda) and (Triangulum) due to their relatively high masses and proximity. So far, neither of these galaxies has been convincingly detected in high-energy gamma rays. was observed by SAS-2 [@fichtel75], COS-B [@pollock81], and EGRET [@sreekumar94], with the most stringent upper limit provided by EGRET being $4.9 \times 10^{-8}$ at a 95% confidence level [see Fig. 3 of @hartman99]. has also been observed by COS-B [@pollock81] and EGRET, providing an upper limit of $3.6 \times 10^{-8}$ [see Fig. 3 of @hartman99].
By comparing M31 properties to those of the MW, @ozel87 estimated that the ratio $\xi$ of the CR density in and in the MW is $\xi \simeq 1$ and computed an expected $>$100 MeV flux from of $2.4 \times 10^{-8} \xi$ . @pavlidou01 made a comparable prediction of $1 \times 10^{-8}$ , based on the assumption that $\xi \approx 0.5$, which they derived by comparing the estimated supernova rate in and in the MW. Using the same approach, they also estimated the $>100$ MeV flux of to be $1.1 \times 10^{-9}$ .
If these estimates are correct, should be detectable by the LAT after 2 years of sky survey observations, while still may fall below the current sensitivity limit. In this letter we report our searches for gamma-ray emission from and with the LAT using almost 2 years of survey data. While we detect for the first time just above the current sensitivity limit, we could only derive an upper limit for the flux from .
Observations and analysis {#sec:observation}
=========================
Data selection and analysis methods
-----------------------------------
The data used in this work have been acquired by the LAT between 8 August 2008 and 30 July 2010, a period of 721 days during which the LAT scanned the sky nearly continuously. Events satisfying the standard low-background event selection [‘Diffuse’ events; @atwood09] and coming from zenith angles $<105\deg$ (to greatly reduce the contribution by Earth albedo gamma rays) were used. Furthermore, we selected only events where the satellite rocking angle was less than $52\deg$. We further restricted the analysis to photon energies above 200 MeV; below this energy, the effective area in the ‘Diffuse class’ is relatively small and strongly dependent on energy. All analysis was performed using the LAT Science Tools package, which is available from the Fermi Science Support Center. Maximum likelihood analysis has been performed in binned mode using the tool [gtlike]{}. We used P6\_V3 post-launch instrument response functions that take into account pile-up and accidental coincidence effects in the detector subsystems.
{#sec:m31}
For the analysis of we selected all events within a rectangular region-of-interest (ROI) of size $10\deg \times 10\deg$ centred on $({\mbox{$\alpha_{\rm J2000}$}}, {\mbox{$\delta_{\rm J2000}$}})=(00^{\rm h}42^{\rm m}44^{\rm s}, +41\deg16'09^{\prime\prime})$ and aligned in Galactic coordinates. The gamma-ray background was modelled as a combination of diffuse model components and 4 significant point sources[^1] that we found within the ROI. Galactic diffuse emission was modelled using an LAT collaboration internal update of the model [gll\_iem\_v02]{} [e.g. @abdo10c] refined by using an analysis of 21 months of LAT data and improved gas template maps with increased spatial resolution. Particular care was taken to remove any contribution from and in the templates by excluding all gas with $V_{\rm LSR}<-50$ km s$^{-1}$ within $2\deg \times 3\deg$ wide boxes around $(l,b)=(121\deg,-21.5\deg)$ and $(l,b)=(133.5\deg,-31.5\deg)$ for and , respectively[^2]. In contrast to [gll\_iem\_v02]{}[^3], we did not include an E(B-V) template in the model because it includes some signal from these galaxies. We verified that the omission of the E(B-V) template did not affect the global fit quality over the ROI. The overall normalization of the Galactic diffuse emission has been left as a free parameter in the analysis. The extragalactic and residual instrumental backgrounds were combined into a single component assumed to have an isotropic distribution and a power-law spectrum with free normalization and free spectral index. The spectra of the 4 point sources were also modelled using power laws with free normalizations and free spectral indices.
{width="9.1cm"} {width="9.1cm"}
Figure \[fig:image\] shows LAT counts maps for the energy range 200 MeV - 20 GeV before (left panel) and after (right panel) subtracting the background model. For the purpose of highlighting emission features on the angular scale of , the counts maps were smoothed using a 2D Gaussian kernel of $\sigma=0.5\deg$. In this representation, an elongated feature that roughly follows the outline of (as indicated by black contours) is already visible in the counts map before background subtraction (left panel). After this subtraction (right panel), this feature becomes the most prominent source of gamma-ray emission in the field. The remainder of the structure in the ‘background subtracted’ map is consistent with statistical fluctuations of the diffuse background emission, which illustrates that the signal from is close to the actual detection sensitivity of the LAT.
To test whether the emission feature is positionally consistent with , we performed maximum likelihood ratio tests for a grid of source positions centred on the galaxy. While the maximum likelihood ratio [or the maximum [*Test Statistic*]{} value TS; cf. @mattox06] over the grid indicates the best-fitting source location, the decrease in TS from the maximum defines uncertainty contours that enclose the true source position at a given confidence level. As usual, TS is defined as twice the difference between the log-likelihood of two alternative models $\mathcal{L}_1$ and $\mathcal{L}_0$, i.e. ${\rm TS} = 2(\mathcal{L}_1 - \mathcal{L}_0)$. Using a point source with a power-law spectrum, we obtain a best-fitting location of $({\mbox{$\alpha_{\rm J2000}$}}, {\mbox{$\delta_{\rm J2000}$}})=(00^{\rm h}42.4^{\rm m} \pm 1.4^{\rm m}, +41\deg10' \pm 11')$ for the gamma-ray source, which encloses the centre of within the $1\sigma$ confidence contour (quoted location uncertainties are at 95% confidence). Using instead of the point source an elliptically shaped uniform intensity region with a semi-major axis of $1.2\deg$, a semi-minor axis of $0.3\deg$ and a position angle of $38\deg$ to approximate the extent and orientation of the galaxy on the sky[^4], we find a best-fitting location of $({\mbox{$\alpha_{\rm J2000}$}}, {\mbox{$\delta_{\rm J2000}$}})=(00^{\rm h}43.9^{\rm m} \pm 1.8^{\rm m}, +41\deg23' \pm 22')$ that again encloses the centre of within the $1\sigma$ confidence contour.
We determined the statistical significance of the detection, as well as its spectral parameters, by fitting a spatial template for to the data on top of the gamma-ray background model that we introduced above. The template was derived from the Improved Reprocessing of the IRAS Survey (IRIS) 100 $\mu$m far infrared map [@miville05]. Far infrared emission can be taken as a first-order approximation of the expected distribution of gamma-ray emission from a galaxy since it traces interstellar gas convolved with the recent massive star formation activity. The spatial distributions of diffuse gamma-ray emission from our own Galaxy or the are indeed traced by far-infrared emission to the first order. From the IRIS 100 $\mu$m map, we removed any pedestal emission, which we estimated from an annulus around , and we clipped the image beyond a radius of $1.6\deg$.
Using this IRIS 100 $\mu$m spatial template for M31 and assuming a power-law spectral shape led to a detection above the background at TS $=28.8$, which corresponds to a detection significance of $5.0\sigma$ for 2 free parameters. We obtained a $>100$ MeV photon flux of $(11.0 \pm 4.7_{\rm stat} \pm 2.0_{\rm sys}) \times 10^{-9}$ and a spectral index of $\Gamma=2.1 \pm 0.2_{\rm stat} \pm 0.1_{\rm sys}$ using this model. Systematic errors include uncertainties in our knowledge of the effective area of the LAT and uncertainties in the modelling of diffuse Galactic gamma-ray emission. As an alternative we fitted the data using the IRIS 60 $\mu$m, IRIS 25 $\mu$m, a template based on H$\alpha$ emission [@finkbeiner03] or the geometrical ellipse shape we used earlier for source localization. All these templates provide results that are close to (and consistent with) those obtained using the IRIS 100 $\mu$m map. Fitting the data using a point source at the centre of provided a slightly smaller TS ($25.5$) and a steeper spectral index ($\Gamma=2.5 \pm 0.2_{\rm stat} \pm 0.1_{\rm sys}$), which provides marginal evidence (at the $1.8\sigma$ confidence level) of a spatial extension of the source beyond the energy-dependent LAT point spread function.
Using the gamma-ray luminosity spectrum determined from a GALPROP model of the MW that was scaled to the assumed distance of $780$ kpc of [@strong10][^5] instead of a power law allows determination of the $>$100 MeV luminosity ratio between and the MW. We obtain ${\mbox{$r_{\gamma}$}}= 0.55\pm0.11_{\rm stat}\pm0.10_{\rm sys}$ where we linearly added uncertainties in the assumed halo size of the model to the systematic errors in the measurement. The luminosity of is thus about half that of the MW. The model gives TS $=28.9$, which is comparable to the value obtained using a power law, yet now with only one free parameter, the detection significance rises to $5.3\sigma$. According to this model, the $>$100 MeV photon flux of is $(9.1 \pm 1.9_{\rm stat} \pm 1.0_{\rm sys}) \times 10^{-9}$ .\
![ Spectrum of the emission obtained using the IRIS 100 $\mu$m spatial template. Red error bars are statistical, black error bars are systematic uncertainties. The solid line shows an MW gamma-ray luminosity model scaled to and the dashed one a possible contribution of (see text). \[fig:spectrum\] ](arxiv-3.eps){width="9.1cm"}
[lccccccc]{} Galaxy & $d$ & $M_{\rm HI}$ & $M_{{\rm H}_2}$ & SFR & & & \
& kpc & $10^8$ & $10^8$ & yr$^{-1}$ & $10^{-8}$ & $10^{41}$ phs$^{-1}$ & $10^{-25}$ phs$^{-1}$H-atom$^{-1}$\
MW & ... & $35\pm4^{(7)}$ & $14\pm2^{(7)}$ & $1-3^{(19)}$ & ... & $11.8 \pm 3.4^{(28)}$ & $2.0\pm0.6$\
& $780\pm33^{(1)}$ & $73\pm22^{(8)}$ & $3.6\pm1.8^{(14)}$ & $0.35-1^{(19)}$ & $0.9 \pm 0.2$ & $6.6\pm1.4$ & $0.7\pm0.3$\
& $847\pm60^{(2)}$ & $19 \pm 8^{(9)}$ & $3.3 \pm 0.4^{(9)}$ & $0.26-0.7^{(20)}$ & $<0.5$ & $<5.0$ & $<2.9$\
& $50 \pm 2^{(3)}$ & $4.8\pm0.2^{(10)}$ & $0.5\pm0.1^{(15)}$ & $0.20-0.25^{(21)}$ & $26.3\pm2.0^{(25)}$ & $0.78\pm0.08$ & $1.2\pm0.1$\
& $61 \pm 3^{(4)}$ & $4.2\pm0.4^{(11)}$ & $0.25\pm0.15^{(16)}$ & $0.04-0.08^{(22)}$ & $3.7\pm0.7^{(26)}$ & $0.16\pm0.04$ & $0.31\pm0.07$\
& $3630 \pm 340^{(5)}$ & $8.8\pm2.9^{(12)}$ & $5\pm4^{(17)}$ & $13-33^{(23)}$ & $1.6\pm0.5^{(27)}$ & $252\pm91$ & $158\pm75$\
& $3940 \pm 370^{(6)}$ & $64\pm14^{(13)}$ & $40\pm8^{(18)}$ & $3.5-10.4^{(24)}$ & $0.6\pm0.4^{(27)}$ & $112\pm78$ & $9\pm6$\
[**References.**]{} (1) @stanek98; (2) @galleti04; (3) @pietrzynski09; (4) @hilditch05; (5) @karachentsev02; (6) @karachentsev03; (7) @paladini07; (8) @braun09; (9) @gratier10; (10) @staveley03; (11) @stanimirovic99; (12) @chynoweth08; (13) @combes77; (14) @nieten06; (15) @fukui08; (16) @leroy07; (17) @mao00; (18) @houghton97; (19) @yin09; (20) @gardan07; (21) @hughes07; (22) @wilke04; (23) @forster03; (24) @lenc06; (25) @abdo10b; (26) @abdo10a; (27) @abdo10d; (28) @strong10: range based on GALPROP models with various halo sizes.
We determined the spectrum of the gamma-ray emission from independently of any assumption about the spectral shape by fitting the IRIS 100 $\mu$m template in five logarithmically spaced energy bins covering the energy range 200 MeV – 50 GeV to the data. Figure \[fig:spectrum\] shows the resulting spectrum on which we superimposed the GALPROP model of the MW for ${\mbox{$r_{\gamma}$}}= 0.55$. Overall, the agreement between the observed spectrum of and the model is very satisfactory. The upturn in the spectrum at high energies, though not significant, could possibly be attributed to emission from the BL Lac object , the only known blazar in the line of sight towards . In a dedicated analysis above 5 GeV, we found a cluster of 6–7 counts that are positionally consistent with coming from that blazar. Adding as a point source to our model and extending the energy range for the fit to 200 MeV – 300 GeV results in a TS$=16-20$ for the source, where the range reflects uncertainties in modelling the spectrum of the isotropic background component at energies $>$100 GeV. The fit suggests a hard power-law spectral index ($\Gamma=1.2 \pm 0.4$), which explains why the source is only seen at high energies. Within 200 MeV – 20 GeV, however, the source contributes only $\sim$8 counts, a number that is tiny compared to the $\sim$240 counts that are attributed to . The impact of on the flux and gamma-ray luminosity estimates for is thus negligible.
We also repeated our analysis for a larger ROI of size $20\deg \times 20\deg$ in which we found 14 point sources in our LAT internal source list. Searching for the faint signal from in such a large ROI relies on the accurate modelling of the spatial distribution of the diffuse gamma-ray background over a large area, which is an important potential source of systematic uncertainties. Nevertheless, results obtained for this large ROI were consistent with those obtained for the $10\deg \times 10\deg$ ROI.
{#sec:m33}
For the analysis of we selected all events within a rectangular ROI of size $10\deg \times 10\deg$ centred on $({\mbox{$\alpha_{\rm J2000}$}}, {\mbox{$\delta_{\rm J2000}$}})=(01^{\rm h}33^{\rm m}51^{\rm s}, +30\deg39'37^{\prime\prime})$ and aligned in Galactic coordinates. Within this field we detected 3 background point sources[^6] that we included in the background model. The remainder of the analysis was similar to what was done for .
We did not detect any significant signal towards the direction of . Using a spatial template based on the IRIS 100 $\mu$m map of and taking the GALPROP models of @strong10 for the spectral shape, we derived an upper $>$100 MeV flux limit of $5.1 \times 10^{-9}$ ($2\sigma$).
Discussion {#sec:discussion}
==========
Based on the flux measured for and the flux upper limit for , we computed the $>$100 MeV photon luminosities ${\mbox{$L_{\gamma}$}}= 4 \pi d^2 {\mbox{$F_{\gamma}$}}$ and average emissivities ${\mbox{$\bar{q}_{\gamma}$}}= {\mbox{$L_{\gamma}$}}/ {\mbox{$N_{\rm H}$}}$, which we compare to the values obtained for the MW, the , and the (see Table \[tab:properties\]). Here, $d$ is the distance of the galaxy and ${\mbox{$N_{\rm H}$}}= 1.19 \times 10^{57} (M_{\rm HI} + M_{{\rm H}_2})$ is the total number of hydrogen atoms in a galaxy, with $M_{\rm HI}$ and $M_{{\rm H}_2}$ in units of . Quoted uncertainties in and include uncertainties in distance and hydrogen mass of the galaxies. The variations in and from one galaxy to another may inform us about how the CR population is affected by global galactic properties. From the values, we estimate the ratio $\xi$ of the average CR density in and in the MW to $\xi=0.35\pm0.25$, consistent with the estimate of @pavlidou01. On the other hand, the flux upper limit for allows for an average CR density in that galaxy that is above the MW value, hence up to a few times greater than the $\xi=0.2$ estimated by @pavlidou01.
![ Gamma-ray $>100$ MeV luminosity versus total number of hydrogen atoms (top panel) and star formation rate (bottom panel) for Local Group galaxies and the starbursts and . In the bottom panel, the lines are power-law fits to the data for the MW, , the , and the , for which the slope was free (solid) or fixed to 1 (dashed). \[fig:lsfr\] ](arxiv-4.eps "fig:"){width="\columnwidth"} ![ Gamma-ray $>100$ MeV luminosity versus total number of hydrogen atoms (top panel) and star formation rate (bottom panel) for Local Group galaxies and the starbursts and . In the bottom panel, the lines are power-law fits to the data for the MW, , the , and the , for which the slope was free (solid) or fixed to 1 (dashed). \[fig:lsfr\] ](arxiv-5.eps "fig:"){width="\columnwidth"}
By comparing the of our sample of Local Group galaxies to their total hydrogen masses and SFRs, we find a close correlation between and SFR and greater scatter between and gas mass (see Fig. \[fig:lsfr\]). In the bottom panel of Fig. \[fig:lsfr\], the ranges of SFR values, which have been rescaled to the distances $d$ adopted here, reflect uncertainties in the SFR estimates based on the various methods used to determine them (see Table \[tab:properties\]). There is a clear trend toward increasing with increasing SFR, with ${\mbox{$L_{\gamma}$}}= (7.4 \pm 1.6) \times {\rm SFR}^{1.4\pm0.3}$ when fitted by a power law, where and SFR are in units of $10^{41}$ phs$^{-1}$ and yr$^{-1}$, respectively. We also added the luminosities derived by @abdo10d for and to this plot, illustrating that the relation obtained for Local Group galaxies also holds for nearby starburst galaxies. Assuming that it also holds for allows estimation of the luminosity of ${\mbox{$L_{\gamma}$}}\sim (1-4) \times 10^{41}$ phs$^{-1}$ for this galaxy, corresponding to a $>$100 MeV flux of $(1-4) \times 10^{-9}$ . thus may be within reach of the LAT within the next few years.
The -SFR plot does suggest a correlation in common for Local Group and starburst galaxies. Although it is premature to draw conclusions about any strong correlation over such a wide range of galaxy properties because of the small size of our sample, if such a correlation exists, it would be analogous to the well-known tight correlation between radio and far-infrared emission over a wide range of galaxy types [e.g. @murphy06]. The latter is linked to the relation between CRs and SFR, and although not yet fully understood, it is thought to result to some extent from CR electron calorimetry. While proton calorimetry clearly can be excluded as an explanation of the -SFR correlation because the intermediate-size galaxies of the Local Group are thought to be very inefficient at retaining CR protons, the dominant CR component [@strong10], a correlation may relate to the contribution of CR leptons to the gamma-ray emission. Depending on the ISM and CR transport conditions, CR leptons may lose their energy predominantly through gamma-ray-emitting processes (like inverse-Compton or Bremsstrahlung, as opposed to ionization and synchrotron) and dominate the total gamma-ray luminosity[^7]. This could drive the correlation between and SFR for galactic systems with high lepton calorimetric efficiency. Whatever the explanation for this global correlation, it is worthwhile noting that it holds despite the fact that conditions may vary considerably within a galaxy (e.g. the peculiar 30 Doradus region in the LMC, or the very active cores of starbursts).
The vs SFR plane therefore seems to hold potential for defining constraints on CR production and transport processes. The inferred values are, however, not uniquely due to CR-ISM interactions but include a contribution of individual galactic sources such as pulsars and their nebulae. The relative contributions of discrete sources and CR-ISM interactions to the total gamma-ray emission very likely vary with galaxy properties like SFR, which may complicate the interpretation of any trend in terms of CR large-scale population and transport.
Also more exotic processes, such as annihilation or decay of WIMPs (weakly interacting massive particles), might contribute to the overall signal from . Several extensions of the Standard Model of particle physics naturally predict the existence of WIMPs (e.g. supersymmetry, universal extra dimensions). Rather than focusing on a specific scenario, we estimate a conservative upper bound on this contribution in the case of a generic 100 GeV WIMP annihilating exclusively into bottom quarks, which is one of the leading tree level annihilation channels of a WIMP predicted by supersymmetric theories. The normalization of the predicted spectrum is initially set to zero and is increased until it just meets, but does not exceed, the 95% confidence upper limit on the measured spectrum at any energy. We find that when assuming an Einasto dark matter halo profile [@navarro10] that matches the kinematic data [@klypin02], this contribution corresponds to a 95% confidence upper limit on the annihilation cross section of approximately 5 $\times$ 10$^{-25}$ cm$^{3}$s$^{-1}$.
The *Fermi* LAT Collaboration acknowledges support from a number of agencies and institutes for both development and the operation of the LAT, as well as for scientific data analysis. These include NASA and DOE in the United States, CEA/Irfu and IN2P3/CNRS in France, ASI and INFN in Italy, MEXT, KEK, and JAXA in Japan, and the K. A. Wallenberg Foundation, the Swedish Research Council, and the National Space Board in Sweden. Additional support from INAF in Italy and CNES in France for science analysis during the operations phase is also gratefully acknowledged.
Abdo, A.A., Ackermann, M., Ajello, M., et al. 2009a, PRL, 103, 251101
Abdo, A.A., Ackermann, M., Ajello, M., et al. 2010a, A&A, in press
Abdo, A.A., Ackermann, M., Ajello, M., et al. 2010b, A&A, 512, A7
Abdo, A.A., Ackermann, M., Ajello, M., et al. 2010c, ApJS, 188, 405
Abdo, A.A., Ackermann, M., Ajello, M., et al. 2010d, ApJ, 709, L152
Atwood, W.B., Abdo, A.A., Ackermann, M., et al. 2009, ApJ, 697, 1071
Braun, R., Thilker, D.A., Walterbos, R.A.M., & Corbelli, E. 2009, ApJ, 695, 937
Chynoweth, K.M., Langston, G.I., Yun, M.S., et al. 2008, AJ, 135, 1983
Combes, F., Gottesman, S.T., & Weliachew, L. 1977, A&A, 59, 181
Digel, S.W., Moskalenko, I.V., Ormes, J.F., & Sreekumar, P. 2000, AIPC, 528, 449
Fichtel, C.E., Hartman, R.C., Kniffen, D.A., et al. 1975, ApJ, 198, 163
Finkbeiner, D. 2003, ApJS, 146, 407
Förster Schreiber, N.M., Genzel, R., & Lutz, D. 2003, ApJ, 599, 193
Fukui, Y., Kawamura, A., Minamidani, T., et al. 2008, ApJS, 178, 56
Galleti, S., Bellazzini, M., & Ferraro, F.R. 2004, A&A, 423, 925
Gardan, E., Braine, J., Schuster, K.F., Brouillet, N., & Sievers, A. 2007, A&A, 473, 91
Gratier, P., Braine, J., Rodriguez-Fernandez, N.J., et al. 2010, A&A, in press
Hartman, R.C., Bertsch, D.L., Bloom, S.D., et al. 1999, ApJS, 123, 79
Hilditch, R.W., Howarth, I.D., & Harries, T.J. 2005, MNRAS, 357, 304
Houghton, S., Whiteoak, J.B., Koribalski, B., et al. 1997, A&A, 325, 923
Hughes, A., Staveley-Smith, L., Kim, S., Wolleben, M., & Filipovic, M. 2007, MNRAS, 382, 543
Karachentsev, I.D., Dolphin, A.E., Geisler, D., et al. 2002, A&A, 383, 125
Karachentsev, I.D., Grebel, E.K., Sharina, M.E., et al. 2003, A&A, 404, 93
Kennicutt, R.C. 1989, ApJ, 344, 685
Klypin, A., Zhao, H., Somerville, R.S. 2003, ApJ, 573, 597
Lenc, E., & Tingay, S.J. 2006, AJ, 132, 1333
Leroy, A., Bolatto, A., Stanimirovic, S., et al. 2007, ApJ, 658, 1027
Mao, R.Q., Henkel, C., Schulz, A., et al. 2000, A&A, 358, 433
Mattox, J.R., Bertsch, D.L., Chiang, J., et al. 2006, ApJ, 461, 396
Miville-Deschênes, M.-A., & Lagache, G. 2005, ApJS, 157, 302
Murphy, E.J., Helou, G., Braun, R., et al. 2006, ApJ, 651, L111
Navarro, J.F., Ludlow, A., Springel, V., et al. 2010, MNRAS, 402, 21
Nieten, Ch., Neininger, N., Guélin, M., et al. 2006, A&A, 453, 450
Özel, M., & Berkhuijsen, E.M. 1987, A&A, 172, 378
Özel, M., & Fichtel, C.E. 1988, ApJ, 335, 135
Paladini, R., Montier, L., Giard, M., et al. 2007, A&A, 465, 839
Pavlidou, V., & Fields, B.D. 2001, ApJ, 558, 63
Pietrzynski, G., Thompson, I.B., Gravzyk, D., et al. 2009, ApJ, 697, 862
Pollock, A.M.T., Bignami, G.F., Hermsen, W., et al. 1981, A&A, 94, 116
Sreekumar, P., Bertsch, D.L., Dingus, B.L., et al. 1994, ApJ, 426, 105
Stanek, K.Z., & Garnavich, P.M. 1998, ApJ, 503, L131
Stanimirović, S., Staveley-Smith, L., Dickey, J.M., Sault, R.J., & Snowden, S.L. 1999, MNRAS, 302, 417
Staveley-Smith, L., Kim, S., Calabretta, M.R., Haynes, F.R., & Kesteven, M.J. 2003, MNRAS, 339, 87
Strong, A.W., Porter, T.A., Digel, S.W., et al. 2010, ApJ, 722, L58
Strong, A.W., Moskalenko, I.V., & Ptuskin, V.S. 2007, ARNPS, 57, 285
Wilke, K., Klaas, U., Lemke, D., et al. 2004, A&A, 414, 69
Yin, J., Hou, J.L., Prantzos, N., et al. 2009, A&A, 505, 497
Gamma-ray spectrum of {#sec:spectrum}
======================
Table \[tab:spectrum\] provides the intensity values of the gamma-ray spectrum that is shown in Fig. \[fig:spectrum\]. Statistical errors are at the $1\sigma$ confidence level, and the upper limit for the 16.6 – 50.0 GeV energy bin at the $2\sigma$ confidence level. Systematic errors include uncertainties in our knowledge of the effective area of the LAT and uncertainties in the modelling of diffuse Galactic gamma-ray emission. The former were determined using modifications of the instrument response functions that bracket the uncertainties in our knowledge of the LAT effective area. The latter were determined by deriving spectra for variations of the diffuse Galactic models that make use of either an E(B-V) template or for which the gas templates have been replaced by the IRIS 100 $\mu$m map, from which emission associated to has been removed. Both types of systematic uncertainties were added linearily.
The last column gives the number of counts attributed to in each of the energy bins from the fit of a spatial model to the present data.
[lcccc]{} Energy & Intensity & Stat. error & Sys. error & Counts\
MeV & &\
200 – 603 & 1.46 & 0.71 & 0.23 & 118.9\
603 – 1821 & 1.60 & 0.49 & 0.14 & 69.5\
1821 – 5493 & 1.02 & 0.47 & 0.15 & 15.9\
5493 –16572 & 1.27 & 0.71 & 0.26 & 7.0\
16572 – 50000 & $<12.5$ & ... & 3.6 & $<24.4$\
[^1]: , , [@abdo10c], and a hard source () located at $({\mbox{$\alpha_{\rm J2000}$}}, {\mbox{$\delta_{\rm J2000}$}})=(00^{\rm h}39^{\rm m}16^{\rm s}, +43\deg27'07^{\prime\prime})$.
[^2]: For , the velocity cut left some residual in the template owing to overlap in velocity with the MW along one side of .
[^3]: See the Galactic diffuse model description at http://fermi.gsfc.nasa.gov/ssc/data/access/lat/BackgroundModels.html.
[^4]: We estimated these parameters by adjusting an ellipse to the IRIS 100 $\mu$m map of [@miville05].
[^5]: We use throughout this work a representative model of the MW from @strong10 with a halo size of 4 kpc and that assumes diffusive reacceleration. The model is based on cosmic-ray, [*Fermi*]{}-LAT and other data, and includes interstellar pion-decay, inverse Compton and Bremsstrahlung. Varying the halo size between 2 and 10 kpc affects the $>$100 MeV luminosity and photon flux by less than 10% and 3%, respectively.
[^6]: , , and [@abdo10c].
[^7]: Some variants of the GALPROP MW model actually predict that leptons can be responsible for up to $\sim$50% of its $>$100 MeV gamma-ray photon flux [@strong10].
|
---
abstract: 'We present the design, fabrication, modeling and feedback control of an earthworm-inspired soft robot that crawls on flat surfaces by actively changing the frictional forces acting on its body. Earthworms are segmented and composed of repeating units called *metameres*. During crawling, muscles enable these metameres to interact with each other in order to generate peristaltic waves and retractable *setae* (bristles) produce variable traction. The proposed robot crawls by replicating these two mechanisms, employing pneumatically-powered soft actuators. Using the notion of controllable subspaces, we show that locomotion would be impossible for this robot in the absence of friction. Also, we present a method to generate feasible control inputs to achieve crawling, perform exhaustive numerical simulations of feedforward-controlled locomotion, and describe the synthesis and implementation of suitable real-time friction-based feedback controllers for crawling. The effectiveness of the proposed approach is demonstrated through analysis, simulations and locomotion experiments.'
author:
- 'Joey Z. Ge, Ariel A. Calderón, and Néstor O. Pérez-Arancibia[^1][^2]'
bibliography:
- 'REF.bib'
title: '**An Earthworm-Inspired Soft Crawling Robot Controlled by Friction**'
---
Introduction {#sec01}
============
Animal locomotion has long been a source of inspiration for robotic research. In particular, the study of limbless crawling has attracted significant attention during the past few years as the most effective method of traveling on unstructured terrains [@ref01; @ref02]. One of the most studied species that travel with a limbless gait is the *nightcrawler*, a type of earthworm (*Lumbricus terrestris*). A typical nightcrawler remains underground during the day and crawls above ground at night. As result of this behavior, they have evolved locomotive mechanisms that enable them to maneuver through their labyrinthine underground burrows and crawl over complex terrains. Specifically, nightcrawlers locomote by employing peristalsis, a motion pattern produced by the coordinated and repeated successive contraction and relaxation of the longitudinal and circular muscles embedded in the animals’ *metameres* (independent body segments). This periodic pattern can be thought of as a retrograde wave that travels along an earthworm’s body to propel it forward using friction-induced traction [@ref03; @ref04; @ref05]. In the case of nightcrawlers, traction is modulated employing microscopic bristle-like skin structures called *setae* [@ref06; @ref07].
Versatility, robustness and spatial efficiency make the nightcrawler’s peristaltic gait a very attractive natural model for robotic locomotion development. Numerous research projects have focused on creating robots that can replicate these earthworms’ peristalsis-based locomotion, adopting a variety of different actuation technologies, including *shape memory alloys* (SMAs) [@ref08; @ref09; @ref10], magnetic fluids [@ref11] and electric motors [@ref12; @ref13; @ref14]. Additionally, recent innovations in fabrication methods have enabled the development of biologically-inspired soft actuators, soft sensors and flexible electronics [@ref15; @ref16; @ref17]. An earthworm-inspired burrowing robot that incorporates these technologies is presented in [@ref18]. That artificial worm was designed to inspect and clean pipes, so its movements and functionalities are constrained to the interior of tubes with diameters in a limited predefined range. As expected, those prototypes are not capable of crawling on open surfaces, which is the problem addressed by the research presented in this paper.
Here, we introduce a new soft robot capable of crawling on flat surfaces, whose basic conceptual design is inspired by the functionality of the *abstract notion* of a two-metamere earthworm. In this design, in order to produce the peristalsis-based retrograde waves required for crawling, a single central linear pneumatic actuator produces the deformations and forces that emulate the axial actions of metameres during earthworm locomotion. Two *extremal* pneumatic actuators produce and modulate the friction forces necessary to alternately anchor the robot’s *extremes* to the ground, which is the crucial action in the generation of friction-based crawling.
The essential mechanism underlying most forms of terrestrial locomotion is friction. Drawing inspiration from nature, researchers have developed several different methods to exploit friction forces, including gecko-inspired adhesives [@ref19], microspine-based anchors [@ref20; @ref21] and anisotropic friction mechanisms [@ref22; @ref23]. In the context of soft robotics, [@ref24] presents a robot that employs materials with different coefficients of friction and a pair of unidirectional clutches to manipulate frictional forces to generate locomotion. In the robot presented here, each extremal actuator, made of silicone rubber, varies the friction coefficient between its surface of contact and the ground by expanding and contracting inside a hard 3D-printed smooth casing. This device is designed such that when the actuator is inflated, its silicone-rubber surface touches the ground, producing high friction. Conversely, when the actuator is deflated, its surface does not touch the ground and only the smooth edges of the casing make contact with the supporting surface, thus producing low friction.
Friction is a nonlinear phenomenon, and consequently, the complete dynamics of the system presented here is both nonlinear and time-varying. However, by treating the forces generated by the central actuator and the friction forces as inputs, the robot’s dynamics can be described by a *linear time-invariant* (LTI) state-space model. This reduced-complexity model enables analysis of the system’s controllability and is instrumental in determining that locomotion is not feasible in the absence of friction. We explicitly show that the controllability subspace associated with the zero-friction case contains only states that define a constant position of the system’s center of mass with respect to the inertial frame of reference. Further analysis shows that if actuation and friction forces were to be chosen at will, the system would become fully controllable. This finding, despite being based on physically unattainable assumptions, indicates that there exists an infinite number of theoretically feasible gaits, and that biologically-inspired locomotion modes represent only a small set of what is possible to achieve with this framework.
![**(a) Metamere:** A metamere expands radially when its longitudinal muscles contract and expand longitudinally when its circular muscles contract. When a metamere is undergoing radial expansion, the 4 pairs of setae on its ventral and lateral surfaces will protrude and anchor it to the ground. **(b) Peristaltic crawling motion:** A *stride* is defined as a complete cycle of peristalsis and the *stride length* is the total distance advanced during one stride. The *protrusion time* is defined as the time span during which an earthworm moves forward within a stride. The head of the earthworm covers the stride length by the end of the protrusion time. Correspondingly, the *stance time* is the period during which the head of a earthworm remains anchored to the ground while the rest of the animal body recovers to the initial state. The sum of the protrusion time and stance time is the *stride period*. The thin dotted lines track the retrograde wave. **(c) Earthworm-inspired crawling robot:** The robot consists of two hard casings, a central actuator, a pair of front and rear actuators constrained by elastomeric o-rings, two machined steel plates, and pneumatic components. \[fig01\]](fig01.pdf){width="46.00000%"}
The rest of the paper is organized as follows. Section \[sec02\] introduces the major concepts unique to earthworm-inspired locomotion. Section \[sec03\] presents a reduced-complexity dynamic model of the proposed robot and a set of locomotion simulations. Section \[sec04\] explains the design and fabrication processes of the soft-robotic components. Section \[sec05\] describes the locomotion planning and associated control strategy. Experimental results are presented and discussed in Section \[sec06\]. Lastly, Section \[sec07\] states the main conclusions of the presented research and provides directions for the future.
Earthworm-Inspired Locomotion {#sec02}
=============================
Earthworms belong to the phylum *annelida*, characterized by their segmented body structures. During locomotion, each ring-shaped segment (metamere) is actively reconfigured by the actions of layers of both longitudinal and circular muscle, as illustrated in Fig. \[fig01\]-(a). Internal sealed cavities in earthworms’ bodies, referred to as *coeloms*, are filled with incompressible fluid so that the volume of each metamere remains constant while reshaping and the structural integrity of the animal is continually preserved. Anatomical schemes of this type are known as hydrostatic skeletons. Also, the fluid inside each coelom is constrained within each metamere, partitioned by *septa* so that there is no movement of fluid across body segments [@ref05]. Such segmentation preserves, to some extent, the locomotion independence of each metamere, enhancing earthworms’ overall mobility [@ref03]. Thus, in order for an earthworm to locomote, the longitudinal and circular muscles of each segment contract alternately, causing each segment to shorten (expanding radially) and elongate (shrinking radially) according to the sequential pattern depicted in Fig. \[fig01\]-(b). Such motions from head to tail create the retrograde peristaltic gaits characteristic of worms. It can be proved mathematically that peristalsis-based crawling requires sufficient traction between anchoring metameres and the ground. In the case of *oligochaetas*, the subclass of *annelida* to which earthworms belong, traction is produced and modulated by retractable setae, as depicted in Fig. \[fig01\]-(a).
Some species of earthworm are both geophagous (earth-eaters) and surface-feeders [@ref06]. That is the case of nightcrawlers, which emerge from their burrows and crawl on ground at night and remain underground during daytime [@ref25]. To transition and adapt to these two different surroundings, they switch between peristalsis-based crawling and burrowing locomotion modes. A worm-inspired burrowing soft robot was presented in [@ref18] and here we extend that work to the crawling case, which requires the active control of friction. This friction-based control strategy is loosely inspired by the morphology of nightcrawlers, which have evolved setae only on the ventral and lateral surfaces of each metamere to facilitate traction during surface crawling, as illustrated in Fig. \[fig01\]-(a). On the other hand, most purely geophagous earthworms have setae arranged in a ring around each body segment [@ref05; @ref07]. During crawling, setae protrude from radially expanding metameres (longitudinal muscle contraction) and anchor into the substratum to provide traction, thus preventing slipping while adjacent body segments contract or expand. Once a metamere’s circular muscle starts to contract, the longitudinal muscle relaxes and the setae retract from the ground to allow for the segment to slide forward.
The basic crawling gait of the robot presented in this paper is loosely based on the nightcrawler’s crawling mechanism, depicted in Fig. \[fig01\]-(b). Following [@ref06], we define a *stride* as one cycle of peristalsis and describe the crawling kinematics as a function of four variables: *stride length*, *protrusion time*, *stance time* and *stride period* (illustrated in Fig. \[fig01\]-(b)). For simplicity, despite the fact that earthworms have numerous segments with staggered stride periods, we define these kinematic variables in relation to an earthworm’s first segment. A prototype of the proposed robot is shown in Fig. \[fig01\]-(c). This system can be thought of as a two-metamere crawling artificial worm composed of pneumatic soft actuators that emulate earthworms’ muscle structures as well as hard casings employed in friction regulation. The processes of locomotion modeling, robotic design, fabrication and controller development are discussed in the next sections.
![Reduced-complexity mass-spring-damper model of the robot in Fig. \[fig01\]-(c). A controllability analysis is carried out for two cases: without friction and with frictions $f_1$ and $f_2$ included as inputs. The values of $f_1$ and $f_2$ are positive when the associated vector forces act in the same direction as ${\boldsymbol{i}}$, and are negative, when the vector forces act in the opposite direction as ${\boldsymbol{i}}$. \[fig02\]](fig02.pdf){width="48.00000%"}
Dynamic Modeling and Simulations {#sec03}
================================
Robot Dynamics and Controllability Analysis {#sec03a}
-------------------------------------------
Several of the existing earthworm-inspired robots consist of three sections: a pair of posterior and anterior actuators that serve as artificial circular muscles, and an axial central actuator, which is the analogue of an earthworm’s longitudinal muscle [@ref10; @ref18; @ref26]. Limited by their configurations, those robots can only travel inside pipes with predetermined diameters. Thus, to locomote, a robot of that type replicates the peristaltic burrowing gaits of earthworms according to a scheme in which its anterior and posterior actuators alternately provide anchoring by pressure to the internal surface of a pipe, while its longitudinal actuator extends and contracts to generate displacements along the pipe’s axial axis.
In this section, employing a reduced-complexity dynamic model, linear system theory and experimental data from [@ref18], we develop the analytical tools necessary to generate a conceptual design for an earthworm-inspired pneumatically-driven soft robot capable of crawling on flat surfaces. An abstraction of this system is shown in Fig. \[fig02\]. In this case, given its function and elastic characteristics, a longitudinal actuator is modeled as a massless elastic spring with stiffness constant $k$ and two forces with opposite directions and identical magnitudes $f_\textrm{a}$ (shown in Fig. \[fig02\]). The anterior and posterior actuators are modeled as two blocks with masses $m_1$ and $m_2$, capable of varying their friction coefficients with the ground in real time in order to modulate the resulting values of the friction forces $f_1$ and $f_2$. For a pneumatically-driven axial actuator of the type in [@ref18] and Fig. \[fig01\]-(c), the magnitude of the produced driving force can be estimated as $$\begin{aligned}
f_{{\textrm{a}}}(t) = s_{{\textrm{a}}} p_{{\textrm{a}}}(t),\end{aligned}$$ where $p_{{\textrm{a}}}$ and $s_{{\textrm{a}}}$ are the instantaneous internal air pressure and constant cross-sectional area of the actuator, respectively. Note that in this model, in agreement with the experimental data presented in [@ref18], $k$ is considered to be constant. This approximation is sufficiently accurate for purposes of design, controllability analysis and controller synthesis. However, the true stiffness of the actuator is nonlinear, time-varying and depends on the air pressure inside the soft structure. Lastly, energy dissipation is modeled by a damper with a constant $c$ to be empirically identified.
To address the problem of controllability, we first consider the frictionless case, in which the force magnitude $f_\textrm{a}$ is the sole input to the system. Thus, by defining $x_1$ and $x_2$ as the position variations of $m_1$ and $m_2$ with respect to an inertial frame of reference, and the corresponding speeds $v_1 = \dot{x}_1$ and $v_2 = \dot{x}_2$, we describe the system with the *single-input–multi-output* (SIMO) state-space realization $$\begin{aligned}
\begin{split}
\dot{x} (t) &= Ax(t)+B_0u_0(t), \\
y(t) &= Cx (t)+Du_0(t),
\end{split}
\label{eqn:eqlabel{2}}\end{aligned}$$ where $$\begin{aligned}
A &=
\left[
\begin{array}{cccc}
~0 & ~1 & ~0 & ~0 \\
-\frac{k}{m_1} & -\frac{c}{m_1} & ~\frac{k}{m_1} & ~\frac{c}{m_1} \\
~0 & ~0 & ~0 & ~1 \\
~\frac{k}{m_2} & ~\frac{c}{m_2} & -\frac{k}{m_2} & -\frac{c}{m_2} \\
\end{array}
\right],~
B_0 = \left[\begin{array}{c} ~0 \\ -\frac{1}{m_1} \\ ~0 \\ ~\frac{1}{m_2} \end{array} \right],\\
C &=
\left[
\begin{array}{cccc}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 \\
\end{array}
\right] ,
~D = \left[\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \end{array} \right],
~x = y = \left[\begin{array}{c} x_1 \\ \dot{x}_1 \\ x_2 \\ \dot{x}_2 \end{array} \right],
\label{eqn:eqlabel{3}}\end{aligned}$$ and $u_0 = f_{{\textrm{a}}}$. In this case, the controllability matrix $\mathcal{C}_{0} = \left[ \begin{array}{cccc} B_0 & AB_0 & A^2B_0 & A^3B_0 \end{array} \right]$ has rank $2$, thus the system is not controllable, meaning that there exists a set of states that cannot be reached from any possible initial state by the action of input signals [@ref27]. Associated with $\mathcal{C}_{0}$ is the controllable subspace, defined as $\mathcal{C}_{AB_0} = {\textrm{Image}} \left\{ \mathcal{C}_{0} \right\}$, which is equivalent to the set of reachable states from the initial condition $x(0) = 0$, $\mathcal{R}_{{\textrm{t}}}$ [@ref28]. It follows that $\mathcal{R}_{{\textrm{t}}} = \mathcal{C}_{AB_0} = {\textrm{Span}} \left\{ \chi_1, \chi_2\right\}$, where $$\begin{aligned}
\chi_1 = \left[ \begin{array}{c}
~1 \\ ~0 \\ -\frac{m_1}{m_2} \\ ~0 \end{array} \right],~
\chi_2 = \left[ \begin{array}{c}
~0 \\ ~1 \\ ~0 \\ -\frac{m_1}{m_2} \end{array} \right].\end{aligned}$$ Therefore, every state in $\mathcal{C}_{AB_0}$ can be written as $\alpha_1\chi_1~+~\alpha_2 \chi_2$ for some $\alpha_1,\alpha_2 \in \mathbb{R}$, which implies that all the reachable positions for the masses take the form $\left\{ x_1 = \alpha_1, x_2 = -\alpha_1 \frac{m_1}{m_2} \right\}$. Thus, we conclude that for all possible inputs and initial state $x(0)=0$, the location of the system’s center of mass with respect to the inertial frame remains constant because the variation $$\begin{aligned}
x_{{\textrm{CM}}} = \frac{m_1 x_1 + m_2 x_2}{m_1 + m_2} = 0,\end{aligned}$$ for all $x \in \mathcal{R}_{{\textrm{t}}}$. The main implication of this analysis is that in the absence of friction, for a robot of the type in Fig. \[fig01\]-(c), locomotion is impossible. This result is consistent with generalizable physical intuition and biological observations [@ref29].
In the presence of friction, a simple model for the generation of traction forces in the system of Fig. \[fig02\] is $$\begin{aligned}
f_i(t)~=~{\textrm{sign}}\left[\dot{x}_i(t) \right]\mu_i(t) m_i g,~\textrm{for}~i=1,2,
\label{EQN05}\end{aligned}$$ where $\mu_i \in \mathbb{R}^+$ are kinetic friction coefficients [@ref30] and $g$ is the acceleration of gravity. Given this structure, the only possible way in which $f_1$ and $f_2$ can be modulated is by varying $\mu_1$ and $\mu_2$. From linear-systems-theory-based analysis it is not possible to determine if the system becomes fully controllable when the inputs are $\left\{f_{{\textrm{a}}},\mu_1,\mu_2 \right\}$. However, we explain the importance of friction for the control of locomotion by analyzing the simplified dynamics resulting from assuming unrestricted inputs $\left\{f_{{\textrm{a}}},f_1,f_2 \right\}$. This case can be modeled by the *multi-input–multi-output* (MIMO) state-space representation $\left\{ A,B_1,C,D\right\}$, where the new input-state matrix and input signal are given by $$\begin{aligned}
B_1 =
\left[ \begin{array}{ccc}
~0 & ~0 & ~0 \\
-\frac{1}{m_1} & -\frac{1}{m_1} & ~0 \\
~0 & ~0 & ~0 \\
~\frac{1}{m_2} & ~0 & -\frac{1}{m_2}
\end{array} \right],~
u_1 =
\left[ \begin{array}{c}
f_{{\textrm{a}}} \\
f_1 \\
f_2
\end{array} \right].\end{aligned}$$
For this augmented state-space realization, the associated controllability matrix $\mathcal{C}_1=\left[ \begin{array}{cccc} B_1 & AB_1 & A^2B_1 & A^3B_1 \end{array} \right]$ has rank $4$, and therefore, the controllable subspace $\mathcal{C}_{AB_1}~=~\textrm{Image} \left\{ \mathcal{C}_1 \right\}$ spans $\mathbb{R}^4$. This analysis implies that if the input $u_1$ could be chosen without restriction, any desired state, and consequently, any position of the system’s center of mass could be reached in a finite amount of time. In reality, however, $u_1$ is highly restricted by actuator limitations and the nonlinear nature of friction. Despite of these restrictions, controlled locomotion can be achieved by varying the friction coefficients $\mu_1$ and $\mu_2$. This fact is explained using numerical examples in the next section.
![Discrete-time model used to implement numerical simulations. $\hat{G}(z)$ is the discretized version of $\left\{A, B_1, C, D\right\}$. Sign operators ensure opposing signs between velocities and frictional forces. In this case, both friction coefficients $\mu_i$, $i=1,2$, switch between $\underline{\mu_i} = 0.1$ and $\overline{\mu_i} = 1$. Zero initial conditions are set at the beginning of the simulations. \[fig03\]](fig03.pdf){width="48.00000%"}
Locomotion Simulations {#sec03b}
----------------------
Through numerical simulations, we illustrate how the robot achieves locomotion with the use of feedforward-controlled time-varying friction. Here, a set of feasible control inputs is chosen via an exhaustive search and iteration process. As described in (\[EQN05\]), the values of the frictional forces $f_i$ are functions of kinematic coefficients of friction $\mu_i$ and the normal forces between the contact surfaces, $m_i g$ (assuming a perfectly flat supporting surface). In the proposed locomotion strategy, normal forces remain constant and friction is regulated by the active variation in real time of the friction coefficients. Specifically, the anterior and posterior actuators of the robot in Fig. \[fig02\] are designed and fabricated to switch their coefficients of friction between a small positive value, $\underline{\mu_i}$, and a larger positive value, $\overline{\mu_i}$, in order to produce friction forces with square-wave-signal shapes. This phenomenon is created by actively switching the surfaces of contact between the actuators and supporting ground. The magnitudes of friction coefficients depend on the materials of the surfaces in contact and range from $\sim \hspace{-0.4ex} 0.04$ for Teflon on steel to $\sim \hspace{-0.4ex} 0.8$ for rubber on concrete [@ref31]. According to experimental tests performed on the extremal friction-varying actuators of the robot in Fig. \[fig01\]-(c), the measured transition between $\underline{\mu_i}$ and $\overline{\mu_i}$ can be as fast as $0.4~\textrm{s}$, which enables the design and implementation of control strategies based on low-frequency *pulse width modulation* (PWM). For the simulations, we assume that these transitions are instantaneous.
![**(a)** Simulated displacements and instantaneous velocities of $m_1$ and $m_2$ when the frequencies of $f_\textrm{a}$, $f_1$ and $f_2$ are set to 1 Hz and $\phi = 0.4\pi~\textrm{rad}$ ($m_1 = m_2 = 0.2~\textrm{Kg},~k = 200~\textrm{N}\cdot \textrm{m}^{-1},~c=0$). **(b)** Simulated displacement of $m_1$ at 60 s across a variety of frequency combinations for $f_\textrm{a}$ (C. freq.) and $f_1$ (F. freq.). The forces $f_1$ and $f_2$ oscillate at the same frequency and $\phi$ is held at $0.4\pi~\textrm{rad}$ ($m_1 = m_2 = 0.2~\textrm{Kg},~k = 200~\textrm{N}\cdot \textrm{m}^{-1},~c=0$). **(c)** Relationship between displacement and phase difference $\phi$ when $f_\textrm{a}$, $f_1$ and $f_2$ are synchronized (at 1 Hz). It can be observed that the direction of locomotion can be reversed by controlling the phase difference between $f_\textrm{a}$ and $f_1$, $f_2$. Trial 1 and Trial 2 correspond to mass values of 0.1 Kg and 0.2 Kg, respectively ($k = 200~\textrm{N}\cdot \textrm{m}^{-1},~c=0$). Heavier masses correspond to higher frictions. These simulation results suggest that higher friction produces faster locomotion. \[fig04\]](fig04.pdf){width="44.00000%"}
Thus, by combining the actuation model for the generation and control of friction forces with the system dynamics discussed in Subsection \[sec03a\], we implement numerical simulations aimed to study the dynamic behavior of the soft robot during surface crawling. This study is relevant for the search of feasible and, eventually, optimal locomotion patterns. The basic simulation scheme is shown in Fig. \[fig03\], where $\hat{G}(z)$ is the discretized version of $\hat{G}(s) = C \left(sI - A \right)^{-1}B_1 + D$, with the state-space representation $\left\{A_{{\textrm{D}}},{B_1}_{{\textrm{D}}},C_{{\textrm{D}}},D_{{\textrm{D}}} \right\}$, obtained with the *zero-order hold* (ZOH) method and employing a sampling frequency of $1~\hspace{-1.6ex}~{\textrm{KHz}}$. Consistently, the sequences $f_{{\textrm{a}}}[n]$, $f_1[n]$, $f_2[n]$, $x_1[n]$, $x_2[n]$, $v_1[n]$ and $v_2[n]$ are the discrete-time versions of the functions $f_{{\textrm{a}}}(t)$, $f_1(t)$, $f_2(t)$, $x_1(t)$, $x_2(t)$, $v_1(t)$ and $v_2(t)$.
{width="100.00000%"}
Assuming a periodic oscillation of the robot’s axial actuator, a sinusoidal signal, with amplitude and bias determined by the actuator’s minimum and maximum internal pressures, is chosen as input $f_{{\textrm{a}}}$. The extremum air-pressure values are estimated from the experimental data published in [@ref18]. The signals $f_1$ and $f_2$ are chosen to have square-wave shapes with amplitudes and biases given by the lower and upper bounds of the frictional forces associated with the lowest and highest friction coefficients, $\underline{\mu_i}$ and $\overline{\mu_i}$, respectively. For consistency with the experimental behaviors of the robot’s pneumatic actuators, the simulation inputs are limited to a frequency of $1~{\textrm{Hz}}$. Also, $f_1$ and $f_2$ are set to have the same frequency but set apart with a phase difference $\phi$ that can be varied between $0$ and $2\pi~{\textrm{rad}}$. Additionally, because kinetic friction always opposes an actuator’s motion, two sign operators are inserted in a feedback configuration, introduced to ensure opposing signs between velocities and frictional forces, as shown in Fig. \[fig03\].
A set of simulation results is presented in Fig. \[fig04\]. Here, for all the cases, we set $k = 200~{\textrm{N}} \cdot {\textrm{m}}^{-1}$, $c = 0 $, $\underline{\mu_1} = \underline{\mu_2} = 0.1$ and $\overline{\mu_1} = \overline{\mu_2} = 1$. Fig. \[fig04\]-(a) shows the displacements and velocities of the two mass-blocks ($m_1 = m_2 = 0.2~{\textrm{Kg}}$), when $f_{{\textrm{a}}}$, $f_1$ and $f_2$ oscillate at $1~{\textrm{Hz}}$ and $\phi = 0.4\pi~{\textrm{rad}}$. Both masses travel in an approximately linear motion at an average speed of $6.31~{\textrm{m}} \cdot {\textrm{min}}^{-1}$ ($10.52~{\textrm{cm}} \cdot {\textrm{s}}^{-1}$). Fig. \[fig04\]-(b) shows the total distance traveled by the robot in $60~{\textrm{s}}$ versus the frequencies of $f_\textrm{a}$ and $f_1$, $f_2$. All the simulations in this plot were run with a constant phase difference $\phi=0.4\pi~{\textrm{rad}}$ and $m_1 = m_2 = 0.2~{\textrm{Kg}}$. These simulation results suggest that for this specific selection of inputs, substantial locomotion can only be attained when the input frequencies are equal, with exception of a few frequency combinations. Also, in general, faster inputs generate faster locomotion. Fig. \[fig04\]-(c) shows the final position reached by the robot after $60~{\textrm{s}}$ across all $\phi$, when all the input frequencies are held at $1~{\textrm{Hz}}$, for two different choices of the pair $\left\{m_1,m_2 \right\}$. This plot suggests that, for this particular type of inputs, $\phi$ is critical for locomotion generation and direction reversal can be realized simply by varying the phase difference between friction inputs. Similar results have been observed in repeated simulations for friction forces with different amplitudes. These findings are limited to the specific set of inputs employed in the discussed cases, but nonetheless exemplify the challenges and potential of friction-controlled crawling. A more comprehensive study of input signals and control strategies to optimize locomotion is a matter of current and future research.
{width="100.00000%"}
Design and Fabrication {#sec04}
======================
The work presented in this paper extends that of [@ref18]. The earthworm-inspired soft robot introduced therein can only function constrained by the specific geometric configuration of pipes, employing a burrowing gait. Here, we develop the design, fabrication and control tools necessary to create an earthworm-inspired soft robot capable of crawling on flat surfaces. The key design innovation introduced in this work is the switching of friction forces by alternating the actuators’ surfaces of contact with the ground. To achieve such objective, we design the soft robot shown in Fig. \[fig01\]-(c), composed of a central longitudinal actuator, a pair of extremal longitudinal actuators, and a pair of hard casings that enclose the extremal actuators. In addition, a pair of soft modules, shown in Fig. \[fig05\]-(c), are employed to connect the central actuator with the extremal actuators. These two connecting modules are also enclosed within the hard casings.
In the proposed robotic design, actuators are driven pneumatically. The central and extremal actuators are designed to emulate the earthworm’s longitudinal and circular muscles, respectively. All actuators are built to expand and contract axially as functions of their internal pressures, unlike those in [@ref18]. Both front and rear actuators are fixed to the upper interior surface of the hard casings and remain above the ground when deflated as the hard casings support the robot’s weight. When inflated, the front and rear actuators elongate and make contact with the surface. The hard casings provide low friction while the actuators yield high friction with the supporting surface. Thus, in this scheme, switching between high and low frictional force values is made possible by a simple inflation and deflation sequence. This actuation method is inspired by the traction variable mechanism employed by nightcrawlers, discussed in Section \[sec02\]. To see this, recall that, when crawling, their contracted longitudinal muscles (coupled with relaxed circular muscles) will cause a metamere to expand radially, pushing the setae into the ground to anchor and prevent backward slippage. Note that, even though the extremal actuators together with their casings are inspired by earthworm’s circular muscles and setae, the underlying working principles are significantly different. In addition, deformation of natural muscles is achieved through active contraction and passive elongation as opposed to the artificial actuators discussed here that elongate actively but contract passively.
The methods and construction sequences employed to fabricate the soft robot are depicted in Fig. \[fig05\]. Fig. \[fig05\]-(a), Fig. \[fig05\]-(b) and Fig. \[fig05\]-(c) illustrate the fabrication processes of the front and rear actuators, central actuator and the connecting modules, respectively. Fig. \[fig05\]-(d) explains the steps leading to the final assembly of the robot. The parts fabricated and materials used to build this robot include 3D-printed *acrylonitrile butadiene styrene* (ABS) molds and casings, silicone elastomer (Ecoflex^^ 00-50, Smooth-On), butadiene rubber elastomeric o-rings, fiberglass sheets and pneumatic components. All actuators measure $35~{\textrm{mm}}$ in diameter, the central actuator measures $83~{\textrm{mm}}$ in length and the extremal actuators combined with the connecting modules measure $26~{\textrm{mm}}$ in height. The wall thickness of the soft components range between $2.5$ and $3~{\textrm{mm}}$. These dimensions were chosen based on the robot design in [@ref18], and were modified to accommodate off-the-shelf pneumatic components.
To drive the system, an Elemental $\textrm{O}_2$ commercial air pump and a 12-V ROB-10398 vacuum pump are employed to inflate and deflate all actuators through a manifold (SMC VV3Q12). Three high speed solenoid valves (SMC VQ110-6M) and three Honeywell ASDX Series digital serial silicon pressure sensors provide regulation and measurement of each actuator’s internal pressure. Data acquisition and signal processing are performed with an AD/DA board (National Instruments PCI-6229) mounted on a target PC which communicates with a host PC via xPC Target 5.5 (P2013b).
Locomotion Planning and Control {#sec05}
===============================
In Section \[sec03b\], using simulations, we demonstrated that fast locomotion is contingent upon perfectly-shaped periodic driving and frictional forces, with perfectly-matched relatively high frequencies. These conditions are not realizable with pneumatically-powered soft actuators as those of the robot in Fig. \[fig01\]-(c) (discussed in Section \[sec04\]). Thus, replicating the high-speed simulated locomotion behaviors on the actual robot is, at this moment, not an attainable objective. However, we can implement bio-inspired locomotion strategies that are compatible with lower frequencies. It is easy to see from Fig. \[fig02\], that $m_1$ will remain stationary (anchored to the ground) and $m_2$ can slide forward as the central actuator inflates if $$\begin{aligned}
|f_1|\geqslant|f_\textrm{a}|>|f_2|
\label{eq07}. \end{aligned}$$ The signal $f_1$ corresponds to static friction while $f_2$ is considered to be kinetic friction. Similarly, $m_2$ will be anchored to the ground and $m_1$ will slide forward as the central actuator deflates if $$\begin{aligned}
|f_2|\geqslant|f_\textrm{a}|>|f_1|
\label{eq08}. \end{aligned}$$ Here, $f_2$ is a static friction force and $f_1$ is instead, considered to be kinetic friction. Thus, locomotion can be induced by actuating each actuator following a pattern such that the conditions defined in (\[eq07\]) and (\[eq08\]) are satisfied in an alternating sequence. In this way, a four-phase actuation sequence is designed to generate one complete stride for the robot as illustrated in Fig. \[fig06\]. Before implementing a locomotion sequence, an actuator characterization test is performed to determine a proper set of values for the robot’s stride length, stance time and protrusion time.
[l\*[4]{}[c]{}r]{} **Phase** & **1** & **2** & **3** & **4**\
Rear Actuator & 1.2 & 1.2 & 1.2 & 0\
Central Actuator & 0 & 3 & 3 & 0\
Front Actuator & 0 & 0 & 1.2 & 1.2\
\[tab01\]
![Example of pressure-tracking experimental results. The continuous lines represent measurements and dashed lines represent references. These data were obtained employing the PID scheme in Fig. \[fig07\] to control the central (upper plot), frontal (middle plot) and rear (bottom plot) actuators, during locomotion. The *protrusion time* is $1.6~{\textrm{s}}$, the *stance time* is $2.4~{\textrm{s}}$ and the *stride period* is $4~{\textrm{s}}$. \[fig08\]](fig08.pdf){width="46.00000%"}
To characterize each actuator, three *proportional-integral-derivative* (PID) controllers $\hat{K}_j,~j = 1, 2, 3$, depicted in Fig. \[fig07\], are implemented to regulate internal pressure. Both pumps are maintained at a constant flow rate and output pressure, and the response of each actuator is controlled by solenoid valves using PWM. The valves are normally closed, a state during which the manifold allows for the vacuum pump to deflate the actuators. The PWM duty cycle excites the valves to open and allows for each actuator to inflate individually. In this structure (Fig. \[fig07\]), the output of $\hat{K}_j$ is the duty cycle input to each valve. Every PID controller is tuned online in an exhaustive manner.
The experimental characterization process follows the procedure introduced in [@ref18]. For the central actuator, a range of pressure values that produce substantial elongations without causing significant radial expansions is identified. For the front and rear actuators, the minimum pressure threshold for which firm contact between the actuators and supporting surface is established is chosen to be the reference pressure. Additionally, two 130-gram machined steel plates are fixed onto the top of both casings, as shown in Fig. \[fig01\]-(c), to increase frictional force and damp the vibration from the valves during actuation. Table \[tab01\] presents a set of reference pressures for individual actuators during the four phases described in Fig. \[fig06\]. Robot locomotion is achieved by controlling each actuator to track the reference pressure during each phase. In reference to the earthworm crawling kinematics described in Section \[sec02\], we define the protrusion time as the period during which the central actuator expands (phase 2). Similarly, the stance time is defined as the time duration after protrusion time during which the front actuator remains static horizontally and completes a cycle of inflation and deflation (phase 3 + phase 4 + phase 1). Protrusion time and stance time are prescribed in experiments.
![Photographic sequence showing the soft robot while crawling on a laboratory benchtop. Locomotion is achieved by tracking the actuators’ pressure references in Table \[tab01\]. In this case, a total distance of $52.4~{\textrm{cm}}$ is covered within $75~{\textrm{s}}$ at an average speed of $0.7~{\textrm{cm}}\cdot {\textrm{s}}^{-1}$. The complete set of locomotion experiments can be found in the supporting movie S1.mp4, also available at <http://www.uscamsl.com/resources/ROBIO2017/S1.mp4>. \[fig09\]](fig09.pdf){width="46.00000%"}
To implement the described locomotion method, low-level PID controllers (Fig. \[fig07\]), tuned during the characterization process, are used to control the actions of each actuator.
Experimental Results and Discussion {#sec06}
===================================
Experiments were conducted to validate the locomotion sequence proposed in Section \[sec05\]. The first set of tests aims to optimize the crawling speed of the robot on a single uniform surface. The effect of different variables, including the duration of each phase and reference pressures for each actuator, is examined across a broad spectrum of values. Fig. \[fig08\] presents the pressure tracking signals of each actuator for the test that produced the fastest locomotion, in which the protrusion time and stance time were $1.6$ and $2.4~{\textrm{s}}$, respectively. A stride length of $2.79~{\textrm{cm}}$ and an average speed of $0.7~{\textrm{cm}} \cdot {\textrm{s}}^{-1}$ were observed and recorded, as shown in Fig. \[fig09\]. Since these experiments adopt a different actuation approach to that of the simulations in Section \[sec03\], the large differences between simulated speeds and experimental locomotion speeds are not surprising. As observed in Fig. \[fig08\], the front and rear actuators were able to track the reference pressures with minor overshoots. However, the central actuator was unable to deflate completely. Lower pressure references for the central actuator and longer protrusion times were found to produce better pressure tracking at the cost of overall locomotion speed. No obvious slippage was observed in any of the tests.
The second set of tests was designed to validate the notion that the robot can travel on surfaces with different coefficients of friction. Using the same actuation sequence than that of Fig. \[fig09\], we proved that the robot can generate peristaltic locomotion on multiple surfaces, including a laboratory benchtop, plywood, *high-density polyethylene* (HDPE), aluminum and a foam pad. Furthermore, we showed that this robot is capable of traversing surfaces with different coefficients of friction by letting it crawl from a foam pad to an HDPE plate. The complete set of all the described tests can be found in the supporting movie S1.mp4, also available at <http://www.uscamsl.com/resources/ROBIO2017/S1.mp4>.
The experiments presented in this section proved friction manipulation to be an effective way to generate peristaltic crawling in the proposed robot. During locomotion, pressure sensors provide feedback to regulate the elongation of each actuator, and therefore, displacement control was achieved indirectly. Direct displacement control can be implemented in the future by employing a motion-capture system or soft sensors. Also, note that actuator characterization in this case is performed empirically. An analytical model that can capture the nonlinear relationships between an actuator’s internal pressure and deformation is needed to improve the control strategy and optimize locomotion.
Conclusion and Future Work {#sec07}
==========================
We presented an earthworm-inspired soft crawling robot capable of locomoting on surfaces by manipulating friction. The robot consists of modular actuators and mechanisms that emulate the functionalities of an earthworm’s longitudinal and circular muscles as well as its bristle-like setae structures. We modeled the robot as a mass-spring-damper system and described its crawling dynamics with an LTI state-space representation. We proved mathematically that frictional forces can be employed as inputs that lead to system controllability. This finding was tested and validated through simulations. Experimentally, we demonstrated that the robot is capable of locomoting on surfaces with different coefficients of friction, emulating an earthworm’s peristaltic crawling.
The modular structure of the robot makes it easily scalable, which leaves great potential for creating longer and more versatile robotic structures. Such complex modular systems will provide an ideal platform to develop and test novel decentralized control strategies. In this work, we empirically explored the feasibility of friction-controlled locomotion on flat surfaces. We anticipate that future research will further explore the proposed robotic concept, employing only soft materials and enabling steering and locomotion on uneven terrains. Additionally, the robot presented here is tethered to both the power source and feedback-control module. To achieve autonomy, novel sensing and wireless communication systems must be implemented. Also, portable sources of energy are required. Feasible options are electrolysis and combustion. These topics are a matter of future research.
[^1]: This work was partially supported by the USC Viterbi School of Engineering through graduate fellowships to J. Z. Ge and A. A. Calderón, and a start-up fund to N. O. Pérez-Arancibia. Additional support was provided by the Chilean National Office of Scientific and Technological Research (CONICYT) through a graduate fellowship to A. A. Calderón.
[^2]: The authors are with the Department of Aerospace and Mechanical Engineering, University of Southern California (USC), Los Angeles, CA 90089-1453, USA (e-mail: [zaoyuang@usc.edu]{}; [aacalder@usc.edu]{}; [perezara@usc.edu]{}).
|
---
abstract: 'We propose a model-independent analysis of the neutrino mass matrix through an expansion in terms of the eigenvectors defining the lepton mixing matrix, which we show can be parametrized as small perturbations of the tribimaximal mixing eigenvectors. This approach proves to be powerful and convenient for some aspects of lepton mixing, in particular when studying the sensitivity of the mass matrix elements to departures from their tribimaximal form. In terms of the eigenvector decomposition, the neutrino mass matrix can be understood as originating from a tribimaximal dominant structure with small departures determined by data. By implementing this approach to cases when the neutrino masses originate from different mechanisms, we show that the experimentally observed structure arises very naturally. We thus claim that the observed deviations from the tribimaximal mixing pattern might be interpreted as a possible hint of a “hybrid” nature of the neutrino mass matrix.'
author:
- 'D. Aristizabal Sierra'
- 'I. de Medeiros Varzielas'
- 'E. Houet'
title: An eigenvector based approach to neutrino mixing
---
There are two main approaches to describe lepton flavor mixing. One is based on assuming the mixing is governed by a fundamental organizing principle, such as a flavor symmetry, which dictates the structure of the lepton mixing pattern and might eventually account for quark mixing as well (see e.g. [@Altarelli:2010gt; @King:2013eh]). The other, usually referred to as the anarchy approach, postulates that lepton mixing originates from a random distribution of unitary $3\times 3$ matrices [@Hall:1999sn]. In either case these approaches are far from providing an ultimate solution to the lepton flavor puzzle. Before the striking measurements of $\theta_{13}$ [@An:2012eh; @Ahn:2012nd], even though global fits had hinted to a non-vanishing $\theta_{13}$ [@Fogli:2008jx], lepton mixing was well described by the Tribimaximal mixing (TBM) pattern [@Harrison:2002er] defined by $\sin^2\theta_{12}=1/3$, $\sin^2\theta_{23}=1/2$ and $\sin^2\theta_{13}=0$. The TBM pattern was for almost a decade a paradigm since its regularity is very suggestive of an underlying principle at work. With a vanishing $\theta_{13}$ now excluded at more than $10\sigma$ [@Tortola:2012te] the situation has changed somewhat. The advent of experimental data proving non-vanishing $\theta_{13}$ and deviations of the best-fit-point values (BFPVs) of the other angles (particularly $\theta_{23}$) from their TBM values [@Tortola:2012te; @GonzalezGarcia:2012sz; @Fogli:2012ua] has greatly motivated the search for possible mechanisms yielding the required deviations from the TBM pattern, which almost without exception are induced by effective operators. In this way, flavor models unable to produce “large” deviations on $\theta_{13}$ from its TBM value have been ruled out, and often deviations on the TBM pattern must be sourced from next-to-leading order non-renormalizable operators, constraining model building.
Majorana neutrino masses can be incorporated in the standard model Lagrangian through the dimension five effective operator ${\cal O}_5\sim LLHH$ [@Weinberg:1979sa], the type-I seesaw [@seesaw] being the most popular and simplest realization of this operator. Other realizations have been considered as pathways to neutrino masses but often the resulting neutrino mass matrix is solely sourced by a single set of lepton number violating parameters, e.g. in type-I seesaw the right-handed neutrino masses. However, given the multiple realizations of ${\cal O}_5$ a conceivable possibility is that in which the neutrino mass matrix involves several independent sets of lepton number breaking parameters, a situation we refer to generically as “hybrid neutrino masses”, as would be the case e.g. in a scheme involving interplay between type-I and type-II seesaw [@Schechter:1980gr]. Already with two contributions sourcing the neutrino mass matrix several scenarios for neutrino mixing arising from interplay between them can be envisaged.
In this letter we start by using an expansion of the neutrino mass matrix in terms of the eigenvectors of the lepton mixing matrix as an alternative model-independent parametrization of the experimentally known values. We show the usefulness of this treatment by studying the constraints on the different mass matrix elements imposed by deviating from TBM, which become evident when using this approach due to the [*TBM + deviations*]{} structure the mass matrix exhibits. We proceed by harnessing this parametrization to analyze the different possibilities that arise with hybrid neutrino masses, and presenting a very appealing scenario where one of the contributions exhibits a purely TBM form that would be well motivated by a flavor symmetry, while the corresponding deviations, required by data are naturally accounted for by the other contribution. In this way we present a paradigm for neutrino masses that matches the qualitative features required by neutrino data and in which the deviations from TBM are interpreted as proof of the existence of hybrid neutrino masses. In the flavor basis (where charged leptons are diagonal) the light neutrino mass matrix can be written as (henceforth we will denote matrices and vectors in boldface) $$\label{eq:neu-mm}
\boldsymbol{m_\nu}=
\boldsymbol{U}^*\,\boldsymbol{\hat m_\nu}\,\boldsymbol{U}^\dagger\,,$$ where $\boldsymbol{\hat
m_\nu}=\mbox{diag}(m_{\nu_1},m_{\nu_2},m_{\nu_3})$ and with $\boldsymbol{U}$ the lepton mixing matrix.
For any experimentally allowed point in parameter space, one can define the eigenvector $\boldsymbol{v_i}$ associated to the eigenvalue $m_{\nu_i}$ and thus $$\label{eq:lepton-mix-eigenvectors}
\boldsymbol{U}=\{\boldsymbol{v_1},\boldsymbol{v_2},\boldsymbol{v_3}\}
=
\begin{pmatrix}
U_{11} & U_{12} & U_{13}\\
U_{21} & U_{22} & U_{23}\\
U_{31} & U_{32} & U_{33}
\end{pmatrix}\,,$$ where each $\boldsymbol{v_i}$ is a column and the neutrino mass matrix can be expressed as the outer (tensor) product of the eigenvectors $$\label{eq:neutrino-mm-eigenvectors}
\boldsymbol{m_\nu}=\sum_{i=1}^3m_{\nu_i}\,\boldsymbol{v_i}\otimes
\boldsymbol{v_i}\,.$$ Consistency with data requires at least two eigenvectors to be present in the above decomposition, and we refer to those cases as “minimal”. In the normal hierarchy a viable minimal setup involves $\boldsymbol{v_{2,3}}$, with $\boldsymbol{v_{1,2}}$ necessary in the inverted case.
We parametrize the mixing angles starting from TBM: $$\begin{aligned}
\label{eq:neutrino-mix-angles-TBM-pert}
\sin\theta_{12}&=\sin\theta_{12}^\text{TBM}-\epsilon_{12}
=\frac{1}{\sqrt{3}}-\epsilon_{12}\,,\\
\label{eq:neutrino-mix-angles-TBM-pert1}
\sin\theta_{23}&=\sin\theta_{23}^\text{TBM}-\epsilon_{23}
=\frac{1}{\sqrt{2}}-\epsilon_{23}\,,\\
\label{eq:neutrino-mix-angles-TBM-pert2}
\sin\theta_{13}&=\sin\theta_{13}^\text{TBM}+\epsilon_{13}
=\epsilon_{13}\,.\end{aligned}$$ This is useful as according to neutrino data [@Tortola:2012te; @GonzalezGarcia:2012sz; @Fogli:2012ua], the $\epsilon_{ij}$ parameters are small: at the $3\sigma$ level, for the normal hierarchy data according to [@Tortola:2012te], we extract their ranges as $$\begin{aligned}
\label{eq:epsilon-ranges}
\epsilon_{12}&\subset[-0.0309,0.0577]\,,\quad \\
\epsilon_{23}&\subset[-0.117,0.107]\,,\quad \\
\epsilon_{13}&\subset[0.130,0.181]\,.\end{aligned}$$ Using this parametrization the eigenvectors $\boldsymbol{v_i}$ can be expressed in terms of the $\epsilon_{ij}$ parameters. We write $$\begin{aligned}
\label{eq:eigenvectors}
\boldsymbol{v_i}=\boldsymbol{v_i^\text{TBM}} + \boldsymbol{\varepsilon_i}\,,\end{aligned}$$ with the TBM eigenvectors in $\boldsymbol{U_\text{TBM}}$ given by $$\label{eq:TBM-eigenvectors}
\{
\boldsymbol{v_1}^\text{TBM},
\boldsymbol{v_2}^\text{TBM},
\boldsymbol{v_3}^\text{TBM} \}=
\begin{pmatrix}
\sqrt{2/3} & 1/\sqrt{3} & 0\\
-1/\sqrt{6} & 1/\sqrt{3} & -1/\sqrt{2}\\
-1/\sqrt{6} & 1/\sqrt{3} & 1/\sqrt{2}
\end{pmatrix}\,.$$ The perturbation vectors $\boldsymbol{\varepsilon_i}$ can be simplified by expanding the trigonometric functions entering in $\boldsymbol{U}$ up to second order in $\epsilon_{ij}$ [^1]. By fixing $\delta=0$ for illustration (this does not affect the main conclusions), they read $$\begin{aligned}
\label{eq:varepsilon-expressions}
\boldsymbol{\varepsilon_1}&=
\begin{pmatrix}
\epsilon_{12}/\sqrt{2}\\
\epsilon_{12}/\sqrt{2} - (\epsilon_{13}+\epsilon_{23})/\sqrt{3}\\
\epsilon_{12}/\sqrt{2} + (\epsilon_{13}+\epsilon_{23})/\sqrt{3}
\end{pmatrix},
\\
\label{eq:varepsilon-expressions1}
\boldsymbol{\varepsilon_2}&=
\begin{pmatrix}
-\epsilon_{12}\\
\epsilon_{12}/2 - \epsilon_{13}/\sqrt{6} + \sqrt{2}\epsilon_{23}/\sqrt{3}\\
\epsilon_{12}/2 + \epsilon_{13}/\sqrt{6} - \sqrt{2}\epsilon_{23}/\sqrt{3}
\end{pmatrix},
\\
\label{eq:varepsilon-expressions2}
\boldsymbol{\varepsilon_3}&=
\begin{pmatrix}
-\epsilon_{13}\\
\epsilon_{23}\\
\epsilon_{23}
\end{pmatrix}.\end{aligned}$$ With the eigenvectors written as perturbations of the $\boldsymbol{v_i}^\text{TBM}$, the neutrino mass matrix can be conveniently interpreted as originating from a TBM structure with modifications that are fixed whenever a given point in the corresponding experimental data range is selected, that is to say $$\label{eq:neutrino-mm-full-eigvecs}
\boldsymbol{m_\nu}=\sum_{i=1}^3
m_{\nu_i}
\left[
\left(
\boldsymbol{v_i}^\text{TBM}\otimes\boldsymbol{v_i}^\text{TBM}
\right)
+
\boldsymbol{{\cal V}_i}
\right]\,,$$ with $$\label{eq:mass-matrix-pert}
\boldsymbol{{\cal V}_i}=
\left[
\left(
\boldsymbol{v_i}^\text{TBM}\otimes \boldsymbol{\varepsilon_i}
\right)
+
\left(
\boldsymbol{\varepsilon_i}\otimes\boldsymbol{v_i}^\text{TBM}
\right)
+
\left(
\boldsymbol{\varepsilon_i}\otimes\boldsymbol{\varepsilon_i}
\right)
\right]\,.$$
{width="7.5cm" height="6.1cm"} {width="7.5cm" height="6.1cm"}
It is clear that consistency with data at a certain confidence level requires some entries of the mass matrix to significantly deviate from the TBM structure. In order to quantify these deviations we calculated the different mass matrix elements through eqs. (\[eq:neutrino-mm-full-eigvecs\]) and (\[eq:mass-matrix-pert\]). In fig. \[fig:deviations-from-TBM\] we display numerical results of the different mass matrix entries (normalized to the corresponding TBM entries), $R_{ij}=m_{\nu_{ij}}/m_{\nu_{ij}}^\text{TBM}$, varying with $\theta_{13}$ and $\theta_{23}$ (the variation with $\theta_{12}$ is weaker, so we do not display it). We used the $3\sigma$ ranges of the angles for the normal hierarchical spectrum according to reference [@Tortola:2012te], with the remaining parameters fixed to their BFPVs and the lightest neutrino mass set to $10^{-3}$ eV (the results are quite insensitive to this parameter). These results indicate clearly that deviating from $\theta_{13}^\text{TBM}=0$ requires $m_{\nu_{12}}$ and $m_{\nu_{13}}$ to have sizable departures from their TBM values together with small departures in the $m_{\nu_{11}}$ entry. It can be seen in fig. \[fig:deviations-from-TBM\] that the other elements remain flat, not playing a relevant role. With $\theta_{23}$ the situation is different: deviating from $\theta_{23}^\text{TBM}=\pi/4$ demands large deviations from the TBM structure in $m_{\nu_{13}}$ and $m_{\nu_{12}}$, but $m_{\nu_{22}}$ and $m_{\nu_{23}}$ need also to differ from their TBM values. For $\theta_{12}$ the results look similar in the sense that deviations from $\theta_{12}^\text{TBM}$ are mostly determined by variations of $m_{\nu_{12}}$ and $m_{\nu_{13}}$ entries. Overall in the $\delta=0$ case deviations of the neutrino mixing angles from their TBM values require neutrino mass matrices with sizable deviations from the TBM structure mainly in the $m_{\nu_{12}}$ and $m_{\nu_{13}}$ elements, and this conclusion holds independently of the neutrino mass spectrum. We emphasize that these results apply in the flavor basis, and that we obtained these conclusions following from the useful eigenvector decomposition approach in a model-independent way.
We now apply our formalism to hybrid neutrino masses (i.e. receiving contributions from physically distinct sources, such as different seesaw mechanisms). In the case with two sources with superindices $A, B$ the effective light neutrino mass matrix reads [^2] $$\label{eq:nmm-several-sources}
\boldsymbol{m_\nu}=\boldsymbol{m_\nu^{(A)}}
+ \boldsymbol{m_\nu^{(B)}}\,.$$ From equation (\[eq:neu-mm\]) we have $$\label{eq:diagonalizing-ApB}
\boldsymbol{\hat m_\nu}=
\boldsymbol{U}^T
\left(
\boldsymbol{m_\nu^{(A)}}
+\boldsymbol{m_\nu^{(B)}}
\right)
\boldsymbol{U}\,.$$ and in general situation $\boldsymbol{U}$ diagonalizes the sum but not the individual matrices $\boldsymbol{m_\nu^{(A,B)}}$. The contributions must add up to (\[eq:neutrino-mm-full-eigvecs\]) but their individual structure needs not be determined by the eigenvectors of $\boldsymbol{U}$. The most general decomposition can be written as $$\label{eq:general-mA-mB-decomposition}
\boldsymbol{m_\nu^{(X)}}=\sum_i
\left[
m_{\nu_i}^{(X)}
\boldsymbol{v_i}^\text{TBM}\otimes \boldsymbol{v_i}^\text{TBM}
+
\delta m_{\nu_i}^{(X)}\boldsymbol{{\cal V}_i}
\right]\,,$$ where $X=A,B$ and by definition, (\[eq:diagonalizing-ApB\]) requires $m_{\nu_i}^{(A)} +
m_{\nu_i}^{(B)}=\delta m_{\nu_i}^{(A)} + \delta
m_{\nu_i}^{(B)} =m_{\nu_i}$. We stress that any realization of hybrid neutrino masses can be defined according to the terms entering in each contribution. Consistency requires that when combining $\boldsymbol{m_\nu^{(A,B)}}$ through (\[eq:nmm-several-sources\]) the eigenvectors entering in the full mass matrix sum up to (\[eq:eigenvectors\]), or in other words that the matrix associated with the generation index $i$ has at the end a structure like (\[eq:neutrino-mm-full-eigvecs\]). Regardless of which eigenvectors appear in each individual contribution, the orthogonality relation $\boldsymbol{v_i}\cdot \boldsymbol{v_j}=\delta_{ij}$ guarantees the lepton mixing matrix diagonalizes the resulting $\boldsymbol{m_\nu}$ (due to $\boldsymbol{v_i}$ being approximate, this holds up to corrections at most of order $v_{kl}^\text{TBM}\epsilon_{ij}\sim 10^{-1}$). Indeed, it is useful to apply the eigenvector decomposition to each contribution as it is made clear that the only way for the eigenvectors building either $\boldsymbol{m_\nu^{(X)}}$ to appear in $U$ unchanged is if they are already orthogonal, which in general will not be the case. For illustration, we consider a minimal setup (minimal in terms of the number of parameters defining the full neutrino mass matrix) $$\begin{aligned}
\label{eq:minimal-setup}
\boldsymbol{m_\nu^{(A)}}&=m_{\nu_2}\boldsymbol{v_2}^\text{TBM}\otimes
\boldsymbol{v_2}^\text{TBM} \,,\\
\boldsymbol{m_\nu^{(B)}}&=m_{\nu_3}\boldsymbol{v_3}\otimes
\boldsymbol{v_3} + m_{\nu_2}\boldsymbol{{\cal V}_2}\,.\end{aligned}$$ In this case the vector $\boldsymbol{v_2}$ becomes “completed” through the combination of $\boldsymbol{m_\nu^{(A)}}$ and the second term in $\boldsymbol{m_\nu^{(B)}}$. This can be seen explicitly as it results in a particular case of (\[eq:neutrino-mm-full-eigvecs\]). Minimal setups are appealing as there are only 6 defining parameters (with a phase in addition to the 3 $\epsilon_{ij}$), so there is no room for arbitrariness on the parameter space in order to match the neutrino oscillation observables ($\Delta m_{21,32}$, $\theta_{ij}$ and $\delta$). Although at the expense of introducing more parameters, going beyond minimal cases can lead to other possibilities. A classification according to the number of eigenvectors included in each mechanism can be done in analogy to the one shown in ref [@AristizabalSierra:2011ab] for the exact TBM pattern.
Another appealing setup (minimal or not) is one where one of the structures (e.g. $\boldsymbol{m_\nu^{(A)}}$) is chosen to involve only TBM eigenvectors (as in (\[eq:minimal-setup\])). In non-minimal scenarios of this type we can have the TBM pattern arise solely from one of the contributions while the other entirely accounts for the observed deviations. This corresponds to setting $\delta m_{\nu_i}^{(A)}=m_{\nu_i}^{(B)}=0$ in (\[eq:general-mA-mB-decomposition\]): $$\begin{aligned}
\label{eq:TBM+deviations-case}
\boldsymbol{m_\nu^{(A)}}&=\sum_i m_{\nu_i}
\boldsymbol{v_i}^\text{TBM}\otimes\boldsymbol{v_i}^\text{TBM}\,, \nonumber\\
\boldsymbol{m_\nu^{(B)}}&=\sum_i m_{\nu_i} \boldsymbol{{\cal V}_i}\,, \end{aligned}$$ which is very suggestive that the experimentally observed small deviations from the TBM pattern may be interpreted as a hint that nature is described by hybrid neutrino masses. With the measured deviations from TBM such a scenario is extremely natural: in general we expect the eigenvectors associated with different mechanisms to not be orthogonal, and the smallness of the deviations would simply be due to a moderate hierarchy in the scales associated with each mechanism.
In conclusion, we parametrized the neutrino mixing angles by small perturbations of the TBM pattern, and expressed the eigenvectors of the neutrino mass matrix (in the flavor basis) in terms of a dominant TBM structure with small perturbations. To very good approximation the TBM deviations are simple eigenvectors depending linearly in the deviations of the mixing angles. This approach was used first to clarify which neutrino mass matrix elements are associated with the deviations of each angle. We then applied the same approach to “hybrid” neutrino mass matrices, where it conveniently describes how the eigenvectors from different sources of neutrino masses combine into the observed mixing. We identified some particularly appealing cases, starting with the minimal ones where a small number of parameters makes the scheme predictive and then considering cases where one of the sources had a mass matrix with the exact TBM form. We argued that given the data, the latter are extremely natural, and claim that neutrino mixing data might therefore be the first hint of the presence of several mechanisms generating neutrino masses in nature. Such a framework is a novel perspective for neutrino mixing which deserves further theoretical and phenomenological scrutiny.
Acknowledgments: The work of I.d.M.V. was supported by DFG grant PA 803/6-1 and partially through PTDC/FIS/098188/2008. DAS is supported by the Belgian FNRS agency through a “Chargé de recherche” contract.
[99]{} G. Altarelli and F. Feruglio, Rev. Mod. Phys. [**82**]{}, 2701 (2010) \[arXiv:1002.0211 \[hep-ph\]\]. S. F. King and C. Luhn, arXiv:1301.1340 \[hep-ph\]. L. J. Hall, H. Murayama and N. Weiner, Phys. Rev. Lett. [**84**]{}, 2572 (2000) \[hep-ph/9911341\]; A. de Gouvea and H. Murayama, arXiv:1204.1249 \[hep-ph\]. F. P. An [*et al.*]{} \[DAYA-BAY Collaboration\], Phys. Rev. Lett. [**108**]{}, 171803 (2012) \[arXiv:1203.1669 \[hep-ex\]\]. J. K. Ahn [*et al.*]{} \[RENO Collaboration\], Phys. Rev. Lett. [**108**]{}, 191802 (2012) \[arXiv:1204.0626 \[hep-ex\]\]. P. F. Harrison, D. H. Perkins and W. G. Scott, Phys. Lett. B [**530**]{}, 167 (2002) \[hep-ph/0202074\]. G. L. Fogli, E. Lisi, A. Marrone, A. Palazzo and A. M. Rotunno, Phys. Rev. Lett. [**101**]{}, 141801 (2008) \[arXiv:0806.2649 \[hep-ph\]\]. D. V. Forero, M. Tortola and J. W. F. Valle, Phys. Rev. D [**86**]{}, 073012 (2012) \[arXiv:1205.4018 \[hep-ph\]\]. S. Weinberg, Phys. Rev. Lett. [**43**]{}, 1566 (1979). P. Minkowski, [*Phys. Lett.*]{} B [**67**]{} 421 (1977); T. Yanagida, in [*Proc. of Workshop on Unified Theory and Baryon number in the Universe*]{}, eds. O. Sawada and A. Sugamoto, KEK, Tsukuba, (1979) p.95; M. Gell-Mann, P. Ramond and R. Slansky, in [*Supergravity*]{}, eds P. van Niewenhuizen and D. Z. Freedman (North Holland, Amsterdam 1980) p.315; P. Ramond, [*Sanibel talk*]{}, retroprinted as hep-ph/9809459; S. L. Glashow, in[*Quarks and Leptons*]{}, Cargèse lectures, eds M. Lévy, (Plenum, 1980, New York) p. 707; R. N. Mohapatra and G. Senjanović, [*Phys. Rev. Lett.*]{} [**44**]{}, 912 (1980); J. Schechter and J. W. F. Valle, Phys. Rev. D [**22**]{} (1980) 2227; Phys. Rev. D [**25**]{} (1982) 774. M. C. Gonzalez-Garcia, M. Maltoni, J. Salvado and T. Schwetz, JHEP [**1212**]{}, 123 (2012) \[arXiv:1209.3023 \[hep-ph\]\]. G. L. Fogli, E. Lisi, A. Marrone, D. Montanino, A. Palazzo and A. M. Rotunno, Phys. Rev. D [**86**]{}, 013012 (2012) \[arXiv:1205.5254 \[hep-ph\]\]. [@Schechter:1980gr] J. Schechter and J. W. F. Valle, Phys. Rev. D [**22**]{}, 2227 (1980); G. Lazarides, Q. Shafi and C. Wetterich, Nucl. Phys. B [**181**]{}, 287 (1981); R. N. Mohapatra and G. Senjanovic, Phys. Rev. D [**23**]{}, 165 (1981); C. Wetterich, Nucl. Phys. B [**187**]{}, 343 (1981); D. Aristizabal Sierra, F. Bazzocchi and I. de Medeiros Varzielas, Nucl. Phys. B [**858**]{}, 196 (2012) \[arXiv:1112.1843 \[hep-ph\]\].
[^1]: When compared with the exact expressions this approximation deviates at the permille level, and unitarity of $U$ is guaranteed up to corrections of order $\epsilon^3$.
[^2]: It is straightforward to generalize to hybrid cases with more contributions.
|
---
abstract: 'Two-photon collisions at the e$^+$e$^-$ colliders allow to investigate the formation and the properties of resonant states in a very clean experimental environment. A remarkable number of new results have been recently obtained giving important contributions to meson spectroscopy and glueball searches. The most recent results from the LEP collider at CERN and CESR at Cornell are reviewed here.'
address: |
University of Geneva,\
24, Quai Ernest-Ansermet, CH-1211 Genève 4, Switzerland\
[*E-mail: Saverio.Braccini@cern.ch*]{}
author:
- 'Saverio BRACCINI[^1]'
title: |
RESONANCE FORMATION\
IN TWO-PHOTON COLLISIONS
---
=by -1
Introduction
============
Two-photon collisions at electron positron storage rings are a good laboratory to investigate the properties of meson resonances and play a crucial role in glueball searches.
A resonant state R can be formed by the collision of two photons via the reaction $\epem\ra\epem \mbox{R}$ (fig. \[fig:ggr\]). The outgoing electron and positron are usually scattered at very small angles and are not detected (no-tag mode). In this case the two photons are quasi real and the resonant state R must be neutral and unflavoured with C=1 and J$\neq$1. If one of the two photons is highly virtual, the outgoing electron or positron can be detected at low angle and the spin of the resonant state is allowed to be one (single-tag mode). In both cases the two outgoing particles carry nearly the full beam energy and the mass of the resonant state is much smaller than the $\epem$ centre of mass energy. This fact allows a clean separation between the two-photon and the annihilation process by using a cut in the visible energy. Since there are no particles produced other than R, the reconstruction of the final state can be performed in a very clean experimental environment.
The cross section for this process is given by the convolution of the QED calculable luminosity function $\cal{L}$, giving the flux of the virtual photons, with the two-photon cross section $\sigma(\gamma\gamma\ra\mbox{R})$ that is expressed by the Breit-Wigner function $$\sigma
(\gamma\gamma\ra\mbox{R})
= 8 \pi (2\mbox{J}+1)
\frac{\Gamma_{\gamma\gamma}(\mbox{R})\Gamma(\mbox{R})}
{(W_{\gamma \gamma}^2-m_R^2)^2+m_R^2\Gamma^2(\mbox{R})}
\label{eq:sgg}$$ where $W_{\gamma\gamma}$ is the invariant mass of the two-photon system, $m_R$, J, $\Gamma_{\gamma\gamma}(\mbox{R})$ and $\Gamma($R) are the mass, spin, two-photon partial width and total width of R, respectively. This leads to the proportionality relation $$\sigma(\epem\ra\epem\mbox{R})=\mbox{$\cal{K}$}\cdot\Gamma_{\gamma \gamma}(\mbox{R})$$ that allows to extract the two-photon width from the cross section. The proportionality factor $\cal{K}$ is evaluated by a Monte Carlo integration.
In the single tag-mode the high virtuality of one of the two photons is taken into account by multiplying the Breit-Wigner function by a VDM pole transition form factor $$F^2(Q^2) = \left(\frac{1}{1+Q^2 / \Lambda^2} \right)^2$$ where $Q^2$ is the four vector squared of the virtual photon and $\Lambda$ is a parameter to be measured experimentally.
Since gluons do not couple directly to photons, the two photon width of a glueball is expected to be very small. A state that can be formed in a gluon rich environment but not in two photon fusion has the typical signature of a glueball. According to lattice QCD predictions [@LatticeQCD], the ground state glueball has J$^{PC}$= 0$^{++}$ and a mass between 1400 and 1800 MeV. The 2$^{++}$ tensor glueball is expected in the mass region around 2200 MeV while the 0$^{-+}$ pseudoscalar glueball is predicted to be heavier. Since several 0$^{++}$ states have been observed in the 1400-1800 MeV mass region, the scalar ground state glueball can mix with nearby quarkonia, making the search for the scalar glueball and the interpretation of the scalar meson nonet a complex problem [@Amsler] [@Minkowski] [@Gastaldi]. The results from two-photon formation represent therefore a fundamental piece of information for glueball searches.
In order to distinguish ordinary quarkonia from glueballs, a parameter called stickiness has been introduced [@Chanowitz]. The stickiness is an estimate of the ratio $|<R|gg>|^2 / |<R|\gamma\gamma>|^2$ evaluated from the the ratio $\Gamma(J/\psi\rightarrow\gamma R) / \Gamma(R\rightarrow\gamma\gamma)$ corrected by a phase space factor. The $s\bar{s}$ mesons have the largest stickiness among quarkonia (14.7 for the f$_2'$(1525)) while much larger values are expected for glueballs.
Because of the large mass of the charm quark, the study of the formation of charmonium states allows to test non-relativistic perturbative QCD calculations and to measure $\alpha_s$ at the charm scale.
Two e$^+$e$^-$ colliders have collected a large amount of data in the last few years. The four LEP experiments ALEPH, DELPHI, L3 and OPAL at CERN have collected approximately 150, 55, 175, 240 pb$^{-1}$ each at $\sqrt{s} \sim $ 91, 183, 189, 191-202 GeV respectively. The CLEO experiment at CESR (Cornell) has collected about 3000 pb$^{-1}$ at $\sqrt{s} \sim $ 10.6 GeV. Since the luminosity function $\cal{L}$ increases with the beam energy, the higher energy allows LEP to partially compensate the smaller luminosity by a larger cross section.
In this paper the most recent results on resonance formation and glueball searches obtained at LEP and CESR are reviewed.
The $\pi^+\pi^-\pi^0$ final state
=================================
A study of the reaction $\gamma\gamma\rightarrow\pi^+\pi^-\pi^0$ is performed by L3 [@L3-a2] [@Saverio-Frascati] using only untagged events. The mass spectrum (fig. \[fig:a2\]) is dominated by the formation of the a$_2$(1320) tensor meson. A clear enhancement is visible around 1750 MeV where the study of the total transverse momentum distribution shows evidence for an exclusive process. The study of the angular distributions shows that the a$_2$ formation is dominated by a J$^{PC}$=2$^{++}$ helicity 2 wave. The radiative width is found to be $\Gamma_{\gamma\gamma}(\mbox{a}_2)=0.98\pm 0.05 \pm 0.09$ keV. A spin-parity analysis in the mass region above the a$_2$(1320) shows that also this region is dominated by a J$^{PC}$=2$^{++}$ helicity 2 wave, confirming the observation of the CERN-IHEP collaboration [@Serpukov] and in contradiction with CELLO [@Cello] and Crystal Ball [@CB] measurements. This can be interpreted as the formation of a radial recurrence of the a$_2$ for which $\Gamma_{\gamma\gamma}(\mbox{a}'_2(1765))\times$BR$(\mbox{a}'_2(1765)\rightarrow\pi^+\pi^-\pi^0)=0.29\pm 0.04 \pm 0.02$ keV in agreement with theoretical predictions [@Munz]. The J$^{PC}$=2$^{-+}$ wave contribution is found compatible with zero.
Pseudoscalar mesons and their form factors
==========================================
The reaction $\gamma\gamma\rightarrow\eta'\rightarrow\pi^+\pi^-\gamma$ is studied by L3 [@L3-etap] in both the no-tag and single-tag mode. The $\pi^+\pi^-\gamma$ mass spectrum (fig. \[fig:etap\]) shows a prominent peak due to the formation of the $\eta'(958)$ while the enhancement around 1250 MeV is due to the process $\gamma\gamma\rightarrow$a$_2$(1320)$\rightarrow\pi^+\pi^-\pi^0$ when one photon from the $\pi^0$ goes undetected. For the two-photon width, $\Gamma_{\gamma\gamma}(\eta')=4.17\pm 0.10 \pm 0.27$ keV is measured. The electromagnetic form factor of the $\eta'$ is studied using tagged and untagged events. For the untagged events $Q^2$ can be measured as $(\Sigma p_t)^2$, as demonstrated by a Monte Carlo study. A low gluonic component in the $\eta'(958)$ is found by comparing the data with the theoretical predictions [@Anisovitch-ff]. The value $0.900\pm0.046\pm0.022$ GeV is obtained for the parameter $\Lambda$.
The transition form factors for the three pseudoscalar mesons $\pi^0$, $\eta$ and $\eta'$ are studied by CLEO [@CLEO-ff] using only the single-tag mode. The values $\Lambda_{\pi^0}$=0.776$\pm$0.010$\pm$0.012$\pm$0.016 GeV, $\Lambda_{\eta }$=0.774$\pm$0.011$\pm$0.016$\pm$0.022 GeV, $\Lambda_{\eta'}$=0.859$\pm$0.009$\pm$0.018$\pm$0.020 GeV are measured. Data are consistent with a similar wave function for the $\pi^0$ and $\eta$. The non-perturbative properties of the $\eta'(958)$ are found to be different from those of the $\pi^0$ and $\eta$. According to T. Feldmann [@Feldmann], another interpretation of these results leads to the conclusion that $\pi^0$, $\eta'$ and $\eta'$ mesons behave similarly in hard exclusive reactions.
Interesting new preliminary results on the K$_s^0$K$^\pm\pi^\mp$ and the $\eta\pi^+\pi^-$ final states are obtained by L3 [@Igor]. The K$_s^0$K$^\pm\pi^\mp$ mass spectrum is studied as a function of $Q^2$ (fig. \[fig:k0kp\]). A prominent signal is present at 1470 MeV at low and at high $Q^2$. At very high $Q^2$ another signal appears around 1300 MeV due to the formation of the f$_1$(1285). The study of the cross section as a function of $Q^2$ in the 1470 MeV region reveals that both the 0$^{-+}$ and 1$^{++}$ waves are needed to fit the data. The 0$^{-+}$ wave is due to the formation of the $\eta(1440)$ and largely dominates at low $Q^2$ while at high $Q^2$ the formation of the f$_1$(1420) is found to be dominant. The value $\Gamma_{\gamma\gamma}(\eta(1440))\times \mbox{BR}(\eta(1440)\rightarrow \mbox{K}\bar{\mbox{K}}\pi)$ = 234 $\pm$ 55 $\pm$ 17 eV is obtained by using data at low $Q^2$. This first observation of the $\eta(1440)$ in untagged two-photon collisions disfavours its interpretation as the 0$^{-+}$ glueball in agreement with the lattice QCD calculations. The $\eta(1440)$ can therefore be interpreted as a radial excitation [@Anisovitch-1]. The $\eta\pi^+\pi^-$ final state shows no evidence for the formation of the $\eta(1440)$ at low and at high $Q^2$ (fig. \[fig:etapipi\]). A prominent signal due to the formation of the $\eta'$(958) is present in the two spectra while the f$_1$(1285) is visible only at high $Q^2$. The upper limits $\Gamma_{\gamma\gamma}(\eta(1440))\times\mbox{BR}(\eta(1440)\rightarrow
\eta\pi\pi)<$ 88 eV and $\Gamma_{\gamma\gamma}(\eta(1295))\times\mbox{BR}(\eta(1295)\rightarrow
\eta\pi\pi)<$ 61 eV at 90% C.L. are obtained.
![The $\eta\pi^+\pi^-$ mass spectrum for $Q^2<$0.02 GeV$^2$(left) and $Q^2>$0.02 GeV$^2$(right).[]{data-label="fig:etapipi"}](fig5.eps "fig:"){width="45.00000%"} ![The $\eta\pi^+\pi^-$ mass spectrum for $Q^2<$0.02 GeV$^2$(left) and $Q^2>$0.02 GeV$^2$(right).[]{data-label="fig:etapipi"}](fig6.eps "fig:"){width="45.00000%"}
Glueball searches in the K$_s^0$K$_s^0$ and $\pi^+\pi^-$ final states
======================================================================
![The K$_s^0$K$_s^0$ mass spectra measured by L3 (left) and CLEO (right).[]{data-label="fig:k0k0"}](fig7.eps "fig:"){width="45.00000%"} ![The K$_s^0$K$_s^0$ mass spectra measured by L3 (left) and CLEO (right).[]{data-label="fig:k0k0"}](fig8.eps "fig:"){width="45.00000%"}
A study of the reaction $\gamma\gamma\rightarrow$ K$^0_s$K$^0_s$ is performed by L3 [@Saverio-Frascati] [@Saverio]. The mass spectrum is shown in fig. \[fig:k0k0\](left). The 1100-1400 MeV mass region shows destructive f$_2$(1270) – a$_2$(1320) interference in agreement with theoretical predictions [@Lipkin]. The spectrum is dominated by the formation of the f$_2\,\!\!\!'$(1525) tensor meson in helicity 2 state as clearly shown by the angular distribution in the $\kos\kos$ center of mass. The preliminary value $\Gamma_{\gamma\gamma}(f_2'(1525))\times \mbox{BR}(f_2'(1525)\rightarrow
\mbox{K}\bar{\mbox{K}})$= 0.076 $\pm$ 0.006 $\pm$ 0.011 keV is obtained. A clear signal is present in the 1750 MeV mass region due to the formation of the f$_J$(1710). The presence of a 0$^{++}$ $s\bar{s}$ meson would support the glueball interpretation of the f$_0$(1500) [@Amsler]. The study of the angular distribution in the 1750 MeV mass region favours the presence of a 2$^{++}$, helicity 2 wave. This is consistent with the interpretation of the f$_J$(1710) as a radial recurrence of the f$_2\,\!\!\!'$(1525) [@Munz]. The presence of a 0$^{++}$ wave cannot however be excluded. The BES Collaboration [@BESKPKM] reported the presence of both 2$^{++}$ and 0$^{++}$ waves in the 1750 MeV region in K$^+$K$^-$ in the reaction $\epem\ra\mbox{J}/\psi\ra\mbox{K}^+\mbox{K}^-\gamma$. No signal for the formation of the $\xi$(2230) [@xi] tensor glueball candidate is observed. The upper limit $\Gamma_{\gamma\gamma}(\xi(2230))\times $BR$(\xi(2230)\ra\kos\kos)<1.4$ eV at 95% C.L. is obtained. The stickiness is found to be $S_{\xi(2230)} >$ 73 at 95% C.L.
The $\xi(2230)$ is searched by CLEO in the K$_s^0$K$_s^0$ [@CLEO-k0k0] and $\pi^+\pi^-$ [@CLEO-pipi] final states. The K$_s^0$K$_s^0$ mass spectrum (fig. \[fig:k0k0\](right)) shows similar features respect to the L3 data. The upper limits $\Gamma_{\gamma\gamma}(\xi(2230))\times\mbox{BR}(\xi(2230)\ra\kos\kos)<$ 1.3 eV and $\Gamma_{\gamma\gamma}(\xi(2230))\times\mbox{BR}(\xi(2230)\rightarrow \pi^+\pi^-)<$ 2.5 eV at 95% C.L. are obtained. Combining these two results the stickiness is found to be $S_{\xi(2230)} >$ 102 at 95% C.L. The very large lower limits for $S_{\xi(2230)}$ obtained by CLEO and L3 give a strong support to the interpretation of the $\xi$(2230) as the tensor glueball. A confirmation of its existence in gluon rich environments becomes now very important.
The $\pi^+\pi^-$ final state is studied by ALEPH [@ALEPH-pipi]. The mass spectrum (fig. \[fig:pipialeph\]) shows a signal due to the formation of the f$_2$(1270) tensor meson. No other signals are present. Assuming the f$_0$(1500) and the f$_J$(1710) to be scalars, the upper limits $\Gamma_{\gamma\gamma}(f_0(1500))\times\mbox{BR}(f_0(1500)\rightarrow
\pi^+\pi^-)<$ 310 eV and $\Gamma_{\gamma\gamma}(f_J(1710))\times\mbox{BR}(f_J(1710)\rightarrow
\pi^+\pi^-)<$ 550 eV at 95% C.L. are obtained. Interference effects with the $\pi^+\pi^-$ continuum are not taken into account. According to A.V.Anisovitch et al. [@Anisovitch], interference with the $\pi^+\pi^-$ continuum should make the f$_0$(1500) appear as a dip.
Charmonium formation
=====================
The formation of the $\eta_c(2980)$ is studied by L3 [@L3-etac]. Since the $\eta_c$ decays in many different final states with small branching fractions, the simultaneous study of several decay channels is mandatory. The mass spectrum shown in fig. \[fig:etac\] is obtained by summing nine different final states. The value $\Gamma_{\gamma\gamma}(\eta_c)$ = 6.9 $\pm$ 1.7 (stat.) $\pm$ 0.8 (sys.)$\pm$ 2.0 (BR) keV is measured. Despite the limited statistics, the study of the formation of the $\eta_c(2980)$ as a function of $Q^2$ allows to exclude a VDM $\rho$ pole transition form factor. Data are consistent with a J/$\psi$ VDM pole form factor, as expected.
From the reaction $\gamma\gamma\rightarrow\chi_{c2}(3555)\rightarrow J/\psi \gamma\rightarrow l^+l^-\gamma$ with $l=e,\mu$, the two-photon width of the $\chi_{c2}$ is measured by OPAL[@OPAL-chic]. The signal is seen in the distribution of the mass difference m$(l^+l^-\gamma)$ – m$(l^+l^-)$ when m$(l^+l^-)$ is compatible with the mass of the J$/\psi$ (fig. \[fig:chic\_opal\]). The value $\Gamma_{\gamma\gamma}(\chi_{c2})$ = 1.76 $\pm$ 0.47 (stat.) $\pm$ 0.37 (sys.)$\pm$ 0.15 (BR) keV is obtained. The value $\Gamma_{\gamma\gamma}(\chi_{c2})$ = 1.02 $\pm$ 0.40 (stat.) $\pm$ 0.15 (sys.)$\pm$ 0.09 (BR) keV is measured by L3 [@L3-chic] using the same method.
The measurements of the two-photon width of the $\eta_c$ performed in two-photon collisions are in good agreement with the ones obtained in $p\bar{p}$ annihilations [@PDG]. For the $\chi_{c2}$ the agreement is not good and the two-photon measurements are significantly higher than the value $\Gamma_{\gamma\gamma}(\chi_{c2})$ = 0.31 $\pm$ 0.05 $\pm$ 0.04 keV measured by E835 at Fermilab [@Stancari] in $p\bar{p}$ annihilations. This value is in agreement with a previous measurement by E760 [@E760-chic]. The reason for this is not known but it is interesting to remark that all the two-photon measurements are performed by using the same final state and the same experimental method.
No signal for the formation of the $\eta_c'$ is observed at LEP. Five different decay channels are examined by DELPHI [@DELPHI-etacp] as shown in fig. \[fig:etacp\_delphi\]. The formation of the $\eta_c(2980)$ is clearly observed while no signal is present in the $\eta_c'$ mass region. The upper limit $\frac{\Gamma_{\gamma\gamma}(\eta_c')}{\Gamma_{\gamma\gamma}(\eta_c)}<0.34$ at 90% C.L. is obtained. The upper limit $\Gamma_{\gamma\gamma}(\eta_c')<2.0$ keV at 95% C.L. is obtained by L3 [@L3-etac] using nine different decay modes.
Conclusions
============
Resonance Experiment Final state J$^{PC}$ $\Gamma_{\gamma\gamma}$ Ref.
------------------- ------------ ----------------------- -------------- ----------------------------------- ----------------
$\eta'$(958) L3 $\pi^+ \pi^- \gamma$ $ 0^{-+} $ 4.17$\pm$0.10$\pm$0.27 keV [@L3-etap]
$a_2$(1320) L3 $\pi^+ \pi^- \pi^0 $ $ 2^{++} $ 0.98$\pm$0.05$\pm$0.09 keV [@L3-a2]
$f_2^{'}$(1525) L3 K$^0_s$K$^0_s$ $ 2^{++} $ 0.085$\pm$0.007$\pm$0.012 keV [@Saverio]
$\eta _{c}$(2980) L3 9 chan. $ 0^{-+} $ 6.9$\pm$1.7.$\pm$0.8 keV [@L3-etac]
$\eta _{c}'$ L3 9 chan. $ 0^{-+} $ $<2.0$ keV [@L3-etac]
$\chi_{c2}$(3555) L3 $ l^+ l^- \gamma $ $ 2^{++} $ 1.02$\pm$0.40$\pm$0.15 keV [@L3-chic]
$\chi_{c2}$(3555) OPAL $ l^+ l^- \gamma $ $ 2^{++} $ 1.76$\pm$0.47$\pm$0.37 keV [@OPAL-chic]
$\eta$(1440) L3 K$^0_s$K$^\pm\pi^\mp$ $ 0^{-+} $ 234$^{\dag}\pm$55$\pm$17 eV [@Igor]
f$_J$(1710) L3 K$^0_s$K$^0_s$ $ (?)^{++} $ [@Saverio]
$a_2'$(1752) L3 $\pi^+ \pi^- \pi^0 $ $ 2^{++} $ 0.29$^{\dag}\pm$0.04$\pm$0.02 keV [@L3-a2]
f$_0$(1500) ALEPH $\pi^+ \pi^-$ $ 0^{++} $ $<310^{\dag}$ eV [@ALEPH-pipi]
f$_0$(1710) ALEPH $\pi^+ \pi^-$ $ 0^{++} $ $<550^{\dag}$ eV [@ALEPH-pipi]
$\xi$(2230) CLEO $\pi^+ \pi^-$ $ 2^{++} $ $<2.5^{\dag}$ eV [@CLEO-pipi]
$\xi$(2230) CLEO K$^0_s$K$^0_s$ $ 2^{++} $ $<1.3^{\dag}$ eV [@CLEO-k0k0]
$\xi$(2230) L3 K$^0_s$K$^0_s$ $ 2^{++} $ $<1.4^{\dag}$ eV [@Saverio]
: The most recent results on the two-photon width of mesons, charmonia, radial excitations and glueball candidates. ($\dag$ the value is given times the decay branching ratio)[]{data-label="summary"}
A remarkable progress on the study of resonance formation in two-photon collisions has been achieved in the last few years. Data from the LEP collider at CERN and CESR at Cornell allowed to improve significantly the precision on the two-photon widths of several resonances, to study the transition form factors, to identify some radial excitations and to search for glueball candidates. All these results are summarised in Table \[summary\]. They represent an important contribution to meson spectroscopy and glueball searches.
Acknowledgements
================
I would like to acknowledge the two-photon physics groups of the ALEPH, CLEO, DELPHI, L3 and OPAL collaborations. I would like to thank M.N. Focacci-Kienzle, J.H. Field, M. Wadhwa, I. Vodopianov, A. Buijs, H. P. Paar, C. Amsler, L. Montanet, U. Gastaldi and P. Minkowski for the very constructive discussions and suggestions. I would like to express my gratitude to B. Monteleoni, recently deceased.
[9]{}
C. Michael, (1999) 12 and references therein.
C. Amsler and F. E. Close, (1996) 295.
P. Minkowski and W. Ochs, Frascati Physics Series Vol.XV (1999) 245 and hep-ph/9905250.
U. Gastaldi et al., LNL-INFN(rep) 148/99 and QCD99 proceedings, Montpellier, France.
M. Chanowitz,[*“Resonances in Photon-Photon Scattering”*]{}, Proceedings of the VI$^{th}$ International Workshop on Photon-Photon Collisions, World Scientific, 1984.
L3 Collab., (1997) 147.
S. Braccini, [*“The 1500-1700 MeV Mass Region and Glueball Searches in Two-Photon Collisions with the L3 Detector at LEP”*]{}, Frascati Physics Series Vol.XV (1999) 53.
CERN-IHEP Collab., (1973) 153.
CELLO Collab., (1990) 583.
Crystal Ball Collab., (1990) 561.
C. R. Münz, (1996) 364.
L3 Collab., (1998) 399.
V. V. Anisovitch et al., (1997) 166.
CLEO Collab., (1998) 33.
T. Feldmann, [*“Phenomenology of $\eta\gamma$ and $\eta'\gamma$ Transition Form Factors at Large Momentum Transfers ”*]{}, (Proc. Suppl.) (2000) 331.
I. Vodopianov, L3 Note 2564, submitted to ICHEP2000, Osaka, Japan, June 2000.
A. V. Anisovitch et al., (1999) 247.
L3 Collab., (1995) 118;\
S. Braccini,[*“The $\kos\kos$ Final State in Two-Photon Collisions and Some Implications for Glueball Searches”*]{}, (1999) 143 and hep-ex/9811017;\
S. Braccini, L3 Note 2557, submitted to ICHEP2000, Osaka, Japan, June 2000.
H. J. Lipkin, (1968) 321.
BES Collab., (1996) 3959.
BES Collab., (1996) 3502;\
Mark III Collab., (1986) 107.
CLEO Collab., 79 (1997) 3829;\
H. P. Paar, [*“Study of the glueball candidate $f_J(2220)$ at CLEO”*]{}, (Proc. Suppl.) (2000) 337.
CLEO Collab., 81 (1998) 3328.
ALEPH Collab., (2000) 189.
A. V. Anisovitch et al., (1999) 289.
L3 Collab., (1999) 155.
OPAL Collab., (1998) 197.
L3 Collab., (1999) 73.
Particle Data Group, (1998) 1.
M. Stancari, [*“Two-Photon Decay Widths of Charmonium States Studied by FNAL Experiment E835”*]{}, (Proc. Suppl.) (2000) 306.
E760 Collab., (1993) 2988.
DELPHI Collab., (1998) 479.
[^1]: Talk given at Meson2000, Cracow, Poland, May 2000.
|
---
abstract: 'Cadmium arsenide (Cd$_3$As$_2$) – a time-honored and widely explored material in solid-state physics – has recently attracted considerable attention. This was triggered by a theoretical prediction concerning the presence of 3D symmetry-protected massless Dirac electrons, which could turn Cd$_3$As$_2$ into a 3D analogue of graphene. Subsequent extended experimental studies have provided us with compelling experimental evidence of conical bands in this system, and revealed a number of interesting properties and phenomena. At the same time, some of the material properties remain the subject of vast discussions despite recent intensive experimental and theoretical efforts, which may hinder the progress in understanding and applications of this appealing material. In this review, we focus on the basic material parameters and properties of Cd$_3$As$_2$, in particular those which are directly related to the conical features in the electronic band structure of this material. The outcome of experimental investigations, performed on Cd$_3$As$_2$ using various spectroscopic and transport techniques within the past sixty years, is compared with theoretical studies. These theoretical works gave us not only simplified effective models, but more recently, also the electronic band structure calculated numerically using ab initio methods.'
author:
- 'I. Crassee'
- 'R. Sankar'
- 'W.-L. Lee'
- 'A. Akrap'
- 'M. Orlita'
title: '3D Dirac semimetal $\mbox{Cd}_3 \mbox{As}_2$: a review of material properties'
---
Introduction
============
Cadmium arsenide (Cd$_3$As$_2$) is an old material for condensed-matter physics, with its very first investigations dating back to the thirties [@StackelbergZP35]. Research on this material then continued extensively into the sixties and seventies, as reviewed in Ref. [@ZdanowiczARMS75]. This is when the physics of semiconductors, those with a sizeable, narrow, but also vanishing energy band gap, strongly developed.
In the early stages of research on Cd$_3$As$_2$, it was the extraordinarily high mobility of electrons, largely exceeding 10$^{4}$ cm$^{2}$/(V.s) at room temperature [@RosenbergJAP59; @TurnerPR61; @RosenmanJPCS69], that already attracted significant attention to this material. Even today, with a declared mobility well above 10$^{6}$ cm$^{2}$/(V.s) [@LiangNatureMater14] at low temperatures, cadmium arsenide belongs to the class of systems with the highest electronic mobilities, joining materials such as graphene, graphite, bismuth and 2D electron gases in GaAs/GaAlAs heterostructures [@MichenaudJPC72; @Brandt88; @HwangPRB08; @BolotinPRL08; @NeugebauerPRL09]. Another interesting – and in view of recent developments, crucial – observation was the strong dependence of the effective mass of electrons on their concentration, which implies a nearly conical conduction band, in other words the energy dispersion is linear in momentum, see Refs. [@ArmitagePLA68; @RosenmanJPCS69; @RogersJPD71] and Fig. \[Rosenman\].
To understand the electronic properties of Cd$_3$As$_2$, and its extraordinarily large mobility of electrons in particular, simple theoretical models for the electronic band structure have been proposed in the standard framework of semiconductors physics, and compared with optical and transport experiments. In the initial phase of research, cadmium arsenide was treated as an ordinary Kane-like semiconductor or semimetal [@CaronPRB77; @Bodnar77]. It was seen as a material with an electronic band structure that closely resembled zinc-blende semiconductors with a relatively narrow or vanishing band gap. However, no clear consensus was achieved concerning the particular band structure parameters, Refs. [@ZdanowiczARMS75] or [@Blom80]. Most notably, there was disagreement about the ordering of electronic bands and the presence of another conduction band at higher energies.
Renewed interest in the electronic properties of Cd$_3$As$_2$ was provoked by a theoretical study where Wang *et al.* [@WangPRB13] invoked the presence of a pair of symmetry-protected 3D Dirac cones. This way, Cd$_3$As$_2$ would fit into an emergent class of materials which are nowadays referred to as 3D Dirac semimetals, for review see, *e.g.*, Ref. [@ArmitageRMP18]. These systems host 3D massless electrons described by the Dirac equation for particles with a vanishing rest mass, thus implying a conical band twice degenerated due to spin. The possible lack of inversion symmetry may lift this degeneracy and transform Cd$_3$As$_2$ into a 3D Weyl semimetal, with two pairs of Weyl cones. The lack or presence of inversion symmetry in Cd$_3$As$_2$ is still not resolved [@SteigmannACB68; @AliIC14; @DesratPRB18].
The theoretical prediction by Wang *et al.* [@WangPRB13] can be illustrated using a simple, currently widely accepted cartoon-like picture depicted in Fig. \[Scheme\]. It shows two 3D Dirac nodes, the points where the conical conduction and valence bands meet, located at the tetragonal $z$ axis of the crystal. Their protection is ensured by $C_4$ rotational symmetry [@YangNatureComm14]. With either increasing or decreasing energy, these two Dirac cones approach each other and merge into a single electronic band centered around the $\Gamma$ point, passing through a saddle-like Lifshitz point.
The prediction of a 3D Dirac semimetallic phase in a material, which is not only well-known in solid-state physics but also relatively stable under ambient conditions, stimulated a considerable experimental effort. The first angular-resolved photoemission spectroscopy (ARPES) and scanning tunneling microscopy/spectroscopy (STM/STS) experiments [@LiuNatureMater14; @JeonNatureMater14; @BorisenkoPRL14; @NeupaneNatureComm14] confirmed the presence of widely extending conical features in the band structure, creating a large wave of interest. This wave gave rise to a number of experimental and theoretical studies, complementing those performed in the past, forming a rich knowledge of this material. As a result, cadmium arsenide now is among the most explored materials in current solid-state physics.
In addition to the extraordinarily high electronic mobility, cadmium arsenide exhibits a very strong linear magneto-resistance [@LiangNatureMater14], an anomalous Nernst effect [@LiangPRL17], and quantum Hall effect signatures when thinned down [@UchidaNC17; @SchumannPRL18]. Furthermore, indications of the chiral anomaly, planar Hall effect, and electron transport through surface states have been reported [@LiNatureComm15; @MollNature16; @JiaNatureComm16; @WuPRB18]. Currently, these effects are all considered to be at least indirectly related to the specific relativistic-like band structure of this material. Yet surprisingly, consensus about the complex electronic band structure of Cd$_3$As$_2$ has not yet been fully established.
A detailed understanding of the basic material parameters of Cd$_3$As$_2$ became essential for the correct interpretation of a wide range of observed, and yet to be discovered, physical phenomena. In this paper we review the the current knowledge of Cd$_3$As$_2$. We start with the properties of the crystal lattice and continue with the technological aspects of Cd$_3$As$_2$ growth. This is followed by a discussion of the theoretical and experimental investigation results, providing different possible views of this material’s electronic bands. In particular, we focus on Dirac-like nodes, the most relevant aspect of the band structure when taking into consideration recent developments in this material’s field.
Crystal lattice
===============
Cadmium arsenide has a relatively complex crystal structure, with 160 atoms in the unit cell. It has been the subject of several $x$-ray studies, resulting in somewhat contradicting conclusions [@StackelbergZP35; @SteigmannACB68; @PietraszkoPSS73; @AliIC14]. The current consensus implies that, at room and lower temperatures, which are the most relevant ones for fundamental and applied research conducted on this material, Cd$_3$As$_2$ is a tetragonal material with a unit cell of $a = b \approx 1.26$ nm and $c \approx 2.54$ nm. The particular space group remains a subject of discussion. More recent investigations favor a centrosymmetric group I4$_1$/acd (No. 142) [@PietraszkoPSS73; @AliIC14] over a non-centrosymmetric space group I4$_1$cd (No. 110), as previously suggested [@SteigmannACB68]. The correct space group assignment is of considerable importance for the complete understanding of Cd$_3$As$_2$. When space inversion symmetry is not present, spin degeneracy is lifted, and the Dirac nodes possibly split into pairs of Weyl nodes.
It is also important to note that the Cd$_3$As$_2$ crystal lattice, though clearly tetragonal, remains nearly cubic ($2a=2b\approx c$) [@AliIC14]. The lattice may therefore be seen, in the simplest approach, as being composed of antifluorite (cubic) cells with two missing cadmium cations (Fig. \[Lattice\]). Due to the cadmium vacancy ordering, a very large unit cell of Cd$_3$As$_2$ is formed. Composed of $2\times2\times 4$ of antifluorite cells, it is oriented along the tetragonal $c$-axis, and contains 160 atoms (96 cadmium and 64 arsenic). Additionally, each single anti-fluorite cell is tetragonally distorted, with a small elongation along the $c$-axis ($c/2a \approx 1.006$) [@ArushanovPCGC80; @AliIC14]. This simplified image of nearly cubic antifluourite cells, serving as building blocks for the entire Cd$_3$As$_2$ lattice, appears as a useful starting point for simplified effective models and [*ab initio*]{} calculations. Both are briefly discussed below.
Finally, let us mention that above room temperature the crystal lattice of Cd$_3$As$_2$ undergoes a sequence of polymorphic phase transitions. The corresponding space group remains tetragonal but changes to P4$_2$/nbc (No. 133) around 220$^\circ$C and to P4$_2$/nmc (No. 137) around 470$^\circ$C [@PietraszkoPSS73; @ArushanovPCGC80]. At temperatures above 600$^\circ$C, the symmetry of the crystal changes to a cubic one, characterized by the space group Fm$\bar{3}$m (No. 225) [@AliIC14]. Each of these phase transitions is accompanied by an abrupt change in lattice constants, leading to potential microcracks in the crystal. Notably, crystals of Cd$_3$As$_2$ can only be grown at temperatures above 425$^\circ$C, regardless of the growth method.
The complexity of the crystal lattice directly impacts the physical properties of Cd$_3$As$_2$, as well as how we understand them. For instance, the large number of unit cell atoms may partly complicate [*ab initio*]{} calculations of the electronic band structure. The complex crystal lattice, with a number of cadmium vacancies, may also be susceptible to small changes in ordering, possibly impacting distinct details in the electronic band structure. Some of the existing controversies about the electronic bands in Cd$_3$As$_2$ may thus stem from differences in the investigated samples’ crystal structure, currently prepared using a wide range of crystal growth methods. The crystal lattice complexity is also directly reflected in the optical response of Cd$_3$As$_2$, characterised by a large number of infrared and Raman-active phonon modes [@JandlJRS84; @NeubauerPRB16; @HoudeSSC86].
[c]{}
Technology of sample growth
===========================
The technology of Cd$_3$As$_2$ growth encompasses a broad range of methods dating back over 50 years. These provide us with a multiplicity of sample forms: bulk or needle-like crystals, thin films, microplatelets, and nanowires. Not only mono- or polycrystalline samples exist, but also amorphous [@ZdanowiczARMS75] samples, all exhibiting different quality and doping levels. In the past, various techniques have been used, such as the Bridgman [@RogersJPD71] and Czochralski [@SilveyJES61; @HiscocksJMS69] methods, sublimation in vacuum or in a specific atmosphere [@Zdanowiczpss64; @PawlikowskiTSF75], pulsed-laser deposition [@DubowskiAPL84], and directional crystallisation in a thermal gradient [@RosenbergJAP59]. The results of these techniques were summarised in review articles dedicated to the growth of II$_3$V$_2$ materials [@ArushanovPCGC80; @ArushanovPCGC92].
\[t\]
[c]{}
Most recently, the fast developing field of Cd$_3$As$_2$ has been largely dominated by experiments [@LiangNatureMater14; @JeonNatureMater14; @BorisenkoPRL14; @NeupaneNatureComm14; @LiangPRL17; @MollNature16; @AkrapPRL16] performed on samples prepared by either growth from a Cd-rich melt [@AliIC14], or self-selecting vapor growth (SSVG) [@SankarSR15], which are below described in greater detail. With Cd-rich melt, Cd$_3$As$_2$ is synthesized from a Cd-rich mixture of elements sealed in an evacuated quartz ampoule, heated to 825$^\circ$C, and kept there for 48 hours. After cooling to 425$^\circ$C at a rate of 6 $^\circ$C/h, Cd$_3$As$_2$ single crystals with a characteristic pseudo-hexagonal (112)-oriented facets are centrifuged from the flux.
The SSVG method comprises several steps [@SankarSR15]. In the first, the compound is synthesized from a stoichiometric mixture of elements in an evacuated sealed ampoule, and heated for 4 hours 50 $^\circ$C above the Cd$_3$As$_2$ melting point. The resulting ingot is then purified using a sublimation process in an evacuated closed tube (kept around 800 $^\circ$C), refined with small amounts of excess metal or chalcogen elements (also at 50 $^\circ$C above the melting point), water quenched, annealed (around 700 $^\circ$C) and subsequently air cooled. Afterwards, the ingot is crushed and then sieved (targeting a particle size of 0.1-0.3 mm), with the obtained precursors then sealed in an evacuated ampoule. This is then inserted for about 10 days into a horizontal alumina furnace, whose temperature profile is shown in Fig. \[SSVG\]. The resulting crystals are plate-like or needle-like monocrystals, or polycrystals with large grains. The crystals have primarily (112)-oriented, but occasionally also ($n$00)-oriented facets, with $n$ an even number, see Fig. \[OM\].
Crystals prepared using the two methods discussed above, as well as past methods, display $n$-type conductivity with a relatively high density of electrons. As-grown and without specific doping, the crystals rarely show an electron density below $10^{18}$ cm$^{-3}$. This translates into typical Fermi levels in the range of $E_F=100-200$ meV measured form the charge-neutrality point. In literature this rather high doping is usually associated with the presence of arsenic vacancies [@SpitzerJAP66]. The doping of Cd$_3$As$_2$ has been reported to vary with conventional annealing [@RamboCJP79]. It seems to decrease after thermal cycling between room and helium temperatures [@CrasseePRB18]. A combined optical and transport study also revealed a relatively thick (10-20 $\mu$m) depleted layer on the surface of Cd$_3$As$_2$ crystals [@SchleijpeIJIMW84]. Optical studies also show relatively large inhomogeneities of up to 30% in the electron density, at the scale of hundred microns, in crystals prepared using different growth methods, with $x$-ray studies indicating the presence of systematic twinning [@CrasseePRB18].
Recently, there has been progress in other growth methods. These include the CVD technique, employed to fabricate Cd$_3$As$_2$ nanowires [@WangNC16]. Other methods, such as pulsed laser deposition in combination with solid phase epitaxy [@UchidaNC17] and molecular beam epitaxy [@YuanNL17; @SchumannPRL18], have been successfully employed to prepare Cd$_3$As$_2$ layers with a thickness down to the nanometer scale. Most importantly, when the thickness of Cd$_3$As$_2$ films is reduced down to the tens of nanometers, ambipolar gating of Cd$_3$As$_2$ becomes possible [@GallettiPRB18].
Electronic band structure: theoretical views
============================================
Soon after the first experimental studies appeared [@RosenbergJAP59; @Zdanowiczpss64; @Haidemenakis66; @Sexerpss67; @RosenmanJPCS69], the band structure of Cd$_3$As$_2$ was approached theoretically. In this early phase, the electronic band structure was described using simple effective models, developed in the framework of the standard $k.p$ theory. Such models were driven by the similarity between the crystal lattice – and presumably also electronic states – in Cd$_3$As$_2$, and conventional binary semiconductors/semimetals such as GaAs, HgTe, CdTe or InAs [@CardonaYu]. In all of these materials, the electronic bands in the vicinity to the Fermi energy are overwhelmingly composed of cation $s$-states and anion $p$-states. In the case of Cd$_3$As$_2$, those are cadmium-like cations and arsenic-like anions.
In the first approach, the electronic band structure of Cd$_3$As$_2$ has been described using the conventional Kane model [@KaneJPCS57], which is widely applied in the field of zinc-blende semiconductors and which has also been successfully expanded to account for effects due to quantizing magnetic fields [@BowersPRB59; @PidgeonPR66]. To the best of our knowledge, the very first attempt to interpret the experimental data collected on Cd$_3$As$_2$ using the Kane model was presented by Armitage and Goldsmid [@ArmitagePLA68] in 1968. Later on, similar studies appeared [@RogersJPD71; @CaronPRB77; @AubinPRB77; @Jay-GerinJLTP77] but with no clear consensus on the particular band structure parameters.
The size and nature of the band gap, describing the separation between the $p$-like and $s$-like states at the $\Gamma$ point, remained a main source of controversy [@ZdanowiczARMS75]. Most often, relatively small values of either inverted or non-inverted gaps have been reported, based on the analysis of different sets of experimental data [@ArmitagePLA68; @RosenmanJPCS69; @CaronPRB77]. Notably, when the band gap is significantly smaller than the overall energy scale of the considered band structure, the Kane model implies approximately conical conduction and valence bands, additionally crossed at the apex by a relatively flat band (Fig. \[Kane\]) [@Kacmanppsb71; @OrlitaNaturePhys14]. This band is usually referred to as being heavy-hole-like in semiconductor physics, and may be considered as flat only in the vicinity of the $\Gamma$ point. At larger momenta, it disperses with a characteristic effective mass close to unity.
Importantly, the presence of a 3D conical band in the Kane model is a result of an approximate accidental degeneracy of $p$-like and $s$-like states at the $\Gamma$ point, therefore making this cone not protected by any symmetry. Additionally, this cone is described by the Kane Hamiltonian [@Kacmanppsb71], which is clearly different from the Dirac Hamiltonian. It should therefore not be confused with 3D Dirac cones subsequently predicted for Cd$_3$As$_2$ [@WangPRB13]. The term *massless Kane electron* was recently introduced [@OrlitaNaturePhys14; @MalcolmPRB15; @TeppeNatureComm16; @AkrapPRL16] to distinguish those two types of 3D massless charge carriers. Therefore, at the level of the strongly simplifying Kane model, Cd$_3$As$_2$ does not host any massless 3D Dirac electrons. A nearly conical band, the presence of which was deduced from early transport experiments indicating energy-dependent effective mass [@ArmitagePLA68; @RosenmanJPCS69] (Fig. \[Rosenman\]), was then interpreted in terms of the Kane model assuming a nearly vanishing band gap.
An improved effective model, which takes into account the tetragonal nature of Cd$_3$As$_2$, not included in the conventional Kane model for zinc-blende semiconductors, has been proposed by Bodnar [@Bodnar77]. The tetragonal distortion of a nearly cubic lattice lifts the degeneracy of $p$-type states (light and heavy hole bands) at the $\Gamma$ point; this degeneracy is typical of all zinc-blende semiconductors [@CardonaYu]. A closer inspection of the theoretical band structure reveals two specific points at the tetragonal axis. At these points the conduction and flat heavy-hole valence bands meet and form two highly anisotropic and tilted cones (Fig. \[Bodnar\]). These may be associated with symmetry-protected 3D Dirac cones. Nevertheless, it was not until the work by Wang *et al.* [@WangPRB13] that such a prediction appeared explicitly in the literature. The crystal-field splitting parameter $\delta$ [@KildalPRB74] is employed by Bodnar [@Bodnar77] to quantify the impact of the tetragonal distortion of the cubic lattice. This parameter directly corresponds to the energy scale $E_D$ of the symmetry-protected massless Dirac electrons in Cd$_3$As$_2$ (cf. Figs. \[Scheme\] and \[Bodnar\]). The work of Bodnar also inspired other theorists, who used the proposed model for electronic band structure calculations in quantizing magnetic fields [@Wallacepss79; @SinghJPC83; @SinghJPC84].
The above described effective models have been applied with some success to explain experimental data available those times. However, these models should be confronted with the much broader experimental knowledge acquired recently. Similar to all other models based on the $k.p$ expansion, their validity is limited to the near vicinity of the Brillouin zone center, and to the number of spin-degenerate bands taken into account (restricted to 4 in the Kane/Bodnar model). Moreover, the number of band structure parameters is strongly restricted in these models. This ensures the models’ relative simplicity, but at the same time, limits the potential to describe the band structure in greater detail, even in the immediate vicinity of the $\Gamma$ point.
It is therefore important to reconcile the implications of such effective models with other theoretical approaches. Such band structure calculations already appeared in the early stages of research on Cd$_3$As$_2$. These were based on either pseudopotential calculations [@Lin-ChungPRB69; @Lin-Chungpss71; @Dowgiallo-Plenkiewiczpps79] or the semiempirical tight-binding method [@SieranskiPRB94]. More recently, [*ab initio*]{} calculations have been performed, with their main focus on the Dirac-like states. To the best of our knowledge, the first [*ab initio*]{} study of Cd$_3$As$_2$ predicting the presence of 3D massless Dirac electrons was presented by Wang *et al.* [@WangPRB13]. Nevertheless, the considered space groups did not comprise the most probable one, I4$_1$/acd (No. 110) [@PietraszkoPSS73; @AliIC14]. Other [*ab initio*]{} calculations may be found in Refs. [@LiuNatureMater14; @NeupaneNatureComm14; @BorisenkoPRL14; @ConteSR17; @ShchelkachevIM18], often carried out in support of experimental findings.
Even though the results of [*ab initio*]{} calculations may differ in some details – most likely related to the particular approximation of exchange and correlation functionals and the size of the unit cell considered – they provide us with a rather consistent theoretical picture of the electronic bands in Cd$_3$As$_2$. They predict that Cd$_3$As$_2$ is a semimetal, with a pair of well-defined 3D Dirac cones, emerging at relatively low energies. In line with symmetry arguments, the Dirac nodes are found to be located at the tetragonal axis, with an exception of Ref. [@LiuNatureMater14].
--------------------------------------------------------------- ----------------- ------------------- -------------
$E_D$ (meV) $k_D$ (nm$^{-1}$)
*ab initio*, GGA, I4$_1$cd [@WangPRB13] 40 0.32 \[001\]
*ab initio*, GGA, I4$_1$/acd [@AliIC14] 45 0.4 \[001\]
*ab initio*, GGA, I4$_1$/acd [@ConteSR17] 20 0.23 \[001\]
*ab initio*, GGA, I4$_1$cd [@LiuNatureMater14] 200 1.2 \[112\][^1]
ARPES [@LiuNatureMater14] several hundred 1.6 \[112\]
ARPES [@NeupaneNatureComm14] several hundred \[001\]
ARPES [@BorisenkoPRL14] several hundred \[001\]
STM/STS [@JeonNatureMater14] 20 0.04 \[001\]
Magneto-optics [@HaklPRB18; @AkrapPRL16] $<$40 $<$0.05 \[001\]
Magneto-transport & Bodnar model [@RosenmanJPCS69; @Bodnar77] 85 0.15 \[001\]
Magneto-transport [@ZhaoPRX15] $<$200 \[001\]
--------------------------------------------------------------- ----------------- ------------------- -------------
When the non-centrosymmetric I4$_1$cd space group is considered, the double degeneracy of the Dirac cones due to spin may be lifted [@WangPRB13]. The loss of inversion symmetry then transforms the 3D Dirac semimetal into a 3D Weyl semimetal with two pairs of Weyl nodes. The Dirac cones are dominantly formed from $p$-type arsenic states, and are well separated from the bands lying at higher or lower energies. Notably, the $s$-like cadmium states are found well below the Fermi energy. The band structure is therefore inverted and may be formally described by a negative band gap ($E_g\approx-0.5$ eV). To certain extent, the electronic bands in Cd$_3$As$_2$ resemble those in HgTe, which is another semimetal with an inverted band structure [@WeilerSaS81].
To illustrate the typical results of [*ab initio*]{} calculations, two examples have been selected from Refs. and and plotted in Figs. \[Ab initio\]a and b, respectively. The 3D Dirac cones are located in close vicinity to the $\Gamma$ point, with the charge neutrality (Dirac) points at the $\Gamma$-Z line and the Fermi level. The parameters of these cones derived in selected [*ab initio*]{} studies are presented in Tab. \[Table\]. For instance, Conte *et al.* [@ConteSR17] deduced, using the GGA approximation, that the energy scale of massless Dirac electrons reaches $E_D\approx 20$ meV, the strongly anisotropic velocity parameter is of the order of $10^5$ m/s, and two Dirac nodes are located at $k_D=\pm 0.23$ nm$^{-1}$.
Let us now explore the experiments carried out on Cd$_3$As$_2$ using various techniques. These provided us with unambiguous evidence for the conical features in this material. However, let us clearly note from the beginning that the scale of the conical bands deduced – using ARPES [@LiuNatureMater14; @NeupaneNatureComm14; @BorisenkoPRL14] and STM/STS [@JeonNatureMater14] techniques, or from the optical response [@AliIC14; @NeubauerPRB16; @AkrapPRL16] – is not consistent with the energy scale predictions for massless Dirac electrons based on [*ab initio*]{} calculations. While the theoretically expected scale of massless Dirac electrons rarely exceeds $E_D\sim 100$ meV (Tab. \[Table\]), the experimentally observed cones extend over a significantly broader interval of energies.
In the following sections, we employ, for the sake of brevity, a simplified notation and refer to the Kane and Dirac models, which both may – at least from the theoretical viewpoint – explain the presence of massless electrons in Cd$_3$As$_2$. In the case of the Dirac model, we always consider 3D massless electrons, which are described by the Dirac equation with the zero rest mass and the presence of which is protected by the discrete ($C_4$) rotational symmetry [@WangPRB13]. In the case of Kane model, zero or vanishing band gap is implicitly assumed.
Angular-resolved photoemission spectroscopy
===========================================
The ARPES technique provided us with a solid piece of evidence for conical features in the electronic band structure of Cd$_3$As$_2$, soon after predictions of Dirac-like states by Wang *et al.* [@WangPRB13]. This largely contributed to the renewed interest in the electronic properties of this material. Such initial observations were made by several groups [@LiuNatureMater14; @BorisenkoPRL14; @NeupaneNatureComm14; @YiSR14], and elaborated further later on [@RothPRB18].
![The valence bands in Cd$_3$As$_2$ visualized by low-temperature ARPES technique by Liu *et al.* [@LiuNatureMater14]. The data was collected on the (112)-terminated surface. The widely extending 3D conical band – characterized by a velocity parameter of $1.3\times10^6$ m/s and interpreted in terms of 3D massless Dirac electrons – coexists with another hole-like weakly dispersing parabolic band, also observed in Ref. [@NeupaneNatureComm14]. Reprinted by permission from Springer Nature: Nature Materials [@LiuNatureMater14], copyright (2014). \[ARPES-Liu\]](ARPES-Liu.jpg "fig:"){width="35.00000%"}\[b\]
Characteristic data collected in ARPES experiments on Cd$_3$As$_2$ [@LiuNatureMater14; @NeupaneNatureComm14] are plotted in Figs. \[ARPES-Liu\] and \[ARPES-Neupane\], and respectively show well-defined conical features for both valence and conduction bands. The conical features in Fig. \[ARPES-Liu\] and \[ARPES-Neupane\] were interpreted in terms of bulk states. Liu *et al.* [@LiuNatureMater14] and Neupane *et al.* [@NeupaneNatureComm14] concluded the presence of a pair of 3D Dirac nodes at the \[112\] and \[001\] axes, respectively. However, it is worth noting that the orientation along the \[112\] axis – or alternatively, along the \[111\] axis when an approximately cubic unit cell is considered like it is the case in Ref. [@LiuNatureMater14] – is not consistent with expectations based on symmetry arguments [@YangNatureComm14], which only allow the Dirac nodes to be present at the tetragonal axis (the \[001\] direction). The velocity parameter was found to be close to $10^6$ m/s in the plane perpendicular to the axis connecting the Dirac nodes, and reduced down to $3\times 10^5$ m/s [@LiuNatureMater14] along this axis. The ARPES data in Refs. [@LiuNatureMater14; @NeupaneNatureComm14] do not directly show any signatures of Dirac cones merging via the corresponding Lifshitz points. Nevertheless, the indicated velocity parameters, and the position of the cones ($k_D$), allow us to estimate the scale of massless Dirac electrons $E_D$ to be several hundred meV or more.
ARPES data similar to Refs. [@LiuNatureMater14; @NeupaneNatureComm14], obtained on the \[112\]-terminated surface of Cd$_3$As$_2$, were also presented by Borisenko *et al.* [@BorisenkoPRL14], who primarily focused on the conical feature in the conduction band. The presence of a pair of symmetry-protected 3D Dirac cones, with the corresponding nodes at the \[001\] axis, has been concluded, and the electron velocity parameter $v\approx0.8\times10^6$ m/s deduced. The Fermi energy in the studied $n$-doped sample exceeded $E_F\approx 200$ meV, and may serve as a lower bound for the $E_D$ parameter. Since the shape of the observed conical band does not provide any signature of the approaching upper Lifshitz point, one may conclude that $E_D\gg E_F$.
The basic parameters of Dirac-like conical bands deduced from the above cited ARPES experiments have been compared in Tab. \[Table\] with results of other experimental techniques, which are discussed in detail later on. One may see that there is a considerable spread of the reported values for both the energy scale, and the position of the Dirac cones in Cd$_3$As$_2$. For instance, the energy scale consistent with the ARPES data exceeds by almost two orders of magnitude the estimate based on STM/STS measurements [@JeonNatureMater14]. At present, the reason for this disagreement remains unclear.
More recently, other ARPES experiments on Cd$_3$As$_2$ have been performed by Roth *et al.* [@RothPRB18]. They provided experimental data similar to previous studies, but differed in their interpretation. From their experiment realized on a sample with a (112)-terminated surface, they concluded that a part of observed conical features does not come from bulk, but instead originate from the surface states (cf. Ref. [@YiSR14]). Let us note that, since each Dirac node is composed of two Weyl nodes with opposite chiralities [@PotterNatureComm14], such surface states may have the form of the Fermi arcs in a 3D symmetry-protected Dirac semimetal.
To the best of our knowledge, the electronic band structure observed in ARPES experiments has not yet been in detail compared with expectations of the Kane/Bodnar model. Nevertheless, it is interesting to note that the conical valence band observed in the ARPES data (Fig. \[ARPES-Liu\]) coexists with another weakly dispersing hole-like parabolic band [@NeupaneNatureComm14], which has a characteristic effective mass close to unity. Such a band is not expected in the model of 3D massless Dirac electrons, nevertheless, it could be straightforwardly explained within the Kane/Bodnar picture with a small or vanishing band gap.
![The conduction band of Cd$_3$As$_2$ visualized by ARPES by Neupane *et al.* [@NeupaneNatureComm14]. The data were collected on the (001)-terminated surface of Cd$_3$As$_2$ at 14 K and show the dispersion in the direction perpendicular to the $\Gamma$-Z line. The observed 3D conical band was interpreted in terms of 3D massless Dirac electrons, implying a velocity parameter of $1.5\times10^6$ m/s. Reprinted by permission from Springer Nature: Nature Communications [@NeupaneNatureComm14], copyright (2014). \[ARPES-Neupane\]](ARPES-Neupane.jpg){width="35.00000%"}
Scanning tunneling spectroscopy and microscopy
==============================================
Similar to ARPES, the STS/STM technique also played an important role in the recent revival of Cd$_3$As$_2$. The data collected in STS experiments performed in magnetic fields [@JeonNatureMater14] revealed, via the characteristic $\sqrt{B}$-dependence of Landau levels, the presence of a single widely extending conical band located at the center of the Brillouin zone. In the literature, this observation is often taken as experimental evidence for the symmetry-protected 3D massless Dirac electrons in Cd$_3$As$_2$. However, according to Jeon *et al.* [@JeonNatureMater14], the existence of “this extended linearity is not guaranteed by the Dirac physics around the band inversion”.
Such a conclusion agrees with theoretical expectations based on symmetry arguments. These exclude the existence of a symmetry-protected Dirac cone located at the center of the Brillouin zone [@YangNatureComm14]. In addition, this observation is in line with the Kane/Bodnar models used to explain the band structure of Cd$_3$As$_2$ in the past. Nevertheless, Jeon *et al.* [@JeonNatureMater14] also conclude – by extrapolating their Landau level spectroscopy data to vanishing magnetic fields – that a pair of symmetry-protected Dirac cones emerges at low energies. They give a rough estimate of $E_D\approx20$ meV for the characteristic Dirac energy scale.
Beyond insights into the bulk electronic states of Cd$_3$As$_2$, the natural sensitivity of the STM/STS technique to the surface of explored systems may provide us with deeper knowledge about their surface states. A 3D symmetry-protected Dirac semimetal like Cd$_3$As$_2$ can be viewed as two copies of a 3D Weyl semimetal with nodes having opposite chiralities; two sets of Fermi arcs are expected on the surface, see Ref. [@KargarianPNAS16] for details. The recent STS/STM study [@ButlerPRB17] dedicated to the surface reconstruction of cadmium vacancies in Cd$_3$As$_2$ may be considered as the initial step in such investigations.
Optical properties
==================
The optical and magneto-optical properties of Cd$_3$As$_2$ have been a topic of study for more than 50 years, and resulted in a series of works [@TurnerPR61; @Haidemenakis66; @ZdanowiczPSS67]. These were often interpreted in terms of Kane/Bodnar models [@WagnerJPCSS71; @RogersJPD71; @RadoffPRB72; @AubinPRB77; @GeltenSSC80; @AubinPRB81; @Jay-GerinSSC83; @SinghPB83; @SinghJPC83; @HoudeSSC86; @LamraniCJP87; @AkrapPRL16; @HaklPRB18; @CrasseePRB18], and more recently using the picture of 3D massless Dirac electrons [@NeubauerPRB16; @JenkinsPRB16; @YuanNL17; @UykurPRB18].
Early optical studies were focused on the basic character of the electronic band structure in Cd$_3$As$_2$. They aimed at clarifying the existence of a band gap and at determining its size. Using infrared reflectivity and transmission techniques, Turner *et al.* [@TurnerPR61] concluded that Cd$_3$As$_2$ should be classified as a narrow-gap semiconductor with a direct band gap of 0.16 eV. Similarly, an indirect band gap around 0.2 eV, was concluded by Zdanowicz *et al.* [@ZdanowiczPSS67] based on transmission experiments. Much lower values were found in magneto-optical studies. Haidemenakis *et al.* [@Haidemenakis66] concluded $E_g<30$ meV, suggesting a semimetallic nature of Cd$_3$As$_2$. The difference in conclusions between optical and magneto-optical studies may be related to the large doping of the explored samples. In that case, the onset of interband absorption, often referred to as the optical band gap, appears due to the Moss-Burstein shift (Pauli blocking) [@BursteinPR54], at photon energies significantly exceeding the size of the energy band gap.
Further series of optical and magneto-optical Cd$_3$As$_2$ studies were performed on various mono- or polycrystalline samples during the seventies and eighties. The collected data was primarily analyzed within the framework of the Kane model [@WagnerJPCSS71; @RogersJPD71; @RadoffPRB72; @AubinPRB77; @GeltenSSC80], and later on the Bodnar model [@AubinPRB81; @Jay-GerinSSC83; @LamraniCJP87]. The authors of these works concluded that the electronic band structure at the $\Gamma$ point is fairly well described using these models: it is inverted, with the $p$-type arsenic states above the $s$-like cadmium states, and characterized by a relatively small negative band gap. A schematic view of such a band structure is plotted in Figs. \[Kane\] and \[Bodnar\]. The deduced value of the band gap reached $E_g \approx -0.2$ eV [@WagnerJPCSS71; @RadoffPRB72; @AubinPRB77] at room or low temperatures, but also values as low as $E_g \approx -0.1$ eV have been reported [@AubinPRB81; @LamraniCJP87; @Jay-GerinSSC83]. Particular attention has been paid to the profile and position of the weakly dispersing band, which is nearly flat in the vicinity of the $\Gamma$ point. Several times, the possibility of the band maxima located at non-zero momenta has been discussed [@AubinPRB77; @GeltenSSC80]. At the same time, Lamrani and Aubin concluded a surprisingly flat heavy-hole band, when the theoretically expected Landau-quantized Bodnar band structure [@SinghPB83; @SinghJPC83] was confronted with the experimentally determined energies of interband inter-Landau level excitations [@LamraniCJP87].
The renewed interest in Cd$_3$As$_2$ motivated several groups to take a fresh look at the optical, magneto-optical, and ultra fast optical properties of this material [@WeberAPL15; @NeubauerPRB16; @JenkinsPRB16; @AkrapPRL16; @YuanNL17; @SharafeevPRB17; @HaklPRB18; @CrasseePRB18; @WeberAPL15; @ZhuAPL15; @LuPRB17; @LuPRB18]. Thanks to this, to the best of our knowledge, the optical conductivity of Cd$_3$As$_2$ (Fig. \[Neubauer\]) has been extracted from the experimental data for the very first time [@NeubauerPRB16]. At low energies, the optical conductivity is characterized by a pronounced Drude peak due to the presence of free charge carriers, and a rich set of phonon excitations. Similar to the Raman response [@JandlJRS84; @SharafeevPRB17], the complexity of the phonon-related response directly reflects the relatively high number of atoms in the Cd$_3$As$_2$ unit cell. Above the onset of interband absorption, optical conductivity increases with a slightly superlinear dependence on photon energy. Such behavior is not far from the expectation for 3D massless charge carriers, $\sigma(\omega)\propto \omega$ [@GoswamiPRL11; @TimuskPRB13]. The observed optical response was thus interpreted in terms of 3D massless Dirac electrons, with a low anisotropy and a velocity parameter lying in the range of $1.2-3.0\times10^{5}$ m/s. No clear indications of Dirac cones merging at the Lifshitz points were found. Reflectivity data similar to Ref. [@NeubauerPRB16] were also presented by Jenkins *et al.* [@JenkinsPRB16], with basically the same conclusions.
The Dirac model was similarly used to interpret the classical-to-quantum crossover of cyclotron resonance observed in magneto-transmission data collected on thin MBE-grown Cd$_3$As$_2$ layers [@YuanNL17] as well as pump-probe experiments in the visible spectral range, which revealed the transient reflection [@WeberAPL15] and transmission [@ZhuAPL15]. These latter experiments show that a hot carrier distribution is obtained after 400–500 fs, after which the charge carriers relax by two processes. Subsequent pump-probe experiments using mid-infrared [@LuPRB17] and THz probe [@LuPRB18] confirm this two-process relaxation, which can be qualitatively reproduced using a two temperature model.
A different view was proposed in a recent magneto-reflectivity study of Cd$_3$As$_2$ by Akrap et al. [@AkrapPRL16] (see Fig. \[CR\]). In high magnetic fields, when the samples were pushed into their corresponding quantum limits with all electrons in the lowest Landau level, the $\sqrt{B}$ dependence of the observed cyclotron mode was found to be inconsistent with 3D massless Dirac electrons. The data was interpreted in terms of the Kane/Bodnar model, with a vanishing band gap. This model also leads to the appearance of 3D massless electrons, and consequently, to a magneto-optical response linear in $\sqrt{B}$. In this model, the $\sqrt{B}$ dependence of the cyclotron mode is also expected in the quantum limit. This is in contrast to 3D massless Dirac electrons, which host characteristic so-called zero-mode Landau levels. These are independent of magnetic field, disperse linearly with momentum along the direction of the applied field, and imply a more complex cyclotron resonance dependence in the quantum limit. An approximately isotropic velocity parameter was found for the nearly conical conduction band: $v \approx 0.9\times 10^6$ m/s [@AkrapPRL16]. Estimates of the band gap and crystal field splitting parameter at low temperatures were obtained in a subsequent magneto-transmission study performed on thin Cd$_3$As$_2$ slabs [@HaklPRB18]: $E_g=-(70\pm20)$ meV and $|\delta|=E_D<40$ meV.
Lately, spatially resolved infrared reflectivity has been used to characterize the homogeneity of Cd$_3$As$_2$ crystals [@CrasseePRB18]. In all the studied samples, independently of how they were prepared and how they were treated before the optical experiments, conspicuous fluctuations in the carrier density up to 30% have been found. These charge puddles have a characteristic scale of 100 $\mu$m. They become more pronounced at low temperatures, and possibly, they become enhanced by the presence of crystal twinning. Such an inhomogeneous distribution of electrons may be a generic property of all Cd$_3$As$_2$ crystals, and should be considered when interpreting experimental data collected using other techniques.
Magneto-transport properties and quantum oscillations
=====================================================
Renewed interest in Cd$_3$As$_2$ brought upon an explosion of interest in magneto-transport studies. Many of these studies focused on the previously reported very high carrier mobility [@RosenbergJAP59], and its possibilities for device application. Often, newer transport results are interpreted within the scenario of two 3D Dirac cones, which are well separated in $k$-space and with the Fermi level lying below the Lifshitz transition.
The first detailed magneto-transport study of Cd$_3$As$_2$ dates back to the 1960s [@RosenmanPL66; @ArmitagePLA68; @RosenmanJPCS69]. The geometry of the Fermi surface was for the first time addressed by Shubnikov-de Haas (SdH) measurements. Rosenman [@RosenmanPL66] explored such quantum oscillations on a series of $n$-doped samples, concluding that the Fermi surface is a simple ellipsoid symmetric around the $c$-axis, and inferred a low anisotropy factor of 1.2. These very first papers also show nearly conical shape of the conduction band (Fig. \[Rosenman\]). A decade later, Zdanowicz *et al.* [@ZdanowiczTSF79] studied SdH oscillations in thin films and single crystals of Cd$_3$As$_2$, confirming the ellipsoidal geometry of the Fermi surface. In addition, they reported a striking linear magnetoresistance (MR).
In a new bout of activity, several groups confirmed that the Fermi surface consists of a simple, nearly spherical ellipsoid, with an almost isotropic Fermi velocity [@LiangNatureMater14]. Such a simple Fermi surface was questioned by Zhao *et al.* [@ZhaoPRX15]. They found that, for particular directions of the magnetic field with respect to the main crystal axes, the magnetoresistance (MR) shows two oscillation periods which mutually differ by 10-25%, pointing to a dumbbell-shaped Fermi surface. This was interpreted to originate from two nested Fermi ellipsoids arising from two separated Dirac cones, where $E_F$ is placed just above the Lifshitz transition. Another picture was proposed by Narayanan *et al.* [@NarayananPRL15], who found single-frequency SdH oscillations, and concluded two nearly isotropic Dirac-like Fermi surfaces. In subsequent studies, Desrat *et al.* followed the SdH oscillations as a function of the field orientation, and always found two weakly separated frequencies (5-10%). Their results were interpreted within the picture of two ellipsoids that are separated in $k$-space due to the possible absence of inversion symmetry [@DesratPRB18]. Clear beating patterns, indicating a multiple frequency in SdH oscillations, were also reported in Nernst measurements [@LiangPRL17], and magnetoresistance [@GuoSR16] on single crystals of Cd$_3$As$_2$. Such beating patterns were attributed to the lifting of spin degeneracy due to inversion symmetry breaking either by an intense magnetic field [@XiangPRL15; @LiangPRL17], or by Cd-antisite defects [@GuoSR16], which may turn a Dirac node into two Weyl nodes.
Several transport studies reported on strikingly linear nonsaturating MR in Cd$_3$As$_2$ [@NarayananPRL15; @FengPRB15; @LiangNatureMater14] being more pronounced in samples with lower mobilities. Since linear MR is observed at magnetic fields far below the quantum limit, the standard Abrikosov’s theory [@AbrikosovJPAMG03] – referring to the transport in the lowest Landau level – cannot be applied, and a different explanation was needed. Liang *et al.* [@LiangNatureMater14] therefore suggest there is a mechanism that protects from backscattering in zero field. This protection is then rapidly removed in field, leading to a very large magnetoresistance. They propose an unconventional mechanism, caused by the Fermi surface splitting into two Weyl pockets in an applied magnetic field. Similarly, Feng *et al.* [@FengPRB15] assigned the large non-saturating MR to a lifting of the protection against backscattering, caused by a field-induced change in the Fermi surface. The authors judged that linear MR cannot be due to disorder, as the Cd$_3$As$_2$ samples are high-quality single crystals. They instead conclude that it is due to the Dirac node splitting into two Weyl nodes. In contrast, Narayanan *et al.* [@NarayananPRL15] find that the Fermi surface does not significantly change up to 65 T, except for Zeeman splitting caused by a large $g$-factor. Through comparing quantum and transport relaxation times, they conclude that transport in Cd$_3$As$_2$ is dominated by small-angle scattering, which they trace back to electrons scattered on arsenic vacancies, and that the linear MR is linked to mobility fluctuations.
The Potter *et al.* theory [@PotterNatureComm14] predicts the existence of specific closed cyclotron orbits in a Dirac or Weyl semimetal (Fig.\[Moll\]a). These orbits are composed of two Fermi arcs located on opposite surfaces of the sample, which are then interconnected via zero-mode Landau levels. Moll *et al.* [@MollNature16] report on the SdH oscillations measured in mesoscopic devices. These were prepared using the focused ion beam technique, allowing Cd$_3$As$_2$ crystals to be cut into sub-micron-thick platelets. They report two series of oscillations, and via specific angle dependence, they associate them with surface-related and bulk-related orbits (Fig. \[Moll\]b).
The chiral anomaly is yet another theoretically expected signature of the field-induced splitting of a Dirac point into a pair of Weyl nodes. The parallel application of an electric and magnetic field is predicted to transfer electrons between nodes with opposite chiralities. Such a transfer should be associated with lowering resistivity (negative MR). For Cd$_3$As$_2$, there are indeed several reports of a negative MR and a suppression of thermopower for particular magnetic-field directions [@LiNatureComm15; @JiaNatureComm16; @LiNatureComm16]. Typically, such behavior was observed in microdevices (platelets or ribbons). It should be noted that a negative MR can also emerge in a topologically trivial case due to so-called current jetting [@dosReisNJP16; @LiangPRX18]. This is a simple consequence of a high transport anisotropy when a magnetic field is applied. In such a case, the current is jetting through a very narrow part of the sample. The measured MR then strongly depends on the distance of the voltage contacts from the current path.
The analysis of the quantum oscillation phase represents a unique way to identify the nature of probed charge carriers. For conventional Schrödinger electrons, one expects so-called Berry phase $\beta=0$, whereas massless Dirac electrons should give rise to $\beta = \pi$. Indeed, several reports based on SdH oscillations in Cd$_3$As$_2$ indicate that $\beta \sim \pi$ [@ZdanowiczTSF79; @HePRL14; @DesratJPCS15; @PariariPRB15]. A weak deviation from the ideal non-trivial Berry phase, $\beta=(0.8-0.9)\pi$ was reported by Narayanan [@NarayananPRL15], who also discussed the influence of Zeeman splitting on the phase of quantum oscillations [@MikitikPRB03]. Notably, the electron $g$-factor in Cd$_3$As$_2$ is relatively large and anisotropic, and it implies Zeeman splitting comparable to the cyclotron energy [@BlomSSC80; @Blom80].
Contrasting results were obtained by Xiang *et al.* [@XiangPRL15], who found the non-trivial Berry phase only when the field is applied along the tetragonal $c$-axis of Cd$_3$As$_2$. When the magnetic field is rotated to be parallel with the $a$ or $b$ axis, the measured Berry phase becomes nearly trivial (see Fig. \[Xiang\] and discussion in Ref. [@HeCPB16]). This result is interpreted in terms of 3D Dirac-phase symmetry-breaking effects, when the magnetic field is tilted away from the $c$-axis. A certain angle dependence of the Berry phase has also been reported by Zhao *et al.* [@ZhaoPRX15], reporting the Berry phase in between 0 and $\pi$. Another study by Cheng *et al.* [@ChengNJP16] was dedicated to the thickness-dependence of the Berry phase in MBE-grown thin films of Cd$_3$As$_2$, implying a non-trivial to trivial transition of the Berry phase with changing the layer thickness.
The partly contradicting results in determining the Berry phase illustrate that even though the phase of quantum oscillation in principle identifies the nature unambiguously, the practical analysis of this phase is straightforward only in well-defined system such as graphene [@NovoselovNature05]. In more complex materials such as Cd$_3$As$_2$, where quantum oscillations are superimposed on the much stronger effect of linear magneto-resistance [@LiangNatureMater14], the precise and reliable determination of the phase may represent a more challenging task. This may be illustrated with the example of bulk graphite, which is another high-mobility system with a strong and approximately linear magneto-resistance. There, the Dirac-like or normal massive nature of hole-type carriers has been a subject of intensive discussion in literature [@LukyanchukPRL04; @LukyanchukPRL06; @SchneiderPRL09; @LukyanchukPRL10; @SchneiderPRL10].
More recently, the very first experiments showing the quantum Hall effect (QHE) in thin films of Cd$_3$As$_2$ appeared. Zhang *et al.* measured SdH oscillations on a series of Cd$_3$As$_2$ nanoplates [@ZhangNatureComm17]. They report multiple cyclotron orbits, distinguishing both 3D and 2D Fermi surfaces. They also observe a quantized Hall effect (QHE), which they attribute to the surface states of Cd$_3$As$_2$, linked to the Weyl orbits. Schumann *et al.* [@SchumannPRL18] studied MBE-grown, 20 nm thick films of Cd$_3$As$_2$, and observed the QHE at low temperatures (Fig. \[Schumann\]). Similarly, they attribute the QHE to surface states, and conclude that the bulk states are weakly gapped at low temperatures. In continuation of this work, Galletti *et al.* [@GallettiPRB18] tuned the carrier concentration in thin films across the charge neutrality point using a gate voltage and concluded that the observed magneto-transport response is in line with expectations for a 2D electron gas of massless Dirac electrons.
The planar Hall effect has also been reported in Cd$_3$As$_2$, an effect in which a longitudinal current and an in-plane magnetic field give rise to a transverse current or voltage. Li *et al.* investigated microribbons of Cd$_3$As$_2$, and found an anisotropic MR and planar Hall effect, which they attribute to the physics of Berry curvature [@LiPRB18]. Guo *et al.* also uncovered unusually large transverse Hall currents in needle-like single crystals of Cd$_3$As$_2$ [@GuoSR16]. Wu *et al.* carried out similar measurements on Cd$_3$As$_2$ nanoplatelets, finding a large negative longitudinal MR, and a planar Hall effect with non-zero transverse voltage when the magnetic field is tilted away from the electric field. These observations are interpreted as transport evidence for the chiral anomaly [@WuPRB18].
Summary
=======
Cadmium arsenide is a prominent member of the topological materials class, widely explored using both theoretical and experimental methods. At present, there is no doubt that Cd$_3$As$_2$ hosts well-defined 3D massless charge carriers. These are often associated with the symmetry-protected 3D Dirac phase, but may also be interpreted using alternative approaches developed in the past. This calls for further investigations of Cd$_3$As$_2$, preferably using a combination of different experimental techniques. Such investigation may resolve the currently existing uncertainties about the electronic band structure, which slow down the overall progress in physics of Cd$_3$As$_2$, and may contribute to our understanding of rich phenomena associated with this appealing material.
Acknowledgements {#acknowledgements .unnumbered}
----------------
The authors acknowledge discussions with F. Bechstedt, S. Borisenko, C. C. Homes, S. Jeon, N. Miller, B. A. Piot, A. V. Pronin, O. Pulci, A. Soluyanov, S. Stemmer, Z. Wang, H. Weng and F. Xiu. This work was supported by ANR DIRAC3D projects and MoST-CNRS exchange programme (DIRAC3D). A. A. acknowledges funding from The Ambizione Fellowship of the Swiss National Science Foundation. I. C. acknowledges support from the postdoc mobility programme of the Suisse National Science Foundation. A. A. acknowledges the support SNF through project PP00P2\_170544.
[126]{}ifxundefined \[1\][ ifx[\#1]{} ]{}ifnum \[1\][ \#1firstoftwo secondoftwo ]{}ifx \[1\][ \#1firstoftwo secondoftwo ]{}““\#1””@noop \[0\][secondoftwo]{}sanitize@url \[0\][‘\
12‘\$12 ‘&12‘\#12‘12‘\_12‘%12]{}@startlink\[1\]@endlink\[0\]@bib@innerbibempty @noop [****, ()]{} [****, ()](https://doi.org/10.1146/annurev.ms.05.080175.001505) [****, ()](\doibase 10.1063/1.1735019) [****, ()](\doibase 10.1103/PhysRev.121.759) [****, ()](\doibase
http://dx.doi.org/10.1016/0022-3697(69)90200-5) [****, ()](\doibase
doi.org/10.1038/nmat4143) [****, ()](http://stacks.iop.org/0022-3719/5/i=21/a=011) in @noop [**]{} (, ) [****, ()](\doibase 10.1103/PhysRevB.77.235437) [****, ()](\doibase
10.1103/PhysRevLett.101.096802) [****, ()](\doibase
10.1103/PhysRevLett.103.136403) [****, ()](\doibase https://doi.org/10.1016/0375-9601(68)90440-4) [****, ()](http://stacks.iop.org/0022-3727/4/i=6/a=310) [****, ()](\doibase 10.1103/PhysRevB.15.3879) in [**](https://arxiv.org/abs/1709.05845), (, ) pp. in @noop [**]{}, Vol. , (, ) [****, ()](\doibase
10.1103/PhysRevB.88.125427) [****, ()](\doibase 10.1103/RevModPhys.90.015001) [****, ()](\doibase 10.1107/S0567740868003705) [****, ()](\doibase
10.1021/ic403163d) [****, ()](\doibase 10.1103/PhysRevB.97.245203) [****, ()](http://www.nature.com/ncomms/2014/140915/ncomms5898/full/ncomms5898.html) [****, ()](http://dx.doi.org/10.1038/nmat3990) [****, ()](\doibase doi:10.1038/nmat4023) [****, ()](\doibase
10.1103/PhysRevLett.113.027603) [****, ()](http://dx.doi.org/10.1038/ncomms4786) [****, ()](\doibase 10.1103/PhysRevLett.118.136601) [****, ()](\doibase
10.1038/s41467-017-02423-1) [****, ()](\doibase
10.1103/PhysRevLett.120.016801) [****, ()](\doibase 10.1038/ncomms10137) [****, ()](http://dx.doi.org/10.1038/nature18276) [****, ()](\doibase 10.1038/ncomms13013) [****, ()](\doibase 10.1103/PhysRevB.98.161110) [****, ](\doibase 10.1002/pssa.2210180234) [****, ()](\doibase
http://dx.doi.org/10.1016/0146-3535(80)90020-9) [****, ()](\doibase 10.1002/jrs.1250150215) [****, ()](\doibase 10.1103/PhysRevB.93.121202) [****, ()](\doibase https://doi.org/10.1016/0038-1098(86)90149-3) [****, ()](\doibase 10.1038/srep12966) [****, ()](\doibase 10.1149/1.2428183) [****, ()](\doibase 10.1007/BF00551073) [****, ()](\doibase 10.1002/pssb.19640060340) [****, ()](\doibase 10.1016/0040-6090(75)90309-0) [****, ()](\doibase 10.1063/1.94752) [****, ()](\doibase
http://dx.doi.org/10.1016/0960-8974(92)90030-T) [****, ()](\doibase
10.1103/PhysRevLett.117.136401) in [**](\doibase 10.1016/S0080-8784(08)60130-1), , Vol. , (, ) pp. @noop [**]{} (, , ) [****, ()](\doibase 10.1063/1.1707927) [****, ()](\doibase 10.1139/p79-284) [****, ()](\doibase
10.1103/PhysRevB.97.125204) [****, ()](\doibase 10.1007/BF01417648) [****, ()](\doibase
10.1038/ncomms10769) [****, ()](\doibase
10.1021/acs.nanolett.6b04778) [****, ()](\doibase
10.1103/PhysRevB.97.115132) in @noop [**]{}, (, ) p. [****, ()](\doibase 10.1002/pssb.19670210121) [****, ()](\doibase 10.1016/0022-3697(57)90013-6) [****, ()](\doibase 10.1103/PhysRev.115.1165) [****, ()](\doibase 10.1103/PhysRev.146.575) [****, ()](\doibase 10.1103/PhysRevB.15.3872) [****, ()](\doibase 10.1007/BF00658953) [****, ()](\doibase 10.1002/pssb.2220470228) [****, ()](http://dx.doi.org/10.1038/nphys2857) [****, ()](\doibase 10.1103/PhysRevB.92.035118) [****, ()](\doibase {10.1038/ncomms12576}) [****, ()]({https://www.nature.com/articles/srep45500}) [****, ()](\doibase 10.1103/PhysRevB.10.5082) [****, ()](\doibase 10.1002/pssb.2220920106) [****, ()](\doibase 10.1088/0022-3719/16/20/013) [****, ()](http://stacks.iop.org/0022-3719/17/i=8/a=013) [****, ()](\doibase 10.1103/PhysRev.188.1272) [****, ()](\doibase 10.1002/pssb.2220470103) [****, ()](\doibase 10.1002/pssb.2220940153) [****, ()](\doibase
10.1103/PhysRevB.50.7331) [****, ()](\doibase 10.1134/S0020168518110110) [****, ()](\doibase
10.1103/PhysRevB.97.115206) [****, ()](\doibase
10.1103/PhysRevX.5.031037) [****, ()](\doibase 10.1038/srep06106) [****, ()](\doibase 10.1103/PhysRevB.97.165439) [****, ()](\doibase 10.1038/ncomms6161) [****, ()](\doibase 10.1073/pnas.1524787113) [****, ()](\doibase 10.1103/PhysRevB.95.081410) [****, ()](\doibase 10.1002/pssb.19670200208) @noop [**** ()]{} [****, ()](\doibase 10.1103/PhysRevB.5.442) [****, ()](\doibase
http://dx.doi.org/10.1016/0038-1098(80)91199-0) [****, ()](\doibase 10.1103/PhysRevB.23.3602) [****, ()](\doibase https://doi.org/10.1016/0038-1098(83)90685-3) [****, ()](\doibase https://doi.org/10.1016/0378-4363(83)90553-3) @noop [****, ()]{} [****, ()](\doibase
10.1103/PhysRevB.94.085121) [****, ()](\doibase 10.1103/PhysRevB.97.195134) [****, ()](\doibase 10.1103/PhysRev.93.632) [****, ()](\doibase
doi:10.1063/1.4922528) [****, ()](\doibase 10.1103/PhysRevB.95.235148) [****, ()](\doibase
10.1063/1.4985688) [****, ()](\doibase
10.1103/PhysRevB.95.024303) [****, ()](\doibase
10.1103/PhysRevB.98.104310) [****, ()](\doibase 10.1103/PhysRevLett.107.196803) [****, ()](\doibase 10.1103/PhysRevB.87.235121) [****, ()](\doibase https://doi.org/10.1016/0031-9163(66)90291-5) [****, ()](http://www.sciencedirect.com/science/article/pii/004060907990498X) [****, ()](\doibase 10.1103/PhysRevLett.114.117201) [****, ()](\doibase 10.1038/srep27487) [****, ()](\doibase
10.1103/PhysRevLett.115.226401) [****, ()](\doibase
10.1103/PhysRevB.92.081306) [****, ()](http://stacks.iop.org/0305-4470/36/i=35/a=301) [****, ()](http://dx.doi.org/10.1038/ncomms10301) [****, ()](http://stacks.iop.org/1367-2630/18/i=8/a=085006) [****, ()](\doibase 10.1103/PhysRevX.8.031002) [****, ()](\doibase 10.1103/PhysRevLett.113.246402) [****, ()](http://stacks.iop.org/1742-6596/647/i=1/a=012064) [****, ()](\doibase 10.1103/PhysRevB.91.155139) [****, ()](\doibase 10.1103/PhysRevB.67.115114) [****, ()](\doibase
https://doi.org/10.1016/0038-1098(80)90698-5) [****, ()](http://stacks.iop.org/1674-1056/25/i=11/a=117105) [****, ()](http://stacks.iop.org/1367-2630/18/i=8/a=083003) [****, ()](\doibase
doi.org/10.1038/nature04233) [****, ()](\doibase 10.1103/PhysRevLett.93.166402) [****, ()](\doibase 10.1103/PhysRevLett.97.256801) [****, ()](\doibase 10.1103/PhysRevLett.102.166403) [****, ()](\doibase 10.1103/PhysRevLett.104.119701) [****, ()](\doibase 10.1103/PhysRevLett.104.119702) [****, ()](https://www.nature.com/articles/s41467-017-01438-y) [****, ()](\doibase
10.1103/PhysRevB.97.201110)
[^1]: equivalent to the \[111\] direction when the cubic cell approximation is considered as in Ref. [@LiuNatureMater14]
|
---
abstract: 'We investigate the cosmology of mini Primordial Black Holes (PBHs) produced by large density perturbations. The mini PBHs evaporate promptly in the early universe and we assume that a stable remnant is left behind. The PBHs remnants can constitute the entire dark matter of the universe for a wide range of remnant masses. We build inflationary models, in the framework of $\alpha$-attractors utilizing exponential functions, in which the PBHs are produced during matter, radiation and kination domination eras. The advantage of these inflationary models is that the spectral index takes values favorable by the Planck 2018 data. The PBH production from runaway inflaton models has the unique and very attractive feature to automatically reheat the universe. In these models the PBHs are produced during the kination stage and their prompt evaporation efficiently produces the required entropy. Such runaway models are remarkably economic having interesting implications for the early universe cosmology, possibly giving rise to a wCDM late time cosmology as well.'
author:
- Ioannis Dalianis$^1$
- George Tringas$^1$
title: 'PBH remnants as dark matter produced in thermal, matter and runaway-quintessence post-inflationary scenarios'
---
\[introsection\] Introduction
=============================
The primordial origin of the black holes (PBH) is rather attractive scenario because PBHs can constitute the entire cosmological dark matter (DM) or some significant fraction of it [@PBH1; @PBH2; @Carr:1975qj]. Contrary to stellar black holes, the mass range of the PBHs can be very wide, spanning over thirty decades of mass, from $10^{-18} M_\odot$ to $10^{15} M_\odot$. A synergy of observations, including the CMB [@Ricotti:2007au], the stochastic gravitational wave background [@Sasaki:2018dmp], Lyman $\alpha$ forest [@Murgia:2019duy], lensing events [@Barnacka:2012bm; @Niikura:2017zjd; @Tisserand:2006zx] and dynamical studies of bound astrophysical systems [@Capela:2013yf; @Capela:2012jz; @Brandt:2016aco; @Graham:2015apa; @Gaggero:2016dpq] derive a combination of constraints on the PBH abundance for nearly the entire PBH mass range. Neglecting, possible enhanced merging rates or an extended PBH mass distribution the current allowed mass windows for the dominant PBH dark matter scenario are quite narrow and have central values $M_\text{} \sim 10^{-15} M_\odot$ and $M_\text{} \sim 10^{-12} M_\odot$. Remarkably, although triggered by the LIGO events [@TheLIGOScientific:2016pea; @Abbott:2016blz; @Abbott:2016nmj], the PBH research has been extended into scenarios with vastly different mass scales. PBHs with very light masses are anticipated to Hawking radiate energetically and this places strong constraints on their abundance. The lightest PBHs that can constitute a non-negligible part of the cosmic dark matter have mass $M\sim 10^{-17}M_\odot$. PBHs with smaller mass are prone to evaporation and hence much constrained from the extra galactic gamma-ray background [@Carr:2016hva], CMB and BBN [@Carr:2009]. Mini PBHs with masses $M \ll 10^{-23}M_\odot \simeq 10^{10}$g will have promptly evaporated in the very early universe potentially leaving no observational traces.
However, the scenario of mini PBHs is of major interest due to the theoretical expectation that black holes cannot evaporate into nothing, see e.g. [@Bowick:1988xh; @Coleman:1991ku; @Chen:2014jwq]. If the black hole evaporation halts at some point then a stable state, called black hole remnant, will survive. Remnants from the PBHs prompt evaporation have important cosmological consequences, with the most notable one being that PBH remnants can comprise the entire dark matter in the universe [@Barrow; @Carr:1994ar]. The cosmological scenario of mini PBHs evaporation and PBH remnants has been studied in several contexts in [@Alexander:2007gj; @Scardigli:2010gm; @Lennon:2017tqq; @Raidal:2018eoo; @Rasanen:2018fom; @Nakama:2018utx; @Morrison:2018xla].
The generation of mini PBHs implies that the ${\cal P_R}(k)$ has to feature a peak at very large wavenumbers $k$. The most attractive mechanism to generate PBHs is inflation. Due to the natural generation of large scale perturbations from quantum fluctuations, inflation is the dominant paradigm that cosmologists follow to explain the origin of the large scale structure and has been, so far, successfully tested by the CMB precision measurements [@Akrami:2018odb]. Nevertheless inflation does not seed large scale perturbations only, it seeds perturbations in all scales. Hence, PBHs can form if perturbations strong enough to collapse are produced at scales $k^{-1} \ll k^{-1}_\text{cmb}$ characteristic of the PBH mass.
There have been a numerous inflationary models constructed to predict a significant abundance of PBHs, for recent proposal see e.g. [@Kawaguchi:2007fz; @Drees:2011hb; @Kawasaki:2016pql; @Garcia-Bellido:2017mdw; @Ballesteros:2017fsr; @Hertzberg:2017dkh; @Dalianis; @Gao:2018pvq; @Kawasaki:2016pql; @Cicoli:2018asa; @Ballesteros:2018wlw; @Kannike:2017bxn; @Gong:2017qlj; @Pi:2017gih; @Cai:2018tuh; @Dimopoulos:2019wew]. Their common feature is that the spectrum of the curvature perturbations, ${\cal P_R}(k)$, turns from red into blue in small scales. Since inflation models primarily generate the CMB anisotropies, these new inflationary proposals, though successful at generating PBHs, might fail at the $k^{-1}_\text{cmb}$ scale. In general the predicted spectral index $n_s$ and running $\alpha_s$ values are in best accordance with the CMB measured values when the PBHs have masses of minimal size. Light PBHs imply that the ${\cal P_R}(k)$ shape has to be modified at $k$ values, far beyond the scales probed by the CMB. From a model building perspective, achieving an enhancement of the ${\cal P_R}(k)$ at the very end of the spectrum might seem an attractive feature. This is an extra motivation to examine the mini PBH scenario.
In this work we build inflationary models that generate mini PBH and examine the early and late universe cosmology of the PBHs remnants. Our inflationary models belong in the family of the $\alpha$-attractors [@Kallosh:2013yoa] and the building blocks we use are exponential potentials. The PBHs are generated by the presence of an inflection point at small field values where the inflaton velocity decreases significantly producing a spike in the ${\cal P_R}(k)$. Since, PBHs form during the very early post-inflationary cosmic stage and reheating might have not been completed, it is natural to examine the evolution of the PBHs and their remnants for different backgrounds and expansion rates.
We derive expressions for the relic abundance of the PBH remnants for an arbitrary barotropic parameter $w$ and remnant mass, $M_\text{rem}$. These expressions are general and applicable to the stable PBH scenario as well. Afterwards, we examine explicitly the radiation domination, the matter domination and the kination domination cases. We explicitly construct and analyze three different inflationary potentials and we compute the power spectrum of the comoving curvature perturbation, solving numerically the Mukhanov-Sasaki equation, and we estimate the fractional PBH remnant abundance. A great advantage of our inflationary models is that the predicted values for the $n_s$ and $\alpha_s$ are placed inside the 1-$\sigma$ CL region of the Planck 2018 data.
Moreover, we introduce the scenario of PBH production during the kination domination regime (called also stiff phase) that has interesting cosmological implications. Kination is driven by the kinetic energy of the inflaton field itself. The duration of the kination regime is solely specified by the mass and the abundance of the PBHs produced. The fact that the PBHs promptly evaporate means that the universe is automatically and successfully reheated without the need of special couplings or tailor made resonance mechanisms. In addition, the inflaton field might play the role of the quintessence at late times giving rise to a testable wCDM cosmology, where the cold dark matter, comprised of PBHs remnants, is produced by the primordial fluctuations of the very same field. Apparently, in terms of ingredients, this is a maximally economic cosmological scenario.
The analysis in this paper is structured as follows. In Section 2 we discuss the bounds on the masses of the PBHs and of their remnants reviewing briefly theoretical considerations and deriving the associated cosmological bounds. In Section 3 the cosmology of the PBH remnants is presented for a general expansion rate and the main formulas are derived. In Section 4 we turn to the inflationary model building and formulate the constraints that the ${\cal P_R}(k)$ has to satisfy. In Section 5 we examine inflationary models based on the $\alpha$-attractors that generate mini PBHs, we compute numerically the ${\cal P_R}(k)$, and construct explicit examples that the PBH remnants comprise the entire dark matter in the universe. We present and illustrate our results with several plots and tables. In section 6 we conclude.
PBH evaporation remnants
========================
Hawking predicted that black holes radiate thermally with a temperature [@Hawking:1974sw; @Hawking:1974rv] $$\begin{aligned}
\label{HawkT}
T_\text{BH}=\frac{\hbar c^3}{8\pi G M k_B} \sim 10^8 \left( \frac{M}{10^{5}\text{g}}\right)^{-1} \text{GeV}\,,\end{aligned}$$ and are expected to evaporate on a time scale $t_\text{evap} \sim G^2M^3/(\hbar c^4)$, that is found to be [@Carr:2009] $$\begin{aligned}
\label{tevap}
t_\text{evap}= 407 \tilde{f}(M) \left( \frac{M}{10^{10} \text{g}} \right)^3 \text{s},\end{aligned}$$ where $M$ the mass of the PBH formed. We see that PBHs in the mass range $M\sim 10^{9}-10^{12}$ g evaporate during or after the BBN cosmic epoch and the $\beta$ is much constrained by the abundance of the BBN relics. For PBH in the mass range $M\sim 10^{13}-10^{14}$ g the evaporation takes place during the cosmic epoch of recombination and the CMB observations put the stringent constraints on the $\beta(M)$. Larger PBH masses contribute to the extra galactic gamma-ray background, for a review see [@Carr:2009]. Hence, the scenario of the PBH remnants as dark matter is motivated for $M<10^9$ g. In particular, the remnants from the evaporation of the PBHs can constitute the entire or a significant portion of the cosmic dark matter if the PBH mass is smaller than $$\begin{aligned}
M < \left( \frac{ \kappa \, m^2_\text{Pl} \, M^{1/2}_\text{eq} }{1+w}\right)^{2/5}\,,
\end{aligned}$$ as we will show in the Section 3. The Planck mass is $m_\text{Pl}=2.2 \times 10^{-5}$ g and $M_\text{eq}$ is the horizon mass at the moment of radiation-matter equality. The $w$ stands for the equation of state of the background cosmic fluid. For reasons that will we explain later, we call the upper mass bound $M_\text{inter}$ and it has size, roughly, $ 2 \, \kappa^{2/5} 10^6$ g. Remnants from PBHs with mass $M>M_\text{inter}$ contribute only a small fraction to the total dark matter.
Theoretical considerations about the PBH remnant mass
-----------------------------------------------------
There are several theoretical reasons for anticipating that black holes do not evaporate completely but leave behind a stable mass state. The Hawking radiation is derived by treating matter fields quantum mechanically, while treating the space-time metric classically. When the mass of an evaporating black hole becomes comparable to the Planck scale such a treatment would breakdown, and quantum gravitational effects would become relevant. Energy conservation [@Torres:2013kda], extra spatial dimensions [@ArkaniHamed:1998rs; @Suranyi:2010yt], higher order corrections to the action of general relativity [@Barrow], the information loss paradox [@Chen:2014jwq] could be sufficient to prevent complete evaporation. Higher order correction to the Hawking radiation emerging from some quantum gravity theory are also expected to modify the evaporation rate. The mass of the final state of the evaporation, i.e the PBH remnants, $M_\text{rem}$ can be written in terms of the Planck mass $$\begin{aligned}
M_\text{rem}=\kappa \, m_\text{Pl}\,.\end{aligned}$$ The $\kappa$ is a factor that parameterizes our ignorance. Different theories predict stable black hole relics of different mass. The $\kappa$ may be of order one, with relic black hole masses characterized by the fundamental scale of gravity, $m_\text{Pl}=G^{-2}$, but other values for the $\kappa$ are also admitted. If black holes have quantum hair, e.g if possess discrete electric and magnetic charges, the remnant mass depends on the value of the charge, $M_\text{rem}\sim m_\text{Pl}/g$, where $g$ the corresponding coupling constant [@Coleman:1991ku]. Thus, the $M_\text{rem}$ can be orders of magnitudes larger than one, $\kappa \gg 1$, for weakly coupled theories. Other theories, such as those where a generalized uncertainty principle is applied [@Adler:2001vs] the mass of the black hole remnants can be much smaller that the $m_\text{Pl}$, see e.g [@Carr:2015nqa], hence it is $\kappa \ll 1$. In our analysis and expressions the $\kappa$ is a free parameter. This is a justified approach since we know nearly next to nothing about the physics at that energy scales. In explicit examples we will pick up the benchmark $\kappa \sim1$ value and we will comment on the cosmology of different $\kappa$ values, see Fig. \[betawk\].
This work aims at the cosmology of the PBH remnants and we will remain agnostic about the fundamental physics that prevents black holes (or holes more properly) from complete evaporation. We will not enter into the details regarding the modification of the Hawking temperature with respect to the black hole mass either. Nevertheless, we remark that the formation of mini PBHs with mass $M<M_\text{inter} \sim \kappa^{2/5} 10^6$ g takes place in the very early universe, at the cosmic time $t_\text{form}$, and we expect these PBHs to evaporate promptly with their temperature reaching a maximum value contrary to what the standard expression (\[HawkT\]) dictates. If the temperature of the PBHs is initially smaller that the background cosmic temperature, $T_\text{BH}(t_\text{form})<T_\text{}(t_\text{form})$, the accretion effects should be taken into account. Although the accretion decreases the temperature of the PBH, the decrease of the cosmic temperature due to the expansion is much faster and the amount of matter that a PBH can accrete is small. Hence the PBH lifetime will not be modified and once the cosmic temperature falls below the value $T_\text{BH} \sim 10^8(M/10^5 \text{g})^{-1}$ GeV the PBHs heat up and evaporate. Concerning the black hole temperature, the $T_\text{BH}$ is expected to reach a maximal value and afterwards decrease as $M\rightarrow M_\text{rem}$. In this last stage of the PBH evaporation the rate $dM/dT_\text{BH}$ turns into positive. The PBHs are expected to exist in stable equilibrium with the background only when their mass is already close to the remnant mass [@Barrow], thus possible related corrections can be considered negligible for the scope of this work.
Cosmological constraints on the mass of the PBH remnants
---------------------------------------------------------
The examination of the PBHs remnants cosmology provides us with observational constraints on the $\kappa$ value. The corresponding analysis is presented in detail in the Section 3 and here, in advance, we will use part of the results to report the cosmological allowed values for the PBH remnants masses.
Let us first examine the minimal possible value for the $M_\text{rem}$. In the inflationary framework the formation of PBHs with mass $M$ can be realized only if the horizon mass right after inflation, $M_\text{end}=4\pi M^2_\text{Pl}/H_\text{end} $ is smaller than $M/\gamma$. The $\gamma$ parameter is the fraction of the Hubble mass that finds itself inside the black hole. In terms of the Hubble scale at the end of inflation, $H_\text{end}$, the bound reads $$\begin{aligned}
\label{Hcon}
{M}> \gamma\, 10^{5}\, \text{g} \,\frac{10^9 \, \text{GeV}}{H_\text{end}}\, .\end{aligned}$$ The above inequality yields a lower bound for the PBH mass. The upper bound on the tensor-to-scalar ratio $r_*<0.064$ [@Akrami:2018odb] and the measured value of the scalar power spectrum amplitude constrain the $H_* \simeq (\pi^2 A_s r_*/2)^{1/2} M_\text{Pl}$. It is $H_\text{end}<H_*<2.6\times 10^{-5} M_\text{Pl}\simeq 6.5 \times 10^{13}$ GeV, hence the minimal PBH mass that can be generated is $M/\gamma > {\cal O}(1) (r_*/0.06)^{-1/2}$ grams. Assuming that a radiation domination phase follows inflation then the fractional abundance of the PBH remnants, given by Eq. (\[fRD2\]), is maximal $f_\text{rem}=1$ for $\kappa \gtrsim 10^{-18.5}$. Hence, the PBH remnants are possible to have a significant relic abundance only if they have mass $$\begin{aligned}
M_\text{rem} > 1 \, \text{GeV} \simeq 1.8\times 10^{-24}\text{g}\,.\end{aligned}$$ This lower bound has been derived assuming the minimum possible PBH mass, $M\sim 1$ g, and the maximum possible formation rate, $\beta \sim 1$, see Eq. (\[fRD2\]). It is also valid for non-thermal post-inflationary cosmic evolution. PBHs remnants with smaller mass can constitute only a negligible amount of the total dark matter energy density.
On the other side, the $\kappa$ value has a maximum value, $\kappa_\text{max}=M/m_\text{Pl}$. Apparently for $\kappa =\kappa_\text{max}$ there is no Hawking radiation and one does not talk about PBH remnants. For $\kappa \ll M/m_\text{Pl}$ the Hawking radiation is important and it might affect the BBN and CMB observables for light enough PBHs. Assuming again a radiation domination phase after inflation the Eq. (\[fRD2\]) implies that $\kappa \propto 1/ \beta$ for $f_\text{rem}=1$. Increasing the $\kappa$ smaller $\beta$ values are needed. According to the Eq. (\[tevap\]), for $M \gtrsim 10^{9}$ g the PBH evaporate after the timescale of one second and the BBN constrains $\beta<10^{-22}$, see Fig. \[FigbetaRD\]. However the BBN $\beta$ upper bounds cannot be satisfied for $\kappa \ll \kappa_\text{max}$. Hence, for $M_\text{rem} \ll M$, we conclude that only the remnants with mass in the window $$\begin{aligned}
10^{-24}\, \text{g} \, < M_\text{rem} \ll 10^{8} \text{g}
\end{aligned}$$ can have a sufficient abundance to explain the observed dark matter in the universe. The upper bound is determined by the BBN constraints on the parent PBH mass. It might be satisfied for $M_\text{rem}$ about one order of magnitude less than $10^8$ grams; its exact value depends on how the Eq. (\[HawkT\]) and (\[tevap\]) are modified and the equation of state $w$ after inflation. In the following we derive the expressions for the relic abundance of the PBH remnants for a general expansion rate and $M_\text{rem}$ parameter, and examine separately the cases of radiation, matter and kination domination eras.
The early universe cosmology of the PBH remnants
================================================
The cosmology of the PBH remnants originating from large primordial inhomogeneities has been studied in detail in Ref. [@Barrow; @Carr:1994ar] where the basic expressions have been derived. PBH remnants might also originate form micro black holes produced from high energy collisions in the early universe [@Nakama:2018lwy]. In the following we will consider the formation of PBHs due to large inhomogeneities and generalize the key expressions for arbitrary barotropic parameter $w$, introducing also the PBH - stiff fluid (kination) scenario.
Let us suppose that at the early moment $t_\text{form}$ a fraction $\beta $ of the energy density of the universe collapses and forms primordial black holes. The mass density of the PBHs is $\rho_\text{PBH}\simeq \gamma \beta \rho_\text{tot}$ at the moment of formation, where $\rho_\text{tot}=3H^2M^2_\text{Pl}$ with $M_\text{Pl}=m_\text{Pl}/\sqrt{8\pi}$, the reduced Planck mass. The formation probability is usually rather small, $\beta\ll 1$, and the background energy density $\rho_\text{bck}=(1-\gamma\beta)\rho_\text{tot}$ is approximately equal to $\rho_\text{tot}$.
The PBHs are pressureless non-relativistic matter and their number density $n_\text{PBH}$ scales like $a^{-3}$. The background energy density scales as $\rho \propto a^{-3(1+w)}$ where $a(t)\propto t^{\frac{2}{3(1+w)}}$ and $w$ equation of state of the background fluid. The perturbations evolve inside the curvature scale $1/H$, that has mass $M_H=(3/4) (1+w) m^2_\text{Pl} t$, called the Hubble scale mass. In the approximation of instantaneous evaporation, the moment right before evaporation, that we label $t^<_\text{evap}$, the energy density of the PBHs over the background energy density is $$\begin{aligned}
\frac{\rho_\text{PBH}(t^<_\text{evap})}{\rho_\text{bck}(t^<_\text{evap})}= \gamma \, \beta\, \gamma^\frac{2w}{1+w}\,\left( \frac{M_H(t^<_\text{evap})}{M} \right)^\frac{2w}{1+w} \tilde{g}(g_*, t_\text{evap})\,,\end{aligned}$$ where $\tilde{g}(g_*, t_\text{evap})$ is equal to one unless the universe is radiation dominated; in that case it is $\tilde{g}(g_*, t_\text{evap})\equiv (g_*(t_\text{form})/g_*(t_\text{evap}))^{-1/4}$ where $g_*$, the thermalized degrees of freedom, that we took equal to the entropic degrees of freedom, $g_s$. Substituting the Hubble mass at the evaporation moment of a PBH with mass $M$, $M_H(t_\text{evap})=3 M^3(1+w)/4m^2_\text{Pl}$, a threshold $\beta(M)$ value is found. For $$\begin{aligned}
\label{betaMax}
\beta < \tilde{h}^{-1}(\gamma, w, t_\text{evap}) \left(\frac{m_\text{Pl}}{M} \right)^\frac{4w}{1+w}\,,\end{aligned}$$ where $\tilde{h}(\gamma, w, t_\text{evap})\equiv \gamma^\frac{1+3w}{1+w}(\frac{3}{4}(1+w))^\frac{2w}{1+w} \tilde{g}(g_*, t_\text{evap})$, the universe has never become PBH dominated.
The moment right after the evaporation, that we label $t^>_\text{evap}$, the energy density of the PBHs has decreased $(\kappa m_\text{Pl}/ M)^{-1}$ times. This factor is much larger than one thus nearly the entire energy density of the initial PBHs turns into radiation apart from a tiny amount, reserved by the PBH remnants. The present density of the PBH remnants depends on the equation of state of the universe after the PBH evaporation. If we assume that a radiation domination phase follows the PBH evaporation the fractional abundance of the PBH remnants over the total DM abundance today is $$\begin{aligned}
\label{f-reldom}
f_\text{rem}(M)=\tilde{c} (\gamma, w, t_\text{eq})\left(\frac{M_\text{eq}}{M_H(t_\text{evap})} \right)^{1/2} \frac{\kappa\, m_\text{Pl}}{M}\, \left( \frac{M}{m_\text{Pl}}\right)^\frac{4w}{1+w}\end{aligned}$$ where $\tilde{c}(\gamma, w, t_\text{eq})=2^{1/4} \tilde{h}(\gamma, w, t_\text{eq}) {\Omega_\text{m}}/{\Omega_\text{DM}}$. However, the assumption that there is a radiation domination phase after the PBH evaporation holds either when the universe has become PBH dominated at the moment $t^<_\text{evap}$ or when the equation of state of the background fluid is $w=1/3$. Otherwise, one has to replace the $M_H(t_\text{evap})$ with the $M_\text{rh}$, that is the Hubble radius mass at the completion of reheating, and include a $w$-dependent factor to account for the different expansion rate. Next, particular cases will be examined.
If the universe has become PBH dominated at the moment $t^<_\text{evap}$ and we ask for $f_\text{rem}(M)=1$ we get the mass $$\begin{aligned}
\label{Minter}
M_\text{inter}= \tilde{\alpha}^{2/5}(w) \left( \kappa \, m^2_\text{Pl} \, M^{1/2}_\text{eq} \right)^{2/5}\,,
\end{aligned}$$ where $\tilde{\alpha}(w)=2^{1/4} (\Omega_\text{m}/\Omega_\text{DM}) (\sqrt{3}(1+w))^{-1}$. This is the intersection mass of Eq. (\[betaMax\]), and the $f_\text{rem}(M)=1$ line, given by Eq. (\[f-reldom\]). We see that the $M_\text{inter}$ slightly depends on the $w$. This means that there is a single PBH mass that the early universe becomes PBH dominated and the evaporation remnants account for the total DM, for any positive value of the equation of state. Plugging in values, $M_\text{eq}=6.9\times 10^{50}$g, $g_*(T_\text{eq})=3.36$ we obtain that $M_\text{inter} \simeq 2 \, \kappa^{2/5} 10^6$ g. The intersection mass is the maximum $M$ value in the Fig. \[betaw1\] and \[betawk\]. For masses $M\geq M_\text{inter}$ the upper bound on $\beta$ is practically removed. Turning to the $\beta$, the $\beta_\text{inter}$ value that yields $f_\text{rem}=1$ and momentarily PBH domination phase is $w$-dependent, $$\begin{aligned}
\beta_\text{inter}(w)=\tilde{h}^{-1}(\gamma, w)\,\tilde{\alpha}^{\frac{-2w}{5+5w}} \left(\frac{m_\text{Pl}}{M_\text{eq}} \right)^\frac{4w}{5+5w}\,.\end{aligned}$$ Larger values for $w$ require smaller $\beta$, hence minimal $\beta$ values are achieved for $w=1$, see Fig. \[betaw1\], \[betawk\].
For masses $M>M_\text{inter}$ the relic abundance of the PBH remnants is always smaller than the total dark matter abundance even if PBHs dominate the early universe. Hence, for $M\geq M_\text{inter}$ there is no constraint on the $\beta$ from the $f_\text{rem}(M)$. This can be understood as follows. Let us assume that $M=M_\text{inter}$ and $\beta=\beta_\text{inter}$ such that $\Omega_\text{rem}=\Omega_\text{DM}$. This means that right before evaporation the PBH number density is $n_\text{PBH}(t^<_\text{evap})\simeq \rho_\text{tot}/M_\text{inter}$. If it had been $\beta>\beta_\text{inter}$ the PBH-domination phase would have started at times $t<t_\text{evap}$, but the number density of the PBH relics at the moment $t_\text{evap}$ would have been the same. Hence, the value of the $f_\text{rem}(M_\text{inter})$ does not increase for $\beta>\beta_\text{inter}$. Also, for $M>M_\text{inter}$ the number density of the PBHs is always smaller than $\rho_\text{tot}/M_\text{inter}$ at the moment $t^<_\text{evap}\sim G^2 M^3_\text{max}$ even if $\beta(M)\sim 1$. Since the PBHs will evaporate into PBH remnants with the universal mass $\kappa m_\text{Pl}$, the conclusion to be drawn is that the relic energy density parameter of PBHs remnants with mass $M>M_\text{inter}$ is always less than $\Omega_\text{DM}$. Summing up, any constraint on the $\beta(M)$ for $M_\text{inter}<M<10^{17}$g comes only from the Hawking radiation of the PBHs, not from the abundance of the PBH remnants.
Let us now turn to the $M=M(k)$ relation assuming a one-to-one correspondence between the scale $k^{-1}$ and the mass $M$. Following the Press-Schechter formalism [@Press:1973iz], there is a probability $\beta$ an overdensity with wavenumber $k$ to collapse when it enters inside the Hubble radius (or some time later if the universe is matter dominated). The mass $M$ of the PBH is related to the wavenumber $k=aH$ as $$\begin{aligned}
\label{ka1}
\frac{M}{M_\text{rh}} =\gamma \frac{H^{-1}}{H_\text{rh}^{-1}}
=\gamma \left( \frac{k}{k_\text{rh}} \right)^{\frac{-3(1+w)}{3w+1}}\,,\end{aligned}$$ where we utilized the relation between the wavenumber and the scale factor, $$\begin{aligned}
\label{ka}
\frac{k}{k_\text{rh}}=\left(\frac{a}{a_\text{rh}} \right)^{-\frac12(3w+1)}\,.\end{aligned}$$ The horizon mass at the completion of reheating, $M_\text{rh}=4\pi \left({\pi^2 g_*}/{90}\right)^{-1/2} {M^3_\text{Pl}}/{T^2_\text{rh}}$, reads $$\begin{aligned}
M_\text{rh} \simeq 10^{12} \, \text{g}\, \left( \frac{T_\text{rh}}{10^{10}\, \text{GeV}}\right)^{-2} \left(\frac{g_*}{106.75} \right)^{-1/2}\,.\end{aligned}$$ If PBHs form during radiation domination era it is $M/\gamma>M_\text{rh}$ whereas if they form before the completion of the thermalization of the universe it is $M/\gamma<M_\text{rh}$. We will return to the relation between the PBH mass and the wavenumber $k$ in the Section 4, where we will explicitly write the $M=M(k, T_\text{rh}, w)$ formula in order to connect the PBH mass with the ${\cal P_R}(k)$ peak.
Let us note the formation of PBHs with mass $M$ is possible only if the horizon mass right after inflation is smaller than $M/\gamma$, see Eq. (\[Hcon\]). Equivalently, a PBH with mass $M$ will form due to superhorizon perturbations only if the corresponding wavelength $k^{-1}$ is larger than the Hubble scale at the end of inflation. Thus, a different way to express the condition (\[Hcon\]) is $k_\text{end}>k$.
Next, we examine separately the interesting cosmological scenarios with barotropic parameter $w=0$, $w=1/3$ and $w=1$.
PBH production during radiation domination
------------------------------------------
Let us assume that the bulk energy density is in the form of radiation. Thus, it is $\rho_\text{PBH}\simeq \gamma_\text{} \beta \rho_\text{rad}$ after the approximation $\rho_\text{rad}=(1-\beta)\rho_\text{tot} \simeq \rho_\text{tot}$, that is legitimate for $\beta \ll 1$. The PBH mass is $M=\gamma_\text{} M_H$ where $M_H=m^2_\text{Pl}/(2H) \simeq m^2_\text{Pl} t$ is the Hubble radius mass during radiation domination (RD).
Assuming a RD phase until the moment of the evaporation and making the approximation of instantaneous evaporation, the energy density of the PBHs at the moment right before evaporation is $\rho_\text{PBH}(t^<_\text{evap})=\gamma^{3/2}_\text{}\beta M\,m^{-1}_\text{Pl} \rho_\text{rad}(t^<_\text{evap})$. Thus, the assumption of a radiation dominated phase is valid for $\gamma^{3/2}_\text{} \beta M \,m^{-1}_\text{Pl} <1$. In the opposite case the universe becomes PBH dominated before the moment of evaporation. At the moment right after the PBH evaporation the energy density of the PBH relics is $\kappa \gamma^{3/2}_\text{} \beta$ times the energy density of the radiation background. For an RD phase until the epoch of matter-radiation equality, $t_\text{eq}$, it is $\rho_\text{rem}(t_\text{eq}) \simeq (\kappa m_\text{Pl}/M)\gamma \beta \rho_\text{rad} T_\text{evap}/T_\text{eq}$ and the fractional abundance of the PBH remnants is found, $$\begin{aligned}
\label{fRD}
f_\text{rem}(M) =\tilde{c}_\text{R}\,\gamma^{3/2} \,\frac{\kappa\, m_\text{Pl}}{M} \beta \, \left(\frac{M_\text{eq}}{M_\text{}}\right)^{1/2} \end{aligned}$$ where $\tilde{c}_\text{R}=2^{1/4} \left({g(T_f)}/{g(T_\text{eq})} \right)^{-1/4}{\Omega_\text{m}}/{\Omega_\text{DM}}$. The $T_\text{evap}$ and $T_\text{eq}$ are the cosmic temperatures at the moment of evaporation and the epoch of matter-radiation equality. The effectively massless degrees of freedom for the energy and entropy densities were taken to be equal. The maximum value $f_\text{rem}=1$, gives the maximum value for the $\beta_\text{max}(M)$, see Fig. \[betaw1\] and \[betawk\]. The Eq. (\[fRD\]) rewrites after inserting benchmark values, $$\label{fRD2}
f_\text{rem} (M)\, \simeq\,\kappa \left(\frac{\beta_\text{}}{10^{-12}}\right) \,
\Big(\frac{\gamma_\text{}}{0.2}\Big)^{\frac{3}{2}}
\left(\frac{M}{10^{5}\text{g}}\right)^{-3/2}\,$$ where we omitted the factor $0.95 \left({g(T_{k})}/{106.75}\right)^{-\frac{1}{4}} $ from the r.h.s. and took $\Omega_{\text{DM}}h^2=0.12$.
Assuming Gaussian statistics, the black hole formation probability for a spherically symmetric region is $$\label{brad}
\beta_\text{}(M)=\int_{\delta_c}d\delta\frac{1}{\sqrt{2\pi\sigma^2(M)}} e^{-\frac{\delta^2}{2\sigma^2(M)}}\,$$ that is approximately equal to $\beta \sqrt{2\pi} \simeq \sigma(M)/\delta_c e^{-\frac{\delta^2_c}{2\sigma^2(M)}} $. The PBH abundance has an exponential sensitivity to the variance of the perturbations $\sigma(k)$ and to the threshold value $\delta_c$. In the comoving gauge Ref. [@Harada:2013epa] finds that the $\delta_c$ has the following dependence on the $w$, $$\label{dc}
\delta_c = \frac{3(1+w)}{5+3w}\sin^2 \frac{\pi \, \sqrt{w}}{1+3w}\,.$$ For $w=1/3$ it is $\delta_c=0.41$. The variance of the density perturbations in a window of $k$ is given by the relation $\sigma^2 \sim {\cal P}_{\delta}$ where the ${\cal P}_{\delta}$ is related to the power spectrum of the comoving curvature perturbation as $$\begin{aligned}
{\cal P_R}=\left( \frac{5+3w}{2(1+w)}\right)^2 {\cal P}_{\delta}\,,\end{aligned}$$ hence, it is $\sigma^2 \sim (4/9)^2 {\cal P_R}$. From the approximation of the Eq. (\[brad\]) we get that ${\cal P_R} \sim (9/4)^2 (\delta^2_c/2) \ln(1/(\sqrt{2 \pi} \beta))^{-1}$. Benchmark values $\beta=10^{-12}$, $\delta_c=0.41$, $\kappa=1$ yield the required value for the power spectrum ${\cal P_R} \sim 1.6\times 10^{-2}$. Increasing the value of $\kappa$ by one orders of magnitude or more gives only a slight decrease in the required value of the ${\cal P_R}$.
Finally, let us note that PBHs are expected to form with mass $M=\gamma_\text{} M_H$ when the cosmic temperature is $$\begin{aligned}
\label{TM}
T(M) \simeq 10^{11} \, \text{GeV}\, \gamma^{1/2}_\text{}\, \left(\frac{M}{10^{10}\,\text{g}}\right)^{-1/2}
\left( \frac{g_*}{106.75}\right)^{-1/4}\,\,.
\end{aligned}$$ For example, formation of PBHs with mass $M\sim 10^5$ g requires reheating temperatures $T_\text{rh}>10^{13}$ GeV. If the reheating temperature is lower than $T(M)$ then PBHs with mass $M$ form during a non-thermal phase.
A particular example that yields $f_\text{rem}=1$ is described in the Section 5 and the $\beta(M)$ is depicted in Fig. [\[FigbetaRD\]]{}.
PBH production during matter domination
---------------------------------------
For PBH formation during matter domination era (MD) the expression (\[fRD\]) has to be multiplied with $(t_\text{form}/t_\text{rh})^{1/2}$ to account for the absence of a relative redshift of the $\rho_\text{PBH}$ with respect to the background energy density. It is, $$\begin{aligned}
\label{fMD}
f_\text{rem}(M, M_\text{rh}) =\tilde{c}_\text{M}\,\frac{\kappa\, m_\text{Pl}}{M} \gamma_\text{} \, \beta \, \left(\frac{M_\text{eq}}{M_\text{rh}}\right)^{1/2} \,,
\end{aligned}$$ where $\tilde{c}_\text{M}=2^{1/4} \left({g(T_\text{rh})}/{g(T_\text{eq})} \right)^{-1/4} \Omega_\text{m}/\Omega_\text{DM}$ and for $M<M_\text{rh}$. The Hubble scale mass at the completion of reheating reads $M_\text{rh}=M_H (T_\text{rh}, g_*)=4\pi \left( {\pi^2 g_*}/{90}\right)^{-1/2} {M^3_\text{Pl}}/{T^2_\text{rh}} $. The Eq. (\[fMD\]) rewrites after normalizing the $M$, $M_\text{rh}$ with benchmark values, $$\begin{aligned}
f_\text{rem}(M, M_\text{rh}) \nonumber
\simeq 3 \, \kappa \,& \gamma_\text{} \, \left(\frac{\beta}{10^{-9}}\right) \,
\left(\frac{M_\text{rh}}{10^{10}\text{g}} \right)^{-1/2} \\
&\times \left( \frac{M}{10^5 \text{g}} \right)^{-1 } \left( \frac{g_*}{106.75} \right)^{-1/4}\end{aligned}$$ and the mass $M$ is related to the scale $k^{-1}$ of the inhomogeneity as $$\begin{aligned}
\label{kmd1}
k_\text{}=k_\text{end}\, \left( \frac{4\pi M^2_\text{Pl}}{H_\text{end}}\right)^{1/3} \left(\frac{M}{\gamma_\text{}}\right)^{-1/3}\,, \quad \text{for} \,\quad
k_\text{}>k_\text{rh}\,.\end{aligned}$$ In a pressureless background overdensities can easier grow and collapse if the MD era is sufficiently long. Contrary to the RD case non-sphericity and spin effects suppress the formation probability. Ref. [@Harada:2016mhb] examined the PBH production in MD era and found that for not very small $\sigma$ the PBH production rate tends to be proportional to $\sigma^5$, $$\label{bmat}
\beta_\text{}(M)\, = \, 0.056 \, \sigma^5\,,$$ If the collapsing region has angular momentum the formation rate is further suppressed and reads [@Harada:2017fjm], $$\label{bspin}
\beta_\text{}(M)=2\times 10^{-6}f_q(q_c) {\cal I}^6 \sigma^2 e^{-0.147\frac{ \, {\cal I}^{4/3}}{ \sigma^{2/3}}}\,.$$ Benchmark values are $q_c=\sqrt{2}$, ${\cal I}=1$, $f_q \sim 1$. According to [@Harada:2017fjm] the expression (\[bspin\]) applies for $\sigma \lesssim 0.005$ whereas Eq. (\[bmat\]) applies for $0.005 \lesssim \sigma \lesssim 0.2$.
During MD era, an additional critical parameter is the duration of the gravitational collapse. Ref. [@Harada:2017fjm] concluded that the finite duration of the PBH formation can be neglected if the reheating time $t_\text{rh}$ satisfies $t_\text{rh} >\left( \frac{2}{5} {\cal I} \, \sigma\right)^{-1} t_k$ where $t_k$ is the time of the horizon entry of the scale $k^{-1}$ (it does not coincide with the formation time $t_\text{form}$). In terms of temperatures this condition rewrites $T_\text{rh}< \left( \frac{2}{5} {\cal I} \, \sigma\right)^{1/2} T_k$ where $T_k$ the temperature that the scale $k^{-1}$ would enter the horizon during RD. Let us define the temperature $$\begin{aligned}
T^{\text{MD}}_\text{form}= \left( \frac{2}{5} {\cal I} \, \sigma\right)^{1/2} T_k\,.\end{aligned}$$ If the reheating temperature is smaller than $T^{\text{MD}}_\text{form}$ then PBHs form during MD era. Unless this conditions is fulfilled the time duration for the overdensity to grow and enter the nonlinear regime is not adequate. Hence, the formation rates (\[bmat\]), (\[bspin\]) apply only for the scales $k$ that experience a variance of the comoving density contrast at horizon entry that is larger than $$\begin{aligned}
\label{sigmaCr}
\sigma > \sigma_\text{cr} \equiv \frac52 {\cal I}^{-1} \left( \frac{k_\text{rh}}{k}\right)^3\,.
\end{aligned}$$ In terms of temperature this translates into $\sigma>5/2 \, {\cal I}^{-1} (T_\text{rh}/T)^2$. If $\sigma<\sigma_\text{cr}$ one should consider the radiation era formation rate. A particular example that yields $f_\text{rem}=1$ is described in the Section 5 and the $\beta(M)$ is depicted in Fig. \[FigbetaMspin\]. The $\sigma$ for that example is less than $0.005$ and larger than $\sigma_\text{cr}$, thus the overdensity collapses during the matter era with the spin effects being crucial.
PBH production during stiff fluid domination
--------------------------------------------
Let us assume that the bulk energy density is in the form of stiff fluid (SD era), that is a fluid with barotropic parameter $w=1$, also called kination phase. A non-oscillatory inflaton can give rise to a kination phase. It is $\rho_\text{PBH}\simeq \gamma_\text{} \beta \rho_\text{S}$ where we approximated $\rho_\text{S}=(1-\beta)\rho_\text{tot} \simeq \rho_\text{tot}$ for $\beta \ll 1$. The PBH mass is $M=\gamma M_H$ where $M_H=(3/2)M^3/m^2_\text{Pl}$ is the Hubble scale mass for stiff fluid domination.
Assuming that the SD era lasts at least until the moment of the evaporation and making the approximation of instantaneous evaporation, the energy density of the PBHs at the moment right before evaporation is $$\begin{aligned}
\label{KinEvap}
\frac{ \rho_\text{PBH}(t^<_\text{evap})}{\rho_\text{S}(t^<_\text{evap})}=\frac{3}{2}\gamma^{2}_\text{} \beta \frac{M^2}{ m^{2}_\text{Pl}} \,.\end{aligned}$$ The assumption of a kination phase is valid roughly for $\gamma^{2}_\text{} \beta M^2 \,m^{-2}_\text{Pl} <1$, otherwise the universe becomes PBH dominated before the moment of evaporation.
Let us assume that $\gamma^{2}_\text{} \beta M^2 \,m^{-2}_\text{Pl} <1$. At the moment right after the PBH evaporation the energy density of the PBH remnants is $\rho_\text{rem}(t^>_\text{evap}) =(3/2)\kappa \gamma^{2}_\text{} \beta (M/m_\text{Pl}) \rho_\text{tot}$. The background energy density is now partitioned between the stiff fluid, $\rho_\text{S}$, and the entropy produced by the PBH evaporation, $\rho_\text{rad}$. The later is about $M/(\kappa m_\text{Pl})$ times larger than the $\rho_\text{rem}(t^>_\text{evap})$. Assuming that the evaporation products thermalize fast, the radiation redshifts like $\rho_\text{rad} \propto g_* g_s^{-4/3} a^{-4}$ whereas the stiff fluid background redshifts like $\rho_\text{S} \propto a^{-6}$. At some moment the radiation dominates the background energy density and we define it as the [*reheating*]{} moment $t_\text{rh}$. The scale factor is $$\begin{aligned}
\frac{a(t_\text{rh})}{a(t_\text{evap})} =\left( \frac{2}{3}\frac{m^2_\text{Pl}}{M^2} \frac{1}{\gamma^2_\text{} \beta}\, \frac{g_*^{1/3}(t_\text{rh})}{g_*^{1/3}(t_\text{evap})}
\right)^{1/2}\,.\end{aligned}$$ At that moment we also define the reheating temperature of the universe that reads $$\begin{aligned}
\label{Tkination}
T_\text{rh} \equiv 6.3\, \text{MeV} \left( \frac{\beta}{10^{-28}} \right)^{3/4} \gamma^{3/2}_\text{} g_*^{-1/2} \,.\end{aligned}$$ Until the moment $t_\text{rh}$ the energy density of the PBH remnants increases relatively to the stiff fluid dominated background as $\rho_\text{rem}/\rho_\text{S}\propto a^3$ and afterwards, that radiation dominates, it increases as $\rho_\text{rem}/\rho_\text{rad}\propto T^{-1}$. It is $$\begin{aligned}
\label{frelS1}
f_\text{rem}(M)= \tilde{c}_\text{S} \frac32 \, \gamma^{2}_\text{} \, \beta \, \frac{\kappa \,M}{m_\text{Pl}}\left(\frac{a(t_\text{rh})}{a(t_\text{evap})} \right)^3 \left(\frac{M_\text{eq}}{M_\text{rh}}\right)^{1/2}\end{aligned}$$ where $\tilde{c}_\text{S}=2^{1/4} \left({g(T_\text{rh})}/{g(T_\text{eq})} \right)^{-1/4} \Omega_\text{m}/\Omega_\text{DM}$ and $M_\text{rh}=M_H(t_\text{rh})$. For times $t<t_\text{rh}$ the Hubble radius mass increases like $M_H \propto a^3$ and, given that $M_H(t_\text{evap})=(3/2)M^3/m^2_\text{Pl}$, we find the $M_\text{rh}$ mass $$\begin{aligned}
M_\text{rh}=\sqrt{\frac{2}{3}}\, \gamma^{-3}_\text{} \, \beta^{-3/2} \, g^{1/2}_* \, m_\text{Pl}\,.\end{aligned}$$ Therefore, the Eq. (\[frelS1\]) rewrites $$\begin{aligned}
\label{frelS2}
f_\text{rem}(M)= \sqrt{\frac23}\,\tilde{c}_\text{} \left(\frac32 \gamma^{2}_\text{} \, \beta\right)^{1/4} \kappa \left(\frac{m_\text{Pl}}{M} \right)^{3/2}
\left(\frac{M_\text{eq}}{M}\right)^{1/2}\end{aligned}$$ and normalizing with benchmark values we attain $$\begin{aligned}
\label{fremKIN}
f_\text{rem}(M)
\simeq 4 \, \kappa \, \sqrt{\gamma_\text{}} \, \left(\frac{\beta}{10^{-32}}\right)^{1/4} \,
\left( \frac{M}{10^5 \text{g}} \right)^{-2 }.\end{aligned}$$ For $\kappa \sim 1$ and $M\sim 10^5$g, $\beta$ values as small as $10^{-32}$ can explain the observed dark matter in the universe. Pressure is maximal and we expect the overdense regions to be spherically symmetric. Utilizing the relation (\[dc\]) the PBH formation occurs when the density perturbation becomes larger than $\delta_c=0.375$ and for the formation probability $\beta(M)$ given by the Eq. (\[brad\]) we find that power spectrum values ${\cal P_R} \lesssim3.5 \times 10^{-3}$, for $\kappa \gtrsim 1$ and $M\sim 10^{5}$g, can yield $f_\text{rem}=1$.
A particular example that yields $f_\text{rem}=1$ is described in the Section 5 and the $\beta(M)$ is depicted in Fig. \[betaw2\].
### BBN constraints {#bbn-constraints .unnumbered}
A kination regime has to comply with the BBN constraints. Let us assume that a runaway inflaton $\varphi$ is responsible for the kination regime. The energy density during BBN is partitioned between the kinetic energy of the $\varphi$ field and the background radiation. Any modification to the simple radiation domination regime is parameterized by an equivalent number of additional neutrinos and the Hubble parameter has to satisfy the constraint [@Simha:2008zj] $$\begin{aligned}
\left. \left( \frac{H}{H_\text{rad}}\right)^2 \right|_{T=T_\text{BBN}} \leq 1+\frac{7}{43}\Delta N_{\nu_\text{eff}}\simeq 1.038\end{aligned}$$ where $H$ the actual Hubble parameter and $H_\text{rad}$ the Hubble parameter if the total energy density was equal to the radiation. The $\Delta N_{\nu_\text{eff}}=3.28-3.046$ is the difference between the cosmologically measured value and the SM prediction for the effective number of neutrinos. In order to prevent the universe from expanding too fast during BBN due the extra energy density $\dot{\varphi}^2/2$ the reheating temperature has to be larger than [@Artymowski:2017pua] $$\begin{aligned}
\nonumber
T_\text{rh}>&(\alpha-1)^{-1/2} \left( \frac{g_*(T_\text{BBN})}{T_\text{rh}}\right)^{1/4} T_\text{BBN} \\
&= {\cal O}(10)\, \text{MeV}\end{aligned}$$ where $\alpha \equiv 1+7/43 \Delta N_{\nu_\text{eff}} \simeq 1.038$.
An additional issue is that the gravitational wave energy in the GHz region gets enhanced during the kination regime [@Giovannini:1999bh; @Riazuelo:2000fc; @Yahiro:2001uh; @Boyle:2007zx]. The energy density of the gravitational waves does not alter BBN predictions if $$\begin{aligned}
I \equiv h^2 \int^{k_\text{end}}_{k_\text{BBN}} \Omega_\text{GW}(k)d \ln k \leq 10^{-5}\end{aligned}$$ which is written as [@Dimopoulos:2017zvq] $$\begin{aligned}
I=\frac{2\epsilon h^2 \Omega_\text{rad}(t_0)}{\pi^{2/3}}\left(\frac{30}{g(T_\text{reh})} \right)^{1/3}\frac{h^2_\text{GW} V^{1/3}_\text{end}}{T^{4/3}_\text{rh}}\end{aligned}$$ where $\epsilon\sim 81/(16\pi^3)$, $ h^2 \Omega_\text{rad}(t_0)=2.6 \times 10^{-5}$ and $h^2_\text{GW}=H^2_\text{end}/(8\pi M^2_\text{Pl})$. Substituting numbers, the observational constraint $I \lesssim 10^{-5}$ gives a lower bound on the reheating temperature, $$\begin{aligned}
\label{TGW}
T_\text{rh} \gtrsim 10^{6}\, \text{GeV} \left( \frac{106.75}{g_*}\right)^{1/4}
\left(\frac{H^2_\text{end}}{10^{-6}M_\text{Pl}} \right)^2\,.\end{aligned}$$ Substituting the reheating temperature predicted by kination-PBH models, Eq. (\[Tkination\]), into the bound (\[TGW\]) we obtain a lower bound on the formation rate $$\begin{aligned}
\beta \gtrsim 2\times 10^{-17} \left( \frac{106.75}{g_*}\right)^{1/3} \left(\frac{H_\text{end}}{10^{-6} M_\text{Pl}} \right)^{8/3}\,.\end{aligned}$$ Smaller values for the $\beta$ mean that the radiation produced from the PBH evaporation dominates later during the early cosmic evolution and the kination regime is dangerously extended. Asking for $f_\text{rem}=1$ the lower bound on the $\beta$ yields a lower bound on the ratio $$\begin{aligned}
\frac{M}{\sqrt{\kappa}} \, \gtrsim 1. 6\times 10^7 \text{g} \,\gamma^{1/4} \left( \frac{ H_\text{end}}{10^{-6}\,M_\text{Pl}}\right)^{1/3} \left( \frac{g_*}{106.75}\right)^{1/12}\,.\end{aligned}$$ We recall that the mass $M$ has to satisfy the upper bound given by the Eq. (\[Minter\]), $M<M_\text{inter} \simeq 2 \kappa^{2/5} 10^6$g, otherwise it is always $f_\text{rem}<1$. This bound gives a maximum value for the $\kappa$, $$\begin{aligned}
\frac{\kappa}{10^{-10}} \lesssim 8.5 \, \gamma^{-5/2} \left( \frac{ H_\text{end}}{10^{-6}\,M_\text{Pl}}\right)^{-10/3} \left( \frac{g_*}{106.75}\right)^{-5/6}\,.\end{aligned}$$ Unless $H_\text{end} \ll 10^{-6}M_\text{Pl}$ it must be $\kappa <1$, hence for high scale inflation the PBH remnants must have subplanckian masses. For $\kappa=\kappa_\text{max}$ a maximum value for the mass of the PBHs, $M=M_\text{inter}$, is obtained for the kination regime in order the remnants to saturate the $\Omega_\text{DM}$. PBH remnants with $\kappa \geq 1$ require $H_\text{end} \lesssim 2 \times 10^{-9} \gamma_\text{}^{-3/4} M_\text{Pl}$, that can be achieved either in small field inflation model or by models where the CMB and PBHs potential energy scales have a large difference, so that the high frequency GWs have a smaller amplitude. We underline that the above results are valid only if the post-inflationary equation of state of the inflaton field satisfies $w_\text{}\simeq 1$, at least until the BBN epoch. If it is $w_\text{}< 1$ the derived bounds get relaxed.
Building a ${\cal P_R}(k)$ peak in accordance with observations
===============================================================
The position of the ${\cal P_R}(k)$ peak
----------------------------------------
The wavenumber that inflation ends is $$\begin{aligned}
k_\text{end} = k_* \frac{H_\text{end}}{H_*} e^{N_*}\end{aligned}$$ where $N_*$ are the e-folds of the observable inflation and given by the expression $$\label{Nrh}
N_* \simeq 57.6 +\frac14 \ln \epsilon_* +\frac14 \ln \frac{V_*}{\rho_\text{end}} -\frac{1-3w_\text{}}{4} \tilde{N}_\text{rh} \,.$$ The $\epsilon_*$, $H_*$, $V_*$ are respectively the first slow-roll parameter, the Hubble scale and the potential energy when the CMB pivot scale exits the Hubble radius, while $H_\text{end}$, $\rho_\text{end}$ are the Hubble scale and the energy density at the end of inflation.
The $N_*$ value is related to the postinflationary reheating efolds $\tilde{N}_\text{rh}$ and the corresponding (averaged) equation of state ${w}_\text{}$. We have implicitly assumed that $w$ refers to the postinflationary equation of state until the moment reheating completes. The number of efolds until the completion of the reheating $\tilde{N}_\text{rh}$ are $$\begin{aligned}
\tilde{N}_\text{rh}&(T_\text{rh},H_\text{end}, g_*, {w}_\text{})= \nonumber
\\
& -\frac{4}{3(1+ {w}_\text{})} \ln\left[\left(\frac{\pi^2 g_*}{90} \right)^{1/4} \frac{T_\text{rh}}{(H_\text{end}M_\text{Pl})^{1/2}} \right]\,.\end{aligned}$$ An inhomogeneity of size $k^{-1}$ crosses inside the horizon during radiation domination if $k<k_\text{rh}$ where $$\begin{aligned}
k_\text{rh}=k_\text{end}e^{-\frac{3w+1}{2}\tilde{N}_\text{rh}}\,.\end{aligned}$$ For a general expansion rate determined by the effective equation of state value ${w}_\text{rh}$ the $k(M)$ relation reads, $$\begin{aligned}
\label{kMI}
k_\text{} (M, w_\text{})= k_\text{end}\, \left(\frac{M/\gamma}{M_\text{end}}\right)^{-\frac{3w_\text{}+1}{3(1+w_\text{})}}\,.\end{aligned}$$ The $M_\text{end}$, $k_\text{end}$ depend on the details of the inflationary model with the later becoming larger for larger values of the $w_\text{}$. The $k(M, w_\text{})$ can be written using the reheating completion moment as the reference period replacing respectively the $k_\text{end}, M_\text{end}$ with the $k_\text{rh}, M_\text{rh}$ in Eq. (\[kMI\]). Then we attain the more general $k(M, T_\text{rh}, w_\text{})$ relation $$\begin{aligned}
\label{kMgen}
k (M, T_\text{rh}, & w_\text{}) \simeq 2 \times 10^{17} \text{Mpc}^{-1} \left( \frac{T_\text{rh}}{10^{10}\text{GeV}} \right)^{\frac{1-3w}{3(1+w)}} \nonumber \\
& \left(\frac{M/\gamma}{10^{12} \text{g}} \right)^{-\frac{3w+1}{3(1+w)}} \left(\frac{g_*}{106.75}\right)^{\frac{1}{4}\frac{1-3w}{3(1+w)}}\end{aligned}$$ For the case of kination domination the minor correction, $T_\text{rh} \rightarrow 2^{1/4} T_\text{rh}$ should be added due to the equipartition of the energy density between the radiation and the scalar field.
Assuming a one-to-one correspondence between the scale of perturbation and the mass of PBHs, an inflationary model builder who aims at generating PBHs with mass $M$ has to produce a ${\cal P_R}(k)$ peak at the wavenumber $k (M, T_\text{rh}, w_\text{})$. Next we briefly discuss the additional observational constraints, regarding the width of the peak, that one has to take into account in order the inflationary model to be viable.
Observational constraints on the ${\cal P_R}(k)$ at small scales
----------------------------------------------------------------
A power spectrum peak at large wavenumbers, $k
\gg k_*$, is welcome for not spoiling the $n_s$ and $\alpha_s$ values measured at $k_*$. Also, such a peak can generate PBHs abundant enough to comprise the entire dark matter in the universe, either as long lived PBHs or as PBHs remnants. However, shifting the peak at large wavenumbers does not render the ${\cal P_R}(k)$ free from constraints. The impact of the Hawking radiation on the BBN and CMB observables and the extragalactic $\gamma$-ray background put strong upper bounds on the ${\cal P_R}(k)$ at large $k$-bands. These bounds rule out a great part of the PBH mass spectrum with range $10^9 \text{g}<M< 10^{17}$g from explaining the $\Omega_\text{DM}$ with PBH remnants (PBHs remnants from holes with mass $M\sim 10^{10}$ g or larger could explain the $\Omega_\text{DM}$ if $\kappa \gg 1$.) Moreover, even if the power spectrum peak produces PBHs with masses $M\sim 10^{5}$g, $M\sim 10^{18}$g or $M\sim 10^{22}$g where the PBH abundance can be maximal, the width of the peak has to be particularly narrow. The stringent constraint comes from the CMB, at the mass scale $M\sim 10^{13}$g, where the electrons and positron produced by PBHs evaporation after the time of recombination scatter off the CMB photons and heat the surrounding matter damping small-scale CMB anisotropies contrary to observations. The next stringent constraint applies at the mass range $M=10^{10}-10^{13}$ g that evaporate affects the BBN relics via hadrodissociation and photodissociation processes [@MacGibbon:1990zk; @MacGibbon:1991tj; @Kohri:1999ex; @Carr:2009].
In Ref. [@Dalianis:2018ymb] the observational constraints have been explicitly translated into ${\cal P_R}(k)$ bounds. In a radiation dominated early universe, utilizing the Eq. (\[kMgen\]) the CMB constraint for $M_\text{cmb} \equiv 2.5\times 10^{13}$g and $w_\text{rh} =1/3$ yields the bound $$\begin{aligned}
\label{kspacCMB}
\sigma\,(3 \times 10^{17}\, k_*) \, \lesssim \, 0.035 \,
\left(\frac{\delta_c}{0.41}\right) \,.\end{aligned}$$ The $\sigma$ is the variance of the comoving density contrast, $\sigma^2 \sim (4/9)^2 {\cal P_R}$ and the bound ${\cal P_R}(4 \times 10^{17}\, k_*)\lesssim {\cal O}(10^{-3})$ is derived.
Turning to a matter dominated early universe reheated at temperatures $T_\text{rh} \lesssim 10^7\,\text{GeV}$ the variance of the density perturbations has to satisfy the CMB bound, $$\begin{aligned}
\label{spinCMB}
& \sigma (k(M_\text{cmb}, T_\text{rh})) \, \lesssim \,
\text{Exp}\left[ \right.
-6.9- 0.09\, \ln \frac{T_\text{rh}}{\text{GeV}} \\
&+ 2\times10^{-3}\,\left(\ln\frac{T_\text{rh}}{\text{GeV}}\right)^2
\left.
- 3\times10^{-5}\,
\left( \ln \frac{T_\text{rh}}{\text{GeV}}\right)^3
\right]
\nonumber\end{aligned}$$ where $k(M_\text{cmb}, T_\text{rh})\simeq 3\times 10^{17}\,k_*\, \gamma_\text{}^{1/3}\left( T_\text{rh}/{10^{7}\, \text{GeV}}\right)^{1/3}$ $(g_*/106.75)^{1/12} $, according to the Eq. (\[kMgen\]) for $w_\text{rh} =0 $. During matter domination era it is $\sigma \sim (2/5){\cal P_R}^{1/2}$ and for $\gamma_\text{} =0.1$, $T_\text{rh}=10^7$ GeV the constraint on the power spectrum reads ${\cal P_R}( 1.4 \times 10^{17}\,k_*)\lesssim {\cal O}(9\times 10^{-7})$. If the reheating temperature is $10^7 \text{GeV }\lesssim T_\text{rh} \lesssim 4\times10^8\,\text{GeV}$ then the BBN constraint on the power spectrum applies. For $M_\text{bbn} \equiv 5\times 10^{10}$ and $T_\text{rh}=10^8$ GeV the constraint reads ${\cal P_R}(10^{18}\,k_*)\lesssim {\cal O}(4\times 10^{-6})$, that is a bit weaker than the CMB. For larger reheating temperatures, $T_\text{rh} \gtrsim 10^9\,\text{GeV}$ the constraints get significantly relaxed.
For the case of kination domination the CMB constraint applies on the scale with wavenumber $k^{}(M_\text{cmb}, T_\text{rh}) $ and reads, $$\begin{aligned}
\label{KB}
\sigma\,(k(M_\text{cmb}, T_\text{rh}) \, \lesssim \, 0.032 \,
\left(\frac{\delta_c}{0.375}\right) \,.\end{aligned}$$ It is $k(M_\text{cmb}, T_\text{rh}) \simeq 5 \times 10^{18}\, k_* \, \gamma_\text{}^{2/3}\left( T_\text{rh}/{10^{7}\, \text{GeV}}\right)^{-1/3}$ $(g_*/106.75)^{-1/12}$, according to the Eq. (\[kMgen\]) for $w_\text{rh}=1$ and $M_\text{cmb} =2.5\times 10^{13}$g. Note here that the CMB bound (\[KB\]) applies for reheating temperatures $T_\text{rh} \lesssim 2\times 10^{9}$ GeV since the gravitational collapse can be considered instantaneous, contrary to the case of the matter era, [@Dalianis:2018ymb].
For smaller PBH masses that evaporate in less than a second there are limits on the amount of the thermal radiation from the PBHs evaporation due to the production of entropy, that may be in conflict with the cosmological photon-to-baryon ratio, [@Zel'dovich3]. There are also constraints from the abundance of dark matter produced by the evaporation, e.g. the lightest supersymmetric particle (LSP). In our models the abundance of the PBH remnants saturates the dark matter density parameter $\Omega_\text{DM}h^2=0.12$, thus the LSP constraint does not apply. Finally, for ultra small PBH masses, the constraint comes only from the relic abundance of the Planck-mass remnants [@MacGibbon:1987my; @Barrow; @Carr:1994ar]. These constraints are labeled [*entropy, LSP*]{} (with dotted-dashed line due to fact that in our models this constraint is raised) and [*Planck*]{} respectively in our figures.
PBHs from the $\alpha$-attractors inflation models \[PSP\]
===========================================================
\[potential\] The inflaton potential and the computation of the ${\cal P_R}(k)$
-------------------------------------------------------------------------------
The above constraints imply that, even in large wavenumbers, the power spectrum peak has to be positioned in a particular range of $k$ and, additionally, be sufficiently narrow. In this Section our goal is to generate PBHs that will evaporate fast enough in the early universe without affecting the BBN and CMB observables and, at the same time, leave behind mass remnants that will saturate the dark matter abundance. In order to implement this scenario we employ the machinery of $\alpha$-attractors and build inflationary models with inflection point at large $k$.
If the inflaton potential features an inflection point a large amplification in the power spectrum ${\cal P_R}(k)$ can be achieved due to the acceleration and deceleration of the inflaton field in the region around the inflection point as was pointed out in [@Garcia-Bellido:2017mdw]. The presence of an inflection point requires $V'\approx0$ and $V''=0$. In the context of supergravity such a model may arise from $\alpha$–attractors, by choosing appropriate values for the parameters in the superpotential as described in Ref. [@Dalianis].
We focus on the effective Lagrangian for the inflaton field $\varphi$ in the $\alpha$–attractors scenario that turns out to be $$\label{la11}
e^{-1}{\cal L}= \frac{1}{2}R-\frac{1}{2}
\Big(\partial_\mu \varphi\Big)^2-f^2\Big(\tanh\frac{\varphi}{\sqrt{6\alpha}}\Big)\,,$$ where Re$\Phi=\phi=\sqrt{3}\tanh({\varphi}/{\sqrt{6\alpha}})$ is a chiral superfield.
Polynomial and trigonometric forms for the function $f(\phi)$ can feature an inflection point plateau sufficient to generate a significant dark matter abundance in accordance with the observational constraints [@Dalianis]. Nevertheless, other forms for the function $f(\phi)$ are plausible. Exponential potentials enjoy a theoretical motivation in several BSM frameworks and their cosmology has been extensively studied, see e.g [@Copeland:1997et; @Kolda:2001ex; @Kehagias:2004bd; @Russo:2004ym; @Dalianis:2014nwa; @Geng:2017mic; @Basilakos:2019dof]. In the following we will examine the PBH formation scenario from $\alpha$-attractor inflationary potentials built by exponential functions.
The form of the potential fully determines the subsequent adiabatic evolution of the universe. Firstly, the number of efolds $N_*$, that follow the moment the $k_*^{-1}$ scale exits the quasi-de Sitter horizon, determine the duration of the non-thermal stage after inflation. Secondly, the position and the features of the inflection point plateau determine the mass and the abundance of the PBHs that form and, in particular, the moment the overdensities reenter the horizon. If PBHs of a given mass $M$ form during radiation era a specific inflationary potential has to be designed. On the contrary, PBHs production during matter era requires a different potential. Furthermore, an inflationary potential might be a runaway without minimum at all. Such a potential is acceptable if it can realize an inflationary exit and a sufficient reheating of the universe. Remarkably, both of these conditions can be satisfied in our modes with the generated mini PBHs to guarantee a successful reheating via their evaporation. Last, but not least, at large scales $k\sim k_*$ we demand the CMB observables, as they are specified by the Planck 2018 data, to remain intact.
The PBH abundance is found only after the computation of the value of the comoving curvature perturbation ${\cal R}_k$. In the comoving gauge we have $\delta \varphi=0$ and $g_{ij}=a^2 \left[(1-2{\cal R}) \delta_{ij} +h_{ij}\right]$, Expanding the inflaton-gravity action to second order in ${\cal R}$ one obtains $$S_{(2)}= \frac12 \int {\rm d}^4x \sqrt{-g} a^3 \frac{\dot{\varphi}^2}{H^2} \left[\dot{{\cal R}}^2 -\frac{(\partial_i {\cal R})^2}{a^2}\right]\,.$$ After the variable redefinition $v=z{\cal R}$ where $z^2=a^2\dot{\phi}^2/H^2=2a^2\epsilon_1$ and switching to conformal time $\tau$ (defined by $d\tau=dt/a$), the action is recast into $$S_{(2)} =\frac12 \int {\rm d}\tau {\rm d}^3x \left[(v')^2 -(\partial_i v)^2 +\frac{z''}{z}v^2 \right]\,.$$ The evolution of the Fourier modes $v_k$ of $v(x)$ are described by the Mukhanov-Sasaki equation $$\label{MS}
v''_k +\left(k^2-\frac{z''}{z}\right) v_k =0 ,$$ where $z''/z$ is expressed in terms of the functions $$\label{eek}
\epsilon_1 \equiv -\frac{\dot{H}}{H^2}, \quad
\epsilon_2 \equiv \frac{\dot{\epsilon}_1}{H\epsilon_1}, \quad
\epsilon_3 \equiv \frac{{\dot{\epsilon}}_2}{H\epsilon_2},$$ as $$\frac{z''}{z} =(aH)^2 \left[2-\epsilon_1 +\frac32 \epsilon_2 - \frac12\epsilon_1 \epsilon_2 +\frac14 \epsilon^2_2+\frac{1}{2} \epsilon_2 \epsilon_3 \right].$$ We are interested in the the super-Hubble evolution of the curvature perturbation, that is for $k^2 \ll z''/z$. The power of ${\cal R}_k$ on a given scale is obtained once the solution $v_k$ of the Mukhanov-Sasaki equation is known and estimated at a time well after it exits the horizon and its value freezes out, $$\left. {\cal P_ R} =\frac{k^3}{2\pi^2}\frac{|v_k|^2}{z^2} \right|_{k\ll aH } \,. \label{ppp}$$ After the numerical computation of the Mukhanov-Sasaki equation the ${\cal P_ R}$ at all the scales is obtained. As it is required, the ${\cal P_R}(k)$ of our models satisfy the constraints qiven by Eq. (\[kspacCMB\]), (\[spinCMB\]) and (\[KB\]) for radiation, matter and kination eras respectively. From the ${\cal P_ R}(k)$ we compute the $f_\text{rem}$ as described in Section 3. We note that we neglected possible impacts on the power spectrum from non-Gaussianities [@Franciolini:2018vbk; @Atal:2018neu; @DeLuca:2019qsy] and quantum diffusion effects [@Pattison:2017mbe; @Biagetti:2018pjj; @Ezquiaga:2018gbw; @Cruces:2018cvq].
Inflaton potential for PBHs production during radiation and matter era
----------------------------------------------------------------------
### Radiation era
A function $f(\phi)$ built by exponentials can feature a proper inflection point plateau. The form of the potential is chosen to produce PBHs of the right abundance. We ask for large reheating temperatures so that the large inhomogeneities reenter the horizon after the thermalization of the universe. This is achieved by sufficiently strong couplings of the inflaton field to the visible sector. We also demand values for the $n_s$ and $\alpha_s$ that are favorable by Planck 2018 data [@Akrami:2018odb]. An example of a combination of exponentials that can fulfill the above requirements is of the form, $$\begin{aligned}
\label{fphi}
f(\phi/ \sqrt{3}) =f_0\, ( c_0 + c_1 e^{\lambda_1 \phi/\sqrt{3}} + c_2 e^{\lambda_2 \phi^2/3})\end{aligned}$$ that generates the potential $$\begin{aligned}
\label{VRM}
V(\varphi)= f^2_0\left( c_0 + c_1 e^{\lambda_1 \tanh\varphi/\sqrt{6}} + c_2 e^{\lambda_2 (\tanh \varphi/\sqrt{6})^2} \right)^2\end{aligned}$$ having taken $\alpha=1$. The determination of the parameters values requires a subtle numerical process that we outline. Firstly, a central PBH mass $M$ has to be chosen and from Eq. (\[fRD2\]) the $\beta$ value that saturates the $f_\text{PBH}$ is specified. The $M$ is the parameter that spots the $k$-position of the ${\cal P_R}(k)$ peak. The $\beta$ is exponentially sensitive to the amplitude of the peak and its exact value is found after a delicate selection of the potential parameters. The ${\cal P_R}(k)$ is produced by solving numerically the Mukhanov-Sasaki equation, following the method described in Ref. [@Dalianis]. At the same time consistency with the CMB normalization, the measured $n_s$ and $\alpha_s$ values, as well as large enough $N_*$ values are required. For $\phi \rightarrow \sqrt{3}$ the potential drives the early universe cosmic inflation and the CMB normalization gives the first constraint for the parameters. We also demand zero potential energy at the minimum of the potential that gives a second constraint. We also note that the $f_0$ is a redundant parameter since it can be absorbed by the $c_0$, $c_1$ and $c_2$. We keep only for numeric convenience. An example of parameter values that realize the PBH production during RD era is listed in the Table III and the potential is depicted in the left panel of the Figure [\[Vplots\]]{}.
### Matter era
An early universe matter domination era can be realized if the shape of the inflationary potential around the minimum is approximated with a quadratic potential. For moderately suppressed inflaton couplings the inflaton decays after a large number of oscillations. The inhomogeneities that reenter the horizon during the stage of the inflaton oscillations might collapse in a pressureless environment.
The calculation of the PBH production during matter era involves the same numerical steps with the case of radiation plus some extra conditions that have to be taken into account. Firstly, the PBH mass $M$ value is not adequate to specify the $k$-position and the amplitude of the ${\cal P_R}(k)$ peak, since there is a crucial dependence on the reheating temperature. Hence, after choosing the PBH mass $M$, the required $\beta$ is fixed for a particular reheating temperature. In turn, the $T_\text{rh}$ fixes the number of efolds that constrain the inflaton excursion in the field space. Moreover, the amplitude of the peak has an additional dependence on the reheating temperature, namely the variance of the perturbations has to satisfy the bound (\[sigmaCr\]), $\sigma>\sigma_\text{cr}(T_\text{rh})$, in order the inhomogeneities to fully collapse during the matter domination era.
An inflationary example that predicts PBH formation during matter era is given by Eq. (\[fphi\]) after proper parameter values are chosen. The set of the parameters, listed in Table III, yields an amplitude for the ${\cal P_R}(k)$ that spin effects have to be considered in the estimation of the formation probability.
\[tab2\]
$\textbf{\textit{era}}$ $\boldsymbol{\beta}$ $\boldsymbol{T_\text{rh}}$ (GeV) $\boldsymbol{\tilde{N}_\text{rh}}$ $\boldsymbol{M_\text{peak} (\text{g})}$ $\boldsymbol{f_\text{rem}}$
------------------------- ----------------------- ---------------------------------- ------------------------------------ ----------------------------------------- -----------------------------
**RD** 1.25$\times 10^{-13}$ $2.13\times 10^{15}$ 0 3.9 $\times 10^{4}$ $\sim 1$
$\textbf{MD}$ 6.88$\times 10^{-16}$ $5.6\times 10^{12}$ 7.9 61 $\sim 1$
**KIN** 1.07$\times 10^{-14}$ $1.8\times 10^{6}$ 13.57 $2 \times 10^{2}$ $\sim 1$
\[potpar1\]
------------------------- --------------------------------------- --------------------- --------------------
$\textbf{\textit{era}}$ $\boldsymbol{{\cal P_R}^\text{peak}}$ $k_{\text{end}}$ $\boldsymbol{N_*}$
\[0.5ex\] **RD** 4$\times 10^{-2}$ 3.45$\times10^{22}$ 55.5
$\textbf{MD}$ 6.6 $\times 10^{-5}$ 5.06$\times10^{21}$ 53.6
**KIN** 2.7$\times10^{-2}$ 2.$\times10^{25}$ 63.
------------------------- --------------------------------------- --------------------- --------------------
\[potpar2\]
------------------------- ------------------------- --------------------- --------------------- -------------------------- -------------------------- --
$\textbf{\textit{era}}$ $ \boldsymbol{c_0}$ $ \boldsymbol{c_1}$ $ \boldsymbol{c_2}$ $\boldsymbol{\lambda_1}$ $\boldsymbol{\lambda_2}$
\[0.5ex\] **RD** -1.856 1.173 -0.14 -0.987 95.5904
**MD** -1.856 1.173 -0.13 -0.987 124.555
**KIN** $-8.70\times 10^{-27} $ 0.1045 $-4\times 10^{25}$ 62.2 -4430.973
------------------------- ------------------------- --------------------- --------------------- -------------------------- -------------------------- --
Inflaton potential for PBHs production during kination era
----------------------------------------------------------
A period of kination domination has an interesting and distinct cosmology. It is possible to be realized after inflation if the potential does not have a vacuum, see [@Dimopoulos:2017zvq; @Dimopoulos:2017tud] for $\alpha$-attractor kination models. A non-oscillatory inflaton field will runaway without decaying resulting in a period where the kinetic energy dominates over the potential energy. The attractive feature of such models is that the inflaton can survive until today and might play the role of quintessence. Moreover, such models are attractive because they lead to a different early universe phenomenology since the effective equation of state is $w_\text{}\sim 1$ and the expansion rate is reduced. This is the so-called stiff fluid or kination era that gives rise to different prediction regarding some early universe observables such as the spectrum of the tensor perturbations, a fact that renders such an era testable.
The kination scenarios usually suffer from radiation shortage since the inflaton field does not decay and special mechanisms have to be introduced. A source of radiation comes from the Hawking temperature of de Sitter space, called gravitational reheating, but this is very inefficient [@Ford:1986sy; @Chun:2009yu]. On the other hand, the Hawking radiation from mini PBHs formed by a runaway inflaton automatically reheat the universe. So, in our models radiation is produced by the evaporation of the PBHs that can be efficient enough. According to Eq. (\[Tkination\]), common values for the $\beta$ imply large enough reheating temperatures.
The construction of kination inflation models that induce the PBH production is very challenging. Firstly, the inflaton runs away until it freezes at some value $\varphi_F$ and this residual potential energy of the inflaton must not to spoil the early and late time cosmology. The inflaton potential energy at $\varphi_F$ has to be tuned to values $V(\varphi_F) \lesssim 10^{-120} M^4_\text{Pl}$, similarly to all the quintessence models. Secondly, the kination inflaton model parameters are self-constrained. A particular PBH mass $M$ specifies the $k$ of the ${\cal P_R}(k)$ only if the reheating temperature is known. However, the reheating temperature is not a free parameter, as e.g in matter or radiation cases where the $T_\text{rh}$ depends on the inflaton decay rate. In the kination scenario the $T_\text{rh}$ depends on the $\beta$. The $\beta$ is found by the condition $f_\text{rem}=1$ and this fixes the reheating temperature. Hence, the characteristics of the peak in the power spectrum determines
- the mass of the evaporating PBHs,
- the dark matter abundance, and
- the reheating temperature of the universe.
In addition, the tail of the potential might lead to the observed late time acceleration of the universe. Undoubtedly, this scenario is remarkably economic.
To be explicit, let us introduce the model $$\begin{aligned}
\label{runf}
f(\phi/\sqrt{3})=f_0\,\left(c_0+ c_1 e^{\lambda_1 \phi/\sqrt{3}} + c_2 e^{\lambda_2 (\phi-\phi_\text{P})^2/3} \right) \end{aligned}$$ that generates the potential $$\begin{aligned}
\label{Vrun} \nonumber
V(\varphi)= f^2_0 \, \big[ c_0 + & c_1 e^{\lambda_1 \tanh \varphi/\sqrt{6}} \, +
\\
& c_2 e^{\lambda_2 \left(\tanh(\varphi/\sqrt{6})-\tanh(\varphi_\text{P} /\sqrt{6})\right)} \big]^2\,.\end{aligned}$$ The $\varphi_\text{P}$ is a fixed value in the field space that determines the position of the inflection point. Again here the $f_0$ can be absorbed in $c_0$, $c_1$ and $c_2$. For $\phi \rightarrow \sqrt{3}$ the early universe cosmic inflation takes place and the CMB normalization gives the first constraint for the parameters. For $\phi \rightarrow -\sqrt{3}$ we demand zero potential energy, thus we get the second constraint $$\begin{aligned}
\label{c0}
c_0=-c_1 e^{-\lambda_1} - c_2 e^{\lambda_2 (\sqrt{3}+\phi_\text{P}))^2/3}\,.\end{aligned}$$ The kination stage lasts until the moment that the radiation produced by the PBH evaporation dominates the energy density. Later the field freezes at some value $\phi_F$ and defreezes at the present universe. The runaway potential is flat enough to lead to the currently observed accelerated expansion, hence implement a wCDM cosmology as a quintessence model.
Let us pursue some approximate analytic expressions that describe the post-inflationary evolution of the field $\varphi$. After inflation the $\varphi$ rolls fast the potential and a stage of kination commends, where $\dot{\varphi}^2/2 \gg V(\varphi)$. The Klein-Gordon equation for the $\varphi$ for negligible potential energy is $\ddot{\varphi}+3H\dot{\varphi}\simeq 0$. During kination it is $a\propto t^{1/3}$ and for $t\gg t_\text{end}$ the field value evolves as $$\begin{aligned}
\varphi-\varphi_\text{end} \simeq -\sqrt{\frac{2}{3}} M_\text{Pl} \ln\left( \frac{t}{t_\text{end}}\right)\,\end{aligned}$$ where we considered negative initial velocity for the $\varphi$. At the moment $t_\text{form}$ the PBHs form and later at $t_\text{evap}$ they evaporate. Later, at the moment $t_\text{rh}$ the universe becomes radiation dominated and the kination regime ends. Until reheating it is $a \propto t^{1/3}$ and one finds that $t_\text{rh}=(\Omega_\text{rad}(t_\text{evap}))^{-3/2} t_\text{evap}$, where $\Omega_\text{rad}(t_\text{evap})=(3/2)\gamma^2_\text{} \beta M^2/m^2_\text{Pl}$, given by Eq. (\[KinEvap\]). At the moment of reheating the field value, $\varphi_\text{rh}$, is $$\begin{aligned}
\varphi_\text{rh}\simeq \varphi_\text{end}-\sqrt{\frac{2}{3}} \left(-\frac{3}{2} \ln \Omega_\text{rad}(t_\text{evap}) + \ln\left(\frac{t_\text{evap}}{t_\text{end}}\right)\right)M_\text{Pl}\end{aligned}$$ After reheating it is $a \propto t^{1/2}$ and the field evolution slows down, $$\begin{aligned}
\varphi-\varphi_\text{rh} \simeq -\frac{2}{\sqrt{3}} M_\text{Pl} \left(1-\sqrt{ \frac{t_\text{rh}}{t}}\right)\,.\end{aligned}$$ For $t \gg t_\text{rh}$ the field gets displaced $2 M_\text{Pl}/\sqrt{3}$ from $\varphi_\text{rh}$ and thus, at some late moment $t_F$ the field freezes at the value $\varphi_F$, $$\begin{aligned}
\varphi_{F}\simeq \varphi_\text{end}-\sqrt{\frac{2}{3}} \left(\sqrt{2}-\frac{3}{2} \ln \Omega_\text{rad}(t_\text{evap}) + \ln\left(\frac{t_\text{evap}}{t_\text{end}}\right)\right)M_\text{Pl}\end{aligned}$$
Asking for $f_\text{rem}=1$ we find from Eq. (\[fremKIN\]) that $ \Omega_\text{rad}(t_\text{evap})=3\times 10^{-13} (M/10^5 \text{g})^{10} (4\kappa)^{-4}$. Also it is $t_\text{evap} \sim 4\times 10^2(M/10^{10} \text{g})^3$ s and $t_\text{end} \simeq t_\text{Pl}(m_\text{Pl}/H_\text{end})$. Therefore we obtain an expression for the $\varphi_F$ that depends only on the initial mass of the PBH $M$ and the mass of the PBH remnant $\kappa m_\text{Pl}$, $$\begin{aligned}
\label{fF}
\varphi_{F}\simeq \varphi_\text{end} - \sqrt{\frac{2}{3}}
\left[19 +13 \ln(M/10^5 \text{g})+4 \ln(1/\kappa) \right]M_\text{Pl}\,.\end{aligned}$$ This is a general approximate expression for any runaway potential that predicts PBH remnants as dark matter. It is general because we have omitted the potential $V(\varphi)$ both from the Friedman and the Klein-Gordon equations as negligible. The $\varphi_F$ value depends only on the mass $M$ and the parameter $\kappa$. For $\kappa=1$ and $M=10^5$g it is $\varphi_F -\varphi_\text{end} \sim -15M_\text{Pl}$. For $\kappa=10^{-10}$ and $M=10^2$g it is $\varphi_F -\varphi_\text{end} \sim -17M_\text{Pl}$. We note that the exact value of the $\varphi_F$ is found after the numerical solution of the Klein-Gordon and Friedman equations and the $|\varphi_F -\varphi_\text{end}| $ is a bit less than the value of the Eq. (\[fF\]) for we neglected the potential $\varphi$ and considered instant transitions between the kination and radiation regime.
If we want to identify the dark energy as the energy density of the scalar field $\varphi$ then we have to tune the potential energy value at $\varphi_F$. For our model (\[Vrun\]) we impose the condition, $$\begin{aligned}
\label{third}
\frac{\rho_\text{inf}}{\rho_0} \simeq \frac{V(\varphi\gg 1)}{V(\varphi_F)}\sim \frac{e^{2\lambda_1}}{e^{-2\lambda_1}} \sim 10^{108}\,,\end{aligned}$$ dictated by the hierarchy of energy scales between the $\alpha$-attractors inflation and the dark energy. This condition gives a third constraint to the parameters of the potential, together with the CMB normalization and the requirement for zero vacuum energy as $\varphi \rightarrow -\infty$, Eq. (\[c0\]). The Eq. (\[third\]) gives a rough relation for the size of the exponent parameter $\lambda_1$, $$\begin{aligned}
4\lambda_1 \sim 108\ln(10)\,.\end{aligned}$$
In the Table III we list a set of parameters that the kination model (\[Vrun\]) generates PBHs which after evaporation leave behind remnants with $f_\text{rem}=1$ and acts as quintessence.
CMB observables
----------------
The $n_s$ and $r$ values in the standard $\alpha$-attractors are expressed as the analytic relations, $n_s\sim1-2/N_*$ and $r\sim12\alpha/N_*^2$. These expressions still apply in $\alpha$-attractor models that feature an inflection point, with the essential difference that the $N_*$ is replaced by the number of efolds $\Delta N$ that separate the moments of horizon exit of the CMB scale $k^{-1}_*$ and the PBH scale $k^{-1}_\text{}$. Thus it is $n_s\sim1-2/\Delta N$ and $r\sim12\alpha/\Delta N^2$. In our models we get $\Delta N \gtrsim 50$ hence the spectral index value is predicted to be $$\begin{aligned}
n_s \gtrsim 0.96\end{aligned}$$ and the tensor-to-scalar ratio $$\begin{aligned}
r< 0.048\end{aligned}$$ placing the prediction of our models in the 68% CL region of the Planck 2018 data [@Akrami:2018odb] without assuming running of the running for the $n_s$. Generally, the $n_s$ value becomes larger than 0.96 if the PBHs have mass less than about $10^5$ grams.
\[concsection\] Conclusions
===========================
In this work we investigated the cosmology of mini primordial black holes. The very motivation of examining this scenario is the theoretical postulation that a stable or long lived remnant is left behind after the evaporation of the “black” holes. The mass of the remnant is expected to depend on the unknown physics that operates at the Planck energy scale. Therefore we examined the cosmology of PBH remnants with arbitrary mass $M_\text{rem}=\kappa \, m_\text{Pl}$ and $\kappa$ a free parameter that might be orders of magnitude larger or smaller than one. The PBH remnants can comprise the entire dark matter of the universe if the mass of the parent PBH is roughly $M \lesssim \, \kappa^{2/5} 10^6$ g. We computed the general relic abundance of the PBHs remnants and found the conditions that they comprise the entire cold dark matter in the universe. We found that the PBH remnants have a significant cosmological abundance only if they have mass $M_\text{rem}> 1$ GeV. Also the mass of the remnants must have mass $M_\text{rem} \ll 10^8$ grams; otherwise the parent PBH affects the BBN or the CMB observables.
Mini PBHs imply that the comoving curvature perturbation is enhanced at the extreme end of the ${\cal P_R}(k)$. This is a rather attractive feature since the required large primordial inhomogeneities can be produced by the inflationary phase without spoiling the spectral index value $n_s$. The PBHs form in the very early universe after the inflationary phase, hence the primordial inhomogeneities are expected to collapse during a non thermal phase unless the inflaton field decays very fast.
In this work we built inflationary models in the framework of $\alpha$-attractors. We produced a peak in power spectrum by constructing an inflection point and computed the numerically the ${\cal P_R}(k)$ by solving the Mukhanov-Sasaki equation. Our models yield a spectral index value $n_s>0.96$, that places them in the $68\%$ CL contour region of Planck 2018 data. The building blocks of the inflationary potentials are exponential functions. We examined the PBHs production for three different inflationary scenarios. In the first, the inflaton field decays nearly instantaneously after inflation reheating the universe at very large temperatures. In this scenario the mini PBHs are produced and evaporate during the radiation phase. In the second scenario the inflaton field decays a bit later, after oscillating several times about the minimum of its potential resulting in a post-inflationary stage of pressureless matter domination. During matter domination the primordial inhomogeneities collapse into PBHs. After the inflaton decay the universe is reheated and the mini PBHs evaporate.
In the third scenario the PBH are produced during a kination regime. This is a novel scenario, hence we examined it in more detail. A kination regime takes place if the inflaton potential has no minimum and the inflaton runs away after the end of inflation. The radiation is produced by the PBH evaporation that gradually dominates the energy density and reheats the universe. The resulting reheating temperature can be larger than $10^6$ GeV terminating fast enough the kination era in accordance with the BBN constraints. The PBHs remnants can account for the entire dark matter of the universe. Interestingly enough, the non-decaying inflaton can additionally act as quintessence field giving rise to the observed late time accelerated expansion implementing a wCDM cosmological model. Actually this model is remarkably economic in terms of ingredients.
Nowadays, that the existence of black holes and dark matter are unambiguous, the investigation of the PBH dark matter scenario is very motivated. Here we examined the less studied mini PBH scenario and derived general expression complementing older results and put forward new and testable cosmological scenarios for the early and late universe.
Acknowledgments {#acknowledgments .unnumbered}
===============
The work of I.D. is supported by the IKY Scholarship Programs for Strengthening Post Doctoral Research, co-financed by the European Social Fund ESF and the Greek government.
[99]{}
B. J. Carr, “The Primordial black hole mass spectrum,” Astrophys. J. [**201**]{}, 1 (1975).\
B. J. Carr and S. W. Hawking, “Black holes in the early Universe,” Mon. Not. Roy. Astron. Soc. [**168**]{}, 399 (1974).\
P. Meszaros, “The behaviour of point masses in an expanding cosmological substratum,” Astron. Astrophys. [**37**]{}, 225 (1974).\
M. Ricotti, J. P. Ostriker and K. J. Mack, “Effect of Primordial Black Holes on the Cosmic Microwave Background and Cosmological Parameter Estimates,” Astrophys. J. [**680**]{}, 829 (2008) \[arXiv:0709.0524 \[astro-ph\]\].\
M. Sasaki, T. Suyama, T. Tanaka and S. Yokoyama, “Primordial black holes—perspectives in gravitational wave astronomy,” Class. Quant. Grav. [**35**]{} (2018) no.6, 063001 \[arXiv[1801.05235]{}\[[astro-ph.CO]{}\]\].\
R. Murgia, G. Scelfo, M. Viel and A. Raccanelli, “Lyman-$\alpha$ forest constraints on Primordial Black Holes as Dark Matter,” \[arXiv:1903.10509 \[astro-ph.CO\]\].\
H. Niikura [*et al.*]{}, “Microlensing constraints on primordial black holes with the Subaru/HSC Andromeda observation,” \[arXiv:1701.02151 \[astro-ph.CO\]\].\
A. Barnacka, J. F. Glicenstein and R. Moderski, “New constraints on primordial black holes abundance from femtolensing of gamma-ray bursts,” Phys. Rev. D [**86**]{}, 043001 (2012) doi:10.1103/PhysRevD.86.043001 \[arXiv:1204.2056 \[astro-ph.CO\]\].\
P. Tisserand [*et al.*]{} \[EROS-2 Collaboration\], “Limits on the Macho Content of the Galactic Halo from the EROS-2 Survey of the Magellanic Clouds,” Astron. Astrophys. [**469**]{}, 387 (2007) \[astro-ph/0607207\].\
T. D. Brandt, “Constraints on MACHO Dark Matter from Compact Stellar Systems in Ultra-Faint Dwarf Galaxies,” Astrophys. J. [**824**]{}, no. 2, L31 (2016) \[arXiv:1605.03665 \[astro-ph.GA\]\].\
P. W. Graham, S. Rajendran and J. Varela, “Dark Matter Triggers of Supernovae,” Phys. Rev. D [**92**]{}, no. 6, 063007 (2015) \[arXiv:1505.04444 \[hep-ph\]\].\
D. Gaggero, G. Bertone, F. Calore, R. M. T. Connors, M. Lovell, S. Markoff and E. Storm, “Searching for Primordial Black Holes in the radio and X-ray sky,” Phys. Rev. Lett. [**118**]{}, no. 24, 241101 (2017) \[arXiv:1612.00457 \[astro-ph.HE\]\].\
F. Capela, M. Pshirkov and P. Tinyakov, “Constraints on Primordial Black Holes as Dark Matter Candidates from Star Formation,” Phys. Rev. D [**87**]{}, no. 2, 023507 (2013) \[arXiv:1209.6021 \[astro-ph.CO\]\].\
F. Capela, M. Pshirkov and P. Tinyakov, “Constraints on primordial black holes as dark matter candidates from capture by neutron stars,” Phys. Rev. D [**87**]{}, no. 12, 123524 (2013) \[arXiv:1301.4984 \[astro-ph.CO\]\].\
B. P. Abbott [*et al.*]{} \[LIGO Scientific and Virgo Collaborations\], “Binary Black Hole Mergers in the first Advanced LIGO Observing Run,” Phys. Rev. X [**6**]{}, no. 4, 041015 (2016) Erratum: \[Phys. Rev. X [**8**]{}, no. 3, 039903 (2018)\] \[arXiv:1606.04856 \[gr-qc\]\].\
B. P. Abbott [*et al.*]{} \[LIGO Scientific and Virgo Collaborations\], “Observation of Gravitational Waves from a Binary Black Hole Merger,” Phys. Rev. Lett. [**116**]{}, no. 6, 061102 (2016) \[arXiv:1602.03837 \[gr-qc\]\].\
B. P. Abbott [*et al.*]{} \[LIGO Scientific and Virgo Collaborations\], “GW151226: Observation of Gravitational Waves from a 22-Solar-Mass Binary Black Hole Coalescence,” Phys. Rev. Lett. [**116**]{}, no. 24, 241103 (2016) \[arXiv:1606.04855 \[gr-qc\]\].\
B. J. Carr, K. Kohri, Y. Sendouda and J. Yokoyama, “Constraints on primordial black holes from the Galactic gamma-ray background,” Phys. Rev. D [**94**]{}, no. 4, 044029 (2016) \[arXiv:1604.05349 \[astro-ph.CO\]\].\
B. J. Carr, K. Kohri, Y. Sendouda and J. Yokoyama, “New cosmological constraints on primordial black holes,” Phys. Rev. D [**81**]{}, 104019 (2010) \[arXiv:0912.5297 \[astro-ph.CO\]\].\
M. J. Bowick, S. B. Giddings, J. A. Harvey, G. T. Horowitz and A. Strominger, “Axionic Black Holes and a Bohm-Aharonov Effect for Strings,” Phys. Rev. Lett. [**61**]{}, 2823 (1988).\
S. R. Coleman, J. Preskill and F. Wilczek, “Quantum hair on black holes,” Nucl. Phys. B [**378**]{}, 175 (1992) \[hep-th/9201059\].\
P. Chen, Y. C. Ong and D. h. Yeom, “Black Hole Remnants and the Information Loss Paradox,” Phys. Rept. [**603**]{} (2015) 1 \[arXiv:1412.8366 \[gr-qc\]\].\
J. D. Barrow, E. J. Copeland and A. R. Liddle, “The Cosmology of black hole relics,” Phys. Rev. D [**46**]{}, 645 (1992).\
B. J. Carr, J. H. Gilbert and J. E. Lidsey, “Black hole relics and inflation: Limits on blue perturbation spectra,” Phys. Rev. D [**50**]{}, 4853 (1994) \[astro-ph/9405027\].\
S. Alexander and P. Meszaros, “Reheating, Dark Matter and Baryon Asymmetry: A Triple Coincidence in Inflationary Models,” \[hep-th/0703070 \[HEP-TH\]\].\
F. Scardigli, C. Gruber and P. Chen, “Black Hole Remnants in the Early Universe,” Phys. Rev. D [**83**]{} (2011) 063507 \[arXiv:1009.0882 \[gr-qc\]\].\
O. Lennon, J. March-Russell, R. Petrossian-Byrne and H. Tillim, “Black Hole Genesis of Dark Matter,” JCAP [**1804**]{} (2018) no.04, 009 \[arXiv:1712.07664 \[hep-ph\]\].\
M. Raidal, S. Solodukhin, V. Vaskonen and H. Veermäe, “Light Primordial Exotic Compact Objects as All Dark Matter,” Phys. Rev. D [**97**]{} (2018) no.12, 123520 \[arXiv:1802.07728 \[astro-ph.CO\]\].\
S. Rasanen and E. Tomberg, “Planck scale black hole dark matter from Higgs inflation,” JCAP [**1901**]{}, no. 01, 038 (2019) \[arXiv:1810.12608 \[astro-ph.CO\]\].\
T. Nakama and Y. Wang, “Do we need fine-tuning to create primordial black holes?,” Phys. Rev. D [**99**]{}, no. 2, 023504 (2019) \[arXiv:1811.01126 \[astro-ph.CO\]\].\
L. Morrison, S. Profumo and Y. Yu, “Melanopogenesis: Dark Matter of (almost) any Mass and Baryonic Matter from the Evaporation of Primordial Black Holes weighing a Ton (or less),” \[arXiv:1812.10606 \[astro-ph.CO\]\].\
Y. Akrami [*et al.*]{} \[Planck Collaboration\], “Planck 2018 results. X. Constraints on inflation,” \[arXiv:1807.06211 \[astro-ph.CO\]\].\
T. Kawaguchi, M. Kawasaki, T. Takayama, M. Yamaguchi and J. Yokoyama, “Formation of intermediate-mass black holes as primordial black holes in the inflationary cosmology with running spectral index,” Mon. Not. Roy. Astron. Soc. [**388**]{} (2008) 1426 \[arXiv[0711.3886]{} \[[astro-ph]{}\]\].\
M. Drees and E. Erfani, “Running-Mass Inflation Model and Primordial Black Holes,” JCAP [**1104**]{} (2011) 005 \[arXiv[1102.2340]{} \[[hep-ph]{}\]\].\
M. Kawasaki, A. Kusenko, Y. Tada and T. T. Yanagida, “Primordial black holes as dark matter in supergravity inflation models,” Phys. Rev. D [**94**]{} (2016) no.8, 083523 \[arXiv[1606.07631]{} \[[astro-ph.CO]{}\]\].\
J. Garcia-Bellido and E. Ruiz Morales, “Primordial black holes from single field models of inflation,” Phys. Dark Univ. [**18**]{}, 47 (2017) \[arXiv:1702.03901 \[astro-ph.CO\]\].\
G. Ballesteros and M. Taoso, “Primordial black hole dark matter from single field inflation,” Phys. Rev. D [**97**]{}, no. 2, 023501 (2018) \[arXiv:1709.05565 \[hep-ph\]\].\
M. P. Hertzberg and M. Yamada, “Primordial Black Holes from Polynomial Potentials in Single Field Inflation,” Phys. Rev. D [**97**]{}, no. 8, 083509 (2018) \[arXiv:1712.09750 \[astro-ph.CO\]\].\
I. Dalianis, A. Kehagias and G. Tringas, “Primordial Black Holes from $\alpha$-attractors,” JCAP [**1901**]{}, 037 (2019) \[arXiv:1805.09483 \[astro-ph.CO\]\].\
T. J. Gao and Z. K. Guo, “Primordial Black Hole Production in Inflationary Models of Supergravity with a Single Chiral Superfield,” Phys. Rev. D [**98**]{}, no. 6, 063526 (2018) \[arXiv:1806.09320 \[hep-ph\]\].\
M. Cicoli, V. A. Diaz and F. G. Pedro, “Primordial Black Holes from String Inflation,” JCAP [**1806**]{}, no. 06, 034 (2018) \[arXiv:1803.02837 \[hep-th\]\].\
G. Ballesteros, J. Beltran Jimenez and M. Pieroni, “Black hole formation from a general quadratic action for inflationary primordial fluctuations,” \[arXiv:1811.03065 \[astro-ph.CO\]\].\
K. Kannike, L. Marzola, M. Raidal and H. Veermäe, “Single Field Double Inflation and Primordial Black Holes,” JCAP [**1709**]{}, no. 09, 020 (2017) \[arXiv:1705.06225 \[astro-ph.CO\]\].\
Y. Gong and Y. Gong, “Primordial black holes and second order gravitational waves from ultra-slow-roll inflation,” JCAP [**1807**]{}, no. 07, 007 (2018) \[arXiv:1707.09578 \[astro-ph.CO\]\].\
S. Pi, Y. l. Zhang, Q. G. Huang and M. Sasaki, “Scalaron from $R^2$-gravity as a Heavy Field,” JCAP [**1805**]{} (2018) no.05, 042 \[arXiv[1712.09896]{} \[[astro-ph.CO]{}\]\].\
Y. F. Cai, X. Tong, D. G. Wang and S. F. Yan, “Primordial Black Holes from Sound Speed Resonance during Inflation,” Phys. Rev. Lett. [**121**]{}, no. 8, 081306 (2018) \[arXiv:1805.03639 \[astro-ph.CO\]\].\
K. Dimopoulos, T. Markkanen, A. Racioppi and V. Vaskonen, “Primordial Black Holes from Thermal Inflation,” \[arXiv:1903.09598 \[astro-ph.CO\]\].\
R. Kallosh, A. Linde and D. Roest, “Superconformal Inflationary $\alpha$-Attractors,” JHEP [**1311**]{}, 198 (2013) \[arXiv:1311.0472 \[hep-th\]\].\
S. W. Hawking, “Particle Creation by Black Holes,” Commun. Math. Phys. [**43**]{} (1975) 199 Erratum: \[Commun. Math. Phys. [**46**]{} (1976) 206\].\
S. W. Hawking, “Black hole explosions,” Nature [**248**]{} (1974) 30.\
R. Torres, F. Fayos and O. Lorente-Espín, “The mechanism why colliders could create quasi-stable black holes,” Int. J. Mod. Phys. D [**22**]{} (2013) no.14, 1350086 \[arXiv:1309.6358 \[gr-qc\]\].\
N. Arkani-Hamed, S. Dimopoulos and G. R. Dvali, “The Hierarchy problem and new dimensions at a millimeter,” Phys. Lett. B [**429**]{} (1998) 263 \[hep-ph/9803315\].\
P. Suranyi, C. Vaz and L. C. R. Wijewardhana, \[arXiv:1006.5072 \[hep-ph\]\].\
R. J. Adler, P. Chen and D. I. Santiago, “The Generalized uncertainty principle and black hole remnants,” Gen. Rel. Grav. [**33**]{} (2001) 2101 \[gr-qc/0106080\].\
B. J. Carr, J. Mureika and P. Nicolini, “Sub-Planckian black holes and the Generalized Uncertainty Principle,” JHEP [**1507**]{} (2015) 052 \[arXiv:1504.07637 \[gr-qc\]\].\
T. Nakama and J. Yokoyama, “Micro black holes formed in the early Universe and their cosmological implications,” \[arXiv:1811.05049 \[gr-qc\]\].\
W. H. Press and P. Schechter, “Formation of galaxies and clusters of galaxies by selfsimilar gravitational condensation,” Astrophys. J. [**187**]{}, 425 (1974).\
T. Harada, C. M. Yoo and K. Kohri, “Threshold of primordial black hole formation,” Phys. Rev. D [**88**]{}, no. 8, 084051 (2013) Erratum: \[Phys. Rev. D [**89**]{}, no. 2, 029903 (2014)\] \[arXiv:1309.4201 \[astro-ph.CO\]\].\
T. Harada, C. M. Yoo, K. Kohri, K. i. Nakao and S. Jhingan, “Primordial black hole formation in the matter-dominated phase of the Universe,” Astrophys. J. [**833**]{}, no. 1, 61 (2016) \[arXiv:1609.01588 \[astro-ph.CO\]\].\
T. Harada, C. M. Yoo, K. Kohri and K. I. Nakao, “Spins of primordial black holes formed in the matter-dominated phase of the Universe,” Phys. Rev. D [**96**]{}, no. 8, 083517 (2017) \[arXiv:1707.03595 \[gr-qc\]\].\
V. Simha and G. Steigman, “Constraining The Early-Universe Baryon Density And Expansion Rate,” JCAP [**0806**]{} (2008) 016 \[arXiv:0803.3465 \[astro-ph\]\].\
M. Artymowski, O. Czerwinska, Z. Lalak and M. Lewicki, “Gravitational wave signals and cosmological consequences of gravitational reheating,” JCAP [**1804**]{} (2018) no.04, 046 \[arXiv:1711.08473 \[astro-ph.CO\]\].\
M. Giovannini, “Production and detection of relic gravitons in quintessential inflationary models,” Phys. Rev. D [**60**]{} (1999) 123511 \[astro-ph/9903004\].\
A. Riazuelo and J. P. Uzan, “Quintessence and gravitational waves,” Phys. Rev. D [**62**]{} (2000) 083506 \[astro-ph/0004156\].\
M. Yahiro, G. J. Mathews, K. Ichiki, T. Kajino and M. Orito, “Constraints on cosmic quintessence and quintessential inflation,” Phys. Rev. D [**65**]{} (2002) 063502 \[astro-ph/0106349\].\
L. A. Boyle and A. Buonanno, “Relating gravitational wave constraints from primordial nucleosynthesis, pulsar timing, laser interferometers, and the CMB: Implications for the early Universe,” Phys. Rev. D [**78**]{} (2008) 043531 \[arXiv:0708.2279 \[astro-ph\]\].\
K. Dimopoulos and C. Owen, “Quintessential Inflation with $\alpha$-attractors,” JCAP [**1706**]{} (2017) no.06, 027 \[arXiv:1703.00305 \[gr-qc\]\].\
J. H. MacGibbon and B. R. Webber, “Quark and gluon jet emission from primordial black holes: The instantaneous spectra,” Phys. Rev. D [**41**]{}, 3052 (1990).\
J. H. MacGibbon, “Quark and gluon jet emission from primordial black holes. 2. The Lifetime emission,” Phys. Rev. D [**44**]{}, 376 (1991).\
K. Kohri and J. Yokoyama, “Primordial black holes and primordial nucleosynthesis. 1. Effects of hadron injection from low mass holes,” Phys. Rev. D [**61**]{}, 023501 (2000) \[astro-ph/9908160\].\
I. Dalianis, “Constraints on the curvature power spectrum from primordial black hole evaporation,” arXiv:1812.09807 \[astro-ph.CO\].\
Y. B. Zel’dovich and A. A. Starobinskii, J. Exp. Theor. Phys. Lett. 24, 571 (1976)\
J. H. MacGibbon, “Can Planck-mass relics of evaporating black holes close the universe?,” Nature [**329**]{}, 308 (1987).\
E. J. Copeland, A. R. Liddle and D. Wands, “Exponential potentials and cosmological scaling solutions,” Phys. Rev. D [**57**]{} (1998) 4686 \[gr-qc/9711068\].\
C. F. Kolda and W. Lahneman, “Exponential quintessence and the end of acceleration,” \[hep-ph/0105300\].\
A. Kehagias and G. Kofinas, “Cosmology with exponential potentials,” Class. Quant. Grav. [**21**]{} (2004) 3871 \[gr-qc/0402059\].\
J. G. Russo, “Exact solution of scalar tensor cosmology with exponential potentials and transient acceleration,” Phys. Lett. B [**600**]{} (2004) 185 \[hep-th/0403010\].\
I. Dalianis and F. Farakos, “Exponential potential for an inflaton with nonminimal kinetic coupling and its supergravity embedding,” Phys. Rev. D [**90**]{} (2014) no.8, 083512 \[arXiv:1405.7684 \[hep-th\]\].\
C. Q. Geng, C. C. Lee, M. Sami, E. N. Saridakis and A. A. Starobinsky, “Observational constraints on successful model of quintessential Inflation,” JCAP [**1706**]{} (2017) no.06, 011 \[arXiv:1705.01329 \[gr-qc\]\].\
S. Basilakos, G. Leon, G. Papagiannopoulos and E. N. Saridakis, “Dynamical system analysis at background and perturbation levels: Quintessence in severe disadvantage comparing to $\Lambda$CDM,” \[arXiv:1904.01563 \[gr-qc\]\].\
G. Franciolini, A. Kehagias, S. Matarrese and A. Riotto, “Primordial Black Holes from Inflation and non-Gaussianity,” JCAP [**1803**]{} (2018) no.03, 016 \[arXiv[1801.09415]{}\[[astro-ph.CO]{}\]\].\
V. Atal and C. Germani, Phys. Dark Univ. 100275 \[arXiv:1811.07857 \[astro-ph.CO\]\].\
V. De Luca, G. Franciolini, A. Kehagias, M. Peloso, A. Riotto and C. Ünal, “The Ineludible non-Gaussianity of the Primordial Black Hole Abundance,” \[arXiv:1904.00970 \[astro-ph.CO\]\].\
C. Pattison, V. Vennin, H. Assadullahi and D. Wands, “Quantum diffusion during inflation and primordial black holes,” JCAP [**1710**]{} (2017) no.10, 046 \[arXiv[1707.00537]{}\[[hep-th]{}\]\].\
M. Biagetti, G. Franciolini, A. Kehagias and A. Riotto, “Primordial Black Holes from Inflation and Quantum Diffusion,” JCAP [**1807**]{} (2018) no.07, 032 \[arXiv:1804.07124 \[astro-ph.CO\]\].\
J. M. Ezquiaga and J. García-Bellido, “Quantum diffusion beyond slow-roll: implications for primordial black-hole production,” JCAP [**1808**]{} (2018) 018 \[arXiv:1805.06731 \[astro-ph.CO\]\].\
D. Cruces, C. Germani and T. Prokopec, “Failure of the stochastic approach to inflation beyond slow-roll,” JCAP [**1903**]{} (2019) no.03, 048 \[arXiv:1807.09057 \[gr-qc\]\].\
K. Dimopoulos, L. Donaldson Wood and C. Owen, “Instant preheating in quintessential inflation with $\alpha$-attractors,” Phys. Rev. D [**97**]{} (2018) no.6, 063525 \[arXiv:1712.01760 \[astro-ph.CO\]\].\
L. H. Ford, “Gravitational Particle Creation and Inflation,” Phys. Rev. D [**35**]{}, 2955 (1987).\
E. J. Chun, S. Scopel and I. Zaballa, “Gravitational reheating in quintessential inflation,” JCAP [**0907**]{} (2009) 022 \[arXiv:0904.0675 \[hep-ph\]\].\
|
---
abstract: |
In earlier papers we showed unpredictability beyond quantum uncertainty in atomic clocks, ensuing from a proven gap between given evidence and explanations of that evidence. Here we reconceive a clock, not as an isolated entity, but as enmeshed in a self-adjusting communications network adapted to one or another particular investigation, in contact with an unpredictable environment. From the practical uses of clocks, we abstract a clock enlivened with the computational capacity of a Turing machine, modified to transmit and to receive numerical communications. Such “live clocks” phase the steps of their computations to mesh with the arrival of transmitted numbers. We lift this phasing, known in digital communications, to a principle of *logical synchronization*, distinct from the synchronization defined by Einstein in special relativity.
Logical synchronization elevates digital communication to a topic in physics, including applications to biology. One explores how feedback loops in clocking affect numerical signaling among entities functioning in the face of unpredictable influences, making the influences themselves into subjects of investigation. The formulation of communications networks in terms of live clocks extends information theory by expressing the need to actively maintain communications channels, and potentially, to create or drop them.
We show how networks of live clocks are presupposed by the concept of coordinates in a spacetime. A network serves as an organizing principle, even when the concept of the rigid body that anchors a special-relativistic coordinate system is inapplicable, as is the case, for example, in a generic curved spacetime.
author:
- |
F. Hadi Madjid and John M. Myers 82 Powers Road, Concord, MA 01742, USA;\
Harvard School of Engineering and Applied Sciences, Cambridge, MA 02138, USA
title: |
Clocking in the face of unpredictability\
beyond quantum uncertainty
---
Introduction
============
Discoveries in physics involve unpredicted events, whether of the imagination of theorists or the surprises of unpredicted experimental evidence. Still, one can wonder: if scientists were smarter, could they have predicted the events that have come as surprises? Within the framework of quantum theory, the answer is “not on the basis of experimental evidence.” As is explained in Sec. \[sec:2\], quantum explanations depend on guesses that are subject to revision in the face of future evidence, not only as a practical matter but as a matter of principle, demonstrating an inescapable unpredictability beyond the probabilistic uncertainty of quantum mechanics.
The proven dependence of explanations on guesswork promotes the aspiration, never quite attainable, to separate explanations from evidence, and to attend to the sources of both. I find a path to discovery when I recognize that my explanation cannot be the only one possible. This recognition of a certain freedom in choosing explanations may be important to philosophy, but does it matter to physics? To begin with, because investigations draw on surprises, both from external outcomes and from acts of our own guessing and making of assumptions, we who investigate are not interchangeable: You experience what I do not, and I can know what you find out only to the extent of our communication. Today investigators are often linked to sensors, to actuators, and to each other by a computer-mediated communications network. Experimental activity, including encounters with unpredictability, is reflected in such a network, both in the records held in the computers of the network and in the timing of communications within the network. Networks involve feedback. Indeed, lots of biological and engineered mechanisms involve networks that respond to unpredicted deviations from some aiming point. We want to lift the feedback that takes place in a computer-mediated system from the province of engineering to a form suitable for application in theoretical physics.
As an application of feedback in computer-mediated networks, we consider investigations of motion. The motion of an object can be thought of or measured only in relation to organized motions of other objects that serve as a reference background against which to see the motion of the object under investigation. Here we argue that recognizing the gap between evidence and its explanations forces a reconception of reference motions, away from the a rigid system of equations and measurement units, conceptually tied to a single investigator, to a pattern of motion maintained by investigators and their devices, linked in some chosen communications network that the investigators create and maintain, operating in an environment of surprise and guesswork.
To develop the concept of a networked reference for motion we start by retracing the topic, opened up a century ago by Einstein, of spatially and temporally locating events of interest. At root, this is a problem for every person from infancy on. Right now I feel my fingers on the keyboard, the reach of my arm for the coffee cup, the chair I sit in; I see the walls of the office and, through the window, the bare trees and the snow lingering in this late-coming season of spring. Or think of Galileo comparing the motion of a rolling ball against himself, e.g. his own heart beat; then as science progresses he (and we who follow after him) delegate the beats of our reference to a more stable external clock. A clock and a flat floor make a background for motion; however, reading a distant clock is an issue.
The prevalent way to think of a background against which to define motion is a coordinate system, but the abstraction that generates a coordinate system obscures the reference motions that necessarily compose such a background. To recover the reference motions intrinsic to a coordinate system, we review a little history. Today’s notion of a coordinate system stems from the theories of special and general relativity. Commenting on how he developed those theories, Einstein spoke of the need to “drag down from the Olympian fields of Plato the fundamental ideas of thought in natural science, and to attempt to reveal their earthly lineage …” [@einstein61]. The earthly lineage leading to the theory of special relativity lies in the railroad age of the 1800’s, in which the development of telegraphy provided the means to synchronize clocks to tell “time” spread across the span of telegraphy. Einstein abstracted the practice of synchronizing clocks by the exchange of telegraph signals into the theory of special relativity. Building on his bold hypothesis about light speed, he defined the synchronization of imagined proper clocks in terms of clock readings at the transmission and reception of idealized light signals. He defined a coordinate frame relative to an imagined rigid body in terms of signals exchanged among synchronized proper clocks fixed to the body. This definition tied the concept of a rigid body to a pattern of signal exchange among proper clocks, thereby relativizing the notion of *rigid*. Without clocks and the motion of the signals by which their synchronization is defined, there can be no concept of a Lorentz coordinate system.
In general relativity, no extended body can be rigid, but the equations that define a metric tensor field are still imagined as fixed. For astronomy and geodesy, the International Astronomical Union (IAU) employs a notion of a ‘reference system’ incorporating fixed equations defining a metric tensor field [@soffel03]. Although obtained by dragging down ideas “from the Olympian fields of Plato,” Einstein’s synchronization criterion for special relativity, as well as the general-relativistic reference systems, are elevated to a blackboard of Plato, suitable for armchair contemplation, but remote from any active experimental setting. From the perspective of a physics that faces the unpredictable, the blackboard on which equations are written needs to be split up and made dynamic, with the equations of a reference system copied into the separate memories of the computers of the network that links the investigators and their automated devices. Copied into these memories, the equations become available to distributed real-time computations essential to the feedback that maintains the network; furthermore, on occasion, the equations written in one or another computer memory can be modified in response to unpredictable happenings.
What can be investigated depends in part on how the computers of a network can communicate. In Sec. \[sec:3\] we represent a real-time process-control computer that takes part in a network as a modified Turing machine stepped by a clock, and we explore the form of synchronization, distinct from Einstein’s, necessary to the transmission of numbers from one Turing machine to another. To deal with communications between Turing machines, it will be necessary for the clock that steps a Turing machine to tick at a rate that can be adjusted by commands issuing from that machine. We call such an adjustable clock in combination with the Turing machine that regulates its rate a *live clock*. (Such rate adjustment is found in the cesium clocks that generate International Atomic Time (TAI)[@aop14].) Because the evidence recorded by one live clock differs unpredictably from that recorded by another, unlike proper clocks postulated in relativity theory, each live clock is unique.
A live clock can receive a number from a detector, such as a photodetector, that acts from a computational point of view as an oracle, the workings of which are unknowable[@turingTh]. Other oracle numbers show up as clock readings made by one live clock and transmitted to another live clock. Both human and automated communication of numbers involves cycling through phases. Think of the cycle of a bucket brigade, in which a person turns to receive a bucket just as a another person offers the bucket. This need for cycling through phases structures the communications in which live clocks participate, leading to a definition of the *logical synchronization* of a channel from one live clock to another, distinct from the synchronization defined by Einstein. The need for logical synchronization of its channels constrains the design of a network of live clocks.
To maintain logical synchronization of the channels of a network of live clocks requires feedback because, among other reasons, at the level of precision with which atomic clocks can be measured, “no two clocks tick alike.”[@aop14] In this feedback, the live clocks of a network steer their tick rates and accelerations in response to unpredictable measured deviations from an aiming point. The aiming point is chosen to express desired communications channels, and steering toward that aiming point is determined by control equations programmed into the live clocks.
Designing and operating the channels of a network and the corresponding control equations requires a hypothesis concerning how signals propagate from one live clock to another. It is a great convenience in making such a hypothesis to draw on the concept of a spacetime manifold; however, evidence in the form of clock readings is independent of any hypothesis about signal propagation, and therefore can be used to test a hypothesis, regardless of whether the hypothesis invokes the concept of a spacetime manifold. Thus there is great freedom in the choosing of hypotheses about signal propagation. In addition to serving as a background against which to think about and measure motion, a network of live clocks serves as a tool to investigate the propagation of the signals by which the live clocks of the network communicate.
Among candidate hypotheses about propagation, those based on a general-relativistic spacetime manifold commend themselves as an appropriate first case for theoretical purposes. It is especially interesting that a generic curved spacetime manifold rules out a fixed general-purpose pattern of signal exchange, and so calls for tailoring a network of live clocks to whatever situation is under investigation. In Sec. \[sec:4\] we discuss a network of live clocks, designed assuming a spacetime manifold and intended to serve as an adaptive background for motion and as a tool to investigate gravitation. The operation of the network involves a cycle of provisionally assuming a metric tensor field, using it to construct an aiming point in terms of relations among clock readings toward which to steer the live clocks, implementing the corresponding control equations, and evaluating the resulting deviations from the aiming point. If the deviations cannot be held within tolerable bounds, the investigators, drawing on the measured clock readings, guess a different metric tensor field, and the cycle continues.
Sec. \[sec:5\] offers a brief perspective on how recognizing essential unpredictability impacts the application of physical concepts, based on the dependence of both explanations and experiments on a dynamically evolving “tree of assumptions.”
Unpredictability implied by quantum theory {#sec:2}
==========================================
Given a quantum state and a measurement operator, quantum theory sets up predictions of a probability measure over a set of possible outcomes, and quantum uncertainty denotes a spread in the probability measure. Quantum theory, however, implies something beyond uncertainty, namely a kind of unpredictability, in some cases involving the introduction of possibilities previously unforeseen. In earlier work we proved that:
1. Infinitely many explanations are consistent with any given evidence. While each explanation fits the given evidence, the explanations have conflicting implications for evidence obtainable from experiments not yet performed. Once candidate explanations are proposed, one can bet on them—in effect, assigning rough probabilities to them. But in the face of an experimental surprise, before any explanation is proposed, no evidence can determine the probability to assign to any of the numberless not-yet-known explanations that fit given evidence. For this reason, explanations as hypotheses chosen for testing come as surprises, unpredictable even to the person who chooses them. [@ams02; @aop05; @CUP].
2. To choose one or a few explanations requires reaching beyond any predictive logic and beyond the guidance of probabilities to make a guess [@aop05].
3. Because it excludes conflicting explanations that fit the given evidence equally well, any guessed explanation is susceptible to refutation by surprising results coming from future experiments, prompting its replacement by some other explanation [@tyler07].
4. Choosing explanations as hypotheses to be tested by experiments takes place in an open cycle, alternating between subjective surprises of guessing hypotheses and objective surprises of unexpected experimental results [@tyler07]. Thus not only do explanations depend on assumptions but so do experiments: their design and the interpretation of their results depend on explanations that rest on layers of guesswork.
5. In addition to the unpredictability of guesses that choose explanations, quantum uncertainty makes individual occurrences of outcomes of generic measurements unpredictable. Two situations are to be distinguished:
1. An experimenter tests a predicted probability measure by repeating a measurement many times and comparing measured relative frequencies against predicted probabilities. In this situation individual occurrences of outcomes matter only as they enter tallies, ratios of which become relative frequencies.
2. As in applications of quantum decision theory [@helstrom; @holevo], an experimenter has a prior probability measure, sometimes only a guess, for the state being measured and uses a single unpredictable result or a short run of results to together with Bayes rule to decide on a parameter value that characterizes the state, acting promptly on that decision. (Such prompt action takes place in the feedback loops by which atomic clocks must be steered in frequency [@aop14].)
Although surprises in physics are familiar as a practical matter, before the proof one might try to relegate them to the status of exceptions to the progress of science. The proof gives theoretical force to the unpredictability of surprises and guesses, so that the implications of this unpredictability can be explored. If a guess is necessary for an explanation, what does that mean for discovery? Who or what makes a guess? In particular, the dependence of an explanation of given evidence on guesswork motivates more attention to two things: (a) the distinction between explanations and the experiences that one explains, and (b) the entity, long neglected in physics, in which evidence, calculation, and guesswork meet.
Clocks that compute and communicate {#sec:3}
===================================
When investigators and their devices are linked to one another by a computer-mediated network, the network serves as a tool for the investigation; it also houses the results of the investigation in records distributed in the memories of its computers. We model a computer in a network by a clock joined to a Turing machine that is modified to permit communication with other such machines and with an unpredictable environment containing sensors such a photodetectors. The clock has provision for its rate of ticking to be adjusted on the fly by commands from the Turing machine. We call such a clock together with the Turing machine a *live clock*. The live clock models an entity in which evidence, guesswork, and calculation meet. It is the key component in any of various communications networks against which to reference the motion of objects under investigation. Live clocks enable a networked reference for motion to adapt to the needs of particular investigations.
Turing represented mathematically the activity of calculating by what is now called a Turing machine, imagined as operating in a sequence of ‘moments’ interspersed by ‘moves’. The Turing machine has a working memory in the form of a tape. At any moment the machine scans one square of the tape, on which it can read or write a single character of some alphabet that need not be more than the binary set $\{0,1\}$ [@turing]. A move as defined in the mathematics of Turing machines consists only of the logical relation between the machine at one moment and the machine at the next moment. Thus the Turing machine models the logic of computation by relating a program and a record in memory to an output in memory, without regard to timing. Two computations executing at different speeds can be represented in their logic by the same sequence of moments and moves. Because the Turing machine is indifferent to timing, it cannot calculate the tick that steps its moves, and so cannot by itself express the physical motion of computation in the way that the live clock does.
People or machines as calculators do lots of things, such as taking a break or fetching new supplies, that Turing abstracted out of sight. In so doing he sealed the Turing machine off from receiving or transmitting communications to anything outside its computational activity. He did, however, open the door to communication a crack by briefly discussing a variant that he called a *choice machine* which, on occasion, could wait for an external input [@turing]. In adapting it as a component of a live clock, we modify the Turing machine further so that the live clock can not only receive characters of its alphabet from an external environment, but can also transmit characters to that environment. The environment can include other live clocks.
Cycle of moves and moments of computation
-----------------------------------------
A live clock operates in a cycle of receiving unpredictable information from an environment, storing that information in memory, computing a response, and issuing that response to the environment. The cycle has subcycles, and at the finest level is composed of moments and moves of the clock-driven Turing machine that makes up the live clock. For a live clock to take part in communication, its moments and moves have to be regulated to avoid the logical conflict of a collision between writing into memory and reading from memory. (In human terms this is the collision between trying to speak and listen at the same time.) To avoid this conflict, the modified Turing machine is driven by the adjustable clock through a cycle with two phases of moves and two phases of moments, with reading from memory taking place in a phase separated from a phase of writing into memory.
A cycle of the live clock corresponds to a unit interval of the readings of its adjustable clock. A reading $t$ of a live clock can be expressed in the form $m.\phi_m$ where an integer $m$ indicates the count of cycles and $\phi_m$ is the phase within the cycle. We choose the convention that $-1/2<\phi_m\le
1/2$. (It is not necessary to think of the signals as points in time; it suffices to think of a point reference within the signal.)
Guesses, such as guessed explanations, enter the memory of a live clock as inputs from outside of the live clock, and are written into its memory during phases of writing. We give no further explanation of this “outside” from which guesses are assumed to come. Because live clocks have memories, they can record and make use of readings of other live clocks and indeed their own readings—a mechanical analog of “self-awareness.”
Logically synchronizing the communications between live clocks
--------------------------------------------------------------
To express the clocking of actual or contemplated communications between one live clock and another, we follow Shannon in speaking of a communications channel; however we augment his information-theoretic concept of a channel [@shannon48] with the live-clock readings at the transmission and reception of character-bearing signals [@aop1; @4]. Each character transmitted from a live clock $A$ to a live clock $B$ is associated with a reading of live clock $A$ of the form $m.\phi_m$ at the transmission and with a reading of live clock $B$ of the form $n.\phi_n$ at the reception. A channel from $A$ to $B$ includes a set of such pairs of readings of the transmitting and the receiving live clocks. The necessity of avoiding a conflict between reading and writing imposes a constraint on the phases of reception.
> **Proposition:** A character can propagate from one live clock to another only if the character arrives within the writing phase of the receiving live clock.
When this phase constraint is met for a channel between a transmitting live clock and a receiving live clock, we say the receiving live clock is *logically synchronized* to the transmitting live clock. Logical synchronization is analogous to the coordination between neighboring people in a bucket brigade, or that between players tossing a ball back and forth, where the arrival of the ball must be within a player’s ‘phase of catching’. In this way the notion of a channel is expanded to include the clock readings that indicate phases of signal arrivals that have to be controlled in order for the logical synchronization of the channel to be maintained. (While in many cases the integers in clock readings that count cycles can be definitely specified, the phases are never exactly predictable.) We model the phase of writing at which a live clock can receive a character as corresponding to $$\label{eq:ls}
|\phi| < (1-\eta)/2,$$ where $\eta$ (with 0 $< \eta < 1$) is a phase interval that makes room for reading.
Logically synchronizing a channel means bringing about the condition (\[eq:ls\]) on phases at which signals arrive. Once logical synchronization is acquired, maintaining it typically requires more or less continually adjusting the rates of ticking and the acceleration of one or both of the live clocks, in order to steer the phases of arriving characters toward a suitable aiming point, say some $\phi_0$. In the simplest case, this aiming point $\phi_0$ is 0. When a live clock receives signals over more than one channel, it measures its own clock readings at all the receptions and takes all the phase deviations into account to steer its own tick rate and acceleration. The clock readings made for the purpose of maintaining logical synchronization are not interpreted as indicating “the time”; instead, like the readings of a clock that is being adjusted by a clockmaker, they indicate the amount by which the live clock needs to be adjusted.
Relations among readings of live clocks that contribute to an aiming point for a network include what we call echo counts, closely related to distances defined by radar:
> **Definition of echo count:** Suppose that at its reading $m.0$ a live clock $A$ transmits a signal at to a live clock $B$, and the first signal that $B$ can transmit back to $A$ after receiving $A$’s signal reaches $A$ at $m'.\phi'$; then the quantity $m'.\phi'- m.0$ will be called the echo count $\Delta_{ABA}$ at $m$.
The need for logical synchronization of the channels strongly constrains the design of a network of live clocks. In Sec. \[sec:4\] we will see how the ticks of the clocks are allowable only within intersections of “stripes in spacetime” [@aop14].
Steering toward an aiming point involving echo counts depends on one live clock receiving signals that convey readings of other live clocks. The possibility of wireless communication requires that such a signal from a live clock $A$ to a live clock $B$ carry an identifier of $A$, so that the receiver $B$ can tell the source of the signal. With these identifiers, the channels of a network of live clocks correspond to directed edges of a graph, leading to a data structure for readings of live clocks of a network illustrated in \[\] and discussed at more length in terms of marked graphs and Petri nets in \[\]. Such data structures for a network can reside in the memory of a live clock, so the live clock has a picture of the network in which it participates.
Numerical communication as a frame of reference {#sec:4}
===============================================
With unpredictability established in Sec. \[sec:2\] and the consequent need for feedback to support logical synchronization established in Sec. \[sec:3\], we now show how networks of live clocks can offer reference patterns of motion against which to think about and measure the motion of some object of interest. The following subsections tell:
1. how the concept of a coordinate system as the prevalent reference for motion depends on Einstein’s synchronization criterion and thereby implicitly depends on a network of live clocks;
2. how, assuming a flat spacetime, the requirement for logical synchronization constrains a network of live clocks that approximates a Lorentz frame;
3. how logical synchronization can be maintained in cases that preclude Einstein synchronization;
4. how, in the theory of curved spacetime, there can be no large rigid body and no all-purpose network of live clocks to realize a background to motion;
5. how a network of live clocks serves as a tool to discover features of its unpredictable environment, including features of gravitation;
6. how predictability meets unpredictabilty on a Turing tape.
Coordinate frames presuppose communication between ‘self-aware’ clocks
----------------------------------------------------------------------
To see that a coordinate system presupposes a network of live clocks, recall Einstein’s definition of synchronization in special relativity. Einstein conceived of the location of an arbitrary event as a clock tick coincident with the event. This makes a spacetime of possible locations of events into a potential for clock ticks, so that for any event one can think of a tick coincident with it. But the tick is significant only in relation to other ticks. By imagining a system of light signals propagating between imagined *proper* clocks, Einstein defined the synchronization of proper clocks fixed to a non-rotating, rigid body in free fall (i.e., a Lorentz frame) and co-defined “time” as the readings of such proper clocks, with the implications that distance from proper clock $A$ to proper clock $B$ is defined, as in radar, in terms of the duration at $A$ from the transmission of a light signal to the return of its echo from $B$. (If we imagine proper clocks as live clocks, this difference in readings of clock $A$ is an echo count.) Specifically, according to Einstein’s definition of the synchronization of proper clocks fixed to a Lorentz frame [@einstein05], clock $B$ is *synchronous* to clock $A$ if at any $A$-reading $t_A$, $A$ could send a signal reaching $B$ at $B$-reading $t_B$, such that an echo from $B$ would reach $A$ at $A$-reading $t'_A$, satisfying the criterion $$\label{eq:es}
t_B=\half(t_A+t'_A).$$ A Lorentz coordinate system presupposes Einstein’s synchronization and thereby presupposes the possibility of proper clocks that exchange light signals—an abstraction of a grid-like network of live clocks and their signals, in which every pair of live clocks exhibits constant echo counts and a constant Einstein synchronization relation with its neighbors.
The clock readings along the history of a moving object constitute a description of its motion, but how does that description become knowable to the investigators? The concept of a Lorentz coordinate system obscures the information processing needed for clock readings to become known to investigators. In contrast, live clocks allow for the local recording of their readings and the subsequent transmission of readings from one live clock to another. They also give expression to the necessity, stemming from quantum theory, to respond to unpredictable deviations from desired relations.
Although special relativity imagines proper clocks free of drift, quantum theory asserts an irreducible drift in tick rates and echo counts [@aop14], so that the readings of nearby, unadjusted clocks wander apart without bound unless their ticking is adjusted. When we recognize the inescapability of this drift, the Einstein synchronization relations (\[eq:es\]) and the echo counts can no longer be taken as externally supplied facts, but instead work as aiming points chosen by investigators, toward which to steer the operation of a network of live clocks. This steering, if it can be accomplished, must depend on feedback of unpredictable measured deviations from the aiming point into the steering of the tick rates and the accelerations of the clocks of a network.
To be steerable on the basis of measured deviations, the clocks must act as live clocks. Furthermore, feedback that steers toward Einstein’s synchronization requires the communication of numerals from one live clock to another. For example, consider a network of live clocks, subject to unpredictable drift, in which each live clock aims to satisfy an aiming point expressed as a set of relations among readings of itself and neighboring live clocks. Suppose the aiming point is comprised of the Eq. (\[eq:es\]) along with constant echo counts for selected channels to nearby live clocks. Because of drift, measured clock readings deviate from the relations that constitute the aiming point. In response to the deviations from the aiming point, each live clock promptly adjusts its tick rate and acceleration. A live clock cannot wait for an external intelligence to compute the deviations exhibited by measured, unpredictable clock readings; it must itself compute the relevant deviations and its own response to these. For example, a clock $B$ computes a deviation that involves the difference between its own reading at its reception of a signal from a nearby clock $A$ and the reading of $A$ at the transmission of that signal. Thus $B$ can know how it deviates from being Einstein-synchronous with $A$ only if readings of $A$’s clock are communicated to $B$. It is not enough for a light signal from $A$ to reach $B$; in addition the light signal has to convey the numerical reading of $A$ at the transmission of that signal. Hence steering toward Einstein’s synchronization criterion of Eq. (\[eq:es\]) requires the communication of numbers, which requires logical synchronization.
Constraints of logical synchronization limit realizations of coordinates
------------------------------------------------------------------------
The requirement for logical synchronization constrains the arrangements of clocks that can satisfy Einstein’s synchronization criterion. To show this, it is instructive to consider the case of ideal logical synchronization in which all phases at signal receptions are zero. Take the theoretical case of 8 live clocks located at the corners of a cube in a flat or conformally flat spacetime. Let each edge of the cube correspond to logically synchronized channels in both directions, all with zero phases. In that case channels across the face diagonals cannot have zero phases, simply because the ratio of the length of a face diagonal to an edge is irrational. But if the cube is replaced by a rectangular brick, the brick can be chosen as an Euler brick for which edges and face diagonals can both have integer lengths [@brick], allowing a pattern of signaling having receptive phases for signals along edges of the brick and along its face diagonals to all be zero. In effect the requirement for logical synchronization puts stripes on spacetime for allowable configurations networks of live clocks [@aop14]. Nonetheless, within any theory that assumes a flat or conformally flat spacetime, a three dimensional grid of live clocks exchanging signals along edges and face diagonals of Euler bricks serves as a universally applicable pattern of motion for a coordinate system. We will soon see a difference in this regard between a conformally flat spacetime and a generic curved spacetime.
Logical synchronization where Einstein synchronization fails
------------------------------------------------------------
While realizing Einstein’s synchronization requires logical synchronization among pairs of live clocks, the converse does not hold: cases of logically synchronized networks can be demonstrated for which Einstein synchronization is impossible. These cases include live clocks in relative motion, subject to the Doppler effect, live clocks on a rotating platform, subject to the Sagnac effect, and live clocks in the presence of gravitational fields. For this and other reasons it becomes an interesting scientific topic to explore possible patterns of signal exchange among logically synchronized live clocks. The topic includes the application of logical synchronization to the study of gravitation.
Live clocks under the assumption of spacetime curvature
-------------------------------------------------------
Bringing hypotheses of one or another curved metric tensor field into consideration reveals a conceptual challenge to the definition and measurement of motion. As amply confirmed by experiments with space vehicles, by astronomical observations, and by experience with the Global Positioning System (GPS), the theory of general relativity asserts that the flat metric tensor field of special relativity and its concomitant realization by a pattern of light signals between clocks can only be approximated over a region the size of which must decrease as the instability of clocks decreases. If generic curvature is significant, there can be no arbitrarily fine, three-dimensional grid of signal-exchanging live clocks that pairwise satisfy Eq. (\[eq:es\]), so that the universally applicable pattern of motion for a coordinate system noted for flat spacetime is ruled out [@perlick]. This effect of generic spacetime curvature is a consequence of general relativity alone without invoking logical synchronization. Because the concept of a rigid body hinges on maintaining Einstein synchronization among the proper clocks thought of as elements of that body, it is apparent that the theory of general relativity is incompatible with any exactly rigid body.
### How to accept “no rigid bodies”
A pattern of motion taken as a tool for determining my own motion serves as a tool for navigation, perhaps best appreciated in circumstances of its absence, as when, perhaps on a hike, I am lost. The thought of being lost is scary. It is soothing to imagine a flat coordinate system, relative to which the distant and the immediate are brought into relation, the ground under my feet stays put, and navigation would be straightforward. But if the earth shifts, as at some level of precision it always does, how is one to think? And at levels of precision made possible by modern clocks, there can be no rigid body on which to stand, nor can there be a fixed background of communication. It becomes necessary to search for a background pattern and to adjust it if increased demands for precision or unforeseen circumstances make the sought pattern unrealizable. If needs to adjust are in the cards, it is perhaps better to be nimble. Recognizing unpredictability can be a first step toward that nimbleness.
Logical synchronization opens up possibilities for discovery
------------------------------------------------------------
While we have spoken of spacetime coordinates, the concept of a network of live clocks as a reference for motion makes no assumption of a spacetime manifold or of a coordinate system. To design an aiming point of desired relations among clock readings for a network of live clocks, however, one needs to invoke some hypothesis about the propagation of signals from one live clock to another live clock. What makes a workable hypothesis depends on circumstances, for example whether the signal is conveyed by light in vacuum, by light in some medium such as optical fiber, or by nerve pulses, etc.
As a first specific case appropriate to theoretical physics, consider live clocks on space vehicles linked in a communications network for an experiment aimed at discovering features of gravitation that affect the network. Suppose the experiment employs a theory of of signals as light-like geodesics in some specified spacetime manifold. Then the theory of propagations of signals among live clocks depends on the curvature of a metric tensor field hypothesized for the spacetime manifold. To maintain logical synchronization, the live clocks employ control equations that depend not only on the investigators’ basic assumption of a (possibly curved) spacetime manifold, but also on an additional provisional hypothesis of a particular metric tensor field, from which they arrive at a model of signal propagation, necessary to designing an aiming point specified in terms of relations among readings of live clocks at the transmission and the reception of light signals. The control equations depend on the provisional metric tensor field, and, together with that metric tensor field, are subject to revision when difficulties in maintaining logical synchronization are encountered.
For example, suppose the investigators guess that their space vehicles behave as if they were in a region of spacetime that has a metric that is flat to within the tolerances they can achieve with their live clocks. To test this guess, the investigators choose relations among clock readings as an aiming point under the provisional hypothesis of a flat spacetime. If, counter to their guess, the live clocks deviate too much from the aiming point, that deviation indicates the effect of a Weyl curvature of the spacetime.
In \[\] a specific case of a cluster of 5 space vehicles, each equipped with a live clock, is analyzed for its capacity to measure unpredictable changes in gravitation by utilizing feedback to adjust clock rates and clock accelerations to generate a background motion maintained by computer-mediated feedback in response to unpredictable events. Gravitational effects are then detectable as they influence the channels found to be implementable. The investigation of gravitational effects thus involves a cycle of hypothesizing a metric tensor field and testing the implications of that hypothesis for clock readings against those recorded by the 5 live clocks.
Under hypotheses defining the second as a measuring unit in the International System of Units (SI) [@aop14], and with sufficiently stable live clocks, such a spaceborne network can potentially measure ripples in gravitation, something already pursued by other but related means by the Laser Interferometer Gravitational Wave Observatory (LIGO) [@ligo]. We suspect that LIGO in fact illustrates how the use of feedback that responds to unpredictable events allows a precision in the background of measurement otherwise unattainable, and hence opens experimental inquiry into effects otherwise invisible. The complexity of LIGO, however, led us to point to the above arrangement of devices that, although likely difficult to implement, is conceptually simpler in its use of feedback that responds to the unpredictable.
Predictability meets unpredictabilty on a Turing tape
------------------------------------------------------
As represented in the memory of a live clock a variable can be thought of as a stretch of a Turing tape. A live clock engaged in a feedback loop houses separate variables—separate stretches of its Turing tape—for an aiming point derived from a hypothesis as distinct from a measured, unpredictable deviation from that aiming point. While both variables for aiming points and variables for deviations can experience changes in their values, typically the rate of change is relatively high for a deviation and low for an aiming point.
Important to logical synchronizaton is the distinction between a count of cycles recorded on tape at the receipt of a signal over a logically synchronized channel and the recorded phase of the cycle at which the signal arrives. The count is so to speak a “public number” that ought to be the same if the live clock were interchanged with another live clock; the phase however, is essentially an analog rather than a digital business; the phase is idiosyncratic in the sense that there must be a tolerance within which it would differ in one live clock were substituted for another. Idiosyncratic phases are indispensable to the control that permits the logically coherent transmission of numerals from one live clock to another, so that a “7” on a tape of live clock $A$ becomes a “7” on live clock $B$, and arithmetic functions across a communications channel.
When we set aside the concept of a coordinate system in favor of a network of live clocks, with its feedback loops that act on its patterns of signal exchange in response to unpredictable influences, do we have anything to hold onto in place of the fixed relations that are most vivid in the rigid body that anchors coordinates in special relativity? What a network of live clocks offers is both a communications facility and a conceptual framework to think about connections among entities of interest. This conceptual framework enriches the notion of a coordinate system in circumstances in which a coordinate system is appropriate. It also serves in situations in which unpredictability makes a coordinate system unavailable. What one holds onto are the records of the communications channels and of the unpredictable adjustments necessary to their maintenance found in the memories of live clocks.
Discussion {#sec:5}
==========
This paper builds on the more detailed analysis of unpredictability in clocks reported in ref. \[\], in which live clocks are called “open machines.”
Guesswork as a necessity has a long history of proponents. For example, Einstein endorses Hume’s showing that experimental evidence can never establish a causal connection, and Einstein goes further to say that all our concepts depend on “freie Schöpfungen des Denkens”—free creations of thought—i.e. guesses [@weltbild]. The virtue of the proofs noted in Sec. \[sec:2\] in not to bring a new thought, but to demonstrate it from within physics.
For computation to work over a communications channel, it is necessary for the steps of computation to be adjusted in response to unpredictable, idiosyncratic phases of signal arrivals over logically synchronized channels. Channels in networks get put into operation, are maintained for a while, and then perish. The memory of a live clock can house a picture of the network at some moment, and this picture may or may not agree with a picture held in the memory of another live clock; indeed, communications delays can make two such pictures held in separated memories incommensurate. The coming into operation and the perishing of channels are almost entirely outside the scope of this report: the maintenance of logically synchronized channels alone, while only the simplest part of the story, shows how, by invoking guesses of how to respond to unpredictable measured detections, live clocks step their computations in a way that allows the spread of arithmetic across a network. Attention to the networks of live clocks as references for motion shows ways in which the unpredictable and the calculated work in combination during scientific discovery, and how that “working together” rules out the view that what is discovered is out there independent of we who, with our dependence on guesswork, go about looking for it.
Contact with the physical world is reflected in records held in the memories of computers, seen here as live clocks. Such memories include traditional libraries holding the extant physics literature. The records in memories distributed over networks of computers depend on assumptions, as does the use of these records in making predictions and in designing experiments. The assumptions that reside in memories and are applied, for example in managing logical synchronization, vary in what one might call depth. At root are community-wide assumptions, including conventions of language and logic. From these assumptions, that investigators are slow to change, there range more flexible assumptions, up to those that an individual can make today and drop tomorrow. Under the impact of responses to surprises to which assumptions are vulnerable, this tangle of axioms, postulates, hypothes and whatever else we call it that comes into our heads without a logical justification, which we call a “tree of assumptions,” evolves unpredictably; it can be glimpsed by any person only in a small part, and a person’s capacity to notice an assumption depends on what other assumptions that person carries or rejects at the moment.
Maintaining logical synchronization among live clocks that contribute to a reference background of motion must draw on a tree of assumptions that puts a gap between evidence and its explanations. The numbers that come as evidence into a network affect the network and are distinct from explanations in terms of particles and fields, also numeric structures, which presuppose a coordinate system. Pretending that particles or fields can enter records directly as evidence obscures the gap between evidence and explanation. Accepting the unpredictability in a reference background that stems from the guesswork intrinsic to the tree of assumptions engenders a deeper sense of order, resident in memories of live clocks, and operative in situations beyond those for which the notion of a coordinate system finds application.
Accepting unpredictability in the backgrounds against which all else is seen takes a certain courage. The reward is the opportunity to learn to work with unpredictability in physics, thereby entering a world in which the joy of an answer can be the new question it enables.
Acknowledgment {#acknowledgment .unnumbered}
==============
We thank Prof. Tai Tsun Wu for discussions of the topic of discovery.
[99]{} A. Einstein *Relativity: The Special and General Theory* (Crown Publishers, New York, 1961) App. V, p. 142. M. Soffel et al., “The IAU resolutions for astrometry, celestial mechanics, and metrology in the relativistic framework: explanatory supplement,” The Astronomical Journal, **126**, 2687–2706 (2003). J. M. Myers and F. H. Madjid, “Distinguishing between evidence and its explanations in the steering of atomic clocks,” Annals of Physics 350, 29–49 (2014); arXiv:1407.8020. The invention of the notion of an oracle is due to A. M. Turing, and can be found in his 1938 thesis, reprinted in *Alan Turing’s Systems of Logic, the Princeton Thesis*, Princeton University Press, Princeton, NJ (2012). J. M. Myers and F. H. Madjid, “A proof that measured data and equations of quantum mechanics can be linked only by guesswork,” in S. J. Lomonaco Jr. and H.E. Brandt (Eds.) *Quantum Computation and Information*, Contemporary Mathematics Series, vol. 305, American Mathematical Society, Providence, 2002, pp. 221–244; arXiv:quant-ph/0003144. F. H. Madjid and J. M. Myers, “Matched detectors as definers of force,” Ann. Physics **319** (2005), 251–273; arXiv:quant-ph/0404113.. J. M. Myers and F. H. Madjid, “What probabilities tell about quantum systems, with application to entropy and entanglement,” in A. Bokulich and G. Jaeger, eds., *Quantum Information and Entanglement*, Cambridge University Press, Cambridge UK, pp. 127–150 (2010); arXiv:1409.5100. J. M. Myers and F. H. Madjid, “Ambiguity in quantum-theoretical descriptions of experiments,” in K. Mahdavi and D. Koslover, eds., *Advances in Quantum Computation*, Contemporary Mathematics Series, vol. 482 (American Mathematical Society, Providence, I, 2009), pp. 107–123; arXiv:1409.5678. C. W. Helstrom, *Quantum Detection and Estimation Theory*, (Academic Press, New York, 1976). A. S. Holevo, *Probabilistic and Statistical Aspects of Quantum Theory*, (North-Holland Publishing Co., Amsterdam, 1982) A. M. Turing, “On computable numbers with an application to the Entscheidungsproblem,” Proc. London Math. Soc., Series 2, **42**, 230–265 (1936). C. E. Shannon, “A mathematical theory of communication,” The Bell System Technical Journal, Vol. 27, pp. 379–423, 623–656, July, October, 1948. J. M. Myers and F. H. Madjid, “Rhythms of Memory and Bits on Edge: Symbol Recognition as a Physical Phenomenon,” arXiv/1106.1639, 2011. A. Einstein, “Zur Elektrodynamik bewegter Körper,” Annalan der Physik, **17**, 891–921 (1905). J. F. Sawyer and C. A. Reiter, “Perfect parallelepipeds exist,” Mathematics of Computation **80**, 1037–1040 (2011) (also arXiv:0907.0220v2). V. Perlick, “On the radar method in general-relativistic spacetimes,” in H. Dittus, C. Lämmerzahl, and S. Turyshev, eds., *Lasers, Clocks and Drag-Free Control: Expolation of Relativistic Gravity in Space*, (Springer, Berlin, 2008); also arXiv:0708.0170v1. The LIGO Scientific Collaboration (http://www.ligo.org) “LIGO: the Laser Interferometer Gravitational-Wave Observatory,” Rep. Prog. Phys. **72**, 076901 (2009) doi:10.1088/0034-4885/72/7/076901 http://www.ligo.org A. Einstein, “Bertrand Russell und das philosophische Denken,” in *Mein Weltbild*, (Ullsteinverlag, Berlin, 2005); translation available in *Ideas and Opinions*, (Crown Publishers, New York, 1954).
|
---
abstract: |
Open clusters can be the key to deepen our knowledge on various issues involving the structure and evolution of the Galactic disk and details of stellar evolution because a cluster’s properties are applicable to all its members. However the number of open clusters with detailed analysis from high resolution spectroscopy and/or precision photometry imposes severe limitation on studies of these objects.
To expand the number of open clusters with well-defined chemical abundances and fundamental parameters, we investigate the poorly studied, anticenter open cluster Tombaugh 1.
Using precision $uvbyCa$H$\beta$ photometry and high resolution spectroscopy, we derive the cluster’s reddening, obtain photometric metallicity estimates and, for the first time, present detailed abundance analysis of 10 potential cluster stars (9 clump stars and 1 Cepheid).
Using radial position from the cluster center and multiple color indices, we have isolated a sample of unevolved probable, single-star members of Tombaugh 1. From 51 stars, the cluster reddening is found to be $E(b-y)$ = 0.221 $\pm$ 0.006 or $E(B-V)$ = 0.303 $\pm$ 0.008, where the errors refer to the internal standard errors of the mean. The weighted photometric metallicity from $m_1$ and $hk$ is \[Fe/H\] = $-$0.10 $\pm$ 0.02, while a match to the Victoria-Regina Strömgren isochrones leads to an age of 0.95 $\pm$ 0.10 Gyr and an apparent modulus of $(m-M)$ = 13.10 $\pm$ 0.10. Radial velocities identify 6 giants as probable cluster members and the elemental abundances of Fe, Na, Mg, Al, Si, Ca, Ti, Cr, Ni, Y, Ba, Ce, and Nd have been derived for both the cluster and the field stars.
Tombaugh 1 appears to be a typical inner thin disk, intermediate-age open cluster of slightly subsolar metallicity, located just beyond the solar circle, with solar elemental abundance ratios except for the heavy s-process elements, which are a factor of two above solar. Its metallicity is consistent with a steep metallicity gradient in the galactocentric region between 9.5 and 12 kpc. Our study also shows that Cepheid XZ CMa is not a member of Tombaugh 1, and reveals that this Cepheid presents signs of barium enrichment, making it a probable binary star.
author:
- 'João V. Sales Silva$^{1,2}$, Giovanni Carraro$^{1,3}$, Barbara J. Anthony-Twarog$^4$, Christian Moni Bidin$^5$, Edgardo Costa$^6$ and Bruce A. Twarog$^4$'
title: |
Properties of the open cluster Tombaugh 1\
from high resolution spectroscopy and uvbyCaH$\beta$ photometry[^1]
---
Introduction
============
Tombaugh 1 is a star cluster in the Canis Majoris constellation at $\alpha= 07^{h}00^{m}29^{s}$, $\delta= -20^{o}34^{\prime}00^{\prime\prime}$, and $\ell,b=232^{\circ}.22,-7^{\circ}.32$ (2000.0 equinox), discovered in 1938 by Clyde Tombaugh (Tombaugh 1938). A quick glance at any sky image shows only a small enhancement of stars in a rich Galactic field, a primary reason why Tombaugh 1 has been studied very little until now. As detailed in Sect. 2, estimates of its fundamental parameters vary strongly from author to author, photometric data are scanty, and spectroscopic data totally absent.
There are, however, several reasons to consider this cluster particularly interesting. With a Galactic latitude of $-7^{o}.32$ and an assumed a distance of 2–3 kpc (Piatti et al. 2004; Carraro & Patat 1995), Tombaugh 1 is situated $\sim$300–400 pc below the Galactic plane, a location rare among open clusters of the presumed age of Tombaugh 1 ($\lesssim 10^9$ yr). The direction to Tombaugh 1 intersects the Perseus and Orion arms in the third Galactic quadrant, and the cluster was for some time associated with the putative Canis Major dwarf galaxy (Bellazzini et al. 2004). Although the discussion of this topic has been dormant for some time (Carraro et al. 2008), it would still be valuable to determine the cluster distance with enough precision to associate it with either the Perseus or, possibly, the Orion arm extension in the third Galactic quadrant (Vásquez et al. 2008). Moreover, a distance of 2–3 kpc places Tombaugh 1 at a galactocentric distance of 10–11 kpc, a location where some observations indicate an abrupt change in the abundance gradient of the disk (see, e.g. Twarog et al. 1997; Lepine et al. 2011). Improved cluster distance and metallicity estimates would be of paramount importance in probing this picture of the outer Galactic disk chemical properties. Metallicity, in particular, has never been measured reliably, but only inferred either from photometric indices or via comparison with isochrones (Piatti et al. 2004).
With these goals in mind, we present new Strömgren $uvbyCa$H$\beta$ photometry for a large field around the cluster and high resolution spectroscopy of 10 potential cluster stars (9 red giant clump stars and 1 Cepheid). This unique dataset will be used to provide precise estimates of the cluster basic parameters, in particular, reddening, distance, age, overall and elemental metallicity.
The paper is organized as follows: Section 2 gives details on the limited investigation of the cluster to date. In Section 3 we present the photometric and spectroscopic observations and their reduction, as well as a description of the radial-velocity determinations for membership. Section 4 details the derivation of the cluster properties from photometry, while Section 5 is devoted to the spectroscopic abundance analysis. In Section 6 we discuss the results of our spectroscopic analysis in detail. Section 7 interprets Tombaugh 1 in the context of galactic evolution and summarizes our findings.
Tombaugh 1: Background
======================
Tombaugh discovered Tombaugh 1 and Tombaugh 2 during the trans-Neptunian planet search at Lowell Observatory (Tombaugh 1938, 1941), reporting an apparent diameter for Tombaugh 1 of $\sim5^{\prime}$, with typical cluster stars having a visual magnitude of 14-15. His short description of the cluster notes that the field in the direction of Tombaugh 1 is extremely rich, suggesting that this might be the reason why the cluster went undetected for so long.
As discussed in Turner (1983); Haffner (1957) and Tifft (1959) independently rediscovered Tombaugh 1, reporting discrepant values for the cluster declination; the Haffner (1957) position is incorrect. Tifft (1959) noted the cluster because the Cepheid XZ CMa lies about one cluster diameter northward of the cluster center. Turner (1983) provided the first estimates for the cluster fundamental parameters from analysis of $UBV$ photoelectric photometry of 26 stars, including 5 likely members and 10 possible members selected by radial location in the cluster and position in the two-color diagram, the $UBV$ diagram also indicating $E(B-V)$ = 0.27 $\pm$ 0.01 mag. Turner (1983) measured a cluster diameter of $\sim$10$^{\prime}$ and estimated a distance and age of 1.26 kpc ($(m-M)$ = 11.34 $\pm$ 0.04) and $\sim$800 Myr, respectively, using the Hyades cluster adjusted to solar metallicity as a reference, though the sparse photometry extended barely 1.5 mag below the top of the turnoff. Lastly, Turner (1983) suggested that Tombaugh 1 hosts a probable blue straggler star (BSS), later confirmed by Ahumada & Lapasset (1995).
Turner (1983) also investigated the membership of Cepheid XZ CMa with Tombaugh 1. Turner suggested that XZ CMa is not a member of Tombaugh 1. However, this conclusion was never subjected to a more rigorous analysis based in high-resolution spectroscopy. So, we also analyzed spectroscopically XZ CMa to confirm or exclude the cluster membership. In fact, cluster Cepheids are important to fix the distance scale (see e.g., Majaess et al. 2013a; Majaess et al. 2013b).
The first CCD study of Tombaugh 1, limited to $VI$, was carried out by Carraro & Patat (1995), covering an area 6$^{\prime}$ on a side, essentially the cluster core (Turner 1983). Very different values for some of the cluster basic parameters were found: reddening, distance, and age of $E(B-V)$ = 0.40 $\pm$ 0.05, 3 kpc ($(m-M)$ = 13.60 $\pm$ 0.2), and 1 Gyr, respectively, tied to color-magnitude diagram (CMD) matches to theoretical isochrones. While this study dealt with only two filters and field star contamination makes it difficult to identify the cluster turnoff clearly, the parameter differences are not unexpected given the sparse sample of the earlier study.
Piatti et al. (2004) presented a more extensive CCD study using Washington photometry, covering a large area around the cluster. The cluster was found to be 1.3 Gyr old from a combination of CMD morphology and isochrone fits, assuming \[Fe/H\] = $-$0.40, with distance and reddening estimates intermediate between the Turner (1983) and Carraro & Patat (1995) values. An attempt was also made to directly measure the metallicity using Washington photometry, obtaining \[Fe/H\] = $-$0.30 with a large uncertainty of $\pm$0.25 dex.
Finally, on the basis of stellar data from PPMXL[^2] and 2MASS, Kharchenko et al. (2013) obtained some spatial, structural, kinematic, and astrophysical parameters of Tombaugh 1. In particular, they determined for Tombaugh 1 an age of 1.16 Gyr, a reddening $E(B-V)$ = 0.281 and a distance of 2642 pc ($(m-M)$ = 12.98), values similar to those obtained by Piatti et al. (2004). Kharchenko et al. (2013) also estimated average proper motion in right ascension ($-$0.99 mas/yr) and in declination (3.97 mas/yr), but didn’t estimate the average radial and galactic space velocities of Tombaugh 1.
Observations and Data Reduction
===============================
Photometry
----------
Photometry for Tombaugh 1 was secured in December 2010, during a 5-night run using the Cerro Tololo Inter-American Observatory 1.0m telescope operated by the SMARTS consortium[^3]. The telescope is equipped with an STA $4064\times4064$ CCD camera[^4] with 15-$\mu$m pixels, yielding a scale of 0.289$^{\prime\prime}$/pixel and a field-of-view (FOV) of $20^{\prime} \times 20^{\prime}$ at the Cassegrain focus of the telescope.
In Table 1 we present the log of our Strömgren observations, together with exposure times and airmasses. A total of 75 images were acquired for Tombaugh 1. All observations were carried out under photometric conditions with good-seeing (0.8–1.2 arc sec). A sample image of the covered field is shown in Fig. 1.
Basic calibration of the CCD frames was done using the Yale/SMARTS y4k reduction script based on the IRAF[^5] package <span style="font-variant:small-caps;">ccdred</span>, and the photometry was performed using IRAF’s <span style="font-variant:small-caps;">daophot</span> and <span style="font-variant:small-caps;">photcal</span> packages. Instrumental magnitudes were extracted following the point spread function (PSF) method (Stetson 1987) using a quadratic, spatially-variable master PSF (PENNY function). Finally, the PSF photometry was aperture-corrected using corrections determined from aperture photometry of bright, isolated stars in the field.
Standard stars for the extended Strömgren system were observed on one of the photometric nights on which Tombaugh 1 was observed. We additionally employed observations of secondary standard fields in several open clusters, using the same telescope and instrument one year later, to derive the form of the calibration equations. The clusters observed in December 2011 were M67 (Nissen et al. 1987), NGC 2287 (Schmidt 1984) and NGC 2516 (Snowden 1975). The zeropoint for each calibration equation applied to Tombaugh 1 was anchored by observations of eight field star standards obtained on 7 December, 2010. Standard values were obtained from the catalogs of Olsen (1983, 1993, 1994) for $uvby$, from Hauck & Mermilliod (1998) for H$\beta$ values and from Twarog & Anthony-Twarog (1995) for $hk$ index values for the field star standards.
Table 2 summarizes the calibration equations’ slopes and color terms. Following a standard practice for Strömgren photometry, calibration of $(b-y)$ would require that separate slopes be determined for cooler dwarfs as distinct from warmer dwarfs and giants. Insufficient cool dwarf standards were observed to accomplish this. The slope listed in Table 2 is appropriate for giants and dwarfs with $(b-y)_0 \leq 0.42$; application of this slope to dwarfs redder than this is an unavoidable extrapolation. Bluer dwarfs represent the only class for which $m_1$ and $c_1$ calibrations could be established with any confidence. The errors listed in Table 2 represent the standard deviation of the calibrated values about the standard values for the field star standards, indicating the external precision of the zeropoints of the calibration equations. The final calibrated catalog was then cross-correlated with 2MASS to convert pixel (i.e., detector) coordinates into RA and DEC for J2000.0 equinox, thus providing 2MASS-based astrometry. An excerpt of the optical photometric table used in this investigation is illustrated in Table 3. Fig. 2 shows the trend of errors with $V$ magnitude. The $V$ and $b-y$ data remain below 0.02 mag uncertainty to the limit of Table 3 ($V$ = 18.5), while errors in the remaining indices begin to rise above this value at $V$ = 16.5, 17.25, 17.25, and 17.5 for $c_1$, $m_1$, $hk$, and H$\beta$, respectively.
Spectroscopy and Radial Velocities
----------------------------------
Over the night of January 5, 2010, we observed ten potential cluster stars (nine clump stars and one Cepheid, see Sec. 4.1) with the *Inamori-Magellan Areal Camera & Spectrograph* (IMACS, Dressler et al. 2006) attached to the Magellan telescope (6.5 m) located at Las Campanas, Chile. The spectra were obtained using Multi-Object Echelle (MOE) mode with two exposures, one of 900s and other of 1200s. Our spectra have a resolution of R$\sim$20000, while the spectral coverage depends on the location of the star on the multi-slit mask, but generally goes from 4200 Å to 9100 Å. The detector consists of a mosaic with eight CCDs with gaps of about 0.93 mm between the CCDs, causing small gaps in stellar spectra.
The spectra were reduced following the standard procedures using IRAF, which includes CCD bias and flat-fielding correction, spectrum extraction, wavelength calibration and sky subtraction using the tasks <span style="font-variant:small-caps;">ccdproc</span>, <span style="font-variant:small-caps;">doecslit</span>, <span style="font-variant:small-caps;">ecidentify</span> and <span style="font-variant:small-caps;">background</span>, respectively. For each CCD, we performed bias and flat-fielding correction separately, after which we used the IRAF tasks <span style="font-variant:small-caps;">imcreate</span> and <span style="font-variant:small-caps;">imcopy</span> to join the CCDs and create the mosaic. The cosmic rays were removed with the IRAF Laplacian edge-detection routine (van Dokkum 2001), and the radial velocities were obtained from the wavelength shift of the unblended absorption lines of Fe covering the entire wavelength range. The values of wavelength shift were measured via line-by-line comparison between observed and laboratory wavelength with the lines center of observed wavelength being determined through the task <span style="font-variant:small-caps;">splot</span> in IRAF. To derived the final radial velocities we applied a zero-point offset correction using the task <span style="font-variant:small-caps;">fxcor</span> in IRAF to cross-correlate the telluric lines of the observed spectra with telluric lines of the high-resolution FEROS solar spectrum collected by us in a previous run (Moni Bidin et al. 2012). To calculate the heliocentric velocities and combine the spectra of different exposures, we used the IRAF tasks <span style="font-variant:small-caps;">rvcorrect</span> and <span style="font-variant:small-caps;">scombine</span>, respectively. We took the star’s heliocentric radial velocity to be the average of the two epochs measured and the error to be the difference between the two values multiplied by 0.63 (small sample statistics; see Keeping 1962). The nominal S/N ratio was evaluated by measuring with IRAF the rms flux fluctuation in selected continuum windows. The values at 6000 Å are shown in Table \[sample\].
Table \[sample\] gives some information about the observed stars: IDs (Carraro & Patat 1995), right ascension, declination, $V$ and $b-y$ from Table 3 and $V-I$ photometry (Carraro & Patat 1995), heliocentric radial velocities (RV$_{1}$ and RV$_{2}$) at two epochs and their mean values ($\langle$RV$\rangle$), projected rotational velocities (vsini) and spectral signal-to-noise at 6000 Å. We estimated the projected rotational velocities, vsini, by a spectral synthesis technique using unblended Fe lines analyzed with model atmospheres, a macroturbulent velocity of 3 km s$^{-1}$, limb darkening and instrumental broadening corresponding to IMACS spectral resolution. For some stars with low vsini it was possible to determine only an upper limit because of the insensitivity of the spectral synthesis to vsini below 2.7 km s$^{-1}$ .
In the literature, there is no information about the radial velocity of Tombaugh 1. So, to determine the membership of stars, we first found for a group of stars with similar heliocentric radial velocities (RV) in the sample, to have a preliminary cluster radial velocity and a list of members, that could be iteratively refined. The stars with RV within 2$\sigma$ of the cluster mean heliocentric radial velocity were classified as member of Tombaugh 1. The membership of the Cepheid XZ CMa (star 806) was not defined by its heliocentric radial velocity, because its RV is variable due to pulsations. So, we classified XZ CMa as non-member of Tombaugh 1 because its metallicity (\[Fe/H\]=$-$0.53) is much lower than metallicity of stars classified as member of Tombaugh 1 (see Table 9). We identify six red clump giants belonging to Tombaugh 1 and derive a mean cluster heliocentric radial velocity of 81.1 $\pm$ 2.5 km s$^{-1}$.
Cluster Parameters from Photometry
==================================
As discussed previously, one of the primary challenges in identifying and studying Tombaugh 1 is isolating the moderately populated cluster from the rich stellar background. This is particularly important for defining directly the fundamental cluster parameters of reddening and metallicity and indirectly the distance and age. The challenge is illustrated in Fig. 3, where the $V, b-y$ CMD for the entire field of study is presented. Red giants observed as part of this investigation and found to be probable radial-velocity members are plotted as red starred points; probable nonmember are open red triangles. The complexity of the stellar population mix in this region of the galaxy is evident and will be discussed in detail in a future paper. To enhance our definition of the cluster, we first reinvestigate the radial profile of the cluster.
Star Counts and Cluster Size
----------------------------
To quantify the amount of field star contamination, we performed star counts to derive an estimate of the cluster center and size. Using an array of positions covering the field of the CCD, we derived a density contour map and calculated the density inside each grid step by a kernel estimate (Carraro et al. 2014c). This is shown in Fig. 4, which confirms the appearance of Fig. 1 that Tombaugh 1 is far from being a symmetric object. The cluster looks elongated in the direction NE to SW, and the highest peak does not represent the center of a uniform star distribution. The largest peak is located at RA = 105$^{o}$.11, DEC = $-$20$^{o}$.58, while the nominal center of the cluster is clearly displaced to the northeast direction at RA = 105$^{o}$.13, DEC = $-$20$^{o}$.54. The loose and irregular shape of Tombaugh 1 may be the result of its dynamical evolution due to its tidal interaction with the Milky Way. However, little kinematic information beyond the cluster radial velocity exists to confirm this scenario. High quality proper motions could go a long way to defining the direction of the cluster motion and test if this coincides with the direction of the apparent cluster deformation, indicating if Tombaugh 1 has indeed been tidally disturbed.
To isolate probable cluster members, assumed to be those stars which lie within the cluster boundaries, we derive the cluster radial surface density profile shown in Fig. 5. This has been computed by drawing concentric rings centered on the nominal cluster center. This is motivated by the fact that, while the densest central regions look distorted, the cluster halo still retains a more spherical profile. Star counts level off at $\sim$4$^{\prime}$ from the cluster nominal center, close to the value reported by Turner (1983). The mean density in the field surrounding the cluster is 5 stars/arcmin$^2$ (see also Fig. 4), and our survey covers the whole cluster area. As a consequence, in the following we will adopt 4$^{\prime}$ as the cluster radius and refer to this area as the cluster area, while the area outside 4$^{\prime}$ from the cluster center will be referred to as the offset field.
Photometric Reddening and Metallicity
-------------------------------------
In the absence of membership information for any stars beyond those in Table 4, we can enhance the likelihood of including cluster members in our sample by restricting the analysis to stars within 3.5$\arcmin$ of the cluster center, just short of the transition region from the cluster to the field based upon star counts. Fig. 6 shows the $V, b-y$ CMD for stars within this core; all red giant members, independent of radial location, are plotted as stars, while open triangles are probable nonmembers. The cluster’s turnoff region and the blue edge of the main sequence are well-defined to the limit of the survey. The color spread at the top of the turnoff and the color and magnitude differential between the turnoff and the giant branch are very reminiscent of NGC 5822, a cluster of slightly subsolar metallicity with an age of 0.9 Gyr (Carraro et al. 2011). An additional means of demonstrating the cluster population comes from the $V, hk$ CMD for the core region, shown in Fig. 7. In the likely probability that there is a modest reddening range across the face of the cluster, this CMD minimizes the impact due to the weak sensitivity of $hk$ to reddening but a strong sensitivity to temperature and metallicity changes, factors we will make effective use of below. The tight vertical band defining the cluster turnoff reflects this fact, while the steep slope in $V$ with $hk$ is indicative of the cluster age; the trend in $hk$ with decreasing $b-y$ plateaus as the stellar sample moves from F to A stars, leading to an almost vertical turnoff in the $V, hk$ CMD for clusters of intermediate age, as illustrated by NGC 5822 in Fig. 13 of Carraro et al. (2011). Among the giants, the positions of the two faintest radial-velocity members, 663 and 1349, place them redward of the majority of the probable clump stars and indicates that, despite their similar velocities, they are probable field stars. In the absence of more information, they will be retained in the discussions below.
To further isolate probable members for defining the reddening and metallicity, we restrict our sample to stars populating the blue edge of the cluster main sequence between $V$ = 15.50 and 17.0. The bright cutoff eliminates the evolved stars that populate the color spread at the top of the main sequence while the faint boundary defines the magnitude range where errors in the color indices begin to increase for $m_1$, $hk$, and especially $c_1$. The blue edge of the main sequence in this magnitude range was used to define a single-star boundary and any star within $\sim$0.4 mag of the boundary was classified as a single star (blue circles in Fig. 8). Stars between 0.4 and 0.8 mag were classed as probable binaries (black squares in Fig. 8), if members, while all stars more than 0.8 mag beyond the main sequence were tagged as likely field stars (red crosses in Fig. 8).
We can check this classification using the $V, hk$ CMD of Fig. 9. Stars can be located off the main sequence for a variety of reasons: bad and/or contaminated photometry, binarity, excessive reddening compared to the typical cluster star and/or nonmembership. As noted earlier, the $hk$ index is very sensitive to color changes due to temperature and only weakly impacted by reddening. In fact, increased reddening moves a star blueward in $hk$. As shown in Fig. 9, the separation by class as defined by Fig. 8 is well corroborated. With only four obvious exceptions, the single stars form a well-defined turnoff band covering a modest range in $hk$. The stars redward of the main sequence band in Fig. 9 are dominated by the stars tagged as redder in Fig. 8, indicating that these are truly redder than the cluster sequence and that the majority are probable nonmembers. Three red crosses which sit on the main sequence in Fig. 9 deserve some explanation. These are either highly reddened field stars or, more likely, field stars in the direction of the cluster with significantly lower metallicity than Tombaugh 1. For metallicity and reddening estimation, we will limit the sample to the single stars (blue circles), with the four points which deviate from the cluster main sequence in Fig. 9 excluded.
For consistency with past cluster work, we will derive the reddening from two Strömgren relations from Olsen (1988) and Nissen (1988), a slightly modified version of the original relations derived by Crawford (1975, 1979). Reddening estimates are derived in an iterative fashion. The indices are corrected using an initial guess at the cluster reddening and the intrinsic $b-y$ is derived using the reddening-free H$\beta$ adjusted for metallicity and evolutionary state. A new reddening is derived by comparing the observed and intrinsic colors and the procedure repeated. The reddening estimate invariably converges after 2–3 iterations. To derive the reddening, one needs to correct $b-y$ for metallicity, so a fixed \[Fe/H\] is adopted for the cluster and the reddening derived under a range of \[Fe/H\] assumption that bracket the final value. The complementary procedure is to vary the mean reddening value for the cluster and derive the mean \[Fe/H\]. Ultimately, only one combination of $E(b-y)$ and \[Fe/H\] will be consistent.
For Tombaugh 1, the metallicity from $m_1$ was varied between \[Fe/H\] = $-$0.28 and $+$0.12, generating a range of $E(b-y)$ = 0.224 to 0.214 for the relation of Olsen (1988) and 0.223 to 0.216 for Nissen (1988) from 51 stars within the H$\beta$ calibration range. In all cases, the standard error of the mean for the final $E(b-y)$ is $\pm$ 0.006 mag. For \[Fe/H\] from $m_1$ equal to $-$0.16, the reddening from the two relations is virtually identical at $E(b-y)$ = 0.221 $\pm$ 0.006; the difference between the two reddening values is statistically insignificant compared to the standard errors of the mean. If $E(b-y)$ = 0.73\*$E(B-V)$, the reddening estimate from Strömgren data alone is $E(B-V) = 0.303 \pm 0.008$. There is weak evidence for a variation in $E(b-y)$ across the face of the cluster, with the reddening being higher on average by 0.03 mag in the southwest and lower by a comparable amount in the northeast. Without more membership information, for purposes of deriving the cluster parameters, we will adopt the cluster mean for all stars.
With the reddening set, we can derive the metallicity from both $m_1$ and $hk$, using H$\beta$ as the primary temperature index. From 51 stars, \[Fe/H\] = $-$0.165 $\pm$ 0.027 from $m_1$ and $-$0.086 $\pm$ 0.013 from $hk$. If one anomalous measurement located more than three sigma from the cluster mean is removed from the $m_1$ analysis, the revised \[Fe/H\] becomes $-$0.153 $\pm$ 0.025. The greater uncertainty in the metallicity estimate from $m_1$ relative to $hk$ is a reflection of the greater sensitivity of $m_1$ to reddening changes and a lower sensitivity to metallicity variation; the small difference in \[Fe/H\] can be entirely explained by a zero-point offset of 0.005 mag in $m_1$. Weighting the two photometric estimates by the inverse square of the errors leads to a final value of \[Fe/H\] = $-$0.10 $\pm$ 0.02.
Age and Distance Estimation
---------------------------
One of the rare sets of available isochrones which include models transformed to the Strömgren system is the Victoria-Regina (VR) set of isochrones (VandenBerg et al. 2006). Fig. 10 shows the scaled-solar models for \[Fe/H\] = $-$0.11, ages 0.8, 0.9, and 1.0 Gyr, adjusted for $E(b-y)$ = 0.221 and $(m-M)$ = 13.10. Symbols have the same meaning as in Fig. 6. The already noted similarity of Tombaugh 1 to NGC 5822 is confirmed. In addition to the similar scatter in $b-y$ at the top of the turnoff, the best fit age estimate is between 0.9 and 1.0 Gyr; the best fit to a different set of broad-band isochones for NGC 5822 produced an age of 0.9 $\pm$ 0.1 Gyr for the more populated cluster (Carraro et al. 2011). The reddening-corrected true distance modulus is $(m-M)_o$ = 12.15, in excellent agreement with the most recent work of Kharchenko et al. (2013).
Atmospheric Parameters and Abundances Analysis
==============================================
The equivalent width measurements of absorption lines of Na, Mg, Al, Si, Ca, Ti, Cr, Ni and Fe were used to obtain their respective chemical abundances while the abundances of Y, Ba, Ce and Nd were derived through spectral synthesis. The equivalent widths were measured manually using the task <span style="font-variant:small-caps;">splot</span> in IRAF to fit a Gaussian profile to the observed absorption line. We rejected the absorption lines with equivalent widths greater than 160 mÅ because these lines are saturated, which prevents a Gaussian fit to the absorption lines (Pereira et al. 2011). All equivalent widths used to obtain the atmospheric parameters and chemical abundances are shown in the Tables \[tabelFea\], \[tabelFeb\], \[tabellinesa\] and \[tabellinesb\].
The atomic-line list adopted in this work is the same as the one used by Santrich et al. (2013) and Sales Silva et al. (2014). For Ba II line, the hyperfine structure (HFS) was taken into account and we used the line list of Carraro et al. (2014b). In Tables \[tabelFea\] and \[tabelFeb\] we describe the line list with excitation potential ($\xi$) and oscillator strength ($gf$) for absorption lines of Fe[i]{} and Fe[ii]{}. The values of the oscillator strength adopted for the Fe[i]{} and Fe[ii]{} lines were taken from Lambert et al. (1996) and Castro et al. (1997). Tables \[tabellinesa\] and \[tabellinesb\] show the atomic parameters ($gf$ and $\xi$ values) of the absorption lines of the elements Na, Mg, Al, Si, Ca, Ti, Cr and Ni with their respective references (column 5). Atomic parameters for several transitions of Ti, Cr, and Ni were retrieved from the National Institute of Science and Technology Atomic Spectra Database (Martin et al. 2002). For Na we used only two absorption lines, 6154.226 Å and 6160.747 Å. These Na lines have a clean profile which makes it possible to calculate the chemical abundance of Na through the equivalent width (Smiljanic 2012). The absorption lines used to obtain s-process elements abundances were 5289 Å and 5402 Å for Y, 5853 Å for Ba, 5117 Å and 5187 Å for Ce and 4914 Å for Nd.
The atmospheric parameters and chemical abundances were obtained in the same manner as in Pereira et al. (2011), Santrich et al. (2013), and Sales Silva et al. (2014) using the local thermodynamic equilibrium (LTE) model atmospheres of Kurucz (1993) and the spectral analysis code MOOG (Sneden 1973). Excitation equilibrium was used to derive the effective temperature ($T_{\rm eff}$) as defined by a zero slope of the trend between the iron abundance derived from Fe[i]{} lines and the excitation potential of the measured lines. Microturbulent velocity was adjusted until both the strong and weak Fe[i]{} lines (represented by reduced equivalent width, $W_{\lambda}/{\lambda}$) gave the same abundance. Finally, the surface gravity was determined using the ionization equilibrium found from the equality of the abundances of Fe[i]{} and Fe[ii]{}. The final adopted atmospheric parameters are given in Table \[tab:atmparam\].
The uncertainty in the slopes of the Fe[i]{} abundance versus excitation potential and Fe[i]{} abundance versus reduced equivalent width were used to derive the uncertainties in our adopted effective temperatures ($T_{\rm eff}$) and microturbulent velocities ($\xi$), respectively. The standard deviation in $\log g$ was set by changing this parameter around the adopted solution until the difference between Fe[i]{} and Fe[ii]{} mean abundance differed by exactly one standard deviation of the \[Fe[i]{}/H\] mean value. We estimated typical uncertainties in atmospheric parameters of the order of $\pm$180K, $\pm$0.3 dex, and $\pm$0.3 kms$^{-1}$ for $T_{\rm eff}$, $\log g$ and $\xi$, respectively.
We also calculated the photometric effective temperature and photometric gravity to compare with our spectroscopic temperature and gravity. Photometric temperatures were calculated using the calibration of Alonso et al. (1999) and our values of $(b-y)$ with $E(b-y)$ = 0.221. The photometric gravity for each star was obtained from the equation:
$$\begin{aligned}
\log g_{\star}\; & = & \log \frac{M_{\star}}{M_{\odot}}
+ 0.4\left(V-A_{ V}+BC\right) \nonumber \\
& & {{\,}\atop{\,}} + 4\log T_{\rm eff} - 2 \log r\; ({\rm pc}) - 10.62.\end{aligned}$$
Where $T_{eff}$ is the photometric effective temperature and $M$ is the mass. Based upon the VR isochrones and an age of 0.95 Gyr, the typical mass for a star in the color range of the likely member red giants is $2.15 M_\odot$. The photometric data of Table 4 were combined with an adopted distance of $r = 2700$ pc and bolometric corrections ($BC$) defined by the relations of Alonso et al. (1999). For the Sun we adopted $M_{bol \odot} = 4.74$ mag (Bessell et al. 1998), $T_{\rm eff \odot} = 5700$ K and $\log g_\odot = 4.3$ dex.
It should be emphasized that for the nonmembers stars, adoption of the cluster parameters for distance, reddening, and metallicity will likely generate discordant results when compared to the spectroscopic parameters. For the six probable members, the temperature difference, in the sense (spectroscopic - photometric), is 52 $\pm$ 196 K, while the residuals in $\log g$ are 0.22 $\pm$ 0.33, consistent with the probable uncertainties in the estimates from the spectra, discussed above, and from the photometry. The modest offsets in temperature and gravity between the spectroscopic and photometric approaches are typical of such comparisons. Different methods are known to produce systematic offsets from each other, but there is no consensus on the source these offsets (e.g. Allende Prieto et al. 1999; Frebel et al. 2013).
The determination of the atmospheric parameters (Table \[tab:atmparam\]) and the knowledge of the atomic parameters of the absorption lines enables us to obtain the chemical abundance by measuring the equivalent widths or by spectral synthesis. In the case of equivalent widths, MOOG uses atmospheric and atomic parameters, as well as equivalent widths measurements, to calculate the chemical abundance. For spectral synthesis, as input for MOOG we supply the atmospheric and atomic parameters and an estimate of the chemical abundance of the elements that influence the absorption line studied. Thereafter MOOG generates a synthetic spectrum which is compared with the observed spectrum, iterating until we find a chemical abundance that makes the synthetic spectra and observed identical.
Tables \[abunda-Na\] and \[abunda-Ni\] show the chemical abundances of Na, Al, Fe-peak, alpha and s-process elements in the notation \[X/Fe\] and its standard deviation. We analyzed a high-resolution FEROS solar spectrum to obtain the atmospheric parameters and solar abundance with the same methodology applied to red clump stars of Tombaugh 1. We found the following values for solar atmospheric parameters: $T_{\rm eff \odot}$ = 5700 K, $\log g_\odot$ = 4.3 dex and $\xi_\odot$ = 0.9 kms$^{-1}$. Pavlenko et al. (2012) found similar values of 5777 K, 4.44 dex and 0.8 kms$^{-1}$ for the effective temperature, surface gravity and microturbulent velocity, respectively. In Table \[sun\] we show our solar chemical abundances together with those given by Grevesse & Sauval (1998) and Asplund et al. (2009) for comparison. The adopted abundances for the elements analyzed in this work were normalized to our solar abundances. In the seventh row of Tables \[abunda-Na\] and \[abunda-Ni\] we show the mean chemical abundance of Tombaugh 1 for each element with their respective standard deviations.
The approach to estimate the uncertainties in abundance consists in determining how the abundances for each element react to the errors associated with each atmospheric parameter, independent of the others. After that we combine quadratically all these errors and set this result as the total abundance uncertainty. These total uncertainties are given in the 5th column of Table \[error\] for star 769. We chose the star 769 to determine the abundance uncertainties for being one of the cluster giants that had the greatest number of elements with derived chemical abundance. The uncertainties for the aluminium weren’t obtained for star 769 because their absorption lines are located in the spectral gaps, so we used star 663 to calculate the aluminium uncertainties. The uncertainties in abundance for the other stars generate similar values.
Results of Abundance Analysis
=============================
In this section we discuss the results of our chemical analysis via comparison with the chemical abundances of field giants stars and open clusters from the literature.
Metallicty and Iron-peak Elements
---------------------------------
In Table \[tab:atmparam\] we show the metallicities obtained for our giants. The range of metallicity for the six stars classified as members of Tombaugh 1 is $-$0.16 to 0.10 dex, with the mean of $-$0.02 $\pm$ 0.05 dex. The spectroscopic values are consistent with photometric value of $-$0.10 $\pm$ 0.02. A weighted average of the two approaches gives a final \[Fe/H\] = $-$0.08 for Tombaugh 1. Comparison with past abundance estimates provides little insight given the large uncertainty in previous published estimates of this cluster parameter.
In Figures 11, 12 and 13 we show the abundance ratio of \[X/Fe\] versus metallicity for our sample of giants, for giants from Mishenina et al. (2006) and Luck & Heiter (2007), and also for the open clusters: NGC 6192, NGC 6404 and NGC 6583 (Magrini et al. 2010); NGC 3114 (Santrich et al. 2013); NGC 2527, NGC 2682, NGC 2482, NGC 2539, NGC 2335, NGC 2251 and NGC 2266 (Reddy et al. 2013); Trumpler 20 (Carraro et al. 2014b); NGC 4337 (Carraro et al. 2014d); NGC 4815 and NGC 6705 (Magrini et al. 2014); Cr 110, Cr 261, NGC 2477, NGC 2506 and NGC 5822 (Mishenina et al. 2015). For the s-process elements of Fig. 13, we added data of the open clusters Berkeley 25, Berkeley 73, Berkeley 75, Ruprecht 4, Ruprecht 7, NGC 6192, NGC 6404 and NGC 6583 from Mishenina et al. (2013).
From our chemical analysis of Tombaugh 1 we derive the following mean abundance ratios \[X/Fe\] for Cr and Ni: 0.10$\pm$0.06 and $-$0.04$\pm$0.02 dex, respectively. Our \[Cr/Fe\] and \[Ni/Fe\] of Tombaugh 1 are in good agreement with disk field giants and open clusters from literature as demonstrated in Figure 11.
Na, Al and alpha elements
-------------------------
Na is synthesized during hydrostatic carbon-burning in massive stars and also through the NeNa cycle during H-burning through the CNO-cycle in intermediate-mass and massive stars (Woosley & Weaver 1995; Denisenkov & Denisenkova 1990). The chemical analysis of Sodium must be performed taking into account NLTE effects, these effects being greater for higher equivalent widths and lower gravities (Gratton et al. 1999; Lind et al. 2011; Smiljanic 2012). In order to account for the NLTE effects in the Na abundances we used the corrections of Gratton et al. (1999). These corrections were typically smaller than 0.10 dex, with higher values for giants with lower $\log g$ (stars 784 and 1534). With this NLTE correction the range in the abundance ratio \[Na/Fe\] for the red clump stars of Tombaugh 1 goes from 0.38 to 0.05 dex, with a mean value of 0.17 $\pm$ 0.06. Star 1534, classified as a field giant, showed the strongest NLTE effects with a correction of 0.22 dex, mainly due to its low surface gravity ($\log g$ = 2.0).
Chemical mixtures in the stellar interior can significantly modify the surface \[Na/Fe\] (e. g. Charbonnel & Lagarde 2010). Comparing Tombaugh 1 with the models of Charbonnel & Lagarde (2010), the mean cluster overabundance of \[Na/Fe\] = 0.17 among the giants is in excellent agreement with the values expected for models with thermohaline and rotation-induced mixing: \[Na/Fe\] = 0.18 for M = 2.0 M$_\odot$ and rotational velocities of 110 kms$^{-1}$ on the ZAMS. The range of \[Na/Fe\] among the giants of Tombaugh 1 could be explained by a range of rotation velocities among the stars in ZAMS which produced the giants (Charbonnel & Lagarde 2010).
The production of Al, Mg, Si, Ca and Ti occurs mainly in massive stars whereas the production of the iron-peak elements is dominated by SN Type Ia (e. g. Woosley & Weaver 1995; Iwamoto et al. 1999). Thus, the chemical ratio of Al and alpha-elements with Fe can give us important information about the SNIa and SNII contributions to the galactic components (bulge, disk and halo). The mean abundances of Mg, Si and Ca relative to Fe for Tombaugh 1 show essentially solar values of $+$0.03$\pm$0.05, $+$0.01$\pm$0.07 and $+$0.01$\pm$0.03, respectively. In the case of \[Ti/Fe\], we found for Tombaugh 1 a slightly overabundant value relative to the sun with a mean of $+$0.11$\pm$0.04 dex. Our values of \[X/Fe\] for alpha elements in Tombaugh 1 are consistent with the disk giants of Luck & Heiter (2007) and Mishenina et al. (2006) and also with open clusters with similar metallicity of literature (Figure 12). The decay of the \[X/Fe\] ratio to alpha elements with increasing of metallicity in the disk stars, as observed in the Figure 12, can be explained by the SNIa yields (Iwamoto et al. 1999), i.e. by high creation of Fe and low generation of alpha-elements. \[Al/Fe\] for Tombaugh 1 is similar to \[Ti/Fe\], with a mean of $+$0.15, in agreement with the chemical pattern of Al in the galactic disk (Figure 12).
Neutron-capture elements
------------------------
The elements Y, Ba, Ce and Nd are formed mainly in the stellar interior by slow neutron-capture process (s-process) during the asymptotic giant branch (AGB) phase and are transported to the stellar surface by the third dredge-up (Busso et al. 1999). In Tombaugh 1, the light s-process element,Y, has a near solar \[X/Fe\] mean of $+$0.06 $\pm$ 0.04 dex while \[X/Fe\] for heavy s-process elements (Ba, Ce and Nd) shows an excess compared to the sun, with a mean of $+$0.35 $\pm$ 0.03 for Ba, $+$0.25 $\pm$ 0.06 for Ce, and $+$0.37 $\pm$ 0.05 for Nd. The difference between light and heavy s-process elements is an indicator of s-process efficiency (e. g. Luck & Bond 1981, 1991; Busso et al. 2001; Pereira et al. 2011), implying a high s-process efficiency for Tombaugh 1. Other open clusters exhibiting this same behavior include the Hyades (De Silva et al. 2006), Berkeley 18, Berkeley 21, Berkeley 22 and Berkeley 32 (Yong et al. 2012), Ruprecht 4, Ruprecht 7, NGC 6192 and NGC 6404 (Mishenina et al. 2013), among others. The s-process efficiency is an important observational constraint to stellar evolutionary models (e.g. Busso et al. 2001) and is affected by metallicity, stellar mass and rotational velocity (e.g. Lugaro et al. 2003; Herwig et al. 2003).
Abundance measurements for s-process elements from the literature are highly inhomogeneous and difficult to compare with our results due to the use of different absorption lines, atomic parameters, and analysis methods (see. e.g., Yong et al. 2012 for a detailed discussion). Nevertheless, our s-process abundances for Tombaugh 1 agree with published s-process abundances for open clusters, as shown in Figure 13. Only our neodymium abundances show a slight overabundance with respect to open clusters and disk field giants from the literature.
The peculiar Tombaugh 1 field Cepheid XZ CMa
--------------------------------------------
XZ CMa (star 806 in Table 4) is a short-period Cepheid (P=2$^{d}$.56, Caldwell & Coulson, 1987) situated within of coronal region of Tombaugh 1 but classified as not cluster member (see section 3.2). Three papers in the literature analyzed in detail the Cepheid XZ CMa (Turner, 1983; Diethelm, 1990; Yong et al. 2006). Turner (1983) and Diethelm (1990) conducted a photometric analysis of XZ CMa, while Yong et al. (2006) analyzed XZ CMa with a high-resolution spectroscopy. Turner (1983), via UBV photoelectric photometry, defined XZ CMa as unlikely member of Tombaugh 1 and found that XZ CMa probably has an unresolved blue companion which is aprox. 2.5 magnitudes fainter in V, due the phase of minimum in the U-V curve is shifted from the phase of minimum light by roughly 0.2 to 0.3 of the star’s period. Subsequently, based in Walraven VBLUW photometric system, Diethelm (1990) derived the mean atmospheric parameters ($T_{\rm eff}$, $\log g$ and \[Fe/H\]) of XZ CMa, obtaining $T_{\rm eff}$=6000 K, $log g$=2.3 (dex) and \[Fe/H\]=$-$0.50$\pm$0.10, with differences between our atmospheric parameters and Diethelm (1990) values of $\Delta T_{\rm eff}$=0 K, $\Delta log g$=0.4 (dex) and $\Delta [Fe/H]$=0.03 (dex).
Lastly, Yong et al. (2006) determined the atmospheric parameters and the chemical abundances of three alpha-elements (Si, Ca and TiII) to Cepheid XZ CMa, using same method but different line-list that used in this work. Our atmospheric parameters $T_{\rm eff}$, $\log g$ and $\xi$ exhibit different values from those found by Yong et al. (2006), with differences of 750 K, 1.12 (dex) and 1.58 kms$^{-1}$, respectively. However, we and Yong et al. (2006) obtained similiar values of metallicity to XZ CMa ($\Delta [Fe/H]$=0.04). Probably, the difference of $T_{\rm eff}$, $\log g$ and $\xi$ displayed in this work and in Yong et al. (2006) is due to observation of Cepheid XZ CMa in distinct pulsation phase, which causes the determination of different values of atmospheric parameters ($T_{\rm eff}$, $\log g$ and $\xi$) and similar metallicity. Finally, in both studies an overabundance of alpha elements in XZ CMa was found, with mean of alpha elements in our analysis of \[$\alpha$/Fe\]=0.13 and in Yong et al. (2006) of \[$\alpha$/Fe\]=0.21, characteristic of Cepheid stars in the outer disk (e.g., see Fig 15 of Yong et al. 2006).
Our results show that Cepheid XZ CMa has a chemical pattern similar to that presented by disk field stars and open clusters (see Fig. 11 and 12). However, in Figure 13 we note that the Tombaugh 1 field star XZ CMa exhibits a high overabundance of Ba compared with field giants from literature. To demonstrate the high Ba abundance in this star, in Figure 14 we present the observed and synthetic spectra in the region around the absorption line of Ba II 5853 Å. Classical Cepheids, like XZ CMa, are not expected to present a high overabundance of s-process elements, as Ba, since such stars not evolved to AGB; e.g., cepheids FO Cas, EW Aur, EE Mon and FF Aur with similar metallicity of XZ CMa presents the ratio \[Ba/Fe\] of 0.17, 0.24, 0.03 and 0.13, respectively (Andrievsky et al., 2014). The chemical abundances of Ba in disk Cepheids is known to suffer from NLTE effects (Andrievsky et al. 2013; Andrievsky et al. 2014). However, the NLTE correction for Ba II line 5853 Å is not especially large, averaging around $-$0.1 dex (Andrievsky et al. 2013), does not having any significant effect in the high overabundance obtained for XZ CMa. We will discuss the case of this star in the final section.
Discussion and Conclusions
==========================
In this paper we have presented the first study of Tombaugh 1 using both high-resolution spectroscopy and precision $uvbyCa$H$\beta$ photometry. Our results for the abundance ratios of elements from Na to Ni and the cluster fundamental parameters of distance and age tag this open cluster as an intermediate-age (0.95 Gyr) cluster belonging to the galactic thin disk. As such, it allows the addition of one more data point to the census of star clusters used to map the chemical history of the disk, falling within a galactocentric zone where there is universal agreement that a significant change in mean metallicity occurs among all classes of objects populating the thin disk. Where disagreement arises is in the exact form and location of the transition region. Does Tombaugh 1 lie along a uniform linear gradient extending from R$_{GC}$ = 5 kpc to 20 kpc, or does the gradient change slope beyond the solar circle? If it changes, where does the transition occur and why? The growing evidence from studies of distant anticenter open clusters and Cepheids (e.g. Magrini et al. (2009); Lepine et al. (2011); Yong et al. (2012); Korotin et al. (2014), among others) is that the metallicity gradient beyond R$_{GC}$ = 13 kpc is considerably flatter than that between 9 and 13 kpc (see Fig. 15).
In Figure 15 we show the radial metallicity gradient from Magrini et al. (2009) (blue points), with the addition of our spectroscopic results for Tombaugh 1 (red point) with \[Fe/H\] =$-$0.02$ \pm$ 0.05 and R$_{GC}$ = 10.46 kpc. If we use the lower photometric value of \[Fe/H\] =$-$0.10, R$_{GC}$ would be reduced to 10.36 kpc, a negligible shift in distance on this scale. Also plotted are additional open clusters analyzed with high-resolution spectroscopy (green squares): IC 4725 and NGC 6087 (Gratton 2000); NGC 6603, NGC 2539, NGC 2447, IC 2714 and NGC 5822 (Santos et al. 2009); NGC 6192, NGC 6404 and NGC 6583 (Magrini et al. 2010); NGC 7160 (Monroe & Pilachowski 2010); Cr 110, NGC 2099, NGC 2420 and NGC 7789 (Pancino et al. 2010); Tombaugh 2 (Villanova et al. 2010); NGC 3114 (Santrich et al. 2013); NGC 4815 and NGC 6705 (Magrini et al. 2014); NGC 4337 (Carraro et al. 2014d); Trumpler 20 (Carraro et al. 2014b). The use of the spectroscopic value alone is tied to an apparent offset between the photometric abundance scale, for Strömgren photometry tied to high dispersion spectroscopy of F dwarfs, and the red giant high-dispersion spectroscopic scale, often distantly coupled to the sun. The issue is apparent in Fig. 15 where, inside R$_{GC}$ = 9.8 kpc, no cluster has \[Fe/H\] below -0.1 and, more important, even ignoring the super-metal-rich outliers, the typical cluster \[Fe/H\] at all ages is $+$0.1. While a virtually identical pattern was found by Twarog et al. (1997), the lower limit and mean abundances from photometry and medium-resolution spectroscopy of cluster red giants were \[Fe/H\] =$-$0.2 and 0.0, respectively. Similar offsets between spectroscopic abundances of red giants and the photometry of F dwarfs have been found in NGC 3680 (Anthony-Twarog et al. 2009), NGC 5822 (Carraro et al. 2011), NGC 6819 (Anthony-Twarog et al. 2014), and NGC 752 (Twarog et al. 2015). In the cases of NGC 3680, NGC 6819, and NGC 752, high dispersion spectroscopic analysis of the F dwarfs agrees with the photometric abundances. If this offset to the spectroscopic scale applies to giants across all metallicities, the trend in Fig. 15 remains correct, even if the curve is shifted vertically by 0.1 dex.
We observe that Tombaugh 1 is consistent with the trend defined by Magrini et al. (2009) for the metallicity gradient, with Tombaugh 1 located in the inner disk (R$_{GC}\lesssim$ 12 kpc). The existence of an apparent transition zone ranging from R$_{GC}$ = 10 to 12 kpc between an inner and outer disk lends support to the contention that metallicity evolution in these two regions occurs in different ways (Magrini et al. 2009; Lepine et al. 2011). According to Lepine et al. (2011), this behavior is due to a barrier created by a void in the interstellar gas in the region of the corotation radius of the main spiral structure. This dynamical interaction produces an inward flow of the gas on the inside of the corotation zone of the Galaxy but an outward flow in the outer disk regions.
In recent years the abundances of the s-process elements in open clusters have become a target of intense study (e.g. D’Orazi et al. 2009, 2012; Jacobson et al. 2011; Maiorca et al. 2011; Jacobson & Friel 2013; Mishenina et al. 2013, 2015). This recent interest was sparked by the unexpected results of D’Orazi et al. (2009) for a sample of twenty open clusters. D’Orazi et al. (2009) found that \[Ba/Fe\] increases as cluster age decreases, contrary to the predictions of yields for Ba from AGB stars (e. g. Travaglio et al. 1999; Busso et al. 2001). Later work supplied confirmation for other s-process elements from unevolved stars in open clusters: Ba (Mishenina et al. 2013; Jacobson & Friel 2013), Ba and La (Jacobson et al. 2011), and Y, Zr, La, and Ce (Maiorca et al. 2011). However, Jacobson & Friel (2013) didn’t find a trend for \[X/Fe\] for La and Zr versus age for their sample of 19 open clusters, which could indicate that the source of the s-process abundance trend with age doesn’t affect all s-process elements equally. Among field stars, some s-process elements, Zr (Reddy et al. 2003) and Ba (Bensby et al. 2005), also show an increase in \[X/Fe\] with the decreasing age, while others, Y (Bensby et al. 2005), Ba and Ce (Reddy et al. 2003) do not.
In this context our photometric and spectroscopic analysis classifies Tombaugh 1 as intermediate age (0.95 Gyr), with an enrichment of heavy s-process elements (Ba with $+$0.35 $\pm$ 0.03 dex, Ce with $+$0.25 $\pm$ 0.06 dex and Nd with $+$0.37 $\pm$ 0.05 dex) and solar values to Y ($+$0.06 $\pm$ 0.04), indicating a high efficiency in the synthesis of the s-process elements. Some open clusters with similar ages show enrichment of the s-process elements similar to that found for Tombaugh 1, e.g. NGC 5822 (0.9 Gyr) (Carraro et al. 2011) and NGC 3680 (1.7 Gyr) (Anthony-Twarog et al. 2009) with \[Ce/Fe\] = 0.25 and \[Ce/Fe\] = 0.26, respectively (Maiorca et al. 2011).
The reason why open clusters younger than $\sim$1.5 Gyr (Maiorca et al. 2011) contain an overabundance of some s-process elements (mainly Ba) compared to the old open clusters still isn’t understood. D’Orazi et al. (2009) and Maiorca et al. (2011) have proposed a scenario with models of extra-mixing phenomena with high efficiency in the production of the neutron source $^{13}$C in stars with M $\leq$ 1.5M$_\odot$ (Busso et al. 2007; Nordhaus et al. 2008; Trippella et al. 2014; Nucci & Busso 2014). Very recently, Mishenina et al. (2015) suggested that the Ba overabundance in open clusters could be due to action from the intermediate neutron-capture process, or i-process (Cowan & Rose 1977). However, as Mishenina et al. (2015) pointed out, it remains difficult to know which open cluster stars would be the host of the i-process; low-metallicity stars are a more probable example of these hosts (Bertolli et al. 2013; Dardelet et al. 2015). Indeed, confirmation of the enrichment of s-process elements in young clusters requires the analysis of a large and homogeneous sample of young and old open clusters with well-determined s-process abundances.
The low number of open clusters with both reliable photometric and spectroscopic parameters, about 13.2% of the known open clusters as defined by the 2014 update of the Dias et al. (2002) catalog, is just one of the factors that hinder a definitive characterization of the galactic metallicity gradient, as well as its variation over time and azimuthally within the disk for individual elements. Studies of other poorly known open clusters like Tombaugh 1 using high-resolution spectroscopy and precision photometry to define reliably all of the key parameters that influence plots like Figs. 11, 12, 13 and 15 remain the key to forward progress in disentangling the complex system known as the galactic disk. The next step in this direction is being conducted by large surveys like Gaia-ESO mapping the chemistry of all the components of the Galaxy.
Finally, the overabundance of barium in Cepheid XZ CMa can be explained by an enhancement of s-process elements in the interstellar medium (ISM) which produced XZ CMa or by mass transfer in a multiple-star system. Yong et al. (2006) found an enhancement of La for a Cepheid sample in the outer disk and suggested that asymptotic giant branch stars have contributed to the chemical evolution of the outer Galactic disk. XZ CMa is situated at the beginning of the outer disk ($R_{GC}$=13 Kpc, Yong et al. 2006), which makes XZ CMa one of the cepheid candidates rich in s-process elements formed by this ISM suggested by Yong et al. (2006).
In a binary system, like Ba and CH stars, the enrichment of Ba is a consequence of mass transfer through stellar winds or through Roche-lobe overflow from an AGB star (now the white dwarf) to a less evolved companion. Turner (1983) suggested the presence of an unresolved blue companion B star to the Cepheid XZ CMa. However, the enrichment of Ba indicates a white dwarf companion to XZ CMa. Thus, we suggest that XZ CMa can belong to a binary system with a white dwarf or a triple system comprising a white dwarf and a B star. About one-third of Galactic Cepheids are known to have companions, and about 44% of those have more than one companion (Evans et al. 2005). Recently, in the study of the occurrence of classical cepheids in binary systems, Neilson et al. (2015) pointed out that a fraction of binary systems may evolve to a system composed of a Cepheid with a white dwarf companion. Harris & Welch (1989) commented that due the occurrence of mass transfer in binary Cepheids an evolutionary connection between Ba stars and binary Cepheids would be possible. In addition, Gonzalez & Wallerstein (1996) found significant similarities between binaries Cepheids, and Ba and CH stars, as orbital parameters and mass range.
UV observations of XZ CMa can be used to confirm its binarity and reveal the nature of its companion (e.g. Evans 1992). In the case of a hot companion to XZ CMa like B main-sequence star suggested by Turner (1983), the presence of a strong Balmer line, H$\epsilon$ (3970.07 Å), in the Cepheid spectrum also can be interpreted as the signature of this blue companion (Kovtyukh et al. 2015). Because of the wavelength coverage of our XZ CMa spectrum (4200 Åto 9000 Å) was not possible to perform this investigation. The discovery of binaries Cepheids is important because unresolved companions is one of the factors that contribute to the scatter around the ridge-line period-luminosity relationship (Szabados & Klagyivik, 2012). In particular, the detection of a Cepheid-white-dwarf binary will give important constraint regarding the most massive progenitors of white dwarfs (Landsman et al. 1996).
Extensive use was made of the WEBDA database maintained by E. Paunzen at the University of Vienna, Austria (http://www.univie.ac.at/webda). The filters used in the program were obtained by BJAT and BAT through NSF grant AST-0321247 to the University of Kansas. NSF support for this project was provided to BJAT and BAT through NSF grant AST-1211621. J.V. Sales Silva acknowledges the support provided by CNPq/Brazil Science without Borders program (project No. 249122/2013-8). C. Moni Bidin acknowledges support by the Fondo Nacional de Investigación Científica y Tecnológica (Fondecyt), project No. 1150060. E. Costa acknowledges support by the Fondo Nacional de Investigación Científica y Tecnológica (projecto No. 1110100, Fondecyt) and the Chilean Centro de Excelencia en Astrofísica y Tecnologías Afines (PFB 06).
[*Facilities:*]{} , .
Ahumada, J. E. & Lapasset, E. 1995, , 109, 375 Allende Prieto, C., Garc[í]{}a L[ó]{}pez, R. J., Lambert, D. L., & Gustafsson, B. 1999, , 527, 879 Alonso, A., Arribas, S., Martínez-Roger, C. 1999, , 140, 261 Asplund, M., Grevesse, N., Sauval, A. J., & Scott, P. 2009, , 47, 481 Andrievsky, S. M., L[é]{}pine, J. R. D., Korotin, S. A., et al. 2013, , 428, 3252 Andrievsky, S. M., Luck, R. E., & Korotin, S. A. 2014, , 437, 2106 Anthony-Twarog, B. J., Deliyannis, C. P., Twarog, B. A., Croxall, K. W., & Cummings, J. D. 2009, , 138, 1171 Anthony-Twarog, B. J., Deliyannis, C. P., & Twarog, B. A. 2014, , 148, 151 Bellazzini, M., Ibata, R., Monaco, L., et al. 2004, , 354, 1263 Bensby, T., Feltzing, S., Lundstr[ö]{}m, I., & Ilyin, I. 2005, , 433, 185 Bertolli, M. G., Herwig, F., Pignatari, M., & Kawano, T. 2013, arXiv:1310.4578 Bessell, M. S., Castelli, F., & Plez, B. 1998, , 333, 231 Blackwell, D. E., Booth, A. J., Menon, S. L. R., & Petford, A. D. 1986, , 220, 289 Busso, M., Gallino, R., & Wasserburg, G. J. 1999, , 37, 239 Busso, M., Gallino, R., Lambert, D. L., Travaglio, C., & Smith, V. V. 2001, , 557, 802 Busso, M., Wasserburg, G. J., Nollett, K. M., & Calandra, A. 2007, , 671, 802 Caldwell, J. A. R., & Coulson, I. M. 1987, , 93, 1090 Carraro, G. & Patat, F. 1995, , 276, 563 Carraro, G., Moitinho, A., & Vázquez, R. A. 2008, , 385, 1597 Carraro, G., Anthony-Twarog, B. J., Costa, E., Jones, B., & Twarog, B. A. 2011, , 142, 127 Carraro, G., Giorgi, E. E., Costa, E., & Vázquez, R.A. 2014a, , 441, L36 Carraro, G., Villanova, S., Monaco, L., et al. 2014b, , 562, 39 Carraro, G., de Silva, G., Monaco, L., Milone, A. P., & Mateluna, R. 2014c, , 566, 39 Carraro, G., Monaco, L., & Villanova, S. 2014, , 568, A86 Carretta, E., Bragaglia, A., & Gratton, R. G. 2007, , 473, 129 Castro, S., Rich, R. M., Grenon, M., Barbuy, B., McCarthy, J. K. 1997, , 114, 376 Charbonnel, C., & Lagarde, N. 2010, , 522, A10 Cowan, J. J., & Rose, W. K. 1977, , 212, 149 Crawford, D. L. 1975, , 80, 955 Crawford, D. L. 1979, , 84, 1858 Dardelet, L., Ritter, C., Prado, P., et al. 2015, arXiv:1505.05500 Depagne, E., Hill, V., Spite, M., et al. 2002, , 390, 187 Denisenkov, P. A., & Denisenkova, S. N. 1990, Soviet Astronomy Letters, 16, 275 De Silva, G. M., Sneden, C., Paulson, D. B., et al. 2006, , 131, 455 Dias, W. S., Alessi, B. S., Moitinho, A., & L[é]{}pine, J. R. D. 2002, , 389, 871 Diethelm, R. 1990, , 239, 186 D’Orazi, V., Magrini, L., Randich, S., et al. 2009, , 693, L31 D’Orazi, V., Biazzo, K., Desidera, S., et al. 2012, , 423, 2789 Drake, J. J., & Smith, G. 1991, , 250, 89 Dressler, A., Hare, T., Bigelow, B. C., & Osip, D. J. 2006, SPIE, 6269, 62690F Edvardsson, B., Andersen, J., Gustafsson, B., et al. 1993, , 275, 101 Evans, N. R. 1992, , 389, 657 Evans, N. R., Carpenter, K. G., Robinson, R., Kienzle, F., & Dekas, A. E. 2005, , 130, 789 Frebel, A., Casey, A. R., Jacobson, H. R., & Yu, Q. 2013, , 769, 57 Gonzalez, G., & Wallerstein, G. 1996, , 280, 515 Gratton, R. 2000, Stellar Clusters and Associations: Convection, Rotation, and Dynamos, 198, 225 Gratton, R. G., & Sneden, C. 1988, , 204, 193 Gratton, R. G., Carretta, E., Eriksson, K., & Gustafsson, B. 1999, , 350, 955 Grevesse, N. & Sauval, A. J. 1998, SSRv, 85, 161 Haffner, H. 1957, , 43, 89 Hauck, B. & Mermilliod, M. 1998, , 129, 431 Harris, H. C., & Welch, D. L. 1989, , 98, 981 Herwig, F., Langer, N., & Lugaro, M. 2003, , 593, 1056 Iwamoto, K., Brachwitz, F., Nomoto, K., et al. 1999, , 125, 439 Jacobson, H. R., Friel, E. D., & Pilachowski, C. A. 2011, Bulletin of the American Astronomical Society, 43, \#152.39 Jacobson, H. R., & Friel, E. D. 2013, , 145, 107 Keeping, E. S. 1962, Introduction to Statistical Inference. London: Van Nostrand Kharchenko, N. V., Piskunov, A. E., Schilbach, E., Röser, S., & Scholz, R.-D. 2013, , 558, 53 Korotin, S. A., Andrievsky, S. M., Luck, R. E., et al. , 444, 3301 Kovtyukh, V., Szabados, L., Chekhonadskikh, F., Lemasle, B., & Belik, S. 2015, , 448, 3567 Kurucz, R. L. 1993, CD-ROM 13, Atlas9 Stellar Atmosphere Programs and 2 km/s Grid (Cambridge, MA: Smithsonian Astrophys. Obs) Lambert, D. L., Heath, J. E., Lemke, M., & Drake, J. 1996, , 103, 183 Landsman, W., Simon, T., & Bergeron, P. 1996, , 108, 250 Lépine, J. R. D., Cruz, P., Scarano, Jr., S., et al. 2011, , 417, 698 Lind, K., Asplund, M., Barklem, P. S., & Belyaev, A. K. 2011, , 528, A103 Luck, R. E., & Bond, H. E. 1981, , 244, 919 Luck, R. E., & Bond, H. E. 1991, , 77, 515 Luck, R. E., & Heiter, U. 2007, , 133, 2464 Lugaro, M., Herwig, F., Lattanzio, J. C., Gallino, R., & Straniero, O. 2003, , 586, 1305 Magrini, L., Sestito, P., Randich, S., & Galli, D. 2009, , 494, 95 Magrini, L., Randich, S., Zoccali, M., et al. 2010, , 523, 11 Magrini, L., Randich, S., Romano, D., et al. 2014, , 563, 44 Maiorca, E., Randich, S., Busso, M., Magrini, L., & Palmerini, S. 2011, , 736, 120 Majaess, D., Sturch, L., Moni Bidin, C., et al. 2013a, , 347, 61 Majaess, D., Carraro, G., Moni Bidin, C., et al. 2013b, , 560, A22 Martin, W. C., Fuhr, J. R., Kelleher, D. E., et al. 2002, NIST Atomic Spectra Database (Version 2.0; Gaithersburg, MD: NIST) McWilliam, A. & Rich, R. M. 1994, , 91, 749 Mishenina, T. V., Bienaym[é]{}, O., Gorbaneva, T. I., et al. 2006, , 456, 1109 Mishenina, T., Korotin, S., Carraro, G., Kovtyukh, V. V., & Yegorova, I. A. 2013, , 433, 1436 Mishenina, T., Pignatari, M., Carraro, G., et al. 2015, , 446, 3651 Moni Bidin, C., Carraro, G., & M[é]{}ndez, R. A. 2012, , 747, 101 Monroe, T. R., & Pilachowski, C. A. 2010, , 140, 2109 Neilson, H. R., Schneider, F. R. N., Izzard, R. G., Evans, N. R., & Langer, N. 2015, , 574, A2 Nissen, P. E. 1988, , 199, 146 Nissen, P. E., Twarog, B. A., & Crawford, D. L. 1987, , 93, 634 Nordhaus, J., Busso, M., Wasserburg, G. J., Blackman, E. G., & Palmerini, S. 2008, , 684, L29 Nucci, M. C., & Busso, M. 2014, , 787, 141 Olsen, E. H. 1983, , 54, 55 Olsen, E. H. 1988, , 189, 173 Olsen, E. H. 1993, , 102, 89 Olsen, E. H. 1994, , 106, 257 Pancino, E., Carrera, R., Rossetti, E., & Gallart, C. 2010, , 511, A56 Pavlenko, Y. V., Jenkins, J. S., Jones, H. R. A., Ivanyuk, O., & Pinfield, D. J. 2012, , 422, 542 Pereira, C. B., Sales Silva, J. V., Chavero, C., et al. 2011, , 533, A51 Piatti, A. E., Claría, J. J., & Ahumada, A. V. 2004, , 421, 991 Preston, G. W., & Sneden, C. 2001, , 122, 1545 Reddy, B. E., Bakker, E. J., & Hrivnak, B. J. 1999, , 524, 831 Reddy, B. E., Tomkin, J., Lambert, D.L., & Allende Prieto, C. 2003, , 340, 304 Reddy, A. B. S., Giridhar, S., & Lambert, D. L. 2013, , 431, 3338 Sales Silva, J. V., Peña Suárez, V. J., Katime Santrich, O. J., et al. 2014, , 148, 83 Santos, N. C., Lovis, C., Pace, G., Melendez, J., & Naef, D. 2009, , 493, 309 Santrich, O. J. K., Pereira, C. B., & Drake, N. A. 2013, , 554, A2 Schmidt, E. G. 1984, , 55, 455 Smiljanic, R. 2012, , 422, 1562 Sneden, C. A. 1973, Ph.D. Thesis, Univ. Texas Snowden, M. S. 1975, , 87, 721 Stetson, P. B. 1987, , 99, 191 Szabados, L., & Klagyivik, P. 2012, , 341, 99 Tifft, W. G. 1959, , 129, 241 Tombaugh, C. W. 1938, , 50, 171 Tombaugh, C. W. 1941, , 53, 219 Travaglio, C., Galli, D., Gallino, R., et al. 1999, , 521, 691 Trippella, O., Busso, M., Maiorca, E., K[ä]{}ppeler, F., & Palmerini, S. 2014, , 787, 41 Turner, D. G. 1983, Journal of the Royal Astronomical Society of Canada, 77, 31 Twarog, B. A. & Anthony-Twarog, B. J. 1995, , 109, 2828 Twarog, B. A., Ashman, K., & Anthony-Twarog, B. A. 1997, , 114, 2556 Twarog, B. A., Anthony-Twarog, B. J., Deliyannis, C. P., & Thomas, D. 2015, , in press VandenBerg, D. A., Bergbusch, P. A., & Dowler, P. D. 2006, , 162, 375 van Dokkum, P. G. 2001, , 113, 1420 Vásquez, R. A., May, J., Carraro, G., et al. 2008, , 672, 930 Villanova, S., Randich, S., Geisler, D., Carraro, G., & Costa, E. 2010, , 509, A102 Wiese, W. L., Smith, M. W., & Miles, B. M. 1969, NSRDS-NBS, Washington, D.C.: US Department of Commerce, National Bureau of Standards, |c 1969, Woosley, S. E., & Weaver, T. A. 1995, , 101, 181 Yong, D., Carney, B. W., Teixera de Almeida, M. L., & Pohl, B. L. 2006, , 131, 2256 Yong, D., Carney, B. W., & Friel, E. D. 2012, , 144, 95
{width="\columnwidth"}
{width="\columnwidth"}
{width="\columnwidth"}
{width="\columnwidth"}
{width="\columnwidth"}
{width="\columnwidth"}
{width="\columnwidth"}
{width="\columnwidth"}
{width="\columnwidth"}
![CMD of Fig. 6 superposed on the VR isochrones with \[Fe/H\] = $-$0.11, assuming $E(B-V)$ = 0.303 and $(m-M)$ = 13.10. The isochrones have ages of 0.8 Gyr (blue), 0.9 Gyr (black) and 1.0 Gyr (red).](Fig10.pdf){width="\columnwidth"}
\[alField\] ![Abundance ratios \[X/Fe\] vs. \[Fe/H\]. Blue crosses: field giants of Luck & Heiter (2007); Black crosses: clump giants of Mishenina et al. (2006); Yellow squares: our sample of field giants stars; Red square: our mean abundances of Tombaugh 1; Green squares: open clusters from literature (NGC 6192, NGC 6404 and NGC 6583 of Magrini et al. 2010; NGC 3114 of Santrich et al. 2013; NGC 2527, NGC 2682, NGC 2482, NGC 2539, NGC 2335, NGC 2251 and NGC 2266 of Reddy et al. 2013; NGC 4337 of Carraro et al. 2014d; Trumpler 20 of Carraro et al. 2014b; NGC 4815 and NGC 6705 of Magrini et al. 2014; Cr 110, Cr 261, NGC 2477, NGC 2506 and NGC 5822 of Mishenina et al. 2015.](alField.pdf "fig:"){width="\columnwidth"}
\[singlealphaField\] ![Abundance ratios \[X/Fe\] vs. \[Fe/H\]. Symbols have the same meaning as in Figure 11.](singlealphaField.pdf "fig:"){width="\columnwidth"}
\[sprocessTomb1\] ![Abundance ratios \[X/Fe\] vs. \[Fe/H\]. Symbols have the same meaning as in Figure 11. In the Y and Ba panel, we added the results of Mishenina et al. (2013) of the open clusters Berkeley 25, Berkeley 73, Berkeley 75, Ruprecht 4, Ruprecht 7, NGC 6192, NGC 6404 and NGC 6583 (green squares). One field star of our sample (806) exhibits Ba enrichment.](sprocessTomb1.pdf "fig:"){width="\columnwidth"}
\[O6300-final\] ![Observed (dotted red line) and synthetic spectra (solid blue lines) in the region of the Ba II line at 5853 Å for the field giant star 806. The synthetic spectra were calculated with the barium ratio abundances of \[Ba/Fe\]= 0.33, 0.73 and 1.13.](O6300-final.pdf "fig:"){width="\columnwidth"}
\[FeHxRgc2\] {width="\columnwidth"}
0.1truecm
[lcccc]{} Date & Field & Filter & Exposures (sec) & airmass (X)\
Dec 05, 2010 & Tombaugh 1 & *y* & 60, 600 &1.03\
& & *b* & 60, 600 & 1.02\
& & *H$\beta_{wide}$* & 60, 600 & 1.01$-$1.02\
& & *Ca* & 120, 1200 & 1.01\
& & *H$\beta_{narrow}$* & 120,1200 & 1.02\
Dec 06, 2010 & Tombaugh 1 & *y* & 2x60, 900 & 1.03\
& & *b* & 2x60,900 & 1.02\
& & *v* & 60, 900 & 1.01$-$1.03\
& & *Ca* & 120,1500 & 1.04-1.08\
Dec 07 , 2010 & Tombaugh 1 & *u* & 10, 300 & 1.02\
& & *b* & 10, 60 & 1.01\
& & *v* & 10, 100 & 1.01\
& & *Ca* & 10, 200 & 1.01\
& & *H$\beta_{wide}$* & 10, 60 & 1.01$-$1.01\
& & *H$\beta_{narrow}$* & 10, 200 & 1.02\
Dec 08, 2010 & Tombaugh 1 & *u* & 2x20, 200 2000 & 1.02$-$1.03\
& & *v* & 20, 90, 900 & 1.01$-$1.03\
& & *H$\beta_{narrow}$* & 20, 150, 1500 & 1.05\
Dec 09, 2010 & Tombaugh 1&*y* & 10, 60, 120, 600 & 1.46$-$1.51\
& &*b* & 60,180,900 & 1.35$-$1.39\
& & *v* & 100, 200, 1200 & 1.25$-$1.28\
& & *Ca* & 100, 300,1800 & 1.04$-$1.05\
& & *H$\beta_{wide}$* & 100, 200, 1200 & 1.16$-$1.18\
& & *H$\beta_{narrow}$* & 10, 300, 1800 & 1.09$-$1.11\
Dec 09, 2010 & Tombaugh 1&*y* & 60, 600 & 1.47$-$1.49\
& &*b* & 180, 900 & 1.38$-$1.40\
& & *v* & 200, 900 & 1.30\
& & *Ca* & 300, 1500 & 1.07$-$1.08\
& & *H$\beta_{wide}$* & 200, 900 & 1.30\
& & *H$\beta_{narrow}$* & 300, 1500 & 1.13$-$1.14\
& & *u* & 400, 1800 & 1.03$-$1.04\
0.3truecm
[crrr]{} Index & slope & color term & Std. deviation\
$V$ & 1.00 & 0.05 & 0.010\
$b-y$ & 1.01 & — & 0.002\
$hk$ & 1.07 & — & 0.009\
H$\beta$& 1.18 & — & 0.015\
$m_1$ & 0.92 & $-$0.075 & 0.025\
$c_1$ & 1.06 & — & 0.021\
0.1truecm
[llrccccccccccc]{} $\alpha(2000)$ & $\delta(2000)$ & $V$ & $b-y$ & $hk$ & $H\beta$ & $m_1$ & $c_1$ & $\sigma_V$ & $\sigma_{by}$ & $\sigma_{m1}$ & $\sigma_{c1}$ & $\sigma_{hk}$ & $\sigma_{\beta}$\
105.1726& $-$20.6754& 7.179& 0.826& 1.377& 2.532& 0.453& 0.187& 0.0195& 0.0204& 0.0205& 0.0067& 0.0213& 0.0074\
105.0252& $-$20.6121& 9.426& 0.232& 0.326& 2.769& 0.119& 0.793& 0.0021& 0.0030& 0.0037& 0.0035& 0.0040& 0.0026\
105.2605& $-$20.4113& 10.227& 0.044& 0.234& 2.915& 0.145& 1.012& 0.0023& 0.0034& 0.0042& 0.0039& 0.0046& 0.0026\
105.1471& $-$20.6414& 10.326& 0.323& 0.550& 2.673& 0.158& 0.406& 0.0023& 0.0039& 0.0055& 0.0057& 0.0058& 0.0042\
105.0961& $-$20.4473& 10.400& 0.268& 0.473& 2.716& 0.209& 0.613& 0.0018& 0.0026& 0.0034& 0.0034& 0.0037& 0.0027\
105.2796& $-$20.5640& 10.463& 1.232& 1.976& 2.574& 0.661& -0.319& 0.0016& 0.0023& 0.0038& 0.0076& 0.0037& 0.0022\
104.9458& $-$20.5607& 10.502& 0.410& 0.585& 2.582& 0.167& 0.251& 0.0016& 0.0023& 0.0030& 0.0030& 0.0032& 0.0019\
105.0569& $-$20.5582& 10.641& 1.387& 1.216& 2.754& 0.050& 0.134& 0.0017& 0.0024& 0.0035& 0.0056& 0.0035& 0.0020\
105.0750& $-$20.6804& 10.733& 1.084& 1.836& 2.550& 0.639& -0.147& 0.0017& 0.0023& 0.0033& 0.0046& 0.0034& 0.0019\
105.1990& $-$20.4917& 10.947& 0.379& 0.568& 2.620& 0.134& 0.411& 0.0016& 0.0024& 0.0032& 0.0035& 0.0034& 0.0020\
105.0211& $-$20.4531& 11.051& 0.797& 1.338& 2.550& 0.471& 0.199& 0.0017& 0.0023& 0.0032& 0.0037& 0.0033& 0.0020\
0.2truecm
[lcccccccccc]{}\
ID & RA(2000.0) & DEC(2000.0) & $V$ & $b-y$ & $V-I$ & RV$_{1}$ & RV$_{2}$ & $\langle$RV$\rangle$ & vsini & S/N\
& hh:mm:ss & dd:mm:ss & mag & mag & mag & (km s$^{-1}$) & (km s$^{-1}$) & (km s$^{-1}$) & (km s$^{-1}$)&\
395 &$07:00:05.7$ & $-20:35:20.5$ & $13.72$ & $0.73$ &$1.19$& 81.6$\pm$1.8 & 82.7$\pm$2.4 & 82.1$\pm$0.7 & $<$2.7 & 100\
663 &$07:00:18.7$ & $-20:31:31.0$ & $14.09$ & $0.75$ &$1.19$& 79.9$\pm$2.6 & 79.8$\pm$2.8 & 79.8$\pm$0.1 & $<$2.7 & 60\
769 &$07:00:23.4$ & $-20:32:59.2$ & $13.20$ & $0.73$ &$1.20$& 81.6$\pm$1.4 & 81.5$\pm$1.5 & 81.5$\pm$0.0 & $<$2.7 & 110\
784 &$07:00:24.1$ & $-20:35:45.1$ & $14.20$ & $0.79$ &$1.26$& 99.0$\pm$2.2 & 98.8$\pm$1.3 & 98.9$\pm$0.1 & 5.6$\pm$0.9 & 55\
806$^a$&$07:00:24.8$ & $-20:25:54.1$ & $13.01$ & $0.51$ &$1.12$&100.0$\pm$2.1 & 99.0$\pm$3.8 & 99.5$\pm$0.6 & — & 95\
1110 &$07:00:36.1$ & $-20:35:47.1$ & $13.60$ & $0.74$ &$1.27$& 84.0$\pm$2.4 & 84.7$\pm$2.4 & 84.3$\pm$0.5 & 6.1$\pm$0.5 & 100\
1118 &$07:00:36.5$ & $-20:38:57.4$ & $13.74$ & $0.79$ &$1.32$& 82.2$\pm$2.3 & 81.1$\pm$2.9 & 81.7$\pm$0.7 & $<$2.7 & 74\
1349 &$07:00:46.3$ & $-20:28:55.7$ & $14.16$ & $0.83$ &$1.32$& 77.2$\pm$3.8 & 76.5$\pm$3.3 & 76.8$\pm$0.5 & 5.1$\pm$0.4 & 44\
1534 &$07:00:54.5$ & $-20:24:30.0$ & $13.94$ & $0.78$ &$1.31$& 93.8$\pm$3.6 & 91.8$\pm$4.6 & 92.8$\pm$1.3 & — & 50\
1616 &$07:00:58.1$ & $-20:33:24.6$ & $13.68$ & $0.67$ &$1.11$& 42.7$\pm$2.4 & 44.8$\pm$2.9 & 43.7$\pm$1.3 & 4.2$\pm$0.4 & 100\
\
**Notes.** The columns inform, from left to right: star identification, right ascension, declination, V and b-y from this paper, and V-I from Carraro & Patat (1995), two epoch heliocentric radial velocities (RV$_{1}$ and RV$_{2}$) and their mean values ($\langle$RV$\rangle$), projected rotational velocities (vsini) and spectral signal-to-noise at 6000 Å. (a) The classical Cepheid XZ CMa.
[ccccccccc]{}\
\[tabelFea\] & & & &\
& & & &\
Element & $\lambda$(Å) & $\chi$(eV) & log $gf$ & 769 & 806 & 1110 & 1118 & 395\
Fe[i]{} & 5159.06 & 4.28 &$-$0.650 & — & — & —& 83& 110\
& 5162.27 & 4.18 & 0.079 & 158 & 148 & —& —& 153\
& 5198.71 & 2.22 &$-$2.140 & 132 & 140 & 107& —& 125\
& 5242.49 & 3.63 &$-$0.970 & 105 & — & —& —& 111\
& 5250.21 & 0.12 &$-$4.920 & — & — & —& —& 136\
& 5288.52 & 3.69 &$-$1.510 & — & 37 & 63& 79& 78\
& 5307.36 & 1.61 &$-$2.970 & — & 94 & 111& 125& 133\
& 5315.05 & 4.37 &$-$1.400 & — & — & 59& —& 51\
& 5321.11 & 4.43 &$-$1.190 & 60 & — & —& —& —\
& 5322.04 & 2.28 &$-$2.840 & — & 31 & 78& —& 103\
& 5364.87 & 4.45 & 0.230 & 136 & — & 126& 150& 143\
& 5367.47 & 4.42 & 0.439 & 141 & — & 131& 135& 142\
& 5373.71 & 4.47 &$-$0.710 & 85 & — & 70& 77& —\
& 5410.91 & 4.47 & 0.400 & — & — & 140& —& —\
& 5417.03 & 4.42 &$-$1.530 & 53 & — & 33& 39& 61\
& 5441.34 & 4.31 &$-$1.580 & 47 & 18 & 46& 58& 65\
& 5522.45 & 4.21 &$-$1.400 & 67 & — & 47& 58& 60\
& 5531.98 & 4.91 &$-$1.460 & 37 & — & 31& 29& —\
& 5554.90 & 4.55 &$-$0.380 & — & 73 & —& —& —\
& 5560.21 & 4.43 &$-$1.040 & 70 & — & 63& 64& 72\
& 5567.39 & 2.61 &$-$2.560 & — & — & 76& —& 110\
& 5576.09 & 3.43 &$-$0.850 & 135 & — & —& —& —\
& 5633.95 & 4.99 &$-$0.120 & 96 & 68 & —& 81& —\
& 5635.82 & 4.26 &$-$1.740 & 58 & — & —& 47& 56\
& 5638.26 & 4.22 &$-$0.720 & — & — & —& —& 96\
& 5691.50 & 4.30 &$-$1.370 & — & 23 & 66& 54& 79\
& 5705.47 & 4.30 &$-$1.360 & 64 & — & 68& 57& —\
& 5731.76 & 4.26 &$-$1.150 & — & — & —& —& 76\
& 5806.73 & 4.61 &$-$0.900 & 73 & — & —& 65& 62\
& 5852.22 & 4.55 &$-$1.180 & 62 & — & 48& —& 60\
& 5883.82 & 3.96 &$-$1.210 & 90 & — & 69& —& —\
& 5934.65 & 3.93 &$-$1.020 & 97 & — & —& —& 92\
& 6020.17 & 4.61 &$-$0.210 & — & 101 & —& —& —\
& 6024.06 & 4.55 &$-$0.060 & — & — & 109& 119& 126\
& 6027.05 & 4.08 &$-$1.090 & 103 & 44 & 80& 87& 95\
& 6056.01 & 4.73 &$-$0.400 & 93 & — & 78& 81& 86\
& 6065.48 & 2.61 &$-$1.530 & 157 & — & 125& 154& 156\
[ccccccccc]{}\
& & & &\
& & & &\
Element & $\lambda$(Å) & $\chi$(eV) & log $gf$ & 769 & 806 & 1110 & 1118 & 395\
Fe[i]{} & 6079.01 & 4.65 &$-$0.970 & — & — & —& 71& —\
& 6096.66 & 3.98 &$-$1.780 & 56 & — & 50& 49& 62\
& 6120.25 & 0.91 &$-$5.950 & — & — & 24& —& —\
& 6151.62 & 2.18 &$-$3.290 & 92 & — & 75& 90& 92\
& 6157.73 & 4.08 &$-$1.110 & 98 & 36 & 89& 94& 89\
& 6165.36 & 4.14 &$-$1.470 & 59 & — & 59& 73& 59\
& 6170.51 & 4.79 &$-$0.380 & — & 36 & —& —& —\
& 6173.34 & 2.22 &$-$2.880 & 107 & 39 & 92& 112& 122\
& 6187.99 & 3.94 &$-$1.570 & 73 & — & 60& 72& 80\
& 6200.31 & 2.60 &$-$2.440 & 118 & — & —& 107& 103\
& 6213.43 & 2.22 &$-$2.480 & 123 & — & —& —& —\
& 6265.13 & 2.18 &$-$2.550 & 128 & 95 & —& 132& 135\
& 6311.50 & 2.83 &$-$3.230 & — & — & —& 57& —\
& 6322.69 & 2.59 &$-$2.430 & — & — & 88& —& 106\
& 6380.74 & 4.19 &$-$1.320 & 87 & — & 65& 86& 70\
& 6392.54 & 2.28 &$-$4.030 & — & — & 29& —& —\
& 6411.65 & 3.65 &$-$0.660 & 142 & 150 & 125& 138& —\
& 6421.35 & 2.28 &$-$2.010 & — & 152 & 135& 153& 150\
& 6430.85 & 2.18 &$-$2.010 & 155 & — & 126& 155& 159\
& 6436.41 & 4.19 &$-$2.460 & 25 & — & —& —& —\
& 6469.19 & 4.83 &$-$0.620 & 85 & 29 & 74& 79& 71\
& 6593.87 & 2.44 &$-$2.420 & 124 & — & —& 117& 129\
& 6597.56 & 4.79 &$-$0.920 & 71 & — & —& 55& 57\
& 6608.03 & 2.28 &$-$4.030 & 48 & — & 28& 39& 37\
& 6646.93 & 2.61 &$-$3.990 & 45 & — & —& —& 30\
& 6653.85 & 4.14 &$-$2.520 & 26 & — & —& —& —\
& 6703.57 & 2.76 &$-$3.160 & 81 & — & —& 64& 59\
& 6739.52 & 1.56 &$-$4.950 & 37 & — & —& 41& —\
& 6750.15 & 2.42 &$-$2.620 & 106 & — & 95& —& 102\
& 6752.71 & 4.64 &$-$1.200 & 69 & — & 48& 59& 59\
& 6806.85 & 2.73 &$-$3.210 & 63 & — & 63& —& 64\
& 6810.26 & 4.61 &$-$0.990 & 58 & — & —& 73& 74\
& 6820.37 & 4.64 &$-$1.170 & 62 & — & —& 58& 72\
& 6851.64 & 1.61 &$-$5.320 & — & — & —& —& 24\
& 6858.15 & 4.61 &$-$0.930 & 77 & — & 68& 63& —\
& 7130.92 & 4.22 &$-$0.700 & 115 & 85 & —& —& 103\
& 7132.99 & 4.08 &$-$1.610 & 63 & 25 & 49& —& 65\
Fe[ii]{} & 5132.66 & 2.81 & -4.000 & — & — & 44& —& —\
& 5425.25 & 3.20 &$-$3.210 & — & — & —& 63& 67\
[ccccccccc]{}\
& & & &\
& & & &\
Element & $\lambda$(Å) & $\chi$(eV) & log $gf$ & 769 & 806 & 1110 & 1118 & 395\
Fe[ii]{} & 5991.37 & 3.15 &$-$3.560 & — & — & 43& —& —\
& 6084.10 & 3.20 &$-$3.800 & — & — & 35& 45& —\
& 6149.25 & 3.89 &$-$2.720 & — & — & 43& —& 60\
& 6247.55 & 3.89 &$-$2.340 & 81 & 139 & —& 74& 82\
& 6416.92 & 3.89 &$-$2.680 & 54 & 96 & 49& —& 52\
& 6432.68 & 2.89 &$-$3.580 & 61 & — & 50& —& 63\
[ccccccccc]{}\
\[tabelFeb\] & & & &\
& & & &\
Element & $\lambda$(Å) & $\chi$(eV) & log $gf$ & 1534 & 1616 & 1349 & 784 & 663\
Fe[i]{} & 5159.06 & 4.28 &$-$0.650 & — & 98 & 106& 109& —\
& 5253.03 & 2.28 &$-$3.790 & — & — & —& —& 91\
& 5288.52 & 3.69 &$-$1.510 & — & — & —& 93& 116\
& 5307.36 & 1.61 &$-$2.970 & — & 109 & 147& 148& —\
& 5315.05 & 4.37 &$-$1.400 & — & — & —& —& 70\
& 5321.11 & 4.43 &$-$1.190 & — & — & —& —& 98\
& 5322.04 & 2.28 &$-$2.840 & 121 & — & —& —& —\
& 5364.87 & 4.45 & 0.230 & 158 & — & 148& 154& —\
& 5367.47 & 4.42 & 0.439 & 157 & — & —& —& —\
& 5373.71 & 4.47 &$-$0.710 & 116 & — & 113& 89& 116\
& 5417.03 & 4.42 &$-$1.530 & 53 & 52 & 60& 61& 80\
& 5441.34 & 4.31 &$-$1.580 & 44 & 40 & 57& —& —\
& 5522.45 & 4.21 &$-$1.400 & — & — & 68& 71& 92\
& 5554.90 & 4.55 &$-$0.380 & — & 103 & 122& —& —\
& 5560.21 & 4.43 &$-$1.040 & — & — & 104& —& —\
& 5567.39 & 2.61 &$-$2.560 & 122 & — & —& —& 155\
& 5576.09 & 3.43 &$-$0.850 & — & — & —& 154& —\
& 5624.02 & 4.39 &$-$1.330 & — & — & —& 76& 103\
& 5633.95 & 4.99 &$-$0.120 & — & 85 & 103& —& —\
& 5635.82 & 4.26 &$-$1.740 & — & — & —& —& 58\
& 5638.26 & 4.22 &$-$0.720 & — & 84 & —& 106& —\
& 5691.50 & 4.30 &$-$1.370 & 85 & 51 & —& 74& 104\
& 5705.47 & 4.30 &$-$1.360 & — & 46 & —& 59& 93\
& 5717.83 & 4.28 &$-$0.979 & 116 & — & 122& —& —\
& 5806.73 & 4.61 &$-$0.900 & — & — & —& 65& 99\
& 5814.81 & 4.28 &$-$1.820 & — & — & —& —& 62\
& 5852.22 & 4.55 &$-$1.180 & — & 63 & 78& 70& 95\
& 5883.82 & 3.96 &$-$1.210 & 94 & — & —& 101& —\
& 5916.25 & 2.45 &$-$2.990 & 104 & — & —& —& —\
& 5934.65 & 3.93 &$-$1.020 & 120 & — & 142& 117& 140\
& 6024.06 & 4.55 &$-$0.060 & — & 102 & —& —& —\
& 6027.05 & 4.08 &$-$1.090 & — & 72 & —& —& —\
& 6056.01 & 4.73 &$-$0.400 & — & 91 & —& —& 123\
& 6065.48 & 2.61 &$-$1.530 & — & 151 & —& —& —\
& 6096.66 & 3.98 &$-$1.780 & — & 62 & 95& 71& 83\
& 6151.62 & 2.18 &$-$3.290 & — & 79 & —& —& —\
& 6157.73 & 4.08 &$-$1.110 & 120 & 95 & —& 102& 120\
[ccccccccc]{}\
& & & &\
& & & &\
Element & $\lambda$(Å) & $\chi$(eV) & log $gf$ & 1534 & 1616 & 1349 & 784 & 663\
Fe[i]{} & 6165.36 & 4.14 &$-$1.470 & — & 56 & —& 75& 97\
& 6173.34 & 2.22 &$-$2.880 & — & 98 & —& —& 145\
& 6187.99 & 3.94 &$-$1.570 & — & — & —& 81& 102\
& 6200.31 & 2.60 &$-$2.440 & — & 111 & 145& 126& —\
& 6265.13 & 2.18 &$-$2.550 & 155 & — & —& 143& —\
& 6380.74 & 4.19 &$-$1.320 & — & 75 & —& —& 119\
& 6392.54 & 2.28 &$-$4.030 & — & — & —& 65& 84\
& 6411.65 & 3.65 &$-$0.660 & — & 145 & —& 157& —\
& 6421.35 & 2.28 &$-$2.010 & — & 133 & —& —& —\
& 6430.85 & 2.18 &$-$2.010 & — & 148 & —& —& —\
& 6436.41 & 4.19 &$-$2.460 & — & — & —& 21& 42\
& 6469.19 & 4.83 &$-$0.620 & 70 & 82 & 102& 99& 117\
& 6551.68 & 0.99 &$-$5.790 & — & — & 56& 41& 65\
& 6591.31 & 4.59 &$-$2.070 & — & — & —& 23& —\
& 6593.87 & 2.44 &$-$2.420 & 137 & — & —& —& —\
& 6597.56 & 4.79 &$-$0.920 & 52 & 40 & —& 62& 73\
& 6608.03 & 2.28 &$-$4.030 & — & 40 & —& 66& 76\
& 6609.11 & 2.56 &$-$2.690 & — & — & 140& —& —\
& 6646.93 & 2.61 &$-$3.990 & 41 & — & —& —& 63\
& 6703.57 & 2.76 &$-$3.160 & — & — & 66& 79& —\
& 6739.52 & 1.56 &$-$4.950 & — & — & —& 70& 48\
& 6750.15 & 2.42 &$-$2.620 & — & — & 150& —& —\
& 6752.71 & 4.64 &$-$1.200 & — & 38 & 86& 73& 101\
& 6806.85 & 2.73 &$-$3.210 & — & 59 & 90& 87& 104\
& 6810.26 & 4.61 &$-$0.990 & — & — & 98& 91& 94\
& 6820.37 & 4.64 &$-$1.170 & — & 58 & 65& —& 83\
& 6858.15 & 4.61 &$-$0.930 & — & 65 & —& —& 103\
& 7130.92 & 4.22 &$-$0.700 & — & — & —& 130& —\
& 7132.99 & 4.08 &$-$1.610 & — & — & 77& 64& 90\
Fe[ii]{} & 5425.25 & 3.20 &$-$3.210 & — & — & —& 69& —\
& 5534.83 & 3.25 &$-$2.770 & — & 87 & —& 102& —\
& 6084.10 & 3.20 &$-$3.800 & — & 25 & —& —& —\
& 6149.25 & 3.89 &$-$2.720 & 73 & 54 & 80& 63& —\
& 6247.55 & 3.89 &$-$2.340 & — & 68 & —& 82& 73\
& 6416.92 & 3.89 &$-$2.680 & 78 & 63 & 69& 59& 56\
& 6432.68 & 2.89 &$-$3.580 & — & — & —& 68& 53\
[cccccccccc]{} \[tabellinesa\]\
& & & & &\
&\
Element & $\lambda$ & $\chi$(eV) & $\log gf$ & Ref & 769 & 806 & 1110 & 1118 & 395\
Na[i]{} & 6154.22 & 2.10 & $-$1.51 & PS & 59 & 17 & 63 & 72 & 55\
Na[i]{} & 6160.75 & 2.10 & $-$1.21 & R03 & 89 & — & 72 & 86 & 78\
Mg[i]{} & 4730.04 & 4.34 & $-$2.39 & R03 & 81 & — & — & — & —\
Mg[i]{} & 5711.10 & 4.34 & $-$1.75 & R99 & 115 & 77 & 107 & — & —\
Mg[i]{} & 6318.71 & 5.11 & $-$1.94 & Ca07 & — & — & 45 & 58 & 55\
Mg[i]{} & 6965.41 & 5.75 & $-$1.72 & MR94 & — & — & — & 43 & 40\
Mg[i]{} & 7387.70 & 5.75 & $-$0.87 & MR94 & 85 & — & 105 & — & 80\
Mg[i]{} & 8717.83 & 5.91 & $-$0.97 & WSM & 62 & — & — & 83 & —\
Mg[i]{} & 8736.04 & 5.94 & $-$0.34 & WSM & — & — & — & 128 & —\
Si[i]{} & 5793.08 & 4.93 & $-$2.06 & R03 & 52 & 17 & — & 61 & 63\
Si[i]{} & 6125.03 & 5.61 & $-$1.54 & E93 & 44 & 24 & 29 & 39 & 42\
Si[i]{} & 6131.58 & 5.62 & $-$1.68 & E93 & — & 14 & — & — & —\
Si[i]{} & 6155.14 & 5.62 & $-$0.77 & E93 & 91 & 77 & — & 87 & 89\
Si[i]{} & 7800.00 & 6.18 & $-$0.72 & E93 & 69 & — & 58 & — & 68\
Si[i]{} & 8728.01 & 6.18 & $-$0.36 & E93 & 89 & — & — & — & —\
Ca[i]{} & 6102.73 & 1.88 & $-$0.79 & D2002& — & — & — & 149 & 141\
Ca[i]{} & 6161.30 & 2.52 & $-$1.27 & E93 & 93 & — & 78 & 92 & 97\
Ca[i]{} & 6166.44 & 2.52 & $-$1.14 & R03 & 83 & 43 & 88 & 99 & 86\
Ca[i]{} & 6169.04 & 2.52 & $-$0.80 & R03 & 114 & — & 93 & 113 & 112\
Ca[i]{} & 6169.56 & 2.53 & $-$0.48 & DS91 & 126 & — & 121 & 125 & 127\
Ca[i]{} & 6455.60 & 2.51 & $-$1.29 & R03 & 79 & — & 71 & — & 89\
Ca[i]{} & 6471.66 & 2.51 & $-$0.69 & S86 & 132 & 98 & — & — & 125\
Ca[i]{} & 6493.79 & 2.52 & $-$0.11 & DS91 & 152 & — & — & — & —\
Ti[i]{} & 4534.78 & 0.84 & 0.280 &D2002& 149 & — & — & — & —\
Ti[i]{} & 4758.12 & 2.25 & 0.420 & MFK & 80 & — & — & — & —\
Ti[i]{} & 4759.28 & 2.25 & 0.514 & MFK & 81 & — & — & — & —\
Ti[i]{} & 4820.41 & 1.50 & $-$0.439 & MFK & 83 & — & — & — & 83\
Ti[i]{} & 4999.51 & 0.83 & 0.250 & MFK & — & — & 119 & — & —\
Ti[i]{} & 5009.66 & 0.02 & $-$2.259 & MFK & — & — & — & 71 & —\
Ti[i]{} & 5022.87 & 0.83 & $-$0.434 & MFK & — & — & — & — & 109\
Ti[i]{} & 5039.96 & 0.02 & $-$1.130 & MFK & 118 & — & 96 & — & —\
Ti[i]{} & 5043.59 & 0.84 & $-$1.733 & MFK & 63 & — & — & — & —\
Ti[i]{} & 5087.06 & 1.43 & $-$0.840 & E93 & 60 & — & — & 56 & 56\
Ti[i]{} & 5113.45 & 1.44 & $-$0.880 & E93 & — & — & — & 59 & —\
Ti[i]{} & 5145.47 & 1.46 & $-$0.574 & MFK & 82 & — & 62 & — & 67\
Ti[i]{} & 5147.48 & 0.00 & $-$2.012 & MFK & 82 & — & — & 97 & 94\
Ti[i]{} & 5173.75 & 0.00 & $-$1.120 & MFK & — & — & — & 136 & —\
Ti[i]{} & 5210.39 & 0.05 & $-$0.883 & MFK & — & — & 102 & — & —\
Ti[i]{} & 5219.71 & 0.02 & $-$2.292 & MFK & 81 & — & — & — & 79\
Ti[i]{} & 5223.63 & 2.09 & $-$0.559 & MFK & — & — & 29 & — & —\
Ti[i]{} & 5295.78 & 1.05 & $-$1.633 & MFK & — & — & — & 39 & 44\
Ti[i]{} & 5490.16 & 1.46 & $-$0.937 & MFK & 70 & — & — & — & —\
Ti[i]{} & 5689.48 & 2.30 & $-$0.469 & MFK & 34 & — & 35 & — & 40\
Ti[i]{} & 5866.46 & 1.07 & $-$0.871 & E93 & — & — & 77 & 82 & 93\
Ti[i]{} & 5922.12 & 1.05 & $-$1.465 & MFK & 62 & — & — & 59 & —\
Ti[i]{} & 5978.55 & 1.87 & $-$0.496 & MFK & 63 & — & 52 & 50 & —\
Ti[i]{} & 6091.18 & 2.27 & $-$0.370 & R03 & 43 & — & — & 42 & —\
[cccccccccc]{}\
& & & & &\
&\
Element & $\lambda$ & $\chi$(eV) & $\log gf$ & Ref & 769 & 806 & 1110 & 1118 & 395\
Ti[i]{} & 6126.22 & 1.07 & $-$1.370 & R03 & 61 & 10 & 45 & — & 60\
Ti[i]{} & 6258.11 & 1.44 & $-$0.355 & MFK & 99 & — & 81 & 81 & —\
Ti[i]{} & 6261.11 & 1.43 & $-$0.480 & B86 & 96 & — & 76 & 83 & 89\
Ti[i]{} & 6554.24 & 1.44 & $-$1.219 & MFK & — & — & — & 55 & —\
Cr[i]{} & 4836.85 & 3.10 & $-$1.137 & MFK & 40 & — & — & — & 32\
Cr[i]{} & 5200.18 & 3.38 & $-$0.650 & MFK & 40 & — & — & — & —\
Cr[i]{} & 5296.70 & 0.98 & $-$1.390 & GS & 133 & 82 & — & 134 & 132\
Cr[i]{} & 5304.18 & 3.46 & $-$0.692 & MFK & 35 & — & — & — & —\
Cr[i]{} & 5345.81 & 1.00 & $-$0.980 & GS & — & 145 & 140 & — & —\
Cr[i]{} & 5348.32 & 1.00 & $-$1.290 & GS & — & 85 & 118 & 135 & 147\
Cr[i]{} & 5783.07 & 3.32 & $-$0.500 & MFK & 49 & — & 47 & 60 & 62\
Cr[i]{} & 5783.87 & 3.32 & $-$0.290 & GS & 74 & — & 64 & 68 & 76\
Cr[i]{} & 5787.93 & 3.32 & $-$0.080 & GS & 71 & — & 69 & — & 70\
Cr[i]{} & 6330.09 & 0.94 & $-$2.920 & R03 & — & — & 58 & 61 & 70\
Ni[i]{} & 4904.42 & 3.54 & $-$0.170 & MFK & 116 & — & 102 & 100 & —\
Ni[i]{} & 4935.83 & 3.94 & $-$0.360 & MFK & 82 & — & 60 & 85 & —\
Ni[i]{} & 4953.21 & 3.74 & $-$0.660 & MFK & 87 & — & — & 72 & 81\
Ni[i]{} & 4967.52 & 3.80 & $-$1.570 & MFK & 40 & — & — & — & —\
Ni[i]{} & 5010.94 & 3.63 & $-$0.870 & MFK & — & — & 62 & 73 & 63\
Ni[i]{} & 5084.11 & 3.68 & $-$0.180 & E93 & 100 & — & 86 & — & 95\
Ni[i]{} & 5094.42 & 3.83 & $-$1.080 & MFK & 51 & — & 38 & — & —\
Ni[i]{} & 5115.40 & 3.83 & $-$0.280 & R03 & 101 & — & — & — & 90\
Ni[i]{} & 5157.98 & 3.61 & $-$1.590 & MFK & — & — & 33 & — & —\
Ni[i]{} & 5578.73 & 1.68 & $-$2.640 & MFK & 93 & — & — & — & 91\
Ni[i]{} & 5589.37 & 3.90 & $-$1.140 & MFK & 42 & — & 35 & 45 & 37\
Ni[i]{} & 5593.75 & 3.90 & $-$0.840 & MFK & 56 & — & — & 61 & 68\
Ni[i]{} & 5760.84 & 4.11 & $-$0.800 & MFK & 68 & — & — & 63 & —\
Ni[i]{} & 5805.23 & 4.17 & $-$0.640 & MFK & 56 & 37 & — & — & 52\
Ni[i]{} & 5996.74 & 4.24 & $-$1.060 & MFK & 30 & — & — & — & 30\
Ni[i]{} & 6053.69 & 4.24 & $-$1.070 & MFK & — & — & — & 39 & —\
Ni[i]{} & 6086.29 & 4.27 & $-$0.510 & MFK & 66 & — & 43 & 66 & 66\
Ni[i]{} & 6108.12 & 1.68 & $-$2.440 & MFK & 106 & — & 90 & — & —\
Ni[i]{} & 6111.08 & 4.09 & $-$0.870 & MFK & 68 & — & 48 & 49 & —\
Ni[i]{} & 6128.98 & 1.68 & $-$3.320 & MFK & 66 & — & 51 & — & —\
Ni[i]{} & 6176.82 & 4.09 & $-$0.264 & R03 & — & — & — & 77 & —\
Ni[i]{} & 6186.72 & 4.11 & $-$0.960 & MFK & 48 & — & 40 & 46 & —\
Ni[i]{} & 6204.61 & 4.09 & $-$1.150 & MFK & 52 & — & 23 & — & 42\
Ni[i]{} & 6223.99 & 4.11 & $-$0.980 & MFK & — & — & — & — & 41\
Ni[i]{} & 6230.10 & 4.11 & $-$1.260 & MFK & 40 & — & — & 26 & 37\
Ni[i]{} & 6322.17 & 4.15 & $-$1.170 & MFK & 33 & — & 31 & — & —\
Ni[i]{} & 6327.60 & 1.68 & $-$3.150 & MFK & 84 & — & — & 79 & 81\
Ni[i]{} & 6378.26 & 4.15 & $-$0.900 & MFK & 60 & — & — & — & 44\
Ni[i]{} & 6482.81 & 1.94 & $-$2.630 & MFK & — & 35 & — & — & —\
Ni[i]{} & 6532.88 & 1.94 & $-$3.390 & MFK & 62 & — & — & — & —\
Ni[i]{} & 6586.32 & 1.95 & $-$2.810 & MFK & 74 & — & 71 & 92 & —\
Ni[i]{} & 6635.14 & 4.42 & $-$0.830 & MFK & — & — & 27 & — & —\
Ni[i]{} & 6643.64 & 1.68 & $-$2.030 & MFK & 142 & — & 109 & 150 & 136\
Ni[i]{} & 6767.78 & 1.83 & $-$2.170 & MFK & 111 & 76 & 100 & 118 & 113\
[cccccccccc]{}\
& & & & &\
&\
Element & $\lambda$ & $\chi$(eV) & $\log gf$ & Ref & 769 & 806 & 1110 & 1118 & 395\
Ni[i]{} & 6772.32 & 3.66 & $-$0.970 & R03 & — & — & 55 & 79 & —\
Ni[i]{} & 6842.04 & 3.66 & $-$1.477 & E93 & 49 & — & — & 42 & 53\
Ni[i]{} & 7788.93 & 1.95 & $-$1.990 & E93 & — & — & — & — & 122\
\
References: B86: Blackwell D.E. et al. (1986); Ca07: Carretta et al. (2007);
D2002; Depagne et al. (2002); DS91: Drake & Smith (1991);
E93: Edvardsson et al. (1993); GS: Gratton & Sneden (1988);
MFK: Martin et al., (2002); MR94: Mcwilliam & Rich (1994);
PS: Preston & Sneden (2001); R03: Reddy et al. (2003);
R99: Reddy et al. (1999); WSM: Wiese, Smith & Miles (1969).
[cccccccccc]{} \[tabellinesb\]\
& & & & &\
&\
Element & $\lambda$ & $\chi$(eV) & $\log gf$ & Ref & 1534 & 1616 & 1349 & 784 & 663\
Na[i]{} & 6154.22 & 2.10 & $-$1.51 & PS & 47 & 68 & — & 68 & 105\
Na[i]{} & 6160.75 & 2.10 & $-$1.21 & R03 & — & — & — & 86 & 122\
Mg[i]{} & 4730.04 & 4.34 & $-$2.39 & R03 & 88 & — & — & 79 & —\
Mg[i]{} & 5711.10 & 4.34 & $-$1.75 & R99 & 140 & 124 & 124 & 138 & 152\
Mg[i]{} & 6318.71 & 5.11 & $-$1.94 & Ca07 & — & 58 & — & 74 & 91\
Mg[i]{} & 7387.70 & 5.75 & $-$0.87 & MR94 & — & 102 & 98 & 113 & 120\
Mg[i]{} & 8736.04 & 5.94 & $-$0.34 & WSM & 133 & — & — & — & —\
Al[i]{} & 6698.67 & 3.14 & $-$1.63 & R03 & — & — & 58 & — & —\
Al[i]{} & 7835.32 & 4.04 & $-$0.58 & R03 & — & 39 & — & 69 & 78\
Al[i]{} & 7836.13 & 4.02 & $-$0.40 & R03 & — & 42 & — & 80 & 102\
Al[i]{} & 8772.88 & 4.02 & $-$0.25 & R03 & 99 & — & 99 & — & —\
Al[i]{} & 8773.91 & 4.02 & $-$0.07 & R03 & 114 & — & — & — & 140\
Si[i]{} & 5793.08 & 4.93 & $-$2.06 & R03 & 53 & 32 & 48 & 76 & 86\
Si[i]{} & 6125.03 & 5.61 & $-$1.54 & E93 & 31 & 26 & — & 50 & —\
Si[i]{} & 6145.02 & 5.61 & $-$1.43 & E93 & — & — & — & 59 & 64\
Si[i]{} & 6155.14 & 5.62 & $-$0.77 & E93 & 79 & 81 & — & 96 & —\
Si[i]{} & 7800.00 & 6.18 & $-$0.72 & E93 & 40 & — & — & 75 & —\
Si[i]{} & 8728.01 & 6.18 & $-$0.36 & E93 & 75 & — & — & — & 122\
Si[i]{} & 8742.45 & 5.87 & $-$0.51 & E93 & — & — & 106 & — & 119\
Ca[i]{} & 6102.73 & 1.88 & $-$0.79 & D2002& — & — & — & 160 & —\
Ca[i]{} & 6161.30 & 2.52 & $-$1.27 & E93 & 82 & — & — & 101 & 137\
Ca[i]{} & 6166.44 & 2.52 & $-$1.14 & R03 & 95 & 80 & 118 & 101 & 128\
Ca[i]{} & 6169.04 & 2.52 & $-$0.80 & R03 & — & — & — & 114 & —\
Ca[i]{} & 6169.56 & 2.53 & $-$0.48 & DS91 & — & — & — & 142 & —\
Ca[i]{} & 6455.60 & 2.51 & $-$1.29 & R03 & — & 80 & 101 & 90 & —\
Ca[i]{} & 6471.66 & 2.51 & $-$0.69 & S86 & 122 & 114 & — & — & —\
Ca[i]{} & 6493.79 & 2.52 & $-$0.11 & DS91 & 144 & — & — & — & —\
Ti[i]{} & 4759.28 & 2.25 & 0.514 & MFK & — & 86 & — & 88 & —\
Ti[i]{} & 4820.41 & 1.50 & $-$0.439 & MFK & — & 65 & — & 92 & —\
Ti[i]{} & 5009.66 & 0.02 & $-$2.259 & MFK & — & — & 112 & 88 & —\
Ti[i]{} & 5039.96 & 0.02 & $-$1.130 & MFK & — & 123 & — & — & —\
Ti[i]{} & 5043.59 & 0.84 & $-$1.733 & MFK & — & — & 73 & — & 111\
Ti[i]{} & 5062.10 & 2.16 & $-$0.464 & MFK & 35 & — & — & — & 64\
Ti[i]{} & 5113.45 & 1.44 & $-$0.880 & E93 & — & — & — & — & 106\
Ti[i]{} & 5147.48 & 0.00 & $-$2.012 & MFK & — & 89 & — & — & —\
Ti[i]{} & 5210.39 & 0.05 & $-$0.883 & MFK & — & — & — & 156 & —\
Ti[i]{} & 5219.71 & 0.02 & $-$2.292 & MFK & — & — & 117 & — & —\
Ti[i]{} & 5295.78 & 1.05 & $-$1.633 & MFK & — & 36 & — & 56 & 64\
Ti[i]{} & 5689.48 & 2.30 & $-$0.469 & MFK & — & — & — & 38 & 67\
Ti[i]{} & 5866.46 & 1.07 & $-$0.871 & E93 & — & 91 & — & — & 155\
Ti[i]{} & 5922.12 & 1.05 & $-$1.465 & MFK & — & — & — & — & 101\
Ti[i]{} & 5978.55 & 1.87 & $-$0.496 & MFK & — & 43 & — & 76 & —\
Ti[i]{} & 6126.22 & 1.07 & $-$1.370 & R03 & — & 44 & — & 71 & —\
Ti[i]{} & 6258.11 & 1.44 & $-$0.355 & MFK & — & 96 & — & — & 149\
Ti[i]{} & 6261.11 & 1.43 & $-$0.480 & B86 & 95 & 86 & 124 & 105 & —\
Ti[i]{} & 6554.24 & 1.44 & $-$1.219 & MFK & — & — & — & 51 & 84\
[cccccccccc]{}\
& & & & &\
&\
Element & $\lambda$ & $\chi$(eV) & $\log gf$ & Ref & 1534 & 1616 & 1349 & 784 & 663\
Cr[i]{} & 4836.85 & 3.10 & $-$1.137 & MFK & — & — & 49 & — & —\
Cr[i]{} & 5193.50 & 3.42 & $-$0.720 & MFK & — & — & — & 23 & —\
Cr[i]{} & 5196.45 & 3.45 & $-$0.270 & MFK & — & — & 92 & — & —\
Cr[i]{} & 5214.13 & 3.37 & $-$0.740 & MFK & — & — & — & 32 & —\
Cr[i]{} & 5296.70 & 0.98 & $-$1.390 & GS & — & 129 & — & — & —\
Cr[i]{} & 5348.32 & 1.00 & $-$1.290 & GS & — & 126 & — & 154 & —\
Cr[i]{} & 5702.32 & 3.45 & $-$0.666 & MFK & 21 & — & 63 & — & —\
Cr[i]{} & 5783.07 & 3.32 & $-$0.500 & MFK & — & — & — & 54 & 75\
Cr[i]{} & 5783.87 & 3.32 & $-$0.290 & GS & 49 & 60 & 95 & 57 & 90\
Cr[i]{} & 5787.92 & 3.32 & $-$0.080 & GS & 51 & — & 100 & — & 103\
Cr[i]{} & 6330.09 & 0.94 & $-$2.920 & R03 & — & 67 & — & — & —\
Ni[i]{} & 4904.42 & 3.54 & $-$0.170 & MFK & 117 & — & 139 & — & —\
Ni[i]{} & 4935.83 & 3.94 & $-$0.360 & MFK & — & — & 99 & 78 & —\
Ni[i]{} & 4953.21 & 3.74 & $-$0.660 & MFK & 61 & 65 & — & — & 105\
Ni[i]{} & 4967.52 & 3.80 & $-$1.570 & MFK & — & — & — & 35 & —\
Ni[i]{} & 5010.94 & 3.63 & $-$0.870 & MFK & — & — & 78 & 68 & —\
Ni[i]{} & 5048.85 & 3.85 & $-$0.370 & MFK & 94 & — & — & — & —\
Ni[i]{} & 5084.11 & 3.68 & $-$0.180 & E93 & — & 94 & — & — & —\
Ni[i]{} & 5094.42 & 3.83 & $-$1.080 & MFK & 51 & 53 & — & — & —\
Ni[i]{} & 5115.40 & 3.83 & $-$0.280 & R03 & — & 95 & — & — & —\
Ni[i]{} & 5157.98 & 3.61 & $-$1.590 & MFK & — & — & 44 & — & —\
Ni[i]{} & 5578.73 & 1.68 & $-$2.640 & MFK & — & — & — & 105 & —\
Ni[i]{} & 5589.37 & 3.90 & $-$1.140 & MFK & — & — & 46 & — & —\
Ni[i]{} & 5593.75 & 3.90 & $-$0.840 & MFK & — & 51 & 68 & — & 84\
Ni[i]{} & 5643.09 & 4.17 & $-$1.250 & MFK & — & — & 27 & — & —\
Ni[i]{} & 5748.36 & 1.68 & $-$3.260 & MFK & — & 64 & — & — & 113\
Ni[i]{} & 5760.84 & 4.11 & $-$0.800 & MFK & — & 59 & 61 & — & —\
Ni[i]{} & 5805.23 & 4.17 & $-$0.640 & MFK & — & — & — & — & 59\
Ni[i]{} & 5996.74 & 4.24 & $-$1.060 & MFK & — & — & — & — & 57\
Ni[i]{} & 6086.29 & 4.27 & $-$0.510 & MFK & — & — & — & 77 & —\
Ni[i]{} & 6108.12 & 1.68 & $-$2.440 & MFK & 124 & 90 & 140 & — & —\
Ni[i]{} & 6111.08 & 4.09 & $-$0.870 & MFK & — & — & — & — & 75\
Ni[i]{} & 6128.98 & 1.68 & $-$3.320 & MFK & — & 48 & — & 69 & 105\
Ni[i]{} & 6176.82 & 4.09 & $-$0.264 & R03 & — & — & — & 86 & 114\
Ni[i]{} & 6177.25 & 1.83 & $-$3.510 & MFK & — & — & — & 45 & 67\
Ni[i]{} & 6186.72 & 4.11 & $-$0.960 & MFK & 38 & — & — & 46 & —\
Ni[i]{} & 6204.61 & 4.09 & $-$1.150 & MFK & — & 34 & — & 49 & 56\
Ni[i]{} & 6223.99 & 4.11 & $-$0.980 & MFK & — & — & — & 50 & —\
Ni[i]{} & 6230.10 & 4.11 & $-$1.260 & MFK & — & 27 & — & — & —\
Ni[i]{} & 6327.60 & 1.68 & $-$3.150 & MFK & 60 & 74 & — & 93 & 99\
Ni[i]{} & 6482.81 & 1.94 & $-$2.630 & MFK & — & 82 & 112 & 96 & —\
Ni[i]{} & 6586.32 & 1.95 & $-$2.810 & MFK & 76 & — & 92 & 76 & —\
Ni[i]{} & 6635.14 & 4.42 & $-$0.830 & MFK & — & — & — & 31 & 43\
Ni[i]{} & 6643.64 & 1.68 & $-$2.030 & MFK & — & 129 & — & 139 & —\
Ni[i]{} & 6767.78 & 1.83 & $-$2.170 & MFK & 131 & — & — & — & 155\
[cccccccccc]{}\
& & & & &\
&\
Element & $\lambda$ & $\chi$(eV) & $\log gf$ & Ref & 1534 & 1616 & 1349 & 784 & 663\
Ni[i]{} & 6772.32 & 3.66 & $-$0.970 & R03 & — & — & — & 91 & 90\
Ni[i]{} & 7788.93 & 1.95 & $-$1.990 & E93 & — & 118 & 156 & 142 & —\
\
References: B86: Blackwell D.E. et al. (1986); Ca07: Carretta et al. (2007);
D2002; Depagne et al. (2002); DS91: Drake & Smith (1991);
E93: Edvardsson et al. (1993); GS: Gratton & Sneden (1988);
MFK: Martin et al., (2002); MR94: Mcwilliam & Rich (1994);
PS: Preston & Sneden (2001); R03: Reddy et al. (2003);
R99: Reddy et al. (1999); WSM: Wiese, Smith & Miles (1969).
\[tab:atmparam\]
[ccccccccc]{}\
ID & $T_{\rm eff, ph}$ & log $g_{ph}$ & $T_{\rm eff, sp}$ & log $g_{sp}$ & $\xi$ & $[FeI/H]\pm$ $\sigma$ (\#) & $[FeII/H]\pm$ $\sigma$ (\#) & Comment\
& (K) & dex & (K) & dex & kms$^{-1}$ & & &\
395 & 5205 & 2.74 & 5100 & 2.7 & 1.6 & $-$0.15$\pm$0.15(53) & $-$0.15$\pm$0.12(5) & Member\
663 & 5123 & 2.85 & 4900 & 3.2 & 2.3 & 0.07$\pm$0.14(37) & 0.06$\pm$0.13(3) & Member\
769 & 5196 & 2.53 & 5200 & 3.0 & 1.3 & 0.10$\pm$0.14(53) & 0.08$\pm$0.17(3) & Member\
784 & 4955 & 2.82 & 5000 & 2.5 & 1.7 & $-$0.08$\pm$0.13(39) & $-$0.11$\pm$0.12(6) & Non-Member\
806 & 6324 & 2.86 & 6000 & 2.7 & 4.9 & $-$0.53$\pm$0.12(20) & $-$0.52(2) & Non-Member/binary Cepheid?\
1110 & 5161 & 2.68 & 5350 & 3.4 & 1.0 & 0.03$\pm$0.15(44) & 0.01$\pm$0.09(6) & Member\
1118 & 4958 & 2.63 & 5100 & 2.6 & 1.5 & $-$0.16$\pm$0.12(44) & $-$0.17$\pm$0.10(3) & Member\
1349 & 4796 & 2.72 & 5100 & 2.6 & 2.1 & 0.01$\pm$0.19(25) & 0.01(2) & Member\
1534 & 4982 & 2.73 & 5000 & 2.0 & 2.2 & $-$0.30$\pm$0.16(15) & $-$0.29(2) & Non-Member\
1616 & 5465 & 2.83 & 5450 & 3.5 & 1.7 & $-$0.07$\pm$0.16(31) & $-$0.09$\pm$0.14(5) & Non-Member\
\
**Notes.** For \[Fe I/H\] and \[Fe II/H\], we also show the standard deviation and the number of lines employed.
0.33truecm \[sun\]
[lccc]{}\
Element & This & Grevesse & & Asplund\
$_{\rule{0pt}{8pt}}$ & work & Sauval (1998)& et al. (2009)\
Fe & 7.50 & 7.50 & 7.50\
Na & 6.26 & 6.33 & 6.24\
Mg & 7.55 & 7.58 & 7.60\
Al & 6.31 & 6.47 & 6.45\
Si & 7.61 & 7.55 & 7.51\
Ca & 6.37 & 6.36 & 6.34\
Ti & 4.93 & 5.02 & 4.95\
Cr & 5.65 & 5.67 & 5.64\
Ni & 6.29 & 6.25 & 6.22\
Y & 2.04 & 2.24 & 2.21\
Ba & 2.18 & 2.13 & 2.18\
Ce & 1.48 & 1.58 & 1.58\
Nd & 1.42 & 1.50 & 1.42\
\
\[abunda-Na\]
[lccccccc]{}\
\
ID & \[Na/Fe\]NLTE & \[Mg/Fe\] & \[Al/Fe\] & \[Si/Fe\] & \[Ca/Fe\] & \[Ti/Fe\] & \[Cr/Fe\]\
395 & +0.15(2) &$+$0.10$\pm$0.13(3) & — &$+$0.13$\pm$0.09(4) &$+$0.02$\pm$0.13(7) &$+$0.07(1) &$+$0.02$\pm$0.14(7)\
663 & +0.12(2) &$+$0.03$\pm$0.12(3) &$+$0.15$\pm$0.04(3) &$+$0.23$\pm$0.11(4) &$-$0.06(2) &$+$0.04$\pm$0.14(9) &$-$0.10$\pm$0.02(3)\
769 & +0.04(2) &$-$0.12$\pm$0.13(4) & — &$-$0.05$\pm$0.04(5) &$-$0.05$\pm$0.15(7) &$+$0.10$\pm$0.11(18) &$+$0.04$\pm$0.12(7)\
1110 & +0.13(2) &$+$0.03$\pm$0.14(3) & — &$-$0.14(2) &$+$0.04$\pm$0.11(5) &$+$0.09$\pm$0.13(11) &$+$0.18$\pm$0.06(6)\
1118 & +0.38(2) &$+$0.23$\pm$0.07(4) & — &$+$0.07$\pm$0.09(3) &$+$0.12$\pm$0.08(5) &$+$0.08$\pm$0.13(13) &$+$0.14$\pm$0.13(6)\
1349 & — &$-$0.11(2) &$+$0.16(2) &$-$0.19(2) &$-$0.01(2) &$+$0.28$\pm$0.11(4) &$+$0.30$\pm$0.09(5)\
Tombaugh 1$^a$& +0.17$\pm$0.06 &$+$0.03$\pm$0.05 &$+$0.15 &$+$0.01$\pm$0.07 &$+$0.01$\pm$0.03 &$+$0.11$\pm$0.04 &$+$0.10$\pm$0.06\
\
ID & \[Na/Fe\]NLTE & \[Mg/Fe\] & \[Al/Fe\] & \[Si/Fe\] & \[Ca/Fe\] & \[Ti/Fe\] & \[Cr/Fe\]\
784 & +0.20(2) &$+$0.17$\pm$0.13(4) &$+$0.27(2) &$+$0.14$\pm$0.10(5) &$-$0.06$\pm$0.09(6) &$+$0.03$\pm$0.07(10) &$-$0.13$\pm$0.10(5)\
806$^b$ & +0.27(1) &$+$0.02(1) & — &$+$0.14$\pm$0.11(4) &$+$0.05(2) &$+$0.32(1) &$-$0.12$\pm$0.08(3)\
1534 & +0.21(1) &$+$0.25$\pm$0.02(3) & — &$-$0.15$\pm$0.09(5) &$-$0.20$\pm$0.15(4) &$+$0.01(2) &$-$0.22$\pm$0.11(3)\
1616 & +0.34(1) &$+$0.18$\pm$0.03(3) &$-$0.10(2) &$-$0.21$\pm$0.08(3) &$-$0.01$\pm$0.06(3) &$+$0.27$\pm$0.13(10) &$+$0.17$\pm$0.14(4)\
\
**Notes.** For all abundances ratios, we also show the standard deviation and the number of lines employed. \[Na/Fe\] accounts for the NLTE effects calculated as in Gratton et al. (1999), see text. (a) Mean abundance ratio for each element for Tombaugh 1. (b) The classical Cepheid XZ CMa.
\[abunda-Ni\]
[lccccc]{}\
\
ID & \[Ni/Fe\] & \[Y/Fe\] & \[Ba/Fe\] & \[Ce/Fe\] & \[Nd/Fe\]\
395 &$-$0.06$\pm$0.13(18)&$-$0.10(1) &$+$0.28(1) &$+$0.25(1) &$+$0.35(1)\
663 &$-$0.04$\pm$0.15(14)&$+$0.00(1) & — & — & —\
769 &$+$0.02$\pm$0.15(28)&$+$0.07(1) &$+$0.38(1) &$+$0.10(1) &$+$0.30(1)\
1110 &$-$0.05$\pm$0.13(19)&$+$0.14(1) &$+$0.37(1) &$+$0.37(1) &$+$0.47(1)\
1118 &$+$0.04$\pm$0.12(19)&$+$0.13(1) &$+$0.36(1) &$+$0.29(1) & —\
1349 &$-$0.14$\pm$0.09(12)& — & — & — & —\
Tombaugh 1$^a$&$-$0.04$\pm$0.02 &$+$0.06$\pm$0.04 &$+$0.35$\pm$0.03 &$+$0.25$\pm$0.06 &$+$0.37$\pm$0.05\
\
ID & \[Ni/Fe\] & \[Y/Fe\] & \[Ba/Fe\] & \[Ce/Fe\] & \[Nd/Fe\]\
784 &$-$0.14$\pm$0.14(18)&$+$0.30(1) &$-$0.02(1) &$+$0.20(1) &$+$0.32(1)\
806$^b$ &$+$0.20$\pm$0.16(3) & — &$+$0.73(1) & — & —\
1534 &$-$0.27$\pm$0.13(9) &$-$0.07(1) &$-$0.05(1) & — & —\
1616 &$+$0.02$\pm$0.13(15)&$-$0.11(1) &$+$0.39(1) & — &$+$0.04(1)\
\
**Notes.** For all abundances ratios, we also show the standard deviation and the number of lines employed. (a) Mean abundance ratio for each element for Tombaugh 1. (b) The classical Cepheid XZ CMa.
0.33truecm \[error\]
[lcccc]{}\
Element & $\Delta T_{eff}$ & $\Delta\log g$ & $\Delta\xi$ & $\left( \sum \sigma^2 \right)^{1/2}$\
$_{\rule{0pt}{8pt}}$ & $+$180 K & $+$0.3 & $+$0.3 kms$^{-1}$ &\
Fe[i]{} & $+$0.14 & 0.00 & $-$0.14 & 0.20\
Fe[ii]{} & $-$0.11 & $+$0.19 & $-$0.11 & 0.25\
Na[i]{} & $+$0.12 & $-$0.01 & $-$0.05 & 0.13\
Mg[i]{} & $+$0.09 & $-$0.01 & $-$0.07 & 0.11\
Al[i]{} & $+$0.09 & $-$0.04 & $-$0.05 & 0.11\
Si[i]{} & 0.00 & $+$0.05 & $-$0.05 & 0.07\
Ca[i]{} & $+$0.16 & $-$0.04 & $-$0.15 & 0.22\
Ti[i]{} & $+$0.23 & $-$0.02 & $-$0.12 & 0.26\
Cr[i]{} & $+$0.15 & $-$0.02 & $-$0.09 & 0.18\
Ni[i]{} & $+$0.11 & $+$0.03 & $-$0.12 & 0.17\
Y[ii]{} & $-$0.02 & $+$0.08 & $-$0.05 & 0.10\
Zr[i]{} & $+$0.01 & $-$0.03 & $-$0.06 & 0.07\
Ba[ii]{} & $+$0.12 & $+$0.17 & $-$0.14 & 0.25\
Ce[ii]{} & $-$0.02 & $+$0.05 & $-$0.10 & 0.11\
Nd[ii]{} & $+$0.05 & $+$0.15 & $-$0.05 & 0.17\
\
**Notes.** Each column gives the variation of the abundance caused by the variation in $T_{\rm eff}$, $\log g$ and $\xi$. The last column gives the compounded rms uncertainty of the second to fourth columns. Abundance uncertainties of aluminium were calculated using the star 663.
[^1]: Based on observations carried out at Las Campanas Observatory (program ID: CN2009B-042) and Cerro Tololo Inter-American Observatory.
[^2]: PPMXL is a catalog of positions, proper motions, 2MASS, and optical photometry of 900 million stars and galaxies. For more information: <http://vo.uni-hd.de/ppmxl>
[^3]: http://http://www.astro.yale.edu/smarts
[^4]: `http://www.astronomy.ohio-state.edu/Y4KCam/detector`
[^5]: IRAF is distributed by the National Optical Astronomy Observatory, operated by the Association of Universities for Research in Astronomy, Inc., under cooperative agreement with the National Science Foundation.
|
---
abstract: 'We present a practical scheme for performing error estimates for Density Functional Theory calculations. The approach which is based on ideas from Bayesian statistics involves creating an ensemble of exchange-correlation functionals by comparing with an experimental database of binding energies for molecules and solids. Fluctuations within the ensemble can then be used to estimate errors relative to experiment on calculated quantities like binding energies, bond lengths, and vibrational frequencies. It is demonstrated that the error bars on energy differences may vary by orders of magnitude for different systems in good agreement with existing experience.'
author:
- 'J. J. Mortensen'
- 'K. Kaasbjerg'
- 'S. L. Frederiksen'
- 'J. K. N[ø]{}rskov'
- 'J. P. Sethna'
- 'K. W. Jacobsen'
title: Bayesian Error Estimation in Density Functional Theory
---
Over the past few decades the use of Density Functional Theory (DFT)[@Hoh64] to predict structures, energetics, and other properties of atomic-scale systems has spread to many different fields and the number of applications has grown enormously. Today applications may vary from studies of chemical reactions in heterogeneous catalysis[@Hon05] through geophysical investigations of melting at the physical conditions of the Earth’s core[@Lai00; @Alf99] to studies of biomolecular systems like DNA[@Sky05; @Art03]. The general usefulness of the calculations lies in their unbiased “first principles” character and the relatively high degree of predictive power and reliability which has been established. With respect to the latter, it is however often difficult to assess directly to which extent a calculated quantity – this being a molecular bond length or some other property – is to be trusted. In practice the evaluation often falls back exclusively on the experience and acquired insight of the person performing the calculation.
In this letter we present a scheme for systematic error evaluation within the generalized-gradient approximation (GGA) DFT. The scheme is based on ideas from Bayesian statistics[@Jay03] in which an ensemble of models or model parameters are assigned probabilities by comparing to a database of experimental results. The ensemble generated can then subsequently be used to estimate error bars on model predictions within the considered class of models. The scheme is simple to apply and amounts in the end to only a few additional non-self-consistent evaluations of exchange-correlation functionals. In spirit, it is in fact close to a rather common practice within the field of DFT-GGA calculations: To asses the validity of a calculated DFT-GGA result it is common to try out different versions of the GGA-functional or perhaps to compare with a local-density approximation (LDA) result. The scheme presented here provides a systematic framework for such an approach.
The statistical approach we use is inspired by Bayesian statistics [@Jay03] and was further developed in the context of modeling complex biomedical networks [@Bro03] and for construction of interatomic potentials [@Fre04]. The main ingredients in the approach are a model $M$ which is given by a set of parameters $\theta$, and a database $D$. In our case the model will be a GGA-type exchange-correlation functional described through a number of parameters $\theta$ and the database $D$ will consist of experimental atomization/cohesive energies $E_k^{\text{exp}}$ for a collection of molecules and solids (details below). We now define a probability distribution in the space of model parameters (given the choice of model and the database) through $$\label{eq:prob}
P(\theta|MD) \sim \exp (-C(\theta)/T),$$ where $C$ denotes a standard least-squares cost function $C(\theta) =
\frac{1}{2} \sum_k (E_k(\theta) - E^{\text{exp}}_k)^2$ with $E_k(\theta)$ being the atomization/cohesive energy of system $k$ in the database calculated with the parameters $\theta$. The “effective temperature” $T$ determines the spread of the ensemble. In simple fitting procedures only the best-fit parameters $\theta_\text{b.f.}$, which are obtained when the cost function takes on its minimal value $C_\text{b.f.}$, are considered. Here, in contrast, a whole ensemble of parameter sets are considered leading to a spread of values on model predictions. Following Ref. , we take the value of the effective temperature to be given by the minimal (best fit) value of the cost function $C_\text{b.f.}$ as $T=2 C_\text{b.f.}/N_\text{p}$, where $N_\text{p}$ is the number of parameters. For a harmonic cost function each parameter will then on the average contribute an additional cost of $T/2 =
C_\text{b.f.}/N_\text{p}$ so that the ensemble will probe model parameters where the cost function is in the range from the minimal value $C_\text{b.f.}$ up to of the order a few times this value. This choice was demonstrated to work well in the case of error estimation for interatomic potentials [@Fre04].
[*The model*]{} we shall consider is GGA-DFT [@Lan80] where the exchange functional is a local function of the density $n$ and its dimensionless gradient $s=|\nabla n|/(2k_\text{F}n)$ ($n=k_\text{F}^3/(3\pi^2)$). Several suggestions for different mathematical forms of the exchange-functional within GGA exist[@Lan80; @Boe00; @Per96]. A commonly used class of these including PW-91[@Per92], PBE[@Per96], revPBE[@Zha98] and RPBE[@Ham99] differ by only the choice of the so-called enhancement factor $F_\text{x}(s)$ in the exchange energy $E_\text{x}$: $$E_{\text{x}}[n] =
\int d\mathbf{r}\, n(\mathbf{r}) \epsilon_{\text{x}}^{\text{LDA}}(n(\mathbf{r}))
F_{\text{x}}(s),$$ where $\epsilon_{\text{x}}^{\text{LDA}}(n)=-3k_\text{F}/(4\pi)$ (for a spin polarized density we have $E_{\text{x}}[n_\uparrow, n_\downarrow]
= (E_{\text{x}}[2n_\uparrow] + E_{\text{x}}[2n_\downarrow]) / 2$). The enhancement factors for PBE and RPBE are shown in Fig. \[fig:enhancement\]. In the following we shall expand the enhancement factor as $$\label{eq:enhancement}
F_{\text{x}}(s) = \sum_{i=1}^{N_\text{p}} \theta_i \left ( \frac{s}{1 +
s}\right )^{2i-2},$$ regarding the $\theta$’s as free parameters. We use three parameters ($N_\text{p}=3$) which a train/test check for our database has shown to give the optimal fit without over-fitting. The model space could be extended in future work to include a fraction of exact exchange as for the B3LYP functional[@Bec93].
[*The database*]{} we use consists of the experimental atomization energies of the molecules H$_2$, LiH, CH$_4$, NH$_3$, OH, H$_2$O, HF, Li$_2$, LiF, Be$_2$, C$_2$H$_2$, C$_2$H$_4$, HCN, CO, N$_2$, NO, O$_2$, F$_2$, P$_2$ and Cl$_2$ and the experimental cohesive energies (per atom) of the solids Na, Li, Si, C, SiC, AlP, MgO, NaCl, LiF, Cu and Pt. In the cost function all systems in the database appear with the same unit weight.
All calculations are performed with a real-space multigrid DFT code[@Mor05] using the projector augmented wave method[@Blo03] to describe the core regions. All calculated energy differences have been converged to an accuracy better than 50 meV with respect to number of grid points, unit-cell size (for the molecules and atoms) and number of k-points (for the solids). The electron density is calculated self-consistently using the PBE-functional and the evaluation of the exchange-correlation energy for other enhancement factors are performed using the PBE density. This is a very good approximation due to the variational principle. Energies are calculated at the calculated equilibrium bond distances.
Since Eq. (\[eq:enhancement\]) is linear in the parameters $\theta$, the total energy of a given system will also be linear in $\theta$: $$\label{eq:linear}
E(\theta) = E_0 + \sum_{i=1}^3 \Delta E_i \theta_i,$$ where the coefficients $E_0$ and $\Delta E_i$ only have to be calculated once for each system. It is therefore very fast to calculate energy values for different enhancement factors in the ensemble.
Minimizing the cost function with respect to the three parameters leads to the best-fit enhancement factor shown in Fig. \[fig:enhancement\] corresponding to the parameters $\theta_\text{b.f.} = (1.0008, 0.1926, 1.8962)$. The function is seen to follow quite closely the PBE enhancement factor at low values of the gradient $s$. In particular it is nearly one in the homogeneous limit ($s=0$) which is exclusively a result of the fitting. For $s$-values greater than 1.5 the best-fit enhancement factor increases more steeply than PBE being more similar to the RPBE factor. In Table \[tab:errors\] the resulting errors are shown for LDA, PBE, RPBE and for the best fit. RPBE performs better on the molecules and PBE is better for the solids; the best fit represents a compromise between the two. We would like to stress that the main point of this letter is not to derive an improved functional. Much experience has been acquired concerning how well different GGA functionals work for different systems [@Kur99; @Sta03; @Sta04] and we do not expect to obtain a large overall improvement within this simple GGA framework. But as we shall see in the following the ensemble construction allows for realistic evaluation of the error bars on calculated quantities.
error LDA PBE RPBE best fit
------------ ------------------ -------------- --------------- ---------------
molecules:
mean abs. 1.46 0.35 0.21 0.24
mean 1.38 0.28 -0.01 0.12
max. (-) -0.35(H$_2$) -0.22(H$_2$) -0.32(CH$_4$) -0.26(Li$_2$)
max. (+) 3.07(C$_2$H$_4$) 0.89(O$_2$) 0.46(O$_2$) 0.71(O$_2$)
solids:
mean abs. 1.35 0.16 0.40 0.27
mean 1.35 -0.09 -0.40 -0.24
max. (-) -0.72(Pt) -1.37(Pt) -0.94(Pt)
max. (+) 2.73(Pt) 0.36(C) 0.15(C)
all:
mean abs. 1.42 0.28 0.28 0.25
mean 1.37 0.14 -0.16 -0.02
: Errors in DFT atomization energies and cohesive energies (in eV) relative to experiment. All calculations are based on self-consistent PBE densities. Experimental numbers are taken from Refs. .
\[tab:errors\]
The cost function appearing in the probability distribution Eq. (\[eq:prob\]) is very nearly quadratic in the model parameters in the relevant range of parameter space. We can therefore expand the exponent in the probability distribution as $C(\theta)/T = \text{const.} + \frac{1}{2}\Delta\theta^{T} A
\Delta\theta$, where $A$ is a symmetric matrix. With $U$ being the unitary matrix that diagonalizes $A$ ($AU=U\Lambda$), we can finally write the parameters of the enhancement factors in the ensemble as $$\begin{aligned}
\label{eq:parameters}
\theta & = & \theta_\text{b.f.} + U\Lambda^{-1/2}\alpha \nonumber\\
& = & \theta_\text{b.f.} +
\begin{pmatrix}
0.066 & 0.055 & -0.034 \\
-0.812 & 0.206 & 0.007 \\
1.996 & 0.082 & 0.004
\end{pmatrix}
\begin{pmatrix}
\alpha_1 \\
\alpha_2\\
\alpha_3
\end{pmatrix},\end{aligned}$$ where the $\alpha_1$, $\alpha_2$, and $\alpha_3$ are stochastic variables which are Gaussian distributed with unit width: $\mathcal{P}(\alpha_i) \sim \exp(-\alpha_i^2/2)$. Using this formula it is very easy to generate a properly distributed ensemble of enhancement factors as shown in Fig. \[fig:enhancement\].
The key suggestion of this letter is that for a given calculated observable, say a bond length of a molecule, the variation of the calculated value of this observable within the ensemble of enhancement factors provides a useful estimate of how large the error of the best fit value is compared to experiment. From an ensemble of parameters, $\theta^1, \theta^2, \ldots, \theta^N$, generated from Eq. (\[eq:parameters\]), the standard deviation $\sigma_\text{BEE}(O)$ which we shall refer to as the Bayesian error estimate (BEE) of the observable $O$ can be determined. Considering $O$ as a function of $\theta$ the BEE is evaluated as $$\label{eq:bee}
\sigma_\text{BEE}(O) = \sqrt{\frac{1}{N} \sum_{\mu=1}^N
\left ( O(\theta^\mu)- O_\text{best-fit} \right )^2 },$$ In the simple case where the observable is approximately linear in the parameters $\theta$ the BEE can be calculated without explicitly generating an ensemble through $\sigma_\text{BEE}(O) = (
\sum_{i=1}^{3} \left(
\partial O / \partial \alpha_i \right)^2)^{1/2}$ and Eq. (\[eq:parameters\]). Further details can be found in our implementation of the approach in the Atomistic Simulation Environment[@ASE].
The ensemble in Eq. (\[eq:bee\]) is around the best-fit enhancement factor corresponding to $\alpha_i=0$. However, considering that the ensemble of enhancement factors (Fig. \[fig:enhancement\]) is quite wide compared with the difference between for example the PBE and the best-fit functional it seems reasonable to alternatively apply the fluctuations around either the PBE or RPBE functional.
It is useful to consider the ratio $(O_\text{best-fit} -
O_\text{exp})/ \sigma_\text{BEE}(O)$ of the actual error relative to the estimate. For any given observable in a particular system this ratio is just a single number so in order to assess whether our approach produces reliable error estimates from a statistical point of view we need to look at the distribution of ratios for several observables and systems.
In Fig. \[fig:ae\] we show histograms of the relative error for different observables for all the (molecular and solid) systems in the database. The upper panel shows the distribution in the case of the atomization/cohesive energies. As can be seen the distribution agrees quite well with a Gaussian distribution of unit width indicating that the error estimates are in fact reasonable for the binding energies. The individual standard deviations for the cohesive energies are in the range from 0.09 eV for Na to 0.75 eV for Al and for the atomization energies the range is from 0.07 eV for Li$_2$ to 0.60 eV for C$_2$H$_4$. As an example, the GGA estimate of the cohesive energy of Na (experimental value is 1.11 eV) is $1.02\pm0.09$ eV. The middle panel in Fig. \[fig:ae\] shows the relative error histogram for the bond lengths (for both molecules and solids) and relative errors for the molecular frequencies and solid bulk moduli is shown in the lower panel[@footnote2]. For both distributions, we see that the BEE’s give reasonable estimates of the actual errors.
It is well-known for experienced users of DFT calculations that the reliability with which energy differences can be calculated vary dramatically depending on the particular system. The BEE catches this behavior as can be seen for example by comparing the cohesive energy and the bcc-fcc structural energy difference for bulk copper (Fig. \[fig:Cu\]). The BEE for the cohesive energy is 0.5 eV while the error bar on the structural energy difference is two orders of magnitudes smaller (4 meV). The small structural energy difference is seen to be significantly greater than zero. The high reliability with which small structural energy differences can be calculated for bulk metals is confirmed by the fact that for almost all metals the correct equilibrium structures are predicted from the calculations[@Skr85].
For chemisorption systems the BEE for the energy difference between chemisorption at two difference surface sites may also be somewhat smaller than the error bar for the total chemisorption energy as illustrated in the case of CO on Cu(100) in Fig. \[fig:Cu\]. However as can be seen from the figure the error bar on the site-preference is so large that the preferred chemisorption site cannot be reliably determined. This is in good agreement with the fact that for a number of CO-metal chemisorption systems DFT-GGA calculations do in fact predict a wrong chemisorption site[@Fei01].
It should be noted that in general the error estimates depend on both the choice of database and the class of models (here the GGA’s). This means that if some piece of physics is not present in the database or if the model is completely unable to describe a particular feature unrealistic estimates may occur. For example all GGA’s are unable to properly describe the long-range van der Waals interactions and hence the BEE’s will be unrealistic for that type of interactions.
We further note, that it may be possible to reduce the error bars in some cases by picking a database focused on a particular type of systems. Using for example only the subset of molecular systems in our database a different best-fit functional is obtained where the mean absolute error is only 0.15 eV compared to the 0.24 eV in Table \[tab:errors\]. With this functional the fluctuations and hence the BEEs will be reduced by almost a factor of two. However, applying this functional to bulk metals would lead to larger errors which would be underestimated.
Summarizing, we propose a simple way to estimate a large portion of the systematic error for DFT-GGA calculations, requiring only a few extra non-self-consistent energy calculations to calculate the error bars of any observable which is a function of energies.
We acknowledge support from the Carlsberg Foundation and the Danish Center for Scientific Computing through Grant No. HDW-1101-05.
[10]{}
P. Hohenberg and W. Kohn, Phys. Rev. [**136**]{}, B864 (1964).
K. Honkala [*et al.*]{}, Science [**307**]{}, 555 (2005).
A. Laio [*et al.*]{}, Science [**287**]{}, 1027 (2000).
D. Alfe, M. J. Gillan, and G. D. Price, Nature [**401**]{}, 462 (1999).
C. Skylaris, P. D. Haynes, A. A. Mostofi, and M. C. Payne, J. Chem. Phys. [ **122**]{}, 084119 (2005).
E. Artacho [*et al.*]{}, Molecular Physics [**101**]{}, 1587 (2003).
E. T. Jaynes, [*Probability Theory*]{} (Cambridge University Press, Cambridge, United Kingdom, 2003).
K. S. Brown and J. P. Sethna, Phys. Rev. E [**68**]{}, 021904 (2003).
S. L. Frederiksen, K. W. Jacobsen, K. S. Brown, and J. P. Sethna, Phys. Rev. Lett. [**93**]{}, 165501 (2004).
D. C. Langreth and J. P. Perdew, Phys. Rev. B [**21**]{}, 5469 (1980).
J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. [**77**]{}, 3865 (1996).
A. D. Boese, N. L. Doltsinis, N. C. Handy, and M. Sprik, J. Chem. Phys. [ **112**]{}, 1670 (2000).
J. P. Perdew [*et al.*]{}, Phys. Rev. B [**46**]{}, 6671 (1992).
Y. Zhang and W. Yang, Phys. Rev. Lett. [**80**]{}, 890 (1998).
B. Hammer, L. B. Hansen, and J. K. N[ø]{}rskov, Phys. Rev. B [**59**]{}, 7413 (1999).
A. D. Becke, J. Chem. Phys. [**98**]{}, 5648 (1993).
J. J. Mortensen, L. B. Hansen, and K. W. Jacobsen, Phys. Rev. B [**71**]{}, 035109 (2005), http://www.camp.dtu.dk/campos/gridpaw.
P. E. Bl[ö]{}chl, C. J. Forst, and J. Schimpl, Bulletin of Materials Science [**26**]{}, 33 (2003).
S. Kurth, J. P. Perdew, and P. Blaha, Int. J. Quantum Chem. [**75**]{}, 889 (1999).
V. N. Staroverov, G. E. Scuseria, J. Tao, and J. P. Perdew, J. Chem. Phys. [ **119**]{}, 12129 (2003).
V. N. Staroverov, G. E. Scuseria, J. Tao, and J. P. Perdew, Phys. Rev. B [ **69**]{}, 075102 (2004).
C. Kittel, [*Introduction to Solid State Physics*]{}, 7th ed. (John Wiley [&]{} Sons, New York, 1996).
I.-H. Lee and R. M. Martin, Phys. Rev. B [**56**]{}, 7197 (1997).
B. Paulus, P. Fulde, and H. Stoll, Phys. Rev. B [**54**]{}, 2556 (1996).
I. P. Daykov, T. A. Arias, and T. D. Engeness, Phys. Rev. Letters [**90**]{}, 216402 (2003).
R. N. Schmid [*et al.*]{}, Adv. Quant. Chem. [**33**]{}, 209 (1998).
http://www.camp.dtu.dk/campos/ASE.
B. G. Johnson, P. M. W. Gill, and J. A. Pople, J. Chem. Phys. [**98**]{}, 5612 (1993).
P. Rodriguez-Hern[á]{}ndez and A. Mu[ñ]{}oz, Semicond. Sci. Technol. [ **7**]{}, 1437 (1992).
Vibrational frequencies are only calculated for a subset of the molecules.
S. Vollmer, G. Witte, and C. W[ö]{}ll, Catal. Lett. [**77**]{}, 97 (2001).
H. L. Skriver, Phys. Rev. B [**31**]{}, 1909 (1985).
P. J. Feibelman [*et al.*]{}, J. Phys. Chem. B [**105**]{}, 4018 (2001).
|
**Dynamic Multi-Factor Bid-Offer Adjustment Model**
**A Feedback Mechanism for Dealers (Market Makers) to Deal (Grapple) with the Uncertainty Principle of the Social Sciences**
**Ravi Kashyap**
**City University of Hong Kong / Gain Knowledge Group**
**Originally Created, October 2008; Revised, October 2013**
****
Abstract
========
The objective is to come up with a model that alters the Bid-Offer, currently quoted by market makers, that varies with the market and trading conditions. The dynamic nature of financial markets and trading, as the rest of social sciences, where changes can be observed and decisions can be taken by participants to influence the system, means that our model has to be adaptive and include a feedback loop that alters the bid offer adjustment based on the modifications we are seeing in the market and trading conditions, without a significant time delay. We will build a sample model that incorporates such a feedback mechanism and also makes it possible to check the efficacy of the changes to the quotes being made, by gauging the impact on the Profits. The market conditions here refer to factors that are beyond the direct control of the market maker and this information is usually available publicly to other participants. Trading conditions refer to factors that can be influenced by the market maker and are dependent on the trading book being managed and will be privy only to the market maker and will be mostly confidential to others. The factors we use to adjust the spread are the price volatility, which is publicly observable; and trade count and volume, which are generally only known to the market maker, in various instruments over different historical durations in time. The contributions of each of the factors to the bid-offer adjustment are computed separately and then consolidated to produce a very adaptive bid-offer quotation. The ensuing discussion considers the calculations for each factor separately and the consolidation in detail.
Any model that automatically updates the quotes is more suited for instruments that have a high number of transactions within short intervals, making it hard for traders to manually monitor and adjust the spread; though this is by no means a stringent requirement. We can use similar models for illiquid instruments as well and use the quotations provided by the model as a baseline for further human refinement.
We have chosen currency markets to build the sample model since they are extremely liquid, Over the Counter (OTC), and hence trading in them is not as transparent as other financial instruments like equities. The nature of currency trading implies that we do not have any idea on the actual volumes traded and the number of trades. We simulate the number of trades and the average size of trades from a log normal distribution. The parameters of the log normal distributions are chosen such that the total volume in a certain interval matches the volume publicly mentioned by currency trading firms. This methodology can be easily extended to other financial instruments and possibly to any product with an ability to make electronic price quotations or even be used to periodically perform manual price updates on products that are traded non-electronically.
Thankfully, we are not at a stage where Starbucks will sell coffee using such an algorithm, since it can possibly lead to certain times of the day when it can be cheaper to have a cup of coffee and as people become wary of this, there can be changes to their buying habits, with the outcome that the time for getting a bargain can be constantly changing; making the joys of sipping coffee, a serious decision making affair.
Motivation for Multi-Factor Bid-Offer Models
============================================
At the outset, let us look at some fundamentals that govern all financial instruments and then delve into the nuances which apply to instruments that are more amenable to adaptive bid-offer models. It is also worthwhile to mention here that for most assertions made below, numerous counter examples and alternate hypothesis can be produced. These are strictly attempts at tracing the essentials rather than getting bogged down with a specific instance. However, building a model for empirical usage requires forming a conceptual framework based on the more common observations, yet being highly attuned to any specifics that can stray from the usual. Also, for the sake of brevity, a number of finer points have been omitted and certain simplifying assumptions have been made.
The various financial instruments that exist today can be broadly viewed upon as vehicles for providing credit and a storage for wealth, for both individuals and institutions alike. The different instruments, both in terms of their nomenclature and their properties, then merely become manifestations of which and how many parties are involved in a transaction and the contractual circumstances or the legal clauses that govern the transaction.
Despite the several advances in the social sciences and in particular economic and financial theory, ***we have yet to discover an objective measuring stick of value, a so called, True Value Theory***. While some would compare the search for such a theory, to the medieval alchemists obsession with turning everything into gold, for our present purposes, the lack of such an objective measure means that the difference in value as assessed by different participants can effect a transfer of wealth. This forms the core principle that governs all commerce that is not for immediate consumption in general, and also applies specifically to all investment related traffic which forms a great portion of the financial services industry and hence the mainstay of market making.
Although, some of this is true for consumption assets; because ***the consumption ability of individuals and organizations is limited and their investment ability is not***, the lack of an objective measure of value affects investment assets in a greater way and hence investment assets and related transactions form a much greater proportion of the financial services industry. Consumption assets do not get bought and sold, to an inordinate extent, due to fluctuating prices, whereas investment assets will. Hull [\[]{}1999[\]]{} has a description of consumption and investment assets, specific to the price determination of futures and forwards. The price effect on consumptions assets affects the quantity bought and consumed, whilst with investment assets, the cyclical linkage between vacillating prices and increasing number of transactions becomes more apparent.
***Another distinguishing feature of investment assets is the existence or the open visibility of bid and ask prices.*** Any market maker for investment assets quotes two prices, one at which he is willing to buy and one at which he is willing to sell. Consumption assets either lack such an outright two sided quote; or it is hard to painlessly infer viewable buy and sell prices, since it involves some conversion from a more basic form of the product into the final commodity being presented to consumers. Examples for consumption assets are a mug of hot coffee, that requires a certain amount of processing from other rudimentary materials before it can be consumed; or a pack of raw almonds which is almost fit for eating. Coffee shops that sell coffee do not quote a price at which they buy ready drinkable coffee; the price at which a merchant will buy almonds is not readily transparent. Gold is an example of both, a consumption and an investment asset. A jewellery store will sell gold and objects made of gold; but it will also buy gold reflecting its combined consumption and investment trait. This leaves us with financial securities like stocks and bonds that are purely investment assets.
A number of disparate ingredients contribute to this price effect; like how soon the product expires and the frequent use of technology to facilitate a marketplace. EBay is an example of a business where certain consumption goods are being bought and sold. This can happen even if goods are only being sold, through the increased application of technology in the sales process. While not implying that the use of technology is bad, technology, or almost anything else, can be put to use that is bad. Thankfully, we are not at a stage where Starbucks will buy and sell coffee, since it can possibly lead to certain times of the day when it can be cheaper to have a cup of coffee and as people become wary of this, there can be changes to their buying habits, with the outcome that the time for getting a bargain can be constantly changing; making the joys of sipping coffee, a serious decision making affair. Even though this is an extreme example, we will overlook some of these diverse influences for now, since our attempt is to exemplify the principal differences between the varieties of financial transactions and the underlying types of assets that drive these deals.
This lack of an objective measure of value, (henceforth, value will be synonymously referred to as the price of an instrument), makes prices react at varying degrees and at varying speeds to the pull of different macro and micro factors. The greater the level of prevalence of a particular instrument (or even a particular facet of an instrument) the more easily it is affected by macro factors. This also means that policies are enforced by centralized institutions, (either directly by the government or by institutions acting under the directive of a single government or a coalition of governments), to regulate the impact of various factors on such popular instruments. Examples for this would be interest rate dependent instruments, which are extremely sensitive to rates set by central banks since even governments issue such instruments; dividends paid by equity instruments which are clearly more sensitive to the explicit taxation laws that govern dividends than to the level of interest rates; and commodities like oil, which are absolutely critical for the smooth functioning of any modern society and hence governments intervene directly to build up supplies and attempt to control the price. See Tuckman [\[]{}1995[\]]{} for interest rate instruments; Bodie, Kane and Marcus [\[]{}2002[\]]{} for equity instruments.
Lastly, it is important that we lay down some basics regarding the efficiency of markets and the equilibrium of prices. Surely, a lot of social science principles and methodologies are inspired from similar counterparts in the natural sciences. A central aspect of our lives is uncertainty and our struggle to overcome it. Over the years, it seems that we have found ways to understand the uncertainty in the natural world by postulating numerous physical laws.
These physical laws are deductive and are based on three statements - a specific set of initial conditions, a specific set of final conditions or explicanda and universally valid generalizations. Combining a set of generalizations with known initial conditions yields predictions; combining them with known final conditions yields explanations; and matching known initial with known final conditions serves as a test of the generalizations involved. The majority of the predictions in the physical world hold under a fairly robust set of circumstances and cannot be influenced by the person making the observation and they stay unaffected if more people become aware of such a possibility.
In the social sciences, the situation is exactly the contrary. Popper [\[]{}2002[\]]{} gives a critique and warns of the dangers of historical prediction in social systems. In their manifesto, Derman and Wilmott [\[]{}2009[\]]{}, mention the need to combine art and science in the discipline of finance. While it is possible to declare that, ***Art is Science that we dont know about; and Science is Art restricted to a set of symbols governed by a growing number of rules***, our current state of affairs necessitate that we remain keenly cognizant of the shortcomings of forecasting. A set of initial conditions yielding a prediction based on some generalization, ceases to hold, as soon as many participants become aware of this situation and act to take advantage of this situation. This means that predictions in the social sciences are valid only for a limited amount of time and we cannot be sure about the length of this time, since we need to constantly factor in the actions of everyone that can potentially influence a prediction, making it an extremely hard task.
***All attempts at prediction, including both the physical and the social sciences, are like driving cars with the front windows blackened out*** and using the rear view mirrors, that give an indication of what type of path has been encountered and using this information to forecast, what might be the most likely type of terrain that lies ahead for us to traverse. The path that has been travelled then becomes historical data that has been collected through observation and we make estimates on the future topography based on this. Best results generally occur, when we combine the data we get in the rear view mirror with the data we get from the side windows, which is the gauge of the landscape we are in now, to get a better comprehension of what lies ahead for us. The quality of the data we gather and what the past and the present hold then give an indication to what the future might be. So if the path we have treaded is rocky, then the chances of it being a bumpy ride ahead are higher. If it has been smooth, then it will be mostly smooth. Surely, the better our predictions, the faster we can move; but then again, it is easy to see that the faster we travel, the more risk we are exposed to, in terms of accidents happening, if the constitution of the unseen scenery in front of us shifts drastically and without much warning.
A paramount peculiarity of the social sciences is that passage on this avenue is part journey and part race. The roads are muddy, rocky and more prone to have potholes. This means being early or ahead on the road brings more winnings. We also have no easy way of knowing how many people are traveling on this path, either with us, ahead of us or even after us. As more people travel on the path, it starts falling apart, making it harder to travel on it, a situation which is accentuated considering we don’t have any vision out front. On the other hand, let us say, physical science roads, being well paved and well-constructed using concrete, hold steady for much longer time durations, so what has been observed in the past can be used to make durable forecasts that hold for lengthier amounts of time in the future.
Paich and Sterman [\[]{}1993[\]]{} inquire into decision making in complex environments and conduct an experiment where subjects must manage a new product from launch through maturity, and make pricing and capacity decisions. They demonstrate that decision making in complex dynamic environments tends to be flawed in specific ways by not accounting sufficiently for feedback loops, time delays and nonlinearities. Even with a decent amount of experience, there is no evidence that environments with high feedback complexity can produce improved decision making ability.
The Sweeney and Sweeney [\[]{}1977[\]]{} anecdote about the Capitol Hill baby-sitting crisis exposits the mechanics of inflation, setting interest rates and monetary policies required to police the optimal amount of money. The creation of a monetary crisis in a small simple environment of good hearted people expounds that even with near ideal conditions, things can become messy; then in a large labyrinthine atmosphere, disaster could be brewing without getting noticed and can strike without much premonition. Taleb [\[]{}2005[\]]{} is an entertaining narrative of the role of chance in life and in the financial markets. Taleb [\[]{}2010[\]]{} calls our attention to Black Swan events, which are extremely hard to detect, highlighting the perils of the prediction business.
This inability to make consistent predictions in the social sciences and the lack of an objective measure of value or a True Price Theory means that is almost impossible for someone to know what a real state of equilibrium is. Elton, Gruber, Brown and Goetzmann [\[]{}2009[\]]{} review the concepts related to efficient markets and other aspects of investing; Kashyap [\[]{}2014[\]]{} explained the pleasures and pitfalls of managing a portfolio, while emphasizing the cyclical nature of the investment process. The efficient market hypothesis in spite of being a very intriguing proposition, can at best claim that markets have a tendency to move towards being efficient, though a state of equilibrium is never fully attained since no one has an idea what that state of equilibrium is and the actions of the participants serves only to displace any state of equilibrium, if it did exist. The analogy for this would be a pendulum with perpetual motion; it swings back and forth around its place of rest with decreasing amplitude and the place of rest keeps changing with time, starting a new cycle of movement with reinforced vigour.
We can then summarize the above with the ***Uncertainty Principle of the Social Sciences***, which can be stated as, ***Any generalization in the social sciences cannot be both popular and continue to yield accurate predictions or in other words, the more popular a particular generalization, the less accurate will be the predictions it yields***. This is because as soon as any generalization and its set of conditions become common knowledge, the entry of many participants shifts the equilibrium or the dynamics, such that the generalization no longer applies to the known set of conditions.
All our efforts as professionals in the field of financial services, will then be to study uncertainty and uncover quasi-generalizations; understand its limitations in terms of what can be the closest states of pseudo-equilibrium; how long can such a situation exist; what factors can tip the balance to another state of temporary equilibrium; how many other participants are aware of this; what is their behaviour and how is that changing; etc., making our professions a very interesting, challenging and satisfying career proposition.
Application to Currency Market Making
======================================
With the above discussion in mind, we can turn specifically to how it applies to market making in financial assets. The increasing use of algorithms and automation has increased the frequency of trading for most securities that trade in high volumes. Dempster, M. A. H., & Jones, C. M. [\[]{}2001[\]]{}; Avellaneda and Stoikov [\[]{}2008[\]]{}; Chiu, Lukman, Modarresi and Velayutham [\[]{}2011[\]]{} and Chlistalla, Speyer, Kaiser and Mayer [\[]{}2011[\]]{} provide detailed accounts of high frequency trading and the evolution of various algorithms used towards that end. The increased frequency of trading means that the bid and offer quoted for a security also need to be constantly changing. It is common practice for market makers to set the bid and offer to depend on the size of the inventory and revise it as the inventory builds up in either direction. This clearly comes with a number of drawbacks, primary among which is the lack of change in the quotes due to the rapidly changing market and the wide variety of variables that capture the trading conditions. The other participants in this market making system, which in this case are the counterparties of the market maker, can observe the quotes and take decisions that will influence the system and the quoting mechanism may not register these new conditions till much later.
Hence to deal with the dynamic nature of the trading and market conditions, our model has to be adaptive and include a feedback loop that alters the bid offer adjustment based on the modifications in the market and trading conditions, without a significant time delay. The market conditions here refer to factors that are beyond the direct control of the market maker and this information is usually available publicly to other participants. Trading conditions refer to factors that can be influenced by the market maker and are dependent on the trading book being managed and will be privy only to the market maker and will be mostly confidential to others.
The market maker has access to a rich set of trading metrics, which are not immediately available to other participants. These metrics can affect the future direction of the price and hence using them to alter the quote leads to better profits. But given that the trading conditions are constantly changing, we need to revise the parameters of the alteration mechanism based on the conditions from the recent past. This forms a feedback loop that keeps changing the model dynamically based on what is happening in the market makers trading book. As discussed earlier, prediction is a perilous business; hence it is important to keep the number of parameters to a minimum while not ignoring any significant causes of change. With this motivation, we include the changes coming in from different sources by using adequate yet relatively simple econometrics techniques. This leads to changes in the model outputs that aid the quotation process and the constant revision of the model parameters is geared to deal with shifting regimes.
Any model that automatically updates the quotes is more suited for instruments that have a high number of transactions within short intervals, making it hard for traders to manually monitor and adjust the spread; though this is by no means a stringent requirement. We can use similar models for illiquid instruments as well and use the quotations provided by the model as a baseline for further human refinement. We have chosen currency markets to build the sample model since they are extremely liquid, Over the Counter (OTC), and hence trading in them is not as transparent as other financial instruments like equities. Copeland [\[]{}2008[\]]{} provides a rich discussion on exchange rates and currencies. The nature of currency trading implies that participants other than the market marker do not have any idea on the actual volumes traded and the number of trades. For the purposes of building our model, we simulate the number of trades and the average size of trades from a log normal distribution. Norstad [\[]{}1999[\]]{} proves key propositions regarding normal and log normal distributions. The parameters of the log normal distributions are chosen such that the total volume in a certain interval matches the volume publicly mentioned by currency trading firms. This methodology can be easily extended to other financial instruments and possibly to any product with an ability to make electronic price quotations or even be used to periodically perform manual price updates on products that are traded non-electronically.
The factors we incorporate in our model to adjust the currency bid-offer spread are
1. The Exchange Rate Volatility
2. The Trade Count
3. The Volume
The exchange rate volatility is publicly observable; and the trade count and volume, are generally only known to the market maker, in various instruments over different historical durations in time. The contributions of each of the factors to the bid-offer adjustment are computed separately and then consolidated to produce a very adaptive bid-offer quotation. The subsequent sections consider the calculations for each factor separately and the consolidation in detail.
Exchange Rate Volatility Factor
--------------------------------
This factor is calculated based on the conditional standard deviation of the exchange rate returns as a function of the lagged conditional standard deviations and the lagged innovations.
$$P_{f}\Longleftrightarrow\sigma_{t}=\alpha\sigma_{t-1}+\beta\varepsilon_{t-1}$$ $$P_{f}\text{ is the Price Factor; }\sigma_{t}\text{ is volatility at time }t\text{ ; }\varepsilon_{t-1}\text{ is the innovation at time }t-1\text{ ; }0<\alpha,\beta<1.$$
Numerous variations to the above formula are possible by extending it to the GARCH$(p,q)$ type of models. Engle [\[]{}1982[\]]{} is the seminal work on modeling heteroscedastic variance. Bollerslev [\[]{}1986[\]]{} extends this technique to a more generalized approach and Bollerslev [\[]{}2008[\]]{} lists an exhaustive glossary of the various kinds of autoregressive variance models that have mushroomed over the years. Hamilton [\[]{}1994[\]]{} and Gujarati [\[]{}1995[\]]{} are classic texts on econometrics methods and time series analysis that accentuate the need for parsimonious models. We prefer the simple nature of the sample model, since we wish to keep the complexity of the system as minimal as possible, while ensuring that the different sources of variation contribute to the modification. This becomes important since we are constantly checking the feedback loop for the system performance. When such a model is being used empirically, less number of parameters eases the burden of monitoring; isolating the causes of feedback failure becomes relatively straight forward; and corrective measures can be quickly implemented, which could involve tweaking the model parameters. Since volatility is mean reverting and has a clustering behavior, it is better to use a model similar to our sample, instead of simply taking the deviation from a historical average as we use for the other factors below. A more common variant that is comparable in simplicity to the one used above is by taking the absolute value of the lagged innovations. It is left to the practitioner to decide on the exact nature of the model to use depending on the suitability for their trading needs and the results they are getting.
The $t=0$ value of the volatility is calculated based on the standard deviation of the rate of change of the exchange rates from a historical period. We use a 30 day historical period to calculate the initial volatility.
We model the innovation, $\varepsilon$, as the rate of the change of the exchange rates with respect to time. This is calculated as the natural logarithm of the ratio of the exchange rates at two consecutive time periods. In the sample designed to demonstrate the model, we use the time interval between consecutive rates to be 60 seconds. $$\varepsilon_{t-1}=\ln\left(\frac{R_{t-1}}{R_{t-2}}\right)$$ $$\varepsilon_{t-1}\text{ is the innovation at time }t-1\text{ ; }R_{t-1}\text{ is the exchange at t-1}.$$
Trade Count Factor
-------------------
We first calculate the historical average of the trade count during a certain time interval. In the sample model, the historical average is based on a 30 day rolling window. The time interval is 60 seconds. We measure how the trade count for the latest time interval differs from the historical average. This is measured as the natural logarithm of the ratio of the trade count for the latest time interval to the historical average of the trade count. $$TC_{f}\Longleftrightarrow\ln\left(\frac{TC_{i}}{TC_{avg}}\right)*\gamma$$
$TC_{f}$ is the Trade Count Factor; $TC_{i}$ is the Trade Count during minute $i$ or during a certain interval of consideration;
$TC_{avg}=\left(\text{Number of Trades in a Month}\right)/\left(\text{Number of Trading Days in the Month}*\text{Number of Minutes in a Day}\right)$. It is calculated as a rolling average; $\gamma$ is the parameter that is used to scale the trade count factor into a similar size as the price factor. It is the average of the price factor over a suitable historical range. We use the average over the last thirty days.
Note: A 30 day rolling window results in the historical averages getting updated every trading day.
Volume Factor
--------------
We first calculate the historical average of the volume during a certain time interval. In the sample model, the historical average is based on a 30 day rolling window. The time interval is 60 seconds. We measure how the volume for the latest time interval differs from the historical average. This is measured as the natural logarithm of the ratio of the volume for the latest time interval to the historical average of the volume. $$V_{f}\Longleftrightarrow\ln\left(\frac{V_{i}}{V_{avg}}\right)*\gamma$$
$V_{f}$ is the Volume Factor; $V_{i}$ is the Volume in USD during minute $i$ or during a certain interval of consideration; $V_{avg}=\left(\text{Volume in a Month}\right)/\left(\text{Number of Trading Days in the Month}*\text{Number of Minutes in a Day}\right)$. It is calculated as a rolling average; $\gamma$ is the parameter that is used to scale the volume factor into a similar size as the price factor. It is the average of the price factor over a suitable historical range. We use the average over the last thirty days.
Note: A 30 day rolling window results in the historical averages getting updated every trading day.
Consolidation of the Three Factors
===================================
The three factors are consolidated by using a weighted sum. In the sample model, all three factors are equally weighted. Henceforth, the consolidated factor will be referred to as the spread factor. Where required, depending on the financial instrument, each of the three individual factors can be scaled down to be in the order of the magnitude of the adjustment we want to make to the bid and the offer. We do not require this step for our sample model, since the order of magnitude of the spread factor is in the same region as the adjustment to the spread we wish to make. We also calculate the historical average and standard deviation of the spread factor. In the sample model, the historical average and standard deviation are based on a 30 day rolling window. We consider the spread factor to be a normal distribution with mean and standard deviation equal to the 30 day historical average and standard deviation. When the spread factor is more than a certain number of standard deviations to the right of the historical average of the spread factor, we increase the bid-offer spread. If the spread factor is more than a certain number of standard deviations to the left of the historical average, we decrease the bid-offer spread. In the sample model, we consider half a standard deviation to the right and a third of a standard deviation to the left of the mean. The increase or decrease of the bid-offer spread is proportional to the magnitude of the spread factor. The maximum spread change is limited to an appropriate pre-set threshold for both the upper and lower limit. $$S_{rf}\Longleftrightarrow w_{p}*P_{f}+w_{tc}*TC_{f}+w_{v}*V_{f}$$ $$\begin{aligned}
S_{f}:\left\{ \left.S_{rf}\right|\text{if }\left[S_{rf}\leq\left(\mu_{S_{rf}}+\frac{\sigma_{S_{rf}}}{m}\right)\right]\text{ then}\right.\\
\text{if }\left[S_{rf}<\left(\mu_{S_{rf}}-\frac{\sigma_{S_{rf}}}{n}\right)\right]\text{ then}\\
\left[\mu_{S_{rf}}-\frac{\sigma_{S_{rf}}}{n}\right]\\
\text{else}\\
\left[S_{rf}\right]\\
\text{end if}\\
\text{else}\\
\left[\mu_{S_{rf}}+\frac{\sigma_{S_{rf}}}{m}\right]\\
\left.\text{end if}\vphantom{\left.S_{rf}\right|\text{if }\left[S_{rf}\leq\left(\mu_{S_{rf}}+\frac{\sigma_{S_{rf}}}{m}\right)\right]\text{ then}}\right\} \end{aligned}$$
$S_{rf}$ is the raw Spread Factor; $\mu_{S_{rf}}$ is the rolling average of the raw Spread factor; $\sigma_{S_{rf}}$ is the rolling standard deviation of the raw Spread Factor; $S_{f}$ is the spread factor after adjusting for the upper and lower bounds; $m,n\in\mathcal{R}$; we have set $m=2$ and $n=3$; $w_{p}$ is the weight for the Price Factor; $P_{f}$ is the Price Factor; $w_{tc}$ is the weight for the Trade Count Factor; $TC_{f}$ is the Trade Count Factor; $w_{v}$ is the weight for the Volume Factor; $V_{f}$ is the Volume Factor.
Note: A 30 day rolling window results in the historical average and standard deviation getting updated every trading day.
Dataset Construction
=====================
To construct a sample model, we need the following data items: the price, the trade count and the volume of the security over different time intervals. We have chosen the currency markets since it is an ideal candidate for a dynamic quotation model, but the price is not publicly disclosed as in the equity markets. We take the average of the high, low, open and close prices over a certain interval as a proxy for the trade price. Many market making firms disclose such a data set at different intervals facilitating the creation of a reasonable hypothetical price. The data is available over our chosen interval of one minute as well.
The trade count and trade volume over a minute are not publicly available. But many providers disclose total quarterly, total monthly and average daily volumes. The volume over a minute is the product of the number of trades and the size of each trade during that minute. We can pick random samples from a log normal distribution to get the trade count and trade size for each minute. The mean and standard deviation of the log normal distributions can be set such that the total volume will match the publicly disclosed figure. We can make an assumption that there will be sixty trades on average in a minute and set the average trade size based on the total volume. Please see endnote [\[]{}1[\]]{} and [\[]{}2[\]]{} in the references for further details on the publicly available data sets and Appendix \[sub:Model-Parameters-and\] for details of the model parameters. Any market maker wishing to use this model can easily substitute the simulated variables with the actual values they observe.
Model Testing Results
======================
1. The model was tested on a time horizon between 24-Jul-2013 to 24-Oct-2013. The currency pair used was the EUR-USD currency pair and the hypothetical trade price is the average of the high, low, open and close during a certain interval, which in our case was a minute. The high, low, open and close is publicly available from a number of providers.
2. The ideal starting historical values are to be calculated based on data from the month preceding this period. Other shorter time intervals can be considered as appropriate to the needs of the specific trading desks.
3. The P&L increase for this time period was USD \$513,050. P&L breakdown by trading day and by trading hour are included in Appendix \[sub:USD-Profit-=000026\], \[sub:USD-Profit-=000026-1\], \[sub:Key-Metrics-by\] and \[sub:Key-Metrics-by-1\]. It is important to keep in mind that most liquid currencies trade continuously from Monday morning Asia time to Friday evening US time.
4. The spread was increased 47,347 times; decreased 48,244 times; the spread factor was greater than the upper bound on 19,605 times and lower than the lower bound on 27,535 times.
5. The volume that was affected by the increased spread was approximately 444.95 Billion; volume affected by the decreased spread was 443.19 Billion.
6. More detailed results are included in the Appendices.
Improvements to the Model
==========================
1. We can skew the change in the bid offer spread to be more on the bid or the offer side based on the buy and sell volumes. We have not considered this exclusively in our model since we only look at the change in the spread and not on which side of the quote the change happens. It is simple to adjust both sides equally or be cleverer in how we split the total spread change into the bid or the offer side.
2. The assumption of normality and the use of a log normal distribution can be relaxed in favor of other distributions. It is also possible to use different distributions that change over time, as a result of the feedback we receive from the system. This is a more realistic portrayal of empirical data which tend to fall into different distributions as regimes change.
3. Each of the variables can be modeled using more advanced econometric techniques like the GARCH$(p,q)$ model. Care needs to be taken that the additional parameters do not impact the feedback loop and when results are not satisfactory, we can easily investigate the reason for issues.
4. For simplicity, we have ignored the question of negative spreads or reverse quotes, where the bid is greater than the offer, resulting in a crossed market. This can happen when the magnitude of the spread factor is greater than the difference between the bid and the offer. This can be handled easily by reducing the size of the spread factor when such an event occurs. Additional ways to handle this are considered in the below points.
5. The model can be made to adapt its scaling factors, the alpha and the beta so that the difference in the average of the increase and the average of the decrease in the spread are equal over a certain time period. With this, the overall spread change stays the same and the market maker is seen to be quoting competitive spreads, though this results in better profitability based on the volume and price movements it is experiencing. See Appendix-I for details on the model parameters.
6. In our current model, we limit the size of the spread change on both the positive and negative sides depending on the value of the spread factor. A variation to this can be to change the spread only when the spread factor lies above or below a certain threshold. The spread change can be a constant value; or two constant values, one for the increment and one for the decrement or it can be made to depend on the spread factor as well.
7. The consolidated spread factor computed as the weighted sum of the exchange rate volatility, trade count and volume factors can be made to depend more on the volatility and trade count by adjusting the corresponding weights.
8. The time interval considered for the factors is 60 seconds. Smaller time intervals will result in better performance for currency markets. Larger time intervals might be more suited for other securities.
9. The rolling average can be taken over shorter or longer intervals depending on the results and the security under consideration. It is also possible to weight different contributors to the average differently resulting in a Moving Average model.
10. The trade count and volume factors can also be modeled similar to the Exchange Rate Volatility Factor. The point to bear in mind is that the exchange rate volatility is mean reverting and the trade count and volume factors have always had an upward trend. This is because we expect more trading to happen and all trading desks are bullish about their activities. Given the volume projections, we can expect the upward trajectory for these two factors to continue. For 30 day rolling windows, we can assume that the trade count and volume follow a mean reverting property. For our purpose, the deviations from the 30 day historical average for the trade count and volume factors produce satisfactory results.
11. A central question is whether the changing spread will have a negative impact on the volumes traded and hence on the overall profitability of the desk. This needs to be monitored closely and the size of the changes need to be adjusted accordingly.
12. Other factors can be included, like the percentage of flow handled by the market marker to the average flow in that currency pair over the course of a trading day. This factor indicates the extent of monopoly that the market maker enjoys and indicates pricing power. This ratio can be used to adjust the spread in the favor of the market maker or in the feedback loop to tweak the parameters that are used for other factors.
Conclusion
===========
The need for a dynamic quotation model comes from the feature of the social sciences and trading, where observations coupled with decision making can impact the system. This aspect was illustrated in detail and summarized as the uncertainty principle of the social sciences. To deal with this phenomenon, we need a feedback mechanism, which incorporates trading conditions into the quotation process, without too much of a temporal lag.
A model was constructed, using price, trade count and volume factors over one minute intervals, to vary the quotes being made. The models constructed are rich enough to capture the effect of the various relevant factors, yet simple enough to accord constant monitoring and to ensure the effectiveness of the feedback loop. The real test of any financial model or trading strategy is the effect on the bottom line and hence when we looked at the performance of our methodology, we found the positive effect on the P&L to be significant, without too much of a change to the way the trading happens or an accompanying increasing in risk or leverage of the trading desk.
Numerous improvements to the model are possible and can be considered depending on the type of instrument being traded and the technology infrastructure available for trading. Future iterations of this study will look to extend this methodology to other asset classes.
References and Notes
=====================
1. For further details on the publicly available datasets, see [http://forexmagnates.com/fxcm-posts-records-quarterly-revenues-and-july-volume-metrics/ and http://ir.fxcm.com/releasedetail.cfm?ReleaseID=797967](http://forexmagnates.com/fxcm-posts-records-quarterly-revenues-and-july-volume-metrics/ and http://ir.fxcm.com/releasedetail.cfm?ReleaseID=797967)
2. The author has utilized similar algorithms for market making in various OTC as well as exchange traded instruments for more than the last ten years. As compared to the sample model, the interval trade count and volumes used in the empirical model were the actual observations; yet the overall results are somewhat similar.
3. Avellaneda, M., & Stoikov, S. (2008). High-frequency trading in a limit order book. Quantitative Finance, 8(3), 217-224.
4. Bodie, Z., Kane, A., & Marcus, A. J. (2002). Investments. International Edition.
5. Bollerslev, T. (1986). Generalized autoregressive conditional heteroskedasticity. Journal of econometrics, 31(3), 307-327.
6. –. (2008). Glossary to arch (garch). CREATES Research Paper, 49.
7. Chiu, J., Lukman, D., Modarresi, K., & Velayutham, A. (2011). High-frequency trading. Standford, California, US: Stanford University.
8. Chlistalla, M., Speyer, B., Kaiser, S., & Mayer, T. (2011). High-frequency trading. Deutsche Bank Research, 1-19.
9. Copeland, L. S. (2008). Exchange rates and international finance. Pearson Education.
10. Dempster, M. A. H., & Jones, C. M. (2001). A real-time adaptive trading system using genetic programming. Quantitative Finance, 1(4), 397-413.
11. Derman, E., & Wilmott, P. (2009). The Financial Modelers’ Manifesto. In SSRN: http://ssrn. com/abstract (Vol. 1324878).
12. Elton, E. J., Gruber, M. J., Brown, S. J., & Goetzmann, W. N. (2009). Modern portfolio theory and investment analysis. John Wiley & Sons.
13. Engle, R. F. (1982). Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation. Econometrica: Journal of the Econometric Society, 987-1007.
14. Gujarati, D. N. (1995). Basic econometrics, 3rd. International Edition.
15. Hamilton, J. D. (1994). Time series analysis (Vol. 2). Princeton university press.
16. Hull, J. C. (1999). Options, futures, and other derivatives. Pearson Education India.
17. Kashyap, R. The Circle of Investment. International Journal of Economics and Finance, Vol. 6, No. 5 (2014), pp. 244-263.
18. Marcus, A. J., Bodie, Z., & Kane, A. (2002). Investments. McGraw Hill.
19. Norstad, J. (1999). The normal and lognormal distributions.
20. Paich, M., & Sterman, J. D. (1993). Boom, bust, and failures to learn in experimental markets. Management Science, 39(12), 1439-1458.
21. Popper, K. R. (2002). The poverty of historicism. Psychology Press.
22. Sweeney, J., & Sweeney, R. J. (1977). Monetary theory and the great Capitol Hill Baby Sitting Co-op crisis: comment. Journal of Money, Credit and Banking, 86-89.
23. Taleb, N. (2005). Fooled by randomness: The hidden role of chance in life and in the markets. Random House LLC.
24. –. (2010). The Black Swan: The Impact of the Highly Improbable Fragility.
25. Tuckman, B. (1995). Fixed Income Securities-Tools for Todays Markets.
Appendix
========
\[sub:Model-Parameters-and\]Model Parameters and Key Metrics
-------------------------------------------------------------
In Figure \[fig:Model-Parameters-and\], values in blue are model parameters that can be used to optimize the model. In the sample model, these act as user inputs and can be changed to see how the model behaves under different conditions. Values in green are common categories that apply to different metrics. Values in bright yellow are important metrics, some of which form a key part of the feedback loop and it would be good to monitor these closely. Alpha and Beta are the parameters used to model the volatility of the price. Gamma is the parameter that is used to scale the trade and volume factors into a similar size as the price factor.
![Model Parameters and Key Metrics\[fig:Model-Parameters-and\]](Model_Parameters_Key_Metrics){width="15cm" height="20cm"}
\[sub:USD-Profit-=000026\]USD Profit & Loss by Trading Day
----------------------------------------------------------
{width="15cm" height="20cm"}
\[sub:USD-Profit-=000026-1\]USD Profit & Loss by Trading Hour
-------------------------------------------------------------
{width="15cm" height="20cm"}
\[sub:Key-Metrics-by\]Key Metrics by Trading Day
------------------------------------------------
{width="16cm" height="22cm"}
\[sub:Key-Metrics-by-1\]Key Metrics by Trading Hour
---------------------------------------------------
{width="15cm" height="20cm"}
|
---
abstract: 'This paper is concerned with the theory and applications of varifolds to the representation, approximation and diffeomorphic registration of shapes. One of its purpose is to synthesize and extend several prior works which, so far, have made use of this framework mainly in the context of submanifold comparison and matching. In this work, we instead consider deformation models acting on general varifold spaces, which allows to formulate and tackle diffeomorphic registration problems for a much wider class of geometric objects and lead to a more versatile algorithmic pipeline. We study in detail the construction of kernel metrics on varifold spaces and the resulting topological properties of those metrics, then propose a mathematical model for diffeomorphic registration of varifolds under a specific group action which we formulate in the framework of optimal control theory. A second important part of the paper focuses on the discrete aspects. Specifically, we address the problem of optimal finite approximations (quantization) for those metrics and show a $\Gamma$-convergence property for the corresponding registration functionals. Finally, we develop numerical pipelines for quantization and registration before showing a few preliminary results for one and two-dimensional varifolds.'
address:
- 'Department of Applied Mathematics and Statistics, Johns Hopkins University'
- 'Department of Applied Mathematics and Statistics, Johns Hopkins University'
author:
- 'Hsi-Wei Hsieh'
- Nicolas Charon
bibliography:
- 'biblio.bib'
title: 'Metrics, quantization and registration in varifold spaces'
---
.
Introduction
============
Shape is a bewildering notion: while simultaneously intuitive and ubiquitous to many scientific areas from pure mathematics to biomedicine, it remains very challenging to pin down and analyze in a systematic way. The goal of the research field known as shape/pattern analysis is precisely to provide solid mathematical and algorithmic frameworks for tasks such as automatic comparison or statistical analysis in ensembles of shapes, which is key to many applications in computer vision, speech and motion recognition or computational anatomy, among many others.
What makes shape analysis such a difficult and still largely open problem is, on the one hand, the numerous modalities and types of objects that can fall under this generic notion of shape but also the fundamental nonlinearity that is an almost invariable trait to most of the shape spaces encountered in applications. As a result, the seemingly simple issue of defining and computing distances or means on shapes is arguably a research topic of its own, which has generated countless works spanning several decades and involving concepts from various subdisciplines of mathematics. Among many important works, the model of shape space laid out by Grenander in is especially relevant to the present paper. The underlying principle is to build distances between shapes which are induced by metrics on some deformation groups acting on those shapes. This approach has the advantage (at a theoretical level at least) of shifting the problem of metric construction from the many different cases of shape spaces to the single setting of deformation groups. One of the fundamental requirement is the right-invariance of the metrics on those groups; finding the induced distance between two given shapes then reduces to determining a deformation of minimal cost in the group, in other words to solving a *registration* problem.
Besides usual finite-dimensional groups like rigid of affine transformations, there is in fact a lot of practical interest in applying such an approach with groups of “large deformations”, specifically groups of *diffeomorphisms*. This has triggered the exploration of right-invariant metrics over diffeomorphism groups. The Large Deformation Diffeomorphic Metric Mapping (LDDMM) model pioneered in is one of such framework that defines Riemannian metrics for diffeomorphic mappings obtained as flows of time-dependent vector fields (c.f. the brief presentation of Section \[ssec:diffeom\_reg\]). In this setting, registering two shapes can be generically formulated as an *optimal control* problem, the functionals to optimize being typically a combination of a deformation regularization term given by the LDDMM metric on the group and a fidelity term that enforces (approximate) matching between the two shape objects. Applications of this model have been widespread in particular within the field of computational anatomy, due to the ability to adapt it to various data structures including landmarks, 2D and 3D images, tensor fields... see e.g. for recent reviews.
Interestingly, this line of work has also been drawing many useful concepts from the seemingly distant area of mathematics known as *geometric measure theory* . The key idea of representing shapes (submanifolds) as measures or distributions has been instrumental in the theoretical study of Plateau’s problem on minimal surfaces and more generally in calculus of variations. It can also prove effective for computational purposes, in problems such as discrete curvature approximations or estimation of shape medians . With regard to the aforementioned deformation analysis problems, the potential interest of geometric measure theory has been identified early on in the works of . Indeed, LDDMM registration of objects like geometric curves or surfaces requires fidelity terms independent of the parametrization of either of the two shapes. On the practical side, this means that one cannot usually rely on predefined pointwise correspondences between the vertices of two triangulated surfaces for instance, which makes the registration problem significantly harder than in the case of labelled objects such as landmarks or images.
The embedding of unparametrized shapes into measure spaces provides one possible way to address the issue, by constructing parametrization-invariant fidelity metrics as restrictions of metrics on those measure spaces themselves. Several competing approaches have been introduced, each relying on embeddings into different spaces of generalized measures: are based on the representation of oriented curves and surfaces as *currents*, and extended this model to the setting of unoriented and oriented *varifolds*, while considers the higher-order representation of *normal cycles*. One common feature to all those works, however, is that they are focused primarily on registration of curves or surfaces. In other words, the use of current, varifolds or normal cycles confines to the computation of a fidelity metric to guide registration algorithms but the deformation model itself remains tied to the curve/surface setting or equivalently, in the discrete situation, to objects described by point set meshes.
The guiding theme and main objective of this paper is to investigate an alternative framework that, in contrast with those prior works, would formulate the deformation model as well as tackle the registration problem directly in these generalized measure spaces: we focus specifically on the (oriented) varifold setting of . There are several arguments for the interest of such an approach but in our point of view, the primary motivation lies in the fact that, varifolds being more general than submanifolds, the proposed framework allows to extend large deformation analysis methods to a range of new geometric objects while giving more flexibility to deal with some of the flaws which are commonplace in shapes segmented from raw data. As a proof of concept, our recent work considered the simple case of registration of discrete one-dimensional varifolds. Building on these preliminary results, the present paper intends to provide a thorough and general study of the framework.
The specific contributions and organization of this paper are the following. First, we propose a comprehensive study of the class of kernel metrics on varifold spaces initiated in , in particular by examining the required conditions to recover true distances between all varifolds (as opposed to the subset of rectifiable varifolds) and comparing the resulting topologies with some standard metrics on measures. This is presented in Section \[sec:metrics\_varifold\] after the brief introduction to the notion of oriented varifold of Section \[sec:varifold\_space\]. In Section \[diff\_matching\], we discuss the action of diffeomorphisms and from there derive a formulation of LDDMM registration of general varifolds, for which we show the existence of solutions and derive the Hamiltonian equations associated to the corresponding optimal control problem. Section \[sec:approximation\_discrete\_var\] addresses the issue of quantization in varifold space, namely of approximating any varifold as a finite sum of Dirac masses. We consider a novel approach in this context, that consists in computing projections onto particular cones of discrete varifolds. We then prove the $\Gamma$-convergence of the corresponding approximate registration functionals. In Section \[sec:numerics\], we derive the discrete version of the optimal control problem and optimality equations, from which we deduce a geodesic shooting algorithm for the diffeomorphic registration of discrete varifolds. Finally, results on $1$- and $2$- varifolds are presented in Section \[sec:results\], emphasizing the potentiality of the approach to tackle data structures which are typically challenging for previous algorithms that are designed for point sets and meshes.
The space of oriented varifolds {#sec:varifold_space}
===============================
The concept of varifold was originally developed in the context of geometric measure theory by [@Young42], [@almgren1966plateau] and [@allard1972first] for the study of Plateau’s problem on minimal surfaces. The interest in registration and shape analysis was evidenced in [@Charon2; @Charon2017]. In those works, varifolds provide a convenient representation of geometric shapes such as rectifiable curves and surfaces and an efficient approach to define and compute fidelity terms for registration, or to perform clustering, classification in those shape spaces. The main purpose of this section is to introduce varifolds in this latter context. The case of non-oriented shapes was thoroughly investigated in [@Charon2]. Later on, the generalized framework of oriented varifold was proposed in [@Charon2017] but only for objects of dimension or co-dimension one. In the following, we provide a fully general presentation of oriented varifolds and their properties, that also does not specifically focus on the case of rectifiable varifolds as these previous works did. Although we assume here that all the considered shapes are oriented, we emphasize that the non-oriented framework of [@Charon2] can be recovered almost straightforwardly through adequate choices of orientation-invariant kernels as we shall briefly point out later on.
Definition {#subsec:defofvf}
----------
The underlying principle of varifolds is to extend measures of $\R^n$ by incorporating an additional tangent space component. In this work, we will consider such spaces to be oriented. Thus, for a given dimension $0\leq d \leq n$, we first need to introduce the set of all possible $d$-dimensional oriented tangent spaces in $\mathbb{R}^n$:
The $d$-dimensional oriented Grassmannian $\widetilde{G}_d^n$ is the set of all oriented $d$-dimensional linear subspaces of $\mathbb{R}^n$.
The oriented Grassmannian is a compact manifold of dimension $d(n-d)$ which can be identified to the quotient $SO(n)/(SO(d) \times SO(n-d))$. It is also a double cover of the (non-oriented) Grassmannian $G_d^n$ of $d$-dimensional subspaces of $\R^n$. For practical purposes, a more convenient representation of $\widetilde{G}_d^n$ is the one detailed in the following remark.
\[rem:Grassmannian\] Given $T \in \widetilde{G}_d^n$, there exists a basis $\{u_i \}_{i=1,\ldots,d} \in \R^{n\times d}$ of $T$ such that $[u_1,\cdots,u_d]$ has consistent orientation with $T$. Then the following map, called the oriented Plücker embedding, is well defined and injective, $$\begin{aligned}
i_P: \ \widetilde{G}_d^n &\mapsto \{\xi \in \Lambda^d(\mathbb{R}^n) : |\xi| =1\} \\
T &\mapsto \frac{u_1 \wedge \cdots \wedge u_d}{|u_1 \wedge \cdots \wedge u_d |}.\end{aligned}$$ This allows to identify $\widetilde{G}^n_d$ as a subset of the unit sphere of $\Lambda^d(\mathbb{R}^n)$ which inherits the topology of the inner product on $\Lambda^d(\mathbb{R}^n)$. We remind that this inner product is defined for any $\xi=\xi_1 \wedge \ldots \wedge \xi_d$, $\eta=\eta_1 \wedge \ldots \wedge \eta_d$ in $\Lambda^d(\mathbb{R}^n)$ by the determinant of the Gram matrix: $$\begin{aligned}
\label{eq:wedge_inner_product}
\langle \xi,\eta \rangle = \det(\xi_i \cdot \eta_j)_{i,j=1,\ldots,d} \end{aligned}$$ Through this identification, one can also define the action of linear transformations on $\widetilde{G}^n_d$ as follows $$\begin{aligned}
\label{eq:act_lt_grass}
A \cdot T := \frac{A u_1 \wedge \cdots \wedge Au_d}{|A u_1 \wedge \cdots \wedge Au_d |}\end{aligned}$$ for any $T \in \widetilde{G}^n_d$ and $A: \mathbb{R}^n \mapsto \mathbb{R}^n$ a linear invertible map.
Similar to the definition of classical varifolds in [@Simon], we define oriented varifolds as measures on $\mathbb{R}^n \times \widetilde{G}^n_d$.
An oriented $d$-varifold $\mu$ on $\mathbb{R}^n$ is a nonnegative finite Radon measure on the space $\mathbb{R}^n \times \widetilde{G}^n_d$. Its weight measure $|\mu|$ is defined by $|\mu|(A) := \mu(A\times \widetilde{G}^n_d)$ for all Borel subset $A$ of $\R^n$. We denote by $\mathcal{V}_d$ the space of all oriented $d$-varifolds.
In the rest of the paper, with a slight abuse of vocabulary, we will often use the word varifold instead of oriented varifold for the sake of concision. Recall that from the Riesz representation theorem, we can alternatively view any varifold $\mu$ as a distribution, i.e. an element of the dual space $C_0(\mathbb{R}^d \times \widetilde{G}^n_d)^*$, where $C_0(\mathbb{R}^d \times \widetilde{G}^n_d)$ denotes the set of continuous functions vanishing at infinity on $\mathbb{R}^d \times \widetilde{G}^n_d$. It is defined for any test function $\omega \in C_0(\mathbb{R}^d \times \widetilde{G}^n_d)$ by: $$\label{eq:var_mu_distribution}
(\mu |\omega) \doteq \int_{\mathbb{R}^n \times \widetilde{G}^n_d} \omega(x,T) d\mu(x,T).$$ As an additional note, another useful representation of a general varifold in $\mathcal{V}_d$ can be obtained by the disintegration theorem (see Chap. 2). Namely, if $\mu \in \mathcal{V}_d$, for $|\mu|$-almost every $x$ in $\R^n$, there exists a probability measure $\nu_x$ on $\widetilde{G}^n_d$ such that $x \mapsto \nu_x$ is $|\mu|$-measurable and we can write $$\label{eq:var_mu_disintegration}
(\mu |\omega) = \int_{\R^n} \int_{\widetilde{G}^n_d} \omega(x,T) d\nu_x(T) d|\mu|(x).$$ In other words, the varifold $\mu$ can be decomposed as its weight measure on $\R^n$ together with a family of tangent space probability measures on the Grassmannian at the different points in the support of $|\mu|$. This is usually referred to as the Young measure representation of $\mu$.
Diracs and rectifiable varifolds
--------------------------------
There are a few important families of varifolds which will be relevant for the following. First of those are the Diracs. For $x \in \mathbb{R}^n$ and $T \in \widetilde{G}^n_d$, the associated Dirac varifold $\delta_{(x,T)}$ acts on functions of $C_0(\mathbb{R}^n \times \widetilde{G}^n_d)$ by the relation $$\begin{aligned}
(\delta_{(x,T)}|\omega) = \omega(x,T), \ \forall \omega \in C_0(\mathbb{R}^n \times \widetilde{G}^n_d).\end{aligned}$$ $\delta_{(x,T)}$ can be viewed as a singular particle at position $x$ that carries the oriented $d$-plane $T$.
A second particular class is the one of *rectifiable varifolds*, which are in essence the varifolds representing an oriented shape of dimension $d$. More precisely, given an oriented $d$-dimensional submanifold $X$ of $\mathbb{R}^n$ of finite total $d$-volume, denoting by $T_X(x) \in \widetilde{G}^n_d$ the oriented tangent space at $x \in X$, one can associate to $X$ the varifold $\mu_X$, which is defined for all Borel subset $B\subset \R^n \times \widetilde{G}^n_d$ by $\mu_X(B) = \mathcal{H}^d(\{x \in X | (x,T_X(x)) \in B \})$. Here, $\mathcal{H}^d$ is the $d$-dimensional Hausdorff measure on $\R^n$, i.e. the measure of $d$-volume of subsets of $\R^n$ (we refer the reader to [@Simon] for the precise construction and properties of Hausdorff measures). It is then not hard to see that, as an element of $C_0(\mathbb{R}^n \times \widetilde{G}^n_d)^*$, $$\begin{aligned}
\label{eq:vfasdistri}
(\mu_X|\omega) &= \int_{\mathbb{R}^d \times \widetilde{G}^n_d} \omega(x,T) d \mu_X(x,T) \nonumber \\
&= \int_X \omega(x,T_X(x)) d \mathcal{H}^d(x).\end{aligned}$$ Such a representation $X \mapsto \mu_X$ can be extended to slightly more general objects known as *oriented rectifiable sets*. A subset $X$ of $\mathbb{R}^n$ is said to be a *countably* $\mathcal{H}^d$-*rectifiable set* if $\mathcal{H}^d(X \setminus \cup_{j=1}^{\infty}F_j(\mathbb{R}^d)) = 0$, where $F_j: \mathbb{R}^d \mapsto \mathbb{R}^n$ are Lipschitz function for all $j$ (c.f. [@Simon]). We say that $(X,T_X)$ is an *oriented rectifiable set* if $X$ is a countably $d$-rectifiable set and $T_X:X \mapsto \widetilde{G}^n_d$ is a $\mathcal{H}^d$-measurable function such that for $\mathcal{H}^d-a.e.\ x \in X$, $T_X(x)$ is the approximate tangent space of $X$ at $x$ with specified orientation. Rectifiable subsets include both usual submanifolds but also piecewise smooth objects like polyhedra. Given any oriented rectifiable set $(X, T_X)$, we can associate a varifold that we also write $\mu_X$ given again by . The set of those $\mu_X$ will be referred to as the rectifiable oriented varifolds in this paper (note that this is actually more restrictive than the standard definition of rectifiable varifold in the literature which also incorporates an additional multiplicity function).
Rectifiable varifolds still make a very “small” subset of $\mathcal{V}_d$: indeed, in the Young measure representation of , we have in this case the very particular constraint that probability measures $\nu_x$ are Dirac masses, specifically $\nu_x = \delta_{T_X(x)}$.
Metrics on varifolds {#sec:metrics_varifold}
====================
In this section, we address the issue of defining adequate metrics on the space $\mathcal{V}_d$. After reviewing some classical metrics and their limitations for the specific applications of this work, we turn to metrics defined through positive definite kernels, for which we extend previous constructions introduced in e.g. [@Charon2; @Charon2017] and derive the most relevant properties of this class of distances.
Standard topologies and metrics on $\mathcal{V}_d$ {#subsec:standard_metrics}
--------------------------------------------------
As a measure/distribution space, $\mathcal{V}_d$ can be equipped with various topologies and metrics, several of which have been regularly used in various contexts. We discuss a few of those below.
- *mass norm*: with the previous identification of measures in $\mathcal{V}_d$ with elements of the dual $C_0(\mathbb{R}^n \times \widetilde{G}^n_d)^*$, one can define the following dual metric on $\mathcal{V}_d$: $$\begin{aligned}
d_{op}(\mu,\nu) \doteq \sup\limits_{|\omega|_{\infty} \leq 1} (\mu -\nu|\omega), \ \forall \mu \in \mathcal{V}_d.\end{aligned}$$ where $|\omega|_{\infty} \doteq \sup_{\mathbb{R}^n \times \widetilde{G}^n_d} |\omega|$. This metric is generally too strong for applications in shape analysis and leads to a discontinuous behavior. Indeed, one can easily verify that for any two Dirac masses $\delta_{(x,T)}$ and $\delta_{(x',T')}$, $d_{op}(\delta_{(x,T)},\delta_{(x',T')}) = 2$ whenever $(x,T) \neq (x',T')$.
- *weak-\** topology: a sequence of $d$-varifolds $\{\mu_i\}_i$ converges to $\mu \in \mathcal{V}_d$ in the weak-\* topology (denoted by $\mu_i \overset{\ast}{\rightharpoonup} \mu$) if and only if for all $\omega \in C_c(\mathbb{R}^d \times \widetilde{G}^n_d)$ (continuous compactly supported function) $$\begin{aligned}
\lim_{i \rightarrow \infty}(\mu_i|\omega) = (\mu|\omega).\end{aligned}$$ In fact, the weak-\* topology on $\mathcal{V}_d$ can be metrized by the following distance: $$\begin{aligned}
d_*(\mu,\nu) = \sum_{k \in \mathbb{N}} 2^{-k} |(\mu-\nu|\omega_k)|,\end{aligned}$$ where $\{\omega_k\}_{k\in \mathbb{N}}$ is a dense sequence in $C_c(\mathbb{R}^n \times \widetilde{G}^n_d)$.
- *Wasserstein metric*: the Wasserstein-1 distance of optimal transport can be expressed in its Kantorovitch dual formulation as $$d_{Wass^1}(\mu,\nu) \doteq \sup\limits_{\textrm{Lip}(\omega) \leq 1} |(\mu-\nu|\omega)|.$$ where the sup is taken over all Lipschitz regular functions on $\mathbb{R}^n \times \widetilde{G}^n_d$ with Lipschitz constant smaller than one. This metric is however well-suited for measures with the same total mass. Several recent works have instead proposed generalized Wasserstein distances derived from unbalanced optimal transport.
- *Bounded Lipschitz metric*: similar to the previous, the bounded Lipschitz distance (sometimes referred to as the *flat metric*) on $\mathcal{V}_d$ is defined by $$d_{BL}(\mu,\nu) \doteq \sup\limits_{\|\omega\|_{\infty}, \textrm{Lip}(\omega) \leq 1} |(\mu-\nu|\omega)|.$$ It can be shown (cf. Ch 8 in [@bogachev2007measure]) that $d_{BL}$ metrizes the *narrow topology* on $\mathcal{V}_d$, namely the topology for which a sequence $(\mu_i)$ converges to $\mu$ if and only if $\lim_{i \rightarrow \infty}(\mu_i|\omega) = (\mu|\omega)$ for all bounded continuous functions $\omega$.
Clearly, the narrow topology is stronger than the weak-\* topology. Furthermore, it is also well known that $d_{BL}$ locally metrizes the weak-\* topology on $\mathcal{V}_d$, namely:
\[prop:dBL\_weakstar\] Let $\mu$ and $\{\mu_i\}_{i}$ be varifolds such that the sequence $\{\mu_i\}_{i}$ is tight. Then $\mu_i \overset{\ast}{\rightharpoonup} \mu$ if and only if $d_{BL}(\mu_i,\mu) \rightarrow 0$.
Since $d_{BL}$ metrizes the narrow topology, it suffices to show that $\mu_i$ converges to $\mu$ in the narrow topology. Let $\omega$ be a bounded continuous function defined on $\mathbb{R}^n \times \widetilde{G}^n_d$ and $\varepsilon>0$. By the tightness property, we may choose a compact set $K\subset \mathbb{R}^n \times \widetilde{G}^n_d$ such that $\mu(K^c)+ \sup_i \mu_i(K^c)< \varepsilon/2\|\omega\|_{\infty}$. Let $B$ be an open ball that contains $K$. Define $$\begin{aligned}
\eta(x,T) \doteq \left\{ \begin{array}{lc}
\omega(x,T), & \ \text{if} \ (x,T) \in K \\
0 , & \ \text{if} \ (x,T) \in B^c
\end{array} \right. \end{aligned}$$ From Tietz extension theorem, there exists a continuous extension $\tilde{\omega}$ of $\eta$ on $\mathbb{R}^n \times \widetilde{G}^n_d$ such that $\tilde{\omega}|_K =\omega|_K$ and $\tilde{\omega} \in C_c(\mathbb{R}^n \times \widetilde{G}^n_d)$. This implies that $$\begin{aligned}
&\left| \int_{\mathbb{R}^n \times \widetilde{G}^n_d} \omega d(\mu_i-\mu) \right| \\
&\leq \int_{\mathbb{R}^n \times \widetilde{G}^n_d}|\omega- \tilde{\omega}| d (\mu_i+\mu)
+ \left| \int_{\mathbb{R}^n \times \widetilde{G}^n_d} \tilde{\omega} d (\mu_i-\mu) \right|\\
&\leq \left| \int_{\mathbb{R}^n \times \widetilde{G}^n_d} \tilde{\omega} d (\mu_i-\mu) \right| + \varepsilon.\end{aligned}$$ Taking $\limsup$ on both sides, we see that $$\begin{aligned}
\limsup_{i \rightarrow \infty} \left| \int_{\mathbb{R}^n \times \widetilde{G}^n_d} \omega d(\mu_i-\mu) \right| < \varepsilon.\end{aligned}$$ Since $\varepsilon$ is arbitrary, we obtain that $\mu_i$ converges to $\mu$ in the narrow topology.
As a direct consequence of Proposition \[prop:dBL\_weakstar\], we have in particular that weak-\* convergence and convergence in $d_{BL}$ are equivalent if one restricts to varifolds that are supported in a fixed compact subset of $\mathbb{R}^n \times \widetilde{G}^n_d$. Note also that a very similar result to Proposition \[prop:dBL\_weakstar\] holds when replacing the bounded Lipschitz distance by generalized Wasserstein metrics, as proved in [@Piccoli2014].
The above metrics on varifolds all originate from classical ones in standard measure theory. Unlike the mass norm, Wasserstein and bounded Lipschitz metrics have nice theoretical properties in terms of shape comparison. However, for the purpose of diffeomorphic registration that we shall tackle below, one needs metrics that are easy to evaluate numerically. This is typically not the case of $d_{Wass^1}$ and $d_{BL}$ expressed above as there is no straightforward way to compute the corresponding suprema over the respective sets of test functions. One line of work has been considering approximations of optimal transport distances with e.g. entropic regularizers for which Sinkhorn-based algorithms can be derived, see for instance the recent work . In this paper, we focus on the alternative approach previously developed for currents in and unoriented varifolds in which instead relies on particular Hilbert spaces of test functions, as we detail in the next section.
Kernel metrics {#ssec:kernel_metrics}
--------------
In this section, we start by defining a general class of pseudo-metrics on $\mathcal{V}_d$ based on positive definite kernels and their corresponding *reproducing kernel Hilbert space* (RKHS). We will then study sufficient conditions on such kernels to recover true metrics before examining the relationship between those kernel metrics and the ones of Section \[subsec:standard\_metrics\].
### Kernels for varifolds
We refer the reader to for a presentation of the construction and main properties of positive kernels and Reproducing Kernel Hilbert Spaces which we do not recall in detail here for the sake of concision. In the context of varifolds, we are interested in defining positive definite kernels on the product $\mathbb{R}^n \times \widetilde{G}^n_d$. Along the lines of previous works like , we restrict to separable kernels for which we have:
\[prop:ctru\_kernel\] Let $k^{pos}$ and $k^{G}$ be continuous positive definite kernels on $\mathbb{R}^n$ and $\widetilde{G}^n_d$ respectively. Assume in addition that for any $x \in \mathbb{R}^n $, $k^{pos}(x,\cdot) \in C_0(\mathbb{R}^n)$. Then $k:= k^{pos} \otimes k^G$ is a positive definite kernel on $\mathbb{R}^n \times \widetilde{G}^n_d$ and the RKHS $W$ associated to $k$ is continuously embedded in $C_0(\mathbb{R}^n \times \widetilde{G}^n_d)$ i.e. there exists $c_{W}>0$ such that for any $\omega \in W$, we have $\|\omega\|_{\infty} \leq c_W \|\omega\|_{W}$.
We recall that the tensor product kernel has the exact expression $k((x,T),(x',T')) = k^{pos}(x,x') k^{G}(T,T')$. The proof of Proposition \[prop:ctru\_kernel\] is a straightforward adaptation of the same result for unoriented varifolds (cf. Proposition 4.1).
\[rk:assumption\_kernel\] To simplify the rest of the presentation and in the perspective of later numerical considerations, we will also assume specific forms for $k^{pos}$ and $k^G$, namely that $k^{pos}$ is a translation/rotation invariant radial kernel $k^{pos}(x,y) = \rho(|x-y|^2), \ \forall x,y \in \mathbb{R}^n$, with $\rho(0)>0$, and $k^G(S,T) = \gamma(\langle S,T \rangle), \ \forall S,T \in \widetilde{G}^n_d$ where $\langle \cdot, \cdot \rangle$ is the inner product on $\widetilde{G}^n_d$ inherited from $\Lambda^d(\mathbb{R}^n)$ introduced in remark \[rem:Grassmannian\]. These assumptions are quite natural as they will eventually induce metrics on varifolds invariant to the action of rigid motion, as we shall explain later. Note that the unoriented framework of can be also recovered in this setting by simply restricting to orientation-invariant kernels $k^{G}$ i.e. such that $\gamma(-t)=\gamma(t)$ for all $t$.
Now, if we let $\iota_W:W \hookrightarrow C_0(\mathbb{R}^d \times \widetilde{G}^n_d)$ be the continuous embedding given by Proposition \[prop:ctru\_kernel\] and $\iota_W^*$ its adjoint, for any $\mu \in C_0(\mathbb{R}^n \times \widetilde{G}^n_d)^*$, we have $$\begin{aligned}
\label{eq:vf_as_wstar}
(\iota^* \mu| \omega) = \int_{\mathbb{R}^d \times \widetilde{G}^n_d} \omega(x,T) d\mu(x,T), \ \forall \omega \in W.\end{aligned}$$ With , we may identify $\mu$ as an element of the dual RKHS $W^*$. Note that $\iota_W^*$ is not injective in general, in other words one can have $\mu = \mu'$ in $W^*$ but $\mu \neq \mu'$ in $C_0(\mathbb{R}^n \times \widetilde{G}^n_d)^*$.
In any case, one can compare any two varifolds $\mu,\mu' \in \mathcal{V}_d$ through the Hilbert norm of $W^*$ by defining: $$d_{W^*}(\mu,\mu')^2 = \|\mu-\mu'\|_{W^*}^2 = \|\mu\|_{W^*}^2 -2\langle \mu,\mu' \rangle_{W^*} + \|\mu'\|_{W^*}^2$$ where we use the small abuse of notation of writing $\mu$ and $\mu'$ instead of $\iota_W^* \mu$ and $\iota_W^* \mu'$ on the two right hand sides. Due to the potential non-injectivity of $\iota_W^*$, in general $d_{W^*}$ only induces a pseudo-metric on $\mathcal{V}_d$.
The main advantage of this construction is that $d_{W^*}$ can be expressed more explicitly based on the reproducing kernel property of $W$. Indeed, given any $\mu$ and $\nu$ in $\mathcal{V}_d$, the inner product between them is given by $$\begin{aligned}
\label{eq:rkp2}
&\langle \mu, \mu' \rangle_{W^*} \nonumber \\
&= \int_{(\mathbb{R}^d \times \widetilde{G}^n_d)^2} k^{pos}(x,x') k^G(T,T') d\mu(x,T) d \mu'(x',T') \nonumber \\
&= \int_{(\mathbb{R}^d \times \widetilde{G}^n_d)^2} \rho(|x-x'|^2) \gamma(\langle T,T' \rangle) d\mu(x,T) d \mu'(x',T')\end{aligned}$$ for kernels selected as in Remark \[rk:assumption\_kernel\].
### Characterization of distances
As mentioned above, $d_{W^*}$ is a priori a pseudo-distance between varifolds. It’s a natural question to ask under which conditions it leads to an actual distance.
Most past works have addressed this question focusing on the case of varifolds representing submanifolds and reunion of submanifolds . We can first provide an extension of these results to the general case of oriented rectifiable varifolds. A key notion for the rest of this section is the one of $C_0$-universality of kernels:
A positive definite kernel $k$ on a metric space $\mathcal{M}$ is called $C_0$-universal when its RKHS is dense in $C_0(\mathcal{M})$ for the uniform convergence topology.
$C_0$-universality has been studied in great length in such works as . In particular, one can provide characterizations of $C_0$-universality for certain classes of kernels and spaces $\mathcal{M}$. In the case of translation-invariant kernels on $\mathcal{M}=\R^n$ for instance, it has been established that $C_0$-universal kernels are the ones which can be expressed through the Fourier transform of finite Borel measures with full support on $\R^n$, which includes: compactly-supported kernels, Gaussian kernels, Laplacian kernels... With the previous definition, we have the following sufficient condition:
\[thm:dist\_rectifiable\_var\] Suppose $k^{pos}$ is a $C_0$-universal kernel on $\R^n$, $\gamma(1)>0$ and $\gamma(t) \neq \gamma(-t), \ \forall t \in [-1,1]$. Let $(X,T(\cdot))$ and $(Y,S(\cdot))$ be two oriented $\mathcal{H}^d$-rectifiable sets with $\mathcal{H}^d(X)$, $\mathcal{H}^d(Y)< \infty$. If $\left\| \mu_X-\mu_Y \right\|_{W'}=0$, then $\mathcal{H}^d(X\bigtriangleup Y) =0$ and $T = S \ \mathcal{H}^d$-$a.e$.
The full proof can be found in the Appendix. Note that the first part of the proof directly gives an equivalent statement for unoriented rectifiable varifolds (if one instead assumes $\gamma(t) = \gamma(-t)$ for all $t$), generalizing the result of .
However, the previous proposition does not necessarily lead to a distance on the full space $\mathcal{V}_d$. Counter-examples in the case $d=1$ are discussed for example in . To recover a true distance on $\mathcal{V}_d$, one needs the previous map $\iota_W^{*}$ or equivalently the map $$\begin{aligned}
\label{eq:featuremap}
\mu \mapsto \int_{\mathbb{R}^d \times \widetilde{G}^n_d} k(\cdot,(y,T)) d \mu(y,T), \ \mu \in C_0(\mathbb{R}^d \times \widetilde{G}^n_d)^*\end{aligned}$$ to be injective. As follows from Theorem 6 in , this is in fact guaranteed when the kernel $k$ on the product space $\mathbb{R}^n \times \widetilde{G}^n_d$ is $C_0$-universal, specifically
\[thm:dist\_general\_var\] The pseudo-distance $d_{W^*}$ induces a distance between signed measures of $\mathbb{R}^n \times \widetilde{G}^n_d$ if and only if $k$ is $C_0$-universal on $\mathbb{R}^n \times \widetilde{G}^n_d$. In particular, a sufficient condition for $d_{W^*}$ to be a distance on $\mathcal{V}_d$ is that $k^{pos}$ and $k^G$ are $C_0$-universal kernels on $\R^n$ and $\widetilde{G}^n_d$ respectively.
Note that these conditions are more restrictive than in Theorem \[thm:dist\_rectifiable\_var\]. To our knowledge, there is no simple characterization for general $C_0$-universal kernels on the Grassmannian. However, within the setting of Remark \[rk:assumption\_kernel\], one easily constructs $C_0$-universal kernels by restriction (based on the Plücker embedding) of $C_0$-universal kernels defined on the vector space $\Lambda^d(\R^n)$.
### Comparison with classical metrics
We now study more precisely the topology induced by the (pseudo) distance $d_{W^*}$ on $\mathcal{V}_d$ in comparison with the ones defined in Section \[subsec:standard\_metrics\]. First of all, we observe that, for any $\omega \in W$ with $\|\omega\|_{W} \leq 1$, one must have $\| \omega \|_{\infty} \leq c_W$, where $c_W$ is the embedding constant of Proposition \[prop:ctru\_kernel\]. Thus, for any $\mu$ and $\mu'$ in $\mathcal{V}_d$, we have $$\label{eq:dbl_wstar}
\| \mu-\mu'\|_{W^*} = \sup_{\omega \in W,\ \|\omega\|_{W \leq 1}} \int_{\mathbb{R}^d \times \widetilde{G}^n_d } \omega \ d(\mu-\mu') \leq c_W d_{op}(\mu,\mu').$$ From the above inequalities we see that convergence in $d_{op}$ implies convergence in $d_{W^*}$.
\[rmk:dom\_W\_BL\] With more assumptions on the regularity of the kernel $k$, namely if $W$ is continuously embedded in $C_0^1(\mathbb{R}^d \times \widetilde{G}^n_d)$, following a similar reasoning as above, one obtains the bound $\| \mu-\mu'\|_{W^*} \leq c_{W} d_{BL}(\mu,\mu')$.
Suppose $\mu_i$ converges to $\mu$ in narrow topology. Since the map $(\nu_1,\nu_2) \mapsto \nu_1 \otimes \nu_2$ is continuous with respect to the narrow topology, we have $$\begin{aligned}
\|\mu_i\|_{W^*}^2 &= \int_{(\mathbb{R}^d \times \widetilde{G}^n_d)^2} k((x,S),(y,T)) d\mu_i(x,S) d \mu_i(y,T) \\
&\rightarrow \int_{(\mathbb{R}^d \times \widetilde{G}^n_d)^2} k((x,S),(y,T)) d\mu(x,S) d \mu(y,T) \\
&= \|\mu\|_{W^*}^2,\end{aligned}$$ as $i \rightarrow \infty$. Also, it’s clear that $\lim_{i \rightarrow \infty}\langle \mu_i, \mu \rangle_{W^*} \rightarrow \|\mu\|_{W^*}^2$ and hence $\mu_i \rightarrow \mu$ with respect to $d_{W^*}$. To summarize the discussion above:
\[prop:top\_comp\] Let $\{\mu_i\}_i$ and $\mu$ be varifolds in $\mathcal{V}_{d}$ and assume that $\mu_i \rightarrow \mu$ with respect to the operator norm or the narrow topology, then $\mu_i \rightarrow \mu$ in $W^*$.
We emphasize that the result of Proposition \[prop:top\_comp\] only requires the assumptions of Proposition \[prop:ctru\_kernel\] and thus holds whether $\iota$ is injective or not.
As for the weak-\* topology, with the $C_0$-universality assumption of Theorem \[thm:dist\_general\_var\] and restricting to varifolds with bounded total mass, we show that $d_{W^*}$ induces a topology stronger than weak-\* convergence:
\[prop:dW\_finer\_weakstar\] If $k$ is $C_0$-universal, then the topology induced by $d_{W^*}$ is finer than the weak-\* topology on $\mathcal{V}_{d,M} \doteq \{\mu \in \mathcal{V}_d \ \text{s.t} \ |\mu|(\mathbb{R}^n) \leq M \}$ for any fixed $M>0$.
Let $\{\mu_i\}_i$ and $\mu$ be varifolds in $\mathcal{V}_{d,M}$ and assume that $\lim_{i \rightarrow \infty}d_{W^*}(\mu_i,\mu) = 0$. For any $f \in C_0(\mathbb{R}^d \times \widetilde{G}^n_d)$ and $\varepsilon>0$, there exists a $g \in W$ such that $\|g-f\| < \varepsilon/2M$. Then we obtain that $\mu_i \overset{\ast}{\rightharpoonup} \mu$ from the following inequalities: $$|(\mu_i-\mu|f)| \leq |(\mu_i| f-g)| + |(\mu| g-f)| + |(\mu_i - \mu|g)| \leq \varepsilon + \|\mu_i -\mu \|_{W^*} \|g\|_W.$$
Note that the topology induced by $d_{W^*}$ may be strictly finer on $\mathcal{V}_{d,M}$. Indeed, if $\rho(0),\gamma(1)>0$, consider $\mu_i = \delta_{(x_i,S)}$, where $\lim_{i \rightarrow \infty}|x_i| = \infty$ and $S\in \widetilde{G}^n_d$ fixed. Then $\mu_i \overset{\ast}{\rightharpoonup} 0$ while $\|\mu_i\|_{W^*}^2 = \rho(0) \gamma(1) > 0$ for all $i$. Yet, by combining Propositions \[prop:dBL\_weakstar\], \[prop:top\_comp\] and \[prop:dW\_finer\_weakstar\], we have the following
Let $M>0$ and $K \subset \R^n \times \widetilde{G}^n_d$ be a compact subset. If $k$ is $C_0$-universal, then $d_{W^*}$ metrizes the weak-\* convergence of varifolds on $\mathcal{V}_{d,M,K} \doteq \{\mu \in \mathcal{V}_d \ \text{s.t} \ |\mu|(\mathbb{R}^n) \leq M, \ \text{supp}(\mu)\subset K \}.$
In summary, $C_0$-universality provides a sufficient condition to obtain actual distances between varifolds that can be expressed based on the kernel function. Furthermore, the resulting topology is locally equivalent to the weak-\* topology as well as the topology induced by the bounded Lipschitz distance. This equivalence will be of importance in Section \[sec:approximation\_discrete\_var\].
Deformation and registration of varifolds {#diff_matching}
=========================================
Having defined a way of comparing general varifolds through the above kernel metrics $d_{W^*}$, our goal is now to focus on deformation models for those objects in order to formulate and study the diffeomorphic registration problem on $\mathcal{V}_d$.
Deformation models {#ssec:deformation_models}
------------------
In this section, we discuss different models for how varifolds can be transported by a diffeomorphism of $\R^n$, in other words what are possible group actions of the diffeomorphism group $\textrm{Diff}(\R^n)$ on $\mathcal{V}_d$.
Let us start by considering the case of an oriented rectifiable subset $(X,T_X)$. A diffeomorphism $\phi \in \textrm{Diff}(\mathbb{R}^n)$ transports $(X,T_X)$ as $$\begin{aligned}
\phi \cdot (X,T_X) \doteq (\phi(X), T_{\phi(X)}),\end{aligned}$$ where the transported orientation map writes $$\begin{aligned}
T_{\phi(X)}(y) \doteq d_{\phi^{-1}(y)} \phi \cdot T_X(\phi^{-1}(y))\end{aligned}$$ the above term being well-defined from . This suggests introducing the following *pushforward action* on $\mathcal{V}_d$, which is defined for all $\mu \in \mathcal{V}_d$ and $\phi \in \textrm{Diff}(\mathbb{R}^n)$ by: $$\label{eq:def_pushforward}
(\phi_{\#} \mu | \omega) \doteq \int_{\mathbb{R}^d \times \widetilde{G}^n_d} \omega(\phi(x),d_{x} \phi \cdot T) J_T \phi(x) d \mu(x,T)$$ in which $J_T \phi(x)$ denotes the determinant of the Jacobian of $\phi$ along $T$ (i.e. the change of d-volume induced by $\phi$ along $T$ at $x$) which is given by $$J_T \phi(x) = \det\left((d_x\phi(e_i) \cdot d_x\phi(e_j))_{i,j=1,\ldots,d} \right)$$ for $(e_1,\ldots,e_d)$ an orthonormal basis of $T$. One easily verifies that $(\phi,\mu) \mapsto \phi_{\#} \mu$ defines a group action which commutes with the action on oriented rectifiable sets, namely
\[prop:pushforward\_rectifiable\] For any oriented rectifiable set $(X,T_X)$ and diffeomorphism $\phi \in \textrm{Diff}(\mathbb{R}^n)$, $\phi_{\#} \mu_X = \mu_{\phi(X)}$.
This follows from the area formula for integrals over rectifiable sets, c.f. Chapter 2.
This pushforward action also extends the diffeomorphic transport of measures with densities on $\R^n$. Indeed if $\mu=\theta(x). \mathcal{L}^n$ with $\theta$ a measurable density function on $\R^n$ and $\mathcal{L}^n$ the Lebesgue measure, we can extend $\mu$ to a n-varifold in $\mathcal{V}_n$ by taking a constant global orientation in $\widetilde{G}_n^n = \{\pm 1\}$. Then, for any orientation-preserving diffeomorphism $\phi$, writes in this case: $\phi_{\#} \mu = |J\phi(x)| \theta(x) . \mathcal{L}^n$ with $J\phi(x)$ is the full Jacobian determinant of $\phi$, leading to the usual action on densities $\phi \cdot \theta(x) = |J\phi(x)| \theta(x)$.
However, in contrast with past works on submanifold registration, this is not the only possible group action that could be considered on the space $\mathcal{V}_d$. For instance, one can define another action by removing the above volume change term, taking instead $$\begin{aligned}
(\phi_{*} \mu | \omega) := \int_{\mathbb{R}^d \times \widetilde{G}^n_d} \omega(\phi(x),d_{x} \phi \cdot T) d \mu(x,T).\end{aligned}$$
This *normalized action* has the property of preserving the total mass of the varifold, i.e., $$\begin{aligned}
| \phi_{*} \mu | (\mathbb{R}^n) = | \mu | (\mathbb{R}^n), \ \forall \mu \in \mathcal{V}_d \textrm{ and } \phi \in \textrm{Diff}(\mathbb{R}^n).\end{aligned}$$ Although this action is not consistent with the action on rectifiable sets as in Proposition \[prop:pushforward\_rectifiable\], this model may be more adequate in applications to certain types of data in which mass preservation is natural.
We refer the interested reader to [@Hsieh2018] for a more in depth discussion on the properties (orbits, isotropy subgroups...) of these group actions in the simpler case of 1-varifolds. In the rest of the paper, we will restrict ourselves to the pushforward action model of , although we expect the following derivations to adapt to other cases as well, which precise study is for now left as future work.
The diffeomorphic registration problem {#ssec:diffeom_reg}
--------------------------------------
With the group action defined above, we are now ready to introduce the mathematical formulation of the diffeomorphic registration problem for general varifolds in $\mathcal{V}_d$. As deformation model, we will rely on the Large Deformation Diffeomorphic Metric Mapping (LDDMM) setting mentioned in the introduction.
Let us briefly sum up the basic construction of LDDMM, which details can be found in . In this framework, deformations consist of diffeomorphisms generated by flowing time-dependent vector fields. Let $V$ be a fixed RKHS of vector fields on $\R^n$ and $L^2([0,1],V)$ be space of time dependent velocity fields $v$ such that for all $t\in [0,1]$, $v_t$ belongs to $V$. The flow map $t \mapsto \varphi_t^v$ is defined for all $t\in[0,1]$ by $\varphi_0^v = \text{id}$ and the ODE $\dot{\varphi}^v_t = v_t \circ \varphi^v_t$. If $V$ is continuously embedded in $C_0^1(\R^n,\R^n)$, one can show that for all $t$, $\varphi_t^v$ is a $C^1$-diffeomorphism of $\R^n$. Moreover, on the subgroup $\text{Diff}_V =\{\varphi_1^v \ | \ v \in L^2([0,1],V)\}$ of $\text{Diff}(\R^n)$, one can define the following right-invariant Riemannian metric: $$d_{G_V}(\text{id},\phi) = \inf \left\{ \int_0^1 \|v_t\|_V^2 dt \ | \ \varphi_1^v = \phi \right\}$$
Let us now consider a source (or template) varifold $\mu_0 \in \mathcal{V}_d$ as well as a target $\mu_{tar} \in \mathcal{V}_d$. With the above deformation model and metric, registering $\mu_0$ to $\mu_{tar}$ consists in finding a deformation $\phi$ that minimizes $d_{G_V}(\text{id},\phi)$ with the constraint that $\phi_{\#} \mu_0$ is close to $\mu_1$ in the sense of a kernel metric $\|\cdot\|_{W^*}$ defined in Section \[ssec:kernel\_metrics\]. This can be reformulated as the following optimal control problem: $$\label{eq:matching_var}
{\mathop{\mathrm{argmin}}\limits}_{v \in L^2([0,1],V)} \bigg\{E(v) = \frac{1}{2}\int_{0}^{1} \|v_t\|_V^2 dt + \lambda \|\mu(1) - \mu_{tar} \|_{W^*}^2 \bigg\}$$ with $v$ being the control, $E$ the total cost and the state equation is given by $\mu(t) \doteq (\phi_t^v)_{\#} \mu_0$ for the pushforward model. The first term in is the regularization term that constrains the regularity of the estimated deformation paths. The second term measures the similarity between the deformed varifold $\mu(1)$ and the target varifold $\mu_{tar}$. $\lambda$ is a weight parameter between the regularization and fidelity terms. Note that this is consistent with the generic inexact registration problem formulation in LDDMM that was proposed for objects like images, landmarks, submanifolds...
The well-posedness of the optimal control problem holds under the following assumptions:
\[thm:exist\_opt\_control\] If $V$ is continuously embedded in $C_0^2(\mathbb{R}^n,\mathbb{R}^n)$, $W$ is continuously embedded in $C_0^1(\mathbb{R}^n \times \widetilde{G}^n_d)$ and $\rm{supp}(\mu_0) \subset K$, for some compact subset $K$ of $\mathbb{R}^n \times \widetilde{G}^n_d$, then there exists a global minimizer to the problem .
The proof is similar to previous results of the same type on rectifiable currents and varifolds. We give it in Appendix for the sake of completeness.
One can derive necessary and sufficient conditions on the kernels of $W$ and $V$ for the two embedding assumptions of Theorem \[thm:exist\_opt\_control\] to hold (see for instance Theorem 2.11 in ). In our context, in order to get $W \hookrightarrow C_0^1(\mathbb{R}^n \times \widetilde{G}^n_d)$ for instance, it is enough to assume that $\rho$ and $\gamma$ are $C^2$ functions such that all derivatives of $\rho$ up to order 2 vanish as $x \rightarrow +\infty$.
As an important note, the formulation of extends registration of submanifolds or rectifiable subsets in the sense that if $\mu_0 = \mu_{X_0}$ and $\mu_{tar}=\mu_{X_{tar}}$ for two oriented d-rectifiable subsets of $\R^n$ then becomes equivalent, thanks to Proposition \[prop:pushforward\_rectifiable\], to registering rectifiable subsets, i.e. to the problem $${\mathop{\mathrm{argmin}}\limits}_{v \in L^2([0,1],V)} \bigg\{\frac{1}{2}\int_{0}^{1} \|v_t\|_V^2 dt + \lambda \|\mu_{X(1)} - \mu_{X_{tar}} \|_{W^*}^2 \bigg\}$$ with $X(t) = \varphi_t^v \cdot X_0$, which is the setting of many past works as for instance .
General optimality conditions {#ssec:general_PMP}
-----------------------------
A last important question we address in this section is the derivation of necessary optimality conditions for the solutions of . In standard finite-dimensional optimal control problems, these are provided by the Pontryagin Maximum Principle (PMP) introduced originally in . The approach generalizes, with a certain number of technicalities, to a broad class of infinite-dimensional shape matching problems, as developed in .
We follow the same setting as well as related works such as by first rewriting the above problem as an optimal control problem on diffeomorphisms, i.e. $${\mathop{\mathrm{argmin}}\limits}_{v \in L^2([0,1],V)} \bigg\{\frac{1}{2}\int_{0}^{1} \|v_t\|_V^2 dt + g(\varphi_1^v) \ | \ \text{s.t.} \ \dot{\varphi}_t^v = v_t \circ \varphi_t^v \bigg\}$$ with $g(\varphi_1^v) \doteq \lambda \|(\varphi_1^v)_{\#} \mu_0 - \mu_{tar} \|_{W^*}^2$. The state variables are now given by the deformations $\varphi_t^v$ which we view as elements of the Banach space $\mathcal{B} \doteq \text{id} + C_0^1(\R^n,\R^n)$. Let us denote, for $\phi \in \text{Diff}(\R^n)$, $\xi_{\phi} : V \rightarrow C_0^1(\R^n,\R^n)$ the mapping $v \mapsto v \circ \phi$. We then introduce the Hamiltonian functional $H: \ C_0^1(\R^n,\R^n)^* \times \mathcal{B} \times V \rightarrow \R$ defined by: $$\label{eq:Hamiltonian_cont}
H(p,\phi,v) = (p|v\circ \phi) - \frac{1}{2}\|v\|_V^2$$ where $p$ is the costate variable which is a vector distribution of $C_0^1(\R^n,\R^n)^*$ and $(p|v\circ \phi)$ denotes the duality bracket in $C_0^1(\R^n,\R^n)^*$. With the assumptions of Theorem \[thm:exist\_opt\_control\], it follows from the maximum principle shown in that if $(v_t,\varphi_t^v)$ is a global minimum of the optimal control problem, there exists a path of costates $p \in H^1([0,1],C_0^1(\R^n,\R^n)^*)$ such that the following equations hold: $$\left\{\begin{array}{ll}
&\dot{\varphi}_t^v = \partial_p H(p_t,\varphi_t^v,v_t) \\
&\dot{p}_t = -\partial_{\phi} H(p_t,\varphi_t^v,v_t) \\
&\partial_v H(p_t,\phi_t^v,v_t) = 0
\end{array}\right.
\label{eq:Hamiltonian_eqn_cont}$$ with the end time boundary conditions $p_1 = -\partial_{\phi} g(\varphi_1^v)$. From the last equation in , we can attempt to deduce the form of the optimal $v$. Introducing the Riesz isometry operator $\bold{K}_V : V^*\rightarrow V$ and its inverse $\bold{L}_V=\mathbf{K}_V^{-1}: V \rightarrow V^*$, we get: $$\label{eq:opt_v_cont}
\xi_{\varphi_t^v}^* p_t - \bold{L}_V v_t = 0 \ \Rightarrow \ v_t = \bold{K}_V \xi^*_{\varphi_t^v} p_t.$$
One additional consequence of is the following conservation of momentum again proved in [@arguillere14:_shape]: for all $u \in C_0^1(\R^n,\R^n)$ and $t\in[0,1]$, $$\label{eq:conservation_momentum}
(p_t | d\varphi_t^v u) = (p_0 | u).$$ Note that , and are generic to the LDDMM model and so far independent of the nature of the deformed objects and of the term $g(\varphi_1^v)$ in the cost. This dependency is entirely encompassed by the boundary condition $p_1 = -\partial_{\phi} g(\varphi_1^v)$ which we may describe a little more precisely based on the following:
\[prop:variation\_g\] The end-time momentum $p_1$ is a vector distribution in $C_0^1(\R^n,\R^n)^*$ of the form $$\begin{aligned}
(p_1|u) = &\int_{\R^n} \alpha(x) \cdot u(x) \ d|\mu_0|(x) \\
+&\int_{\R^n \times \widetilde{G}^n_d} \beta(x,T) du|_T(x) \ d\mu_0(x,T) \\
+&\int_{\R^n \times \widetilde{G}^n_d} \gamma(x,T) \text{div}_T u(x) \ d\mu_0(x,T)
\end{aligned}$$ where $\alpha: \R^n \rightarrow \R^n$, $\beta: \R^n \times \widetilde{G}^n_d \rightarrow (\R^{n\times d})^*$ and $\gamma: \R^n \times \widetilde{G}^n_d \rightarrow \R$ are continuous fields and for all $T \in \widetilde{G}^n_d$, $\text{div}_T u$ and $du|_{T}$ denote the divergence and differential of $u$ restricted to $T$.
A condensed proof of this proposition can be found in the Appendix, although we have left aside the technical derivations related to differential calculus on the Grassmannian (this will be discussed further in Section \[sec:numerics\] in the discrete setting). This result extends in a way first variation formulas for varifolds proved in which considered variations of rectifiable varifolds resulting from variations of the underlying rectifiable sets. This corresponds to the special case in which $\mu_0 = \mu_{X_0}$. In that case, one can show, after some derivations, that the above expression of $p_1$ can be rewritten in the form of a vector distribution $u \mapsto \int_{\varphi_1^v(X_0)} u(x) \cdot h(x) d \mathcal{H}^d$ in $C_0^0(\R^n,\R^n)^*$ with vectors $h(x)$ normal to $\varphi_1^v(X_0)$ at each $x$. In our more general situation, this is however not possible and $p_1$ is a priori a distribution that involves first order derivatives of the test function $u$.
Now, the conservation law of gives that for all $t \in [0,1]$, $$(p_t | d\varphi_t^v u) = (p_1 | d\varphi_1^v u) = (p_0 | u).$$ Using the expression of $p_1$ in Proposition \[prop:variation\_g\], and grouping all $0$-th and $1$-st order terms in the resulting expressions, we may write $p_t$ in the general form: $$\begin{aligned}
(p_t | u) &= \int_{\R^n \times \widetilde{G}_d^n} \alpha_t(x,T) \cdot u(x) \ d\mu_0(x,T) + \int_{\R^n \times \widetilde{G}_d^n} B_t(x,T) d_x u|_{T} \ d\mu_0(x,T)\end{aligned}$$ where $\alpha_t: \R^n \times \widetilde{G}_d^n \rightarrow \R^n$ and $B_t: \R^n \times \widetilde{G}_d^n \rightarrow (\R^{n\times d})^*$ are continuous fields, with $\alpha_1(x,T)=\alpha(x)$ and $B_1(x,T) du|_{T}(x) = \beta(x,T) du|_T(x) + \gamma(x,T) \text{div}_T u(x)$. Furthermore, optimal vector fields satisfy $v_t = K_V \xi^*_{\varphi_t^v} p_t$ and we have $$\begin{aligned}
(\xi^*_{\varphi_t^v} p_t | u) &= (p_t | u\circ \varphi_t^v) \\
&= \int_{\R^n \times \widetilde{G}_d^n} \alpha_t(x,T) \cdot u(\varphi_t^v(x)) \ d\mu_0(x,T) + \int_{\R^n \times \widetilde{G}_d^n} B_t(x,T) d_{\varphi_t^v(x)}u|_{d_x\varphi_t^v \cdot T} \ d\mu_0(x,T).\end{aligned}$$ Denoting $K_V: \R^n \times \R^n \rightarrow \R^{n\times n} $ the reproducing kernel of $V$, the reproducing kernel property implies that for all $u\in V$ and $x,h \in \R^n$, $u(x) \cdot h = \langle K_V(x,\cdot) h, u \rangle_V$. Moreover, the similar property on the kernel first order derivatives gives that for any $h,h' \in \R^n$, $$d_x u (h) \cdot h' = \langle \partial_1 K_V(x,\cdot)(h) \cdot h', u \rangle_V.$$ Then, we rewrite the linear maps $B_t$ as $B_t(x,T) H = \sum_{i=1}^d b_{t,i}(x,T) \cdot H_i$ for any $H=(H_1,\ldots,H_d) \in \R^{n\times d}$ and where $b_i(x,T) \in \R^n$ are the component vector fields of $B_t$. By the above and the linearity of $\bold{K}_V$, we obtain the following general expression for optimal vector fields $$\begin{aligned}
v_t &= \int_{\R^n \times \widetilde{G}_d^n} K_V(\varphi_t^v(x),\cdot) \alpha_t(x,T) \ d\mu_0(x,T) & \nonumber\\
&+ \int_{\R^n \times \widetilde{G}_d^n} \left(\sum_{i=1}^d \partial_1 K_V(\varphi_t^v(x),\cdot)(d_x\varphi_t^v(t_i))\cdot b_{t,i}(x,T)\right) d\mu_0(x,T). \end{aligned}$$ In contrast with LDDMM registration of submanifolds or point clouds, the expression of optimal deformation fields involves in general both the kernel function and its first order derivatives. We do not explicit the vector fields $\alpha$ and $b_i$ at this point, it will be specified later in the discrete setting, see Section \[ssec:discrete\_registration\].
Approximations by discrete varifolds {#sec:approximation_discrete_var}
====================================
The previous derivations were so far conducted for completely general measures in the space $\mathcal{V}_d$ which include objects of widely different natures. In the perspective of implementing numerically the above approach, which is the subject of Section \[sec:numerics\], we first need to build an adequate discretization framework in $\mathcal{V}_d$ with approximation guarantees, and even more importantly investigate the consistency of the discretized registration problems (Theorem \[thm:convergence\_sol\]), which is the main result of this section.
Discrete approximations {#ssec:discrete_approx}
-----------------------
In what follows, we will consider the specific class of varifolds which can be written as finite combinations of Dirac masses: $$\begin{aligned}
\label{eq:discrete_varifolds}
\mu = \sum_{i=1}^N r_i \delta_{(x_i,T_i)},\ r_i \in \mathbb{R}_+,\ x_i \in \mathbb{R}^n, \ T_i \in \tilde{G}^n_d.\end{aligned}$$ for some $N\geq 1$. Throughout this paper, varifolds of this form will be called *discrete varifolds*. It is quite natural to consider this type of varifolds for the purpose of representing discrete shapes, which has been exploited in previous works on piecewise linear curves and surfaces. For example, if $X = \bigcup_{i=1}^N X_i$ is a triangulated surface, with $X_i$ being the mesh triangles with specified orientations, one can write $\mu_X = \sum_{i=1}^N \mu_{X_i}$ and for each $i \in \{1,\cdots,N\}$ approximate $\mu_{X_i}$ by $r_i \delta_{(x_i,T_i)}$, where $x_i$ is the center of $X_i$, $T_i$ the oriented plane containing $X_i$ and $r_i = \mathcal{H}^d(X_i)$. This leads to the approximation $\widetilde{\mu}_{X} := \sum_{i=1}^N r_i \delta_{(x_i,d_i)}$. As proved in [@Charon2017], this approximation provides an acceptable error bound for $d_{W^*}$: $$\begin{aligned}
d_{W^*}(\mu_X,\widetilde{\mu}_{X}) \leq Cte \ \mathcal{H}^d(X) \max_{i} \textrm{diam}(X_i).\end{aligned}$$ The main interest of such discrete varifold approximations is that the expression of the metric becomes particularly simple to compute numerically. Indeed, given two discrete varifolds $\mu = \sum_{i=1}^{N} r_i \delta_{(x_i,S_i)} $ and $\mu' = \sum_{j=1}^{M} r'_j \delta_{(x'_j,T'_j)}$, we have as a particular case of : $$\begin{aligned}
\label{eq:norm_discrete_varifolds}
\langle \mu,\mu' \rangle_{W^*} = \sum_{i=1}^N\sum_{j=1}^M r_i r'_j \rho(|x_i-x'_j|^2) \gamma(\langle T_i,T'_j \rangle). \end{aligned}$$
The above approximation scheme only applies to the case of piecewise linear shapes given by meshes such as polygonal curves or triangulated surfaces. In the more general context of this work, a key issue is to construct similar discrete varifold approximations for more general and less structured objects. Specifically, given a varifold $\mu$ with finite total weight, can it be approximated by discrete varifolds and will approximations converge as $N\rightarrow +\infty$? This is the problem known as *quantization*, which has been studied intensively in the case of probability measures over Euclidean spaces or manifolds , under specific regularity assumptions on those measures. In the situation of varifolds, an interesting recent work on this question is . The authors prove that any rectifiable varifold with finite mass can be approximated by a sequence of discrete varifolds for the bounded Lipschitz distance and propose a numerical approach to approximate mean curvature measures based on discrete varifolds.
In this section, we first wish to extend approximation results to general oriented varifolds of finite mass for both $d_{BL}$ and $d_{W^*}$ metrics.
\[thm:varifold\_approximation\] Let $$\begin{aligned}
\mathcal{V}^N_d := \left\{ \sum_{i=1}^N r_i \delta_{(x_i,T_i)} | r_i \in \mathbb{R}_+,\ x_i \in \mathbb{R}^n, \ T_i \in \widetilde{G}^n_d \right\}
\end{aligned}$$ be the (non-convex) cone of discrete varifolds with at most $N$ Diracs. For any oriented varifold $\mu \in \mathcal{V}_d$ with $|\mu |(\mathbb{R}^n) < \infty$, there exists a sequence $\mu_N \in \mathcal{V}_d^N$such that $\lim_{N \rightarrow \infty} d_{BL}(\mu_N,\mu) =0$. Moreover, if $\mu$ has compact support, then we can assume that for all $N$, $\rm{supp}(\mu_N) \subset K$ for some compact set $K \subset \mathbb{R}^n \times \widetilde{G}^n_d$ and $$d_{BL}(\mu_N,\mu) < \frac{C}{N^{1/(n+d(n-d))}},$$ where $C$ is a constant that only depends on $n$, $d$ and $\rm{supp}(\mu)$.
We first tackle the case of compactly supported $\mu$. Without loss of generality, we may also assume that $\mu$ is a probability measure. Let $D = n+ d(n-d)$ and $B \subset \mathbb{R}^n$ be a closed ball that contains $\rm{supp}|\mu|$. For brevity, we write $M \doteq B \times \widetilde{G}^n_d$. Since we can view $M$ as a compact $D$-dimensional submanifold of $\R^n \times \Lambda^d(\mathbb{R}^n)$ (using Plücker embedding), M is also regular of dimension $D$ (cf. [@graf2007foundations]), i.e., $0<\mathcal{H}^D(M)< \infty$ and there exist $c,r_0>0$, such that $$\begin{aligned}
\frac{1}{c}r^D \leq \mathcal{H}^D {\mathbin{\vrule height 1.6ex depth 0pt width
0.13ex\vrule height 0.13ex depth 0pt width 1.3ex}}M(B_r(a)) \leq c r^D,\ \forall a \in M, \ r \in (0,r_0).
\end{aligned}$$ Given $\varepsilon \in (0,5 r_0)$, by the 5-Times Covering Lemma (cf. [@Simon]), these exists a subset $\mathcal{I} \subset M$, such that $M \subset \cup_{x \in \mathcal{I}} B_{\varepsilon}(x)$ and $B_{\varepsilon/5}(x) \cap B_{\varepsilon/5}(y) = \varnothing$ for all $x \neq y \in \mathcal{I}$. Therefore, $$\begin{aligned}
\mathcal{H}^D(M) \geq \sum_{x \in \mathcal{I}} \mathcal{H}^D(M \cap B_{\varepsilon/5}(x))
\geq \frac{|\mathcal{I}| \varepsilon^D}{c 5^D}.
\end{aligned}$$ We can thus obtain a partition $\{A_i\}_{i=1,\cdots,|\mathcal{I}|}$ of $M$ from the the collection $\{B_{\varepsilon}(x) \cap M\}_{x\in\mathcal{I}}$ which satisfies $\sup_i \textrm{diam}(A_i) < \varepsilon$ and $$|\mathcal{I}| \leq \frac{c 5^D \mathcal{H}^D(M)}{ \varepsilon^D}.$$ Let $r_i = \mu(A_i)$ and $(x_i,T_i) \in A_i$ and define $\nu=\sum_{i=1}^{|\mathcal{I}|} r_i \delta_{(x_i,T_i)}$. For any $\varphi \in \textrm{Lip}_1(\mathbb{R}^n \times \widetilde{G}^n_d)$, with $\|\varphi\|_{\infty} \leq 1$, we have
$$\begin{aligned}
\left| \int_{\mathbb{R}^n \times G_d^n} \varphi(x,T)d \nu - \int_{\mathbb{R}^n \times G_d^n} \varphi(x,T) d\mu\right|
&= \left| \sum_{i=1}^N \left( \mu(A_i) \varphi(x_i,T_i) - \int_{A_i}\varphi(x,T) d\mu \right) \right| \\
&\leq \sum_{i=1}^N \int_{A_i}| \varphi(x_i,T_i) - \varphi(x,T)| d\mu \\
&<\sum_{i=1}^N \varepsilon \mu(A_i) = \varepsilon.\end{aligned}$$
Taking the supremum over all $\omega \in \textrm{Lip}_1(\mathbb{R}^n \times \widetilde{G}^n_d)$ with $\|\varphi\|_{\infty} \leq 1$, we obtain $d_{BL}(\mu,\nu)< \varepsilon$. Then for each $N \in \mathbb{N}$, we can choose $\varepsilon_N = 5(C \mathcal{H}^D(M)/N)^{1/D}$ and we obtain $\mu_N \in \mathcal{V}_d^N$ such that $$d_{BL}(\mu,\mu_N)< \frac{5 C^{1/D} (\mathcal{H}^D(M))^{1/D}}{N^{1/D}}$$ and in particular $\lim_{N \rightarrow +\infty} d_{BL}(\mu,\mu_N) = 0$.
Suppose now that $\rm{supp}(\mu)$ is not compact: we show that for any $\varepsilon>0$, there exists a discrete varifold $\nu$ such that $d_{BL}(\mu,\nu)< \varepsilon$. Choose a compact set $K \subset \mathbb{R}^n \times \widetilde{G}^n_d$ such that $\mu(\mathbb{R}^n \times \widetilde{G}^n_d \setminus K)< \varepsilon/2$. From the previous case, we can find a discrete varifold $\nu$ such that $d_{BL}(\mu {\mathbin{\vrule height 1.6ex depth 0pt width
0.13ex\vrule height 0.13ex depth 0pt width 1.3ex}}K,\nu )<\varepsilon/2$, and hence $d_{BL}(\mu,\nu )<\varepsilon$.
Note that the proposition clearly holds for non-oriented varifolds as well. Another direct consequence, thanks to proposition \[prop:top\_comp\] and remark \[rmk:dom\_W\_BL\], is the following corresponding statement for $d_{W^*}$:
\[cor:var\_approx\_W\] With the assumptions from proposition \[prop:ctru\_kernel\], one also has $\lim_{N \rightarrow \infty} d_{W^*}(\mu_N, \mu) =0$. If in addition $W \hookrightarrow C_0^1(\R^n\times \tilde{G}_d^n)$, an equivalent upper bound as in Theorem \[thm:varifold\_approximation\] holds for $d_{W^*}(\mu,\mu_N)$.
We should point out that the asymptotic convergence rate given by the previous upper bound is rather slow, especially as the dimensions $d$ and $n$ grow. This is however under very mild assumptions on the varifold $\mu$. We expect much better convergence properties for certain specific classes of varifolds, for instance assuming Alfors regularity as in , although we leave such questions for future investigation.
Optimal approximating sequence
------------------------------
In addition to the asymptotic approximation results of the previous section, we now want to construct such sequences of discrete approximating varifolds. Given any $\mu \in \mathcal{V}_d$ and $N \in \N$, a natural idea is to look for the optimal discrete varifold in $\mathcal{V}_d^N$ that approximates $\mu$ in terms of the metric $d_{W^*}$. Due to the intricate structure of the set $\mathcal{V}_d^N$ (infinite-dimensional non-convex cone), this is far from a straightforward problem. Several different approaches in some simpler contexts have been proposed to circumvent this issue, which we briefly recap. One possibility is to restrict to finite-dimensional vector spaces of $\mathcal{V}_d$ (e.g. generated by finite sets of Diracs). Works such as for instance, which are focused on the model of currents, consider dictionaries of Diracs defined on a predefined grid of point positions in $\R^n$. Then the problem can be recast as the one of finding sparse approximations of $\mu$ in such a dictionary. It remains a NP hard problem but solutions can be approached either through greedy algorithms like *orthogonal matching pursuit* as proposed in or by considering the $L^1$ relaxation formulation leading to a standard convex LASSO program. Such ideas apply well to the specific situation of currents mainly as a result of the inherent linearity of this model: indeed, at any iteration of a matching pursuit procedure, once the optimal position of a Dirac is found, the corresponding direction vector and weight are explicitly determined. This allows to limit the search over grid of points in the spatial domain only. Unfortunately, for the general oriented varifold metrics we consider in this paper, such a property no longer holds and, as a result, these methods would involve very large dictionaries defined on grids on the product $\R^n \times \widetilde{G}_d(\R^n)$. Such an increase in dimension makes the approach numerically impractical as soon as $n\geq 3$ and $d\geq2$. Another downside is that the use of finite dictionaries and greedy algorithms like matching pursuit is not guaranteed to give an optimal approximation of varifolds for a given number $N$ of Diracs. The approach we develop in this section consists instead in directly tackling the non-convex problem of computing the projection onto $\mathcal{V}_d^N$ for the class of kernel metrics $d_{W^*}$. It shares some connection with the recent work of that considers a related problem for standard measures defined on the torus $\R^n / \mathbb{Z}^n$.
Fix a varifold $\mu_* \in \mathcal{V}_d$. For any $N \in \mathbb{N}, \ N\geq 1$, we seek $\mu_N \in \mathcal{V}_d^N$ that is closest to $\mu_*$ for $d_{W^*}$, namely $$\begin{aligned}
\label{eq:projection_problem}
\mu_{N} = \operatorname*{argmin}_{\mu \in \mathcal{V}^N_d} \| \mu - \mu_* \|_{W^*}\end{aligned}$$
By construction, if $|\mu|(\mathbb{R}^n)< \infty$ then Corollary \[cor:var\_approx\_W\] will imply that $(\mu_N)$ converges to $\mu$ in the metric $d_{W^*}$. We only need to ensure that such a projection is well defined, which is the object of the following proposition:
\[thm:exist\_min\_proj\] Suppose all assumptions in proposition \[prop:ctru\_kernel\] and remark \[rk:assumption\_kernel\] hold. We further assume that the functions $\rho$ and $\gamma$ defining the kernels are non-negative. Then for any $\mu \in \mathcal{V}_d$ and $N \in \mathbb{N}$, there exists $\mu_{N} \in \mathcal{V}^N_d$ such that $\mu_{N} = \operatorname*{argmin}_{\nu \in \mathcal{V}^N_d} \| \mu - \mu_* \|_{W^*}$
Let $\mu_m = \sum_{i=1}^N r_i^m \delta_{(x_i^m,T_i^m)}$ be a minimizing sequence, $K_W: W^* \mapsto W$ be the dual operator and $f := K_W(\mu) $. Without loss of generality, we may assume that there is a $N_1 \leq N$ such that $\sup_{1\leq i \leq N_1} |x_i^m| $ remains bounded and $\inf_{ M+1\leq i \leq N} |x_i^m|$ tends to $\infty$ as $m \rightarrow \infty$.
Observe that $\sup_{1 \leq i \leq N} \{r_i^m\}$ must be bounded. If it’s not bounded, then from the assumptions that $\rho, \gamma \geq 0$, we obtain
$$\begin{aligned}
\|\mu_m -\mu_* \|_{W^*}
&\geq \| \mu_m \|_{W^*} -\| \mu_* \|_{W^*}
= \sqrt{\sum_{i,j=1}^N r_i^m r_j^m \rho(|x_i^m - x_j^m|^2) \gamma(\langle T_i^m,T_j^m \rangle)} - \| \mu_* \|_{W^*} \\
&\geq \sqrt{\sum_{i=1}^N (r_i^m)^2 \rho(0) \gamma(1)} - \| \mu_* \|_{W^*} \rightarrow \infty\end{aligned}$$
as $m \rightarrow \infty$, which is absurd.
Since $r_i^m,$ $T_i^m$, $f(x_i^m,T_i^m)$ and $$\begin{aligned}
A_m := (\rho(|x_i^m -x_j^m|^2) \gamma(\langle T_i^m, T_j^m \rangle))_{1 \leq i,j \leq N}\end{aligned}$$ are all bounded sequences of $m$, we may replace them by convergent subsequences, thus we could assume that
$$\begin{aligned}
&\lim_{m \rightarrow \infty} r_i^m = r_i, \ \ \lim_{m \rightarrow \infty} T_i^m = T_i, \
\lim_{m \rightarrow \infty} f(x_i^m,T_i^m) = f_i, \ \ \lim_{m \rightarrow \infty} A_m = A.\end{aligned}$$
Since $\rho \in C_0(\mathbb{R})$, the matrix $A$ must has the following form:
$$\begin{aligned}
A = \left(\begin{array}{cc}
B_1 & \boldsymbol{0} \\
\boldsymbol{0} & B_2
\end{array} \right),\end{aligned}$$
where $B_1$ and $B_2$ are $N_1$-by-$N_1$ and $N-N_1$-by-$N-N_1$ semi-positive definite matrices. Combining this with the assumption $f \in C_0(\mathbb{R}^n \times \widetilde{G}^n_d)$, we obtain
$$\begin{aligned}
&\lim_{m \rightarrow \infty} \| \mu_m -\mu_* \|_{W^*}^2
= \boldsymbol{r'}^T B_1 \boldsymbol{r'} + \boldsymbol{r''}^T B_2 \boldsymbol{r''} - 2 \boldsymbol{f'}^T \boldsymbol{r'} + \| \mu_* \|_{W^*}^2,\end{aligned}$$
where $\boldsymbol{r'} = (r_1,\cdots,r_{N_1})$, $\boldsymbol{r''} = (r_{N_1+1},\cdots,r_{N})$, and $\boldsymbol{f'} = (f_1,\cdots,f_{N_1})$. Since $\sup_{1 \leq i \leq N_1} |x_i^m|$ is bounded we can assume that $\lim_{m \rightarrow \infty} x_i^m = x_i, \ 1 \leq i \leq N_1$. Let $\mu := \sum_{i=1}^{N_1} r_i \delta_{(x_i,u_i)}$, then $$\begin{aligned}
\| \mu -\mu_* \|_{W^*}^2 &= \boldsymbol{r'}^T B_1 \boldsymbol{r'} - 2 \boldsymbol{f'}^T \boldsymbol{r'} + \| \mu_* \|_{W^*}^2
\leq \lim_{m \rightarrow \infty} \| \mu_m -\mu_* \|_{W^*}^2.\end{aligned}$$ Hence $\mu$ is a minimizer.
However, in general this projection is not unique. We also point out that the existence is a priori not guaranteed if kernels $\rho$ and $\gamma$ take negative values. It is so far an open question to determine to what extent one could generalize the result of Proposition \[thm:exist\_min\_proj\], one particular but important case being the one of current metrics obtained for $\gamma(t)=t$ which is not covered by our result.
As written in the proof of Proposition \[thm:exist\_min\_proj\], is equivalent to the optimization problem: $$\begin{aligned}
(r_i,x_i,T_i) = \operatorname*{argmin}_{(w_i,y_i,S_i)} \| \sum_{i=1}^N w_i \delta_{(y_i,S_i)} - \mu_* \|_{W^*}^2\end{aligned}$$ Any solution must satisfy first order optimality conditions obtained by differentiating $\|\mu_N-\mu\|_{W^*}$ with respect to the $(r_k,x_k,T_k)$. In particular, we have $$\begin{aligned}
0 &= \frac{\partial \| \mu_N-\mu_* \|_{W^*}^2}{\partial r_k} \\
&= 2 \bigg( \sum_{i=1}^N r_i \rho(|x_i-x_k|^2) \gamma(\langle T_i,T_k \rangle) - \int_{\mathbb{R}^n \times \widetilde{G}^n_d} \rho(|x_k-x|^2) \gamma(\langle T_k,T \rangle) d \mu_*(x,T) \bigg).\end{aligned}$$ which gives $\langle \mu_N-\mu_*,\mu_N \rangle_{W^*} = 0$. It shows that for any $N \in \N$, $\|\mu_N\|_{W^*} \leq \|\mu_*\|_{W^*}$.
$\Gamma$-convergence of registration functionals
------------------------------------------------
Ultimately, our purpose is to use the previous approximating discrete varifolds $\mu_N$ to approximate the diffeomorphic registration problem . The natural question that arises is whether replacing the source varifold $\mu_0$ by its projections $\mu_N$ in still leads to reasonable approximations (at least asymptotically) of optimal deformation fields for the original problem. In this section, we address this by showing a $\Gamma$-convergence property for these variational problems. We point out that our setting and the following proof differ quite a bit from previous results of the same type that were dealing with the specific case of surface triangulations such as .
To obtain such convergence results for solutions of variational problems, one usually requires the approximating sequence to possess certain nice properties. Specifically, assuming $\mu \in \mathcal{V}_d$ with compact support and finite mass and $\{\mu_N\} \subset \mathcal{V}^N_d$ such that $\lim_{N \rightarrow \infty} \|\mu_N - \mu \|_{W^*} = 0$, we will need that $\bigcup_{N} \textrm{supp}(\mu_N) \subset K$ for some compact set $K \subset \mathbb{R}^n$ or that $\sup_{N} |\mu_N|(\mathbb{R}^n) < \infty$. Unfortunately, this does not hold in general since convergence in $d_{W^*}$ does not allow to control the support nor the total mass of the sequence $\mu_N$.
Yet, provided that $\bigcup_{N} \textrm{supp}(\mu_N) \subset K$, we can actually retrieve the boundedness of the total mass. We assume in what follows that the kernels are such that $\rho(0)>0$ and $\gamma(1)>0$.
\[lemma:unif\_bdd\_weight\] Let $\{\mu_N\}$ be a sequence of discrete varifolds with finite mass such that there exists a compact $K \subset \R^n$ with $\rm{supp}(|\mu_N|) \subset K$ for all $N$. We assume that $\{\|\mu_N\|_{W^*}\}$ is bounded. Then $\{|\mu_N|(\mathbb{R}^n)\}$ is bounded.
We prove it by contradiction. Assume that $(|\mu_N|(\mathbb{R}^n))_{N \geq 1}$ is unbounded. Then, up to extracting a subsequence, we can assume that $|\mu_N|(\mathbb{R}^n) \rightarrow +\infty$. Let’s write $\mu_N = \sum_{i=1}^{p_N} r_{i,N} \delta_{(x_{i,N},T_{i,N})}$. Thus $|\mu_N|(\mathbb{R}^n) = \sum_{i=1}^{p_N} r_{i,N} \rightarrow +\infty$. Since $\rho$ and $\gamma$ are continuous and $\rho(0),\gamma(1)>0$, we can find compact subsets $A \subset K$ and $B \subset \tilde{G}^n_d$ with diameters small enough, so that: $\inf_{x,y \in A} \rho(|x-y|^2)>m>0$, $\inf_{u,v \in B} \gamma(\langle u, v \rangle) > m' >0 $ and $\lim_{N \rightarrow \infty} \sum_{i\in \mathcal{I}_N} r_{i,N} = \infty$, where $\mathcal{I}_N := \{i: (x_{i,N},u_{i,N}) \in A \times B\}$. It follows that, as $N \rightarrow \infty$, $$\begin{aligned}
&\|\mu_N\|_{W^*}^2 \\
&= \sum_{i=1}^{p_N} \sum_{j=1}^{p_N} r_{i,N} r_{j,N} \rho(|x_{i,N}-x_{j,N}|^2) \gamma(\langle T_{i,N},T_{j,N} \rangle) \\
&\geq mm' \sum_{i,j \in \mathcal{I}_N}r_{i,N} r_{j,N} = mm' \left( \sum_{i \in \mathcal{I}_N}r_{i,N} \right)^2 \rightarrow \infty,\end{aligned}$$ which is a contradiction.
Lemma \[lemma:unif\_bdd\_weight\] suggests that one should enforce the uniform compactness of the supports of the $\mu_N$. To do so in the context of the projection approach of the previous sections, we consider solving the optimization problem with the additional constraint that the support of $\mu_N$ stays in a compact set containing $\rm{supp}(|\mu|)$. We still have to verify the convergence of the resulting sequence:
\[prop:nice\_app\_seq\] Let $\mu_0$ be a varifold with finite mass and $K$ be a compact set in $\mathbb{R}^n$ which contains $\rm{supp}(|\mu_0|)$. Construct the approximating sequence of $\mu_0$ by solving the following constrained optimization problem: $$\begin{aligned}
&\mu_{K,N} = \operatorname*{argmin}_{\nu \in \mathcal{V}^N_d} \| \nu - \mu \|_{W^*} \\
&subject \ to \ \rm{supp}(|\nu|) \subset K.
\end{aligned}$$ Then $\mu_{K,N}$ converges to $\mu_0$ in $d_{W^*}$ and, if the kernel $k$ is $C_0$-universal, it also converges in $d_{BL}$.
Thanks to Theorem \[thm:varifold\_approximation\] and Lemma \[lemma:unif\_bdd\_weight\], we immediately get that $\|\mu_{K,N} - \mu_0\|_{W^*} \rightarrow 0$ as $N \rightarrow \infty$ and $\sup_N(|\mu_{K,N}|(\mathbb{R}^n))< \infty$. Moreover, if $k$ is $C_0$-universal, then by Proposition \[prop:dW\_finer\_weakstar\] it implies that $\mu_{K,N} \overset{\ast}{\rightharpoonup} \mu_0$. Since $ \bigcup_N \rm{supp}(|\mu_{K,N}|) \subset K$ and $\sup_N(|\mu_{K,N}|(\mathbb{R}^n))< \infty$, weak-\* convergence implies that $\mu_{K,N}$ converges to $\mu_0$ in $d_{BL}$ by Proposition \[prop:dBL\_weakstar\].
We are now able to state the main result of this section. We assume that the source/template varifold $\mu_0$ is compactly supported and we fix $K$ is a compact subset of $\R^n$ that contains $\rm{supp}(|\mu_0|)$. Then for any $N \in \N, \ N\geq 1$, $\mu_{K,N}$ is defined as in Proposition \[prop:nice\_app\_seq\] and we introduce the following energy functionals $E_N : \ L^2([0,1],V) \rightarrow \R_+$: $$\begin{aligned}
\label{eq:energy_optcontrol_approx}
&E_N(v) \doteq \frac{1}{2} \int_{0}^{1} \|v_t\|_V^2 dt + \lambda \|\mu_{K,N}(1) - \mu_{tar} \|_{W^*}^2 \nonumber\\
&subj \ to \ \left\{ \begin{array}{l}
\partial_t \varphi^v_t = v_t \circ \varphi^v_t, \ \varphi^v_0 = id \\
\mu_{K,N}(t) = ( \varphi^v_t)_{\#} \mu_{K,N}
\end{array} \right.\end{aligned}$$ which are the equivalent to the energy $E$ of the original problem but replacing the template varifold $\mu_0$ by its approximations $\mu_{K,N}$.
\[thm:convergence\_sol\] With the above notations, we assume that the reproducing kernel $k$ of $W$ is $C_0$-universal and satisfies all the conditions of Proposition \[thm:exist\_min\_proj\]. We also assume the continuous embedding $V \hookrightarrow C_0^2(\mathbb{R}^n,\mathbb{R}^n)$. Then, the sequence of functionals $E_N$ $\Gamma$-converges to $E$ for the weak topology on $L^2([0,1],V)$. Consequently, if $v_N$ is a global minimizer of $E_N$ for each $N\geq 1$, then $(v^N)$ is bounded in $L^2([0, 1],V)$ and every cluster point for the weak topology of $L^2([0,1],V)$ is a global minimum of $E$.
We first show that whenever $v^N$ converges to $\bar{v}$ weakly in $L^2([0,1],V)$, we have $$\begin{aligned}
E(\bar{v}) \leq \liminf_{N \rightarrow \infty} E_N(v^N).\end{aligned}$$ Since $v \mapsto \int_0^1 \|v\|_{V}^2 dt$ is lower semicontinuous with respect to the weak topology, we only need to prove the following, $$\begin{aligned}
\label{eq:conv_deform_1}
\lim_{N \rightarrow \infty} \|(\varphi_1^{v^N})_{\#} \mu_{K,N} - \varphi_1^{\bar{v}} \cdot \mu_0 \|_{W^*} = 0.\end{aligned}$$ For all $\omega \in W$ with $\| \omega \|_{W} \leq 1$, we have
$$\begin{aligned}
\left|\left((\varphi_1^{v^N})_{\#} \mu_{K,N} - (\varphi_1^{v^N})_{\#} \mu_0 | \omega \right) \right|
&= \bigg{|} \int_{K \times \tilde{G}^n_d} J_S \varphi_1^{v^N}(x)
\omega (\varphi_1^{v^N}(x),d_x \varphi_1^{v^N} \cdot S ) d (\mu_{K,N} - \mu_0) \bigg{|} \\
&\leq C_1 \int_{K \times \tilde{G}^n_d} \sup_{N \geq 1} J_S \varphi_1^{v^N} d (\mu_{K,N} - \mu_0) \\
&\leq C_1 \int_{\mathbb{R}^n \times \tilde{G}^n_d} g(x,T) d (\mu_{K,N} - \mu_0),\end{aligned}$$
where $g \in C_c(\mathbb{R}^n \times \tilde{G}^n_d)$ and $\sup_{N \geq 1} J_S \varphi_1^{v^N} \leq g(x,T)$, for all $(x,T) \in K \times \tilde{G}^n_d$. Similar to the computation done in the proof of Theorem \[thm:exist\_opt\_control\], we see that
$$\begin{aligned}
&\left|\left((\varphi_1^{v^N})_{\#} \mu_0 - (\varphi_1^{\bar{v}})_{\#} \mu_0 | \omega \right) \right|
\leq C_2 \| (\varphi_1^{v^N} - \varphi_1^{\bar{v}})|_K \|_{1,\infty}.\end{aligned}$$
Taking supremum over all $\omega \in W$ with $\|\omega\|_W \leq 1$, we obtain the following inequality, $$\begin{aligned}
&\|(\varphi_1^{v^N})_{\#} \mu_{K,N} - (\varphi_1^{\bar{v}})_{\#} \mu_0 \|_{W^*}
\leq C_1 (\mu_N-\mu| g)
+C_2 \| (\varphi_1^{v^N} - \varphi_1^{\bar{v}})|_K \|_{1,\infty}.\end{aligned}$$ From Proposition \[prop:nice\_app\_seq\], $\mu_{K,N}$ converges to $\mu_0$ in the narrow topology. Hence the right hand side in the equation above converges to $0$ as $N \rightarrow \infty$. This proves .
Second, we need to show that for each $\bar{v} \in L^2([0,1],V)$, there exists a sequence $v^N$ converging to $\bar{v}$ weakly such that $$\begin{aligned}
E(\bar{v}) \geq \limsup_{N \rightarrow \infty} E_N(v^N).\end{aligned}$$ In fact, it suffices here to take $v^N$ to be the constant sequence $v^N = \bar{v}$ since, by a similar argument to the proof of , it leads to $$\begin{aligned}
\label{eq:conv_deform_2}
\lim_{N \rightarrow \infty}\|(\varphi_1^{\bar{v}})_{\#} \mu^N - (\varphi_1^{\bar{v}})_{\#} \cdot \mu \|_{W^*} = 0
\end{aligned}$$ and thus implies that $$\begin{aligned}
\limsup_{N \rightarrow \infty} E_N(v^N) = \lim_{N \rightarrow \infty} E_N(\bar{v}) = E(\bar{v}).
\end{aligned}$$
Note that we stated the result of Theorem \[thm:convergence\_sol\] in the situation where only the source varifold $\mu_0$ is approximated by the projection approach that we presented in the previous sections but one can easily extend it to the scenario in which both source and target are replaced by discrete approximating sequences, the conclusion being the same in that case.
Numerical considerations {#sec:numerics}
========================
Having introduced a variational formulation for the varifold registration problem together with an approach for projecting onto the space of discrete varifolds with fixed number of Diracs, we now turn more specifically to the numerical implementation of methods for solving those problems. The first hurdle, which we start by addressing in Section \[ssec:frame\_compression\], is to define an adequate framework for representing and computing with elements of the oriented Grassmannian.
Frame representation for metric computation and quantization {#ssec:frame_compression}
------------------------------------------------------------
In order to come up with a computationally effective representation of $\widetilde{G}_d(\R^n)$ and by extension of discrete oriented varifolds, we consider a slightly different setting than the Plücker embedding idea of Remark \[rem:Grassmannian\], primarily because the dimension of the embedding vector space $\Lambda^d(\R^n)$ may become prohibitively large in practice. We may instead choose to represent an element $T \in \widetilde{G}_d(\R^n)$ by an oriented frame $(u^{(1)},\ldots,u^{(d)}) \in \R^{n \times d}$ of independent vectors for which $T=\text{Span}(u^{(1)},\ldots,u^{(d)})$. Such a representation is of course not unique since elements of $\widetilde{G}_d(\R^n)$ are equivalence classes of oriented frames but we leave to the next section the more thorough analysis of the additional invariances that this representation will imply.
We will in fact go one step further by also incorporating the weight of Dirac varifolds in this frame representation itself, which is done as follows. Let $\mu$ be a discrete varifold of the form $\mu = \sum_{i=1}^{N} r_i \delta_{(x_i,T_i)}$. For each $i$, we consider a frame $\{u_i^{(1)},\cdots,u_i^{(d)}\}$ such that $$\label{eq:frame_representation_mu}
T_i = \frac{u_i^{(1)} \wedge \cdots \wedge u_i^{(d)}}{|u_i^{(1)} \wedge \cdots \wedge u_i^{(d)}|} \textrm{ and } r_i = |u_i^{(1)} \wedge\cdots\wedge u_i^{(d)}|.$$ In other words, the oriented space spanned by the frame $\{u_i^{(1)},\cdots,u_i^{(d)}\}$ corresponds to $T_i$ while its $d$-volume matches the weight $r_i$. Given such a choice of frame for each $i$, we can then identify $\mu$ with the (non-unique) state variable $q = (x_i,u^{(1)}_i,\cdots,u^{(d)}_i)_{i=1,\cdots,N}$ in the vector space $\mathbb{R}^{Nn(d+1)}$. Conversely, such a frame $q$ with $(u^{(1)}_i,\cdots,u^{(d)}_i)$ a matrix of rank $k$ for all $i$, corresponds to the (unique) discrete oriented varifold defined by the relations of ; we will denote it by $\mu^q$ in what follows.
In this representation, the kernel metrics for discrete varifolds expressed in can be explicitly written as $$\label{eq:norm_discrete_varifolds_frame}
\langle \mu, \mu' \rangle_{W^*}^2 =\sum_{i=1}^N\sum_{j=1}^M r_i r'_j \rho(|x_i-x'_j|^2) \gamma\left(\frac{1}{r_i r'_j}\det(u_i^{(k)} \cdot {u'}_j^{(l)})_{k,l}\right)$$ where $r_i = |u_i^{(1)} \wedge\cdots\wedge u_i^{(d)}| = \sqrt{\det(u_i^{(k)} \cdot u_i^{(l)})_{k,l}}$. Note that this expression does not depend on the choice of frames that satisfy the conditions of for $\mu$ (and similarly for $\mu'$). In the case where $\mu'$ is a more general non-discrete varifold in $\mathcal{V}_d$, the computation of $\langle \mu, \mu' \rangle_{W^*}^2$ involves integrals over $\R^n \times \widetilde{G}_d(\R^n)$ of the kernel functions, which requires introducing specific quadrature schemes for approximating them. We do not address those issues in more details in this work as it needs particular discussion depending on the nature, regularity and dimension of the varifolds under consideration. Provided such adequate quadrature schemes have been defined, the $W^*$ metric then formally reduces to an expression equivalent to in which the $x_j',u'_j$ and $r'_j$ are now the quadrature nodes and associated weights of the scheme.
In this setting, the solution to the projection problem can be computed by an iterative descent strategy on the vector $q=(x_i,u^{(1)}_i,\cdots,u^{(d)}_i)_{i=1,\cdots,N}$. The gradient of $q \mapsto \|\mu^q-\mu_*\|_{W^*}^2$ can be computed by direct differentiation of expressions like with respect to the $x_i$ and $u_i^{(l)}$. In practice, computations of varifold kernel metrics for different classes of kernels and gradients of the metrics can be conveniently implemented with automatic differentiation pipelines. In our MATLAB implementation, we make use of the recent KeOps library which allows to generate CUDA functions for the low-level kernel sum evaluations and their automatic differentiation. The optimization itself is done using a limited memory BFGS algorithm from the HANSO library which we typically initialize by taking a random subset of $N$ Diracs composing the varifold $\mu_*$. Note that one of the main downside of this projection algorithm, in contrast with the previously mentioned approach of fixing a dictionary and solving a convex sparse decomposition problem, is that we can provide no general guarantees of convergence to a global minimum of . Results of this algorithm are discussed below in Section \[ssec:results\_approx\_reg\].
Discrete registration model {#ssec:discrete_registration}
---------------------------
This frame representation also provides a convenient setting to express the diffeomorphism action and registration problem on discrete varifolds. Indeed, let $\varphi$ be a diffeomorphism of $\R^n$ and $\mu \in \mathcal{V}_d^N$, the pushforward action $\varphi_{\#} \mu$ in is equivalent to the following action in the frame model: $$\begin{aligned}
\varphi_{\#} q := (\varphi(x_i),d_x \varphi (u^{(1)}_i),\cdots,d_x \varphi (u^{(d)}_i))_{i=1,\cdots,N} .\end{aligned}$$ Now, this allows us to rewrite the former infinite-dimensional optimal control problem by considering instead the finite-dimensional state variable $q \in \R^{Nn(d+1)}$. In the next paragraphs, we give a direct derivation of the optimality conditions in this discrete setting, in order to arrive at simpler and more explicit equations than the general abstract derivations presented in Section \[ssec:general\_PMP\]. Note that the resulting Hamiltonian equations we obtain are eventually very similar to the ones appearing in the 1st-order jets model studied in , although there are a few notable differences due to the specific extra invariances attached to the varifold framework (c.f. for a more detailed discussion in the $d=1$ case).
Following once again the Pontryagin maximum principle approach, the Hamiltonian for this discrete representation is given by: $$\begin{aligned}
&H(q,p,v) \doteq
\sum_{i=1}^N \left[ p_i^{x}\cdot v(x_i) + \sum_{k=1}^d p_i^{u_k}\cdot d_{x_i} v(u_i^{(k)}) \right] - \frac{1}{2}\|v\|^2_V\end{aligned}$$ with $p^x,p^{u_k} \in \R^n$ denoting respectively the costates for the position $x$ and frame vector $u^{(k)}$ variables. The PMP then shows that optimal trajectories of the registration problem are governed by the dynamical system: $$\begin{aligned}
\label{eq:ham_eq_discrete}
\left\{ \begin{array}{l}
\dot{x}_i = v_t(x_i) \\
\dot{u}_i^{(k)} = d_{x_i} v(u_i^{(k)}) \\
\dot{p}_i^{x} = - d_{x_i}v^T p_i^{x} - \sum_{k=1}^d d_{x_i}^{(2)}v(\cdot,u_i^{(k)})^T p_i^{u_k} \\
\dot{p}_i^{u_k} = - d_{x_i}v^T p_i^{u_k} \\
\end{array} \right.\end{aligned}$$ while optimal vector fields $v$ satisfy $$\begin{aligned}
\label{eq:optimal_discrete_v}
v_t(\cdot) &= \sum_{i=1}^N K(x_i(t),\cdot)p^x_i(t)
+ \sum_{j=1}^d \partial_1 K(x_i(t),\cdot)(u_i^{(j)}(t)) \cdot p_i^{u_j} . \end{aligned}$$ Plugging this expression of $v$ in the above Hamiltonian system gives the reduced Hamiltonian equations which is a coupled system in the variables $q$ and $p$. Consequently, the set of optimal paths is entirely determined by the initial values $(q(0),p(0))$.
There are in addition several conserved quantities in such a system. One of those is naturally the reduced Hamiltonian function. But in this particular case, we have actually many others as shown by the following lemma.
\[lemma:conservation\_forward\_eq\] For any $i=1,\ldots,N$, the matrix $$\begin{aligned}
D^i(t) \doteq \left( \langle u_i^{(k)}(t), p_i^{u_{\ell}}(t) \rangle \right)_{1 \leq k, \ell \leq d},\end{aligned}$$ is constant in time.
Using the Hamiltonian equations written above, we have for all $k,l=1,\ldots,d$ $$\frac{d}{dt} \left(D^i(t) \right)_{k,\ell} = \langle d_{x_i}v(u_i^{(k)}(t)), p_i^{u_{\ell}} \rangle - \langle u_i^{(k)}, d_{x_i}v^T p_i^{u_{\ell}} \rangle = 0.$$ Hence $D^i(t)$ is a constant matrix.
Note that, at this point, all those equations are fundamentally modelling the deformation of the frames $\{x_i,(u_i^{(k)})\}$ but are not yet taking into account the invariances that result from the representation of the discrete oriented varifolds as oriented frames. Those extra invariances can be derived from the boundary conditions of the PMP: $$\label{eq:bd_condition_discrete_PMP}
p(1) = - \partial_q g(q)|_{q=q(1)}, \ \text{with} \ g(q) = \lambda \|\mu^q - \mu_{tar}\|_{W^*}^2.$$ As a clear consequence of , $\mu^q$ and thus $g(q)$ are independent of the choices of the frame vectors $(u_i^{(k)})_{k=1,\dots,d}$ that span the same oriented vector spaces $T_i$ with the same $d$-volumes $r_i$. This in turn leads to a set of conditions satisfied by the different components of the final costate $p(1)$ and, with Lemma \[lemma:conservation\_forward\_eq\], of the full path $p(t)$. These are summed up by the following result:
\[prop:invariance\_momentum\] Let $(q(t),p(t))$ be optimal trajectory, then for all $i$, the matrices $D^i(t)$ as defined above are constant scalar matrices. In particular, we have $p_i^{u_k}(t) \perp \text{Span}(\{u_i^{(\ell)}(t)\}_{\ell\neq k})$ for all $t \in [0,1]$, $i=1,\ldots,N$ and $k=1,\ldots,d$.
This result, which proof can be found in Appendix, is particularly interesting from a computational point of view as it allows to partly alleviate the redundancy introduced by the frame representation of Grassmannians. Indeed, we see that the costates $p(t)$ actually lie in affine subspaces of $\R^{Nn(d+1)}$ of lower dimensions $N(n + d(n-d))$, which is precisely the dimension of the ’true’ state space $(\R^n \times \widetilde{G}_d(\R^n))^N$.
Registration algorithm {#ssec:registration_algorithm}
----------------------
Based on the optimality equations of the previous section, we can now easily design an algorithm to solve the discrete registration problem. As mentioned earlier, optimal trajectories are completely determined, through the Hamiltonian equations and , by the initial conditions $q(0)=q_0$, which is known, and $p(0)$. One of the standard class of methods in optimal control, known as *shooting methods*, consist in directly optimizing the cost function over $p(0)$, which has been the approach of choice in many past works on shape registration such as . We adopt a similar strategy for our particular problem.
The main issue is to compute the gradient of the fidelity metric $g(q(1))$ with respect to the initial costate $p(0)$. As standard for this type of optimal control problems, c.f. or , this can be computed by flowing backward in time the adjoint Hamiltonian system with end-time condition $-\partial_q g(q)|_{q=q(1)}$. Then, at high level, our registration algorithm consists of essentially the same steps as the aforementioned works:
From $(q(0),p(0))$ compute $(q(t),p(t))$ by forward integration of the reduced Hamiltonian system given by and . Compute $g(q(1))$ and $-\partial_q g(q)|_{q=q(1)}$. Integrate backward the adjoint Hamiltonian equations to obtain $\partial_{p(0)} g(q(1))$. Deduce the gradient of the full cost function with respect to $p(0)$. Update $p(0)$.
For the numerical ODE integration steps of lines 2 and 4, we use a standard RK4 scheme with regular time samples in $[0,1]$. The optimization update in line 6 follows the limited memory BFGS algorithm. One can take additional advantage of the dimensionality reduction provided by Proposition \[prop:invariance\_momentum\] by restricting each of the components $p_i^{u_k}(0)$ to the linear subspace $\text{Span}(\{u_i^{(\ell)}(0)\}_{\ell\neq k})^{\perp}$. Lastly, as in Section \[ssec:frame\_compression\], all kernel summation and differentiation operations appearing in both the varifold fidelity terms and Hamiltonian equations are coded in CUDA using the KeOps library .
Results {#sec:results}
=======
We now present some results of the previous algorithms on discrete varifolds of dimension $d=1$ and $d=2$. In all these experiments, we choose the deformation kernel $K$ of $V$ to be a diagonal Gaussian kernel $K(x,y)=\exp(-\frac{|x-y|^2}{\sigma_V^2}) Id$. The kernel function $\rho$ is a Gaussian of scale $\sigma_{\rho}$. The choice of these scales is adapted to the sizes of the shapes in each of the experiment. We will not discuss these questions more in detail here, since this is not our main topic and it has been more thoroughly analyzed in previous works such as . The function $\gamma$ is chosen, depending on the situation, among the different classes of functions discussed in detail in , the main distinction being whether the considered varifolds are rectifiable or not according to the conditions given by Theorem \[thm:dist\_rectifiable\_var\] and Theorem \[thm:dist\_general\_var\]. All simulations are run on a desktop computer equipped with a NVIDIA Quadro P5000 graphics card.
-- ------- --------- ---------- ---------- ---------
$t=0$ $t=1/5$ $t=7/15$ $t= 4/5$ $t = 1$
-- ------- --------- ---------- ---------- ---------
Diffeomorphic registration
--------------------------
-- ------------------------------------ -- ---------------------------------------
2-varifold associated to curve set 2-varifold associated to mesh surface
-- ------------------------------------ -- ---------------------------------------
3ex
-- ------- --------- --------- --------
$t=0$ $t=0.3$ $t=0.6$ $t= 1$
-- ------- --------- --------- --------
We start with results of registration obtained from the algorithm of Section \[ssec:registration\_algorithm\]. In this section, we will mostly focus on examples involving 2-varifolds, the reader may refer to for additional examples in the case $d=1$. First, as a sanity check, we compare our 2-varifold registration approach applied to triangulated surfaces with the previous LDDMM mesh surface matching implementation of [@Charon2; @Charon2017] using the same kernel size parameters, in which case we expect both approaches to be theoretically equivalent as pointed out in the last paragraph of Section \[ssec:diffeom\_reg\]. Shown in Fig. \[fig:Amygdala\] are triangulated surfaces of amygdala segmented from two different subjects of the BIOCARD database , containing 563 vertices, 1122 triangles and 488 vertices, 972 triangles respectively. Following the simple procedure outlined at the beginning of Section \[ssec:discrete\_approx\], we obtain discrete 2-varifolds (one Dirac for each triangle). The first row in the figure shows the optimal deformation estimated with our approach through the evolution of the discrete varifold of the source shape (red) to the target varifold (blue). Discrete varifolds are here displayed in the form of tangent patches and normal vectors (instead of 2-frames) for the purpose of better visualization. Now, the estimated vector fields $v_t$ define a path of dense deformations of the full space which we can also apply to deform the original triangulated surface, which we show on the second row of Fig. \[fig:Amygdala\]. This is very comparable to the result of the surface mesh LDDMM registration approach displayed on the third row. In terms of computation times, the varifold registration takes a total of 494s (0.99s per iteration of BFGS) against 92.5s (0.18s per iterations) for the surface LDDMM algorithm. This difference comes from mainly two factors: the fact that the numerical complexities are quadratic in the number of Diracs (i.e. triangles) for varifold matching as opposed to the number of vertices for surface LDDMM, and from the increased dimensionality of the Hamiltonian systems in our model.
In Fig. \[fig:heart\_curves\_surf\], we consider a more challenging registration scenario which was originally studied in . Here, one of the two shape is a triangulated surface of a heart membrane segmented from high resolution CT imaging while the second one only consists of a sparse set of cross-sectional curves of the heart contour obtained from lower resolution clinical cardiac MRI data. The varifold framework of this paper leads to an alternative registration approach to the one proposed in that relies on a tailored closest point fidelity cost for the surface to curve set comparison. In our case, we instead represent both shapes as 2-varifolds and register them using the exact same varifold registration algorithm as in the previous example. The triangulated surface is again associated to a discrete 2-varifold in the same way as above. As for the set of cross-sectional curve set, we first obtain its 1-varifold representation $\{x_i,u_i^{(1)}\}$ which involve the tangent vectors $u_i^{(1)}$ to the curve that passes through $x_i$. We then complete it into a 2-varifold by adding a second “vertical” (i.e. inter-sectional) frame vector $u_i^{(2)}$, which can be estimated in this case by simply finding the projection of $x_i$ onto the corresponding curve in the section immediately above (note that this does involve any attempt to estimate an actual surface mesh of the data). We show the 2-varifolds associated to each shape in the first row of Fig. \[fig:heart\_curves\_surf\] and as well as the result of the 2-varifold registration both from curve set to surface and surface to curve set. In each case, we have again applied the estimated deformation between varifolds on the original shapes for visualization.
-- -- --
-- -- --
Along the same lines, we finally look into the case of even less structured data objects. Specifically, we consider two point clouds which are obtained as noisy samples drawn around two groundtruth surfaces, as displayed on the first row of Fig. \[fig:pt cloud bunny\]. A first possible registration approach could be to treat such point clouds as standard measures of $\R^3$ (i.e. 0-varifolds) and follow the simple point distribution LDDMM algorithm for unlabelled point sets proposed in . The result shown on the third row of Fig. \[fig:pt cloud bunny\] illustrates the shortcomings of such a model for this type of data. Indeed, one can see that, in the absence of any tangential information, many details of the target shape are not well-recovered. Furthermore, this point set model is not robust to sampling changes and imbalances which results in the mismatches observed below the ear region. An arguably more adequate method would be to exploit the fact that these point clouds are close to their underlying surfaces. However, due to noise and the presence of outliers, estimating triangulations of the point clouds with standard meshing algorithms can prove particularly challenging and inefficient. Instead, our approach consists in directly learning the 2-varifold structure from the point clouds based on the geometric multi-resolution analysis (GMRA) framework developed in . Here, we fix a specific scale and GMRA then provides local partitions with estimates of tangent planes to the point clouds which eventually gives us an approximate representation as a 2-varifold illustrated on the second row of Fig. \[fig:pt cloud bunny\]. Besides its robustness and numerical efficiency, such manifold learning algorithm is also particularly well suited for our proposed registration framework since it naturally leads to approximations in the form of 2-varifolds (and generally not meshes). In the last row of Fig. \[fig:pt cloud bunny\], we show the deformed point cloud resulting from the deformation estimated by the 2-varifold registration algorithm. It obviously outperforms the direct point cloud registration described above both in terms of quality of matching but also computation time (10 mins vs 39 mins in total).
--------------------------------------------------- --------------------------- ----------------------- ---------------------------
{width="4.9cm"}
$N=25$, rel err=$12.19\%$ $N=40$, rel err=$1\%$ $N=150$, rel err=$0.01\%$
--------------------------------------------------- --------------------------- ----------------------- ---------------------------
3ex
-------------------------------------------------------------- -------- -------- ---------
{width="4.9cm"}
$N=25$ $N=40$ $N=150$
-------------------------------------------------------------- -------- -------- ---------
Approximation and registration {#ssec:results_approx_reg}
------------------------------
In this second part, we examine some results of the varifold quantization procedure proposed in Section \[sec:approximation\_discrete\_var\], and in particular its interplay with the registration algorithm. Specifically, we wish to numerically validate the statements of Corollary \[cor:var\_approx\_W\] and Theorem \[thm:convergence\_sol\]. We shall consider the following protocol. Starting from a highly sampled shape (that we treat as the groundtruth) for which the associated varifold $\mu_0$ is composed of a very high number of Diracs, we compute the compressed varifolds given by the $\mu_N$ of for increasing values of $N$ and evaluate the resulting quantization error in terms of the $d_{W^*}$ metric. Then we solve the registration problems to a fixed target $\mu_{tar}$ from the source varifolds given by the $\mu_N$ in lieu of $\mu_0$, and compare the estimated solutions to the registration of the groundtruth. For comparison, we will evaluate the total energy $E(v^N)$ of the estimated deformation fields $v^N$ for the original problem, i.e. $$E(v^N) = \int_0^1 \|v^N_t\|_{V}^2 dt + \lambda \|(\varphi_1^{v^N})_{\#} \mu_0 - \mu_{tar} \|_{W^*}^2.$$ We shall also compare this overall approach against the alternative idea of directly subsampling the original meshes and registering those subsampled shapes with point set mesh LDDMM.
-- ---------------------------------- -- ----------------------------------
Source surface (42448 triangles) Target surface (50352 triangles)
-- ---------------------------------- -- ----------------------------------
3ex
-- ----------------------------------------------------- ------------------------ -------------------------- ---------------------------
{width="5.5cm"}
Relative quantization error plot $N=65$, rel err=$28\%$ $N=125$, rel err=$7.1\%$ $N=375$, rel err=$0.07\%$
-- ----------------------------------------------------- ------------------------ -------------------------- ---------------------------
3ex
---------------------------------------------------------
{width="7cm"}
Total registration energy differences: $E(v^n)-E(v^*)$.
---------------------------------------------------------
We begin with a 1-varifold toy example given by the curves shown in Fig. \[fig:Bone\_Bottle\_quantization\_registration\] from the Kimia database. These very simple curves segmented from binary images have a relatively high number of points to start with (368 vertices and edges). We look first at how well they can be approximated with smaller number of Diracs through the quantization approach described above. The upper row shows the plot of the relative approximation error $\|\mu_N - \mu_0\|_{W^*}/\|\mu_0\|_{W^*}$ of the source curve as a function of $N$ (blue) as well as the same error in varifold norm when instead the curve is uniformly subsampled (green). Consistent with the fact that varifold quantization should provide the optimal error rate at a given $N$, we observe that the error is indeed smaller than with the subsampling approach. We also display a few of the quantized $\mu_N$ for several values of $N$. As a second step, we compute the optimal deformations from the reduced shapes to the fixed target and compare their registration energies to the “groundtruth” $E(v^*)$ estimated from the full resolution source shape. The corresponding plots for the quantization versus subsampling methods are shown on the lower row in blue and green respectively. It suggests again a faster convergence to the optimal energy $E(v^*)$ with the quantization strategy, although the difference between the two methods is rather tenuous in this example.
Those effects can be much more significant in the two-dimensional case. We emphasize it with the triangulated heart surfaces of Fig. \[fig:heart\_quantization\_registration\] (data courtesy of C. Chnafa, S. Mendez and F. Nicoud, University of Montpellier). The source surface has a total of 42448 triangles leading to the same number of Diracs for the source 2-varifold $\mu_0$ and thus compressing the representation may be in that case quite critical from a computational standpoint. Indeed, computing the groundtruth matching at full resolution takes more than 7 hours (68s per iteration) in this case. We again compare two approaches: our quantization algorithm applied to $\mu_0$ versus directly subsampling the triangulated surface itself (we use here the reducepatch function in MATLAB to reduce the initial mesh to a given number of triangles). For both methods, we compute the relative approximation error $\|\mu_N - \mu_0\|_{W^*}/\|\mu_0\|_{W^*}$ with different values of $N$, the number of Diracs (resp. triangles) of the compressed varifold (resp. mesh). This is shown on the left second row in Fig. \[fig:heart\_quantization\_registration\]. Unsurprisingly, we see that the quantization approach leads to a much faster decrease in the error as a function of $N$ but that in addition we obtain a very good approximation of $\mu_0$ with only a small fraction of the initial number of Diracs. Some of the quantized varifolds $\mu_N$ are displayed in the figure. We also evaluate how well the solution of the registration problem to the target varifold or surface can be approximated based on the quantized source shapes. With $v^*$ being a numerical solution for the groundtruth and $v^N$ the solutions based on the quantized source shapes, the third row of Fig. \[fig:heart\_quantization\_registration\] shows the difference of the energies $E(v^N)-E(v^*)$. We observe again a faster convergence towards the groundtruth optimal energy with the varifold quantization than with mesh subsampling.
Discussion
==========
In this paper, we proposed a registration framework between varifolds that goes beyond the previous restrictions of such models to the registration of discrete or smooth submanifolds of $\R^n$. To achieve so, we studied a general class of distances between oriented varifolds based on reproducing kernels and derived a deformation model on the space $\mathcal{V}_d$, which are combined into an optimal control formulation of the registration problem between any two varifolds. We also examined the possibility to couple this approach with a quantization/compression methodology in order to eventually tackle the registration problem, in practice, on discrete varifolds with a relatively low number of Dirac masses.
We showed that first of all this setting leads to an equivalent yet alternative formulation to the diffeomorphic registration of rectifiable sets such as continuous or discrete curves and surfaces; the resulting higher-order Hamiltonian systems in our model provides richer local patterns for the deformations but at the price of a higher numerical cost. From an application standpoint, however, the main advantage we expect from this framework is that it applies very naturally to more general geometric objects, in particular to typical situations where well-defined and reliable meshes are not available. We gave a taste of it through some of the examples of Section \[sec:results\], although future work on a larger scale will be needed in order to evaluate such benefits more thoroughly. Besides the cases mentioned here, there are also several types of data that could constitute interesting test applications for this setting. This includes for instance high-angular resolution diffusion MRI in which the data is effectively modeled as spatially distributed orientation probability distribution functions consistent with the Young measure representation of varifolds in , or the case of contrast-invariant image registration c.f. .
At the theoretical level, there are several questions left open by this work which we believe can constitute interesting tracks for future work. One is to study the possibility of extending all or part of the results of Section \[sec:approximation\_discrete\_var\] to more general kernel metrics (in particular currents) and determining tighter quantization error bounds. Moreover, the registration model at play in this paper is based on the pushforward group action of $\text{Diff}(\R^n)$ on $\mathcal{V}_d$. Yet, other group actions could be have been considered, as briefly evoked in Section \[ssec:deformation\_models\], that involve different choices of reweighing factor, for which we could expect very different properties of the solutions to the registration problem.
Lastly, some additional work on the numerical side is likely needed for potential future applications to large scale databases, most notably to generalize this work to the estimation of means and atlases over populations of many high resolution shapes. Indeed, as we pointed out, even with the ability to compress the size of varifolds in the registration pipeline using the quantization approach, the higher complexity of the dynamical equations involved in the registration model has a non-negligible numerical toll. This could be improved in the future by using more efficient computational schemes for the repeated evaluations of sums of kernels and derivative of kernels appearing in the Hamiltonian equations, possibly along the lines of fast multiple methods.
Acknowledgements {#acknowledgements .unnumbered}
================
The authors would like to thank Benjamin Charlier, Siamak Ardekani, Laurent Younes and the BIOCARD team for sharing the data used in some of the examples of this paper. This work was supported by NSF grant No 1819131.
Appendix {#appendix .unnumbered}
========
**Proof of Theorem \[thm:dist\_rectifiable\_var\]** 1ex We first prove that $\mathcal{H}^d(X\bigtriangleup Y) =0$. Let us denote by $W^{pos}$ and $W^G$ the RKHS associated to kernels $k^{pos}$ and $k^{G}$ respectively. Suppose that $X$ and $Y$ are rectifiable sets as above such that $\left\| \mu_X-\mu_Y \right\|_{W^*}=0$ and $\mathcal{H}^d(X\bigtriangleup Y) >0$. Without loss of generality, we may assume that $\mathcal{H}^d(X \setminus Y) >0$. From Lusin’s theorem, there exists a subset $U$ of $X$ such that $T|_{U}$ is continuous and $\mathcal{H}^d(X\setminus U) < \mathcal{H}^d(X\setminus Y)$. Let us denote by $E := U \cap (X \setminus Y)$, we see that $\mathcal{H}^d(E)>0$. Since for $\mathcal{H}^d \ a.e. \ x\in E$,
$$\begin{aligned}
\limsup\limits_{r \rightarrow 0} \frac{\mathcal{H}^d(B_r(x)\cap E )}{\frac{\pi^{\frac{d}{2}}}{\Gamma(\frac{d}{2}+1)}r^d} \geq \frac{1}{2^d},\end{aligned}$$
(cf [@evans2018measure]), there exists $x_0 \in E, \ \mathcal{H}^d(B_r(x_0) \cap E)>0$ for any $r >0$.
Let $g:\widetilde{G}^n_d \rightarrow \R$ be defined by $g(\cdot) = k^G(T(x_0),\cdot)$. Since $x \longmapsto g(T(x))$ is continuous on $E$ and $g(T(x_0))>0$, there exists $r_0>0$ such that $\forall \ x \in B_{r_0}(x_0) \cap E, \ g(T(x))>0$. Let $A \doteq B_{r_0}(x_0) \cap E$ and $h(x) := \mathbf{1}_A(x)$, then $\mathcal{H}^d(A)>0$ and $g(T(x))>0, \ \forall \ x \in A$. Using the density of $C_c(\mathbb{R}^n)$ in $L^1(\mathbb{R}^n,\mathcal{H}^d {\mathbin{\vrule height 1.6ex depth 0pt width
0.13ex\vrule height 0.13ex depth 0pt width 1.3ex}}(X \cup Y))$ together with the fact that $k^{pos}$ is $C_0$-universal, there exist $\{f_j\}_{j=1}^{\infty} \subset C_c(\mathbb{R}^n)$ and $\{h_j\}_{j=1}^{\infty} \subset W^{pos}$ such that $\lim\limits_{j \rightarrow \infty} f_j =h$ in $L^1(\mathbb{R}^n,\mathcal{H}^d {\mathbin{\vrule height 1.6ex depth 0pt width
0.13ex\vrule height 0.13ex depth 0pt width 1.3ex}}(X \cup Y))$ and $\|f_j-h_j \|_{\infty}< \frac{1}{j}$. Now, since $h_j \otimes g \in W$ and $\mu_X = \mu_Y$ in $W^*$, we have
$$0 = (\mu_X-\mu_Y)(h_j \otimes g) = \int_{X} h_j(x) g(T(x)) d \mathcal{H}^d(x) - \int_{Y} h_j(y) g(S(y)) d \mathcal{H}^d(y) \rightarrow \int_A g(T(x)) d \mathcal{H}^d(x) >0,$$
which is a contradiction. Hence we have $\mathcal{H}^d(X\bigtriangleup Y) = 0$
Next, we show that $T(x) = S(x)$ $\mathcal{H}^d$-$a.e.$. Let $F := \{x \in X | T(x) = -S(x)\}$ and assume that $\mathcal{H}^d(F)>0$. From Lusin’s theorem, there exists subset $F' \subset F$ such that $T|_{F'}$ is continuous and $\mathcal{H}^d(F')>0$. Using the upper density argument as above, we can find $z_0 \in F'$ such that $\mathcal{H}^d(B_r(z_0) \cap F')>0$ for all $r>0$. Since the map $x \mapsto \langle T(x),T(z_0) \rangle$ restricted to $F'$ is continuous, there exists a $\delta_0>0$ satisfying: $$\begin{aligned}
\langle T(x),T(z_0) \rangle >0, \ \forall x \in B_{\delta_0}(z_0) \cap F'.\end{aligned}$$ Define $B:= B_{\delta_0}(z_0) \cap F'$, $\eta(\cdot):= \gamma(\langle \cdot,T(z_0) \rangle)$ and $u(x) := \eta(T(x)) - \eta(S(x))$. Observe that, from the assumption $\gamma(t) \neq \gamma(-t), \ \forall t \in [-1,1]$, $$\begin{aligned}
u(x) = \eta(T(x)) - \eta(-T(x)) \neq 0, \ \forall x \in F'.\end{aligned}$$ From this, we may assume that $u(x)>0, \ \forall x \in F'$. Let $\{f_j'\}_j$ and $\{h_j'\}_j$ be sequences in $C_c(\mathbb{R}^n)$ and $W_{pos}$ such that $f_j'$ converges to $\mathbf{1}_B$ in $L^1(\mathbb{R}^n,\mathcal{H}^d {\mathbin{\vrule height 1.6ex depth 0pt width
0.13ex\vrule height 0.13ex depth 0pt width 1.3ex}}F)$ and $\|f_j'-h_j'\|_{\infty} < 1/j$. We obtain
$$0 = (\mu_X - \mu_Y| h_j' \otimes \eta) = \int_X h_j'(x) u(x) d\mathcal{H}^d(x) \rightarrow \int_{B} u(x) d\mathcal{H}^d(x) > 0,$$
which is impossible.
3ex
**Proof of Theorem \[thm:exist\_opt\_control\]** 1ex Thanks to the first term in $E$, any minimizing sequence of $E$ is bounded in $L^2([0,1],V)$. Let $\{v_j\}$ be a subsequence of such minimizing sequence which converges weakly to some $\bar{v}$ in $L^2([0,1],V)$. Using the results of [@Younes] Chapter 8.2, we know that $$\begin{aligned}
\lim_{j \rightarrow \infty} \|\varphi_1^{v_j} - \varphi_1^{\bar{v}} \|_{1,\infty} = 0.
\end{aligned}$$ Furthermore, for any $\omega \in W$, we have $$\begin{aligned}
\left|\left( (\varphi_1^{v_j})_{\#} \mu_0 - (\varphi_1^{\bar{v}})_{\#} \mu_0 | \omega \right) \right|
&=\bigg{|} \int_{K} J_S \varphi_1^{v_j}(x) \omega(\varphi_1^{v_j}(x),d_x \varphi_1^{v_j} \cdot S)
- J_S\varphi_1^{\bar{v}}(x) \omega(\varphi_1^{\bar{v}}(x),d_x \varphi_1^{\bar{v}} \cdot S) d \mu_0 \bigg{|} \\
&\leq \int_{K} |J_S \varphi_1^{v_j}(x)|
\left|\omega(\varphi_1^{v_j}(x),d_x \varphi_1^{v_j} \cdot S) - \omega(\varphi_1^{\bar{v}}(x),d_x \varphi_1^{\bar{v}} \cdot S) \right| d \mu_0 \\
&+ \int_{K} \left|J_S \varphi_1^{v_j}(x) - J_S\varphi_1^{\bar{v}}(x) \right| \left|\omega(\varphi_1^{\bar{v}}(x),d_x \varphi_1^{\bar{v}} \cdot S) \right| d \mu_0 \\\end{aligned}$$ Now, using the embedding $W \hookrightarrow C_0^1(\mathbb{R}^n \times \widetilde{G}^n_d)$ $$\begin{aligned}
\left|\left( (\varphi_1^{v_j})_{\#} \mu_0 - (\varphi_1^{\bar{v}})_{\#} \mu_0 | \omega \right) \right|
&\leq \left( \int_{K} |J_S \varphi_1^{v_j}(x)| d \mu_0 \right) \| \omega\|_{1,\infty} \| (\varphi_1^{v^N} - \varphi_1^{\bar{v}})|_K \|_{1,\infty}
+ C \| (\varphi_1^{v_j} - \varphi_1^{\bar{v}})|_K \|_{1,\infty} \\
&\leq C' \| (\varphi_1^{v_j} - \varphi_1^{\bar{v}})|_K \|_{1,\infty}.\end{aligned}$$ Taking supremum over all $\omega \in W$ with $\|\omega\|_W \leq 1$, we obtain that $$\begin{aligned}
\|(\varphi_1^{v_j})_{\#} \mu_0 - (\varphi_1^{\bar{v}})_{\#} \mu_0 \|_{W^*} \leq C' \| (\varphi_1^{v_j} - \varphi_1^{\bar{v}})|_K \|_{1,\infty} \rightarrow 0\end{aligned}$$ as $j \rightarrow \infty$. Combining this with lower semicontinuity of $v \mapsto \|v\|_{L^2([0,1],V)}^2$, we finally obtain that $$\begin{aligned}
E(\bar{v}) \leq \liminf_{j \rightarrow \infty} E(v_j)\end{aligned}$$ and hence $\bar{v}$ is a global minimizer.
3ex
**Proof of Proposition \[prop:variation\_g\]** 1ex Recall that for all $\phi \in \text{Diff}(\R^n)$, $g(\phi) = \lambda \|\phi_{\#} \mu_0 - \mu_{tar} \|_{W^*}^2$ which we may rewrite as $$g(\phi) = \lambda (\phi_{\#} \mu_0 | K_W(\phi_{\#} \mu_0 -2\mu_{tar})) + \lambda \|\mu_{tar}\|_{W^*}^2.$$ Thus, the variation with respect to $\phi$ in the Banach space $\mathcal{B}$ writes $$\partial_{\phi} g(\phi) = \partial_{\phi} (\phi_{\#} \mu_0 | \omega_0)$$ where $\omega_0 \doteq 2\lambda K_W(\phi_{\#} \mu_0 -\mu_{tar}) \in W$. Moreover $$(\phi_{\#} \mu_0 | \omega_0) = \int_{\R^n \times \widetilde{G}_d^n} \omega_0(\phi(x),d_x \phi \cdot T) J_T \phi(x) d \mu_0(x,T).$$ Taking the variation with respect to $\phi$ along any $u \in C_0^1(\R^n,\R^n)$, we obtain: $$\begin{aligned}
\label{eq:variation_g_1}
(\partial_{\phi} g(\phi) | u) &= \int_{\R^n \times \widetilde{G}_d^n} \partial_x \omega_0(\phi(x),d_x \phi \cdot T) \cdot u(x) J_T \phi(x) d \mu_0(x,T) \nonumber \\
&+\int_{\R^n \times \widetilde{G}_d^n} \partial_T \omega_0(\phi(x),d_x \phi \cdot T) \cdot (d_x u |_T) J_T \phi(x) d \mu_0(x,T) \nonumber \\
&+\int_{\R^n \times \widetilde{G}_d^n} \omega_0(\phi(x),d_x \phi \cdot T). \text{div}_T u(x). J_T \phi(x) d \mu_0(x,T)\end{aligned}$$ where the last term follows from the differentiation of Gram determinant matrices while the notation $\partial_T$ in the second term is a shortcut notation for differentiation on the Grassmannian which we do not explicit further here, we however refer to the similar computations done in and to the developments in Section \[sec:numerics\] for more details. For the first term, we can rely on the Young measure decomposition $\mu_0 = |\mu_0| \otimes \nu_x$ introduced at the end of Section \[subsec:defofvf\] which gives: $$(1) = \int_{\R^n} \tilde{\alpha}(\phi,x) \cdot u(x) \ d |\mu_0|(x), \ \ \text{where} \ \ \tilde{\alpha}(\phi,x) = \int_{\widetilde{G}_d^n} \partial_x \omega_0(\phi(x),d_x \phi \cdot T) J_T \phi(x) d\nu_x(T).$$ We can also rewrite the third term as: $$(3) = \int_{\R^n \times \widetilde{G}_d^n} \tilde{\gamma}(\phi,x,T) \ \text{div}_T u(x) \ d \mu_0(x,T), \ \ \text{with} \ \ \tilde{\gamma}(\phi,x,T) = \omega_0(\phi(x),d_x \phi \cdot T) \ J_T \phi(x).$$ As for the second term in , for each $(x,T)$ the integrand involves a linear combination (depending on $\phi$) of the partial derivatives of $u$ along the subspace $T$ i.e. of the elements of the matrix $d_x u|_T \in \R^{n\times d}$. Thus, without attempting to specify this term explicitly, we can in general write it as $\tilde{\beta}(\phi,x,T) d_x u|_T$ where $\tilde{B}$ is a continuous map from $\mathcal{B} \times \R^n \times \tilde{G}_d^n$ into $\mathcal{L}(\R^{n\times d},\R)$ giving us $$(2) = \int_{\R^n \times \widetilde{G}_d^n} \tilde{B}(\phi,x,T) d_x u|_T \ d \mu_0(x,T).$$ The result of the theorem then follows by setting $\alpha(x) \doteq \tilde{\alpha}(\varphi_1^v,x)$, $\beta(x,T) \doteq \tilde{\beta}(\varphi_1^v,x,T)$ and $\gamma(x,T)=\tilde{\gamma}(\varphi_1^v,x,T)$.
3ex
**Proof of Proposition \[prop:invariance\_momentum\]** 1ex We can treat the case of each particle $i$ separately and thus, without loss of generality, we may directly assume that $N=1$. We write $q(t)=(x(t),u^{(1)}(t),\cdots,u^{(d)}(t))$, $p(t)=(p^x(t),p^{u_1}(t),\ldots,p^{u_d}(t))$ for the state and costate variables along an optimal trajectory and $$\begin{aligned}
U \doteq \textrm{Span}\{u^{(1)}(1),\cdots,u^{(d)}(1)\}.\end{aligned}$$ Consider the group of linear transformations, $G \doteq {\rm SL} (U) \oplus {\rm GL} (U^{\perp})$, i.e., for any $\mathrm{g} \in G$, $$\begin{aligned}
\mathrm{g}(x) = \mathrm{g}_{{\mathbin{\!/\mkern-5mu/\!}}}(x_{U}) + \mathrm{g}_{\perp}(x_{U^{\perp}}),\end{aligned}$$ where $x_{U}$ and $x_{U^{\perp}}$ are the orthogonal projections of $x$ on $U$ and $U^{\perp}$, with $\mathrm{g}_{{\mathbin{\!/\mkern-5mu/\!}}} \in {\rm SL}(\Omega)$ and $\mathrm{g}_{\perp} \in {\rm GL}(\Omega)$. The Lie algebra of $G$ is $\mathfrak{g} = \mathfrak{sl}(U) \times \mathcal{L}(U^\perp)$ and $\mathfrak{sl}(U)$ is the set of all zero trace linear transformations of $U$. Now, consider the action of $G$ on $\mathbb{R}^{(d+1)n}$ defined as: $$\begin{aligned}
\mathrm{g} \cdot q := (q_0,\mathrm{g}(q_1),\cdots,\mathrm{g}(q_d)).\end{aligned}$$ for any $q=(q_0,\ldots,q_{d}) \in \mathbb{R}^{(d+1)n}$. We see that $\mu^{\mathrm{g}\cdot q(1)} = \mu^{q(1)}$ for all $\mathrm{g} \in G$ and therefore $g(\mathrm{g}\cdot q(1)) = g(q(1))$.
Now, if we let $\{\mathrm{g}_t\}$ be a smooth curve in $G$ that satisfies $\mathrm{g}_0 = id$ and $\frac{d}{d\tau}|_{\tau=0} \mathrm{g}_\tau = h \in \mathfrak{g}$, differentiating the equality $g(\mathrm{g}_\tau \cdot q(1)) = g(q(1))$ shows that for any $h \in \mathfrak{g}$, we have $$0 = (p(1)|h \cdot q(1)) = \sum_{k=1}^{d} \langle p^{u_k}(1) , h(u^{(k)}(1)) \rangle$$ Since $h \in \mathfrak{g}$, we must have that $h|_{U}$ is a zero trace linear map. For any $1 \leq i < j \leq d$, we may choose $h$ such that $h(u^{(i)}(1)) = -h(u^{(j)}(1))$ and $h(u^{(k)}(1)) = 0 , \ \forall k \notin\{i,j\}$, which leads to $\langle u^{(i)}(1), p^{u_i}(1) \rangle = \langle u^{(j)}(1), p^{u_j}(1) \rangle$. Consequently, $$\begin{aligned}
\langle u^1(1), p^{u_1}(1) \rangle = \cdots = \langle u^d(1), p^{u_d}(1) \rangle =\alpha \end{aligned}$$ for some constant $\alpha$. In addition, for any $i \neq j$, we can also choose $h$ such that $h(u^{(i)}(1)) = u^{(j)}(1)$ and $h(u^{(k)}(1))= 0 , \ \forall k \notin\{i,j\}$, which gives $\langle u^{(i)}(1),p^{u_j}(1) \rangle = 0$. It results that $D(1) = \alpha.I_{d \times d}$.
Finally, since $D(t)$ is constant by Lemma \[lemma:conservation\_forward\_eq\], we obtain that $$\begin{aligned}
D(t) = \left( \begin{array}{ccc}
\alpha & & 0 \\
& \ddots & \\
0 & & \alpha
\end{array} \right),\end{aligned}$$ for all $t \in [0,1]$.
|
---
abstract: 'In this paper, we prove that the position vector of every space curve satisfies a vector differential equation of fourth order. Also, we determine the parametric representation of the position vector $\psi=\Big(\psi_1,\psi_2,\psi_3\Big)$ of general helices from the intrinsic equations $\kappa=\kappa(s)$ and $\tau=\tau(s)$ where $\kappa$ and $\tau$ are the curvature and torsion of the space curve $\psi$, respectively. Our result extends some knwown results. Moreover, we give four examples to illustrate how to find the position vector from the intrinsic equations of general helices.'
author:
- |
Ahmad T. Ali\
Mathematics Department\
Faculty of Science, Al-Azhar University\
Nasr City, 11448, Cairo, Egypt\
email: atali71@yahoo.com
title: 'Determination of the position vectors of general helices from intrinsic equations in $\e^3$ '
---
*MSC:* 53C40, 53C50
*Keywords*: Classical differential geometry; Frenet equations; general helix; Intrinsic equations.
Introduction
=============
[*Helix*]{} is one of the most fascinating curves in science and nature. Scientist have long held a fascinating, sometimes bordering on mystical obsession, for helical structures in nature. Helices arise in nano-springs, carbon nano-tubes, $\alpha$-helices, DNA double and collagen triple helix, lipid bilayers, bacterial flagella in salmonella and escherichia coli, aerial hyphae in actinomycetes, bacterial shape in spirochetes, horns, tendrils, vines, screws, springs, helical staircases and sea shells [@choua; @lucas; @watson]. Also we can see the helix curve or helical structures in fractal geometry, for instance hyperhelices [@toledo]. In the field of computer aided design and computer graphics, helices can be used for the tool path description, the simulation of kinematic motion or the design of highways, etc. [@yang]. From the view of differential geometry, a helix is a geometric curve with non-vanishing constant curvature $\kappa$ and non-vanishing constant torsion $\tau$ [@barros]. The helix may be called a [*circular helix*]{} or [*W-curve*]{} [@ilarslan; @mont1].
Indeed a helix is a special case of the [*general helix*]{}. A curve of constant slope or general helix in Euclidean 3-space $\e^3$ is defined by the property that the tangent makes a constant angle with a fixed straight line called the axis of the general helix. A classical result stated by Lancret in 1802 and first proved by de Saint Venant in 1845 (see [@struik] for details) says that: [*A necessary and sufficient condition that a curve be a general helix is that the ratio $$\dfrac{\kappa}{\tau}$$ is constant along the curve, where $\kappa$ and $\tau$ denote the curvature and the torsion, respectively*]{}.
A general helices or [*inclined curves*]{} are well known curves in classical differential geometry of space curves [@milm] and we refer to the reader for recent works on this type of curves [@barros; @ba1; @gl1; @mont2; @sc; @tur2]. Many important results in the theory of the curves in $\e^3$ were initiated by G. Monge and G. Darboux pioneered the moving frame idea. Thereafter, F. Frenet defined his moving frame and his special equations which play important role in mechanics and kinematics as well as in differential geometry [@boyer]. For unit speed curve with non-vanishing curvature $\kappa \neq 0$, it is well-known the following result [@hacis]:
\[th-main\] A curve is defined uniquely by its curvature and torsion as function of a natural parameters.
The equations $$\kappa=\kappa(s),\,\,\,\,\,\tau=\tau(s)$$ which give the curvature and torsion of a curve as functions of $s$ are called the [*natural*]{} or [*intrinsic equations*]{} of a curve, for they completely define the curve.
Given two functions of one parameter (potentially curvature and torsion parameterized by arc-length) one might like to find an arc-length parameterized curve for which the two functions work as the curvature and the torsion. This problem, known as [*solving natural equations*]{}, is generally achieved by solving a [*Riccati equation*]{} [@struik]. Barros et. al. [@ba2] showed that the general helices in Euclidean 3-space $\e^3$ and in the three-sphere $\bold{S}^3$ are geodesic either of right cylinders or of Hopf cylinders according to whether the curve lies in $\e^3$ or $\bold{S}^3$, respectively.
In classical differential geometry, The problem of the determination of parametric representation of the position vector of an arbitrary space curve according to the intrinsic equations is still open [@eisenh; @lips]. This problem is solved in the case of a plane curve $(\tau=0)$ and in the case of circular helix ($\kappa$ and $\tau$ are both non-vanishing constants). However, This problem is not solved in the case of the general helix ($\dfrac{\tau}{\kappa}$ is constant).
Our main result in this work is to proven that the components of the position vector of every space curve satisfies a vector differential equation of forth order and determined the parametric representation of the position vector $\psi$ from intrinsic equations in $\e^3$ for a general helix $\dfrac{\tau}{\kappa}=\cot[\alpha]$, where the constant $\alpha$ is the angle between the tangent of the curve $\psi$ and the constant vector $\bold{U}$ called the axis of a general helix.
Preliminaries
==============
In Euclidean space $\e^3$, it is well known that to each unit speed curve with at least four continuous derivatives, one can associate three mutually orthogonal unit vector fields $\t$, $\n$ and $\b$ are respectively, the tangent, the principal normal and the binormal vector fields [@hacis].
We consider the usual metric in Euclidean 3-space $\e^3$, that is, $$\langle,\rangle=dx_1^2+dx_2^2+dx_3^2,$$ where $(x_1,x_2,x_3)$ is a rectangular coordinate system of $\e^3$. Let $\psi:I\subset\r\rightarrow\e^3$, $\psi=\psi(s)$, be an arbitrary curve in $\e^3$. The curve $\psi$ is said to be of unit speed (or parameterized by the arc-length) if $\langle\psi'(s),\psi'(s)\rangle=1$ for any $s\in I$. In particular, if $\psi(s)\not=0$ for any $s$, then it is possible to re-parameterize $\psi$, that is, $\alpha=\psi(\phi(s))$ so that $\alpha$ is parameterized by the arc-length. Thus, we will assume throughout this work that $\psi$ is a unit speed curve.
Let $\{\t(s),\n(s),\b(s)\}$ be the moving frame along $\psi$, where the vectors $\t, \n$ and $\b$ are mutually orthogonal vectors satisfying $\langle\t,\t\rangle=\langle\n,\n\rangle=\langle\b,\b\rangle=1$. The Frenet equations for $\psi$ are given by ([@struik]) $$\label{u2}
\left[
\begin{array}{c}
\t' \\
\n' \\
\b' \\
\end{array}
\right]=\left[
\begin{array}{ccc}
0 & \kappa & 0 \\
-\kappa & 0 & \tau \\
0 & -\tau & 0 \\
\end{array}
\right]\left[
\begin{array}{c}
\t \\
\n \\
\b \\
\end{array}
\right].$$
If $\tau(s)=0$ for any $s\in I$, then $\b(s)$ is a constant vector $V$ and the curve $\psi$ lies in a $2$-dimensional affine subspace orthogonal to $V$, which is isometric to the Euclidean $2$-space $\e^{2}$.
We observe that the Frenet equations form a system of three vector differential equations of the first order in $\t, \n$ and $\b$. It is reasonable to ask, therefore, given arbitrary continuous functions $\kappa$ and $\tau$, whether or not there exist solutions $\t, \n, \b$ of the Frenet equations, and hence, since $\psi^{\prime}=\t$, a curve $$\psi=\int\,\t\,ds+\bold{C}$$ which the prescribed curvature and torsion. The answer is in the affirmative and is given by
[**(Fundamental existence and uniqueness theorem for space curve).**]{}\[th-main\] Let $\kappa(s)$ and $\tau(s)$ be arbitrary continuous function on $a\leq s \leq b$. Then there exists, except for position in space, one and only one curve $C$ for which $\kappa(s)$ is the curvature, $\tau(s)$ is the torsion and $s$ is a natural parameter along $C$.
Position vectors of space curves
=================================
\[th-main\] Let $\psi=\psi(s)$ be an unit speed curve. Then, position $\psi$ satisfies a vector differential forth order as follows $$\label{u21}
\dfrac{d}{ds}\Big[\dfrac{1}{\tau}\dfrac{d}{ds}\Big(\dfrac{1}{\kappa}\dfrac{d^2\psi}{ds^2}\Big)\Big]+
\Big(\dfrac{\kappa}{\tau}+\dfrac{\tau}{\kappa}\Big)\dfrac{d^2\psi}{ds^2}+
\dfrac{d}{ds}\Big(\dfrac{\kappa}{\tau}\Big)\dfrac{d\psi}{ds}=0.$$
[**Proof.**]{} Let $\psi=\psi(s)$ be an unit speed curve. If we substitute the first equation of (\[u2\]) to the second equation of (\[u2\]), we have $$\label{u3}
\b=\dfrac{1}{\tau}\dfrac{d}{ds}\Big(\dfrac{1}{\kappa}\dfrac{d\t}{ds}\Big)+\dfrac{\kappa}{\tau}\t.$$ The last equation of (\[u2\]) takes the form $$\label{u4}
\dfrac{d}{ds}\Big[\dfrac{1}{\tau}\dfrac{d}{ds}\Big(\dfrac{1}{\kappa}\dfrac{d\t}{ds}\Big)\Big]+
\Big(\dfrac{\kappa}{\tau}+\dfrac{\tau}{\kappa}\Big)\dfrac{d\t}{ds}+
\dfrac{d}{ds}\Big(\dfrac{\kappa}{\tau}\Big)\t=0.$$ Denoting $\dfrac{d\psi}{ds}=\t$, we have a vector differential equation of fourth order (\[u21\]) as desired.
The equation (\[u4\]) can be written in the following simple form: $$\label{u5}
\dfrac{d}{d\theta}\Big(f\,\dfrac{d^2\t}{d\theta^2}\Big)+
\Big(\dfrac{f^2+1}{f}\Big)\dfrac{d\t}{d\theta}+
\dfrac{df}{d\theta}\t=0,$$ where $f=f(\theta)=\dfrac{\kappa(\theta)}{\tau(\theta)}$ and $\theta=\int\kappa(s)ds$. By means of solution of the above equation, position vector of an arbitrary space curve can be determined. However, for general helices, we have
\[th-main2\] The position vector of a general helix are computed in the natural parameter form $$\label{u211}
\psi(s)=\sin[\alpha]\int\Big(\cos\Big[\csc[\alpha]\int\kappa(s)ds\Big],\sin\Big[\csc[\alpha]\int\kappa(s)ds\Big], \cot[\alpha]\Big)ds+\bold{C}$$ or in the parametric form $$\label{u212}
\psi(\phi)=\int\dfrac{\sin^2[\alpha]}{\kappa(\phi)}\Big(\cos[\phi],\sin[\phi], \cot[\alpha]\Big)d\phi+\bold{C},\,\,\,\phi=\csc[\alpha]\int\kappa(s)ds.$$
[**Proof:**]{} If $\psi$ is a general helix whose tangent vector $\psi^{\prime}$ makes an angle $\alpha$ with the axis $U$, then we can write $f(\theta)=\tan[\alpha]$. Therefore the equation (\[u212\]) becomes $$\label{u6}
\dfrac{d^3\t}{d\theta^3}+\csc^2[\alpha]\dfrac{d\t}{d\theta}=0.$$ or $$\label{u61}
\dfrac{d^3\t}{d\phi^3}+\dfrac{d\t}{d\phi}=0, \,\,\,\,\,\phi=\csc[\alpha]\theta.$$
If we write the tangent vector $\t=\Big(T_1, T_2, T_3\Big)$ the general solution of (\[u61\]) takes the form $$\label{u7}
\t(\phi)=T_i(\phi)\bold{e}_i=\Big(a_i\cos[\phi]+b_i\sin[\phi]+
c_i\Big)\bold{e}_i,\,\,\,i=1,2,3,$$ where $a_i, b_i, c_i\in R$ for i=1,2,3.
Hence the curve $\psi$ is general helix, i.e. the tangent vector $\t$ makes an constant angle $\alpha$ with the constant vector called the axis of the helix. So, with out loss of generality, we take the axis of helix is parallel to $\bold{e}_3$. Then $T_3=\langle\t,\bold{e}_3\rangle=\cos[\alpha]$ which leads to $a_3=b_3=0$ and $c_3=\cos[\alpha]$.
On other hand the tangent vector $\t$ is a unit vector, so the following condition is satisfied $$\label{u8}
T_1^2+T_2^2+T_3^2=1,$$ which leads to $$\label{u9}
\begin{array}{ll}
&\Big(a_1\cos[\phi]+b_1\sin[\phi]+c_1\Big)^2+\Big(a_2\cos[\phi]+b_2\sin[\phi]+
c_2\Big)^2=\sin^2[\alpha].
\end{array}$$ The above equation can be written in the form $$\label{u10}
A_0+\sum_{i=1}^2\Big[A_i\cos[i\,\phi]+B_i\sin[i\,\phi]\Big]=0,$$ where $$\label{u11}
\left\{\begin{array}{ll}
A_2&=\dfrac{1}{2}\Big(a_1^2+b_1^2-a_2^2-b_2^2\Big)\\
B_2&=a_1a_2+b_1b_2\\
A_1&=2(a_1a_3+b_1b_3)\\
B_1&=2(a_2a_3+b_2b_3)\\
A_0&=\dfrac{1}{2}\Big[a_1^2+b_1^2+a_2^2+b_2^2+2\Big(a_3^2+b_3^2-\sin^2[\alpha]\Big)\Big].
\end{array}\right.$$ If equation (\[u10\]) is satisfied, then all coefficients must be zero, so we have the following set of algebraic equations in the six unknowns $a_1, a_2, a_3, b_1, b_2$ and $b_3$. $$\label{u12}
A_i=0,\,\,\, \forall\,\,\, i=1,2,...,5.$$ Solving the five algebraic equations above we obtain four cases of solutions as the following: $$\label{u13}
\left\{\begin{array}{ll}
a_3=b_3=0,\,\,\,\,b_2=a_1,\,\,\,\, a_2=-b_1,\,\,\,\,b_1=\pm\sqrt{\sin^2[\alpha]-a_1^2}\\
a_3=b_3=0,\,\,\,\,b_2=-a_1,\,\,\,\, a_2=b_1,\,\,\,\,b_1=\pm\sqrt{\sin^2[\alpha]-a_1^2}.
\end{array}\right.$$ The four cases above leas to the one general form solution, so the equation (\[u7\]) takes the form: $$\label{u14}
\begin{array}{ll}
\t(\phi)=&\Big(a_1\cos[\phi]-\sqrt{\sin^2[\alpha]-a_1^2}\sin[\phi]\Big)\bold{e}_1\\
&+
\Big(\sqrt{\sin^2[\alpha]-a_1^2}\cos[\phi]+a_1\sin[\phi]\Big)\bold{e}_2+
\cos[\alpha]\bold{e}_3,
\end{array}$$ or in the following form: $$\label{u15}
\begin{array}{ll}
\t(\phi)=\Big(\sin[\alpha]\cos[\phi+\varepsilon],
\sin[\alpha]\sin[\phi+\varepsilon], \cos[\alpha]\Big).
\end{array}$$ where $\varepsilon=\arctan\Big[\sqrt{\dfrac{\sin^2[\alpha]}{a_1^2}-1}\Big]$. Without loss of generality we can written: $$\label{u16}
\begin{array}{ll}
\t(\phi)=\Big(\sin[\alpha]\cos[\phi],
\sin[\alpha]\sin[\phi], \cos[\alpha]\Big).
\end{array}$$ By Integrating the above equation with respect to $s$ along with $\phi=\csc[\alpha]\int\kappa(s)ds$, we have the two equations (\[u211\]) and (\[u212\]) which it completes the proof.
Examples
========
In this section, we take several choices for the curvature $\kappa$ and torsion $\tau$, and next, we apply Theorem \[th-main2\].
[**Example 1.**]{} The case of plane curve $\tau=0, \kappa=\kappa(s)$, i.e., $\alpha=\dfrac{\pi}{2}$. Then the tangent vector takes the form: $$\label{u17}
\begin{array}{ll}
\t(\phi)=\Big(\cos[\phi],
\sin[\phi], 0\Big).
\end{array}$$ Integrate the above equation with respect to $s$ along with $\phi=\int\kappa(s)ds$, we have the natural form of the plane curve as the following: $$\label{u213}
\psi(s)=\int\Big(\cos\Big[\int\kappa(s)ds\Big], \sin\Big[\int\kappa(s)ds\Big], 0\Big)ds+\bold{C}$$ or in the parametric form $$\label{u214}
\psi(\phi)=\int\dfrac{1}{\kappa(\phi)}\Big(\cos[\phi], \sin[\phi], 0\Big)d\phi+\bold{C},\,\,\,\phi=\int\kappa(s)ds,$$ which is the well-known equation of a plane curve with an arbitrary curvature $\kappa=\kappa(s)$.
[**Example 2.**]{} The case of a curve when both of the curvature and torsion are constants, i.e., $\kappa=\dfrac{\sin[\alpha]}{a},\,\tau=\dfrac{\cos[\alpha]}{a}$. Then position vector takes the form: $$\label{u18}
\begin{array}{ll}
\psi=\sin[\alpha]\int\Big(\cos\Big[\dfrac{s}{a}\Big],
\sin\Big[\dfrac{s}{a}\Big],\cot[\alpha]\Big)ds+\bold{C}.
\end{array}$$ Integrating the above equation and putting $s=a\,\phi$, we obtain the parametric representation of this curve as the following: $$\label{u191}
\begin{array}{ll}
\psi=a\,\sin[\alpha]\Big(\sin[\phi],-\cos[\phi], \cot[\alpha]\,\phi\Big)+\bold{C}.
\end{array}$$ which is the equation of the circular helix.
[**Example 3.**]{} The case of a general helix with $\kappa=\dfrac{\sin[\alpha]}{a\,s}$ and $\tau=\dfrac{\cos[\alpha]}{a\,s}$. Then position vector takes the form: $$\label{u20}
\begin{array}{ll}
\psi=a\sin[\alpha]\int\exp[a\phi]\Big(\cos[\phi],
\sin[\phi], \cot[\alpha]\Big)d\phi+\bold{C}.
\end{array}$$ Integrating the above equation, we obtain the parametric representation of this curve as the following: $$\label{u21}
\begin{array}{ll}
\psi=\dfrac{a\sin[\alpha]}{1+a^2}\exp[a\,\phi]\Big(
\sin[\phi]+a\cos[\phi], a\sin[\phi]-\cos[\phi], \dfrac{(1+a^2)\cot[\alpha]}{a}\Big)+\bold{C},
\end{array}$$ which is the equation of a helix on a cone of revolution.
[**Example 4.**]{} The case of a general helix with $\kappa=\dfrac{a\,\sin[\alpha]}{a^2+s^2}$ and $\tau=\dfrac{a\,\cos[\alpha]}{a^2+s^2}$. Then position vector takes the form: $$\label{u20}
\begin{array}{ll}
\psi=a\sin[\alpha]\int\sec^2[\phi]\Big(\cos[\phi],
\sin[\phi], \cot[\alpha]\Big)d\phi+\bold{C}.
\end{array}$$ After some computations, we can obtain the parametric representation of this curve as the following: $$\label{u21}
\begin{array}{ll}
\psi=a\,\sin[\alpha]\Big(
\theta, \cosh[\theta], \cot[\alpha]\sinh[\theta]\Big)+\bold{C},
\end{array}$$ where $\theta=\sinh^{-1}[\tan[\phi]]$.
[99]{} Barros M; *General helices and a theorem of Lancret*, Proc. Amer. Math. Soc., [**125**]{}, (1997), 1503–1509.
Barros M, Ferrandez A, Lucas P and Merono MA, General helices in the three-dimensional Lorenzian space forms, Rocky Mount. J. Math., [**31**]{}, (2001), 373–388.
Barros M, Ferrandez A, Lucas P and Merono MA, Hopf cylinders, B-scrolls and solitons of the Betchov-Da Rios equation in the three-dimensional anti-de Sitter space, C. R. Acad. Sci. Paris Sér. I Math., [**321**]{}, (1995), 505–509.
Boyer C, A History of Mathematics, Wiley, New york, 1968.
Chouaieb N, Goriely A and Maddocks HH; ACastro I. and Urbano F.; [*Helices*]{}, PANS, [**[103]{}**]{}, (2006), 9398–9403.
Eisenhart LP, A Treatise on the Differential Geometry of Curves and Surfaces, Ginn and Co., 1909.
Gluck H, Higher curvatures of curves in Eulidean space, Amer. Math. Monthly, [**73**]{}, (1996), 699–704.
Hacisalihoglu HH, Differential Geometry, Ankara University, Faculty of Science Press, 2000.
Ilarslan, K and Boyacioglu O, [*Position vectors of a spacelike W-cuerve in Minkowski space $\e_1^3$*]{}, Bull. Korean Math. Soc., [**[44]{}**]{}, (2007), 429–438.
Lipschutz MM, Schum$^,$s Outline of Theory and Problems of Differential Geometry, McGraw-Hill Book Company, New York, 1969.
Lucas AA and Lambin P, [*Diffraction by DNA, carbon nanotubes and other helical nanostructures*]{}, Rep. Prog. Phys., [**[68]{}**]{}, (2005), 1181–1249.
Milman RS and Parker GD, Elements of Differential Geometry, Prentice-Hall Inc., Englewood Cliffs, New Jersey, 1977.
Monterde J, Salkowski curves revisted: A family of curves with constant curvature and non-constant torsion, Comput. Aided Geomet. Design, [**26**]{}, (2009), 271–278.
J. Monterde, Curves with constant curvature ratios, [**13**]{}, (2007), arXiv:math/04/2323v1.
P.D. Scofield, Curves of constant precession, Amer. Math. Monthly, [**102**]{}, (1995), 531–537.
D.J. Struik, Lectures in Classical Differential Geometry, Addison,-Wesley, Reading, MA, 1961.
Toledo-Suarez CD; [*On the arithmetic of fractal dimension using hyperhelices*]{}, Chaos Solitons and Fractals, [**[39]{}**]{}, (2009), 342–349.
M. Turgut, S. Yilmaz, Contributions to classical differential geometry of the curves in $\e^3$, Scientia Magna, [**4**]{}, (2008), 5–9.
Watson JD and Crick FHC; [*Generic implications of the structure of deoxyribonucleic acid*]{}, Nature, [**171**]{}, (1953), 964–967.
Yang X; [*High accuracy approximation of helices by quintic curve*]{}, Comput. Aided Geomet. Design, [**[20]{}**]{}, (2003), 303–317.
|
---
abstract: |
Using first-principles plane-wave calculations systematic study of magnetic properties of doped two-dimensional honeycomb structures of III-V binary compounds have been conducted, either for magnetic or nonmagnetic dopants. Calculations show, that all cases where magnetic moment is non-zero are energetically more favorable. For such cases band structure and (partial) density of states were calculated and analyzed in detail. The possible applications of these structures were also discussed.\
author:
- Krzysztof Zberecki$^1$
title: 'Magnetism in doped two-dimensional honeycomb structures of III-V binary compounds.'
---
INTRODUCTION
============
Since its discovery in 2004 graphene [@nov1] draws much attention beacuse of uniqe features of this two-dimensional system. Graphene is composed of a sp$^ {2}$-bonded carbon atoms forming honeycomb structure. It has very interesting electronic structure with characteristic, linear energy dispersion near K point of Brilloin zone. Summary of the subject can be found for example in [@graph1]. Shortly after, experimental techniques allowed fabrication of other new two-dimensional materials, like BN and MoS$_2$ honeycomb structures [@nov2] or ZnO monolayers [@zno1]. The discovery of such stable two-dimensional material like graphene triggered search for similar structures made from different compounds. Up to now many of these hypotetical structures constructed from silanene (2D Si) and germanene (2D Ge) [@sil1; @sil2], III-V compounds [@III-V1], SiC [@sic1] or ZnO [@zno2] have been studied theoretically. On the other hand graphene and other nano-scale materials are recognized as future building blocks of new electronics technologies [@nano1], including spintronics (e.g. [@spin1]). In the case of low (one- and two-) dimensional structures problem arises because of famous Mermin-Wagner theorem [@mermin1], which prevents ferro- or antifoerromagnetic order to occur in finite temperatures, which is essential for spintronics and other modern applications. This started the theoretical and experimental search for magnetism in graphene and other structures. One of the most promising directions is emergence of magnetism in such structures as an effect of presence of local defects [@exp1]. According to works of Palacios et al. [@palacios1] and, independently, of Yazev [@yazyev1] single-atom defects can induce ferromagnetism in graphene based materials. In both cases, the magnetic order arises as an effect of presence of single-atom defects in combination with a sublattice discriminating mechanism, in agreement with Lieb’s theorem [@lieb1]. Based on these findings several theoretical studies have been conducted in search for magnetism in low-dimensional structures either for graphene and BN [@bn1] or other (hypothetical) structures like SiC [@sic1]. In this paper influence of local defects on magnetic structure of two-dimensional honeycomb structures of GaN, AlN and InN have been analysed by means of $ab$-$initio$ calculations. Since bulk versions of these compounds are very important semiconductors in todays electronics it would be interesting to check whether such two-dimensional materials could have non-zero magnetic moment. Despite of the fact that neither of them have been yet synthesized, calculated cohesion energies [@III-V1] suggest that such structures would be stable and their experimental procurement is highly probable. Next section contains computational details followed by results. Last section concludes this work.
COMPUTATIONAL DETAILS
=====================
To investigate magnetic properties of GaN, AlN and InN honeycomb structures a series of $ab$-$initio$ calculations have been conducted with use of DFT VASP code [@vasp1; @vasp2] with PAW potentials [@vasp3]. For both spin-unpolarized and spin-polarized cases exchange-correlation potential has been approximated by generalized gradient approximation (GGA) using PW91 functional [@pw91]. Kinetic energy cutoff of 500 eV for plane-wave basis set has been used. Supercells of size 4x4x1 have been checked to be large enough to prevent defects interact with its periodic image. In all cases for self-consistent structure optimizations, the Brillouin zone (BZ) was sampled by 20x20x1 special k points. All structures have been optimized for both, spin-unpolarized and spin-polarized cases unless Feynman-Hellman forces acting on each atom become smaller than 10$^{-4}$ eV/${\rm\AA}$. A vacuum spacing of 12 [Å]{} was applied to hinder the interactions between monolayers in adjacent supercells. Calculated lattice constants are in agreement with [@III-V1].
RESULTS
=======
As mentioned, non-magnetic honeycomb sheets can attain spin polarized states due to presence of local defects. In this work two kinds of defects have been analysed - vacancies and subsitutions. For all three compounds a vacancy was generated first by removing a single atom, Al, Ga, In or N from each supercell, then the atomic structure was optimized. In all cases structures with single N vacancy are non-magnetic, while Al, Ga or In vacancies induce non-zero magnetic moment, equal to 3.00 $\mu_{B}$. This is in disagreement with Lieb’s theorem, which states that magnetic moment should be equal to 1.00 $\mu_{B}$ when one of sublattices has exacly one atom more/less than the other (e.g. N$_{Al}$ - N$_{N}$ = $\pm$ 1). This discrepancy can be addressed to charge transfer from Al(Ga,In) to N. Fig. \[fig1\] shows density of states (DoS) for spin-polarized Al-vacant AlN, on which difference between majority spin (up) and minority spin (dn) in the vicinity of Fermi level (horizontal line) can be observed which is the main source of non-zero magnetic moment. Analysis of calculated partial magnetisation shows that almost all magnetic moment is situated on p-states of N atoms located in the area of vacancy. This is in full agreement with previous studies of vacancies in SiC [@sic1].
In the case of substitution the procedure was as follows. For all three compounds various single foreign atoms have been substituted, then structure has been optimized. In the case of AlN and GaN, Al or Ga has been subsituted by atoms from 4-th period of periodic table from K to Zn, including Na and Mg for AlN. In the case of InN, In has been subsituted by atoms from 5-th period of periodic table from Rb to Cd, excluding Tc. In all three compounds, N has been subsituted by C, B and P atoms. Table I shows calculated magnetic moments and differences between total energy of spin-unpolarized and spin-polarized states, $\Delta$E = E$_{nsp}$ - E$_{sp}$, i.e. positive value of $\Delta$E means that spin-polarized state is more energetically favored. This is the case for all compounds with non-zero magnetic moment - the largest energy differences (up to 1.25 eV) are for instances with highest values of magnetic moment. For AlN highest values of induced magnetic moment are in case of Mn, Co (4.00 $\mu_{B}$) and for Fe (4.26 $\mu_{B}$). Similar situation can be observed for GaN doped with Mn, Co and Fe. With decreasing number of d-shell electrons value of magnetic moment drops as well as for the case of Zn which has d-shell closed. In the case of InN this tendency holds although values of magnetic moments are much smaller, being the highest for Ru and Rh. Figs. \[fig2a\] and \[fig2b\] show mechanism of generation of magnetic moment in the case of GaN doped with Ni. Left plot of Fig. \[fig2a\] shows bandstructure of Ni-doped GaN vs. undoped one (which is a semiconductor with bandgap equal to 2.30 eV, calculated within GGA) in spin-unpolarized case. One can see the formation of doping bands in the vicinity of Fermi level. The top right plot shows bandstructure of GaN+Ni in the spin-polarized case, where can be observed quite large splitting of these bands between spin up and down bands.
Top plots of Fig. \[fig2b\] show density of states for spin polarized case. Right one shows total DoS vs. DoS of spin up and down. Fermi energy is almost exactly in the middle of splitted up and down DoS. Left plot shows vicinity of Fermi energy more closely, where large splitting of up and down DoS can be observed. Since almost all electrons occupying vicinity of Fermi level are d-shell electrons, which can be read from partial density of states (bottom left plot) the mechanism of magnetic moment emergence becomes clear. Calculations show that this mechanism is universal for all structures having non-zero magnetic moment doped with transition metal elements. In case of doping with alkali metal elements and alkaline earth metal elements only in Na- and K-doped AlN calculations show non-zero magnetic moment (1.88 and 1.70 $\mu_{B}$, respectively for Na and K). Mechanism of formation of magnetic moment is similar to the case of vacant structures (since Na and K have only one valence electron). Fig. \[fig3\] shows DoS for spin-polarized Na-doped AlN, on which difference between spin up and spin down in the vicinity of Fermi level can be observed.
This is similar to situation depicted on Fig. \[fig1\] altough in the case of Na-doped structure splitting is smaller. In case of substitution of N atom by C, B and P, only C-doped structures had non-zero magnetic moment, which was equal to 1.00 $\mu_{B}$ in all compounds. Magnetic states in all cases are lower by about 0.1 eV than nonmagnetic states.
-- --
-- --
-- --
-- --
AlN
----------------- ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------
for Al Na Mg K Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn
$\mu (\mu_{B})$ 1.88 0.00 1.70 0.00 0.00 0.96 0.00 2.98 3.99 4.26 3.96 2.98 1.85 0.00
$\Delta$E (eV) 0.06 0.00 0.04 0.00 0.00 0.11 0.00 0.70 0.98 1.09 0.94 0.38 0.06 0.00
GaN
for Ga K Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn
$\mu (\mu_{B})$ 0.00 0.00 0.00 0.90 1.72 2.99 3.99 4.51 3.97 2.97 1.32 0.00
$\Delta$E (eV) 0.00 0.00 0.00 0.12 0.07 0.66 1.15 1.25 0.91 0.30 0.01 0.00
InN
for In Rb Sr Y Zr Nb Mo Tc Ru Rh Pd Ag Cd
$\mu (\mu_{B})$ 0.00 0.00 0.00 0.00 0.99 0.39 - 1.38 1.91 0.42 0.00 0.00
$\Delta$E (eV) 0.00 0.00 0.00 0.00 0.16 0.01 - 0.04 0.10 0.01 0.00 0.00
: Magnetic moments and total energy differences between spin up and spin down states for different transition metal elements dopants.
AlN
----------------- ------ ------ ------ -- --
for N B C P
$\mu (\mu_{B})$ 0.00 1.00 0.00
$\Delta$E (eV) 0.00 0.14 0.00
GaN
for N B C P
$\mu (\mu_{B})$ 0.00 1.00 0.00
$\Delta$E (eV) 0.00 0.11 0.00
InN
for N B C P
$\mu (\mu_{B})$ 0.00 1.00 0.00
$\Delta$E (eV) 0.00 0.10 0.00
: Magnetic moments and total energy differences between spin up and spin down states for different dopants.
CONCLUSIONS
===========
$Ab$-$initio$ calculations have been conducted for vacancy and subsitution defects in honeycomb AlN, GaN and InN compounds. Calculations show that in all three compounds vacancy of Al, Ga or In respectively gives magnetic moment of 3.00 $\mu_{B}$, which is interesting conclusion from application point of view. On the other hand substitution of Al or Ga by transition metal elements (Mn, Fe, Co) can give even higher value of magnetic moment (4.00 $\mu_{B}$). Since techniqe of implantation of metal atoms into 2D surface has been recently reported [@nano2], it is also very promising direction. On the other hand substitution by non-metallic atoms or substitution of nitrogen atoms by IV or V group atoms does not give significant magnetic moment. Thes results may give a hint for experimentalists seeking for two-dimensional magnetic materials.
Numerical calculations were performed at the Interdisciplinary Centre for Mathematical and Computational Modelling (ICM) at Warsaw University.
[99]{} K. S. Novoselov et al., Science 306, 666 (2004). K. S. Novoselov et al., Proc. natl. Acad. Sci. USA, 102, 10451 (2005). A. H. Castro Neto et al., Rev. Mod. Phys. 81, 109 (2009). S. Lebegue et al., Phys. Rev. B 79, 115409 (2009). S. Cahangirov et al., Phys. Rev. Lett. 102, 236804 (2009). H. Sahin et al., Phys. Rev. B 80, 155453 (2009). E. Bakaroglu et al., Phys. Rev. B 81, 075433 (2010). C. Tusche et. al. Phys. Rev. Lett. 99, 026102 (2007). M. Topsakal et al., Phys Rev. B 80, 235119 (2009). P. Avouris et al., Nat. Nanotech. 2, 605 (2007). I. Zutic et al., Rev. Mod. Phys. 76, 323 (2004). N. D. Mermin, H. Wagner Phys. Rev. Lett. 17, 1133 (1966). P. Esquinazi, Phys. Rev. Lett. 91, 227201 (2003). J. J. Palacios et al., Phys. Rev. B 77, 195428 (2008). O. Yazev, Phys. Rev. Lett. 101, 037203 (2008). O. Yazev et al., Phys. Rev. B 75, 125408 (2007). E. H. Lieb, Phys. Rev. Lett. 62, 1201 (1989); 62, 1927(E) (1989). C. Ataca et al., Phys. Rev. B 82, 165402 (2010). G. Kresse, J. Hafner, Phys. Rev. B 47, 558 (1993). G. Kresse, J. Furthmuller, Phys. Rev. B 54, 11169 (1996). E. Blochl, Phys. Rev. B 50, 17953 (1994). J. P. Perdew, et al., Phys. Rev. B 46, 6671 (1992). J. Lahiri et al., Nat. Nanotech. 5, 326 (2010).
|
---
author:
- 'M. Pierre'
- 'R. Wacquez'
- 'X. Jehl'
- 'M. Sanquer'
- 'M. Vinet'
- 'O. Cueto'
title: Single donor ionization energies in a nanoscale CMOS channel
---
[**One consequence of the continued downwards scaling of transistors is the reliance on only a few discrete atoms to dope the channel, and random fluctuations of the number of these dopants is already a major issue in the microelectonics industry [@Intel45nm]. While single-dopant signatures have been observed at low temperature [@Schofield03; @Simmons06; @Zhong05; @Ono07; @Khalafalla07; @Khalafalla09; @Delft08], studying the impact of only one dopant up to room temperature requires extremely small lengths. Here, we show that a single arsenic dopant dramatically affects the off-state behavior of an advanced microelectronics field effect transistor (FET) at room temperature. Furthermore, the ionization energy of this dopant should be profoundly modified by the close proximity of materials with a different dielectric constant than the host semiconductor [@Diarra07; @Bjork09]. We measure a strong enhancement, from 54meV to 108meV, of the ionization energy of an arsenic atom located near the buried oxide. This enhancement is responsible for the large current below threshold at room temperature and therefore explains the large variability in these ultra-scaled transistors. The results also suggest a path to incorporating quantum functionalities into silicon CMOS devices through manipulation of single donor orbitals.**]{}
Progresses in nanotechnology made it possible to observe signatures of single dopants in semiconducting nanostructures, with for instance STM technique[@Schofield03; @Simmons06] or grown nanowires [@Zhong05]. Single dopants have been detected as a hump in the sub-threshold slope of a capacitively coupled surface channel [@Ono07; @Khalafalla07; @Khalafalla09]. Resonant tunneling transport via an impurity state has been studied in structures markedly larger than a typical donor orbital [@Bending85; @Fowler86; @Geim94; @Savchenko95; @Calvet08]. Resonant transport spectroscopy has even been performed in a transistor geometry favoring a strong hybridization of a donor electronic wavefunction with a surface channel [@Delft08]. Both a very small volume and a small number of dopants in the channel are required to observe resonant transport through the first isolated dopant. Otherwise hybridization with surface states, polarization of nearby dopants, or many-dopant problem [@kuznetsov97] will dramatically modify the wavefunction and energy spectrum. The distance between the source and drain is also of crucial importance because the barriers will have an exponential dependence on this length divided by the characteristic wavefunction extension.
Therefore a very natural way to study transport through only one dopant is to take advantage of CMOS technology to build a very short, narrow and doped transistor. We fabricated silicon-on-insulator (SOI) transistors (Fig. \[fig1\]a) with a geometry and process designed to take advantage of the inhomogeneous dopant profile after ion-implantation of the source and drain. A simulation of this doping profile, based on the real process and continuous approximation, is shown in Fig. \[fig1\]b. A similar simulation taking into account discrete dopants with a Kinetic Monte Carlo approach for diffusion is shown in Fig. \[fig1\]c, with the As atom size set to the mean Bohr radius in bulk Si, $a_{B}=2.2\,\mathrm{nm}$. All the regions with a concentration above the Mott transition of bulk Si, that is $n_{c} = 8\times 10^{18} As\,\mathrm{cm^{-3}}$ \[\], must be considered as part of the source and drain. Based on these simulations the effective electrical channel length is of the order of 10nm. Therefore the remaining volume available for As dopants below $n_c$ is roughly $V=$10$\times$20$\times$50nm$\mathrm{^3}$. The probabilities of finding one dopant near the buried oxide (for maximum ionization energy enhancement) or a dopant cluster, most likely a donor pair, can be evaluated and compared. For $N$ dopants the probability of finding a pair with separation $a_{B}$ or less is given by [@Geim94]: $$P_{pair}=1-\prod_{n=1}^N{(1-nz)}$$ where $z$ is the ratio $\frac{4}{3}\pi {a_{B}}^3/V\approx0.005$ . For 5 dopants $P_{pair}$ is only 5%, but it reaches 90% for 30 dopants. In the same time the probability for having at least one donor out of five at 1nm or less from the BOX interface is already 25%.
These ultra-scaled dimensions, corresponding to a 10nm technology node transistor [@Intel45nm], naturally yield variability. A set of 25 nominally identical samples was measured. We found similar on-state current at saturation (V$_{\mathrm g}$=+2V) but very large sub-threshold dispersion (see Supplementary Information, Fig. S1). Two extreme room temperature characteristics, featuring the highest and lowest off-state currents (at V$_{\mathrm g}$=-2V) are shown in Fig.\[fig1\]d. A threshold voltage of -0.5V $\pm 0.05$ is obtained after extrapolating the point of maximum transconductance in the linear I$_{\mathrm d}$-V$_{\mathrm g}$ characteristics of the samples with the lowest off-state current and steepest subthreshold swing (arrows in Fig. 1d). This phenomenological criterion commonly used in microelectronics is well adapted to such mesoscopic samples where the usual definitions (relying on self-averaging) are not valid. At T=4.2K these samples exhibit a pattern of large drain current oscillations above V$_{\mathrm g}$=-0.2V and the drain conductance quickly increases up to or above the quantum of conductance $\frac{e^2}{h}\approx 25.8\,k\Omega$ that fixes the onset of a diffusive channel at low temperature (not shown). The shift of the onset voltage from $-0.5\,V$ to $-0.2\,V$ is due to localization of carriers at low temperature. Above V$_{\mathrm g} \simeq$-0.2V resonant states observed at low temperature result from hybridization of several electronic states, for instance a donor state hybridized with a surface state[@Delft08].
New and strikingly different features are observed at low temperature for the samples with large off-state current at 300K. A very few resonances appear at gate voltages down to -1.3V, i.e. when electrons are pushed away far from the gate. A 2D plot of differential conductance at 4.2K versus drain-source voltage is shown in Fig. \[rhombus4K\]. The gate voltage is translated in energy (referenced from the band edge) using the lever arm parameter $\alpha$=0.16 extracted from the Coulomb diamonds. This rather small value for $\alpha$ indicates that the orbital responsible for this resonance is less coupled to the gate than to source and drain, i.e. in our geometry, that the electronic state is rather on the BOX side of the nanowire. The value $V_{g1}=-1.3\,V$ combined with $\alpha$=0.16 and a threshold voltage of -0.5V yields an ionization energy for the first donor of 108meV$\pm$10meV, a value markedly larger than the 53.7meV expected [@Kohn55] for bulk Si. Note that the other sample shown in Fig. \[fig1\]d shows a value 98meV$\pm$10meV, i.e. also a strong enhancement. Fig. \[rhombus4K\]a shows that the drain current in excess at low V$_{\mathrm g}$ and temperature up to 300K is due to thermal broadening of these resonances. For instance for $V_{g}\le V_{g1}$ the current is given by thermal activation to the first resonance: $ I_{ds}\propto exp(-(e\alpha (V_{g}-V_{g1})/k_{B}T$). The sub-threshold variability is the most affected by the presence of a single centered donor, which increases the drain conductance by four orders of magnitude at V$_{\mathrm g} $=-1.5V and at room temperature (see Fig.\[fig1\]d), a much larger effect than reported before [@Ono07]. Detailed transport spectroscopy of the first resonance is shown in Fig. \[1erpic\]. No electronic states exist below $V_{g1}$ since no distortion of the diamond occurs at large drain voltage, up to 100mV (see Fig. \[rhombus4K\]). The absence of electrons in the channel at lower energy is crucial to ensure that the spectroscopy of the state at $V_{g1}$ is not modified by electronic correlations with other electrons weakly bound on extra donors (D$^{0}$ states). We observe many lines of differential conductance parallel to both edges of the diamond and successively positive and negative (see Fig. \[1erpic\]), due to fluctuations of the local density-of-states (LDOS) in the source and drain [@Falko97; @Falko01]. Lines with different slopes correspond to probing the source or the drain, depending on the exact balance of the tunneling rates (see Supplementary Information, Fig. S2). The large observed LDOS fluctuations originate from the extremely small non-invasive As doped contacts with a finite transverse doping gradient. There is no evidence of any differential conductance line due to an excited state of the donor at least up to $V_d$=40mV.
Donor states in bulk silicon are well described [@Kohn55] by a hydrogenic model extended by variational methods to account for the anisotropy of the effective mass of Si. The ground state lies 53.7meV below the conduction band and the first excited state at 32.6meV. More recently strong corrections have been predicted for silicon nanowires, either because of quantum confinement, relevant for diameters below 5nm \[\], or due to dielectric confinement, more important for larger diameters [@Diarra07]. Compared to bulk Si case, an enhanced ionization energy is expected for donors close to a Si/SiO$\mathrm{_2}$ interface, and a reduction for dopants close to the gate, due to the long range screening of the positive core donor charge potential for electrons. This has a dramatic impact on dopant ionization even at room temperature [@Diarra07; @Bjork09]. In our data the first resonance occuring 108meV below the conduction band (Fig. \[rhombus4K\]) illustrates the sensitivity of the ionization energy with the dielectric environment. Following ref. , we can estimate the correction to the ionization energy for a single dopant at position $r_{0}$ and distance $z$ from the dielectric interface as: $$\Delta E_{I} \simeq \langle \psi \mid V_{s}(r,r_{0}) \mid \psi \rangle,$$ where $V_{s}(r,r_{o})$ is the image potential at position $r$ and $\psi$ the bound state on the donor. The approximation $V_{s}(r,r_{o}) \approx V_{s}(r_{o},r_{o})=V_{s}(z)$ yields $$\Delta E_{I} \approx \frac{1}{4 \pi \epsilon_{0}} \frac{e^{2}}{2z} \frac{\epsilon_{\mathrm{Si}}-\epsilon_{\mathrm{SiO_2}}}{\epsilon_{\mathrm{Si}}(\epsilon_{\mathrm{Si}}+\epsilon_{\mathrm{SiO_2}})} \simeq \frac{30.5\,\mathrm{meV}}{z\,(\mathrm{nm})}$$
The effects of screening by the source and drain, as well as quantum confinement [@Niquet06] are neglected. This estimation indicates that a large effect as we observe is realistic for a dopant very near the buried oxide interface ($\leq$1nm). Theory also predicts an extension of the bound state function along the nanowire axis and a reduction in the transverse direction [@Diarra07]. This should increase the energy difference between the ground and first excited state. Indeed our experiment gives a lower bound of 40meV for this energy, significantly larger than the expected 21meV for As in bulk Si. The shift of electrostatic potential $V_{s}(z)$ due to the positive image charge in the BOX is nearly equivalent to the bare Coulomb potential of another ionized donor located at $4z$ in bulk silicon. The mean distance between As donors is 20nm resulting in a bare shift of $\Delta E_{I}(20\,\mathrm{nm}) \simeq 6\,\mathrm{meV}$ for each extra ionized donor. However donors are located less than $d$=5nm from either the source or drain which efficiently screen the bare potential by a factor $2(\frac{d}{z})^2$ \[\]. $\Delta E_{I}(20\,\mathrm{nm})$ is then reduced to less than 1meV, a value much smaller than the observed energy enhancement. The shift due to the ionization of the second donor is in fact directly measured ($\simeq$-2.5meV) as the shift of the first diamond as it crosses the second one, which corresponds to the $D^{+} \rightarrow D^{0}$ transition for the second donor (see Fig. \[rhombus4K\]c). Fig. \[champ\] shows the evolution of the first peak with parallel magnetic field. A Zeeman splitting with a Landé factor g=2 is observed, both as a shift of the ground state down to lower energy and as an extra line of differential conductance (no adjustable parameter). This is in agreement with our interpretation of the first peak corresponding to the $ D^{+} \rightarrow D^{0} $ transition, i.e. a spin change of $\frac{1}{2}$.
The second resonance appears near V$\mathrm{_g}$=-1V, i.e. $\approx$50meV below the conduction band. Because the lever arm parameter is different from the first resonance and the lines of differential conductance cross the first diamond with only a slight modification (see Fig. \[rhombus4K\]c), the second resonance does not correspond to the double occupation of the first As donor. Indeed confinement makes it very unlikely for a second electron to sit on the same dopant. The corresponding charging energy is certainly larger than the ionization energy of another As atom unavoidably present in the channel. Therefore the second peak corresponds to a second dopant located closer to the gate oxide as it is sensed at higher gate voltage. This should have four main consequences: $\Delta E_{I}$ should be strongly reduced, the excited state should be lower in energy, double occupancy could be observed (albeit very close to the conduction band [@Delft08]), and the tunneling rate is likely to be strongly enhanced due to a larger orbital extension. All these features are observed, although the physics involved is much more complex than for the first resonance, because of electronic correlations. Fig. \[cotunneling\] shows the transport spectroscopy for the second and third resonances. In addition to the lines of negative differential conductance due to the LDOS fluctuations, a clear line of positive conductance is detected 8meV above the ground state. This is lower than the 21meV expected for the bulk [@Kohn55], but much larger than in surface donor states measured in ref. . The line is unambiguously identified as an excited state of the donor by the presence of an inelastic cotunneling line in the Coulomb blockade region [@defranceschi]. The third resonance shows features correlated to the second one, proving that it corresponds to the double occupancy of the same donor [@defranceschi]: the inelastic cotunneling lines are continued by lines due to transfer $2e^{-} \rightarrow 1e^{-}$ from the lower energy state to the drain. The charging energy is then $\approx $18meV, which, added to the 8meV level spacing gives an addition energy for the second electron of 26meV. This charging energy is comparable to the 32meV deduced in ref. . As expected the tunneling rates are very high: the second resonance is well fitted below T=2K by a Lorentzian profile (Breit-Wigner resonance) of intrinsic width $h \Gamma = k_{B}\times$2K, and the conductance at resonance is $\approx$ 0.6 $e^{2}/h$, making cotunneling well visible. The small lines inside the Coulomb diamond are due to elastic cotunneling. This effect is proportionnal to both tunneling rates to source and drain and fluctuates with bias due to LDOS variations.
In conclusion, we have observed a doubling of the ionization energy of a single As donor because of dielectric confinement near an oxide interface. This measurement on a single isolated dopant requires extremely small samples, a condition achieved in ultra-scaled microelectronics devices. This shift in ionization energy is responsible for the very large off-state current up to room temperature, and is therefore a major source of variability for the off-state current in ultra scaled CMOS devices. Since dopants closer to the gate are much less affected by this shift, gate-all-around structures are the most suitable choice to avoid unacceptable sample to sample variations. On the other hand the next step for taking advantage of this effect is to manipulate single donor electronic orbitals to implement new quantum functionalities in silicon CMOS devices.
[19]{} Kuhn, K. [*et al.*]{}, managing process variation in Intel’s 45nm CMOS technology, [*Intel Technology Journal*]{} [**12**]{}, 93-109 (2008).
Schofield, S. R. [*et al.*]{}, Atomically precise placement of single dopants in Si, [*Phys. Rev. Lett.*]{} [**91**]{}, 136104 (2003).
Rueß, F. J. [*et al.*]{}, Realization of atomically controlled dopant devices in silicon, [*Small*]{}, [**3**]{}, 563-567 (2007).
Zhong, Z. [*et al.*]{}, Coherent single charge transport in molecular-scale silicon nanowires, [*Nano Lett.*]{}, [**5**]{}, 1143-1146 (2005).
Ono, Y. [*et al.*]{}, Conductance modulation by individual acceptors in Si nanoscale field-effect transistors, [*Appl. Phys. Lett.*]{}, [**90**]{}, 102106 (2007).
Khalafalla M. A. H. [*et al.*]{}, Identification of single and coupled acceptors in silicon nano-field-effect transistors, [*Appl. Phys. Lett.*]{}, [**91**]{}, 263513 (2007).
Khalafalla M. A. H. [*et al.*]{}, Horizontal position analysis of single acceptors in Si nanoscale field-effect transistors, [*Appl. Phys. Lett.*]{}, [**94**]{}, 223501 (2009).
Lansbergen, G. [*et al.*]{}, Gate-induced quantum-confinement transition of a single dopant atom in a silicon FinFET, [*Nature Phys.*]{}, [**4**]{}, 656-661 (2008).
Diarra, M. [*et al.*]{}, Ionization energy of donor and acceptor impurities in semiconductor nanowires: Importance of dielectric confinement, [*Phys. Rev. B*]{} [**75**]{}, 045301 (2007).
Bjork, M. T. [*et al.*]{}, Donor deactivation in silicon nanostructures, [*Nat. Nanotech.*]{} [**4**]{}, 103-107 (2009).
Bending, S. J. and Beasley, M. R., Transport processes via localized states in thin $\alpha$-Si tunnel barriers, [*Phys. Rev. Lett.*]{} [**55**]{}, 324-327 (1985).
Fowler, A. B. [*et al.*]{}, Observation of resonant tunneling in silicon inversion layers, [*Phys. Rev. Lett.*]{} [**57**]{}, 138-141 (1986).
Geim, A. K. [*et al.*]{}, Resonant tunneling through donor molecules, [*Phys. Rev. B*]{} [**50**]{}, 8074-8077 (1994).
Savchenko, A. K. [*et al.*]{}, Resonant tunneling through two impurities in disordered barriers, [*Phys. Rev. B*]{} [**52**]{}, 17021-17024 (1995).
Calvet, L. [*et al.*]{}, Excited-state spectroscopy of single Pt atoms in Si, [*Phys. Rev. B*]{} [**78**]{}, 195309 (2008).
Kuznetsov, K. [*et al.*]{}, Resonant tunneling spectroscopy of interacting localized states: Observation of the correlated current through two impurities, [*Phys. Rev. B*]{} [**56**]{}, 15533-15536 (1997).
Newman P. F. and Holcomb D. F., Metal-insulator transition in Si:As, [*Phys. Rev. B*]{} [**28**]{}, 638-640 (1983).
Kohn, W. and Luttinger J. M., Theory of Donor States in Silicon, [*Phys. Rev.*]{} [**98**]{} 915-922 (1955).
Schmidt, T. [*et al.*]{}, Observation of the Local Structure of Landau Bands in a Disordered Conductor, [*Phys. Rev. Lett.*]{} [**78**]{}, 1540-1543 (1997).
Koenemann, J. [*et al.*]{}, Correlation-function spectroscopy of inelastic lifetime in heavily doped GaAs heterostructures, [*Phys. Rev. B*]{} [**64**]{}, 155314 (2001).
Niquet, Y. M. [*et al.*]{}, Electronic structure of semiconductor nanowires, [*Phys. Rev. B*]{} [**73**]{}, 165319 (2006).
Sanquer M. [*et al.*]{}, Coulomb blockade in low-mobility nanometer size MOSFETs, [*Phys. Rev. B*]{} [**61**]{}, 7249-7252 (2000).
De Franceschi, S. [*et al.*]{}, Electron Cotunneling in a Semiconductor Quantum Dot, [*Phys. Rev. Lett.*]{} [**86**]{}, 878-881 (2001).
Sellier, H. [*et al.*]{}, Transport Spectroscopy of a Single Dopant in a Gated Silicon Nanowire, [*Phys. Rev. Lett.*]{} [**97**]{}, 206805 (2006).
{width="85.00000%"}
{width="85.00000%"}
![[**Coulomb blockade spectroscopy of the first dopant atom. a,**]{} Differential conductance versus bias voltage V$\mathrm{_d}$ at $V_{\mathrm g}$=-1.3V. [**b,**]{} Close-up of the first peak of Figure \[rhombus4K\], recorded at T=100mK. The differential conductance $ \propto {d \nu_{D}\over dE}$ (parallel to the right edge) or $ \propto {d \nu_{s}\over dE}$ (parallel to the left edge) depending on the exact balance between tunneling rates to source and drain. The typical 1meV correlation energy for the lines is the inverse diffusion time in the electrodes before an inelastic event (corresponding diffusion length about 10nm).[]{data-label="1erpic"}](figure3.eps){width="0.6\columnwidth"}
![[**Magnetic field dependence of the first peak. a,**]{} Coulomb diamond spectroscopy at zero field. [**b,**]{} Magnetic field dependence at V$\mathrm{_g}$=-1.2895V (dashed line in a), the magnetic field being applied parallel to the wire. [**c,**]{} Coulomb diamond at 6T. The dotted lines in [**b**]{} are the predicted Zeeman shifts for $\Delta$S$_z=\pm \frac{1}{2}$ and g=2. The different slopes correspond to different level arm factors extracted from the Coulomb diamonds (no adjustable parameter). For negative drain voltage the positive differential conductance line associated to the spin up level is clearly seen. For positive drain voltage the corresponding line crosses negative differential conductance lines due to the spectroscopy of the drain, resulting in a cancellation of the differential conductance rather than a positive conductance line.[]{data-label="champ"}](figure4.eps){width="0.6\columnwidth"}
![[**The second and third peaks of Fig. \[rhombus4K\] recorded at T=100mK,**]{} respectively corresponding to the $D^{+} \rightarrow D^{0}$ transition (second) and $D^{0} \rightarrow D^{-}$ (third) . Solid lines indicate the diamond edges while dashed lines highlight features due to inelastic co-tunneling via the first excited state of the donor. The measured difference in energy between the ground and first excited state is 8meV, and the charging energy is 18meV. Lines inside the diamonds are due to LDOS in the electrodes revealed by elastic cotunneling.[]{data-label="cotunneling"}](figure5.eps){width="0.6\columnwidth"}
![figure S1: Room temperature characteristics (drain-source current versus gate voltage) for a set of nominally identical samples from the same wafer. Sample to sample variations are very important, especially below the threshold voltage (-0.5V for the best device, in red). In the most extreme case (blue curve) the many decades of excess current compared to the best device are attributed to direct transport through individual arsenic dopants which have diffused into the channel after doping and annealing steps. The source-drain voltage is 10mV for these data.[]{data-label="figS1"}](figureS1.eps){width="0.6\columnwidth"}
![figure S2: Sketch for negative or positive differential conductance lines arising from the spectroscopy of the source and drain local density of states (LDOS) probed by a single dopant level. A single dopant level (red line) has a slightly unbalanced tunneling rate to source and drain: the barrier to drain is less transmissive at zero drain bias, represented by a thicker barrier. The drain source differential conductance is dominated by the less transmissive barrier (red arrow, green arrow for the more transmissive ones) and proportional to the LDOS in the corresponding reservoir. At finite bias around point B the lines are due to LDOS fluctuations in the drain, because the drain barrier is less transmissive. On the opposite, at finite bias in A the drain barrier becomes more transmissive than the source barrier due to the electric field (Fowler-Nordheim tunneling): differential conductance lines are then proportional to the LDOS fluctuations in the source.[]{data-label="figS2"}](figureS2.eps){width="0.6\columnwidth"}
|
---
abstract: 'We study Gamow-Teller strength distributions of $^{76}$Ge and $^{76}$Se within a deformed QRPA formalism, which includes residual spin-isospin forces in the particle-hole and particle-particle channels. We consider two different methods to construct the quasiparticle basis, a selfconsistent approach based on a deformed Hartree-Fock calculation with density-dependent Skyrme forces and a more phenomenological approach based on a deformed Woods-Saxon potential. Both methods contain pairing correlations in the BCS approach. We discuss the sensitivity of Gamow-Teller strength distributions to the deformed mean field and residual interactions.'
address:
- |
Instituto de Estructura de la Materia, Consejo Superior de Investigaciones Científicas,\
Serrano 123, E-28006 Madrid, Spain
- 'Institute für Theoretische Physik, Universität Tübingen, D-72076 Tübingen, Germany'
- 'Department of Nuclear Physics, Comenius University, SK-842 15, Bratislava, Slovakia'
- |
Department of Theoretical Physics and Mathematics, Bucharest University, P.O.Box MG11,\
and Institute of Physics and Nuclear Engineering, Bucharest, P.O.Box MG6, Romania
author:
- 'P. Sarriguren, E. Moya de Guerra'
- 'L. Pacearescu, Amand Faessler'
- 'F. Šimkovic'
- 'A.A. Raduta'
title: 'Gamow-Teller strength distributions in $^{76}$Ge and $^{76}$Se from deformed QRPA'
---
Introduction
============
The nuclear double $\beta$-decay process is widely considered [@2breview] as one of the most important sources of information about fundamental issues, such as lepton number nonconservation and massive neutrinos, that can be used to test the Standard Model.
Theoretically, a condition to obtain reliable estimates for the limits of the double $\beta$-decay half-lives is that the nuclear structure involved in the process through the nuclear matrix elements can be calculated correctly. The proton-neutron quasiparticle random phase approximation (pnQRPA or QRPA in short) is one of the most reliable and extended microscopic approximations for calculating the correlated wave functions involved in $\beta$ and double $\beta$ decay processes. The method was first studied in ref. [@halb] to describe the $\beta $ strength. It was developed on spherical single-particle wave functions and energies with pairing and residual interactions.
The QRPA method was also successfully applied to the description of double $\beta$-decay [@2bqrpa] after the inclusion of a particle-particle ($pp$) residual interaction, in addition to the particle-hole ($ph$) usual channel. Many more extensions of the QRPA method have been proposed in the literature, see ref. [@extension] and references therein.
An extension of the pnQRPA method to deal with deformed nuclei was done in ref. [@kru], where a Nilsson potential was used to generate single particle orbitals. Subsequent extensions including Woods-Saxon type potentials [@moll], residual interactions in the particle-particle channel [@homma], selfconsistent deformed Hartree-Fock mean fields with consistent residual interactions [@sarr] and selfconsistent approaches in spherical neutron-rich nuclei [@doba], can also be found in the literature. Nevertheless, the effect of deformation on the double $\beta$-decay processes has not been sufficiently studied [@raduta; @larisa].
In ref. [@sarr], ground state and $\beta $-decay properties of exotic nuclei were studied on the basis of a deformed selfconsistent HF+BCS+QRPA calculation with density dependent effective interactions of Skyrme type. This is a well founded approach that has been very successful in the description of spherical and deformed nuclei within the valley of stability [@flocard]. In this work we extend those calculations to the study of the dependence on deformation of the single $\beta$ branches that build up the double $\beta$ process. We focus on the example of the double $\beta$-decay of $^{76}$Ge and study $\beta^-$ Gamow-Teller (GT) transitions to the intermediate nucleus as well as the $\beta^+$ Gamow-Teller transitions of the daughter nucleus $^{76}$Se to the same intermediate nucleus. We discuss the similarities and differences of using different single particle mean fields of Woods-Saxon (WS) and Hartree-Fock (HF) types.
In sect. II, we present a brief summary containing the basic points in our theoretical description. Section III contains the results obtained for the bulk properties of $^{76}$Ge and $^{76}$Se and a comparison of our results to the experimental available information. In sect. IV we present our results for the GT strength distributions and discuss their dependence on the deformed mean field and residual interactions. The conclusions are given in sect. V.
Theoretical approach
====================
In this Section we describe the QRPA formalism used in this work, which is based on two different assumptions for the deformed mean field, a Woods-Saxon potential and a selfconsistent mean field obtained from a Hartree-Fock procedure with Skyrme forces.
In the first approach we use a deformed WS potential with axial symmetry to generate single particle energies and wave functions. The parameters of this potential are taken from the work of Tanaka [*et al.*]{} [@tanaka]. This parametrization was proposed originally for spherical nuclei ranging from $^{16}$O to $^{208}$Pb, but the derived isospin dependence of the parameters allows an extension to deformed nuclei as well. Previous QRPA calculations have shown that this parametrization provides realistic levels also for deformed nuclei and good results on $M1$ excitations were obtained [@m1] for nuclei in various mass regions as well.
In these calculations, the quadrupole deformation of the WS potential $\beta_2$ is usually determined by fitting the microscopically calculated ground state quadrupole moment to the corresponding experimental value. The hexadecapole deformation $\beta_4$ is expected to be small for these nuclei and we assume it is equal to zero.
On the other hand, we also perform selfconsistent microscopic calculations based on a deformed HF method with density-dependent Skyrme interactions. We consider in this paper the force Sk3 [@beiner] and the force SG2 [@giai] that has been successfully tested against spin and isospin excitations in spherical [@giai] and deformed nuclei [@sarr; @sarrnoj]. For the solution of the HF equations we follow the McMaster procedure that is based in the formalism developed in ref. [@vautherin] as described in ref. [@vallieres]. Time reversal and axial symmetry are also assumed here.
In both schemes, WS and HF, the single-particle wave functions are expanded in terms of the eigenstates of an axially symmetric harmonic oscillator in cylindrical coordinates, which are written in terms of Laguerre and Hermite polynomials. The single-particle states $\left| i\right\rangle $ and their time reversed $\left| \bar{i}\right\rangle $ are characterized by the eigenvalues $\Omega $ of $J_{z}$, parity $\pi _i$, and energy $\epsilon _{i}$
$$\left| i\right\rangle =\sum_{N}\frac{\left( -1\right) ^{N}+ \pi _{i}}{2}
\sum_{n_{r},n_{z},\Lambda \geq 0,\Sigma } C_{Nn_{r}n_{z}\Lambda
\Sigma }^{i}\left| Nn_{r}n_{z}\Lambda \Sigma \right\rangle \; \label{sp_i}$$
with $\Omega _{i}=\Lambda +\Sigma \geq \frac{1}{2}$, and
$$\left| \bar{i}\right\rangle =\sum_{N}\frac{\left( -1 \right) ^{N}+
\pi _{i}}{2} \sum_{n_{r},n_{z},\Lambda \geq 0, \Sigma }C_{Nn_{r}n_{z}
\Lambda \Sigma }^{i}\left( -1 \right) ^{\frac{1}{2}-\Sigma }\left|
Nn_{r}n_{z}-\Lambda -\Sigma \right\rangle \label{sp_ibar}$$
with $\Omega _{\bar{i}}=-\Omega _{i}=-\Lambda -\Sigma \leq -\frac{1}{2}$. For each $N$ the sum over $n_{r},n_{z},\Lambda \geq 0$ is extended to the quantum numbers satisfying $2n_{r}+n_{z}+\Lambda =N.$ The sum over $N$ goes from $N=0$ to $N=10$ in our calculations.
Pairing correlations between like nucleons are included in both cases in the BCS approximation with fixed gap parameters for protons $\Delta _{\pi},$ and neutrons $\Delta _{\nu}$.
The number equation in the neutron sector reads
$$2\sum_{i}v_{i}^{2}=N \label{numeq}$$
where $v_{i}^{2}$ are the occupation probabilities
$$v_{i}^{2}=\frac{1}{2}\left[ 1-\frac{\epsilon _{i}- \lambda _{\nu}}{E_{i}}
\right] \;;\;\;u_{i}^{2}=1-v_{i}^{2} \label{occu}$$
in terms of the quasiparticle energies
$$E_{i}=\sqrt{\left( \epsilon _{i}-\lambda _{\nu} \right) ^{2}+
\Delta _{\nu}^{2}}
\label{qpener}$$
These equations are solved iteratively for the WS and HF single-particle energies to determine the Fermi level $\lambda _{\nu}$ and the occupation probabilities. Similar equations are used to determine the Fermi level and occupation probabilities for protons by changing $N$ into $Z$, $\Delta _{\nu}$ into $\Delta _{\pi}$, and $\lambda_{\nu} $ into $\lambda _{\pi}$.
The fixed gap parameters are determined phenomenologically from the odd-even mass differences through a symmetric five term formula involving the experimental binding energies [@audi]:
$$\begin{aligned}
\Delta _{\nu} &=&\frac{1}{8}\left[ B\left( N-2,Z\right) -4B\left(
N-1,Z\right) +6B\left( N,Z\right) \right. \nonumber \\
&&\left. -4B\left( N+1,Z\right) +B\left( N+2,Z\right) \right]
\label{gaps}\end{aligned}$$
A similar expression is found for the proton gap $\Delta _{\pi}$ by changing $N $ by $Z$ and vice versa. For $^{76}$Ge we obtain $\Delta _{\nu}= 1.54$ MeV, $\Delta_{\pi}=1.56 $ MeV and for $^{76}$Se we obtain $\Delta _{\nu}= 1.71$ MeV and $\Delta_{\pi}=1.75 $ MeV.
Therefore, at the quasiparticle mean field level, we can observe several differences with respect to the treatment of the mean field in terms of HF or WS potentials. The most important is that the quadrupole deformation of the ground state is determined selfconsistently in HF and no explicit input parameter is needed. Other differences come from the structure of the two-body density-dependent Skyrme force that contains terms absent in the WS potential, such as a spin-spin interaction in the selfconsistent mean field through the spin exchange operators of the Skyrme force.
Now, we add to the mean field a spin-isospin residual interaction, which is expected to be the most important residual interaction to describe GT transitions. This interaction contains two parts. A particle-hole part, which is responsible for the position and structure of the GT resonance [@homma; @sarr] and a particle-particle part, which is a neutron-proton pairing force in the $J^\pi=1^+$ coupling channel.
$$V^{ph}_{GT} = 2\chi ^{ph}_{GT} \sum_{K=0,\pm 1} (-1)^K \beta ^+_K
\beta ^-_{-K}, \qquad
\beta ^+_K = \sum_{\pi\nu } \left\langle \nu | \sigma _K |
\pi \right\rangle a^+_\nu a_\pi \, ;$$
$$V^{pp}_{GT} = -2\kappa ^{pp}_{GT} \sum_K (-1)^K P ^+_K P_{-K}, \qquad
P ^+_K = \sum_{\pi\nu} \left\langle \pi \left| \left( \sigma_K\right)^+
\right|\nu \right\rangle a^+_\nu a^+_{\bar{\pi}} \, .$$
The two forces $ph$ and $pp$ are defined with a positive and a negative sign, respectively, according to their repulsive and attractive character, so that the coupling strengths $\chi$ and $\kappa $ take positive values.
The particle-hole residual interaction could in principle be obtained consistently from the same Skyrme force used to create the mean field as it was done in refs. [@sarr] to study exotic nuclei. However, in this paper we use as a first attempt the coupling strengths from ref. [@homma]. In this reference, the strengths $\chi ^{ph}_{GT}$, and $\kappa ^{pp}_{GT}$ are considered to be smooth functions of the mass number $A$, proportional to $A^\mu$. The strength of the $ph$ force is determined by adjusting the calculated positions of the GT giant resonances for $^{48}$Ca, $^{90}$Zr and $^{208}$Pb. This gives a mass dependence with $\mu =0.7$. The same mass dependence is assumed for the $pp$ force and the coefficient is determined by a fitting procedure to $\beta$-decay half-lives of nuclei with $Z\le 40$. The result found in ref. [@homma] is $\chi ^{ph}_{GT}= 5.2 \; /A^{0.7}$ MeV and $\kappa ^{pp}_{GT}= 0.58\; /A^{0.7}$ MeV. A word of caution is in order concerning this parametrization of the residual forces. It serves to our purpose of comparing the effects of different deformed mean fields on the GT strength distributions, but one should keep in mind that the coupling strengths obtained in this way depend in particular, on the model used for single particle wave functions and on the set of experimental data considered. In ref. [@homma] a Nilsson potential was used and the set of experimental data did not include the nuclei under study here. Therefore, the coupling strengths of ref. [@homma] cannot be safely extrapolated and are not necessarily the best possible choices. As we shall see in the next sections, the strengths from ref. [@homma] reproduce well the data when using the WS potential, but one needs a somewhat smaller value of $\chi ^{ph}_{GT}$ to reproduce the GT resonance with the HF mean field.
The proton-neutron quasiparticle random phase approximation phonon operator for GT excitations in even-even nuclei is written as
$$\Gamma _{\omega _{K}}^{+}=\sum_{\pi\nu}\left[ X_{\pi\nu}^{\omega _{K}}
\alpha _{\nu}^{+}\alpha _{\bar{\pi}}^{+}+Y_{\pi\nu}^{\omega _{K}}
\alpha _{\bar{\nu}} \alpha _{\pi}\right]\, , \label{phon}$$
where $\alpha ^{+}\left( \alpha \right) $ are quasiparticle creation (annihilation) operators, $\omega _{K}$ are the RPA excitation energies, and $X_{\pi\nu}^{\omega _{K}},Y_{\pi\nu}^{\omega _{K}}$ the forward and backward amplitudes, respectively. The solution of the QRPA equations can be found solving first a dispersion relation [@hir], which is of fourth order in the excitation energies $\omega_K$.
In the intrinsic frame the GT transition amplitudes connecting the QRPA ground state $\left| 0\right\rangle \ \ \left( \Gamma _{\omega _{K}}
\left| 0 \right\rangle =0 \right)$ to one phonon states $\left| \omega _K \right\rangle \ \ \left( \Gamma ^+ _{\omega _{K}} \left| 0
\right\rangle = \left| \omega _K \right\rangle \right)$, are given by $$\left\langle \omega _K | \sigma _K t^{\pm} | 0 \right\rangle = \mp
M^{\omega _K}_\pm \, .
\label{intrin}$$
The functions $M^{\omega _K}_\pm $ can be found for instance in [@hir]. The basic ingredients in their structure are the spin matrix elements connecting neutron and proton states with spin operators $$\Sigma _{K}^{\nu\pi}=\left\langle \nu\left| \sigma _{K}\right|
\pi\right\rangle \, ,
\label{sigma}$$ which can be written in terms of the coefficients of the expansion in eqs.(\[sp\_i\])-(\[sp\_ibar\]).
$$\Sigma _{K}^{\nu\pi}=\sum_{Nn_{z}\Lambda \Sigma } C_{Nn_{z}\Lambda
\Sigma +K}^{\nu}C_{Nn_{z}\Lambda \Sigma }^{\pi}\left( 2\Sigma \right)
\sqrt{1+\left| K\right| } \label{sig1}$$
$$\Sigma _{K=1}^{\nu\bar{\pi}}=\sum_{Nn_{z}}C_{Nn_{z}0
\frac{1}{2}}^{\nu}C_{Nn_{z}0\frac{1}{2}}^{\pi}\left( -\sqrt{2}\right)
\label{sig2}$$
Once the intrinsic amplitudes are calculated according to eq. (\[intrin\]), the GT strength $B(GT)_\pm$ in the laboratory system for a transition $I_iK_i (0^+0) \rightarrow I_fK_f (1^+K_f)$ can be obtained as $$B(GT)_\pm = \frac{g_A^2}{4\pi}\left[ \delta_{K_f,0}\left\langle \phi_{K_f}
\left| \sigma_0t^\pm \right| \phi_0 \right\rangle ^2 +2\delta_{K_f,1}
\left\langle \phi_{K_f} \left| \sigma_1t^\pm \right| \phi_0 \right\rangle ^2
\right],
\label{bgt}$$ where we have used the initial and final states in the laboratory frame expressed in terms of the intrinsic states $\left| \phi_K \right\rangle $ using the Bohr-Mottelson factorization [@bohr].
In the simple uncorrelated 2qp approximation, neglecting the residual $ph$ and $pp$ forces, the functions $M^{\omega _K}_\pm$ reduce to the following expressions
$$M^{\omega _K}_+ = u_{\nu} v_{\pi} \Sigma _{K}^{\nu\pi} ;\qquad
M^{\omega _K}_- = v_{\nu} u_{\pi} \Sigma _{K}^{\nu\pi} ,
\label{mspm}$$
where the excitation energies are the bare two quasiparticle energies $\omega_K^{\rm 2qp}=E_\nu +E_\pi$.
The Ikeda sum rule is always fulfilled in our calculations
$$\sum_\omega \left[ \left( M^{\omega }_{-}\right) ^2-
\left( M^{\omega }_{+}\right) ^2 \right] = 3(N-Z) \, .
\label{ikedaeq}$$
Bulk properties
===============
In this Section we present results for the bulk properties of $^{76}$Ge and $^{76}$Se obtained from WS and HF descriptions.
First, we analyze the energy surfaces as a function of deformation. In the case of WS, this is simply done by varying the quadrupole deformation of the potential $\beta_2$, which is an input parameter. In the case of HF, we perform constrained calculations [@constraint], minimizing the HF energy under the constraint of keeping fixed the nuclear deformation.
We can see in fig. 1 the total energy plotted versus the microscopically calculated mass quadrupole moment. The results correspond to HF calculations with the forces SG2 (solid line) and Sk3 (dashed line), as well as to calculations with the WS potential (dotted line). The origin of the energy axis is different in each case but the distance between ticks corresponds always to 1 MeV.
We observe that the HF calculation predicts the existence of two energy minima close in energy, giving rise to shape isomers in these nuclei, while the WS potential originates a single energy minimum, which is in agreement with the absolute prolate minimum in the case of $^{76}$Ge and close to the prolate HF solution in the case of $^{76}$Se.
We can see in Table 1 the experimental and the microscopically calculated charge root mean square radii $r_c$, quadrupole moments $Q_p$, and quadrupole deformations $\beta$ ($\beta = \sqrt{\frac{\pi}{5}} \frac{Q_p}{Zr_c^2}$). In the case of $^{76}$Se, the calculated values correspond to prolate/oblate deformations. The input WS prolate deformation is chosen to be $\beta_2=0.10$ in both nuclei $^{76}$Ge and $^{76}$Se. In the oblate case of the nucleus $^{76}$Se, the WS deformation chosen is $\beta_2=-0.20$. With these values we guarantee that the intrinsic deformations of the ground state are similar in HF and WS and therefore the differences in their predictions will have their origin in the structure of mean fields having the same deformation.
The values obtained for the charge radii are in good agreement with the experimental values from ref. [@radiiexp], which are also shown in Table 1. They are also in good agreement with the results obtained from relativistic mean field calculations [@ring]: $r_{c ({\rm rel})} (^{76}{\rm Ge})=4.057$ fm and $r_{c ({\rm rel})} (^{76}{\rm Se})= 4.119$ fm.
The charge quadrupole moments quoted in Table 1 have been derived microscopically from the deformed potentials as ground state expectations of the $Q_{20}$ operator. We can compare again with the results from relativistic mean field calculations of ref. [@ring]: $Q_{p ({\rm rel})} (^{76}{\rm Ge})= 111.4$ fm$^2$ and $Q_{p ({\rm rel})} (^{76}{\rm Se})= -146.8$ fm$^2$. These relativistic results are in perfect accordance with our calculated results. They can also be compared with experimental intrinsic quadrupole moments from ref. [@raghavan]. The empirical intrinsic moments are derived from the laboratory moments assuming a well defined deformation. These values are shown in Table 1 in the first place: $Q_{p ({\rm exp})} (^{76}{\rm Ge})=66(21)$ fm$^2$ and $Q_{p ({\rm exp})} (^{76}{\rm Se})= 119(25)$ fm$^2$. Experimental quadrupole moments can also be derived [@raman] from the experimental values of $B(E2)$ strengths, although in this case the sign cannot be extracted. Assuming that the intrinsic electric quadrupole moments are given by $Q=\sqrt{16\pi B(E2)/5e^2}$, then $|Q_{p ({\rm exp})}| (^{76}{\rm Ge})=164(24)$ fm$^2$ and $|Q_{p ({\rm exp})}| (^{76}{\rm Se})= 205(24)$ fm$^2$.
Gamow-Teller strength distributions
===================================
In this Section we show and discuss the Gamow-Teller strength distributions obtained from different choices of the deformed mean fields and residual interactions.
As a general rule, the following figures showing the GT strength distributions are plotted versus the excitation energy of daughter nucleus. The distributions of the GT strength have been folded with Breit-Wigner functions of 1 MeV width to facilitate the comparison among the various calculations, so that the original discrete spectrum is transformed into a continuous profile. These distributions are given in units of $g_A^2/4\pi$ and one should keep in mind that a quenching of the $g_A$ factor, typically $g_{A,{\rm eff}}=(0.7-0.8)\ g_{A,{\rm free}}$, is expected on the basis of the observed quenching in charge exchange reactions.
First of all, we discuss in figs. 2 and 3, the dependence of the GT strength distributions on the deformed quasiparticle mean field of $^{76}$Ge and $^{76}$Se, respectively. To make the discussion meaningful we show the results obtained at the two-quasiparticle level without including the spin-isospin residual interactions. In these figures we can see the $B(GT_-)$ and $B(GT_+)$ strength distributions in the upper and lower panels, respectively. One should notice that the relevant strength distributions for the double $\beta$-decay of $^{76}$Ge, as it can be seen schematically in fig. 4, are the $B(GT_-)$ distribution of the parent $^{76}$Ge and the $B(GT_+)$ distribution of daughter $^{76}$Se, but for completeness we show both distributions for each nucleus. Solid lines in figs. 2 and 3 correspond to the results obtained from the Skyrme force SG2 within a HF scheme, dashed lines correspond to the results obtained with the WS potential. The deformation of the mean fields are as indicated in Table 1, using the prolate shape in $^{76}$Se. Pairing correlations are included in HF and WS cases in a similar way with the gap parameters for neutrons and protons mentioned earlier. Then, the only source of discrepancy between HF and WS comes from the different single particle wave functions and energies.
In general, we observe that WS and HF produce a similar structure of three peaks in the $B(GT_-)$ profiles of $^{76}$Ge and $^{76}$Se, although the WS results are somewhat displaced to lower energies with respect to the HF peaks. The strengths contained in the peaks are also comparable. In the case of the $B(GT_+)$ distributions, we first observe the different scale, which is about one order of magnitude lower than the $B(GT_-)$ scale. This is a consequence of the Pauli blocking. We can see from eq. (\[mspm\]) that while the occupation amplitudes $u's$ and $v's$ favor $M_-$ strengths, they are very small factors in $M_+$ strengths when connecting similar proton and neutron states. The difference between total $B(GT_-)$ and $B(GT_+)$ strengths (Ikeda sum rule (\[ikedaeq\]), which is fulfilled in our calculations), is a large number $3(N-Z)=36$ in $^{76}$Ge and $3(N-Z)=24$ in $^{76}$Se, reflecting the different magnitude of the $B(GT_-)$ and $B(GT_+)$ strengths shown in figs. 2 and 3.
The profiles of the $B(GT_+)$ distributions with WS and HF present some discrepancies that are amplified by the scale. In particular, it is remarkable the large strength produced by WS in the region of high excitation energies in $^{76}$Ge that we discuss later in terms of the single particle wave functions.
In order to clarify the origin of the various peaks in the strength distributions we have added in fig. 2 labels showing some of the leading transitions generating the strength. The labels stand for $pK^{\pi} - nK^{\prime\pi}$ of the orbitals connected by the spin operator in eq. (\[sigma\]) and a number that identifies the transition. In both cases, $B(GT_-)$ and $B(GT_+)$, the same type of transitions are connected by the GT operator but the occupation probabilities, weighting the matrix elements, enhance or reduce them accordingly. We can see from fig. 2 that the structure of the profiles in both WS and HF are generated by the same type of GT transitions.
This can be further illustrated by looking at fig. 5, where we show the single particle energies for protons and neutrons obtained in HF(SG2) and WS in $^{76}$Ge. In the left part of the figure corresponding to the HF calculation we have plotted the occupation probabilities $v_\nu^2$ and $v_\pi^2$ and the Fermi energies $\lambda_\nu$ and $\lambda_\pi$. We can also see for completeness the spherical levels labeled by their $\ell_j$ values. We have indicated by arrows the most relevant Gamow-Teller transitions in the $\beta^-$ and $\beta^+$ directions that are labeled by the same numbers used in fig. 2 to identify the peaks. To be more precise, we can see in Table 2 the correspondence between these labels and the transitions connecting the proton and neutron states using the asymptotic quantum number notation $[Nn_z\Lambda ]K^\pi$.
Now, looking at fig. 2, we can understand that the two first peaks in $B(GT_-)$ are generated mainly by transitions between neutrons and protons dominated by contributions within the $N=3$ shell and that the third peak is generated by transitions between neutrons and protons with main contributions coming from the $N=4$ shell. The different energies of the peaks are due to the different concentration of energy levels in HF and WS.
In the case of $B(GT_+)$, the strength below 8 MeV is mainly generated by transitions within the $N=3$ shells. Beyond 8 MeV the strength, which is negligible in HF, is generated by transitions between the proton shell $N=2$ and the neutron shell $N=4$ as well as between the proton shell $N=3$ and the neutron shell $N=5$, always understood as the main components of the wave functions. Then, very deep inside protons $(v_p=1)$ are connected with very unoccupied neutron states $(u_n=1)$, giving rise to maximum occupation factors. The different behavior in this high energy region between HF and WS is therefore due, other factors like deformation and occupations being equal, to the structure of the deformed orbitals.
To illustrate the role of the different single particle wave functions in the development of the peak structure, we consider in detail the case of the last peak observed in the $B(GT_+)$ distribution of the WS potential. As we can see it is mainly due to a transition between the proton state $[303]$ in the $N=3$ shell with $K^{\pi}=7/2^-$ and the neutron state $[523]$ in the $N=5$ shell with $K^{\pi}=5/2^-$. The structure of the single particle wave functions, according to the expansion in eq. (\[sp\_i\]), of these two states in the cases of HF and WS can be seen in Table 3. With these coefficients we can construct the matrix elements in eq. (\[sig1\]). The resulting strength is almost two orders of magnitude in favor of WS, which explains the huge discrepancy observed between WS and HF in the higher energy domain.
Nevertheless, these discrepancies are smaller in the case of the $B(GT_+)$ of $^{76}$Se, which is the relevant branch for the double $\beta$-decay of the parent nucleus $^{76}$Ge.
Figures 6 and 7 contain the strength distributions obtained from QRPA calculations for $^{76}$Ge and $^{76}$Se, respectively. The data in Fig. 6 are from ref. [@geexp] and were obtained from charge exchange $^{76}$Ge(p,n)$^{76}$As reactions. The thick line in Fig. 6 corresponds to these data folded by the same procedure used for the theoretical results. The data in Fig. 7 and 8 are from ref. [@seexp] and were obtained from charge exchange $^{76}$Se(n,p)$^{76}$As reactions.
The coupling constants of the $ph$ and $pp$ residual interactions used in Figs 6-8 are from ref. [@homma] in the case of WS. In our case with $A=76$, these parameters are $\chi ^{ph}_{GT}=0.25$ MeV and $\kappa ^{pp}_{GT}=0.027$ MeV. In the case of the HF calculations with the Skyrme forces Sk3 and SG2 better agreement with the measured location of the $B(GT_-)$ resonance in $^{76}$Ge is obtained with a somewhat smaller value of the $ph$ strength. The curves shown in Figs. 6-8 for the HF results have been obtained using $\chi ^{ph}_{GT}=0.16$ MeV and the same $\kappa ^{pp}_{GT}=0.027$ MeV.
In Fig. 6 we have used the prolate deformations for $^{76}$Ge given in Table 1. We can see that WS follows the structure of the experimental $B(GT_-)$ strength distribution with two peaks at low energies ($E_{ex}=$5 and 8 MeV) and the resonance at 11 MeV. The HF calculations produce also a few peaks at low excitation energies and a resonance between 10 and 13 MeV. We can see that the structure of the strength distributions is qualitatively similar for the two Skyrme forces and that the difference with the WS curves can be traced back to the discrepancies found at the two quasiparticle level.
Fig. 7 contains similar calculations for $^{76}$Se. The coupling strengths of the residual forces are as indicated for $^{76}$Ge. The results in the HF cases are obtained with the oblate deformation of $^{76}$Se that produces the absolute minimum of the energy and agrees better with the experimental quadrupole moment. In general, comparison with experiment is reasonable and should not be stressed too much since, as stated in ref. [@seexp], the experimental results, especially above 6 MeV, must be considered to be of a qualitative nature only.
The role of the residual interactions on the GT strengths was already studied in ref. [@sarr], where it was shown that the repulsive $ph$ force introduces two types of effects: A shift of the GT strength to higher excitation energies with the corresponding displacement of the position of the GT resonance and a reduction of the total GT strength. The residual $pp$, being an attractive force, shifts the strength to lower excitation energies, reducing the total GT strength as well. Also shown in ref. [@sarr] was the effect of the BCS correlations on the GT strength distribution. The main effect of pairing correlations is to create new transitions that are forbidden in the absence of such correlations. The main effect of increasing the Fermi diffuseness is to smooth out the profile of the GT strength distribution, increasing the strength at high energies and decreasing the strength at low energies.
The role of deformation was also studied in ref. [@sarr], showing that the GT strength distributions corresponding to deformed nuclear shapes are much more fragmented than the corresponding spherical ones, as it is clear because deformation breaks down the degeneracy of the spherical shells. It was also shown that the crossing of deformed energy levels that depends on the magnitude of the quadrupole deformation as well as on the oblate or prolate character, may lead to sizable differences between the GT strength distributions corresponding to different shapes.
We can see in fig. 8 the GT strength distributions in $^{76}$Se obtained from spherical, prolate and oblate shapes. They correspond to QRPA calculations performed with the HF basis obtained with the force Sk3. In the spherical case, the only possible transitions (see fig. 5) are those connecting spherical $\ell_j$ partners with $\Delta \ell=0,\; \Delta j=0,1$, in allowed approximation. Therefore, there is GT strength only at a few excitation energies. The strength we observe in fig. 8 is the result of the folding procedure performed at these energies. On the other hand, in the deformed cases we can observe a stronger fragmentation, which is the result of all possible connections among the deformed states (see fig. 5). Thus, the spherical peaks become broader when deformation is present.
We can see in Table 4 the total GT strengths in $^{76}$Se contained below an energy cut of 60 MeV. We show the results obtained for $\beta^+$ and $\beta^-$ strengths with oblate, spherical, and prolate shapes. The Ikeda sum rule $3(N-Z)=24$, is fulfilled at this energy cut within a $0.3\%$ accuracy. We can see from Table 4 that deformation increases both $\beta^+$ and $\beta^-$ strengths in a similar amount in order to preserve the Ikeda sum rule ($\beta^- - \beta^+$). We also show for comparison the results obtained in 2qp approximation. We can see the reduction of the strength introduced by the QRPA correlations, which is again similar in absolute terms for $\beta^+$ and $\beta^-$ strengths in order to keep the Ikeda sum rule conserved in QRPA. Since the $\beta^-$ strength is much larger than the $\beta^+$ strength, the relative effect of the QRPA correlations is much stronger for $\beta^+$, where the total strength is reduced by a 50%.
Comparing the results for $^{76}$Se obtained at different deformations with the selfconsistent mean fields (HF with Sk3) in Fig. 8 and Table 4, we see that there is a strong dependence on deformation in the strength distributions as a function of the energy. However, the total strength does not depend so much on deformation. There is an increase of a few percent in going from the spherical to the oblate and prolate shapes. The latter observation enters in contradiction with SU(3) and shell model calculations by previous authors [@auerbach] on the dependence on deformation of the GT strengths in $^{20}$Ne and $^{44}$Ti. We think that this is due to the much larger and richer single particle basis used in the present calculations. In our case each single particle state contains mixtures from many harmonic oscillator shells (up to $N=10$), while in the above mentioned calculations [@auerbach], the single particle basis is restricted to a single harmonic oscillator major shell (the $sd$ shell in $^{20}$Ne and the $fp$ shell in $^{44}$Ti). On the other hand, one may question whether in the deformed cases the total strengths calculated here may contain spurious contributions from higher angular momentum components in the initial and final nuclear wave functions. Since the matrix elements of the transition operator, which is a dipole tensor operator, are taken between the states considered in the laboratory frame (see eq. (\[bgt\])), the effect of angular momentum projection is to a large extent taken into account. We have calculated an upper bound to such contributions using angular momentum projection techniques [@elvira]. We find that this upper bound is less than $1\%$ ($\sim \left\langle J_\perp ^2\right\rangle ^{-2}$, with $\left\langle J_\perp ^2\right\rangle=19$ for the oblate shape in $^{76}$Se). Thus, exact angular momentum projection would not wash out the small increase of the total strength with deformation.
Concluding remarks
==================
We have studied the GT strength distributions for the two decay branches $\beta^-$ and $\beta^+$ in the double $\beta$-decay of $^{76}$Ge. This has been done within a deformed QRPA formalism, which includes $ph$ and $pp$ separable residual interactions. The quasiparticle mean field includes pairing correlations in BCS approximation and it is generated by two different methods, a deformed HF approach with Skyrme interactions and a phenomenological deformed WS potential. One difference is that with HF and Sk3 we get the minimum and stable deformation for $^{76}$Se to be oblate, while the prolate minimum is comparable to that obtained with WS and it is higher in energy.
We have studied the similarities and differences observed in the GT strength distributions with these two methods. Among the similarities we can mention the structure of peaks found in the strength distributions and among the differences the displacement in the excitation energies found between HF and WS results. This discrepancy has its origin in the structure of the single particle wave functions and energies generated by the deformed mean fields. This also implies that different mean fields require different residual interactions to reproduce the experimental GT resonances.
Therefore, in order to obtain reliable GT strength distributions and consequently reliable estimates for double $\beta$-decay half-lives, it is important to have not only the proper residual interactions but also a good deformed single particle basis as a starting point. In the case of HF we have seen that standard Skyrme forces, such as SG2 or Sk3, give a good description of the GT strength distributions provided the proper residual interactions are included. Even though the selfconsistent HF approach is a more sophisticated type of calculation, the deformed WS potential produces comparable results when the parameters of the potential and the residual interactions for a given mass region are chosen properly.
There is work in progress to extend these calculations to the double $\beta$-decay process studying the dependence on deformation of the half-lives.
[**Acknowledgments**]{}
This work was supported by Ministerio de Ciencia y Tecnología (Spain) under contract numbers PB98/0676 and BFM2002-03562 and by International Graduiertenkolleg GRK683, by the “Land Baden-Wuerttemberg” within the “Landesforschungsschwerpunkt: Low Energy Neutrinos”, and by the DFG under 436SLK 17/2/98.
[00]{}
J. Suhonen and O. Civitarese, Phys. Rep. [**300**]{} (1998) 123; A. Faessler and F. Šimkovic, J. Phys. G: Nucl. Part. Phys. [**24**]{} (1998) 2139; H.V. Klapdor-Kleingrothaus, [*Sixty Years of Double Beta Decay*]{}, World Scientific, Singapore (2001).
J.A. Halbleib and R.A. Sorensen, Nucl. Phys. A [**98**]{} (1967) 542.
P. Vogel and M.R. Zirnbauer, Phys. Rev. Lett. [**57**]{} (1986) 3148; D. Cha, Phys. Rev. C [**27**]{} (1987) 2269; T.Tomoda and A. Faessler, Phys. Lett. B [**199**]{} (1987) 475; K. Muto, E. Bender and H.V. Klapdor, Z. Phys. A [**334**]{} (1989) 177.
J. Hirsch, E. Bauer and F. Krmpotić, Nucl. Phys. A [**516**]{} (1990) 304; A.A. Raduta, A. Faessler, S. Stoica, and W.A. Kaminski, Phys. Lett. B [**254**]{} (1991) 7; J. Toivanen and J. Suhonen, Phys. Rev. Lett. [**75**]{} (1995) 410; J. Schwieger, F. Šimkovic, and A. Faessler, Nucl. Phys. A [**600**]{} (1996) 179; A.A. Raduta, M.C. Raduta, A. Faessler, and W.A. Kaminski, Nucl. Phys. A [**634**]{} (1998) 497; A.A. Raduta, F. Šimkovic, and A. Faessler, J. Phys. G [**26**]{} (2000) 793.
J. Krumlinde and P. Möller, Nucl. Phys. A [**417**]{} (1984) 419.
P. Moller and J. Randrup, Nucl. Phys. A [**514**]{} (1990) 1.
H. Homma, E. Bender, M. Hirsch, K. Muto, H.V. Klapdor-Kleingrothaus and T. Oda, Phys. Rev. C [**54**]{} (1996) 2972.
P. Sarriguren, E. Moya de Guerra, A. Escuderos, and A.C. Carrizo, Nucl. Phys. A [**635**]{} (1998) 55; P. Sarriguren, E. Moya de Guerra, and A. Escuderos, Nucl. Phys. A [**658**]{} (1999) 13; Nucl. Phys. A [**691**]{} (2001) 631; Phys. Rev. C [**64**]{} (2001) 064306.
J. Engel, M. Bender, J. Dobaczewski, W. Nazarewicz, and R. Surnam, Phys. Rev. C [**60**]{} (1999) 014302; M. Bender, J. Dobaczewski, J. Engel, and W. Nazarewicz, Phys. Rev. C [**65**]{} (2002) 054322.
A.A. Raduta, A. Faessler, and D.S. Delion, Nucl. Phys. A [**564**]{} (1993) 185; Phys. Lett. B [**312**]{} (1993) 13; Nucl. Phys. A [**617**]{} (1997) 176.
F. Šimkovic, L. Pacearescu, and A. Faessler, preprint [*Two-neutrino double beta decay of $^{76}$Ge within deformed QRPA*]{}.
H. Flocard, P. Quentin and D. Vautherin, Phys. Lett. B [**46**]{} (1973) 304; P. Quentin and H. Flocard, Annu. Rev. Nucl. Part. Sci. [**28**]{} (1978) 253; P. Bonche, H. Flocard, P.H. Heenen, S.J. Krieger, and M.S. Weiss, Nucl. Phys. A [**443**]{} (1985) 39.
Y. Tanaka, Y. Oda, F. Petrovich and R.K. Sheline, Phys. Lett. B [**83**]{} (1979) 279
R. Nojarov, A. Faessler, P. Sarriguren, E. Moya de Guerra and M. Grigorescu, Nucl. Phys. A [**563**]{} (1993) 349; P. Sarriguren, E. Moya de Guerra, R. Nojarov and A. Faessler, J. Phys. G: Nucl. Part. Phys. [**20**]{} (1994) 315; J.M. Udias, R. Nojarov and A. Faessler, J. Phys. G: Nucl. Part. Phys. [**23**]{} (1997) 1673.
M. Beiner, H. Flocard, N. Van Giai, and P. Quentin, Nucl. Phys. A [**238**]{} (1975) 29.
N. Van Giai and H. Sagawa, Phys. Lett. B [**106**]{} (1981) 379.
P. Sarriguren, E. Moya de Guerra, and R. Nojarov, Phys. Rev. C [**54**]{} (1996) 690; P. Sarriguren, E. Moya de Guerra, and R. Nojarov, Z. Phys. A [**357**]{} (1997) 143.
D. Vautherin and D. M. Brink, Phys. Rev. C [**5**]{} (1972) 626; D. Vautherin, Phys. Rev. C [**7**]{} (1973) 296.
M. Vallières and D.W.L. Sprung, Can. J. Phys. [**56**]{} (1978) 1190.
G. Audi and A.H. Wapstra, Nucl. Phys. A [**595**]{} (1995) 409; G. Audi, O. Bersillon, J. Blachot, and A.H. Wapstra, Nucl. Phys. A [**624**]{} (1997) 1.
M. Hirsch, A. Staudt, K. Muto, and H.V. Klapdor-Kleingrothaus, Nucl. Phys. A [**535**]{} (1991) 62; K. Muto, E. Bender, and H.V. Klapdor, Z. Phys. A [**333**]{} (1989) 125; K. Muto, E. Bender, T. Oda and H.V. Klapdor-Kleingrothaus, Z. Phys. A [**341**]{} (1992) 407.
A. Bohr and B. Mottelson, [*Nuclear Structure*]{} (Benjamin, New York, 1975).
H. Flocard, P. Quentin, A.K. Kerman, and D. Vautherin, Nucl. Phys. A [**203**]{} (1973) 433.
H. de Vries, C.W. de Jager and C. de Vries, At. Data Nucl. Data Tables [**36**]{} (1987) 495; G. Fricke, C. Bernhardt, K. Heilig, L.A. Schaller, L. Schellenberg, E.B. Shera, C.W. de Jager, At. Data and Nucl. Data Tables [**60**]{} (1995) 177.
G.A. Lalazissis, S. Raman and P. Ring, At. Data and Nucl. Data Tables [**71**]{} (1999) 1.
P. Raghavan, At. Data Nucl. Data Tables [**42**]{} (1989) 189; N.J. Stone, Oxford University, preprint (2001).
S. Raman, C.H. Malarkey, W.T. Milner, C.W. Nestor Jr. and P.H. Stelson, At. Data Nucl. Data Tables [**36**]{} (1987) 1.
R. Madey, B.S. Flanders, B.D. Anderson, A.R. Baldwin, J.W. Watson, S.M. Austin, C.C. Foster, H.V. Klapdor and K. Grotz, Phys. Rev. C [**40**]{} (1989) 540.
R.L. Helmer [*et al.*]{}, Phys. Rev. C [**55**]{} (1997) 2802.
N. Auerbach, D.C. Zheng, L. Zamick and B.A. Brown, Phys. Lett. B [**304**]{} (1993) 17; D. Troltenier, J.P. Draayer and J.G. Hirsch, Nucl. Phys. A [**601**]{} (1996)89.
E. Moya de Guerra, Phys. Rep. [**138**]{} (1986) 293.
[**Table 1.**]{} Charge root mean square radii $r_c$ \[fm\], intrinsic charge quadrupole moments $Q_p$ \[fm$^2$\], and quadrupole deformations $\beta$ for $^{76}$Ge and $^{76}$Se calculated with various assumptions for the mean field. In the case of $^{76}$Se we show theoretical values corresponding to the prolate shape in first place and to the oblate shape in second place. Experimental values for $r_c$ are from [@radiiexp] and for $Q_p$ from [@raghavan] the first value and from [@raman] the second (see text).
0.5cm
[llccc]{} & & $r_c$ & $Q_p$ & $\beta$\
\
$^{76}$Ge & exp. & 4.080 - 4.127 & 66(21) - 164(24) & 0.10 - 0.24\
\
& Sk3 & 4.130 & 111.0 & 0.161\
& SG2 & 4.083 & 105.9 & 0.157\
& WS & 3.950 & 110.9 & 0.176\
\
$^{76}$Se & exp. & 4.088 - 4.162 & 119(25) - 205(24) & 0.16 - 0.29\
\
& & prol / obl & prol / obl & prol / obl\
& Sk3 & 4.170 / 4.180 & 117.5 / -136.0 & 0.158 / -0.181\
& SG2 & 4.113 / 4.143 & 35.2 / -140.6 & 0.049 / -0.191\
& WS & 3.991 / 4.138 & 81.6 / -141.4 & 0.119 / -0.193\
0.5cm
[**Table 2.**]{} Correspondence of the labels used in figs. 2 and 5 with the asymptotic quantum numbers notation $[Nn_z\Lambda ]K^\pi$.
[ccccc]{} & $\beta^-$ & & $\beta^+$\
\
(1) & $\nu [301]1/2^- \rightarrow \pi [301]3/2^-$ & & $\pi [303]7/2^- \rightarrow \nu [303]5/2^-$\
(2) & $\nu [301]3/2^- \rightarrow \pi [301]1/2^-$ & & $\pi [312]5/2^- \rightarrow \nu [303]5/2^-$\
(3) & $\nu [303]7/2^- \rightarrow \pi [303]5/2^-$ & & $\pi [312]5/2^- \rightarrow \nu [312]3/2^-$\
(4) & $\nu [312]5/2^- \rightarrow \pi [312]3/2^-$ & & $\pi [202]3/2^+ \rightarrow \nu [413]5/2^+$\
(5) & $\nu [420]1/2^+ \rightarrow \pi [440]1/2^+$ & & $\pi [330]1/2^- \rightarrow \nu [530]1/2^-$\
(6) & & & $\pi [303]7/2^- \rightarrow \nu [523]5/2^-$\
[**Table 3.**]{} Main coefficients $C^i_{\alpha}$ in the expansion of eq. (\[sp\_i\]) for the proton state $[303]$ with $K^\pi =7/2^-$ and the neutron state $[523]$ with $K^\pi =5/2^-$. This is the main contribution to the peak at 15 MeV in the $B(GT_+)$ strength distribution of $^{76}$Ge with the Woods-Saxon potential. The basis states are labeled by $\left| N n_z \Lambda \right\rangle$ quantum numbers. The table contains also the contributions from these basis states to the spin matrix elements in eqs. (\[sigma\])-(\[sig1\]).
0.5cm
$\left| 303\right\rangle$ $\left| 503\right\rangle$ $\left| 523\right\rangle$ $\left| 703\right\rangle$ $\left| 723\right\rangle$ $\left| 903\right\rangle$
---------------------------------------- --------- --------------------------- --------------------------- --------------------------- --------------------------- --------------------------- ---------------------------
$7/2^-$ proton
HF(SG2) -0.9742 0.2204 -0.0061 0.0219 -0.0272 0.0122
WS 0.9876 -0.1400 0.0563 -0.0233 0.0107 -0.0295
$5/2^-$ neutron
HF(SG2) 0.1369 0.5933 -0.5031 -0.3928 0.2349 0.2385
WS -0.2397 -0.5173 0.5049 0.3794 -0.2596 -0.2056
contribution to $\Sigma _{K}^{\nu\pi}$
HF(SG2) -0.1333 0.1308 0.0031 -0.0107 -0.0064 0.0029
WS -0.2367 0.0724 0.0284 -0.0088 -0.0028 0.0061
[**Table 4.**]{} Total Gamow-Teller strength in $^{76}$Se calculated with the force Sk3. Results correspond to $\beta^+$ and $\beta^-$ strengths for the oblate, spherical, and prolate shapes calculated in 2qp and QRPA approximations. All the GT strength contained below an excitation energy of 60 MeV has been included.
0.5cm
[lcccccccccc]{} && && &&\
&& $\beta^+$ & $\beta^-$ && $\beta^+$ & $\beta^-$ && $\beta^+$ & $\beta^-$\
\
RPA && 2.420 & 26.331 && 1.846 & 25.765 && 2.599 & 26.524\
2qp && 4.387 & 28.298 && 3.816 & 27.736 && 4.971 & 28.892
|
---
abstract: |
We say that two graphs $H_1,H_2$ on the same vertex set are $G$-creating ($G$-different in other papers, this difference is explained in the introduction) if the union of the two graphs contains $G$ as a subgraph. Let $H(n,k)$ be the maximal number of pairwise $C_k$-creating paths (of arbitrary length) on $n$ vertices. The behaviour of $H(n,2k+1)$ is much better understood than the behaviour of $H(n,2k)$, the former is an exponential function of $n$ while the latter is larger than exponential, for every fixed $k$. We study $H(n,k)$ for fixed $k$ and $n$ tending to infinity. The only non trivial upper bound on $H(n,2k)$ was in the case where $k=2$ $$H(n,4)\leq n^{\left(1-\frac{1}{4} \right) n-o(n)},$$ this was proved by Cohen, Fachini and Körner. In this paper, we generalize their method to prove that for every $k \geq 2$,
$$H(n,2k) \leq n^{\left( 1- \frac{2}{3k^2-2k} \right)n-o(n)}.$$ Our proof uses constructions of bipartite, regular, $C_{2k}$-free graphs with many edges by Reiman, Benson, Lazebnik, Ustimenko and Woldar. For some special values of $k$ we can have slightly denser such bipartite graphs than for general $k$, this results in having better upper bounds on $H(n,2k)$ than stated above for these special values of $k$.
address: 'MTA, Rényi Institute'
author:
- Daniel Soltész
title: Even cycle creating paths
---
[^1]
Introduction
============
The problem of determining the maximal number of pairwise $G$-creating paths on $n$ vertices has a code theoretic flavour. Indeed, we wish to have as many objects as possible (paths in this case) with the restriction that every pair of objects is different in a prescribed way (having $G$ in their union). The original motivation for these problems ultimately came from a desire to understand Shannon capacity of graphs [@gabortoldme]. In previous papers [@original; @triangle; @komesi] instead of $G$-creating, the name $G$-different was used to highlight the connection with code theory. After multiple talks about the subject this name turned out to be confusing or not satisfactory for a large portion of the audiences, hence in this paper we use $G$-creating.
Observe that in the definition of $H(n,k)$, we can safely assume that each path is of maximal length. Indeed, given a set of pairwise $C_k$-creating paths, if one of the paths $P$ is not of maximal length, we can add extra edges to it until its length reaches $n-1$. This new maximal length path was not in the original family of paths since its union with $P$ does not contain any cycle.
The study of $H(n,k)$ was initiated in [@komesi]. The authors of [@komesi] were motivated by a question concerning permutations. Hence they defined $H(n,k)$ using Hamiltonian paths of the complete graph $K_n$. They observed that the maximal number of Hamiltonian paths of $K_n$ so that every pairwise union contains an odd cycle is the number of balanced bipartitions of $[n]$. (The requirement that each union contains an odd cycle is equivalent to the requirement that no union can be bipartite. Since every Hamiltonian path is a balanced bipartite graph we cannot have more than the number of balanced bipartitions of $[n]$. And since a Hamiltonian path has a unique bipartition as a bipartite graph, any system of Hamiltonian paths with pairwise different balanced bipartitions satisfies our conditions. ) They asked whether the answer remains the same if we insist on having a triangle in every union. This was answered affirmatively in [@triangle].
\[thm:triangle\] For every integer $n \geq 3$, $$H(n,3)= \begin{cases} \binom{n}{\left \lfloor \frac{n}{2} \right \rfloor} & \text{ when } n \equiv 1 \mod{2} \\
\frac{1}{2}\binom{n}{\left \lfloor \frac{n}{2} \right \rfloor} & \text{ when } n \equiv 0 \mod{2}.
\end{cases}$$
Theorem \[thm:triangle\] implies that $H(n,3)= 2^{n-o(n)}$. The hard part of Theorem \[thm:triangle\] is the construction of a suitable set of Hamiltonian paths. The method of the construction in [@triangle] was generalized in [@oddcycle].
For every integer $ \ell \geq 1$, $$H(n,2^{\ell}+1)= 2^{n-o(n)}.$$
It is conjectured in [@oddcycle] that the behaviour of $H(n,2k+1)$ is similar for other values of $k$.
\[conj:oddcycle\] For every integer $k \geq 1$, $$H(n,2k+1)=2^{n-o(n)}.$$
The behaviour of $H(n,2k)$ is very different from the behaviour of $H(n,2k+1)$. Upper and lower bounds for $H(n,4)$ were established in [@original].
\[thm:original\] For every $n$, $$n^{\frac{1}{2}n-o(n)} \leq H(n,4) \leq n^{\frac{3}{4}n-o(n)}.$$
An easy generalization of the construction in [@original] gives $n^{\frac{1}{k}-o(n)} \leq H(n,2k)$. In this paper we generalize the upper bound of Cohen, Fachini and Körner for longer even cycles.
\[thm:main\] For every positive integer $k \geq 2$,
------------------------------------------------------------- -------------------------------------
$H(n,2k)\leq n^{\left( 1- \frac{1}{k^2} \right)n-o(n)}$ when $k=2,3,5$
$H(n,2k)\leq n^{\left( 1- \frac{2}{3k^2-2k} \right)n-o(n)}$ when $k \neq 2,3,5$ and $k$ is even
$H(n,2k)\leq n^{\left( 1- \frac{2}{3k^2-3k} \right)n-o(n)}$ when $k \neq 2,3,5$ and $k$ is odd
------------------------------------------------------------- -------------------------------------
Observe that the case when $k=2$ gives the upper bound of Theorem \[thm:original\]. The reason why the upper bounds are different in the three cases is that for the proof we need the existence of bipartite, regular, $C_{2k}$-free graphs with many edges, and for different values of $k$, the order of magnitude of the number of edges for the known constructions is different, see Table \[results\].
The paper is organized as follows. In the short second section we present simple constructions for lower bounds on $H(n,2k)$. In Section \[sec:main\] we prove Theorem \[thm:main\], in Section \[sec:reversing\] we elaborate on the connection between $M(n,4)$ and the maximal number of pairwise reversing permutations (to be defined later). Finally in Section \[sec:concluding\] we highlight the similarities between the proof of the present paper and the results of Cibulka, Cohen, Fachini and Körner. Also in Section \[sec:concluding\] we elaborate on the strong connections between $M(n,2k)$ and $H(n,2k)$.
Lower bound {#sec:lower}
===========
In this short section we present a simple lower bound on $H(n,2k)$.
For every $k$, $n^{\frac{1}{k}n-o(n)} \leq H(n,2k).$
It is enough to construct a suitable family of Hamiltonian paths when $n \equiv 1 \mod{k}$ since $k$ is fixed and we can safely ignore a constant number of vertices. We build directed Hamiltonian paths the sole reason for this is that we can refer to the first or second etc. vertex of the path. For every $t=1,\ldots, (n-k-1)/k$, the $(tk+1)$-th vertex of every Hamiltonian path will be the vertex $tk+1 \in [n]$, we call these the *fixed vertices*. Moreover, every Hamiltonian path will contain the following set of paths: $(2,3,\ldots,k), \ldots, (tk+2,tk+3, \ldots (t+1)k), \ldots (n-k+1,\ldots,n-1)$ directed towards the larger elements of $[n]$, we call these the *fixed paths*. The only difference between the Hamiltonian paths will be the order of the fixed paths between the fixed vertices. For each permutation of these paths we will associate a Hamiltonian path which traverses the fixed paths in this order. (Recall that the fixed vertices have their positions fixed in every path.) Since the number of fixed paths is $(n-1)/k$, the number of Hamiltonian paths is $n^{\frac{1}{k}n-o(n)}.$ Two such paths are $C_{2k}$-creating by the following reasoning. If their permutations differ on the $i$-th coordinate then the vertices $(i-1)k$ and $ik$ are connected by a different fixed path of length $k$, in each Hamiltonian path. This results in a cycle of length $2k$ in the union.
Note that when $k=2$ the fixed paths consist of a single vertex. Also note that when $k>2$, the fixed paths of length $k-1$ can be used to enlarge the set of $C_{2k}$-creating Hamiltonian paths in the following way: If we have two sets of paths of length $k-1$ that have a $C_{2k}$ in their union, on the non fixed vertices, the two sets of paths can both be used as fixed paths. The resulting system of Hamiltonian paths will be $C_{2k}$-creating altogether.
The author was not able to improve more than an exponential factor with this construction method. But he was not able to prove that this method can only improve an exponential factor either, see Question \[q:1\] in Section \[sec:concluding\].
Upper bound {#sec:main}
===========
The proof of the upper bound mimics the proof of the non-trivial upper bound by Cohen, Fachini and Körner for $C_4$-creating Hamiltonian paths [@original]. Their proof takes a large set of $C_4$-creating Hamiltonian paths and produces a still large set of so called pairwise flipful permutations. (Two permutations are flipful if there are two coordinates where the two permutations have the same two elements but the order of these elements is different in the two permutations.) Then they use a theorem of Cibulka [@cib] to have an upper bound for the maximal number of pairwise flipful permutations. We will proceed similarly but instead of flipful permutations we will use $C_{2k}$-creating perfect matchings. (In section \[sec:reversing\] we show that there is a connection between flipful permutations and pairwise $C_{4}$-creating perfect matchings. In Section \[sec:concluding\] we further discuss the similarities between [@original], [@cib] and the proof of Theorem \[thm:main\]. )
Let $M(n,2k)$ be the maximal number of pairwise $C_{2k}$-creating perfect matchings of the complete graph $K_n$.
First let us establish a connection between $M(n,2k)$ and $H(n,2k)$.
\[intomatching\] For every fixed $k$, $$n^{-(1-1/k)n-o(n)} H(n,2k) \leq M(2n/k,2k).$$
We first deal with the case where $n$ is even and divisible by $3k$ (we will reduce everything else to this case later). Let $\mathcal{H}$ be a set of pairwise $C_{2k}$-creating Hamiltonian paths of size $H(n,2k)$, on $n$ vertices. For every $1 \leq i \leq n$ we denote the $i$-th vertex of the Hamiltonian path $H \in \mathcal{H}$ by $\pi_H(i)$. For each Hamiltonian path $H$ we associate a triple of its subgraphs $(X_H^1,X_H^2,X_H^3)$ as follows. For every $1 \leq j \leq 3$ we define $X_H^j$ to be the induced subgraph of $H$ on the vertices $$\bigcup_{i=0}^{n/(3k)-1} \{ \pi_H\big((j-1+3i)k+1\big),\pi_H\big((j-1+3i)k+2\big),\ldots, \pi_H\big((j-1+3i)k+k\big) \}.$$ Informally, if we partition the vertices of $H$ into consecutive subsets of size $k$, then $X_H^1$ is the induced subgraph of $H$ on the first plus the fourth plus the seventh etc. set of $k$ vertices. The useful feature of these associated triples will turn out to be that two paths with the same associated triple can only be $C_{2k}$-creating in a very specific way. The number of possible triples $(X_H^1,X_H^2,X_H^3)$ is $$\binom{n}{\underbrace{k,k,\ldots,k}_{n/k}} ((n/k)!)^{-1} \left(k! \right)^{n/k} \binom{n/k}{n/(3k),n/(3k),n/(3k)}$$ which can be seen by the following reasoning: We can partition the ground set into $n/k$ unordered parts of size $k$ in exactly $\binom{n}{k,k,\ldots,k} ((n/k)!)^{-1}$ ways, then we can choose a directed path of length $k$ in each partition in $\left(k! \right)^{n/k}$ ways, then we can partition these paths into three classes in $\binom{n/k}{n/(3k),n/(3k),n/(3k)}$ ways. It is a routine calculation that
$$\binom{n}{\underbrace{k,k,\ldots,k}_{n/k}} ((n/k)!)^{-1} \left(k! \right)^{n/k} \binom{n/k}{n/(3k),n/(3k),n/(3k)} = n^{\left(1-1/k \right)n+o(n)}.$$
By the pigeon-hole principle, there is a subset $\mathcal{M}' \subseteq \mathcal{H}$ so that for every pair of Hamiltonian paths $H_1, H_2 \in \mathcal{M}'$, their associated triples are identical: $$\big(X_{H_1}^1,X_{H_1}^2,X_{H_1}^3\big)=\big(X_{H_2}^1,X_{H_2}^2,X_{H_2}^3\big)$$ and $$\label{eq:pidgeon}
n^{-(1-1/k)n-o(n)}|\mathcal{H}|\leq |\mathcal{M}'|.$$
Let $H \in \mathcal{M}'$ and let $F$ be the union of the three graphs $(X_H^1,X_H^2,X_H^3)$, thus $F$ is the disjoint union of $n/k$ directed paths on $k$ vertices. $F$ can be thought of the set of fixed edges since every edge of $F$ is contained in every Hamiltonian path in $\mathcal{M}'$. Since $\mathcal{M}'$ is a subset of $\mathcal{H}$, it consists of $C_{2k}$-creating Hamiltonian paths. We claim that in the union of any two Hamiltonian paths from $\mathcal{M}'$, no edge of $F$ is used in a cycle of length $2k$.
\[notused\] Let $H_1, H_2 \in \mathcal{M}'$ if $C$ is a (not necessarily circularly) directed cycle of length $2k$ in $H_1 \cup H_2$ then no edge of $C$ is in $F$.
Recall that $F$ is a subgraph of $H_1 \cup H_2$ and $F$ is the disjoint union of $n/k$ paths, each on $k$ vertices, see Figure \[unionstructure\]. Furthermore, every edge of $H_1 \cup H_2$ that is not in $F$, connects one endpoint of a path in $F$ from a set $X_H^j$ to a first point of a path in $F$ from a set $X_H^{j+1}$, for some $j$ (where $j+1$ is understood modulo $3$). See Figure \[unionstructure\].
Suppose to the contrary that $e$ is an edge in both $C$ and $F$. Since $e$ is in $F$, it must be in $X_H^{j}$ for some $j \in \{1,2,3\}$. Thus a whole path $P(e)$ on $k$ vertices from $X_H^{j}$ must be in $C$. Therefore $C \setminus P(e)$ is a path on $k+2$ vertices that is edge disjoint from $P(e)$ and connects the endpoints of $P(e)$. Such a path must either contain an other whole path from $X_H^{j}$, or one path from $X_H^{j+1}$ and another from $X_H^{j-1}$. In the later case, $C \setminus P(e)$ contains at least $2k+2$ vertices, a contradiction. In the former case $C \setminus P(e)$ contains at least $k+4$ vertices: the two endpoints of $P(e)$, $k$ vertices from the other path in $X_H^{j}$ and two additional vertices since no edges connect two starting or two endpoints of different paths in $X_H^{j}$, a contradiction.
at (3.2,2) [$e$]{}; at (4.7,2.4) [paths in $X_H^{j}$]{}; at (2,-1) [paths in $X_H^{j+1}$]{}; at (-0.9,2.2) [paths in $X_H^{j-1}$]{};
(0,0) circle (25pt); (2,3) circle (25pt); (4,0) circle (25pt);
(0,0.5) – (1.7/4,0.5+2.5/4); (1.7/4,0.5+2.5/4) – (2\*1.7/4,0.5+2\*2.5/4); (2\*1.7/4,0.5+2\*2.5/4)–(3\*1.7/4,0.5+3\*2.5/4); (3\*1.7/4,0.5+3\*2.5/4)–(1.7,3);
(0.2,0.3) – (0.2+1.7/4,0.3+2.5/4); (0.2+1.7/4,0.3+2.5/4)–(0.2+2\*1.7/4,0.3+2\*2.5/4); (0.2+2\*1.7/4,0.3+2\*2.5/4)–(0.2+3\*1.7/4,0.3+3\*2.5/4); (0.2+3\*1.7/4,0.3+3\*2.5/4)–(0.2+4\*1.7/4,0.3+4\*2.5/4);
(0.4,0.1) – (0.4+1.7/4,0.1+2.5/4);
(0.4+1.7/4,0.1+2.5/4) – (0.4+2\*1.7/4,0.1+2\*2.5/4); (0.4+2\*1.7/4,0.1+2\*2.5/4) – (0.4+3\*1.7/4,0.1+3\*2.5/4); (0.4+3\*1.7/4,0.1+3\*2.5/4) – (0.4+4\*1.7/4,0.1+4\*2.5/4);
(0,0.5) – (1.7/4,0.5+2.5/4); (1.7/4,0.5+2.5/4) – (2\*1.7/4,0.5+2\*2.5/4); (2\*1.7/4,0.5+2\*2.5/4)–(3\*1.7/4,0.5+3\*2.5/4); (3\*1.7/4,0.5+3\*2.5/4)–(1.7,3);
(0.2,0.3) – (0.2+1.7/4,0.3+2.5/4); (0.2+1.7/4,0.3+2.5/4)–(0.2+2\*1.7/4,0.3+2\*2.5/4); (0.2+2\*1.7/4,0.3+2\*2.5/4)–(0.2+3\*1.7/4,0.3+3\*2.5/4); (0.2+3\*1.7/4,0.3+3\*2.5/4)–(0.2+4\*1.7/4,0.3+4\*2.5/4);
(0.4,0.1) – (0.4+1.7/4,0.1+2.5/4);
(0.4+1.7/4,0.1+2.5/4) – (0.4+2\*1.7/4,0.1+2\*2.5/4); (0.4+2\*1.7/4,0.1+2\*2.5/4) – (0.4+3\*1.7/4,0.1+3\*2.5/4); (0.4+3\*1.7/4,0.1+3\*2.5/4) – (0.4+4\*1.7/4,0.1+4\*2.5/4);
(0,0.5) – (1.7/4,0.5+2.5/4); (1.7/4,0.5+2.5/4) – (2\*1.7/4,0.5+2\*2.5/4); (2\*1.7/4,0.5+2\*2.5/4)–(3\*1.7/4,0.5+3\*2.5/4); (3\*1.7/4,0.5+3\*2.5/4)–(1.7,3);
(0.2,0.3) – (0.2+1.7/4,0.3+2.5/4); (0.2+1.7/4,0.3+2.5/4)–(0.2+2\*1.7/4,0.3+2\*2.5/4); (0.2+2\*1.7/4,0.3+2\*2.5/4)–(0.2+3\*1.7/4,0.3+3\*2.5/4); (0.2+3\*1.7/4,0.3+3\*2.5/4)–(0.2+4\*1.7/4,0.3+4\*2.5/4);
(0.4,0.1) – (0.4+1.7/4,0.1+2.5/4);
(0.4+1.7/4,0.1+2.5/4) – (0.4+2\*1.7/4,0.1+2\*2.5/4); (0.4+2\*1.7/4,0.1+2\*2.5/4) – (0.4+3\*1.7/4,0.1+3\*2.5/4); (0.4+3\*1.7/4,0.1+3\*2.5/4) – (0.4+4\*1.7/4,0.1+4\*2.5/4);
Let $$\mathcal{M}:= \{H \setminus F : H \in \mathcal{M}' \}.$$ $\mathcal{M}$ is a set of perfect matchings on $2n/k-2$ vertices. The union of every pair of matchings in $\mathcal{M}$ contains a cycle of length $2k$ since their original Hamiltonian paths had such a cycle in their union and by Claim \[notused\] we only deleted edges that are not used in a cycle of length $2k$. Therefore $|\mathcal{M}|\leq M(2n/k-2,2k) \leq M(2n/k,2k)$, this combined with (\[eq:pidgeon\]) yields
$$n^{-(1-1/k)n-o(n)}|\mathcal{H}|\leq |\mathcal{M}'|= |\mathcal{M}| \leq M(2n/k,2k)$$ as claimed. Thus the proof is complete when $n$ is even and divisible with $3k$.
We deal with the case where $3k$ does not divide $n$, by proving $$\label{eq:reducing} M(n,2k) \leq (n-1)M(n-2,2k).$$ Since applying (\[eq:reducing\]) at most a constant number of times, we can ensure that the ground set is even and divisible by $3k$. We prove (\[eq:reducing\]) as follows. Let $\mathcal{M}$ be a set of pairwise $C_{2k}$-creating perfect matchings on $n$ vertices. Every perfect matchings connects the vertex $1 \in [n]$, to an other vertex from the remaining $(n-1)$ ones. By the pigeon-hole principle, there is a vertex $i \in [n] \setminus \{1\}$ that is the neighbor of $1$ in at least $|\mathcal{M}|/(n-1)$ perfect matchings. Let $\mathcal{M}'$ be the subset of $\mathcal{M}$, that consists of those perfect matchings that connect $1$ to $i$. Observe that in the union of two perfect matchings from $\mathcal{M}'$, there must be a $C_{2k}$, since $\mathcal{M}' \subseteq \mathcal{M}$. Finally observe that in the union of two perfect matchings from $\mathcal{M}'$, $1$ and $i$ always form a connected component of size two, hence they can be deleted without destroying the $C_{2k}$-creating property. Therefore the proof is complete.
Now we aim for an upper bound on $M(n,2k)$. Let $G_{PM}(C_{2k}) $ be the graph whose vertices correspond to perfect matchings on $[n]$ and two vertices of $G$ are adjacent if the corresponding perfect matchings are $C_{2k}$-creating. Clearly $\omega(G_{PM}(C_{2k}))=M(n,2k)$. It is well known that for every vertex transitive graph $G$, $\alpha(G) \omega(G) \leq |V(G)|$, see [@fractional]. (It is easy to prove that the fractional chromatic number $\chi_f (G)$ of such a graph is exactly $|V(G)|/\alpha(G)$ and clearly $\omega(G) \leq \chi_f(G)$.) Since $G_{PM}(C_{2k})$ is vertex transitive we have $$\alpha(G_{PM}(C_{2k})) \omega(G_{PM}(C_{2k})) \leq |V(G_{PM}(C_{2k}))|$$ or equivalently $$\label{eq:vertextrans}
M(n,2k)=\omega(G_{PM}(C_{2k})) \leq \frac{|V(G_{PM}(C_{2k}))|}{\alpha(G_{PM}(C_{2k}))}.$$
Thus we will prove an upper bound on $M(n,2k)$ by proving a lower bound to the number of pairwise non-$C_{2k}$-creating perfect matchings on $n$ vertices and using (\[eq:vertextrans\]). We construct a large set of pairwise non-$C_{2k}$-creating perfect matchings by constructing a $C_{2k}$-free graph and proving that there are many perfect matchings in this graph. For this we will need bipartite, regular, $C_{2k}$-free graphs with many edges.
Such constructions are often used to give lower bounds to the Turán number of even cycles. The bipartiteness and regularity properties are not required when one aims to give lower bounds to the Turán number of an even cycle. But for our method they will be essential! These constructions have an algebraic nature and they require that the number of vertices is special in some way. In Table \[results\] we summarize the current best constructions for bipartite, regular, $C_{2k}$-free graphs.
authors 2k degrees density
------------------------------------------- -------------- ------------------------ ----------------------------------------------------------
Reiman, see below 4 $n^{\frac{1}{k}-o(1)}$ $n/2=\sum_{i=0}^{2}q^i$, for $q$ a prime power
Benson [@benson] 6 $n^{\frac{1}{k}-o(1)}$ $n/2=\sum_{i=0}^{3}q^i$, for $q$ a prime power
Lazebnik, Ustimenko, Woldar [@ingredient] 8 $n^{\frac{2}{3k-2}}$ $n/2=q^{3}$
Benson [@benson] 10 $n^{\frac{1}{k}-o(1)}$ $n/2=\sum_{i=0}^{5}q^i$ , for $q$ an odd prime power
Lazebnik, Ustimenko, Woldar [@ingredient] $2(2\ell)$ $n^{\frac{2}{3k-2}}$ $n/2=q^{2k-4-\left\lfloor \frac{2k-3}{4}\right\rfloor}$
Lazebnik, Ustimenko, Woldar [@ingredient] $2(2\ell+1)$ $n^{\frac{2}{3k-3}}$ $n/2=q^{2k-4-\left\lfloor \frac{2k-3}{4}\right\rfloor}$
: The order of magnitude of the degrees in regular, bipartite, $C_{2k}$-free graphs on $n$ vertices. The density column indicates that such constructions are known only for special ground sets. But in all cases, the set of numbers for which there are such constructions will turn out to be dense enough for all our purposes.[]{data-label="results"}
We sketch the construction for the $C_4$-free case.
If $n/2=q^2+q+1$ then there is a bipartite, $n^{\frac{1}{2}-o(n)}$ regular, $C_{4}$-free graph on $n$ vertices.
A finite projective plane of order $N$ has $N^2+N+1$ points and the same number of lines. Every point is incident to $N+1$ lines and every line contains $N+1$ points. Every pair of points is contained in exactly one line. Projective planes exist when $N=q^2+q+1$ where $q$ is a prime power. A bipartite regular $C_4$-free graph can be obtained from a projective plane as follows: Let the vertices of one of the color classes be the points of the plane, the vertices of the other class be the lines of the plane. Two vertices corresponding to a point and a line are adjacent when the point is contained in the line. The graph has exactly $2N^2+2N+2$ vertices and is $N+1$ regular. This graph is $C_4$-free as every pair of vertices is contained in a single line.
We introduce a notation so that we can refer to the results of Table \[results\] in a simple, unified way.
Let $t(x)$ denote the exponent of $n$ in the third column of Table \[results\] in the row where $k=x$.
For example $t(4)=2/(3k-2)$. To show that these graphs contain many perfect matchings, we need the following results.
\[waerden\](van der Waerden’s conjecture, Gyires-Egorychev-Falikman theorem [@egor] [@falik] [@gyires]) If $A$ is an $n \times n$ matrix where the sum of every row and column is $1$ (a doubly stochastic matrix) then $$Per(A)\geq \frac{n!}{n^n}$$ where $Per(A)$ is the permanent of $A$.
\[manymatchings\] If $c$ is a constant, $r=n^{c-o(1)}$ and $G=(A,B,E)$ is an $r$-regular bipartite graph on $n$ vertices, then $G$ contains at least $n^{\frac{c}{2}n-o(n)}$ perfect matchings.
In both color classes of $G$ let us fix an ordering of the vertices. Let $A$ be an $\frac{n}{2} \times \frac{n}{2}$ matrix where $a_{i,j}=1$ if and only if the $i$-th vertex of $A$ is adjacent to the $j$-th vertex of $B$. Clearly the number of perfect matchings of $G$ is equal to $Per(A)$. Let $A'=r^{-1}A$, clearly
$$Per(A)= (n^{c-o(1)})^{\frac{n}{2}} Per(A').$$ Since the matrix $A'$ is doubly stochastic as $G$ was regular, by Theorem \[waerden\] we have $$Per(A)= (n^{c-o(1)})^{\frac{n}{2}} Per(A') \geq (n^{c-o(1)})^{\frac{n}{2}} \frac{(n/2)!}{(n/2)^{(n/2)}}=n^{\frac{c}{2}n-o(n)}$$ as claimed.
There are many theorems that can be used for our “the set of primes is dense enough” type argument. We (following in the footsteps of Cibulka) choose to use the most recent and most powerful one.
\[dense\](Baker-Harman-Pintz [@pintz]) For all large enough $n$, there is a prime in the interval $[n-n^{0.525},n].$
In the next lemma we prove that although we can only construct dense, regular, bipartite, $C_{2k}$-free graphs on vertex sets of special size, these sizes are dense enough. Observe that in Table \[results\], for every $k$, the requirement for $n$ can be strengthened into the form: ’$n=g(p)$ for some polynomial $g(x)$ and prime $p$’ (not prime power!). For example, when $2k=4$, a suitable choice is $g(x)=2x^2+2x+2$. For $2k=10$, there is a single exception since there the prime $2$ cannot be used, but this only gives a single error $n=31$ which does not influence our asymptotic results.
\[matchingbound\_alpha\] Let $k$ be an integer and $g(x)=g_{k}(x)$ a polynomial for which $\lim_{x \rightarrow \infty}g(x)=\infty$. Suppose that whenever $n=g(p)$ for some prime $p$, there is a bipartite, $C_{2k}$-free, $n^{t(k)}$-regular graph on $n$ vertices. In this case, there is a family $\mathcal{M}$ of pairwise non $C_{2k}$-creating perfect matchings on $n$ vertices satisfying $n^{\frac{1}{2t(k)}n-o(n)}\leq |\mathcal{M}|.$
We say that a number $n$ is suitable when $n=g(p)$ for a prime $p$. Let $m>n_0$ be large enough for Theorem \[dense\], furthermore let $m$ be so large that $m-m^{0.525}$ is larger than the largest root of $g(x)$. By Theorem \[dense\], there is a prime $p$ in the interval $[m-m^{0.525},m]$. Since in this interval $g(x)$ is monotone increasing, there is a suitable $n$ in the interval $[g(m-m^{0.525}),g(m)]$ for every large enough $m$. $$\label{eq:dense}
g(m)-g(m-m^{0.525})=o(g(m))$$ since for every fixed $k$, and $x$ tending to infinity $x^k-(x-x^{0.525})^k=o(x^k)$. Let now $n$ be large. Since $n$ is between $g(m)$ and $g(m+1)$ for some $m$, and $g(m+1)-g(m)=o(g(m))$ we have $n-g(m)=o(n)$. By Theorem \[dense\], there is a prime in the interval $[m-m^{0.525},m]$, thus there is a suitable integer in the interval $[g(m-m^{0.525}),g(m)]$. By (\[eq:dense\]) the length of this interval is $o(g(m))=o(n)$. Therefore there is a suitable integer $n'$ such that $n'=n-o(n)$.
By our assumptions, there is a bipartite, $C_{2k}$-free, $(n')^{t(k)}$-regular graph $G$ on $n'$ vertices. By Lemma \[manymatchings\] there are at least $n'^{\frac{t(k)}{2}n'-o(n')}=n^{\frac{t(k)}{2}n-o(n)}$ perfect matchings in $G$. Since $n'=n-o(n)$, by adding $n-n'$ new vertices and a fixed matching on the new vertices to these, the proof is complete.
\[corr\] $$M(n,2k) \leq n^{\left(\frac{1}{2}-\frac{1}{2t(k)} \right)n-o(n)}.$$
By Table \[results\] and Lemma \[matchingbound\_alpha\] there is a set of pairwise not $C_4$-creating perfect matchings of size $n^{\frac{1}{2t(k)}n-o(n)}$ on $n$ vertices. It is well known that the number of perfect matchings on $n$ vertices is $n^{\frac{1}{2}n-o(n)}$, thus by (\[eq:vertextrans\]) we have $$M(n,2k)\leq \frac{n^{\frac{1}{2}n-o(n)}}{n^{\frac{1}{2t(k)}n-o(n)}}=n^{\left(\frac{1}{2}-\frac{1}{2t(k)} \right)n-o(n)}.$$
Finally we are ready to prove our main theorem.
\[thm:main2\] For every fixed $k$ $$H(n,2k) \leq n^{\left(1-\frac{1}{k t(k)} \right)n-o(n)}.$$
By Lemma \[intomatching\] we have $$\label{eq:main1} n^{-\left(1-\frac{1}{k}\right)n-o(n)} H(n,2k) \leq M(2n/k,2k).$$
By Corollary \[corr\] $$\label{eq:main2}
M(2n/k,2k) \leq (2n/k)^{\left(\frac{1}{2}-\frac{1}{2t(k)} \right)2n/k-o(n)}=n^{\left(\frac{1}{k}-\frac{1}{k t(k)} \right)n-o(n)}.$$
Equations (\[eq:main1\]) and (\[eq:main2\]) together yield the claimed upper bound.
Theorem \[thm:main\] follows from Theorem \[thm:main2\] and Table \[results\].
Connection with reversing permutations {#sec:reversing}
======================================
We say that two permutations $\pi_1 , \pi_2$ of the elements $[n]$ are reversing if there are two coordinates $1 \leq i < j \leq n$ for which $\pi_1(i)=\pi_2(j)$ and $\pi_1(j)=\pi_2(i)$. Let $RP(n)$ denote the maximal number of pairwise reversing permutations of $[n]$.
In this short section we establish a connection between $M(n,4)$ and $RP(n/2)$. In [@original] the authors prove Theorem \[thm:original\] using a relation between $RP(n/2)$ and $H(n,4)$, for more details see Section \[sec:concluding\]. The following lemma states that ignoring exponential factors, the values $M(n,4)$ and $RP(n/2)$ are the same.
\[claim:permcorrespondence\] When $n$ is even, $$2^{\frac{n}{2}}\binom{n}{\frac{n}{2}}^{-1}M(n,4) \leq RP\left(\frac{n}{2}\right) \leq M(n,4).$$
For a permutation $\pi$ of the elements $[n/2]$, let us associate the perfect matching $M(\pi)$ on the vertices $[n]$ that consists of the edges $(i,\pi(i)+n/2)$. Observe that two permutations of $[n/2]$ are reversing if and only if their associated matchings are $C_4$-creating. This proves the second inequality. Observe that this correspondence is a bijection between the set of permutations of $[n/2]$ and the set of perfect matchings on $[n]$ which have all of their edges between the sets $\{1, \ldots n/2\}$ and $\{n/2+1, \ldots n\}$. The first inequality follows from the observation that given a set of $M(n,4)$ pairwise $C_4$-creating perfect matchings on $n$ vertices, the average number of perfect matchings that have their edges between $S \subset [n]$ and $[n] \setminus S$, averaging over every $|S|=n/2$ is $2^{\frac{n}{2}} \binom{n}{n/2}^{-1}M(n,4)$. And we can have a similar bijection between these perfect matchings and permutations of $[n/2]$ that takes a pair of $C_4$-creating perfect matchings into a pair of reversing permutations.
Thus from the proof of Claim \[claim:permcorrespondence\] we see that $RP(n/2)$ can be viewed as a version of $M(n,4)$ where we restrict our matchings to have all their edges between two fixed subsets of $[n]$.
Concluding remarks {#sec:concluding}
==================
It might not be immediately apparent that the proof of Theorem \[thm:main2\] in the case when $k=2$ is essentially equivalent to the proof of Theorem \[thm:original\]. Let us elaborate on this equivalence. The proof of Theorem \[thm:main2\] follows the following steps.
- Lemma \[intomatching\] establishes a connection between $H(n,2k)$ and $M(n,2k)$.
- (\[eq:vertextrans\]) gives a very rough upper bound on $M(n,2k)$ using a lower bound on $\alpha(G_{PM}(C_{2k}))$.
- We give a lower bound on $\alpha(G_{PM}(C_{2k}))$ using $C_{2k}$-free graphs that contain many perfect matchings.
The essential equivalence in the $k=2$ case can be seen as follows. In [@original] the authors establish a connection between $H(n,4)$ and $RP(n/2)$ (recall that by Claim \[claim:permcorrespondence\] we already know that $RP(n/2)$ is only an exponential factor away from $M(n,4)$). Their proof is generalized to Lemma \[intomatching\]. Then the authors of [@original] refer to the upper bound on $RP(n/2)$ proved in [@cib] to conclude that $H(n,4) \leq n^{\frac{3}{4}n-o(n)}$. In [@cib] the author actually proves that the maximal number of pairwise non-reversing permutations of $[n/2]$ is equal to $n^{\frac{1}{4}n-o(n)}$. Then he uses (\[eq:vertextrans\]) to prove an upper bound on $RP(n/2)$. Observe that for the upper bound on $RP(n/2)$ (and thus $H(n,4)$) we only need the lower bound on the number of pairwise non-reversing permutations! (The proof of the upper bound is much longer and much more difficult.) Our lower bound on the number of pairwise non $C_{2k}$-creating perfect matchings is a natural generalization of the lower bound in [@cib]. Therefore the main result of the present paper (Theorem \[thm:main2\]) should be considered a natural generalization of the ideas of Cibulka, Cohen, Fachini and Körner which led to Theorem \[thm:original\].
In [@cib], it is proven that the maximal number of pairwise non-reversing permutations of $[n/2]$ is $n^{\frac{1}{4}n-o(n)}$. Therefore using only \[eq:vertextrans\] we cannot get a smaller upper bound on $M(n,4)$ than $n^{\frac{1}{4}n-o(n)}$. Or in other terms, the fractional clique number of $G_{PM}(C_{4})$ is $n^{\frac{1}{4}n-o(n)}$ (since for vertex transitive graphs $G$ the fractional clique number is $|V(G)|/\alpha(G)$) thus if the clique number is actually smaller, no method can prove it which would also work for the fractional clique number.
We saw in Section \[sec:lower\] that $H(n,2k)$ is larger than any exponential function of $n$, for every fixed $k$. In the constructions presented there, for every $k>2$, every Hamiltonian path contains a set of roughly $\frac{n}{k}$ paths on exactly $k-1$ vertices. Since every Hamiltonian path constructed there contains these fixed paths, we can add any Hamiltonian path which forms a $C_{2k}$ with these fixed paths. It is natural to try to add a set of Hamiltonian paths that is constructed similarly but the set of paths of length $k-1$ is different, moreover $C_{2k}$-creating from the original set of fixed paths. Let $\mathcal{P}_k(n)$ be the set of graphs on $n$ vertices that is the disjoint union of paths of length $k-1$. This is the motivation behind the following questions.
\[q:1\] Let $k>2$. What is the maximal number of pairwise $C_{2k}$-creating graphs from $\mathcal{P}_k(n)$? Is the answer an exponential function of $n$?
The first non-trivial case of Question \[q:1\] is: what is the maximal number of pairwise $C_6$-creating perfect matchings on $n$ vertices? Or in other words, what is the value of $M(n,6)$? Although for larger $k$, $\mathcal{P}_k(n)$ contains longer paths it is not hard to see that for $k>3$, a larger than exponential lower bound for $M(n,2k)$ would result in larger than exponential lower bound for the number of pairwise $C_{2k}$-creating graphs from $\mathcal{P}_k(n)$. This, and Lemma \[intomatching\] means that for $k$ at least $3$, better lower bounds for $M(n,2k)$ would lead to better constructions for $H(n,2k)$, and better upper bounds for $M(n,2k)$ would lead to better upper bounds on $H(n,2k)$.
acknowledgement
===============
The author would like to thank Gábor Simonyi and Kitti Varga for their valuable comments and suggestions that improved the quality of this manuscript.
[^1]: This research of the author was supported by the Hungarian Foundation for Scientific Research Grant (OTKA) No. 108947 and by the National Research, Development and Innovation Office NKFIH, No. K-120706.
|
---
abstract: 'Let $K$ be a complete discrete valuation field of mixed characteristic $(0,p)$ with perfect residue field. Let $(\pi_n)_{n\ge 0}$ be a system of $p$-power roots of a uniformizer $\pi=\pi_0$ of $K$ with $\pi^p_{n+1}=\pi_n$, and define $G_s$ (resp. $G_{\infty}$) the absolute Galois group of $K(\pi_s)$ (resp. $K_{\infty}:=\bigcup_{n\ge 0} K(\pi_n)$). In this paper, we study $G_s$-equivatiantness properties of $G_{\infty}$-equivariant homomorphisms between torsion (potentially) crystalline representations.'
author:
- 'Yoshiyasu Ozeki[^1]'
title: On Galois equivariance of homomorphisms between torsion potentially crystalline representations
---
Introduction
============
Let $p$ be a prime number and $r\ge 0 $ an integer. Let $K$ be a complete discrete valuation field of mixed characteristic $(0,p)$ with perfect residue field and absolute ramification index $e$. Let $\pi=\pi_0$ be a uniformizer of $K$ and $\pi_n$ a $p^n$-th root of $\pi$ such that $\pi^p_{n+1}=\pi_n$ for all $n\ge 0$. For any integer $s\ge 0$, we put $K_{(s)}=K(\pi_s)$. We also put $K_{\infty}=\bigcup_{n\ge 0}K_{(n)}$. We denote by $G_K, G_s$ and $G_{\infty}$ absolute Galois groups of $K$, $K_{(s)}$ and $K_{\infty}$, respectively. By definition we have the following decreasing sequence of Galois groups: $$G_K=G_0\supset G_1\supset G_2\supset\cdots \supset G_{\infty}.$$ Since $K_{\infty}$ is a strict APF extension of $K$, the theory of fields of norm implies that $G_{\infty}$ is isomorphic to the absolute Galois group of some field of characteristic $p$. Therefore, representations of $G_{\infty}$ has easy interpretations via Fontaine’s étale ${\varphi}$-modules. Hence it seems natural to have the following question:
\[que1\] Let $T$ be a representation of $G_K$. For a “small” integer $s\ge 0$, can we reconstruct various information of the $G_s$-action on $T$ from that of the $G_{\infty}$-action?
Nowadays there is an interesting insight of Breuil; he showed that representations of $G_K$ arising from finite flat group schemes or $p$-divisible groups over the integer ring of $K$ is “determined” by its restriction to $G_{\infty}$. Moreover, for ${\mathbb}{Q}_p$-representations, Kisin proved the following theorem in [@Kis] (which was a conjecture of Breuil): the restriction functor from the category of crystalline ${\mathbb}{Q}_p$-representations of $G_K$ into the category of ${\mathbb}{Q}_p$-representations of $G_{\infty}$ is fully faithful.
In this paper, we give some partial answers to Question \[que1\] for [*torsion crystalline representations*]{}, moreover, [*torsion potentially crystalline representations*]{}. Our first main result is as follows. Let ${\mathrm}{Rep}^{r,{\mathrm}{ht},{\mathrm}{pcris}(s)}_{{\mathrm}{tor}}(G_K)$ be the category of torsion ${\mathbb}{Z}_p$-representations $T$ of $G_K$ which satisfy the following: there exist free ${\mathbb}{Z}_p$-representations $L$ and $L'$ of $G_K$, of height $\le r$, such that
- $L|_{G_s}$ is a subrepresentation of $L'|_{G_s}$. Furthermore, $L|_{G_s}$ and $L'|_{G_s}$ are lattices in some crystalline ${\mathbb}{Q}_p$-representation of $G_s$ with Hodge-Tate weights in $[0,r]$;
- $T|_{G_s} \simeq (L'|_{G_s})/(L|_{G_s})$.
\[Main1\] Suppose that $p$ is odd and $e(r-1)<p-1$. Let $T$ and $T'$ be objects of ${\mathrm}{Rep}^{r,{\mathrm}{ht},{\mathrm}{pcris}(s)}_{{\mathrm}{tor}}(G_K)$. Then any $G_{\infty}$-equivariant homomorphism $T\to T'$ is in fact $G_s$-equivariant.
We should remark that the condition $e(r-1)<p-1$ in the above does not depend on $s$. We put ${\mathrm}{Rep}^{r, {\mathrm}{cris}}_{{\mathrm}{tor}}(G_K)={\mathrm}{Rep}^{r, {\mathrm}{ht}, {\mathrm}{pcris}(0)}_{{\mathrm}{tor}}(G_K)$. By definition, a torsion ${\mathbb}{Z}_p$-representation $T$ of $G_K$ is contained in this category if and only if it can be written as the quotient of lattices in some crystalline ${\mathbb}{Q}_p$-representation of $G_K$ with Hodge-Tate weights in $[0,r]$. We call the objects in ${\mathrm}{Rep}^{r, {\mathrm}{cris}}_{{\mathrm}{tor}}(G_K)$ [*torsion crystalline representations with Hodge-Tate weights in $[0,r]$*]{}. In the case $r=1$, such representations are equivalent to finite flat representations. (Here, a torsion ${\mathbb}{Z}_p$-representation of $G_K$ is finite flat if it arises from the generic fiber of some $p$-power order finite flat commutative group scheme over the integer ring of $K$.) Combining Theorem \[Main1\] with results of [@Kim], [@La], [@Li3] (the case $p=2$) we obtain the following full faithfulness theorem for torsion crystalline representations.
\[FFTHMtorcris\] Suppose $e(r-1)<p-1$. Then the restriction functor ${\mathrm}{Rep}^{r, {\mathrm}{cris}}_{{\mathrm}{tor}}(G_K)\to {\mathrm}{Rep}_{{\mathrm}{tor}}(G_{\infty})$ is fully faithful.
Before this work, some results are already known. First, the full faithfulness theorem was proved by Breuil for $e=1$ and $r<p-1$ via the Fontaine-Laffaille theory ([@Br2], the proof of Théorèm 5.2). He also proved the theorem under the assumptions $p>2$ and $r\le 1$ as a consequence of a study of the category of finite flat group schemes ([@Br3 Theorem 3.4.3]). Later, his result was extended to the case $p=2$ in [@Kim], [@La], [@Li3] (proved independently). In particular, the full faithfulness theorem for $p=2$ is a consequence of their works. On the other hand, Abrashkin proved the full faithfulness in the case where $p>2, r<p$ and $K$ is a finite unramified extension of ${\mathbb}{Q}_p$ ([@Ab2 Section 8.3.3]). His proof is based on calculations of ramification bounds for torsion crystalline representations. In [@Oz2], a proof of Corollary \[FFTHMtorcris\] under the assumption $er<p-1$ is given via $({\varphi},\hat{G})$-modules (which was introduced by Tong Liu [@Li2] to classify lattices in semi-stable representations).
Our proof of Theorem \[Main1\] is similar to the proof for the main result of [@Oz2], but we need more careful considerations for $({\varphi},\hat{G})$-modules. Indeed we need special base change arguments to study some potential crystalline representations. In fact, we prove a full faithfulness theorem for torsion representations arising from certain classes of $({\varphi},\hat{G})$-modules (cf. Theorem \[FFTHM\]), which immediately gives our main theorem. In addition, our study gives a result as below which is the second main result of this paper (here, we define ${\mathrm}{log}_p(x):=-\infty$ for any real number $x\le 0$).
\[Main2\] Suppose that $p$ is odd and $s> n-1 + {\mathrm}{log}_p(r-(p-1)/e)$. Let $T$ and $T'$ be objects of ${\mathrm}{Rep}^{r, {\mathrm}{cris}}_{{\mathrm}{tor}}(G_K)$ which are killed by $p^n$. Then any $G_{\infty}$-equivariant homomorphism $T\to T'$ is in fact $G_s$-equivariant.
For torsion semi-stable representations, a similar result was shown in Theorem 3 of [@CL2], which was a consequence of a study of ramification bounds. The bound appearing in their theorem was $n-1 + {\mathrm}{log}_p(nr)$. By applying our arguments given in this paper, we can obtain a generalization of their result; our refined condition is $n-1 + {\mathrm}{log}_pr$ (see Theorem \[Main3\]). Some other consequences of our study are described in subsection \[consequences\]. Motivated by the full faithfulness theorem (= Corollary \[FFTHMtorcris\]) and Theorem \[Main2\], we raise the following question.
Is any $G_{\infty}$-equivariant homomorphism in the category ${\mathrm}{Rep}^{r, {\mathrm}{cris}}_{{\mathrm}{tor}}(G_K)$ in fact $G_s$-equivariant for $s>{\mathrm}{log}_p(r-(p-1)/e)$?
On the other hand, there exist counter examples of the full faithfulness theorem when we ignore the condition $e(r-1)< p-1$. Let ${\mathrm}{Rep}_{{\mathrm}{tor}}(G_1)$ be the category of torsion ${\mathbb}{Z}_p$-representations of $G_1$.
\[nonfull\] Suppose that $K$ is a finite extension of ${\mathbb}{Q}_p$, and also suppose $e\mid (p-1)$ or $(p-1)\mid e$. If $e(r-1)\ge p-1$, the restriction functor ${\mathrm}{Rep}^{r, {\mathrm}{cris}}_{{\mathrm}{tor}}(G_K)\to
{\mathrm}{Rep}_{{\mathrm}{tor}}(G_1)$ is not full $($in particular, the restriction functor ${\mathrm}{Rep}^{r, {\mathrm}{cris}}_{{\mathrm}{tor}}(G_K)\to
{\mathrm}{Rep}_{{\mathrm}{tor}}(G_{\infty})$ is not full$)$.
In particular, if $p=2$, then the full faithfulness never hold for any finite extension $K$ of ${\mathbb}{Q}_2$ and any $r\ge 2$. Theorem \[nonfull\] implies that the condition “$e(r-1)<p-1$” in Corollary \[FFTHMtorcris\] is the best possible for many finite extensions $K$ of ${\mathbb}{Q}_p$.
Now we describe the organization of this paper. In Section 2, we setup notations and summarize facts we need later. In Section 3, we define variant notions of $({\varphi},\hat{G})$-modules and give some basic properties. They are needed to study certain classes of potentially crystalline representations and restrictions of semi-stable representations. In Section 4, we study technical torsion $({\varphi},\hat{G})$-modules which are related with torsion (potentially) crystalline representations. The key result in this section is the full faithfulness result Theorem \[FFTHM\] on them, which allows us to prove our main results immediately. Finally, in Section 5, we calculate the smallest integer $r$ for a given torsion representation $T$ such that $T$ admits a crystalline lift with Hodge-Tate weights in $[0,r]$. We mainly study the rank two case. We use our full faithfulness theorem to assure the non-existence of crystalline lifts with small Hodge-Tate weights. Theorem \[nonfull\] is a consequence of studies of this section.
The author would like to thank Shin Hattori, Naoki Imai and Yuichiro Taguchi who gave him many valuable advice. This work was supported by JSPS KAKENHI Grant Number 25$\cdot$173.
[**Notation and convention:**]{} Throughout this paper, we fix a prime number $p$. Except Section 5, we always assume that [*$p$ is odd*]{}.
For any topological group $H$, we denote by ${\mathrm}{Rep}_{{\mathrm}{tor}}(H)$ (resp. ${\mathrm}{Rep}_{{\mathbb}{Z}_p}(H)$, resp. ${\mathrm}{Rep}_{{\mathbb}{Q}_p}(H)$) the category of torsion ${\mathbb}{Z}_p$-representations of $H$ (resp. the category of free ${\mathbb}{Z}_p$-representations of $H$, resp. the category of ${\mathbb}{Q}_p$-representations of $H$). All ${\mathbb}{Z}_p$-representations (resp. ${\mathbb}{Q}_p$-representations) in this paper are always assumed to be finitely generated over ${\mathbb}{Z}_p$ (resp. ${\mathbb}{Q}_p$).
For any field $F$, we denote by $G_F$ the absolute Galois group of $F$ (for a fixed separable closure of $F$).
Preliminaries
=============
In this section, we recall definitions and basic properties for Kisin modules and $({\varphi},\hat{G})$-modules. Throughout Section 2, 3 and 4, we always assume that $p$ is an odd prime.
Basic notations
---------------
Let $k$ be a perfect field of characteristic $p$, $W(k)$ the ring of Witt vectors with coefficients in $k$, $K_0=W(k)[1/p]$, $K$ a finite totally ramified extension of $K_0$ of degree $e$, $\overline{K}$ a fixed algebraic closure of $K$. Throughout this paper, we fix a uniformizer $\pi$ of $K$. Let $E(u)$ be the minimal polynomial of $\pi$ over $K_0$. Then $E(u)$ is an Eisenstein polynomial. For any integer $n\ge 0$, we fix a system $(\pi_n)_{n\ge 0}$ of a $p^n$-th root of $\pi$ in $\overline{K}$ such that $\pi^p_{n+1}=\pi_n$. Let $R={\varprojlim}{\mathcal{O}}_{\overline{K}}/p$, where ${\mathcal{O}}_{\overline{K}}$ is the integer ring of $\overline{K}$ and the transition maps are given by the $p$-th power map. For any integer $s\ge 0$, we write $\underline{\pi_s}:=(\pi_{s+n})_{n\ge 0}\in R$ and $\underline{\pi}:=\underline{\pi_0}\in R$. Note that we have $\underline{\pi_s}^{p^s}=\underline{\pi}$.
Let $L$ be the completion of an unramified algebraic extension of $K$ with residue field $k_L$. Then $\pi_s$ is a uniformizer of $L_{(s)}:=L(\pi_s)$ and $L_{(s)}$ is a totally ramified degree $ep^s$ extension of $L_0:=W(k_L)[1/p]$. We set $L_{\infty}:=\bigcup_{n\ge 0}L_{(n)}$. We put $G_{L,s}:=G_{L_{(s)}}={\mathrm}{Gal}(\overline{L}/L_{(s)})$ and $G_{L,\infty}:=G_{L_{\infty}}={\mathrm}{Gal}(\overline{L}/L_{\infty})$. By definitions, we have $L=L_{(0)}$ and $G_{L,0}=G_L$. Put ${\mathfrak{S}}_{L,s}=W(k_L)[\![u_s]\!]$ (resp. ${\mathfrak{S}}_L=W(k_L)[\![u]\!]$) with an indeterminate $u_s$ (resp. $u$). We equip a Frobenius endomorphism $\varphi$ of ${\mathfrak{S}}_{L,s}$ (resp. ${\mathfrak{S}}_L$) by $u_s\mapsto u_s^p$ (resp. $u\mapsto u^p$) and the Frobenius on $W(k_L)$. We embed the $W(k_L)$-algebra $W(k_L)[u_s]$ (resp. $W(k_L)[u]$) into $W(R)$ via the map $u_s\mapsto [\underline{\pi_s}]$ (resp. $u\mapsto [\underline{\pi}]$), where $[\ast]$ stands for the Teichmüller representative. This embedding extends to an embedding ${\mathfrak{S}}_{L,s}\hookrightarrow W(R)$ (resp. ${\mathfrak{S}}_L\hookrightarrow W(R)$). By identifying $u$ with $u_s^{p^s}$, we regard ${\mathfrak{S}}_L$ as a subalgebra of ${\mathfrak{S}}_{L,s}$. It is readily seen that the embedding ${\mathfrak{S}}_L\hookrightarrow {\mathfrak{S}}_{L,s} \hookrightarrow W(R)$ is compatible with the Frobenius endomorphisms. If we denote by $E_s(u_s)$ the minimal polynomial of $\pi_s$ over $K_0$, with indeterminate $u_s$, then we have $E_s(u_s)=E(u_s^{p^s})$. Therefore, we have $E_s(u_s)=E(u)$ in ${\mathfrak{S}}_{L,s}$. We note that the minimal polynomial of $\pi_s$ over $L_0$ is $E_s(u_s)$.
Let $S^{{\mathrm}{int}}_{L_0,s}$ (resp. $S^{{\mathrm}{int}}_{L_0})$) be the $p$-adic completion of the divided power envelope of $W(k_L)[u_s]$ (resp. $W(k_L)[u]$) with respect to the ideal generated by $E_s(u_s)$ (resp. $E(u)$). There exists a unique Frobenius map ${\varphi}\colon S^{{\mathrm}{int}}_{L_0,s}\to S^{{\mathrm}{int}}_{L_0,s}$ (resp. ${\varphi}\colon S^{{\mathrm}{int}}_{L_0}\to S^{{\mathrm}{int}}_{L_0}$) and monodromy $N\colon S^{{\mathrm}{int}}_{L_0,s}\to S^{{\mathrm}{int}}_{L_0,s}$ defined by $\varphi(u_s)=u_s^p$ (resp. $\varphi(u)=u^p$) and $N(u_s)=-u_s$ (resp. $N(u)=-u$). Put $S_{L_0,s}=S^{{\mathrm}{int}}_{L_0,s}[1/p]=L_0\otimes_{W(k_L)} S^{{\mathrm}{int}}_{L_0,s}$ (resp. $S_{L_0}=S^{{\mathrm}{int}}_{L_0}[1/p]=L_0\otimes_{W(k_L)} S^{{\mathrm}{int}}_{L_0}$). We equip $S^{{\mathrm}{int}}_{L_0,s}$ and $S_{L_0,s}$ (resp. $S^{{\mathrm}{int}}_{L_0}$ and $S_{L_0}$) with decreasing filtrations ${\mathrm}{Fil}^iS^{{\mathrm}{int}}_{L_0,s}$ and ${\mathrm}{Fil}^iS_{L_0,s}$ (resp. ${\mathrm}{Fil}^iS^{{\mathrm}{int}}_{L_0,s}$ and ${\mathrm}{Fil}^iS_{L_0,s}$) by the $p$-adic completion of the ideal generated by $E^j_s(u_s)/j!$ (resp. $E^j(u)/j!$) for all $j\ge 0$. The inclusion $W(k_L)[u_s]\hookrightarrow W(R)$ (resp. $W(k_L)[u]\hookrightarrow W(R)$) via the map $u_s\mapsto [\underline{\pi_s}]$ (resp. $u\mapsto [\underline{\pi}]$) induces ${\varphi}$-compatible inclusions ${\mathfrak{S}}_{L,s}\hookrightarrow S^{{\mathrm}{int}}_{L_0,s}\hookrightarrow A_{{\mathrm}{cris}}$ and $S_{L_0,s}\hookrightarrow B^+_{{\mathrm}{cris}}$ (resp. ${\mathfrak{S}}_L\hookrightarrow S^{{\mathrm}{int}}_{L_0}\hookrightarrow A_{{\mathrm}{cris}}$ and $S_{L_0}\hookrightarrow B^+_{{\mathrm}{cris}}$). By these inclusions, we often regard these rings as subrings of $B^+_{{\mathrm}{cris}}$. By identifying $u$ with $u_s^{p^s}$ as before, we regard $S^{{\mathrm}{int}}_{L_0}$ (resp. $S_{L_0}$) as a ${\varphi}$-stable (but not $N$-stable) subalgebra of $S^{{\mathrm}{int}}_{L_0,s}$ (resp. $S_{L_0,s}$). By definitions, we have ${\mathfrak{S}}_L={\mathfrak{S}}_{L,0},\ S^{{\mathrm}{int}}_{L_0,0}=S^{{\mathrm}{int}}_{L_0}$ and $S_{L_0,0}= S_{L_0}$.\
[**Convention:**]{} For simplicity, if $L=K$, then we often omit the subscript “$L$” from various notations (e.g. $G_{K_s}=G_s$, $G_{K_{\infty}}=G_{\infty}$, ${\mathfrak{S}}_K={\mathfrak{S}}, {\mathfrak{S}}_{K,s}={\mathfrak{S}}_s$).\
$\displaystyle \xymatrix{
& W(R) \ar@{-}[rr]
&
& A_{{\mathrm}{cris}} \ar@{-}[rr]
&
& B^+_{{\mathrm}{cris}}
\\
&
{\mathfrak{S}}_{L,s} \ar@{-}[rr] \ar@{-}[u]
&
& S^{{\mathrm}{int}}_{L_0,s}\ar@{-}[rr] \ar@{-}[u]
&
& S_{L_0,s} \ar@{-}[u]
\\
{\mathfrak{S}}_L \ar@{-}[rr] \ar@{-}[ru]
& \ar@{-}[u]
& S^{{\mathrm}{int}}_{L_0}\ar@{-}[rr] \ar@{-}[ru]
& \ar@{-}[u]
& S_{L_0} \ar@{-}[ru]
&
\\
&
{\mathfrak{S}}_s \ar@{-}[r] \ar@{-}[u]
& \ar@{-}[r]
& S^{{\mathrm}{int}}_{K_0,s}\ar@{-}[r] \ar@{-}[u]
& \ar@{-}[r]
& S_{K_0,s} \ar@{-}[uu]
\\
{\mathfrak{S}}\ar@{-}[rr] \ar@{-}[uu] \ar@{-}[ru]
&
& S^{{\mathrm}{int}}_{K_0}\ar@{-}[rr] \ar@{-}[uu] \ar@{-}[ru]
&
& S_{K_0} \ar@{-}[uu] \ar@{-}[ru]
&
}$
Kisin modules
-------------
Let $r, s\ge 0$ be integers. A [*${\varphi}$-module*]{} [*over ${\mathfrak{S}}_{L,s}$*]{} is an ${\mathfrak{S}}_{L,s}$-module ${\mathfrak{M}}$ equipped with a ${\varphi}$-semilinear map ${\varphi}\colon {\mathfrak{M}}\to {\mathfrak{M}}$. A morphism between two ${\varphi}$-modules $({\mathfrak{M}}_1,{\varphi}_1)$ and $({\mathfrak{M}}_2,{\varphi}_2)$ over ${\mathfrak{S}}_{L,s}$ is an ${\mathfrak{S}}_{L,s}$-linear map ${\mathfrak{M}}_1\to {\mathfrak{M}}_2$ compatible with ${\varphi}_1$ and ${\varphi}_2$. Denote by $'{\mathrm}{Mod}^r_{/{\mathfrak{S}}_{L,s}}$ the category of ${\varphi}$-modules $({\mathfrak{M}},{\varphi})$ over ${\mathfrak{S}}_{L,s}$ [*of height $\le r$*]{} in the sense that ${\mathfrak{M}}$ is of finite type over ${\mathfrak{S}}_{L,s}$ and the cokernel of $1\otimes {\varphi}\colon {\mathfrak{S}}_{L,s}\otimes_{{\varphi},{\mathfrak{S}}_{L,s}}{\mathfrak{M}}\to {\mathfrak{M}}$ is killed by $E_s(u_s)^r$.
Let ${\mathrm}{Mod}^r_{/{\mathfrak{S}}_{L,s}}$ be the full subcategory of $'{\mathrm}{Mod}^r_{/{\mathfrak{S}}_{L,s}}$ consisting of finite free ${\mathfrak{S}}_{L,s}$-modules. We call an object of ${\mathrm}{Mod}^r_{/{\mathfrak{S}}_{L,s}}$ a [*free Kisin module*]{} [*of height $\le r$ $($over ${\mathfrak{S}}_{L,s})$*]{}.
Let ${\mathrm}{Mod}^r_{/{\mathfrak{S}}_{L,s,\infty}}$ be the full subcategory of $'{\mathrm}{Mod}^r_{/{\mathfrak{S}}_{L,s}}$ consisting of finite ${\mathfrak{S}}_{L,s}$-modules which are killed by some power of $p$ and have projective dimension $1$ in the sense that ${\mathfrak{M}}$ has a two term resolution by finite free ${\mathfrak{S}}_{L,s}$-modules. We call an object of ${\mathrm}{Mod}^r_{/{\mathfrak{S}}_{L,s,\infty}}$ a [*torsion Kisin module of height $\le r$ $($over ${\mathfrak{S}}_{L,s})$*]{}.
For any free or torsion Kisin module ${\mathfrak{M}}$ over ${\mathfrak{S}}_{L,s}$, we define a ${\mathbb}{Z}_p[G_{L,\infty}]$-module $T_{{\mathfrak{S}}_{L,s}}({\mathfrak{M}})$ by $$\begin{aligned}
T_{{\mathfrak{S}}_{L,s}}({\mathfrak{M}}):=
\left\{
\begin{array}{ll}
{\mathrm}{Hom}_{{\mathfrak{S}}_{L,s},{\varphi}}({\mathfrak{M}},W(R))\hspace{21mm} {\rm if}\ {\mathfrak{M}}\ {\rm is}\ {\rm free},
\cr
{\mathrm}{Hom}_{{\mathfrak{S}}_{L,s},{\varphi}}({\mathfrak{M}},{\mathbb}{Q}_p/{\mathbb}{Z}_p\otimes_{{\mathbb}{Z}_p} W(R))\quad {\rm if}\ {\mathfrak{M}}\ {\rm is}\ {\rm torsion}.
\end{array}
\right.\end{aligned}$$ Here a $G_{L,\infty}$-action on $T_{{\mathfrak{S}}_{L,s}}({\mathfrak{M}})$ is given by $(\sigma.g)(x)=\sigma(g(x))$ for $\sigma\in G_{L,\infty}, g\in T_{{\mathfrak{S}}}({\mathfrak{M}}), x\in {\mathfrak{M}}$.\
[**Convention:**]{} For simplicity, if $L=K$, then we often omit the subscript “$L$” from various notations (e.g. ${\mathrm}{Mod}^r_{/{\mathfrak{S}}_{K,s,\infty}}={\mathrm}{Mod}^r_{/{\mathfrak{S}}_{s,\infty}}$, $T_{{\mathfrak{S}}_{K,s}}=T_{{\mathfrak{S}}_s}$ ). Also, if $s=0$, we often omit the subscript “$s$” from various notations (e.g. ${\mathrm}{Mod}^r_{/{\mathfrak{S}}_{L,0,\infty}}={\mathrm}{Mod}^r_{/{\mathfrak{S}}_{L,\infty}}$, $T_{{\mathfrak{S}}_{L,0}}=T_{{\mathfrak{S}}_L}$, ${\mathrm}{Mod}^r_{/{\mathfrak{S}}_{K,0,\infty}}={\mathrm}{Mod}^r_{/{\mathfrak{S}}_{\infty}}$, $T_{{\mathfrak{S}}_{K,0}}=T_{{\mathfrak{S}}}$ ).\
\[Kisinfunctor\] $(1)$ [([@Kis Corollary 2.1.4 and Proposition 2.1.12])]{} The functor $T_{{\mathfrak{S}}_{L,s}}\colon {\mathrm}{Mod}^r_{/{\mathfrak{S}}_{L,s}}\to
{\mathrm}{Rep}_{{\mathbb}{Z}_p}(G_{\infty})$ is exact and fully faithful.
$(2)$ [([@CL1 Corollary 2.1.6, 3.3.10 and 3.3.15])]{} The functor $T_{{\mathfrak{S}}_{L,s}}\colon {\mathrm}{Mod}^r_{/{\mathfrak{S}}_{L,s,\infty}}\to
{\mathrm}{Rep}_{{\mathrm}{tor}}(G_{\infty})$ is exact and faithful. Furthermore, it is full if $er<p-1$.
$({\varphi},\hat{G})$-modules {#Liumodule:section}
-----------------------------
The notion of $({\varphi},\hat{G})$-modules are introduced by Tong Liu in [@Li2] to classify lattices in semi-stable representations. We recall definitions and properties of them. We continue to use same notations as above.
Let $L_{p^{\infty}}$ be the field obtained by adjoining all $p$-power roots of unity to $L$. We denote by $\hat{L}$ the composite field of $L_{\infty}$ and $L_{p^{\infty}}$. We define $H_L:={\mathrm}{Gal}(\hat{L}/L_{\infty})$, $H_{L,\infty}:={\mathrm}{Gal}(\overline{K}/\hat{L})$ $G_{L,p^{\infty}}:={\mathrm}{Gal}(\hat{L}/L_{p^{\infty}})$ and $\hat{G}_L:={\mathrm}{Gal}(\hat{L}/L)$. Furthermore, putting $L_{(s),p^{\infty}}=L_{(s)}L_{p^{\infty}}$, we define $\hat{G}_{L,s}={\mathrm}{Gal}(\hat{L}/L_{(s)})$ and $G_{L,s,p^{\infty}}:={\mathrm}{Gal}(\hat{L}/L_{(s), p^{\infty}})$.
$$\xymatrix{
& & & & & \bar{K} \\
& & & & & \hat{L} \ar@{-}[u] \ar@{-}[u] \ar@/_1pc/@{-}[u]_{H_{L,\infty}}
\\
& & & L_{p^{\infty}}
\ar@/^1pc/@{-}[rru] ^{G_{L,p^{\infty}}}
\ar@{-}[rru]
& & & & \\
& & & & &
L_{\infty} \ar@/_1pc/@{-}[uu] _{H_L}
\ar@{-}[uu]
\ar@/_4pc/@{-}[uuu] _{G_{L,\infty}}
& & & & \\
& & & L \ar@{-}[rru]
\ar@/^1pc/@{-}[rruuu] ^{\hat{G}_L}
\ar@{-}[rruuu]
\ar@/^6pc/@{-}[rruuuu] ^{G_L}
\ar@{-}[uu]
& & & & \\
}$$
Since $p>2$, it is known that $L_{(s),p^{\infty}}\cap L_{\infty}=L_{(s)}$ and thus $\hat{G}_{L,s}=G_{L,s,p^{\infty}}\rtimes H_{L,s}$. Furthermore, $G_{L,s,p^{\infty}}$ is topologically isomorphic to ${\mathbb}{Z}_p$.
\[easylemma\] A natural map $G_{L,s,p^{\infty}}\to G_{K,s,p^{\infty}}$ defined by $g\mapsto g|_{\hat{K}}$ is bijective.
By replacing $L_s$ with $L$, we may assume $s=0$. It suffices to prove $\hat{K}\cap L_{p^{\infty}}=K_{p^{\infty}}$. Since $G_{K,p^{\infty}}$ is isomorphic to ${\mathbb}{Z}_p$, we know that any finite subextension of $\hat{K}/K_{p^{\infty}}$ is of the form $K_{(s),p^{\infty}}$ for some $s\ge 0$. Assume that we have $\hat{K}\cap L_{p^{\infty}}\not=K_{p^{\infty}}$. Then we have $K_{(1)}\subset \hat{K}\cap L_{p^{\infty}}\subset L_{p^{\infty}}$. Thus $\pi_1$ is contained in $L_{p^{\infty}}\cap L_{\infty}=L$. However, since $L$ is unramified over $K$, this contradicts the fact that $\pi$ is a uniformizer of $L$.
We fix a topological generator $\tau$ of $G_{K,p^{\infty}}$. We also denote by $\tau$ the pre-image of $\tau\in G_{K,p^{\infty}}$ for the bijection $G_{L,p^{\infty}}\simeq G_{K,p^{\infty}}$ of the above lemma. Note that $\tau^{p^s}$ is a topological generator of $G_{L,s,p^{\infty}}$.
For any $g\in G_K$, we put $\underline{{\varepsilon}}(g)=g(\underline{\pi})/\underline{\pi}\in R$, and define $\underline{{\varepsilon}}:=\underline{{\varepsilon}}(\tilde{\tau})$. Here, $\tilde{\tau}\in G_K$ is any lift of $\tau\in \hat{G}_K$ and then $\underline{{\varepsilon}}(\tilde{\tau})$ is independent of the choice of the lift of $\tau$. With these notation, we also note that we have $g(u)=[\underline{{\varepsilon}}(g)]u$ (recall that ${\mathfrak{S}}$ is embedded in $W(R)$). An easy computation shows that $\tau(\underline{\pi})/\underline{\pi}=\tau^{p^s}(\underline{\pi_s})/\underline{\pi_s}=\underline{{\varepsilon}}$. Therefore, we have $\tau(u)/u=\tau^{p^s}(u_s)/u_s=[\underline{{\varepsilon}}]$.
We put $t=-{\mathrm}{log}([\underline{{\varepsilon}}])\in A_{{\mathrm}{cris}}$. Denote by $\nu\colon W(R)\to W(\overline{k})$ a unique lift of the projection $R\to \overline{k}$, which extends to a map $\nu \colon B^+_{{\mathrm}{cris}}\to W(\overline{k})[1/p]$. For any subring $A\subset B^+_{{\mathrm}{cris}}$, we put $I_+A={\mathrm}{Ker}(\nu\ {\mathrm}{on}\ B^+_{{\mathrm}{cris}})\cap A$. For any integer $n\ge 0$, let $t^{\{n\}}:=t^{r(n)}\gamma_{\tilde{q}(n)}(\frac{t^{p-1}}{p})$ where $n=(p-1)\tilde{q}(n)+r(n)$ with $\tilde{q}(n)\ge 0,\ 0\le r(n) <p-1$ and $\gamma_i(x)=\frac{x^i}{i!}$ is the standard divided power. We define a subring ${\mathcal}{R}_{L_0,s}$ (resp. ${\mathcal}{R}_{L_0}$) of $B^+_{{\mathrm}{cris}}$ as below: $${\mathcal}{R}_{L_0,s}:=\{\sum^{\infty}_{i=0} f_it^{\{i\}}\mid f_i\in S_{L_0,s}\
{\mathrm}{and}\ f_i\to 0\ {\mathrm}{as}\ i\to \infty\}$$ $$({\rm resp}.\quad {\mathcal}{R}_{L_0}:=\{\sum^{\infty}_{i=0} f_it^{\{i\}}\mid f_i\in S_{L_0}\
{\mathrm}{and}\ f_i\to 0\ {\mathrm}{as}\ i\to \infty\}).$$
Put ${\widehat}{{\mathcal}{R}}_{L,s}={\mathcal}{R}_{L_0,s}\cap W(R)$ (resp. ${\widehat}{{\mathcal}{R}}_{L}={\mathcal}{R}_{L_0}\cap W(R)$) and $I_{+,L,s}=I_+{\widehat}{{\mathcal}{R}}_{L,s}$ (resp. $I_{+,L}=I_+{\widehat}{{\mathcal}{R}}_L$). By definitions, we have ${\mathcal}{R}_{L_0,0}={\mathcal}{R}_{L_0}$, ${\widehat}{{\mathcal}{R}}_{L,0}={\widehat}{{\mathcal}{R}}_{L}$ and $I_{+,L,0}=I_{+,L}$. Lemma 2.2.1 in [@Li2] shows that ${\widehat}{{\mathcal}{R}}_{L,s}$ $($resp. ${\mathcal}{R}_{L_0,s})$ is a ${\varphi}$-stable ${\mathfrak{S}}_{L,s}$-subalgebra of $W(R)$ $($resp. $B^+_{{\mathrm}{cris}})$, and $\nu$ induces ${\mathcal}{R}_{L_0,s}/I_+{\mathcal}{R}_{L_0,s}\simeq L_0$ and ${\widehat}{{\mathcal}{R}}_{L,s}/I_{+,L,s}\simeq S^{{\mathrm}{int}}_{L_0,s}/I_+S^{{\mathrm}{int}}_{L_0,s}
\simeq {\mathfrak{S}}_{L,s}/I_+{\mathfrak{S}}_{L,s}\simeq W(k_L)$. Furthermore, ${\widehat}{{\mathcal}{R}}_{L,s}, I_{+,L,s}, {\mathcal}{R}_{L_0,s}$ and $I_+{\mathcal}{R}_{L_0,s}$ are $G_{L,s}$-stable, and $G_{L,s}$-actions on them factors through $\hat{G}_{L,s}$. For any torsion Kisin module ${\mathfrak{M}}$ over ${\mathfrak{S}}_{L,s}$, we equip ${\widehat{\mathcal{R}}}_{L,s}\otimes_{{\varphi},{\mathfrak{S}}_{L,s}} {\mathfrak{M}}$ with a Frobenius by ${\varphi}_{{\widehat}{{\mathcal}{R}}_{L,s}}\otimes {\varphi}_{{\mathfrak{M}}}$. It is known that the natural map $
{\mathfrak{M}}\rightarrow {\widehat}{{\mathcal}{R}}_{L,s}\otimes_{{\varphi}, {\mathfrak{S}}_{L,s}} {\mathfrak{M}}$ given by $x\mapsto 1\otimes x$ is an injection (cf. [@Oz1 Corollary 2.12]). By this injection, we regard ${\mathfrak{M}}$ as a ${\varphi}({\mathfrak{S}}_{L,s})$-stable submodule of ${\widehat{\mathcal{R}}}_{L,s}\otimes_{{\varphi},{\mathfrak{S}}_{L,s}} {\mathfrak{M}}$.
\[Liumod\] A [*free*]{} (resp. [*torsion*]{}) [*$({\varphi}, \hat{G}_{L,s})$-module of height*]{} $\le r$ over ${\mathfrak{S}}_{L,s}$ is a triple $\hat{{\mathfrak{M}}}=({\mathfrak{M}}, {\varphi}_{{\mathfrak{M}}}, \hat{G}_{L,s})$ where
1. $({\mathfrak{M}}, {\varphi}_{{\mathfrak{M}}})$ is a free (resp. torsion) Kisin module of height $\le r$ over ${\mathfrak{S}}_{L,s}$,
2. $\hat{G}_{L,s}$ is an ${\widehat}{{\mathcal}{R}}_{L,s}$-semilinear $\hat{G}_{L,s}$-action on ${\widehat{\mathcal{R}}}_{L,s}\otimes_{{\varphi}, {\mathfrak{S}}_{L,s}} {\mathfrak{M}}$ which induces a continuous $G_{L,s}$-action on $W({\mathrm}{Fr}R)\otimes_{{\varphi}, {\mathfrak{S}}_{L,s}} {\mathfrak{M}}$ for the weak topology[^2],
3. the $\hat{G}_{L,s}$-action commutes with ${\varphi}_{{\widehat}{{\mathcal}{R}}_{L,s}}\otimes {\varphi}_{{\mathfrak{M}}}$,
4. ${\mathfrak{M}}\subset ({\widehat{\mathcal{R}}}_{L,s}\otimes_{{\varphi},{\mathfrak{S}}_{L,s}} {\mathfrak{M}})^{H_L}$,
5. $\hat{G}_{L,s}$ acts on the $W(k_L)$-module $({\widehat{\mathcal{R}}}_{L,s}\otimes_{{\varphi},{\mathfrak{S}}_{L,s}} {\mathfrak{M}})/
I_{+,L,s}({\widehat{\mathcal{R}}}_{L,s}\otimes_{{\varphi},{\mathfrak{S}}_{L,s}} {\mathfrak{M}})$ trivially.
A morphism between two $({\varphi}, \hat{G}_{L,s})$-modules $\hat{{\mathfrak{M}}}_1=({\mathfrak{M}}_1, {\varphi}_1, \hat{G})$ and $\hat{{\mathfrak{M}}}_2=({\mathfrak{M}}_2, {\varphi}_2, \hat{G})$ is a morphism $f\colon {\mathfrak{M}}_1\to {\mathfrak{M}}_2$ of ${\varphi}$-modules over ${\mathfrak{S}}_{L,s}$ such that ${\widehat}{{\mathcal}{R}}_{L,s}\otimes f\colon {\widehat{\mathcal{R}}}_{L,s}\otimes_{{\varphi},{\mathfrak{S}}_{L,s}} {\mathfrak{M}}_1\to
{\widehat{\mathcal{R}}}_{L,s}\otimes_{{\varphi},{\mathfrak{S}}_{L,s}} {\mathfrak{M}}_2$ is $\hat{G}_{L,s}$-equivariant. We denote by ${\mathrm}{Mod}^{r,\hat{G}_{L,s}}_{/{\mathfrak{S}}_{L,s}}$ (resp. ${\mathrm}{Mod}^{r,\hat{G}_{L,s}}_{/{\mathfrak{S}}_{L,s,\infty}}$) the category of free (resp. torsion) $({\varphi}, \hat{G}_{L,s})$-modules of height $\le r$ over ${\mathfrak{S}}_{L,s}$. We often regard ${\widehat{\mathcal{R}}}_{L,s}\otimes_{{\varphi},{\mathfrak{S}}_{L,s}} {\mathfrak{M}}$ as a $G_{L,s}$-module via the projection $G_{L,s}\twoheadrightarrow \hat{G}_{L,s}$.
For any free or torsion $({\varphi}, \hat{G}_{L,s})$-module $\hat{{\mathfrak{M}}}$ over ${\mathfrak{S}}_{L,s}$, we define a ${\mathbb}{Z}_p[G_{L,s}]$-module $\hat{T}_{L,s}(\hat{{\mathfrak{M}}})$ by $$\begin{aligned}
\hat{T}_{L,s}(\hat{{\mathfrak{M}}})=
\left\{
\begin{array}{ll}
{\mathrm}{Hom}_{{\widehat}{{\mathcal}{R}}_{L,s},{\varphi}}
({\widehat{\mathcal{R}}}_{L,s}\otimes_{{\varphi},{\mathfrak{S}}_{L,s}} {\mathfrak{M}}, W(R))
\hspace{21mm} {\rm if}\ {\mathfrak{M}}\ {\rm is}\ {\rm free},
\cr
{\mathrm}{Hom}_{{\widehat}{{\mathcal}{R}}_{L,s},{\varphi}}
({\widehat{\mathcal{R}}}_{L,s}\otimes_{{\varphi},{\mathfrak{S}}_{L,s}} {\mathfrak{M}}, {\mathbb}{Q}_p/{\mathbb}{Z}_p\otimes_{{\mathbb}{Z}_p}W(R))
\hspace{3.5mm} {\rm if}\ {\mathfrak{M}}\ {\rm is}\ {\rm torsion}.
\end{array}
\right.\end{aligned}$$ Here, $G_{L,s}$ acts on $\hat{T}_{L,s}(\hat{{\mathfrak{M}}})$ by $(\sigma.f)(x)=\sigma(f(\sigma^{-1}(x)))$ for $\sigma\in G_{L,s},\ f\in \hat{T}_{L,s}(\hat{{\mathfrak{M}}}),\
x\in {\widehat{\mathcal{R}}}_{L,s}\otimes_{{\varphi},{\mathfrak{S}}_{L,s}}{\mathfrak{M}}$.\
Then, there exists a natural $G_{L,\infty}$-equivariant map $$\theta_{L,s}\colon T_{{\mathfrak{S}}_{L,s}}({\mathfrak{M}})\to \hat{T}_{L,s}(\hat{{\mathfrak{M}}})$$ defined by $\theta(f)(a\otimes x)=a{\varphi}(f(x))$ for $f\in T_{{\mathfrak{S}}_{L,s}}({\mathfrak{M}}),\ a\in {\widehat}{{\mathcal}{R}}_{L,s}, x\in {\mathfrak{M}}$. We have
The map $\theta_{L,s}$ is an isomorphism of ${\mathbb}{Z}_p[G_{L,\infty}]$-modules.
[**Convention:**]{} For simplicity, if $L=K$, then we often omit the subscript “$L$” from various notations (e.g. “a $({\varphi}, \hat{G}_{K,s})$-module” = “a $({\varphi}, \hat{G}_s)$-module”, ${\mathrm}{Mod}^{r,\hat{G}_{K,s}}_{/{\mathfrak{S}}_{K,s}}
={\mathrm}{Mod}^{r,\hat{G}_s}_{/{\mathfrak{S}}_s}$, ${\mathrm}{Mod}^{r,\hat{G}_{K,s}}_{/{\mathfrak{S}}_{K,s,\infty}}
={\mathrm}{Mod}^{r,\hat{G}_s}_{/{\mathfrak{S}}_{s,\infty}}$, $\hat{T}_{K,s}=\hat{T}_s$, $\theta_{K,s}=\theta_s$). Furthermore, if $s=0$, we often omit the subscript “$s$” from various notations (e.g. ${\mathrm}{Mod}^{r,\hat{G}_{L,0}}_{/{\mathfrak{S}}_{L,0}}
={\mathrm}{Mod}^{r,\hat{G}_L}_{/{\mathfrak{S}}_L}$, ${\mathrm}{Mod}^{r,\hat{G}_{L,0}}_{/{\mathfrak{S}}_{L,0,\infty}}
={\mathrm}{Mod}^{r,\hat{G}_L}_{/{\mathfrak{S}}_{L,\infty}}$, $\hat{T}_{L,0}=\hat{T}_L$, ${\mathrm}{Mod}^{r,\hat{G}_{K,0}}_{/{\mathfrak{S}}_{K,0}}
={\mathrm}{Mod}^{r,\hat{G}}_{/{\mathfrak{S}}}$, “a $({\varphi}, \hat{G}_{K,0})$-module” = “a $({\varphi}, \hat{G})$-module”, $\hat{T}_{K,0}=\hat{T}$, $\theta_{K,0}=\theta$).\
Let ${\mathrm}{Rep}^{r,{\mathrm}{st}}_{{\mathbb}{Q}_p}(G_{L,s})$ (resp. ${\mathrm}{Rep}^{r,{\mathrm}{cris}}_{{\mathbb}{Q}_p}(G_{L,s})$, resp. ${\mathrm}{Rep}^{r,{\mathrm}{st}}_{{\mathbb}{Z}_p}(G_{L,s})$, resp. ${\mathrm}{Rep}^{r,{\mathrm}{cris}}_{{\mathbb}{Z}_p}(G_{L,s})$) be the categories of semi-stable ${\mathbb}{Q}_p$-representations of $G_{L,s}$ with Hodge-Tate weights in $[0,r]$ (resp. the categories of crystalline ${\mathbb}{Q}_p$-representations of $G_{L,s}$ with Hodge-Tate weights in $[0,r]$, resp. the categories of lattices in semi-stable ${\mathbb}{Q}_p$-representations of $G_{L,s}$ with Hodge-Tate weights in $[0,r]$, resp. the categories of lattices in crystalline ${\mathbb}{Q}_p$-representations of $G_{L,s}$ with Hodge-Tate weights in $[0,r]$).
There exists ${\mathfrak{t}}\in W(R)\smallsetminus pW(R)$ such that ${\varphi}({\mathfrak{t}})=pE(0)^{-1}E(u){\mathfrak{t}}$. Such ${\mathfrak{t}}$ is unique up to units of ${\mathbb}{Z}_p$ (cf. [@Li2 Example 2.3.5]). Now we define the full subcategory ${\mathrm}{Mod}^{r,\hat{G},{\mathrm}{cris}}_{/{\mathfrak{S}}}$ (resp. ${\mathrm}{Mod}^{r,\hat{G},{\mathrm}{cris}}_{/{\mathfrak{S}}_{\infty}}$) of ${\mathrm}{Mod}^{r,\hat{G}}_{/{\mathfrak{S}}}$ (resp. ${\mathrm}{Mod}^{r,\hat{G}}_{/{\mathfrak{S}}_{\infty}}$) consisting of objects $\hat{{\mathfrak{M}}}$ which satisfy the following condition; $
\tau(x)-x\in u^p{\varphi}({\mathfrak{t}})(W(R)\otimes_{{\varphi},{\mathfrak{S}}} {\mathfrak{M}})
$ for any $x\in {\mathfrak{M}}$.
The following results are important properties for the functor $\hat{T}_{L,s}$.
\[Thm1\] $(1)$ [([@Li2 Theorem 2.3.1 (2)])]{}The functor $\hat{T}$ induces an anti-equivalence of categories between ${\mathrm}{Mod}^{r,\hat{G}}_{/{\mathfrak{S}}}$ and ${\mathrm}{Rep}^{r,{\mathrm}{st}}_{{\mathbb}{Z}_p}(G_K)$.
$(2)$ [([@GLS Proposition 5.9], [@Oz2 Theorem 19])]{} The functor $\hat{T}$ induces an anti-equivalence of categories between ${\mathrm}{Mod}^{r,\hat{G},{\mathrm}{cris}}_{/{\mathfrak{S}}}$ and ${\mathrm}{Rep}^{r,{\mathrm}{cris}}_{{\mathbb}{Z}_p}(G_K)$.
$(3)$ [([@Oz1 Corollary 2.8 and 5.34])]{}The functor $\hat{T}_{L,s}\colon {\mathrm}{Mod}^{r,\hat{G}_{L,s}}_{/{\mathfrak{S}}_{L,s,\infty}}
\to {\mathrm}{Rep}_{{\mathrm}{tor}}(G_{L,s})$ is exact and faithful. Furthermore, it is full if $er<p-1$.
$({\varphi}, \hat{G})$-modules, Breuil modules and filtered $({\varphi},N)$-modules {#relations}
-----------------------------------------------------------------------------------
We recall some relations between Breuil modules and $({\varphi},\hat{G})$-modules. Here we give a rough sketch only. For more precise information, see [@Br1 Section 6], [@Li1 Section 5] and the proof of [@Li2 Theorem 2.3.1 (2)].
Let $\hat{{\mathfrak{M}}}$ be a free $({\varphi},\hat{G}_{L,s})$-module over ${\mathfrak{S}}_{L,s}$. If we put ${\mathcal}{D}:=S_{L_0,s}\otimes_{{\varphi},{\mathfrak{S}}_{L,s}} {\mathfrak{M}}$, then ${\mathcal}{D}$ has a structure of a Breuil module over $S_{L_0,s}$ which corresponds to the semi-stable representation ${\mathbb}{Q}_p\otimes_{{\mathbb}{Z}_p} \hat{T}_{L,s}(\hat{{\mathfrak{M}}})$ of $G_{L,s}$ (for the definition and properties of Breuil modules, see [@Br1]). Thus ${\mathcal}{D}$ is equipped with a Frobenius ${\varphi}_{{\mathcal}{D}}(={\varphi}_{S_{L_0,s}}\otimes {\varphi}_{{\mathfrak{M}}})$, a decreasing filtration $({\mathrm}{Fil}^i{\mathcal}{D})_{i\ge 0}$ of $S_{L_0,s}$-submodules of ${\mathcal}{D}$ and a $L_0$-linear monodromy operator $N\colon {\mathcal}{D}\to {\mathcal}{D}$ which satisfy certain properties (for example, Griffiths transversality).
Putting $D={\mathcal}{D}/I_+S_{L_0,s}{\mathcal}{D}$, we can associate a filtered $({\varphi},N)$-module over $L_{(s)}$ as following: ${\varphi}_{D}:={\varphi}_{{\mathcal}{D}}\ {\mathrm}{mod}\ I_+S_{L_0,s}{\mathcal}{D}$, $N_D:= N_{{\mathcal}{D}}\ {\mathrm}{mod}\ I_+S_{L_0,s}{\mathcal}{D}$ and ${\mathrm}{Fil}^iD_{L_{(s)}}:=f_{\pi_s}(Fil^i({\mathcal}{D}))$. Here, $f_{\pi_s}\colon {\mathcal}{D}\to D_{L_{(s)}}$ is the projection defined by ${\mathcal}{D}\twoheadrightarrow {\mathcal}{D}/{\mathrm}{Fil}^1S_{L_0,s}{\mathcal}{D}\simeq D_{L_{(s)}}$. Proposition 6.2.1.1 of [@Br1] implies that there exists a unique ${\varphi}$-compatible section $s\colon D\hookrightarrow {\mathcal}{D}$ of ${\mathcal}{D}\twoheadrightarrow D$. By this embedding, we regard $D$ as a submodule of ${\mathcal}{D}$. Then we have $N_{{\mathcal}{D}}|_{D}=N_D$ and $N_{{\mathcal}{D}}=N_{S_{L_0,s}}\otimes {\mathrm}{Id}_D + {\mathrm}{Id}_{S_{L_0,s}}\otimes N_D$ under the identification ${\mathcal}{D}=S_{L_0,s} \otimes_{L_{(s)}} D$.
The $G_{L,s}$-action on ${\widehat{\mathcal{R}}}_{L,s}\otimes_{{\varphi},{\mathfrak{S}}_{L,s}}{\mathfrak{M}}$ extends to $B^+_{{\mathrm}{cris}}\otimes_{{\widehat{\mathcal{R}}}_{L,s}} ({\widehat{\mathcal{R}}}_{L,s}\otimes_{{\varphi},{\mathfrak{S}}_{L,s}}{\mathfrak{M}})
\simeq B^+_{{\mathrm}{cris}}\otimes_{S_{L_0,s}}{\mathcal}{D}$. This action is in fact explicitly written as follows: $$\label{explicit}
g(a\otimes x)=\sum^{\infty}_{i=0}g(a)\gamma_i
(-{\mathrm}{log}(\frac{g[\underline{\pi_s}]}{[\underline{\pi_s}]}))\otimes N^i_{{\mathcal}{D}}(x)\quad
{\rm for}\ g\in G_{L,s}, a\in B^+_{{\mathrm}{cris}}, x\in {\mathcal}{D}.$$ By this explicit formula, we can obtain an easy relation between $N_{{\mathcal}{D}}$ and $\tau^{p^s}$-action on $\hat{{\mathfrak{M}}}$ as follows: first we recall that $t=-{\mathrm}{log}(\tau([\underline{\pi}])/[\underline{\pi}])
=-{\mathrm}{log}(\tau^{p^s}([\underline{\pi_s}])/[\underline{\pi_s}])$. By the formula, for any $n\ge 0$ and $x\in {\mathcal}{D}$, an induction on $n$ shows that we have $$(\tau^{p^s}-1)^n(x)=\sum^{\infty}_{m=n}(\sum_{i_1+\cdots i_n=m, i_j\ge 0}\frac{m!}{i_1!\cdots i_n!})
\gamma_m(t)\otimes N^m_{{\mathcal}{D}}(x)
\in B^{+}_{{\mathrm}{cris}}\otimes_{S_{L_0,s}} {\mathcal}{D}$$ and in particular we see $\frac{(\tau^{p^s}-1)^n}{n}(x)\to 0$ $p$-adically as $n\to \infty$. Hence we can define $${\mathrm}{log}(\tau^{p^s})(x):=
\sum^{\infty}_{n=1}(-1)^{n-1}\frac{(\tau^{p^s}-1)^n}{n}(x) \in B^{+}_{{\mathrm}{cris}}\otimes_{S_{L_0,s}} {\mathcal}{D}.$$ It is not difficult to check the equation $$\label{eq1}
{\mathrm}{log}(\tau^{p^s})(x)=t\otimes N_{{\mathcal}{D}}(x).$$
Base changes for Kisin modules
------------------------------
Let ${\mathfrak{M}}$ be a free or torsion Kisin module of height $\le r$ over ${\mathfrak{S}}_L$ (resp. over ${\mathfrak{S}}$). We put ${\mathfrak{M}}_{L,s}={\mathfrak{S}}_{L,s}\otimes_{{\mathfrak{S}}_L} {\mathfrak{M}}$ (resp. ${\mathfrak{S}}_L={\mathfrak{S}}_L\otimes_{{\mathfrak{S}}} {\mathfrak{M}}$) and equip ${\mathfrak{M}}_{L,s}$ (resp. ${\mathfrak{M}}_L$) with a Frobenius by ${\varphi}={\varphi}_{{\mathfrak{S}}_{L,s}}\otimes {\varphi}_{{\mathfrak{M}}}$ (resp. ${\varphi}={\varphi}_{{\mathfrak{S}}_L}\otimes {\varphi}_{{\mathfrak{M}}}$). Then it is not difficult to check that ${\mathfrak{M}}_{L,s}$ (resp. ${\mathfrak{M}}_L$) is a free or torsion Kisin module of height $\le r$ over ${\mathfrak{S}}_{L,s}$ (resp. over ${\mathfrak{S}}_L$) (here we recall that $E_s(u_s)=E(u^{p^s}_s)=E(u)$). Hence we obtained natural functors $${\mathrm}{Mod}^r_{/{\mathfrak{S}}_L}\to {\mathrm}{Mod}^r_{/{\mathfrak{S}}_{L,s}} \quad {\rm and}\quad
{\mathrm}{Mod}^r_{/{\mathfrak{S}}_{L,\infty}}\to {\mathrm}{Mod}^r_{/{\mathfrak{S}}_{L,s,\infty}}$$ $${\rm (resp.}\quad
{\mathrm}{Mod}^r_{/{\mathfrak{S}}}\to {\mathrm}{Mod}^r_{/{\mathfrak{S}}_L} \quad {\rm and}\quad
{\mathrm}{Mod}^r_{/{\mathfrak{S}}_{\infty}}\to {\mathrm}{Mod}^r_{/{\mathfrak{S}}_{L,\infty}}).$$ By definition, we immediately see that we have $T_{{\mathfrak{S}}_L}({\mathfrak{M}})\simeq T_{{\mathfrak{S}}_{L,s}}({\mathfrak{M}}_{L,s})$ (resp. $T_{{\mathfrak{S}}}({\mathfrak{M}})|_{G_{L_{\infty}}}\simeq T_{{\mathfrak{S}}_L}({\mathfrak{M}}_L)$). In particular, it follows from Proposition \[Kisinfunctor\] (1) that the following holds:
\[basechange1:Kisin\] The functor ${\mathrm}{Mod}^r_{/{\mathfrak{S}}_L}\to {\mathrm}{Mod}^r_{/{\mathfrak{S}}_{L,s}}$ is fully faithful.
Base changes for $({\varphi},\hat{G})$-modules
----------------------------------------------
Let $\hat{{\mathfrak{M}}}$ be a free or torsion $({\varphi},\hat{G}_L)$-module (resp. $({\varphi},\hat{G})$-module) of height $\le r$ over ${\mathfrak{S}}_L$ (resp. over ${\mathfrak{S}}$). The $G_{L,s}$ action on ${\widehat{\mathcal{R}}}_L\otimes_{{\varphi},{\mathfrak{S}}_L} {\mathfrak{M}}$ (resp. the $G_L$ action on ${\widehat{\mathcal{R}}}\otimes_{{\varphi},{\mathfrak{S}}} {\mathfrak{M}}$) extends to ${\widehat{\mathcal{R}}}_{L,s}\otimes_{{\widehat{\mathcal{R}}}_L}({\widehat{\mathcal{R}}}_L\otimes_{{\varphi},{\mathfrak{S}}_L} {\mathfrak{M}})
\simeq {\widehat{\mathcal{R}}}_{L,s}\otimes_{{\varphi},{\mathfrak{S}}_{L,s}} {\mathfrak{M}}_{L,s}$ (resp. ${\widehat{\mathcal{R}}}_L\otimes_{{\widehat{\mathcal{R}}}}({\widehat{\mathcal{R}}}\otimes_{{\varphi},{\mathfrak{S}}} {\mathfrak{M}})
\simeq {\widehat{\mathcal{R}}}_L\otimes_{{\varphi},{\mathfrak{S}}_L} {\mathfrak{M}}_L$), which factors through $\hat{G}_{L,s}$ (resp. $\hat{G}_L$). Then it is not difficult to check that ${\mathfrak{M}}_{L,s}$ (resp. ${\mathfrak{M}}_L$) has a structure of a $({\varphi},\hat{G}_{L,s})$-module (resp. $({\varphi},\hat{G}_L)$-module). Hence we obtained natural functors $${\mathrm}{Mod}^{r,\hat{G}_L}_{/{\mathfrak{S}}_L}\to {\mathrm}{Mod}^{r,\hat{G}_{L,s}}_{/{\mathfrak{S}}_{L,s}} \quad {\rm and}\quad
{\mathrm}{Mod}^{r,\hat{G}_L}_{/{\mathfrak{S}}_{L,\infty}}\to {\mathrm}{Mod}^{r,\hat{G}_{L,s}}_{/{\mathfrak{S}}_{L,s,\infty}}$$ $${\rm (resp.}\quad
{\mathrm}{Mod}^{r,\hat{G}}_{/{\mathfrak{S}}}\to {\mathrm}{Mod}^{r,\hat{G}_L}_{/{\mathfrak{S}}_L} \quad {\rm and}\quad
{\mathrm}{Mod}^{r,\hat{G}}_{/{\mathfrak{S}}_{\infty}}\to {\mathrm}{Mod}^{r,\hat{G}_L}_{/{\mathfrak{S}}_{L,\infty}}).$$ By definition, we immediately see that we have $\hat{T}_L(\hat{{\mathfrak{M}}})|_{G_{L,s}}\simeq \hat{T}_{L,s}(\hat{{\mathfrak{M}}}_{L,s})$ (resp. $\hat{T}(\hat{{\mathfrak{M}}})|_{G_L}\simeq \hat{T}_L(\hat{{\mathfrak{M}}}_L)$). Similar to Proposition \[basechange1:Kisin\], we can prove the following.
The functor ${\mathrm}{Mod}^{r,\hat{G}_L}_{/{\mathfrak{S}}_L}\to {\mathrm}{Mod}^{r,\hat{G}_{L,s}}_{/{\mathfrak{S}}_{L,s}} $ is fully faithful.
The proposition immediately follows from the full faithfulness property of Theorem \[Thm1\] (1) and the lemma below.
\[totst\] Let $K'$ is a finite totally ramified extension of $K$. Then the restriction functor from the category of semi-stable ${\mathbb}{Q}_p$-representations of $G_K$ into the category of semi-stable ${\mathbb}{Q}_p$-representations of $G_{K'}$ is fully faithful.
Let $V$ and $V'$ be semi-stable ${\mathbb}{Q}_p$-representations of $G_K$ and let $f\colon V\to V'$ be a $G_{K'}$-equivariant homomorphism. Considering the morphism of filtered $({\varphi}, N)$-modules over $K'$ corresponding to $f$, we can check without difficulty that $f$ is in fact a morphism of filtered $({\varphi}, N)$-modules over $K$. This is because $K'$ is totally ramified over $K_0$ as same as $K$. This gives the desired result.
Variants of free $({\varphi},\hat{G})$-modules
==============================================
In this section, we define some variant notions of $({\varphi},\hat{G})$-modules. We continue to use same notation as in the previous section. In particular, $p$ is odd.
Definitions {#vardef}
-----------
We start with some definitions which are our main concern in this and the next section.
\[varLiumod\] We define the category ${{\mathrm}{Mod}}^{r,\hat{G}_{L,s}}_{/{\mathfrak{S}}_L}$ (resp. ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_{L,s}}_{/{\mathfrak{S}}_L}$) as follows. An object is a triple $\hat{{\mathfrak{M}}}=({\mathfrak{M}}, {\varphi}_{{\mathfrak{M}}}, \hat{G}_{L,s})$ where
1. $({\mathfrak{M}}, {\varphi}_{{\mathfrak{M}}})$ is a free Kisin module of height $\le r$ over ${\mathfrak{S}}_L$,
2. $\hat{G}_{L,s}$ is an ${\widehat}{{\mathcal}{R}}_L$-semilinear $\hat{G}_{L,s}$-action on ${\widehat{\mathcal{R}}}_L\otimes_{{\varphi}, {\mathfrak{S}}_L} {\mathfrak{M}}$ (resp. an ${\widehat}{{\mathcal}{R}}_{L,s}$-semilinear $\hat{G}_{L,s}$-action on ${\widehat{\mathcal{R}}}_{L,s}\otimes_{{\varphi}, {\mathfrak{S}}_L} {\mathfrak{M}}$) which induces a continuous $G_{L,s}$-action on $W({\mathrm}{Fr}R)\otimes_{{\varphi}, {\mathfrak{S}}_L} {\mathfrak{M}}$ for the weak topology,
3. the $\hat{G}_{L,s}$-action commutes with ${\varphi}_{{\widehat}{{\mathcal}{R}}_L}\otimes {\varphi}_{{\mathfrak{M}}}$ (resp. ${\varphi}_{{\widehat}{{\mathcal}{R}}_{L,s}}\otimes {\varphi}_{{\mathfrak{M}}}$),
4. ${\mathfrak{M}}\subset ({\widehat{\mathcal{R}}}_L\otimes_{{\varphi},{\mathfrak{S}}_L} {\mathfrak{M}})^{H_L}$ (resp. ${\mathfrak{M}}\subset ({\widehat{\mathcal{R}}}_{L,s}\otimes_{{\varphi},{\mathfrak{S}}_L} {\mathfrak{M}})^{H_L}$),
5. $\hat{G}_{L,s}$ acts on the $W(k_L)$-module $({\widehat{\mathcal{R}}}_L\otimes_{{\varphi},{\mathfrak{S}}_L} {\mathfrak{M}})/
I_{+,L}({\widehat{\mathcal{R}}}_L\otimes_{{\varphi},{\mathfrak{S}}_L} {\mathfrak{M}})$ (resp. $({\widehat{\mathcal{R}}}_{L,s}\otimes_{{\varphi},{\mathfrak{S}}_L} {\mathfrak{M}})/
I_{+,L,s}({\widehat{\mathcal{R}}}_{L,s}\otimes_{{\varphi},{\mathfrak{S}}_L} {\mathfrak{M}})$) trivially.
Morphisms are defined by the obvious way. By replacing “free” of (1) with “torsion”[^3], we define the category ${{\mathrm}{Mod}}^{r,\hat{G}_{L,s}}_{/{\mathfrak{S}}_{L,\infty}}$ (resp. ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_{L,s}}_{/{\mathfrak{S}}_{L,\infty}}$).
For any object $\hat{{\mathfrak{M}}}$ of ${{\mathrm}{Mod}}^{r,\hat{G}_{L,s}}_{/{\mathfrak{S}}_L}$ or ${{\mathrm}{Mod}}^{r,\hat{G}_{L,s}}_{/{\mathfrak{S}}_{L,\infty}}$, we define a ${\mathbb}{Z}_p[G_{L,s}]$-module $\hat{T}_{L,s}(\hat{{\mathfrak{M}}})$ by $$\begin{aligned}
\hat{T}_{L,s}(\hat{{\mathfrak{M}}})=
\left\{
\begin{array}{ll}
{\mathrm}{Hom}_{{\widehat}{{\mathcal}{R}}_L,{\varphi}}
({\widehat{\mathcal{R}}}_L\otimes_{{\varphi},{\mathfrak{S}}_L} {\mathfrak{M}}, W(R))
\hspace{21mm} {\rm if}\ {\mathfrak{M}}\ {\rm is}\ {\rm free},
\cr
{\mathrm}{Hom}_{{\widehat{\mathcal{R}}}_L,{\varphi}}
({\widehat{\mathcal{R}}}_L\otimes_{{\varphi},{\mathfrak{S}}_L} {\mathfrak{M}}, {\mathbb}{Q}_p/{\mathbb}{Z}_p\otimes_{{\mathbb}{Z}_p}W(R))
\hspace{3.5mm} {\rm if}\ {\mathfrak{M}}\ {\rm is}\ {\rm torsion}.
\end{array}
\right.\end{aligned}$$ Here, $G_{L,s}$ acts on $\hat{T}_{L,s}(\hat{{\mathfrak{M}}})$ by $(\sigma.f)(x)=\sigma(f(\sigma^{-1}(x)))$ for $\sigma\in G_{L,s},\ f\in \hat{T}_{L,s}(\hat{{\mathfrak{M}}}),\
x\in {\widehat{\mathcal{R}}}_L\otimes_{{\varphi},{\mathfrak{S}}_L}{\mathfrak{M}}$. Similar to the above, for any object $\hat{{\mathfrak{M}}}$ of ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_{L,s}}_{/{\mathfrak{S}}_L}$ or ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_{L,s}}_{/{\mathfrak{S}}_{L,\infty}}$, we define a ${\mathbb}{Z}_p[G_{L,s}]$-module $\hat{T}_{L,s}(\hat{{\mathfrak{M}}})$ by $$\begin{aligned}
\hat{T}_{L,s}(\hat{{\mathfrak{M}}})=
\left\{
\begin{array}{ll}
{\mathrm}{Hom}_{{\widehat{\mathcal{R}}}_{L,s},{\varphi}}
({\widehat{\mathcal{R}}}_{L,s}\otimes_{{\varphi},{\mathfrak{S}}_L} {\mathfrak{M}}, W(R))
\hspace{21mm} {\rm if}\ {\mathfrak{M}}\ {\rm is}\ {\rm free},
\cr
{\mathrm}{Hom}_{{\widehat}{{\mathcal}{R}}_{L,s},{\varphi}}
({\widehat{\mathcal{R}}}_{L,s}\otimes_{{\varphi},{\mathfrak{S}}_L} {\mathfrak{M}}, {\mathbb}{Q}_p/{\mathbb}{Z}_p\otimes_{{\mathbb}{Z}_p}W(R))
\hspace{3.5mm} {\rm if}\ {\mathfrak{M}}\ {\rm is}\ {\rm torsion}.
\end{array}
\right.\end{aligned}$$
On the other hand, we obtain functors $
{{\mathrm}{Mod}}^{r,\hat{G}_L}_{/{\mathfrak{S}}_L}\to {{\mathrm}{Mod}}^{r,\hat{G}_{L,s}}_{/{\mathfrak{S}}_L}\to
{\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_{L,s}}_{/{\mathfrak{S}}_L}\to {{\mathrm}{Mod}}^{r,\hat{G}_{L,s}}_{/{\mathfrak{S}}_{L,s}}
$ and $
{{\mathrm}{Mod}}^{r,\hat{G}_L}_{/{\mathfrak{S}}_{L,\infty}}\to {{\mathrm}{Mod}}^{r,\hat{G}_{L,s}}_{/{\mathfrak{S}}_{L,\infty}}\to
{\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_{L,s}}_{/{\mathfrak{S}}_{L,\infty}}\to {{\mathrm}{Mod}}^{r,\hat{G}_{L,s}}_{/{\mathfrak{S}}_{L,s,\infty}}
$ by natural manners and it is readily seen that these functors are compatible with $\hat{T}_L$ and $\hat{T}_{L,s}$. In particular, the essential images of the functors $\hat{T}_{L,s}$ on ${{\mathrm}{Mod}}^{r,\hat{G}_{L,s}}_{/{\mathfrak{S}}_L}$ and ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_{L,s}}_{/{\mathfrak{S}}_L}$ has values in ${\mathrm}{Rep}^{r,{\mathrm}{st}}_{{\mathbb}{Z}_p}(G_{L,s})$ since we have an equivalence of categories $\hat{T}_{L,s}\colon {{\mathrm}{Mod}}^{r,\hat{G}_{L,s}}_{/{\mathfrak{S}}_{L,s}}
\overset{\sim}{\rightarrow} {\mathrm}{Rep}^{r,{\mathrm}{st}}_{{\mathbb}{Z}_p}(G_{L,s})$ by Theorem \[Thm1\].
In the rest of this section, we study free cases. We leave studies for torsion cases to the next section.\
[**Convention:**]{} For simplicity, if $L=K$, then we often omit the subscript “$L$” from various notations (e.g. ${{\mathrm}{Mod}}^{r,\hat{G}_{K,s}}_{/{\mathfrak{S}}_K}={{\mathrm}{Mod}}^{r,\hat{G}_s}_{/{\mathfrak{S}}},
{\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_{K,s}}_{/{\mathfrak{S}}_K}={\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s}_{/{\mathfrak{S}}}$). Furthermore, if $s=0$, we often omit the subscript “$s$” from various notations (e.g. ${{\mathrm}{Mod}}^{r,\hat{G}_{L,0}}_{/{\mathfrak{S}}_L}={{\mathrm}{Mod}}^{r,\hat{G}_L}_{/{\mathfrak{S}}_L},
{\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_{L,0}}_{/{\mathfrak{S}}_{L,0}}={\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_L}_{/{\mathfrak{S}}_L}$).\
The functors $
{{\mathrm}{Mod}}^{r,\hat{G}}_{/{\mathfrak{S}}}\to {{\mathrm}{Mod}}^{r,\hat{G}_s}_{/{\mathfrak{S}}}\to
{\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s}_{/{\mathfrak{S}}}\to {{\mathrm}{Mod}}^{r,\hat{G}_s}_{/{\mathfrak{S}}_s}
$
------------------------------------------------------------------------------------------------------------------
Now we consider the functors $
{{\mathrm}{Mod}}^{r,\hat{G}}_{/{\mathfrak{S}}}\to {{\mathrm}{Mod}}^{r,\hat{G}_s}_{/{\mathfrak{S}}}\to
{\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s}_{/{\mathfrak{S}}}\to {{\mathrm}{Mod}}^{r,\hat{G}_s}_{/{\mathfrak{S}}_s}.
$ At first, by Proposition \[basechange1:Kisin\], we see that the functor ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s}_{/{\mathfrak{S}}}\to {{\mathrm}{Mod}}^{r,\hat{G}_s}_{/{\mathfrak{S}}_s}$ is fully faithful. It follows from this fact and Theorem \[Thm1\] (1) that the functor $\hat{T}_s\colon {\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s}_{/{\mathfrak{S}}}
\to {\mathrm}{Rep}^{r,{\mathrm}{st}}_{{\mathbb}{Z}_p}(G_s)$ is fully faithful. It is clear that the functor ${{\mathrm}{Mod}}^{r,\hat{G}_s}_{/{\mathfrak{S}}}\to
{\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s}_{/{\mathfrak{S}}}$ is fully faithful and thus so is $\hat{T}_s\colon {{\mathrm}{Mod}}^{r,\hat{G}_s}_{/{\mathfrak{S}}}
\to {\mathrm}{Rep}^{r,{\mathrm}{st}}_{{\mathbb}{Z}_p}(G_s)$. Combining this with Theorem \[Thm1\] (1) and Lemma \[totst\], we obtain that the functor ${{\mathrm}{Mod}}^{r,\hat{G}}_{/{\mathfrak{S}}}\to {{\mathrm}{Mod}}^{r,\hat{G}_s}_{/{\mathfrak{S}}}$ is also fully faithful. Furthermore, we prove the following.
\[equal\] The functor ${{\mathrm}{Mod}}^{r,\hat{G}_s}_{/{\mathfrak{S}}}\to
{\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s}_{/{\mathfrak{S}}}$ is an equivalence of categories.
Summary, we obtained the following commutative diagram.
$\displaystyle \xymatrix{
{{\mathrm}{Mod}}^{r,\hat{G}}_{/{\mathfrak{S}}}
\ar@{^{(}->}[r] \ar[d]_{\wr} \ar^{\hat{T}}[d]
&
{{\mathrm}{Mod}}^{r,\hat{G}_s}_{/{\mathfrak{S}}}
\ar^{\sim}[r] \ar@{^{(}->}[rrd] \ar^{\hat{T}_s}[rrd]
&
{\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s}_{/{\mathfrak{S}}}
\ar@{^{(}->}[r] \ar@{^{(}->}[rd] \ar^{\hat{T}_s}[rd]
&
{{\mathrm}{Mod}}^{r,\hat{G}_s}_{/{\mathfrak{S}}_s}
\ar[d]_{\wr} \ar^{\hat{T}_s}[d]\\
{\mathrm}{Rep}^{r,{\mathrm}{st}}_{{\mathbb}{Z}_p}(G_K)
\ar@{^{(}->}[rrr] \ar^{{\mathrm}{restriction}}[rrr]
&
&
&
{\mathrm}{Rep}^{r,{\mathrm}{st}}_{{\mathbb}{Z}_p}(G_s).
}$
The functor ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s}_{/{\mathfrak{S}}}\hookrightarrow {{\mathrm}{Mod}}^{r,\hat{G}_s}_{/{\mathfrak{S}}_s}$ may not be possibly essentially surjective. In fact, under some conditions, there exists a representation of $G_K$ which is crystalline over $K_s$ but not of finite height. For more precise information, see [@Li2 Example 4.2.3].
Before a proof of Proposition \[equal\], we give an explicit formula such as (\[explicit\]) for an object of ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s}_{/{\mathfrak{S}}}$. The argument below follows the method of [@Li2]. Let $\hat{{\mathfrak{M}}}$ be an object of ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s}_{/{\mathfrak{S}}}$. Let $\hat{{\mathfrak{M}}}_s$ be the image of $\hat{{\mathfrak{M}}}$ for the functor ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s}_{/{\mathfrak{S}}}\to {{\mathrm}{Mod}}^{r,\hat{G}_s}_{/{\mathfrak{S}}_s}$. Put ${\mathcal}{D}=S_{K_0}\otimes_{{\varphi},{\mathfrak{S}}} {\mathfrak{M}}$ and also put ${\mathcal}{D}_s=S_{K_0,s}\otimes_{{\varphi},{\mathfrak{S}}_s} {\mathfrak{M}}_s
=S_{K_0,s}\otimes_{S_{K_0}} {\mathcal}{D}$. Then ${\mathcal}{D}_s$ has a structure of a Breuil module and also $D={\mathcal}{D}_s/I_+S_{K_0,s}{\mathcal}{D}_s$ has a structure of a filtered $({\varphi},N)$-module corresponding to ${\mathbb}{Q}_p\otimes_{{\mathbb}{Z}_p} \hat{T}_s(\hat{{\mathfrak{M}}}_s)$ (see subsection \[relations\]), which is isomorphic to ${\mathcal}{D}/I_+S_{K_0}{\mathcal}{D}$ as a ${\varphi}$-module over $K_0$. By [@Li1 Lemma 7.3.1], we have a unique ${\varphi}$-compatible section $D\hookrightarrow {\mathcal}{D}$ and we regard $D$ as a submodule of ${\mathcal}{D}\subset {\mathcal}{D}_s$ by this section. Then we have ${\mathcal}{D}=S_{K_0} \otimes_{K_0} D$ and ${\mathcal}{D}_s=S_{K_0,s} \otimes_{K_0} D$. By the explicit formula (\[explicit\]) for $\hat{{\mathfrak{M}}}_s$, we know that $$\hat{G}_s(D)\subset (K_0[\![t]\!]\cap {\mathcal}{R}_{K_0,s})\otimes_{K_0} D.$$ Hence, taking any $K_0$-basis $e_1,\dots ,e_d$ of $D$, there exist $A_s(t)\in M_{d\times d}(K_0[\![t]\!])$ such that $\tau^{p^s}(e_1,\cdots ,e_d)=(e_1,\dots ,e_d)A_s(t)$. Since $A_s(0)={\mathrm}{I}_d$, we see that ${\mathrm}{log}(A_s(t))\in M_{d\times d}(K_0[\![t]\!])$ is well-defined. On the other hand, choose $g_0\in G_s$ such that $\chi_p(g_0)\not=1$, where $\chi_p$ is the $p$-adic cyclotomic character. Since $g_0\tau^{p^s}=(\tau^{p^s})^{\chi_p(g_0)}g_0$, we have $A_s(\chi_p(g_0)t)=A_s(t)^{\chi_p(g_0)}$ and thus we also have ${\mathrm}{log}(A_s(\chi_p(g_0)t))=\chi_p(g_0){\mathrm}{log}(A_s(t))$. Since ${\mathrm}{log}(A_s(0))={\mathrm}{log}(I_d)=0$, we can write ${\mathrm}{log}(A_s(t))$ as $tB(t)$ for some $B(t)\in M_{d\times d}(K_0[\![t]\!])$. Then we have $\chi_p(g_0)tB(\chi_p(g_0)t)=\chi_p(g_0)tB(t)$, that is, $B(\chi_p(g_0)t)=B(t)$. Hence the assumption $\chi_p(g_0)\not= 1$ implies that $B(t)$ is a constant. Putting $N_s=B(t)\in M_{d\times d}(K_0)$, we obtain $$\tau^{p^s}(e_1,\cdots ,e_d)=(e_1,\cdots ,e_d)(\sum_{i\ge 0}N^i_s\gamma_i(t)).$$ Now we define $N_D\colon D\to D$ by $N(e_1,\cdots ,e_d)=(e_1,\cdots ,e_d)p^{-s}N_s$ and also define $N_{{\mathcal}{D}}:=N_{S_{K_0}}\otimes {\mathrm}{Id}_D+{\mathrm}{Id}_{S_{K_0}}\otimes N_D$. (Note that we have $N_D{\varphi}_D=p{\varphi}_D N_D$ and thus $N_D$ is nilpotent.) It is a routine work to check the following: $$\label{explicit''}
g(a\otimes x)=\sum^{\infty}_{i=0}g(a)\gamma_i
(-{\mathrm}{log}([\underline{{\varepsilon}}(g)]))\otimes N^i_D(x)\quad
{\rm for}\ g\in G_s, a\in B^+_{{\mathrm}{cris}}, x\in D.$$ Since we have $$\label{easyeq}
g(f)=\sum_{i\ge 0} \gamma_i(-{\mathrm}{log}([\underline{{\varepsilon}}(g)]))N^i_{S_{K_0}}(f)$$ for any $g\in G_K$ and $f\in S_{K_0}$, we obtain the following explicit formula: $$\label{explicit'}
g(a\otimes x)=\sum^{\infty}_{i=0}g(a)\gamma_i
(-{\mathrm}{log}([\underline{{\varepsilon}}(g)]))\otimes N^i_{{\mathcal}{D}}(x)\quad
{\rm for}\ g\in G_s, a\in B^+_{{\mathrm}{cris}}, x\in {\mathcal}{D}.$$ In particular, as in subsection \[relations\], we can show that $$\label{eq2}
{\mathrm}{log}(\tau^{p^s})(x)=p^st\otimes N_{{\mathcal}{D}}(x)$$ for any $x\in {\mathcal}{D}$.
We continue to use the above notation. It suffices to prove that the $G_s$-action on ${\widehat{\mathcal{R}}}_s\otimes_{{\varphi},{\mathfrak{S}}} {\mathfrak{M}}$ preserves ${\widehat{\mathcal{R}}}\otimes_{{\varphi},{\mathfrak{S}}}{\mathfrak{M}}$. Take any $g\in G_s$. We know that $g({\mathfrak{M}})\subset {\widehat{\mathcal{R}}}_s\otimes_{{\varphi},{\mathfrak{S}}} {\mathfrak{M}}\subset W(R)\otimes_{{\varphi},{\mathfrak{S}}} {\mathfrak{M}}$. Hence it is enough to prove that $g({\mathcal}{D})\in {\mathcal}{R}_{K_0}\otimes_{{\varphi},{\mathfrak{S}}} {\mathfrak{M}}$. Let $s\in S^{{\mathrm}{int}}_{K_0}$ and $y\in D$ and put $x=s\otimes y\in S^{{\mathrm}{int}}_{K_0}\otimes_{W(k)} D={\mathcal}{D}$. By (\[explicit”\]) or (\[explicit’\]), we have $$g(x)=\sum_{i\ge 0}\sum_{0\le j\le i} \gamma_i(-{\mathrm}{log}([\underline{{\varepsilon}}(g)]))
\binom{i}{j}N^{i-j}_{S_{K_0}}(s)\otimes N^j_D(y)$$ On the other hand, we know that $N_D$ is nilpotent, that is, there exists $j_0>0$ such that $N^{j_0}_D=0$. Then we obtain $$g(x)=\sum_{0\le j\le j_0} \sum^{\infty}_{i=j} \gamma_i(-{\mathrm}{log}([\underline{{\varepsilon}}(g)]))
\binom{i}{j}N^{i-j}_{S_{K_0}}(s)\otimes N^j_D(y).$$ Therefore, it suffices to show that $\sum^{\infty}_{i=j} \gamma_i(-{\mathrm}{log}([\underline{{\varepsilon}}(g)]))
\binom{i}{j}N^{i-j}_{S_{K_0}}(s)$ is contained in ${\mathcal}{R}_{K_0}$ for each $0\le j\le j_0$. Taking $\alpha(g)\in {\mathbb}{Z}_p$ such that ${\mathrm}{log}([\underline{{\varepsilon}}(g)])=-\alpha(g)t$, we have $$\sum^{\infty}_{i=j} \gamma_i(-{\mathrm}{log}([\underline{{\varepsilon}}(g)]))
\binom{i}{j}N^{i-j}_{S_{K_0}}(s)
=
\sum^{\infty}_{i=j} (\alpha(g))^i\frac{\tilde{q}(i)!p^{\tilde{q}(i)}}{i!}
\binom{i}{j}N^{i-j}_{S_{K_0}}(s)t^{\{i\}}.$$ Since $\frac{\tilde{q}(i)p^{\tilde{q}(i)}}{i!}\to 0$ ($p$-adically) as $i\to \infty$, we finish a proof.
Relations with crystalline representations
------------------------------------------
We know that ${\mathbb}{Q}_p\otimes_{{\mathbb}{Z}_p} \hat{T}_s(\hat{{\mathfrak{M}}})$ is semi-stable over $K_s$ for any object $\hat{{\mathfrak{M}}}$ of ${{\mathrm}{Mod}}^{r,\hat{G}_s}_{/{\mathfrak{S}}}$ or ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s}_{/{\mathfrak{S}}}$. This subsection is devoted to prove a criterion, for $\hat{{\mathfrak{M}}}$, that describes when ${\mathbb}{Q}_p\otimes_{{\mathbb}{Z}_p} \hat{T}_s(\hat{{\mathfrak{M}}})$ becomes crystalline.
Following [@Fo2 Section 5] we set $I^{[m]}B^+_{{\mathrm}{cris}}:=\{x\in B^+_{{\mathrm}{cris}} \mid {\varphi}^n(x)
\in {\mathrm}{Fil}^mB^+_{{\mathrm}{cris}}\ {\rm for\ all}\ n\ge0 \}$. For any subring $A\subset B^+_{{\mathrm}{cris}}$, we put $I^{[m]}A=A\cap I^{[m]}B^+_{{\mathrm}{cris}}$. Furthermore, we also put $I^{[m+]}A=I^{[m]}A.I_+A$ (here, $I_+A$ is defined in Subsection \[Liumodule:section\]). By [@Fo2 Proposition 5.1.3] and the proof of [@Li2 Lemma 3.2.2], we know that $I^{[m]}W(R)$ is a principal ideal which is generated by ${\varphi}({\mathfrak{t}})^m$.
Now we recall Theorem \[Thm1\] (2): if ${\mathfrak{M}}$ is an object of ${{\mathrm}{Mod}}^{r,\hat{G}_s}_{/{\mathfrak{S}}_s}$, then ${\mathbb}{Q}_p\otimes_{{\mathbb}{Z}_p} \hat{T}_s(\hat{{\mathfrak{M}}})$ is crystalline if and only if $\tau^{p^s}(x)-x\in u_s^p(I^{[1]}W(R)\otimes_{{\varphi},{\mathfrak{S}}_s} {\mathfrak{M}})$ for any $x\in {\mathfrak{M}}$. However, if such ${\mathfrak{M}}$ descends to a Kisin module over ${\mathfrak{S}}$, then we can show the following.
\[cris\] Let $\hat{{\mathfrak{M}}}$ be an object of ${{\mathrm}{Mod}}^{r,\hat{G}_s}_{/{\mathfrak{S}}}$ or ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s}_{/{\mathfrak{S}}}$. Then the following is equivalent:
1. ${\mathbb}{Q}_p\otimes_{{\mathbb}{Z}_p} \hat{T}_s(\hat{{\mathfrak{M}}})$ is crystalline,
2. $\tau^{p^s}(x)-x\in u^p(I^{[1]}W(R)\otimes_{{\varphi},{\mathfrak{S}}} {\mathfrak{M}})$ for any $x\in {\mathfrak{M}}$,
3. $\tau^{p^s}(x)-x\in I^{[1+]}W(R)\otimes_{{\varphi},{\mathfrak{S}}} {\mathfrak{M}}$ for any $x\in {\mathfrak{M}}$.
\(1) $\Rightarrow$ (2): The proof here mainly follows that of [@GLS Proposition 4.7]. We may suppose $\hat{{\mathfrak{M}}}$ is an object of ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s}_{/{\mathfrak{S}}}$. Put ${\mathcal}{D}=S_{K_0}\otimes_{{\varphi},{\mathfrak{S}}} {\mathfrak{M}}$ and $D={\mathcal}{D}/I_+S_{K_0} {\mathcal}{D}$ as in the previous subsection. We fix a ${\varphi}({\mathfrak{S}})$-basis $(\hat{e}_1,\dots ,\hat{e}_d)$ of ${\mathfrak{M}}\subset {\mathcal}{D}$ and denote by $(e_1,\dots ,e_d)$ the image of $(\hat{e}_1,\dots ,\hat{e}_d)$ for the projection ${\mathcal}{D}\to D$. Then $(e_1,\dots ,e_d)$ is a $K_0$-basis of $D$. As described before the proof of Proposition \[equal\], we regard $D$ as a ${\varphi}$-stable submodule of ${\mathcal}{D}$, and we have $N_D\colon D\to D$ and $N_{{\mathcal}{D}}\colon D_{{\mathcal}{D}}\to D_{{\mathcal}{D}}$.
Now we consider a matrix $X\in GL_{d\times d}(S_{K_0})$ such that $(\hat{e}_1,\dots ,\hat{e}_d)=(e_1,\dots ,e_d)X$. We define $\tilde{S}=W(k)[\![u^p, u^{ep}/p]\!]$ as in Section 4 of [@GLS], which is a sub $W(k)$-algebra of $S^{{\mathrm}{int}}_{K_0}$ with the property $N_{S_{K_0}}(\tilde{S})\subset u^p\tilde{S}$. By an easy computation we have $U=X^{-1}BX+X^{-1}N_{S_{K_0}}(X)$. Here, $B\in M_{d\times d}(K_0)$ and $U\in M_{d\times d}(S_{K_0})$ are defined by $N_D(e_1,\dots ,e_d)=(e_1,\dots ,e_d)B$ and $N_{\mathcal}{D}(\hat{e}_1,\dots ,\hat{e}_d)=(\hat{e}_1,\dots ,\hat{e}_d)U$. By the same proof as in the former half part of the proof of [@GLS Proposition 4.7], we obtain $X,X^{-1}\in M_{d\times d}(\tilde{S}[1/p])$. On the other hand, let $\hat{{\mathfrak{M}}}_s$ be the image of $\hat{{\mathfrak{M}}}$ for the functor ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s}_{/{\mathfrak{S}}}\to {{\mathrm}{Mod}}^{r,\hat{G}_s}_{{\mathfrak{S}}_s}$. Now we recall that ${\mathcal}{D}_s=S_{K_0,s}\otimes_{{\varphi},{\mathfrak{S}}_s} {\mathfrak{M}}_s$ has a structure of the Breuil module corresponding to ${\mathbb}{Q}_p\otimes_{{\mathbb}{Z}_p} \hat{T}_s(\hat{{\mathfrak{M}}}_s)$ Denote by $N_{{\mathcal}{D}_s}$ its monodromy operator. By the formula (\[eq1\]) for $\hat{{\mathfrak{M}}}_s$ and the formula (\[eq2\]) for $\hat{{\mathfrak{M}}}$, we see that $p^sN_{{\mathcal}{D}}=N_{{\mathcal}{D}_s}$ on ${\mathcal}{D}$. Therefore, ${\mathbb}{Q}_p\otimes_{{\mathbb}{Z}_p} \hat{T}_s(\hat{{\mathfrak{M}}})$ is crystalline if and only if $N_{{\mathcal}{D}_s}$ mod $I_+S_{K_0,s}{\mathcal}{D}_s$ is zero, which is equivalent to say that $N_D=(N_{{\mathcal}{D}}$ mod $I_+S_{K_0}{\mathcal}{D})$ is zero, that is, $B=0$. Therefore, the latter half part of the proof [@GLS Proposition] gives the assertion (2).
\(2) $\Rightarrow$ (3): This is clear.
\(3) $\Rightarrow$ (1): Suppose that (3) holds. We denote by $\hat{{\mathfrak{M}}}_s$ the image of $\hat{{\mathfrak{M}}}$ for the functor ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s}_{/{\mathfrak{S}}}\to {{\mathrm}{Mod}}^{r,\hat{G}_s}_{{\mathfrak{S}}_s}$ as above. We claim that, for any $x\in {\mathfrak{M}}_s$, we have $\tau^{p^s}(x)-x\in I^{[1+]}W(R)\otimes_{{\varphi}_s} {\mathfrak{M}}_s$. Let $x=a\otimes y\in {\mathfrak{M}}_s={\mathfrak{S}}_s\otimes_{{\mathfrak{S}}} {\mathfrak{M}}$ where $a\in {\mathfrak{S}}_s$ and $y\in {\mathfrak{M}}$. Then $$\tau^{p^s}(x)-x=\tau^{p^s}({\varphi}(a))(\tau^{p^s}(y)-y)+(\tau^{p^s}({\varphi}(a))-{\varphi}(a))y$$ and thus it suffices to show $\tau^{p^s}({\varphi}(a))-{\varphi}(a)\in I^{[1+]}W(R)$. This follows from the lemma below and thus we obtained the claim. By the claim and Theorem \[Thm1\] (2), we know that ${\mathbb}{Q}_p\otimes_{{\mathbb}{Z}_p} \hat{T}_s(\hat{{\mathfrak{M}}}_s)\simeq
{\mathbb}{Q}_p\otimes_{{\mathbb}{Z}_p} \hat{T}_s(\hat{{\mathfrak{M}}})$ is crystalline.
\[cryslem\] $(1)$ We have $I^{[1]}W(R)\cap u^{\ell}B^+_{{\mathrm}{cris}}
=u^{\ell}I^{[1]}W(R)$ for $\ell\ge 0$.
$(2)$ We have $g(a)-a\in uI^{[1]}W(R)$ for $g\in G$ and $a\in {\mathfrak{S}}$.
This is due to [@GLS the proof of Proposition 7] but we write a proof here.
\(1) Take $x=u^{\ell}y\in I^{[1]}W(R)$ with $y\in B^+_{{\mathrm}{cris}}$. By Lemma 3.2.2 of [@Li3] we have $y\in W(R)$. Now we remark that $uz\in {\mathrm}{Fil}^nW(R)$ with $z\in W(R)$ implies $z\in {\mathrm}{Fil}^nW(R)$ since $u$ is a unit of $B^+_{{\mathrm}{dR}}$. Hence $u^{\ell}y\in I^{[1]}W(R)$ implies $y\in I^{[1]}W(R)$.
\(2) By the relation (\[easyeq\]), we see that $g(a)-a\in I^{[1]}W(R)$. On the other hand, if $i>0$, we can write $N^i_{S_{K_0}}(a)=ub_i$ for some $b_i\in {\mathfrak{S}}$. Thus by the relation (\[easyeq\]) again we obtain $g(a)-a\in uB^+_{{\mathrm}{cris}}$. Then the result follows from (1).
Variants of torsion $({\varphi},\hat{G})$-modules
=================================================
In this section, we mainly study full subcategories of ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s}_{/{\mathfrak{S}}_{\infty}}$ defined below and also study representations associated with them. As a consequence, we prove theorems in Introduction. We use same notation as in Section 2 and 3. In particular, $p$ is odd. In below, let $v_R$ be the valuation of $R$ normalized such that $v_R(\underline{\pi})=1/e$ and, for any real number $x\ge 0$, we denote by ${\mathfrak{m}}^{\ge x}_R$ the ideal of $R$ consisting of elements $a$ with $v_R(a)\ge x$.
Let $J$ be an ideal of $W(R)$ which satisfies the following conditions:
- $J\not\subset pW(R)$,
- $J$ is a principal ideal,
- $J$ is ${\varphi}$-stable and $G_s$-stable in $W(R)$.
By the above first and second assumptions for $J$, the image of $J$ under the projection $W(R)\twoheadrightarrow R$ is of the form ${\mathfrak{m}}^{\ge c_J}_{R}$ for some real number $c_J\ge 0$.
We denote by ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s,J}_{/{\mathfrak{S}}_{\infty}}$ the full subcategory of ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s}_{/{\mathfrak{S}}_{\infty}}$ consisting of objects $\hat{{\mathfrak{M}}}$ which satisfy the following condition: $$\tau^{p^s}(x)-x\in JW(R)\otimes_{{\varphi}, {\mathfrak{S}}} {\mathfrak{M}}\quad {\rm for\ any}\ x\in {\mathfrak{M}}.$$ Also, we denote by ${\widetilde}{{\mathrm}{Rep}}^{r,\hat{G}_s, J}_{{\mathrm}{tor}}(G_s)$ the essential image of the functor $\hat{T}_s\colon {\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s}_{/{\mathfrak{S}}_{\infty}}\to {\mathrm}{Rep}_{{\mathrm}{tor}}(G_s)$ restricted to ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s,J}_{/{\mathfrak{S}}_{\infty}}$.
By definition, we have ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s,J}_{/{\mathfrak{S}}_{\infty}}\subset {\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s,J'}_{/{\mathfrak{S}}_{\infty}}$ and ${\widetilde}{{\mathrm}{Rep}}^{r,\hat{G}_s,J}_{{\mathrm}{tor}}(G_s)
\subset {\widetilde}{{\mathrm}{Rep}}^{r,\hat{G}_s,J'}_{{\mathrm}{tor}}(G_s)$ for $J\subset J'$.
Full faithfullness for ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s,J}_{/{\mathfrak{S}}_{\infty}}$
------------------------------------------------------------------------------------------------
For the beginning of a study of ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s,J}_{/{\mathfrak{S}}_{\infty}}$, we prove the following full faithfullness result.
\[FFTHMMOD\] Let $r$ and $r'$ be non-negative integers with $c_J> pr/(p-1)$. Let $\hat{{\mathfrak{M}}}$ and $\hat{{\mathfrak{N}}}$ be objects of ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s,J}_{/{\mathfrak{S}}_{\infty}}$ and ${\widetilde{{\mathrm}{Mod}}}^{r',\hat{G}_s,J}_{/{\mathfrak{S}}_{\infty}}$, respectively. Then we have ${\mathrm}{Hom}(\hat{{\mathfrak{M}}}, \hat{{\mathfrak{N}}})={\mathrm}{Hom}({\mathfrak{M}}, {\mathfrak{N}})$. $($Here, two “${\mathrm}{Hom}$”s are defined by obvious manners.$)$
In particular, if $c_J> pr/(p-1)$, then the forgetful functor ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s,J}_{/{\mathfrak{S}}_{\infty}}\to {{\mathrm}{Mod}}^r_{/{\mathfrak{S}}_{\infty}}$ is fully faithful.
A very similar proof of [@Oz2 Lemma 7] proceeds, and hence we only give a sketch here. Let $\hat{{\mathfrak{M}}}$ and $\hat{{\mathfrak{N}}}$ be objects of ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s,J}_{/{\mathfrak{S}}_{\infty}}$ and ${\widetilde{{\mathrm}{Mod}}}^{r',\hat{G}_s,J}_{/{\mathfrak{S}}_{\infty}}$, respectively. Let $f\colon {\mathfrak{M}}\to {\mathfrak{N}}$ be a morphism of Kisin modules over ${\mathfrak{S}}$. Put $\hat{f}=W(R)\otimes f\colon W(R)\otimes_{{\varphi},{\mathfrak{S}}}{\mathfrak{M}}\to W(R)\otimes_{{\varphi},{\mathfrak{S}}}{\mathfrak{M}}$. Choose any lift of $\tau\in \hat{G}$ to $G_K$; we denote it also by $\tau$. Since the $\hat{G}_s$-action for $\hat{{\mathfrak{M}}}$ is continuous, it suffices to prove that $\Delta(1\otimes x)=0$ for any $x\in {\mathfrak{M}}$ where $\Delta:=\tau^{p^s}\circ \hat{f}-\hat{f}\circ \tau^{p^s}$. We use induction on $n$ such that $p^n{\mathfrak{N}}=0$.
Suppose $n=1$. Since $\Delta=(\tau^{p^s}-1)\circ \hat{f}-\hat{f}\circ (\tau^{p^s}-1)$, we obtain the following:
$(0)$:For any $x\in {\mathfrak{M}}$, $\Delta(1\otimes x)\in
{\mathfrak{m}}^{\ge c(0)}_R(R\otimes_{{\varphi},{\mathfrak{S}}} {\mathfrak{N}})$
where $c(0)=c_J$. Since ${\mathfrak{M}}$ is of height $\le r$, we further obtain the following for any $i\ge 1$ inductively:
$(i)$:For any $x\in {\mathfrak{M}}$, $\Delta(1\otimes x)\in
{\mathfrak{m}}^{\ge c(i)}_R(R\otimes_{{\varphi},{\mathfrak{S}}} {\mathfrak{N}})$
where $c(i)=pc(i-1)-pr
=(c_J-pr/(p-1))p^i+pr/(p-1)$. The condition $c_J>pr/(p-1)$ implies that $c(i)\to \infty$ as $i\to \infty$ and thus $\Delta(1\otimes x)=0$.
Suppose $n>1$. Consider the exact sequence $0\to {\mathrm}{Ker}(p)\to {\mathfrak{N}}\overset{p}{\to} p{\mathfrak{N}}\to 0$ of ${\varphi}$-modules over ${\mathfrak{S}}$. It is not difficult to check that ${\mathfrak{N}}':={\mathrm}{Ker}(p)$ and ${\mathfrak{N}}'':=p{\mathfrak{N}}$ are torsion Kisin modules of height $\le r'$ over ${\mathfrak{S}}$ (cf. [@Li1 Lemma 2.3.1]). Moreover, we can check that ${\mathfrak{N}}'$ and ${\mathfrak{N}}''$ have natural structures of objects of ${\widetilde{{\mathrm}{Mod}}}^{r',\hat{G}_s}_{/{\mathfrak{S}}_{\infty}}$ (which are denoted by $\hat{{\mathfrak{N}}}'$ and $\hat{{\mathfrak{N}}}''$, respectively) such that the sequence $0\to {\mathfrak{N}}'\to {\mathfrak{N}}\overset{p}{\to} {\mathfrak{N}}''\to 0$ induces an exact sequence $0\to \hat{{\mathfrak{N}}}'\to \hat{{\mathfrak{N}}}\to \hat{{\mathfrak{N}}}''\to 0$. By the lemma below, we know that $\hat{{\mathfrak{N}}}'$ and $\hat{{\mathfrak{N}}}''$ are in fact contained in ${\widetilde{{\mathrm}{Mod}}}^{r',\hat{G}_s,J}_{/{\mathfrak{S}}_{\infty}}$. By the induction hypothesis, we see that $\Delta(1\otimes x)$ has values in $(W(R)\otimes_{{\varphi}, {\mathfrak{S}}} {\mathfrak{N}}')\cap
(JW(R)\otimes_{{\varphi}, {\mathfrak{S}}} {\mathfrak{N}})$. By Lemma 6 of [@Oz2] and the assumption that $J\not\subset pW(R)$ is principal, we obtain that $\Delta(1\otimes x)\in JW(R)\otimes_{{\varphi}, {\mathfrak{S}}} {\mathfrak{N}}'$. Since $p{\mathfrak{N}}'=0$, an analogous argument in the case $n=1$ proceeds and we have $\Delta(1\otimes x)=0$ as desired.
\[speciallemma\] Let $0\to \hat{{\mathfrak{M}}}'\to \hat{{\mathfrak{M}}}\to \hat{{\mathfrak{M}}}''\to 0$ be an exact sequence in ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s}_{/{\mathfrak{S}}_{\infty}}$. Suppose that $\hat{{\mathfrak{M}}}$ is an object of ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s,J}_{/{\mathfrak{S}}_{\infty}}$. Then $\hat{{\mathfrak{M}}}'$ and $\hat{{\mathfrak{M}}}''$ are also objects of ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s,J}_{/{\mathfrak{S}}_{\infty}}$.
The fact $\hat{{\mathfrak{M}}}''\in {\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s,J}_{/{\mathfrak{S}}_{\infty}}$ is clear. Take any $x\in {\mathfrak{M}}'$. Then we have $\tau^{p^s}(x)-x\in (JW(R)\otimes_{{\varphi},{\mathfrak{S}}}{\mathfrak{M}})\cap
(W(R)\otimes_{{\varphi},{\mathfrak{S}}}{\mathfrak{M}}')$. Since $J$ is a principal ideal which is not contained in $pW(R)$, we obtain $\tau^{p^s}(x)-x\in JW(R)\otimes_{{\varphi},{\mathfrak{S}}}{\mathfrak{M}}'$ by Lemma 6 of [@Oz2]. This implies $\hat{{\mathfrak{M}}}'\in {\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s,J}_{/{\mathfrak{S}}_{\infty}}$.
The category ${\widetilde}{{\mathrm}{Rep}}^{r,\hat{G}_s,J}_{{\mathrm}{tor}}(G_s)$
---------------------------------------------------------------------------------
In this subsection, we study some categorical properties of ${\widetilde}{{\mathrm}{Rep}}^{r,\hat{G}_s,J}_{{\mathrm}{tor}}(G_s)$.
Let $\hat{{\mathfrak{M}}}$ be an object of ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s}_{/{\mathfrak{S}}_{\infty}}$. Following Section 3.2 of [@Li2] (note that arguments in [@Li2] is the “free case”), we construct a map $\hat{\iota}_s$ which connects $\hat{{\mathfrak{M}}}$ and $\hat{T}_s(\hat{{\mathfrak{M}}})$ as follows. Observe that there exists a natural isomorphism of ${\mathbb}{Z}_p[G_s]$-modules $$\hat{T}_s(\hat{{\mathfrak{M}}})
\simeq
{\mathrm}{Hom}_{W(R),{\varphi}}(W(R)\otimes_{{\varphi},{\mathfrak{S}}}{\mathfrak{M}}, {\mathbb}{Q}_p/{\mathbb}{Z}_p\otimes_{{\mathbb}{Z}_p} W(R))$$ where $G_s$ acts on ${\mathrm}{Hom}_{W(R),{\varphi}}(W(R)\otimes_{{\varphi},{\mathfrak{S}}}{\mathfrak{M}},{\mathbb}{Q}_p/{\mathbb}{Z}_p\otimes_{{\mathbb}{Z}_p} W(R))$ by $(\sigma.f)(x)=\sigma(f(\sigma^{-1}(x)))$ for $\sigma\in G_s,
f\in {\mathrm}{Hom}_{W(R),{\varphi}}(W(R)\otimes_{{\varphi},{\mathfrak{S}}}{\mathfrak{M}},{\mathbb}{Q}_p/{\mathbb}{Z}_p\otimes_{{\mathbb}{Z}_p} W(R)),
x\in W(R)\otimes_{{\varphi},{\mathfrak{S}}}{\mathfrak{M}}=W(R)\otimes_{{\widehat{\mathcal{R}}}_s} ({\widehat{\mathcal{R}}}_s\otimes_{{\varphi},{\mathfrak{S}}}{\mathfrak{M}})$. Thus we can define a morphism $\hat{\iota}'_s\colon W(R)\otimes_{{\varphi}, {\mathfrak{S}}}{\mathfrak{M}}\to
{\mathrm}{Hom}_{{\mathbb}{Z}_p}(\hat{T}_s(\hat{{\mathfrak{M}}}),{\mathbb}{Q}_p/{\mathbb}{Z}_p\otimes_{{\mathbb}{Z}_p} W(R))$ by $$x\mapsto (f\mapsto f(x)),\quad
x\in W(R)\otimes_{{\varphi},{\mathfrak{S}}}{\mathfrak{M}}, f\in \hat{T}_s(\hat{{\mathfrak{M}}}).$$ Since $\hat{T}_s(\hat{{\mathfrak{M}}})\simeq \oplus_{i\in I}{\mathbb}{Z}_p/p^{n_i}{\mathbb}{Z}_p$ as ${\mathbb}{Z}_p$-modules, we have a natural isomorphism ${\mathrm}{Hom}_{{\mathbb}{Z}_p}(\hat{T}_s(\hat{{\mathfrak{M}}}),{\mathbb}{Q}_p/{\mathbb}{Z}_p\otimes_{{\mathbb}{Z}_p} W(R))\simeq
W(R)\otimes_{{\mathbb}{Z}_p}\hat{T}_s^{\vee}(\hat{{\mathfrak{M}}})$ where $\hat{T}_s^{\vee}(\hat{{\mathfrak{M}}})={\mathrm}{Hom}_{{\mathbb}{Z}_p}(\hat{T}_s(\hat{{\mathfrak{M}}}),{\mathbb}{Q}_p/{\mathbb}{Z}_p)$ is the dual representation of $\hat{T}_s(\hat{{\mathfrak{M}}})$. Composing this isomorphism with $\hat{\iota}'_s$, we obtain the desired map $$\hat{\iota}_s\colon W(R)\otimes_{{\varphi},{\mathfrak{S}}}{\mathfrak{M}}\to
W(R)\otimes_{{\mathbb}{Z}_p}\hat{T}_s^{\vee}(\hat{{\mathfrak{M}}}).$$ It follows from a direct calculation that $\hat{\iota}_s$ is ${\varphi}$-equivariant and $G_s$-equivariant. If we denote by $\hat{{\mathfrak{M}}}_s$ the image of $\hat{{\mathfrak{M}}}$ for the functor ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s}_{/{\mathfrak{S}}_{\infty}}\to
{{\mathrm}{Mod}}^{r,\hat{G}_s}_{/{\mathfrak{S}}_{s,\infty}}$ (cf. Section \[vardef\]), then the above $\hat{\iota}_s$ is isomorphic to “$\hat{\iota}$ for $\hat{{\mathfrak{M}}}_s$ in Section 4.1 of [@Oz1]”. Hence Lemma 4.2 (4) in [*loc*]{}. [*cit*]{}. implies that $$W({\mathrm}{Fr}\ R)\otimes \hat{\iota}_s\colon
W({\mathrm}{Fr}\ R)\otimes_{W(R)}(W(R)\otimes_{{\varphi},{\mathfrak{S}}}{\mathfrak{M}})
\to
W({\mathrm}{Fr}\ R)\otimes_{W(R)}(W(R)\otimes_{{\mathbb}{Z}_p}\hat{T}_s^{\vee}(\hat{{\mathfrak{M}}}))$$ is bijective.
Let $(R) \colon 0\to T'\to T\to T''\to 0$ be an exact sequence in ${\mathrm}{Rep}_{{\mathrm}{tor}}(G_s)$. Assume that there exists $\hat{{\mathfrak{M}}}\in {\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s,J}_{/{\mathfrak{S}}_{\infty}}$ such that $\hat{T}_s(\hat{{\mathfrak{M}}})\simeq T$. Then there exists an exact sequence $(M) \colon 0\to \hat{{\mathfrak{M}}}''\to \hat{{\mathfrak{M}}}\to \hat{{\mathfrak{M}}}'\to 0$ in ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s,J}_{/{\mathfrak{S}}_{\infty}}$ such that $\hat{T}_s((M))\simeq (R)$.
The same proof as [@Oz1 Theorem 4.5], except using not $\hat{\iota}$ in the proof of [*loc.*]{} [*cit.*]{} but $\hat{\iota}_s$ as above, gives an exact sequence $(M) \colon 0\to \hat{{\mathfrak{M}}}''\to \hat{{\mathfrak{M}}}\to \hat{{\mathfrak{M}}}'\to 0$ in ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s}_{/{\mathfrak{S}}_{\infty}}$ such that $\hat{T}_s((M))\simeq (R)$. Therefore, Lemma \[speciallemma\], gives the desired result.
\[stability\] The full subcategory ${\widetilde}{{\mathrm}{Rep}}^{r,\hat{G}_s,J}_{{\mathrm}{tor}}(G_s)$ of ${\mathrm}{Rep}_{{\mathrm}{tor}}(G_s)$ is stable under subquotients.
Let $L$ be as in Section 2, that is, the completion of an unramified algebraic extension of $K$ with residue field $k_L$. We prove the following base change lemma.
\[bc\] Assume that $J\supset u^pI^{[1]}W(R)$ or $L$ is a finite unramified extension of $K$. If $T$ is an object of ${\widetilde}{{\mathrm}{Rep}}^{r,\hat{G}_s,J}_{{\mathrm}{tor}}(G_s)$, then $T|_{G_{L,s}}$ is an object of ${\widetilde}{{\mathrm}{Rep}}^{r,\hat{G}_{L,s},J}_{{\mathrm}{tor}}(G_{L,s})$.
By an obvious way, we define a functor ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s}_{/{\mathfrak{S}}_{\infty}}\to
{\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_{L,s}}_{/{\mathfrak{S}}_{L,\infty}}$. The underlying Kisin module of the image of $\hat{{\mathfrak{M}}}\in {\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s}_{/{\mathfrak{S}}_{\infty}}$ for this functor is ${\mathfrak{M}}_L={\mathfrak{S}}_L\otimes_{{\mathfrak{S}}} {\mathfrak{M}}$. Lemma \[bc\] immediately follows from the lemma below.
Assume that $J\supset u^pI^{[1]}W(R)$ or $L$ is a finite unramified extension of $K$. Then the functor ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s}_{/{\mathfrak{S}}_{\infty}}\to {\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_{L,s}}_{/{\mathfrak{S}}_{L,\infty}}$ induces a functor ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s,J}_{/{\mathfrak{S}}_{\infty}}\to {\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_{L,s},J}_{/{\mathfrak{S}}_{L,\infty}}$.
Let $\hat{{\mathfrak{M}}}$ be an object of ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s}_{/{\mathfrak{S}}_{\infty}}$ and let $\hat{{\mathfrak{M}}}_L$ be the image of $\hat{{\mathfrak{M}}}$ for the functor ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s}_{/{\mathfrak{S}}_{\infty}}\to {\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_{L,s}}_{/{\mathfrak{S}}_{L,\infty}}$. In the rest of this proof, to avoid confusions, we denote the image of $x\in {\mathfrak{M}}_L$ in $W(R)\otimes_{{\varphi},{\mathfrak{S}}_L} {\mathfrak{M}}_L$ by $1\otimes x$. Recall that we abuse notations by writing $\tau$ for the pre-image of $\tau\in G_{K,p^{\infty}}$ via the bijection $G_{L,p^{\infty}}\simeq G_{K,p^{\infty}}$ of lemma \[easylemma\]. Then $\tau^{p^s}$ is a topological generator of $G_{L,s,p^{\infty}}$. It suffices to show the following: if $\hat{{\mathfrak{M}}}$ is an object of ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s,J}_{/{\mathfrak{S}}_{\infty}}$, then we have $\tau^{p^s}(1\otimes x)-(1\otimes x)\in JW(R)\otimes_{{\varphi},{\mathfrak{S}}_L} {\mathfrak{M}}_L$ for any $x\in {\mathfrak{M}}_L$. Now we suppose $\hat{{\mathfrak{M}}}\in {\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s,J}_{/{\mathfrak{S}}_{\infty}}$. Take any $a\in {\mathfrak{S}}_L$ and $x\in {\mathfrak{M}}$. Note that we have $\tau^{p^s}(1\otimes ax)-(1\otimes ax)
=\tau^{p^s}({\varphi}(a))(\tau^{p^s}(1\otimes x)-(1\otimes x))
+(\tau^{p^s}({\varphi}(a))-{\varphi}(a))(1\otimes x)$ in $W(R)\otimes_{{\varphi},{\mathfrak{S}}_L} {\mathfrak{M}}_L$. Since $\hat{{\mathfrak{M}}}$ is an object of ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s,J}_{/{\mathfrak{S}}_{\infty}}$, we have $\tau^{p^s}({\varphi}(a))(\tau^{p^s}(1\otimes x)-(1\otimes x))\in
JW(R)\otimes_{{\varphi},{\mathfrak{S}}_L} {\mathfrak{M}}_L$. Therefore, it is enough to show $(\tau^{p^s}({\varphi}(a))-{\varphi}(a))(1\otimes x)\in JW(R)\otimes_{{\varphi},{\mathfrak{S}}_L} {\mathfrak{M}}_L$. This follows from Lemma \[cryslem\] immediately in the case where $J\supset u^pI^{[1]}W(R)$. Next we consider the case where $L$ is a finite unramified extension of $K$. Let $c_1,\dots ,c_{\ell}\in W(k_L)$ be generators of $W(k_L)$ as a $W(k)$-module. Then we have ${\mathfrak{S}}_L=\sum^{\ell}_{j=1} c_j {\mathfrak{S}}$ and thus we can write $a=\sum^{\ell}_{j=1}a_jc_j$ for some $a_j\in {\mathfrak{S}}$. Hence it suffices to show $(\tau^{p^s}({\varphi}(a_j))-{\varphi}(a_j))(1\otimes x)\in JW(R)\otimes_{{\varphi},{\mathfrak{S}}_L} {\mathfrak{M}}_L$ but this in fact immediately follows from the equation $(\tau^{p^s}({\varphi}(a_j))-{\varphi}(a_j))(1\otimes x)
=(\tau^{p^s}(1\otimes a_jx)-(1\otimes a_jx))-
(\tau^{p^s}({\varphi}(a_j))(\tau^{p^s}(1\otimes x)-(1\otimes x)))$.
Full faithfulness theorem for ${\widetilde}{{\mathrm}{Rep}}^{r,\hat{G}_s,J}_{{\mathrm}{tor}}(G_s)$
--------------------------------------------------------------------------------------------------
Our goal in this subsection is to prove the following full faithfulness theorem, which plays an important roll in our proofs of main theorems.
\[FFTHM\] Assume that $J\supset u^pI^{[1]}W(R)$ or $k$ is algebraically closed. If $p^{s+2}/(p-1)\ge c_J>pr/(p-1)$, then the restriction functor ${\widetilde}{{\mathrm}{Rep}}^{r,\hat{G}_s,J}_{{\mathrm}{tor}}(G_s)\to {\mathrm}{Rep}_{{\mathrm}{tor}}(G_{\infty})$ is fully faithful.
First we give a very rough sketch of the theory of maximal models for Kisin modules (cf. [@CL1]). For any ${\mathfrak{M}}\in {\mathrm}{Mod}^r_{/{\mathfrak{S}}_{\infty}}$, put ${\mathfrak{M}}[1/u]={\mathfrak{S}}[1/u]\otimes_{{\mathfrak{S}}} {\mathfrak{M}}$ and denote by $F^r_{{\mathfrak{S}}}({\mathfrak{M}}[1/u])$ the (partially) ordered set (by inclusion) of torsion Kisin modules ${\mathfrak{N}}$ of height $\le r$ which are contained in ${\mathfrak{M}}[1/u]$ and ${\mathfrak{N}}[1/u]={\mathfrak{M}}[1/u]$ as ${\varphi}$-modules. The set $F^r_{{\mathfrak{S}}}({\mathfrak{M}}[1/u])$ has a greatest element (cf. [*loc*]{}. [*cit*]{}., Corollary 3.2.6). We denote this element by ${\mathrm{Max}}^r({\mathfrak{M}})$. We say that ${\mathfrak{M}}$ is [*maximal of height $\le r$*]{} (or, [*maximal*]{} for simplicity) if it is the greatest element of $F^r_{{\mathfrak{S}}}({\mathfrak{M}}[1/u])$. The implication ${\mathfrak{M}}\mapsto {\mathrm{Max}}^r({\mathfrak{M}})$ defines a functor “${\mathrm{Max}}^r$” from the category ${\mathrm}{Mod}^r_{/{\mathfrak{S}}_{\infty}}$ of torsion Kisin modules of height $\le r$ into the category ${\mathrm{Max}}^r_{/{\mathfrak{S}}_{\infty}}$ of maximal Kisin modules of height $\le r$. The category ${\mathrm{Max}}^r_{/{\mathfrak{S}}_{\infty}}$ is abelian (cf. [*loc*]{}. [*cit*]{}., Theorem 3.3.8). Furthermore, the functor $T_{{\mathfrak{S}}}\colon {\mathrm{Max}}^r_{/{\mathfrak{S}}_{\infty}}\to {\mathrm}{Rep}_{{\mathrm}{tor}}(G_{\infty})$, defined by $T_{{\mathfrak{S}}}({\mathfrak{M}})={\mathrm}{Hom}_{{\mathfrak{S}},{\varphi}}({\mathfrak{M}},{\mathbb}{Q}_p/{\mathbb}{Z}_p\otimes_{{\mathbb}{Z}_p} W(R))$, is exact and fully faithful (cf. [*loc*]{}. [*cit*]{}., Corollary 3.3.10). It is not difficult to check that $T_{{\mathfrak{S}}}({\mathrm{Max}}^r({\mathfrak{M}}))$ is canonically isomorphic to $T_{{\mathfrak{S}}}({\mathfrak{M}})$ as representations of $G_{\infty}$ for any torsion Kisin module ${\mathfrak{M}}$ of height $\le r$.
\[Def1\] Let $d$ be a positive integer. Let ${\mathfrak{n}}=(n_i)_{i\in {\mathbb}{Z}/d{\mathbb}{Z}}$ be a sequence of non-negative integers of smallest period $d$. We define a torsion Kisin module ${\mathfrak{M}}({\mathfrak{n}})$ as below:
- as a ${k[\![u]\!]}$-module, ${\mathfrak{M}}({\mathfrak{n}})=\bigoplus_{i\in {\mathbb}{Z}/d{\mathbb}{Z}} {k[\![u]\!]}e_i$;
- for all $i\in {\mathbb}{Z}/d{\mathbb}{Z}$, ${\varphi}(e_i)=u^{n_i}e_{i+1}$.
We denote by ${\mathcal}{S}^r_{{\mathrm}{max}}$ the set of sequences ${\mathfrak{n}}=(n_i)_{i\in {\mathbb}{Z}/d{\mathbb}{Z}}$ of integers $0\le n_i\le {\mathrm}{min}\{er, p-1\}$ with smallest period $d$ for some integer $d$ except the constant sequence with value $p-1$ (if necessary). By definition, we see that ${\mathfrak{M}}({\mathfrak{n}})$ is of height $\le r$ for any ${\mathfrak{n}}\in {\mathcal}{S}^r_{{\mathrm}{max}}$. Putting $r_0={\mathrm}{max}\{r'\in {\mathbb}{Z}_{\ge 0};e(r'-1)<p-1 \}$, we also see that ${\mathfrak{M}}({\mathfrak{n}})$ is of height $\le r_0$ for any ${\mathfrak{n}}\in {\mathcal}{S}^r_{{\mathrm}{max}}$. It is known that ${\mathfrak{M}}({\mathfrak{n}})$ is maximal for any ${\mathfrak{n}}\in {\mathcal}{S}^r_{{\mathrm}{max}}$ ([@CL1 Proposition 3.6.7]). If $k$ is algebraically closed, then ${\mathfrak{M}}({\mathfrak{n}})$ is simple in ${\mathrm{Max}}^r_{/{\mathfrak{S}}_{\infty}}$ for any ${\mathfrak{n}}\in {\mathcal}{S}^r_{{\mathrm}{max}}$ (cf. [*loc. cit.*]{}, Propositions 3.6.7 and 3.6.12) and furthermore, the converse holds; any simple object in ${\mathrm{Max}}^r_{/{\mathfrak{S}}_{\infty}}$ is of the form ${\mathfrak{M}}({\mathfrak{n}})$ for some ${\mathfrak{n}}\in {\mathcal}{S}^r_{{\mathrm}{max}}$ (cf. [*loc. cit.*]{}, Propositions 3.6.8 and 3.6.12).
\[Lem1\] Assume that $p^{s+2}/(p-1)\ge c_J$. Let $d$ be a positive integer. Let ${\mathfrak{n}}=(n_i)_{i\in {\mathbb}{Z}/d{\mathbb}{Z}}$ be a sequence of non-negative integers of smallest period $d$. If ${\mathfrak{M}}({\mathfrak{n}})$ is of height $\le r$, then ${\mathfrak{M}}({\mathfrak{n}})$ has a structure of an object of ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s,J}_{/{\mathfrak{S}}_{\infty}}$.
Choose any $(p^d-1)$-th root $\eta\in R$ of $\underline{{\varepsilon}}$. Since $[\eta]\cdot {\mathrm}{exp}(t/(p^d-1))$ is a $(p^d-1)$-th root of unity, it is of the form $[a]$ for some $a\in {\mathbb}{F}^{\times}_{p^d}$. Replacing $\eta a^{-1}$ with $\eta$, we obtain $[\eta]={\mathrm}{exp}(-t/(p^d-1))\in {\widehat{\mathcal{R}}}^{\times}$. Put $x_i=[\eta]^{m_i}\in {\widehat{\mathcal{R}}}^{\times}$ and $\bar{x}_i=\eta^{m_i}\in ({\widehat{\mathcal{R}}}/p{\widehat{\mathcal{R}}})^{\times}\subset R^{\times}$ for any $i\in {\mathbb}{Z}/d{\mathbb}{Z}$, where $m_i=\sum^{d-1}_{j=0}n_{i+j}p^{d-j}$. We see that $x_i-1$ is contained in $I_+{\widehat{\mathcal{R}}}$. In the rest of this proof, to avoid confusions, we denote the image of $x\in {\mathfrak{M}}({\mathfrak{n}})$ in ${\widehat{\mathcal{R}}}_s\otimes_{{\varphi},{\mathfrak{S}}} {\mathfrak{M}}({\mathfrak{n}})\subset R\otimes_{{\varphi},k[\![u]\!]} {\mathfrak{M}}({\mathfrak{n}})$ by $1\otimes x$. Now we define a $\hat{G}_s$-action on ${\widehat{\mathcal{R}}}_s\otimes_{{\varphi},{\mathfrak{S}}} {\mathfrak{M}}({\mathfrak{n}})$ by $\tau^{p^s}(1\otimes e_i):=x^{p^s}_i(1\otimes e_i)$ for the basis $\{e_i\}_{i\in {\mathbb}{Z}/d{\mathbb}{Z}}$ of ${\mathfrak{M}}({\mathfrak{n}})$ as in Definition \[Def1\]. It is not difficult to check that ${\mathfrak{M}}({\mathfrak{n}})$ has a structure of an object of ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s}_{/{\mathfrak{S}}_{\infty}}$ via this $\hat{G}_s$-action; we denote it by $\hat{{\mathfrak{M}}}({\mathfrak{n}})$. It suffices to prove that $\hat{{\mathfrak{M}}}({\mathfrak{n}})$ is in fact an object of ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s,J}_{/{\mathfrak{S}}_{\infty}}$. Recall that $v_R$ is the valuation of $R$ normalized such that $v_R(\underline{\pi})=1/e$. Define $\tilde{{\mathfrak{t}}}={\mathfrak{t}}\ {\mathrm}{mod}\ pW(R)$ an element of $R$. We denote by $v_p$ the usual $p$-adic valuation normalized by $v_p(p)=1$. Note that we have $v_R(\underline{{\varepsilon}}-1)=p/(p-1)$ and $v_R(\tilde{{\mathfrak{t}}})=1/(p-1)$ (here, the latter equation follows from the relation ${\varphi}({\mathfrak{t}})=pE(0)^{-1}E(u){\mathfrak{t}}$). We see that $$v_R(\bar{x}^{p^s}_i-1)=p^{s+v_p(m_i)}\cdot p/(p-1)\ge p^{s+2}/(p-1).$$ Since $p^{s+2}/(p-1)\ge c_J$ and the image of $J$ in $R$ is ${\mathfrak{m}}_R^{\ge c_J}$, we obtain $$\tau^{p^s}(1\otimes e_i)-(1\otimes e_i)\in
{\mathfrak{m}}_R^{\ge c_J}R\otimes_{{\varphi},k[\![u]\!]} {\mathfrak{M}}({\mathfrak{n}})
\simeq JW(R)\otimes_{{\varphi},{\mathfrak{S}}} {\mathfrak{M}}({\mathfrak{n}}).$$ Finally we have to show that $\tau^{p^s}(1\otimes ae_i)-(1\otimes ae_i)\in
{\mathfrak{m}}_R^{\ge c_J}R\otimes_{{\varphi},k[\![u]\!]} {\mathfrak{M}}({\mathfrak{n}})$ for any $a\in k[\![u]\!]$. Since $\tau^{p^s}(1\otimes ae_i)-(1\otimes ae_i)=\tau^{p^s}({\varphi}(a))
(\tau^{p^s}(1\otimes e_i)-(1\otimes e_i))+
(\tau^{p^s}({\varphi}(a))-{\varphi}(a))(1\otimes e_i)$, it suffices to show $\tau^{p^s}({\varphi}(a))-{\varphi}(a)\in {\mathfrak{m}}_R^{\ge c_J}$. Write ${\varphi}(a)=\sum_{i\ge 0}a_iu^{pi}$ for some $a_i\in k$. Then we have $\tau^{p^s}({\varphi}(a))-{\varphi}(a)=
\sum_{i\ge 1}a_i(\underline{{\varepsilon}}^{p^{s+1}i}-1)u^{pi}$. Since we have $$v_R((\underline{{\varepsilon}}^{p^{s+1}i}-1)u^{pi})
=p^{s+1}v_R(\underline{{\varepsilon}}^i-1)+v_R(u^{pi})
> p^{s+2}/(p-1)\ge c_J$$ for any $i\ge 1$, we have done.
Recall that $r_0={\mathrm}{max}\{r'\in {\mathbb}{Z}_{\ge 0}; e(r'-1)<p-1\}$. Put $r_1:={\mathrm}{min}\{r,r_0\}$.
\[Cor1\] Assume that $p^{s+2}/(p-1)\ge c_J$. If ${\mathfrak{n}}\in {\mathcal}{S}^r_{{\mathrm}{max}}$, then ${\mathfrak{M}}({\mathfrak{n}})$ has a structure of an object of ${\widetilde{{\mathrm}{Mod}}}^{r',\hat{G}_s,J}_{/{\mathfrak{S}}_{\infty}}$ for any $r'\ge r_1$. Furthermore, if $c_J>pr_1/(p-1)$, it is uniquely determined. We denote this object by $\hat{{\mathfrak{M}}}({\mathfrak{n}})$.
We should remark that ${\mathfrak{M}}({\mathfrak{n}})$ is of height $\le r_1$ for any ${\mathfrak{n}}\in {\mathcal}{S}^r_{{\mathrm}{max}}$. The uniqueness assertion follows from Proposition \[FFTHMMOD\].
\[tameres\] The functor from tamely ramified ${\mathbb}{Z}_p$-representations of $G_K$ to ${\mathbb}{Z}_p$-representations of $G_{\infty}$, obtained by restricting the action of $G_K$ to $G_{\infty}$, is fully faithful.
The result immediately follows from the fact that $G_K$ is topologically generated by $G_{\infty}$ and the wild inertia subgroup of $G_K$.
We remark that any semi-simple ${\mathbb}{F}_p$-representation of $G_K$ is automatically tame.
\[FFLEM\] Assume that $J\supset u^pI^{[1]}W(R)$ or $k$ is algebraically closed. Let $T\in {\mathrm}{Rep}_{{\mathrm}{tor}}(G_s)$ and $T'\in {\widetilde}{{\mathrm}{Rep}}^{r,\hat{G}_s,J}_{{\mathrm}{tor}}(G_s)$. Suppose that $T$ is tame, $pT=0$ and $T|_{G_{\infty}}\simeq T_{{\mathfrak{S}}}({\mathfrak{M}})$ for some ${\mathfrak{M}}\in {{\mathrm}{Mod}}^r_{/{\mathfrak{S}}_{\infty}}$. Furthermore, we suppose $p^{s+2}/(p-1)\ge c_J>pr/(p-1)$. Then all $G_{\infty}$-equivariant homomorphisms $T\to T'$ are in fact $G_s$-equivariant.
Let $L$ be the completion of the maximal unramified extension $K^{{\mathrm}{ur}}$ of $K$. By identifying $G_L$ with $G_{K^{{\mathrm}{ur}}}$, we may regard $G_L$ as a subgroup of $G_K$. Note that $L_{(s)}=K_{(s)}L$ is the completion of the maximal unramified extension of $K_{(s)}$, and $G_s$ is topologically generated by $G_{L,s}$ and $G_{\infty}$. Consider the following commutative diagram:
$\displaystyle \xymatrix{
{\mathrm}{Hom}_{G_{L,s}}(T,T')\ar@{^{(}->}[rr] & &
{\mathrm}{Hom}_{G_{L,\infty}}(T,T') \\
{\mathrm}{Hom}_{G_s}(T,T') \ar@{^{(}->}[u] \ar@{^{(}->}[rr] & &
{\mathrm}{Hom}_{G_{\infty}}(T,T'). \ar@{^{(}->}[u]
}$
Since $T'|_{G_{L,s}}$ is contained in ${\widetilde}{{\mathrm}{Rep}}^{r,\hat{G}_{L,s},J}_{{\mathrm}{tor}}(G_{L,s})$ if $J\supset u^pI^{[1]}W(R)$ (cf. Lemma \[bc\]), the above diagram allows us to reduce a proof to the case where $k$ is algebraically closed. In the rest of this proof, we assume that $k$ is algebraically closed. Under this assumption, an ${\mathbb}{F}_p$-representation of $G_s$ is tame if and only if it is semi-simple by Maschke’s theorem. Thus we may also assume that $T$ is irreducible (here, we remark that any subquotient of $T$ is tame and, also remark that the essential image of $T_{{\mathfrak{S}}}\colon {{\mathrm}{Mod}}^r_{/{\mathfrak{S}}_{\infty}}\to {\mathrm}{Rep}_{{\mathrm}{tor}}(G_{\infty})$ is stable under subquotients in ${\mathrm}{Rep}_{{\mathrm}{tor}}(G_{\infty})$). We claim that $T|_{G_{\infty}}$ is also irreducible. If not, there exists a non-zero irreducible ${\mathbb}{F}_p[G_{\infty}]$-submodule $W$ of $T|_{G_{\infty}}$. Let $K^{{\mathrm}{t}}_{(s)}$ be the maximal tamely ramified extension of $K_{(s)}$ and $I_{p,s}:={\mathrm}{Gal}(\overline{K}/K^{{\mathrm}{t}}_{(s)})$ the wild inertia subgroup of $G_s$. We see that $K^{{\mathrm}{t}}_{(s)}\cap K_{\infty}=K_{(s)}$. Since $G_{\infty}\cap I_{p,s}$ acts on $W$ trivially, the $G_{\infty}$-action on $W$ extends to $G_s$ via the composition map $G_s\twoheadrightarrow {\mathrm}{Gal}(K^{{\mathrm}{t}}_{(s)}/K_{(s)})
\simeq G_{\infty}/(G_{\infty}\cap I_{p,s})$. Thus we can regard $W$ as an irreducible ${\mathbb}{F}_p[G_s]$-module. By Lemma \[tameres\], we see that $W$ is a sub ${\mathbb}{F}_p[G_s]$-module of $T$. This contradicts the irreducibility of $T$ and the claim follows.
By the assumption on $T$, we have $T|_{G_{\infty}}\simeq T_{{\mathfrak{S}}}({\mathfrak{M}})\simeq T_{{\mathfrak{S}}}({\mathrm{Max}}^r({\mathfrak{M}}))$ for some ${\mathfrak{M}}\in {{\mathrm}{Mod}}^r_{/{\mathfrak{S}}_{\infty}}$. Since $T|_{G_{\infty}}$ is irreducible and $T_{{\mathfrak{S}}}\colon {\mathrm{Max}}^r_{/{\mathfrak{S}}_{\infty}}\to {\mathrm}{Rep}_{{\mathrm}{tor}}(G_{\infty})$ is exact and fully faithful, we know that ${\mathrm{Max}}^r({\mathfrak{M}})$ is a simple object in the abelian category ${\mathrm{Max}}^r_{/{\mathfrak{S}}_{\infty}}$. Therefore, since $k$ is algebraically closed, we have ${\mathrm{Max}}^r({\mathfrak{M}})\simeq {\mathfrak{M}}({\mathfrak{n}})$ for some ${\mathfrak{n}}\in {\mathcal}{S}^r_{{\mathrm}{max}}$ (cf. [@CL1 Propositions 3.6.8 and 3.6.12]). Let $\hat{{\mathfrak{M}}}({\mathfrak{n}})$ be the object of ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G},J}_{/{\mathfrak{S}}_{\infty}}$ as in Corollary \[Cor1\]. We recall that $T_{{\mathfrak{S}}}({\mathfrak{M}}({\mathfrak{n}}))$ is isomorphic to $\hat{T}_s(\hat{{\mathfrak{M}}}({\mathfrak{n}}))|_{G_{\infty}}$ (see Theorem \[Thm1\] (1)), and hence we have an isomorphism $T|_{G_{\infty}}\simeq \hat{T}_s(\hat{{\mathfrak{M}}}({\mathfrak{n}}))|_{G_{\infty}}$. Here, we note that $T$ and $\hat{T}_s(\hat{{\mathfrak{M}}}({\mathfrak{n}}))$ are irreducible as representations of $G_s$ (cf. [@CL1 Theorem 3.6.11]). Applying Lemma \[tameres\] again, we obtain an isomorphism $T\simeq \hat{T}_s(\hat{{\mathfrak{M}}}({\mathfrak{n}}))$ as representations of $G_s$. On the other hand, we can take $\hat{{\mathfrak{M}}}'=({\mathfrak{M}}',{\varphi},\hat{G}_s)\in {\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s,J}_{/{\mathfrak{S}}_{\infty}}$ such that $T'\simeq \hat{T}_s(\hat{{\mathfrak{M}}}')$. We consider the following commutative diagram:
$\displaystyle \xymatrix{
{\mathrm}{Hom}_{G_s}(T,T')\ar@{^{(}->}[rr] & &
{\mathrm}{Hom}_{G_{\infty}}(T,T') \\
{\mathrm}{Hom}(\hat{{\mathfrak{M}}}',\hat{{\mathfrak{M}}}({\mathfrak{n}})) \ar^{\hat{T}_s}[u] \ar^{{\mathrm}{forgetful}\ }[r] &
{\mathrm}{Hom}({\mathfrak{M}}',{\mathfrak{M}}({\mathfrak{n}})) \ar^{{\mathrm{Max}}^r\quad \ \ }[r] & {\mathrm}{Hom}({\mathrm{Max}}^r({\mathfrak{M}}'),{\mathfrak{M}}({\mathfrak{n}})).
\ar^{T_{{\mathfrak{S}}}}[u].
}$
Here, ${\mathrm}{Hom}(\hat{{\mathfrak{M}}}',\hat{{\mathfrak{M}}}({\mathfrak{n}}))$ (resp. ${\mathrm}{Hom}({\mathfrak{M}}',{\mathfrak{M}}({\mathfrak{n}}))$, resp. ${\mathrm}{Hom}({\mathrm{Max}}^r({\mathfrak{M}}'),{\mathfrak{M}}({\mathfrak{n}}))$) is the set of morphisms $\hat{{\mathfrak{M}}}'\to \hat{{\mathfrak{M}}}({\mathfrak{n}})$ (resp. ${\mathfrak{M}}'\to {\mathfrak{M}}({\mathfrak{n}})$, resp. ${\mathrm{Max}}^r({\mathfrak{M}}')\to {\mathfrak{M}}({\mathfrak{n}})$) in ${\widetilde{{\mathrm}{Mod}}}^{r_1,\hat{G}_s,J}_{/{\mathfrak{S}}_{\infty}}$ (resp. ${{\mathrm}{Mod}}^{r_1}_{/{\mathfrak{S}}_{\infty}}$, resp. ${\mathrm{Max}}^r_{/{\mathfrak{S}}_{\infty}}$). The first bottom horizontal arrow is bijective by Theorem \[Thm1\] (3) and so is the second (this follows from the fact that ${\mathfrak{M}}({\mathfrak{n}})$ is maximal by [@CL1 Proposition 3.6.7]). Since the right vertical arrow is bijective, the top horizontal arrow must be bijective.
Now we are ready to prove Theorem \[FFTHM\].
Let $T$ and $T'$ be objects of ${\widetilde}{{\mathrm}{Rep}}^{r,\hat{G}_s,J}_{{\mathrm}{tor}}(G_s)$. Take any Jordan-Höllder sequence $0=T_0\subset T_1\subset \cdots \subset T_n=T$ of $T$ in ${\mathrm}{Rep}_{{\mathrm}{tor}}(G_s)$. By Corollary \[stability\], we know that $T_i$ and $T_i/T_{i-1}$ are contained in ${\widetilde}{{\mathrm}{Rep}}^{r,\hat{G}_s,J}_{{\mathrm}{tor}}(G_s)$ for any $i$. By Corollary \[stability\] again, the category ${\widetilde}{{\mathrm}{Rep}}^{r,\hat{G}_s,J}_{{\mathrm}{tor}}(G_s)$ is an exact category in the sense of Quillen ([@Qu Section 2]). Hence short exact sequences in ${\widetilde}{{\mathrm}{Rep}}^{r,\hat{G}_s,J}_{{\mathrm}{tor}}(G_s)$ give rise to exact sequences of Hom’s and Ext’s in the usual way. (This property holds for any exact category.) On the other hand, by Lemma \[FFLEM\], if an exact sequence $0\to T'\to V\to T_i/T_{i-1}\to 0$ in ${\widetilde}{{\mathrm}{Rep}}^{r,\hat{G}_s,J}_{{\mathrm}{tor}}(G_s)$ splits as representation of $G_{\infty}$, then it splits as a sequence of representations of $G_s$. Therefore, comparing exact sequences of Hom’s and Ext’s arising from $0\to T_{i-1}\to T_i\to T_i/T_{i-1}\to 0$ in the category ${\widetilde}{{\mathrm}{Rep}}^{r,\hat{G}_s,J}_{{\mathrm}{tor}}(G_s)$ with that in the category ${\mathrm}{Rep}_{{\mathrm}{tor}}(G_{\infty})$, we obtain the following implication (here, use Lemma \[FFLEM\] again): if we have ${\mathrm}{Hom}_{G_s}(T_{i-1}, T')={\mathrm}{Hom}_{G_{\infty}}(T_{i-1}, T')$, then it gives the equality ${\mathrm}{Hom}_{G_s}(T_i, T')={\mathrm}{Hom}_{G_{\infty}}(T_i, T')$. Hence a dévissage argument works and the desired full faithfulness follows.
Proof of Theorem \[Main1\]
--------------------------
Now we are ready to prove our main theorems. First we prove Theorem \[Main1\].
Recall that ${\mathrm}{Rep}^{r,{\mathrm}{ht},{\mathrm}{pcris}(s)}_{{\mathrm}{tor}}(G_K)$ is the category of torsion ${\mathbb}{Z}_p$-representations $T$ of $G_K$ which satisfy the following: there exist free ${\mathbb}{Z}_p$-representations $L$ and $L'$ of $G_K$, of height $\le r$, such that
- $L|_{G_s}$ is a subrepresentation of $L'|_{G_s}$. Furthermore, $L|_{G_s}$ and $L'|_{G_s}$ are lattices in some crystalline ${\mathbb}{Q}_p$-representation of $G_s$ with Hodge-Tate weights in $[0,r]$;
- $T|_{G_s} \simeq (L'|_{G_s})/(L|_{G_s})$.
We apply our arguments given in previous subsections with the following $J$: $$J=u^pI^{[1]}W(R)=u^p{\varphi}({\mathfrak{t}})W(R).$$ Then we have $c_J=p/e+p/(p-1)$ and thus the inequalities $p^{s+2}/(p-1)\ge c_J>pr/(p-1)$ are satisfied if $e(r-1)<p-1$. Therefore, Theorem \[Main1\] is an easy consequence of the following proposition and Theorem \[FFTHM\].
\[proofMain1\] If $T$ is an object of ${\mathrm}{Rep}^{r,{\mathrm}{ht},{\mathrm}{pcris}(s)}_{{\mathrm}{tor}}(G_K)$, then $T|_{G_s}$ is contained in ${\widetilde}{{\mathrm}{Rep}}^{r,\hat{G}_s,J}_{{\mathrm}{tor}}(G_s)$.
Take free ${\mathbb}{Z}_p$-representations $L$ and $L'$ of $G_K$, of height $\le r$, such that
- $L|_{G_s}$ is a subrepresentation of $L'|_{G_s}$. Furthermore, $L|_{G_s}$ and $L'|_{G_s}$ are lattices in some crystalline ${\mathbb}{Q}_p$-representation of $G_s$ with Hodge-Tate weights in $[0,r]$;
- $T|_{G_s} \simeq (L'|_{G_s})/(L|_{G_s})$.
By Theorem \[Thm1\] (1), there exists an injection $\hat{{\mathfrak{L}}}'\hookrightarrow \hat{{\mathfrak{L}}}$ of $({\varphi},\hat{G}_s)$-modules over ${\mathfrak{S}}_s$ which corresponds to the injection $L|_{G_s}\hookrightarrow L'|_{G_s}$. On the other hand, there exist ${\mathfrak{N}}$ and ${\mathfrak{N}}'$ in ${{\mathrm}{Mod}}^r_{/{\mathfrak{S}}}$ such that $T_{{\mathfrak{S}}}({\mathfrak{N}})\simeq L|_{G_{\infty}}$ and $T_{{\mathfrak{S}}}({\mathfrak{N}}')\simeq L'|_{G_{\infty}}$. Then Proposition \[Kisinfunctor\] (1) implies that ${\mathfrak{S}}_s\otimes_{{\mathfrak{S}}} {\mathfrak{N}}\simeq {\mathfrak{L}}$ and ${\mathfrak{S}}_s\otimes_{{\mathfrak{S}}} {\mathfrak{N}}'\simeq {\mathfrak{L}}'$ as ${\varphi}$-modules over ${\mathfrak{S}}_s$. Therefore, we see that ${\mathfrak{N}}$ and ${\mathfrak{N}}'$ have structures of objects of ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s}_{/{\mathfrak{S}}}$; denote them by $\hat{{\mathfrak{N}}}$ and $\hat{{\mathfrak{N}}}'$, respectively. Since the functor ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s}_{/{\mathfrak{S}}}\to {{\mathrm}{Mod}}^{r,\hat{G}_s}_{/{\mathfrak{S}}_s}$ is fully faithful (cf. Section \[vardef\]), the injection $\hat{{\mathfrak{L}}}'\hookrightarrow \hat{{\mathfrak{L}}}$ descends to an injection $\hat{{\mathfrak{N}}}'\hookrightarrow \hat{{\mathfrak{N}}}$. Now we put ${\mathfrak{M}}={\mathfrak{N}}/{\mathfrak{N}}'$. Since $(L'|_{G_s})/(L|_{G_s})$ is killed by a power of $p$, it is an object of ${{\mathrm}{Mod}}^r_{/{\mathfrak{S}}_{\infty}}$. We equip a $\hat{G}_s$-action with ${\widehat{\mathcal{R}}}_s\otimes_{{\varphi},{\mathfrak{S}}} {\mathfrak{M}}$ by a natural isomorphism ${\widehat{\mathcal{R}}}_s\otimes_{{\varphi},{\mathfrak{S}}} {\mathfrak{M}}\simeq
({\widehat{\mathcal{R}}}_s\otimes_{{\varphi},{\mathfrak{S}}} {\mathfrak{N}})/({\widehat{\mathcal{R}}}_s\otimes_{{\varphi},{\mathfrak{S}}} {\mathfrak{N}}')$. Then we see that ${\mathfrak{M}}$ has a structure of an object of ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s}_{/{\mathfrak{S}}_{\infty}}$; denote it by $\hat{{\mathfrak{M}}}$. Moreover, Theorem \[cris\] implies that $\hat{{\mathfrak{M}}}$ is in fact contained in ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s,J}_{/{\mathfrak{S}}_{\infty}}$. By a similar argument to the proof of Lemma 3.1.4 of [@CL2], we have an exact sequence $0\to \hat{T}_s(\hat{{\mathfrak{N}}})\to \hat{T}_s(\hat{{\mathfrak{N}}}')\to \hat{T}_s(\hat{{\mathfrak{M}}})\to 0$ in ${\mathrm}{Rep}_{{\mathrm}{tor}}(G_s)$, which is isomorphic to $0\to L|_{G_s}\to L'|_{G_s}\to T|_{G_s}\to 0$. This finishes a proof.
Proof of Theorem \[Main2\]
--------------------------
We give a proof of Theorem \[Main2\]. If $s\ge n-1$, then we put $$J=u^pI^{[p^{s-n+1}]}W(R)=u^p{\varphi}({\mathfrak{t}})^{p^{s-n+1}}W(R).$$ Note that we have $c_J=p/e+p^{s-n+2}/(p-1)$ and thus the inequalities $p^{s+2}/(p-1)\ge c_J>pr/(p-1)$ are satisfied if $s>n-1+{\mathrm}{log}_p(r-e/(p-1))$.
\[Main2Lem\] Suppose $s\ge n-1$. If $T$ is an object of ${\mathrm}{Rep}^{r,{\mathrm}{cris}}_{{\mathrm}{tor}}(G_K)$ which is killed by $p^n$, then $T|_{G_s}$ is contained in ${\widetilde}{{\mathrm}{Rep}}^{r,\hat{G}_s,J}_{{\mathrm}{tor}}(G_s)$.
Let $L$ be an object of ${\mathrm}{Rep}^{r,{\mathrm}{cris}}_{{\mathbb}{Z}_p}(G_K)$. Take a $({\varphi},\hat{G})$-module $\hat{{\mathfrak{L}}}$ over ${\mathfrak{S}}$ such that $L\simeq \hat{T}(\hat{{\mathfrak{L}}})$. It is known that $(\tau-1)^i(x)\in u^pI^{[i]}W(R)\otimes_{{\varphi},{\mathfrak{S}}} {\mathfrak{L}}$ for any $i\ge 1$ and any $x\in {\mathfrak{L}}$ (cf. the latter half part of the proof of [@GLS Proposition 4.7]). Take any $x\in {\mathfrak{L}}$. Since $(\tau^{p^s}-1)(x)=\sum^{p^s}_{i=1}\binom{p^s}{i}(\tau-1)^i(x)$, we obtain that $$\label{relat}
(\tau^{p^s}-1)(x)\in \sum^{p^s}_{i=1} p^{s-v_p(i)}u^pI^{[i]}W(R)\otimes_{{\varphi},{\mathfrak{S}}} {\mathfrak{L}}.$$ Now let $T$ be an object of ${\mathrm}{Rep}^{r,{\mathrm}{cris}}_{{\mathrm}{tor}}(G_K)$ which is killed by $p^n$. Take an exact sequence $(R)\colon 0\to L_1\to L_2\to T\to 0$ of ${\mathbb}{Z}_p$-representations of $G_K$ with $L_1,L_2\in {\mathrm}{Rep}^{r,{\mathrm}{cris}}_{{\mathbb}{Z}_p}(G_K)$. By Theorem 3.1.3 and Lemma 3.1.4 of [@CL2], there exists an exact sequence $(M)\colon 0\to \hat{{\mathfrak{L}}}_2\to \hat{{\mathfrak{L}}}_1\to \hat{{\mathfrak{M}}}\to 0$ of $({\varphi},\hat{G})$-modules over ${\mathfrak{S}}$ such that $\hat{T}((M))\simeq (R)$. By (\[relat\]), we see that $$(\tau^{p^s}-1)(x)\in \sum^{p^s}_{i=1} p^{s-v_p(i)}u^pI^{[i]}W(R)\otimes_{{\varphi},{\mathfrak{S}}} {\mathfrak{M}}$$ for any $x\in {\mathfrak{M}}$. Since ${\mathfrak{M}}$ is killed by $p^n$ and $s\ge n-1$, we have $$\begin{aligned}
\sum^{p^s}_{i=1} p^{s-v_p(i)}u^pI^{[i]}W(R)\otimes_{{\varphi},{\mathfrak{S}}} {\mathfrak{M}}&= \sum_{i=1,\dots ,p^s, s-v_p(i)<n} p^{s-v_p(i)}u^pI^{[i]}W(R)\otimes_{{\varphi},{\mathfrak{S}}} {\mathfrak{M}}\\
&= \sum^{n-1}_{\ell=0} p^{\ell}u^pI^{[p^{s-\ell}]}W(R)\otimes_{{\varphi},{\mathfrak{S}}} {\mathfrak{M}}\\
& \subset u^pI^{[p^{s-n+1}]}W(R)\otimes_{{\varphi},{\mathfrak{S}}} {\mathfrak{M}}.\end{aligned}$$ Therefore, we obtained the desired result.
By Corollary \[FFTHMtorcris\], we may suppose ${\mathrm}{log}_p(r-(p-1)/e)\ge 0$, that is, $e(r-1)\ge p-1$ . Suppose $s> n-1+{\mathrm}{log}_p(r-(p-1)/e)$. Note that the condition $s\ge n-1$ is now satisfied. Let $T$ and $T'$ be as in the statement of Theorem \[Main2\]. Let $f\colon T\to T'$ be a $G_{\infty}$-equivariant homomorphism. Denote by $L$ the completion of $K^{{\mathrm}{ur}}$ and identify $G_L$ with the inertia subgroup of $G_K$. We note that $T|_{G_L}$ and $T'|_{G_L}$ are object of ${\mathrm}{Rep}^{r,{\mathrm}{cris}}_{{\mathrm}{tor}}(G_L)$. By Proposition \[Main2Lem\], $T|_{G_{L,s}}$ and $T'|_{G_{L,s}}$ are objects of ${\widetilde}{{\mathrm}{Rep}}^{r,\hat{G}_{L,s},J}_{{\mathrm}{tor}}(G_{L,s})$. Hence we have that $f$ is $G_{L,s}$-equivariant by Theorem \[FFTHM\]. Since $G_s$ is topologically generated by $G_{L,s}$ and $G_{\infty}$, we see that $f$ is $G_s$-equivariant.
Galois equivariance for torsion semi-stable representations {#torsemi}
-----------------------------------------------------------
In this subsection, we prove a Galois equivariance theorem for torsion semi-stable representations. A torsion ${\mathbb}{Z}_p$-representation $T$ of $G_K$ is [*torsion semi-stable with Hodge-Tate weights in $[0,r]$*]{} if it can be written as the quotient of lattices in some semi-stable ${\mathbb}{Q}_p$-representation of $G_K$ with Hodge-Tate weights in $[0,r]$. We denote by ${\mathrm}{Rep}^{r, {\mathrm}{st}}_{{\mathrm}{tor}}(G_K)$ the category of them. Note that ${\mathrm}{Rep}^{0, {\mathrm}{st}}_{{\mathrm}{tor}}(G_K)=
{\mathrm}{Rep}^{0, {\mathrm}{cris}}_{{\mathrm}{tor}}(G_K)$. Similar to Theorem \[Main2\], we show the following, which is the main result of this subsection.
\[Main3\] Suppose that $s> n-1 + {\mathrm}{log}_pr$. Let $T$ and $T'$ be objects of ${\mathrm}{Rep}^{r, {\mathrm}{st}}_{{\mathrm}{tor}}(G_K)$ which are killed by $p^n$. Then any $G_{\infty}$-equivariant homomorphism $T\to T'$ is in fact $G_s$-equivariant.
If $s\ge n-1$, then we put $$J=I^{[p^{s-n+1}]}W(R)={\varphi}({\mathfrak{t}})^{p^{s-n+1}}W(R).$$ Then we have $c_J=p^{s-n+2}/(p-1)$. To show Theorem \[Main3\], we use similar arguments to those in the proof of Theorem \[Main2\].
\[Main3Lem\] Suppose $s\ge n-1$. If $T$ is an object of ${\mathrm}{Rep}^{r,{\mathrm}{st}}_{{\mathrm}{tor}}(G_K)$ which is killed by $p^n$, then $T|_{G_s}$ is contained in ${\widetilde}{{\mathrm}{Rep}}^{r,\hat{G}_s,J}_{{\mathrm}{tor}}(G_s)$.
Let $L$ be a lattice in a semi-stable ${\mathbb}{Q}_p$-representation of $G_K$ with Hodge-Tate weights in $[0,r]$. Take a $({\varphi},\hat{G})$-module $\hat{{\mathfrak{L}}}$ over ${\mathfrak{S}}$ such that $L\simeq \hat{T}(\hat{{\mathfrak{L}}})$. It is known that $(\tau-1)^i(x)\in I^{[i]}W(R)\otimes_{{\varphi},{\mathfrak{S}}} {\mathfrak{L}}$ for any $i\ge 1$ and any $x\in {\mathfrak{L}}$ (cf. the proof of [@Li4 Proposition 2.4.1]). Thus the same proof proceeds as that of Proposition \[Main2Lem\].
We have the equality ${\mathrm}{Rep}^{0, {\mathrm}{st}}_{{\mathrm}{tor}}(G_K)={\mathrm}{Rep}^{0, {\mathrm}{cris}}_{{\mathrm}{tor}}(G_K)$ and thus Theorem \[Main2\] for $r=0$ is an easy consequence of Corollary \[FFTHMtorcris\]. Hence we may assume $r\ge 1$. The rest of a proof is similar to the proof of Theorem \[Main2\].
Some consequences {#consequences}
-----------------
In this subsection, we generalize some results proved in Section 3.4 of [@Br3]. First of all, we show the following elementary lemma, which should be well-known to experts, but we include a proof here for the sake of completeness.
\[stability’\] The full subcategories ${\mathrm}{Rep}^{r,{\mathrm}{cris}}_{{\mathrm}{tor}}(G_K)$ and ${\mathrm}{Rep}^{r,{\mathrm}{st}}_{{\mathrm}{tor}}(G_K)$ of ${\mathrm}{Rep}_{{\mathrm}{tor}}(G_K)$ are stable under formation of subquotients, direct sums and the association $T\mapsto T^{\vee}(r)$. Here $T^{\vee}={\mathrm}{Hom}_{{\mathbb}{Z}_p}(T,{\mathbb}{Q}_p/{\mathbb}{Z}_p)$ is the dual representation of $T$.
We prove the statement only for ${\mathrm}{Rep}^{r,{\mathrm}{cris}}_{{\mathrm}{tor}}(G_K)$. Let $T\in {\mathrm}{Rep}^{r,{\mathrm}{cris}}_{{\mathrm}{tor}}(G_K)$ be killed by $p^n$ for some $n>0$. Assertions for quotients and direct sums are clear. We prove that $T^{\vee}(r)$ is contained in ${\mathrm}{Rep}^{r,{\mathrm}{cris}}_{{\mathrm}{tor}}(G_K)$. There exist lattices $L_1\subset L_2$ in some crystalline ${\mathbb}{Q}_p$-representation of $G_K$ and an exact sequence $0\to L_1\to L_2\to T\to 0$ of ${\mathbb}{Z}_p[G_K]$-modules. This exact sequence induces an exact sequence $0\to T\to L_1/p^nL_1\to L_2/p^nL_2\to T\to 0$ of finite ${\mathbb}{Z}_p[G_K]$-modules. By duality, we obtain an exact sequence $0\to T^{\vee}\to (L_2/p^nL_2)^{\vee}\to
(L_1/p^nL_1)^{\vee}\to T^{\vee}\to 0$ of finite ${\mathbb}{Z}_p[G_K]$-modules. Then we obtain a $G_K$-equivariant surjection $L_1^{\vee}\twoheadrightarrow T^{\vee}$ by the composite $L_1^{\vee}\twoheadrightarrow L_1^{\vee}/p^nL_1^{\vee}\overset{\sim}{\to}
(L_1/p^nL_1)^{\vee}\twoheadrightarrow T^{\vee}$ of natural maps (here, for any free ${\mathbb}{Z}_p$-representation $L$ of $G_K$, $L^{\vee}:={\mathrm}{Hom}_{{\mathbb}{Z}_p}(L,{\mathbb}{Z}_p)$ stands for the dual of $L$). Therefore, we obtain $L_1^{\vee}(r)\twoheadrightarrow T^{\vee}(r)$ and thus $T^{\vee}(r)\in {\mathrm}{Rep}^{r,{\mathrm}{cris}}_{{\mathrm}{tor}}(G_K)$. Finally, we prove the stability assertion for subobjects. Let $T'$ be a $G_K$-stable submodule of $T$. We have a $G_K$-equivariant surjection $f\colon L_1^{\vee}\twoheadrightarrow T^{\vee}\twoheadrightarrow (T')^{\vee}$. Let $L'_2$ be a free ${\mathbb}{Z}_p$-representation of $G_K$ such that its dual is the kernel of $f$. We have an exact sequence $0\to (L'_2)^{\vee}\to L^{\vee}_1\overset{f}{\to} (T')^{\vee}\to 0$ of ${\mathbb}{Z}_p[G_K]$-modules. Repeating the construction of the surjection $L_1^{\vee}\twoheadrightarrow T^{\vee}$, we obtain a $G_K$-equivariant surjection $L'_2=(L'_2)^{\vee \vee}\twoheadrightarrow (T')^{\vee \vee}=T'$ and thus we have $T'\in {\mathrm}{Rep}^{r,{\mathrm}{cris}}_{{\mathrm}{tor}}(G_K)$.
In the case where $r=1$, the assertion (1) of the following corollary was shown in Theorem 3.4.3 of [@Br3].
\[imagestable\] Let $T$ be an object of ${\mathrm}{Rep}^{r,{\mathrm}{cris}}_{{\mathrm}{tor}}(G_K)$ which is killed by $p^n$ for some $n>0$. Let $T'$ be a $G_{\infty}$-stable subquotient of $T$.
$(1)$ If $e(r-1)<p-1$, then $T'$ is $G_K$-stable $($with respect to $T)$.
$(2)$ If $s>n-1+{\mathrm}{log}_p(r-(p-1)/e)$, then $T'$ is $G_s$-stable $($with respect to $T)$.
By the duality assertion of Lemma \[stability’\], it is enough to show the case where $T'$ is a $G_{\infty}$-stable submodule of $T$. Take any sequence $T'=T_0\subset T_1\subset \cdots \subset T_m=T$ of torsion $G_{\infty}$-stable submodules of $T$ such that $T_i/T_{i-1}$ is irreducible for any $i$. As explained in the proof of Proposition \[FFLEM\], the $G_{\infty}$-action on $T_i/T_{i-1}$ can be (uniquely) extended to $G_K$. By Theorem \[tamelift\] given in the next section, we know that $T_i/T_{i-1}$ is an object of ${\mathrm}{Rep}^{r_0,{\mathrm}{cris}}_{{\mathrm}{tor}}(G_K)$ where $r_0:={\mathrm}{max}\{ r'\in {\mathbb}{Z}_{\ge 0}; e(r'-1)<p-1 \}$.
\(1) We may suppose $r=r_0$. The $G_{\infty}$-equivariant projection $T=T_m\twoheadrightarrow T_m/T_{m-1}$ is $G_K$-stable by the full faithfulness theorem (= Corollary \[FFTHMtorcris\]). Thus we know that $T_{m-1}$ is $G_K$-stable in $T$, and also know that $T_{m-1}$ is contained in ${\mathrm}{Rep}^{r,{\mathrm}{cris}}_{{\mathrm}{tor}}(G_K)$ by Lemma \[stability’\]. By the same argument for the $G_{\infty}$-equivariant projection $T_{m-1}\twoheadrightarrow T_{m-1}/T_{m-2}$, we know that $T_{m-2}$ is $G_K$-stable in $T$, and also know that $T_{m-2}$ is contained in ${\mathrm}{Rep}^{r,{\mathrm}{cris}}_{{\mathrm}{tor}}(G_K)$. Repeating this argument, we have that $T'=T_0$ is $G_K$-stable in $T$.
\(2) Put $J=u^pI^{[p^{s-n+1}]}W(R)$. By (1) we may assume $e(r-1)\ge p-1$. Under this assumption we have $r\ge r_0$ and $s>n-1+{\mathrm}{log}_p(r-(p-1)/e)\ge n-1$. In particular, $T|_{G_s}$ and $(T_i/T_{i-1})|_{G_s}$, for any $i$, are contained in ${\widetilde}{{\mathrm}{Rep}}^{r,\hat{G}_s,J}_{{\mathrm}{tor}}(G_s)$ by Proposition \[Main2Lem\]. First we consider the case where $k$ is algebraically closed. By Theorem \[FFTHM\], the $G_{\infty}$-equivariant projection $T=T_m\twoheadrightarrow T_m/T_{m-1}$ is $G_s$-stable. Thus we know that $T_{m-1}$ is $G_s$-stable in $T$, and also know that $T_{m-1}$ is contained in ${\widetilde}{{\mathrm}{Rep}}^{r,\hat{G}_s,J}_{{\mathrm}{tor}}(G_s)$ by Corollary \[stability\]. By the same argument for the $G_{\infty}$-equivariant projection $T_{m-1}\twoheadrightarrow T_{m-1}/T_{m-2}$, we know that $T_{m-2}$ is $G_s$-stable in $T$, and also know that $T_{m-2}$ is contained in ${\widetilde}{{\mathrm}{Rep}}^{r,\hat{G}_s,J}_{{\mathrm}{tor}}(G_s)$. Repeating this argument, we have that $T'=T_0$ is $G_s$-stable in $T$. Next we consider the case where $k$ is not necessary algebraically closed. Let $L$ be the completion of the maximal unramified extension $K^{{\mathrm}{ur}}$ of $K$, and we identify $G_L$ with the inertia subgroup of $G_K$. Clearly $T|_{G_L}$ is contained in ${\mathrm}{Rep}^{r,{\mathrm}{cris}}_{{\mathrm}{tor}}(G_L)$ and $T'$ is $G_{L_{\infty}}$-stable submodule of $T$. We have already shown that $T'$ is $G_{L,s}$-stable in $T$. Since $G_s$ is topologically generated by $G_{L,s}$ and $G_{\infty}$, we conclude that $T'$ is $G_s$-stable in $T$.
Now let $V$ be a ${\mathbb}{Q}_p$-representation of $G_K$ and $T$ a ${\mathbb}{Z}_p$-lattice of $V$ which is stable under $G_{\infty}$. Then we know that $T$ is automatically $G_s$-stable for some $s\ge 0$. Indeed we can check this as follows. Take any $G_K$-stable ${\mathbb}{Z}_p$-lattice $T'$ of $V$ which contains $T$, and take an integer $n>0$ with the property that $p^nT'\subset T$. Furthermore, we take a finite extension $K'$ of $K$ such that $G_{K'}$ acts trivially on $T'/p^nT'$. Then $T/p^nT'$ is $G_{\infty}$-stable and also $G_{K'}$-stable in $T'/p^nT'$. If we take any integer $s\ge 0$ with the property $K'\cap K_{\infty}\subset K_{(s)}$, we know that $T/p^nT'$ is $G_s$-stable. This implies that $T$ is $G_s$-stable in $T'$.
The following corollary, which was shown in Corollary 3.4.4 of [@Br3] in the case where $r=1$, is related with the above property.
\[stablelattice\] Let $V$ be a crystalline ${\mathbb}{Q}_p$-representation of $G_K$ with Hodge-Tate weights in $[0,r]$ and $T$ a finitely generated ${\mathbb}{Z}_p$-submodule of $V$ which is stable under $G_{\infty}$. If $e(r-1)<p-1$, then $T$ is stable under $G_K$.
We completely follow the method of the proof of [@Br3 Corollary 3.4.4]. Take any $G_K$-stable ${\mathbb}{Z}_p$-lattice $T'$ of $V$ which contains $T$. Since $T'/p^nT'$ is contained in ${\mathrm}{Rep}^{r,{\mathrm}{cris}}_{{\mathrm}{tor}}(G_K)$ for any $n>0$, Corollary \[imagestable\] (1) implies that any $G_{\infty}$-stable submodule of $T'/p^nT'$ is in fact $G_K$-stable. Thus $(T+p^nT')/p^nT'$ is $G_K$-stable in $T'/p^nT'$. Therefore, we obtain $g(T)\subset \bigcap_{n>0}\ (T+p^nT')=T$ for any $g\in G_K$.
Crystalline lifts and c-weights
===============================
We continue to use the same notation except for that we may allow $p=2$. We remark that a torsion ${\mathbb}{Z}_p$-representation of $G_K$ is torsion crystalline with Hodge-Tate weights in $[0,r]$ if there exist a lattice $L$ in some crystalline ${\mathbb}{Q}_p$-representation of $G_K$ with Hodge-Tate weights in $[0,r]$ and a $G_K$-equivariant surjection $f\colon L\twoheadrightarrow T$. We call $f$ a [*crystalline lift*]{} ([*of $T$*]{}) [*of weight $\le r$*]{}. Our interest in this section is to determine the minimum integer $r$ (if it exists) such that $T$ admits crystalline lifts of weight $\le r$. We call this minimum integer the [*c-weight of $T$*]{} and denote it by $w_c(T)$. If $T$ does not have crystalline lifts of weight $\le r$ for any integer $r$, then we define the c-weight $w_c(T)$ of $T$ to be $\infty$. Motivated by [@CL2 Question 5.5], we raise the following question.
For a torsion ${\mathbb}{Z}_p$-representation $T$ of $G_K$, is the c-weight $w_c(T)$ of $T$ finite? Furthermore, can we calculate $w_c(T)$?
General properties of c-weights
-------------------------------
We study general properties of c-weights. At first, by ramification estimates, it is known that c-weights may have infinitely large values ([@CL2 Theorem 5.4]); for any $c>0$, there exists a torsion ${\mathbb}{Z}_p$-extension $T$ of $G_K$ with $w_c(T)>c$. In this paper, we mainly consider representations with “small” c-weights. If c-weights are “small”, they are closely related with [*tame inertia weights*]{}. Now we recall the definition of tame inertia weights. Let $I_K$ be the inertia subgroup of $G_K$. Let $T$ be a $d$-dimensional irreducible ${\mathbb}{F}_p$-representation of $I_K$. Then $T$ is isomorphic to $${\mathbb}{F}_{p^d}(\theta^{n_1}_{d,1}\cdots \theta^{n_d}_{d,d})$$ for one sequence of integers between $0$ and $p-1$, periodic of period $d$. Here, $\theta_{d,1},\dots , \theta_{d,d}$ are the fundamental characters of level $d$. The integers $n_1/e,\dots ,n_d/e$ are called the tame inertia weights of $T$. For any ${\mathbb}{F}_p$-representation $T$ of $G_K$, the tame inertia weights of $T$ are the tame inertia weights of the Jordan-Hölder quotients of $T|_{I_K}$.
Let $\chi_p\colon G_K\to {\mathbb}{Z}^{\times}_p$ be the $p$-adic cyclotomic character and $\bar{\chi}_p\colon G_K\to {\mathbb}{F}^{\times}_p$ the mod $p$ cyclotomic character. It is well-known that $\bar{\chi}_p|_{I_K}=\theta_1^e$ where $\theta_1\colon I_K\twoheadrightarrow {\mathbb}{F}^{\times}_p$ is the fundamental character of level $1$. In particular, denoting by $K^{{\mathrm}{ur}}$ the maximal unramified extension of $K$, we have $[K^{{\mathrm}{ur}}(\mu_p):K^{{\mathrm}{ur}}]=(p-1)/{\mathrm}{gcd}(e,p-1)$.
\[poly\] $(1)$ Minimum c-weights are invariant under finite unramified extensions of the base field $K$.
$(2)$ The c-weight of an unramified torsion ${\mathbb}{Z}_p$-representation of $G_K$ is $0$.
$(3)$ Put $\nu=(p-1)/{\mathrm}{gcd}(e,p-1)$. If $\nu (s-1)<w_c(T)\le \nu s$, then we have $\nu (s-1)<w_c(T^{\vee})\le \nu s$. In particular, if $(p-1)\mid e$, then we have $w_c(T)=w_c(T^{\vee})$.
$(4)$ Let $T$ be an ${\mathbb}{F}_p$-representation of $G_K$ and $i$ the largest tame inertia weight of $T$. Then we have $w_c(T)\geq i$.
\(1) Let $T$ be a torsion ${\mathbb}{Z}_p$-representations of $G_K$. Let $K'$ be a finite unramified extension of $K$. It suffices to prove that $T$ has crystalline lifts of weight $\le r$ if and only if $T|_{G_{K'}}$ has crystalline lifts of weight $\le r$. The “only if” assertion is clear and thus it is enough to prove the “if” assertion. Let $f\colon L\twoheadrightarrow T|_{G_{K'}}$ be a crystalline lift of $T|_{G_{K'}}$ of weight $\le r$. Since $K'/K$ is unramified, ${\mathrm}{Ind}^{G_K}_{G_{K'}}L$ is a lattice in some crystalline ${\mathbb}{Q}_p$-representation of $G_K$ with Hodge-Tate weights in $[0,r]$. Furthermore, the map $${\mathrm}{Ind}^{G_K}_{G_{K'}}L={\mathbb}{Z}_p[G_K]\otimes_{{\mathbb}{Z}_p[G_{K'}]} L\to T,\quad
\sigma\otimes x\mapsto \sigma(f(x))$$ is a $G_K$-equivariant surjection and hence we have done.
\(2) The result follows from (1) immediately.
\(3) Taking a finite unramified extension $K'$ of $K$ with the property $[K^{{\mathrm}{ur}}(\mu_p):K^{{\mathrm}{ur}}]=[K'(\mu_p):K']$, it follows from Lemma \[stability’\] that we have $\nu (s-1)<w_c(T|_{G_K'})\le \nu s$ if and only if we have $\nu (s-1)<w_c((T^{\vee})|_{G_K'})\le \nu s$. Thus the result follows from the assertion (1).
\(4) If $ew_c(T)\ge p-1$, then there is nothing to prove, and thus we may suppose that $ew_c(T)< p-1$. Let $L\twoheadrightarrow T$ be a crystalline lift of $T$ of weight $\le w_c(T)$. Since the tame inertia polygon of $L$ lies on the Hodge polygon of $L$ ([@CS Théorème 1]), the largest slope of the former polygon is less than or equal to that of the latter polygon. This implies $w_c(T)\geq i$.
\[tamelift\] Let $T$ be a tamely ramified ${\mathbb}{F}_p$-representation of $G_K$. Let $i$ be the largest tame inertia weight of $T$. Then we have $w_c(T)={\mathrm}{min}\{ h\in {\mathbb}{Z}_{\ge 0}; h\ge i \}$.
The proof below is essentially due to Caruso and Liu [@CL2 Theorem 5.7], but we give a proof here for the sake of completeness. Put $i_0={\mathrm}{min}\{ h\in {\mathbb}{Z}_{\ge 0}; h\ge i \}$. By Proposition \[poly\] (4), we have $w_c(T)\ge i_0$. Thus it suffices to show $w_c(T)\le i_0$. We note that $T|_{I_K}$ is semi-simple. Any irreducible component $T_0$ of $T|_{I_K}$ is of the form $
{\mathbb}{F}_{p^d}(\theta^{n_1}_{d,1}\cdots \theta^{n_d}_{d,d})
$ for one sequence of integers between $0$ and $p-1$, periodic of period $d$. We decompose $n_j=em_j+n'_j$ by integers $0\le m_j\le i_0$ and $0\le n'_j<e$. Now we define an integer $k_{j, \ell}$ by $$\begin{aligned}
k_{j, \ell}:=
\left\{
\begin{array}{ll}
e\quad \hspace{2.0mm} {\rm if}\ 1\le \ell\le m_j,
\cr
n'_j\quad {\rm if}\ \ell=m_j+1,
\cr
0\quad \hspace{2.0mm} {\rm if}\ \ell>m_j+1.
\end{array}
\right.\end{aligned}$$ Note that we have $n_j=\sum^{i_0}_{\ell=1} k_{j, \ell}$, and also have an $I_K$-equivariant surjection $$T_0={\mathbb}{F}_{p^d}(\theta^{n_1}_{d,1}\dots \theta^{n_d}_{d,d})
=\bigotimes_{\ell=1,\dots i_0, {\mathbb}{F}_{p^d}}
{\mathbb}{F}_{p^d}(\theta^{k_{1,\ell}}_{d,1}\dots \theta^{k_{d,\ell}}_{d,d})
\twoheadleftarrow \bigotimes_{\ell=1,\dots i_0, {\mathbb}{F}_p}
{\mathbb}{F}_{p^d}(\theta^{k_{1,\ell}}_{d,1}\dots \theta^{k_{d,\ell}}_{d,d}).$$ By a classical result of Raynaud, each ${\mathbb}{F}_{p^d}(\theta^{k_{1,\ell}}_{d,1}\cdots \theta^{k_{d,\ell}}_{d,d})$ comes from a finite flat group scheme defined over $K^{{\mathrm}{ur}}$. We should remark that such a finite flat group scheme is in fact defined over a finite unramified extension of $K$. Since any finite flat group scheme can be embedded in a $p$-divisible group, the above observation implies the following: there exist a finite unramified extension $K'$ over $K$, a lattice $L$ in some crystalline ${\mathbb}{Q}_p$-representation of $G_{K'}$ with Hodge-Tate weights in $[0, i_0]$ and an $I_K$-equivariant surjection $f\colon L\twoheadrightarrow T$. The map $f$ induces an $I_K$-equivariant surjection $\tilde{f}\colon L/pL\twoheadrightarrow T$. Since $L/pL$ and $T$ is finite, we see that $\tilde{f}$ is in fact $G_{K''}$-equivariant for some finite unramified extension $K''$ over $K'$, and then so is $f$. Therefore, we obtain $w_c(T|_{G_{K''}})\le i_0$. By Proposition \[poly\] (1), we obtain $w_c(T)\le i_0$.
Rank $2$ cases
--------------
We give some computations of c-weights related with torsion representations of rank $2$. We prove the following lemma by an almost identical method with [@GLS Lemma 9.4].
\[2liftlem\] Let $K$ be a finite extension of ${\mathbb}{Q}_p$. Let $E$ be a finite extension of ${\mathbb}{Q}_p$ with residue field ${\mathbb}{F}$. Let $i$ and $\nu$ be integers such that $\nu$ is divisible by $[K(\mu_p):K]$. Suppose that $T$ is an ${\mathbb}{F}$-representation of $G_K$ which sits in an exact sequence $(\ast)\colon 0\to {\mathbb}{F}(i)\to T\to {\mathbb}{F}\to 0$ of ${\mathbb}{F}$-representations of $G_K$. Then there exist a ramified degree at most $2$ extension $E'$ over $E$, with integer ring ${\mathcal{O}}_{E'}$, and an unramified continuous character $\chi\colon G_K\to {\mathbb}{F}^{\times}$ with trivial reduction such that $(\ast)$ is the reduction of some exact sequence $0\to {\mathcal{O}}_{E'}(\chi \chi_p^{i+\nu})\to \Lambda\to {\mathcal{O}}_{E'}\to 0$ of free ${\mathcal{O}}_{E'}$-representations of $G_K$. Furthermore, we have the followings:
$(1)$ If $i+\nu=1$ or $\bar{\chi}_p^{1-i}\not= 1$, then we can take $E'=E$ and $\chi=1$.
$(2)$ If $i+\nu=0$ and $T$ is unramified, then we can take $E'=E$, $\chi=1$ and $\Lambda$ to be unramified.
Suppose $i+\nu=1$ (resp. $\bar{\chi}_p^{1-i}\not= 1$). Then the map $H^1(K, {\mathcal{O}}_E(i+\nu))\to H^1(K,{\mathbb}{F}(i))$ arising from the exact sequence $0\to {\mathcal{O}}_E(i+\nu)\overset{\varpi}{\to} {\mathcal{O}}_E(i+\nu)\to {\mathbb}{F}(i)\to 0$ is surjective since $H^2(K, {\mathcal{O}}_E(1))\simeq {\mathcal{O}}_E$ (resp. $H^2(K, {\mathcal{O}}_E(i+\nu))=0$), where $\varpi$ is a uniformizer of $E$. Hence we obtained a proof of (1). The assertion (2) follows immediately from the fact that the natural map $H^1(G_K/I_K, {\mathcal{O}}_E)\to H^1(G_K/I_K, {\mathbb}{F})$ is surjective.
In the rest of this proof, we always assume that $i+\nu\not=1$ and $\bar{\chi}_p^{1-i}= 1$. Let $L\in H^1(K,{\mathbb}{F}(i))$ be a $1$-cocycle corresponding to $(\ast)$. We may suppose $L \not=0$. For any unramified continuous character $\chi\colon G_K\to {\mathbb}{F}^{\times}$ with trivial reduction, we denote by $$\begin{aligned}
&\delta^1_{\chi}\colon H^1(K,{\mathbb}{F}(i))\to H^2(K, {\mathcal{O}}_E(\chi\chi_p^{i+\nu}))\\
({\mathrm}{resp.}\ &\delta^0_{\chi}\colon
H^0(K, E/{\mathcal{O}}_E(\chi^{-1}\chi_p^{1-i-\nu}))\to H^1(K, {\mathbb}{F}))\end{aligned}$$ the connection map arising from the exact sequence $0\to {\mathcal{O}}_E(\chi\chi_p^{i+\nu})\overset{\varpi}{\to}
{\mathcal{O}}_E(\chi\chi_p^{i+\nu}) \to {\mathbb}{F}(i)\to 0$ (resp. $0\to {\mathbb}{F}\to E/{\mathcal{O}}_E(\chi^{-1}\chi_p^{1-i-\nu})\overset{\varpi}{\to}
E/{\mathcal{O}}_E(\chi^{-1}\chi_p^{1-i-\nu})\to 0$) of ${\mathcal{O}}_E[G_K]$-modules. Consider the following commutative diagram:
$\displaystyle \xymatrix{
H^1(K,{\mathbb}{F}(i)) \ar_{\delta^1_{\chi}}[d]
& \times
& H^1(K,{\mathbb}{F}) \ar[rrr]
& &
& E/{\mathcal{O}}_E \ar@{=}[d] \\
H^2(K, {\mathcal{O}}_E(\chi\chi_p^{i+\nu}))
& \times
& H^0(K, E/{\mathcal{O}}_E(\chi^{-1}\chi_p^{1-i-\nu})) \ar^{\delta^0_{\chi}}[u] \ar[rrr]
& &
& E/{\mathcal{O}}_E
}$
Since the above two pairings are perfect, we see that $L$ lifts to $H^1(G_K, {\mathcal{O}}_E(\chi\chi_p^{i+\nu}))$ if and only if $H$ is contained in the image of $\delta^0_{\chi}$. Here, $H\subset H^1(K,{\mathbb}{F})$ is the annihilator of $L$ under the local Tate pairing $H^1(K,{\mathbb}{F}(i)) \times H^1(K,{\mathbb}{F}) \to E/{\mathcal{O}}_E$. Let $n\ge 1$ be the largest integer with the property that $\chi^{-1}\chi_p^{1-i-\nu}\equiv 1\ {\mathrm}{mod}\ \varpi^n$ (such $n$ exists since $\bar{\chi}_p^{1-i}=1$ and $1-i-\nu\not=0$). We define $\alpha_{\chi}\colon G_K\to {\mathcal{O}}_E$ by the relation $\chi^{-1}\chi_p^{1-i-\nu}=1+\varpi^n\alpha_{\chi}$, and denote $(\alpha_{\chi}\ {\mathrm}{mod}\ \varpi)\colon G_K\to {\mathbb}{F}$ by $\bar{\alpha}_{\chi}$. By definition, $\bar{\alpha}_{\chi}$ is a non-zero element of $H^1(K,{\mathbb}{F})$, and it is not difficult to check that the image of $\delta^0_{\chi}$ is generated by $\bar{\alpha}_{\chi}$. If $\bar{\alpha}_{\chi}$ is contained in $H$ for some $\chi$, we are done. Suppose this is not the case.
Suppose that $H$ is not contained in the unramified line in $H^1(K,{\mathbb}{F})$. We claim that we can choose $\chi$ such that $\bar{\alpha}_{\chi}$ is ramified. Let $m$ be the largest integer with the property that $(\chi^{-1}\chi_p^{1-i-\nu})|_{I_K}\equiv 1\ {\mathrm}{mod}\ \varpi^n$. Clearly, we have $m\ge n$. If $m=n$, then we are done and thus we may assume $m>n$. Fix a lift $g\in G_K$ of the Frobenius of $K$. We see that $\bar{\alpha}_{\chi}(g)\not= 0$. Let $\chi'$ be the unramified character sending $g$ to $1+\varpi^n\alpha_{\chi}(g)$. Then $\chi'$ has trivial reduction. After replacing $\chi$ with $\chi\chi'$, we reduce the case where $m=n$ and thus the claim follows. Suppose $\bar{\alpha}_{\chi}$ is ramified. Then there exists a unique $\bar{x}\in {\mathbb}{F}^{\times}$ such that $\bar{\alpha}_{\chi}+u_{\bar x}\in H$ where $u_{\bar x}\colon G_K\to {\mathbb}{F}$ is the unramified character sending $g$ to $\bar x$. Denote by $\chi''$ the unramified character sending $g$ to $1+\varpi^n\alpha_{\chi}(g)$. Replacing $\chi$ with $\chi\chi''$, we have done.
Suppose that $H$ is contained in the unramified line in $H^1(K,{\mathbb}{F})$ (thus $H$ and the unramified line coincide with each other). By replacing $E$ with $E(\sqrt{\varpi})$, we may assume that $n>1$. Let $\chi_0$ be a character defined by $\chi$ times the unramified character sending our fixed $g$ to $1+\varpi$. Since $n>1$, we see that $\chi_0^{-1}\chi_p^{1-i-\nu}\equiv 1\ {\mathrm}{mod}\ \varpi$ and $\chi_0^{-1}\chi_p^{1-i-\nu}\not\equiv 1\ {\mathrm}{mod}\ \varpi^2$. We define $\alpha_{\chi_0}\colon G_K\to {\mathcal{O}}_E$ by the relation $\chi_0^{-1}\chi_p^{1-i-\nu}=1+\varpi\alpha_{\chi_0}$, and denote $(\alpha_{\chi_0}\ {\mathrm}{mod}\ \varpi)\colon G_K\to {\mathbb}{F}$ by $\bar{\alpha}_{\chi_0}$. By definition and the assumption $n>1$, $\bar{\alpha}_{\chi_0}$ is a non-zero unramified element of $H^1(K,{\mathbb}{F})$, hence it is contained in $H$. Therefore, we have done.
Let $K$ be a finite extension of ${\mathbb}{Q}_p$, $n\ge 2$ an integer and $\chi\colon G_K\to E^{\times}$ an unramified character. Since any $E$-representation of $G_K$ which is an extension of $E$ by $E(\chi\chi_p^n)$ is automatically crystalline, we obtain the following.
\[rank2\] Suppose $p>2$. Let $K$ be a finite unramified extension of ${\mathbb}{Q}_p$. Let $T\in {\mathrm}{Rep}_{{\mathrm}{tor}}(G_K)$ be killed by $p$ and sit in an exact sequence $0\to {\mathbb}{F}_p(i)\to T\to {\mathbb}{F}_p\to 0$ of ${\mathbb}{F}_p$-representations of $G_K$. Then we have the followings:
$(1)$ If $i=0$ and $T$ is unramified, then we have $w_c(T)=0$.
$(2)$ If $i=0$ and $T$ is not unramified, then we have $w_c(T)=p-1$.
$(3)$ If $i=2,\dots, p-2$, then we have $w_c(T)=i$.
(1), (2) By Lemma \[2liftlem\] (2), it suffices to prove that $T$ is not torsion crystalline with Hodge-Tate weights in $[0,p-2]$ if $T$ is not unramified. Let $K_T$ be the definition field of the representation $T$ of $G_K$ and put $G={\mathrm}{Gal}(K_T/K)$. Let $G^j$ be the upper numbering $j$-th ramification subgroup of $G$ (in the sense of [@Se]). Since $T$ is not unramified and killed by $p$, we see that $K_T$ is a totally ramified degree $p$ extension over $K$. Thus $G^1$ is the wild inertia subgroup of $G$ and $G^1=G$, which does not act on $T$ trivial by the definition of $G$. Thus we obtain the desired result by ramification estimates of [@Fo1] (or [@Ab1]) for torsion crystalline representations with Hodge-Tate weights in $[0,p-2]$: if $T$ is torsion crystalline with Hodge-Tate weights in $[0,p-2]$, then $G^j$ acts on $T$ trivial for any $j>(p-2)/(p-1)$.
\(3) The result follows immediately from Proposition \[poly\] (4) and Lemma \[2liftlem\].
\[2lift\] Let $K$ be a finite unramified extension of ${\mathbb}{Q}_p$. Then any $2$-dimensional ${\mathbb}{F}_p$-representation of $G_K$ is torsion crystalline with Hodge-Tate weights in $[0,2p-2]$.
If $T$ is irreducible, the result follows from Theorem \[tamelift\]. Assume that $T$ is reducible. Since $K$ is unramified over ${\mathbb}{Q}_p$, any continuous character $G_K\to {\mathbb}{F}^{\times}_p$ is of the form $\chi \bar{\chi}^i_p$ for some unramified character $\chi$ and some integer $i$. Replacing $K$ with its finite unramified extension, we may assume that $T$ sits in an exact sequence $0\to {\mathbb}{F}_p(i)\to T\to {\mathbb}{F}_p(j)\to 0$ of ${\mathbb}{F}_p$-representations of $G_K$, where $i$ and $j$ are integers in the range $[0,p-2]$ (we remark that $w_c(T)$ is invariant under unramified extensions of $K$ by Proposition \[poly\] (1)). It follows from Lemma \[2liftlem\] that $w_c(T(-j))\le p$. Therefore, we obtain $w_c(T)=w_c(T(-j)\otimes_{{\mathbb}{F}_p} {\mathbb}{F}_p(j))
\le w_c(T(-j))+ w_c({\mathbb}{F}_p(j))\le p+(p-2)
=2p-2$.
Extensions of ${\mathbb}{F}_p$ by ${\mathbb}{F}_p(1)$ and non-fullness theorems
-------------------------------------------------------------------------------
By Lemma \[2liftlem\], we know that the c-weight $w_c(T)$ of an ${\mathbb}{F}_p$-representation $T$ of $G_K$ which sits in an exact sequence $0\to {\mathbb}{F}_p(1)\to T\to {\mathbb}{F}_p\to 0$ of ${\mathbb}{F}_p$-representations of $G_K$, is less than or equal to $p$. Let us calculate $w_c(T)$ for such $T$ more precisely. We should remark that such $T$ is written as $p$-torsion points of a Tate curve. Hence we consider torsion representations coming from Tate curves.
Let $v_K$ be the valuation of $K$ normalized such that $v_K(K^{\times})={\mathbb}{Z}$, and take any $q\in K^{\times}$ with $v_K(q)>0$. Let $E_q$ be the Tate curve over $K$ associated with $q$ and $E_q[p^n]$ the module of $p^n$-torsion points of $E_q$ for any integer $n>0$. It is well-known that there exists an exact sequence $$(\#)\quad 0\to \mu_{p^n}\to E_q[p^n]\to {\mathbb}{Z}/p^n{\mathbb}{Z}\to 0$$ of ${\mathbb}{Z}_p[G_K]$-modules. Here, $\mu_{p^n}$ is the group of $p^n$-th roots of unity in $\overline{K}$. Let $x_n\colon G_K\to \mu_{p^n}$ be the $1$-cocycle defined to be the image of $1$ for the connection map $H^0(K, {\mathbb}{Z}/p^n{\mathbb}{Z})\to H^1(K, \mu_{p^n})$ arising from the exact sequence $(\#)$. Then $x_n$ corresponds to $q$ mod $(K^{\times})^{p^n}$ via the isomorphism $K^{\times}/(K^{\times})^{p^n}\simeq H^1(K,\mu_{p^n})$ of Kummer theory. Thus the exact sequence $(\#)$ splits if and only if $q\in (K^{\times})^{p^n}$.
First we consider the case $p\mid v_K(q)$ (i.e. [*peu ramifié*]{} case).
\[easycase\] Let $K$ be a finite extension of ${\mathbb}{Q}_p$. If $p\mid v_K(q)$, then $E_q[p]$ is the reduction modulo $p$ of a lattice in some $2$-dimensional crystalline ${\mathbb}{Q}_p$-representation with Hodge-Tate weights in $[0,1]$.
Since $p\mid v_K(q)$, there exists $q'\in K^{\times}$ such that $v_K(q'-1)>0$ and $q\equiv q'$ mod $(K^{\times})^p$. Considering the exact sequence $0\to {\mathbb}{Z}_p(1)\to L\to {\mathbb}{Z}\to 0$ of ${\mathbb}{Z}_p$-representations of $G_K$ corresponding to $q'$ via the isomorphism $H^1(K,{\mathbb}{Z}_p(1))\simeq {\varprojlim}_{n} K^{\times}/(K^{\times})^{p^n}$ of Kummer theory, we obtain the desired result.
\[minwt\] Suppose that $K$ is a finite extension of ${\mathbb}{Q}_p$, $(p-1)\nmid e$ and $p\mid v_K(q)$. Then we have $w_c(E_q[p])=1$.
By the assumption $(p-1)\nmid e$, we know that the largest tame inertia weight of $E_q[p]$ is positive. Thus Proposition \[poly\] (4) shows $w_c(E_q[p])\ge 1$. The inequality $w_c(E_q[p])\le 1$ follows from Lemma \[easycase\].
Next we consider the case $p\nmid v_K(q)$ (i.e. [*très ramifié*]{} case).
\[Tatecurve\] If $e(r-1)<p-1$ and $p\nmid v_K(q)$, then $E_q[p^n]$ is not torsion crystalline with Hodge-Tate weights in $[0,r]$ for any $n>0$.
If $e=1$, the fact that $E_{\pi}[p^n]$ is not torsion crystalline with Hodge-Tate weights in $[0,p-1]$ immediately follows from the theory of ramification bound as below. We may suppose $n=1$. Suppose $E_{\pi}[p]$ is torsion crystalline with Hodge-Tate weights in $[0,p-1]$. Then the upper numbering $j$-th ramification subgroup $G^j_K$ of $G_K$ (in the sense of [@Se]) acts trivially on $E_{\pi}[p]$ for any $j>1$ ([@Ab1 Section 6, Theorem 3.1]). However, this contradicts the fact that the upper bound of the ramification of $E_{\pi}[p]$ is $1+1/(p-1)$.
We may suppose $n=1$. We choose any uniformizer $\pi'$ of $K$. Putting $v_K(q)=m$, we can write $q=(\pi')^mx$ with some unit $x$ of the integer ring of $K$. Since $m$ is prime to $p$, we have a decomposition $x=\zeta_{\ell}y^m$ in $K^{\times}$ for some $\ell>0$ prime to $p$ and $y\in K$ with $v_K(y-1)>0$. Here $\zeta_{\ell}$ is a (not necessary primitive) $\ell$-th root of unity. Since $\ell$ is prime to $p$, we have $\zeta_{\ell}=\zeta^{ps}_{\ell}$ for some integer $s$. We put $\pi=\pi'y$. This is a uniformizer of $K$. Choose any $p$-th root $\pi_1$ of $\pi$ and put $q_1=\zeta^s_{\ell}\pi^m_1\in K(\pi_1)^{\times}$. Then we have $q=q^p_1\in (K(\pi_1)^{\times})^p$ and in particular, the exact sequence $(\#)$ (for $n=1$) splits as representations of ${\mathrm}{Gal}(\overline{K}/K(\pi_1))$. Now assume that $E_q[p]$ is torsion crystalline with Hodge-Tate weights in $[0,r]$. Then $(\#)$ (for $n=1$) splits as representations of $G_K$ by Corollary \[FFTHMtorcris\]. This contradicts the assumption $p\nmid v_K(q)$ (and hence $q\notin (K^{\times})^p$).
Now we put $r'_0={\mathrm}{min}\{r\in {\mathbb}{Z}_{\ge 0} ; e(r-1)\ge p-1\}$. Recall that we have $[K^{{\mathrm}{ur}}(\mu_p):K^{{\mathrm}{ur}}]=(p-1)/{\mathrm}{gcd}(e,p-1)$.
\[roughbound\] Let $K$ be a finite extension of ${\mathbb}{Q}_p$. Then $E_q[p]$ is torsion crystalline with Hodge-Tate weights in $[0,1+(p-1)/{\mathrm}{gcd}(e,p-1)]$.
Taking a finite unramified extension $K'$ of $K$ such that $[K^{{\mathrm}{ur}}(\mu_p):K^{{\mathrm}{ur}}]=[K'(\mu_p):K']$, we obtain $w_c((E_q[p])|_{G_{K'}})\le 1+(p-1)/{\mathrm}{gcd}(e,p-1)$ by Lemma \[2liftlem\]. Thus we have $w_c(E_q[p])\le 1+(p-1)/{\mathrm}{gcd}(e,p-1)$ by Proposition \[poly\] (1).
\[wctr1\] Suppose that $K$ is a finite extension of ${\mathbb}{Q}_p$, and also suppose $e\mid (p-1)$ or $(p-1)\mid e$. We further suppose that $p\nmid v_K(q)$. Then we have $w_c(E_q[p])=r'_0$.
We have $w_c(E_q[p])\le r'_0$ by Lemma \[roughbound\]. In addition, we also have $w_c(E_q[p])\ge r'_0$ by Proposition \[Tatecurve\].
Lemma \[roughbound\] gives some non-fullness results on torsion crystalline representations.
\[nonfullthm\] Suppose that $K$ is a finite extension of ${\mathbb}{Q}_p$. If $r\ge 1+(p-1)/{\mathrm}{gcd}(e,p-1)$, then the restriction functor ${\mathrm}{Rep}^{r, {\mathrm}{cris}}_{{\mathrm}{tor}}(G_K)\to
{\mathrm}{Rep}_{{\mathrm}{tor}}(G_1)$ is not full.
Two representations $E_{\pi}[p]$ and ${\mathbb}{F}_p(1)\oplus {\mathbb}{F}_p$ are objects of ${\mathrm}{Rep}^r_{{\mathrm}{tor}}(G_K)$ by Lemma \[roughbound\]. They are not isomorphic as representations of $G_K$ but isomorphic as representations of $G_1$. Thus the desired non-fullness follows.
\[p2\] Suppose that any one of the following holds:
- $p=2$ and $K$ is a finite extension of ${\mathbb}{Q}_2$ $($in this case $r'_0=2)$;
- $K$ is a finite unramified extension of ${\mathbb}{Q}_p$ $($in this case $r'_0=p)$;
- $K$ is a finite extension of ${\mathbb}{Q}_p(\mu_p)$ $($in this case $r'_0=2)$.
Then the restriction functor ${\mathrm}{Rep}^{r, {\mathrm}{cris}}_{{\mathrm}{tor}}(G_K)\to
{\mathrm}{Rep}_{{\mathrm}{tor}}(G_1)$ is not full.
[1000]{}
Victor Abrashkin, *Modular representations of the Galois group of a local field and a generalization of a conjecture of Shafarevich*, Izv. Akad. Nauk SSSR Ser. Mat. [**53**]{} (1989), no. 6, 1135–1182 (Russian), Math. USSR-Izv. [**35**]{}, no. 3, 469–518 (English).
Victor Abrashkin, *Group schemes of period $p>2$*, Proc. Lond. Math. Soc. (3), [**101**]{} (2010), 207–259.
Christophe Breuil, *Représentations [$p$]{}-adiques semi-stables et transversalité de Griffiths*, Math. Ann. [**307**]{} (1997), 191–224.
Christophe Breuil, *Une application du corps des normes*, Compos. Math. [**117**]{} (1999) 189–203.
Christophe Breuil, *Integral $p$-adic Hodge theory*, in Algebraic geometry 2000, Azumino (Hotaka), Adv. Stud. Pure Math., vol. 36, Math. Soc. Japan, 2002, 51–80.
Xavier Caruso, [Représentations semi-stables de torsion dans le case $er < p-1$]{}, J. Reine Angew. Math. [**594**]{} (2006), 35–92.
Xavier Caruso and Tong Liu, *Quasi-semi-stable representations*, Bull. Soc. Math. France, [**137**]{} (2009), no. 2, 185–223.
Xavier Caruso and Tong Liu, *Some bounds for ramification of $p^n$-torsion semi-stable representations*, J. Algebra, [**325**]{} (2011), 70–96.
Xavier Caruso and David Savitt, *Polygons de Hodge, de Newton et de línertie moderee des representations semi-stables*, Math. Ann. [**343**]{} (2009): 773–789.
Jean-Marc Fontaine, *Scémas propres et lisses sur ${\mathbb}{Z}$*, Proceedings of the Indo-French Conference on Geometry Bombay, 1989, (Hindustan Book Agency 1993), 43–56.
Jean-Marc Fontaine, *Le corps des périodes $p$-adiques*, Astérisque (1994), no. 223, 59–111, With an appendix by Pierre Colmez, Périodes $p$-adiques (Bures-sur-Yvette, 1988).
Toby Gee, Tong Liu and David Savitt, *The Buzzard-Diamond-Jarvis conjecture for unitary groups*, J. Amer. Math. Soc. [**27**]{} (2014), 389-435.
Mark Kisin, *Crystalline representations and [$F$]{}-crystals*, Algebraic geometry and number theory, Progr. Math. [**253**]{}, Birkhäuser Boston, Boston, MA (2006), 459–496.
Wansu Kim, *The classification of $p$-divisible groups over 2-adic discrete valuation rings*, Math. Res. Lett. [**19**]{} (2012), no. 1, 121–141.
Eike Lau, *A relation between Dieudonné displays and crystalline Dieudonné theory*, preprint, arXiv:1006.2720v3
Tong Liu, *Torsion $p$-adic Galois representations and a conjecture of Fontaine*, Ann. Sci. École Norm. Sup. (4) [**40**]{} (2007), no. 4, 633–674.
Tong Liu, *A note on lattices in semi-stable representations*, Math. Ann. [**346**]{} (2010), 117–138.
Tong Liu, *Lattices in filtered $({\varphi},N)$-modules*, J. Inst. Math. Jussieu 2, Volume 11, Issue 03 (2012), 659-693.
Tong Liu, *The correspondence between Barsotti-Tate groups and Kisin modules when $p=2$*, appear at Journal de Théroie des Nombres de Bordeaux.
Daniel Quillen, *Higher algebraic $K$-theory: I*, in Algebraic $K$-theory, I: Higher $K$-theories (Seattle, 1972), Lecture Notes in Math. [**341**]{}, Springer-Verlag, New York, 1973, 85–147.
Jean-Pierre Serre, *Corps locaux*, Hermann, Paris, 1968.
Yoshiyasu Ozeki, *Torsion representations arising from $({\varphi},\hat{G})$-modules*, J. Number Theory [**133**]{} (2013), 3810–3861.
Yoshiyasu Ozeki, *Full faithfulness theorem for torsion crystalline representations*, preprint, arXiv:1206.4751v4
[^1]: Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, JAPAN. e-mail: [yozeki@kurims.kyoto-u.ac.jp]{} Supported by the JSPS Fellowships for Young Scientists.
[^2]: This continuity condition is needed to assure that the $G_{L,s}$-action on $\hat{T}_{L,s}(\hat{{\mathfrak{M}}})$, defined after Definition \[Liumod\], is continuous (here, note that $\hat{T}_{L,s}(\hat{{\mathfrak{M}}})$ is the dual of $(W({\mathrm}{Fr}R)\otimes_{{\varphi}, {\mathfrak{S}}_{L,s}} {\mathfrak{M}})^{{\varphi}=1}$; see Corollary 3.12 of [@Oz1]).
[^3]: As explained in the footnote of Definition \[Liumod\], we may ignore the continuity condition of (2) in the case where ${\mathfrak{M}}$ is torsion.
|
---
abstract: 'We investigate the effect of strain-induced gauge fields on statistical distribution of energy levels of triangular graphene nanoflakes with zigzag edges. In the absence of strain fields but in the presence of weak potential disorder such systems were found in Ref. [@Ryc12] to display the spectral statistics of the Gaussian unitary ensemble (GUE) due to the effective time-reversal ([*symplectic*]{}) symmetry breaking. Here show that, in the absence of disorder, strain fields may solely lead to spectral fluctuations of GUE providing a nanoflake is deformed such that all its geometric symmetries are broken. In a particular case when a single mirror symmetry is preserved the spectral statistics follow the Gaussian orthogonal ensemble (GOE) rather then GUE. The corresponding transitions to quantum chaos are rationalized by means of additive random-matrix models and the analogy between strain-induced gauge fields and real magnetic fields is discussed.'
author:
- Adam Rycerz
title: ' Strain-induced transitions to quantum chaos and effective time-reversal symmetry breaking in triangular graphene nanoflakes '
---
=1
Introduction
============
Graphene, a two-dimensional form of carbon, provides an intriguing condensed-matter analogue of $(2\!+\!1)$ dimensional quantum electrodynamics [@Sem84]. Several unique features of graphene, not found in conventional metals, semimetals, and insulators, follow from the effective description for low-energy excitations given by the Dirac equation for spin-$1/2$ fermions with zero rest mass [@Cas09; @Gei09; @Das11]. Unlike a massless neutrino described by a similar Weyl equation [@Abe04], each effective quasiparticle in graphene has the electric charge $-e$ and is coupled to external electromagnetic fields via scalar and vector potential terms [@Bee08], offering several ways to tune the quantum states in order to control graphene-based electronic devices. Additionally, the dynamics of carriers in graphene can be affected by mechanical deformations inducing effective gauge fields [@Cas09; @Suz02; @Voz10]. As strains exceeding $10\%$ can be applied to graphene in a reversible manner [@Lee08], this new way of controlling the electronic structure has attracted significant theoretical [@Fog08; @Per09; @Gui10a; @Cho10; @Gui10b; @Jua11] and experimental [@Boo08; @Bun08; @Bao09; @Hua09; @Moh09; @Lev10] attention. Remarkably, the Landau quantization signalling the presence of strain-induced pseudomagnetic fields greater than $300\,$T was recently demonstrated [@Lev10]. For the opposite limit of weak strain fields, the existence of a zero magnetic field analogue of the Aharonov-Bohm effect in graphene [@Sch12] is predicted theoretically [@Jua11].
Surprisingly, the influence of strain fields on quantum chaotic behavior of electrons confined in graphene quantum dots [@Pon08] has not been discussed so far. Quantum chaotic behavior appears generically for systems, whose classical dynamics are chaotic, and manifests itself via the fact that energy levels show statistical fluctuations following those of Gaussian ensembles of random matrices [@Haa10]. In particular, if such a system posses the time-reversal symmetry (TRS), its spectral statistics follow the Gaussian orthogonal ensemble (GOE). A system with TRS and half-integer spin has the symplectic symmetry and, in turn, shows spectral fluctuations of the Gaussian symplectic ensemble (GSE). If TRS is broken, as in the presence of nontrivial gauge fields, and the system has no other antiunitary symmetry [@Rob86], spectral statistics follow the Gaussian unitary ensemble (GUE). For a particular case of massless spin-$1/2$ particles, it was pointed out by Berry and Mondragon [@Ber87], that the confinement may break TRS in a persistent manner (i.e., even in the absence of gauge fields), leading to the spectral fluctuations of GUE [@XNi12].
When applying the above symmetry classification to graphene nanosystems [@Ryc12; @Wur09] one needs, however, to take into account that Dirac fermions in graphene appear in the two valleys, $K$ and $K'$, coupled by TRS [@valefoo]. If the valley pseudospin is conserved, a special (symplectic) time-reversal symmetry (STRS) becomes relevant, playing a role of an effective TRS in a single valley [@Wur09]. Both real magnetic and strain-induced gauge fields may break STRS leading to the spectral fluctuations of GUE. As demonstrated numerically in Ref. [@Ryc12], such fluctuations also appear for particular [*closed*]{} nanosystems in graphene in the presence of random scalar potentials slowly varying on the scale of atomic separation. Such nanosystems include equilateral triangles with zigzag or Klein edges, i.e., with terminal atoms belonging to one sublattice. Generic graphene nanoflakes with irregular edges show spectral fluctuations of GOE [@Wur09; @Lib09], as strong intervalley scattering restores TRS in the absence of gauge fields. In contrast, the boundary effects are suppressed in [*open*]{} graphene systems, for which signatures of the symplectic symmetry class were reported [@Tik08].
In this paper, we analyze numerically the spectral statistics of triangular graphene nanoflakes with zigzag edges bent in-plane according to the strain geometry proposed by Guinea [*at al.*]{} [@Gui10b] (see Fig. \[strageo\]). In the absence of strain fields and the disorder, such systems were found in Ref. [@Ryc12] to show the Poissonian distribution of energy levels, which gradually evolves towards GUE when increasing the disorder strength indicating the transition to quantum chaos. Although the experimental energy resolution seems not sufficient as yet to discuss spectral statistics, regular graphene nanoflakes, including triangular ones bounded entirely with zigzag edges, were recently fabricated on metallic substrates [@Ham11]. Also, strain-induced gauge fields were shown to affect quantum states of similar nanosystems [@Lev10]. Here, we demonstrate numerically, that experimentally realistic strains may lead to clear signatures of quantum chaos even in a nanosystem which is otherwise perfect, i.e., not subjected to substrate-induced disorder, atomic-scale defects, etc. We also discuss the role of a particular strain geometry applied, which may lead to the [*true*]{} or [*false*]{} STRS breaking, in a similar way as the geometry of chaotic Schrödinger systems leads to the true or false TRS breaking in the presence of magnetic fields [@Rob86].
The paper in organized as follows. In Sec. \[strainga\], we briefly discuss how magnetic fields and geometric strains contribute to the effective Dirac Hamiltonian for low-energy spectrum of graphene. In Sec. \[specsta\], we present our main results concerning the statistical distribution of energy levels of triangular graphene nanoflakes with zigzag edges. A quantitative comparison of the effect of uniform magnetic field and the effects of different strain-induced fields on such a distribution is given in Sec. \[reremag\]. The conclusions are given in Section \[conclu\].
![\[strageo\] Systems studied numerically in the paper. Bottom left: Triangular graphene nanoflake with zigzag edges characterized by the height $H$. Remaining plots: The same system bent in-plain employing the strain geometry proposed by Guinea [*et al.*]{} [@Gui10b] (top left) in the variant breaking all geometric symmetries ([*ac-strain*]{}), and (bottom right) in the variant preserving the mirror symmetry ([*zz-strain*]{}). The radii of arcs limiting the flake area are $R\pm{}H/\sqrt{3}$ for ac-strain or $R\pm{}H/2$ for zz-strain. The ratio $H/R=2$ in both cases. ](strageom){width="0.9\linewidth"}
![\[weakstr\] Strain-induced gauge fields appearing on the edges of triangular nanoflake for in-plane deformations of Fig. \[strageo\] in the limit of $R\gg{}H$ (schematic). Red \[blue\] arrows are for $\tilde{A}_x>0$ \[$\tilde{A}_x<0$\]. Left: ac-strain leaves only to the approximate $S_x$ symmetry, which is broken by a small $\tilde{A}_y$ term; see Eq. (\[aaprop\]). Right: zz-strain breaks both ${\cal T}_{\rm sl}$ and $S_x$, but it leaves the exact invariance under the reflection $\mbox{\boldmath$\tilde{A}$}\rightarrow{}-\!\mbox{\boldmath$\tilde{A}$}$ combined with $S_x$, corresponding to the antiunitary symmetry ${\cal T}_0S_x$ ([*false*]{} symplectic time-reversal symmetry breaking). ](weakstra){width="0.9\linewidth"}
Strain-induced gauge fields and Dirac fermions in graphene \[strainga\]
=========================================================================
We start from the tight-binding Hamiltonian, including the nearest-neighbor hopping-matrix elements between $\pi$ orbitals on a honeycomb lattice [@Cas09; @dilafoo] $$\begin{gathered}
\label{hamtba}
{\cal H}_{\rm TB}= -t_0\sum_{\langle{}ij\rangle}
\left(1-\beta\frac{\delta{}d_{ij}}{d_0}\right) \\
\times
\left[ \exp\left(i\frac{2\pi}{\Phi_0}\int_i^j{\bf A}\cdot{}d{\bf l}\right)
|i\rangle\langle{}j| + {\rm h.c.} \right],\end{gathered}$$ where $\delta{}d_{ij}/d_0$ is the relative change in bond length, with $d_0=a/\sqrt{3}$ being the equilibrium bond length defined via the lattice spacing in graphene $a=0.246\,$nm, and ${\bf A}$ is the vector potential related to the real magnetic field by ${\bf B}=\mbox{rot}{\bf A}$ and incorporated as a Peierls phase (with the flux quantum $\Phi_0=h/e$). Remaining parameters are the equilibrium hopping integral $t_0\simeq{}3\,$eV and the dimensionless electron-phonon coupling $\beta=-\partial\log(t)/\partial\log(d)|_{d=d_0}\simeq{}2-3$ [@Voz10].
The effective Hamiltonian and its symmetries
--------------------------------------------
It can be shown, that for low-energy excitations, low magnetic fields, and deformations slowly varying on the scale of atomic separation $a$, ${\cal H}_{\rm TB}$ (\[hamtba\]) reduces to the effective Dirac Hamiltonian [@Sem84; @Suz02; @Voz10] $$\label{hameff}
{\cal H}_{\rm eff}=
v_F\tau_0\mbox{\boldmath$\sigma$}\cdot
\left(\mbox{\boldmath$p$}\!+\!e{\bf A}\right)
- v_F\tau_z\mbox{\boldmath$\sigma$}\cdot\mbox{\boldmath$\tilde{A}$},
% {\cal H}_{\rm Dirac}=
% v_F\sigma_x\!\otimes\!\left[ \tau_z (p_x\!+\!e{\rm A}_x)
% - \tau_0\tilde{A}_x \right] \\
% +v_F\sigma_y\!\otimes\!\left[ \tau_0 (p_y\!+\!e{\rm A}_y)
% - \tau_z\tilde{A}_y \right],$$ where $v_F=3t_0d_0/(2\hbar)\simeq{}10^6\,$m/s is the energy-independent Fermi velocity, $\mbox{\boldmath$\sigma$}=(\sigma_x,\sigma_y)$, $\sigma_i$ and $\tau_i$ ($i=1,2,3$) are the Pauli matrices acting on sublattice and valley degrees of freedom (respectively), $\sigma_0$ ($\tau_0$) denotes the unit matrix, and $\mbox{\boldmath$p$}=-i\hbar(\partial_x,\partial_y)$ is the in-plane momentum operator. ${\cal H}_{\rm eff}$ (\[hameff\]) is written in the so-called valley-isotropic representation [@Bee08]; namely, it acts on spinors $\psi\equiv[\psi_A,\psi_B,-\psi_B',-\psi_A']^T$, where $A/B$ is the the sublattice index, the primed and unprimed entries correspond to $K$ and $K'$ valleys. The strain-induced gauge fields can be written as $$\label{tildeaa}
\mbox{\boldmath$\tilde{A}$}\equiv
\left(\begin{array}{c} \tilde{A}_x \\ \tilde{A}_y \end{array}\right)=
\frac{c\beta}{d_0}
\left(\begin{array}{c} u_{xx}-u_{yy} \\ -2u_{xy} \end{array}\right),$$ where $c$ is a dimensionless coefficient of the order of unity, $u_{ij}=\frac{1}{2}(\partial_iu_j+\partial_ju_i)$ ($i,j=1,2$) is the strain tensor for in-plane deformations [@Lan59], and we have chosen the coordinate system $(x,y)$ such that the $x$ axis corresponds to a zigzag direction of a honeycomb lattice.
In the absence of gauge fields (${\bf A}=\mbox{\boldmath$\tilde{A}$}=0$), the Hamiltonian (\[hameff\]) is invariant upon the antiunitary operations ${\cal T}_k^{-1}{\cal H}_{\rm eff}{\cal T}_k$ with [@Wur11] $${\cal T}_k= \sigma_y\otimes\tau_k{\cal C},\ \ \ \ k=0,1,2,$$ where ${\cal C}$ denotes complex conjugation. We notice that ${\cal T}_y^2=1$, and thus ${\cal T}_y$ represents the true time reversal coupling the two valleys; whereas ${\cal T}_0^2={\cal T}_x^2=-1$, leading to the Kramer’s degeneracy of the two valleys and to the additional Kramer’s degeneracy in each valley. It is easy to see that real magnetic fields ${\bf A}\neq{}0$ break all the symmetries associated with ${\cal T}_k$-s. To the contrary, introducing the strain fields $\mbox{\boldmath$\tilde{A}$}\neq{}0$ and keeping ${\bf A}=0$, one may only break the invariance under the symplectic time reversal ${\cal T}_0$ (STRS), leaving TRS and the invariance under the valley exchange ${\cal T}_x$ unaffected [@edgefoo].
The above discussion is complete, providing we ignore noncollinear local magnetization (which may appear on the edges of a graphene nanoflake [@Sep10; @Pot12]), so one can assume that the boundary condition to the effective Dirac equation ${\cal H}_{\rm eff}\psi=E\psi$ preserves TRS [@Bee08]. Recent first-principle study by Potasz [*et al.*]{} [@Pot12] shows, that local magnetization decays relatively fast when small graphene flake is charged out of the neutrality point. Although the extrapolation of this result onto much larger flakes as considered here may require some further analysis, it seems natural to expect, that the magnetization does not affect statistical properties of energy levels significantly [@magnfoo]. Such an assumption if further supported by the fact, that no disambiguous effects of edge magnetization were observed experimentally even for the lowest-lying energy levels of graphene nanoflakes so far. For these reasons, the effects of possible edge magnetization are neglected in the remaining parts of the paper.
Deformations considered in the paper
------------------------------------
We further limit our discussion to the deformations earlier considered in Ref. [@Gui10b], namely $$\label{acstra}
\left(\begin{array}{c} u_x \\ u_y \end{array}\right)_{\rm ac} =
\left[\begin{array}{c}
(x-R)\cos\theta_{\rm ac}(y)+R \\
(R-x)\sin\theta_{\rm ac}(y)
\end{array}\right]$$ with $\theta_{\rm ac}(y)=\left(2y/H\right)\arcsin\left[H/(2R)\right]$, and $$\label{zzstra}
\left(\begin{array}{c} u_x \\ u_y \end{array}\right)_{\rm zz} =
\left[\begin{array}{c}
(R-y)\sin\theta_{\rm zz}(x) \\
(y-R)\cos\theta_{\rm zz}(x)+R
\end{array}\right]$$ with $\theta_{\rm zz}(x)=\left(\sqrt{3}\,x/H\right)\arcsin\left[H/(\sqrt{3}\,R)\right]$. $R$ is the bending radius of the deformation applied to equilateral triangle of the height $H$; see Fig. \[strageo\]. The labels ’ac’ and ’zz’ indicate that the strain following from Eq. (\[acstra\]), hereinafter referred as [*ac-strain*]{}, predominantly affects the bonds oriented along an armchair direction, whereas the strain following from Eq. (\[zzstra\]) ([*zz-strain*]{}) predominantly affects the bonds oriented along a zigzag direction of a honeycomb lattice. The maximal strain in the triangle area is $$\label{maxdelij}
\mbox{max}\left(\frac{\delta{}d_{ij}}{d_0}\right) \simeq
% \begin{cases}
% H/(\sqrt{3}\,R) & \text{for \ ac-strain}, \\
% H/(2R) & \text{for \ zz-strain}.
% \end{cases}
\frac{H}{R}\times
\begin{cases}
\frac{1}{3}\sqrt{3} & \text{for ac-strain}, \\
\frac{1}{2} & \text{for zz-strain}.
\end{cases}$$
We notice, that ac-strain breaks all geometric symmetries of the triangular nanoflake for any $H/R>0$, while zz-strain preserves the symmetry with respect to a mirror reflection $S_x:\ (x,y)\rightarrow(-x,y)$. Such symmetries of the deformations map onto the symmetries of effective gauge fields $\mbox{\boldmath$\tilde{A}$}$ in the Hamiltonian (\[hameff\]). Keeping only the terms of the order of $R^{-1}$ and $R^{-2}$ one can express the strain fields following from Eqs. (\[acstra\]) and (\[zzstra\]) for $R\gtrsim{}H$ as $$\label{aaprop}
\mbox{\boldmath$\tilde{A}$}_{\rm ac}\propto
\left(\begin{array}{c} R^{-1}x \\ R^{-2}xy \end{array}\right)
\ \ \text{and}\ \
\mbox{\boldmath$\tilde{A}$}_{\rm zz}\propto
\left(\begin{array}{c} -R^{-1}y \\ R^{-2}xy \end{array}\right).$$ In Fig. \[weakstr\], we plot $\mbox{\boldmath$\tilde{A}$}_{\rm ac}$ and $\mbox{\boldmath$\tilde{A}$}_{\rm zz}$ on the flake edges only, as the fields inside the flake smoothly interpolates between extreme values which are reached at the system boundaries. It is clear that ac-strain breaks the symmetry with respect to $S_x$ (depicted with dashed vertical lines) due to the term $(\tilde{A}_y)_{\rm ac}\propto{}R^{-2}$. For zz-strain, $S_x$ symmetry is also broken by the term $(\tilde{A}_x)_{\rm zz}\propto{}R^{-1}$. Both strain distributions break STRS, as they are not invariant under the transformation $\mbox{\boldmath$\tilde{A}$}\rightarrow -\mbox{\boldmath$\tilde{A}$}$. However, $\mbox{\boldmath$\tilde{A}$}_{\rm zz}$ exhibits the symmetry under the antiunitary operation ${\cal T}_0S_x$ (with $({\cal T}_0S_x)^2=-1$). In the absence of intervalley scattering and in the limit of quantum chaos, such a symmetry shall lead to false STRS breaking and spectral fluctuations following GOE rather then GUE [@Ber87; @Rob86].
Spectral statistics \[specsta\]
=================================
This section presents the central results of the paper, concerning statistical distribution of energy levels for Dirac fermions confined in graphene nanoflakes, in the presence of gauge fields introduced in Sec. \[strainga\]. For the numerical illustration, we took an equilateral triangle with zigzag edges containing $N_{\rm C}=32758$ carbon atoms, corresponding to $H=270\,d_0$ and the physical sample area of ${\cal A}_{\rm S}\simeq{}(29\,\mbox{nm})^2$. Such a system was previously found to show negligibly weak intervalley scattering in the absence of magnetic field, as well as spectral fluctuations following GUE in the presence of disorder smoothly varying on the length scale of $a$ (see Ref. [@Ryc12]).
![\[levrep\] Evolution of energy levels for a triangular nanoflake containing $N_{\rm C}=32758$ carbon atoms with varying magnetic field (a), ac-strain (b), and zz-strain (c). ](levrepFR){width="\linewidth"}
The effects of magnetic and strain fields
-----------------------------------------
In Fig. \[levrep\], we plot energy levels of our model system obtained by numerical diagonalization of ${\cal H}_{\rm TB}$ (\[hamtba\]) as functions of the magnetic flux $\Phi={\cal A}_{\rm S}B$ \[Fig. \[levrep\](a)\], and the parameter $H/R$ characterizing ac-strain or zz-strain \[Figs. \[levrep\](b) or \[levrep\](c)\]. We limit ourselves to extreme cases of uniform magnetic field applied in the absence of geometric deformations, and two distinct deformations given by Eqs. (\[acstra\]) and (\[zzstra\]) in the absence of magnetic fields, as intermediate situations simply combine the features of these extreme cases. First, it is clear from Fig. \[levrep\](a), that uniform magnetic field splits the valley degeneracy of electronic levels [@Rec07] and leads to level crossings characteristic for integrable systems. To the contrary, both ac-strain and zz-strain preserve the valley degeneracy and lead to avoided crossings \[see Figs. \[levrep\](b) and \[levrep\](c)\] characteristic for chaotic quantum systems [@Haa10]. A secondary difference between the effects of ac-strain and zz-strain on the electronic structure of the system considered is related to the fact, that different deformations lead to different pseudomagnetic fields. Namely, Eq. (\[aaprop\]) leads to $$\label{bsprop}
B_{{\rm s},\xi}=\hat{\bf e}_z\cdot\mbox{rot}\mbox{\boldmath$\tilde{A}$}_{\xi}\propto
\frac{H}{R}\left(\eta_{\xi}+\frac{y}{R}\right),$$ with $\eta_\xi=0$ if $\xi={\rm ac}$ or $\eta_\xi=1$ if $\xi={\rm zz}$. It is clear from Eq. (\[bsprop\]), that the strain-induced Landau quantization may be observed in small systems [@delnfoo] for zz-strain only (see also Ref. [@Gui10b]). As this issue is beyond the scope of the paper, we only notice that the systematic drift of lowest-lying electronic levels towards the zero-energy Landau level is totally absent in Fig. \[levrep\](b) and visible in Fig. \[levrep\](c). For this reason, the arrangement of ac-strain seems particularly interesting when studying strained graphene nanosystems. In principle, ac-strain may allow one to discuss physical effects of strain-induced gauge fields in the absence of pseudomagnetic fields, in analogy to the Aharonov-Bohm effect appearing for real gauge fields in the absence of magnetic fields, providing the system geometry is modified appropriately [@Sch12].
![\[csfirh\] Integrated level-spacing distributions $C^{(1)}(S)$ for the same system as in Fig. \[levrep\], in the presence of uniform magnetic field (a)–(c) or ac-strain (d)–(f). The values of total flux $\Phi$ or the strain parameter $H/R$ are specified for each panel. Insets show nearest-neighbor spacings distributions $P^{(1)}(S)$. Numerical results are shown with black solid lines. Remaining lines are for Poisson distribution with the degeneracy $g=2$ of each level \[blue dashed\], with $g=1$ \[blue dotted\], and for the Wigner surmise for GUE \[blue dash-dotted\]. ](csfirh3){width="\linewidth"}
![\[psfirh\] Second-neighbor level-spacing distributions $P^{(2)}(S)$ for the same system in same physical situations as in Fig. \[csfirh\]. Red solid lines show the best-fitted approximating distributions $P_{\rm Poi-2xPoi}(\lambda;S)$ (\[pspoi2poi\]) \[panels (b), (c)\] or $P_{\rm Poi-GUE}(\lambda_{\rm fit};S)$ (\[pspoigue\]) \[panels (e), (f)\] with $\lambda_{\rm fit}$ specified for each plot. Remaining lines are same as in Fig. \[csfirh\]. ](psfirh3){width="\linewidth"}
![\[psrhab\] Level-spacing distributions $P^{(1)}(S)$ (a)–(c) and $P^{(2)}(S)$ (d)–(f) for the same system as in Figs. \[levrep\]–\[psfirh\] in the presence of zz-strain (with the parameter $H/R$ specified for each panel). Numerical results are shown with black solid lines. Remaining lines are for Poisson distribution with the degeneracy $g=2$ of each level \[blue dashed\], the Wigner surmise for GOE \[blue dotted\], or for the best-fitted approximating distributions $P_{\rm Poi-GOE}(\lambda_{\rm fit};S)$ (\[pspoigoe\]) \[red solid lines on panels (e), (f)\] with $\lambda_{\rm fit}$ specified for each plot. ](psrhabtri3){width="\linewidth"}
The above-mentioned basic signatures of quantum chaos, accompanying strained-induced gauge fields, are further supported with spectral statistics presented in Figs. \[csfirh\]–\[del3th\].
Level-spacing distributions
---------------------------
First, we discuss the level-spacing distributions $P^{(1)}(S)$ and their integrals $C^{(1)}(S)\equiv\int_0^SP^{(1)}(S')dS'$ (see Fig. \[csfirh\]), with $P^{(k)}(S)$ ($k=1,2,\dots$) being the probability that the quantity $\langle\rho(E)\rangle(E_{n+k}\!-\!E_n)/k$ is located in the interval $(S,S\!+\!dS)$. $E_{n+k}\!-\!E_n$ is the distance between $k$-th neighbors in the level sequence $E_1\leqslant{}E_2\leqslant\dots$, and $\langle\rho(E)\rangle$ is the average density of levels in the energy interval $(E,E+dE)$, which can be approximated by $\langle\rho(E)\rangle\simeq{\pi}^{-1}{\cal A}_{\rm S}|E|/(\hbar{}v_F)^2$ for $|E|\ll{}t_0$. The numerical results, shown with black solid lines in Fig. \[csfirh\], are obtained for about $800$ energy levels $0<E_n<0.5\,t_0$ [@unfofoo]. The theoretical curves (blue lines) are given by $$\label{psxg12}
P_{X,g}^{(1)}(S)=\begin{cases}
P_X(S), & \text{if } g=1, \\
\frac{1}{2}\delta(S)+\frac{1}{4}P_X(S/2), & \text{if } g=2,
\end{cases}$$ with $g=1,2$ the level degeneracy, and $$\label{psxwig}
P_X(S)=\begin{cases}
\exp(-S), & \!\!\text{for Poisson}, \\
(\pi/2)S\exp\!\left(-\pi{}S^2\!/4\right), & \!\!\text{for GOE}, \\
(32/\pi^2)S^2\!\exp\!\left(-4S^2\!/\pi\right), & \!\!\text{for GUE},
\end{cases}$$ where we have used the Wigner surmise approximating $P^{(1)}(S)$ for the relevant ensemble of random matrices [@Haa10]. The evolution of spectral statistics $C^{(1)}(S)$ and $P^{(1)}(S)$ with the magnetic field, illustrated in Figs. \[csfirh\](a)–(c), confirms that the system energy spectrum gradually transforms from Poissonian with the twofold valley degeneracy (manifesting itself for $\Phi\simeq{}0$) towards Poissonian without such a degeneracy (approached for $\Phi\gtrsim\Phi_0$), showing no signatures of quantum chaos in the absence of strain fields. The evolution of same statistics with the varying ac-strain and fixed $\Phi=0$ \[Figs. \[csfirh\](d)–(f)\] unveils the spectral fluctuations characteristic for GUE, starting from $H/R\gtrsim{}0.1$.
The corresponding results for the second-neighbor spacing distributions $P^{(2)}(S)$ are presented in Fig. \[psfirh\]. The theoretical expectations depicted with blue lines follow from Eq. (\[psxwig\]) via $$P_{X,g}^{(2)}(S)=\begin{cases}
2\int_0^{2S}dS'P_X(2S\!-\!S')P_X(S'), & \text{if } g=1, \\
P_X(S), & \text{if } g=2.
\end{cases}$$ \[For instance, $P^{(2)}_{{\rm Poi},\,g=1}(S)=4S\exp(-2S)$.\] Additionally, we have utilized additive random-matrix models of the form [@Ryc12; @Zyc93] $$\label{hamlam}
H(\lambda)=\frac{H_0+\lambda{}V}{\sqrt{1+\lambda^2}},
% H(\lambda)=\left(H_0+\lambda{}V\right)/{\color{blue}\sqrt{1+\lambda^2}},$$ allowing one to generate (for $0<\lambda<\infty$) the spacing distributions interpolating between that of two distinct ensembles of random matrices.
Three particular choices of $H_0$ and $V$ lead to:
- A distribution interpolating between Poisson with $g=2$ (reproduced for $\lambda=0$) and Poisson with $g=1$ (reproduced for $\lambda=\infty$) \[see Figs. \[psfirh\](a)–(c)\] $$\label{pspoi2poi}
P_{{\rm Poi-2xPoi}}(\lambda;S)=
\left[\frac{1+a(\lambda)}{1-a(\lambda)}\right]\exp(-S)
% \\ \times
\left\{ \exp[-Sa(\lambda)]
- \exp\left[-\frac{S}{a(\lambda)}\right] \right\}.$$
- A distribution interpolating between Poisson and GOE \[see Fig. \[psrhab\]\] $$\label{pspoigoe}
P_{\rm Poi-GOE}(\lambda;S)=
\left[\frac{u(\lambda)^2S}{\lambda}\right]
\exp\left[{-\frac{u(\lambda)^2S^2}{4\lambda^2}}\right]
% \\ \times
\int_0^\infty\!{d\eta}\,\exp(-\eta^2-2\lambda\eta)
I_0\left[\frac{\eta{u(\lambda)}S}{\lambda}\right].$$
- A distribution interpolating between Poisson and GUE \[see Figs. \[psfirh\](d)–(f)\] $$\label{pspoigue}
P_{\rm Poi-GUE}(\lambda;S)=
\sqrt{\frac{2}{\pi}}\left[\frac{c(\lambda)^2S}{\lambda}\right]
\exp\left[{-\frac{c(\lambda)^2S^2}{2\lambda^2}}\right]
% \\ \times
\int_0^\infty\!\frac{d\eta}{\eta}\,\exp\left(-\lambda\eta-\frac{\eta^2}{2}\right)\sinh\left[\frac{\eta{c(\lambda)}S}{\lambda}\right].$$
The coefficients $a(\lambda)$, $u(\lambda)$, and $c(\lambda)$ are chosen such that $\langle{}S\rangle_X=\int_0^\infty{}SP_X(\lambda;S)dS=1$ for any value of $\lambda$ [@abcfoo]; $I_0(x)$ in Eq. (\[pspoigoe\]) is the modified Bessel function of the first kind.
Formally, Eq. (\[pspoi2poi\]) represents the exact expression for second-neighbor spacing distribution in case $H_0$ is diagonal random matrix with twofold degeneracy of each eigenvalue and $V$ is diagonal random matrix without such a degeneracy. Similarly, Eqs. (\[pspoigoe\]) and (\[pspoigue\]) represent the exact level-spacing distributions for $2\times{}2$ random matrices of the form $H(\lambda)$ (\[hamlam\]), with $H_0$ chosen as diagonal random matrix and $V$ being a member of GOE or GUE (respectively). It was found numerically, however, that such distributions approximate, with a suprising accuracy, the actual level-spacings distributions of large random matrices [@Zyc93] as well as distributions obtained for various dynamic systems undergoing transitions between symmetry classes [@Haa10; @Ber86; @Zyc93].
The distribution $P_{\rm Poi-2xPoi}(\lambda;S)$ (\[pspoi2poi\]), with the best-fitted parameter $\lambda=\lambda_{\rm fit}$, is capable of reproducing the evolution of $P^{(2)}(S)$ when the valley degeneracy is being split by external magnetic field in the absence of strain fields \[see red lines in Figs. \[psfirh\](b) and \[psfirh\](c)\]. Similarly, the distribution $P_{\rm Poi-GUE}(\lambda_{\rm fit};S)$ (\[pspoigue\]) is capable of reproducing $P^{(2)}(S)$ when the system undergoes transition to quantum chaos induced by ac-strain \[see red lines in Figs. \[psfirh\](e) and \[psfirh\](f)\]. In the latter case, $P^{(2)}(S)$ exhibits transition Poisson-GUE analogous to the transition demonstrated numerically in Ref. [@Ryc12] for the similar system with smooth potential disorder.
The evolution of spacing distributions $P^{(1)}(S)$ and $P^{(2)}(S)$ with zz-strain is illustrated in Fig. \[psrhab\]. This time, transition to quantum chaos manifests itself by a systematic crossover of the actual spectral statistics obtained numerically (black solid lines) between the theoretical predictions for Poisson and GOE (blue lines) with the twofold valley degeneracy preserved. Also, the approximating distribution $P_{\rm Poi-GOE}(\lambda_{\rm fit};S)$ (\[pspoigoe\]) is capable of reproducing $P^{(2)}(S)$ during the transition Poisson-GOE \[see red lines in Figs. \[psrhab\](e) and \[psrhab\](f)\].
![\[del3th\] Spectral rigidity $\Delta_3(L)$ for the same system as in Figs. \[levrep\]–\[psrhab\] in the presence of ac-strain (a) or zz-strain (b). Datapoints show the results obtained numerically for different values of the strain parameter $H/R$ (specified for each dataset). Lines are the theoretical expectations for the relevant ensembles of random matrices. ](del3thab){width="\linewidth"}
Spectral rigidity
-----------------
The fluctuations of more distant spacings between energy levels of quantum system can be described in a compact way by the spectral rigidity [@Dys63] $$\label{del3def}
\Delta_3(L)=\frac{1}{L}\Big<
\underset{\scriptsize (a,b)}{\mbox{Min}}
\int_{-L/2}^{L/2}\!dx\left[{\cal N}(x_0\!+\!x)-ax-b\right]^2
\Big>,$$ where $x\equiv\langle{\cal N}(E)\rangle$ and ${\cal N}(E)$ denotes the number of energy levels such that $0<E_n\leqslant{}E$. Theoretical expectations $\Delta_3^{\rm (Poi)}(L)$, $\Delta_3^{\rm (GOE)}(L)$, and $\Delta_3^{\rm (GUE)}(L)$ are given explicitly in Refs. [@Ryc12; @Dys63]. For the case of $g$-fold degeneracy of each energy level one finds immediately from Eq. (\[del3def\]) that the corresponding theoretical expression needs to be rescaled via $\Delta_3^{(g,X)}(L)=g^2\Delta_3^{(X)}(L/g)$.
The numerical values of $\Delta_3(L)$ for our model system are shown in Fig. \[del3th\] together with theoretical expectations for the relevant ensembles of random matrices. Similarly as for the statistics $P^{(1,2)}(S)$ discussed above, the evolution of $\Delta_3(L)$ with ac-strain clearly exhibits transition Poisson-GUE \[see Fig. \[del3th\](a)\] whereas the evolution of $\Delta_3(L)$ with zz-strain exhibits transition Poisson-GOE \[see Fig. \[del3th\](b)\], with the valley degeneracy ($g=2$) preserved in the absence of magnetic field.
![\[psfirhab\] Level-spacing distributions $P^{(1)}(S)$ for the same system as in Figs. \[levrep\]–\[del3th\] in the presence of uniform magnetic field (with the flux $\Phi$ varied between the panels) and persistent ac-strain \[panels (a)–(c)\] or zz-strain \[panels (d)–(f)\], with the strain parameter fixed at $H/R=0.2$ in both cases. Numerical results are shown with black solid lines. Remaining lines are for GUE with the degeneracy $g=2$ of each level \[blue dashed\], or for two independent GUEs \[blue dotted\]. ](psfirhab3){width="\linewidth"}
![\[del3firh\] Same as Fig. \[del3th\] but in the presence uniform magnetic field ($\Phi/\Phi_0$ is specified for each dataset) and persistent ac-strain (a) or zz-strain (b); $H/R=0.2$ in both cases. ](del3firhab){width="\linewidth"}
Systems with persistent strains in external magnetic fields
-----------------------------------------------------------
For a sake of completeness, we consider now the systems of Fig. \[strageo\] with the strain parameter fixed at $H/R=0.2$ (the cases of ac-strain and zz-strain are studied separately) placed in uniform magnetic fields varying in the range $0\leqslant{}\Phi/\Phi_0\leqslant{}4$ \[see Figs. \[psfirhab\] and \[del3firh\]\]. Although the discussion is still limited to the deformations given by Eqs. (\[acstra\]) and (\[zzstra\]), the universal nature of spectral fluctuations in the limit of quantum chaos allows us to believe that the effects which we demonstrate numerically in this subsection may be observable for graphene nanoflakes with persistent (for instance, substrate-induced) strains, such as studied experimentally in Refs. [@Lev10; @Ham11].
The numerical results for level-spacing distributions $P^{(1)}(S)$ are shown in Fig. \[psfirhab\] (black solid lines). The theoretical predictions for GOE and GUE (with the valley degeneracy) are given by Eqs. (\[psxg12\],\[psxwig\]) and drawn with blue dashed lines. The predictions for level sequences following from two statistically-independent GOEs or GUEs (blue dotted lines) are given by expressions derived by Robnik and Berry [@Rob86] $$\label{psd2es}
P^{(1)}_{2\times{}X}(S) = \frac{d^2}{dS^2}\left[ E_{X}(S/2) \right]^2,$$ with $$\label{esxwig}
E_X(S)=\begin{cases}
1-\mbox{erf}\left(S\sqrt{\pi}/2\right), & \text{for GOE}, \\
\exp\left(-4S^2/\pi\right) \\
\ \ \ \ \ \ -S+S\,\mbox{erf}\left(2S/\sqrt{\pi}\right), & \text{for GUE},
\end{cases}$$ being the probability that interval S contains no energy level of a single sequence following GOE or GUE. The error function $\mbox{erf}(x)=(2/\sqrt{\pi})\int_0^x\exp(-t^2)dt$. Hereinafter, we suppose identical densities of energy levels in both components of a combined sequence.
The datasets presented in Figs. \[psfirhab\](a)–(c) illustrates a systematic crossover of the actual spectral statistics $P^{(1)}(S)$ between the theoretical predictions for GUE with the valley degeneracy $g=2$ and two independent GUEs driven by external magnetic field in the presence of persistent ac-strain. Similarly, a field-driven crossover between GOE (with $g=2$) and two independent GOEs is clearly visible in Figs. \[psfirhab\](d)–(f) in the situation with persistent zz-strain. Additionally, both crossovers between the relevant ensembles of random matrices are visualized with the spectral rigidity $\Delta_3(L)$; see respectively Figs. \[del3firh\](a) and \[del3firh\](b) for the cases of persistent ac-strain and persistent zz-strain. \[Notice that theoretical expectations for two independent GOEs or GUEs are $\Delta_3^{(2\times{}X)}(L)=2\Delta_3^{(X)}(L/2)$.\] Remarkably, either $P^{(1)}(S)$ or $\Delta_3(L)$ approaches the theoretical predictions for two independent GOEs (or GUEs) when $\Phi\gtrsim{}\Phi_0$, and the (approximate) valley degeneracy no longer affects the electronic spectrum of the system.
It is worth to stress that, apart from lifting up the valley degeneracy, weak magnetic fields do not alter the symmetry classes of chaotic Dirac systems we consider here. In the absence of intervalley scattering, turning on the magnetic field simply transforms a system into a pair of two independent chaotic systems (one at each valley) each of which is showing the same symmetry class as the original system at zero field: Namely, the unitary if strain fields break STRS (the case of ac-strain of a generic nature) or the orthogonal if a mirror symmetry leads to false STRS breaking (the special case of zz-strain). Chaotic graphene systems with strong intervalley scattering, earlier considered in Refs. [@Wur09; @Lib09], show standard transition GOE-GUE, exhibiting another striking difference between Dirac billiards and generic graphene flakes (with irregular edges). Nevertheless, the former still can be modelled effectively within particular graphene nanoflakes, with terminal atoms belonging predominantly to one sublattice.
![\[lbfirh\] Least-squares fitted parameters $\lambda_{\rm fit}$ for transitions between random ensembles demonstrated numerically in Sec. \[specsta\] \[see Eqs. (\[pspoi2poi\])–(\[pspoigue\])\], as functions of the magnetic flux $\Phi$ \[solid symbols\], or effective fluxes $\Phi_{{\rm eff},\xi}$ \[open symbols\] defined by Eqs. (\[phieffxi\]) and (\[cxivals\]). Dashed line represent the best-fitted power-low relation given by Eq. (\[lbfitsim\]). Inset shows the data for strained nanosystems directly as functions of the strain parameter $H/R$, compared with the data obtained for real magnetic fields, in the log-log scale. ](lbfitfirh){width="0.9\linewidth"}
Relation between strain fields and real magnetic fields \[reremag\]
=====================================================================
We supplement our numerical study of quantum chaos in strained graphene nanoflakes by comparing, in a quantitative manner, the effects of geometric deformations given by Eqs. (\[acstra\]) and (\[zzstra\]) with the effects of real magnetic fields on the spacing distribution $P^{(2)}(S)$.
As discussed in Sec. \[specsta\], a significant pseudomagnetic field *per se*, $B_{\rm s}\propto{}H/R$ (\[bsprop\]), appears for zz-strain only. However, as the actual spacing distributions $P^{(2)}(S)$ can be rationalized by $P_X(\lambda_{\rm fit},S)$ given (respectively) by Eqs. (\[pspoi2poi\]), (\[pspoigoe\]), or (\[pspoigue\]) for the cases of real magnetic field, zz-strain, or ac-strain, the numerical comparison of the best-fitted parameters $\lambda_{\rm fit}$ (provided in Fig. \[lbfirh\]) allows one to define the effective fluxes $$\label{phieffxi}
\frac{\Phi_{{\rm eff},\xi}}{\Phi_0}=c_\xi\frac{H}{R}\ \ \ \
(\text{with }\xi={\rm ac, zz})$$ for either the cases of ac-strain and zz-strain. The coefficients $c_\xi$ are adjusted such that $\lambda_{\rm fit}$ for strained systems \[open symbols in Fig. \[lbfirh\]\] follow the approximating power-law relation \[red dashed lines\] found for the case of real magnetic field $$\label{lbfitsim}
\lambda_{\rm fit}\simeq 0.41(1)\times\left[ \Phi/\Phi_0 \right]^{1.53(7)},$$ with the numerical values of parameters obtained via least-squares fitting (the standard deviation of a last digit are specified by numbers in parenthesis). This leads to $$\label{cxivals}
c_{\rm ac}=10.5(2)\ \ \ \ \text{and}\ \ \ \ c_{\rm zz}=8.8(1).$$ We notice, that the dimensionless parameter $c_{\rm zz}$ given by Eq. (\[cxivals\]) corresponds to $c_{\rm zz}\Phi_0/{\cal A}_{\rm S}\simeq{}43\,$T, what is numerically very close to the value reported by the second paper of Ref. [@Gui10b] for the limit of Landau quantization. Therefore, the relation between uniform magnetic and strain-induced pseudomagnetic fields, following from the statistical description of the evolution of energy levels in weak fields by means of additive random-matrix models, appears to be consistent with the corresponding relation arising from transport properties in strong fields.
Conclusions \[conclu\]
========================
We have discussed the selected spectral statistics of triangular graphene nanoflakes with zigzag edges in the presence of strain-induced gauge fields. Such systems may show the spectral fluctuations following GUE of random matrices, providing the link between chaotic graphene flakes [@Ryc12; @Wur09; @Lib09] and Dirac billiards for massless spin-$1/2$ fermions [@Ber87; @XNi12].
In the absence of disorder, strain fields associated with moderate in-plain deformations drive a highly-symmetric nanosystem into chaotic regime. Moreover, our results show, that the system symmetry class is related to the particular arrangement the deformation: In a generic case of the deformation breaking all geometric symmetries the unitary symmetry class is observed, as the strain field also breaks the effective (symplectic) time reversal symmetry (STRS) in a single valley. To the contrary, if a single mirror symmetry is preserved, we have only the false STRS breaking leading to the orthogonal symmetry class. (Such a physical situation has no analogue in graphene nanoflakes with substrate-induced disorder [@Ryc12] or irregular edges [@Wur09; @Lib09], where all geometric symmetries are naturally broken.) It is worth to stress, that although particular strain arrangements studied here are different from those in existing experiments [@Boo08; @Bun08; @Bao09; @Hua09; @Moh09; @Lev10], the nature of the results allows one to expect, that the system symmetry class remains unitary (orthogonal) for an arbitrary strain arrangement leading to the true (false) STRS breaking, providing the deformations are sufficiently small that the effective Dirac theory for low-energy excitations can be applied.
In both cases of the unitary and the orthogonal symmetry classes, spectral statistics obtained numerically follow those of the relevant ensembles of random matrices (GUE or GOE), with the twofold valley degeneracy of each energy level. When the real magnetic and strain fields are applied simultaneously, the valley degeneracy no longer applies, and our system displays spectral fluctuations following two statistically-independent GUEs (or GOEs), one per each valley.
The evolution of spectral statistics with weak strain fields, exhibiting the transition to quantum chaos, is rationalized using additive random-matrix models. The functional relations between the best-fitted model parameters and the deformation strength allow one to compare the effects of strain fields with the effects of real magnetic fields in a quantitative manner. The resulting numerical relation between uniform magnetic and strain-induced pseudomagnetic fields stays in agreement with the similar relation, earlier obtained for strong fields leading to the appearance of Landau quantization (see the second paper of Ref. [@Gui10b]).
Although our work was primarily motivated by the fabrication of regular graphene nanoflakes [@Ham11], it may also be possible to observe the effects which we describe in artificial graphenes, such as the arrays of GaAs/AlGaAs quantum wells [@Sin11], or in the recently discussed acoustic analog of graphene [@Tor12].
Acknowledgments {#acknowledgments .unnumbered}
===============
I thank to Ari Harju for the correspondence. The work was supported by the National Science Centre of Poland (NCN) via Grant No. N–N202–031440, and partly by Foundation for Polish Science (FNP) under the program TEAM [*“Correlations and coherence in quantum materials and structures (CCQM)”*]{}. Computations were partly performed using the PL-Grid infrastructure.
A. Rycerz, Phys. Rev. B [**85**]{}, 245424 (2012).
G.W. Semenoff, Phys. Rev. Lett. [**53**]{}, 2449 (1984); D.P. DiVincenzo and E.J. Mele, Phys. Rev. B [**29**]{}, 1685 (1984).
A.H. Castro Neto, F. Guinea, N.M.R. Peres, K.S. Novoselov, and A.K. Geim, Rev. Mod. Phys. **81**, 109 (2009).
A.K. Geim, Science **324**, 1530 (2009).
S. Das Sarma, Sh. Adam, E.H. Hwang, and E. Rossi, Rev. Mod. Phys. **83**, 407 (2011).
E.S. Abers, [*Quantum Mechanics*]{}, Pearson Ed. (Addison Wesley, Prentice Hall, 2004).
C.W.J. Beenakker, Rev. Mod. Phys. **80**, 1337 (2008). H. Suzuura and T. Ando, Phys. Rev. B [**65**]{}, 235412 (2002); J.L. Mañes, [*ibid.*]{} [**76**]{}, 045430 (2007).
For a comprehensive review of the topic, see: M.A.H. Vozmediano, M.I. Katsnelson, and F. Guinea, Phys. Rep. [**496**]{}, 109 (2010).
C. Lee, X. Wei, J.W. Kysar, and J. Hone, Science [**321**]{}, 385 (2008).
M.M. Fogler, F. Guinea, and M.I. Katsnelson, Phys. Rev. Lett. [**101**]{}, 226804 (2008).
V.M. Pereira, A.H. Castro Neto, and N.M.R. Peres, Phys. Rev. B [**80**]{}, 045401 (2009).
F. Guinea, M.I. Katsnelson, and A.K. Geim, Nature Phys. [**6**]{}, 30 (2010).
S. Choi, S. Jhi, Y. Son, Phys. Rev. B [**81**]{}, 081407(R) (2010).
F. Guinea, A.K. Geim, M.I. Katsnelson, and K.S. Novoselov, Phys. Rev. B [**81**]{}, 035408 (2010); T. Low and F. Guinea, Nano. Lett. [**10**]{}, 3551 (2010).
F. de Juan, A. Cortijo, M.A.H. Vozmediano, and A. Cano, Nature Phys. [**7**]{}, 811 (2011).
T.J. Booth, P. Blake, R.R. Nair, D. Jiang, E.W. Hill, U. Bangert, A. Bleloch, M. Gass, K.S. Novoselov, M.I. Katsnelson, and A.K. Geim, Nano Lett. [**8**]{}, 2442 (2008).
J.S. Bunch, S.S. Verbridge, J.S. Alden, A.M. van der Zande, J.M. Parpia, H.G. Craighead, and P.L. McEuen, Nano Lett. [**8**]{}, 2458 (2008).
W. Bao, F. Miao, Z. Chen, H. Zhang, W. Jang, C. Dames, and C.N. Lau, Nat. Nanotechnol. [**4**]{}, 562 (2009).
M. Huang, H. Yan, C. Chen, D. Song, T.F. Heinz, and J. Hone, Proc. Natl. Acad. Sci. U.S.A. [**106**]{}, 7304 (2009).
T.M.G. Mohiuddin, A. Lombardo, R.R. Nair, A. Bonetti, G. Savini, R. Jalil, N. Bonini, D.M. Basko, C. Galiotis, N. Marzari, K.S. Novoselov, A.K. Geim, and A.C. Ferrari, Phys. Rev. B [**79**]{}, 205433 (2009).
N. Levy, S.A. Burke,, K.L. Meaker, M. Panlasigui, A. Zettl, F. Guinea, A.H. Castro Neto, and M.F. Crommie, Science [**329**]{}, 544 (2010).
For a recent review, see: J. Schelter, P. Recher, and B. Trauzettel, Solid State Comm. [**152**]{}, 1411 (2012).
L.A. Ponomarenko, F. Schedin, M.I. Katsnelson, R. Yang, E.W. Hill, K.S. Novoselov, and A.K. Geim, Science [**320**]{}, 356 (2008); C. Stampfer, E. Schurtenberger, F. Molitor, J. Güttinger, T. Ihn, and K. Ensslin, Nano Lett. [**8**]{}, 2378 (2008).
F. Haake, [*Quantum Signatures of Chaos*]{}, 3rd ed. (Springer, Berlin, 2010).
M. Robnik and M.V. Berry, J. Phys. A: Math. Gen. [**19**]{}, 669 (1986).
M.V. Berry and R.J. Mondragon, Proc. R. Soc. A [**412**]{}, 53 (1987).
For a recent numerical study of generic Dirac billiards, see: X. Ni, L. Huang, Y.-Ch. Lai, and C. Grebogi, Phys. Rev. E [**86**]{}, 016702 (2012).
J. Wurm, A. Rycerz, I. Adagideli, M. Wimmer, K. Richter, and H.U. Baranger, Phys. Rev. Lett. [**102**]{}, 056806 (2009).
In particular, real magnetic field break TRS and has the same sign in the two valleys, whereas the strain-induced gauge field preserves TRS and has opposite signs in the two valleys, see Ref. [@Voz10].
F. Libisch, C. Stampfer, and J. Burgdörfer, Phys. Rev. B [**79**]{}, 115423 (2009); I. Amanatidis and S.N. Evangelou, Phys. Rev. B [**79**]{}, 205420 (2009); L. Huang, Y.-Ch. Lai, and C. Grebogi, Phys. Rev. E [**81**]{}, 055203(R) (2010); M. Wimmer, A.R. Akhmerov, and F. Guinea, Phys. Rev. B [**82**]{}, 045409 (2010).
F.V. Tikhonenko, D.W. Horsell, R.V. Gorbachev, and A.K. Savchenko, Phys. Rev. Lett. **100**, 056802 (2008); A.N. Pal, V. Kochat, and A. Ghosh [*ibid.*]{} **109**, 196601 (2012).
S.K. Hämäläinen, Z. Sun, M.P. Boneschanscher, A. Uppstu, M. Ijäs, A. Harju, D. Vanmaekelbergh, and P. Liljeroth, Phys. Rev. Lett. [**107**]{}, 236803 (2011); D. Subramaniam, F. Libish, Y. Li, C. Pauly, V. Geringer, R. Reiter, T. Mashoff [*et al.*]{}, [*ibid.*]{} [**108**]{}, 046801 (2012); M. Olle, G. Ceballos, D. Serrate, and P. Gambardella, Nano Lett. [**12**]{}, 4431 (2012).
Hereinafter, we neglect the effects of scalar potential induced by lattice dilatation, see Ref. [@Gui10b]
L.D. Landau and E.M. Lifschitz, [*Theory of Elasticity*]{} (Pergamon Press, Oxford, 1959).
J. Wurm, K. Richter, and I. Adagideli, Phys. Rev. B [**84**]{}, 205421 (2011); J. Wurm, M. Wimmer, and K. Richter, [*ibid.*]{} [**85**]{}, 245418 (2012).
In a closed nanosystem, symmetries associated with ${\cal T}_0$ and ${\cal T}_x$ become relevant if the intervalley scattering due to the boundaries is insignificant, see Refs. [@Ryc12; @Wur11].
M. Sepioni, R.R. Nair, S. Rablen, J. Narayanan, F. Tuna, R. Winpenny, A.K. Geim, and I.V. Grigorieva, Phys. Rev. Lett. [**105**]{}, 207205 (2010).
P. Potasz, A.D. Güçlü, A. Wójs, and P. Hawrylak, Phys. Rev. B [**85**]{}, 075431 (2012); J. Kunstmann, C. Özdogan, A. Quandt, and H. Fehske, [*ibid.*]{} [**83**]{}, 045414 (2011).
Notice also, that a similar edge magnetism very weakly affects the energetic structure of graphene nanoribbons apart the lowest electronic and the highest hole subbands, see: Y.-W. Son, M.L. Cohen, and S. Louie, Nature (London) [**444**]{}, 347 (2006); O.V. Yazyev and M.I. Katsnelson, Phys. Rev. Lett. [**100**]{}, 047209 (2008); M. Wimmer, I. Adagideli, S. Berber, D. Tománek, and K. Richter, [*ibid.*]{} [**100**]{}, 177207 (2008). P. Recher, B. Trauzettel, A. Rycerz, Ya.M. Blanter, C.W.J. Beenakker, and A.F. Morpurgo, Phys. Rev. B [**76**]{}, 235404 (2007); M. Zarenia, A. Chaves, G.A. Farias, and F.M. Peeters, [*ibid.*]{} [**84**]{}, 245403 (2011).
For a given strain-induced pseudomagnetic field $B_{\rm s}$, Landau quantization is observable when the average number of quantum states per one Landau level $\langle\Delta{\cal N}\rangle={\cal A}_S/(\pi{}l_s^2)\gg{}1$, with $l_s\equiv{}\sqrt{\hbar/(e|B_{\rm s}|)}$.
We use the empirical density of states in a form $
\rho_{\rm eff}(E)=\rho_0+{\pi}^{-1}{\cal A}_{\rm eff}|E|/(\hbar{}v_F)^2,
$ with least-squares fitted parameters $\rho_0$ and ${\cal A}_{\rm eff}$, and find that it provides a resonable approximation of $\langle{}\rho(E)\rangle$ for the whole energy range of $0<E<0.5\,t_0$ even for the strain parameters as high as $H/R\simeq{}0.5$. Remaining details of calculating level-spacing distributions $P^{(k)}(S)$ are same as in Ref. [@Ryc12].
K. Życzkowski, Acta Phys. Polon. B [**24**]{}, 967 (1993).
Namely, $a(\lambda)=\lambda/(\lambda+1)$, $u(\lambda)=\sqrt{\pi}U(-\frac{1}{2},0,\lambda^2)$ with $U(a,b,x)$ the confluent hypergeometric function [@Abram], and $
c(\lambda)=\sqrt{2/\pi}\,\lambda+
\exp\left(\lambda^2/2\right)
\left[1-\mbox{erf}\left(\lambda/\sqrt{2}\right)\right]
+ \lambda^2\int_0^\infty{d\chi}\,\chi^{-1}\exp(-\lambda{}\chi)
\mbox{erf}\left(\chi/\sqrt{2}\right)
$ with the error function $\mbox{erf}(x)=(2/\sqrt{\pi})\int_0^x\exp(-t^2)dt$.
M. Abramowitz and I.A. Stegun, eds., *Handbook of Mathematical Functions* (Dover Publications, Inc., New York, 1965), Chapter 13.
M.V. Berry and M. Robnik, J. Phys. A: Math. Gen. [**19**]{}, 649 (1986); G. Lenz and K. Życzkowski, [*ibid.*]{} [**25**]{}, 5539 (1992).
F.J. Dyson and M.L. Mehta, J. Math. Phys. [**4**]{}, 701 (1963); M.L. Mehta, [*Random Matrices*]{}, 3rd ed. (Elsevier, Amsterdam, 2004), Appendix A40.
A. Singha, M. Gibertini, B. Karmakar, S. Yuan, M. Polini, G. Vignale, M.I. Katsnelson [*et al.*]{}, Science [**332**]{}, 1176 (2011); M. Gibertini, A. Singha, V. Pellegrini, M. Polini, G. Vignale, A. Pinczuk, L.N. Pfeiffer, and K.W. West, Phys. Rev. B [**79**]{}, 241406(R) (2009).
D. Torrent and J. Sánchez-Dehesa, Phys. Rev. Lett. [**108**]{}, 174301 (2012).
|
---
abstract: |
Using the Near Infrared Camera and Multi-Object Spectrometer (NICMOS) on board the Hubble Space Telescope, we have obtained $F110W$ ($\sim
J$) and $F160W$ ($\sim H$) images of three fields in NGC3379, a nearby normal giant elliptical galaxy. These images resolve individual red giant stars, yielding the first accurate color-magnitude diagrams for a normal luminous elliptical. The photometry reaches $\sim$ 1 magnitude below the red giant branch tip with errors of $\lesssim
0.2$ mags in $F110W-F160W$. A strong break in the luminosity function at $F160W = 23.68 \pm 0.06$ is identified as the tip of the red giant branch (RGB); comparison with theoretical isochrones implies a distance of $10.8 \pm 0.6$ Mpc, in good agreement with a number of previous estimates using various techniques. The mean metallicity is close to solar, but there is an appreciable spread in abundance, from at least as metal poor as \[Fe/H\]$\approx -1.5$ to as high as $+0.8$. There is a significant population of stars brighter than the RGB tip by up to $\sim 1$ magnitude. The observations of each field were split over two epochs, separated by $2-3$ months, allowing the identification of candidate long period variables; at least $40\%$ of the stars brighter than the RGB tip are variable. Lacking period determinations, the exact nature of these variables remains uncertain, but the bright AGB stars and variables are similar to those found in metal rich globular clusters and are not luminous enough to imply an intermediate age population. All of the evidence points to a stellar population in NGC3379 which is very similar to the bulge of the Milky Way, or an assortment of Galactic globular clusters covering a large metallicity spread.
author:
- |
Michael D. Gregg, Henry C. Ferguson, Dante Minniti,\
Nial Tanvir, and Robin Catchpole
title: '**Resolving the Stellar Population of the Standard Elliptical Galaxy NGC3379**'
---
Introduction
============
Knowledge of the present stellar content and star formation histories of early type galaxies is essential for theories of galaxy evolution and has important consequences for the use of ellipticals in the distance ladder. A major barrier to divining the nature of luminous elliptical galaxies is the dearth of nearby examples whose stellar populations can be resolved and studied star-by-star. Consequently, considerable effort has been expended in spectral synthesis modeling of the integrated light of early type galaxies (e.g. Leitherer, et al. 1996), but spectral modeling has not yet clarified the nature of the stellar population of normal giant elliptical galaxies. There is often significant disagreement over even the most fundamental issues of mean age and metallicity among various methods, as was documented by Arimoto (1996) in a blind test of many spectral synthesis methods by independent researchers. Resolved stellar population analyses of elliptical galaxies is essential for making real progress.
Ground-based color-magnitude diagrams (CMDs) in the near-infrared by Freedman (1989; 1992) and Elston & Silva (1992) for the Local Group high surface brightness, compact dwarf elliptical M32 (NGC221) appeared to exhibit an excess of stars well above the tip of the red giant branch (RGB), evidence for an intermediate age population. More recent V-I CMDs from the Hubble Space Telescope (HST) using WFPC2 (Grillmair et al. 1996), however, find no direct evidence for this population, apparently an artifact of crowding in the ground-based studies, as was predicted by Renzini (1992). After M32, the next nearest early type galaxy stand-in is the spheroid of NGC5128. Soria et al. (1996), Harris et al. (1999), and Marleau et al. (2000) have constructed CMDs using HST WFPC2 V + I or NICMOS J + H images of the outer halo of NGC5128, detecting the tip of the RGB and plentiful bright asymptotic giants, the latter strongly suggesting an intermediate age population. Maffei 1, at a distance of $\sim 4$ Mpc (Luppino & Tonry 1993), is probably the nearest normal elliptical galaxy (Buta & McCall 2003), but at a Galactic latitude of $-0\fdg55$, it is severely obscured by large and highly variable Galactic extinction, $A_V\approx5$, making it a very difficult object for study at optical and IR wavelengths. Using ground-based adaptive optics, Davidge (2002) has resolved AGB stars in Maffei 1 in the $H$ and $K$ bands, reaching $K\approx22$ with errors of $0.25$ magnitudes, but not yet good enough to detect the RGB tip or study the RGB population in detail.
These results are important for understanding early type galaxy populations, but neither M32 nor NGC5128 is a normal, luminous, elliptical galaxy. With M$_{\rm V} = -16$, M32 is at the low luminosity extreme of high surface brightness ellipticals and its proximity to M31 has almost certainly influenced its development (Faber 1973; Nieto & Prugniel 1983; Bekki et al. 2001). NGC5128 is quite peculiar, a probable recent merger harboring an active galaxy nucleus (Soria et al. 1996) and stars as young as 10 Myr (Rejkuba et al. 2001; 2002). Study of Maffei 1 is greatly complicated and compromised by the variable foreground Galactic dust across its field. Color-magnitude diagrams for these unusual objects cannot, without further investigation, be considered representative of the class of standard, luminous ellipticals.
The nearest normal giant elliptical galaxy which can be studied free of complications is NGC3379 in the Leo I galaxy group. It is an E0, with typical early type colors, Mg$_2$ index, and velocity dispersion (Davies et al. 1987). It is well fit by an R$^{\onequarter}$ law (de Vaucouleurs & Capaccioli 1979) and appears to have no large scale morphological peculiarities (Schweizer and Seitzer 1992). There is some evidence that NGC3379 could be a face-on S0 (Statler & Smecker-Hane 1999; Capaccioli et al. 1991), but the evidence is equivocal. It contains a tiny nuclear dust ring, $R \approx 1\farcs5$, and small ionized gas ring, $R \approx 8$ (Pastoriza et al. 2000; Statler 2001). This amount of visible interstellar material is small compared to that in the 40% of typical bright ellipticals with gas and dust (Faber et al. 1997; Sadler & Gerhard 1985). The Leo I group, which also includes the S0 NGC3384 and the Sc NGC3389, is distinguished by its large partial ring of neutral hydrogen (Schneider 1989).
The HST Extragalactic Distance Scale Key Project (Freedman et al.2001) arrives at a distance of $\sim 9.5$ Mpc ($m-M = 29.90\pm0.10$) for the Leo I group, somewhat lower than most other estimates (Tanvir et al. 1999, Graham et al. 1997; Tonry et al. 1990; Sakai, Freedman, & Madore 1996; see Gregg 1997 for a short summary). At this distance, the resolved stellar population of NGC3379 is all but impossible to study with present ground-based instrumentation. Sakai et al. (1996) used WFPC2 to obtain I-band images just deep enough to locate the tip of the RGB in NGC 3379 in a field $5-6\arcmin$ from the center. While providing a distance estimate of $11.5\pm1.6$ Mpc ($m-M
= 30.30\pm0.14$), marginally consistent with the Key Project distance, their single filter observations do not constrain the metallicity of the RGB or probe the nature of the AGB.
Apart from the detection of the RGB tip (TRGB) by Sakai et al., everything that is known about the stellar population of NGC3379 has been inferred from spectroscopy and photometry of its integrated light. Compiling the spectral line data from a number of recent studies, Terlevich & Forbes (2002) have derived age and metallicity estimates for 150 nearby elliptical galaxies, using the stellar population models of Worthey & Ottaviani (1997). The mean results for NGC3379 indicate values of $t = 9.3$ Gyr and \[Fe/H\] $=+0.16$, \[Mg/Fe\] $=+0.24$, consistent with its having a canonical old, relatively metal-rich stellar population. Rose (1985), however, has presented arguments based on detailed analysis of integrated light spectral line strengths, that luminous ellipticals, NGC3379 included, have a “substantial” intermediate age component, $6-7$ Gyr old, though the fraction is not quantified. These conclusions from integrated spectra are based on single-metallicity population models and so must be viewed with some reservations; NGC3379, like most ellipticals, exhibits line strength and color gradients (Davies, Sadler, & Peletier 1993), indicative of abundance variations (Peletier et al. 1999) and hence a composite metallicity population. A metal-poor component can plausibly account for the relatively young turnoff age reported by Rose (1985).
Using HST with NICMOS Camera 2 (Thompson et al. 1998), we have resolved the bright RGB and AGB stars in three fields in the halo of NGC3379 in the $F110W$ ($\sim J$) and $F160W$ ($\sim H$) filters. These images have yielded CMDs, the first such data for a normal, luminous elliptical galaxy which probe below the RGB tip with some precision. Analysis of these data allow us to place constraints on the mean metallicity of the giant stars and to compare the CMDs to those for nearer, relatively well-observed early type populations, such as the Milky Way bulge and globular clusters, as a first attempt in understanding the star formation history of a typical bright, normal elliptical. By dividing our NICMOS observations into two epochs separated by many weeks, we have identified candidate variable stars in its halo, opening another window on elliptical galaxy stellar populations.
The analysis is summarized as follows: the $F160W$ luminosity function yields the TRGB apparent magnitude ($\S 5.1$). Comparison of the CMD to theoretical isochrones yields a mean abundance which must be near solar $\S 6.1$), irrespective of age. With the abundance determined, the isochrones provide a distance estimate via the TRGB, again with very little age sensitivity; the largest sources of error are the choice of whose isochrones to use and the binning of the isochrone points at the tip of the RGB, rather than the age or abundance of the stars, or the photometric and statistical uncertainty in the TRGB location. With the distance estimate, a more detailed comparison of the CMDs to the isochrones reveals a large abundance spread of $-2 <
[{\rm Fe/H}] < +0.8$ ($\S 6.1.3$) while no young or intermediate age stars ($t \lesssim 5$ Gyr) are required. The contribution of any population with age $\lesssim 5$ Gyr is put on more quantitative footing by the number of very bright AGB stars, which sets an upper limit of $\sim 20\%$ coming from such a population ($\S 6.1.4$). Comparison of the NGC3379 CMDs to the Milky Way Bulge and globular clusters and NGC5128 supports this analysis ($\S6.2$). While a large age spread in the range 8 to 15 Gyrs cannot be ruled out by our data, there is no compelling evidence for any significant contribution from stars with ages $< 8$ Gyrs.
Observations
============
The locations of our three NICMOS Camera-2 fields are shown in Figure 1, on the Digitized Sky Survey image of NGC3379. The positioning was driven by two factors. First was to sample a large range in distance from the center so as to be sensitive to any radial variations in stellar population. The fields are 3 (8.7 kpc), 45 (13.0 kpc), and 6 (17.5 kpc) from the nucleus. The half-light radius $R_e$ of NGC3379 is 54, so these fields are at 3.33, 5.0, and 6.67 $R_e$ with $B$-band surface brightnesses of 25.2, 26.4, and 27.2 mags/, respectively (Capaccioli et al.1990). The somewhat scattered locations were chosen in part to maximize the usefulness of the parallel STIS and WFPC2 observations. Unfortunately, the parallel observations were not executed as efficiently as we had hoped and are not deep enough to probe the resolved stellar population of NGC3379.
Observations of each of the three fields were divided evenly between two epochs, spaced 2-3 months apart to identify candidate variable stars. The interval was set by HST scheduling constraints and the desire to have the same orient for follow-up visits, while obtaining all data in a single observing cycle. A drawback in observing at two epochs was the significant increase in zodiacal light during the second visits. The overall background was roughly twice as high, completely consistent with the variation in zodiacal light due to the object-HST-Sun viewing angle changing from $\sim 145\arcdeg$ to $\sim
56\arcdeg$.
Data were taken using $\sim 1400$s MULTIACCUM SPARS64 integration sequences; the fields were dithered by a few pixels between sequences. The total integration time spent in each filter for each field is 11.2 ksec (8 MULTIACCUM sequences). We initially planned to divide the exposure times between $F160W$ and $F110W$ with a 5:3 ratio. After the first epoch observations, however, it was evident that we were not obtaining sufficient depth in $F110W$, so the second visits were used to equalize the total exposure times in the two filters.
[lllllllll]{} Apr 02 & $F160W$ & 1343.9 & May 02 & $F160W$ & 1343.9 & May 03 & $F160W$ & 1343.9\
& & 1407.9 & & & 1407.9 & & & 1407.9\
& & 1407.9 & & & 1407.9 & & & 1407.9 SAA\
& & 1407.9 & & & 1407.9 & & & 1407.9\
& & 1407.9 & & & 1407.9 & & & 1407.9 SAA\
& $F110W$ & 1407.9 & & $F110W$ & 1407.9 & & $F110W$ & 1407.9\
& & 1407.9 & & & 1407.9 & & & 1407.9\
& & 1407.9 & & & 1407.9 & & & 1407.9 SAA\
Jul 04 & $F160W$ & 1407.9 & Jul 03 & $F160W$ & 1407.9 & Jul 05 & $F160W$ & 1407.9\
& & 1407.9 & & & 1407.9 saa & & & 1407.9 SAA\
& & 1407.9 & & & 1407.9 SAA & & & 1407.9 saa\
& $F110W$ & 1343.9 & & $F110W$ & 1343.9 & & $F110W$ & 1343.9\
& & 1407.9 & & & 1407.9 & & & 1407.9\
& & 1407.9 & & & 1407.9 & & & 1407.9\
& & 1407.9 & & & 1407.9 & & & 1407.9 saa\
& & 1407.9 & & & 1407.9 SAA & & & 1407.9 SAA\
Only the innermost field is uncompromised by cosmic ray (CR) persistence during South Atlantic Anomaly (SAA) passage. Severely affected data were omitted from the final images, resulting in reduced total exposure times. The combination of higher background and CR persistence has rendered the second epoch $F160W$ images for Field 2 nearly useless. The outer field which overlaps with the previous WFPC2 data is also seriously impacted, with effective exposures of only 6975s in $F160W$ and 8383s in $F110W$, and even these have noticeable CR persistence. A log of the observations is given in Table 1.
Data Reduction
==============
The raw NICMOS images were reduced using IRAF scripts kindly provided by M. Dickinson (1998, private comm.) These scripts are very similar to the now-public versions released in mid-1999 which improve on the original CALNICA pipeline procedures, primarily in the removal of the quadrant-dependent bias level or “pedestal” signature. Updated dark frames which included the modeled temperature dependent variable bias level (“shading”) correction were supplied by E. Bergeron. A correction was also made to remove excess shot noise from non-optimum dark subtraction; this was done by subtracting a noise-free map of the difference between the non-optimal pipeline dark correction and a more accurate, very deep blank field frame; this “delta-dark” was also supplied by M. Dickinson.
The 8 reprocessed individual [MULTIACCUM]{} exposures in each filter were then registered using [xregister]{} in [IRAF]{}. At this point it was possible to assess the impact of the SAA on individual exposures. The unaffected [MULTIACCUM]{}s were then combined using the “drizzle” technique (Fruchter & Hook 2002) to produce images subsampled by a factor of 2 using a 0.9 drop size. A very low order smooth surface was fit to the individual frames and subtracted to remove the background to bring all the frames to the same mean value, necessary to achieve good results when drizzling. A mean sky level was added back to the final frame to preserve background counts for computing photometric errors. To look for variable stars, separate images were made for each of the two epochs in the same manner.
The final drizzled $F160W$ images are shown in Figure 2-4. The substantial gradient in stellar density from field to field is immediately apparent. A string of background galaxies is visible in the middle field. The outermost field is of very limited value because it is so sparsely populated with stars, partly because of its extreme outer location but also because it is most impacted by the higher noise levels and reduced effective exposure times due to SAA CR persistence. This is quite unfortunate because this field overlaps with the previous WFPC2 $I$-band observations. This NICMOS location detects so few stars, in fact, that comparison with the optical data is useless.
Photometry
==========
PSF Fitting and Calibration
---------------------------
Point-spread-function (PSF) fitting photometry was performed using standard [iraf/daophot]{} procedures, after masking out obvious background galaxies and the NICMOS coronograph hole. Attempts to construct a reliable PSF model from the data were not successful primarily because crowding raised the noise level in the wings of the bright, well-exposed stars to unworkable levels. Instead, we used the [tiny tim]{} package (Krist 1993) to construct NICMOS $F110W$ and $F160W$ PSFs. We added “jitter” to the [tiny tim]{} models until we reproduced the well-determined cores of the empirical PSFs in the final drizzled images. Because of the better-behaved profile wings, the model PSFs gave much better results when used in the [daophot]{} routines, judging from the lower residuals in the subtracted images and the much lower photometric errors.
We used apertures of 2.5 ($F110W$) and 3.0 ($F160W$) pixels radius to measure the instrumental magnitudes in each drizzled image. Objects were found independently in $F110W$ and $F160W$ and then matched with a cutoff distance of two drizzled pixels, allowing for possible small registration errors, geometric distortions, and noise. Using modeled PSFs for the photometry made determination of the aperture corrections straightforward; from the measurement aperture to a 05 radius (13.33 drizzled pixels) required an $F110W$ correction of $-0.65$ magnitudes while the $F160W$ correction is $-0.78$. These magnitudes were then corrected to “infinite” apertures with the standard factor of 1.15. We adopted the NICMOS Data Handbook “Vega magnitude” zeropoints of 1775.0 mJy in $F110W$ and 1040.7 mJy in $F160W$. The photometry has also been corrected for the small Galactic extinction towards NGC3379, 0.022 and 0.014 magnitudes in $J$ and $H$ (Schlegel, Finkbeiner, & Davis 1998).
To ensure that only the best measured stars are used in the analysis, the photometry was then filtered against several criteria reported by the PSF fitting task. Objects with $\chi^2 > 2.5$, “sharpness” parameter outside the range $-0.9$ to $1.$, and photometric error $ >
0.5$ in $F110W-F160W$ were discarded. The first two criteria eliminate nearly all extended objects, residual bad pixels, and edge effects. Total numbers of objects with both $F110W$ and $F160W$ photometry meeting the above criteria are 1751, 477, and 144 in Fields 1, 2, and 3, respectively. The statistics of the photometric errors are reported in Table 2.
[cccccc]{} 22.50 & 0.020 & & 1.350 & 0.032 & 0.22\
23.00 & 0.024 & & 1.436 & 0.048 & 0.21\
23.50 & 0.033 & & 1.362 & 0.056 & 0.21\
24.00 & 0.048 & & 1.358 & 0.080 & 0.24\
24.50 & 0.072 & & 1.275 & 0.111 & 0.30\
25.00 & 0.107 & & 1.174 & 0.161 & 0.43\
Identification of Variable Star Candidates
------------------------------------------
[cccc]{} $22.2-23.65$ & 98 & 38 & 0.388\
$23.65-24.0$ & 197 & 24 & 0.122\
$24.0-24.5$ & 503 & 33 & 0.066\
$24.5-25.0$ & 569 & 29 & 0.051\
$25.0-25.5$ & 329 & 2 & 0.006\
$22.2-25.5$ & 1696 & 121 & 0.071
The observations were split between two epochs for the express purpose of detecting variable stars. The intervals for fields 1, 2, and 3, are 93, 62, and 63 days, respectively. Blinking the reduced images immediately revealed many variables (Figure 5). To quantitatively identify and characterize the variable star candidates, we separately ran [daophot]{} on the reduced and combined images for each epoch. The positions of stars determined in the total integration time images were used to constrain the photometry for the noisier separate epoch images. With just two epochs, the problem of identifying variable star candidates reduces to gauging the significance of the change in brightness of a star from first to second epoch, which can be done in myriad ways. For Field 1, we show $\Delta(H) = (F160W_1 - F160W_2)$ divided by the photometric errors summed in quadrature from each epoch, plotted against the first epoch $H$ magnitude (Figure 6). Because stars may vary below the detection threshold in one epoch, some of the most extreme variables are potentially missed, but, as it happens, all of the stars in the combined image brighter than $F160W
\approx 24$ were detected in both epochs.
The dotted lines at the 6.5 and 10 “sigma” levels are drawn to help evaluate the diagram, but can also be taken as liberal and conservative criteria for variability. In Field 2, no variables were reliably detected because the second epoch images are so shallow due to SAA problems. A few stars qualified as variable in Field 3.
Many of the brightest stars with the tiniest photometric errors are among the most significantly variable (Figures 5, 6), providing confidence that the majority of the brightest stars are not due to crowding of ordinary red giants. The fractions of variables as a function of $F160W$ magnitude are listed in Table 3.
Luminosity Functions
====================
The $F160W$ luminosity function (LF) histograms for the combined epoch imaging data for Fields 1 and 2 are shown in Figure 7; bin size is 0.1 magnitudes. Field 3 has such poor statistics – no stars at all within 0.25 magnitudes of what will turn out to be the RGB tip – that we will not discuss its LF further. The Field 1 and 2 distributions both begin to turn over in the $F160W = 24.7$ bin, suggesting that this is where incompleteness becomes important. This is confirmed by the artificial star tests which indicate $\sim 80\%$ completeness at this brightness (see Appendix A). The LF for Field 1 (shown with and without the variable candidates) shows a definite jump in the number of stars in the bin spanning $23.65 - 23.75$, which we attribute to the brightness limit of the helium-burning RGB tip. Omitting the variables, nearly all of which are probably AGB rather than RGB stars (see $\S$6.1.4), increases the significance of this break in the luminosity function; the RGB tip bin has more than 2.5 times as many non-variables as the next brighter bin.
The Field 2 distribution shows a similar but not as prominent break in the next fainter bin, $23.75 - 23.85$ magnitudes. As both fields are at the same distance, this shift, if real, could indicate a modest metallicity difference of $\sim 0.1$ dex (see $\S 6$), but with the small number of stars in Field 2, this is not highly significant. The summed LF is shown in the bottom panel; the two histograms again being with and without the variables from Field 1. In the combined LF, the significance of the RGB tip break remains roughly the same as for Field 1 alone (2.50 times as many nonvariable stars as the next brighter bin), even though we are unable to flag variables in Field 2. The slope of the LF is close to linear over the brightest magnitude of the RGB, as shown by the dashed line fit to the nonvariables in Figure 7. This fit predicts that the first bin brighter than the RGB tip should contain more than twice as many stars as observed (37 vs.18).
Pinpointing the Magnitude of the Tip of the RGB
-----------------------------------------------
A blow-up histogram of the break region is shown in the inset; the bin size is reduced to 0.01 magnitudes. Even at this fine scale, there is an obvious increase in the numbers of stars in each bin beginning at $F160W=23.65$. In the inset, the median number of stars per bin brighter than this is 2; fainter is 6, firmly placing the RGB tip magnitude at a level where the formal photometric errors are only $\pm
0.04$ magnitudes per star. There are about a dozen stars within 0.01 magnitudes of the tip of the RGB, so the error budget for the RGB tip magnitude is dominated by the systematic uncertainty in the photometric calibration of NICMOS – $5\%$ – over any other source of error. Artificial star tests show that crowding effects for Field 1 introduce a median systematic error of only $\sim 0.01$ magnitudes per star at this brightness (Appendix A). In the face of 0.04 magnitude errors, the extremely sharp break at $F160W=23.65 \pm 0.01$ must be partly fortuitous, but certainly suggests an unambiguous placement of the RGB tip. To obtain a completely objective estimate of the RGB tip apparent magnitude, we employed a maximum likelihood approach. Figure 8 shows the composite field 1+2 luminosity function (excluding the identified variable stars) on a logarithmic scale with 0.04 magnitude binning. We have determined the magnitude of the TRGB by fitting a two power-law model:
$$N(f) = \left\{ \begin{array}{ll}
f^{\gamma_1} & \mbox{for $f > f({\rm TRGB})$};\\
A f^{\gamma_2} & \mbox{for $f < f({\rm
TRGB})$}. \end{array} \right.$$
The free parameters are $f({\rm TRGB})$, the flux corresponding to the TRGB magnitude, the amplitude of the TRGB discontinuity, $A$, and the two power-law indices $\gamma_1$ and $\gamma_2$. To constrain the models, we perform a maximum-likelihood fit on the [*unbinned*]{} magnitudes. As a simplification in our maximum-likelihood fit, we have not convolved the model with the magnitude errors. At the TRGB the median error is 0.035 mag. We verify through Monte-Carlo simulations that this simplification does not significantly bias the results (see below). For each set of model parameters we compute the probability distribution of star fluxes. We then vary the parameters of the model, maximizing the log of the likelihood of observing the data. Fitting data in the range $22.5 < F160W < 24.5$ (905 stars) we find a best-fit TRGB magnitude of F160W = 23.68. The best-fit model is shown as a solid line in Figure 8.
We have assessed uncertainties in two ways: (1) by bootstrap resampling the data and re-performing the fit and (2) by creating simulated data sets with realistic magnitude errors and known LFs and carrying maximum-likelihood fits to verify that the input parameters can be recovered. The 68% confidence interval spans the range $ 23.63 < F160W < 23.70$ and the 95% confidence interval is $23.62 < F160W < 23.76$. The parameters $\gamma_1$ and $\gamma_2$ have been allowed to vary in these experiments — the best-fit TRGB magnitude is remarkably robust.
The Monte-Carlo simulations allow us to determine if the fitting procedure introduces any bias. In our simulations, data points are drawn from a model distribution function and scattered with a magnitude-dependent error derived from the real data data: $\sigma(m)
= 10^{0.03449 m - 9.605}.$ Ten thousand data sets with the same number of data points as the observed sample are created and best-fit model parameters determined using the identical maximum-likelihood procedure that was applied to the real data. We tried input models with a range of parameters similar to the true data. As a typical example, for a model with parameters $m({\rm TRGB}) = 23.65, A = 2.0,
\gamma_1 = -2.3$ and $\gamma_2 = -4.3$, we recover in 10000 Monte-Carlo iterations $m({\rm TRGB}) = 23.64 \pm 0.02.$ Ignoring the magnitude errors in performing the maximum-likelihood fits thus does not appear to introduce any significant bias in the results. The dominant source of error in the RGB tip magnitude is likely to be the 5% systematic uncertainty in the NICMOS photometric calibration. Departure of the true shape of the LF relative our the simple model of a sharp discontinuity — due to binaries, stellar rotation, variability, and the finite metallicity spread, for example — is another possible source of systematic error for the TRGB-distance technique.
While in principle the LF in $F110W$ might provide additional information on the TRGB location, the tip is much less pronounced in $F110W$. In $\S6$, it will be shown that NGC3379 has a large abundance spread. Theoretical isochrones show that the TRGB is a strong function of abundance in the $F160W$ band, with the solar and near-solar metallicity TRGBs rising well above the more metal-poor TRGBs, effectively producing a nearly single-abundance TRGB to measure. In $F110W$, however, the RGB tips of different metallicity populations are closer together in luminosity, and the large abundance spread of NGC3379 blurs the discontinuity so evident at $F160W = 23.68$ (Figure 7).
Other Luminosity Function Details
---------------------------------
In Field 1, there are 98 stars in the bins brighter than the tip; 38 of these (39%) are variable candidates. These will be discussed further in the next section, along with the implications for the distance and stellar population of NGC3379. Also in Field 1, there is an excess of stars in the bin centered at $F160W = 24$, which, in the summed LF becomes another apparent break: the LF jumps by a factor of 1.65, then falls again. This excess is not as statistically significant as the brighter RGB tip break, and occurs in just one or two bins. Taking the average number of nonvariables in the bins on either side of this feature predicts 85 stars whereas 109 are seen, about a $2.5\sigma$ difference. This second break can be interpreted as an AGB contribution and is discussed further in $\S 6.1.4$.
Color Magnitude Diagrams
========================
The NICMOS CMDs for each of the three fields are compared in Figure 9. The variable star candidates identified in Figure 6 from Fields 1 and 3 are circled, double circles indicating the variables with significance levels $> 10$. The colored solid lines trace the smoothed, running median color ridge lines, with the dotted lines indicating one standard deviation; the ridges are computed using stars with $F110W-F160W \leq 2.0$ only. Objects with redder colors, especially faint objects, are probably galaxies, consistent with the roughly constant numbers of such objects from field to field. For reference, the horizontal dotted lines mark the location of the RGB tip found from analysis of the LF histograms ($\S5.1$, Figure 7).
NGC3379 is at Galactic coordinates $l=133^\circ, b=57^\circ$. There are two obvious bright foreground stars in the images, one each in Fields 1 and 3 (Figures 2 and 4). Scaling from star counts in the Hubble Deep Field (Williams et al. 1996; Méndez & Minniti 1999) and infrared Subaru Deep Field (Maihara et al. 2001), we estimate that over the magnitude range displayed in Figure 9, $22.3 < F160W <
25.7$, the expected number of Galactic foreground stars is zero.
Two sets of $F110W-F160W$ error bars are shown: the inner are the median $F160W$ photometric errors reported by [daophot]{}, determined in half-magnitude bins. The outer error bars show the $1\sigma$ widths of the $F110W-F160W$ color distribution at each magnitude level, again determined in half-magnitude intervals. The bright limits of the ridge lines are determined solely by the extent of the data and are not meant to suggest the locations of the RGB tip or any other stellar evolutionary phase. The three ridge and dispersion lines are over-plotted for comparison in the lower right panel of Figure 9. There is no significant shift of the main RGB locus from one field to another, and, given the small number of stars in Field 3 and the similarity of the loci for Fields 1 and 2, these data suggest that there are no substantial differences in the stellar populations of the three fields. Most of our subsequent analysis is based on Field 1 alone, but by implication applies to the other fields as well.
Comparison to Theoretical Isochrones
------------------------------------
Determination of the metallicity and spread in abundance of the stars in NGC3379 was a primary motivation of this project. Using the best-fit RGB tip magnitude of $F160W = 23.68$ found above, it is possible to derive a distance to NGC3379, in turn allowing constraints on its age and metallicity by comparing the NICMOS CMDs to theoretical isochrones. We have considered two independent sets of isochrones: the latest version of the Bruzual & Charlot (1993) models (BC; see also Liu, Charlot & Graham 2000), kindly provided in advance of publication by S. Charlot, plus the isochrones of Girardi et al.(2002). The BC isochrones are derived largely from the “Geneva” evolutionary tracks (Maeder & Meynet 1989) while the Girardi set is based on the “Padova” evolutionary tracks (Bertelli et al. 1994).
### Mean Metallicity and Age Constraints
In Figure 10, upper RGB isochrones for 3, 5, 8, 10 and 15 (or 14) Gyr populations with \[Fe/H\]=$-0.7$, 0.0, and $+0.4$ or $+0.2$ are over-plotted on the ridge lines for Field 1. Also plotted are the standard deviation lines (dashed) of the observed NGC3379 giant branch. The isochrones have been shifted to match the observed RGB tip magnitude, $F160W = 23.68$, in NGC3379. The resulting individual distance moduli are listed in the figure. For the Girardi isochrones, the AGB loci are also plotted (dotted lines). The AGB in the BC isochrones do not rise more than a few hundredths of a magnitude above the RGB tip.
Both sets of isochrones demonstrate that the location of the RGB in an infrared CMD depends primarily on metallicity while age is a second order effect. Neither set of \[Fe/H\]=$-0.7$ isochrones for any age reproduces the mean color location of the RGB in NGC3379; the data demand a much higher abundance. The younger super-solar isochrones are perhaps a reasonable match in principle, but the relatively bright and blue turnoff of a majority 3-5 Gyr population would be incompatible with the optical broad band colors and spectrophotometry of NGC3379 (Terlevich & Forbes 2002), although strictly, the integrated light analysis applies only to the nuclear regions. This leaves open the somewhat unlikely possibility that the halo comprises mainly very metal rich intermediate age stars. Additional age constraints based on the CMD come from consideration of the observed AGB, which is better fit by the older ($>8$ Gyr) isochrones of Girardi et al. (2002), as will be shown in $\S 6.1.4$.
The solar abundance, 8, 10 and 14/15 Gyr isochrones of either set bracket the RGB ridge line down to $F160W = 25$. At this magnitude, the observed RGB begins to get bluer. The blueward trend is from incompleteness in the $F110W$ band and the onset of confusion close to the limit of the data (Appendix A). The brightest 0.5 magnitude of the RGB of the solar BC isochrones do not track the observed ridge line as well as the corresponding Girardi isochrones.
The isochrones thus constrain the [*mean*]{} abundance of NGC3379 to be close to solar metallicity, independent of the age of the stellar population. While the usual age-metallicity degeneracy still affects the conclusions at some level, the [*mean*]{} age is likely to be in the range $8-15$ Gyr, probably very close to 10 Gyr. Given the relatively small offsets in color due to age in the $F110W-F160W$ colors, and the relatively large observed width of the RGB, a considerable spread in age above $\sim 8$ Gyr cannot be ruled out.
A metallicity close to solar is perhaps surprising so far out in the halo, even of a giant elliptical. One possible explanation for the high abundance is that NICMOS is measuring the brightest, most metal rich stars. Looking at the CMDs and isochrones in Figures 9 and 11, the metal poor populations are fainter and bluer. Stars with \[Fe/H\]$<-1.7$ will drop below the sensitivity range of the observations. The mean abundance of the [*measured*]{} stars is close to solar, but a deeper census might reveal the existence of a bluer, much lower abundance population with \[Fe/H\]$<-2$. This effect is the opposite of what happens in the optical where $V$ (and even $I$) band RGBs become fainter with increasing metallicity, leaving just the metal poor stars to be measured in shallow observations, resulting in a lower mean abundance. With the present NICMOS data we may be overestimating the mean metallicity. Complementary deep $V$ and $I$ band observations with the Advanced Camera for Surveys (ACS), especially leveraged against these and/or additional NICMOS fields, are required to document the full extent of the metallicity spread in NGC3379.
### An Infrared RGB Tip Distance to NGC3379
The sharpness of the break in the LF and the small photometric errors at the RGB tip ($\pm 0.035$ magnitudes per star), provide a robust location of the RGB tip from the maximum-likelihood analysis. Adopting the solar metallicity, 10 Gyr isochrone as the best fit, the implied distance modulus to NGC3379 is 30.10 (10.4 Mpc) for the BC isochrone, $+0.08$ greater for the Girardi isochrone. The major contributors to the uncertainty in distance are the NICMOS photometric calibration and the isochrone differences. The NICMOS absolute calibration is good to $\pm0.05$ magnitudes according to the NICMOS website. The Girardi et al. isochrones are systematically brighter and redder than those of BC, the solar 10 Gyr RGB tip in particular by 0.08 magnitudes. Much of this difference could arise from the coarseness of the isochrone sampling of the RGB in steps of $\sim 0.05
- 0.07$ magnitudes. Both the NICMOS calibration and the isochrone differences are systematic in nature and we conservatively add them linearly to derive a total uncertainty. The total error in the distance estimate is $\pm 0.14$ magnitudes, only 7% in distance, almost entirely from the uncertainties due to systematic differences between the theoretical isochrones and the NICMOS absolute photometric calibration, not the data.
Averaging of the two isochrone sets leads to $m-M=30.14 \pm 0.14$ (10.67 Mpc $\pm 0.74$) as the distance for NGC3379. Our estimated $1\sigma$ error includes the full covariance of the TRGB magnitude with respect to the other parameters in the model, but does not include possible systematic effects due to the abundance spread, which we have not included in the maximum-likelihood analysis in $\S5.1$. The mean abundance of NGC3379 is very near solar, and, fortuitously, the solar abundance isochrones of either BC or Girardi et al. (2002) have the most luminous (or nearly the most luminous) TRGB, so modest contributions of stars from non-solar abundance populations will not significantly affect the detection of the brightest (roughly solar abundance) TRGB location.
The better match of the upper RGB and the better representation of the brighter AGB stars by the Girardi isochrones supports a preference for an increase in the distance to $m-M = 30.18$ (10.86 Mpc); a positive choice of one set of isochrones over the other also removes a contribution of 0.08 magnitudes to the systematic errors, which would reduce the error estimate to $< 0.1$ magnitudes. Regardless, the derived distance is in good agreement with estimates by other techniques (Gregg 1997; Hjorth & Tanvir 1997), and consistent at the $1\sigma$ with the results of Sakai et al. (1997) who found $m-M=30.30\pm0.14$ (random errors) $\pm 0.23$ (systematic) from the $I$-band luminosity of the RGB tip using WFPC2 imaging.
### Metallicity Spread
The metal poor isochrones are not a good match to the mean RGB of NGC3379, but provide constraints on the metallicity spread. From examining the isochrone sets in Figure 10, it is clear that it is impossible to account for the RGB width solely with an age spread; most or all of the intrinsic width must be due to a spread in abundance. In Figure 11, over-plotted on the Field 1 ridge and dispersion lines are BC and Girardi 10 Gyr isochrones for a range of abundance, all shifted to the same distance modulus, 30.10 for BC and 30.18 for Girardi. In both isochrone sets, the lower abundance RGB tips have lower luminosity and are shifted to bluer colors, naturally tracing the blue envelope of the observed RGB stars. A range of metallicity will thus increase the intrinsic (and observed) width of the RGB at fainter magnitudes, even without a contribution from increasing photometric errors, because there simply are no extremely metal-poor RGB stars brighter than $F160W\approx 24.5$. The super-solar BC isochrone also has a somewhat fainter RGB tip luminosity, perhaps signaling the same effect on the high metallicity side.
Consequently, the isochrones predict that for a heterogeneous abundance stellar population, the [*spread*]{} in metallicity will be a function of magnitude over the upper $\sim 1.5$ magnitudes of the RGB. The color distributions in four 0.5 magnitude intervals beginning at the RGB tip are plotted in Figure 12. Over-plotted on each is a best-fit gaussian with the same standard deviation and median, normalized to the total number of stars. The upper abscissa is an approximate metallicity scale computed simply by dividing the isochrone metallicity range (2.1 dex) by the color difference at the central $F160W$ in each interval. This is approximate and plotted merely for reference; the change in color is not strictly a linear function of abundance, nor constant as one goes down the RGB.
For comparison, we also plot gaussians at the mean color location of the 10 Gyr BC isochrones with \[Fe/H\] = $-1.7, -0.7, 0.0, +0.4$. Each has a $1\sigma$ width equal to the spread in color caused by crowding and photometric errors, as determined by our artificial star tests (Appendix A). The peaks are simply scaled to the local value of the total gaussian fit and so these are meant to be suggestive only; the abundance distribution of the stellar population is, as far as we can tell from our data, continuous. The near absence of any contribution from extremely metal-poor blue stars in the brightest interval is evident, as discussed above. Such blue stars become numerous at magnitudes $> 24.5$, once the tip of the fainter metal-poor RGB enters the plots. The presence of stars redder than the \[Fe/H\]$=+0.4$ isochrone implies a very metal-rich population. The intrinsic width of the color distributions increases gradually from 0.21 to 0.31 magnitudes over the three brightest intervals, but does not increase further in the faintest, suggesting that we may be seeing the full extent of the abundance spread, but deeper data are required to confirm this. The faintest interval is essentially confusion limited (see Appendix A), however, so while it is consistent with the next brighter interval, it cannot bear much weight in the analysis.
Using the color-to-metallicity conversions computed from the isochrones, the metallicity spreads in each 0.5 magnitude interval on the RGB are $\pm 0.6$, $\pm 0.8$, $\pm 1.2$, and $\pm 1.2$ dex. The tail to very red objects, $F110W-F160W\gtrsim2.$ is perhaps due to unresolved galaxies. There is a suggestion of a break in the middle two color histograms at $F110W - F160W \approx 1.65$, perhaps an indication of the transition between the stellar population and the background galaxies. The blueward trend of the most metal-rich isochrones with fainter magnitudes combined with photometric errors may produce this break on the red side. Extrapolating from the BC isochrones, we estimate that this color corresponds to \[Fe/H\]$ \approx
+0.8$. Stars this metal-rich have been shown to exist in the inner Milky Way halo (Rich 1986; Minniti et al. 1995; Sadler, Rich, & Terndrup 1996), coexisting with extremely metal poor objects of $[Fe/H] = -2$ (Blanco 1984), similar to the situation we find in NGC3379.
### The Asymptotic Giant Branch Population
The characteristics of the AGB population evolve with age and are a function of metallicity, so in principle the AGB can be a stellar populations diagnostic. To be useful, though, the AGB contribution must somehow be separated from the RGB, which is problematic when photometric errors and stellar physics cause them to completely overlap fainter than the RGB tip.
Stars significantly brighter than the RGB tip, however, can be relatively easy to isolate; work by Mould & Aaronson (1979) showed that such AGB stars mark the presence of intermediate age populations, $ \lesssim 3$ Gyr. Yet the existence of stars up to $\sim 1$ magnitude brighter than the RGB tip in metal rich Galactic globular clusters (Guarnieri, Renzini, & Ortolani 1997), advises caution in interpreting such stars as an unambiguous signal of younger populations. The behavior of the AGB in the Girardi isochrones (Figures 10, 11) also shows that stars brighter than the RGB tip by up to $\sim 1$ magnitude can be expected even in ancient populations. We find 98 stars above the tip of the RGB in Field 1 (Table 3, Figure 9), 38% of which are variable, and an additional 19 bright stars in Field 2, plus one more in Field 3. All but 13 of these are within 0.6 magnitudes of the RGB tip.
To test the conclusions from $\S 6.1.1$ that the population of NGC3379 is old and solar metallicity, we computed theoretical LFs using a Salpeter initial mass function (IMF) using the Girardi 5, 10, and 14 Gyr solar isochrones. The resulting comparison with the observed $F160W$ LF is necessarily approximate as the isochrones are sampled in uneven mass bins determined by theoretical considerations while the observed LF has been computed in equal magnitude bins. We have not interpolated the isochrones because this smooths out sharp features such as the RGB tip and also because the isochrones are not monotonic functions in luminosity in regions of interest, making interpolation problematic. Fortuitously, the isochrone mass intervals correspond to magnitude intervals of 0.1 to 0.15 along the RGB. We have, however, rebinned the theoretical AGB LF to the same magnitude intervals as the theoretical RGB to permit easy summing and have scaled the total LFs by eye to give an approximate match to the observed LF (Figure 13).
The 10 or 14 Gyr isochrones, or a composite, can account for the AGB stars up to 0.6 magnitudes brighter than the RGB tip. These stars then do not require an intermediate age or younger population in NGC3379. Any claim of such a population would rest on the 13 yet brighter AGB stars ($F160W < 23.0$). Such stars are predicted by the 5 Gyr isochrone, but the numbers of observed AGB stars above the TRGB limit any 5 Gyr contribution to $\sim 20\%$ of the total. The brightest AGB stars are more naturally explained by a contribution from the substantial \[Fe/H\]$=-0.7$ component needed to account for the metallicity spread (c.f. Figure 11); this population will generate AGB stars up to 0.8 magnitudes above the RGB tip. Blends of stars in the images, or even evolving blue straggler or mass transfer binaries are also viable explanations for these few extremely bright AGB stars. We conclude that there is no compelling evidence for any substantial population with an age $< 10$ Gyr.
An explanation for the second LF break at $F160W=24$ also emerges: it is the top of the stably evolving AGB. Between $F160W=24$ and the RGB tip, the AGB stars in the Girardi isochrones evolve rapidly, thinning out the AGB LF, and it is in this region that they exhibit non-monotonic luminosity behavior. The 10 and 14 Gyr ages reproduce this feature somewhat better than the 5 Gyr model, suggesting that the structure of the LF may be useful in constraining the age of the observed population. Detailed comparisons require isochrones more finely sampled in mass (luminosity) and perhaps also better statistics in the observed stars.
Comparison to Other Stellar Systems
-----------------------------------
Comparison of the NICMOS data for NGC3379 to the theoretical isochrones of BC and Girardi has resulted in a consistent, perhaps even plausible, picture of the stellar populations in the halo of a giant elliptical. We next compare the data to similar observations of various well-studied stellar systems.
### Galactic Globular Clusters and the Bulge
The stellar population of Galactic globular clusters is simple, well understood, and has much IR data available. Though the Milky Way Bulge population is less well-determined, high quality IR data exists and provides another point of comparison for a system which is perhaps more similar to NGC3379. Comparison with the extensive Milky Way ground-based data requires color transformations from the CIT $J \& H$ bands to NICMOS $F110W \& F160W$. We found it necessary to derive our own transformation relation for this purpose; details are given in Appendix B.
In Figure 14, we compare the transformed ground-based CIT $J$ and $H$ photometry of the Milky Way Bulge (Frogel et al. 1990; Tiede et al.1995) and several representative Galactic globular clusters (Frogel et al. 1981, 1983; Davidge & Simons 1994) to the Field 1 NGC3379 RGB/AGB ridge line and variable star distribution. The data have been shifted to the distance modulus of NGC3379; individual distance and reddening estimates are taken from the photometry sources. The globulars range from \[Fe/H\]$ = -2.2$ (M92) to $-0.2$ (NGC6553). Known Bulge and globular variables are circled.
Given the wide spread in abundance known to exist in the MW Bulge (Rich 1986; Sadler et al. 1996; Minniti et al. 1995; Blanco 1984), it is puzzling that its RGB/AGB locus is so tight, implying a relatively narrow range of metallicity. Comparison with Figure 10 reveals that the older \[Fe/H\]$ = -0.7$ isochrones are a good match for the bulge giants, similar to 47 Tuc, consistent with the analysis of (Frogel et al. 1990). The Bulge clearly has AGB stars much brighter than those in NGC3379, up to 2 magnitudes brighter than the RGB tip; in fact, we have omitted a number of Bulge stars still brighter than those plotted here. Frogel et al. (1990) concluded that no intermediate age stars were necessary to account for the bright AGB population of the Bulge. The Girardi isochrones, however, indicate that an intermediate age ($\lesssim 5$ Gyr) population is required to account for the AGB stars more than 1 magnitude above the RGB tip, which exist in some of the Bulge fields. It is not clear from the literature accounts whether blending of stars in the crowded fields or perhaps a large spread in distance can account for all of these very bright AGB stars.
The collection of Galactic globular clusters is a better approximation to the data for NGC3379. The width of the RGB, the distribution of variables, and the luminosity of stars brighter than the RGB tip all point to a purely old population in NGC3379 with a wide metallicity spread, consistent with the analysis based on theoretical isochrones above. The variables up to 1 magnitude brighter than the RGB tip in NGC3379 attest to its having a relatively metal rich component, similar to 47 Tuc or greater, as such stars are not found in the metal poor globulars with \[Fe/H\]$ < -1$ (Frogel 1983). The lack of yet brighter AGB stars in our NICMOS data amounts to lack of evidence for a population $\lesssim 5$ Gyr, as seen in regions known to harbor intermediate age populations, such as the Magellanic Clouds (Frogel et al. 1990).
### NGC5128
The peculiar elliptical NGC5128 hosts an active galactic nucleus (Soria et al. 1996) and powerful radio source (Centaurus A). Its prominent dust lane and stellar and gaseous shells are only part of the strong evidence of recent and perhaps multiple merger events, possibly the trigger for the nuclear activity and radio emission. NICMOS observations in F110W and F160W have been carried out by Marleau et al. (2000) for a field in the halo of NGC5128, $8\arcmin50\arcsec$ (9 kpc) from its nucleus, reaching slightly greater effective depth relative to the RGB as our data for NGC3379.
In Figure 15 we plot both the NGC3379 Field 1 RGB/AGB ridge lines and the mean $F110W-F160W$ color locations from Marleau et al. (2000) for NGC5128, shifted from the TRGB distance modulus of NGC5128 ($m-M=27.98$, Harris, Harris, & Poole 1999) to our derived NICMOS TRGB distance modulus for NGC3379. The photometric errors of the two data sets are comparable at a given RGB luminosity. Despite the uncertainties in extinction corrections and distances to the two objects, plus the independent and different natures of the reductions and analysis – they used an empirical PSF and did not employ drizzling – the mean loci of the lower RGB points differ in color by only $\sim 0.02$ magnitudes. This is reassuring evidence that there are no serious systematic calibration differences between the studies.
Two differences in the populations of the galaxies emerge, however. The observed color spread of the NGC3379 RGB is considerably greater, $\sigma = 0.2$ magnitudes for NGC5128 compared to $\sigma = 0.4$ magnitudes for NGC3379 over the fainter half of the data; this cannot be accounted for entirely by our slightly greater observational errors. It is most easily understood as an abundance range only half as great in NGC5128. Second, the brighter NGC5128 RGB points lie to the blue of the ridge line of NGC3379. Marleau et al. argue for a significant contribution from an intermediate age ($1-5$ Gyr) population, based on the presence of numerous stars, probably belonging to the AGB, up to almost 2 magnitudes brighter and several tenths bluer than the RGB tip, much like those of the Milky Way Bulge. The stars above the RGB tip in NGC3379 are concentrated within 0.6 magnitudes of the tip, completely consistent with older, metal rich stars. An increased population of these brighter and bluer AGB stars from a younger component in NGC5128 can easily account for the 0.1 to 0.25 magnitude shift of its upper RGB/AGB locus relative to NGC3379. The number of bright AGB stars seen in NGC5128 is $\sim 10-20$, so the contrast with NGC3379 is probably not simply small number statistics, underscoring the conclusion that NGC3379 has a significantly smaller, if any, intermediate age population. Such a difference is certainly consistent, perhaps expected, from the respective morphologies, NGC3379 the quintessential normal elliptical and NGC5128 being quite peculiar.
The coincidence of the NGC3379 and NGC5128 RGB loci, along with our recalibration of the ground-based IR to NICMOS photometry transformation, implies that the mean abundance of NGC5128 in the field studied by Marleau et al. (2000) should be revised upward from their \[Fe/H\]=$-0.76$ to near solar. In support of this, Harris et al. (1999), using $V$ and $I-$band WFPC2 data find that the peak of the abundance distribution for halo stars in NGC5128 is at \[Fe/H\]$=-0.3$ in a field twice as far from the nucleus, where one might expect the abundance to be lower than in the Marleau et al.field. Having suffered a recent merger, NGC5128 is a complex system probably not in equilibrium nor well-mixed, so perhaps such field-to-field differences are real. Harris et al. (1999) also find a large abundance spread in NGC5128, with stars at least as metal poor as \[Fe/H\]$=-2$ up to at least \[Fe/H\]$=+0.2$, not unlike what we find for NGC3379 (Figure 12). Their photometry reaches about 1 magnitude farther down the RGB, allowing them to discern that the metallicity distribution of NGC5128 has two components. Such a conclusion is not ruled out for NGC3379 by our data, but a deeper exploration of its giant branch is needed to explore the similarities with NGC5128 in greater detail.
Rejkuba, Minniti, & Silva (2003) have found large numbers of long period variables (LPV) in NGC5128, with periods ranging from 150d to $>
800$. The existence of variables with P$ > 300$d is consistent with NGC5128 having a significant intermediate age population. Variables with periods greater than several hundred days have ages $1-5$ Gyr, whereas AGB variables with periods of $< 250$ days are much older (Frogel 1983; Hughes & Wood 1990). With just two epochs, we cannot constrain the periods of the variables in NGC3379; however, the necessary observations are well within reach of either the refurbished NICMOS camera or ground-based IR cameras with adaptive optics. Period determinations for the LPV stars in NGC3379 is perhaps the most sensitive test for detecting and quantifying any intermediate age population in this normal elliptical galaxy.
A Note on the Morphological Type of NGC3379
===========================================
There has been extensive discussion in the literature of the true morphological type of NGC3379 (e.g., Van den Bergh 1989; Capaccioli et al. 1991; Statler 1994). Is it a bona fide normal elliptical galaxy or is it a face-on S0? On one hand, this is an extremely important issue, for the present picture of the origin and evolution of elliptical galaxies is significantly different from that supposed for S0s (Gregg 1989), thus the morphology of NGC3379 has ramifications for the conclusions we can draw for galaxies in general. On the other hand, the question of its morphological type is moot: with its standard colors, spectrum, and appearance, it might as well be considered a prototypical elliptical. If after such detailed investigations, we are unable to discern the morphological type of NGC3379, at a distance of only 10 Mpc, then it is practically impossible to establish the “true” morphology of other early type galaxies at greater distances in clusters such as Coma, let alone at high redshift. There will be many NGC3379 clones in these more-distant samples, so the population of NGC3379 is bound to be representative of an appreciable fraction galaxies taken to be early type, fundamental plane objects.
Conclusions
===========
Our conclusions are that NGC3379 has:
- a distance of $10.8\pm0.7$ Mpc, derived from comparison of the tip of the RGB in the F160W ($H$) band to the theoretical isochrones of Bruzual & Charlot (1993) and Girardi et al. (2002);
- a mean metallicity of roughly solar, perhaps slightly less,
- a large abundance spread with \[Fe/H\] ranging from $-2$ to +0.8;
- no significant change in the mean abundance over almost a factor of two in distance from the nucleus, 9 kpc to 18 kpc.
- a relatively old mean age, in the range 8 to 15 Gyr;
- no significant intermediate age population $\lesssim 5$ Gyr, but a large age spread from 8 to 15 Gyr cannot be ruled out
- a large population of bright AGB and RGB variables, similar to old stellar populations in the Galactic Bulge and globular clusters.
The overall stellar population of NGC3379 resembles a collection of old Galactic globular clusters having a wide distribution of metallicities, but including a significant contribution from super-solar abundance stars. If the large abundance spread and the significant population of metal poor stars extends to the central regions of NGC3379, then this must be taken into account when modeling and interpreting the integrated spectrum, where the infamous age-metallicity degeneracy problem is acute. This investigation has merely scratched the surface of what is now possible to glean from detailed, star-by-star analyses of stellar populations at this distance. Exploring the metallicity range in more detail, especially the metal poor stellar population, awaits deep optical imaging with the Advanced Camera for Surveys on HST. Determining the periods and period distribution of the luminous IR variable stars is the most sensitive way to better constrain any intermediate age population, and this can be done with NICMOS or, eventually, with ground-based adaptive optics observations. Both optical and IR investigations can build on the work described here, refining our understanding of the formation and evolution of early type galaxies.
Support for this work was provided by NASA through grant number GO-7878 from the Space Telescope Science Institute, which is operated by AURA, Inc., under NASA contract NAS5-26555. Part of the work reported here was done at the Institute of Geophysics and Planetary Physics, under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under contract No. W-7405-Eng-48. DM is supported by FONDAP Center for Astrophysics 15010003. We thank Stephane Charlot and Leo Girardi for supplying isochrones in the NICMOS filters and also for helpful discussions.
Appendix A: Artificial Star Tests
=================================
Our three target fields were selected to span a large range of surface brightness, with the innermost still having low enough stellar density so as not to be crippled by confusion of sources as bright as the RGB tip, following the analysis and prescription laid out by Renzini (1998). To assess the impact of crowding on our data, we used the [daophot]{} routine [addstar]{} to implant artificial stars in the reduced images of Field 1. In each trial, 100 stars of a given $F160W$ magnitude and having a color of 1.1 – about $1\sigma$ to the blue of the mean RGB color – were inserted on a $10\times10$ grid, roughly evenly spaced over the image. Fifty sets of $F110W$, $F160W$ image pairs were made for each input $F160W$ magnitude, ranging from 22.75 to 26.5 in steps of 0.25 magnitudes. The locations of the artificial stars were randomly perturbed within a 20 pixel radius for each run, but held fixed within each $F110W, F160W$ image pair. Each of the 13 magnitude intervals then has 5000 artificial stars of known input brightness and color. A second and equally extensive trial was made with stars having a color of 1.5, about $1\sigma$ to the red of the mean RGB.
The same reduction procedure described above for the real data was used to find and measure all the stars in each artificial star image; and then compared to the input artificial star lists. Table 4 summarizes the the input star magnitudes and the means and dispersions of the recovered magnitudes and colors. These can be compared to the photometric errors reported by [daophot]{} (Table 2).
From these tests, we can conclude the following: At F160W=24.68, one magnitude below the RGB tip, we are 80% complete. The mean recovered $F160W$ magnitudes at this level are too bright by $0.12$ magnitudes because of faint stars blending to increase the measured brightnesses. The mean recovered colors of the inserted $F160W=24.68$ stars are somewhat less affected: those with $F110W-F160W = 1.1$ ($\sim 0.2$ magnitudes bluer than the mean RGB), are on average measured to be too red by 0.05, while those with $F110W-F160W = 1.5$ ($\sim 0.2$ magnitudes redder than the mean RGB) are found to be too blue on average by 0.07 magnitudes. At the RGB tip itself, the systematic error due to crowding is only 0.01 magnitudes in brightness or color, thus crowding has no significant effect on the distance estimate. At faint magnitudes, the mean recovered color of both the red and blue star tests converge at $\sim 1.28$, which we take to be the color of the unresolved faint star background.
Fainter than an input magnitude of $F160W = 25$, the recovered magnitudes saturate at $\sim 24.7$, signaling the confusion limited reliability level of the data.
The spread in color of the recovered artificial stars is nearly as great as the observed spread, but this results from the combined effects of crowding and the intrinsic color spread of the real stars. To determine the amount of spread in the RGB directly caused by crowding, we created two new artificial images. Beginning with the residual noise images left after PSF subtraction by [allstar]{}, we added back artificial stars at the same locations as all the detected stars; for the $F160W$ image, we used the detected magnitudes, effectively reconstructing the original $F160W$ image, but with a little extra Poisson noise which [addstar]{} includes. For the $F110W$ image, we added stars at all the locations of the detected stars, but adjusted their magnitudes so that [*every*]{} star had the same color, $F110W-F160W = 1.3$, about the mean of the RGB. Running the [daophot]{} suite on these images in a manner identical to that used on the real data produces the CMD in Figure A1; this figure should be compared to the real data CMD of Figure 9. The recovered spread of the artificial RGB is greater than the median photometric errors, but is quite a bit less than the observed RGB width of NGC3379.
In Figure A2, we compare the artificial data color spread to the actual observed color spread, using the same magnitude intervals as in Figure 12. The quadrature difference in color spread between the real and artificial images is our estimate of the intrinsic width of the RGB. The intrinsic color spread increases from 0.20 at the RGB tip to 0.30 at the limit of our data; this is consistent with the presence of a very metal poor population entering at fainter magnitudes, as predicted by the isochrones in a composite metallicity population (c.f. Figure 11).
The artificial RGB should have a mean color of exactly 1.3, but the measured color moves 0.06 magnitudes bluer at $F160W=24.65$, where we are 80% complete, and 0.16 magnitudes bluer in the confusion-limited regime. The reddest faint stars go undetected in $F110W$, which causes the shift to bluer colors and accounts for the difference between the mean location of the observed RGB and the Girardi isochrones at faint magnitudes (Figures 10, 11).
[cccccccccc]{} 22.75 & 23.85 & & 99.7 & 1.102 & 0.061 & 22.750 & 0.053 & 23.848 & 0.055\
23.00 & 24.10 & & 99.7 & 1.104 & 0.077 & 22.998 & 0.068 & 24.096 & 0.069\
23.25 & 24.35 & & 99.8 & 1.102 & 0.091 & 23.243 & 0.084 & 24.340 & 0.086\
23.50 & 24.60 & & 99.4 & 1.106 & 0.115 & 23.493 & 0.101 & 24.589 & 0.102\
23.75 & 24.85 & & 99.0 & 1.107 & 0.143 & 23.735 & 0.128 & 24.836 & 0.132\
24.00 & 25.10 & & 97.4 & 1.111 & 0.172 & 23.977 & 0.156 & 25.073 & 0.162\
24.25 & 25.35 & & 94.8 & 1.113 & 0.217 & 24.214 & 0.194 & 25.310 & 0.198\
24.50 & 25.60 & & 88.3 & 1.121 & 0.248 & 24.427 & 0.237 & 25.519 & 0.239\
24.75 & 25.85 & & 77.3 & 1.156 & 0.280 & 24.609 & 0.276 & 25.709 & 0.293\
25.00 & 26.10 & & 63.0 & 1.198 & 0.323 & 24.735 & 0.328 & 25.861 & 0.366\
25.25 & 26.35 & & 51.8 & 1.230 & 0.326 & 24.762 & 0.429 & 25.921 & 0.421\
25.50 & 26.60 & & 45.4 & 1.231 & 0.321 & 24.695 & 0.547 & 25.895 & 0.517\
25.75 & 26.85 & & 43.2 & 1.246 & 0.312 & 24.634 & 0.580 & 25.837 & 0.560\
& &\
22.75 & 24.25 & & 99.8 & 1.496 & 0.071 & 22.750 & 0.053 & 24.245 & 0.078\
23.00 & 24.50 & & 99.7 & 1.496 & 0.091 & 22.998 & 0.068 & 24.492 & 0.098\
23.25 & 24.75 & & 99.7 & 1.490 & 0.112 & 23.244 & 0.084 & 24.731 & 0.124\
23.50 & 25.00 & & 99.4 & 1.488 & 0.140 & 23.493 & 0.101 & 24.977 & 0.147\
23.75 & 25.25 & & 98.8 & 1.486 & 0.169 & 23.735 & 0.128 & 25.222 & 0.186\
24.00 & 25.50 & & 97.1 & 1.476 & 0.209 & 23.977 & 0.157 & 25.447 & 0.228\
24.25 & 25.75 & & 94.0 & 1.465 & 0.256 & 24.214 & 0.194 & 25.668 & 0.273\
24.50 & 26.00 & & 87.0 & 1.437 & 0.280 & 24.426 & 0.237 & 25.846 & 0.319\
24.75 & 26.25 & & 75.7 & 1.424 & 0.311 & 24.601 & 0.280 & 25.987 & 0.390\
25.00 & 26.50 & & 61.5 & 1.393 & 0.340 & 24.711 & 0.347 & 26.057 & 0.460\
25.25 & 26.75 & & 51.0 & 1.364 & 0.328 & 24.712 & 0.468 & 26.024 & 0.525\
25.50 & 27.00 & & 45.9 & 1.322 & 0.340 & 24.656 & 0.564 & 25.948 & 0.588\
25.75 & 27.25 & & 42.9 & 1.292 & 0.313 & 24.603 & 0.574 & 25.868 & 0.571\
Appendix B: NICMOS Color Transformations
========================================
To compare NICMOS data with ground-based IR photometry for stars in globular clusters and the Milky Way bulge requires a rather severe transformation because $F110W$ is much wider and extends well to the blue of standard $J$-band filters. A number of transformations have been published, but none were suitable for our purposes, either because they did not reliably extend to cool giants (Origlia & Leitherer 2000; Marleau et al. 2000) or were based on outdated NICMOS photometry zeropoints (Stephens et al. 2000).
To derive an improved transformation, we used the Girardi isochrones themselves, which are available in both ground-based Bessell-Brett (1988) and NICMOS IR colors. In keeping with our finding that the stellar population of NGC3379 is relatively old and metal rich, and to obtain the reddest possible points along a well-developed giant branch, we base the transformation on the 14 Gyr isochrones of various abundances. Comparison of the isochrones in the two photometric systems shows that there is little dependence of the transformation on metallicity or age and also that the transformation relation clearly differs from the linear approximation of Marleau et al. (2000) by up to nearly 0.2 magnitudes for $J-H > 0.7$ (Figure B1). To derive our improved transformation, we fit a high order polynomial to the isochrone-isochrone comparison and then extended the transformation linearly using the two reddest NICMOS photometric standards, OPH-S1 and CSKD-12 (Figure B1). Our reliance on a linear extrapolation for colors redder than $F110W-F160W = 1.4$ is again probably an oversimplification, but there are no additional data that can be called into play.
Comparison with the four blue NICMOS standard star calibration data ($J-H < 1.$) shows that the isochrone-derived transformation yields a good relation. There has been some uncertainty concerning the correct ground-based $J$-band photometry of the NICMOS standard BRI B0021-02. We have placed it in Figure B1 using the $J-H$ color from the 2MASS data base, transformed to the BB system using equations obtained at the 2MASS website (http://www.ipac.caltech.edu/2mass). The star’s location is consistent with the isochrones. The ground-based photometry listed at the NICMOS STScI website, however, places BRI B0021-02 about 0.15 magnitudes to the blue in the BB system (open triangle in Figure B1). Marleau et al. (2000) use an even bluer color for this star in deriving their linear transformation equation (see their Figure 6 and Appendix), which perhaps led them to adopt a color transformation considerably steeper than ours.
To test our transformation, we compared the IR photometry data of Frogel et al. (1981) for 47 Tuc to the appropriate Girardi isochrones. The Frogel et al. data are on the CIT photometric system, so it is necessary to apply the mild 10% transformation from CIT to the Bessell-Brett system given in Marleau et al. (2000). The left panel of Figure B2 shows the resulting excellent fit of the 47 Tuc IR giant branch by the 14 Gyr, \[Fe/H\]$=-0.7$ Girardi isochrone, adopting $m-M = 13.27$ for 47 Tuc; these numbers are consistent with prevailing views of the stellar population of this cluster (e.g. Hesser et al. 1987). In the right panel, we have transformed the cluster data points to the NICMOS system using our derived transformation (Figure B1) and over-plotted them on the same Girardi isochrone for the NICMOS bandpasses. Using our isochrone-derived transformation, the 47 Tuc data, perhaps not surprisingly, are still well-described by the same age and metallicity isochrone in the NICMOS system. This is not a trivial result, however; we have not applied the transformation to the isochrones, only to the cluster data. The isochrones in the different photometric systems are produced by direct convolution of theoretical spectral energy distributions with filter transmission curves. The transformation we have derived is in this sense a wholly theoretical one, but it works well and is consistent with the four NICMOS standards that overlap the isochrone color range. Other published empirical ground-based–to–NICMOS photometric transformations, however, result in large discrepancies of the 47 Tuc data with the proper isochrone at these red colors. The Marleau et al. transformation, for instance, makes 47 Tuc appear to be solar metallicity (Figure B2), in conflict with all published results for this cluster.
Arimoto, N. 1996, in [*From Stars to Galaxies*]{}, ASP Conference Series 98, Leitherer, C., Fritze-von Alvensleben, U., & Huchra, J., eds. Bekki, K., Couch, W. J., Drinkwater, M. J., & Gregg, M. D. 2001, ApJL, 557, 39. Bertelli, G., Bressan, A., Chiosi, C., Fagotto, F., & Nasi, E. 1994, A&AS, 106, 275 Blanco, B. M. 1984, AJ, 89, 1836 Buta, R. J. & McCall, M. L. 2003, AJ, 125, 1150 Capaccioli, M., Held, E.V., Lorenz, H. & Vietri, M., 1990, AJ, 99, 1813 Capaccioli, M., Vietri, M., Held, E. V., & Lorenz, H. 1991, ApJ, 371, 535 Davidge, T. J. 2002, AJ, 124, 2012 Davies, R. L., Burstein, D., Dressler, A., Faber, S. M., Lynden-Bell, D., Terlevich, R. J. and Wegner, G. 1987, ApJS, 64, 581 Davies, R. L., Sadler, E. M., & Peletier, R. F. 1993, MNRAS, 262, 650 de Vaucouleurs, G. & Capaccioli, M. 1979, ApJS, 40 699 Elston, R. & Silva, D. 1992, AJ, 104, 1360 Faber, S. M., 1973, ApJ, 179, 423 Faber, S. M. et al. 1997, AJ, 114, 1771 Freedman, W. L. 1989, AJ 98, 1285 Freedman, W. L. 1992, AJ 104, 1349 Freedman, W. L. et al. 2001, ApJ 553, 47 Frogel, J. A., Persson, S. E., & Cohen, J.G. 1981 ApJ, 246, 842 Frogel, J. A., Persson, S. E., & Cohen, J.G. 1983 ApJS, 53, 749 Frogel, J. A. 1983 ApJ, 272, 167 Frogel, J. A., Terndrup, D. M., Blanco, V. M., & Whitford, A.E. 1990, ApJ, 353, 494 Frogel, J. A. & Whitford, A. 1987 ApJ, 320, 199 Fruchter, A. S. & Hook, R. N. 2002, PASP, 114, 144 Gibson, B. K. et al. 2000, ApJ, 529, 723 Girardi, L., Bertelli, G., Bressan, A., Chiosi, C., Groenewegen, M. A. T., Marigo, P., Salasnich, B., & Weiss, A. 2002, A&A, 391, 195 Graham, J. A. et al. 1997, ApJ, 477, 535 Gregg, M. D. 1992, ApJ, 384, 43 Gregg, M. D. 1995, ApJ, 443, 527 Gregg, M. D. 1997, New Astronomy, 1, 363 Grillmair, C. et al. 1996, AJ, 112, 1975 Guarnieri, M. D., Renzini, A., & Ortolani, S. 1997, ApJL, 477, 21 Harris, G. L. H., Harris, W. E., & Poole, G. B. 1999, AJ, 117, 855 Hesser, J. E., Harris, W. E., Vandenberg, D. A., Allwright, J. W. B., Shott, P., & Stetson, P. B. 1987 PASP, 99, 739 Hjorth, J. & Tanvir, N. R. 1997, ApJ, 482, 68 Hughes, S. M. G., Wood, P. R. 1990, AJ, 99, 784 Krist, J. 1993, in ASP Conf. Ser. 52: Astronomical Data Analysis Software and Systems II, ed. R. J. Hanisch, R. J. V. Brissendon, & J. Barnes, (San Francisco: ASP), 536 Liu, M. C., Charlot, S. & Graham, J. R. 2000, ApJ, 543, 644 Leitherer, C., et al. 1996, in ASP Conf. Ser. 98: From Stars to Galaxies, ed. Leitherer, C., Fritze-von Alvensleben, U., & Huchra, J., (San Francisco: ASP) Luppino, G. A. & Tonry, J. L. 1993, ApJ, 410, 81 Maihara, T., et al. 2001, PASJ, 53, 25 Maeder, A., Meynet, G. 1989, A&A, 210, 155 Méndez, R. A.; Minniti, D. 2000, ApJ, 529, 911 Minniti, D., Olszewski, E.W., Liebert, J. W., White, S. D. M., Hill, J. M., & Irwin, M. 1995, M.N.R.A.S, 277, 1293. Nieto, J.-L. & Prugniel, P. 1987, A&A, 186, 30 Pastoriza, M. G., Winge, C., Ferrari, F., Macchetto, F. D., & Caon, N. 2000, ApJ, 529, 866 Peletier, R. F. et al. 1999, MNRAS, 310, 863 Rejkuba, M., Minniti, D., Silva, D. R., & Bedding, T. R. 2001, A&A, 379, 781 Rejkuba, M., Minniti, D., Courbin, F., & Silva, D. R. 2002, ApJ, 564, 688 Rejkuba, M., Minniti, D., & Silva, D. R. 2003, A&A, in press (astro-ph/0305432) Renzini, A. 1998, AJ, 115, 2459 Rich, R. M. 1988, AJ, 95, 828 Rose, J. A. 1985, AJ, 90, 1927 Sadler, E. M., Rich, R. M., Terndrup, D. M. 1996, AJ, 112, 171 Sadler, E. M. & Gerhard, O. E. 1985, MNRAS, 214, 177 Sakai, S., Madore, B. F., Freedman, W. L., Lauer, T. R., Ajhar, E. A., & Baum, W. A. 1997, ApJ, 478, 49 Schlegel, D. J., Finkbeiner, D. P., & Davis, M. 1998, ApJ, 500, 525 Schneider, S. E. 1989, ApJ, 343, 94 Soria, R., et al. 1996, ApJ 465, 79 Schweizer, F. & Seitzer P. 1992, AJ, 104, 1039 Statler, T. S. 1994, AJ, 108, 111 Statler, T. S. 2001, AJ, 121 244 Statler, T. S. & Smecker-Hane, T. 1999, ApJ, 117, 839 Tanvir, N. R., Ferguson, H. C., Shanks, T., 1999, MNRAS, 310, 175 Terlevich, A. I. & Forbes, D. A. 2002, MNRAS, 330, 547 Thompson, R. I., Rieke, M., Schneider, G., Hines, D. C. & Corbin, M. R. 1998, ApJL, 492, L95 Tiede, G. P., Frogel, J. A., & Terndrup, D. M. 1995, AJ, 110, 2788 Tonry, J. L., Ajhar, E. A., & Luppino, G. A. 1990, AJ, 100, 1416 Van den Bergh, S. 1989, PASP, 101, 1072 Williams, R. E., et al. 1996, AJ, 112, 1335 Worthey, G. & Ottaviani, D. L. 1997, ApJS, 111, 377
|
---
author:
- |
Naofumi Akimoto$^1$ Huachun Zhu$^2$ Yanghua Jin$^2$ Yoshimitsu Aoki$^1$\
$^1$ Keio University $^2$ Preferred Networks\
[nakimoto@aoki-medialab.jp {zhu, jinyh}@preferred.jp aoki@elec.keio.ac.jp]{}
bibliography:
- 'egbib.bib'
title: Fast Soft Color Segmentation
---
{width="\linewidth"}
Introduction
============
Related Work
============
**Layer decomposition**, actively studied in both computer vision and computer graphics, is the task of decomposing a single image into multiple RGB or RGBA images. For example, reflection removal [@fan2017generic; @Wen_2019_CVPR; @zhang2018single], haze removal [@Gandelsman_2019_CVPR; @Ren-ECCV-2016; @Zhang_2017_CVPR; @7128396], and rain removal [@8099669; @derain_zhang_2018; @Li_2019_CVPR] estimate the foreground and background layers and the mixture ratios for each pixel. Segmentation is also one of the layer decomposition tasks. Part segmentation includes human parsing [@7053923; @Yang_2019_CVPR] or vector graphics layer generation [@FLB17; @sbai2018vector]. Semantic segmentation has been actively addressed [@chen2017deeplab; @liu2019auto; @long2015fully; @ronneberger2015u; @zhao2017pyramid]. The segmentation tasks mentioned above classify each pixel into a single class. In contrast to these hard segmentation tasks, [*soft*]{} segmentation are studied [@aksoy16; @pan2016soft; @sss; @tai07] to expresses transitions between regions of classes using alpha value. For instance, matting [@chen2013knn; @ifm; @levin2008spectral; @samplenet; @wang2018deep; @Xu_2017_CVPR; @zhang2019late] obtains transitions between a foreground and a background. Soft segmentation is more suitable than hard segmentation for handling motion blur, transparency, illumination, and so on. **Soft color segmentation** is a specific soft segmentation task of decomposing a single image into multiple RGB or RGBA color layers, each of which contains homogeneous colors. The state-of-the-art method proposed by [Aksoy [*et al.*]{} ]{}[@aksoy17] improved the method in [Aksoy [*et al.*]{} ]{}[@aksoy16] by adding a sparsity term to the color unmixing formulation. Research similar to [Aksoy [*et al.*]{} ]{}[@aksoy17] includes [Koyama [*et al.*]{} ]{}[@koyama] and [Tan [*et al.*]{} ]{}[@tan18]. [Koyama [*et al.*]{} ]{}[@koyama] generalized [Aksoy [*et al.*]{} ]{}[@aksoy17] to enable advanced color blending. [Tan [*et al.*]{} ]{}[@tan18] are a geometric approach that finds a RGBXY convex hull, extending [Tan [*et al.*]{} ]{}[@tan16]. Whereas [Koyama [*et al.*]{} ]{}[@koyama] and [Tan [*et al.*]{} ]{}[@tan16] consider the order of layer stacking, [Aksoy [*et al.*]{} ]{}[@aksoy17] and [Tan [*et al.*]{} ]{}[@tan18] deal with a [*linear additive model*]{}, which does not consider the order. [Aksoy [*et al.*]{} ]{}[@aksoy17] have shown that color unmixing based segmentation can be achieved by solving the constrained minimization problem, as explained in Section \[section\_optimization\_approach\].
Motivation {#section_optimization_approach}
==========
Several works [@aksoy17; @koyama; @tan16] adopt an optimization-based approach to decompose an image into multiple color-considering RGBA layers. In particular, the state-of-the-art method from [Aksoy [*et al.*]{} ]{}[@aksoy17] proposes the color unmixing formulation, in which the objective function consists of a color constraint, an alpha constraint, a box constraint, and a sparse color unmixing energy function.
To be specific, the color constraint requires that the decomposed layers restore the original image. For a pixel $p$ in the $i^{\rm th}$ layer, $$\sum_i \alpha_i^p \bm{u}_i^p = \bm{c}^p \quad \forall p,
\label{eq:color-constraint}$$ where $\bm{c}^p$ denotes an original color at a pixel $p$, and $\bm{u}_i^p$ denotes a layer color at $p$. We omit the superscript $p$ in the remainder of the paper for convenience.
The alpha constraint enforces the sum of the decomposed layers to be an opaque image: $$\sum_i \alpha_i = 1.
\label{eq:alpha-unity}$$
The box constraint requires that alpha and color values should be in bound: $$\alpha_i, \bm{u}_i \in [0,1] \quad \forall i.
\label{eq:box-constraint}$$
The sparse color unmixing energy function is the weighted sum of the distances between colors and the color distribution in each layer, plus a sparsity term: $$\mathcal{F}_S = \sum_i \alpha_i \mathcal{D}_i(\bm{u}_i) + \sigma \left( \frac{\sum_i \alpha_i}{\sum_i \alpha_{i}^2} - 1 \right),
\label{eq:energy-function}$$ where the layer cost $\mathcal{D}_i(\bm{u}_i)$ is defined as the squared Mahalanobis distance of the layer color $\bm{u}_i$ to the layer distribution $\mathcal{N}(\bm{u}_i, \Sigma_i)$, and $\sigma$ is the sparsity weight.
There is space for improvement of the above approach [@aksoy17] in terms of inference speed. Based on an input image, the iterative optimization of the energy function tends to be slow and running time scales up linearly with the number of pixels, as shown in Section \[section\_quantitative\_evaluation\].
The success of optimzation-based methods [@aksoy17; @koyama; @tan16] and its disadvantage in speed inspire us to train neural networks on the dataset. Without any on-the-fly iteration, the networks decompose the original image into soft color layers in a significant lower inference time. Partly inspired by the minimization of the energy function, we come up with an objective function that is designed for training our neural networks system, as detailed in the next section.
Methods {#section_methods}
=======
Our proposed method consists of three stages: palette color selection, alpha layer estimation and color layer estimation. The input of the proposed system is a single RGB image, and the outputs are RGBA soft color layers.
Palette Color Selection
-----------------------
The inputs of the methods of both [Aksoy [*et al.*]{} ]{}and [Koyama [*et al.*]{} ]{}[@aksoy17; @koyama] are a set of color models that represent the means and the (co)variances of colors of the desired output layers. Although covariances provide additional controllability, a user has to understand the definition of the covariance of colors, how to adjust it, and how it interacts with the system to produce the final color layers. Consequently, the user may not find it intuitive to manipulate covariances. Aiming for an easy-to-use user experience, we believe that a user should not be exposed to more parameters than necessary. Therefore, we make the design choice to take palette colors (means only) instead of color models (means + covariances) as inputs. A palette color has the simple interpretation as the mean value of the colors that should be included in a color layer.
During training, we use K-means algorithm to partition pixels in an input image into $K$ clusters in the 3-dimensional RGB space. We pick the center RGB values of the clusters as the palette colors. The number of palette colors $K$ is fixed throughout the training of the networks. During inference, the palette colors can be specified manually, in addition to automatic selection using K-means.
Alpha Layer Estimation
----------------------
We adopt the U-Net architecture [@ronneberger2015u] for our [*alpha predictor*]{}. The inputs of the alpha predictor are the original RGB image and $K$ single-color images. Each of the single-color image is simply an RGB image filled with a single palette color. The outputs are $K$ alpha layers, which are single-channel images of alpha values. In the subsequent [*alpha layer processing*]{} in Figure \[fig:system\], the outputs of the network are normalized by: $$\alpha_i = \frac{\alpha_i}{\sum_k \alpha_k},
\label{eq:normalization}$$ where $\alpha_i$ is the alpha value (opacity) at a certain pixel position. This normalization step ensures that the output satisfies the alpha-add condition (Eq. (\[eq:alpha-unity\])). For inference, we can add various kinds of alpha layer processing in addition to normalization, which we detail in Section 4.4.
Color Layer Estimation
----------------------
The palette colors and the alpha layers are not sufficient to reconstruct the original image, as shown in Figure (\[fig:composition\]). Although a user only needs to specify the palette colors, each color layer should contains a more variety of colors than a single palette color.
To introduce color variations, we add a [*residue predictor*]{} to estimate color residues from the palette colors. The inputs of the residue predictor are the original image, $K$ single-color images, and $K$ normalized alpha layers. The residue predictor network has an identical architecture as the alpha predictor, except for the number of input and output channels. We add the output residues to palette colors to compute $K$ RGB layers. The final RGBA layers can be obtained by concatenating RGB layers and normalized alpha layers along the channel axis.
{width="\linewidth"}
Network Training
----------------
The above two networks are trained end-to-end with an objective composed of several losses. The main training objective, inspired by the color constraint (Eq. (\[eq:color-constraint\])), is to minimize a self-supervised reconstruction loss between the input and the output: $$L_r = \| \sum_i \alpha_i \bm{u_i} - \bm{c} \|_1.
\label{eq:r_loss}$$
To regularize the training of the alpha predictor, we propose a novel loss for regularization, formulated as the reconstruction loss between the original image and the reconstructed image [*without*]{} color residues: $$\label{eq:regularization_loss}
L_a = \| \sum_i \alpha_i \bm{p_i} - \bm{c} \|_1,$$ where $\bm{p_i}$ denotes the palette color of the $i^{\rm th}$ layer. In other words, $\sum_i \alpha_i \bm{p_i}$ is an image that is reconstructed only using the normalized alpha layers and the palette colors, as shown in the middle left of Figure (\[fig:composition\]). To gather only homogeneous colors in each color layer, we propose a novel distance loss, in reminiscence of Eq. (\[eq:energy-function\]), formulated as $$\label{eq:distance_loss}
L_d = \sum_i \alpha_i \| \bm{p_i} - \bm{u_i} \|_2.$$ We use Euclidean distance in RGB space because our inputs are simply $K$ palette colors. Now we are ready to formulate our total loss as follows: $$L_{total} = L_r + \lambda_a L_a + \lambda_d L_d,
\label{eq:total-loss}$$ where $\lambda_a$ and $\lambda_d$ are coefficients for regularization loss and distance loss, respectively. In comparison with the method of [Aksoy [*et al.*]{} ]{}[@aksoy17] which minimizes the proposed energy function for a given input image, our method trains neural networks to minimize the total loss on a training dataset.
We note that 1) the outputs of the networks automatically satisfy the box constraint (Eq. (\[eq:box-constraint\])) because of the sigmoid functions and clip operations appended to the networks. And 2) we do not enforce sparsity like the sparsity term in sparse color unmixing energy function (Eq. (\[eq:energy-function\])). In preliminary experiments we introduced such a sparsity loss with a coefficient to control its weight in the total loss. When its weight is high, there are no soft transition on the boundary of regions, resulting in nearly hard segmentation. When we decrease the weight, however, the sparsity loss no longer promotes sparsity. We believe that the novel regularization loss $L_a$ and the normalization step in alpha layer processing (Eq. (\[eq:normalization\])) have collectively encouraged the sparsity of alpha layers, and therefore it is redundant to introduce an extra sparsity loss.
Experiments
===========
For training, we use Places365-Standard validation images [@places]. All images in the training dataset have a resolution of $256 \times 256$. We use the Adam optimizer with $lr = 0.2$, $\beta_1 = 0.0$, and $\beta_2 = 0.99$. For all experiments, we set $\lambda_a = 1$ and $\lambda_d = 0.5$.
For inference, since our networks are fully-convolutional, we can apply the model to decompose images of various resolutions. Both of the networks are U-Net with five convolutions, three transposed convolutions, and skip connections at each scale. We discuss the network structures in detail in the supplementary material.
Qualitative Evaluation
----------------------
**Comparison with state-of-the-art methods.** Figure \[fig:qualitative\] shows a qualitative comparison between our results and results of [Aksoy [*et al.*]{} ]{}[@aksoy17] or [Koyama [*et al.*]{} ]{}[@koyama]. Note that the inputs are different: their inputs are distributions of colors, while ours are simply colors.
We would like to mention that there is no optimal solution for soft color segmentation, because an image can be decomposed into meaningful and high-quality color layers in various way. As an attempt to make the comparison as intuitive as possible, we manually select palette colors for our method so that the result color layers look similar to the result in [Aksoy [*et al.*]{} ]{}and [Koyama [*et al.*]{} ]{}[@aksoy17; @koyama].
Quantitative Evaluation {#section_quantitative_evaluation}
-----------------------
**Speed.** Figure \[fig:speed\] shows the running time of our algorithm, algorithm of [Aksoy [*et al.*]{} ]{}and [Tan [*et al.*]{} ]{}[@aksoy17; @tan18], depending on the size of the input image. Both [Aksoy [*et al.*]{} ]{}[@aksoy17] and our method use a palette size of 7, and [Tan [*et al.*]{} ]{}[@tan18] use an average palette size of 6.95, and the median palette size is 7. It takes 2.9 ms for the proposed method to decompose a 1080p resolution (about 2 MP) image, 3.0 ms for a 5 MP image and 3.4 ms for a 4K resolution (about 8 MP) image, averaged from 20 experiments at each resolution. [Aksoy [*et al.*]{} ]{}[@aksoy17] reports that their algorithm takes 1,000 s to decompose a 5 MP image. Their running time scales up linearly as the number of pixels grows due to the per-pixel optimization. [Tan [*et al.*]{} ]{}[@tan18] improve the execution speed of [Aksoy [*et al.*]{} ]{}, utilizing RGBXY convex hull computation and layer updating. We collect their running time (about 50 s to 100 s) in Figure 9 in their paper [@tan18] because exact values are not reported. See supplementary material for detail experimental settings.
The comparison shows that our neural network based method has a significant speed improvement over the state-of-the-art methods. Furthermore, only our method can decompose a video in a practical amount of time. Specifically, when the video consists of 450 frames (1080p, 30 fps, 15 s), our method takes 1.35 seconds, while the method of [Aksoy [*et al.*]{} ]{}[@aksoy17] takes around 50 hours.
![Speed comparison with the state-of-the-art methods. Note that the time axis is in logarithmic scale. []{data-label="fig:speed"}](figs/speed.pdf){width="0.8\linewidth"}
**Reconstruction error.** In Table \[table:quantitative-comparison\], We use pixel-level mean squared error to evaluate the difference between an input image and the reconstructed image. Althought we minimize the reconstruction loss (Eq. (\[eq:r\_loss\])) on a training dataset instead of the input image as [Aksoy [*et al.*]{} ]{}[@aksoy17] do, our reconstruction errors are sufficiently low, indicating that our networks generalize well on the test images. We note that the reconstruction error also depends on the palette colors selected. In particular, the error increases if some palette colors never appear in the original image at all.
Method Reconst. $\downarrow$ PSNR $\uparrow$ SSIM $\uparrow$ Sparsity $\downarrow$
------------------------------- ----------------------- ----------------- ----------------- -----------------------
Aksoy [*et al.*]{} [@aksoy17] [**0.00050**]{} - - -
Ours 0.00088 31.07 0.9740 1.456
w/o $L_r$ 0.00308 25.70 0.9158 1.279
w/o $L_a$ 0.00090 31.17 [**0.9750**]{} 1.959
w/o $L_d$ 0.00076 31.72 0.9743 1.640
w/o Skip 0.00350 27.37 0.9366 [**1.149**]{}
w/o Zero-centered 0.00073 [**31.82**]{} 0.9710 1.450
Single network 0.00104 30.71 0.9633 NaN
: Quantitative comparison. Our method is the only setting in the ablation study that achieves similar reconstruction error as [Aksoy [*et al.*]{} ]{}[@aksoy17] with both low sparsity score and high image quality. See \[section\_quantitative\_evaluation\] for details of the quantitative evaluation, and \[section\_ablation\_study\] for details of the ablation study. []{data-label="table:quantitative-comparison"}
{width="0.8\linewidth"}
Method Aksoy [*et al.*]{} [@aksoy17] Ours w/o $L_a$ w/o $L_d$ w/o Zero-centered
------------------------- ------------------------------- --------------- ----------- ----------- -------------------
Color Var. $\downarrow$ 0.005 [**0.003**]{} 0.007 0.006 0.162
Ablation Study {#section_ablation_study}
--------------
In this section, we validate the losses and architectures of the neural networks. Figure \[fig:ablation-quality\] shows a sample for qualitative comparison. The reconstruction error and the sparsity scores shown in Table \[table:quantitative-comparison\] are the averaged scores of 100 images of 1 MP or more. The sparsity score is calculated as $$L_s = \frac{\sum_i \alpha_i}{\sum_i \alpha_{i}^2} - 1.$$ A lower value of $L_s$ means the decomposed layers are sparser. The color variance score in Table \[table:color-var\] represents the score corresponding to the results in Figure \[fig:ablation-quality\]. This score is the sum of individual variances of the RGB channel averaged over all decomposed layers. For fair comparison, we used same palette colors to decompose an input for each setting.
**Ours versus ours without $\bm{L_r}$.** Table \[table:quantitative-comparison\] and Figure \[fig:ablation-quality\] show that our method without $L_r$ cannot properly reconstruct an input image. $L_a$ trains the only [*alpha predictor*]{}, so the [*residue predictor*]{} cannot function properly.
**Ours versus ours without $\bm{L_a}$.** Although the SSIM score is marginally better without $\bm{L_a}$, the sparsity score is significantly higher, which suggests that excessive overlapping exists between alpha layers. On the third row in Figure \[fig:ablation-quality\], the blue plate on the person’s hand wrongly shows up in the white layer (highlighted in red), causing overlapping between white and blue layers. Overlapping is intended to occur only sparsely, e.g. at the boundary of a region, because excessive overlapping is not suitable for application to image editing.
Moreover, the color variance score is higher without $\bm{L_a}$, indicating that some of the layer might be contaminated, i.e. containing colors that is much different from the corresponding palette color. We can observe such contamination on the top-left corner of the blue layer. We believe this is because $L_a$ improves the performance alpha predictor.
**Ours versus ours without $\bm{L_d}$.** Without $\bm{L_d}$ suppressing the variance of colors in each layer, the reconstruction error decreases. However, a large color variance causes the same color to spread across multiple layers, which is not desirable for color-based editing, as shown in Table \[table:color-var\] and Figure \[fig:ablation-quality\].
**Ours versus ours without skip connection.** Without skip connections in the [*alpha predictor*]{} and the [*residue predictor*]{}, alpha layers are not accurate, leading to a higher the reconstruction error.
**Ours versus ours without zero-centered residues.** It is easier to train a neural network with zero-centered output, and we can make sure the palette color is the mean value of each layer. Without zero-centering, the PSNR increases, but the color variance increases even more. **Ours versus a single network only.** If we use a plain single network, the reconstruction error increases. We believe that accurate alpha layers as inputs enhance the performance of the residue predictor. Specifically, in our method, we apply smoothing filters to remove checkerboard artifacts from alpha layers, as shown in Figure \[fig:same-color\], and predict the RGB channels based on the processed alpha layers. We doubt that there is a way to incorporate smoothing filter processing in a neural network that predicts both alpha and RGB channels simultaneously.
![ Comparison with hard segmentation and color-based soft segmentation on images with ambiguous object boundaries. In (a), hard segmentation produces a background layer that is tainted by the moving object in the foreground. In (b), hard semantic segmentation divides the image into semantic areas, but soft color segmentation enable us to change simultaneously the reflected light on the rock along with the lighting condition of the sky. []{data-label="fig:usefulcase"}](figs/usefulcase.pdf){width="\linewidth"}
Applications
------------
**Decomposition of images with blurry boundary.** Soft color segmentation is useful not only for decomposing an image into soft color layers, but for any case where the resulting layers are preferred to have blurry boundary. Figure \[fig:usefulcase\] shows the results of our decomposition and the benefits of soft color segmentation. In the case of an image with a foreground object with motion blur, although both the foreground and the background are visible at a single pixel, hard segmentation has to assign that pixel into either of the classes. In such a case, soft segmentation can recognize the pixel as a mixture of fore/background and thus has an advantage over conventional hard segmentation.
**Natural image editing.** Figure \[fig:teaser\] and Figure \[fig:image-editing\] show examples of recoloring and compositing. These editing results are created by editing each decomposed layer with the [*alpha add*]{} mode in Adobe After Effects.
**Video decomposition.** Figure \[fig:video-decomposition\] shows our method performing video soft color segmentation. We decompose the video frame-by-frame, without any constraints for temporal consistency. Nevertheless, the decomposed layers do not flicker. It can be partly attributed to applying smoothing filter and fixing color palettes, both of which encourage consistent alpha layers, and regularization loss $\bm{L_d}$, which encourages consistent color layers. Compared to other methods, only our method can decompose video in practical time. See the video material for detailed results.
**Alpha layer processing.** In the [*alpha layer processing*]{} stage, a user can edit the predicted alpha layers, and subsequently use these edited alpha layers for color estimation, thanks to the fact that the estimation of alpha layers is independent from color layers. As shown in Figure \[fig:alpha-processing\], we can use the guided filter [@gfilter; @fastgfilter] to smooth the image, or manipulate a mask to change the alpha region. We can prepare the mask manually or automatically, capitalizing on state-of-the-art methods (e.g. semantic segmentation and depth estimation). Therefore, our method can be complementary to various image editing techniques.
![An example when the palette contains a duplicated color. It shows that, although the number of layers is fixed during training, our model tolerates the duplication of colors and is thus capable of decomposing an image into fewer layers.[]{data-label="fig:same-color"}](figs/image-editing.pdf){width="0.95\linewidth"}
![An example when the palette contains a duplicated color. It shows that, although the number of layers is fixed during training, our model tolerates the duplication of colors and is thus capable of decomposing an image into fewer layers.[]{data-label="fig:same-color"}](figs/video-decomposition.pdf){width="0.95\linewidth"}
![An example when the palette contains a duplicated color. It shows that, although the number of layers is fixed during training, our model tolerates the duplication of colors and is thus capable of decomposing an image into fewer layers.[]{data-label="fig:same-color"}](figs/alpha-processing.pdf){width="0.95\linewidth"}
![An example when the palette contains a duplicated color. It shows that, although the number of layers is fixed during training, our model tolerates the duplication of colors and is thus capable of decomposing an image into fewer layers.[]{data-label="fig:same-color"}](figs/same-color.pdf){width="0.95\linewidth"}
Limitations
-----------
**Memory limit.** Because we use a GPU for inference, we cannot handle high-resolution images that exceed the GPU memory limit. Also, sufficient GPU memory is needed to hold the intermediate features of the encoder-decoder networks with skip connections. Concretely, when an input is 1080p (4K) resolution and computation is based on 32-bit floating point, about 12 (21) GB is consumed. Countermeasures include using 16-bit floating point and discarding unnecessary intermediate features.
**Fixed palette size.** In our approach, the number of decomposed layers for each trained model is fixed. To handle various numbers of layers, one solution is to train a model that decomposes an image into the sufficient number of layers, and use palettes with duplicated colors, as shown in Figure \[fig:same-color\]. We merge layers of duplicated colors after decomposition.
Conclusion
==========
Acknowledgement {#acknowledgement .unnumbered}
===============
We are grateful for Yingtao Tian and Prabhat Nagarajan for helpful advice on writing the paper. We thank Edgar Simo-Serra and all reviewers for helpful discussions and comments.
|
---
abstract: 'The measurement of the pion form factor and, more generally, of the cross section for electron–positron annihilation into hadrons through the radiative return has become an important task for high luminosity colliders such as the $\Phi$- or $B$-meson factories. This quantity is crucial for predictions of the hadronic contributions to $(g-2)_\mu$, the anomalous magnetic moment of the muon, and to the running of the electromagnetic coupling. But the radiative return opens also the possibility of many other physical applications. The physics potential of this method at high luminosity meson factories is discussed and recent results are reviewed.'
author:
- 'Germán Rodrigo [^1]'
title: 'RITORNO RADIATIVO PER LA MISURA DI R: COME E PERCHÉ [^2]'
---
Electron–positron annihilation into hadrons is one of the basic reactions of particle physics, crucial for the understanding of hadronic interactions. At high energies, around the $Z$ resonance, the measurement of the inclusive cross section and its interpretation within perturbative QCD give rise to one of the most precise and theoretically founded determinations of the strong coupling constant $\alpha_s$. Also, measurements in the intermediate energy region, between 3 GeV and 11 GeV can be used to determine $\alpha_s$ and at the same time give rise to precise measurements of charm and bottom quark masses. The low energy region is crucial for predictions of the hadronic contributions to $a_\mu=(g-2)_\mu/2$, the anomalous magnetic moment of the muon, and to the running of the electromagnetic coupling from its value at low energy up to $M_Z$. Last, but not least, the investigation of the exclusive final states at large momenta allows for tests of our theoretical understanding of form factors within the framework of perturbative QCD. Beyond the intrinsic interest in this reaction, these studies may provide important clues for the interpretation of exclusive decays of B-mesons, a topic of evident importance for the extraction of CKM matrix elements.
The main uncertainty to $a_\mu$ and $\alpha_\mathrm{QED}$ is driven by their respective hadronic contributions, which are not calculable perturbatively, but can be estimated though dispersion integrals $$a_{\mu}^{\mathtt{had},\mathrm{LO}}
= \left( \frac{\alpha m_\mu}{3 \pi} \right)^2 \int_{4m_{\pi}^2}^{\infty}
\frac{ds}{s^2} \; \hat{K}(s) \; R(s)~, \nonumber$$ $$\Delta \alpha_{\mathtt{had}}(m_Z^2) = -\frac{\alpha m_Z^2}{3 \pi} \
\mathrm{Re} \int_{4m_{\pi}^2}^{\infty}
\frac{ds}{s} \; \frac{ R(s)}{s-m_Z^2-i\eta}~,$$ where the spectral function $R(s)$ is obtained from experimental data of the reaction $e^+e^- \to$ hadrons. The most recent experimental result for $a_\mu$ [@Bennet] shows a 2$\sigma$ discrepancy with respect to the SM prediction for this quantity [@Davier:2002dy; @HMNT02; @Ghozzi:2003yn]. Alternatively, one can also use current conservation (CVC) and isospin symmetry to obtain $R(s)$ from $\tau$ decays. In the latter, a 0.7$\sigma$ discrepancy is found [@Davier:2002dy], which however is incompatible with the $e^+e^-$ based result. Unaccounted isospin breaking corrections due to the difference of the mass and width of the neutral to the charged $\rho$-meson could explain this discrepancy [@Ghozzi:2003yn], leaving the $e^+e^-$ based analysis as the most reliable.
The recent advent of $\Phi$- and $B$-meson factories allows us to exploit the radiative return to explore the hadronic cross section in the whole energy region from threshold up to the nominal energy of the collider in one homogeneous data sample [@Binner:1999bt; @Zerwas]. The radiative suppression factor ${\cal O}(\alpha/\pi)$ is easily compensated at these factories by their enormous luminosity.
In principle, the reaction $e^+e^- \to \gamma + {\mathrm {hadrons}}$ receives contributions from both initial- and final-state radiation (Fig. \[fig1\]), ISR and FSR respectively. Only the former is of interest for the radiative return. A variety of methods to disentangle FSR from the ISR contribution have been described in detail in [@Binner:1999bt; @Kuhn:2002xg; @Rodrigo:2002hk; @Czyz:2002np], among them the employment of suitable kinematical cuts to suppress FSR, or the identification of different distributions, e.g. angular distributions, charge asymmetry, for independent tests of the FSR model amplitude. Notice however that at $B$-factories the $\pi^+\pi^-\gamma$ final state is completely dominated by ISR.
The proper analysis requires necessarily the construction of Monte Carlo event generators. The event generators EVA [@Binner:1999bt] and EVA4$\pi$ [@Czyz:2000wh] were based on a leading order treatment of ISR and FSR, supplemented by an approximate inclusion of additional collinear radiation based on structure functions. Subsequently, the event generator PHOKHARA was developed; it is based on a complete next-to-leading order (NLO) treatment of radiative corrections [@Kuhn:2002xg; @Rodrigo:2002hk; @Czyz:2002np]. In its version 2.0 it included ISR at NLO and FSR at LO for $\pi^+ \pi^-$ and $\mu^+ \mu^-$ final states, and four-pion final states (without FSR) with some improvements with respect to the formulation described in [@Czyz:2000wh].
The most recent version of PHOKHARA, version 3.0 [@Czyz:2002np], allows for the simultaneous emission of one photon from the initial and one photon from the final state. This includes in particular the radiative return to $\pi^+ \pi^- (\gamma)$ and thus the measurement of the (one-photon) inclusive $\pi^+ \pi^-$ cross section, an issue closely connected to the question of $\pi^+ \pi^- (\gamma)$ contributions to $a_{\mu}$.
Recently, a new Monte Carlo event generator, EKHARA [@Czyz:2003gb], has been constructed to simulate the reaction $e^+e^- \to \pi^+\pi^-e^+e^-$, a potential background of the radiative return specially at lower energies. Future developments of PHOKHARA include the simulation of FSR at NLO and the narrow resonances for $\mu^+\mu^-$, as well as many other hadronic channels: $K^+K^-$, $K^0\bar{K}^0$, $3\pi$, $KK\pi$, $p\bar{p}$, and the simulation of the continuum $q\bar{q}$ supplemented by some hadronization model. Encouraging preliminary experimental results from KLOE, BABAR and BELLE [@unknown:2003jn; @Pisa] demonstrate the power of the method and its physics potential.\
[**Acknowledgements:**]{} This contribution is based on work performed in collaboration with H. Czyż, A. Grzelińska and J.H. Kühn. I’m very grateful to P. Gambino for his kind invitation and to P. Ciafaloni for the pleasant organization of the workshop. Partial support from Generalitat Valenciana under grant CTIDIB/2002/24, and MCyT under grant FPA-2001-3031.
[99]{}
M. Davier, S. Eidelman, A. Höcker and Z. Zhang, hep-ph/0308213. K. Hagiwara, A.D. Martin, D. Nomura and T. Teubner, Phys. Lett. [**B 557**]{} (2003) 69 \[hep-ph/0209187\]. T. Teubner, HEP2003 Aachen. S. Ghozzi and F. Jegerlehner, hep-ph/0310181. G.W.Bennett [*et al.*]{} \[Muon $g-2$ Collaboration\], Phys. Rev. Lett. [**89**]{} (2002) 101804; Erratum, ibid. [**89**]{} (2002) 129903, \[hep-ex/0208001\]. S. Binner, J. H. Kühn and K. Melnikov, Phys. Lett. B [**459**]{} (1999) 279 \[hep-ph/9902399\]. Min-Shih Chen and P. M. Zerwas, Phys. Rev. D [**11**]{} (1975) 58. H. Czyż and J. H. Kühn, Eur. Phys. J. C [**18**]{} (2001) 497 \[hep-ph/0008262\]. J. H. Kühn and G. Rodrigo, Eur. Phys. J. C [**25**]{} (2002) 215 \[hep-ph/0204283\]. G. Rodrigo, H. Czyż, J.H. Kühn and M. Szopa, Eur. Phys. J. C [**24**]{} (2002) 71 \[hep-ph/0112184\]. G. Rodrigo, Acta Phys. Polon. B [**32**]{} (2001) 3833 \[hep-ph/0111151\]. G. Rodrigo, A. Gehrmann-De Ridder, M. Guilleaume and J. H. Kühn, Eur. Phys. J. C [**22**]{} (2001) 81 \[hep-ph/0106132\]. G. Rodrigo, H. Czyż and J. H. Kühn, hep-ph/0205097; Nucl. Phys. Proc. Suppl. [**123**]{} (2003) 167 \[hep-ph/0210287\]; Nucl. Phys. Proc. Suppl. [**116**]{} (2003) 249 \[hep-ph/0211186\]. H. Czyż and A. Grzelińska, hep-ph/0310341. H. Czy[ż]{}, A. Grzeli[ń]{}ska, J. H. K[ü]{}hn and G. Rodrigo, hep-ph/0308312; Eur. Phys. J. C [**27**]{} (2003) 563 \[hep-ph/0212225\].\
[http://cern.ch/german.rodrigo/phokhara]{}
H. Czyż and E. Nowak, hep-ph/0310335. A. Denig \[KLOE Collaboration\], hep-ex/0311012. S. Di Falco \[KLOE Collaboration\], hep-ex/0311006. A. Aloisio [*et al.*]{} \[KLOE Collaboration\], hep-ex/0307051. B. Aubert [*et al.*]{} \[BABAR Collaboration\], hep-ex/0310027. B. Valeriani, [http://www.pi.infn.it/congressi/sighad03/]{}.\
M. Davier, ibid.. S. Eidelman, ibid..
[^1]: Supported by EC 5th Framework Programme under contract HPMF-CT-2000-00989. E-mail: [german.rodrigo@cern.ch]{}
[^2]: Talk given in italian at Incontri sulla Fisica delle Alte Energie, Lecce, Italy, Apr 2003.
|
---
date: 'June 30, 2009'
title: 'Spectrum and variability of the Galactic Center VHE $\gamma$-ray source HESS J1745$-$290'
---
[A detailed study of the spectrum and variability of the source HESS J1745$-$290 in the Galactic Center (GC) region using new data from the H.E.S.S. array of Cherenkov telescopes is presented. Flaring activity and quasi periodic oscillations (QPO) of HESS J1745$-$290 are investigated.]{} [The image analysis is performed with a combination of a semi-analytical shower model and the statistical moment-based Hillas technique. The spectrum and lightcurves of HESS J1745$-$290 are derived with a likelihood method based on a spectral shape hypothesis. Rayleigh tests and Fourier analysis of the H.E.S.S. GC signal are used to study the periodicity of the source.]{} [With a three-fold increase in statistics compared to previous work, a deviation from a simple power law spectrum is detected for the first time. The measured energy spectrum over the three years 2004, 2005 and 2006 of data taking is compatible with both a power law spectrum with an exponential cut-off and a broken power law spectrum. The curvature of the energy spectrum is likely to be intrinsic to the photon source, as opposed to effects of interstellar absorption. The power law spectrum with an exponential cut-off is characterized by a photon index of 2.10 $\pm$ 0.04$_{\mathrm{stat}}$ $\pm$ 0.10$_{\mathrm{syst}}$ and a cut-off energy at 15.7 $\pm$ 3.4$_{\mathrm{stat}}$ $\pm$ 2.5$_{\mathrm{syst}}$ TeV. The broken power law spectrum exhibits spectral indices of 2.02 $\pm$ 0.08$_{\mathrm{stat}}$ $\pm$ 0.10$_{\mathrm{syst}}$ and 2.63 $\pm$ 0.14$_{\mathrm{stat}}$ $\pm$ 0.10$_{\mathrm{syst}}$ with a break energy at 2.57 $\pm$ 0.19$_{\mathrm{stat}}$ $\pm$ 0.44$_{\mathrm{syst}}$ TeV. No significant flux variation is found. Increases in the $\gamma$-ray flux of HESS J1745$-$290 by at least a factor of two would be required for a 3$\sigma$ detection of a flare with time scales of an hour. Investigation of possible QPO activity at periods claimed to be detected in X-rays does not show any periodicities in the H.E.S.S. signal.]{}
Introduction
============
The discovery of very high energy (VHE) $\gamma$-rays from the Galactic Center (GC) has been reported by CANGAROO (Tsuchiya et al [@Tsuchiya04]), VERITAS (Kosack et al. [@Kosack]), H.E.S.S. (HESS J1745$-$290, Aharonian et al. [@Aharonian2]) and MAGIC (Albert et al. [@Albert]). Possible associations of the source with the Sgr A East supernova remnant (Crocker et al. [@Crocker]) and more recently with the newly detected plerion G359.95$-$0.04 (Wang et al. [@Wang]) have been widely discussed in the literature. However, with the reduced systematic pointing error obtained using H.E.S.S. data up to 2006 (van Eldik et al. [@VanEldik]), Sgr A East is now ruled out as being associated with the VHE emission of HESS J1745$-$290. The interpretation of the GC TeV signal as annihilation products of dark matter (DM) particles has been discussed in Aharonian et al. ([@PRL]). It is unlikely that the bulk of the signal comes from DM annihilations. One possibility is that the supermassive black hole Sgr A\* located at the center of the Milky Way is responsible for the VHE emission of the detected HESS J1745$-$290 source. In this case, a cut-off in the high energy part of the spectrum might be expected (Ballantyne et al. [@Ballantyne], Aharonian and Neronov [@Neronov1]). Time variability as seen in X-rays and IR might also appear, along with flares, and QPO frequencies such as those found in Aschenbach et al. ([@Aschenbach]). QPO periods of $\approx$ 100 s, 219 s, 700 s, 1150 s and 2250 s have been observed simultaneously in two different datasets collected with the Chandra (Baganoff et al. [@Baganoff]) and the XMM-Newton (Porquet et al. [@Porquet]) observatories in 2000 and 2002, respectively. The power density spectra of the 2003 infrared flare also shows three distinct peaks at $\approx$ 214 s, 733 s and 1026 s (Genzel et al. [@Genzel]), fully consistent with three of the five X-ray detected periods. Recently however, the validity of the detection of these periods has been disputed by infrared observations on the Keck II telescope (see Meyer et al. ([@Meyer]) for references).\
The search for a curvature in the TeV energy spectrum and time variability in the TeV signal is strongly motivated by GC wideband emission models and multi-wavelength data, respectively. In this paper, updated results on the energy spectrum and variability of HESS J1745$-$290 are presented, based on a dataset collected in 2004, 2005 and 2006. Preliminary results on the source position and morphology were published in van Eldik et al. ([@VanEldik]).
H.E.S.S. observations and analysis
==================================
The H.E.S.S. instrument
-----------------------
The H.E.S.S. (High Energy Stereoscopic System) instrument (Hofmann et al. [@Hofmann]) consists of four Imaging Atmospheric Cherenkov Telescopes (IACTs) located in the Khomas Highland of Namibia at an altitude of 1800 m above sea level. H.E.S.S. is dedicated to very high energy $\gamma$-ray astronomy (defined as E $\geq$ 100 GeV), beyond the energy range accessible to satellite-based detectors. The IACTs have a large mirror area of 107 m$^{2}$, reflecting the Cherenkov light emitted by $\gamma$-induced air showers onto a camera of 960 photomultiplier tubes (PMTs). Each PMT covers a field of view of 0.16. The total field of view of the camera is 5in diameter. The telescopes are positioned at the corners of a square, of side 120 m, which allows for an accurate reconstruction of the direction and energy of the $\gamma$-rays using the stereoscopic technique. The energy threshold of H.E.S.S. is approximately 100 GeV at zenith. More details on the H.E.S.S. instrument can be found in Aharonian et al. ([@Crab]).
Dataset and analysis
--------------------
The previously published H.E.S.S. observations on the GC (Aharonian et al. [@PRL]) was on a 48.7 hours (live time) data sample collected in 2004. In this paper new results on the spectral analysis of the GC are derived, using data from subsequent observation campaigns carried out in 2005 and 2006, with zenith angles ranging up to 70. The runs taken at high zenith angles are sensitive to higher $\gamma$-ray energies and allow us to probe the high energy part of the HESS J1745$-$290 spectrum. The datasets are described in Table \[table1\].
------ ---------------------- ---------------------- --------------- ---------------------- --------------
Year $\theta_z$[^1] range $\bar{\theta_z}^{a}$ T$_{obs}$[^2] $\rm N_{\gamma}$[^3] $\sigma$[^4]
() () (hrs)
2004 0-60 21.3 28.5 1516 35.6
2005 0-70 26.6 51.7 2062 41.2
2006 0-50 19.1 12.7 607 23.0
All 0-70 23.0 92.9 4185 60.7
------ ---------------------- ---------------------- --------------- ---------------------- --------------
: Details of the observation of the GC region with H.E.S.S. for the 2004, 2005 and 2006 observation campaigns.[]{data-label="table1"}
Most of the data were taken in “wobble mode” where the telescope pointing is typically shifted by $\pm$ 0.7from the nominal target position. The dataset used for the analysis was selected using the standard quality criteria, excluding runs taken under bad and variable weather conditions, which would lead to large systematic errors. An additional quality selection was applied using the cosmic ray rate during data collection. Runs with rates deviating by more than 3$\sigma$ from the cosmic ray rate averaged over the three years of observation were removed in the subsequent analysis. After the run selection procedure, the dataset amounts to 93 hours of live time.\
Two independent techniques are commonly used to select $\gamma$-ray events and derive energy spectra and lightcurves. The first technique computes the “Hillas geometrical moments” of the shower image (Aharonian et al. [@Hillas]) and the second one is based on a semi-analytical model of showers, which predicts the expected intensity in each pixel of the camera (de Naurois et al. [@deNaurois]). The shower direction, the impact point and the primary particle energy are then derived with a likelihood fit of the model to match the data. Both analysis techniques provide an energy resolution of 15-20$\%$ with an angular resolution better than 0.1above the analysis energy threshold. Results described in this paper are obtained with a combination of these two analysis techniques, the so-called combined Hillas/Model analysis (de Naurois et al. [@deNaurois]), which yields an improved background rejection, and were cross-checked against either technique alone. The background is calculated at each position in the field of view using the acceptance corrected background rate from an annulus around that position (the OFF-source region). More details on this so-called ring background method are given in Berge et al. ([@Berge]). The OFF-source regions do not overlap with known TeV sources in the GC field of view (Aharonian et al. [@GalacticScanPaper]) and areas of diffuse emission (Aharonian et al. [@DE]).\
The analysis of the whole 2004-2006 dataset shows an excess above the background of 4185 $\gamma$-events in a 0.11radius region centered on the GC (Table \[table1\]). The total significance, calculated according to the method of Li and Ma ([@LiMa]), is 60.7$\sigma$. Diffuse $\gamma$-ray emission along the galactic plane is visible as shown by the distribution in $\theta$, $\theta$ being the angle of a reconstructed $\gamma$-ray relative to Sgr A\* (Fig. \[fig6\], excess above the background level outside the ON source region). A discussion in Aharonian et al. ([@DE]) gives a possible interpretation of the emission as cosmic ray interactions in the central molecular zone (CMZ). A point-source + linear background fit to the $\theta$ distribution has been performed to estimate the diffuse emission contribution to the signal. Extrapolating the linear component inside the ON-source region gives a diffuse emission contribution of 13$\%$ $\pm$ 1$\%$ within 0.11. Diffuse $\gamma$-ray emission located in the ON-source region was not subtracted in the following analysis, since its spectral feature (a simple power law with the same spectral index as the central source) would not influence the result (Aharonian et al. [@DE]). Any deviation from a power law spectrum or variability in the lightcurve would be solely due to the point source.
Energy spectrum of HESS J1745$-$290
===================================
Spectral reconstruction
-----------------------
The energy spectrum of HESS J1745$-$290 has been derived following the forward folding method. The forward folding method is based on a spectral shape hypothesis for the source spectrum (Piron et al. [@Piron1], Djannati-Atai et al. [@Djannati]). Spectral points are then calculated using the adjusted spectral shape and the 1$\sigma$ error bars are computed using the error matrix of the fit procedure. Relative systematic errors on the reconstructed spectral indices coming from the presence of broken pixels in the camera are less than 5$\%$, whereas those coming from variations of the atmospheric conditions are negligible. Thus, systematic errors on the spectral indices derived in the following analysis are taken to be 5$\%$. Systematic errors on the integrated fluxes mainly come from the variations of the atmospheric conditions and the absolute calibration of the response of the telescopes. They amount to 10 to 20$\%$ (Aharonian et al. [@Crab]). The systematic errors on the integrated fluxes above 1 TeV are taken to be 20$\%$. A Monte-Carlo study of the reconstruction of the cut-off energy revealed a systematic bias linearly increasing with the cut-off energy: $$\mathrm{E_{cut}} = (0.92 \pm 0.01) \times \mathrm{E_{cut,true}} + (0.25 \pm 0.05)\,\mathrm{TeV}.$$ The systematic errors on the reconstruction of the cut-off energy amount to 17$\%$.
Results
-------
The data taken in 2004, 2005 and 2006 were compared to the folding of three distributions with the detector response: a power law (Eq. \[Eq1\]), a power law with a high energy exponential cut-off (Eq. \[Eq2\]) and a smoothed broken power law (Eq. \[Eq3\]): $$\begin{aligned}
\frac{dN}{dE}&=\Phi_{0}\times\Big(\frac{E}{1 \rm{TeV}}\Big)^{-\Gamma}\label{Eq1}\\
\frac{dN}{dE}&=\Phi_{0}\times\Big(\frac{E}{1 \rm{TeV}}\Big)^{-\Gamma}\times e^{-(\frac{E}{E_{\rm{cut}}})^{\beta}}\label{Eq2}\\
\frac{dN}{dE}&=\Phi_{0}\times\Big(\frac{E}{1 \rm{TeV}}\Big)^{-\Gamma_1}\times\frac{1}{\Big(1+\Big(\frac{E}{E_{\rm{break}}}\Big)^{(\Gamma_2-\Gamma_1)}\Big)}\label{Eq3}\end{aligned}$$ where $\Phi_{0}$ is the flux normalisation in TeV$^{-1}$ cm$^{-2}$ s$^{-1}$, $\Gamma_i$ the spectral indices. E$_{\rm{cut}}$ is the cut-off energy in Eq. \[Eq2\], and E$_{\rm{break}}$ is the break energy in Eq. \[Eq3\]. In Eq. \[Eq2\], $\beta$ is the strength of the cut-off. Except at the end of paragraph 3.2.1., $\beta$ is taken equal to one.\
The measured spectrum for the whole three-year dataset ranges from 160 GeV, the energy threshold of the analysis, to 70 TeV (Fig. \[fig7\]). For the first time, with additional statistics, a deviation from a pure power law starts to be visible. The spectrum is well described by either Eq. \[Eq2\] (equivalent $\chi^{2}$ of 23/26 d.o.f.) or Eq. \[Eq3\] (equivalent $\chi^{2}$ of 20/19 d.o.f.). Fig. \[fig7\] shows the HESS J1745$-$290 spectra with fits to Eq. \[Eq2\] and Eq. \[Eq3\]. The power law with an exponential cut-off fit yields $\Phi_0$ = (2.55 $\pm$ 0.06$_{\mathrm{stat}}$ $\pm$ 0.40$_{\mathrm{syst}}$) $\times$ 10$^{-12}$ TeV$^{-1}$ cm$^{-2}$ s$^{-1}$, $\Gamma$ = 2.10 $\pm$ 0.04$_{\mathrm{stat}}$ $\pm$ 0.10$_{\mathrm{syst}}$, a cut-off energy E$_{\rm{cut}}$ = (15.7 $\pm$ 3.4$_{\mathrm{stat}}$ $\pm$ 2.5$_{\mathrm{syst}}$) TeV and an integrated flux above 1 TeV of (1.99 $\pm$ 0.09$_{\mathrm{stat}}$ $\pm$ 0.40$_{\mathrm{syst}}$) $\times$ 10$^{-12} $cm$^{-2}$ s$^{-1}$. The broken power law fit yields $\Phi_0$ = (2.57 $\pm$ 0.07$_{\mathrm{stat}}$ $\pm$ 0.40$_{\mathrm{syst}}$) $\times$ 10$^{-12}$ TeV$^{-1}$ cm$^{-2}$ s$^{-1}$, $\Gamma_1$ = 2.02 $\pm$ 0.08$_{\mathrm{stat}}$ $\pm$ 0.10$_{\mathrm{syst}}$, $\Gamma_2$ = 2.63 $\pm$ 0.14$_{\mathrm{stat}}$ $\pm$ 0.10$_{\mathrm{syst}}$, a break energy E$_{\rm{break}}$ = (2.57 $\pm$ 0.19$_{\mathrm{stat}}$ $\pm$ 0.44$_{\mathrm{syst}}$) TeV and an integrated flux above 1 TeV of (1.98 $\pm$ 0.38$_{\mathrm{stat}}$ $\pm$ 0.40$_{\mathrm{syst}}$) $\times$ 10$^{-12} $cm$^{-2}$ s$^{-1}$. The values of the cut-off energy E$_{\rm{cut}}$ and break energy E$_{\rm{break}}$ are those corrected for the systematic bias mentioned in the previous section. By comparison, a power law spectrum gives a worse equivalent $\chi^{2}$ of 64/27 d.o.f. and yields a flux normalisation of (2.40 $\pm$ 0.05$_{\mathrm{stat}}$ $\pm$ 0.40$_{\mathrm{syst}}$) $\times$ 10$^{-12}$ TeV$^{-1}$ cm$^{-2}$ s$^{-1}$, a spectral index of 2.29 $\pm$ 0.02$_{\mathrm{stat}}$ $\pm$ 0.10$_{\mathrm{syst}}$ with an integrated flux above 1 TeV of (1.87 $\pm$ 0.05$_{\mathrm{stat}}$ $\pm$ 0.40$_{\mathrm{syst}}$) $\times$ 10$^{-12} $cm$^{-2}$ s$^{-1}$. The effect of introducing a high cut-off energy in the power law spectrum does not change the integrated flux by more than 10$\%$, less than systematic errors. Integrated fluxes found either with a power law with an exponential cut-off shape or a smoothed broken power law one are consistent with the values given in the previously published H.E.S.S. analysis (Aharonian et al. [@PRL]). The spectral parameters for the two curved spectral shapes, for year-wise data, are given in the next two subsections.
### Power law with an exponential cut-off
Spectral parameters for the different years (differential flux normalisation $\Phi_0$, spectral index $\Gamma$, cut-off energy and integrated flux above 1 TeV, I($\geq$1 TeV)) are summarized in Table \[table2\]. The cut-off energy corrected for the systematic bias mentioned in section 3.1 is also shown. The energy ranges of the 2004 and 2006 spectra do not allow the determination of a significant cut-off. The 2004 and 2006 datasets contain fewer runs collected at high zenith angles than the 2005 one. High zenith angle observations provide an increased effective area at high energies and contribute to the high energy part of the spectrum. The 2005 spectral index value is smaller than the values for 2004 and 2006, because of the correlation between the reconstructed spectral index $\Gamma$ and the cut-off energy induced by the fit procedure. Fig. \[fig11\] shows a 2-D plot of the fitted photon index $\Gamma$ against the cut-off energy for each year’s dataset. It can be seen that the spectral parameter values are compatible. Moreover, when fixing the cut-off energy to the uncorrected cut-off energy of 14.7 TeV (see Table \[table2\]), the spectral fits give $\Gamma$ = 2.14 $\pm$ 0.07$_{\mathrm{stat}}$ $\pm$ 0.10$_{\mathrm{syst}}$, $\Gamma$ = 2.04 $\pm$ 0.04$_{\mathrm{stat}}$ $\pm$ 0.10$_{\mathrm{syst}}$ and $\Gamma$ = 2.11 $\pm$ 0.09$_{\mathrm{stat}}$ $\pm$ 0.10$_{\mathrm{syst}}$ for the 2004, 2005 and 2006 datasets, respectively. Spectral indices are thus compatible with each other.
------ ------------------------------------------ ----------------- -------------------- ------------------------- --------------------------------- ----------------
Year $\Phi_0$[^5] $\Gamma$[^6] E$_{\rm{cut}}$[^7] E$_{\rm{cut,true}}$[^8] I($\geq$1 TeV)[^9] $\chi^{2}$/dof
$\rm (10^{-12} TeV^{-1} cm^{-2} s^{-1})$ (TeV) (TeV) $\rm (10^{-12} cm^{-2} s^{-1})$
2004 2.40 $\pm$ 0.10 2.20 $\pm$ 0.07 20.70 $\pm$ 11.80 22.20 $\pm$ 11.80 1.81 $\pm$ 0.14 25/26
2005 2.56 $\pm$ 0.09 1.94 $\pm$ 0.07 9.09 $\pm$ 2.13 9.61 $\pm$ 2.13 2.09 $\pm$ 0.16 33/25
2006 2.35 $\pm$ 0.16 2.16 $\pm$ 0.11 32.90 $\pm$ 39.50 35.50 $\pm$ 39.50 1.88 $\pm$ 0.22 17/23
All 2.55 $\pm$ 0.06 2.10 $\pm$ 0.04 14.70 $\pm$ 3.41 15.70 $\pm$ 3.41 1.99 $\pm$ 0.09 23/26
------ ------------------------------------------ ----------------- -------------------- ------------------------- --------------------------------- ----------------
The spectral shape given by Eq. \[Eq2\] with $\beta$ = 0.5 is motivated by some shock acceleration scenarios (see discussion in section 5.). A fit over the whole three years with $\beta$ = 0.5 has been performed. It gives a reasonable $\chi^{2}$/dof of 30/23, $\Phi_0$ = (2.55 $\pm$ 0.07$_{\mathrm{stat}}$ $\pm$ 0.40$_{\mathrm{syst}}$) $\times$ 10$^{-12}$ TeV$^{-1}$ cm$^{-2}$ s$^{-1}$, $\Gamma$ = 1.91 $\pm$ 0.08$_{\mathrm{stat}}$ $\pm$ 0.09$_{\mathrm{syst}}$, a low value for the cut-off energy (corrected from the systematic bias) of E$_{\rm{cut}}$ = (4.0 $\pm$ 1.9$_{\mathrm{stat}}$ $\pm$ 0.7$_{\mathrm{syst}}$) TeV and an integrated flux above 1 TeV of (1.98 $\pm$ 0.04$_{\mathrm{stat}}$ $\pm$ 0.40$_{\mathrm{syst}}$) $\times$ 10$^{-12} $cm$^{-2}$ s$^{-1}$.
### Smoothed broken power law
Spectral parameters (differential flux normalisation $\Phi_0$, spectral indices $\Gamma_1$ and $\Gamma_2$, break energy and integrated flux above 1 TeV, I($\geq$1 TeV)) are summarized in Table \[table3\]. The break energy corrected for the systematic bias mentioned in section 3.1 is also given. Parameters show a large dispersion (at most at the 3$\sigma$ level) but remain compatible with each other. The larger value of $\Gamma_2$ for the 2005 data may reflect a possible steepening of the spectrum in the very high energy part.
------ ------------------------------------------ ----------------- ----------------- ----------------------- ---------------------------- --------------------------------- ---------------- --
Year $\Phi_0$[^10] $\Gamma_1$[^11] $\Gamma_2^{b}$ E$_{\rm{break}}$[^12] E$_{\rm{break,true}}$[^13] I($\geq$1 TeV)[^14] $\chi^{2}$/dof
$\rm (10^{-12} TeV^{-1} cm^{-2} s^{-1})$ (TeV) (TeV) $\rm (10^{-12} cm^{-2} s^{-1})$
2004 2.39 $\pm$ 0.12 2.18 $\pm$ 0.13 2.51 $\pm$ 0.23 2.40 $\pm$ 0.28 2.33 $\pm$ 0.28 1.79 $\pm$ 0.58 25/19
2005 2.64 $\pm$ 0.10 1.73 $\pm$ 0.10 3.07 $\pm$ 0.26 3.35 $\pm$ 0.37 3.37 $\pm$ 0.37 2.11 $\pm$ 0.60 28/18
2006 2.28 $\pm$ 0.18 2.27 $\pm$ 0.24 2.19 $\pm$ 0.28 2.04 $\pm$ 0.38 1.94 $\pm$ 0.38 1.85 $\pm$ 0.87 14/18
All 2.57 $\pm$ 0.07 2.02 $\pm$ 0.08 2.63 $\pm$ 0.14 2.61 $\pm$ 0.19 2.57 $\pm$ 0.19 1.98 $\pm$ 0.38 21/19
------ ------------------------------------------ ----------------- ----------------- ----------------------- ---------------------------- --------------------------------- ---------------- --
### Spectral variability
A refined study of the variations of the spectral index with time over the whole three years has been carried out using time intervals of roughly 5 hours, comprising ten consecutive runs. Each data subset has been fitted independently with a power law shape. The spectral index light curve has 25 points. A fit to a constant gives a $\chi^{2}$ of 29/24 d.o.f. confirming that the spectral index did not change significantly over the three years.
Correction for absorption of very high energy $\gamma$-rays
-----------------------------------------------------------
A recent calculation of the Galactic interstellar radiation field (Porter and Strong [@Porter]) has shown that the infra-red radiation field near the GC is considerably enhanced compared to what was previously thought. Thus, some attenuation of very high energy $\gamma$-rays might occur at TeV energies. The attenuation coefficient accounting for very high energy $\gamma$-ray absorption by e$^{+}$e$^{-}$ pair production on the interstellar radiation field (ISRF) toward the GC was derived in Zhang et al. ([@Zhang]) and can be used to correct the measured spectrum: $$F(E)=F_0(E)\times\exp(-\tau(E))$$ where F(E) represents the observed spectrum after attenuation, F$_0$(E) the intrinsic spectrum and $\tau$(E) the optical depth of the $\gamma$-rays toward the GC as a function of the energy. The optical depth $\tau$(E) depends on the number density of the ISRF photons and on the pair production cross section.\
Typically, 1$\%$ and 10$\%$ of $\gamma$-rays are absorbed at 10 TeV and 50 TeV, respectively, with an increase of the attenuation factor at higher energies. As expected, the spectral parameters do not change significantly after applying the correction. Thus, the curvature of the spectrum does not seem to be caused by $\gamma$-ray absorption and the cut-off (Eq. \[Eq2\]) or the spectral break (Eq. \[Eq3\]) is intrinsic to the source spectrum.
Search for time variability
===========================
As mentioned in the introduction, flaring of Sgr A\* and QPOs were detected in various bands such as X-rays and IR. In this section, the “run by run” and “night by night” light curves are used to search for any variablility in the GC source activity.\
The run by run light curves are obtained by dividing the data into 28 minutes intervals and by computing the integrated fluxes above 1 TeV for each of these time intervals. The integrated flux is computed from the results of a power law fit with exponential cut-off whose normalization was varied in the likelihood minimization while the spectral index and cut-off energy were fixed to the values obtained for the 2004-2006 dataset. The flux normalisation is adjusted for each 28 minutes time slice. The integrated fluxes above a fixed energy of 1 TeV are then calculated. The night by night light curves are derived in the same way as for the 28 minutes light curves except that the time intervals comprise runs that were recorded in the same night.
Light curves and flare sensitivity
----------------------------------
The run by run integrated flux light curves of the GC covering the 2004, 2005 and 2006 observation seasons are displayed in Fig. \[fig1\]. The fit of a constant to the data taken over the whole three years gives a $\chi^{2}$ of 233/216 d.o.f. and does not reveal any variability on time scales longer than 28 minutes.\
Because of the large error bars of the light curve points implied by the low statistics, the H.E.S.S. signal is only sensitive to relatively large amplitude flares. The flare sensitivity was estimated by adding an artificial Gaussian with variable duration $\sigma_t$, a time of maximum amplification t$_0$ and an amplification at maximum A to the H.E.S.S. light curve (LC): $$\mbox{LC}_{mod}\mbox{(t)} = \mbox{LC(t)}\times\biggl(1+A\times e^{\frac{(t-t_0)^{2}}{2\sigma_t^{2}}}\biggr).$$ When varying the time of maximum amplification t$_0$ along the LC, a distribution of the $A$ values compatible with a flare detection at a given significance is obtained. A fit to this distribution by a Landau function gives the most probable value of $A$ and its corresponding 1$\sigma$ variations. Fig. \[fig2\] shows the maximum amplification factor $A$ compatible with no flare detection at the 3$\sigma$ confidence level as a function of the flare duration. Long flares involve a larger number of LC points in the constant fit and then increase the $\chi^{2}$ over the number of d.o.f. much more easily, so that $A$ decreases with increasing flare duration. Fig. \[fig2\] shows that an increase of the flux by at least a factor of two is necessary to detect flares of hour time scales. An increase of the $\gamma$-ray flux of a factor of 2 or greater was excluded at a confidence level of 99$\%$ during a Chandra flare night by the H.E.S.S. collaboration (Aharonian et al. [@Hinton]), which is fully consistent with this result.
{height="30.00000%"} {height="30.00000%"} {height="30.00000%"} {height="30.00000%"}
![Maximum amplification factor $A$ for a 3$\sigma$ flare detection as a function of the flare duration. $A$ is obtained by a Landau fit (see text). The error bars are the corresponding 1$\sigma$ variations, depending on the time of the assumed flare. The maximum amplification factor decreases with increasing flare duration.[]{data-label="fig2"}](11569fg5.eps){width="50.00000%"}
Search for QPOs
---------------
Four oscillation frequencies ranging from 100 s to 2250 s have been observed in the X-ray light curve of Sgr A\* (Aschenbach et al. [@Aschenbach]). These frequencies, if confirmed, are likely to correspond to gravitational cyclic modes associated with the accretion disk of Sgr A\*. The occurence of these frequencies was searched for in the data. First, the coherence time of the disk precession is assumed to be less than 28 minutes. A Rayleigh test (de Jager et al. [@deJager]) is then performed on photon arrival time distributions for continuous observations of 28 minutes. Each 28 minutes observation gives a Rayleigh power spectrum. The Rayleigh power averaged over 2004-2006 data is shown in Fig. \[fig3\] as a function of the frequency. Error bars are estimated by calculating the variance of the Rayleigh power distribution plotted at the corresponding frequency. The probed frequencies range from 1/28 min$^{-1}$ to the inverse of the average time spacing between two consecutive events of 1.2 min$^{-1}$. No Rayleigh power is significantly higher than expected for noise at any probed frequencies. No significant peaks are seen at the 100 s, 219 s, 700 s and 1150 s periods observed in X-rays.
![Rayleigh power plotted as a function of the frequency. Rayleigh power is normalized such that a pure noise spectrum results in unit power. The dotted line shows the fit to a constant of the Rayleigh spectrum. The arrows denote the 100 s, 219 s, 700 s and 1150 s periods observed in X-rays.[]{data-label="fig3"}](11569fg6.eps){width="50.00000%"}
\
In a second analysis, the coherence time of oscillations is assumed to be of the order of a few hours. The Fourier power distribution using Lomb-Scargle periodograms (Scargle [@Scargle]) for each night of the dataset is then constructed. Data are binned into 5 minutes intervals. The Fourier power averaged over the 2004-2006 data is displayed in Fig. \[fig4\] as a function of the frequency. Frequencies tested range from 10$^{-2}$ min$^{-1}$ to 0.1 min$^{-1}$. No significant oscillation frequencies are detected.
![Top panel: \[10$^{-2}$ min$^{-1}$ - 0.1 min$^{-1}$\] Lomb-Scargle periodogram of the H.E.S.S. Sgr A\* light curve averaged over the 2004-2006 nights of observation. Bottom panel: Fourier power distribution derived from the averaged Lomb-Scargle periodogram. No significant peak is visible from the Lomb-Scargle periodogram and the $\chi^{2}$ of the exponential fit to the Fourier power distribution is 72/55 d.o.f.[]{data-label="fig4"}](11569fg7a.eps "fig:"){width="48.00000%"} ![Top panel: \[10$^{-2}$ min$^{-1}$ - 0.1 min$^{-1}$\] Lomb-Scargle periodogram of the H.E.S.S. Sgr A\* light curve averaged over the 2004-2006 nights of observation. Bottom panel: Fourier power distribution derived from the averaged Lomb-Scargle periodogram. No significant peak is visible from the Lomb-Scargle periodogram and the $\chi^{2}$ of the exponential fit to the Fourier power distribution is 72/55 d.o.f.[]{data-label="fig4"}](11569fg7b.eps "fig:"){width="50.00000%"}
Discussion and conclusions
==========================
A strong signal has been detected from the GC region with the H.E.S.S. instrument. An energy spectrum has been measured with a differential spectrum well described either by a power law with slope $\Gamma$=2.10 $\pm$ 0.04$_{\mathrm{stat}}$ $\pm$ 0.10$_{\mathrm{syst}}$ with an exponential cut-off at 15.7 $\pm$ 3.4$_{\mathrm{stat}}$ $\pm$ 2.5$_{\mathrm{syst}}$ TeV, or a smoothed broken power law with photon indices $\Gamma_1$=2.02 $\pm$ 0.08$_{\mathrm{stat}}$ $\pm$ 0.10$_{\mathrm{syst}}$, $\Gamma_2$=2.63 $\pm$ 0.14$_{\mathrm{stat}}$ $\pm$ 0.10$_{\mathrm{syst}}$ and a break energy at 2.57 $\pm$ 0.19$_{\mathrm{stat}}$ $\pm$ 0.44$_{\mathrm{syst}}$ TeV. An integrated flux above 1 TeV of (1.99 $\pm$ 0.09$_{\mathrm{stat}}$ $\pm$ 0.40$_{\mathrm{syst}}$) $\times$ 10$^{-12} $cm$^{-2}$ s$^{-1}$ is derived using the power law spectrum with an exponential cut-off fit. No indication for variability, flaring or QPOs has been found in the H.E.S.S. data, suggesting a non-variable emission of the GC region in the VHE regime.\
Different mechanisms have been suggested to explain the broadband spectrum of the GC (Fig. \[fig10\]). Firstly, the stochastic acceleration of electrons interacting with the turbulent magnetic field in the vicinity of Sgr A\*, as discussed by Liu et al. ([@Liu]), has been advocated to explain the millimeter and sub-millimeter emission. This model would also reproduce the IR and X-ray flaring (Atoyan and Dermer [@Atoyan]). In addition, it assumes that charged particles are accreted onto the black hole, and predicts the escape of protons from the accretion disk and their acceleration (Liu et al. [@Liu]). These protons produce $\pi^{0}$ mesons by inelastic collisions with the interstellar medium in the central star cluster of the Galaxy.\
The cut-off energy found in the $\gamma$-ray spectrum could reflect a cut-off E$_{\mathrm{cut,p}}$ in the primary proton spectrum. In that case, one would expect a cut-off in the $\gamma$-ray spectral shape at E$_{\mathrm{cut}}$ $\simeq$ E$_{\mathrm{cut,p}}$/30. The measured value given in section 3.2.1., with the parameter $\beta$ equal to 0.5, would correspond in this scenario to a low cut-off energy in the primary proton spectrum of roughly 100 TeV. It would correspond to a larger value of $\mathrm{E_{cut,p}}$ of $\mathrm{\sim 400\,TeV}$ if $\beta$ = 1.\
Energy-dependent diffusion models of protons to the outside of the central few parsecs of the Milky Way (Aharonian and Neronov [@Neronov2]) are alternative plausible mechanisms to explain the TeV emission observed with the H.E.S.S. instrument. They would lead to a spectral break as in the measured spectrum due to competition between injection and escape of protons outside the vicinity of the GC. A similar mechanism would explain the diffuse emission detected by H.E.S.S. along the galactic plane (Aharonian et al. [@DE]).\
The recent discovery of the G359.95$-$0.04 pulsar wind nebulae (PWN) located at 8” from Sgr A\* also provides interesting models as discussed by Wang et al. ([@Wang]) and Hinton and Aharonian ([@HintonAharonian]) to explain the steepening of the measured spectrum of HESS J1745$-$290. Inverse Compton emission due to a population of electrons whose energies extend up to 100 TeV might be responsible for at least a fraction of the TeV emission. The PWN model of Wang et al. and Hinton and Aharonian would imply a constant flux with time since the time scale for global PWN changes is typically much longer than a few years (more like centuries to millennia).
\
The absence of variability in the TeV data suggests that the emission mechanisms and emission regions differ from those invoked for the variable IR and X-ray emission. Moreover, the above-mentioned models can both accomodate a cut-off in the $\gamma$-ray energy spectrum and predict the absence of variability in the TeV emission. They are thus viable scenarios to explain the strong TeV signal detected by H.E.S.S. in the GC region.
The support of the Namibian authorities and of the University of Namibia in facilitating the construction and operation of H.E.S.S. is gratefully acknowledged, as is the support by the German Ministry for Education and Research (BMBF), the Max Planck Society, the French Ministry for Research, the CNRS-IN2P3 and the Astroparticle Interdisciplinary Programme of the CNRS, the U.K. Particle Physics and Astronomy Research Council (PPARC), the IPNP of the Charles University, the Polish Ministry of Science and Higher Education, the South African Department of Science and Technology and National Research Foundation, and by the University of Namibia. We appreciate the excellent work of the technical support staff in Berlin, Durham, Hamburg, Heidelberg, Palaiseau, Paris, Saclay, and in Namibia in the construction and operation of the equipment.
Aharonian, F., et al. 2004a, Astropart.Phys., 22, 109 Aharonian, F., et al. 2004b, , 425, L13 Aharonian, F., et al. 2005, 430, 865 Aharonian, F., et al. 2006a, , 439, 695 Aharonian, F., et al. 2006b, , 636, 777 Aharonian F., et al. 2006c, , 97, 221102 Aharonian, F., et al., 2006d, , 457, 899 Aharonian, F., et al. 2008, , 492, L25 Aharonian, F. & Neronov, A. 2005a, , 619, 306 Aharonian, F. & Neronov, A. 2005b, Ap&SS, 300, 255 Albert, J,. et al. 2006, , 638, L101 Aschenbach, B., et al. 2004, , 417, 71 Atoyan, A. & Dermer, C. 2004, , 617, L123 Baganoff, F., Bautz, M., Brandt, W., et al. 2001, , 413, 45 Baganoff, F., et al. 2003, , 591, 891 Ballantyne, D., Melia, F., Liu, S. & Crocker, R. 2007, , 657, L13 Belanger, G., et al. 2006, , 636, L275 Berge, D., Funk, S. & Hinton, J. 2004, , 466, 1219 Crocker, R., et al. 2005, , 622, 892 de Jager, O., Swanepoel, J. & Raubenheimer, B. 1989, , 221, 180 de Naurois, M., et al. 2003, Proc. of 28th ICRC (Tsubaka), Vol.5, p.2907 Djannati-Atai, A., et al. 1999, , 350, 17 Falcke, H., et al. 1998, , 499, 731 Genzel, R., Schödel, R., Ott, T., et al. 2003, , 425, 934 Ghez, A., et al. 2004, , 602, L159 Hinton, J. & Aharonian, F. 2007, , 657, 302 Hofmann, W., et al. 2003, Proc. of the 28th ICRC (Tsubuka), Vol.1, p.2811 Hornstein, S., et al. 2002, , 577, L9 Kosack, K., et al. 2004, , 608, L97 Li, T. & Ma, Y. 1983, , 272, 317 Liu, S., Melia, F., Petrosian, V. & Fatuzzo, M. 2006, , 647, 1099 Meyer, L., Do, T., Ghez, A. & Morris, M. 2008, , 688, L17 Porquet, D., Predehl, P., Aschenbach, B., et al. 2003, , 407, L17 Porter, T. & Strong, A. 2006, , 449, 641 Piron, F., et al. 2001, , 374, 895 Scargle, J. 1982, , 263, 835 Tsuchiya, K., et al. 2004, , 606, L115 van Eldik, C., Bolz, 0., Braun, I., et al. 2007, Proc. of the 30th ICRC (Merida) Wang, Q., Lu, F. & Gotthelf, E. 2006, , 367, 937 Zhang, J.-L, et al. 2006, , 449, 641 Zylka, R., et al. 1995, , 297, 83
[^1]: $\theta_z$ denotes the zenith angle
[^2]: Live time of the data sample
[^3]: Number of detected $\gamma$-rays after background substraction
[^4]: Significance of the $\gamma$-ray excess N$_{\gamma}$ calculated according to the prescription of Li and Ma ([@LiMa])
[^5]: Flux normalisation
[^6]: Reconstructed spectral index
[^7]: Reconstructed cut-off energy without corrections for the systematic bias
[^8]: Reconstructed cut-off energy corrected for the systematic bias
[^9]: Integrated $\gamma$-ray flux above 1 TeV
[^10]: Flux normalisation
[^11]: Reconstructed spectral index
[^12]: Reconstructed break energy without corrections for the systematic bias
[^13]: Reconstructed break energy corrected for the systematic bias
[^14]: Integrated $\gamma$-ray flux above 1 TeV
|
---
abstract: 'By sculpting the magnetic field applied to magneto-acoustic materials, phonons can be used for information processing. Using a combination of analytic and numerical techniques, we demonstrate designs for diodes (isolators) and transistors that are independent of their conventional, electronic formulation. We analyze the experimental feasibility of these systems, including the sensitivity of the circuits to likely systematic and random errors.'
author:
- 'Sophia R. Sklan'
- 'Jeffrey C. Grossman'
title: 'Phonon Diodes and Transistors from Magneto-acoustics'
---
Heat is ubiquitous. It accompanies almost any form of energy loss in real systems, but is one of the most difficult phenomena to control precisely. The most successful utilizations of heat (e.g. heat engines and heat pumps) essentially treat it as a homogenous current. However, consider crystals, where heat is often transported by electrons, photons, and phonons. There exist impressive arrays of devices for controlling both electrons and photons (down to specific modes and locations), but no equivalent toolkit for phonons. Perhaps the starkest example of this is computing, where strict control of a signal’s state is compulsory and commonplace. Recent efforts have sought to extend this degree of control to phonons, to realize devices like diodes, transistors, logic, and memory [@Phononics; @Phn; @Logic; @Phn; @Diode; @Phn; @Transistor; @Phn; @Mem]. Throughout this process the assumption that all computers should be the strict analog of electronic computers has been implicit. Since information in electronics is scalar (high or low voltage = 1 or 0), it has been assumed that information from phonons would be encoded in temperature (hot or cold = 1 or 0). Similarly, since electronics uses pn junctions for constructing circuits, interface effects have been considered for phonon diodes. Hence, research has thus far focused on nano-structures [@CNT; @Rect; @VO; @Rect] or 1D materials [@Phononics; @Phn; @Logic; @Phn; @Diode; @Therm; @Rect], where interface effects are strong, but fabrication was difficult.
Abandoning the assumption that phononic and electronic computing are strictly analogous presents a host of new opportunities. Here, we make an analogy to optical computing. We encode information in the polarization of a phonon current (transverse vertical or horizontal = 1 or 0). Our operators therefore modify some generic elliptic polarization, i.e. gyrators (which rotate the polarization angle) and polarizers (which project the polarization) from which we can construct diodes and transistors. The relationship between devices used in electronics, optics, and phononics and the abstract logic elements is shown in Figure \[fig:compare\] [^1]. To make these, we require systems that break time-reversal, rather than reflection, symmetry $-$ that is, we require a magnetic field. For the magnetic field to have a measurable effect upon the phonon current, we focus on magneto-acoustic (MA) materials. These materials were first described by Kittel, who noted that they could be used to create “gyrators, isolators \[diodes\], and other nonreciprocal acoustic elements" [@Kittel], but subsequent research on MA focused on other applications (e.g. acoustic control of magnetization) [@GG; @Book; @AFM; @Th; @TGG; @Exp; @Luthi; @Book; @Magnons]. MA coupling is a bulk effect found in commercially available materials, so fabrication is easier compared to the nano-structures of the electronic analogy.
![\[fig:compare\]Constitutive construction of logic elements in electronics, optics, and phononics. Circles represent classes of signals: electronic, optical, phononic, and logical. Elementary devices for controlling these signals are listed in each circle. For electronics, optics, and phononics, the basic elements are typically a single material or interface. Logical signals are an abstraction, so devices are defined by their effect on signals. Arrows indicate which basic elements are required to construct these logic elements for a given signal.](Architecture_Comparison.pdf)
In this article, we employ a combination of analytic and numerical methods to demonstrate that phononic logic elements (diodes and transistors) can be designed outside of an electronic computing paradigm. Our results confirm that MA polarizers and gyrators, when combined with a means of generating and measuring phonon currents, are sufficient to realize logic elements. Further, we show how present experimental techniques are likely sufficient to actualize logic elements that are reliably insensitive to errors. Taken together, these results reveal the potential for an under-explored class of phonon logic gates.
The goal of this work is to explore the feasibility of frequency-dependent phonon computing. In order to tackle this, knowledge of the phonon dispersion’s dependence on fixed (e.g. length) and tunable (e.g. magnetic field) parameters is necessary. Hence, we begin with the dispersion relation for two special geometries.
When the magnetic field is oriented along the length of the MA ($\vec{H}\Vert\vec{k}$, where $H$ is the applied field and $k$ the phonon wavevector), we have the circular birefringence (acoustic Faraday effect (AFE)) necessary for a gyrator [@GG; @Book]. The dispersion relation is: $$k_\pm(\omega) = \frac{T}{l}(1-\frac{A}{B\pm T\mp i})^{-1/2},$$ where $k_\pm$ are the right and left circularly polarized wavevectors (at fixed frequency $\omega$), $l$ the natural length scale $\tau \sqrt{c_{1313}/\rho}$ ( where $\tau$ the shortest relevant lifetime, typically the magnon lifetime, $c_{1313}$ the stiffness constant, and $\rho$ the density), $T$ the dimensionless frequency $\omega\tau$, $B$ the dimensionless field strength $\gamma\tau H$ ($\gamma$ is the gyroscopic ratio), and $A$ the dimensionless coupling constant $\gamma b_2^2 \tau/c_{1313}M_0$ ($b_2$ is the MA constant and $M_0$ the saturation magnetization of the MA, assuming a net ferromagnetic moment exists). Conversely, when the magnetic field is oriented perpedicular to the length of the MA ($\vec{H}\perp\vec{k}$), we have linear birefringence (Cotton-Mouton effect), necessary for a polarizer [@GG; @Book]. The dispersion relation for the mode polarized along the magnetic field is: $$k_\Vert(\omega) = \frac{T}{l}(1-\frac{AB}{(1-iT)^2+B(B+4\pi\gamma\tau M_0)})^{-1/2},$$ whereas the mode polarized perpendicular to the magnetic field is unaffected by the magnetic field ($k_\perp(\omega)=T/l=\omega\sqrt{\rho/c_{1313}}$). In both cases, there will be both real and imaginary components to the dispersion, corresponding to birefringence and dichroism [^2].
In optics, diodes are constructed by sandwiching a $\pi/4$ gyrator between two linear polarizers (oriented by $\pi/4$ with respect to each other (see Fig. \[fig:diode\])) [@Pht; @Isolator]. A signal entering in the forward mode, passes through the first polarizer, acquires a rotated polarization from the gyrator, and emerges polarized along the second polarizer. Conversely, a signal in the reverse direction is polarized and then acquires the same rotation in polarization, emerging orthogonal to the second polarizer. Both polarizers and gyrators can be constructed from MA by tuning the magnetic field. For a diode, one must select magnetic field strengths (at fixed frequency) that yield weak dichroism for the gyrator (even weak circular dichroism can prevent complete destructive interference, as we see below) and strong dichroism for the polarizer.
 
Our independent parameters for designing the gyrators and polarizers are field strength, phonon frequency, and MA length. We assume that the properties of the MA are fixed, taking values from representative experiments [@EXP]. The Phonon frequency is selected to be 10 GHz (slightly larger than in [@EXP]). $k(H)$ is then calculated for each dispersion and used to select reasonable magnetic fields that give desirable ratios of birefringe to dichroism (0.01T for the polarizers and 0.1T for the gyrator). Lastly, lengths are selected such that the gyration ($\theta=L(k_+^\prime-k_-^\prime)/2$ ) and filtering ($\alpha=\mathrm{exp}(-k_\Vert^{\prime\prime}L)$) are effective. The resulting circuit is then modeled by numerically evaluating the phase acquired by the phonon current at each stage. The results are plotted in Figure \[fig:diode\], where we find the circuit successfully blocks ($>95\%$ loss of intensity) all signals except the desired polarization and direction. Because the AFE’s solutions have opposite signs in their imaginary components, the amplification found in the forward mode is expected (the polarizers suppress it in the reverse).
Turning to the transistor, we require a more complicated approach. Firstly our transistor requires a measurement apparatus. This type of measurement remains difficult, but we show a heuristic approach in Fig. \[fig:measurement\]. While there may be more efficient experimental realizations, the form presented here benefits from its conceptual simplicity. The different stages of detection and transduction (piezoelectric), rectification (electronic diode), amplification (op amp), and application (electronic transistor, electromagnet) are all differentiated and are in principle realizable.
![\[fig:measurement\]Measurement operator. Phonon current passes through a piezoelectric, transducing an electronic signal proportional to the polarized phonon amplitude. This voltage is then rectified and amplified (via a diode and an op amp). The resulting voltage is used to switch between driving a current through an electromagnet (producing a magnetic field) or not ($R_{high}\gg R_{low}$). Operation shown here is for a magnetic field withheld when a signal is detected.](Magnetic_Control.pdf)
Given a measurement operator, we send a fixed logical 0 signal into a gyrator, and then determine if a magnetic field should be applied by measuring the amplitude of the phonon current for one polarization. If this polarization exceeds some threshold, a magnetic field is supressed (the gyration is strongest as $B\to0$ [^3]; remanence magnetization provides the necessary magnetic field to keep the gyrator working). Conversely, when the threshhold is not reached, then a magnetic field will be applied, suppressing the birefringence and partially cancelling the gyration (perfect cancellation requires $B \to \infty$). These two operations are summarized in Fig \[fig:transistor\]A,B.
For the transistor to work as a logic operation, the gyration should be $\pi/2$. Using the same process as in the design of the diode (magnetic field of $10^{-4}$T for off and 0.5T for on), we model the transistor in Figure \[fig:transistor\]C,D. In doing so we abstract the measurement device, focusing instead on the effect of applying or suppressing a magnetic field.
 
We find that there are imperfections in each operating regime. When the gyrator is off (field applied), the relatively modest size of the field implies a small gyration is still present. Whereas, when the gyrator is on, circular dichroism prevents perfect cancellation of the left and right circularly polarized modes, resulting in a small horizontally polarized remnant. For the specific case of an incoming signal at $\theta_{in}=0$ and the length optimized for $\theta_{out}=\pi/2$, the outgoing polarization angle is limited by $$\tan\theta = \coth \frac{\pi}{2} \frac{\mathrm{Im}[k_+-k_-]}{\mathrm{Re}[k_+-k_-]}.$$ [^4] While this can be accounted for by allowing some fuzziness to the range of polarizations that are deemed logical 0 or 1 (indeed, the piezoelectric transduction in Fig \[fig:measurement\] is relatively insensitive to the undesired polarization), there is a more stringent limit implied by these errors. Since the undesireable gyrations in the off state will accumulate, there exists a maximum total length of transistors that can be chained in series while maintaining well-separated logic states. This problem can be surmounted in practice by applying a repeater circuit (which maps a noisy input to a desired, less noisy value, as occurs in our transistor design when the signal is sent to the gate, not the source). If we think of each gyrator in a series as tied to a separate gate input, then this also limits the number of independent inputs in a logic operation that can be performed without using a repeater. We can exceed this limit because multiple phonon currents can superimpose, but practical difficulties in distinguishing between different inputs for super-imposed signals make it unlikely that this distinction will do more than double the number of logical inputs. To estimate the practical implications of this limitation, we consider the following encoding. Logical 0 is \[0,$\pi$/5\] and logical 1 is \[3$\pi$/10,$\pi$/2\] (other quadrants mapping to the 1st by reflection symmetry). In this case, using our previous independent parameter values, we find that the number of (fixed length) gyrators goes as $$N = \mathrm{floor}[6.4H^2-0.059|H|-0.0047],$$ where $N$ is the maximum number of gyrators and $H$ the applied field strength in Tesla. The minimum allowable field strength for the off state is therefore 0.4T. While a similar limit for the on state exists, the insistence on $B\approx0$ for this regime makes it a weaker constraint on the number of stages and field.
The presence of circular dichroism in the AFE produces a systematic error that limits computational power. In addition to systematic errors, random errors can also corrupt a circuit’s operation (be it diode or transistor). While sufficiently thick polarizers are relatively insensitive to such errors (the damping is exponential), gyrators can be quite sensitive. In general, this sensitivity depends upon frequency and field strength. To assess the sensitivity for an arbitrary case, we use the linearized equation of uncertainty propagation. Expressing the result in fractional uncertainties gives: $$|\frac{\sigma_{\Delta kL}}{\Delta kL}|^2 =(\frac{\sigma_L}{L})^2+|\frac{\partial \Delta k}{\partial H}|^2(\frac{\sigma_H}{H})^2+|\frac{\partial \Delta k}{\partial \omega}|^2(\frac{\sigma_\omega}{\omega})^2.$$ This method overestimates the effects of random errors since it does not distinguish between contributions to the real and imaginary parts of the dispersion. To determine the maximum tolerance for a given error, we consider each error acting alone. The results of this calculation are summarized in Table \[tbl:tolerances\]. The dramatically worse tolerances for the polarizers in the diode are due to reliance on resonant losses, which constrains $B(T)$. However, the operation of the polarizers is perhaps the least important part of the diode. So long as they produce appreciable losses, their precise magnitude is unimportant. Hence we can more easily accept errors here than other parts of the circuit. Moreover, we can always improve polarizers by increasing their thickness.
Polarizer Gyrator Transistor
---------------- ------------ ------------ ------------
$\delta H$ 81.5$\mu$G 3.42G 25.0G
$\delta\omega$ 3.32kHz 34.2MHz 49.5MHz
$\delta L$ 30.0$\mu$m 12.0$\mu$m 10.0$\mu$m
: \[tbl:tolerances\]Maximum allowable tolerances for errors in independent parameters, assuming 1% operational error. Each calculation assumes that other errors are 0. The “polarizer” and “gyrator” columns refer to parts of the diode. Tolerances for the transistor are calculated in the off state, which are more stringent.
This trade-off between performance and thickness is a common feature in our circuits. Ergo, it is worth considering some of the problems that might hinder circuit minimization. Here, we considered systems with length scales in the mm-cm range because this possessed the most robust body of experimental literature [@GG; @Book; @TGG; @Exp; @Magnons; @Luthi; @Book; @YIG; @Exp; @EXP]. However, for practical computers, working with smaller feature sizes is preferable. This has several difficulties for our approach. The most fundamental limit is that, for 10GHz phonons in YIG (as we consider here), the wavelength is about 2.5$\mu$m. For feature sizes smaller than a wavelength, the assumption that the device can simply be treated as a continuous medium ceases to be applicable and we are forced to treat our devices as defects in a background medium. To exceed this limit, would likely require even higher frequency phonons, where techniques to prepare and measure shear waves are less developed [@Pico; @Phn; @Pico; @Phn2]. Even before we reach this limit, shrinking the system while maintaining the same effect (i.e. $k_{new}(L_{new})
=k_{old}(L_{old})$) is a non-trivial demand. For gyrators, in the off-state limit ($B\to\infty$), the phase acquired is proportional to $LT^2/B^2$. Since we don’t care about decreasing the phase acquired, then we can simply allow $L$ to decrease without needing to modify any other parameters. In the on-state ($B\to0$), however, the phase is proportional to $LT^{3/2}$ (for small $T$). Shrinking $L$ therefore requires a concomitant increase in $T$ (and only result in an approximate invarience) or a modification of the material used. Finally, for the polarizers, assuming that we’re on-resonance ($B=B^*(T)$), then the requirement of phase invariance is quite similar to the active gyrator (although not as strict, since a more effective polarizer is still acceptable). To modify the MA material is therefore likely necessary for miniaturizing our circuits. This could be done in several ways. The most promising modifications of this approach would be to use single molecule magnets, which also show MA properties [@SMM], or a bulk MA with a reduced speed of sound (exposing the phonons to the MA for longer).
We have demonstrated the operation and limitations of phonon logic circuits outside of the electronic circuit paradigm. Diodes and transistors remain difficult to construct for phonons, but the MA approach presented here avoids many of the problems found in other techniques. While it faces challenges not present in previous approaches (e.g. minimization), here we demonstrate that proof-of-concept realizations are feasible. We find that, not only are the requisite experimental conditions within an accessible range, but also that such circuit elements should be sufficiently robust that noise should not effect them.
This material is based upon work supported by the National Science Foundation Graduate Research Fellowship under Grant No. 1122374.
[25]{}
N. Li, J. Ren, L. Wang, G. Zhang, P. Hänggi, and B. Li, *Rev. Mod. Phys.* **84**, 1045-1066 (2012).
L. Wang and B. Li, *Phys. Rev. Lett.* **99**, 177208 (2007).
B. Li, L. Wang, and G. Casati, *Phys. Rev. Lett.* **93**, 184301 (2004).
W.C. Lo, L. Wang, and B. Li, *J. Phys. Soc. Japan* **77**, 054402 (2008).
L. Wang and B. Li, *Phys. Rev. Lett.* **101**, 267203 (2008).
C. Chang, D. Okawa, A. Majumdar, and A. Zettl, *Science* **314**, 1121 (2006).
R.-G. Xie, C-T. Bui, B. Varghese, M.-G. Xia, Q.-X. Zhang, C.-H. Sow, B. Li, and J.T.L. Thong, *Adv. Funct. Mater.* **21**, 1602 (2011).
M. Terraneo, M. Peyrard, and G. Casati, *Phys. Rev. Lett.* **88**, 094302 (2002).
C. Kittel, *Phys. Rev.* **110**, 836 (1958).
J.D. Gavenda, V.V. Gudkov, *Magnetoacoustic Polarization Phenomena in Solids* (Springer, New York, NY, 2000).
M. Boiteux, P. Doussineau, B. Ferry, J. Joffrin, and A. Levelut, *Phys. Rev. B* **4**, 3077 (1971).
A. Sytcheva, U. Löw, S. Yasin, J. Wosnitza, S. Zherlitsyn, P. Thalmeier, T. Goto, P. Wyder, and B. Lüthi , *Phys. Rev. B* **81**, 214415 (2010).
B. Lüthi, *Physical Acoustics in the Solid State* (Springer, New York, NY, 2007).
M. Bombeck *et al.*, *Phys. Rev. B* **85**, 195324 (2012).
R. Guermeur, J. Joffrin, A. Levelut, and J. Penne, *Solid State Commun.* **5**, 369 (1967)
D. Hurley, O.B. Wright, O. Matsudaa, V.E. Gusevc, and O.V. Kolosov, *Ultrasonics* **38**, 470 (2000).
O. Matsuda, O. B. Wright, D. H. Hurley, V. E. Gusev, and K. Shimizu, *Phys. Rev. Lett.* **93**, 095501 (2004).
H. Matthews and R.C. LeCraw, *Phys. Rev. Lett.* **8**, 397 (1962).
J. Ballato and E. Snitzer, *Appl. Opt.* **34**, 6848-6854 (1995).
I. D. Tokman, V. I. Pozdnjakova, and A. I. Beludanova, *Phys. Rev. B* **83**, 014405 (2011).
[^1]: In optics isolators are essentially diodes [@Pht; @Isolator].
[^2]: Dichroism can have any sign in MA, as energy can be added or extracted from the magnetic field
[^3]: The model used here does not include thermal fluctuations reducing the MA’s magnetization at remanence that of saturation. A small, non-zero field is likely prefereable.
[^4]: Numerical calculations slightly exceed this limit, since we find numerically that the maximum $\theta$ occurs for a slightly thinner transistor than would be predicted by $L=\pi/(k_+^\prime-k_-^\prime)$. This gain is not large enough to merit abandoning the proposed limit.
|
---
abstract: 'Quantum computers hold promise to enable efficient simulations of the properties of molecules and materials; however, at present they only permit *ab initio* calculations of a few atoms, due to a limited number of qubits. In order to harness the power of near-term quantum computers for simulations of larger systems, it is desirable to develop hybrid quantum-classical methods where the quantum computation is restricted to a small portion of the system. This is of particular relevance for molecules and solids where an active region requires a higher level of theoretical accuracy than its environment. Here we present a quantum embedding theory for the calculation of strongly-correlated electronic states of active regions, with the rest of the system described within density functional theory. We demonstrate the accuracy and effectiveness of the approach by investigating several defect quantum bits in semiconductors that are of great interest for quantum information technologies. We perform calculations on quantum computers and show that they yield results in agreement with those obtained with exact diagonalization on classical architectures, paving the way to simulations of realistic materials on near-term quantum computers.'
author:
- |
He Ma,$^{1, 3}$ Marco Govoni$^{2, 3}$, Giulia Galli$^{1, 2, 3\ast}$\
\
\
\
\
bibliography:
- 'ref.bib'
title: 'Quantum simulations of materials on near-term quantum computers'
---
Introduction {#introduction .unnumbered}
============
In the last three decades, atomistic simulations based on the solution of the basic equation of quantum mechanics have played an increasingly important role in predicting the properties of functional materials, encompassing catalysts and energy storage systems for energy applications, and materials for quantum information science. Especially in the case of complex, heterogeneous materials, the great majority of first-principles simulations are conducted using density functional theory (DFT), which is in principle exact but in practice requires approximations to enable calculations. Within its various approximations, DFT has been extremely successful in predicting numerous properties of solids, liquids and molecules, and in providing key interpretations to a variety of experimental results; however it is often inadequate to describe so-called strongly-correlated electronic states [@Cohen2008; @Su2018]. We will use here the intuitive notion of strong correlation as pertaining to electronic states that cannot be described by mean-field theories. Several theoretical and computational methods have been developed over the years to treat systems exhibiting strongly-correlated electronic states, including dynamical mean-field theory [@Georges1996; @Kotliar2006] and quantum Monte-Carlo [@Ceperley1986; @Wagner2016]; in addition, *ab initio* quantum chemistry methods, traditionally developed for molecules, have been recently applied to solid state problems as well [@Sun2017]. Unfortunately, these approaches are computationally demanding and it is still challenging to apply them to complex materials containing defects and interfaces, even using high-performance computing architectures.
Quantum computers hold promise to enable efficient quantum mechanical simulations of weakly and strongly-correlated molecules and materials alike [@Aspuru-Guzik2005; @Bravyi2017; @Babbush2018; @Kivlichan2018; @Motta2019; @Ollitrault2019; @Cao2019; @Bauer2020]; in particular when using quantum computers, one is able to efficiently simulate systems of interacting electrons, with an exponential speedup over exact diagonalization on classical computers (full configuration interaction, FCI, calculations). Thanks to decades of successful experimental efforts, we are now entering the noisy intermediate-scale quantum (NISQ) era [@Preskill2018], with quantum computers expected to have on the order of 100 quantum bits (qubits); unfortunately this limited number of qubits still prevents straightforward quantum simulations of realistic molecules and materials, whose description requires hundreds of atoms and thousands to millions of degrees of freedom to represent the electronic wavefunctions. An important requirement to tackle complex chemistry and material science problems using NISQ computers is the reduction of the number of electrons treated explicitly at the highest level of accuracy [@Rubin2016; @Yamazaki2018]. For instance, building on the idea underpinning dynamical mean field theory [@Georges1996; @Kotliar2006], one may simplify complex molecular and material science problems by defining active regions (or building blocks) with strongly-correlated electronic states, embedded in an environment that may be described within mean-field theory [@Bauer2016; @Kreula2016; @Rungger2019].
In this work, we present a quantum embedding theory built on DFT, which is scalable to large systems and which includes the effect of exchange-correlation interactions of the environment on active regions, thus going beyond commonly adopted approximations. In order to demonstrate the effectiveness and accuracy of the theory, we compute ground and excited state properties of several spin-defects in solids including the negatively charged nitrogen-vacancy (NV) center [@Davies1976; @Rogers2008; @Doherty2011; @Maze2011; @Choi2012; @Doherty2013; @Goldman2015], the neutral silicon-vacancy (SiV) center [@Haenens2011; @Gali2013; @Green2017; @Rose2018; @Green2019; @Thiering2019] in diamond, and the Cr impurity (4+) in 4H-SiC [@Son1999; @Koehl2017; @Diler2019]. These spin-defects are promising platforms for solid-state quantum information technologies, and they exhibit strongly-correlated electronic states that are critical for the initialization and read-out of their spin states [@Weber2010; @Seo2016; @Seo2017; @Ivady2018; @Dreyer2018; @Anderson2019]. Our quantum embedding theory yields results in good agreement with existing measurements. In addition, we present the first theoretical predictions for the position and ordering of the singlet states of SiV and of Cr, and we provide an interpretation of experiments which have so far remained unexplained.
Importantly, we report calculations of spin-defects using a quantum computer [@Qiskit; @Yorktown]. Based on the effective Hamiltonian derived from the quantum embedding theory, we investigated the strongly-correlated electronic states of the NV center in diamond using quantum phase estimation algorithm (PEA) [@Abrams1997; @Aspuru-Guzik2005] and variational quantum eigensolvers (VQE) [@Peruzzo2014; @McClean2016; @Kandala2017], and we show that quantum simulations yield results in agreement with those obtained with classical FCI calculations. Our findings pave the way to the use of near term quantum computers to investigate the properties of realistic heterogeneous materials with first-principles theories.
Results {#results .unnumbered}
=======
**General strategy**
We summarize our strategy in Fig. \[strategy\]. Starting from an atomistic structural model of materials (e.g. obtained from DFT calculations or molecular dynamics simulations), we identify active regions with strongly-correlated electrons, which we describe with an effective Hamiltonian that includes the effect of the environment on the active region. This effective Hamiltonian is constructed using the quantum embedding theory described below, and its eigenvalues can be obtained by either classical algorithms such as exact diagonalization (FCI) or quantum algorithms.
**Embedding Theory**
A number of interesting quantum embedding theories have been proposed over the past decades [@Sun2016]. For instance, density functional embedding theory has been developed to improve the accuracy and scalability of DFT calculations [@Huang2011; @Goodpaster2014; @Jacob2014; @Genova2014; @Wen2019]. Density matrix embedding theory (DMET) [@Knizia2012; @Wouters2016; @Pham2019] and various Green’s function based approaches [@Nguyen2016; @Dvorak2019], e.g. dynamical mean field theory (DMFT), have been developed to describe systems with strongly-correlated electronic states. At present, *ab initio* calculations of materials using DMET and DMFT have been limited to relatively small unit cells (a few tens of atoms) of [*pristine*]{} crystals, due to their high computational cost [@Zhu2019; @Cui2019]. In this work, we present a quantum embedding theory that is applicable to strongly-correlated electronic states in realistic [*heterogeneous*]{} materials and we apply it to systems with hundreds of atoms. The theory, inspired by the constrained random phase approximation (cRPA) approach [@Aryasetiawan2004; @Miyake2009; @Hirayama2017], does not require the explicit evaluation of virtual electronic states [@Wilson2008; @Govoni2015], thus making the method scalable to materials containing thousands of electrons. Furthermore, cRPA approaches contain a specific approximation (RPA) to the screened Coulomb interaction, which neglects exchange-correlation effects and may lead to inaccuracies in the description of dielectric screening. Our embedding theory goes beyond the RPA by explicitly including exchange-correlation effects, which are evaluated with a recently developed finite-field algorithm [@Ma2018; @Nguyen2019].
The embedding theory developed here aims at constructing an effective Hamiltonian operating on an active space (A), defined as a subspace of the single-particle Hilbert space: $$\label{Heff}
H^{\mathrm{eff}} = \sum_{ij}^{\mathrm{A}} t^{\mathrm{eff}}_{ij} a_{i}^{\dagger} a_{j} + \frac{1}{2} \sum_{ijkl}^{\mathrm{A}} V^{\mathrm{eff}}_{ijkl} a_{i}^{\dagger} a_{j}^{\dagger} a_{l} a_{k}.$$ Here $t^{\mathrm{eff}}$ and $V^{\mathrm{eff}}$ are one-body and two-body interaction terms that take into account the effect of all the electrons that are part of the environment (E) in a mean-field fashion, at the DFT level. An active space can be defined, for example, by solving the Kohn-Sham equations of the full system and selecting a subset of eigenstates among which electronic excitations of interest take place (e.g. defect states within the gap of a semiconductor or insulator). To derive an expression for $V^{\mathrm{eff}}$ that properly accounts for all effects of the environment including exchange and correlation interactions, we define the environment density response function (reducible polarizability) $\chi^\mathrm{E} = \chi_0^E + \chi_0^\mathrm{E} f \chi^E$, where $\chi_0^\mathrm{E} = \chi_0 - \chi_0^\mathrm{A}$ is the difference between the polarizability of the Kohn-Sham system $\chi_0$ and its projection onto the active space $\chi_0^\mathrm{A}$ (see Supplementary Information (SI)). $\chi^\mathrm{E}$ thus represents the density response outside the active space. The term $f = V + f_{\mathrm{xc}}$ is often called the Hartree-exchange-correlation kernel, where $V$ is the Coulomb interaction and the exchange-correlation kernel $f_{\mathrm{xc}}$ is defined as the derivative of the exchange-correlation potential with respect to the electron density. We define the effective interactions between electrons in A as $$\label{Veff}
V^{\mathrm{eff}} = V + f \chi^\mathrm{E} f,$$ given by the sum of the bare Coulomb potential and a polarization term arising from the density response in the environment E. When the RPA is adopted, the exchange-correlation kernel $f_{\mathrm{xc}}$ is neglected in Eq. \[Veff\] and the expression derived here reduces to that used within cRPA. We represent $\chi^\mathrm{E}$ and $f$ on a compact basis obtained from a low-rank decomposition of the dielectric matrix [@Wilson2008; @Govoni2015] that allows us to avoid the evaluation and summation over virtual electronic states. Once $V^{\mathrm{eff}}$ is defined, the one-body term $t^{\mathrm{eff}}$ can be computed by subtracting from the Kohn-Sham Hamiltonian a term that accounts for Hartree and exchange-correlation effects in the active space (see SI).
**Embedding theory applied to spin-defects**
The embedding theory presented above is general and can be applied to a variety of systems for which active regions, or building blocks, with strongly-correlated electronic states may be identified: for example active sites in inorganic catalysts or organic molecules or defects in solids and liquids (e.g. solvated ions in water). Here we apply the theory to spin-defects including NV and SiV in diamond and Cr in 4H-SiC. Most of these defects’ excited states are strongly-correlated (they cannot be represented by a single Slater determinant of single-particle orbitals), as shown e.g. for the NV center in diamond by Bockstedte et al.[@Bockstedte2018]. We demonstrate that our embedding theory can successfully describe the many-body electronic structure of different types of defects including transition metal atoms; our results not only confirm existing experimental observations but also provide a detailed description of the electronic structure of defects not presented before, which sheds light into their optical cycles.
We first performed spin-restricted DFT calculations using hybrid functionals [@Skone2014] to obtain a mean-field description of the defects and of the whole host solid. The spin restriction ensures that both spin channels are treated on an equal footing and that there is no spin-contamination when building effective Hamiltonians. Based on our DFT results, we then selected active spaces that include single-particle defect wavefunctions and relevant resonant and band edge states. We verified that the size of the chosen active spaces yields converged excitation energies (see SI). We then constructed effective Hamiltonians (Eq. \[Heff\]-\[Veff\]) by taking into account exchange-correlation effects, and we obtained many-body ground and excited states using classical (FCI) and, for selected cases, quantum algorithms (PEA, VQE). All calculations were performed at the spin triplet ground state geometries obtained by spin-unrestricted DFT calculations, thus obtaining vertical excitation energies (equal to the sums of zero phonon line (ZPL) and Stokes energies). It is straightforward to extend the current approach to compute potential energy surfaces at additional geometries, so as to include relaxations and Jahn-Teller effects [@Bockstedte2018; @Thiering2019]. In Fig. \[defects\] we present atomistic structures, single-particle defect levels and the many-body electronic structure of three spin-defects. Several relevant vertical excitation energies are reported in Table \[excitation\_energies\], and additional ones are given in the SI. In the following discussion, lower-case symbols represent single-particle states obtained from DFT and upper-case symbols represent many-body states.
For the NV in diamond, we constructed effective Hamiltonians (Eq. \[Heff\]) by using an active space that includes $a_1$ and $e$ single-particle defect levels in the band gap and states near the valence band maximum (VBM). Our FCI calculations correctly yield the symmetry and ordering of the low-lying $^3A_2$, $^3E$, $^1E$ and $^1A_1$ states. The vertical excitation energies reported in Table 1 show that including exchange-correlation effects yields results in better agreement with experiments than those obtained within the RPA.
In the case of the SiV in diamond, we built effective Hamiltonians using an active space with the $e_u$ and $e_g$ defect levels and states near the VBM, including resonant $e'_u$ and $e'_g$ states. Effective Hamiltonians including or neglecting exchange-correlation effects yield similar results, with the excitation energies obtained beyond RPA being slightly higher. We predicted the first optically-allowed excited state to be a $^3E_{u}$ state with vertical excitation energy of 1.59 eV, in good agreement with the sum of 1.31 eV ZPL measured experimentally [@Haenens2011] and 0.258 eV Stokes shift estimated using an electron-phonon model [@Thiering2019]. Remarkably, our calculations predicted a $^3A_{2u}$ state 11 meV below the $^3E_{u}$ state, supporting a recent experimental observation by Green et al. [@Green2019], which proposed a $^3A_{2u}$-$^3E_{u}$ manifold with 7 meV separation in energy. The small difference in energy splitting between our results and experiment is likely due to geometry relaxation effects not yet taken into account in our study. In addition to states of $u$ symmetry generated by $e_u \rightarrow e_g$ excitations, we observed a number of optically dark states of $g$ symmetry (grey levels in Fig. \[defects\]b) originating from the excitation from the $e'_g$ level and the VBM states to the $e_g$ level.
Despite significant efforts [@Green2017; @Rose2018; @Green2019; @Thiering2019], several important questions on the singlet states of SiV remain open. These states are crucial for a complete understanding of the optical cycle of the SiV center. Our predicted ordering of singlet states of SiV is shown in Fig. \[defects\]b. We find the vertical excitation energies of the $^1A_{1u}$ state to be slightly higher than that of the $^3A_{2u}$-$^3E_{u}$ triplet manifold, suggesting that the intersystem crossing (ISC) from $^3A_{2u}$ or $^3E_{u}$ to singlet states may be energetically unfavorable (first-order ISC to lower $^1E_{g}$ and $^1A_{1g}$ states are forbidden). We note that the $^1E_{u}$ and $^1A_{2u}$ states are much higher in energy than $^1A_{1u}$ and are not expected to play a significant role in the optical cycle. In addition the first-order ISC process from the lowest energy singlet state $^1E_{g}$ to the $^3A_{2g}$ ground state is forbidden by symmetry. Overall our results indicate that the $^3A_{2g}$ state is populated through higher-order processes and therefore the spin-selectivity of the full optical cycle is expected to be low. While more detailed studies including spin-orbit coupling are required for definitive conclusions, our predictions shed light on the strongly-correlated singlet states of SiV and provide a possible explanation for the experimental difficulties in measuring optically-detected magnetic resonance (ODMR) of SiV.
We now turn to Cr in 4H-SiC, where we considered the hexagonal configuration. We constructed effective Hamiltonians with the half-filling $e$ level in the band gap and states near the conduction band minimum (CBM) including resonance states. Upon solving the effective Hamiltonian, we predict the lowest excited state to be a $^1E$ state arising from $e \rightarrow e$ spin-flip transition, with excitation energy of 1.09 (0.86) eV based on embedding calculations beyond (within) the RPA. Results including exchange-correlation effects are in better agreement with the measured ZPL of 1.19 eV [@Son1999], where the Stokes energy is expected to be small given the large Debye-Waller factor[@Diler2019]. There is currently no experimental report for the triplet excitation energies of Cr in 4H-SiC, but our results are in good agreement with existing experimental measurements for Cr in GaN, a host material with a crystal field strength similar to that of 4H-SiC [@Koehl2017]. We predict the existence of a $^3E+{}^3A_1$ manifold at $\simeq$ 1.4 eV and a $^3E'+{}^3A'_2$ manifold at $\simeq$ 1.7 eV above the ground state (Fig. \[defects\]c), resembling the $^3T_2$ manifold (1.2 eV) and $^3T_1$ manifold (1.6 eV) for Cr in GaN observed experimentally [@Heitz1995]. We note that in many cases it is challenging to study materials containing transition metal elements with DFT [@Anisimov1997]. The agreement between FCI results and experimental measurements clearly demonstrates that the embedding theory developed here can effectively describe the strongly-correlated part of the system, while yielding at the same time a quantitatively correct description of the environment.
**Quantum simulations of spin-defects**
The results presented in the previous section were obtained using classical algorithms. We now turn to the use of quantum algorithms. To perform quantum simulations with PEA and VQE, we constructed a minimum model of an NV center including only $a_1$ and $e$ orbitals in the band gap. This model (4 electrons in 6 spin orbitals) yields excitation energies within 0.2 eV of the converged results using a larger active space. In Fig. \[quantum\] we show the results of quantum simulations.
We first performed PEA simulations with a quantum simulator (without noise) [@Qiskit] to compute the energy of $^3A_2$, $^3E$, $^1E$ and $^1A_1$ states. We used molecular orbital approximations of these states derived from group theory [@Doherty2011] as initial states for PEA, which are single Slater determinant for $^3A_2$ ($M_S = 1$) and $^3E$ ($M_S = 1$) states, and superpositions of two Slater determinants for $^1E$ and $^1A_1$ states. As shown in Fig. \[quantum\]a, PEA results converge to classical FCI results with an increasing number of ancilla qubits.
We then performed VQE simulations with a quantum simulator and with the IBM Q 5 Yorktown quantum computer [@Yorktown]. We estimated the energy of the $^3A_2$ ground state manifold by performing VQE calculations for both the single-Slater-determinant $M_{S} = 1$ component and the strongly-correlated $M_{S} = 0$ component. Within a molecular orbital notation, $M_{S} = 1$ and $M_{S} = 0$ ground states can be represented as $\ket{a \bar{a} e_x e_y}$ and $\frac{1}{2}\left( \ket{a \bar{a} e_x \bar{e}_y} + \ket{a \bar{a} \bar{e}_x e_y} \right)$, respectively, where $a$, $e_x$, $e_y$ (spin-up) and $\bar{a}$, $\bar{e}_x$, $\bar{e}_y$ (spin-down) denote $a_1$ and $e$ orbitals. To obtain the $M_{S} = 0$ ground state, we used a closed-shell Hartree-Fock state $\ket{a \bar{a} e_x \bar{e}_x}$ as reference; the $M_{S} = 1$ ground state is itself an open-shell Hartree-Fock state, so we started with a higher energy reference state $\ket{a e_x \bar{e}_x e_y}$ in the $^3E$ manifold. We used unitary coupled-cluster single and double (UCCSD) ansatzs [@Peruzzo2014] to represent the trial wavefunctions. Fig. \[quantum\]b and Fig. \[quantum\]c show the estimated ground state energy as a function of the number of VQE iterations, where VQE calculations performed with the quantum simulator correctly converges to the ground state energy in both the $M_{S} = 1$ and $M_{S} = 0$ case. Despite the presence of noise, whose characterization and study will be critical to improve the use of quantum algorithms [@Kandala2019], the results obtained with the quantum computer converge to the ground state energy within a 0.2 eV error. Calculations of excited states with quantum algorithms will be the focus of future works.
Discussion {#discussion .unnumbered}
==========
With the goal of providing a strategy to solve complex materials problems on NISQ computers, we proposed a first-principles quantum embedding theory where appropriate active regions of a material and their environment are described with different levels of accuracy, and the whole system is treated quantum mechanically. In particular, we used hybrid density functional theory for the environment, and we built a many-body Hamiltonian for the active space with effective electron-electron interactions that include dielectric screening and exchange-correlation effects from the environment. Our method overcomes the commonly used random phase approximation, which neglects exchange-correlation effects; importantly it is applicable to heterogeneous materials and scalable to large systems, thanks to the novel algorithms used here to compute response functions [@Ma2018; @Nguyen2019]. We emphasize that the embedding theory presented here provides a flexible framework where multiple effects of the environment may be easily incorporated. For instance, dynamical screening effects can be included by considering a frequency-dependent screened Coulomb interaction, evaluated using the same procedure as the one outlined here for static screening; electron-phonon coupling effects can be incorporated by including phonon contributions in the screened Coulomb interactions. Furthermore, for systems where the electronic structure of the active region is expected to influence that of the host material, a self-consistent cycle in the calculation of the screened Coulomb interaction of the environment can be easily added to the approach.
We presented results for spin-defects in semiconductors obtained with both classical and quantum algorithms, and we showed excellent agreement between the two sets of techniques. Importantly, for selected cases we showed results obtained using a quantum simulator and a quantum computer, which agree within a relatively small error, in spite of the presence of noise in the quantum hardware. We predicted several excited states not observed before, in particular our results for the SiV in diamond and Cr in SiC represent the very first theoretical predictions of their singlet states, and provide important insights into their full optical cycle. We also demonstrated that a treatment of the dielectric screening beyond the random phase approximation leads to an accurate prediction of excitation energies.
The method proposed in our work enables calculations of realistic, heterogeneous materials using the resources of NISQ computers. We demonstrated quantum simulations of strongly-correlated electronic states in considerably larger systems (with hundreds of atoms) than previous studies combining quantum simulation and quantum embedding [@Rubin2016; @Yamazaki2018; @Bauer2016; @Kreula2016; @Rungger2019]. We have studied solids with defects, not just pristine materials, which are of great interest for quantum technologies. The strategy adopted here is general and may be applied to a variety of problems, including the simulation of active regions in molecules and materials for the understanding and discovery of catalysts and new drugs, and of aqueous solutions containing complex dissolved species. We finally note that our approach is not restricted to strongly-correlated active regions and will be useful also in the case of weakly correlated systems, where different regions of a material may be treated with varying levels of accuracy. Hence we expect the strategy presented here to be widely applicable to carry out quantum simulations of materials on near-term quantum computers.
Methods {#methods .unnumbered}
=======
**Density Functional Theory**
All ground state DFT calculations are performed with the Quantum Espresso code [@Giannozzi2009] using the plane-wave pseudopotential formalism. Electron-ion interactions are modeled with norm-conserving pseudopotentials from the SG15 library [@Schlipf2015]. A kinetic energy cutoff of 50 Ry is used. All geometries are relaxed with spin-unrestricted DFT calculations using the Perdew–Burke-Ernzerhof (PBE) functional [@Perdew1996] until forces acting on atoms are smaller than 0.013 $\mathrm{eV} / \mathrm{\AA}$. NV and SiV in diamond are modeled with 216-atom supercells; Cr in 4H-SiC is modeled with a 128-atom supercell. The Brillouin zone is sampled with the $\Gamma$ point.
**Construction of effective Hamiltonians**
Construction of effective Hamiltonians is performed with the WEST code [@Govoni2015], starting from wavefunctions of spin-restricted DFT calculations. For this step, we remark that the use of hybrid functional is important for an accurate mean-field description of defect levels, even though the geometry of defects are well represented at the PBE level. We used a dielectric dependent hybrid (DDH) functional [@Skone2014] which self-consistently determines the fraction of exact exchange based on the dielectric constant of the host material. The DDH functional was shown to yield accurate band gaps of diamond and silicon carbide [@Skone2014], as well as optical properties of defects [@Seo2016; @Seo2017; @Pham2017; @Smart2018; @Gerosa2018]. After hybrid functional solutions of the Kohn-Sham equations are obtained, iterative diagonalizations of $\chi_0$ are performed, and density response functions and $f_{\mathrm{xc}}$ of the system are represented on a basis consisting of the first 512 eigenpotentials of $\chi_0$. Finite field calculations of $f_{\mathrm{xc}}$ are performed by coupling the WEST code with the Qbox [@Gygi2008] code. FCI calculations [@Knowles1984] on the effective Hamiltonian are carried out using the PySCF [@Sun2017] code.
**Quantum simulations**
In order to carry out quantum simulations, a minimum model of the NV center is constructed by applying the embedding theory with $a_1$ and $e$ orbitals beyond the RPA.
In PEA simulations, the Jordan-Wigner transformation [@Jordan1928] is used to map the fermionic effective Hamiltonian to a qubit Hamiltonian, and Pauli operators with prefactors smaller than $10^{-6}$ a.u. are neglected to reduce the circuit depth, which results in less than $10^{-4}$ a.u. (0.003 eV) change in eigenvalues. In order to achieve optimal precision, the Hamiltonian is scaled such that 0 and 2.5 eV are mapped to phases $\phi = 0$ and $\phi = 1$ of the ancilla qubits, respectively. We used the first-order Trotter formula to split time evolution operators into 4 time slices.
In VQE simulations, the parity transformation [@Bravyi2017] is adopted. For the simulation of the $M_{S} = 1$ state, the resulting qubit Hamiltonian acts on 4 qubits and there are 2 variational parameters in the UCCSD ansatz. For the simulation of the $M_{S} = 0$ state, we fixed the occupation of the $a$ orbital and the resulting qubit Hamiltonian acts on 2 qubits. We replicated the exponential excitation operator twice, with parameters in both replicas variationally optimized. Such a choice results in 6 variational parameters, providing a sufficient number of degrees of freedom for an accurate representation of the strongly-correlated $M_{S} = 0$ state. Parameters in the ansatz are optimized with the COBYLA algorithm [@Powell1994].
Quantum simulations are performed with the QASM simulator and the IBM Q 5 Yorktown quantum computer using the IBM Qiskit package [@Qiskit]. Each quantum circuit is executed 8192 times to obtain statistically reliable sampling of the measurement results.
Acknowledgments {#acknowledgments .unnumbered}
===============
We thank C. P. Anderson, D. D. Awschalom, T. C. Berkelbach, B. Diler, S. Dong, D. Freedman, L. Gagliardi, F. Gygi, F. J. Heremans, L. Jiang, A. M. Lewis, A. Mezzacapo, P. J. Mintun, H. Seo, S. Sullivan, S. J. Whiteley, and G. Wolfowicz for fruitful discussions and comments on the manuscript. We also thank the Qiskit Slack channel for generous help. This work was supported by MICCoM, as part of the Computational Materials Sciences Program funded by the U.S. Department of Energy, Office of Science, Basic Energy Sciences, Materials Sciences and Engineering Division through Argonne National Laboratory, under contract number DE-AC02-06CH11357. This research used resources of the National Energy Research Scientific Computing Center (NERSC), a DOE Office of Science User Facility supported by the Office of Science of the US Department of Energy under Contract No. DE-AC02-05CH11231, resources of the Argonne Leadership Computing Facility, which is a DOE Office of Science User Facility supported under Contract DE-AC02-06CH11357, and resources of the University of Chicago Research Computing Center.
Author contributions {#author-contributions .unnumbered}
====================
H.M., M.G., and G.G. designed the research. H.M. implemented the quantum embedding theory and performed simulations with classical and quantum algorithms, with supervision by M.G. and G.G. All authors wrote the manuscript.
Competing interests {#competing-interests .unnumbered}
===================
Authors declare no competing interests.
Figures and tables {#figures-and-tables .unnumbered}
==================
![General strategy for quantum simulations of materials using quantum embedding. The full system is separated into an active space and its environment, with the electronic states in the active space described by an effective Hamiltonian solved with either classical (e.g. full configuration interaction, FCI) or quantum algorithms (e.g. phase estimation algorithm (PEA), variational quantum eigensolver (VQE)). The effective interaction between electrons in the active space includes the bare Coulomb interaction and a polarization term arising from the dielectric screening of the environment (see text), which is evaluated including exchange-correlation interactions.[]{data-label="strategy"}](strategy.png){width="6in"}
![Electronic structure of the negatively-charged nitrogen vacancy (NV) in diamond (a), the neutral silicon vacancy (SiV) in diamond (b) and the Cr impurity (4+) in 4H-SiC (c). Left panels show spin densities obtained from spin unrestricted DFT calculations. Middle panels show the position of single-particle defect levels computed by spin restricted DFT calculations. States included in active spaces (see text) are indicated by blue vertical lines. Right panels show the symmetry and ordering of the low-lying many-body electronic states obtained by exact diagonalization (FCI calculations) of effective Hamiltonians constructed with exchange-correlation interactions included.[]{data-label="defects"}](defects_new.png){width="5.5in"}
System Excitation RPA Beyond-RPA Expt.
-------- ----------------------------------------- ------- ------------ ------------------------------------
NV ${}^3A_{2} \leftrightarrow {}^3E_{}$ 1.921 2.001 2.180$^*$ (1.945$^*$)
${}^3A_{2} \leftrightarrow {}^1A_{1}$ 1.376 1.759
${}^3A_{2} \leftrightarrow {}^1E_{}$ 0.476 0.561
${}^1E_{} \leftrightarrow {}^1A_{1}$ 0.900 1.198 (1.190$^\dagger$)
${}^1A_{1} \leftrightarrow {}^3E_{}$ 0.545 0.243 (0.344-0.430$^\ddagger$)
SiV ${}^3A_{2g} \leftrightarrow {}^3E_{u}$ 1.590 1.594 1.568$^\mathsection$ (1.31$^{||}$)
${}^3A_{2g} \leftrightarrow {}^3A_{1u}$ 1.741 1.792
${}^3A_{2g} \leftrightarrow {}^1E_{g}$ 0.261 0.336
${}^3A_{2g} \leftrightarrow {}^1A_{1g}$ 0.466 0.583
${}^3A_{2g} \leftrightarrow {}^1A_{1u}$ 1.608 1.623
${}^3A_{2g} \leftrightarrow {}^1E_{u}$ 2.056 2.171
${}^3A_{2g} \leftrightarrow {}^1A_{2u}$ 2.365 2.515
${}^3A_{2u} \leftrightarrow {}^3E_{u}$ 0.003 0.011 (0.007$^{||}$)
Cr ${}^3A_{2} \leftrightarrow {}^3E_{}$ 1.365 1.304
${}^3A_{2} \leftrightarrow {}^3A_{1}$ 1.480 1.406
${}^3A_{2} \leftrightarrow {}^3E_{}'$ 1.597 1.704
${}^3A_{2} \leftrightarrow {}^3A_{2}'$ 1.635 1.755
${}^3A_{2} \leftrightarrow {}^1E_{}$ 0.860 1.090 (1.190$^\mathparagraph$)
${}^3A_{2} \leftrightarrow {}^1A_{1}$ 1.560 1.937
\[excitation\_energies\]
$^*$Ref [@Davies1976]. $^\dagger$Ref [@Rogers2008]. $^\ddagger$Estimated by Ref [@Goldman2015] with a model for intersystem crossing. $^\mathsection$Computed with Stokes energy from Ref [@Thiering2019]. $^{||}$Ref [@Green2019]. $^\mathparagraph$Ref [@Son1999].
![Quantum simulations of a minimum model of the NV center in diamond using the phase estimation algorithm (PEA) and a variational quantum eigensolver (VQE). The energy of the $^3A_2$ ground state manifold is set to zero for convenience. (a) PEA estimation of ground and excited states of the NV center. Error bars represent the uncertainties due to the finite number of ancilla qubits used in the simulations; dashed lines show classical FCI results. (b) VQE estimation of ground state energy, starting from $\ket{a e_x \bar{e}_x e_y}$ state ($M_S = 1$, see text). (c) VQE estimation of ground state energy, starting from $\ket{a \bar{a} e_x \bar{e}_x}$ state ($M_S = 0$); strongly-correlated $\frac{1}{2}\left( \ket{a \bar{a} e_x \bar{e}_y} + \ket{a \bar{a} \bar{e}_x e_y} \right)$ state ($M_S = 0$ state in the $^3A_2$ manifold) is obtained with VQE. []{data-label="quantum"}](quantum.png){width="3.5in"}
|
---
abstract: |
Since the early 1970s, stellar population modelling has been one of the basic tools for understanding the physics of unresolved systems from observation of their integrated light. Models allow us to relate the integrated spectra (or colours) of a system with the evolutionary status of the stars of which it is composed and hence to infer how the system has evolved from its formation to its present stage. On average, observational data follow model predictions, but with some scatter, so that systems with the same physical parameters (age, metallicity, total mass) produce a variety of integrated spectra. The fewer the stars in a system, the larger is the scatter. Such scatter is sometimes much larger than the observational errors, reflecting its physical nature. This situation has led to the development in recent years (especially since 2010) of Monte Carlo models of stellar populations. Some authors have proposed that such models are more realistic than state-of-the-art standard synthesis codes that produce the mean of the distribution of Monte Carlo models.
In this review, I show that these two modelling strategies are actually equivalent, and that they are not in opposition to each other. They are just different ways of describing the probability distributions intrinsic in the very modelling of stellar populations. I show the advantages and limitations of each strategy and how they complement each other. I also show the implications of the probabilistic description of stellar populations in the application of models to observational data obtained with high-resolution observational facilities. Finally, I outline some possible developments that could be realized in stellar population modelling in the near future.
address:
- 'Instituto de Astrofísica de Andlucía (IAA-CSIC), Placeta de la Astronomía s/n, 18028, Granada, Spain'
- 'Instituto de Astrofísica de Canarias (IAC), 38205 La Laguna, Tenerife, Spain'
- 'Departamento de Astrofísica, Universidad de La Laguna (ULL), E-38206 La Laguna, Spain'
author:
- Miguel Cerviño
title: The stochastic nature of stellar population modelling
---
stars: evolution ,galaxies: stellar content ,Hertzprung–Russell (HR) and C-M diagrams ,methods: data analysis
-------------------------------------------------
[You only get a measure of order and control]{}
[when you embrace randomness.]{}
[(N.N. Taleb, *Antifragile*) ]{}
-------------------------------------------------
Introduction
============
Motivation
----------
Open your window and take a look at the night sky on a clear night. You can see lots of stars, lots of different types of stars. You work in astrophysics so you can count the stars and measure their light (you can use different photometric filters or measure spectra), sum the individual observations, and obtain the integrated luminosity (magnitudes or spectra). After that, ask some colleagues to do the same experiment, observing the same number of stars, $\cal N$, you have counted in distant places, let us say $n_\mathrm{sam}$ colleagues. In fact, they will see different regions of the sky so they will observe different stars; hence, you are sampling the sky with $n_\mathrm{sam}$ elements, each with $\cal N$ stars. Compare the integrated luminosities and, almost certainly, they will differ. But you are looking in the Sun’s neighbourhood where you can define the set of physical conditions that define the stars that you would observe (initial mass function, star formation history, age, metallicity); hence, your results and those of your colleagues should be consistent with these physical conditions, although they differ from each other, right?
Now take your favourite synthesis code, include the physical conditions, and obtain the integrated luminosity, magnitude, and spectra. In addition, perform millions ($n_\mathrm{sam}$ is millions) of Monte Carlo simulations with those physical conditions using $\cal N$ stars in each simulation. Almost certainly, neither the integrated luminosities obtained by the code nor any of those obtained by Monte Carlo simulations equal the ones you or any of your colleagues have obtained. However, such scatter is an inherent result of nature and, as it should be, is implicit in the modelling. Therefore, you see scatter in both observations by you and your colleagues and the Monte Carlo simulations; but where is the scatter in the standard results provided by synthesis codes? After all, most codes only produce a single result for given physical conditions.
After some time thinking about this (several years in my case), you realize that the results of the Monte Carlo simulations are [*distributed*]{}, and you can glimpse the shape of such distributions. Moreover, you realize that your observations are inside the distribution of the Monte Carlo results once binned. Maybe they are not in the most populated bin, maybe some of them are in a low-frequency bin, but they are inside the distribution of simulations, exactly as if they were additional simulations. You then obtain the mean value of the distribution and you realize that it is suspiciously similar to the value obtained by the synthesis code using $\cal N$ as a scaling factor. It is also applied to the mean value obtained from your observational set. Furthermore, if you obtain the variance (a measure of scatter) for the Monte Carlo set and the observational set and divide them by the mean values you obtained before, the results are similar to each other and to the (so-called) surface brightness fluctuations, SBF, the synthesis code would produce. It is valid for ${\cal N} = 1$ and ${\cal N} = \infty$, although the larger that $n_\mathrm{sam}$ is, the more similar the results are.
From this experiment you realize that the shape of the integrated luminosity distribution changes with $\cal N$ and would actually be power law-like, multimodal, bimodal, or Gaussian-like, but [*synthesis models, the standard synthesis models you have known for years, always produce the correct mean value of the simulation /observational set. Standard models are also able to obtain a correct measure of the dispersion of possible results (some codes have actually computed it since the late 1980s)*]{}. A different issue is that most standard synthesis models provide only the mean value of the distribution of possible luminosities; hence, they lose part of their predicting power.
If you perform this experiment using different photometric filters or integrated spectra, you also realize that, depending on the wavelength, you obtain different values of the mean [*and the variance*]{}. You will certainly find that the scatter is greater for red than for blue wavelengths. But this result is consistent with the SBF values obtained from the population synthesis code, which does not include observational errors, so such variation in the scatter with wavelength is a physical result valid for ${\cal N} = 1$ and ${\cal N} = \infty$. This is somewhat confusing since it implies that knowledge of the emission in a spectral region does not directly provide knowledge regarding other spectral regions. There is no perfect correlation between different wavelengths, only partial correlations. You might then think that a $\chi^2$ fitting including only the observational error and not the theoretical scatter might not be very good; it might be better to include the physical dispersion obtained from the models. It would be even better to include the theoretical correlation coefficients among different wavelengths.
You then realize that the meaning of sampling effects are primarily related to $n_\mathrm{sam}$ and only secondarily to $\cal N$, and you begin to think about how to take advantage of this. We know that $n_\mathrm{sam} \times {\cal N}$ is the total number of stars in the system, $N_\mathrm{tot}$; hence, we can establish that the analysis of resolved populations using colour–magnitude diagrams (CMDs) uses ${\cal N} = 1$ and $n_\mathrm{sam} = N_\mathrm{tot}$. In fact, CMDs are the best option for inferring stellar population parameters since you have information about all the stars [*and about how their luminosities are distributed*]{}. Analysis of the integrated properties of a fully unresolved system uses ${\cal N} = N_\mathrm{tot}$ and $n_\mathrm{sam} = 1$. Systems that are not fully resolved or not fully unresolved are becoming the norm with new and future observational facilities; they have $\cal N$ and $n_\mathrm{sam}$ values in the range 1 and $N_\mathrm{tot}$. The stars are not resolved, but you have more information than only the integrated properties: you have more than one event to sample the distribution of possible luminosities. If you apply this idea to integral field units (IFUs), you realize that IFU observations (the overall set of IFU observations, not the individual observation of an IFU) can be used to sample the distribution of possible luminosities and to obtain more accurate inferences about stellar populations in the system. If you understand how the distributions of integrated luminosities vary with $\cal N$ and the wavelength, going from power-law-like, multimodal, bimodal and Gaussian-like when $\cal N$ increases, you can apply this to any set of systems. For instance, it can be applied to the globular cluster system and to predict $\cal N$, age, metallicity, and star formation history ranges, where you would observe bimodal colour distributions.
Stellar population modelling is intrinsically probabilistic by construction and is independent whether we are aware of it or not. It describes all the possible situations once physical conditions and the number of stars, $\cal N$, are defined. It is actually the most accurate description we have of stellar populations. Mother Nature is intrinsically stochastic, playing with the whole range of available possibilities for given physics of the system. If you realize that, you will also realize that the observed scatter (once corrected for observational errors) contains physical information, and you will look for such scatter.
The study of stochasticity in the modelling of stellar populations is not new. However, in recent years, with the increasing resolution of observational facilities, the subject has become more and more relevant. There are several papers that address partial aspects related to the stochasticity of stellar populations, but almost none that address the issue from a general point of view and exploit the implications. This last point is the objective of this paper. To achieve this, I assume that all synthesis models are equivalent and correct since comparison of synthesis model results is beyond the scope of this paper. I begin with a brief historical outline of the evolution of stellar population modelling. I continue with an analysis of the origin of stochasticity in modelling in Section \[sec:origin\]. Parametric and Monte Carlo descriptions of stellar population modelling are presented in Sections \[sec:param\] and \[sec:montecarlo\], respectively. I describe the implications of stochasticity in the use of stellar population codes in Section \[sec:implications\]. This section includes some rules of thumb for the use of synthesis models. I outline an unexplored area in which stochasticity could play a role in Section \[sec:open\]. My conclusions are presented in Section \[sec:conclu\].
A short historical review {#sec:histo}
-------------------------
The stellar population concept can be traced back to the work of @Baa44 through the empirical characterization of CMDs in different systems by direct star counting. Similarities and differences in star cluster CMD structure allowed them to be classified as a function of stellar content (their stellar population). Closely related to CMD studies is the study of the density distribution of stars with different luminosities (the stellar luminosity function). In fact, luminosity functions are implicit in CMDs when the density of stars in each region of the CMD is considered, that is, the Hess diagram [@Hess24]. The 3D structure provided by the Hess diagram contains information about the stellar content of the system studied.
It was the development of stellar evolution theory in the 1950s [e.g. @SS52] that allowed us to relate the observed structures in CMDs to the age and metallicity of the system, and explain the density of stars in the different areas of the Hess diagram according to the lifetime of stellar evolutionary phases (and hence the nuclear fuel in each phase). CMDs are divided into the main sequence (MS) region, with hydrogen-burning stars, and post-main sequence (PMS) stars. The stellar density of MS stars in a CMD depends on the stars formed at a given time with a given metallicity. The density of PMS stars in different areas depends on the lifetime of different evolutionary phases after MS turn-off. Stellar evolution theory allows us to transform current observable quantities into initial conditions and hence provides the frequency distribution of the properties of stars at birth. This frequency distribution, when expressed in probability terms, leads to the stellar birth rate, ${\cal B}(m_0,t,Z)$, which provides the probability that [*a star*]{} was born with a given initial mass, at a given time, and with a given metallicity.
By applying stellar evolution to the stellar luminosity function observed in the solar neighbourhood, @Sal55 inferred the distribution of initial masses, the so-called initial mass function (IMF, $\phi(m_0)$). To quote @Sal55, ‘This luminosity function depends on three factors: (i) $\phi(m_0)$, the relative probability for the creation of stars near $m_0$ at a particular time; (ii) the rate of creation of stars as a function of time since the formation of our galaxy; and (iii) the evolution of stars of different masses.’ There are various implications in this set of assumptions that merit detailed analysis. The first is that the luminosity function used in the work is not directly related to a stellar cluster, where a common physical origin would be postulated, but to a stellar ensemble where stars would have been formed under different environmental conditions. Implicitly, the concept of the stellar birth rate is extended to any stellar ensemble, independently of how the ensemble has been chosen. The second implication is that the stellar birth rate is decomposed into two different functions, the IMF and the star formation history (SFH, $\psi(t,Z)$). This is the most important assumption made in the modelling of stellar populations since it provides the definition of SFH and IMF, and our current understanding of galaxy evolution is based on such an assumption. Third, the IMF is a probability distribution; that is, the change from a (discrete) frequency distribution obtained from observations to a theoretically continuous probability distribution [@Sal55; @Math59; @Sca86 among others]. The direct implication of the IMF definition is that, to quote @Sca86, there is ‘no means of obtaining an empirical mass distribution which corresponds to a consistent definition of the IMF and which can be directly related to theories of star formation without introducing major assumptions.’ I refer interested readers to @Ceretal13a for further implications that are usually not considered in the literature.
Following the historical developments, @Hod89 defined the concept of a ‘population box’, a 3D histogram in which the $x$–$y$ plane is defined by the age of the stars and their metallicity. The vertical axis denotes the number of stars in each $x$–$y$ bin, or the sum of the initial masses of the stars in the bin. It is related to the star formation history, $\psi(t,Z)$, and, ultimately, to the stellar birth rate. Fig. \[fig:CMDcarma\] shows the population box for a field in the Large Magellanic Cloud disc taken from @Mesetal13.
The population box of a system is the very definition of its stellar population; it comprises information relating to the different sets of stars formed at a given time with a given metallicity. The objective of any stellar population model is to obtain such population boxes. The model is obtained at a global level, restricted to subregions of the system, or restricted to particular $x$–$y$ bins of the population box.
A stellar ensemble with no resolved components does not provide a CMD, but the sum of all the stars in the CMD. However, we can still infer the stellar population of the system by using just this information if we can characterize the possible stellar luminosity functions that sum results in the integrated luminosity of the system. The problem of inferring the stellar content from the integrated light of external galaxies was formalized by @Whi35. The idea is to take advantage of different photometric or particular spectral characteristics of the stars in the different regions of a CMD and combine them in such a way that we can reproduce the observations. The only requirement is to establish, on a physical basis, the frequency distribution of the spectral types and absolute magnitudes of the stars in the CDM. Studies by Sandage, Schwarzschild and Salpeter provide the probability distribution (instead of a frequency distribution) needed for proper development of Whipple’s ideas, and in the late 1960s and the 1970s, mainly resulting from the work of @Tins68 [@Tins72; @TG76; @Tins80], a framework for evolutionary population synthesis and chemical evolution models was established.
In the 1980s, there were several significant developments related to stellar populations. The first was the definition of the single-age single-metallicity stellar population (SSP) by @Ren81 [see also @RB83]. An SSP is the stellar birth rate when the star formation history is a Dirac delta distribution [see @Buzz89 for an extensive justification of such an approximation]. In addition, an SSP is each one of the possible points in the age–metallicity plane of the population box. The second development resulted from work by @CB91 and @BC93 using isochrone synthesis. The density of PMS stars in an SSP depends mainly on the lifetime of each PMS phase, which is linked to the amount of fuel in the phase. In addition, all PMS stars in an SSP have a mass similar to that of the MS turn-off, $m_\mathrm{TO}$. Isochrone synthesis assumes that the similar recipes used to interpolate tracks in main sequence stars (homology relations in polytropic models) are valid for PMS stars as long as interpolations are between [*suitable*]{} equivalent evolutionary points. Hence, each PMS phase at a given age is related to stars with initial mass $m_\mathrm{TO}+\Delta m$, and the density of stars in the phase is given by the integral of the IMF over the $\Delta m$ interval. I refer interested readers to @MG01 for additional aspects of isochrone synthesis and the lifetime of PMS phases. The connection of the SSP concept to isochrones in terms of CMD diagrams and stellar luminosity functions implies that an SSP is the set of stars defined by an isochrone [*including*]{} the density of stars at each point of the isochrone when weighted by the IMF.
In the 1980s there were also several advances in the study of the so-called sampling effects in population synthesis, especially in relation to the number of PMS stars in a stellar population. @BB77 investigated how evolutionary model results vary depending on the number of stars in a synthetic cluster. The authors reported that sampling effects originate from the way the IMF is sampled [*and*]{} large changes in the effective temperature of a star during the PMS evolution, when there are rapid evolutionary phases (situations in which small variations in $m_0$ produce wide variations in luminosity). Similar studies have been carried out by @BM83 [@CBB88; @GB93; @GCBB95; @SF97]. A common characteristic in these studies, besides the application to LMC clusters, is the identification of sampling effects with the occurrence by number of luminous PMS stars dominating the integrated light.
From a more global perspective, @Buzz89 established a direct analytical formalism to evaluate sampling effects in clusters of different sizes by defining the effective number of stars, $N_\mathrm{eff}$, which contributes to the integrated luminosity[^1] (which varies with the age and wavelength considered). To do so, he assumed that the number of stars with a given mass follows a Poisson statistic; hence, dispersion of the total luminosity is the sum of the independent Poisson statistics for the number of stars with a given luminosity. Obviously, the main contribution to global dispersion of the given luminosity is caused by sparse but luminous stars. Independently, @TS88, using similar arguments about Poisson statistics, proposed the use of dispersion of the total flux of a galaxy image as a primary distance indicator (so-called surface brightness fluctuation, SBF), which is an [*observational*]{} quantity. In fact, $N_\mathrm{eff}$ and SBFs are related, as shown by @Buzz93. However, we must recall that theoretical SBFs are a measure of possible fluctuations around a mean value, a more general concept than that used by @TS88.
SBFs observations are one of the smoking guns of the stochasticity of stellar populations at work in nature, and the first case to take advantage of such stochasticity to draw inferences about the physical quantities of stellar systems. Another application of SBFs is the breaking of age–metallicity degeneracy in SSP results for old ages [@W94]; additional applications have been described by @Buz05.
Another aspect of SBFs is that they had been extensively studied and used in systems with [*old*]{} stellar populations. It would be surprising if we related stochasticity to IMF sampling: the integrated light in old stellar populations is dominated by low-mass stars, which are more numerous than high-mass ones. Hence, if stochasticity is related to the IMF alone, we would expect dispersion to fade out as a system evolves. In fact, this is an erroneous interpretation of sampling effects: as cited before, the number of stars in a given PMS phase is defined by the $\Delta m$ interval and the size of such an interval does not depend on the IMF.
To explain stochasticity in terms of Poisson statistics was the norm in the modelling of the physical scatter of stellar populations up to 2005 [@LM00; @Cetal01; @CVGLMH02; @GLB04 among others]. The Poisson distribution, since it is discrete, is easily related to a natural number of stars and to interpretation of the IMF as a frequency distribution (as opposed to a probability density function). But it was modelling in the X-ray domain by @GGS04 that established the key point of stochastic modelling by using the stellar luminosity function. The ideas of @GGS04 were expanded to the optical domain by @CLCL06, who thereby established a unified formalism in the modelling of stellar populations that can be applied from CMDs to the integrated light of galaxies. This work showed that the Poisson statistic is invalid in so far as the total number of stars in a system are correlated to each other by the ${\cal B}(m_0,t,Z)$ probability distribution and introduced covariance terms in the computation of the scatter by synthesis models. Such covariance terms are especially relevant in SBF computation, as shown by @CLJ08. That study also showed that probability distributions, when expressed as frequency distributions, follow a multinomial distribution, which is the natural result of binning of the stellar luminosity function.
In a phenomenological approach, the scatter of synthesis models has been studied in Monte Carlo simulations, particularly since the late 1990s, although previous studies can be found in the literature [e.g. @MHK91; @CMH94 as well as work related to LMC clusters quoted earlier]. There is a wide variety of such studies, ranging from simulations applied to specified targets or wavelength domains to general studies of Monte Carlo simulation results. Examples of specific domains include globular clusters [old populations with ${\cal N} \sim 10^6$ stars by @Yoetal06; @CB07 among others], $\gamma$-ray and optical emission from young clusters in our galaxy [@CLC00; @Knoetal02; @Vosetal09] and the study of SBF [@BCC98; @RBCC05]. Examples of general studies include work by @Broetal99 [@Brutuc; @BC03; @CLCL06; @FL10; @massclean; @Pisetal11; @SVL11; @slug; @bin], among others. There are currently some public Monte Carlo synthesis codes for further research in the area, such as the [*SLUG*]{} package () [@slug] and the [*MASSCLEAN*]{} packages ([%[7Ebogdan/massclean.html]{}]{}) [@massclean; @masscleancolors; @masscleannage].
A natural effect of Monte Carlo modelling is direct visualization of the range of scatter in model results. However, the relevant results in Monte Carlo sets are the [*distribution of results*]{} instead of their range (I discuss this point later). Whatever the case, these phenomenological studies have opened up several questions on the modelling of stellar populations and their applications. Some of these questions are as follows. (a) What are the limitations of traditional synthesis models, especially for low values of $\cal N$ [@CVG03; @CL04; @SVL11 among others]? (b) What is actually computed by traditional synthesis models, how are they linked to Monte Carlo modelling, and are they the limit for ${\cal N} \rightarrow \infty$ (e.g. use of terms such as [*discrete population synthesis*]{} vs. [*continuous population synthesis*]{} by @FL10, or [*discrete IMF*]{} vs. [*continuous IMF*]{} models by @Pisetal11) ? (c) How can Monte Carlo modelling be used to make inferences about stellar systems [@FLCW12; @PHE12]? In this study I aim to solve some of these questions and provide additional uses of stellar population modelling. I begin with the modelling itself.
The origin of stochasticity in stellar population modelling {#sec:origin}
===========================================================
The stellar mass distribution in the fragmentation of a molecular cloud, even though it produces a discrete number of stars, has a continuous range of possible outcomes. Although we do not know the details of fragmentation and they vary from one cloud to another, we observe that the larger the stellar mass considered, the lower is the number of stars in the mass range near such a stellar mass. From the observed frequency distributions we make an abstraction to a continuous probability density distribution (statistical inference) such as the IMF or the stellar birth rate, which has proved to be a useful approach in characterizing this physical situation. By using such probability distributions, we construct the theory of stellar populations to obtain the possible luminosities of stellar ensembles (probabilistic description, the forward problem of [*predicting results*]{}) and produce a continuous family of probability distributions that vary according to certain parameters (mainly $\cal N$, $t$ and $Z$). Finally, we compare these predictions with particular observations aimed at [*false* ]{} regions of the parameter space; that is, to obtain combinations of $\cal N$, $t$ and $Z$ that are [*not compatible*]{} with observations (hypothesis testing in the space of observable luminosities).
The previous paragraph summarizes the three different steps in which stellar populations are involved. These three steps require stochastic (or random) variables, but, although related, each step needs different assumptions and distributions that should not be mixed up. Note that in the mathematical sense, a random variable is one that does not have a single fixed value, but a set of possible values, each with an associated probability distribution.
1. [*IMF inference.*]{} This is a statistical problem: inference of an unknown underlying probability distribution from observational data with a discrete number of events. This aspect is beyond the scope of our review. However, IMF inferences require correct visualization of the frequency distribution of stellar masses to avoid erroneous results [@DAS86; @MAU05]. Such a problem in the visualization of distributions is general for inferences independently of the frequency distribution (stellar masses, distribution of globular clusters in a galaxy, or a set of integrated luminosities from Monte Carlo simulations of a system with given physical parameters).
2. [*Prediction of stellar population observables for given physical conditions*]{}. This is a probabilistic problem for which the underlying probability distribution, IMF and/or ${\cal B}(m_0,t,Z)$ is obtained by hypothesis, since it defines the initial physical conditions. Such probability distributions are modified (evolved) to obtain a new set of probability distributions in the observational domain. The important point is that, in so far as we are interested in generic results, we must renounce particular details. We need a description that covers [*all possible*]{} stellar populations quantitatively, but not necessarily any particular one. This is reflected in the inputs used, which can only be modelled as probability distributions. Hence, we have two types of probability distributions: one defining the initial conditions and one defining observables.
Regarding the initial conditions, the stellar mass is a continuous distribution and hence it cannot be described as a frequency distribution but as a continuous probability distribution, that is, a continuous probability density function (PDF). Hence, the IMF does not provide ‘the number of stars with a given initial mass’, but, after integration, the probability that a star had an initial mass within a given mass range. If the input distribution refers to the probability that [*a star*]{} has a given initial mass, the output distribution must refer to the probability that [*a star*]{} has a given luminosity (at a given age and metallicity).
For the luminosity of an ensemble of stars, we must consider [*all possible combinations of the possible luminosities of the individual stars in the ensemble*]{}. Again, this situation must be described by a PDF. In fact, the PDF of integrated luminosities is intrinsically related to the PDF that describes the luminosities of individual stars. A different question is how such a PDF of integrated luminosities is provided by synthesis models: these can produce a parametric description of the PDFs (traditional modelling), a set of luminosities from which the [*shape*]{} of the PDF can be recovered (Monte Carlo simulation), or an explicit computation of the PDF (self-convolution of the stellar PDF; see below).
3. [*Inference about the physical conditions from observed luminosities*]{}. This is a hypothesis-testing problem, which differs from statistical problems in the sense that it is not possible to define a universe of hypotheses. This implies that we are never sure that the best-fit solution is the actual solution. It is possible that a different set of hypotheses we had not identified might produce an even better fit. The only thing we can be sure of in hypothesis-testing problems is which solutions are incompatible with observed data. We can also evaluate the degree of compatibility of our hypothesis with the data, but with caution. In fact, the best $\chi^2$ obtained from comparison of models with observational data would be misleading; the formal solution of any $\chi^2$ fit is the whole $\chi^2$ distribution. @tarantola provide a general view of the problem, and @FLCW12 (especially Section 7.4 and Fig. 16) have described this approach for stellar clusters.
Assuming the input distributions is correct and we aim to obtain only the evolutionary status of a system, we must still deal with the fact that the possible observables are distributed. An observed luminosity (or a spectrum) corresponds to a different evolutionary status and the distribution of possible luminosities (or spectra) is defined by the number of stars in our resolution element, $\cal N$. Since we have an accurate description of the distribution of possible luminosities as a PDF, the best situation corresponds to the case in which we can sample such a distribution of possible luminosities with a larger number of elements, $n_\mathrm{sam}$, with the restriction that the total number of stars, $N_\mathrm{tot}$, is fixed. As pointed out before, it is CMDs that have ${\cal N} = 1$ and $n_\mathrm{sam} = N_\mathrm{tot}$. We can understand now that the so-called sampling effects are related to $n_\mathrm{sam}$ and how we can take advantage of this by managing the trade-off between $n_\mathrm{sam}$ and $\cal N$. Note that the literature on sampling effects usually refers to situations in which $N_\mathrm{tot}$ has a low value; obviously, this implies that $n_\mathrm{sam}$ is low. However, that explanation loses the advantages that we can obtain by analysis of the scatter for the luminosities of stellar systems.
In summary, [*any time a probability distribution is needed, and such a probability distribution is reflected in observational properties, the description becomes probabilistic.*]{} Stochasticity, involving descriptions in terms of probability distributions, is intrinsic to nature and is implicit in the modelling of stellar populations. It can be traced back to the number of available stars sampled for the IMF or the ${\cal B}(m_0,t,Z)$ distributions, but it is misleading to talk about IMF or stellar-birth-rate sampling effects, since these input distributions are only half of the story.
From the stellar birth rate to the stellar luminosity function
--------------------------------------------------------------
The aim of stellar population modelling is to obtain the evolution of observable properties for given initial conditions ${\cal{B}}(m_0,t,Z)$. The observable property may be luminosities in different bands, spectral energy distributions, chemical yields, enrichment, or any parameter that can be related to stellar evolution theory. Thus, we need to transform the initial probability distribution ${\cal{B}}(m_0,t,Z)$ in the PDF of the observable quantities, $\ell_i$, at a given time, $t_\mathrm{mod}$. In other words, we need to obtain the probability that in a stellar ensemble of age $t_\mathrm{mod}$, a randomly chosen star has given values of $\ell_1, \dots, \ell_n$. Let us call such a PDF $\varphi(\ell_1, \dots,\ell_n; t_\mathrm{mod})$, which is the theoretical stellar luminosity function. [*The stellar luminosity function is the distribution needed to describe the stellar population of a system.*]{}
Any stellar population models (from CMD to integrated luminosities or spectra), as well as their applications, have $\varphi(\ell_1, \dots,\ell_n;t_\mathrm{mod})$ as the underlying distribution. ${\cal{B}}(m_0,t,Z)$ and stellar evolution theory are the gateway to obtaining $\varphi(\ell_1, \dots,\ell_n; t_\mathrm{mod})$. However, in most stellar population studies, $\varphi(\ell_1, \dots,\ell_n;t_\mathrm{mod})$ is not computed explicitly; it is not even taken into consideration, or even mentioned. Neglecting the stellar luminosity function as the underlying distribution in stellar population models produces tortuous, and sometimes erroneous, interpretations in model results and inferences from observational data.
There are several justifications for not using $\varphi(\ell_1, \dots,\ell_n; t_\mathrm{mod})$; it is difficult to compute explicitly (see below) and it is difficult to work in an $n$-dimensional space in both the mathematical and physical senses. We can work directly with ${\cal{B}}(m_0,t,Z)$ and stellar evolution theory and isochrones, $\ell_i(m_0;\tau,Z)$, where $\tau$ is the time measured since the birth of the star, without explicit computation of $\varphi(\ell_1, \dots,\ell_n;t_\mathrm{mod})$. In the following paragraphs I present a simplified description of a simple luminosity $\varphi(\ell; t_\mathrm{mod})$. Such a description is enough for understanding most of the results and applications of stellar population models.
First, the physical conditions are defined by ${\cal{B}}(m_0,t,Z)$ but we must obtain the possible luminosities at a given $t_\mathrm{mod}$. When $t_\mathrm{mod}$ is greater than the lifetime of stars above a certain initial mass, the first component of $\varphi(\ell; t_\mathrm{mod})$ is a Dirac delta function at $\ell = 0$ that contains the probability that a randomly chosen star taken from ${\cal{B}}(m_0,t,Z)$ is a dead star at $t_\mathrm{mod}$. With regard to isochrones, dead stars are rarely in the tables in so far as only luminosities are included. However, they are included if the isochrone provides the cumulative ejection of chemical species from stars of different initial masses (as needed for chemical evolution models).
Second, stars in the MS are described smoothly by $\ell(m_0;\tau,Z)$, which are monotonic continuous functions. In this region there is a one-to-one relation between $\ell$ and $m_0$, so $\varphi(\ell; t_\mathrm{mod})$ resembles ${\cal{B}}(m_0,t,Z)$. For an SSP case with a power law IMF, $\varphi(\ell; t_\mathrm{mod})$ is another power law with a different exponent.
Third, discontinuities in $\ell(m_0;\tau,Z)$ lead to discontinuities or bumps in $\varphi(\ell; t_\mathrm{mod})$. Depending on luminosities before $\ell(m_{0-}; \tau, Z)$ and after $\ell(m_{0+}; \tau, Z)$, they lead to a gap in the distribution (when $\ell(m_{0-}; \tau, Z) < \ell(m_{0+}; \tau, Z)$), or an extra contribution to $\ell(m_{0+}; \tau, Z)$ (when $\ell(m_{0-}; \tau, Z) > \ell(m_{0+}; \tau, Z)$). Discontinuities are related to changes in stellar evolutionary phases as a function of the initial mass, and to intrinsic discontinuities of stellar evolution (e.g. stars on the horizontal branch).
Fourth, variations in the derivative of $\ell(m_0;\tau,Z)$ led to depressions and bumps in $\varphi(\ell; t_\mathrm{mod})$. Consider a mass range for which a small variation in $m_0$ leads to a large variation in $\ell$ (the so-called fast evolutionary phases). We must cover a large $\ell$ range with a small range of $m_0$ values; hence, there will be a lower probability of finding stars in the given luminosity range (and in such an evolutionary phase), and hence a depression in $\varphi(\ell; t_\mathrm{mod})$. The steeper the slope, the deeper is the depression. Conversely, when the slope of $\ell(m_0;\tau,Z)$ is flat, there is a large mass range sharing the same luminosity; hence, it is easier to find stars with the corresponding $\ell$ values, and the probability for such $\ell$ values increases. Both situations are present in PMS phases.
In summary: (a) $\varphi(\ell_1, \dots,\ell_n; t_\mathrm{mod})$ has three different regimes, a Dirac delta component at zero luminosity because of dead stars, a low luminosity regime corresponding to the MS that resembles ${\cal{B}}(m_0,t,Z)$, and a high luminosity regime primarily defined by the PMS evolution that may overlap the MS regime; and (b) the lower the possible evolutionary phases in the PMS, the simpler is $\varphi(\ell_1, \dots,\ell_n; t_\mathrm{mod})$. In addition, the greater the possible evolutionary phases, the greater is the possibility of the incidence of bumps and depressions in the high-luminosity tail of the distribution.
From resolved CMD to integrated luminosities and the dependence on $\cal N$
---------------------------------------------------------------------------
\
The previous section described the probability distribution that defines the probability that a randomly chosen star in a given system with defined ${\cal B}(m_0,t,Z)$ at time $t_\mathrm{mod}$ has given values of luminosity $\ell_1, \dots,\ell_n$. Let us assume that we now have a case in which we do not have all the stars resolved, but that we observe a stellar system for which stars have been grouped (randomly) in pairs. This would be a cluster in which single stars are superposed or blended. The probability of observing an integrated luminosity $L_{i,{\cal N}=2}$ from the sum for two stars is given by the probability that the first star has luminosity of $\ell_i$ multiplied by the probability that the second star has luminosity of $L_{i,{\cal N}=2} - \ell_i$ considering all possible $\ell_i$ values; that is, convolution of the stellar luminosity function with itself [@CLCL06]. Following the same argument, it is trivial to see that the PDF describing the integrated luminosity of a system with $\cal N$ stars, $\varphi_{\cal N}(L_1, \dots,L_n; t_\mathrm{mod})$, is the result of $\cal N$ self-convolutions of $\varphi(\ell_1, \dots,\ell_n; t_\mathrm{mod})$.
The situation in a $U-V$ versus $M_V$ Hess diagram is illustrated in Fig. \[fig:jma34\] for ${\cal N} = 1, 2, 64$ and 2048. The figure is from @Maiz08, who studied possible bias in IMF inferences because of crowding and used the convolution process already described. The figure uses logarithmic quantities; hence, it shows the [*relative*]{} scatter, which decreases as $\cal N$ increases. An important feature of the plot is that the case with a larger number of stars shows a banana-like structure caused by correlation between the $U$ and $V$ bands. In terms of individual stars, those that dominate the light in the $V$ band are not exactly the same as those that dominate the light in the $U$ band. However, there is partial correlation between the two types of stars resulting from ${\cal B}(m_0,t,Z)$. Such partial correlation is present when stars are grouped to obtain integrated luminosities. These figures also illustrate how CMD studies (${\cal N} = 1$) can be naturally linked to studies of integrated luminosities (${\cal N} > 2$) as long as we have enough observations ($n_{\mathrm{sam}}$) to sample the CMD diagram of integrated luminosities.
We now know that we can describe stellar populations as $\varphi_{\cal N}(L_1, \dots,L_n; t_\mathrm{mod})$ for all possible $\cal N$ values. The question now is how to characterize such a PDF. We can do this in several ways. (a) We can obtain PDFs via the convolution process, which is the most exact way. Unfortunately, this involves working in the $n$D space of observables and implementation of $n$-dimensional convolutions, which are not simple numerically. (b) We can bypass the preceding issues using a brute force methodology involving Monte Carlo simulations. This has the advantage (among others; see below) that the implicit correlations between the $\ell_i$ observables are naturally included and is thus a suitable phenomenological approach to the problem. The drawbacks are that the process is time-consuming and requires large amounts of storage and additional analysis for interpretation tailored to the design of the Monte Carlo simulations. This is discussed below. (c) We can obtain parameters of the PDFs as a function of $\cal N$. In fact, this is the procedure that has been performed in synthesis codes since their very early development: mean values and sometimes the variance of the PDFs expressed as SBF or $N_\mathrm{eft}$ are computed. The drawback is that a mean value and variance are not enough to establish probabilities (confidence intervals or percentiles) if we do not know the shape of the PDF.
Stellar population modelling: the parametric approach and its limitations {#sec:param}
=========================================================================
The parametric description of the stellar luminosity PDF is the one used by far the most often, although usually only the mean value of the distribution is computed. In fact, when we weight the stellar luminosity along an isochrone with an IMF in an SSP, it is such a mean value, $\mu_1'(\ell_i)$, that is obtained [see @CLCL06 for more details].
We can also evaluate by how much the possible $\ell_i$ values differ from the mean value. For example, the variance $\mu_2(\ell_i)$, which is the average of the square of the distance to the mean (i.e. e integral of $(\ell_i - \mu_1'(\ell_i))^2 \, \varphi(\ell_i)$ over the possible $\ell_i$ values). In general, we can compute the difference between the mean and the possible $\ell_i$ elements of the distribution using any power, $(\ell_i - \mu_1'(\ell_i))^n$, and the resulting parameter is called the central $n$-moment, $\mu_n(\ell_i)$. We can also obtain the covariance for two different luminosities $\ell_i$ and $\ell_j$ by computing $(\ell_i - \mu_1'(\ell_i))^n(\ell_j - \mu_1'(\ell_j))^m$ integrated over $\varphi(\ell_i,\ell_j)$, where linear covariance coefficients are obtained for the case $n = m = 1$.
Parametric descriptions of PDFs usually use few parameters: the mean, variance, skewness, $\gamma_1(\ell_i) = \mu_3/\mu_2^{3/2}$, and kurtosis $\gamma_2(\ell_i) = \mu_4/\mu_2^2 -3$. Skewness is a measure of the asymmetry of the PDF. Kurtosis can be interpreted as a measure of how flat or peaked a distribution is (if we focus the comparison on the central part of the distribution) or how fat the tails of the distribution are (if we focus on the extremes) when compared to a Gaussian distribution. For reference, a Gaussian distribution has $\gamma_1 = \gamma_2 = 0$. Typical values of the four parameters and their evolution with time for an SSP case are shown in Fig. \[fig:clcl06\] taken from @CLCL06.
Large positive $\gamma_1$ values indicate that stellar luminosity PDFs in the SSP case are L-shaped, and large positive $\gamma_2$ values indicate that they have fat tails. In fact, we noted in the previous section that the stellar luminosity function is an L-shaped distribution composed of a power-law-like component resulting from MS stars and a fat tail at large luminosities because of PMS stars. However, $\gamma_1$ and $\gamma_2$ computation provides us with a [*quantitative*]{} characterization of the distribution shape without an explicit visualization.
The parameters that describe the distribution of integrated luminosities, $\varphi_{\cal N}(L_1, \dots,L_n; t_\mathrm{mod})$, are related to those of the stellar luminosity function by simple scale relations [@CLCL06]:
$$\begin{aligned}
\mu'_{1;{\cal N}}(L_i) & = & {\cal N} \times \mu_1'(\ell_i),\label{eq:mean}\\
\mu_{2;{\cal N}}(L_i) & = & {\cal N} \times \mu_2(\ell_i),\label{eq:variance}\\
\gamma_{1;{\cal N}}(L_i) & = & \frac{1}{\sqrt{\cal N}} \times \gamma_1(\ell_i), \label{eq:ske}\\
\gamma_{2;{\cal N}}(L_i) & = & \frac{1}{\cal N} \times \gamma_2(\ell_i). \label{eq:kur}\end{aligned}$$
We can then obtain additional scale relations for the effective number of stars at a given luminosity, $N_{\mathrm{eff};{\cal N}}(L_i)$ [@Buzz89], the SBF, $\bar{L}$ [@TS88; @Buzz93], and the correlation coefficients between two different luminosities, $\rho(L_i,L_j)$:
$$\begin{aligned}
N_{\mathrm{eff};{\cal N}}(L_i) & = & {\cal N} \times N_\mathrm{eff}(\ell_i) = {\cal N} \times \frac{\mu'^2_1(\ell_i)}{\mu_2(\ell_i)},\label{eq:neff}\\
\bar{L}_{i} & = & = \frac{\mu_2(\ell_i)}{\mu'_1(\ell_i)} ~~\forall {\cal N}, \label{eq:sbf}\\
\rho(L_i,L_j) & = & \rho(\ell_i,\ell_j) = \frac{\mathrm{cov}(\ell_i,\ell_j)}{\sqrt{\mu_2(\ell_i) \mu_2(\ell_j)}} ~~~\forall {\cal N}. \label{eq:rho}\end{aligned}$$
Direct (and simple) computation of the parameters of the distribution provides several interesting results. First, SBFs are a measure of the scatter that is independent of ${\cal N}$ and can be applied to any situation (from stellar clusters to galaxies) in fitting techniques. Second, correlation coefficients are also invariant about ${\cal N}$ and can be included in any fitting technique. Third, since the inverse of $N_{\mathrm{eff};{\cal N}}(L_i)$ is the relative dispersion, when ${\cal N} \rightarrow \infty$ the relative dispersion goes to zero, although the absolute dispersion (square root of the variance, $\sigma$) goes to infinity. Fourth, when ${\cal N} \rightarrow \infty$, $\gamma_{1;{\cal N}}(L_i) $ and $\gamma_{2;{\cal N}}(L_i)$ goes to zero, and hence the shape of the distribution of integrated luminosities becomes a Gaussian-like distribution (actually an $n$-dimensional Gaussian distribution including the corresponding $\rho(L_i,L_j)$ coefficients). We can also obtain the range of $\gamma_{1;{\cal N}}(L_i)$ and $\gamma_{2;{\cal N}}(L_i)$ values for which the distribution can be approximated by a Gaussian or an expansion of Gaussian distributions (such an Edgeworth distribution) for a certain luminosity interval. As reference values, the shape of PDFs where $\gamma_{1;{\cal N}} < 0.3$ or $\gamma_{2;{\cal N}} < 0.1$ are well described with these four parameters by an Edgeworth distribution; when $\gamma_{1;{\cal N}} < 0.03$ and $\gamma_{2;{\cal N}} < 0.1$, the PDFs are well described by a Gaussian distribution with the corresponding mean and variance. A Monte Carlo simulation or a convolution process is needed in other situations. The possible situations are shown in Fig. \[fig:gammas\], taken from @CLCL06.
Another possibility that covers situations for higher $\gamma_{1;{\cal N}}$ and $\gamma_{2;{\cal N}}$ values quoted here (i.e. asymmetric PDFs) is to approximate $\varphi(\ell_i)$ by gamma distributions, as done by @MA09. This approach can be used in a wide variety of situations as long as the PDF has no bumps or the bumps are smooth enough and an accurate description of the tails of the distribution is not required.
The mean and variance obtained using standard models
----------------------------------------------------
We have seen that using Eq. (\[eq:mean\]), the mean value can be expressed for any possible $\cal N$ value or for any quantity related to $\cal N$. Most population synthesis codes use the mass of gas transformed into stars or the star formation rate (also expressed as the amount of mass transformed into stars over a time interval) instead of referring to the number of stars. Hence, the typical unit of luminosity is \[erg s$^{-1}$ M$_\odot^{-1}$\] or something similar. However, here I argue that the computed value actually refers to the mean value of $\varphi_\ell(\ell)$, so the units of the luminosity obtained by the codes should be \[erg s$^{-1}$\] and refer to individual stars.
In fact, the difference is in the interpretation and algebraic manipulation of ${\cal B}(m_0,t,Z)$ and the IMF. The usual argument has two distinct steps. (1) The integrated luminosity and total stellar mass in a system are the sum of the luminosities and masses of all the individual stars in the system. Thus, the ratio of luminosity to mass produces the mass–luminosity relation for the system. (2) ${\cal B}(m_0,t,Z)$ (or the IMF) provides the [*actual*]{} mass of the individual stars in the system, and since the shape of such functions is independent of the number of stars in the system, the previous mass–luminosity relations are valid for any ensemble of stars with similar ${\cal B}(m_0,t,Z)$ functional form. The first step is always true as long as we know the masses and evolutionary status of all the stars in the system (actually it is the way that each [*individual*]{} Monte Carlo simulation obtains observables). The second step is false: [*we do not know the individual stars in the system*]{}. We can describe the set using a probability distribution, and hence we can describe the integrated luminosity of [*all*]{} possible combinations of a sample of ${\cal N}$ individual stars.
It is trivial to see that the mass normalisation constant used in most synthesis codes is actually the mean stellar mass $\mu'_1(m_0)$ obtained using the IMF as a PDF.[^2] Equivalently, total masses or star formation rates obtained using inferences from synthesis models are actually ${\cal N} \times \mu'_1(m_0)$ and ${\cal N} \times \mu'_1(m_0) \times t^{-1}$. Usually, this difference has no implication; however, it is different to say that a galaxy has a formation rate of, say, 0.1 stars per year (it forms, on average, a star every 10 years whatever its mass) than 0.1 M$_\odot$ per year (does this mean that, on average, $10^3$ years are needed to form a 100-M $_\odot$ star without forming any other star?).
Different renormalisations are performed on a physical basis, depending on the system we are interested in. Low-mass stars in young starbursts make almost no contribution to the UV integrated luminosity, so we can exclude low-mass stars from the modelling. Massive stars are not present in old systems; hence, we do not need to include massive stars in these SSP models. The use of different normalisations can be solved easily using a renormalisation process. Hence, we can compare the mean values obtained from two synthesis codes that use different constraints. However, we must be aware that underlying any such renormalisation there are different constraints on ${\cal B}(m_0,t,Z)$, and hence there are changes in the shape of the possible $\varphi(\ell_1, \dots, \ell_n; t_\mathrm{mod})$, such as the absence or presence of a Dirac delta function at $\ell_i = 0$. This affects the possible values of the mean, variance (SBF or $N_\mathrm{eft}$), skewness, and kurtosis used to describe $\varphi_{\cal N}(L_1, \dots,L_n; t_\mathrm{mod})$.
Stellar populations using Monte Carlo modelling {#sec:montecarlo}
===============================================
Monte Carlo simulations have the advantage (and the danger) that they are easy to do and additional distributions and constraints can be included in the modelling. Fig. \[fig:mon1\] shows the results of $10^7$ Monte Carlo simulations of SSP models with ages following a power-law distribution between 40 Myr and 2.8 Gyr, and $\cal N$ distributed following a (discrete) power-law distribution in the range between 2 and $10^6$ stars. The figure also shows the standard modelling result using the mean value of the corresponding observable in the age range from 40 Myr to 10 Gyr.
The first impression obtained from the plot is psychologically depressing. We had developed population synthesis codes to draw inferences from observational data, and the Monte Carlo simulations show so large a scatter that we wonder if our inferences are correct. Fig. \[fig:massclean\], from @PHE12, compares the age inferences for LMC clusters using Monte Carlo simulations to sample the PDF of integrated luminosities and traditional $\chi^2$ fitting to the mean value of the PDF (traditional synthesis model results). The figure shows a systematic discrepancy at young ages. Such an effect was also found by @FLCW12 for M83 stellar clusters and by @FL10 [@SVL11] (among others) when trying to recover the inputs of the Monte Carlo simulations using $\chi^2$ fitting to the mean value obtained by parametric models. This requires stepwise interpretation. First, we must understand Monte Carlo simulation results. Second, we must figure out how to use Monte Carlo simulations to make inferences about observational data (hypothesis testing).
Understanding Monte Carlo simulations: the revenge of the stellar luminosity function
-------------------------------------------------------------------------------------
The usual first step is to analyse (qualitatively) the results for Monte Carlo simulations that implicitly represent how $\varphi_{\cal N}(L_1, \dots,L_n; t_\mathrm{mod})$ varies with $\cal N$ [@Brutuc; @CVG03; @massclean; @masscleancolors; @masscleannage; @Pisetal11; @FL10; @SVL11; @slug; @bin]. In general, researchers have found that at very low $\cal N$ values, most simulations are situated in a region away from the mean obtained by parametric models; at intermediate $\cal N$ values, the distribution of integrated luminosities or colours are (sometimes) bimodal ; at large $\cal N$ values, the distributions become Gaussian.
The different situations are shown quantitatively in Fig. \[fig:MonlocationC03\], taken from @Cer03, which can be fully understood if we take advantage of the Monte Carlo simulations and the parametric description of synthesis models [after @GGS04; @CLCL06]. For ages greater than 0.1 Gyr, the figure compares when the mean bolometric luminosity of a cluster with $\cal N$ stars, $L_\mathrm{bol}^{\mathrm{clus}} ={\cal N}Ê\times \mu_1'(\ell_\mathrm{bol}; t_\mathrm{mod})$, equals the maximum value of $\varphi(\ell_\mathrm{bol}; t_\mathrm{mod})$, $L_{*,\mathrm{bol}}^{\mathrm{max}}$. This can be interpreted as the situation in which the light of a cluster is dominated by a single star, called the ‘lowest luminosity limit’ by @CL04. In addition, the $\cal N$ value needed to have, on average, at least 1 PMS star is also computed; it can be obtained using a binomial distribution over the IMF for stars below and above $m_\mathrm{TO}$. The corresponding $\cal N$ values are expressed as the mean total mass of the cluster, $\cal M$.
From the description of $\varphi(\ell;t_\mathrm{mod})$ we know that the distribution is L-shaped with a modal value (maximum of the distribution) in the MS region and a mean located somewhere between the MS and PMS regions. The mean number of PMS will be less than one at low $\cal N$ values. This means that most simulations are low-luminosity clusters composed of only MS stars, and a few simulations are high-luminosity clusters dominated by PMS stars. Here, the mode of $\varphi_{\cal N}(L ;t_\mathrm{mod})$ is defined by MS stars and is biased with respect to the mean of $\varphi_{\cal N}(L ;t_\mathrm{mod})$. We can also be sure that the distribution $\varphi_{\cal N}(L ;t_\mathrm{mod})$ is not Gaussian-like if its mean is lower than the maximum of $\varphi(\ell ;t_\mathrm{mod})$, since the shape of the distribution is sensitive to how many luminous stars are present in each simulation (e.g. 0, 1, 2, …) and this leads to bumps (and possibly multi-modality) in $\varphi_{\cal N}(L ;t_\mathrm{mod})$. Finally, when $\cal N$ is large, $\varphi_{\cal N}(L ;t_\mathrm{mod})$ becomes Gaussian-like and the mean and mode coincide. @Cer03 [@CL04] established the maximum of $\varphi(\ell ;t_\mathrm{mod})$ 10 times to reach this safe result. Curiously, the value of $\cal N$ that can be inferred from Fig. \[fig:MonlocationC03\] can be used in combination with $\gamma_1$ and $\gamma_2$ values from Fig. \[fig:clcl06\]. The resulting $\gamma_{1:{\cal N}}$ and $\gamma_{2;{\cal N}}$ values are approximately 0.1, the values required for Gaussian-like distributions obtained by the parametric analysis of the stellar luminosity distribution function.
This exercise shows that interpretation of Monte Carlo simulations can be established [*quantitatively*]{} when the parametric description is also taken into account. Surprisingly, however, except for Monte Carlo simulations used to obtain SBFs [@Broetal99; @RBCC05; @GLB04], most studies do not obtain the mean or compare it with the result obtained by parametric modelling. Usually, Monte Carlo studies only verify that for large $\cal N$ values, using relative values, the results coincide with standard models. In addition, hardly any Monte Carlo studies make explicit reference to the stellar luminosity function, nor do they compute simulations for the extreme case of ${\cal N} = 1$. Thus, psychological bias is implicit for interpretation of ${\cal B}(m_0,t,Z)$ referring only to stellar [*clusters*]{} and synthesis models referring only to [*integrated*]{} properties.
Another application of the stellar luminosity function is definition of the location of simulations/observations in colour–colour diagrams (actually in any diagnostic diagram using indices that do not depend on $\cal N$). Any integrated colour is a combination of the contributions of different stars; hence, individual stars define an envelope of possible colours of simulations/observations. The situation is illustrated in Fig. \[fig:Monlocation\] taken from @BdGC08, where the possible range of colours of individual stars and the mean of parametric synthesis models are compared.
The final application of parametric descriptions of the integrated luminosity function to Monte Carlo simulations is a back-of-the-envelope estimation of how many Monte Carlo simulations are needed for reliable results. This number depends on the simulation objective, but a minimal requirement is that, for a fixed $\cal N$, the mean value obtained from the simulations (the sample mean $\left< \tilde{L} \right>$) must be consistent to within an error of $\epsilon$ of the mean obtained by the parametric modelling (the population mean). Statistics textbooks show that, independent of the shape of the distribution, the sampling mean is distributed according to a sampling distribution with variance equal to the variance of the population distribution variance divided by the sample size:
$$\sigma^2(\left< \tilde{L} \right>) = {\frac{\mu_{2;{\cal N}}(L)}{n_\mathrm{sam}}} = \frac{\cal N}{n_\mathrm{sam}} \, \mu_{2}(\ell).$$
Hence, expressed in relative terms, we can require that
$$\frac{\sigma(\left< \tilde{L} \right>)}{\mu_1'(L)} = \frac{1}{\sqrt{n_\mathrm{sam} {\cal N}}} \frac{\sqrt{\mu_{2}(\ell)}}{\mu_1'(\ell)} < \epsilon.$$
We can impose a similar requirement for the variance obtained from the distribution, which results in
$$\frac{\sigma(\left< \tilde{\sigma}^2(L) \right>)}{\mu_{2;{\cal N}}(L)} \approx \frac{1}{\sqrt{n_\mathrm{sam} {\cal N}}} \sqrt{\frac{1}{{\cal N}} \gamma_2 +2} < \epsilon.
\label{eq:samvarvar}$$
An interesting result is that, for relative dispersion, the relevant parameter is the total number of stars $N_\mathrm{tot}$ used in the overall simulation set. Hence, the number of simulations needed to sample the distribution of integrated luminosities decreases when $\cal N$ increases. However, for absolutes values, the ratio of ${\cal N}$ to $n_\mathrm{sam}$ is the relevant quantity.
What we can learn from Monte Carlo simulations?
------------------------------------------------
The principal advantage of Monte Carlo simulations is that they allow us to include constraints that are difficult to manage with a parametric description. Examples are constraints in the inputs (e.g. considering only simulations with a given number of stars in a given (observed) mass range, @Knoetal02) or constraints in the outputs (e.g. considering only simulations that verify certain observational constraints in luminosities, @Luretal03). The issue here is to compute Monte Carlo simulations and only consider those that are consistent with the desired constraints. Note that once a constraint is included, the process requires transformation of the associated probability distributions and hence a change in the parameters of the distribution, and some constraints cannot be expressed analytically as a function of input parameters. Another application that can easily be performed with Monte Carlo simulations is testing of hypotheses by comparing the distribution of the simulations and observations; examples have been described by @FdSK11 [@bin]. Hence, this approach is ideal when fine-tuned to observational (or theoretical) constraints.
However, at the beginning of this section I stated that one of the dangers of Monte Carlo simulations is that they are easy to do. They are so easy that we can include additional distributions, such as the distribution of numbers of stars in clusters, the total mass distribution, an age distributions, without a fine-tuned specific purpose. Here, the danger is that we must understand the questions that we are addressing and how such additional distributions affect the possible inferences. For instance, @Yoetal06 used Monte Carlo simulations with the typical number of stars in a globular cluster to explain bimodal distributions in globular clusters. They argued that bimodality is the result of a non-linear relation between metallicity and colour transformation. The transformation undoubtedly has an effect on the bimodality; however, the number of stars used (actually the typical number of stars that globular clusters have) is responsible for the bimodal distributions. Monte Carlo simulations using large $\cal N$ values must converge to a Gaussian distribution.
When Monte Carlo simulations are used to obtain parameters for stellar clusters, the usual approach is to apply large grids covering the parameter space for $\cal N$ and age. However, the situation is not as simple as expected. First, a typical situation is to use a distribution of total masses. Hence, $\cal N$ is described by an unknown distribution. Even worse, $\cal M$ fluctuates since the simulations must include the constraint that $\cal N$ is an integer. Therefore, the mean values of such simulations diverge from the mean values obtained in the parametric description noted before, since they include additional distributions. Second, the inferences depend on the input distributions. For instance, @PHE12 produced a grid assuming a flat distribution for total mass and a flat distribution in $\log t$. They compared observational colours with the Monte Carlo grid results and obtained a distribution of the parameters (age and mass) of the Monte Carlo simulations compatible with the observations. When expressed in probabilistic terms, the grid of Monte Carlo simulations represents the probability that a cluster has a given luminosity set for given age and mass, ${\cal P}(L | t, {\cal M})$, and comparison of observational data with such a set represents the probability that a cluster has a given age and mass for a given luminosity set ${\cal P}(t, {\cal M} | L)$. The method seems to be correct, but it is a typical fallacy of conditional probabilities: it would be correct only if the resulting distribution of ages were flat in $\log t$ and if the distribution of masses were flat. The set of simulations has a prior hypothesis about the distribution of masses and ages, and the results are valid only so far as the prior is realistic. In fact, @PHE12 obtained a distribution of ages and total masses that differs from the input distributions used in the Monte Carlo simulation set.
The situation was also illustrated by @FLCW12, who computed a Monte Carlo set with a flat distribution in both $\log t$ and $\log {\cal M}$ as priors. The authors were aware of the Bayes theorem, which connects conditional probabilities:
$${\cal P}(t, {\cal M} | L) = \frac{{\cal P}(L | t, {\cal M}) \, {\cal P}({\cal M})}{{\cal P}(L)}.
\label{eq:bayes}$$
They obtained age and mass distributions for observed clusters using a similar methodology to that of @PHE12. In fact, they found that the distribution of total masses follows a power law with an exponent $-2$; hence, the distribution used as a prior is false. However, they claimed that the real distribution of total masses is the one obtained, similar to @PHE12. Unfortunately, the authors were unaware that such a claim is valid only as long as a cross-validation is performed. This should involve repetition of the Monte Carlo simulations with a total mass distribution following an exponent of $-2$ and verification that the resulting distribution is compatible with such a prior [@tarantola].
Apart from problems in using the Bayes theorem to make false hypotheses, both studies can be considered as major milestones in the inference of stellar populations using Monte Carlo modelling. Their age and total mass inferences are more realistic since they consider intrinsic stochasticity in the modelling, which is undoubtedly better than not considering stochasticity at all. The studies lack only the final step of cross-validation to obtain robust results.
Monte Carlo simulations provide two important lessons in the modelling of stellar populations that also occur in the Gaussian regime and can apply to systems of any size. The first is the problem of priors and cross-validation (i.e. an iteration of the results). The second and more important lesson is that there are no unique solutions; the best solution is actually the distribution of possible solutions. In fact, we can take advantage of the distribution of possible solutions to obtain further results [@FLCW12].
Implications of probabilistic modelling {#sec:implications}
=======================================
We have seen the implications of probabilistic modelling for a low-$\cal N$ regime when Monte Carlo simulations (or covolution of the stellar luminosity function) are required. However, we have shown that some characteristics of stochasticity are present independently of $\cal N$ (as in SBFs and partial correlations). In addition, we have seen that we can combine different probability distributions to describe new situations, as for integrated luminosities at a given $\cal N$ or integrated luminosities of clusters that follow a $\cal N$ or age distribution. Let us explore the implications of such results.
Metrics of fitting {#sec:metrics}
------------------
The first implication of the modelling of stellar populations is that the redder the wavelength, the greater is the scatter, since fewer stars contribute to red wavelengths than to blue wavelengths in absolute and relative terms. In fact, each wavelength can fluctuate around the mean of the corresponding distribution of integrated luminosities in a different way, even though it is correlated with the other wavelengths. This naturally implies that for each age and metallicity, each model has its own fitting metrics.
We can take advantage of $N_\mathrm{eff}$ or SBF definitions to theoretically define the weight for each wavelength in a $\chi^2$ fit. In fact, a good $\chi^2$ fit cannot be better than the physical dispersion of the model, which is a physical limit. Exceedance of this physical limit (overfitting) leads to a more precise but erroneous result. Fig. \[fig:metrics\], taken from @CL09, illustrates the physical dispersion used to identify overfitting for SBFs.
An additional advantage of including the physical weight in the fit is that it breaks degeneracies that are present when observational data are fitted only to the mean value for parametric models. For instance, Fig. \[fig:agemet\], taken from @Buz05, illustrates breaking of age–metallicity degeneracy using $N_\mathrm{eff}$. Given that the allowed scatter depends on age and metallicity, the resulting $\chi^2$ defines the probability that a fit will produce different results.
Unfortunately, implementation of these ideas is not straightforward. Use of $N_\mathrm{eff}$ directly provides the theoretical weight for a wavelength in a $\chi^2$ fit, but it is dependent on $\cal N$. The use of SBF is independent of $\cal N$. However, it cannot be implemented directly, but requires an iterative process involving a standard fit, use of the SBF to identify overfitted results, and iteration of the process until convergence.
An unexplored area involves taking full advantage of $\rho(L_i,L_j)$, which can be obtained theoretically. In fact, the covariance coefficients and variance define a covariance matrix that can be directly implemented in a $\chi^2$ fit. However, as far as I know, computation of $\rho(L_i,L_j)$ has not been implemented and is not considered in any synthesis code [exceptions are @Cetal01; @CVGLMH02; @CVG03; @GLB04 but they use a Poisson approximation of the stellar luminosity function and the covariance coefficients obtained are not correct].
The population and the sample definition
----------------------------------------
The second implication is related to the definition of the population described by computed distributions. We have seen that we can define populations composed of ${\cal N} = 1$ stars, which are CMD diagrams, and that we can analyse such populations as long as we have enough events $n_\mathrm{sam}$ to describe the distribution.
We can also define populations composed of events with a similar number of stars, $\cal N$, taking advantage of additional information about the source. For instance, we can take the luminosity profile of a galaxy. We can assume that for a given radius, the number of stars is roughly the same (additional information in the form of the geometry of the galaxy is required). Hence, we can evaluate the scatter for the assumed profile, which, independent of observational errors, must be wavelength-dependent. At each galactocentric radius, we are sampling different $\varphi_{\cal N}(L_1\dots L_n)$ distributions with different $n_\mathrm{sam}$ elements. It is even possible that in the outer parts, where $\cal N$ is lower, we find pixels forming a binomial or extremely asymmetric distribution. However, the better our sampling of such distributions, the better will be our characterisation of the population parameters at this radius. Finally, our results for stellar populations must be independent of the radius range chosen once corrected for the radial profile [see @CLJ08 for details on such corrections]. Hence, we can perform a cross-validation of our results by repeating the analysis by integration over large radial ranges; this means that we reduce $n_\mathrm{sam}$ and increase $\cal N$. The only requirements are: (1) $\cal N$ must be kept constant in each of the elements of the sample ; and (2) sufficient $n_\mathrm{sam}$ elements are required for correct evaluation of the variance (c.f. Eq. (\[eq:samvarvar\])). The results obtained must also be consistent with the stellar population obtained using the integral light for the whole galaxy. Note that if we use SBF we do not need to know the value of $\cal N$, but just need to ensure that it is constant (but unknown) in the $n_\mathrm{sam}$ elements used.
A similar study can be performed using different ways to divide the image. For example, we can take slices of a spherical system and use each slice to compute the variance of the distribution [see @Buz05 for an example]. A similar technique can be applied to IFU observations. The problem is to obtain $n_\mathrm{sam}$ elements with a similar number of stars and stellar populations that allow us to estimate the scatter (SBF) for comparison with model results. In summary, we can include additional information about the system (geometry, light profile, etc.) in inferences about the stellar populations.
Finally, we can modify the $\varphi_{\cal N}(L_1\dots L_n)$ distributions to include other distributions representing different objects. For instance, the globular cluster distribution of a galaxy [assuming they have the same age and metallicity, in agreement with @Yoetal06] implicitly includes a distribution of possible $\cal N$ values. Since these globular clusters have intrinsically low $\cal N$ values, it is possible that some clusters will be in the biased regime described in Fig. \[fig:MonlocationC03\]. Since the few clusters dominated by PMS stars are luminous, they will be observed and will be extremely red in colour, even redder than the mean colour of parametric models (Fig. \[fig:Monlocation\]). In addition, there would be a blue tail corresponding to faint clusters with low $\cal N$ comprising mainly clusters with low-mass MS stars [see @CL04 for more details].
Some rules of thumb {#sec:hurisitcs}
-------------------
I finish this section with some rules of thumb that can be extracted from the modelling of stellar populations when applied to the inference of physical properties of stellar systems.
First, the relevant quantity in describing possible luminosities is not the total mass of the system, but the total mass (or number of stars) observed [*for your resolution elements*]{}, $\cal N$. The other relevant quantity is the number of resolution elements $n_\mathrm{sam}$ for a given $\cal N$. The lower the ratio of $\cal N$ to $n_\mathrm{sam}$, the better. In the limit, the optimal case is a CMD analysis.
Second, the scatter depends not only on $\cal N$ but also on the age and wavelength considered. Fig. \[fig:resumen\] shows $N_\mathrm{eff}$ values for SSP models with metallicity $Z=0.020$ for different ages and wavelengths. As I showed earlier, the lower $N_\mathrm{eff}$ is, the greater is the scatter. Fig. \[fig:resumen\] shows that blue optical wavelengths with $\lambda < 5000$ Å have intrinsically lower scatter than red wavelengths, independent of $\cal N$. The range 5000–8000 Å has intermediate scatter and scatter increases for wavelengths longer than $\sim 8000$ Å, depending on the age. In comparison to age determinations that do not consider intrinsic dispersion (i.e. all wavelengths have a similar weight), the safest age inferences correspond to ages between 8 and $\sim$ 200 Myr since variation of $N_\mathrm{eff}$ values with wavelength is lower in this mass range.
Fig. \[fig:resumen\] also shows the mean total mass for $\gamma_{1,{\cal N}} \le 0.1$ as a function of age and wavelength for $Z=0.020$ SSP models. This is a quantitative visualization of how the average total mass needed to reach a Gaussian-like regime varies with age and wavelength. It is evident that blue optical wavelengths reach a Gaussian regime at a lower average total mass when compared to other wavelengths. In contrast, red wavelengths not only have greater scatter, but this scatter is also associated with non-Gaussian distributions for a wider range of average total mass. Thus, we can strengthen the statement about age inferences: the safest age inferences correspond to ages between 8 and $\sim$ 200 Myr if the cluster mass is greater than $10^5 \mathrm{M}_\odot$.
Third, the input distributions ${\cal B}(m_0,t,Z)$ or IMF and SFH define the output distributions. It should be possible to obtain better fits (more precise, but not necessarily more accurate) by changing the input distributions. However, we must be aware that we have explored the possible output distributions before any such changes.
For instance, low-mass clusters have strong fluctuations around the IMF; each cluster (each IMF realization) could produce an excess or deficit of massive stars. Using the mean value obtained by parametric models, a top-heavy or bottom-heavy IMF would produce a better fit of models and data. Such IMF variations would undoubtedly be linked to variations in age and the total mass/number of stars for a system. However, Monte Carlo simulations can also produce a better fit without invoking any IMF variation. It is possible that Monte Carlo simulations using distributions of different IMFs (e.g. combining IMFs with a variable lower or upper mass limit with a distribution of possible lower or upper mass limits) would produce even better fits. I am sure that the approach using only the mean value is methodologically erroneous. However, it is not clear which of the two solutions obtained by Monte Carlo simulations is the best unless one of the hypotheses (fixed IMF or a distribution combining different IMFs) is incompatible with observational data.
As a practical rule, before exploring or claiming variations in the input parameters, a check is required to ensure that such input parameters are actually incompatible with observational data.
Another issue is how to evaluate scatter outside the SSP hypothesis. Formally, we may consider any SFH as a combination of SSPs. Hence, for any SFH scenario evaluated at time $t_\mathrm{mod}$, we can assume that the scatter can be evaluated using the most restrictive SSP situation in the time range from 0 to $t_\mathrm{mod}$. For instance, Fig. \[fig:resumen\] shows that the ionising flux ($\lambda \lesssim 912$ Å) requires an average total mass greater than $10^5 \mathrm{M}_\odot$ to reach a Gaussian-like regime. Hence, we must ensure that at least $10^5 \mathrm{M}_\odot$ has been formed in each SSP comprising the SFH. Assume that 1 Myr is the time interval used to define a star formation rate. This implies that there would be no Gaussian-like distributions for SFR less than $0.1 \mathrm{M}_\odot \mathrm{year}^{-1}$ (i.e. there would be a bias in the inferences obtained using the mean obtained by parametric models). However, a more quantitative study of this subject is required. @slug have suggested additional ideas on the evaluation of scatter including the SFH.
Fourth, regarding the output parameters and inferences for time and the total mass/number of stars, we can summarize the following rules:
1. [*Use all available information for the system, including previous inferences*]{}. However, wavelength ranges used for inferences in the literature must be considered. Additional criteria, such as those in the following points, can be used to evaluate roughly which inferences are more reliable.
Additional information on the system can be obtained from images and other data that, although not used directly in the inferences, constrain the possible range of solutions. Recovery of a complete picture of the system compatible with all the available information should be the aim, and not just a partial picture that can be drawn from particular data.
It is particularly useful to look for the ‘smoking guns’ for age inference: for example, emission lines in star-forming systems imply an age less than 10 Myr ; Wolf–Rayet stars imply an age less than 6 Myr (neglecting binary systems); supernova emission or supernova remnants (from optical, radio or X-ray observations) imply an age less than 50 Myr; high-mass X-ray binaries imply previous supernova events, and hence an age greater than 3 Myr. For instance, @FLCW12 showed that the use of broad-band photometry with narrow-band H$_\alpha$ photometry greatly improves the quality of inferences. However, note that the presence of a ‘smoking gun’ helps to define age ranges, but the absence of smoking guns does not provide information if the mass/number of stars in observations is not known.
2. [*Always obtain an estimate of the mass/number of stars in the resolution element*]{}. The confidence of an age inference cannot be evaluated unless an estimate of the mass/number of stars of such an age has been obtained [see @masscleannage for additional implications of this point].
3. [*Identify the integrated luminosity distribution regime for the system considered* ]{}. $\chi^2$ fitting including the physical variance and covariance coefficients is optimal for the Gaussian regime, but it fails for other distributions. Fig. \[fig:resumen\] can be used to identify Gaussianity. For large wavelength coverage, for which different regimes would be present, rejection of some parts of the spectra in the fit can be considered; it is better to obtain a less precise but more accurate result than a very precise but erroneous result caused by overfitting; in any case, such information can be used as a guide to obtain a complete picture of the system.
4. [*When using diagnostic diagrams (indices), compare the location of the parametric model, individual stars and observations.*]{} It is especially useful to identify the origin of outliers in diagnostic diagrams, and to evaluate the expected range of scatter in the model.
5. In general, [*CMDs analyses are more robust than analyses of integrated spectra*]{}. If such information is available, use it and do not try surpass CMDs, which is simply impossible. The sum of (unknown) elements in a sample cannot provide more information than knowledge for all the particular elements of the sample.
6. [*Blue wavelengths (3000–5000 Å) are robust*]{}. We have shown that blue wavelengths have intrinciscally lower physical scatter than any other wavelengths. Hence, when comparing different inferences in the literature, those based on just blue wavelengths are the most accurate. They may not be the most precise, but blue wavelengths always provide better fitting than any other wavelength. Hence, take the greatest possible advantage of blue wavelengths (e.g. when using normalized spectra to a given wavelength to obtain SFH, use blue wavelengths).
7. [*A good solution is the distribution of possible solutions*]{}. I discussed this in Section \[sec:metrics\], but I would like to emphasise the point. The best $\chi^2$ value would be a numerical artefact (e.g. local numerical fluctuations). For instance, it seems surprising that codes that infer the SFH using the mean value of parametric models do not usually quote any age–metallicity degeneracy, although it is present at a spectral level (c.f. Fig. \[fig:agemet\]). In fact, it is an artefact of using only the best $\chi^2$ fit when results are presented. Again, I refer to @FLCW12 as an example of how the use of the distribution of possible solutions improve inferences.
The (still) poorly explored arena {#sec:open}
=================================
In the preceding sections I raised several questions on the application of population synthesis models, ranging from their use in CMD diagrams for semi-resolved systems to fitting metrics and the still unexplored arena of covariance coefficients. In fact, most of the applications presented here have not been fully developed and have been presented in conference papers. In this respect I refer to work by @Buz05, which has been a source of inspiration to most of the applications described here and some additional applications not discussed. However, let me finish with two comments about additional unexplored areas in which stochasticity could play a role.
The first is chemical evolution. At the start of this paper I noted that any modelling that makes use of the stellar birth rate (IMF and SFH) is intrinsically probabilistic. Hence, the present discussion also applies to chemical evolution models. Here, there is no stellar luminosity function, but a stellar yield function, which is the amount of material that a star has ejected (instantaneously or accumulatively). Obviously, such a stellar yield function refers mainly to dead stars, stars with $\ell =0$ in the stellar luminosity function, and is hence correlated with live stars. The application of such ideas is suggestive, but not simple. Chemical evolution affects the metallicity and depends on the SFH; hence, metallicity and SFH cannot be separable functions in the stellar birth rate. Even worse, the actual metallicity of a system depends on its previous metallicity evolution, which is itself described as a probability distribution. Determination of the evolution of a system would require a change in the ordinary differential equations defining chemical evolution to stochastic differential equations. Interested readers can find very preliminary approaches to the subject in work by @Ceretal00 [@Cetal01; @CM02], and more detailed studies by @SF95 [@CH08].
The second question is the inclusion of highly variable phases such as thermal pulses in synthesis models. The present idea is actually a re-elaboration of the so-called fuel consumption theorem proposed by @RB83 and @Buzz89 applied over isochrone synthesis. A requirement of isochrone synthesis is that evolutionary tracks must be smooth enough to allow interpolations and compute isochrones. In particular, variation is not allowed for a star with a given mass and a given age. However, we can include variability as far as we can model it by a probability distribution function. In fact, we can define such a probability as proportional to the period of variation and include it directly in the stellar luminosity function. I refer readers interested in this subject to the paper by @GGS04, whose ideas provide a formal development of the probabilistic modelling of stellar populations.
Conclusions {#sec:conclu}
===========
We studied the modelling of stochasticity for stellar populations. Such stochasticity is intrinsic in the modelling in that we do not directly know the individual stars in a [*generic*]{} stellar system. We must use probability distributions and assign probabilities for the presence of each individual star. The only possible result of the modelling is a probabilistic description including [*all*]{} possible situations.
The input probability distribution is the stellar birth rate ${\cal B}(m_0,t,Z)$, which represents the probability that an individual star was born with a given initial mass $m_0$ at a given time $t$ with a given metallicity $Z$. This probability distribution is modified by the transformations provided by stellar evolution theory and evaluated at different times $t_\mathrm{mod}$. The resulting distribution, the stellar luminosity function, represents the probability that a star had a given luminosity set (e.g. a set of colours or wavelengths), $\ell_1 \dots \ell_n$, at a given time.
The probability distribution of the integrated luminosity of a system with $\cal N$ stars can be obtained directly from the stellar luminosity function in an exact way via a self-convolution process. It can also be obtained from a set of $n_\mathrm{sim}$ Monte Carlo simulations. Finally, it can be described as a function of parameters of the distribution (mean, variance, skewness, and kurtosis) that is related to the stellar luminosity function by simple scale relations. Standard synthesis models use a parametric description, although, with some exceptions, they only compute the mean value of the distribution.
Hence, the results of standard (parametric) modelling can be used in any situation, including ${\cal N} =1$ (i.e. CMDs), as long as we understand that it is only a mean value of possible luminosities. However, mean values are not useful unless we know the shape of the distribution. In the most optimistic case, for large $\cal N$ values, the distribution is Gaussian and is hence defined by the mean and variance of each luminosity, and the covariance between the different luminosities. The value of $\cal N$ when this Gaussian regime is reached can be obtained by analysis of the skewness and kurtosis of the stellar luminosity function.
Monte Carlo modelling (or the self-convolution process) is a useful approach outside the Gaussian regime. However, Monte Carlo simulations can only be fully understood if they are analysed with a parametric description of the possible probability distributions. In addition, Monte Carlo simulations are an ideal tool for including constraints in fine-tuned modelling of particular objects or situations. A drawback is that large sets of Monte Carlo simulations implicitly include additional priors, such as the distribution of ages or $\cal N$ values. Such priors must be chosen carefully in any application inferring physical parameters from observations since possible inferences depend on them according to the Bayes theorem. Finally, it is not possible to evaluate the reliability of Monte Carlo simulations without explicit knowledge of the number of simulations $n_\mathrm{sim}$, $\cal N$, and all the assumed priors.
The fact that even in the most optimistic case of ${\cal N} = \infty$, the distribution of integrated luminosities is Gaussian (where the parameters of the Gaussian distribution depend on size, age, and metallicity) has major implications for the use of synthesis models to obtain inferences. The principal one is that each model and wavelength has its own physical scatter. This defines a metric of fitting for each model (i.e. not all wavelengths are equivalent in the fit and they must be weighted according their own physical scatter). An advantage of the use of such fitting metrics is that it breaks degeneracies in physical properties, such as age–metallicity degeneracy. A second implication is that we can obtain physical information from observed scatter.
Finally, throughout the paper we explored additional implications and applications of the probabilistic modelling of stellar populations. Most of them are only tentative ideas, but they increase the predictive power of synthesis models and provide more accurate inferences of physical parameters from observational data.
[*Addendum after publication (only in astro-ph version):*]{} After the publication of this review, appear in the literature an additional paper dealing with modeling stellar populations by means of Monte Carlo simulations by [@Andersetal13]. Models results and programs related with that work will available at: [http://data.galev.org/models/anders13]{}.
Acknowledgements
================
My work is supported by MICINN (Spain) through AYA2010-15081 and AYA2010-15196 programmes. I acknowledge the development of TopCat software [@TopCat] since it helped me greatly in obtaining Figs. \[fig:mon1\] and \[fig:resumen\]. This review compiles the efforts of many years of thinking and working with stellar population models and stochasticity, and it would not have been possible without the collaboration of Valentina Luridiana. I also acknowledge Alberto Buzzoni and Steve Shore for suggestions, comments and discussions on the modelling and applications of these ideas, most of then outlined here. I acknowledge Roberto and Elena Terlevich, David Valls Gabaud, Sandro Bressan, Mercedes Molla, Jesús Ma[í]{}z Apellaniz, and Angel Bongiovani for useful discussion. Also, I acknowledge Carma Gallart and J. Ma[í]{}z Appelaniz for providing me some figures for this review, and Sally Oey for her feedback and to give me the opportunity of write this review. Finally, I acknowledge Nuno and Carlos (twins), Dario and Eva who show experimentally that Nature fluctuates around theoretical expectations (no-twins), and that such fluctuations make life (and science) more interesting.
[00]{}
Anders, P., Kotulla, R., de Grijs, R., & Wicker, J. 2013, , 778, 138
Baade, W., 1944. 100, 137.
Barbaro, C., Bertelli, C., 1977. 54, 243.
Barker, S., de Grijs, R., Cervi[ñ]{}o, M., 2008. 484, 711.
Becker, S.A., Mathews, G. J., 1983. 270, 155.
Brocato, E., Capaccioli, M., Condelli, M., 1998. Mem. Soc. Astron. Ital. 69, 155.
Brocato, E., Castellani, V., Raimondo, G., Romaniello, M., 1999. 136, 65.
Buzzoni, A., 1989. 71, 817.
Buzzoni, A., 1993. 275, 433.
Buzzoni, A., 2005. arXiv:astro-ph/0509602.
Bruzual, G., Charlot, S., 1993. 405, 538.
Bruzual, G., 2002. arXiv: astro-ph/ 0110245.
Bruzual, G., Charlot, S., 2003. 344, 1000.
Cantiello, M., Blakeslee, J. P., 2007. 669, 982.
Carigi, L., Hernandez, X., 2008. 390, 582.
Cervi[ñ]{}o, M., Mas-Hesse, J. M., 1994. 284, 749.
Cervi[ñ]{}o, M., Kn[ö]{}dlseder, J., Schaerer, D., von Ballmoos, P., Meynet, G., 2000a. 363, 970.
Cervi[ñ]{}o, M., Luridiana, V., Castander, F. J., 2000b. 360, L5.
Cervi[ñ]{}o, M., G[ó]{}mez-Flechoso, M.A., Castander, F.J., Schaerer, D., Moll[á]{}, M., Kn[ö]{}dlseder, J., Luridiana, V., 2001. 376, 422.
Cerviño, M., Valls-Gabaud, D., Luridiana, V., Mas-Hesse, J.M., 2002. 381, 51.
Cervi[ñ]{}o, M., Moll[á]{}, M., 2002. 394, 525.
Cerviño, M., 2003. ASP 297, 43.
Cervi[ñ]{}o, M., Valls-Gabaud, D., 2003. 338, 481.
Cervi[ñ]{}o, M., Luridiana, V.. 2004. 3, 145.
Cervi[ñ]{}o, M., Luridiana, V., 2006. 451, 475.
Cervi[ñ]{}o, M., Luridiana, V., Jamet, L., 2008. 491, 693.
Cervi[ñ]{}o, M., Luridiana, V., 2009. arXiv:0711.1355.
Cervi[ñ]{}o, M., Rom[á]{}n-Z[ú]{}[ñ]{}iga, C., Luridiana, V., et al., 2013. 553, A31.
Charlot, S., Bruzual, A. G., 1991. 367, 126.
Chiosi, C., Bertelli, G., Bressan, A., 1988. 196, 84.
D’Agostino, R.B., Stephens, M.A., 1986. Goodness-of-Fit Techniques (New York: Marcel Dekker).
Eldridge, J. J., 2012. 422, 794.
Fouesneau, M., Lan[ç]{}on, A., 2010. 521, 22.
Fouesneau, M., Lan[ç]{}on, A., Chandar, R., Whitmore, B. C., 2012. 750, 60.
Fumagalli, M., da Silva, R.L., Krumholz, M. R., 2011. 741, L26.
Girardi, L., Chiosi, C., Bertelli, G., Bressan, A., 1995. 298, 87.
Girardi, L., Bica, E., 1993. 274, 279.
Gilfanov, M., Grimm, H. J., Sunyaev, R., 2004. 351, 1365.
Gonz[' a]{}lez, R.A., Liu, M.C., Bruzual, G., 2004. 611, 270.
Gonz[á]{}lez Delgado, R.M., Cervi[ñ]{}o, M., Martins, L.P., Leitherer, C., Hauschildt, P. H., 2005. 357, 945.
Hess, R., 1924. Probleme der Astronomie. Festschrift fur Hugo v. Seeliger. Springer, Berlin. p. 265
Hodge, P., 1989. 27, 139.
Kn[ö]{}dlseder, J., Cervi[ñ]{}o, M., Le Duigou, J. M., et al., 2002. 390, 945.
Lançon, A., Mouhcine, M., 2000. ASP Conf. Ser. 211, 34.
Luridiana, V., Peimbert, A., Peimbert, M., Cervi[ñ]{}o, M., 2003. 592, 846.
Mas-Hesse, J.M., Kunth, D., 1991. 88, 399.
Ma[í]{}z Apell[á]{}niz, J., [Ú]{}beda, L., 2005. 629, 873.
Ma[í]{}z Apell[á]{}niz, J., 2008. 677, 1278.
Ma[í]{}z Apell[á]{}niz, J., 2009. 699, 1938.
Marigo, P., Girardi, L., 2001. 377, 132.
Mathis, J. S., 1959. 129, 259.
Meschin et al., 2013. (submitted).
Piskunov, A.E., Kharchenko, N.V., Schilbach, E.,Röser, S., Scholz, R.D., Zinnecker, H., 2011. 525, 122.
Popescu, B., Hanson, M. M., 2009. 138, 1724.
Popescu, B., Hanson, M. M., 2010a. 713, L21.
Popescu, B., Hanson, M. M., 2010b. 724, 296.
Popescu, B., Hanson, M.M., Elmegreen, B. G., 2012. 751, 122.
Salpeter, E. E., 1955. 121, 161.
Sandage, A.R., Schwarzschild, M., 1952. 116, 463.
Scalo, J.M., 1986. Fund. Cosm. Phys. 11, 1.
Santos, J. F. C., Jr., Frogel, J. A., 1997. 479, 764.
Raimondo, G., Brocato, E., Cantiello, M., Capaccioli, M., 2005. 130, 2625.
Renzini, A., 1981. Ann. Phys. 6, 87.
Renzini, A., Buzzoni, A., 1983. 54, 739.
da Silva, R.L., Fumagalli, M., Krumholz, M., 2012. 745, 145.
Silva-Villa, E., Larsen, S. S., 2011. 529, A25.
Shore, S.N., Ferrini, F., 1995. 16, 1.
Tarantola, A., 2006. Nat. Phys. 2, 492.
Taylor, M. B., 2005. Astronomical Data Analysis Software and Systems XIV, 347, 29.
Tinsley, B. M., 1968. 151, 547.
Tinsley, B. M., 1972. 20, 383.
Tinsley, B. M., 1980. 5, 287.
Tinsley, B.M., Gunn, J. E., 1976. 203, 52.
Tonry, J., Schneider, D. P., 1988. 96, 807.
Voss, R., Diehl, R., Hartmann, D.H., et al., 2009. 504, 531.
Whipple, F. L., 1935. Harvard College Observatory Circular 4, 1.
Worthey, G., 1994. 95, 107.
Yoon, S.-J., Yi, S.K., Lee, Y.-W., 2006. Science 311, 1129.
[^1]: Monochromatic $N_\mathrm{eff}$ values for old stellar populations can be found at [http://www.bo.astro.it/$\sim$eps/home.html]{} and [http://www.iaa.es/$\sim$rosa/research/synthesis/HRES/ESPS-HRES.html]{} [@GDetal05] for young and intermediate SSPs.
[^2]: For reference, a Salpeter IMF in the mass range 0.08–120 M$_\odot$ has $\mu'_1(m_0) = 0.28$ M$_\odot$, $\mu_2(m_0) = 1.44$, $\gamma_1(m_0) = 1691.47$, and $\gamma_2(m_0) = 2556.66$. This implies that the distribution of the total mass becomes Gaussian-like for ${\cal N} \sim 3\times 10^7$ stars when the average total mass is approximately $9\times 10^6$ M$_\odot$.
|
---
abstract: 'We report the most precise measurements to date of the strong-phase parameters between $D^0$ and $\bar{D}^0$ decays to $K^0_{S,L}\pi^+\pi^-$ using a sample of 2.93 fb$^{-1}$ of $e^+e^-$ annihilation data collected at a center-of-mass energy of 3.773 GeV with the BESIII detector at the BEPCII collider. Our results provide the key inputs for a binned model-independent determination of the Cabibbo-Kobayashi-Maskawa angle $\gamma/\phi_3$ with $B$ decays. Using our results, the decay model sensitivity to the $\gamma/\phi_3$ measurement is expected to be between 0.7$^{\circ}$ and 1.2$^{\circ}$, approximately a factor of three smaller than that achievable with previous measurements. The improved precision of this work ensures that measurements of $\gamma/\phi_3$ will not be limited by knowledge of strong phases for the next decade. Furthermore, our results provide critical input for other flavor-physics investigations, including charm mixing, other measurements of $CP$ violation, and the measurement of strong-phase parameters for other $D$-decay modes.'
author:
- |
M. Ablikim$^{1}$, M. N. Achasov$^{10,d}$, P. Adlarson$^{59}$, S. Ahmed$^{15}$, M. Albrecht$^{4}$, M. Alekseev$^{58A,58C}$, D. Ambrose$^{51}$, A. Amoroso$^{58A,58C}$, F. F. An$^{1}$, Q. An$^{55,43}$, Anita$^{21}$, Y. Bai$^{42}$, O. Bakina$^{27}$, R. Baldini Ferroli$^{23A}$, I. Balossino$^{24A}$, Y. Ban$^{35,l}$, K. Begzsuren$^{25}$, J. V. Bennett$^{5}$, N. Berger$^{26}$, M. Bertani$^{23A}$, D. Bettoni$^{24A}$, F. Bianchi$^{58A,58C}$, J Biernat$^{59}$, J. Bloms$^{52}$, I. Boyko$^{27}$, R. A. Briere$^{5}$, H. Cai$^{60}$, X. Cai$^{1,43}$, A. Calcaterra$^{23A}$, G. F. Cao$^{1,47}$, N. Cao$^{1,47}$, S. A. Cetin$^{46B}$, J. Chai$^{58C}$, J. F. Chang$^{1,43}$, W. L. Chang$^{1,47}$, G. Chelkov$^{27,b,c}$, D. Y. Chen$^{6}$, G. Chen$^{1}$, H. S. Chen$^{1,47}$, J. Chen$^{16}$, J. C. Chen$^{1}$, M. L. Chen$^{1,43}$, S. J. Chen$^{33}$, Y. B. Chen$^{1,43}$, W. Cheng$^{58C}$, G. Cibinetto$^{24A}$, F. Cossio$^{58C}$, X. F. Cui$^{34}$, H. L. Dai$^{1,43}$, J. P. Dai$^{38,h}$, X. C. Dai$^{1,47}$, A. Dbeyssi$^{15}$, D. Dedovich$^{27}$, Z. Y. Deng$^{1}$, A. Denig$^{26}$, I. Denysenko$^{27}$, M. Destefanis$^{58A,58C}$, F. De Mori$^{58A,58C}$, Y. Ding$^{31}$, C. Dong$^{34}$, J. Dong$^{1,43}$, L. Y. Dong$^{1,47}$, M. Y. Dong$^{1,43,47}$, Z. L. Dou$^{33}$, S. X. Du$^{63}$, J. Z. Fan$^{45}$, J. Fang$^{1,43}$, S. S. Fang$^{1,47}$, Y. Fang$^{1}$, R. Farinelli$^{24A,24B}$, L. Fava$^{58B,58C}$, F. Feldbauer$^{4}$, G. Felici$^{23A}$, C. Q. Feng$^{55,43}$, M. Fritsch$^{4}$, C. D. Fu$^{1}$, Y. Fu$^{1}$, Q. Gao$^{1}$, X. L. Gao$^{55,43}$, Y. Gao$^{56}$, Y. Gao$^{45}$, Y. G. Gao$^{6}$, Z. Gao$^{55,43}$, B. Garillon$^{26}$, I. Garzia$^{24A}$, E. M. Gersabeck$^{50}$, A. Gilman$^{51}$, K. Goetzen$^{11}$, L. Gong$^{34}$, W. X. Gong$^{1,43}$, W. Gradl$^{26}$, M. Greco$^{58A,58C}$, L. M. Gu$^{33}$, M. H. Gu$^{1,43}$, S. Gu$^{2}$, Y. T. Gu$^{13}$, A. Q. Guo$^{22}$, L. B. Guo$^{32}$, R. P. Guo$^{36}$, Y. P. Guo$^{26}$, A. Guskov$^{27}$, S. Han$^{60}$, X. Q. Hao$^{16}$, F. A. Harris$^{48}$, K. L. He$^{1,47}$, F. H. Heinsius$^{4}$, T. Held$^{4}$, Y. K. Heng$^{1,43,47}$, M. Himmelreich$^{11,g}$, Y. R. Hou$^{47}$, Z. L. Hou$^{1}$, H. M. Hu$^{1,47}$, J. F. Hu$^{38,h}$, T. Hu$^{1,43,47}$, Y. Hu$^{1}$, G. S. Huang$^{55,43}$, J. S. Huang$^{16}$, X. T. Huang$^{37}$, X. Z. Huang$^{33}$, N. Huesken$^{52}$, T. Hussain$^{57}$, W. Ikegami Andersson$^{59}$, W. Imoehl$^{22}$, M. Irshad$^{55,43}$, Q. Ji$^{1}$, Q. P. Ji$^{16}$, X. B. Ji$^{1,47}$, X. L. Ji$^{1,43}$, H. L. Jiang$^{37}$, X. S. Jiang$^{1,43,47}$, X. Y. Jiang$^{34}$, J. B. Jiao$^{37}$, Z. Jiao$^{18}$, D. P. Jin$^{1,43,47}$, S. Jin$^{33}$, Y. Jin$^{49}$, T. Johansson$^{59}$, N. Kalantar-Nayestanaki$^{29}$, X. S. Kang$^{31}$, R. Kappert$^{29}$, M. Kavatsyuk$^{29}$, B. C. Ke$^{1}$, I. K. Keshk$^{4}$, A. Khoukaz$^{52}$, P. Kiese$^{26}$, R. Kiuchi$^{1}$, R. Kliemt$^{11}$, L. Koch$^{28}$, O. B. Kolcu$^{46B,f}$, B. Kopf$^{4}$, M. Kuemmel$^{4}$, M. Kuessner$^{4}$, A. Kupsc$^{59}$, M. Kurth$^{1}$, M. G. Kurth$^{1,47}$, W. Kühn$^{28}$, J. S. Lange$^{28}$, P. Larin$^{15}$, L. Lavezzi$^{58C}$, H. Leithoff$^{26}$, T. Lenz$^{26}$, C. Li$^{59}$, Cheng Li$^{55,43}$, D. M. Li$^{63}$, F. Li$^{1,43}$, F. Y. Li$^{35,l}$, G. Li$^{1}$, H. B. Li$^{1,47}$, H. J. Li$^{9,j}$, J. C. Li$^{1}$, J. W. Li$^{41}$, Ke Li$^{1}$, L. K. Li$^{1}$, Lei Li$^{3,53}$, P. L. Li$^{55,43}$, P. R. Li$^{30}$, Q. Y. Li$^{37}$, W. D. Li$^{1,47}$, W. G. Li$^{1}$, X. H. Li$^{55,43}$, X. L. Li$^{37}$, X. N. Li$^{1,43}$, Z. B. Li$^{44}$, Z. Y. Li$^{44}$, H. Liang$^{55,43}$, H. Liang$^{1,47}$, Y. F. Liang$^{40}$, Y. T. Liang$^{28}$, G. R. Liao$^{12}$, L. Z. Liao$^{1,47}$, J. Libby$^{21}$, C. X. Lin$^{44}$, D. X. Lin$^{15}$, Y. J. Lin$^{13}$, B. Liu$^{38,h}$, B. J. Liu$^{1}$, C. X. Liu$^{1}$, D. Liu$^{55,43}$, D. Y. Liu$^{38,h}$, F. H. Liu$^{39}$, Fang Liu$^{1}$, Feng Liu$^{6}$, H. B. Liu$^{13}$, H. M. Liu$^{1,47}$, Huanhuan Liu$^{1}$, Huihui Liu$^{17}$, J. B. Liu$^{55,43}$, J. Y. Liu$^{1,47}$, K. Liu$^{1}$, K. Y. Liu$^{31}$, Ke Liu$^{6}$, L. Y. Liu$^{13}$, Q. Liu$^{47}$, S. B. Liu$^{55,43}$, T. Liu$^{1,47}$, X. Liu$^{30}$, X. Y. Liu$^{1,47}$, Y. B. Liu$^{34}$, Z. A. Liu$^{1,43,47}$, Zhiqing Liu$^{37}$, Y. F. Long$^{35,l}$, X. C. Lou$^{1,43,47}$, H. J. Lu$^{18}$, J. D. Lu$^{1,47}$, J. G. Lu$^{1,43}$, Y. Lu$^{1}$, Y. P. Lu$^{1,43}$, C. L. Luo$^{32}$, M. X. Luo$^{62}$, P. W. Luo$^{44}$, T. Luo$^{9,j}$, X. L. Luo$^{1,43}$, S. Lusso$^{58C}$, X. R. Lyu$^{47}$, F. C. Ma$^{31}$, H. L. Ma$^{1}$, L. L. Ma$^{37}$, M. M. Ma$^{1,47}$, Q. M. Ma$^{1}$, X. N. Ma$^{34}$, X. X. Ma$^{1,47}$, X. Y. Ma$^{1,43}$, Y. M. Ma$^{37}$, F. E. Maas$^{15}$, M. Maggiora$^{58A,58C}$, S. Maldaner$^{26}$, S. Malde$^{53}$, Q. A. Malik$^{57}$, A. Mangoni$^{23B}$, Y. J. Mao$^{35,l}$, Z. P. Mao$^{1}$, S. Marcello$^{58A,58C}$, Z. X. Meng$^{49}$, J. G. Messchendorp$^{29}$, G. Mezzadri$^{24A}$, J. Min$^{1,43}$, T. J. Min$^{33}$, R. E. Mitchell$^{22}$, X. H. Mo$^{1,43,47}$, Y. J. Mo$^{6}$, C. Morales Morales$^{15}$, N. Yu. Muchnoi$^{10,d}$, H. Muramatsu$^{51}$, A. Mustafa$^{4}$, S. Nakhoul$^{11,g}$, Y. Nefedov$^{27}$, F. Nerling$^{11,g}$, I. B. Nikolaev$^{10,d}$, Z. Ning$^{1,43}$, S. Nisar$^{8,k}$, S. L. Niu$^{1,43}$, S. L. Olsen$^{47}$, Q. Ouyang$^{1,43,47}$, S. Pacetti$^{23B}$, Y. Pan$^{55,43}$, M. Papenbrock$^{59}$, P. Patteri$^{23A}$, M. Pelizaeus$^{4}$, H. P. Peng$^{55,43}$, K. Peters$^{11,g}$, J. Pettersson$^{59}$, J. L. Ping$^{32}$, R. G. Ping$^{1,47}$, A. Pitka$^{4}$, R. Poling$^{51}$, V. Prasad$^{55,43}$, H. R. Qi$^{2}$, M. Qi$^{33}$, T. Y. Qi$^{2}$, S. Qian$^{1,43}$, C. F. Qiao$^{47}$, N. Qin$^{60}$, X. P. Qin$^{13}$, X. S. Qin$^{4}$, Z. H. Qin$^{1,43}$, J. F. Qiu$^{1}$, S. Q. Qu$^{34}$, K. H. Rashid$^{57,i}$, K. Ravindran$^{21}$, C. F. Redmer$^{26}$, M. Richter$^{4}$, A. Rivetti$^{58C}$, V. Rodin$^{29}$, M. Rolo$^{58C}$, G. Rong$^{1,47}$, Ch. Rosner$^{15}$, M. Rump$^{52}$, A. Sarantsev$^{27,e}$, M. Savrié$^{24B}$, Y. Schelhaas$^{26}$, K. Schoenning$^{59}$, W. Shan$^{19}$, X. Y. Shan$^{55,43}$, M. Shao$^{55,43}$, C. P. Shen$^{2}$, P. X. Shen$^{34}$, X. Y. Shen$^{1,47}$, H. Y. Sheng$^{1}$, X. Shi$^{1,43}$, X. D Shi$^{55,43}$, J. J. Song$^{37}$, Q. Q. Song$^{55,43}$, X. Y. Song$^{1}$, S. Sosio$^{58A,58C}$, C. Sowa$^{4}$, S. Spataro$^{58A,58C}$, F. F. Sui$^{37}$, G. X. Sun$^{1}$, J. F. Sun$^{16}$, L. Sun$^{60}$, S. S. Sun$^{1,47}$, X. H. Sun$^{1}$, Y. J. Sun$^{55,43}$, Y. K Sun$^{55,43}$, Y. Z. Sun$^{1}$, Z. J. Sun$^{1,43}$, Z. T. Sun$^{1}$, Y. T Tan$^{55,43}$, C. J. Tang$^{40}$, G. Y. Tang$^{1}$, X. Tang$^{1}$, V. Thoren$^{59}$, B. Tsednee$^{25}$, I. Uman$^{46D}$, B. Wang$^{1}$, B. L. Wang$^{47}$, C. W. Wang$^{33}$, D. Y. Wang$^{35,l}$, K. Wang$^{1,43}$, L. L. Wang$^{1}$, L. S. Wang$^{1}$, M. Wang$^{37}$, M. Z. Wang$^{35,l}$, Meng Wang$^{1,47}$, P. L. Wang$^{1}$, R. M. Wang$^{61}$, W. P. Wang$^{55,43}$, X. Wang$^{35,l}$, X. F. Wang$^{1}$, X. L. Wang$^{9,j}$, Y. Wang$^{55,43}$, Y. Wang$^{44}$, Y. F. Wang$^{1,43,47}$, Y. Q. Wang$^{1}$, Z. Wang$^{1,43}$, Z. G. Wang$^{1,43}$, Z. Y. Wang$^{1}$, Zongyuan Wang$^{1,47}$, T. Weber$^{4}$, D. H. Wei$^{12}$, P. Weidenkaff$^{26}$, H. W. Wen$^{32}$, S. P. Wen$^{1}$, U. Wiedner$^{4}$, G. Wilkinson$^{53}$, M. Wolke$^{59}$, L. H. Wu$^{1}$, L. J. Wu$^{1,47}$, Z. Wu$^{1,43}$, L. Xia$^{55,43}$, Y. Xia$^{20}$, S. Y. Xiao$^{1}$, Y. J. Xiao$^{1,47}$, Z. J. Xiao$^{32}$, Y. G. Xie$^{1,43}$, Y. H. Xie$^{6}$, T. Y. Xing$^{1,47}$, X. A. Xiong$^{1,47}$, Q. L. Xiu$^{1,43}$, G. F. Xu$^{1}$, J. J. Xu$^{33}$, L. Xu$^{1}$, Q. J. Xu$^{14}$, W. Xu$^{1,47}$, X. P. Xu$^{41}$, F. Yan$^{56}$, L. Yan$^{58A,58C}$, W. B. Yan$^{55,43}$, W. C. Yan$^{2}$, Y. H. Yan$^{20}$, H. J. Yang$^{38,h}$, H. X. Yang$^{1}$, L. Yang$^{60}$, R. X. Yang$^{55,43}$, S. L. Yang$^{1,47}$, Y. H. Yang$^{33}$, Y. X. Yang$^{12}$, Yifan Yang$^{1,47}$, Z. Q. Yang$^{20}$, M. Ye$^{1,43}$, M. H. Ye$^{7}$, J. H. Yin$^{1}$, Z. Y. You$^{44}$, B. X. Yu$^{1,43,47}$, C. X. Yu$^{34}$, J. S. Yu$^{20}$, T. Yu$^{56}$, C. Z. Yuan$^{1,47}$, X. Q. Yuan$^{35,l}$, Y. Yuan$^{1}$, A. Yuncu$^{46B,a}$, A. A. Zafar$^{57}$, Y. Zeng$^{20}$, B. X. Zhang$^{1}$, B. Y. Zhang$^{1,43}$, C. C. Zhang$^{1}$, D. H. Zhang$^{1}$, H. H. Zhang$^{44}$, H. Y. Zhang$^{1,43}$, J. Zhang$^{1,47}$, J. L. Zhang$^{61}$, J. Q. Zhang$^{4}$, J. W. Zhang$^{1,43,47}$, J. Y. Zhang$^{1}$, J. Z. Zhang$^{1,47}$, K. Zhang$^{1,47}$, L. Zhang$^{45}$, L. Zhang$^{33}$, S. F. Zhang$^{33}$, T. J. Zhang$^{38,h}$, X. Y. Zhang$^{37}$, Y. Zhang$^{55,43}$, Y. H. Zhang$^{1,43}$, Y. T. Zhang$^{55,43}$, Yang Zhang$^{1}$, Yao Zhang$^{1}$, Yi Zhang$^{9,j}$, Yu Zhang$^{47}$, Z. H. Zhang$^{6}$, Z. P. Zhang$^{55}$, Z. Y. Zhang$^{60}$, G. Zhao$^{1}$, J. W. Zhao$^{1,43}$, J. Y. Zhao$^{1,47}$, J. Z. Zhao$^{1,43}$, Lei Zhao$^{55,43}$, Ling Zhao$^{1}$, M. G. Zhao$^{34}$, Q. Zhao$^{1}$, S. J. Zhao$^{63}$, T. C. Zhao$^{1}$, Y. B. Zhao$^{1,43}$, Z. G. Zhao$^{55,43}$, A. Zhemchugov$^{27,b}$, B. Zheng$^{56}$, J. P. Zheng$^{1,43}$, Y. Zheng$^{35,l}$, Y. H. Zheng$^{47}$, B. Zhong$^{32}$, L. Zhou$^{1,43}$, L. P. Zhou$^{1,47}$, Q. Zhou$^{1,47}$, X. Zhou$^{60}$, X. K. Zhou$^{47}$, X. R. Zhou$^{55,43}$, Xiaoyu Zhou$^{20}$, Xu Zhou$^{20}$, A. N. Zhu$^{1,47}$, J. Zhu$^{34}$, J. Zhu$^{44}$, K. Zhu$^{1}$, K. J. Zhu$^{1,43,47}$, S. H. Zhu$^{54}$, W. J. Zhu$^{34}$, X. L. Zhu$^{45}$, Y. C. Zhu$^{55,43}$, Y. S. Zhu$^{1,47}$, Z. A. Zhu$^{1,47}$, J. Zhuang$^{1,43}$, B. S. Zou$^{1}$, J. H. Zou$^{1}$\
(BESIII Collaboration)\
title: 'Determination of strong-phase parameters in $D\rightarrow K^0_{S,L}\pi^+\pi^-$ '
---
The mechanism of $CP$ violation in particle physics is of primary importance because of its impact on cosmological baryogenesis and matter-antimatter asymmetry in the universe. In the standard model (SM), $CP$ violation is studied by measuring the elements of the Cabibbo-Kobayashi-Maskawa (CKM) matrix [@CKM], using the convenient representation given by the unitarity triangle (UT) formed in the complex plane. The angle $\gamma$ (also denoted $\phi_3$) of the UT is of particular interest since it is the only one that can be extracted from tree-level processes, for which the contribution of non-SM effects is expected to be very small. Therefore, measurement of $\gamma$ provides a benchmark for the SM with minimal theoretical uncertainty [@jhep1401_051; @epjc79_159]. A precision measurement of $\gamma$ is an essential ingredient in comprehensive testing of the SM description of $CP$ violation and probing for evidence of new physics. Direct measurements of $\gamma$ have not yet achieved the required precision, with a world-average value of $\gamma=(73.5^{+4.2}_{-5.1})^{\circ}$ [@pdg18], to be compared to the indirect determination of $\gamma=(65.8^{+1.0}_{-1.7})^{\circ}$ [@CKMfitter]. These different determinations deviate by 1.5$\sigma$. It has been predicted that new physics at the tree level could introduce a deviation in $\gamma$ up to $4^{\circ}$ [@prd92_033002], which is close to the current experimental precision. Achieving sub-degree precision in the determination of $\gamma$ is clearly a top priority for current and future flavor-physics experiments.
One of the most sensitive decay channels for measuring $\gamma$ is $B^-\rightarrow DK^-$ with $D\rightarrow
K^0_S\pi^+\pi^-$ [@prd68_054018], where $D$ represents a superposition of $D^0$ and $\bar{D}^0$ mesons. (Throughout this paper, charge conjugation is assumed unless otherwise explicitly noted.) The model-independent approach [@epjc47_347] requires a binned Dalitz plot analysis of the amplitude-weighted average cosine and sine of the relative strong-phase ($\Delta \delta_D$) between $D^0$ and $\bar{D}^0\rightarrow K^0_S\pi^+\pi^-$ to determine $\gamma$. These strong-phase parameters were first studied by the CLEO collaboration using 0.82 fb$^{-1}$ of data [@prd80_032002; @prd82_112006]. The limited precision of CLEO’s results contributes a systematic uncertainty of approximately 4$^{\circ}$ to the $\gamma$ measurement [@jhep08_176], currently the dominant systematic limitation in this determination. In the coming decades, the statistical uncertainties of measuring $\gamma$ will be greatly reduced by LHCb and Belle II, potentially to $1^{\circ}$ or less. The model-independent approach provides the most precise stand-alone $\gamma$ measurement [@jhep08_176], and therefore improved measurements of the $D$ strong-phase parameters are essential in maximizing the precision of $\gamma$ from these future data sets.
In this Letter, we use the model-independent approach of Ref. [@epjc47_347] for the determination of the strong-phase parameters between $D^0$ and $\bar{D}^0\rightarrow K^0_{S,L}\pi^+\pi^-$. More details are presented in a companion paper submitted to Physical Review D [@PRD]. Our data sample was collected from $e^+e^-$ annihilations at $\sqrt{s}=3.773$ GeV, just above the energy threshold for production of $D \bar{D}$ events. At this energy we take advantage of unique quantum correlations afforded by production through the $\psi(3770)$ resonance. The total integrated luminosity of our sample is 2.93 fb$^{-1}$ [@lumi], 3.6 times that of the CLEO measurement. The expected improvement in precision of the strong-phase parameters will significantly reduce the uncertainties of determinations of $\gamma$ [@plb718_43; @jhep10_097; @jhep06_131; @jhep08_176; @prd85_112014] that utilize $D \to K^0_{S,L} \pi^+\pi^-$. Additionally, improved knowledge of these strong-phase parameters will have significant impact in other applications, including measurements of the CKM angle $\beta$ (also denoted $\phi_1$) through time-dependent analyses of $B^0\rightarrow Dh^0$ [@prd94_052004] (where $h$ is a light meson) and $B^0\rightarrow D\pi^+\pi^-$ [@jhep03_195], as well as measurements of charm mixing and $CP$ violation [@jhep10_185; @jhep04_033; @prd99_012007; @prl122_231802].
For this study we analyze the $D \rightarrow K^0_S \pi^+ \pi^-$ Dalitz plot phase space of $m^2_-$ vs. $m^2_+$, where $m^2_-$ and $m^2_+$ are the squared invariant masses of the $K^0_S \pi^-$ and $K^0_S \pi^+$, respectively. The phase space is partitioned into eight pairs of irregularly shaped bins following the three schemes defined in Ref. [@prd82_112006], which are divided according to regions of similar strong-phase difference $\Delta\delta_D$ or maximum sensitivity to $\gamma$ in the presence of negligible (significant) background; here these schemes are referred to as “equal $\Delta\delta_D$" and “(modified) optimal", respectively. The bin index $i$ ranges from $-8$ to $8$ (excluding 0), with the bins symmetric under the exchange $m^2_-\leftrightarrow m^2_+$ ($i\leftrightarrow -i$). The strong-phase parameters are denoted $c_i$ and $s_i$, where $c_i$ is the amplitude-weighted average of cos$\Delta\delta_D$ in the $i$th region of the Dalitz plot ($\mathcal{D}_i$) and is given by $$c_i=\frac{\int_{\mathcal{D}_i}|\mathcal{A}||\bar{\mathcal{A}}|{\rm cos}\Delta\delta_Dd\mathcal{D}}{\sqrt{\int_{\mathcal{D}_i}|\mathcal{A}|^2d\mathcal{D}\int_{\mathcal{D}_i}|\bar{\mathcal{A}}|^2d\mathcal{D}}},$$ where $\mathcal{A}$ is the amplitude for $D^0\rightarrow
K^0_S\pi^+\pi^-$ over the Dalitz plot. The term $s_i$ is defined analogously, with cos$\Delta\delta_D$ replaced by sin$\Delta\delta_D$. Because the effects of charm mixing and $CP$ violation in the $D$ decay are negligible, we take $c_i=c_{-i}$ and $s_i=-s_{-i}$. The measurement involves studying the density of the correlated $D\rightarrow K^0_S\pi^+\pi^-$ vs. $D\rightarrow K^0_{S,L}\pi^+\pi^-$ Dalitz plots, as well as decays of a $D$ meson tagged in a $CP$ eigenstate decaying to $K^0_{S,L}\pi^+\pi^-$. The expected yields can be expressed in terms of the parameters $K_i$, $c_i$ and $s_i$ for $D^0\rightarrow K^0_S\pi^+\pi^-$, and $K^{\prime}_i$, $c^{\prime}_i$ and $s^{\prime}_i$ for $D\rightarrow K^0_L\pi^+\pi^-$, where $K_i^{(\prime)}$ is determined from the distribution of the flavor-tagged $D^0\rightarrow K^0_{S,L}\pi^+\pi^-$ decays across the bins of the Dalitz plot as $K_i^{(\prime)}=h_D\int_{\mathcal{D}_i}|\mathcal{A}|^2d\mathcal{D}$ and $h_D$ is a normalization factor. Therefore, the strong-phase parameters $c_i$, $s_i$, $c_i^{\prime}$, and $s_i^{\prime}$ can be determined by minimizing the likelihood function constructed from the observed and expected yields of these decays.
Details about the BESIII detector design and performance are provided in Ref. [@Ablikim:2009aa]. Signal efficiencies and background yields are determined from simulated Monte Carlo (MC) events that are processed identically to data. The generation of simulated samples for the signal processes $D^0\rightarrow K^0_{S}\pi^+\pi^-$ and $D^0\rightarrow K^0_{L}\pi^+\pi^-$ are based on measured isobar resonance amplitudes from the Dalitz plot of $D^0\rightarrow
K^0_{S}\pi^+\pi^-$.
To measure strong-phase parameters, we select “single-tag” (ST) and “double-tag” (DT) samples as listed in Table \[tab:numST\]. STs are $D$ mesons reconstructed from their daughter particles in one of 17 decay modes, of which four are flavor-specific, five are $CP$-even, seven are $CP$-odd, and one ($K^0_S\pi^+\pi^-$) is $CP$-mixed. Note that we count $D\rightarrow \pi^+\pi^-\pi^0$ as a $CP$-even eigenstate while explicitly correcting for its small $CP$-odd component [@Malde:2015mha]. DTs are events with an ST and a second $D$ meson reconstructed as either $K^0_S\pi^+\pi^-$ or $K^0_L\pi^+\pi^-$. The $K^0_{L}$ mesons are not directly reconstructed and their presence is inferred by partial reconstruction technique where one particle is identified by the missing energy and mass in the event. DTs are only formed in combinations where there is a maximum of one unreconstructed particle.
The selection and yield determination procedures of ST and DT candidates are described in the companion article [@PRD] and are summarized below. The ST $D$ signals are identified using the beam-constrained mass ${\rm M}_{\rm BC}=\sqrt{(\sqrt{s}/2)^2-|\overrightarrow{p}_{D_{\rm tag}}|^2}$ and $\Delta E=\sqrt{s}/2-E_{D_{\rm tag}}$, where $\overrightarrow{p}_{D_{\rm tag}}$ and $E_{D_{\rm tag}}$ are the momentum and energy of the $D$ candidate, respectively. The ST yields are obtained by performing maximum likelihood fits to ${\rm M}_{\rm BC}$ distributions. For decay modes that cannot be fully reconstructed ($K^+e^-\bar{\nu}_e$, $K^0_L\pi^0$ and $K^0_L\pi^0\pi^0$), the ST yields are estimated with the number of neutral $D\bar{D}$ events in data [@cpc42_083001], the decay branching fractions [@pdg18] and ST efficiencies determined with MC simulation. The ST yields, $N_{\rm ST}$, are listed in the second column of Table \[tab:numST\].
[l|ccc]{} Mode & $N_{\rm ST}$ & $N^{\rm DT}_{K^0_S\pi^+\pi^-}$ & $N^{\rm DT}_{K^0_L\pi^+\pi^-}$\
Flavor tags & & &\
$K^+\pi^-$ &$549373\pm756$ & $4740\pm71$ & $9511\pm115$\
$K^+\pi^-\pi^0$ &$1076436\pm1406$ & $5695\pm78$ & $11906\pm132$\
$K^+\pi^-\pi^-\pi^+$ & $712034\pm1705$ & $8899\pm95$ & $19225\pm176$\
$K^+e^-\bar{\nu}_e$ & $458989\pm5724$ & $4123\pm75$ &\
$CP$-even tags & & &\
$K^+K^-$ & $57050\pm231$ & $443\pm22$ &$1289\pm41$\
$\pi^+\pi^-$ & $20498\pm263$ & $184\pm14$ & $531\pm28$\
$K^0_S\pi^0\pi^0$ & $22865\pm438$ & $198\pm16$ & $612\pm35$\
$\pi^+\pi^-\pi^0$ &$107293\pm716$ & $790\pm31$ & $2571\pm74$\
$K^0_L\pi^0$ & $103787\pm7337$ & $913\pm41$ &\
$CP$-odd tags & & &\
$K^0_S\pi^0$ & $66116\pm324$ & $643\pm26$ & $861\pm46$\
$K^0_S\eta_{\gamma\gamma}$ & $9260\pm119$ & $89\pm10$ & $105\pm15$\
$K^0_S\eta_{\pi^+\pi^-\pi^0}$ & $2878\pm81$ & $23\pm5$ & $40\pm 9$\
$K^0_S\omega$ & $24978\pm448$ & $245\pm17$ & $321\pm25$\
$K^0_S\eta^{\prime}_{\pi^+\pi^-\eta}$ & $3208\pm88$ & $24\pm6$ & $ 38\pm 8$\
$K^0_S\eta^{\prime}_{\gamma\pi^+\pi^-}$ & $9301\pm139$ & $81\pm10$ & $120\pm14$\
$K^0_L\pi^0\pi^{0}$ & $50531\pm6128$ & $620\pm32$ &\
Mixed $CP$ tags & & &\
$K^0_S\pi^+\pi^-$ &$188912\pm756$ & $899\pm31$ & $3438\pm72$\
$K^0_S\pi^+\pi^-_{\rm miss}$ & & $224\pm17$ &\
$K^0_S(\pi^0\pi^0_{\rm miss})\pi^+\pi^-$ & & $710\pm34$ &\
\[tab:numST\]
The yields of DT candidates consisting of $K^0_S\pi^+\pi^-$ vs. fully reconstructed final states are determined with a two-dimensional unbinned maximum-likelihood fit to the ${\rm M}^{\rm sig}_{\rm BC}$ (signal) vs. ${\rm M}^{\rm tag}_{\rm BC}$ (tag) distribution. The DT candidates with an undetectable neutrino or $K^0_L$ are reconstructed by combining a $K^0_S\pi^+\pi^-$ candidate with the remaining charged or neutral particles, that are assigned to the other $D$ decay. The variable ${\rm U}_{\rm miss} = E_{\rm miss}-|\vec{p}_{\rm miss}|$ (for $K^+e^-\bar{\nu}_e$) or missing-mass squared (${\rm M}^2_{\rm miss}$) are calculated from the missing energy and momentum in the event. To reduce background contributions, events with excess neutral energy or charged tracks are rejected.
The $K^0_{S}\pi^+\pi^-$ vs. $K^0_{S}\pi^+\pi^-$ DTs are crucial for determining the $s_i$ values, and thus in order to increase the yield for these events, we include two types of partially reconstructed events, which more than doubles the yield. The first ($K^0_S\pi^{\pm}\pi^{\mp}_{\rm miss}$) allows for one pion originating from the $D$ meson to be unreconstructed in the detector. For these events, which have only three charged tracks recoiling against the $D\rightarrow K^0_S\pi^+\pi^-$ ST, the missing pion is inferred from the ${\rm M}^2_{\rm miss}$ of the event. The second ($K^0_{S}(\pi^0\pi^0_{\rm miss})\pi^+\pi^-$) is the case where one $K^0_S$ meson decays to $\pi^0\pi^0$, with only one $\pi^0$ detected while the other $\pi^0$ is undetected. We select events with only two additional oppositely charged tracks recoiling against the $D\rightarrow K^0_S\pi^+\pi^-$ ST and identify these as the $\pi^+$ and $\pi^-$ from the other $D$ meson. The resulting distributions of ${\rm M}^2_{\rm miss}$ show clear signals with minimal background, and signal yields are obtained with unbinned maximum-likelihood fits, as is shown in Fig. \[fig:MM2\].
![ Fits to ${\rm M}_{\rm miss}^2$ distributions in data. Points with error bars are data, long-dashed (blue) curves are the fitted combinatorial backgrounds. The shaded areas (pink) show MC estimates of the peaking backgrounds from (a) $D\rightarrow \pi^+\pi^-\pi^+\pi^-$ and (b) $D\rightarrow
\pi^+\pi^-\pi^0\pi^0$, and the red solid curves are the total fits.[]{data-label="fig:MM2"}](MM.eps){width="\linewidth"}
![Dalitz plots of $K^0_S\pi^+\pi^-$ events in data. The effect of the quantum correlation is clearly visible. The approximate locations of events from $K^0_S\rho(770)^0$ are indicated by arrows for clarity.[]{data-label="fig:Dalitz"}](DALITZ.eps){width="\linewidth"}
The DT yields of $K^0_S\pi^+\pi^-$ and $K^0_L\pi^+\pi^-$ tagged by different channels are shown in the third and fourth columns of Table \[tab:numST\], respectively. Overall, the DT yields of $D \to
K^0_{S(L)}\pi^+\pi^-$ involving a $CP$ eigenstate are a factor of 5.3 (9.2) larger than those in Ref. [@prd82_112006], and the DT yields of $K^0_S\pi^+\pi^-$ tagged with $D \to K^0_{S(L)}\pi^+\pi^-$ decays are a factor of 3.9 (3.0) larger than those in Ref. [@prd82_112006]. These increases come not only from the larger data set available at BESIII but also from the additional tag modes and the application of partial-reconstruction techniques. Figure \[fig:Dalitz\] shows the Dalitz plots of $CP$-even and $CP$-odd tagged $D\rightarrow K^0_S\pi^+\pi^-$ events selected in the data. The effect of quantum correlations arising from production through $\psi(3770)\rightarrow D^0\bar{D}^0$ is demonstrated by the differences between these plots. Most noticeably, the $CP$-odd component $K^0_S \rho(770)^0$ is visible in $CP$-even tagged $K^0_S\pi^+\pi^-$ samples but absent from $CP$-odd samples.
The DT yield for the $i$th bin of the Dalitz plot of each tagged $D\rightarrow K^0_{S(L)}\pi^+\pi^-$ sample, $N^{\rm obs}_{i}$, can be determined by fitting the DT events observed in this bin. Here the yield includes the signal and any peaking background component. The expected DT yields in the $i$th bin of Dalitz plot of each tagged $D\rightarrow K^0_{S(L)}\pi^+\pi^-$ sample, $N^{\rm exp}_{i}$, are sums of the expected signal yields and the expected peaking backgrounds. It should be noted that detector resolution effects can cause individual events to migrate between Dalitz plot bins after reconstruction. Such migration effects vary among bins due to the irregular bin shapes, coupled with the rapid variations of the Dalitz plot density. Furthermore, migrations differ between $D\rightarrow
K^0_S\pi^+\pi^-$ and $D\rightarrow K^0_L\pi^+\pi^-$ decays due to different resolutions in the Dalitz plots (0.0068 GeV$^2/c^4$ for $D\rightarrow K^0_S\pi^+\pi^-$ and 0.0105 GeV$^2/c^4$ for $D\rightarrow K^0_L\pi^+\pi^-$). The resultant bin migrations range within (3-12)% and (3-18)% for the $K^0_S\pi^+\pi^-$ and $K^0_L\pi^+\pi^-$ signals, respectively. Therefore, in the determination of the DT yields, simulated efficiency matrices are introduced to account for bin migration and reconstruction efficiencies [@PRD]. Studies indicate that neglecting bin migration introduces biases in the determination of $c_i~(s_i)$ that average a factor of 0.7 (0.3) times the statistical uncertainty of this analysis, so it is important to correct for this effect. The values of $K_i$ and $K_i^{\prime}$ that are used to evaluate $N^{\rm exp}_{i}$ are determined from the flavor-tagged DT yields, where corrections from doubly Cabibbo-suppressed decays, efficiency and migration effects have been applied, which are explained in detail in Ref. [@PRD].
The values of $c^{(\prime)}_i$ and $s^{(\prime)}_i$ are obtained by minimizing the negative log-likelihood function constructed as $$\begin{aligned}
-2{\rm log}\mathcal{L}&=&-2\sum\limits_{i}\sum\limits_{j} {\rm ln}P(N^{\rm obs}_{ij},\langle N^{\rm exp}_{ij}\rangle)_{K^0_S\pi^+\pi^-,K^0_{S(L)}\pi^+\pi^-} \nonumber \\
&&-2\sum\limits_{i} {\rm ln}P(N^{\rm obs}_i,\langle N_i^{\rm exp}\rangle)_{CP,K^0_{S(L)}\pi^+\pi^-} +\chi^2, \nonumber
\label{eq:likelihood}\end{aligned}$$ where $P(N^{\rm obs},\langle N^{\rm exp} \rangle)$ is the Poisson probability to observe $N^{\rm obs}$ events given the expected number $\langle N^{\rm exp} \rangle$. Here the sums are over the bins of the $D^0\to K^{0}_{S(L)}\pi^+\pi^-$ Dalitz plots. The $\chi^2$ term is used to constrain the difference $c_i^{\prime}-c_i$ ($s_i^{\prime}-s_i$) to the predicted quantity $\Delta c_i$ ($\Delta
s_i$). The values of $\Delta c_i$ and $\Delta s_i$ are estimated based on the decay amplitudes of $D^0\rightarrow
K^0_S\pi^+\pi^-$ [@prd98_112012] and $D^0\rightarrow
K^0_L\pi^+\pi^-$, where the latter is constructed by adjusting the $D^0\rightarrow K^0_S\pi^+\pi^-$ model taking the $K^0_S$ and $K^0_L$ mesons to have opposite $CP$, as is discussed in Refs. [@prd80_032002; @prd82_112006]. The details of assigning $\Delta c_i$ ($\Delta s_i$) and their uncertainties $\delta\Delta c_i$ ($\delta\Delta s_i$) are presented in Table VI of Ref. [@PRD].
The measured strong-phase parameters $c^{(\prime)}_i$ and $s^{(\prime)}_i$ are presented in Fig. \[fig:cisi\] and Table \[tab:cisifit\_final\]. Their systematic uncertainties arise from the evaluation of $K_i$ and $K_i^{\prime}$, the estimation of ST yields, the MC statistics, subtraction of the DT peaking background, the fitting procedure for determining DT yields, the difference in momentum resolution between data and MC simulation, and the effects of $D^0\bar{D}^0$ mixing. The estimation of these systematic uncertainties is described in detail in Ref. [@PRD]. In addition to our results, Fig. \[fig:cisi\] includes the predictions of Ref. [@prd98_112012] and the results from Ref. [@prd82_112006], which show reasonable agreement.
{width="1.0\linewidth"}
In summary, measurements of the strong-phase parameters between $D^0$ and $\bar{D}^0\rightarrow K^0_{S,L}\pi^+\pi^-$ in bins of phase space have been performed using 2.93 fb$^{-1}$ of data collected at $\sqrt{s}$=3.773 GeV with the BESIII detector. Compared to the previous CLEO measurement [@prd82_112006], two main improvements have been incorporated. First, additional tag decay modes are used. In particular the inclusion of the $\pi^+\pi^-\pi^0$ tag improves the sensitivity to $c_i$ and the addition of the $K^0_{S}(\pi^0\pi^0_{\rm miss})\pi^+\pi^-$ improves the sensitivity to $s_i$. Second, corrections for bin migration have been included, as their neglect would lead to uncertainties comparable to the statistical uncertainty. The results presented in this Letter are on average a factor of 2.5 (1.9) more precise for $c_i$ ($s_i$) and a factor of 2.8 (2.2) more precise for $c^{\prime}_i$ ($s^{\prime}_i$) than has been achieved previously. The strong-phase parameters provide an important input for a wide range of $CP$ violation measurements in the beauty and charm sectors, and also for measurements of strong-phase parameters in other $D$ decays where $D \to K^0_S\pi^+\pi^-$ is used as a tag [@plb747_9; @plb757_520; @jhep01_082; @jhep01_144; @prd85_092016].
To assess the impact of our $c_i$ and $s_i$ results on a measurement of $\gamma$, we use a large simulated data set of $B^-\rightarrow DK^-$, $D\rightarrow K^0_S\pi^+\pi^-$ events. These are generated with the expected distribution given for our measured central values of $K_i$, $c_i$, and $s_i$ and the input values $\gamma=73.5^{\circ}$, $r_B=0.103$, and $\delta_B=136.9^{\circ}$ [@epjc77_895], where $r_B$ is the ratio of the suppressed amplitude to the favored amplitude and $\delta_B$ is their strong-phase difference. The simulated data are fitted 10,000 times to determine $\gamma$, $\delta_B$ and $r_B$. The values of $c_i$ and $s_i$ used in each fit are sampled from the measured values smeared by their uncertainties accounting for any correlations. Based on this study, the uncertainty in $\gamma$ associated with our uncertainties for $c_i$ and $s_i$ is found to be 0.7$^{\circ}$, 1.2$^{\circ}$ and 0.8$^{\circ}$ for the equal $\Delta\delta_D$, optimal and modified optimal binning schemes, respectively. For comparison, the corresponding results from CLEO are 2.0$^{\circ}$, 3.9$^{\circ}$ and 2.1$^{\circ}$ [@prd82_112006]. Therefore, the uncertainty on $\gamma$ arising from knowledge of the charm strong phases is approximately a factor of three smaller than was possible with the CLEO measurements.
For the first time, the dominant systematic uncertainty for $\gamma$ measurement from the strong-phase parameters will be constrained to around $1^{\circ}$, or less, for $\gamma$ measurements with future $B$ experiments [@plb718_43; @jhep10_097; @jhep06_131; @jhep08_176; @prd85_112014]. The predicted statistical uncertainties on $\gamma$ from LHCb prior to the start of High-Luminosity LHC operation in the mid 2020s, and from Belle II are expected to be around 1.5$^\circ$ [@1808.08865; @1808.10567]. The improved precision achieved here will ensure that measurements of $\gamma$ from LHCb and Belle II over the next decade are not limited by the knowledge of these strong-phase parameters.
\[tab:cisifit\_final\]
The BESIII collaboration thanks the staff of BEPCII and the IHEP computing center for their strong support. This work is supported in part by National Key Basic Research Program of China under Contract No. 2015CB856700; National Natural Science Foundation of China (NSFC) under Contracts Nos. 11625523, 11635010, 11735014, 11775027, 11822506, 11835012; the Chinese Academy of Sciences (CAS) Large-Scale Scientific Facility Program; Joint Large-Scale Scientific Facility Funds of the NSFC and CAS under Contracts Nos. U1532257, U1532258, U1732263, U1832207; CAS Key Research Program of Frontier Sciences under Contracts Nos. QYZDJ-SSW-SLH003, QYZDJ-SSW-SLH040; 100 Talents Program of CAS; INPAC and Shanghai Key Laboratory for Particle Physics and Cosmology; ERC under Contract No. 758462; German Research Foundation DFG under Contracts Nos. Collaborative Research Center CRC 1044, FOR 2359; Istituto Nazionale di Fisica Nucleare, Italy; Koninklijke Nederlandse Akademie van Wetenschappen (KNAW) under Contract No. 530-4CDP03; Ministry of Development of Turkey under Contract No. DPT2006K-120470; National Science and Technology fund; STFC (United Kingdom); The Knut and Alice Wallenberg Foundation (Sweden) under Contract No. 2016.0157; The Royal Society, UK under Contracts Nos. DH140054, DH160214; The Swedish Research Council; U. S. Department of Energy under Contracts Nos. DE-FG02-05ER41374, DE-SC-0010118, DE-SC-0012069; University of Groningen (RuG) and the Helmholtzzentrum fuer Schwerionenforschung GmbH (GSI), Darmstadt; This paper is also supported by Beijing municipal government under Contract No CIT&TCD201704047, and by the Royal Society under Contract No. NF170002.
[99]{}
N. Cabibbo, Phys. Rev. Lett. [**10**]{}, 531 (1963); M. Kobayashi and T. Maskawa, Prog. Theor. Phys. [**49**]{}, 652 (1973). J. Brod and J. Zupan, J. High Energy Phys. [**01**]{} (2014) 051. M. Blanke and A. Buras, Eur. Phys. J. C [**79**]{}, 159 (2019).
M. Tanabashi $et~al.$ (Particle Data Group), Phys. Rev. D [**98**]{}, 030001 (2018).
J. Charles [*et al.*]{} (CKMfitter Group), Eur. Phys. J. C [**41**]{}, 1 (2005); and updates at <http://ckmfitter.in2p3.fr>; M. Bona [*et al.*]{} (UTfit Collaboration), J. High Energy Phys. [**07**]{} (2005) 028; and updates at <http://utfit.org/UTfit/WebHome>.
J. Brod, A. Lenz, G. Tetlalmatzi-Xolocotzi and M. Wiebusch, Phys. Rev. D [**92**]{}, 033002 (2015).
A. Giri, Y. Grossman, A. Soffer, and J. Zupan, Phys. Rev. D [**68**]{}, 054018 (2003). A. Bondar and A. Poluektov, Eur. Phys. J. C [**47**]{}, 347 (2006); Eur. Phys. J. C [**55**]{}, 51 (2008).
R. A. Briere $et$ $al$. (CLEO Collaboration), Phys. Rev. D [**80**]{}, 032002 (2009). J. Libby $et$ $al$. (CLEO Collaboration), Phys. Rev. D [**82**]{}, 112006 (2010).
R. Aaij $et$ $al$. (LHCb Collaboration), J. High Energy Phys. [**08**]{} (2018) 176; Erratum: J. High Energy Phys. [**08**]{} (2018) 107. A. Cerri, V.V. Gligorov, S. Malvezzi, J. M. Camalich, J. Zupan $et$ $al$.,CERN Yellow Rep. Monogr. [**7**]{}, (2019) 867; arXiv:1812.07638. M. Ablikim $et$ $al.$ (BESIII Collaboration), submitted to PRD, arXiv:xxxx. M. Ablikim, $et~al.$ (BESIII Collaboration), Chin. Phys. C [**37**]{}, 123001 (2013); Phys. Lett. B [**753**]{}, 629 (2016).
R. Aaij $et$ $al$. (LHCb Collaboration), Phys. Lett. B [**718**]{}, 43 (2012). R. Aaij $et$ $al$. (LHCb Collaboration), J. High Energy Phys. [**10**]{} (2014) 097. R. Aaij $et$ $al$. (LHCb Collaboration), J. High Energy Phys. [**06**]{} (2016) 131.
H. Aihara $et$ $al$. (Belle Collaboration), Phys. Rev. D [**85**]{}, 112014 (2012). V. Vorobyev $et$ $al$. (Belle Collaboration), Phys. Rev. D [**94**]{}, 052004 (2016). A. Bondar, A. Kuzmina and V. Vorobyev, J. High Energy Phys. [**03**]{} (2018) 195.
C. Thomas and G. Wilkinson, J. High Energy Phys. [**10**]{} (2012) 185. R. Aaij $et$ $al$. (LHCb Collaboration), J. High Energy Phys. [**04**]{} (2016) 033. A. Di. Canto, J. Garra Ticó, T. Gershon, N. Jurik, M. Martinelli, T. Pilař, S. Stahl and D. Tonelli, Phys. Rev. D [**99**]{}, 012007 (2019). R. Aaij $et$ $al$. (LHCb Collaboration), Phys. Rev. Lett. [**122**]{}, 231802 (2019).
M. Ablikim $et~al.$ (BESIII Collaboration), Nucl. Instrum. Methods Phys. Res., Sect. A [**614**]{}, 345 (2010).
S. Malde [*et al.*]{}, Phys. Lett. B [**747**]{}, 9 (2015).
M. Ablikim, $et~al.$ (BESIII Collaboration), Chin. Phys. C [**42**]{}, 083001 (2018).
I. Adachi $et$ $al$. (BaBar and Belle Collaborations), Phys. Rev. D [**98**]{}, 110212 (2018).
M. Nayak $et$ $al$., Phys. Lett. B [**747**]{}, 9 (2015). S. Harnew, P. Naik, C. Prouve, J. Rademacker and D. Asner, J. High Energy Phys. [**01**]{} (2018) 144.
T. Evans, S.T. Harnew, J. Libby, S. Malde, J. Rademacker and G. Wilkinson, Phys. Lett. B. [**757**]{}, 520 (2016).
J. Insler $et~al.$ (CLEO Collaboration), Phys. Rev. D [**85**]{}, 092016 (2012).
P. K. Resmi, J. Libby, S. Malde and G. Wilkinson, J. High Energy Phys. [**01**]{} (2018) 082. Y. Amhis $et~al.$, (Heavy Flavor Averaging Group), Eur. Phys. J. C [**77**]{}, 895 (2017); and updates at https://hflav.web.cern.ch.
I. Bediaga $et~al.$ (LHCb Collaboration), HCB-PUB-2018-009, CERN-LHCC-2018-027 \[arXiv:1808.08865\]. E. Kou $et~al.$ (Belle II Collaboration), KEK Preprint 2018-27, BELLE2-PUB-PH-2018-001, FERMILAB-PUB-18-398-T, JLAB-THY-18-2780, INT-PUB-18-047, UWThPh 2018-26 \[arXiv:1808.10567\].
|
---
author:
- 'H. Kusumaatmaja'
- 'M. L. Blow'
- 'A. Dupuis'
- 'J. M. Yeomans'
title: The collapse transition on superhydrophobic surfaces
---
Introduction
============
It is well-known that the hydrophobic nature of a surface is amplified by its roughness [@Quere1; @Quere3]. This can happen in two different ways. When the liquid drop occupies the spaces between the surface projections, and is everywhere in contact with the surface, it is said to be in the collapsed or Wenzel state [@Wenzel]. The contact angle is $$\begin{aligned}
& \cos{\theta_{W}} = r \cos{\theta_e} \, \label{eq1}\end{aligned}$$ where $r$ is the ratio between the real surface area and its projection onto the horizontal plane and $\theta_e$ is the equilibrium contact angle of the flat surface. On the other hand, if penetration does not occur and the drop remains balanced on the surface projections with air beneath it, it is in the suspended or Cassie-Baxter state [@Cassie] with contact angle $$\begin{aligned}
& \cos{\theta_{CB}} = \Phi \cos{\theta_e} - (1-\Phi) \, , \label{eq2} \end{aligned}$$ with $\Phi$ the solid fraction of the surface. Both states are (local) minimum of the free energy, but there is often a finite energy barrier opposing the transition between them. The magnitude of the energy barrier has been shown to depend on both the size of the drop and the roughness of the surface [@Ishino1; @Patankar1].
The main aim of this paper is to explore the mechanisms by which the drop spontaneously collapses [@Mchale1; @Quere2]. We consider micron-scale drops, sufficiently large that we can ignore thermal fluctuations but smaller than the capillary length so that gravity is not important. We focus on the limit where the evaporation timescale is much longer than the timescale for drop equilibration so that the drop is always in thermodynamic equilibrium. This is normally the physically relevant situation for experiments on micron scale drops. The question of how and when collapse occurs is important because, even though both states show high values of the contact angle, many of their other physical properties, for example, contact angle hysteresis are very different [@Kusumaatmaja1].
We first consider a drop on a two dimensional, superhydrophobic surface and present analytic results for how it collapses as its volume is decreased. We argue that there are two mechanisms for collapse. For short posts, as the curvature of the drop increases, it touches the surface below the posts, thus breaching the free energy barrier. For longer posts the free energy barrier is removed when the surface free energy gained by the drop as it collapses wins over the surface free energy lost by increased contact with the hydrophobic posts. However, importantly, the collapse transition is usually preempted by the contact line of the drop retreating across the surface. Therefore collapse for drops on long posts will normally occur only when the drop covers a very small number of the posts.
In three dimensions anaytical calculations are not feasible so we use numerical simulations to follow the behaviour of the shrinking drop. A new feature is that the base of the drop tends to form a bowl-shape, where the lines of contact depin and move down all but the outermost posts. We further argue that the tendency for the contact line to prefer to retreat across the surface than to collapse is even more pronounced in three dimensions than in two. A conclusion summarises our results and compares them to experiments.
Drop collapse in two dimensions: analytical results
===================================================
Consider a two dimensional drop suspended on a regular array of hydrophobic posts as shown in Fig. \[fig1\]. The posts have width $a$, spacing $b$ and height $l$, and the substrate material has an intrinsic contact angle $\theta_e>90^{\mathrm{o}}$. The drop forms a circular cap with a contact angle $\theta$, cross-sectional area $S$, radius of curvature $R$ and base length $2r$.
![Schematic diagram of a suspended drop.[]{data-label="fig1"}](Suspended.eps)
We consider a drop with contact line that is pinned at the outer edges of two posts and we first assume that the contact line pinning persists as the drop spontaneously collapses. Considering the motion of a retreating contact line across a superhydrophobic surface (as, say, the volume of the drop is slowly decreased) shows that the line is pinned for $180^{\mathrm{o}}>\theta>\theta_e$ [@Kusumaatmaja1]. After we have explained the possible collapse mechanisms we shall return to a discussion of when they are preempted by depinning.
Given pinning, the base radius $r$ is fixed and takes discrete values $$r = (m+1/2) \, a+m \, b \nonumber
\label{geometry}$$ where $2m+1 = 1, 2, 3, \ldots$ is the number of posts beneath the drop. The cross-sectional area of the drop, which is constant, can be written $$S = r^2 \, \frac{\theta-\sin{\theta}\cos{\theta}}{\sin^2{\theta}} + 2 \,
m \, b \, h + \frac{2 \, m \, b^2}{4} \, \frac{\phi-\sin{\phi}\cos{\phi}}{\sin^2{\phi}}.
\, \label{eq3}$$ The last term in Eq. (\[eq3\]) is due to the curved interface underneath the drop and $\phi = \theta_p -
90^{\mathrm{o}}$ where $\theta_p$ is the angle this interface makes with the sides of the posts.
Our aim is to investigate when and how the collapse transition occurs. We do this by considering the behaviour of the drop free energy as a function of $h$, the distance it penetrates into the substrate (see Fig. \[fig1\]). The non-constant contributions to the drop free energy $F$ come from three terms. The first two correspond to the liquid–gas interfacial free energy above and beneath the surface and the third term is the free energy required by the liquid drop to wet the posts to a depth $h$ $$\begin{aligned}
& f \equiv F/\gamma = \frac{2\,r\,\theta}{\sin{\theta}} + \frac{2\,m\,b\,\phi}{\sin{\phi}} -
4\,m\,h\,\cos{\theta_e} \, \label{eq5} \end{aligned}$$ where $\gamma$ is the liquid–gas interfacial tension.
We now consider the variation of the free energy with $h$. The drop will start to penetrate the posts if $\frac{df}{dh}<0$ at $h=0$, or equivalently $\frac{df}{d\theta}>0$, since $\frac{dh}{d\theta}<0$. Using the constraint of constant area to eliminate $dh$ gives $$\begin{aligned}
df &=& \frac{2r\,(\sin{\theta} - \theta\,\cos{\theta})}{\sin^3{\theta}} \, (\sin{\theta}
+ \frac{2r}{b}\cos{\theta_e}) \, d\theta + \nonumber \\
&& \frac{2mb\,(\sin{\phi} - \phi\,\cos{\phi})}{\sin^3{\phi}} \, (\sin{\phi} +
\cos{\theta_e}) \, d\phi \, . \label{eq13}\end{aligned}$$
Consider first $d\phi=0$. Since $2\,r\,(\sin{\theta} - \theta\,\cos{\theta})/\sin^3{\theta} > 0$, the condition for the drop to start collapsing is $$\sin{\theta} + \frac{2r}{b}\cos{\theta_e} > 0 |_{h=0} \, . \label{eq6}$$ The corresponding critical drop radius of curvature and contact angle are [@Barrat] $$\begin{aligned}
& R_c = - \frac{b}{2\cos{\theta_e}} \, , \label{eq23} \\
& \sin{\theta_c} = - \frac{2r}{b}\cos{\theta_e} \, . \label{eq24}\end{aligned}$$ $\theta$ gets smaller and $\sin{\theta}$ gets larger as the drop penetrates the posts. As a result, once Eq. (\[eq6\]) is satisfied it will always be satisfied and once the drop has started to move it collapses fully, into the Wenzel state.
The drop will be in equilibrium at $h=0$ on the threshold of the collapse transition. Therefore we may combine Eq. (\[eq24\]) and the Laplace pressure condition to show that $\theta_p$ = $\theta_e$, or $\phi = \theta_e - 90^{\mathrm{o}}$ as expected from the Gibbs’ criterion [@Gibbs]. Hence, from Eq. (\[eq13\]) the free energy is at an extremum with respect to changes in $\phi$. Calculating the second derivative confirms that this is a minimum and hence that the assumption $d \phi =0$ is appropriate.
Typical plots of the free energy of a drop against $h$, the distance it penetrates into the substrate are shown in Fig. \[fig2\], where for simplicity we have neglected the corrections due to the curvature of the interfaces in the grooves. In Fig. \[fig2\](a), where we have used $m=3$, $b/a=1.5$, $\theta_e = 95^{\mathrm{o}}$, and $\theta
|_{h=0} = 111^{\mathrm{o}} < \theta_c = 111.6^{\mathrm{o}}$ the free energy is a smoothly decreasing function of $h$ and the drop will collapse. In Fig. \[fig2\](b) on the other hand, for $\theta|_{h=0} =
112^{\mathrm{o}} > \theta_c = 111.6^{\mathrm{o}}$, there is a free energy barrier and therefore no collapse. The peak of the free energy barrier occurs at $\theta = \theta_c$ and the magnitude of the barrier is $$\begin{aligned}
\Delta{f} &=& \frac{2\,r\,\theta_c}{\sin{\theta_c}} + \frac{2\,r^2\,\cos{\theta_e}}{b}\,\frac{\theta_c-\sin{\theta_c}\cos{\theta_c}}{\sin^2{\theta_c}} \label{eq16} \\
&-& \left[\frac{2\,r\,\theta}{\sin{\theta}} + \frac{2\,r^2\,\cos{\theta_e}}{b}\,\frac{\theta-\sin{\theta}\cos{\theta}}{\sin^2{\theta}}\right]_{\theta\equiv\theta |_{h=0}} \, . \nonumber\end{aligned}$$
![ Normalised drop free energy against penetration depth when (a) the collapse transition occurs and (b) there is a free energy barrier between Cassie-Baxter and Wenzel states. $F_0$ is the drop free energy in the Cassie-Baxter state, $\gamma$ is the liquid–gas surface tension and $S$ is the drop area. $m=3$, $b/a=1.5$, $\theta_e =
95^{\mathrm{o}}$ and $\theta |_{h=0} = 111^{\mathrm{o}}$ and $112^{\mathrm{o}}$ for (a) and (b) respectively.[]{data-label="fig2"}](2DEnergyCurve.eps)
We have argued that, for $R < R_c$, there is no free energy barrier to drops penetrating hydrophobic posts. The critical radius depends on the post width $a$, the post separation $b$, the base radius $r$, and the equilibrium contact angle $\theta_e$. It does not, however, depend on the post height $l$. There is, however, another route to drop collapse [@Quere2], which will pre-empt this mechanism for shallow posts.
Prior to collapse the liquid drop has not penetrated the posts, the system is in mechanical equilibrium, and the Laplace pressure is the same everywhere. Thus the liquid–gas interface between the posts bows out with a radius of curvature equal to that of the circular cap $R$. The centre of the curved interface reaches a distance $d$ into the posts: $$d = R \, (1- \cos{\phi}) \simeq \frac{b^2}{8R} \label{eq18}$$ for small $\phi$. As $R$ gets smaller, $d$ increases. When $d=l$ the liquid–gas interface touches the base surface initiating the transition between the Cassie-Baxter and Wenzel states. At this point there is a considerable free energy release because the drop is replacing two interfaces (liquid–gas and gas–solid) with a single liquid–solid interface. Consequently this transition is irreversible and for the opposite transition to occur (Wenzel to Cassie-Baxter) an external force is needed to overcome the free energy barrier. For this mechanism to be possible it is apparent from simple geometry that $d<b/2$.
Regions of parameter space where there is (i) collapse due to the contact line sliding down the posts, (ii) collapse due to the centre of the interface touching the base surface, (iii) no collapse are distiguished in Fig. \[fig7\]. The crossover between regions (i) and (ii) occurs when $$\cos{\theta_e} < -\frac{4\,l}{b}. \label{eq17}$$ It is interesting to note that the crossover point between the two regimes (Eq. \[eq17\]) will slide to larger $l$ as the posts are made more hydrophobic.
![The crossover between the two different drop collapse mechanisms in two dimensions.[]{data-label="fig7"}](CrossOver.eps)
We now revisit the assumption that the contact line is pinned at the outer edges of the posts. The (theoretical) advancing contact angle is $180^\mathrm{o}$ [@Kusumaatmaja1] and therefore the line will not move outwards. The receding angle in the quasi-static limit is $\theta_e$ [@Kusumaatmaja1] and therefore it will not jump inwards if $\theta_c > \theta_e$ or, equivalently, $\sin^2{\theta_c} < 1 - \cos^2{\theta_e}$. Using Eqs. (\[geometry\]) and (\[eq24\]) this is equivalent to $$\cos^2{\theta_e} < \left[ 4\left(\frac{m+1/2}{b/a}+m\right)^2+1 \right]^{-1}. \label{eq15}$$
Fig \[fig4\](c) shows the maximum value of $\theta_e$ at which collapse will occur for different $m$ and $b/a$. For small $b/a$ collapse is strongly suppressed and only occurs for tiny drops on slightly hydrophobic surfaces. Even for $b/a >> 1$ the tendency to depin is strong and the collapse occurs for small value of $m$ unless $\theta_e$ is close to $90^{\mathrm{o}}$. By setting $r=b$ in Eq. (\[eq6\]), we conclude that the transition will never occur spontaneously for $\theta_e > 120^{\mathrm{o}}$.
In Fig. \[fig4\](a) and (b), we plot the critical radius of curvature at which a transition occurs $R_c/a$ and the corresponding drop area (which we present as $\sqrt{S/\pi a^2}$) as a function of $\theta_e$, $m$ and $b/a$. As expected the critical radius of curvature does not depend on $m$; it is a function of $b/a$ and $\theta_e$ only. The critical base area of the drop does, however, depend on $m$ and is smaller for larger values of $m$ and $a/b$. The curves for increasing $m$ terminate at decreasing values of $\theta_e$ corresponding to the contact line receding inwards before the drop is able to penetrate the posts.
Simulations of drop collapse
============================
We now describe the details of a numerical model which will allow us to explore the collapse transition in both two and three dimensions. We describe the equilibrium properties of the drop by a continuum free energy [@Briant1] $$\Psi = \int_V (\psi_b(n)+\frac{\kappa}{2} (\partial_{\alpha}n)^2) dV + \int_S \psi_s(n_s) dS . \label{eq103}$$ $\psi_b(n)$ is a bulk free energy term which we take to be [@Briant1] $$\psi_b (n) = p_c (\nu_n+1)^2 (\nu_n^2-2\nu_n+3-2\beta\tau_w) \, ,$$ where $\nu_n = {(n-n_c)}/{n_c}$, $\tau_w = {(T_c-T)}/{T_c}$ and $n$, $n_c$, $T$, $T_c$ and $p_c$ are the local density, critical density, local temperature, critical temperature and critical pressure of the fluid respectively. This choice of free energy leads to two coexisting bulk phases of density $n_c(1\pm\sqrt{\beta\tau_w})$, which represent the liquid drop and surrounding gas respectively. Varying $\beta$ has the effects of varying the densities, surface tension, and interface width; we typically choose $\beta = 0.1$.
![ Critical drop area (presented as $\sqrt{S/\pi a^2}$) as a function of contact angle: comparison between the two–dimensional theory and simulations for $m=3$, $a = 8$, $b = 12$. The circles and crosses represent drops which collapse and remain suspended respectively. The solid line is the theoretical prediction.[]{data-label="fig8"}](LBCollapse2D.eps)
The second term in Eq. (\[eq103\]) models the free energy associated with any interfaces in the system. $\kappa$ is related to the liquid–gas surface tension and interface width via $\sigma_{lg} = {(4\sqrt{2\kappa p_c}
(\beta\tau_w)^{3/2} n_c)}/3$ and $\xi = (\kappa n_c^2/4\beta\tau_w p_c)^{1/2}$ [@Briant1]. We use $\kappa = 0.0018$, $p_c = 1/8$, $\tau_w = 0.3$, and $n_c = 3.5$.
The last term in Eq. (\[eq103\]) describes the interactions between the fluid and the solid surface. Following Cahn [@Cahn] the surface energy density is taken to be $\psi_s (n) = -\lambda \, n_s$, where $n_s$ is the value of the fluid density at the surface. The strength of interaction, and hence the local equilibrium contact angle, is parameterised by the variable $\lambda$. Minimising the free energy (\[eq3\]) leads to a boundary condition at the surface, $\partial_{\perp}n = -\lambda/\kappa$, and a relation between $\lambda$ and the equilibrium contact angle $\theta_e$ [@Briant1] $$\lambda = 2\beta\tau_w\sqrt{2p_c\kappa} \,\,
\mathrm{sign}(\frac{\pi}{2}-\theta_e)\sqrt{\cos{\frac{\alpha}{3}}(1-\cos{\frac{\alpha}{3}})} \, , \label{eq106}$$ where $\alpha=\cos^{-1}{(\sin^2{\theta_e})}$ and the function sign returns the sign of its argument. Similar boundary conditions can be used for surfaces that are not flat: a way to treat the corners and ridges needed to model superhydrophobic surfaces is described in [@Dupuis2].
The equations of motion of the drop are the continuity and the Navier-Stokes equations $$\begin{aligned}
&\partial_{t}n+\partial_{\alpha}(nu_{\alpha})=0 \, , \label{eq104}\\
&\partial_{t}(nu_{\alpha})+\partial_{\beta}(nu_{\alpha}u_{\beta}) = - \partial_{\beta}P_{\alpha\beta} + \nonumber \\
&\nu \partial_{\beta}[n(\partial_{\beta}u_{\alpha} + \partial_{\alpha}u_{\beta} + \delta_{\alpha\beta}
\partial_{\gamma} u_{\gamma}) ] \label{eq105}\end{aligned}$$ where $\mathbf{u}$, $\mathbf{P}$, and $\nu$ are the local velocity, pressure tensor, and kinematic viscosity respectively. The thermodynamic properties of the drop appear in the equations of motion through the pressure tensor $\mathbf{P}$ which can be calculated from the free energy [@Briant1; @Dupuis2] $$\begin{aligned}
&P_{\alpha\beta} = (p_{\mathrm{b}}-\frac{\kappa}{2} (\partial_{\alpha}n)^2 - \kappa n
\partial_{\gamma\gamma}n)\delta_{\alpha\beta} + \kappa (\partial_{\alpha}n)(\partial_{\beta}n), \nonumber \\
&p_{\mathrm{b}} = p_c (\nu_n+1)^2 (3\nu_n^2-2\nu_n+1-2\beta\tau_w) .\end{aligned}$$ When the drop is at rest $\partial_{\alpha}P_{\alpha\beta} = 0$ and the free energy (\[eq103\]) is minimised. As we are considering the quasi–static problem when the drop is in equilibrium until the point of collapse details of its dynamics should not affect the results. However we choose to implement physical equations of motion as this helps the drop to reach equilibrium quickly as its volume is decreased and for comparison to possible work on non-equilibrium collapse.
We use a lattice Boltzmann algorithm to solve Eqs. (\[eq104\]) and (\[eq105\]). No-slip boundary conditions on the velocity are imposed on the surfaces adjacent to and opposite the drop and periodic boundary conditions are used in the two perpendicular directions. Details of the lattice Boltzmann approach and of its application to drop dynamics are given in [@Briant1; @Dupuis2; @Succi1; @Kwok1; @Sbragaglia].
To implement evaporation we need to slowly decrease the drop volume. To do this we vary the liquid density by $- 0.1\%$ every $2\times 10^5$ time steps to ensure that the evaporation timescale is well separated from the drop equilibration timescale. This in turn affects the drop volume as the system relaxes back to its coexisting equilibrium densities.
Results for two dimensions are compared to the analytic solution in Fig. \[fig8\], where we have used $a=8$, $b=12$, and $\theta_e = 95^{\mathrm{o}}$. The critical drop area at which the collapse transition occurs is close to the theoretical value but critical radii obtained from simulations are typically too large by $\sim 2$ lattice spacings. This is because the liquid–gas interface is diffuse ($\sim 3-4$ lattice spacings). We checked that, as expected, $R_c$ is independent of the post height and that $h$ is the same everywhere underneath the drop.
Drop collapse in three dimensions: numerical results
====================================================
Analytic calculations in three dimensions are, in general, not possible for several reasons. Firstly, the drop shape is not a spherical cap but is influenced by the underlying topological patterning. Secondly, the shape of the liquid–gas interface spanning the posts is complicated. Thirdly, $h$, the distance the drop penetrates the substrate, is not neccesarily the same everywhere. Therefore we need to use the numerical approach presented in the last section to explore collapse. We consider a square array of posts of widths $a=3$ and spacing $b=9$. We present results for both spherical drops and ‘cylindrical’ drops which demonstrate the relevant physics but are less demanding in computer time.
![ (Color online) Equilibrium drop configurations. (a) $\theta_e = 95^{\mathrm{o}}$ and varying volume. (b) Fixed volume and contact angles, $\theta=93^{\mathrm{o}}$ (black), $95^{\mathrm{o}}$ (purple), $97^{\mathrm{o}}$ (blue), $100^{\mathrm{o}}$ (green), and $110^{\mathrm{o}}$ (red).[]{data-label="fig6"}](3DVariation.eps)
A new feature in three dimensions is that for a spherical (or cylindrical) drop on a square array of posts the base of the drop can form a bowl-shape where the lines of contact with the top of all but the peripheral posts depin and move down the posts leaving the drop suspended by just its outer rim. This was seen in simulations for drops of both cylindrical and spherical symmetry, and has recently been reported experimentally [@Moulinet1]. The depinning occurs to reduce the distortion of the interface from spherical. Depinning is favoured for smaller drops and for contact angles close to $90^{\mathrm{o}}$.
![ (Color online) Evolution with time of a cylindrical drop on a square array of posts of width $a=3$, spacing $b=9$ and height $l=15$. (a–c) Time evolution before collapse showing depinning of the receding contact line (note the scale change between (b) and (c)). (d–f) Motion of the collapsing drop: (d) cross sections in the plane bisecting the posts. (e) same times as (d), but in the plane bisecting the gap between the posts. (f) cross sections in the plane bisecting the gap, but with $l=45$ to enable the collapse to be followed to later times. []{data-label="figQuasi2D"}](Quasi2D.eps)
This is apparent in Fig. \[fig6\](a) which shows the equilibrium profile of the drop as its volume is varied with $\theta_e = 95^{\mathrm{o}}$. As expected the drop penetrates further into the posts as the radius is decreased (corresponding to increasing curvature). Fig. \[fig6\](b) shows cross sections of the final states of 5 drops with volumes $V
\simeq 4.5 \times 10^{5}$ (in the units of lattice spacing) equal to within $\sim 1\%$ but varying equilibrium contact angles in the range $\theta_e = 93^{\mathrm{o}}$ to $110^{\mathrm{o}}$. Although the drop penetrates deeper into the posts as the intrinsic contact angle approaches $90^{\mathrm{o}}$ there is no collapse (these values would give a collapse transition in two dimensions). Instead, for $\theta_e=110^{\mathrm{o}}$, the contact line depins and moves to cover 9 rather than 21 posts. Note that this jump also corresponds to a transition to the state when the drop is suspended on all the posts beneath it, not just those around its rim.
To explore the depinning further, and to try to find a collapse transition, we turned to the geometry of a cylindrical drop on a square array of posts. This preserves the physics whilst allowing us to exploit the quasi-two-dimensional geometry to run larger simulations. Results are shown in Fig. \[figQuasi2D\] for $\theta_e=93^{\mathrm{o}}$. Successive frames show how the drop profile evolves as its volume is quasi-statically decreased (note that they are drawn on different scales). Initially, the contact line is pinned at the edges of the posts and the drop penetrates further beneath the posts as the radius is decreased. However, as the drop continues to decrease in size, the drop contact angle reaches the receding angle and the contact line depins. As it depins we observe that the penetration into the posts decreases (because the drop is approximately spherical and the base area is reduced), thus moving the system away from the point where either a curvature or a free energy driven collapse is favourable. Eventually collapse is seen but only, for this example, when the drop spans just three posts. Note that for $l=45$ the drop stops moving once it is fully inside the posts as its free energy becomes independent of height: it forms a liquid bridge connecting several neighbouring posts.
Indeed, we expect from the two dimensional calculations that collapse is preempted by depinning for posts with $b/a \sim 1$ until the drops are very small. In three dimensions depinning will be even more important because the receding contact angle is larger than $\theta_e$, its value in two dimensions, because the distortion of the interface makes it more favourable for the drop to depin. From Fig. \[figQuasi2D\](a) and (b), we obtain $\theta_R \sim 120^\mathrm{o}$.
Summary
=======
To conclude, we have investigated the behaviour of an evaporating drop on a superhydrophobic surface. As the drop volume decreases quasi-statically it can move in three ways: (i) the drop attains its receding contact angle and the contact line moves inwards across the surface (ii) the free energy barrier to collapse vanishes and the drop moves smoothly down the posts (iii) the drop touches the base of the surface patterning and immediately collapses. The depinning (i) is predominant and, unless the posts are widely spaced, or the surface is only very weakly hydrophobic, collapse occurs only for drops spanning a very small number of posts.
This suggests strategies that could be used to suppress transitions to the Wenzel state. Long enough posts are needed to prevent curvature-driven collapse, i.e. $l \gtrsim b^2/R$, and the free energy barrier to the transition can be enhanced by choosing $\theta_e$ as large as possible and using closely spaced posts, i.e. $b \lesssim a$. A mobile contact line will also help as this will relax any build up of curvature.
Our results are in line with recent experiments [@Mchale1; @Quere2; @Moulinet2]. In [@Quere2], for long posts, the contact line retreated as the drop shrank and collapsed only at the very end of evaporation. For short posts, a few depinning events were followed by collapse at a radius consistent with a curvature-driven mechanism, $R_c \propto b^2/l$. It is not clear, however, whether the drop interface was suspended on all the posts or just those at the rim at the point of collapse: this detail is important in determining the constant of proportionality. In [@Moulinet1], the various drop configurations found here are also observed, including the depinning of the drop from all but the outer posts. In [@Moulinet2], for the somewhat different situation of drops bounced onto a surface, the critical pressure for impalement varied linearly with post height for short posts, as expected for curvature-driven collapse, and showed a clear crossover to a length-independent regime for longer posts, consistent with a drop overcoming a free energy barrier.
We thank D. Quéré for bringing this problem to our attention and for pointing out the collapse mechanism due to the centre of the interface touching the base surface. We appreciate useful discussions with G. Alexander, G. McHale and S. Moulinet. HK acknowledges support from a Clarendon Bursary and the INFLUS project.
[0]{}
.
.
.
.
.
.
.
*Europhys. Lett.* in press.
.
This condition is also obtained in .
.
.
.
.
.
.
.
submitted (2007).
.
|
---
abstract: 'We show that the first order theory of $H_{\omega_1}$ is the model companion of the first order theory of the universe of sets assuming the existence of class many Woodin cardinals, and working in a signature with predicates for all universally Baire sets of reals. We also outline some basic conditions granting the model completeness of the first order theory of $H_{\omega_2}$ and of the axiom system $\ZF+V=L$ in an appropriate language.'
author:
- Giorgio Venturi and Matteo Viale
bibliography:
- 'Biblio.bib'
title: The model companions of set theory
---
Introduction {#introduction .unnumbered}
============
This paper outlines a deep connection between two important threads of mathematical logic: the notion of model companionship, a central concept in model theory due to Robinson, and the notion of generic absoluteness, which plays a fundamental role in the current meta-mathematical investigations of set theory.
In order to unveil this connection, we proceed as follows: we enrich the first order language in which to formalize set theory by predicates whose meaning is as “clear” as that of the $\in$-relation, specifically we add predicates for $\Delta_0$-formulae and predicates for universally Baire sets of reals[^1]. In this extended language we are able to apply Robinson’s notions of model completeness and model companionship to argue that (assuming large cardinals) the first order theory of $H_{\omega_1}$ (the family of all hereditarily countable sets) is model complete and is the model companion of the first theor theory of $V$ (the universe of all sets). The study of model companionship goes back to the work of Abraham Robinson from the period 1950–1957 [@MacCompl], and gives an abstract model-theoretic characterization of key closure properties of algebraically closed fields. Robinson introduced the notion of model completeness to characterize the closure properties of algebraically closed fields, and the notion of model companionship to describe the relation existing between these fields and the commutative rings without zero-divisors. Robinson then showed how to extend these notions and results to a variety of other classes of first order structures.
On the other hand, generic absoluteness characterizes exactly those set theoretic properties whose truth value cannot be changed by means of forcing.
In [@VenturiRobinson] the first author found the first indication of a strict connection existing between these two apparently unrelated concepts. In this paper we will enlighten this connection much further.
Recall that a first order theory $T$ in a signature $\tau$ is model-complete if whenever $\mathcal{M}\sqsubseteq\mathcal{N}$ are models of $T$ with one a substructure of the other, we get that $\mathcal{M}\prec\mathcal{N}$; i.e. being a substructure amounts to be an elementary substructure.
The theory of algebraically closed fields has this property, as it occurs for all theories admitting quantifier-elimination, however it is the case that many natural theories not admitting quantifier-elimination are model-complete. Robinson regarded model-completeness as a strong indication of tameness for a first order theory.
A weak point of this notion is that model completeness of a theory is very sensitive to the signature in which the theory is formalized: for all theories $T$ in a signature $\tau$ there is a conservative extension to a theory $T'$ in a signature $\tau'$ which admits quantifier elimination (it suffices to add symbols and axioms for Skolem functions to $\tau$ and $T$, [@TENZIE Thm. 5.1.8]). In particular we can always extend a first order language $\tau$ to a language $\tau'$ so to make a $\tau$-theory $T$ model-complete with respect to $\tau'$. However if model-completeness of $T$ is shown with respect to a “natural” language in which $T$ can be formalized, then it brings many useful informations on the combinatorial-algebraic properties of models of $T$.
Recall also that for a first order signature $\tau$, a $\tau$-theory $T$ is the model companion of a $\tau$-theory $S$ if $T$ is model complete, and every model of $T$ can be embedded in a model of $S$ and conversely.
Robinson’s infinite forcing is loosely inspired by Cohen’s forcing method and gives an elegant formulation of the notion of model companionship: a theory $T$ is the model companion of a theory $S$ in the same first order signature if it is model complete and the models of $T$ are exactly the infinitely generic structures for Robinson’s infinite forcing applied to models of $S$. In [@VenturiRobinson] we describe a fundamental connection between the notion of being an infinitely generic structure and that of being a structure satisfying certain types of forcing axioms. This suggests an interesting parallel between a semantic approach *à la Robinson* to the study of the models of set theory and generic absoluteness results.
The main result of this paper (Thm. \[thm:Vmodcompl\]) shows that, modulo a natural extension of the language of set theory (given by the addition of predicates for all universally Baire sets of reals), the existence of class many Woodin cardinals implies that the model companion of the theory of the universe of all sets is the theory of $H_{\omega_1}$. We consider our extension natural because the predicates so added are exactly those whose truth value is unaffected by the forcing method, and for which, therefore, we have a concrete and stable understanding of their behaviour; for example Borel sets of reals are universally Baire, all sets of reals defined by a $\Delta_0$-formula are universally Baire, and (assuming large cardinals) all universally Baire sets of reals have all the desirable regularity properties such as: Baire property, Lebesgue measurability, perfect set property, determinacy, etc; moreover (assuming large cardinals) such sets form a point-class closed under projections, countable unions and intersections, complementation, continous images, etc.
We also remark that:
- On the one hand Hirschfeld [@Hir] showed that any extension of $\ZF$ has a model companion in the signature $\bp{\in}$. His result however is uninformative (a consideration he himself made in [@Hir]), since the model companion of $\ZF$ for the signature $\bp{\in}$ turns out to be (a small variation of) the theory of dense linear orders, a theory for a binary relation which has not much to do with the true meaning of the $\in$-relation. We consider this fact another indication of the naturalness of our choice of the first order language in which we formalize set theory: in a first order language containing just the $\in$-relation, there are many basic concepts whose formalization in first order logic is syntactically too complex (for example being a surjective function is a $\Delta_0$-property, but it is only $\Pi_2$-expressible in the signature $\bp{\in}$), this discrepancy causes the “anomaly” of Hirschfeld’s result, which is here resolved by adding predicates for all the concepts which are sufficiently simple and stable across the different models of set theory, i.e. the $\Delta_0$-properties and the universally Baire predicates.
- On the other hand (unlike Hirschfeld’s result) our results have a highly non-constructive flavour and require to embrace a fully platonistic perspective on the onthology of sets to be meaningfully formulated: we assume that the universe of sets $V$ and the family of hereditarily countable sets $H_{\omega_1}$ are rightful elements of our semantics, which —whenever endowed with suitably defined predicates and constants– give well-defined first order structures for the appropriate signature. Of course it is possible to reformulate our results so to make them compatible with a formalist approach to set theory *à la Hilbert*, but in this case their meaning would be much less transparent, hence we refrain here from pursuing this matter further.
The main philosophical thesis we draw form the results of the present paper is that the success of large cardinals in solving problems of second-order arithmetic[^2] via determinacy is due to the fact that these axioms make (in the appropriate language) the theory of $H_{\omega_1}$ the model-companion of the theory of $V$, and in particular a model complete theory. Similar considerations can be drawn for other axioms (such as forcing axioms or the constructibility axiom $V=L$) which are able to decide most of the problems which cannot be settled on the basis of $\ZFC$ alone. In particular we show that if one has a simply definable well-order of $H_{\omega_2}$ (which is the case assuming the bounded proper forcing axioms hold), then one has simply definable Skolem functions producing witnesses of $\Delta_0$-properties. In which case one can easily prove that the first order theory of $H_{\omega_2}$ is the model companion of the universe of sets in a signature with parameters for all elements of $H_{\omega_2}$, predicates for all bounded formulae, and Skolem functions for such predicates. We can see this result as a companion to the various generic absoluteness results for the theory of $H_{\omega_2}$ assuming forcing axioms the second author has recently presented in [@VIAASP; @VIAAUD14; @VIAMM+++; @VIAMMREV; @VIAUSAX]. We prove as well that $\ZFC+V=L$ is model complete with respect to a natural appropriate first order language.
The paper is structured as follows:
- §\[sec:boolvalmod\] recalls few important results on boolean-valued structures and generic absoluteness.
- §\[sec:modeltheoreticcompl\] recalls the basic facts on model companionship and on Robinson’s infinite forcing.
- In §\[sec:modelcompsetth\] we perform and justify the extension of the first order language of set theory, roughly described above, so to include predicates for all $\Delta_0$-formulae; after relativizing the notion of model completeness to the generic multiverse, Theorem \[thm:omegaVmodcompl\] shows that (assuming large cardinals) the theory of $H_{\omega_1}$ is the model companion of the theory of $V$ relative to the generic multiverse for the language admitting predicates for all $\Delta_0$-formulae.
- In §\[sec:modcompletUBpred\] we offer reasons for the necessity of a further expansion of the language of set theory, which includes all universally Baire predicates.
- §\[sec:modcompanUBpred\] gives the proof of Theorem \[thm:Vmodcompl\] showing that in a language admitting predicates for all the universally Baire sets, the theory of $H_{\omega_1}$ is the model companion of the theory of $V$, if we assume the existence of class many Woodin cardinals.
- §\[sec:modcomplforcax\] extends the above result to the theory of $H_{\omega_2}$ assuming forcing axioms, and to the theory $\ZFC+V=L$.
Boolean valued models and generic absoluteness {#sec:boolvalmod}
==============================================
Our first aim is to outline which first order properties are first order invariant with respect to the forcing method. Toward this aim we recall some standard facts on boolean-valued models for set theory, giving appropriate references for the relevant proofs (in particular [@BELLSTBVM], or [@VIAAUDSTE13], the forthcoming [@VIAAUDSTEBOOK], the notes [@viale-notesonforcing]), we assume below that the reader is familiar with the basic theory of boolean valued models, else we invite him to consult one of the above references (for example [@viale-notesonforcing Chapter 4]). Recall that $V$ denotes the universe of all sets and for any complete boolean algebra $\bool{B}\in V$ $$V^{\bool{B}}=\bp{\tau: \, \tau:X\to \bool{B} \text{ is a function with $X\subseteq V^{\bool{B}}$ a set}}$$ is the boolean valued model for set theory generated by forcing with $\bool{B}$.
$V^{\bool{B}}$ is endowed with the structure of a $\bool{B}$-valued model for the language of set theory $\mathcal{L}=\bp{\in,\subseteq}$, letting (see [@viale-notesonforcing Def. 5.1.1] for details) $$\Qp{\tau_1\in\tau_2}_{\bool{B}}=\bigvee_{\sigma\in\dom(\tau_2)}
(\Qp{\tau_1=\sigma}_{\bool{B}}\wedge\tau_2(\sigma)),$$ $$\Qp{\tau_1\subseteq\tau_2}_{\bool{B}}=\bigwedge_{\sigma\in\dom(\tau_1)}
(\neg\tau_1(\sigma)\vee\Qp{\sigma\in\tau_2}_{\bool{B}}),$$ $$\Qp{\tau_1=\tau_2}_{\bool{B}}=
\Qp{\tau_1\subseteq\tau_2}_{\bool{B}}\wedge
\Qp{\tau_2\subseteq\tau_1}_{\bool{B}}.$$
The boolean value $\Qp{\phi(\tau_1,\dots,\tau_n}_{\bool{B}}$ of formulae $\phi(x_1,\dots,x_n)$ with assignment $\tau_1,\dots,\tau_n$ are given according to the standard rules of boolean valued semantics (see for example [@viale-notesonforcing Section 4.1]); concretely: atomic formulae of type $\tau_1\mathrel{R}\tau_2$ are given the boolean value $\Qp{\tau_1\mathrel{R}\tau_2}_{\bool{B}}$; the boolean operations allows to define the boolean value associated to a conjunction/disjunction/negation of formulae; completeness of $\bool{B}$ allows to define $$\Qp{\exists x\phi(x,\vec{\tau})}_{\bool{B}}=\bigvee_{\sigma\in V^{\bool{B}}}\Qp{\phi(\sigma,\vec{\tau})}_{\bool{B}}.$$
The class of models we will analyze is given by the generic extensions of initial segments of $V$. To make this precise we need a couple of definitions.
Let $\bool{B}$ be a complete boolean algebra. and $\dot{\kappa}\in V^{\bool{B}}$ be such that $\Qp{\dot{\kappa}\text{ is a regular cardinal}}_{\bool{B}} =1_{\bool{B}}$. Given $\kappa\geq\bool{B}$ least regular cardinal in $V$ such that $\Qp{\dot{\kappa}\leq\check{\kappa}}=1_{\bool{B}}$ and $\bool{B}$ is $<\kappa$-CC, let $$H_{\dot{\kappa}}^\bool{B}=\bp{\tau\in V^{\bool{B}}\cap H_\kappa^V:
\Qp{ \tau\text{ has transitive closure of size less than }\dot{\kappa}}_\bool{B}=1_{\bool{B}}}$$
It can be shown that $\Qp{\tau_1\in\tau_2}_\bool{B}$, $\Qp{\tau_1=\tau_2}_{\bool{B}}$, $\Qp{\tau_1\subseteq\tau_2}_{\bool{B}}$ are well defined $\bool{B}$-valued relations on $H_{\dot{\kappa}}^\bool{B}$ making it a $\bool{B}$-valued model, the interpretation of all formulae follow the same rules given for $V^\bool{B}$, except that in evaluating quantifiers now we let $\sigma$ range just over the appropriate domain $H_{\dot{\kappa}}^\bool{B}$.
It is the case that for all $G$ $V$-generic for $\bool{B}$ $$H_{\dot{\kappa}}^\bool{B}[G]=\bp{\tau_G: \tau\in H_{\dot{\kappa}}^\bool{B}}=H_{\dot{\kappa}_G}^{V[G]},$$ i.e. $H_{\dot{\kappa}}^\bool{B}$ is a canonical family of $\bool{B}$-names to denote the $H_{\dot{\kappa}_G}^{V[G]}$ of the generic extension. A key property of $V^{\bool{B}}$ and and of the models $H_{\dot{\kappa}}^\bool{B}$ defined above is fullness:
A $\bool{B}$-valued model $\mathcal{M}$ for the signature $\mathcal{L}$ is *full* if for any $\mathcal{L}$-formula $\phi(x_0,\dots,x_n)$ and $\tau_1,\dots,\tau_n\in \mathcal{M}$ $$\Qp{\exists x\phi(x,\tau_1,\dots,\tau_n)}^{\mathcal{M}}_{\bool{B}}=\Qp{\phi(\sigma,\tau_1,\dots,\tau_n)}^{\mathcal{M}}_{\bool{B}}$$ for some $\sigma\in \mathcal{M} $.
$V^{\bool{B}}$ and $H_{\dot{\kappa}}^\bool{B}$ are full $\bool{B}$-valued model for any cba $\bool{B}$ and any $\dot{\kappa}\in V^{\bool{B}}$ such that $\Qp{\dot{\kappa}\text{ is a regular cardinal}}_{\bool{B}} =1_{\bool{B}}$.
See [@viale-notesonforcing Thm. 5.1.34] for the case of $V^{\bool{B}}$. The same proof can be easily adapted for $H_{\dot{\kappa}}^\bool{B}$ since all the predense subsets needed in the proof have size less than the $\kappa$ chosen for the definition of $H_{\dot{\kappa}}^\bool{B}$.
For any ultrafilter $G$ on $\bool{B}$ and $\mathcal{M}$ any structure among $V^{\bool{B}}$ or $H_{\dot{\kappa}}^\bool{B}$, $\mathcal{M}/_G$ stands for the class (or set) $\bp{[\tau]_G:\tau\in \mathcal{M}}$, where $[\tau]_G=\bp{\sigma\in V^{\bool{B}}: \Qp{\sigma=\tau}_\bool{B}\in G}$. We make $\mathcal{M}/_G$ a first order structure for the language $\bp{\in,\subseteq}$, letting $[\tau]_G\mathrel{R}/_G[\sigma]_G$ if and only if $\Qp{\tau\mathrel{R}\sigma}\in G$ for $R$ among $\in,\subseteq$.
The forcing theorem states that:
- [@viale-notesonforcing Thm 4.3.2, Thm 5.1.34] (Łoś theorem for full boolean valued models) For all ultrafilter $G$ on $\bool{B}$, $\tau_1,\dots,\tau_n\in V^{\bool{B}}$, and $\phi(x_1,\dots,x_n)$ $$(V^{\bool{B}}/G,\in/_G)\models\phi([\tau_1]_G,\dots,[\tau_n]_G)\text{ if and only if } \Qp{\phi(\tau_1,\dots,\tau_n)}_{\bool{B}}\in G.$$
- The same conclusion holds with $H_{\dot{\kappa}}^\bool{B}$ in the place of $V^{\bool{B}}$.
- [@viale-notesonforcing Thm. 5.2.3] Whenever $G$ is $V$-generic for $\bool{B}$ the map $$[\tau]_G\mapsto \tau_G=\bp{\sigma_G: \exists b\in G\,\ap{\sigma,b}\in\tau}$$ is the Mostowski collapse of the class $V^{\bool{B}}/G$ defined in $V[G]$ onto $V[G]$ and its restriction to $H_{\dot{\kappa}}^\bool{B}/_G$ maps the latter onto $H_{\dot{\kappa}_G}^{V[G]}$.
When $\bool{B}\in V$ is a $<\kappa$-cc complete boolean algebra, then $\Qp{\check{\kappa}\text{ is a regular cardinal}}=1_{\bool{B}}$. Therefore $H_{\check{\kappa}}^\bool{B}$ is a canonical set of $\bool{B}$-names which describes the $H_\kappa$ of a generic extension of $V$ by $\bool{B}$.
The choice to work with $H_{\dot{\kappa}}^\bool{B}$, instead of $V^\bool{B}$, is motivated also by the fact that the former is a set definable in $V$ using the parameters $\bool{B}$ and $\dot{\kappa}$, while the latter is just a definable class in parameter $\bool{B}$.
Having defined the structures we will be interested in (the structures $H_{\dot{\kappa}}^\bool{B}/_G$) we now turn to the definition of the relevant morphisms between them.
Given $i:\bool{B}\to\bool{C}$ complete homomorphism of complete boolean algebras, $i$ extends to a map $\hat{i}:V^{\bool{B}}\to V^{\bool{C}}$ defined by transfinite recursion by $$\hat{i}(\tau)=\bp{\ap{\hat{i}(\sigma),i(b)}: \,\ap{\sigma,b}\in\tau}.$$ Given $\tau_1,\dots,\tau_n\in V^{\bool{B}}$, $\phi(\tau_1,\dots,\tau_n)$ is generically absolute for $i$ if $$i(\Qp{\phi(\tau_1,\dots,\tau_n}_\bool{B})=\Qp{\phi(\hat{i}(\tau_1),\dots,\hat{i}(\tau_n)}_\bool{C}.$$
It is well known that $\Delta_1$-properties[^3] are generically absolute (see for example [@VIAAUDSTEBOOK Prop. 4.1.2]); but it can be argued that $\Sigma_1$-properties in real parameters are also generically absolute. Indeed, we can prove the following Lemma:
\[lem:Cohengen\] Assume that $\phi(x,y)$ is a $\Delta_1$-property. Let $i:\bool{B}\to\bool{C}$ be a complete homomorphism. Then $\exists x\phi(x,y)\wedge y\subseteq\hat{\omega}$ is generically absolute for $i$.
[@VIAMMREV Lemma 1.2] states that $H_{\omega_1}^M\prec_{\Sigma_1} N$ for any $M$ (eventually non-transitive) model of $\ZFC$ and any $N$ superstructure of $M$ obtained by forcing over $M$ (i.e for some $\bool{B}$ in $M$ such that $M$ models $\bool{B}$ is a complete boolean algebra, and some $G\in St(\bool{B})$, we have that $N=(V^{\bool{B}})^M/_G$). Apply the Lemma to the case $M=V^\bool{B}/_{i^{-1}[G]}$ and $N=V^{\bool{C}}/_G$ for any $G\in St(\bool{C})$. Then conclude by the forcing theorem.
The following is a major achievement of Woodin [@LARSON Thm 3.1.7], conveniently reformulated in a weaker form and in a slightly different terminology for the purposes of this paper.
\[thm:Woodingen\] In the presence of class many Woodin cardinals, the structures of the form $H_{\omega_1^{\bool{B}}}^\bool{B}/_G$ are all models of the theory $\text{Th}(H_{\omega_1}^V)$ with parameters for elements of $H_{\omega_1}^V$.
Model theoretic completions {#sec:modeltheoreticcompl}
===========================
In what follows we are interested in studying certain classes of first order structures in a given first order signature $\tau$; we will be interested just in theories consisting of sentences. To fix notation, if $T$ is a first order theory in the signature $\tau$, $\mathcal{M}_T$ denotes the $\tau$-structures which are models of $T$.
A theory $T$ is *model complete* if for all models $\mathcal{M}$ and $\mathcal{N}$ of $T$ we have that $\mathcal{M} \sqsubseteq \mathcal{N}$ ($\mathcal{M}$ is a substructure of $\mathcal{N}$) implies $\mathcal{M} \prec \mathcal{N}$ ($\mathcal{M}$ is an elementary substructure of $\mathcal{N}$).
Let $\tau$ be a first order signature and $T$ be a theory for $\tau$. Given two models $\mathcal{M}$ and $\mathcal{N}$ of a theory $T$
- $\mathcal{M}$ is *existentially closed* in $\mathcal{N}$ ($\mathcal{M} \prec_1 \mathcal{N}$) if the existential and universal formula with parameters in $\mathcal{M}$ have the same truth value in $\mathcal{M}$ and $\mathcal{N}$.
- $\mathcal{M}$ is existentially closed for $T$ if it is existentially closed in all its $\tau$-superstructures which are models of $T$.
$\mathcal{E}_T$ denotes the class of $\tau$-models which are existentially closed for $T$.
Note that in general models in $\mathcal{E}_T$ need not be models[^4] of $T$. Model completeness describes exactly when this is the case.
[@TENZIE Lemma 3.2.7] (Robinson’s test) Let $T$ be a theory. The following are equivalent:
1. $T$ is model complete.
2. $\mathcal{E}_T=\mathcal{M}_T$.
3. Each $\tau$-formula is equivalent, modulo $T$, to a universal $\tau$-formula.
Model completeness comes in pair with another fundamental concept which generalizes to arbitrary first order theories the relation existing between algebraically closed fields and commutative rings without zero-divisors. As a matter of fact, the case described below occurs when $T^*$ is the theory of algebraically closed fields and $T$ is the the theory of comutative rings with no zero divisors.
Given two theories $T$ and $T^*$, in the same language $\tau$, $T^*$ is the *model companion* of $T$ if the following conditions holds:
1. Each model of $T$ can be extended to a model of $T^*$.
2. Each model of $T^*$ can be extended to a model of $T$.
3. $T^*$ is model complete.
The model companion of a theory does not necessarily exist, but, if it does, it is unique.
[@TENZIE Thm. 3.2.9] A theory $T$ has, up to equivalence, at most one model companion $T^*$.
Different theories can have the same model companion, for example the theory of fields and the theory of commutative rings with no zero-divisors which are not fields both have the theory of algebraically closed fields as their model companion.
Using the fact that a theory $T$ is mutually consistent with its model companion $T^*$, i.e. the models of one theory can be extended to a model of the other theory and vice-versa, together with the fact that universal theories are closed under sub-models it is easy to show that a theory and its model companion agree on their universal sentences.
In what follows, given a theory $T$, $T_{\forall}$ denotes the collection of all $\Pi_1$-sentences which are logical consequences of $T$. Similarly $T_{\exists}$ and $T_{\forall \exists}$ denote, respectively, the $\Sigma_1$ and the $\Pi_2$-theorems of $T$.
Let $T$ be a first order theory. If its model companion $T^*$ exists, then
1. $T_{\forall} = T^*_{\forall}$.
2. $T^*$ is the theory of the existentially closed models of $T_{\forall}$.
3. $T^*$ is axiomatized by $T_{\forall \exists}$.
Possibly inspired by Cohen’s forcing method, Robinson introduced what is now called Robinson’s infinite forcing [@HIRWHE75]. In this paper we are interested in a slight generalization of Robinson’s definition which makes the class of models over which we define infinite forcing an additional parameter.
Given a class of structure $\mathcal{C}$ for a signature $\tau$, *infinite forcing for $\mathcal{C}$* is recursively defined as follows for a $\tau$-formula $\phi(x_1,\dots,x_n)$, a structure $\mathcal{M} \in \mathcal{C}$ with domain $M$ and $a_1,\dots,a_n\in M$:
- For $\phi(x_1,\dots,x_n)$ atomic, $\mathcal{M} \VDash_\mathcal{C}\varphi(a_1,\dots,a_n)$ if and only if $\mathcal{M} \models\varphi(a_1,\dots,a_n)$;
- $\mathcal{M} \VDash_\mathcal{C} \varphi(a_1,\dots,a_n)\land \psi(a_1,\dots,a_n)$ if and only if $\mathcal{M} \VDash_\mathcal{C} \varphi(a_1,\dots,a_n)$ and $\mathcal{M} \VDash_\mathcal{C} \psi(a_1,\dots,a_n)$;
- $\mathcal{M} \VDash_\mathcal{C} \varphi(a_1,\dots,a_n) \lor \psi(a_1,\dots,a_n)$ if and only if $\mathcal{M} \VDash_\mathcal{C} \varphi(a_1,\dots,a_n)$ or $\mathcal{M} \VDash_\mathcal{C} \psi(a_1,\dots,a_n)$;
- $\mathcal{M} \VDash_\mathcal{C} \forall x\varphi(x,a_1,\dots,a_n)$ if and only if (expanding $\tau$ with constant symbols for all elements of $M$) $\mathcal{M} \VDash_\mathcal{C} \varphi(a,a_1,\dots,a_n)$, for every $a \in M$;
- $\mathcal{M} \VDash_\mathcal{C} \neg \varphi(a_1,\dots,a_n)$ if and only if $\mathcal{N} \not\VDash_\mathcal{C}\varphi(a_1,\dots,a_n)$ for all $\mathcal{N}\in \mathcal{C}$ superstructures of $\mathcal{M}$.
Robinson’s infinite forcing consider only the case in which $\mathcal{C}=\mathcal{M}_T$. We are interested in considering Robinson’s infinite forcing also in case $\mathcal{C}$ is not of this type.
As in the case of Cohen’s forcing, this method produces objects that are generic. In this case generic models.
Given a class of structure $\mathcal{C}$ for a signature $\tau$ A structure $\mathcal{M}\in\mathcal{C}$ is *infinitely generic for $\mathcal{C}$* whenever satisfaction and infinite forceability coincide: i.e., for every formula $\varphi(x_1,\dots,x_n)$ and $a_1,\dots,a_n\in M$, we have $$\mathcal{M} \vDash \varphi(a_1,\dots,a_n) \iff \mathcal{M} \VDash_\mathcal{C} \varphi(a_1,\dots,a_n).$$ By $\mathcal{F}_\mathcal{C}$, we indicate the class of infinitely generic structures for $\VDash_{\mathcal{C}}$.
Generic structures capture semantically the syntactic notion of model companionship.
Let $T$ be a theory in a signature $\tau$. The following are equivalent:
1. $T^*$ exists.
2. $\mathcal{E}_T$ is an elementary class.
3. $\mathcal{F}_T$ is an elementary class.
4. $\mathcal{E}_T=\mathcal{F}_{\mathcal{M}_{T_\forall}}$ (i.e. the existentially closed structures for $T$ are the generic structures for Robinson’s infinite forcing applied to the class $\mathcal{M}_{T_\forall}$).
The model companion of set theory for the generic multiverse {#sec:modelcompsetth}
============================================================
We already outlined that the model completeness of a theory is sensitive to the language in which that theory is expressed. We now embark in the task of selecting the right first order language to use for the construction of the model companion of (extensions of) $\ZFC$. We will first argue that (at least for our purposes) this is neither the language $\{\in\}$ nor the language $\{\in,\subseteq\}$, even if these are the languages in which set theory is usually formalized in almost all textbooks.
As a preliminary result, we have that the model companion of $\ZF$ for the language $\{\in\}$ has been already fully described.
(Hirschfeld [@Hir Thm. 1, Thm. 5]) The universal theory of any $T\supseteq \ZF$ in the signature $\bp{\in}$ is the theory $$S = \bp{\forall x_1\dots\forall x_n(x_1\notin x_2\vee x_2\notin x_3\vee\dots\vee
x_{n-1}\notin x_n\vee x_n\notin x_1): \,n\in\mathbb{N}}.$$ Letting for $A\subseteq n$ $$\delta_A(x_1,\dots,x_n,y)=\bigwedge_{i\in A} x_i\in y\wedge\bigwedge_{i\not\in A} x_i\not\in y,$$ the model companion of $\ZF$ is the theory $$\begin{aligned}
S^*=&\bp{\forall x_1\dots x_n \exists y\,\delta_A(x_1,\dots,x_n,y): n\in\omega, \,A\subseteq n}\cup\\
&\cup\bp{\forall x,y\,\exists z[x=y\vee (x\in z\wedge z\in y)\vee(y\in z\wedge z\in x)]}.\end{aligned}$$
In particular $S^*$ is also the model companion of $\ZFC$, given that $S$ is the universal theory of any $T\supseteq\ZF$, among which $\ZFC$.
Notice that $S$ only says that the graph of the $\in$-relation has no loops, while Hirschefeld also shows that in every model of $S^*$ the interpretation of $\in$ defines a dense linear order without endpoints [@Hir Thm. 3]. In particular there is no apparent relation between the meaning of the $\in$-relation in a model of $\ZF$ (in its standard models it is a well-founded relation not linearly ordered) and the meaning of the $\in$-relation in models of $S^*$ (it is a dense linear order without end-points).
We believe (as Hirschfeld) that the above result gives a clear mathematical insight of why the language $\bp{\in}$ is not expressive enough to describe the “right” model companion of set theory. A key issue is the following: we are inclined to consider concepts and properties which can be formalized by formulae with bounded quantifiers much simpler and concrete than those which can only be formalized by formulae which make use of unrestricted quantification. This is reflected by the fact that properties formalizable by means of formulae with bounded quantifiers are absolute between transitive models of $\ZFC$. This fact fails badly for properties defined by means of unbounded quantification.
For example the property *$f$ is a function* is expressible using only bounded quantification, while the property *$\kappa$ is a cardinal* is not. It is well known that the former is a property that is absolute between transitive models of $\ZFC$ containing $f$, while the latter is not. It is also a matter of fact that absolute properties are regarded as “tame” set theoretic properties (as their truth value cannot be changed by forcing, e.g *$f$ is a function* remains true in any transitive model to which $f$ belongs), while non-absolute ones are more difficult to control (they are not immune to forcing, e.g whenever $\kappa$ is an uncountable cardinal of the ground model, it will not be anymore so in a generic extension by $\Coll(\omega,\kappa)$).
Hence it is necessary to formalize set theory in a first order language able to recognize syntactically the different semantic complexity of absolute and non-absolute concepts. As Hirschfeld has shown this is not the case for the $\ZF$-axioms in the language $\bp{\in}$. In Kunen and Jech’s books the solution adopted is that of passing from first order logic to a logic with bounded quantifiers $\exists x\in y$ and $\forall x\in y$ binding the variable $x$ so that $\exists x\in y\phi(x,y,\vec{z})$ is logically equivalent to $\exists x(x\in y\wedge\phi(x,y,\vec{z}))$ and $\forall x\in y\phi(x,y,\vec{z})$ is logically equivalent to $\forall x(x\in y\rightarrow\phi(x,y,\vec{z}))$. In this new logic *$f$ is a function* is expressible by a formula with only bounded quantifiers, while *$\kappa$ is a cardinal* is expressible by a formula of type $\forall x\phi(x,\kappa)$ with $\phi$ having only bounded quantifiers. On the other hand Jech and Kunen’s solution is not convenient for the scopes of this paper, because it formalizes set theory outside first order logic, making less transparent how we could use model theoretic techniques (designed expressly for first order logic) to isolate what is the correct model companion of set theory. The alternative solution we adopt in this paper is that of expressing set theory in a first order language with relation symbols for any bounded formula.
Given the first order signature $$\mathcal{L}^*=\bp{R_\phi: \phi\text{ logically equivalent to a bounded formula in the signature $\bp{\in}$}},$$ $\ZFC^*$ is the $\mathcal{L}^*$-theory obtained adding to $\ZFC$ the axioms $$\forall\vec{x}(\phi(\vec{x})\leftrightarrow R_\phi(\vec{x}))$$ for all formulae $\phi(\vec{x})$ logically equivalent to a bounded formula.
In $\ZFC^*$ we now obtain that many absolute concepts (such as that of being a function) are now expressed by an atomic formula, while certain more complicated ones (for example those defined by means of transfinite recursion over an absolute property, such as *$x$ is the transitive closure of $y$*) can still be expressed by means of $\Delta_1$-properties of $\mathcal{L}^*$ (i.e. properties which are formalized at the same time by a $\Pi_1$-formula and by a $\Sigma_1$-formula), hence are still absolute between any two models (even non-transitive) $\mathcal{M}$, $\mathcal{N}$ of $\ZFC^*$ of which one is a substructure of the other. On the other hand many definable properties have truth values which may vary depending on which model of $\ZFC^*$ we work in (for example *$\kappa$ is an uncountable cardinal* is a $\Pi_1\setminus\Sigma_1$-property in $\ZFC^*$ whose truth value may depend on the choice of the model of $\ZFC^*$ to which $\kappa$ belongs).
Our first aim is to identify what is $\ZFC^*_\forall$. To this aim, first recall that Levy’s absoluteness gives that $H_{\omega_1}\prec_{\Sigma_1}V$, and that for any set $X$ there is a forcing extension in which $X$ is countable (just force with $\Coll(\omega,X)$). In particular one can argue that the $\Pi_2$-assertion $\forall X\exists f:\omega\to X\emph{ surjectve}$ is generically true for Robinson’s infinite forcing applied to the forcing extensions of $V$. Notice that $H_{\omega_1}\models\forall X\exists f:\omega\to X\emph{ surjectve}$.
The natural conjecture is to infer that the first order theory of $H_{\omega_1}$ is the model companion of the first order theory of $V$. We now show exactly to which extent the conjecture is true, while proving that it is false.
We first relativize the notion of model completeness to this new setting.
Given a theory $T$ and a category $(\mathbb{M},\to_{\mathbb{M}})$ with $\mathbb{M}$ a class of models of $T$ and $\to_{\mathbb{M}}$ a class of morphisms between them, $T$ is *model complete with respect to $\mathbb{M}$* if for all models $\mathcal{M}$ and $\mathcal{N}$ in $\mathbb{M}$ we have that $\mathcal{M} \prec_1 \mathcal{N}$, whenever there is a morphism $f: \mathcal{M} \to \mathcal{N}$ in $\to_{\mathbb{M}}$.
In order to define the class of structures and morphisms $\mathbb{M}$ we need the following useful results (see for example [@VIAAUDSTEBOOK Prop. 4.1.2])
Let $\bool{B}$ and $\bool{C}$ be complete boolean algebras.
1. Given $k:\bool{B}\to\bool{C}$ complete homomorphism of complete boolean algebras, define $$\hat{k}:V^{\bool{B}}\to V^{\bool{C}}$$ by transfinite recursion letting $$\hat{k}(\sigma)=\bp{\ap{\hat{k}(\tau),k(b)}: \ap{\tau,b}\in\sigma}.$$
Then for any $\Delta_1$-property $P(x_1,\dots,x_n)$ in $\mathcal{L}^*$ and every $\tau_1,\dots,\tau_n\in V^{\bool{B}}$ $$k(\Qp{P(\tau_1,\dots,\tau_n)}_\bool{B})=\Qp{P(\hat{k}(\tau_1),\dots,\hat{k}(\tau_n))}_\bool{C}.$$
2. Moreover whenever $f:\bool{B}\to\bool{C}$ is a complete homomorphism, for any $H\in St(\bool{C})$ such that such that $\Qp{\hat{f}(\dot{\kappa})\leq\dot{\delta}}_{\bool{C}}\in H$, letting $G\in St(\bool{B})$ be $f^{-1}[H]$, the map $$\begin{aligned}
\hat{f}/_H:&H_{\dot{\kappa}}^{\bool{B}}/_G\to H_{\dot{\delta}}^{\bool{C}}/_H\\
&[\tau]_G\mapsto [\hat{f}(\tau)]_H\end{aligned}$$ is an $\mathcal{L}^*$-morphism.
The *generic multiverse* $(\Omega(V),\to_{\Omega(V)})$ is the collection: $$\bp{H_{\dot{\kappa}}^{\bool{B}}/_G:\,\Qp{\dot{\kappa}\text{ is a regular cardinal}}_{\bool{B}}=1_{\bool{B}},\,G\in St(\bool{B})}.$$ its morphism are the $\mathcal{L}^*$-morphisms of the form $\hat{f}/_H:H_{\dot{\kappa}}^{\bool{B}}/_G\to H_{\dot{\delta}}^{\bool{C}}/_H$ for some complete homomorphism $f:\bool{B}\to\bool{C}$ with $H\in St(\bool{C})$, $G=f^{-1}[H]$, $\Qp{\hat{f}(\dot{\kappa})\leq\dot{\delta}}_{\bool{C}}\in H$.
Notice[^5] that $\Omega(V)$ is a definable class in $V$. $\Omega(V)$ is a formulation in the language of boolean valued models of the notion of generic multiverse.
This is the first result we want to bring forward:
\[thm:omegaVmodcompl\] The first order theory with parameters for elements of $H_{\omega_1}^V$ of the $\mathcal{L}^*$-structure $(H_{\omega_1}^V,R_\phi^V: R_\phi\in\mathcal{L}^*)$ in the signature $\mathcal{L}^*\cup H_{\omega_1}$ is the model companion of $\ZFC^*+$ *there exist class many Woodin cardinals* with respect to $(\Omega(V),\to_{\Omega(V)})$.
We prove the Theorem in a series of lemmas. By $\omega_1^{\bool{B}}$ we denote a $\bool{B}$-name such that $$\llbracket \omega_1^{\bool{B}}\text{ is the first uncountable cardinal}\rrbracket_{\bool{B}}=1_{\bool{B}}.$$
\[lem:Cohen\] $H_{\omega_1^{\bool{B}}}^\bool{B}/_G$ is existentially closed with respect to its superstructures in $\Omega(V)$.
It is a reformulation of Cohen’s absoluteness Lemma, that is Lemma \[lem:Cohengen\].
\[fac:embed\] Given any structure in $\Omega(V)$, there is a natural morphism that embeds it in a structure of the form $H_{\omega_1^{\bool{C}}}^\bool{C}/_G$.
Let $H_{\dot{\kappa}}^{\bool{B}}/_G\in \Omega(V)$. Find a regular $\delta>2^\kappa$ and consider the forcing notion $\Coll(\omega,<\delta)$. By a classical forcing result (see for example [@JECH Lemma 26.9]), we have that $\bool{B}$ is isomorphic to a complete sub-algebra of the boolean completion $\bool{C}$ of $\Coll(\omega,<\delta)$, i.e there is a (even injective) complete homomorphism $f:\bool{B}\to\bool{C}$.
Moreover it is well known (see [@JECH Thm. 15.17(iii)]) that $$\Qp{\check{\delta}\text{ is the first uncountable cardinal}}_{\bool{C}}=1_{\bool{C}}.$$ Hence $\omega_1^\bool{C}=\check{\delta}$.
Extend the prefilter $f[G]$ on $\bool{C}$ to an ultrafilter $H$ on $\bool{C}$. Then $f^{-1}[H]=G$. Since $H_{\check{\delta}}^{\bool{C}}=H_\delta \cap V^{\bool{C}}$ (by the $<\delta$-CC of $\bool{C}$), and $\Qp{\hat{f}(\dot{\kappa})<\check{\delta}}_{\bool{C}}=1_{\bool{C}}\in H$, $\hat{f}[H_{\dot{\kappa}}^{\bool{B}}]\subseteq H_{\check{\delta}}^{\bool{C}}$. Hence $\hat{f}/_H$ is a morphism in $\to_{\Omega(V)}$.
This completes the proof of Theorem \[thm:omegaVmodcompl\]: by Theorem \[thm:Woodingen\] the models of $\text{Th}(H_{\omega_1}^V)$ with parameters for elements of $H_{\omega_1}^V$ in $\Omega(V)$ are of the form $H_{\omega_1^\bool{B}}^\bool{B}/_G$. By Lemma \[lem:Cohen\] $\text{Th}(H_{\omega_1})$ is model complete with respect to $\Omega(V)$. Finally Fact \[fac:embed\] provides the mutual consistency between arbitrary models in $\Omega(V)$ and models of $\text{Th}(H_{\omega_1})$ in $\Omega(V)$.
Two natural questions arise:
- is the $\mathcal{L}^*$-theory $T=\mathrm{Th}(\ap{H_{\omega_1}^V,R_\phi^V:R_\phi\in\mathcal{L}^*,H_{\omega_1}})$ model complete?
- Can we embed any set sized model $\mathcal{L}^*$-model of $S=\mathrm{Th}(\ap{V,R_\phi^V:R_\phi\in\mathcal{L}^*,H_{\omega_1}})$ into some model of $\Omega(V)$ and conversely?
If we could answer positively to both questions we would have that $T$ is the model companion of $\mathrm{Th}(\ap{V,R_\phi^V:R_\phi\in\mathcal{L}^*,H_{\omega_1}})$, since $H_{\omega_1}$ is $\Sigma_1$-elementary in $V$ with respect to $\mathcal{L}^*$, hence the two structures have the same universal theory and we can apply Robinson’s test to the two theories.
In the forthcoming [@VIAPAR] the second author and Parente show that the answer to the second question is positive (assuming large cardinal axioms). This is already quite interesting: it outlines that any set sized $\mathcal{L}^*$-model of the theory of (an initial segment of) $V$ (obtained by whatever means model theory gives us) is in fact a substructure of a $\mathcal{L}^*$-model of the theory of (an initial segment of) $V$ obtained by forcing.
Nonetheless in the next section we argue that the first question has a negative answer. This will bring us to further expand the language of set theory, including predicates for universally Baire sets, in order to argue that Woodin’s generic absoluteness results for this type of sets bring, as a byproduct, the model completeness of the theory of $H_{\omega_1}$ with predicates for the universally Baire sets.
Second order arithmetic and $\text{Th}(H_{\omega_1})$ {#sec:modcompletUBpred}
=====================================================
We define second order number theory as the first order theory of the structure $$(\mathcal{P}(\mathbb{N})\cup\mathbb{N},\in,\subseteq,=,\mathbb{N}).$$
$\Pi^1_n$-sets (respectively $\Sigma^1_n$, $\Delta^1_n$) are the subsets of $\mathcal{P}(\mathbb{N})\equiv 2^{\mathbb{N}}$ defined by a $\Pi_n$-formula (respectively by a $\Sigma_n$-formula, at the same time by a $\Sigma_n$-formula and a $\Pi_n$-formula in the appropriate language), if the formula defining a set $A\subseteq (2^{\mathbb{N}})^n$ has some parameter $r\in 2^{\mathbb{N}}$ we accordingly say that $A$ is $\Pi^1_n(r)$ (respectively $\Sigma^1_n(r)$, $\Delta^1_n(r)$).
Given $a\in H_{\omega_1}$, $r\in 2^{\mathbb{N}}$ codes $a$, if (modulo a recursive bijection of $\mathbb{N}$ with $\mathbb{N}^2$), $r$ codes a well-founded extensional relation on $\mathbb{N}$ whose transitive collapse is the transitive closure of $\{a\}$.
- $\mathrm{Cod}:2^{\mathbb{N}}\to H_{\omega_1}$ is the map assigning $a$ to $r$ if and only if $r$ codes $a$ and assigning the emptyset to $r$ otherwise.
- $\mathrm{WFE}$ is the set of $r\in 2^{\mathbb{N}}$ which (modulo a recursive bijection of $\mathbb{N}$ with $\mathbb{N}^2$) are a well founded extensional relation.
The following are well known facts[^6].
The map $\mathrm{Cod}$ is defined by a provably $\Delta_1$-property over $H_{\omega_1}$ and is surjective. Moreover $\mathrm{WFE}$ is a $\Pi^1_1$-subset of $2^{\mathbb{N}}$.
Assume $A\subseteq 2^{\mathbb{N}}$ is $\Sigma^1_{n+1}$. Then $A$ is $\Sigma_{n}$-definable in $H_{\omega_1}$ in the language $\mathcal{L}^*$.
Assume $A$ is $\Sigma_n$-definable in $H_{\omega_1}$ in the language $\mathcal{L}^*$. Then $A=\textrm{Cod}^{-1}[\textrm{Cod}[A]]$, and $\textrm{Cod}[A]$ is $\Sigma^1_{n+1}$.
We can now easily conclude the following:
$T=\text{Th}(\ap{H_{\omega_1},R_\phi^V:R_\phi\in\mathcal{L}^*,H_{\omega_1}})$ is not model complete.
For all $n$ there is some $A_n\in \Sigma^1_{n+1}\setminus\Pi^1_n$ (see for a proof [@kechris:descriptive Thm. 22.4]). Therefore $A_2$ is $\Sigma_2$-definable but not $\Pi_1$-definable in $H_{\omega_1}$. Consequently, Robinson test fails, and $T$ is not model complete.
Model completeness for set theory with predicates for universally Baire sets {#sec:modcompanUBpred}
============================================================================
Given a topological space $(X,\tau)$, $A\subseteq X$ is nowhere dense if its closure has a dense complement, meager if it is the countable union of nowhere dense sets, with the Baire property if it has meager symmetric difference with an open set.
(Feng, Magidor, Woodin) $A\subseteq 2^{\mathbb{N}}$ is *universally Baire* if for every compact Hausdorff space $X$ and every continuous $f:X\to 2^{\mathbb{N}}$ we have that $f^{-1}[A]$ has the Baire property in $X$.
\[thm:UBsetsgenabs\] Let $T$ be the $\mathcal{L^*}$-theory $\mathsf{ZFC^*}+$*there are class many Woodin cardinals*.
1. [@LARSON Thm. 3.3.9, Thm. 3.3.14] Assume $V$ models $T$. Then every set of reals in $L(\mathbb{R})$ is universally Baire.
2. [@LARSON Thm. 3.4.17] Assume $V\models T$ and is obtained as a generic extension of $W$ such that for some $\delta$ which is supercompact in $W$, we have that $(2^\delta)^W$ is countable in $V$. Let $\mathsf{UB}$ be the family of universally Baire sets in $V$. Then every subset of $2^{\mathbb{N}}$ in $L(\mathsf{Ord}^\omega,\mathsf{UB})^V$ is universally Baire.
Let $T$ be the theory $\ZFC^*+$*there are class many Woodin cardinals*. Assume $V$ models $T$ and condition (2) of Thm. \[thm:UBsetsgenabs\] holds. Let $\mathcal{L}^{**}=\mathcal{L}^*\cup\bp{B:\overline{B}\in\mathsf{UB}}$. Then the $\mathcal{L}^{**}$-theory $T_1$ of $$\mathcal{M}=(H_{\omega_1},\overline{R}_\phi:\text{ $\phi$ bounded},\overline{B}: \overline{B}\in\mathsf{UB})$$ is model complete.
Let $A\subseteq H_{\omega_1}$ be defined as the extension in $\mathcal{M}$ of some $\mathcal{L}^{**}$-formula $\phi(x,r_1,\dots,r_n)$ with $r_i\in 2^{\mathbb{N}}$.
Then $B=\mathrm{Cod}^{-1}[A]\cap \mathsf{WFE}$ is a definable subset of $2^{\mathbb{N}}$ in $$(H_{\omega_1},\overline{R}_\phi:\text{ $\phi$ bounded},\overline{B}: \overline{B}\in\mathsf{UB}),$$ hence it belongs to $L(\mathsf{Ord}^\omega,\mathsf{UB})$, therefore $B\in\mathsf{UB}$.
Now $$A=\bp{a\in H_{\omega_1}: \forall y (\langle y,a\rangle \in\mathrm{Cod}\rightarrow y\in B)}.$$ Since $(\langle y,a\rangle\in\mathrm{Cod})$ can be expressed by a $\Sigma_1$-formula in the structure $$(H_{\omega_1},\overline{R}_\phi:\text{ $\phi$ bounded},\,\overline{B}:\overline{B}\in\mathsf{UB}),$$ we have that $A$ is the extension of a $\Pi_1$-formula $\psi(x)$ using the universally Baire predicate $B$ in the structure $$(H_{\omega_1},\overline{R}_\phi:\text{ $\phi$ bounded},\,\overline{B}:\overline{B}\in\mathsf{UB}).$$ By the third criterion of Robinson’s test we conclude that $T_1$ is model complete.
\[thm:Vmodcompl\] Let $T$ be the theory $\mathsf{ZFC}+$*there are class many Woodin cardinals*. Assume $V$ models $T$ and condition (2) of Thm. \[thm:UBsetsgenabs\] holds. Let:
- $T_0$ be the $\mathcal{L}^{**}$-theory of $V$ with parameters in $H_{\omega_1}$ and predicates for all elements of $\mathsf{UB}$,
- $T_1$ be the $\mathcal{L}^{**}$-theory with parameters of $$(H_{\omega_1},\overline{R}_\phi:\text{ $\phi$ bounded},\overline{B}: \overline{B}\in\mathsf{UB}).$$
Then $T_1$ is the model companion of $T_0$.
By (a slight variation of the proof of) Levy’s absoluteness we have that $$(H_{\omega_1},\overline{R}_\phi:\text{ $\phi$ bounded},\,\overline{B}:\overline{B}\in\mathsf{UB})
\prec_1
(V,\overline{R}_\phi:\text{ $\phi$ bounded},\,\overline{B}:\overline{B}\in\mathsf{UB}).$$ In particular $T_1$ and $T_0$ satisfy the same universal sentences.
It is now a standard result in model theory [@TENZIE Lemma 3.1.2] that in this case it is possible to embed any model $\mathcal{M}$ of each theory into some model $\mathcal{N}$ of the other theory by choosing $\mathcal{N}$ saturated enough so to realize all existential types of $\mathcal{M}$. The conclusion follows by model completeness of $T_1$.
Minimal variations of the above argument yield the following result:
Let $T$ be the theory $\mathsf{ZFC}+$*there are class many Woodin cardinals*. Assume $V$ models $T$ and condition (1) of Thm. \[thm:UBsetsgenabs\] holds. Let:
- $T_0$ be the $\mathcal{L}^{**}$-theory of $V$ with parameters in $H_{\omega_1}$ and predicates for all sets of reals definable in $L(\mathbb{R})$,
- $T_1$ be the $\mathcal{L}^{**}$-theory with parameters of $$(H_{\omega_1},\overline{R}_\phi:\text{ $\phi$ bounded},\overline{B}: \overline{B}\in L(\mathbb{R})\cap \pow{2^\omega}).$$
Then $T_1$ is the model companion of $T_0$.
Model completeness for the theory of $H_{\omega_2}$ assuming forcing axiom and for the theory of $V$ assuming $V=L$ {#sec:modcomplforcax}
===================================================================================================================
We can show that mild forcing axioms such as the bounded proper forcing axiom $\bool{BPFA}$ already entail model completeness for the $\mathcal{L}^*$-theory of $H_{\omega_2}$ expanded by absolutely definable Skolem functions. Similarly we will argue that $\ZFC^*+V=L$ is model complete in the appropriate natural language. This is a rather straightforward consequence of the existence of simply definable well-orders of $H_{\omega_2}$ in the presence of forcing axioms and of a simply definable well order of $L$. We investigte in some details the model completeness of the theory of $H_{\omega_2}$ assuming forcing axioms and brieflly discuss the model completeness of $\ZFC+V=L$ in the appropriate natural language.
We will use the following result:
[@CAIVEL06 Thm. 2] Assume $\BPFA$ and let $A\subseteq\omega_1$ be a ladder system on $\omega_1$. There is a $\ZF\setminus\textrm{Power-set}$-provably $\Delta_1$-definable property $P(x,y,z)$ in the signature $\mathcal{L}^*$ such that $P(x,y,A)$ provides a well-order of $H_{\omega_2}$ in type $\omega_2$.
We now expand $\mathcal{L}^*$ to the signature $\mathcal{L}^{**}$ obtained adding constant symbols for $\omega$, and a function symbol $f_\phi$ of ariety $n_\phi$ for each $R_\phi\in\mathcal{L}^*$ of ariety $n_\phi+1$. We then extend $\ZFC^*$ to a $\mathcal{L}^{**}\cup\bp{\omega_1, A,H_{\omega_2}}$-theory (with new constant symbols $\omega_1,A,H_{\omega_2}$) obtained by adding:
- The axiom (expressible in the signature $\mathcal{L}^*\cup\bp{H_{\omega_2},\omega_1,\omega}$) stating that $\omega_1$ is the first uncountable cardinal $$\forall f (f\text{ is a function with domain $\omega$}\rightarrow \omega_1\not\subseteq\ran(f)).$$
- The axiom (expressible in the signature $\mathcal{L}^*\cup\bp{H_{\omega_2},\omega_1,\omega}$) stating that $H_{\omega_2}$ is the set of all sets with transitive closure of size $\omega_1$ $$\forall x (x\in H_{\omega_2}\leftrightarrow x\emph{ has transitive closure of size at most $\omega_1$})$$ (remark that $x$ *has transitive closure of size at most $\omega_1$* is a $\Delta_1(\omega,\omega_1)$-property for $\ZFC^*$).
- The axiom (expressible in the signature $\mathcal{L}^*\cup\bp{A,\omega_1,\omega}$) $$\text{\emph{$A\subseteq\omega_1$ codes a ladder system on $\omega_1$}.}$$ (A ladder system on $\omega_1$ is a sequence $\ap{C_\alpha:\alpha<\omega_1}$ such that $C_\alpha\subseteq\alpha$ and $C_\alpha$ is cofinal in $\alpha$ of order type $\omega$ whenever $\alpha$ is a limit ordinal).
- The axioms (expressible in the signature $\mathcal{L}^{**}\cup\bp{H_{\omega_2},\omega_1,\omega}$) $$\begin{aligned}
\forall x_1\dots x_{n_\phi}[&\\
&(\bigwedge_{i=1}^n x_i\in H_{\omega_2})\rightarrow&\\
&[\\
&(\forall y\neg R_\phi(y,x_1,\dots,x_{n_\phi})\wedge f_\phi(x_1,\dots,x_{n_\phi})=A) \\
&\vee\\
&(R_\phi(f_\phi(x_1,\dots,x_{n_\phi}),x_1,\dots,x_{n_\phi})\wedge
\forall u (P(u,f_\phi(x_1,\dots,x_{n_\phi}),A)\rightarrow \neg R_\phi(u,x_1,\dots,x_{n_\phi})))\\
&]\\
]\end{aligned}$$ stating that for any $x_1\dots,x_n\in H_{\omega_2}$ $f_\phi(x_1,\dots,x_n)$ is the least element $y$ such that $R_\phi(y,x_1,\dots,x_n)$ according to the well-order of $H_{\omega_2}$ defined by $P(x,z,A)$ (if such a $y$ exists), and is $A$ otherwise.
Remark that all the above axioms are universal statements in the language $\mathcal{L}^{**}$. We can immediately prove the following:
\[ref:H2modcompl\] Let $T$ be any complete extension of $\ZFC^{**}+\bool{BPFA}$ and $M$ a model of $T$. Let $S$ be the $\mathcal{L}^{**}$-theory of the structure $H_{\omega_2}^M$. Then $S$ is the model companion of $T$.
The axioms added to $\mathcal{L}^{**}$ yield that $H_{\omega_2}^M$ satisfies $$\begin{aligned}
\forall x_1\dots x_{n_\phi}[&\\
&(\forall y\neg R_\phi(y,x_1,\dots,x_{n_\phi})\wedge f_\phi(x_1,\dots,x_{n_\phi})=A) \\
&\vee\\
&(R_\phi(f_\phi(x_1,\dots,x_{n_\phi}),x_1,\dots,x_{n_\phi})\wedge
\forall u (P(u,f_\phi(x_1,\dots,x_{n_\phi}),A)\rightarrow \neg R_\phi(u,x_1,\dots,x_{n_\phi})))\\
]\end{aligned}$$ for all $R_\phi\in\mathcal{L}^*$. Therefore $S$ admits quantifier elimination, and is a universal $\mathcal{L}^{**}$-theory, by well known standard results on the Skolemization of first order theories (see for example [@TENZIE Thm. 5.1.8, and proof of the Claim in Cor. 5.1.9]). We conclude that $S$ is model complete (by quantifier elimination any substructure of a model $N$ of $S$ which is itself a model of $S$ is an elementary substructure of $N$). Since $H_{\omega_2}^M$ and $M$ satisfy the same universal $\mathcal{L}^{*}$-sentences[^7], we conclude by Robinson’s test.
We have a series of remarks to make.
The above result does not say that $\ZFC^{**}+\bool{BPFA}$ has a model companion. For example assume $M\models\bool{BMM}+$*there exists a reflecting cardinal $\delta$*. Let $N$ be the generic extension of $M$ by standard proper forcing of length $\delta$. Then $M$ and $N$ are both models of $\ZFC^{**}+\bool{BPFA}$ (since $\bool{BMM}$ implies $\bool{BPFA}$). On the other hand in $H_{\omega_2}^M$ it holds that the family of canonical functions on $\omega_1$ is dominating modulo club, while $H_{\omega_2}^N$ models that this family is not dominating. Hence $H_{\omega_2}^M$ is an $\mathcal{L}^{**}$-substructure of $H_{\omega_2}^N$ which is not elementary. The result just says that any *complete extension of* $\ZFC^{**}+\bool{BPFA}$ has a model companion.
In case one assumes $V=L$ we can produce a more constructive result:
Consider the language $\mathcal{L}^{*}\cup\bp{<}$, with $<$ a binary relation symbol. Let $\psi$ be the $\mathcal{L}^{*}\cup\bp{<}$-sentence asserting that $<$ defines one of the provably $\Delta_1$-definable well-order of $L$. The theory $\ZFC^*+V=L+\psi$ is model complete with respect to $\mathcal{L}^{**}\cup\bp{<}$.
We leave the details to the reader; the key fact is that there are $\Delta_1$-definable Skolem functions selecting witnesses for $\Delta_0$-properties by means of a canonical $\Delta_1$-definable well-order of $L$. This gives that in this language the theory of $\ZFC+V=L+\psi$ can be expressed by the existence of Skolem witnesses for all $\Delta_0$-properties. Such axioms are defined by $\Pi_1$-properties of the new language of the form: $$\begin{aligned}
\forall x_1\dots x_{n_\phi}[&\\
&(\forall y\neg R_\phi(y,x_1,\dots,x_{n_\phi})\wedge f_\phi(x_1,\dots,x_{n_\phi})=\emptyset) \\
&\vee\\
&(R_\phi(f_\phi(x_1,\dots,x_{n_\phi}),x_1,\dots,x_{n_\phi})\wedge
\forall u ((u<f_\phi(x_1,\dots,x_{n_\phi}))\rightarrow \neg R_\phi(u,x_1,\dots,x_{n_\phi})))\\
]\end{aligned}$$ for all $R_\phi\in\mathcal{L}^*$.
[^1]: It is a standard result of set theory that $\Delta_0$-formulae define absolute properties for transitive models of $\ZFC$. On the other hand the notion of universal Baireness captures exactly those sets of reals whose first order properties cannot be changed by means of forcing (for example all Borel sets of reals are universally Baire). Therefore these predicates have a meaning which is clear across the different models of set theory. We do not expand further on this matter here, we just remark that: on the one hand a fine classification of which sets of reals are universally Baire and which are not would bring us into rather delicate grounds; on the other hand the results of this paper are based on the closure under first order definability of the class of universally Baire sets (i.e. closure under projections, finite intersections, finite unions, complementation), which is the case if we assume the existence of class many Woodin cardinals.
[^2]: All problems of second order arithmetic are first order properties of $H_{\omega_1}$.
[^3]: I.e. properties which are extension at the same time of a $\Pi_1$-formula and of a $\Sigma_1$-formula
[^4]: For example let $T$ be the theory of commutative rings with no zero divisors which are not algebraically closed fields. Then $\mathcal{E}_T$ is exactly the class of algebraically closed fields and no model in $\mathcal{E}_T$ is a model of $T$.
[^5]: There can be morphisms $h:H_\kappa^{\bool{B}}/_G\to H_\delta^{\bool{C}}/_H$ which are not of the form $\hat{f}/_H$ for some complete homomorphism $f:\bool{B}\to \bool{C}$, even in case $\bool{B}$ preserve the regularity of $\kappa$ and $\bool{C}$ the regularity of $\delta$. We do not spell out the details of such possibilities.
[^6]: See [@JECH Section 25] and in particular the statement and proof of Lemma 25.25, which contains all ideas on which one can elaborate to draw the conclusions below.
[^7]: Notice that $T$ does not admit quantifier elimination because the Skolemization fails for the $n_\phi$-tuples $x_1,\dots,x_{n_\phi}$ which are not all in $H_{\omega_2}$.
|
---
abstract: 'Twisted $T$-adic exponential sums are studied. The Hodge bound for the $T$-adic Newton polygon of the $C$-function is established. As an application, the behavior of the $L$-function under diagonal base change is explicitly given.'
address: 'Department of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai 200240, P.R. China, E-mail: clliu@sjtu.edu.cn'
author:
- Chunlei Liu
title: '$T$-adic exponential sums under diagonal base change'
---
Introduction
============
Preliminaries
-------------
Let $\mathbb{F}_q$ be the field of characteristic $p$ with $q$ elements, and $\mathbb{Z}_q=W(\mathbb{F}_q)$. Let $T$ and $s$ be two independent variables. In this subsection we are concerned with the ring $\mathbb{Z}_q[[T]][[s]]$, elements of which are regarded as power series in $s$ with coefficients in $\mathbb{Z}_q[[T]]$.
Let $\mathbb{Q}_p=\mathbb{Z}_p[\frac1p]$, $\overline{\mathbb{Q}}_p$ the algebraic closure of $\mathbb{Q}_p$, and $\widehat{\overline{\mathbb{Q}}}_p$ the $p$-adic completion of $\overline{\mathbb{Q}}_p$.
A (vertical) specialization is a morphism $T\mapsto t$ from $\mathbb{Z}_q[[T]]$ into $\widehat{\overline{\mathbb{Q}}}_p$ with $0\neq|t|_p<1$.
We shall prove the vertical specialization theorem.
Let $A(s,T)\in 1+s\mathbb{Z}_q[[T]][[s]]$ be a $T$-adic entrie series in $s$. If $0\neq|t|_p<1$, then $$t-adic\text{ NP of
}A(s,t)\geq T-adic\text{ NP of }A(s,T),$$where NP is the short for Newton polygon. Moreover, the equality holds for one $t$ iff it holds for all $t$.
By the vertical specialization, the Newton polygon of a $T$-adic entire series in $1+s\mathbb{Z}_q[[T]][[s]]$ goes up under vertical specialization, and is stable under all specializations if it is stable under one specialization.
A $T$-adic entire series in $1+s\mathbb{Z}_q[[T]][[s]]$ is said to be stable if its Newton polygon is stable under specialization.
If $$A(s,T)=\exp(-\sum\limits_{k=1}^{+\infty}a_k(T)\frac{s^k}{k}),$$ and $$B(s,T)=\exp(-\sum\limits_{k=1}^{+\infty}b_k(T)\frac{s^k}{k}),$$ we define $$A\otimes B(s,T)=\exp(-\sum\limits_{k=1}^{+\infty}a_k(T)b_k(T)\frac{s^k}{k}).$$
We have the distribution law $$(A_1A_2)\otimes B=(A_1\otimes B)(A\otimes B).$$ So, equipped with the usual multiplication and the new tensor operation, the set of $T$-adic entire series in $1+s\mathbb{Z}_q[[T]][[s]]$ becomes a ring. We shall prove that the stable $T$-adic entire series form a subring.
The set of stable $T$-adic entire series $1+s\mathbb{Z}_q[[T]][[s]]$ is closed under multiplication and tensor operation.
Twisted $T$-adic exponential sums
---------------------------------
In this subsection we introduce $L$-functions of twisted $T$-adic exponential sums. The theory of $T$-adic exponential sums without twists was developed by Liu-Wan [@LW].
Let $\mu_{q-1}$ be the group of $(q-1)$-th roots of unity in $\mathbb{Z}_q$, $\omega:x\mapsto\hat{x}$ the Teichmuller character of $\mathbb{F}_q^{\times}$ into $\mu_{q-1}$, $\chi=\omega^{-d}$ with $s\in\mathbb{Z}^n/(q-1)$ a character of $(\mathbb{F}_q^{\times})^n$ into $\mu_{q-1}$, and $\chi_k=\chi\circ{\rm
Norm}_{\mathbb{F}_{q^k}/\mathbb{F}_q}$. Let $\psi(x)=(1+T)^x$ be the quasi-character from $\mathbb{Z}_p$ to $\mathbb{Z}_p[[T]]^{\times}$, and $\psi_q=\psi\circ\text{Tr}_{\mathbb{Z}_{q}/\mathbb{Z}_p}$. Let $f\in\mu_{q-1}[x_1^{\pm1},\cdots,x_n^{\pm1}]$ be a non-constant polynomial in $n$-variables with coefficients in $\mu_{q-1}$.
The sum $$S_{f}(T,\chi)=S_{f}(T,\chi,\mathbb{F}_q)=
\sum\limits_{x\in\mu_{q-1}^n}x^{-d} \psi_q\circ f(x)$$ is called a twisted $T$-adic exponential sum. And the function $$L_{f,\chi}(s,T)=L_{f,\chi}(s,T,\mathbb{F}_q)=
\exp(\sum\limits_{k=1}^{+\infty}S_{f}(T,\chi_k,\mathbb{F}_{q^k})\frac{s^k}{k})$$ is called an $L$-function of twisted exponential sums.
We have $$L_{f,\chi}(s,T)=\prod_{x\in|\mathbb{G}_m^n\otimes\mathbb{F}_q|}
\frac{1}{1-\chi_{\deg(x)}(x)\psi_{\deg(x)}\circ
f(\hat{x})s^{\deg(x)}},$$ where $\mathbb{G}_m$ is the multiplicative group $xy=1$.
That Euler product formula gives $$L_{f,\chi}(s,T)\in 1+s\mathbb{Z}_q[[T]][[s]].$$
Define $$C_{f,\chi}(s,T)=
\exp(\sum\limits_{k=1}^{+\infty}\frac{-1}{(q^k-1)^n}
S_{f}(T,\chi_k)\frac{s^k}{k}).$$ Call it a $C$-function of twisted $T$-adic exponential sums. We have $$L_{f,\chi}(s,T)=\prod_{i=0}^n
C_{f,\chi}(q^is,T)^{(-1)^{n-i+1}{n\choose i}} ,$$ and $$C_{f,\chi}(s,T)=\prod_{j=0}^{+\infty}
L_{f,\chi}(q^js,T)^{(-1)^{n-1}{n+j-1\choose j}}.$$ So we have $$C_{f,\chi}(s,T)\in1+s\mathbb{Z}_q[[T]][[s]].$$
We shall prove the analytic continuation of $C_{f,\chi}(s,T)$.
The series $C_{f,\chi}(s,T)$ is $T$-adic entire in $s$.
The analytic continuation of $C_{f,\chi}(s,T)$ immediately gives the meromorphic continuation of $L_{f,\chi}(s,T)$.
The series $L_{f,\chi}(s,T)$ is $T$-adic meromorphic.
The Laurent polynomial $f$ is said to be $\chi$-twisted stable if $C_{f,\chi}(s,T)$ is stable $T$-adic entire series in $s$.
Let $\zeta_{p^m}$ denote a primitive $p^m$-th root of unity. The specialization $L_{f,\chi}(s,\zeta_{p^m}-1)$ is the $L$-function of twisted algebraic exponential sums $S_{f}(\zeta_{p^m},\chi_k)$. These sums were studied by Liu [@L], with the $m=1$ case studied by Adolphson-Sperber [@AS].
Let $f(x)=\sum\limits_{u\in I}a_ux^u$ with $I\subseteq\mathbb{Z}$ and $a_u^{q-1}=1$. We define $\triangle(f)$ to be the convex polytope in $\mathbb{R}^n$ generated by the origin and the vectors $u\in I$.
We call $f$ non-degenerate if $\triangle(f)$ is of dimension $n$, and for every closed face $\sigma\not\ni0$ of $\triangle(f)$, the system $$\frac{\partial f_{\sigma}}{\partial x_1} \equiv\cdots \equiv\frac{\partial f_{\sigma}}{\partial
x_n} \equiv0(\mod p)$$ has no common zeros in $(\overline{\mathbb{F}}_q^{\times})^n$, where $f_{\sigma}=\sum\limits_{u\in\sigma}a_ux^u$.
Gelfand-Kapranov-Zelevinsky proved the following.
Let $\triangle\ni0$ be an integral convex polytope in $\mathbb{R}^n$. If $p$ is sufficiently large, and $f$ is a generic Laurent polynomial in $\triangle(f)=\triangle$, then $f$ is non-degenerate.
By the above theorem, we are mainly concerned with non-degenerate $f$. We have the following.
If $f$ is non-degenerate, then $L_{f,\chi}(s,\zeta_{p^m}-1,\mathbb{F}_q)^{(-1)^{n-1}}$ is a polynomial of degree $p^{n(m-1)}{\rm Vol}(\triangle(f))$.
For non-degenerate $f$, the determination of the Newton polygon of $L_{f,\chi}(s,\zeta_{p^m}-1,\mathbb{F}_q)^{(-1)^{n-1}}$ is a challenging problem. The case $m=1$ is already very difficult, let alone the case $m>1$. However, from the vertical specialization theorem, one can prove the following.
Suppose that $f$ is non-degenerate and $\chi$-twisted stable. Let $\lambda_1,\cdots,\lambda_r$ be the slopes of the $q$-adic Newton polygon of $L_{f,\chi}(s,\zeta_p-1)^{(-1)^{n-1}}$. Then the slopes of the $q$-adic Newton polygon of $L_{f,\chi}(s,\zeta_{p^m}-1)^{(-1)^{n-1}}$ are the numbers $$\frac{\lambda_i+j_1+j_2+\cdots+j_n}{p^{m-1}},$$ where $i=1,\cdots, r$, and each $j_k=0,1,\cdots,p^{m-1}-1$.
The non-degenerate condition for $f$ can be replaced by the condition that the functions $L_{f,\chi}(s,\zeta_{p}-1)^{(-1)^{n-1}}$ and $L_{f,\chi}(s,\zeta_{p^m}-1)^{(-1)^{n-1}}$ are polynomials. The above theorem reduces the determination of the Newton polygon of $L_{f,\chi}(s,\zeta_{p^m}-1)$ to the $m=1$ case, provided that $f$ is non-degenerate and $\chi$-twisted stable. So, for non-degenerate $f$, we are mainly concerned with the stability of $f$ and the determination of the Newton polygon of $L_{f,\chi}(s,\zeta_{p}-1)^{(-1)^{n-1}}$. We shall prove the following stability criterion.
If $f$ is $\chi$-twisted ordinary, then it is $\chi$-twisted stable.
We now recall the notion of $\chi$-twisted ordinary Laurent polynomial. Let $\triangle\ni0$ be an integral convex polytope in $\mathbb{R}^n$, $C(\triangle)$ the cone generated by $\triangle$, $M(\triangle)=C(\triangle)\cap\mathbb{Z}^n$, and $\deg_{\triangle}$ the degree function on $C(\triangle)$, which is $\mathbb{R}_+$ linear and takes the value $1$ on each face $\delta\not\ni0$. Let $d\in\mathbb{Z}^n/(q-1)$, and $$M_{d}(\triangle):=\frac{1}{q-1}(M(\triangle)\cap d).$$
Let $b$ be the least positive integer such that $p^bd=d$. Order elements of $\cup_{i=0}^{b-1}M_{p^id}(\triangle)$ so that $$\deg_{\triangle}(x_1)\leq\deg_{\triangle}(x_2)\leq\cdots.$$ The infinite $d$-twisted Hodge polygon $H_{\triangle,d}^{\infty}$ of $\triangle$ is the convex function on $\mathbb{R}_+$ with initial value $0$ which is linear between consecutive integers and whose slopes (between consecutive integers) are $$\frac{\deg_{\triangle}(x_{bi+1})+\deg_{\triangle}(x_{bi+2})+\cdots+\deg_{\triangle}(x_{b(i+1)})}{b},\ i=0,1,\cdots.$$
If $$T-adic \text{ NP of }
C_{f,\omega^{-d}}(s,T,\mathbb{F}_q)={\rm
ord}_p(q)(p-1)H_{\triangle(f),d}^{\infty},$$ then $f$ is called $\omega^{-d}$-twisted ordinary.
Exponential sums under diagonal base change
-------------------------------------------
In this subsection we introduce the exponential sums associated to the tensor product of two Laurent polynomials.
If $g=\sum\limits_{v}b_vy^v\in\mu_{q-1}^n[y_1^{\pm1},\cdots,y_m^{\pm1}],$ we define $$f\otimes g=\sum\limits_{u,v}a_ub_v^uz^{u\otimes
v}\in\mu_{q-1}[z_{ij}^{\pm1},i=1,\cdots,n,j=1,\cdots,m],$$ and call it a diagonal base change of $f$.
The number $q$ acts on the $m$-tuples $u=(u_1,\cdots,u_m)$ of vectors in $\mathbb{Z}_{(p)}^n/\mathbb{Z}^n$ by multiplication. The length of the orbit $u$ is denoted by $|u|$. The congruences $u_1\otimes v_1+\cdots +u_m\otimes v_m\equiv \frac{d}{q-1}$, and $(q^k-1)u_j\equiv0$ are defined on the orbit space $q\setminus(\mathbb{Z}_{(p)}^n/\mathbb{Z}^n)^m$.
We shall prove the following.
Let $v_1,\cdots,v_m$ be an integral basis of $\mathbb{R}^m$, and $g(y)=\sum\limits_jb_jy^{v_j}$ with $b_j\in\mu_{q-1}^n$. Then $$C_{f\otimes g,\omega^{-d}}(s,T)=\prod_{\stackrel{(u_1,\cdots,u_m)\in
q\setminus(\mathbb{Z}_{(p)}^n/\mathbb{Z}^n)^m}{u_1\otimes v_1+\cdots
+u_m\otimes v_m\equiv \frac{d}{q-1}}}
\otimes_{j=1}^mC_{f,\omega^{-u_j(q^{|u|}-1)}}(s^{|u|}\prod_{j=1}^m
b_j^{-u_j(q^{|u|}-1)},T,\mathbb{F}_{q^r}).$$
As a function of $(u_1,\cdots,u_m)$, the tensor product on the right-hand side of the equality is defined on the orbit space. And, as the solutions of $u_1\otimes v_1+\cdots +u_m\otimes v_m\equiv0$, under the map $$(u_1,\cdots,u_m)\mapsto u_1\otimes v_1+\cdots +u_m\otimes v_m,$$ can be embedded into $\mathbb{Z}^{n}\otimes\mathbb{Z}^m/\mathbb{Z}^n\otimes\sum\limits_{j=1}^m\mathbb{Z}v_j$, the product on the right-hand side of the equality is a finite product.
The above theorem has the following equivalent form.
Let $v_1,\cdots,v_m$ be an integral basis of $\mathbb{R}^m$, and $g(y)=\sum\limits_jb_jy^{v_j}$ with $b_j\in\mu_{q-1}^n$. Then
$$L_{f\otimes g,\omega^{-d}}(s,T)^{(-1)^{mn-1}}=\prod_{\stackrel{(u_1,\cdots,u_m)\in
q\setminus(\mathbb{Z}_{(p)}^n/\mathbb{Z}^n)^m}{u_1\otimes v_1+\cdots
+u_m\otimes v_m\equiv \frac{d}{q-1}}}
\otimes_{j=1}^mL_{f,\omega^{-u_j(q^{|u|}-1)}}(s^{|u|}\prod_{j=1}^m
b_j^{-u_j(q^{|u|}-1)},T,\mathbb{F}_{q^{|u|}})^{(-1)^{n-1}}.$$
By the above theorem, the Newton polygon of the $L$-function or $C$-function of $T$-adic (resp. algebraic) exponential sums of $f\otimes g$ is determined by that of $f$.
Combine the above with theorem the fact that the set of stable $T$-adic entire series in $1+s\mathbb{Z}_q[[T]][[s]]$ is closed under multiplication and tensor operation, we get the following.
Let $v_1,\cdots,v_m$ be an integral basis of $\mathbb{R}^m$, and $g(x)=\sum\limits_{j=1}^mb_jx^{v_j}$ with $b_j\in\mu_{q-1}^n$. If $f$ is $\chi$-twisted stable for all $\chi$, then so is $f\otimes
g$.
Analytic continuation
=====================
In this section, we prove the analytic continuation of $C_{f,\psi}(s,T,\mathbb{F}_q)$.
Define a new variable $\pi$ by the relation $E(\pi)=1+T$, where $$E(\pi)=\exp(\sum_{i=0}^{\infty}\frac{\pi^{p^i}}{p^i}) \in 1+\pi{\mathbb Z}_p[[\pi]]$$ is the Artin-Hasse exponential series. Thus, $\pi$ is also a $T$-adic uniformizer of ${\mathbb Q}_p((T))$.
Let $\triangle=\triangle(f)$, and $D$ the least common multiple of the denominators of $\deg(\triangle)$. Write $$L_d(\triangle)=\{\sum\limits_{u\in M_d(\triangle)}c_u\pi^{\deg(u)}x^u :
c_u\in\mathbb{Z}_{q}[[\pi^{\frac{1}{D(q-1)}}]]\},$$ and $$B_d(\triangle)=\{\sum\limits_{u\in M_d(\triangle)}c_u\pi^{\deg(u)}x^u :
c_u\in\mathbb{Z}_{q}[[\pi^{\frac{1}{D(q-1)}}]],\
\text{ord}_T(c_u)\rightarrow+\infty
\text{ if }\deg(u)\rightarrow+\infty\}.$$ Note that $L_d(\triangle)$ is stable under multiplication by elements of $L_0(\triangle)$, and $$E_f(x) =\prod\limits_{a_u\neq0}E(\pi \hat{a}_u\hat{x}^u)\in L_0(\triangle).$$
Define $$\phi:L_d(\Delta)\rightarrow L_{dp^{-1}}(\Delta),\
\sum\limits_{u\in
M_d(\Delta)} c_ux^u\mapsto\sum\limits_{u\in M_{dp^{-1}}(\Delta)}
c_{pu}x^u.$$ Then the map $\phi\circ E_f$ sends $L_d$ to $B_{dp^{-1}}$.
If $x^{p^a-1}=1$, then $$E(\pi)^{x+x^p+\cdots+x^{p^{a-1}}}=E(\pi x)E(\pi x^p)\cdots E(\pi x^{p^{a-1}}).$$
Since $$\sum\limits_{j=0}^{a-1}x^{p^j}=\sum\limits_{j=0}^{a-1}x^{p^{j+i}},$$ we have $$E(\pi)^{x+x^p+\cdots+x^{p^{a-1}}}
=\exp(\sum_{i=0}^{\infty}\frac{\pi^{p^i}}{p^i}\sum\limits_{j=0}^{a-1}x^{p^{j+i}})=E(\pi
x)E(\pi x^p)\cdots E(\pi x^{p^{a-1}}).$$
The Galois group $\text{Gal}(\mathbb{Q}_q/\mathbb{Q}_p)$ is generated by the Frobenius element $\sigma$, whose restriction to $(q-1)$-th roots of unity is the $p$-power map. That Galois group can act on $L(\Delta)$ by fixing $\pi^{\frac{1}{D(q-1)}}$ and $x_1,\cdots,x_n$.
If $q=p^a$, and $x\in(\mathbb{F}_{q^k}^{\times})^n$, then $$E(\pi)^{\text{Tr}_{\mathbb{Z}_{q^k}/\mathbb{Z}_p}
(\sum\limits_u\hat{a}_u\hat{x}^u)}
=\prod\limits_{i=0}^{ak-1}E_f^{\sigma^i}(\hat{x}^{p^i}).$$
We have $$E(\pi)^{\text{Tr}_{\mathbb{Z}_{q^k}/\mathbb{Z}_p}
(\sum\limits_u\hat{a}_u\hat{x}^u)}
=\prod\limits_{a_u\neq0}E(\pi)^{\text{Tr}_{\mathbb{Z}_{q^k}/\mathbb{Z}_p}(\hat{a}_u\hat{x}^u)}$$ $$=\prod\limits_{a_u\neq0}\prod\limits_{i=0}^{ak-1}E(\pi(\hat{a}_u\hat{x}^u)^{p^i})=
\prod\limits_{i=0}^{ak-1}E_f^{\sigma^i}(\hat{x}^{p^i}).$$
Define $c(u,v)=\deg(u)+\deg(v)-\deg(u+v)\text{ if }u,v\in
C(\triangle)$. Then $c(u,v)\geq0$, and is zero if and only if $u$ and $v$ are cofacial. We call $c(u,v)$ the cofacial defect of $u$ and $v$.
Write $$E_f(x) =\sum\limits_{u\in
M(\triangle)}\alpha_u(f)\pi^{\deg(u)}x^u.$$ Then, for $u\in
M_{d}(\Delta)$, we have $$\phi\circ E_f(\pi^{\deg(u)}x^u)=\sum\limits_{w\in
M_{dp^{-1}}(\Delta)}\alpha_{pw-u}(f)\pi^{c(pw-u,u)}\pi^{(p-1)\deg(w)}\pi^{\deg(w)}x^w.$$
Obvious.
Define $\phi_p:=\sigma^{-1}\circ\phi\circ E_f$, and $\phi_{p^a}=\phi_p^{a}$. Then $\phi_{p^a}$ sends $B_d$ to $B_{dp^{-a}}$, and $$\phi_{p^{a}}=\sigma^{-a}\circ\phi^{a}\circ
\prod\limits_{i=0}^{a-1}E_{f}^{\sigma^i}(x^{p^i}).$$ It follows that $\phi_q$ operates on $B_d$, and is linear over $\mathbb{Z}_q[[\pi^{\frac{1}{D(q-1)}}]]$. Moreover, by the last lemma, it is completely continuous in the sense of [@Se].
\[trace-formula\]Suppose that $\chi=\omega^{-d}$. Then $$S_{f,\chi}(T,\mathbb{F}_{q^k})
=(q^k-1)^n\text{Tr}_{B_d/\mathbb{Z}_{q}[[\pi^{\frac{1}{D(q-1)}}]]}(\phi_{q}^{k}),\
k=1,2\cdots.$$
Suppose that $q=p^a$. Let $g(x)\in B_d$. We have $$\phi_q^{k}(g)=\phi^{ak}(g\prod\limits_{i=0}^{ak-1}E_f^{\sigma^i}(x^{p^i})).$$Write $\prod\limits_{i=0}^{ak-1}E_f^{\sigma^i}(x^{p^i})=\sum\limits_{u\in
M(\Delta)}\beta_u x^u $. Then $$\phi_q^{k}(\pi^{\deg(v)}x^v)=\sum\limits_{u\in
M_d(\Delta)}\beta_{q^ku-v}\pi^{\deg(v)}x^u.$$ So the trace of $\phi_q^{k}$ on $B_d$ over $\mathbb{Z}_{q}[[\pi^{\frac{1}{D(q-1)}}]]$ equals $\sum\limits_{u\in
M_d(\Delta)} \beta_{(q^k-1)u}$. But, by Dwork’s splitting lemma, we have $$S_{f,\chi}(T,\mathbb{F}_{q^k})
=\sum\limits_{x_1^{q^k-1}=1,\cdots,x_n^{q^k-1}=1}
x^{-d(1+q+\cdots+q^{k-1})} \prod\limits_{i=0}^{ak-1}
E_f^{\sigma^i}(x^{p^i})=(q^k-1)^n\sum\limits_{u\in M_d(\Delta)}
\beta_{(q^k-1)u}.$$ The theorem now follows.
\[analytic-trace-formula\] If $\chi=\omega^{-d}$, then $$C_{f,\chi}(s,T,\mathbb{F}_q)={\rm det}_{\mathbb{Z}_{q}[[\pi^{\frac{1}{D(q-1)}}]]}(1-\phi_qs\mid
B_d).$$ In particular, $C_{f,\chi}(s,T,\mathbb{F}_q)$ is $T$-adic analytic in $s$.
This follows from the last theorem and the identity $${\rm det}_{\mathbb{Z}_{q}[[\pi^{\frac{1}{D(q-1)}}]]}(1-\phi_qs\mid B_d)
=\exp(-\sum\limits_{k=1}^{+\infty}\text{Tr}_{B_d/\mathbb{Z}_{q}[[\pi^{\frac{1}{D(q-1)}}]]}(\phi_q^{k})\frac{s^k}{k}).$$
Hodge bound
===========
In this section, we prove the Hodge bound for the Newton polygon of $C_{f,\chi}(s,T,\mathbb{F}_q)$. It will play an important role in establishing the stability of ordinary Laurent polynomial.
Let the Galois group $\text{Gal}(\mathbb{Q}_q/\mathbb{Q}_p)$ act on $\mathbb{Z}_q[[T]][[s]]$ by fixing $s$ and $T$.
We have $$C_{f,\chi}(s,T,\mathbb{F}_q)^{\sigma}=C_{f,\chi^p}(s,T,\mathbb{F}_q).$$
Obvious.
Suppose that $q=p^a$ and $\chi=\omega^{-d}$. Let $b$ be the least positive integer such that $p^bd=d$. Then, as power series in $s$ with coefficients in $\mathbb{Z}_q[[T]]$, $$\text{NP of }C_{f,\chi}(s,T,\mathbb{F}_q)^{ab}=
\text{NP of }{\rm
det}_{\mathbb{Z}_{p}[[\pi^{\frac{1}{D(q-1)}}]]}(1-\phi_qs\mid
\oplus_{i=0}^{b-1}B_{p^id}).$$
In fact, we have $${\rm det}_{\mathbb{Z}_p[[\pi^{\frac{1}{D(q-1)}}]]}(1-\phi_qs\mid
\oplus_{i=0}^{b-1}B_{p^id}) =\prod_{j=0}^{a-1}{\rm
det}_{\mathbb{Z}_q[[\pi^{\frac{1}{D(q-1)}}]]}(1-\phi_qs\mid
\oplus_{i=0}^{b-1}B_{p^id})^{\sigma^j}$$$$=\prod_{j=0}^{a-1}\prod_{i=0}^{b-1}{\rm
det}_{\mathbb{Z}_q[[\pi^{\frac{1}{D(q-1)}}]]}(1-\phi_qs\mid
B_{d})^{\sigma^{i+j}}.$$ The corollary now follows.
Suppose that $q=p^a$ and $\chi=\omega^{-d}$. Let $b$ be the least positive integer such that $p^bd=d$. Then, as power series in $s$ with coefficients in $\mathbb{Z}_q[[T]]$, $$\text{NP of }C_{f,\chi}(s^a,T,\mathbb{F}_q)^{b}=
\text{NP of }{\rm
det}_{\mathbb{Z}_{p}[[\pi^{\frac{1}{D(q-1)}}]]}(1-\phi_ps\mid
\oplus_{i=0}^{b-1}B_{p^id}).$$
This follows from the identity $${\rm det}_{\mathbb{Z}_p[[\pi^{\frac{1}{D(q-1)}}]]}(1-\phi_qs^a\mid
\oplus_{i=0}^{b-1}B_{p^id}) =\prod_{\zeta^a=1}{\rm
det}_{\mathbb{Z}_p[[\pi^{\frac{1}{D(q-1)}}]]}(1-\phi_p\zeta s\mid
\oplus_{i=0}^{b-1}B_{p^id}).$$
\[hodge-for-psi\]Suppose that $q=p^a$ and $b$ is the least positive integer such that $p^bd=d$. Then, as a power series in $s$ with coefficients in $\mathbb{Z}_q[[T]]$, the $T$-adic Newton polygon of ${\rm
det}_{\mathbb{Z}_{p}[[\pi^{\frac{1}{D(q-1)}}]]}(1-\phi_ps\mid
\oplus_{i=0}^{b-1}B_{p^id})$ lies above the convex polygon with initial point $(0,0)$ and slopes $(p-1)\deg(w)$, where $w$ runs through elements of $\cup_{i=0}^{b-1}M_{p^id}$ with multiplicity $a$.
Choose $\zeta\in\mathbb{Z}_q^{\times}$ such that $\zeta^{\sigma^i}$, $i=0,1,\cdots, a-1$ be a basis of $\mathbb{Z}_q$ over $\mathbb{Z}_p$. Write $$\alpha_u(f)=\sum\limits_{i=0}^{a-1}\alpha_{u,i}(f)\zeta^{\sigma^i},\
\alpha_{u,i}(f)\in\mathbb{Z}_p[[\pi^{1/D}]].$$ Then, for $u\in
M_{p^id}(\triangle)$, we have $$\phi_p(\zeta^{\sigma^j}\pi^{\deg(u)}x^u)=\sum\limits_{i=0}^{a-1}\sum\limits_{w\in
M_{dp^{i-1}}(\Delta)}
\alpha_{pw-u,i-j+1}(f)\pi^{c(pw-u,u)}\pi^{(p-1)\deg(w)}\zeta^{\sigma^i}\pi^{\deg(w)}x^w.$$ So, the matrix of $\phi_p$ over $\mathbb{Z}_p[[\pi^{\frac{1}{D(q-1)}}]]$ with respect to the basis $\{\zeta^{q^j}\pi^{\deg(u)}x^u\}_{0\leq j<a,u\in
\oplus_{i=0}^{b-1}B_{p^id}(\Delta)}$ is $$A=(\alpha_{pw-u,i-j+1}(f)\pi^{c(pw-u,u)}\pi^{(p-1)\deg(w)})_{(i,w),(j,u)}.$$ It follows that, the $T$-adic Newton polygon of ${\rm
det}_{\mathbb{Z}_{p}[[\pi^{\frac{1}{D(q-1)}}]]}(1-\phi_ps\mid
\oplus_{i=0}^{b-1}B_{p^id})$ lies above the convex polygon with initial point $(0,0)$ and slopes $(p-1)\deg(w)$, where $w$ runs through elements of $\cup_{i=0}^{b-1}M_{p^id}$ with multiplicity $a$.
\[hodge2\]Suppose that $q=p^a$ and $\chi=\omega^{-d}$. Let $b$ be the least positive integer such that $p^bd=d$. Then, as a power series in $s$ with coefficients in $\mathbb{Z}_q[[T]]$, the $T$-adic Newton polygon of $C_{f,\chi}(s,T,\mathbb{F}_q)^{b}$ lies above the convex polygon with initial point $(0,0)$ and slopes $$a(p-1)\deg(w),\ w\in\cup_{i=0}^{b-1}M_{p^id}.$$
Obvious.
\[hodge-bound\]Suppose that $\chi=\omega^{-d}$, and $q=p^a$. Then $$T-adic \text{ NP of } C_{f,\chi}(s,T)\geq
a(p-1)H_{\triangle(f),d}^{\infty},$$ where NP is the short for Newton polygon, and $H_{\triangle(f),d}^{\infty}$ is the infinite $d$-twisted Hodge polygon of $\triangle(f)$.
Obvious.
If $\chi=\omega^{-d}$, $q=p^a$, $0\neq|t|_p<1$, and $$t-adic\text{ NP of
}C_{f,\chi}(s,t)=a(p-1)H_{\triangle(f),d}^{\infty},$$ then $f$ is said to be $\chi$-twisted ordinary.
Vertical specialization and stability
=====================================
In this section we prove the vertical specialization theorem, the theorem for the Newton polygon of stable Laurent polynomials, and the stability of ordinary Laurent polynomials.
Let $A(s,T)\in 1+s\mathbb{Z}_q[[T]][[s]]$ be a $T$-adic entrie series in $s$. If $0\neq|t|_p<1$, then $$t-adic\text{ NP of
}A(s,t)\geq T-adic\text{ NP of }A(s,T),$$where NP is the short for Newton polygon. Moreover, the equality holds for one $t$ iff it holds for all $t$.
Write $$A(s,T)=\sum\limits_{i=0}^{\infty}a_i(T)s^i.$$ The inequality follows from the fact that $a_i(T)\in\mathbb{Z}_q[[T]]$. Moreover, $$t-adic\text{ NP of
}A(s,t)=T-adic\text{ NP of }A(s,T)$$if and only if $$a_i(T)\in T^e\mathbb{Z}_q[[T]]^{\times}$$ for every turning point $(i,e)$ of the $T$-adic Newton polygon of $A(s,T)$. It follows that the equality holds for one $t$ iff it holds for all $t$.
Suppose that $f$ is non-degenrate and $\chi$-twisted stable. Let $\lambda_1,\cdots,\lambda_r$ be the slopes of the $q$-adic Newton polygon of $L_{f,\chi}(s,\zeta_p-1)^{(-1)^{n-1}}$. Then the $q$-adic orders of the reciprocal roots of $L_{f,\chi}(s,\zeta_{p^m}-1)^{(-1)^{n-1}}$ are the numbers $$\frac{\lambda_i+j_1+j_2+\cdots+j_n}{p^{m-1}},$$ where $i=1,\cdots, r$, and each $j_k=0,1,\cdots,p^{m-1}-1$.
Apply the relationship between the $L$-function and the $C$-function, we see that the $q$-adic orders of the reciprocal roots of $C_{f,\chi}(s,\zeta_{p}-1)$ are the numbers $$\lambda_i+j_1+j_2+\cdots+j_n,$$ where $i=1,\cdots, r$, and each $j_k=0,1,\cdots$. So the $(\zeta_p-1)$-adic orders of the reciprocal roots of $C_{f,\chi}(s,\zeta_{p}-1)$ are the numbers $${\rm ord}_{\zeta_p-1}(q)(\lambda_i+j_1+j_2+\cdots+j_n),$$ where $i=1,\cdots, r$, and each $j_k=0,1,\cdots$. Apply the $\chi$-twisted stability of $f$, we see that the $(\zeta_{p^m}-1)$-adic orders of the reciprocal roots of $C_{f,\chi}(s,\zeta_{p^m}-1)$ are the numbers $${\rm ord}_{\zeta_p-1}(q)(\lambda_i+j_1+j_2+\cdots+j_n),$$ where $i=1,\cdots, r$, and each $j_k=0,1,\cdots$. So the $q$-adic orders of the reciprocal roots of $C_{f,\chi}(s,\zeta_{p^m}-1)$ are the numbers $$\frac{\lambda_i+j_1+j_2+\cdots+j_n}{p^{m-1}},$$ where $i=1,\cdots,r$, and each $j_k=0,1,\cdots$. Apply the relationship between the $C$-function and the $L$-function, we see that the $q$-adic orders of the reciprocal roots of $L_{f,\chi}(s,\zeta_{p^m}-1)$ are the numbers $$\frac{\lambda_i+j_1+j_2+\cdots+j_n}{p^{m-1}},$$ where $i=1,\cdots,r$, and each $j_k=0,1,\cdots,p^{m-1}-1$.
If $\chi=\omega^{-d}$, $q=p^a$, and $0\neq|t|_p<1$, then $$t-adic\text{ NP of
}C_{f,\chi}(s,t)\geq T-adic\text{ NP of }C_{f,\chi}(s,T)\geq
a(p-1)H_{\triangle(f),d}^{\infty}.$$Moreover, the equalities hold for one $t$ iff they hold for all $t$.
Just combine the Hodge bound for the Newton polygon of $C_{f,\chi}(s,T)$ with the vertical specialization theorem.
If $f$ is $\chi$-twisted ordinary, then it is $\chi$-twisted stable, and $\chi$-twisted $T$-adic ordinary.
Obvious.
Let $\alpha_1,\alpha_2,\cdots$ be the slopes of the infinite $d$-twisted Hodge polygon of $\triangle$. Then $$(1-t)^n\sum\limits_{i}t^{\alpha_i}=\sum\limits_{i=1}^{n!\text{Vol}(\triangle)}t^{w_i}.$$ The $d$-twisted Hodge polygon $H_{\triangle,d}$ of $\triangle$ is the convex function on $[0,n!\text{Vol}(\triangle)]$ with initial value $0$ which is linear between consecutive integers and whose slopes (between consecutive integers) are $w_i$, $i=1,\cdots,n!\text{Vol}(\triangle)$.
Let $f$ be non-degenerate, $\chi=\omega^{-d}$ and $m\geq1$. Then $f$ is $\chi$-twisted ordinary if and only if $$q-adic \text{ NP of }
L_{f,\chi}(s,\zeta_{p^m}-1)^{(-1)^{n-1}}=H_{p^{m-1}\triangle(f),d}.$$
The non-degenerate condition for $f$ can be replaced by the condition that the function $L_{f,\chi}(s,\zeta_{p^m}-1)^{(-1)^{n-1}}$ is a polynomial. In fact, $f$ is $\chi$-twisted ordinary if and only if $$(\zeta_p-1)-adic\text{ NP of }C_{f,\chi}(s,\zeta_p-1)={\rm ord}_p(q)(p-1)H_{\triangle(f),d}^{\infty},$$ if and only if $$(\zeta_{p^m}-1)-adic\text{ NP of
}C_{f,\chi}(s,\zeta_{p^m}-1)={\rm
ord}_p(q)(p-1)H_{\triangle(f),d}^{\infty},$$ if and only if $$q-adic\text{ NP of }C_{f,\chi}(s,\zeta_{p^m}-1)
=\frac{1}{p^{m-1}}H_{\triangle(f),d}^{\infty}
=H_{p^{m-1}\triangle(f),d}^{\infty},$$ if and only if $$q-adic \text{
NP of }
L_{f,\chi}(s,\zeta_{p^m}-1)^{(-1)^{n-1}}=H_{p^{m-1}\triangle(f),d}.$$.
$$L_{x,\chi}(s,\zeta_p,\mathbb{F}_{q})=1+G(\zeta_p,\chi,\mathbb{F}_q)s.$$
This follows from the the following classical formulation:$$G(\zeta_p,\chi_k,\mathbb{F}_{q^k})=(-1)^{k-1}G(\zeta_p,\chi,\mathbb{F}_{q})^k.$$
Let $q=p^a$, and $d\in\mathbb{Z}/(q-1)$. We define $$\sigma_q(d)=(p-1)\sum\limits_{i=0}^{a-1}\{\frac{p^id}{q-1}\}.$$
The polynomial $f(x)=x$ is $\chi$-twisted ordinary for all $\chi$.
We have $\triangle=\triangle(f)=[0,1]$, $C(\triangle)=\mathbb{R}_+$, and $\deg_{\triangle}(u)=u$. Let $q=p^a$, and $\chi=\omega^{-d}$. We have $M(\triangle)=\mathbb{N}$, and $M_{p^id}(\triangle)=\{\frac{p^id}{q-1}\}+\mathbb{N}$. It follows that the infinite $d$-twisted Hodge polygon of $\triangle$ has slopes $$\frac{\sigma_q(d)}{a(p-1)}+k,\ k=0,1,\cdots.$$ So the finite $d$-twisted Hodge polygon of $\triangle$ has only one slope $\frac{\sigma_q(d)}{a(p-1)}$. By the Hasse-Davenport relation and the classical Stickelberger theorem, the Newton polygon of the $L$-function $L_{x,\chi}(s,\zeta_p-1)$ also has only one slope $\frac{\sigma_q(d)}{a(p-1)}$. Therefore the polynomial $f(x)=x$ is $\chi$-twisted ordinary.
Let $q=p^a$, and $\chi=\omega^{-d}$. Then $q$-adic orders of the reciprocal zeros of the $L$-function $L_{x,\chi}(s,\zeta_{p^m}-1)$ of the Gauss-Heilbronn sums $G(\zeta_{p^m}-1,\chi_k,\mathbb{F}_{q^k})$ are $$\frac{\sigma_q(d)}{a(p-1)p^{m-1}}+\frac{k}{p^{m-1}},\ k=0,1,\cdots,p^{m-1}-1.$$
The above theorem was proved by Blache [@B], and Liu [@L]. But the proof here is much simpler.
Apply the Stickelberger theorem for Gauss sums and the theorem on the Newton polygon for ordinary $p$-power order exponential sums, we get $$q-adic \text{ NP of }
L_{x,\chi}(s,\zeta_{p^m}-1)=H_{p^{m-1}\triangle,d}$$ with $\triangle=[0,1]$. In the proof of the Stickelberger theorem for Gauss sums, we show that $$M_{p^id}(\triangle)=\{\frac{p^id}{q-1}\}+\mathbb{N}.$$ It follows that the infinite $d$-twisted Hodge polygon of $p^{m-1}\triangle$ has slopes $$\frac{\sigma_q(d)}{a(p-1)p^{m-1}}+\frac{k}{p^{m-1}},\ k=0,1,\cdots.$$ So the finite $d$-twisted Hodge polygon of $p^{m-1}\triangle$ has slopes $$\frac{\sigma_q(d)}{a(p-1)p^{m-1}}+\frac{k}{p^{m-1}},\
k=0,1,\cdots,p^{m-1}-1.$$ The theorem now follows.
Stable $T$-adic entire series
=============================
In this section we prove that the set of stable $T$-adic entire series in $1+s\mathbb{Z}_q[[T]][[s]]$ is closed under multiplication and tensor operation.
Let $A(s,T)\in\mathbb{Z}_q[[T]]\langle s\rangle$ be a $T$-adically strictly convergent power series in $s$ with unitary constant term. Suppose that $A(s,T)(\mod
T)\in\mathbb{Z}_q[s]$ is a unitary polynomial of degree $n$. Then $$A(s,T)=u(s,T)B(s,T),$$ where $u(s,T)\in1+T\mathbb{Z}_q[[T]]\langle
s\rangle$, and $B(s,T)\in\mathbb{Z}_q[[T]][s]$ is a monic polynomial of degree $n$ with unitary constant term.
Let $A_0(s)=A(s,T)(\mod T)$, and $\alpha={\rm
ord}_T(A(s,T)-A_0(s))$. Then $$\mathbb{Z}_q[[T]]\langle
s\rangle/(T^{\alpha},A(s,T))=\mathbb{Z}_q[[T]]\langle
s\rangle/(T^{\alpha},A_0(s)),$$ and is generated as $\mathbb{Z}_q[[T]]$-module by $1,s,\cdots,s^{n-1}$. In particular. $$v_0=s^n=\sum\limits_{j=0}^{n-1}a_{1j}s^j+v_1T^{\alpha}+w_1A(s,T),\
v_1\in\mathbb{Z}_q[[T]]\langle
s\rangle,w_1\in1+T\mathbb{Z}_q[[T]]\langle s\rangle.$$ By induction, we can construct sequences $v_i,w_i\in\mathbb{Z}_q[[T]]\langle
s\rangle$ so that$$v_{i-1}=\sum\limits_{j=0}^{n-1}a_{ij}s^j+v_iT^{\alpha}+w_iA(s
,T).$$ We have $$\sum\limits_{i=1}^{\infty}v_{i-1}T^{(i-1)\alpha}
=\sum\limits_{j=0}^{n-1}s^j\sum\limits_{i=1}^{\infty}a_{ij}T^{(i-1)\alpha}
+\sum\limits_{i=1}^{\infty}v_iT^{i\alpha}+A(s
,T)\sum\limits_{i=1}^{\infty}w_iT^{(i-1)\alpha}.$$ So $$s^n-\sum\limits_{j=0}^{n-1}s^j\sum\limits_{i=1}^{\infty}a_{ij}T^{(i-1)\alpha}=A(s
,T)\sum\limits_{i=1}^{\infty}w_iT^{(i-1)\alpha}.$$ Since $w=\sum\limits_{i=1}^{\infty}w_iT^{(i-1)\alpha}\in1+T\mathbb{Z}_q[[T]]\langle
s\rangle$, we have $u(s,T)=w^{-1}\in1+T\mathbb{Z}_q[[T]]\langle
s\rangle$. Set $B(s,T)=s^n-\sum\limits_{j=0}^{n-1}s^j\sum\limits_{i=1}^{\infty}a_{ij}T^{(i-1)\alpha}\in\mathbb{Z}_q[[T]][s]$, we get $$A(s,T)=u(s,T)B(s,T).$$
Let $A(s,T)\in1+s\mathbb{Z}_q[[T]]\langle s\rangle$ be a $T$-adically entire power series in $s$, whose Newton polygon has slopes $\lambda_i$ of horizontal length $n_i$. Then $$A(s,T)=\prod_{i=1}^{\infty}A_i(s),$$ where $A_i(s)\in\mathbb{Z}_q[[T^{\lambda_1},\cdots,T^{\lambda_i}]][s]$ is polynomial of degree $n_i$ with unitary constant term and linear Newton polygon.
Apply the Weierstrass preparation theorem to construct $A_i(s)$ inductively.
Let $A(s,T)$ and $B(s,T)$ be two $T$-adic entire power series in $1+s\mathbb{Z}_q[[T]][[s]]$. Suppose that $$A(s,T)=\prod_{\alpha\in I}(1-\alpha s),$$ and $$B(s,T)=\prod_{\beta\in J}(1-\beta s).$$ Then $$A\otimes
B(s,T)=\prod_{\alpha\in I,\beta\in J}(1-\alpha\beta s).$$
We have $$A(s,T)=\exp(-\sum\limits_{k=1}^{+\infty}\frac{s^k}{k}
\sum\limits_{\alpha\in I}\alpha^k),$$ and $$B(s,T)=\exp(-\sum\limits_{k=1}^{+\infty}\frac{s^k}{k}
\sum\limits_{\beta\in J}\beta^k).$$ So $$A\otimes B(s,T)=\exp(-\sum\limits_{k=1}^{+\infty}\frac{s^k}{k}
\sum\limits_{\alpha\in I,\beta\in
J}\alpha^k\beta^k)=\prod_{\alpha\in I,\beta\in J}(1-\alpha\beta
s).$$
Let $A(s,T)$ and $B(s,T)$ be two $T$-adic entire power series in $1+s\mathbb{Z}_q[[T]][[s]]$. Then $A(s,T)B(s,T)$ is stable iff both $A(s,T)$ and $B(s,T)$ are stable.
Write $$A(s,T)=\sum\limits_nA_n(T)s^n,$$ $$B(s,T)=\sum\limits_nB_n(T)s^n,$$ and $$A(s,T)B(s,T)=\sum\limits_nC_n(T)s^n.$$Let $\{\alpha\}$ be the set of the reciprocal zeros of $A(s,T)$, and $\{\beta\}$ the set of reciprocal zeros of $B(s,T)$. Let $(n,e)$ be a turning points of the Newton polygon of $A(s,T)B(s,T)$. Then $n=\sum\limits_{{\rm
ord}_T(\alpha)\leq r}1+\sum\limits_{{\rm ord}_T(\beta)\leq r}1$ for some $r\in\mathbb{R}_+$, and $$C_n(T)\equiv (-1)^n\prod_{{\rm
ord}_T(\alpha)\leq r}\alpha\prod_{{\rm ord}_T(\beta)\leq
r}\beta\equiv A_{n_1}(T)B_{n_2}(T)(\mod T^{>e}),$$ where $n_1=\sum\limits_{{\rm ord}_T(\alpha)\leq r}1$, and $n_2=\sum\limits_{{\rm ord}_T(\beta)\leq r}1$. Note that the $T$-adic order of $C_n(T)$ is stable under specialization iff both the $T$-adic orders of $A_{n_1}(T)$ and $B_{n_2}(T)$ are stable under specialization. The lemma now follows.
The set of stable $T$-adic entire series in $1+s\mathbb{Z}_q[[T]][[s]]$ is closed under tensor operation.
Let $A(s,T)$ and $B(s,T)$ be two stable $T$-adic entire power series in $1+s\mathbb{Z}_q[[T]][[s]]$. By the Weierstrass factorization theorem, and the last lemma, we may assume that $A$ and $B$ are polynomials with linear Newton polygon. Let $\{\alpha_1,\cdots,\alpha_m\}$ be the set of the reciprocal zeros of $A(s,T)$, and $\{\beta_1,\cdots,\beta_n\}$ the set of reciprocal zeros of $B(s,T)$. Then the leading term of $A\otimes B$ is $$C_{mn}(T)=\prod_{i=1}^m\prod_{j=1}^n(-\alpha_i\beta_j)=
(-1)^{mn}A_m(T)^nB_{n}^m(T),$$ where $A_m$ is the leading coefficient of $A(s,T)$, and $B_n$ is the leading coefficient of $B(s,T)$. Since the $T$-adic orders of $A_{m}(T)$ and $B_{n}(T)$ do not go up under specialization, so does the $T$-adic order of $C_{mn}(T)$. The theorem is proved.
Exponential sums under the tensor operation
===========================================
In this section we explore Wan’s method [@W], and study the exponential sums associated to the tensor product of two Laurent polynomials.
Let $v_1,\cdots,v_m$ be an integral basis of $\mathbb{R}^m$, and $g(y)=\sum\limits_jb_jy^{v_j}$ with $b_j\in\mu_{q-1}^n$. Then
$$S_{f\otimes g}(T,\omega^{-d})
=\sum\limits_{\stackrel{u_1,\cdots,u_m\in\mathbb{Z}^n/(q-1)}{u_1\otimes
v_1+\cdots +u_m\otimes v_m\equiv d}}
\prod_{j=1}^mb_j^{u_j}\prod_{j=1}^m
S_f(T,\omega^{-u_j},\mathbb{F}_q)$$
Since $$S_{f}(T,\omega^{-u},\mathbb{F}_q)
=\sum\limits_{\alpha\in\mu_{q-1}^n}\alpha^{-u}\psi_q\circ
f(\alpha),$$ we have $$\psi_q\circ f(\alpha)=\frac{1}{(q-1)^n}\sum\limits_{u\in\mathbb{Z}^n/(q-1)}
\alpha^uS_f(T,\omega^{-u},\mathbb{F}_q).$$ So $$\psi_q(f\otimes b_jy^{v_j}))
=\frac{1}{(q-1)^n}\sum\limits_{u\in\mathbb{Z}^n/(q-1)}
b_j^uz^{u\otimes v_j}S_f(T,\omega^{-u},\mathbb{F}_q).$$ Thus, $$\psi_q(f\otimes g(z))
=\frac{1}{(q-1)^{mn}}\prod_j\sum\limits_{u\in\mathbb{Z}^n/(q-1)}
b_j^uz^{u\otimes v_j}S_f(T,\omega^{-u},\mathbb{F}_q).$$ Therfore $$S_{f\otimes g}(T,\omega^{-d})=\sum\limits_{z\in\mu_{q-1}^{mn}}
\frac{z^{-d}}{(q-1)^{mn}}\sum\limits_{u_1,\cdots,u_m\in\mathbb{Z}^n/(q-1)}
z^{u_1\otimes v_1+\cdots+u_m\otimes
v_m}\prod_{j=1}^mb_j^{u_j}S_f(T,\omega^{-u_j}).$$ Change the order of summation, we get the desired formula.
Let $v_1,\cdots,v_m$ be an integral basis of $\mathbb{R}^m$, and $g(y)=\sum\limits_jb_jy^{v_j}$ with $b_j\in\mu_{q-1}^n$. Then
$$S_{f\otimes g}(T,\omega^{-\frac{d}{q-1}(q^k-1)},\mathbb{F}_{q^k})
=\sum\limits_{\stackrel{u_1,\cdots,u_m\in\mathbb{Z}_{(p)}^n/\mathbb{Z}^n}{u_1\otimes
v_1+\cdots +u_m\otimes v_m\equiv \frac{d}{q-1},(q^k-1)u_j\equiv0}}
\prod_{j=1}^mb_j^{u_j(q^k-1)}\prod_{j=1}^m
S_f(T,\omega^{-u_j(q^k-1)},\mathbb{F}_{q^k}).$$
Just scale the variables $u_j$ in the last lemma.
We have $$S_f(T,\chi^q,\mathbb{F}_{q^k})=S_f(T,\chi,\mathbb{F}_{q^k}).$$
Since $\sigma:x\mapsto x^q$ is an automorphism of $\mu_{q^k-1}$, and extends to be an element of ${\rm
Gal}(\mathbb{Q}_{q^k}/\mathbb{Q}_q)$, we have $$S_f(T,\chi^q,\mathbb{F}_{q^k})
=\sum\limits_{x\in\mu_{q^k-1}^n}\chi(x^q)\psi_{q^k}\circ f(x)
=\sum\limits_{x\in\mu_{q^k-1}^n}\chi(x)\psi_{q^k}\circ
f(x^{\sigma^{-1}}).$$ Note that $$\psi_{q^k}\circ f(x^{\sigma^{-1}})
=\psi_q\circ{\rm
Tr}_{\mathbb{Q}_{q^k}/\mathbb{Q}_q}(f(x)^{\sigma^{-1}})=\psi_q\circ{\rm
Tr}_{\mathbb{Q}_{q^k}/\mathbb{Q}_q}(f(x))=\psi_{q^k}\circ f(x).$$The lemma now follows.
By the above lemma, the product $$\prod_{j=1}^mb_j^{-u_j(q^k-1)}\prod_{j=1}^m
S_f(T,\omega^{-u_j(q^k-1)},\mathbb{F}_{q^k})),$$ as a function of $(u_1,\cdots,u_m)$, is also defined on the orbit space $q\setminus(\mathbb{Z}_{(p)}^n/\mathbb{Z}^n)^m$. So we can restate the last corollary as follows.
Let $v_1,\cdots,v_m$ be an integral basis of $\mathbb{R}^m$, and $g(y)=\sum\limits_jb_jy^{v_j}$ with $b_j\in\mu_{q-1}^n$. Then
$$S_{f\otimes g}(T,\omega^{-\frac{d}{q-1}(q^k-1)},\mathbb{F}_{q^k})
=\sum\limits_{\stackrel{(u_1,\cdots,u_m)\in
q\setminus(\mathbb{Z}_{(p)}^n/\mathbb{Z}^n)^m}{u_1\otimes v_1+\cdots
+u_m\otimes v_m\equiv \frac{d}{q-1},(q^k-1)u_j\equiv0}}
|u|\prod_{j=1}^mb_j^{u_j(q^k-1)}\prod_{j=1}^m
S_f(T,\omega^{-u_j(q^k-1)},\mathbb{F}_{q^k}).$$
Let $v_1,\cdots,v_m$ be an integral basis of $\mathbb{R}^m$, and $g(y)=\sum\limits_jb_jy^{v_j}$ with $b_j\in\mu_{q-1}^n$. Then
$$C_{f\otimes g,\omega^{-d}}(s,T)=\prod_{\stackrel{(u_1,\cdots,u_m)\in
q\setminus(\mathbb{Z}_{(p)}^n/\mathbb{Z}^n)^m}{u_1\otimes v_1+\cdots
+u_m\otimes v_m\equiv \frac{d}{q-1}}}
\otimes_{j=1}^mC_{f,\omega^{-u_j(q^{|u|}-1)}}(s^{|u|}\prod_{j=1}^m
b_j^{-u_j(q^{|u|}-1)},T,\mathbb{F}_{q^{|u|}}).$$
We have $$\sum\limits_{k=1}^{\infty}\frac{-1}{(q^k-1)^{mn}}S_{f\otimes g}(T,\omega^{-\frac{d}{q-1}(q^k-1)},\mathbb{F}_{q^k})\frac{s^k}{k}$$$$=\sum\limits_{k=1}^{\infty}\frac{-1}{(q^k-1)^{mn}}\frac{s^k}{k}\sum\limits_{\stackrel{(u_1,\cdots,u_m)\in
q\setminus(\mathbb{Z}_{(p)}^n/\mathbb{Z}^n)^m}{u_1\otimes v_1+\cdots
+u_m\otimes v_m\equiv \frac{d}{q-1},(q^k-1)u_j\equiv0}}
|u|\prod_{j=1}^mb_j^{u_j(q^k-1)}\prod_{j=1}^m
S_f(T,\omega^{-u_j(q^k-1)},\mathbb{F}_{q^k})$$ $$=\sum\limits_{\stackrel{(u_1,\cdots,u_m)\in
q\setminus(\mathbb{Z}_{(p)}^n/\mathbb{Z}^n)^m}{u_1\otimes v_1+\cdots
+u_m\otimes v_m\equiv \frac{d}{q-1}}}
\sum\limits_{k=1}^{\infty}\frac{-1}{(q^{k|u|}-1)^{mn}}\frac{s^{k|u|}}{k}
\prod_{j=1}^mb_j^{u_j(q^{k|u|}-1)}\prod_{j=1}^m
S_f(T,\omega^{-u_j(q^{k|u|}-1)},\mathbb{F}_{q^{k|u|}})$$ So $$C_{f\otimes g,\omega^{-d}}(s,T)
=\exp(\sum\limits_{k=1}^{\infty}\frac{-1}{(q^k-1)^{mn}}
S_f(T,\omega^{-\frac{d}{q-1}(q^k-1)})\frac{s^k}{k})$$ $$=\prod_{\stackrel{(u_1,\cdots,u_m)\in
q\setminus(\mathbb{Z}_{(p)}^n/\mathbb{Z}^n)^m}{u_1\otimes v_1+\cdots
+u_m\otimes v_m\equiv \frac{d}{q-1}}}
\otimes_{j=1}^mC_{f,\omega^{-u_j(q^{|u|}-1)}}(s^{|u|}\prod_{j=1}^m
b_j^{-u_j(q^{|u|}-1)},T,\mathbb{F}_{q^{|u|}}).$$
[99]{}
A. Adolphson and S. Sperber, Twisted exponential sums and Newton polyhedra, J. reine angew. Math. 443 (1993), 151-177. R. Blache, Stickelberger’s theorem for $p$-adic Gauss sums, Acta Arith., 118(2005), no.1, 11-26. I.M. Gelfand, M.M. Kapranov and A.V. Zelevinsky, *Discriminatns, Resultants and Multidimensional Determinants, Birkhüser Boston, Inc., Boston, MA, 1994. S. Hong, Twisted exponential sums of diagonal forms, Proc. Amer. Math. Soc., C. Liu, The $L$-functions of twisted Witt coverings, J. Number Theory, 125 (2007), 267-284. $T$-adic exponential sums, Algebra & Number Theory, to appear. J-P. Serre, Endomorphismes complétement continus des espaces de Banach $p$-adiques, Publ. Math., IHES., 12(1962), 69-85.*
D. Wan, Variation of $p$-adic Newton polygons for L-functions of exponential sums, Asian J. Math., Vol 8, 3(2004), 427-474.
|
---
abstract: 'Homophily and social influence are the fundamental mechanisms that drive the evolution of attitudes, beliefs and behaviour within social groups. Homophily relates the similarity between pairs of individuals’ attitudinal states to their frequency of interaction, and hence structural tie strength, while social influence causes the convergence of individuals’ states during interaction. Building on these basic elements, we propose a new mathematical modelling framework to describe the evolution of attitudes within a group of interacting agents. Specifically, our model describes sub-conscious attitudes that have an activator-inhibitor relationship. We consider a homogeneous population using a deterministic, continuous-time dynamical system. Surprisingly, the combined effects of homophily and social influence do not necessarily lead to group consensus or global monoculture. We observe that sub-group formation and polarisation-like effects may be transient, the long-time dynamics being quasi-periodic with sensitive dependence to initial conditions. This is due to the interplay between the evolving interaction network and Turing instability associated with the attitudinal state dynamics.'
address: 'Centre for the Mathematics of Human Behaviour, Department of Mathematics and Statistics, University of Reading, Whiteknights, UK'
author:
- 'Jonathan A. Ward'
- Peter Grindrod
title: Aperiodic dynamics in a deterministic model of attitude formation in social groups
---
Social dynamics ,Cultural dissemination ,Coevolving networks ,Adaptive networks ,Social influence ,Homophily ,Activator-inhibitor
37N99 ,97M70 ,91D30
Introduction {#sec:introduction}
============
Our attitudes and opinions have a reciprocal relationship with those around us: who we know depends on what we have in common, while simultaneously our beliefs influence, and are influenced by, those of our peers. These two mechanisms—homophily and social influence—underpin a wide range of social phenomena, including the diffusion of innovations [@Valente96; @Watts02; @Bettencourt06; @Onnela10; @Krapivsky11], complex contagions [@Dodds05; @Centola07a; @LopezPintado08], collective action [@Granovetter78; @Fowler10; @Baldassarri07], opinion dynamics [@Friedkin99; @Hegselmann02; @Ben-Naim03; @Holme06; @Gil06; @Kozma08; @Mobilia11; @McCullen11; @Durrett11] and the emergence of social norms [@Shoham97; @Friedkin01; @Centola05]. Thus homophily and social influence represent the atomistic ingredients for models of social dynamics [@Castellano09]. Starting with these basic elements, we investigate a new type of modelling framework intended to describe the coevolution of sub-conscious attitudinal states and social tie strengths in a population of interacting agents.
The first ingredient in our modelling framework, homophily, relates the similarity of individuals to their frequency of interaction [@McPherson01]. Thus homophily is structural, affecting the strength of ties between people and hence the underlying social network. Homophily has been observed over a broad range of sociodemographics: implicit characteristics, such as age, gender and race; acquired characteristics, such as education, religion and occupation; and internal states that govern attitudes and behaviour [@Lazarsfeld54; @McPherson01]. Homophily inextricably links state dynamics with the evolution of social tie strength, and consequently a faithful model must be coevolutionary, connecting both the dynamics *of* the social network and the dynamics *on* the social network [@Centola07]. Such network models are known as *coevolving* or *adaptive*; see [@Gross07] for a review. There has been a recent surge of interest in coevolving networks, particularly models of opinion dynamics [@Holme06; @Gil06; @Kozma08; @Durrett11], which build on simple models of voting behaviour.
The second ingredient is social influence, which affects people’s attitudinal state through typically dyadic interactions. It is a fundamental result of social psychology that people tend to modify their behaviour and attitudes in response to the opinions of others [@Sherif36; @Campbell61; @Collins70], sometimes even when this conflicts sharply with what they know to be true [@Asch51] or believe to be morally justifiable [@Milgram65]. Similarly to Flache and Macy [@Flache11], we use diffusion to model social influence: agents adjust their state according to a weighted sum of the differences between their state and their neighbours’. The weights, which represent the strength of influence between pairs of agents, are the corresponding elements of the undirected (dynamic) social network, whose evolution is driven by homophily. Although our model is built on the notions of homophily and social influence described above, we point out that differentiating between the effects of these processes, particularly in observational settings, may be very difficult [@Aral09; @Shalizi11].
Social scientists have developed ‘agent-based’ models that incorporate homophily and social influence in order to examine a variety of social-phenomena, including group stability [@Carley91], social differentiation [@Mark98] and cultural dissemination [@Axelrod97], where a culture is defined as an attribute that is subject to social influence. In such models, an agent’s state is typically described by a vector of discrete cultures and the more similar (according to some metric) two agent’s states are, the higher the probability of dyadic interactions between them (homophily) in which one agent replicates certain attributes of the other (social influence). Surprisingly, the feedback between homophily and social influence does not necessarily lead to a global monoculture [@Axelrod97]. In fact, the dissolution of ties between culturally distinct groups, or equivalently the creation of ‘structural holes’ [@Macy03], may lead to *cultural polarisation*—equilibrium states that preserve diversity. However, such multi-cultural states are not necessarily stable when there is ‘cultural drift’, i.e. small, random perturbations or noise, which inevitably drive the system towards monoculture [@Klemm05]. There have been a number of attempts to develop models with polarised states that are stable in the presence cultural drift [@Centola07; @Flache11], but this is still an open area of research [@Castellano09].
Two key features differentiate our approach from those described above. Firstly, we specifically focus on *sub-conscious* attitude formation driven by a general class of activator-inhibitor processes. This is motivated by neuropsychological evidence that the activation of emotional responses are associated with the (evolutionarily older) regions of the brain know as the limbic system and our inhibitions are regulated by the (evolutionarily younger) prefrontal cortex [@Morgane05]. This has led psychologists to develop theories in which various personality traits (such as extraversion, impulsivity, neuroticism and anxiety) form an independent set of dimensions along which different types of behaviour may be excited or regulated [@Mathews99; @Eysenck67; @Gray87]. Thus it is natural in our modelling framework for these processes to be communicated independently and in parallel through distinct transmission channels and hence via distinct diffusion coefficients. The consequence of activator-inhibitor attitudinal state dynamics is that we would expect to encounter Turing instability, since the rates at which social influence can change homophilious attributes may differ dramatically.
Secondly, and in sharp contrast to recent models of cultural dissemination [@Axelrod97; @Centola07; @Flache11] and indeed many other types of behavioural model [@Castellano09] that are stochastic or probabilistic[^1], we consider a *deterministic*, continuous-time dynamical systems formulation. While this does not reflect the mercurial nature seemingly ingrained in human interaction, it allows us to probe the underlying mechanisms driving dynamical phenomena. In fact, our principle observation is that the tension between Turing instability and the coevolution of the social network and attitudinal states gives rise to aperiodic dynamics that are sensitive to initial conditions and surprisingly unpredictable. This begs the question, are the mechanisms that govern our behaviour the cause of its volatility? For parsimony, we also consider systems of homogeneous agents. This allows us to identify parameters that destabilise the global monocultural steady state, giving rise to transient sub-group formation.
This paper is organised as follows: in Section \[sec:model\], we describe our model in detail, analyse the stability of global monoculture and describe the underlying dynamical mechanisms; in Section \[sec:examples\] we illustrate typical numerical results from both a large population of individuals and a simple example consisting of just two agents; in Section \[sec:conclusion\] we summarise our work and finally in Section \[sec:discussion\] we discuss our results in the context of other models of cultural dynamics and polarisation phenomena.
A deterministic model of cultural dynamics {#sec:model}
==========================================
Consider a population of $N$ identical individuals (agents/actors), each described by a set of $M$ real attitude state variables that are continuous functions of time $t$. Let ${\mathbf{x}}_i (t) \in \mathbb{R}^M $ denote the $i$th individual’s attitudinal state. In the absence of any influence or communication between agents we assume that each individual’s state obeys an autonomous rate equation of the form $$\label{one}
\dot{{\mathbf{x}}}_{i}={\mathbf{f}}({\mathbf{x}}_i),\ \ i=1,...,N,$$ where ${\mathbf{f}}$ is a given smooth field over $\mathbb{R}^M$, such that ${\mathbf{f}}({\mathbf{x}}^*)=0$ for some ${\mathbf{x}}^*$. Thus (\[one\]) has a uniform population equilibrium ${\mathbf{x}}_i={\mathbf{x}}^*$, for $i=1,...,N$, which we shall assume is locally asymptotically stable. As discussed in the introduction, we shall more specifically assume that (\[one\]) is drawn from a class of activator-inhibitor systems.
Now suppose that the individuals are connected up by a dynamically evolving weighted network. Let $A(t)$ denote the $N \times N$ weighted adjacency matrix for this network at time $t$, with the $ij$th term, $A_{ij}(t)$, representing the instantaneous weight (frequency and/or tie strength) of the mutual influence between individual $i$ and individual $j$ at time $t$. Throughout we assert that $A(t)$ is symmetric, contains values bounded in \[0,1\] and has a zero diagonal (no self influence). We extend (\[one\]) and adopt a first order model for the coupled system: $$\label{two}
\dot{{\mathbf{x}}}_{i}={\mathbf{f}}({\mathbf{x}}_i)+ D \sum_{j=1}^NA_{ij}\left({\mathbf{x}}_{j}-{\mathbf{x}}_{i}\right),\ \ \ i=1,...,N.$$ Here $D$ is a real, diagonal and non-negative matrix containing the maximal transmission coefficients (diffusion rates) for the corresponding attitudinal variables between neighbours. Thus some of the attitude variables may be more easily or readily transmitted, and are therefore influenced to a greater extent by (while simultaneously being more influential to) those of neighbours. Note that ${\mathbf{x}}_i={\mathbf{x}}^*$, for $i=1,...,N$, is also a uniform population equilibrium of the augmented system.
Let ${\mathbf{X}}(t)$ denote the $M \times N$ matrix with $i$th column given by ${\mathbf{x}}_i(t)$, and ${\mathbf{F}}({\mathbf{X}})$ be the $M \times N$ matrix with $i$th column given by ${\mathbf{f}}({\mathbf{x}}_i(t))$. Then (\[two\]) may be written as $$\dot{{\mathbf{X}}}={\mathbf{F}}({\mathbf{X}}) - D {\mathbf{X}}\Delta.
\label{eq:Xmat}$$ Here $\Delta(t)$ denotes the weighted graph Laplacian for $A(t)$, given by $\Delta(t)={\rm diag}(\mathbf{k}(t))-A(t)$, where $\mathbf{k}(t)\in\mathbb{R}^N$ is a vector containing the degrees of the vertices ($k_{i}(t)=\sum_{j=1}^N
A_{ij}(t)$). Equation (\[eq:Xmat\]) has a rest point at ${\mathbf{X}}={\mathbf{X}}^*$, where the $i$th column of ${\mathbf{X}}^*$ is given by ${\mathbf{x}}^*$ for all $i=1,...,N$.
To close the system, consider the evolution equation for $A(t)$ given by $$\dot{A}=\alpha A\circ({\bf 1}-A)\circ\left({\varepsilon}{\bf 1}-\Phi({\mathbf{X}})\right).
\label{eq:A}$$ Here ${\bf 1}$ denotes the adjacency matrix of the fully weighted connected graph (with all off-diagonal elements equal to one and all diagonal elements equal to zero); $\circ$ denotes the element-wise ‘Hadamard’ matrix product; $\alpha>0$ is a rate parameter; ${\varepsilon}>0$ is a homophily scale parameter; and $\Phi:\mathbb{R}^{N\times
N}\rightarrow\mathbb{R}^{N\times N}$ is a symmetric matrix function that incorporates homophily effects. We assume $\Phi$ to be of the form $\Phi_{ij}:=\phi({\lvert{\mathbf{x}}_{i}-{\mathbf{x}}_{j}\rvert})\ge0$, where ${\lvert\cdot\rvert}$ is an appropriate norm or semi-norm, and the real function $\phi$ is monotonically increasing with $\phi(0)=0$. Note that the sign of the differences held in ${\varepsilon}{\bf 1}-\Phi(X)$ controls the growth or decay of the corresponding coupling strengths. All matrices in (\[eq:A\]) are symmetric, so in practice we need only calculate the super-diagonal terms. For the $ij$th edge, from (\[eq:A\]), we have $$\dot{A}_{ij}=\alpha{A}_{ij}(1-{A}_{ij})({\varepsilon}-\phi({\lvert{\mathbf{x}}_i-{\mathbf{x}}_j\rvert}).$$ The nonlinear “logistic growth"-like term implies that the weights remain in \[0,1\], while we refer to the term ${\varepsilon}-\phi({\lvert{\mathbf{x}}_i-{\mathbf{x}}_j\rvert})$ as the [*switch*]{} term.
Stability analysis {#sec:stability}
------------------
By construction, there are equilibria at ${\mathbf{X}}={\mathbf{X}}^*$ with either $A=0$ or $A={\bf 1}$. To understand their stability, let us assume that $\alpha\to 0$ so that $A(t)$ evolves very slowly. We may then consider the stability of the uniform population, ${\mathbf{X}}^*$, under the fast dynamic (\[eq:Xmat\]) for any fixed network $A$. Assuming that $A$ is constant, writing ${\mathbf{X}}(t)={\mathbf{X}}^* + \tilde{{\mathbf{X}}}(t)$ and Linearising (\[eq:Xmat\]) about ${\mathbf{X}}^*$, we obtain $$\label{thic}
\dot{\tilde{{\mathbf{X}}}}={\mathbf{df}}({\mathbf{x}}^*) \tilde{{\mathbf{X}}}- D \tilde{ {\mathbf{X}}} \Delta.$$ Here ${\mathbf{df}}({\mathbf{x}}^*)$ is an $M \times M$ matrix given by the linearisation of ${\mathbf{f}}({\mathbf{x}})$ at ${\mathbf{x}}^*$. Letting $(\lambda_i, {\mathbf{w}}_i)\in [0,\infty)
\times \mathbb{R}^N,\ i=1,...,N$, be the eigen-pairs of $\Delta$, then we may decompose uniquely [@nakao10]: $$\tilde{{\mathbf{X}}}(t) =\sum_{i=1}^N {\mathbf{u}}_i(t) {\mathbf{w}}_i^T,$$ where each ${\mathbf{u}}_i(t)\in \mathbb{R}^M$. The stability analysis of (\[thic\]) is now trivial since decomposition yields $$\dot {{\mathbf{u}}}_i=( {\mathbf{df}}({\mathbf{x}}^*)- D \lambda_i ) {\mathbf{u}}_i.$$ Thus the uniform equilibrium, ${\mathbf{X}}^*$, is asymptotically stable if and only if all $N$ matrices, $( {\mathbf{df}}({\mathbf{x}}^*)- D \lambda_i )$, are simultaneously stability matrices; and conversely is unstable in the $i$th mode of the graph Laplacian if $( {\mathbf{df}}({\mathbf{x}}^*)- D \lambda_i )$ has an eigenvalue with positive real part.
Consider the spectrum of $( {\mathbf{df}}({\mathbf{x}}^*)- D \lambda)$ as a function of $\lambda$. If $\lambda$ is small then this is dominated by the stability of the autonomous system, $ {\mathbf{df}}({\mathbf{x}}^*)$, which we assumed to be stable. If $\lambda$ is large then this is again a stability matrix, since $D$ is positive definite. The situation, dependent on some collusion between choices of $D$ and ${\mathbf{df}}({\mathbf{x}}^*)$, where there is a [*window of instability*]{} for an intermediate range of $\lambda$, is know as a Turing instability. Turing instabilities occur in a number of mathematical applications and are tied to the use of activator-inhibitor systems (in the state space equations, such as (\[one\]) here), where inhibitions diffuse faster than activational variables.
Now we can see the possible tension between homophily and Turing instability in the attitude dynamics when the timescale of the evolving network, $\alpha$, is comparable to the changes in agents’ states. There are two distinct types of dynamical behaviour at work. In one case, $\Delta(t)$ has presently no eigenvalues within the window of instability and each individual’s states ${\mathbf{x}}_i(t)$ approach the mutual equilibrium, ${\mathbf{x}}^*$; consequently all switch terms become positive and the edge weights all grow towards unity, i.e. the fully weighted clique. In the alternative case, unstable eigen-modes cause the individual states to diverge from ${\mathbf{x}}^*$, and subsequently some of the corresponding switch terms become negative, causing those edges to begin losing weight and hence partitioning the network.
The eigenvalues of the Laplacian for the fully weighed clique, $A={\bf
1}$, are at zero (simple) and at $N$ (with multiplicity $N-1$). So the interesting case is where the system parameters are such that $\lambda=N$ lies within the window of instability. Then the steady state $({\mathbf{X}}, A)=({\mathbf{X}}^*, {\bf 1})$ is unstable and thus state variable patterns will form, echoing the structure of (one or many of) the corresponding eigen-mode(s). This Turing driven symmetry loss may be exacerbated by the switch terms (depending upon the choice of ${\varepsilon}$ being small enough), and then each sub-network will remain relatively well intra-connected, while becoming less well connected to the other sub-networks. Once relatively isolated, individuals within each of these sub-networks may evolve back towards the global equilibrium at ${\mathbf{x}}^*$, providing that $A(t)$ is such that the eigenvalues of $\Delta$ have by that time left the window of instability. Within such a less weightily connected network, all states will approach ${\mathbf{x}}^*$, the switch terms will become positive, and then the whole qualitative cycle can begin again.
Thus we expect aperiodic or pseudo-cyclic emergence and diminution of patterns, representing transient variations in attitudes in the form of different [*norms*]{} adopted by distinct sub-populations. As we shall see though, the trajectory of any individual may be sensitive and therefore effectively unpredictable, while the dynamics of the global behaviour is qualitatively predictable.
In the next section we introduce a specific case of the more general setting described here.
Examples {#sec:examples}
========
We wish to consider activator-inhibitor systems as candidates for the attitudinal dynamics in (\[one\]) and hence (\[eq:Xmat\]). The simplest such system has $M=2$, with a single inhibitory variable, $x(t)$, and a single activational variable, $y(t)$. Let ${\mathbf{x}}_i(t)=(x_i(t), y_i(t))^T$ in (\[two\]), and consider the Scnackenberg dynamics defined by the field $${\mathbf{f}}({\mathbf{x}}_i)=(p-x_i y_i^2,\, q-y_i+x_iy_i^2)^T,$$ where $p > q\ge 0$ are constants. The equations have the required equilibrium point at $${\mathbf{x}}^*=\left(\frac{p}{(p+q)^2},\, p+q\right)^T,$$ and in order that ${\mathbf{df}}$ be a stability matrix, we must have $$p-q<(p+q)^3.$$ We employ $\phi_{ij}=(x_i-x_j)^2$ as the homophily function and we must have $D={\rm diag}(D_1, D_2)$ in (\[eq:Xmat\]).
When $M=2$, the presence of Turing instability depends on the sign of the determinant of $( {\mathbf{df}}({\mathbf{x}}^*)-D \lambda)$, which is quadratic in $\lambda$. For the Schnakenberg dynamics defined above, the roots of this quadratic are given by $$\lambda_{\pm}= \frac{ (p-q) - \frac{D_2}{D_1} (p+q)^3 \pm \sqrt{ \left[(p-q)-\frac{D_2}{D_1} (p+q)^3 \right]^2 - 4\frac{D_2}{D_1} (p+q)^4}
}{2 D_2 (p+q)}
>0.$$ It is straightforward to show that if $$\frac{D_2}{D_1}<\frac{3p+q-2\sqrt{2p(p+q)}}{(p+q)^3}:=\sigma_{\rm c},$$ then $\lambda_{\pm}$ are real positive roots and hence $( {\mathbf{df}}({\mathbf{x}}^*)-
D \lambda)$ is a stability matrix if and only if $\lambda$ lies outside of the interval $(\lambda_-, \lambda_+)$, the [*window of instability*]{}. Inside there is always one positive and one negative eigenvalue, and the equilibrium ${\mathbf{X}}^*$ is unstable for any fixed network $A$. Note that, as is well known, it is the *ratio* of the diffusion constants that determines whether there is a window of instability.
Group dynamics {#sec:group}
--------------
We now present simulations of the Schnakenberg dynamics with $N=10$. Parameter values are $p=1.25$, $q=0.1$, $\alpha=10^{4}$, $\varepsilon=10^{-6}$, $D_1\approx0.571$ and $D_2\approx0.037$. The ratio of the diffusion constants is $D_2/D_1:=\sigma=0.9\sigma_{\rm
c}$, and to ensure that the window of instability is centred on the fully coupled system we have $$D_1=\frac{(p-q)-\sigma(p+q)^3 }{2\sigma N(p+q)}.
\label{eq:centre}$$ The initial coupling strengths were chosen uniformly at random between 0.1 and 0.5. The initial values of $x$ and $y$ were chosen at equally spaced intervals on a circle of radius $10^{-3}$ centred on the uniform equilibrium.
In Figure \[fig:duA\] we illustrate the trajectories of $\delta_{ij}:=x_i-x_j$ and the corresponding coupling strengths $A_{ij}$ up to $t\approx440$.
![Trajectories of $\delta_{ij}:=x_i-x_j$ and $A_{ij}$ for all $(i,j)$ pairs for unstable parameters integrated until $t\approx440$. Parameter values and initial conditions are described in the main text. In the light grey shaded region, $\delta_{ij}<\varepsilon$ and the direction of trajectories are indicated with arrows. The grey horizontal line indicates the scaled stability threshold (unstable above, stable below). \[fig:duA\]](duA.eps){width="55.00000%"}
The shaded region corresponds to $\delta_{ij}<\varepsilon$, within which the $A_{ij}$ increase and outside of which they decrease, indicated by the dark grey arrows. The horizontal grey line marks the scaled instability threshold $\lambda_-/N$, which is indicative of the boundary of instability, above being unstable and below being stable. Because agents are only weakly coupled initially, their attitudes move towards the steady state ${\mathbf{x}}^*$, which causes the differences $\delta_{ij}$ to decrease. The switch terms subsequently become positive and hence the coupling strengths increase, along with the eigenvalues of the Laplacian $\lambda_i$. When one or more of the $\lambda_i$ are within the window of instability, some of the differences $\delta_{ij}$ begin to diverge. However, this eventually causes their switch terms to become negative, reducing the corresponding coupling strengths and hence some of the $\lambda_i$. This then affects the differences $\delta_{ij}$, which start to decrease, completing the qualitative cycle. As the system evolves beyond $t>440$, this quasi-cyclic behaviour becomes increasingly erratic.
Although the long term behaviour of any given agent is unpredictable, the behaviour of the mean coupling strength of the system fluctuates around the instability boundary $k_{-}/N$. In Figure \[fig:network\](a), we plot the time series of the mean coupling strength, $\bar{A}(t)$, between $t=5\times 10^3$ and $10^{4}$.
![Panel (a): mean coupling strength, $\bar{A}(t)$, time series. Panels (b)–(g): network snapshots at sequential time intervals. Node positions are plotted in the rotated coordinates $(x^{\prime},y^{\prime})$, shading illustrates coupling strength for edges and mean coupling strength for nodes. \[fig:network\]](network.eps){width="79.00000%"}
The dashed line indicates the instability boundary $k_{-}/N\approx0.6372$, which is very close to the time-averaged mean coupling strength $\langle A \rangle\approx0.6343$. Also plotted in Figure \[fig:network\] are snapshots of the network at six sequential times. To improve the visualisation of the network, the positions of nodes have been rotated by approximately $72^{\circ}$, since the differences in diffusion rates mean that the unrotated coordinates, $(x,y)$, become contracted in one direction. The shading of the nodes corresponds to their average coupling strength and the shading of the edges correspond to their weight. The sequence of figures illustrate the general scenario: agents’ trajectories cycle around the origin with the network repeatedly contracting and expanding as agents become more and less similar in attitude respectively.
We now illustrate how the quasi-equilibrium end state changes as the window of instability is moved. We fix all parameters as above, but consider a range of values of $D_1$ whilst keeping the ratio $D_2/D_1=0.9\sigma_{\rm c}$ held fixed. This has the effect of shifting the window of instability from above $\langle A\rangle=1$ to below as $D_1$ increases. We integrate until $t=1.5\times10^{5}$ and then compute the mean coupling strength $\langle A\rangle$ for $t\ge10^{4}$. We compute 50 realisations for each set of parameters, the results of which are plotted in Figure \[fig:bif\].
![Mean coupling strength at large times for different values of $D_1$. Shading indicates the window of instability, the dashed grey line indicates where $A={\bf 1}$. The black line is the median of 50 realisations and the black markers are the mean coupling strengths for each of the realisations. \[fig:bif\]](bif.eps){width="55.00000%"}
The shaded region corresponds to the (scaled) region of instability, the dashed grey line is where $\langle A\rangle=1$, the black line is the median of the fifty realisations and the dots are the values from each of the realisations. At low values of $D_1$, where the $A={\bf
1}$ equilibrium first becomes unstable, the mean coupling strength fits tightly to the lower edge of the instability boundary at $\lambda_{-}/N$. When the $A={\bf 1}$ equilibrium restabilises (at $D_1\approx0.78$), the long time behaviour of the mean coupling strength changes, moving away from the $\lambda_{-}/N$ boundary. In the region between $D_1\approx 1.5$ and $D_1\approx2.3$, some realisations return to the fully coupled equilibrium $A={\bf 1}$, but not all. We would expect that simulating for longer would result in more realisations reaching the fully coupled equilibrium, although it is possible that its basin of attraction does not include every initial condition in the set that we are sampling from.
Dyad dynamics {#sec:dyad}
-------------
To probe the mechanism driving the aperiodic dynamics illustrated in Section \[sec:group\], we consider a simpler dynamical setting consisting of just two agents. This reduces the coupling strength evolution (\[eq:A\]) to a single equation, and hence five equations in total, $$\begin{aligned}
\dot{x}_1&=p-x_1y_1^2-D_1a(x_1-x_2),\label{eq:dyadx1}\\
\dot{x}_2&=p-x_2y_2^2+D_1a(x_1-x_2),\\
\dot{y}_1&=q-y_1+x_1y_1^2-D_2a(y_1-y_2),\\
\dot{y}_2&=q-y_2+x_2y_2^2+D_2a(y_1-y_2),\label{eq:dyady2}\\
\dot{a}&=\alpha a(1-a)\left[\varepsilon-(x_1-x_2)^2\right].\label{eq:dyada}\end{aligned}$$
In Figure \[fig:chaos\], we plot the trajectories for each of the two agents (black and grey lines) in $(x,y)$ space, and in $(x,y,a)$ space in the upper-right inset.
![Main: Trajectories of dyadic system in $(x,y)$ space with unstable parameters. Upper-right inset: Trajectories in $(x,y,a)$ space. Lower-left inset: zoom of boxed region in the main plot. Parameters described in the main text. \[fig:chaos\]](chaos.eps){width="99.00000%"}
The parameter values are $p=1.25$, $q=0.1$, $\alpha=10^{4}$, $\varepsilon=10^{-6}$, $D_1\approx2.857$ and $D_2\approx0.184$. Again, the diffusion constants have the ratio $D_2/D_1=0.9\sigma_{\rm c}$ and the window of instability is centred on the fully coupled system via (\[eq:centre\]). The initial conditions are chosen near to the uniform equilibrium $\mathbf{x}^*=(x^*,y^*)^T$, specifically $x_1(0)=x^*+1.5\times10^{-4}$, $x_2(0)=x^*-1.5\times10^{-4}$, $y_1(0)=y^*-1\times10^{-6}$, $y_2(0)=y^*+1\times10^{-6}$; the initial coupling strength is $a(0)=0.1$. This system is numerically stiff—on each cycle, trajectories get very close to the equilibrium ${\mathbf{x}}^*$ and the invariant planes $a=0$ and $a=1$—thus very low error tolerances are necessary in order to accurately resolve the trajectories.
The mechanisms driving the near cyclic behaviour illustrated in Figure \[fig:chaos\] are qualitatively similar to those described in Section \[sec:stability\], but the present case is much simpler since the coupling constant, $a$, is a scalar. If we consider $a$ as a parameter in the attitudinal dynamics (\[eq:dyadx1\]–\[eq:dyady2\]), then a Turing instability occurs as a pitchfork bifurcation at some $a=a_*$, where $0<a_*<1$. The equilibria at $({\mathbf{x}},a)=({\mathbf{x}}^*,0)$ and $({\mathbf{x}}^*,1)$ are both saddle-foci, where the unstable manifolds are respectively parallel to the $a$-axis and entirely within the attitudinal state space, ${\mathbf{x}}$. Near to the $({\mathbf{x}}^*,1)$ equilibrium, a given trajectory tracks the unstable manifold of $({\mathbf{x}}^*,1)$ in one of two opposing directions, the choice of which is sensitively dependent on its earlier position when $a\approx a_*$. The combination of this sensitivity together with the spiral dynamics around the unstable manifold of $({\mathbf{x}}^*,0)$, leads to an orbit switching sides unpredictably on each near-pass of $({\mathbf{x}}^*,1)$ (c.f. the top-right inset of Figure \[fig:chaos\]). The mechanism by which this chaotic behaviour arises is not standard (e.g. via a Shilnikov bifurcation or homoclinic explosion) and warrants its own study, which we address in an article currently in preparation.
Conclusions {#sec:conclusion}
===========
We have proposed a new modelling framework to describe the evolution of sub-conscious attitudes within social groups. We based this framework on the fundamental mechanisms of homophily and social influence, but it differs from previous approaches in two respects. Firstly, we have focused on sub-conscious attitudes, where it is natural to consider dynamics described by a class of activator-inhibitor processes. Secondly, we have formulated a deterministic system, enabling us to highlight (via mathematical analysis and simulation) the mechanisms driving dynamical phenomena. Specifically, we have illustrated that the tension between Turing instability and the evolving network topology gives rise to behaviour that at the system level is qualitatively predictable — sub-group formation and dissolution — yet at the level of individual agent journeys is entirely unpredictable. We point out that a stochastic model based on similar principles to those described in this paper is presented in [@Parsons12], where qualitatively similar dynamical phenomena are also observed. Thus we might conclude that even if stochasticity is entirely absent, the mechanisms that govern human behaviour seem to give rise to unpredictable dynamics.
Discussion {#sec:discussion}
==========
While we have differentiated our modelling framework from other models of attitudinal dynamics [@Axelrod97; @Centola07; @Flache11], we now discuss our findings in the context of more general cultural models and cultural polarisation.
Current interest in cultural models largely stems from the work of Axelrod [@Axelrod97], who demonstrated that local convergence could lead to cultural polarisation. This topic has particular resonance in our digital society: will global connectivity accelerate a descent into monoculture, or can diversity persist? Models such as Axelrod’s provide us with an optimistic outlook, suggesting that even the most basic mechanisms that model social influence and homophily can lead to cultural diversity. But by no means is there presently a completely satisfactory understanding of this phenomena. The polarised states of the Axelrod model are fragile; even low rates of random perturbations to cultural traits can reinstate global monoculture [@Klemm05]. Thus additional dynamical rules have been investigated in this context. The variant of the Axelrod model proposed by Centola et al. [@Centola07] allows agents to disassociate themselves from neighbours that have no similar traits and select a new neighbour at random. Similarly, a number of adaptive network models of opinion dynamics have also found absorbing polarised states, in which groups with differing opinions are completely disconnected [@Gil06; @Kozma08; @Durrett11]. Such polarised states seem artificial and we conjecture that a form of cultural drift, characterised by random rewiring of a small number of edges, would destabilise these states.
This touches on another key issue: in the absence of noise, the mechanisms employed by cultural dissemination models typically *reduce* diversity. It is not surprising then that these types of models can be perturbed in such a way that the eventual result is monoculture. An approach that allows diversity to increase has been suggested by Flache and Macy [@Flache11]. They model social influence via diffusion, whereby an agent adjusts their cultural state, described by a vector of continuous real variables, according to a weighted sum of the differences between their state and their neighbours’. The weights are dynamic and their evolution is driven by homophily. In some sense, the corresponding elements of our model are like a continuous-time version of the Flache and Macy model. However, Flache and Macy consider the weights embedded on a clustered network and, more importantly, allow their weights to be negative, representing *xenophobia*. It is this feature that allows diversity to both decrease and increase via diffusion and convergence respectively. The effects of cultural drift on polarised states in the Flache and Macy model have not been investigated in detail, but perturbations can cause agents to switch groups [@Flache11] and so we would expect that sustained noise would erode smaller groups. Monoculture is also a stable fixed point of their model.
The key element that differentiates our model is that agents’ cultural states have an activator-inhibitor dynamic that is independent of other agents. The presence of diffusion allows for Turing instability and hence means that diversity can increase. Moreover, we can identify regions in which global monoculture is *unstable*. For fixed or slowly evolving networks, instability gives rise to stable ‘Turing patterns’ [@nakao10], which could be interpreted as culturally polarised states. However, one would expect inter-group connections to be weaker than intra-group connections within polarised states. But if homophily dissolves such inter-group ties then the patterned or polarised states can no longer be stable, since it is precisely the differences in culture that balance individuals’ attitudinal dynamics with diffusion. If non-trivial stable equilibria were to exist in our model, they would involve a delicate balance of cultural differences within the switching terms. However, we have seen no evidence of this occurring in numerical simulations. Thus in its present form, sub-group formation and polarisation are transient phenomena in our model.
It is possible however that extensions to our model could produce stable polarised states. For example, introducing agent heterogeneity, in the form of distinct uncoupled equilibria, offers some promise. Agents could then adopt a state close to their uncoupled equilibrium, allowing distinct groups to form, but Turing instability could still destabilise the monocultural equilibrium. Alternatively, the network evolution equations could include higher order effects such as edge snapping [@DeLellis10] or triangulation. These ideas will be investigated further in follow-up work and we hope that our model may provide a new paradigm from which to explore cultural polarisation phenomena.
Acknowledgements {#acknowledgements .unnumbered}
================
JAW and PG acknowledge the EPSRC for support through MOLTEN (EP/I016058/1). We would like to thank Jon Dawes, Michael Macy and those at the Centre for Mathematics of Human Behaviour for their valuable input, discussion and comments.
[48]{} natexlab\#1[\#1]{}\[2\][\#2]{} , , () . , , () . , , , , , () . , , , () . , , , , () . , , , () . , , , , () . , , , () . , , () . , , , () . , , , () . , , , () . , , , (). , , , , , () . , , , () . , , , () . , , , () . , , () . , , , , , () . , , , , , , , , , () . , , , () . , , () . , , , , () . , , , , () . , , , , () . , , , , , . , , , , , () . , , , () . , , , , . , , in: , , , . , , , , , . , , in: , , , . , , () . , , , () . , , , , () . , , , () . , , () . , , () . , , () . , , , , , in: , , , . , , , , , () . , , , , () . , , , () . , , , , . , , , , edition, . , , , () . , , , . , , , , , () .
[^1]: A notable exception is the deterministic, discrete-time model of Friedkin and Johnsen [@Friedkin99]; see also [@Hegselmann02] and references therein.
|
---
author:
- 'Delphine Hypolite$^{1}$, Stéphane Mathis$^{1}$, Michel Rieutord $^{2}$'
bibliography:
- 'mabib.bib'
title: 2D dynamics of the radiation zone of low mass stars
---
Introduction
============
It has long been known that rotation plays a central role in the dynamical and chemical evolution of stars ([@meynetmaeder00], [@maedermeynet09]). Through the rotationnal mixing driven by the differential rotation, the meridional circulation and the turbulence it sustains, chemical elements and angular momentum are transported through the radiation zones of stars (e.g. [@zahn92], [@maederzahn98], [@mathiszahn04], [@MR06], [@ELR07]) in a way that needs to be precisely characterized and modeled.
![Squared Brunt-Väisälä frequency from a 1D $1 M_\odot$ MESA model near the ZAMS (with $Z=0.02$, $\alpha_{\rm MLT}=2$).[]{data-label="fig2"}](profilesBV1Msun.eps){width="1\linewidth"}
Indeed, helioseismology provides the internal rotation profile deep within the Sun (until $0.2R_\odot$) where the radiation zone has a quasi solid rotation under the tachocline ([@couvidat03], [@garcia07]) which 1D rotating stellar models fail to reproduce ([@TC05], [@TC10]). One reason probably lies in the need for other physical mechanisms such as internal gravity waves ([@zahn97], [@TC05]) or an internal magnetic field ([@GM98], [@spruit99]) or/and in the multidimensional nature of the differential rotation and of the meridional circulation [@MR06]. Moreover, one wants to go beyond the slow rotation hypothesis underlying the 1D modeling of rotating stellar radiation zones to properly describe the dynamics occuring during the early stages of the evolution of stars during which they are rapidly rotating ([@GB13], [@GB15]). To tackle this issue, we build a 2D numerical model of a fast rotating sphere representing such a radiative core at the top of which we impose a latitudinal shear so as to reproduce the conical differential rotation applied by the convective zone.
Numerical simulations ([@matt11], [@kapyla14], [@gastine14], [@varela16]) reveal that the differential rotation of the convective envelope of low mass stars may be solar (with slow pole and fast equatorial regions) or anti-solar (with fast pole and slow equator) depending on the value of the convective Rossby number, which quantifies the ratio of the rotation period and of the convective turnover time (a small Rossby number corresponds to the rapidly rotating regime; see e.g. [@brun15]). We propose to perform a systematic parameter study over the boundary condition describing the convective differential rotation applied at the top of the radiation core between the pole and the equator. We provide a full 2D description of the differential rotation and the associated meridional circulation computed self-consistently using the Boussinesq approximation. Since previous 2D modelings ([@friedlander76], [@garaud02]) are limited to the solar case, this setting lays a first important step for the next generation of rotating models of low mass stars in 2D.
Description of the model
========================
Scaled equations of the dynamics
--------------------------------
We consider a viscous fluid enclosed within a spherical shell, see Fig. \[fig1\], and aim to describe the velocity field and the temperature field resulting from the combined action stable stratification and rotation. The core is rotating and we assume that it is not deformed, meaning that the effects of the centrifugal acceleration are neglected. We focus on the case of fast rotation, which leads to the damping of the baroclinic modes on a time scale shorter than in the slow rotation case [@busse81]. It allows us to describe the steady state which arises from this configuration and which is the solution of the system of equations (6) from [@MR06] (hereafter R06) in the Boussinesq approximation, which reads $$\left \{
\begin{array}{lcl}
\vec{\nabla} \times (\vec{e}_z \wedge \vec{u} - \theta_T \vec{r} - E\Delta\vec{u}) = - n^2(r) \sin{\theta}\cos{\theta}\vec{e}_{\varphi}\; ,\\
\left(\frac{n^2(r)}{r} \right)u_r = \frac{E}{Pr}\Delta\theta_T\; ,\\
\vec{\nabla}\cdot\vec{u} = 0\; ,\\
\end{array}
\right.
\label{eq1}$$ where $\vec{e}_z$ is the axis of rotation of the fluid, $\vec{u}$ is the scaled velocity field using the baroclinic scale $V=\frac{\Omega_0\mathcal{N}^2R_c^2}{2g_S}$ from R06 where $\mathcal{N}^2=\alpha T_\star g_S/R_c$ is the scale of the squared Brunt-Väisälä frequency profile, $\alpha$ is the fluid dilation coefficient and $g_S$ is the surface gravity. The dimensionless temperature field $\theta_T$ is scaled using $\frac{\Omega_0^2R_c}{g_S}T_\star$ as a temperature scale with $T_\star$ an arbitrary stellar temperature scale. In the same way, the scaled squared Brunt-Väisälä frequency profile $n^2(r)$ using is scaled $\mathcal{N}^2$. We choose to work within the corotating frame with the pole of the shell rotating at the rate $\Omega_0$. The first equation is known as the vorticity equation. Its first term is the Coriolis acceleration, the second one is the buoyancy force, the third one the viscous force and on the right hand side is the baroclinic torque arising in stably stratified fluid. The second equation is the energy equation where the heat advection balances its diffusion and where we do not take into account as a first step any nuclear heating. The last equation is the continuity equation when using the Boussinesq approximation.
The dimensionless numbers that characterize such a system are:
- The Ekman number $E=\displaystyle{\frac{\nu}{2\Omega_0R_c^2}}$.
It quantifies the importance of the kinematic viscosity $\nu$ over the Coriolis effects where $R_c$ is the radius of the radiative core. Using realistic values from ZAMS 1D MESA stellar models ([@paxton10]) for the microscopic viscosity and the radius $R_c$, we show that in the solar case ($M=1M_\odot$, $Z=0.02$, $\alpha_{\rm{MLT}}=2$), the Ekman number is around $\sim10^{-11}-10^{-13}$. This value is very small and numerical simulations cannot reach it. We therefore study the case $E=10^{-6}$, which still describes qualitatively the physical behavior of the solution (the asymptotic regime is reached).
- The Prandtl number $Pr=\displaystyle{\frac{\nu}{\kappa}}$.
This number is the ratio of the viscosity to the thermal diffusivity. We use the solar value $Pr=2.10^{-6}$ (see e.g. [@ruediger14]) for our simulations.
{width="0.3\linewidth"} {width="0.3\linewidth"} {width="0.3\linewidth"}
{width="0.3\linewidth"} {width="0.3\linewidth"} {width="0.3\linewidth"}
Dynamical boundary conditions
-----------------------------
We impose a shear at the surface $r=R_c$ in order to account for a differentially rotating convective zone lying at the top of the radiation core such as $$\Omega_{cz}(r=R_c,\theta)= \Omega_0 + \Delta\Omega\sin^2\theta\; ,$$ which is the simplest expression that simulations of the dynamics of stellar convective envelopes inspire ([@matt11], [@kapyla14], [@gastine14]). When using dimensionless quantities in the corotating frame with the pole of the radiation core, the azimuthal velocity reads $$u_\varphi(r=1,\theta)= b \sin^3\theta\; .$$ The dimensionless number $b=\frac{R_c \Delta\Omega}{V}=\frac{1}{\mathcal{R}o}\frac{\Delta\Omega}{2\Omega_0}$ quantifies the shear at the top of the radiation core. Differential rotation is solar-like when the equatorial regions rotate faster than the pole ($b>0$) and anti-solar otherwise ($b<0$). The Rossby number has been written $\mathcal{R}o=\frac{V}{2\Omega_0R_c}$ and quantifies the amplitude of the non-linearity of the system. This number is always less than $10^{-2}$ in the solar radiation core and therefore allows to focus on the linear case as long as $b<10^2$.
The meridional components of the velocity fields are set to zero at the upper boundary $r=R_c$ since the convection envelope applies stresses at the top of the radiative core and act as a non-penetrative condition. At the center of the core ($r=0$), the azimuthal velocity, the radial velocity, its first radial derivative and the first radial derivative of the temperature perturbation are set to zero.
In the next section, we perform a systematic parameter study over the shear parameter $b$.
Numerical results
=================
We use a spectral method to solve numerically the set of Eqs. (\[eq1\]). On the horizontal directions, the solution is described by vectorial spherical harmonics on the $(\vec{R}_l^m,\vec{S}_l^m,\vec{T}_l^m)$ basis [@R87] where $$\vec{R}_l^m=Y_l^m(\theta,\varphi) \vec{e_r},\qquad \vec{S}_l^m=\vec{\nabla}_H Y_l^m, \qquad \vec{T}_l^m=\vec{\nabla}_H \wedge \vec{R}_l^m\; .$$ $Y_l^m$ are the normalized spherical harmonics, functions of colatitude and azimuth ($\theta$,$\varphi$), $\vec{e_r}$ is the unit radial vector and the horizontal gradient $\vec{\nabla}_H=\partial_\theta \vec{e_\theta}+\frac{1}{\sin\theta}\partial_\varphi \vec{e_\varphi}$ is defined on the unity sphere.
Radially, the solution is projected on Chebyshev polynomials on a Gauss-Lobatto grid (e.g. [@canuto06]). This grid has more points on the edge allowing a good description of the boundary layer at the top of the domain.
We here use a profile from a ZAMS[^1] $1M_\odot$ 1D MESA model (with $Z=0.02$, $\alpha_{\rm MLT}=2$) for interpolating the Brunt-Väisälä frequency profile on the Gauss-Lobatto grid. We show this profile within the radiation zone of the model on Fig. \[fig2\]. Changing the mass considered within the range of low mass stars ($0.5-1.1M_\odot$) does not affect the behavior of the solutions presented next.
On Fig. \[fig3\], we compute and show the resulting differential rotation $\delta\Omega$ relative to the pole rotation for $b=\{-10, 10^{-2}, 10\}$. The extremum value normalizes the field in each case. When $|b|\leq 10^{-2}$, the solution is the baroclinic solution described by R06. The shear is too weak to be felt. This solution is called the thermal wind with a roughly shellular differential rotation close to the equator. When $|b|>10^{-2}$, the amplitude of the geostrophic flow arising from the shear applied at the upper boundary is larger than the baroclinic solution and the Taylor-Proudman balance tends to be restored (quasi columnar structure). Indeed, the azimuthal velocity at the equator has an $\mathcal{O}(b)$ amplitude when we subtract the baroclinic solution and we show numerically that the baroclinic solution is overcomed for $|b|>10^{-2}$. The associated differential rotation thus tends towards a cylindrical profile.
On Fig. \[fig4\], the meridional circulation is computed for $b=\{-10, 10^{-2}, 10$}. When the baroclinic solution dominates, the number of cells of the meridional circulation is equal to the number of inflection points of the Brunt-Väisälä frequency profile plus one, here two. The rotation aligns the cells with the cylindrical z-direction (i.e. along the axis of rotation). When $|b|>10^{-2}$, the dynamics is dominated by the geostrophic flow arising from the shear and the meridional circulation is dominated by a single, global circulation cell in each hemisphere. For $b$ positive (solar-like differential rotation), the meridional circulation is counterclockwise and clockwise for negative $b$. At the radiation/convection interface, the fluid moves toward the pole which rotates slower than the equator (for $b>0$ and vice-versa for $b<0$).
Discussion and perspectives
===========================
In this work, we propose a first 2D description of the dynamics of fast rotating radiative cores of low mass stars undergoing the shear induced by the differential rotation of the convective envelope at its upper boundary.
This simple Boussinesq model shows the necessity to resort to a 2D approach. Indeed, the arising flow from the shear is cylindrical, which must be described using a large number of spherical harmonics. Such a setting also permits to take into account the Brunt-Väisälä frequency from selected evolutionary stages and stellar masses allowing the exploration of the HR diagram in 2D.
In the solar case, the shear is evaluated close to $b_\odot\simeq10$. Therefore, the third plot of Fig. \[fig3\] exhibits the best fast rotating solar model we may provide within the current setting. We compute a core to surface rotation rate ratio lower than $1$ which is in agreement with what [@benomar15] deduced from observations. But it also tells us to expect the differential rotation to be cylindrical until half of the radiative core depth with a fast equatorial region. If we compare to the actual solar rotation profile, this calls for other physical processes responsible for transport of angular momentum deep within the internal regions, which is not taken into account in the present setting. Internal gravity waves [@zahn97], anisotropic turbulence ([@zahn92], [@maeder03], [@mathiszahn04]) and magnetic fields ([@GM98], [@spruit99], [@strugarek11], [@AGW13]) are the best candidates and will be introduced in our 2D model in future works.
Acknowledgments {#acknowledgments .unnumbered}
===============
[D.H. and S.M. thank funding by the European Research Council through ERC SPIRE grant 647383, the CNES PLATO grant at CEA Saclay and the MESA website.]{}
[^1]: We use the stopping condition *stop\_near\_zams* of the MESA code to produce this model (http://mesa.sourceforge.net).
|
---
abstract: 'We study semiprojective, subhomogeneous $C^*$-algebras and give a detailed description of their structure. In particular, we find two characterizations of semiprojectivity for subhomogeneous $C^*$-algebras: one in terms of their primitive ideal spaces and one by means of special direct limit structures over one-dimensional NCCW complexes. These results are obtained by working out several new permanence results for semiprojectivity, including a complete description of its behavior with respect to extensions by homogeneous $C^*$-algebras.'
address: 'Dominic Enders Department of Mathematical Sciences, University of Copenhagen Universitetsparken 5, DK-2100 Copenhagen Ø, Denmark'
author:
- Dominic Enders
date: 'July 14, 2014'
title: |
A characterization of semiprojectivity\
for subhomogeneous $C^*$-algebras
---
Introduction
============
The concept of semiprojectivity is a type of perturbation theory for $C^*$-algebras which has become a frequently used tool in many different aspects of $C^*$-algebra theory. Due to a certain kind of rigidity, semiprojective $C^*$-algebras are technically important in various situations. In particular, the existence and comparison of limit structures via approximate interwinings, which is an integral part of the Elliott classification program, often relies on perturbation properties of this type. This is one of the reasons why direct limits over semiprojective $C^*$-algebras, e.g., AF- or A$\mathbb{T}$-algebras, are particularly tractable and one therefore constructs models preferably from semiprojective building blocks. The most popular of those are without doubt the non-commutative CW-complexes (NCCWs) introduced by Eilers, Loring and Pedersen. These are in fact semiprojective in dimension one ([@ELP98], but see also [@End14]). In this paper, we study semiprojectivity for general subhomogeneous $C^*$-algebras and see whether there exist more interesting examples, i.e., besides the one-dimensional NCCW complexes (1-NCCWs), that could possibly serve as useful building blocks in the construction of ASH-algebras. In Theorem \[thm structure\], we give two characterizations of semiprojectivity for subhomogenous $C^*$-algebras: an abstract one in terms of primitive ideal spaces and a concrete one by means of certain limit structures. These show that it is quite a restriction for a subhomogeneous $C^*$-algebra to be semiprojective, though many examples beyond the class of 1-NCCWs exist. On the other hand, a detailed study of the structure of these algebras further reveals that they can always be approximated by 1-NCCWs in a very strong sense, see Corollary \[cor approx nccw\], and hence essentially share the same properties.
The work of this paper is based on the characterization of semiprojectivity for commutative $C^*$-algebras, which was recently obtained by S[ø]{}rensen and Thiel in [@ST12]. They showed that a commutative $C^*$-algebra ${\mathcal{C}}(X)$ is semiprojective if and only if $X$ is an absolute neighborhood retract of dimension at most 1 (a 1-ANR), thereby confirming a conjecture of Blackadar and generalizing earlier work of Chigogidze and Dranishnikov on the projective case ([@CD10]). Their characterization further applies to trivially homogeneous $C^*$-algebras, i.e. to algebras of the form ${\mathcal{C}}(X,{\mathbb{M}}_n)$. In a first step, we generalize their result to general homogeneous $C^*$-algebras. The main difficulty, however, is to understand which ways of ’gluing together’ several homogeneous $C^*$-algebras preserve semiprojectivity, or more precisely: Which extensions of semiprojective, homogeneous $C^*$-algebras are again semiprojective? Conversely, is semiprojectivity preserved when passing to a homogeneous subquotient? These questions essentially ask for the permanence behavior of semiprojectivity along extensions of the form $0\rightarrow{\mathcal{C}}_0(X,{\mathbb{M}}_n)\rightarrow A\rightarrow B\rightarrow 0$. While it is known that the permanence properties of semiprojectivity with respect to extensions are rather bad in general, we are able to work out a complete description of its behavior in the special case of extensions by homogeneous ideals, see Theorem \[thm 2 out of 3\]. With this permanence result at hand, it is then straightforward to characterize semiprojectivity for subhomogeneous $C^*$-algebras in terms of their primitive ideal spaces. In particular, it is a necessary condition that the subspaces corresponding to a fixed dimension are all 1-ANRs. Combining this with the structure result for one-dimensional ANR-spaces from [@ST12], we further obtain a more concrete description of semiprojective, subhomogeneous $C^*$-algebras by identifying them with certain special direct limits of 1-NCCWs.\
This paper is organized as follows. In section \[section preliminaries\], we briefly recall some topological definitions and results that will be used troughout the paper. We further remind the reader of some facts about semiprojectivity, subhomogeneous $C^*$-algebras and their primitive ideal spaces. We then start by constructing a lifting problem which is unsolvable for strongly quasidiagonal $C^*$-algebras. This lifting problem then allows us to extend the results of [@ST12] from the commutative to the homogeneous case.
Section \[section semiprojectivity\] contains a number of new contructions for semiprojective $C^*$-algebras. We first introduce a technique to extend lifting problems, a method that can be used to show that in certain situations semiprojectivity passes to ideals. After that, we introduce a class of maps which give rise to direct limits that preserve semiprojectivity. Important examples of such maps are given and discussed.
Section \[section extensions\] is devoted to the study of extensions by homogeneous $C^*$-algebras, i.e. extensions of the form $0\rightarrow{\mathcal{C}}_0(X,{\mathbb{M}}_n)\rightarrow A\rightarrow B\rightarrow 0$. In \[section retract maps\], we define and study a certain set-valued retract map $R\colon {\operatorname{Prim}}(A)\rightarrow 2^{{\operatorname{Prim}}(B)}$ associated to such an extension. We discuss regularity concepts for $R$, i.e. continuity and finiteness conditions, and show how regularity of $R$ relates to lifting properties of the corresponding Busby map and, by that, to splitting properties of the extension itself. In particular, we identify conditions under which regularity of $R$ implies the existence of a splitting map $s\colon B\rightarrow A$ with good multiplicative properties. After that, we verify the required regularity properties for $R$ in the case of a semiprojective extension $A$. In section \[section existence limit structures\] it is shown how certain limit structures for the space $X$ give rise to limit structures for the extension $A$, again provided that the associated retract map $R$ is sufficiently regular. Putting all these results together in \[section keeping track\], we find a ’2 out of 3’-type statement, Theorem \[thm 2 out of 3\], which gives a complete description for the behavior of semiprojectivity along extensions of the considered type.
In section \[section main\], we use this permanence result to work out two characterizations of semiprojectivity for subhomogeneous $C^*$-algebras. These are presented in Theorem \[thm structure\], the main result of this paper. Based on this, we find a number of consequences for the structure of these algebras, e.g. information about their $K$-theory and dimension. Further applications, such as closure and approximation properties, are discussed in \[section applications\]. We finish by illustrating how this also gives a simple method to exclude semiprojectivity and show that the higher quantum permutation algebras are not semiprojective.
Preliminaries {#section preliminaries}
=============
The structure of 1-dimensional ANR-spaces {#section 1-ANR}
-----------------------------------------
We are particularly interested in ANR-spaces of dimension at most one. The structure of these spaces has been studied and described in detail in [@ST12 section 4]. Here we recall the most important notions and results. More information about ANR-spaces can be found in [@Bor67]. For proofs and further reading on the theory of continua, we refer the reader to Nadler’s book [@Nad92].
\[def anr\] A compact, metric space $X$ is an absolute retract (abbreviated AR-space) if every map $f\colon Z\rightarrow X$ from a closed subspace $Z$ of a compact, metric space $Y$ extends to a map $g\colon Y\rightarrow X$, i.e. $g\circ\iota=f$ with $\iota\colon Z\rightarrow Y$ the inclusion map: $$\xymatrix{
& Y \ar@{-->}[dl]_g \\
X & Z \ar[l]^f \ar[u]_\iota
}$$ If every map $f\colon Z\rightarrow X$ from a closed subspace $Z$ of a compact, metric space $Y$ extends to a map $g\colon V\rightarrow X$ on a closed neighborhood $V$ of $Z$ $$\xymatrix{
& Y \\
& V \ar@{-->}[dl]_g \ar[u] \\
X & Z \ar[l]^f \ar[u]_\iota
}$$ then $X$ is called an absolute neighborhood retract (abbreviated ANR-space).
A compact, locally connected, metric space is called a Peano space. A connected Peano space is called a Peano continuum. Now given an ANR-space $X$, we can embed it into the Hilbert cube $\mathcal{Q}$ and obtain a retract from a neighborhood of $X$ in $\mathcal{Q}$ onto $X$. Hence an ANR-space inherits all local properties of the Hilbert cube which are preserved under retracts. These properties include local connectedness, so that all ANR-spaces are Peano spaces. The converse, however, is not true in general. But as we will see, it is possible to identify the ANR-spaces among all Peano spaces, at least in the one-dimensional case.
A closed subspace $Y$ of a space $X$ is a retract of $X$ if there exists a continuous map $r\colon X\rightarrow Y$ such that $r_{|Y}={\operatorname{id}}_Y$. If the retract map $r\colon X\rightarrow Y$ regarded as a map to $X$ is homotopic to the identity, then $Y$ is called a deformation retract of $X$. It is a strong deformation retract if in addition the homotopy can be chosen to fix the subspace $Y$. The following concept of a core continuum is due to Meilstrup. It is crucial for understanding the structure of one-dimensional ANR-spaces.
\[def core\] Let $X$ be a non-contractible one-dimensional Peano continuum. Then there exists a unique strong deformation retract which contains no further proper deformation retract. We call it the core of $X$ and denote it by ${\operatorname{core}}(X)$.
As in [@ST12], we define the core of a contractible, one-dimensional Peano continuum to be any fixed point. Many questions about one-dimensional Peano continua can be reduced to questions about their cores. This reduction step uses a special retract map, the so-called first point map:
\[def fpm\] Let $X$ be a one-dimensional Peano continuum and $Y$ a subcontinuum with ${\operatorname{core}}(X)\subset Y$. For each $x\in X\backslash Y$ there is a unique point $r(x)\in Y$ such that $r(x)$ is a point of an arc in $X$ from $x$ to any point of $Y$. Setting $r(x)=x$ for all $x\in Y$, we obtain a map $r\colon X\rightarrow Y$. This map is called the first point map, it is continuous and a strong deformation retract from $X$ onto $Y$.
The following follows directly from the proof of [@ST12 Lemma 4.14].
\[lemma arc to core\] Let $X$ be a one-dimensional Peano continuum, $Y\subseteq X$ a subcontinuum containing ${\operatorname{core}}(X)$ and $r\colon X\rightarrow Y$ the first point map onto $Y$. Then the following is true:
1. For every point $x\in X\backslash Y$ there exists an arc from $x$ to $r(x)\in Y$ which is unique up to reparametrization.
2. If $\alpha$ is a path from $x\in X\backslash Y$ to $y\in Y$, then $r({\operatorname{im}}(\alpha))\subseteq{\operatorname{im}}(\alpha)$.
The simplest example of a one-dimensional Peano space is a graph, i.e. a finite, one-dimensional CW-complex. The order of a point $x$ in a graph $X$ is defined as the smallest number $n\in\mathbb{N}$ such that for every neighborhood $V$ of $x$ there exists an open neighborhood $U\subseteq V$ of $x$ with $|\partial U|=|\overline{U}\backslash U| \leq n$. We denote the order of $x$ in $X$ by ${\operatorname{order}}(x,X)$.
Given a one-dimensional Peano continuum $X$, one can reconstruct the space $X$ from its core by ’adding’ the arcs which connect points of $X\backslash{\operatorname{core}}(X)$ with the core as described in \[lemma arc to core\]. This procedure yields a limit structure for one-dimensional Peano spaces which first appeared as Theorem 4.17 of [@ST12]. In the case of one-dimensional ANR-spaces, the core is a finite graph and hence the limit structure entirely consists of finite graphs.
\[thm ST\] Let $X$ be a one-dimensional Peano continuum. Then there exists a sequence $\{Y_k\}_{k=1}^\infty$ such that
1. each $Y_k$ is a subcontinuum of $X$.
2. $Y_k\subset Y_{k+1}$.
3. $\lim_kY_k=X$.
4. $Y_1={\operatorname{core}}(X)$ and for each $k$, $Y_{k+1}$ is obtained from $Y_k$ by attaching a line segment at a single point, i.e., $\overline{Y_{k+1}\backslash Y_k}$ is an arc with end point $p_k$ such that $\overline{Y_{k+1}\backslash Y_k}\cap Y_k=\{p_k\}$.
5. letting $r_k\colon X\rightarrow Y_k$ be the first point map for $Y_k$ we have that $\{r_k\}_{k=1}^\infty$ converges uniformly to the identity map on $X$.
If $X$ is also an ANR, then all $Y_k$ are finite graphs. If $X$ is even contractible (i.e. an AR), then ${\operatorname{core}}(X)$ is just some point and all $Y_k$ are finite trees.
We will need a local criterion for identifying one-dimensional ANR-spaces among general Peano spaces. It was observed by Ward how to get such a characterization in terms of embeddings of circles.
\[def small circles\] Let $X$ be a compact, metric space, then $X$ does not contain small circles if there is an $\epsilon>0$ such that ${\operatorname{diam}}(\iota(S^1))\geq\epsilon$ for every embedding $\iota\colon S^1\rightarrow X$.
Note that the property of containing arbitrarily small circles does not depend on the particular choice of metric.
\[thm ward\] For a Peano space $X$ the following are equivalent:
1. $X$ does not contain small circles.
2. $X$ is an ANR-space of dimension at most one.
This statement can also be interpreted as follows. As was shown independently by Bing ([@Bin49]) and Moise ([@Moi49]), every Peano continuum $X$ admits a geodesic metric $d$. Now non-embeddability of circles into $X$ is the same as uniqueness of geodesics in $X$. More precisely, a Peano continuum is a one-dimensional AR-space if and only if there is no embedding $S^1\hookrightarrow X$ if and only if $X$ admits unique geodesics. Similarly, Theorem \[thm ward\] can be read as: A Peano continuum $X$ is a one-dimensional ANR-space if and only if it has locally unique geodesics, meaning that there exists $\epsilon>0$ such that any two points with distance smaller then $\epsilon$ can be joined by a unique geodesic.
Subhomogeneous $C^*$-algebras {#section subhomogeneous}
-----------------------------
In this section we collect some well known results on subhomogeneous $C^*$-algebras. In particular, we recall some facts on their primitive ideal spaces. More detailed information can be found in [@Dix77 Chapter 3] and [@Bla06 Section IV.1.4].
Let $N\in\mathbb{N}$. A $C^*$-algebra $A$ is $N$-homogeneous if all its irreducible representations are of dimension $N$. $A$ is $N$-subhomogeneous if every irreducible representation of $A$ has dimension at most $N$.
The standard example of a $N$-homogeneous $C^*$-algebra is ${\mathcal{C}}_0(X,{\mathbb{M}}_N)$ for some locally compact space $X$. As the next proposition shows, subhomogeneous $C^*$-algebras can be characterized as subalgebras of such. A proof of this fact can be found in [@Bla06 IV.1.4.3-4].
\[prop sub homogeneous\] A $C^*$-algebra $A$ is $N$-subhomogeneous if and only if it is isomorphic to a subalgebra of some $N$-homogeneous $C^*$-algebra ${\mathcal{C}}(X,{\mathbb{M}}_N)$. If $A$ is separable, we may choose $X$ to be the Cantor set $K$.
\[ex 1-nccw\] One of the most important examples of subhomogeneous $C^*$-algebras is the class of non-commutative CW-complexes (NCCWs) defined by Eilers, Loring and Pedersen in [@ELP98]. The one-dimensional NCCWs, which we will abbreviate by 1-NCCWs, are defined as pullbacks of the form $$\xymatrix{
\text{1-NCCW} \ar@{-->}[r] \ar@{-->}[d] & G \ar[d] \\
{\mathcal{C}}([0,1],F) \ar[r]^(.58){{\operatorname{ev}}_0\oplus{\operatorname{ev}}_1} & F\oplus F
}$$ with $F$ and $G$ finite-dimensional $C^*$-algebras. These are particularly interesting since they are semiprojective by [@ELP98 Theorem 6.2.2].
For a subhomogeneous $C^*$-algebra $A$, the primitive ideal space ${\operatorname{Prim}}(A)$, i.e. the set of kernels of irreducible representations endowed with the Jacobson topology, contains a lot of information. Another useful decription of the topology on ${\operatorname{Prim}}(A)$ is given by the folllowing lemma which will make use of regularly. For an ideal $J$ in a $C^*$-algebra $A$ we write $\|x\|_J$ to denote the norm of the image of the element $x\in A$ in the quotient $A/J$.
\[lemma top prim\] Let $A$ be a $C^*$-algebra.
1. If $x\in A$, define $\check{x}\colon {\operatorname{Prim}}(A)\rightarrow\mathbb{R}_{\geq 0}$ by $\check{x}(J)=\|x\|_J$. Then $\check{x}$ is lower semicontinuous.
2. If $\{x_i\}$ is a dense set in the unit ball of $A$, and $U_i=\{J\in{\operatorname{Prim}}(A)\colon\check{x_i}(J)>1/2\}$, then $\{U_i\}$ forms a base for the topology of ${\operatorname{Prim}}(A)$.
3. If $x\in A$ and $\lambda>0$, then $\{J\in{\operatorname{Prim}}(A)\colon\check{x}(J)\geq\lambda\}$ is compact (but not necessarily closed) in ${\operatorname{Prim}}(A)$.
Since we will mostly be interested in finite-dimensional representations, we consider the subspaces $${\operatorname{Prim}}_n(A)=\{\ker(\pi)\in{\operatorname{Prim}}(A)\colon{\operatorname{dim}}(\pi)=n\}$$ for each finite $n$. Similarly, we write $${\operatorname{Prim}}_{\leq n}(A)=\{\ker(\pi)\in{\operatorname{Prim}}(A)\colon{\operatorname{dim}}(\pi)\leq n\}=\bigcup_{k\leq n} {\operatorname{Prim}}_k(A).$$ The following theorem describes the structure of these subspaces of ${\operatorname{Prim}}(A)$ and the relations between them.
\[thm prim\] Let $A$ be a $C^*$-algebra. The following holds for each $n\in\mathbb{N}$:
1. ${\operatorname{Prim}}_{\leq n}(A)$ is closed in ${\operatorname{Prim}}(A)$.
2. ${\operatorname{Prim}}_n(A)$ is open in ${\operatorname{Prim}}_{\leq n}(A)$.
3. ${\operatorname{Prim}}_n(A)$ is locally compact and Hausdorff.
Now assume that $A$ is a $N$-subhomogeneous $C^*$-algebra. In this case Theorem \[thm prim\] gives a set-theoretical (but in general not a topological) decomposition of its primitive spectrum $${\operatorname{Prim}}(A)=\bigsqcup_{n=1}^N {\operatorname{Prim}}_n(A).$$ While each subspace in this decomposition is nice, in the sense that it is Hausdorff, ${\operatorname{Prim}}(A)$ itself is typically non-Hausdorff. In the subhomogeneous setting it is at least a $T_1$-space, i.e. points are closed. If we further assume $A$ to be separable and unital, the space ${\operatorname{Prim}}(A)$ will also be separable and quasi-compact.
Given a general $C^*$-algebra $A$, there is a one-to-one correspondence between (closed) ideals $J$ of $A$ and closed subsets of ${\operatorname{Prim}}(A)$. More precisely, one can identify ${\operatorname{Prim}}(A/J)$ with the closed subset $\{K\in{\operatorname{Prim}}(A)\colon J\subseteq K\}$. In particular, we can consider the quotient $A_{\leq n}$ corresponding to the closed subset ${\operatorname{Prim}}_{\leq n}(A)\subseteq{\operatorname{Prim}}(A)$. This quotient is the maximal $n$-subhomogeneous quotient of $A$ and has the following universal property: Any $^*$-homomorphism $\varphi\colon A\rightarrow B$ to some $n$-subhomogeneous $C^*$-algebra $B$ factors uniquelythrough $A_{\leq n}$: $$\xymatrix{
A \ar[rr]^\varphi \ar@{->>}[dr]&& B \\
& A_{\leq n} \ar@{-->}[ur]
}$$
Semiprojective $C^*$-algebras {#section sp}
-----------------------------
We recall the definition of semiprojectivity for $C^*$-algebras, the main property of study in this paper. More detailed information about lifting properties for $C^*$-algebras can be found in Loring’s book [@Lor97].
\[def sp\] A separable $C^*$-algebra $A$ is semiprojective if for every $C^*$-algebra $B$ and every increasing chain of ideals $J_n$ in $B$ with $J_\infty=\overline{\bigcup_n J_n}$, and for every $^*$-homomorphism $\varphi\colon A\rightarrow B/J_\infty$ there exists $n\in\mathbb{N}$ and a $^*$-homomorphism $\overline{\varphi}\colon A\rightarrow B/J_n$ making the following diagram commute: $$\xymatrix{
& B \ar@{->>}[d]^{\pi_0^n} \\ & B/J_n \ar@{->>}[d]^{\pi_n^\infty} \\
A \ar[r]^\varphi \ar@{-->}[ur]^{\overline{\varphi}} & B/J_\infty
}$$ In this situation, the map $\overline{\varphi}$ is called a partial lift of $\varphi$. The $C^*$-algebra $A$ is projective if, in the situation above, we can always find a lift $\overline{\varphi}\colon A\rightarrow B$ for $\varphi$.
Let $\mathcal{C}$ be a class of $C^*$-algebras. A $C^*$-algebra $A$ is (semi)projective with respect to $\mathcal{C}$ if it satisfies the definitions above with the restriction that the $C^*$-algebras $B,B/J_n$ and $B/J_\infty$ all belong to the class $\mathcal{C}$.
One may also define semiprojectivity as a lifting property for maps to certain direct limits: an increasing sequence of ideals $J_n$ in $B$ gives an inductive system $(B/J_n)_n$ with surjective connecting maps $\pi_n^{n+1}\colon B/J_n\rightarrow B/J_{n+1}$ and limit (isomorphic to) $B/J_\infty$. On the other hand, it is easily seen that every such system gives an increasing chain of ideals $(\ker(\pi_1^n))_n$. Hence, semiprojectivity is equivalent to being able to lift maps to $\varinjlim D_n$ to a finite stage $D_n$ provided that all connecting maps of the system are surjective. It is sometimes more convenient to work in this picture.
### An unsolvable lifting problem {#section unsolvable}
In order to show that a $C^*$-algebra does not have a certain lifting property, we need to construct unsolvable lifting problems. One such construction by Loring ([@Lor97 Proposition 10.1.8]) uses the fact that normal elements in quotient $C^*$-algebras do not admit normal preimages in general, e.g. Fredholm operators of non-zero index. Here, we generalize Loring’s construction and obtain a version which also works for almost normal elements. Combining this with Lin’s theorem on almost normal matrices, we are able to construct unsolvable lifting problems not only for commutative $C^*$-algebras, as in Loring’s case, but for the much larger class of strongly quasidiagonal $C^*$-algebras.
First we observe that almost normal elements in quotient $C^*$-algebras always admit (almost as) almost normal preimages. Given an element $x$ of some $C^*$-algebra and $\epsilon>0$, we say that $x$ is $\epsilon$-normal if $\|x^*x-xx^*\|\leq\epsilon\|x\|$ holds.
\[lemma preimage\] Let $A$, $B$ be $C^*$-algebras and $\pi\colon A\rightarrow B$ a surjective $^*$-homomorphism. Then for every $\epsilon$-normal element $y\in B$ there exists a $(2\epsilon)$-normal element $x\in A$ with $\pi(x)=y$ and $\|x\|=\|y\|$.
Let $(u_\lambda)_{\lambda\in\Lambda}$ denote an approximate unit for $\ker(\pi)$ which is quasicentral for $A$. Pick any preimage $x_0$ of $x$ with $\|x_0\|=\|x\|$ and set $x:=(1-u_{\lambda_0})x_0$ for a suitable $\lambda_0\in\Lambda$.
The next lemma is due to Halmos. A short proof using the Fredholm alternative can be found in [@BH74 Lemma 2].
\[lemma Halmos\] Let $S\in\mathcal{B}(H)$ be a proper isometry, then $${\operatorname{dist}}\left(S,\{N+K\,|\,N,K\in \mathcal{B}(H),\; N\,\text{normal},\; K\,\text{compact}\}\right)=1.$$
It is a famous result by H. Lin that in matrix algebras almost normal elements are uniformly close to normal ones ([@Lin97]). A short, alternative proof involving semiprojectivity arguments can be found in [@FR01].
\[thm Lin\] For every $\epsilon>0$, there is a $\delta>0$ so that, for any $d$ and any $X$ in ${\mathbb{M}}_d$ satisfying $$\|XX^*-X^*X\|\leq\delta\;\;\;\text{and}\;\;\;\|X\|\leq 1$$ there is a normal $Y$ in ${\mathbb{M}}_d$ such that $$\|X-Y\|\leq\epsilon.$$
The following is the basis for most of our unsolvable lifting problems appearing in this paper. Recall that a $C^*$-algebra $A$ is strongly quasidiagonal if every representation of $A$ is quasidiagonal. See [@Bla06 Section V.4.2] or [@Bro00] for more information on quasidiagonality.
In the following, let $\mathcal{T}$ denote the Toeplitz algebra $C^*(S|S^*S=1)$ and $\varrho\colon\mathcal{T}\rightarrow{\mathcal{C}}(S^1)$ the quotient map given by mapping $S$ to the canonical generator $z$ of ${\mathcal{C}}(S^1)$.
\[prop nslp\] There exists $\delta>0$ such that the following holds for all $n\in\mathbb{N}$: If $A$ is strongly quasidiagonal and $\varphi\colon A\rightarrow{\mathcal{C}}(S^1)\otimes{\mathbb{M}}_n$ is any $^*$-homomorphism with ${\operatorname{dist}}(z\otimes 1_n,{\operatorname{im}}(\varphi))<\delta$, then $\varphi$ does not lift to a $^*$-homomorphism from $A$ to $\mathcal{T}\otimes{\mathbb{M}}_n$: $$\xymatrix{
&& \mathcal{T}\otimes {\mathbb{M}}_n \ar@{->>}[d]^(0.4){\varrho\otimes{\operatorname{id}}}\\
A \ar[rr]^\varphi \ar@{-->}@/^1pc/[urr]^(.4)\nexists && {\mathcal{C}}(S^1)\otimes {\mathbb{M}}_n
}$$
Choose $\delta'>0$ corresponding to $\epsilon=1/6$ as in Theorem \[thm Lin\] and set $\delta=\delta'/14$. Let $a'\in A$ be such that $\|\varphi(a')-z\otimes 1_n\|<\delta$, then $\|[\varphi(a'),\varphi(a')^*]\|\leq 2\delta(\|\varphi(a')\|+1)<5\delta\|\varphi(a')\|$. Hence by Lemma \[lemma preimage\] there exists a $(10\delta)$-normal element $a\in A$ with $\varphi(a)=\varphi(a')$ and $5/6<\|a\|=\|\varphi(a')\|<6/5$. Now if $\psi$ is a $^*$-homomorphism with $(\varrho\otimes{\operatorname{id}})\circ\psi=\varphi$ as indicated, we regard $\psi$ as a representation on $\mathcal{H}^{\oplus n}$ with $\mathcal{T}$ generated by the unilateral shift $S$ on $\mathcal{H}$. By assumption, $\psi$ is then a quasidiagonal representation. In particular, $\psi(a)$ can be approximated arbitrarily well by block-diagonal operators ([@Bro00 Theorem 5.2]). We may therefore choose a $(11\delta)$-normal block-diagonal operator $B$ with $5/6\leq\|B\|\leq 6/5$ within distance at most $1/3$ from $\psi(a)$. Applying Lin’s Theorem to the normalized, $(14\delta)$-normal block-diagonal operator $\|B\|^{-1} B$ shows the existence of a normal element $N\in\mathcal{H}^{\oplus n}$ with $\|\psi(a)-N\|\leq 2/3$. But then we find $$\begin{array}{rl}
& \|(N-S\otimes 1_n)+\mathcal{K}(\mathcal{H}^{\oplus n})\| \\
\leq & \|N-\psi(a)\|+\|(\varrho\otimes{\operatorname{id}})(\psi(a)-S\otimes 1_n)\| \\
\leq & \frac{2}{3}+\|\varphi(a')-z\otimes 1_n\| \\
\leq & \frac{2}{3}+\delta<1
\end{array}$$ in contradiction to Lemma \[lemma Halmos\].
### The homogeneous case {#section homogeneous}
In [@ST12], A. Sørensen and H. Thiel characterized semiprojectivity for commutative $C^*$-algebras. Moreover, they gave a description of semiprojectivity for homogeneous trivial fields, i.e. $C^*$-algebras of the form ${\mathcal{C}}_0(X,{\mathbb{M}}_N)$. Note that the projective case was settled earlier by A. Chigogidze and A. Dranishnikov in [@CD10]. Their result is as follows.
\[thm comm case\] Let $X$ be a locally compact, metric space and $N\in\mathbb{N}$. Then the following are equivalent:
1. ${\mathcal{C}}_0(X,{\mathbb{M}}_N)$ is (semi)projective.
2. The one-point compactification $\alpha X$ is an A(N)R-space and ${\operatorname{dim}}(X)\leq 1$.
The work of S[ø]{}rensen and Thiel will be the starting point for our analysis of semiprojectivity for subhomogeneous $C^*$-algebras. In this section, we reduce the general $N$-homogeneous case to their result by showing that semiprojectivity for homogeneous, locally trivial fields implies global triviality. We further obtain some information about parts of the primitive ideal space for general semiprojective $C^*$-algebras.
\[lemma ideal Peano\] Let $I$ be a $N$-homogeneous ideal in a $C^*$-algebra $A$. If $A$ is semiprojective with respect to $N$-subhomogeneous $C^*$-algebras, then the one-point compactification $\alpha{\operatorname{Prim}}(I)$ is a Peano space. If $A$ is semiprojective, we further have ${\operatorname{dim}}(\alpha{\operatorname{Prim}}(A))\leq 1$.
Let $A_{\leq N}$ be the maximal $N$-subhomogeneous quotient of $A$, then $I$ is also an ideal in $A_{\leq N}$. Being $N$-homogeneous, the ideal $I$ is isomorphic to the section algebra $\Gamma_0(E)$ of a locally trivial ${\mathbb{M}}_N$-bundle $E$ over the locally compact, second countable, metrizable Hausdorff space ${\operatorname{Prim}}(I)$ by [@Fel61 Theorem 3.2]. Since $A_{\leq N}$ is separable and $N$-subhomogenous, we can embed it into $C(K,{\mathbb{M}}_N)$ with $K$ the Cantor set by Proposition \[prop sub homogeneous\]. Using the well known middle-third construction of $K=\varprojlim_k (\bigsqcup^{2^k} [0,1])$, we can apply semiprojectivity of $A_{\leq N}$ with respect to $N$-subhomogenous $C^*$-algebras to obtain an embedding of $A_{\leq N}$ into $C([0,1]^{\oplus 2^k},{\mathbb{M}}_N)$ for some $k$. The restriction of this embedding to $I$ induces a continuous surjection $\pi$ of $\bigsqcup^{2^k} [0,1]$ onto $\alpha{\operatorname{Prim}}(I)$. By the Hahn-Mazurkiewicz Theorem ([@Nad92 Theorem 8.18]), this shows that $\alpha{\operatorname{Prim}}(I)$ is a Peano space. Furthermore, we find a basis of compact neighborhoods consisting of Peano continua for any point $x$ of $\alpha{\operatorname{Prim}}(I)$ by [@Nad92 Theorem 8.10].
Now let $A$ be semiprojective and assume that ${\operatorname{dim}}({\operatorname{Prim}}(I))={\operatorname{dim}}(\alpha{\operatorname{Prim}}(I))>1$. Arguing precisely as in [@ST12 Proposition 3.1], we use our basis of neighborhoods for points of ${\operatorname{Prim}}(I)$ to find arbitrarily small circles around a point $x\in{\operatorname{Prim}}(I)$. Using triviality of $E$ around $x$, we obtain a lifting problem for $A$: $$\xymatrix{
& A \ar[d] \ar@{-->}[r] & \left(\left(\bigoplus_\mathbb{N}\mathcal{T}\right)^+/\left(\bigoplus_1^n\mathbb{K}\right)\right)\otimes{\mathbb{M}}_N \ar@{->>}[d] \\
I \ar[r] \ar[ur]^\subseteq&\left(\bigoplus_\mathbb{N}{\mathcal{C}}(S^1)\right)^+\otimes{\mathbb{M}}_N \ar@{=}[r] & \left(\left(\bigoplus_\mathbb{N}\mathcal{T}\right)^+/\left(\bigoplus_\mathbb{N}\mathbb{K}\right)\right)\otimes{\mathbb{M}}_N
}$$ Semiprojectivity of $A$ allows us to solve this lifting problem. Now restrict a partial lift to the ideal $I$ and consider its coordinates to obtain a commutative diagram $$\begin{xy}\xymatrix{
& \hspace{0.65cm}\mathcal{T}\otimes{\mathbb{M}}_N \ar@<0.15cm>@{->>}[d] \\
I \ar@{->>}[r] \ar@{-->}[ur] & {\mathcal{C}}(S^1)\otimes{\mathbb{M}}_N.
}\end{xy}$$ The map on the bottom is surjective since it is induced by the inclusion of one of the circles around $x$. But a diagram like this does not exist by Proposition \[prop nslp\] because $I$ is homogeneous and by that strongly quasidiagonal.
\[cor sp peano\] Let $A$ be a semiprojective $C^*$-algebra, then $\alpha{\operatorname{Prim}}_n(A)$ is a Peano space for every $n\in\mathbb{N}$.
If $A$ is semiprojective, each $A_{\leq n}$ is semiprojective with respect to $n$-subhomogeneous $C^*$-algebras. Hence we can apply Lemma \[lemma ideal Peano\] to the $n$-homogeneous ideal $\ker(A_{\leq n}\rightarrow A_{\leq n-1})$ in $A_{\leq n}$ whose primitive ideal space is homeomorphic to ${\operatorname{Prim}}_n(A)$.
It is known to the experts that there are no non-trivial ${\mathbb{M}}_n$-valued fields over one-dimensional spaces and we are indebted to L. Robert for pointing this fact out to us. Since we couldn’t find a proof in the literature, we include one here.
\[lemma trivial bundles\] Let $E$ be a locally trivial field of $C^*$-algebras over a separable, metrizable, locally compact Hausdorff space $X$ with fiber ${\mathbb{M}}_N$ and $\Gamma_0(E)$ the corresponding section algebra. If ${\operatorname{dim}}(X)\leq 1$, then $\Gamma_0(E)$ is ${\mathcal{C}}_0(X)$-isomorphic to ${\mathcal{C}}_0(X,{\mathbb{M}}_N)$.
First assume that $X$ is compact. One-dimensionality of $X$ implies that that the Dixmier-Douady invariant $\delta\in \check{H}^3(X,\mathbb{Z})$ corresponding to $\Gamma_0(E)$ vanishes. Therefore $\Gamma_0(E)$ is stably ${\mathcal{C}}(X)$-isomorphic to ${\mathcal{C}}(X,{\mathbb{M}}_N)$ by Dixmier-Douady classification (see e.g. [@RW98 Corollary 5.56]). Let $\psi\colon\Gamma(E)\otimes\mathcal{K}\rightarrow {\mathcal{C}}(X,{\mathbb{M}}_N)\otimes\mathcal{K}$ be such an isomorphism and note that $\Gamma(E)\cong {\operatorname{her}}(\psi(1_{\Gamma(E)}\otimes e))$ via $\psi$ with $e$ a minimal projection in $\mathcal{K}$. Equivalence of projections over one-dimensional spaces is completely determined by their rank by [@Phi07 Proposition 4.2]. Since $\psi(1_{\Gamma(E)}\otimes e)$ and $1_{{\mathcal{C}}(X,{\mathbb{M}}_N)}\otimes e$ share the same rank $N$ everywhere we therefore find $v\in {\mathcal{C}}(X,{\mathbb{M}}_N)\otimes\mathcal{K}$ with $v^*v=\psi(1_{\Gamma(E)}\otimes e)$ and $vv^*=1_{{\mathcal{C}}(X,{\mathbb{M}}_N)}\otimes e$. But then ${\operatorname{Ad}}(v)$ gives a $C(X)$-isomorphism from ${\operatorname{her}}(\psi(1_{\Gamma(E)}\otimes e))$ onto ${\operatorname{her}}(1_{{\mathcal{C}}(X,{\mathbb{M}}_N)}\otimes e)={\mathcal{C}}(X,{\mathbb{M}}_N)$.
Now consider the case of non-compact $X$. Since $X$ is $\sigma$-compact, it clearly suffices to prove the following: Given compact subsets $X_1\subseteq X_2$ of $X$ and a ${\mathcal{C}}(X_1)$-isomorphism $\varphi_1\colon \Gamma(E_{|X_1})\rightarrow {\mathcal{C}}(X_1,{\mathbb{M}}_N)$ there exists a ${\mathcal{C}}(X_2)$-isomorphism $\varphi_2\colon\Gamma(E_{|X_2})\rightarrow {\mathcal{C}}(X_2,{\mathbb{M}}_N)$ extending $\varphi_1$. By the first part of the proof there is a ${\mathcal{C}}(X_2)$-isomorphism $\psi_2\colon\Gamma(E_{|X_2})\rightarrow {\mathcal{C}}(X_2,{\mathbb{M}}_N)$. One-dimensionality of $X_1$ implies $\check{H}^2(X_1,\mathbb{Z})=0$, which means that every ${\mathcal{C}}(X_1)$-automorphism of ${\mathcal{C}}(X_1,{\mathbb{M}}_N)$ is inner by [@RW98 Theorem 5.42]. In particular, $\varphi_1\circ(\psi^{-1}_2)_{|X_1}$ is of the form ${\operatorname{Ad}}(u)$ for some unitary $u\in {\mathcal{C}}(X_1,{\mathbb{M}}_N)$. It remains to extend $u$ to a unitary in ${\mathcal{C}}(X_2,{\mathbb{M}}_N)$. This, however, follows from one-dimensionality of $X$ and [@HW48 Theorem VI.4].
We are now able to extend the results of [@ST12] to the case of general $N$-homogeneous $C^*$-algebras:
\[thm homogeneous case\] Let $A$ be a $N$-homogeneous $C^*$-algebra. The following are equivalent:
1. $A$ is (semi)projective.
2. $A\cong{\mathcal{C}}_0({\operatorname{Prim}}(A),{\mathbb{M}}_N)$ and $\alpha{\operatorname{Prim}}(A)$ is an A(N)R-space of dimension at most 1.
By Lemma \[lemma ideal Peano\] and Lemma \[lemma trivial bundles\], we know that $(1)$ implies $A\cong{\mathcal{C}}_0({\operatorname{Prim}}(A),{\mathbb{M}}_N)$. The remaining implications are given by Theorem \[thm comm case\].
Constructions for semiprojective $C^*$-algebras {#section semiprojectivity}
===============================================
Unfortunately, the class of semiprojective $C^*$-algebras lacks good permanence properties. In fact, semiprojectivity is not preserved by most $C^*$-algebraic standard constructions and the list of positive permanence results, most of which can be found in [@Lor97], is surprisingly short. Here, we extend this list by a few new results.
Extending lifting problems
--------------------------
In this section, we introduce a technique to extend lifting problems from ideals to larger $C^*$-algebras. This technique can be used to show that in many situations lifting properties of a $C^*$-algebra pass to its ideals.
\[lemma extended lifting\] Given a surjective inductive system of short exact sequences $$\xymatrix{
0\ar[r] & C_n \ar[r]^{\iota_n} \ar@{->>}[d]_{\pi_n^{n+1}}& D_n \ar[r]^{\varrho_n} \ar@{->>}[d]_{\overline{\pi}_n^{n+1}}& E_n \ar[r] \ar@{->>}[d]_{\overline{\overline{\pi}}_n^{n+1}}& 0 \\
0 \ar[r] & C_{n+1} \ar[r]^{\iota_{n+1}} & D_{n+1} \ar[r]^{\varrho_{n+1}} & E_{n+1} \ar[r] & 0
}$$ and a commutative diagram of extensions $$\xymatrix{
0 \ar[r] & \varinjlim C_n \ar[r]^{\iota_\infty} & \varinjlim D_n \ar[r]^{\varrho_\infty} & \varinjlim E_n \ar[r] & 0 \\
0 \ar[r] & I \ar[r]^i \ar[u]^\varphi & A \ar[r]^p \ar[u]^{\overline{\varphi}} & B \ar[r] \ar[u]^{\overline{\overline{\varphi}}} & 0
}$$ the following holds: If both $A$ and $B$ are semiprojective, then $\varphi$ lifts to $C_n$ for some $n$. If both $A$ and $B$ are projective, then $\varphi$ lifts to $C_1$.
First observe that we may assume the $^*$-homomorphism $\overline{\overline{\varphi}}$ to be injective since otherwise we simply pass to the system of extensions $$\xymatrix{0\ar[r]&C_n\ar[r]^(.4){\iota_n}&D_n\oplus B\ar[r]^{\varrho_n\oplus{\operatorname{id}}}&E_n\oplus B\ar[r]&0}$$ and replace $\overline{\varphi}$ by $\overline{\varphi}\oplus p$ and $\overline{\overline{\varphi}}$ by $\overline{\overline{\varphi}}\oplus{\operatorname{id}}$. Using semiprojectivity of $B$, we can find a partial lift $\psi\colon B\rightarrow E_{n_0}$ of $\overline{\overline{\varphi}}$ for some $n_0$, i.e. $\overline{\overline{\pi}}_{n_0}^\infty\circ\psi=\overline{\overline{\varphi}}$. Now consider the $C^*$-subalgebras $$D_n':=\varrho_n^{-1}((\overline{\overline{\pi}}_{n_0}^n\circ\psi)(B))\subseteq D_n$$ and observe that the restriction of $\overline{\pi}_n^{n+1}$ to $D_n'$ surjects onto $D_{n+1}'$. We also find that the direct limit $\varinjlim D_n'=\overline{\pi}_{n_0}^\infty(D'_{n_0})$ of this new system contains $\overline{\varphi}(A)$. Hence semiprojectivity of $A$ allows us to lift $\overline{\varphi}$ (regarded as a map to $\varinjlim D_n'$) to $D_n'$ for some $n\geq n_0$. Let $\sigma\colon A\rightarrow D_n'$ be a suitable partial lift, i.e. $\overline{\pi}_n^{\infty}\circ\sigma=\overline{\varphi}$, then the restriction of $\sigma$ to the ideal $I$ will be a solution to the original lifing problem for $\varphi$: The only thing we need to check is that the image of $I$ under $\sigma$ is in fact contained in $C_n$. But we know that $\overline{\overline{\pi}}_n^\infty$ is injective on $(\varrho_n\circ\sigma)(A)\subseteq (\overline{\overline{\pi}}_{n_0}^n\circ\psi)(B)$ since $\overline{\overline{\varphi}}=\overline{\overline{\pi}}_n^\infty\circ(\overline{\overline{\pi}}_{n_0}^n\circ\psi)$ was assumed to be injective. Hence the identity $$(\overline{\overline{\pi}}_n^\infty\circ\varrho_n\circ\sigma)(i(I))=(\varrho_\infty\circ\overline{\pi}_n^\infty\circ\sigma)(i(I))=(\varrho_\infty\circ\overline{\varphi})(i(I))=(\varrho_\infty\circ\iota_\infty)(\varphi(I))=0$$ confirms that $\sigma(i(I))\subseteq i_n(C_n)$ holds.
Now assume that we are given an inductive system $$\xymatrix{
\cdots \ar@{->>}[r] & C_n \ar@{->>}[r]^{\pi_n^{n+1}} & C_{n+1} \ar@{->>}[r] & \cdots
}$$ of separable $C^*$-algebras with surjective connecting homomorphisms. Then each connecting map $\pi_n^{n+1}$ canonically extends to a surjective $^*$-homomorphism $\overline{\pi}_n^{n+1}$ on the level of multiplier $C^*$-algebras ([@WO93 Theorem 2.3.9]), i.e., we automatically obtain a surjective inductive system of extensions $$\xymatrix{
0\ar[r] & C_n \ar[r] \ar@{->>}[d]_{\pi_n^{n+1}}& \mathcal{M}(C_n) \ar[r] \ar@{->>}[d]_{\overline{\pi}_n^{n+1}}& \mathcal{Q}(C_n) \ar[r] \ar@{->>}[d]_{\overline{\overline{\pi}}_n^{n+1}}& 0 \\
0 \ar[r] & C_{n+1} \ar[r] & \mathcal{M}(C_{n+1}) \ar[r] & \mathcal{Q}(C_{n+1}) \ar[r] & 0
}.$$ We would like to apply Lemma \[lemma extended lifting\] to such a system of extensions. However, the reader should be really careful when working with multipliers and direct limits at the same time since these constructions are not completely compatible: Each $\pi_n^\infty\colon C_n\rightarrow\varinjlim C_n$ extends to a $^*$-homomorphism $\mathcal{M}(C_n)\rightarrow\mathcal{M}(\varinjlim C_n)$. The collection of these maps induces a $^*$-homomorphism $p_\mathcal{M}\colon \varinjlim\mathcal{M}(C_n)\rightarrow\mathcal{M}(\varinjlim C_n)$ which is always surjective but only in trivial cases injective. The same occurs for the quotients, i.e. for the system of corona algebras $\mathcal{Q}(C_n)$. The situation can be summarized in the commutative diagram with exact rows $$\xymatrix{
0 \ar[r] & C_n \ar@{->>}[d] \ar[r] & \mathcal{M}(C_n) \ar@{->>}[d] \ar[r] & \mathcal{Q}(C_n) \ar@{->>}[d] \ar[r] & 0\\
0 \ar[r] &\varinjlim C_n \ar[r] \ar@{=}[d] & \varinjlim\mathcal{M}(C_n) \ar@{->>}[d]^{p_\mathcal{M}} \ar[r] & \varinjlim\mathcal{Q}(C_n) \ar@{->>}[d]^{p_\mathcal{Q}} \ar[r] & 0 \\
0 \ar[r]& \varinjlim C_n \ar[r] & \mathcal{M}(\varinjlim C_n) \ar[r] & \mathcal{Q}(\varinjlim C_n) \ar[r] & 0
}$$ where the quotient maps $p_\mathcal{M}$ and $p_\mathcal{Q}$ are the obstacles for an application of Lemma \[lemma extended lifting\]. The following proposition makes these obstacles more precise.
\[prop busby lifting\] Let $A$ and $B$ be semiprojective $C^*$-algebras and $$\xymatrix{
0\ar[r]&I\ar[r]&A\ar[r]&B\ar[r]&0&[\tau]
}$$ a short exact sequence with Busby map $\tau\colon B\rightarrow\mathcal{Q}(I)$. Let $I\xrightarrow{\sim}\varinjlim C_n$ be an isomorphism from $I$ to the limit of an inductive system of separable $C^*$-algebras $C_n$ with surjective connecting maps. If the Busby map $\tau$ can be lifted as indicated $$\xymatrix{
&\hspace{1.3cm}\varinjlim\mathcal{Q}(C_n) \ar@<0.7cm>@{->>}[d]^{p_\mathcal{Q}}\\
B \ar[r]^(0.32)\tau \ar@{-->}[ur]&\mathcal{Q}(I)\cong\mathcal{Q}(\varinjlim C_n)
},$$ then $I\rightarrow\varinjlim C_n$ lifts to $C_n$ for some $n$. If both $A$ and $B$ are projective, we can obtain a lift to $C_1$.
Keeping in mind that $p_{\mathcal{Q}}$ is the Busby map associated to the extension $0\rightarrow\varinjlim C_n\rightarrow\varinjlim\mathcal{M}(C_n)\rightarrow\varinjlim\mathcal{Q}(C_n)\rightarrow0$, the claim follows by combining Theorem 2.2 of [@ELP99] with Lemma \[lemma extended lifting\].
One special case, in which the existence of a lift for the Busby map $\tau$ as in Proposition \[prop busby lifting\] is automatic, is when the quotient $B$ is a projective $C^*$-algebra. Hence we obtain a new proof for the permanence result below which has the advantage that it does not use so-called corona extendability (cf. [@Lor97 Section 12.2]).
\[cor projective ideal\] Let $0\rightarrow I\rightarrow A\rightarrow B\rightarrow 0$ be short exact. If $A$ is (semi)projective and $B$ is projective, then $I$ is also (semi)projective.
Another very specific lifting problem for which Proposition \[prop busby lifting\] applies, is the following mapping telescope contruction due to Brown.
\[lemma telescope\] Let a sequence $(C_k)_k$ of separable $C^*$-algebras be given and consider the telescope system $(T_n,\varrho_n^{n+1})$ associated to $\bigoplus_{k=0}^\infty C_k=\varinjlim_n \bigoplus_{k=0}^n C_k$, i.e. $$T_n=\left\{f\in{\mathcal{C}}\left([n,\infty],\bigoplus_{k=0}^\infty C_k\right)\colon\;t\leq m\Rightarrow\; f(t)\in\bigoplus_{k=1}^mC_k\right\}$$ with $\varrho_n^{n+1}\colon T_n\rightarrow T_{n+1}$ the (surjective) restriction maps, so that $\varinjlim_n(T_n,\varrho_n^{n+1})\cong\bigoplus_{k=1}^\infty C_k$. Then both canonical quotient maps in the diagram $$\xymatrix{
0 \ar[r] & \varinjlim T_n \ar[r] \ar@{=}[d] & \varinjlim\mathcal{M}(T_n) \ar[r] \ar@{->>}[d]^{p_\mathcal{M}} & \varinjlim\mathcal{Q}(T_n) \ar[r] \ar@{->>}[d]^{p_\mathcal{Q}} & 0 \\
0 \ar[r] & \varinjlim T_n \ar[r] & \mathcal{M}(\varinjlim T_n) \ar[r] \ar@/^1pc/@{..>}[u]& \mathcal{Q}(\varinjlim T_n) \ar[r] \ar@/^1pc/@{..>}[u] & 0
}$$ split.
It suffices to produce a split for $p_\mathcal{M}$ which is the identity on $\varinjlim T_n$. Under the identification $\varinjlim T_n \cong \bigoplus_{k=0}^\infty C_k$ we have $\mathcal{M}(\varinjlim T_n)\cong\prod_{k=0}^\infty\mathcal{M}(C_k)$. One checks that $$T_n=\bigoplus_{k=0}^n{\mathcal{C}}([n,\infty],C_k)\oplus\bigoplus_{k>n}{\mathcal{C}}_0((k,\infty],C_k)$$ and hence $$\prod_{k=0}^\infty{\mathcal{C}}([\max\{n,k\},\infty],\mathcal{M}(C_k))\subset\mathcal{M}(T_n).$$ It follows that the sum of embeddings as constant functions $$\prod_{k=0}^\infty\mathcal{M}(C_k)\rightarrow\prod_{k=0}^\infty{\mathcal{C}}([\max\{n,k\},\infty],\mathcal{M}(C_k))\subset\mathcal{M}(T_n)$$ defines a split for the quotient map $\varinjlim\mathcal{M}(T_n)\rightarrow\mathcal{M}(\varinjlim T_n)$. It is easily verified that this split is the identity on $\bigoplus_{k=1}^\infty C_k$.
Given an extension $0\rightarrow I\rightarrow A\rightarrow B\rightarrow 0$ with both $A$ and $B$ semiprojective, the associated Busby map does in general not lift as in \[prop busby lifting\]. However, there are a number of interesting situations where it does lift and we therefore can use Propostion \[prop busby lifting\] to obtain lifting properties for the ideal $I$. One such example is studied in [@End14], where it is (implicitly) shown that the Busby map lifts if $B$ is a finite-dimensional $C^*$-algebra. This observation leads to the fact that semiprojectivity passes to ideals of finite codimension. Further examples will be given in section \[section extensions\], where we study Busby maps associated to extensions by homogeneous ideals and identify conditions which guarantee that \[prop busby lifting\] applies.
Direct limits which preserve semiprojectivity {#section limits}
---------------------------------------------
### Weakly conditionally projective homomorphisms {#section wcp}
The following definition characterizes $^*$-homomorphisms along which lifting solutions can be extended in an approximate manner. This type of maps is implicitly used in [@CD10] and [@ST12] in the special case of finitely presented, commutative $C^*$-algebras.
\[def wcp\] A $^*$-homomorphism $\varphi\colon A\rightarrow B$ is weakly conditionally projective if the following holds: Given $\epsilon>0$, a finite subset $F\subset A$ and a commuting square $$\xymatrix{
A \ar[d]_\varphi \ar[r]^\psi & D\ar@{->>}[d]^\pi \\
B \ar[r]^\varrho & D/J,
}$$ there exists a $^*$-homomorphism $\psi'\colon B\rightarrow D$ as indicated $$\xymatrix{
A \ar[d]_\varphi \ar[r]^\psi & D\ar@{->>}[d]^\pi \\
B \ar[r]^\varrho \ar@{-->}[ur]^{\psi'}& D/J
}$$ which satisfies $\pi\circ\psi'=\varrho$ and $\|(\psi'\circ\varphi)(a)-\psi(a)\|<\epsilon$ for all $a\in F$.
The definition above is a weakening of the notion of conditionally projective morphisms, as introduced in section 5.3 of [@ELP98], where one asks the homomorphism $\psi'$ in \[def wcp\] to make both triangles of the lower diagram to commute exactly. While conditionally projective morphisms are extremely rare (even when working with projective $C^*$-algebras, cf. the example below), there is a sufficient supply of weakly conditionally projective ones, as we will show in the next section.
The inclusion map ${\operatorname{id}}\oplus\, 0\colon{\mathcal{C}}_0(0,1]\rightarrow{\mathcal{C}}_0(0,1]\oplus{\mathcal{C}}_0(0,1]$ is weakly conditionally projective but not conditionally projective. This can be illustrated by considering the commuting square $$\xymatrix{
{\mathcal{C}}_0(0,1] \ar[r]^\psi \ar[d]_{{\operatorname{id}}\oplus\, 0} & {\mathcal{C}}_0(0,3) \ar@{->>}[d]^\pi \\
{\mathcal{C}}_0(0,1]\oplus{\mathcal{C}}_0[2,3) \ar@{=}[r] & {\mathcal{C}}_0(0,1]\oplus{\mathcal{C}}_0[2,3)
}$$ where $\pi$ is the restriction map and $\psi$ is given by sending the canonical generator $t$ of ${\mathcal{C}}_0(0,1]$ to the function $$(\psi(t))(s)=\begin{cases}s & \text{if}\quad s\leq 1\\ 1-s & \text{if}\quad 1<s\leq 2\\ 0 & \text{if}\quad 2\leq s\end{cases}.$$ It is clear that there is no lift for the generator of ${\mathcal{C}}_0[2,3)$ which is orthogonal to $\psi(t)$. This shows that the map ${\operatorname{id}}\oplus\, 0$ is not conditionally projective. However, after replacing $\psi(t)$ with $(\psi(t)-\epsilon)_+$ for any $\epsilon>0$, finding an orthogonal lift for the generator of the second summand is no longer a problem. Using this idea, it will be shown in Proposition \[prop wcp examples\] that ${\operatorname{id}}\oplus\, 0$ is in fact weakly conditionally projective,
If $A$ is a (semi)projective $C^*$-algebra and $\varphi\colon A\rightarrow B$ is weakly conditionally projective, then $B$ is of course also (semi)projective. The next lemma shows that (semi)projectivity is even preserved along a sequence of such maps. Its proof is of an approximate nature and relies on a one-sided approximate intertwining argument (cf. section 2.3 of [@Ror02]), a technique borrowed from the Elliott classification program.
\[lemma limit criterium\] Suppose $\xymatrix{A_1\ar[r]^{\varphi_1^2} & A_2 \ar[r]^{\varphi_2^3} & A_3 \ar[r]^{\varphi_3^4} & \cdots}$ is an inductive system of separable $C^*$-algebras. If $A_1$ is (semi)projective and all connecting maps $\varphi_n^{n+1}$ are weakly conditionally projective, then the limit $A_\infty=\varinjlim (A_n,\varphi_n^{n+1})$ is also (semi)projective.
We will only consider the projective case, the statement for the semiprojective case is proven analogously with obvious modifications. Choose finite subsets $F_n\subset A_n$ with $\varphi_n^{n+1}(F_n)\subseteq F_{n+1}$ such that the union $\bigcup_{m=n}^\infty (\varphi_n^m)^{-1}(F_m)$ is dense in $A_n$ for all $n$. Further let $(\epsilon_n)_n$ be a sequence in $\mathbb{R}_{>0}$ with $\sum_{n=1}^\infty\epsilon_n<\infty$. Now let $\varrho\colon A_\infty\rightarrow D/J$ be a $^*$-homomorphism to some quotient $C^*$-algebra $D/J$. By projectivity of $A_1$ there is a $^*$-homomorphism $s_1\colon A_1\rightarrow D$ with $\pi\circ s_1=\varrho\circ\varphi_1^\infty$. Since the maps $\varphi_n^{n+1}$ are weakly conditionally projective, we can inductively choose $s_{n+1}\colon A_{n+1}\rightarrow D$ with $\pi\circ s_{n+1}=\varrho\circ\varphi_{n+1}^\infty$ such that $$\|s_n(a)-(s_{n+1}\circ\varphi_n^{n+1})(a)\|<\epsilon_n$$ holds for all $a\in F_n$. It is now a standard computation (and therefore ommited) to check that $((s_m\circ\varphi_n^m)(x))_m$ is a Cauchy sequence in $D$ for every $x\in F_n$. Furthermore, the induced map $\varphi_n^\infty(x) \mapsto \lim_m(s_m\circ\varphi_n^m)(x)$ extends from the dense subset $\bigcup_n\varphi_n^\infty(F_n)$ to a $^*$-homomorphism $s\colon A_\infty\rightarrow D$. $$\xymatrix{
A_n \ar[r]^{s_n} \ar[d]_{\varphi_n^{n+1}} & D \ar@{->>}[ddd]^\pi \\
A_{n+1} \ar[dd]_{\varphi_{n+1}^\infty} \ar@{-->}[ur]^{s_{n+1}} \\ \\
A_\infty \ar[r]^\varrho \ar@{..>}[uuur]^s & D/J
}$$ Since each $s_n$ lifts $\pi$, the same holds for their pointwise limit, i.e. the limit map $s$ satisfies $\pi\circ s=\varrho$. This shows that $A_\infty$ is projective.
### Adding non-commutative edges {#section adding edges}
In order to make Lemma \[lemma limit criterium\] a useful tool for constructing semiprojective $C^*$-algebras, we have to ensure the existence of weakly projective $^*$-homomorphisms as defined in \[def wcp\]. The examples we work out in this section arise in special pullback situations where one ’adds a non-commutative edge’ to a given $C^*$-algebra $A$. By this we mean that we form the pullback of $A$ and ${\mathcal{C}}([0,1])\otimes {\mathbb{M}}_n$ over a $n$-dimensional representation of $A$ and the evaluation map ${\operatorname{ev}}_0$. In the special case of $A={\mathcal{C}}(X)$ being a commutative $C^*$-algebra and $n=1$ this pullback construction already appeared in [@CD10] and [@ST12] where it indeed corresponds to attaching an egde $[0,1]$ at one point to the space $X$. Here we show that the map obtained by extending elements of $A$ as constant functions onto the attached non-commutative edge gives an example of a weakly conditionally projective $^*$-homomorphism. As an application, we observe that the AF-telescopes studied in [@LP98] arise from weakly projective $^*$-homomorphisms and hence projectivity of these algebras is a direct consequence of Lemma \[lemma limit criterium\].
Adapting notation from [@ELP98], we set $$T(\mathbb{C},G)=\{f\in{\mathcal{C}}_0((0,2],G):\quad t\leq 1\Rightarrow f(t)\in\mathbb{C}\cdot 1_G\},$$ $$S(\mathbb{C},G)=\{f\in{\mathcal{C}}_0((0,2),G):\quad t\leq 1\Rightarrow f(t)\in\mathbb{C}\cdot 1_G\}$$ for $G$ a unital $C^*$-algebra. We further write $$T(\mathbb{C},G,F)=\left\{f\in{\mathcal{C}}_0((0,3],F):\begin{array}{l} t\leq 2\Rightarrow f(t)\in G \\ t\leq 1\Rightarrow f(t)\in\mathbb{C}\cdot 1_G \end{array}\right\}$$ with respect to a fixed inclusion $G\subseteq F$. We have the diagram $$\xymatrix{
T(\mathbb{C},G,F) \ar[r] \ar[d] & {\mathcal{C}}([2,3],F) \ar[d]^{{\operatorname{ev}}_2} \\
T(\mathbb{C},G) \ar[r]^{{\operatorname{ev}}_2} & F
}$$ which is a special case of the pullback situation considered in the next proposition. However, this example is in some sense generic and implementing it into the general situation is an essential part of proving the following.
\[prop wcp examples\] Given a (semi)projective $C^*$-algebra $Q$ and a $^*$-homomorphism $\tau\colon Q\rightarrow {\mathbb{M}}_n$, the following holds:
1. The pullback $P$ over $\tau$ and ${\operatorname{ev}}_0\colon{\mathcal{C}}([0,1],{\mathbb{M}}_n)\rightarrow{\mathbb{M}}_n$, i.e. $$P=\{(q,f)\in Q\oplus{\mathcal{C}}([0,1],{\mathbb{M}}_n):\;\tau(q)=f(0)\},$$ is (semi)projective.
2. The canonical split $s\colon Q\rightarrow P$, $q\mapsto (q,\tau(q)\otimes 1_{[0,1]})$ is weakly conditionally projective.
\(1) Semiprojectivity of the pullback $P$ follows from [@End14 Corollary 3.4]. Since $P$ is homotopy equivalent to $Q$, the projective statement follows from the semiprojective one using [@Bla12 Corollary 5.2].\
(2) For technical reasons we identify the attached interval $[0,1]$ with $[2,3]$ and consider the pullback $$\xymatrix{
P \ar@{-->}[r] \ar@{-->}[d] & {\mathcal{C}}([2,3],{\mathbb{M}}_n) \ar[d]^{{\operatorname{ev}}_2} \\
Q \ar[r]^\tau & {\mathbb{M}}_n
}$$ with $s\colon Q\rightarrow P$, $q\mapsto(q,\tau(q)\otimes 1_{[2,3]})$ instead. Denote by $G\subseteq{\mathbb{M}}_n$ the image of $\tau$. According to [@ELP98 Theorem 2.3.3], we can find a $^*$-homomorphism $\overline{\varphi}\colon T(\mathbb{C},G)\rightarrow Q$ such that $$\xymatrix{
0 \ar[r] & \ker(\tau) \ar[r] & Q \ar[r]^(0.6)\tau & G \ar[r] & 0 \\
0 \ar[r] & S(\mathbb{C},G) \ar[r] \ar[u] & T(\mathbb{C},G) \ar[u]^{\overline{\varphi}} \ar[r]^(0.6){ev_2} & G \ar@{=}[u] \ar[r] & 0
}$$ commutes and $\overline{\varphi}_{|S(\mathbb{C},G)}$ is a proper $^*$-homomorphism to $\ker(\tau)$ (meaning that the hereditary subalgebra generated by its image is all of $\ker(\tau)$). Using the pullback property of $P$, $\overline{\varphi}$ can be extended to $\varphi\colon T(\mathbb{C},G,{\mathbb{M}}_n)\rightarrow P$ such that $$\xymatrix{
0 \ar[r] & {\mathcal{C}}_0((2,3],{\mathbb{M}}_n) \ar[r] & P \ar[r] & Q \ar[r] \ar@/_/[l]_s& 0 \\
0 \ar[r] & {\mathcal{C}}_0((2,3],{\mathbb{M}}_n) \ar@{=}[u] \ar[r] & T(\mathbb{C},G,{\mathbb{M}}_n) \ar[u]^\varphi \ar[r] & T(\mathbb{C},G) \ar[u]^{\overline{\varphi}} \ar[r] \ar@/_/[l]_{s'}& 0
}$$ commutes. In particular we have $\varphi\circ s'=s\circ\overline{\varphi}$, where $s'$ is the canonical split which simply extends functions constantly onto $[2,3]$.
Choose generators $f_1,...,f_l$ of norm 1 for ${\mathcal{C}}_0((2,3],{\mathbb{M}}_n)$ and generators $g_1,...,g_k$ of norm 1 for $T(\mathbb{C},G)$. We need the following ’softened’ versions of $P$: For $\delta>0$ we consider the universal $C^*$-algebra $$P_\delta=C^*\left(\left\{f^\delta,q^\delta:f\in {\mathcal{C}}_0((2,3],{\mathbb{M}}_n),q\in Q\right\}|\mathcal{R}_{{\mathcal{C}}_0((2,3],{\mathbb{M}}_n)}\&\mathcal{R}_Q\&\mathcal{R}_\delta\right)$$ which is generated by copies of ${\mathcal{C}}_0((2,3],{\mathbb{M}}_n)$ and $Q$ (here $\mathcal{R}_{{\mathcal{C}}_0((2,3],{\mathbb{M}}_n)},\mathcal{R}_Q$ denote all the relations from ${\mathcal{C}}_0((2,3],{\mathbb{M}}_n)$ resp. from $Q$) and additional, finitely many relations $$\mathcal{R}_\delta=\left\{\|f_i^\delta(\overline{\varphi}(g_j))^\delta-(f_i(g_j(2)\otimes 1_{[2,3]}))^\delta\|\leq\delta\right\}_{\begin{subarray}{l}1\leq i\leq l \\ 1\leq j\leq k \end{subarray}}.$$ Note that $P=\varinjlim P_\delta$ with respect to the canonical surjections $p_{\delta,\delta'}\colon P_\delta\rightarrow P_{\delta'}$ (for $\delta>\delta'$) and denote the induced maps $P_\delta\rightarrow P,f^\delta\mapsto f,q^\delta\mapsto s(q)$ by $p_{\delta,0}$. Since $P$ is semiprojective by part (1) of this proposition, we can find a partial lift $j_\delta\colon P\rightarrow P_\delta$ for some $\delta>0$, i.e. $p_{\delta,0}\circ j_\delta={\operatorname{id}}_P$.
Now let a finite set $F=\{x_1,...,x_m\}\subseteq Q$ and $\epsilon>0$ and be given. Denoting the inclusions $Q\rightarrow P_\delta,q\mapsto q^\delta$ by $s_\delta$, we can (after decreasing $\delta$ if necessary) assume that $\|s_\delta(x_i)-(j_\delta\circ s)(x_i)\|\leq\epsilon$ holds for all $1\leq i\leq m$. Now given any commuting square $$\xymatrix{
Q \ar[d]^s \ar[r]^\psi &D \ar@{->>}[d]^\pi \\
P \ar[r]^\varrho & D/J
}$$ it only remains to construct a $^*$-homomorphism $\psi_\delta\colon P_\delta\rightarrow D$ such that in the diagram $$\xymatrix{
Q \ar[rr]^\psi \ar[dd]_s \ar[dr]^{s_\delta} & & D \ar@{->>}[dd]^\pi \\
& P_\delta \ar@{-->}[ur]^{\psi_\delta} \ar[dl]_(0.4){p_{\delta,0}}\\
P \ar@/_1pc/[ur]_(0.6){j_\delta} \ar[rr]_\varrho && D/J
}$$ the upper central triangle and the lower right triangle commute.
We consider the following subalgebras of $T(\mathbb{C},G)$ and $S(\mathbb{C},G)$ for any $\eta>0$: $$\begin{array}{rl}
T_\eta(\mathbb{C},G) & =\{f\in T(\mathbb{C},G):f\;\text{is constant on}\;(0,\eta]\cup[2-\eta,2]\} \\
S_\eta(\mathbb{C},G) &=\{f\in S(\mathbb{C},G):f\;\text{is constant (=0) on}\;(0,\eta]\cup[2-\eta,2]\}
\end{array}$$ Since $$T(\mathbb{C},G)=\overline{\bigcup\limits_{\eta>0}T_\eta(\mathbb{C},G)}$$ we find $0<\eta<\frac{1}{2}$ and elements $\tilde{g}_j\in T_\eta(\mathbb{C},G)$ with $\tilde{g}_j(2)=g_j(2)$ and $\|g_j-\tilde{g}_j\|<\delta$ for every $1\leq j\leq k$. Let $h\in T(\mathbb{C},G)$ be the scalar-valued function which equals $1_G$ on $[\eta,2-\eta]$, satisfies $h(0)=h(2)=0$ and is linear in between. Consider the hereditary $C^*$-subalgebra $D'=\overline{(1-(\psi\circ\overline{\varphi})(h))D(1-(\psi\circ\overline{\varphi})(h))}$ and define $$D'':=(\psi\circ\overline{\varphi})(T_\eta(\mathbb{C},G))+ D'\subseteq D.$$ Then $(\psi\circ\overline{\varphi})(S_\eta(\mathbb{C},G))$ and $D'$ are orthogonal ideals in $D''$ because $h$ is central in $T(\mathbb{C},G)$. We further have $(\varrho\circ\varphi)({\mathcal{C}}_0((2,3],{\mathbb{M}}_n))\subseteq\pi(D')$ and hence obtain a commutative diagram $$\scalebox{0.88}{\xymatrix{
0\ar[r]&(\psi\circ\overline{\varphi})(S_\eta(\mathbb{C},G))\ar[r]\ar@{->>}[d]^\pi&D''\ar[r]\ar@{->>}[d]^\pi&H_D+D'\ar[r]\ar@{-->>}[d]&0\\
0\ar[r]&(\varrho\circ\varphi\circ s')(S_\eta(\mathbb{C},G))\ar[r]&\pi(D'')\ar[r]&H_{D/J}+\pi(D')\ar[r]&0\\
0\ar[r]&s'(S_\eta(\mathbb{C},G))\ar[r]\ar[u]^{\varrho\circ\varphi}&s'(T_\eta(\mathbb{C},G))+{\mathcal{C}}_0((2,3],{\mathbb{M}}_n)\ar[u]^{\varrho\circ\varphi}\ar[r]&\hat{T}(G,{\mathbb{M}}_n)\ar[r]\ar@{-->}[u]&0
}}$$ where $H_D$ and $H_{D/J}$ are finite-dimensional $C^*$-algebras given by $$H_D=(\psi\circ\overline{\varphi})(T_\eta(\mathbb{C},G))/(\psi\circ\overline{\varphi})(S_\eta(\mathbb{C},G)),$$ $$H_{D/J}=(\varrho\circ\varphi\circ s')(T_\eta(\mathbb{C},G))/(\varrho\circ\varphi\circ s')(S_\eta(\mathbb{C},G))$$ and $\hat{T}(G,{\mathbb{M}}_n)$ denotes what is called a crushed telescope in [@ELP98]: $$\hat{T}(G,{\mathbb{M}}_n)=\{f\in{\mathcal{C}}([2,3],{\mathbb{M}}_n)\colon f(2)\in G\}$$ By [@ELP98 Proposition 6.1.1], the embedding $G\rightarrow\hat{T}(G,{\mathbb{M}}_n)$ as constant functions is a conditionally projective map (in the sense of [@ELP98 Section 5.3]). It is hence possible to extend the map $\xymatrix{G\ar[r]^(0.22)\sim & T_\eta(\mathbb{C},G)/S_\eta(\mathbb{C},G)\ar[r] & H_D\subset H_D+D'}$ to a $^*$-homomorphism $\psi'\colon \hat{T}(G,{\mathbb{M}}_n)\rightarrow H_D+D'$ such that the diagram with exact rows $$\xymatrix{
0\ar[r]&D'\ar[r]\ar@{->>}[d]_\pi&H_D+D'\ar[r]\ar@{->>}[d]&H_D\ar[r]\ar@{->>}[d]\ar@/_/[l]&0\\
0\ar[r]&\pi(D')\ar[r]&H_{D/J}+\pi(D')\ar[r]|(0.69)\hole&H_{D/J}\ar[r]&0\\
0\ar[r]&{\mathcal{C}}_0((2,3],{\mathbb{M}}_n)\ar[r]\ar[u]^{\varrho\circ\varphi}&\hat{T}(G,{\mathbb{M}}_n)\ar[r]_{{\operatorname{ev}}_{2}}\ar@{-->}@/^2pc/[uu]^(0.7){\psi'}|(0.45)\hole|(0.5)\hole|(0.55)\hole \ar[u]&G\ar[u]\ar@/_/[l]\ar[r]\ar@/^2pc/[uu]&0
}$$ commutes. In particular, $\psi'$ restricts to a $^*$-homomorphism ${\mathcal{C}}_0((2,3],{\mathbb{M}}_n)\rightarrow D'$ which we will also denote by $\psi'$. But then a diagram chase confirms that $$\psi'(f_i)\cdot(\psi\circ\overline{\varphi})(\tilde{g}_j)=\psi'(f_i\cdot (\tilde{g}_j(2)\otimes 1_{[2,3]}))$$ holds for every $i,j$. Finally, define $\psi_\delta\colon P_\delta\rightarrow D$ by $$\begin{tabular}{ccc}$q^\delta\mapsto\psi(q)$&and&$f_i^\delta\mapsto\psi'(f_i)$.\end{tabular}$$ It needs to be checked that $\psi_\delta$ is well-defined, i.e. that the elements $\psi_\delta(f_i^\delta)$ and $\psi_\delta(\overline{\varphi}(g_j)^\delta)$ satisfy the relations $\mathcal{R}_\delta$: $$\begin{array}{rl}
& \|\psi_\delta(f_i^\delta)\psi_\delta(\overline{\varphi}(g_j)^\delta)-\psi_\delta((f_i(g_j(2)\otimes 1_{[2,3]}))^\delta)\|\\
=& \|\psi'(f_i)((\psi\circ\overline{\varphi})(g_j))-\psi'(f_i(g_j(2)\otimes 1_{[2,3]}))\| \\
\leq & \|\psi'(f_i)(\psi\circ\overline{\varphi})(\tilde{g}_j)-\psi'(f_i(\tilde{g}_j(2)\otimes 1_{[2,3]}))\|+\|f_i\|\cdot\|g_j-\tilde{g}_j\|<\delta
\end{array}$$ Since we also have $\psi_\delta\circ s_\delta=\psi$ and $\pi\circ\psi_\delta=\varrho\circ p_{\delta,0}$, the proof is hereby complete.
One example, where pullbacks as in \[prop wcp examples\] show up, is the class of so-called AF-telescopes defined by Loring and Pedersen:
\[def af-telescope\] Let $A=\overline{\bigcup A_n}$ be the inductive limit of an increasing union of finite-dimensional $C^*$-algebras $A_n$. We define the AF-telescope associated to this AF-system as $$T(A)=\{f\in{\mathcal{C}}_0((0,\infty],A):\;t\leq n\Rightarrow f(t)\in A_n\}.$$
We have an obvious limit structure for $T(A)=\varinjlim T(A_k)$ over the finite telescopes $$T(A_k)=\{f\in{\mathcal{C}}_0((0,k],A_k)):\;t\leq n\Rightarrow f(t)\in A_n\}.$$ Now the embedding of $T(A_k)$ into $T(A_{k+1})$ is given by extending the elements of $T(A_k)$ constantly onto the attached interval $[k,k+1]$. This is nothing but a finite composition of maps as in part (2) of \[prop wcp examples\]. Hence the connecting maps in the system of finite telescopes are weakly conditionally projective and using Lemma \[lemma limit criterium\] we recover [@LP98 Theorem 7.2]:
\[cor af-telescopes\] All $AF$-telescopes are projective.
In contrast to the original proof we didn’t have to work out any description of the telescopes by generators and relations. Such a description would have to encode the structure of each $A_n$ as well as the inlusions $A_n\subset A_{n+1}$ (i.e., the Bratteli-diagram of the system). Showing that such an infinite set of generators and relations gives rise to a projective $C^*$-algebra is possible but complicated. Instead we showed that these algebras are build up from the projective $C^*$-algebra $T(A_0)$=0 using operations which preserve projectivity.
Extensions by homogeneous $C^*$-algebras {#section extensions}
========================================
In this section we study extensions by (trivially) homogeneous $C^*$-algebras, i.e. extensions of the form $$\xymatrix{0 \ar[r] & {\mathcal{C}}_0(X,{\mathbb{M}}_N) \ar[r] & A \ar[r] & B \ar[r] & 0.}$$ Our final goal is to understand the behavior of semiprojectivity along such extensions, and we will eventually achieve this in Theorem \[thm 2 out of 3\].
Associated retract maps {#section retract maps}
-----------------------
Identifying $X$ with an open subset of ${\operatorname{Prim}}(A)$, we make the following definition of an associated retract map. This map will play a key role in our study of extensions.
\[def R\] Let $X$ be locally compact space with connected components $(X_i)_{i\in I}$ and $$\xymatrix{0 \ar[r] & {\mathcal{C}}_0(X,{\mathbb{M}}_N) \ar[r] & A \ar[r] & B \ar[r] & 0}$$ a short exact sequence of $C^*$-algebras. We define the (set-valued) retract map $R$ associated to the extension to be the map $$R\colon{\operatorname{Prim}}(A)\rightarrow 2^{{\operatorname{Prim}}(B)}$$ given by $$R(z)=\begin{cases}
\;z & \text{if}\;z\in{\operatorname{Prim}}(B), \\
\partial X_i=\overline{X_i}\backslash X_i & \text{if}\;z\in X_i\subseteq X.
\end{cases}$$
Note that $R$ defined as above takes indeed values in $2^{{\operatorname{Prim}}(B)}$ because the connected components $X_i$ are always closed in $X$. However, in our cases of interest the components $X_i$ will actually be clopen in $X$ (e.g. if $X$ is locally connected) so that we have a topological decomposition $X=\bigsqcup_i X_i$.
### Regularity properties for set-valued maps
Let $X,Y$ be sets and $S\colon X \rightarrow 2^{Y}$ a set-valued map. We say that $S$ has pointwise finite image if $S(x)\subseteq Y$ is a finite set for every $x\in X$. If furthermore $X$ and $Y$ are topological spaces, we will use the following notion of semicontinuity for $S$ (cf. [@AF90 Section 1.4]).
\[def lsc set map\] Let $X,Y$ be topological spaces. A set-valued map $S\colon X\rightarrow 2^Y$ is lower semicontinuous if one of the following equivalent conditions holds:
1. $\{x\in X\colon S(x)\subseteq B\}$ is closed in $X$ for every closed $B\subseteq Y$.
2. For every neighborhood $N(\overline{y})$ of $\overline{y}\in S(\overline{x})$ there exists a neighborhood $N(\overline{x})$ of $\overline{x}$ with $S(x)\cap N(\overline{y})\neq\emptyset$ for every $x\in N(\overline{x})$.
3. For every net $(x_\lambda)_{\lambda\in\Lambda}\subset X$ with $x_\lambda\rightarrow x$ and every $y\in S(x)$ there exists a net $(y_\mu)_{\mu\in M}\subset\{S(x_\lambda)\colon\lambda\in\Lambda\}$ such that $y_\mu\rightarrow y$.
$(i)\Rightarrow (ii)$: Let $N(\overline{y})$ be an open neighborhood of $\overline{y}\in S(\overline{x})$. Then $\{x\in X: S(x)\subset Y\backslash N(\overline{y})\}$ is closed and does not contain $\overline{x}$. Hence we find an open neighborhood $N(\overline{x})$ of $\overline{x}$ in $X\backslash\{x\in X: S(x)\subset Y\backslash N(\overline{y})\}=\{x\in X:S(x)\cap N(\overline{y})\neq\emptyset\}$.
$(ii)\Rightarrow (iii)$: Denote by $\mathcal{N}$ the family of neighborhoods of $y$ ordered by reversed inclusion. Set $M=\{(\lambda, N)\in\Lambda\times\mathcal{N}\colon S(x_{\lambda'})\cap N\neq\emptyset\;\forall\;\lambda'\geq\lambda\}$, then by assumption $M$ is nonempty and directed with respect to the partial order $(\lambda_1,N_1)\leq(\lambda_2,N_2)$ iff $\lambda_1\leq\lambda_2$ and $N_2\subseteq N_1$. Now pick a $y_{(\lambda,N)}\in S(x_\lambda)\cap N$ for each $(\lambda,N)\in M$, then $(y_\mu)_{\mu\in M}$ constitutes a suitable net converging to $y$.
$(iii)\Rightarrow (i)$: Let a closed set $B\subseteq Y$ and $(x_\lambda)_{\lambda\in\Lambda}\subset\{x\in X:S(x)\subseteq B\}$ with $x_\lambda\rightarrow\overline{x}$ be given. Then for any $\overline{y}\in S(\overline{x})$ we find a net $y_\mu\rightarrow\overline{y}$ with $(y_\mu)\subset\{S(x_\lambda):\lambda\in\Lambda\}\subset B$. Since $B$ is closed we have $\overline{y}\in B$ showing that $S(\overline{x})\subset B$.
An ordinary (i.e. a single-valued) map is evidently lower semicontinuous in the sense above if and only if it is continuous. If both spaces $X$ and $Y$ are first countable, we may use sequences instead of nets in condition $(iii)$.
Examples of set-valued maps that are lower semicontinuous in the sense above arise from split extensions by homogeneous $C^*$-algebras as follows.
\[ex split\] Let a split-exact sequence of separable $C^*$-algebras $$\xymatrix{
0 \ar[r]&{\mathcal{C}}_0(X,{\mathbb{M}}_n)\ar[r]&A \ar[r]_\pi&B\ar[r]\ar@/_1pc/[l]_s&0
}$$ be given and consider the set-valued map $R_s\colon {\operatorname{Prim}}(A)\rightarrow 2^{{\operatorname{Prim}}(B)}$ given by $$R_s(z)=\begin{cases} \hspace{1.2cm} z&\text{if}\;z\in {\operatorname{Prim}}(B) \\ \left\{[\pi_{z,1}],...,[\pi_{z,r(z)}]\right\}& \text{if}\;z\in X \end{cases}$$ where $\pi_{z,1}\oplus ...\oplus\pi_{z,r(z)}$ is the decomposition of $B\xrightarrow{s} A\rightarrow{\mathcal{C}}_b(X,{\mathbb{M}}_n)\xrightarrow{{\operatorname{ev}}_z}{\mathbb{M}}_n$ into irreducible summands. Then $R_s$ is lower semicontinuous in the sense of \[def lsc set map\].
We verify condition $(ii)$ of \[def lsc set map\]: Let $z_n\rightarrow \overline{z}$ in ${\operatorname{Prim}}(A)$ and a neighborhood $N(\overline{y})$ of $\overline{y}\in R_s(\overline{z})$ in ${\operatorname{Prim}}(B)$ be given. By Lemma \[lemma top prim\] we may assume that $N(\overline{y})$ is of the form $\{z\in{\operatorname{Prim}}(B)\colon\check{b}(z)>1/2\}$ for some $b\in B$. By definition of $R_s$, we find $\check{s(b)}(z)=\max_{y\in R_s(z)} \check{b}(y)$ for all $z\in{\operatorname{Prim}}(A)$. Hence $N(\overline{z})=\{z\in{\operatorname{Prim}}(A)\colon\check{s(b)}>1/2\}$ constitutes a neighborhood of $\overline{z}$ in ${\operatorname{Prim}}(A)$ which satisfies \[def lsc set map\] $(ii)$.
Note that the retract map $R_s$ in \[ex split\] highly depends on the choice of splitting $s$ while the retract map $R$ from \[def R\] is associated to the underlying extension in a natural way. It is the goal of section \[section lifting busby\] to find a splitting $s$ such that $R=R_s$ holds. This is, however, not always possible. It can even happen that the underlying extension splits while $R$ is not of the form $R_s$ for any splitting $s$ (cf. remark \[remark comm retract\]). Under suitable conditions, we will at least be able to arrange $R=R_s$ outside of a compact set $K\subset X$, i.e. we can find a (not necessarily multiplicative) splitting map $s$ such that $B\xrightarrow{s} A\rightarrow{\mathcal{C}}_b(X,{\mathbb{M}}_n)$ is multiplicative on $X\backslash K$ so that $R_s(x)$ is still well-defined and coincides with $R(x)$ for all $x\in{\operatorname{Prim}}(A)\backslash K$.
### Lifting the Busby map {#section lifting busby}
In this section we identify conditions on an extension $$\xymatrix{
0 \ar[r] & {\mathcal{C}}_0(X,{\mathbb{M}}_N) \ar[r] & A \ar[r] & B \ar[r] \ar@{..>}@/_1pc/[l]^? & 0 & [\tau]
}$$ which allow us to contruct a splitting $s\colon B\rightarrow A$. This is evidently the same as asking for a lift of the corresponding Busby map $\tau$ as indicated on the left of the commutative diagram $$\xymatrix{
& {\mathcal{C}}(\beta X,{\mathbb{M}}_N) \ar@{->>}[d]^\varrho \ar@{=}[r] & \prod\limits_{i\in I}{\mathcal{C}}(\beta X_i,{\mathbb{M}}_N) \ar@{->>}[d] \\
B \ar[r]^(0.3)\tau \ar@/^1pc/@{..>}[ur]^s \ar@/_2pc/[rr]^{\oplus\tau_i} & {\mathcal{C}}(\chi(X),{\mathbb{M}}_N) \ar@{->>}[r] & \prod\limits_{i\in I}{\mathcal{C}}(\chi(X_i),{\mathbb{M}}_N).
}$$ We will produce a suitable lift of $\tau$ in two steps:
1. For every component $X_i$ of $X$, we trivialize the map $\tau_i\colon B\rightarrow{\mathcal{C}}(\chi(X_i),{\mathbb{M}}_N)$, i.e. we conjugate it to a constant map, so that it can be lifted to ${\mathcal{C}}(\beta X_i,{\mathbb{M}}_N)$. This step requires the associated retract map $R$ from \[def R\] to have pointwise finite image and the spaces $\chi(X_i)$ to be connected and low-dimensional.
2. We extend the collection of lifts for the $\tau_i$’s to a lift for $\tau$. Here we need the associated retract map $R$ to be lower semicontinuous.
In many cases of interest, the spaces $\chi(X_i)$ will not be connected, so that we have to modify the first step of the lifting process. This results in the fact that we cannot find a (multiplicative) split $s$ in general. Instead we will settle for a lift $s$ of $\tau$ with slightly weaker multiplicative properties.
First we give the connection between the retract map $R$ and the Busby map $\tau$ of the extension.
\[lemma Busby vs boundary\] Let a short exact sequence $$\xymatrix{
0 \ar[r] & I \ar[r] & A \ar[r]^\pi & B \ar[r] & 0
}$$ with Busby map $\tau\colon B\rightarrow \mathcal{Q}(I)$ be given. Identifying ${\operatorname{Prim}}(I)$ with the open subset $\{J|I\nsubseteq J\}$ of ${\operatorname{Prim}}(A)$, the following statements hold:
1. $J\in\partial{\operatorname{Prim}}(I)\Leftrightarrow I+I^\bot\subseteq J$ for every $J\in{\operatorname{Prim}}(A)$,
2. $\partial{\operatorname{Prim}}(I)={\operatorname{Prim}}(\tau(B))$.
If in addition $I$ is subhomogeneous, we further have
3. $\left|\partial{\operatorname{Prim}}(I)\right|<\infty\Leftrightarrow{\operatorname{dim}}(\tau(B))<\infty$.
For (i) it suffices to check that ${\operatorname{Prim}}(I^\bot)={\operatorname{Prim}}(A)\backslash\overline{{\operatorname{Prim}}(I)}$ where $I^\bot$ denotes the annihilator of $I$ in $A$. But this follows directly from the definition of the Jacobson topology on ${\operatorname{Prim}}(A)$: $$\begin{array}{rcl}
J\notin\overline{{\operatorname{Prim}}(I)} & \Leftrightarrow & \bigcap\limits_{K\in {\operatorname{Prim}}(I)}K\nsubseteq J \\
& \Leftrightarrow & \exists x\in A\colon x\notin J\;\text{while}\;\left\|x\right\|_K=0\;\forall\; K\in{\operatorname{Prim}}(I) \\
& \Leftrightarrow & \exists x\in I^\bot\colon x\notin J \\
& \Leftrightarrow & I^\bot\nsubseteq J \\
& \Leftrightarrow & J\in{\operatorname{Prim}}(I^\bot).
\end{array}$$ Now (i) and $\ker(\tau\circ\pi)=I+I^\bot$ imply $$\begin{array}{rl}
{\operatorname{Prim}}(\tau(B))= &{\operatorname{Prim}}((\tau\circ\pi)(A)) \\
= & \{J\in{\operatorname{Prim}}(A)\colon\ker(\tau\circ\pi)\subseteq J\} \\
= &\{J\in{\operatorname{Prim}}(A)\colon I+I^\bot\subseteq J\} \\
= &\partial{\operatorname{Prim}}(I).
\end{array}$$ For the last statement note that if all irreducible representations of $I$ have dimension at most $n$, the same holds for all irreducible representations $\pi$ of $A$ with $\ker(\pi)$ contained in $\overline{{\operatorname{Prim}}(I)}$. So by the correspondence described in (ii), irreducible representations of $\tau(B)$ are also at most $n$-dimensional. Hence, in this case, finitenesss of $\partial{\operatorname{Prim}}(I)$ is equivalent to finite-dimensionality of $\tau(B)$.
For technical reasons we would prefer to work with unital extensions. However, it is not clear whether unitization preserves the regularity of $R$, i.e. whether the retract map associated to a unitized extension $0\rightarrow {\mathcal{C}}_0(X,{\mathbb{M}}_N)\rightarrow A^+ \rightarrow B^+ \rightarrow 0$ is lower semicontinuous provided that the retract map associated to the original extension is. As the next lemma shows, this is true and holds in fact for more general extensions.
\[lemma finite extension\] Let a locally compact space $X$ with clopen connected components and a commutative diagram $$\xymatrix{
&& 0 \ar[d] & 0 \ar[d] \\
0 \ar[r] & {\mathcal{C}}_0(X,{\mathbb{M}}_N) \ar[r] \ar@{=}[d] & A \ar[r] \ar[d] & B \ar[r] \ar[d] & 0 \\
0 \ar[r] & {\mathcal{C}}_0(X,{\mathbb{M}}_N) \ar[r] & C \ar[r] \ar[d]^\pi & D \ar[r] \ar[d] & 0 \\
&& F \ar@{=}[r] \ar[d] & F \ar[d] \\
&& 0 & 0
}$$ of short exact sequences of separable $C^*$-algebras be given. Let $R\colon {\operatorname{Prim}}(A)\rightarrow 2^{{\operatorname{Prim}}(B)}$ (resp. $S\colon {\operatorname{Prim}}(C)\rightarrow 2^{{\operatorname{Prim}}(D)}$) be the set-valued retract map associated to the upper (resp. the lower) horizontal sequence as in \[def R\]. If the quotient $F$ is a finite-dimensional $C^*$-algebra, then the following holds:
1. If $R$ has pointwise finite image, then so does $S$.
2. If $R$ is lower semicontinuous, then so is $S$.
\(1) This is immediate since ${\operatorname{Prim}}(F)$ is a finite set and one easily verfies $S(x)\subseteq R(x)\cup{\operatorname{Prim}}(F)$ for all $x\in X$.
\(2) We may assume that $F$ is simple and hence $\pi$ is irreducible. Note that $S(J)=R(J)$ for all $J\in{\operatorname{Prim}}(B)\subset{\operatorname{Prim}}(D)$, while for $x\in X$ we have either $S(x)=R(x)$ or $S(x)=R(x)\cup\{[\pi]\}$. Given a closed subset $K\subseteq{\operatorname{Prim}}(D)$, we need to verify that $\{J\in{\operatorname{Prim}}(C)\colon S(J)\subseteq K\}$ is closed in ${\operatorname{Prim}}(C)$. If $[\pi]\in K$, then $\{J\in{\operatorname{Prim}}(C)\colon S(J)\subseteq K\}=\{J\in{\operatorname{Prim}}(A)\colon R(J)\subseteq K\}\cup\{[\pi]\}$ is closed in ${\operatorname{Prim}}(C)$ because $\{J\in{\operatorname{Prim}}(A)\colon R(J)\subseteq K\}$ is closed in ${\operatorname{Prim}}(A)$ by semicontinuity of $R$. Now if $[\pi]\notin K$, the only relevant case to check is a sequence $x_n\subset X$ converging to $\overline{x}\in{\operatorname{Prim}}(D)$ with $S(x_n)\subseteq K$ for all $n$. We then need to show that $S(\overline{x})=\overline{x}\in K$ as well. Decompose $X=\bigcup_{i\in I}X_i$ into its clopen connected components and write $x_n\in X_{i_n}$ for suitable $i_n\in I$. We may assume that $i_n\neq i_m$ for $n\neq m$ since otherwise $\overline{x}\in\partial X_{i_n}=S(x_n)$ for some $n$. Since $R$ is lower semicontinuous, we know that the boundary of $\bigcup_n X_{i_n}$ in ${\operatorname{Prim}}(A)$ is contained in $K\cap{\operatorname{Prim}}(A)$ and hence $\partial\left(\bigcup_n X_{i_n}\right)\subset K\cup\{[\pi]\}$ in ${\operatorname{Prim}}(C)$.
Let $p$ denote the projection of ${\mathcal{C}}_0(X,{\mathbb{M}}_N)$ onto ${\mathcal{C}}_0(\bigcup_n X_{i_n},{\mathbb{M}}_N)$. This map canonically extends to $\overline{p}$ and $\overline{\overline{p}}$ making the diagram $$\xymatrix{
0 \ar[r] & {\mathcal{C}}_0(X,{\mathbb{M}}_k) \ar[r] \ar@{->>}[d]^p & C \ar[r] \ar[d]^{\overline{p}} & D \ar[r]\ar[d]^{\overline{\overline{p}}} & 0\\
0 \ar[r] & \bigoplus\limits_n{\mathcal{C}}_0(X_{i_n},{\mathbb{M}}_N) \ar[r] \ar[d]^\subseteq & \prod\limits_n{\mathcal{C}}(\beta X_{i_n},{\mathbb{M}}_N) \ar[r] \ar@{=}[d]& \frac{\prod_n{\mathcal{C}}(\beta X_{i_n},{\mathbb{M}}_N)}{\bigoplus_n{\mathcal{C}}_0(X_{i_n},{\mathbb{M}}_N)} \ar[r] \ar@{->>}[d]^q&0\\
0\ar[r] & \prod\limits_n{\mathcal{C}}_0(X_{i_n},{\mathbb{M}}_N) \ar[r] & \prod\limits_n{\mathcal{C}}(\beta X_{i_n},{\mathbb{M}}_N) \ar[r] & \prod\limits_n{\mathcal{C}}(\chi(X_{i_n}),{\mathbb{M}}_N) \ar[r] & 0
}$$ commute. Using Lemma \[lemma Busby vs boundary\], we can indentify the boundary of $\bigcup_n X_{i_n}$ in ${\operatorname{Prim}}(C)$ with ${\operatorname{Prim}}\left(\overline{\overline{p}}(D)\right)$. We already know that $\overline{\overline{p}}$ factors through $D_K\oplus F$, where $D_K$ denotes the quotient corresponding to the closed subset $K$ of ${\operatorname{Prim}}(D)$, and denote the induced map by $\varphi$: $$\xymatrix{
D \ar[rr]^{\overline{\overline{p}}} \ar[dr]_{\pi_K\oplus \pi}&& \frac{\prod_n{\mathcal{C}}(\beta X_{i_n},{\mathbb{M}}_N)}{\bigoplus_n{\mathcal{C}}_0(X_{i_n},{\mathbb{M}}_N)} \\
& D_K\oplus F \ar@{-->}[ur]_\varphi
}$$ We further know that the composition $q\circ\varphi_{|F}\colon F\rightarrow\prod_n{\mathcal{C}}(\chi(X_{i_n}),{\mathbb{M}}_N)$ vanishes because $[\pi]\notin\partial X_{i_n}=R(x_n)\subseteq K$ for all $n$. Hence the image of $F$ under $\varphi$ is contained in $\ker(q)=\frac{\prod_n{\mathcal{C}}_0(X_{i_n},{\mathbb{M}}_N)}{\bigoplus_n{\mathcal{C}}_0(X_{i_n},{\mathbb{M}}_N)}$. But since this $C^*$-algebra is projectionless and $F$ is finite-dimensional, we find $\varphi_{|F}=0$. Consequently, $\overline{\overline{p}}$ factors through $D_K$ which means nothing but $\overline{x}\in\partial\left(\bigcup_n X_{i_n}\right)={\operatorname{Prim}}\left(\overline{\overline{p}}(D)\right)\subseteq K$.
\[lemma diagonal\] Let $X$ be a connected, compact space of dimension at most 1. For every finite-dimensional $C^*$-algebra $F\subseteq{\mathcal{C}}(X,{\mathbb{M}}_n)$ there exists a unitary $u\in{\mathcal{C}}(X,{\mathbb{M}}_n)$ such that $uFu^*$ is contained in the constant ${\mathbb{M}}_n$-valued functions on $X$.
Since ${\operatorname{dim}}(X)\leq 1$, equivalence of projections in ${\mathcal{C}}(X,{\mathbb{M}}_n)$ is completely determined by their rank ([@Phi07 Proposition 4.2]). In particular, the $C^*$-algebra ${\mathcal{C}}(X,{\mathbb{M}}_n)$ has cancellation. Hence [@RLL00 Lemma 7.3.2] shows that the inclusion $F\subset{\mathcal{C}}(X,{\mathbb{M}}_n)$ is unitarily equivalent to any constant embedding $\iota\colon F\rightarrow{\mathbb{M}}_n\subseteq{\mathcal{C}}(X,{\mathbb{M}}_n)$ with ${\operatorname{rank}}(\iota(p))={\operatorname{rank}}(p)$ for all minimal projections $p\in F$.
\[lemma lift unitary\] Let $X$ be a connected, locally compact, metrizable space of dimension at most 1. Then every unitary in ${\mathcal{C}}(\chi(X),{\mathbb{M}}_n)$ lifts to a unitary in ${\mathcal{C}}(\beta X,{\mathbb{M}}_n)$.
By [@Phi07 Proposition 4.2], we have $K_0({\mathcal{C}}(\alpha X,{\mathbb{M}}_n))\cong{\mathbb{Z}}$ via $[p]\mapsto{\operatorname{rank}}(p)$. Using the 6-term exact sequence in $K$-theory, this shows that the induced map $K_1({\mathcal{C}}(\beta X,{\mathbb{M}}_n))\rightarrow K_1({\mathcal{C}}(\chi(X),{\mathbb{M}}_n))$ is surjective. Combining this with $K_1$-bijectivity of ${\mathcal{C}}(\beta X,{\mathbb{M}}_n)$, which is guaranteed by ${\operatorname{dim}}(\beta X)={\operatorname{dim}}(X)\leq 1$ ([@Nag70 Thm. 9.5]) and [@Phi07 Theorem 4.7], the claim follows.
\[prop diagonal Busby\] Let a short exact sequence of separable $C^*$-algebras $$\xymatrix{
0 \ar[r] & {\mathcal{C}}_0(X,{\mathbb{M}}_N) \ar[r] & A \ar[r] & B \ar[r] & 0 & [\tau]
}$$ with Busby invariant $\tau$ be given. Assume that $X$ is at most one-dimensional, has clopen connected components $(X_i)_{i\in I}$ and that every corona space $\chi(X_i)$ has only finitely many connected components. If the associated set-valued retract map $R$ as in \[def R\] has pointwise finite image, then there is a unitary $U\in{\mathcal{C}}(\beta X,{\mathbb{M}}_N)$ such that for each $i\in I$ the composition $$B \xrightarrow{\tau} {\mathcal{C}}(\chi(X),{\mathbb{M}}_n) \xrightarrow{{\operatorname{Ad}}(\varrho(U))}{\mathcal{C}}(\chi(X),{\mathbb{M}}_N)\rightarrow {\mathcal{C}}(\chi(X_i),{\mathbb{M}}_n)$$ has image contained in the locally constant ${\mathbb{M}}_N$-valued functions on $\chi(X_i)$.
By Lemma \[lemma Busby vs boundary\], the image of each $\tau_i\colon B \xrightarrow{\tau} {\mathcal{C}}(\chi(X),{\mathbb{M}}_N) \rightarrow {\mathcal{C}}(\chi(X_i),{\mathbb{M}}_N)$ is finite-dimensional. Since by [@Nag70 Thm. 9.5] furthermore ${\operatorname{dim}}\chi(X_i)\leq {\operatorname{dim}}\beta X_i={\operatorname{dim}}X_i\leq{\operatorname{dim}}X\leq 1$, we can apply Lemma \[lemma diagonal\] to obtain unitaries $u_i\in{\mathcal{C}}(\chi(X_i),{\mathbb{M}}_N)$ such that $u_i\tau_i(B)u_i^*$ is contained in the locally constant functions on $\chi(X_i)$. These unitaries can be lifted to unitaries $U_i\in{\mathcal{C}}(\beta X_i,{\mathbb{M}}_N)$ by Lemma \[lemma lift unitary\]. Now $U=\oplus_i U_i\in\prod_i{\mathcal{C}}(\beta X_i,{\mathbb{M}}_N)={\mathcal{C}}(\beta X,{\mathbb{M}}_N)$ has the desired property.
\[lemma approx unitary\] Let a short exact sequence of separable $C^*$-algebras $$\xymatrix{
0 \ar[r] & {\mathcal{C}}_0(X,{\mathbb{M}}_N) \ar[r] & A \ar[r] & B \ar[r] & 0 & [\tau]
}$$ with Busby map $\tau$ be given. Assume that $X$ is at most one-dimensional and that the connected components $(X_i)_{i\in I}$ of $X$ are clopen. Further assume that the image of $\tau$ is constant on each $\chi(X_i)\subseteq\chi(X)$. Denote by $\iota\colon A \rightarrow \mathcal{M}({\mathcal{C}}_0(X,{\mathbb{M}}_N))={\mathcal{C}}(\beta X,{\mathbb{M}}_N)$ the canonical map. If the set-valued retract map $R\colon {\operatorname{Prim}}(A)\rightarrow 2^{{\operatorname{Prim}}(B)}$ as defined in \[def R\] is lower semicontinuous, the following statement holds:
For every finite set $\mathcal{G}\subset A$, every $\epsilon>0$ and almost every $i\in I$ there exists a unitary $U_i \in{\mathcal{C}}(\alpha X_i,{\mathbb{M}}_N)\subset{\mathcal{C}}(\beta X_i,{\mathbb{M}}_N)$ such that $$\left\|(U_i\iota(a)_{|\beta X_i}U_i^*)(x)-\iota(a)(y)\right\|<\epsilon$$ holds for all $a\in\mathcal{G}$, $x\in\beta X_i$ and $y\in\chi(X_i)$.
We may assume that $A$ is unital by Lemma \[lemma finite extension\]. Let a finite set $\mathcal{G}\subset A$ and $\epsilon>0$ be given. For each $x\in\beta X$, we write $F_x={\operatorname{im}}({\operatorname{ev}}_x\circ\iota)\subseteq{\mathbb{M}}_N$ and $$T_1(F_x)=\{f\in{\mathcal{C}}([0,1],F_x)\colon f(0)\in\mathbb{C}\cdot 1_{F_x}\},$$ $$S_1(F_x)=\{f\in{\mathcal{C}}_0([0,1),F_x)\colon f(0)\in\mathbb{C}\cdot 1_{F_x}\}.$$ Further let $h_\eta\in{\mathcal{C}}_0[0,1)$ denote the function $t\mapsto \max\{1-t-\eta,0\}$. Using the Urysohn-type result [@ELP98 Theorem 2.3.3], we find for each $x\in\beta X$ a commuting diagram $$\xymatrix{
0 \ar[r] & J_x \ar[r] & A \ar[r]^{{\operatorname{ev}}_x\circ\iota} & F_x \ar[r] & 0 \\
0 \ar[r] & S(\mathbb{C},F_x) \ar[u]^{\varphi_x} \ar[r] & T(\mathbb{C},F_x) \ar[u]^{\overline{\varphi}_x} \ar[r]_{{\operatorname{ev}}_1} & F_x \ar@{=}[u] \ar[r] \ar@/_1pc/@{..>}[l]_{s_x}& 0
}$$ such that $\overline{\varphi}_x$ is unital and $\varphi_x$ is proper. Let $s_x$ be any split for the lower sequence satisfiying $s_x(b)(t)=b$ for $t\geq 1/2$.
Now consider $$V_{x,\delta}=\{y\in\beta X\colon\quad ({\operatorname{ev}}_y\circ\iota)(\overline{\varphi}_x(h_\delta))=0\}$$ which is, for $\delta>0$, a closed neighborhood of $x$ in $\beta X$. Note that by assumption $\chi(X_i)\cap V_{x,\delta}\neq\emptyset$ implies $\chi(X_i)\subseteq V_{x,\delta}$. We further claim the following: For almost every $i\in I$ the inclusion $\chi(X_i)\subseteq V_{x,\delta}$ implies $X_i\subset V_{x,2\delta}$. Assume otherwise, then we find pairwise different $i_n\in I$, points $x_n\in X_{i_n}$ and some $1\leq j\leq m$ such that $\chi(X_{i_n})\subseteq V_{x,\delta}$ while $x_n\notin V_{x,2\delta}$ for all $n$. We may assume that ${\operatorname{ev}}_{x_n}\circ\iota$ converges pointwise to a representation $\pi$. Then $$\|\pi(\overline{\varphi}_x(h_\delta))\|=\lim_n\|({\operatorname{ev}}_{x_n}\circ\iota\circ\overline{\varphi}_x)(h_\delta)\|\geq\delta$$ since $x_n\neq V_{x,2\delta}$ implies that ${\operatorname{ev}}_{x_n}\circ\iota\circ\overline{\varphi}$ contains irreducible summands corresponding to evaluations at points $t$ with $t<1-2\delta$. On the other hand, since the retract map $R$ is lower semicontinuous, we find each irreducible summand of $\pi$ to be the limit of irreducible subrepresentations $\varrho_n$ of ${\operatorname{ev}}_{y_n}\circ\iota$ where $y_n\in\chi(X_{i_n})\subseteq V_{x,\delta}$. Hence $$\|\pi(\overline{\varphi}_x(h_\delta))\|\leq\liminf_n\|\varrho_n(\overline{\varphi}_x(h_\delta))\|=0$$ by \[lemma top prim\], giving a contradiction and thereby proving our claim.
Since $\varphi_x$ is proper, we have $J_x=\overline{\bigcup_{\eta>0}{\operatorname{her}}(\varphi_x(h_\eta))}$. Hence there exists $1/2>\delta(x)>0$ such that $$\inf\left\{\|(a-(\overline{\varphi}_x\circ s_x\circ{\operatorname{ev}}_x\circ\iota)(a))-b\|\colon b\in{\operatorname{her}}(\varphi_x(h_{2\delta(x)}))\right\}<\frac{\epsilon}{2}$$ for all $a\in\mathcal{G}$. By compactness of $\chi(X)$, we find $x_1,...,x_m$ such that $$\chi(X)\subseteq\bigcup_{j=1}^mV_{x_j,\delta(x_j)}.$$ Then by the claim proved earlier, for almost every $i$ with $\chi(X_i)\subseteq V_{x_j,\delta(x_j)}$ we have a factorization as indicated $$\xymatrix{
A \ar[d]_{{\operatorname{ev}}_{x_j}\circ\iota} \ar@{->>}[drr]^{\pi_j} \ar[rr]^\iota && \prod_i{\mathcal{C}}(\alpha X_i,{\mathbb{M}}_N) \ar[r] & {\mathcal{C}}(\alpha X_i,{\mathbb{M}}_N), \\
F_{x_j} \ar[rr]_{\pi_j\circ\overline{\varphi}_{x_j}\circ s_{x_j}} && A/\langle\overline{\varphi}_{x_j}(h_{2\delta(x_j)})\rangle \ar@{-->}[ur]_{\overline{\iota}_i}
}$$ where $\langle\overline{\varphi}_{x_j}(h_{2\delta(x_j)})\rangle$ denotes the ideal generated by $\overline{\varphi}_{x_j}(h_{2\delta(x_j)})$ and $\pi_j$ the corresponding quotient map. By the choice of $\delta(x_j)$, the lower left triangle commutes up to $\epsilon/2$ on the finite set $\mathcal{G}$. Also note that the map $\pi_j\circ\overline{\varphi}_{x_j}\circ s_{x_j}$ is multiplicative.
Finally, by Lemma \[lemma diagonal\] there exists a unitary $U_i\in{\mathcal{C}}(\alpha X_i,{\mathbb{M}}_N)$ such that ${\operatorname{Ad}}(U_i)\circ(\overline{\iota}_i\circ\pi_j\circ\overline{\varphi}_{x_j}\circ s_{x_j})$ is a constant embedding. Of course, we may arrange $U(\infty)=1$. We then verify $$\begin{array}{rl}
& \|(U_i\iota(a)_{|\beta X_i}U_i^*)(x)-\iota(a)(y)\| \\
\leq & \|(U_i(\overline{\iota}_i\circ\pi_j\circ\overline{\varphi}_{x_j}\circ s_{x_j})(({\operatorname{ev}}_{x_j}\circ\iota)(a))U_i^*)(x)-(\overline{\iota}_i\circ\pi_j)(a)(y)\| + \frac{\epsilon}{2}\\
\leq & \|(U_i(\overline{\iota}_i\circ\pi_j\circ\overline{\varphi}_{x_j}\circ s_{x_j})(({\operatorname{ev}}_{x_j}\circ\iota)(a))U_i^*)(y)-(\overline{\iota}_i\circ\pi_j)(a)(y)\| + \frac{\epsilon}{2}\\
= & \|(\overline{\iota}_i\circ(\pi_j\circ\overline{\varphi}_{x_j}\circ s_{x_j})\circ({\operatorname{ev}}_{x_j}\circ\iota))(a)(y)-(\overline{\iota}_i\circ\pi_j)(a)(y)\| + \frac{\epsilon}{2}\\
\leq & \epsilon. \\
\end{array}$$ Applying this procedure to each of the finitely many points $x_1,...,x_m$, the statement of the lemma follows.
Using Lemma \[lemma approx unitary\] we can now construct a split for our sequence of interest - at least in the case of $\tau(B)$ being constant on each $\chi(X_i)$.
\[cor projective split\] If $0\rightarrow{\mathcal{C}}_0(X,{\mathbb{M}}_N)\rightarrow A\rightarrow B\rightarrow 0$ is a short exact sequence of separable $C^*$-algebras such that the assumptions of Lemma \[lemma approx unitary\] hold, then this sequence splits.
Let $\tau\colon B\rightarrow\mathcal{Q}({\mathcal{C}}_0(X,{\mathbb{M}}_N))={\mathcal{C}}(\chi(X),{\mathbb{M}}_N)$ denote the Busby map of the sequence. We have the canonical commutative diagram $$\xymatrix{
0 \ar[r] & \bigoplus_i {\mathcal{C}}_0(X_i,{\mathbb{M}}_N) \ar[r] & {\mathcal{C}}(\beta X,{\mathbb{M}}_N) \ar[r]^\varrho & {\mathcal{C}}(\chi(X),{\mathbb{M}}_N) \ar[r] & 0 \\
0 \ar[r] & {\mathcal{C}}_0(X,{\mathbb{M}}_N) \ar[r] \ar@{=}[u] & A \ar[r]^\pi \ar[u]^\iota & B \ar[r] \ar[u]^\tau & 0.
}$$ Choose points $y_i\in\chi(X_i)$ for every $i\in I$. Using separability of $A$ and Lemma \[lemma approx unitary\], we find a unitary $U\in\prod_i{\mathcal{C}}(\alpha X_i,{\mathbb{M}}_N)\subset\prod_i{\mathcal{C}}(\beta X_i,{\mathbb{M}}_N)={\mathcal{C}}(\beta X,{\mathbb{M}}_N)$ with $$U\iota(a)U^*-\prod_i\iota(a)(y_i)\cdot 1_{\alpha X_i}\in\bigoplus_i{\mathcal{C}}_0(X_i,{\mathbb{M}}_N)$$ for all $a\in A$ (where $\iota(a)(y_i)\cdot 1_{\alpha X_i}$ denotes the function on $\alpha X_i$ with constant value $\iota(a)(y_i)$). By setting $s(\pi(a))=U^*\left(\prod_i(\iota(a)(y_i)\cdot 1_{\alpha X_i}\right)U$ we find $s\colon B\rightarrow{\mathcal{C}}(\beta X,{\mathbb{M}}_N)$ with $(\varrho\circ s)(\pi(a))=(\varrho\circ\iota)(a)=\tau(\pi(a))$ by the formula above. Identifying $A$ with the pullback over $\varrho$ and $\tau$, we can regard $s$ as a map from $B$ to $A$ with $\pi\circ s={\operatorname{id}}_B$, i.e. we have constructed a split for the sequence.
As the example $0 \rightarrow {\mathcal{C}}_0(0,1) \rightarrow {\mathcal{C}}[0,1] \rightarrow \mathbb{C}^2 \rightarrow 0$ shows, we cannot expect extensions by ${\mathcal{C}}_0(X,M_N)$ to split if the corona space of $X$ (or of one of its components) is not connected. We will now deal with these components and show that one can still obtain a split $s\colon B\rightarrow A$ which, though not multiplicative in general, has still good multiplicative properties.
\[lemma almost split\] Let $0 \rightarrow {\mathcal{C}}_0(X,{\mathbb{M}}_N) \rightarrow A \rightarrow B \rightarrow 0$ be a short exact sequence with Busby map $\tau$. Assume that the corona space $\chi(X)$ of $X$ has only finitely many connected components and that the image of $\tau$ is contained in the locally constant functions on $\chi(X)$. Then there exists a compact set $K\subset X$ and a completely positive split $s\colon B \rightarrow {\mathcal{C}}(\beta X,{\mathbb{M}}_N)$ which is multiplicative outside of an open set $U\subset K$.
Let $\chi(X)=\bigcup_{k=1}^K Y_k$ be the decomposition of the corona space into its connected components. By assumption $\tau$ decomposes as $\oplus_{k=1}^K \tau_k$ with ${\operatorname{im}}(\tau_k)\subset{\mathbb{M}}_N\cdot 1_{Y_k}\subseteq{\mathcal{C}}(\chi(X),{\mathbb{M}}_N)$. Lift the indicator functions $1_{Y_1},\cdots,1_{Y_K}$ to pairwise orthogonal contractions $h_1,\cdots,h_K$ in ${\mathcal{C}}(\beta X,\mathbb{C}\cdot 1_{M_N})$ and let $f\colon[0,1]\rightarrow[0,1]$ be the continuous function which equals $1$ on $\left[\frac{1}{2},1\right]$, satisfies $f(0)=0$ and is linear in between. We define a completely positive map $s\colon B\rightarrow{\mathcal{C}}(\beta X,{\mathbb{M}}_N)$ by $s(b)(x)=\sum_{k=1}^K \tau_k(b)\cdot f(h_k)(x)$ and check that in the diagram $$\xymatrix{
A \ar[r]^(0.3)\iota \ar[d] & {\mathcal{C}}(\beta X,{\mathbb{M}}_N) \ar[d] \\
B \ar[r]^(0.3)\tau \ar@{-->}[ur]^s & {\mathcal{C}}(\chi(X),{\mathbb{M}}_N)
}$$ the right triangle commutes. Set $K=\bigcap_{k=1}^K h_k^{-1}([0,\frac{1}{2}])\subset X$, then $s$ is multiplicative outside of the open set $U=\bigcap_{k=1}^K h_k^{-1}([0,\frac{1}{2}))\subset K\subset X$.
\[prop almost split 2\] Let $0 \rightarrow {\mathcal{C}}_0(X,{\mathbb{M}}_N) \rightarrow A \rightarrow B \rightarrow 0$ be a short exact sequence of separable $C^*$-algebras with Busby map $\tau$. Assume that $X$ is at most one-dimensional and has clopen connected components $(X_i)_{i\in I}$. Further assume that each corona space $\chi(X_i)$ has only finitely many connected components and that $\chi(X_i)$ is connected for almost all $i\in I$. If for each $i\in I$ the image of $\tau_i\colon B \xrightarrow{\tau} {\mathcal{C}}(\chi(X),{\mathbb{M}}_N)\rightarrow{\mathcal{C}}(\chi(X_i),{\mathbb{M}}_N)$ is locally constant on $\chi(X_i)$ and the set-valued retract map $R\colon{\operatorname{Prim}}(A)\rightarrow 2^{{\operatorname{Prim}}(B)}$ as in \[def R\] is lower semicontinuous, the following holds: There exists a compact set $K\subset X$ and a completely positive split $s\colon B\rightarrow{\mathcal{C}}(\beta X,{\mathbb{M}}_N)$ which is multiplicative outside of an open set $U\subset K$.
Let $I_0\subseteq I$ be a finite set such that $\chi(X_i)$ is connected for every $i\in I_1:=I\backslash I_0$. We may then study the extensions ($*=0$ or 1) $$\xymatrix{
0 \ar[r] & \bigoplus\limits_{i\in I}{\mathcal{C}}_0(X_i,{\mathbb{M}}_N) \ar[r] \ar@{->>}[d]^{pr_{I_*}} & A \ar[r] \ar@{=}[d]& B \ar[r] \ar[d]^{\varphi_*} & 0 & [\tau] \\
0 \ar[r] & \bigoplus\limits_{i\in I_*}{\mathcal{C}}_0(X_i,{\mathbb{M}}_N) \ar[r] & A \ar[r] \ar[d]^{\iota_*}& A/\bigoplus\limits_{i\in I_*}{\mathcal{C}}_0(X_i,{\mathbb{M}}_N) \ar[r] \ar[d]^{\tau_*} \ar@{-->}[dl]_{s_*} & 0 & [\tau_*] \\
0 \ar[r] & \bigoplus\limits_{i\in I_*}{\mathcal{C}}_0(X_i,{\mathbb{M}}_N) \ar[r] \ar@{=}[u]& \prod\limits_{i\in I_*}{\mathcal{C}}(\beta X_i,{\mathbb{M}}_N) \ar[r] & \frac{\prod_{i\in I_*}{\mathcal{C}}(\beta X_i,{\mathbb{M}}_N)}{\bigoplus_{i\in I_*} {\mathcal{C}}_0(X_i,{\mathbb{M}}_N)} \ar[r] & 0
}$$ with Busby maps $\tau_*$. Denote the map $B\rightarrow A/\bigoplus_{i\in I_*}{\mathcal{C}}_0(X_i,{\mathbb{M}}_N)$ induced by the projection $pr_{I_*}$ by $\varphi_*$. It is now easy to check that for $*=1$ the short exact sequence in the middle row satisfies the assumptions of Lemma \[lemma approx unitary\] and hence admits a splitting $s_1$ by Corollary \[cor projective split\]. For $*=0$, we apply Lemma \[lemma almost split\] to obtain a compact set $K\subset\bigsqcup_{i\in I_0}X_i$ and a completely positive split $s_0$ which is multiplicative outside of an open set $U\subset K\subset\bigsqcup_{i\in I_0}X_i$. Setting $s=s_0\circ\varphi_0\oplus s_1\circ\varphi_1$, we now get a split for the original sequence. In particular, $\varrho\circ s=\tau$ holds due to the commutative diagram $$\xymatrix{
0 \ar[r] & {\mathcal{C}}_0(X,{\mathbb{M}}_N) \ar[r] \ar@{=}[d] & A \ar[r] \ar[d]^{\iota_0\oplus\iota_1} & B \ar[r] \ar[d]_{\tau_0\oplus\tau_1} \ar@{-->}[dl]_s \ar@/^2pc/[dd]^(0.7)\tau|(0.37)\hole|(0.4)\hole|(0.45)\hole|(0.5)\hole|(0.55)\hole & 0 \\
0 \ar[r] & \bigoplus\limits_{*=0,1}\bigoplus\limits_{i\in I_*}{\mathcal{C}}_0(X_i,{\mathbb{M}}_N) \ar[r] \ar@{=}[d] & \bigoplus\limits_{*=0,1}\prod\limits_{i\in I_*}{\mathcal{C}}(\beta X_i,{\mathbb{M}}_N) \ar[r] \ar@{=}[d] & \bigoplus\limits_{*=0,1}\frac{\prod_{i\in I_*}{\mathcal{C}}(\beta X_i,{\mathbb{M}}_N)}{\bigoplus_{i\in I_*}{\mathcal{C}}_0(X_i,{\mathbb{M}}_N)} \ar[r] \ar@{=}[d] & 0 \\
0 \ar[r] & \bigoplus\limits_{i\in I}{\mathcal{C}}_0(X_i,{\mathbb{M}}_N) \ar[r] & \prod\limits_{i\in I}{\mathcal{C}}(\beta X_i,{\mathbb{M}}_N) \ar[r]^\varrho & \frac{\prod_{i\in I}{\mathcal{C}}(\beta X_i,{\mathbb{M}}_N)}{\bigoplus_{i\in I}{\mathcal{C}}_0(X_i,{\mathbb{M}}_N)} \ar[r] & 0.
}$$
Summarizing the results of this section, we obtain following.
\[thm semiprojective split\] Let a short exact sequence of separable $C^*$-algebras $$\xymatrix{
0 \ar[r] & {\mathcal{C}}_0(X,{\mathbb{M}}_N) \ar[r] & A \ar[r] & B \ar[r] & 0 & [\tau]
}$$ with Busby map $\tau$ be given. Assume that $X$ satisfies the conditions
1. ${\operatorname{dim}}X\leq 1$,
2. the connected components $(X_i)_{i\in I}$ of $X$ are clopen,
3. each $\chi(X_i)$ has finitely many connected components,
4. almost all $\chi(X_i)$ are connected,
then the following holds: If the associated set-valued retract map $R\colon {\operatorname{Prim}}(A)\rightarrow 2^{{\operatorname{Prim}}(B)}$ given as in \[def R\] by $$R(z)=
\begin{cases}
z\;\text{if}\;z\in {\operatorname{Prim}}(B) \\
\partial X_i=\overline{X_i}\backslash X_i\;\text{if}\;z\in X_i\subseteq X
\end{cases}$$ is lower semicontinuous and has pointwise finite image, then there exists a compact set $K\subset X$ and a completely positive split $s\colon B\rightarrow A$ for the sequence such that the composition $$\xymatrix{
B \ar[r]^s & A \ar[r] & \mathcal{M}({\mathcal{C}}_0(X,{\mathbb{M}}_N))={\mathcal{C}}_b(X,{\mathbb{M}}_N)
}$$ is multiplicative outside of an open set $U\subset K$.
Note that we can replace the given extension by any strongly unitarily equivalent one (in sense of [@Bla06 II.8.4.12]) without changing the retract map $R$. Hence, by Proposition \[prop diagonal Busby\], we may assume that the image of $\tau$ is locally constant on each $\chi(X_i)$. Now Proposition \[prop almost split 2\] provides a split $s$ with the desired properties.
### Retract maps for semiprojective extensions
We now verify the regularity properties for the set-valued retract map $R\colon {\operatorname{Prim}}(A)\rightarrow 2^{{\operatorname{Prim}}(B)}$ associated to an extension $0\rightarrow{\mathcal{C}}_0(X,{\mathbb{M}}_N)\rightarrow A\rightarrow B\rightarrow 0$ in the case that both the ideal ${\mathcal{C}}_0(X,{\mathbb{M}}_N)$ and the extension $A$ are semiprojective $C^*$-algebras.\
First we need the following definition which is an adaption of \[def core\] and \[def fpm\] to the setting of pointed spaces.
\[def extended core\] Let $(X,x_0)$ be a pointed one-dimensional Peano continuum and $r\colon X\rightarrow {\operatorname{core}}(X)$ the first point map onto the core of $X$ as in \[def fpm\] (where we choose ${\operatorname{core}}(X)$ to be any point $x\neq x_0$ if $X$ is contractible). Denote the unique arc from $x_0$ to $r(x_0)$ by $[x_0,r(x_0)]$, then we say that $${\operatorname{core}}(X,x_0):={\operatorname{core}}(X)\cup[x_0,r(x_0)]$$ is the core of $(X,x_0)$. It is the smallest subcontinuum of $X$ which contains both ${\operatorname{core}}(X)$ and the point $x_0$.
Now let $X$ be a non-compact space with the property that its one-point compactification $\alpha X=X\cup\{\infty\}$ is a one-dimensional ANR-space. We are interested in the structure of the space $X$ at around infinitity (i.e. outside of large compact sets) which is reflected in its corona space $\chi(X)=\beta X\backslash X$. At least some information about $\chi(X)$ can be obtained by studying neighborhoods of the point $\infty$ in $\alpha X$. The following lemma describes some special neighborhoods which relate nicely to the finite graph ${\operatorname{core}}(\alpha X,\infty)$.
\[lemma 1-ANR at infinity\] Let $X$ be a connected, non-compact space such that its one-point compactification $\alpha X=X\cup\{\infty\}$ is a one-dimensional ANR-space.Fix a geodesic metric $d$ on $\alpha X$, then for any compact set $C\subset X\backslash\{x_0\}$ there exists a closed neighborhood $V$ of $\infty$ with the following properties:
1. $\{x\in X\colon d(x,\infty)\leq\epsilon\}\subseteq V\subseteq X\backslash C$ for some $\epsilon>0$.
2. $V\cap {\operatorname{core}}(\alpha X,\infty)$ is homeomorphic to the space of $K$ many intervals $[0,1]$ glued together at the $0$-endpoints with $K={\operatorname{order}}(\infty,{\operatorname{core}}(\alpha X,\infty))$. The gluing point corresponds to $\infty$ under this identification.
Let $D^{(k)}\subseteq V$ denote the $k$-th copy of $[0,1]$ under the identification described above and let $r$ be the first point map onto ${\operatorname{core}}(\alpha X,\infty)$. We can further arrange:
3. $V=\bigcup_{k=1}^K r^{-1}\left(D^{(k)}\right)$ and $r^{-1}\left(D^{(k\phantom{'})}\right)\cap r^{-1}\left(D^{(k')}\right)=\{\infty\}$ for $k\neq k'$.
4. The connected components of $V\backslash\{\infty\}$ are given by $V^{(k)}:=r^{-1}\left(D^{(k)}\backslash\{\infty\}\right)$.
5. Every path in $V$ from $x\in V^{(k)}$ to $x'\in V^{(k')}$ with $k\neq k'$ contains $\infty$.
We first note that $r^{-1}(\{\infty\})\cap X$ is open. Assume there is $x\in X$ with $r(x)=\infty$ and $d(x,\infty)=r>0$. Then given any $y\in X$ with $d(x,y)< r$ we choose an isometric arc $\alpha\colon [0,d(x,y)]\rightarrow \alpha X$ from $x$ to $y$. Now the arc from $y$ to $\infty$ given by first following $\alpha$ in reverse direction and then going along the unique arc from $x$ to $\infty$ must run through $r(y)$ by \[def fpm\]. Since every point on the second arc gets mapped to $\infty$ by $r$, we find either $r(y)=\infty$ or there is $0<t<d(x,y)$ such that $\alpha(t)=r(y)\in {\operatorname{core}}(\alpha X,\infty)$. In the second case, the arc $\alpha_{|[0,t]}$ must run through $r(x)=\infty$ which, using the fact that $\alpha$ was isometric, gives the contradiction $d(x,\infty)<t<d(x,y)<r$. Since $r^{-1}(\{\infty\})$ is also closed, connectedness of $X$ implies in fact $r^{-1}(\{\infty\})=\{\infty\}.$
By definition of $K$ (see section \[section 1-ANR\]), the closed set $\{x\in{\operatorname{core}}(\alpha X,\infty)\colon d(x,\infty)\leq\epsilon\}$ satisfies the description in (ii) for all sufficiently small $\epsilon>0$. We set $$V=\{x\in \alpha X\colon d(r(x),\infty)\leq\epsilon\},$$ then $V\cap {\operatorname{core}}(\alpha X,\infty)=r(V)=\{x\in {\operatorname{core}}(\alpha X,\infty)\colon d(x,\infty)\leq\epsilon\}$ so that condition (ii) is satisfied. For (i), we observe that $d(x,\infty)\leq\epsilon$ implies $d(r(x),\infty)\leq\epsilon$ since $d$ is geodesic and every arc from $x$ to $\infty$ runs through $r(x)$. Since $\infty\notin r(C)$, we have $\min\{d(r(x),\infty)\colon x\in C\}>0$ and therefore $V\cap C=\emptyset$ for $\epsilon$ sufficiently small. Condition (iii) follows immediately from the definition of $V$. The sets $V^{(k)}$ are connected and open by construction, so that (iv) holds. (v) follows from (iv).
We now collect some information about the corona space $\chi(X)$ in the case of connected $X$. These observations are mostly based on the work of Grove and Pedersen in [@GP84] and the graph-like structure of one-dimensional ANR-spaces.
\[lemma corona\] Let $X$ be a connected, non-compact space such that its one-point compactification $\alpha X$ is a one-dimensional ANR-space. Then the corona space $\chi(X)$ has covering dimension at most 1 and its number of connected components is given by $K={\operatorname{order}}(\infty,{\operatorname{core}}(\alpha X,\infty))<\infty$. In particular, if $\alpha X$ is a one-dimensional AR-space, then $\chi(X)$ is connected.
Apply Lemma \[lemma 1-ANR at infinity\] to $(\alpha X,\infty)$. It is straightforward to check that the map $${\mathcal{C}}(\chi(X))={\mathcal{C}}_b(X)/{\mathcal{C}}_0(X)\rightarrow\bigoplus_{k=1}^K{\mathcal{C}}_b(V^{(k)})/{\mathcal{C}}_0(V^{(k)})=\bigoplus_{k=1}^K{\mathcal{C}}(\chi(V^{(k)}))$$ is an isomorphism. Therefore we find $\chi(X)=\bigsqcup_{k=1}^K\chi(V^{(k)})$ and it suffices to check that each $\chi(V^{(k)})$ is connected. By Proposition 3.5 of [@GP84], it is now enough to show that each $V^{(k)}$ is connected at infinity. So let a compact set $C_1\subset V^{(k)}$ be given and denote by $r\colon V^{(k)}\cup\{\infty\}\rightarrow D^{(k)}$ the first point map. Using the identification $[0,1]\cong D^{(k)}$ where the point $0$ corresponds to the point $\infty$, we find $t>0$ such that $r(C_1)\subset [t,1]$. But $C_2:=r^{-1}([t,1])$ is easily seen to be compact while $V^{(k)}\backslash C_2=r^{-1}((0,t))$ is pathconnected by definition of $r$. For the dimension statement we note that ${\operatorname{dim}}(\chi(X))\leq{\operatorname{dim}}(\beta X)={\operatorname{dim}}(X)\leq 1$ by [@Nag70 Theorem 9.5].
\[remark component\] The assumption that $X$ is connected in \[lemma corona\] is necessary. If we drop it, the corona space $\chi(X)$ may no longer have finitely many connected components, but the following weaker statement holds: If $\alpha X$ is a one-dimensional ANR-space, so will be $\alpha X_i$ for any connected component $X_i$ of $X$. However, it follows from \[thm ward\] that all but finitely many components lead to contractible spaces $\alpha X_i$, i.e. to one-dimensional AR-spaces. Since in this case ${\operatorname{core}}(\alpha X_i,\infty)$ is just an arc $[x,\infty]$ for some $x\in X_i$, we see from Lemma \[lemma corona\] that $\chi(X_i)$ is connected for almost every component $X_i$ of $X$.
We will now see that, in the situation described in the beginning of this section, the set-valued retract map $R$ has pointwise finite image, i.e. $|R(z)|<\infty$ for all $z\in {\operatorname{Prim}}(A)$. The cardinality of these sets is in fact uniformly bounded and we give an upper bound which only depends on $N$ and the structure of the finite graph ${\operatorname{core}}(\alpha X,\infty)$.
\[prop finite boundary\] Let $A$ be a semiprojective $C^*$-algebra containing an ideal of the form ${\mathcal{C}}_0(X,{\mathbb{M}}_N)$. If $\alpha X=X\cup\{\infty\}$ is a one-dimensional ANR-space, then every connected component $C$ of $X$ has finite boundary $\partial C=\overline{C}\backslash C$ in ${\operatorname{Prim}}(A)$. More precisely, we find $$|\partial C|\leq N\cdot {\operatorname{order}}(\infty,(\alpha C,\infty))<\infty.$$
Since $X$ is locally connected, the connected components of $X$ are clopen and $\alpha C$ is again a one-dimensional ANR-space for every component $C$ of $X$. Hence we may assume that $C=X$. Fix a geodesic metric $d$ on $\alpha X=X\cup\{\infty\}$ and let $V$ be a neighborhood of $\infty$ as constructed in Lemma \[lemma 1-ANR at infinity\], satisfying $\{x\in\alpha X\colon d(x,\infty)\leq\epsilon\}\subseteq V$ for some $\epsilon>0$. We further choose sequences $(x_n^{(k)})_n\subseteq D^{(k)}\backslash\{\infty\}$ converging to $\infty$ and write $x_\infty^{(1)}=\dots=x_\infty^{(K)}=\infty$. By compactness of the unit ball in ${\mathbb{M}}_N$ and separability of $A$, we may assume that the representation $$\pi^{(k)}\colon A\rightarrow{\mathbb{M}}_N,\quad a\mapsto\lim_{n\rightarrow\infty}a (x_{n}^{(k)})$$ exists for all $1\leq k\leq K$. Here, $a(x)$ denotes the image of $a\in A$ under the extension of the point evaluation ${\operatorname{ev}}_x\colon{\mathcal{C}}_0(X,{\mathbb{M}}_N)\rightarrow {\mathbb{M}}_N$ to $A$. For a sequence $(x_n)_n$ in $X\subseteq{\operatorname{Prim}}(A)$ we write ${\operatorname{Lim}}(x_n)=\{z\in {\operatorname{Prim}}(A)\colon x_n\rightarrow z\}$. Our goal is then to show that there exists a finite set $S\subset{\operatorname{Prim}}(A)$ such that ${\operatorname{Lim}}(x_n)\subset S$ for every sequence $(x_n)_n\subset X$ with $x_n\rightarrow\infty$ in $\alpha X$. We will show that each $S^{(k)}:={\operatorname{Lim}}(x_n^{(k)})$ consists of at most $N$ elements and that $S:=\bigcup_{k=1}^K S^{(k)}$ has the desired property described above. First observe that $$S^{(k)}=\left\{\left[\pi_1^{(k)}\right],\dots,\left[\pi_{r(k)}^{(k)}\right]\right\}$$ holds, where $\pi^{(k)}\simeq\pi_1^{(k)}\oplus\cdots\oplus\pi_{r(k)}^{(k)}$ is the decomposition of $\pi^{(k)}$ into irreducible summands. The $\supseteq$-inclusion is immediate, for the other direction assume that $x_n^{(k)}\rightarrow\ker(\varrho)$ for some irreducible representation $\varrho$ with $\varrho\not\simeq\pi_i^{(k)}$ for all $i$. Since all $x_n^{(k)}$ correspond to $N$-dimensional representations, we also have ${\operatorname{dim}}(\varrho)\leq N$. Therefore all $\pi_i^{(k)}$ and $\varrho$ drop to irreducible representations of the maximal $N$-subhomogeneous quotient $A_{\leq N}$ of $A$ (cf. section \[section subhomogeneous\]). Because ${\operatorname{Prim}}(A_{\leq N})$ is a $T_1$-space, the finite set $\{[\pi_1^{(k)}],\dots,[\pi_{r(k)}^{(k)}]\}$ is closed and $[\varrho]$ can be separated from it. In terms of \[lemma top prim\], this means that there exists $a\in A$ such that $\|\varrho(a)\|>1$ while $\|\pi_i^{(k)}(a)\|\leq 1$ for all $i$. On the other hand, we find $$\|\varrho(a)\|\leq\liminf_{n\rightarrow\infty}\left\|a(x_n^{(k)})\right\|=\left\|\pi^{(k)}(a)\right\|=\max\limits_{i=1...r(k)}\left\|\pi_i^{(k)}(a)\right\|\leq1,$$ using \[lemma top prim\] again. Hence $[\varrho]=[\pi_i^{(k)}]$ for some $i$ and in particular $\left|S^{(k)}\right|=r(k)\leq N$ for every $k$.
It now suffices to show that ${\operatorname{Lim}}(x_n)\subseteq S^{(k)}$ for sequences $(x_n)\subset X$ with $x_n\rightarrow\infty$ such that $r(x_n)\in D^{(k)}$ for some fixed $k$ and all $n$. Let such a sequence $(x_n)_n$ for some fixed $k$ be given and pick $z\in{\operatorname{Lim}}(x_n)$. In order to show that $z\in S^{(k)}$, we consider the compact spaces $$Y_n:=\left\{(t,t)|0\leq t\leq\frac{1}{n}\right\}\cup\bigcup_{m\geq n} \left(\left\{\frac{1}{m}\right\}\times\left[0,\frac{1}{m}\right]\right)\subset\mathbb{R}^2.$$ Note that $Y_{n+1}\subset Y_n$ and $\bigcap_n Y_n=(0,0)$. We will now ’glue’ ${\mathcal{C}}(Y_n,{\mathbb{M}}_N)$ to $A$ in the following way: As before, we may assume that $x_n\rightarrow z$ in ${\operatorname{Prim}}(A)$ and that $\pi_\infty(a)=\lim_n a(x_n)$ exists for every $a\in A$. In particular, we find $z=[\pi_{i,\infty}]$ for some $i$ where $\pi_\infty\simeq\pi_{1,\infty}\oplus\cdots\oplus\pi_{r_\infty,\infty}$ is the decomposition of $\pi_\infty$ into irreducible summands. Let $c$ denote the $C^*$-algebra of convergent ${\mathbb{M}}_N$-valued sequences, we can then form the pullback $A_n:=A\oplus_c{\mathcal{C}}(Y_n,{\mathbb{M}}_N)$ over the two $^*$-homomorphisms $$\begin{array}{ccc}
A\longrightarrow c &\text{and}&{\mathcal{C}}(Y_n,{\mathbb{M}}_N)\longrightarrow c . \\
a\mapsto (a(x_n),a(x_{n+1}),a(x_{n+2}),\dots)&& f\mapsto f((\frac{1}{n},0),f(\frac{1}{n+1},0),f(\frac{1}{n+2},0),\dots)
\end{array}$$ These pullbacks form an inductive system in the obvious way. Further note that the connecting maps $A_n\rightarrow A_{n+1}$ are all surjective. The limit $\varinjlim A_n$ can be identified with $A$ via the isomorphism induced by the projections $A_n=A\oplus_c{\mathcal{C}}(Y_n,{\mathbb{M}}_N)\rightarrow A$ onto the left summand. Using semiprojectivity of $A$, we can find a partial lift to some finite stage $A_n$ of this inductive system: $$\xymatrix{
&&A_n=A\oplus_c{\mathcal{C}}(Y_n,{\mathbb{M}}_N) \ar[r] \ar@{->>}[d] & {\mathcal{C}}(Y_n,{\mathbb{M}}_N) \\
{\mathcal{C}}_0(X,{\mathbb{M}}_N) \ar[r]^(.6)\subseteq & A \ar[r]^{\cong} \ar@{-->}[ur] & \varinjlim A_n
}$$ Let $\varphi\colon A\rightarrow{\mathcal{C}}(Y_n,{\mathbb{M}}_N)$ be the composition of this lift with the projection $A_n\rightarrow{\mathcal{C}}(Y_n,{\mathbb{M}}_N)$ to the right summand. The restriction of $\varphi$ to the ideal ${\mathcal{C}}_0(X,{\mathbb{M}}_N)$ then induces a continuous map $\varphi^*\colon Y_n\rightarrow\alpha X$ with $\varphi^*\left(\frac{1}{m},0\right)=x_m$ for all $m\geq n$ and $\varphi^*(0,0)=\infty$. Denote by $h$ the strictly positive element of ${\mathcal{C}}_0(X,{\mathbb{M}}_N)$ given by $h(x)=d(x,\infty)\cdot 1_{{\mathbb{M}}_N}$. After increasing $n$, we may assume that $\|\varphi(h)\|<\epsilon$ holds. For $m\geq n$, we consider the paths $\alpha_m\colon\left[0,\frac{2}{m}\right]\rightarrow Y_n$ given by $$\alpha_m(t)=\begin{cases}
\left(\frac{1}{m},t\right) & \text{if}\;\;0\leq t\leq\frac{1}{m} \\
\left(\frac{2}{m}-t,\frac{2}{m}-t\right) & \text{if}\;\;\frac{1}{m} \leq t\leq\frac{2}{m}.
\end{cases}$$ Set $t_{\infty,m}=\min\{t\colon\varphi(h)(\alpha_m(t))=0\}$, then $0<t_{\infty,m}\leq\frac{2}{m}$ because of $\|\varphi(h)(\alpha_m(0))\|=\|\varphi(h)(\frac{1}{m},0)\|=\|h(x_m)\|=d(x_m,\infty)>0$ and $\varphi(h)(\alpha_m(\frac{2}{m}))=\varphi(h)(0,0)=h(\infty)=0$. By setting $\beta_m(t)=\varphi^*(\alpha_m(t))$ we obtain paths $\beta_m\colon[0,t_{\infty,m}]\rightarrow\alpha X$ which have the properties $$\begin{array}{l}
(1)\;\beta_m(0)=x_m,\\
(2)\;\beta_m(t)=\infty\;\text{if and only if}\;t=t_{\infty,m},\\
(3)\;{\operatorname{im}}(\beta_m)\subseteq V^{(k)}\;\text{for all}\;m,\\
(4)\;x_l^{(k)}\in{\operatorname{im}}(\beta_m)\;\text{for fixed}\; m \;\text{and all sufficiently large}\;l.
\end{array}$$ The first property is clear while the second one follows directly from the definition of $t_{\infty,m}$. In order to verify properties $(3)$ and $(4)$ we have to involve the structure of the neighborhood $V$ and by that the special structure of $\alpha X$ as a one-dimensional ANR-space. From $\|\varphi(h)\|<\epsilon$ we obtain ${\operatorname{im}}(\beta_m)\subseteq{\operatorname{im}}(\alpha_m) \subseteq \{x\in\alpha X\colon d(x,\infty)\leq\epsilon\}\subseteq V$, it then follows from (1), (2) and property (v) in Lemma \[lemma 1-ANR at infinity\] that ${\operatorname{im}}(\beta_m)\subseteq V^{(k)}$. For (4), observe that ${\operatorname{im}}(\beta_m)$ contains $r({\operatorname{im}}(\beta_m))$ by part (ii) of Lemma \[lemma arc to core\], where $r$ is the first-point map $\alpha X\rightarrow{\operatorname{core}}(\alpha X,\infty)$. Under the identification $D^{(k)}\cong [0,1]$, the connected set $r({\operatorname{im}}(\beta_m))$ corresponds to a proper interval containing the $0$-endpoint and hence it contains $x_l^{(k)}$ for almost every $l$.
Now set $\pi_m={\operatorname{ev}}_{\beta(t_{\infty,m})}\circ\varphi\colon A\rightarrow{\mathbb{M}}_N$ and let $\pi_m\simeq\pi_{1,m}\oplus\dots\oplus\pi_{r_m,m}$ be the decomposition into irreducible summands. We claim that the identity $$S^{(k)}=\left\{\left[\pi_{1,m}\right],\cdots,\left[\pi_{r_m,m}\right]\right\}$$ holds for all $m$. Involving property (4) for the path $\beta_m$, we find $$\|\pi_m(a)\|=\lim\limits_{t\nearrow t_{\infty,m}}\left\|({\operatorname{ev}}_{\beta(t)}\circ\varphi)(a)\right\|=\lim\limits_{l\rightarrow\infty}\left\|a\left(x_l^{(k)}\right)\right\|=\left\|\pi^{(k)}(a)\right\|$$ for every fixed $m$ and all $a\in A$. Now the same separation argument as in the beginning of the proof shows that the finite-dimensional representations $\pi^{(k)}$ and $\pi_m$ share the same irreducible summands for every $m$. Since $\beta_m(t_{\infty,m})\rightarrow (0,0)$ in $Y_n$, we find $\pi_m={\operatorname{ev}}_{\beta(t_{\infty,m})}\circ\varphi\rightarrow {\operatorname{ev}}_{(0,0)}\circ\varphi=\pi_\infty$ pointwise. Hence by the above identity, $\pi_\infty$ and $\pi^{(k)}$ also share the same irreducible summands. In particular, we find $z\in S^{(k)}$ which finishes the proof.
Next, we show that in our situation the set-valued retract map $R$ is also lower semicontinuous in the sense of \[def lsc set map\].
\[prop R is lsc\] Let $0\rightarrow{\mathcal{C}}_0(X,{\mathbb{M}}_N)\rightarrow A\rightarrow B\rightarrow 0$ be a short exact sequence of separable $C^*$-algebras. If $\alpha X$ is a one-dimensional ANR-space and $A$ is semiprojective, then the associated retract map $R$ as in \[def R\] is lower semicontinuous.
Let $X=\bigsqcup_{i\in I}X_i$ denote the decomposition of $X$ into its connected components. By separability of $A$ it suffices to verify condition (iii) of Lemma \[def lsc set map\] for a given sequence $x_n\rightarrow z$ in ${\operatorname{Prim}}(A)$. The case $z\in X$ is trivial since $X$ is locally connected and therefore has open connected components. The critical case is when $x_n\in X$ for all $n$ but $z\in{\operatorname{Prim}}(B)$. In this case, we write $x_n\in X_{i_n}$ and we may assume that $\pi_\infty(a)=\lim_n a(x_n)$ is well defined for all $a\in A$. In particular, $z$ corresponds to the kernel of an irreducible summand $\pi_{j,\infty}$ of $\pi_\infty\simeq\pi_{1,\infty}\oplus\cdots\oplus\pi_{r,\infty}$, as we have already seen in the beginning of the proof of Proposition \[prop finite boundary\]. Using exactly the same construction of ’gluing the space $Y$ to $A$ along the sequence $(x_n)$’ as in the proof of \[prop finite boundary\], one now shows that $$\{[\pi_{1,\infty}],\cdots,[\pi_{r,\infty}]\}\subseteq\overline{\bigcup\limits_n \partial X_{i_n}}.$$ Hence we find $y_n\in\partial X_{i_n}=R(x_n)$ with $y_n\rightarrow [\pi_{j,\infty}]=z$ showing that the retract map $R$ is in fact lower semicontinuous.
Existence of limit structures {#section existence limit structures}
-----------------------------
Consider an extension of separable $C^*$-algebras $$0\rightarrow{\mathcal{C}}_0(X,{\mathbb{M}}_N)\rightarrow A\rightarrow B\rightarrow 0$$ where the one-point compactification of $X$ is assumed to be a one-dimensional ANR-space. We know from Theorem \[thm ST\] that in this case $\alpha X$ comes as a inverse limit of finite graphs over a surprisingly simple system of connecting maps. Here we show that under the right assumptions on the set-valued retract map $R\colon {\operatorname{Prim}}(A)\rightarrow 2^{{\operatorname{Prim}}(B)}$ associated to the sequence above, this limit structure for $\alpha X$ is compatible with the extension of $B$ by ${\mathcal{C}}_0(X,{\mathbb{M}}_N)$ in the following sense: We prove the existence of a direct limit structure for $A$ which describes it as the $C^*$-algebra $B$ with a sequence of non-commutative finite graphs (1-NCCW’s) attached. The connecting maps of this direct system are obtained from the limit structure for $\alpha X$ and hence can be described in full detail.
\[lemma limit structure\] Let a short exact sequence of separable $C^*$-algebras $0\rightarrow{\mathcal{C}}_0(X,{\mathbb{M}}_N)\rightarrow A\rightarrow B\rightarrow 0$ with Busby map $\tau$ be given. Assume that $\alpha X$ is a one-dimensional ANR-space and that the associated set-valued retract map $R\colon {\operatorname{Prim}}(A)\rightarrow 2^{{\operatorname{Prim}}(B)}$ as in \[def R\] is lower semicontinuous and has pointwise finite image. Then $A$ is isomorphic to the direct limit $B_\infty$ of an inductive system $$\xymatrix{
B_0 \ar[r]_{s_0^1}& B_1 \ar[r]_{s_1^2} \ar@/_1pc/@{->>}[l]_{r_1^0}& B_2 \ar[r] \ar@/_1pc/@{->>}[l]_{r_2^1} &
\cdots \ar[r]_{s_{i-1}^i} \ar@/_1pc/@{->>}[l]& B_i \ar[r] \ar@/_1pc/[rr]_{s_i^\infty} \ar@/_1pc/@{->>}[l]_{r_i^{i-1}}& \cdots \ar[r] &
B_\infty=\varinjlim \left(B_i,s_i^{i+1}\right) \ar@/_1pc/@{->>}[ll]_{r_\infty^i}
}$$ where
- $B_0$ is given as a pullback $B\oplus_F D$ with $D$ a 1-NCCW and ${\operatorname{dim}}(F)<\infty$. Furthermore, if $\alpha X$ is contractible, we may even arrange $B_0\cong B$.
and
- for every $i\in\mathbb{N}$ there is a representation $\pi_i\colon B_i\rightarrow{\mathbb{M}}_N$ such that $B_{i+1}$ is defined as the pullback $\xymatrix{
B_{i+1} \ar@{..>>}[d]^{r_{i+1}^i} \ar@{..>}[r]& {\mathcal{C}}([0,1],{\mathbb{M}}_N) \ar@{->>}[d]^{ev_0} \\
B_i \ar@/^1pc/[u]^{s_i^{i+1}} \ar[r]^{\pi_i}& {\mathbb{M}}_N.
}$\
The map $s_i^{i+1}\colon B_i\rightarrow B_{i+1}$ is given by $a\mapsto (a,\pi_i(a)\otimes 1_{[0,1]})$ and hence satisfies $r_{i+1}^i\circ s_i^{i+1}={\operatorname{id}}_{B_i}$.
Let $X=\bigsqcup_{j\in J}C_j$ be the decomposition of $X$ into its clopen connected components. Denote by $J_1\subseteq J$ the subset of those indices for which the corona space $\chi(C_j)$ is connected and note that $J_0:=J\backslash J_1$ is finite by Remark \[remark component\]. We have the canonical commutative diagram $$\xymatrix{
0 \ar[r] & {\mathcal{C}}_0(X,{\mathbb{M}}_N) \ar[r] \ar@{=}[d] & A \ar[r] \ar[d]^{\iota_0\oplus\iota_1} & B \ar[r] \ar[d]_{\tau_0\oplus\tau_1} & 0 \\
0 \ar[r] & \bigoplus\limits_{*=0,1}\bigoplus\limits_{j\in J_*}{\mathcal{C}}_0(C_j,{\mathbb{M}}_N) \ar[r] &
\bigoplus\limits_{*=0,1}\prod\limits_{j\in J_*}{\mathcal{C}}(\beta C_j,{\mathbb{M}}_N) \ar[r]^{q_0\oplus q_1} & \bigoplus\limits_{*=0,1}\frac{\prod_{j\in J_*}{\mathcal{C}}(\beta C_j,{\mathbb{M}}_N)}{\bigoplus_{j\in J_*}{\mathcal{C}}_0(C_j,{\mathbb{M}}_N)} \ar[r] & 0
}$$ where $\tau_0\oplus\tau_1$ is the Busby map $\tau$ and the right square is a pullback diagram. Since we can pass to any strongly unitarily equivalent extension (in the sense of [@Bla06 II.8.4.12]) without changing the retract map $R$, we can, by Proposition \[prop diagonal Busby\] and the finiteness condition on $R$, assume that for every $j$ the image of $$\tau_j\colon B\xrightarrow{\tau}\frac{\prod_{j'}{\mathcal{C}}(\beta C_{j'},{\mathbb{M}}_N)}{\bigoplus_{j'}{\mathcal{C}}_0(C_{j'},{\mathbb{M}}_N)}\rightarrow\frac{{\mathcal{C}}(\beta C_j,{\mathbb{M}}_N)}{{\mathcal{C}}_0(C_j,{\mathbb{M}}_N)}={\mathcal{C}}(\chi(C_j),{\mathbb{M}}_N)$$ is locally constant on $\chi(C_j)$, and even constant if $j\in J_1$. Furthermore, using lower semicontinuity of $R$ and arguing as in the proof of Corollary \[cor projective split\], we may assume that $$\iota_1(A)\subseteq \prod_{j\in J_1}{\mathbb{M}}_N\cdot 1_{\beta C_j}+\bigoplus_{j\in J_1}{\mathcal{C}}_0(C_j,{\mathbb{M}}_N).$$
Next, we write $\alpha X=X\cup\{\infty\}$ as a limit of finite graphs. By Theorem \[thm ST\] we can find a sequence of finite graphs $X_i\subset X_{i+1} \subset \alpha X$ such that $X_0={\operatorname{core}}(\alpha X,\infty)$ (in the sense of \[def extended core\]) and each $X_{i+1}$ is obtained from $X_i$ by attaching a line segment $[0,1]$ at the $0$-endpoint to a single point $y_{i}$ of $X_i$. Furthermore we have $\varprojlim X_i = \alpha X$ along the sequence of first point maps $\varrho_\infty^i\colon\alpha X\rightarrow X_i$. We need to fix some notation: Denote the inclusion of $X_i$ into $X_{i+1}$ by $\iota_i^{i+1}$ and the retract from $X_{i+1}$ to $X_i$ by collapsing the attached interval to the attaching point $y_i$ by $\varrho_{i+1}^i$. An analogous notation is used for the inclusion $X_i\subseteq\alpha X$: $$\xymatrix{
X_i \ar[r]_{\iota_i^{i+1}}& X_{i+1}\ar@{->>}@/_1pc/[l]_{\varrho_{i+1}^i} \ar[r]_{\iota_{i+1}^{\infty}}& \alpha X\ar@{->>}@/_1pc/[l]_{\varrho_{\infty}^{i+1}}
}$$
Now for every pair of indices $i,j$ we have $X_i\cap C_j$ sitting inside $C_j$. Note that $X_{i+1}\backslash X_i\cap C_{j(i)}\neq\emptyset$ for a unique $j(i)\in J$ since $\infty\in X_0$. We define suitable compactifications $\alpha_j(X_i\cap C_j)$ of $X_i\cap C_j$ as follows: if $X_0\cap C_j=\emptyset$, we let $\alpha_j(X_i\cap C_j)=\alpha(X_i\cap C_j)$ be the usual one-point compactification for any $i\in\mathbb{N}$. In the case $X_0\cap C_j\neq\emptyset$, which will occur only finitely many times, we have an inclusion ${\mathcal{C}}_b(X_i\cap C_j)\subseteq {\mathcal{C}}_b(C_j)$ induced by the surjective retract $\varrho^i_{\infty|C_j}\colon C_j\rightarrow X_i\cap C_j$ and we define $\alpha_j(X_i\cap C_j)$ via $${\mathcal{C}}(\alpha_j(X_i\cap C_j))=\left\{f\in{\mathcal{C}}_b(X_i\cap C_j)\subseteq{\mathcal{C}}_b(C_j)={\mathcal{C}}(\beta C_j)\colon \begin{array}{c}f\;\text{is locally} \\ \text{constant on }\;\chi(C_j)\end{array}\right\}.$$ Since the corona space $\chi(C_j)$ has only finitely many connected components by Lemma \[lemma corona\], $\alpha_j(X_i\cap C_j)$ will be a finite-point compactification of $X_i\cap C_j$ (meaning that $\alpha_j(X_i\cap C_j)\backslash(X_i\cap C_j)$ is a finite set). In particular, $\alpha_j(X_i\cap C_j)$ is a finite graph for any pair of indices $i$ and $j$. We are now ready to iteratively define the $C^*$-algebras $B_i$ as the pullbacks over $$\xymatrix{
B_i \ar@{..>}[r] \ar@{..>}[d] & B \ar[d]^\tau\\
\prod_j{\mathcal{C}}(\alpha_j(X_i\cap C_j),{\mathbb{M}}_N) \ar[r]^(0.6)q & \frac{\prod_j{\mathcal{C}}(\beta C_j,{\mathbb{M}}_N)}{\bigoplus_j{\mathcal{C}}_0(C_j,{\mathbb{M}}_N)}
}$$ with respect to the inclusions $(\varrho^i_{\infty|C_j})^*\otimes{\operatorname{id}}_{{\mathbb{M}}_N}\colon{\mathcal{C}}(\alpha_j(X_i\cap C_j),{\mathbb{M}}_N)\subseteq{\mathcal{C}}(\beta C_j,{\mathbb{M}}_N)$. Let us first simplify the description of $B_i$. For every fixed $i$, the set $X_i\cap C_j$ is empty for almost every $j\in J$ so that ${\mathcal{C}}(\alpha_j(X_i\cap C_j),{\mathbb{M}}_N)={\mathbb{M}}_N\cdot 1_{\beta C_j}$ for almost every $j$. Given $((f_j)_j,b)\in B_i$, this implies $f_j=\tau_j(b)\cdot 1_{\beta C_j}$ for almost every $j$. Hence $B_i$ is isomorphic to the pullback $$\xymatrix{
B_i \ar@{..>}[r] \ar@{..>}[d] & B \ar[d]^{\mathop{\oplus}\limits_{j\in J(i)} \tau_j} \\
\bigoplus\limits_{j\in J(i)} {\mathcal{C}}(\alpha_j (X_i\cap C_j),{\mathbb{M}}_N) \ar[r]^(0.6)q & \bigoplus\limits_{j\in J(i)}\frac{{\mathcal{C}}(\beta C_j,{\mathbb{M}}_N)}{{\mathcal{C}}_0(C_j,{\mathbb{M}}_N)}
}$$ for the finite set $J(i)=\{j\in J\colon X_i\cap C_j\neq\emptyset\}\subseteq J$. Since every $\alpha(X_i\cap C_j)$ is a finite graph, the $C^*$-algebra on the lower left side is a 1-NCCW. One also checks that the pullbacks are taken over finite-dimensional $C^*$-algebras because $(\oplus_{j\in J(i)}\tau_j)(B)$ consists of locally constant functions on the space $\bigsqcup_{j\in J(i)}\chi(C_j)$ which has only finitely many connected components by Lemma \[lemma corona\].
Next, we specify the inductive structure, i.e. the connecting maps $s_i^{i+1}\colon B_i\rightarrow B_{i+1}$ and retracts $r_{i+1}^i\colon B_{i+1}\rightarrow B_i$. By definition, we find $B_i\subseteq B_{i+1}\subseteq A$ with the inclusions coming from $(\varrho^i_{i+1})^*\otimes{\operatorname{id}}_{{\mathbb{M}}_N}$ resp. by $(\varrho^{i+1}_\infty)^*\otimes{\operatorname{id}}_{{\mathbb{M}}_N}$. We denote them by $s^{i+1}_i$ resp. by $s_i^\infty$. Since $\overline{\bigcup_i{\mathcal{C}}(\alpha_j(X_i\cap C_j),{\mathbb{M}}_N)}\supseteq\overline{\bigcup_i{\mathcal{C}}_0(X_i\cap C_j,{\mathbb{M}}_N)}={\mathcal{C}}_0(X\cap C_j,{\mathbb{M}}_N)$ for every $j\in J$, we find ${\mathcal{C}}_0(X,{\mathbb{M}}_N)\subseteq\overline{\bigcup_i B_i}$. One further checks that $\bigoplus_{j\in J_0}{\mathcal{C}}(\alpha_j(X_0\cap C_j),{\mathbb{M}}_N)$ surjects via $q$ onto the locally constant functions on $\bigsqcup_{j\in J_0}\chi(C_j)$. Together with $\tau_1(B)\subseteq q_1(\prod_{j\in J_1}{\mathbb{M}}_N\cdot 1_{\beta C_j})\subseteq q_1(\prod_{j\in J_1}{\mathcal{C}}(\alpha_j(X_0\cap C_j),{\mathbb{M}}_N))$ it follows that $\overline{\bigcup_iB_i}$ is the pullback over $\tau$ and $q$, and hence $\overline{\bigcup_i B_i}=A$.
It remains to verify the description of the connecting maps $s_i^{i+1}$. We have $X_i\cap C_j=X_{i+1}\cap C_j$ if $j\neq j(i)$ and $\alpha_j(X_i\cap C_{j(i)})\subseteq \alpha_j(X_{i+1}\cap C_{j(i)})\cong \alpha_j(X_i\cap C_{j(i)})\cup_{\{y_i\}=\{0\}}[0,1]$. This means there is a pullback diagram $$\xymatrix{
{\mathcal{C}}(\alpha_j(X_{i+1}\cap C_{j(i)}),{\mathbb{M}}_N) \ar@{..>}[r] \ar@{..>>}[d] & {\mathcal{C}}([0,1],{\mathbb{M}}_N) \ar@{->>}[d]^{{\operatorname{ev}}_0} \\
{\mathcal{C}}(\alpha_j(X_i\cap C_{j(i)}),{\mathbb{M}}_N) \ar[r]^(0.6){{\operatorname{ev}}_{y_i}} \ar@/^1pc/[u]^{(\varrho_{i+1}^i)^*\otimes {\operatorname{id}}_{{\mathbb{M}}_N}}& {\mathbb{M}}_N
}$$ where $(\varrho_{i+1}^i)^*\otimes {\operatorname{id}}_{{\mathbb{M}}_N}$ corresponds to $f\mapsto(f,f(y_i)\otimes 1_{[0,1]})$ in the pullback picture and the downward arrow on the left side comes from the inclusion $\alpha_j(X_i\cap C_{j(i)})\subseteq\alpha_j(X_{i+1}\cap C_{j(i)})$. This map induces a surjection $B_{i+1}\rightarrow B_i$ which will be denoted by $r_{i+1}^i$ and gives the claimed pullback diagram.
Finally, if $\alpha X$ is an AR-space, the core $X_0={\operatorname{core}}(\alpha X,\infty)=[x_0,\infty]$ is nothing but an arc from some point $x_0\in X$ to $\infty$. In this case the finite set $J(0)$ consists of a single element $j(0)$, namely the index corresponding to the component containing $x_0$. By definition, $B_0$ comes as a pullback $$\xymatrix{
B_0 \ar@{..>}[r] \ar@{..>}[d] & B \ar[d]^{\tau_{j(0)}} \\
{\mathcal{C}}([x_0,\infty],{\mathbb{M}}_N) \ar[r]^{{\operatorname{ev}}_\infty} & {\mathbb{M}}_N\cdot 1_{\chi(C_{j(0)})}
}$$ and hence an index shift allows us to start with $B_0\cong B$.
The procedure of forming extensions by $C^*$-algebras of the form ${\mathcal{C}}_0(X,{\mathbb{M}}_N)$ can of course be iterated. The next proposition shows that, if all the attached spaces $X$ are one-dimensional ANRs up to compactification, the limit structures which we get from Lemma \[lemma limit structure\] for each step can be combined into a single one.
\[prop iteration\] Let a short exact sequence of separable $C^*$-algebras $0\rightarrow{\mathcal{C}}_0(X,{\mathbb{M}}_N)\rightarrow A\rightarrow B\rightarrow 0$ be given. Assume that $\alpha X$ is a one-dimensional ANR-space and that the associated set-valued retract map $R\colon {\operatorname{Prim}}(A)\rightarrow 2^{{\operatorname{Prim}}(B)}$ as in \[def R\] is lower semicontinuous and has pointwise finite image. Further assume that there exists a direct limit structure for $B$ $$\xymatrix{
B_0 \ar[r]_{s_0^1}& B_1 \ar[r]_{s_1^2} \ar@/_1pc/@{->>}[l]_{r_1^0}& B_2 \ar[r] \ar@/_1pc/@{->>}[l]_{r_2^1} &
\cdots \ar[r]_{s_{i-1}^i} \ar@/_1pc/@{->>}[l]& B_i \ar[r] \ar@/_1pc/[rr]_{s_i^\infty} \ar@/_1pc/@{->>}[l]_{r_i^{i-1}}& \cdots \ar[r] & B \ar@/_1pc/@{->>}[ll]_{r_\infty^i}
}$$ such that all $B_i$ are 1-NCCWs and at each stage there is a representation $p_i\colon B_i\rightarrow{\mathbb{M}}_{n_i}$ such that $B_{i+1}$ is defined as the pullback $$\xymatrix{
B_{i+1} \ar@{..>>}[d]^{r_{i+1}^i} \ar@{..>}[r]^(0.3){t_{i+1}}& {\mathcal{C}}([0,1],{\mathbb{M}}_{n_i}) \ar@{->>}[d]^{{\operatorname{ev}}_0} \\
B_i \ar@/^1pc/[u]^{s_i^{i+1}} \ar[r]^{p_i}& {\mathbb{M}}_{n_i}
}$$ and $s_i^{i+1}\colon B_i\rightarrow B_{i+1}$ is given by $a\mapsto (a,p_i(a)\otimes 1_{[0,1]})$.
Then $A$ is isomorphic to the limit $A_\infty$ of an inductive system $$\xymatrix{
A_0 \ar[r]_{\sigma_0^1}& A_1 \ar[r]_{\sigma_1^2} \ar@/_1pc/@{->>}[l]_{\varrho_1^0}& A_2 \ar[r] \ar@/_1pc/@{->>}[l]_{\varrho_2^1} &
\cdots \ar[r]_{\sigma_{i-1}^i} \ar@/_1pc/@{->>}[l]& A_i \ar[r] \ar@/_1pc/[rr]_{\sigma_i^\infty} \ar@/_1pc/@{->>}[l]_{\varrho_i^{i-1}}& \cdots \ar[r] & A_\infty \ar@/_1pc/@{->>}[ll]_{\varrho_\infty^i}
}$$ where all $A_i$ are 1-NCCWs and at each stage there is a representation $\pi_i\colon A_i\rightarrow{\mathbb{M}}_{m_i}$ such that $A_{i+1}$ is defined as the pullback $$\xymatrix{
A_{i+1} \ar@{..>>}[d]^{\varrho_{i+1}^i} \ar@{..>}[r]& {\mathcal{C}}([0,1],{\mathbb{M}}_{m_i}) \ar@{->>}[d]^{{\operatorname{ev}}_0} \\
A_i \ar@/^1pc/[u]^{\sigma_i^{i+1}} \ar[r]^{\pi_i}& {\mathbb{M}}_{m_i}
}$$ and $\sigma_i^{i+1}\colon A_i\rightarrow A_{i+1}$ is given by $a\mapsto (a,\pi_i(a)\otimes 1_{[0,1]})$. Furthermore, if $\alpha X$ is an AR-space we may even arrange $A_0\cong B_0$.
By Lemma \[lemma limit structure\], we know that $A$ can be written as an inductive limit $$\xymatrix{
\overline{A}_0 \ar[r]_{\overline{s}_0^1}& \overline{A}_1 \ar[r]_{\overline{s}_1^2} \ar@/_1pc/@{->>}[l]_{\overline{r}_1^0}& \overline{A}_2 \ar[r] \ar@/_1pc/@{->>}[l]_{\overline{r}_2^1} &
\cdots \ar[r]_{\overline{s}_{i-1}^i} \ar@/_1pc/@{->>}[l]& \overline{A}_i \ar[r] \ar@/_1pc/[rr]_{\overline{s}_i^\infty} \ar@/_1pc/@{->>}[l]_{\overline{r}_i^{i-1}}& \cdots \ar[r] &
A \ar@/_1pc/@{->>}[ll]_{\overline{r}_\infty^i}
}$$ with a pullback structure $$\xymatrix{
\overline{A}_{i+1} \ar@{..>>}[d]^{\overline{r}_{i+1}^i} \ar@{..>}[r]& {\mathcal{C}}([0,1],{\mathbb{M}}_N) \ar@{->>}[d]^{{\operatorname{ev}}_0} \\
\overline{A}_i \ar@/^1pc/[u]^{\overline{s}_i^{i+1}} \ar[r]^{\overline{p}_i}& {\mathbb{M}}_N
}$$ at every stage and $\overline{s}_i^{i+1}\colon\overline{A}_i\rightarrow \overline{A}_{i+1}$ given by $a\mapsto (a,\overline{p}_i(a)\otimes 1_{[0,1]})$. The starting algebra $\overline{A}_0$ comes as a pullback $$\xymatrix{
\overline{A}_0 \ar[r] \ar[d] & D \ar[d]^\varphi \\
B \ar[r]^\psi & F
}$$ with $D$ a 1-NCCW and ${\operatorname{dim}}(F)<\infty$. In the case of $\alpha X$ being an AR-space, we may choose $\overline{A}_0=B$, i.e. $D=0$. For $j\in\mathbb{N}$ we now define $A_{0,j}$ to be the pullback $$\xymatrix{
A_{0,j} \ar@{..>}[r] \ar@{..>}[d]_{\varrho_{0,j}} & D \ar[d]^\varphi \\
B_j \ar[r]^{\psi\circ s_j^{\infty}} & F.
}$$ The maps $s_j^{j+1}$,$s_j^\infty$ induce compatible homomorphisms $\sigma_{0,j}^{0,j+1}\colon A_{0,j}\rightarrow A_{0,j+1}$ and $\sigma_{0,j}^{0,\infty}\colon A_{0,j}\rightarrow\overline{A}_0$, leading to an inductive limit structure with $\varinjlim_j(A_{0,j},\sigma_{0,j}^{0,j+1})=\overline{A}_0$. We proceed iteratively, defining $A_{i+1,j}$ to be the pullback $$\xymatrix{
A_{i+1,j} \ar@{..>}[r] \ar@{..>}[d]^{\varrho_{i+1,j}^{i,j}} & {\mathcal{C}}([0,1],{\mathbb{M}}_N) \ar[d]^{{\operatorname{ev}}_0} \\
A_{i,j} \ar[r]_{\overline{p}_i\circ\sigma_{i,j}^{i,\infty}} \ar@/^1pc/@{..>}[u]^{\sigma_{i,j}^{i+1,j}}& {\mathbb{M}}_N
}$$ with $\sigma_{i,j}^{i+1,j}\colon A_{i,j}\rightarrow A_{i+1,j}$ given by $a\mapsto (a,(\overline{p}_i\circ\sigma_{i,j}^{i,\infty})(a)\otimes 1_{[0,1]})$. It is then checked that $\sigma_{i,j}^{i,j+1}$ and $\sigma_{i,j}^{i,\infty}$ induce compatible homomorphisms $\sigma_{i+1,j}^{i+1,j+1}\colon A_{i+1,j}\rightarrow A_{i+1,j+1}$ and $\sigma_{i+1,j}^{i+1,\infty}\colon A_{i+1,j}\rightarrow \overline{A}_{i+1}$ with $\varinjlim_j (A_{i+1,j},\sigma_{i+1,j}^{i+1,j+1})=\overline{A}_{i+1}$. Observing that for every $i$ and $j$ $$\xymatrix{
A_{i,j+1} \ar[rrr]^{t_{j+1}\circ\varrho_{0,j+1}\circ\varrho_{i,j+1}^{0,j+1}} \ar[d]^{\varrho_{i,j+1}^{i,j}} &&& {\mathcal{C}}([0,1],{\mathbb{M}}_{n_j}) \ar[d]^{{\operatorname{ev}}_0} \\
A_{i,j} \ar[rrr]_{p_j\circ\varrho_{0,j}\circ\varrho_{i,j}^{0,j}} \ar@/^1pc/[u]^{\sigma_{i,j}^{i,j+1}} &&& {\mathbb{M}}_{n_j}
}$$ is indeed a pullback diagram, we get the desired limit structure for $A$ by following the diagonal in the commutative diagram $$\xymatrix{
A_{00} \ar[r] \ar@{..>}[d] & A_{01} \ar[r] \ar[d] & A_{02} \ar[r] \ar[d] & A_{03} \ar[r] \ar[d] & \cdots \ar[r] & \overline{A}_0 \ar[d]\\
A_{10} \ar@{..>}[r] \ar[d] & A_{11} \ar[r] \ar@{..>}[d] & A_{12} \ar[r] \ar[d] & A_{13} \ar[r] \ar[d] & \cdots \ar[r] & \overline{A}_1 \ar[d] \\
A_{20} \ar[r] \ar[d] & A_{21} \ar@{..>}[r] \ar[d] & A_{22} \ar[r] \ar@{..>}[d] & A_{23} \ar[r] \ar[d] & \cdots \ar[r] & \overline{A}_2 \ar[d] \\
A_{30} \ar[r] \ar[d] & A_{31} \ar[r] \ar[d] & A_{32} \ar@{..>}[r] \ar[d] & A_{33} \ar[r] \ar@{..>}[d] & \cdots \ar[r] & \overline{A}_3 \ar[d]\\
\vdots & \vdots & \vdots & \vdots & \ddots &\vdots
}$$ as indicated. Note that, since all connecting maps are injective, the limit over the diagonal equals $\varinjlim\overline{A}_n=A$.
Keeping track of semiprojectivity {#section keeping track}
---------------------------------
We now reap the fruit of our labor in the previous sections and work out a ’2 out of 3’-type statement describing the behavior of semiprojectivity with respect to extensions by homogeneous $C^*$-algebras. While for general extensions the behavior of semiprojectivity is either not at all understood or known to be rather bad, Theorem \[thm 2 out of 3\] gives a complete and satisfying answer in the case of homogeneous ideals. It is the very first result of this type and allows to understand semiprojectivity for $C^*$-algebras which are built up from homogeneous pieces, see chapter \[section main\].
\[prop ideal ANR\] Let $0\rightarrow{\mathcal{C}}_0(X,{\mathbb{M}}_N)\rightarrow A\rightarrow B\rightarrow 0$ be a short exact sequence of $C^*$-algebras. If both $A$ and $B$ are (semi)projective, then the one-point compactification $\alpha X$ is a one-dimensional A(N)R-space.
The projective case follows directly from Corollary \[cor projective ideal\] and Theorem \[thm comm case\] while the semiprojective case requires some more work. By Lemma \[lemma ideal Peano\] we know that $\alpha X$ is a Peano space of dimension at most 1. The proof of \[lemma ideal Peano\] further shows that there are no small circles accumulating in $X$. However, in order to show that $\alpha X$ is an ANR-space we have to exclude the possibility of smaller and smaller circles accumulating at $\infty\in\alpha X$, see Theorem \[thm ward\]. Assume that we find a sequence of circles with diameters converging to $0$ (with respect to some fixed geodesic metric) at around $\infty\in\alpha X$. After passing to a subsequence, there are two possible situations: either each circle contains the point $\infty$ or none of them does. Both cases are treated exactly the same, for the sake of brevity we only consider the situation where $\infty$ is contained in all circles. In this case have a $^*$-homomorphism $\varphi\colon{\mathcal{C}}_0(X,{\mathbb{M}}_N)\rightarrow\bigoplus_{n=1}^\infty{\mathcal{C}}_0((0,1)_n,{\mathbb{M}}_N)$ where $(0,1)_n\cong(0,1)$ is the part of the $n$-th circle contained in $X$. Note that each coordinate projection gives a surjection $\varphi_n\colon{\mathcal{C}}_0(X,{\mathbb{M}}_N)\rightarrow{\mathcal{C}}_0((0,1),{\mathbb{M}}_N)$ while $\varphi$ itself is not necessarily surjective (because the circles might intersect in $X$). We make use of Brown’s mapping telescope associated to $\bigoplus_{n=1}^\infty{\mathcal{C}}_0((0,1)_n,{\mathbb{M}}_N)$, i.e. $$T_k=\left\{f\in{\mathcal{C}}([k,\infty],\bigoplus_{n=1}^\infty{\mathcal{C}}_0((0,1)_n,{\mathbb{M}}_N))\colon t\leq l\Rightarrow f(t)\in\bigoplus_{n=1}^l{\mathcal{C}}_0((0,1)_n,{\mathbb{M}}_N)\right\}$$ with the obvious (surjective) restrictions as connecting maps giving $\varinjlim T_k=\bigoplus_{n=1}^\infty{\mathcal{C}}_0((0,1)_n,{\mathbb{M}}_N)$. Using Lemma \[lemma telescope\], we find a commutative diagram $$\xymatrix{
0 \ar[r] & \varinjlim T_k \ar[r] & \varinjlim\mathcal{M}(T_k) \ar[r] & \varinjlim\mathcal{Q}(T_k) \ar[r] & 0 \\
0 \ar[r] & {\mathcal{C}}_0(X,{\mathbb{M}}_N) \ar[r] \ar[u]^\varphi & A \ar[r] \ar[u]^{\overline{\varphi}} & B \ar[r] \ar[u]^{\overline{\overline{\varphi}}} & 0
}.$$ It now follows from the semiprojectivity assumptions and Lemma \[lemma extended lifting\] that $\varphi$ lifts to $T_k$ for some $k$, which is equivalent to a solution of the original lifting problem $$\xymatrix{
& \bigoplus\limits_{n=1}^k {\mathcal{C}}_0((0,1)_n,{\mathbb{M}}_N) \ar[d]^\subseteq \\
{\mathcal{C}}_0(X,{\mathbb{M}}_N) \ar[r]^(0.4)\varphi \ar@/^1pc/@{..>}[ur] & \bigoplus\limits_{n=1}^\infty{\mathcal{C}}_0((0,1)_n,{\mathbb{M}}_N)
}$$ up to homotopy. This, however, implies $${\operatorname{im}}(K_1(\varphi))\subseteq K_1\left(\bigoplus_{n=1}^k{\mathcal{C}}_0((0,1)_n,{\mathbb{M}}_N)\right)=\sum_{n=1}^k\mathbb{Z}\subset\sum_{n=1}^\infty\mathbb{Z} =K_1\left(\bigoplus_{n=1}^\infty{\mathcal{C}}_0((0,1)_n,{\mathbb{M}}_N)\right)$$ which gives a contradiction as follows. Because $\varphi_{k+1}$ is surjective and ${\operatorname{dim}}(\alpha X)\leq 1$, we can extend the canonical unitary function from $\alpha((0,1)_n)$ to a unitary $u$ on all of $\alpha X$ by [@HW48 Theorem VI.4]. This unitary then satisfies $u-1\in{\mathcal{C}}_0(X)$ and $K_1(\varphi)([u\otimes 1_{{\mathbb{M}}_N}])=N\in\mathbb{Z}=K_1({\mathcal{C}}_0((0,1)_{k+1},{\mathbb{M}}_N))$. This shows that there are no small circles at around $\infty$ in $\alpha X$ and hence that $\alpha X$ is a one-dimensional ANR-space by Theorem \[thm ward\].
\[thm 2 out of 3\] Let a short exact sequence of $C^*$-algebras $0\rightarrow I\rightarrow A\rightarrow B\rightarrow 0$ be given and assume that the ideal $I$ is a $N$-homogeneous $C^*$-algebra with ${\operatorname{Prim}}(I)=X$. Denote by $(X_i)_{i\in I}$ the connected components of $X$ and consider the following statements:
1. $I$ is (semi)projective.
2. $A$ is (semi)projective.
3. $B$ is (semi)projective and the set-valued retract map $R\colon{\operatorname{Prim}}(A)\rightarrow 2^{{\operatorname{Prim}}(B)}$ given as in \[def R\] by $$R(z)=\begin{cases}
\;z & \text{if}\;z\in{\operatorname{Prim}}(B), \\
\partial X_i=\overline{X_i}\backslash X_i & \text{if}\;z\in X_i\subseteq X={\operatorname{Prim}}(I)
\end{cases}$$ is lower semicontinuous and has pointwise finite image.
If any two of these statements are true, then the third one also holds.
(I)+(II)$\Rightarrow$(III): By Theorem \[thm homogeneous case\], we know that the sequence is isomorphic to an extension $$\xymatrix{0\ar[r]&{\mathcal{C}}_0(X,{\mathbb{M}}_N)\ar[r]&A\ar[r]^\pi&B\ar[r]&0}$$ with the one-point compactification of $X$ a one-dimensional A(N)R-space. The set-valued retract map $R$ is then lower semicontinuous by Proposition \[prop R is lsc\] and has pointwise finite image by Proposition \[prop finite boundary\]. But now Theorem \[thm semiprojective split\] applies and shows that there is a completely positive split $s$ for the quotient map $\pi$ such that the composition $B\xrightarrow{s} A\xrightarrow{\iota}{\mathcal{C}}_b(X,{\mathbb{M}}_N)$ is multiplicative outside of an open set $U\subset K\subset X$ where $K$ is compact.
Let a lifting problem $\varphi\colon B\xrightarrow{\sim} D/J=\varinjlim D/J_n$ be given. Since $A$ is semiprojective, we can solve the resulting lifting problem for $A$, meaning we find $\psi\colon A\rightarrow D/J_n$ for some $n$ with $\pi_n\circ\psi=\varphi\circ\pi$. Restricting to ${\operatorname{her}}_{D/J_n}(\psi({\mathcal{C}}_0(X,{\mathbb{M}}_N)))+\psi(A)\subseteq D/J_n$, we may assume that $\psi_{|{\mathcal{C}}_0(X,{\mathbb{M}}_N)}$ is proper as a $^*$-homomorphism to $J/J_n$ and hence induces a map $\mathcal{M}(\psi)$ between multiplier algebras. Since the restriction of $\pi_n\circ\psi$ to the ideal ${\mathcal{C}}_0(X,{\mathbb{M}}_N)$ vanishes, we may use compactness of $K$ to assume that $\psi$ maps ${\mathcal{C}}_0(U,{\mathbb{M}}_N)$ to $0$ (after increasing $n$ if necessary). This further implies that $\mathcal{M}(\psi)$ factors through $r\colon{\mathcal{C}}_b(X,{\mathbb{M}}_N)\rightarrow{\mathcal{C}}_b(X\backslash U,{\mathbb{M}}_N)$. We then find $s':=r\circ\iota\circ s$ to be multiplicative and hence a $^*$-homomorphism: $$\xymatrix{
A\ar[dr]^\iota \ar[ddd]^\pi \ar[rrr]^\psi & & & D/J_n \ar[dr]^{\iota_n}\ar@{->>}[ddd]^(0.6){\pi_n}|!{[dll];[dr]}\hole|!{[ddl];[dr]}\hole \\
& {\mathcal{C}}_b(X,{\mathbb{M}}_N) \ar@{->>}[dr]+(-12.5,3)^(.65)r \ar[rrr]^{\mathcal{M}(\psi)} & & & \mathcal{M}(J/J_n) \ar@{->>}[dd]^{\varrho_n}\\
& & {\mathcal{C}}_b(X\backslash U,{\mathbb{M}}_N)\ar@{-->}[urr]^(0.35){\mathcal{M}(\psi)'} \\ B \ar[rrr]^\varphi \ar@/^2pc/[uuu]^s \ar@{-->}[urr]^{s'}& & & D/J \ar[r]^(0.4){\tau_n}& \mathcal{Q}(J/J_n)
}$$ The inclusion of $J/J_n$ as an ideal in $D/J_n$ gives canonical $^*$-homomorphisms $\iota_n$ and $\tau_n$ as in the diagram above. One now checks that $\varrho_n\circ(\mathcal{M}(\psi)'\circ s')=\tau_n\circ\varphi$ holds. Combining this with the fact that the trapezoid on the right is a pullback diagram, we see that the pair $(\varphi,(\mathcal{M}(\psi)'\circ s'))$ defines a lift $B\rightarrow D/J_n$ for $\varphi$. This shows that the quotient $B$ is semiprojective.
For the projective version of the statement, one uses Corollary \[cor projective split\] to see that the sequence admits a multiplicative split $s\colon B\rightarrow A$ rather than just a completely positive one.\
(I)+(III)$\Rightarrow$(II): We know that $I\cong{\mathcal{C}}_0(X,{\mathbb{M}}_N)$ with $\alpha X$ a one-dimensional A(N)R-space by Theorem \[thm homogeneous case\]. Now Lemma \[lemma limit structure\] applies and we obtain a limit structure for $A$ $$\xymatrix{
B_0 \ar[r]_{s_0^1}& B_1 \ar[r]_{s_1^2} \ar@/_1pc/@{->>}[l]_{r_1^0}& B_2 \ar[r] \ar@/_1pc/@{->>}[l]_{r_2^1} & \cdots \ar[r]_{s_{i-1}^i} \ar@/_1pc/@{->>}[l]& B_i \ar[r] \ar@/_1pc/[rr]_{s_i^\infty} \ar@/_1pc/@{->>}[l]_{r_i^{i-1}}& \cdots \ar[r] & \varinjlim \left(B_i,s_i^{i+1}\right)\cong A \ar@/_1pc/@{->>}[ll]_{r_\infty^i}
}$$ with $B_0$ given as a pullback of $B$ and a 1-NCCW $D$ over a finite-dimensional $C^*$-algebra. In particular, $B_0$ is semiprojective by [@End14 Corollary 3.4]. In the projective case, we can take $B_0=B$ to be projective. In both cases, the connecting maps in the system above arise from pullback diagrams $$\xymatrix{
B_{i+1} \ar@{..>>}[d]^{r_{i+1}^i} \ar@{..>}[r]& {\mathcal{C}}([0,1],{\mathbb{M}}_N) \ar@{->>}[d]^{{\operatorname{ev}}_0} \\
B_i \ar@/^1pc/[u]^{s_i^{i+1}} \ar[r]^{\pi_i}& {\mathbb{M}}_N
}$$ with $s_i^{i+1}(a)=(a,\pi_i(a)\otimes 1_{[0,1]})$. Since these maps are weakly conditionally projective by Proposition \[prop wcp examples\], we obtain (semi)projectivity of $A$ from Lemma \[lemma limit criterium\].\
(II)+(III)$\Rightarrow$(I): This implication holds under even weaker hypothesis. More precisely, we show that (semi)projectivity of both $A$ and $B$ implies $I$ to be (semi)projective. The assumption on the retract map $R$ is not needed here.
First we apply Lemma \[lemma ideal Peano\] to find the one-point compactification of ${\operatorname{Prim}}(I)$ to be a Peano space of dimension at most 1, and hence $I$ is trivially homogeneous by Lemma \[lemma trivial bundles\]. Now Proposition \[prop ideal ANR\] shows that $\alpha X$ is in fact an ANR-space which, together with Theorem \[thm homogeneous case\], means that $I$ is semiprojective. The projective version is Corollary \[cor projective ideal\].
\[remark comm retract\] Theorem \[thm 2 out of 3\] shows that regularity properties of the retract map $R\colon{\operatorname{Prim}}(A)\rightarrow 2^{{\operatorname{Prim}}(B)}$ are crucial for semiprojectivity to behave nicely with respect to extensions by homogeneous $C^*$-algebras. This can already be observed and illustrated in the commutative case. Given an extension of commutative $C^*$-algebras $$0\rightarrow{\mathcal{C}}_0(X)\rightarrow{\mathcal{C}}_0(Y)\rightarrow{\mathcal{C}}_0(Y\backslash X)\rightarrow 0,$$ the following holds: If both the ideal ${\mathcal{C}}_0(X)$ and the quotient ${\mathcal{C}}_0(Y\backslash X)$ are (semi)projective, then the extension ${\mathcal{C}}_0(Y)$ is (semi)projective if and only if the retract map $R\colon Y\rightarrow 2^{Y\backslash X}$ is lower semicontinuous and has pointwise finite image. The following examples show that both properties for $R$ are not automatic:
\(a) An examples with $R$ not having pointwise finite image is contained as example 5.5 in [@LP98], we include it here for completeness. Let $X=\{(x,\sin x^{-1}):0<x\leq 1\}\subset\mathbb{R}^2$ and $Y=X\cup\{(0,y)\colon -1\leq y< 2\}$, then we get an extension isomorphic to $$0\rightarrow{\mathcal{C}}_0(0,1]\rightarrow{\mathcal{C}}_0(Y)\rightarrow{\mathcal{C}}_0(0,1]\rightarrow 0.$$ Here both the ideal and the quotient are projective, but the extension ${\mathcal{C}}_0(Y)$ is not (because $\alpha Y$ is not locally connected and hence not an AR-space). In this example, we find $R(x)=\{(0,y)\colon -1\leq y\leq 1\}$ to be infinite for all $x\in X$.
\(b) The following is an example where $R$ fails to be lower semicontinuous. Consider $Y=\{(x,0)\colon 0\leq x<1\}\cup\bigcup_n C_n\subset\mathbb{R}^2$ with $C_n=\{(t,(1-t)/n)\colon 0\leq t<1\}$ the straight line from $(0,1/n)$ to $(1,0)$ with the endpoint $(1,0)$ removed. With $X=\bigcup_n C_n\subset Y$ we obtain an extension isomorphic to $$0\rightarrow\bigoplus_n{\mathcal{C}}_0(0,1]\rightarrow {\mathcal{C}}_0(Y)\rightarrow{\mathcal{C}}_0(0,1]\rightarrow 0.$$ Here both the ideal and the quotient are projective while the extension ${\mathcal{C}}_0(Y)$ is not (again because $\alpha Y$ is not locally connected). We also find $(0,1/n)\rightarrow (0,0)$ in $Y$ but $R((0,1/n))=\emptyset$ for all $n$, which shows that $R$ is not lower semicontinuous. The descriptive reason for ${\mathcal{C}}_0(Y)$ not being projective in this case is that the length of the attached intervals $C_n$ does not tend to $0$ as $n$ goes to infinity.
The structure of semiprojective subhomogeneous $C^*$-algebras {#section structure}
=============================================================
The main result {#section main}
---------------
With Theorem \[thm 2 out of 3\] at hand, we are now able to keep track of semiprojectivity when decomposing a subhomogeneous $C^*$-algebra into its homogeneous subquotients. On the other hand, Theorem \[thm 2 out of 3\] also tells us in which manner homogeneous, semiprojective $C^*$-algebras may be combined in order to give subhomogeneous $C^*$-algebras which are again semiprojective. This leads to the main result of this chapter, Theorem \[thm structure\], which gives two characterizations of projectivity and semiprojectivity for subhomogeneous $C^*$-algebras.
\[lemma Prim\_max\] Let $A$ be a $N$-subhomogeneous $C^*$-algebra. If $A$ is semiprojective, then the maximal $N$-homogeneous ideal of $A$ is also semiprojective.
By Lemma \[lemma ideal Peano\] we know that the one-point compactification of $X={\operatorname{Prim}}_N(A)$ is a one-dimensional Peano space. Since any locally trivial ${\mathbb{M}}_N$-bundle over $X$ is globally trivial by Lemma \[lemma trivial bundles\], we are concerned with an extension of the form $$\xymatrix
{0\ar[r]&{\mathcal{C}}_0(X,{\mathbb{M}}_N)\ar[r]&A\ar[r]^(0.4)\pi&A_{\leq N-1}\ar[r]&0
}$$ where $A_{\leq N-1}$ denotes the maximal ($N$-1)-subhomogeneous quotient of $A$. Since $A$ is semiprojective, $A_{\leq N-1}$ will be semiprojective with respect to ($N$-1)-subhomogeneous $C^*$-algebras. In order to show that ${\mathcal{C}}_0(X,{\mathbb{M}}_N)$ is semiprojective, it remains to show that $\alpha X=X\cup\{\infty\}$ does not contain small circles at around $\infty$, cf. Theorem \[thm ward\]. The proof for this is similar to the one of \[prop ideal ANR\]. We use notations from \[lemma ideal Peano\] and follow the proof there to arrive a commutative diagram $$\xymatrix{
0\ar[r]&\varinjlim T_k \ar[r] & \varinjlim\mathcal{M}(T_k) \ar[r] & \varinjlim\mathcal{Q}(T_k) \ar[r] & 0 \\
0 \ar[r] & {\mathcal{C}}_0(X,{\mathbb{M}}_N) \ar[r] \ar[u]^\varphi & A \ar[r]^(0.4)\pi \ar[u]^{\overline{\varphi}} & A_{\leq N-1} \ar[r] \ar[u]^{\overline{\overline{\varphi}}} & 0
}.$$ We may not solve the lifting problem for $A_{\leq N-1}$ directly since the algebras $\mathcal{Q}(T_k)$ are not ($N$-1)-subhomogeneous. Instead we will replace the $\mathcal{Q}(T_k)$ by suitable ($N$-1)-subhomogeneous subalgebras which will then lead to a solvable lifting problem for $A_{\leq N-1}$. Let $\iota_n$ denote the $n$-th coordinate of the map $A\rightarrow{\mathcal{C}}_b(X,{\mathbb{M}}_N)\rightarrow\prod_n{\mathcal{C}}_b((0,1)_n,{\mathbb{M}}_N)$. We then have a lift of $\overline{\varphi}$ given by $$\begin{array}{rcccl}
A & \rightarrow & {\mathcal{C}}([k,\infty],\prod_n{\mathcal{C}}_b((0,1),{\mathbb{M}}_N)) & \rightarrow & \mathcal{M}(T_k) \\
a & \mapsto & 1_{[k,\infty]}\otimes(\iota_n(a))_{n=1}^\infty
\end{array}$$ where the map on the right is induced by the inclusion of $T_k$ as an ideal in ${\mathcal{C}}([k,\infty],\prod_n{\mathcal{C}}_b((0,1),{\mathbb{M}}_N))$. Consider in there the central element $f=(f_n)_{n=1}^\infty$ with $f_n$ the scalar function that equals 0 on $[k,n]$, 1 on $[n+1,\infty]$ and which is linear in between. Then $$\begin{array}{rcccl}
\psi\colon A & \rightarrow & {\mathcal{C}}([k,\infty],\prod_n{\mathcal{C}}_b((0,1),{\mathbb{M}}_N)) & \rightarrow & \mathcal{M}(T_k) \\
a & \mapsto & (f_n\otimes\iota_n(a))_{n=1}^\infty
\end{array}$$ is a completely positive lift of $\overline{\varphi}$ which sends ${\mathcal{C}}_0(X,{\mathbb{M}}_N)$ to $T_k$. Hence $\psi$ induces a completely positive lift $\psi'\colon A_{\leq N-1}\rightarrow \mathcal{Q}(T_k)$ of $\overline{\overline{\varphi}}$. We claim that $C^*(\psi'(A_{\leq N-1}))$ is in fact ($N$-1)-subhomogeneous. To see this, we use the algebraic characterization of subhomogenity as described in [@Bla06 IV.1.4.6]. It suffices to check that $\gamma(C^*(\psi'(A_{\leq N-1})))$ satisfies the polynomial relations $p_{r(N-1)}$ for every irreducible representation $\gamma$ of $\mathcal{Q}(T_k)$. By definition of $\psi$, we find $\gamma\circ\psi'(\pi(a))=t\cdot\gamma'(\iota(a))$ for some representation $\gamma'$ of $\iota(A)$, some $t\in[0,1]$ and every $a\in A$. Moreover, since $\psi'$ maps ${\mathcal{C}}_0(X,{\mathbb{M}}_N)$ to 0, we obtain $\gamma\circ\psi'(\pi(a))=t\cdot\gamma''(\pi(a))$ for some representation $\gamma''$ of $A_{\leq N-1}$. Using ($N$-1)-subhomogeneity of $A_{\leq N-1}$, it now follows easily that the elements of $\gamma(C^*(\psi'(A_{\leq N-1})))$ satisfy the polynomial relations $p_{r(N-1)}$ from [@Bla06 IV.1.4.6]. Knowing that the image of $\overline{\overline{\varphi}}$ has a ($N$-1)-subhomogenous preimage in $\mathcal{Q}(T_k)$, we may now solve the lifting problem for $A_{\leq N-1}$. It then follows from Lemma \[lemma extended lifting\] (and its proof) that $\varphi$ lifts to $T_k$ for some $k$. The remainder of the proof is exactly the same as the one of Proposition \[prop ideal ANR\].
We now present two characterizations of projectivity and semiprojectivity for subhomogeneous $C^*$-algebras. The first one describes semiprojectivity of these algebras in terms of their primitive ideal spaces. The second description characterizes them as those $C^*$-algebras which arise from 1-NCCWs by adding a sequence of non-commutative edges (of bounded dimension), cf. section \[section adding edges\].
\[thm structure\] Let $A$ be a $N$-subhomogeneous $C^*$-algebra, then the following are equivalent:
1. $A$ is semiprojective (resp. projective).
2. For every $n=1,...,N$ the following holds:
- The one-point compactification of ${\operatorname{Prim}}_n(A)$ is an ANR-space (resp. an AR-space) of dimension at most 1.
- If $(X_i)_{i\in I}$ denotes the family of connected components of ${\operatorname{Prim}}_n(A)$, then the set-valued retract map $$R_n\colon{\operatorname{Prim}}_{\leq n}(A)\rightarrow 2^{{\operatorname{Prim}}_{\leq n-1}(A)}$$ given by $$\begin{array}{rl}
z&\mapsto
\begin{cases}
z & \;\text{if}\;z\in {\operatorname{Prim}}_{\leq n-1}(A) \\
\partial X_i & \;\text{if}\;z\in X_i\subset {\operatorname{Prim}}_n(A)
\end{cases}
\end{array}$$ is lower semicontinuous and has pointwise finite image.
3. $A$ is isomorphic to the direct limit $\varinjlim_k(A_k,s_k^{k+1})$ of a sequence of 1-NCCW’s $$\xymatrix{\cdots \ar[r] & A_k \ar[r]_{s_k^{k+1}} \ar@/_1pc/@{->>}[l] & A_{k+1} \ar[r] \ar@/_1pc/@{->>}[l]_{r_{k+1}^k} & \cdots \ar@/_1pc/@{->>}[l]}$$ (with $A_0=0$) such that for each stage there is a pullback diagram $$\xymatrix{
A_{k+1} \ar@{..>}[r] \ar@{..>>}[d]^{r_{k+1}^k}& {\mathcal{C}}([0,1],{\mathbb{M}}_n) \ar[d]^{{\operatorname{ev}}_0} \\
A_k \ar[r]^{\pi_k} \ar@/^1pc/[u]^{s_k^{k+1}} & {\mathbb{M}}_n
}$$ with $n\leq N$ and $s_k^{k+1}$ given by $a\mapsto(a,\pi_k(a)\otimes 1_{[0,1]})$.
$(1)\Rightarrow (2)$: We prove the implication by induction over $N$. The base case $N=1$ is given by Theorem \[thm comm case\]. Now given a $N$-subhomogeneous, (semi)projective $C^*$-algebra $A$, we know by Lemma \[lemma Prim\_max\] that the maximal $N$-homogeneous ideal $A_N$ of $A$ is (semi)projective as well. This forces $\alpha{\operatorname{Prim}}_N(A)$ to be a one-dimensional A(N)R-space by Theorem \[thm homogeneous case\]. Applying Theorem \[thm 2 out of 3\] to the sequence $$0\rightarrow A_N\rightarrow A\rightarrow A_{\leq N-1}\rightarrow 0$$ now shows that the retract map $R_N\colon{\operatorname{Prim}}_N(A)\rightarrow 2^{{\operatorname{Prim}}_{\leq N-1}(A)}$ is lower semicontinuous, has pointwise finite image and that the maximal ($N$-1)-subhomogeneous quotient $A_{\leq N-1}$ is (semi)projective. The remaining statements follow from the induction hypothesis applied to $A_{\leq N-1}$.
$(2)\Rightarrow (3)$: By Lemma \[lemma trivial bundles\], we know that the maximal $N$-homogeneous ideal $A_N$ of $A$ is of the form ${\mathcal{C}}_0({\operatorname{Prim}}_N(A),{\mathbb{M}}_N)$. Using induction over $N$, the statement then follows from Proposition \[prop iteration\] applied to the sequence $$0\rightarrow{\mathcal{C}}_0({\operatorname{Prim}}_N(A),{\mathbb{M}}_N)\rightarrow A\rightarrow A_{\leq N-1}\rightarrow 0.$$ The base case $N=1$ is given by Theorem \[thm ST\].
$(3)\Rightarrow (1)$: Note that the connecting maps are weakly conditionally projective by Proposition \[prop wcp examples\], then apply Lemma \[lemma limit criterium\].
The most prominent examples of subhomogeneous, semiprojective $C^*$-algebras are the one-dimensional non-commutative CW-complexes (1-NCCWs, see Example \[ex 1-nccw\]). The structure theorem \[thm structure\] shows that these indeed play a special role in the class of all subhomogeneous, semiprojective $C^*$-algebras. By part (2) of \[thm structure\], they are precisely those subhomogeneous, semiprojective $C^*$-algebras for which the spaces $\alpha{\operatorname{Prim}}_n$ are all finite graphs rather than general one-dimensional ANR-spaces. Hence 1-NCCWs should be thought of as the elements of ’finite type’ in the class of subhomogeneous, semiprojective $C^*$-algebras. Moreover, part (3) of \[thm structure\] shows that every subhomogeneous, semiprojective $C^*$-algebra can be constructed from 1-NCCWs in a very controlled manner. Therefore these algebras share many properties with 1-NCCWs, as we will see in section \[section properties\] in more detail.
Applications {#section applications}
------------
Now we discuss some consequences of Theorem \[thm structure\]. First we collect some properties of semiprojective, subhomogeneous $C^*$-algebras which follow from the descriptions in \[thm structure\]. This includes information about their dimension and $K$-theory as well as details about their relation to 1-NCCWs and some further closure properties.
At least in principle one can use the structure theorem \[thm structure\] to test any given subhomogeneous $C^*$-algebra $A$ for (semi)projectivity. Since this would require a complete computation of the primitive ideal space of $A$, it is not recommended though. Instead one might use \[thm structure\] as a tool to disprove semiprojectivity for a candidate $A$. In fact, showing directly that a $C^*$-algebra $A$ is not semiprojective can be surprisingly difficult. One might therefore take one of the conditions from \[thm structure\] which are easier to verify and test $A$ for those instead. We illustrate this strategy in section \[section A4\] by proving the quantum permutation algebras to be not semiprojective. This corrects a claim in [@Bla04] on semiprojectivity of universal $C^*$-algebras generated by finitely many projections with order and orthogonality relations.
### Further structural properties {#section properties}
By part (3) of Theorem \[thm structure\], we know that any semiprojective, subhomogeneous $C^*$-algebra comes as a direct limit of 1-NCCWs. Since the connecting maps are explicitly given and of a very special nature, it is possible to show that these limits are approximated by 1-NCCWs in a very strong sense. The following corollary makes this approximation precise.
\[cor approx nccw\] Let $A$ be a subhomogeneous $C^*$-algebra. If $A$ is semiprojective, then for every finite set $\mathcal{G}\subset A$ and every $\epsilon>0$ there exist a 1-NCCW $B\subseteq A$ and a $^*$-homomorphism $r\colon A\rightarrow B$ such that $\mathcal{G}\subset_\epsilon B$ and $r$ is a strong deformation retract for $B$, meaning that there exists a homotopy $H_t$ from $H_0={\operatorname{id}}_A$ to $H_1=r$ with $H_{t|B}={\operatorname{id}}_B$ for all $t$. In particular, $A$ is homotopy equivalent to a one-dimensional non-commutative CW-complex.
Use part (3) of Theorem \[thm structure\] to write $A=\varinjlim A_n$ and find a suitable 1-NCCW $B=A_{n_0}$ which almost contains the given finite set $\mathcal{G}$. It is straightforward to check that the strong deformation retracts $r_n^{n_0}\colon A_n\rightarrow A_{n_0}$ give rise to a strong deformation retract $r\colon \varinjlim A_n\rightarrow A_{n_0}$.
In particular, 1-NCCWs and semiprojective, subhomogeneous $C^*$-algebras share the same homotopy invariant properties. For example, we obtain the following restrictions on the $K$-theory of these algebras:
\[cor k-theory\] Let $A$ be a subhomogeneous $C^*$-algebra. If $A$ is semiprojective, then its $K$-theory is finitely generated and $K_1(A)$ is torsion free.
Another typical phenomenon of (nuclear) semiprojective $C^*$-algebras is that they appear to be one-dimensional in some sense. In the context of subhomogeneous $C^*$-algebras, we can now make this precise, using the notion of topological dimension given by ${\operatorname{topdim}}(A)=\max_n {\operatorname{dim}}({\operatorname{Prim}}_n(A))$.
\[cor dimension\] Let $A$ be a subhomogeneous $C^*$-algebra. If $A$ is semiprojective, then $A$ has stable rank 1 and ${\operatorname{topdim}}(A)\leq 1$.
The statement on the stable rank of $A$ follows from Corollary \[cor approx nccw\], while the topological dimension can be estimated using part (2) of Theorem \[thm structure\].
Our structure theorem can also be used to study permanence properties of semiprojectivity when restricted to the class of subhomogeneous $C^*$-algebras. In fact, these turn out to be way better then in the general situation. This can be illustrated by the following longstanding question by Blackadar and Loring: Given a short exact sequence of $C^*$-algebras $$\xymatrix{0 \ar[r] & I \ar[r] & A \ar[r] & F \ar[r] & 0}$$ with finite-dimensional $F$, does the following hold? $$I\;\text{semiprojective}\Leftrightarrow A\;\text{semiprojective}$$ While we showed the ’$\Leftarrow$’-implication to hold in general in [@End14], S. Eilers and T. Katsura proved the ’$\Rightarrow$’-implication to be wrong ([@EK]), even in the case of split extensions by $\mathbb{C}$. We refer the reader to [@Sor12] for counterexamples which involve infinite $C^*$-algebras. However, when one restricts to the class of subhomogeneous $C^*$-algebras, this implication holds:
\[cor blalor\] Let a short exact sequence of $C^*$-algebras $$\xymatrix{0 \ar[r]&I\ar[r]&A\ar[r]^\pi &F\ar[r]&0}$$ with finite-dimensional $F$ be given. If $I$ is subhomogeneous and semiprojective, then $A$ is also semiprojective.
We verify condition (2) in Theorem \[thm structure\] for $A$. By assumption, each ${\operatorname{Prim}}_k(I)$ is a one-dimensional ANR-space after compactification and the same holds for any space obtained from ${\operatorname{Prim}}_k(I)$ by adding finitely many points ([@ST12 Theorem 6.1]). Hence the one-point compactifications of ${\operatorname{Prim}}_k(A)$ are 1-dimensional ANRs for all $k$. If we assume $F={\mathbb{M}}_n$, then the set-valued retract maps $R_k$ are unchanged for $k<n$. For $k=n$, regularity of $R_k$ follows from regularity of the retract map for $I$ and the fact that $\{[\pi]\}$ is closed in ${\operatorname{Prim}}_{\leq k}(A)={\operatorname{Prim}}_{\leq k}(I)\cup\{[\pi]\}$. For $k>n$, we apply Lemma \[lemma finite extension\] to $$\xymatrix{
&& 0 \ar[d] & 0 \ar[d] \\
0 \ar[r] & {\mathcal{C}}_0({\operatorname{Prim}}_k(I),{\mathbb{M}}_k) \ar[r] \ar@{=}[d] & I_{\leq k} \ar[r] \ar[d] & I_{\leq k-1} \ar[r] \ar[d] & 0 \\
0 \ar[r] & {\mathcal{C}}_0({\operatorname{Prim}}_k(A),{\mathbb{M}}_k) \ar[r] & A_{\leq k} \ar[r] \ar[d] ^\pi & A_{\leq k-1} \ar[r] \ar[d] & 0 \\
&& F \ar[d] \ar@{=}[r] & F \ar[d] \\
&& 0 & 0
}$$ and see that $R_k\colon{\operatorname{Prim}}_{\leq k}(A)\rightarrow 2^{{\operatorname{Prim}}_{\leq k-1}(A)}$ is again lower semicontinuous and has pointwise finite image.
### Quantum permutation algebras {#section A4}
We are now going to demonstrate how the structure theorem \[thm structure\] can be used to show that certain $C^*$-algebras fail to be semiprojective. We would like to thank T. Katsura for pointing out to us the quantum permutation algebras ([@Wan98], [@BC08]) as a testcase:
\[definition A4\] For $n\in\mathbb{N}$, the quantum permutation algebra $A_s(n)$ is the universal $C^*$-algebra generated by $n^2$ elements $u_{ij}$, $1\leq i,j\leq n$, with relations $$\begin{array}{rcl}u_{ij}=u_{ij}^*=u_{ij}^2 & \& & \sum_j u_{ij}=\sum_i u_{ij} =1.\end{array}$$
It is not clear from the definition whether the $C^*$-algebras $A_s(n)$ are semiprojective or not. For $n\in\{1,2,3\}$ one easily finds $A_s(n)\cong\mathbb{C}^{n!}$ so that we have semiprojectivity in that cases. For higher $n$ one might expect semiprojectivity of $A_s(n)$ because of the formal similarity to graph $C^*$-algebras. In fact, their definition only involves finitely many projections and orthogonality resp. order relations between them. Since graph $C^*$-algebras associated to finite graphs are easily seen to be semiprojective, one might think that we also have semiprojectivity for the quantum permutation algebras. This was even erroneously claimed to be true in [@Bla04 example 2.8(vi)]. In this section we will show that the $C^*$-algebras $A_s(n)$ are in fact not semiprojective for all $n\geq 4$.
One can reduce the question for semiprojectivity of these algebras to the case $n=4$. The following result of Banica and Collins shows that the algebra $A_s(4)$ is 4-subhomogeneous, so that our machinery applies. The idea is to get enough information about the primitive spectrum of $A_s(4)$ to show that it contains closed subsets of dimension strictly greater than 1. This will then contradict part (2) of \[thm structure\], so that $A_s(4)$ cannot be semiprojective.\
We follow notations from [@BC08] and denote the Pauli matrices by $$\begin{array}{cccc}
c_1=\begin{pmatrix}1 & 0 \\ 0 & 1\end{pmatrix},&c_2=\begin{pmatrix}i & 0 \\0 & -i\end{pmatrix},&c_3=\begin{pmatrix}0&1\\-1&0\end{pmatrix},&c_4=\begin{pmatrix}0&i\\i&0\end{pmatrix}.
\end{array}$$ Set $\xi_{ij}^x=c_ixc_j$ and regard ${\mathbb{M}}_2$ as a Hilbert space with respect to the scalar product $<a|b>={\operatorname{tr}}(b^*a)$. Then for any $x\in SU(2)$ we find $\{\xi^x_{ij}\}_{j=1..4}$ and $\{\xi^x_{ij}\}_{i=1..4}$ to be a basis for ${\mathbb{M}}_2$. Under the identification ${\mathbb{M}}_4\cong\mathcal{B}({\mathbb{M}}_2)$, Banica and Collins studied the following representation of $A_s(4)$:
\[prop pauli rep\] The $^*$-homomorphism given by $$\begin{array}{rl}
\pi\colon A_s(4)\longrightarrow & {\mathcal{C}}(SU(2),{\mathbb{M}}_4) \\ u_{ij}\mapsto & \left(x\mapsto\text{rank one projection onto}\; \mathbb{C}\cdot \xi^x_{ij}\right)
\end{array}$$ is faithful. It is called the Pauli representation of $A_s(4)$.
For the remainder of this section let $S$ denote the following subset of $SU(2)$: $$S:=\left\{\begin{pmatrix}\lambda & -\overline{\mu} \\ \mu &\overline{\lambda}\end{pmatrix}\in SU(2)\colon\min\left\{|{\operatorname{Re}}(\lambda\mu)|,|{\operatorname{Im}}(\lambda\mu)|,|{\operatorname{Re}}(\overline{\lambda}\mu)|,|{\operatorname{Im}}(\overline{\lambda}\mu)|,\left||\lambda|-|\mu|\right|\right\}=0\right\}$$ We will now study the representations of $A_s(4)$ obtained by composing the Pauli representation with a point evaluation. As we will see, most points of $SU(2)$ lead to irreducible representations which are furthermore locally pairwise inequivalent.
\[lemma A4 irreducible\] The representation $\pi_x={\operatorname{ev}}_x\circ\pi\colon A_s(4)\rightarrow{\mathbb{M}}_4$ is irreducible for every $x\in SU(2)\backslash S$.
Let $x=\begin{pmatrix}\lambda &-\overline{\mu}\\ \mu & \overline{\lambda}\end{pmatrix}\in SU(2)\backslash S$ be given, we show that the commutant of $\pi_x(A_s(4))$ equals the scalars. Therefore we will check the matrix entries of the elements $\pi_x(u_{ij})$ with respect to the orthonormal basis $\left\{\frac{1}{\sqrt{2}}\xi_{11}^x,\frac{1}{\sqrt{2}}\xi_{12}^x,\frac{1}{\sqrt{2}}\xi_{13}^x,\frac{1}{\sqrt{2}}\xi_{14}^x\right\}$ of $M_2\cong\mathbb{C}^4$. Since in this picture $\pi_x(U_{1i})$ equals the elementary matrix $e_{ii}$, every element in $\left(\pi_x(A_s(4))\right)'$ is diagonal. But we also find $$\begin{array}{rl}
\left(\pi_x(U_{23})\right)_{12} &=\frac{1}{2}<\pi_x(U_{23})\xi^x_{12}|\xi^x_{11}> \\ &=\frac{1}{4}<\xi^x_{12}|\xi^x_{23}><\xi^x_{23}|\xi^x_{11}>=4\cdot {\operatorname{Re}}(\lambda\mu)Im(\lambda\mu)\neq 0, \\
\left(\pi_x(U_{22})\right)_{13} &=\frac{1}{4}<\xi^x_{13}|\xi^x_{22}><\xi^x_{22}|\xi^x_{11}>=2\cdot {\operatorname{Re}}(\lambda\mu)(|\lambda|^2-|\mu|^2)\neq 0, \\
\left(\pi_x(U_{22})\right)_{14} &=\frac{1}{4}<\xi^x_{14}|\xi^x_{22}><\xi^x_{22}|\xi^x_{11}>=-2\cdot {\operatorname{Im}}(\lambda\mu)(|\lambda|^2-|\mu|^2)\neq 0.
\end{array}$$ So the only elements of ${\mathbb{M}}_4$ commuting with all of $\pi_x(A_s(4))$ are the scalars.
\[prop A4 inequivalent\] Every $x\in SU(2)\backslash S$ admits a small neighborhood $V\subseteq SU(2)\backslash S$ such that for all distinct $y,y'\in V$ the representations $\pi_y$ and $\pi_{y'}$ are not unitarily equivalent.
Let $x=\begin{pmatrix}\lambda_0&-\overline{\mu_0}\\ \mu_0&\overline{\lambda_0}\end{pmatrix}\in SU(2)\backslash S$ be given, then $$\epsilon:=\min\left\{|{\operatorname{Re}}(\lambda_0\mu_0)|,|{\operatorname{Im}}(\lambda_0\mu_0)|,|{\operatorname{Re}}(\overline{\lambda_0}\mu_0)|,|{\operatorname{Im}}(\overline{\lambda_0}\mu_0)|,\left||\lambda_0|-|\mu_0|\right|,|\lambda_0|\right\}>0.$$ Define a neighborhood $V\subseteq SU(2)\backslash S$ of $x$ by $$V=\left\{\begin{pmatrix}\lambda &-\overline{\mu}\\ \mu & \overline{\lambda}\end{pmatrix}\in SU(2)\backslash S:|\lambda-\lambda_0|<\frac{\epsilon}{3},|\mu-\mu_0|<\frac{\epsilon}{3}\right\}.$$ Now let $y,y'\in V$ with unitarily equivalent representations $\pi_y$ and $\pi_{y'}$ be given. We compute the value $$\begin{array}{rl}
\|\pi_{y}(U_{11}U_{22})\| &=\frac{1}{4}\|(<-\;|\xi^{y}_{11}>\xi^{y}_{11})\circ(<-\;|\xi^{y}_{22}>\xi^{y}_{22})\| \\
&=\frac{1}{4}|<\xi^{y}_{22}|\xi^{y}_{11}>|\cdot\|(<-\;|\xi^{y}_{22}>\xi^{y}_{11})\| \\ &=\frac{1}{4}|<\xi^{y}_{22}|\xi^{y}_{11}>|\cdot\|\xi^{y}_{22}\|\|\xi^{y}_{11}\| \\ &=\left||\lambda|^2-|\mu|^2\right|
\end{array}$$ which is invariant under unitary equivalence. So we find $\left||\lambda|^2-|\mu|^2|\right|=\left||\lambda'|^2-|\mu'|^2\right|$. This implies $$\begin{array}{rcl}
\left(|\lambda|=|\lambda'|\wedge |\mu|=|\mu'|\right) &\vee &\left(|\lambda|=|\mu'|\wedge |\mu|=|\lambda'|\right)
\end{array}$$ because of $|\lambda|^2+|\mu|^2=1=|\lambda'|^2+|\mu'|^2$. By definition of $V$ we have $$\left||\lambda|-|\mu'|\right|\geq\left||\lambda_0|-|\mu_0|\right|-\left||\lambda|-|\lambda_0|\right|-\left||\mu'|-|\mu_0|\right|>\frac{\epsilon}{3}>0,$$ so that we can exclude the second case. Analogously, computing the invariants $\|\pi_{y}(U_{13}U_{22})\|$ and $\|\pi_{y}(U_{14}U_{22})\|$ gives $$\begin{tabular}{rcl}
$|{\operatorname{Re}}(\lambda\mu)|=|{\operatorname{Re}}(\lambda'\mu')|$&and&$|{\operatorname{Im}}(\lambda\mu)|=|{\operatorname{Im}}(\lambda'\mu')|$
\end{tabular}$$ and checking $\|\pi_{y}(U_{11}U_{42})\|$ and $\|\pi_{y}(U_{11}U_{32})\|$ shows $$\begin{tabular}{rcl}
$|{\operatorname{Re}}(\overline{\lambda}\mu)|=|{\operatorname{Re}}(\overline{\lambda'}\mu')|$&and&$|{\operatorname{Im}}(\overline{\lambda}\mu)|=|{\operatorname{Im}}(\overline{\lambda'}\mu')|$
\end{tabular}.$$ The last four equalities imply $\lambda\mu=\lambda'\mu'$ and $\overline{\lambda}\mu=\overline{\lambda'}\mu'$ by the choice of $V$. Together with $|\lambda|=|\lambda'|$ and $|\mu|=|\mu'|$ we find $(\lambda,\mu)=(\lambda',\mu')$ or $(\lambda,\mu)=(-\lambda',-\mu')$. In the second case we get $|\lambda-\lambda'|=2|\lambda|\geq 2|\lambda_0|-2|\lambda-\lambda_0|\geq \frac{4\epsilon}{3}$ contradicting $|\lambda-\lambda'|\leq|\lambda-\lambda_0|+|\lambda'-\lambda_0|<\frac{2\epsilon}{3}$ by the choice of $V$. It follows that $y=y'$.
By now we have obtained enough information about ${\operatorname{Prim}}(A_s(4))$ to show that it does not satisfy condition (2) of Theorem \[thm structure\]. Hence we find:
\[thm A4 not sp\] The $C^*$-algebra $A_s(4)$ is not semiprojective.
Choose a point $x_0\in SU(2)\backslash S$ and a neighborhood $V$ of $x_0$ as in Proposition \[prop A4 inequivalent\]. Since $SU(2)$ is a real 3-manifold, there is a neighborhood of $x_0$ contained in $V$ which is homeomorphic to $\mathbb{D}^3=\{x\in\mathbb{R}\colon\|x\|\leq 1\}$. The restriction of the Pauli representation $\pi$ to this neighborhood gives a $^*$-homomorphism $\varphi\colon A_s(4)\rightarrow{\mathcal{C}}(\mathbb{D}^3,{\mathbb{M}}_4)$ with the property that ${\operatorname{ev}}_x\circ\varphi$ and ${\operatorname{ev}}_y\circ\varphi$ are irreducible but not unitarily equivalent for all distinct $x,y\in\mathbb{D}^3$. The pointwise surjectivity of $\varphi$ given by Lemma \[lemma A4 irreducible\] and a Stone-Weierstra[ß]{} argument ([@Kap51 Theorem 3.1]) show that $\varphi$ is in fact surjective. This implies that ${\operatorname{Prim}}_4(A_s(4))$ contains a closed 3-dimensional subset and hence ${\operatorname{dim}}({\operatorname{Prim}}_4(A_s(4)))\geq 3$. As a consequence, $A_s(4)$ cannot be semiprojective because it is subhomogeneous by Proposition \[prop pauli rep\] but fails to satisfy condition $(2)$ of Theorem \[thm structure\].
It is not hard to show that semiprojectivity of $A_s(n)$ for some $n>4$ would force $A_s(4)$ to be semiprojective. Since we have just shown that this is not the case, we obtain:
\[cor An not sp\] The $C^*$-algebras $A_s(n)$ are not semiprojective for $n\geq 4$.
For $n\geq 4$ there is a canonical surjection $\varrho_n\colon A_s(n)\rightarrow A_s(4)$ given by $$u^{(n)}_{ij}\mapsto\begin{cases}u^{(4)}_{ij} & \text{if}\; 1\leq i,j\leq 4\\ 1 & \text{if}\; i=j>4 \\ 0 & \text{otherwise}\end{cases}.$$ The kernel of $\varrho_n$ is generated by the finite set of projections $\left\{u_{ij}^{(n)}\colon\varrho_n\left(u_{ij}^{(n)}\right)=0\right\}$. It follows from [@Sor12 Proposition 3], which extends the idea of [@Neu00 Proposition 5.19], that semiprojectivity of $A_s(n)$ would imply semiprojectivity of $\varrho_n(A_s(n))=A_s(4)$. Since this is not the case by Theorem \[thm A4 not sp\], $A_s(n)$ cannot be semiprojective for all $n\geq 4$.
[3]{}
J.-P. Aubin and H. Frankowska. [*Set-valued analysis*]{}, Birkh[ä]{}user, Boston, 1990. T. Banica and B. Collins. Integration over the Pauli quantum group, [*J. Geom. Phys.*]{}, 58(8):942-961, 2008. J.J. Bastian and K.J. Harrison. Subnormal weighted shifts and asymptotic properties of normal operators, [*Proc. Amer. Math. Soc.*]{}, 42(2):475-479, 1974. R.H. Bing. Partitioning a set, [*Bull. Amer. Math. Soc.*]{}, 55:1101-1110, 1949. B. Blackadar. Shape theory for $C^*$-algebras, [*Math. Scan.*]{}, 56(2):249-275, 1985. B. Blackadar. Semiprojectivity in simple $C^*$-algebras, in [*Operator algebras and applications*]{}, volume 38 of [*Adv. Stud. Pure Math.*]{}, pages 1-17. Math. Soc. Japan, Tokyo, 2004. B. Blackadar. [*Operator algebras*]{}, volume 122 of [*Encyclopedia of Mathematical Sciences*]{}, Springer-Verlag, Berlin, 2006. B. Blackadar. The homotopy lifting theorem for semiprojective $C^*$-algebras, arXiv preprint [math.OA/1207.1909v3](http://arxiv.org/abs/1207.1909v3), 2012. K. Borsuk. [*Theory of retracts*]{}, Monografie Matematyczne, 44, Warszawa: PWN - Polish Scientific Publishers, 1967. N.P. Brown. On quasidiagonal $C^*$-algebras, arXiv preprint [math.OA/0008181](http://arxiv.org/abs/math/0008181), 2000. A. Chigogidze and A.N. Dranishnikov. Which compacta are noncommutative ARs?, Topology Appl. 157(4):774-778, 2010. J. Dixmier. [*$C^*$-algebras*]{}, North-Holland, Amsterdam and New York, 1977. S. Eilers, T. A. Loring and G. K. Pedersen. Stability of anticommutation relations: An application of noncommutative CW complexes, [*J. reine angew. Math.*]{}, 499:101-143, 1998. S. Eilers, T. A. Loring and G. K. Pedersen. Morphisms of extensions of $C^*$-algebras: Pushing forward the Busby invariant, [*Advances in Mathematics*]{}, 147:74-109, 1999. D. Enders. Subalgebras of finite codimension in semiprojective $C^*$-algebras, arXiv preprint [math.OA/1405.2750](http://arxiv.org/abs/1405.2750), 2014. S. Eilers and T. Katsura. Semiprojectivity and properly infinite projections in graph $C^*$-algebras, in preparation. J.M.G. Fell. The structure of algebras of operator fields, [*Acta Math.*]{} 106:233-280, 1961. P. Friis and M. Rørdam. Approximation with normal operators with finite spectrum and an elementary proof of a Brown-Douglas-Fillmore theorem, [*Pacific J. Math.*]{}, 199(2):347-366, 2001. K. Grove and G.K. Pedersen. Sub-stonean spaces and corona sets, [*Journal of Functional Analysis*]{}, 56:124-143, 1984. W. Hurewicz and H. Wallman, [*Dimension theory*]{}, Princeton University Press, Princeton, 1948. I. Kaplansky. The structure of certain operator algebras, [*Trans. Amer. Math. Soc.*]{}, 70:219-255, 1951. H. Lin. Almost commuting selfadjoint matrices and applications, [*Operator algebras and their applications*]{} (Waterloo, ON, 1994/1995), 193-233, Fields Inst. Commun. 13, American Mathematical Society, Providence, RI, 1997. T.A. Loring. [*Lifting solutions to perturbing problems in $C^*$-algebras*]{}, volume 8 of [*Fields Institute Monographs*]{}, American Mathematical Society, Providence, RI, 1997. T.A. Loring and G.K. Pedersen. Projectivity, transitivity and AF-telescopes, [*Trans. Amer. Math. Soc.*]{}, 350(11):4313-4339, 1998. M. Meilstrup. Classifying homotopy types of one-dimensional Peano continua, Thesis for Master of Science at the Department of Mathematics at Brigham Young University, 2005. E. Moise, Grille decomposition and convexification theorems for compact metric locally connected continua, [*Bull. Amer. Math. Soc*]{}, 55:1111-1121. S.B.jun. Nadler. [*Continuum theory: An introduction*]{}, volume 158 of [*Pure and Applied Mathematics*]{}, Marcel Dekker, New York, 1992. K. Nagami. [*Dimension Theory*]{}, Pure Appl. Math., No. 37, Academic Press, London, New York, 1970. B. Neub[ü]{}ser. Semiprojektivit[ä]{}t und Realisierungen von rein unendlichen $C^*$-Algebren, Ph.D. thesis, Universit[ä]{}t M[ü]{}nster, 2000. N.C. Phillips, Recursive subhomogeneous algebras, [*Trans. Amer. Math. Soc.*]{}, 359:4595-4623, 2007. M. Rørdam, F. Larsen and N. Lautsen. [*An introduction to K-theory for $C^*$-algebras*]{}, volume 49 of [*London Mathematical Society Student Texts*]{}. Cambridge University Press, Cambridge, 2000. M. Rørdam. Classification of nuclear, simple $C^*$-algebras, in [*Classification of nuclear $C^*$-algebras. Entropy in operator algebras*]{}, volume 126 of [*Encyclopaedia Math. Sci.*]{}, pages 1-145, Springer, Berlin, 2002. A.P.W. Sørensen. On a counterexample to a conjecture by Blackadar, Operator Algebra and Dynamics, Springer Proceedings in Mathematics & Statistics, Vol. 58, 2013. I. Raeburn and D.P. Williams. [*Morita equivalence and continuous-trace $C^*$-algebras*]{}, vol. 60 of [*Mathematical Surveys and Monographs*]{}, AMS,Providence, RI, 1998. A.P.W. Sørensen and H. Thiel, A characterization of semiprojectivity for commutative $C^*$-algebras, [*Proc. Lond. Math. Soc.*]{}, 105(5):1021-1046, 2012. S. Wang. Quantum symmetry groups of finite spaces, [*Comm. Math. Phys.*]{}, 195:195-211, 1998. L.E. Ward. On local trees, [*Proc. Amer. Math. Soc.*]{}, 11:940-944, 1960. N.E. Wegge-Olsen. [*K-Theory and $C^*$-algebras: a friendly approach*]{}, Oxford University Press, 1993.
|
---
author:
- 'Alexander Kurz[^1]'
- 'Tao Liu[^2]'
- 'Peter Marquard[^3]'
- 'Alexander V. Smirnov[^4]'
- |
\
Vladimir A. Smirnov[^5]
- 'Matthias Steinhauser[^6]'
title: 'Higher order hadronic and leptonic contributions to the muon $g-2$'
---
Introduction
============
The anomalous magnetic moment of the muon, $a_\mu$, is among the most precise measured quantities in particle physics. It is measured to a precision of 0.54 parts per million which matches the precision of the Standard Model theory prediction [@Bennett:2006fi; @Roberts:2010cj]. However, since many years one observes a discrepancy of about three to four standard deviations which survives persistent all improvements. This concerns both the experimental data and theoretical calculations entering the prediction.
Currently a new experiment is built at FERMILAB with the aim to increase the accuracy of the measured value by about a factor four [@Carey:2009zzb; @Herzog_fccp15]. In the upcoming years also improvements on the theory side can be expected. On the one hand this is connected to improved measurements of $R(s)$ at low energies (see, e.g., Refs. [@Aubert:2009ad; @Ambrosino:2010bv; @Eidelman_fccp15]). On the other hand it can be expected that within the next few years results from lattice simulations become available both for the hadronic vacuum polarization and hadronic light-by-light contributions (see, e.g., Refs. [@DellaMorte:2011aa; @Burger:2013jya; @Blum:2014oka; @Blum:2015gfa; @Lehner_fccp15]).
The by far dominant numerical contribution to $a_\mu$ originates from QED corrections which are known to five-loop order [@Aoyama:2012wk]. Note, however, that the four- and the five-loop corrections have only been computed by a single group.[^7] For this reason we have recently started to systematically check the four-loop results of [@Aoyama:2012wk]. In Ref. [@Lee:2013sx] analytic results for the gauge-invariant subsets with two or three closed electron loops have been obtained neglecting power corrections of the form $m_e/m_\mu$. All contributions involving a $\tau$ lepton have been computed in Ref. [@Kurz:2013exa]. After including three (analytic) expansion terms in $m_\mu^2/m_\tau^2$ a better precision has been obtained than in the numerical approach of Ref. [@Aoyama:2012wk]. The numerically most important QED contributions at four-loop level arise from light-by-light-type diagrams (i.e. the external photon does not couple to the external muon line) containing a closed electron loop. This well-defined subset has been considered in Ref. [@Kurz:2015bia] where an asymptotic expansion for $m_e \ll m_\mu$ has been performed to compute four expansion terms.
We adopt the notation from Ref. [@Aoyama:2012wk] and parametrize the anomalous magnetic moment in the form $$\begin{aligned}
a_\mu &=& \sum_{n=1}^\infty a_\mu^{(2n)} \left( \frac{\alpha}{\pi} \right)^n
\,,
\label{eq::amu}\end{aligned}$$ where the four-loop contribution can be written as $$\begin{aligned}
a_\mu^{(8)} &=& A_1^{(8)} + A_2^{(8)}(m_\mu / m_e) + A_2^{(8)}(m_\mu / m_\tau)
\nonumber\\&&\mbox{}
+ A_3^{(8)}(m_\mu / m_e, m_\mu / m_\tau)
\,.
\label{eq::Amu}\end{aligned}$$ $A_1^{(8)}$ contains only contributions from photons and muons, $A_2^{(8)}(m_\mu / m_e)$ and $A_2^{(8)}(m_\mu / m_\tau)$ involve closed electron or tau loops, and each Feynman diagram which contributes to $A_3^{(8)}(m_\mu / m_e, m_\mu / m_\tau)$ contains all three lepton flavours simultaneously. In Sections \[sec::electron\] and \[sec::tau\] we describe the calculation of the light-by-light type QED contribution to $A_2^{(8)}(m_\mu / m_e)$ (see also [@Kurz:2015bia]) and the computation of $A_2^{(8)}(m_\mu / m_\tau)$ (see also [@Kurz:2013exa]), respectively. Afterwards we summarize in Section \[sec::hadr\] the computation of the next-to-next-to-leading order (NNLO) hadronic vacuum polarization contribution published in Ref. [@Kurz:2014wya]. A brief summary and an outlook is given in Section \[sec::summary\].
\[sec::electron\]Four-loop electron contribution
================================================
The numerically most important contribution to $a_\mu^{(8)}$ originates from diagrams involving a closed electron loop (denoted by $A_2^{(8)}(m_\mu / m_e)$ in Eq. (\[eq::Amu\]). This contribution contains a gauge invariant subset where the external photon does not couple to the external muon line but to a closed fermion loop, the so-called leptonic light-by-light-type diagrams. Due to Furry’s theorem such diagrams do not contribute at two but only start at three loops where four photons can be attached to the closed fermion loop. Here we discuss the four-loop result which can be sub-divided into three gauge invariant and finite contributions which we denote by IV(a), IV(d) and IV(c). Sample Feynman diagrams are shown in Fig. \[fig::FDs\]. Case IV(a) can be further subdivided according to the flavour of the leptons in the closed fermion loops. The contribution with two electron loops is denoted by IV(a0), the one with one muon and one electron loop and the coupling of the external photon to the electron by IV(a1), and the remaining one with one muon and one electron loop by IV(a2). We do not consider the case with two muon loops since this contribution is part of $A_1^{(8)}$.
  \
IV(a) IV(b) IV(c)
The light-by-light-type diagrams are numerically dominant and provide about 95% of the four-loop electron loop contribution. The main reason for this are $\log(m_e/m_\mu)$ terms which are even present in the limit $m_e\to0$. In fact, IV(a0) even has quadratic logarithms which makes this part the most important one.
Our calculation is based on an asymptotic expansion [@Beneke:1997zp; @Smirnov:2013] for $m_e\ll m_\mu$ which is implemented with the help of [asy]{} [@Pak:2010pt; @Jantzen:2012mw] and in-house [Mathematica]{} programs. Similar to the hard mass procedure applied in Section \[sec::tau\] we obtain a factorization of the original two-scale integrals into products of one-scale integrals. The latter are either vacuum or on-shell integrals or integrals containing eikonal propagators of the form $1/(p\cdot q)$ (see Ref. [@Kurz:2015bia] for more details). For each integral class we perform a reduction to master integrals and obtain analytic results expressed as a linear combination of about 150 so-called master integrals. About 50% of them we know analytically or to high numerical precision. The remaining ones are computed with the help of the package [FIESTA]{} [@Smirnov:2013eza] which is the source of the numerical uncertainty in our final result. We would like to stress that in our approach a systematic improvement is possible if it is required to improve the accuracy.
$A_2^{(8)}\left(\frac{m_\mu}{m_e}\right)$
------------------------------------------- ------------------- --------------------------------------------------
IV(a0) $116.76 \pm 0.02$ $116.759183 \pm 0.000292$
IV(a1) $2.69 \pm 0.14$ $2.697443 \pm 0.000142$
IV(a2) $4.33 \pm 0.17$ $4.328885 \pm 0.000293$
IV(a) $123.78\pm 0.22$ $123.78551\hphantom{2} \pm 0.00044\hphantom{2}$
IV(b) $-0.38 \pm 0.08$ $-0.4170\hphantom{22} \pm 0.0037\hphantom{22}$
IV(c) $2.94 \pm 0.30$ $2.9072\hphantom{22} \pm 0.0044\hphantom{22}$
: \[tab::a8\]Summary of the final results for the individual four-loop light-by-light-type contributions and their comparison with results presented in Refs. [@Kinoshita:2004wi; @Aoyama:2012wk].
For all five cases we compute terms up to order $(m_e/m_\mu)^3$ (i.e. four expansion terms) and check that the cubic corrections only provide a negligible contribution. Our final results can be found in Tab. \[tab::a8\] where we compare to the findings of Refs. [@Kinoshita:2004wi; @Aoyama:2012wk]. Note that results for IV(a0) have also been obtained in Refs. [@Calmet:1975tw; @Chlouber:1977dr], though with significantly larger uncertainty. In all cases good agreement is found with [@Kinoshita:2004wi; @Aoyama:2012wk]. Although our numerical uncertainty, which amounts to approximately $0.4 \times (\alpha/\pi)^4 \approx
1.2 \times 10^{-11}$, is larger, the final result is nevertheless sufficiently accurate as can be seen by the comparison to the difference between the experimental result and theory prediction which is given by $$\begin{aligned}
a_\mu({\rm exp}) - a_\mu({\rm SM}) &\approx& 249(87) \times 10^{-11}
\,.
\label{eq::amu_diff}\end{aligned}$$ This result is taken from Ref. [@Aoyama:2012wk]. Note that the uncertainty in Eq. (\[eq::amu\_diff\]) receives approximately the same amount from experiment and theory. Even after a projected reduction of the uncertainty by a factor four both in $a_\mu({\rm exp})$ and $a_\mu({\rm SM})$ our numerical precision is a factor ten below the uncertainty of the difference.
\[sec::tau\]Four-loop tau lepton contribution
=============================================
In this section we discuss the gauge invariant and finite subset of Feynman diagrams involving a closed heavy tau lepton loop. In the limit of infinitely heavy $m_\tau$ this contribution has to vanish. Thus $A_2^{(8)}(m_\mu /
m_\tau)$ has a parametric dependence $m_\mu^2/m_\tau^2$ which is of order $10^{-3}$. Note, that $\alpha/\pi \approx 2\cdot 10^{-3}$ and thus one can expect that the four-loop tau lepton contribution is of the same order as the universal five-loop result [@Aoyama:2012wk].
We compute this contribution by applying an asymptotic expansion in the limit $m_\tau^2 \gg m_\mu^2$. This is realized with the help of the program [ exp]{} [@Harlander:1997zb; @Seidensticker:1999bb] which is written in [ C++]{}. As a result the two-scale four-loop integrals factorize into one-scale vacuum ($m_\tau$) and on-shell ($m_\mu$) integrals. Both integral classes are well studied in the literature (for references see [@Kurz:2013exa]). This concerns both the reduction to master integrals and the analytic evaluation of the latter.
-------
=0.85
-------
In the first line of Fig. \[fig::ae\] a sample Feynman diagram is shown where the thick solid lines represent the tau leptons. Rows two and three of Fig. \[fig::ae\] show the result of the asymptotic expansion where the graphs left of the symbol $\otimes$ have to be expanded in all small quantities, i.e., the external momenta and the muon mass. Thus, the only mass scale of the remaining vacuum integral is the tau lepton mass. The result of the Taylor expansion is inserted into the effective vertex (thick blob) present in the diagram to the right of $\otimes$. Afterwards the remaining loop integrations, which are of on-shell type, are performed.
As a final result we obtain an expansion in $m_\mu^2/m_\tau^2$ with analytic coefficients containing $\log(m_\mu^2/m_\tau^2)$ terms. Note that with the help of this method a better accuracy has been obtained than with the numerical approach of Ref. [@Aoyama:2012wk]. Inserting numerical values for the lepton masses leads to $$\begin{aligned}
A^{(8)}_{2,\mu}(m_\mu/m_\tau) &=&
0.0421670 + 0.0003257
\nonumber\\&&\mbox{}
+ 0.0000015 + \ldots
\,,
\label{eq::A8}\end{aligned}$$ where the ellipsis indicates terms of order $(m_\mu^2/m_\tau^2)^4$ which are expected to contribute at order $10^{-8}$ to $A^{(8)}_{2,\mu}(m_\mu/m_\tau)$.
$a_\mu$ receives contribution from $\tau$ lepton loops starting at two-loop order. Their numerical impact is given by $$\begin{aligned}
10^{11} \times a_\mu\Big|_{\tau \rm loops}
&=& 42.13 + 0.45 + 0.12
\,,
\label{eq::amu_tau}\end{aligned}$$ where the numbers on the right-hand side correspond to the two, three and four loops. It is interesting to note that the three-loop term is only less than a factor four larger than the four-loop counterpart. Furthermore, it is worth comparing the numbers in Eq. (\[eq::amu\_tau\]) to the universal contributions contained in $A_{1}$ which read [@Aoyama:2012wk] $$\begin{aligned}
10^{11} \times a_\mu\Big|_{\rm univ.} \!\!\!&=&\!\!\!
116\,140\,973.21 - 177\,230.51
\nonumber\\&&\mbox{}
+ 1\,480.42 - 5.56 + 0.06
\,,\end{aligned}$$ where the individual terms on the right-hand side represent the results from one to five loops. Note that the four-loop tau lepton term is twice bigger than the five-loop photonic contribution.
\[sec::hadr\]NNLO hadronic contribution
=======================================
The LO hadronic contribution to the anomalous magnetic moment of the muon is obtained from diagram (a) in Fig. \[fig::FD\_nnlo\]. One parametrizes the hadronic contribution (represented by the blob) by the polarization function $\Pi(q^2)$ which appears as a factor in the integrand of the one-loop diagram. In a next step one exploits analyticity of $\Pi(q^2)$ and uses a dispersion integral to introduce its imaginary part, $$\begin{aligned}
R(s) &=& \frac{ \sigma(e^+e^-\to\mbox{hadrons}) }{ \sigma_{pt} }
\,,\end{aligned}$$ with $\sigma_{pt} = 4\pi\alpha^2/(3s)$. Note that $\sigma(e^+e^-\to\mbox{hadrons})$ does note include initial state radiative or vacuum polarization corrections. At that point the loop integration and the dispersion integral are interchanged and one obtains $$\begin{aligned}
a_\mu^{(1)} &=& \frac{1}{3} \left(\frac{\alpha}{\pi}\right)^2
\int_{m_\pi^2}^\infty {\rm d} s \frac{R(s)}{s} K^{(1)}(s)
\,,
\label{eq::aLO}\end{aligned}$$ A convenient integral representation for the kernel function $K^{(1)}(s)$, which is the result of the loop integration, is given by $$\begin{aligned}
K^{(1)}(s) &=&
\int_0^1 {\rm d} x \frac{x^2(1-x)}{x^2 + (1-x) \frac{s}{m_\mu^2}}
\,.
\label{eq::K1}\end{aligned}$$ At one-loop order it is possible to obtain analytic results (see Refs. [@BroRaf68; @Eidelman:1995ny]). Nevertheless, it is promising to consider $K^{(1)}(s)$ in the limit $m_\mu^2 \ll s$ which is justified since the lower integration limit in Eq. (\[eq::aLO\]) is $m_\pi^2$ which is bigger than $m_\mu^2$. The expansion of $K^{(1)}(s)$ is easily obtained by remembering that it originates from the vertex diagram similar to Fig. \[fig::FD\_nnlo\](a) where the hadronic blob (including the external photon lines) is replaced by a massive photon with mass $\sqrt{s}$. The expansion $m_\mu^2 \ll s$ is easily implemented with the help of the program [exp]{} [@Harlander:1997zb; @Seidensticker:1999bb] which implements the rules of asymptotic expansions involving a large internal mass (see, e.g., Ref. [@Smirnov:2013]). As a result the original two-scale integral is represented as a sum of one-scale integrals which are simple to compute. Using this approach several expansion terms in $m_\mu^2/s$ can be computed. One observes that an excellent approximation for $a_\mu^{(1)}$ is obtained by including terms up to order $(m_\mu^2/s)^5$.
The approach described in detail for the one-loop diagram can also be applied at two and three loops where exact calculations of the kernel functions are either very difficult or even impossible. In Ref. [@Kurz:2014wya] four expansion terms have been computed which provides an approximation at the per mil level.
A slight complication arises for the contributions involving more than one hadronic insertion, see Figs. \[fig::FD\_nnlo\](d,h,i,j,l). In case they are present in the same photon line formulas similar to Eq. (\[eq::K1\]) can be derived with two- and three-dimensional integrations. Diagrams of type $(3c)$ in Fig. \[fig::FD\_nnlo\] are more involved. Here, we apply a multiple asymptotic expansion in the limits $s\gg s^\prime \gg m_\mu^2$, $s \approx
s^\prime\gg m_\mu^2$ and $s^\prime\gg s\gg m_\mu^2$ ($s$ and $s^\prime$ are the integration variables) and construct an interpolating function by combining the results from the individual limits.
----------------------- ----------------------- ------------------------- -----------------------
   
\(a) LO \(b) $2a$ \(c) $2b$ \(d) $2c$
   
\(e) $3a$ \(f) $3b$ \(g) $3b$ \(h) $3c$
   
\(i) $3c$ \(j) $3c$ \(k) $3b$,lbl \(l) $3d$
----------------------- ----------------------- ------------------------- -----------------------
The LO result for the hadronic vacuum polarization contribution to $a_\mu$ can be found in Refs. [@Davier:2010nc; @Hagiwara:2011af; @Jegerlehner:2011ti; @Benayoun:2012wc; @Jegerlehner:2015stw] and NLO analyses have been performed in Refs. [@Krause:1996rf; @Greynat:2012ww; @Hagiwara:2003da; @Hagiwara:2011af]. Our NLO results for the three contributions read $$\begin{aligned}
a_\mu^{(2a)} &=& -20.90 \times 10^{-10}\,, \nonumber\\
a_\mu^{(2b)} &=& 10.68 \times 10^{-10}\,, \nonumber\\
a_\mu^{(2c)} &=& 0.35 \times 10^{-10}\,,\end{aligned}$$ which leads to $$\begin{aligned}
a_\mu^{\rm had,NLO} &=& -9.87 \pm 0.09 \times 10^{-10}\,,\end{aligned}$$ in a good agreement with Refs. [@Hagiwara:2003da; @Hagiwara:2011af]. Note that in our analyses no correlated uncertainties are taken into account. Such a rough treatment should not be done at LO but is certainly acceptable at NNLO.
For the individual NNLO contributions we obtain the results $$\begin{aligned}
a_\mu^{(3a)} &=& 0.80 \times 10^{-10} \,,\nonumber\\
a_\mu^{(3b)} &=& -0.41 \times 10^{-10} \,,\nonumber\\
a_\mu^{(3b,\rm lbl)} &=& 0.91 \times 10^{-10} \,,\nonumber\\
a_\mu^{(3c)} &=& -0.06 \times 10^{-10} \,,\nonumber\\
a_\mu^{(3d)} &=& 0.0005 \times 10^{-10} \,,\end{aligned}$$ which leads to $$\begin{aligned}
a_\mu^{\rm had,NNLO} &=& 1.24 \pm 0.01 \times 10^{-10}\,.
\label{eq::amuNNLO}\end{aligned}$$ It is interesting to note that similar patterns are observed at two and three loops: multiple hadronic insertions are small and the contributions of type (b) involving closed electron two-point functions reduce the contributions of type (a) by about 50%. However, at three-loop order there is a new type of diagram where the external photon couples to a closed electron loop ($a_\mu^{(3b,\rm lbl)}$) which provides the largest individual contribution. This is in analogy to the three-loop QED corrections where the light-by-light type diagrams dominate the remaining contributions. In fact, due to $a_\mu^{(3b,\rm lbl)}$ the NNLO hadronic vacuum polarization contribution has a non-negligible impact. It has the same order of magnitude as the current uncertainty of the leading order hadronic contribution and should thus be included in future analyses.
An important contribution to $a_\mu$ is provided by the so-called hadronic light-by-light diagrams where the external photon is connected to the hadronic blob. The NLO part of this contribution is of the same perturbative order as the corrections in Eq. (\[eq::amuNNLO\]). A first-principle calculation of this part is currently not available, however, in [@Colangelo:2014qya] it has been estimated to $a_\mu^{\rm lbl-had,NLO}=0.3 \pm 0.2 \times 10^{-10}$.
We want to mention that there is a further hadronic contribution where four internal photons couple to the hadronic blob and the external photon couples to the muon line (“internal hadronic light-by-light”). This contribution, which is formally of the same perturbative order as $a_\mu^{\rm had,NNLO}$, is currently unknown.
\[sec::summary\]Summary and conclusions
=======================================
For more than a decade the measured and predicted results for the anomalous magnetic moment of the muon show a discrepancy of three to four standard deviations. This circumstance has triggered many publications which try to interpret the deviation with the help of beyond-SM theories. However, before drawing definite conclusions it is necessary to cross check the experimental result by performing an independent high-precision determination of $a_\mu$. Furthermore, all ingredients of the theory prediction should be computed by at least two groups independently.
In this contribution we describe the calculation of two classes of four-loop QED contributions to $a_\mu$, which up to date only have been computed by one group: the contribution involving tau leptons and the one involving light-by-light-type closed electron loops. Good agreement with the results in the literature is found. To complete the cross check of the four-loop result the non-light-by-light electron contribution, the diagrams involving simultaneously electrons and taus, and the pure-muon contribution have to be computed. From the technical point of view the missing diagram classes have the same complexity as those described in Sections \[sec::electron\] and \[sec::tau\].
As a further topic we have discussed in Section \[sec::hadr\] the calculation of the NNLO hadronic vacuum polarization contribution.
Acknowledgments {#acknowledgments .unnumbered}
===============
M.S. would like to thank the organizers of “Flavour changing and conserving processes 2015” for the pleasant atmosphere during the conference. We thank the High Performance Computing Center Stuttgart (HLRS) and the Supercomputing Center of Lomonosov Moscow State University [@LMSU] for providing computing time used for the numerical computations with [FIESTA]{}. P.M. was supported in part by the EU Network HIGGSTOOLS PITN-GA-2012-316704. This work was supported by the DFG through the SFB/TR 9 “Computational Particle Physics”. The work of V.S. was supported by the Alexander von Humboldt Foundation (Humboldt Forschungspreis).
G. W. Bennett [*et al.*]{} \[Muon g-2 Collaboration\], Phys. Rev. D [**73**]{} (2006) 072003 \[hep-ex/0602035\]. B. L. Roberts, Chin. Phys. C [**34**]{} (2010) 741 \[arXiv:1001.2898 \[hep-ex\]\]. R. M. Carey [*et al.*]{}, FERMILAB-PROPOSAL-0989. D. Hertzog, these proceedings.
B. Aubert [*et al.*]{} \[BaBar Collaboration\], Phys. Rev. Lett. [**103**]{} (2009) 231801 \[arXiv:0908.3589 \[hep-ex\]\]. F. Ambrosino [*et al.*]{} \[KLOE Collaboration\], Phys. Lett. B [**700**]{} (2011) 102 \[arXiv:1006.5313 \[hep-ex\]\]. S. Eidelman, these proceedings.
M. Della Morte, B. Jager, A. Juttner and H. Wittig, JHEP [**1203**]{} (2012) 055 \[arXiv:1112.2894 \[hep-lat\]\]. F. Burger [*et al.*]{} \[ETM Collaboration\], JHEP [**1402**]{} (2014) 099 \[arXiv:1308.4327 \[hep-lat\]\]. T. Blum, S. Chowdhury, M. Hayakawa and T. Izubuchi, Phys. Rev. Lett. [**114**]{} (2015) 1, 012001 \[arXiv:1407.2923 \[hep-lat\]\]. T. Blum, N. Christ, M. Hayakawa, T. Izubuchi, L. Jin and C. Lehner, arXiv:1510.07100 \[hep-lat\]. C. Lehner, these proceedings.
T. Aoyama, M. Hayakawa, T. Kinoshita and M. Nio, Phys. Rev. Lett. [**109**]{} (2012) 111808 \[arXiv:1205.5370 \[hep-ph\]\]. S. Laporta, Phys. Lett. B [**312**]{} (1993) 495 \[hep-ph/9306324\]. P. A. Baikov and D. J. Broadhurst, \[hep-ph/9504398\]. J. -P. Aguilar, D. Greynat and E. De Rafael, Phys. Rev. D [**77**]{} (2008) 093010 \[arXiv:0802.2618 \[hep-ph\]\]. R. Lee, P. Marquard, A. V. Smirnov, V. A. Smirnov and M. Steinhauser, JHEP [**1303**]{} (2013) 162 \[arXiv:1301.6481 \[hep-ph\]\]. P. A. Baikov, K. G. Chetyrkin, J. H. Kuhn, C. Sturm, Nucl. Phys. B [**867**]{} (2013) 182 \[arXiv:1207.2199 \[hep-ph\]\]. P. A. Baikov, A. Maier and P. Marquard, Nucl. Phys. B [**877**]{} (2013) 647 \[arXiv:1307.6105 \[hep-ph\]\]. A. Kurz, T. Liu, P. Marquard and M. Steinhauser, Nucl. Phys. B [**879**]{} (2014) 1 \[arXiv:1311.2471 \[hep-ph\]\]. A. Kurz, T. Liu, P. Marquard, A. V. Smirnov, V. A. Smirnov and M. Steinhauser, arXiv:1508.00901 \[hep-ph\]. A. Kurz, T. Liu, P. Marquard and M. Steinhauser, Phys. Lett. B [**734**]{} (2014) 144 \[arXiv:1403.6400 \[hep-ph\]\]. M. Beneke and V. A. Smirnov, Nucl. Phys. B [**522**]{} (1998) 321 \[hep-ph/9711391\]. V. A. Smirnov, [*Analytic tools for Feynman integrals*]{}, Springer Tracts Mod. Phys. [**250**]{} (2012) 1. A. Pak and A. Smirnov, Eur. Phys. J. C [**71**]{} (2011) 1626 \[arXiv:1011.4863 \[hep-ph\]\]. B. Jantzen, A. V. Smirnov and V. A. Smirnov, Eur. Phys. J. C [**72**]{} (2012) 2139 \[arXiv:1206.0546 \[hep-ph\]\]. A. V. Smirnov, Comput. Phys. Commun. [**185**]{} (2014) 2090 \[arXiv:1312.3186 \[hep-ph\]\]. T. Kinoshita and M. Nio, Phys. Rev. D [**70**]{} (2004) 113001 \[hep-ph/0402206\]. J. Calmet and A. Peterman, Phys. Lett. B [**56**]{} (1975) 383. C. Chlouber and M. A. Samuel, Phys. Rev. D [**16**]{} (1977) 3596. R. Harlander, T. Seidensticker and M. Steinhauser, Phys. Lett. B [**426**]{} (1998) 125, arXiv:hep-ph/9712228. T. Seidensticker, arXiv:hep-ph/9905298. S. J. Brodsky and E. De Rafael, Phys. Rev. [**168**]{} (1968) 1620. S. Eidelman and F. Jegerlehner, Z. Phys. C [**67**]{} (1995) 585 \[hep-ph/9502298\]. M. Davier, A. Hoecker, B. Malaescu and Z. Zhang, Eur. Phys. J. C [**71**]{} (2011) 1515 \[Erratum-ibid. C [**72**]{} (2012) 1874\] \[arXiv:1010.4180 \[hep-ph\]\]. K. Hagiwara, R. Liao, A. D. Martin, D. Nomura and T. Teubner, J. Phys. G [**38**]{} (2011) 085003 \[arXiv:1105.3149 \[hep-ph\]\]. F. Jegerlehner and R. Szafron, Eur. Phys. J. C [**71**]{} (2011) 1632 \[arXiv:1101.2872 \[hep-ph\]\]. M. Benayoun, P. David, L. DelBuono and F. Jegerlehner, Eur. Phys. J. C [**73**]{} (2013) 2453 \[arXiv:1210.7184 \[hep-ph\]\]. F. Jegerlehner, these proceedings, arXiv:1511.04473 \[hep-ph\]. B. Krause, Phys. Lett. B [**390**]{} (1997) 392 \[hep-ph/9607259\]. D. Greynat and E. de Rafael, JHEP [**1207**]{} (2012) 020 \[arXiv:1204.3029 \[hep-ph\]\]. K. Hagiwara, A. D. Martin, D. Nomura and T. Teubner, Phys. Rev. D [**69**]{} (2004) 093003 \[hep-ph/0312250\]. G. Colangelo, M. Hoferichter, A. Nyffeler, M. Passera and P. Stoffer, Phys. Lett. B [**735**]{} (2014) 90 \[arXiv:1403.7512 \[hep-ph\]\]. V. Sadovnichy, A. Tikhonravov, V. Voevodin, and V. Opanasenko, “‘Lomonosov’: Supercomputing at Moscow State University.” In Contemporary High Performance Computing: From Petascale toward Exascale (Chapman & Hall/CRC Computational Science), pp.283-307, Boca Raton, USA, CRC Press, 2013.
[^1]:
[^2]:
[^3]:
[^4]:
[^5]:
[^6]:
[^7]: Partial four-loop corrections have been obtained in Refs. [@Laporta:1993ds; @Baikov:1995ui; @Aguilar:2008qj; @Lee:2013sx; @Baikov:2012rr; @Baikov:2013ula].
|
---
abstract: |
Multi robot systems have the potential to be utilized in a variety of applications. In most of the previous works, the trajectory generation for multi robot systems is implemented in known environments. To overcome that we present an online trajectory optimization algorithm that utilizes communication of robots’ current states to account to the other robots while using local object based maps for identifying obstacles. Based upon this data, we predict the trajectory expected to be traversed by the robots and utilize that to avoid collisions by formulating regions of free space that the robot can be without colliding with other robots and obstacles. A trajectory is optimized constraining the robot to remain within this region.The proposed method is tested in simulations on Gazebo using ROS.\
*Keywords*- Multi-Robot System, Collision Avoidance, Trajectory Optimization
author:
- 'Vijay Arvindh, Govind Aadithya R, Shravan Krishnan and Sivanathan K [^1] [^2] [^3] [^4]'
bibliography:
- 'references.bib'
- 'refer.bib'
title: '**Collision-Free Multi Robot Trajectory Optimization in Unknown Environments using Decentralized Trajectory Planning**'
---
Introduction
============
Multi-robots systems are a group of individual entities working together so as to maximize their own performance while accounting for some higher goals. The trajectories generated in such scenarios will have to ensure that the robots don’t collide with one another while keeping up with their dynamic limits. The trajectory generation process in multi agent systems has long since been done in a centralized manner wherein the trajectories are generated before hand and transmitted across to individual robots. These methods show excellent performance and safety when the number of robots are known beforehand and limited in number as scalability is a problem in them. Recently, this has branched out to decentralized approaches that attempt to plan trajectories in known environments using a variety of different approaches, but it is important to mitigate disturbances and unmodeled errors by modifying the trajectories during run time, making it more like an implicit feedback.
![Six AscTec Fireflys and two Neos at an Intersection-like Environment[]{data-label="firefly6"}](Firefly){width="50.00000%"}
Our approach attempts to solve this problem using two steps, i) Collision-free regions that the robot can be safe within to avoid collisions with other robots and ii) Generate a trajectory for the robot constraining it within the safe region. Other obstacles also have to be considered while planning the trajectory. To achieve this the obstacles are stored using primitive model representations and used as soft constraint for the objective with the obstacles approximated as circles of appropriate radii. The approach exploits differential flatness [@fliess1995flatness] based trajectory generation for n^th^ order systems property of robots [@mellinger2011minimum], [@optimalwalambe2016]. and constrains the generated trajectory to stay within a safe region at specified discrete time points in two dimensional environments.
The contributions can be stated as
1. A decentralized trajectory optimization algorithm for multi robot systems.
2. A simple method for obstacle representation in 2D environments based on Lidar data under assumption of no uncertainty
3. Extensive simulation experiments of the proposed algorithm
The algorithm requires a sharing of data amongst the agents and assumes that the robots are equipped with depth sensors (RADAR,LiDAR). An advantage of the proposed algorithm is a continuous time parametrization of the trajectory generation problem with discrete inter-robot collision avoidance. Moreover,the trajectory optimization is solved as a convex optimization problem.
The rest of the paper is organised as: Related works are presented in Section \[Related works\]. A formal problem definition and assumptions are provided in Section \[Problem formulation\]. Section \[Convex\] details the formulation of the safe region following while the obstacle representation is explained in \[local map\]. The trajectory optimisation formulation is given in Section \[trajectory generation\]. The results are discussed in Section \[Results\] The paper is concluded in Section \[Conclusions\].
Related Work {#Related works}
============
In [@tang2018hold], a centralized multi robot trajectory planner for obstacle free environments was proposed utilizing tools from non linear optimization and calculus of variation and exploiting a two step process with first step accounting for the collision avoidance. A centralized mixed integer programming based approach to multi robot path planning was proposed by [@schouweenars2001mixed] with collision avoidance accounted for using binary integer constraints.
In [@sutorius2017decentralized], a decentralized method was proposed using polygonal representations for obstacles utilizing a switched systems to achieve collision-avoidance. Generating regions of free space for the robots was attempted using voronoi cells was tried out by [@2017fast]. They utilize a Receding Horizon control based approach which they are able to formulate as a convex quadratic program.
Distributed collision avoidance for multi robot systems have also been attempted [@Berg2008] in a method called reciprocal velocity obstacle, exploiting the concept of Velocity obstacles proposed by [@Fiorini1998]. This approach assumed other agents continued movement in a straight line with collisions accounted in future by relative velocities.[@mora2018cooperative] proposed a collaborative collision avoidance for non holonomic robots with grid based mapped environments whilst respecting the vehicular constraints and also accounting for potential tracking error bounds for the robot that is planning.
A fully distributed algorithm for navigation in unknown environments was proposed by [@Zhou2017real] using incremental sequential convex optimisation for trajectory generation in a model predictive control setting. Distributed re-planning for multiple robots with each robots having different planning cycles in known obstacle filled environments was attempted by [@Bekris2017Safe].The transmission of previously generated trajectories and plan trajectories while avoiding collisions with these trajectories and utilize conservative approximations to account for deviations from these expected trajectories by the other robots.
Problem formulation {#Problem formulation}
===================
Consider $N \in \mathbb{N} $ robots in a 2 dimensional workspace with an unknown number of obstacles and their sizes. The position of i^th^ robot is represented by $ P_i \in \mathbb{R}^2 $. Each of the N robots have a set of time stamped desired poses. We also assume every robot gets to know the current time stamped state of other robots in the vicinity at frequent intervals. We take the state as $x_i = [P_i \hspace{2mm} v_i \hspace{2mm} a_i ]$ . Where $ v_i$ is the velocity of i^th^ along the two axes, $ a_i$ is acceleration of i^th^ robot. Therefore, a common reference frame for all the robots is a requisite. Then it is required that the robots go from their current positions through all their desired poses at as close as possible to the specified timestamps as possible. That is for all the robots, a trajectory is to be planned that ensures that robot traverses from its current position to a within a region from the desired position within the specified time while not colliding with any of the other robots and/or obstacles in the environment.
Furthermore, we assume that each robot is equipped with a rangefinding based sensor that can give the depth information of the obstacles and that the depth sensor only perceives obstacles within a sensing region. We also assume that the robots do not know the number of robots in the environment and their desired poses and just utilize the states received by them for forward prediction.
The constraints on the overall system are:
1. The current state of all the robots
2. The states remain within the feasible set of the respective robots
3. The positions of any robots at any time from $t_1$ to $t_2$ should not coincide.(Inter-Agent Collision Avoidance)
4. The position of an obstacle and a robot should not overlap (Obstacle Avoidance)
Obstacle Detection {#local map}
==================
A local map representation is implemented for obstacles in the environment. The robot is assumed to be employed with a 2D laser range finder, whose data is utilized to find the shape of the object and its center. The rangefinder data from the lidar provides the distance of the obstacles.
The detection of reflection pattern is achieved by using a search through the data that is available in the lidar, that isn’t infinity(no reflection back). The set of reflections together between a non reflection form a single obstacle. as the resolution of a lidar is known beforehand and hence, using the resolution and distance to the obstacle from the robots current position, the obstacle point’s position can be comprehended by :
$$\begin{split}
x_{obs}= l*cos(\theta) + P_{x_{ego}} \\
y_{obs}= l*sin(\theta) + P_{y_{ego}}
\end{split}
\label{distance}$$
This results in a primitive but a simple structure of the robot’s obstacles to be noted at. The selection of circle is motivated by the reason that higher sided convex polygons can be approximated accurately by circles of appropriate radius. Moreover, it is easier to approximate from observing a shape partially.
The robot’s obstacles with a threshold formed by their radius are utilized for formulating the obstacle avoidance. That is the obstacle’s center and it’s radius are used for obstacle from the lidar data. This method while being primitive, allows for simpler obstacle representations for a robot functioning in a 2 dimensional environment. Hence, the obstacles can be easily stored as with their center and sizes. It also has a drawback of over approximating the obstacles at times by formulating drastic radii.
Convex Region {#Convex}
=============
For the formulation of the safe regions for the robots to plan trajectories within, We forward simulate other robot’s trajectories utilizing their current states.
Forward Simulation
------------------
Forward simulation is done utilizing the current state of the robot. The forward simulation of the robots is done by:
$$P_i(t) = P_i(t_\delta) + v_i(t_\delta)(t-t_\delta) + a_i(t_\delta)(t-t_\delta)^2
\label{prediction}$$
where $ t_\delta $ is the time stamp of the robot’s transmitted state.
The forward simulation is done for specified time horizon $t_h$ based upon the discretization $\tau$.
Accommodating the size of robots {#size}
--------------------------------
At each of the discretized time points, utilizing the transmitted size of the robots, we formulate regions depending on the transmitted data. For simplicity that if three numbers are sent, The robot is modeled as a cuboid or a cube, two numbers a cylinder and a single number as a sphere. In the case of three numbers, robots are modeled as a square of diagonal of largest side, thereby allowing the robot to rotate freely. In case of two or one, the robot is modeled as a circle. A robot inflated to its size is represented as $\mathsf{R}_i$ . $$\mathsf{R}_i =
\begin{cases}
A_i P_i(t) \leq B & Square \\
{\left\lVert{P}_i(t)-\textbf{P}_i(t)\right\rVert} \leq r & Circle
\end{cases}$$
Where $B $ is formulated as $P_i(t) \pm \sqrt[]{2}\max(l)$ and $r$ is the radius of the robot
{width="50.00000%"} \[convex region\]
\[euclid\]
**Given:** Agents’ current state and size Number of robots (N) Moving Volume at specified time points
$P_i(t) = P_i(t_\delta) + v_i(t_\delta)(t-t_\delta) + a_i(t_\delta)(t-t_\delta)^2$ Approximate size of the robots according to \[size\] Get the intersection of hyperplanes
Safe Regions
Hyperplane
----------
Considering a time $t_\tau $ between $t$ and $t_h$, supporting hyperplanes are formulated for all $\mathsf{R}_i$ that are within their appropriate moving volumes and as $\mathsf{R}_i$ convex and a supporting hyperplane hence, exists. [@boyd2004convex]. A support hyperplane for $\mathsf{R}_i$ can be formulated as $$\eta_i(\mathsf{R}_i) \leq \eta_i(\mathfrak{r}_i^0)$$ where $\mathfrak{r}_i^0$ is the boundary of the set $ \mathsf{R}_i $
The intersection of all the support hyperplanes is convex polyhedron as an intersection of convex hyperplane is convex. We Constrain the robot to remain within the generated polyhedron at the discretized time points with polyhedron at each time point represented as $\mathcal{H}(t_{disc}) \leq h $. But as this region constrains the overall robot as a point we subtract the robot’s dimensions from the $h$ to constrain the robot to be within the region. the convex regions at specific time points. As the robot’s modeled so that it’s size is invariant to rotation, dilating the robot according to it’s shape of the robot.
Trajectory Generation {#trajectory generation}
=====================
The generation of trajectory by the robot can be formulated as an optimization problem that tries to optimize the system while ensuring that
[***x***]{}[C\_[int]{} + C\_[final]{} + C\_[collision]{} ]{} \[cost functional\]
While the above mentioned problem is a continuous time problem and thereby a infinite dimensional problem. To transform the problem into a finite dimension, the trajectory is represented by polynomial in each dimension.
$$x(t)= \sum_{j=0}^{2n-1}\alpha_j t^j\label{decision with degree}$$
$$\mathcal{D}_i = [\alpha_0 \hspace{1mm} \alpha_1 \hspace{1mm} \alpha_2 \hspace{1mm} \cdots \alpha_{2n-1}]^T
\label{per_vehicle}$$
Objective Function
------------------
$C_{int}$ is the integral cost functional that specifies the objective for the derivative over the integral. $C_{final}$ is the cost at the end of the time horizon. $ C_{collision} $ is the collision cost for static obstacles along the trajectory.
### Derivative Cost
Derivative penalty utilized is square of the integral of n^th^ derivative and n-1^th^ squared over time horizon t~h~: It is represented as: $$C_{int}=\int_{t}^{t_h} Q_{\text{n-1\textsuperscript{th}}}{\left\lVert\frac{d^{n-1}\textbf{\textit{x}}}{dt^{n-1}}\right\rVert}^2 + Q_{\text{n\textsuperscript{th}}} {\left\lVert\frac{d^{n}\textbf{\textit{x}}}{dt^{n}} \right\rVert}^2 \hspace{1mm} dt
\label{Derivative Cost}$$
Where $Q_{\text{n\textsuperscript{th}}}, Q_{\text{n-1\textsuperscript{th}}}$ are tuning weights for the objective.
As the time horizon is known before hand and the initial time and position are known, We can formulate this cost with a closed form solution as a Quadratic Objective with the decision vector as: $$D^T H(t+t_h) D$$ With $H(t+t_h)$ formulated by integrating Equation \[Derivative Cost\] and substituting $ t$, $t_h $ and separating according to the coefficients of the polynomial.
### End Cost
We add an end point quadratic cost for the final position along the trajectory as a soft constraint for two reasons. One, to allow to robot to plan appropriate trajectory if a robot or an obstacle is occupying or blocking the path directly to the end point. Two, in scenarios where the robot’s end pose’s time stamp is beyond it’s trajectory planning horizon, this cost tries to drive the robot as close as possible to it, while ensuring the dynamic limits aren’t violated by the hard dynamic constraint
$$C_{end}=(x_{des}-\textbf{\textit{x}}(t_h))^2 Q_{final}$$
The final position is penalized but if required addition penalties on velocity, acceleration can be added. The cost can be reformulated with respect to the decision variables resulting in:
$$D^T H(\text{Fin}) D + F(\text{Fin})^T D$$
### Collision Cost
For the generated trajectory to be collision free with respect to the obstacles in the environment,the following penalty is used:
$$C_{collision}=\int_{t}^{t_h} Q_{Obs}c(x(t))v(t)\hspace{1mm} dt
\label{Obstacle Cost}$$
Where $$c(x) = \frac{x(t)-x_{obs}}{\exp^{K_p(d(x)-\rho)}d(x)}
\label{Pieceswise obstacle}$$
Here $d(x)$ is euclidean distance to each obstacle. Similar cost functions for collision avoidance have been utilized for collision avoidance for autonomous cars [@pmpc], aerial robots [@rob].$ K_p$ is a smoothness tuning parameter that allows to increase or decrease the smoothness of the collision cost.
For efficient optimization, a quadratic approximation of the obstacle cost around the previous optimized trajectory is used resulting in:
$$D^T H(\text{Obs}) D + F(\text{Obs})^T D$$
Constraints
-----------
The trajectory optimisation is constrained by the derivatives of the trajectory staying within the feasible limits(Dynamic Constraints), Staying within the convex region and way-points.
### Waypoint constraints
The trajectory has to also pass through the given time stamped poses along the trajectory, This results in linear equality constraints on the trajectory.
$$A_{way}D=P$$
where $P$ is the stack of poses at their time.
Moreover, as the end pose’s time is given in this scenario, the polynomial while minimizing the costs only for the specific time horizon also ensures that the robot reaches the end goal at the desired time stamp
### Convex Region {#convex-region}
We require that the generated trajectory also remains within the feasible convex region generated at the specific time samples. This constrain is formulated as $$\sum_{t_{disc}=1}^{\frac{t_h}{\tau}}\mathcal{H}_i(t_{disc}) \mathcal{T}_i D \leq h$$
where $\mathcal{T}$ is the map from the polynomial coefficients to the positions. As both $\mathcal{H} \& \mathcal{T} $ are linear with the polynomial coefficients this results in a convex constraint.
### Dynamic constraints
Dynamic constraint on the robot is an infinite dimensional and hence we apply the constrains at specific points on the trajectory. This results in added Inequality constrain on the system
$$\sum_{i=1}^n \underline{d} \leq A_{dyn}D \leq \bar{d}$$
Where $n$ is the number of discrete points wherein the constraints are added and $\underline{d} \& \bar{d}$ represent the minimal and maximal limits of the derivatives.
![The success rates of the algorithm for Turtlebot in the intersection like environment[]{data-label="graph10"}](Turtlebot){width="50.00000%"}
The resulting optimization problem can be formulated as a Non Linear Program
[D]{}[D\^T H\_[net]{} D + F\_[net]{}\^T D]{} \[trajectory NLP\]
Where $H_{net}$ is formulated by $H(\text{Obs}) + H(\text{Fin})+H(t+t_h)$ and $F_{net}$ by $F(\text{Fin})+ F(\text{Obs})$
The Non Linear Program in Equation \[trajectory NLP\] is a Convex QP and can be solved using available solvers
Results and Discussion {#Results}
======================
The algorithm was implemented in C++ and integrated into Robot Operating System(ROS) and tested on a workstation with Intel Xeon E5 1630v5 processor, 32GB of RAM and a Nvidia Quadro M4000 GPU. A degree of $2n$-1 was utilized for the polynomials with $t_{h}$ being three seconds. We utilized a $\tau$ of 0.1 seconds for the other agents’ prediction. The threshold distance for the obstacle is kept at 0.75m.The algorithm was run at frequency of 25Hz. For solving the QP, qpOASES[@qpoases] was used.
To handle infeasible QPs that arise due to the inequality constraints, We utilize a two step process for the same. In the initial step, we apply the previous solution for some time and in second step relax the dynamic constraints.
The proposed algorithm was tested with two sets of robots one a Turtlebot 3 Burger and Waffle (utilizing the native sensor suite available on these robots) and two, AscTec Firefly and Neo using the a high fidelity simulator [@rotors] with robots mounted with Velodyne Puck. The weight of the Velodyne Puck was modified to allow firefly to fly with it. Moreover, Velodyne Puck is a 3D sensor but we limited the sensing to a 2D region. The experiments included both homogeneous and heterogeneous interactions in an intersection like environment, a cross road-like structure which was formed with the help of walls. We utilize an intersection-like environment as it is an important usage of the labeled multi robot problem.
The robots were spawned randomly. The desired timestamped poses are predefined in for every robot. For the multirotors, the altitude was fixed at 1. For the ground robots, the generated trajectory is tracked using the MPC proposed in [@mpc] which was solved using qpOASES [@qpoases] with a time horizon of 2 secs and a discretization of 10Hz. The aerial robots tracked trajectories with the controller proposed in [@taecontroller]. The failure rates are shown in Fig \[graph10\] and Fig \[graph6\]. The trajectories of Aerial robots at the intersection-like environment is shown in Fig \[traj\].
![The Success rates of the algorithm for Aerial Robots in the intersection like environment[]{data-label="graph6"}](aerial){width="50.00000%"}
Discussions
-----------
During the course of simulation, in a few experiments, the robots collided with the obstacles. The collisions in some cases are due to the inaccurate representation of the obstacles and also the unaccounted sensors noise. Moreover, the LIDAR measurements are available at 5Hz whereas the trajectory optimization algorithm runs at 25Hz. The usage of a single polynomial is also a potential cause for this problem. The algorithm also shows some unnecessary non smoothness in its transition towards the final end pose. The safe region generation is conservative but also results in available free-space being neglected. This at times results in the QP being infeasible.
Conclusion {#Conclusions}
==========
A decentralized algorithm for collision free navigation of multiple robots in unknown two dimensional environments was proposed in this work. The proposed algorithm parametrized trajectories by a time parameterized polynomial and generated safe regions based on the prediction of other robots in the environment. A method for obstacle representation was utilized that allowed for simpler collision avoidance with the obstacles. The proposed method was tested extensively in simulations using gazebo for up to eight aerial robots and ten turtlebots.
Utilizing piecewise spline representations of non uniform B-Splines or bezier curves are another avenue for research. The collision representation is a discrete time representation and a continuous time representation of the collision is an avenue for future research. Moreover, sophisticated models of prediction can be utilized for a better prediction accuracy. The sensors used had low update rates. Utilizing faster sensors or utilizing RGB-D or Stereo cameras for local maps is an another important possible extension. Moreover, incorporating a higher variety of primitives for the obstacles will allow for a much more accurate obstacle representation. Another avenue for future research would be test the capabilities of the algorithm in an unstructured environment.
Supplementary Material {#supplementary-material .unnumbered}
======================
The experiments in Gazebo can be found at <https://youtu.be/wGu0GMOTeH8>\
and <https://www.youtube.com/watch?v=JRrxJCXMD_I>
[^1]: This work was supported by SRM Institute of Science and Technology
[^2]: The authors are with Autonomous Systems Lab, SRM Institute of Science and Technology,India
[^3]: Email:*vijay\_arvindh,govinda\_adithya,shravan\_krishnan)@srmuniv.edu.in, sivanathan.k@ktr.srmuniv.ac.in*
[^4]: *Corresponding Author: Shravan Krishnan*
|
---
abstract: 'We propose (), a framework for semantically parsing documents in specific domains. Basically, reads a document and parses it into a predesigned object-oriented data structure (referred to as *ontology* in this paper) that reflects the domain-specific semantics of the document. An parser models semantic parsing as a decision process: a neural net-based Reader sequentially goes through the document, and during the process it builds and updates an intermediate ontology to summarize its partial understanding of the text it covers. supports a rich family of operations (both symbolic and differentiable) for composing the ontology, and a big variety of forms (both symbolic and differentiable) for representing the state and the document. An parser can be trained with supervision of different forms and strength, including supervised learning (SL) , reinforcement learning (RL) and hybrid of the two. Our experiments on both synthetic and real-world document parsing tasks have shown that can learn to handle fairly complicated ontology with training data of modest sizes.'
author:
- |
Zhengdong Lu$^1$ Haotian Cui$^2$\* Xianggen Liu$^2$\* Yukun Yan$^2$\* Daqi Zheng$^{1}$\
$^1$DeeplyCurious.ai\
[{luz,da}@deeplycurious.ai]{}\
$^2$Department of Bio-medical Engineering, Tsinghua University\
[{cht15, liuxg16, yanyk13}@mails.tsinghua.edu.cn]{}\
bibliography:
- 'cite.bib'
title: |
Object-oriented Neural Programming (OONP)\
for Document Understanding
---
|
Photometry of millions of stars collected during the regular long term monitoring of the targets of the Optical Gravitational Lensing Experiment (OGLE) is a unique observational material that can be used for many astrophysical applications. Since the second phase, OGLE-II, the OGLE group has regularly released the OGLE Photometric Maps of Dense Stellar Regions containing calibrated [*BVI*]{} or [*VI*]{} photometry and precise astrometry of millions of stars from the observed fields. They included astrophysically important objects like the Large and Small Magellanic Clouds and the Galactic Center.
The most recent version of the OGLE Maps comes from the third phase of the OGLE project – OGLE-III. OGLE-III observations covered an area of the sky larger by an order of magnitude as compared to the original OGLE-II maps, and contain photometry of about ten times more stars. So far OGLE-III Maps of the Large and Small Magellanic Clouds were released (Udalski 2008ab).
OGLE-III maps have already been widely used by astronomers to many projects (Subramanian and Subramaniam 2010, Szczygie³ 2010). They are also widely used as a huge set of secondary photometric standards for calibrating photometry.
The OGLE-III target list included a set of fields from the Galactic disk. These fields located in dense stellar regions at low Galactic lattitude and longitudes between $280\arcd$ and $310\arcd$ were extensively monitored with high cadence (order of 15 minutes) for low luminosity object and planetary transits leading to the discovery of the first transiting exoplanets (Udalski 2002, Bouchy 2004).
Precise photometry of the Galactic disk fields can be a very useful tool for studying the Galactic structure. Large area of the sky around the Galactic equator has been currently monitored by the OGLE-IV survey in the optical, [*VI*]{}, domain, as well as in the near infrared by the VVV project conducted on the VISTA telescope at ESO Paranal Observatory, Chile. OGLE-III photometry of selected Galactic disk fields can be then a good anchoring point for these larger scale surveys of the Galactic disk.
This paper is the next in the OGLE-III Map series. We present here OGLE-III photometric maps of the Galactic disk fields covering about 7 square degrees in the sky and containing photometry and astrometry of about 9 million stars. The maps are available to the astrophysical community from the OGLE Internet archive. The photometric data presented in this paper were collected during the OGLE-III phase between February 2002 and May 2009 with the 1.3-m Warsaw Telescope at Las Campanas Observatory, Chile, operated by the Carnegie Institution of Washington. The telescope was equipped with the eight chip mosaic camera (Udalski 2003) covering approximately $35\times35$ arcmin in the sky with the scale of 0.26 arcsec/pixel.
Observations were carried out in [*V*]{}- and [*I*]{}-band filters closely resembling the standard bands. One should be however aware, that the OGLE glass [*I*]{}-band filter approximates well the standard one for $V-I<3$ mag colors. For very red objects the transformation to the standard band is less precise. The vast majority of observations were obtained through the [*I*]{}-band filter. Typically up to $\sim2700$ images for each field were collected in this band and just a few in the [*V*]{}-band. The exposure time was 180 s or 120 s for the [*I*]{}-band and 240 s for the [*V*]{}-band.
Observations were conducted only in good seeing (less than 18) and transparency conditions. The median seeing of the [*I*]{}-band images is equal to $1\zdot\arcs2$.
The Galactic disk fields observed during OGLE-III phase as well as the equatorial and galactic coordinates of their centers and number of stars detected in the [*I*]{}-band are listed in Table 1. The area of the sky covered by OGLE-III observations of these fields exceeds 7 square degrees.
[ccccrr]{} Field & RA & DEC & $l_{II}$ & & $N_{\rm Stars}$\
& (2000) & (2000) & & &\
CAR100 & 110700& $-61$0630& 2906544 & $-0$7510 & 382528\
CAR104 & 105730& $-61$4000& 2898439 & $-1$7249 & 475893\
CAR105 & 105220& $-61$4000& 2892911 & $-1$9906 & 459781\
CAR106 & 110300& $-61$5000& 2905054 & $-1$6063 & 401528\
CAR107 & 104715& $-62$0025& 2889089 & $-2$5647 & 318932\
CAR108 & 104715& $-61$2435& 2886343 & $-2$0343 & 379610\
CAR109 & 104210& $-62$1025& 2884607 & $-2$9904 & 307029\
CAR110 & 104215& $-61$3435& 2881846 & $-2$4606 & 371079\
CAR111 & 104715& $-60$4845& 2883599 & $-1$5037 & 334938\
CAR112 & 105220& $-61$0415& 2890276 & $-1$4562 & 370525\
CAR113 & 105720& $-61$0408& 2895717 & $-1$1922 & 369094\
CAR114 & 105720& $-60$2818& 2893178 & $-0$6516 & 344169\
CAR115 & 104030& $-62$0900& 2882783 & $-3$0629 & 349173\
CAR116 & 103700& $-62$4500& 2882176 & $-3$7841 & 299996\
CAR117 & 104205& $-62$4500& 2887274 & $-3$5017 & 316077\
CAR118 & 103830& $-63$2050& 2886602 & $-4$2207 & 261809\
CEN106 & 113230& $-60$5000& 2934598 & $ 0$5784 & 500453\
CEN107 & 115400& $-62$0000& 2962458 & $ 0$1238 & 557251\
CEN108 & 133300& $-64$1500& 3074281 & $-1$7417 & 845638\
MUS100 & 131500& $-64$5100& 3054335 & $-2$0928 & 735652\
MUS101 & 132500& $-64$5800& 3064749 & $-2$3261 & 766910\
The construction of the OGLE-III maps was presented in detail in Udalski (2008a). We followed this procedure for the Galactic disk fields as well.
The OGLE-III maps contain the mean photometry of all detected stellar objects. The mean photometry was obtained for all objects with minimum of 6 observations in the [*I*]{}-band by averaging all observations after removing $5\sigma$ outliers. Because of small number of [*V*]{}-band epochs for some of the fields even a single [*V*]{}-band observation entered the database. In the case of more [*V*]{}-band data points they were also averaged with the $5\sigma$ outliers rejection.
[ r@ c@ c@ r@ c@ c@ r@ c@ c@ r@ c@ r@ c@ c]{} ID & RA & DEC & $X$ & $Y$ & $V$ & $V-I$ & $I$ & $N_V$ & $N^{\rm bad}_V$ & $\sigma_V$ & $N_I$ & $N^{\rm bad}_I$ & $\sigma_I$\
&(2000) & (2000)&&&&&&&&&&&\
1 & 11070450 & $-61\arcd13\arcm44\zdot\arcs5$ & 456.05 & 53.37 & 15.624 & 1.631 & 13.993 & 3 & 0 & 0.001 & 1749 & 0 & 0.008\
2 & 11070655 & $-61\arcd13\arcm43\zdot\arcs3$ & 460.44 & 110.06 & 14.076 & 0.507 & 13.568 & 3 & 0 & 0.001 & 2533 & 10 & 0.009\
3 & 11070711 & $-61\arcd15\arcm00\zdot\arcs8$ & 162.74 & 123.92 & 18.553 & 4.137 & 14.416 & 3 & 0 & 0.022 & 2547 & 1 & 0.015\
4 & 11070869 & $-61\arcd13\arcm25\zdot\arcs4$ & 528.71 & 169.98 & 14.647 & 0.842 & 13.805 & 3 & 0 & 0.003 & 2667 & 12 & 0.011\
5 & 11071078 & $-61\arcd13\arcm26\zdot\arcs0$ & 525.92 & 227.89 & 14.392 & 9.999 & 99.999 & 3 & 0 & 0.003 & 0 & 0 & 9.999\
6 & 11071268 & $-61\arcd14\arcm47\zdot\arcs5$ & 213.08 & 278.48 & 16.273 & 2.599 & 13.674 & 3 & 0 & 0.007 & 2684 & 0 & 0.010\
7 & 11071388 & $-61\arcd14\arcm17\zdot\arcs0$ & 329.61 & 312.49 & 14.197 & 0.420 & 13.777 & 3 & 0 & 0.003 & 2682 & 2 & 0.006\
8 & 11071527 & $-61\arcd15\arcm09\zdot\arcs3$ & 129.01 & 349.62 & 15.406 & 1.741 & 13.665 & 3 & 0 & 0.002 & 2514 & 2 & 0.005\
9 & 11071633 & $-61\arcd15\arcm27\zdot\arcs8$ & 57.57 & 378.53 & 14.735 & 9.999 & 99.999 & 3 & 0 & 0.004 & 0 & 0 & 9.999\
10 & 11070624 & $-61\arcd15\arcm01\zdot\arcs8$ & 159.36 & 99.77 & 15.930 & 1.116 & 14.814 & 3 & 0 & 0.004 & 2371 & 3 & 0.007\
11 & 11070673 & $-61\arcd13\arcm47\zdot\arcs6$ & 443.65 & 115.12 & 16.710 & 1.101 & 15.609 & 3 & 0 & 0.009 & 2561 & 6 & 0.007\
12 & 11071087 & $-61\arcd14\arcm07\zdot\arcs8$ & 365.50 & 229.39 & 16.291 & 0.826 & 15.465 & 3 & 0 & 0.008 & 2681 & 3 & 0.006\
13 & 11071111 & $-61\arcd13\arcm57\zdot\arcs4$ & 405.35 & 236.04 & 16.061 & 0.980 & 15.081 & 3 & 0 & 0.005 & 2684 & 0 & 0.005\
14 & 11071169 & $-61\arcd13\arcm56\zdot\arcs7$ & 408.00 & 252.21 & 16.296 & 1.001 & 15.296 & 3 & 0 & 0.004 & 2674 & 10 & 0.009\
15 & 11071418 & $-61\arcd15\arcm20\zdot\arcs2$ & 87.43 & 318.99 & 17.938 & 2.442 & 15.496 & 3 & 0 & 0.002 & 2010 & 0 & 0.177\
16 & 11071533 & $-61\arcd15\arcm02\zdot\arcs8$ & 153.89 & 351.27 & 16.080 & 0.960 & 15.120 & 3 & 0 & 0.003 & 2603 & 0 & 0.006\
17 & 11071537 & $-61\arcd13\arcm27\zdot\arcs2$ & 520.71 & 354.97 & 15.932 & 0.926 & 15.006 & 3 & 0 & 0.005 & 2683 & 1 & 0.007\
18 & 11071708 & $-61\arcd14\arcm20\zdot\arcs9$ & 314.19 & 400.92 & 17.147 & 1.461 & 15.686 & 3 & 0 & 0.002 & 2683 & 1 & 0.007\
19 & 11071914 & $-61\arcd15\arcm13\zdot\arcs8$ & 111.06 & 456.46 & 16.629 & 1.004 & 15.625 & 3 & 0 & 0.004 & 2388 & 4 & 0.008\
20 & 11072006 & $-61\arcd13\arcm53\zdot\arcs3$ & 419.50 & 484.11 & 16.996 & 1.524 & 15.472 & 3 & 0 & 0.006 & 2682 & 2 & 0.006\
21 & 11072039 & $-61\arcd13\arcm36\zdot\arcs2$ & 485.25 & 493.84 & 16.486 & 0.898 & 15.587 & 3 & 0 & 0.002 & 2683 & 0 & 0.007\
22 & 11072054 & $-61\arcd14\arcm57\zdot\arcs2$ & 174.38 & 495.87 & 16.584 & 1.164 & 15.420 & 3 & 0 & 0.005 & 2653 & 1 & 0.008\
23 & 11072110 & $-61\arcd14\arcm19\zdot\arcs4$ & 319.17 & 512.36 & 16.438 & 0.844 & 15.593 & 3 & 0 & 0.005 & 2681 & 3 & 0.008\
24 & 11070420 & $-61\arcd15\arcm24\zdot\arcs0$ & 74.40 & 42.86 & 16.865 & 9.999 & 99.999 & 3 & 0 & 0.007 & 0 & 0 & 9.999\
25 & 11070422 & $-61\arcd14\arcm05\zdot\arcs1$ & 377.10 & 45.19 & 17.009 & 1.081 & 15.928 & 3 & 0 & 0.009 & 1589 & 0 & 0.009\
In Table 2 we present a sample of the data – the first 25 entries from the map of the CAR100.2 subfield. The columns contain: (1) ID number; (2,3) equatorial coordinates J2000.0; (4,5) $X,Y$ pixel coordinates in the [*I*]{}-band reference image; (6,7,8) photometry: [*V*]{}, $V-I$, [*I*]{}; (9,10,11) number of points for average magnitude, number of $5\sigma$ removed points, $\sigma$ of magnitude for [*V*]{}-band; (12,13,14) same as (9,10,11) for the [*I*]{}-band. 9.999 or 99.999 markers mean “no data”. Value of $-1$ in column (9) indicates multiple [*V*]{}-band cross-identification (when the [*V*]{}-band counterpart was detected in more than one field in overlaping areas; the average magnitude is the mean of the merged datasets).
The full set of the OGLE-III Photometric Maps of the Galactic Disk Fields is available from the OGLE Internet archive (see Section 5).
The OGLE-III Photometric Maps of the Galactic Disk Fields contain entries for about 9 million stars located in 21 OGLE-III fields. Figs. 1 and 2 show the typical accuracy of the OGLE-III Photometric Maps of these targets: standard deviation of magnitudes as a function of magnitude in the [*V*]{}- and [*I*]{}-band for the field CAR100.2 and denser MUS100.2.
The completeness of the photometry can be assessed from the histograms presented in Fig. 3 and 4 for the same fields as in Figs. 1 and 2. It reaches $I\approx20.5$ mag and $V\approx 21$ mag.
Figs. 5–8 present color–magnitude diagrams (CMDs) constructed for a few selected subfields from different Galactic disk fields observed by OGLE-III survey.
The OGLE-III Photometric Maps of the SMC are available to the astronomical community from the OGLE Internet Archive:
[*http://ogle.astrouw.edu.pl*]{}\
[*ftp://ftp.astrouw.edu.pl/ogle3/maps/gd/*]{}
The archives include, besides the tables with photometric data and astrometry for each of the subfields, also the [*I*]{}-band reference images. Usage of the data is allowed under the proper acknowledgment to the OGLE project.
We also plan to build an on-line, interactive access to the photometric maps database, allowing to search for objects fulfilling user-defined set of criteria. The availability of such an access will be announced on the OGLE project WWW page.
List of figures.
Fig. . Standard deviation of magnitudes as a function of magnitude for the field CAR100.2.
Fig. . Standard deviation of magnitudes as a function of magnitude for the field MUS100.2.
Fig. . Histogram of stellar magnitudes in the Galactic disk field CAR100.2.
Fig. . Histogram of stellar magnitudes in the Galactic disk field MUS100.2.
Fig. . Color–magnitude diagram of the Galactic disk field CAR100.2.
Fig. . Color–magnitude diagram of the Galactic disk field CAR110.2.
Fig. . Color–magnitude diagram of the Galactic disk field CEN108.2.
Fig. . Color–magnitude diagram of the Galactic disk field MUS100.2.
|
---
abstract: 'We report on the extensive multi-wavelength observations of the blazar Markarian 421 (Mrk 421) covering radio to $\gamma$-rays, during the 4.5 year period of ARGO-YBJ and Fermi common operation time, from August 2008 to February 2013. These long-term observations extending over an energy range of 18 orders of magnitude provide a unique chance to study the variable emission of Mrk 421. In particular, thanks to the ARGO-YBJ and $Fermi$ data, the whole energy range from 100 MeV to 10 TeV is covered without any gap. In the observation period, Mrk 421 showed both low and high activity states at all wavebands. The correlations among flux variations in different wavebands were analyzed. The X-ray flux is clearly correlated with the TeV $\gamma$-ray flux, while GeV $\gamma$-rays only show a partial correlation with TeV $\gamma$-rays. Radio and UV fluxes seem to be weakly or not correlated with the X-ray and $\gamma$-ray fluxes. Seven large flares, including five X-ray flares and two GeV $\gamma$-ray flares with variable durations (3$-$58 days), and one X-ray outburst phase were identified and used to investigate the variation of the spectral energy distribution with respect to a relative quiescent phase. During the outburst phase and the seven flaring episodes, the peak energy in X-rays is observed to increase from sub-keV to few keV. The TeV $\gamma$-ray flux increases up to 0.9$-$7.2 times the flux of the Crab Nebula. The behavior of GeV $\gamma$-rays is found to vary depending on the flare, a feature that leads us to classify flares into three groups according to the GeV flux variation. Finally, the one-zone synchrotron self-Compton model was adopted to describe the emission spectra. Two out of three groups can be satisfactorily described using injected electrons with a power-law spectral index around 2.2, as expected from relativistic diffuse shock acceleration, whereas the remaining group requires a harder injected spectrum. The underlying physical mechanisms responsible for different groups may be related to the acceleration process or to the environment properties.'
author:
- |
B. Bartoli, P. Bernardini, X.J. Bi, Z. Cao, S. Catalanotti, S.Z. Chen, T.L. Chen, S.W. Cui, B.Z. Dai, A. D’Amone, Danzengluobu, I. De Mitri, B. D’Ettorre Piazzoli, T. Di Girolamo, G. Di Sciascio, C.F. Feng, Zhaoyang Feng, Zhenyong Feng, Q.B. Gou, Y.Q. Guo, H.H. He, Haibing Hu, Hongbo Hu, M. Iacovacci, R. Iuppa, H.Y. Jia, Labaciren, H.J. Li, C. Liu, J. Liu, M.Y. Liu, H. Lu, L.L. Ma, X.H. Ma, G. Mancarella, S.M. Mari, G. Marsella, S. Mastroianni, P. Montini, C.C. Ning, L. Perrone, P. Pistilli, P. Salvini, R. Santonico, P.R. Shen, X.D. Sheng, F. Shi, A. Surdo, Y.H. Tan, P. Vallania, S. Vernetto, C. Vigorito, H. Wang, C.Y. Wu, H.R. Wu, L. Xue, Q.Y. Yang, X.C. Yang, Z.G. Yao, A.F. Yuan, M. Zha, H.M. Zhang, L. Zhang, X.Y. Zhang, Y. Zhang, J. Zhao, Zhaxiciren, Zhaxisangzhu, X.X. Zhou, F.R. Zhu, and Q.Q. Zhu\
(The ARGO-YBJ Collaboration)
title: '4.5 years multi-wavelength observations of Mrk 421 during the ARGO-YBJ and Fermi common operation time'
---
Introduction
============
Active galactic Nuclei (AGNs), one of the most luminous sources of electromagnetic radiation in the universe, are galaxies with a strong and variable non-thermal emission, believed to be the result of accretion of mass onto a supermassive black hole (with a mass ranging from $\sim$10$^6$ to $\sim$10$^{10}$ M$_{\bigodot}$) lying at the center of the galaxy. In some cases ($\leq$10%) AGNs show powerful and highly collimated relativistic jets shooting out in opposite directions, perpendicular to the accretion disc. The jets emanate from the vicinity of the black hole ($\sim$0.1 pc) and extend up to $\sim$ 1 Mpc. They are usually associated to several bright superluminal knots, which appear related to the episodic ejection of plasmoid blobs (see for example the case of the active galaxy M87 [@chenug07]). The origin of the AGN jets is one of the open problems in astrophysics.
AGNs viewed at a small angle to the axis of the jet are called blazars. They usually show flat radio spectra, strong variability, optical polarization and $\gamma$-ray emission. Blazars include BL Lac objects, which have a lower luminosity and lack of strong emission lines in the optical band, and Flat-Spectrum Radio Quasars (FSRQ), which show a higher luminosity with strong and broad emission lines. The strongly Doppler-boosted radiation makes blazars the most extreme class of AGNs, where the boosted emission overwhelms all other emissions from the source. Therefore, the observation of blazars allows a deep insight into the physical conditions and emission processes of relativistic jets.
Blazars are the dominant extragalactic source class in $\gamma$-rays, as revealed by the $Fermi$ Large Area Telescope (LAT) survey at GeV energies [@nolan12]. Moving to very high energies (VHE, $>$0.1 TeV), the BL Lac objects dominate the extragalactic sky. Up to now, 60 AGNs have been established as VHE $\gamma$-ray emitters, including 52 BL Lac objects[^1]. Although the $\gamma$-ray emission from blazars has been studied for about two decades, it is still unclear where and how the emission originates. Observations of the misaligned radio galaxy M87 indicate that VHE $\gamma$-rays, at least during flaring periods, seem to originate within the jet collimation region, in the immediate vicinity of the black hole [@accia09; @abram12]. The high energy particles responsible for the nonthermal emission are generally believed to be accelerated in the relativistic shock front, described by the theory of diffusive acceleration (for a review, see [@kirk99]).
The radiation of a blazar is a broadband continuum ranging from radio through X-rays to $\gamma$-rays. The spectral energy distributions (SEDs) are characterized by two distinct bumps, which are believed to be dominated by non-thermal emission. The lower energy component, which peaks in the optical through X-ray, is caused by the synchrotron radiation from relativistic electrons (and positrons) within the jet. The origin of the high energy $\gamma$-ray component is still debated. The general view attributes it to inverse Compton scattering of the synchrotron (synchrotron self-Compton, SSC) or external photons (external Compton, EC) by the same population of relativistic electrons [@ghise98; @dermer92; @sikor94]. However, the hadronic scenario, which attributes the $\gamma$-ray emission to proton-initiated cascades and/or to proton-synchrotron emission in a magnetic field-dominated jet [@aharo00], cannot be excluded.
In this panorama, multi-wavelength observations are of fundamental importance. According to the present measurements, X-rays and VHE $\gamma$-rays are correlated during the flaring periods (for reviews, see [@wagner08; @chen13]). Recently, a long-term continuous monitoring of Mrk 421 performed by the Astrophysical Radiation with Ground-based Observatory at YangBaJing (ARGO-YBJ) experiment and different satellite-borne X-ray detectors [@barto11] showed a good correlation in terms of flux and spectral index. All the observational features indicate that $\gamma$-rays and X-rays have a common origin, supporting the leptonic models. The tight correlation is a challenge for models based on hadronic processes. According to a recent collective evidence [@meyer12], the SSC mechanism seems to dominate the emission of BL Lac objects, while the EC component becomes important for FSRQ. The lack of strong emission lines in the radiation of BL Lac objects is also taken as an evidence for a minor role of ambient photons (e.g., [@krawczy04]), favouring the SSC model. In this sense we can assume that BL Lac objects are less affected by the circumambient background radiation and can be considered as ideal targets for the study of the physical processes within the jets. However, even in the framework of the SSC model, the fundamental question of the origin of the flux and spectral variability, observed on timescales from minutes to years, is still open.
Mrk 421 (z=0.031), classified as a BL Lac object, is one of the brightest VHE $\gamma$-ray blazars known. It is a very active blazar with major outbursts, composed of many short flares, about once every two years, in both X-rays and $\gamma$-rays [@chen13; @aielli10; @barto11]. Actually it is considered an excellent candidate to study the physical processes within the AGN jets. During the last decade, several coordinated multi-wavelength campaigns focusing on Mrk 421 were conducted, both in response to strong outbursts or as part of dedicated observation campaigns. Complex relations between X-rays and VHE $\gamma$-rays spectra were observed in many flares (see our previous review in [@barto11]). However, due to the sparse multi-frequency data during long periods of time, no systematic studies on flux variation and SED evolution were achieved, especially in the $\gamma$-ray band. In the beginning of 2009, a multi-frequency observational campaign of Mrk 421 was carried out for 4.5 months with an excellent temporal (except at VHE) and energy coverage from radio to VHE $\gamma$-rays. During the whole campaign, however, Mrk 421 showed a low activity at all wavebands [@abdo11].
To understand the emission variability and the underlying acceleration and radiation mechanisms in jets, continuous multi-wavelength observations, particularly in X-rays and VHE $\gamma$-rays, are crucial. A simultaneous SED could provide a snapshot of the emitting population of particles and also constrain the model parameters at a given time [@yan14; @zhang12]. The shape of particle energy distribution could bring information on the underlying acceleration processes (e.g. [@yan13; @cao13; @peng14; @chen14]). In the VHE band, Cherenkov telescopes cannot regularly monitor AGNs, because of their limited duty cycle and narrow field of view (FOV). Wide-FOV Extensive Air Showers (EAS) arrays, with high duty cycles, are more suitable for this purpose. A review on EAS arrays and their observations of AGNs can be found in [@chen13].
ARGO-YBJ is an EAS array with an energy threshold for primary $\gamma$-rays of $\sim$300 GeV. During 5 years ARGO-YBJ continuously monitored the blazar Mrk 421, extending at higher energies the multi-wavelength survey carried out by the Owens Valley Radio Observatory (OVRO), the satellite-borne X-ray detectors $Swift$, the Rossi X-ray Timing Explorer (RXTE), the Monitor of All-sky X-ray Image (MAXI) and the GeV $\gamma$-ray detector $Fermi$-LAT. In particular, thanks to the ARGO-YBJ and $Fermi$-LAT data, the high energy component of Mrk 421 SED has been completely covered without any gap from 100 MeV to 10 TeV. In this paper we report on the 4.5-year multi-wavelength data recorded from August 2008 to February 2013, a period that includes several large flares of Mrk 421. Such a long-term multi-wavelength observation is rare and provides a unique opportunity to investigate on the emission variability of Mrk 421 from radio frequencies to TeV $\gamma$-rays.
This work is organized as follows:In Section 2 we summarize the data at different wavelengths collected by both satellite-borne and ground-based detectors. In Section 3 the light curves and SEDs observed by the different detectors are presented. In Section 4 the key parameters of the one-zone SSC emission model are obtained by fitting the data, then the astrophysical implications are discussed in Section 5. A summary is given in Section 6. The cosmology parameters used in this paper are: $H_0$ = 70 km $s^{-1}$ $Mpc^{-1}$, $\Omega_{M}$ = 0.3, and $\Omega_{\Lambda}$ = 0.7. The redshift of Mrk 421 corresponds to a luminosity distance of 135.9 Mpc.
Multi-wavelength observations and analysis
==========================================
The present section reviews the available data sets. We briefly summarize the energy range in which each detector works and the data processing steps. More details can be found in the cited references.
ARGO-YBJ VHE $\gamma$-ray data
------------------------------
ARGO-YBJ is an EAS detector located at an altitude of 4300 m a.s.l. (atmospheric depth 606 g cm$^{-2}$) at the Yangbajing Cosmic Ray Laboratory (30.11 N, 90.53 E) in Tibet, P.R. China. It is mainly devoted to $\gamma$-ray astronomy [@barto12a; @barto12b; @barto12c; @barto13b; @barto14; @barto15] and cosmic ray physics [@barto12d; @barto12e; @barto13c]. The detector consists of a carpet ($\sim$74$\times$ 78 m$^2$) of resistive plate chambers (RPCs) with $\sim$93% of active area, surrounded by a partially instrumented area ($\sim$20%) up to $\sim$100$\times$110 m$^2$. Each RPC is read out by 10 pads (55.6 cm$\times$61.8 cm) representing the space-time pixels of the detector. Details of the detector layout can be found in [@aielli06]. Due to the full coverage configuration and the location at high altitude, the detector energy threshold is $\sim$300 GeV, much lower than any previous EAS array, as Tibet AS$\gamma$ [@ameno05] and Milagro .
The ARGO-YBJ experiment, with a $\sim$ 2 sr FOV, is able to monitor the sources in the sky with a zenith angle less than 50$^{\circ}$. At the ARGO-YBJ site, Mrk 421 culminates with a zenith angle of 8.$^{\circ}$1, and is observable for 8.1 (4.7) hours per day with a zenith angle less than 50$^{\circ}$ (30$^{\circ}$).
The detector, in its full configuration, has been in stable data taking since November 2007 to February 2013, with a 4% of dead time and an average duty cycle higher than $86\%$. The detector performance and the analysis techniques are extensively discussed in [@barto13a]. The detector angular resolution depends on the number of triggered pads N$_{pad}$, ranging from $1^{\circ}.7$ for N$_{pad} >$ 20 to $0^{\circ}.2$ for N$_{pad} >$ 1000. The median primary energy of $\gamma$-rays is 0.36 TeV for events with N$_{pad} >$ 20 and 8.9 TeV for N$_{pad} >$ 1000 [@barto13a]. The light curves presented here are obtained selecting the events with N$_{pad}>$60, corresponding to a photon median energy of $\sim$1.1 TeV. The cosmic ray background around the source direction is estimated using the *direct integral method* [@fley04]. The spectrum is estimated as described in [@barto11], by comparing the detected and the expected signal (i.e. the number of events) as a function of N$_{pad}$. Five intervals, N$_{pad}$=20-59, 60-99, 100-199, 200-499, and $>$500, are considered, corresponding to a $\gamma$-ray energy range between 0.3 TeV and 10 TeV. The spectrum is assumed to follow a single Power Law (PL): $f(E)=J_{0} \cdot E^{-\alpha}$. The simulated events are sampled in the energy range from 10 GeV to 100 TeV. The ARGO-YBJ detector response has been evaluated using a custom Montecarlo simulation (see [@guo10]).
$Fermi$-LAT HE $\gamma$-ray data
--------------------------------
$Fermi$-LAT [@atwood09] is a pair-conversion telescope, with a FOV of over 2 sr, operating in the energy range above 100 MeV. $Fermi$-LAT started science data taking in August 2008. The data used in this work have been downloaded through the Fermi science support center[^2]. A circular region of $15^{\circ}$ radius centered on Mrk 421 was chosen for the event reconstruction. The analysis was performed using the ScienceTool and the corresponding threads, provided by the $Fermi$-LAT collaboration (Version v9r33p0)[^3]. Events with zenith angles $<100^{\circ}$ were selected from the *Source* class events, which have the highest probability of being photons. The light curve was created through aperture photometry, which allows a model independent determination of the flux, including both background and source emission. To build the SED of the source we used the suggested *gtlike* tool, based on a binned maximum likelihood method. The instrument response function is $P7REP_{-}SOURCE_{-}V15$, and the Galactic emission was reproduced using the model of $gll_{-}iem_{-}v05_{-}rev1.fit$. The model for the extragalactic isotropic diffuse emission was $iso_{-}source05_{-}rev1.txt$. All sources within $20^{\circ}$ from the Mrk 421 position were taken into account. The spectral parameters are kept free for the sources within $10^{\circ}$, while are fixed to the values given in the second $Fermi$-LAT catalogue [@nolan12] for other sources.
To describe the source spectrum from 100 MeV to 500 GeV, we use two different models: the Power Law model and the LogParabolic model (LP). The latter is described by the expression $f(E)=J_{0} \cdot (E/E_{0})^{-(a+b \cdot log(E/E_{0})}$, where $E_{0}$ indicates the normalization energy, $J_{0}$ the flux at $E_{0}$, $b$ the curvature around the SED peak and $a$ the spectral index below the SED peak. Following [@nolan12], these two models are compared by defining the curvature test statistic TS$_{curve}$=(TS$_{LP}$ - TS$_{PL}$). The significance of the curvature can be approximately estimated as $\sqrt{TS_{curve}}$.
$Swift$-BAT hard X-ray data
---------------------------
$Swift$ Burst Alert Telescope (BAT) [@gehre04] is a coded aperture mask imaging telescope (1.4 sr FOV) operating since February 2005. It orbits the Earth every 1.5 hours and monitors the whole sky at hard X-rays once per day. The daily flux from Mrk 421 at energy 15$-$50 keV is provided by $Swift$-BAT[^4] [@krimm13] and is used here to build the light curve. To obtain the SED, we downloaded all the available data of Mrk 421 through the HEASARC data archive[^5]. The data analysis includes the recipes presented in [@ajello08; @tueller10]. The spectrum was obtained as a weighted average of the source spectra using a six-channel binning in the 14$-$195 keV energy range, i.e., 14$-$22, 22$-$30, 30$-$47, 47$-$71, 71$-$121, and 121$-$195 KeV. A power law function was used to fit the measured spectrum.
$MAXI$-GSC X-ray data
---------------------
The $MAXI$ Gas Slit Camera (GSC) [@matsu09] detector, made of twelve one-dimensional position sensitive proportional counters, operating in the 2$-$20 keV range, started data taking in August 2009. The experiment achieves 97% of sky coverage per day. The light curves for specific sources are publicly available [^6] in three energy bands: 2$-$4, 4$-$10, and 10$-$20 keV. These data were used in this work to build the X-ray spectrum of Mrk 421, comparing the measured counting rate in each band with the one of the Crab Nebula, used as standard candle, as proposed by the $Swift$-BAT collaboration [@tueller10].
$RXTE$-ASM soft X-ray data
--------------------------
$RXTE$ All Sky Monitor (ASM) [@levine96] consists of three proportional counters, each one with a field of view of $6^{\circ}\times90^{\circ}$. It covers about 80% of the sky during one full revolution in about 1.5 hr. The $RXTE$-ASM data in the (2$-$12) keV range are publicly available[^7]. The light curves are given in three energy bands: 1.5$-$3, 3$-$5, and 5$-$12 keV, which were used here to build the X-ray spectrum. For Mrk 421 the daily flux is provided from 1995 up to the middle of 2010.
$Swift$-XRT soft X-ray data
---------------------------
$Swift$ X-ray Telescope (XRT) [@burro05] is a focusing X-ray telescope with an energy range from 0.2 to 10 keV. The light curves at 0.3$-$10 keV for Mrk 421 (available here[^8]) were directly used in this work. To obtain the SED all the $Swift$-XRT Windowed Timing (WT) Observations, available at HEASARC[^9], were downloaded. The $Swift$-XRT data set are calibrated using the calibration files available in the Swift data base (CALDB[^10]) and processed with the XRTDAS software package (distributed by HEASARC within the HEASoft package (v.6.16)[^11]) using the *xrtpipeline* task. Events for the spectral analysis were selected within a circle of 30 pixels ($~$71$^{\prime\prime}$) radius centered on the source position. The background was extracted from an annular region with a 40 pixels inner radius and 80 pixels outer radius, also centered on the source position. The count rate is less than 100 Hz for all the observations considered in this work, so the WT mode data should not be affected by pile-up effects. The average spectrum in the 0.3$-$10 keV energy band was fitted using the XSPEC package [^12] (v.12.8.2), assuming a LP model (fixing $E_{0}$=1 keV), with an absorption hydrogen-equivalent column density set to the Galactic value in the direction of the source, namely $1.92 \times 10^{20}$ $cm^{-2}$ [@kalber05]. For such a spectrum model, the energy of the SED peak is estimated as $E_{peak}=10^{(2-a)/2b}$. In addition, a small energy offset ($\sim$40 eV) was applied to the observed energy spectrum, according to [@abdo11].
$Swift$-UVOT ultraviolet data
-----------------------------
$Swift$ Ultraviolet/Optical Telescope (UVOT) [@roming05] is the ultraviolet and optical Telescope onboard the satellite. All the $Swift$-UVOT observations of Mrk 421, at the three ultraviolet bands (UVW1, UVM2 and UVW2) available at the HEASARC data archive, were included in our analysis. The level 2 UVOT images from the archive, produced by a custom UVOT pipeline with data screening and coordinate transformation, were directly used in this work. The photometry was computed using a 8$^{\prime\prime}$ source region centered on the Mrk 421 position, performing the calibrations presented in [@poole08], which also convert UVOT magnitudes to flux units. The latest UVOT calibration files released on January 18th 2013 were used here. The background was extracted from an annular region (with radius of 20$^{\prime\prime}$ $-$ 50$^{\prime\prime}$) centered on the source position. The flux has been corrected for the Galactic extinction using the [@card89] parameterization, with E$_{B-V}$ = 0.013 mag [@schlaf11].
OVRO radio data
---------------
The OVRO [@richard11] is a 40-m radio telescope working at 15.0 GHz with 3 GHz bandwidth. Mrk 421 was observed by OVRO as part of the blazar monitoring programme, which observed a sample of over 1800 AGNs twice per week. Mrk 421 has been included since the end of 2007. The light curve for Mrk 421 is publicly available [^13] and is directly used in this work. The systematic error is estimated to be about 5 percent of the flux density, which is not included in the error bars.
Results
=======
Figure 1 summarizes the temporal and energy coverage of the different instruments considered in this analysis. To monitor efficiently the HE and VHE components of Mrk 421 spectrum, we limit the observation time to the $\sim$4.5 years in which the data of $Fermi$-LAT (100 MeV$<$E$<$500 GeV) and ARGO-YBJ (E$>$300 GeV) overlap, i.e. since 2008 August 5 (start time of $Fermi$-LAT science data acquisition) to 2013 February 7 (end time of ARGO-YBJ data taking).
![ Time and energy coverage of different detectors in 4.5 years of Mrk 421 observation. []{data-label="fig1"}](Fig1.eps){width="3.5in" height="2.4in"}
In the following, we firstly determine the light curves of Mrk 421 in the energy ranges explored by the different detectors. Then, by inspecting the light curves, we define the flaring and steady phases of the source. Finally, we analyze the general features of the corresponding SEDs.
Light curves
------------
Figure 2 shows the light curves of Mrk 421, as obtained by the data of the previously described experiments, covering the entire energy range from radio to the TeV band. The time integration is chosen taking into account the sensitivity of the instruments. For ARGO-YBJ each point corresponds to one month (30 days) of data, while for $Fermi$-LAT, $Swift$-BAT, $RXTE$-ASM and $MAXI$-GSC the data are averaged over one week. For $Swift$-XRT and $Swift$-UVOT each point is the result of each dwell, which last about hundreds of seconds. Note that since the $Swift$-UVOT light curves in the three photometric bands (UVW1, UVM2, and UVW2) show similar behaviors, only the light curve of UVW1 is considered here. For OVRO each point is the result of each observation. The data presented in Figure 2 are also listed in Table 1.
{width="7in" height="1.4in"} {width="7in" height="1.4in"} {width="7in" height="1.4in"} {width="7in" height="1.4in"} {width="7in" height="1.4in"} {width="7in" height="1.4in"} {width="7in" height="1.4in"}
[ccccc]{} MJD & $\bigtriangleup$T (Day) & Flux$^a$ & $\bigtriangleup$Flux$^a$ & Detector\
54698.00 & 15.00 & 2.231e+01 & 1.560e+01 & ARGO-YBJ\
54728.00 & 15.00 & 3.486e+01 & 1.527e+01 & ARGO-YBJ\
54758.00 & 15.00 & -1.327e+00 & 1.570e+01 & ARGO-YBJ\
54788.00 & 15.00 & 1.026e+01 & 1.973e+01 & ARGO-YBJ\
54818.00 & 15.00 & 1.047e+01 & 1.737e+01 & ARGO-YBJ\
54848.00 & 15.00 & 3.495e+01 & 1.492e+01 & ARGO-YBJ\
54878.00 & 15.00 & 2.322e+01 & 1.723e+01 & ARGO-YBJ\
54908.00 & 15.00 & 3.786e+01 & 1.477e+01 & ARGO-YBJ\
54938.00 & 15.00 & 1.242e+01 & 1.493e+01 & ARGO-YBJ\
54968.00 & 15.00 & -2.462e+00 & 1.544e+01 & ARGO-YBJ\
54998.00 & 15.00 & 2.471e+01 & 1.505e+01 & ARGO-YBJ\
55028.00 & 15.00 & 4.428e+01 & 1.520e+01 & ARGO-YBJ\
55058.00 & 15.00 & 2.294e+01 & 2.037e+01 & ARGO-YBJ\
55088.00 & 15.00 & 6.377e+01 & 1.677e+01 & ARGO-YBJ\
55118.00 & 15.00 & 4.981e+01 & 1.594e+01 & ARGO-YBJ\
55148.00 & 15.00 & 2.032e+01 & 2.004e+01 & ARGO-YBJ\
55178.00 & 15.00 & 4.956e+01 & 1.539e+01 & ARGO-YBJ\
55208.00 & 15.00 & 2.683e+01 & 1.561e+01 & ARGO-YBJ\
55238.00 & 15.00 & 4.596e+01 & 1.528e+01 & ARGO-YBJ\
55268.00 & 15.00 & 2.997e+01 & 1.557e+01 & ARGO-YBJ\
\
\
\
\
\
According to the long-term light curves presented in Figure 2, Mrk 421 showed both low and high activity phases at all wavebands during the 4.5 years considered in this work. To quantify the variability amplitudes in each energy band, the normalized variability amplitude (F$_{var}$), defined according to [@edelson96], was computed as $$F_{var}=\frac{\sqrt{\sigma_{tot}^2-\sigma_{err}^2}}{\bar{F}}$$ where $\sigma_{tot}$ is the standard deviation of the flux, $\sigma_{err}$ is the mean error of the flux points, and $\bar{F}$ is the mean flux. To facilitate the comparison of F$_{var}$ for different bands, we rebinned all light curves shown in Figure 2 with the same bin size, i.e., 7 days per bin. The ARGO-YBJ data with 7-day bin size are presented in Table 1. The F$_{var}$ as a function of band energy is shown in Figure 3. The variability amplitude increases from 21% in radio to 137% in hard X-rays. The amplitude is 39% for GeV $\gamma$-rays, and it increases to 84% at TeV energies. It should be noted that the light curve of GeV $\gamma$-rays is obtained through the aperture photometry method, which includes a contribution from the background emission at a 18% level according to our estimation. Since the background only affects the average flux and not the variability amplitude, the effective amplitude of GeV $\gamma$-ray variability is 47%.
![ The normalized variability amplitude F$_{var}$ for different energy bands. []{data-label="fig3"}](Fig3.eps){width="3.5in" height="2.4in"}
[c|ccc]{} & r$_{UVOT}^a$ & r$_{BAT}^b$ & r$_{LAT}^c$\
ORVO & 0.62$\pm$0.14$^d$ & -0.26$\pm$0.06 &0.38$\pm$0.11$^e$\
$Swift$-UVOT & ... &-0.35$\pm$0.08 &0.33$\pm$0.11\
$Swift$-XRT & -0.39$\pm$0.08 & 0.85$\pm$0.21 &0.27$\pm$0.15\
$RXTE$-ASM & -0.47$\pm$0.10 & 0.87$\pm$0.17 &0.38$\pm$0.14\
$MAXI$-GSC & -0.34$\pm$0.12 & 0.89$\pm$0.22 &0.30$\pm$0.10\
$Swift$-BAT & -0.35$\pm$0.08 & ... &-0.01$\pm$0.08\
$Fermi$-LAT & 0.33$\pm$0.11 &-0.01$\pm$0.08 &...\
ARGO-YBJ & -0.90$\pm$0.33 & 0.76$\pm$0.25 &0.61$\pm$0.22\
\
\
\
\
\
According to Figure 2, only the light curves of $Swift$-BAT, $Fermi$-LAT and ARGO-YBJ continuously sampled the whole 4.5 years period considered here. Several large X-ray and GeV $\gamma$-ray flares are visible from the light curves of $Swift$-BAT and $Fermi$-LAT. The flux variability in the $Swift$-BAT and $Fermi$-LAT energy bands seems not to be correlated during flares. The variability of VHE $\gamma$-ray flux is roughly correlated both with X-ray and GeV flares. The variability of radio and UV flux seems not to be correlated with that of X-rays and $\gamma$-rays. To be more rigorous, the discrete correlation function (DCF; [@edelson88]) is used to quantify the degree of correlation between the light curve of $Swift$-BAT ($Fermi$-LAT, $Swift$-UVOT) with the others. To uniform the data, the cross-correlation analysis was performed, using weekly binned light curves. No significant time lag (within \[-200,200\] days) was measured in this analysis except between $Fermi$-LAT and ORVO, where it was found that the GeV $\gamma$-rays lead the radio by 42 days, with a correlation coefficient $r$=0.83$\pm$0.27. This result is consistent with [@hovatta15] who also measured a 40$\pm$9 days time lag using 4 years ORVO and $Fermi$-LAT data. Beside this, a possible time lag is measured between $Swift$-UVOT and ORVO data. The UV flux seems to lead the radio by 21 days with a correlation coefficient $r$=0.79$\pm$0.17, that however is comparable to the coefficient $r$=0.62$\pm$0.14 obtained for a time lag equal to zero. The correlation coefficients for a time lag zero are listed in Table 2, for all the datasets. According to our analysis, the $Swift$-BAT hard X-ray flux is weakly anti-correlated with the radio and UV flux, while is significantly correlated with the soft X-ray flux, not correlated with GeV $\gamma$-rays and clearly correlated with VHE $\gamma$-rays. The $Fermi$-LAT GeV $\gamma$-ray flux is weakly correlated with radio, UV and soft X-rays, and moderately correlated with VHE $\gamma$-rays. The UV flux appears to be moderately correlated with radio, weakly anti-correlated with X-rays, and clearly anti-correlated with VHE $\gamma$-rays. It should be noted however that the observation time of $Swift$-UVOT has several long gaps, which would affect the cross-correlation analysis. In particular, the anti-correlation with VHE $\gamma$-rays needs to be checked by future observations.
Source state definition
-----------------------
In this paper we will focus on the large X-ray and GeV $\gamma$-ray flares, with the aim to investigate the spectral variation at different wavebands, compared to the low activity states. We will define different states of activity for Mrk 421 mainly basing on the light curves of $Fermi$-LAT and $Swift$-BAT, partially taking into account the curves of $RXTE$-ASM and $MAXI$-GSC. For X-ray flares, we will only select the flares which show a large increase both in hard and soft X-rays.
From August 2008 (MJD=54683) to June 2009 (MJD=55000), Mrk 421 shows a low activity at all wavebands. We mark this period as Steady 1 (S1) phase. It should be noted that, during this period, a 4.5 months long multi-frequency campaign was organized [@abdo11]. Afterwards, according to the X-ray light curves of $Swift$-BAT, $RXTE$-ASM and $MAXI$-GSC, the source entered a long-lasting outburst phase starting in June 2009 and ending in June 2010 (MJD=55350), which we denoted as Outburst (OB). The X-ray flux is higher than in the S1 period and also varies with time. During this active phase, three large flares, named Flare 1, Flare 2 and Flare 3 (F1, F2 and F3) are clearly detected both by $Swift$-BAT and $RXTE$-ASM. During F1, the flux starts to increase on 2009 November 9 (MJD=55144), reaches the maximum on November 12, then decreases to a quasi-steady state on November 14. The average flux is about 3 and 14 times higher than in the S1 phase, in the 2$-$12 keV and 15$-$50 keV ranges, respectively. The F2 (from 2010 February 15 (MJD=55242) to 17) and F3 (from 2010 February 18 to March 16 (MJD=55271)) states were reported by MAXI at 2$-$10 keV [@isobe10]. A zoom view of the light curves during both flares is shown in Figure 4, with a 3-day binning for ARGO-YBJ and one-day binning for the other experiments. F2 is a very fast flare reaching the peak flux in one day and then decaying in one day too. This flare is associated with the huge VHE $\gamma$-ray flare with a flux around 10 times the Crab Nebula flux (I$_{crab}$) detected by VERITAS on February 17 (MJD=55244) [@ong10]. The $\gamma$-ray flux enhancement is also evident in $Fermi$-LAT and ARGO-YBJ data. F3 follows flare 2. Note that for the OB state, the embedded durations of flares F1, F2 and F3 are excluded.
{width="7in" height="7.in"}
After May 2010, Mrk 421 entered a low steady phase that ended on October 6 (MJD=55475), when the flux of both hard and soft X-rays started gradually to increase for two weeks. The whole period lasted about one month and is marked as Flare 4 (F4). Then Mrk 421 came to a long and steady phase (S2), both in X-rays and $\gamma$-rays, which lasted about 1.6 years, since November 2010 (MJD=55516) to June 2012 (MJD=56106). This is the longest steady phase during the monitored period, therefore it has been picked out as a baseline reference to all the other states. The embedded strong flare denoted as Flare 5 (F5), occurred in September 2011 (MJD=55811) and lasting $\sim$7 days, has been excluded from S2.
In the whole year 2012 the flux in hard X-rays was almost stable, while the GeV $\gamma$-ray flux measured by $Fermi$-LAT entered into a high flux level from 2012 July 9 (MJD=56117) to September 17 (MJD=56187). This is the first long-term GeV $\gamma$-ray flare from Mrk 421 ever detected, reported by both $Fermi$-LAT [@amman12] and ARGO-YBJ [@barto12f]. According to the GeV $\gamma$-ray light curve, two peaks are selected, marked as Flare 6 (F6, from 2012 July 9 to 21 (MJD=56129) and Flare 7 (F7, from 2012 July 22 to September 16), during which also the VHE $\gamma$-ray flux detected by ARGO-YBJ seems to be enhanced. Some hints of enhancement were partly observed by $MAXI$-GSC in the soft X-ray energy range.
The light curves during F1, F4, F5, F6, F7 are shown in Figure 5, where the flux measured by ARGO-YBJ is averaged over the different flare durations, and a 3-day binning is used for the other experiments. The duration of all the selected states are summarized in Table 3. Note that $Swift$-XRT data are only available for F1, F2, F3, S1, S2 and OB states.
{width="7.02in" height="1.4in"} {width="7in" height="1.4in"} {width="7in" height="1.4in"} {width="7in" height="1.4in"}
[c|c|cc|cc|cc|cc]{} State & MJD& F$_{2-20keV}$ & $\alpha$ & F$_{14-195keV}$ & $\alpha$ & F$_{0.1-500GeV}$ & $\alpha$ & F$_{>1TeV}$ & $\alpha$\
& & ($\times 10^{-2}$) & & ($\times 10^{-3}$) & & ($\times 10^{-7}$) & & (I$_{crab}^{a}$) &\
Flare 1 &55144$-$55149 & 18.3$\pm$1.8 &2.00$\pm$0.16 & 23.1$\pm$2.6 &2.78$\pm$0.27 &1.11$\pm$0.36 &1.57$\pm$0.15 &3.2$\pm$0.9 &2.78$\pm$0.36\
Flare 2 &55242$-$55245 & 41.4$\pm$4.2 &2.38$\pm$0.15 & 19.2$\pm$1.9 &2.76$\pm$0.18 &3.48$\pm$0.60 &1.50$\pm$0.08 &7.2$\pm$1.5 &2.61$\pm$0.27\
Flare 3 &55245$-$55272 & 19.9$\pm$1.6 &2.20$\pm$0.11 & 12.9$\pm$0.9 &2.61$\pm$0.13 &1.84$\pm$0.21 &1.79$\pm$0.09 &1.4$\pm$0.5 &2.42$\pm$0.46\
Flare 4 &55475$-$55503 & 10.0$\pm$1.1 &2.17$\pm$0.22 & 5.82$\pm$0.83 &3.02$\pm$0.30 &2.16$\pm$0.19 &1.74$\pm$0.05 &1.9$\pm$0.5 &2.85$\pm$0.26\
Flare 5 &55811$-$55818 & 17.0$\pm$1.6 &2.13$\pm$0.15 & 13.3$\pm$1.4 &2.74$\pm$0.23 &2.23$\pm$0.33 &1.79$\pm$0.09 &2.1$\pm$0.8 &2.75$^{b}$\
Flare 6 &56117$-$56130 & 6.15$\pm$0.96 &2.05$\pm$0.35 & 1.45$\pm$0.40 &2.47$\pm$0.62 &6.05$\pm$0.39 &1.68$\pm$0.04 &1.7$\pm$0.6 &2.84$\pm$0.39\
Flare 7 &56130$-$56187 & 8.70$\pm$0.91 &2.97$\pm$0.18 & 1.53$\pm$0.36 &3.09$\pm$0.61 &5.20$\pm$0.19 &1.75$\pm$0.02 &1.1$\pm$0.4 &3.22$\pm$0.24\
Outburst&55000$-$55350$^{c}$ &8.91$\pm$0.74 &2.41$\pm$0.11 & 4.58$\pm$0.59 &2.97$\pm$0.23 &1.92$\pm$0.06 &1.76$\pm$0.02 &0.91$\pm$0.14 &2.67$\pm$0.16\
Steady 1& 54683$-$55000& ... & ... & 1.50$\pm$0.12 &2.51$\pm$0.16 &1.53$\pm$0.05 &1.75$\pm$0.02 &0.56$\pm$0.13 &2.64$\pm$0.27\
Steady 2& 55516$-$56106$^{d}$& 1.78$\pm$0.18 & 2.38$\pm$0.19 & 0.318$\pm$0.087 & 3.0$^{b}$ & 1.69$\pm$0.04$^{e}$ & 1.77$\pm$0.01 & 0.33$\pm$0.10 & 2.75$^{b}$\
Detector & & & & &\
\
\
\
\
\
Spectral Energy Distribution
----------------------------
In this section we report the multi-wavelength SEDs observed by the running experiments in the outlined states.
To model the spectral energy distribution f(E), we assume a simple power law spectrum for $Swift$-BAT, $RXTE$-ASM, $MAXI$-GSC, $Fermi$-LAT and ARGO-YBJ, while for $Swift$-XRT we assume a logparabolic function. In the following, F represents the integral flux over the detector energy range, $\alpha$ the spectral index of the power law function, while $a$ and $b$ the parameters of the logparabolic model (see Section 2 for details). During S2, the $Swift$-BAT data are not significant enough to fit both the flux and the spectral index. For such a situation, the spectral index is fixed to 3.0. A similar assumption has been chosen for ARGO-YBJ data by fixing the spectral index to 2.75 during F5 and S2. The time-averaged SEDs for the different activity states, obtained by fitting the data of all the experiments, are summarized in Table 3 and 4. The flux at each energy is shown in Figure 7 and also listed in Table 5. Note that in the following text, the first and the second components refer to the two SED bumps of the SED, as foreseen in the SSC model.
### $Swift$-XRT SED
---------- ------------------- ----------------- ----------------- ----------------- -------------------- ---------------
State F$_{0.3-10keV}$ a b E$_{peak}$ F$_{2-12keV}$ $\alpha$
(keV) ($\times 10^{-2}$)
Flare 1 1.227$\pm$0.003 1.420$\pm$0.004 0.412$\pm$0.008 5.06$\pm$0.18 18.40$\pm$0.73 1.86$\pm$0.08
Flare 2 1.251$\pm$0.004 1.656$\pm$0.004 0.390$\pm$0.009 2.76$\pm$0.07 27.89$\pm$0.76 2.02$\pm$0.06
Flare 3 0.903$\pm$0.001 1.690$\pm$0.002 0.393$\pm$0.003 2.48$\pm$0.02 13.74$\pm$0.29 1.89$\pm$0.04
Flare 4 ... ... ... 10.15$\pm$0.67 1.65$\pm$0.12
Outburst 0.5746$\pm$0.0003 1.864$\pm$0.001 0.448$\pm$0.002 1.419$\pm$0.004 7.19$\pm$0.13 2.19$\pm$0.04
Steady 1 0.4526$\pm$0.0004 2.104$\pm$0.001 0.462$\pm$0.003 0.771$\pm$0.003 4.86$\pm$0.11 2.41$\pm$0.06
Steady 2 0.229$\pm$0.0003 2.352$\pm$0.001 0.434$\pm$0.003 0.394$\pm$0.003 0.68$\pm$0.18 2.16$\pm$0.49
Detector
---------- ------------------- ----------------- ----------------- ----------------- -------------------- ---------------
According to the $Swift$-XRT data at 0.3$-$10 keV, the peak energy of the first component $E_{peak}$ is 0.394$\pm$0.003 keV and 0.771$\pm$0.003 keV during S2 and S1, respectively. It increases to 1.429$\pm$0.004 keV during the OB phase and even up to 2.4$-$5.1 keV during F1, F2, and F3. The correlation between the flux and $E_{peak}$ is shown in Table 4 and Figure 6. A power-law function is adopted to fit their relation, yielding $f(E_{peak})=(0.2056\pm0.004) \cdot E_{peak}^{-1.266\pm 0.004}$ keV$^{-1}$ cm$^{-2}$ s$^{-1}$ with $\chi^{2}/dof$=2113/4. The S1, S2, OB, F3 roughly follow this function, while F1 and F2 clearly deviate, indicating a different behavior of F1 and F2 with respect to the other flaring states. It is worth to note that the $Swift$-XRT observations during Flare 1 only cover the period with the maximum flux. For this reason, the measured flux is higher than those of $RXTE$-ASM and $MAXI$-GSC, as shown in Figure 7.
![ The peak energy and the corresponding flux for different Mrk 421 states, as measured by $Swift$-XRT at 0.3$-$10 keV. The solid line is fitting result using a power-law function, which yields $f(E_{peak})=(0.2056\pm0.004) \cdot E_{peak}^{-1.266\pm 0.004}$ keV$^{-1}$ cm$^{-2}$ s$^{-1}$ with $\chi^{2}/dof$=2113/4. []{data-label="fig6"}](Fig6.eps){width="3.5in" height="2.4in"}
### $MAXI$-GSC and $Swift$-BAT SEDs
According to $MAXI$-GSC measurements at 2$-$20 keV, the flux during the flaring periods increases by about a factor 4$-$20, compared to S2. The spectral indexes for most of the states are in the 2$-$2.4 range while the spectrum softens (index 2.97$\pm$0.18) during F7. In the X-ray band (14$-$195 keV) the $Swift$-BAT data shows an even larger variation (4$-$70 times) with spectral indexes ranging from 2.5 to 3.1.
### $Fermi$-LAT and ARGO-YBJ SEDs
As stated in Section 2, we also test the significance of the spectrum curvature in the $Fermi$-LAT data. During the S2 phase, the TS$_{curve}$ value is found to be 32.6, corresponding to 5.7 standard deviations (s.d.). An evidence for a curvature is then observed in S2, with a peak energy E$_{peak}$=(60$\pm$11) GeV. The TS$_{curve}$ values for F2 and F7 are 8.0 and 9.4, respectively, corresponding to 2.8 and 3.1 s.d., only showing a hint of curvature. The TS$_{curve}$ values for the other seven states are less than 3.4. No curvature are detected in these cases. These features are visible in Figure 7. Note that the data points of $Fermi$-LAT are the result of the analysis made in differential energy ranges, and are independent of the assumed spectral models.
In the GeV $\gamma$-ray band, the F3, F4, F5, OB, S1 and S2 phases have similar spectral indexes (ranging from 1.74 to 1.80) and fluxes (within 32%), as shown in Table 3. Compared to the S2 state, the spectral index of F1 shows a moderate hardening ( $\Delta \alpha=$0.20$\pm$0.15) and a flux decrease of (34$\pm$21)%. The spectral index of F2 hardens more significantly ($\Delta \alpha=$0.27$\pm$0.08) with a flux increase of a factor 2.06$\pm$0.36. A flux enhancement by a factor 3 is observed during F6 and F7, with a harder spectral index during F6 ($\Delta \alpha=$0.09$\pm$0.04) and a negligible index variation during F7.
In VHE $\gamma$-ray band the S2 flux is estimated to be (0.33$\pm$0.10) I$_{crab}$, assuming a fixed spectral index, $\alpha=$2.75. This result is comparable to the baseline flux of Mrk 421 obtained using a 20 years long-term combined ACT data [@tlucz10], which is estimated to be less than 0.33 I$_{crab}$ above 1 TeV. The averaged measured flux is (0.56$\pm$0.13) and (0.91$\pm$0.14) I$_{crab}$ during S1 and OB phase, respectively. F2 is the largest flare, achieving a flux of (7.2$\pm$1.5) I$_{crab}$. The flux of the remaining flares is around (1$-$3) I$_{crab}$. The spectral index of F7 ($\alpha=$3.22$\pm$0.24) marks the softest spectrum of the observed flares. The flux modulations appears in coincidence with the X-ray observations.
Summarizing the above results, we can conclude that the flux enhancements are detected in both X-rays and VHE $\gamma$-rays during all the nine states, compared to the baseline S2. The behavior in the GeV band is actually different. Accordingly, a phenomenological classification of 3 types of SEDs (T1, T2 and T3) is here introduced, i.e., (I) flares with no or little GeV flux and photon index variations, (II) flares with $\gamma$-ray spectral hardening, irrespective of the flux variations, and (III) flares with flux enhancements, irrespective of spectral behavior. Type T1 includes phases S1, S2, F3, F4, F5 and OB. Type T2 includes the F1 and F2 states and also the day (MJD=56124), corresponding to the F6 maximum flux, during which the spectral index significantly hardens to $\alpha=$1.60$\pm$0.04. It is worth to note that this variation is fast: hardening phase only lasts two days and recovers soon, indicating an unstable state. Type T3 includes phases F6 and F7. Actually the spectral index of F7 becomes softer above the peak energy for both low and high energy component. The previous flare on May 7th, 2008, reported by [@accia09b], not included in this present discussion, may also belong to this type.
During flares of types T1 and T2, the peak energies of both the low and high energy components shift to higher energy with respect to the baseline state S2. This tendency is consistent with most of the previous measurements [@massa08; @albert07] based on fragmented observations. This indicates that the modulation of Mrk 421 flux follows these types in most of the cases. During flares of type T3, the peak energies could shift to lower energy with respect to S2, but this must be determined by future observations of similar flares.
{width="7in" height="9.in"}
{width="7in" height="5.5in"}
[cccccc]{} State & E & E$^{2}$dN/dE & $\bigtriangleup$(E$^{2}$dN/dE) & 95% u.l. & Detector\
& (TeV) & (erg cm$^{-2}$s$^{-1}$ ) & (erg cm$^{-2}$s$^{-1}$ ) &(erg cm$^{-2}$s$^{-1}$ ) &\
S1 & 4.470e-01 & 1.698e-11 & 2.625e-11 & 6.386e-11 & ARGO-YBJ\
S1 & 8.910e-01 & 3.632e-11 & 1.494e-11 & 6.094e-11 & ARGO-YBJ\
S1 & 1.413e+00 & 3.713e-11 & 9.739e-12 & 0 & ARGO-YBJ\
S1 & 2.818e+00 & 9.018e-12 & 6.650e-12 & 2.024e-11 & ARGO-YBJ\
S1 & 4.467e+00 & 7.560e-12 & 7.384e-12 & 2.028e-11 & ARGO-YBJ\
S1 & 1.520e-04 & 2.099e-11 & 1.840e-12 & 0 & Fermi-LAT\
S1 & 3.080e-04 & 2.143e-11 & 1.447e-12 & 0 & Fermi-LAT\
S1 & 6.270e-04 & 3.193e-11 & 1.669e-12 & 0 & Fermi-LAT\
S1 & 1.280e-03 & 3.228e-11 & 1.963e-12 & 0 & Fermi-LAT\
S1 & 2.590e-03 & 4.266e-11 & 2.837e-12 & 0 & Fermi-LAT\
S1 & 5.270e-03 & 5.517e-11 & 4.431e-12 & 0 & Fermi-LAT\
S1 & 1.070e-02 & 5.411e-11 & 6.199e-12 & 0 & Fermi-LAT\
S1 & 2.180e-02 & 7.172e-11 & 1.020e-11 & 0 & Fermi-LAT\
S1 & 4.440e-02 & 7.453e-11 & 1.478e-11 & 0 & Fermi-LAT\
S1 & 9.020e-02 & 9.906e-11 & 2.503e-11 & 0 & Fermi-LAT\
S1 & 1.830e-01 & 1.883e-10 & 5.048e-11 & 0 & Fermi-LAT\
S1 & 3.730e-01 & 6.174e-11 & 4.369e-11 & 1.640e-10 & Fermi-LAT\
S1 & 1.800e-08 & 4.853e-11 & 3.441e-12 & 0 & Swift-BAT\
S1 & 2.605e-08 & 3.283e-11 & 4.348e-12 & 0 & Swift-BAT\
S1 & 3.845e-08 & 2.889e-11 & 4.074e-12 & 0 & Swift-BAT\
\
Cherenkov detectors VHE $\gamma$-ray data
-----------------------------------------
{width="7in" height="2.5in"}
During the 4.5 years period from August 2008 to February 2013, Cherenkov detectors (e.g. VERITAS, MAGIC), also observed Mrk 421 in the VHE band. Even if these detectors cannot monitor Mrk 421 day by day, they can provide more precise measurements for short periods, due to their excellent sensitivity.
To give an example of the timing coverage of a Cherenkov detector, a dummy instrument located at 30$^{\circ}$ N (close to the latitude of VERITAS, 32$^{\circ}$ N, and MAGIC, 28$^{\circ}$ N) is here considered to estimate the allowable observation time for Mrk 421. Figure 9 shows the allowable time, requiring the Sun zenith angle be greater than 105$^{\circ}$, the Moon zenith angle greater than 100$^{\circ}$, and the Mrk 421 zenith angle less than 50$^{\circ}$.
Among the 7 flares presented in the last section, F4, F5, F6 and F7 occurred during the period from July to October, forbidden for Cherenkov detector observations, since Mrk 421 is close to the direction of the Sun. The moonlight completely hampered Cherenkov telescope observations during F1 and partially during F3. Only flare F2 could be eventually observed every night by Cherenkov detectors.
Actually VERITAS observed Mrk 421 during the last day of F2, i.e. MJD=55244. The preliminary spectrum is shown in Figure 7 and the corresponding flux above 1 TeV is 16.8 I$_{crab}$ [@fort12]. TACTIC observed Mrk 421 every night during F2 [@singh15]. The peak flux measured on February 16 (MJD=55243) and the corresponding spectrum during the peak flux day is also shown in Figure 7. The flux above 1 TeV is 2.7 I$_{crab}$. HAGAR also observed Mrk 421 every night during F2 [@shukla12]. The peak flux on February 17 (MJD=55244) is about 7 I$_{crab}$. It is worth to recall that the average flux detected by ARGO-YBJ during the three days of F2 is (7.2$\pm$1.5) I$_{crab}$.
Both VERITAS and HESS observed the first three days of flare F3. VERITAS preliminary results show that the flux decreased from 5.7 I$_{crab}$ to 2.9 I$_{crab}$ [@fort12]. HESS preliminary results also shows a decreasing flux from 4.8 to 1.4 I$_{crab}$ [@tlucz11].
The S1 period extends over years 2008$-$2009 and during this time the Whipple telescope monitored the Mrk 421 emission. The total observation time is 130.6 hours and the mean $\gamma$-ray rate is 0.55$\pm$0.03 I$_{crab}$ [@accia14], which is consistent with the ARGO-YBJ result, i.e., 0.56$\pm$0.13 I$_{crab}$. During this period, MAGIC also observed Mrk 421 for about 27.7 hours [@abdo11; @aleksi15]. The photon fluxes for the individual observations gave an average flux of about 50% that of the Crab Nebula, with relatively mild (typically less than a factor of two) flux variations. The spectrum, shown in Figure 7, is consistent with the ARGO-YBJ data.
The OB period lasts over the years 2009$-$2010. The spectrum measured by TACTIC, shown in Figure 7, is lower than that by ARGO-YBJ. The preliminary spectrum reported by VERITAS [@galan11] is also shown in Figure 7, which is higher than the one of ARGO-YBJ. These differences may be caused by the different observation times, considering that Mrk 421 was in an active and variable phase.
The SSC model for Mrk 421
=========================
The different types of Mrk 421 flux variations can be associated to the intrinsic astrophysical mechanisms of the emission. In the following, we will investigate the major parameters correlated to these variations in the framework of the one-zone SSC emission model.
In this paper the one-zone homogeneous SSC model proposed by [@krawczy04] is adopted to fit the multi-wavelength SEDs measured during different states. In this model a spherical blob of plasma with a co-moving radius $R$ is assumed. The relativistic Doppler Factor of the emitting plasma is defined as $\delta =[\Gamma(1-\beta cos\theta)]^{-1}$, where $\Gamma$ is the bulk Lorentz factor of the emitting plasma, $\beta$ is its bulk velocity in units of the speed of light, and $\theta$ is the angle between the jet axis and the line of sight, as measured in the observer frame. The emission volume is filled with an isotropic population of electrons and a randomly oriented uniform magnetic field $B$. The SED of the injected electrons in the jet frame is assumed to follow a broken power law with indexes $p1$ and $p2$ below and above the break energy $E_{break}$. The electron distribution is normalized by a factor $u_{e}$ (in units of erg cm$^{-3}$). To reduce the free parameters in the model, the low limit for the electron energy $E_{min}$ is fixed to be 500$\cdot m_{e}c^{2}$ and the high limit $E_{max}$ is assumed to be 10$\cdot E_{break}$. The radius of the emission zone is constrained by the variability of the time scale as $R<ct_{var}\delta/(1+z)$. In the multi-wavelength data considered in this analysis, the fastest variability has a time scale of $\sim$1 day, observed during F2, in X-rays and GeV $\gamma$-rays, as shown in Figure 4. In this work, the radius $R$ for all phases are arbitrarily set to be 10$^{14}$ m, being $t_{var}>4.8(20/\delta)$ hours the allowed time variability range. We have so far still six free parameters: $p1$, $p2$, E$_{break}$, $u_{e}$, B and $\delta$ to be determined experimentally by fitting the data presented in Figure 7.
In this work, we use the least-square method to determine the best values of the parameters $p1$, $p2$, E$_{br}$, B, $\delta$, and $u_{e}$. For the $Swift$-XRT flux data, we added a 3% systematic error besides the statistical error listed in Table 5. The ultraviolet and radio data points were not used for the fit, as will be discussed later. The Extragalactic Background Light (EBL) absorption of the VHE $\gamma$-ray is included in the calculation, according to the [@franc08] model.
Since the parameter $p1$ is related to the spectral shape at energies below the synchrotron and inverse Compton peaks, it is mainly determined by $Swift$-XRT data in X-rays and by $Fermi$-LAT data in GeV $\gamma$-rays. For five out of ten states the $Swift$-XRT data around the synchrotron peak are not available, therefore in this cases $p1$ is mainly determined by the $Fermi$-LAT data. The synchrotron peak energies of states S1 and S2 are close to the low energy limit of the $Swift$-XRT data, hence the measurement below the synchrotron peak cannot strongly constrain the $p1$ value, that also in this case will be mainly determined by the $Fermi$-LAT data. For the remaining three states, the statistical accuracy of the $Swift$-XRT data is much better than that of other detectors, however the low energy $Swift$-XRT measurements (at energies below $\sim$1 keV) are biased by the uncertainty in the absorption of hydrogen-equivalent column density and by detector systematics [^14](such as the CCD charge trapping, generated by radiation and high-energy proton damage, affecting mostly the low energy events). For this reason we prefer to use for all the 10 states only $\gamma$-ray data when fitting the parameter $p1$. The result and chi squares ($\chi^{2}_{\gamma}$/ndf) obtained by the fitting procedure are listed in Table 6. The derived $p1$ values for type T1 and T3 states are consistent within errors, ranging from 2.2 to 2.4. The results for T2 flares indicate a harder electron spectrum, with $p1=1.7\pm0.3$ for F1 and $p1=1.85\pm0.20$ for F2.
To obtain the remaining 5 parameters, the whole SED data above 0.3 keV are used. The $Swift$-XRT observation during F1 is not taken into account. The parameter $p2$ is determined by the spectral shape above the synchrotron and inverse Compton peak energies. Therefore, the parameter $p2$ is mainly determined by the $Swift$-XRT, $RXTE$-ASM, $MAXI$-GSC and $Swift$-BAT data in X-rays, and by the ARGO-YBJ data in TeV $\gamma$-rays. The accuracy of $Swift$-XRT measurements is much higher than that of other detectors, therefore $p2$ can be well determined for states in which $Swift$-XRT data are available. The parameter $\delta$ is mainly determined by both the synchrotron and inverse Compton peak energies. There are no data data around the synchrotron peak for F5, F6 and F7, therefore in these cases $\delta$ cannot be well constrained. In particular, we cannot find a best value for F7 within a reasonable range. For flare F7, we arbitrarily set the value of $E_{break}$ equal to the one derived for the S2 state.
According to the fit results, the SSC model reasonably describes all the SEDs, as shown in Figure 7 and 8. The fact that $Swift$-XRT data below 0.5 keV are not perfectly described during OB, F2 and F3, may be explained by the systematic errors previously discussed. The obtained parameters and chi squares ($\chi^{2}_{all}$/ndf) are summarized in Table 6. The Doppler Factor $\delta$, ranging from 10 to 41, is similar to the ones found in previous investigations [@abdo11; @barto11; @shukla12; @zhang12]. The values of the magnetic field, $B$ $\sim$0.1 G, are almost constant within a factor 2. The jet power in electrons ($10^{46}-10^{47}$ erg) is much higher than that in magnetic field, as indicated by the electron energy density to the magnetic field energy density ratio $u_{e}/u_{B}$, listed in Table 6.
[c|ccc|cc|ccc|cc]{} State& $p_1$ & $p_2$ & log(E$_{break}$) & $\delta$ & B & $u_{e}$ & $u_{e}/u_{B}$ $^{a}$ &W$_{e}$ $^{b}$ &$\chi^{2}_{\gamma}$/ndf & $\chi^{2}_{all}$/ndf\
& & & (eV) & &(G) &($10^{3}$erg cm$^{-3}$) & &($10^{46}$erg) & &\
Steady 1& 2.30$_{-0.04}^{+0.06}$ &4.70$_{-0.03}^{+0.07}$ & 11.00$_{-0.03}^{+0.04}$ &38$_{-4}^{+6}$ &0.048$_{-0.012}^{+0.012}$ &6.65$_{-0.15}^{+0.15}$ &70.6 &2.8 &23.0/11 &238/501\
Steady 2& 2.22$_{-0.06}^{+0.09}$ &4.68$_{-0.04}^{+0.02}$ & 10.78$_{-0.04}^{+0.05}$ &15$_{-2}^{+4}$ &0.17$_{-0.05}^{+0.07}$ &12.7$_{-0.5}^{+0.4}$ &10.6 &5.3 &13.2/11 &236/467\
Outburst& 2.30$_{-0.05}^{+0.08}$ &4.51$_{-0.02}^{+0.03}$ & 11.13$_{-0.05}^{+0.02}$ &35$_{-5}^{+3}$ &0.054$_{-0.005}^{+0.026}$ &7.74$_{-0.14}^{+0.24}$ &65.7 &3.2& 23.5/12&600.8/643\
Flare 1 & 1.7$_{-0.3}^{+0.3}$ &4.7$_{-0.5}^{+1.2}$ & 11.51$_{-0.03}^{+0.09}$ &10$_{-2}^{+2}$ &0.14$_{-0.04}^{+0.07}$ &31$_{-5}^{+7}$ &41.3 &12.9& 1.6/2 & 7.3/14\
Flare 2 & 1.85$_{-0.20}^{+0.20}$ &4.30$_{-0.12}^{+0.05}$ & 11.27$_{-0.03}^{+0.03}$ &17$_{-2}^{+3}$ &0.092$_{-0.024}^{+0.028}$ &24$_{-3}^{+2}$ &73.0 &10.1& 3.7/3& 179/308\
Flare 3 & 2.40$_{-0.15}^{+0.15}$ &4.60$_{-0.09}^{+0.08}$ & 11.21$_{-0.03}^{+0.02}$ &41$_{-3}^{+5}$ &0.080$_{-0.017}^{+0.011}$ &4.40$_{-0.10}^{+0.07}$ &17.3 &1.8 &13.2/4 & 1055.7/574\
Flare 4 & 2.30$_{-0.15}^{+0.15}$ &5.6$_{-0.6}^{+0.9}$ & 11.49$_{-0.07}^{+0.08}$ &35$_{-7}^{+10}$ &0.033$_{-0.013}^{+0.019}$ &12.3$_{-1.0}^{+1.3}$ &289.2&5.2 & 6.1/5 &23.4/15\
Flare 5 & 2.3$_{-0.2}^{+0.4}$ &4.6$_{-0.5}^{+0.7}$ & 11.35$_{-0.12}^{+0.13}$ &31$_{-13}^{+21}$ &0.072$_{-0.047}^{+0.108}$ &8.0$_{-1.2}^{+1.4}$ &39.2 &3.4&6.8/4 &12.1/13\
Flare 6 & 2.20$_{-0.17}^{+0.11}$ &5.1$_{-0.7}^{+1.8}$ & 11.17$_{-0.43}^{+0.24}$ &15$_{-5}^{+24}$ &0.085$_{-0.033}^{+0.053}$ &40$_{-30}^{+28}$ &120.8 &16.8& 12.1/8 & 17.5/17\
Flare 7 & 2.20$_{-0.07}^{+0.12}$ &5.2$_{-0.2}^{+0.3}$ & 10.78 &30$_{-5}^{+7}$ &0.115$_{-0.032}^{+0.038}$ &8.77$_{-1.32}^{+1.40}$ &16.6 &3.7 &2.45/3 & 14.1/20\
\
\
Discussion
==========
Using the long-term multi-wavelength data from radio to VHE $\gamma$-rays, we have shown that Mrk 421 is active at all wavebands during the 4.5 years considered in this work. The variability of Mrk 421 increases with energy for both the low and high energy SED components. The source is highly variable in the X-ray and VHE $\gamma$-ray bands, with the normalized variability amplitudes greater than 70% (see Figure 3). An overall cross-correlation analysis (see Table 2) between $Swift$-BAT and ARGO-YBJ data shows that the variabilities in X-rays and VHE $\gamma$-rays are generally correlated. The correlation is also clearly visible in the light curves during the large X-ray flares (see Figure 2, 4 and 5). Our previous observations during the outburst of 2008 also show that X-rays and VHE $\gamma$-rays were tightly correlated with the peak times in good agreement with each other [@barto11]. A clear correlation between X-rays and VHE $\gamma$-rays has also been reported in many observations in the past decades [@blaze05; @chen13; @accia14]. All these results firmly support the idea that the X-ray and VHE $\gamma$-ray emissions of Mrk 421 originate from the same zone.
The variability amplitude in GeV $\gamma$-rays is 39% (see Figure 3), which is less than that in VHE $\gamma$-rays. In fact, the amplitude of the GeV $\gamma$-ray variability is very low during most of the time, being only about 20% if the large GeV $\gamma$-ray flares F6 and F7 are excluded. The overall cross-correlation analysis (see Table 2) shows that GeV $\gamma$-rays are moderately correlated with VHE $\gamma$-rays. The GeV and VHE $\gamma$-ray light curves reported in Figure 4 and 5 show an evident correlation only during F2, F6 and F7. According to the measurements shown in Table 3, during flares F3, F4, F5 and OB phase, the flux above 0.1 GeV increases by 20%-46% with respect to the S1 steady state, while the VHE flux increases by 63%-275%. With these results, we could not therefore exclude the possibility that GeV and TeV $\gamma$-rays are produced in different emission zones. A possible scenario would be that the observed emission is the superposition of one stationary zone with a steady GeV emission and one active zone responsible for the VHE flux variation (e.g. [@cao13; @aleksi15b]). However, during flares F1, F2, F6, and F7, GeV and TeV $\gamma$-rays show clear correlated variations, according to the SEDs reported in Figure 7. This could indicate that GeV and TeV emissions are generated in the same zone, at least during these phases. In particular during flare F1 the flux above 0.1 GeV decreases of about (27$\pm$24)%. If this decrease is not due to a statistical fluctuation, the stationary zone would be excluded, and the above hypothesis of two emission zones could be excluded too. Moreover, during F6 and F7, the GeV $\gamma$-ray flux increases by a large amount (240%$-$295%), while the spectral indices were about consistent. Also this result does not support the hypothesis of two zones, since the predicted spectrum at GeV energies for the active zone is much harder than that for the steady zone, according to [@cao13] and [@aleksi15b].
Compared to other wavebands, the flux variation at radio frequencies is much weaker. A radio flare is observed in 2012, which is the largest radio flare ever observed in Mrk 421 [@hovatta15]. The cross-correlation analysis shows that this radio flare follows the GeV $\gamma$-ray flare with a time-lag of about 42 days (see Table 2), which is consistent with [@hovatta15]. If the radio and GeV $\gamma$-ray flares are physically connected [@hovatta15], then the GeV $\gamma$-ray emission could originate upstream of the radio emission. The distance between radio and GeV $\gamma$-ray emission sites and their distances to the central black hole have been discussed by [@max14].
The variability amplitude in the UV band is 33%, similar to that in GeV $\gamma$-rays, however according to the cross-correlation analysis (see Table 2) and to the light curves (see Figure 2 and 4) the UV flux appears not to be correlated with GeV $\gamma$-rays. The UV flux does not even shown an evident correlation with X-rays. Moreover, according to the SED of flare F1 shown in Figure 7, the UV and X-ray data cannot be fitted together with a unique component. These results seem to indicate that the UV and X-ray emissions do not share the same origin. Instead, the moderate correlation observed between UV and radio suggests that their emission regions could be the same, possibly located downstream of the X-ray and $\gamma$-ray emission zone. Therefore, the radio and UV data points were removed from the SSC model fitting. This choice is different from other authors, e.g. [@abdo11; @shukla12] who included them when fitting the SED. Finally, the SSC model generally under-predicts the UV and radio emissions. This supports our hypothesis that, at least partially, the radio and UV emission comes from regions in the jet not emitting X-rays and $\gamma$-rays.
For the type T1 and T3 states, the derived spectral indices $p1$ are generally consistent with $p1$=2.2, i.e. the canonical particle spectral index predicted for relativistic diffuse shock acceleration (for a review see [@kirk99]). This suggests that this process is active in Mrk 421. The change of the electron index $\Delta p$ = $p2$ - $p1$ is larger than the expected typical cooling break $\Delta p$ =1 [@kino02], indicating that the break is not induced by radiative cooling. The cooling mechanism may be less important, as suggested in [@accia11]. [@abdo11] speculated that the steep break is a characteristic of the acceleration process which is not yet understood. The break energy $E_{break}$ should represent the maximum energy that can be achieved in the acceleration process, depending on the acceleration time of the particle in the shock area. Accordingly, the flares of T1 type should be mainly caused by the variation of the maximum energy of the electrons reached within the shock area. Concerning the type T3 flares (i.e., F6 and F7), these flares might be due to the increase of the magnetic field (B) or comoving particle density ($u_e$) compared to S1.
Up to now, it is not yet clear how shocks work in a jet. If the different states detected in this work are caused by different shocks, we could expect different features of emission zones, such as B, $\delta$, R, electron density and spectrum. If the different states are caused by similar shocks moving down the jet, we would expect emission zones with the same R, for all the states, as we assumed when fitting the SED using the SSC model. In such a hypothesis, we could guess that the underlying mechanism responsible for the flares of T1 and T3 types may be due to the variation of the ambient medium. In the theory of diffusive acceleration, the acceleration time scale of particles is related to the strength of the magnetic field both in upstream and downstream regions [@drury83]. The upstream magnetic field could be related to the ambient medium. The variation of the acceleration time scale may change the maximum energy of the particles achieved within the shock. When the density and the magnetic field of the ambient medium are different, the number of particles and the corresponding maximum energy can be different. In such a hypothesis, the variability time scale for each state would characterize the size of each medium block that is crossed by the shock.
The T2 flares require a harder injected electron spectrum than the other types. This change would be caused by the acceleration processes. The slopes of F1 and F2 go beyond the predictions of a spectral index of 2.0 given by the canonical non-relativistic diffuse shock acceleration. According to [@steck07], particle spectra with spectral indexes less than 2 can be realized within the relativistic shock using extreme parameters, i.e., large scattering angles. F1 and F2 flares only last a few days, which means that the produced electrons with spectral index less than 2.2 is a transient state and cannot last for a long time. So, we cannot exclude the possibility that such short flares are due to extreme parameters. Another alternative mechanism for such a hard spectrum would be that the particles accelerated by the shock are subsequently accelerated by the stochastic process in the downstream region, which is able to produce spectra that are significantly harder than the limits of the first-order mechanism within a short time [@virtan05]. Recently, [@guo14] also predicts a hard spectral index resulting from relativistic magnetic reconnection.
Summary
=======
In this paper, we have presented a 4.5-year continuous multi-wavelength monitoring of Mrk 421, from 2008 August 5 to 2013 February 7, a period that includes both steady states and episodes of strong flaring activity. The observations concern a wide energy range, from radio to TeV $\gamma$-rays. In particular, thanks to the ARGO-YBJ and $Fermi$ data, the whole energy range from 100 MeV to 10 TeV is covered without any gap. These extensive datasets are essential for studying the origin of the flux variability and to investigate the underlying emission mechanism. The main results of this work can be summarized as follows:
1\. Mrk 421 showed both low and high activity phases at all wavebands during the 4.5 year period analyzed in this work. The variability increases with energy for both the SED components. Concerning the synchrotron component, the variability amplitude increases from 21% in radio and 33% in UV to 71%-73% in soft X-rays and 103%-137% in hard X-rays. For the Inverse Compton component, the amplitude is 39% at GeV energies and increases to 84% at TeV energies.
2\. The time correlation among the flux variation in different wavebands was analyzed. The variation of the X-ray flux is clearly correlated with the TeV $\gamma$-ray flux. This result is consistent with many previous observations in the past decades, supporting the idea that the X-ray and VHE $\gamma$-ray emissions originate from the same zone. The GeV $\gamma$-ray flux appears to be moderately correlated with the TeV $\gamma$-ray flux. This result is new compared to previous results. The correlation is mainly due to the large GeV $\gamma$-ray flares occurred in 2012 and a large X-ray/TeV $\gamma$-ray flare in 2010. Taking into account the spectral shape during this flares, we can conclude that the GeV and TeV $\gamma$-rays also originate from the same zone, at least during these flares. On the contrary, X-ray and $\gamma$-ray fluxes are weakly or not correlated with radio and UV fluxes.
3\. According to the observed light curves, ten states (two steady periods, S1 and S2, one outburst period, OB, and seven large flares) have been selected and analyzed. Five flares have been identified in X-rays, and two in GeV $\gamma$-rays. The duration of the flares ranges between 3 and 58 days. X-ray and TeV $\gamma$-ray fluxes increase during all the active states. In X-rays, the flux increases by a factor 4$-$70 and the peak energy increases from 0.4 keV to 1.4$-$5.1 keV. At TeV energies, the flux has been observed to vary from 0.33 I$_{crab}$ to 7 I$_{crab}$.
4\. According to the behavior of GeV $\gamma$-rays, the activity states can be classified into three groups. (I) flares with no or little GeV flux and photon index variations, e.g. S1, S2, OB, F3, F4 and F5; (II) flares with $\gamma$-ray spectral hardening, irrespective of the flux variations, e.g. F1 and F2; (III) flares with flux enhancements, irrespective of spectral behavior, e.g. F6 and F7.
5\. A simple one-zone SSC model is adopted to fit the multi-wavelength SED state by state. The SSC model can satisfactorily reproduce all the SED measurement except the $Swift$-XRT data below 0.5 keV, probably due to a detector systematics affecting the low energy events. For type I and III states, the derived injected electron spectral indices are around 2.2, as expected from relativistic diffuse shock acceleration, indicating that this process can be active in Mrk 421. According to the derived parameters, the variation of these states may be caused by the variation of environment properties. Instead, type II flares require harder injected electron spectra, with spectral indices around 1.7$-$1.9. The underlying physical mechanisms responsible for this type of flares may be related to the acceleration process itself.
This work is supported in China by NSFC (No.11205165, No.11575203, No.11375210), the Chinese Ministry of Science and Technology, the Chinese Academy of Sciences, the Key Laboratory of Particle Astrophysics, CAS, and in Italy by the Istituto Nazionale di Fisica Nucleare (INFN). We are grateful to Dahai Yan and Liang Chen for their helpful suggestions that improved the paper. We also acknowledge the essential supports of W.Y. Chen, G. Yang, X.F. Yuan, C.Y. Zhao, R. Assiro, B. Biondo, S. Bricola, F. Budano, A. Corvaglia, B. D’Aquino, R. Esposito, A. Innocente, A. Mangano, E. Pastori, C. Pinto, E. Reali, F. Taurino and A. Zerbini, in the installation, debugging and maintenance of the detector of ARGO-YBJ. This research has made use of data and software provided by the High Energy Astrophysics Science Archive Research Center (HEASARC), which is a service of the Astrophysics Science Division at NASA/GSFC and the High Energy Astrophysics Division of the Smithsonian Astrophysical Observatory. We acknowledge the use of the ASM/RXTE quick-look results provided by the ASM/RXTE team, the MAXI data provided by RIKEN, JAXA and the MAXI team, Swift/BAT transient monitor results provided by the Swift/BAT team, and the 15 GHz data provided by the OVRO team.
Abdo, A. A., Ackermann, M., Ajello, M., et al. 2011, , 736, 131 Abdo, A. A., Abeysekara, A. U., Allen, B. T., et al. 2014, , 7824, 110 Abramowski, A., Acero, F., Aharonian, F., et al. 2012, , 746, 151 Acciari, V. A., Aliu, E., Arlen, T., et al. 2009a, Science, 325, 444 Acciari, V. A., Aliu, E., Aune, T., et al. 2009b, , 703, 169 Acciari, V. A., Aliu, E., Arlen, T., et al. 2011, , 738, 25 Acciari, V. A., Arlen, T., Aune, T., et al. 2014, Astrop. Phys., 54, 1 Aharonian, F.A. 2000, New Astron, 5, 377 Aielli, G., et al. 2006, Nucl. Instrum. Meth. A, 562, 92 Aielli, G., Bacci, C., Bartoli, B., et al. 2010, , 714, L208 Ajello, M., Rau, A., Greiner, J., et al. 2008, , 673, 96 Albert, J., Aliu, E., Anderhubet, H., et al. 2007, , 663, 125 Aleksic, J., Ansoldi, S., Antonelli, L. A., et al. 2015a, A&A, 576, 126 Aleksic, J., Ansoldi, S., Antonelli, L. A., et al. 2015b, A&A, 578, 22 Amenomori, M., Ayabe, S., Chen, D., et al. 2005, , 633, 1005 Ammando F. D., Orienti, M. 2012, ATel\#4261 Atwood, W. B., Abdo, A. A., Ackermann, M., et al. 2009, , 697, 1071 Bartoli, B., Bernardini, P., Bi, X. J., et al. 2011a, , 734, 110 Bartoli, B., Bernardini, P., Bi, X. J., et al. 2012a, , 745, L22 Bartoli, B., Bernardini, P., Bi, X. J., et al. 2012b, , 758, 2 Bartoli, B., Bernardini, P., Bi, X. J., et al. 2012c, , 760, 110 Bartoli, B., Bernardini, P., Bi, X. J., et al. 2012d, Phys. Rev. D, 85, 022002 Bartoli, B., Bernardini, P., Bi, X. J., et al. 2012e, Phys. Rev. D, 85, 092005 Bartoli, B., Bernardini, P., Bi, X. J., et al. ATel\#4272 Bartoli, B., Bernardini, P., Bi, X. J., et al. 2013a, , 779, 27 Bartoli, B., Bernardini, P., Bi, X. J., et al. 2013b, , 767, 99 Bartoli, B., Bernardini, P., Bi, X. J., et al. 2013c, Phys. Rev. D, 88, 082001 Bartoli, B., Bernardini, P., Bi, X. J., et al. 2014, , 790, 152 Bartoli, B., Bernardini, P., Bi, X. J., et al. 2015, , 798, 119 Blazejowski, M., et al. 2005, , 630, 130 Burrows, D. N., Hill, J. E., Nousek, J. A., et al. 2005, Space Sci. Rev., 120, 165 Cardelli, J. A., Clayton, G. C., and Mathis, J. S. 1989, , 345, 245 Cao, G, Wang, J. C. 2013, Publications of the Astronomical Society of Japan, 65,109 Chen, L. 2014, , 788, 179 Chen, S. Z. 2013, SCIENCE CHINA Physics, Mechanics & Astronomy, 56, 1454 Cheung, C. C., Harris, D. E., and Stawarz, ${\L}$ 2007, , 663, L65 Dermer, C. D., et al. 1992, A&A, 256, L27 Drury, L. O., 1983, Rep. Prog. Phys., 46, 973 Edelson, R. R., & Krolik, J. H. 1988, , 333, 646 Edelson, R. A., et al. 1996, , 470, 364 Fleysher, R., Fleysher, L., Nemethy, P., & Mincer, A. I. 2004, , 603, 355 Fortson, L., VERITAS Collaboration, AIP Conf. Proc. 1505 (2012) 514. Franceschini, A., et al. 2008, A & A, 487, 837 Galante, N., 2011, in Proc. 32nd ICRC,(or arXiv:1109.6059) Gehrels, N., et al. 2004, , 611, 1005 Ghisellini, G., et al. 1998, MNRAS, 301, 451 Guo, F., Li, H., Daughton, W., & Liu Y. H. 2014, PRL, 113, 155005 Guo, Y. Q., Zhang, X. Y., Zhang, J. L., et al. 2010, Chin. Phys. C (HEP & NP), 34, 555 Hovatta, T., et al. 2015, MNRAS, 448, 3121 Isobe, N., Sugimori, K., Kawai, N., et al. 2010, Publications of the Astronomical Society of Japan, 62, L55 Kalberla, P. M. W., Burton, W. B., Hartmann, Dap, et al. 2005, A&A, 440, 775 Kino, M., Takahara, F., and Kusunose, M., 2002, , 564, 97 Kirk, J. G., Duffy, P., 1999, Journal of Physics G Nuclear Physics, 25, 163 Krawczynski, H. 2004, NewA Rev., 48, 367 Krimm, H.A., Holland1, S.T., Corbet, R. H. D., et al. 2013 , 209, 14 Levine, A. M., Bradt, H., Cui, W., et al. 1996, , 469, L33 Massaro, F., Tramacere, A., Cavaliere, A., Perri, M., and Giommi, P. 2008, A & A, 478, 395 Matsuoka, M., Kawasaki, K., Ueno, S., et al. 2009, PASJ, 61, 999 Max-Moerbeck, W., Hovatta, T., Richardsm J. L., et al. 2014, MNRAS, 445, 428 Meyer, E. T., Fossati, G., Georganopoulos, M., and Lister, M. L. 2012, , 752, L4 Nolan, P. L., Abdo, A. A., Ackermann, M. et al. 2012, , 199, 31 Ong, R.A. 2010, ATel\#2443 Peng, Y. P., Yan, D. H., & Zhang, L. 2014, MNRAS, 442, 2357 Poole, T.S., Breeveld, A. A., Page,M. J. et al. 2008, MNRAS, 383, 627 Richards, J. L. et al., 2011, , 194, 29 Roming, P. W. A., Kennedy, T. E., Mason, K. O., et al. 2005, Space Sci. Rev., 120, 95 Schlafly, E. F., Finkbeiner, D. P. 2011, 737, 103 Sikora, M., Begelman, M.C., and Rees, M.J. 1994, , 421, 153 Singh, K.K, et al. 2015, Astrop. Phys., 61, 32 Shukla, A., Chitnis, V. R., Vishwanath, P. R., et al. 2012, A&A, 541, 140 Stecker, F. W., Baring, M. G., and Summerlin, E. J. 2007, , 667, L29 Tluczykont, M., Bernardini, E., Satalecka, K., Clavero, R., Shayduk, M., & Kalekin, O. 2010, A&A, 524, 48 Tluczykont, M., H.E.S.S. Collaboration, 2011, arXiv:1106.1035. Tueller, J., Baumgartner,W. H., Markwardt, C. B., et al. 2010, , 186, 378 Virtanen, J. J. P., & Vainio, R. 2005, , 621:313 Wagner, R. 2008, arXiv:0808.2483v1 Yan, D. H., Zhang, L., Yuan, Q., et al. 2013, , 765,122 Yan, D. H., Zeng, H. D., Zhang, L. 2014, MNRAS, 439, 2933 Zhang, J., Liang, E. W., Zhang, S. N., and Bai, J. M. 2012, , 752, 157
[^1]: http://tevcat.uchicago.edu/(Version: 3.400, as of March 2015).
[^2]: http://fermi.gsfc.nasa.gov/ssc/
[^3]: http://fermi.gsfc.nasa.gov/ssc/
[^4]: Transient monitor results provided by the $Swift$-BAT team: http://heasarc.gsfc.nasa.gov/docs/swift/results/\
transients/weak/.
[^5]: http://heasarc.nasa.gov/docs/archive.html
[^6]: The MAXI data are provided by RIKEN, JAXA and the MAXI team: http://maxi.riken.jp/top/
[^7]: Quick-look results provided by the $RXTE$-ASM team: http://xte.mit.edu/ASM\_lc.html.
[^8]: http://www.swift.psu.edu/monitoring/
[^9]: http://heasarc.nasa.gov/docs/archive.html
[^10]: http://heasarc.gsfc.nasa.gov/docs/heasarc/caldb/caldb$_{-}$supported$_{-}$miss -ions.html
[^11]: http://heasarc.nasa.gov/lheasoft/
[^12]: http://heasarc.gsfc.nasa.gov/xanadu/xspec/
[^13]: http://www.astro.caltech.edu/ovroblazars/
[^14]: http://heasarc.gsfc.nasa.gov/docs/heasarc/caldb/swift/docs/xrt/
|
---
abstract: |
Let X be a finite type simply connected rationally elliptic CW-complex with Sullivan minimal model $(\Lambda V, d)$ and let $k \geq 2$ the biggest integer such that $d=\sum \limits_{\underset{}{i\geq k}}d_i$ with $d_i(V ) \subseteq \Lambda^iV$.
In [@murillo02] the authors showed that if $(\Lambda V,d_k)$ is morever elliptic then $cat(\Lambda V,d)=(k-2)dimV^{even} + dimV^{odd}.$
Our work focuses on the estimation of L.S.-category of such spaces in the case when $k=3$ and when $(\Lambda V,d_3)$ is not necessarily elliptic.
address: |
Département de Mathématiques & Informatique,\
Université My Ismail, B. P. 11 201 Zitoune, Meknès, Morocco,
author:
- 'K. Boutahir and, Y. Rami'
date: '11/10/2013'
title: 'On L.S.-category of a family of rational elliptic spaces\'
---
Introduction \[intro\]
======================
Let X be a finite type simply connected CW- complex with Sullivan minimal model $(\Lambda V, d)$ and let $k \geq 2$ the biggest integer such that $d=\sum \limits_{\underset{}{i\geq k}}d_i$ with $d_i(V ) \subseteq \Lambda^iV$ and $dim(V ) < \infty$.\
Consider on $(\Lambda V, d)$ the filtration given by $$F^p = \Lambda^{\geq (k-1)p}V =\bigoplus_{i=(k-1)p}^{\infty}\Lambda^i V.$$ $F^p$ is preserved by the differential d and satisfies $F^p(\Lambda V)\otimes F^q(\Lambda V)\subseteq F^{p+q}(\Lambda V)$, $\forall p, q \geq 0$, so it is a filtration of differential graded algebras. Also, since $F^0=\Lambda V$ and $F^{p+1}\subseteq F^p$ this filtration is decreasing and bounded, so it induces a convergent spectral sequence. Its $0^{th}$-term is $$E_0^{p,q}=\bigg(\frac{F^p}{F^{p+1}}\bigg)^{p+q} = \bigg(\frac{\Lambda^{\geq (k-1)p}V}{\Lambda^{\geq (k-1)(p+1)}V}\bigg)^{p+q}.$$ Hence, we have the identification: $$\label{ssk}
E_0^{p,q}=
\big(\Lambda^{p(k-1)}V\oplus\Lambda^{p(k-1)+1}V\oplus...\oplus\Lambda^{p(k-1)+k-2}V
\big)^{p+q}\,\,\,\,\,\,\,\,\,$$
In this general situation, the $1^{th}$-term is the graded algebra $\Lambda V$ proveded with a differential $\delta $, which is’nt necessarely a derivation on the set $V$ of generators (see $\S3$). That is $(\Lambda V, \delta )$ is a commutative differential graded algebra, but it is not a Sullivan algebra. The spectral sequence is therefore: $$H^{p,q}(\Lambda V, \delta)\Rightarrow H^{p+q}(\Lambda V, d).$$ Hence if $dim(V)< \infty$ and $(\Lambda V,\delta)$ has finite dimensional cohomology, then $(\Lambda V,d)$ is elliptic. This gives a new family of rationally elliptic spaces for which $d=\sum \limits_{\underset{}{i\geq k}}d_i$ Recall first that in [@murillo02] the authors gives the explicit formula $cat(\Lambda V,d) = dimV^{odd} + (k-1)dimV^{even}$ of L.-S. category for a minimal Sullivan model $(\Lambda V,d)$ satisfying the restrictive condition : $(\Lambda V,d_k)$ is also elleptic.
It is important to note also that their algorithm that induces the fundamental class of $(\Lambda V,d)$ from that of $(\Lambda V,d_k)$ corresponds to the progress of a cocycle that survive to term $E_{\infty}$ (cf. Remark 1).
The main result of this work is a project of determination of an explicit formula for $cat(\Lambda V,d)$ with $(\Lambda V,d)$ being elliptic and $(\Lambda V,d_k)$ not elliptic, completing the formula given by L. Lechuga and A. Murillo in [@murillo02].
In what follow, we consider the case where $d=\sum \limits_{\underset{}{i\geq 3}}d_i$, that is where $k=3$ and $N$ designate the formal dimention of $(\Lambda V, d)$. With the notation as above, our first result reads:
If $(\Lambda V,d)$ is elleptic and $H^N(\Lambda V, \delta ) = \mathbb{Q}.\alpha$ is one dimentional, then $ cat_0(X)=cat(\Lambda V,d)= sup\{ k\geq 0, \; \alpha = [\omega _0] \; with \; \omega _0 \in \Lambda^{\geq k} V\}.$
Let $(\Lambda W,d)$ a minimal Sullivan model of $(\Lambda V,\delta )$. If $dim(W)<\infty $ then ([@murillo94]) $(\Lambda W,d)$ is a Gorenstein algebra and so is $(\Lambda V,\delta )$. If additionaly $dim H(\Lambda V,\delta )<\infty$, then ([@felix82]) $(\Lambda W,d)$ is elliptic and so its L.S. category is finite. It follows ([@felix01 Th. 29.15]) that $Mcat(\Lambda V, \delta )<\infty $. Hence ([@felix88 Th. 3.6]) $H(\Lambda V, \delta )$ is a Poincaré Duality algebra. There follow the
Let $(\Lambda W,d)$ a minimal Sullivan model of $(\Lambda V,\delta )$. If $dim(W)<\infty $ and $dimH(\Lambda V, \delta )<\infty$ then $cat_0(X)= sup\{ k\geq 0, \; \alpha = [\omega _0] \; with \; \omega _0 \in \Lambda^{\geq k} V\}.$
Now if $dimH^N(\Lambda V, \delta ) > 1$, the technique used to show Theorem 1.1 can be adapted to have a similar result under this general hypothesis. The procedure is as follows:
Note first that in the proof of Theorem 1.1, the algorithm applied to the representative $\omega _0$ of the generating class of $H^N(\Lambda V,\delta)$ resulted in one of the fundamental class of $H(\Lambda V, d)$ because $\omega _0$ is a cocycle which survives to $E_{\infty}$ in the spectral sequence.
On the other hand, since $dim(V)<\infty $, we have $dimH^N(\Lambda V, \delta )<\infty $, with $N$ being the formal dimension of $(\Lambda V,d)$. Since the filtration induces on cohomology a graduation such that $H^N(\Lambda V,\delta) = \oplus _{p+q=N}H^{p,q}(\Lambda V,\delta)$, there is a basis $\{\alpha_1,...,\alpha_m\}$ of $H^N(\Lambda V,\delta)$ with $\alpha _i\in H^{p_i,q_i}(\Lambda V,\delta)$, $(1\leq i \leq m)$. That is, $\alpha _i = [(\omega _0^i,\omega _1^i)], \; \hbox{where}\; (\omega _0^i,\omega _1^i)\in \Lambda ^{2p_i}V \oplus \Lambda ^{2p_i+1}V$. Also since $(\Lambda V,d)$ is elleptic, there exist a unique $j$ such that some $\alpha _j\in H^{p_j,q_j}(\Lambda V,\delta)$ survives to $E_{\infty}$ and consequently induces an representative of the fundamental class of $(\Lambda V,d)$. Explicitly, the corresponding obstructions $[a_2^0]=0$, $[a_3^1]= 0$, $\ldots $, $[a_{t_j +l_j }^{t_j +l_j -2}] =0$ are necessary satisfied. Now, applying to $(\omega _0^j,\omega _1^j)\in \Lambda ^{2p_j}V \oplus \Lambda ^{2p_j+1}V$, the same role as that applied to $ \omega_0$ in the case of the first inequality (see §4) we obtain an $\omega _{l_j + t_j -1}\in \Lambda ^{\geq 2p_j}V$ (resp. $\omega _{l_j + t_j -1}\in \Lambda ^{\geq 2p_j +1}V$) if $\omega _0^j\not = 0$ (resp. if $\omega _0^j = 0$) representing the fundamental class of $(\Lambda V,d)$. It follows that $e_0 (\Lambda V,d)\geq 2p_j$ (resp. $e_0 (\Lambda V,d)\geq 2p_j+1$).
For the other inequality, any representative $ \omega\in\Lambda^{\geq s}V$ (where $s=e_0(\Lambda V,d)$) of the fundamental class of $(\Lambda V,d)$ induces by the same way, a representative $(\omega_0, \omega_1)\in \Lambda ^{\geq s}V$ of a certain non zero class in $H^N(\Lambda V, \delta) = \oplus _{p+q=N}H^{p,q}(\Lambda V,\delta)$. By convergence of the spectral sequence $[\omega]$ correspond to a basis element of $E_{\infty}^{s,N-s}$ whch is one-dimentional, by ellepticity. It follow that in $E_{2}^{s,N-s}$ there is an element which survives to $E_{\infty}^{s,N-s}$. Hence with notations above, $s=2p_j$ or $s=2p_j +1$. Therefore $e_0 (\Lambda V,d) = 2p_j$ or $e_0 (\Lambda V,d) = 2p_j +1$.
With the notation of the previous remark we can therefore state the following generalization of the previous theorem
If $(\Lambda V,d)$ is elleptic and $dimH^N(\Lambda V,d)=m$ with basis $\{\alpha _1, \ldots , \alpha _m\}$. Then $cat_0(X)= cat(\Lambda V,d) =r_j$ with $r_j=2p_j$ or else $r_j=2p_j +1$.
Basic facts and properties
==========================
Let $\mathbb{K}$ be a field of characteristic $\neq$ 2.
A Sullivan algebra is a free commutative differential graded algebra (cdga for short) ($\Lambda V$, d) (where $\Lambda V$ =Exterior($V^{odd}$)$\otimes$ Symmetric($V^{even}$)) generated by the graded $\mathbb{K}$-vector space $V=\bigoplus_{i=0}^{i=\infty}V^i$ which has a well ordered basis $\{x_\alpha\}$ such that $dx_\alpha$ $\in$ $\Lambda V_{<\alpha}$. Such algebra is said minimal if $deg(x_\alpha)< deg(x_\beta)$ implies $\alpha <\beta$. If $V^0 = V^1$ = 0 this is equivalent to saying that $d(V )\subseteq \bigoplus_{i=2}^{i=\infty}\Lambda^i
V$.\
A Sullivan model for a commutative differential graded algebra (A, d) is a quasi- isomorphism (morphism inducing isomorphism in cohomology) $(\Lambda V, d)\longrightarrow (A, d)$ with source, a Sullivan algebra. If $H^0(A) = K$, $H^1(A) = 0$ and $dim(H^i (A, d))
<\infty$ for all $i\geq 0$, then [@halperin92 Th.7.1], this minimal model exists. If X is a topological space any (minimal) model of the algebra $C^*(X,\mathbb{K})$ is said a Sullivan (minimal) model of X.\
The differential $d$ of any element of V is a “polynomial” in $\Lambda V$ with no linear term. A model ($\Lambda V$,$d$) is $elliptic$ if both V and $H^*$($\Lambda
V$,$d$) are finite dimentional spaces (see for example [@felix01]) .\
For an elliptic space with model ($\Lambda V$, d) the formal dimension $N$, i.e., the largest $n$ for which $H^n(\Lambda V,
d)\neq 0$, is given by [@halperin77] $$N = dim V^{even} - \sum _{i =1}^{dim V}(-1)^{|x_i|}|x_i|$$ An element $0\neq \omega \in H^N(\Lambda V, d)$ is called a fundamental or top class of $(\Lambda V, d)$.\
In [@halperin92] S.Halperin associated to any minimal model ($\Lambda V$, d) a pure model $(\Lambda V, d_{\sigma})$ defined as follows:\
If $Q=V^{even}$ and $P=V^{odd}$ then
$(\Lambda V, d_{\sigma})=(\Lambda Q\otimes\Lambda P, d_{\sigma})$; $d_{\sigma}(Q)=0 $ and $(d-d_{\sigma})(P)\subseteq\Lambda Q\otimes\Lambda^+P$
This model is related to $(\Lambda V, d)$ via the odd spectral sequence $$H^{p,q}(\Lambda V, d_{\sigma})\Rightarrow H^{p+q}(\Lambda V, d)$$
The main result using this algebra and due to S. Halperin ([@halperin77]) shows that in the rational case, if $dim(V)<\infty$, then: $$dim(H(\Lambda V,d))<\infty \Leftrightarrow dim(H(\Lambda V,d_{\sigma}))<\infty$$
If X is a topological space, cat(X) is the least integer n such that X is covered by n+1 open subset $U_i$, each contractible in X. It is an invariant of homotopy type (c.f. [@felix01]). In [@felix] Y. Félix, S. Halperin and J.M. Lemaire showed that for Poincaré duality spaces, the rational LS-category coincide with the rational Toomer invariant denoted $e_0(X)$.\
By [@felix82 Lemma 10.1] the Toomer invariant of a minimal model $e_0(\Lambda V, d)$ is the largest integer s for which there is a non trivial cohomology class in $H^*(\Lambda V, d)$ represented by a cycle in $\Lambda^{\geq s}
V$. As usual, $\Lambda^sV$ denotes the elements in $\Lambda V$ of $"wordlength" s$. For more details [@felix01], [@halperin83], [@sullivan78] are standard references.\
In [@murillo93] A. Murillo gave an expression of the fondamental class of $H(\Lambda V,d)$ in the case where $(\Lambda V,d)$ is a pure model. We recall it here:\
Assume $dim V <\infty$, choose homogeneous basis $\{x_1,...,x_n\}$, $\{y_1,...,y_m\}$ of $V^{even}$ and $V^{odd}$ respectively, and write $$dy_j = a^1_j x_1 + a^2_j x_2+...+a^{n-1}_j x_{n-1}+a^n_j x_n\,\,\,\,\,\, j=1,2,...m,$$ where each $a^i_j$ is a polynomial in the variables $x_i,x_{i+1},...,x_n$, and consider the matrix,
$$A=\begin{pmatrix}
\begin{tikzpicture}
\node (a) at (0,0) {$a_1^1$}; \node (b) at (0.5,0) {$a_1^2$}; \node
(c) at (1.5,0) {$a_1^n$};
\node (d) at (0,-0.5) {$a_2^1$}; \node (e) at
(0.5,-0.5) {$a_2^2$}; \node (f) at (1.5,-0.5) {$a_2^n$};
\node (g) at
(0,-1.5) {$a_m^1$}; \node (h) at (0.5,-1.5) {$a_m^2$}; \node (i) at
(1.5,-1.5) {$a_m^n$};
\draw[dotted] (b.east) -- (c.west);
\draw[dotted] (d.south) -- (g.north);
\draw[dotted] (e.east) -- (f.west);
\draw[dotted] (e.south) -- (h.north);
\draw[dotted] (h.east) -- (i.west);
\draw[dotted] (f.south) -- (i.north);
\end{tikzpicture}
\end{pmatrix}$$
For any $1 \leq j_1<...<j_n\leq m$, denote by $P_{j_1...j_n}$ the determinant of the matrix of order n formed by the columns $i_1, i_2, ..., i_n$ of A:
$$\begin{pmatrix}
\begin{tikzpicture}
\node (a) at (0,0) {$a_{j_1}^1$}; \node (b) at (1.5,0)
{$a_{j_1}^n$}; \node (c) at (0,-1.5) {$a_{j_n}^1$}; \node (d) at
(1.5,-1.5) {$a_{j_n}^n$};
\draw[dotted] (a.east) -- (b.west);
\draw[dotted] (b.south) -- (d.north);
\draw[dotted] (c.east) -- (d.west);
\draw[dotted] (a.south) -- (c.north);
\draw[dotted] (a.south east) -- (d.north west);
\end{tikzpicture}
\end{pmatrix}$$
Then (see [@murillo93]) if $dim H^*(\Lambda V,d)<\infty$ the element $\omega\in\Lambda V$
$$\label{omega}
\omega=\sum\limits_{\underset{}{1 \leq j_1<...<j_n\leq
m}}(-1)^{j_1+...+j_n}P_{j_1...j_n}y_1...\hat{y}_{j_1}...\hat{y}_{j_n}...y_m,\,\,\,$$
is a cycle representing the fundamental class of the cohomology algebra.
The spectral sequence
=====================
In what follows, we give the expression for $\delta$ in the case where k=3.
As mentioned in the introdction, our filtration is one of filterd differentail graded algebras, hence in this case the identification (\[ssk\]) becomes : $$E_0^{p,q} = \big(\Lambda^{2p}V\oplus\Lambda^{2p+1}V\big)^{p+q}$$ with the product given by: $$(u,v)\otimes(u',v')=(uu',uv'+vu'),\;\;\; \forall (u,v)\in E_0^{p,q}, \forall (u',v')\in E_0^{p',q'}.$$
On the other hand, since $d_1=d_2=0$ the diffential on $ E_0$ is zero , hence $E_1^{p,q} = E_0^{p,q}$ and so the identification obove gives the following diagram
$$\begin{tikzpicture}
\matrix (m) [matrix of math nodes,row sep=3em,column sep=8em,minimum width=2em]
{
E_1^{p,q} & \big(\Lambda^{2p}V\oplus\Lambda^{2p+1}V\big)^{p+q} \\
E_1^{p+1,q} & \big(\Lambda^{2(p+1)}V\oplus\Lambda^{2(p+1)+1}V\big)^{p+q+1} \\};
\path[-stealth]
(m-1-1) edge node [left] {$\delta$} (m-2-1)
edge node [above] {$\cong$} (m-1-2)
(m-2-1.east|-m-2-2) edge
node [above] {$\cong$} (m-2-2)
(m-1-2) edge node [left] {$\delta$} (m-2-2)
;
\draw[color=red][<-,dashed] (0.5,-0.5) -- (1,0.5) node[pos=0.5,below] {$d_3$};
\draw[color=red][<-,dashed] (1.7,-0.5) -- (1.2,0.5) node[pos=0.5,below] {$d_4$};
\draw[color=red][<-,dashed] (1.9,-0.5) -- (2.4,0.5) node[pos=0.5,below] {$d_3$};
\end{tikzpicture}$$ with $\delta$ defined as follows, $$\delta(u,v)=(d_3u,d_3v+d_4u)$$
Let $E_1^p=E_1^{p,\ast}=\bigoplus\limits_{\underset{}{q\geq0}}E_1^{p,q}$ and $E_1^{\ast}=\bigoplus\limits_{\underset{}{p\geq0}}E_1^{p,\ast}$. This gives a commutative differential graded algebra $(E_1^{\ast}, \delta)$ wich is the first term of our spectral sequence: $$E_2^{p,q} = H^{p,q}(\Lambda V,\delta)\Rightarrow H^{p+q}(\Lambda V,d).$$
Proof of the theorem 1.1
========================
Recall that we restrict ourself to the case $k=3$. The approch used here is inspered by that used in [@murillo02]. Note also that the subsequent notations imposed us to replace certain somes by pairs and vice-versa.\
**For the first inequality**\
We note first that since by hypothesis, $dimH^N(\Lambda V,d)=1$, the class $\alpha \in E_2^{*,*}$ must survive to $E_{\infty}$.
In what follow we put : $r = sup\{ k\geq 0, \; \alpha = [\omega _0] \; with \; \omega _0 \in \Lambda^{\geq k} V\}.$
Let then $\omega_0\in \Lambda^{\geq r}V$. We may suppose that $r=2p$ is even (inded, if $r = 2p+1$ is odd, it suffice to rewrite $\omega _0$ with the coordinate in $\Lambda ^{2p}V$ being $0$). More explicily $\omega_0\in(\Lambda^{2p} V\oplus\Lambda^{2p+1} V)\oplus(\Lambda^{2p+2} V\oplus\Lambda^{2p+3} V)\oplus...$,
Since $\mid \omega _0 \mid = N$, there is an integer $l$ such that: $$\omega_0 = \omega^0_0+\omega^1_0+...+\omega^l_0 \,\,\,\,\,\,
with \,\,\,\,\,\,\omega^i_0=(\omega^{i,1}_0,\omega^{i,2}_0)\in\Lambda^{2(p+i)} V\oplus\Lambda^{2(p+i)+1} V$$ We hace successivly: $$\delta(\omega^i_0)=\delta(\omega^{i,1}_0,\omega^{i,2}_0)=(d_3\omega^{i,1}_0,d_3\omega^{i,2}_0+d_4\omega^{i,1}_0)$$ $$\delta(\omega_0)=\sum_{i=0}^l\delta(\omega^{i,1}_0,\omega^{i,2}_0)=\sum_{i=0}^l(d_3\omega^{i,1}_0,d_3\omega^{i,2}_0+d_4\omega^{i,1}_0)$$ Also, we have $ d\omega_0=d\omega_0^0+d\omega_0^1+ ... +d\omega_0^l$, with:
$$d\omega_0^0=d(\omega_0^{0,1},\omega_0^{0,2})=(d_3\omega_0^{0,1},d_3\omega_0^{0,2}+d_4\omega_0^{0,1})+ ...\in(\Lambda^{2p+2} V\oplus\Lambda^{2p+3} V)\oplus...$$ $$d\omega_0^1=d(\omega_0^{1,1},\omega_0^{1,2})=(d_3\omega_0^{1,1},d_3\omega_0^{1,2}+d_4\omega_0^{1,1})+ ...\in(\Lambda^{2p+4} V\oplus\Lambda^{2p+5} V)\oplus...$$ $$........$$ $$d\omega_0^i=d(\omega_0^{i,1},\omega_0^{i,2})=(d_3\omega_0^{i,1},d_3\omega_0^{i,2}+d_4\omega_0^{i,1})+ ...\in(\Lambda^{2p+2i} V\oplus\Lambda^{2p+2i+1} V)\oplus...$$ Therfore $$\begin{aligned}
d\omega_0&=(d_3(\omega_0^{0,1}+\omega_0^{1,1}+\omega_0^{2,1}+...)+d_4\omega_0^{0,2}+d_5\omega_0^{0,1}+...,d_3(\omega_0^{0,2}+\omega_0^{1,2}+...)\\
&+d_4(\omega_0^{0,1}+\omega_0^{1,1}+\omega_0^{2,1}+...)+d_5\omega_0^{0,2}+d_6\omega_0^{0,1}+...)\end{aligned}$$ that is: $d\omega_0=\delta(\omega_0)+(d_4\omega_0^{0,2}+d_5\omega_0^{0,1}+...,d_5\omega_0^{0,2}+d_6\omega_0^{0,1}+...)$. As $\delta(\omega_0)=0$\
we can rewrite: $$d\omega_0=a_2^0 + a_3^0 + ... + a_{t+l}^0 \,\,\,\,\, with \,\,\,\,\,a_i^0=(a_i^{0,1},a_i^{0,2})\in\Lambda^{2(p+i)} V\oplus\Lambda^{2(p+i)+1} V$$
Note also that t is a fix integer. Indeed the degree of $a_{t+l}^0$ is greater or equal than $2(2(p+t+l)+1)$ and it coincides with N + 1, being N the formal dimension.\
Then $$N+1\geq 2(2(p+t+l)+1)$$ Hence $$t\leq \frac{1}{4}(N-4p-4l-1).$$
In what follows, we take $t$ the largest integer satisfying this enequality.
Now, we have: $$\begin{aligned}
d^2\omega_0&= da_2^0+da_3^0+ ... +da_{t+l}^0\\
&= d(a_2^{0,1},a_2^{0,2})+d(a_3^{0,1},a_3^{0,2})+...+d(a_{t+l}^{0,1},a_{t+l}^{0,2})\end{aligned}$$ with; $$\begin{aligned}
d(a_2^{0,1},a_2^{0,2})&= d_3(a_2^{0,1},a_2^{0,2})+d_4(a_2^{0,1},a_2^{0,2})+d_5(a_2^{0,1},a_2^{0,2})+...\\
&= (d_3a_2^{0,1},d_3a_2^{0,2}+d_4a_2^{0,1})+(d_5a_2^{0,1}+d_4a_2^{0,2},d_6a_2^{0,1}+d_5a_2^{0,2})+...\end{aligned}$$ $$\begin{aligned}
d(a_3^{0,1},a_3^{0,2})&= d_3(a_3^{0,1},a_3^{0,2})+d_4(a_3^{0,1},a_3^{0,2})+d_5(a_23^{0,1},a_3^{0,2})+...\\
&= (d_3a_3^{0,1},d_3a_3^{0,2}+d_4a_3^{0,1})+(d_5a_3^{0,1}+d_4a_3^{0,2},d_6a_3^{0,1}+d_5a_3^{0,2})+...\end{aligned}$$ $$........$$
It follows that: $$d^2\omega_0= (d_3a_2^{0,1},d_3a_2^{0,2}+d_4a_2^{0,1})+(d_5a_2^{0,1}+d_4a_2^{0,2}+d_3a_3^{0,1},d_6a_2^{0,1}+d_5a_2^{0,2}+d_4a_3^{0,1}+d_3a_3^{0,2})+...$$
Since $d^2\omega_0=0$, we have $(d_3a_2^{0,1},d_3a_2^{0,2}+d_4a_2^{0,1})=\delta(a_2^0)=0$ with $a_2^0\in \Lambda^{2(p+2)} V\oplus\Lambda^{2(p+2)+1} V $. Hence $a_2^0$ is a $\delta$-boundary, i.e., there is $b_2\in \Lambda^{2(p+2)-2} V\oplus\Lambda^{2(p+2)-1} V $ such that $a_2^0=\delta(b_2)$. Otherwise the cocycle will not survive to $E_3$ and a fortiori to $E_{\infty}$.
Consider $\omega_1=\omega_0-b_2$ and reconsider the previous calculation: $$\begin{aligned}
d\omega_1&=d\omega_0-db_2\\
&=(a_2^0+a_3^0+ ... +a_{t+l}^0)-(d_3b_2+d_4b_2+...+d_{t+3}b_2)\end{aligned}$$ With\
$d_3b_2=d_3(b_2^1,b_2^2)=(d_3b_2^1,d_3b_2^2)\in \Lambda^{2p+4} V\oplus\Lambda^{2p+5} V$\
$d_4b_2=d_4(b_2^1,b_2^2)=(d_4b_2^1,d_4b_2^2)\in \Lambda^{2p+5} V\oplus\Lambda^{2p+6} V $\
$ .............$
This imply that: $$\begin{aligned}
d\omega_1&=a_2^0+a_3^0+ ... +a_{r+l}^0-(d_3b_2^1,d_3b_2^2+d_4b_2^1)+...\\
&=a_2^0-\delta b_2+a_3^0+ ... +a_{r+l}^0-(d_5b_2^1+d_4b_2^2,d_5b_2^2+...)-...\\
&=a_3^0-(d_5b_2^1+d_4b_2^2,d_5b_2^2+...)+...\end{aligned}$$ and then: $$d\omega_1=a_3^1+a_4^1+...+a_{t+l}^1,\,\,\,\,\, with \,\,\,\,\, a_i^1\in \Lambda^{2(p+i)} V\oplus\Lambda^{2(p+i)+1} V$$ So, $$\begin{aligned}
d^2\omega_1&=da_3^1+da_4^1+...+da_{t+l}^1\\
&=d(a_3^{1,1},a_3^{1,2})+d(a_4^{1,1},a_4^{1,2})+...+d(a_{t+l}^{1,1},a_{t+l}^{1,2})\\
&=(d_3a_3^{1,1},d_3a_3^{1,2}+d_4a_3^{1,1})+(d_5a_3^{1,1}+d_4a_3^{1,2}+d_3a_4^{1,1},d_5a_3^{1,2}+...)+...\end{aligned}$$
Since $d^2\omega_1=0$, by wordlength reasons, $(d_3a_3^{1,1},d_3a_3^{1,2}+d_4a_3^{1,1})=\delta(a_3^1)=0$. Hence ( for the same reason as before ) $a_3^1$ is a $\delta$-boundary, i.e., there is $b_3\in \Lambda^{2(p+3)-2} V\oplus\Lambda^{2(p+3)-1} V $ such that $\delta(b_3)=a_3^1.$\
Consider $\omega_2=\omega_1-b_3.$\
By the same way we show that $$d\omega_2=a_4^2+a_5^2+...+a_{t+l}^2,\,\,\,\,\, with \,\,\,\,\,a_i^2\in \Lambda^{2(p+i)} V\oplus\Lambda^{2(p+i)+1} V.$$
We continue this process defining inductively $\omega_j=\omega_{j-1}-b_{j+1}$, $j< r+l$ such that: $$d\omega_j=a_{j+2}^j+a_{j+3}^j+...+a_{t+l}^j,\,\,\,\,\, with \,\,\,\,\,a_i^j\in \Lambda^{2(p+i)} V\oplus\Lambda^{2(p+i)+1} V$$ Also, we have: $$\omega_{t+l-2}=\omega_{t+l-3}-b_{t+l-1},\,\,\,\,\, with\,\,\,\,\,b_{t+l-1}\in \Lambda^{2(p+t+l-1)-2} V\oplus\Lambda^{2(p+t+l-1)-1} V$$ $$d\omega_{t+l-2}=a_{t+l}^{t+l-2}=\delta(b_{t+l-1})\in \Lambda^{2(p+t+l)} V\oplus\Lambda^{2(p+t+l)+1} V$$ $$d^2\omega_{t+l-2}=da_{t+l}^{t+l-2}=(d_3a_{t+l}^{t+l-2,1},d_3a_{t+l}^{t+l-2,2}+d_4a_{t+l}^{t+l-2,1})+...$$ Since $d^2\omega_{t+l-2}=0$, by wordlength reasons, $$(d_3a_{t+l}^{t+l-2,1},d_3a_{t+l}^{t+l-2,2}+d_4a_{t+l}^{t+l-2,1})=\delta(a_{t+l}^{t+l-2})=0$$ Hence $a_{t+l}^{t+l-2}$ is a $\delta$-boundary, i.e., there is $b_{t+l}\in \Lambda^{2(p+t+l)-2} V\oplus\Lambda^{2(p+t+l)-1} V $ such that $\delta(b_{t+l})=a_{t+l}^{t+l-2}$.\
Consider $\omega_{t+l-1}=\omega_{t+l-2}-b_{t+l}$.\
Note that $|d\omega_{t+l-1}|=|d\omega_{t+l-2}|=N+1$, but by the hypothesis on $t$, we have: $$|d(\omega_{t+l-2}-b_{t+l})|=|a_{t+l}^{t+l-2}-\delta(b_{t+l})-(d-\delta)b_{t+l}|=|-(d-\delta)b_{t+l}|>N+1,$$ then $d\omega_{t+l-1}=0$ and so $\omega_{t+l-1}$ can’t be a d-boundary. Indeed suppose that $\omega_{t+l-1}=(\omega_0^0+\omega_0^1+...+\omega_0^l)-(b_2+b_3+...+b_{t+l})$ were a d-boundary, By wordlength reasons, $\omega_0^0$ would be a $\delta$-boundary, i.e., there is $x\in \Lambda^{2p-2} V\oplus\Lambda^{2p-1} V$ such that $\delta(x)=\omega_0^0$. Then $$\omega_0=\delta(x)+\omega_0^1+...+\omega_0^l$$ Since $\delta(\omega_0)=0$ , we would have $\delta(\omega_0^1+...+\omega_0^l)=0$, but $\omega_0^1+...+\omega_0^l$ is not a $\delta$-boundary.\
Thus $\omega_{t+l-1}$ is a non trivial cocycle of degree $N$, the formal dimension, and therefore it represents the fundamental class.\
Finaly, since $\omega_{t+l-1}\in \Lambda^{\geq r} V$ we have;
$$e_0(\Lambda V,d)\geq r$$
**For the second inequality**\
Denote $s=e_0(\Lambda V,d)$ and let $\omega\in\Lambda^{\geq s} V$ be a cocycle representing the generating class $\alpha$ of $H^*(\Lambda V,d)$ . Write $ \omega=\omega_0+\omega_1+...+\omega_t, \,\, \,\,\,\,\omega_i\in\Lambda^{s+i} V$. We deduce that: $$\begin{aligned}
d\omega&=(d_3\omega_0+d_3\omega_1+...+d_3\omega_r)+(d_4\omega_0+d_4\omega_1+...+d_4\omega_t)+...\\
&=\delta(\omega_0 , \omega_1)+...\end{aligned}$$
Since $d\omega=0$, by wordlength reasons, it follows that $\delta(\omega_0 , \omega_1)=0$.\
If $(\omega_0 , \omega_1)$ were a $\delta$-boundary, i.e., $(\omega_0 , \omega_1)$=$\delta(x)$, then $$\begin{aligned}
\omega-dx&=(\omega_0 , \omega_1) +...+ \omega_r-(d_3x+d_4x+...)\\
&=(\omega_0 , \omega_1)-\delta(x)+(\omega_2+\omega_3+...+\omega_t)-...
\end{aligned}$$
so $\omega-dx\in \Lambda^{\geq s+2} V$ which contradicts the fact $s=e_0(\Lambda V,d)$.\
Hence $(\omega_0 , \omega_1)$ represents the generating class of $H^N(\Lambda V,\delta)$.\
Since $(\omega_0 ,\omega_1)\in \Lambda^{\geq s} V$ we will have $s\leq r$
Hence $$e_0(\Lambda V,d)\leq r$$ We conclude that $$e_0(\Lambda V,d)= r$$
Some examples and remarks
=========================
Let $(\Lambda V,d)$ be the pure model defined by $V^{even}=<x_2,x_6>$,\
$V^{odd}=<y_5, y_{15}, y_{23}>$ , $dx_2=dx_6=0$, $dy_5=x_2^3$, $dy_{15}= x_2^2x_6^2$ and $dy_{23}=x_6^4 $.
Clearly we have $dimH(\Lambda V,d_3) = \infty $ and $dimH(\Lambda V,d) < \infty $.
We note also that, since $N = 37$ is odd, then any representative of the fundamental class of $(\Lambda V, d)$ will be of the form: $n_1x_2^kx_6^ly_5 + n_2x_2^{k'}x_6^{l'}y_{15} + n_3x_2^{k''}x_6^{l''}y_{23}$, with $n_1, \; n_2 \; \hbox{and}\; n_3 \in \mathbb{N}$.
Using A. Murillo’s algorithm (cf. §2) the matrix determining the fundamental class is: $$A= \begin{pmatrix}
\begin{tikzpicture}
\node (a) at (0,0) {$x_2^2$}; \node (b) at (1,0) {0}; \node
(c)at(0,-0.5){$x_2x_6^2$};
\node (d) at (1,-0.5) {0};
\node (e) at (0,-1) {0}; \node (f) at (1,-1) {$x_6^3$};
\end{tikzpicture}
\end{pmatrix}$$
So $\omega _0 =
-x_2^2x^{3}_6y_{15} + x_2x_6^5y_5 \in \Lambda^{\geq6} V$ is an generator of this fundamental cohomology class. As in the first example, it is straightforward to verify that there is only two representatives, with $\omega _1 = x_2^4x_6y_{23} - x_2^2x_6^3y_{15}$ being the second one. It follow that $e_0(\Lambda V,d) = 6$.
Remark also that for this model, $(\omega _0^0, \omega _0^1)=(-x_2^2x^{3}_6y_{15} , x_2x_6^5y_5)\in \Lambda ^6V \oplus \Lambda ^7V$ is a $\delta$-cocycle and in fact $[(\omega _0^0, \omega _0^1)]\in H^N(\Lambda V,\delta)$ is non zero. $[(x_2x_6^2y_{23} ,0)]$ is another generating class, hence $dimH^N(\Lambda V,\delta) >1$. The algorithm described in remark 1. is applied to $(\omega _0^0, \omega _0^1)$.
On the other hand, $\omega _0$ is not an $d_3$-cocycles, but $0\not =[\omega _1]\in H^N(\Lambda V,d_3)$. Also $0 \not = [x_2x_6^2y_{23}]$ is another generating class of $H^N(\Lambda V,d_3)$, hence $dimH^N(\Lambda V,d_3)>1$. Application of the algorithm in the proof of Theorem 5. in [@murillo02] to $\omega _1$ (which is a homogenious $d_3$-cocycle) gives immediatly $\omega _1$ as a representative of the fundamental class of $(\Lambda V,d)$.
Finaly we note also that $e_0(\Lambda V,d) = 6 \not = (k-2)dimV^{even} + dimV^{odd} =5$.
Let $(\Lambda V,d)$ be the pure model defined by $V^{even}=<x_2,x_6>$,\
$V^{odd}=<y_5, y_{13}, y_{23}>$ , $dx_2=dx_6=0$, $dy_5=x_2^3$, $dy_{13}= x_2x_6^2$ and $dy_{23}=x_6^4 $.
Clearly we have $dimH(\Lambda V,d_3) = \infty $ and $dimH(\Lambda V,d) < \infty $.
We note also that, since $N = 35$ is odd, then any representative of the fundamental class of $(\Lambda V, d)$ will be of the form: $n_1x_2^kx_6^ly_5 + n_2x_2^{k'}x_6^{l'}y_{13} + n_3x_2^{k''}x_6^{l''}y_{23}$, with $n_1, \; n_2 \; \hbox{and}\; n_3 \in \mathbb{N}$.
Using A. Murillo’s algorithm (cf. §2) the matrix determining the fundamental class is: $$A= \begin{pmatrix}
\begin{tikzpicture}
\node (a) at (0,0) {$x_2^2$}; \node (b) at (1,0) {0}; \node
(c)at(0,-0.5){$x_6^2$};
\node (d) at (1,-0.5) {0};
\node (e) at (0,-1) {0}; \node (f) at (1,-1) {$x_6^3$};
\end{tikzpicture}
\end{pmatrix}$$
So $\omega _0 =
-x_2^2x^{3}_6y_{13} + x_6^5y_5 \in \Lambda^{\geq6} V$ is an generator of this fundamental cohomology class. Another representative of this class is $\omega _1 = - x_2^3x_6y_{23} + x_2^2x_6^3y_{13}$. It is a straightforward calculation to prouve that they are the uniques representatives. We conclude that $e_0(\Lambda V,d)= 6$.
On the other hand $H^N(\Lambda V, \delta)$ has at least tow generators: $(\omega _0, 0) \in \Lambda ^6 V\oplus \Lambda ^7V$ and $[(0,x_6^2y_{23})]$, hence $dimH^N(\Lambda V, \delta)>1$. We have also $dimH^N(\Lambda V,d_3) >1$ with $[\omega _0]$ and $[x_6^2y_{23}]$ being two generators of $H^N(\Lambda V,d_3)$. Here the algorithm is applied to $(\omega _0, 0)$ and the one of [@murillo02] is applied to $[\omega _0]$.
Note finaly that $e_0(\Lambda
V,d)=6\neq (k-2)dimV^{even} + dimV^{odd} = 5.$
It should be noted that the algorithms that are described in [@murillo02] and in Remark 1. are both valid in the previous examples. The previlege of one or the other depends on $dimH^N(\Lambda V, \delta)$ and $dimH^N(\Lambda V, d_3)$ and also in the expressions of there basis.
On the other hand all the lower bounds for $e_0(\Lambda V,d)$ known up to now can be used to relax the application of the algorithm.
[9]{}
Y. Félix, S. Halperin, *Rational LS-category and its applications*. Trans. Amer. Math. Soc. 273 (1982) 1-37.
Y. Félix, S. Halperin and J.-C. Thomas, *Rational homotopy theory*. Graduate Texts in Mathematics 205, Springer-Verlag, 2001.
Y. Félix, S. Halperin, and J. M. Lemaire, *The Rational LS-category of Products and Poincaré Duality Camplexes*, Topology 37.
Y. Félix, S. Halperin and J.-C. Thomas, *Gorenstein spaces*. Academic Press; New Yprk and London Vol.71, No.1, September 1988
S. Halperin , *Finiteness in the minimal models of Sullivan*. Trans. Amer. Math. Soc. 230 (1977) 173-199
S. Halperin , *Lectures on minimal models*. Mém. Soc. Math. France 9/10 (1983).
S. Halperin , *Universal enveloping algebras and loop space homology*. Journal of Pure and Applied Algebra, 83 (1992), 237-282. A. Murillo, *The top cohomology class of certain spaces*. Journal of Pure and Applied. Algebra, 84:209-214, 1993. A. Murillo, *The evaluation map of some Gorenstein algebras*. ournal of Pure and Applied Algebra, 91:209-218, 1994.
L. Lechuga, A. Murillo, *The fundamental class of a rational space, the graph coloring problem and other classical decision problems*. Bull. Belgian Math. Soc. 8 (2001),451-467.
L. Lechuga, A. Murillo, *A formula for the rational LS-category of certain spaces*. Ann. L’inst. Fourier, 2002.
D. Sullivan , *Infinitesimal computations in topology*. Publ. Math. IHES (1978) 269-331.
|
---
bibliography:
- 'Bibli.bib'
---
[**Simple examples of perfectly invisible\
and trapped modes in waveguides**]{}
<span style="font-variant:small-caps;">Lucas Chesnel</span>$^1$, <span style="font-variant:small-caps;">Vincent Pagneux</span>$^2$\
$^1$ INRIA/Centre de mathématiques appliquées, École Polytechnique, Université Paris-Saclay, Route de Saclay, 91128 Palaiseau, France;\
$^2$ Laboratoire d’Acoustique de l’Université du Maine, Av. Olivier Messiaen, 72085 Le Mans, France.\
E-mails: `lucas.chesnel@inria.fr`, `vincent.pagneux@univ-lemans.fr`\
– –
**Abstract.** We consider the propagation of waves in a waveguide with Neumann boundary conditions. We work at low wavenumber focusing our attention on the monomode regime. We assume that the waveguide is symmetric with respect to an axis orthogonal to the longitudinal direction and is endowed with a branch of height $L$ whose width coincides with the wavelength of the propagating modes. In this setting, tuning the parameter $L$, we prove the existence of simple geometries where the transmission coefficient is equal to one (perfect invisibility). We also show that these geometries, for possibly different values of $L$, support so called trapped modes (non zero solutions of finite energy of the homogeneous problem) associated with eigenvalues embedded in the continuous spectrum.\
**Key words.** Waveguides, invisibility, trapped modes, scattering matrix, asymptotic analysis.
Introduction
============
In this work, we consider a problem of wave propagation, governed by the Helmholtz equation, in a waveguide unbounded in one direction, the longitudinal direction $(Ox)$, in frequency regime. We supplement it with Neumann boundary conditions. Such a problem appears for example in the theory of water-waves, in acoustics or in electromagnetism. We shall assume that the wavenumber $k$ is sufficiently small so that only one mode (the piston mode) propagates. We will be particularly interested in the scattering of the piston mode coming from $-\infty$ by the structure. To describe such a process, we introduce two complex coefficients, the so-called *reflection* and *transmission* coefficients, denoted ${\mathcal{R}}$ and ${\mathcal{T}}$, such that ${\mathcal{R}}$ (resp. ${\mathcal{T}}-1$) corresponds to the amplitude of the scattered field at $x=-\infty$ (resp. $x=+\infty$) (see (\[DefScatteringCoeff\])). According to the conservation of energy, we have $$\label{NRJconservation}
|{\mathcal{R}}|^2+|{\mathcal{T}}|^2=1.$$ In the first part of the article, we explain how to construct waveguides, different from the straight (reference) geometry, such that ${\mathcal{R}}=0$, ${\mathcal{T}}=1$. In this case, we shall say that the total field is a *perfectly invisible mode*. The reason is that in this situation, the scattered field is exponentially decaying both at $\pm\infty$ and for an observer with measurement devices located far from the geometrical defect, everything happens like in the reference waveguide. In other words, the geometrical perturbation is invisible from far field measurements. The problem of imposing ${\mathcal{R}}=0$, ${\mathcal{T}}=1$ seems quite new in literature, at least when one looks for proofs. A simpler problem consists in finding *non reflecting* geometries such that ${\mathcal{R}}=0$ (and so $|{\mathcal{T}}|=1$ according to (\[NRJconservation\])). For such waveguides, all the energy is transmitted but there is a possible shift of phase for the field at $x=+\infty$. However, one can speak of backscattering invisibility. The latter problem has been investigated numerically ($|{\mathcal{R}}|$ small) for example in [@PoNe14; @EvMP14] for water wave problems and in [@AlSE08; @EASE09; @NgCH10; @OuMP13; @FuXC14], with strategies based on the use of new “zero-index” and “epsilon near zero” metamaterials, in electromagnetism (see [@FlAl13] for an application to acoustic).\
A perturbative approach, relying on the fact that ${\mathcal{R}}=0$ in the straight geometry, has been proposed in [@BoNa13] to construct waveguides such that ${\mathcal{R}}=0$, $|{\mathcal{T}}|=1$ (see also [@BLMN15; @ChNa16; @BoCNSu; @ChHS15] for applications in related contexts). It has been adapted in [@BoCNSu] to solve the problem of imposing ${\mathcal{R}}=0$, ${\mathcal{T}}=1$. The technique, based on the proof of the implicit function theorem, allows one to design invisible perturbations which *a priori* are small with respect to the wavelength. An alternative method has been developed in [@ChNPSu] to obtain larger invisible defects. In the latter work, it is explained how to get ${\mathcal{R}}=0$, $|{\mathcal{T}}|=1$ in waveguides which are symmetric with respect to the $(Oy)$ axis (perpendicular to the direction of propagation). The idea consists in using the symmetry properties and to play with a branch of finite height $L$ making $L\to+\infty$. In [@ChNPSu], it is also shown how to work with three branches to impose ${\mathcal{R}}=0$, ${\mathcal{T}}=1$. In the present article, we use a similar idea but, importantly, we work with only one branch whose width coincides with the wavelength of the incident wave. We provide examples of very simple geometries where ${\mathcal{R}}=0$, ${\mathcal{T}}=1$ (which is not obvious to attain in general).\
In the second part of the paper, we will be concerned with so-called *trapped modes*. We remind the reader that trapped modes are non zero solutions of the homogeneous problem (\[PbInitial\]) (without source term) which are of finite energy. It has been known for a while that trapped modes play a key role in physical systems and they have been widely studied. In particular, literature concerning trapped modes is much more developed than the one concerning invisible modes. We refer the reader for example to [@Urse51; @Evan92; @EvLV94; @DaPa98; @LiMc07; @Naza10c; @NaVi10; @CPMP11; @Pagn13]. More precisely, in our work, we will be interested in trapped modes associated with eigenvalues embedded in the continuous spectrum. Such trapped modes are also called *Bound States in the Continuum* (BSCs or BICs) in quantum mechanics (see for example [@SaBR06; @Mois09; @GPRO10] as well as the recent review [@HZSJS16]). Such objects are difficult to observe. In particular they are unstable with respect to the geometry and a small perturbation transforms them in general into complex resonances [@AsPV00]. A classical method to construct trapped modes associated with eigenvalues embedded in the continuous spectrum consists in working with waveguides which are symmetric with respect to the $(Ox)$ axis [@Evan92; @DaPa98; @LiMc07; @Pagn13]. This is interesting because it allows one to decouple symmetric and skew-symmetric modes. Then the idea is to design a defect such that trapped modes exist below the continuous spectrum for the problem with mixed boundary conditions satisfied by the skew-symmetric component of the field. Back to the original waveguide, this guarantees the existence of trapped modes. What we will do in the second part of the article is to propose a simple alternative mechanism to construct trapped modes associated with an eigenvalue embedded in the continuous spectrum. We emphasize that it will be based on symmetry arguments with respect to $(Oy)$ and not with respect to $(Ox)$. It will also involve the *augmented scattering matrix*, a convenient tool introduced in [@NaPl94bis; @KaNa02; @Naza06; @Naza11], permitting to detect the presence of trapped modes.\
The outline is as follows. In Section \[SectionPerfectInvisibility\], we exhibit geometries admitting perfectly invisible modes. In Section \[SectionExistenceOfTrappedModes\], we provide examples of geometries supporting trapped modes. In Section \[SectionNumExpe\], we give numerical illustrations of the results showing also that other geometrical shapes can be considered. We end the paper with a short conclusion (Section \[SectionConclusion\]) and an Annex where we give the proof of a classical result used in the analysis. The main results of this work are Theorem \[MainThmPart1\] (existence of perfectly invisible modes) and Theorem \[MainThmPart2\] (existence of trapped modes).
Perfectly invisible modes {#SectionPerfectInvisibility}
=========================
Setting
-------
(-2,1) rectangle (2,2); (-5,1) rectangle (-2,2); (-2,1.8) rectangle (-1,3.8); (-2,1)–(2,1); (-5,2)–(-2,2)–(-2,3.8)–(-1,3.8)–(-1,2)–(2,2); (-5,1)–(-2,1); (3,1)–(2,1); (3,2)–(2,2); (-6,2)–(-5,2); (-6,1)–(-5,1); (-2,4)–(-1,4); (-0.8,1.95)–(-0.8,3.85); at (-1.5,4.2)[$2\ell$]{}; at (-0.4,2.9)[$L-1$]{}; (-1.5,1)–(-1.5,3.8); (3,1.2)–(3.6,1.2); (3.1,1.1)–(3.1,1.7); at (3.65,1.3)[$x$]{}; at (3.25,1.6)[$y$]{};
plot\[domain=0:pi/4,samples=100\] (,[0.2\*sin(20\*r)]{}) node\[anchor=west\] [$1$]{};
plot\[domain=0:pi/4,samples=100\] (,[0.2\*sin(20\*r)]{}) node\[anchor=west\] [$\hspace{-2.4cm}\mathcal{R}$]{};
plot\[domain=0:pi/4,samples=100\] (,[0.2\*sin(20\*r)]{}) node\[anchor=west\] [$\mathcal{T}$]{};
Pick a wavenumber $k\in(0;\pi)$. Set $\ell=\pi/k$ and for $L>1$, define the waveguide (see Figure \[DomainOriginal\]) $$\label{defOriginalDomain}
{\Omega}_L:=\{ (x,y)\in{\mathbb{R}}\times(0;1)\ \cup\ (-\ell;\ell)\times [1;L)\}.$$ The value $2\pi/k$ for the width of the vertical branch of ${\Omega}_L$ is very important in the analysis we develop below as we will see later. For the main exposition, we will stick with this simple waveguide ${\Omega}_L$. However the method we propose works in other geometries as the one described in (\[defDomainBis\]) (see Figure \[figResultT1Gros\])). We consider the Helmholtz problem with Neumann boundary conditions (sound hard walls in acoustics) $$\label{PbInitial}
\begin{array}{|rcll}
\Delta v + k^2 v & = & 0 & \mbox{ in }{\Omega}_L\\[3pt]
\partial_nv & = & 0 & \mbox{ on }\partial{\Omega}_L.
\end{array}$$ In (\[PbInitial\]), $\Delta$ denotes the ${\mathrm{2D}}$ Laplace operator and $n$ refers to the outer unit normal vector to $\partial{\Omega}_{L}$. Set $$w^{\pm}_{1}(x,y)=\cfrac{1}{\sqrt{2k}}\,e^{\mp i k x}\quad\mbox{ and }\quad \hat{w}^{\pm}_{1}(x,y)=\cfrac{1}{\sqrt{2k}}\,e^{\pm i k x}.$$ Let $\chi_l\in\mathscr{C}^{\infty}({\mathbb{R}}^2)$ (resp. $\chi_r\in\mathscr{C}^{\infty}({\mathbb{R}}^2)$) be a cut-off function equal to one for $x\le -2\ell$ (resp. $x\ge2\ell$) and to zero for $x\ge -\ell$ (resp. $x\le \ell$). In order to describe the scattering process of the incident piston modes $w_1^{-}$ coming from $x=-\infty$ by the structure, we introduce the following solution of Problem (\[PbInitial\]) $$\label{DefScatteringCoeff}
v= \chi_l\,(w^{-}_{1}+{\mathcal{R}}\,w^{+}_{1})+\chi_r\,{\mathcal{T}}\,\hat{w}^{+}_{1}+\tilde{v},$$ where ${\mathcal{R}},\,{\mathcal{T}}\in{\mathbb{C}}$ and where $\tilde{v}$ decays exponentially as $O(e^{-\sqrt{\pi^2-k^2}|x|})$ for $x\to\pm\infty$. The *reflection coefficient* ${\mathcal{R}}$ and *transmission coefficient* ${\mathcal{T}}$ in (\[DefScatteringCoeff\]) are uniquely defined. Note that the cut-off functions $\chi_l$, $\chi_r$ in (\[DefScatteringCoeff\]) are just a convenient way to write radiation conditions at $x=\pm\infty$. Physically, the function $v$ defined in (\[DefScatteringCoeff\]) corresponds to the so called total field associated to the incident piston wave propagating from $-\infty$ to $+\infty$. According to the energy conservation, we have $$\label{ConservationOfNRJ}
|{\mathcal{R}}|^2+|{\mathcal{T}}|^2=1.$$ The coefficients ${\mathcal{R}}$ and ${\mathcal{T}}$ depend on the features of the geometry, in particular on $L$. From time to time, we shall write ${\mathcal{R}}(L)$, ${\mathcal{T}}(L)$ instead of ${\mathcal{R}}$, ${\mathcal{T}}$. In this section, we show that, there are some $L>1$ such that ${\mathcal{R}}=0$, ${\mathcal{T}}=1$ (*perfect invisibility*). To obtain such particular values for the scattering coefficients, we will use the fact that the geometry is symmetric with respect to the $(Oy)$ axis.
Decomposition in two half-waveguide problems
--------------------------------------------
Define the half-waveguide $$\label{defHalfWaveguide}
{\omega}_L:=\{(x,y)\in{\Omega}_L\,|\,x<0\}$$ (see Figure \[LimitDomain\] left). Introduce the problem with Neumann boundary conditions $$\label{PbChampTotalSym}
\begin{array}{|rcll}
\Delta u +k^2 u & = & 0 & \mbox{ in }{\omega}_L\\[3pt]
\partial_nu & = & 0 & \mbox{ on }\partial{\omega}_L
\end{array}$$ as well as the problem with mixed boundary conditions $$\label{PbChampTotalAntiSym}
\begin{array}{|rcll}
\Delta U + k^2 U & = & 0 & \mbox{ in }{\omega}_L\\[3pt]
\partial_nU & = & 0 & \mbox{ on }\partial{\omega}_L\cap\partial{\Omega}_L \\[3pt]
U & = & 0 & \mbox{ on }\Sigma_L:=\{0\}\times(0;L).
\end{array}$$ Problems (\[PbChampTotalSym\]) and (\[PbChampTotalAntiSym\]) admit respectively the solutions $$\begin{aligned}
\label{defZetaLsym} u &=& w^+_1+{r}\,w^-_1 + \tilde{u},\qquad\hspace{2mm}\mbox{ with }\tilde{u}\in{{\mathrm{H}}}^1({\omega}_L),\\[3pt]
\label{defZetaLanti}U &=& w^+_1+{R}\,w^-_1 + \tilde{U},\qquad\mbox{ with }\tilde{U}\in{{\mathrm{H}}}^1({\omega}_L),\end{aligned}$$ where ${r}$, ${R}\in{\mathbb{C}}$ are uniquely defined. Moreover, due to conservation of energy, one has $$\label{NRJHalfguide}
|{r}|=|{R}|=1.$$ Direct inspection shows that if $v$ is a solution of Problem (\[PbInitial\]) then we have $v(x,y)=(u(x,y)+U(x,y))/2$ in ${\omega}_L$ and $v(x,y)=(u(-x,y)-U(-x,y))/2$ in ${\Omega}_L\setminus\overline{{\omega}_L}$ (up possibly to a term which is exponentially decaying at $\pm\infty$ if there is a trapped mode at the given wavenumber $k$). We deduce that the scattering coefficients ${\mathcal{R}}$, ${\mathcal{T}}$ appearing in the decomposition (\[DefScatteringCoeff\]) of $v$ are such that $$\label{Formulas}
{\mathcal{R}}=\frac{{r}+{R}}{2}\qquad\mbox{ and }\qquad {\mathcal{T}}=\frac{{r}-{R}}{2}.$$
Consequence of the choice $\ell=\pi/k$
--------------------------------------
In the particular geometry considered here, one observes that $u':=w^+_1+w^-_1=\sqrt{2/k}\cos(kx)$ is a solution to (\[PbChampTotalSym\]) for all $L>1$. Note that this property is based on the fact that $\ell=\pi/k$, the width of the vertical branch of ${\omega}_L$, coincides with the half wavelength of the waves $w^{\pm}_1$. By uniqueness of the definition of the coefficient ${r}$ in (\[defZetaLsym\]), we deduce that $$\label{PartRelation}
{r}={r}(L)=1\qquad\mbox{ for all $L>1$}.$$ From this important remark, we see that to get ${\mathcal{T}}=1$, it remains to find $L$ such that $R=R(L)=-1$. This is the goal of the next section.
Asymptotic behaviour of the reflection coefficient for the problem with mixed boundary conditions {#paragraphAsymptoAnalysis}
-------------------------------------------------------------------------------------------------
(-4,0) rectangle (0,1); (-0.4,0) rectangle (0,2.5); (-4,1)–(-0.4,1)–(-0.4,2.5)–(0.02,2.5); (0.02,0)–(-4,0); (0,0)–(0,2.5); (-4.5,1)–(-4,1); (-4.5,0)–(-4,0); at (0.3,1.2)[$\Sigma_{L}$]{}; at (-2.4,0.2)[${\omega}_{L}$]{}; (-0.5,2.7)–(0.1,2.7); (0.7,-0.1)–(0.7,2.6); at (0.9,1.2)[$L$]{}; at (-0.2,2.9)[$\ell$]{};
plot\[domain=0:pi/4,samples=100\] (,[0.2\*sin(20\*r)]{}) node\[anchor=west\] [$1$]{};
plot\[domain=0:pi/4,samples=100\] (,[0.2\*sin(20\*r)]{}) node\[anchor=west\] [$\hspace{-2.6cm}r/R$]{};
(-4,0) rectangle (0,1); (-0.4,0) rectangle (0,3); (-4,0)–(0.05,0); (0,0)–(0,3); (-4,1)–(-0.4,1)–(-0.4,3); (-4.5,1)–(-4,1); (-4.5,0)–(-4,0); (0,3.5)–(0,3); (-0.4,3.5)–(-0.4,3); at (0.3,1.6)[$\Sigma_{\infty}$]{}; at (-3.4,0.2)[${\omega}_{\infty}$]{};
In this paragraph, we study the behaviour of the reflection coefficient ${R}={R}(L)$ of the half-waveguide problem with mixed boundary conditions (\[PbChampTotalAntiSym\]) as $L\to\infty$. We follow the approach proposed in [@ChNPSu]. In the analysis, the properties of Problem (\[PbChampTotalAntiSym\]) set in the limit geometry ${\omega}_{\infty}$ (see Figure \[LimitDomain\] right) obtained formally taking $L\to+\infty$, play a key role. Denote $$\label{DefModeVert}
w^{\pm}_{2}(x,y)=\cfrac{1}{\sqrt{\gamma\ell}}\,e^{\pm i \gamma y}\sin( \frac{\pi x}{2\ell}),\qquad \mbox{with }\gamma:=\sqrt{k^2-(\pi/(2\ell))^2}=k\sqrt{3}/2,$$ the modes propagating in the vertical branch of ${\omega}_{\infty}$. In ${\omega}_{\infty}$, there are the solutions $$\label{defMatrixScaLim}
\begin{array}{lcl}
U_{1}^{\infty}&=& \chi_l(w^-_{1}+S^{\infty}_{11}\,w^+_{1})+\chi_t\,S^{\infty}_{12}\,w^+_{2}+\tilde{U}_{1}^{\infty}\\[4pt]
U_{2}^{\infty}&=&\chi_l\,S^{\infty}_{21}\,w^+_{1}+\chi_t\,(w^-_{2}+S^{\infty}_{22}\,w^+_{2})+\tilde{U}_{2}^{\infty},
\end{array}$$ where $\tilde{U}_{1}^{\infty}$, $\tilde{U}_{2}^{\infty}$ decay exponentially at infinity. Here $\chi_t\in\mathscr{C}^{\infty}({\mathbb{R}}^2)$ is a cut-off function equal to one for $y\ge 1+2\tau$ and to zero for $y\le 1+\tau$, where $\tau>0$ is a given constant. The scattering matrix $$\label{UnboundedScatteringMatrix}
\mathcal{S}^{\infty}:=\left(\begin{array}{cc}
S^{\infty}_{11} & S^{\infty}_{12} \\
S^{\infty}_{21} & S^{\infty}_{22} \\
\end{array}\right)\in\mathbb{C}_{2\times 2}$$ is uniquely defined. Moreover, as shown in Proposition \[PropositionUnitary\] in Annex (this is a classical result), $\mathcal{S}^{\infty}$ is unitary ($\mathcal{S}^{\infty}\overline{\mathcal{S}^{\infty}}^{\top}={\mathrm{Id}}_{2\times2}$) and symmetric. For the function $U$ defined in (\[defZetaLanti\]), following for example [@MaNP00 Chap. 5, §5.6], we make the ansatz $$\label{DefAnsatzs}
U = U_{1}^{\infty}+A(L)\,U_{2}^{\infty}+\dots$$ where $A(L)$ is a gauge function to determine and where the dots correspond a to small remainder. On $(-\ell;0)\times\{L\}$, the condition $\partial_n U=0$ leads to choose $A(L)$ such that $$S^{\infty}_{12}\,e^{i\gamma L}+A(L)\,(-e^{-i\gamma L}+S^{\infty}_{22}\,e^{i\gamma L})=0\qquad\Leftrightarrow\qquad A(L)={\displaystyle}\cfrac{S^{\infty}_{12}}{e^{-2i\gamma L}-S^{\infty}_{22}}\ .$$ We shall consider ansatz (\[DefAnsatzs\]) when $|S^{\infty}_{22}|\ne1$. Since $\mathcal{S}^{\infty}$ is unitary and symmetric, this is equivalent to assume that $S^{\infty}_{12}=S^{\infty}_{21}\ne 0$. This assumption is needed so that a coupling exists between the two channels of ${\omega}_{\infty}$. In this case, the gauge function $A(L)$ is well-defined for all $L>1$. If $|S^{\infty}_{22}|=1 \Leftrightarrow S^{\infty}_{12}=0$, we can show that as $L\to+\infty$, the reflection transmission ${R}(L)$ defined in (\[defZetaLanti\]) tends to $S^{\infty}_{11}$. This exceptional case is not interesting for our analysis and therefore, we discard it assuming that $k\in(0;\pi)$ is such that $|S^{\infty}_{22}|\ne1 \Leftrightarrow S^{\infty}_{12}\ne0$ (we shall see in §\[paragraphPerfectRef\] below an example of situation where numerically $S_{12}^{\infty}\ne0$). Then we can prove that ${R}(L)={R}^{{\mathrm{asy}}}(L)+\dots$, with $$\label{defCoeffLim}
{R}^{{\mathrm{asy}}}(L)=S^{\infty}_{11}+A(L)\,S^{\infty}_{21}=S^{\infty}_{11}+{\displaystyle}\cfrac{S^{\infty}_{12}\,S^{\infty}_{21}}{e^{-2i\gamma L}-S^{\infty}_{22}}\ .$$ Here the dots stand for exponentially small terms. More precisely, we can establish (work as in [@MaNP00 Chap. 5, §5.6]) an error estimate of the form $|{R}(L)-{R}^{{\mathrm{asy}}}(L)| \le C\,e^{-\gamma L}$ where $C$ is a constant independent of $L>1$ and $\gamma=k\sqrt{3}/2$ (see Definition (\[DefModeVert\])). Observe that $L\mapsto {R}^{{\mathrm{asy}}}(L)$ is a periodic function of period $\pi/\gamma=2\pi/(k\sqrt{3})=2\ell/\sqrt{3}$. As a consequence, $L\mapsto {R}(L)$ is “almost periodic” of period $2\pi/(k\sqrt{3})$. Denote $\mathscr{C}:=\{z\in{\mathbb{C}}\,|\,|z|=1\}$ the unit circle. As $L$ tends to $+\infty$, the coefficient ${R}^{{\mathrm{asy}}}(L)$ runs on the set $$\label{setSPlusMoins}
\{S^{\infty}_{11}+{\displaystyle}\frac{S^{\infty}_{12}\,S^{\infty}_{21}}{z-S^{\infty}_{22}} \,|\,z\in\mathscr{C}\}.$$ Using classical results concerning the Möbius transform (see e.g. [@Henr74 Chap. 5]), one finds that this set is the circle centered at $$\label{eqnCenters}
\alpha_{{R}}:=S^{\infty}_{11}+{\displaystyle}\frac{S^{\infty}_{12}\,\overline{S^{\infty}_{22}}\,S^{\infty}_{21}}{1-|S^{\infty}_{22}|^{2}}\quad\mbox{ of radius }\quad \rho_{{R}}:={\displaystyle}\frac{|S^{\infty}_{12}\,S^{\infty}_{21}|}{1-|S^{\infty}_{22}|^{2}}.$$
\[PropositionGoesThroughZero\] Assume that $S_{12}^{\infty}\ne0$. Then the set (\[setSPlusMoins\]) is nothing else but the unit circle $\mathscr{C}$.
Since $\mathcal{S}^{\infty}$ is unitary, we have $|S^{\infty}_{12}|^2+|S^{\infty}_{22}|^2=1$ and $S^{\infty}_{11}\overline{S^{\infty}_{12}}+S^{\infty}_{12}\overline{S^{\infty}_{22}}=0$. We deduce that $\alpha_{{R}}=0$ and $\rho_{{R}}=1$.
Proposition \[PropositionGoesThroughZero\] together with the error estimate $|{R}(L)-{R}^{{\mathrm{asy}}}(L)| \le C\,e^{-\gamma L}$ and the energy conservation relation $|{R}(L)|=1$ show that $L\mapsto{R}(L)$ runs on the unit circle as $L\to+\infty$. As a consequence, the reflection coefficient for the problem with mixed boundary conditions $L\mapsto{R}(L)$ passes (exactly) through the point of affix $-1+0i$ an infinite number of times.
Existence of perfectly invisible modes {#paragraphConcluPart1}
--------------------------------------
In Formula (\[Formulas\]), we found that the transmission coefficient ${\mathcal{T}}$ in the full waveguide ${\Omega}_L$ satisfies ${\mathcal{T}}=({r}-{R})/2$. Since ${r}=1$ (Formula (\[PartRelation\])), from the analysis of the previous paragraph for the map $L\mapsto R(L)$, we deduce that $L\mapsto {\mathcal{T}}(L)$ runs continuously and almost periodically on the circle of radius $1/2$ centered at $1/2+0i$ in the complex plane. In particular, $L\mapsto {\mathcal{T}}(L)$ passes through the point of affix $1+0i$ almost periodically. The period is $2\pi/(k\sqrt{3})$. We summarize this result in the following theorem, the main result of the section.
\[MainThmPart1\] Assume that the coefficient $S_{12}^{\infty}$ in (\[defMatrixScaLim\]) satisfies $S_{12}^{\infty}\ne0$. Then the complex curve $L\mapsto {\mathcal{T}}(L)$ for the transmission coefficient passes through the point of affix $1+0i$ an infinite number of times as $L\to+\infty$.
Again, we mention that in §\[paragraphPerfectRef\] below, we provide an example of situation where numerically $S_{12}^{\infty}\ne0$.
\[RmkMirorEffect\] The analysis above also shows that there are some $L>1$ such that ${\mathcal{T}}(L)=0$ (*perfect reflection*) and ${\mathcal{R}}(L)=1$ (see an illustration with Figure \[MirrorEffect\] below). In that case, all the energy sent in the waveguide is backscattered at $x=-\infty$. But this is not surprising because it has been proved in [@ChPaSu] that this mirror effect appears naturally, even in waveguides which are not symmetric with respect to the $(Oy)$ axis and for almost all widths of the vertical branch of ${\Omega}_L$.
Trapped modes {#SectionExistenceOfTrappedModes}
=============
In this section, we prove for that certain values of $L>1$, trapped modes exist for the half-waveguide problem with Neumann boundary conditions defined in (\[PbChampTotalSym\]). We remind the reader that we say that $u$ is a trapped mode for Problem (\[PbChampTotalSym\]) if $u$ belongs to the Sobolev space ${{\mathrm{H}}}^1({\omega}_L)$ and satisfies (\[PbChampTotalSym\]). Using a symmetry argument with respect to the line $\{x=0\}$, this will prove the existence of trapped modes for the Neumann problem set in the original domain ${\Omega}_L$ defined in (\[defOriginalDomain\]).
Setting
--------
We shall use the same notation as in the previous section. Additionally, set $\beta=\sqrt{\pi^2-k^2}$ and define $$W^{\pm}_{2}(x,y)=\cfrac{1}{\sqrt{2\beta}}\,(e^{-\beta x}\mp i e^{\beta x })\cos(\pi y).$$ Note the particular definition of the functions $W^{\pm}_{2}$ which are “wave packets”, combinations of exponentially decaying and growing modes as $x\to-\infty$. The normalisation coefficient for $W^{\pm}_{2}$ is chosen so that the matrix defined in (\[AugmentedScatteringDef\]) is unitary. In [@NaPl94bis; @KaNa02; @Naza06; @Naza11], it is proved that in the half-waveguide ${\omega}_L$ (unbounded in the left direction), there are the solutions $$\label{defu1u2}
\begin{array}{lcl}
u_{1}&=&w^-_{1}+s_{11}\,w^+_{1}+s_{12}\,W^+_{2}+\tilde{u}_{1}\\[4pt]
u_{2}&=&W^-_{2}+s_{21}\,w^+_{1}+s_{22}\,W^+_{2}+\tilde{u}_{2}
\end{array}$$ where $\tilde{u}_{1}$, $\tilde{u}_{2}$ decay as $O(e^{\sqrt{4\pi^2-k^2}x})$ when $x\to-\infty$. The complex constants $s_{ij}$, $i,j\in\{1,2\}$ in (\[defu1u2\]) are uniquely defined. They allow us to define the augmented scattering matrix introduced in [@NaPl94bis; @KaNa02; @Naza06; @Naza11] $$\label{AugmentedScatteringDef}
\mathbb{S}:=\left(\begin{array}{cc}
s_{11} & s_{12}\\
s_{21} & s_{22}
\end{array}\right)\in\mathbb{C}_{2\times 2}.$$ Working exactly as in the proof of Proposition \[PropositionUnitary\] in Annex, one shows that the matrix $\mathbb{S}$ is unitary ($\mathbb{S}\,\overline{\mathbb{S}}^{\top}={\mathrm{Id}}_{2\times2}$) and symmetric ($s_{21}=s_{12}$). This augmented scattering matrix turns out be a very efficient tool to detect the presence of trapped modes. Indeed, we have the following algebraic criterion (see e.g. [@Naza11 Thm. 2]).
\[LemmaExistenceTrappedMode\] If $s_{22}=-1$, then $u_{2}$ is a trapped mode for Problem (\[PbChampTotalSym\]) set in ${\omega}_L$.
Note that $s_{22}=-1$ is only a sufficient criterion of existence of trapped modes. Indeed the geometry ${\omega}_L$ can support trapped modes for Problem (\[PbChampTotalSym\]) with $s_{22}\ne-1$. In this case, this trapped mode must decay as $O(e^{\sqrt{4\pi^2-k^2}x})$ when $x\to-\infty$.
If $s_{22}=-1$, since $\mathbb{S}$ is unitary, then $s_{21}=0$. In such a situation, according to (\[defu1u2\]), we have $u_2=-i\sqrt{2/\beta}\,e^{\beta x}\cos(\pi y)+O(e^{\sqrt{4\pi^2-k^2}x})$ as $x\to-\infty$. This shows that $u_2\not\equiv0$ belongs to ${{\mathrm{H}}}^1({\omega}_L)$. In other words $u_2$ is a trapped mode.
Asymptotic behaviour of the augmented scattering matrix and existence of trapped modes
--------------------------------------------------------------------------------------
In this paragraph, we study the asymptotic behaviour of $\mathbb{S}=\mathbb{S}(L)$ as $L\to+\infty$.\
$\star$ As in the previous section, we first observe that $u'=w^+_1+w^-_1=\sqrt{2/k}\cos(kx)$ is a solution to (\[PbChampTotalSym\]) for all $L>1$. From the uniqueness of the definition of the coefficients $s_{ij}$ in (\[defu1u2\]), we deduce that for all $L>1$, we have $s_{11}(L)=1$ and $s_{12}(L)=s_{21}(L)=0$. Since $\mathbb{S}$ is unitary, we infer that $|s_{22}(L)|=1$ for all $L>1$.\
$\star$ It remains to investigate the behaviour of $s_{22}=s_{22}(L)$ as $L\to+\infty$. We adapt a bit what has been done in §\[paragraphAsymptoAnalysis\]. In the vertical branch of ${\omega}_{\infty}$, the limit geometry of ${\omega}_L$ as $L\to+\infty$, the following functions $$w^{\pm}_{3}(x,y)=\cfrac{1}{\sqrt{2k\ell}}\,e^{\pm i k y},\qquad w^{\pm}_{4}(x,y)=\cfrac{1}{\sqrt{\ell}}\,(y\mp i )\cos(\pi \frac{x}{\ell})$$ are propagating modes for Problem (\[PbChampTotalSym\]). Note that the width $\ell>0$ has been chosen so that the frequency $k=\pi/\ell$ is a threshold frequency for Problem (\[PbChampTotalSym\]) in the vertical branch of ${\omega}_{\infty}$. This explains the special form of the modes $w_4^{\pm}$. The normalisation coefficients for $w^{\pm}_{3}$, $w^{\pm}_{4}$ are chosen so that the matrix $\mathbb{S}^{\infty}$ defined in (\[DefAugScaMat\]) is unitary. In ${\omega}_{\infty}$, there are the solutions $$\begin{array}{lcl}
u_{1}^{\infty}&=& \chi_l(w^-_{1}+s^{\infty}_{11}\,w^+_{1}+s^{\infty}_{12}\,W^+_{2})+\chi_t\,(s^{\infty}_{13}\,w^+_{3}+s^{\infty}_{14}\,w^+_{4})+\tilde{u}_{1}^{\infty}\\[4pt]
u_{2}^{\infty}&=& \chi_l(W^-_{2}+s^{\infty}_{21}\,w^+_{1}+s^{\infty}_{22}\,W^+_{2})+\chi_t\,(s^{\infty}_{23}\,w^+_{3}+s^{\infty}_{24}\,w^+_{4})+\tilde{u}_{2}^{\infty}\\[4pt]
u_{3}^{\infty}&=& \chi_l(s^{\infty}_{31}\,w^+_{1}+s^{\infty}_{32}\,W^+_{2})+\chi_t\,(w^-_{3}+s^{\infty}_{33}\,w^+_{3}+s^{\infty}_{34}\,w^+_{4})+\tilde{u}_{3}^{\infty}\\[4pt]
u_{4}^{\infty}&=& \chi_l(s^{\infty}_{41}\,w^+_{1}+s^{\infty}_{42}\,W^+_{2})+\chi_t\,(w^-_{4}+s^{\infty}_{43}\,w^+_{3}+s^{\infty}_{44}\,w^+_{4})+\tilde{u}_{4}^{\infty},
\end{array}$$ where $\tilde{u}_{1}^{\infty}$, $\tilde{u}_{2}^{\infty}$, $\tilde{u}_{3}^{\infty}$, $\tilde{u}_{4}^{\infty}$ decay as $O(e^{\sqrt{4\pi^2-k^2}x})$ for $x\to-\infty$ and as $O(e^{-\sqrt{4\pi^2/\ell^2-k^2}y})$ for $y\to+\infty$. The augmented scattering matrix $$\label{DefAugScaMat}
\mathbb{S}^{\infty}:=\left(\begin{array}{cccc}
s^{\infty}_{11} & s^{\infty}_{12} & s^{\infty}_{13}& s^{\infty}_{14}\\
s^{\infty}_{21} & s^{\infty}_{22} & s^{\infty}_{23}& s^{\infty}_{24} \\
s^{\infty}_{31} & s^{\infty}_{32} & s^{\infty}_{33}& s^{\infty}_{34}\\
s^{\infty}_{41} & s^{\infty}_{42} & s^{\infty}_{43}& s^{\infty}_{44}
\end{array}\right)\in\mathbb{C}_{4\times 4}$$ is uniquely defined, unitary ($\mathbb{S}^{\infty}\overline{\mathbb{S}^{\infty}}^{\top}={\mathrm{Id}}_{4\times4}$) and symmetric (again, work as in the proof of Proposition \[PropositionUnitary\] in Annex to prove the two latter properties). For $u_{2}$, we make the ansatz (see [@MaNP00 Chap. 5, §5.6]) $$\label{ansatzTrapped}
\begin{array}{lcl}
u_{2} &=& u_{2}^{\infty}+a(L)\,u_{3}^{\infty}+b(L)\,u_{4}^{\infty}+\dots \ ,
\end{array}$$ where $a(L)$, $b(L)$ are gauge functions to determine and where the dots stand for small remainders. On $(-\ell;0)\times\{L\}$, the condition $\partial_nu_{2}=0$ leads to choose $a(L)$, $b(L)$ such that $$\label{system}
\begin{array}{c}
s^{\infty}_{23}\,e^{ikL}+a(L)\,(-e^{-ikL}+s^{\infty}_{33}\,e^{ikL})+b(L)\,s^{\infty}_{43}\,e^{ikL}=0\\[4pt]
s^{\infty}_{24}+a(L)\,s^{\infty}_{34}+b(L)(1+s^{\infty}_{44})=0.
\end{array}$$ This yields $$a(L)=\cfrac{s^{\infty}_{24}s^{\infty}_{43}-s^{\infty}_{23}(1+s^{\infty}_{44})}{(1+s^{\infty}_{44})(-e^{-2ikL}+s^{\infty}_{33})-s^{\infty}_{34}s^{\infty}_{43}}\quad\mbox{ and }\quad b(L)=\cfrac{s^{\infty}_{24}(e^{-2ikL}-s^{\infty}_{33})+s^{\infty}_{23}s^{\infty}_{34}}{(1+s^{\infty}_{44})(-e^{-2ikL}+s^{\infty}_{33})-s^{\infty}_{34}s^{\infty}_{43}}.$$ We shall consider ansatz (\[ansatzTrapped\]) for $u_2$ when $s^{\infty}_{44}\ne-1$. If $s^{\infty}_{44}=-1$, according to relations (\[relPart\]) below, then $s^{\infty}_{11}=1$. Since $\mathbb{S}^{\infty}$ is unitary and symmetric, we deduce that $s^{\infty}_{12}=s^{\infty}_{13}=s^{\infty}_{14}=s^{\infty}_{24}=s^{\infty}_{34}=s^{\infty}_{21}=s^{\infty}_{31}=s^{\infty}_{41}=s^{\infty}_{42}=s^{\infty}_{43}=0$. In such a situation, an analysis similar to what has been done in §\[paragraphAsymptoAnalysis\] can be developed when $|s^{\infty}_{33}|\ne 1$ allowing us to show the existence of trapped modes for certain $L>1$. We will not consider this rather exceptional case in the following. Instead, we will assume that $k\in(0;\pi)$ is such that $$\label{AssumptionsCoef}
s^{\infty}_{14}\ne0\qquad\mbox{ and }\qquad |s^{\infty}_{33}-\cfrac{s^{\infty}_{13}s^{\infty}_{34}}{s^{\infty}_{14}}|\ne1.$$ The first assumption of (\[AssumptionsCoef\]) implies that $s^{\infty}_{44}\ne-1$. The second one is needed so that we can solve system (\[system\]) with respect to $a(L)$ and $b(L)$ (again we use relations (\[relPart\]) below to get this statement). The authors do not know how to proceed without the latter assumption.\
When (\[AssumptionsCoef\]) is true, for all $L>1$ the denominators appearing in the definition of $a(L)$, $b(L)$ are not null. Then replacing in (\[ansatzTrapped\]) $a(L)$, $b(L)$ by its expression derived above, we obtain $s_{22}(L)=s^{{\mathrm{asy}}}_{22}(L)+\dots$ with $$\label{DefSetComplex}
s^{{\mathrm{asy}}}_{22}(L) = s^{\infty}_{22}+\cfrac{s^{\infty}_{24}s^{\infty}_{43}s^{\infty}_{32}-s^{\infty}_{23}(1+s^{\infty}_{44})s^{\infty}_{32}}{(1+s^{\infty}_{44})(-e^{-2ikL}+s^{\infty}_{33})-s^{\infty}_{34}s^{\infty}_{43}}+\cfrac{s^{\infty}_{24}(e^{-2ikL}-s^{\infty}_{33})s^{\infty}_{42}+s^{\infty}_{23}s^{\infty}_{34}s^{\infty}_{42}}{(1+s^{\infty}_{44})(-e^{-2ikL}+s^{\infty}_{33})-s^{\infty}_{34}s^{\infty}_{43}}.$$ Below we prove the following result.
\[LemmaCaracCircle\] Assume that the coefficients of the matrix $\mathbb{S}^{\infty}\in{\mathbb{C}}_{4\times4}$ satisfy Assumptions (\[AssumptionsCoef\]). Then $\{s^{{\mathrm{asy}}}_{22}(L)\,|\,L\in(1;+\infty)\}$ is the unit circle $\mathscr{C}:=\{z\in{\mathbb{C}}\,|\,|z|=1\}$.
From the error estimate $|s_{22}(L)-s^{{\mathrm{asy}}}_{22}(L)| \le C\,e^{-\beta_{\ell} L}$ where $\beta_{\ell}:=\sqrt{(\pi/\ell)^2-k^2}$, the relation $|s_{22}(L)|=1$ for all $L>1$, and the fact that $L\mapsto s_{22}(L)$ is continuous, we deduce that we have $\{s_{22}(L)\,|\,L\in(1;+\infty)\}=\mathscr{C}$. And in particular, we infer that $L \mapsto s_{22}(L)$ passes through the point of affix $-1+i0$ almost periodically (the period is equal to $\pi/k$). From Lemma \[LemmaExistenceTrappedMode\], we deduce the following theorem, the main result of the section.
\[MainThmPart2\] Assume that the coefficients of the matrix $\mathbb{S}^{\infty}\in{\mathbb{C}}_{4\times4}$ satisfy Assumptions (\[AssumptionsCoef\]). Then there is an infinite number of $L>1$ such that there are trapped modes for the Neumann Problem (\[PbChampTotalSym\]) in ${\omega}_L$.
Note that the period of the function $L\mapsto s^{{\mathrm{asy}}}_{22}(L)$ is equal to $\pi/k$ while the period of $L\mapsto{R}^{{\mathrm{asy}}}(L)$ (see (\[defCoeffLim\])) is equal to $2\pi/(k\sqrt{3})$. Thus in this geometry, trapped modes occur more frequently (with respect to $L\to+\infty$) than perfectly invisible modes.
If $u$ is a trapped mode for the Neumann Problem (\[PbChampTotalSym\]) in ${\omega}_L$, then the function $v$ such that $v=u$ in ${\omega}_L$ and $v(x,y)=u(-x,y)$ in ${\Omega}_L\setminus\overline{{\omega}_L}$ is a trapped mode for the Neumann Problem (\[PbInitial\]) in the unfold geometry ${\Omega}_L$.
Proof of Lemma \[LemmaCaracCircle\]
-----------------------------------
In this paragraph, we prove Lemma \[LemmaCaracCircle\] which ensures that the set $\{s^{{\mathrm{asy}}}_{22}(L)\,|\,L\in(1;+\infty)\}$ defined in (\[DefSetComplex\]) coincides with the unit circle.\
The already met function $u'=w^-_{1}+w^+_{1}=\sqrt{2/k}\cos(kx)$ is a solution of Problem (\[PbChampTotalSym\]) set in the limit geometry ${\omega}_{\infty}$ admitting the decomposition $$u' = \chi_l\,(w^-_{1}+w^+_{1})+\cfrac{\sqrt{\ell}}{i\sqrt{2k}}\,\chi_t\,(w^-_{4}-w^+_{4})+\tilde{u}'.$$ where $\tilde{u}'$ decays as $O(e^{\sqrt{4\pi^2-k^2}x})$ for $x\to-\infty$ and as $O(e^{-\sqrt{4\pi^2/\ell^2-k^2}y})$ for $y\to+\infty$. Set $\lambda=\sqrt{\ell}/(i\sqrt{2k})$. Observing that $u'-u^{\infty}_1+\lambda\,u^{\infty}_4$ has the same decay as $\tilde{u}'$, we deduce $$\label{relPart}
s^{\infty}_{11}+\lambda\,s^{\infty}_{41}=1,\qquad s^{\infty}_{12}+\lambda\, s^{\infty}_{42}=0,
\qquad s^{\infty}_{13}+\lambda\,s^{\infty}_{43}=0,\qquad s^{\infty}_{14}+\lambda\,s^{\infty}_{44}=-\lambda.$$ This allows one to write $$\begin{array}{lcl}
s^{{\mathrm{asy}}}_{22} &=& s^{\infty}_{22}+\cfrac{s^{\infty}_{24}s^{\infty}_{43}s^{\infty}_{32}-s^{\infty}_{23}(1+s^{\infty}_{44})s^{\infty}_{32}}{(1+s^{\infty}_{44})(-e^{-2ikL}+s^{\infty}_{33})-s^{\infty}_{34}s^{\infty}_{43}}+\cfrac{s^{\infty}_{24}(e^{-2ikL}-s^{\infty}_{33})s^{\infty}_{42}+s^{\infty}_{23}s^{\infty}_{34}s^{\infty}_{42}}{(1+s^{\infty}_{44})(-e^{-2ikL}+s^{\infty}_{33})-s^{\infty}_{34}s^{\infty}_{43}}\\[14pt]
&= & s^{\infty}_{22}+\cfrac{2s^{\infty}_{24}s^{\infty}_{13}s^{\infty}_{32}-s^{\infty}_{23}s^{\infty}_{14}s^{\infty}_{32}-s^{\infty}_{12}(-e^{-2ikL}+s^{\infty}_{33})s^{\infty}_{42}}{s^{\infty}_{14}(-e^{-2ikL}+s^{\infty}_{33})-s^{\infty}_{13}s^{\infty}_{34}}\\[14pt]
&= & s^{\infty}_{22}-\cfrac{s^{\infty}_{12}s^{\infty}_{42}}{s^{\infty}_{14}}+\cfrac{1}{s^{\infty}_{14}}\cfrac{2s^{\infty}_{24}s^{\infty}_{13}s^{\infty}_{32}-s^{\infty}_{23}s^{\infty}_{14}s^{\infty}_{32}-\cfrac{s^{\infty}_{12}s^{\infty}_{42}s^{\infty}_{13}s^{\infty}_{34}}{s^{\infty}_{14}}}{-e^{-2ikL}+s^{\infty}_{33}-\cfrac{s^{\infty}_{13}s^{\infty}_{34}}{s^{\infty}_{14}}}\ .
\end{array}$$ Set $$a=s^{\infty}_{33}-\cfrac{s^{\infty}_{13}s^{\infty}_{34}}{s^{\infty}_{14}},\quad b=\cfrac{1}{s^{\infty}_{14}}\,(\,2s^{\infty}_{24}s^{\infty}_{13}s^{\infty}_{32}-s^{\infty}_{23}s^{\infty}_{14}s^{\infty}_{32}-\cfrac{s^{\infty}_{12}s^{\infty}_{42}s^{\infty}_{13}s^{\infty}_{34}}{s^{\infty}_{14}}\,)\quad\mbox{ and }\quad c=s^{\infty}_{22}-\cfrac{s^{\infty}_{12}s^{\infty}_{42}}{s^{\infty}_{14}}.$$ One can check that $b=-d^2$ with $$d=s^{\infty}_{23}-\cfrac{s^{\infty}_{24}s^{\infty}_{13}}{s^{\infty}_{14}}.$$ With this notation, we have $s^{{\mathrm{asy}}}_{22}=c-d^2/(-e^{-2ikL}+a)$. Thus, as $L\to+\infty$, the coefficient $s^{{\mathrm{asy}}}_{22}(L)$ runs on the set $\{c-d^2/(z+a)\,|\,z\in\mathscr{C}\}$. Working with the Möbius transform (same result as in the previous section), we deduce that $\{c-d^2/(z+a)\,|\,z\in\mathscr{C}\}$ coincides with the circle centered at $$\label{caracCircle}
\alpha:=c+\cfrac{d^2\,\overline{a}}{1-|a|^2}\qquad\mbox{ of radius }\qquad\rho:=\cfrac{|d|^2}{1-|a|^2}.$$ Therefore, to complete the proof of Lemma \[LemmaCaracCircle\], it remains to show that $\alpha=0$ and $\rho=1$.\
$\star$ First we establish that $\rho=1$. This is equivalent to show that $|a|^2+|d|^2=1 \Leftrightarrow I=|s^{\infty}_{14}|^2$ with $$\label{quantiteI}
I:=(|s^{\infty}_{33}|^2+|s^{\infty}_{23}|^2)|s^{\infty}_{14}|^2
+(|s^{\infty}_{24}|^2+|s^{\infty}_{34}|^2)|s^{\infty}_{13}|^2-2\,\Re e\,(s^{\infty}_{14}\overline{s^{\infty}_{13}}\,(s^{\infty}_{23}\overline{s^{\infty}_{24}}+s^{\infty}_{33}\overline{s^{\infty}_{34}})).$$ Since $\mathbb{S}^{\infty}$ is unitary and symmetric, we have the identity $$\label{rel6}
s^{\infty}_{13}\overline{s^{\infty}_{14}}+s^{\infty}_{23}\overline{s^{\infty}_{24}}+s^{\infty}_{33}\overline{s^{\infty}_{34}}+s^{\infty}_{34}\overline{s^{\infty}_{44}}=0.$$ Using (\[rel6\]) in (\[quantiteI\]), we get $$\label{quantiteIbis}
\begin{array}{lcl}
I&=&(|s^{\infty}_{23}|^2+|s^{\infty}_{33}|^2)|s^{\infty}_{14}|^2
+(|s^{\infty}_{24}|^2+|s^{\infty}_{34}|^2)|s^{\infty}_{13}|^2+2\,\Re e\,(s^{\infty}_{14}\overline{s^{\infty}_{13}}\,(s^{\infty}_{13}\overline{s^{\infty}_{14}}+s^{\infty}_{34}\overline{s^{\infty}_{44}}))\\[4pt]
&=&(|s^{\infty}_{13}|^2+|s^{\infty}_{23}|^2+|s^{\infty}_{33}|^2)|s^{\infty}_{14}|^2
+(|s^{\infty}_{14}|^2+|s^{\infty}_{24}|^2+|s^{\infty}_{34}|^2)|s^{\infty}_{13}|^2+2\,\Re e\,(s^{\infty}_{14}\overline{s^{\infty}_{13}} s^{\infty}_{34}\overline{s^{\infty}_{44}}).
\end{array}$$ Using (\[relPart\]), we can write $$\label{quantiteInterBis}
\begin{array}{lcl}
2\,\Re e\,(s^{\infty}_{14}\overline{s^{\infty}_{13}}s^{\infty}_{34}\overline{s^{\infty}_{44}})=-2\,|s^{\infty}_{34}|^2|\Re e\,(s^{\infty}_{14}\,\overline{\lambda}\,\overline{s^{\infty}_{44}}) &=&-|s^{\infty}_{34}|^2(|\lambda|^2-|s^{\infty}_{34}|^2-|\lambda|^2|s^{\infty}_{44}|^2)\\[5pt]
&=&-|s^{\infty}_{13}|^2+|s^{\infty}_{34}|^2|s^{\infty}_{14}|^2+|s^{\infty}_{13}|^2|s^{\infty}_{44}|^2.
\end{array}$$ Plugging (\[quantiteInterBis\]) in (\[quantiteIbis\]), we obtain $$\label{conclusionRho}
I=(|s^{\infty}_{13}|^2+|s^{\infty}_{23}|^2+|s^{\infty}_{33}|^2+|s^{\infty}_{34}|^2)|s^{\infty}_{14}|^2
+(|s^{\infty}_{14}|^2+|s^{\infty}_{24}|^2+|s^{\infty}_{34}|^2+|s^{\infty}_{44}|^2-1)|s^{\infty}_{13}|^2=|s^{\infty}_{14}|^2.$$ To derive the second equality in (\[conclusionRho\]), we used again the fact $\mathbb{S}^{\infty}$ is unitary. Identity (\[conclusionRho\]) ensures that $\rho=1$.\
$\star$ Now, we prove that $\alpha=c+d^2\overline{a}/(1-|a|^2)=0$. Since $\rho=|a|^2+|d|^2=1$, this is equivalent to show that $c\overline{d}+d\overline{a}=0$. We have $$\label{TermToAssess}
\begin{array}{lcl}
|s^{\infty}_{14}|^2(c\overline{d}+d\overline{a}) &=&\phantom{-}|s^{\infty}_{14}|^2(s^{\infty}_{22}\overline{s^{\infty}_{23}}+s^{\infty}_{23}\overline{s^{\infty}_{33}})+|s^{\infty}_{24}|^2s^{\infty}_{12}\overline{s^{\infty}_{13}}+|s^{\infty}_{13}|^2s^{\infty}_{24}\overline{s^{\infty}_{34}}\\[4pt]
& & -s^{\infty}_{14}\overline{s^{\infty}_{13}}(s^{\infty}_{23}\overline{s^{\infty}_{34}}+s^{\infty}_{22}\overline{s^{\infty}_{24}})
-s^{\infty}_{24}\overline{s^{\infty}_{14}}(s^{\infty}_{12}\overline{s^{\infty}_{23}}+s^{\infty}_{13}\overline{s^{\infty}_{33}}).
\end{array}$$ Since $\mathbb{S}^{\infty}$ is unitary and symmetric, we have $$\begin{array}{lcl}
s^{\infty}_{12}\overline{s^{\infty}_{13}}+s^{\infty}_{22}\overline{s^{\infty}_{23}}+s^{\infty}_{23}\overline{s^{\infty}_{33}}+s^{\infty}_{24}\overline{s^{\infty}_{34}}&=&0\\[4pt]
s^{\infty}_{12}\overline{s^{\infty}_{14}}+s^{\infty}_{22}\overline{s^{\infty}_{24}}+s^{\infty}_{23}\overline{s^{\infty}_{34}}+s^{\infty}_{24}\overline{s^{\infty}_{44}}&=&0\\[4pt]
s^{\infty}_{11}\overline{s^{\infty}_{31}}+s^{\infty}_{12}\overline{s^{\infty}_{23}}+s^{\infty}_{13}\overline{s^{\infty}_{33}}+s^{\infty}_{14}\overline{s^{\infty}_{34}}&=&0.
\end{array}$$ Using these three identities in (\[TermToAssess\]), we find $$\label{lastEquality}
\begin{array}{lcl}
|s^{\infty}_{14}|^2(c\overline{d}+d\overline{a}) &=&-|s^{\infty}_{14}|^2(s^{\infty}_{12}\overline{s^{\infty}_{13}}+s^{\infty}_{24}\overline{s^{\infty}_{34}})+|s^{\infty}_{24}|^2s^{\infty}_{12}\overline{s^{\infty}_{13}}+|s^{\infty}_{13}|^2s^{\infty}_{24}\overline{s^{\infty}_{34}}\\[4pt]
& & +s^{\infty}_{24}\overline{s^{\infty}_{13}}(s^{\infty}_{11}\overline{s^{\infty}_{14}}+s^{\infty}_{14}\overline{s^{\infty}_{44}})
+|s^{\infty}_{14}|^2 (s^{\infty}_{12}\overline{s^{\infty}_{13}}+ s^{\infty}_{24}\overline{s^{\infty}_{34}})\\[8pt]
& =& |s^{\infty}_{24}|^2s^{\infty}_{12}\overline{s^{\infty}_{13}}+|s^{\infty}_{13}|^2s^{\infty}_{24}\overline{s^{\infty}_{34}}-|s^{\infty}_{24}|^2s^{\infty}_{12}\overline{s^{\infty}_{13}}+|s^{\infty}_{13}|^2s^{\infty}_{24}\overline{s^{\infty}_{34}}=0.
\end{array}$$ To derive the last line of the above equality, we used the relation $$s^{\infty}_{11}\overline{s^{\infty}_{14}}+s^{\infty}_{12}\overline{s^{\infty}_{24}}+s^{\infty}_{13}\overline{s^{\infty}_{34}}+s^{\infty}_{14}\overline{s^{\infty}_{44}}=0.$$ This gives $\alpha=0$ and allows us to conclude that the set $\{s^{{\mathrm{asy}}}_{22}(L)\,|\,L\in(1;+\infty)\}$ is indeed the unit circle.
One can note that the special form of the matrix $\mathbb{S}^{\infty}$ (see relations (\[relPart\])) due to the particular choice of the geometry is used only in the proof of $\rho=1$. Once this property is established, the fact that $\mathbb{S}^{\infty}$ is unitary suffices to conclude that $\alpha=0$.
Numerical experiments {#SectionNumExpe}
=====================
Perfectly invisible modes {#paragraphPerfectRef}
-------------------------
In the first series of experiments, we exhibit some $L>1$ such that perfect invisibility holds in the geometry ${\Omega}_L$ defined in (\[defOriginalDomain\]). For each $L$ in a given range, we compute numerically the transmission coefficient ${\mathcal{T}}$ defined in (\[DefScatteringCoeff\]). To proceed, we use a ${\mathrm{P}}2$ finite element method in a truncated waveguide. On the artificial boundary created by the truncation, a Dirichlet-to-Neumann operator with ${\mathrm{15}}$ terms serves as a transparent condition. We take $k=0.8\pi$ so that the width of the vertical branch of ${\Omega}_L$ is equal to $2\ell=2\pi/k=2.5$. In Figure \[figResult3\] left, we display the curve $L\mapsto {\mathcal{T}}(L)$ for $L\in(1.3;8)$. In accordance with the results obtained in §\[paragraphConcluPart1\], we observe that when $L\to+\infty$, $L\mapsto {\mathcal{T}}(L)$ runs on the circle of radius $1/2$ centered at $1/2+0i$ in the complex plane. In particular, $L\mapsto {\mathcal{T}}(L)$ passes through the point of affix $1+0i$ (Theorem \[MainThmPart1\]). In Figure \[figResult3\] right, we display the curve $L\mapsto -\ln |{\mathcal{T}}(L)-1|$ for $L\in(1.3;8)$. The picks correspond to the values of $L$ such that ${\mathcal{T}}(L)=1$. According to the proof of Theorem \[MainThmPart1\], we expect that the picks are almost periodic with a distance between two picks tending to $2\pi/(k\sqrt{3})=2.5/\sqrt{3}\approx1.44$ as $L\to+\infty$. The numerical results we get are coherent with this value. In Figure \[figResultT1\], we represent the real part of the total field $v$ defined in (\[DefScatteringCoeff\]) as well as $v-w^-_1$ for $L=2.5756$ (first pick of Figure \[figResult3\] right). Finally in Figure \[figResultT1Gros\], we display $v$ and $v-w^-_1$ in another geometry where ${\mathcal{T}}(L)=1$. Here the domain is $$\label{defDomainBis}
\tilde{{\Omega}}_L=\{(x,y)\in{\mathbb{R}}^2\,|\,0<y<g_L(x)\}$$ where $g_L:{\mathbb{R}}\to{\mathbb{R}}$ is the even staircase function such that $g_L(x)=L$ for $ 0 \le x <\ell$, $g_L(x)=2.5$ for $\ell < x <2\ell$, $g_L(x)=2$ for $2\ell < x <3\ell$, $g_L(x)=1.5$ for $3\ell < x <4\ell$ and $g_L(x)=1$ for $4\ell<x$ (see Figure \[figResultT1Gros\]). The two important points are to choose $g_L$ so that $u'=w^-_1+w^+_1=\sqrt{2/k}\cos(kx)$ satisfies the initial problem (\[PbInitial\]) set in $\tilde{{\Omega}}_L$ for all $L>1$ and to preserve the symmetry of the geometry with respect to the $(Oy)$ axis. Then playing with $L$ as explained in Section \[SectionPerfectInvisibility\], one can get ${\mathcal{T}}(L)=1$ (theoretically and numerically). Of course other staircase geometries satisfy the two mentioned properties.
\[ExplanationRealPart\] In the numerical experiments leading to Figures \[figResultT1\], \[figResultT1Gros\], one finds that $\Re e\,(v-w^-_1)\equiv 0$. This can be proved. Indeed, observing that $v-w^-_1+\overline{v-w^-_1}=v+\overline{v}-(w^+_1+w^-_1)$, we deduce that $v-w^-_1+\overline{v-w^-_1}$ is a solution of Problem (\[PbInitial\]). Since $v-w^-_1+\overline{v-w^-_1}$ is exponentially decaying as $|x|\to+\infty$ (because ${\mathcal{T}}=1$), we infer that if trapped modes do not exist, there holds $v-w^-_1+\overline{v-w^-_1}\equiv0\Leftrightarrow \Re e\,(v-w^-_1)\equiv 0$. Note that in the proof, we use again the fact that in the particular geometries considered in the present work, $w^+_1+w^-_1$ is a solution of Problem (\[PbInitial\]).
![Left: coefficient $L\mapsto {\mathcal{T}}(L)$ for $L\in [1.3;8]$. According to the conservation of energy (\[ConservationOfNRJ\]), we know that the scattering coefficient ${\mathcal{T}}$ is located inside the unit disk delimited by the black bold line. Right: curve $L\mapsto -\ln|{\mathcal{T}}(L)-1|$ for $L\in [1.3;8]$.\[figResult3\]](TransmissionCoefficient.pdf "fig:"){width="48.00000%"}![Left: coefficient $L\mapsto {\mathcal{T}}(L)$ for $L\in [1.3;8]$. According to the conservation of energy (\[ConservationOfNRJ\]), we know that the scattering coefficient ${\mathcal{T}}$ is located inside the unit disk delimited by the black bold line. Right: curve $L\mapsto -\ln|{\mathcal{T}}(L)-1|$ for $L\in [1.3;8]$.\[figResult3\]](logTMoins1.pdf "fig:"){width="48.00000%"}
![$\Re e\,v$ (top), $\Im m\,v$ (middle) and $\Im m\,(v-w^-_1)$ (bottom) for $L=2.5756$ where $v$ is the function introduced in (\[DefScatteringCoeff\]). One finds that $\Re e\,(v-w^-_1)\equiv 0$. The latter result is specific to the geometry considered here (see Remark \[ExplanationRealPart\]). \[figResultT1\]](uReal.png "fig:"){width="95.00000%"}\
![$\Re e\,v$ (top), $\Im m\,v$ (middle) and $\Im m\,(v-w^-_1)$ (bottom) for $L=2.5756$ where $v$ is the function introduced in (\[DefScatteringCoeff\]). One finds that $\Re e\,(v-w^-_1)\equiv 0$. The latter result is specific to the geometry considered here (see Remark \[ExplanationRealPart\]). \[figResultT1\]](uImag.png "fig:"){width="95.00000%"}\
![$\Re e\,v$ (top), $\Im m\,v$ (middle) and $\Im m\,(v-w^-_1)$ (bottom) for $L=2.5756$ where $v$ is the function introduced in (\[DefScatteringCoeff\]). One finds that $\Re e\,(v-w^-_1)\equiv 0$. The latter result is specific to the geometry considered here (see Remark \[ExplanationRealPart\]). \[figResultT1\]](usImag.png "fig:"){width="95.00000%"}
![$\Re e\,v$ (top), $\Im m\,v$ (middle) and $\Im m\,(v-w^-_1)$ (bottom) in the waveguide $\tilde{{\Omega}}_L$ defined in (\[defDomainBis\]) for $L=4.5808$. One finds that $\Re e\,(v-w^-_1)\equiv 0$. The latter result is specific to the geometry considered here (see Remark \[ExplanationRealPart\]). \[figResultT1Gros\]](uRealGros.png "fig:"){width="\textwidth"}\
![$\Re e\,v$ (top), $\Im m\,v$ (middle) and $\Im m\,(v-w^-_1)$ (bottom) in the waveguide $\tilde{{\Omega}}_L$ defined in (\[defDomainBis\]) for $L=4.5808$. One finds that $\Re e\,(v-w^-_1)\equiv 0$. The latter result is specific to the geometry considered here (see Remark \[ExplanationRealPart\]). \[figResultT1Gros\]](uImagGros.png "fig:"){width="\textwidth"}\
![$\Re e\,v$ (top), $\Im m\,v$ (middle) and $\Im m\,(v-w^-_1)$ (bottom) in the waveguide $\tilde{{\Omega}}_L$ defined in (\[defDomainBis\]) for $L=4.5808$. One finds that $\Re e\,(v-w^-_1)\equiv 0$. The latter result is specific to the geometry considered here (see Remark \[ExplanationRealPart\]). \[figResultT1Gros\]](usImagGros.png "fig:"){width="\textwidth"}
Perfect reflection
------------------
As indicated in Remark \[RmkMirorEffect\], the method proposed in Section \[SectionPerfectInvisibility\] allows one also to exhibit situations where ${\mathcal{T}}(L)=0$ (perfect reflection) and ${\mathcal{R}}(L)=1$. In Figure \[MirrorEffect\], we display this mirror effect, with the energy completely backscattered, in the geometry $\tilde{{\Omega}}_L$ defined in (\[defDomainBis\]) for a well chosen $L$. More precisely, we display the real part of the total field and we observe that it is indeed exponentially decaying as $x\to+\infty$.
\[ExplanationTzero\] In Figure \[MirrorEffect\], we observe that $\Im m\,v\equiv 0$ when ${\mathcal{R}}=1$. This is a general result that holds without assumption on the geometry of the waveguide as soon as ${\mathcal{R}}=1$. Indeed, if ${\mathcal{T}}=0$ then $|{\mathcal{R}}|=1$ and one can check that $v-\overline{{\mathcal{R}}}\,\overline{v}$ is a solution of Problem (\[PbInitial\]) which is exponentially decaying as $|x|\to+\infty$. Therefore, if trapped modes do not exist, we deduce that $v-\overline{{\mathcal{R}}}\,\overline{v}\equiv0\Leftrightarrow v=\overline{{\mathcal{R}}}\,\overline{v}$. In particular, if ${\mathcal{R}}=1$, we obtain $\Im m\,v\equiv 0$.
![$\Re e\,v$ in the waveguide $\tilde{{\Omega}}_L$ defined in (\[defDomainBis\]) for $L=4.3758$. One finds that $\Im m\,v\equiv 0$. The latter result actually holds in any waveguide where ${\mathcal{R}}=1$ (see Remark \[ExplanationTzero\]). \[MirrorEffect\]](MirrorEffect.png){width="\textwidth"}
Trapped modes {#trapped-modes}
-------------
In the third series of experiments, we give examples of geometries supporting trapped modes for the initial Helmholtz problem (\[PbInitial\]) with Neumann boundary condition. For each $L$ in a given range, this time we compute numerically the coefficients of the augmented scattering matrix $\mathbb{S}\in{\mathbb{C}}_{2\times2}$ defined in (\[AugmentedScatteringDef\]). Again we use a ${\mathrm{P}}2$ finite element method set in a truncated waveguide. We emphasize here that we need to work with a well-suited Dirichlet-to-Neumann map to deal with the wave packet $W_2^+$ appearing in the decompositions of $u_1$, $u_2$ in (\[defu1u2\]). In Figure \[AugmentedScatteringMatrix\] left, we display the coefficients $L\mapsto s_{ij}(L)$ for $i,j\in\{1,2\}$ and $L\in(1.3;8)$. The wavenumber is set to $k=0.8\pi$. As shown by the theory, indeed we have $s_{11}(L)=1$ and $s_{12}(L)=s_{21}(L)=0$ for all $L>1$. Moreover, we observe that $L\mapsto s_{22}(L)$ runs on the unit circle. Consequently, it indeed goes through the point of affix $-1+i0$ which guarantees the existence of trapped modes for certain $L>1$ (Theorem \[MainThmPart2\]). In Figure \[AugmentedScatteringMatrix\] right, we display the curve $L\mapsto -\ln |s_{22}(L)+1|$ for $L\in(1.3;8)$. Indeed, it has some picks indicating values of $L$ such that $s_{22}(L)=-1$. According to the proof of Theorem \[MainThmPart2\], we expect a distance between two picks approximately equal to $\pi/k=1.25$. The numerical results are in good agreement with this value. In Figure \[TrappedMode\], we display the real part of a trapped mode in ${\omega}_L$ for $L=2.5524$ (first pick of Figure \[AugmentedScatteringMatrix\] right). Finally, in Figure \[TrappedModeFancy\], we display the real part of a trapped mode in the waveguide $\tilde{{\Omega}}_L$ defined in (\[defDomainBis\]) for $L=3.8273$. This trapped mode has been obtained symmetrising the trapped mode of the half-waveguide problem with Neumann boundary conditions with respect to the $(Oy)$ axis. Following the analysis of Section \[SectionExistenceOfTrappedModes\], we can construct trapped modes in any staircase geometry satisfying the two properties mentioned at the end of §\[paragraphPerfectRef\].
![Left: coefficients of the augmented scattering matrix defined in (\[AugmentedScatteringDef\]) for $L\in [1.3;8]$. Right: curve $L\mapsto -\ln|s_{22}(L)+1|$ for $L\in [1.3;8]$.\[AugmentedScatteringMatrix\]](ScatteringAugmentee.pdf "fig:"){width="48.00000%"}![Left: coefficients of the augmented scattering matrix defined in (\[AugmentedScatteringDef\]) for $L\in [1.3;8]$. Right: curve $L\mapsto -\ln|s_{22}(L)+1|$ for $L\in [1.3;8]$.\[AugmentedScatteringMatrix\]](logs22Plus1.pdf "fig:"){width="48.00000%"}
![Real part of a trapped mode in the geometry ${\omega}_L$ defined in (\[defHalfWaveguide\]) for $L=2.5524$. \[TrappedMode\]](TrappedMode.png)
![Real part of a trapped mode in the waveguide $\tilde{{\Omega}}_L$ defined in (\[defDomainBis\]) for $L=3.8273$. \[TrappedModeFancy\]](TrappedModePyramid.png)
Conclusion {#SectionConclusion}
==========
In this article, we proved the existence of perfectly invisible and trapped modes in simple particular geometries for the Helmholtz problem with Neumann boundary conditions. Importantly, the waveguide has to be symmetric with respect to the $(Oy)$ axis and endowed with a branch of tunable height whose width coincides with the wavelength of the incident wave. The analysis has been done in the domain introduced in (\[defOriginalDomain\]). Other examples of geometrical situations where the method can be performed have been presented in Section \[SectionNumExpe\]. Our technique seems specific to the problem with Neumann boundary conditions and it does not look simple to modify it to consider for example the case of Dirichlet boundary conditions (appearing e.g. in the analysis of quantum waveguides). Finally, what we did in ${\mathrm{2D}}$ can be adapted to higher dimension.
Annex {#annex .unnumbered}
=====
In this Annex, for the convenience of the reader, we provide the detail of the proof of a well-known proposition.
\[PropositionUnitary\] The scattering matrix $\mathcal{S}^{\infty}$ defined in (\[UnboundedScatteringMatrix\]) is unitary and symmetric.
Define the symplectic (sesquilinear and anti-hermitian ($q(\varphi,\psi)=-\overline{q(\psi,\varphi)}$)) form $q(\cdot,\cdot)$ such that for all $\varphi,\psi\in{{\mathrm{H}}}^1_{{\mbox{\scriptsize loc}}}({\omega}_\infty)$ $$q(\varphi,\psi)=\int_{\Sigma} \cfrac{\partial \varphi}{\partial n}\overline{\psi}-\varphi\cfrac{\partial \overline{\psi}}{\partial n}\,d\sigma.$$ Here $\Sigma:=\{-2\ell\}\times(0;1)\cup (-\ell;0)\times\{1+2\tau\}$, $\partial_n=-\partial_x$ on $\{-2\ell\}\times(0;1)$, $\partial_n=\partial_y$ on $(-\ell;0)\times\{1+2\tau\}$. Moreover, ${{\mathrm{H}}}^1_{{\mbox{\scriptsize loc}}}({\omega}_\infty)$ refers to the Sobolev space of functions $\varphi$ such that $\varphi|_{\mathcal{O}}\in{{\mathrm{H}}}^1(\mathcal{O})$ for all bounded domains $\mathcal{O}\subset{\omega}_\infty$. Integrating by parts and using that the functions $U^{\infty}_1$, $U^{\infty}_2$ satisfy the Helmholtz equation, we obtain $q(U^{\infty}_i,U^{\infty}_j)=0$ for $i,\,j\in\{1,2\}$. On the other hand, decomposing $U^{\infty}_1$, $U^{\infty}_2$ in Fourier series on $\Sigma$, we find $$\begin{array}{c}
q(U^{\infty}_1,U^{\infty}_1) = (-1+|S^{\infty}_{11}|^2+|S^{\infty}_{12}|^2)\,i,\quad q(U^{\infty}_2,U^{\infty}_2) = (-1+|S^{\infty}_{22}|^2+|S^{\infty}_{21}|^2)\,i\\[5pt]
q(U^{\infty}_1,U^{\infty}_2)=-\overline{q(U^{\infty}_2,U^{\infty}_1)}=S^{\infty}_{11}\overline{S^{\infty}_{21}}+S^{\infty}_{12}\overline{S^{\infty}_{22}}.
\end{array}$$ These relations allow us to prove that $\mathcal{S}^{\infty}\,\overline{\mathcal{S}^{\infty}}^{\top}={\mathrm{Id}}_{2\times 2}$, that is to conclude that $\mathcal{S}^{\infty}$ is unitary. On the other hand, one finds $q(U^{\infty}_1,\overline{U^{\infty}_2})=0=-S^{\infty}_{21}+S^{\infty}_{12}$. We deduce that $\mathcal{S}^{\infty}$ is symmetric.
|
---
abstract: |
We combine the extended source catalogue and the 2dF galaxy redshift survey to produce an infrared-selected galaxy catalogue with measured redshifts. We use this extensive dataset to estimate the galaxy luminosity functions in the J- and -bands. The luminosity functions are fairly well fit by Schechter functions with parameters $\Mstarj-5 \log h= -22.36 \pm 0.02$, $\alphaj= -0.93\pm 0.04$, $\phij=0.0104 \pm 0.0016 h^3$ Mpc$^{-3}$ in the J-band and $\Mstark- 5 \log h= -23.44 \pm 0.03$, $\alphak=
-0.96\pm 0.05$, $\phik=0.0108 \pm 0.0016 h^3$ Mpc$^{-3}$ in the -band ( Kron magnitudes). These parameters are derived assuming a cosmological model with $\Omega_0=0.3$ and $\Lambda_0=0.7$. With datasets of this size, systematic rather than random errors are the dominant source of uncertainty in the determination of the luminosity function. We carry out a careful investigation of possible systematic effects in our data. The surface brightness distribution of the sample shows no evidence that significant numbers of low surface brightness or compact galaxies are missed by the survey. We estimate the present-day distributions of $-$ and J$-$ colours as a function of absolute magnitude and use models of the galaxy stellar populations, constrained by the observed optical and infrared colours, to infer the galaxy stellar mass function. Integrated over all galaxy masses, this yields a total mass fraction in stars (in units of the critical mass density) of $\Omega_{\rm stars}h= (1.6 \pm 0.24) \times 10^{-3}$ for a Kennicutt IMF and $\Omega_{\rm stars}h= (2.9 \pm 0.43)\times 10^{-3}$ for a Salpeter IMF. These values are consistent with those inferred from observational estimates of the total star formation history of the universe provided that dust extinction corrections are modest.
author:
- |
\
$^1$Department of Physics, University of Durham, Science Laboratories, South Road, Durham DH1 3LE, United Kingdom\
$^2$Anglo-Australian Observatory, P.O. Box 296, Epping, NSW 2121, Australia\
$^3$Research School of Astronomy & Astrophysics, The Australian National University, Weston Creek, ACT 2611, Australia\
$^4$Astrophysics Research Institute, Liverpool John Moores University, Twelve Quays House, Egerton Wharf, Birkenhead, L14 1LD, UK\
$^5$Department of Astrophysics, University of New South Wales, Sydney, NSW2052, Australia\
$^6$School of Physics and Astronomy, North Haugh, St Andrews, Fife, KY16 9SS, United Kingdom\
$^7$Department of Physics, Keble Road, Oxford OX1 3RH, United Kingdom\
$^8$Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge CB3 0HA, United Kingdom\
$^9$Department of Astronomy, California Institute of Technology, Pasadena, CA 91125, USA\
$^{10}$Department of Physics & Astronomy, Johns Hopkins University, 3400 North Charles Street Baltimore, MD 212182686, USA\
$^{11}$Department of Physics & Astronomy, E C Stoner Building, Leeds LS2 9JT, United Kingdom\
$^{12}$School of Physics and Astronomy, University of Nottingham, University Park, Nottingham, NG7 2RD, United Kingdom\
$^{13}$Institute of Astronomy, University of Edinburgh, Royal Observatory, Edinburgh EH9 3HJ, United Kingdom\
title: 'The 2dF Galaxy Redshift Survey: Near Infrared Galaxy Luminosity Functions[^1] '
---
galaxies: luminosity function estimators
------------------------------------------------------------------------
Introduction
============
The near-infrared galaxy luminosity function is an important characteristic of the local galaxy population. It is a much better tracer of evolved stars, and hence of the total stellar content of galaxies, than optical luminosity functions which can be dominated by young stellar populations and are also strongly affected by dust extinction. Hence, infrared luminosities can be much more directly related to the underlying stellar mass of galaxies and so knowledge of the present form and evolution of the infrared galaxy luminosity function places strong constraints on the history of star formation in the universe and on galaxy formation models (Cole and references therein).
The local K-band luminosity function has been estimated from optically selected samples by Mobasher, Sharples and Ellis (), Szokoly () and Loveday () and from K-band surveys by Glazebrook (), and Gardner (). The existing K-band surveys are small. The largest, by Gardner et al., covers only 4 deg$^2$ and contains only 510 galaxies. The recent survey of Loveday covers a much larger solid angle. In this survey the redshifts were known in advance of measuring the K-band magnitudes and this was exploited by targetting bright and faint galaxies resulting in an effective sample size much larger than the 345 galaxies actually measured. However, like all optically selected samples, it suffers from the potential problem that galaxies with extremely red infrared to optical colours could be missed. In this paper we combine the 2-Micron All Sky Survey () with the 2dF galaxy redshift survey () to create an infrared selected redshift survey subtending deg$^2$. Currently the sky coverage of both surveys is incomplete, but already the overlap has an effective area of deg$^2$. Within this area the redshift survey is complete to the magnitude limit of the catalogue and so constitutes a complete survey which is 50 times larger than the previous largest published infrared selected redshift survey. A new catalogue of a similarly large area, also based on , has very recently been analysed by Kochanek (). They adopt isophotal rather than total magnitudes and concentrate on the dependence of the luminosity function on galaxy morphology.
In Section \[sec:selection\] we briefly describe the relevant properties of the and catalogues. Section \[sec:matching\] is a detailed examination of the degree to which the matched – galaxies are a complete and representative subset of the catalogue. Section \[sec:mags\] examines the calibration of the total magnitudes and Section \[sec:compl\] demonstrates that the catalogue and the inferred luminosity functions are not affected by surface brightness selection effects. In Section \[sec:pop\] we present the method by which we compute k-corrections and evolutionary corrections and relate the observed luminosities to the underlying stellar mass. The estimation methods and normalization of the luminosity functions are described briefly in Section \[sec:lf\_est\]. Our main results are presented and discussed in Section \[sec:results\]. These include estimates of the J and (K-short) luminosity functions, the $-$ and J$-$ colour distributions as a function of absolute magnitude and the distribution of spectral type. We also estimate the stellar mass function of galaxies, which can be integrated to infer the fraction of baryons in the universe which are in the form of stars. We conclude in Section \[sec:conc\].
The Dataset {#sec:data}
===========
The data that we analyze are the extended source catalogue from the second incremental release of the 2-Micron All Sky Survey (http://pegasus.phast.umass.edu) and the galaxy catalogue of the 2dF galaxy redshift survey ( http://www.mso.anu.edu.au/2dFGRS). Here, we present the relevant properties of these two catalogues and investigate their selection characteristics and level of completeness.
Selection Criteria {#sec:selection}
------------------
The is a ground-based, all-sky imaging survey in the J, H and bands. Details of how extended sources are identified and their photometric properties measured are given by Jarrett (). The detection sensitivity (10$\sigma$) for extended sources is quoted as 14.7, 13.9 and 13.1 magnitudes in J, H and respectively. The complete survey is expected to contain 1 million galaxies of which approximately 580,000 are contained in the second incremental data release made public in March 2000.
The is selected in the photographic band from the APM galaxy survey ([@apmI],,) and subsequent extensions to it, that include a region in the northern galactic cap (Maddox in preparation). The survey covers approximately deg$^2$ consisting of two broad declination strips. The larger is centred on the SGP and approximately covers $-22^\circ\negthinspace.5$$>$$\delta$$>$$-37^\circ\negthinspace.5$, $21^{\rm h}40^{\rm m}$$<$$\alpha$$<$$3^{\rm h}30^{\rm m}$; the smaller strip is in the northern galactic cap and covers $2^\circ\negthinspace.5$$>$$\delta$$>$$-7^\circ\negthinspace.5$, $9^{\rm h}50^{\rm
m}$$<$$\alpha$$<$$14^{\rm h}50^{\rm m}$. In addition, there are a number of randomly located circular 2-degree fields scattered across the full extent of the low extinction regions of the southern APM galaxy survey. There are some gaps in the sky coverage within these boundaries due to small regions that have been excluded around bright stars and satellite trails. The aims to measure the redshifts of all the galaxies within these boundaries with extinction-corrected magnitudes brighter than 19.45. When complete, at the end of 2001, 250,000 galaxy redshifts will have been measured. In this paper we use the redshifts obtained prior to .
The overlap of the two surveys is very good. There are some gaps in the sky coverage due to strips of the sky that were not included in the second incremental release, but overall a substantial fraction of the deg$^2$ of the is covered by . The homogeneity and extensive sky coverage of the combined dataset make it ideal for studies of the statistical properties of the galaxy population.
=8 truecm
The Completeness of the Matched – Catalogue {#sec:matching}
-------------------------------------------
Here we consider whether all the galaxies within the survey region have counterparts and assess the extent to which the fraction of galaxies with measured redshifts represents an unbiased sub-sample.
The astrometry in both and is, in general, very good and it is an easy matter to match objects in the two catalogues. We choose to find the closest pairs within a search radius equal to three quarters of the semi-major axis of the J-band image (denoted j\_r\_e in the database). Scaling the search radius in this way helps with the matching of large extended objects. This procedure results in the identification of counterparts for of the objects, when at random one would only expect to find a handful of such close pairs. Moreover, the distribution of separations shown in Fig. \[fig:seps\] peaks at 0.5 arcsec, with only 3% having separations greater than 3 arcsec. A significant part of this tail comes from the most extended objects as is evident from the dotted histogram in Fig. \[fig:seps\] which shows objects with semi-major axes larger than 12 arcsec. Thus, we can be very confident in these identifications.
The objects for which we have found secure counterparts amount to % of the extended sources that fall within the boundary of the . As discussed below, a more restrictive criterion that includes only sources fainter than J$=$$12$ that are confidently classified as galaxies by , increases the fraction with matches to %. The remaining % are missed for well understood reasons (star-galaxy classification: %; merged or close images: %; miscellaneous: %), none of which ought to introduce a bias. This is confirmed explicitly, in the middle row of Fig. \[fig:hists\], by the close correspondence between the photometric properties of the missed % and those of the larger matched sample. Hence, in estimating luminosity functions no significant bias will be introduced by assuming the matched sample to be representative of the full population. Furthermore, the distribution shown in the bottom row of Fig. \[fig:hists\] shows that the subset of galaxies for which we have measured redshifts is a random sample of the full matched – catalogue. This summary is the result of a thorough investigation, which we describe in the remainder of this section, into the reasons why % of the sources are missed and whether their omission introduces a bias in the properties of the matched sample.
=15.0 truecm
We first consider objects in the catalogue which based on their images and colours are not confidently classified as galaxies. In the database a high e\_score or g\_score indicates a high probability that the object is either not an extended source or not a galaxy. A cc\_flag$\ne$0 indicates an artifact or contaminated and/or confused source. For detailed definitions of these parameters we refer the reader to Jarrett (). Rejecting all objects which have either e\_score$>$1.4, g\_score$>$1.4 or cc\_flag$\ne$0 removes just % of the total. However, removing these reduces significantly the fraction of the sample that does not match with the catalogue, from % to %. Thus, it is likely that about 30% of the objects which have e\_score$>$1.4, g\_score$>$1.4 or cc\_flag$\ne$0 are not galaxies.
The may contain a tail of very red objects that are too faint in the -band to be included in the $<$19.45 sample. Fig. \[fig:colours\] shows the distribution of -J colours for the matched objects with J$<$14.7. (Here, the J-band magnitude we are using is the default magnitude denoted j\_m in the database. In Section \[sec:mags\] we will consider the issue of what magnitude definition is most appropriate for estimating the luminosity function.) The vertical dashed line indicates the colour at which this sample starts to become incomplete due to the $<$19.45 magnitude limit of . The colour distribution cuts off sharply well before this limit, suggesting that any tail of missed very red objects is extremely small. In other words the is sufficiently deep that even the reddest objects detected at the faintest limits of ought to be detected in .
In the top row of Fig. \[fig:hists\] we compare the distributions of magnitude, colour and surface brightness for the matched and missed objects. In general, the properties of the missed subset overlap well with those of the much larger matched subset. However, we do see that the distributions for missed objects contain tails of bright and blue objects. It is quite likely that this is due to the extended source catalogue being contaminated by a small population of saturated or multiple stars. The dotted histograms in the top row of Fig. \[fig:hists\] show the distributions of magnitude, colour and surface brightness for the bright subset of the missed objects with J$<$12. Here we clearly see bimodal colour and surface brightness distributions. The blue peak of the colour distribution is consistent with that expected for stars (see Jarrett ). Excluding these bright, J$<$12, objects which are clearly contaminated by stars reduces the fraction of missed objects from % to %. The magnitude, colour and surface brightness distributions for this remaining % are shown in the middle panel of Fig. \[fig:hists\]. We see that the missed objects are slightly under-represented at the faintest magnitudes and also slightly bluer on average than the matched sample, while the distribution of surface brightness is almost indistinguishable for the two sets of objects. These differences are small and so will introduce no significant bias in our luminosity function estimates. To elucidate the reasons for the remaining missed % of objects we downloaded 100 1$\times$1 arcmin images from the STScI Digitized Sky Survey (DSS) centred on the positions of a random sample of the missed objects. In each image we plotted a symbol to indicate the position of any galaxies within the 1$\times$1 arcmin field. We also plotted symbols to indicate the positions and classifications of all images identified in the APM scans from which the catalogue was drawn, down to a magnitude limit of $\approx 20.5$. These images are classified as galaxies, stars, merged images (galaxy+galaxy, galaxy+star or star+star) or noise. This set of plots allows us to perform a census of the reasons why some objects are not present in the survey.
=8.5 truecm
The main cause for the absence of objects in the is that the APM has classified these objects as stars. These amount to % of the missed sample (% of the full sample). In some cases, the DSS image shows clearly that these are stars and in others that they are galaxies. However, the majority of these objects cannot easily be classified from the DSS images. Thus, they could be galaxies that the APM has falsely classified as stars or stars that has falsely classified as galaxies. The first possibility is not unexpected since the parameters used in the APM star-galaxy separation algorithm were chosen as a compromise between high completeness and low contamination such that the expected completeness is around 95% with 5% stellar contamination ([@apmI]). It is hard to rule out the possibility that this class of object does not include a substantial fraction of stars, but if so, their presence appears not to distort the distribution of colours shown in Fig \[fig:colours\]. Another % of the random sample (% of the full sample) are classified by the APM as mergers or else consist of two close images in the DSS but are classified by the APM as a single galaxy offset from the position. The remaining % of the random sample (% of the full sample) are missed for a variety of reasons including proximity to the diffraction spikes of very bright stars and poor astrometry caused by the presence of a neighbouring unclassified image.
Magnitude Definitions and Calibration {#sec:mags}
--------------------------------------
=9.0 truecm
=18.0 truecm
The extended source database provides a large selection of different magnitude measurements. In the previous section we used the default magnitudes (denoted j\_m and k\_m in the database). These are magnitudes defined within the same circular aperture in each waveband. For galaxies brighter than $=$$14$, the aperture is the circular -band isophote of 20 mag arcsec$^{-2}$ and for galaxies fainter than $=$14 it is the circular J-band isophote of 21 mag arcsec$^{-2}$. These are not the most useful definitions of magnitude for determining the galaxy luminosity function. Since we are interested in measuring the total luminosity and ultimately the total stellar mass of each galaxy, we require a magnitude definition that better represents the total flux emitted by each galaxy. We consider Kron magnitudes (Kron ) and extrapolated magnitudes. Kron magnitudes (denoted j\_m\_e and k\_m\_e in the database) are measured within an aperture, the Kron radius, defined as 2.5 times the intensity-weighted radius of the image. The extrapolated magnitudes (denoted j\_m\_ext and k\_m\_ext in the database) are defined by first fitting a modified exponential profile, $f(r) = f_0
\exp [-(\alpha r)^{1/\beta}]$, to the image from 10 arcsec to the 20 mag/arcsec$^2$ isophotal radius, and extrapolating this from the Kron radius to 4 times this radius or 80 arcsec if this is smaller (Jarrett private communication). Note that improvements are being made to the extended source photometry algorithms developed and employed team and so in the final data release the definitions of the Kron and extrapolated magnitudes may be slightly modified (Jarrett private communication).
Fig. \[fig:mags\] compares the default, Kron and extrapolated magnitudes in the J and bands for the matched – catalogue. The upper panel shows that while the median offset between the J-band isophotal default magnitudes and the pseudo-total Kron magnitudes is small there is a large spread with some galaxies having default magnitudes more than 0.5 magnitudes fainter than the Kron magnitude. The Kron magnitudes are systematically fainter than the extrapolated magnitudes by between approximately 0.1 and 0.3 magnitudes. This offset is rather larger than expected: if the Kron radius is computed using a faint isophote to define the extent of the image from which the intensity weighted radius is measured, then the Kron magnitudes should be very close to total. For an exponential light profile ($\beta=1$), the Kron radius should capture 96% of the flux, while for an $r^{1/4}$ law ($\beta=4$), 90% of the flux should be enclosed. In other words, the Kron magnitude should differ from the total magnitude by only 0.044 and 0.11 magnitudes in these two cases. However, the choice of isophote is a compromise between depth and statistical robustness. In the case of the second incremental release, an isophote of 21.7(20.0) mag arcsec$^{-2}$ in J() was adopted (Jarrett private communication). These relatively bright isophotes, particularly the -band isophote, could lead to underestimates of the Kron radii and fluxes for lower surface brightness objects and plausibly accounts for much of the median offset of 0.3 magnitudes seen in Fig. \[fig:mags\] between the -band Kron and extrapolated magnitudes. This line of reasoning favours adopting the extrapolated magnitudes as the best estimate of the total magnitudes, but, on the other hand, the extrapolated magnitudes are model-dependent and have larger measurement errors.
To understand better the offset and scatter in the magnitudes we have compared a subset of the data with the independent K-band photometry of Loveday (2000). The pointed observations of Loveday have better resolution than the 2MASS images and good signal-to-noise to a much deeper isophote. This enables accurate, unbiased Kron magnitudes to be measured. Note that the offset between the -band and the standard K-band used by Loveday is expected to be almost completely negligible (see Carpenter ). The left hand panels of Fig. \[fig:loveday\] compare these measurements with the corresponding Kron and extrapolated magnitudes. The right hand panels show -band Kron and extrapolated magnitudes computed by taking the J-band Kron and extrapolated magnitudes and subtracting the J$-$ colour measured within the default aperture. These indirect estimates are interesting to consider as they combine the profile information from the deeper J-band image with the J$-$colour measured within the largest aperture in which there is good signal-to-noise. The straight lines plotted in Fig. \[fig:loveday\] show simple least squares fits and the slope and zero-point offset of these fits are indicated on each panel along with bootstrap error estimates. Also shown in the inset panels is the distribution of residual magnitude differences about each of the fits and a gaussian fit to this distribution. The rms of these residuals and a bootstrap error estimate is also given in each panel.
From these comparisons we first see that all the fits have slopes entirely consistent with unity, but that their zeropoints and scatters vary. The zero-point offsets, $\CalkDelta$, between both the Kron magnitude measurements and those of Loveday confirm that the Kron magnitudes systematically underestimate the galaxy luminosities. In the case of the direct -band magnitudes the offset is $\CalkDelta=0.164$ magnitudes. In the case of the Kron magnitudes inferred from the deeper J-band image profiles, the offset is reduced to $\CalkDelta=0.061$ magnitudes. Conversely the extrapolated magnitudes are systematically brighter than the Loveday Kron magnitudes by $-\CaleDelta=0.137$ and $0.158$ magnitudes, where one would expect an offset of only $\KDelta=0.044$ to $0.11$ due to the difference in definition between ideal Kron and true total magnitudes. For both estimates of the extrapolated magnitude and for the directly estimated Kron magnitude the scatter about the correlation is approximately 0.14 magnitudes and we note a slight tendency for the scatter to increase at faint magnitudes. The magnitude estimate that best correlates with the Loveday measurements is the Kron magnitude estimated from the J-band Kron magnitude and the default aperture J$-$ colour. Here the distribution of residuals has a much reduced scatter of only 0.1 magnitudes and has very few outliers.
Our conclusions from the comparison of Kron magnitudes is that it is preferable to adopt the -band magnitude inferred from the J-band Kron or extrapolated magnitude by converting to the -band using default aperture colour, rather than to use the noisier and more biased direct -band estimates. With this definition, we find that the Kron magnitudes slightly underestimate the galaxy luminosities while the extrapolated magnitudes slightly overestimate the luminosities, particularly at faint fluxes. We will present results for both magnitude definitions, but we note that to convert to total magnitudes we estimate that the Kron magnitudes should be brightened by $\CalkDelta+\KDelta=0.1$–$0.17$ magnitudes and the extrapolated magnitudes dimmed by $-\CaleDelta-\KDelta=0.05$–$0.11$ magnitudes.
Completeness of the Catalogue {#sec:compl}
------------------------------
Here we define the magnitude limited samples which we will analyze in Section \[sec:lf\_est\] and test them for possible incompleteness in both magnitude and surface brightness. For the Kron and extrapolated magnitudes, the catalogue has high completeness to the nominal limits of J$<$14.7 and $<$13.9. However, to ensure very high completeness and avoid any bias in our luminosity function estimates, we made the following more conservative cuts. For the Kron magnitudes, we limited our sample to either J$<$14.45 or $<$13.2, and for the extrapolated magnitudes to either J$<$14.15 or $<$12.9. These choices are motivated by plots such as the top panel of Fig. \[fig:mags\]. Here the isophotal default magnitude limit of J$<$$14.7$ is responsible for the right hand edge to the distribution of data points. One sees that this limit begins to remove objects from the distribution of Kron magnitudes for J$\gsim$$14.5$. An indication that the survey is complete to our adopted limits is given by the number counts shown in Fig \[fig:counts\], which only begin to roll over at fainter magnitudes.
=8.5 truecm
=8.0 truecm
More rigorously, we have verified that the samples are complete to these limits by examining their $V(z_i)/V(z_{{\rm
max},i})$ distributions. Here, $z_i$ is the redshift of a galaxy in the sample, $z_{{\rm max},i}$ is the maximum redshift at which this galaxy would satisfy the sample selection criteria, and $V(z)$ is the survey volume that lies at redshift less than $z$. If the sample is complete and of uniform density, $V(z_i)/V(z_{{\rm max},i})$ is uniformly distributed within the interval 0 to 1. To evaluate $z_{\rm max}$ we made use of the default k+e corrections described in the following section, but the results are not sensitive to reasonable variations in the assumed corrections or in the cosmology. The solid histograms in Fig. \[fig:v\_vmax\] show these distributions for each of our four magnitude limited samples. Note that the -band magnitudes are those inferred from the J-band magnitudes and aperture colours. The dashed histograms in the lower panels show the corresponding distributions for the directly measured -band magnitudes. In all these cases we have computed $V_{\rm max}$ simply from the imposed apparent magnitude limits and have ignored any possible dependence of the catalogue completeness on surface brightness.
If the samples were incomplete the symptom one would expect to see is a deficit in the $V/V_{\rm max}$ distributions at large $V/V_{\rm max}$ and hence a mean $\langle V/V_{\rm max} \rangle$$<$$0.5$. There is no evidence for such a deficit in these distributions. In fact each has a mean $\langle V/V_{\rm max} \rangle$ slightly greater than $0.5$. The slight gradient in the $V/V_{\rm max}$ distribution is directly related to the galaxy number counts shown in Fig \[fig:counts\], which are slightly steeper than expected for a homogeneous, non-evolving galaxy distribution. A similar result has been found in the bright -band counts ([@apmcounts]). The -band result has variously been interpreted as evidence for rapid evolution, systematic errors in the magnitude calibration, or a local hole or underdensity in the galaxy distribution ([@apmcounts; @metcalfe; @shanks]). Here we note that the gradient in the $V/V_{\rm max}$ distributions (and also in the galaxy counts) becomes steeper both as one switches from Kron to the less reliable extrapolated magnitudes and as one switches from the J-band data to the lower signal-to-noise -band data. This gives strong support to our decision to adopt the -band magnitudes derived from the J-band Kron and extrapolated magnitudes and aperture J$-$ colours. It also cautions that the mean $\langle V/V_{\rm max} \rangle$$>$$0.5$ cannot necessarily be taken as a sign of evolution or a local underdensity, but may instead be related to the accuracy of the magnitude measurements. The comparison to the observations of Loveday () shows no evidence for systematic errors in the magnitudes, but does not constrain the possibility that the distribution of magnitude measurement errors may become broader or skewed at fainter magnitudes. Such variations would affect the $V/V_{\rm max}$ distributions and could produce the observed behaviour. We conclude by noting that while the shift in the mean $\langle V/V_{\rm max} \rangle$ is statistically significant, it is nevertheless quite small for the samples we analyze and has little effect on the resulting luminosity function estimates.
=8.0 truecm
We now investigate explicitly the degree to which the completeness of the catalogue depends on surface brightness by estimating $V_{\rm max}$ as a function of both absolute magnitude and surface brightness. This is an important issue: if the catalogue is missing low-surface brightness galaxies our estimates of the luminosity function will be biased. The approach we have taken follows that developed in Cross () for the . We estimate an effective central surface brightness, $\mu_0^z$, for each observed galaxy assuming an exponential light distribution, that the Kron magnitudes are total and that the Kron radii are exactly five exponential scale-lengths. This is then corrected to redshift $z=0$ using $$\mu_0=\mu_0^z - 10\log(1+z) - k(z) - e(z)$$ to account for redshift dimming and k+e corrections (c.f Section \[sec:pop\]). Note that in the catalogue, galaxies with estimated Kron radii less than $7$ arcsec, have their Kron radii set to 7 arcsec. This will lead us to underestimate the central surface brightnesses of these galaxies, but this will only affect high surface brightness objects and will not affect whether a galaxy can or cannot be seen. The distribution in the $M_{\rm J}$–$\mu_0$ plane of our Kron J-band selected sample is shown by the points in Fig. \[fig:sb\].
Cross () use two different methods to estimate the value of $V_{\rm max}$ associated with each position in this plane. The first method uses the visibility theory of Phillipps, Davies & Disney (). We model the selection characteristics of the extended source catalogue by a set of thresholds. The values appropriate in the J-band are a minimum isophotal diameter of $8.5$ arcsec at an isophote of $20.5$ mag arcsec$^{-2}$, and an isophotal magnitude limit of J$<$$14.7$ at an isophote of $21.0$ mag arcsec$^{-2}$ (Jarrett et al. 2000). In addition, we impose the limits in the Kron magnitude of $11$$<$J$<$$14.45$ that define the sample we analyze. We then calculate for each point on the $M_{\rm J}$–$\mu_0$ plane the redshift at which a such a galaxy will drop below one or other of these selection thresholds and hence compute a value of $V_{\rm max}$. The results of this procedure are shown by the contours of constant $V_{\rm max}$ plotted in Fig. \[fig:sb\]. Note that these estimates of $V_{\rm max}$ are only approximate since we have made the crude assumption that all the galaxies are circular exponential disks. In addition, the diameter and isophotal limits are only approximate and vary with observing conditions.
The second method developed by Cross () consists of making an empirical estimate of $V_{\rm max}$ in bins in the $M_{\rm J}$–$\mu_0$ plane. They look at the distribution of observed redshifts in a given bin and adopt the 90$^{\rm th}$ percentile of this distribution to define $z_{\rm max}$ and hence $V_{\rm max}$. It is more robust to use the 90$^{\rm th}$ percentile rather than the 100$^{\rm th}$ percentile and the effect of this choice can easily be compensated for when estimating the luminosity function ([@cross]). Note that in our application to the data we do not apply corrections for incompleteness or the effects of clustering. The result of this procedure is to confirm that for the populated bins, the $V_{\rm max}$ values given by the visibility theory are a good description of the data.
=8.5 truecm
In Fig. \[fig:sb\] we see that the distribution of galaxies in the $M_{\rm J}$–$\mu_0$ plane is well separated from the low surface brightness limit of approximately $20.5$ mag arcsec$^{-2}$ where the $V_{\rm max}$ contours indicate that the survey has very little sensitivity. Thus, there is no evidence that low-surface brightness galaxies are missing from the catalogue. Furthermore, in the region occupied by the observed data, the $V_{\rm max}$ contours are close to vertical indicating that there is little dependence of $V_{\rm max}$ on surface brightness. The way in which the $V/V_{\rm max}$ distribution is modified by including this estimate of the surface brightness dependence is shown by the dotted histogram in the top-left panel of Fig. \[fig:v\_vmax\]. Its effect is to increase the mean $V/V_{\rm max}$ slightly, suggesting that this estimate perhaps overcorrects for the effect of surface brightness selection. Even so, the change in the estimated luminosity function is negligible as confirmed by the three estimates of the Kron J-band luminosity function shown in Fig. \[fig:lf\_sb\]. These are all simple $1/V_{\rm max}$ estimates, but with $V_{\rm max}$ computed either ignoring surface brightness effects or using one of the two methods described above. These luminosity functions differ negligibly, indicating that no bias is introduced by ignoring surface brightness selection effects.
Modelling the Stellar Populations {#sec:pop}
=================================
=8.5 truecm
=8.5 truecm
The primary aim of this paper is to determine the present-day J and -band luminosity functions and also the stellar mass function of galaxies. Since the survey spans a range of redshift (see Fig. \[fig:dn\_dz\]), we must correct for both the redshifting of the filter bandpass (k-correction) and for the effects of galaxy evolution (e-correction). In practice, the k and e-corrections at these wavelengths are both small and uncertainties in them have little effect on the estimated luminosity functions. This is because these infrared bands are not dominated by young stars and also because the survey does not probe a large range of redshift. We have chosen to derive individual k and e-corrections for each galaxy using the stellar population synthesis models of Bruzual & Charlot (). We have taken this approach not because such detailed modelling is necessary to derive robust luminosity functions, but because it enables us to explore the uncertainties in the derived galaxy stellar mass functions, which are, in fact, dominated by uncertainties in the properties of the stellar populations.
The latest models of Bruzual & Charlot () provide, for a variety of different stellar initial mass functions (IMFs), the spectral energy distribution (SED), $l_\lambda(t,Z)$, of a single population of stars formed at the same time with a single metallicity, as a function of both age, $t$, and metallicity, $Z$. We convolve these with an assumed star formation history, $\psi(t^\prime)$, to compute the time-evolving SED of the model galaxy, $$L_\lambda(t) = \int_0^t l_\lambda(t-t^\prime,Z)
\ \psi(t^\prime)
\ dt^\prime.$$ We take account of the effect of dust extinction on the SEDs using the Ferrara () extinction model normalized so that the V-band central face-on optical depth of the Milk-Way is 10. This value corresponds to the mean optical depth of $L_\star$ galaxies in the model of Cole () which employs the same model of dust extinction. We assume a typical inclination angle of 60 degrees which yields a net attenuation factor of 0.53 in the V-band and 0.78 in the J-band. By varying the assumed metallicity, $Z$, and star formation history, we build up a two-dimensional grid of models. Then, for each of these models, we extract tracks of $-$ and J$-$ colours and stellar mass-to-light ratio as a function of redshift.
Our standard set of tracks assumes a cosmological model with $\Omega_0=0.3$, $\Lambda_0=0.7$, Hubble constant $H_0=70$ km s$^{-1}$ Mpc$^{-1}$, and star formation histories with an exponential form, $\psi(t) \propto \exp(-[t(z)-t(z_{\rm f})]/\tau)$. Here, $t(z)$ is the age of the Universe at redshift $z$ and the galaxy is assumed to start forming stars at $z_{\rm f}=20$ . For these tracks, we adopt the Kennicutt IMF ([@kenn83]) and include the dust extinction model. The individual tracks are labelled by a metallicity, $Z$, which varies from $Z=0.0001$ to $Z=0.05$ and a star formation timescale, $\tau$, which varies from $\tau=1$ Gyr to $\tau=50$ Gyr. Examples of these tracks are shown in Fig. \[fig:tracks\], along with the observed redshifts and colours of the galaxies. We can see that the infrared J$-$ colour depends mainly on metallicity while the $-$ colour depends both on metallicity and star formation timescale. Thus, the use of both colours allows a unique track to be selected. Note from the bottom panel that, for all the tracks, the k+e correction is always small for the range of redshift spanned by our data.
We can gauge how robust our results are by varying the assumptions of our model. In particular, we vary the IMF, the dust extinction and cosmological models, and include or exclude the evolutionary contribution to the k+e correction. Also, we consider power-law star formation histories, $\psi(t)
\propto [t(z)/t(z_{\rm f})]^{-\gamma}$, as an alternative to the exponential model. The results are discussed at beginning of Section \[sec:results\].
The procedure for computing the individual galaxy k+e corrections is straightforward. At the measured redshift of a galaxy, we find the model whose $-$ and J$-$ colours most closely match that of the observed galaxy. Having selected the model we then follow it to $z=0$ to predict the galaxy’s present-day J and -band luminosities and also its total stellar mass. We also use the model track to follow its k+e correction to higher redshift in order to compute $z_{\rm max}$, the maximum redshift at which this galaxy would have passed the selection criteria for inclusion into the analysis sample.
Luminosity Function Estimation {#sec:lf_est}
==============================
We use both the simple $1/V_{\rm max}$ method and standard maximum likelihood methods to estimate luminosity functions. We present Schechter function fits computed using the STY method (Sandage, Tammann & Yahil ) and also non-parametric estimates using the stepwise maximum likelihood method (SWML) of Efstathiou, Ellis & Peterson (). Our implementation of each of these methods is described and tested in Norberg (). The advantage of the maximum likelihood methods is that they are not affected by galaxy clustering (provided that the galaxy luminosity function is independent of galaxy density). By contrast the $1/V_{\rm max}$ method, which makes no assumption about the dependence of the luminosity function with density, is subject to biases produced by density fluctuations.
The two maximum likelihood methods determine the shape of the luminosity function, but not its overall normalization. We have chosen to normalize the luminosity functions by matching the galaxy number counts of Jarrett (in preparation). These were obtained from a 184 deg$^2$ area selected to have low stellar density and in which all the galaxy classifications have all been confirmed by eye. The counts are reproduced in Fig. \[fig:counts\]. By using the same 7 arcsec aperture magnitudes as Jarrett (in preparation) and scaling the galaxy counts in our redshift survey, we deduce that the effective area of our redshift catalogue is $\areaeff\pm25$ deg$^2$. Note that normalizing in this way by-passes the problem of whether or not some fraction of the missed objects are stars. Fig. \[fig:counts\] also shows the Kron and extrapolated magnitude J and counts of the – redshift survey. In the lower panel, these counts are seen to be in agreement with the published K-band counts of Gardner () and Glazebrook ().
We also checked the normalization using the following independent estimate of the effective solid angle of the redshift survey. For galaxies in the parent catalogue brighter than $<B_{\rm limit}$, we computed the fraction that have both measured redshifts and match a galaxy. For a faint $B_{\rm limit}$ this fraction is small as the catalogue is much deeper than the catalogue, but as $B_{\rm limit}$ is made brighter, the fraction asymptotes to the fraction of the area of the parent catalogue covered by the joint –redshift survey. By this method we estimate that the effective area of our redshift catalogue is $\areaeffa\pm22$ deg$^2$, which is in good agreement with the estimate from the counts of Jarrett .
It should be noted that for neither of these estimates of the effective survey area do the quoted uncertainties take account of variations in the number counts due to large scale structure. To estimate the expected variation in the galaxy number counts within the combined - survey due to large scale structure we constructed an ensemble of mock catalogues from the $\Lambda$CDM Hubble volume simulation of the VIRGO consortium (Evrard ; Evrard in preparation; http://www.physics.lsa.umich.edu/hubble-volume). Mock catalogues constructed from the VIRGO Hubble Volume simulations (Baugh in preparation) can be found at http://star-www.dur.ac.uk/$\tilde{\hphantom{n}}$cole/mocks/hubble.html . We simply took these catalogues and sampled them to the depth of over a solid angle of $\areaeff$ deg$^2$. To this magnitude limit we found an rms variation in the number of galaxies of 15%. We took this to be a realistic estimate of the uncertainty in the number counts and propagated this error through when computing the error on the normalization of the luminosity function.
Results {#sec:results}
=======
Luminosity Functions
--------------------
----- ----- -------- ----------------- ---------------- ------------------------------ ----------------- ---------------- ------------------------------ -- -- -- -- -- -- -- -- -- --
0.3 0.7 k$+$e -22.36$\pm$0.02 -0.93$\pm$0.04 $1.04\pm0.16 \times 10^{-2}$ -23.44$\pm$0.03 -0.96$\pm$0.05 $1.08\pm0.16 \times 10^{-2}$
0.3 0.7 k only -22.47$\pm$0.02 -0.99$\pm$0.04 $0.90\pm0.14 \times 10^{-2}$ -23.51$\pm$0.03 -1.00$\pm$0.04 $0.98\pm0.15 \times 10^{-2}$
0.3 0.0 k$+$e -22.29$\pm$0.03 -0.89$\pm$0.04 $1.16\pm0.18 \times 10^{-2}$ -23.36$\pm$0.03 -0.93$\pm$0.05 $1.21\pm0.18 \times 10^{-2}$
0.3 0.0 k only -22.38$\pm$0.03 -0.95$\pm$0.04 $1.02\pm0.15 \times 10^{-2}$ -23.43$\pm$0.03 -0.96$\pm$0.05 $1.10\pm0.16 \times 10^{-2}$
1.0 0.0 k$+$e -22.22$\pm$0.02 -0.87$\pm$0.03 $1.26\pm0.19 \times 10^{-2}$ -23.28$\pm$0.03 -0.89$\pm$0.05 $1.34\pm0.20 \times 10^{-2}$
1.0 0.0 k only -22.34$\pm$0.02 -0.93$\pm$0.04 $1.08\pm0.16 \times 10^{-2}$ -23.38$\pm$0.03 -0.93$\pm$0.05 $1.18\pm0.17\times 10^{-2}$
----- ----- -------- ----------------- ---------------- ------------------------------ ----------------- ---------------- ------------------------------ -- -- -- -- -- -- -- -- -- --
\[tab:lfpar\]
-------- --------------------------------- --------------------------------- -- -- --
-18.00 (5.73$\pm$3.58)$\times 10^{-3}$ (3.13$\pm$3.64)$\times 10^{-3}$
-18.25 (5.38$\pm$3.34)$\times 10^{-3}$ (8.26$\pm$6.68)$\times 10^{-3}$
-18.50 (7.60$\pm$3.75)$\times 10^{-3}$
-18.75 (7.94$\pm$3.59)$\times 10^{-3}$ (4.65$\pm$4.10)$\times 10^{-3}$
-19.00 (1.11$\pm$3.82)$\times 10^{-2}$ (5.76$\pm$4.32)$\times 10^{-3}$
-19.25 (6.98$\pm$2.26)$\times 10^{-3}$ (9.16$\pm$5.67)$\times 10^{-3}$
-19.50 (8.14$\pm$1.80)$\times 10^{-3}$ (1.12$\pm$0.64)$\times 10^{-2}$
-19.75 (8.17$\pm$1.45)$\times 10^{-3}$ (1.05$\pm$0.57)$\times 10^{-2}$
-20.00 (7.16$\pm$1.12)$\times 10^{-3}$ (8.58$\pm$4.63)$\times 10^{-3}$
-20.25 (6.62$\pm$0.88)$\times 10^{-3}$ (8.82$\pm$3.86)$\times 10^{-3}$
-20.50 (7.30$\pm$0.76)$\times 10^{-3}$ (6.94$\pm$2.44)$\times 10^{-3}$
-20.75 (7.07$\pm$0.64)$\times 10^{-3}$ (6.09$\pm$1.63)$\times 10^{-3}$
-21.00 (5.84$\pm$0.48)$\times 10^{-3}$ (9.26$\pm$1.69)$\times 10^{-3}$
-21.25 (4.97$\pm$0.39)$\times 10^{-3}$ (6.96$\pm$1.18)$\times 10^{-3}$
-21.50 (5.69$\pm$0.35)$\times 10^{-3}$ (7.29$\pm$0.98)$\times 10^{-3}$
-21.75 (5.15$\pm$0.28)$\times 10^{-3}$ (6.99$\pm$0.79)$\times 10^{-3}$
-22.00 (4.89$\pm$0.21)$\times 10^{-3}$ (5.98$\pm$0.61)$\times 10^{-3}$
-22.25 (4.49$\pm$0.17)$\times 10^{-3}$ (5.93$\pm$0.52)$\times 10^{-3}$
-22.50 (3.41$\pm$0.12)$\times 10^{-3}$ (5.39$\pm$0.42)$\times 10^{-3}$
-22.75 (2.37$\pm$0.09)$\times 10^{-3}$ (5.85$\pm$0.37)$\times 10^{-3}$
-23.00 (1.59$\pm$0.06)$\times 10^{-3}$ (5.24$\pm$0.28)$\times 10^{-3}$
-23.25 (1.06$\pm$0.04)$\times 10^{-3}$ (4.96$\pm$0.22)$\times 10^{-3}$
-23.50 (5.41$\pm$0.27)$\times 10^{-4}$ (4.18$\pm$0.17)$\times 10^{-3}$
-23.75 (2.66$\pm$0.17)$\times 10^{-4}$ (2.72$\pm$0.11)$\times 10^{-3}$
-24.00 (1.19$\pm$0.10)$\times 10^{-4}$ (1.88$\pm$0.08)$\times 10^{-3}$
-24.25 (4.69$\pm$0.54)$\times 10^{-5}$ (1.21$\pm$0.06)$\times 10^{-3}$
-24.50 (1.20$\pm$0.22)$\times 10^{-5}$ (6.54$\pm$0.37)$\times 10^{-4}$
-24.75 (5.40$\pm$1.34)$\times 10^{-6}$ (3.46$\pm$0.23)$\times 10^{-4}$
-25.00 (5.42$\pm$3.88)$\times 10^{-7}$ (1.48$\pm$0.13)$\times 10^{-4}$
-25.25 (5.55$\pm$0.65)$\times 10^{-5}$
-25.50 (2.13$\pm$0.33)$\times 10^{-5}$
-25.75 (9.42$\pm$1.96)$\times 10^{-6}$
-26.00 (1.09$\pm$0.56)$\times 10^{-6}$
-------- --------------------------------- --------------------------------- -- -- --
: The SWML J and -band luminosity functions for Kron magnitudes as plotted in Fig. \[fig:lf\_jk2\]. The units of both $\phi$ and its uncertainty $\Delta \phi$ are number per $h^{-3}$ Mpc$^3$ per magnitude.
\[tab:lfswml\]
Fig. \[fig:lf\_jk1\] shows SWML estimates of the Kron J and luminosity functions. The points with errorbars show results for our default choice of k+e corrections, namely those obtained for an $\Omega_0=0.3$, $\Lambda_0=0.7$, $H_0=70$ km s$^{-1}$ Mpc$^{-1}$ cosmology with a Kennicutt IMF and including dust extinction. The figure also illustrates that the luminosity functions are very robust to varying this set of assumptions. The various curves in each plot are estimates made neglecting dust extinction and/or switching to a Salpeter IMF and/or changing the Hubble constant to $H_0=50$ km s$^{-1}$Mpc$^{-1}$ and/or adopting power-law star formation histories and/or making a k-correction but no evolution correction. The systematic shifts caused by varying these assumptions are all comparable with or smaller than the statistical errors. The biggest shift results from applying or neglecting the evolutionary correction. In terms of the characteristic luminosity in the STY Schechter function fit, the estimates which include evolutionary corrections are $0.05$ to $0.1$ magnitudes fainter than those that only include k-corrections (see Table \[tab:lfpar\]).
In Fig. \[fig:lf\_jk2\] we compare $1/V_{\rm max}$ and SWML Kron luminosity function estimates (for our default choice of k+e corrections) with STY Schechter function estimates. In general, the luminosity functions are well fit by Schechter functions, but there is marginal evidence for an excess of very luminous galaxies over that expected from the fitted Schechter functions. We tabulate the SWML estimates in Table \[tab:lfswml\]. Integrating over the luminosity function gives luminosity densities in the J and -bands of $\rho_J=(2.75\pm 0.41) \times 10^8 h \Lsun \Mpc^{-3}$ and $\rho_{K_S}=(5.74\pm 0.86) \times 10^8 h \Lsun \Mpc^{-3}$ respectively, where we have adopted $M^{\odot}_J=3.73$ and $M^\odot_{K_S}=3.39$ (Allen ; Johnson ). In this analysis, we have not taken account of the systematic and random measurement errors in the galaxy magnitudes. In the case of the STY estimate, the random measurement errors can be accounted for by fitting a Schechter function which has been convolved with the distribution of magnitude errors. However, for the Kron magnitudes, the rms measurement error is only 0.1 magnitudes, as indicated by the comparison in the top right hand panel of Fig \[fig:loveday\], and such a convolution has only a small effect on the resulting Schechter function parameters. We find that the only parameter that is affected is ${M^\star}$ which becomes fainter by just $\ConvDelta=0.02$ magnitudes. The comparison to the Loveday () data also indicates a systematic error in the Kron magnitudes of $\CalkDelta=0.061\pm0.031$. Combining these two systematic errors results in a net brightening of ${M^\star}$ by $\CalkDelta-\ConvDelta=0.041\pm0.031$ magnitudes. As this net systematic error is both small and uncertain we have chosen not to apply a correction to our quoted Kron magnitude luminosity function parameters. We recall also that to convert from Kron to total magnitudes requires brightening ${M^\star}$ by between $\KDelta=0.044$ and $0.11$ depending on whether the luminosity profile of a typical galaxy is fit well by an exponential or $r^{1/4}$-law.
Fig. \[fig:lf\_jk3\] shows the SWML and STY luminosity function estimates for samples defined by the extrapolated, rather than Kron, magnitudes. With this definition of magnitude, the luminosity functions differ significantly from those estimated using Kron magnitudes. In particular, the characteristic luminosities are $0.34$ and $0.28$ magnitudes brighter in J and respectively. Most of this difference is directly related to the systematic offset in the J-band Kron and extrapolated magnitudes, which can be seen in either the middle panel of Fig. \[fig:mags\] or the right hand panels of Fig. \[fig:loveday\] to be approximately $0.23$ magnitudes. Note that even in the -band, it is this J-band offset that is relevant as the -band magnitudes we use are derived from the J-band values using the measured aperture colours. We have argued in Section \[sec:mags\] that this offset is caused by the J-band Kron magnitudes being fainter than true total magnitudes by between $\CalkDelta+\KDelta=0.1$ and $0.17$ and the extrapolated magnitudes being systematically too bright by $-\CaleDelta-\KDelta=0.05$ to $0.11$ magnitudes. Subtracting this $0.23$ magnitude offset results in Kron and extrapolated luminosity functions that differ in $M^\star$ by only 0.11 magnitudes. In the -band, the faint end slope of the best-fit Schechter function is significantly steeper in the extrapolated magnitude case, but note that this function is not a good description of the faint end of the luminosity function since the SWML and $1/V_{\rm max}$ estimates lie systematically below it. The Schechter function fit is constrained mainly around $M_\star$ and in this case the $\chi^2$ value indicates it is not a good fit overall.
The residual differences between the Kron and extrapolated magnitude luminosity functions arise from the scatter in the relation between extrapolated and Kron magnitudes. If this scatter is dominated by measurement error, then these differences represent small biases, which are largest for the less robust, extrapolated magnitudes. However, it is possible that the scatter is due to genuine variations in galaxy morphology and light profiles. To assess which of these alternatives is correct requires independent deep photometry of a sample of galaxies to quantify the accuracy of the extrapolated magnitudes. However, we note that the $V/V_{\rm max}$ distributions for the extrapolated magnitudes shown in Fig. \[fig:v\_vmax\] have mean $\langle V/V_{\rm max}\rangle$ values significantly greater than $0.5$, which is probably an indication that the extrapolated magnitudes are not robust. Thus, overall we favour adopting Kron magnitudes, noting the small offset of $\CalkDelta+\KDelta-\ConvDelta=0.08$ to $0.15$ required to convert to total magnitudes and correct for the convolving effect of measurement errors.
The parameters of the STY Schechter function fits shown in Fig. \[fig:lf\_jk2\] are listed in the first row of Table \[tab:lfpar\]. The subsequent rows illustrate how the best-fit parameters change when the cosmological model is varied and the evolutionary correction is included or excluded. The $M^\star$ values are approximately $0.14$ magnitudes fainter for the $\Omega_0=1$ case than for our standard $\Omega_0=0.3$, $\Lambda_0=0.7$ cosmology. This shift is largely due to the difference in distance moduli between the two cosmologies at the median redshift of the survey. This, and the difference in the volume-redshift relation, cause $\phi_\star$ to change in order to preserve the same galaxy number counts.
=15.0 truecm
------------------------ ------------------- -------------------------------------- ----------------------------------------- -- -- -- -- -- -- -- -- -- -- -- -- -- -- --
Mobasher et al. 1993 $-23.37\pm0.30$ $1.0\hphantom{0}\pm 0.3\hphantom{0}$ $1.12\pm0.16\times 10^{-2}$
Glazebrook et al. 1995 $-23.14\pm0.23$ $1.04\pm 0.3\hphantom{0}$ $2.22\pm0.53\times 10^{-2}$
Gardner et al. 1997 $-23.30\pm0.17$ $1.0\hphantom{0}\pm 0.24$ $1.44\pm0.20\times 10^{-2}$
Szokoly et al. 1998 $-23.80 \pm 0.30$ $1.3\hphantom{0}\pm 0.2\hphantom{0}$ $0.86\pm0.29\times 10^{-2}$
Loveday et al. 2000 $-23.58\pm 0.42$ $1.16\pm 0.19$ $1.20\pm0.08 \times 10^{-2}$
Kochanek et al. 2000 $-23.43\pm 0.05$ $1.09\pm 0.06$ $1.16\pm0.1\hphantom{0} \times 10^{-2}$
This paper $-23.44 \pm 0.03$ $0.96\pm 0.05$ $1.08\pm 0.16\times 10^{-2}$
------------------------ ------------------- -------------------------------------- ----------------------------------------- -- -- -- -- -- -- -- -- -- -- -- -- -- -- --
\[tab:lfpar\_old\]
= 9 truecm = 9 truecm
Fig. \[fig:lf\_k4\] compares our estimates of the -band luminosity function for our standard $\Omega_0=0.3$, $\Lambda_0=0.7$ cosmology, with the estimates of Mobasher (1993), Glazebrook (1995), Gardner (1997), Szokoly (1998), Loveday (2000) and Kochanek (). In general, these authors assumed different cosmological models when analysing their data. We have therefore modified the estimates from each survey. First, we apply a shift in magnitude reflecting the difference in distance moduli, at the median redshift, between the assumed cosmological model and our standard $\Omega_0=0.3$, $\Lambda_0=0.7$ model. We then apply a shift in number density so as to keep fixed the surface density of galaxies per square degree at the survey magnitude limit. In the case of the Kochanek () luminosity function we have shifted the data points brightwards by 0.05 magnitudes to account for the mean difference between the isophotal magnitudes used by Kochanek () and the Kron magnitudes we have adopted. Schechter function parameters scaled and adjusted in this manner are given for each survey in Table \[tab:lfpar\_old\]. Note that due to the correlations between the Schechter function parameters it is better to judge the agreement between the different estimates by reference to Fig. \[fig:lf\_k4\] rather than by the parameter values in Table \[tab:lfpar\_old\]. Our new estimate of the -band luminosity function is in excellent agreement with the independent estimates and has the smallest statistical errors at all magnitudes brighter than $M_{\rm {K_S}} - 5 \log
h = -22$. For very faint magnitudes, from $-20$ to $-16$, the sparsely sampled survey of Loveday (2000) has smaller statistical errors. Note that many previous analyses of the K-band luminosity function ignored the contribution of large scale structure to the error in $\Phi_\star$, and so the errors in Table \[tab:lfpar\_old\] are likely to be underestimated.
Also shown on the lower panel of Fig. \[fig:lf\_k4\] is an estimate of the K-band luminosity function inferred from the Sloan Digital Sky Survey (SDSS) near infrared, z$^*$-band luminosity function of Blanton (). To convert from z$^*$ (AB system) to standard K we have simply subtracted 2.12 magnitudes from the SDSS z$^*$ magnitudes. This offset consists of a contribution of 0.51 magnitudes to convert from AB magnitudes to the standard Vega system, and a mean z$^*$- colour of 1.61, which we find is typical of the model spectra discussed in Section \[sec:pop\] that match our observed - colours. As has been noted by Wright () the luminosity function inferred from the SDSS data is offset compared to our estimate. One suggestion put forward by Wright () is that the magnitudes could be systematically too faint. The systematic error would have to amount to 0.5 magnitudes to reconcile the luminosity density inferred from the SDSS data with that which we infer from the - catalogue. Such an error is comprehensively excluded by the very small offset that was found in Section \[sec:mags\] between the Kron magnitudes and the data of Loveday (). Also, a direct galaxy-by-galaxy comparison of the z$^*$-J and z$^*$-colours computed using the SDSS Petrosian and Kron magnitudes produced galaxy colours in good accord with expectations based on model spectra (Ivezic, Blanton and Loveday private communication). Finally, we note that a good match to our estimate of the -band luminosity function cannot be achieved by simply moving the SDDS curve in Fig. \[fig:lf\_k4\] horizontally. If slid by 0.5 magnitudes to match the luminosity density then it falls well below our estimate at bright magnitudes. However if the SDSS curve is moved vertically, by a factor of 1.6, then the two estimates come into reasonable agreement at all magnitudes. Thus, the most likely explanation of the difference between the SDSS and - luminosity functions is the uncertainty in the overall normalization induced by large-scale density fluctuations. It is to be hoped that as the sky coverage of the SDSS and - surveys increases this discrepancy will be reduced.
Colour Distributions {#sec:col}
--------------------
Since our combined – catalogue includes -band and infrared magnitudes, it is also possible to estimate the -band optical luminosity function and the optical/infrared bivariate luminosity function. We do not present the -band optical luminosity function here as estimates from the are discussed in detail in Norberg () and decomposed into luminosity functions of different spectral types in Folkes () and Madgwick (). Instead, we present the bivariate / and J/ luminosity functions in Fig. \[fig:col\], in the form of the rest-frame $-$ and J$-$ colour distributions, split by -band absolute magnitude. Results are shown for just our default set of k+e corrections, but the colour distributions are extremely insensitive to this choice and to whether evolutionary corrections are ignored or included. The shape of $-$ colour distribution varies systematically with -band luminosity. At fainter magnitudes there is an increasingly large population of bluer, star-forming galaxies. The star formation rate has less effect on the infrared J$-$ colours. Here, the shape of the J$-$colour distribution varies little with luminosity, but the position of the peak moves gradually redder with increasing luminosity. Colour distributions such as these are sensitive to both the distribution of stellar age and the metallicity, and therefore provide important constraints on models of galaxy formation (for example, see Cole ).
Spectral Type Distribution {#sec:sp_type}
--------------------------
=8.5 truecm
Another interesting issue that we can address with our data is the distribution of spectral types in the catalogue. For this, we make use of the spectral information in the 2dF galaxies extracted by a principal component analysis (Folkes ). Specifically, we use the new continuous variable introduced in Madgwick, Lahav & Taylor () which is defined by a linear combination of the first 2 principal component projections, $\eta
\equiv 0.44 pc_1 - pc_2$. This variable was chosen to be robust to instrumental uncertainties whilst, at the same time, preserving physical information about the galaxy. The dominant influence on the $\eta$ parameter is the relative strength of absorption and emission lines ($\eta$$<$$0$ implies less than average emission-line strength while $\eta$$>$$0$ implies stronger than average emission-line strength). A more detailed description will be presented in Madgwick et al. (in preparation).
We can now gain insight into the population mix of our sample by simply creating a histogram of the $\eta$ values for the corresponding – matched galaxies with $J$$<$$14.45$ (Kron). We plot this in Fig. \[fig:sp\_type\] where we also show data for the entire sample as comparison. Also shown in Fig. \[fig:sp\_type\] (bottom panel) is the morphology-$\eta$ relation derived from a sample of galaxy spectra from the Kennicutt Atlas (Kennicutt ).
It can be clearly seen from Fig. \[fig:sp\_type\] that the predominant population in the sample is has $\eta$$<$$-2$. By contrasting this with values of $\eta$ obtained from the spectra of galaxies with known morphological type (Kennicutt 1992), we can see that this corresponds to galaxies of E/S0 morphologies. More precisely, the fraction of galaxies in our matched sample with spectral types corresponding to E/S0 morphologies is 62% (compared with $\sim35\%$ in the full ). Sa-Sb galaxies make up a further 22% and the remaining 16% are galaxies of later morphological types.
The Galaxy Stellar Mass Function {#sec:smf}
--------------------------------
In contrast to optical light, near-infrared luminosities are relatively insensitive to the presence of young stars and can be more accurately related to the underlying stellar mass. Thus, with relatively few model assumptions, we can derive the distribution of galaxy stellar masses. The integral of this distribution is the total mass density in stars, which can be expressed in units of the critical density as $\Omega_{\rm stars}$. Attempts to estimate this quantity date back many decades, but even recent estimates such as those by Persic & Salucci (), Gnedin & Ostriker (), Fukugita, Hogan & Peebles () and Salucci & Persic () have very large uncertainties because they are based on B-band light and require uncertain B-band mass-to-light ratios. The much more accurate estimate that we provide here should prove very useful for a variety of purposes.
To estimate the galaxy stellar mass function, we use the modelling of the stellar populations described in Section \[sec:pop\] to obtain estimates of the present luminosity and stellar mass-to-light ratio for each galaxy in the survey. This is done on a galaxy-by-galaxy basis as described in Section \[sec:pop\]. The sample we analyze is defined by the $11$$<$J$<$$14.45$ (Kron) apparent magnitude limits. The stellar mass that we estimate for each galaxy is the mass locked up in stars and stellar remnants. This differs from the time integral of the star formation rate because some of the material that goes into forming massive stars is returned to the interstellar medium via winds and supernovae. For a given IMF, this recycled fraction, $R$, can be estimated reasonably accurately from stellar evolution theory. Here, we adopt the values $R=0.42$ and $0.28$ for the Kennicutt () and Salpeter () IMFs respectively, as described in Section 5.2 of Cole () who made use of the models of Renzini & Voli () and Woosley & Weaver (). Hence, the stellar masses we choose to estimate are $(1$$-$$R)$ times the time integral of the star formation rate to the present day. Note that the IMFs we consider assume that only stars with mass greater than 0.1M$_\odot$ ever form and so we are not accounting for any mass that may be locked up in the form of brown dwarfs.
=8 truecm
------- --------------------------------- --------------------------------- -- -- --
9.06 (4.24$\pm$2.62)$\times 10^{-2}$ (1.37$\pm$1.05)$\times 10^{-2}$
9.16 (3.42$\pm$1.80)$\times 10^{-2}$ (2.41$\pm$1.35)$\times 10^{-2}$
9.26 (3.01$\pm$1.31)$\times 10^{-2}$ (2.06$\pm$1.13)$\times 10^{-2}$
9.36 (3.33$\pm$1.11)$\times 10^{-2}$ (3.01$\pm$1.13)$\times 10^{-2}$
9.46 (4.21$\pm$1.04)$\times 10^{-2}$ (3.25$\pm$0.92)$\times 10^{-2}$
9.56 (2.75$\pm$0.67)$\times 10^{-2}$ (2.87$\pm$0.67)$\times 10^{-2}$
9.66 (2.70$\pm$0.55)$\times 10^{-2}$ (3.10$\pm$0.56)$\times 10^{-2}$
9.76 (2.31$\pm$0.42)$\times 10^{-2}$ (3.30$\pm$0.47)$\times 10^{-2}$
9.86 (2.20$\pm$0.35)$\times 10^{-2}$ (2.67$\pm$0.34)$\times 10^{-2}$
9.96 (2.21$\pm$0.31)$\times 10^{-2}$ (2.51$\pm$0.27)$\times 10^{-2}$
10.06 (1.77$\pm$0.23)$\times 10^{-2}$ (2.03$\pm$0.20)$\times 10^{-2}$
10.16 (1.91$\pm$0.20)$\times 10^{-2}$ (1.93$\pm$0.17)$\times 10^{-2}$
10.26 (1.77$\pm$0.16)$\times 10^{-2}$ (1.86$\pm$0.15)$\times 10^{-2}$
10.36 (1.46$\pm$0.12)$\times 10^{-2}$ (1.62$\pm$0.11)$\times 10^{-2}$
10.46 (1.11$\pm$0.08)$\times 10^{-2}$ (1.49$\pm$0.09)$\times 10^{-2}$
10.56 (8.15$\pm$0.61)$\times 10^{-3}$ (1.61$\pm$0.08)$\times 10^{-2}$
10.66 (5.62$\pm$0.43)$\times 10^{-3}$ (1.30$\pm$0.06)$\times 10^{-2}$
10.76 (3.39$\pm$0.29)$\times 10^{-3}$ (1.06$\pm$0.04)$\times 10^{-2}$
10.86 (2.08$\pm$0.20)$\times 10^{-3}$ (7.40$\pm$0.30)$\times 10^{-3}$
10.96 (1.07$\pm$0.12)$\times 10^{-3}$ (5.50$\pm$0.22)$\times 10^{-3}$
11.06 (5.95$\pm$0.82)$\times 10^{-4}$ (3.29$\pm$0.15)$\times 10^{-3}$
11.16 (2.75$\pm$0.49)$\times 10^{-4}$ (2.02$\pm$0.10)$\times 10^{-3}$
11.26 (1.05$\pm$0.26)$\times 10^{-4}$ (1.13$\pm$0.07)$\times 10^{-3}$
11.36 (2.77$\pm$1.11)$\times 10^{-5}$ (5.56$\pm$0.40)$\times 10^{-4}$
11.46 (9.51$\pm$5.65)$\times 10^{-6}$ (2.90$\pm$0.26)$\times 10^{-4}$
11.56 (2.05$\pm$2.38)$\times 10^{-6}$ (9.87$\pm$1.26)$\times 10^{-5}$
11.66 (6.87$\pm$13.6)$\times 10^{-7}$ (3.73$\pm$0.66)$\times 10^{-5}$
(8.46$\pm$2.58)$\times 10^{-6}$
(2.22$\pm$1.20)$\times 10^{-6}$
------- --------------------------------- --------------------------------- -- -- --
: The SWML stellar mass functions as plotted in Fig. \[fig:smf\]. The units of both $\phi$ and its uncertainty $\Delta \phi$ are number per $h^{-3}$ Mpc$^3$ per decade of mass.
\[tab:smfswml\]
Our results are presented in Fig. \[fig:smf\] which shows both SWML and Schechter function estimates of the present-day galaxy stellar mass function for two choices of IMF. The SWML estimates are tabulated in Table \[tab:smfswml\]. Just as for the luminosity functions, the stellar mass function is quite well described by the Schechter functional form. Integrating over these Schechter functions to determine the total stellar mass gives $\Omega_{\rm stars}h= (1.4
\pm 0.21) \times 10^{-3}$ for the Kennicutt IMF and $\Omega_{\rm
stars}h= (2.6 \pm 0.39)\times 10^{-3}$ for the Salpeter IMF. Note that the integral converges rapidly at both limits and, in particular, the contribution to $\Omega_{\rm stars}$ from objects with $M_{\rm
stars}<10^9$ h$^{-2}$ M$_\odot$ is negligible. We find that these values vary by less than the quoted errors when we alter the assumed (k+e)-corrections by either ignoring evolution, ignoring dust or changing $\Omega_0$. Taken together with our estimates of the -band luminosity density these, estimates imply mean stellar mass-to-light ratios of $0.73\, \Msun/\Lsun$ in the case of the Kennicutt IMF and $1.32\, \Msun/\Lsun$ for the Salpeter IMF. If we apply the correction we estimated in Section \[sec:mags\] to transform Kron into total magnitudes, then these estimates and their uncertainties increase to $\Omega_{\rm stars}h= (1.6 \pm 0.24) \times 10^{-3}$ for the Kennicutt IMF and $\Omega_{\rm stars}h= (2.9 \pm 0.43)\times
10^{-3}$ for the Salpeter IMF. Both of these estimates are consistent with the value, $\Omega_{\rm stars}= (3.0 \pm 1.0)\times 10^{-3}$ , derived by Salucci & Persic () but have fractional statistical errors which are several times smaller. With our method, the uncertainty in $\Omega_{\rm stars}$ is clearly dominated by the uncertainty in the IMF. For some purposes, it is not possible to improve upon this without a more precise knowledge of the true IMF – assuming there is a universal IMF. However, for other applications, such as modelling the star formation history of the universe, it is necessary to assume a specific IMF to convert the observational tracers of star formation to star formation rates. Hence, in this case, it is the much smaller statistical errors that are relevant.
------------- ----------------------- ----------------------- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- --
E(B-V)=0.05 $0.80 \times 10^{-3}$ $1.30 \times 10^{-3}$
E(B-V)=0.10 $1.17 \times 10^{-3}$ $1.86 \times 10^{-3}$
E(B-V)=0.15 $1.63 \times 10^{-3}$ $2.66 \times 10^{-3}$
------------- ----------------------- ----------------------- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- --
: Estimates of the present-day mass in stars and stellar remnants obtained by integrating over observational estimates of the star formation history of the universe. We express this stellar mass density in terms of the critical density and give values of $\Omega_{\rm stars} h^2$ estimated for different assumed IMF’s and dust corrections. All values are for an $\Omega_0=0.3$, $\Lambda_0=0.7$ cosmology and assume stellar populations of solar metallicity.
\[tab:madau\]
It is interesting to compare our values with what is inferred by integrating the observational estimates of the mean star formation history of the universe. Fig. \[fig:madau\] shows observational estimates for one particular choice of cosmology and IMF and illustrates how the rates are sensitive to the assumed dust extinction. By fitting a smooth curve through these estimates, we can calculate the mass of stars formed by the present day and how this depends on the IMF and assumed dust extinction. The upper smooth curve shown in Fig. \[fig:madau\] is of the form $\dot \rho_\star = (a + bz)/(1 + (z/c)^d) h M_\odot $yr$^{-1}$Mpc$^{-3}$, where $(a,b,c,d) = (0.0166,0.1848,1.9474,2.6316)$. The data points uncorrected for dust extinction are fit with $(a,b,c,d) = (0.0,0.0798,1.658,3.105)$. As for our estimates above, we assume that no mass goes into forming brown dwarfs and multiply the star formation rate by $1$$-$$R$, where $R$ is the recycled fraction, so as to form an estimate of the mass locked up in stars. Values of $\Omega_{\rm stars} h^2$ estimated in this way are listed in Table \[tab:madau\]. The values in this table are for an $\Omega_0=0.3$, $\Lambda_0=0.7$ cosmology, but they are insensitive to this choice. They depend slightly on the assumed metallicity of the stellar population and would be 10% lower if half solar, rather than solar metallicity were assumed. Note that the $\Omega_{\rm stars}$ values inferred from the star formation history of the universe scale differently with the assumed Hubble constant than those inferred above from the IR luminosity functions. For $h=0.7$ our estimates from become $\Omega_{\rm stars}h^2= (1.12 \pm 0.16)\times 10^{-3} $ for the Kennicutt IMF and $\Omega_{\rm stars}h^2= (2.03 \pm 0.30)
\times 10^{-3} $ for the Salpeter IMF. Comparison with Table \[tab:madau\] shows that these values are consistent with those inferred from the cosmic star formation history only if the dust correction assumed in the latter is modest, E(B-V)$\approx0.1$. This value is 50% smaller than the value preferred by Steidel ().
Conclusions {#sec:conc}
===========
The new generation of very large surveys currently underway make it possible to characterize the galaxy population with unprecedented accuracy. In this paper, we have combined two such large surveys, the infrared imaging 2MASS and the 2dF Galaxy Redshift Survey[^2] to obtain a complete dataset which is more than an order of magnitude larger than previous datasets used for statistical studies of the near-infrared properties of the local galaxy population. We have used this combined catalogue to derive the most precise estimates to date of the galaxy J and -band luminosity functions and of the galaxy stellar mass function.
Characterizing the near-infrared properties of galaxies offers several advantages. Firstly, the near-infrared light is dominated by established, old stellar populations rather than by the recent star formation activity that dominates the blue light. Thus, the J and K-band luminosity functions reflect the integrated star formation history of a galaxy and, as a result, provide particularly important diagnostics of the processes of galaxy formation. For the same reason, the distribution of stellar mass in galaxies –the galaxy stellar mass function– can be derived from the near-infrared luminosities in a relatively straightforward way, with only a weak model dependence. Finally, corrections for dust extinction as well as k-corrections are much smaller in the near-infrared than in the optical.
Due to the size of our sample, our determination of the J- and -band galaxy luminosity functions have, for the most part, smaller statistical errors than previous estimates. Furthermore, since our sample is infrared-selected, our estimates are free from any potential biases that might affect infrared luminosity functions derived from optically-selected samples. We find that the J- and -band galaxy luminosity functions are fairly well described by Schechter functions, although there is some evidence for an excess of bright galaxies relative to the best-fit Schechter functional form. In general, the SWML estimates are a truer representation of the luminosity functions. Our K-band estimates are in overall agreement with most previous determinations, but have smaller statistical errors.
The exception is the K-band luminosity function inferred from the near infrared SDDS photometry ([@blanton]). The difference between the K-band luminosity function we infer from their data and our own estimate is too large to be explained by photometric differences. The difference between the two estimates is better described by a difference in overall number density of a factor of $1.6$. A similar discrepancy is seen in the -band between the SDSS and luminosity function estimates (see [@norberg00] and [@blanton]). The suspicion is that the uncertainty in the overall normalization of the luminosity functions induced by large-scale structure within the large, but finite, survey volumes could be to blame. However, the errors that we quote for the -luminosity functions already include an estimate of this sampling uncertainty as derived from realistic mock catalogues. A similar exercise for a catalogue with the same area and depth as that of Blanton () indicates that the required overdensity of a factor of $1.6$ is unlikely. So probably there is more than one contributory factor at work and the hope is that these will be identified as the surveys progress.
Using our J-band luminosity function, $-$ and J$-$ colours and simple galaxy evolutionary tracks, we have obtained the first estimate of the galactic stellar mass function derived directly from near-infrared data. We find that this mass function is also fairly well described by a Schechter form. An integral over the stellar mass function gives $\Omega_{\rm stars}$, the universal mass density locked up in luminous stars and stellar remnants, expressed in terms of the critical density. $\Omega_{\rm stars}$ is a key component of the overall inventory of baryons in the universe. An accurate determination of this quantity is essential for detailed comparisons with other quantities of cosmological interest such as the total baryonic mass, $\Omega_{\rm baryon}$, inferred from Big Bang nucleosynthesis considerations (e.g. Burles & Tytler 1998), the cosmic star formation rate (e.g. Steidel ), and the cosmic evolution of the gas content of the universe (Storrie-Lombardi & Wolfe 2000). The statistical uncertainty in our estimate of $\Omega_{\rm stars}$ is about 15%, several times smaller than the best previous determination by Salucci & Persic (). In fact, the errors in $\Omega_{\rm stars}$ are dominated by systematic uncertainties associated with the choice of stellar IMF in the galaxy evolution model. A Kennicutt IMF gives $\Omega_{\rm stars}h= (1.6 \pm 0.24)\times 10^{-3}$ while a Salpeter IMF gives $\Omega_{\rm stars}h= (2.9 \pm 0.43)\times 10^{-3}$. For $h=0.7$, these values correspond to less than 11% of the baryonic mass inferred from Big Bang Nucleosynthesis ([@burles]). Our values of $\Omega_{\rm stars}$ today are only consistent with recent determinations of the integrated cosmic star formation if the correction for dust extinction is modest.
Acknowledgements {#acknowledgements .unnumbered}
================
We thank John Lucey for the initial suggestion to look at the database. We thank Tom Jarrett for advice and detailed explanations of the 2MASS data reduction procedure. We also thank Ian Smail for help in manipulating DSS images, Jon Loveday for supplying his luminosity function data in electronic form and Simon White for useful suggestions. We are grateful to the VIRGO consortium for allowing us to use the Hubble Volume simulation data prior to publication.The redshift data used here were obtained with the 2-degree field facility on the 3.9m AngloAustralian Telescope (AAT). We thank all those involved in the smooth running and continued success of the 2dF and the AAT. SMC acknowledges a PPARC Advanced Fellowship, IPRN an SNSF and ORS Studentship and CSF a Leverhulme Research Fellowship.
The 2dF Galaxy Redshift Survey was made possible through the dedicated efforts of the staff at the Anglo-Australian Observatory, both in creating the two-degree field facility and supporting it on the telescope.
Allen, C.W. 1973, Astrophysical Quantities, London: Athlone Press, University of London. (3rd edition).
Blanton, M.R., , 2001, ApJ, in press.
Bruzual, A.G., Charlot, S., 1993, ApJ, 405, 538
Bruzual, A.G., Charlot, S., 2001, in preparation.
Burles, S. & Tytler, D., 1998, ApJ, 507, 732
Burles, S., Nollet, K.M., Truran, J.N., Turner, M.S., 1999, Phys. Rev. Lett. 82, 4176.
Calzetti, D. 1999, UV emission and dust properties of high-z galaxies, Workshop on ultraluminous galaxies: Monsters or Babies. Kluwer in press. (astro-ph/9902107)
Carpenter, J.M., 2001, AJ, in press. (astro-ph/0101463)
Cole, S., Lacey, C.G., Baugh, C.M., Frenk, C.S., 2000, MNRAS, 319, 168
Cross, N., ( Team), 2001, MNRAS in press (astro-ph/0012165)
Efstathiou, G., Ellis, R.S., Peterson, B.A., 1988, MNRAS, 232, 431
Evrard, A.E., 1998, in From Recombination to Garching eds T. Banday and R. Sheth (astro-ph/9812377)
Ferrara, A., Bianchi, S., Cimatti, A., Giovanardi C., 1999 ApJS, 123, 423
Folkes, S., ( Team), 1999, MNRAS, 308, 459
Fukugita, M., Hogan, C.J., Peebles, P.J.E., 1998, ApJ., 503, 518
Gardner J.P., Sharples, R.M., Carrasco, B.E., Frenk, C.S., 1996, MNRAS, 282, 1
Gardner J.P., Sharples, R.M., Frenk, C.S., Carrasco, B.E., 1997, ApJL, 480, 99
Glazebrook, K., Peacock, J.A., Collins, C.A., Miller, L., 1994, MNRAS, 266, 65
Glazebrook, K., Peacock, J.A., Miller, L., Collins, C.A., 1995, MNRAS, 275, 169
Gnedin, Ostriker, 1992, ApJ., 400, 108, 459
Jarrett, T.H., Chester, T., Cutri, R., Schneider, S., Skrutskie, M., 2000, ApJ, 119, 2498
Johnson, H.L., 1966, ARA&A, 4,193
Kochanek, C.S., 2001, ApJ, submitted. (astro-ph/0011456)
Kennicutt, R.C., 1983, ApJ, 272, 54
Kennicutt, R.C., 1992, ApJS, 79, 255
Kron, R.G., 1980, ApJS, 43, 305
Loveday, J., 2000, MNRAS, 312, 517
Maddox, S.J., Efstathiou, G., Sutherland, W.J., Loveday, J., 1990a, MNRAS, 243, 692
Maddox, S.J., Efstathiou, G., Sutherland, W.J., 1990b, MNRAS, 246, 433
Maddox, S.J., Sutherland, W.J., Efstathiou, G., Loveday, J., Peterson, B.A., 1990c, MNRAS, 247, 1P
Maddox, S.J., Efstathiou, G., Sutherland, W.J., Loveday, J., 1996, MNRAS 283, 1227
Madgwick, D., ( Team), in preparation.
Madgwick, D., Lahav, O., Taylor, K., 2001, Springer-Verlag series “ESO Astrophysics Symposia", in press. (astro-ph/0010307)
Metcalfe, N., Fong, R., Shanks, T., 1995, MNRAS, 274, 769
Mobasher, B., Sharples, R.M., Ellis, R.S., 1993, MNRAS, 263, 560
Norberg, P., ( Team), 2001, in preparation.
Persic, M., Salucci, P., 1992, MNRAS, 258, 14P
Phillipps, S., Davies, J., Disney M. 1990, MNRAS, 242, 235.
Renzini, A., Voli, M., 1981, A&A, 94, 175
Salpeter, E.E., 1955, ApJ. 121, 61
Salucci, P., Persic, M., 1999, MNRAS, 309, 923
Sandage, A., Tammann, G.A., Yahil, A., 1978, ApJ, 232, 352
Shanks, T., 1990, in Galactic and Extragalactic Background Radiation, (ed. K. Mattilla), Reidel, Dordrecht, p269.
Steidel, C.C., Adelberger, K. L.; Giavalisco, M., Dickinson, M., Pettini, M., 1999, ApJ, 519, 1
Storrie-Lombardi, L. & Wolfe. A., 2000, ApJ, 543, 552. (astro-ph/0006044).
Szokoly, G.P., Subbarao, M.U., Connolly, A.J., Mobasher, B., 1998, ApJ, 492, 452
Woosley, S.E., Weaver, T.A., 1995, ApJS, 101, 181
Wright, E.L., 2001, ApJ, submitted. (astro-ph/0102053)
[^1]: This publication makes use of data products from the Two Micron All Sky Survey (2MASS), which is a joint project of the University of Massachusetts and the Infrared Processing and Analysis Center/California Institute of Technology, funded by the National Aeronautics and Space Administration and the National Science Foundation.
[^2]: A table containing the positions, 2MASS infra-red magnitudes and redshifts used in this paper and electronic versions of Tables \[tab:lfswml\] and \[tab:smfswml\] are available at http://star-www.dur.ac.uk/$\tilde{\hphantom{n}}$cole/2dFGRS-2MASS .
|
---
abstract: 'We show that correlations between the phases of the galaxy density field in redshift space provide additional information about the growth rate of large-scale structure that is complementary to the power spectrum multipoles. In particular, we consider the multipoles of the line correlation function (LCF), which correlates phases between three collinear points, and use the Fisher forecasting method to show that the LCF multipoles can break the degeneracy between the measurement of the growth rate of structure $f$ and the amplitude of perturbations $\sigma_8$ that is present in the power spectrum multipoles at large scales. This leads to an improvement in the measurement of $f$ and $\sigma_8$ by up to 220 per cent for $k_{\rm max} = 0.15 \, h\mathrm{Mpc}^{-1}$ and up to 50 per cent for $k_{\rm max} = 0.30 \, h\mathrm{Mpc}^{-1}$, with respect to power spectrum measurements alone for the upcoming generation of galaxy surveys like DESI and Euclid.'
author:
- |
Joyce Byun,$^{1}$[^1] Felipe Oliveira Franco,$^{1}$ Cullan Howlett,$^{2}$ Camille Bonvin$^{1}$ and Danail Obreschkow$^{3}$\
$^{1}$Département de Physique Théorique and Center for Astroparticle Physics (CAP), University of Geneva, 24 quai Ernest Ansermet,\
CH-1211 Geneva, Switzerland\
$^{2}$School of Mathematics and Physics, The University of Queensland, Brisbane, QLD 4072, Australia\
$^{3}$International Centre for Radio Astronomy Research (ICRAR), University of Western Australia, 35 Stirling Highway, Crawley WA\
6009, Australia
bibliography:
- 'references.bib'
date: 'Accepted XXX. Received YYY; in original form ZZZ'
title: Constraining the growth rate of structure with phase correlations
---
\[firstpage\]
Cosmology: theory – large-scale structure of the Universe
Introduction
============
Galaxy clustering is a powerful cosmological probe that provides a way of measuring the growth of structure and constraining cosmological parameters. The standard way of extracting information from galaxy clustering is to measure the two-point correlation function, or its Fourier-space counterpart, the power spectrum. Since observations are performed in redshift-space, the power spectrum is sensitive not only to the matter density distribution, but also to the peculiar velocities of galaxies, through the so-called redshift-space distortions (RSD). These distortions generate anisotropies in the power spectrum, in the form of a quadrupole and hexadecapole. Measuring these multipoles along with the monopole provides a measurement of three key variables in our Universe: the growth rate of structure $f$, the amplitude of fluctuations $\sigma_8$, and the linear galaxy bias $b_1$ [@Kaiser:1987qv]. However, these quantities are not measured independently: at large scales the power spectrum is sensitive to $f\sigma_8$ and $b_1\sigma_8$. Although in theory these degeneracies can be overcome by including information from non-linear scales, in practice additional data like the Cosmic Microwave Background (CMB) or gravitational lensing are used.
Instead, in this paper we show that by using Fourier phase information it is possible to break the degeneracies and constrain $f, b_1$ and $\sigma_8$ separately, without relying on external data sets. This method has the strong advantage of being model-independent, as it does not depend on a model for how the linear growth factor and growth rate evolve with redshift. This is not the case when probes at different redshifts are combined, for example the power spectrum and the CMB, where translating the measured primordial amplitude of perturbations $A_s$ into a measurement of $\sigma_8$ requires assuming a cosmological model, like $\Lambda$CDM and general relativity (GR), to describe the growth of density perturbations [@Alam:2016hwk]. Similarly, lensing data provide a measurement of $S_8\equiv \sigma_8\sqrt{\Omega_m/0.3}$, which requires a knowledge of the matter density parameter $\Omega_m$ to extract $\sigma_8$ [@Abbott:2017wau; @Hildebrandt:2016iqg; @Hildebrandt:2018yau; @Hikage:2018qbn; @Hamana:2019etx]. Aiming for a more model-independent determination of $f, b_1$ and $\sigma_8$ is particularly important in the current situation where tensions exist between different data sets [@Verde:2019ivm].
Various estimators have been proposed in the past to measure information contained in Fourier phases $\epsilon({\mathbf{k}}) \equiv \delta({\mathbf{k}})/|\delta({\mathbf{k}})|$, where $\delta({\mathbf{k}})$ is the density field. Here we focus on the line correlation function (LCF) that measures a type of three-point correlation function restricted to phases along a line [@Obreschkow:2012yb]. Geometrically, the LCF can be interpreted as a quantification of the amount of filamentary structure in the cosmic web on different scales (see @Obreschkow:2012yb for details). The LCF has been modelled in real space using both perturbation theory and effective non-linear kernels and was shown to agree well with measurements from N-body simulations over a wide range of scales [@Wolstenhulme:2014cla]. [@Eggemeier:2016asq] demonstrated that the LCF used in combination with the power spectrum improves the constraints on cosmological parameters by up to a factor of 2 compared with the power spectrum alone. [@Ali:2018sdk] then extended these forecasts to models beyond $\Lambda$CDM, including warm dark matter models and modified gravity models, and showed that the LCF also improves parameter constraints in these cases.
These previous works have shown the potential of the LCF to improve parameter constraints in real space. The goal of this work is to extend these analyses to redshift space, where observations are made. In [@Eggemeier:2015ifa], the LCF was modified to be more sensitive to RSD, and the resulting anisotropic LCF estimator was applied to toy mocks. More recently, [@Franco:2018yag] used tree-level perturbation theory to model the LCF multipoles in redshift space. They showed that even though in principle an infinite number of multipoles are generated by RSD, nearly all of the information about the velocity phase correlations is contained in the monopole, quadrupole and hexadecapole. In this work we explore how these multipoles may be used in combination with the power spectrum multipoles to break the degeneracy between $f, b_1$ and $\sigma_8$, with data from the upcoming generation of large-scale structure surveys like the Dark Energy Spectroscopic Instrument (DESI; @Aghamousa:2016zmz) and Euclid [@Laureijs2011].
We note that this study of the information content of the LCF multipoles is within the larger context of how higher-order statistics can complement and strengthen cosmological constraints from power spectrum analyses. In real-space, [@Byun:2017fkz] showed that even though the LCF does not constrain $\Lambda$CDM parameters as well as the full bispectrum or the modal bispectrum, it still provides an efficient way to compress information and gives improvements in cosmological parameter constraints ranging from a few per cent to 70 per cent, depending on the parameter. In redshift space, analyses of the Baryon Oscillation Spectroscopic Survey (BOSS) have combined the bispectrum monopole with power spectrum multipoles [@Gil-Marin:2016wya] and detected the bispectrum quadrupole [@Sugiyama:2018yzo], while for future surveys, [@Gagrani:2016rfy], [@Yankelevich:2018uaz] and [@Gualdi:2020ymf] have explored the additional benefit of using the anisotropic bispectrum beyond the monopole for cosmological constraints. We defer a comparison of the information in the redshift-space LCF and redshift-space bispectrum to a future work.
The main results of this work are Fisher forecasts for how well the LCF multipoles may further improve constraints on $f$ and $\sigma_8$ from the power spectrum multipoles that will be measured by upcoming surveys such as DESI and Euclid. To provide guidance for our forecasts, we compare our theoretical model for the power spectrum and LCF multipoles with simulations and find that the model reproduces well the measured power spectrum multipoles up to $k_{\rm max} = 0.3\,h {\mathrm{Mpc}}^{-1}$ and the LCF multipoles down to $r_{\rm min} = 20\,h^{-1}{\mathrm{Mpc}}$. We also use perturbation theory to calculate the full covariance of the power spectrum and LCF multipoles, and we compare it with the covariance measured from 500 <span style="font-variant:small-caps;">l-picola</span> simulations at redshift $z=0$. We find that using the theoretical covariance matrix underestimates the forecasted errors on cosmological parameters by 20 per cent at most, compared with constraints obtained using the simulated covariance matrix, and we find a similar level of agreement when comparing constraints with measured or predicted values of the Fingers-of-God velocity dispersions. We also show that neglecting the cross-covariance between the power spectrum and LCF changes the resulting constraints by less than 10 per cent. To estimate the future constraints that may be achievable with a joint power spectrum and LCF multipoles analysis, at present we must rely more heavily on our theoretical models, but the encouraging results of these checks support the extension of our forecasting pipeline to upcoming surveys.
For upcoming surveys like DESI and Euclid, we calculate Fisher forecasted constraints for $f$ and $\sigma_8$ in each redshift bin and compare constraints from only the power spectrum multipoles with constraints from the combined power spectrum and LCF multipoles. Our forecast estimates that the LCF multipoles could strengthen the constraints on $f$ and $\sigma_8$ from the power spectrum multipoles by up to 220 per cent if scales up to $k_{\rm max}=0.15\,h{\mathrm{Mpc}}^{-1}$ are included. If $k_{\rm max}=0.3\,h{\mathrm{Mpc}}^{-1}$, the improvement from the LCF multipoles is smaller but still significant, up to 50 per cent. If CMB measurements from the Planck survey [@Aghanim:2018eyx] are included through a prior on $\sigma_8$, the improvement brought by the LCF multipoles becomes marginal, less than $\sim 10$ per cent. However, we note again that the advantage of using the LCF multipoles instead of external CMB data is that translating the CMB constraint on $A_s$ into a constraint on $\sigma_8$ depends on an assumed cosmological model, such as $\Lambda$CDM and general relativity.
The outline for this paper is as follows. In Section \[sec:modeling\], we summarise the theoretical models we use for the halo power spectrum and LCF multipoles. Section \[sec:sims\] describes the simulation data and the power spectrum and LCF multipole estimators. Section \[sec:theoretical\_covariance\] summarises the theoretical models we use for the covariance matrices, while the detailed derivations are in Appendices \[app:Qn\_cov\] and \[app:PlQn\_crosscov\]. In Section \[sec:compare\], we compare our model predictions with measurements from simulations, and in Section \[sec:fisher\] we validate our forecasting pipeline with Fisher forecasts based on the simulation boxes. We present our forecasts for surveys like DESI and Euclid in Section \[sec:surveys\], and we end with our conclusions in Section \[sec:conclusions\].
Theoretical modeling {#sec:modeling}
====================
In this section, we specify the models we use for the power spectrum and line correlation function (LCF) multipoles in redshift space. Galaxy clustering observables measure the correlations between galaxy overdensities, $$\Delta(\mathbf{x})\equiv\frac{N(\mathbf{x})-\bar N}{\bar N},$$ where $N(\mathbf{x})$ is the number density of galaxies at position $\mathbf{x}$ and $\bar N$ is the average number density. The LCF is defined as the correlation between the phases of $\Delta$ at three collinear points, $$\begin{aligned}
\ell\left(\mathbf{r}\right) & \equiv V^3 \left(\frac{r^{3}}{V}\right)^{3/2}\Braket{\epsilon\left(\mathbf{x}+\mathbf{r}\right)\epsilon\left(\mathbf{x}\right)\epsilon\left(\mathbf{x}-\mathbf{r}\right)} \nonumber\\
& = \frac{V^{3}}{\left(2\pi\right)^{9}}\left(\frac{r^{3}}{V}\right)^{3/2}\iiintop_{k_{1},k_{2},k_{3}\leq\frac{2\pi}{r}}{\ensuremath{\text{d}^3 k_1}}{\ensuremath{\text{d}^3 k_2}}{\ensuremath{\text{d}^3 k_3}}e^{i\mathbf{x}\cdot\left(\mathbf{k}_{1}+\mathbf{k}_{2}+\mathbf{k}_{3}\right)}e^{i\mathbf{r}\cdot\left(\mathbf{k}_{1}-\mathbf{k}_{2}\right)}\Braket{\epsilon\left(\mathbf{k}_{1}\right)\,\epsilon\left(\mathbf{k}_{2}\right)\,\epsilon\left(\mathbf{k}_{3}\right)},
\label{eq:LCF definition fourier}\end{aligned}$$ where $\epsilon(\mathbf{x})$ is the inverse Fourier transform of $\epsilon(\mathbf{k}) \equiv \frac{\Delta(\mathbf{k})}{|\Delta(\mathbf{k})|}$ and $\Delta(\mathbf{k})$ is the Fourier transform of $\Delta(\mathbf{x})$.
In principle the 3-point correlation function of phases depends on the bispectrum and all higher-order cumulants, but in the mildly non-Gaussian regime, it can be expressed in terms of the power spectrum and bispectrum of $\Delta$ using the Edgeworth expansion as [@Matsubara:2003te; @Wolstenhulme:2014cla] $$\Braket{\epsilon\left(\mathbf{k}_{1}\right)\,\epsilon\left(\mathbf{k}_{2}\right)\,\epsilon\left(\mathbf{k}_{3}\right)} = \frac{(2\pi)^3}{V} \left( \frac{\sqrt{\pi}}{2}\right)^3 \frac{B(\mathbf{k}_{1},\mathbf{k}_{2},\mathbf{k}_{3})}{\sqrt{V P(\mathbf{k}_{1})P(\mathbf{k}_{2})P(\mathbf{k}_{3})}} \delta_D(\mathbf{k}_{1}+\mathbf{k}_{2}+\mathbf{k}_{3}).
\label{eq:3eps}$$ In [@Franco:2018yag], the ratio in eq. was calculated using tree-level perturbation theory, where two types of contributions arise: an intrinsic contribution due to the fact that the density, velocity and bias are non-linear, and a mapping contribution generated by the non-linear mapping between real space and redshift space. In this paper, we modify the intrinsic contribution in the bispectrum and use a non-linear model for the power spectrum to better describe the LCF down to smaller scales where tree-level perturbation theory breaks down. In the following subsections, we summarise how we model these quantities using established models in the literature.
The LCF multipoles are then given by $$Q_n(r) = \frac{2n+1}{2} \int_{-1}^1 d\nu\, \ell (r,\nu) L_n(\nu)\, , \label{eq:Qn}$$ where $\nu$ depends on the orientation of the line with respect to the direction of observation, $\nu \equiv \hat{\mathbf{r}} \cdot \hat{{\mathbf{n}}}$, and $L_n$ is the Legendre polynomial of degree $n$.
Power spectrum modeling
-----------------------
For the redshift-space power spectrum, we use a variant of the model from [@Vlah:2013lia], as implemented in [@Howlett:2019bky]. To summarise, the model accounts for the large-scale coherent motions of galaxies towards overdensities (the Kaiser effect), random non-linear motions of these galaxies within virialised halos, and the non-linear and non-local bias of galaxies with respect to the underlying matter field. It does this using a distribution function approach, where different contributions to the full anisotropic power spectrum are broken down into terms containing unique powers of the mass-weighted velocity (momentum) field. Each of these contributions is then evaluated using standard (one-loop) perturbation theory for biased tracers.
For consistency with the Fingers-of-God modeling used in the bispectrum and LCF, we remove the terms dependent on the velocity dispersion from within the various components of the model and replace them with a single, global damping term, $D^{P}_{{\textrm{FoG}}}$. We adopt three different forms of the damping term, which are described in Section \[sec:fog\].
Following this, we write the anisotropic power spectrum as $$P(k,\mu) = D^{P}_{{\textrm{FoG}}}(k,\mu,\sigma_P)\biggl[P_{00}(k) + \mu^{2}(2P_{01}(k) + P_{02}(k,\mu)) + P_{11}(k,\mu) + \mu^{4}(P_{12}(k,\mu) + 1/4P_{22}(k,\mu))\biggl],
\label{eq:Pvlah}$$ where $\mu=\hat{{\mathbf{k}}}\cdot\hat{{\mathbf{n}}}$ and $$\begin{aligned}
P_{00} & = b_{1}^{2}P_{\delta \delta} + 2b_{1}(b_{2}K_{00} + b_{s}K^{s}_{00} + b_{3nl}\sigma^{2}_{3}P_{m}) + 1/2b_{2}^{2}K_{01} + 1/2b^{2}_{s}K^{s}_{01} + b_{2}b_{s}K^{s}_{02}\, , \\
P_{01} & = fb_{1}(P_{\delta \theta} + 2b_{1}I_{10} + 6k^{2}P_{m}b_{1}J_{10} - b_{2}K_{11} - b_{s}K^{s}_{11}) - f(b_{2}K_{10} + b_{s}K^{s}_{10} + b_{3nl}\sigma^{2}_{3}P_{m})\, , \\
P_{02} & = f^{2}b_{1}(I_{02} + \mu^{2}I_{20} + 2k^{2}P_{m}(J_{02} + \mu^{2}J_{20})) - f^{2}(b_{2}(K_{20}+\mu^{2}K_{30}) + b_{s}(K^{s}_{20}+\mu^{2}K^{s}_{30}))\, , \\
P_{11} & = f^{2}(\mu^{2}(P_{\theta \theta} + 4b_{1}I_{22} + b_{1}^{2}I_{13} + 12k^{2}P_{m}b_{1}J_{10}) + b_{1}^{2}I_{31})\, , \\
P_{12} & = f^{3}(I_{12} + \mu^{2}I_{21} - b_{1}(I_{03} + \mu^{2}I_{30}) + 2k^{2}P_{m}(J_{02} + \mu^{2}J_{20}))\, , \\
P_{22} & = 1/4f^{4}(I_{23} + 2\mu^{2}I_{32} + \mu^{4}I_{33})\, .\end{aligned}$$ Here $f$ is the growth rate of structure, and $b_{1}$, $b_{2}$, $b_{s}$ and $b_{3nl}$ are the galaxy bias parameters under a Eulerian bias expansion relating the galaxy overdensity $\delta_{g}$ and matter overdensity $\delta$, $$\delta_{g}({\boldsymbol{x}}) = b_{1}\delta({\boldsymbol{x}}) + \frac{b_{2}}{2}(\delta^{2}({\boldsymbol{x}})-\langle \delta \rangle) + \frac{b_{s}}{2}(s^{2}({\boldsymbol{x}})-\langle s \rangle) + \frac{b_{3nl}}{6}\delta^{3}({\boldsymbol{x}}).$$ In deriving the power spectrum model, the above bias expansion is renormalised following [@Saito:2014qha] and from which we also adopt $b_{s}=-4/7(b_{1}-1)$ and $b_{3nl}=32/315(b_{1}-1)$. The terms $I_{ij}$, $J_{ij}$, $K_{ij}$ and $\sigma_{3}$ are integrals over the matter power spectrum $P_{m}$ and are presented in the appendices of [@Vlah:2012ni] and [@Howlett:2019bky]. $s(\boldsymbol{x})$ is the Fourier transform of $s(\boldsymbol{k})$, which is also presented in [@Vlah:2013lia]. Finally, $P_{\delta \delta}$, $P_{\delta \theta}$ and $P_{\theta \theta}$ are the non-linear density-density, density-velocity divergence and velocity divergence-velocity divergence power spectra. In [@Vlah:2013lia] these were computed using one-loop standard perturbation theory, but in this work we use the non-linear real-space matter power spectrum measured from the simulations in Section \[sec:sims\] for $P_{\delta \delta}$ and to compute the non-linear RSD terms. As such we also set $P_{m}=P_{\delta\delta}$ in the above equations. For $P_{\delta \theta}$ and $P_{\theta \theta}$, we use the fitting formulae from [@Jennings:2012ej], again with the measured simulation matter power spectrum as input. Although the use of the simulated non-linear power spectrum in the above model is not strictly consistent from a theoretical point of view, this gave the best fit to the simulation redshift-space power spectrum and is suitable for the forecasts in this work.
Bispectrum modeling
-------------------
As our model for the redshift-space halo bispectrum, we have implemented the fitting formula in [@Gil-Marin:2014pva] that extends the tree-level expression for the bispectrum to smaller scales by modifying the perturbation theory kernels that describe the non-linear density and peculiar velocity.
The standard perturbation theory expression for the tree-level halo bispectrum in redshift space is $$\begin{aligned}
B_{\rm tree}({\mathbf{k}}_1,{\mathbf{k}}_2) &= 2 Z_1({\mathbf{k}}_1) Z_1({\mathbf{k}}_2) Z_2({\mathbf{k}}_1,{\mathbf{k}}_2)P_m(k_1)P_m(k_2) + 2\;\mathrm{cyclic\;permutations}\,,\label{eq:B}\\
Z_1({\mathbf{k}}_i) &\equiv b_1 + f \mu_i^2, \label{eq:Z1}\\
Z_2({\mathbf{k}}_1,{\mathbf{k}}_2) &\equiv b_1 F_2({\mathbf{k}}_1,{\mathbf{k}}_2) + f \mu^2 G_2({\mathbf{k}}_1,{\mathbf{k}}_2) + \frac{b_2}{2} + \frac{b_{s^2}}{2} S_2({\mathbf{k}}_1,{\mathbf{k}}_2)+ \frac{b_1f \mu k}{2} \left( \frac{\mu_1}{k_1} + \frac{\mu_2}{k_2} \right)
+ \frac{f^2 \mu k}{2} \mu_1 \mu_2 \left( \frac{\mu_2}{k_1} + \frac{\mu_1}{k_2} \right),\label{eq:Z2}\end{aligned}$$ where $\mu_i\equiv \hat{{\mathbf{k}}}_i \cdot \hat{{\mathbf{n}}}$, $\mu \equiv (\mu_1k_1+\mu_2k_2)/k$, and $k^2 \equiv ({\mathbf{k}}_1 + {\mathbf{k}}_2)^2$. $F_2$ and $G_2$ are the non-linear density and velocity kernels at second-order in perturbation theory [@Bernardeau:2001qr]. The first four terms in eq. give rise to the intrinsic kernel of the 3-point phase correlations, called $W_2^{\rm int}$ in [@Franco:2018yag], while the last two terms give rise to the mapping kernel, $W_2^{\rm map}$. Here we modify the intrinsic part following [@Gil-Marin:2014pva], by replacing the $F_2$ and $G_2$ kernels in the above expression with the fitted effective kernels $F_2^{\mathrm{eff}}$ and $G_2^{\mathrm{eff}}$ given in eqs. (2.19), (2.20), (5.1), (5.2), and Appendix A of that paper. Each effective kernel depends on nine free parameters that are fitted using a subset of triangle configurations of the matter bispectrum monopole measured from simulations at $0 \leq z \leq 1.5$. As shown in [@Gil-Marin:2014pva], these parameters depend only weakly on cosmology, and therefore we do not vary them in our Fisher forecasts.
Similarly to the power spectrum model in eq. , we also multiply the bispectrum by a global damping factor $D^B_{{\textrm{FoG}}}$ to include the Fingers-of-God effect. The three different forms of this damping factor that we implement in this work are described in Section \[sec:fog\]. Thus the bispectrum model we use is $$\begin{aligned}
B({\mathbf{k}}_1,{\mathbf{k}}_2) &= D_{{\textrm{FoG}}}^B({\mathbf{k}}_1,{\mathbf{k}}_2,\sigma_B)
\biggl[ 2 Z_1({\mathbf{k}}_1) Z_1({\mathbf{k}}_2) Z_2^{\rm eff}({\mathbf{k}}_1,{\mathbf{k}}_2)P_m(k_1)P_m(k_2) + 2\;\mathrm{cyclic\;permutations}\biggl], \label{eq:Bgm}\\
Z_2^{\rm eff}({\mathbf{k}}_1,{\mathbf{k}}_2) &\equiv b_1 F_2^{\rm eff}({\mathbf{k}}_1,{\mathbf{k}}_2) + f \mu^2 G_2^{\rm eff}({\mathbf{k}}_1,{\mathbf{k}}_2) + \frac{b_2}{2} + \frac{b_{s^2}}{2} S_2({\mathbf{k}}_1,{\mathbf{k}}_2)+ \frac{b_1f \mu k}{2} \left( \frac{\mu_1}{k_1} + \frac{\mu_2}{k_2} \right)
+ \frac{f^2 \mu k}{2} \mu_1 \mu_2 \left( \frac{\mu_2}{k_1} + \frac{\mu_1}{k_2} \right).\end{aligned}$$ Strictly speaking, for the tree-level halo bispectrum in perturbation theory, the matter power spectrum $P_m$ in eq. is the *linear* matter power spectrum, since non-linearities are encoded in $Z_2$. However, as in [@Gil-Marin:2014pva], we will use the average real-space *non-linear* matter power spectrum measured from simulations for $P_m$ in eq. .
Fingers-of-God {#sec:fog}
--------------
In both the power spectrum and bispectrum models summarised above, we multiplied a phenomenological damping factor to account for the Fingers-of-God (FoG) effect on small scales. Damping factors for the power spectrum and bispectrum have taken different analytic forms in the literature, each with free parameters representing the velocity dispersions of galaxies. Previous works that have jointly analysed simulation power spectra and bispectra include a damping factor for each [@Gil-Marin:2014pva; @Hashimoto:2017klo], and we follow the same procedure to create a damping factor for the LCF by appropriately combining damping factors that have been used previously for the power spectrum and bispectrum.
We explore three choices for the functional forms of the damping factors. The first is a Gaussian function [@Hashimoto:2017klo; @Yankelevich:2018uaz], $$\begin{aligned}
D_{{\textrm{FoG}}}^P(k,\mu,\sigma_P) &= \exp \left( - \frac{1}{2} k^2 \mu^2 \sigma_P^2 \right),
&D_{{\textrm{FoG}}}^B(k_i,\mu_i,\sigma_B) &= \exp \left( - \frac{1}{2} [ k_1^2 \mu_1^2 + k_2^2 \mu_2^2 + k_3^2 \mu_3^2 ] \sigma_B^2 \right),
\label{eq:gaussianfog}\end{aligned}$$ while the second is from [@Gil-Marin:2014pva], $$\begin{aligned}
D_{{\textrm{FoG}}}^P(k,\mu,\sigma_P) &= \left( 1 + \frac{1}{2} k^2 \mu^2 \sigma_P^2 \right)^{-2},
&D_{{\textrm{FoG}}}^B(k_i,\mu_i,\sigma_B) &= \left( 1 + \frac{1}{2} [ k_1^2 \mu_1^2 + k_2^2 \mu_2^2 + k_3^2 \mu_3^2 ]^2 \sigma_B^2 \right)^{-2},
\label{eq:gmfog}\end{aligned}$$ and the third form is a Lorentzian function [@Hashimoto:2017klo], $$\begin{aligned}
D_{{\textrm{FoG}}}^P(k,\mu,\sigma_P) &= \left( 1 + \frac{1}{2} k^2 \mu^2 \sigma_P^2 \right)^{-1},
&D_{{\textrm{FoG}}}^B(k_i,\mu_i,\sigma_B) &= \left( 1 + \frac{1}{2} [ k_1^2 \mu_1^2 + k_2^2 \mu_2^2 + k_3^2 \mu_3^2 ] \sigma_B^2 \right)^{-1}.
\label{eq:lorentzianfog}\end{aligned}$$
The velocity dispersions $\sigma_P$ and $\sigma_B$ generally depend on the halo populations and will evolve with redshift, so we do not adopt the exact values of the velocity dispersions from the previously mentioned papers. Instead, we either fit their values to the <span style="font-variant:small-caps;">l-picola</span> simulations at $z=0$ that we use in this work, or for the survey forecasts at higher redshifts, we use the linear prediction for the velocity dispersion, given by $$\sigma_P^2 = \frac{2f^2}{3} \int \frac{{\ensuremath{\text{d}^3 k}}}{(2\pi)^3} \, \frac{P_m(k)}{k^2},
\label{eq:sigmav2}$$ where, consistently with the rest of our analysis, we use the non-linear real-space matter power spectrum for $P_m$.
In perturbation theory, assuming that the power spectrum and bispectrum are treated consistently, one might expect that the velocity dispersions $\sigma_P$ and $\sigma_B$ are the same [@Hashimoto:2017klo]. However, in this work we conservatively treat $\sigma_P$ and $\sigma_B$ as two separate nuisance parameters that reflect our lack of knowledge about clustering on very small scales. For this reason, we allow the damping factors for the power spectrum and bispectrum to have independently varying velocity dispersions and constrain both at the same time. For the survey forecasts, we choose the fiducial values such that $\sigma_B = \sigma_P$, but treat $\sigma_P$ and $\sigma_B$ as two independently varying nuisance parameters.
Shot noise {#sec:shot noise}
----------
Lastly, our models of the halo power spectrum and LCF must include the effect of shot noise. To calculate the shot noise effect on the LCF, we include Poissonian shot noise in the power spectrum and bispectrum in eq. by adding the following $P_{\rm noise}$ and $B_{\rm noise}$ terms to the halo power spectrum and bispectrum, respectively [@Eggemeier:2016asq]: $$\begin{aligned}
P_{\rm noise} &= \frac{1}{\overline{n}}\,,
&B_{\rm noise}(\textbf{k}_1,\textbf{k}_2,\textbf{k}_3) &= \frac{1}{\overline{n}}\left[ P(\textbf{k}_1) + P(\textbf{k}_2) + P(\textbf{k}_3)\right] + \frac{1}{\overline{n}^2}\,,
\label{eq:PBnoise}\end{aligned}$$ where $P(\textbf{k}_i)$ here is the redshift-space halo power spectrum without shot noise. Replacing $P$ and $B$ in eq. by $P+P_{\rm noise}$ and $B+B_{\rm noise}$ gives $$\Braket{\epsilon\left(\mathbf{k}_{1}\right)\epsilon\left(\mathbf{k}_{2}\right)\epsilon\left(\mathbf{k}_{3}\right)} = \frac{(2\pi)^3}{V} \left( \frac{\sqrt{\pi}}{2}\right)^3 \sqrt{\frac{\nu_{\rm eff}({\mathbf{k}}_1)\nu_{\rm eff}({\mathbf{k}}_2)\nu_{\rm eff}({\mathbf{k}}_3)}{V P(\mathbf{k}_{1})P(\mathbf{k}_{2})P(\mathbf{k}_{3})}}
\left[B(\mathbf{k}_{1},\mathbf{k}_{2},\mathbf{k}_{3})+\frac{1}{\overline{n}}\left[ P(\textbf{k}_1) + P(\textbf{k}_2) + P(\textbf{k}_3)\right] + \frac{1}{\overline{n}^2}\right] \delta_D(\mathbf{k}_{1}+\mathbf{k}_{2}+\mathbf{k}_{3})\,,
\label{eq:3epsshot}$$ where $$\nu_{\rm eff}({\mathbf{k}})\equiv\frac{\overline{n}P({\mathbf{k}})}{1+\overline{n}P({\mathbf{k}})}\,.
\label{eq:nu eff}$$
Shot noise therefore has two effects on the LCF. First, it suppresses the signal coming from the bispectrum, i.e. the term proportional to the bispectrum $B$, due to the shot noise suppression factor $\nu_{\rm eff}$. This term is called the effective LCF in [@Eggemeier:2016asq]. Second, shot noise adds a noise contribution to the signal, made of the last two terms in eq. . The predicted effective LCF and total LCF are plotted as dashed and solid curves in Figure \[fig:lcffits\], where the theoretical LCF multipoles are compared with measurements from simulations.
As in the case of the bispectrum, the shot noise contribution to the LCF depends on cosmology. As discussed in [@Yankelevich:2018uaz], there are then two possible ways to treat the shot noise terms when performing the forecasts. The first option is to conservatively assume that the shot noise modeling is not accurate enough to extract information from its cosmology dependence, and therefore not to vary this term in the Fisher forecast derivatives. In this case, the observable is the effective LCF, as in [@Eggemeier:2016asq]. The second option is to assume that the cosmology-dependence of the shot noise is accurately modeled by the form of $B_{\rm noise}$, such that we can use the cosmology-dependence of the shot noise as a source of information about the parameters of interest. In this case, the observable is the total LCF. In Section \[sec:fisher\], we compare Fisher forecasts for both options, to demonstrate how much constraining power can be gained through the cosmology-dependence of the shot noise.
Summary of theoretical modeling {#sec:model_summary}
-------------------------------
We use the halo power spectrum model in eq. , which is originally from [@Vlah:2013lia] but modified according to [@Howlett:2019bky], and further modified in this work to accommodate a global Fingers-of-God damping factor. Including the additive shot noise term in eq. , our theoretical model for the power spectrum multipoles is then given by $$P_\ell(k) \equiv \frac{2n+1}{2} \int_{-1}^1 {\ensuremath{\text{d}\mu}}\, \left[ P(k,\mu) + P_{\rm noise} \right] L_n(\mu).$$
Our model for the LCF multipoles is built from a combination of eq. and the bispectrum model from [@Gil-Marin:2014pva] in eq. . To obtain our predictions for the LCF multipoles, we combine eqs. , and . This results in an expression for $Q_n$ that contains a 7-dimensional integral. To simplify this expression, we first note that $\ell(r,\nu)$ depends only on the angle $\nu=\hat{\mathbf{r}}\cdot\hat{\mathbf{n}}$ and not on the angle $\phi$, which describes a rotation of $\mathbf{r}$ around $\hat{\mathbf{n}}$. We can therefore average the multipoles in eq. over $\phi$. We then rewrite the exponential in eq. and the Legendre polynomial in eq. using $$e^{i\mathbf{r}\cdot\boldsymbol{\kappa}_{1}}=4\pi\sum_{l=0}^{\infty}\sum_{m=-l}^{l}i^{l}j_{l}\left(\kappa_1 r\right)Y_{lm}\left(\hat{\boldsymbol{\kappa}}_1\right)Y_{lm}^{*}\left(\hat{\mathbf{r}}\right),
\label{eq:exp as Ylm}$$ where $\boldsymbol{\kappa}_{1}\equiv \mathbf{k}_{1}-\mathbf{k}_{2}$, and $$L_{n}\left(\nu\right)=\frac{4\pi}{2n+1}\sum_{p=-n}^{n}Y_{np}\left(\hat{\mathbf{r}}\right)Y_{np}^*\left(\hat{\mathbf{n}}\right).
\label{eq:L as Ylm}$$ Using these substitutions and noting the orthonormality of the spherical harmonic functions, $$\int d\Omega_{\hat{\mathbf{r}}} \,Y_{lm}\left(\hat{\mathbf{r}}\right)Y_{np}^{*}\left(\hat{\mathbf{r}}\right) = \delta_{nl}^{K}\delta_{pm}^{K}\,,
\label{eq:Ylm orth}$$ we obtain our theoretical prediction for the LCF multipoles as $$\begin{aligned}
Q_n(r) &= \left(2n+1\right)i^{n}\frac{V^2}{(2\pi)^6}\left(\frac{r^{3}}{V}\right)^{3/2} \left( \frac{\sqrt{\pi}}{2}\right)^3 \iintop_{k_{1},k_{2},|\mathbf{k}_{1}+\mathbf{k}_{2}|\leq\frac{2\pi}{r}} {\ensuremath{\text{d}^3 k_1}}{\ensuremath{\text{d}^3 k_2}}\, j_{n}\left(\kappa_1r\right)L_{n}\left(\hat{\boldsymbol{\kappa}}_1\cdot\hat{\mathbf{n}}\right) \nonumber \\
&\times \sqrt{\frac{\nu_{\rm eff}({\mathbf{k}}_1)\nu_{\rm eff}({\mathbf{k}}_2)\nu_{\rm eff}({\mathbf{k}}_3)}{V P(\mathbf{k}_{1})P(\mathbf{k}_{2})P(\mathbf{k}_{3})}} \left[B(\mathbf{k}_{1},\mathbf{k}_{2},\mathbf{k}_{3})+\frac{1}{\overline{n}}\left[ P(\textbf{k}_1) + P(\textbf{k}_2) + P(\textbf{k}_3)\right] + \frac{1}{\overline{n}^2}\right].\end{aligned}$$
We note the LCF modeling in this work is comprised of a combination of existing models in the literature for the power spectrum and bispectrum which are themselves derived from different assumptions and frameworks. Thus they are not fully consistent from a perturbation theory point of view. However, for the purposes of the first forecast using the redshift-space LCF that is in this work, we implement this model and check how it compares with simulations in Section \[sec:compare\] before proceeding with the Fisher forecasts.
We have previously mentioned that the LCF multipoles aid in breaking parameter degeneracies, and here we discuss this point in more detail. [@Franco:2018yag] derived explicit expressions for the multipoles of the LCF using tree-level perturbation theory and found that they depend on different combinations of the growth rate $f$ and the amplitude of perturbations $\sigma_8$. For example, eq. (40) in that work showed that the intrinsic part of the monopole takes the simple form $$\begin{aligned}
Q^{\rm int}_0(r)=&\frac{r^{9/2}}{8\pi\sqrt{2}}\int_{0}^{2\pi/r}\hspace{-0.2cm}dk_{1}k_{1}^{2}\int_{0}^{2\pi/r}\hspace{-0.2cm}dk_{2}k_{2}^{2}\int_{-1}^{\alpha_{\text{cut}}}\hspace{-0.1cm}d\alpha \sqrt{\frac{P_{L}\left(\left|\mathbf{k}_{1}+\mathbf{k}_{2}\right|\right)P_{L}\left(k_{1}\right)}{P_{L}\left(k_{2}\right)}} \Bigg\{F_{2}(-{\mathbf{k}}_1-{\mathbf{k}}_2,{\mathbf{k}}_1) \nonumber\\
+&\left(F_{2}(-{\mathbf{k}}_1-{\mathbf{k}}_2,{\mathbf{k}}_1)-G_2(-{\mathbf{k}}_1-{\mathbf{k}}_2,{\mathbf{k}}_1)\right)\left(\frac{\arctan\sqrt{f/b_1}}{\sqrt{f/b_1}}-1\right)+\left(\frac{b_2}{2b_1}+\frac{b_{s^2}}{2b_1}S_2(-{\mathbf{k}}_1-{\mathbf{k}}_2,{\mathbf{k}}_1)\right)\left(\frac{\arctan\sqrt{f/b_1}}{\sqrt{f/b_1}}\right)\Bigg\}\sum_{i=1}^3j_0(\kappa_i r)\, ,
\label{Qmono}\end{aligned}$$ where $\alpha\equiv\hat{{\mathbf{k}}}_1\cdot\hat{{\mathbf{k}}}_2$, $\alpha_{\rm cut}\equiv\min\{1,\max\{-1,\big[(2\pi/r)^2-k_1^2-k_2^2\big]/[2k_1k_2]\}\}$ enforces the condition $|{\mathbf{k}}_1 + {\mathbf{k}}_2| \leq 2\pi/r$, $\boldsymbol{\kappa}_{1}\equiv\mathbf{k}_{1}-\mathbf{k}_{2}$, $\boldsymbol{\kappa}_{2}\equiv\mathbf{k}_{1}+2\mathbf{k}_{2}$ and $\boldsymbol{\kappa}_{3}\equiv-2\mathbf{k}_{1}-\mathbf{k}_{2}$. The monopole is directly proportional to $\sigma_8$ through the ratio of power spectra in the first line of eq. , but it depends on the growth rate $f$ through the arctangents and the square roots in the second line of eq. . This explicitly shows that the monopole of the LCF can be used in conjunction with the power spectrum multipoles to break the degeneracy between the measurements of $f$ and $\sigma_8$. Similar expressions have been derived for the intrinsic and mapping parts of the other multipoles in [@Franco:2018yag]. Going beyond tree-level perturbation theory, eq. must be modified; in particular, the integrals over ${\mathbf{k}}_1, {\mathbf{k}}_2$ and $\nu$ in eq. can no longer be reduced to three integrals over $k_1, k_2$ and $\alpha$, as in eq. . However, the fact that the $Q_n$ multipoles can break the degeneracy between $f$ and $\sigma_8$ remains valid.
Finally, we end this section on the theoretical modeling of the power spectrum and LCF by discussing the broader modelling assumptions in this work. Traditional analyses of the redshift-space power spectrum typically use a fixed template for the linear power spectrum to measure $f\sigma_8$, which is then used to constrain potential deviations from the assumed cosmology that generated the fixed template, e.g. $\Lambda$CDM and GR. We note that CMB measurements strongly constrain the shape of the linear power spectrum at early times, independently of late-time acceleration and structure growth (as discussed in Section 3.3 of [@Jelic-Cizmek:2020pkh]), such that the galaxy power spectrum multipoles in the *linear* regime can subsequently constrain $f\sigma_8$ in a model-independent way, without assuming that the linear growth factor or the growth rate match that of $\Lambda$CDM and GR. However, in this work, we use the *non-linear* power spectrum and LCF that depend on the non-linear matter power spectrum and the perturbation theory kernels, which we fix to those predicted by $\Lambda$CDM and GR. This means that our forecasted constraints on $f$ and $\sigma_8$ represent a consistency test of $\Lambda$CDM and GR. We apply this test at many individual redshift bins, allowing for the redshift evolution of the growth of structure to deviate from $\Lambda$CDM and GR in a model-independent way.
However, it is important to note that these modelling assumptions are not a prerequisite for using the LCF for cosmological constraints, and the joint analysis of the power spectrum and LCF can also provide constraints on other non-$\Lambda$CDM or non-GR models as long as theoretical predictions for the power spectrum and LCF within these scenarios are available [@Ali:2018sdk]. For example, to constrain modified gravity models where the perturbation theory kernels are changed or the growth rate is scale-dependent, the power spectrum and LCF could be used to simultaneously constrain $f$, $\sigma_8$ and the additional free parameters that are specific to the model.
Simulations and estimators {#sec:sims}
==========================
In this work, we rely on simulations to check multiple components of our forecasting pipeline. Here, we describe the simulation data that we use, as well as the power spectrum and LCF multipole estimators. The LCF multipole estimator in eq. is presented and implemented for the first time in this work.
Simulations {#subsec:sims}
-----------
The simulations are the same as those of [@Ali:2018sdk] and consist of 500 unique dark matter realisations generated using the <span style="font-variant:small-caps;">l-picola</span> approximate simulation code [@Tassev:2013pn; @Howlett:2014opa; @Howlett:2015hfa]. The authors of these previous works have demonstrated that the COLA algorithm is able to reproduce well the two- and three-point clustering, and the LCF, for scales $k<0.3 \,h{\mathrm{Mpc}}^{-1}$ and $r>10 \,h^{-1}\mathrm{Mpc}$, which are the scales of interest to this work and next generation surveys.
The simulations were generated using $256^{3}$ particles in a box of length $512\,h^{-1}\mathrm{Mpc}$. This corresponds to a mass resolution of $6.7\times 10^{11}h^{-1}M_{\odot}$. Halos were then identified in each of the realisations using a Friends-of-Friends algorithm [@Davis:1985rj] limited to a minimum of 10 particles per halo. We do not expect the <span style="font-variant:small-caps;">l-picola</span> simulations to accurately reproduce the mass function of halos down to these scales, nor the internal properties of the halos. We also do not extract subhalos from the simulations. Nonetheless, they are adequate for checking the theoretical modelling presented in Section \[sec:modeling\]. The average halo number density in the simulations is $\overline{n} = 3.57 \times 10^{-4}\,h^{3}\mathrm{Mpc^{-3}}$. Details of how measurements of the power spectra and LCF were made from the simulated halo catalogs are given below.
Power spectrum estimator
------------------------
The estimator for the bin-averaged isotropic power spectrum is $$\begin{aligned}
\hat{P}\left(k\right)=\frac{1}{V}\int_{k} {\ensuremath{\text{d}^3 q}}
\frac{\Delta\left(\mathbf{q}\right)\Delta\left(-\mathbf{q}\right)}{V_{P}\left(k\right)}\,,\end{aligned}$$ where the integrals run over the interval $q\in\left[k-\Delta k/2,k+\Delta k/2\right]$, $\Delta k$ is the bin width, and $V_{P} \equiv 4\pi k^{2}\Delta k$ is the volume of modes in a spherical shell. Similarly, assuming the global plane-parallel approximation, one can define an estimator for the power spectrum multipoles as $$\begin{aligned}
\hat{P}_{n}\left(k\right)=\frac{2n+1}{V}\int_{k}{\ensuremath{\text{d}^3 q}}\frac{\Delta\left(\mathbf{q}\right)\Delta\left(-\mathbf{q}\right)}{V_{P}\left(k\right)}L_{n}\left(\hat{\mathbf{q}}\cdot\hat{\mathbf{n}}\right).
\label{eq:angular power spectrum estimator}\end{aligned}$$
LCF estimator
-------------
The LCF multipoles are defined by eq. . To derive an estimator for $Q_n$ we first use the fact that $\ell (r,\nu)$ depends only on $\nu \equiv \hat{\mathbf{r}} \cdot \hat{{\mathbf{n}}}$, and not on the angle $\phi$ which describes a rotation of $\mathbf{r}$ around $\hat{{\mathbf{n}}}$. We can therefore average the multipoles over $\phi$ to obtain $$Q_n(r) = \frac{2n+1}{4\pi} \int_0^{2\pi} {\ensuremath{\text{d}\phi}} \int^{1}_{-1} {\ensuremath{\text{d}\nu}}\,\ell(r,\nu) L_n(\nu). \label{eq:Qest}$$ The estimator for $\ell(\mathbf{r})$ is given by $$\ell(\mathbf{r}) = \frac{V^2}{(2\pi)^6} \left(\frac{r^{3}}{V}\right)^{3/2} \iintop_{k_{1},k_{2},|\mathbf{k}_{1}+\mathbf{k}_{2}|\leq\frac{2\pi}{r}}
{\ensuremath{\text{d}^3 k_1}}{\ensuremath{\text{d}^3 k_2}} \,
e^{i\mathbf{r}\cdot\left(\mathbf{k}_{1}-\mathbf{k}_{2}\right)}\Braket{\epsilon(-\mathbf{k}_{1}-\mathbf{k}_{2})\epsilon(\mathbf{k}_{1})\epsilon(\mathbf{k}_{2})}. \label{eq:lcontinuous}$$ The exponential in eq. and the Legendre polynomial in eq. can be written in terms of spherical harmonics using the identities in eqs. and . Then the orthonormality of the spherical harmonic functions in eq. gives the final expression for the estimator, $$\begin{aligned}
\hat{Q}_n(r) = \left(2n+1\right)i^{n}\frac{V^2}{(2\pi)^6}\left(\frac{r^{3}}{V}\right)^{3/2} \iintop_{k_{1},k_{2},|\mathbf{k}_{1}+\mathbf{k}_{2}|\leq\frac{2\pi}{r}} {\ensuremath{\text{d}^3 k_1}}{\ensuremath{\text{d}^3 k_2}}\, j_{n}\left(\kappa_1r\right)L_{n}\left(\hat{\boldsymbol{\kappa}}_1\cdot\hat{\mathbf{n}}\right)\Braket{\epsilon(-\mathbf{k}_{1}-\mathbf{k}_{2})\epsilon(\mathbf{k}_{1})\epsilon(\mathbf{k}_{2})}.
\label{eq:Qn_estimator}\end{aligned}$$
Estimating means and covariance matrix
--------------------------------------
We estimate the mean of the power spectrum and LCF multipoles as $$\overline{X}_i = \frac{1}{N_{\rm sims}} \sum_{n=1}^{N_{\rm sims}} \hat{X}_i^{(n)},$$ while the covariance matrix is estimated as $$\hat{{\textbf{\textsf{C}}}}_{*,ij} = \frac{1}{N_{\rm sims}-1} \sum_{n=1}^{N_{\rm sims}}
\left( \hat{X}_i^{(n)} - \overline{X}_i \right)
\left( \hat{X}_j^{(n)} - \overline{X}_j \right).$$ $N_{\rm sims} = 500$ is the total number of independent realisations of the <span style="font-variant:small-caps;">l-picola</span> simulations, and $\hat{X}_i^{(n)}$ is the measured data vector in the $n$-th realisation.
For calculating signal-to-noise ratios and the Fisher forecasts that follow, we require estimates of the inverse covariance matrix. Taking the direct inverse of $\hat{{\textbf{\textsf{C}}}}_*$ results in a biased estimate of the true inverse covariance. To remedy this, we apply an approximate correction by multiplying the inverse of $\hat{{\textbf{\textsf{C}}}}_*$ with the Anderson-Hartlap factor [@Anderson2003; @Hartlap2007] and estimate the inverse covariance as $$\hat{{\textbf{\textsf{C}}}}^{-1} = \frac{N_{\rm sims} - N_{\rm bins} - 2}{N_{\rm sims} - 1} \hat{{\textbf{\textsf{C}}}}_*^{-1},$$ where $N_{\rm bins}$ is the length of the data vector.
Theoretical covariance matrices {#sec:theoretical_covariance}
===============================
We first review expressions for the leading-order power spectrum multipoles covariance. Then we present for the first time the leading-order contributions to the LCF multipoles covariance and the cross-covariance between the power spectrum multipoles and the LCF multipoles. In this section we briefly summarise the equations and defer the detailed derivations to Appendices \[app:Qn\_cov\] and \[app:PlQn\_crosscov\].
For the DESI and Euclid galaxy surveys that we consider in this work, the redshift bins have widths of $\Delta z \geq 0.1$, so we expect that the covariance between different redshift bins is negligible. All covariance expressions we present are therefore implicitly for one fixed redshift bin.
Power spectrum covariance
-------------------------
The Gaussian covariance for the power spectrum multipoles is $$\begin{aligned}
{\rm cov}\big[P_{n_1}(k_i),P_{n_2}(k_j)\big]= \delta_{ij} \frac{(2n_1+1)(2n_2+1)}{V_P(k_i)}
\frac{(2\pi)^3}{V}
\int_{-1}^{1}d\mu \left[ P(k_i,\mu)+ \frac{1}{\overline{n}} \right]^2L_{n_1}(\mu)L_{n_2}(\mu),
\label{eq:covP}\end{aligned}$$ where $P(k_i,\mu)$ is given by eq. .
LCF covariance
--------------
The Gaussian covariance for the LCF multipoles is $$\begin{aligned}
\text{cov}\left[Q_{n_{1}}\left(r_{i}\right),Q_{n_{2}}\left(r_{j}\right)\right] =& \delta^K_{n_1n_2} \frac{\left(2n_{1}+1\right)(-1)^{n_{1}}\left(r_{i}r_{j}\right)^{9/2}}{4\pi^{4}V}\int_{0}^{2\pi/R}dk_{1}k_{1}^{2}\int_{0}^{2\pi/R}dk_{2}k_{2}^{2}\int_{-1}^{\alpha_{\text{cut}}}d\alpha j_{n_{1}}\left(\kappa_{1}r_{i}\right)\nonumber \\
& \times\left[j_{n_{1}}\left(\kappa_{1}r_{j}\right)+2j_{n_{1}}\left(\kappa_{2}r_{j}\right)L_{n_{1}}\left(\hat{\boldsymbol{\kappa}}_{1}\cdot\hat{\boldsymbol{\kappa}}_{2}\right)\right],
\label{eq:covQn}
\end{aligned}$$ with $R\equiv\max\left(r_{i},r_{j}\right)$, $\alpha\equiv\hat{{\mathbf{k}}}_1\cdot\hat{{\mathbf{k}}}_2$, $\boldsymbol{\kappa}_{1}\equiv \mathbf{k}_{1}-\mathbf{k}_{2}$, and $\boldsymbol{\kappa}_{2}\equiv \mathbf{k}_{1}+2\mathbf{k}_{2}$. $\alpha_{\text{cut}}=\min\{ 1,\max\{ -1,[\left(2\pi/R\right)^{2}-k_{1}^{2}-k_{2}^{2}]/[2k_{1}k_{2}]\} \}$ is imposed to keep $\left|\mathbf{k}_{1}+\mathbf{k}_{2}\right|\leq2\pi/R$ satisfied. The argument of the Legendre polynomial is given by $$\hat{\boldsymbol{\kappa}}_{1}\cdot\hat{\boldsymbol{\kappa}}_{2}=\frac{k_1^2+k_1k_2\alpha-2k_2^2}{\sqrt{\big(k_1^2-2k_1k_2\alpha+k_2^2\big)\big(k_1^2+4k_1k_2\alpha+4k_2^2\big)}}\, .$$ The covariance between different multipoles of the LCF is zero. A complete derivation of this covariance is presented in Appendix \[app:Qn\_cov\].
We note that this covariance in eq. is independent of both cosmology and shot noise, like the Gaussian covariance of the real-space LCF. This is due to the fact that the Gaussian covariance is proportional to products of two-point functions of the phases, $$\langle\epsilon({\mathbf{k}}_1)\epsilon({\mathbf{k}}_2) \rangle=\frac{(2\pi)^3}{V}\delta_D({\mathbf{k}}_1+{\mathbf{k}}_2)\,,$$ which do not depend on the power spectrum and are therefore unaffected by shot noise and cosmology. The only dependence it has on the survey of interest is through the survey volume. Cosmology dependence would appear in the LCF covariance through additional non-Gaussian terms that we do not include here, but we expect that these contributions would be small, because in Section \[sec:theory\_cov\] and Figure \[fig:halo\_theorycov\] we find that the Gaussian covariance is a good approximation for the LCF scales of interest in this work.
Power spectrum and LCF cross-covariance
---------------------------------------
The covariance between the power spectrum and LCF multipoles is intrinsically non-Gaussian because it is a 5-point correlator. As shown in Appendix \[app:PlQn\_crosscov\], there are two contributions: one containing the product of the bispectrum and power spectrum and one containing a connected 5-point correlator term, $$\text{cov}\left[P_{n_{1}}\left(k_{i}\right),Q_{n_{2}}\left(r_{j}\right)\right]=\frac{V}{\left(2\pi\right)^{3}}\left(\frac{\sqrt{\pi}}{2}\right)^{3}\left(\frac{r_{j}^{3}}{V}\right)^{3/2}\left[\mathcal{C}^{\left(n_{1},n_{2}\right)}_{PB}+\mathcal{C}^{\left(n_{1},n_{2}\right)}_{P_{5}}\right].
\label{eq:covPQ}$$ The expression for $\mathcal{C}^{\left(n_{1},n_{2}\right)}_{PB}$ derived in Appendix \[app:PlQn\_crosscov\], and modified to include shot noise, is $$\begin{aligned}
\mathcal{C}^{\left(n_{1},n_{2}\right)}_{PB} & = 2\left(2n_{1}+1\right)\left(2n_{2}+1\right)i^{n_2}
\left(\frac{2}{\sqrt{\pi}}\right)^3
\int_{k_{i}}\frac{{\ensuremath{\text{d}^3 k_2}}}{V_{P}\left(k_{i}\right)}
\left[P\left(\mathbf{k}_{2}\right) + \frac{1}{\overline{n}}\right]
L_{n_{1}}\left(\hat{\mathbf{k}}_{2}\cdot\hat{{\mathbf{n}}}\right)\Theta\left(1-\frac{k_{2}r_{j}}{2\pi}\right) \nonumber \\
&\times\underset{k_{1},\left|\mathbf{k}_{1}+\mathbf{k}_{2}\right|\leq
\frac{2\pi}{r_{j}}
}{\int}{\ensuremath{\text{d}^3 k_1}}
\Braket{\epsilon\left(\mathbf{k}_{2}\right)\,\epsilon\left(-\mathbf{k}_{1}-\mathbf{k}_{2}\right)\,\epsilon\left(\mathbf{k}_{1}\right)}
\left[j_{n_{2}}\left(\kappa_{1}r_{j}\right)L_{n_{2}}\left(\hat{\boldsymbol{\kappa}}_{1}\cdot\hat{{\mathbf{n}}}\right)+j_{n_{2}}\left(\kappa_{2}r_{j}\right)L_{n_{2}}\left(\hat{\boldsymbol{\kappa}}_{2}\cdot\hat{{\mathbf{n}}}\right)+j_{n_{2}}\left(\kappa_{3}r_{j}\right)L_{n_{2}}\left(\hat{\boldsymbol{\kappa}}_{3}\cdot\hat{{\mathbf{n}}}\right)\right],
\label{eq:CPBfinal}\end{aligned}$$ where for the 3-point phase correlations we use eq. . The expression for $\mathcal{C}^{\left(n_{1},n_{2}\right)}_{P^5}$ is in eq. . $\boldsymbol{\kappa}_{1}$ and $\boldsymbol{\kappa}_{2}$ are defined in the same way as before, while $\boldsymbol{\kappa}_{3}\equiv-2\mathbf{k}_{1}-\mathbf{k}_{2}$.
The contribution from $\mathcal{C}^{\left(n_{1},n_{2}\right)}_{P^5}$ is expected to be subdominant for combined power spectrum and bispectrum galaxy clustering analyses on large scales [@Sefusatti:2006pa], and it is also more difficult to compute this term, so in our theoretical model for the cross-covariance we do not include it. We assess the impact of this assumption to the power spectrum–LCF cross-covariance in Section \[sec:theory\_cov\] by comparing the forecasts obtained using the full measured covariance from simulations with the forecasts obtained using the theoretical covariance matrix. We will see that the impact of ignoring the cross-covariance between the power spectrum and the LCF is similarly minimal in both cases, implying that it is a good approximation to neglect the $\mathcal{C}^{\left(n_{1},n_{2}\right)}_{P^5}$ term or even the cross-covariance $\text{cov}\left[P_{n_{1}}\left(k_{i}\right),Q_{n_{2}}\left(r_{j}\right)\right]$ entirely.
Comparing models and simulations {#sec:compare}
================================
In this section we compare theoretical predictions for the power spectrum and LCF multipoles, and their covariance, with measurements from 500 <span style="font-variant:small-caps;">l-picola</span> simulations. This comparison, described in detail below, validates our implementation of the theoretical predictions and estimators, and gives valuable guidance on the modelling choices, range of scales and fiducial values to use in the Fisher forecasts that we present later in Sections \[sec:fisher\] and \[sec:surveys\].
Mean power spectrum and LCF at fiducial cosmology {#sec:means}
-------------------------------------------------
In Section \[sec:modeling\], we presented the models that we use for the halo power spectrum and LCF multipoles. Here we compare these models to the simulations by fitting them to the mean of 500 <span style="font-variant:small-caps;">l-picola</span> simulation measurements obtained using the estimators in Section \[sec:sims\]. This comparison checks that our theoretical predictions and measurements are in good agreement before proceeding to the Fisher forecasts in the rest of this work.
In the ideal scenario, we would perform a fully joint fit to both the $P_\ell$ and $Q_n$ multipoles at the same time. However, the LCF predictions are computationally very demanding, so for each model we perform the fit in two stages. In the first stage, we fit the $P_\ell$ multipoles by running an MCMC chain with a Gaussian likelihood and the $P_\ell$ covariance matrix measured from simulations. We fit the shot noise-corrected $P_0$, while $P_2$ and $P_4$ do not have any shot noise contribution in our model. We include all $k$ bins up to some $k_{\rm max}$ and consider the 3-dimensional parameter space of $b_1$, $b_2$ and $\sigma_P$. We leave out $\sigma_B$ because it does not appear in the power spectrum, and for simplicity we fix the values of $f$ and $\sigma_8$ to the values that correspond to the known $\Lambda$CDM cosmology of the simulations. Then in the second stage, we fix $b_1$, $b_2$ and $\sigma_P$ to their best-fit values from the power spectrum fit, and only fit the LCF data for the last remaining parameter, $\sigma_B$. We note that we fit the total LCF, which includes both the effective and shot noise terms discussed in Section \[sec:shot noise\]. The LCF fitting procedure is not performed using MCMC chains because the LCF predictions are computed too slowly; instead, we maximize the likelihood interpolated over a grid in the 1-dimensional $\sigma_B$ parameter space. Assuming that the likelihood is approximately Gaussian, the error $\sigma_\theta$ on the parameter $\theta$ is estimated as $$\begin{aligned}
\frac{1}{\sigma_\theta^2} = \frac{1}{2} \frac{\partial^2 \chi^2}{\partial \theta^2},\end{aligned}$$ where the second derivative is evaluated numerically at the maximum likelihood point.
[lccccccc]{} & &\
Model & $k_{\rm max}$ & $b_{1}$ & $b_{2}$ & $\sigma_P$ & min $\chi^2(P_\ell)$ & $\sigma_B$ & min $\chi^2(Q_n)$\
(lr)[1-1]{} (lr)[2-6]{} (lr)[7-8]{}\
Vlah + GM fit + Gaussian FoG & 0.30 & $1.642\pm 0.023$ & $0.406^{+0.063}_{-0.080}$ & $4.18\pm 0.10$ & 11.5 & $2.73\pm 0.90$ & 4.3\
\
Vlah + GM fit + GM FoG & 0.30 & $1.650\pm 0.023$ & $0.406^{+0.062}_{-0.078}$ & $3.169\pm 0.088$ & 9.2 & $8.11 \pm 2.54$ & 5.7\
\
Vlah + GM fit + Lorentzian FoG & 0.30 & $1.657\pm 0.023$ & $0.408^{+0.060}_{-0.078}$ & $4.81\pm 0.15$ & 9.1 & $3.17\pm 1.01$ & 4.3\
\
The results from this fitting procedure with $k_{\rm max} = 0.30 \, h{\mathrm{Mpc}}^{-1}$ are shown in Table \[tab:fits\] for all three models, where the only difference between them is whether the analytic form of the Fingers-of-God damping factor is a Gaussian, the same as in [@Gil-Marin:2014pva], or a Lorentzian.
For the power spectrum multipoles, we find that all three models perform well up to $k_{\rm max} = 0.30 \, h{\mathrm{Mpc}}^{-1}$, but the Lorentzian form returns the lowest value of the minimum $\chi^2$.[^2] The best-fit $P_\ell$ with Lorentzian FoG are compared with the data in the left column of Figure \[fig:pkfits\], where we show that the model for the monopole and quadrupole is consistent with the measurements to within $1\sigma$ up to $k_{\rm max} = 0.3 \,h{\mathrm{Mpc}}^{-1}$. The hexadecapole is at most 2$\sigma$ away from the measurements for the same $k$ bins.
The choice of $k_{\rm max} = 0.3 \,h{\mathrm{Mpc}}^{-1}$ is our optimistic case, but in later sections we also explore a more conservative forecast by using $k_{\rm max} = 0.15 \,h{\mathrm{Mpc}}^{-1}$. However, we do not show fits separately for this lower $k_{\rm max}$ because the MCMC analysis showed that $b_2$ cannot be constrained independently from the other parameters using this limited range of scales. The MCMC chains resulted in a bimodal posterior, with one of the modes being consistent with the best-fit from $k_{\rm max} = 0.3 \,h{\mathrm{Mpc}}^{-1}$.
{width="\textwidth"}
{width="\textwidth"}
Once the power spectrum fits are complete, we fit the halo LCF multipoles using each model. For this we fix the parameter values for $b_1$, $b_2$ and $\sigma_P$ to the best-fit values from the power spectrum, and subsequently only fit the LCF multipoles for $\sigma_B$ using the $r_{\rm min}$ value that corresponds to $k_{\rm max} = 0.3 \,h{\mathrm{Mpc}}^{-1}$.
As seen in eq. , the LCF integrates modes up to $k=2\pi/r$, and indeed as shown in [@Wolstenhulme:2014cla], the integral is dominated by this upper bound. Therefore, to trust the LCF at a scale $r_{\rm min}$ we need to trust the modelling of the power spectrum and bispectrum up to $k_{\rm max}=2\pi/r_{\rm min}$. We use this to determine the cut-off scale, i.e. $r_{\min} = 21 \, h^{-1}{\mathrm{Mpc}}$ if $k_{\rm max}=0.30\,h{\mathrm{Mpc}}^{-1}$. Similarly, for the more conservative forecast with $k_{\rm max}=0.15\,h{\mathrm{Mpc}}^{-1}$, we use $r_{\min} = 43 \, h^{-1}{\mathrm{Mpc}}$. We note, however, that the cutoff for $\mathbf{k}_i$ in the integrals of eq. , and the choice of corresponding $k_{\rm max}$ for a chosen $r_{\rm min}$, sometimes differ in previous works on the LCF. For example, in the Fisher forecasts of [@Eggemeier:2016asq] and [@Byun:2017fkz], though the cutoff was also $k_i \leq 2\pi/r$ as we use here, the LCF bins down to $r_{\rm min} = 10 \,h^{-1}{\mathrm{Mpc}}$ were combined with power spectrum bins up to $k_{\rm max} \approx 0.30\,h{\mathrm{Mpc}}^{-1}$. Both works used $k_{\rm max} = \pi/r_{\rm min}$, which comes from arguing that a density perturbation with wavelength $\lambda=2\pi/k$ corresponds to an overdensity of size $r=\lambda/2=\pi/k$. In this work, our different way of choosing $k_{\rm max}$ and our more conservative choices for $r_{\rm min}$ ensure that both the power spectrum and the LCF only have access to Fourier modes up to $k_{\rm max} = 0.30\,h{\mathrm{Mpc}}^{-1}$ in both the theoretical modeling and the estimation of these observables.
The resulting best-fit values for $\sigma_B$ are shown in Table \[tab:fits\], where we find that the Gaussian and Lorentzian FoG give the same minimum $\chi^2$ value that is lower than from the Gil-Marin FoG. The best-fit model using the Lorentzian FoG is shown in the left column of Figure \[fig:lcffits\], where we see that the model for the LCF multipoles is in good agreement with the simulations down to $r_{\rm min} \approx 20\,h^{-1}{\mathrm{Mpc}}$: the quadrupole and hexadecapole are within 1$\sigma$ agreement with the measurements, while the monopole is at most 1.2$\sigma$ away from the measurements.
The panels on the right sides of Figures \[fig:pkfits\] and \[fig:lcffits\] are only different from the left panels in that, rather than using the best-fit values of $\sigma_P$ and $\sigma_B$, they use the linear theory prediction in eq. , which for $z=0$ is $\sigma_P = \sigma_B = 4.5 \, h^{-1}{\mathrm{Mpc}}$. This changes the power spectrum at the highest $k$-bins, while the LCF monopole is $2\sigma$ away from the measurement in the smallest $r \approx 20 \,h^{-1}{\mathrm{Mpc}}$ bin. In Section \[subsec:theory\_veldisp\], we explore in more detail the impact of using the theoretical velocity dispersions in the Fisher forecasts.
\
Given these results from the power spectrum and LCF fits, we choose to use the model with Lorentzian FoG for our Fisher forecasts, since it has a lower minimum $\chi^2$ value for both the power spectrum and LCF, while also providing best-fit $\sigma_P$ and $\sigma_B$ that are more in agreement. In the following sections, we present the forecasts for both an optimistic $k_{\rm max} = 0.30 \,h{\mathrm{Mpc}}^{-1}$ ($r_{\rm min}=20\,h^{-1}{\mathrm{Mpc}}$) and a conservative $k_{\rm max} = 0.15 \,h{\mathrm{Mpc}}^{-1}$ ($r_{\rm min}=40\,h^{-1}{\mathrm{Mpc}}$). When the fiducial values of the parameters are set to their best-fit values, we will take these to be the best-fit values from the full range of scales, i.e. up to $k_{\rm max} = 0.3 \,h{\mathrm{Mpc}}^{-1}$, even for the forecasts where only the $k$ bins up to $k_{\rm max} = 0.15 \,h{\mathrm{Mpc}}^{-1}$ are used.
Lastly, in Figure \[fig:onions\] we show the two-dimensional 2-point correlation function and LCF from both simulations and the best-fit model with Lorentzian FoG. In all panels, we only show contours in the regions where $r \geq 20 \, h^{-1}{\mathrm{Mpc}}$ to correspond to the range of scales that we use in the fits. The simulation 2PCF is the average of the 2PCF measured in 500 <span style="font-variant:small-caps;">l-picola</span> simulations using the direct pair counting method implemented in `nbodykit` [@Hand:2017pqn], while the theoretical 2PCF is calculated by taking the inverse Fourier transform of the best-fit anisotropic power spectrum. In both the simulation and theoretical 2PCF panels, we see the characteristic Kaiser squashing along the line of sight that is indicative of linear redshift-space distortions. The non-linear Fingers-of-God effect resulting from virialized subhalos typically appears as a strong elongation along the line of sight, but we do not see it here because the halos in the <span style="font-variant:small-caps;">l-picola</span> simulations are larger than $6.7\times 10^{12} \,h^{-1}M_{\odot}$ and do not include subhalos.
In the bottom panels of Figure \[fig:onions\], the simulation LCF is reconstructed from the average $Q_n$ measured in the simulations, while the theoretical LCF is reconstructed from the best-fit $Q_n$. In both the measured and theoretical LCF panels, we see a slight elongation along the line of sight which comes from the positive quadrupole $Q_2$. This higher correlation along the line of sight is a general feature of RSD in the LCF on these scales: it is present even for tree-level matter densities in redshift-space when we do not include halo biasing, shot noise, or the Fingers-of-God effect. The counter-intuitive enhancement of the LCF along the line of sight is qualitatively explained by the *inverse* relationship between the LCF amplitude and the number density of filaments, as discussed Section 3.4 of [@Obreschkow:2012yb] and Section 3.3 of [@Eggemeier:2015ifa]. Briefly, the addition of spatially uncorrelated filamentary structure aligned with a particular direction reduces the LCF signal along the same direction due to the increased random phase noise from the different filaments. In the case of RSD, the Kaiser effect works in the opposite way: it reduces the apparent density of filamentary structure along the line of sight, and the resulting decreased phase noise boosts the LCF along the line of sight.
Covariance matrices at fiducial cosmology {#sec:cov}
-----------------------------------------
Here we present the covariance matrix measured from simulations and compare it with the theoretical covariance matrix in Section \[sec:theoretical\_covariance\], evaluated at the best-fit model from the previous subsection.
Figure \[fig:var\] compares the standard deviations of the power spectrum and LCF multipoles measured from simulations (solid lines) with theoretical predictions (dashed lines). We find that the predictions for the power spectrum standard deviations match the measured ones to within approximately 10 per cent in all $k$ bins. For the LCF multipoles, the bins with $r > 30 \, h^{-1}{\mathrm{Mpc}}$ have standard deviations that are also predicted to within 10 per cent, but this grows up to 30 per cent for the smallest $r$ bin shown in Figure \[fig:var\].
{width="\textwidth"}
In Figure \[fig:cov\] we compare the correlation coefficients of the simulation and theoretical covariance matrices, $$C_{ij}\equiv\frac{{\rm cov}[X_i,X_j]}{\sqrt{{\rm cov}[X_i,X_i]{\rm cov}[X_j, X_j]}}\, ,$$ where $X_i$ are elements of the data vector containing $P_0(k),P_2(k), P_4(k), Q_0(r), Q_2(r)$ and $Q_4(r)$. We note that the theoretical covariance here is not internally fully consistent, since for the power spectrum multipoles and the LCF multipoles we include only the Gaussian contribution, while we include a non-Gaussian contribution for the cross-covariance between the power spectrum and the LCF multipoles. Still, Figure \[fig:cov\] shows that the theoretical covariance matrix captures some of the most prominent features in the simulation covariance matrix: for example, the covariance between different $P_\ell$ multipoles in the same $k$ bins, the covariance between neighboring $r$ bins of $Q_n$, and the cross-covariance between $P_0$ and $Q_0$.
![Correlation coefficients for the simulation covariance (lower left) compared to the theoretical covariance (upper right). Bins for $k_{\rm max}=0.30 \,h{\mathrm{Mpc}}^{-1}$ are shown, and the grid of black vertical and horizontal lines delineate the different multipoles. Starting from the upper left corner of each matrix and going down or to the right, $k$ increases for $P_\ell$ and $r$ increases for $Q_n$.[]{data-label="fig:cov"}](halo_covs.pdf){width="50.00000%" height="50.00000%"}
{width="\textwidth"}
In Figure \[fig:SN\], we show the cumulative signal-to-noise ratio as a function of $k_{\rm max}$ for the power spectrum, LCF, and their combination. Each panel is computed using either the simulation covariance matrix on the left or the theoretically predicted covariance matrix on the right. In all cases, the signal is fixed to the average measurement from the simulations, so differences between the two panels are due only to differences in the covariance matrices. We see that the total signal-to-noise from the $P+Q$ combination begins to be overestimated by the theoretical covariance by more than 10 per cent for $k_{\rm max} \gtrsim 0.12 \, h{\mathrm{Mpc}}^{-1}$. We also note that the signal-to-noise does not appear to be sensitive to the cross-covariance between the power spectrum and LCF; in both panels, neglecting the cross-covariance changes the total signal-to-noise for the combined probes by less than 5 per cent for all $k_{\rm max}$ shown. Interestingly, the LCF multipoles do not appear to contribute noticeably to the total signal-to-noise, but we expect that how the available signal-to-noise is translated into parameter constraints depends on the modelling of the signal and the parameters under consideration. In the next section, we will present the results of Fisher forecasts based on the simulation data that explore the parameter constraints in more detail.
Forecasts based on simulations {#sec:fisher}
==============================
We present power spectrum and LCF Fisher forecasts which are based on our simulation data. We first present our benchmark forecast in Section \[sec:benchmark\], which relies most heavily on the simulations and least on theoretical assumptions. Then in Sections \[sec:theory\_cov\] and \[subsec:theory\_veldisp\], we test the modelling of the covariance matrix and the velocity dispersions by comparing forecasts with more theoretical modelling to the benchmark forecast. We end this section by discussing how we obtain predictions for the galaxy bias that we will use for the survey forecasts in Section \[sec:surveys\].
Benchmark forecast {#sec:benchmark}
------------------
In this work, we use the Fisher forecasting method to estimate parameter constraints [@Tegmark:1996bz]. Assuming the likelihood is a multivariate Gaussian and the data covariance does not vary with the parameters of interest, the Fisher information matrix is $${\textbf{\textsf{F}}} = \frac{\partial{\boldsymbol{D}}}{\partial{\boldsymbol{\theta}}}^T \cdot {\textbf{\textsf{C}}}^{-1} \cdot \frac{\partial{\boldsymbol{D}}}{\partial{\boldsymbol{\theta}}},$$ where our data vector ${\boldsymbol{D}}$ is comprised of the power spectrum and LCF multipoles, ${\boldsymbol{\theta}}$ are the parameters of interest, and ${\textbf{\textsf{C}}}$ is the data covariance matrix. Both the partial derivatives and the data covariance are evaluated at a fiducial cosmology. Then ${\textbf{\textsf{F}}}^{-1}$ is the forecasted parameter covariance matrix.
For the data vector, we use the same data bins that were used in Section \[sec:means\]. This means that for the power spectrum forecast, we include $P_0$, $P_2$ and $P_4$, with all $k$-modes between $k_{\rm min}=0.018\,h{\mathrm{Mpc}}^{-1}$ and $k_{\rm max}=0.15 \,h{\mathrm{Mpc}}^{-1}$ or $k_{\rm max}=0.3 \,h{\mathrm{Mpc}}^{-1}$. Equivalently, for the LCF forecast, we include all separations between $r_{\rm min}=40\,h^{-1}{\mathrm{Mpc}}$ or $r_{\rm min}=20\,h^{-1}{\mathrm{Mpc}}$ and $r_{\rm max}=100\,h^{-1}{\mathrm{Mpc}}$.
We calculate all of our Fisher matrices for six parameters, ${\boldsymbol{\theta}} = (f,\sigma_8,b_1,b_2,\sigma_P,\sigma_B)$, where the $\sigma_P$ and $\sigma_B$ velocity dispersions are considered nuisance parameters and marginalised over. Fiducial values of these parameters are evaluated at the known cosmology or best-fit parameter values discussed in Section \[sec:means\]. All other cosmological parameters are kept fixed to their fiducial values, which match those of the simulations, since our goal is to determine how the LCF multipoles can break degeneracies between $f$, $\sigma_8$, $b_1$ and $b_2$. In $\Lambda$CDM, $f$ is fully determined by $\Omega_m$. Therefore, by keeping $\Omega_m$ fixed and varying $f$, we promote $f$ to a free parameter that is used to test models beyond $\Lambda$CDM. We evaluate the derivatives of the power spectrum and LCF multipoles with respect to the parameters numerically.
The covariance matrix measured from simulations is representative of a survey at redshift $z=0$ that has volume $V_{\rm sim} = 0.13 \, h^{-3} \mathrm{Gpc}^3$, but upcoming surveys will be much larger than this by covering large sky areas in multiple redshift bins. Therefore, for the forecasts in this section, we rescale the simulation covariance to a comoving volume of $V = 3 \, h^{-3} \mathrm{Gpc}^3$, which approximates more closely the effective volume of a single redshift slice of width $\Delta z = 0.1$ from upcoming surveys at $z \approx 0.75$. More explicitly, the inverse covariance matrix we use in the benchmark forecast is ${\textbf{\textsf{C}}}^{-1} = \hat{{\textbf{\textsf{C}}}}^{-1} \, V/V_{\rm sim}$, where $\hat{{\textbf{\textsf{C}}}}^{-1}$ is the inverse covariance matrix that we estimate from the simulations.
The results from the benchmark forecast for both $k_{\rm max} = 0.15 \,h{\mathrm{Mpc}}^{-1}$ and $0.30 \,h{\mathrm{Mpc}}^{-1}$ are shown in Figure \[fig:halo\_fisher\_woPlanck\] and Table \[tab:fisher\]. The per cent values in the table show the amount of improvement in the forecasted error that is brought by the LCF multipoles. We see that the constraints from the LCF give significant improvements to the power spectrum for both $k_{\rm max}$, but the improvement is significantly larger for $k_{\rm max} = 0.15 \,h{\mathrm{Mpc}}^{-1}$. This is due to the fact that if $k_{\rm max}$ is smaller, the power spectrum is more degenerate in the parameters, so the LCF has a larger opportunity to help by breaking some of these degeneracies. In the case where $k_{\rm max}$ is higher, the mild non-linearities in the power spectrum at smaller scales also help in breaking the degeneracies, so as a consequence, the LCF gives less improvement. Still, for both $k_{\rm max}$ values, adding the LCF multipoles provides noticeable improvements in constraining these parameters.
[cS\[table-format=1.4\]S\[table-format=1.4\]cS\[table-format=1.4\]S\[table-format=1.4\]c]{}\
& [$P$]{} & & [$\mathrm{Planck}+P$]{} &\
(lr)[2-2]{} (lr)[3-4]{} (lr)[5-5]{} (lr)[6-7]{} [$b_1$]{} & 0.676 & 0.286 & (137%) & 0.0164 & 0.0160 & (2.1%)\
[$b_2$]{} & 0.571 & 0.271 & (10%) & 0.0754 & 0.0725 & (4.0%)\
[$f$]{} & 0.218 & 0.103 & (113%) & 0.0332 & 0.0327 & (1.3%)\
[$\sigma_8$]{} & 0.331 & 0.141 & (134%) & 0.00600 & 0.00599 & (0.1%)\
\
& [$P$]{} & & [$\mathrm{Planck}+P$]{} &\
(lr)[2-2]{} (lr)[3-4]{} (lr)[5-5]{} (lr)[6-7]{} [$b_1$]{} & 0.120 & 0.0779 & (54%) & 0.0139 & 0.0135 & (2.8%)\
[$b_2$]{} & 0.0962 & 0.0683 & (41%) & 0.0220 & 0.0216 & (1.7%)\
[$f$]{} & 0.0458 & 0.0340 & (35%) & 0.0171 & 0.0167 & (2.1%)\
[$\sigma_8$]{} & 0.0577 & 0.0382 & (51%) & 0.00597 & 0.00593 & (0.7%)\
In Figure \[fig:halo\_fisher\_corrmats\_kmax0.3\], we show the correlation coefficients of the parameter covariance matrix from the power spectrum, LCF and their combination for $k_{\rm max} = 0.3 \,h{\mathrm{Mpc}}^{-1}$. Compared to the power spectrum (left panel), which has strong degeneracies for all pairs of parameters, the LCF (central panel) exhibits a very different pattern of correlations. In particular, as expected, $\sigma_8$ is much less degenerate with $f$ and $b_1$ in the LCF than in the power spectrum, showing that the LCF contains new, complementary information with respect to the power spectrum. The amount of new information is, however, lessened by the fact that the overall constraining power of the power spectrum is much stronger than that of the LCF. As a consequence, the correlation coefficients from the joint constraint (right panel) are more similar to the power spectrum than to the LCF, except now with less severe degeneracies between parameters. Another way to see that the main advantage of the LCF is to break degeneracies is by forecasting the constraints on $(f,\sigma_8, \sigma_P, \sigma_B)$ while fixing the values of $b_1$ and $b_2$. In this case, we find that the improvement from the LCF is negligible—less than 3 per cent for both $k_{\rm max}$ values. This is due to the fact that when $b_1$ and $b_2$ are fixed, the power spectrum multipoles are sensitive to different combinations of $\sigma_8$ and $f\sigma_8$. As a consequence, the power spectrum multipoles alone can break the degeneracy between $f$ and $\sigma_8$, and there is little benefit to including the LCF multipoles.
{width="\textwidth"}
What impact does the cross-covariance between the power spectrum and LCF multipoles have on the parameter constraints? In Section \[sec:cov\], we saw that the impact of the cross-covariance on the signal-to-noise ratio was small. Similarly, the impact of the cross-covariance on the benchmark forecast is also small. For both $k_{\rm max}$ values, we confirm that neglecting the cross-covariance between the power spectrum and the LCF changes the forecasted constraints minimally, by less than 7 per cent. This confirms the expectation that an estimator targeted at measuring the phases of the density and velocity fields is minimally correlated with the power spectrum, which is only sensitive to the amplitude of these fields. As a consequence, the information in the LCF multipoles is minimally redundant with that in the power spectrum multipoles.
To determine which multipoles of the LCF are most relevant for breaking the degeneracies between parameters, we compare forecasts where the LCF multipoles are included cumulatively: first, only the monopole of the LCF is combined with the power spectrum multipoles, then the LCF monopole and quadrupole are added, and finally all three LCF multipoles are included. The results in Figure \[fig:halo\_fisher\_Qn\_contribs\] show that most of the improvement comes from the first two multipoles, $Q_0$ and $Q_2$. In the left panel of the figure, we find that for $k_{\rm max} = 0.15 \,h{\mathrm{Mpc}}^{-1}$ combining only the monopole of the LCF, $Q_0$, with the power spectrum gives an improvement of 60 to 70 per cent, depending on the parameter, with respect to the power spectrum alone. Further adding the quadrupole of the LCF, $Q_2$, nearly doubles the per cent improvement, while adding the hexadecapole, $Q_4$, changes the constraints by less than 1 per cent. The right panel of Figure \[fig:halo\_fisher\_Qn\_contribs\] is for $k_{\rm max} = 0.30 \,h{\mathrm{Mpc}}^{-1}$, where we find again that nearly all of the improvement from the LCF multipoles is contained in $Q_0$ and $Q_2$. In this case, $Q_0$ gives an improvement of 25 to 45 per cent and further including $Q_2$ adds another 10 per cent improvement. Again, including $Q_4$ does not strengthen the constraints any further.
![Percentage improvement in the forecasted constraints from including LCF multipoles one by one, relative to the $P_\ell$-only forecast, when $k_{\rm max} = 0.15 \,h{\mathrm{Mpc}}^{-1}$ (left panel) or $0.30 \,h{\mathrm{Mpc}}^{-1}$ (right panel). For each parameter, the bars show the improvement in the forecasted constraint from adding only the monopole $Q_0$ (red bar on the left), adding the monopole $Q_0$ and quadrupole $Q_2$ (blue bar in the middle), and adding all three LCF multipoles $Q_0$, $Q_2$ and $Q_4$ (turquoise bar on the right), to the $P_\ell$ multipoles. Most of the constraining power of the LCF comes from the monopole $Q_0$ and the quadrupole $Q_2$, while adding the hexadecapole $Q_4$ provides less than 1 per cent of additional improvement.[]{data-label="fig:halo_fisher_Qn_contribs"}](halo_fisher_Qn_contribs_barplot.pdf){width="\textwidth"}
As discussed in Section \[sec:shot noise\], the forecasts can be done either with or without the shot noise term in the LCF. Our benchmark forecast includes the shot noise term, which implicitly assumes that the cosmology-dependence of the shot noise is modelled accurately enough to extract information from it. On the other hand, calculating the forecast *without* the shot noise term requires both assuming that the shot noise is modelled accurately enough to isolate the effective term, and that the true cosmology is already known very accurately. When we compare the forecasts for the joint power spectrum and LCF multipoles with and without the shot noise term in the LCF, we find that the constraints are almost identical. This is due to the fact that most of the constraining power comes from the power spectrum. On the other hand, if only the LCF multipoles are used to constrain the parameters, then the inclusion of the shot noise term makes a noticeable difference, as shown in Figure \[fig:halo\_fisher\_sn\_effect\]. The constraints are stronger for all parameters if the shot noise term is included. This is what we would expect, since when the shot noise term is included, we also get information from the cosmology-dependence in the LCF shot noise.
{width="50.00000%"}
Finally, we check the impact of including a CMB prior from the Planck 2018 results [@Aghanim:2018eyx]. The forecasts with a Gaussian $1\sigma$ prior on $\sigma_8$ of $\Delta\sigma_8 = 0.0060$ are shown in the last three columns of Table \[tab:fisher\].[^3] In this case, the improvement from the LCF multipoles is very small, less than 4 per cent, because the parameter degeneracies in the power spectrum are broken by the external constraint on $\sigma_8$, such that there is less opportunity for the LCF to further improve the constraints. As discussed in the introduction, however, an important benefit of extracting additional information from the spatial distribution of galaxies to break the parameter degeneracies is that the combination of large-scale structure data is model-independent. In contrast, the combination of large-scale structure data with CMB analyses is model-dependent, since CMB data constrains the primordial amplitude of perturbations, and in order to translate this into a prior on $\sigma_8$, a cosmological model, such as $\Lambda$CDM and general relativity, must be assumed. The growth rate $f$ measured in this way is therefore not model-independent and cannot be consistently used to test models beyond $\Lambda$CDM.[^4]
Theoretical covariance matrices {#sec:theory_cov}
-------------------------------
To calculate Fisher forecasts for upcoming surveys, we require theoretical predictions for the forecasting ingredients that we have so far measured in simulations. Previously, we measured the full covariance matrix in simulations and fitted simulation data to find our fiducial values of $b_1$, $b_2$, $\sigma_P$ and $\sigma_B$. Here, and in the rest of Section \[sec:fisher\], we examine how the benchmark forecasts presented in Section \[sec:benchmark\] change when these quantities are replaced with theoretical predictions.
First, we check how well our theoretical model for the covariance matrix can recover the benchmark forecast. We compute the Fisher matrix as in the benchmark forecast, with the only difference that we use the theoretically predicted covariance matrix (shown in the upper right part of Figure \[fig:cov\]) instead of the covariance matrix estimated from simulations. The results are shown in Figure \[fig:halo\_theorycov\] as the bars labelled “Fitted $\sigma_{PB}$” (in the left side of each panel) to indicate that the forecast is evaluated at the best-fit values of $\sigma_P$ and $\sigma_B$ from Section \[sec:means\]. The height of the bars corresponds to the per cent difference in the forecasted constraints relative to the benchmark forecast with the simulation covariance matrix. For $k_{\rm max} = 0.30 \,h{\mathrm{Mpc}}^{-1}$ (bottom row in the figure), we find that using the theoretical covariance instead of the simulated one for the power spectrum-only constraints underestimates the forecasted parameter error by up to 10 per cent. A similar difference is seen for the LCF-only constraints. For the power spectrum-LCF joint constraints, using the theory covariance matrix underestimates the constraints by up to 20 per cent. This larger mismatch for the joint constraints does not seem to be due to our model for the cross-covariance; we have checked that ignoring the cross-covariance between the power spectrum and LCF in the theoretical covariance matrix only changes the constraints by less than 3 per cent for both $k_{\rm max}$ values, which is similar to the behaviour we found for the simulation covariance matrix in the benchmark forecast. If $k_{\rm max} = 0.15 \,h{\mathrm{Mpc}}^{-1}$ (top row of Figure \[fig:halo\_theorycov\]), the agreement is better: using the theoretical covariance matrix changes the power spectrum-only constraints by 4 per cent, the LCF-only constraints by 10 per cent, and the joint constraints by 10 per cent.
Figure \[fig:halo\_theorycov\_improvement\] shows the per cent improvement that is gained in the forecasted constraints by adding the LCF to the power spectrum. The red bars correspond to the same per cent improvements that are in Table \[tab:fisher\] (without the Planck prior). The blue bars show that when the theoretical covariance is used with the fitted $\sigma_P$ and $\sigma_B$, the per cent improvement is overestimated by 9 to 14 per cent when $k_{\rm max} = 0.15 \,h{\mathrm{Mpc}}^{-1}$ and by 14 to 17 per cent when $k_{\rm max} = 0.30 \,h{\mathrm{Mpc}}^{-1}$.
The comparisons between the theoretical and simulation covariance matrices in this section show that the modelling for the covariance matrices presented in Section \[sec:theoretical\_covariance\] is accurate enough to return forecasted constraints to within $\sim 10$ per cent of the benchmarks for $k_{\rm max} = 0.15 \,h{\mathrm{Mpc}}^{-1}$ and within $\sim 20$ per cent of the benchmarks for $k_{\rm max} = 0.30 \,h{\mathrm{Mpc}}^{-1}$. We note that the comparisons in this work are necessarily done at redshift $z=0$ and for the halo catalogs which we are able to identify with the <span style="font-variant:small-caps;">l-picola</span> simulations available to us. The modelling may perform differently for different number densities and halos, but in general we would expect that the theoretical modelling becomes more accurate at the higher redshifts that are more relevant for upcoming surveys, since non-linearities are less important at high redshift. In the next section, we consider the impact of assuming the linear predictions for the fiducial velocity dispersions, in place of the fitted values of $\sigma_P$ and $\sigma_B$ that we have used here.
{width="\textwidth"}
{width="\textwidth"}
Theoretical velocity dispersions {#subsec:theory_veldisp}
--------------------------------
In addition to theoretical predictions for the covariance matrices, the Fisher forecasts for upcoming surveys will also require fiducial values for the velocity dispersions $\sigma_P$ and $\sigma_B$, so we now check the impact of using the linear theory prediction for the velocity dispersions, rather than their fitted values. In Section \[sec:means\], we found that the best-fit $\sigma_P$ and $\sigma_B$ values in the Lorentzian FoG model are consistent with each other and with the linear prediction from eq. , which at $z=0$ is $\sigma_P = 4.5 \,h^{-1}\mathrm{Mpc}$. Therefore, in our forecast with theoretical velocity dispersions, we choose as fiducial values $\sigma_P = \sigma_B = 4.5 \,h^{-1}\mathrm{Mpc}$, but we still treat these as two separate nuisance parameters which vary independently. We note that this requires not only changing where in the parameter space the Fisher derivatives are evaluated, but we also recompute the full theoretical covariance, which depends on the fiducial $\sigma_P$ and $\sigma_B$.
We compare the constraints with the benchmark forecast in Figure \[fig:halo\_theorycov\] as the bars labeled “Theory $\sigma_{PB}$” (right side of each panel) to indicate that the forecast is evaluated at the values of $\sigma_P$ and $\sigma_B$ predicted by linear theory. For $k_{\rm max} = 0.3 \,h{\mathrm{Mpc}}^{-1}$ (bottom row of the figure), we find that the power spectrum-only constraints with the theoretical covariance underestimates the forecasted parameter error from simulated covariances by up to 15 per cent, and the LCF-only constraints agree to within 20 per cent. For the joint power spectrum-LCF constraints, using the theory covariance makes the constraints match to within 10 per cent. If $k_{\rm max} = 0.15 \,h{\mathrm{Mpc}}^{-1}$ (top row of the figure), the agreement is roughly similar: within 10 per cent for power spectrum-only constraints, 30 per cent for LCF-only constraints, and 10 per cent for the joint constraints. Figure \[fig:halo\_theorycov\_improvement\] shows that for both $k_{\rm max}$, the theory covariance with the theory prediction for the velocity dispersions returns a per cent improvement that is within a 12 per cent difference with the simulation forecasts.
These results show a remarkable agreement with the benchmark forecast, considering that they are different by both covariance matrix modelling and fiducial values of $\sigma_P$ and $\sigma_B$. We consider this agreement good enough for our purposes of forecasting constraints from future surveys, and in particular, estimating the benefit of combining the LCF multipoles with the power spectrum, so we proceed to use the theoretical covariance matrix modelling with the linear predictions for the velocity dispersions for the survey forecasts in Section \[sec:surveys\].
We note that we also calculated a forecast where the two velocity dispersions $\sigma_P$ and $\sigma_B$ were treated as if they were the same single nuisance parameter, following the discussion in [@Hashimoto:2017klo] and [@Yankelevich:2018uaz]. This results in much stronger constraints—the per cent improvement from the LCF is roughly 2 to 6 times larger. However, since this requires a very strong assumption about the relationship between the Fingers-of-God damping factors in the power spectrum and bispectrum, we do not make this assumption in this work.
Theoretical galaxy bias {#subsec:theory_bias}
-----------------------
We now briefly discuss how we obtain theoretical predictions for $b_2$ when performing the survey forecasts presented in the next section. For dark matter halos, $b_1$ and $b_2$ depend on the halo mass, and fits for $b_2$ as a function of $b_1$, calibrated to N-body simulations, have been presented in [@Lazeyras:2015lgp] and [@Hoffmann:2016omy]. In this work, we use the fit from [@Lazeyras:2015lgp] to obtain our fiducial values of the quadratic bias: $b_2 = 0.412 - 2.143 b_1 + 0.929 b_1^2 + 0.008 b_1^3$. In general, this relation is not guaranteed to hold for galaxies, and to predict the galaxy $b_1$ and $b_2$ from the halo biases requires modeling how galaxies populate halos using prescriptions such as subhalo abundance matching or a halo occupation distribution. In [@Yankelevich:2018uaz], the latter was used to calculate the galaxy $b_2$ for the $H\alpha$ galaxies that will be observed by Euclid. In that work, the galaxy $b_2$ was very well approximated by the halo $b_2$, which implies that the galaxy bias is insensitive to the details of the HOD. Our forecasts in this work will assume this is also the case for the DESI galaxy samples, and we leave it to future work to include more precise modeling of the galaxy-halo connection for these surveys.
We note that when the fit for $b_2$ is applied to the best-fit $b_1$ from the <span style="font-variant:small-caps;">l-picola</span> simulations in Section \[sec:means\], we obtain a value of $b_2 = -0.55$, which is not in agreement with our best-fit $b_2 = 0.41$. This is most likely due to the fact that our halo catalogs from the <span style="font-variant:small-caps;">l-picola</span> simulations contain all halos above a minimum halo mass of $6.7\times 10^{12}h^{-1}M_{\odot}$, so our fitted values of the halo biases are effective values that cover a large population of halos of different masses, whereas the fit derived in [@Lazeyras:2015lgp] has been calibrated on halos that fall within narrow mass bins.
Forecasts for upcoming surveys {#sec:surveys}
==============================
In this section, we apply our forecasting method to the upcoming DESI and Euclid galaxy surveys using the theoretical predictions that were validated in the previous section.
For DESI, we consider the Bright Galaxy Sample (BGS), Emission Line Galaxies (ELGs), Luminous Red Galaxies (LRGs), and quasars (QSOs) with $14{,}000 \,{\rm deg}^2$ of sky coverage and the redshift bins and galaxy number densities in Tables 2.3 and 2.5 of [@Aghamousa:2016zmz]. All redshift bins have width $\Delta z=0.1$, and the bin centers are: $z_{\rm BGS}=0.05-0.45$, $z_{\rm ELG}=0.65-1.65$, $z_{\rm LRG}=0.65-1.15$, and $z_{\rm QSO}=0.65-1.85$. As in that work, we set the fiducial linear bias for each sample by imposing constant $b_1(z)D(z)$, where $D(z)$ is the linear growth factor that is normalized to one at $z=0$: $b_{\rm BGS}(z)D(z)=1.34$, $b_{\rm ELG}(z)D(z)=0.84$, $b_{\rm LRG}(z)D(z)=1.7$, and $b_{\rm QSO}(z)D(z)=1.2$. For Euclid, we assume the survey parameters for the H$\alpha$ emitting galaxies from Table 3 of [@Blanchard:2019oqi], where there are four redshift bins centered around $z_{\rm H \alpha} = 1.00$, 1.20, 1.40, 1.65, and the bin widths are $\Delta z = 0.2$ for the first three bins and $\Delta z = 0.3$ for the highest redshift bin.
In each redshift bin, we forecast the constraints on $f(z)$ and $\sigma_8(z)$, marginalised over $b_1(z), b_2(z), \sigma_P(z)$, and $\sigma_B(z)$. We compute a theoretical covariance matrix for each redshift bin, and we assume that there is no cross-covariance between the power spectrum and LCF multipoles, since we found in Section \[sec:benchmark\] that the cross-covariance made a difference of less than 3 per cent in the forecasted constraints, compared to the theoretical covariance with cross-covariance included.
The constraints on $f(z)$ and $\sigma_8(z)$ are shown in Figure \[fig:surveys1zbinOnly\] for $k_{\rm max}=0.15 \,h{\mathrm{Mpc}}^{-1}$ (top row) and $k_{\rm max}=0.30 \,h{\mathrm{Mpc}}^{-1}$ (bottom row). For DESI, we choose to focus on the forecasts from the BGS and ELG samples, because these two samples together span a large range of redshifts, $z=0.05-1.65$, and where the ELG bins overlap with those of LRGs and QSOs, the ELG forecasts generally give stronger constraints on $f(z)$ and $\sigma_8(z)$ from the power spectrum only. The figure shows a pair of error bars for each redshift bin and galaxy sample, where within each pair, the error bar on the left is obtained from the power spectrum multipoles only and the smaller error bar on the right is from the joint power spectrum-LCF multipoles analysis. We have not combined the constraints from different redshift bins, because doing so would require assuming a model for how $f$ and $\sigma_8$ evolve with redshift.
\
To put the size of the error bars into context, we have also plotted the predictions for $f(z)$ and $\sigma_8(z)$ for models with different values for the growth rate index $\gamma$, which is defined by $$f(z) = \Omega_m(z)^\gamma.$$ The growth rate index is often used as a simple parametrisation of the growth rate used in searches for modifications to gravity [@Linder:2005in; @Linder:2007hg]. In general relativity, $\gamma \approx 0.55$, but it can take different values modified gravity model, and the predictions for $\gamma = 0.40$ and 0.68 shown in Figure \[fig:surveys1zbinOnly\] roughly approximate the range of values that are consistent with the data to within $\sim 2\sigma$ in recent analyses combining multiple low redshift probes and Planck CMB data [@Mueller:2016kpu; @Sanchez:2016sas; @Grieb:2016uuo; @Wang:2017wia; @Zhao:2018gvb]. The predictions of $\sigma_8(z)$ for different $\gamma$ are calculated as $$\begin{aligned}
\sigma_8(\gamma,z) = \sigma_8(z)
\frac{D_{\rm GR}(z_*)}{D_{\rm GR}(z)}
\frac{D_{\gamma}(z)}{D_{\gamma}(z_*)}\,,\end{aligned}$$ where $$\frac{D_\gamma(a)}{D_\gamma(a_*)} = \exp \left[ \int_{\ln a_*}^{\ln a} {\ensuremath{\text{d}\ln{a'}}} \, \Omega_m(a')^\gamma \right].$$ We take $z_*=500$ to be a high redshift at which the linear growth factor was very close to that of general relativity.
The panels showing $\sigma_P/\sigma_{P+Q}$ in Figure \[fig:surveys1zbinOnly\] indicate the factor of improvement in the constraint from including the LCF measurements, which is equal to the ratio of the error bars. The improvement brought by including the LCF multipoles can be significant. For a fixed $k_{\rm max}$, the DESI BGS sample in the lower redshift bins (red circles) gives a larger improvement than in the higher redshift bins populated by the DESI ELG and Euclid H$\alpha$ galaxies. For a fixed galaxy sample and redshift bin, the relative improvement from the LCF is larger when $k_{\rm max}$ is lower, though the absolute size of the errors is larger. In particular, our forecast for the DESI BGS sample shows that the LCF multipoles can strengthen the constraints on $f(z)$ and $\sigma_8(z)$ by up to $\sim 220$ per cent for $k_{\rm max}=0.15 \,h{\mathrm{Mpc}}^{-1}$ or $\sim 50$ per cent for $k_{\rm max}=0.30 \,h{\mathrm{Mpc}}^{-1}$. These forecasts show that the LCF may be very useful for further improving the constraints on the growth rate $f$ with the upcoming generation of galaxy surveys. The fact that the LCF helps to break the degeneracy between $f$ and $\sigma_8$ within each individual redshift bin is highly relevant for performing model-independent analyses that do not need to assume any modelling for the evolution of $f(z)$ and $\sigma_8(z)$ with redshift.
We also consider combining the forecast with the Planck 2018 prior on $\sigma_8(z=0)$, as we did in Section \[sec:benchmark\]. Assuming that $\sigma_8(z)$ evolves with redshift according to our fiducial $\Lambda$CDM cosmology, we combine the prior with our forecast in each individual redshift bin. In this case, the improvement from measuring the LCF is minimal, and gives less than 12 per cent reduction in the errors on $f$ and $\sigma_8$ for all redshift bins, galaxy samples, and $k_{\rm max}$. This is because the CMB prior strongly breaks the degeneracy between $\sigma_8(z)$ and $f(z)$, such that there is little degeneracy left for the LCF to break further. However, the resulting $f(z)$ measured in this way is not model-independent and can therefore not be used to consistently test models beyond $\Lambda$CDM.
Conclusions {#sec:conclusions}
===========
In this work, we have shown that the correlations between phases provide a powerful way to test general relativity, by improving the constraint on the growth rate of structure, $f$, with the coming generation of galaxy large-scale structure surveys like DESI and Euclid. We have focused on a specific estimator of phase correlations, the line correlation function (LCF), and studied how the multipoles of the LCF can be used in combination with the multipoles of the power spectrum to improve the measurement of $f$. The key property of the LCF which makes it complementary to the power spectrum is the fact that it contains different combinations of $f$ and $\sigma_8$, allowing it to break the degeneracy between these parameters that are present in a power spectrum-only analysis. We have argued that this method has the advantage of not relying on an assumed cosmological model for how the growth of structure evolves with redshift, which is not the case for joint clustering-CMB analyses or joint clustering-weak lensing analyses that require a specific cosmological model to break the degeneracy between $f$ and $\sigma_8$.
We have constructed a model of the LCF multipoles and of their covariance, which goes beyond linear perturbation theory and is valid in the non-linear regime. We have tested this model using a large suite of <span style="font-variant:small-caps;">l-picola</span> realizations, and found that it agrees well with the simulations down to separations of $20 \,h^{-1}{\mathrm{Mpc}}$. Using Fisher matrices we have forecasted the constraints expected on $f$ and $\sigma_8$ for surveys like DESI and Euclid and found that adding the LCF leads to an improvement of up to 220 per cent for $k_{\rm max}=0.15 \, h{\mathrm{Mpc}}^{-1}$ and up to 50 per cent for $k_{\rm max}=0.30 \, h{\mathrm{Mpc}}^{-1}$, depending on the redshift bin and galaxy sample.
This result is a strong motivation to use phase correlations in future analyses of redshift-space distortions, but further work is necessary before this new estimator can be applied to more realistic data. For example, there are several observational effects which our modelling and forecast did not include, such as the effect of complex survey window functions. Furthermore, the numerical calculation of theoretical predictions for the LCF must be improved before it would be fast enough to be part of a standard MCMC likelihood analysis. In parallel, it would be interesting to study other configurations, beyond correlations restricted to a line, to see if the impact of redshift-space distortions may be enhanced in specific triangular configurations.
Acknowledgements {#acknowledgements .unnumbered}
================
We thank William Wright for useful discussions. JB acknowledges support from the SNSF Sinergia grant No. 173716. FOF and CB acknowledge support from the SNSF. CH was supported by the Australian Government through the Australian Research Council’s Laureate Fellowship funding scheme (project FL180100168). DO is a recipient of an Australian Research Council Future Fellowship (FT190100083) funded by the Australian Government. Computations were performed at the University of Geneva on the Baobab computing cluster and on the OzSTAR national facility at the Swinburne University of Technology. OzSTAR is funded by Swinburne University of Technology and the National Collaborative Research Infrastructure Strategy (NCRIS).
We acknowledge the use of the `emcee` [@ForemanMackey:2012ig], `GetDist` [@Lewis:2019xzd], `COLOSSUS` [@Diemer:2017bwl], `EuclidEmulator` [@Knabenhans:2018cng], and `nbodykit` [@Hand:2017pqn] Python packages.
Covariance of the LCF multipoles {#app:Qn_cov}
================================
Given the form of the estimator for the LCF multipoles in eq. , we calculate the covariance of the $Q_n$ at lowest order (i.e. at Gaussian order) in the Edgeworth expansion. We have $$\text{cov}\left[Q_{n_{1}}(r_{i}),Q_{n_{2}}(r_{j})\right]=\Braket{\hat{Q}_{n_{1}}(r_{i})\hat{Q}_{n_{2}}(r_{j})},
\label{eq:covmult}$$ since $\Braket{\hat{Q}_{n}(r)}=0$ at Gaussian order. Eq. contains the six-point phase correlation $$\mathcal{E}_{\ell\ell}=\Braket{\epsilon_{-\mathbf{k}_{1}-\mathbf{k}_{2}}\epsilon_{\mathbf{k}_{1}}\epsilon_{\mathbf{k}_{2}}\epsilon_{-\mathbf{k}_{3}-\mathbf{k}_{4}}\epsilon_{\mathbf{k}_{3}}\epsilon_{\mathbf{k}_{4}}}_{G},$$ which can be split into a sum of products of $\Braket{\epsilon_{\mathbf{k}}\epsilon_{\mathbf{q}}}_{G}$ through Wick’s theorem. Statistical homogeneity implies that the two-point phase correlation is given by $\Braket{\epsilon_{\mathbf{k}}\epsilon_{\mathbf{q}}}_{G}=\delta_{\mathbf{k}+\mathbf{q}}^{K}$ [@Wolstenhulme:2014cla; @Eggemeier:2016asq]. Then, neglecting all terms that give rise to background modes (${\mathbf{k}}=0$), $\mathcal{E}_{\ell\ell}$ has six terms $$\mathcal{E}_{\ell\ell} = \delta_{\mathbf{k}_{1}+\mathbf{k}_{2}+\mathbf{k}_{3}+\mathbf{k}_{4}}^{K}\delta_{\mathbf{k}_{1}+\mathbf{k}_{3}}^{K}\delta_{\mathbf{k}_{2}+\mathbf{k}_{4}}^{K}+\delta_{-\mathbf{k}_{1}-\mathbf{k}_{2}+\mathbf{k}_{3}}^{K}\delta_{\mathbf{k}_{1}-\mathbf{k}_{3}-\mathbf{k}_{4}}^{K}\delta_{\mathbf{k}_{2}+\mathbf{k}_{4}}^{K}+
\delta_{-\mathbf{k}_{1}-\mathbf{k}_{2}+\mathbf{k}_{4}}^{K}\delta_{\mathbf{k}_{1}-\mathbf{k}_{3}-\mathbf{k}_{4}}^{K}\delta_{\mathbf{k}_{2}+\mathbf{k}_{3}}^{K}
+\left(\mathbf{k}_{1}\leftrightarrow\mathbf{k}_{2}\right).
\label{eq:kron}$$ Inserting eq. into eq. yields $$\begin{aligned}
\text{cov}\left[Q_{n_{1}}(r_{i}),Q_{n_{2}}(r_{j})\right] & = \frac{\left(2n_{1}+1\right)\left(2n_{2}+1\right)i^{n_{1}+n_{2}}\left(r_{i}r_{j}\right)^{9/2}}{32\pi^{6}V}\underset{k_{1},k_{2},\left|\mathbf{k}_{1}+\mathbf{k}_{2}\right|\leq\frac{2\pi}{\max(r_{i},r_{j})}}{\iint}\!\!\!\!{\ensuremath{\text{d}^3 k_1}}{\ensuremath{\text{d}^3 k_2}}\,
j_{n_{1}}\left(\kappa_{1}r_{i}\right)L_{n_{1}}\left(\hat{\boldsymbol{\kappa}}_{1}\cdot\hat{\mathbf{n}}\right)\nonumber \\
& \times\left[j_{n_{2}}\left(\kappa_{1}r_{j}\right)L_{n_{2}}\left(\hat{\boldsymbol{\kappa}}_{1}\cdot\hat{\mathbf{n}}\right)+j_{n_{2}}\left(\kappa_{2}r_{j}\right)L_{n_{2}}\left(\hat{\boldsymbol{\kappa}}_{2}\cdot\hat{\mathbf{n}}\right)+j_{n_{2}}\left(\kappa_{3}r_{j}\right)L_{n_2}\left(\hat{\boldsymbol{\kappa}}_{3}\cdot\hat{\mathbf{n}}\right)\right],
\label{eq:covQ}\end{aligned}$$ with $\boldsymbol{\kappa}_{1}\equiv \mathbf{k}_{1}-\mathbf{k}_{2}$, $\boldsymbol{\kappa}_{2}\equiv \mathbf{k}_{1}+2\mathbf{k}_{2}$ and $\boldsymbol{\kappa}_{3}\equiv-2\mathbf{k}_{1}-\mathbf{k}_{2}$. Moreover, since $n_{1}$ and $n_{2}$ are even, the third term is equal to the second one through the transformation $\mathbf{k}_{1}\leftrightarrow \mathbf{k}_{2}$.
This expression contains a six-dimensional integral, over the modulus of ${\mathbf{k}}_1$ and ${\mathbf{k}}_2$ and over their directions, which we denote respectively by $(\theta_1, \varphi_1)$ and $(\theta_2, \varphi_2)$. Three of these integrals can be done analytically. To do this, we first note that since the multipoles and their covariance do not depend on the line-of-sight direction ${\hat{\mathbf{n}}}$, we can integrate eq. over ${\hat{\mathbf{n}}}$ and divide by $4\pi$. Since the Legendre polynomials are the only contributions that contain ${\hat{\mathbf{n}}}$, the integral over ${\hat{\mathbf{n}}}$ reduces to $$\begin{aligned}
\frac{1}{4\pi}\int d\Omega_{{\hat{\mathbf{n}}}} L_{n_{1}}\left(\hat{\boldsymbol{\kappa}}_{1}\cdot\hat{\mathbf{n}}\right)L_{n_{2}}\left(\hat{\boldsymbol{\kappa}}_{i}\cdot\hat{\mathbf{n}}\right)&=\frac{4\pi}{(2n_1+1)(2n_2+1)}\sum_{m_1=-n_1}^{n_1}\sum_{m_2=-n_2}^{n_2}Y_{n_1m_1}(\hat{\boldsymbol{\kappa}}_{1})Y^*_{n_2m_2}(\hat{\boldsymbol{\kappa}}_{i})\int d\Omega_{{\hat{\mathbf{n}}}} Y^*_{n_1m_1}({\hat{\mathbf{n}}})Y_{n_2m_2}({\hat{\mathbf{n}}})\nonumber\\
&=\frac{\delta_{n_1n_2}}{2n_1+1}L_{n_1}(\hat{\boldsymbol{\kappa}}_{1}\cdot\hat{\boldsymbol{\kappa}}_i) \quad\mbox{for}\quad i=1,2.\end{aligned}$$ We insert this into eq. and do the following coordinate transformation: $\left\{ \theta_{1},\phi_{1},\theta_{2},\phi_{2}\right\} \rightarrow\left\{ \gamma,\phi,\theta_{2},\phi_{2}\right\} $, where $\cos\gamma\equiv\hat{\mathbf{k}}_{1}\cdot\hat{\mathbf{k}}_{2}$ and $\phi$ is the azimuthal angle of $\mathbf{k}_{1}$ around $\mathbf{k}_{2}$. The Jacobian of this transformation is 1, since it is a rotation. In this coordinate system, the product $\hat{\boldsymbol{\kappa}}_1\cdot \hat{\boldsymbol{\kappa}}_i$ depends only on $k_1, k_2$ and $\gamma$. Therefore the integral over $\phi, \theta_2$ and $\phi_2$ can be performed and gives rise to a factor $8\pi^2$. We obtain $$\begin{aligned}
\text{cov}\left[Q_{n_{1}}\left(r_{i}\right),Q_{n_{2}}\left(r_{j}\right)\right] =& \frac{\left(2n_{1}+1\right)(-1)^{n_{1}}\left(r_{i}r_{j}\right)^{9/2}}{4\pi^{4}V}\int_{0}^{2\pi/R}dk_{1}k_{1}^{2}\int_{0}^{2\pi/R}dk_{2}k_{2}^{2}\int_{-1}^{\alpha_{\text{cut}}}d\alpha j_{n_{1}}\left(\kappa_{1}r_{i}\right)\nonumber \\
& \times\left[j_{n_{1}}\left(\kappa_{1}r_{j}\right)+2j_{n_{1}}\left(\kappa_{2}r_{j}\right)L_{n_{1}}\left(\hat{\boldsymbol{\kappa}}_{1}\cdot\hat{\boldsymbol{\kappa}}_{2}\right)\right]\cdot\delta^K_{n_1n_2}\,,\label{eq:covfinal}
\end{aligned}$$ where $R\equiv\max\left(r_{i},r_{j}\right)$. $\alpha$ is the cosine of the angle between $\hat{{\mathbf{k}}}_1$ and $\hat{{\mathbf{k}}}_2$, and $\alpha_{\text{cut}}\equiv\min\{ 1,\max\{ -1,[\left(2\pi/R\right)^{2}-k_{1}^{2}-k_{2}^{2}]/[2k_{1}k_{2}]\} \}$ is imposed to keep $\left|\mathbf{k}_{1}+\mathbf{k}_{2}\right|\leq2\pi/R$ satisfied. The argument of the Legendre polynomial is given by $$\hat{\boldsymbol{\kappa}}_{1}\cdot\hat{\boldsymbol{\kappa}}_{2}=\frac{k_1^2+k_1k_2\alpha-2k_2^2}{\sqrt{\big(k_1^2-2k_1k_2\alpha+k_2^2\big)\big(k_1^2+4k_1k_2\alpha+4k_2^2\big)}}\, .$$ From eq. , we see that the different multipoles are not correlated.
Cross-covariance between the power spectrum and LCF multipoles {#app:PlQn_crosscov}
==============================================================
We now calculate the cross-covariance between the power spectrum multipoles and the LCF multipoles $$\text{cov}\left[P_{n_{1}}\left(k_{i}\right),Q_{n_{2}}\left(r_{j}\right)\right]=\Braket{\hat{P}_{n_{1}}\left(k_{i}\right)\hat{Q}_{n_{2}}\left(r_{j}\right)}-\Braket{\hat{P}_{n_{1}}\left(k_{i}\right)}\Braket{\hat{Q}_{n_{2}}\left(r_{j}\right)}.
\label{eq:covcross}$$ The Gaussian contribution to eq. exactly vanishes since the first term is a five-point correlation, which is zero for a Gaussian field, and the second term contains a three-point correlation which is also zero for a Gaussian field. Therefore, to account for any non-zero correlation between the LCF and the power spectrum, we need to compute the non-Gaussian contribution to eq. .
Eq. contains the mixed five-point correlator of phases and amplitudes, $$\mathcal{E}_{P\ell}=
\Braket{\Delta(\mathbf{q})\Delta(-\mathbf{q})\epsilon(-\mathbf{k}_{1}-\mathbf{k}_{2})\epsilon(\mathbf{k}_{1})\epsilon(\mathbf{k}_{2})}
-\Braket{\Delta(\mathbf{q})\Delta(-\mathbf{q})}\Braket{\epsilon(-\mathbf{k}_{1}-\mathbf{k}_{2})\epsilon(\mathbf{k}_{1})\epsilon(\mathbf{k}_{2})}.$$ Neglecting the background modes as in Appendix \[app:Qn\_cov\], we split this expression into its connected correlators using Wick’s theorem. We find two kinds of contributions $$\begin{aligned}
\mathcal{E}_{PB} & = \Braket{\Delta(-\mathbf{q})\epsilon(\mathbf{k}_{1})}_{c}\Braket{\Delta(\mathbf{q})\epsilon(\mathbf{k}_{2})\epsilon(-\mathbf{k}_{1}-\mathbf{k}_{2})}_{c}+\Braket{\Delta(-\mathbf{q})\epsilon(\mathbf{k}_{2})}_{c}\Braket{\Delta(\mathbf{q})\epsilon(\mathbf{k}_{1})\epsilon(-\mathbf{k}_{1}-\mathbf{k}_{2})}_{c}+\Braket{\Delta(-\mathbf{q})\epsilon(-\mathbf{k}_{1}-{\mathbf{k}}_2)}_{c}\Braket{\Delta(\mathbf{q})\epsilon(\mathbf{k}_{1})\epsilon({\mathbf{k}}_2)}_{c}\nonumber\\
&+\left(\mathbf{q}\leftrightarrow-\mathbf{q}\right),
\label{eq:EPB}\\
\mathcal{E}_{P_{5}} & = \Braket{\epsilon(-\mathbf{k}_{1}-\mathbf{k}_{2})\epsilon(\mathbf{k}_{1})\epsilon(\mathbf{k}_{2})\Delta(\mathbf{q})\Delta(-\mathbf{q})}_{c}.
\label{eq:EP5}\end{aligned}$$ These connected mixed-correlators can be evaluated using the joint PDF of Fourier modes, and at lowest order they are given by [@Eggemeier:2016asq] $$\begin{aligned}
\Braket{\Delta(\mathbf{q})\epsilon(\mathbf{k})}_{c} & = \frac{\left(2\pi\right)^{3}}{V}\frac{\sqrt{\pi}}{2}\sqrt{VP\left(\mathbf{q}\right)}\delta_{D}\left(\mathbf{k}+\mathbf{q}\right),
\label{eq:del ep}\\
\Braket{\Delta(\mathbf{q})\epsilon(\mathbf{k}_{1})\epsilon(\mathbf{k}_{2})}_{c} & = \frac{\left(2\pi\right)^{3}}{V}\left(\frac{\sqrt{\pi}}{2}\right)^{2}\sqrt{VP\left(\mathbf{q}\right)}p^{\left(3\right)}\left(\mathbf{q},\mathbf{k}_{1},\mathbf{k}_{2}\right)\delta_{D}\left(\mathbf{k}_{1}+\mathbf{k}_{2}+\mathbf{q}\right),
\label{eq:del ep ep}\\
\Braket{\Delta(\mathbf{q}_{1})\Delta(\mathbf{q}_{2})\epsilon(\mathbf{k}_{1})\epsilon(\mathbf{k}_{2})\epsilon(\mathbf{k}_{3})}_{c} & = \left(2\pi\right)^{3}\left(\frac{\sqrt{\pi}}{2}\right)^{3}\sqrt{P\left(\mathbf{q}_{1}\right)P\left(\mathbf{q}_{2}\right)}p^{\left(5\right)}\left(\mathbf{q}_{1},\mathbf{q}_{2},\mathbf{k}_{1},\mathbf{k}_{2},\mathbf{k}_{3}\right)\delta_{D}\left(\mathbf{k}_{1}+\mathbf{k}_{2}+\mathbf{k}_{3}+\mathbf{q}_{1}+\mathbf{q}_{2}\right),
\label{eq:del del ep ep ep}\end{aligned}$$ where the $N$th order cumulants $p^{\left(N\right)}$ are related to the ordinary $N$th order spectra $P^{\left(N\right)}$ by $$p^{\left(N\right)}\left(\mathbf{k}_{1},\ldots,\mathbf{k}_{N}\right)\equiv V^{1-\frac{N}{2}}\frac{P^{\left(N\right)}\left(\mathbf{k}_{1},\ldots,\mathbf{k}_{N}\right)}{\sqrt{P\left(\mathbf{k}_{1}\right)\ldots P\left(\mathbf{k}_{N}\right)}}\,.
\label{eq:pN cumulants}$$ With this the covariance becomes $$\text{cov}\left[P_{n_{1}}\left(k_{i}\right),Q_{n_{2}}\left(r_{j}\right)\right]=\frac{V}{\left(2\pi\right)^{3}}
\left(\frac{\sqrt{\pi}}{2}\right)^{3}\left(\frac{r_{j}^{3}}{V}\right)^{3/2}\left[\mathcal{C}^{\left(n_{1},n_{2}\right)}_{PB}+\mathcal{C}^{\left(n_{1},n_{2}\right)}_{P_{5}}\right],$$ where the contribution from the connected 5-point correlation function is $$\begin{aligned}
\mathcal{C}^{\left(n_{1},n_{2}\right)}_{P^{5}} & = \left(2n_{1}+1\right)\left(2n_{2}+1\right)i^{n_2}\int_{k_{i}}\frac{{\ensuremath{\text{d}^3 q}}}{V_{P}\left(k_{i}\right)}L_{n_{1}}\left(\hat{\mathbf{q}}\cdot\hat{{\mathbf{n}}}\right)P\left(\mathbf{q}\right)\nonumber \\
& \times\underset{k_{1},k_{2},\left|\mathbf{k}_{1}+\mathbf{k}_{2}\right|\leq\frac{2\pi}{r_{j}}}{\iint}{\ensuremath{\text{d}^3 k_1}}{\ensuremath{\text{d}^3 k_2}}\,j_{n_{2}}\left(\kappa_{1}r_{j}\right)L_{n_{2}}\left(\hat{\boldsymbol{\kappa}}_{1}\cdot\hat{{\mathbf{n}}}\right)p^{\left(5\right)}\left(\mathbf{q},-\mathbf{q},-\mathbf{k}_{1}-\mathbf{k}_{2},\mathbf{k}_{1},\mathbf{k}_{2}\right),
\label{eq:CP5}\end{aligned}$$ and the $\mathcal{C}^{\left(n_{1},n_{2}\right)}_{PB}$ contribution is $$\mathcal{C}^{\left(n_{1},n_{2}\right)}_{PB} = \frac{\left(2n_{1}+1\right)\left(2n_{2}+1\right)i^{n_2}}{\pi^{9/2}}\int_{k_{i}}\frac{{\ensuremath{\text{d}^3 q}}}{V_{P}\left(k_{i}\right)}L_{n_{1}}\left(\hat{\mathbf{q}}\cdot\hat{{\mathbf{n}}}\right)\underset{k_{1},k_{2},\left|\mathbf{k}_{1}+\mathbf{k}_{2}\right|\leq\frac{2\pi}{r_{j}}}{\iint}{\ensuremath{\text{d}^3 k_1}}{\ensuremath{\text{d}^3 k_2}}\,j_{n_{2}}\left(\kappa_{1}r_{j}\right)L_{n_{2}}\left(\hat{\boldsymbol{\kappa}}_{1}\cdot\hat{{\mathbf{n}}}\right)\mathcal{E}_{PB}.
\label{eq:CPB}$$
Since $n_1$ is even, $L_{n_1}(\hat{\mathbf{q}}\cdot\hat{{\mathbf{n}}})=L_{n_1}(-\hat{\mathbf{q}}\cdot\hat{{\mathbf{n}}})$, and the last three terms in eq. have the same contribution as the first three terms. Moreover, one can rewrite the first term by relabelling the dummy wavenumbers as $$\begin{cases}
\mathbf{k}_{1} & \rightarrow\mathbf{k}_{2}\\
\mathbf{k}_{2} & \rightarrow-\mathbf{k}_{1}-\mathbf{k}_{2}\\
\boldsymbol{\kappa}_{1} & \rightarrow\boldsymbol{\kappa}_{2},
\end{cases}$$ while doing a different relabelling for the third term, $$\begin{cases}
\mathbf{k}_{1} & \rightarrow-\mathbf{k}_{1}-\mathbf{k}_{2}\\
\mathbf{k}_{2} & \rightarrow\mathbf{k}_{1}\\
\boldsymbol{\kappa}_{1} & \rightarrow\boldsymbol{\kappa}_{3}.
\end{cases}$$ Then eq. takes the form $$\begin{aligned}
\mathcal{C}^{\left(n_{1},n_{2}\right)}_{PB} & = \frac{2\left(2n_{1}+1\right)\left(2n_{2}+1\right)i^{n_2}}{\pi^{9/2}}\int_{k_{i}}\frac{{\ensuremath{\text{d}^3 q}}}{V_{P}\left(k_{i}\right)}L_{n_{1}}\left(\hat{\mathbf{q}}\cdot\hat{{\mathbf{n}}}\right)\underset{k_{1},k_{2},\left|\mathbf{k}_{1}+\mathbf{k}_{2}\right|\leq\frac{2\pi}{r_{j}}}{\iint}\!\!\!{\ensuremath{\text{d}^3 k_1}}{\ensuremath{\text{d}^3 k_2}}\Braket{\Delta(-\mathbf{q})\epsilon(\mathbf{k}_{2})}_{c}\Braket{\Delta(\mathbf{q})\epsilon(-\mathbf{k}_{1}-\mathbf{k}_{2})\epsilon(\mathbf{k}_{1})}_{c}\nonumber\\
& \times\left[j_{n_{2}}\left(\kappa_{1}r_{j}\right)L_{n_{2}}\left(\hat{\boldsymbol{\kappa}}_{1}\cdot\hat{{\mathbf{n}}}\right)+j_{n_{2}}\left(\kappa_{2}r_{j}\right)L_{n_{2}}\left(\hat{\boldsymbol{\kappa}}_{2}\cdot\hat{{\mathbf{n}}}\right)+j_{n_{2}}\left(\kappa_{3}r_{j}\right)L_{n_{2}}\left(\hat{\boldsymbol{\kappa}}_{3}\cdot\hat{{\mathbf{n}}}\right)\right].\end{aligned}$$ Using eqs. and , one of the Dirac delta functions allows us to integrate over ${\ensuremath{\text{d}^3 k_2}}$, generating a Theta function $\Theta\left(1-k_{2}r_{j}/2\pi\right)$. The other Dirac delta function is redundant, so it contributes a factor of $\delta_{D}\left(\mathbf{0}\right)=V/\left(2\pi\right)^{3}$. Then relabelling $\mathbf{q}$ as $\mathbf{k}_{2}$, we obtain $$\begin{aligned}
\mathcal{C}^{\left(n_{1},n_{2}\right)}_{PB} & = 2\left(2n_{1}+1\right)\left(2n_{2}+1\right)i^{n_2}\int_{k_{i}}\frac{{\ensuremath{\text{d}^3 k_2}}}{V_{P}\left(k_{i}\right)}P\left(\mathbf{k}_{2}\right)L_{n_{1}}\left(\hat{\mathbf{k}}_{2}\cdot\hat{{\mathbf{n}}}\right)\Theta\left(1-\frac{k_{2}r_{j}}{2\pi}\right)\underset{k_{1},\left|\mathbf{k}_{1}+\mathbf{k}_{2}\right|\leq
\frac{2\pi}{r_{j}}
}{\int}{\ensuremath{\text{d}^3 k_1}}p^{\left(3\right)}\left(\mathbf{k}_{2},-\mathbf{k}_{1}-\mathbf{k}_{2},\mathbf{k}_{1}\right)\nonumber\\
& \times\left[j_{n_{2}}\left(\kappa_{1}r_{j}\right)L_{n_{2}}\left(\hat{\boldsymbol{\kappa}}_{1}\cdot\hat{{\mathbf{n}}}\right)+j_{n_{2}}\left(\kappa_{2}r_{j}\right)L_{n_{2}}\left(\hat{\boldsymbol{\kappa}}_{2}\cdot\hat{{\mathbf{n}}}\right)+j_{n_{2}}\left(\kappa_{3}r_{j}\right)L_{n_{2}}\left(\hat{\boldsymbol{\kappa}}_{3}\cdot\hat{{\mathbf{n}}}\right)\right].\end{aligned}$$
Finally, we rewrite the expression in terms of the power spectrum and 3-point phase correlations. Since the $P^{(3)}$ in $p^{(3)}$ is simply the bispectrum, we use eq. to write $$\begin{aligned}
\mathcal{C}^{\left(n_{1},n_{2}\right)}_{PB} & = 2\left(2n_{1}+1\right)\left(2n_{2}+1\right)i^{n_2}
\left(\frac{2}{\sqrt{\pi}}\right)^3
\int_{k_{i}}\frac{{\ensuremath{\text{d}^3 k_2}}}{V_{P}\left(k_{i}\right)}P\left(\mathbf{k}_{2}\right)L_{n_{1}}\left(\hat{\mathbf{k}}_{2}\cdot\hat{{\mathbf{n}}}\right)\Theta\left(1-\frac{k_{2}r_{j}}{2\pi}\right) \underset{k_{1},\left|\mathbf{k}_{1}+\mathbf{k}_{2}\right|\leq
\frac{2\pi}{r_{j}}}{\int}{\ensuremath{\text{d}^3 k_1}}
\Braket{\epsilon\left(\mathbf{k}_{2}\right)\,\epsilon\left(-\mathbf{k}_{1}-\mathbf{k}_{2}\right)\,\epsilon\left(\mathbf{k}_{1}\right)}\nonumber\\
&\times
\left[j_{n_{2}}\left(\kappa_{1}r_{j}\right)L_{n_{2}}\left(\hat{\boldsymbol{\kappa}}_{1}\cdot\hat{{\mathbf{n}}}\right)+j_{n_{2}}\left(\kappa_{2}r_{j}\right)L_{n_{2}}\left(\hat{\boldsymbol{\kappa}}_{2}\cdot\hat{{\mathbf{n}}}\right)+j_{n_{2}}\left(\kappa_{3}r_{j}\right)L_{n_{2}}\left(\hat{\boldsymbol{\kappa}}_{3}\cdot\hat{{\mathbf{n}}}\right)\right].\end{aligned}$$
\[lastpage\]
[^1]: E-mail: joyce.byun@unige.ch
[^2]: We note that because we are fitting the mean of 500 simulations, the fitted data has very little statistical noise, which results in reduced $\chi^2$ values that are very small (much less than 1). However, here we only use the minimum $\chi^2$ values to compare between models with the same number of data bins and free parameters, so the absolute magnitude of the reduced $\chi^2$ is not especially important.
[^3]: We use the data combination called `base_plikHM_TTTEEE_lowl_lowE_lensing_post_BAO`, which is the baseline model in Section 2.18 of [`https://wiki.cosmos.esa.int/planck-legacy-archive/images/4/43/Baseline_params_table_2018_68pc_v2.pdf`](https://wiki.cosmos.esa.int/planck-legacy-archive/images/4/43/Baseline_params_table_2018_68pc_v2.pdf).
[^4]: In this work we consider constraints on the growth in multiple redshift bins to allow for any $z$-dependence. However, CMB data can be used to constrain beyond $\Lambda$CDM models given a model-specific parametrisation of the growth rate.
|
---
abstract: 'Within the recently proposed structured FRW model universe the averaged Einstein equations are derived. The backreaction turns out to have an interesting behavior. Its equivalent density and pressure, being proportional, are negative at early times of the dark ages of the universe, and change sign near our present time in our local patch. In addition to explaining the observed dimming of the SNIa it leads to new effects for small cosmic redshifts and also to the difference between the local and global Hubble parameter. Interpreting the backreaction in the FRW-[*picture*]{}, it is equivalent to a time dependent dark energy with $w = -1$.'
author:
- Reza Mansouri
title: 'Illuminating the dark ages of the universe: the exact backreaction in the SFRW model and the acceleration of the universe '
---
Take the universe as it is: homogeneous at large scales and inhomogeneous at small scales within structured patches. How are the deviations from the standard homogeneous cosmic fluid in the matter dominated phase of the universe reflected in the cosmological data? Recent observational data on SNIa imply a larger distance to supernovae than predicted by the conventional FRW universe [@r98; @p99; @sh], leading to the term acceleration of the universe[@r04; @f04], and to the concept of the dark energy.\
We have recently proposed[@m] the [*Structured*]{} FRW(SFRW) model of the universe as a first step to incorporate the local inhomogeneity of the cosmic fluid into a model universe in accordance with the observational needs. In the SFRW model of the universe the local patches, grown out of the primordial perturbations, and their backreaction on the homogeneous background are modelled exactly as a truncated flat Lomaitre-Tolman-Bondi (LTB) manifold embedded in a FRW universe from which a sphere of the same extent as the LTB patch is removed. As a result of the junction conditions the mean density of any such inhomogeneous patch, with over- and under-dense regions, has to be equal to the density of the FRW bulk[@m; @km]. Therefore, the Copernican principle is in no way violated and we are led to a model universe where the local patches are distributed homogeneously in the bulk having the same mass as a local FRW patch would have, accounting for all the structures we see grown out of the primordial perturbations within a FRW universe. The analysis of the luminosity distance relation in our structured FRW model showed explicitly a dimming of objects within a patch relative to what it would be inferred from a standard FRW universe[@m].\
The local inhomogeneous matter dominated patch has a geometrical domain denoted by $D$ and a hypersurface $\Sigma$ as its boundary to the FRW bulk. Our calculation is based on an exact general relativistic formulation of gluing manifolds. The inhomogeneous patch containing dust matter is represented by a flat LTB metric embedded in a pressure-free FRW background universe with the uniform density $\rho_{b}$. We choose the general LTB metric to be written in the synchronous comoving coordinates in the form[@km]: $$ds^2 = -dt^2 + a^2\big[\big(1 + \frac{a' r}{a}\big)^2 \frac{dr^2}{1-k(r)r^2}+
r^2d\Omega^2 \big],$$ where the familiar LTB metric function is now defined as $R(r,t) = a(t,r) .r$.The similarity to the Robertson-Walker metric as now obvious. The overdot and prime denote partial differentiation with respect to $t$ and $r$, respectively, and $k(r)$ is an arbitrary real function such that $k(r)r^2 < +1 $ playing the role of the curvature scalar $k$ in the FRW universe. Hence, the flat LTB is defined by the vanishing of $k(r)$. For a homogeneous universe, $a$ and $k$ don’t depend on $r$ and we get the familiar Robertson-Walker metric. In our SFRW universe, the metric outside the inhomogeneous patch, is Robertson-Walker again. The corresponding field equations and the solution for the flat case $k(r) = 0$, can be written in the following familiar form: $$\begin{aligned}
\big (\frac{\dot a}{a} \big) = {1\over 3} \varrho, \\
\frac{\ddot a}{a} = -\frac{1}{6}\varrho,\\
a(r) = (\frac{3}{4}\varrho)^{1\over 3}(t- t_n(r))^{2\over3}, \end{aligned}$$ where we have introduced $\varrho \equiv \frac{6M(r)}{r^3}$. The mass $M(r)$ is defined as $$M(r) = \int^{R(r,t)}_{0}\rho(r,t)R^{2}dR
= \frac{1}{6} \overline{\rho}(r,t) R^3,$$ where $\rho(r,t)$ is the density and $\overline {\rho}$, as a function of $r$ and $t$, is an average density up to the radius $R(r,t)$. Note that the volume element in the integral above is not in general equal to the proper volume element of the metric, except for the flat case $k(r) = 0$ we will consider[@s]. In contrast, the average density in the patch defined by using the proper volume element will be different from the average above, except for the flat LTB case. The field equations (2-5) are very similar to the familiar Friedmann equations, except for the $r$-dependence of the different quantities. Furthermore, we assume $R'(r,t) = a + ra'>0$ to avoid shell crossing of dust matter during their radial motion. $t_n (r)$ is an arbitrary function of $r$ appearing as an integration ’constant’. This arbitrary function has puzzled different authors who give it the name of ’bang time function’ corresponding to the big bang singularity[@ce; @bo; @he]. It has, however, a simple astrophysical meaning within our structured FRW universe. As $R(r,t)$ is playing the role of the radius of our local patch, the time $t = t_n$, leading to $R = 0$, means the time of the onset of the mass condensation or nucleation within the homogeneous cosmic fluid. That is why we have preferred to use the subscript $n$ for it indicating the time of nucleation. As was pointed out in [@m], for a realistic density profile, $t_n$ is a decreasing function of the coordinate $r$ having a maximum at $r = 0$, i.e. at the center of our patch. This means, contrary to the usual interpretation in the literature, that $t > t_n$ for all $0 < r < r_{\Sigma} \equiv L$. Therefore, for all times after the onset of mass condensation within our patch $R(t, r)$ is non-vanishing and for times $t < t_n(r =0)$ we have the full FRW without any structure.\
Now, without going into the detailed discussion(see [@km; @m]), we know already that, assuming there is no thin shell at the boundary of the matching, we must have $$\overline{\rho} \stackrel{\Sigma}{=} \rho_b,$$ where $\stackrel{\Sigma}{=}$ means the quantities are to be taken at the boundary to the FRW bulk. We, therefore, are left with the only case imposed by the dynamics of the Einstein equations in which the mean density of a local patch is exactly equal to the density of the background FRW universe: a desired exact dynamical result reflecting the validity of the cosmological principle at large, meaning each nucleated patch within the FRW universe have the same average mass density as the bulk. The total mass in a local patch, being equal to the background density times the volume of the patch, is distributed individually due to its self-gravity, leading to overdense structures and voids to compensate it. Assuming again the matter inside each patch to be smoothed out in the form of an inhomogeneous cosmic fluid, we expect it to be overdense at the center, decreasing smoothly to an underdense compensation region, a void, up to the point of matching to the background.\
The density distribution within a patch must be such that the overdensities of structures are compensated by voids. The nucleation time signals the onset of condensation in the patch which- at least partially- opposes the overall expansion. The running of the function $t_n$ is crucial for the expansion history of the patch and therefore will influence the luminosity of the structures growing within the patch. So far it was shown that $t_n' < 0$[@m]. Of course, the nucleation time function is related to the actual mass distribution for which, taking into account the fine structure of the patch including the substructures, we have to rely on the overall observations and the matter power spectrum[@zehavi; @goodwin; @dekel; @kocevski].\
We envisage now an averaging process in which the inhomogeneities within the local patch are smoothed out and we have again a FRW-type homogeneous modeling of our local patch. The traditional way of doing cosmology is to take the average of the matter distribution in the universe and write down the Einstein equations for it, adding some symmetry requirement. One then solves the equations $G_{\mu \nu} = \langle T_{\mu \nu} \rangle$, assuming homogeneity and isotropy of the mass distribution as the underlying symmetry. This is based on the simplicity principle much used in theoretical physics. As far as the precision of the observations allow, we may go ahead with this simplification. The more exact equation, however, is $\langle G_{\mu \nu} \rangle = \langle T_{\mu \nu} \rangle$. Calling the difference $G_{\mu \nu} - \langle G_{\mu \nu} \rangle = Q_{\mu \nu}$, one may write the correct equation as $G_{\mu \nu} = \langle T_{\mu \nu} \rangle + Q_{\mu \nu}$. The backreaction term $Q$ has so far been neglected in cosmology because of its smallness. Now that measuring $Q$ is within the range of observational capabilities we have to take it into account. There is, therefore, no need yet to change the underlying general relativity or introduce any mysterious dark energy to mimic $Q$. Of course, the averaging process is neither trivial nor unambiguous, but it is the art of physics to master it. Fortunately, there is an averaging formalism, developed mainly by Thomas Buchert[@b; @b03; @b057; @b05], which can easily be adapted to our LTB patch, having the same mass as the the FRW sphere cut out of it. In this formalism the space-average of any function $f(t,r)$ is defined by $$\langle f\rangle \equiv {1 \over V_D} \int_D dV f,$$ where $dV$ is the proper volume element of the 3-dimensional domain $D$ of the patch we are considering and $V_D$ is its volume. It has been shown[@be; @b] that in such a mass preserving patch the space-volume average of any function $f(r,t)$ does not commute with its time derivative: $$\langle f\rangle^{\cdot} - \langle \dot f\rangle = \langle f\theta\rangle -
\langle f\rangle \langle \theta \rangle,$$ where the expansion scalar $\theta$, being equal to the minus of the trace of the second fundamental form of the hypersurface $t = const.$, is now a function of $r$ and $t$. The right hand side trivially vanishes for a FRW universe because of the homogeneity. This fact has far-reaching consequences for observational cosmology in our non-homogeneous neighborhood. The variation of the Hubble function with respect to the red-shift is not so simple any more as in the simple case of FRW universe[@m]. This affects a lot of observational data processing which so far has been done assuming homogeneity of the universe. Depending on the smoothing width $\Delta z$, the bins, and the matter power spectrum there may be huge effects due to the non-commutativity of the averaging process[@e].\
The averaged scale factor is defined using the volume of our patch $D$ by $a_D \equiv V(t)_D^{1 \over 3}$. Now it can be shown that[@b; @be] $$\theta_D \equiv \langle \theta\rangle \equiv {\dot V \over V} =
3{\dot a_D \over a_D} = 3 H_D.$$ where we have used the notation $\dot a_D \equiv {d\over dt}a_D$, and denoted the average Hubble function as $H_D$. Averaging over the local patch means we are taking it as an effective FRW patch. Therefore all the derived quantities should be based on the average value $a_D$. This is why we take the above definition for the mean Hubble parameter and not $\langle \frac{\dot a}{a} \rangle$, which is different from $\frac{{\dot a}_D}{a_D}$. A similar difference holds for the second derivative of $a$: $$\langle\frac{\ddot a}{a} \rangle \not = \frac{\langle \ddot a \rangle}
{\langle a \rangle} \not = {\ddot a_D \over a_D}.$$ Therefore, the definition of the averaged deceleration parameter is not without ambiguity, specially because there is no nice relation like (9) for the deceleration parameter. To choose the most appropriate definition, we make recourse to the fact that in the averaging process we are taking our patch to be homogeneous and FRW-like. Therefore, in averaging the redshift as a function $a$, we always encounter $a_D$ and its time derivatives $\dot a_D$ and $\ddot a_D$. This justifies the above definition of the mean Hubble parameter and motivates us to make the following definition for the deceleration parameter: $$q_D = - \frac{{\ddot a}_D a_D}{{\dot a_D}^2} = - \frac{{\ddot a}_D}{a_D}\frac{1}{H_D^2},$$ as was done in the literature so far[@b; @hs; @nt; @kmr; @s]. Now, we are ready to take the average of the Einstein equations in our local patch to see how the mean field equations will look like and what are the differences to the simple FRW field equations. The emergence of a crucial term in the mean Einstein equations, the so called [*backreaction*]{} term, is interesting. Buchert’s backreaction term is defined by[@b; @b05] $$\begin{aligned}
Q = \langle \sigma^2 \rangle - {1\over3}\langle (\theta - \langle \theta \rangle)^2 \rangle \\
= \langle \sigma^2 \rangle - {1\over3}[\langle \theta^2\rangle - \theta_D^2],\end{aligned}$$ where $\sigma$ is the shear scalar and $\theta$ is the expansion. Although $\theta_D$ and $H_D$ are proportional, $\langle \theta^2 \rangle$ and $\langle H^2 \rangle$ are not. Hence, the relations (13, 14) can not be written in terms of $H$, as was done in[@nt]. The averages of the Einstein equations using the Hamiltonian constraint and the Raychaudhuri equation, taking into account the subtleties of the observation just mentioned, is then written in the following form[@b; @b05]: $$\begin{aligned}
\big({\dot a_D \over a_D} \big)^2 = {1\over3} (\rho_b + Q) \\
\frac{\ddot a_D}{a_D} = -{1\over6}(\rho_b + 4Q),\end{aligned}$$ where we have set $\langle \rho \rangle = \rho_b$, the density of the background FRW universe, as a result of the junction conditions reflected in the eq.(6). Note that in the so-called Friedmann equation (15) the local Hubble parameter enters instead of the global one $H_b$. The effect of the backreaction within the local patch is realized as an effective perfect fluid with the equation of state $$\begin{aligned}
\rho_Q = {4\over3} p_Q. \end{aligned}$$ The backreaction term $Q$ can not yet be considered as representing dark energy in the FRW-[*picture*]{}. The $\rho_b$ appearing in the field equation of our SFRW model is the total background energy density, i.e. $\rho_b = \rho_M + \Lambda$, where we have chosen $\rho_M$ for the matter density. The curvature term is set equal zero to have a simple flat universe. But let us first investigate the sign of $Q$ which is crucial for the interpretation of these averaged equations. As the running of the density and the nucleation time $t_n$ influence the mean values of the Hubble parameter and the shear scalar, the sign of $Q$ is determined by the balance between the mean values of the shear and the term related to the mean values of the Hubble parameter and the expansion scalar in a complex manner depending of the running of the density and the nucleation time. Given this complex behavior of the backreaction term, let us approximate $t_n$ in the following way: $$t_n = t_0 -{\beta \over 2} r^2.$$ For $\beta > 0$ the above expansion satisfies all the necessary conditions to be fullfilled by $t_n$ within the SFRW model universe[@m]. It happens that this behavior corresponds to the special parabolic case of [@s] formulated in their step 5 and illustrated in their figure 7.a. The calculation of different terms in $Q$ is messy but exactly doable. The result is $$Q = {6\over L^3}\frac{A}{B} + {4\over 3}\frac{C}{B^2},$$ where $$\nonumber
A = -5.13L^3 +9.12{2t\over \beta}L - 0.95({2t\over \beta})^{3\over 2} \arctan(\sqrt
{\beta \over 2t} L)$$ $$\nonumber
+ 0.08 ({2t \over \beta})^{3\over 2} \arctan \sqrt {7\over 3}(\sqrt {\beta\over 2t} L),$$ $$\nonumber B = 2t^2 + 2 \beta tL^2 +{1\over 2} \beta^2 L^4,\hspace{1 cm}
C = (2t + \beta L^2)^2.$$ To understand its behavior, we determine its sign for two limiting cases: at the onset of nucleation, i.e. $t - t_0 \ll \beta L^2$, where the coordinate dependence of different quantities has the biggest effect, and at the present time $t - t_0 \gg \beta L^2$. From the exact result of $Q$ we obtain $$\begin{aligned}
t - t_0 \ll \beta L^2: Q \simeq -50.26 \frac{1}{\beta^2 L^4} < 0, \\
t - t_0 \gg \beta L^2: Q \simeq +49.75 \frac{1}{\beta L^2}\frac{1}{t} > 0.\end{aligned}$$ We, therefore, conclude that the backreaction has its strongest effect on the onset of mass condensation at the beginning of the cosmic dark ages and after the density contrast increases to a proper value, where its effective density and pressure are negative. It then changes somewhere, probably at the end of the dark ages, the sign and behaves as a normal fluid.\
Negative values of $Q$ at the early stages of mass condensation is a novel effect, even though it is just for a local patch. It has the effect of reducing the Hubble parameter and producing a negative pressure for a range of expansion time with the effect of dimming of the cosmic objects in our vicinity. Sometime near our present cosmic time the effect reverses and has to lead to new effects for small redshifts, which are somehow opposed to the acceleration of the universe. Remember that these effects are just within our local patch and in interpreting data along the light cone one must be cautious. The mere fact that local Hubble parameter, $H_D$, may be less or greater than the global one, depending on the behavior of $Q$, is interesting and should be taken into account in announcing the $H$ values.\
A comparison to the dark energy concept is possible if we switch from the SFRW-picture to the FRW-picture. Let us denote $ -Q = \Lambda$, where $\Lambda$ is now a function of time and space being most of the time positive. Then we have the following equations at our disposal $$\begin{aligned}
H_D^2 = \big({\dot a_D \over a_D} \big)^2 = {1\over3} \rho_M \\
H_b^2 = \big({\dot a \over a} \big)^2 = {1\over3} (\rho_M + \Lambda) \\
\frac{\ddot a_D}{a_D} = -{1\over6}(\rho_M - 3\Lambda).\end{aligned}$$ Observational cosmologist, using the FRW-picture are used to $H_b$ equation, but taking both the values of $H_b$ and $\rho_b = \rho_M + \Lambda$ from observational data. For the interpretation of the dimming of cosmic objects, however, there is no other way than to use the third equation above. In this picture our backreaction can be interpreted as a time dependent cosmological constant having $w = -1$. The actual SFRW picture, however, gives us a much wider spectrum of information we should be aware of. In the same picture we may say that the backreaction produces a dissipative pressure or anti-frictional force $Q = -\Lambda$ along the line of reasoning in[@ds]. A realistic structured FRW universe not only explains the dimming of cosmological objects but also leads to new effects which should be looked for in the huge data already existent.\
I would like to thank Thomas Buchert for indicating the non-vanishing character of the shear in an early draft of the paper. It is also a pleasure to thank members of the cosmology group at McGill University, specially Robert Brandenberger and McGill Physics department for the hospitality.
A. G. Riess et. al., Astron. J., 116, 10-09 (1998); astro-ph/ 9805201 S. Perlmutter et. al., Astrophys.J., 517, 565 (1999); astro-ph/981. Charles A. Shapiro and Michel S. Turner, astro-ph/0512586. A.G. Riess et. al., Astrophys. J., 607, 665 (2004). A. V. Fillipenko, astro-ph/0410609. Reza Mansouri, astro-ph/0512605. S. Khakshournia and R. Mansouri, Phys. Rev. D65, 027302, 2003; R. A. Sussman and Luis Garcia Trujillo, Class.Quant.Grav. 19, 2897, 2001; gr-qc/0105081.
M. Celerier, A&A, 353, 63, 2000; astro-ph/0512103. K. Bolejko, astro-ph/0512103. C. Hellaby and A. Krasinsky, gr-qc/0510093 (to be published in Phys. Rev. D). gr-qc/0307023. I. Zehavi et. al. Astrophys. J. 503, 483, 1998; astro-ph/9802252. S.P. Goodwin, P. A. Thomas, A. J. Barber, J. Gibbon, and L. I. Onuora, astro-ph/9906187. A. Dekel, ARA&A, 32, 371, 1994. D. D. Kocevski, H. Ebeling, Ch. R. Mullis, and R. Brent Tully, astro -ph/0512321. T. Buchert, Gen. Rel. Grav. 32, 105, 2000 and 33, 1381, 2001; gr-qc/9906015 and gr-qc/0102049; T. Buchert, Phys. Rev. Lett. 90, 031101, 2003; gr-qc/0210045; T. Buchert, Class. Quant. Grav. 22, L113, 2005; gr-qc/0507028; gr-qc/0509124. Thomas Buchert and Juergen Ehlers, A&A, 320, 1, 1997. D. J. Eisenstein et.al., Astrophys.J. 633, 560, 2005; astro-ph/0501171. E. W. Kolb, S. Matarrese, and A. Riotto, astro-ph/0506534. Christopher M. Hirata and Uros Seljak, astro-ph/0503582. Y. Nambu and M. Tanimoto, gr-qc/0507057. Dominik J. Schwarz, astro-ph/0209584 (talk given at CERN-TH-2002-246).
|
---
abstract: 'Defocus Blur Detection (DBD) aims to separate in-focus and out-of-focus regions from a single image pixel-wisely. This task has been paid much attention since bokeh effects are widely used in digital cameras and smartphone photography. However, identifying obscure homogeneous regions and borderline transitions in partially defocus images is still challenging. To solve these problems, we introduce depth information into DBD for the first time. When the camera parameters are fixed, we argue that the accuracy of DBD is highly related to scene depth. Hence, we consider the depth information as the approximate soft label of DBD and propose a joint learning framework inspired by knowledge distillation. In detail, we learn the defocus blur from ground truth and the depth distilled from a well-trained depth estimation network at the same time. Thus, the sharp region will provide a strong prior for depth estimation while the blur detection also gains benefits from the distilled depth. Besides, we propose a novel decoder in the fully convolutional network (FCN) as our network structure. In each level of the decoder, we design the Selective Reception Field Block (SRFB) for merging multi-scale features efficiently and reuse the side outputs as Supervision-guided Attention Block (SAB). Unlike previous methods, the proposed decoder builds reception field pyramids and emphasizes salient regions simply and efficiently. Experiments show that our approach outperforms 11 other state-of-the-art methods on two popular datasets. Our method also runs at over 30 fps on a single GPU, which is 2x faster than previous works. The code is available at: https://github.com/vinthony/depth-distillation'
author:
- Xiaodong Cun
- 'Chi-Man Pun'
bibliography:
- 'egbib.bib'
title: Defocus Blur Detection via Depth Distillation
---
|
---
abstract: 'The hot intracluster plasma in clusters of galaxies is weakly magnetized. Mergers between clusters produce gas compression and motions which can increase the magnetic field strength. In this work, we perform high-resolution non-radiative magnetohydrodynamics simulations of binary galaxy cluster mergers with magnetic fields, to examine the effects of these motions on the magnetic field configuration and strength, as well as the effect of the field on the gas itself. Our simulations sample a parameter space of initial mass ratios and impact parameters. During the first core passage of mergers, the magnetic energy increases via gas compression. After this, shear flows produce temporary, Mpc-scale, strong-field “filament” structures. Lastly, magnetic fields grow stronger by turbulence. Field amplification is most effective for low mass ratio mergers, but mergers with a large impact parameter can increase the magnetic energy more via shearing motions. The amplification of the magnetic field is most effective in between the first two core passages of each cluster merger. After the second core passage, the magnetic energy in this region gradually decreases. In general, the transfer of energy from gas motions to the magnetic field is not significant enough to have a substantial effect on gas mixing and the subsequent increase in entropy which occurs in cluster cores as a result. In the absence of radiative cooling, this results in an overall decrease of the magnetic field strength in cluster cores. In these regions, the final magnetic field is isotropic, while it can be significantly tangential at larger radii.'
author:
- Bryan Brzycki
- John ZuHone
title: 'A Parameter Space Exploration of Galaxy Cluster Mergers II: Effects of Magnetic Fields'
---
Introduction {#sec:intro}
============
The largest gravitationally bounded objects in our universe are galaxy clusters. Most of the mass in galaxy clusters is comprised of dark matter (DM), which is believed to be largely collisionless [@zwi37; @bah77]. Most of the baryonic material in galaxy clusters is comprised of a hot diffuse plasma called the intracluster medium [hereafter ICM; @sar88], which emits in X-rays. The last and smallest component of mass is that of the galaxies themselves, which have an effect on the cluster as a whole via feedback from stars and active galactic nuclei (AGN). Galaxy clusters allow us to study the interplay of these different forms of matter in a gravitationally bound system close to cosmological length scales.
There is strong observational evidence that the ICM is weakly magnetized [@car02; @fer08; @fer12]. Synchrotron radio emission has been observed from sources such as radio halos and radio relics [@fer01; @gov01; @bur92; @bac03; @ven07; @git07; @gov09; @gia11]. Furthermore, Faraday rotation of polarized emission can be measured for galaxy clusters in radio. Rotation measure (RM) studies place magnetic field strengths in clusters on the order of 0.1-10 $\mu$G, going up to tens of $\mu$G in cluster cool cores [@per91; @tay93; @fer95; @fer99; @tay02; @tay06; @tay07; @bon10]. Such field strengths imply that the magnetic field itself is dynamically weak; this is typically parameterized using the plasma parameter $\beta = p_{\rm th}/p_{\rm B} \sim 100-1000$, where $p_{\rm th}$ and $p_{\rm B}$ are the thermal and magnetic pressures, respectively. RM maps have been developed for some clusters, from which it appears that the coherence length of cluster magnetic fields is on the order of 10 kpc or less. Studies have also used RM maps to infer the cluster magnetic field power spectrum, which indicate that the magnetic field power spectrum is similar to a Kolmogorov type ($P_B(k) \propto k^{-5/3}$, where $k$ is the wavenumber), depending on the assumed value for the coherence length of the field fluctuations [@vogt03; @vogt05; @mur04; @gov06; @gov09; @gui08]. Despite the dynamical weakness of this field, it has important effects on the microphysical properties of the cluster plasma. The Larmor radii of electrons and ions ($\rho_{\rm L} \sim$ npc) are many orders of magnitude smaller than their mean free paths ($\lambda_{\rm mfp} \sim$ kpc), with the result that momentum and heat fluxes from the dissipative processes of viscosity and thermal conduction are highly anisotropic [@bra65; @nar01; @rob16]. Finally, cosmic-ray electrons radiate in the radio band via synchrotron emission in radio relics, radio halos, and radio mini-halos [@brunetti2014].
Merging between galaxy clusters is responsible for forming new clusters and changing the state of both baryonic and dark matter within clusters. As many observed galaxy clusters show evidence of current or recent merging, it is important to understand the internal physics and observable properties of cluster mergers. Mergers compress the gas and generate shocks, cold fronts, and turbulent motions in the ICM. Because the magnetic field is effectively “frozen-in” the plasma, this compression and gas motions amplify and/or stretch magnetic field lines and increase the energy in the magnetic field. A number of recent studies have used magnetohydrodynamic (MHD) simulations in the cosmological context to investigate the mechanisms and efficiency by which gas compression, bulk flows, and turbulent gas motion can amplify magnetic fields in clusters [e.g. @dol99; @dolag2002; @dolag2005; @xu2009; @xu2010; @xu2011; @vazza2014; @marinacci2015; @egan2016; @vazza2018; @domfern19]. For an extensive review of processes which amplify the magnetic field in clusters, see also @donnert2018.
Such magnetic field amplification may have important observable effects. It is well-known that magnetic fields stretched parallel to cold fronts by shear flows can stabilize them against Kelvin-Helmholtz instabilities [@zuh11b hereafter ZML11] and at least partially suppress thermal conduction across them [@zuh13a; @zuh15]. If the magnetic field strength is increased enough, it may produce a “plasma depletion layer” with a high magnetic field and low gas density and X-ray emissivity (ZML11). Evidence for such layers has been tenatively observed in a few clusters, including Virgo [@wer16], A520 [@wang16], and A2142 [@wang18]. The regions of amplified magnetic field also coincide with the same regions of increased turbulence, enhancing radio halo and radio mini-halo emission [@donnert2013; @zuh13b; @marinacci2018].
It is also possible that an increased magnetic field can suppress gas mixing. During a cluster merger, the gas from the two clusters mixes. For “cool-core” clusters (characterized by temperature inversions, high gas densities, and low central entropies and cooling times), such mixing can increase the entropy of the core gas substantially [@mitchell2009; @zuh11a hereafter Z11]. If the magnetic pressure and tension are comparable to turbulent gas motions, they could resist this mixing and prevent this increase of the entropy of the gas (ZML11).
In this work, we analyze MHD simulations of a parameter space of idealized binary cluster mergers with magnetic fields, and examine the properties of the gas and magnetic fields throughout the merger. In particular, we seek to determine the effect of various merger scenarios on the structure and strength of the magnetic field, and also determine its effects on the hot plasma. These simulations are complementary to cosmological simulations, since these controlled setups allow for a finer degree of control over cluster conditions. The simulations we use span a range of mass ratios and impact parameters. Though a number of past studies have included the effect of magnetic fields on idealized mergers or cluster substructure simulations [e.g. @roe99; @asai04; @tak08; @donnert2013; @suz13; @lage2014; @vij17a; @vij17b], ours includes the most expansive study of different merger scenarios to date. We will also compare these simulations to the otherwise identical “unmagnetized” simulations from Z11, which do not contain magnetic fields.
This paper is organized as follows: In Section \[sec:methods\], we describe the methods, including the relevant physics, the simulation code, and the initial conditions. In Section \[sec:results\], we present our results. In Section \[sec:conclusions\], we summarize these results and their implications for physics within the cores of galaxy cluster mergers. We assume a $\Lambda$CDM cosmology with $h = 0.7$, $\Omega_m = 0.3$, and $\Omega_\Lambda = 0.7$.
Methods {#sec:methods}
=======
Physics {#subsec:physics}
-------
Our simulations solve the ideal MHD equations. Written in conservation form in Gaussian units, they are: $$\frac{\partial \rho_g}{\partial t} + \nabla\cdot(\rho_g\mathbf{v})=0$$ $$\frac{\partial(\rho_g \mathbf{v})}{\partial t} + \nabla\cdot\left(\rho_g\mathbf{v}\mathbf{v}-\frac{\mathbf{B}\mathbf{B}}{4\pi}\right)+\nabla p=\rho_g \mathbf{g}$$ $$\frac{\partial E}{\partial t} + \nabla\cdot\left[\mathbf{v}(E+p)-\frac{\mathbf{B}(\mathbf{v}\cdot\mathbf{B})}{4\pi}\right]=\rho_g\mathbf{g}\cdot\mathbf{v}$$ $$\label{eqn:induction}
\frac{\partial B}{\partial t} + \nabla\cdot(\mathbf{v}\mathbf{B}-\mathbf{B}\mathbf{v})=0,$$ where $\rho_g$ is the gas density, $\textbf{v}$ is the gas velocity, and $\textbf{B}$ is the magnetic field strength. The total energy $E$, total pressure $p$, and gravitational acceleration $\textbf{g}$ have the usual definitions: $$p = p_{\text{th}}+\frac{B^2}{8\pi}$$ $$E = \frac{1}{2}\rho v^2+\epsilon+\frac{B^2}{8\pi}$$ $$\mathbf{g}=-\nabla\phi$$ $$\nabla^2\phi=4\pi G(\rho_g+\rho_{\text{DM}})$$ where $\epsilon$ is the gas internal energy per unit volume, and $\phi$ is the gravitational potential. We assume an ideal gas equation of state with $\gamma=5/3$.
Simulation Code {#subsec:simulationcode}
---------------
We performed our simulations using FLASH, a parallel hydrodynamics/N-body astrophysical simulation code developed at the Center for Astrophysical Thermonuclear Flashes at the University of Chicago [@fry00; @dub09]. FLASH employs adaptive mesh refinement (AMR), a method of partitioning a grid throughout the simulation box such that higher resolutions (smaller cell sizes) are only used where needed, such as in the cores of clusters and at the gas discontinuities formed in cluster mergers such as shocks and cold fronts.
FLASH solves the equations of magnetohydrodynamics using a directionally unsplit staggered mesh algorithm [USM; @lee09]. The USM algorithm used in FLASH is based on a finite-volume, high-order Godunov scheme combined with a constrained transport method (CT), which guarantees that the evolved magnetic field satisfies the divergence-free condition [@eva88]. In our simulations, the order of the USM algorithm corresponds to the Piecewise-Parabolic Method (PPM) of @col84, which is ideally suited for capturing shocks and contact discontinuties (such as the cold fronts that appear in our simulations). FLASH also includes an $N$-body module which uses the particle-mesh method to solve for the forces on gravitating particles. The gravitational potential is computed using a multigrid solver included with FLASH [@ric08].
Initial Conditions {#subsec:initialconditions}
------------------
We carry out 9 simulations of idealized binary mergers of two spherically symmetric clusters in hydrostatic and virial equilibrium. Our initial galaxy clusters are initialized in the same way as in Z11, choosing initial conditions based on cosmological simulations and cluster observations, with the only difference that we now initialize the clusters with magnetic fields included. Since the simulations are otherwise identical to this previous set, we can compare each magnetized simulation to its corresponding unmagnetized version.
These simulations have mass ratios $M_1/M_2$ of 1, 3, and 10, in which the primary cluster has an initial mass of $M_{200} = 6\times10^{14}~M_\odot$. From the mass, the rest of the parameters for each cluster can be derived using the scaling relations determined by @vik09a. In Table \[table1\], we list these parameters for each cluster, including some which are derived below.
In the rest of this section, we will describe the setup of our simulations, following closely the discussion from Z11.
### Gas and Dark Matter {#subsec:gas_dm}
Cluster $M_{200}$ ($M_\odot$)$^a$ $r_{200}$ (kpc)$^b$ $c_{200}$$^c$ $f_{g,500}$$^d$ $T_X$ (keV)$^e$ $S_0$ (keV cm$^2$)$^f$ $S_1$ (keV cm$^2$)$^g$ $N_p$$^h$
--------- --------------------------- --------------------- --------------- ----------------- ----------------- ------------------------ ------------------------ -----------
C1 $6\times 10^{14}$ 1552.25 4.5 0.1056 4.97 9.62 192.40 5,000,000
C2 $2\times 10^{14}$ 1076.27 4.7 0.0879 2.42 5.08 101.60 1,684,119
C3 $6\times 10^{13}$ 720.49 5.1 0.0686 1.10 2.73 54.60 513,137
\[table1\]\
$^a$Virial mass; $^b$Virial radius; $^c$Concentration parameter; $^d$Gas mass fraction; $^e$X-ray temperature; $^f$Core entropy; $^g$Scale entropy; $^h$Number of DM particles
Simulation $M_1/M_2$ $b/r_{200}$
------------ ----------- -------------
MS1, S1 1 0
MS2, S2 1 0.3
MS3, S3 1 0.6
MS4, S4 3 0
MS5, S5 3 0.3
MS6, S6 3 0.6
MS7, S7 10 0
MS8, S8 10 0.3
MS9, S9 10 0.6
: Initial Merger Parameters
\[table2\]
For the total mass distribution, we use the Navarro-Frenk-White (NFW) density profile [@nav97]: $$\rho_{\text{tot}}(r)=\frac{\rho_s}{r/r_s\left(1+r/r_s\right)^2},$$ with $$r_s = r_{200}/c_{200},$$ $$\rho_s = \frac{200}{3}c_{200}^3\rho_{\text{crit}}\left[\log\left(1+r/r_s\right)-\frac{r/r_s}{1+r/r_s}\right]^{-1}.$$
We carry the NFW profile out to the virial radius $r=r_{200}$, the radius at which the average density is 200 times the critical density of the universe $\rho_{\text{crit}}$. Then, for $r>r_{200}$, we employ an exponentially decreasing density prescription: $$\rho_{\text{tot}}(r)=\frac{\rho_s}{c_{200}(1+c_{200})^2}\left(\frac{r}{r_{200}}\right)^{\kappa}\exp\left({-\frac{r-r_{200}}{0.1r_{200}}}\right),$$ where $\kappa$ is a constant such that $\rho_{\text{tot}}$ and its first derivative are continuous at the boundary $r=r_{200}$. We employ this exponential cutoff, since the NFW mass profile does not converge as $r$ goes to infinity.
We take the gas to be in hydrostatic equilibrium in the DM-dominated potential well. A key observable quantity for the ICM is the gas entropy, defined as $S = k_BTn_e^{-2/3}$, where $k_BT$ is the gas temperature in keV and $n_e$ is the electron number density. The entropy profiles of galaxy clusters can be well-modeled by a “baseline” power-law profile combined with a constant floor value to represent the entropy of the core [@voi05; @cav09], which can be written as: $$S(r)=S_0+S_1\left(\frac{r}{0.1r_{200}}\right)^{\alpha} \label{eqn:S_powerlaw}$$ where $S_0$ is the core entropy and $\alpha \sim 1.0-1.3$. We start off with small core entropies $S_0$ and set $\alpha=1.1$ in order to make our initial models consistent with relaxed, “cool-core” galaxy clusters.
Using the above definition of entropy, we can take the equation of hydrostatic equilibrium and derive an equivalent expression in terms of the gas entropy and temperature: $$\begin{aligned}
{dP \over dr} &=& -\rho_g{d\phi \over dr} \\
{k_B \over {\mu}m_p}{d({\rho_g}T) \over dr} &=& -\rho_g{d\phi \over dr} \\
{k_B \over {\mu}m_p}{d \over dr}\left[{T{\left(T \over S\right)}^{3/2}}\right] &=& -\left({T \over S}\right)^{3/2}{d\phi \over dr}\end{aligned}$$
We solve this equation using standard numerical integration methods and by imposing two conditions: the gas mass fraction $f_{g,500} = M_{\text{g}}(r_{500})/M_{\text{tot}}(r_{500})$ (see Table \[table1\]) using the scaling relation from @vik09a and setting $T(r_{200})=\frac{1}{2} T_{200}$, where $$k_B T_{200} \equiv \frac{ G M_{200} \mu m_p}{2r_{200}}$$ is the “virial temperature” of the cluster [@poo06]. From the temperature and entropy, we determine the gas density profile. The DM density profile is then given by $$\rho_{\text{DM}}(r)=\rho_{\text{tot}}(r)-\rho_g(r).$$
After determining the initial radial profiles, we set up the distribution of positions and velocities for the DM particles, following the procedure outlined in @kaz04. For the positions, we uniformly sample a random deviate $u\in [0,1]$, and we invert the function $u=M_{\text{DM}}(r)/M_{\text{DM}}(r_{\text{cut}})$ to calculate the radius of each particle from the center of the DM halo, where $r_{\text{cut}}$ is the radius of the halo at which the gas density reaches the mean gas density of the universe, which we take to be the boundary of the halo, and is typically a few $r_{200}$. We choose to directly calculate the velocity distribution using the energy distribution function [@edd16]: $$\mathcal{F} = \frac{1}{\sqrt{8} \pi^2}\left[\int_0^{\mathcal{E}} \frac{d^2\rho}{d\psi^2}\frac{d\psi}{\sqrt{\mathcal{E}-\psi}}+\frac{1}{\sqrt{\mathcal{E}}}\left(\frac{d\rho}{d\psi}\right)_{\psi=0}\right],$$ where $\psi = -\phi$ is the relative potential and $\mathcal{E}=\psi - \frac{1}{2} v^2$ is the relative energy of the particle. Particle speeds are determined using this distribution function using the acceptance-rejection method. With the particle radii and speeds determined, we find the position and velocity vectors by choosing random unit vectors isotropically distributed in $\mathbb{R}^3$. The number of DM particles we use for each cluster is shown in Table \[table1\].
### Magnetic Fields
The magnetic field of the cluster is set up after the manner of ZML11. This procedure is designed to produced a tangled magnetic field with a magnetic pressure roughly proportional to the thermal pressure, satisfying the condition $\nabla\cdot\mathbf{B}=0$. A Gaussian random magnetic field $\tilde{\bf B}({\bf k})$ is set up in $\bf{k}$-space on a uniform grid using independent normal random deviates for the real and imaginary components of the field. We adopt a dependence of the magnetic field amplitude $B(k)$ on the wavenumber $|\bf{k}|$ similar to (but not the same as) @rus07 and @rus10: $$B(k) \propto k^{-11/6}{\rm exp}[-(k/k_0)^2]{\rm exp}[-(k_1/k)^2]$$ which corresponds to a Kolmogorov power spectrum with exponential cutoffs at scales of $k_0$ and $k_1$. The cutoff at high wavenumber $k_0 = 2\pi/\lambda_0$ is set to $\lambda_0$ = 1 kpc, though the finest cell size of $\Delta{x} \approx 7$ kpc effectively sets the smallest scale. The cutoff at low wavenumber (large length scale) $k_1 = 2\pi/\lambda_1$ corresponds to $\lambda_1 \approx$ 500 kpc. This field is then Fourier transformed to yield ${\bf B}({\bf x})$, which is rescaled to have an average value of $\sqrt{8\pi{p}/\beta}$ to yield a field that has a pressure that scales with the gas pressure, i.e. to have a spatially uniform $\beta$ for the initial field. The value $\beta = 200$ is chosen to produce magnetic fields which agree with typical field measurements from Faraday rotation measurements [@bon10] and simulations [@dol99; @dub08]. Recently, @walker17 [@walker18] showed that simulations with an initial $\beta = 200$ from ZML11 provide the best match to conditions seen in the Perseus Cluster, further motivating our choice.
### Merger Trajectories
In each merger simulation, we set up two clusters centered within a cubical simulation box of $\sim$14.29 Mpc on a side. The boundary conditions of the simulation box are such that matter may flow into and out of the box. We find that mass loss through these boundaries is negligible and does not affect the evolution of our cluster mergers, which occur in the central $\sim$(6 Mpc)$^3$ region.
The cluster centers are initialized in the $x-y$ coordinate plane at $z$ = 0, and the initial distance between them is given by the sum of their respective $r_{200}$. @vit02 demonstrated from cosmological simulations that the average infall velocity for merging clusters is $v_{\rm in}(r_{\rm vir}) = 1.1V_c$, where $V_c = \sqrt{GM(r_{\rm vir})/r_{\rm vir}}$ is the circular velocity at the virial radius $r_{\rm vir}$ for the primary cluster. For all of our simulations, this is chosen as the initial relative velocity ($v_{\rm in} \approx 1200$ km/s). In addition to varying the mass ratio between the simulation, we also vary the impact parameter of each merger simulation between $b$ = 0 (head-on), $0.3r_{200}$, and $0.6r_{200}$. In Table \[table2\], we list all of the simulations, referring to the magnetized simulations with prefix “MS” and unmagnetized with prefix “S”.
Results {#sec:results}
=======
Slices in Density and Magnetic Field Strength {#subsec:slices}
---------------------------------------------
{width="0.94\linewidth"}
{width="0.94\linewidth"}
{width="0.94\linewidth"}
{width="0.94\linewidth"}
{width="0.94\linewidth"}
{width="0.94\linewidth"}
{width="0.94\linewidth"}
{width="0.94\linewidth"}
{width="0.94\linewidth"}
{width="0.94\linewidth"}
{width="0.94\linewidth"}
{width="0.94\linewidth"}
{width="0.94\linewidth"}
{width="0.94\linewidth"}
{width="0.94\linewidth"}
{width="0.94\linewidth"}
{width="0.94\linewidth"}
{width="0.94\linewidth"}
We first seek to obtain a qualitative picture of how magnetic fields evolve during the different merger stages. Figures \[fig:1to1\_b0\_density\]-\[fig:1to10\_b1\_magfield\] show slices in density and magnetic field strength perpendicular to the $z$-axis and through the merger plane. In addition to these figures, slices of the temperature and DM density are shown in the Appendix in Figures \[fig:1to1\_b0\_kT\]-\[fig:1to10\_b1\_all\_cic\]. The evolution of the gas thermodynamical properties within each cluster merger for our simulations is essentially identical to that in the simulations in Z11, and have also been described in previous binary merger simulation investigations [e.g. @ric01; @poo06]. Here, we focus on the evolution of the magnetic field structures during the merger. Many of the magnetic field structures we note below were noticed in similar simulations in @roe99 and @tak08, though we have many more combinations of mass ratio and impact parameter than those earlier works.
For each simulation MS1-MS9, we make slices at 6 different points in time. Time $t = 0$ Gyr is the initial state, where the virial radii of both clusters are just touching. The next epoch is chosen to approximately mark the first core passage for each simulation. In MS1-MS3, this is at about $t\approx1.4$ Gyr; in MS4-MS6, $t\approx1.2$ Gyr; in MS7-MS9, $t\approx1.1$ Gyr. After that, the plotted epochs are $t$ = 2.0, 3.0, 5.0, and 7.0 Gyr. These epochs are simply chosen to show relevant epochs from the first two core passages until late in the simulation.
The equal-mass mergers have a qualitatively different evolution from the unequal-mass cases, so we will discuss each of these separately.
### Equal-mass Mergers
In the $M_1/M_2 = 1, b = 0$ merger (simulation MS1, Figure \[fig:1to1\_b0\_density\]), the first core passage occurs at about $t = 1.4$ Gyr. Building up to the core passage, the gas in the midplane of the merger becomes shocked and compressed and forms a flat, “pancake”-like structure perpendicular to the line of centers between the cluster cores. The compressed gas amplifies the magnetic field in these regions. The magnetic field strength is also increased behind the shock fronts as they propagate outward. The DM cores, unimpeded by ram pressure, pass through and then subsequently oscillate about each other. These rapid and violent changes in the gravitational potential drag the surrounding gas back and forth, driving smaller shocks and turbulence. At first, these motions create long, thin, and laminar strong “filament” structures in the field along the merger axis with strengths up to $\sim$10 $\mu$G (Figure \[fig:1to1\_b0\_magfield\]), but as the gas is violently stirred by the DM cores, a turbulent magnetic field is generated in the center. Turbulent fields are also generated from the gravitational infall of the pancake structure into the center.
{width="0.95\linewidth"}
In the $M_1/M_2 = 1$ mergers with nonzero impact parameter (simulations MS2 and MS3, Figures \[fig:1to1\_b0.5\_density\]-\[fig:1to1\_b1\_magfield\]), the gas and DM cores “sideswipe,” ram-pressure stripping gas through the two cores. Two cold fronts $\sim$1-2 Mpc in length emerge from the center at $t \sim 3$ Gyr, expanding mostly radially outward from their respective cores. The magnetic field is stretched and amplified along these fronts by their associated velocity shears up to $\sim$10 $\mu$G, which produces the same “filament” structures in the field as seen in the $M_1/M_2 = 1, b = 0$ merger simulation. These structures are short-lived, however, as the DM cores undergo their second and third core passages, driving yet more turbulence. Similar to the $M_1/M_2 = 1, b = 0$ case, long, straight field structures can persist for longer at larger radii $r {\,\hbox{\lower0.6ex\hbox{$\sim$}\llap{\raise0.6ex\hbox{$>$}}}\,}1.5$ Mpc.
In all of the $M_1/M_2 = 1$ simulations, at large radii ($r {\,\hbox{\lower0.6ex\hbox{$\sim$}\llap{\raise0.6ex\hbox{$>$}}}\,}1.5$ Mpc), long, mostly straight magnetic field lines are also stretched and amplified between gas which is moving radially outward, on either side of the cores, and gas which is falling back into them. These structures, Mpc in length, persist until the end of each simulation at $t = 10$ Gyr, but within $r \sim 1.5$ Mpc random turbulent motions have produced a mostly turbulent magnetic field.
### Unequal-mass Mergers
At first core passage in the unequal-mass, $b = 0$ mergers (simulations MS4 and MS7, Figures \[fig:1to3\_b0\_density\], \[fig:1to3\_b0\_magfield\], \[fig:1to10\_b0\_density\], and \[fig:1to10\_b0\_magfield\]), the core of the secondary punches through that of the primary, completely disrupting it. At the same time, gas is ram-pressure stripped from the secondary and mixed in with the primary’s gas via Kelvin-Helmholtz instabilities. Roughly 2 Gyr after the first core passage (t = 3.0 Gyr), the remaining core gas from the secondary (and its DM core as well) begin to fall back into the secondary, forming a cold inflow of gas that is shock-heated as it enters the core region. As the secondary cluster passes through the primary and returns back, it stretches the magnetic field behind it via shear amplification into similar filament features. As both gas from the primary and the stripped gas from the secondary start to fall back onto the oscillating cores, the magnetic field lines wrap inwards. These magnetic field structures are initially fairly laminar, but quickly become tangled within the core region due to Kelvin-Helmholtz instabilities and turbulence.
In the non-zero impact parameter, unequal mass cases (simulations MS5, MS6, MS8, and MS9), the primary cluster core produces sloshing cold fronts, and the secondary cluster develops a dense, cold, and long ($\sim$ a few Mpc) plume of gas which trails behind it and is stripped as it leaves the primary’s core region and later returns, also producing a cold front (Figures \[fig:1to3\_b0.5\_density\], \[fig:1to3\_b1\_density\], \[fig:1to10\_b0.5\_density\], and \[fig:1to10\_b1\_density\]). Along these cold fronts and plume, the magnetic field is similarly stretched and amplified as in the equal-mass, non-zero impact parameter cases (Figures \[fig:1to3\_b0.5\_magfield\], \[fig:1to3\_b1\_magfield\], \[fig:1to10\_b0.5\_magfield\], and \[fig:1to10\_b1\_magfield\]). Because the second core passage occurs much later in these simulations, these magnetic field structures can persist for a few Gyr longer than in the other simulations.
The Evolution of the Magnetic Energy Over Time {#sec:energyovertime}
----------------------------------------------
Figure \[fig:emag\_evol\] shows the evolution of the magnetic energy within the central $V = (8~{\rm{Mpc}})^3$ in the simulations, scaled by the value of the magnetic energy at $t = 0$ Gyr. This volume is large enough to contain the two clusters out to their respective $r_{200}$ initially and their subsequent evolution. The first and second core passages are marked in each panel with vertical dashed lines with the same colors as the lines in the legend.
The stages in the evolution of the magnetic energy can be explained by reference to the events detailed in Section \[subsec:slices\]. After a short period of decline as the initial field relaxes, it increases rapidly in all simulations at the first core passage due to compression of the gas. A slower increase after the core passage occurs due to stretching of the field lines due to shear motions, which are more significant for larger impact parameters. After this, the magnetic energy increase flattens out, as the cluster cores move away from each other and the gas re-expands. Another increase occurs at the second core passage, with more gas compression and stretching of field lines. Throughout the period between the first and second core passages, additional energy is transferred to the magnetic field through the stretching of field lines from turbulence. The increase in the magnetic energy is less significant in simulations with smaller subclusters, but for a given mass ratio it is more significant for larger impact parameters, as the shearing motions generated after the first core passage strongly amplify the magnetic field.
At the end of every simulation, the magnetic energy gradually decreases, as the magnitude of the compression and stretching diminishes. We have verified that a negligible amount of magnetic energy is being advected through the boundaries of our $V = (8~{\rm{Mpc}})^3$ volume, thus the main reasons for the decrease in the magnetic energy after the second core passage are the expansion of the core due to the increase of its entropy (see Section \[subsec:radialprofiles\]) and the relaxation of the magnetic tension in the turbulent field. This late-time decrease in magnetic field strength appears to be in conflict with the results of previous simulations of cluster mergers (idealized or cosmological) with magnetic fields, where the magnetic field increases continuously and gradually without decreasing [e.g. @dolag2002; @xu2009; @marinacci2015; @vazza2018; @domfern19]. Needless to say, our binary merger simulations are very different in the respect that only one merger between two clusters is happening, whereas in cosmological simulations there is constant merging and accretion of material. Sustaining such an increase in the magnetic energy requires a setting where the drivers of turbulent and compressive motions are being constantly replenished [@sub06]. Our simulations also lack radiative cooling, which would compress the core gas and amplify the field in these regions after the effects of the merger subside. In @roe99, another work involving idealized mergers such as ours, the magnetic field does increase for 5 Gyr after the first core passage, but the low spatial resolution in that work may have prevented the gas mixing that drives the expansion of the core gas and hence decreases the magnetic field in the core (see their Figure 5 and Section \[subsec:radialprofiles\] of this work).
Radial Profiles of the Final State {#subsec:radialprofiles}
----------------------------------
By the end of each simulation ($t$ = 10 Gyr), the clusters have fully merged and are nearly relaxed. It is instructive to examine radial profiles of the gas properties. For our purposes, the most relevant quantities to examine are those related to the velocity and magnetic fields, as well as the gas entropy. The radial profiles are taken in radial bins centered on the cluster potential minimum.
### Radial Profiles of Velocity Fields {#sec:velocity_profiles}
The local velocity dispersion is a measurement of the turbulent kinetic energy: $$\begin{aligned}
\sigma_v^2 &=& \langle v^2\rangle-\langle v\rangle^2\end{aligned}$$ We want to compare the relative contributions to the energy from the internal (IE), kinetic (KE), and magnetic (ME) energies, so we also compute profiles of the sound and Alfvén speeds: $$\begin{aligned}
c_s^2 &=& \frac{\gamma{P}}{\rho} \\
v_A^2 &=& \frac{\langle B^2\rangle}{4\pi\rho},\end{aligned}$$ where $\gamma=5/3$. These are the characteristic speeds for the internal and magnetic energies, respectively. Taking ratios of these squared velocities essentially yields the ratios between these different forms of energy: $$\begin{aligned}
\label{eqn:energy_ratios}
\frac{\sigma_v^2}{c_s^2} &=& {\cal M_{\rm turb}}^2 = \frac{\sigma_v^2}{\gamma{P}/\rho} = \frac{\rho\sigma_v^2/2}{\gamma(\gamma-1)\rho{\epsilon}} \sim \frac{\rm KE}{\rm IE} \\
\frac{\sigma_v^2}{v_A^2} &=& {\cal M_{\rm A,turb}}^2 = \frac{\sigma_v^2}{{\langle B^2\rangle}/{4\pi\rho}}=\frac{\rho\sigma_v^2/2}{\langle B^2\rangle/8\pi} = \frac{\rm KE}{\rm ME}\end{aligned}$$ where ${\cal M}_{\rm turb}$ ${\cal M}_{\rm A,turb}$ are the Mach and Alfvénic Mach numbers of the turbulent gas motions, respectively. These ratios quantify the physical relevance of one type of energy to another.
Figures \[fig:sigma\_vA\] and \[fig:sigma\_cs\] show profiles of these ratios at the final state of all of our simulations. First, Figure \[fig:sigma\_vA\] shows profiles of $\sigma_v^2/v_A^2$. Within the core region of $r \sim 100-300$ kpc, this ratio is generally on the order of $\sigma_v^2/v_A^2 \sim 10-100$. In the equal mass mergers, increasing impact parameter has only a mild effect on this ratio, since each case is exceptionally turbulent. In the equal-mass, zero-impact parameter case, given the symmetry of the simulation, the gas components of two clusters do not significantly penetrate one anothers’ atmospheres but instead “stick together”. Hence, there is not nearly as much shear amplification in this case, and the magnetic field is weaker in the center than in the equal-mass, off-center collisions. In the high mass ratio mergers, there is a mild trend towards higher ratios of $\sigma_v^2/v_A^2$ with increasing impact parameter, since in those cases, turbulence has gone more into heating the core and expanding it (thus lowering the magnetic field strength) than into amplifying and stretching the magnetic field. For example, in MS9, the magnetic field has only been mildly amplified by turbulence since the subcluster is small and passes by at a large distance. Though there is still turbulence in the core, it transfers little energy to the magnetic field. This result is consistent with that of Section \[sec:energyovertime\].
Beyond the core, this ratio is closer to $\sim$10 or even lower, with the magnetic energy more comparable to the kinetic energy, but still lower. It should be noted again that our idealized simulations began with no turbulent velocities but with a turbulent magnetic field, highlighting the efficiency with which the bulk motion of the merger is converted into turbulent motion and the relative inefficiency with which that energy is transferred to the magnetic field [see also @min15; @vazza2018].
Figure \[fig:sigma\_cs\] shows profiles of $\sigma_v^2/c_s^2$ for the magnetized and unmagnetized simulations. In all cases, there are only modest differences in the turbulent kinetic energy profiles between the magnetized and unmagnetized simulations, again consistent with the results of Section \[sec:energyovertime\].
\[tp!\]
{width="\linewidth"}
\[tp!\]
{width="\linewidth"}
### Radial Profiles of Entropy {#sec:entropy_profiles}
Z11 showed that turbulent mixing from cluster mergers can increase the entropy of the core region substantially for a range of mass ratios and impact parameters. Magnetic fields are capable of preventing this mixing if they are strong enough (see Figure 24 of ZML11 for an illustration of this effect in a sloshing cluster core). If this effect is strong, we expect the core entropies in the magnetic simulations to be significantly lower than their unmagnetized counterparts.
Figure \[fig:entropy\] shows the entropy profiles for the magnetized and unmagnetized simulations. In each case, the magnetized simulations have slightly lower core entropies than the corresponding unmagnetized simulations in most cases, but this effect is very modest. Given that our previous results have shown that the magnetic field has little effect on the turbulent velocities in our merger simulations compared to their unmagnetized versions from Z11, this should come as no surprise. As was seen in Z11, higher levels of core entropy tend to be correlated with higher turbulence in the core region, as seen in Figure \[fig:sigma\_cs\].
### Radial Profiles of Magnetic Field Strength {#sec:bfield_profiles}
Figure \[fig:magfield\_final\], shows the magnetic field strength at the initial and final states of each simulation. Within the inner region where the entropy is constant with radius (Figure \[fig:entropy\]), the magnetic field strength is an order of magnitude or more weaker than that in the initial condition. In these regions, turbulent mixing has lowered the gas density considerably, flattening the core (see Figure 15 from Z11), and due to the effect of flux freezing this causes the magnetic energy to decrease. In the large mass ratio, large impact parameter simulations, the magnetic field strength decreases steeply towards the center of the cluster, because there is not enough turbulence to mix the gas and produce large isentropic cores, and not enough of this energy goes into amplifying the magnetic field in these regions. Outside of the core regions, the magnetic field strength has instead increased, with larger values for smaller mass ratios/larger secondary clusters and larger impact parameters.
\[tp!\]
{width="\linewidth"}
\[t!\]
{width="\linewidth"}
### Radial Profiles of Velocity and Magnetic Field Anisotropy {#sec:beta_profiles}
Lastly, we examine the anisotropy of the velocity and magnetic fields at the final state. To do this, we employ a common paramerization for the anisotropy in spherical coordinates: $$\begin{aligned}
\beta_v &=& 1-\frac{\sigma_{vt}^2}{2\sigma_{vr}^2} \\
\beta_B &=& 1-\frac{\sigma_{Bt}^2}{2\sigma_{Br}^2}\end{aligned}$$ where $\sigma_{vr}^2$ and $\sigma_{Br}^2$ are the variances of the radial components of the velocity and magnetic fields, respectively, and $\sigma_{vt}^2 = \sigma_{v\theta}^2 + \sigma_{v\phi}^2$ and $\sigma_{Bt}^ 2= \sigma_{B\theta}^2 + \sigma_{B\phi}^2$ are the same for the tangential components. With this parameterization, perfect isotropy of a vector field is $\beta = 0$, preferentially radial vectors have $1 \geq \beta > 0$, and preferentially tangential vectors have $\beta < 0$.
Figure \[fig:beta\_final\] shows the velocity and magnetic field anisotropy radial profiles for the final state. In the core regions, both the velocity and magnetic field anisotropies are very nearly isotropic or mildly radially anisotropic. In these regions, the cores are isentropic and not stratified, and thus bouyancy effects do not prevent the radial gas motions from becoming as large as the tangential ones. The $b = 0$ simulations (MS1, MS4, MS7) have more radial anisotropy ($\beta > 0$) within and outside of this core region due to the fact that merger proceeds along an entirely radial trajectory. The magnetic field strength is less radially anisotropic than the velocity, however.
Conversely, there is a considerable amount of tangential anisotropy ($\beta < 0$) in both fields in the simulations with nonzero $b$. This is most noticeable in the large mass ratio simulations MS6, MS8, and MS9. There are two reasons for this effect. These simulations have high angular momentum in the initial merger configuration, and the resulting tangential gas motions will stretch the magnetic field lines in a more tangential direction.
In general, then, cluster mergers produce turbulent magnetic fields which are either mostly isotropic or have a tangential bias. Since conductive heat fluxes are parallel to magnetic field lines in the ICM due to the previously mentioned fact that $\lambda_{\rm mfp} \gg \rho_{\rm L}$, the isotropic and tangential magnetic fields caused by merger-driven turbulence will more efficiently prevent thermal conduction from hotter regions in the outer cluster to the core region in a cool-core cluster than radial fields would [@par10; @rus10]. As noted in previous works, if the thermal conductivity of the ICM parallel to magnetic field lines is efficient, cool-core clusters with positive temperature gradients are susceptible to the heat-flux buoyancy instability, which can rearrange magnetic fields tangentially and thereby suppress radial thermal conduction [@qua08; @par08; @par09; @bog09]. On the other hand, the gas in the outer regions of the cluster are susceptible to the magnetothermal instability, which rearranges magnetic fields radially, thereby enhancing radial thermal conduction [@bal00; @par05; @par07]. Subsequent work has shown that even relatively gentle turbulence driven by galaxy motions and minor mergers can dominate these instabilities and arrange the magnetic field more isotropically [@par10; @rus10; @zuh13a]. Since in our simulations the turbulent velocity dispersion dominates over the Alfvén speed even at late times (Figure \[fig:sigma\_vA\]), we expect the effect of these instabilities to be swamped by the merger-driven turbulence, though simulations similar to ours including anisotropic thermal conduction would be required to verify this.
\[t!\]
{width="\linewidth"}
Conclusions {#sec:conclusions}
===========
In this work, we have analyzed a parameter space over mass ratio and impact parameter of idealized binary cluster mergers including the effects of magnetic fields. Our main conclusions are as follows:
- The bulk and turbulent flows created by merging clusters amplify magnetic fields in distinct ways. Compression of gas during core passages and behind shock fronts increases the magnetic energy in the core region overall. Shear flows created shortly after the first core passage of a merger along cold fronts and ram-pressure stripped gas amplify and stretch magnetic field lines, producing long, laminar magnetic structures which can stretch for $\sim$1-2 Mpc, which are easier to create and sustain in off-axis mergers. These structures are transient, however, and only last for a few Gyr at most, due to the turbulence driven by the second and following core passages. Otherwise, the velocities generated are turbulent in character, which generate turbulent magnetic fields.
- The magnetic energy of the clusters increases during the first and second core passages. At the first core passage, the dominant mechanism is the compression of magnetic field lines from the deepening of the gravitational potential and by shock fronts. For simulations with a non-zero impact parameter, shearing motions also increase the magnetic energy after this time. At later times, the dominant mechanism of amplification is stretching of magnetic fields by shear motions and turbulence. At the end of each simulation, as the merger remnant relaxes, the magnetic energy of the cluster gradually decreases. In general, simulations with smaller subclusters (increased mass ratio) generate less compressive motions and turbulence, and hence generate less amplification of the magnetic field, but simulations with larger impact parameters for a given mass ratio generate more magnetic energy via shearing motions.
- At the final merged state of each simulation, the turbulent kinetic energy in the core region is $\sim$1-3 orders of magnitude higher than the magnetic energy, but only a factor of 10 or less higher outside the core region. The ratio between these quantities is largest in simulations with high mass ratio and impact parameter, since these cores have been less affected by strong turbulence and hence have less amplification of the magnetic field.
- The turbulent velocity dispersion of the gas in the cluster is very similiar in both merger simulations which include a magnetic field and those which do not. Due to this fairly insignificant effect of the magnetic field on the turbulent motions in the cluster, the final core entropies in each merger with magnetic fields included are nearly indistinguishable from those in the unmagnetized simulations.
- In the absence of radiative processes, the turbulent-driven mixing which increases the entropy of the core also decreases the density in the core region substantially. Since the magnetic field is frozen into the fluid, the average magnetic field strength also decreases substantially in the core region. Outside of the core region, the magnetic field of the final merger remnant is increased over the initial value. In the core region, the velocity and magnetic fields are very nearly isotropic, but outside of the cores these fields can become preferentially radial in on-axis mergers or preferentially tangential in off-axis mergers.
These results confirm in detail the fact stated simply at the outset: magnetic fields are generally not dynamically significant in the ICM of merging clusters of galaxies. The compressive and stretching actions on the magnetic field driven by the merger, though they do increase the magnetic field strength, do not increase it to the extent that it has a significant effect on the kinematic or thermodynamic properties of the merger remnant. This is in some contrast to the results of ZML11, who found that magnetic fields can have a dynamically significant effect on the cluster core in a relatively relaxed system with sloshing gas motions. However, the gas in those simulations was comparatively “gently stirred” by a gasless subcluster, and the typical turbulent velocities which resulted are much slower than encountered in these simulations, and thus comparable to the Alfvén velocity of the magnetic field. It is important to note that this conclusion holds [*on average*]{} in the bulk of the ICM–at regions where strong shear flows exist such as cold fronts, magnetic fields are still amplified to dynamically significant strengths ($\beta \sim$ 3-10, ZML11).
Our binary merger simulations highlight the importance of gas compression, shear motions, and turbulence in amplifying the magnetic field during the most violent stages of cluster mergers. However, unlike what has been observed in cosmological simulations of cluster formation, during the period of relaxation after the formation of the merger remnant the magnetic energy decreases. This is likely due to a combination of the lack of radiative cooling in our simulations and the fact that real clusters undergo multiple mergers and are continuously stirred by substructure. Supporting this conclusion, @sub06 argued that after a major merger the magnetic field would decay after the turbulence which drives its further growth had diminished, which is exactly what is seen in Figure \[fig:emag\_evol\].
This investigation leaves room for further study. In the absence of a high angular-resolution X-ray observatory equipped with a microcalorimeter which can directly measure gas motions, a number of studies have used [*Chandra*]{} observations of surface brightness fluctuations in the ICM to indirectly probe the power spectrum of turbulent gas motions over a large range of scales [e.g. @zhu15; @chu16; @zhu18]. Though this work shows that the effects of magnetic fields are not dynamically significant in the ICM during cluster mergers overall, it remains to be seen if localized field amplification on small scales can have an effect on the properties of observed X-ray surface brightness fluctuations.
We did not include the effects of viscosity or thermal conduction in our simulations. As already mentioned, these processes will be highly anisotropic due to the wide separation in scales between the Larmor radii and mean free paths of the electrons and ions. The effects that these processes will have on the merger-driven gas motions may also have a non-negligible effect on the results described here. This is also left for future work.
Finally, our non-radiative simulations neglect the important effects of radiative cooling, star formation, and stellar and AGN feedback. Without radiative cooling, the mixing of hot and cold gas produces a non-cool-core cluster as the final merger remnant, and the average magnetic field strength in its core region is lower than in the inital state, since the field is frozen in and the field lines become more spread apart as the gas becomes more dilute. In a real cluster with radiative cooling, the gas density would increase again and the field would become stronger. Future work will include the effects of cooling, star formation, and feedback in idealized cluster mergers to provide a more complete picture.
This work required the use and integration of a number of Python software packages for science, including Matplotlib [@hun07][^1], NumPy [^2], SciPy [^3], and yt [@tur11][^4]. We are thankful to the developers of these packages. The authors thank Paul Nulsen and Grant Tremblay for useful comments. JAZ acknowledges support through Chandra Award Number G04-15088X issued by the Chandra X-ray Center, which is operated by the Smithsonian Astrophysical Observatory for and on behalf of NASA under contract NAS8-03060. The numerical simulations were performed using the computational resources of the Advanced Supercomputing Division at NASA/Ames Research Center.
Asai, N., Fukuda, N., & Matsumoto, R. 2004, , 606, L105 Bacchi, M., Feretti, L., Giovannini, G., & Govoni, F. 2003, A&A, 400, 465 Bahcall, J. N., & Sarazin, C. L. 1977, ApJ, 213, L99 Balbus, S. A. 2000, ApJ, 534, 420 Bogdanovi[ć]{}, T., Reynolds, C. S., Balbus, S. A., et al. 2009, , 704, 211 Bonafede, A., Feretti, L., Murgia, M., et al. 2010, A&A, 513, A30 Braginskii, S. I. 1965, Reviews of Plasma Physics, 1, 205 Brunetti, G., & Jones, T. W. 2014, International Journal of Modern Physics D, 23, 1430007-98 Burns, J. O., Sulkanen, M. E., Gisler, G. R., & Perley, R. A. 1992, ApJ, 388, L49 Carilli, C. L., & Taylor, G. B. 2002, , 40, 319 Cavagnolo, K. W., Donahue, M., Voit, G. M., & Sun, M. 2009, ApJS, 182, 12 Churazov, E., Arevalo, P., Forman, W., et al. 2016, , 463, 1057 Colella, P., & Woodward, P. R. 1984, Journal of Computational Physics, 54, 174 Dolag, K., Bartelmann, M., & Lesch, H. 1999, A&A, 348, 351 Dolag K., Bartelmann M., Lesch H., 2002, A&A, 387, 383 Dolag, K., Grasso, D., Springel, V., et al. 2005, Journal of Cosmology and Astro-Particle Physics, 2005, 9. Dom[í]{}nguez-Fern[á]{}ndez, P., Vazza, F., Br[ü]{}ggen, M., et al. 2019, arXiv e-prints , arXiv:1903.11052. Donnert, J., Dolag, K., Brunetti, G., & Cassano, R. 2013, , 429, 3564 Donnert, J., Vazza, F., Br[ü]{}ggen, M., & ZuHone, J. 2018, , 214, 122 Dubey, A., et al. 2009, Parallel Comput., 35, 512 Dubois, Y., & Teyssier, R. 2008, A&A, 482, L13 Eddington, A. S. 1916, , 76, 572 Egan, H., O’Shea, B. W., Hallman, E., et al. 2016, arXiv e-prints , arXiv:1601.05083. Evans, C. R., & Hawley, J. F. 1988, , 332, 659 Feretti, L., Dallacasa, D., Giovannini, G., & Tagliani, A. 1995, A&A, 302, 680 Feretti, L., Dallacasa, D., Govoni, F., et al. 1999, A&A, 344, 472 Feretti, L., Fusco-Femiano, R., Giovannini, G., & Govoni, F. 2001, A&A, 373, 106 Feretti, L., Giovannini, G., Govoni, F., Murgia, M. 2012, A&A Rev., 20, 54 Ferrari, C., Govoni, F., Schindler, S., Bykov, A. M., & Rephaeli, Y. 2008, , 134, 93 Fryxell, B., et al. 2000, ApJS, 131, 273 Giacintucci, S., Markevitch, M., Brunetti, G., Cassano, R., & Venturi, T. 2011, A&A, 525, L10 Gitti, M., Ferrari, C., Domainko, W., Feretti, L., & Schindler, S. 2007, A&A, 470, L25 Govoni, F., En[ß]{}lin, T. A., Feretti, L., & Giovannini, G. 2001, A&A, 369, 441 Govoni, F., Murgia, M., Feretti, L., et al. 2006, A&A, 460, 425 Govoni, F., Murgia, M., Markevitch, M., et al. 2009, A&A, 499, 371 Guidetti, D.,Murgia, M., Govoni, F., et al. 2008, A&A, 483, 699 Hunter, J. D. 2007, Computing In Science & Engineering, 9, 90-95 Kazantzidis, S., Magorrian, J., & Moore, B. 2004, , 601, 37 Lage, C., & Farrar, G. 2014, , 787, 144. Lee, D., & Deane, A. E. 2009, Journal of Computational Physics, 228, 952 Marinacci, F., Vogelsberger, M., Mocz, P., & Pakmor, R. 2015, , 453, 3999 Marinacci, F., Vogelsberger, M., Pakmor, R., et al. 2018, , 480, 5113 Miniati F., Beresnyak A., 2015, Nature, 523, 59 Mitchell, N. L., McCarthy, I. G., Bower, R. G., Theuns, T., & Crain, R. A. 2009, , 395, 180 Murgia, M., Govoni, F., Feretti, L., et al. 2004, A&A, 424, 429 Narayan, R., & Medvedev, M. V. 2001, ApJ, 562, 2, L129-L132 Navarro, J. F., Frenk, C. S., & White, S. D. M. 1997, ApJ, 490, 493 Parrish, I. J., & Stone, J. M. 2005, , 633, 334 Parrish, I. J., & Stone, J. M. 2007, , 664, 135 Parrish, I. J., & Quataert, E. 2008, ApJ, 677, L9 Parrish, I. J., Quataert, E., & Sharma, P. 2009, , 703, 96 Parrish, I. J., Quataert, E., & Sharma, P. 2010, , 712, L194 Perley, R. A., & Taylor, G. B. 1991, AJ, 101, 1623 Poole, G. B., Fardal, M. A., Babul, A., et al. 2006, , 373, 881 Quataert, E. 2008, ApJ, 673, 758 Roberg-Clark, G. T., Drake, J. F., Reynolds, C. S., Swisdak, M. 2016, , 830, 1 Ricker, P. M., & Sarazin, C. L. 2001, , 561, 621 Ricker, P.M. 2008, ApJS, 176, 293 Roettiger, K., Stone, J. M., & Burns, J. O. 1999, , 518, 594 Ruszkowski, M., En[ß]{}lin, T. A., Br[ü]{}ggen, M., Heinz, S., & Pfrommer, C. 2007, , 378, 662 Ruszkowski, M., & Oh, S. P. 2010, , 713, 1332 Sarazin, C. L. 1988, X-ray Emission from Clusters of Galaxies (Cambridge Astrophysics Series 11; Cambridge: Cambridge Univ. Press) Subramanian K., Shukurov A., Haugen N. E. L., 2006, MNRAS, 366, 1437 Suzuki, K., Ogawa, T., Matsumoto, Y., & Matsumoto, R. 2013, , 768, 175 Takizawa, M. 2008, , 687, 951 Taylor, G. B., & Perley, R. A. 1993, ApJ, 416, 554 Taylor, G. B., Fabian, A. C., & Allen, S.W. 2002, MNRAS, 334, 769 Taylor, G. B., Gugliucci, N. E., Fabian, A. C., et al. 2006, MNRAS, 368, 1500 Taylor, G. B., Fabian, A. C., Gentile, G., et al. 2007, MNRAS, 382, 67 Turk, M. J., et al. 2011, ApJS, 192, 9 Vazza, F., Br[ü]{}ggen, M., Gheller, C., et al. 2014, , 445, 3706. Vazza, F., Brunetti, G., Br[ü]{}ggen, M., et al. 2018, , 474, 1672. Venturi, T., Giacintucci, S., Brunetti, G., et al. 2007, A&A, 463, 937 Vijayaraghavan R., Sarazin C., 2017, ApJ, 841, 22 Vijayaraghavan, R., & Ricker, P. M. 2017, , 841, 38 Vikhlinin, A., et al. 2006, ApJ, 640, 691-709 Vikhlinin, A., et al. 2009a, , 692, 1033 Vitvitska, M., Klypin, A. A., Kravtsov, A. V., Wechsler, R. H., Primack, J. R., & Bullock, J. S. 2002, , 581, 799 Vogt, C., & En[ß]{}lin, T. A. 2003, A&A, 412, 373 Vogt, C., & En[ß]{}lin, T. A. 2005, A&A, 434, 67 Voit, G. M., Kay, S. T., Bryan, G. L. 2005, MNRAS, 364, 909 Walker, S. A., Hlavacek-Larrondo, J., Gendron-Marsolais, M., et al. 2017, , 468, 2506 Walker, S. A., ZuHone, J., Fabian, A., & Sanders, J. 2018, Nature Astronomy, 2, 292 Wang, Q. H. S., Markevitch, M., & Giacintucci, S. 2016, , 833, 99 Wang, Q. H. S., & Markevitch, M. 2018, , 868, 45 Werner, N., ZuHone, J. A., Zhuravleva, I., et al. 2016, , 455, 846 Xu, H., Li, H., Collins, D. C., et al. 2009, , 698, L14. Xu, H., Li, H., Collins, D. C., et al. 2010, , 725, 2152. Xu, H., Li, H., Collins, D. C., et al. 2011, , 739, 77. Zhuravleva, I., Churazov, E., Ar[é]{}valo, P., et al. 2015, , 450, 4184 Zhuravleva, I., Allen, S. W., Mantz, A., & Werner, N. 2018, , 865, 53 ZuHone, J.A., 2011, ApJ, 728, 54 (Z11) ZuHone, J. A., Markevitch, M., & Lee, D. 2011, , 743, 16 (ZML11) ZuHone, J. A., Markevitch, M., Ruszkowski, M., & Lee, D. 2013, , 762, 69 ZuHone, J. A., Markevitch, M., Brunetti, G., & Giacintucci, S. 2013, , 762, 78 ZuHone, J. A., Kunz, M. W., Markevitch, M., Stone, J. M., & Biffi, V. 2015, , 798, 90 Zwicky, F. 1937, ApJ, 86, 217
Supplemental Slice Plots
========================
We include additional slices of temperature and DM density in Figures \[fig:1to1\_b0\_kT\]-\[fig:1to10\_b1\_all\_cic\], which correspond to the same simulations and epochs as Figures \[fig:1to1\_b0\_density\]-\[fig:1to10\_b1\_magfield\].
\[th!\]
{width="0.9\linewidth"}
\[h!\]
{width="0.9\linewidth"}
\[t!\]
{width="0.9\linewidth"}
\[t!\]
{width="0.9\linewidth"}
\[t!\]
{width="0.9\linewidth"}
\[t!\]
{width="0.9\linewidth"}
\[t!\]
{width="0.9\linewidth"}
\[t!\]
{width="0.9\linewidth"}
\[t!\]
{width="0.9\linewidth"}
\[t!\]
{width="0.9\linewidth"}
\[t!\]
{width="0.9\linewidth"}
\[t!\]
{width="0.9\linewidth"}
\[t!\]
{width="0.9\linewidth"}
\[t!\]
{width="0.9\linewidth"}
\[t!\]
{width="0.9\linewidth"}
\[t!\]
{width="0.9\linewidth"}
\[t!\]
{width="0.9\linewidth"}
\[t!\]
{width="0.9\linewidth"}
[^1]: <http://matplotlib.org>
[^2]: <http://www.numpy.org>
[^3]: <http://www.scipy.org/scipylib/>
[^4]: <http://yt-project.org>
|
---
author:
- 'Héctor Martínez^\*^'
- Joaquín Tárraga^^
- Ignacio Medina^^
- 'Sergio Barrachina^\*^'
- 'Maribel Castillo^\*^'
- Joaquín Dopazo^^
- 'Enrique S. Quintana-Ortí^\*^'
bibliography:
- 'enrique.bib'
- 'hector.bib'
title: |
Concurrent and Accurate RNA Sequencing\
on Multicore Platforms
---
{width="3cm"}
[Technical Report ICC 2013-03-01]{}
****
^\*^ Dpto. de Ingeniería y Ciencia de los Computadores\
Universidad Jaume I\
12006 - Castellón\
Spain\
[{martineh,barrachi,castillo,quintana}@uji.es]{}
^^ Computational Genomics Institute\
Centro de Investigación Príncipe Felipe\
46012 - Valencia\
Spain\
[{jtarraga,imedina,jdopazo}@cipf.es]{}
date
Acknowledgments {#acknowledgments .unnumbered}
===============
The researchers from the Universidad Jaume I (UJI) were supported by project TIN2011-23283 and FEDER.
|
---
author:
- |
[Abbas Razaghpanah]{}\
[Stony Brook University]{}
- |
[Anke Li]{}\
[Stony Brook University]{}
- |
[Arturo Filast[ò]{}]{}\
[The Tor Project]{}
- |
[Rishab Nithyanand]{}\
[Stony Brook University]{}
- |
[Vasilis Ververis]{}\
[Humboldt University Berlin]{}
- |
[Will Scott]{}\
[University of Washington]{}
- |
[Phillipa Gill]{}\
[Stony Brook University]{}
- '\[Paper \#, 6 pages (Max: 6 pages)\]'
bibliography:
- 'oni.bib'
- 'main.bib'
title: Exploring the Design Space of Longitudinal Censorship Measurement Platforms
---
|
---
abstract: 'With the rapid growth of the express industry, intelligent warehouses that employ autonomous robots for carrying parcels have been widely used to handle the vast express volume. For such warehouses, the warehouse layout design plays a key role in improving the transportation efficiency. However, this work is still done by human experts, which is expensive and leads to suboptimal results. In this paper, we aim to automate the warehouse layout designing process. We propose a two-layer evolutionary algorithm to efficiently explore the warehouse layout space, where an auxiliary objective fitness approximation model is introduced to predict the outcome of the designed warehouse layout and a two-layer population structure is proposed to incorporate the approximation model into the ordinary evolution framework. Empirical experiments show that our method can efficiently design effective warehouse layouts that outperform both heuristic-designed and vanilla evolution-designed warehouse layouts.'
author:
- Haifeng Zhang
- Zilong Guo
- Han Cai
- Chris Wang
- Weinan Zhang
- Yong Yu
- Wenxin Li
- Jun Wang
bibliography:
- 'storage-env.bib'
title: Layout Design for Intelligent Warehouse by Evolution with Fitness Approximation
---
Introduction
============
The global express delivery industry has been a trillion market, serving the people’s daily life around the world. In 2017, the industry revenue is 248 billion USD [@ibis2018global] and in China, particularly, the annual gross express volume has surpassed 30 billion USD since 2016 [@fan2017considerable]. During the recent two years, a new type of shipping warehouses, with intelligent robots sorting thousands of parcels per hour, emerged [@ChinaDaily2017robots]. As shown in Figure \[fig:real-warehouse\] and \[fig:warehouse-env\], autonomous robots carry parcels across the warehouse and unload the parcels into the target holes which connect to the vehicles heading to the target destinations. The layout of the warehouse, i.e. the matching of the holes and the target destinations, is usually designed by human experts. It can be challenging and also likely to be suboptimal, especially when the number of holes is large as shown in Figure \[fig:warehouse-env\]. Moreover, the demand of such warehouse layout design is not one-off, since the distribution of the parcel destinations is not fixed and the warehouse layout design should be adaptive to achieve the best performance.
In this paper, we present an evolution-based method for automatically designing warehouse layout. To tackle the efficiency issue arising from time-consuming evaluation of each designed warehouse layout, we consider to train a neural network to predict outcomes of layouts without actually running agents in it, which is known as fitness approximation in the context of evolution [@jin2005comprehensive]. We further propose a novel two-layer population structure to incorporate the prediction model into the evolution framework for improving efficiency, which can be categorised as multiple-deme parallel genetic algorithms[@cantu1998survey]. Particularly, the higher layer consists of layouts that are actually evaluated and occupies a small fraction of the whole population while the lower layer contains layouts whose fitnesses are predicted by the learned model. Compared to existing methods for combining fitness approximation with evolution [@de2004hierarchical; @hong2003acceleration], the proposed two-layer evolutionary algorithm explicitly manages evaluated individuals and predicted individuals separately in two sub-populations and trains the approximation model online using the samples evaluated by the original fitness function. As such, the proposed method incorporates fitness function approximation into the multiple-deme parallel genetic algorithm naturally. Moreover, within an evaluation of a designed warehouse layout, we can observe not only the final outcome but also additional agent trajectories that comprise hidden information about the causes of the outcome. To take advantage of such additional information to improve the quality of the prediction model, we construct an auxiliary objective, i.e. to predict the heatmap of the environment where each individual value is the total number of visits of a point.
Our experiments of designing warehouse layouts demonstrate improved efficiency and better performance compared to both manual design and vanilla evolution-based methods without fitness approximation. Such a two-layer evolution-based environment optimization framework is promising to be applied onto various environment design tasks.
Related Work
============
There are many real-world scenarios that can be regarded as environment design problems, ranging from game-level design with a desired level of difficulty [@togelius2011search], shopping space design for impulsing customer purchase and long stay [@penn2005complexity] to traffic signal control for improving transportation efficiency [@ceylan2004traffic]. In a recent work, [@zhang2018learning] formulates these environment design problems using a reinforcement learning framework. In this paper, we focus on a new environment design scenario, i.e. warehouse layout design, emerging from the rapidly growing express industry.
Traditional warehouse design problems can be categorised to three levels, strategic level, tactical level and operational level [@rouwenhorst2000warehouse]. At the strategic level, long-term decisions are considered, including the size of a warehouse [@roll1989determining] and the selection of component systems [@oser1996design; @keserla1994analysis]. At the tactical level, medium term decisions are made, such as the layout of a conventional warehouse [@bassan1980internal; @berry1968elements]. At the operational level, detailed control policies are studied, e.g. batching [@elsayed1983computerized] and storage policies [@goetschalckx1991optimal]. The problem discussed in this paper is about warehouse layout design, which is at the tactical level traditionally. However, in the era of big data, the layout of warehouse could be adaptive to the changes of the external environment. Specifically, the layout of the warehouse could be redesigned at intervals according to the changing destination distribution of the parcels. Thus, this problem is better to be categorised as a operational level problem.
For solving this problem, we adopt evolutionary algorithms. As getting a guiding signal means evaluating the designed objective in the target task, which would result in unacceptable computational resource requirement for scenarios where evaluation is expensive. To reduce the amount of expensive evaluations on real data needed before a satisfying result can be obtained, some works propose to learn a model to predict the outcome of a designed objective without actually running on real data [@baker2018accelerating; @liu2017progressive]. Similar idea has been explored in the field of evolution and is known as fitness approximation [@jin2005comprehensive]. Due to the inaccuracy of fitness approximation, it is essential to use the approximation model together with the original fitness function [@grierson1993optimal; @ratle1998accelerating]. To incorporate the fitness model into the simulation-based evolutionary algorithms, individual-based [@bull1999model] and generation-based [@ratle1998accelerating] methods are studied. Differently, our approach explicitly manages two sub-populations whose individuals are evaluated by the approximation model and the original fitness function respectively. Similar approaches are known as multiple-deme parallel genetic algorithms [@cantu1998survey]. Our work can be classified as a multiple-deme parallel genetic algorithm with a two-layer sub-population topology to balance exploitation and exploration.
Problem Definition {#sec:problem}
==================
In this section, we formulate the environment design problem and introduce the particular robotic warehouse environment. We fix the agent policy in the robotic warehouse environment and focus on the remaining task, assigning destinations to the holes, which can be viewed as an environment design problem.
Environment Design {#sec:env-design}
------------------
In many scenarios, there are $n$ agents taking actions in a designable environment, such as cars running in a transportation system, consumers shopping in a mall, and so on. Denote the $i^{th}$ agent’s policy as $\pi_i$ and the environment is parametrized as $\mathcal{M}_\theta=\langle S, A, T_\theta, R_\theta, \lambda \rangle$, where $S,A,T_\theta,R_\theta,\lambda$ denote state space, action space, transition function, reward function and reward discount respectively. After the agents play in the environment in an episode, a joint trajectory $H= \langle s_1, a_1, s_2, a_2, ... \rangle$ is produced and a cumulative reward $G_i$ is given to the $i^{th}$ agent, where $s_t$ and $a_t$ denote state and joint action respectively. Moreover, the objective of the environment designer is given as $O(H)$, whose function form can be defined specifically, and the designer intends to design an optimal environment to maximize the expectation of its objective $$\begin{aligned}
\theta^* = {\underset{\theta}{\operatorname{arg}\operatorname{max}}\;} \mathbb{E}[O(H)|\mathcal{M}_\theta; \pi_{1\ldots n}].
\label{eq:original-general-problem}\end{aligned}$$ Note that the randomness of $H$ is derived from the possible randomness of $\pi_i$ when selecting actions.
[0.35]{} ![(a) Real-world robotic warehouse for parcel sorting (screenshot from [@ChinaDaily2017robots]). (b) Robotic warehouse environment. The triangles stand for the sources where parcels emerge. The circles stand for the robots carrying the parcels. The squares stand for the holes for the agents to put into the parcels. The squares are colored according to which destination the parcels coming into will go to. The agents repeatedly take a parcel with a color (destination) from a source to a hole with the same color. The objective is to maximize the total number of the parcels processed by the agents in a fixed period.](figs/warehouse-small "fig:"){height=".98\columnwidth"}
[0.6]{} ![(a) Real-world robotic warehouse for parcel sorting (screenshot from [@ChinaDaily2017robots]). (b) Robotic warehouse environment. The triangles stand for the sources where parcels emerge. The circles stand for the robots carrying the parcels. The squares stand for the holes for the agents to put into the parcels. The squares are colored according to which destination the parcels coming into will go to. The agents repeatedly take a parcel with a color (destination) from a source to a hole with the same color. The objective is to maximize the total number of the parcels processed by the agents in a fixed period.](figs/environment-desc-32 "fig:"){height="0.6\columnwidth"}
Robotic Warehouse Environment
-----------------------------
In this paper, we consider a robotic warehouse environment abstracted from a real-world express system as shown in Figure \[fig:real-warehouse\], where there is a warehouse for sorting parcels from a mixed input stream to separate output streams according to their respective destinations. The sorting process is done by the robots carrying parcels from the input positions (sources) to the appropriate output positions (holes) in the plane warehouse as Figure \[fig:warehouse-env\] illustrates. In order to maximize the efficiency of sorting, we should set the robots’ cooperative pathfinding algorithm and assign the destinations to the holes. In this task, the agents share a common reward $G$ and the environment also takes $G$ as its design objective, i.e. $O(H) = G$. We set $\pi_\phi$ as a joint policy model for the agents. As such, the problem is formulated as $$\begin{aligned}
\theta^*, \phi^* = {\underset{\theta, \phi}{\operatorname{arg}\operatorname{max}}\;} \mathbb{E}[G|\mathcal{M}_\theta, \pi_{\phi}].
\label{eq:general-problem}\end{aligned}$$ For solving Eq. (\[eq:general-problem\]), we should firstly set a sound cooperative pathfinding algorithm $\pi_{\phi^*}$ for the robots. After, we focus on optimizing the environment parameter $\theta$, i.e. optimizing the layout of the warehouse (the assignment of the destinations to the holes) via $$\begin{aligned}
\theta^* = {\underset{\theta}{\operatorname{arg}\operatorname{max}}\;} \mathbb{E}[G|\mathcal{M}_\theta; \pi_{\phi^*}].\end{aligned}$$ Note that the demand of such environment layout design is not one-off. Since the external variables (such as the destination distribution of the parcels) may be changing, the best layout of the warehouse is changing accordingly. Thus, the layout of the warehouse should be redesigned at intervals, which gives a reason to find an efficient layout design approach.
Detailed Environment Description {#sec:experiment-desc}
--------------------------------
The warehouse is abstracted as a grid containing $h \times w$ cells. Among them, $n_s$ cells are sources and $n_h$ cells are holes, whose locations $l_{s}^{1..n_s}, l_{h}^{1..n_h}$ are given. There are $n_r$ robots available to carrying parcels from sources to holes. Each cell is only for one robot to stand.
In each time-step, each robot is able to take a move to an adjacent cell. When an empty robot moves into a source, it loads a new parcel whose destination follows a distribution over $n_d$ destinations (cities) with the proportions $p_1,p_2,...,p_{n_d}$. On the other hand, when a loaded robot moves into a hole with the destination that is as the same as the loading parcel’s, it unloads the parcel into that hole. That is to say, the rates of input and output flows are not restricted in our setting. Parcels are always sufficient when a robot moves into a source.
Our objective is to sort as many parcels as possible in a given time period $T$. We could achieve this objective by designing the layout of the warehouse, i.e. assigning the proper destinations to the holes. Specifically, we should determine the parameter $\theta=\langle \theta_1,\theta_2,...,\theta_{n_h} \rangle$ of the environment $\mathcal{M}_\theta$, where $\theta_i \in \{1..n_d\}$ for $i=1..n_h$. Intuitively, the assignment of the destinations to the holes will affect the robots’ paths and hence the efficiency of the whole warehouse.
The notations defined in this section are listed in Table \[table:notation\].
Problem Complexity
------------------
For the problem defined above, the scale of the layout assignment space is $n_d^{n_h}$, where $n_h$ denotes the number of the holes and $n_d$ denotes the number of the parcel destinations. Since the robot pathfinding algorithm works like a black box to evaluate each layout assignment, it is hard to determine a global optimum without exploring the solution space completely. Thus, this optimization problem is an exponential time problem. Even for a small setting, such as $n_h=20, n_d=5$, the number of the assignments is as large as about $100$ trillion, which is hard to be explored completely.
Robot Pathfinding Algorithms {#sec:problem-ware-design}
----------------------------
In our problem, the robot pathfinding algorithm is fixed. As the robots are quite dense in the real-world warehouse, jam prevention is the key point. We considered two cooperative pathfinding algorithms with jam prevention design. The first one adopts WHCA\* [@silver2005cooperative] as a planner, which searches the shortest path from an origin to a destination for each robot in turn and ensures non-collision. The second algorithm is a greedy one, which guides the robots by a look-up table in each position and reduces conflicts by setting one-way roads in the map as illustrated in Figure \[fig:road\]. We studied these two algorithms and the results showed that the greedy one has a significant advantage on time complexity and a minor disadvantage on performance. Due to the large simulation demand for testing environment parameter, we selected the time-saving greedy algorithm as the agent policy in our experiments. However, the proposed warehouse layout design solution can work with other robot pathfinding algorithm as well.
Solution {#sec:solution}
========
In this section, we first introduce an evolution framework for automatically designing warehouse layout, and then present the auxiliary objective fitness approximation and the two-layer population structure for improving the efficiency.
[0.32]{} {height="0.98\columnwidth"}
[0.32]{} {height="0.98\columnwidth"}
[0.32]{} {height="0.98\columnwidth"}
Evolution with Robot Policy Simulation {#sec:solution-baseline}
--------------------------------------
Under the evolution framework, we maintain a population containing $n$ warehouse layout individuals, i.e. assignments of the destinations to the holes (Figure \[fig:ea-sample\]), and evolve the population for $n_g$ generations. Within each generation, we perform crossover, mutation and selection in order:
- In the **crossover** phase, we randomly select $c$ pairs of samples. For each pair of samples, we splice their holes from two matrices to two lines respectively. Then, we randomly select a common breakpoint for both lines and cross the two lines just like chromosomal crossover. Finally we generate two square matrices by reshaping the two lines.
- In the **mutation** phase, we randomly select $m_1$ samples generated in the crossover phase. For each sample, we randomly select $m_2$ holes and randomly permute their destinations.
- In the **selection** phase, we evaluate the generated samples in the crossover and mutation phases by robot policy simulations, then merge the original and the generated samples. The best $n$ ones are selected for the next generation.
Two-layer Evolutionary Algorithm with Fitness Approximation {#sec:solution-proposed}
-----------------------------------------------------------
In this section, we propose a novel evolutionary algorithm that trains an auxiliary objective fitness function to evaluate a large population for providing promising individuals to a small population evaluated by simulations.
### Auxiliary Objective Fitness Approximation
![An illustration of the process of evaluating an assignment sample $\theta$. First, the latent representation $X$ is learned via shared deep layers. Then based on $X$, separated layers are built to predict heatmap $\hat{I}$ and reward $\hat{G}$ respectively. Two loss functions are calculated based on the difference between the prediction and the simulated results.[]{data-label="fig:aux-obj-network"}](figs/heatmap-g-loss.pdf){width="1\linewidth"}
In practise, the simulation of robots performing in the environment is time-consuming. A promising way to reducing the simulation time is to use an approximation function to compute fitness: $$\begin{aligned}
f_G(\theta) = \hat{G} \approx \mathbb{E}[G|\mathcal{M}_\theta; \pi_{\phi^*}],\end{aligned}$$ where $f_G$ is the fitness approximation function, $\theta$ is a sample of environment parameter and $\hat{G}$ is the predicted fitness of $\theta$, whose learning target is the expectation of the reward $G$.
Moreover, since a simulation generates a trajectory $H$ in addition to the reward $G$, we consider utilizing $H$ to help training fitness function $f_G$. Although $G$ is the exact objective for fitness function to learn, we may extract additional information $I(H)$ from $H$ that helps training the fitness function, under the assumption that $G$ and $I$ are correlated. We set an auxiliary training objective and use a neural network to capture this: $$\begin{aligned}
f(\theta) &= \langle f_I(f_X(\theta)), f_G(f_X(\theta)) \rangle = \langle \hat{I}, \hat{G} \rangle \\ \nonumber
&\approx \langle \mathbb{E}[I(H)|\mathcal{M}_\theta, \pi_{\phi^*}], \mathbb{E}[G|\mathcal{M}_\theta, \pi_{\phi^*}] \rangle,\end{aligned}$$ where $f$ is a neural network consisting of three sub-networks: $f_X$ is the bottom network that captures the common features and outputs $X$; $f_I$ and $f_G$ are the two separate networks on the top of $X$ that predict $\hat{I}$ and $\hat{G}$ respectively.
In the robotic warehouse layout design problem, $\theta$ represents the assignment of the destinations to the holes and $H$ represents the movements of the robots. Furthermore, we define $I$ as the heatmap of the movements as Figure \[fig:heatmap\] shows. Intuitively, the distribution of busy areas should be correlated with the efficiency of sorting and the reward. The process of learning the fitness function in the warehouse layout problem is illustrated in Figure \[fig:aux-obj-network\].
Since obtaining simulation samples is time-consuming, we train the fitness model online. Specifically, the fitness model is trained with the samples simulated along the process of the evolutionary algorithm. There is no pre-training in our approach.
### Two-layer Population
![The process of the two-layer population evolutionary algorithm in a single generation. The yellow and grey squares stand for the populations who have been (or will be) evaluated by simulation and fitness model respectively.[]{data-label="fig:two-layer-ea"}](figs/two-layer-ea.pdf){width="1\linewidth"}
The fitness model provides a less accurate but more speedy evaluation than the simulation. These property indicates that the simulation is better to find the local optimum exactly and the fitness model is better to explore the global space speedily. For the standard simulation-based evolution, mutation rate is usually set small enough to ensure convergence within an acceptable time, thus the search space is relatively local. Therefore, we consider incorporating the fitness model into the standard simulation-based evolution as an additional part for exploring the global space.
Specifically, we maintain two sub-populations. The first one is of the same size as the population set in the standard simulation-based evolution. Also, the individuals in the first sub-population are evaluated by simulations. The second sub-population is multiple times larger than the first one and the samples in it are evaluated by the fitness model. We view the second sub-population as a candidate population whose top individuals have a chance of joining the first sub-population. On the other hand, the bottom individuals in the first sub-population may be moved to the second sub-population. We name the first-layer sub-population noble and the second civilian. Noble population and civilian population evolve separately while keeping a channel for migration.
noble population $N$, civilian population $C$, untrained fitness model $f$, empty simulation sample set $S$ generate $N_1$ from $N$ by crossover and mutation; generate $C_1$ from $C$ by crossover and mutation; rank $C \cup C_1$ by $f$ to generate top population $C_2$, middle population $C_3$ and bottom population $C_4$; evaluate $N_1$ and $C_2$ by simulation and add the results to $S$; rank $N \cup N_1 \cup C_2$ by the simulation score to generate top population $N_2$ and bottom population $N_3$; generate random population $R$ and discard $C_4$; pass $N_2$ to the next generation as $N$; pass $N_3 \cup C_3 \cup R$ to the next generation as $C$; update $f$ using $S$.
In detail, the two-layer population evolves as Figure \[fig:two-layer-ea\] and Algorithm \[algo:tlea\] show. In general, $N$ and $C$ maintain individuals evaluated by the simulation and the fitness model respectively. In each generation, migration takes place. Specifically, $C_2$ from the civilian layer go up to the noble layer and $N_3$ from the noble layer go down to the civilian layer. In addition, the civilian layer discards the worst population $C_4$ and absorbs randomly generated population $R$.
There are $9$ parameters related to the proposed two-layer evolutionary algorithm. They are noble population number $|N|$, civilian population number $|C|$, crossover rate $c_N, c_C$, mutation rate $m_N, m_C$, $|C_2|$ for the number of civilian individuals migrate to the noble layer, $|R|$ for the number of the randomly generated individuals, and $n_u$ for the number of model updates in each generation. Other variables can be determined by these parameters. In each generation, $|N_1|+|C_2|$ simulations, $n_u$ model updates and $|C_1|$ model predictions are performed. Since the time cost of training the network and use it to predict is negligible compared to the simulations (see Table \[table:time-cost\]), the time complexity of the two-layer evolutionary algorithm for $n_g$ generations is $O(n_g(|N_1|+|C_2|))$.
Experiment {#sec:experiment}
==========
We set up a virtual intelligent warehouse environment based on real-world settings and test our proposed approach comparing to the baselines. Our experiment is repeatable and the source code is provided in the supplementary.
Experiment Settings
-------------------
*Environment.* We test our proposed approach in $20 \times 20$ maps. The positions of the sources and holes are set as the real-world scenarios. The detailed parameters are given in Table \[table:warehouse-parameter\]. The destination distributions are set according to long-tail functions to reflect reality. In our experiments, the reward is defined as the sum of parcel loading times and unloading times (roughly two times as the number of parcels processed).
$h$ $w$ $n_s$ $n_h$ $n_r$ $n_d$ $T$ $p_{1..n_d}$
----- ----- ------- ------- ------- ------- ------ -----------------------------------
20 20 12 20 60 5 1000 $0.367, 0.267, 0.2, 0.133, 0.033$
: Environment parameter settings.[]{data-label="table:warehouse-parameter"}
*Robots.* As introduced, we adopt a greedy algorithm as the cooperative pathfiding algorithm for the robots. Firstly, we set one-way roads in the map as Figure \[fig:road\] shows to avoid opposite-directional conflicts, while right-angled conflicts are avoided by setting priority. On the one-way roads, the robots decide moves by a look-up table containing $h \times w \times (n_s + n_h)$ records, each of which indicates the first step towards a particular source or hole from a particular cell.
*Baselines.* We test $5$ baselines to compare with our proposed two-layer evolutionary algorithm (**TLEA**). **Random**: The holes are assigned with random destinations uniformly. **Heuristic**: Destinations select holes in turns according to their proportions. For example, if $10\%$ parcels are going to destination A, then A select $10\%$ of the holes. This process start from the destination with the most proportion. Each destination greedily selects each hole that minimizes the sum of the average distance from the sources to the selected holes. **Simu**: The evolutionary algorithm with simulations as introduced in the Solution section. **SimuInd**: An implementation of the individual-based evolution control algorithm [@bull1999model]. This approach maintains a single large population for evolution whose individuals are evaluated by the fitness model. In each generation, the best individuals evaluated by the fitness model are evaluated by the simulation once again. The fitness model is trained online with the samples produced by the simulations. **SimuGen**: An implementation of the generation-based evolution control algorithm [@ratle1998accelerating]. This approach also maintains a single large population as SimuInd. The difference is that SimuGen uses the simulations intensively in a generation and uses the fitness model in the next several generations.
*Hyper-parameters.* To ensure fairness, for Simu, SimuInd, SimuGen and TLEA, the number of generation is set as $60$ and the number of simulations in each generation is set as $200$. The model update and prediction times are also fixed as $5000$ and $10000$ respectively for SimuInd, SimuGen and TLEA. The population of Simu is $100$; in each generation $200$ individuals are generated by crossover; $50$ of them are mutated. For SimuInd and SimuGen, the populations are $5000$; $10000$ are generated by crossover in each generation; $2500$ of them are mutated. For the TLEA, $|N|,|C|,c_N,c_C,m_N,m_C,|C_2|,|R|,n_u$ are set to be $100, 5000, 1, 1, 0.25, 0.25, 50, 2500, 5000$ respectively.
*Fitness model.* Our network is composed of three sub-networks $f_X$, $f_I$, $f_G$. The output of $f_X$ is used for the input of $f_I$ and $f_G$. $f_X$ has two fully connected layers whose output is a vector that can be reshaped to match the size of map. Then, a 2D transposed convolution layer follows.$f_I$ has one transposed convolution layer to generate the heat map. And $f_G$ contains three fully connected layers to predict the reward. All the layers except the output layers have a ReLU activation function. The loss functions for the two outputs are set to be MSE. The first two fully connected layers have 128, 400 units respectively. The first 2D transposed convolution layer have 16 filters. And the second one has one filter. The three fully connected layers for reward prediction have 256, 128 and 1 unit respectively.
*Hardware.* We use two computers with an Intel core i7-4790k and an Intel core i7-6900k respectively. The one with 4790k also has an extra Nvidia Titan X GPU.
Random Heuristic Simu SimuInd SimuGen TLEA
--------- -------- ----------- -------------------- ---------------------- -------------------- ----------
Reward 4757 5386 5572 5605 5499 **5646**
T Score - - 5.8778 2.7708 5.8782 -
P-Value - - $7 \times 10^{-6}$ $6.3 \times 10^{-4}$ $7 \times 10^{-6}$ -
: Performance of Random, Heuristic, Simu, SimuInd, SimuGen and TLEA. The algorithms are repeatedly performed for $10$ runs. The reward samples pass the Shapiro-Wilk test to be normal. T-tests are performed for TLEA against Simu, SimuInd and SimuGen. The statistical results show that the superiority of TLEA is significant. []{data-label="table:exp-result"}
[0.32]{} ![Environments designed by Random, Heuristic, Simu and TLEA.[]{data-label="fig:result"}](figs/result_random_20.pdf "fig:"){height="1\columnwidth"}
[0.32]{} ![Environments designed by Random, Heuristic, Simu and TLEA.[]{data-label="fig:result"}](figs/result_heuristic_20.pdf "fig:"){height="1\columnwidth"}
[0.32]{} ![Environments designed by Random, Heuristic, Simu and TLEA.[]{data-label="fig:result"}](figs/result_simu_20.pdf "fig:"){height="1\columnwidth"}
[0.32]{} ![Environments designed by Random, Heuristic, Simu and TLEA.[]{data-label="fig:result"}](figs/result_simuind_20.pdf "fig:"){height="1\columnwidth"}
[0.32]{} ![Environments designed by Random, Heuristic, Simu and TLEA.[]{data-label="fig:result"}](figs/result_simugen_20.pdf "fig:"){height="1\columnwidth"}
[0.32]{} ![Environments designed by Random, Heuristic, Simu and TLEA.[]{data-label="fig:result"}](figs/result_tlea_20.pdf "fig:"){height="1\columnwidth"}
Results
-------
We perform the baselines and TLEA. The results are shown in Table \[table:exp-result\]. We find Heuristic is fairly high compared to Random but is inferior to evolutionary algorithms. Moreover, TLEA outperforms all the baselines.
Figure \[fig:result\] shows the layouts designed by the baselines and TLEA with the heatmaps. We can see that the tracks of the robots running in the maps of TLEA are better balanced, indicating that there are less traffic jams.
Figure \[fig:converge\] shows the learning curves. Since SimuInd and SimuGen mix the individuals evaluated by the simulation and the fitness model, their current best individuals may be the over-estimated ones by the inaccurate fitness model, which may lead to discarding the real best individuals. TLEA solves this problem by separating the two populations and ensure that the real best individual is always kept in the noble population.
In addition, TLEA and Simu are more stable than SimuInd and SimuGen, because the temporary best individual may be evaluated by the fitness model in SimuInd and SimuGen, which may be corrected by the simulation in later generations. The slight fluctuations of Simu and TLEA are caused by the variance of the simulations, which results in that the best samples can be over-estimated (which is much slighter than the fitness model) and would be averaged by extra simulations in later generations.
Discussions {#sec:exp-analysis}
-----------
*Time cost.* The time costs of the tested algorithms are listed in Table \[table:time-cost\]. It shows that the time cost proportion of the fitness model is less than $5\%$. In out experiment, we just ignore the time difference between Simu and other algorithms.
Simulation Model Update Model Predicting Time
--------- ------------ -------------- ------------------ ---------
Simu $12k$ $0$ $0$ $8.73h$
SimuInd $12k$ $300k$ $600k$ $9.11h$
SimuGen $12k$ $300k$ $600k$ $9.11h$
TLEA $12k$ $300k$ $600k$ $9.11h$
: Time cost comparison. The average time costs for simulation, model update and model predicting are $2.62$s, $2.42$ms and $1.06$ms respectively. The number of generations is $60$ for all the algorithms.[]{data-label="table:time-cost"}
*Effectiveness of heatmap.* We evaluate randomly generated samples by the simulations and use them to train the fitness functions with and without heatmaps as auxiliary objective. We compare MSE and Pearson Correlation of them in Table \[table:exp-heatmap\], which shows that heatmap provides significant improvement to the fitness function.
------- ------ --------------- ------- ----------------
w/o Heatmap w/o Heatmap
5000 4.36 2.89(-33.72%) 0.277 0.519(+87.36%)
10000 2.75 1.59(-42.18%) 0.405 0.687(+69.63%)
20000 1.67 0.69(-58.68%) 0.766 0.908(+18.54%)
------- ------ --------------- ------- ----------------
: Comparison of fitness functions with and without heatmap.[]{data-label="table:exp-heatmap"}
*Simulation allocation.* Since simulations are scarce resources when running evolutionary algorithm, the allocation of simulations between the noble layer and the civilian layer is important. Moreover, it also determines the migration rate between the two layers. We test different $\frac{|N_1|}{|N_1|+|C_2|}$, the ratio of simulations allocated to the noble layer, and find that $0.75$ is a proper setting (see Table \[table:simulation-proportion\]), which means three fourths simulations are allocated to ensure the accuracy of the noble layer and one fourth simulations are allocated to give chances to the civilian layer.
Noble Proportion 0.25 0.5 0.75 1
------------------ ------ ------ ---------- ------
Reward 5629 5634 **5646** 5581
: Simulation allocation analysis.[]{data-label="table:simulation-proportion"}
*Impact of civilian population.* We are interested in how much contribution has the civilian population made to the evolution of the noble population. We calculate a number named purity that measures how much the evolved noble population inherits from the initial noble population. As Figure \[fig:purity\] shows, the purity of the noble population declines rapidly along with the increasing of the reward (fitness). Finally, civilian population contributes more than $70$ percent to the noble population.
[0.49]{} {height="0.7\columnwidth"}
[0.49]{} {height="0.7\columnwidth"}
Conclusion
==========
In this paper, we study the problem of automatic warehouse layout design. The proposed two-layer evolutionary algorithm takes advantage of a fitness approximation model, augmented with an auxiliary objective of predicting the heatmap. Our approach enhances the exploration of the evolutionary algorithm with the help of the fitness model. The experiments demonstrates the superiority of our approach over the heuristic and the traditional evolution-based methods. For future work, we would apply the proposed two-layer evolutionary algorithm to other environment design scenarios, such as shopping mall design, game design and traffic light control.
|
---
abstract: '**The iron-based high temperature superconductors exhibit a rich phase diagram reflecting a complex interplay between spin, lattice, and orbital degrees of freedom [@doping122review; @FernandesNematicPnictides; @Fernandes2012; @FernandesPRLnematicT1]. The nematic state observed in many of these compounds epitomizes this complexity, by entangling a real-space anisotropy in the spin fluctuation spectrum with ferro-orbital order and an orthorhombic lattice distortion [@TanatarTensileStressPRB; @StrainedPnictidesNS2014science; @StrainBa122neutronsPRB2015]. A more subtle and much less explored facet of the interplay between these degrees of freedom arises from the sizable spin-orbit coupling present in these systems, which translates anisotropies in real space into anisotropies in spin space. Here, we present a new technique enabling nuclear magnetic resonance under precise tunable strain control, which reveals that upon application of a tetragonal symmetry-breaking strain field, the magnetic fluctuation spectrum in the paramagnetic phase of BaFe$_{2}$As$_{2}$ also acquires an anisotropic response in spin-space. Our results unveil a hitherto uncharted internal spin structure of the nematic order parameter, indicating that similar to liquid crystals, electronic nematic materials may offer a novel route to magneto-mechanical control.**'
author:
- 'T. Kissikov'
- 'R. Sarkar'
- 'M. Lawson'
- 'B. T. Bush'
- 'E. I. Timmons'
- 'M. A. Tanatar'
- 'R. Prozorov'
- 'S. L. Bud’ko'
- 'P. C. Canfield'
- 'R. M. Fernandes'
- 'N. J. Curro'
bibliography:
- 'NMRstrainBibliography.bib'
title: 'Uniaxial strain control of spin-polarization in multicomponent nematic order of BaFe$_{2}$As$_{2}$'
---
In the absence of external strain, BaFe$_{2}$As$_{2}$ undergoes a weakly first-order antiferromagnetic phase transition at $T_{N}=135$K, accompanied by an orthorhombic structural distortion that breaks the tetragonal symmetry of the unit cell in the paramagnetic phase. The relatively small orthorhombic lattice distortion ($\sim0.3\%$) [@TanatarTensileStressPRB] is driven by a nematic instability [@FradkinNematicReview], whose electronic origin is manifested by the large in-plane resistivity anisotropy ($\sim100\%$) [@TanatarDetwin; @IronArsenideDetwinnedFisherScience2010]. Despite being simultaneous in BaFe$_{2}$As$_{2}$, the nematic and antiferromagnetic transition temperatures, $T_{s}$ and $T_{N}$, split upon doping, giving rise to a regime with long-range nematic order but no antiferromagnetic order, since $T_{N}<T_{s}$ [@doping122review; @FernandesSchmalianNatPhys2014].
![\[fig:multinematic\] Spin fluctuations in momentum space (left) and in real space (right) and polarization directions of the Fe spins for the three nematic components, $\varphi_{xy}$ (a,b), $\varphi_{yx}$ (c,d), and $\varphi_{zz}$ (e,f). The red arrows correspond to the magnetic ordering vector $\mathbf{Q}_{1}=(\pi,0)$ and the blue arrows correspond to $\mathbf{Q}_{2}=(0,\pi)$. The black spheres are the Fe sites, the green sphere is the As site, and the green arrows indicate the direction of the hyperfine field.](MultiComponentNematic4){width="1\linewidth"}
The close relationship between nematicity and the magnetic degrees of freedom can be seen directly from the stripe-like nature of the antiferromagnetic state, which orders with one of two possible wave-vectors related by a $90^{\circ}$ rotation: $\mathbf{Q}_{1}=(\pi,0)$ (corresponding to spins parallel along the $y$ axis and anti-parallel along $x$) and $\mathbf{Q}_{2}=(0,\pi)$ (corresponding to spins parallel along $x$ and anti-parallel along $y$). Below $T_{N}$ nearest neighbor spins are parallel or antiparallel depending on whether they are connected by a short or long bond, however above $T_{N}$ but below $T_{s}$ the magnetic fluctuations centered around $\mathbf{Q}_{1}$ become weaker or stronger than those centered around $\mathbf{Q}_{2}$, depending on whether the $b$ axis is parallel or perpendicular to $\mathbf{Q}_{1}$, respectively. Mathematically, this allows one to define the nematic order parameter $\bar{\varphi}$ in terms of the (spin unpolarized) magnetic susceptibility $\chi\left(\mathbf{q}\right)$ according to $\bar{\varphi}\equiv\chi^{-1}(\mathbf{Q}_{2})-\chi^{-1}(\mathbf{Q}_{1})$ [@FernandesNematicPnictides]. Such an interplay between nematic and spin degrees of freedom has been indeed observed by neutron scattering [@DaiRMP2015; @StrainBa122neutronsPRB2015; @StrainedPnictidesNS2014science; @PengchangPRB2015] and nuclear magnetic resonance (NMR) [@DioguardiNematicGlass2015; @NMRnematicStrainBa122PRB2016; @PhysRevB.89.214511] experiments in detwinned BaFe$_{2}$As$_{2}$ crystals.
However, orbital degrees of freedom also participate actively in the nematic phase. This leads to the well known effect that tetragonal symmetry-breaking is also manifested by a ferro-orbital polarization that makes the occupation of the Fe $d_{xz}$ orbitals different than the occupation of the Fe $d_{yz}$ orbitals. A less explored effect emerges from the relatively sizable spin-orbit coupling (SOC), which converts anisotropies in real space into anisotropies in spin space. On one hand, SOC enforces the spins to point along the ordering vector direction below $T_{N}$. On the other hand, SOC leads to different magnitudes of the diagonal spin susceptibility components, $\chi_{\alpha\alpha}\left(\mathbf{q}\right)$ with $\alpha=(x,y,z)$, in the nematic temperature regime, $T_{N}<T<T_{s}$. As a result, the nematic order parameter naturally acquires an internal spin structure, since generically one must define $\varphi_{\alpha\beta}=\chi_{\alpha\alpha}^{-1}(\mathbf{Q}_{2})-\chi_{\beta\beta}^{-1}(\mathbf{Q}_{1})$. Clearly, the nematic order parameter $\bar{\varphi}$ defined above can be understood as an average over all possible polarizations, $\bar{\varphi}=\frac{1}{9}\sum\limits _{\alpha\beta}\varphi_{\alpha\beta}$. The space-group symmetry of the iron pnictides enforces many of these combinations to vanish, yielding only three non-zero independent components: $\varphi_{xy}$, $\varphi_{yx}$, and $\varphi_{zz}$. The physical meaning of each component is depicted in Fig. \[fig:multinematic\]; for instance, $\varphi_{xy}$ is a measure of the asymmetry between spin fluctuations peaked at $\mathbf{Q}_{1}$ and polarized along the $x$ axis, and spin fluctuations peaked at $\mathbf{Q}_{2}$ and polarized along the $y$ axis.
![\[fig:cartoon\] (a) Crystal structure of BaFe$_{2}$As$_{2}$, with Ba (green), Fe (blue) and As (magenta) sites shown. Lower panel shows the Fe-As plane in the tetragonal phase, with arrows indicating the unit cell axes of the orthorhombic phase ($\mathbf{a}~||~(110)_{tet}$, $\mathbf{b}~||~(1\overline{1}0)_{tet}$). (b,c) Orientation of the magnetic field with respect to the coil ($H_{1}$) and strain axis for $\mathbf{H}_{0}\perp\mathbf{c}$ (b) and $\mathbf{H}_{0}~||~\mathbf{c}$ (c). For positive (tensile) strain $\mathbf{H}_{0}$ is parallel to $\mathbf{b}$, whereas for negative (compressive) strain $\mathbf{H}_{0}$ is along $\mathbf{a}$.](cartoon5){width="\linewidth"}
Elucidating the hitherto unkown spin structure of the nematic order parameter is fundamental to shed light on the intricate interplay between orbital, spin, and lattice degrees of freedom, which are ultimately responsible for the superconducting instability of the system. In this paper, we perform NMR spin-lattice relaxation measurements to probe the anisotropy of the spin fluctuations under fixed strain in the paramagnetic phase of BaFe$_{2}$As$_{2}$. The role of the applied uniaxial strain is to provide a small tetragonal symmetry-breaking field, akin to externally applied magnetic fields in ferromagnets. In contrast to previous works, here we probe the magnetic fluctuations anisotropy both in real space and in spin space more specifically, we determine each of the nematic susceptibilities associated with the three nematic components $\varphi_{xy}$, $\varphi_{yx}$, and $\varphi_{zz}$. This is possible because the magnetic fluctuations associated with each spin polarization pattern generate very different types of fluctuating local fields experienced by the $^{75}$As nuclear spin ($I=3/2$), which couples to the four nearest neighbor Fe spins via a transferred hyperfine interaction (see Fig. \[fig:multinematic\]) [@T1formfactorsArsenides]. Our main result is that the three nematic components respond differently to external strain, i.e. nematic order induces not only real-space anisotropy, but also affects the spin-space anisotropy. In particular, we find that the out-of-plane spin fluctuations centered at $\mathbf{Q}\parallel\hat{a}$ are more strongly enhanced by the strain, as compared to the spin fluctuations polarized along the longer in-plane axis. This raises the interesting possibility of reversing the spin polarization of the system from in-plane to out-of-plane by applying a sufficiently strong in-plane strain. More broadly, our results thus opens a new avenue toward magneto-mechanical manipulation of strongly correlated systems that display nematic order.
Key to this study is our ability to control precisely the uniaxial strain applied in the sample, which is achieved by integrating a novel piezoelectric strain cell with an NMR probe. This new device is based upon a design used previously to investigate the superconducting transition temperature of Sr$_{2}$RuO$_{4}$ [@Hicks2014; @Sr2RuO4strainScience2014], and can achieve both positive and negative strains with large strain homogeneity. This device differs from the horseshoe-clamp [@TanatarDetwin] used previously for NMR [@NMRnematicStrainBa122PRB2016], and offers superior control over the sample alignment and the level of strain applied.
{width="0.6\linewidth"} {width="0.39\linewidth"}
Single crystals of BaFe$_{2}$As$_{2}$ were cut along the tetragonal (110) direction and mounted in the cryogenic strain cell with field oriented both parallel and perpendicular to the crystallographic $c$-axis, as shown in Fig. \[fig:cartoon\]. The strain cell contains two sets of piezoelectric stacks, one inner and two outer. Because the sample is freely suspended between the piezoelectric stacks rather than glued down over a portion of the stack, the full displacement of each stack is transferred to the sample. As a result the device is able to achieve displacements of $\pm6\mu$m at room temperature and $\pm3\mu$m at 4K, corresponding to strains of the order of $10^{-3}$ in this material. A free-standing NMR coil was placed around the sample prior to securing the ends of the crystal in the strain device with epoxy. The radiofrequency field $\mathbf{H}_{1}$ is oriented parallel to the strain axis, which is always perpendicular to the external field, $\mathbf{H}_{0}$. In our device, strain is always applied along the $x$ axis defined in Fig. \[fig:cartoon\]; since the $b$ axis is defined as the shorter axis, positive (i.e. tensile) strain corresponds to $x\parallel a$ and $y\parallel b$, whereas negative (i.e. compressive) strain gives $y\parallel a$ and $x\parallel b$. When the crystal is strained by applying voltage to the piezoelectric stacks, the displacement, $x$, is measured by a capacitive position sensor, and strain is calculated as $\epsilon=(x-x_{0})/L_{0}$, where $L_{0}$ is the unstrained length of the crystal. To account for differential thermal contraction, the zero-strain displacement, $x_{0}$, was determined by the condition that the quadrupolar splitting $\nu_{\alpha\alpha}$ satisfies the tetragonal-symmetry relationship $|\nu_{xx}|=|\nu_{yy}|=|\nu_{zz}|/2$, as described in the supplemental material. The linear relationship between $\nu_{\alpha\alpha}$ and strain (Fig. S1) indicates that both positive and negative strains are achieved, without bowing of the crystal. The field $\mathbf{H}_{0}$ was oriented either along the $z$ direction parallel to the crystalline $c$ axis, or in the plane of the crystal along the $y$-direction, as shown in Fig. \[fig:cartoon\].
The spin lattice relaxation rate $(T_{1}T)_{\mu}^{-1}$ for different field orientations $\mu=z,\,y$ is shown in Fig. \[fig:T1Tinv\] both as a function of strain $\varepsilon$ and temperature $T$. It is striking that while $(T_{1}T)_{z}^{-1}$ increases by approximately 30% at 137K for the largest applied strain (approximatelly $0.3\%$), $(T_{1}T)_{y}^{-1}$ increases by 500%. In both cases, both positive and negative strain increase $(T_{1}T)^{-1}$ in a nonlinear fashion. This behavior is a manifestation of the spin anisotropy induced by nematic order. More precisely, the spin lattice relaxation rate is primarily dominated by the fluctuations of the local hyperfine field at the As site, which in turn is determined by the neighboring iron spins according to: $$\left(\frac{1}{T_{1}T}\right)_{\mu}=\frac{\gamma^{2}}{2}\lim_{\omega\rightarrow0}\sum\limits _{\mathbf{q},\alpha,\beta}\mathcal{F}_{\alpha\beta}^{(\mu)}(\mathbf{q})\frac{\textrm{Im}\chi_{\alpha\beta}(\mathbf{q},\omega)}{\hslash\omega},\label{eqn:dynamical_susceptibility}$$ where $\gamma$ is the nuclear gyromagnetic factor, $\mathcal{F}_{\alpha\beta}^{(\mu)}$ are the hyperfine form factors, which depend on the field direction $\mu$ (see Supplemental Material), $\chi_{\alpha\beta}(\mathbf{q},\omega)$ is the dynamical magnetic susceptibility, and $\alpha,\beta=\left\{ x,y,z\right\} $ [@T1formfactorsArsenides]. Because the system is metallic, spin fluctuations experience Landau damping, resulting in the low-energy dynamics $\chi_{\alpha\beta}^{-1}(\mathbf{q},\omega)=\chi_{\alpha\beta}^{-1}(\mathbf{q})-i\hbar\omega/\Gamma$, where $\Gamma$ is the Landau damping, as seen by neutron scattering experiments. Consequently, $\lim\limits _{\omega\rightarrow0}\frac{\textrm{Im}\chi_{\alpha\beta}(\mathbf{q},\omega)}{\hslash\omega}=\frac{1}{\Gamma}\chi_{\alpha\beta}^{2}(\mathbf{q})$, i.e. the spin-lattice relaxation rate is proportional to the squared susceptibility integrated over the entire Brillouin zone. Since the magnetically ordered state has wave-vectors $\mathbf{Q}_{1}=\left(\pi,0\right)$ and $\mathbf{Q}_{2}=\left(0,\pi\right)$, one expects that the susceptibility is peaked at these two momenta. Indeed, neutron scattering experiments confirm that the magnetic spectral weight is strongly peaked at $\mathbf{Q}_{1}$ and $\mathbf{Q}_{2}$.
Therefore, as an initial step to elucidate the effect of strain on the spin fluctuations anisotropy, we consider that the susceptibility is sharply peaked at these two magnetic ordering vectors. Evaluation of the hyperfine form factors yields:
$$\begin{aligned}
\left(T_{1}T\right)_{x}^{-1} & \propto\chi_{xx}^{2}\left(\mathbf{Q}_{1}\right)+\chi_{yy}^{2}\left(\mathbf{Q}_{2}\right)+\chi_{zz}^{2}\left(\mathbf{Q}_{2}\right)\nonumber \\
\left(T_{1}T\right)_{y}^{-1} & \propto\chi_{xx}^{2}\left(\mathbf{Q}_{1}\right)+\chi_{yy}^{2}\left(\mathbf{Q}_{2}\right)+\chi_{zz}^{2}\left(\mathbf{Q}_{1}\right)\nonumber \\
\left(T_{1}T\right)_{z}^{-1} & \propto\chi_{zz}^{2}\left(\mathbf{Q}_{1}\right)+\chi_{zz}^{2}\left(\mathbf{Q}_{2}\right)\label{eq:T1T}\end{aligned}$$
where the prefactors are approximately the same in all equations (see SM), and proportional to the off-diagonal hyperfine matrix element $\mathcal{F}_{xz}$ coupling in-plane Fe spin fluctuations to out-of-plane As hyperfine fields (and vice-versa). The fact that $\chi_{zz}\left(\mathbf{Q}_{i}\right)$ contributes to $T_{1}$ for all directions of the applied magnetic field is thus consistent with the hyperfine field analysis depicted in Fig. \[fig:multinematic\], since out-of-plane spin fluctuations on the Fe sites produce hyperfine fluctuating fields in the As sites along both in-plane directions. Similarly, the fact that only $\chi_{xx}\left(\mathbf{Q}_{1}\right)$ and $\chi_{yy}\left(\mathbf{Q}_{2}\right)$ contribute to $T_{1}$ for external fields applied along the plane is a consequence of the fact that these spin fluctuations generate hyperfine fields in the As site oriented out of the plane.
Because by symmetry $\left(T_{1}T\right)_{x}^{-1}\left(\varepsilon\right)=\left(T_{1}T\right)_{y}^{-1}\left(-\varepsilon\right)$, the NMR data can be used to extract the strain and temperature dependence of the three polarized spin-susceptibility combinations $\chi_{zz}^{2}\left(\mathbf{Q}_{1}\right)$, $\chi_{zz}^{2}\left(\mathbf{Q}_{2}\right)$, and $\chi_{xx}^{2}\left(\mathbf{Q}_{1}\right)+\chi_{yy}^{2}\left(\mathbf{Q}_{2}\right)$, as shown in Fig. \[fig:T1Tinv\](e). This analysis provides several interesting insights. First, focusing on the out-of-plane fluctuations, in-plane strain enhances spin fluctuations around one of the two ordering vectors ($\chi_{zz}\left(\mathbf{Q}_{1}\right)$ for $\varepsilon>0$ and $\chi_{zz}\left(\mathbf{Q}_{2}\right)$ for $\varepsilon<0$) at the same time as it suppresses the fluctuations around the other ordering vector. Therefore, in-plane strain transfers magnetic spectral weight between the two dominant out-of-plane spin-fluctuation channels. This is consistent with neutron scattering experiments in detwinned pnictides [@StrainedPnictidesNS2014science], which however only probed the unpolarized susceptibility. More importantly, this behavior is a direct manifestation of the response of the nematic order parameter $\varphi_{zz}$ to strain, since $\varphi_{zz}=\chi_{zz}^{-1}(\mathbf{Q}_{2})-\chi_{zz}^{-1}(\mathbf{Q}_{1})$.
Turning now to the average in-plane fluctuations $\chi_{xx}^{2}\left(\mathbf{Q}_{1}\right)+\chi_{yy}^{2}\left(\mathbf{Q}_{2}\right)$, we note that, in contrast to the quantity $\chi_{zz}\left(\mathbf{Q}_{1}\right)-\chi_{zz}\left(\mathbf{Q}_{2}\right)$, it is an even function of the applied strain. This behavior can be attributed to the response of the nematic order parameter $\varphi_{xy}=\chi_{xx}^{-1}(\mathbf{Q}_{2})-\chi_{yy}^{-1}(\mathbf{Q}_{1})$ to strain. Similarly to $\varphi_{zz}$, $\varphi_{xy}$ promotes a transfer of magnetic spectral weight, but now between $x$-polarized spin fluctuations around $\mathbf{Q}_{1}$ and $y$-polarized spin fluctuations around $\mathbf{Q}_{2}$. Since only the combination $\chi_{xx}^{2}\left(\mathbf{Q}_{1}\right)+\chi_{yy}^{2}\left(\mathbf{Q}_{2}\right)$ contributes to the spin-lattice relation rate, the total magnetic spectral weight remains the same to linear order in $\varphi_{xy}$, since what is suppressed in, say, $\chi_{yy}(\mathbf{Q}_{2})$ is tranferred to $\chi_{xx}(\mathbf{Q}_{1})$. Of course, as strain is enhanced, non-linear effects quadratic in $\varphi_{xy}^{2}$ take place, in agreement with the behavior displayed by Fig. \[fig:T1Tinv\](e). Note that the third nematic order parameter, $\varphi_{yx}=\chi_{yy}^{-1}(\mathbf{Q}_{2})-\chi_{xx}^{-1}(\mathbf{Q}_{1})$, does not affect the in-plane fluctuations that contribute the most to the spin-lattice relaxation rate. This is not unexpected, since the spin fluctuations associated with $\chi_{yy}(\mathbf{Q}_{1})$ and $\chi_{xx}(\mathbf{Q}_{2})$ do not generate hyperfine fields in the As sites, as shown in Fig. \[fig:multinematic\].
The most striking feature of Fig. \[fig:T1Tinv\](e) is that the out-of-plane spin fluctuations seem to have a larger response to in-plane strain than the in-plane spin fluctuations. This observation suggests that the nematic susceptibility associated with $\varphi_{zz}$, $\chi_{\mathrm{nem}}^{(zz)}\equiv\partial\varphi_{zz}/\partial\varepsilon$, is larger than the nematic susceptibility associated with $\varphi_{xy}$, $\chi_{\mathrm{nem}}^{(xy)}\equiv\partial\varphi_{xy}/\partial\varepsilon$, and is manifestation of the fact that nematic order induces not only real-space anisotropy, but also spin-space anisotropy. To make this analysis more quantitative, we fit the full temperature, strain, and field orientation dependence of $T_{1}$ to a model that incorporates the fact that the magnetic fluctuations are not infinitely peaked at the ordering vectors $\mathbf{Q}_{1,2}$, since the magnetic correlation length is finite above the magnetic transition. In the tetragonal phase, there are three different magnetic correlation lengths, $\xi_{x}$, $\xi_{y}$, and $\xi_{z}$, associated respectively with the pairs of peaks $\left(\chi_{xx}\left(\mathbf{Q}_{1}\right),\chi_{yy}\left(\mathbf{Q}_{2}\right)\right)$; $\left(\chi_{yy}\left(\mathbf{Q}_{1}\right),\chi_{xx}\left(\mathbf{Q}_{2}\right)\right)$, and $\left(\chi_{zz}\left(\mathbf{Q}_{1}\right),\chi_{zz}\left(\mathbf{Q}_{2}\right)\right)$. This spin anisotropy is intrinsic to the tetragonal crystalline symmetry and is enforced by the spin-orbit coupling even in the absence of nematic order. Nematic order induced by strain breaks the equivalence between these pairs of peaks, splitting the correlation lengths into $\tilde{\xi}_{x}^{-2}=\xi_{x}^{-2}\mp\varphi_{xy}$, $\tilde{\xi}_{y}^{-2}=\xi_{y}^{-2}\mp\varphi_{yx}$, and $\tilde{\xi}_{z}^{-2}=\xi_{z}^{-2}\mp\varphi_{zz}$. This model is similar to the one used previously in [@NMRnematicStrainBa122PRB2016] and is described in the supplemental material.
The fits for $(T_{1}T)_{z}^{-1}$ and $(T_{1}T)_{y}^{-1}$ in the absence of strain are shown as solid gray lines in Figs. \[fig:T1Tinv\](b) and (d) for $\xi_{x}=\xi_{y}$. We find $\xi_{z}/\xi_{x}=0.88$, in agreement with the fact that in the absence of strain the spins point along the plane. Moreover, the temperature dependence of $\xi_{x}(T)$, shown in Fig. \[fig:fitpars\](a), gives values consistent with those measured by inelastic neutron scattering. Having fixed the unstrained parameters, we perform fits in the presence of strain, shown by the solid lines in Fig. \[fig:T1Tinv\](a) and (c). The only parameters introduced in this case are the nematic order parameters $\varphi_{xy}=\varphi_{yx}$ and $\varphi_{zz}$. The good agreement between the fitted and the experimental curves of both $(T_{1}T)_{z}^{-1}$ and $(T_{1}T)_{y}^{-1}$ over a wide temperature-strain regime demonstrates the suitability of the phenomenological model employed in our analysis.
![\[fig:fitpars\] Fit parameters (a) $\kappa_{zz,yx}/\kappa_{xy}$, and (b) $\xi$ and $\kappa_{xy}$ versus temperature, based on the fits (solid lines) shown in Fig. \[fig:T1Tinv\]. Also shown are the nematic susceptibilities measured by Raman and elastoresistance measurements, reproduced from Refs. and , respectively. The solid lines are fits as described in the text.](fitparsVStempV5){width="1\linewidth"}
The temperature and strain behaviors of the nematic order parameters $\varphi_{\alpha\beta}$ allows us to extract the temperature dependence of the nematic susceptibilities $\chi_{\mathrm{nem}}^{(xy)}$ and $\chi_{\mathrm{nem}}^{(zz)}$, as shown in Fig. \[fig:fitpars\](b). It is clear that generally $\chi_{\mathrm{nem}}^{(zz)}>\chi_{\mathrm{nem}}^{(xy)}$, particularly close to the magnetic transition. This quantitative analysis corroborates the qualitative conclusion above, namely that nematic order induces anisotropies in spin-space, and that the out-of-plane spin fluctuations are more strongly enhanced by in-plane strain than the in-plane spin fluctuations. It is interesting to compare $\chi_{\mathrm{nem}}^{(xy)}$ and $\chi_{\mathrm{nem}}^{(zz)}$ with the nematic susceptibility extracted from elastoresistance [@FisherScienceNematic2012] and from electronic Raman spectroscopy experiments [@Ba122RamanPRL2013]. As shown in Fig. \[fig:fitpars\](b), the values are consistent, and the NMR-extracted nematic susceptibilities also follow a Curie-Weiss type of behavior [@Gallais2016], with a Curie temperature $T_{0}=116$ K comparable to that extracted from the elastoresistance [@FisherScienceNematic2012]. Note however that, in contrast to our NMR analysis, the other probes for the nematic susceptibility are not sensitive to the “polarization” of the nematic susceptibility.
To the best of our knowledge, our results are the first to reveal the internal spin structure of the nematic order parameter in iron-based superconductors. This behavior is a clear manifestation of the entanglement between spin, orbital, and lattice degrees of freedom in the normal state of these compounds. Since superconductivity emerges from this unique state, the rich interplay between these different degrees of freedom revealed by our NMR analysis will certainly affect the properties of the superconducting state.
The surprising anisotropic response of different nematic components to in-plane strain reveals that the spin polarization can be controlled by lattice distortions, similar to a piezomagnetic effect. In particular, the result $\chi_{\mathrm{nem}}^{(zz)}>\chi_{\mathrm{nem}}^{(xy)}$ implies that for sufficiently large strain $\varepsilon^{*}$, the dominant spin polarization will shift from in-plane to out-of-plane. The value of $\varepsilon^{*}$ can be estimated from the condition that the out-of-plane magnetic correlation length $\tilde{\xi}_{z}=\xi_{z}-\varepsilon\chi_{\mathrm{nem}}^{(zz)}$ becomes larger than the in-plane magnetic correlation length $\tilde{\xi}_{x}=\xi_{x}-\varepsilon\chi_{\mathrm{nem}}^{(xy)}$, yielding $\varepsilon^{*}\approx0.4\%$ close to the magnetic transition temperature. Such a strain value, which is just beyond the capability of our specific piezo device, can reasonably be achieved by similar types of devices, however. More importantly, this analysis opens a new avenue to control spin polarization in nematic materials without using magnetic fields, but instead by using mechanical strain. Since nematic order has been observed in other correlated materials such as cuprates and ruthenates, it will be interesting to investigate whether similar sizable effects are present in these systems as well.
More broadly, our work demonstrates that precision tunable strain in combination with NMR provides a novel and important method to probe spin and charge degrees of freedom. It provides an intriguing possibility to tune the NMR spin relaxation rate by changing a voltage bias on the piezoelectric stacks. The subtle coupling between the lattice and spin polarizations exhibited by BaFe$_{2}$As$_{2}$ offers the potential for controlling magnetic properties through lattice deformations in next generation materials. Another potential application of our technique is the use of nuclear quadrupolar resonance to image local strains. The large response of the EFG to strain observed in this study would translate into high spatial resolution in a linear strain gradient, so that As NMR may be able to resolve microscopic features such as grain boundaries or defects.
Acknowledgements
================
We thank A. Dioguardi, S. Kivelson, and I. Fisher for enlightening discussions, and P. Klavins, for assistance in the laboratory. Work at UC Davis was supported by the NSF under Grant No. DMR-1506961. RMF is supported by the U. S. Department of Energy, Office of Science, Basic Energy Sciences, under award number DE-SC0012336. R. Sarkar was partially supported by the DFG through SFB 1143 for the project C02. Work done at Ames Lab (SLB, PCC, MT, RP, EIT) was supported by the U.S. Department of Energy, Office of Basic Energy Science, Division of Materials Sciences and Engineering. Ames Laboratory is operated for the U.S. Department of Energy by Iowa State University under Contract No. DE-AC02-07CH11358.
Methods {#sec:methods}
=======
Crystals were grown in self-flux as described in [@CanfieldBa122phasediagram2008] and cut along the $(110)_{{\rm T}}$ direction. Sample A had a mass of 2.52 mg and was mounted with the field parallel to the $\mathbf{c}$ axis, and Sample B had a mass 0.91 mg and was mounted with the field perpendicular to the $\mathbf{c}$ axis (see Fig. \[fig:cartoon\]). The crystals were secured with heat-cured epoxy (UHU Plus 300 epoxy resin). Strain was applied along the $(110)_{{\rm T}}$ direction using the CS100 cryogenic uniaxial strain cell developed by Razorbill Instruments based on a design by Hicks et. al. [@Hicks2014], mounted in a modified probe operating in a Quantum Design PPMS cryostat. The displacement, $x$ was measured by monitoring the capacitance of using a precision capacitance bridge with a resolution of 0.1nm. The strain was computed as $\epsilon=(x-x_{0})/L_{0}$, where $L_{0}=2.052$ mm and $x_{0}=49.5$ $\mu$m for sample A and $L_{0}=1.494$ mm and $x_{0}=51.58$ $\mu$m for sample B. For sample B, positive (tensile) strain corresponds to $\mathbf{H}_{0}~||~\hat{b}$ and negative (compressive) strain corresponds to $\mathbf{H}_{0}~||~\hat{a}$. Because the sample was mounted at room temperature, thermal contraction creates positive strain even at zero piezo bias at low temperatures, making a precise determination of $x_{0}$ difficult. For sample A $x_{0}$ was determined by the minimum in $(T_{1}T)^{-1}$ versus $x$, and for sample B $x_{0}$ was determined by the value $\nu_{bb}(x_{0})=|\nu_{cc}|/2=1.23$ MHz, where $\nu_{\alpha\alpha}$ is the quadrupolar splitting for field along the $\alpha$ direction (see supplemental materials). The maximum/minimum possible applied voltages to the piezoelectric stacks limited the range of strains that could be applied to between approximately $-0.002$ to $+0.003$ in the perpendicular case, and $-0.0015$ to $+0.002$ for the parallel case. The spin-lattice relaxation rate was measured using inversion recovery at the central transition in fixed field, and the data were fit to the expression $M(t)=M_{0}\left[1-2f\left(\frac{9}{10}e^{-6t/T_{1}}+\frac{1}{10}e^{-t/T_{1}}\right)\right]$. The data were well-fit to a single $T_{1}$ value.
Spectral Measurements
=====================
When the crystal is strained by applying voltage to the piezoelectric stacks, the displacement, $x$, is measured by a capacitive position sensor, and strain is calculated as $\epsilon=(x-x_{0})/L_{0}$, where $L_{0}$ is the unstrained length of the crystal. It is crucial to determine the unstrained displacement, $x_{0}$, at cryogenic temperatures due to differential thermal contraction between the strain device and the sample. This value can be obtained by observing the asymmetry of the electric field gradient (EFG) tensor. The spectra were measured by acquiring echoes while sweeping the magnetic field $H_{0}$ at fixed frequency. The quadrupolar satellite resonances occur at fields $H_{sat}=(f_{0}\pm\nu_{\alpha\alpha})/\gamma(1+K_{\alpha\alpha})$, where $f_{0}$ is the radiofrequency, $\gamma=7.29019$ MHz/T is the gyromagnetic ratio, $K_{\alpha\alpha}$ and $\nu_{\alpha\alpha}$ are the Knight shift and EFG tensor components in the $\alpha=(x,y,z)$ direction. The central transition field is given by: $H_{cen}=\frac{f_{0}}{\gamma(1+K_{\alpha\alpha})}\left(\frac{1}{2}+\sqrt{\frac{3f_{0}^{2}-2(\nu_{\beta\beta}+\nu_{\alpha\alpha})^{2}}{12}}\right)$, where $\beta=(y,x,z)$ for $\alpha={x,y,z}$. The center of gravity of each peak was used to determine the resonance field, and hence $K_{\alpha\alpha}$ and $\nu_{\alpha\alpha}$ as a function of strain.
![\[fig:EFGandKS\] (a) Knight shift versus strain at 138K. (b) The $^{75}$As spectrum at 138K for a strain level of 0.0265% at frequency 55.924 MHz. The solid line is a fit to the spectrum as described in the text. (c) The quadrupolar splitting versus strain, and (d) versus temperature. The zero-strain points (diamonds) are reproduced from Ref. .](EFGandKS){width="\linewidth"}
Fig. \[fig:EFGandKS\](b) shows a typical field-swept NMR spectrum of the $^{75}$As, revealing a narrow central transition ($I_{z}=1/2\leftrightarrow-1/2$) and two quadrupolar satellite peaks ($\pm3/2\leftrightarrow\pm1/2$). The spectrum was fit to the sum of three Gaussians to extract both the Knight shift, $K_{\alpha\alpha}$, and the EFG, $\nu_{\alpha\alpha}$. The EFG tensor is given by $\nu_{\alpha\beta}=(eQ/12h)\partial^{2}V/\partial x_{\alpha}\partial x_{\beta}$, where $Q$ is the quadrupolar moment of the $^{75}$As and $V$ is the electrostatic potential at the As site. This quantity is dominated by the occupation of the $d_{xz,yz}$-orbitals of the neighboring Fe atoms, and the EFG asymmetry $\eta=(\nu_{yy}-\nu_{xx})/(\nu_{xx}+\nu_{yy})$ is a measure of the nematic order parameter [@DioguardiPdoped2015; @IyeJPSLorbitalnematicity2015]. Note that the magnetic field lies along the shorter $b$-axis under tensile strain ($\epsilon>0$), and along the longer $a$-axis under compressive strain ($\epsilon<0$), as shown in Fig. 2 of the main text. The EFG enables us to identify the zero-strain displacement, $x_{0}$, by the condition $|\nu_{xx}|=|\nu_{yy}|=|\nu_{zz}|/2$. As shown in Fig. \[fig:EFGandKS\](c), $\nu_{yy}$, and hence $\eta(\epsilon)=(\nu_{yy}(\epsilon)-\nu_{yy}(-\epsilon))/(\nu_{yy}(\epsilon)+\nu_{yy}(-\epsilon))$, varies linearly with strain.
Despite the fact that the EFG varies with strain, we find no significant variation of the satellite linewidth with strain. The strong variation of the EFG with strain explains the quadrupolar broadening observed in Co, Ni or Cu-doped Ba(Fe,M)$_{2}$As$_{2}$ [@ImaiLightlyDoped; @Dioguardi2010; @Takeda:2014ia]. The dopant atoms create an inhomogeneous strain field that gives rise to a distribution of local EFGs. Recently a finite value of $\eta\sim0.1$ was reported in the tetragonal phase of unstrained BaFe$_{2}$(As$_{1-x}$P$_{x}$)$_{2}$ above $T_{s}$ [@IyeJPSLorbitalnematicity2015]. The origin of this finite nematicity is likely due to local defects, and based on our results the strain fields are on the order of $0.05\%$.
The Knight shift is shown versus strain in Fig. \[fig:EFGandKS\](a) for $\mathbf{H}_{0}\perp c$. The in-plane Knight shift shows little or no variation with $\epsilon$, such that $(K_{xx}-K_{yy})/K_{yy}\leq3\%$ at the highest strain levels in this material. This result is surprising because the same quantity is approximately 6% in the nematic phase of FeSe [@Baek2015]. Recent static susceptibility measurements in BaFe$_{2}$As$_{2}$ under strain indicate that $\chi_{xx}$ and $\chi_{yy}$ in the paramagnetic phase differ by only 5% [@MeingastBaFe2As2strain2016]. This result suggests that $\chi_{\alpha\alpha}(\mathbf{q}=0)$ couples only weakly to the strain.
Spin-Lattice Relaxation Rate: Model
===================================
As stated in the main text, the spin-lattice relaxation rate is given by:
$$\left(\frac{1}{T_{1}T}\right)_{\mu}=\frac{\gamma^{2}}{2}\sum\limits _{\mathbf{q},\alpha,\beta}\mathcal{F}_{\alpha\beta}^{(\mu)}(\mathbf{q})\frac{\textrm{Im}\chi_{\alpha\beta}(\mathbf{q},\omega)}{\hslash\omega}$$
where $\gamma$ is the gyromagnetic ratio of the nuclear spin, and $\mathcal{F}_{\alpha\beta}^{(\mu)}$ is a form factor that depends on the direction of the applied field (indicated by $\mu$), and $\alpha,\beta=\left\{ x,y,z\right\} $. The coordinate system is defined such that $x$ and $y$ connect nearest neighbor Fe atoms. Ref. [@T1formfactorsArsenides] derived the form factor for an As nucleus subject to an arbitrary field direction. In the paramagnetic state, one obtains (see also Ref. [@NMRnematicStrainBa122PRB2016]):
$$\left(\frac{1}{T_{1}T}\right)_{\mu}=\frac{\gamma^{2}}{2}\sum_{\mathbf{q}}\sum_{\alpha=1,2}\left[\bar{R}^{(\mu)}\cdot\bar{A}_{\mathbf{q}}\cdot\bar{\tilde{\chi}}\left(\mathbf{q}\right)\cdot\bar{A}_{\mathbf{q}}^{\dagger}\cdot\left(\bar{R}^{(\mu)}\right)^{\dagger}\right]_{\alpha\alpha}\label{def}$$
All quantities with an overbar are $3\times3$ matrices. The matrix $\bar{\tilde{\chi}}\left(\mathbf{q}\right)$ is diagonal; its matrix elements are related to the magnetic susceptibility elements according to:
$$\tilde{\chi}_{\alpha\alpha}\left(\mathbf{q}\right)\equiv\lim_{\omega\rightarrow0}\frac{\mathrm{Im}\chi_{\alpha\alpha}\left(\mathbf{q},\omega\right)}{\hbar\omega}=\frac{1}{\Gamma}\chi_{\alpha\alpha}^{2}(\mathbf{q})\label{aux_def}$$
where $\Gamma$ is the Landau damping term. Furthermore, we have the hyperfine tensor:
$$\bar{A}_{\mathbf{q}}=4\left(\begin{array}{ccc}
A_{xx}\cos\left(\frac{q_{x}}{2}\right)\cos\left(\frac{q_{y}}{2}\right) & -A_{xy}\sin\left(\frac{q_{x}}{2}\right)\sin\left(\frac{q_{y}}{2}\right) & iA_{xz}\sin\left(\frac{q_{x}}{2}\right)\cos\left(\frac{q_{y}}{2}\right)\\
-A_{yx}\sin\left(\frac{q_{x}}{2}\right)\sin\left(\frac{q_{y}}{2}\right) & A_{yy}\cos\left(\frac{q_{x}}{2}\right)\cos\left(\frac{q_{y}}{2}\right) & iA_{yz}\cos\left(\frac{q_{x}}{2}\right)\sin\left(\frac{q_{y}}{2}\right)\\
iA_{zx}\sin\left(\frac{q_{x}}{2}\right)\cos\left(\frac{q_{y}}{2}\right) & iA_{zy}\cos\left(\frac{q_{x}}{2}\right)\sin\left(\frac{q_{y}}{2}\right) & A_{zz}\cos\left(\frac{q_{x}}{2}\right)\cos\left(\frac{q_{y}}{2}\right)
\end{array}\right)$$
and the rotation matrix:
$$\bar{R}^{(\mu)}=\left(\begin{array}{ccc}
\sin^{2}\phi+\cos\theta\,\cos^{2}\phi & -\sin2\phi\,\sin^{2}\frac{\theta}{2} & \cos\phi\,\sin\theta\\
-\sin2\phi\,\sin^{2}\frac{\theta}{2} & \cos^{2}\phi+\cos\theta\,\sin^{2}\phi & \sin\phi\,\sin\theta\\
-\cos\phi\,\sin\theta & -\sin\phi\,\sin\theta & \cos\theta
\end{array}\right)$$
Here, the field direction $\mu$ is described by the angles $\theta,\varphi$ according to $\hat{\mathbf{h}}=\cos\varphi\sin\theta\,\hat{\mathbf{x}}+\sin\varphi\sin\theta\,\hat{\mathbf{y}}+\cos\theta\,\hat{\mathbf{z}}$. Because the lattice distortion is very small, we consider hereafter that the hyperfine tensor remains essentially tetragonal, i.e. $A_{xx}=A_{yy}$, $A_{yz}=A_{xz}$, and $A_{xy}=A_{yx}$.
It is now straightforward to obtain the expressions for $1/\left(T_{1}T\right)_{\mu}$ for different field directions $\mu$. We find:
$$\begin{aligned}
\left(\frac{1}{T_{1}T}\right)_{x} & = & 8\gamma^{2}\sum_{\mathbf{q}}\left[\sin^{2}\left(\frac{q_{x}}{2}\right)\sin^{2}\left(\frac{q_{y}}{2}\right)A_{xy}^{2}+\sin^{2}\left(\frac{q_{x}}{2}\right)\cos^{2}\left(\frac{q_{y}}{2}\right)A_{xz}^{2}\right]\tilde{\chi}_{xx}\left(\mathbf{q}\right)\nonumber \\
& & 8\gamma^{2}\sum_{\mathbf{q}}\left[\cos^{2}\left(\frac{q_{x}}{2}\right)\cos^{2}\left(\frac{q_{y}}{2}\right)A_{yy}^{2}+\cos^{2}\left(\frac{q_{x}}{2}\right)\sin^{2}\left(\frac{q_{y}}{2}\right)A_{yz}^{2}\right]\tilde{\chi}_{yy}\left(\mathbf{q}\right)\nonumber \\
& & 8\gamma^{2}\sum_{\mathbf{q}}\left[\cos^{2}\left(\frac{q_{x}}{2}\right)\sin^{2}\left(\frac{q_{y}}{2}\right)A_{yz}^{2}+\cos^{2}\left(\frac{q_{x}}{2}\right)\cos^{2}\left(\frac{q_{y}}{2}\right)A_{zz}^{2}\right]\tilde{\chi}_{zz}\left(\mathbf{q}\right)\label{T1T_a_recap}\end{aligned}$$
$$\begin{aligned}
\left(\frac{1}{T_{1}T}\right)_{y} & = & 8\gamma^{2}\sum_{\mathbf{q}}\left[\cos^{2}\left(\frac{q_{x}}{2}\right)\cos^{2}\left(\frac{q_{y}}{2}\right)A_{xx}^{2}+\sin^{2}\left(\frac{q_{x}}{2}\right)\cos^{2}\left(\frac{q_{y}}{2}\right)A_{xz}^{2}\right]\tilde{\chi}_{xx}\left(\mathbf{q}\right)\nonumber \\
& & 8\gamma^{2}\sum_{\mathbf{q}}\left[\sin^{2}\left(\frac{q_{x}}{2}\right)\sin^{2}\left(\frac{q_{y}}{2}\right)A_{xy}^{2}+\cos^{2}\left(\frac{q_{x}}{2}\right)\sin^{2}\left(\frac{q_{y}}{2}\right)A_{yz}^{2}\right]\tilde{\chi}_{yy}\left(\mathbf{q}\right)\nonumber \\
& & 8\gamma^{2}\sum_{\mathbf{q}}\left[\sin^{2}\left(\frac{q_{x}}{2}\right)\cos^{2}\left(\frac{q_{y}}{2}\right)A_{xz}^{2}+\cos^{2}\left(\frac{q_{x}}{2}\right)\cos^{2}\left(\frac{q_{y}}{2}\right)A_{zz}^{2}\right]\tilde{\chi}_{zz}\left(\mathbf{q}\right)\label{T1T_b_recap}\end{aligned}$$
and:
$$\begin{aligned}
\left(\frac{1}{T_{1}T}\right)_{z} & =8\gamma^{2}\sum_{\mathbf{q}}\left[\cos^{2}\left(\frac{q_{x}}{2}\right)\cos^{2}\left(\frac{q_{y}}{2}\right)A_{xx}^{2}+\sin^{2}\left(\frac{q_{x}}{2}\right)\sin^{2}\left(\frac{q_{y}}{2}\right)A_{xy}^{2}\right]\tilde{\chi}_{xx}\left(\mathbf{q}\right)\nonumber \\
& 8\gamma^{2}\sum_{\mathbf{q}}\left[\cos^{2}\left(\frac{q_{x}}{2}\right)\cos^{2}\left(\frac{q_{y}}{2}\right)A_{yy}^{2}+\sin^{2}\left(\frac{q_{x}}{2}\right)\sin^{2}\left(\frac{q_{y}}{2}\right)A_{xy}^{2}\right]\tilde{\chi}_{yy}\left(\mathbf{q}\right)\nonumber \\
& 8\gamma^{2}\sum_{\mathbf{q}}\left[\sin^{2}\left(\frac{q_{x}}{2}\right)\cos^{2}\left(\frac{q_{y}}{2}\right)A_{xz}^{2}+\cos^{2}\left(\frac{q_{x}}{2}\right)\sin^{2}\left(\frac{q_{y}}{2}\right)A_{yz}^{2}\right]\tilde{\chi}_{zz}\left(\mathbf{q}\right)\label{T1T_c_recap}\end{aligned}$$
If we approximate the magnetic susceptibility as delta-functions peaked at the magnetic ordering vectors $\mathbf{Q}_{1}=\left(\pi,0\right)$ and $\mathbf{Q}_{2}=\left(0,\pi\right)$, we obtain:
$$\begin{aligned}
(T_{1}T)_{x}^{-1} & = & \frac{8\gamma^{2}A_{xz}^{2}}{\Gamma}\left[\chi_{xx}^{2}\left(\mathbf{Q}_{1}\right)+\chi_{yy}^{2}\left(\mathbf{Q}_{2}\right)+\chi_{zz}^{2}\left(\mathbf{Q}_{2}\right)\right]\\
(T_{1}T)_{y}^{-1} & = & \frac{8\gamma^{2}A_{xz}^{2}}{\Gamma}\left[\chi_{xx}^{2}\left(\mathbf{Q}_{1}\right)+\chi_{yy}^{2}\left(\mathbf{Q}_{2}\right)+\chi_{zz}^{2}\left(\mathbf{Q}_{1}\right)\right]\\
(T_{1}T)_{z}^{-1} & = & \frac{8\gamma^{2}A_{xz}^{2}}{\Gamma}\left[\chi_{zz}^{2}\left(\mathbf{Q}_{1}\right)+\chi_{zz}^{2}\left(\mathbf{Q}_{2}\right)\right]\end{aligned}$$
These equations can be inverted to extract the quantities: $$\begin{aligned}
\chi_{zz}^{2}\left(\mathbf{Q}_{1}\right) & = & \frac{\Gamma}{16\gamma^{2}A_{xz}^{2}}\left[-(T_{1}T)_{y}^{-1}(-\epsilon)+(T_{1}T)_{y}^{-1}(\epsilon)+(T_{1}T)_{z}^{-1}(\epsilon)\right]\label{invert1}\\
\chi_{zz}^{2}\left(\mathbf{Q}_{2}\right) & = & \frac{\Gamma}{16\gamma^{2}A_{xz}^{2}}\left[(T_{1}T)_{y}^{-1}(-\epsilon)-(T_{1}T)_{y}^{-1}(\epsilon)+(T_{1}T)_{z}^{-1}(\epsilon)\right]\label{invert2}\\
\chi_{xx}^{2}\left(\mathbf{Q}_{1}\right)+\chi_{yy}^{2}\left(\mathbf{Q}_{2}\right) & = & \frac{\Gamma}{16\gamma^{2}A_{xz}^{2}}\left[(T_{1}T)_{y}^{-1}(-\epsilon)+(T_{1}T)_{y}^{-1}(\epsilon)-(T_{1}T)_{z}^{-1}(\epsilon)\right],\label{invert3}\end{aligned}$$
using the fact that $(T_{1}T)_{x}^{-1}(\epsilon)=(T_{1}T)_{y}^{-1}(-\epsilon)$. These quantities are plotted in Fig. 3(e) of the main text.
Although useful for a qualitative analysis, this approximation neglects the important fact that the magnetic fluctuations have finite correlation lengths $\xi$. To model this effect, we consider susceptibilities peaked at $\mathbf{Q}_{1}$ and $\mathbf{Q}_{2}$, as seen by neutron scattering experiments (the amplitude $\chi_{0}$ of the susceptibilities is absorbed in $\Gamma$, for convenience) [@StrainedPnictidesNS2014science]:
$$\begin{aligned}
\Gamma\tilde{\chi}_{xx}\left(\mathbf{q}\right) & = & \frac{1}{\left[\left(\xi_{x}^{-2}-\varphi_{xy}\right)+\left(\cos q_{x}-\cos q_{y}+2\right)\right]^{2}}+\frac{1}{\left[\left(\xi_{y}^{-2}+\varphi_{yx}\right)+\left(-\cos q_{x}+\cos q_{y}+2\right)\right]^{2}}\nonumber \\
\Gamma\tilde{\chi}_{yy}\left(\mathbf{q}\right) & = & \frac{1}{\left[\left(\xi_{y}^{-2}-\varphi_{yx}\right)+\left(\cos q_{x}-\cos q_{y}+2\right)\right]^{2}}+\frac{1}{\left[\left(\xi_{x}^{-2}+\varphi_{xy}\right)+\left(-\cos q_{x}+\cos q_{y}+2\right)\right]^{2}}\nonumber \\
\Gamma\tilde{\chi}_{zz}\left(\mathbf{q}\right) & = & \frac{1}{\left[\left(\xi_{y}^{-2}-\varphi_{zz}\right)+\left(\cos q_{x}-\cos q_{y}+2\right)\right]^{2}}+\frac{1}{\left[\left(\xi_{y}^{-2}+\varphi_{zz}\right)+\left(-\cos q_{x}+\cos q_{y}+2\right)\right]^{2}},\nonumber\\\label{chi_def}\end{aligned}$$
Note that we have three different correlation lengths: $\xi_{x}$ corresponds to in-plane spin fluctuations with spins parallel to the ordering vector direction; $\xi_{y}$ corresponds to in-plane spin fluctuations with spins perpendicular to the ordering vector direction; and $\xi_{z}$ corresponds to out-of-plane spin fluctuations. This spin anisotropy originates from the spin-orbit coupling, as shown in Ref. [@Christensen2015]. The nematic order parameters $\varphi_{\alpha\beta}$ split the tetragonal degeneracy between $\chi_{xx}\left(\mathbf{Q}_{1}\right)$ and $\chi_{yy}(\mathbf{Q}_{2})$, between $\chi_{xx}\left(\mathbf{Q}_{2}\right)$ and $\chi_{yy}(\mathbf{Q}_{1})$, and between $\chi_{zz}\left(\mathbf{Q}_{1}\right)$ and $\chi_{zz}(\mathbf{Q}_{2})$. They are related to the external strain $\epsilon$ according to the nematic susceptibilities $\chi_{\mathrm{nem}}^{(\alpha\beta)}$, i.e. $\varphi_{\alpha\beta}=\epsilon\chi_{\mathrm{nem}}^{(\alpha\beta)}$.
Substituting these expressions in Eqs. (\[T1T\_a\_recap\]), (\[T1T\_b\_recap\]), and (\[T1T\_c\_recap\]) give:
$$\begin{aligned}
\frac{\Gamma}{8\gamma^{2}}\left(\frac{1}{T_{1}T}\right)_{x} & = & A_{xy}^{2}\left[J_{1}\left(\xi_{x}^{-2}-\varphi_{xy}\right)+J_{1}\left(\xi_{y}^{-2}+\varphi_{yx}\right)\right]+A_{xz}^{2}\left[J_{3}\left(\xi_{x}^{-2}-\varphi_{xy}\right)+J_{2}\left(\xi_{y}^{-2}+\varphi_{yx}\right)\right]\nonumber \\
& & +A_{yy}^{2}\left[J_{1}\left(\xi_{y}^{-2}-\varphi_{yx}\right)+J_{1}\left(\xi_{x}^{-2}+\varphi_{xy}\right)\right]+A_{yz}^{2}\left[J_{2}\left(\xi_{y}^{-2}-\varphi_{yx}\right)+J_{3}\left(\xi_{x}^{-2}+\varphi_{xy}\right)\right]\nonumber \\
& & +A_{yz}^{2}\left[J_{2}\left(\xi_{z}^{-2}-\varphi_{zz}\right)+
J_{3}\left(\xi_{z}^{-2}+\varphi_{zz}\right)\right]+A_{zz}^{2}\left[J_{1}\left(\xi_{z}^{-2}-\varphi_{zz}\right)+J_{1}\left(\xi_{z}^{-2}+\varphi_{zz}\right)\right]\nonumber \\
\label{eq1}\end{aligned}$$
as well as
$$\begin{aligned}
\frac{\Gamma}{8\gamma^{2}}\left(\frac{1}{T_{1}T}\right)_{y} & = & A_{xx}^{2}\left[J_{1}\left(\xi_{x}^{-2}-\varphi_{xy}\right)+J_{1}\left(\xi_{y}^{-2}+\varphi_{yx}\right)\right]+A_{xz}^{2}\left[J_{3}\left(\xi_{x}^{-2}-\varphi_{xy}\right)+J_{2}\left(\xi_{y}^{-2}+\varphi_{yx}\right)\right]\nonumber \\
& & +A_{xy}^{2}\left[J_{1}\left(\xi_{y}^{-2}-\varphi_{yx}\right)+J_{1}\left(\xi_{x}^{-2}+\varphi_{xy}\right)\right]+A_{yz}^{2}\left[J_{2}\left(\xi_{y}^{-2}-\varphi_{yx}\right)+J_{3}\left(\xi_{x}^{-2}+\varphi_{xy}\right)\right]\nonumber \\
& & +A_{xz}^{2}\left[J_{3}\left(\xi_{z}^{-2}-\varphi_{zz}\right)+J_{2}\left(\xi_{z}^{-2}+\varphi_{zz}\right)\right]+A_{zz}^{2}\left[J_{1}\left(\xi_{z}^{-2}-\varphi_{zz}\right)+J_{1}\left(\xi_{z}^{-2}+\varphi_{zz}\right)\right]\nonumber \\ \label{eq2}\end{aligned}$$
and
$$\begin{aligned}
\frac{\Gamma}{8\gamma^{2}}\left(\frac{1}{T_{1}T}\right)_{z} & =A_{xx}^{2}\left[J_{1}\left(\xi_{x}^{-2}-\varphi_{xy}\right)+J_{1}\left(\xi_{y}^{-2}+\varphi_{yx}\right)\right]+A_{xy}^{2}\left[J_{1}\left(\xi_{x}^{-2}-\varphi_{xy}\right)+J_{1}\left(\xi_{y}^{-2}+\varphi_{yx}\right)\right]\nonumber \\
& +A_{yy}^{2}\left[J_{1}\left(\xi_{y}^{-2}-\varphi_{yx}\right)+J_{1}\left(\xi_{x}^{-2}+\varphi_{xy}\right)\right]+A_{xy}^{2}\left[J_{1}\left(\xi_{y}^{-2}-\varphi_{yx}\right)+J_{1}\left(\xi_{x}^{-2}+\varphi_{xy}\right)\right]\nonumber \\
& +A_{xz}^{2}\left[J_{3}\left(\xi_{z}^{-2}-\varphi_{zz}\right)+J_{2}\left(\xi_{z}^{-2}+\varphi_{zz}\right)\right]+A_{yz}^{2}\left[J_{2}\left(\xi_{z}^{-2}-\varphi_{zz}\right)+J_{3}\left(\xi_{z}^{-2}+\varphi_{zz}\right)\right]\nonumber \\ \label{eq3}\end{aligned}$$
Here, we defined the integrals:
$$\begin{aligned}
J_{1}\left(r\right) & = & \int_{-\pi}^{\pi}\int_{-\pi}^{\pi}\frac{dq_{x}dq_{y}}{\left(2\pi\right)^{2}}\,\frac{\cos^{2}\left(\frac{q_{x}}{2}\right)\cos^{2}\left(\frac{q_{y}}{2}\right)}{\left[r+\left(\cos q_{x}-\cos q_{y}+2\right)\right]^{2}}\equiv\int_{-\pi}^{\pi}\int_{-\pi}^{\pi}\frac{dq_{x}dq_{y}}{\left(2\pi\right)^{2}}\,\frac{\sin^{2}\left(\frac{q_{x}}{2}\right)\sin^{2}\left(\frac{q_{y}}{2}\right)}{\left[r+\left(\cos q_{x}-\cos q_{y}+2\right)\right]^{2}}\nonumber \\
J_{2}\left(r\right) & = & \int_{-\pi}^{\pi}\int_{-\pi}^{\pi}\frac{dq_{x}dq_{y}}{\left(2\pi\right)^{2}}\,\frac{\cos^{2}\left(\frac{q_{x}}{2}\right)\sin^{2}\left(\frac{q_{y}}{2}\right)}{\left[r+\left(\cos q_{x}-\cos q_{y}+2\right)\right]^{2}}\nonumber \\
J_{3}\left(r\right) & = & \int_{-\pi}^{\pi}\int_{-\pi}^{\pi}\frac{dq_{x}dq_{y}}{\left(2\pi\right)^{2}}\,\frac{\sin^{2}\left(\frac{q_{x}}{2}\right)\cos^{2}\left(\frac{q_{y}}{2}\right)}{\left[r+\left(\cos q_{x}-\cos q_{y}+2\right)\right]^{2}}\label{integrals}\end{aligned}$$
In the limit $\xi_{i}^{-2}\pm\varphi_{\alpha\beta}\ll1$, we can approximate the integrals by expanding the integrand near $\left(\pi,0\right)$, yielding:
$$\begin{aligned}
J_{1}\left(r\right) & \approx\frac{1}{4\pi}\ln\left(\frac{\Lambda_{1}}{\sqrt{r}}\right)\nonumber \\
J_{2}\left(r\right) & \approx\frac{1}{8\pi}\left[1-\frac{r}{2}\,\ln\left(\frac{\Lambda_{2}}{\sqrt{r}}\right)\right]\nonumber \\
J_{3}\left(r\right) & \approx\frac{1}{2\pi r}\end{aligned}$$
where $\Lambda_{1}\approx1.45$ and $\Lambda_{2}\approx3.2$ for $r<0.5$, according to numerical evaluations of the integrals. Note that, as expected from symmetry considerations, $\left(T_{1}T\right)_{x}^{-1}\left(-\epsilon\right)=\left(T_{1}T\right)_{y}^{-1}\left(\epsilon\right)$ and $\left(T_{1}T\right)_{z}^{-1}\left(-\epsilon\right)=\left(T_{1}T\right)_{z}^{-1}\left(\epsilon\right)$.
Fitting the Spin Lattice Relaxation Rate Data
=============================================
The expressions for $(T_{1}T)_{\alpha}^{-1}$ given above depend on six parameters: $\xi_{x}$, $\xi_{y}$, $\xi_{z}$, $\varphi_{xy}$, $\varphi_{yx}$, and $\varphi_{zz}$. We first fit the zero-strain data shown in Fig. 3(b) and 3(d) of the main text assuming all the $\varphi_{\alpha\beta}=0$, and that $\xi_{y}=\xi_{x}$. Because the Landau damping term, $\Gamma$, is unknown, one cannot simply extract the $\xi_{x,z}$ directly from the data. However, the ratio of $(T_{1}T)_{x}^{-1}/(T_{1}T)_{z}^{-1}$ does constrain the data and enable us to fit the data using the temperature-dependent correlation lengths shown in Fig. 4(b) of the main text. The hyperfine coupling constants are given by: $A_{xx}=A_{yy}=0.66$ T/$\mu_{B}$, $A_{zz}=0.47$ T/$\mu_{B}$, and $A_{xz}=A_{yz}=0.43$ T/$\mu_{B}$ [@takigawa2008], and we assume the value $A_{xy}=0.33$ T/$\mu_{B}$ [@NMRnematicStrainBa122PRB2016].
Using these values for $\xi_{x,z}$ and assuming that $\xi_{y}=\xi_{x}$, we then proceed to fit the strain-dependent $(T_{1}T)^{-1}$ data to the three nematic order parameters, $\varphi_{xy}=\chi_{\mathrm{nem}}^{(xy)}\epsilon$, $\varphi_{yx}=\chi_{\mathrm{nem}}^{(yx)}\epsilon$, and $\varphi_{zz}=\chi_{\mathrm{nem}}^{(zz)}\epsilon$, where the $\chi_{\mathrm{nem}}^{(\alpha\beta)}$ are the static nematic susceptibilities of the three components of the nematic order. These data are shown in Fig. 4 of the main text as a function of temperature.
|
---
abstract: 'Resonance scattering (RS) is an important process in astronomical objects, because it affects measurements of elemental abundances and distorts surface brightness of the object. It is predicted that RS can occur in plasmas of supernova remnants (SNRs). Although several authors reported hints of RS in SNRs, no strong observational evidence has been established so far. We perform a high-resolution X-ray spectroscopy of the SNR N49 with the Reflection Grating Spectrometer aboard XMM-Newton. The RGS spectrum of N49 shows a high G-ratio of He$\alpha$ lines as well as Ly$\beta$/$\alpha$ and (3s–2p)/(3d–2p) ratios which cannot be explained by the emission from a thin thermal plasma. These line ratios can be well explained by the effect of RS. Our result implies that RS has a large impact particularly on a measurement of the oxygen abundance.'
author:
- Yuki Amano
- Hiroyuki Uchida
- Takaaki Tanaka
- Liyi Gu
- Takeshi Go Tsuru
title: 'Evidence for Resonance Scattering in the X-ray Grating Spectrum of the Supernova Remnant N49'
---
Introduction {#sec:intro}
============
X-ray imaging spectroscopy of astronomical objects provides us with many insights into their chemical evolution and formation mechanism. This is partially because X-ray emitting plasmas are often optically thin, which allows us to directly estimate its elemental abundances and their spatial distributions. However, resonance lines such as He$\alpha$ and L$\alpha$ may suffer from effects of scattering (resonance scattering: RS), as discussed in the cases of galaxies [e.g., @Xu2002], solar active regions [e.g., @Rugge1985], and galaxy clusters [e.g., @Hitomi2018].
RS is an apparent scattering phenomenon due to an absorption and re-emission of line photons by ions. Since the RS effect apparently reduces intensities of some lines and/or distorts profiles of surface brightness [@Shigeyama1998], ignoring its contribution can sometimes lead to, for example, biases in elemental abundance measurements. On the other hand, if RS is significant, quantifying its contribution will allow us to measure several important parameters such as micro-turbulence velocities [@dePlaa2012] and absolute abundances [@Waljeski1994].
[@Kaastra1995] predicted that RS of X-ray photons can occur in a plasma with a large depth along the line of sight such as a rim of supernova remnants (SNRs). Several observational signatures of RS have been reported such as a high forbidden-to-resonance ratio of He$\alpha$ obtained from grating spectra of DEM L71 [@vanderHeyden2003] and N23 [@Broersen2011]. A difference in surface brightness between forbidden and resonance lines also supports the presence of RS [e.g., @vanderHeyden2003]. Based on a Suzaku observation of the Cygnus Loop, [@Miyata2008] claimed that a depleted abundance of O may be partially explained by RS. A recent grating observation of the Loop also hints at a possibility of RS [@Uchida2019]. These studies suggest that the RS effect may potentially be significant in SNRs. However, no strong observational evidence has been established so far.
N49 is a middle-aged [$\sim4800$ yr; @Park2012] SNR located in the Large Magellanic Cloud (LMC). Although the origin of N49 is somewhat controversial, most of recent researches support that it originated from a core-collapse explosion based on a possible association with the soft gamma-ray repeater (SGR) 0526$-$66 [@Cline1982] and the presence of a dense interstellar medium (ISM) [e.g., @Banas1997; @Yamaguchi2014]. The thermal X-ray emission is explained by a mixture of two components; metal-rich ejecta and a shock-heated ISM [@Park2003; @Park2012; @Uchida2015]. It is also notable that N49 is in an overionized state [@Uchida2015] and is interacting with dense molecular clouds on the eastern side [@Banas1997; @Yamane2018]. Due to the interaction with molecular clouds, the thermal X-ray emission of N49 is particularly bright on the southeastern rim. N49 is an attractive object since [@Kaastra1995] pointed out that the effect of RS can show up if an SNR has such a spherically asymmetric structure.
Here, we present a high-resolution X-ray spectroscopy of N49 with the Reflection Grating Spectrometer (RGS) aboard XMM-Newton. The obtained G-ratio of He$\alpha$ and other line ratios as well imply a non-negligible contribution of RS in N49. Throughout the paper, errors are given at a 68% confidence level. We assume the distance to N49 (LMC) to be 50 kpc [@Pietrzy2013].
Observation and Data Reduction {#sec:obs}
==============================
![EPIC-MOS image (0.4–8.0 keV) of N49. The cross-dispersion width of the RGS (5 arcmin) is in between the white solid line. The spectral extraction region is enclosed by the white dashed line. N49B is another SNR near N49.[]{data-label="fig:mosimage"}](20191010_N49_image.eps){width="7.0cm"}
N49 was observed with the XMM-Newton satellite [@Jansen2001] in 2001 (Obs.ID 0113000201) and 2007 (Obs.ID 0505310101). The observation in 2007 was performed with a roll angle which placed N49 and a nearby SNR, N49B, along the dispersion direction of the RGS, making it difficult to extract RGS spectra of N49. We thus analyzed only the data obtained in 2001. For our spectral analysis, we used the RGS [@denHerder2001] and the European Photon Imaging Camera MOS [@Turner2001] data. We reduced the data using XMM Science Analysis Software version 16.1.0. The RGS data were processed with the RGS pipeline tool [rgsproc]{}. To discard periods of background flares, we applied good time intervals based on the count rate in CCD9, which is the closest to the optical axis of the mirror and has the least X-ray counts from the source. and most affected by the background flares. The resulting effective exposure time is 11 ks for both RGS1 and RGS2. Since the second order spectra are of low statistical quality, we analyzed only the first order spectra.
{width="15.0cm"}
{width="15.0cm"}
[llccc]{} Absorption & ${N_{\rm H}({\rm MW})~(10^{22}~{\rm cm^{-2}})}$ & $0.6$ (fixed)& $0.6$ (fixed) & $0.6$ (fixed)\
& ${N_{\rm H}({\rm LMC})~(10^{22}~{\rm cm^{-2}})}$ & $3.1 \pm 0.1$ & $2.8 \pm 0.1$ & $3.3 \pm 0.1$\
ISM & ${kT_{\rm e}}~({\rm keV})$ & $0.204 \pm 0.003$ & $0.230 \pm 0.006$ & $0.205 \pm 0.003$\
& ${n_{\rm e}t~(10^{11}~{\rm cm^{-3}~s}}$) & $>10$ & $>10$ & $>10$\
& ${EM~(10^{56}~{\rm cm^{-3}})}$ & $260 \pm 20$ & $170 ^{+20}_{-10}$ & $322^{+23}_{-22}$\
Ejecta & ${kT_{\rm e}~({\rm keV})}$ & $0.61 \pm 0.01$ & $0.63 \pm 0.01$ & $0.56^{+0.08}_{-0.01}$\
& ${kT_{\rm init}~({\rm keV})}$ & $11$ (fixed) & $11$ (fixed) & $11$ (fixed)\
& ${n_{\rm e}t~(10^{11}~{\rm cm^{-3}~s}}$) & $7.2^{+0.5}_{-0.3}$ & $7.2^{+0.4}_{-0.3}$ & $6.9^{+1.0}_{-0.4}$\
& O (= C = N) & $0.71^{+0.16}_{0.06}$ & $0.67^{+0.20}_{-0.14}$ & $1.3 \pm 0.1$\
& Ne & $0.96^{+0.09}_{-0.07}$ & $0.94^{+0.09}_{-0.06}$ & $1.0 \pm 0,1$\
& Mg & $0.75 \pm 0.07$ & $0.72^{+0.07}_{-0.06}$ & $0.75 {\rm (fixed)}$\
& Si & $0.87^{+0.08}_{-0.06}$ & $0.85^{+0.08}_{-0.06}$ & $0.87 {\rm (fixed)}$\
& S & $1.2 \pm 0.1$ & $1.2 \pm 0.1$ & $1.2 {\rm (fixed)}$\
& Ar & $1.8 \pm 0.5$ & $1.8^{+0.4}_{-0.3}$ & $1.8 {\rm (fixed)}$\
& Fe & $0.32 \pm 0.03$ & $0.29^{+0.03}_{-0.02}$ & $0.32 \pm 0.01$\
& ${EM~(10^{56}~{\rm cm^{-3}})}$ & $58 \pm 4$ & $57^{+5}_{-2}$ & $53 \pm 0.1$\
CX & ${kT_{\rm e}~({\rm keV})}$ & & (= value of the ISM component) &\
& abundance & & (= abundances of the ISM component) &\
& ${v_{\rm collision}}~({\rm km}~{\rm s}^{-1})$ & & $270 \pm 110$ &\
& ${EM~(10^{56}~{\rm cm^{-3}})}$ & & $47^{+25}_{-13}$ &\
Gaussian: He$\alpha$ & Normalization ($10^{44}~{\rm ph~s^{-1}}$) & & & $0.36 \pm 0.09$\
L$\alpha$ ($\sim 15.0~{\rm \AA}$) & & & & $0.92 \pm 0.11$\
L$\alpha$ ($\sim 15.3~{\rm \AA}$) & & & & $< 6.9 \times10^{-2}$\
Ly$\beta$ & & & & $< 1.3 \times10^{-2}$\
L$\alpha$ ($\sim 16.8~{\rm \AA}$) & & & & $0.50 \pm 0.12$\
L$\alpha$ ($\sim 17.1~{\rm \AA}$) & & & & $< 1.5 \times10^{-3}$\
He$\beta$ & & & & $< 5.8 \times10^{-2}$\
Ly$\alpha$ & & & & $4.6 \pm 0.3$\
He$\alpha$(r) & & & & $2.5 \pm 0.5$\
He$\alpha$(i) & & & & $< 3.0 \times10^{-3}$\
& C-statistic/d.o.f. & $4700/3211$ & $4660/3214$ & $4611/3190$\
Analysis {#sec:analysis}
========
We analyzed the spectra using version 3.04.0 of the SRON SPEX package [@Kaastra1996] with the maximum likelihood C-statistic [@Cash1979; @Kaastra2017]. The RGS spectra were fitted simultaneously with those of MOS1 and 2. To account for the spatial broadening of the source, we multiplied spectral models with the SPEX model [lpro]{}, to which we input the MOS1 image of the source. Our models have two absorption models: one for the Milky Way and the other for the LMC. The column density of the former was fixed to $6 \times 10^{20}~{\rm cm^{-2}}$ [@DickeyLockman1990] whereas that of the latter is left free. The elemental abundances for the LMC absorption were fixed to values found in literature [$\sim$ 0.3 solar; @Russell1992; @Schenck2016].
Figure \[fig:spec\] shows MOS1 and combined RGS1$+$2 spectra of N49. Prominent lines are detected at $\sim$12 Å ( Ly$\alpha$), $\sim$13.5 Å ( He$\alpha$), $\sim$15 Å ( L$\alpha$; 3d–2p), $\sim$16 Å ( Ly$\beta$), $\sim$17 Å ( L$\alpha$; 3s–2p), $\sim$20 Å ( He$\beta$, Ly$\alpha$), $\sim$22 Å ( He$\alpha$). We applied a two-component nonequilibrium ionization (NEI) model [[neij]{}; @KaastraJansen1993] absorbed by neutral gas to the spectra. The NEI model consists of emissions from the overionized hot ejecta and an ionizing cool component originating from the swept-up ISM [@Uchida2015]. Although SGR 0526$-$66 cannot be spatially resolved with the RGS, according to the result of [@Uchida2015], the emission is negligible compared to that from N49 in the energy band covered by the RGS (0.3–2.0 keV).
Free parameters of the NEI components include the electron temperature ($kT_{\rm e}$), ionization time scale ($n_{\rm e}t$, where $n_{\rm e}$ and $t$ are the electron number density and the elapsed time since shock heating or rapid cooling, respectively), and emission measure ($n_{\rm e}n_{\rm H}V$). In addition to these parameters, the abundances of O (=N=C), Ne, Mg, Si, S, Ar, and Fe (=Ni) of the ejecta were set free. The [neij]{} model has another parameter $kT_{\rm init}$, which represents an initial temperature before a rapid cooling or a shock heating of the plasma. [@Uchida2015] determined $kT_{\rm init}$ for the ejecta component mainly from the ionization state of the Fe K$\alpha$ emission at $\sim$6.6 keV. Since the line centroid is out of the wavelength band of the RGS data, we fixed $kT_{\rm init}$ of the ejecta component at 11 keV based on [@Uchida2015]. On the other hand, the $kT_{\rm init}$ of the ISM component is fixed at $\sim 0 ~{\rm keV}$.
Figure \[fig:spec\] (a) shows the result of the spectral fitting with the two-component NEI model (hereafter, NEI model). The best-fit parameters are listed in Table \[tab:parameters\]. The model well reproduces the overall MOS spectrum; the best-fit parameters of $kT_{\rm e}$ and $n_{\rm e}t$ of the ejecta component are consistent with those obtained by [@Uchida2015]. Focusing on the RGS spectrum in Figure \[fig:FeOspec\], however, we found discrepancies between the model and the data especially around the triplet, Ly$\beta$, and L$\alpha$ series.
We first focus on the He$\alpha$ line, more specifically on its G-ratio ($F+I)/R$, where $F$, $I$, and $R$ are intensities of the forbidden, intercombination, and resonance lines, respectively. The G-ratio of the He$\alpha$ line strongly depends on $kT_{\rm e}$ as presented in Figure \[fig:gratio\]. We assume here that the emitting plasma is in an ionizing state since most of the He$\alpha$ line emission of N49 is attributed to the shocked ISM (Figure \[fig:spec\] (a)). Fitting with four Gaussians, we estimated the G-ratio of the He$\alpha$ line observed in N49, which is plotted in Figure \[fig:gratio\]. The observed G-ratio obviously requires an unreasonably low $kT_{\rm e}$ ($< 0.15~{\rm keV}$) with any reasonable range of $n_{\rm e}t$, indicating necessity of some physical process to enhance the G-ratio.
![ Close-up views of the RGS spectrum (Figure \[fig:spec\]) around L$\alpha$ and Ly$\beta$ (left) and He$\alpha$ (right). The dark green, cyan, and red solid lines in the top panels represent the best-fit “NEI”, “NEI$+$CX”, and “NEI$-$Gaussians” models, respectively. The middle and bottom panels show residuals from the models with the same color scheme as the top panels. []{data-label="fig:FeOspec"}](20200422_FeO.eps){width="8.0cm"}
![ Relation between $kT_{\rm e}$ and the G-ratio of He$\alpha$. The solid curves represent the G-ratio expected for an ionizing plasma emission predicted by the [neij]{} model in SPEX. The colors of each line indicate $n_{\rm e}t$ assumed. The red hatched area denotes the G-ratio derived from the observed line ratio and its statistical error. The gray lines indicates $kT_{\rm e}$ from the best-fit NEI model. []{data-label="fig:gratio"}](20200126_N49_gratio.eps){width="7.0cm"}
{width="15.0cm"}
Charge exchange (CX) is one of the possible processes to make the G-ratio higher. Recent X-ray spectroscopy studies with gratings found enhanced $F/R$ ratios of in Puppis A [@Katsuda2012], and the Cygnus Loop [@Uchida2019]. They claimed that the anomalous line ratios are explained by CX X-ray emission. As suggested by [@Uchida2019], the CX emission can be enhanced particularly in a region where a shock is interacting with dense gas. CX is therefore promising in the case of N49 since it is interacting with surrounding molecular clouds [@Yamane2018]. We thus added a CX model [@Gu2016a] to the NEI model. All the abundances and $kT_{\rm e}$ for the CX component were coupled to those of the ISM component. The collision velocity $v_{\rm collision}$ ([*i.e.*]{} shock velocity) was allowed to vary. We assumed a multiple collision case in which an ion continuously undergoes CX until it becomes neutral. The best-fit “NEI$+$CX” model is displayed in Figure \[fig:spec\] (b), and the parameters are listed in Table \[tab:parameters\]. Although the addition of the CX component improves the residuals at the triplet, the discrepancies still remain around Ly$\beta$ and L$\alpha$ series (Figure \[fig:FeOspec\]). Therefore, we conclude that the NEI + CX model is insufficient to describe the spectrum of N49.
RS is another possible process that would be responsible for the enhanced G-ratio. If photons of the resonance line are scattered off the line of sight due to RS, observed $R$ would become lower and the G-ratio would be enhanced. The observed high (3s–2p)/(3d–2p) and Ly$\beta$/$\alpha$ ratios can also be caused by RS, as pointed out by previous studies [e.g., @Xu2002; @Hitomi2018]. To quantify the contribution of the scattering effect, we added negative Gaussians at wavelengths where lines of the ISM component are prominent in the best-fit NEI model: the He$\alpha$ intercombination, He$\alpha$ resonance, He$\beta$, Ly$\alpha$, Ly$\beta$, L$\alpha$, and He$\alpha$ resonance lines. Since the oscillator strengths of the and forbidden lines are several orders of magnitude smaller than those of the other lines, we assume that the scattering effect is negligible for the forbidden lines. The free parameters of the NEI components are the electron temperature ($kT_{\rm e}$), ionization time scale ($n_{\rm e}t$), emission measure ($n_{\rm e}n_{\rm H}V$). The abundances of O (=N=C), Ne, and Fe (=Ni) of the ejecta were set free and the abundances of Mg, Si, S, and Ar were fixed to the values obtained in the NEI model fit. The best-fit “${\rm NEI}-{\rm Gaussians}$” model is displayed in Figure \[fig:rsfit\] and the best-fit parameters are listed in Table \[tab:parameters\].
[lccc]{} He$\alpha$ &$1{\rm s}2{\rm p}\, ^1{\rm P}_1 \textrm{--} 1{\rm s}^2 \, ^1{\rm S}_0 $&13.45& $0.79 \pm 0.11$\
L$\alpha$ &$ 2{\rm s}^22{\rm p}^53{\rm d}\, ^1{\rm P}_1 \textrm{--} 2{\rm s}^22{\rm p}^6\,^1{\rm S}_0 $&15.02& $0.70^{+0.04}_{-0.08}$\
L$\alpha$ &$ 2{\rm s}^22{\rm p}^53{\rm d}\,^3{\rm D}_1 \textrm{--} 2{\rm s}^22{\rm p}^6\,^1{\rm S}_0 $&15.26& $>0.85$\
Ly$\beta$ &$ 3{\rm p}\,^2{\rm P}_{3/2} \textrm{--} 1{\rm s}\,^2{\rm S}_{1/2} $&16.01& $>0.82$\
L$\alpha$ &$ 2{\rm s}^22{\rm p}^53{\rm s}\,^3{\rm P}_1 \textrm{--} 2{\rm s}^22{\rm p}^6\,^1{\rm S}_0 $&16.78& $0.73^{+0.11}_{-0.18}$\
L$\alpha$ &$ 2{\rm s}^22{\rm p}^53{\rm s}\,^1{\rm P}_1 \textrm{--} 2{\rm s}^22{\rm p}^6\,^1{\rm S}_0 $&17.05& $>0.90$\
He$\beta$ &$ 1{\rm s}3{\rm p}\,^1{\rm P}_1 \textrm{--} 1{\rm s}^2\,^1{\rm S}_0 $&18.63& $>0.79$\
Ly$\alpha$ &$2{\rm p}\,^2{\rm P}_{3/2} \textrm{--} 1{\rm s}\,^2{\rm S}_{1/2}$&18.97& $0.46^{+0.02}_{-0.03}$\
He$\alpha$(r) &$1{\rm s}2{\rm p}\,^1{\rm P}_1 \textrm{--} 1{\rm s}^2\,^1{\rm S}_0$&21.60& $0.79 \pm 0.08$\
He$\alpha$(i) &$1{\rm s}2{\rm p}\,^3{\rm P}_1 \textrm{--} 1{\rm s}^2\,^1{\rm S}_0$&21.81& $>0.76$\
Discussion {#sec:results}
==========
We compare transmission factors estimated based on the result from the “${\rm NEI}-{\rm Gaussians}$” model fit with those expected for RS. The transmission factor $p$ is defined as $$\label{eq:ep}
p = \frac{A - \Delta A}{A},$$ where $A$ is the total number of photons emitted by the plasma, and $\Delta A$ is the number of photons scattered out of our line of sight. We can derive $p$ for each line from the data, given that the best-fit normalizations of the negative Gaussians and the line intensities of the ISM component correspond to $\Delta A$ and $A$, respectively. The transmission factors derived from the data are listed in Table \[tab:transmission\].
Referring to [@Kaastra1995], we calculate the transmission factors in a case where RS effectively occurs. Under the slab approximation as a simple geometrical model of the SNR rim, [@Kaastra1995] used a single-scattering treatment where a photon completely escapes from the line of sight at every scattering event. As shown by [@Park2003], the ISM plasma of N49 has a particularly bright emission at the southeastern rim. Assuming that the RS occurs dominantly at the southeastern rim, we adopt the same assumption as [@Kaastra1995]. Then, the transmission factor is written as $$\label{eq:ep2}
p = \frac{1}{1 + 0.43 \tau},$$ where $\tau$ is the optical depth of the ISM plasma [@Kastner1990]. The optical depth $\tau$ at the line centroid is given by [@Kaastra1995] as $$\label{eq:RS_tau}
\tau = \frac{4.24 \times 10^{26} f N_{\rm H} \left(\frac{n_{\rm i}} {n_{\rm z}}\right) \left(\frac{n_{\rm z}}{n_{\rm H}}\right) \left(\frac{M}{T_{\rm keV}}\right)^{1/2}}{E_{\rm eV}\left(1 + \frac{0.0522 M v_{100}^2}{T_{\rm keV}}\right)^{1/2}},$$ where $f$ is the oscillator strength of the line, $E_{\rm eV}$ is the line centroid energy in eV, $N_{\rm H}$ is the hydrogen column density in ${\rm cm^{-2}}$, $n_{\rm i}$ is the number density of the ion, $n_{\rm Z}$ is the number density of the element, $M$ is the atomic weight of the ion, $T_{\rm keV}$ is the ion temperature in keV, and $v_{100}$ is the micro-turbulence velocity in units of $100~{\rm km}~{\rm s}^{-1}$. We assumed a thermal equilibrium between all ions and electrons and neglected the micro-turbulence velocity. The oscillator strengths and ion fractions for each element were taken from SPEX. In our case, the absorption column density ($N_{\rm H}$) is the only free parameter.
In Figure \[fig:epplot\], we compare the transmission factors estimated from the data and those calculated with equation (\[eq:ep2\]). They are roughly consistent if $N_{\rm H}$ is (3.0–10)$\times10^{19}~{\rm cm^{-2}}$, which corresponds to a plasma depth of (10–34)$\times(n_{\rm H}/{\rm cm^{-3}})$ pc. Since the plasma depth is comparable to the diameter of N49, $\sim20$ pc, the result supports that RS occurs at the rim of N49. We note that we would underestimate the O abundance by about a factor of 1.8 if we do not take into account the RS effect (Table \[tab:parameters\]). This demonstrates the importance of RS in measuring elemental abundances as already pointed out by, e.g., [@Kaastra1995] and [@Miyata2008].
{width="15.0cm"}
In Figure \[fig:epplot\], we found that Ly$\alpha$ requires a higher column density than the other lines. A possible explanation would be that RS occurs also in the Galactic Halo (GH), as proposed by [@Gu2016b]. According to [@Nakashima2018], GH spectra are represented by a collisional ionization equilibrium (CIE) plasma model with $kT_{\rm e} \sim 0.26$ keV. Since such a plasma has a larger optical depth for the Ly$\alpha$ line than that of the He$\alpha$ resonance line, the GH should selectively reduce the intensity of Ly$\alpha$. We applied the CIE absorption model, [hot]{} [@dePlaa2004; @Steenbrugge2005], to the ${\rm NEI}-{\rm Gaussians}$ model. The electron temperature and Fe abundance of the [hot]{} model are fixed to 0.26 keV and 0.56 solar, respectively (the other elemental abundances are fixed to the solar values), by referring to [@Nakashima2018]. We found that the discrepancy cannot be reduced with the model, where the transmission factors of Ly$\alpha$ and He$\alpha$ are calculated to be $\sim0.43$ and $\sim0.80$, respectively.
The cause of the observed high Ly$\beta$/$\alpha$ ratio (the measured value is 0.18 as an upper limit) is not clear. Uncertainties in the model of the Fe-L lines [e.g., @Gu2019] might partially explain the result since Ly$\beta$ overlaps with the L$\alpha$ line. Another possibility to explain the high Ly$\beta$/$\alpha$ ratio is an effect related to RS. As discussed by [@Chevalier1980], Ly$\beta$ can be converted to H$\alpha$ by a $3p$-$2s$ transition in a collisionless shock through RS. However, this is less plausible because this effect reduces the intensity of Ly$\beta$ rather than that of Ly$\alpha$. RS by the ejecta may selectively reduce Ly$\alpha$, which we did not take into account in our analysis. We will be able to evaluate the ejecta contribution by measuring the intensity ratio of the emission lines in Fe K$\alpha$, which originates only from the ejecta [e.g., @Yamaguchi2014; @Uchida2015]. Since Fe K$\alpha$ is out of the wavelength band of the RGS, further observations with X-ray microcalorimeters [e.g., @Kelley2016] is required to clarify this point.
Conclusions {#sec:conclusions}
===========
We analyzed a high resolution X-ray grating spectrum of LMC SNR N49 obtained with the RGS aboard XMM-Newton. We found that the G-ratio of He$\alpha$ is significantly higher than that expected for a thin thermal plasma emission. The ratios of L$\alpha$ (3s–2p)/(3d–2p) and Ly$\beta$/$\alpha$ also show large residuals from the model. While an extra CX component well reproduces the G-ratio of the He$\alpha$ triplet, the residuals around L$\alpha$ and Ly$\alpha$ still remain. On the other hand, RS can fairly reproduce the RGS spectrum. We estimated the optical depth for the RS from the intensities of the scattered lines and found that the depth is roughly consistent with the size of N49. Our results indicate that RS has a particularly strong effect on the measurement of oxygen abundance. As demonstrated by [@Hitomi2018] with the X-ray microcalorimeter SXS aboard Hitomi [@Kelley2016], future missions such as the X-Ray Imaging and Spectroscopy Mission [XRISM; @Tashiro2018] and the Advanced Telescope for High ENergy Astrophysics [Athena; @Nandra2013] will provide useful means for further studying the effects of RS in SNRs.
We would like to thank Dr. Hiroya Yamaguchi for fruitful discussions. We also thank Dr. John Raymond for helpful advice. We deeply appreciate all the XMM-Newton team members. This work is supported by JSPS/MEXT Scientific Research grant Nos. JP25109004 (T.T. and T.G.T.), JP19H01936 (T.T.), JP26800102 (H.U.), JP19K03915 (H.U.), and JP15H02090 (T.G.T.).
Banas, K. R., Hughes, J. P., Bronfman, L., et al. 1997, , 480, 607
Broersen, S., Vink, J., Kaastra, J., et al. 2011, , 535, A11
Cash, W. 1979, , 228, 939
Chevalier, R. A., Kirshner, R. P., & Raymond, J. C. 1980, , 235, 186
Cline, T. L., Desai, U. D., Teegarden, B. J., et al. 1982, , 255, L45
den Herder, J. W., Brinkman, A. C., Kahn, S. M., et al. 2001, , 365, L7
de Plaa, J., Kaastra, J. S., Tamura, T., et al. 2004, , 423, 49
de Plaa, J., Zhuravleva, I., Werner, N., et al. 2012, , 539, A34
Dickey, J. M., & Lockman, F. J. 1990, , 28, 215
Gu, L., Kaastra, J., & Raassen, A. J. J. 2016, , 588, A52
Gu, L., Mao, J., Costantini, E., et al. 2016, , 594, A78
Gu, L., Raassen, A. J. J., Mao, J., et al. 2019, , 627, A51
Hitomi Collaboration, Aharonian, F., Akamatsu, H., et al. 2018, , 70, 10
Jansen, F., Lumb, D., Altieri, B., et al. 2001, , 365, L1
Kaastra, J. S. & Jansen, F. A. 1993, , 97, 873
Kaastra, J. S., & Mewe, R. 1995, , 302, L13
Kaastra, J. S., Mewe, R., & Nieuwenhuijzen, H. 1996, UV and X-ray Spectroscopy of Astrophysical and Laboratory Plasmas, 411
Kaastra, J. S. 2017, , 605, A51
Kastner, S. O., & Kastner, R. E. 1990, , 44, 275
Katsuda, S., Tsunemi, H., Mori, K., et al. 2012, , 756, 49
Kelley, R. L., Akamatsu, H., Azzarello, P., et al. 2016, , 99050V
Lodders, K., Palme, H., & Gail, H.-P. 2009, Landolt B[ü]{}rnstein, 4B, 712
Miyata, E., Masai, K., & Hughes, J. P. 2008, , 60, 521
Nakashima, S., Inoue, Y., Yamasaki, N., et al. 2018, , 862, 34
Nandra, K., Barret, D., Barcons, X., et al. 2013, arXiv e-prints, arXiv:1306.2307
Park, S., Burrows, D. N., Garmire, G. P., et al. 2003, , 586, 210
Park, S., Hughes, J. P., Slane, P. O., et al. 2012, , 748, 117
Pietrzy[ń]{}ski, G., Graczyk, D., Gieren, W., et al. 2013, , 495, 76
Rugge, H. R., & McKenzie, D. L. 1985, , 297, 338
Russell, S. C., & Dopita, M. A. 1992, , 384, 508
Sanders, J. S., Fabian, A. C., Allen, S. W., et al. 2004, , 349, 952
Schenck, A., Park, S., & Post, S. 2016, , 151, 161
Shigeyama, T. 1998, , 497, 587
Steenbrugge, K. C., Kaastra, J. S., Crenshaw, D. M., et al. 2005, , 434, 569
Str[ü]{}der, L., Briel, U., Dennerl, K., et al. 2001, , 365, L18
Tashiro, M., Maejima, H., Toda, K., et al. 2018, , 1069922
Turner, M. J. L., Abbey, A., Arnaud, M., et al. 2001, , 365, L27
Uchida, H., Koyama, K., & Yamaguchi, H. 2015, , 808, 77
Uchida, H., Katsuda, S., Tsunemi, H., et al. 2019, , 871, 234
van der Heyden, K. J., Bleeker, J. A. M., Kaastra, J. S., et al. 2003, , 406, 141
Waljeski, K., Moses, D., Dere, K. P., et al. 1994, , 429, 909
Xu, H., Kahn, S. M., Peterson, J. R., et al. 2002, , 579, 600
Yamaguchi, H., Badenes, C., Petre, R., et al. 2014, , 785, L27
Yamane, Y., Sano, H., van Loon, J. T., et al. 2018, , 863, 55
|
---
abstract: 'We derive an analytical theory for two interacting electrons in a $d$–dimensional random potential. Our treatment is based on an effective random matrix Hamiltonian. After mapping the problem on a nonlinear $\sigma$ model, we exploit similarities with the theory of disordered metals to identify a scaling parameter, investigate the level correlation function, and study the transport properties of the system. In agreement with recent numerical work we find that pair propagation is subdiffusive and that the pair size grows logarithmically with time.'
address: 'Service de Physique de l’État condensé, Commissariat à l’Energie Atomique Saclay, 91191 Gif-sur-Yvette, France'
author:
- 'Klaus Frahm[@kf], Axel Müller–Groeling[@amg] and Jean-Louis Pichard'
title: 'Effective $\sigma$ Model Formulation for Two Interacting Electrons in a Disordered Metal'
---
[2]{}
In the theory of Anderson localization one is accustomed to characterize localized single particle states by their typical extension. This so–called localization length is also believed to be the relevant length scale separating metallic and insulating behavior. However, it has become clear recently that two interacting electrons can form a pair that propagates up to a scale $L_2$, far beyond the one–particle localization length $L_1$. This effect, originally proposed by Shepeleyansky [@shep1] and subsequently investigated, made more precise and generalized in a series of papers [@imry; @numerics], has added an important aspect to the theory of localization. Inhowfar this phenomenon is amenable to experimental verification is presently under active consideration in the community. The theoretical information accumulated so far is, however, mainly of either qualitative or numerical nature. It is the purpose of this letter to develop an analytical theory of the effect for length scales $L > L_1$, where the interaction–assisted pairs are well defined.
Starting from an effective Hamiltonian we derive a nonlinear $\sigma$ model describing the problem in arbitrary dimensions. This enables us to address a number of interesting issues. We identify a certain scaling parameter, namely an effective “pair conductivity” $\sigma_{eff}$, which coincides in one dimension with the pair localization length $L_2\sim \sigma_{eff}\sim L_1^2$. In the perturbative regime, the $\sigma$ model gives rise to subdiffusive dynamics, where both the diffusion constant ${\cal D}_{eff}(\omega)$ of the pairs and the local density of diffusing pair states $\nu_{eff}(\omega)$ depend on frequency. As a consequence, we find that (for a certain time scale) the pair size grows logarithmically with time and that the diffusion is supressed by a similar logarithmic factor. These results have some very interesting relations with recent numerical findings [@shep3; @shep4] for two “interacting” kicked rotators. Finally, we identify two regimes with different spectral statistics separated by a pair Thouless energy $E_c^{(2)}$.
The first step is to motivate the effective random matrix Hamiltonian (ERMH) that serves as our starting point. We consider two interacting electrons on a lattice in $d$ dimensions with site diagonal disorder and nearest neighbor hopping elements. We assume that all one–electron eigenstates are localized with a typical localization length $L_1\gg 1$ (all length scales are measured in units of the lattice spacing). For $d=1,2$ this condition implies sufficiently weak disorder while for $d>2$ the disorder strength should be slightly above the critical disorder $W_c$. The localized one electron states are denoted by $\varphi_\rho(r)$, where the index $\rho$ refers to the “center” of the state. The Hamiltonian, expressed in the basis of symmetrized (i.e. both electrons have opposite spin) one–electron product states $|\rho_1 \rho_2\rangle$ with $\langle
r_1r_2|\rho_1\rho_2\rangle =
[\varphi_{\rho_1}(r_1)\varphi_{\rho_2}(r_2)+
\varphi_{\rho_1}(r_2)\varphi_{\rho_2}(r_1)]/\sqrt{2}$, reads $$\label{eq:1}
H_{\rho_1\rho_2,\,\rho_3\rho_4}=
(\varepsilon_{\rho_1}+\varepsilon_{\rho_2})\delta_{\rho_1,\rho_2}\,
\delta_{\rho_3,\rho_4}\,
+Q_{\rho_1 \rho_2,\, \rho_3 \rho_4} \quad.$$ Here, the $Q_{\rho_1 \rho_2,\, \rho_3 \rho_4}$ are the interaction matrix elements expressed in the basis of the product states $|\rho_1 \rho_2>$ and $\varepsilon_\rho$ is the one–electron energy of the localized state $\varphi_\rho(r)$. We emphasize that we consider the dynamics of the pairs in the basis of the above product states. Only for scales $L>L_1$ can this be easily interpreted as motion in real space. For a local Hubbard interaction with interaction strength $U$ we have [@shep1]: $$\label{eq:2}
Q_{\rho_1 \rho_2,\, \rho_3 \rho_4}
=2U\sum_r \varphi_{\rho_1}(r)\,\varphi_{\rho_2}(r)\,\varphi_{\rho_3}(r)\,
\varphi_{\rho_4}(r)\quad.$$ These interaction matrix elements become exponentially small whenever any two of the “positions” $\rho_j$ differ by much more than $L_1$. Shepelyansky mapped [@shep1] (for $d=1$) the Hamiltonian (\[eq:1\]) on a random band matrix with a superimposed, strongly fluctuating diagonal matrix. To do so he employed two crucial approximations: First he neglected all badly coupled pair states (those with $|\rho_1-\rho_2|\ge L_1$) and second he assumed the remaining coupling elements (\[eq:2\]) to be independent, equally distributed random variables. With the ansatz $\varphi_\rho(r)\sim e^{-|r-\rho|/ L_1}\,a_\rho(r)
/L_1^{d/2}$, where $a_\rho(r)$ is a random variable with $\langle a_\rho(r)\rangle =0$ and $\langle a_\rho(r)
a_{\tilde\rho}(\tilde r)\rangle =\delta_{\rho \tilde\rho}\,
\delta_{r \tilde r}$, and using the central limit theorem one gets the estimate [@shep1; @imry] $\left\langle Q_{\rho_1 \rho_2,\, \rho_3 \rho_4}^2\right\rangle \sim
U^2 L_1^{-3d}$ for the well–coupled states. Our ERMH defined below only relies on the second of the above assumptions: [*All*]{} pair states are taken into account, but the coupling matrix elements (\[eq:2\]) will still be taken to be independent Gaussian variables. This latter point is indeed quite a serious simplification since the correlations between different coupling matrix elements are neglected. We believe, however, that essential physics can be learnt even without paying attention to this refinement. In particular, the strongly fluctuating diagonal elements tend to eliminate the effect of those correlations.
Let us consider as configuration space a $2d$–dimensional lattice with two coordinate vectors $R\equiv \rho_1+\rho_2$ and $j\equiv(\rho_1-\rho_2)/2$ corresponding to twice the center of mass and half the distance of the two electrons, respectively. Our ERMH, ${\cal H}=\hat\eta+\hat \zeta$, consists of a strongly fluctuating diagonal part $\hat\eta$ with entries $\eta_R^{\,j}$ and of an interaction induced coupling matrix $\hat\zeta$. The $\eta_R^{\,j}$ correspond to the one–electron energies $\varepsilon_{\rho_1}+
\varepsilon_{\rho_2}$ in (\[eq:1\]). We take them to be independent random variables with the distribution function $\rho_0(\eta)$. We typically have $\rho_0(\eta)\simeq 1/(2W_b)$ for $|\eta|\le W_b$, where $W_b$ is the bandwidth of the disordered one–electron Hamiltonian. The matrix elements of $\zeta$ are independent Gaussian random variables with zero mean and variance $$\Bigl \langle \left(\hat\zeta_{R \tilde R}^{j \tilde j}\right)^2\Bigr
\rangle =\frac{1}{2}(1+\delta_{R \tilde R}\,\delta_{j \tilde j})
\,a(|R-\tilde R|)\, v(j)\,v(\tilde j)\quad.$$ Here $a(|R-\tilde R|)$ and $v(j)$ are smooth functions decaying exponentially on the scale $L_1$. We need not specify their particular form, it is sufficient to know their typical behavior $$\begin{aligned}
\label{eq:4}
a(|R|) & \sim &
\left\{\begin{array}{ll}
U^2 L_1^{-3d} &\ ,|R|\lesssim L_1\ , \phantom{\Big|}\\
U^2 L_1^{-3d} e^{-2|R|/L_1} &\ ,
|R|\gg L_1\ , \phantom{\Big|}\\
\end{array}\right.\\
\label{eq:5}
v(j)& \sim &
\left\{\begin{array}{ll}
1 &\ ,|j|\lesssim L_1\ , \phantom{\Big|}\\
e^{-4|j|/L_1} &\ ,|j|\gg L_1\ , \phantom{\Big|}\\
\end{array}\right.\end{aligned}$$ which is justified by (\[eq:2\]) and the exponential decrease (on the scale $L_1$) of the localized eigenfunctions. The function $a(|R-\tilde R|)$ describes how the coupling strength decreases exponentially if the distance of the centers of masse increases, wheras $v(j)$ describes how the increasing size of the pair states reduces the coupling.
To investigate the spectral statistics, the transport and the localization properties of the ERMH we apply the supersymmetric technique [@efetov; @vwz]. In the following, we choose a fixed realization of the diagonal elements $\eta_R^j$ and restrict the ensemble average to the random variables in $\hat\zeta$. This particular average is denoted by $\langle \cdots\rangle_\zeta$. We consider the generating functional $$F(J) =
\Bigl\langle \int {\textstyle
D\psi\,\exp\left[\frac{i}{2}\bar\psi
(E-{\cal H} +(\frac{\omega}{2}+i\varepsilon)\Lambda+J)\psi\right]
}\Bigr\rangle_\zeta \ .$$ Here, $\psi$ is a supervector with components $\psi_j(R)$, which are themselves 8–dimensional supervectors with entries $z_1,\bar z_1,\chi_1,\bar\chi_1,
z_2,\bar z_2,\chi_2,\bar\chi_2$, where the $z_\nu$ ($\chi_\nu$) are complex bosonic (fermionic) variables. The diagonal matrix $\Lambda$ has an equal number of eigenvalues $+1$ and $-1$ and describes the grading into advanced and retarded Greens’ functions. Furthermore, $\omega$ is a frequency, $J$ a source matrix, and $\bar\psi$ is given by $\bar\psi
=\psi^\dagger\Lambda$.
Choosing $J$ appropriately [@vwz; @iwz] and taking derivatives of $F(J)$ with respect to $J$ at $J=0$, one obtains ensemble averages of arbitrary products of Green’s functions, in terms of which the above mentioned properties can be studied. Skipping most of the technical detail (for more information see [@efetov; @vwz]), we derive a nonlinear $\sigma$ model from the functional $F(J)$. The physical implications are then discussed by comparing our formulation of the present problem with the supersymmetric description of a disordered metal given by Efetov [@efetov]. Performing the ensemble average, we get $$\begin{aligned}
\nonumber
\langle{\textstyle \exp(-\frac{i}{2}\bar\psi \zeta\psi)}\rangle_\zeta&=&
\exp\biggl(-\frac{1}{8}\sum_{R,\tilde R}a(|R-\tilde R|)\\
\label{eq:7}
&&\times\mbox{str} [K(R)\, K(\tilde R)]\biggr) \quad,\end{aligned}$$ where $K(R)$ is a $8\times 8$ super matrix given by $K(R)=\sum_j v(j)\,\psi_j(R)\,\bar\psi_j(R)$. The quartic term (\[eq:7\]) can be decoupled in the usual way [@efetov; @vwz] by a Hubbard-Stratonovich transformation. This introduces a functional integration over a field of $8\times 8$ super matrices $\sigma(R)$ with the same symmetries as $K(R)$. Proceeding [@framg2] in analogy to [@framg1; @fyodorov2], we apply a saddle point approximation, which is justified in the limit $L_1\gg 1$. Then we put $\sigma(R)=\Gamma_0
+i\Gamma_1 Q(R)$, where $Q(R)$ is an element of the orthogonal $\sigma$ model space [@efetov; @vwz] and fulfills the nonlinear constraint $Q(R)^2=1$. The quantities $\Gamma_0$ and $\Gamma_1$ are determined by the implicit equation $$\begin{aligned}
\nonumber
\Gamma_0+i\Gamma_1\Lambda & = & -B_0\sum_j v(j) \int d\eta\ \rho_0(\eta)\\
\nonumber&&\times
[E-\eta+i\varepsilon\Lambda+\frac{1}{2}v(j)
(\Gamma_0+i\Gamma_1\Lambda)]^{-1}\quad,\end{aligned}$$ having the approximate solutions (in the limit $L_1\gg 1$): $$\begin{aligned}
\nonumber
\Gamma_0 & \simeq & -B_0 S\,{\cal P}\int d\eta\ \rho_0(\eta)
\frac{1}{E-\eta}\quad,\\
\nonumber
\Gamma_1 & \simeq & \pi B_0 S\,\rho_0(E)\quad.\end{aligned}$$ Here, $B_0=\sum_R a(|R|)$, $S=\sum_j v(j)$, and ${\cal P}\int d\eta\ (\cdots)$ denotes a principal value integral. From this and (\[eq:4\],\[eq:5\]) we find the estimate $\Gamma_1\sim U^2/(W_b\, L_1^{d})\ll W_b$ if $U$ and $W_b$ are of the same order of magnitude. The matrix $A_{R,\tilde R}=a(|R-\tilde R|)$ defines a $d$–dimensional generalization of a random band matrix [@fyodorov] with the typical bandwidth $L_1$. As a consequence of the Hubbard–Stratonovich transformation the coupling in the bilinear term in the $Q$–field is given by the inverse, $(A^{-1})_{R,\tilde R}$. Therefore mainly the slow modes [@efetov; @fyodorov] (i.e. small momenta) contribute to the functional integral. Performing a standard gradient expansion [@efetov; @fyodorov] and going over to the continuum limit, we can express the generating functional as $F(J)=\int DQ\,
\exp[-{\cal L}_2[Q]-\Delta {\cal L}(J)]$. The effective action is given by $$\begin{aligned}
{\cal L}_2[Q] & = & \int dR\ \mbox{str}\left[
-\frac{\Gamma_1^2 B_2}{8 B_0^2}[\nabla_R\, Q(R)]^2
+f_R(Q(R))\right]
\nonumber\\
f_R(Q)&=&\frac{1}{2}\sum_j \mbox{str}\,\ln \Bigl( {\textstyle
E-\eta_R^j +(\frac{\omega}{2}
+i\varepsilon)\Lambda}
\nonumber\\
&&{\textstyle+\frac{1}{2}v(j)\,(\Gamma_0+i\Gamma_1 Q)}\Bigr) \quad,
\label{eq:11}\end{aligned}$$ where $B_2=1/(2d)\sum_R R^2\,a(|R|)$ and $\Delta {\cal L}(J)$ is the part of the action that accounts for the source matrix [@framg2]. The $Q$–dependent “potential” $f_R(Q)$ can be written in a more convenient form [@framg2] $$\label{eq:12}
f_R(Q)\simeq -i(\pi/4)\,\omega\ h(\Gamma_1/\omega)
\,\rho_0(E)\ \mbox{str}(Q\Lambda)$$ with $h(y)=iy\sum_{|j|\lesssim L_c} v(j)/[1+iy\, v(j)]$, where $L_c\approx
L_1\ln L_1$ is a cutoff length to be explained below.
Before discussing this result, we mention that it is also straightforward to derive the $\zeta$–averaged local density of states from the functional $F(J)$. It has the Breit–Wigner form $$\langle \rho_R^j(E)\rangle_\zeta =
\frac{1}{\pi}\ \frac{\frac{1}{2}\Gamma_1\, v(j)}
{[E-\eta_R^j+\frac{1}{2}\Gamma_0\, v(j)]^2 + [\frac{1}{2} \Gamma_1\,
v(j)]^2}$$ with an energy width $\Gamma_1\,v(j)$ [*that depends on the relative coordinate*]{} $j$. The levels $\eta_R^j$ of the product states $|\rho_1 \rho_2>$ acquire a finite width (or inverse life time) $\Gamma_1\,v(j)$ due to the interaction $\hat{\zeta}$. This width is of the order of $\Gamma_1$ for $|j|\lesssim L_1$ and decreases exponentially like $\Gamma_1\,e^{-4|j|/L_1}$ for $|j|\gg L_1$. This result demonstrates that product-states with ($|j|\gg L_1$) have an exponentially large life time because both electrons are simply localized far away from each other without feeling the Hubbard interaction. Let us determine a critical length $L_c$ such that inside a volume of this diameter all pair states are well coupled. Equating the effective level spacing $\Delta_{eff}=W_b\,L_c^{-2d}$ with the smallest possible level width $\Gamma_1\,e^{-2L_c/L_1}$, yields the estimate $L_c\sim L_1\ln L_1$. Product states with $|j|>L_c$ contribute to a discrete point spectrum while product states with $|j|\lesssim L_c$ are well coupled and correspond to interaction-assisted pairs of the size $L_c$.
The dynamics of these well-coupled product-states is conveniently described in terms of the $\sigma$ model (\[eq:11\]). At this point we reiterate that our model describes diffusion (and localization) in the space of product states $|\rho_1\rho_2\rangle$. The translation to coordinate space is straightforward provided we consider scales $L>L_1$, where the $|\rho_1\rho_2\rangle$ are associated with well defined positions. The interpretation of our $\sigma$ model is greatly facilitated by the close similarities between (\[eq:11\]) and the $\sigma$ model for a [*disordered metal*]{} as derived by Efetov [@efetov]. Comparing (\[eq:11\]) and (\[eq:12\]) with the standard $\sigma$ model in [@efetov], one can formally identify an effective diffusion constant ${\cal D}_{eff}$ and an effective local density of (diffusing) states $\nu_{eff}$, both of which depend on the frequency $\omega$: $$\begin{aligned}
\label{eq:14}
\sigma_{eff}=\nu_{eff}(\omega)\,{\cal D}_{eff}(\omega) & = &
\frac{1}{\pi}\,\frac{\Gamma_1^2\,B_2}{B_0^2}
\sim \frac{U^2}{W_b^2} L_1^2 \quad, \\
\label{eq:15}
\nu_{eff}(\omega) & = & \rho_0(E)\,h(\Gamma_1/\omega)\quad.\end{aligned}$$ We have introduced, via the Einstein relation, a formal “pair conductivity” $\sigma_{eff}$, which does [*not*]{} depend on frequency. The function $h(\Gamma_1/\omega)$ can be interpreted as the number of states contributing to the [*diffusion*]{}. A detailed analysis [@framg2] yields the following limiting cases: $$\label{eq:16}
h(\Gamma_1/\omega)\simeq
\left\{\begin{array}{ll}
i(\Gamma_1/\omega)\,S &,\quad \Gamma_1\ll|\omega|\ ,\phantom{\Big|}\\
\left[(L_1/4)\,\ln(\Gamma_1/|\omega|)\right]^d &
,\quad \tau_c^{-1}\lesssim |\omega|\ll \Gamma_1\ ,\phantom{\Big|}\\
\left[(L_1/4)\,\ln(\Gamma_1\,\tau_c)\right]^d &
,\quad |\omega|\ll \tau_c^{-1}\ .\phantom{\Big|}\\
\end{array}\right.$$ Here, $\tau_c^{-1}$ is the effective level spacing $\Delta_{eff}
=\Gamma_1\,e^{-2L_c/L_1}$ inside a blocks of size $L_c$. The physical picture of diffusing pairs makes sense in the regime $|\omega|\ll \Gamma_1$ only. Furthermore, in formal analogy to the condition $|\omega|\ll 1/\tau$ in a disordered metal [@efetov] (where $\tau$ is the elastic scattering time), only those product-states with $|\omega|\lesssim
\Gamma_1\,v(j)$ contribute to the diffusion. These remarks provide a physical interpretation of (\[eq:15\]) and (\[eq:16\]).
The analogy between the present problem and the problem of independent electrons in a disordered metal enables us to draw at least three important conclusions.
First, the coupling constant $\frac{\pi}{8}\nu_{eff}\,{\cal D}_{eff}=
\frac{\pi}{8}\sigma_{eff}$ can be identified as a universal scaling parameter. The corresponding scaling function is precisely the same as that of a disordered metal provided the latter is described by the “standard” $\sigma$ model [@efetov]. In particular, the perturbative evaluation of the $\beta$–function in $2+\varepsilon$ dimensions [@efetov; @wegner; @hikami] is equally valid for our present problem of [*diffusing*]{} or [*localized*]{} electron pairs (the term (\[eq:12\]) is not affected under the renormalization [@efetov]). This first conclusion also provides a rather rigorous justification for Imry’s [@imry] application of the Thouless scaling block picture [@thouless]. For $d=1$ we immediately recover Shepelyansky’s original result [@shep1] for the pair localization length $L_2$, $L_2\sim
\sigma_{eff}\sim (U^2/W_b^2) L_1^2$. It is important to note that this result has been obtained by taking into account [*all*]{}, also the badly coupled ($|j|\gg L_1$), pair states. In their study [@shep3] Borgonovi and Shepelyansky argue that the badly coupled states should lead to a logarithmic correction $L_2\sim L_1^2/\ln L_1$. We cannot confirm this result.
Second, we find that the pair dynamics is subdiffusive in agreement with recent numerical results of Borgonovi and Shepelyansky [@shep3; @shep4] who study two interacting kicked rotators. In the “pair–metallic” regime ($ L_1 < L\lesssim L_2$ for $d=1,2$) the $\sigma$ model can be treated perturbatively as in [@efetov]. The relevant diffusion propagator ${\cal R}(q,\omega)=[\sigma_{eff}\, q^2
- -i\omega\nu_{eff}(\omega)]^{-1}=[{\cal D}_{eff}(\omega)\,
\nu_{eff}(\omega)\, q^2 -i\omega\nu_{eff}(\omega)]^{-1}$ contains both the frequency dependent diffusion constant (\[eq:14\]) and density of states (\[eq:15\]). This $\omega$ dependence gives rise to subtle modifications of standard diffusion. Instead of trying to calculate the diffusion propagator $\tilde{\cal R}(R,t)=(2\pi)^{-(d+1)}\int d\omega\int dq\ {\cal R}(q,\omega)
\,e^{i(qR-\omega t)}$ as a function of $t$ and $R$ we replace in a qualitative approximation $\omega$ by $1/t$. Therefore we expect the pairs to diffuse according to (see also (\[eq:16\])) $$\langle R^2(t)\rangle \sim {\cal D}_{eff}(\frac{1}{t})\,t\sim
\left\{\begin{array}{ll}
L_1^2 &,\ t\ll\Gamma_1^{-1}\phantom{\Big|}\\
D_0\,\ln(\Gamma_1 t)^{-d}\,t & ,\ \Gamma_1^{-1}\ll t
\lesssim \tau_c \phantom{\Big|}\\
D_0\,\ln(L_1)^{-d}\,t &,\ \tau_c\lesssim t\ .\phantom{\Big|}\\
\end{array}\right.$$ We have used the notation $D_0=(U^2/W_b)\,L_1^{2-d}$. Obviously, $\langle R^2(t)\rangle$ increases weaker than linearly with time (subdiffusion). The number of diffusing states given by the function $h(\Gamma_1/\omega)$ in (\[eq:16\]) also depends on time. Putting $h(\Gamma_1/\omega) = [L_{eff}(\omega)]^d$ with $L_{eff}(\omega)$ the effective pair size, we get $L_{eff}(1/t)\sim L_1\,\ln(\Gamma_1 t)$ for $\Gamma_1^{-1}\ll t\lesssim \tau_c$ and $L_{eff}(1/t)\sim L_1 \ln L_1$ for $\tau_c\lesssim t$. This means that the pair size grows logarithmically with time as has also been found numerically in [@shep3; @shep4].
Third, calculating the two point correlation function $Y_2(\omega)$, we can study the level correlations of the well–coupled product-states. The nearly localized pair states are mainly uncorrelated with the diffusing states and among themselves. Therefore their contribution to $Y_2(\omega)$ essentially cancels out. For a finite system of size $L$ the diffusive dynamics determines another energy scale, the “pair Thouless energy” $E_c^{(2)}$. In our case $E_c^{(2)}$ is given by the implicit equation ${\cal D}_{eff}(E_c^{(2)})/L^2=E_c^{(2)}$. For $\omega < E_c^{(2)}$ the second term (\[eq:12\]) of the action ${\cal L}_2[Q]$ dominates the level correlations. The function $Y_2(\omega)$ can be calculated in complete analogy with [@efetov] and we recover the random matrix result: $$Y_2(\omega)=Y_2^{(GOE)}\bigl(\omega/\Delta(\omega)\bigr)
\quad,\quad |\omega|\ll E_c^{(2)} \quad.$$ Here, $\Delta(\omega)=[L^d\,\nu_{eff}(\omega)]^{-1}$ is the frequency dependent effective level spacing and $Y_2^{(GOE)}(r)$ is the universal spectral correlation function of the Gaussian orthogonal ensemble. For higher frequencies, i.e. $E_c^{(2)}
\ll|\omega|\ll \Gamma_1$, also the first (kinetic) term of ${\cal L}_2[Q]$ has to be taken into account. The necessary perturbative evaluation of $Y_2(\omega)$ proceeds analogously to the corresponding diagrammatical calculation of $Y_2(\omega)$ given by Altshuler and Shklovskii [@alt1]. The result is $$Y_2(\omega)\sim \frac{\Delta^2(\omega)}{\omega^2}\,\left(\frac{\omega}
{{\cal D}_{eff}(\omega)/L^2}\right)^{d/2}
,\ E_c^{(2)}\ll|\omega|\ll\Gamma_1 \ ,$$ where ${\cal D}_{eff}(\omega)/L^2$ can be interpreted as a frequency dependent Thouless energy setting the scale of the level correlation function.
In conclusion, starting from an effective Hamiltonian, we have derived a nonlinear $\sigma$ model for two interacting electrons in a random potential in arbitrary dimension. Exploiting the analogy with Efetov’s description of noninteracting electrons in a disordered metal, we identified a scaling parameter and investigated the level correlation function for the well coupled pairs. Furthermore, we analytically confirmed the numerical result that pair propagation is subdiffusive and that the pair size grows logarithmically with time.
Acknowledgments. We are grateful to D. Weinmann, Y. Imry and D.L. Shepelyansky for fruitful discussions. This work was supported by the European HCM program (KF) and a NATO fellowship through the DAAD (AMG).
[99]{}
Present address: Instituut-Lorentz, University of Leiden, P.O. Box 9506, 2300 RA Leiden, The Netherlands.
Present address: Max–Planck–Institut für Kernphysik, Postfach 10 39 80, D–69029 Heidelberg, Germany.
D. L. Shepelyansky, Phys. Rev. Lett. [**73**]{}, 2607 (1994).
Y. Imry, Europhys. Lett. [**30**]{}, 405 (1995).
K. Frahm, A. Müller–Groeling, J.–L. Pichard, and D. Weinmann, Europhys. Lett. [**31**]{}, 169 (1995); D. Weinmann, A. Müller–Groeling, J.–L. Pichard, and K. Frahm, Phys. Rev. Lett. [**75**]{}, 1598 (1995); F. von Oppen, T. Wettig, J. Müller, preprint Heidelberg.
F. Borgonovi and D. L. Shepelyansky, Brescia and Toulouse preprint, cond-mat/9503075.
F. Borgonovi and D. L. Shepelyansky, Brescia and Toulouse preprint, cond-mat/9507107.
K.B. Efetov, Adv. in Phys. [**32**]{}, 53 (1983).
\[vwz\] J. J. M. Verbaarschot, H. A. Weidenmüller, and M. R. Zirnbauer, Phys. Rep. [**129**]{}, 367 (1985).
\[iwz\] S. Iida, H.A. Weidenmüller, and J. A. Zuk, Ann. Phys. (N.Y.) [**200**]{}, 219 (1990).
K. Frahm, A. Müller–Groeling, J.–L. Pichard, in preparation.
K. Frahm and A. Müller–Groeling, Saclay preprint, cond-mat/9507057, Europhys. Lett. (in press).
Y. V. Fyodorov and A. D. Mirlin, Essen and Stuttgart preprint, cond-mat/9507043, Phys. Rev. B (in press).
Y. V. Fyodorov and A. D. Mirlin, Phys. Rev. Lett. [**67**]{}, 2405 (1991); ibid. [**69**]{}, 1093 (1992); ibid. [**71**]{}, 412 (1993); D. Mirlin and Y. V. Fyodorov, J. Phys. A: Math. Gen. [**26**]{}, L551 (1993).
F. J. Wegner, Nucl. Phys. B [**316**]{}, 663 (1989).
Prog. Theor. Phys. [**107**]{}, 213 (1992).
D. C. Thouless, Phys. Rev. Lett. [**39**]{}, 1167 (1977).
B. L. Al’tshuler and B. I. Shklovskii, Zh. Eksp. Teor. Fiz. [**91**]{}, 220 (1986) \[Sov. Phys. – JETP [**64**]{}, 127 (1986)\].
|
[The identity of information: how deterministic dependencies constrain information synergy and redundancy]{}
[Daniel Chicharro $^{1,2,\ast}$, Giuseppe Pica$^{2}$, Stefano Panzeri$^{2}$]{}
\
[$^2$ *Neural Computation Laboratory, Center for Neuroscience and Cognitive Systems@UniTn, Istituto Italiano di Tecnologia, Rovereto (TN), Italy.*]{}\
[$\ast$ daniel\_chicharro@hms.harvard.edu or daniel.chicharro@iit.it]{}\
[Abstract]{}
Understanding how different information sources together transmit information is crucial in many domains. For example, understanding the neural code requires characterizing how different neurons contribute unique, redundant, or synergistic pieces of information about sensory or behavioral variables. Williams and Beer (2010) proposed a partial information decomposition (PID) which separates the mutual information that a set of sources contains about a set of targets into nonnegative terms interpretable as these pieces. Quantifying redundancy requires assigning an identity to different information pieces, to assess when information is common across sources. Harder et al.(2013) proposed an identity axiom stating that there cannot be redundancy between two independent sources about a copy of themselves. However, Bertschinger et al.(2012) showed that with a deterministically related sources-target copy this axiom is incompatible with ensuring PID nonnegativity. Here we study systematically the effect of deterministic target-sources dependencies. We introduce two synergy stochasticity axioms that generalize the identity axiom, and we derive general expressions separating stochastic and deterministic PID components. Our analysis identifies how negative terms can originate from deterministic dependencies and shows how different assumptions on information identity, implicit in the stochasticity and identity axioms, determine the PID structure. The implications for studying neural coding are discussed.
**Keywords**: Information theory, mutual information decomposition, synergy, redundancy
\
Introduction {#s1}
============
The characterization of dependencies between the parts of a multivariate system helps to understand its function and its underlying mechanisms. Within the information-theoretic framework, this problem can be investigated by breaking down into parts the joint entropy of a set of variables [@Amari01; @Schneidman03b; @Ince10] or the mutual information between sets of variables [@Panzeri99b; @Chicharro14b; @Timme14]. These approaches have many applications to study dependencies in complex systems such as genes networks [e.g. @Watkinson09; @Erwin09; @Chatterjee16], neural coding and communication [e.g. @Panzeri08; @Marre09; @Faes16], or interactive agents [e.g. @Katz11; @Flack2012; @Ay2012].
An important aspect of how information is distributed across a set of variables concerns whether different variables provide redundant, unique or synergistic information when combined with other variables. Intuitively, variables share redundant information if each variable carries individually the same information carried by other variables. Information carried by a certain variable is unique if it is not carried by any other variables or their combination, and a group of variables carries synergistic information if some information arises only when they are combined. The presence of these different types of information has implications for example to determine how the information can be decoded [@Latham05], how robust it is to disruptions of the system [@Rauh14b], or how the variables’ set can be compressed without an information loss [@Tishby99].
Characterizing the distribution of redundant, unique, and synergistic information is especially relevant in systems neuroscience, to understand how information is distributed in neural population responses. This requires identifying the features of neural responses that represent sensory stimuli and behavioural actions [@Averbeck06; @Panzeri17] and how this information is transmitted and transformed across brain areas [@Wibral14; @Timme16]. The breakdown of information into these different types of components can determine the contribution of different classes of neurons, and of different spatiotemporal components of population activity [@Panzeri10; @Panzeri15]. Moreover, the identification of synergistic or redundant components of information transfer may help to map dynamic functional connectivity and integration of information across neurons or networks [@Valdes11; @Vicente10; @Ince15; @Deco15].
Despite the notions of redundant, unique, and synergistic information seem at first intuitive, their rigorous quantification within the information-theoretic framework has proven to be elusive. Synergy and redundancy have traditionally been quantified with the measure called interaction information [@McGill54] or co-information [@Bell03], but this measure does not quantify them separately, and the presence of one or the other is associated with positive or negative values, respectively. Synergy has also been quantified using maximum entropy models as the information that can only be retrieved from the joint distribution of the variables [@Amari01; @Olbrich15; @Perrone16].
However, a recent seminal work of [@Williams10] introduced a framework, called Partial Information Decomposition (PID), to more precisely and simultaneously quantify the redundant, unique, and synergistic information that a set of variables (or primary sources) S has about a target X. This decomposition has two cornerstones. The first is the definition of a general measure of redundancy following a set of axioms that impose desirable properties, in agreement with the corresponding abstract notion of redundancy [@Williams10b]. The second is the construction of a redundancy lattice, structured according to these axioms, which reflects a partial ordering of redundancies for different sets of variables [@Williams10].
The PID framework has been adopted and further developed by many others [e.g. @Harder12; @Bertschinger12; @Griffith13; @Ince16; @Rauh17; @Chicharro17b]. However, its concrete implementation and the properties that the PID terms should have continue to be debated [@Rauh17b; @Ince16]. [@Harder12] argued that the original redundancy measure of [@Williams10] quantifies only quantitatively equal amounts of information and not information that is qualitatively the same. They introduced a new axiom, namely the identity axiom, which states that when the target is a copy of two sources redundancy should correspond to the mutual information between them, and cancel for independent sources. Several measures that fulfill the identity axiom have been subsequently proposed in substitution of the original redundancy measure [@Harder12; @Griffith13; @Bertschinger12]. However, [@Bertschinger12b] provided a counterexample illustrating that in the multivariate case the identity axiom is incompatible with ensuring the nonnegativity of the PID terms. Like the target-source copy example used to motivate the axiom, also this counterexample involves deterministic target-sources dependencies.
Here we study in a general way the effect of deterministic target-sources dependencies in the PID decomposition. While the counterexample of [@Bertschinger12b] reveals the inconsistency of nonnegativity and the identity axiom, it does not provide a general clue of why they are incompatible and what has to be modified. Furthermore, while the identity axiom was advocated based on a concrete example, the question of generally determining the identity of different pieces of information has not been addressed independently of proposing specific redundancy measures. As we show in what follows, our analysis addresses more generally and explicitly the question of assigning information identity and the cause of negative PID terms.
We start this work reviewing the PID decompositions (Section \[s2\]). We then introduce two alternative forms, a weak and a strong form, of a stochasticity axiom that imposes constraints to the existence of synergistic information in the presence of deterministic target-source dependencies (Section \[s3\]). Using these axioms, we derive general expressions that separate each PID term into a stochastic and a deterministic component for the bivariate (Section \[s41\]) and trivariate (Section \[s51\]) case. We show how these axioms lead to two alternative generalizations of the identity axiom (Section \[s42\]) and check if several previously proposed redundancy measures conform to these generalizations (Section \[s43\]). We reconsider the examples used by [@Bertschinger12b], characterizing their bivariate and trivariate decompositions and illustrating how in general negative PID terms can occur (Sections \[s44\] and \[s52\]). Finally, comparing the stochasticity and identity axioms, we discuss the implications of assuming a certain criterion to identify pieces of information in the target in the presence of deterministic target-sources dependencies, and concretely of assuming that their identity is related to specific sources (Section \[s45\] and Section \[s53\]).
A review of the PID framework {#s2}
=============================
The seminal work of [@Williams10] introduced a new approach to decompose the mutual information into a set of nonnegative contributions. Let us consider first the bivariate case. Assume that we have a target $X$ formed by one variable or by a set of variables and two variables $1$ and $2$ which information about $X$ we want to characterize. [@Williams10] argued that the mutual information of each variable can be expressed as
$$I(X;1) = I(X;1.2)+I(X;1 \backslash 2),
\label{e1}$$
and similarly for $I(X;2)$. The term $I(X;1.2)$ refers to a redundancy component between variables $1$ and $2$, which can be obtained either by knowing $1$ or $2$ separately. The terms $I(X;1 \backslash 2)$ and $I(X;2 \backslash 1)$ quantify a component that is unique of $1$ and of $2$, respectively, that is, the information that can be obtained from one of the variables alone but that cannot be obtained from the other alone. Furthermore, the joint information of $12$ can be expressed as
$$I(X;12) =I(X;1.2)+I(X;1 \backslash 2)+I(X;2 \backslash 1)+I(X;12 \backslash 1,2),
\label{e2}$$
where the term $I(X;12 \backslash 1,2)$ refers to the synergistic information of the two variables, which is unique for the joint source $12$ with respect to both variables alone. Therefore, given the standard information-theoretic chain rule equalities [@Cover06]
$$\begin{aligned}
I(X;12) &= I(X;1)+I(X;2|1)\\
&= I(X;2)+I(X;1|2),\end{aligned}$$
\[e3\]
the conditional mutual information is decomposed as
$$I(X;2|1) = I(X;2 \backslash 1)+I(X;12 \backslash 1,2),
\label{e4}$$
and analogously for $I(X;1|2)$. Conditioning removes the redundant component but adds the synergistic component so that conditional information is the sum of the unique and synergistic terms.
In this decomposition a redundancy and a synergy component can exist simultaneously. In fact, [@Williams10] showed that the measure of co-information [@Bell03] that previously had been used to quantify synergy and redundancy, defined as
$$C(X;1;2) = I(i;j)-I(i;j|k) = I(i;j)+I(i;k)-I(i;j,k)
\label{e5}$$
for any assignment of $\{X,1,2\}$ to $\{i,j,k\}$, corresponds to the difference between the redundancy and the synergy terms of Eq.\[e2\]:
$$C(X;1;2) = I(X;1.2)-I(X;12 \backslash 1,2).
\label{e5_2}$$
More generally, [@Williams10] defined decompositions of the mutual information about a target $X$ for any multivariate set of variables $S$. This general formulation relies on the definition of a general measure of redundancy and the construction of a redundancy lattice. In more detail, to decompose the information $I(X;S)$, [@Williams10] defined a *source* $A$ as a subset of the variables in $S$, and a *collection* $\alpha$ as a set of sources. They then introduced a measure of redundancy to quantify for each collection the redundancy between the sources composing the collection, and constructed a redundancy lattice which reflects the relation between the redundancies of all different collections. Here we will generically refer to the redundancy of a collection $\alpha$ by $I(X;\alpha)$. Furthermore, following [@Chicharro17], we use a more concise notation than in [@Williams10]: For example, instead of writing $\{1\}\{23\}$ for the collection composed by the source containing variable $1$ and the source containing variables $2$ and $3$, we write $1.23$, that is, we save the curly brackets that indicate for each source the set of variables and we use instead a dot to separate the sources. We will also refer to the single variables in $S$ as *primary* sources when we want to specifically distinguish them from general sources that can contain several variables.
[@Williams10b] argued that a measure of redundancy should comply with the following axioms:
- **Symmetry**: $I(X;\alpha)$ is invariant to the order of the sources in the collection.
- **Self-redundancy**: The redundancy of a collection formed by a single source is equal to the mutual information of that source.
- **Monotonicity**: Adding sources to a collection can only decrease the redundancy of the resulting collection, and redundancy is kept constant when adding a superset of any of the existing sources.
The monotonicity property allows introducing a partial ordering between the collections, which is reflected in the redundancy lattice. Self-redundancy links the lattice to the joint mutual information $I(X;S)$ because at its top there is the collection formed by a single source including all the variables in $S$. Furthermore, the number of collections to be included in the lattice is limited by the fact that adding a superset of any source does not change redundancy. For example, the redundancy between the source $12$ and the source $2$ is all the information $I(X;2)$. Accordingly, the set of collections that can be included in the lattice is defined as
$$\mathcal{A}(S)= \{ \alpha \in \mathcal{P}(S) \backslash \{ \emptyset \}: \forall\ A_i, A_j \in \alpha, A_i \nsubseteq A_j\},
\label{e6}$$
where $\mathcal{P}(S) \backslash \{ \emptyset \}$ is the set of all nonempty subsets of the set of nonempty sources that can be formed from $S$. This domain reflects the symmetry axiom in that it does not distinguish the order of the sources. For this set of collections, [@Williams10] defined a partial ordering relation to construct the lattice:
$$\forall\ \alpha, \beta \in \mathcal{A}(S), (\alpha \preceq \beta \Leftrightarrow \forall B \in \beta, \exists A \in \alpha, A \subseteq B),
\label{e7}$$
that is, for two collections $\alpha$ and $\beta$, $\alpha \preceq \beta$ if for each source in $\beta$ there is a source in $\alpha$ that is a subset of that source. This partial ordering relation is reflexive, transitive, and antisymmetric. In fact, the consistency of the redundancy measures with the partial ordering of the collections, that is, that $I(X;\alpha) \leq I(X;\beta)$ iif $\alpha \preceq \beta$, represents a stronger form of the monotonicity axiom.
The mutual information multivariate decomposition was constructed in [@Williams10] by implicitly defining partial information measures $\Delta(X;\alpha)$ associated with each node $\alpha$ of the redundancy lattice, such that redundancy measures are obtained from the sum of partial information measures:
$$I(X;\alpha) = \sum_{\beta \in \downarrow \alpha} \Delta(X;\beta),
\label{e8}$$
where $\downarrow\alpha$ refers to the set of collections lower than or equal to $\alpha$ in the partial ordering, and hence reachable descending from $\alpha$ in the lattice. The partial information measures are obtained inverting Eq.\[e8\] by applying the principle of inclusion-exclusion to the terms in the lattice [@Williams10]. Redundancy lattices for $S$ being bivariate and trivariate are shown in Figure \[fig1\]. As studied in [@Chicharro17], a mapping exists between the terms of the trivariate and bivariate PID decompositions, as indicated by the colors and labels.
An extra axiom, called the identity axiom, was later introduced by [@Harder12] specifically for the bivariate redundancy measure:
- **Identity axiom:** For two sources $A_1$ and $A_2$, $I(A_1 \cup A_2; A_1.A_2)$ is equal to $I(A_1;A_2)$.
[@Harder12] pointed out that with the original measure of redundancy of [@Williams10] a nonzero redundancy is obtained for two independent variables and a target being a copy of them, and that a measure quantifying the amount of qualitatively common information and not the quantitatively equal amount of information should be zero in this case. [@Ince16] has specifically differentiated between the identity axiom, which assumes the form of redundancy for any degree of dependence between the primary sources when the target is a copy of them, and a more concrete property, namely the Independent Identity property, which only requires that redundancy about the copy target cancels when the primary sources are independent. Several alternative measures have been proposed that fulfill this additional axiom [@Harder12; @Bertschinger12; @Griffith13]. The properties of the PID terms have been characterized, either based on the axioms and the structure of the redundancy lattice [@Chicharro17; @Pica17], or also considering the properties of specific measures [@Bertschinger12b; @Griffith13; @Rauh14; @Griffith14; @Banerjee15; @Rauh17b]. However, only for specific cases such as multivariate Gaussian systems with univariate targets, it has been shown that several of the proposed measures are actually equivalent [@Barret15; @Faes17].
Stochasticity axioms for synergistic information {#s3}
================================================
We start our analysis of deterministic relations between the target $X$ and the set of primary sources $S$ by enunciating two versions of a stochasticity axiom for synergistic information. These axioms impose different constraints on the synergy terms when the dependency of the target upon the sources can be partly deterministic and partly stochastic. We consider first the weak axiom. This axiom is motivated by the idea that if any subset $X'$ of variables comprised in the target $X$ can be completely determined by a source corresponding to a subset $S'$ of $S$ then there cannot be synergistic information about that subset $X'$ between $S'$ and any other sources. This is because $S'$ can already provide all the information about $X'$ without combining it with any other variable. Accordingly, the weak axiom assumes that:
**Synergy weak stochasticity axiom:** *For a target $X$ and a set of variables $S$, if there is a subset $X'= X(S')$ of $X$ such that it can be determined completely from a subset $S'$ of $S$, then*
$$\Delta(X; \alpha) = \Delta(X \backslash X(S'); \alpha)\ \ \forall \alpha \notin \bigcup_{i \in S} \downarrow i,
\label{r0}$$
*where $\downarrow i$ indicates the collections reachable by descending the lattice from node $i$, corresponding to a primary source.*
That is, any synergistic term about $X$ is equal to the synergy about a target $X \backslash X(S')$ that does not include the variables $X(S')$ determined by $S'$.
This weak form of the stochasticity axiom implies that the sources cannot have synergistic information about a part $X’$ of the target that is deterministically related to them. However, the axiom does not restrict that those variables in $S'$ that determine $X'$ may provide information about other parts of the target in a synergistic way. Conversely, a strong form of the stochasticity axiom imposes that the variables in $S'$ can only provide synergistic information to the degree that they are not themselves deterministically related to the variables in $X'$. In particular, it assumes that:
**Synergy strong stochasticity axiom:** *For a target $X$ and a set of variables $S$, if there is a subset $X' = X(S')$ of $X$ such that it can be determined completely from a subset $S'$ of $S$, then*
$$\Delta(X; \alpha) = \Delta(X \backslash X(S'); \alpha | X(S'))\ \ \forall \alpha \notin \bigcup_{i \in S} \downarrow i.
\label{r0b}$$
That is, the synergy about $X$ is equal to the synergy in the lattice associated with the decomposition of the mutual information $I(X \backslash X(S'); S |X(S'))$ that $S$ has about $X \backslash X(S')$ conditioned on $X(S')$. The logic of the strong axiom can be better appreciated when, instead of just a functional relation, some of the primary sources are themselves contained in the target (i.e.$X(S') = S'$). In this case the strong axiom states that there cannot be any synergistic contribution involving variables in $S'$. In contrast to the weak axiom, these contributions cannot be present even if providing information about $X \backslash X(S')$. The motivation is that the primary sources in $S'$ cannot provide other information about the target than the information about themselves, which can be provided without combining them with any other variable. Accordingly, when $X'=S'$,
$$\Delta(X; \alpha) = 0 \ \ \forall \alpha \notin \bigcup_{i \in S} \downarrow i\ : \exists A \in \alpha, S' \cap A \neq \emptyset,
\label{r1b}$$
that is, there is no synergy for those nodes whose collection has a source containing a variable from $S'$.
In this work we will study how, based on these axioms, bivariate and trivariate PID decompositions are affected by deterministic relations between the target and the primary sources. To simplify the derivations we will focus on the case in which the target $X$ contains some of the primary sources themselves. A more general formulation that considers target variables determined as a function of the sources leads to the same main qualitative conclusions. All the derivations follow from the relations characteristic of the redundancy lattice, and we do not need to select any specific measure of redundant, unique or synergistic information.
Bivariate decompositions with deterministic target-sources dependencies {#s4}
=======================================================================
We start with the bivariate case. Consider that the target $X$ may have some overlap $X \cap 12$ with the sources $1$ and $2$. Following the weak stochasticity axiom (Eq.\[r0\]) synergy is expressed as:
$$I(X; 12 \backslash 1,2) = I(X \backslash 12 ; 12 \backslash 1,2).
\label{r2}$$
On the other hand, for the strong stochasticity axiom (Eq.\[r1b\]) we have: $$I(X; 12 \backslash 1,2) = \begin{cases} I(X \backslash 12; 12 \backslash 1,2) \ \ \mathrm{if}\ X \cap 12 = \emptyset \\ 0 \ \ \mathrm{if}\ X \cap 12 \neq \emptyset\end{cases}.
\label{r2b}$$
Given these expressions of the synergistic terms we will now derive how deterministic relations affect the other PID terms.
General formulation {#s41}
-------------------
For both forms of the stochasticity axiom we will derive expressions of unique and redundant information in the presence of a target-sources overlap. These derivations follow the same procedure: First, given that unique and synergistic information are related to conditional mutual information by Eq.\[e4\], the synergy stochasticity axioms determine the form of the unique information terms. Second, once the unique information terms are derived, their relation to the mutual information together with the redundancy term (Eq.\[e1\]) allows identifying redundancy. For both unique and redundant information terms this procedure separates stochastic and deterministic components. However, how these components are combined depends on the order in which stochastic and deterministic target-sources dependencies are partitioned. In particular, using the chain rule [@Cover06] of the mutual information we can separate the information about the target in two different ways:
$$\begin{aligned}
I(X;12) &= I(X \backslash 12; 12) + I(X \cap 12 ; 12| X \backslash 12) \\
&= I(X \cap 12; 12) + I(X \backslash 12 ; 12| X \cap 12).\end{aligned}$$
\[r3\]
The first case considers first the stochastic dependencies and after the conditional deterministic dependencies. In the second case, this order is reversed. We will see that for each axiom only one of these partitioning orders leads to expressions that additively separate stochastic and deterministic components for each PID terms.
### PID decompositions with the weak axiom {#s411}
We start with the PID decomposition of $I(X;12)$ derived from the weak axiom (Eq.\[r2\]). Consider the mutual information partitioning order of Eq.\[r3\]a, which can be reexpressed as
$$\begin{split}
I(X;12) = I(X \backslash 12; 12) + H(X \cap 12| X \backslash 12),
\label{r6}
\end{split}$$
that is, the second summand corresponds to the conditional entropy of the overlapping target variables given the non-overlapping ones. We now proceed analogously for the PID terms. Since conditional mutual informations are the sum of a unique and a synergistic information component (Eq.\[e4\]), we have that
$$\begin{split}
&I(X; 1 \backslash 2) = I(X; 1|2) - I(X; 12 \backslash 1,2)\\
&= I(X \backslash 12 ;1|2) + I(X \cap 12; 1|2, X \backslash 12)- I(X \backslash 12; 12 \backslash 1,2).
\label{r4}
\end{split}$$
The first equality indicates that unique information is conditional information minus synergy. The second equality uses the chain rule to separate the conditional mutual information stochastic and deterministic components, and applies the stochasticity axiom to remove the overlapping part of the target in the synergy term. Using again the relation between conditional mutual information and unique and synergistic terms but now for the target $X \backslash 12$ we get
$$I(X; 1 \backslash 2) = I(X \backslash 12 ;1 \backslash 2)+ H(X \cap 1| 2, X \backslash 12),
\label{r5}$$
where we also used that $I(X \cap 12; 1|2, X \backslash 12)$ equals the entropy $H(X \cap 1| 2, X \backslash 12)$. Accordingly, the unique information of $1$ can be separated into a stochastic component, the unique information about target $X \backslash 12$, and a deterministic component, the entropy $H(X \cap 1| 2, X \backslash 12)$. This last term is zero if the target does not contain source $1$. If it does, it quantifies the entropy that only $1$ as a source can explain about itself as part of the target, which is thus an extra unique information contribution.
Once we have identified the unique information stochastic and deterministic components we can use the relation of unique and redundant information with the mutual information (Eq.\[e1\]) to characterize the redundancy. We get that:
$$I(X; 1.2) = I(X \backslash 12; 1.2) + \begin{cases} 0 \ \ \mathrm{if}\ X \cap 12 = \emptyset \\ I(1;2|X \backslash 12)\ \ \mathrm{if}\ X \cap 12 \neq \emptyset \end{cases}.
\label{r7}$$
Therefore, it suffices that one of the two primary sources overlaps with the target so that their conditional mutual information given the non-overlapping target variables contributes to redundancy.
We can follow the same procedure to derive expressions for the unique and redundant information terms but applying the other mutual information partitioning order of Eq.\[r3\]b. The resulting terms can be compared in Table \[tab1\] and are derived in more detail in Appendix \[a0\], where we also show the consistency between the expressions obtained with each partitioning order. In the upper part of the table we collect the decompositions into stochastic and deterministic contributions for each PID term and for the two partitioning orders. To simplify the expressions, their form is shown only for the case of $X \cap i \neq \emptyset$. With the alternative partitioning order, both the expressions of unique information and redundancy contain a cross-over component, namely the synergy about $X \backslash 12$, instead of being expressed in terms of the unique information and redundancy of $X \backslash 12$, respectively. Furthermore, the separation of the deterministic and stochastic components is not additive. This indicates that, while the chain rule holds for the mutual information, it is not guaranteed that the same type of separation holds separately for each PID term. Only for a certain partitioning order, when stochastic dependencies are considered first, unique and redundant information terms derived from the weak axiom can both be separated additively into a stochastic and a deterministic component without cross-over terms. In the lower part of the table we individuate the deterministic PID components obtained from the partitioning order for which each PID term is separated additively into a stochastic and deterministic component.
Term Decomposition
--------------------------- --------------------------------------------------------------------------------------------------------------------------------------------- --
$I(X; ij \backslash i,j)$ $I(X \backslash ij; ij \backslash i,j)$
$I(X; i \backslash j)$ $\begin{array}{c} I(X \backslash ij; i \backslash j) + H(i|j, X \backslash ij)\\ H(i|j) - I(X \backslash ij; ij \backslash i,j)\end{array}$
$I(X; i.j)$ $\begin{array}{c} I(X \backslash ij; i.j) + I(i;j |X \backslash ij )\\ I(i;j) + I(X \backslash ij; ij \backslash i,j) \end{array} $
Term Measure
$\Delta_d(X; ij)$ $0$
$\Delta_d(X; i)$ $H(i|j, X \backslash ij)$
$\Delta_d(X; i.j)$ $I(i;j|X \backslash ij)$
: Decompositions of synergistic, unique, and redundant information terms into stochastic and deterministic contributions obtained assuming the weak stochasticity axiom. For each term we show the decompositions resulting from two alternative mutual information partitioning orders (Eq.\[r3\]), which are consistent with each other (see Appendix \[a0\]). For the partitioning order leading to an additive separation of each PID term into a stochastic and deterministic component we also individuate the deterministic contributions $\Delta_d(X; \beta)$. Synergy has only a stochastic component, according to the axiom (Eq.\[r2\]). Expressions of unique information come from Eqs.\[r5\] and \[r9\], and the ones of redundancy from Eqs.\[r7\] and \[r12\]. The expressions have been simplified with respect to the equations, indicating their form for the case $X \cap i \neq \emptyset$. The terms $\Delta_d(X; \beta)$ have analogous expressions for $X \cap j \neq \emptyset$ when a symmetry exists between $i$ and $j$ and are zero otherwise.[]{data-label="tab1"}
### PID decompositions with the strong axiom {#s412}
The procedure to derive the unique and redundant PID terms is the same if the strong stochasticity axiom is assumed, but determining synergy with Eq.\[r2b\] instead of Eq.\[r2\]. To simplify the expressions we indicate in advance that if $X \cap 12 = \emptyset$ each PID term with target $X$ is by definition equal to the one with target $X \backslash 12$ and we only provide expressions derived with some target-sources overlap. In contrast to the weak axiom, with the strong axiom an additive separation of stochastic and deterministic components is obtained with the partitioning order of Eq.\[r3\]b. See Appendix \[a0\] for details about the other partitioning order. For the unique information we obtain $$I(X; 1 \backslash 2) = \begin{cases} I(X \backslash 12 ;1 \backslash 2) + I(X \backslash 12; 12 \backslash 1, 2)\ \ \mathrm{if}\ X \cap 1 = \emptyset \\ H(1|2)\ \ \mathrm{if}\ X \cap 1 \neq \emptyset \end{cases},
\label{r31}$$ and for the redundancy $$I(X; 1.2) = I(1;2).
\label{r32}$$ As before, we summarize the PID decompositions in Table \[tab1b\]. Comparing Table \[tab1\] and \[tab1b\] we see that the expressions obtained with the weak and strong axiom differ because of a cross-over contribution, corresponding to the synergy about $X \backslash 12$, which is transferred from redundancy to unique information. This is due to the synergy constraints imposed by each axiom: the strong axiom assumes that there is no synergy, and hence this part of the information has to be transferred to the unique information because the sum of synergy and unique information is constrained to equal the conditional mutual information. As a consequence, redundancy is reduced by an equivalent amount to comply with the constraints that relate unique informations and redundancy to mutual informations. Furthermore, like for the weak axiom, the chain rule property does not generally hold for each PID term separately. PID terms are consistent with the mutual information decompositions obtained applying the chain rule, but depending on the partitioning order and on the version of the axiom assumed, information contributions are redistributed between different PID terms, and between their stochastic and deterministic components. This is in agreement with previous concrete counterexamples provided by [@Bertschinger12b] and [@Rauh14] that showed that the chain rule does not hold in general for each PID term.
Term Decomposition
--------------------------- --------------------------------------------------------------------------------------------------------------------------------------------- --
$I(X; ij \backslash i,j)$ $0$
$I(X; i \backslash j)$ $\begin{array}{c} I(X \backslash ij; i \backslash j)+ I(X \backslash ij; ij \backslash i,j) + H(i|j, X \backslash ij)\\ H(i|j) \end{array}$
$I(X; i.j)$ $\begin{array}{c} I(i;j |X \backslash ij )+ I(X \backslash ij; i . j)- I(X \backslash ij; ij \backslash i, j)\\ I(i;j) \end{array} $
Term Measure
$\Delta_d(X; ij)$ $0$
$\Delta_d(X; i)$ $H(i|j)$
$\Delta_d(X; i.j)$ $I(i;j)$
: Decompositions of synergistic, unique, and redundant information terms into stochastic and deterministic contributions obtained assuming the strong stochasticity axiom. The table is analogous to Table \[tab1\]. Synergy cancels according to the axiom (Eq.\[r2b\]). Expressions of unique information come from Eqs.\[r29\] and \[r31\], and the ones of redundancy from Eqs.\[r30\] and \[r32\]. Again, expressions are shown for the case $X \cap i \neq \emptyset$, with the corresponding symmetries holding for $X \cap j \neq \emptyset$ and with terms $\Delta_d(X; \beta)$ equal to zero otherwise.[]{data-label="tab1b"}
The relation between the synergy stochasticity axioms and the redundancy identity axiom {#s42}
---------------------------------------------------------------------------------------
The two forms of the stochasticity axiom result in different expressions for the redundancy term. We now examine how these expressions are related to the redundancy identity axiom [@Harder12]. This axiom determines redundancy for a very specific deterministic target-sources relation, namely when there are two primary sources $1$ and $2$ and the target is equal to them, $X = 12$. It is straightforward to see that the redundancy identity axiom is subsumed by both stochasticity axioms:
**Proposition:** *The fulfillment of the synergy weak or strong stochasticity axioms implies the fulfillment of the redundancy identity axiom*
*Proof*: If $X = 12$ then $X \cap 12 = 12$ and $X \backslash 12 = \emptyset$. For the weak stochasticity axiom, redundancy (Eq.\[r7\]) reduces to $I(12; 1.2) = I(1;2)$. For the strong stochasticity axiom, Eq.\[r32\] is already $I(12; 1.2) = I(1;2)$. $\Box$
Therefore, the stochasticity axioms represent two alternative generalizations of the redundancy identity axiom: First, they do not only consider a target that is a copy of the primary sources, but a target with any degree of overlap or functional dependence with the sources. Second, they are not restricted to the bivariate case but are formulated for any number of primary sources.
Redundancy terms derived from each axiom coincide for the particular case that is addressed by the identity axiom, but more generally differ. The strong axiom leads to redundancy being equal to $I(1;2)$ not only for the case addressed by the identity axiom but in general when $X \cap 12 \neq \emptyset$ and independently of which are the non-overlapping target variables. Conversely, with the weak axiom redundancy depends on these other variables. We will further discuss these differences below based on concrete examples.
The compliance of the stochasticity axioms by concrete measures {#s43}
---------------------------------------------------------------
We now check for several proposed measures if they conform to the predictions of the stochasticity axioms. In particular we examine the original redundancy measures of [@Williams10], the one based on the pointwise common change in surprisal of [@Ince16], and the one based on maximum entropy of [@Bertschinger12].
It is well-known that the redundancy measure of [@Williams10] does not comply with the identity axiom [@Harder12]. Even if $I(1;2)=0$, a redundancy $I(12; 1.2)>0$ can be obtained. This excess of redundancy leads to less unique information, which in turn produces a nonzero synergistic contribution inconsistently with both the weak and strong stochasticity axioms. Neither the redundancy measure of [@Ince16] complies with the identity axiom, and thus it does not conform to the stochasticity axioms.
Conversely, the redundancy measure of [@Bertschinger12] fulfills the identity axiom. To see how more broadly it compares to the redundancies derived from the weak and strong axioms consider the following example: if there is a target $X = 23$ and two sources $1$ and $2$, according to the weak axiom (Table \[tab1\]) the redundancy $I(23; 1.2)$ should be equal to the sum of a deterministic component $I(1;2)$ and of a stochastic component $I(3; 12 \backslash 1,2)$. Conversely, according to the strong axiom, the redundancy equals only $I(1;2)$ (Eq.\[r32\]). The redundancy measure of [@Bertschinger12] is calculated by minimizing the mutual information that the sources have about the target within the family of distributions which preserves the marginals of the target with each of the sources. In particular, redundancy is calculated as the co-information for the distributions leading to the minimal information within the family. In this example preserving p(1, 23), the marginal of the target $23$ and source $1$, implies preserving the whole joint distribution $p(1,2,3)$ and hence the minimal information within the family is equal to the original information. Accordingly, given Eq.\[e5\], for the [@Bertschinger12] measure
$$\begin{split}
I(23; 1.2) &= I(23;1)+I(23;2)-I(23;1,2)\\
&= I(23;1)+ H(2)-[H(2)+I(3;1|2)]\\
&= I(1;2).
\label{r14b}
\end{split}$$
This redundancy measure coincides with the one predicted from the strong axiom. This holds in general, because it is a property of the co-information that if $X$ overlaps with $1$ or $2$ then $C(X;1;2) = I(1;2)$. Given this matching of the redundancy, it is straightforward to check that the rest of PID terms match as well.
Illustrative systems {#s44}
--------------------
So far we have derived the predictions for the PID decompositions according to each version of the stochasticity axiom, pointed out the relation with the identity axiom, and checked how different previously proposed measures conform to these predictions. We now use concrete examples to further examine the decompositions. In particular, we reconsider two examples that have been previously studied in [@Bertschinger12b] and [@Rauh14], namely the decompositions of the mutual information about a target jointly formed by the inputs and the output of a logical XOR operation or of an AND operation. We first describe below the decompositions obtained and in Section \[s45\] we will discuss them in relation to underlying assumptions on how to assign an identity to different pieces of information of the target. The deterministic components for these examples are derived without assuming any specific measure of redundancy, unique, or synergistic information. The stochastic components have already been previously studied and some of the terms depend on the measures selected. We will indicate previous work examining these terms when required.
### XOR {#s441}
We start with the XOR operation. Consider an output variable $3$ determined through the operation $3=1\ \mathrm{XOR}\ 2$, resulting in the joint probability displayed in Figure \[fig2a\]A. We also indicate the values of the information-theoretic measures needed to calculate the PID bivariate decompositions studied here and that will also serve for the trivariate decompositions addressed in Section \[s52\]. We want to examine the decomposition of $I(123; 1,2)$, where the target is composed by the three variables. For each version of the stochasticity axiom we will focus on the mutual information partitioning order that allows separating additively a stochastic and a deterministic component of each PID term.
Since $X \backslash 12 =3$, for the weak axiom the PID decomposition (Figure \[fig2a\]B) can be obtained by implementing the decomposition of $I(3; 12)$ and separately calculating the deterministic PID components $\Delta_d(123; \beta)$ as collected in Table \[tab1\]. The decomposition of $I(3;12)$ for the XOR operation has been characterized repeatedly [e.g. @Griffith13], showing that all terms are zero except the synergy, which contributes one bit of information. There is no stochastic redundancy or unique information because $I(3;i)=0$ for $i=1,2$. Regarding the deterministic components, redundancy has $1$ bit because $I(1;2|3)=1$. The deterministic unique information components are zero because $H(i|jk)=0$ for $i=1,2$ and, according to the axiom, there is no deterministic synergy.
In the case of the strong axiom (Figure \[fig2a\]C), since both primary sources overlap with the target, only deterministic components appear in the decomposition when selecting the partitioning order that additively separates stochastic and deterministic contributions, as indicated in Table \[tab1b\]. By assumption, there is no synergy. Since $I(1;2)=0$, the redundancy is also zero and all the information is contained in the unique information terms. As pointed out for the generic expressions, the two decompositions differ in the transfer of the stochastic component of synergy to unique information, which in turns forces an equivalent transfer from redundancy to unique information.
We can compare these decompositions with previous analyses of this example [@Bertschinger12b; @Rauh14]. In these studies the PID terms were derived using the identity axiom. In particular, it was argued that, since $12$ totally determines $3$, the target can be reduced from $123$ to $12$ and redundancy is thus $I(123; 1.2)= I(12;1.2)$. Then, using the identity axiom, $I(12;1.2)= I(1;2)$, which is zero. This reasoning leads to the same decomposition derived from the strong stochasticity axiom.
### AND {#s442}
As a second example, we now consider the AND operation. Following the weak axiom, again the decomposition can be obtained by implementing the PID decomposition of $I(3; 12)$ and separately calculating the deterministic PID components from Table \[tab1\], using the joint distribution of inputs and output displayed in Figure \[fig2b\]A. The PID decomposition of $I(3; 12)$ for the AND operation has also been already characterized [e.g. @Harder12]. For $I(123;12)$, each PID term contributes half a bit. Unique contributions come exclusively from the deterministic components. Each unique information has half a bit because the output and one input determine the other input only when not both have a value of $0$. Redundancy is also half a bit, but it comes in part from a stochastic component and in part from a deterministic one. The stochastic component was previously determined by [@Harder12], indicating that this redundancy about the output appears intrinsically because of the AND mechanism, even if the inputs are independent. The deterministic component appears because, although the inputs are independent, conditioned on the output $I(1;2|3)>0$. The synergy $I(3; 12 \backslash 1, 2)=0.5$ was also previously determined by [@Harder12]. This PID decomposition differs from the one obtained with the weak axiom for the XOR example. Conversely, with the strong axiom the decomposition is the same as for the XOR example, because it is completely determined by $I(1;2)=0$. This latter decomposition is again in agreement with the arguments of [@Bertschinger12b] and [@Rauh14] based on the identity axiom.
Implications of the stochasticity axioms for the notions of redundant, unique, and synergistic information {#s45}
----------------------------------------------------------------------------------------------------------
Each version of the stochasticity axiom implies a different quantification of redundancy. Since the value of unconditional and conditional mutual informations does not depend on the PID decomposition, for the bivariate case the extra constraints on synergy of the strong axiom imply assigning more information to unique information, which in turns restricts the amount of redundancy, as compared to the constraints implied by the weak axiom. This restriction imposes that, if there is some target-sources overlap, redundancy only depends on the mutual information between the primary sources and is independent of dependencies between the sources and the other target variables.
We now examine in more detail how these different quantifications are related to the notion of redundancy as common information about the target that can be obtained by observing either source alone. The key point is how identity is assigned to different pieces of information in order to assess which information about the target carried by the sources is qualitatively common. In particular, for the strong axiom, its logic is that if a source is part of the target it cannot provide other information about the target than the information about itself and thus, if the other source does not contain information about it, this information is unique. Implicit in this argument there is the assumption that when a primary source is part of the target we can still identify and separate the bits of information about that source from the information about the rest of the target. This idea regarding the identity of the bits that are shared also motivated the introduction of the identity axiom. Although the identity axiom was formulated for sources with any degree of dependence, its motivation was mainly based on the case in which $I(1;2)=0$, when the sources are independent [@Harder12]. In this case, we can identify the bits of information related to variable $1$ and the ones to variable $2$, and thus redundancy, if it quantifies the qualitatively equal information that is shared and not only common amounts of information, has to cancel.
But assigning an identity to pieces of information in the target is in general less straightforward. For the XOR example, we will now consider different combinations of mutual information partitioning orders for $I(123;1)$ and $I(123;2)$ and show how, if the assignment of identity of the bits in the target $123$ is based on their association with the sources $1$ and $2$, the interpretation of redundant and unique information is ambiguous. First, consider that we decompose the information of each primary source as follows:
$$\begin{split}
I(123;1) &= I(1;1)+I(2;1|1)+I(3;1|12) = I(1;1) = H(1)\\ I(123;2) &= I(2;2)+ I(1;2|2)+I(3;2|12) = I(2;2)= H(2).
\label{r33}
\end{split}$$
If we assume that we can identify the bit of information carried by each primary source about the target, these decompositions would suggest that there is no redundant information, since each source only carries one bit of information about itself and $I(1;2)=0$ for the XOR system. However, keeping the same decomposition of $I(123;1)$, we can consider alternative decompositions of $I(123;2)$:
$$\begin{aligned}
I(123;2) &= I(3;2)+ I(1;2|3)+I(2;2|13) = I(1;2|3)= H(1) \\ &= I(1;2)+ I(3;2|1)+ I(2;2|13) = I(3;2|1) = H(3).\end{aligned}$$
\[r34\]
The redundancy and unique information terms should not depend on how we apply the chain rule to $I(123;2)$. However, in contrast to Eq.\[r33\], the first decomposition of Eq.\[r34\]a suggests, if the identity of the bits in the target is related to the overlapping variables in the sources, that there is redundancy between sources $1$ and $2$. In particular, in $I(1;2|3)$, if variable $1$ as part of the target is associated with source $1$, then the contribution of $I(1;2|3)$ to $I(123;2)$ can be interpreted as redundant with the information $I(123;1) = I(1;1)$ in Eq.\[r33\] that source $1$ has about itself. The second decomposition in Eq.\[r34\]b further challenges the interpretation of redundancy and unique information based on the assignment of an identity to bits of information in the target given their association with the overlapping target variables. Given $I(3;2|1)$, source $2$ provides information about $3$. But the amount of information contained in $3$ is shared with $1$ and $2$, given the conditional dependencies of the XOR system. Moreover, in $I(3;2|1)$ the conditioning on variable $1$ as a target, if this variable is associated with source $1$, would suggest that $I(3;2|1)$ contributes information to $I(123;2)$ by combining the two sources. Accordingly, when using the target-sources correspondence to identify pieces of information, different partitioning orders of the mutual information ambiguously suggest that information can be obtained uniquely, redundantly, or even in a synergistic way. These problems arise because, in contrast to the case of $I(12;1,2)$ with independent sources, in the XOR system the two bits of $123$ cannot be identified as belonging to a certain variable, but can only be distinguished as the bit that any first variable provides alone, and the bit that a second variable provides combined with the first. This lack of correspondence between pieces of information and individual variables is incompatible with the identification of the pieces of information based on the association of target-sources overlapping variables.
The differences in the quantification of redundancy with each stochasticity axiom are related to the alternative interpretations of identity discussed for Eqs.\[r33\] and \[r34\]. A notion of redundancy compatible with the weak axiom considers the common information about the target that can be obtained by observing either source alone or conditioned on variables in the target. Indeed, the deterministic component of redundancy comprises the conditional dependence of the sources given the rest of the target, $I(1;2|X \backslash 12)$, when there is a target-sources overlap, and thus fits to Eq.\[r34\]a. Conversely, with the strong axiom, when there is a target-source overlap, redundancy equals $I(1;2)$ independently of $X \backslash 12$, in agreement with the logic of Eq.\[r33\]. We will further discuss the implications of the axioms about information identity in Section \[s53\] after dealing with the trivariate case.
Trivariate decompositions with deterministic target-sources dependencies {#s5}
========================================================================
We now extend the analysis to the trivariate case. This is relevant because, in contrast to the bivariate case, it has been argued that with more than two sources the PID decompositions that jointly comply with the monotonicity and the identity axiom do not guarantee the nonnegativity of the PID terms [@Bertschinger12b]. In particular, [@Bertschinger12b] used the XOR example we reconsidered above as a counterexample to show that negative terms appear. Therefore, we would like to be able to extend the general formulation of Section \[s41\] to the trivariate case, and thus apply it to further examine the XOR and AND examples by identifying each component of the trivariate decomposition of $I(123;123)$ and not only of the decomposition of $I(123;12)$.
General formulation {#s51}
-------------------
While in the bivariate decomposition there is a single PID term that involves synergistic information, in the trivariate lattice of Figure \[fig1\]B all nodes which are not reached descending from $1$, $2$, or $3$ imply synergistic information, and the nodes of the form $i.jk$ too. The weak and strong axioms impose constraints on these terms given Eqs.\[r0\] and \[r0b\], respectively.
### PID decompositions with the weak axiom {#s511}
We start with the weak stochasticity axiom. Consider that any of the three primary sources is part of the target and how synergy may appear. For example, consider the extra information obtained when observing $12$ together instead of $1$ and $2$ separately. This information is distributed across the nodes reached descending from $12$ that are not already reached descending from $1$ or from $2$. But the axiom states that if any of $1$ and $2$ is contained in the target, $12$ cannot have synergistic information about them. Furthermore, if $3$ is part of the target and $12$ provides some extra information about it not given by $1$ and $2$ alone, this information is redundant with the one that $3$ provides about itself, and hence is contained in the node $3.12$, which is still reachable descending from $3$. Accordingly, the weak stochasticity axiom implies that
$$\Delta_d(X; \alpha) =0\ \ \forall \alpha \notin \bigcup_{i=1,2,3} \downarrow i.
\label{r15}$$
We can then proceed analogously to the bivariate case to characterize the remaining deterministic contributions to PID terms. We again apply the mutual information chain rule to separate stochastic and deterministic dependencies. Again we focus on the partitioning order that considers first the stochastic dependencies, since only this order leads to an additive separation of stochastic and deterministic components for each PID term. With this partitioning order
$$\begin{split}
I(X;123) &= I(X \backslash 123;123)+ I(X \cap 123; 123| X \backslash 123)\\
&= I(X \backslash 123;123)+ H(X \cap 123| X \backslash 123).
\label{r15b}
\end{split}$$
Following derivations analogous to the ones of Section \[s41\] (see Appendix \[a1\]), it can be seen that deterministic contributions are further restricted by
$$\Delta_d(X; \alpha) =0\ \ \forall \alpha \notin \bigcup_{i \in X \cap \{1, 2, 3\}} \downarrow i.
\label{r15c}$$
If a certain primary source $i$ does not overlap with the target, the nodes that can only be reached descending from its corresponding node will not have a deterministic component. This can be understood intuitively. For example, suppose that the target includes $1$ and $2$ but not $3$. Then the entropy in Eq.\[r15b\] is $H(12|X \backslash 123)$. The PID terms that can be reached descending from $3$ and not from $1$ or $2$ are $\Delta(X;3)$ and $\Delta(X;3.12)$ (see Figure \[fig1\]B). The first quantifies information that can only be obtained from $3$, and not from $12$. The second is information that can be obtained from $3$ or from $12$, but not from $1$ or $2$ alone. But since all the information about $12$ can be obtained either from $1$ or $2$, these nodes do not contribute to the decomposition of $H(12|X \backslash 123)$.
Term Measure
---------------------- -------------------------------------------------------------------------------------- --
$\Delta_d(X; i)$ $H(i|jk, X \backslash ijk)$
$\Delta_d(X; i.jk)$ $I(X \backslash jk; jk \backslash j,k) - I(X \backslash ijk; jk \backslash j,k)$
$\Delta_d(X; i.j)$ $I(i;j|k, X \backslash ijk)- \left [ \Delta_d(X; i.jk) + \Delta_d(X; j.ik) \right ]$
$\Delta_d(X; i.j.k)$ $C(i;j;k| X \backslash ijk) + \Delta_d(X; i.jk)+\Delta_d(X; j.ik)+\Delta_d(X; k.ij)$
: Deterministic components of the PID terms for the trivariate decomposition derived from the weak stochasticity axiom. All terms not included in the table have no deterministic component due to the axiom. These expressions correspond to the case in which the primary source $i$ overlaps with the target. If $i$ does not overlap, $\Delta_d(X; i)$ and $\Delta_d(X; i.jk)$ are zero, while the other terms depend on their characteristic symmetry for the other variables $j$ and $k$, and cancel if none of the variables with the corresponding symmetry overlaps with the target. See the main text and Appendix \[a1\] for details.[]{data-label="tab2"}
Using the condition of Eq.\[r15c\], we can use the same procedure as in Section \[s41\] to derive the expressions of all the deterministic PID trivariate components. These terms are collected in Table \[tab2\] and we leave the detailed derivations and discussion for Appendix \[a1\]. Their expressions are indicated for the case in which variable $i$ is part of the target and are symmetric with respect to $j$ or $k$ when this symmetry is characteristic of a certain PID term, or cancel otherwise, consistently with Eq.\[r15c\].
The first two terms $\Delta_d(X;i)$ and $\Delta_d(X;i.jk)$ are nonnegative, the former because it is an entropy and the latter because according to the axiom adding a new source can only reduce synergy. But for the terms $\Delta_d(X; i.j)$ and $\Delta_d(X; i.j.k)$ it is not guaranteed that they are nonnegative. For $\Delta_d(X; i.j)$, we will see examples of negative values below. For $\Delta_d(X; i.j.k)$, the conditional co-information can be negative if there is synergy between the primary sources when conditioning on the non-overlapping target variables, and this can happen when there is no synergy about the target, leading to a negative value. Therefore, following the weak stochasticity axiom, the PID decomposition cannot ensure the nonnegativity of all terms when deterministic target-sources dependencies are in place. We will further discuss this limitation after examining the full trivariate decomposition for the XOR and AND examples.
### PID decompositions with the strong axiom {#s512}
With the strong form of the axiom, not only deterministic but stochastic components of synergy are restricted. There cannot be any synergistic contribution that involves a source overlapping with the target. Eq.\[r1b\] can be applied with $S = 123$. Furthermore, since the cancelation of synergistic terms has to hold not only for the terms $\Delta(X; \alpha)$ of the trivariate lattice but also of any bivariate lattice associated with it, given the mapping of PID terms between these lattices (Figure \[fig1\]), this implies that in the trivariate lattice also the PID terms of the form $i.jk$ are constrained. In fact, there is only one case in which synergistic contributions can be nonzero if there is any target-sources overlap for the trivariate case, and this is when only one variable overlaps. Consider that only variable $1$ is part of the target. Since there cannot be any synergy involving $1$, all synergistic PID terms contained in $I(X;1|2)$, $I(X;1|3)$, or $I(X;1|23)$ have to cancel, and also $\Delta(X; 2.13)$ and $\Delta(X; 3.12)$. It can be checked that this includes all synergistic terms except $\Delta(X; 23)$ and $\Delta(X; 1.23)$. The former quantifies synergy about other target variables and the latter synergy redundant with the information of $1$ itself. With more than one primary source overlapping with the target all synergistic terms have to cancel for the trivariate case.
Term Measure
---------------------- ------------------------------- --
$\Delta_d(X; i)$ $H(i|jk)$
$\Delta_d(X; i.jk)$ $I(i; jk \backslash j,k)$
$\Delta_d(X; i.j)$ $I(i;j|k)-\Delta_d(X; i.jk)$
$\Delta_d(X; i.j.k)$ $C(i;j;k)+ \Delta_d(X; i.jk)$
: Deterministic components of the PID terms for the trivariate decomposition derived from the strong stochasticity axiom. All terms not included in the table have no deterministic component due to the axiom. Again, the expressions shown here correspond to the case in which the source $i$ overlaps with the target. For $\Delta_d(X; i.jk)$ we further consider that neither $j$ nor $k$ overlap with the target, and otherwise this term cancels. If $i$ does not overlap, $\Delta_d(X; i)$ is zero, while the other terms depend on their characteristic symmetry for the other variables $j$ and $k$ and cancel otherwise. See the main text and Appendix \[a1\] for details.[]{data-label="tab3"}
Like for the weak axiom, we now leave the derivations for Appendix \[a1\]. The PID deterministic terms are collected in Table \[tab3\], again for simplicity showing their expressions for the case in which $i$ overlaps with the target. The form of the expressions respects the symmetries of each term. For example, if $j$ instead of $i$ overlaps with the target then $\Delta_d(X; i.j) = I(i;j|k)-\Delta_d(X; j.ik)$. Note however that, because $\Delta_d(X; j.ik)=0$ when $i$ overlaps, if both $i$ and $j$ overlap then $\Delta_d(X; i.j) = I(i;j|k)$. See Appendix \[a1\] for further details.
In comparison to the deterministic components derived from the weak axiom there are two differences: First, the lack of conditioning on $X \backslash ijk$ is due to the reversed partitioning order selected. Like for the bivariate case, the deterministic PID components are independent of the non-overlapping target variables when adopting the strong stochasticity axiom. Second, assuming the strong axiom the terms $\Delta_d(X; i.jk)$ can only be nonzero if $j$ and $k$ are not contained in the target and when more than one source overlaps all terms of the form $\Delta_d(X; i.jk)$ cancel. In that case it is clear that $\Delta_d(X; i.j.k)$ can be negative, since the co-information can be negative. Therefore, also the PID decomposition derived from the strong axiom does not ensure nonnegativity. We will now show examples of negative terms for both PID decompositions.
Illustrative systems {#s52}
--------------------
We now continue the analysis of the XOR and AND examples by decomposing $I(123;123)$. Since now $X \backslash 123 = \emptyset$ the decompositions are completely deterministic and are obtained calculating the PID components described in Table \[tab2\] and Table \[tab3\]. Accordingly, given that deterministic and joint PID terms are equal, we will use $\Delta(X; \beta)$ instead of $\Delta_d(X; \beta)$ to refer to them.
### XOR {#s521}
We start with the XOR example and the decomposition derived from the weak stochasticity axiom (Figure \[fig3\]A). We show the trivariate decomposition of $I(123;123)$ and also again the decomposition of $I(123; 12)$, now indicating the mapping of the nodes with the trivariate decomposition. For the trivariate lattice we only show the nodes lower than the ones of the primary sources because for all others the corresponding terms are zero (Eq.\[r15\]). The PID terms are calculated considering Table \[tab2\] and the information-theoretic quantities displayed in Figure \[fig2a\]A.
The trivariate terms $\Delta(X; i)$ are all zero, because any two variables determine the third. This is also reflected in the terms $\Delta(X; i.jk)$ having $1$ bit. The terms $\Delta(X; i.j)$ are all equal to $-1$ bit. These terms should quantify the redundant information between two variables which is unique with respect to the third, but their interpretation is impaired by the negative values. Furthermore, $\Delta(X; i.j.k)=2$, so that redundancy monotonicity does not hold. However, it can be checked that the values obtained are consistent from the point of view of the constraints linking PID terms and mutual informations. Similarly, the calculated PID components are consistent between the bivariate and trivariate decompositions. In particular, the sum of the nodes with the same color or label in the trivariate lattice equals the corresponding node in the bivariate lattice. This equality holds for the joint bivariate lattice, and not for the deterministic lattice alone, even if in the trivariate case the lattice is uniquely deterministic. This reflects a transfer of stochastic synergy in the bivariate case to deterministic redundancy in the trivariate case (see yellow nodes labeled with $d$).
We now consider the decomposition derived from the strong axiom (Figure \[fig3\]B). In this case also $\Delta(X; i)$ are all zero because any two variables determine the third, but now also $\Delta(X; i.jk)$ are zero. This is because the axiom assumes that there is no synergy involving any of the primary sources overlapping with the target. $\Delta(X; 3.12)=0$ is consistent with the lack synergy for the decomposition of $I(123;12)$, as indicated by the mapping of the yellow nodes labeled with $d$. Also the mapping of all other PID terms is consistent. In particular, the $1$ bit corresponding to the unique informations of the bivariate decomposition are contained in the terms $\Delta(X; i.j)= I(i;j|k)$ of the trivariate one. In comparison to the decomposition from the weak axiom, these terms are not negative, but instead a negative value is obtained for $\Delta(X; i.j.k)$. Therefore nonnegativity is neither fulfilled for this decomposition.
In fact, this decomposition was used by [@Bertschinger12b] and [@Rauh14] as a counterexample to show that with more than two sources there is no decomposition that can simultaneously comply with the redundancy monotonicity axiom and the identity axiom and also lead to global nonnegativity of the PID terms. In Section \[s411\] we pointed out that the bivariate decomposition of $I(123;12)$ derived from the strong stochasticity axiom coincides with the one obtained from the arguments of [@Bertschinger12b] based on the identity axiom. However, the trivariate decomposition we obtain is not the same as they did. The divergence occurs because [@Bertschinger12b], after arguing that $I(123; 1.2)=0$ based on the identity axiom as we discussed in Section \[s441\], further argued that this implies $I(123; 1.2.3)=0$, based on redundancy monotonicity, and hence also $\Delta(X; i.j)=0$. Once having all terms $\Delta(X; i.j)=0$ and $\Delta(X; i.j.k)=0$, this led them to find a negative value for $\Delta(X; 12.13.23)$. However, as we can see in the trivariate decomposition of Figure \[fig3\]B, to respect the monotonicity axiom having $I(123; 1.2)=0$ does not necessarily imply that $\Delta(X; i.j)=0$ and $\Delta(X; i.j.k)=0$. Indeed, monotonicity is respected by already having a negative value $\Delta(X; i.j.k)=-1$, as we get from assuming the strong stochasticity axiom. Accordingly, once recognising this possibility, the results obtained from the strong axiom are compatible with the arguments of [@Bertschinger12b], showing that this decomposition is a counterexample for global nonnegativity, but indicating that a negative value appears already in $\Delta(X; i.j.k)$.
### AND {#s522}
We present the AND decomposition as a further example. All PID terms are derived using the information-theoretic quantities of Figure \[fig2b\]A in combination with Tables \[tab2\] and \[tab3\]. Like for the XOR case, the mapping of trivariate to bivariate decompositions is consistent. Again, both trivariate decompositions contain some negative term. With the strong axiom, while the bivariate decompositions for the XOR and AND example are equal, the trivariate PID terms differ substantially, reflecting the different symmetries of each operation. In particular, for the XOR case there is a symmetry between all three variables while in the AND case only between the inputs.
PID terms nonnegativity and Information identity {#s53}
------------------------------------------------
The analysis of the trivariate PID decompositions derived from the weak and strong versions of the stochasticity axiom shows explicitly how nonnegative PID terms can arise in the presence of deterministic target-sources dependencies. The form of the deterministic components indicated in Tables \[tab2\] and \[tab3\] provides a general understanding of how negative PID terms can occur, beyond the concrete counterexample examined in [@Bertschinger12b]. In particular, the axioms enforce that certain pieces of information are attributed to redundancy or unique information terms because their identity is associated to the sources, and hence deterministic components of the decomposition are bounded to the low part of the redundancy lattice, which leads to negative terms in order to conform to the lattice structure and to the relations between PID terms and mutual informations. Furthermore, as argued by [@Rauh17b] based on continuity arguments for the mutual information, the same problem of obtaining negative PID terms is expected to occur not only when deterministic target-sources dependencies exist, but also in the limit of strong dependencies tending to be deterministic.
Avoiding negative PID terms would require changing the assumptions about how deterministic target-sources dependencies constrain the terms. The common assumption of the weak and strong axioms that information about an overlapping variable can only be redundant or unique may be too restrictive and implies assuming that we can assign an identity to pieces of information in the target as exclusively related to the overlapping variable. Conversely, only in few cases the identity of a bit can be assigned to a single variable, as it is the case for $I(12;12)$ with $1$ and $2$ being independent, which motivated the identity axiom and in particular the Independent Identity property [@Ince16] that requires $I(12;1.2)=0$ for this case.
In general, the overall composition of the target affects the identity of each piece of information. For example, even if $1$ and $2$ are independent and for target $12$ we can identify each piece as associated with a different variable, if we incorporate a third variable $3$ determined by $1$ and $2$, now identity will generally change, and depend on the specific operation that generates $3$ from $1$ and $2$. This is the case, for example, of the XOR system when $123$ is taken as the target. The two bits of $123$ cannot be identified as belonging to a certain variable, but only as the bit that any first variable provides alone, and the bit that a second variable provides combined with the first. Oppositely, on one hand, the strong axiom assumes that each source alone can uniquely provide a bit, corresponding to its own identity, as reflected in the decomposition of $I(123;12)$ (see Figure \[fig3\]B). On the other hand, with the weak axiom, the second bit is classified as synergy, consistently with the idea that retrieving it requires the combination of two variables (Figure \[fig3\]A). However, because the weak axiom still assumes that any information about an overlapping variable has to be redundant or unique, it imposes that the synergy is contained in the terms $\Delta(X; i.jk)$ in the trivariate decomposition and not in terms corresponding to nodes upper than the ones of single variables. This means that the axiom is still not compatible with the identification of the two bits as the one that can be obtained from a single variable and the one that can only be obtained from the combination of two variables.
In Figure \[fig6\]A we show a trivariate decomposition that is consistent with this identification of the bits for the XOR example. The first bit is the one that can be obtained from any variable alone, and is thus redundant to all three variables. None of the variables can provide more information alone, so all the remaining PID terms associated with nodes lower than $1$, $2$, and $3$ are zero. Furthermore, since the second bit can be obtained by the combination of any two variables, its information is redundant to any pair, and is thus contained in the node $12.13.23$. Since in total there are two bits the rest of PID terms are also zero. This decomposition is nonnegative, but does not conform to any version of the stochasticity axiom (neither to the identity axiom) because $\Delta(X; 12.13.23)>0$. In fact, it corresponds to the one obtained using as redundancy measure $\mathrm{min} \{ I(X; A_i)\}$ over the sources $A_i$ of each collection [@Bertschinger12b], which is closely related to the measure of [@Williams10].
With this same measure we also get a nonnegative decomposition for the AND example (Figure \[fig6\]B). Like for the XOR case, when we move from target $12$ to target $123$ there is no qualitative argument to associate the two bits of the target to particular source variables. Identity of the pieces of information is assigned only with a quantitative criterion: Since $H(3)\approx 0.81$ this is the maximum information that can be redundant to all three sources about the target $123$. Both $1$ and $2$ can still provide alone $\approx 0.19$, reaching one bit of information. Any combination of two primary sources provides another half bit. In total this provides already the information corresponding to the entropies $H(13)$ and $H(23)$, which is one and a half bits. The remaining information is unique of the synergystic term combining $12$.
Overall, this analysis highlights that distinguishing redundant, unique, and synergistic information requires a criterion to assign identity to each piece of information in the target. For measures fulfilling the identity axiom [@Bertschinger12] and more generally complying with the stochasticity axioms, the implicit criterion assumes that the identity of the sources is preserved within the target. This criterion respects the Independent Identity property but leads to negative terms. Oppositely, the criterion is quantitative for the redundancy measure used for Figure \[fig6\] or for the measure of [@Williams10] and while it leads to a nonnegative decomposition it does not respect the intuition about qualitative redundancy of the Independent Identity property. It remains an open question if there is a general criterion of identity that accommodates the intuition associated with the Independent Identity property and is also compatible with the relations intrinsic to the redundancy lattice of [@Williams10].
Discussion
==========
Implications for the theoretical definition of redundant, synergistic and unique information
--------------------------------------------------------------------------------------------
The proposal of [@Williams10] of decomposing mutual infor¬mation into nonnegative redundant, unique, and synergistic components has been a fruitful and influential conceptual framework. However, a concrete implementation consistent with a set of axioms formalizing the notions for such types of information has proven to be elusive. The main difficulty stems from determining if redundant sources contain the same qualitative information, which requires assigning an identity to pieces of informa¬tion in the target. [@Harder12] pointed out that the redundancy defined by [@Williams10] only captures quantitatively the common amounts of information shared by the sources. They introduced the identity axiom to ensure that two independent variables cannot have redundant information about a copy of themselves. The lack of redundancy for this particular case has been enunciated as the Independent Identity property by [@Ince16]. However, [@Bertschinger12b] provided a counterexample showing that nonnegativity of the PID terms is not ensured when the identity axiom is assumed. This counterexample also involved a target constituted as a copy of the primary sources, in particular as the inputs and output variables of the XOR logical operation. Since these cases in which a deterministic relation exists between the target and the sources have played an important role both in motivating the identity axiom and raising caveats about the internal consistency of the axioms, in this work we have examined more systematically the effect of deterministic target-sources dependencies in the PID decomposition.
We introduced (Section \[s3\]) two variants of a stochasticity axiom that imposes some constraints to the existence of synergy in the presence of deterministic target-sources dependencies. In presence of a target-sources overlap, the weak form of the axiom states that there cannot be synergistic information about the overlapping target variables. The strong axiom further constrains synergy assuming that the overlapping sources cannot provide other information than about themselves, and thus can neither contribute synergistic information about the non-overlapping part of the target.
We derived (Section \[s41\]) general formulas for the PID terms in the bivariate case, following each version of the stochasticity axiom. We showed that the PID terms can be separated into a stochastic and a deterministic component, which account for the information about the non-overlapping and overlapping target variables, respectively. We indicated that the stochasticity axioms subsume the identity axiom and provide two alternative generalizations to characterize redundancy for any multivariate system with any degree of target-sources overlap (Section \[s42\]). We checked how several previously proposed measures conform to these generalizations (Section \[s43\]). We then examined (Section \[s44\]) two concrete examples based on the XOR and AND logical operations, with variables $1$ and $2$ as inputs and variable $3$ as output, calculating the PID decomposition of the mutual information $I(123;12)$.
Based on these examples we argued that the two axioms imply different interpretations of redundancy as common information about the target that can be obtained by observing either source alone. With the weak axiom each source can be combined with some target variables to provide information about other target variables, even in the presence of a target-sources overlap. Conversely, the strong axiom assumes that any overlapping variable only provides information about itself, and thus redundant information cannot appear because a source is combined with target variables. This leads to redundancy being always equal to the mutual information between the primary sources when there is some target-sources overlap, independently of the non-overlapping target variables. However, while enforcing different constraints, both stochasticity axioms assume that the identity of the overlapping sources is preserved within the target, that is, that some pieces of information have an identity that can be associated only with the corresponding sources, independently of the other variables composing the target.
In Section \[s51\] we extended the general derivations to the trivariate case. This allowed us to understand what originates negative PID terms -as found in the counterexample of [@Bertschinger12b]- from a general perspective. We identified that, following the stochasticity axioms, several PID terms have a deterministic component that is not nonnegatively defined. We resumed the XOR and AND examples in Section \[s52\], now analyzing the decomposition of $I(123;123)$, as was done in the counterexample of [@Bertschinger12b]. [@Bertschinger12b] did not fully characterize this decomposition, but instead used the identity axiom to indicate that at least a negative PID term appears. We confirmed their results indicating that their arguments are consistent with the strong axiom, although we pointed out that a negative value is already present for the redundancy between the three variables, instead of in an upper PID term involving synergy.
Our analysis applying the general derivations from the stochasticity axioms allowed us to expose the relation between the assumptions on information identity and the lack of nonnegativity. In particular, imposing that certain pieces of information can only be attributed to redundancy or unique information terms based on the premise that their identity is associated to the sources enforces that deterministic components of the mutual information are bounded to the low part of the redundancy lattice, and this leads to negative terms in order to conform to the lattice structure and to the relations between PID terms and mutual informations.
The identity axiom was motivated by the especial case of $I(12;1, 2)$ with $1$ and $2$ being independent, for which, as particularly enunciated in the Independent Identity property [@Ince16], redundancy should intuitively cancel because the pieces of information in the target can be separately assigned to each source. However, in general, the overall composition of the target affects the identity of each piece of information. For example, even if $1$ and $2$ are independent, incorporating to the target a third variable $3$ determined by $1$ and $2$ alters the identification of the target information. This is the case for the XOR example with the target formed jointly by the inputs and output variables, since the two bits of $123$ cannot be identified as belonging to any of the three variables, but only as the bit that any first variable provides alone, and the bit that a second variable provides combined with the first. In Section \[s53\] we examined an alternative decomposition consistent with this alternative identification of the two bits of the XOR system. We showed that, for both the XOR and AND example, nonnegative decompositions are attained by admitting nonzero synergistic contributions. However, the identity criterion used in this alternative decomposition is purely quantitative, as the one of [@Williams10], and thus does not respect the desired Independent Identity property.
Although the notion of redundancy as information shared about the same pieces of information is intuitive in plain language, its precise implementation within the information-theoretic framework is not straightforward. The measure of mutual information has applications in many fields, such as communication theory and statistics [@Cover06]. Accordingly, a certain decomposition in terms of redundant, unique, and synergistic contributions may be compatible only with one of its interpretations. Indeed, if information is understood in the context of a communication channel [@Shannon48], nonnegativity is required from its operational interpretation as the number of messages that can be transmitted without errors. Furthermore, semantic content cannot be attributed, and thus information identity should rely only on the statistical properties of the distribution of the target variables. For example, in the case of the target composed by two independent variables, identity is assigned based on independence. Alternatively, if mutual information is used as a descriptor of statistical dependencies [@Kullback1959], nonnegativity is not required since locally negative information, or misinformation [@Wibral14b], simply reflects a certain change in the probability distribution of one variable due to conditioning on another variable. With this interpretation of information based on local dependencies, a criterion of information identity can introduce semantic content in association with the specific value of the variables and common information of two sources can be associated with dependencies that induce coherent modifications of the probability distribution of the target variables [@Ince16]. These local measures of information may be interpreted operationally in terms of changes in beliefs, or in relation to a notion of information more associated with ideal observer analysis than with communication theory [@Wibral14b; @Thomson05]. In this work, we have not considered local versions of mutual information, and we adopted the premise that nonnegativity is a desirable property for the PID terms.
Implications for studying neural codes
--------------------------------------
Determining the proper criterion of information identity to evaluate when information carried by different sources is qualitatively common is essential to interpret the results of the PID decomposition in practical applications, such as in the analysis of the distribution of redundant, unique, and synergistic information in neural population responses. For example, when examining how information about a multidimensional sensory stimulus is represented across neurons, the decomposition should identify information about different features of the stimulus, and not only common amounts of information. The PID terms should reflect the functional properties of the neural population so that we can properly characterize the neural code. On the other hand, nonnegativity of the PID terms facilitates their interpretation not only as a description of statistical dependencies, but as a breakdown of the information content of neural responses, for example to assess the intersection information between sensory and behavioural choice representations [@Panzeri17; @Pica17; @Pica17b].
The underlying criterion of information identity for the PID decomposition is also important when examining information flows among brain areas because, only if redundant and unique information terms correctly separate qualitatively the information, we can interpret the spatial and temporal dynamics of how unique new information is transmitted across areas. It is common to apply dynamic measures of predictability such as Granger causality [@Granger69] to characterize information flows between brain areas [@Wibral14]. The effect of synergistic and redundant information components in the characterization of information flows with Granger causality has been studied [@Stramaglia14; @Stramaglia16], and [@Williams11] applied their PID framework to decompose the information-theoretic measure of Granger causality, namely Transfer entropy [@Marko73; @Schreiber00b], into terms separately accounting for state-independent and state-dependent components of information transfer. Furthermore, they also indicated which terms of the PID decompositions can be associated with information uniquely transmitted at a certain time or information transfer about a specific variable, such as a certain sensory stimulus [@Beer15]. These applications of the PID framework identify meaningful PID terms based on the redundancy lattice, and thus can be applied for any actual definition of the measures, but our considerations highlight the necessity to properly determine information identity in order to fully exploit their explanatory power.
Furthermore, our discussion of how the interpretation of information identity depends on the dependencies between the variables composing the target indicates that the analysis of how redundant, unique, and synergistic information components are distributed across neural population responses can be particularly useful in combination with interventional approaches [@Panzeri17; @Chicharro14]. In particular, the manipulation of neural activity with optogenetics techniques [@Oconnor13; @Otchy15] can disentangle causal effects from other sources of dependencies such as common factors. Although this work illustrates the principled limitations of current PID measures, their combination with these powerful experimental techniques can help to better probe the functional meaning of the PID terms.
Concluding remarks
------------------
Overall, we have studied the effect of deterministic target-sources dependencies in the PID decomposition by enunciating two variants of a new stochasticity axiom, comparing them to the identity axiom [@Harder12], and discussing their implications regarding information identity. Our analysis suggests that, if the redundancy lattice of [@Williams10] is to remain as the backbone of a nonnegative decomposition of the mutual information, a new criterion of information identity should be established that, while conforming to the Independent Identity property, it is less restrictive in the presence of deterministic target-source dependencies than the ones underlying these axioms.
#### **Acknowledgments**
This work was supported by the Fondation Bertarelli.
#### **Authors Contribution**
All authors contributed to the design of the research. The research was carried out by Daniel Chicharro. The manuscript was written by Daniel Chicharro with the contribution of Stefano Panzeri and Giuseppe Pica. All authors have read and approved the final manuscript.
#### **Conflicts of Interest**
The authors declare no conflicts of interest.
Alternative partitioning orders for the bivariate decomposition with target-sources overlap {#a0}
===========================================================================================
We here derive in more detail the alternative expressions for the unique and redundant information terms collected in Table \[tab1\], which are obtained applying the other mutual information partitioning order of Eq.\[r3\]b. Using the relation decomposing conditional mutual information into unique information and synergy we get $$\begin{split}
&I(X; 1 \backslash 2) = I(X; 1|2) - I(X; 12 \backslash 1,2)\\
&= I(X \cap 12 ;1|2) + I(X \backslash 12; 1|2,X \cap 12)- I(X \backslash 12; 12 \backslash 1,2).
\label{r8}
\end{split}$$ This leads to express the unique information of $1$ as $$I(X; 1 \backslash 2) = \begin{cases} I(X \backslash 12; 1 \backslash 2)\ \ \mathrm{if}\ X \cap 1 = \emptyset\\ H(1|2)- I(X \backslash 12; 12 \backslash 1,2)\ \ \mathrm{if}\ X \cap 1 \neq \emptyset \end{cases}.
\label{r9}$$ In this case the unique information is separated into nonadditive terms and involves the synergy about $X \backslash 12 $. This cross-over may seem at odds with the expression obtained with the other partitioning order (Eq.\[r5\]), but on the contrary it reflects the internal consistency of the relations between the information-theoretic quantities: Eqs.\[r5\] and \[r9\] coincide if $1$ is not part of the target. For $X \cap 1 \neq \emptyset$, their equality $$H(1|2)- I(X \backslash 12; 12 \backslash 1,2)= I(X \backslash 12 ;1 \backslash 2)+ H(X \cap 1| 2, X \backslash 12)
\label{r10}$$ is consistent with the definition $I(X \backslash 12; 1|2) = H(1|2) - H(1| 2, X \backslash 12)$, taking into account that conditional information is the sum of the unique and synergistic components.
Proceeding as with the other partitioning order, once we have the expression of the unique information we can use the relation with the mutual information to determine redundancy: $$I(X; 1.2) = \begin{cases} I(X \backslash 12; 1.2) \ \ \mathrm{if}\ X \cap 12 = \emptyset \\ I(1;2)+ I(X \backslash 12; 12 \backslash 1,2) \ \ \mathrm{if}\ X \cap 12 \neq \emptyset \end{cases}.
\label{r12}$$ Also here internal consistency with Eq.\[r7\] holds. In particular, the equality $$I(1;2)+ I(X \backslash 12; 12 \backslash 1,2) = I(X \backslash 12; 1.2) + I(1;2|X \backslash 12)
\label{r13}$$ reflects that $$\begin{split}
C(X \backslash 12 ; 1; 2) &= I(1;2)- I(1;2|X \backslash 12)\\ &= I(X \backslash 12; 1.2)-I(X \backslash 12; 12 \backslash 1,2)
\label{r14}
\end{split}$$ because the co-information is invariant to permutations (Eq.\[e5\]) and also corresponds to the difference of the redundancy and synergistic PID components.
Also following the strong axiom the alternative partitioning order, in this case the one considering first stochastic dependencies with the non-overlapping target variables, can be derived. With overlap, Eq.\[r2b\] implies that $I(X; 1 \backslash 2)= I(X; 1 | 2)$. For the unique information we get
$$I(X; 1 \backslash 2) = I(X \backslash 12 ;1 \backslash 2) + I(X \backslash 12 ;12 \backslash 1, 2) + H(X \cap 1| 2, X \backslash 12),
\label{r29}$$
and for the redundancy
$$I(X; 1.2) = I(X \backslash 12; 1.2) + I(1;2|X \backslash 12) - I(X \backslash 12; 12 \backslash 1, 2).
\label{r30}$$
Like with the weak axiom, internal consistency holds for the expressions obtained with the two partitioning orders.
Derivations of the trivariate decomposition with target-sources overlap {#a1}
=======================================================================
We here derive in more detail the trivariate deterministic PID components. We start with the derivations following the weak stochasticity axiom. If we consider the unique information of one primary source with respect to the other two, for example $I(X; 3 \backslash 12)$, we have that
$$\begin{split}
I(X; 3 \backslash 12) &= \Delta(X;3) \\ &= I(X; 3|12)- \left[ \Delta(X;123)+\Delta(X;13)+\Delta(X;23)+\Delta(X;13.23) \right].
\label{r16}
\end{split}$$
The weak axiom imposes for the trivariate case that synergy deterministic components upper than the single source nodes have to be zero (Eq.\[r15\]). Accordingly, any deterministic component of $I(X; 3|12)$ has to be contained in $\Delta(X;3)$. Decomposing this conditional mutual information with the partitioning order that considers first the dependencies with the non-overlapping target variables
$$I(X; 3 \backslash 12) = I(X \backslash 123; 3 \backslash 12)+ H(X \cap 3|12, X \backslash 123),
\label{r17}$$
and thus in general
$$\Delta_d(X;i) = H(X \cap i|jk, X \backslash ijk).
\label{r18}$$
We now consider the conditional information of two primary sources given the third, for example
$$\begin{split}
I(X;23|1) &= I(X \backslash 123; 23|1) + H(X \cap 23| 1 , X \backslash 123).
\label{r19}
\end{split}$$
The deterministic part $H(X \cap 23| 1, X \backslash 123)$ again can only be contained in the PID terms contributing to $I(X; 23|1)$ that are lower than the single source nodes. This means that it has to be contained in the terms $$\begin{split}
\Delta_d(X;2) +\Delta_d(X;3)+\Delta_d(X;2.3)+\Delta_d(X;3.12)+\Delta_d(X;2.13).
\label{r19b}
\end{split}$$ Furthermore, this conditional entropy can be decomposed considering explicitly the part of the uncertainty associated with conditional entropies of the form of Eq.\[r18\]:
$$\begin{split}
&H(X \cap 23| 1, X \backslash 123) = H(X \cap 3| 1, X \backslash 123)+ H(X \cap 2| 1, X \cap 3, X \backslash 123) \\ &= H(X \cap 3| 12, X \backslash 123) + I(2; X \cap 3)|1, X \backslash 123)+ H(X \cap 2| 1, X \cap 3, X \backslash 123).
\label{r20}
\end{split}$$
Accordingly, using the definition of the terms $\Delta_d(X;i)$ in Eq.\[r18\] and combining Eqs.\[r19\] and \[r19b\] we get the following equalities. First,
$$\Delta_d(X;i)+ \Delta_d(X;i.j) + \Delta_d(X;i.jk)+ \Delta_d(X;j.ik) = H(i|k, X \backslash ijk)\ \mathrm{if}\ X \cap i \neq \emptyset ,
\label{r21}$$
and second
$$\Delta_d(X;i.j) + \Delta_d(X;i.jk)+ \Delta_d(X;j.ik) = I(i; j|k, X \backslash ijk)\ \ \mathrm{if}\ X \cap i \neq \emptyset.
\label{r22}$$
Like in the expressions of the deterministic PID components in the Tables of Sections \[s41\] and \[s51\], we here for simplicity indicate the equalities that hold when the primary source $i$ overlaps with the target. The symmetries of each $\Delta_d(X; \beta)$ term indicate when it can be nonzero. For example, $\Delta_d(X; i.j)$ is constrained by an equality of the form of Eq.\[r22\] both if $i$ or $j$ overlap with the target.
Finally, we consider also how an unconditional mutual information is decomposed in PID terms. For example, again using the partitioning order that considers first stochastic target-source dependencies we have
$$\begin{split}
I(X;3) &= I(X \backslash 123;3)+ I(X \cap 123; 3|X \backslash 123) \\
&= I(X \backslash 123;3)+ H(X \cap 3|X \backslash 123)\ \ \mathrm{if}\ X \cap 3 \neq \emptyset.
\label{r23}
\end{split}$$
When $3$ is part of the target the deterministic part of this information has to be contained in the nodes reached descending from $3$, and thus in general
$$\sum_{\beta \in \downarrow i} \Delta_d(X;\beta)= H(i|X \backslash ijk)\ \ \mathrm{if}\ X \cap i \neq \emptyset .
\label{r24}$$
Combining Eq.\[r24\] with Eq.\[r21\] we get that
$$\begin{split}
\Delta_d(X;i.j) + \Delta_d(X;i.j.k)- \Delta_d(X;k.ij) &= I(i;j|X \backslash 123) \ \ \mathrm{if}\ X \cap i \neq \emptyset \\ &= H(i|X \backslash 123)-H(i|j, X \backslash 123).
\end{split}
\label{r25}$$
Altogether, from Eqs.\[r18\], \[r21\], \[r22\], \[r24\], and \[r25\] we can proceed to obtain expressions of the PID terms as a function of mutual informations and entropies. Doing so, the rest of PID terms remain as a function also of the terms $\Delta_d(X; i.jk)$. These terms can be understood by comparing the trivariate decomposition and a bivariate decomposition with only sources $j$ and $k$. For the latter, if $i$ is part of the target, $I(i; jk \backslash j,k)$ quantifies a stochastic synergistic contribution, because $i$ is not a source. Conversely, in the trivariate decomposition $i$ is a source and this information is now redundant with the information provided by variable $i$ itself. This means that we can identify $\Delta_d(X;i.jk)$ by comparing synergy between these two decompositions. For example, for the bivariate decomposition of $I(X; 12)$, $3$ is not a source and according to the weak axiom synergy can provide information about the non-overlapping part of the target, which can comprise $3$. Moving to the trivariate case by adding $3$ as a source this synergy stochastic component becomes redundant to information source $3$ has about itself, and thus
$$\begin{split}
I(X; 12 \backslash 1,2) &= I(X \backslash 12; 12 \backslash 1,2) \\
&= I(X \backslash 123; 12 \backslash 1,2) + \left [ I(X \backslash 12; 12 \backslash 1,2) - I(X \backslash 123; 12 \backslash 1,2) \right ].
\end{split}
\label{r26}$$
In general, this means that these type of PID terms can be quantified as
$$\begin{split}
\Delta_d(X; i.jk) = I(X \backslash jk; jk \backslash j,k) - I(X \backslash ijk; jk \backslash j,k).
\end{split}
\label{r27}$$
These terms are nonnegatively defined, because according to the axiom adding a new source can only reduce synergy. After calculating these terms we can obtain all the expressions collected in Table \[tab2\].
For the strong stochasticity axiom, instead of repeating all the derivations we proceed by arguing about what has to change with respect to the decomposition obtained for the weak axiom. Changes originate from the difference in the constraints that both versions of the axiom impose on the existence of synergistic components and from the alternative mutual information partitioning order that leads to an additive separation of stochastic and deterministic PID components depending on the axiom. With the strong axiom this additive separation is reached using the partitioning order that first considers deterministic target-sources dependencies. This means that the conditioning of entropies and mutual informations on $X \backslash ijk$ will in this case not be present. Moreover, since the strong axiom restricts also synergy with the non-overlapping target variables, even a stochastic component of $I(X; 12 \backslash 1,2)$ can only be nonzero if $3$, but neither $1$ or $2$, overlap with the target. Since once further adding $3$ to the sources any synergistic component should be zero, the expression of the terms $\Delta_d(X; i.jk)$ in Eq.\[r27\] is reduced to $I(i; jk \backslash j, k)$ when only $i$ overlaps with $X$, and to zero otherwise. Implementing these two modifications, the expressions of Table \[tab3\] are obtained from the ones of Table \[tab2\].
Amari, S. (2001). Information geometry on hierarchy of probability distributions. , 47(5):1701–1711.
Averbeck, B. B., Latham, P. E., and Pouget, A. (2006). Neural correlations, population coding and computation. , 7(5):358–366.
Ay, N., Der, R., and Prokopenko, M. (2012). Information-driven self-organization : the dynamical system approach to autonomous robot behavior. , 131(3):125–127.
Banerjee, P. K. and Griffith, V. (2015). Synergy, redundancy, and common information. .
Barrett, A. B. (2015). Exploration of synergistic and redundant information sharing in static and dynamical gaussian systems. , 91:052802.
Beer, R. D. and Williams, P. L. (2015). Information processing and dynamics in minimally cognitive agents. , 39:1–39.
Bell, A. J. (2003). The co-information lattice. , pages 921–926.
Bertschinger, N., Rauh, J., Olbrich, E., and Jost, J. (2012). Shared information - new insights and problems in decomposing information in complex systems. , pages 251–269.
Bertschinger, N., Rauh, J., Olbrich, E., Jost, J., and Ay, N. (2014). Quantifying unique information. , 16:2161–2183.
Chatterjee, P. and Pal, N. R. (2016). Construction of synergy networks from gene expression data related to disease. , 590(2):250–262.
Chicharro, D. (2014). A causal perspective on the analysis of signal and noise correlations and their role in population coding. , 26:999–1054.
Chicharro, D. (2017). Quantifying multivariate redundancy with maximum entropy decompositions of mutual information. page arXiv:1708.03845v1.
Chicharro, D. and Panzeri, S. (2014). Algorithms of causal inference for the analysis of effective connectivity among brain regions. , 8:64.
Chicharro, D. and Panzeri, S. (2017). Synergy and redundancy in dual decompositions of mutual information gain and information loss. , 19(2):71.
Cover, T. M. and Thomas, J. A. (2006). . John Wiley and Sons, 2nd edition.
Deco, G., Tononi, G., Boly, M., and Kringelbach, M. L. (2015). Rethinking segregation and integration: contributions of whole-brain modelling. , 16:430–439.
Erwin, D. H. and Davidson, E. H. (2009). The evolution of hierarchical gene regulatory networks. , 10:141–148.
Faes, L., Marinazzo, D., Nollo, G., and Porta, A. (2016). An information-theoretic framework to map the spatiotemporal dynamics of the scalp electroencephalogram. , 63(12):2488–2496.
Faes, L., Marinazzo, D., and Stramaglia, S. (2017). Multiscale information decomposition: Exact computation for multivariate gaussian processes. , 19:408.
Flack, J. C. (2012). Multiple time-scales and the developmental dynamics of social systems. , 367:1802–1810.
Granger, C. W. J. (1969). Investigating causal relations by econometric models and cross-spectral methods. , 37(3):424–38.
Griffith, V., Chong, E. K. P., James, R. G., Ellison, C. J., and Crutchfield, J. P. (2014). Intersection information based on common randomness. , 16(4):1985–2000.
Griffith, V. and Koch, C. (2013). Quantifying synergistic mutual information. .
Harder, M., Salge, C., and Polani, D. (2013). Bivariate measure of redundant information. , 87:012130.
Ince, R. A. A. (2017). Measuring multivariate redundant information with pointwise common change in surprisal. , 19(7):318.
Ince, R. A. A., Senatore, R., Arabzadeh, E., Montani, F., Diamond, M. E., and Panzeri, S. (2010). Information-theoretic methods for studying population codes. , 23(6):713–727.
Ince, R. A. A., van Rijsbergen, N. J., Thut, G., Rousselet, G. A., Gross, J., Panzeri, S., and Schyns, P. G. (2015). Tracing the flow of perceptual features in an algorithmic brain network. , 5(1):17681.
Katz, Y., Tunstr[ø]{}m, K., Ioannou, C. C., Huepe, C., and Couzin, I. D. (2011). Inferring the structure and dynamics of interactions in schooling fish. , 108(46):18720–18725.
Kullback, S. (1959). . Dover, Mineola,NY.
Latham, P. E. and Nirenberg, S. (2005). Synergy, redundancy, and independence in population codes, revisited. , 25(21):5195–5206.
Marko, H. (1973). Bidirectional communication theory - generalization of information-theory. , 12:1345–1351.
Marre, O., El Boustani, S., Fregnac, Y., and Destexhe, A. (2009). Prediction of spatiotemporal patterns of neural activity from pairwise correlations. , 102(13):138101.
McGill, W. J. (1954). Multivariate information transmission. , 19:97–116.
O’Connor, D. H., Hires, S. A., Guo, Z. V., Li, N., Yu, J., Sun, Q. Q., Huber, D., and Svoboda, K. (2013). Neural coding during active somatosensation revealed using illusory touch. , 16:958–965.
Olbrich, E., Bertschinger, N., and Rauh, J. (2015). Information decomposition and synergy. , 17(5):3501–3517.
Otchy, T. M., Wolff, S. B. E., Rhee, J. Y., Pehlevan, C., Kawai, R., Kempf, A., Gobes, S. M. H., and Olveczky, B. P. (2015). Acute off-target effects of neural circuit manipulations. , 528:358–363.
Panzeri, S., Brunel, N., Logothetis, N. K., and Kayser, C. (2010). Sensory neural codes using multiplexed temporal scales. , 33(3):111–120.
Panzeri, S., Harvey, C. D., Piasini, E., Latham, P. E., and Fellin, T. (2017). Cracking the neural code for sensory perception by combining statistics, intervention and behavior. , 93(3):491–507.
Panzeri, S., Macke, J. H., Gross, J., and Kayser, C. (2015). Neural population coding: combining insights from microscopic and mass signals. , 19(3):162–172.
Panzeri, S., Magri, C., and Logothetis, N. K. (2008). On the use of information theory for the analysis of the relationship between neural and imaging signals. , 26(7):1015–1025.
Panzeri, S., Schultz, S., Treves, A., and Rolls, E. T. (1999). Correlations and the encoding of information in the nervous system. , 266:1001–1012.
Perrone, P. and Ay, N. (2016). Hierarchical quantification of synergy in channels. , 2:35.
Pica, G., Piasini, E., Chicharro, D., and Panzeri, S. (2017a). Invariant components of synergy, redundancy, and unique information among three variables. , 19(9):451.
Pica, G., Piasini, E., Safaai, H., Runyan, C. A., Diamond, M. E., Fellin, T., Kayser, C., Harvey, C. D., and Panzeri, S. (2017b). Quantifying how much sensory information in a neural code is relevant for behavior. .
Rauh, J. (2017). Secret sharing and shared information. , 19(11):601.
Rauh, J. and Ay, N. (2014). Robustness, canalyzing functions and systems design. , 133(2):63–78.
Rauh, J., Banerjee, P. K., Olbrich, E., Jost, J., and Bertschinger, N. (2017). On extractable shared information. , 19(7):328.
Rauh, J., Bertschinger, N., Olbrich, E., and Jost, J. (2014). Reconsidering unique information: Towards a multivariate information decomposition. , pages 2232–2236.
Schneidman, E., Still, S., Berry, M. J., and Bialek, W. (2003). Network information and connected correlations. , 91(23):238701.
Schreiber, T. (2000). Measuring information transfer. , 85:461–464.
Shannon, C. E. (1948). A mathematical theory of communication. , 27:379–423, 623–656.
Stramaglia, S., Angelini, L., Wu, G., Cortes, J. M., Faes, L., and Marinazzo, D. (2016). Synergetic and redundant information flow detected by unnormalized granger causality: application to resting state fmri. , 63(12):2518–2524.
Stramaglia, S., Cortes, J. M., and Marinazzo, D. (2014). Synergy and redundancy in the granger causal analysis of dynamical networks. , 16(10):105003.
Thomson, E. E. and Kristan, W. B. (2005). Quantifying stimulus discriminability: A comparison of information theory and ideal observer analysis. , 17(4):741–778.
Timme, N., Alford, W., Flecker, B., and Beggs, J. M. (2014). Synergy, redundancy, and multivariate information measures: an experimentalist’s perspective. , 36(2):119–140.
Timme, N. M., Ito, S., Myroshnychenko, M., Nigam, S. Shimono, M., Yeh, F. C., Hottowy, P., Litke, A. M., and Beggs, J. M. (2016). High-degree neurons feed cortical computations. , 12(5):e1004858.
Tishby, N., Pereira, F. C., and Bialek, W. (1999). The information bottleneck method. , pages 368–377.
Valdes-Sosa, P. A., Roebroeck, A., Daunizeau, J., and Friston, K. (2011). Effective connectivity: Influence, causality and biophysical modeling. , 58(2):339–361.
Vicente, R., Wibral, M., Lindner, M., and Pipa, G. (2011). Transfer entropy: A model-free measure of effective connectivity for the neurosciences. , 30:45–67.
Watkinson, J., Liang, K. C., Wang, X., Zheng, T., and Anastassiou, D. (2009). Inference of regulatory gene interactions from expression data using three-way mutual information. , 1158:302–313.
Wibral, M., Lizier, J. T., and Priesemann, V. (2015). Bits from brains for biologically inspired computing. , 2:5.
Wibral, M., Vicente, R., and Lizier, J. T. (2014). . Springer-Verlag, Berlin Heidelberg.
Williams, P. L. (2011). Information dynamics: Its theory and application to embodied cognitive systems. .
Williams, P. L. and Beer, R. D. (2010). Nonnegative decomposition of multivariate information. .
Williams, P. L. and Beer, R. D. (2011). Generalized measures of information transfer. .
|
---
abstract: 'To securely leverage the advantages of Cloud Computing, recently a lot of research has happened in the area of *“Secure Query Processing over Encrypted Data"*. As a concrete use case, many encryption schemes have been proposed for securely processing k Nearest Neighbors (SkNN) over encrypted data in the outsourced setting. Recently Zhu et al.[@zhu1] proposed a SkNN solution which claimed to satisfy following four properties: *(1)Data Privacy, (2)Key Confidentiality, (3)Query Privacy,* and *(4)Query Controllability*. However, in this paper, we present an attack which breaks the Query Controllability claim of their scheme. Further, we propose a new SkNN solution which satisfies all the four existing properties along with an additional essential property of *Query Check Verification*. We analyze the security of our proposed scheme and present the detailed experimental results to showcase the efficiency in real world scenario.'
author:
- Gagandeep Singh
- Akshar Kaul
- Sameep Mehta
bibliography:
- 'sigproc.bib'
title: 'Secure k-NN as a Service Over Encrypted Data in Multi-User Setting'
---
|
---
abstract: 'In this paper we prove a comparison theorem between the category of certain modules with integrable connection on the complement of a normal crossing divisor of the generic fiber of a proper semistable variety over a DVR and the category of certain log overconvergent isocystrals on the special fiber of the same open.'
author:
- Valentina Di Proietto
title: '<span style="font-variant:small-caps;">On $p$-adic differential equations on semistable varieties</span>'
---
Introduction
============
The “théorème d’algébrisation” by Christol and Mebkhout (théorème 5.0-10 of [@CtMeIV]) asserts that on a open of a smooth and proper curve all the overconvergent isocrystals (with some non-Liouville conditions) are algebraic. The aim of this paper is to give a generalized version of this result to the case of a variety of arbitrary dimension that is the special fiber of a semistable variety.\
We first recall the “théorème d’algébrisation” and explain in which sense our result generalizes it. Let $V$ be a complete discrete valuation ring of mixed characteristic $(0,p)$ with uniformizer $\pi$, let $K$ be its fraction field and let $k$ be the residue field. Christol and Mebkhout consider a proper and smooth curve $X$ over $V$ and an affine open $U$ with complement $D$. They define a functor $$\label{CM}
\dag:MICLS(U_K/K) \longrightarrow I^{\dag}((U_k,X_k)/{\mathrm{Spf}}(V))$$ where the first category is the subcategory of $MIC(U_K/K)$ of algebraic modules with connection on $U_K$ which satisfy certain convergent conditions and the second is the category of overconvergent isocrystals as defined by Berthelot in [@Be].\
They prove that under some non-Liouville conditions the functor $\dag$ is essentially surjective, *i. e.* every overconvergent isocrystal is algebraic. Moreover they notice that, always assuming non-Liouville conditions, $\dag$ is fully faithful if one restricts to the category of algebraic modules with connection on $U_K$ with some convergent conditions that are extendable to module with connections on $X_K$ and logarithmic singularities along $D_K$. The image of this restricted functor turns out to be the category of overconvergent isocrystals with slope zero (probl[\`]{}eme 5.0-14 1) of [@CtMeIV] and paragraph 6 of [@CtMeII] ).\
We are interested in the following generalized situation.\
Let $X$ be a proper semistable variety over $V$, which means that locally for the étale topology is ' etale over ${\mathrm{Spec}}V [x_1, \dots ,x_n,y_1,\dots ,y_n]/(x_1 \dotsm x_r - \pi)$ with $D$ a normal crossing divisor, which ' etale locally is given by the equation $\{y_1\dotsm y_s = 0\}$. The divisor $X_k\cup D$ induces on $X$ a logarithmic structure that we denote by $M$; similarly, the closed point induces a logarithmic structure on ${\mathrm{Spec}}(V)$ that we denote by $N$. We assume we have the following diagram of fine log schemes $$\label{diagrammacommintro}
\xymatrix{
\ (X_k,M)\ \ar@{^(->}[r] \ar[d] &\ (X,M)\ \ar[d]& \ (X_K,M)\ \ar@{_(->}[l] \ar[d] \\
\ ({\mathrm{Spec}}(k), N)\ \ar@{^(->}[r]& \ ({\mathrm{Spec}}(V), N) & \ \ ({\mathrm{Spec}}(K),N)\ \ar@{_(->}[l] \\
}$$ where the two squares are cartesian. The log structures denoted again by $M$ on the special fiber $X_k$ and by $N$ on $k$ are defined in such a way that the closed immersions of fine log schemes $(X_k, M)\hookrightarrow (X,M)$ and $({\mathrm{Spec}}(k),N)\hookrightarrow ({\mathrm{Spec}}(V),N)$ are exact. In an analogous way the log structures on $X_K$ and on $K$ are defined in such a way that the open immersions $(X_K,M)\hookrightarrow (X,M)$ and $({\mathrm{Spec}}(K), N)\hookrightarrow ({\mathrm{Spec}}(V),N)$ are strict. Let us note that the log structure on $K$ constructed in this way is isomorphic to the trivial log structure.\
We consider on $X$ the open $U$ defined as the complement of the divisor $D$; then there is an open immersion $$j:U=X\setminus D\hookrightarrow X$$ and it induces on $U$ the log structure $j^*(M)$ that, by abuse of notation, we again denote by $M$.\
So we have a diagram analogous to (\[diagrammacommintro\]) for $U$, with the same notations for the log structures: $$\label{diagrammacommintroU}
\xymatrix{
\ (U_k,M)\ \ar@{^(->}[r] \ar[d] &\ (U,M)\ \ar[d]& \ (U_K,M)\ \ar@{_(->}[l] \ar[d] \\
\ ({\mathrm{Spec}}(k), N)\ \ar@{^(->}[r]& \ ({\mathrm{Spec}}(V), N) & \ \ ({\mathrm{Spec}}(K),N)\ \ar@{_(->}[l] \\
}$$ In this situation we consider the category, denoted by $MIC(U_K/K)^{reg}$, of pairs $(E,\nabla)$ where $E$ is a sheaf of coherent $\mathcal{O}_{U_K}$-modules and $\nabla$ is an integrable connection regular along $D_K$.\
As for the rigid side we look at $I^{\dag}((U_k,X_k)/{\mathrm{Spf}}(V))^{log, \Sigma}$, the category of overconvergent log isocrystals on the log pair given by $((U_k,M),(X_k,M)/({\mathrm{Spf}}(V),N))$ with $\Sigma$-unipotent monodromy along $D_k$, where $\Sigma$ is a subset of $\mathbb{Z}_p^h$, satisfying some non-Liouville hypothesis and $h$ is the number of the irreducible component of $D_k$. A log overconvergent isocrystal is represented by a module with connection on a strict neighborhood $W$ of the tube $]U_k[_{\hat{X}}$ in the tube $]X_k[_{\hat{X}}$. To define $\Sigma$-unipotent monodromy we proceed étale locally and we fix an irreducible component $D^{\circ}_{j,k}$ of $D^{\circ}_{k}$, the smooth locus of $D_k$; then the part of the tube $]D^{\circ}_{j,k}[_{\hat{X}}$ which is contained in $W$ is isomorphic to a product of an annulus of small width times certain rigid space associate to $D^{\circ}_{j,k}$, that we think as a base. A log overconvergent isocrystal $\mathcal{E}$ has $\Sigma$-unipotent monodromy along $D^{\circ}_{j,k}$ if $\mathcal{E}$, restricted to the product described above, admits a filtration such that every successive quotient is the pullback of a module with connection on the base twisted by a module with connection on the annulus depending on $\Sigma$. We say that $\mathcal{E}$ has $\Sigma$-unipotent monodromy along $D_k$ if the above condition holds for every irreducible component of $D^{\circ}_{k}$. The category of overconvergent log isocrystals is defined by Shiho in [@Sh4] as a log version of the category of overconvergent isocrystals defined by Berthelot in [@Be] and the notion of $\Sigma$-unipotent monodromy is introduced by Shiho in [@Sh6] as a generalization of the notion of unipotent monodromy introduced by Kedlaya in [@Ke]. Let us note that the notion of $\Sigma$-unipotent monodromy for a module with connection on an annulus, with $\Sigma$ satisfying some non-Liouville conditions, coincides with the notion of satisfying the Robba condition and having exponent in the sense of Christol and Mebkhout in $\Sigma$ (proposition 1.18 of [@Sh7]).\
Our main result is:
\[mainintro\] There is a natural algebrization functor $$I^{\dag}((U_k,X_k)/{\mathrm{Spf}}(V))^{log, \Sigma}\longrightarrow MIC(U_K/K)^{reg}.$$ It is a fully faithful functor.
Let us note that our functor goes in the opposite direction with respect to Christol and Mebkhout’s functor $\dag$.\
The strategy of our proof is as follows. We start from the category of log overconvergent isocrystals on $((U_k,M),(X_k,M)/({\mathrm{Spf}}(V),N))$ with $\Sigma$-unipotent monodromy and we prove that the restriction functor $$\label{jdagintro}
j^{\dag}:I_{conv}((X_k,M)/({\mathrm{Spf}}(V),N))^{lf,\Sigma}\longrightarrow I^{\dag}((U_k,X_k)/{\mathrm{Spf}}(V))^{log, \Sigma}$$ is an equivalence of categories.\
We denote by $I_{conv}((X_k,M)/({\mathrm{Spf}}(V),N))^{lf,\Sigma}$ the category of locally free log convergent isocrystals on the log convergent site $((X_k,M)/({\mathrm{Spf}}(V),N))_{conv}$ with exponents in $\Sigma$. The category of log convergent isocrystals is defined by Shiho in [@Sh1], as a log version of the category of convergent isocrystal defined by Ogus in [@Og] and by Berthelot in [@Be]. The equivalence of categories in (\[jdagintro\]) is a generalization of theorem 3.16 of [@Sh6], since Shiho proves the same result in the case of $X_k$ smooth and $D_k$ a divisor with simple normal crossing, *i. e.* its components are regular and meet transversally.\
On the other hand we have a fully faithful functor $$\tilde{i} : I_{conv}((X_k,M)/({\mathrm{Spf}}(V),N))^{lf,\Sigma}\longrightarrow MIC((X_K, M_D)/K)^{lf, \Sigma}$$ between locally free log convergent isocrystals with exponents in $\Sigma$ and locally free $\mathcal{O}_{X_K}$-modules with integrable connection on $X_K$, logarithmic singularities on $D_K$ and exponents in $\Sigma$. The theorem of algebraic logarithmic extension of [@AnBa] I,4 gives us a fully faithful functor $$\label{ABintro}
MIC((X_K, M_D)/K)^{lf, \Sigma}\longrightarrow MIC(U_K/K)^{reg}.$$ If we denote by $MICLS(U_K/K)^{reg, \Sigma}$ the essential image of the functor $\tilde{i}$ in $MIC(U_K/K)^{reg}$, we can conclude that we have an equivalence of categories $$I^{\dag}((U_k,X_k)/{\mathrm{Spf}}(V))^{log, \Sigma}\longrightarrow MICLS(U_K/K)^{reg, \Sigma}.$$
Let us now describe in detail the contents of the paper.\
In the first paragraphs we suppose that $(X, M)\rightarrow ({\mathrm{Spec}}(V), N)$ is a general log smooth morphism of fine log schemes (not necessarily semistable), with $X$ proper and $X_k$ reduced. We denote by $(\hat{X}, M)\rightarrow ({\mathrm{Spf}}(V), N)$ the associated morphism of the $p$-adic completions. We recall the definition of modules with integrable log connection on a fine log scheme, *i.e.* the category $MIC((X_K, M)/K)$, the definition of log infinitesimal isocrystals, the category $I_{inf}((\hat{X},M)/({\mathrm{Spf}}(V),N))$, and the definition of log convergent isocrystals, the category $I_{conv}((X_k,M)/({\mathrm{Spf}}(V),N))$. Using the fact that $X$ is proper Shiho (Theorem 3.15, Corollary 3.2.16 of [@Sh1]) shows that $MIC((X_K, M)/K)$ is equivalent to the category $I_{inf}((X_k,M)/({\mathrm{Spf}}(V),N))$ of log infinitesimal isocrystals . We prove that the property of being locally free is stable with respect this equivalence of categories (propositions \[primolibero\] and \[secondolibero\]), so that we have an equivalence of categories between $I_{inf}((X_k,M)/({\mathrm{Spf}}(V),N))^{lf}$ and $MIC((X_K, M)/K)^{lf}$. We use the definition of log convergent site and isocrystals on it given by Shiho in chapter 5 of [@Sh1] and the functor $$\Phi : I_{conv}(X_k,M)/({\mathrm{Spf}}(V),N))\longrightarrow I_{inf}((\hat{X},M)/({\mathrm{Spf}}(V),N)),$$ that he defines between log convergent isocrystals and log infinitesimal isocrystal. We adapt the proof of proposition 5.2.9 of [@Sh1] to show that $$\begin{split}
\tilde{\Phi} : I_{conv}((X_k,M)/({\mathrm{Spf}}(V),N))^{lf}\longrightarrow I_{inf}((\hat{X},M)&/({\mathrm{Spf}}(V),N))^{lf}\\
&\cong MIC((X_K, M)/K)^{lf}
\end{split}$$ is fully faithful (theorem \[ff\]). Shiho proves full faithfulness on the subcategory of log convergent isocrystals that are iterated extensions of the unit object. As in Shiho’s proof, the key tool used is locally freeness. Finally, we give a characterization of the essential image of the functor $\tilde{\Phi}$ in terms of special stratifications, a suitable modification of the special stratifications introduced by Shiho in [@Sh1]. We conclude by proving that requiring a stratification to be special is the same as requiring that the radius of convergence of the stratification is 1. In this last part we do not use log differential calculus, because we prove that we can restrict to the case of trivial log structures (proposition \[U denso\]).\
Then we restrict to the semistable situation: we suppose that the morphism $(X,M)\rightarrow ({\mathrm{Spec}}(V),N)$ is as described before (\[diagrammacommintro\]). After describing the geometric situation, we introduce the definition of log convergent isocrystals with exponents in $\Sigma$, using log-$\nabla$-modules defined by Kedlaya in [@Ke] and used by Shiho in [@Sh6]. We first define the notion of exponents in $\Sigma$ étale locally using coordinates, then we prove that it is independent of the particular étale cover chosen and from the coordinates (lemma \[indepnilpotentresidue\]). After that, following Shiho, we recall the notion of $\Sigma$-unipotence for a log-$\nabla$-module defined over a product of a rigid analytic space and a polyannulus and we recall the extension theorem for $\Sigma$-unipotent log-$\nabla$-modules. Then we analyze log overconvergent isocrystals. Our setting is different from Shiho’s ([@Sh4]), since in our case the base has a non-trivial log structure. To define a log overconvergent isocrystal with $\Sigma$-unipotent monodromy we do it étale locally. Then we recall the three key propositions that we use in the proof of the main theorem. The first property is called “generization of monodromy” and asserts that the property of being $\Sigma$-unipotent is generic on the base in some sense that we make precise (proposition \[Generization of monodromy\]). The second property, called “overconvergent generization", says that the property of being $\Sigma$-unipotent can be extended on strict neighborhoods (proposition \[Overconvergent generization\]). The third one says that under certain conditions a convergent log-$\nabla$-module that has exponents in $\Sigma$ is $\Sigma$-unipotent (proposition \[Convergent plus nilpotent implies unipotent\]). The proofs of these propositions are given in [@Sh6] as a generalization the ones contained in [@Ke]. Using these properties we prove that the notion of $\Sigma$-unipotent monodromy for an overconvergent isocrystal is well posed (lemma \[indepunipmonodromy\]). Finally we prove (theorem \[log extension\]) that the restriction functor $$j^{\dag}:I_{conv}((X_k,M)/({\mathrm{Spf}}(V),N))^{lf,\Sigma}\longrightarrow I^{\dag}((U_k,X_k)/{\mathrm{Spf}}(V))^{log, \Sigma}$$ is an equivalence of categories. The strategy of the proof is the same as in theorem 3.16 of [@Sh6] and theorem 6.4.5 of [@Ke].\
In the last part we verify that the notion of exponents in $\Sigma$ behaves well with respect to the functor $\tilde{i}$, *i. e.* the functor $$I_{conv}((X_k,M)/({\mathrm{Spf}}(V),N))^{lf, \Sigma}\longrightarrow MIC((X_K, M_D)/K)^{lf, \Sigma}$$ is well defined. Finally we adapt André and Baldassarri’s theorem of algebraic logarithmic extension to find theorem \[mainintro\].\
Connections with logarithmic poles
==================================
We will recall the definition of log connection on a fine log scheme or on a $p$-adic fine log formal scheme, which is taken from [@Sh1] definition 3.1.1; see also [@Kz] paragraph 4.\
We suppose the reader familiar with the language of log schemes introduced in [@Ka] and with his version for formal schemes given in [@Sh1] chapter 2.\
We denote by $V$ a discrete valuation ring of mixed characteristic $(0, p)$, complete and separated for the $p$-adic topology, $\pi$ a uniformizer of $V$, $K$ its fraction field and $k$ its residue field. By a formal scheme over ${\mathrm{Spf}}(V)$ or a formal $V$-scheme, we mean a $p$-adic Noetherian and topologically of finite type formal scheme over $V$.\
According to Shiho’s notation we will give the following definition.
If $X$ is a scheme, we denote the category of coherent $\mathcal{O}_X$-modules by Coh$(\mathcal{O}_X)$. If $X$ is formal $V$-scheme we denote by Coh$(K\otimes \mathcal{O}_X)$ the category of sheaves of $K\otimes_V\mathcal{O}_X$-modules that are isomorphic to $K\otimes_V F$ for some coherent $\mathcal{O}_X$-module $F$. We will call an object of Coh$(K\otimes \mathcal{O}_X)$ an isocoherent sheaf.
The category of isocoherent sheaves is introduced in [@Og] and (see [@Og] remark 1.5) it is equivalent to the category of coherent sheaves on $X^{an}$, the rigid analytic space associated to the $p$-adic formal scheme $X$ via Raynaud generic fiber.
\[connections\] Let $f:(X,M)\rightarrow(S,N)$ be a map of fine log schemes (resp. fine log formal $V$-schemes) and let $E$ be a coherent $\mathcal{O}_{X}$-module (resp. $E$ $\in $ Coh$(K\otimes \mathcal{O}_X)$). A log connection on $E$ is an $\mathcal{O}_S$-linear map $$\nabla: E \rightarrow E\otimes \omega^1_{(X,M)/(S,N)}$$ that is additive and satisfies the Leibniz rule: $$\nabla(ae)=a\nabla(e)+e\otimes da$$ for $a$ $\in$ $\mathcal{O}_X$ and $e$ $\in$ $E.$\
Here $\omega^1_{(X,M)/(S,N)}$ denotes the sheaf of log differentials (resp. the sheaf of log formal differentials).\
We can extend $\nabla$ to $\nabla_i$ $$E\otimes \omega^i_{(X,M)/(S,N)}\xrightarrow{\nabla_i}E\otimes \omega^{i+1}_{(X,M)/(S,N)},$$ where $\nabla_i(e\otimes \omega)=e \otimes d\omega+(-1)^i\nabla(e)\wedge \omega$. We say that $\nabla$ is integrable if $\nabla_{i+1} \circ \nabla_i=0$ for all $i$.\
We indicate the category of pairs $(E,\nabla)$ of a coherent sheaf $E$ (resp. an isocoherent sheaf) and an integrable log connection $\nabla$ with $MIC((X,M)/(S,N))$ (resp. $\widehat{MIC}((X,M)/(S,N))$). If $M$ and $N$ are isomorphic to the trivial log structures we will write $M=\textrm{triv}$ and $N=\textrm{triv}$ and we use the notation $MIC(X/S)$ (resp. $\widehat{MIC}(X/S)$) to denote $MIC((X,\textrm{triv})/(S,\textrm{triv}))$ (resp. $\widehat{MIC}((X,\textrm{triv})/(S,\textrm{triv}))$).
From proposition 8.9 of [@Kz] we know that in the smooth case in characteristic zero every coherent module with integrable connection is locally free; more precisely we have the following result.
\[8.9Ka\] If $S$ is the spectrum of a field of characteristic $0$ and $X$ is a smooth scheme over $S$, then, for every object $(E,\nabla)$ in $MIC(X/S)$, $E$ is a locally free $\mathcal{O}_X$-module.
This proposition is not true without the smoothness hypothesis; in fact, there is the following example.
\[esempiodishiho\] Let $X$ be a curve over $K$ and $D$ a closed point, locally defined by the equation $\{t=0\}$. If we call $M_D$ the log structure on $X$ induced by $D$ we can consider the following log connection $$d:\mathcal{O}_{X}\rightarrow \omega^{1}_{(X,M_D)/(K,triv)}.$$ If we consider the subsheaf $\mathcal{O}_{X_K}(-D)$, that consists of sections vanishing on $D$, then $d$ induces a log connection on $\mathcal{O}_{X_K}(-D)$. We can see it locally: every section of $\mathcal{O}_{X_K}(-D)$ can be written as a product of $ft$, with $f$ in $\mathcal{O}_{X_K}$. The induced log connection is: $$\mathcal{O}_{X_K}(-D)\rightarrow \mathcal{O}_{X_K}(-D)\otimes \omega^{1}_{(X,M_D)/(K,triv)}$$ $$ft\mapsto d(ft)=fdt+tdf=ft d\mathrm{log} t+t df$$ We can induce an integrable log connection on the the quotient $\mathcal{O}_{X_K}/\mathcal{O}_{X_K}(-D)=\mathcal{O}_D$ which is a skyscraper sheaf.
As in the case where the log structures are trivial (see for example [@BeOg] chapter 1), even the category of modules with integrable log connections is equivalent to the category of log stratifications: we now describe this equivalence.\
If $(X,M)\rightarrow(S,N)$ is a morphism of fine log schemes (resp. a morphism of fine log formal schemes), we denote by $(X^n,M^n)$ the $n$-th log infinitesimal neighborhood of $(X,M)$ in $(X,M)\times_{(S,N)}(X,M) $(defined in [@Ka] (5.8), [@Sh1] remark 3.2.4 as a log version of the $n$-th infinitesimal neighborhood described in [@BeOg] chapter 1).
\[Stratifications\] Let $(X, M)\rightarrow(S, N)$ be a morphism of fine log schemes (resp. of fine log formal $V$-schemes) and let $E$ be a coherent sheaf (reps. an isocoherent sheaf on $X$). Then, a log stratification (resp. a formal log stratification) on $E$ is a family of morphisms $\epsilon_n:\mathcal{O}_{X^n}\otimes E\rightarrow E\otimes\mathcal{O}_{X^n}$, that satisfy the conditions:
- $\epsilon_n$ is $\mathcal{O}_{X^n}$-linear and $\epsilon_{0}$ is the identity;
- $\epsilon_n$ and $\epsilon_m$ are compatible via the maps $$\mathcal{O}_{X^n}\rightarrow \mathcal{O}_{X^m},\; \textrm{for}\; m\leq n;$$
- (cocycle condition) if we call $p_{i,j}$ (for $i,j=1,2,3$) the projections from the $n$-th log infinitesimal neighborhood $(X^n(2),M(2)^n)$ of $(X,M)$ in $(X,M)\times_{(S,N)} (X,M)\times_{(S,N)} (X,M)$ to the $n$-th log infinitesimal neighborhood $(X^n,M^n)$ of $(X,M)$ in $(X,M)\times_{(S,N)} (X,M)$ $$p_{i,j}:(X^n(2),M^n(2))\rightarrow (X^n,M^n),$$ then for all $n$ $$p_{1,2}^*(\epsilon_n)\circ p_{2,3}^*(\epsilon_n)=p_{1,3}^{*}(\epsilon_n).$$
We denote by $Str((X,M)/(S,M))$ the category of log stratifications (resp. with $\widehat{Str}((X,M)/(S,N))$ the category of log formal stratifications).
Theorem 3.2.15 of [@Sh1] gives us the equivalence of categories that we announced: if $(X,M)\rightarrow(S,N)$ is log smooth morphism of fine log schemes over $\mathbb{Q}$ (resp. a formally smooth morphism of fine log formal $V$-schemes), then $$Str((X,M)/(S,N))\cong MIC((X,M)/(S,N))$$ $$(\mathrm{resp.} \, \widehat{Str}((X,M)/(S,N))\cong \widehat{MIC}((X,M)/(S,N))).$$
Log infinitesimal isocrystals
=============================
We now define the infinitesimal site and log isocrystals on it. These are Shiho’s log formal analogous of Grothendieck’s infinitesimal site and crystals on it defined in [@Gr] or [@BeOg] chapter 1. All the definitions that follow are taken from chapter 3 of [@Sh1]. We define the infinitesimal site only in the case of a morphism of fine log formal schemes, but analogous definition can be given for a morphism of log schemes.
Let $(\mathscr{X},M)\rightarrow(\mathscr{S},N)$ be a morphism of fine log formal $V$-schemes. An object of the log infinitesimal site $((\mathscr{X},M)/(\mathscr{S},N))_{inf}$, or by brevity $(\mathscr{X}/\mathscr{S})^{log}_{inf}$ when the log structures are clear, is a $4$-ple $({\mathscr{U}},{\mathscr{T}},L,\phi)$ such that ${\mathscr{U}}$ is a formal $V$-scheme formally étale over ${\mathscr{X}}$, $(\mathscr{T},L)$ is a fine log formal $V$-scheme over $({\mathscr{S}},N)$ and $\phi: ({\mathscr{U}},M)\rightarrow({\mathscr{T}},L)$ is a nilpotent exact closed immersion of log formal $V$-schemes over $({\mathscr{S}}, N)$. A morphism between $({\mathscr{U}},{\mathscr{T}},L,\phi)$ and $({\mathscr{U}}',{\mathscr{T}}',L',\phi')$ is pair of maps $g:({\mathscr{T}},L)\rightarrow({\mathscr{T}}',L')$ and $f:{\mathscr{U}}\rightarrow {\mathscr{U}}'$ such that $\phi'\circ f=g \circ \phi$. The coverings in this site are the coverings of ${\mathscr{T}}$ for the étale topology $\{{\mathscr{T}}_i\rightarrow{\mathscr{T}}\}_i$ such that ${\mathscr{U}}_i={\mathscr{T}}_i\times_{{\mathscr{T}}}{\mathscr{U}}$. We sometimes denote the $4$-ple $({\mathscr{U}},{\mathscr{T}},L,\phi)$ simply by ${\mathscr{T}}.$
\[loginf\] A log isocrystal on the infinitesimal site $(\mathscr{X}/\mathscr{S})^{log}_{inf}$, or a log infinitesimal isocrystal, is a sheaf $\mathcal{E}$ on $(\mathscr{X}/\mathscr{S})^{log}_{inf}$ such that:
- for every object $({\mathscr{U}},{\mathscr{T}},L,\phi)$ the Zariski sheaf $\mathcal{E}_{{\mathscr{T}}}$ induced on ${\mathscr{T}}$ is an isocoherent sheaf;
- for every morphism $g:{\mathscr{T}}\rightarrow{\mathscr{T'}}$, the map $g^*(\mathcal{E}_{{\mathscr{T}}'})\rightarrow \mathcal{E}_{{\mathscr{T}}}$ is an isomorphism.
We denote the category of log isocrystals on the infinitesimal site $(\mathscr{X}/\mathscr{S})^{log}_{inf}$ by $I_{inf}({\mathscr{X}}/{\mathscr{S}})^{log}$.
\[locally free\] Let ${\mathscr{X}}$ be a formal $V$-scheme and let $\mathcal{F}$ be an isocoherent sheaf. We say that it is locally free module if there is a formal affine covering $\{U_i\}_{i\in I}$ of $\mathscr{X}$ such that for every $U_i={\mathrm{Spf}}{A_i}$, there exists a finitely generated $A_i$-module $F_i$ such that $\mathcal{F}(U_i)=K\otimes F_i$ is a projective $K\otimes A_i$-module.
\[locally free2\] A log isocrystal $\mathcal{E}$ on the infinitesimal site $(\mathscr{X}/\mathscr{S})^{log}_{inf}$ is said to be locally free if for every object $({\mathscr{U}},{\mathscr{T}},L,\phi)$ of the infinitesimal site, the sheaf $\mathcal{E}_{{\mathscr{T}}}$ induced on ${\mathscr{T}}$ is an isocoherent locally free module.\
We will denote the subcategory of $I_{inf}({\mathscr{X}}/{\mathscr{S}})^{log}$ consisting of the locally free infinitesimal log isocrystal by $I_{inf}({\mathscr{X}}/{\mathscr{S}})^{log,lf}$.
Thanks to theorem 3.2.15 of [@Sh1] we can see that if $({\mathscr{X}}, M)\rightarrow ({\mathscr{S}}, N)$ is a formally log smooth morphism of fine log formal $V$-schemes, then there exists a canonical equivalence of categories $$I_{inf}(({\mathscr{X}},M)/({\mathscr{S}},N))\cong \widehat{MIC}(({\mathscr{X}},M)/({\mathscr{S}},N))$$
Log convergent isocrystals
==========================
In this section we define the log convergent site and the isocrystals on it. The following definitions are taken from [@Sh1] paragraph 5.1.
For every log formal $V$-scheme $(\mathscr{Y},M)$ we indicate with ${\mathscr{Y}}_1$ the closed subscheme defined by the ideal $p$ and the associated reduced subscheme of ${\mathscr{Y}}_1$ by ${\mathscr{Y}}_0$.
Let $(\mathscr{X},M)\rightarrow(\mathscr{S},N)$ be a morphism of fine log formal $V$-schemes. An enlargement of $(\mathscr{X},M)$ over $(\mathscr{S}, N)$ is a triple $(\mathscr{T},L,z)$, that we will indicated with $\mathscr{T}$, such that $({\mathscr{T}},L)$ is a fine log formal $V$-scheme over $({\mathscr{S}},N)$ and $z$ is a morphism $({\mathscr{T}}_0,L)\rightarrow({\mathscr{X}},M)$ over $({\mathscr{S}},N)$. A morphism between two enlargements $(\mathscr{T},L,z)$ and $(\mathscr{T}',L',z')$ is a morphism $g:({\mathscr{T}},L)\rightarrow ({\mathscr{T}}',L')$ such that $z=z'\circ g_0$, where $g_0:(\mathscr{T}_0,L)\rightarrow (\mathscr{T}'_0,L')$ is the map induced by $g$.
We define the log convergent site of $(\mathscr{X},M)\rightarrow(\mathscr{S},N)$ to be the site whose objects are enlargements, morphisms are morphisms of enlargements and coverings are given by the étale topology on ${\mathscr{T}}$. We denote it by $((\mathscr{X},M)/(\mathscr{S},N))_{conv}$ or $(\mathscr{X}/\mathscr{S})^{log}_{conv}$ if there is no ambiguity about the log structures.
\[logconvisocrystals\] A log isocrystal on $(\mathscr{X}/\mathscr{S})^{log}_{conv}$, or a log convergent isocrystal, is a sheaf $\mathcal{E}$ on $(\mathscr{X}/\mathscr{S})^{log}_{conv}$ such that:
- for every enlargement $({\mathscr{T}},L,z)$ the Zariski sheaf $\mathcal{E}_{{\mathscr{T}}}$ induced on ${\mathscr{T}}$ is an isocoherent sheaf;
- for every morphism of enlargements $g:({\mathscr{T}},L)\rightarrow({\mathscr{T'}},L')$, the map $g^*(\mathcal{E}_{{\mathscr{T}}'})\rightarrow \mathcal{E}_{{\mathscr{T}}}$ is an isomorphism.
We denote by $I_{conv}({\mathscr{X}}/{\mathscr{S}})^{log}$ the category of log isocrystals on the log convergent site.
\[locally free1\] A log isocrystal $\mathcal{E}$ on the convergent site $((\mathscr{X},M)/(\mathscr{S},N))_{conv}$ is locally free if for every object ${\mathscr{T}}$ on the convergent site the sheaf $\mathcal{E}_{{\mathscr{T}}}$ induced on ${\mathscr{T}}$ is an isocoherent locally free sheaf.\
We denote the subcategory of $I_{conv}({\mathscr{X}}/{\mathscr{S}})^{log}$ consisting of the locally free log isocrystals on the convergent site by $I_{conv}({\mathscr{X}}/{\mathscr{S}})^{log, lf}$.
Relations between algebraic and analytic modules with integrable connections {#Relations between algebraic and analytic modules with integrable connections}
============================================================================
In what follows we consider the following situation: we fix $(X,M)\rightarrow({\mathrm{Spec}}(V),N)$ a log smooth and proper morphism of fine log schemes. We denote by $(X_K,M)\rightarrow ({\mathrm{Spec}}(K),N)$ its generic fiber and $(X_k,M)\rightarrow({\mathrm{Spec}}(k),N)$ its special fiber, that we suppose to be reduced. So we have a commutative diagram $$\label{diagrammacomm}
\xymatrix{
\ (X_k,M)\ \ar@{^(->}[r] \ar[d] &\ (X,M)\ \ar[d]_f& \ (X_K,M)\ \ar@{_(->}[l] \ar[d] \\
\ ({\mathrm{Spec}}(k), N)\ \ar@{^(->}[r]& \ ({\mathrm{Spec}}(V), N) & \ \ ({\mathrm{Spec}}(K),N)\ \ar@{_(->}[l] \\
}$$ The log structures on $X_k$ and ${\mathrm{Spec}}(k)$, that with an abuse of notation we again call $M$ and $N$, are defined in such a way that the inclusions in $(X,M)$ and $({\mathrm{Spec}}(V),N)$ in this diagram are exact closed immersions of log schemes.\
In the same way we define the log structures $M$ on $X_K$ and $N$ on ${\mathrm{Spec}}(K)$ as the log structures defined in such a way that the inclusions to $(X,M)$ and $({\mathrm{Spec}}(V),N)$ are strict.\
We consider the $p$-adic completion of $(X,M)\rightarrow ({\mathrm{Spec}}(V),N)$ and we call it $(\hat{X},M) \rightarrow ({\mathrm{Spf}}(V),N)$. With $(\hat{X},M)$ we mean $\hat{X}=(X_k, \varprojlim \mathcal{O}_{X}/p^n \mathcal{O}_X)$ with the log structure, that we call again $M$ with an abuse of notation, defined as the pull back of $M$ via the canonical morphism $\hat{X}\rightarrow X$.\
Another way to see this is [@ChFo] Definition-Lemma 0.9, where the authors prove that the log structure $M$ over $\hat{X}$ is isomorphic to $\varprojlim_n (M)_n$ with $(M)_n$ the log structure on $\hat{X}_n=(X_k, \mathcal{O}_X/p^n \mathcal{O}_X)$ that is the pull-back of $M$ via the morphism $\hat{X}_n\rightarrow X$.\
Now we want to construct a fully faithful functor $$i : I_{conv}((\hat{X},M)/({\mathrm{Spf}}(V),N))^{lf}\longrightarrow MIC((X_K, M)/(K,N))^{lf}.$$ In corollary 3.2.16 of [@Sh1] we have a useful characterization of algebraic modules with integrable log connection: the following result holds.
\[derhaminf\] Under the above assumptions, there is an equivalence of categories $$\Psi:MIC((X_K, M)/({\mathrm{Spec}}(K),N))\longrightarrow I_{inf}((\hat{X},M)/({\mathrm{Spf}}(V),N)).$$
As we saw in example \[esempiodishiho\] it is not true that every coherent module with integrable connection is locally free, so we will restrict to the category that we call $$MIC((X_K,M)/(K,N))^{lf}$$ and that consists of pairs $(E,\nabla)$ where $\nabla$ is an integrable log connection and $E$ a locally free $\mathcal{O}_{X_{K}}$-module.\
In the next two propositions we will see that the functor $\Psi$ of proposition \[derhaminf\] induces an equivalence of categories $$MIC((X_K,M)/(K,N))^{lf}\longrightarrow I_{inf}((\hat{X},M)/({\mathrm{Spf}}(V),N))^{lf}.$$
\[primolibero\] For every element $\mathcal{E}=(E,\nabla)$ in $MIC((X_K,M)/(K,N))^{lf}$, the corresponding element $\Psi(\mathcal{E})$ $\in $ $I_{inf}((\hat{X},M)/({\mathrm{Spf}}(V),N))$ is a log infinitesimal locally free isocrystal .
For the proof we look carefully at the definition of the functor $\Psi$. The functor is defined as the composition of three functors each one being an equivalence of category. The first functor $\Psi_1$ is the one that gives the equivalence of category between $MIC((X_K,M)/(K,N))$ and the category of log stratifications. So given $\mathcal{E}=(E,\nabla)$, with $E$ a locally free $\mathcal{O}_{X_{K}}$-module, $\Psi_1(\mathcal{E})$ is again the $\mathcal{O}_{X_{K}}$-module $E$ with a collection of isomorphisms $\epsilon_n:\mathcal{O}_{X_{K}^n}\otimes E\rightarrow E\otimes\mathcal{O}_{X_{K}^n}$, where $(X_{K}^{n},M^n)$ is the $n$-th log infinitesimal neighborhood of $(X_{K},M)$ in $(X_{K},M)\times_{(K,N)}(X_{K},M)$, that satisfies the conditions of definition \[Stratifications\]. By lemma 3.2.7 of [@Sh1] we know that all the $\mathcal{O}_{X_{K}^n}$ are free $\mathcal{O}_{X_{K}}$-modules, so $E\otimes\mathcal{O}_{X_{K}^n}$ are locally free $\mathcal{O}_{X_{K}}$-modules.
Now, thanks to example 1.4 of [@Og], that uses a formal version of GAGA principle, we know that the category of coherent $\mathcal{O}_{X_{K}}$-modules on $X_{K}$ is equivalent to the category of isocoherent modules on $\hat{X}$. So we can associate to our $E$ an isocoherent sheaf $\hat{E}$, that we now show to be locally free. The functor that gives this equivalence is defined locally by extension of scalars: if we suppose that $X={\mathrm{Spec}}(B)$ then the functor is $$E\longmapsto E\otimes_{B_{K}}\hat{B}_{K}$$ where $B_{K}=B\otimes_V K$ and $\hat{B}_K=\varprojlim_n B/p^n \otimes K$. If $E$ locally free as coherent $\mathcal{O}_{X_K}$-module, then $E$ is a projective-$B_K$ module. This implies that $E\otimes_{B_{K}}\hat{B}_{K}$ is a projective $\hat{B}_{K}$-module too. Indeed, since $E$ is projective, there exists a $B_K$-module $F$ such that $E\oplus F$ is a free $B_K$-module. Then $(E\oplus F)\otimes_{B_{K}}\hat{B}_{K}$ is a free $\hat{B}_K$-module, which implies that $E\otimes_{B_{K}}\hat{B}_{K}$ is a projective $\hat{B}_K$-module.\
Moreover if we indicate with $(\hat{X}^n, M^n)$ and $(X_{K}^n,M^n)$ the $n$-th log infinitesimal neighborhoods of the diagonal morphisms $(\hat{X},M)\rightarrow (\hat{X},M)\times_{({\mathrm{Spf}}(V),N)}(\hat{X},M)$ and $(X_K,M)\rightarrow (X_K,M)\times_{(K,N)} (X_K,M)$ respectively, then we have an equivalence of categories also between $\textrm{Coh}(\mathcal{O}_{X_{K}^n})$ and $\textrm{Coh}(K\otimes\mathcal{O}_{\hat{X}^n})$.\
This means that we have a functor $\Psi_2$ from the category $Str((X_K,M)/K)$ to $\widehat{Str}((\hat{X},M)/({\mathrm{Spf}}(V),N))$. Moreover $\Psi_2$ is an equivalence of categories which sends locally free objects in locally free objects.
Now we construct the functor $\Psi_3$ between $\widehat{Str}((\hat{X},M)/({\mathrm{Spf}}(V),N))$ and $I_{inf}(((\hat{X},M)/({\mathrm{Spf}}(V),N))$.\
Let us take an element $({\mathscr{U}},{\mathscr{T}},L, \phi)$ of the log infinitesimal site $(\hat{X}/{\mathrm{Spf}}(V))^{log}_{inf}$. We know by definition that $({\mathscr{U}},M)\rightarrow({\mathrm{Spf}}(V),N)$ is formally log smooth over $({\mathrm{Spf}}(V),N)$, because $({\mathscr{U}},M)$ is formally étale over $(\hat{X},M)$, that is formally log smooth over $({\mathrm{Spf}}{V},N)$. Therefore we have a diagram $$\xymatrix{
\ ({\mathscr{U}},M)\ \ar[r] \ar@{^(->}[1,0]_\phi &\ ({\mathscr{U}},M)\ \ar[d] \\
\ ({\mathscr{T}},L)\ \ar[r]& \ ({\mathrm{Spf}}(V), N) \\
}$$ and by proposition 2.2.13 of [@Sh1] we know that étale locally over ${\mathscr{T}}$ there exists a morphism $c:({\mathscr{T}},L)\rightarrow ({\mathscr{U}},M)$ that is a section for $\phi:({\mathscr{U}},M)\rightarrow({\mathscr{T}},L)$.\
If we have ${\mathscr{T}}'$ étale over ${\mathscr{T}}$, then we call $s:({\mathscr{T'}},L)\rightarrow (\hat{X},M)$ the composition of the section $c$ with the morphism $({\mathscr{U}},M)\rightarrow (\hat{X},M)$.\
Then we can define a sheaf over ${\mathscr{T}}'$ using the pullback map $ s^{*}_{K}: \mathrm{Coh}(K \otimes \mathcal{O}_{\hat{X}})\rightarrow \mathrm{Coh}(K \otimes \mathcal{O_{{\mathscr{T}}'}})$, and we call the resulting sheaf $\hat{E}_{{\mathscr{T}}'}=s_K^{*}\hat{E}$. Let us note that $E_{{\mathscr{T}}'}$ is obviously a locally free isocoherent sheaf on ${\mathscr{T'}}$, as soon as $\hat{E}$ is.\
If we have two sections $c$ and $d$ and respectively two morphisms $s$ and $t$, the formally log smooth morphism $s\times t: ({\mathscr{T}}',L)\rightarrow (\hat{X},M)\times_{({\mathrm{Spf}}(V),N)}(\hat{X},M)$ factors through the $n$-th log infinitesimal neighborhood $(\hat{X}^n,M^n)$, for some $n$: $$s\times t: ({\mathscr{T}}',L)\stackrel{u}{\longrightarrow} (\hat{X}^n,M^n)\longrightarrow (\hat{X},M)\times_{({\mathrm{Spf}}(V),N)}(\hat{X},M),$$ because of the universal propriety of the $n$-th infinitesimal neighborhood.\
Now, pulling back by $u$ the isomorphisms $\hat{\epsilon}_n:{\mathcal{O}}_{\hat{X}^n}\otimes \hat{E}\rightarrow \hat{E}\otimes {\mathcal{O}}_{\hat{X}^n}$ given by the stratification, we obtain an isomorphism $t_{K}^{*}\hat{E}\cong s_{K}^*\hat{E}$. We want to descend this sheaf to ${\mathscr{T}}$; this is possible using the cocycle condition and the theorem of faithfully flat descent for isocoherent sheaves of Gabber ([@Og] proposition 1.9.).\
In fact, let us consider the formally étale morphism $({\mathscr{T'}},L)\rightarrow ({\mathscr{T}},L)$ and the two projections $$\xymatrix{
({\mathscr{T}}',L)\times_{({\mathscr{T}},L)}({\mathscr{T}}',L) \ar[d]_{p_1} \ar@{->}[0,1]^(.68){p_2} & ({\mathscr{T'}},L) \\
({\mathscr{T}}',L) & . \\
}$$ Composing $p_i$ with the morphism $s:({\mathscr{T}}',L)\rightarrow (\hat{X},M)$ we find $\pi_i:({\mathscr{T}}',L)\times_{({\mathscr{T}},L)}({\mathscr{T}}',L)\rightarrow (\hat{X},M)$ for $i=1,2$ respectively.\
As before we have a map $\pi_1\times\pi_2:({\mathscr{T}},L')\times_{({\mathscr{T}},L)}({\mathscr{T}}',L)\rightarrow (\hat{X},M)\times_{({\mathrm{Spf}}(V),N)}(\hat{X},M)$ that factors through the $n$-th log infinitesimal neighborhood and we can deduce an isomorphism $\pi_{1,K}^{*}\hat{E}\cong \pi_{2,K}^{*}\hat{E}$ and consequently an isomorphism $p_{1,K}^{*}\hat{E}_{{\mathscr{T}}'}\cong p_{2,K}^{*}\hat{E}_{{\mathscr{T}}'}$, that is a covering datum; to obtain a descent datum we adapt the above argument using the cocycle condition of the stratification. Now using proposition 1.9 of [@Og] we are allowed to descend the sheaf $\hat{E}_{{\mathscr{T}}'}$ to an isocoherent sheaf on ${\mathscr{T}}$, that we call $\hat{E}_{{\mathscr{T}}}$.\
Let us note that $\hat{E}_{{\mathscr{T}}}$ defines a log isocrystal on the infinitesimal site: to check property (ii) of definition \[loginf\] we can use the same arguments as before.\
Now we prove that $\hat{E}_{{\mathscr{T}}}$ is a locally free isocoherent module on ${\mathscr{T}}$.\
We know that $\hat{E}_{{\mathscr{T}}}$ is an isocoherent module on ${\mathscr{T}}$, isomorphic to $K\otimes F$, for some coherent sheaf $F$ of $\mathcal{O}_{{\mathscr{T}}}$-modules, and that there exists an étale covering of ${\mathscr{T}}$ such that the $\hat{E}_{{\mathscr{T}}}$, restricted to every element of the covering, is a locally free isocoherent module. We can restrict ourselves to the affine case; we assume that ${\mathscr{T}}={\mathrm{Spf}}(A)$, ${\mathscr{T}}'={\mathrm{Spf}}(B)$ and ${\mathrm{Spf}}(B)\rightarrow {\mathrm{Spf}}(A)$ ' etale surjective. We can conclude applying the following lemma.
\[Milne\] Let $A$ and $B$ be commutative noetherian $V$-algebras and let $M$ be a finitely generated $A\otimes K=A_K$-module. If we have a map $A\rightarrow B$ that is faithfully flat and $B_K\otimes_{A_K} M$ is projective, then $M$ is a projective $A_K$-module.
We have the following isomorphism for every $A_K$-module $N$: $$B_K\otimes_ {A_K} \mathrm{Ext}^i_{A_K}(M,N) \cong \mathrm{Ext}^i_{B_K}(B_K \otimes_{A_K} M,B_K \otimes_{A_K} N),$$ for every $i\geq 0 $, because $B_K$ is flat over $A_K$. We can conclude that $M$ is projective because $B_K \otimes_{A_K} M$ is assumed to be projective.
It is true also the viceversa of proposition \[primolibero\].
\[secondolibero\] If $\mathcal{E}$ is a log infinitesimal locally free isocrystal, then there exists an object $(E, \nabla)$ $\in$ $MIC((X_K,M)/(K,N))^{lf}$ such that $\Psi ((E, \nabla))=\mathcal{E}.$
From proposition \[derhaminf\] we know that there exists an element $(E,\nabla)$ in $MIC((X_K,M)/(K,N))$ such that $\Psi((E,\nabla))=\mathcal{E}$.\
So we have to show that $E$ is a locally free $\mathcal{O}_{X_{K}}$-module. By proposition \[primolibero\] we are reduced to prove that the equivalence of categories $$\label{GAGA}
j:\textrm{Coh}(\mathcal{O}_{X_K})\rightarrow \textrm{Coh}(K\otimes \mathcal{O}_{\hat{X}})$$ behaves well with respect to locally free objects, in particular that if $\mathcal{F}$ is a locally free isocoherent module then there exists a locally free sheaf of $\mathcal{O}_{X_K}$-modules $F$ such that $j(F)=\mathcal{F}.$\
Let us take $F$ $\in $ Coh$(\mathcal{O}_{X_K})$ such that $\mathcal{F}=j(F)$. We claim that if $F$ is not locally free, then $\mathcal{F}$ is not locally free. Let us assume that $F$ is not locally free. So there exists an open affine, that we can suppose local, $U=\textrm{Spec}(A)$ $\subseteq$ $X_K$ such that $F|_{U}$ is not flat on $U$. By definition of flatness there exists a coherent ideal $\mathcal{I}$ of $A$ such that $\mathcal{I}\otimes F|U\rightarrow F$ is not injective. Let us take a coherent ideal $\mathcal{I}'$ of $\mathcal{O}_{X_K}$ that extends $\mathcal{I}$ ([@Ha] ex. II 5.15). Then $\mathcal{I}'\otimes F \rightarrow F$ is not injective. Therefore, since the functor $j$ is faithful, exact and compatible with tensor products the map $j(\mathcal{I}')\otimes \mathcal{F}=j(\mathcal{I}'\otimes F)\rightarrow j(F)=\mathcal{F}$ is not injective. So the functor $$-\otimes \mathcal{F}:\textrm{Coh}(K\otimes \mathcal{O}_{\hat{X}})\rightarrow \textrm{Coh}(K\otimes \mathcal{O}_{\hat{X}})$$ is not an exact functor and then $\mathcal{F}$ is not locally free.
Now, as in [@Sh1] paragraph 5.2, we construct a functor $$\Phi : I_{conv}((\hat{X},M)/({\mathrm{Spf}}(V),N))\longrightarrow I_{inf}((\hat{X},M)/({\mathrm{Spf}}(V),N)).$$ We remark that the functor $\Phi$ and its restriction to locally free objects, $\tilde{\Phi}$, that we will mention below, can be constructed for $({\mathscr{X}},M)\rightarrow({\mathscr{S}},N)$, morphism of fine log formal schemes.\
Let $({\mathscr{U}},{\mathscr{T}},L, \phi)$ be an object of the infinitesimal site, define an enlargement $\Phi ^*({\mathscr{T}})=({\mathscr{T}},L,z:({\mathscr{T}}_0,L) \cong ({\mathscr{U}}_0,M)\rightarrow (\hat{X},M))$; this is clearly an element of the log convergent site. Let us observe that $\Phi ^*(\hat{X})=(\hat{X}, M, z:(X_k, M)\rightarrow (\hat{X},M)).$\
If we have an isocrystal $\mathcal{E}$ on the log convergent site we define $$\Phi(\mathcal{E})_{{\mathscr{T}}}=\mathcal{E}_{\Phi^*({\mathscr{T}})}.$$ We have already saw in proposition \[primolibero\] and \[secondolibero\] that $\tilde{\Psi}$, the restriction of the functor $\Psi$ to the locally free objects $$\tilde{\Psi} :MIC((X_K, M)/({\mathrm{Spf}}(V),N))^{lf} \longrightarrow I_{inf}((\hat{X},M)/({\mathrm{Spf}}(V),N))^{lf},$$ is well-defined and is an equivalence of categories. Now we want to show that also $\tilde{\Phi}$, the restriction of the functor $\Phi$ to the locally free objects $$\tilde{\Phi} : I_{conv}((\hat{X},M)/({\mathrm{Spf}}(V),N))^{lf}\longrightarrow I_{inf}((\hat{X},M)/({\mathrm{Spf}}(V),N))^{lf},$$ is well-defined. In particular we have the following lemma.
\[secondoliberoperconv\] If $\mathcal{E}$ is a log convergent isocrystal, such that $\Phi(\mathcal{E})$ is a locally free log infinitesimal isocrystal, then also $\mathcal{E}$ is locally free.
We can evaluate $\mathcal{E}$ at the enlargement $(\hat{X},M, z:(X_k,M)\rightarrow (\hat{X},M))$ and we have that $\mathcal{E}_{\hat{X}}=\mathcal{E}_{\Phi^*(\hat{X})}=\Phi(E)_{\hat{X}}$; so $\mathcal{E}_{\hat{X}}$ is a locally free isocoherent module. Now taken an enlargement $({\mathscr{T}},L,z:({\mathscr{T}}_0,L)\rightarrow (\hat{X},M))$, we have the following commutative diagram of log formal schemes $$\xymatrix{
\ ({\mathscr{T}}_0,L)\ \ar[r]^{z} \ar[d]^{i} &\ (\hat{X},M)\ \ar[d]^{f} \\
\ ({\mathscr{T}},L)\ \ar[r]^{b}& \ ({\mathrm{Spf}}(V), N). \\
}$$ As $f$ is formally log smooth, we know that étale locally on ${\mathscr{T}}$ there is a morphism $c:({\mathscr{T}},L)\rightarrow (\hat{X},M)$ such that $c\circ i =z$ and $f \circ c=b$.\
So let us consider ${\mathscr{T}}'$ formally étale over ${\mathscr{T}}$ and the enlargement ${\mathscr{T}}'=({\mathscr{T}}',L,z:({\mathscr{T}}'_{0},L)\rightarrow(\hat{X},M))$, and call again $c$ the morphism $({\mathscr{T}}',L)\rightarrow (\hat{X},M)$; this clearly extends to a morphism of enlargements. So by definition of isocrystal we have an isomorphism $c^{*}(\mathcal{E}_{\hat{X}})\cong \mathcal{E}_{{\mathscr{T'}}}$, from which we know that $\mathcal{E}_{{\mathscr{T'}}}$ is a locally free isocoherent module.\
Then we consider the formally étale morphism $u:({\mathscr{T}}', L)\rightarrow ({\mathscr{T}},L)$ that extends to a morphism of enlargements; again by definition of isocrystal we have $u^{*}(\mathcal{E}_{{\mathscr{T}}})\cong \mathcal{E}_{{\mathscr{T'}}}$. We know that $\mathcal{E}_{{\mathscr{T'}}}$ is a locally free isocoherent module and we want to prove that $\mathcal{E}_{{\mathscr{T}}}$ is locally free: we can proceed as in the last part of proposition \[primolibero\] and conclude.
From the definition of the functor $\Phi$ we see also the viceversa of proposition \[secondoliberoperconv\]; indeed if $\mathcal{E}$ is a locally free log convergent isocrystal, then for every element $\mathscr{T}$ on the log infinitesimal site $$\Phi(\mathcal{E})_{\mathscr{T}}=\mathcal{E}_{\Phi^*(\mathscr{T})},$$ by definition of $\Phi$.\
So we can compose the functors $\tilde{\Phi}^{-1}$ and $\tilde{\Psi}^{-1}$ and obtain a well defined functor $$i:I_{conv}((\hat{X},M)/({\mathrm{Spf}}(V),N))^{lf}\rightarrow MIC((X_K, M)/({\mathrm{Spf}}(V),N))^{lf}.$$ Our next goal is to prove that $\tilde{\Phi}$ is fully faithful, showing first that this can be proved étale locally.
\[discesaperi\] If $\tilde{\Phi}:I_{conv}(\hat{X}/{\mathrm{Spf}}(V))^{log,lf}\rightarrow I_{inf}(\hat{X}/{\mathrm{Spf}}(V))^{log,lf}$ is fully faithful étale locally on $\hat{X}$, then $\tilde{\Phi}:I_{conv}(\hat{X}/{\mathrm{Spf}}(V))^{log,lf}\rightarrow I_{inf}(\hat{X}/{\mathrm{Spf}}(V))^{log,lf}$ is fully faithful.
We suppose that $\coprod_j \hat{X}_j =\hat{X}^e\rightarrow \hat{X}$ is an étale covering of $\hat X$. If $\hat{X}'=\hat{X}^e\times_{\hat{X}} \hat{X}^e$ and $\hat{X}''=\hat{X}^e\times_{\hat{X}} \hat{X}^e\times_{\hat{X}} \hat{X}^e$, we have the following diagram: $$\xymatrix{
\ I_{conv}(\hat{X}/{\mathrm{Spf}}(V))^{log, lf}\ \ar[r]^{\tilde{\Phi}} \ar[d] &\ I_{inf}(\hat{X}/{\mathrm{Spf}}(V))^{log, lf}\ \ar[d] \\
\ I_{conv}(\hat{X}^e/{\mathrm{Spf}}(V))^{log, lf}\ \ar[r]^{\tilde{\Phi}^{e}} \ar[d] \ar@<1ex>[d] &\ I_{inf}(\hat{X}^e/{\mathrm{Spf}}(V))^{log, lf}\ \ar[d] \ar@<1ex>[d]\\
\ I_{conv}(\hat{X}'/{\mathrm{Spf}}(V))^{log, lf}\ \ar[r]^{\tilde{\Phi}'} \ar[d] \ar@<1ex>[d] \ar@<-1ex>[d]&\ I_{inf}(\hat{X}'/{\mathrm{Spf}}(V))^{log, lf}\ \ar[d] \ar@<1ex>[d]\ar@<-1ex>[d]\\
\ I_{conv}(\hat{X}''/{\mathrm{Spf}}(V))^{log, lf}\ \ar[r]^{\tilde{\Phi}''} &\ I_{inf}(\hat{X}''/{\mathrm{Spf}}(V))^{log, lf} \\
}$$ where the vertical arrows are induced by the following morphisms: $$\hat{X}^e\rightarrow \hat{X},$$ $$p_i :\hat{X}'=\hat{X}^e\times_{\hat{X}} \hat{X}^e\rightarrow \hat{X}^e \,\,\,\textrm{for} \,\,\,i=1,2,$$ $$p_{i,j}:\hat{X}''=\hat{X}^e\times_{\hat{X}} \hat{X}^e\times_{\hat{X}} \hat{X}^e\rightarrow \hat{X}^e\times_{\hat{X}}\hat{X}^e\,\,\, \,\,\,\textrm{for} 1\leq i < j \leq 3$$ where the first is the étale morphism defining the étale cover, $p_i$ and $p_{i,j}$ are the natural projections.\
Thanks to étale descent for log convergent isocrystals ([@Sh1] remark 5.1.7), giving $\mathcal{E}$, a locally free log convergent isocrystal on $\hat{X}$ is the same as giving $\mathcal{E}^{e}$, a locally free log convergent isocrystal on $\hat{X}^e$ and $\alpha_{\mathcal{E}}$ isomorphism between $p_2^*\mathcal{E}^{e}\rightarrow p_1^* \mathcal{E}^{e}$, compatible with the usual cocycle conditions.\
A morphism $f$ from $\mathcal{E}$ to $\mathcal{F}$ is the same as a morphism $f^{e}$ from $\mathcal{E}^{e}$ and $\mathcal{F}^{e}$ that satisfies $\alpha_{\mathcal{F}}\circ p^*_2 f^e = p_1^*f^e \circ \alpha_{\mathcal{E}}$ and the compatibility conditions given by cocycle conditions on $\hat{X}''$.\
By hypothesis $\tilde{\Phi}^{e}$, $\tilde{\Phi}'$ and $\tilde{\Phi}''$ are fully faithful, so $f^{e}$ induces a unique morphism $\tilde{\Phi}^{e}(f^e)$ between $\tilde{\Phi}^{e}(\mathcal{E}^{e})\rightarrow \tilde{\Phi}^{e}(\mathcal{F}^{e})$ that satisfies the same compatibility conditions on $\hat{X}'$ and $\hat{X}''$. Moreover this association is surjective.\
Using étale descent for log infinitesimal isocrystals proven in [@Sh1] remark 3.2.20 we can descend $\tilde{\Phi}^e(f^e)$ to a morphism which coincides with $\tilde{\Phi}(f)$ between $\tilde{\Phi}(\mathcal{E})$ and $\tilde{\Phi}(\mathcal{F})$.
Before proving the full faithfulness étale locally we recall the construction of the universal enlargement and of convergent stratifications given by Shiho in [@Sh1] paragraph 5.2.\
Let $\ell:(\mathscr{X},M)\hookrightarrow (\mathscr{Y},M')$ be an exact closed immersion of $p$-adic log formal $V$-schemes over $({\mathscr{S}},N)$ defined by a sheaf of ideals $\mathcal{I}.$ We need a more general notion of enlargement.
We say that the quadruple $(\mathscr{T},L,z,g)$ is an enlargement of $(\mathscr{X},M)$ in $(\mathscr{Y},M')$ if $(\mathscr{T},L,z)$ is an enlargement of $(\mathscr{X},M)$ and $g$ is an $(\mathscr{S},N)$ morphism $(\mathscr{T},L)\rightarrow (\mathscr{Y},M')$ such that the following diagram is commutative $$\xymatrix{
({\mathscr{T}}_0,L) \ar[d]^{z} \ar[r]^{g_0} & ({\mathscr{Y}}_0,M') \ar[d] \\
({\mathscr{X}},M) \ar[r]^{\ell} & ({\mathscr{Y}}, M') \\
}$$
We call $\mathcal{B}_{n,\mathscr{X}}(\mathscr{Y})$ the formal blow up of $(\mathscr{Y}, M')$ with respect to $p\mathcal{O}_{\mathscr{Y}}+\mathcal{I}^{n+1}$ and we denote by $T_{n,\mathscr{X}}(\mathscr{Y})$ the open of $\mathcal{B}_{n,\mathscr{X}}(\mathscr{Y})$ defined by $p$: $$T_{n,\mathscr{X}}(\mathscr{Y}):=\{x \in \mathcal{B}_{n,\mathscr{X}}(\mathscr{Y})|\,\,(p\mathcal{O}_{\mathscr{Y}}+\mathcal{I}^{n+1})\mathcal{O}_{\mathcal{B}_{n,\mathscr{X}}(\mathscr{Y}),x}=p\mathcal{O}_{\mathcal{B}_{n,\mathscr{X}}(\mathscr{Y}),x}\}.$$ We put on $T_{n,\mathscr{X}}(\mathscr{Y})$ the log structure $L_{n,\mathscr{X}}(\mathscr{Y})$ induced by the pull-back of $M'.$
We can see from [@Og] proposition 2.3 that if $\mathscr{Y}={\mathrm{Spf}}(A)$ and $\mathcal{I}^{n+1}=(a_1,\dots a_m)$ then $T_{n,\mathscr{X}}(\mathscr{Y})={\mathrm{Spf}}(A\{t_1, \dots t_m\}/(pt_1-a_1,\dots,pt_m-a_m)$ modulo $p$-torsion). For a local description of the formal blow up see [@Bo] paragraph 2.6.
The map $(T_{n,\mathscr{Xz}}(\mathscr{Y}))_0\rightarrow ({\mathscr{Y}})_0$ factors through ${\mathscr{X}}_0$, so that we can equip the pair $(T_{n,\mathscr{X}}(\mathscr{Y}),L_{n,\mathscr{X}}(\mathscr{Y})) $ with two maps $(z_n,t_n)$ in such a way that the quadruple $((T_{n,\mathscr{X}}(\mathscr{Y}),L_{n,\mathscr{X}}(\mathscr{Y}),z_n,t_n)$ is an enlargement of $({\mathscr{X}},M)$ in $({\mathscr{Y}},M')$ and the set $\{(T_{n,\mathscr{X}}(\mathscr{Y}),L_{n,\mathscr{X}}(\mathscr{Y}), z_n, t_n)\}_{n\in \mathbb{N}}$ is an inductive system of enlargements that is universal in the sense that for every enlargement $(T',L',z',t')$ of $(\mathscr{X},M)$ in $(\mathscr{Y},M')$ there exists a unique morphism to the inductive system given by $\{(T_{n,\mathscr{X}}(\mathscr{Y}),L_{n,\mathscr{X}}(\mathscr{Y}), z_n, t_n)\}_{n\in \mathbb{N}}$ (proposition 5.2.4 of [@Sh1]).\
The fiber product of fine log formal schemes is a log formal scheme that is not necessarily fine, but, thanks to proposition 2.1.6 of [@Sh1], there is a functor $(-)^{int}$ that sends a log formal scheme to a fine log formal scheme that is right adjoint to the natural inclusion of fine formal schemes in the category of log formal schemes. If $f:({\mathscr{X}},M)\rightarrow ({\mathscr{S}},N)$ is a morphism of fine formal schemes, we denote the fiber product in the category of log formal schemes of $({\mathscr{X}},M)$ and $({\mathscr{X}},M)$ over $({\mathscr{S}},N)$ by $({\mathscr{X}},M)\times_{({\mathscr{S}},N)}({\mathscr{X}},M)$ (resp. the fiber product in the category of log formal schemes of $({\mathscr{X}},M)$ with itself three times over $({\mathscr{S}},N)$ with $({\mathscr{X}},M)\times_{({\mathscr{S}},N)}({\mathscr{X}},M)\times_{({\mathscr{S}},N)}({\mathscr{X}},M)$) and the fine log formal scheme associated to this with $(({\mathscr{X}},M)\times_{({\mathscr{S}},N)}({\mathscr{X}},M))^{int}$ (resp. $(({\mathscr{X}},M)\times_{({\mathscr{S}},N)}({\mathscr{X}},M)\times_{({\mathscr{S}},N)}({\mathscr{X}},M))^{int}$).\
We want to construct a log formal scheme that we will indicated by $({\mathscr{X}}(1),M(1))$ (resp. $({\mathscr{X}}(2),M(2))$) which factors the diagonal embedding $\Delta^{int} : ({\mathscr{X}},M) \rightarrow (({\mathscr{X}},M)\times_{({\mathscr{S}},N)}({\mathscr{X}},M))^{int}$ (resp. $\Delta^{int} : ({\mathscr{X}},M) \rightarrow (({\mathscr{X}},M)\times_{({\mathscr{S}},N)}({\mathscr{X}},M)\times_{({\mathscr{S}},N)}({\mathscr{X}},M))^{int}$) in an exact closed immersion $({\mathscr{X}},M)\hookrightarrow ({\mathscr{X}}(1),M(1))$ (resp. the closed immersion $({\mathscr{X}},M)\hookrightarrow ({\mathscr{X}}(2),M(2))$) followed by a formally log étale morphism: following [@Sh1] or [@Ka] proposition 4.10, we can do this if the morphism $f:({\mathscr{X}},M)\rightarrow ({\mathscr{S}},N)$ has a chart.\
We indicate with $(P_{{\mathscr{X}}}\rightarrow M, Q_V\rightarrow N, Q\rightarrow P )$ a chart of $f$, with $\alpha(1)$ (resp. $\alpha(2)$) the homomorphism induced by the map $P\oplus_Q P\rightarrow P$ (resp. $P\oplus_Q P \oplus_Q P \rightarrow P$) and with $R(1)$ the set $(\alpha(1)^{gp})^{-1}(P)$ (resp. with $R(2)$ the set $(\alpha(2)^{gp})^{-1}(P)$).\
With this notation we define ${\mathscr{X}}(1)=({\mathscr{X}}\times_{{\mathscr{S}}}{\mathscr{X}})\times_{{\mathrm{Spf}}(\mathbb{Z}_p\{P\oplus_Q P\})}{\mathrm{Spf}}({\mathbb{Z}_p\{R(1)\}})$ (resp. ${\mathscr{X}}(2)=({\mathscr{X}}\times_{{\mathscr{S}}}{\mathscr{X}}\times_{{\mathscr{S}}}{\mathscr{X}})\times_{{\mathrm{Spf}}(\mathbb{Z}_p\{P\oplus_Q P\oplus_Q P\})}{\mathrm{Spf}}({\mathbb{Z}_p\{R(2)\}})$) equipped with the log structure $M(1)$ (resp $M(2)$) defined as the log structure induced by the canonical log structure on ${\mathrm{Spf}}({\mathbb{Z}_p}\{R(1)\})$ (resp. on ${\mathrm{Spf}}({\mathbb{Z}_p}\{R(2)\})$). Thanks to proposition 4.10 of [@Ka] we have that $({\mathscr{X}}(1),M(1))$ (resp. $({\mathscr{X}}(2),M(2))$) factors the diagonal embedding as we wanted.\
Using the fact that $({\mathscr{X}},M)\hookrightarrow ({\mathscr{X}}(i),M(i))$ are exact closed immersions of log formal schemes for $i=1,2$, we define $\{(T_{{\mathscr{X}},n}({\mathscr{X}}(i)), L_{{\mathscr{X}},n}({\mathscr{X}}(i)),z_n(i),t_n(i))\}_{n\in \mathbb{N}}$ which is the universal system of enlargements associated to this closed immersions. For simplicity of notation we will denote by $(T_n(i),L_n(i))$ the $n$-th universal enlargement $(T_{{\mathscr{X}},n}({\mathscr{X}}(i)), L_{{\mathscr{X}},n}({\mathscr{X}}(i)),z_n(i),t_n(i))$.\
The natural maps $$p_i:({\mathscr{X}},M)\times_{({\mathscr{S}},N)}({\mathscr{X}},M)\rightarrow({\mathscr{X}},M)\;\;\;\;\;\;\;\;\;\textrm{for}\,\,\,i=1,2,$$ $$p_{i,j}:({\mathscr{X}},M)\times_{({\mathscr{S}},N)}({\mathscr{X}},N)\times_{({\mathscr{S}},N)}({\mathscr{X}},N)\rightarrow({\mathscr{X}},N)\times_{({\mathscr{S}},N)}({\mathscr{X}},N)\;\;\;\;\;\;$$ $$\textrm{for}\;\; 1\leq i < j\leq 3$$ $$\Delta: ({\mathscr{X}},M) \rightarrow ({\mathscr{X}},M)\times_{({\mathscr{S}},N)}({\mathscr{X}},M)$$ induce compatible morphisms of enlargements: $$q_{i;n}:(T_n(1),L_n(1))\rightarrow ({\mathscr{X}},M)$$ $$q_{i,j;n}: (T_n(2),L_n(1))\rightarrow (T_n(1),L_n(1))$$ $$\Delta_n: ({\mathscr{X}},M)\rightarrow (T_n(1),L_n(1)).$$ With the same notation we can give the following definition.
\[convstra\] A log convergent stratification on $({\mathscr{X}},M)$ is an isocoherent sheaf $\mathcal{E}_{{\mathscr{X}}}$ on ${\mathscr{X}}$ and a compatible family of isomorphisms $$\epsilon_{n}:q_{2;n}^*\mathcal{E}_{{\mathscr{X}}}\rightarrow q_{1;n}^*\mathcal{E}_{{\mathscr{X}}}$$ such that every $\epsilon_n$ satisfies $$\Delta_n^*(\epsilon_n)=\textrm{id};$$ $$q_{1,2;n}^*(\epsilon_n)\circ q_{2,3;n}^*(\epsilon_n)= q_{1,3;n}^*(\epsilon_n).$$ We denote the category of log convergent stratifications by $Str'(({\mathscr{X}}, M)/({\mathscr{S}},N)).$
As in the case of log infinitesimal isocrystals we can establish an equivalence of categories between log convergent stratifications and convergent log isocrystals: this is the statement of proposition 5.2.6 of [@Sh1].
\[stratconvisoconv\] If the log formal scheme $({\mathscr{X}},M)$ is formally log smooth over the log formal scheme $({\mathrm{Spf}}(V),N)$, then the category $I_{conv}(({\mathscr{X}},M)/({\mathrm{Spf}}(V),N))$ is equivalent to $Str'(({\mathscr{X}},M)/({\mathrm{Spf}}(V),N))$.
Now that we have introduced all the machinery we can finish the proof of full faithfulness of the functor $\tilde{\Phi}$. We need the following result, whose proof is essentially the same as the one of proposition 5.2.9 of [@Sh1].\
\[ff\] The functor $$\tilde{\Phi} : I_{conv}((\hat{X},M)/({\mathrm{Spf}}(V),N))^{lf}\longrightarrow I_{inf}((\hat{X},M)/({\mathrm{Spf}}(V),N))^{lf}$$ is fully faithful étale locally.
Since fine log formal schemes have charts étale locally and the statement is of étale local nature, we can suppose that the morphism $f:(\hat{X},M)\rightarrow ({\mathrm{Spf}}(V),N)$ has a chart globally. Let us note first that for $\mathcal{E}$ and $\mathcal{F}$ in $I_{conv}((\hat{X},M)/({\mathrm{Spf}}(V),N))^{lf}$, we can define $\mathcal{H}om(\mathcal{E}, \mathcal{F})$ $\in$ $I_{conv}((\hat{X},M)/({\mathrm{Spf}}(V),N))^{lf}$ by $$\mathcal{H}om(\mathcal{E}, \mathcal{F})_{{\mathscr{T}}}=\mathcal{H}om(\mathcal{E}_{{\mathscr{T}}}, \mathcal{F}_{{\mathscr{T}}})$$ and moreover that the global sections of $\mathcal{H}om(\mathcal{E}, \mathcal{F})$ are isomorphic to Hom$(\mathcal{E},\mathcal{F})$; these two assertions follow from the fact that $\mathcal{E}$ and $\mathcal{F}$ are locally free.\
Using the local freeness we can conclude that the same holds for the category $I_{inf}((\hat{X},M)/({\mathrm{Spf}}(V),N))^{lf}$.\
So we are reduced to prove that there is an isomorphism $$H^0((\hat{X},M)/({\mathrm{Spf}}(V),N)_{conv},\mathcal{E})\longrightarrow H^{0}((\hat{X},M)/({\mathrm{Spf}}(V),N)_{inf},\tilde{\Phi}(\mathcal{E}))$$ for every $\mathcal{E}$ in $I_{conv}((\hat{X},M)/({\mathrm{Spf}}(V),N))^{lf}$.\
As we noticed before the morphism $f:(\hat{X}, M)\rightarrow ({\mathrm{Spf}}(V),N)$ has a chart globally and so we can construct the scheme $(\hat{X}(1),(M)(1))$ that we described above.\
The equivalence in proposition \[stratconvisoconv\] associates to $\mathcal{E}$ $\in$ $I_{conv}((\hat{X},M)/({\mathrm{Spf}}(V),N))^{lf}$ a log convergent stratification $(\mathcal{E}_{\hat{X}},\epsilon_n)$ given by a locally free isocoherent sheaf $\mathcal{E}_{\hat{X}}$ on $\hat{X}$ and isomorphisms $$\epsilon_n:(K\otimes \mathcal{O}_{T_n(1)})\otimes \mathcal{E}_{\hat{X}}\longrightarrow \mathcal{E}_{\hat{X}}\otimes (K\otimes \mathcal{O}_{T_n(1)})$$ for all $n$.\
So the set $H^0((\hat{X},M)/({\mathrm{Spf}}(V),N)_{conv},\mathcal{E})$ can be characterized in terms of log convergent stratifications as follows: $$H^0((\hat{X},M)/({\mathrm{Spf}}(V),N)_{conv},\mathcal{E})=\{\,\,e \,\,\in \,\, \Gamma(\hat{X},\mathcal{E}_{\hat{X}})|\,\,\,\epsilon_n(1\otimes e)=e\otimes 1\,\,\forall n\}.$$ Let $J$ be the sheaf of ideals that defines the closed immersion $ \hat{X}\hookrightarrow \hat{X}(1)$; we denote by $\mathcal{O}_{\hat{X}(1)^{an}}$ the sheaf $\varprojlim_m K\otimes \mathcal{O}_{\hat{X}(1)}/J^m$. By proposition 5.2.7 (2) of [@Sh1] we know that there is an injective map $K\otimes \mathcal{O}_{T_n(1)}\rightarrow \mathcal{O}_{\hat{X}(1)^{\textrm{an}}}$. If we tensor the isomorphisms $\epsilon_n$ of the convergent stratification $(\mathcal{E}_{\hat{X}},\epsilon_n)$ with this map we obtain a map $$\epsilon': \mathcal{O}_{\hat{X}(1)^{\textrm{an}}}\otimes \mathcal{E}_{\hat{X}}\longrightarrow \mathcal{E}_{\hat{X}}\otimes \mathcal{O}_{\hat{X}(1)^{\textrm{an}}}$$ that coincides with the limit of the isomorphisms of the stratification induced by $\tilde{\Phi}(\mathcal{E})$ through the equivalence of categories $$I_{inf}((\hat{X},M)/({\mathrm{Spf}}(V),N))\cong \widehat{Str}((\hat{X},M)/({\mathrm{Spf}}(V),N)).$$ So we can characterize the set $H^0((\hat{X},M)/({\mathrm{Spf}}(V),N)_{inf}$ as follows $$H^0((\hat{X},M)/({\mathrm{Spf}}(V),N)_{inf}, \tilde{\Phi}(\mathcal{E}))=\{\,\,e \,\,\in \,\, \Gamma(\hat{X},\mathcal{E}_{\hat{X}})|\,\,\,\epsilon '(1\otimes e)=e\otimes 1\}.$$ This means that the claim is reduced to prove that the following diagram $$\label{primotriangolo}
\xymatrix{
\mathcal{O}_{\hat{X}(1)^{\textrm{an}}}\otimes \mathcal{E}_{\hat{X}} \ar[rr]^{\epsilon '} & &
\mathcal{E}_{\hat{X}}\otimes \mathcal{O}_{\hat{X}(1)^{\textrm{an}}} \\
& \mathcal{E}_{\hat{X}}
\ar[ul] \ar[ur] & \\
}$$ is commutative if and only if this is commutative $$\label{secondotriangolo}
\xymatrix{
(K\otimes \mathcal{O}_{T_n(1)})\otimes \mathcal{E}_{\hat{X}} \ar[rr]^{\epsilon_n} & &
\mathcal{E}_{\hat{X}}\otimes (K\otimes \mathcal{O}_{T_n(1)}) \\
& \mathcal{E}_{\hat{X}}
\ar[ul] \ar[ur] & . \\
}$$ Knowing that the following diagram is commutative $$\label{quadrato}
\xymatrix{\mathcal{O}_{\hat{X}(1)^{\textrm{an}}}\otimes \mathcal{E}_{\hat{X}} \ar[r]^{\epsilon'} & \mathcal{E}_{\hat{X}}\otimes \mathcal{O}_{\hat{X}(1)^{\textrm{an}}} \\
(K\otimes \mathcal{O}_{T_n(1)})\otimes \mathcal{E}_{\hat{X}} \ar[r]^{\epsilon_n} \ar[u] &
\mathcal{E}_{\hat{X}}\otimes (K\otimes \mathcal{O}_{T_n(1)}) \ar[u]\\
}$$ and putting together (\[secondotriangolo\]) and (\[quadrato\]), we can conclude that if (\[secondotriangolo\]) is commutative then (\[primotriangolo\]) is commutative.\
Let us suppose instead that (\[primotriangolo\]) is commutative: then using the fact that $\mathcal{E}_{\hat{X}}$ is flat and that the map $K\otimes \mathcal{O}_{T_n(1)}\rightarrow \mathcal{O}_{\hat{X}(1)^{\textrm{an}}}$ is injective we can conclude that also (\[secondotriangolo\]) is commutative.
Shiho in [@Sh1] proposition 5.2.9 proves that the functor $\Phi$ is fully faithful étale locally when it is restricted to the nilpotent part of $I_{conv}((\hat{X},M)/({\mathrm{Spf}}(V),N))$ and $I_{inf}((\hat{X},M)/({\mathrm{Spf}}(V),N))$. Our proof is essentially the same, because the key property of nilpotent objects used in Shiho’s proof is that the nilpotent isocrystals are locally free.\
Putting together theorem \[ff\] and proposition \[discesaperi\] we obtain the following
The functor $\tilde{\Phi}$ is fully faithful.
Characterizations of log convergent isocrystals in terms of stratifications {#Characterizations of log convergent isocrystals in terms of stratifications}
===========================================================================
We want to describe the essential image of the functor $\tilde{\Phi}$. As for the case of the proof of full faithfulness it will be enough to describe it étale locally, then, with descent argument we can conclude as as in proposition \[discesaperi\]. So we can suppose that $(\hat{X},M)\rightarrow({\mathrm{Spf}}(V),N)$ has a chart globally.\
To avoid log differential calculus we will prove that we can restrict to the case of trivial log structures. We need the following lemma whose proof is essentially the same as proposition 5.2.11 of [@Sh1].
\[U denso\] Let $f:(\hat{X},M) \rightarrow ({\mathrm{Spf}}(V),N)$ be a formally smooth morphism of fine log schemes that admits a chart. Let $U$ be a dense open subset of $\hat{X}$ and set $j$ the open immersion $j:U\hookrightarrow \hat{X}$. If $\mathcal{E}_i \in I_{inf}((\hat{X},M)/({\mathrm{Spf}}(V),N))^{lf}$ and there exists $\mathcal{E}_c$ in the category $I_{conv}((U,M)/({\mathrm{Spf}}(V),N))^{lf}$ such that $\tilde{\Phi}(\mathcal{E}_c)=\mathcal{E}_i$, then $\mathcal{E}_i\in$ $\tilde{\Phi}( I_{conv}((\hat{X},M)/({\mathrm{Spf}}(V),N))^{lf}).$
Let us consider the sheaf $$\mathcal{F}=z_{n*}\mathcal{H}om((K\otimes \mathcal{O}_{T_{n}(1)})\otimes \mathcal{E}_i,\mathcal{E}_i \otimes (K\otimes \mathcal{O}_{T_{n}(1)})).$$ The following sequence is exact: $$0\rightarrow \mathcal{F}\xrightarrow{a} (\mathcal{F}\otimes \mathcal{O}_{\hat{X}(1)^{an}})\oplus j_*j^*(\mathcal{F})\xrightarrow{b} j_*j^*(\mathcal{F}\otimes \mathcal{O}_{\hat{X}(1)^{an}}),$$ because $\mathcal{F}$ is a projective $z_{n*}(K\otimes \mathcal{O}_{T_{n}(1)})$-module, and the following sequence is exact by proposition 5.2.8 of [@Sh1] and [@Og] lemma 2.14 $$0\rightarrow z_{n*}(K\otimes \mathcal{O}_{T_{n}(1)})\rightarrow \mathcal{O}_{\hat{X}(1)^{an}}\oplus j_*j^*z_{n*}(K\otimes \mathcal{O}_{T_{n}(1)})\rightarrow j_*j^* \mathcal{O}_{\hat{X}(1)^{an}}.$$ Viewing $\mathcal{E}_c$ as a convergent stratification we have a map $$\epsilon'_n: j_*j^*z_{n*}((K\otimes \mathcal{O}_{T_{n}(1)})\otimes \mathcal{E}_{i,\hat{X}})\rightarrow j_*j^*z_{n*} (\mathcal{E}_{i,\hat{X}}\otimes(K\otimes \mathcal{O}_{T_{n}(1)}));$$ on the other hand the log infinitesimal isocrystal $\mathcal{E}_i$ induces a stratification $$\epsilon' :\mathcal{O}_{\hat{X}(1)^{an}}\otimes \mathcal{E}_{i,\hat{X}}\rightarrow \mathcal{E}_{i,\hat{X}}\otimes \mathcal{O}_{\hat{X}(1)^{an}}.$$ The pair $(\epsilon'_n, \epsilon')$ lies in Ker$(b)$, then there exists an $\epsilon_n$ $\in$ $\mathcal{F}$ such that $a(\epsilon_n)=(\epsilon'_n, \epsilon')$ that defines a convergent stratification, i.e. a convergent isocrystal $\bar{\mathcal{E}_c}$. Moreover one can verify that $\tilde{\Phi}(\bar{\mathcal{E}_c})=\mathcal{E}_i $.
Now we want to apply proposition $\ref{U denso}$ choosing as $U$ the subset $$\hat{X}_{\hat{f}-triv}=\{x\in \hat{X}\;|\;(\hat{f}^{*}N)_{\bar{x}}=(M)_{\bar{x}}\}.$$ Let us prove that it is open and dense in $\hat{X}$. Clearly $\hat{X}_{\hat{f}-triv}$ is homeomorphic to $X_{k,f_k-triv}$ so it will be sufficient to prove that $X_{k,f_k-triv}$ is open dense in $X_k$. But this follows from proposition 2.3.2 of [@Sh1] because the special fiber is reduced.\
Now applying proposition \[U denso\] we can restrict ourselves to the case in which $\hat{f}^{*}N=M$. As Shiho notices the hypothesis $\hat{f}^{*}N=M$ gives an equivalence of categories $$I_{conv}((\hat{X},M)/({\mathrm{Spf}}(V),N))\cong I_{conv}((\hat{X},\textrm{triv})/({\mathrm{Spf}}(V),\textrm{triv})),$$ where with triv we indicate the trivial log structure and $$I_{inf}((\hat{X},M)/({\mathrm{Spf}}(V),N))\cong I_{inf}((\hat{X},\textrm{triv})/({\mathrm{Spf}}(V),\textrm{triv})).$$ So we are reduce to the case of trivial log structures, as we wanted. We will characterize the essential image using certain type of stratification that we call special.
\[special\] Let $(E,\epsilon_n)$ be an object of $\widehat{Str}(\hat{X}/{\mathrm{Spf}}(V))$ and let $\tilde {E}$ be a coherent $p$-torsion-free $\mathcal{O}_{\hat{X}}$-module such that $K\otimes \tilde{E}=E$; we say that $(E,\epsilon_n)$ is special if there exists a sequence of integers $k(n)$ for $n \in \mathbb{N}$ such that:
- $k(n)=o(n)$ for $n\rightarrow \infty$,
- the restriction of the map $p^{k(n)}\epsilon_n$ to $p_{2,n}^{*}(\tilde{E})$ has image contained in $p_{1,n}^{*}(\tilde{E})$ and the restriction of the map $p^{k(n)}\epsilon_n^{-1}$ to $p_{1,n}^{*}(\tilde{E})$ has image contained in $p_{2,n}^{*}(\tilde{E})$.
This definition is a small modification of definition of special stratification given by Shiho ([@Sh1] Definition 5.2.12). Our definition of special is weaker then Shiho’s definition: every special object in the sense of Shiho is special in our sense and allows us to characterize the essential image of the functor $\tilde{\Phi}$.\
Let us see now that the definition of special is well-posed.
Definition \[special\] is independent of the choice of the $p$-torsion-free sheaf $\tilde{E}$.
Suppose that $(E,\epsilon_n)$ is special and that the conditions in definition \[special\] are verified for a given coherent $p$-torsionfree $\mathcal{O}_{\hat{X}}$-module $\tilde{E}$ such that $K\otimes \tilde{E}=E$. We take an other $p$-torsion-free $\mathcal{O}_{\hat{X}}$-module $\tilde{F}$ such that $K\otimes \tilde{F}=E$ and we want to prove that the same conditions are verified. From [@Og] proposition 1.2 we know the following isomorphisms $$K\otimes \mathrm{Hom}_{\mathcal{O}_{\hat{X}}}(\tilde {E},\tilde{F})\cong\mathrm{Hom}_{K\otimes \mathcal{O}_{\hat{X}}}(K\otimes \tilde{E},K\otimes \tilde{F})\cong \mathrm{End}_{K\otimes \mathcal{O}_{\hat{X}}}(E).$$ If we take the identity as endomorphism of $E$ we know that there exists a power of $p$, say $p^{a}$, such that the multiplication by $p^a$ is a morphism from $\tilde{E}$ to $\tilde{F}$; moreover this morphism is injective because $\tilde{F}$ is $p$-torsion-free.\
In the same way we can prove that there exists a $b$ such that the multiplication by $p^b$ is an injective morphism between $\tilde{F}$ and $\tilde{E}$.\
If we consider now the morphism $p^{k(n)+a+b}\epsilon_n$ then we have that the restriction of this to $p_{2,n}^{*}(\tilde{F})$ goes to $p_{1,n}^{*}(\tilde{F})$.\
Arguing analogously for $\epsilon_{n}^{-1}$ we are done.
Let us see, first, that every object in the essential image of the functor $\Phi$ is special.\
Following Shiho [@Sh1], proposition 3.2.14 and proposition 5.2.6, both in the case of trivial log structures, we have the equivalences of categories $$I_{conv}(\hat{X}/{\mathrm{Spf}}(V))={Str'}(\hat{X}/{\mathrm{Spf}}(V)),$$ $$I_{inf}(\hat{X}/{\mathrm{Spf}}(V))=\widehat{Str}(\hat{X}/{\mathrm{Spf}}(V)).$$ The functor $\Phi$ induces the functor $$\alpha:{Str'}(\hat{X}/{\mathrm{Spf}}(V))\rightarrow \widehat{Str}(\hat{X}/{\mathrm{Spf}}(V)).$$ From now on we may work locally.\
We are reduced to the situation where $\hat{X}$ is formally smooth and we call $dx_1, \dots dx_l$ a basis of $\Omega^{1}_{\hat{X}/{\mathrm{Spf}}(V)}$. Let us call $\xi_1, \dots, \xi_l$ the dual basis of $dx_1,\dots dx_l$ where $\xi_j=1\otimes x_j-x_j \otimes 1$; we will indicate $(\xi_1, \dots ,\xi_l)$ with $\boldsymbol{\xi}$ and an $l$-ple of natural numbers $(\beta_1,\dots \beta_l)$ with $\boldsymbol{\beta}$. We will use multi-index notations denoting $\prod_j \xi_j^{\beta_j}$ by $\boldsymbol{\xi}^{\boldsymbol{\beta}}$ and $\beta_1+\dots +\beta_l$ by $|\boldsymbol{\beta}|.$\
We call $\hat{X}^{n}$ the $n$-th infinitesimal neighborhood of $\hat{X}$ in $\hat{X}\times_{{\mathrm{Spf}}{V}}\hat{X}$. By a formal version of proposition 2.6 of [@BeOg] we know that $\mathcal{O}_{\hat{X}^{n}}$ is a free $\mathcal{O}_{\hat{X}}$-module generated by $\{\boldsymbol{\xi}^{\boldsymbol{\beta}}:|\boldsymbol{\beta}| \leq n\}$, so that we can write $\mathcal{O}_{\hat{X}^n}=\mathcal{O}_{\hat{X}}[[\boldsymbol{\xi}]]/(\boldsymbol{\xi}^{\boldsymbol{\beta}},\,\, |\boldsymbol{\beta}|=n+1)$.\
We want to give a local description also for the universal system of enlargement $\{T_n\}_n$ of $\hat{X}$ in $\hat{X}\times_{{\mathrm{Spf}}{V}}\hat{X}$. By [@Og] remark 2.6.1 we know that $T_n$ is isomorphic to the $n$-th universal enlargement of $\hat{X}$ in $\hat{X}\times_{{\mathrm{Spf}}{V}}\hat{X}_{|\hat{X}}$, the formal completion of $\hat{X}\times_{{\mathrm{Spf}}{V}}\hat{X}$ along $\hat{X}$. Using this and the local description given in the proof of proposition 2.3 of [@Og] we can write $\mathcal{O}_{T_n}=\mathcal{O}_{\hat{X}}\{\boldsymbol{\xi},\boldsymbol{\xi}^{\boldsymbol{\beta}}/p\,\,\,(|\boldsymbol{\beta}|=n+1)\}$. By universality of blowing up there exists a unique map $\psi_n$ such that the following diagram is commutative $$\label{phin}
\xymatrix{
\ar@{}|(.7)\cal[dr]&\hat{X}^n\ar@{-->}[dl]_{\psi_n}\ar[d]\\
T_n\ar[r]&\hat{X}\times_{{\mathrm{Spf}}{V}} \hat{X}.
}$$ The functor $\alpha$ is induced by the pull back of $\psi_n$ and in local coordinates is given by $$\mathcal{O}_{T_n}=\mathcal{O}_{\hat{X}}\{\boldsymbol{\xi},\boldsymbol{\xi}^{\boldsymbol{\beta}}/p\,\,\,(|\boldsymbol{\beta}|=n+1) \}\rightarrow \mathcal{O}_{\hat{X}}[|\boldsymbol{\xi}|]/(\boldsymbol{\xi}^{\boldsymbol{\beta}},\,\, |\boldsymbol{\beta}|=n+1).$$
\[immagineessenziale\] If $(E,\epsilon_n)$ is in $\widehat{Str}(\hat{X}/{\mathrm{Spf}}(V))$ and it is in the image of the functor $\alpha$, then it is special.
Let $(E,\epsilon'_n)$ be an element of $Str'(\hat{X}/{\mathrm{Spf}}(V))$ such that $\alpha(E,\epsilon'_n)=(E,\epsilon_n)$, with $$q_{2,n}^*E\xrightarrow{\;\epsilon'_n\;}q_{1,n}^{*}E,$$ where $q_{i,n}$ are the projections from $T_n$ to $\hat{X}$, that exist by definition of convergent stratification.\
We note that $q_{1,n}^{*}E=E\otimes_{\mathcal{O}_{\hat{X}}}\mathcal{O}_{\hat{X}}[|\boldsymbol{\xi}|]\{\boldsymbol{\xi}^{\boldsymbol{\beta}}/p\,\, ,( |\boldsymbol{\beta}|=n+1)\}$ is embedded in $\prod_{\boldsymbol{\beta}} E\boldsymbol{\xi}^{\boldsymbol{\beta}}$ and so we have a map, that we call $\epsilon'$, which is the composition of $$E\rightarrow q_{2,n}^*E\xrightarrow{\;\epsilon'_n\;}q_{1,n}^{*}E\rightarrow \prod_{\boldsymbol{\beta}} E\boldsymbol{\xi}^{\boldsymbol{\beta}}.$$ Let us note that $\epsilon'$ does not depend from $n$, a consequence of the fact that the maps $\epsilon'_n$ coming from the convergent stratification are compatible. Using the isomorphisms $\epsilon_n$ that define the stratification $(E,\epsilon_n)$ $$p_{2,n}^*E\xrightarrow{\;\epsilon_n\;}p_{1,n}^*E=\prod_{|\boldsymbol{\beta}|\leq n}E\boldsymbol{\xi}^{\boldsymbol{\beta}},$$ where $p_{i,n}$ are the projections $p_{i,n}:X^n\rightarrow X$, we can define a map $$\label{daEaq2}
E\rightarrow p_{2,n}^*E\xrightarrow{\;\epsilon_n\;}p_{1,n}^*E=\prod_{|\boldsymbol{\beta}|\leq n}E\boldsymbol{\xi}^{\boldsymbol{\beta}}.$$ The fact that $\alpha(E,\epsilon'_n)=(E,\epsilon_n)$ means that, if we call $pr$ the projection $$\prod_{\boldsymbol{\beta}} E\boldsymbol{\xi}^{\boldsymbol{\beta}}\xrightarrow{pr} \prod_{|\boldsymbol{\beta}|\leq n}E\boldsymbol{\xi}^{\boldsymbol{\beta}},$$ the map in (\[daEaq2\]) coincides with the map $$E\xrightarrow{\;\epsilon'\;}\prod_{\boldsymbol{\beta}} E \boldsymbol{\xi}^{\boldsymbol{\beta}}\xrightarrow{\;pr\;}\prod_{|\boldsymbol{\beta}|\leq n}E\boldsymbol{\xi}^{\boldsymbol{\beta}}.$$ Let $\tilde{E}$ be $p$-torsion-free $\mathcal{O}_{\hat{X}}$-module such that $K\otimes \tilde{E}=E$; so $q_{1,n}^*(\tilde{E})=\tilde{E}\otimes\mathcal{O}_{\hat{X}}[|\boldsymbol{\xi}|]\{\boldsymbol{\xi}^{\boldsymbol{\beta}}/p\,\,\,(|\boldsymbol{\beta}|=n+1) \}$ and this is embedded in $\prod_{\boldsymbol{\beta}} \tilde{E}\boldsymbol{\xi}^{\boldsymbol{\beta}}/p^{\left\lfloor \frac{|{\boldsymbol{\beta}}|}{n+1}\right\rfloor}$.\
Thus there exists $a$ in $\mathbb{N}$ such that $$p^a\epsilon'(\tilde{E})\subset \prod_{\boldsymbol{\beta}} \tilde{E}\frac{\boldsymbol{\xi}^{\boldsymbol{\beta}}}{p^{\left\lfloor \frac{|\boldsymbol{\beta}|}{n+1} \right\rfloor}}$$ and, if we call $\pi_{\boldsymbol{\beta}}$ the projection $\prod_{\boldsymbol{\beta}} E\boldsymbol{\xi}^{\boldsymbol{\beta}}\rightarrow E$, then $$p^{a+{\left\lfloor \frac{|\boldsymbol{\beta}|}{n+1} \right\rfloor}}\pi_{\boldsymbol{\beta}} \circ \epsilon'(\tilde{E})\subset \tilde{E};$$ therefore there exists a sequence $b_n(\boldsymbol{\beta})$ that tends to infinity when $|\boldsymbol{\beta}|$ goes to infinity such that $$\label{conbn}
p^{\left\lfloor \frac{|\boldsymbol{\beta}|}{n} \right\rfloor}\pi_{\boldsymbol{\beta}} \circ \epsilon'(\tilde{E})\subset p^{b_n(\boldsymbol{\beta})}\tilde{E}.$$ If we define now $$a(k):=\mathrm{min}\{a\in \mathbb{N}|\;p^a\pi_{\boldsymbol{\beta}}\circ \epsilon'(\tilde{E})\subset \tilde{E}\; \mathrm{for}\,\,\mathrm{all} \; \boldsymbol{\beta}\,\,\,\textrm{such that }\,\, |\boldsymbol{\beta}|\leq k\},$$ then $p^{a(k)}\epsilon'_k(\tilde{E})\subset \prod_{|\boldsymbol{\beta}|\leq k}\tilde E \boldsymbol{\xi}^{|\boldsymbol{\beta}|}, $ which means that $p^{a(k)}\epsilon'_k$ sends $p_{2,k}^*(\tilde{E})$ into $p_{1,k}^*(\tilde{E})$. So we are left to prove that $a(k)=o(k)$ for $k\rightarrow \infty$ .\
We notice, from the definition of $a(k)$, that $a(k)$ is an increasing sequence. If $a(k)$ is bounded, then we are done. Arguing by contradiction, then $a(k)\rightarrow \infty$ for $k\rightarrow \infty$. This means that there exists a sequence $\{k_i\}_i$ such that $$0<a(k_1)<a(k_2)<...<a(k_i)<a(k_{i+1})<\dots .$$ Then $$p^{a(k_i)}\pi_{\boldsymbol{\beta}}\circ \epsilon '(\tilde{E})\subseteq \tilde{E},$$ for every $\boldsymbol{\beta}$ such that $|\boldsymbol{\beta}|\leq k_i$. Let us prove that $$\label{nonincl}
p^{a(k_i)}\pi_{\boldsymbol{\beta}}\circ \epsilon '(\tilde{E}) \nsubseteq p\tilde{E}$$ for some $\boldsymbol{\beta}$ with $|\boldsymbol{\beta}|=k_i.$ Let us suppose that this is not true; this means that $$\label{-1}
p^{a(k_i)-1}\pi_{\boldsymbol{\beta}}\circ \epsilon '(\tilde{E}) \subseteq \tilde{E}$$ for every $\boldsymbol{\beta}$ such that $|\boldsymbol{\beta}|=k_i.$ Moreover for $\boldsymbol{\beta}$ with $|\boldsymbol{\beta}|<k_i$ we have $$p^{a(k_i)-1}\pi_{\boldsymbol{\beta}}\circ \epsilon '(\tilde{E}) \subseteq p^{a(k_{i-1})}\pi_{\boldsymbol{\beta}}\circ \epsilon' (\tilde{E}) \subseteq \tilde{E}.$$ Hence we have $$p^{a(k_i)-1}\pi_{\boldsymbol{\beta}}\circ \epsilon '(\tilde{E}) \subseteq \tilde{E}$$ for all $\boldsymbol{\beta}$ with $|\boldsymbol{\beta}|\leq k_i$ and this contradicts the definition of $a(k_i)$’s, so (\[nonincl\]) holds. If we now put together the formula (\[conbn\]) with $|\boldsymbol{\beta}|=k_i$ and (\[nonincl\]), we find that $$\lim_{i \to \infty}\left(\left\lfloor \frac{k_i}{n}\right\rfloor-a(k_i)\right)=\infty,$$ so that there exists $i_0$ such that $$0\leq \frac{a(k_i)}{k_i}\leq \frac{1}{n}$$ for all $i\geq i_0$. Then for any $k\geq k_{i_0}$ we can find some $k_i$ with $k_i\leq k \leq k_{i+1}-1$ and then $$0\leq \frac{a(k)}{k}\leq \frac{a(k_i)}{k_i}\leq \frac{1}{n}.$$ Hence we have that $\limsup_{k} \frac{a(k)}{k}\leq \frac{1}{n}.$ Since this is true for any $n$, we have that $a(k)=o(k).$
Now we want to prove the converse: that every special object is in the image of functor $\alpha$. This is proven by Shiho in proposition 5.2.13 of [@Sh1] for his special objects, but the proof works also in our case.
If $(E,\epsilon_n )$ is a special stratification on $\hat{X}$, then there exists $(E', \epsilon'_n)$ $\in$ $Str'(\hat{X},{\mathrm{Spf}}(V))$ such that $\alpha( (E', \epsilon'_n))=(E,\epsilon_n ).$
So we have a complete characterization in term of stratification of the differential equations coming from a log-convergent isocrystal.\
We want to describe the property of being special in term of radius of convergence. We will use the formalism given in [@LS] in the local situation described before proposition \[immagineessenziale\]. If we have an element $(E,\epsilon_n) \in \widehat{Str}(\hat{X}/{\mathrm{Spf}}(V))$, then we can take the inverse limit of the map that we considered in the proof of the proposition \[immagineessenziale\] $$E\rightarrow p_{2,n}^*E\xrightarrow{\;\epsilon_n\;}p_{1,n}^*E=\prod_{|\boldsymbol{\beta}|\leq n}E\boldsymbol{\xi}^{\boldsymbol{\beta}},$$ and we obtain $$\theta:E\rightarrow \varprojlim p_{2,n}^*E\xrightarrow{\;\varprojlim \epsilon_n\;}\varprojlim p_{1,n}^*E=\prod_{\boldsymbol{\beta}}E\boldsymbol{\xi}^{\boldsymbol{\beta}}.$$ According to definition 4.4.1 of [@LS] we can say that a section $s\in \Gamma(\hat{X},E)$ is $\eta$-convergent, with $\eta<1$ for the stratification $(E,\epsilon_n)$ if $$\theta(s)\in \Gamma(\hat{X},E\otimes \mathcal{O}_{\hat{X}}\{\frac{\boldsymbol{\xi}}{\eta}\} ).$$
\[radius\] The radius of convergence of the section $s$ for the stratification $(E,\epsilon_n)$ is defined as $$R(s)=\mathrm{sup}\{\eta| \;s \;\mathrm{is} \;\eta\mathrm{-convergent}\}.$$ And the radius of convergence of the stratification $(E,\epsilon_n)$ is $$R((E,\epsilon_n), \hat{X})=\mathrm{inf}_{s\in \Gamma(\hat{X},E)}R(s).$$
A stratification $(E, \epsilon_n)$ is special if and only if its radius of convergence is equal to 1.
We know that $\widehat{Str}(\hat{X}/{\mathrm{Spf}}(V))$ is equivalent to the category $\widehat{\mathrm{MIC}}(\hat{X}/{\mathrm{Spf}}(V))$. By lemma 5.2.15 of [@Sh1] we can write the map $\theta$ locally. Following the notation that we recalled before proposition \[immagineessenziale\] we denote by $\{D_{\boldsymbol{\beta}}\}_{0\leq|\boldsymbol{\beta}|\leq n}$ the dual basis of $\{\boldsymbol{\xi}^{\boldsymbol{\beta}}\}_{0\leq|\boldsymbol{\beta}|\leq n}$ in $\mathcal{D}\textrm{iff}^n(\mathcal{O}_{\hat{X}},\mathcal{O}_{\hat{X}})$, the differential operators of order $\leq n$ and in particular we indicate with $D_{(i)}:=D_{(0\dots,1,\dots,0)}$ with $1$ at the $i$-th place .\
With this notation $$\theta(e)=\sum_{\boldsymbol{\beta}}\frac{1}{\boldsymbol{\beta}!}\nabla_{\boldsymbol{\beta}}(e)\otimes \xi^{\boldsymbol{\beta}}$$ with $\nabla_{\boldsymbol{\beta}}:=(\textrm{id}\otimes D_{(1)}\circ \nabla )^{\beta_1}\circ \dots \circ (\textrm{id}\otimes D_{(l)}\circ \nabla )^{\beta_l}.$ Given an $\tilde{E}$, as in the definition \[special\], then the fact of being special can be translated as follows: there exists a sequence of integers $a(n)$ such that $a(n)=o(n)$ for $n\rightarrow \infty$ and that $$p^{a(n)}\frac{\nabla_{\boldsymbol{\beta}}(e)}{\boldsymbol{\beta}!}\in \tilde{E}$$ for $e$ in $\tilde{E}$ and for any multi index $\boldsymbol{\beta}$ such that $|\boldsymbol{\beta}|\leq n$ . Let us see that the radius of convergence of a section $e$ $\in \tilde{E}$ is $1$, because $e \in \tilde{E}$ is $\eta$-convergent for every $\eta$, i.e. for every $\eta$ $$\theta(e)=\sum_{\boldsymbol{\beta}}\frac{1}{\boldsymbol{\beta}!}\nabla_{\boldsymbol{\beta}}(e)\otimes \boldsymbol{\xi}^{\boldsymbol{\beta}} \in \Gamma(\hat{X},E\otimes \mathcal{O}_{\hat{X}}\{\frac{\boldsymbol{\xi}}{\eta}\} ).$$ To prove it we have to show that if we denote by $\|\,\,\,\|$ the $p$-adic Banach norm on $E$ such that $\|\tilde{E}\|=1$ $$\|\frac{1}{\boldsymbol{\beta}!}\nabla_{\boldsymbol{\beta}}(e)\|\eta^{|\boldsymbol{\beta}|}\rightarrow 0, \forall \;\eta<1.$$ This is clearly true because, fixed an $n$, the following estimate holds: $$\|\frac{1}{\boldsymbol{\beta}!}\nabla_{\boldsymbol{\beta}}(e)\|\eta^{|\boldsymbol{\beta}|}=p^{a(n)}\|p^{a(n)}\frac{1}{\boldsymbol{\beta}!}\nabla_{\boldsymbol{\beta}}(e)\|\eta^{|\boldsymbol{\beta}|}\leq p^{a(n)}\eta^{|\boldsymbol{\beta}|}$$ for every $\boldsymbol{\beta}$ such that $|\boldsymbol{\beta}|\leq n$, because $p^{a(n)}\frac{1}{\boldsymbol{\beta}!}\nabla_{\boldsymbol{\beta}}(e)\in \tilde{E}$.\
This means that $\|\frac{1}{\boldsymbol{\beta}!}\nabla_{\boldsymbol{\beta}}(e)\|\eta^{|\boldsymbol{\beta}|}\rightarrow 0$ since $$0 \leq \|\frac{1}{\boldsymbol{\beta}!}\nabla_{\boldsymbol{\beta}}(e)\|\eta^{|\boldsymbol{\beta}|}\leq p^{a(n)}\eta^{\boldsymbol{\beta}}$$ and $p^{a(n)}\eta^{\boldsymbol{\beta}}\rightarrow 0$ because $a(n)=o(n)$. So we can say that $$R(e)=1\;\forall e \in \tilde{E},$$ and if we take $s\in E$, then there exists a positive integer $k$ such that $p^{k}s \in \tilde{E}$, so that $R(p^{k}s)=1$; moreover we know that $s$ is $\eta$-convergent if and only if $p^{k}s$ is $\eta $ convergent so $$R(p^{k}s)=R(s)$$ and we can conclude that our stratification has radius of convergence $1$.\
The converse is also true: if $(E,\epsilon_n)$ is such that $R((E,\epsilon_n),\hat{X})=1$, then $(E,\epsilon_n)$ is special. We choose an $\tilde{E}$ coherent $\mathcal{O}_{\hat{X}}$-module $p$-torsion free such that $K\otimes \tilde{E}=E$; for every $e \in \tilde{E}$ let $a(n,e)$ the minimal integer such that $$p^{a(n,e)} \frac{\nabla_{\boldsymbol{\beta}}(e)}{\boldsymbol{\beta}!}\in \tilde{E}$$ for any $\boldsymbol{\beta}$ with $|\boldsymbol{\beta}|\leq n$. Since $R((E,\epsilon_n),\hat{X})=1$, then $$\textrm{max}_{|\boldsymbol{\beta}| \leq n}\left( \|\frac{\nabla_{\boldsymbol{\beta}}(e)}{\boldsymbol{\beta}!}\|\right)\eta^n\leq \textrm{max}_{|\boldsymbol{\beta}| \leq n}\left( \|\frac{\nabla_{\boldsymbol{\beta}}(e)}{\boldsymbol{\beta}!}\|\eta^{\frac{|\boldsymbol{\beta}| }{2}}\right)\eta^{\frac{n}{2}}\leq (\textrm{const}) \eta^{\frac{n}{2}},$$ so that for $n\rightarrow \infty$ $p^{a(n,e)}\eta^{n}\rightarrow 0$ for any $\eta< 1$. This means that $a(n,e)=o(n)$ for any $e$ $\in $ $\tilde{E}.$ Now if $e$ $\in$ $\tilde{E}$, then we can write $e=\sum_{i}^{l}f_i e_i$ where $f_i$ $\in$ $\mathcal{O}_{\hat{X}}$ and $e_i$’s are generator of $\tilde{E}$ which is finitely generated $\mathcal{O}_{\hat{X}}$ module, and we put $a(n):=\textrm{max}_{1\leq i \leq l }a(n,e_i)$ (let us note that $a(n)=o(n)$). If we denote by $d_{\boldsymbol{\beta}}$ the operator $\nabla_{\boldsymbol{\beta}}$ for the trivial stratification $(K\otimes \mathcal{O}_{\hat{X}},\textrm{id})$, then for any $f\in \mathcal{O}_{\hat{X}}$ we have $\frac{d_{\boldsymbol{\beta}}(f)}{\boldsymbol{\beta}!}$ $\in$ $\mathcal{O}_{\hat{X}}$ for any $\boldsymbol{\beta}$. Therefore, for any $e=\sum_{i}^{l}f_i e_i$ $\in $ $\tilde{E}$, we have $$p^{a(n)}\frac{\nabla_{\boldsymbol{\beta}}(e)}{\boldsymbol{\beta}!}=\sum_{i=1}^l \sum_{0\leq \boldsymbol{\gamma} \leq \boldsymbol{\beta}}\frac{d_{\boldsymbol{\gamma}}f_i}{\boldsymbol{\gamma}!}\left(p^{a(n)} \frac{\nabla_{\boldsymbol{\beta}-\boldsymbol{\gamma}}e_i}{(\boldsymbol{\beta}-\boldsymbol{\gamma})!}\right)\,\,\,\in\,\,\,\tilde{E} .$$ Hence $(E,\epsilon_n)$ is special.
Description of the semistable case {#Description of the semistable case}
==================================
In what follows we suppose that $X$ is proper semistable variety over $V$, which means that locally for the étale topology there is an étale map $$X\xrightarrow{\acute{e}t} {\mathrm{Spec}}\frac{V[x_1,\dots,x_n,y_1,\dots,y_m]}{x_1\dotsm x_r -\pi}.$$ We call $M_{X_k}$ the log structure on $X$ induced by the special fiber $X_{k}$ that is a divisor with normal crossing, so locally for the étale topology it admits a chart given by $$\mathbb{N}^r\rightarrow \frac{V[x_1,\dots,x_n,y_1,\dots,y_m]}{x_1\dotsm x_r -\pi}$$ that sends $e_i$ to $x_i$, where $e_i=(0,\dots,1,\dots,0)$ with $1$ at the i-th place.\
We consider the log structure $N$ induced by the closed point of $\textrm{Spec} (V)$ that has a chart given by $$\mathbb{N}\rightarrow V,$$ that maps $1$ to $\pi.$ This is explained in [@Ka] example 2.5 (1) and example 3.7 (2).\
We also consider a normal crossing divisor $D$ on $X$ that locally for the étale topology is defined by the equation $\{ y_1\dotsm y_s=0\}$ and we indicate by $M_D$ the log structure induced by $D$ on $X$.\
We consider on $X$ the log structure $M=M_{X_k}\oplus M_D$, that corresponds to the log structure induced by the divisor with normal crossing $X_k\cup D$ in X (let us remark that with the notation $M_{X_k}\oplus M_D$ we indicate the sum in the category of log structures); the structural morphism extends to a log smooth morphism of log schemes $(X,M)\rightarrow({\mathrm{Spec}}(V),N)$. Moreover the special fiber is reduced, hence the hypothesis stated at the beginning of section \[Relations between algebraic and analytic modules with integrable connections\] are satisfied.\
If we denote by $\hat{D}$ the $p$-adic completion of $D$, then we have a diagram $$\xymatrix{
D_k\ar@{^(->}[r] \ar@{^(->}[d]&\hat{D}\ar@{^(->}[d]\\
X_k \ar@{^(->}[r] \ar[d] & \hat{X} \ar[d]\\
{\mathrm{Spec}}(k )\ar@{^(->}[r] & {\mathrm{Spf}}(V). \\
}$$ We suppose that étale locally on $\hat{X}$ we have the following diagram $$\label{etale locally}
\xymatrix{
\ \hat{D}=\bigcup_{j=1}^{s}D_{j}\ \ar@{^(->}[r] \ar[d] &\ \hat{X} \ar[d] \\
\ \bigcup_{j=1}^s\{y_j=0\}\ \ar@{^(->}[r]& \ {\mathrm{Spf}}(V\{x_1,\dots,x_n,y_1,\dots,y_m\}/(x_1\dotsm x_r-\pi) ) \\
}$$ which is cartesian with the vertical maps that are étale and the horizontal maps closed immersions.\
If $\hat{X}_{sing}$ and $\hat{D}_{sing}$ are the singular loci of $\hat{X}$ and $\hat{D}$ respectively, then we will use the following notations: $$\hat{X}^{\circ}=\hat{X}-(\hat{X}_{sing}\cup \hat{D}_{sing})$$ $$\hat{D}^{\circ}=\hat{D}-(\hat{X}_{sing}\cup \hat{D}_{sing})=\hat{X}^{\circ}\cap \hat{D}.$$ When we consider the situation étale locally and fix a diagram (\[etale locally\]), we have a decomposition of the formal schemes $\hat{X}-\hat{X}_{sing}$, $\hat{X}^{\circ}$ and $\hat{D}^{\circ}$ which will be useful later. First let $\hat{X}_i^{\circ}$ be the open formal scheme of $\hat{X}$ defined by pullback of the open formal scheme of ${\mathrm{Spf}}(V\{x_1,\dots,x_n,y_1,\dots,y_m\}/(x_1\dotsm x_r-\pi))$ on which all the $x_{i'}$’s for $i'\neq i$ are invertible and let $\hat{X}^{\circ}_{i,j}$ be the open formal subscheme of $\hat{X}^{\circ}_i$ defined by étale pullback of the open formal subscheme of $ {\mathrm{Spf}}(V\{x_1,\dots,x_n,y_1,\dots,y_m\}/(x_1\dotsm x_r-\pi))$, where $x_{i'}$ are invertible $\forall i'\neq i, 1 \leq i \leq r$ and $y_{j'}$ are invertible $\forall j'\neq j, 1 \leq j \leq s$.\
Moreover we will indicate with $\hat{D}^{\circ}_{i,j}$ the set $\hat{X}^{\circ}_{i,j}\cap\hat{D}=\hat{X}^{\circ}_{i,j}\cap\hat{D}_j$, that is the open formal subscheme of $\hat{D}_j$ defined by pullback of the open formal subscheme of ${\mathrm{Spf}}(V\{x_1,\dots,x_n,y_1,\dots,\hat{y}_j,\dots ,y_m\}/(x_1\dotsm x_r-\pi))$, where all the $x_{i'}$ and the $y_{j'}$ are invertible for all $ i'\neq i, 1 \leq i \leq r$,$\forall j'\neq j, 1 \leq j \leq s$. In the previous line $\hat{y}_j$ in ${\mathrm{Spf}}(V\{x_1,\dots,x_n,y_1,\dots,\hat{y}_j,\dots ,y_m\}/(x_1\dotsm x_r-\pi))$ means that the coordinate $y_j$ is missing.\
With this notations we have the following relations: $$\coprod_{i}\hat{X}_i^{\circ}=\hat{X}-\hat{X}_{sing},\;\;\;\;\;\;\coprod_{i,j}\hat{X}^{\circ}_{i,j}=\hat{X}^{\circ},$$ $$\coprod_{i,j}\hat{D}^{\circ}_{i,j}=\hat{D}^{\circ}.$$ Note that this decomposition is defined only if we work étale locally and we fix a diagram as (\[etale locally\]). If we denote by the subscript $_K$ the rigid analytic space associate to a formal scheme, then the sets $\hat{D}^{\circ}_{i,j;K}$ and $\hat{X}^{\circ}_{i,j;K}$ can be described as follows: $$\hat{X}^{\circ}_{i,j;K}=\{p\in \hat{X}_{K}| \,\forall i'\neq i \,\,\,|x_{i'}(p)|=1 \,\,\,,\forall j'\neq j \,\,\,|y_{j'}(p)|=1\,\},$$ $$\hat{D}^{\circ}_{i,j;K}=\{p\in \hat{X}_{K}| \,\forall i'\neq i \,\,\,|x_{i'}(p)|=1 \,\,\,,\forall j'\neq j \,\,\,|y_{j'}(p)|=1\,\,\,,y_j(p)=0\}.$$ Finally we will denote by $\hat{U}$ the open formal subscheme complement of $\hat{D}$ in $\hat{X}$.
Log convergent isocrystals with exponents in $\Sigma$
=====================================================
We consider now the category of locally free log convergent isocrystals on $\hat{X}$, that we denote, as before, by $I_{conv}((\hat{X},M)/({\mathrm{Spf}}(V),N))^{lf}$. By remark 5.1.3 of [@Sh1] we know that there is an equivalence of categories between $((X_k,M)/({\mathrm{Spf}}(V),N))_{conv}$, the log convergent site on the special fiber, and $((\hat{X},M)/({\mathrm{Spf}}(V),N))_{conv}$, the log convergent site on the lifting, hence an equivalence of categories between $I_{conv}((\hat{X},M)/({\mathrm{Spf}}(V),N))^{lf}$ and $I_{conv}((X_k,M)/({\mathrm{Spf}}(V),N))^{lf}$. As we saw in section \[Relations between algebraic and analytic modules with integrable connections\], through the functor $\Phi$ we can associate to a locally free convergent log isocrystal $\mathcal{E}$ on $\hat{X}$ a locally free infinitesimal log isocrystal $\tilde{\Phi}(\mathcal{E})$. Using the terminology of [@Ke] and [@Sh6], in the local situation as in (\[etale locally\]), $\tilde{\Phi}(\mathcal{E})$ induces a log-$\nabla$-module $E$ on $\hat{X}_K$ with respect to $y_1,\dots y_s$, that means a locally free coherent module $E$ on $\hat{X}_K$ and an integrable connection $$\nabla: E\rightarrow E\otimes \omega^1_{\hat{X}_K/K},$$ where $\omega^1_{\hat{X}_K/K}$ is the coherent sheaf on $\hat{X}_K$ associated to the isocoherent sheaf $K\otimes \omega^1_{(\hat{X},M)/({\mathrm{Spf}}(V),N)}$ on $\hat{X}$. If we are in the situation of (\[etale locally\]) we can write $\omega^1_{\hat{X}_K/K}$ more explicitly: if we denote by $\Omega^1_{{\hat{X}_K}/K}$ the sheaf of continuous classical 1-differentials on the rigid analytic space $\hat{X}_K$, then $$\omega^1_{\hat{X}_K/K}=(\Omega^1_{\hat{X}_K}/K \oplus \bigoplus_{j=1}^s \mathcal{O}_{\hat{X}_K}\textrm{dlog}y_j)/L,$$ where $L$ is the coherent sub $\mathcal{O}_{\hat{X}_K}$ generated by $(dy_j,0)-(0,y_j \textrm{dlog}y_j)$ for $1\leq j \leq s.$ Fixed a $j'$ $\in$ $\{1,\dots, s\}$, i.e a component $\hat{D}_{j';K}=\{y_j'=0\}$ of $\hat{D}_K$, then there is a natural immersion of $$\Omega^1_{\hat{X}_K}/K \oplus \bigoplus_{j\neq j'}\mathcal{O}_{\hat{X}_K}\textrm{dlog}y_j\rightarrow \omega^{1}_{\hat{X}_K/K}$$ and we call $M_{j'}$ the image. The endomorphism $\textrm{res}_j$ $\in End_{\mathcal{O}_{\hat{D}_{j;K}}} (E\otimes_{\mathcal{O}_{\hat{X}_K}}\mathcal{O}_{D_{j;K}})$ obtained tensoring by $\mathcal{O}_{D_{j;K}}$ the following map $$E\rightarrow E \otimes \omega^1_{\hat{X}_K/K} \rightarrow E\otimes \omega^1_{\hat{X}_K/K}/M_j$$ is called the residue of $E$ along $\hat{D}_{j;K}.$ Thanks to proposition 1.5.3 of [@BaCh] we know that there exists a minimal and monic polynomial $P_j$ $\in$ $K[T]$ such that $P_j(res_j)=0$. The exponents of $(E,\nabla)$ along $\hat{D}_{j;K}$ are the roots of $P_j$.\
We fix a set $\Sigma=\prod_{h=1}^k\Sigma_h$ $\in$ $\mathbb{Z}_p^k$, where $k$ is the number of the irreducible components of $\hat{D}=\bigcup_{h=1}^k\hat{D}^h$ in $\hat{X}$.\
If there exists an étale covering $\coprod_l\phi_l:\coprod_l \hat{X}_l \rightarrow \hat{X}$ such that every $\hat{X}_l$ has a diagram as in (\[etale locally\]), then we can define a function of sets $h:\{1,\dots,r\}\times\{1,\dots,s\}\rightarrow \{1,\dots,k\}$ as follows: with the notation as in the previous paragraph $\phi_l(\hat{D}^{\circ}_{i,j,l})$ is contained in one irreducible component of $\hat{D}$, which we denote by $\hat{D}^{h(i,j)}$. We denote by $\Sigma_{h(i,j)}$ the factor of $\Sigma$ corresponding to the component $\hat{D}^{h(i,j)}$.
\[residueanalytic\] A locally free convergent isocrystal $\mathcal{E}$ has exponents along $\hat{D}_K$ in $\Sigma$ if there exists an étale covering $\coprod_l\phi_l:\coprod_l \hat{X}_l \rightarrow \hat{X}$ such that every $\hat{X}_l$ has a diagram $$\label{etale locally X_l}
\xymatrix{
\ \hat{D}_l=\bigcup_{j=1}^{s}\hat{D}_{j,l}\ \ar@{^(->}[r] \ar[d] &\ \hat{X}_l \ar[d] \\
\ \bigcup_{j=1}^s\{y_{l,j}=0\}\ \ar@{^(->}[r]& \ {\mathrm{Spf}}V\{x_{l,1},\dots,x_{l,n},y_{l,1},\dots,y_{l,m}\}/(x_{l,1}\dots x_{l,r}-\pi) \\
}$$ as in (\[etale locally\]) with $\hat{D}_l:=\phi_l^{-1}(\hat{D})$ such that for every $l$ the log-$\nabla$-module $E_l$ on $\hat{X}_{l;K}$ induced by $\mathcal{E}$ has exponents along $\hat{D}_{j,l;K}$ in $\cap_{i=1}^r\Sigma_{h(i,j)}$, if $\phi_l(\hat{D}^{\circ}_{i,j,l})\subset \hat{D}^{h(i,j)}$.\
We will denote by $I_{conv}((\hat{X},M)/({\mathrm{Spf}}(V),N))^{\Sigma}$ or $I_{conv}(\hat{X}/{\mathrm{Spf}}(V))^{log,\Sigma}$ the category of locally free log convergent isocrystals with exponents in $\Sigma$ .
In the next lemma we prove that the definition of isocrystals with exponents along $\hat{D}_K$ in $\Sigma$ is well posed.
\[indepnilpotentresidue\] The notion of locally free log convergent isocrystal with exponents in $\Sigma$ is independent of the choice of the étale covering and the diagram as in (\[etale locally\]), which are chosen in definition \[residueanalytic\].
Let us suppose that $\mathcal{E}$ is a log convergent isocrystal with exponents along $\hat{D}_{K}$ in $\Sigma$. It is sufficient to prove that for any étale morphism $\phi: \hat{X}^{\prime}\rightarrow \hat{X}$, such that for $\hat{X}^{\prime}$ there exists a diagram $$\label{etale locally X'}
\xymatrix{
\ \hat{D}^{\prime}=\bigcup_{j'=1}^{s'}\hat{D}^{\prime}_{j'}\ \ar@{^(->}[r] \ar[d] &\ \hat{X}^{\prime} \ar[d] \\
\ \bigcup_{j'=1}^{s'}\{y^{\prime}_{j'}=0\}\ \ar@{^(->}[r]& \ {\mathrm{Spf}}V\{x^{\prime}_1,\dots,x^{\prime}_{n'},y^{\prime}_1,\dots,y^{\prime}_{m'}\}/(x^{\prime}_1\dots x^{\prime}_{r'}-\pi) \\
}$$ as in (\[etale locally\]), with $\hat{D}^{\prime}:=\phi^{-1}(\hat{D})$, the log-$\nabla$-module $E^{\prime}$ on $\hat{X}^{\prime}_K$ induced by $\mathcal{E}$ has exponents along $\hat{D}^{\prime}_{j';K}$ in $\cap_{i'=1}^{r'}\Sigma_{h(i',j')}$, if $\phi(\hat{D}^{' \circ}_{i',j'})\subset \hat{D}^{h(i',j')}$ .\
By hypothesis $\mathcal{E}$ has exponents along $\hat{D}_K$ in $\Sigma$, hence there exists an étale covering $\coprod_l\phi_l:\coprod_l \hat{X}_l \rightarrow \hat{X}$ such that every $\hat{X}_l$ has a diagram $$\label{etale locally X_l}
\xymatrix{
\ \hat{D}_l=\bigcup_{j=1}^{s}\hat{D}_{j,l}\ \ar@{^(->}[r] \ar[d] &\ \hat{X}_l \ar[d] \\
\ \bigcup_{j=1}^s\{y_{l,j}=0\}\ \ar@{^(->}[r]& \ {\mathrm{Spf}}V\{x_{l,1},\dots,x_{l,n},y_{l,1},\dots,y_{l,m}\}/(x_{l,1}\dots x_{l,r}-\pi) \\
}$$ as in (\[etale locally\]) with $\hat{D}_l:=\phi_l^{-1}(\hat{D})$ such that for every $l$ the log-$\nabla$-module $E_l$ on $\hat{X}_{l;K}$ induced by $\mathcal{E}$ has exponents along $\hat{D}_{l,j;K}$ in $\cap_{i=1}^{r}\Sigma_{h(i,j)}$, if $\phi_l(\hat{D}^{ \circ}_{i,j})\subset \hat{D}^{h(i,j)}$ .\
Let us denote by $\hat{X}^{\prime}_l$ the fiber product $\hat{X}^{\prime}\times_{\hat{X}} \hat{X}_l$ and by $\hat{D}^{\prime}_l,\,\, \hat{D}^{\prime}_{l,j'},\,\, \hat{D}''_{l,j}$ the inverse image of $\hat{D}_l,\,\, \hat{D}'_{j'},\,\, \hat{D}_{j,l}$ on $\hat{X}'_l$ respectively. With this notation we have two diagrams on $\hat{X}'_l$: $$\label{etale locally X'_l indotto da X'}
\xymatrix{
\ \hat{D}'_l=\bigcup_{j'=1}^{s'}D'_{j',l}\ \ar@{^(->}[r] \ar[d] &\ \hat{X}'_l \ar[d] \\
\ \bigcup_{j'=1}^{s'}\{y'_{j'}=0\}\ \ar@{^(->}[r]& \ {\mathrm{Spf}}V\{x'_{1},\dots,x'_{n'},y'_{1},\dots,y'_{m'}\}/(x'_{1}\dots x'_{r'}-\pi) \\
}$$ and $$\label{etale locally X'_l indotto da X_l}
\xymatrix{
\ \hat{D}'_l=\bigcup_{j=1}^{s}\hat{D}''_{j,l}\ \ar@{^(->}[r] \ar[d] &\ \hat{X}'_l \ar[d] \\
\ \bigcup_{j=1}^s\{y_{j,l}=0\}\ \ar@{^(->}[r]& \ {\mathrm{Spf}}V\{x_{1,l},\dots,x_{n,l},y_{1,l},\dots,y_{m,l}\}/(x_{1,l}\dots x_{r,l}-\pi) .\\
}$$ The diagram (\[etale locally X’\_l indotto da X’\]) is induced by (\[etale locally X’\]) through $p_2$, the projection on the second factor $\mathrm{pr}_2:\hat{X}^{\prime}\times_{\hat{X}} \hat{X}_l\rightarrow \hat{X}_l$; and the diagram (\[etale locally X’\_l indotto da X\_l\]) is induced by (\[etale locally X\_l\]) through $p_1$, the projection on the first factor $\mathrm{pr}_2:\hat{X}^{\prime}\times_{\hat{X}} \hat{X}_l\rightarrow \hat{X}'$.\
The log-$\nabla$-module $E'_l$ induced by $\mathcal{E}$ on $\hat{X}'_{l;K}$ has exponents along $\hat{D}''_{j,l;K}$ which are contained in the set of exponents of $E_l$ along $\hat{D}_{l,j;K}$. This happens because the residue of $E'_l$ along $\hat{D}''_{j,l;K}$, denoted by $\mathrm{res}^{''}_{j,l}$ is the image of the residue of $E_l$ along $\hat{D}_{j,l;K}$, denoted by $\mathrm{res}_{j,l}$, via the map $$\textrm{End}_{\mathcal{O}_{\hat{D}_{j,l;K}}}({E_l}|_{\hat{D}_{j,l;K}})\rightarrow \textrm{End}_{\mathcal{O}_{\hat{D}^{ \prime \prime}_{j,l;K}}}({E'_l}|_{\hat{D}^{\prime \prime }_{j,l;K}}) ,$$ which is induced by the projection $\mathrm{pr}_{2}$. If $P_{j,l}$ is the minimal and monic polynomial such that $P_{j,l}(res_{j,l})=0$, then $P_{j,l}(res^{''}_{j,l})=0$, so if we denoted by $P^{''}_{j,l}$ the minimal and monic polynomial such that $P^{''}_{j,l}(res^{''}_{j,l})=0$, then $P^{''}_{j,l}\mid P_{j,l}$. So the roots of $P^{''}_{j,l}$ are contained in the roots of $P_{j,l}$, which means that the exponents of $E'_l$ along $\hat{D}''_{j,l}$ are a subset of the set of exponents of $E_l$ along $\hat{D}_{j,l;K}$. Since for every $(i,j)$ such that $\phi_l \circ \mathrm{pr}_2(\hat{D}^{\prime \prime \circ}_{i,j,l})$ is contained in $\hat{D}^{h(i,j)}$ also $\phi_l(\hat{D}^{\circ}_{i,j,l})$ is contained in $\hat{D}^{h(i,j)}$ and viceversa, then we proved that for every $(i,j)$ such that $\phi_l \circ \mathrm{pr}_2(\hat{D}^{\prime \prime \circ}_{i,j,l})$ is contained in $\hat{D}^{h(i,j)}$ the exponents along $\hat{D}''_{j,l;K}$ are contained in $\cap_{i=1}^r\Sigma_{h(i,j)}$ .\
Now we want to look at the exponents of $E'_l$ along $\hat{D}'_{j',l;K}$.\
Let us put $\hat{X}^{\prime \circ}_l:=\hat{X}'_l\cap \hat{X}^{\circ}$, $\hat{D}^{\prime \circ }_l:=\hat{D}'_l\cap \hat{X}^{\circ}$, $\hat{D}^{\prime \circ }_{l,j'}:=\hat{D}'_{l,j'}\cap \hat{X}^{\circ}$ and $\hat{D}^{\prime \prime \circ }_{l,j}:=\hat{D}''_{l}\cap \hat{X}^{\circ}$. Since the map $$\textrm{End}_{\mathcal{O}_{\hat{D}'_{l,j';K}}}(E'_l|_{\hat{D}'_{l,j';K}})\rightarrow \textrm{End}_{\mathcal{O}_{\hat{D}^{\prime \circ }_{l,j';K}}}(E'_l|_{\hat{D}^{\prime \circ }_{l,j';K}})$$ is injective it is enough to look at the exponents of $E'_l|_{\hat{X}^{\prime \circ }_{l;K}}$ along $\hat{D}^{'\circ}_{l,j';K}$.
Let us note that $\hat{D}^{'\circ}_{l}$ is a relative normal crossing divisor in a smooth formal $V$-scheme; if $\hat{D}^{'\circ}_{l}=\bigcup_t \hat{C}_{t,l}$ is the decomposition of $\hat{D}^{'\circ}_{l}$ in irreducible components, from [@NaSh] proposition A.0.3 and proposition A.0.7, we can deduce that $\hat{D}^{'\circ}_{i',j',l}$ and $\hat{D}^{'' \circ}_{i,j,l}$, that are irreducible components of $\hat{D}^{'\circ}_{l}$, correspond to some $\hat{C}_{t,l}$’s. Thanks to what we have proven before we know that $E^{'}_l|_{\hat{X}^{' \circ}_{l;K}}$ has exponents along $\hat{D}^{''\circ}_{l,j;K}$ in $\cap_{i=1}^r\Sigma_{h(i,j)}$ where $(i,j)$ are such that $\phi_l \circ \mathrm{pr}_2(\hat{D}^{\prime \prime \circ}_{i,j,l})$ is contained in $\hat{D}^{h(i,j)}$.We now consider $\hat{D}^{'\circ}_{j',l}$, then $\hat{D}^{'\circ}_{i',j',l}$ will coincide with some $C_{t,l}$’s, so will correspond to some $\hat{D}^{''\circ}_{i,j,l}$. If $C_{t,l}$ is such that $\phi \circ\mathrm{pr}_1(C_{t,l})\subset \hat{D}^h$ then also $\phi_{l} \circ\mathrm{pr}_2(C_{t,l})\subset \hat{D}^h$ for the commutativity of the following diagram $$\xymatrix{
\ \hat{X}^{\prime}\times_{\hat{X}} \hat{X}_l \ar[d]^{pr_{1}} \ar[r]^{pr_{2}} & \hat{X}_{l} \ar[d]_{\phi_{l}} \\
\hat{X}^{\prime} \ar[r]^{\phi} & \hat{X}
}.$$ So we can conclude that the exponents of $E'_l$ along $\hat{D}^{'}_{j',l;K}$ are contained in $\cap_{i'=1}^{r'}\Sigma^{h(i',j')}$ with $(i',j')$ such that $\phi \circ \mathrm{pr}_1(\hat{D}^{'\circ}_{i',j',l}) \subset \hat{D}^{h(i',j')}$.\
Finally we prove that $E'$ has exponents along $\hat{D}'_{j';K}$ in $\cap_{i'=1}^{r'}\Sigma_{h(i',j')}$ with $(i',j')$ such that $\phi(\hat{D}^{' \circ}_{i',j'})\subset \hat{D}^{h(i',j')}$. Having a surjective étale map $\coprod_l \hat{X}'_l\rightarrow \hat{X}'$, the thesis is reduced to prove that the induced map $$\label{coprod}
\textrm{End}_{\mathcal{O}_{\hat{D}^{'}_{j';K}}}(E'\otimes \mathcal{O}_{\hat{D}^{\prime }_{j';K}}) \rightarrow \textrm{End}_{\mathcal{O}_{\coprod_l\hat{D}^{'}_{j',l;K}}}(\coprod_l{E'_l\otimes \mathcal{O}_{\coprod_l\hat{D}^{\prime }_{j',l;K}}} )$$ is injective.\
If $\phi(\hat{D}^{' \circ}_{i',j'})\subset \hat{D}^{h(i',j')}$, then $\phi \circ \mathrm{pr}_1(\hat{D}^{'\circ}_{i',j',l}) \subset \hat{D}^{h(i',j')}$ for every $l$ and $\phi \circ \mathrm{pr}_1(\coprod_l\hat{D}^{'\circ}_{i',j',l})$ is contained in $\hat{D}^{h(i',j')}$. One can see that the residue of $E'$ along $\hat{D}^{'}_{j';K}$ goes via the map in (\[coprod\]) into the residue of $\coprod_l{E'_l}$ along $\coprod_l\hat{D}^{'}_{j',l;K}$.\
$$\Gamma(\hat{D}'_{j';K},\mathcal{O}_{\hat{D}'_{j';K}})\rightarrow\Gamma(\coprod_l \hat{D}'_{j',l;K}, \mathcal{O}_{\coprod_l\hat{D}'_{j',l;K}} )$$ is injective, since $\coprod_l \hat{X}'_l\rightarrow \hat{X}'$ is étale surjective and then faithfully flat.
Log-$\nabla$-modules on polyannuli
==================================
We recall in this section the notion of log-$\nabla$-modules on some particular rigid space defined and used by Kedlaya in [@Ke] and by Shiho in [@Sh6].\
An aligned interval is a interval $I$ contained in $[0,\infty)$ such that any end point is contained in $\Gamma^{*}$ with $\Gamma^{*} $ the divisible closure of the image of the absolute value $|\,\,\,|:K^*\rightarrow \mathbb{R}^+$. An aligned interval is said to be quasi open if is open at any non zero end point. For an aligned interval we define a polyannulus as the rigid analytic space $A^n_K(I)=\{(t_1,\dots,t_n) \in \mathbb{A}^{n,rig}_K| |t_i|\in I \,\forall \,\, i=1,\dots,n\}.$\
If $Y$ is a smooth rigid analytic space and $y_1,\dots, y_s$ are global sections such that they are smooth and meet transversally, then for a subset $\Sigma=\prod_{j=1}^{s}\Sigma_{j}$ $\subset$ $\bar{K}^{s}$ we denote by $\mathrm{LNM}_{Y,\Sigma}$ the category of log-$\nabla$-module on $Y$ such that all the exponents along $\{y_{j}=0\}$ are contained in $\Sigma_{j}$ for every $j=1,\dots,s.$\
If $Y$ is a smooth rigid analytic space and $y_1,\dots, y_s$ are global sections such that they are smooth and meet transversally, then we set $\omega_{Y\times A^n_K([0,0])/K}=\omega^1_{Y/K}\oplus \bigoplus_{i=1}^{n}\mathcal{O}_{Y}\textrm{dlog} t_i.$ We define a log-$\nabla$-module $(E,\nabla)$ on $Y\times A^n_K([0,0])/K$ with respect to $y_1\dots, y_s, t_1,\dots t_n$ as a log-$\nabla$-module $(E,\nabla)$ on $Y$ with respect to $y_{1},\dots, y_{s}$ with $n$ commuting endomorphisms $\partial_i=t_i\frac{\partial}{\partial t_i}$ of $(E,\nabla)$ for $i=1,\dots n$. If we fix $\Sigma=\prod_{j=1}^{s}\Sigma_j\times \prod_{i=1}^{n}\Sigma_{i}$ $\subset$ $\bar{K}^{s+n}$, we can define a log-$\nabla$-module $(E,\nabla)$ with respect to $y_1\dots y_s, t_1,\dots, t_n$ on $Y\times A^n_K([0,0])/K$ with exponents in $\Sigma$ if the log-$\nabla$-module $(E,\nabla)$ on $Y$ has exponents along $\{y_{j}=0\}$ in $\Sigma_{j}$ and if the commuting endomorphisms $\partial_i=t_i\frac{\partial}{\partial t_i}$ have eigenvalues in $\Sigma_{i}$ for every $i=1,\dots, n$. Following Shiho we denote the category of locally free log-$\nabla$-modules on $Y\times A^n_K([0,0])/K$ with exponents in $\Sigma$ by $\mathrm{LNM}_{Y\times A^n_K([0,0]),\Sigma}$.\
If $I$ is an aligned interval and $\xi:=(\xi_1,\dots, \xi_n)$ $\in$ $\bar{K}^n$, the log-$\nabla$-module denoted by $(M_{\xi},\nabla_{M_{\xi}})$ is the log-$\nabla$-module on $A_K^n(I)$ given by $(\mathcal{O}_{A_K^n(I)}, d+\sum_{j=1}^n\xi_j\textrm{dlog}t_j).$ We will define now the notion of $\Sigma$-unipotence for log-$\nabla$-modules on a product of a smooth rigid analytic space and a polyannulus ([@Sh6] definition 1.3).
Let $Y$ be a smooth rigid analytic space, $y_1,\dots,y_s$ global sections whose zero loci are smooth and meet transversally, $I$ an aligned interval and $\Sigma=\prod_{j=1}^{s+n}\Sigma_j$ $\subset$ $\bar{K}^{s+n}$. We say that $\mathcal{E}=(E,\nabla)$ $\in$ $\mathrm{LNM}_{Y\times A^n_K(I)}$ is $\Sigma$-unipotent if after some finite extension of $K$ there exists a filtration $$0\subset\mathcal{E}_1\subset \dots \subset \mathcal{E}_n=\mathcal{E}$$ of subobjects such that every successive quotient $\mathcal{E}_i/\mathcal{E}_{i-1}=\pi_1^*{\mathcal{F}}\times \pi_2^*(M_{\xi}, \nabla_{M_{\xi}})$, where by $\pi_1$ we denote the first projection $\pi_1:Y\times A^n_K(I)\rightarrow Y$, by $\pi_2$ the second projection $\pi_2:Y\times A^n_K(I)\rightarrow A^n_K(I)$, by $\mathcal{F}$ a log-$\nabla$-module $\in$ $\mathrm{LNM}_{X,\prod_{j=1}^s \Sigma_j}$ and by $(M_{\xi},\nabla_{M_{\xi}})$ the log -$\nabla$-module we defined before with $\xi$ $\in$ $\prod_{j=s+1}^{s+n}\Sigma_j$.\
We will denote by $\mathrm{ULNM}_{Y\times A^n_K(I), \Sigma}$ the categories of $\Sigma$-unipotent log-$\nabla$-modules on $Y\times A^n_K(I).$
Shiho ([@Sh6] Definition 1.5) defines a functor $$\mathcal{U}_I:\mathrm{LNM}_{Y\times A^n_K([0,0]),\Sigma}\rightarrow \mathrm{LNM}_{Y\times A^n_K(I),\Sigma }$$ that associates to a log-$\nabla$-module $\mathcal{E}$ with respect to $y_1,\dots,y_s$ on $Y$ with $n$ commuting endomorphisms $N_i$ for $i=1,\dots n$ with eigenvalues on $\Sigma_i$ a log-$\nabla$-module $\mathcal{U}_I(\mathcal{E})$ defined as the sheaf $\pi_1^*{\mathcal{E}}$ and the connection $$v\mapsto \pi_1^*(\nabla)v+\sum_{i=1}^n\pi_1^*(N_i)(v)\otimes \frac{dt_i}{t_i}.$$ We recall here the definition of a non Liouville number, which we will use in the sequel.
An element $\alpha$ in $\bar{K}$ is said to be $p$-adically non-Liouville if both the power series $\sum_{n\neq \alpha}\frac{x^n}{\alpha -n}$ and $\sum_{n\neq \alpha}\frac{x^n}{n-\alpha }$ have radius of convergence equal to $1$.\
As in definition 1.8 of [@Sh6] we can define the following.
A set $\Sigma$ $\subset$ $\bar{K}$ is called (NID) (resp. (NLD)) if for any $\alpha$, $\beta$ $\in$ $\Sigma$, $\alpha-\beta$ is a non zero integer (resp. is p-adically non-Liouville). A set $\Sigma=\prod_{j=1}^{s}\Sigma_{j}$ $\subset$ $\bar{K}^{s}$ is called (NID) (resp. (NLD)) if for any $j=1,\dots,s$ $\Sigma_{j}$ is (NID) (resp. (NLD)).\
We will use the following result ([@Ke] 3.3.4, [@Ke] 3.3.6, [@Sh6] corollary 1.15 and [@Sh6] corollary 1.16)
\[unipotentiequivalenti\] Let $Y$ be as before, $I$ a quasi open interval and $\Sigma=\prod_{j=1}^{s+n}\Sigma_{j}$ $\subset$ $\bar{K}^{s+n}$ which is (NID) and (NLD) then the restriction of the functor $\mathcal{U}_I$ to the $\Sigma$-unipotent log-$\nabla$-modules $$\mathcal{U}_I:\mathrm{ULNM}_{Y\times A^n_K([0,0]),\Sigma}\rightarrow \mathrm{ULNM}_{Y\times A^n_K(I),\Sigma}$$ is an equivalence of categories. If $I$ is an interval of positive length, then the functor $\mathcal{U}_I$ is fully faithful.
Log overconvergent isocrystals
==============================
Before defining the category of log overconvergent isocrystals, we recall the notion of log tubular neighborhood given by Shiho in [@Sh2] definition 2.2.5 with some restrictive hypothesis and in [@Sh3] paragraph 2 in full generality. This is the log version of the tubular neighborhood defined by Berthelot in [@Be].\
Given a closed immersion of fine log formal schemes $i:(Z,M_Z)\hookrightarrow ({\mathscr{Z}},M_{{\mathscr{Z}}})$, there exists a fine log formal scheme $({\mathscr{Z}}^{ex},M_{{\mathscr{Z}}^{ex}})$ and an associated homeomorphic closed exact immersion $i^{ex}:(Z,M_Z)\hookrightarrow ({\mathscr{Z}}^{ex},M_{{\mathscr{Z}}^{ex}})$ such that the functor that associates $i^{ex}$ to $i$ is a right adjoint functor to the inclusion functor from the category of homeomorphic closed immersions of log formal schemes in the category of closed immersions of log formal schemes. The functor $i\mapsto i^{ex}$ is called the exactification functor and its existence is proven in [@Sh3] proposition-definition 2.10.\
Let $(Z,M_Z)\hookrightarrow({\mathscr{Z}},M_{{\mathscr{Z}}})$ be a closed immersion of log formal schemes, then the log tubular neighborhood $]Z[^{log}_{{\mathscr{Z}}}$ of $(Z,M_Z)$ in $({\mathscr{Z}},M_{{\mathscr{Z}}})$ is defined as the rigid analytic space ${\mathscr{Z}}^{ex}_K$ associated to the formal scheme ${\mathscr{Z}}^{ex}$. We can define the specialization map $$\textrm{sp}: ]Z[^{log}_{{\mathscr{Z}}}\rightarrow \hat{{\mathscr{Z}}},$$ where $\hat{{\mathscr{Z}}}$ is the completion of ${\mathscr{Z}}$ along $Z$, as the composition of the usual specialization map $ ]Z[^{log}_{{\mathscr{Z}}}={\mathscr{Z}}^{ex}_{K}\rightarrow {\mathscr{Z}}^{ex}$ with the map ${\mathscr{Z}}^{ex}\rightarrow \hat{{\mathscr{Z}}}$ induced by the morphism ${\mathscr{Z}}^{ex}\rightarrow {\mathscr{Z}}$.
We can notice that, if the closed immersion $i:(Z,M_Z)\hookrightarrow({\mathscr{Z}},M_{{\mathscr{Z}}})$ is exact, then ${\mathscr{Z}}^{ex}={\mathscr{Z}}_K$ and the log tubular neighborhood $]Z[^{log}_{{\mathscr{Z}}}$ coincides with the classical tubular neighborhood.\
We define the category of log overconvergent isocrystals for log pairs. Log pairs are defined in paragraph 4 of [@Sh4] and in 2.1 of [@ChTs] in the case of trivial log structures. A log pair is a pair $((X,M_X),(\bar{X},M_{\bar{X}}))$ of fine log schemes in characteristic $p$ endowed with a strict open immersion $(X,M_X)\hookrightarrow (\bar{X},M_{\bar{X}})$. A morphism of log pairs $f:((X,M_X),(\bar{X},M_{\bar{X}}))\rightarrow((Y,M_Y),(\bar{Y},M_{\bar{Y}}))$ is a morphism of log schemes $f :\bar{X}\rightarrow \bar{Y}$ that verifies $f(X)\subset Y$. A log pair $((X,M_X),(\bar{X},M_{\bar{X}}))$ over a log pair $((S,M_S),(\bar{S},M_{\bar{S}}))$ is a log pair endowed with the structural morphism $f:((X,M_X),(\bar{X},M_{\bar{X}}))\rightarrow((S,M_S),(\bar{S},M_{\bar{S}}))$. We assume that all log pairs are log pairs over a given log pair $((S,M_{S}),(S,M_{S}))$. In paragraph 4 of [@Sh4] there is a definition of log overconvergent isocrystals for log pairs over a log pair $((S,M_{S}),(S,M_{S}))$ with $M_{S}$ isomorphic to the trivial log structure; we will give analogous definition in the case of non necessarily trivial $M_S$.\
A log triple is a triple $((X,M_X),(\bar{X},M_{\bar{X}}),({\mathscr{P}},M_{{\mathscr{P}}}))$ which consists of a log pair $((X,M_{X}),(\bar{X},M_{\bar{X}}))$ and a log formal scheme $({\mathscr{P}},M_{{\mathscr{P}}})$ over a log formal scheme $({\mathscr{S}},M_{{\mathscr{S}}})$ endowed with a closed immersion $(\bar{X},M_{\bar{X}})\hookrightarrow ({\mathscr{P}},M_{{\mathscr{P}}}).$ Morphisms of log triples are defined in the natural way, as well as a log triple over an other log triple. We will work only with triples $((X,M_X),(\bar{X},M_{\bar{X}}),({\mathscr{P}},M_{{\mathscr{P}}}))$ over a fixed log triple $((S,M_S),(S,M_{S}),({\mathscr{S}},M_{{\mathscr{S}}}))$.\
As in the classical case, for a log triple $((X,M_X),(\bar{X},M_{\bar{X}}),({\mathscr{P}},M_{{\mathscr{P}}}))$ we can define a strict neighborhood $W$ of $]X[^{log}_{{\mathscr{P}}}$ in $]\bar{X}[^{log}_{{\mathscr{P}}}$ to be an admissible open of $]\bar{X}[^{log}_{{\mathscr{P}}}$ such that $\{W,]\bar{X}[^{log}_{{\mathscr{P}}}-]{X}[^{log}_{{\mathscr{P}}}\}$ is an admissible covering of $]\bar{X}[^{log}_{{\mathscr{P}}}$. Given a sheaf of $O_W$ modules $\mathcal{E}$ we define the sheaf of overconvergent sections as the sheaf $j^{\dag}_W \mathcal{E}=\varinjlim_{W'} \alpha_{W',]\bar{X}[^{log}_{{\mathscr{P}}}*}\alpha_{W',W}^{-1}\mathcal{E},$ where $W'$ varies among the strict neighborhood of $]X[^{log}_{{\mathscr{P}}}$ in $]\bar{X}[^{log}_{{\mathscr{P}}}$ that are contained in $W$ and $\alpha_{T,T'}:T\hookrightarrow T'$ is the natural inclusion. If $W=]\bar{X}[^{log}_{{\mathscr{P}}}$, then we will denote the sheaf of overconvergent sections by $j^{\dag}\mathcal{E}$.\
We suppose that there exists a commutative diagram $$\label{diaconj}
\xymatrix{
\ (\bar{X},M_{\bar{X}})\ \ar[r]^{j} \ar[d]^{g} &\ ({\mathscr{P}},M_{{\mathscr{P}}})\ \ar[d]^{h} \\
\ (\bar{Y}, M_{\bar{Y}})\ \ar[r]^{i}& \ ({\mathscr{Y}} , M_{{\mathscr{Y}}}) \\
}$$ where $j$ is a closed immersion and $h$ is formally log smooth. If we denote by $({\mathscr{P}}(1), M_{{\mathscr{P}}(1)})$ (resp. $({\mathscr{P}}(2), M_{{\mathscr{P}}(2)})$) the fiber product of $({\mathscr{P}}, M_{{\mathscr{P}}})$ with itself over $({\mathscr{Y}},M_{{\mathscr{Y}}})$ (reps. the fiber product of $({\mathscr{P}}, M_{{\mathscr{P}}})$ with itself over $({\mathscr{Y}},M_{{\mathscr{Y}}})$ three times), then the projections and the diagonal induce the following maps: $$p_{i}:]\bar{X}[_{{\mathscr{P}}(1)}^{log}\rightarrow ]\bar{X}[_{{\mathscr{P}}}^{log}\,\,\, \textrm{for} \,\,\,i=1,2,$$ $$p_{i,j}:]\bar{X}[_{{\mathscr{P}}(2)}^{log}\rightarrow ]\bar{X}[_{{\mathscr{P}}(1)}^{log}\,\,\, \textrm{for} \,\,\,1\leq i,j \leq 3,$$ $$\Delta:]\bar{X}[^{log}_{{\mathscr{P}}}\rightarrow ]\bar{X}[_{{\mathscr{P}}(1)}^{log}.$$
With the previous notation a log overconvergent isocrystal is a pair $(\mathcal{E},\epsilon)$ consisting of a coherent $j^{\dag}\mathcal{O}_{]\bar{X}[_{{\mathscr{P}}}^{log}}$-module $\mathcal{E}$ and $\epsilon$ a $j^{\dag}\mathcal{O}_{]\bar{X}[_{{\mathscr{P}}(1)}^{log}}$-linear isomorphism $\epsilon:p_1^*{\mathcal{E}}\rightarrow p_2^*{\mathcal{E}}$ that satisfies $\Delta^*(\epsilon)=\textrm{id}$ and $p_{1,2}^*(\epsilon)\circ p_{2,3}^*(\epsilon)=p_{1,3}^*(\epsilon).$ We denote by $I^{\dag}(((X,\bar{X})/{\mathscr{Y}},{\mathscr{P}})^{log})$ the category of log overconvergent isocrystals on $((X,M_X),(\bar{X},M_{\bar{X}})/{\mathscr{Y}})$ over $({\mathscr{P}},M_{{\mathscr{P}}})$. We say that $(\mathcal{E},\epsilon)$ is a locally free log overconvergent isocrystal if $\mathcal{E}$ is a locally free $j^{\dag}\mathcal{O}_{]\bar{X}[_{{\mathscr{P}}}^{log}}$-module and we indicate the category of locally free log overconvergent isocrystals with $I^{\dag}(((X,\bar{X})/{\mathscr{Y}},{\mathscr{P}})^{log})^{lf}.$
In the case of trivial log structures the previous definition coincides with the definition of overconvergent isocrystals given by Berthelot [@Be].
Shiho in [@Sh4] definition 4.2 defines the category of log overconvergent isocrystals also in a more general situation, but for our purposes the definition we gave is sufficient.
Given a log pair $((X, M_X),(\bar{X}, M_{\bar{X}}))$, we assume the existence of a diagram $$\label{suk}
(\bar{X}, M_{\bar{X}})\xrightarrow{g}(\bar{Y}, M_{\bar{Y}})\xrightarrow{i}({\mathscr{Y}}, M_{{\mathscr{Y}}}),$$ where $(\bar{Y}, M_{\bar{Y}})$ is a log scheme over $(S,M_{S})$, $({\mathscr{Y}}, M_{{\mathscr{Y}}})$ is a $p$-adic log formal scheme over $({\mathscr{S}},M_{{\mathscr{S}}})$ and $i$ is a closed immersion.\
Coming back to the setting discussed in section \[Description of the semistable case\], we consider the log triple $((U_k,M),(X_k,M),(\hat{X}, M))$ over $(({\mathrm{Spec}}(k), N),({\mathrm{Spec}}(k), N),({\mathrm{Spf}}(V),N))$ and the following commutative diagram $$\xymatrix{
\ (U_k,M)\ \ar@{^(->}[r] \ar[d] &\ (X_k,M)\ \ar[d]_{f_k} \ar@{^(->}[r] & \ (\hat{X}, M)\ \ar[d] \\
\ ({\mathrm{Spec}}(k), N)\ \ar@{^(->}[r]& \ ({\mathrm{Spec}}(k), N) \ar@{^(->}[r] & \ \ ({\mathrm{Spf}}(V), N)\ .\\
}$$ The diagram as in (\[suk\]) is given by $$(U_k,M)\xrightarrow{g}({\mathrm{Spec}}k,N)\xrightarrow{i}({\mathrm{Spf}}V,N)$$ and the commutative diagram as in (\[diaconj\]) is $$\xymatrix{
\ (X_k,M)\ \ar[r]^{j} \ar[d]^{g} &\ \ (\hat{X},M)) \ar[d]^{h} \\
\ ({\mathrm{Spec}}(k), N)\ \ar[r]^{i}& \ ({\mathrm{Spf}}(V), N) . \\
}$$ Let us note that since the immersion $(U_k,M)\hookrightarrow(\hat{X},M)$ is strict and the closed immersion $(X_k,M)\hookrightarrow(\hat{X},M)$ is exact, the log tubes in these cases coincide with the classical tubes: $$]X_k[^{log}_{\hat{X}}=]X_k[_{\hat{X}}=\hat{X}_{K}$$ $$]U_k[^{log}_{\hat{X}}=]U_k[_{\hat{X}}=\hat{U}_{K}.$$ Now we want to give a description of integrable connections associated to locally free overconvergent isocrystals in our case. By proposition 2.1.10 of [@Be] we know that there is an equivalence of categories between Coh$(j^{\dag}\mathcal{O}_{]X_k[_{\hat{X}}})$, the category of $j^{\dag}\mathcal{O}_{]X_k[_{\hat{X}}}$-coherent modules, and the inductive limit category of coherent modules over strict neighborhoods of $]U_k[_{\hat{X}}$ in $]X_k[_{\hat{X}}$. Thanks to remark after proposition 2.1.10 of [@Be], if $(\mathcal{E}, \epsilon)$ is a locally free log overconvergent isocrystal, then $\mathcal{E}$ is a locally free $j^{\dag}\mathcal{O}_{]X_k[_{\hat{X}}}$ module, which means that there exists a strict neighborhood $W$ of $]U_k[_{\hat{X}}$ in $]X_k[_{\hat{X}}$ and a locally free $\mathcal{O}_{W}$-module $E$, such that $j^{\dag}_W E=\mathcal{E}$ .\
The log overconvergent isocrystal $(\mathcal{E},\epsilon)$ induces an integrable connection on $\mathcal{E}$ $$\nabla :\mathcal{E}\rightarrow \mathcal{E}\otimes_{j^{\dag}\mathcal{O}_{]X_k[_{\hat{X}}}}j^{\dag}\omega^{1}_{{]X_k[_{\hat{X}}}/K}$$ where $\omega^{1}_{{]X_k[_{\hat{X}}}/K}$ is the restriction of $K\otimes \omega^1_{(\hat{X},M)/({\mathrm{Spf}}(V),N)}$ to $]X_k[_{\hat{X}}$; moreover given a strict neighborhood $W$ of $]U_k[_{\hat{X}}$ in $]X_k[_{\hat{X}}$, as we saw before, there exists $E$ on $W$ such that $j^{\dag}_W E=\mathcal{E}$ and there exists an integrable connection $$\nabla: E\rightarrow E\otimes \omega^1_{W/K}$$ which induces the above connection on $\mathcal{E}$, where $\omega^1_{W/K}$ is the restriction of $K\otimes \omega^1_{(\hat{X},M)/({\mathrm{Spf}}(V),N)}$ to W.\
If étale locally we are in the situation described in (\[etale locally\]), then $W$ contains a subspace of the form $$\{p \in \hat{X}_K\,\,\,|\,\,\,\forall j \,\,\,|y_j(p)| \geq \lambda\}$$ for some $\lambda$ $\in $ $(0,1)\cap \Gamma^{*}$ with $\Gamma^{*} $ the divisible closure of the image of the absolute value $|\,\,\,|:K^*\rightarrow \mathbb{R}^+$.\
Therefore we can restrict $E$ to the space $$\{p \in \hat{X}^{\circ}_{i,j;K}\,\,\,|\,\,\, \lambda \leq |y_j(p)| < 1 \}.$$
There is an isomorphism $$\phi: \hat{D}_{j,i;K}^{\circ}\times A^1_K([\lambda,1)) \rightarrow\{p \in \hat{X}^{\circ}_{i,j;K}\,\,\,|\,\,\, \lambda \leq |y_j(p)| < 1 \},$$ where $A^1_K([\lambda,1)):=\{t\in \mathbb{A}^1_K\,\,\,|\,\,\, |t|\,\,\,\in\,\,\,[\lambda,1) \}$.
If we can prove that $\hat{D}_{j,i;K}^{\circ}\times A^1_K([0,1))\cong \hat{X}^{\circ}_{i,j;K}$, then the isomorphism of the proposition will be clear. To prove this we will apply lemma 4.3.1 of [@Ke]: we consider $A=\Gamma(\hat{X}^{\circ}_{i,j;K}, \mathcal{O}_{X_K})$, and as $B$ the ring $A/y_j A=\Gamma(\hat{D}_{i,j;K}^{\circ},\mathcal{O}_{X_K})$. We can apply lemma 4.3.1 of [@Ke] because $\Gamma(\hat{D}_{i,j;K}^{\circ},\mathcal{O}_{X_K})$ is formally smooth over $K$ and we can conclude that $$\Gamma(\hat{X}_{i,j;K}^{\circ},\mathcal{O}_{X_K})\cong\Gamma(\hat{D}_{i,j;K}^{\circ},\mathcal{O}_{X_K})[|y_j|]$$ i.e. the isomorphism that we wanted.
We fix a set $\Sigma=\prod_{h=1}^k\Sigma_h$ $\in$ $\mathbb{Z}_p^k$, where $k$ is the number of the irreducible components of $\hat{D}=\bigcup_{h=1}^k\hat{D}^h$ in $\hat{X}$, with the same notations as before definition \[residueanalytic\].\
\[unipotentmonodromy\] A log overconvergent isocrystal $\mathcal{E}$ has $\Sigma$-unipotent monodromy if there exists an étale covering $\coprod_l\phi_l:\coprod_l \hat{X}_l \rightarrow \hat{X}$ such that every $\hat{X}_l$ has a diagram $$\xymatrix{
\ \hat{D}_l=\bigcup_{j=1}^{s}\hat{D}_{j,l}\ \ar@{^(->}[r] \ar[d] &\ \hat{X}_l \ar[d] \\
\ \bigcup_{j=1}^s\{y_{j,l}=0\}\ \ar@{^(->}[r]& \ {\mathrm{Spf}}V\{x_{1,l},\dots,x_{n,l},y_{1,l},\dots,y_{m,l}\}/(x_{1,l}\dots x_{r,l}-\pi) \\
}$$ as in (\[etale locally\]) with $\hat{D}_l:=\phi_l^{-1}(\hat{D})$ such that for every $l$ the restriction of the log-$\nabla$-module $E_l$ on $\hat{X}_{l;K}$ to $$\hat{D}^{\circ}_{i,j,l;K}\times A^1_K[\lambda,1),$$ is $\cap_{i=1}^r\Sigma_{h(i,j)}$-unipotent , $\forall$ $(i,j)$ such that $\phi_l(\hat{D}_{i,j,l}^{\circ})\subset \hat{D}^{h(i,j)}$.\
We will denote by $I^{\dag}((U_k, M),(X_k,M))/({\mathrm{Spf}}(V),N))^{\Sigma}= I^{\dag}(((U_k,X_k)/{\mathrm{Spf}}(V))^{log, \Sigma}$the category of log overconvergent isocrystals with $\Sigma$-unipotent monodromy .
In definition \[unipotentmonodromy\] we do not ask any locally freeness hypothesis, because every object in the category $I^{\dag}((U_k, M),(X_k,M))/({\mathrm{Spf}}(V),N))$ is such that $\mathcal{E}$ is locally free. This is clear because $(\mathcal{E},\epsilon)$ induces on a strict neighborhood $W$ an $\mathcal{O}_{W}$-module $E$, such that $j^{\dag}_W E=\mathcal{E}$ endowed with an integrable connection. As $K$ is of characteristic $0$ we can conclude that $E$ is locally free and $\mathcal{E}$ is locally free.
Unipotence, generization and overconvergent generization
========================================================
In this section we recall the three propositions that we will use in the proof of the main theorem. They are proven by Shiho in [@Sh6] as generalization of the analogous propositions proven by Kedlaya in [@Ke] (assuming $\Sigma=0$). We will write the statements only in the cases that we need, that are simplified versions of the propositions given in paragraph 2 of [@Sh6]. The first property that we consider is called by Shiho and Kedlaya generization property for monodromy ([@Ke] proposition 3.4.3 and [@Sh6] proposition 2.4 ).
\[Generization of monodromy\] Let $A$ be an affinoid algebra such that $Y=\textrm{Spm}(A)$ is smooth and endowed with sections $y_1,\dots, y_s$ that are smooth and meet transversally. Suppose that there exists $A\subseteq L$ such that $L$ is an affinoid algebra over $K$, $\textrm{Spm}(L)$ is smooth, all the $y_i$’s are invertible in $L$ and the spectral norm on $L$ restricts to the spectral norm on $A$. Let $I$ be a quasi open interval contained in $[0,1)$ and $A^n_L(I)$ defined as $\textrm{Spm}(L)\times A^n_K(I)$. Let $\Sigma \subset \mathbb{Z}_p^{n+s}$ be a set which is (NLD) and (NID); if $E$ $\in $ $LNM_{Y\times A_K^n(I),\Sigma}$ is such that the induced object $F$ $\in$ $LNM_{A^n_L(I),\Sigma}$ is unipotent, then $E$ is $\Sigma$-unipotent.
The second result that we need is called overconvergent generization and describes the property of extension of unipotence on strict neighborhoods (proposition 2.7 of [@Sh6] and proposition 3.5.3 of [@Ke]).
\[Overconvergent generization\] Let $P$ be a $p$-adic formal affine scheme topologically of finite type over $V$. Let $Y_k\subseteq P_k$ be an open dense subscheme of the special fiber of $P$ such that $P$ is formally smooth over $V$ in a neighborhood of $Y_k$. Let $W$ be a strict neighborhood of $]Y_k[_P$ in $P_K$, $I \subset [0,1)$ a quasi open interval and $\Sigma$ a subset of $\mathbb{Z}_p^n$. Given $E\in $ $LNM_{W\times A^n_K(I),\Sigma}$ such that the restriction of $E$ to $]Y_k[_P\times A^n_K(I)$ is $\Sigma$-unipotent, then for every closed interval $[b,c]\subseteq I$ there exists a strict neighborhood $W'$ of $]Y_k[_P$ in $P_K$ such that $W'$ is contained in $W$ and such that the restriction of $E$ to $W'\times A_K^n[b,c]$ is $\Sigma$-unipotent.
The third property that we need states that, under certain assumptions, a log-$\nabla$-module with exponents in $\Sigma$ that is convergent is $\Sigma$-unipotent (proposition 2.12 of [@Sh6] and lemma 3.6.2 of [@Ke]).
\[Convergent plus nilpotent implies unipotent\] Let $A$ be an integral affinoid algebra such that $Y=\textrm{Spm}(A)$ is smooth. Suppose that there exists $A\subseteq L$ such that $L$ is an affinoid algebra over $K$, $\textrm{Spm}(L)$ is smooth and $y_j$’s are invertible in $L$ and the spectral norm of $L$ restricts to the spectral norm of $A$. Let $\Sigma \subset \mathbb{Z}_p^{n+s}$ be a set which is (NLD) and (NID); if $E$ is an object of $LNM_{Y\times A^n_K([0,1)), \Sigma}$ which is log convergent ([@Sh6], definition 2.9), then it is $\Sigma$-unipotent.
Extension theorem
=================
Now we come back to the semistable situation and we will prove that the definition of $\Sigma$-unipotent monodromy is well posed.
\[indepunipmonodromy\] Let $\mathcal{E}$ be an overconvergent log isocrystal which is in the category $I^{\dag}((U_k,X_k)/{\mathrm{Spf}}(V))^{log, \Sigma}$. The notion of $\Sigma$-unipotent monodromy for $\mathcal{E}$ is independent of the choice of the étale cover and of the diagram in (\[etale locally\]) which we have chosen in definition \[unipotentmonodromy\].
First we will prove that if $\mathcal{E}$ has $\Sigma$-unipotent monodromy for some diagram as in (\[etale locally\]), then it has $\Sigma$-unipotent monodromy for any diagram as in (\[etale locally\]). So we suppose that there exists an étale covering $\coprod_l\phi_l:\coprod_l \hat{X}_l \rightarrow \hat{X}$ such that every $\hat{X}_l$ has a diagram as in (\[etale locally\]). As we saw before $\mathcal{E}$ induces on some $W$ strict neighborhood of $]U_k[_{\hat{X}}$ in $]X_k[_{\hat{X}}$ a locally free $\mathcal{O}_{W}$-module $E$ with an integrable connection. In the situation of (\[etale locally\]) $W$ contains the set $$\{p \in \hat{X}_{;K}\,\,\,|\,\,\,\forall j \,\,\,|y_j(p)| \geq \lambda\}$$ and the restriction of $E$ to $$\hat{D}^{\circ}_{i,j;K}\times A^1_K[\lambda,1)$$ is $\cap_{i=1}^r\Sigma_{h(i,j)}$-unipotent for some $\lambda,$ if $(i,j) $ are such that $\phi_l(\hat{D}^{\circ}_{i,j}) \subset \hat{D}^{h(i,j)}$.\
Using theorem \[unipotentiequivalenti\] we can extend $E$ to a module with connection on $$\hat{D}_{i,j;K}^{\circ}\times A^1_K[0,1)$$ that is $\cap_{i=1}^r\Sigma_{h(i,j)}$-unipotent. This is true for every $(i,j)$ so $E$ can be extended to a locally free isocrystal on $((\coprod_{(i,j)}\hat{X}^{\circ}_{i,j}, M)/({\mathrm{Spf}}(V),N))$ which is convergent because it is convergent on $\hat{U}\cap \coprod_{(i,j)}\hat{X}^{\circ}_{i,j}$ that is an open dense $\hat{X}\cap \coprod_{(i,j)}\hat{X}^{\circ}_{i,j}$ (we are applying here proposition \[U denso\]). Hence we have a locally free isocrystal on $((\hat{X}^{\circ}, M)/({\mathrm{Spf}}(V),N))$ which is convergent and such that has exponents along $\hat{D}_{j}$ in $\cap_{i=1}^r\Sigma_{h(i,j)}$ for every $(i,j)$ such $\phi_l(\hat{D}^{\circ}_{i,j})\subset \hat{D}^{h(i,j)}$.\
For any other diagram as in (\[etale locally\]), $E$ on $\hat{D}_{i,j;K}^{\circ}\times A^1_K[\lambda, 1)$ with $(i,j)$ such that $\phi_l(\hat{D}_{i,j}^{\circ})\subset \hat{D}^{h(i,j)}$ is the restriction of a convergent module with connection on $\hat{D}^{\circ}_{i,j:K}\times A^1_K[0,1)$ with exponents in $\cap_{i=1}^r \Sigma^{h(i,j)}$, so that it is $\cap_{i=1}^r\Sigma^{h(i,j)}$-unipotent on $\hat{D}_{i,j;K}^{\circ}\times A^1_K[\lambda, 1)$ by proposition \[Convergent plus nilpotent implies unipotent\].\
Now we prove that the notion of $\Sigma$-unipotence is independent of the choice of the étale covering. To do this, it suffices to prove that if $\mathcal{E}$ is a log overconvergent isocrystal with $\Sigma$-unipotent monodromy, for any étale morphism $\phi: \hat{X}'\rightarrow \hat{X}$ such that $\hat{X}'$ admits a diagram $$\label{etale locally X'unip}
\xymatrix{
\ \hat{D}^{\prime}=\bigcup_{j=1}^{s}\hat{D}^{\prime}_{j}\ \ar@{^(->}[r] \ar[d] &\ \hat{X}^{\prime} \ar[d] \\
\ \bigcup_{j=1}^s\{y^{\prime}_j=0\}\ \ar@{^(->}[r]& \ {\mathrm{Spf}}V\{x^{\prime}_1,\dots,x^{\prime}_n,y^{\prime}_1,\dots,y^{\prime}_m\}/(x^{\prime}_1\dots x^{\prime}_r-\pi) \\
}$$ as in (\[etale locally\]), with $\hat{D}':=\phi^{-1}(\hat{D})$, the log-$\nabla$-module $E$ induced on $\hat{X}'$ by $\mathcal{E}$ is $\cap_{i=1}^r\Sigma_{h(i,j)}$-unipotent on $\hat{D}^{' \circ}_{i,j;K}\times A^1_K[\lambda,1)$, for every $(i,j)$ such that $\hat{D}^{' \circ}_{i,j}$ is such that $\phi(\hat{D}^{' \circ}_{i,j})\subset \hat{D}^{h(i,j)}$. We may assume that $\hat{D}^{' \circ}_{i,j;K}=\textrm{Spm}A$ is affinoid. As in the proof of proposition \[indepnilpotentresidue\] we know that there exists an étale covering $\coprod_l\hat{X}'_l\rightarrow \hat{X}'$ such that, for any $l$, $\hat{X}'_l$ admits a diagram as in (\[etale locally\]) such that $\mathcal{E}$ has $\Sigma$-unipotent monodromy with respect to this diagram.\
However since we have already showed that the notion of $\Sigma$-unipotent monodromy does not depend on the choice of a diagram as in (\[etale locally\]), we can say that $\mathcal{E}$ has $\Sigma$-unipotent monodromy with respect to a diagram as in (\[etale locally\]) for $\hat{X}'_l$ induced by the diagram (\[etale locally X’unip\]). This means that the log-$\nabla$-module $E$ is $\cap_{i=1}^r\Sigma_{h(i,j)}$-unipotent when it is restricted to $\coprod_l(\hat{D}^{' \circ}_{i,j;K}\times_{\hat{X}_{K}}\hat{X}'_{l,K})\times A^{1}_K[\lambda,1)$ with $(i,j)$ such that $\phi (\coprod_{l}(\hat{D}^{' \circ}_{i,j}\times_{\hat{X}}\hat{X}'_{l}))\subset \hat{D}^{h(i,j)}$.\
Let us take an affine covering $\coprod_h \hat{C}_{h}\rightarrow \coprod_l (\hat{D}^{' \circ}_{i,j}\times_{\hat{X}}\hat{X}'_{l})$ by affine formal schemes and put $\textrm{Spm}L:=\coprod_h \hat{C}_{h;K}$. Then $E$ is $\cap_{i=1}^r\Sigma_{h(i,j)}$-unipotent on $\textrm{Spm}L\times A^1_K[\lambda,1)$ and the affinoid algebras $A$ and $L$ satisfy the assumption of proposition \[Generization of monodromy\]. Applying proposition \[Generization of monodromy\] we can conclude that $E$ is $\cap_{i=1}^r\Sigma_{h(i,j)}$-unipotent on $\hat{D}^{' \circ}_{i,j;K}\times A^1_K[\lambda,1)$ as we wanted.
We can now state the main result, an extension theorem that generalizes theorem 6.4.5 of [@Ke] and theorem 3.16 of [@Sh6]. The strategy of the proof is the same as the one in [@Ke] and [@Sh6] and we will follow step by step the proof of theorem 3.16 of [@Sh6].\
We will need for the proof the following lemma:
\[denso\] If $W$ is an open dense in $P=\textrm{Spf}(A)$, with $A$ a formal $V$ algebra of topologically finite type such that $A_k$ is reduced, then the spectral seminorm on $\mathcal{O}(]W_k[_{P})$ restricts to the spectral seminorm on $\mathcal{O}(]P_k[_{P})$.
We can suppose that $W$ is defined by the equation $\{g\neq 0\},$ in particular that $W=\textrm{Spf}(A\left\{\frac{1}{g}\right\}).$\
So we have a map of $V$-algebras $$A\longrightarrow A \left\{\frac{1}{g}\right\},$$ which is injective modulo $\pi$ because $W_k$ is dense in $A_k$ that is reduced. By the topological Nakayama’s lemma (ex 7.2 of [@Ei]) we can conclude that we have an inclusion of $V$-algebras $$A\hookrightarrow A \left\{\frac{1}{g}\right\}$$ which induces an inclusion of affinoid algebras $$\mathcal{O}(]P_k[_{P})=A\otimes K\hookrightarrow \mathcal{O}(]W_k[_{P})=A \left\{\frac{1}{g}\right\}\otimes K$$ Let us take the Banach norm $|\,\,\,|_P$ on $A\otimes K$ induced by $A$ and the Banach norm $|\,\,\,|_W$ on $A \left\{\frac{1}{g}\right\}\otimes K$ induced by $A\{\frac{1}{g}\}$, then $$|a|_P=|a|_W$$ for any $a$ $\in$ $K\otimes A.$ By the well known formula (see for example [@FvdP] corollary 3.4.6) $$|a|_{P,\textrm{sp}}=\lim_{n\rightarrow \infty}|a^n|_P^{\frac{1}{n}}$$ where with $|\,\,\,|_{P,\textrm{sp}}$ we denote the spectral norm, we are done.
\[log extension\] We fix a set $\Sigma=\prod_{h=1}^k\Sigma_h$ $\in$ $\mathbb{Z}_p^k$, where $k$ is the number of the irreducible components of $\hat{D}=\bigcup_{h=1}^k\hat{D}^h$ in $\hat{X}$ and we require that $\Sigma$ has the properties (NID) and (NLD). Let us suppose that locally for the étale topology we have a diagram as (\[etale locally\]), then the restriction functor $$j^{\dag}:I_{conv}((\hat{X},M)/({\mathrm{Spf}}(V),N))^{\Sigma}\longrightarrow I^{\dag}((U_k, M),(X_k,M))/({\mathrm{Spf}}(V),N))^{\Sigma}$$ is an equivalence of categories.
We will divide the proof in 3 steps.\
**Step 1**: the functor $j^{\dag}$ is well defined.\
Let $\mathcal{E}$ be in the category $I_{conv}((\hat{X},M)/({\mathrm{Spf}}(V),N))^{\Sigma}$, then we have to prove that $j^{\dag}(\mathcal{E})$ in in the category $I^{\dag}((U_k, M),(X_k,M))/({\mathrm{Spf}}(V),N))^{\Sigma}$. Thanks to lemma \[indepnilpotentresidue\] and lemma \[indepunipmonodromy\], we can work étale locally. We suppose that $\phi$ is an étale map to $\hat{X}$ and we call again $\hat{X}$ an étale neighborhood for which we have the diagram as in (\[etale locally\]); in this situation the log convergent isocrystal $\mathcal{E}$ induces a log-$\nabla$-module with respect to $y_1,\dots ,y_s$ on $\hat{X}_K$ such that has exponents along $\hat{D}_{j;K}$ in $\cap_{i=1}^r\Sigma_{h(i,j)}$ if $i$ and $j$ are such that $\phi(\hat{D}^{\circ}_{i,j})\subset \hat{D}^{h(i,j)}$. By the definition of $\Sigma$-unipotent monodromy (definition \[unipotentmonodromy\]) we are reduced to prove that, if we restrict $E$ to $$\hat{D}_{i,j;K}^{\circ}\times A^1_K [0,1),$$ then it is $\cap_{i=1}^r\Sigma_{h(i,j)}$-unipotent if $i, j$ are such that $\phi(\hat{D}_{i,j}^{\circ})\subset \hat{D}^{h(i,j)}$.\
We know by hypothesis that the restriction of $E$ to $\hat{D}^{\circ}_{i,j;K}\times A^1_K [0,1)$ is log convergent and has exponents along $\hat{D}_{j;K}$in $\cap_{i=1}^r\Sigma_{h(i,j)}$, hence we have to prove that this implies $\cap_{i=1}^r\Sigma_{h(i,j)}$-unipotence; from proposition \[Convergent plus nilpotent implies unipotent\] we know that this implies $\cap_{i=1}^r\Sigma_{h(i,j)}$-unipotence.\
**Step 2**: the functor $j^{\dag}$ is fully faithful.\
We have to prove that given $f:\mathcal{E}\rightarrow \mathcal{F}$, a morphism of log overconvergent isocrystals of $\Sigma$-unipotent monodromy, if there exist extensions of $\mathcal{E}$ and $\mathcal{F}$ to log convergent isocrystals with exponents in $\Sigma$ that we call respectively $\tilde{\mathcal{E}}$ and $\tilde{\mathcal{F}}$, then $f$ extends uniquely to $\tilde{f}:\tilde{\mathcal{E}}\rightarrow \tilde{\mathcal{F}}.$\
We can work étale locally. We denote by $\phi$ an étale map to $\hat{X}$ and again by $\hat{X}$ an étale neighborhood that we consider for which there exists a diagram as in (\[etale locally\]).\
Let us take $W$ a strict neighborhood of $]U_k[_{\hat{X}}$ in $]X_k[_{\hat{X}}$; by definition $f$ induces a morphism $\varphi$ of $\nabla$-modules between $E_{\mathcal{E}}$ and $E_{\mathcal{F}}$, the $\nabla$-modules on $W$ that are induced by $\mathcal{E}$ and $\mathcal{F}$ respectively: $$\varphi:E_{\mathcal{E}}\rightarrow E_{\mathcal{F}}.$$ We call $E_{\tilde{\mathcal{E}}}$ and $E_{\tilde{\mathcal{F}}}$ the log-$\nabla$-modules on $\hat{X}_K$ induced by $\tilde{\mathcal{E}}$ and $\tilde{\mathcal{F}}$.\
Let us take the following covering of $$\hat{X}_K=\bigcup_{J\subset \{1\dots s\}}A_J$$ where $$A_J=\{p\in \hat{X}_K|\,\,\, |y_j(p)|<1 \,\,\,(j\in J)\,\,\, |y_j(p)|\geq \lambda\,\,\, (j \notin J) \}$$ and $\lambda \in (0,1)\cup \Gamma^*$ is such that both $\mathcal{E}$ and $\mathcal{F}$ are defined on the following set: $$B=\{p\in \hat{X}_K|\,\,\, |y_j(p)|\geq \lambda \,\,\, \forall j \}.$$ The covering of $\hat{X}_{K}$ given by the $A_J$’s restricts to the following covering of $B=\bigcup_{J\subset\{1,\dots,s\}}B_J$, where $$B_J=\{p\in \hat{X}_K|\,\,\, \lambda \leq |y_j(p)|<1 \,\,\,(j\in J),\,\,\,|y_j(p)|\geq \lambda\,\,\, (j \notin J)\}.$$ The extensions $E_{\tilde{\mathcal{E}}}$ and $E_{\tilde{\mathcal{F}}}$ are log convergent in $$\label{Jfinoa1}
\{p\in \hat{X}_K| y_j(p)=0 \,\,\,(j\in J),\,\,\,|y_j(p)|\geq \lambda\,\,\, (j \notin J)\}\times A^{|J|}[0,1)$$ by proposition 3.6 of [@Sh6] and they have exponents in $\prod_j\cap_{i=1}^r\Sigma_{h(i,j)}.$ They extend the restrictions of $E_{\mathcal{E}}$ and $E_{\mathcal{F}}$ on $$\{p\in \hat{X}_K| y_j(p)=0 \,\,\,(j\in J)\,\,\, |y_j(p)|\geq \lambda\,\,\, (j \notin J)\}\times A^{|J|}[\lambda,1).$$ By theorem \[unipotentiequivalenti\] we can conclude that $\phi$ extends to $$A_J=\{p\in \hat{X}_K| y_j(p)=0 \,\,\,(j\in J)\,\,\, |y_j(p)|\geq \lambda\,\,\, (j \notin J)\}\times A^{|J|}[0,1);$$ this means that on this set there exists a unique $$\tilde{\phi_J}:E_{\tilde{\mathcal{E}}}\rightarrow E_{\tilde{\mathcal{F}}}$$ that extends $\phi$ on $B_J$.\
On $A_I\cap A_J$ we have the extensions $\phi_I$ and $\phi_J$, which glue because they coincide on the set $$\begin{split}
B_I\cap B_J=\{p\in \hat{X}_K|\,\,\, \lambda \leq |y_j(p)| < 1 & \,\,\,(j\in (I\cup J)-(I\cap J)),\\
& |y_j(p)|\geq \lambda \,\,\, (j \notin (I\cup J))\}\times A^{|I\cap J|}[\lambda,1)
\end{split}$$ because they extend the map $\phi$ on $B_I\cap B_J.$\
**Step 3**: the functor $j^{\dag}$ is essentially surjective.\
Since we have the étale descent property for the category of locally free log convergent isocrystals (remark 5.1.7 of [@Sh1]) and the full faithfulness of the functor $j^{\dag}$, we may work étale locally to prove the essential surjectivity.\
If $\mathcal{E} \in I^{\dag}((U_k,X_k),{\mathrm{Spf}}(V))^{log, \Sigma}$, then by definition of log overconvergent isocrystal we know that $\mathcal{E}$ induces a module with connection on the following set $$\{p \in \hat{X}_K|\,\,\, \forall j: |y_j(p)|\geq \lambda\}$$ that we will denote by $E$ that is $\cap_{i=1}^r\Sigma_{h(i,j)}$-unipotent on $$\hat{D}^{\circ}_{i,j,K}\times A^1[\lambda,1)=\{p \in \hat{D}_{j,K}|\,\,\, \forall j'\neq j : |y_j'(p)|=1, \,\,\, \forall i'\neq i : |x_i'(p)|=1\}\times A^1[\lambda,1) ,$$ for $i$, $j$ such that $\phi(\hat{D}^{\circ}_{i,j})\subset \hat{D}^{h(i,j)}$ .\
We will prove that $E$ extends to a log-$\nabla$-module on $C_{a,\lambda}=\{p \in \hat{X}_K|\,\,\, \forall j>a\,\,\,,|y_{j}(p)|\geq \lambda\}$ $\forall a=0, \dots, s$ such that has exponents along $\{y_{j}=0\}$ in $\cap_{i=1}^r\Sigma_{h(i,j)}$ with $i,j$ such that $\phi(\hat{D}^{\circ}_{i,j})\subset \hat{D}^{h(i,j)}$, proceeding by induction on $a$.\
So we suppose, by induction hypothesis, that $E$ extends to the set $C_{a-1,\lambda}=\{p \in \hat{X}_K|\,\,\, \forall j>a-1\,\,\,,|y_j(p)|\geq \lambda\}$ for some $\lambda$ with exponents along $\{y_{j}=0\}$ in $\cap_{i=1}^r\Sigma_{h(i,j)}$ with $i,j$ such that $\phi(\hat{D}^{\circ}_{i,j})\subset \hat{D}^{h(i,j)}$.\
We consider the following admissible covering of $\hat{X}_K=A\cup B$, where $$A=\{p \in \hat{X}_K|\,\,\,|y_{a}(p)|\geq \lambda'\}$$ $$B=\{p \in \hat{X}_K|\,\,\,|y_{a}(p)|< 1 \}$$ with $\lambda' \in [\lambda,1)\cap \Gamma^*.$\
Intersecting the covering $A\cup B$ with $C_{a-1,\lambda'}$ we obtain the following admissible covering: $$\label{C_{a-1}}
C_{a-1,\lambda'}=(C_{a-1,\lambda'}\cap A)\cup (C_{a-1,\lambda'}\cap B)=$$ $$\begin{split}
=\{p \in \hat{X}_K|\,\,\,\forall j>a-1\,\,\,|y_{j}(p)|&\geq \lambda'\}\cup \\
&\{p \in \hat{X}_K|\,\,\,\forall j>a\,\,\,|y_{j}(p)|\geq \lambda',\,\,\,\lambda' \leq |y_a(p)|<1 \}=
\end{split}$$ $$\begin{split}
=\{p \in \hat{X}_K|\,\,\,\forall j>a-1\,\,\,|y_{j}(p)|\geq \lambda'\}\cup\\
&\{p \in \hat{D}_{a,K}|\,\,\,\forall j>a\,\,\,|y_{j}(p)|\geq \lambda'\}\times A^1[\lambda',1),
\end{split}$$ and intersecting with $C_{a,\lambda'}$: $$\label{C_a}
C_{a,\lambda'}=(C_{a,\lambda'}\cap A)\cup (C_{a,\lambda'}\cap B)=$$ $$\{p \in \hat{X}_K|\,\,\,\forall j>a-1\,\,\,|y_{j}(p)|\geq \lambda'\}\cup\{p \in \hat{X}_K|\,\,\,\forall j>a\,\,\,|y_{j}(p)|\geq \lambda',\,\,\, |y_a(p)|<1 \} =$$ $$\{p \in \hat{X}_K|\,\,\,\forall j>a-1\,\,\,|y_{j}(p)|\geq \lambda'\}\cup\{p \in \hat{D}_{a,K}|\,\,\,\forall j>a\,\,\,|y_{j}(p)|\geq \lambda'\}\times A^1[0,1) .$$ Comparing the formulas in (\[C\_[a-1]{}\]) and (\[C\_a\]), we see that it is sufficient to prove that $E$ extends from $$\{p \in \hat{D}_{a,K}|\,\,\, \forall j>a\,\,\,,|y_j(p)|\geq \lambda'\}\times A^1_K[\lambda',1)$$ to $$\{p \in \hat{D}_{a,K}|\,\,\,\forall j>a\,\,\,|y_{j}(p)|\geq \lambda'\}\times A^1[0,1),$$ for some $\lambda'$ $\in$ $[\lambda,1)\cap \Gamma^*$ in a log-$\nabla$-module such that it has exponents along $\{y_{j}=0\}$ in $\cap_{i=1}^r\Sigma_{h(i,j)}$ with $i,j$ for which $\phi(\hat{D}_{i,j} ^{\circ})\subset \hat{D}^{h(i,j)}$.\
As we saw before $E$ is $\cap_{i=1}^r\Sigma_{h(i,a)}$-unipotent on $$\begin{split}
\hat{D}^{\circ}_{i,a,K}\times &A^1_K([\lambda,1))=\\
&\{p \in \hat{D}_{a,K}|\,\,\, \forall j'\neq a : |y_j'(p)|=1, \,\,\, \forall i'\neq i : |x_i'(p)|=1\}\times A^1_K([\lambda,1)),
\end{split}$$ so it is $\cap_{i=1}^r\Sigma_{h(i,a)}$-unipotent also on $$\begin{split}
\coprod_i\hat{D}^{\circ}_{i,a,K}\times & A^1_K([\lambda,1))= \\
&
\coprod_i\{p \in \hat{D}_{a,K}|\,\,\, \forall j'\neq a : |y_j'(p)|=1, \,\,\, \forall i'\neq i : |x_i'(p)|=1\}\times A^1_K([\lambda,1)) .
\end{split}$$ We now want to apply proposition \[Overconvergent generization\]; following the notation given in the proposition in our case we have that $P$ is the pull-back of $$\textrm{Spf}\frac{V\{x_1,\dots,x_n,y_1\dots \hat{y_a} \dots y_m\}}{x_1\dotsm x_r-\pi}\{y_j^{-1}|j<a\}\{\prod_{i,i'\,\,\,i\neq i'}(x_i-x_i')^{-1}\}$$ by the morphism $$\hat{X}\rightarrow {\mathrm{Spf}}V\{x_1,\dots, x_n, y_1, \dots, y_m\}/(x_1\dotsm x_r - \pi),$$ $Y_k$ is the open defined in $P_k$ by the following equation $$\{y_{a+1}\dotsm y_s\neq 0\}$$ and $$\begin{split}
W\times A^1_K(I)=\{p \in \hat{D}_{a,K}|&\,\,\, \forall j < a : |y_j(p)|=1,\\
& \,\,\, \forall j>a\,\,\,,|y_j(p)|\geq \lambda \,\,\, \forall i'\neq i : |x_i'(p)|=1\} \times A^1_K([\lambda,1)).
\end{split}$$ The hypothesis of proposition \[Overconvergent generization\] are fulfilled.\
The restriction of $E$ to $$\{p \in \hat{D}_{a,K}|\,\,\, \forall j>a\,\,\,,|y_j(p)|\geq \lambda \}\times A^1_K([\lambda,1))$$ is a log-$\nabla$-module with exponents in $\prod_j\cap_{i=1}^r\Sigma_{h(i,j)}$, that is $\prod_j\cap_{i=1}^r\Sigma_{h(i,j)}$-unipotent on $$]Y_k[_P=\coprod_i\{p \in \hat{D}_{a,K}|\,\,\, \forall j'\neq a : |y_j'(p)|=1, \,\,\, \forall i'\neq i : |x_i'(p)|=1\}\times A^1_K([\lambda,1));$$ so applying proposition \[Overconvergent generization\] we know that for every $[b,c]$ $\subset [\lambda,1)$ there exists a $\lambda'$ (we suppose that it verifies $\lambda' \in (c,1)$ for gluing reasons) such that $E$ is $\prod_j\cap_{i=1}^r\Sigma_{h(i,j)}$-unipotent on $$\begin{split}
\coprod_i\{p \in \hat{D}_{a,K}|&\,\,\, \forall j < a : |y_j(p)|=1,\\
& \,\,\, \forall j > a : |y_j(p)|\geq \lambda', \,\,\, \forall i'\neq i : |x_i'(p)|=1\}\times A^1([b,c]).
\end{split}$$ Now we apply proposition \[Generization of monodromy\] with $$\begin{split}
\textrm{Spm}(L)=\coprod_i\{p \in \hat{D}_{a,K}|&\,\,\, \forall j < a : |y_j(p)|=1,\\
& \forall j > a : |y_j(p)|\geq \lambda', \,\,\, \forall i'\neq i : |x_i'(p)|=1\}
\end{split}$$ and $$Y=\{p \in \hat{D}_{a,K}| \,\,\, \forall j > a : |y_j(p)|\geq \lambda' \};$$ we are in the hypothesis of that proposition thanks to lemma \[denso\], hence we deduce that $E$ is $\prod_j\cap_{i=1}^r\Sigma_{h(i,j)}$-unipotent on $$\{p \in \hat{D}_{a,K}|\,\,\, \forall j > a : |y_j(p)|\geq \lambda'\}\times A^1((b,c)).$$ By theorem \[unipotentiequivalenti\] we see that $E$ can be extended to a $\prod_j\cap_{i=1}^r\Sigma_{h(i,j)}$-unipotent log-$\nabla$-module, in particular to a log-$\nabla$-module with exponents in $\prod_j\cap_{i=1}^r\Sigma_{h(i,j)}$ on $$\{p \in \hat{D}_{a,K}|\,\,\, \forall j > a : |y_j(p)|\geq \lambda'\}\times A^1([0,c)).$$ Now we glue this with $E$ and we obtain a log-$\nabla$-module with exponents in the set $\prod_j\cap_{i=1}^r\Sigma_{h(i,j)}$ on $$\{p \in \hat{D}_{a,K}|\,\,\, \forall j>a\,\,\,,|y_j(p)|\geq \lambda'\}\times A^1_K([0,1))$$ as we wanted.\
Therefore we have a log-$\nabla$-module defined on the space $\hat{X}_K$ and we now prove that it is convergent.\
We know that the restriction of $E$ to $\hat{U}_K$ is log convergent because it is an extension of an overconvergent log isocrystal on $\hat{U}_K$, hence it belongs to the category $$I_{\textrm{conv}}((\hat{U}, M),({\mathrm{Spf}}(V),N))^{lf}.$$ Since $\hat{U}$ is an open dense in $\hat{X}$, we have a module with log connection defined in the whole space that is convergent on an open dense of the space; we can apply proposition \[U denso\] and conclude that $E$ is convergent.\
Main theorem
============
As we saw in proposition \[log extension\] there is an equivalence of categories $$j^{\dag}:I_{conv}((\hat{X},M)/({\mathrm{Spf}}(V),N))^{lf,\Sigma}\longrightarrow I^{\dag}((U_k,X_k)/{\mathrm{Spf}}(V))^{log, \Sigma}.$$ Now we want to compare this to the category $MIC((X_K,M)/K)^{lf}$ that we defined in definition \[connections\]. We can define the notion of exponents also in the algebraic case, giving the analogous definition that we gave before definition \[residueanalytic\], replacing the rigid analytic space $\hat{X}_K$ with the algebraic space $X_K$, the divisor $\hat{D}_K$ with the divisor $D_K$ and the $\mathcal{O}_{\hat{X}_K}$-module $\omega^1_{(\hat{X}_K,M)/K}$ with the $\mathcal{O}_{X_K}$-module $\omega^1_{(X_K,M)/K}$.\
We fix a set $\Theta=\prod_{p=1}^f \Theta_p \subset \bar{K}^f$, where $f$ is the number of irreducible components of the divisor $D_K=\cup_{p=1}^fD^{p;K}$. We say that (E,$\nabla$) in $MIC((X_K,M_D)/K)$ has residue along $D_{K}$ in $\Theta$ if étale locally there exists a diagram analogous to (\[etale locally\]) such that for every $l$ the log-$\nabla$-module $X_{l;K}$ induced by ($E$, $\nabla$) has exponents along $D_{j,l;K}$ in $\Theta_{p(j)}$ for every $j$ such that $\phi_{l;K}(\hat{D}_{j,l})\subset D^{p(j)}$.\
We will denote the category of locally free module with integrable log connection with exponents in $\Theta$ by $MIC((X_K,M_D)/K)^{\Theta}$. We can prove as in lemma \[indepnilpotentresidue\] that the notion of exponents in $\Theta$ can be given étale locally and that is independent from the choice of a diagram as in (\[etale locally\]). If $(E,\nabla)$ is in $MIC((X_K,M_D)/K)$ we restrict locally étale in a situation for which there exists a diagram analogous to (\[etale locally\]) for the algebraic setting and we look at the exponents of $(E,\nabla)$ along $D_{j;K}$. In particular we consider the log-$\nabla$-module $(\hat{E},\hat{\nabla})$ on $\hat{X}_K$ induced by the log infinitesimal locally free isocrystal $\Psi (E,\nabla)$ (where $\Psi$ is the functor defined in proposition \[derhaminf\]) and the residue of it along $\hat{D}_{j;K}$. We have a map $$\textrm{End}_{\mathcal{O}_{D_{j;K}}}(E|_{D_{j;K}})\rightarrow \textrm{End}_{\mathcal{O}_{\hat{D}_{j,K}}}(\hat{E}|_{\hat{D}_{j;K}}),$$ that sends the residue of $(E,\nabla)$ along $D_{j;K}$ to the residue of $(\hat{E},\hat{\nabla})$ along $\hat{D}_{j;K}.$ Moreover the map is injective because the map $$\Gamma(D_{j;K},\mathcal{O}_{D_{j;K}})\rightarrow \Gamma(\hat{D}_{j,K},\mathcal{O}_{\hat{D}_{j;K}})$$ is injective. This means that étale locally $(E,\nabla)$ and the log-$\nabla$-module $(\hat{E},\hat{\nabla})$ have the same exponents along $D_{j;K}$ and $\hat{D}_{j;K}$ respectively.\
The relation between $\Theta$ and $\Sigma$ is as follows.\
Given a log infinitesimal isocrystal $\mathcal{E}$ with exponents in $\Sigma$ then the functor $\Psi^{-1}$ associates to it a module with integrable log connection $(E,\nabla)$ such that it has exponents in $\Theta=\prod_{p=1}^f\Theta_p$, where the $p$-th component $\Theta_p$ is given by $\cap_{i=1}^r\Sigma^{h(i,j)}$ where $j$ is such that $\phi_{l,K}(D_{j;K})\subset D^{p;K}.$ Viceversa given a module with integrable log connection $(E, \nabla)$ such that it has exponents in $\Theta$, the functor $\Psi$ associates to it a log infinitesimal isocrystal $\mathcal{E}$ with exponents in $\Sigma=\prod_{h=1}^k\Sigma_h$ where the $h$-th component is given by $\Theta^{p(j)}$ where $j$ is such that $\phi(\hat{D}^{\circ}_{i,j})\subset\hat{D}^h$.\
From this it follows that the functor $\Psi$ induces an equivalence of categories $$MIC((X_K,M_D)/(K,\textrm{triv}))^{lf,\Theta}\longrightarrow I_{inf}((\hat{X},M)/({\mathrm{Spf}}V,N))^{lf,\Sigma}.$$ If we start from a log overconvergent isocrystal $\mathcal{E}$ with $\Sigma$-unipotent monodromy as in \[unipotentmonodromy\], we apply the equivalence of category given by the functor $j^{\dag}$ of theorem \[log extension\] and the observations written above, we can conclude that there is fully faithful functor $$I^{\dag}((U_k, M),(X_k,M))/({\mathrm{Spf}}(V),N))^{\Sigma}\longrightarrow MIC((X_K,M_D)/(K,\textrm{triv}))^{lf,\Theta}.$$
The logarithmic extension theorem of Andr[\`]{}e and Baldassarri (theorem 4.9 of [@AnBa]) gives an equivalence of category between $MIC((X_K,M_D)/(K,\textrm{triv}))^{lf,\Theta}$ and the category of coherent modules with connection on $U_K$ regular along $D_K$, that we denote by $MIC(U_K/K)^{reg}$ gives us the general result.\
There is a fully faithful functor $$I^{\dag}((U_k, M),(X_k,M))/({\mathrm{Spf}}(V),N))^{\Sigma}\longrightarrow MIC(U_K/K)^{reg}.$$
Acknowledgements {#acknowledgements .unnumbered}
================
This work is a generalization of the author’s PhD thesis. It is a pleasure to thank her advisors: prof. Francesco Baldassarri and prof. Atsushi Shiho. Much of this work was done while the author was visiting the University of Tokyo; she wants to thank all the people there for their hospitality. In particular she wants to acknowledge prof. Atsushi Shiho for the helpfulness in guiding her work and for the careful reading of her thesis and prof. Takeshi Saito for many useful discussions. The author would like to express her gratitude also to prof. Bruno Chiarellotto and prof. Marco Garuti for valuable advises.
Y. André, F. Baldassarri, *De Rham Cohomology of Differential Modules on Algebraic Varieties*, Progress in Mathematics **189**, Birkhäuser, 2001. Second edition in preparation. F. Baldassarri, B. Chiarellotto, *Formal and $p$-adic theory of differential systems with logarithmic singularities depending upon parameters*, Duke Math. J., **72**, 241-300, (1993).
P. Berthelot, *Cohomologie rigide et cohomologie rigide à support propre. Première partie*, prépublication de l’IRMAR, 96-03, (1993).
P. Berthelot, A. Ogus, *Notes on crystalline cohomology*, Mathematical Notes, Princeton University Press, (1978). S. Bosch, *Lectures on Formal and Rigid Geometry*, Preprintreihe des Mathematischen Instituts, Heft 378, Westfälische Wilhelms-Universität, Münster, (2005). B. Chiarellotto, M. Fornasiero, *Logarithmic de Rham, Infinitesimal and Betti cohomologies*, J. Math. Sci. Univ. Tokyo, **13**, 205-257, (2006). B. Chiarellotto, N. Tsuzuki, *Cohomological descent of rigid cohomology for étale coverings*, Rend. Sem. Mat. Univ. Padova, **109**, 63-215, (2003). G. Christol, Z. Mebkhout, *Sur le théorème de l’indice des équations différentielles $p$-adiques II*, Annals of math., 2nd Ser., **146**, No. 2, 345-410, (1997). G. Christol, Z. Mebkhout, *Sur le théorème de l’indice des équations différentielles $p$-adiques IV*, Invent. math., **143**, 629-672, (2001). P. Deligne, *Catégories Tannakiennes*, in Grothendieck Festschrift II, Progress in Mathematics, Birkhauser, pp. 111-195, (1990). P. Deligne; J. S. Milne, *Tannakian Categories*, in Hodge cycles, motives, and Shimura varieties, Lecture Notes in Mathematics, 900. Springer-Verlag, Berlin-New York, (1982). B. Dwork, G. Gerotto, and F. Sullivan, *An Introduction to G-Functions*, Annals of Math. Studies **133**, Princeton University Press, Princeton, (1994). D.Eisenbud, *Commutative algebra with a view toward algebraic geometry*, **GTM** 150, Springer-Verlag, New York, (1995).
J. Fresnel, M. van der Put, *Rigid analytic geometry and its applications*, Progress in Mathematics **218**, Birkhäuser, (2004). A. Grothendieck, *Crystals and the de Rham cohomology of schemes*, in *Dix Exposes sur la Cohomologie des Schemas*, pp. 306-358, North Holland publishing company, (1968). R.Hartshorne, *Algebraic Geometry*, **GTM** 52, Springer-Verlag, New York, (1977). K. Kato, *Logarithmic structure of Fontaine-Illusie*, in Algebraic Analysis, Geometry and Number Theory, The John Hopkins Univ. Press, 191-224, (1988). N. M. Katz, *Nilpotent connection and the monodromy theorem: an application of a result of Turrittin*, Pub. Math. IHES, **39**, 175-232, (1970). K. S. Kedlaya,*Semistable reduction for overconvergent F-isocrystals,I: Unipotence and logarithmic extensions*, Compositio Math., **143**, 1164-1212, (2007). B. Le Stum, *Rigid Cohomology*, Cambridge Tracts in Mathematics 172, Cambridge University Press, (2007). Y. Nakkajima, A. Shiho, *Weight filtration on log crystalline cohomologies of families of open smooth varieties*, Lecture Notes in Mathematics, **1959**, Springer-Verlag, (2008). A. Ogus, *F-isocrystals and de Rham cohomology II - convergent isocrystals*, Duke Math. **51**, 765-850, (1984). A. Shiho, *Crystalline fundamental groups I- Isocrystals on log crystalline site and log convergent site*, Journal of Mathematical Sciences, University of Tokyo, **7**, 509-656, (2000). A. Shiho, *Crystalline fundamental groups I- Log convergent cohomology and rigid cohomology*, Journal of Mathematical Sciences, University of Tokyo, **9**, 1-163, (2002). A. Shiho, *Relative log convergent cohomology and relative rigid cohomology I*, arXiv:0707.1742v3, (2007). A. Shiho, *Relative log convergent cohomology and relative rigid cohomology II*, arXiv:0707.1743v2, (2007). A. Shiho, *Relative log convergent cohomology and relative rigid cohomology III*, arXiv:0805.3229v1, (2008). A. Shiho, *On logarithmic extension of overconvergent isocrystals*, Math. Ann., published online, (2010). A. Shiho, *Cut-by-curves criterion for the log extendability of overconvergent isocrystals*, Math. Zeit., published online, (2010).
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.