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arXiv:1612.02306 · Physical review. B./Physical review. B

Maximum entropy analytic continuation for frequency-dependent transport coefficients with nonpositive spectral weight

Alexis Reymbaut, Anne-Marie Gagnon, Dominic Bergeron, A.–M. S. Tremblay

Published Mar 13, 2017• 11 citations
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Abstract

The computation of transport coefficients, even in linear response, is a major challenge for theoretical methods that rely on analytic continuation of correlation functions obtained numerically in Matsubara space. While maximum entropy methods can be used for certain correlation functions, this is not possible in general, important examples being the Seebeck, Hall, Nernst, and Reggi-Leduc coefficients. Indeed, positivity of the spectral weight on the positive real-frequency axis is not guaranteed in these cases. The spectral weight can even be complex in the presence of broken time-reversal symmetry. Various workarounds, such as the neglect of vertex corrections or the study of the infinite frequency or Kelvin limits, have been proposed. Here, we show that one can define auxiliary response functions that allow one to extract the desired real-frequency susceptibilities from maximum entropy methods in the most general multiorbital cases with no particular symmetry. As a benchmark case, we study the longitudinal thermoelectric response and corresponding Onsager coefficient in the single-band two-dimensional Hubbard model treated with dynamical mean-field theory and continuous-time quantum Monte Carlo. We thereby extend the maximum entropy analytic continuation with auxiliary functions (MaxEntAux method), developed for the study of the superconducting pairing dynamics of correlated materials, to transport coefficients.

DOI
10.1103/physrevb.95.121104

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The computation of transport coefficients, even in linear response, is a major challenge for theoretical methods that rely on analytic continuation of correlation functions obtained numerically in Matsubara space. While maximum entropy methods can be used for certain correlation functions, this is not possible in general, important examples being the Seebeck, Hall, Nernst, and Reggi-Leduc coefficients. Indeed, positivity of the spectral weight on the positive real-frequency axis is not guaranteed in these cases. The spectral weight can even be complex in the presence of broken time-reversal symmetry. Various workarounds, such as the neglect of vertex corrections or the study of the infinite frequency or Kelvin limits, have been proposed. Here, we show that one can define auxiliary response functions that allow one to extract the desired real-frequency susceptibilities from maximum entropy methods in the most general multiorbital cases with no particular symmetry. As a benchmark case, we study the longitudinal thermoelectric response and corresponding Onsager coefficient in the single-band two-dimensional Hubbard model treated with dynamical mean-field theory and continuous-time quantum Monte Carlo. We thereby extend the maximum entropy analytic continuation with auxiliary functions (MaxEntAux method), developed for the study of the superconducting pairing dynamics of correlated materials, to transport coefficients.
Information about the pairing mechanism for superconductivity is contained in the spectral weight for the anomalous (Gorkov) Green's function. In the most general case, this spectral weight can change sign on the positive real axis or even be complex in the presence of broken time-reversal symmetry. This renders impossible the direct analytic continuation with maximum entropy methods of numerical results obtained for the anomalous Green's function either in Matsubara frequency or in imaginary time. Here we show that one can define auxiliary Green's functions that allow one to extract the anomalous spectral weight in the most general multiorbital spin-singlet or triplet cases with no particular symmetry. As a simple example, we treat the case of lead in Eliashberg theory: Maximum entropy analytic continuation with auxiliary Green's functions (MaxEntAux) allows us to recover spectral functions that change sign on the positive real axis. The approach can be extended to transport quantities.
This thesis is mainly a computational work studying the analytical continuation of Green’s functions using the Maximum Entropy method. A strongly correlated electron system is described with the single-band Hubbard model and paramagnetic solutions are studied using Dynamic Mean Field Theory on a Bethe lattice. Continuous Time Quantum Monte Carlo is used as Impurity solver, for the infinite Anderson model at a finite temperature, to obtain the Matsubara single-particle Green’s function propagator. Both metallic and insulating spectral functions are obtained using the Maximum Entropy Method. General properties of the Maximum Entropy Method as an analytic continuation method from imaginary to real time are also discussed.
Recently, Han and Heary [Phys. Rev. Lett. 99, 236808 (2007)] proposed an approach to steady-state quantum transport through mesoscopic structures, which maps the nonequilibrium problem onto a family of auxiliary quantum impurity systems subject to imaginary voltages. We employ continuous-time quantum Monte-Carlo solvers to calculate accurate imaginary time data for the auxiliary models. The spectral function is obtained from a maximum entropy analytical continuation in both Matsubara frequency and complexified voltage. To enable the analytical continuation we construct a kernel which is compatible with the analytical structure of the theory. While it remains a formidable task to extract reliable spectral functions from this unbiased procedure, particularly for large voltages, our results indicate that the method in principle yields results in agreement with those obtained by other methods.
We present a generalization of the maximum entropy method to the analytic continuation of matrix-valued Green's functions. To treat off-diagonal elements correctly based on Bayesian probability theory, the entropy term has to be extended for spectral functions that are possibly negative in some frequency ranges. In that way, all matrix elements of the Green's function matrix can be analytically continued; we introduce a computationally cheap element-wise method for this purpose. However, this method cannot ensure important constraints on the mathematical properties of the resulting spectral functions, namely positive semidefiniteness and Hermiticity. To improve on this, we present a full matrix formalism, where all matrix elements are treated simultaneously. We show the capabilities of these methods using insulating and metallic dynamical mean-field theory (DMFT) Green's functions as test cases. Finally, we apply the methods to realistic material calculations for LaTiO3{\mathrm{LaTiO}}_{3}LaTiO3​, where off-diagonal matrix elements in the Green's function appear due to the distorted crystal structure.
Diagrammatic expansions are a central tool for treating correlated electron systems. At thermal equilibrium, they are most naturally defined within the Matsubara formalism. However, extracting any dynamic response function from a Matsubara calculation ultimately requires the ill-defined analytical continuation from the imaginary- to the real-frequency domain. It was recently proposed [A. Taheridehkordi et al., Phys. Rev. B 99, 035120 (2019)] that the internal Matsubara summations of any interaction-expansion diagram can be performed analytically by using symbolic algebra algorithms. The result of the summations is then an analytical function of the complex frequency rather than Matsubara frequency. Here we apply this principle and develop a diagrammatic Monte Carlo technique which yields results directly on the real-frequency axis. We present results for the self-energy $\mathrm{\ensuremath{\Sigma}}(\ensuremath{\omega})$ of the doped $32\ifmmode\times\else\texttimes\fi{}32$ cyclic square-lattice Hubbard model in a nontrivial parameter regime, where signatures of the pseudogap appear close to the antinode. We discuss the behavior of the perturbation series on the real-frequency axis and in particular show that one must be very careful when using the maximum entropy method on truncated perturbation series. Our approach holds great promise for future application in cases when analytical continuation is difficult and moderate-order perturbation theory may be sufficient to converge the result.
Correlation functions are central to understanding the behavior of quantum systems, with their spectra giving direct access to quasiparticle and collective excitations. However, the long-time evolution required for sufficient spectral resolution is challenging to reach for both classical and quantum simulations. In this work, we introduce a principled approach for extending time series of quantum correlation functions and reconstructing their spectral densities, based on their connection with covariance sequences of stationary stochastic processes. This connection enables us to apply Burg's maximum-entropy principle to find the least-committal extension compatible with the measured values of the correlation function, which corresponds to an all-pole autoregressive model whose coefficients solve a set of Yule--Walker equations. The resulting extension is positive semi-definite by construction, guaranteeing the positivity of the spectrum and the automatic satisfaction of its total spectral weight sum rule. The approach applies to both scalar and matrix-valued correlation functions, including those with highly structured or continuous spectra. The generalization to the matrix case makes it possible to trade simulation time for additional measured correlation functions, enabling accurate time-series extension and spectral reconstruction from short-time data, even in the presence of noise.