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arXiv:1705.08838 · Physical review. B./Physical review. B

Maximum entropy formalism for the analytic continuation of matrix-valued Green's functions

Gernot J. Kraberger, Robert Triebl, Manuel Zingl, Markus Aichhorn

Published Oct 17, 2017• 109 citations
arXivHTMLPDF

Abstract

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 ${\mathrm{LaTiO}}_{3}$, where off-diagonal matrix elements in the Green's function appear due to the distorted crystal structure.

DOI
10.1103/physrevb.96.155128

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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.
A maximum entropy method for analytic continuation is extended by introducing quantum relative entropy. This method is formulated in terms of matrix-valued functions and therefore invariant under arbitrary unitary transformation of input matrix. As a result, the continuation of off-diagonal elements becomes straightforward. Without introducing any further ambiguity, the Bayesian probabilistic interpretation is maintained just as in the conventional maximum entropy method. The applications of our generalized formalism to a model spectrum and a real material demonstrate its usefulness and superiority.
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.
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.
Finite-temperature quantum field theories are formulated in terms of Green's functions and self-energies on the Matsubara axis. In multiorbital systems, these quantities are related to positive semidefinite matrix-valued functions of the Carath\'eodory and Schur class. Analysis, interpretation, and evaluation of derived quantities such as real-frequency response functions requires analytic continuation of the off-diagonal elements to the real axis. We derive the criteria under which such functions exist for given Matsubara data and present an interpolation algorithm that intrinsically respects their mathematical properties. For small systems with precise Matsubara data, we find that the continuation exactly recovers all off-diagonal and diagonal elements. In real-materials systems, we show that the precision of the continuation is sufficient for the analytic continuation to commute with the Dyson equation, and we show that the truncation of the off-diagonal self-energy elements leads to considerable approximation artifacts.
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.
In this paper, we propose an analytic continuation method to extract real-frequency spectral functions from imaginary-frequency Green's functions of quantum many-body systems. This method is based on the pole representation of Matsubara Green's function and a stochastic sampling procedure is utilized to optimize the amplitudes and locations of poles. In order to capture narrow peaks and sharp band edges in the spectral functions, a constrained sampling algorithm and a self-adaptive sampling algorithm are developed. To demonstrate the usefulness and performance of this method, we at first apply it to study the spectral functions of representative fermionic and bosonic correlators. Then we employ this method to tackle the analytic continuation problems of matrix-valued Green's functions. The synthetic Green's functions, as well as realistic correlation functions from finite-temperature quantum many-body calculations, are used as input. The benchmark results demonstrate that this method is capable of reproducing most of the key characteristics in the spectral functions. The sharp, smooth, and multipeak features in both low- and high-frequency regions of spectral functions could be accurately resolved, which overcomes one of the main limitations of the traditional maximum entropy method. More importantly, it exhibits excellent robustness with respect to noisy and incomplete input data. The causality of spectral function is always satisfied even in the presence of sizable noises. As a by-product, this method could derive a fitting formula for the Matsubara data, which provides a compact approximation to the many-body Green's functions. Hence, we expect that this method could become a pivotal workhorse for numerically analytic continuation and be broadly useful in many applications.