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

Abstract

The computation of transport coefficients, even in linear response, is a major challenge for theoretical methods that rely on analytic continuation of correlations 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 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 (DMFT) and continuous-time quantum Monte Carlo (CTQMC). We thereby extend to transport coefficients the maximum entropy analytic continuation with auxiliary functions (MaxEntAux method), developed for the study of the superconducting pairing dynamics of correlated materials. PACS numbers: 71.27.+a, 72.10.-d, 72.10.Bg, 72.15.Jf, 72.20.Pa a. Introduction. Transport properties are of interest for both fundamental and applied purposes. For example, while thermoelectric power tells us about the nature of charge carriers, materials with large thermoelectric power could lead to various applications, including efficient conversion of heat loss into useful electricity [1 -7]. Unfortunately, computing transport coefficients from numerical results is no simple task. Usually, one starts by computing the corresponding response functions in Matsubara frequency using the Kubo formula. For quantum Monte-Carlo data in particular, the most direct way to extract the real-frequency dependent response functions is then to perform maximum entropy analytic continuations (MEACs) [8,9]. However, as we explain in more details below, MEAC is not always trivial since it requires that the spectral weight of response functions is real and positive, which is not necessarily the case in general.Many approaches have been investigated to circumvent this major problem for the Seebeck coefficient [10][11][12][13][14][15][16][17][18][19], the Hall coefficient [20-24] and the Nernst coefficient [25] for instance, but all of them are either approximations or analytic methods that are exact only in a certain frequency limit [26]. The most common approach consists in neglecting vertex corrections, in which case it is possible to compute transport coefficients directly from the single-particle spectral weight. This is not possible when vertex corrections are included, which seems to be a necessary step in...

Analytical continuation of matrix-valued functions: Carathéodory formalism

“…It is common practice to perform continuations only of the diagonal parts of the self-energy or cumulant, truncating all off-diagonal contributions [37][38][39][40]. In systems where the self-energy can be chosen to be diagonal for all frequencies due to symmetry, such as in rotationally invariant three-orbital systems [68][69][70] and certain model systems, this is exact.…”

“…Unless symmetry dictates that all off-diagonal elements for all frequencies must be zero, this is an uncontrolled approximation and results will depend on the basis chosen. Finally, one may generalize the maximum entropy method to matrix-valued functions and off-diagonal terms [37][38][39][40], using either a positive-negative or a maximum 'quantum' entropy approach. These methods enforce the positive semidefiniteness of the Green's function but eliminate sharp and high-energy features, such as the band structure contained in Green's function data, due to the intrinsic limitations of the fitting procedure.…”

Numerical approaches for calculating the low-field dc Hall coefficient of the doped Hubbard model

“…It remains a challenge to perform analytic continuation directly from Matsubara frequencies or imaginary time data. A promising approach would be to use techniques designed to treat non-positive-definite spectra [51,52], or other methods of analytic continuation [53][54][55][56]. If a reliable method of analytic continuation can be found, then our evaluation of the exact three-current linear-response χ ZF xy through the numerically exact and unbiased DQMC algorithm will allow us to find the exact σ xy (ω) spectra for all frequencies for the Hubbard model.…”

Robust analytic continuation of Green's functions via projection, pole estimation, and semidefinite relaxation

“…Furthermore, most methods have been developed only for diagonal entries of the Green's function. The analytic continuation of off-diagonal entries of the Green's function [16][17][18][19][20] is often attempted by diagonalizing the Green's function at a certain Matsubara frequency, conducting the analytic continuation only of the diagonal entries with respect to the transformed basis at other frequencies, and neglecting the remaining non-diagonal entries.…”