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Stochastische Analytische Fortsetzung: Ein Bayesscher Ansatz

Khaldoon Ghanem

Published Jan 1, 2017• 5 citations
Publisher

Abstract

The stochastic sampling method (StochS) is used for the analytic continuation of quantum Monte Carlo data from the imaginary axis to the real axis. Compared to the maximum entropy method, StochS does not have explicit parameters, and one would expect the results to be unbiased. We present a very efficient algorithm for performing StochS and use it to study the effect of the discretization grid. Surprisingly, we find that the grid affects the results of StochS acting as an implicit default model. We provide a recipe for choosing a reliable StochS grid.To reduce the effect of the grid, we extend StochS into a gridless method (gStochS) by sampling the grid points from a default model instead of having them fixed. The effect of the default model is much reduced in gStochS compared to StochS and depends mainly on its width rather than its shape. The proper width can then be chosen using a simple recipe like we did in StochS.Finally, to avoid fixing the width, we go one step further and extend gStochS to sample over a whole class of default models with different widths. The extended method (eStochS) is then able to automatically relocate the grid points and concentrate them in the important region. Test cases show that eStochS gives good results resolving sharp features in the spectrum without the need for fine tuning a default model.

DOI
10.18154/rwth-2017-06704

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The stochastic sampling method (StochS) is used for the analytic continuation of quantum Monte Carlo data from the imaginary axis to the real axis. Compared to the maximum entropy method, StochS does not have explicit parameters, and one would expect the results to be unbiased. We present a very efficient algorithm for performing StochS and use it to study the effect of the discretization grid. Surprisingly, we find that the grid affects the results of StochS acting as an implicit default model. We provide a recipe for choosing a reliable StochS grid.To reduce the effect of the grid, we extend StochS into a gridless method (gStochS) by sampling the grid points from a default model instead of having them fixed. The effect of the default model is much reduced in gStochS compared to StochS and depends mainly on its width rather than its shape. The proper width can then be chosen using a simple recipe like we did in StochS.Finally, to avoid fixing the width, we go one step further and extend gStochS to sample over a whole class of default models with different widths. The extended method (eStochS) is then able to automatically relocate the grid points and concentrate them in the important region. Test cases show that eStochS gives good results resolving sharp features in the spectrum without the need for fine tuning a default model.
Analytic continuation of imaginary time or frequency data to the real axis is a crucial step in extracting dynamical properties from quantum Monte Carlo simulations. The average spectrum method provides an elegant solution by integrating over all nonnegative spectra weighted by how well they fit the data. In a recent paper, we found that discretizing the functional integral, as in Feynman's path-integrals, does not have a well-defined continuum limit. Instead, the limit depends on the discretization grid whose choice may strongly bias the results. In this paper, we demonstrate that sampling the grid points, instead of keeping them fixed, also changes the functional integral limit and rather helps to overcome the bias considerably. We provide an efficient algorithm for doing the sampling and show how the density of the grid points acts now as a default model with a significantly reduced biasing effect. The remaining bias depends mainly on the width of the grid density, so we go one step further and average also over densities of different widths. For a certain class of densities, including Gaussian and exponential ones, this width averaging can be done analytically, eliminating the need to specify this parameter without introducing any computational overhead.
We report multipronged progress on the stochastic averaging approach to numerical analytic continuation of imaginary-time correlation functions computed by quantum Monte Carlo simulations. After reviewing the conventional maximum-entropy approach and established stochastic analytic continuation methods, we present several new developments in which the configurational entropy of the sampled spectrum plays a key role. Parametrizing the spectrum as a large number of δ-functions in continuous frequency space, an exact calculation of the entropy lends support to a simple goodness-of-fit criterion for the optimal sampling temperature. We also compare spectra sampled in continuous frequency with those from amplitudes sampled on a fixed frequency grid. Insights into the functional form of the entropy in different cases allow us to demonstrate equivalence in a generalized thermodynamic limit (large number of degrees of freedom) of the average spectrum and the maximum-entropy solution, with different parametrizations corresponding to different forms of the entropy in the prior probability. These results revise prevailing notions of the maximum-entropy method and its relationship to stochastic analytic continuation. In further developments of the sampling approach, we explore various adjustable (optimized) constraints that allow sharp low-temperature spectral features to be resolved, in particular at the lower frequency edge. The constraints, e.g., the location of the edge or the spectral weight of a quasi-particle peak, are optimized using a statistical criterion based on entropy minimization under the condition of optimal fit. We show with several examples that this method can correctly reproduce both narrow and broad quasi-particle peaks. We next introduce a parametrization for more intricate spectral functions with sharp edges, e.g., power-law singularities. We present tests with synthetic data as well as with real simulation data for the spin-1/2 Heisenberg chain, where a divergent edge of the dynamic structure factor is due to deconfined spinon excitations. Our results demonstrate that distortions of sharp edges or quasi-particle peaks, which arise with other analytic continuation methods, propagate and cause artificial spectral features also at higher energies. The constrained sampling methods overcome this problem and allow analytic continuation of spectra with sharp edge features at unprecedented fidelity. We present results for S=1/2 Heisenberg 2-leg and 3-leg ladders to illustrate the ability of the methods to resolve spectral features arising from both elementary and composite excitations. Finally, we also propose how the methods developed here could be used as “pre processors” for analytic continuation by machine learning. Edge singularities and narrow quasi-particle peaks being ubiquitous in quantum many-body systems, we expect the new methods to be broadly useful and take numerical analytic continuation to a new quantitative level in many applications.
The analytic continuation of imaginary-time quantum Monte Carlo data to extract real-frequency spectra remains a key problem in connecting theory with experiment. Here we present a fast and efficient stochastic optimization method (FESOM) as a more accessible variant of the stochastic optimization method introduced by Mishchenko et al. [Phys. Rev. B 62, 6317 (2000)], and we benchmark the resulting spectra with those obtained by the standard maximum entropy method for three representative test cases, including data taken from studies of the two-dimensional Hubbard model. We generally find that our FESOM approach yields spectra similar to the maximum entropy results. In particular, while the maximum entropy method yields superior results when the quality of the data is strong, we find that FESOM is able to resolve fine structure with more detail when the quality of the data is poor. In addition, because of its stochastic nature, the method provides detailed information on the frequency-dependent uncertainty of the resulting spectra, while the maximum entropy method does so only for the spectral weight integrated over a finite frequency region. We therefore believe that this variant of the stochastic optimization approach provides a viable alternative to the routinely used maximum entropy method, especially for data of poor quality.
A method for analytic continuation of quantum Monte Carlo data is presented. The spectrum $A(\ensuremath{\omega})$ is parametrized as a sum of $\ensuremath{\delta}$ functions, the weights Ai{A}_{i}Ai​ of which are sampled according to a distribution $p(A)\ensuremath{\sim}\mathrm{exp}(\ensuremath{-}{\ensuremath{\chi}}^{2}/\ensuremath{\Theta})$. It is argued that the calculated entropy SSS provides a criterion for determining the $\ensuremath{\Theta}$ corresponding to the ``best'' averaged spectrum. The appearance of spurious structure is signaled by a sharp drop in $〈S(\ensuremath{\Theta})〉$, which in test cases is preceded by a local maximum. Results for the dynamic spin structure factor of a 161616-site Heisenberg chain obtained at this maximum are in better agreement with exact results than ``classic'' maximum-entropy results.
A method for analytic continuation of imaginary-time correlation functions (here obtained in quantum Monte Carlo simulations) to real-frequency spectral functions is proposed. Stochastically sampling a spectrum parametrized by a large number of δ functions, treated as a statistical-mechanics problem, it avoids distortions caused by (as demonstrated here) configurational entropy in previous sampling methods. The key development is the suppression of entropy by constraining the spectral weight to within identifiable optimal bounds and imposing a set number of peaks. As a test case, the dynamic structure factor of the S=1/2 Heisenberg chain is computed. Very good agreement is found with Bethe ansatz results in the ground state (including a sharp edge) and with exact diagonalization of small systems at elevated temperatures.
We present an algorithm for the analytic continuation of imaginary-time quantum Monte Carlo data which is strictly based on principles of Bayesian statistical inference. Within this framework we are able to obtain an explicit expression for the calculation of a weighted average over possible energy spectra, which can be evaluated by standard Monte Carlo simulations, yielding as by-product also the distribution function as function of the regularization parameter. Our algorithm thus avoids the usual ad hoc assumptions introduced in similar algorithms to fix the regularization parameter. We apply the algorithm to imaginary-time quantum Monte Carlo data and compare the resulting energy spectra with those from a standard maximum-entropy calculation.