We present a method for performing analytical continuation of spectral data from imaginary time to real frequencies based on a statistical sampling method. Compared with the maximum entropy method (MEM), an advantage is that no default model needs to be introduced. For the problems studied here, the statistical sampling method gives comparable or slightly better results than MEM using quite accurate default models.
We present a method for performing analytical continuation of spectral data from imaginary time to real frequencies based on a statistical sampling method. Compared with the maximum entropy method (MEM), an advantage is that no default model needs to be introduced. For the problems studied here, the statistical sampling method gives comparable or slightly better results than MEM using quite accurate default models.
We compare different methods for performing analytical continuation of spectral data from the imaginary time or frequency axis to the real frequency axis for the optical conductivity $\ensuremath{\sigma}(\ensuremath{\omega})$. We compare the maximum entropy (MaxEnt), singular value decomposition (SVD), sampling, and Pad\'e methods for analytical continuation. We also study two direct methods for obtaining $\ensuremath{\sigma}(0)$. For the MaxEnt approach we focus on a recent modification. The data are split up in batches, a separate MaxEnt calculation is done for each batch and the results are averaged. For the problems studied here, we find that typically the SVD, sampling, and modified MaxEnt methods give comparable accuracy while the Pad\'e approximation is usually less reliable.
We study the maximum entropy (MaxEnt) approach for analytical continuation of spectral data from imaginary times to real frequencies. The total error is divided in a statistical error, due to the noise in the input data, and a systematic error, due to deviations of the default function, used in the MaxEnt approach, from the exact spectrum. We find that the MaxEnt approach in its classical formulation can lead to a nonoptimal balance between the two types of errors, leading to an unnecessary large statistical error. The statistical error can be reduced by splitting up the data in several batches, performing a MaxEnt calculation for each batch and averaging. This can outweigh an increase in the systematic error resulting from this approach. The output from the MaxEnt calculation can be used as a default function for a new MaxEnt calculation. Such iterations often lead to worse results due to an increase in the statistical error. By splitting up the data in batches, the statistical error is reduced and the increase resulting from iterations can be outweighed by a decrease in the systematic error. Finally we consider a linearized version to obtain a better understanding of the method.
The average spectrum method is a promising approach for the analytic continuation of imaginary time or frequency data to the real axis. It determines the analytic continuation of noisy data from a functional average over all admissible spectral functions, weighted by how well they fit the data. Its main advantage is the apparent lack of adjustable parameters and smoothness constraints, using instead the information on the statistical noise in the data. Its main disadvantage is the enormous computational cost of performing the functional integral. Here we introduce an efficient implementation, based on the singular value decomposition of the integral kernel, eliminating this problem. It allows us to analyze the behavior of the average spectrum method in detail. We find that the discretization of the real-frequency grid, on which the spectral function is represented, biases the results. The distribution of the grid points plays the role of a default model while the number of grid points acts as a regularization parameter. We give a quantitative explanation for this behavior, point out the crucial role of the default model and provide a practical method for choosing it, making the average spectrum method a reliable and efficient technique for analytic continuation.
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.
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.
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.