[Paper Review] Progress on stochastic analytic continuation of quantum Monte Carlo data

Hui Shao, Anders W. Sandvik|arXiv (Cornell University)|Feb 20, 2022

Spectroscopy and Quantum Chemical Studies6 citations

TL;DR

This paper advances stochastic analytic continuation (SAC) for quantum Monte Carlo data by introducing entropy-based optimization and constrained sampling to resolve sharp spectral features—such as quasi-particle peaks and power-law edges—previously distorted by conventional methods. It demonstrates equivalence between SAC averages and maximum-entropy solutions in the large-Nω limit, enabling high-fidelity reconstruction of spectra with unprecedented resolution and accuracy.

ABSTRACT

We report multipronged progress on the stochastic averaging approach to numerical analytic continuation of quantum Monte Carlo data. With the sampled spectrum parametrized with delta-functions in continuous frequency space, a calculation of the configurational entropy lends support to a simple goodness-of-fit criterion for the optimal sampling temperature. To further investigate entropic effects, we compare spectra sampled in continuous frequency with results of amplitudes sampled on a fixed frequency grid. We demonstrate equivalences between sampling and optimizing spectral functions with the maximum-entropy approach with different forms of the entropy. These insights revise prevailing notions of the maximum-entropy method and its relationship to stochastic analytic continuation. We further explore various adjustable (optimized) constraints that allow sharp 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. We show 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. Tests with synthetic data as well as with real simulation data for the spin-1/2 Heisenberg chain demonstrate that constrained sampling methods can reproduce spectral functions with sharp edge features at unprecedented fidelity. We present new 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.

Motivation & Objective

• To overcome limitations in existing analytic continuation methods that distort sharp spectral features like quasi-particle peaks and power-law edges.
• To establish a rigorous connection between stochastic analytic continuation (SAC) and maximum-entropy (ME) methods through configurational entropy and large-Nω limits.
• To develop optimized constraints—such as peak location and spectral weight—that enhance resolution of low-temperature features and edge singularities.
• To enable accurate reconstruction of complex spectral functions, including those with divergent edges from deconfined spinons in Heisenberg chains and ladders.
• To propose SAC as a preprocessing tool for machine learning-based analytic continuation, enhancing spectral fidelity in quantum many-body systems.

Proposed method

• Parametrize the spectral function as a large number of δ-functions in continuous frequency space, enabling exact calculation of configurational entropy.
• Use a goodness-of-fit criterion based on χ² and entropy to determine the optimal sampling temperature Θ, avoiding reliance on ad hoc choices.
• Implement constrained sampling with adjustable parameters (e.g., peak position, edge location) optimized via entropy minimization under optimal fit conditions.
• Compare spectral parametrizations in continuous frequency space versus fixed frequency grids, showing equivalence in the generalized thermodynamic limit (large Nω).
• Introduce monotonicity and edge-optimization constraints to stabilize sampling and resolve sharp features such as power-law singularities.
• Apply the method to synthetic data and real QMC data from spin-1/2 Heisenberg chains and ladders, validating performance across diverse spectral structures.

Experimental results

Research questions

• RQ1 Can configurational entropy in SAC provide a principled criterion for selecting the optimal sampling temperature Θ?
• RQ2 How do different parametrizations (continuous vs. fixed-grid frequency) affect spectral reconstruction, and are they equivalent in the large-Nω limit?
• RQ3 Can constrained sampling with optimized parameters resolve sharp spectral edges and narrow quasi-particle peaks that are typically blurred by conventional methods?
• RQ4 What is the relationship between SAC averages and maximum-entropy solutions, and under what conditions are they equivalent?
• RQ5 Can SAC be used as a preprocessor to improve machine learning-based analytic continuation of QMC data?

Key findings

• The optimal sampling temperature Θ is determined by a balance between χ² fit quality and configurational entropy, providing a physically motivated alternative to heuristic choices.
• In the large-Nω limit, SAC averages converge to the maximum-entropy solution, with different parametrizations corresponding to different prior entropy forms.
• Constrained sampling with optimized peak and edge parameters successfully resolves both narrow quasi-particle peaks and broad features without artificial distortions.
• The method accurately reconstructs power-law singularities and divergent edges in the dynamic structure factor of the spin-1/2 Heisenberg chain, confirming the presence of deconfined spinon excitations.
• Spectral features such as sharp edges and low-energy peaks are preserved without spurious artifacts, unlike in standard analytic continuation methods.
• The approach is validated on 2-leg and 3-leg Heisenberg ladders, demonstrating resolution of both elementary and composite excitations.