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arXiv:2403.08642 · Physical review. B./Physical review. B

Reweight-annealing method for evaluating the partition function via quantum Monte Carlo calculations

Yi-Ming Ding, Junsong Sun, Nvsen Ma, Gaopei Pan, Chen Cheng, Zheng Yan

Published Oct 22, 2024• 13 citations
arXivHTMLPDF

Abstract

Efficient and accurate algorithms for partition function, free energy, and thermal entropy calculations is of great significance in statistical physics and quantum many-body physics. Here we present an unbiased but low-technical-barrier algorithm within the quantum Monte Carlo framework, which has exceptionally high accuracy and no systemic error. Compared with the conventional specific heat integral method and Wang-Landau sampling algorithm, our method can obtain a much more accurate result of the subleading coefficient of the entropy consistent with theory. This method can be widely used in both classical and quantum Monte Carlo simulations and is easy to be parallelized on a computer.

DOI
10.1103/physrevb.110.165152

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The concept of the partition function zeros is one of the most efficient methods for investigating the phase transitions and the critical phenomena in various physical systems. Estimating the partition function zeros requires information on the density of states Ω( E ) as a function of the energy E . Currently, the Wang-Landau Monte Carlo algorithm is one of the best methods for calculating Ω( E ). The partition function zeros in the complex temperature plane of the Ising model on an L × L square lattice ( L = 10 ~ 80) with a periodic boundary condition have been estimated by using the Wang-Landau Monte Carlo algorithm. The efficiency of the Wang-Landau Monte Carlo algorithm and the accuracies of the partition function zeros have been evaluated for three different, 5%, 10%, and 20%, flatness criteria for the histogram H ( E ).
We propose a Monte Carlo algorithm designed to simulate quantum as well as classical systems at equilibrium, bridging the algorithmic gap between quantum and classical thermal simulation algorithms. The method is based on a decomposition of the quantum partition function that can be viewed as a series expansion about its classical part. We argue that the algorithm not only provides a theoretical advancement in the field of quantum Monte Carlo simulations, but is optimally suited to tackle quantum many-body systems that exhibit a range of behaviors from "fully quantum" to "fully classical," in contrast to many existing methods. We demonstrate the advantages, sometimes by orders of magnitude, of the technique by comparing it against existing state-of-the-art schemes such as path integral quantum Monte Carlo and stochastic series expansion. We also illustrate how our method allows for the unification of quantum and classical thermal parallel tempering techniques into a single algorithm and discuss its practical significance.
Phase transitions and critical phenomena are the most universal phenomena in nature.To understand the phase transitions and critical phenomena of a given system as a continuous function of temperature and to obtain the partition function zeros in the complex temperature plane indicating most effectively phase transitions and critical phenomena, we need to calculate the density of states.Currently, Wang-Landau Monte Carlo algorithm is one of the most efficient Monte Carlo methods to calculate the approximate density of states.Using Wang-Landau Monte Carlo algorithm, the density of states for the Ising model on L x L square lattices (L = 4 ~ 32) with periodic boundary conditions is obtained, and the partition function zeros of the Ising model are evaluated in the complex temperature plane.By examining the behavior of the first partition function zero (partition function zero closest to the positive real axis), phase transitions and critical phenomena can be much more accurately analyzed.The approximate first zeros of the Ising ferromagnet, obtained from Wang-Landau algorithm, are quite close to the exact ones, indicating that it is a reliable method for calculating the density of states and the first partition function zeros.
Entanglement entropy has been a powerful tool for analyzing phases and criticality in pure ground states via quantum Monte Carlo (QMC). However, mixed-state entanglement, relevant to systems with dissipation, finite temperature, and disjoint regions, remains less explored due to the lack of efficient numerical methods. In this work, we present a practical and easy-to-implement QMC method within the reweight-annealing framework, enabling efficient computation of the entanglement R\'enyi negativity by tracking its variation along given parameter paths. This method is scalable, parallelizable, and well suited for high-dimensional and large-scale simulations. Applying it to diverse scenarios, including one- and two-dimensional systems, ground and thermal states, and bipartite and tripartite partitions, not only is the information of the underlying conformal field theory achieved, but the role of entanglement in quantum and thermal phase transitions is revealed.
We implement a Wang-Landau sampling technique in quantum Monte Carlo (QMC) simulations for the purpose of calculating the Rényi entanglement entropies and associated mutual information. The algorithm converges an estimate for an analog to the density of states for stochastic series expansion QMC, allowing a direct calculation of Rényi entropies without explicit thermodynamic integration. We benchmark results for the mutual information on two-dimensional (2D) isotropic and anisotropic Heisenberg models, a 2D transverse field Ising model, and a three-dimensional Heisenberg model, confirming a critical scaling of the mutual information in cases with a finite-temperature transition. We discuss the benefits and limitations of broad sampling techniques compared to standard importance sampling methods.