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arXiv:2306.09366

Loop ensembles in Stochastic Series Expansion of Two-Dimensional Heisenberg Antiferromagnets

Vedant Motamarri

Published Jun 14, 2023• 0 citations
arXivHTMLPDF

Abstract

The Stochastic Series Expansion (SSE) method along with resummation over the spin or flavor values maps the partition function of a quantum antiferromagnet to a closely-packed loop gas model in one higher dimension. Earlier work by Nahum et al. has shown that certain closely-packed three-dimensional loop gas models exhibit phases dominated by macroscopic loops, wherein the corresponding joint distribution of loop lengths is Poisson-Dirichlet. On grounds of universality, the same is expected of the ensemble of loops obtained in (2+1)-dimensional quantum antiferromagnets, albeit the loops emerge from a different microscopic origin. We sample the SSE loop ensemble for SU(N) antiferromagnets on a square lattice using Monte Carlo and study how the joint distribution varies with the degree of representation N and inverse temperature $β$. We observe that, for low temperatures and small N($\leq$ 4), the distribution indeed shows characteristics of Poisson-Dirichlet behaviour when antiferromagnetic correlations dominate the system.

DOI
10.48550/arxiv.2306.09366

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The Stochastic Series Expansion (SSE) method along with resummation over the spin or flavor values maps the partition function of a quantum antiferromagnet to a closely-packed loop gas model in one higher dimension. Earlier work by Nahum et al. has shown that certain closely-packed three-dimensional loop gas models exhibit phases dominated by macroscopic loops, wherein the corresponding joint distribution of loop lengths is Poisson-Dirichlet. On grounds of universality, the same is expected of the ensemble of loops obtained in (2+1)-dimensional quantum antiferromagnets, albeit the loops emerge from a different microscopic origin. We sample the SSE loop ensemble for SU(N) antiferromagnets on a square lattice using Monte Carlo and study how the joint distribution varies with the degree of representation N and inverse temperature βββ. We observe that, for low temperatures and small N(≤\leq≤ 4), the distribution indeed shows characteristics of Poisson-Dirichlet behaviour when antiferromagnetic correlations dominate the system.
For spin rotational symmetric models with a positive-definite high-temperature expansion of the partition function, a stochastic sampling of the series expansion upon partial resummation becomes logically equivalent to sampling an uncolored closely packed loop-gas model in one higher dimension. Based on this, we devise quantum Monte Carlo updates that importance-sample loop configurations for general SU(N)SU(N)SU(N) in fundamental and higher-symmetric representations. The algorithmic performance systematically improves with an increase in (continuous) NNN, allowing the efficient simulation of quantum paramagnets. The underlying reason for the increased efficacy is the correspondence of quantum paramagnetic phases such as valence bond solids to short-loop phases on the loop-gas side rather than the particular value of NNN. This also gives a connection between Sandvik's JQJQJQ model class and classical loop-gas models in the deconfined universality class.
The stochastic series expansion (SSE) algorithm is one of the most powerful quantum Monte Carlo methods and has been extensively applied to the study of quantum many body systems. Its efficiency is particularly enhanced with a deterministic loop update scheme in the study of the S=1/2 quantum spin systems that preserve SU(2) spin rotational symmetry. Once the symmetry is broken, such as by an external field, a directed loop method is typically required, resulting in a significant reduction in efficiency. Inspired by the SSE approach developed for the quantum Ising model, we introduce a deterministic loop SSE method that is particularly suited for antiferromagnetic systems under a staggered magnetic field. This method enables separate investigations of longitudinal and transverse modes in magnetically ordered phases arising from spontaneous symmetry breaking. We benchmark the performance of our algorithm against the standard directed loop approach applied to the antiferromagnetic Heisenberg chain and demonstrate that our method substantially reduces CPU time per Monte Carlo step, thereby can outperform the directed loop algorithm in efficiency.
We investigate the Stochastic Series Expansion (SSE) technique, guided by Sandvik's algorithm of the quantum Monte Carlo simulation for spin-1/2 antiferromagnetic Heisenberg linear chain. The impact of the Monte Carlo steps, lattice sizes, and various temperatures on the variability and fidelity of simulation outcomes are studied. The findings show that the Monte Carlo steps where measurements are made (Msteps), and lattice size are crucial for stabilizing the results. Our results show that for larger lattice sizes, such as L=1024, achieving reliable outcomes necessitates extensive Monte Carlo simulations, specifically. The convergence in the properties is achieved with a high number of Monte Carlo steps starting from 7×10^4. Strong fluctuations are calculated between the results for each magnetic property and the existence of negative and positive in the specific heat at low temperatures.
Background: Antiferromagnetic systems exhibit complex quantum behaviours that require accurate numerical methods to analyse. The Stochastic Series Expansion (SSE) is a quantum Monte Carlo technique that simulates these systems. This study examines the effect of the number of bins on the accuracy of physical property measurements using SSE. Materials and Methods: The SSE simulations were performed using Fortran 90 on a Ryzen 7 processor. The system was initialised with varying lattice sizes (64, 128, 256, and 1024) and dimensionless temperatures (1/32, 1/16, 1/2, and 4) to analyse different configurations. Nbins and Monte Carlo steps were adjusted systematically to investigate their impact on the results. Results were visualised in Origin Lab. Results: Increasing the number of bins (Nbins) reduced fluctuations, leading to reliable results, especially at low temperatures. Lower temperatures lead to higher fluctuations in susceptibility and specific heat of the system. The Neel temperatures were observed around dimensionless temperature. Indicating phase transitions. Conclusions: The study shows that Nbins are crucial for accurate results in SSE simulations. Although the method is effective for most properties, specific heat calculations require higher computational costs and present limitations in precision.
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The Antiferromagnetic Heisenberg Model with spin 1/2 on a square lattice was simulated by using the Stochastic Series Expansion(SSE) Quantum Monte Carlo Method.The temperature dependence of the internal energy,specific heat and uniform susceptibility was observed.The results showed that specific heat has a maximum at kT/J=0.6 and uniform susceptibility saturates at kT/J=1 for the isotropy case.And the size of lattice also affects the temperature dependence of these thermodynamic quantities.For the anisotropy case,energy decreases with anisotropic parameter g increasing.Susceptibility changes slowly with temperature for g1 and exponentially decay for g1 in the low temperature region and its behaviors approach consistent for different g in the high temperature region.