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Physical Review Letters

Monte Carlo Simulation of Quantum Spin Systems

Sudip Chakravarty, Douglas B. Stein

Published Aug 23, 1982• 35 citations
Publisher

Abstract

A Monte Carlo method applicable to a class of noncommuting spin variables is developed. The technique relies on a stochastic procedure which simulates the dominant vacuum-to-vacuum graphs of quantum many-body theory. Computational speed is only weakly dependent on the dimensionality of the system and is fast enough to handle large lattices. This method is tested on the $S=\frac{1}{2}$ ferromagnetic Heisenberg model in one, two, and three dimensions and on the $S=\frac{1}{2}$ $\mathrm{XY}$ model in one dimension where exact results are known.

DOI
10.1103/physrevlett.49.582

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AIP Advances•May 1, 2017•1 citation
A Monte Carlo method applicable to a class of noncommuting spin variables is developed. The technique relies on a stochastic procedure which simulates the dominant vacuum-to-vacuum graphs of quantum many-body theory. Computational speed is only weakly dependent on the dimensionality of the system and is fast enough to handle large lattices. This method is tested on the S=12S=\frac{1}{2}S=21​ ferromagnetic Heisenberg model in one, two, and three dimensions and on the S=12S=\frac{1}{2}S=21​ XY\mathrm{XY}XY model in one dimension where exact results are known.
A simple Monte Carlo method1) which was shown to be applicable to finite spin-systems of rather large spin size ragardless of dimensionality of the system is applied to the spin-1/2 isotropic Heisenberg ferromagnetic system on the square lattice of spin size 10 × 10, 20 × 20 and 30 × 30. Rather accurate values are obtained for thermodynamic quantities for temperature down to k B T / J =0.01. The accuracy and efficiency of the MC method are discussed in detail in comparison with the other works.
These lecture notes introduce quantum spin systems and several computational methods for studying their ground-state and finite-temperature properties. Symmetry-breaking and critical phenomena are first discussed in the simpler setting of Monte Carlo studies of classical spin systems, to illustrate finite-size scaling at continuous and first-order phase transitions. Exact diagonalization and quantum Monte Carlo (stochastic series expansion) algorithms and their computer implementations are then discussed in detail. Applications of the methods are illustrated by results for some of the most essential models in quantum magnetism, such as the S=1/2 Heisenberg antiferromagnet in one and two dimensions, as well as extended models useful for studying quantum phase transitions between antiferromagnetic and magnetically disordered states.
We present a new algorithm for world line Monte Carlo simulations of quantum spin systems. The algorithm consists of a stochastic pairing and a stochastic flipping procedure, and the formula for the pairing probability is explicitly given for Heisenberg models. As an illustration of the algorithm, we calculated autocorrelation times for simulations of the S=1S=1S=1 antiferromagnetic Heisenberg chain. The rapid increase in autocorrelation times which is seen in the conventional world line method as the imaginary-time spacing decreases is eliminated by the present algorithm.
We use a recently developed quantum simulation approach to study the properties of a three-dimensional Ising model consisting of S = 1/2 quantum spins localized at the sites of a simple cubic lattice. We assume nearest-neighbor interaction between spins with an exchange interaction that can be either ferromagnetic or antiferromagnetic. It is found that the computational method quickly converges towards the expected equilibrium spin configurations. The resulting spontaneous magnetization curves corresponding to the two types of magnetic interactions under consideration were found to be almost identical to the ones obtained via quantum mean field theory at all temperatures. The derived total energies, total free energies, magnetic entropies and specific heats per mole of spins show no sizeable differences from known theoretical values. Furthermore, the results of the simulations for two different 3D Ising systems containing 4×4×4 and 20×20×20 spins localized at the sites of a simple cubic lattice were found to be almost identical to each other. This finding suggests that the self-consistent algorithm approach of the current simulation method allows one to obtain the physical bulk properties of a large magnetic system by relying on simulations of a much smaller spin system sample. Therefore, the method presently considered appears to be not only very accurate as gauged by comparison to mean field theory, but also able to greatly increase the speed of simulations.