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Progress of Theoretical Physics

Monte Carlo Simulation for Quantum Spin Systems

Shigeo Homma, H. Matsuda, N. Ogita

Published Dec 1, 1984• 15 citations
Publisher

Abstract

A Simple Monte Carlo simulation is proposed, which we call decoupled cell method, for equilibrium states of quantum spin systems. In one-dimensional XY-model, the result of this simulation is found to give a good approximation for the internal energy at any temperature. A discussion is given of a possibile range of applicability of this method.

DOI
10.1143/ptp.72.1245

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Progress of Theoretical Physics•Nov 1, 1977•271 citations
A Simple Monte Carlo simulation is proposed, which we call decoupled cell method, for equilibrium states of quantum spin systems. In one-dimensional XY-model, the result of this simulation is found to give a good approximation for the internal energy at any temperature. A discussion is given of a possibile range of applicability of this method.
Decoupled cell method is introduced as a method of Monte Carlo simulation for quantum spin systems. Based on this method we calculated the thermodynamic quantities such as an internal energy, in-plane and out-of-plane susceptibilities and spin-spin correlation functions in one dimensional XY model. The agreement with the exact results is excellent above a certain demarcation temperature. This method is also applied to the antiferromagnetic Heisenberg model on the triangular lattice. The results indicate the existence of a sublattice structure composed of three sublattices, although its spatial and temporal behaviours are short ranged.
A new efficient scheme for the finite-size-scaling study of ground-state phase transitions in quantum spin systems is proposed on the basis of the quantum Monte Carlo method. That is, Trotter numbers and simulated temperatures are scaled in accordance with system sizes, and only one simulation is required for each system size. The transition point is estimated independently of the critical exponents using the non-equilibrium relaxation of the order parameter in a very large system. This scheme is confirmed by the estimation of the critical phenomenon of the one-dimensional S =1/2 asymmetric XY model.
The three-dimensional spin-1/2 Heisenberg model on a simple cubic lattice is studied for ferromagnetic and antiferromagnetic interactions using the Decoupled Cell Method for quantum Monte Carlo. Results for the relaxation time τL are determined for both ferromagnetic and antiferromagnetic systems and found to be similar to those found for the classical (s → ∞) Heisenberg model. The scaling of τL with system size is used to extract the dynamical critical exponent z for the two systems. The values of z = 1.98 ± 0.12 for the ferromagnet and z = 1.94 ± 0.09 for the antiferromagnet are in good agreement with theoretical predictions and previous Monte Carlo studies of the classical Heisenberg model.
A general explicit formulation of Monte Carlo simulation for quantum systems is given in this paper on the basis of the previous fundamental proposal by Suzuki. This paper also demonstrates explicitly the possibility of it and gives new interesting physical results on the two-dimensional XY-model. That is, the present preliminary simulation seems to indicate a phase transition with a divergent susceptibility, and a very weak singularity of specific heat if it exists, and without long-range order.