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Progress of Theoretical Physics

A New Monte Carlo Approach to Quantum Spin Systems. II

Shinji Kadowaki, Akira Ueda

Published Sep 1, 1989• 2 citations
Publisher

Abstract

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.

DOI
10.1143/ptp.82.493

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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.
A simple Monte Carlo high-temperature expansion method to evaluate the partition function of quantum spin systems which was proposed in the previous paper l) is improved so that all of the needed expansion-coefficients are obtained from interpolating points of less than ten with an interpolation formula. By this improvement, this method is applicable to the finite spin-systems of rather large spin size regardless of dimensionality of the system. As an example, the method is applied to the one-dimensional spin-l/2 isotropic Heisenberg ferromagnetic system.
The Monte Carlo method provides accurate values for the thermal and magnetic properties of magnetic systems of localized moments. The basic ‘importance sampling’ technique is described and a critique of the method is presented. The relative accuracy of the method is demonstrated by direct comparison with both exact results for the Ising square lattice and with other approximate theories. The wide applicability of the Monte Carlo approach is shown through the presentation of results of general thermodynamic properties as well as both critical and tricritical phenomena in systems with diverse Hamiltonians. In addition, the method has been applied to systems which do not undergo cooperative transitions and to some complex models of several real magnetic systems. Recent work involving investigations of the dynamic behavior of the Ising model is also discussed.
The static critical properties of classical, isotropic Heisenberg ferromagnets have been studied numerically. Using an improved cluster algorithm we performed extensive Monte Carlo simulations on L×L×L simple cubic and body-centered cubic systems with L≤40 over a temperature range covering the critical region. Temperature dependences of thermodynamic quantities were determined with an optimized multiple-histogram method, and both critical temperatures and critical exponents were extracted from finite-size scaling of several different thermodynamic quantities. It was found that within their respective error bars the critical exponents for both ferromagnets agree with each other and with field theoretical predictions.
Thermodynamic properties of the spin-1/2 Heisenberg ferromagnet are calculated by using Handscomb's Monte Carlo method. Several methods of analyses are used to determine critical properties of the model. For a square lattice we find that the susceptibility diverges exponentially at zero temperature. That is, \ensuremath{\chi}\ensuremath{\sim}exp[b(J/kB{\mathit{k}}_{\mathit{B}}kB​T)] with the constant b=4.5\ifmmode\pm\else\textpm\fi{}0.5, which is lower than the prediction (b=2\ensuremath{\pi}) of a modified spin-wave theory. For the simple-cubic lattice, we find that the critical temperature kB{\mathit{k}}_{\mathit{B}}kB​Tc{\mathit{T}}_{\mathit{c}}Tc​/J=1.68\ifmmode\pm\else\textpm\fi{}0.01, and the ratio of the exponents \ensuremath{\gamma}/\ensuremath{\nu}=2.0\ifmmode\pm\else\textpm\fi{}0.05, which are in good agreement with the estimates of the high-temperature series-expansion method.