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Progress of Theoretical Physics

A New Monte Carlo Approach to Quantum Spin Systems: Direct Evaluation of the Partition Function

Shinji Kadowaki, A. Ueda

Published Aug 1, 1987• 2 citations
Publisher

Abstract

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.

DOI
10.1143/ptp.78.224

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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.
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 quantum Monte Carlo simulation scheme for spin systems is presented. The method is a generalization of Handscomb's method but applicable to any length of the spin, i.e., when the spin traces cannot be evaluated analytically. The Monte Carlo sampling is extended to the space of spin vectors in addition to the usual operator-index sequences. An important technical point is that the index sequences are augmented with the aid of unit operators to a constant, self-consistently determined length. The scheme is applied to the one-dimensional antiferromagnetic spin-S Heisenberg model. Results at low temperatures are reported for S=1 and S=3/2 and system sizes up to N=64. The computed magnetic structure factor in the S=1 chain is in agreement with earlier ground-state calculations. For S=3/2 we find the exponent \ensuremath{\gamma}\ifmmode\bar\else\textasciimacron\fi{}=0.49\ifmmode\pm\else\textpm\fi{}0.04 for the divergence of the antiferromagnetic structure factor. Further, the susceptibility as a function of the wave number is computed. For S=1 the staggered susceptibility \ensuremath{\chi}(\ensuremath{\pi}) at T=0 is found to take the value 20.0\ifmmode\pm\else\textpm\fi{}1.5 in units such that \ensuremath{\chi}(q)\ensuremath{\rightarrow}${\mathit{T}}^{\mathrm{\ensuremath{-}}1}$ at high temperatures (with the temperature scale defined by kB{\mathit{k}}_{\mathit{B}}kB​=1). For S=3/2 we obtain the exponent \ensuremath{\gamma}=1.45\ifmmode\pm\else\textpm\fi{}0.05 for the divergence of the staggered susceptibility.
On the basis of the path integral representation of the partition function of a quantum spin system, which is derived by Lieb with use of the Bloch coherent state representation, we propose a new Monte Carlo method for quantum spin systems. In this method, the magnitude S of the spin can be varied freely, because S appears in the path integral only as a parameter. The approximation used in this method can be systematically improved by the 1/n-expansion, where n is the additional dimension in the path integral representation. The formulation and some properties of the method are discussed. Simple examples of the application of the method are also presented.
We have developed a method for evaluating the specific heat of lattice spin systems. It is based on the knowledge of high-temperature series expansions, the total entropy of the system, and the low-temperature expected behavior of the specific heat as well as the ground-state energy. By the choice of an appropriate variable (entropy as a function of energy), a stable interpolation scheme between low and high temperature is performed. Contrary to previous methods, the constraint that the total entropy is log(2S+1)\mathrm{log}(2S+1)log(2S+1) for a spin S on each site is automatically satisfied. We present some applications to quantum spin models on one- and two-dimensional lattices. Remarkably, in most cases, a good accuracy is obtained down to zero temperature.