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Journal of Physics A Mathematical and General

Quantum Monte Carlo simulation of the one-dimensional spin-S xxz model. II. High precision calculations for S=1/2

M Marcu, Jürgen Müller, F -K Schmatzer

Published Nov 11, 1985• 24 citations
Publisher

Abstract

For pt.I see ibid., vol.18, p.2479, (1985). A quantum Monte Carlo method developed in the first paper of this series is used to investigate the spin-1/2 isotropic ferromagnet and antiferromagnet. Critical indices are computed in a high precision simulation. The authors obtain alpha =-0.261+or-0.013, gamma =1.552+or-0.008 for the ferromagnet, and alpha =-1.202+or-0.009, gamma =1.132+or-0.012 for the antiferromagnet.

DOI
10.1088/0305-4470/18/16/020

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For pt.I see ibid., vol.18, p.2479, (1985). A quantum Monte Carlo method developed in the first paper of this series is used to investigate the spin-1/2 isotropic ferromagnet and antiferromagnet. Critical indices are computed in a high precision simulation. The authors obtain alpha =-0.261+or-0.013, gamma =1.552+or-0.008 for the ferromagnet, and alpha =-1.202+or-0.009, gamma =1.132+or-0.012 for the antiferromagnet.
The authors develop a quantum Monte Carlo procedure for the xxz spin chain. The quantum model in one dimension is mapped by a path integral method based on the Trotter formula into the infinite anisotropy limit of a classical two-dimensional spin system. The expectation values of the energy density, susceptibility and static s z -s z structure function are computed in the classical model for finite but increasing anisotropy. The computation method is a Monte Carlo simulation of the classical system. The results are systematically extrapolated to the infinite anisotropy limit.
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.
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.
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.