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Physical review. B, Condensed matter

Ground-state properties of the spin-1/2 antiferromagnetic Heisenberg chain obtained by use of a Monte Carlo method

Mário J. de Oliveira

Published Sep 1, 1993• 10 citations
Publisher

Abstract

A Monte Carlo method is presented to calculate the largest eigenvalue of a non-negative matrix. The method is applied to the spin-1/2 antiferromagnetic Heisenberg chain. The ground-state energy and the singlet-triplet energy gap are calculated for chains up to N=48. The results are in agreement with results obtained by the power method for small values of N.

DOI
10.1103/physrevb.48.6141

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Abstract An efficient projector Monte Carlo method has been used to calculate the energies of the ground state and the first excited state, and hence the gap between them, of the spin-1 Heisenberg antiferromagnetic chains with linear sizes up to <?CDATA N=40N=40N=40?> . We have computed them on chains up to <?CDATA N=40N=40N=40?> . We have found that the size-dependence of the ground state energy per site has the form <?CDATA [e0(N)−e0]∼1/N5/2s[e_0(N)-e_0]\sim 1/N^{5/2s}[e0​(N)−e0​]∼1/N5/2s?> , which implies that there is a gap between the ground state and the first excited state. The value of the ground state energy per site as <?CDATA N→∞N\rightarrow \inftyN→∞?> is <?CDATA e0=−1.3996e_0=-1.3996e0​=−1.3996?> . The direct extrapolation of <?CDATA ΔE(N)\Delta E(N)ΔE(N)?> also shows that a gap of <?CDATA ΔE=0.36\Delta E=0.36ΔE=0.36?> exists as <?CDATA N→∞N\rightarrow \inftyN→∞?> .
An efficient projector Monte Carlo method has been used to calculate the energies of the ground state and the first excited state, and hence the gap between them, of the spin-2 Heisenberg antiferromagnetic chains with linear sizes up to N=32. By direct extrapolation, we found that a Haldane gap exists, \ensuremath{\Delta}E=0.05, and the value of the ground-state energy per site is e0{\mathit{e}}_{0}e0​=-4.7545 as N\ensuremath{\rightarrow}\ensuremath{\infty}.