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Physical Review Letters

Monte Carlo calculation of elementary excitation of spin chains

Minoru Takahashi

Published May 8, 1989• 202 citations
Publisher

Abstract

An efficient Monte Carlo method is proposed to calculate the elementary excitation spectrum of quantum systems. The lowest energy with arbitrary momentum is obtained by the projector Monte Carlo method. This is applied to the spin-(1/2 and -1 Heisenberg antiferromagnetic chain with length 32. For the S=(1/2 case, the spectrum coincides completely with the spectrum of des Cloiseaux and Pearson. For the S=1 case, the spectrum has a gap at momentum \ensuremath{\pi} as was predicted by Haldane. The value of the gap coincides with the calculation of Nightingale and Blote. The spectrum satisfies a variational relation with the structure factor.

DOI
10.1103/physrevlett.62.2313

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We give a full description of a recently developed efficient Monte Carlo Approach to low-lying excitations of one-dimensional quantum spin systems. The idea is in a word expressed as extracting the lower edge of the excitation spectrum from imaginary-time quantum Monte Carlo data at a sufficiently low temperature. First, the method is applied to the antiferromagnetic Heisenberg chains of S=1/2, 1, 3/2, and 2. In the cases of S=1/2 and S=1, comparing the present results with the previous findings, we discuss the reliability of the method. The spectra for S=3/2 and S=2 turn out to be massless and massive, respectively. In order to demonstrate that our method is very good at treating long chains, we calculate the S=2 chain with length up to 512 spins and give a precise estimate of the Haldane gap. Second, we show its fruitful use in studying quantum critical phenomena of bond-alternating spin chains. Using the conformal invariance of the system as well, we calculate the central charge of the critical S=1 chain, which results in the Gaussian universality class. Third, we study an alternating-spin system composed of two kinds of spins S=1 and 1/2, which shows the ferrimagnetic behavior. We find a quadratic dispersion relation in the small-momentum region. The numerical findings are qualitatively explained well in terms of the spin-wave theory. Finally, we argue a possibility of applying the method to the higher excitations, where we again deal with the S=1 Heisenberg antiferromagnet and inquire further into its unique low-energy structure. All the applications demonstrate the wide applicability of the method and its own advantages.
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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→∞?> .
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An efficient projector method is used to calculate the energies of the ground state and the first excited state, and hence the energy gap between them, of the spin- Heisenberg antiferromagnetic chains with linear sizes up to . For the chains up to the exact results are obtained, and for longer chains up to they are estimated by imposing on Monte Carlo technique. It is found that the size dependence of the ground state energy per site has the form of , and the value of the ground state energy per site as is . Our results show that the energy gap trends to zero as , which is consistent with the rigorous proof.