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Physical review. B, Condensed matter

S(q,ω) for theS=1/2 andS=1 one-dimensional Heisenberg antiferromagnet: A quantum Monte Carlo study

J. J. Deisz, M. Jarrell, Daniel Lee Cox

Published Sep 1, 1990• 47 citations
Publisher

Abstract

The dynamic-spin-correlation function S(q,\ensuremath{\omega}) is computed for the one-dimensional antiferromagnetic Heisenberg model for S=1/2 and S=1. Imaginary-time correlation functions are evaluated with the quantum Monte Carlo method and the maximum-entropy method is used to continue these results to real frequencies. This technique produces a value for the S=1 Haldane gap that agrees with previous work and also gives a good description of inelastic neutron scatter- ing data for the one-dimensional S=1/2 antiferromagnet ${\mathrm{CuCl}}_{2}$\ensuremath{\cdot}2N(${\mathrm{C}}_{5}$${\mathrm{D}}_{5}$).

DOI
10.1103/physrevb.42.4869

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Journal of Applied Physics•Apr 15, 1991•1 citation
The dynamic-spin-correlation function S(q,\ensuremath{\omega}) is computed for the one-dimensional antiferromagnetic Heisenberg model for S=1/2 and S=1. Imaginary-time correlation functions are evaluated with the quantum Monte Carlo method and the maximum-entropy method is used to continue these results to real frequencies. This technique produces a value for the S=1 Haldane gap that agrees with previous work and also gives a good description of inelastic neutron scatter- ing data for the one-dimensional S=1/2 antiferromagnet CuCl2{\mathrm{CuCl}}_{2}CuCl2​\ensuremath{\cdot}2N({\mathrm{C}}_{5}$${\mathrm{D}}_{5}).
A combination of world-line quantum Monte Carlo and maximum-entropy methods is used to calculate the dynamic structure factor ${\mathit{S}}_{\mathrm{\ensuremath{\alpha}}\mathrm{\ensuremath{\alpha}}}$(q,\ensuremath{\omega}) for one-dimensional antiferromagnets. A comparison to exact results demonstrates that qualitative features of spectra are reproduced, but are broadened. The broadening is frequency dependent with low-frequency features being reproduced most accurately. Peak positions are well described. Results for the Heisenberg model with S\ensuremath{\le}2 demonstrate the qualitative difference between the dynamics of half-integer and integer spins predicted by Haldane. From the positions of the low-temperature q=\ensuremath{\pi} peaks quantitative estimates are produced for the S=1 and 2 gaps. The q=\ensuremath{\pi} peak position for S=1 increases with temperature in agreement with experiment. When on-site anisotropy is included for S=1 a gap splitting results that resembles that found for NENP [Ni(C2{\mathrm{C}}_{2}C2​H8{\mathrm{H}}_{8}H8​N2{\mathrm{N}}_{2}N2​)2{)}_{2})2​NO2{\mathrm{NO}}_{2}NO2​ClO4{\mathrm{ClO}}_{4}ClO4​].
The spin correlations of the S=1 Heisenberg antiferromagnetic chains in AgVP2{\mathrm{AgVP}}_{2}AgVP2​S6{\mathrm{S}}_{6}S6​ have been studied by neutron scattering at temperatures of the order of T\ensuremath{\sim}${10}^{\mathrm{\ensuremath{-}}2}$ J/kB{\mathit{k}}_{\mathit{B}}kB​. On the polycrystalline samples, with a careful choice of experimental conditions, we have observed a Haldane gap of Eg{\mathit{E}}_{\mathit{g}}Eg​=26 meV at the antiferromagnetic point at q=\ensuremath{\pi}. The excitation spectra are consistent with a mode that has a spin-wave velocity C=150 meV and a correlation length \ensuremath{\xi}\ensuremath{\approxeq}5.5, close to C/Eg{\mathit{E}}_{\mathit{g}}Eg​. These results are in quantitative agreement with the predictions and numerical results on the Haldane state.
We used a combination of the quantum Monte Carlo technique and the maximum entropy method to calculate the dynamic structure factor of the quantum Heisenberg antiferromagnet. The dynamics is governed by renormalized damped spin waves and we calculate their frequencies and linewidths. The computed spin-wave dispersion is in quantitative agreement with inelastic-neutron-scattering experiments on La2{\mathrm{La}}_{2}La2​CuO4{\mathrm{CuO}}_{4}CuO4​. The low-energy sector exhibits scaling behavior with the characteristic frequency scale, which is inversely proportional to the correlation length.
A world-line Monte Carlo approach is proposed to calculate the elementary excitations of quantum systems. The lowest energies as a function of momentum are successfully extracted from an imaginary-time correlation function evaluated at a sufficiently low temperature. The method is applied to the antiferromagnetic Heisenberg chains of S=12,1,32,S\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\frac{1}{2},1,\frac{3}{2},S=21​,1,23​, and 222 with length up to 64. The spectra for S=12S\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\frac{1}{2}S=21​ and 1 coincide well with the result by des Cloizeaux and Pearson and with the result by Takahashi, respectively. The spectra for S=32S\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\frac{3}{2}S=23​ and 2 turn out to be massless and massive, respectively. The S=2S\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}2S=2 Haldane gap is estimated to be $0.049\ifmmode\pm\else\textpm\fi{}0.018$.
The spectral properties, and associated critical singularities, of spin-1 Heisenberg antiferromagnetic chains are of high current interest, both theoretical and experimental. Theoretical studies using the single mode approximation and also a Monte Carlo calculation indicate that the spectrum has a gap at k=π, as predicted by Haldane. They also predict the unusual feature that the excited state continuum is asymmetric about the mid-point of the Brillouin zone, k=π/2. However, neither approach is sufficiently accurate to predict reliably the magnitude of the gap at k=0 which was not discussed by Haldane. Our approach is to use static (spectral) calculations of excited states of chains of N=4 through 14 spins for all values of k, a novel feature not associated with previous finite-size calculations, to discuss the dispersion spectra of the S=1 Heisenberg chain. In addition, we have calculated dynamical properties, in particular S(q,ω), exactly for finite chains up to N=14 spins. This allows us to estimate the gap at k=0 to good accuracy, and we find that to within this accuracy the gap at k=0 is twice the Haldane gap at k=π. This result is important in connection with very recent neutron scattering and magnetization measurements. In addition, we have calculated the correlation length and critical singularities associated with dynamical quantities.