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Physical review. B, Condensed matter

Elementary excitations of S =1 antiferromagnetic Heisenberg chains with bond alternation

Shoji Yamamoto

Published Jun 1, 1995• 26 citations
Publisher

Abstract

A world-line quantum Monte Carlo method is employed to study the elementary excitations of S=1 antiferromagnetic Heisenberg chains with bond alternation described by the Hamiltonian, scrH=J ${\mathit{tsum}}_{\mathit{i}}$ [1-(-1${)}^{\mathit{i}}$\ensuremath{\delta}]${\mathrm{S}}_{\mathit{i}}$\ensuremath{\cdot}${\mathrm{S}}_{\mathit{i}+1}$. Imaginary-time correlation functions of this system at low-enough temperatures exhibit almost single-exponential decay and thus the lowest excitation spectrum can be successfully extracted from them. As \ensuremath{\delta} increases from 0 to 1, the system encounters a continuous phase transition at \ensuremath{\delta}\ensuremath{\simeq}0.25, where the energy(E)-momentum(q) relation is fitted well to the form E(q)=(${\mathit{v}}^{2}$ ${\mathrm{sin}}^{2}$q+${\mathrm{\ensuremath{\Delta}}}^{2}$${)}^{1/2}$ with the effective light velocity v=2.46\ifmmode\pm\else\textpm\fi{}0.08 and the energy gap \ensuremath{\Delta} which vanishes in the long chain limit. The central charge c at the massless critical point is estimated to be 1.02\ifmmode\pm\else\textpm\fi{}0.09, which suggests the universality class of the Gaussian model.

DOI
10.1103/physrevb.51.16128

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A world-line quantum Monte Carlo method is employed to study the elementary excitations of S=1 antiferromagnetic Heisenberg chains with bond alternation described by the Hamiltonian, scrH=J tsumi{\mathit{tsum}}_{\mathit{i}}tsumi​ [1-(-1)i{)}^{\mathit{i}})i\ensuremath{\delta}]Si{\mathrm{S}}_{\mathit{i}}Si​\ensuremath{\cdot}Si+1{\mathrm{S}}_{\mathit{i}+1}Si+1​. Imaginary-time correlation functions of this system at low-enough temperatures exhibit almost single-exponential decay and thus the lowest excitation spectrum can be successfully extracted from them. As \ensuremath{\delta} increases from 0 to 1, the system encounters a continuous phase transition at \ensuremath{\delta}\ensuremath{\simeq}0.25, where the energy(E)-momentum(q) relation is fitted well to the form E(q)=(v2{\mathit{v}}^{2}v2 sin2{\mathrm{sin}}^{2}sin2q+{\mathrm{\ensuremath{\Delta}}}^{2}$${)}^{1/2} with the effective light velocity v=2.46\ifmmode\pm\else\textpm\fi{}0.08 and the energy gap \ensuremath{\Delta} which vanishes in the long chain limit. The central charge c at the massless critical point is estimated to be 1.02\ifmmode\pm\else\textpm\fi{}0.09, which suggests the universality class of the Gaussian model.
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 $0.049\ifmmode\pm\else\textpm\fi{}0.018$.
Low-temperature properties of S=1 antiferromagnetic Heisenberg chains with alternating interaction, scrH=Jtsumi{\mathit{Jtsum}}_{\mathit{i}}Jtsumi​[1-(-1)i{)}^{\mathit{i}})i\ensuremath{\delta}]Si{\mathrm{S}}_{\mathit{i}}Si​\ensuremath{\cdot}Si+1{\mathrm{S}}_{\mathit{i}+1}Si+1​, are studied by a quantum Monte Carlo method. Spatial distributions of the spin correlation function 〈S1z{\mathit{S}}_{1}^{\mathit{z}}S1z​Siz{\mathit{S}}_{\mathit{i}}^{\mathit{z}}Siz​〉 and the local magnetic moment 〈Siz{\mathit{S}}_{\mathit{i}}^{\mathit{z}}Siz​〉 are visualized as a function of \ensuremath{\delta}. The continuous phase transition at \ensuremath{\delta}=0.25, which was indicated by several authors, is confirmed by the \ensuremath{\delta} dependence of the low-lying level structure, the divergence of the correlation length, and also the collapse of the edge states. The critical exponent \ensuremath{\eta}, governing the power-law decay of the spin correlation function, is estimated to be 0.99\ifmmode\pm\else\textpm\fi{}0.05. The present result raises a possibility that the variety of quantum spin chains including not only the half-odd-integer-S ones but also the integer-S ones may be described in terms of the generic critical theory.
Ground-state properties of S =1 antiferromagnetic Heisenberg chains with bond alternation, H = J ∑ i {1-(-1) i δ} S i · S i +1 , are studied by a quantum Monte Carlo method. Spin correlation between chain ends on the open chain appears for δ<0.25, while it vanishes for δ≥0.25. At δ≃0.25, the spin correlation length has a peak and the S =1/2 edge states for the open chain disappear. The fourfold degeneracy of the ground state of the open chain holds for δ<0.25, while it is removed for δ≥0.25. All the results support the continuous phase transition at δ≃0.25 demonstrated in the previous works.
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.
We study the phase transitions in antiferromagnetic quantum spin chains with bond alternation, H=J${\ensuremath{\sum}}_{\mathrm{i}}$[1-(-1)i{)}^{\mathrm{i}})i\ensuremath{\delta}]Si{\mathbf{S}}_{\mathrm{i}}Si​\ensuremath{\cdot}Si+1{\mathbf{S}}_{\mathrm{i}+1}Si+1​, of spin S=1, 3/2, and 2. On the one hand, using a transfer matrix technique, we make a simple variational approach, which results in the better estimates of the transition points than by the O(3) nonlinear-\ensuremath{\sigma}-model quantum field theory. On the other hand, employing a quantum Monte Carlo method, we calculate the generalized string order parameter, Ostringz{\mathrm{O}}_{\mathrm{string}}^{\mathrm{z}}Ostringz​(\ensuremath{\theta})=${\mathrm{lim}}_{\mathrm{L}\ensuremath{\rightarrow}\mathrm{\ensuremath{\infty}}}$Ostringz{\mathrm{O}}_{\mathrm{string}}^{\mathrm{z}}Ostringz​(\ensuremath{\theta};L) with Ostringz{\mathrm{O}}_{\mathrm{string}}^{\mathrm{z}}Ostringz​(\ensuremath{\theta};L)=〈SL/4z{\mathrm{S}}_{\mathrm{L}\mathrm{/}4}^{\mathrm{z}}SL/4z​${\ensuremath{\prod}}_{\mathrm{j}\mathrm{=}\mathrm{L}\mathrm{/}4}^{3\mathrm{L}\mathrm{/}4\mathrm{\ensuremath{-}}1}$exp[i\ensuremath{\theta}Sjz{\mathrm{S}}_{\mathrm{j}}^{\mathrm{z}}Sjz​]S3L/4z{\mathrm{S}}_{3\mathrm{L}\mathrm{/}4}^{\mathrm{z}}S3L/4z​〉. It turns out that the transition points are successfully detected by observing the overall behavior of Ostringz{\mathrm{O}}_{\mathrm{string}}^{\mathrm{z}}Ostringz​(\ensuremath{\theta}) at various values of \ensuremath{\delta}. Investigating the dependences of Ostringz{\mathrm{O}}_{\mathrm{string}}^{\mathrm{z}}Ostringz​(\ensuremath{\theta};L) on \ensuremath{\theta}, L, and S, we discuss the applicability of the valence-bond-solid picture to the ground states of the present Hamiltonian.