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arXiv:cond-mat/9401053 · Journal of the Physical Society of Japan

Numerical Study of the S =1 Antiferromagnetic Spin Chain with Bond Alternation

Yusuke S. Kato, Akihiro Tanaka

Published Apr 15, 1994• 123 citations
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Abstract

We study the $S=1$ quantum spin chain with bond alternation ${\cal H}=\sum _i (1-(-1)^iδ)\vect{S}_i\cdot \vect{S}_{i+1}$ by the density matrix renormalization group method recently proposed by Steven R. White (\PRL{69}{3844}{1993}). We find a massless point at $δ_c =0.25 \pm 0.01$. We also find the edge states in the region $δδ_{c}$. At the massless point, the spin wave velocity $v_s$ is $3.66 \pm 0.10$ and the central charge $c$ is $1.0\pm 0.15$. Our results indicate that a continuous phase transition occurs at the massless point $δ=δ_c $ accompanying breaking of the hidden $Z_2\times Z_2$ symmetry.

DOI
10.1143/jpsj.63.1277

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We study the S=1S=1S=1 quantum spin chain with bond alternation H=∑i(1−(−1)iδ)\vectSi⋅\vectSi+1{\cal H}=\sum _i (1-(-1)^iδ)\vect{S}_i\cdot \vect{S}_{i+1}H=∑i​(1−(−1)iδ)\vectSi​⋅\vectSi+1​ by the density matrix renormalization group method recently proposed by Steven R. White (\PRL{69}{3844}{1993}). We find a massless point at δc=0.25±0.01δ_c =0.25 \pm 0.01δc​=0.25±0.01. We also find the edge states in the region δδcδδ_{c}δδc​. At the massless point, the spin wave velocity vsv_svs​ is 3.66±0.103.66 \pm 0.103.66±0.10 and the central charge ccc is 1.0±0.151.0\pm 0.151.0±0.15. Our results indicate that a continuous phase transition occurs at the massless point δ=δcδ=δ_c δ=δc​ accompanying breaking of the hidden Z2×Z2Z_2\times Z_2Z2​×Z2​ symmetry.
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We study the phase diagram of S=3/2 and S=2 bond-alternating spin chains numerically. In previous papers, the phase diagram of S=1 XXZ spin chain with bond-alternation was shown to reflect the hidden Z2×Z2Z_{2}\times Z_{2}Z2​×Z2​ symmetry. But for the higher S Heisenberg spin chain, the successive dimerization transition occurs, and for anisotropic spin chains the phase structure will be more colorful than the S=1 case. Using recently developed methods, we show directly that the phase structure of the anisotropic spin chains relates to the Z2×Z2Z_{2}\times Z_{2}Z2​×Z2​ symmetry.
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{)}^{1/2})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.
We study ground state properties of the S=2S=2S=2 quantum antiferromagnetic chain with a bond alternation \[ H = \sum_{j} [ 1 + δ(-1)^j ] \mbox{\boldmath $S$}_{j} \cdot \mbox{\boldmath $S$}_{j+1} \] by a Quantum Monte Carlo calculation. We find that the hidden Z2×Z2Z_2 \times Z_2Z2​×Z2​ symmetry is broken for 0.3<∣δ∣<0.50.3 < |δ| < 0.50.3<∣δ∣<0.5 while it is unbroken in the other regions. This confirms the successive dimerization transitions first predicted by Affleck and Haldane. Our result shows that these transitions can be understood in terms of the hidden Z2×Z2Z_2 \times Z_2Z2​×Z2​ symmetry breaking, as was discussed using the Valence-Bond-Solid states. Furthermore, we find that the behavior of the generalized string correlation is qualitatively very similar to that in the Valence-Bond-Solid states, including the location of zeroes as a function of the angle parameter.
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