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Journal of Applied Physics

Numerical studies of the quantum spin chain (abstract)

H. Q. Lin, C. Y. Pan

Published May 1, 1990• 0 citations
Publisher

Abstract

We report the results of recent numerical studies of the antiferromagnetic Heisenberg chain with S=1/2, 1, 3/2, and 2. The ground-state energy and the energy gap between the ground state and the first excited state are obtained by using exact diagonalization method on various finite lattices and an improved real-space renormalization method. We discuss our numerical methods and results. Some of our results on the S=1 biquadratic antiferromagnetic model will also be discussed.

DOI
10.1063/1.345902

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We report the results of recent numerical studies of the antiferromagnetic Heisenberg chain with S=1/2, 1, 3/2, and 2. The ground-state energy and the energy gap between the ground state and the first excited state are obtained by using exact diagonalization method on various finite lattices and an improved real-space renormalization method. We discuss our numerical methods and results. Some of our results on the S=1 biquadratic antiferromagnetic model will also be discussed.
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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→∞?> .
A real space renormalization group method is applied to study the properties of the anisotropic and alternating Heisenberg antiferromagnets with spin 1/2, 1 and 3/2. To improve the accuracy of the numerical estimates larger spin cluster is used. The ground state energy and the energy gap are obtained. For the case of S = 1 at Heisenberg point, the ground state energy per site is E (0)/N = -1.4021 ± 0.0005 and the energy gap is 0.4097 ± 0.0005, in excellent agreement with a recent Monte Carlo calculation. For the cases of spin 1/2 and 3/2 the energy gap is absent. Therefore our results are shown to support Haldane's conjecture. We briefly discuss the properties of the novel phase.
We have studied anisotropic Heisenberg antiferromagnets by using a real space renormalization-group analysis and exact diagonalization approach. For the spin-1 case, the ground-state energy and Haldane gap at the Heisenberg point are estimated, and they are in good agreement with that obtained by other methods. The critical point is identified where the gap phase changes into a gapless phase. For spin-2 case, the calculated Haldane gap supports a recent prediction suggested by a quantum O(N) nonlinear sigma model study.