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International Journal of Modern Physics B

SPIN-1/2 TWO-DIMENSIONAL ANISOTROPIC HEISENBERG ANTIFERROMAGNETS

C. Y. Pan

Published Sep 1, 1989• 4 citations
Publisher

Abstract

A real space renormalization group method is applied to study the spin-1/2 two-dimensional anisotropic Heisenberg antiferromagnets. We carry out the calculation on a 5×5 cluster by using a variational approach. The ground-state energy per site is estimated as e0=−0.6845±0.0005 which is in good agreement with other numerical estimates. We also calculate the different ground state energies when anisotropy changes. By comparing with 3×3 cluster calculation the resulting threshold for long range order drops and the curve of the renormalization group parameter vs. anisotropy shows no hint of discontinuity over a wide range of anisotropic parameters. Hence, our calculation suggests that long-range order exists for the ground-state of the spin-1/2 Heisenberg antiferromagnets in two dimensions.

DOI
10.1142/s0217979289000920

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A real space renormalization group method is applied to study the spin-1/2 two-dimensional anisotropic Heisenberg antiferromagnets. We carry out the calculation on a 5×5 cluster by using a variational approach. The ground-state energy per site is estimated as e0=−0.6845±0.0005 which is in good agreement with other numerical estimates. We also calculate the different ground state energies when anisotropy changes. By comparing with 3×3 cluster calculation the resulting threshold for long range order drops and the curve of the renormalization group parameter vs. anisotropy shows no hint of discontinuity over a wide range of anisotropic parameters. Hence, our calculation suggests that long-range order exists for the ground-state of the spin-1/2 Heisenberg antiferromagnets in two dimensions.
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.
An even-site block renormalization-group method is applied to the spin-1 and spin-(3/2 anisotropic or alternating Heisenberg antiferromagnets. In the spin-1 case it shows an energy gap \ensuremath{\Delta} \ensuremath{\simeq}0.368 166 and ground-state energy E0{E}_{0}E0​/NJ\ensuremath{\simeq}1.449 724, which are in good agreement with the recent Monte Carlo results. In the spin-(3/2 case it shows an energy excitation behavior similar to that in the spin-(1/2 case. Therefore, our study tends to support Haldane's conjecture. In addition, a possible novel property of the spin-1 Heisenberg antiferromagnet has been predicted.
A simple real-space renormalization method yields the ground-state energy of the Heisenberg antiferromagnet. We find the ground-state energy per spin for s=12s=\frac{1}{2}s=21​ (-0.4438 in 1D, -0.6723 in 2D) and s=1s=1s=1 (-1.388 in 1D and -1.907 in 2D) to three-figure accuracy, using properties of relatively small odd-numbered clusters. Our results provide reasonable proof for long-range order in the ground state of the s=12s=\frac{1}{2}s=21​ Heisenberg antiferromagnet in 2D.
We perform a real-space renormalization-group analysis for the two-dimensional antiferromagnetic Heisenberg model. We first derive an effective model, described by the Hamiltonian: H=J Jj=1L{\mathcal{J}}_{\mathit{j}=1}^{\mathit{L}}Jj=1L​ S(j)\ensuremath{\cdot}S(j+1)+ Jj=1L{\mathcal{J}}_{\mathit{j}=1}^{\mathit{L}}Jj=1L​(-1)j{)}^{\mathit{j}})j\ensuremath{\Vert}J(j)\ensuremath{\Vert}S(0)\ensuremath{\cdot}S(j), where S(0) represents a single spin located in the center of the ring. We show that the renormalized coupling constants \ensuremath{\Vert}J(j)\ensuremath{\Vert} tend to nonzero values as one increases the block size by successive renormalization. Using analytic and numerical arguments, we establish the existence of antiferromagnetic long-range order in the effective model and hence, in the original Heisenberg model. We compare our results with those obtained from perturbation theory, spin-wave theory, and exact-diagonalization calculations.
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.