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Journal of the Physical Society of Japan

On the Ground-State Phase Transition of the Spin 1/2 XXZ Model on the Square Lattice

Macoto Kikuchi, Yutaka Okabe, Seiji Miyashita

Published Feb 15, 1990• 9 citations
Publisher

Abstract

We study the effect of anisotropy for the ground state of the spin 1/2 antiferromagnet on the square lattice by diagonalizing the Hamiltonian numerically. The symmetry change of the ground state at the isotropic point is discussed.
DOI
10.1143/jpsj.59.492

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Using the coupled-cluster method for infinite lattices and the exact diagonalization method for finite lattices, we study the influence of an exchange anisotropy Δ on the ground-state phase diagram of the spin-1/2 frustrated J 1 - J 2 XXZ antiferromagnet on the square lattice. We find that increasing Δ > 1 (i.e. an Ising type easy-axis anisotropy) as well as decreasing Δ < 1 (i.e. an XY type easy-plane anisotropy) both lead to a monotonic shrinking of the parameter region of the magnetically disordered quantum phase. Finally, at Δ c ≈ 1.9 this quantum phase disappears, whereas in the pure XY limit (Δ = 0) there is still a narrow region around J 2 = 0.5 J 1 where the quantum paramagnetic ground-state phase exists.
The magnetization process of the spin-1/2 antiferromagnetic XXZ\mathrm{XXZ}XXZ models with Ising-like anisotropy in the ground state is investigated. We show numerically that the Ising-like XXZ\mathrm{XXZ}XXZ models on square and cubic lattices show a first-order phase transition at some critical magnetic field. We estimate the value of the critical field and the magnetization jump on the basis of the Maxwell construction. The magnetization jump in the Ising limit is investigated by means of perturbation theory. Based on our numerical results, we briefly discuss the phase diagram of the extended Bose-Hubbard model in the hard-core limit.
We study the zero-temperature phase diagram of the J 1 XXZ – J 2 XXZ Heisenberg model for spin-1 particles on an infinite square lattice interacting via nearest-neighbour ( ) and next-nearest-neighbour ( J 2 >0) bonds. The two bonds have the same XXZ -type anisotropy in spin space. The effects on the quasiclassical Néel-ordered and collinear stripe-ordered states of varying the anisotropy parameter Δ are investigated using the coupled cluster method carried out up to high orders. By contrast with the case for spin- particles studied previously, no intermediate disordered phase between the Néel and collinear stripe phases, for any value of the frustration J 2 / J 1 , for either the z -aligned (Δ>1) or xy -planar-aligned (0≤Δ<1) states, is predicted here. The quantum phase transition is determined as first order for all values of J 2 / J 1 and Δ. The position of the phase boundary J 2 c (Δ) is determined accurately. It is observed to deviate most from its classical position (for all values of Δ>0) at the Heisenberg isotropic point (Δ = 1), where J 2 c (1) = 0.55 ± 0.01. By contrast, at the XY isotropic point (Δ = 0), we find J 2 c (0) = 0.50 ± 0.01. In the Ising limit ( ), as expected.
We use a variational mean-field-like approach, the coupled cluster method (CCM), and exact diagonalization to investigate the ground-state order–disorder transition for the square lattice spin-half XXZ model with two different nearest-neighbour couplings J and . On increasing the model shows in the isotropic Heisenberg limit a second-order transition from semi-classical Néel order to a quantum paramagnetic phase with enhanced local dimer correlations on the bonds at about . This transition is driven by the quantum competition between and J . On increasing the anisotropy parameter Δ>1 we diminish the quantum fluctuations and thus the degree of competition. As a result the transition point is shifted to larger values. We find indications for a linear increase of with Δ, i.e. the transition disappears in the Ising limit .
Ground-state properties of a spin-\textonehalf{} XXZ\mathrm{XXZ}XXZ antiferromagnet on the Bethe lattice with three nearest neighbors are investigated; especially, the nature of the phase transition driven by the anisotropy is examined. In actual calculations, we employ the density-matrix renormalization-group method which is so far used in the studies on the one-dimensional quantum systems; its applicability to the higher-dimensional system is thus exhibited in our calculations. Numerical data on local spin correlations imply that the model undergoes the first-order phase transition at the Heisenberg point Jxy=Jz{J}_{\mathrm{xy}}={J}_{z}Jxy​=Jz​ and the ground state is conjectured to be in the Ising-like ordered phase (the ordered phase in XY\mathrm{XY}XY plain) for Jxy<Jz{J}_{\mathrm{xy}}<{J}_{z}Jxy​<Jz​ (Jxy>Jz{J}_{\mathrm{xy}}>{J}_{z}Jxy​>Jz​). We also compare our results with those for other systems, say, the XYZ\mathrm{XYZ}XYZ model on the square lattice to discuss a possible type of symmetry breaking.