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Journal of the Physical Society of Japan

The Phase Transition of the Quantum SpinS=1/2-XXZModel for a Square Lattice

Takeshi Kawabe

Published Feb 15, 1989• 0 citations
Publisher

Abstract

The transverse and longitudinal spin-pair correlations are obtained as a function of temperature by the second-order Green's function method. The longitudinal N. N. spin-pair correlation at the ground state becomes negative for anisotropy parameter η between the X Y model (η=0) and quite near the Heisenberg model (η∼1.0). It is predicted from these results that the first-order phase transition occurs near the Heisenberg limit. For η=0.9, it occurs at a temperature τ c restricted by 2.4<τ<6.3.

DOI
10.1143/jpsj.58.348

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The transverse and longitudinal spin-pair correlations are obtained as a function of temperature by the second-order Green's function method. The longitudinal N. N. spin-pair correlation at the ground state becomes negative for anisotropy parameter η between the X Y model (η=0) and quite near the Heisenberg model (η∼1.0). It is predicted from these results that the first-order phase transition occurs near the Heisenberg limit. For η=0.9, it occurs at a temperature τ c restricted by 2.4<τ<6.3.
Nearest neighbour spin-pair correlations for the S = 1/2 Ising-like and XY-like system of a linear chain and for the XY-like system of a square lattice are obtained as a function of temperature by the second-order Green's function method. The transverse correlation and the specific heat for the XY model of a linear chain are in good agreement with rigorous results except for quite low temperatures. The longitudinal correlation for the XY model of a square lattice is given by -0.07311 in the ground state in fair agreement with -0.076 ±0.002 by the finite lattice method and with -0.0714 by the variational method. For the XY-like system of a square lattice the longitudinal nearest neighbour correlation in the ground state becomes always negative. The possibility of a transition of a certain kind is predicted near the Heisenberg limit for a square lattice but not for a linear chain.
The ground-state properties of the spin-1/2 antiferromagnetic Heisenberg model with spatially anisotropic couplings on a square lattice are investigated by a spin-rotation-invariant Green's-function approach and by Lanczos diagonalizations on lattices up to 36 sites supplemented by finite-size scaling. We focus on the anisotropy-driven transition from the N\'eel state to a paramagnetic state with antiferromagnetic short-range order and on the spatial dependence of spin correlation functions. Our principal result is that a rather sharp crossover in the magnetic behavior occurs at the coupling ratio ${R}_{0}\ensuremath{\simeq}0.2 {(R=J}_{y}{/J}_{x}).$
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 .
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.
The phase transition for the spin systems described by the Hamiltonian H =-2 J ∑ < i , j > (ξ S i x S j x +η S i y S j y + S i z S j z )- g µ H ∑ i S i z is studied by means of the high-temperature series expansion of the zero field susceptibility. The first five terms in the susceptibility series are explicitly obtained for the two-dimensional square lattice in the cases of (i) 0≤ξ≤1, η=0 and (ii) 0≤ξ=η≤1. The critical temperature T c and the critical exponent γ (susceptibility) are obtained as functions of the spin S and the anisotropy parameter ξ. The critical temperature has a finite value in the XY model (ξ=1, η=0) as well as the Heisenberg model (ξ=η=1). The simple formulae expressing T c and γ valid for arbitrary S and ξ are proposed.
Using the correlated effective-field theory,the spin-2 transverse Ising model with the crystal field for the square lattice has been investigated.The expressions of magnetization were obtained and by numerical calculation the ground-state magnetization curves were given.It was found that there were the first order-order phase transitions,the first order-disorder phase transitions and the second order phase transitions in the ground state.The second order transitions are widespread,and the first order transitions are occurred in the very narrow range of crystal field and transverse field.