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arXiv:cond-mat/0006272 · Brazilian Journal of Physics

Quantum to classical crossover in the 2D easy-plane XXZ model

Holger Fehske, C. Schindelin, Alexander Weiße, H. Büttner, Dieter Ihle

Published Dec 1, 2000• 3 citations
arXivHTMLPDF

Abstract

Ground-state and thermodynamic properties of the spin-1/2 two-dimensional easy-plane XXZ model are investigated by both a Green's-function approach and by Lanczos diagonalization on lattices with up to 36 sites. We calculate the spatial and temperature dependences of various spin correlation functions, as well as the wave-vector dependence of the spin susceptibility for all anisotropy parameters delta. In the easy plane ferromagnetic region (-1 < delta < 0), the longitudinal correlators of spins at distance r change sign at a nite temperature T0 (delta, r). This transition, observed in the 2D case for the first time, can be interpreted as a quantum to classical crossover.

DOI
10.1590/s0103-97332000000400015

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Ground-state and thermodynamic properties of the spin-1/2 two-dimensional easy-plane XXZ model are investigated by both a Green's-function approach and by Lanczos diagonalization on lattices with up to 36 sites. We calculate the spatial and temperature dependences of various spin correlation functions, as well as the wave-vector dependence of the spin susceptibility for all anisotropy parameters delta. In the easy plane ferromagnetic region (-1 < delta < 0), the longitudinal correlators of spins at distance r change sign at a nite temperature T0 (delta, r). This transition, observed in the 2D case for the first time, can be interpreted as a quantum to classical crossover.
The critical behavior of 2D magnetic easy-plane systems has mainly been studied by the classical XY model. However, the z components of the spins have to be considered in order to describe real systems, and their fluctuations cannot positively be neglected when quantum effects are to be included, quantum spins being intrinsically three-component objects. Therefore, Monte Carlo simulations are performed for the Heisenberg ferromagnet with easy-plane anisotropy (XXZ model) on a two-dimensional square lattice with a twofold aim: first, to obtain accurate quantitative results about the critical behavior of the classical model, showing the relevant role played by the out-of-plane fluctuations; second, to open the way for approaching the quantum thermodynamics by means of the effective Hamiltonian method that reduces the quantum thermodynamics of the XXZ ferromagnet to the investigation of an effective classical model with temperature-dependent renormalized interaction parameters. Specific heat, magnetic susceptibility, and correlation length are calculated in the critical region for lattice sizes up to 128×128. These quantities preserve the Kosterlitz–Thouless behavior of the XY model. For the transition temperature of the classical XX0 model we obtain the estimate kBTc/(JS̃2)=0.69±0.01.
Phase diagram, critical properties and thermodynamic functions of the two-dimensional field-free quantum-spin-1/2 XXZ model has been calculated globally using a numerical renormalisation group theory. The nearest-neighbour spin-spin correlations and entanglement properties, as well as internal energy and specific heat are calculated globally at all temperatures for the whole range of exchange interaction anisotropy, from XY limit to Ising limits, for both antiferromagnetic and ferromagnetic cases. We show that there exists long-range (quasi-long-range) order at low-temperatures, and the low-lying excitations are gapped (gapless) in the Ising-like easy-axis (XY-like easy-plane) regime. Besides, we identify quantum phase transitions at zero-temperature.
We compute, by means of exact diagonalization of systems of N=16N=16N=16 and 181818 spins, the correlation function $〈{\ensuremath{\sigma}}_{0}^{z}{\ensuremath{\sigma}}_{n}^{z}〉$ at nonzero temperature for the XXZ\mathrm{XXZ}XXZ model with anisotropy $\ensuremath{\Delta}.$ In the gapless attractive region $\ensuremath{-}1<\ensuremath{\Delta}<0$ for fixed separation the temperature can always be made sufficiently low so that the correlation is always negative for $n\ensuremath{\ne}0.$ However we find that for sufficiently large temperatures and fixed separation or for fixed temperature greater than some ${T}_{0}(\ensuremath{\Delta})$ and sufficiently large separations the correlations are always positive. This sign changing effect has not been previously seen and we interpret it as a crossover from quantum to classical behavior.
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 present a Green's-function theory of magnetic short-range order in the S=1/2S=1/2S=1/2 easy-plane XXZ\mathrm{XXZ}XXZ chain based on the projection method for the dynamic spin susceptibility and a decoupling of three-spin operator products introducing vertex parameters. The longitudinal and transverse static susceptibilities and two-point correlation functions of arbitrary range are calculated self-consistently for all wave numbers, temperatures, and anisotropy parameters $\ensuremath{-}1<~\ensuremath{\Delta}<~1.$ In the easy-plane ferromagnetic region $(\ensuremath{\Delta}<0),$ the longitudinal correlators of spins at distance n change sign at a finite temperature ${T}_{0}(n,\ensuremath{\Delta}),$ in reasonable agreement with recent data obtained by finite-chain diagonalizations. The temperature dependence of the uniform static susceptibilities exhibits a maximum which is explained as an effect of magnetic short-range order which decreases with increasing temperature.
The quantum easy-plane ferromagnetic two-dimensional XXZ model is approached by the pure-quantum self-consistent harmonic approximation that reduces it to an effective classical model. Quantum fluctuations weaken both the effective exchange, leading to a reduced Berezinskii-Kosterlitz-Thouless transition temperature with respect to the classical model, and the effective easy-plane anisotropy. The latter vanishes when the anisotropy is smaller than a cutoff value, leading to an instability that could be interpreted as a crossover to a strongly quantum regime where a picture of classical-like renormalized vortices is inadequate.