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Physical review. B, Condensed matter

Classical two-dimensional XXZ model: A test of a generalized self-consistent harmonic approximation

Antonio S. T. Pires

Published Sep 1, 1996• 34 citations
Publisher

Abstract

We calculate the Kosterlitz-Thouless transition temperature for the ferromagnetic classical $\mathrm{XXZ}$ model on a square lattice for interactions of the out-of-plane spin components $\ensuremath{\lambda}$ varying between 0 and 1. We compare our theoretical calculations with Monte Carlo results of Cuccoli et al. [Phys. Rev. B 52, 10 221 (1995)]

DOI
10.1103/physrevb.54.6081

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We calculate the Kosterlitz-Thouless transition temperature for the ferromagnetic classical XXZ\mathrm{XXZ}XXZ model on a square lattice for interactions of the out-of-plane spin components \ensuremathλ\ensuremath{\lambda}\ensuremathλ varying between 0 and 1. We compare our theoretical calculations with Monte Carlo results of Cuccoli et al. [Phys. Rev. B 52, 10 221 (1995)]
Results of Monte Carlo simulations of the ferromagnetic classical XXZ model on a square lattice are reported for interactions of the out of plane spin components \ensuremath{\lambda} varying between 0 and 0.99. For all the value of \ensuremath{\lambda} the signature of a Berezinskii-Kosterlitz-Thouless (BKT) phase transition was observed, and the corresponding critical temperature was located both by BKT fits and finite-size scaling methods. Data for in-plane and out of plane susceptibility, correlation length, specific heat, and internal energy are given in a broad range of temperature and for lattice sizes ranging from 32\ifmmode\times\else\texttimes\fi{}32 to 256\ifmmode\times\else\texttimes\fi{}256.
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.
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.
The statistical mechanics of the two-dimensional ferromagnetic model with easy-plane anisotropy is approached by the pure-quantum self-consistent harmonic approximation (PQSCHA), that reduces the calculation of thermodynamic averages to effective classical expressions. In the PQSCHA, the quantum corrections to the classical thermodynamics are reduced to suitable (temperature-dependent) renormalizations of the interaction parameters, so that the full role of the nonlinear excitations is preserved. A particular case is the XX0 model (also known as the quantum XY model), which undergoes a Kosterlitz–Thouless phase transition at some finite temperature Tc. Since it is possible to calculate how much the effective exchange interaction is weakened by quantum fluctuations, we can predict, for instance, the corresponding amount of reduction of Tc for any value of the spin. Even in the extreme quantum case of the spin-1/2 model, our result is compatible with the estimates of Tc obtained by other authors.
Convincing numerical evidence is obtained via extensive quantum Monte Carlo simulations on square lattices as large as 96\ifmmode\times\else\texttimes\fi{}96 that the spin-1/2 XY model undergoes a Kosterlitz-Thouless phase transition at kTc{\mathit{kT}}_{\mathit{c}}kTc​/J=0.350(4). Correlation length and in-plane susceptibility diverge at Tc{\mathit{T}}_{\mathit{c}}Tc​ precisely according to the form predicted by Kosterlitz and Thouless for the classical XY model. The specific heat increases very rapidly near Tc{\mathit{T}}_{\mathit{c}}Tc​ and exhibits a peak around kT/J=0.45.
We study the classical XY (plane rotator) model at the Kosterlitz–Thouless phase transition. We simulate the model using the single-cluster algorithm on square lattices of a linear size up to L = 2048. We derive the finite-size behaviour of the second moment correlation length over the lattice size ξ 2nd / L at the transition temperature. This new prediction and the analogous one for the helicity modulus ϒ are confronted with our Monte Carlo data. This way β KT = 1.1199 is confirmed as inverse transition temperature. Finally, we address the puzzle of logarithmic corrections of the magnetic susceptibility χ at the transition temperature.