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Journal of Applied Physics

Quantum renormalization of the XY model

Alessandro Cuccoli, V. Tognetti, Paola Verrucchi, Ruggero Vaia

Published May 15, 1994• 14 citations
Publisher

Abstract

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.

DOI
10.1063/1.355577

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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.
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 thermodynamic properties of the quantum ferromagnetic chain with easy-plane single-ion anisotropy are evaluated over a wide temperature range. The basic theoretical method is a recently proposed one, called the pure-quantum self-consistent harmonic approximation (PQSCHA), which, starting from the original Hamiltonian operator scr\ifmmode \hat{H}\else \^{H}\fi{}, allows the definition of an effective classical Hamiltonian scrHeff{\mathit{scrH}}_{\mathrm{eff}}scrHeff​. We use the Villain transformation to canonical conjugate variables, in order to treat the quantum renormalizations within this framework. In the resulting effective Hamiltonian we restore classical spin variables, according to the classical counterpart of the Villain transformation. Provided that the quantum coupling parameter (which depends on the spin value and on the anisotropy constant) is not too large, this effective Hamiltonian is able to account for the quantum behavior, by means of classical-like calculations. We present a number of results for various quantum thermodynamic quantities, obtained numerically by the classical transfer-matrix method. Moreover, we also calculate static spin correlation functions. Finally, the case of the real ferromagnet CsNiF3{\mathrm{CsNiF}}_{3}CsNiF3​ is considered; even though the corresponding easy-plane anisotropy is rather weak, the experimental data for this system are shown to be well reproduced.
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 $\ensuremath{\lambda}$ 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)]
Using the semiclassical method called pure-quantum self-consistent harmonic approximation (PQSCHA), we evaluate thermodynamic quantities of one-dimensional Heisenberg ferromagnets and antiferromagnets. Since the PQSCHA reduces their evaluation to classical-like calculations, we take advantage of Fisher's exact solution [M. E. Fisher, Am J. Phys. 32, 343 (1964)] to get all results in an almost fully analytical way. Explicitly considered here are the specific heat, the correlation length, and the susceptibility. Good agreement with available numerical data and Monte Carlo simulations is found for S>1S>1S>1 ferromagnets and antiferromagnets; for the latter it is seen that topological terms and the related Haldane gap are relevant only for the lowest spin values and temperatures.
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.