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Physical review. B, Condensed matter

Interplay between one- and two-dimensional fluctuations for a class ofXYmodels

Peter Olsson, Petter Minnhagen

Published Feb 1, 1991• 6 citations
Publisher

Abstract

A class of two-dimensional XY models, constructed by inserting one-dimensional spin chains between the spins of the usual XY model on a square lattice, is studied with the aid of Monte Carlo simulations. The one-dimensional fluctuations associated with the chains drive the phase transition to zero temperature in the limit of infinite chains. The modifications due to the one-dimensional fluctuations of the vortex unbinding causing the phase transition are obtained. It is found that the transition is of the Kosterlitz-Thouless type also in the limit of infinite chains. The variation of the critical temperature with the average number of nearest-neighboring spins is determined and is interpreted with the aid of an approximate translation to the two-dimensional Coulomb-gas model. Coulomb-gas scaling is found to be obeyed to a good approximation.

DOI
10.1103/physrevb.43.3356

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A class of two-dimensional XY models, constructed by inserting one-dimensional spin chains between the spins of the usual XY model on a square lattice, is studied with the aid of Monte Carlo simulations. The one-dimensional fluctuations associated with the chains drive the phase transition to zero temperature in the limit of infinite chains. The modifications due to the one-dimensional fluctuations of the vortex unbinding causing the phase transition are obtained. It is found that the transition is of the Kosterlitz-Thouless type also in the limit of infinite chains. The variation of the critical temperature with the average number of nearest-neighboring spins is determined and is interpreted with the aid of an approximate translation to the two-dimensional Coulomb-gas model. Coulomb-gas scaling is found to be obeyed to a good approximation.
A new critical point is established through Monte Carlo simulations for a class of two dimensional XY models with a modified nearest-neighbor spin interaction. The critical exponents found are \ensuremath{\nu}\ensuremath{\approxeq}1.4, \ensuremath{\beta}\ensuremath{\approxeq}0.6, and an \ensuremath{\alpha}0, consistent with hyperscaling. The phase transition is interpreted as a vortex-unbinding transition. Comparisons are made with the phase diagram of the two dimensional Coulomb gas and with the spin-1/2 XY model.
We study the two-dimensional XY\mathrm{XY}XY model with quenched random phases by Monte Carlo simulation and finite-size scaling analysis. We determine the phase diagram of the model and study its critical behavior as a function of disorder and temperature. If the strength of the randomness is less than a critical value, ${\ensuremath{\sigma}}_{c},$ the system has a Kosterlitz-Thouless (KT) phase transition from the paramagnetic phase to a state with quasi-long-range order. Our data suggest that the latter exists down to T=0T=0T=0 in contradiction with theories that predict the appearance of a low-temperature reentrant phase. At the critical disorder ${T}_{\mathrm{KT}}\ensuremath{\rightarrow}0$ and for $\ensuremath{\sigma}>{\ensuremath{\sigma}}_{c}$ there is no quasiordered phase. At zero temperature there is a phase transition between two different glassy states at ${\ensuremath{\sigma}}_{c}.$ The functional dependence of the correlation length on $\ensuremath{\sigma}$ suggests that this transition corresponds to the disorder-driven unbinding of vortex pairs.
The two-dimensional spin-1/2 XY model is investigated via an extensive quantum Monte Carlo simulation on square lattices as large as 128\ifmmode\times\else\texttimes\fi{}128. The transverse susceptibility and correlation length show precise thermodynamic scaling behaviors of a phase transition of the Kosterlitz-Thouless type at kTc{\mathit{kT}}_{\mathit{c}}kTc​/J=0.353(3). The correlation functions near the transition point exhibit a universal scaling behavior. Various critical exponents are determined and our results demonstrate that the universality class of the two-dimensional quantum system with continuous symmetry is the same as its classical analog. The specific heat exhibits a finite peak around kT/J=0.45, decreases very rapidly near Tc{\mathit{T}}_{\mathit{c}}Tc​, and falls off as T2{\mathit{T}}^{2}T2 near T=0, validating the spin-wave treatment. The ground-state energy is -0.5543(1)J per spin.
We study the phase diagram of the two-dimensional fully frustrated XY model (FFXY) and of two related models, a lattice discretization of the Landau-Ginzburg-Wilson Hamiltonian for the critical modes of the FFXY model, and a coupled Ising-XY model. We present Monte Carlo simulations on square lattices L × L , L ≲ 10 3 . We show that the low-temperature phase of these models is controlled by the same line of Gaussian fixed points as in the standard XY model. We find that, if a model undergoes a unique transition by varying temperature, then the transition is of first order. In the opposite case we observe two very close transitions: a transition associated with the spin degrees of freedom and, as temperature increases, a transition where chiral modes become critical. If they are continuous, they belong to the Kosterlitz-Thouless and to the Ising universality class, respectively. Ising and Kosterlitz-Thouless behavior is observed only after a preasymptotic regime, which is universal to some extent. In the chiral case, the approach is nonmonotonic for most observables, and there is a wide region in which finite-size scaling is controlled by an effective exponent ν eff ≈ 0.8. This explains the result ν ≈ 0.8 of many previous studies using smaller lattices.
Monte Carlo simulations of the two-dimensional XY model are performed in a square geometry with fixed boundary conditions. Using a conformal mapping, it is very easy to deduce the exponent η σ ( T ) of the order parameter correlation function at any temperature in the critical phase of the model. The temperature behaviour of η σ ( T ) is obtained numerically with a good accuracy up to the Kosterlitz-Thouless transition temperature. At very low temperatures, a good agreement is found with Berezinskii's harmonic approximation. Surprisingly, we show some evidence that there are no logarithmic corrections to the behaviour of the order parameter density profile (with symmetry-breaking surface fields) at the Kosterlitz-Thouless transition temperature.