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Physical review. B, Condensed matter

Monte Carlo studies of the quantum XY model in two dimensions

Eugene Loh, Douglas J. Scalapino, Paul Michael Grant

Published Apr 1, 1985• 70 citations
Publisher

Abstract

Monte Carlo simulations are performed for the spin-\textonehalf{} $\mathrm{XY}$ model in two dimensions for large (up to 24\ifmmode\times\else\texttimes\fi{}24 sites) lattices. Results are obtained over a wide temperature range which includes the critical temperature ${T}_{c}$, estimated to be 0.4-0.5. The energy, specific heat, vortex density, and derivative of the helicity modulus are given as functions of temperature. As the lattice size is increased, the specific heat per spin approaches a finite value and does not diverge.

DOI
10.1103/physrevb.31.4712

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Monte Carlo simulations are performed for the spin-\textonehalf{} XY\mathrm{XY}XY model in two dimensions for large (up to 24\ifmmode\times\else\texttimes\fi{}24 sites) lattices. Results are obtained over a wide temperature range which includes the critical temperature Tc{T}_{c}Tc​, estimated to be 0.4-0.5. The energy, specific heat, vortex density, and derivative of the helicity modulus are given as functions of temperature. As the lattice size is increased, the specific heat per spin approaches a finite value and does not diverge.
We present detailed Monte Carlo results for the susceptibility \ensuremath{\chi}, correlation length \ensuremath{\xi}, and specific heat Cv{\mathit{C}}_{\mathit{v}}Cv​ for the XY model. The simulations are done on 642{64}^{2}642, 1282{128}^{2}1282, 2562{256}^{2}2562, and 5122{512}^{2}5122 lattices over the temperature range 0.98\ensuremath{\le}T\ensuremath{\le}1.43 corresponding to 270. Fits to \ensuremath{\chi} and \ensuremath{\xi} data favor a Kosterlitz-Thouless (KT) singularity over a second-order transition; however, unconstrained four-parameter KT fits do not confirm the predicted values \ensuremath{\nu}=0.5 and \ensuremath{\eta}=0.25. Our best estimate, Tc{\mathit{T}}_{\mathit{c}}Tc​=0.894(5), is obtained using KT fits with \ensuremath{\nu} fixed at 0.5. The exponent \ensuremath{\eta} is calculated as a function of temperature in the spin-wave phase using Monte Carlo renormalization-group and finite-size-scaling methods. Both methods give consistent results and we find \ensuremath{\eta}\ensuremath{\approxeq}0.235 at T=0.894. We also present results for the behavior of vortex density across the transition and exhibit how the dilute-gas approximation breaks down.
A Monte Carlo simulation of the helicity modulus and the specific heat of the classical XY\mathrm{XY}XY model in two dimensions is presented. In particular, we have investigated the dependence of these quantities with the size of the Monte Carlo sample. The height of the specific-heat peak is found to saturate as the infinite-volume limit is approached. The peak height of the derivative of the helicity modulus is found to grow rapidly with the increase in the size of the system and is located only 7% below the temperature at which the specific heat has the maximum.
A simulation of two-dimensional XY spin model has been done using Monte Carlo methods written in C. In this simulation, the behaviour of spins on a certain temperature had been visually learned, and a quantitative result of data containing the values of Hamiltonian energy, specific heat, magnetization, susceptibility, and amount of vortices in a system with temperature below and above the critical temperature Tc has also been obtained. Moreover, different critical temperature value obtained for both lattice size tested; Tc = 1.1 for system with lattice size 16×16 and Tc = 1.0 for system with lattice size 256×256, proving that critical temperature of a finite-size lattice would be shifted. The obtained Tc values for both finite-size systems are close to the ideal critical temperature for Kosterlitz-Thouless transition, TKT = 0.892.
Two-dimensional generalized XY spin model on a triangular lattice is studied by means of Monte-Carlo simulations. The critical temperatures of Berezinskii-Kosterlitz-Thouless (BKT) phase transition are obtained by the method of helicity modulus. It is found that the results are consistent with those obtained by other methods. The vortex density and the vortex-antivortex pair formation energy are also obtained. The result shows that the critical temperature decreases with the increase of the generalization parameter q . While the vortex-antivortex pair formation energy increases with the increase of q when q>1 .
The XXZ spin Hamitonian H = - ( S i x S j x + S i y S j y + λ S i z S j z ) is simulated for quantum spins. The XY model (λ = 0) has a Kosterlitz- Thouless Phase transition T KT = 0.45 ± 0.05, slightly above which the specific heat has a finite peak. The vortex density has a non-zero value in the low-temperature limit, probably due to quantum fluctuations. The critical temperature holds up at least to λ = -0.8. The Ising transition temperature T Ising is measured for Ising-like antiferromagnets λ < -1 and is found to drop off to zero in a manner consistent with the reciprocal-logarithmic behavior T c α-1/log|-λ-1|, as proposed for the classical model.