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Physical review. B, Condensed matter

Critical behavior of the two-dimensional XY model: A Monte Carlo simulation

Julio F. Fernández, Manuel F. Ferreira, Jolanta Stankiewicz

Published Jul 1, 1986• 124 citations
Publisher

Abstract

We have performed Monte Carlo (MC) simulations on systems of L\ifmmode\times\else\texttimes\fi{}L classical planar unit spins on square lattices, for L=6, 15, 30, 60, 90, and 200. The interaction between any two given spins S${\ensuremath{\rightarrow}}_{1}$ and S${\ensuremath{\rightarrow}}_{2}$ is given by -JS${\ensuremath{\rightarrow}}_{1}$\ensuremath{\cdot}S${\ensuremath{\rightarrow}}_{2}$ if ${S}_{1}$ and ${S}_{2}$ are nearest neighbors and vanishes otherwise. In order to make sure that our results correspond to equilibrium values, we have looked into the time-dependent properties of this model in the vicinity of critical temperature (${T}_{c}$). We have found that the diffusion constant for vortex motion is given at ${T}_{c}$ by D\ensuremath{\simeq}0.2 (in units of nearest-neighbor distance squared per MC step per spin). The values of the relaxation times follow from the value of D. Our computer running times were typically ${10}^{5}$ MC steps per spin, larger than any relaxation time for the system sizes we deal with. We use a procedure based on finite-size scaling to establish the value of ${T}_{c}$=0.89J/${k}_{B}$, the value of \ensuremath{\nu}=0.5\ifmmode\pm\else\textpm\fi{}0.1, and the value of ${\ensuremath{\eta}}_{c}$=0.24\ifmmode\pm\else\textpm\fi{}0.03, in agreement with the values predicted by the Kosterlitz-Thouless theory.

DOI
10.1103/physrevb.34.292

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We have performed Monte Carlo (MC) simulations on systems of L\ifmmode\times\else\texttimes\fi{}L classical planar unit spins on square lattices, for L=6, 15, 30, 60, 90, and 200. The interaction between any two given spins S\ensuremath→1{\ensuremath{\rightarrow}}_{1}\ensuremath→1​ and S\ensuremath→2{\ensuremath{\rightarrow}}_{2}\ensuremath→2​ is given by -JS\ensuremath→1{\ensuremath{\rightarrow}}_{1}\ensuremath→1​\ensuremath{\cdot}S\ensuremath→2{\ensuremath{\rightarrow}}_{2}\ensuremath→2​ if S1{S}_{1}S1​ and S2{S}_{2}S2​ are nearest neighbors and vanishes otherwise. In order to make sure that our results correspond to equilibrium values, we have looked into the time-dependent properties of this model in the vicinity of critical temperature (Tc{T}_{c}Tc​). We have found that the diffusion constant for vortex motion is given at Tc{T}_{c}Tc​ by D\ensuremath{\simeq}0.2 (in units of nearest-neighbor distance squared per MC step per spin). The values of the relaxation times follow from the value of D. Our computer running times were typically 105{10}^{5}105 MC steps per spin, larger than any relaxation time for the system sizes we deal with. We use a procedure based on finite-size scaling to establish the value of Tc{T}_{c}Tc​=0.89J/kB{k}_{B}kB​, the value of \ensuremath{\nu}=0.5\ifmmode\pm\else\textpm\fi{}0.1, and the value of \ensuremathηc{\ensuremath{\eta}}_{c}\ensuremathηc​=0.24\ifmmode\pm\else\textpm\fi{}0.03, in agreement with the values predicted by the Kosterlitz-Thouless theory.
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.
We have used Monte Carlo simulations to study the behavior of $L\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}D$ slabs containing classical spins which interact via nearest-neighbor XY\mathrm{XY}XY coupling. The coupling constant JS{J}_{S}JS​ for spins in the surface layer is fixed at 0.5J0.5J0.5J. Finite-size scaling is used to analyze data for D=59D=59D=59 and to extract estimates for the surface critical exponents. We find that ${\ensuremath{\beta}}_{1}$ is in good agreement with theoretical predictions.
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.
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.
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.
Abstract In this paper, we thoroughly examined the Berezinskii-Kosterlitz-Thouless (BKT) phase transition in the two-dimensional XY model on the honeycomb lattice. To address its thermodynamical behavior, we combined standard numerical Monte Carlo simulations with the simulated annealing (SA) protocol and entropic simulations based on the Wang-Landau (WL) algorithm. The transition temperature was determined using the second (ϒ) and fourth-order (ϒ4) helicity modulus as the order parameter. Our best finite-size scaling analysis suggests T BKT = 0.575(8) from SA and T BKT = 0.576(3) from WL. These values deviate significantly from the expected theoretical value of 1 / 2 . We believe that this discrepancy suggests that the theoretical assumptions regarding the analytical calculation may need to be revisited. Additionally, we calculated the vortex density and the formation energy of the vortex-antivortex pairs, where the obtained vortex formation energy is 2μ = 5.80(12). Upon comparison with the square lattice, our results support the notion of instability of the honeycomb lattice to support the spin long-range order and give additional backing to the critical behavior we found.