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Physical Review E

Stiffness jump in the generalized XY model on the square lattice

David M. Hübscher, Stefan Weßel

Published Jun 10, 2013• 40 citations
Publisher

Abstract

We study the thermal phase transitions in the generalized classical XY model on the two-dimensional square lattice using single-cluster Monte Carlo simulations. In particular, we examine the (spin-wave) stiffness (helicity modulus) jump at the transition between the low-temperature algebraic phases and the disordered high-temperature regime. Employing a finite-size scaling ansatz from conformal field theory to estimate the termination of the algebraic phases that does not require knowledge of the critical properties, we provide an unbiased estimate of the stiffness jump. Our results are in full accord with the Berzinskii-Kosterlitz-Thouless scenario, i.e., the jump in the helicity modulus does not depend explicitly on the strength of the nematic coupling, but relates directly to the vorticity of the vortex excitations that drive the phase transition. We comment on previous work on related models, where Berzinskii-Kosterlitz-Thouless transition temperatures were based on scaling assumptions contradicted by our findings.

DOI
10.1103/physreve.87.062112

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We study the thermal phase transitions in the generalized classical XY model on the two-dimensional square lattice using single-cluster Monte Carlo simulations. In particular, we examine the (spin-wave) stiffness (helicity modulus) jump at the transition between the low-temperature algebraic phases and the disordered high-temperature regime. Employing a finite-size scaling ansatz from conformal field theory to estimate the termination of the algebraic phases that does not require knowledge of the critical properties, we provide an unbiased estimate of the stiffness jump. Our results are in full accord with the Berzinskii-Kosterlitz-Thouless scenario, i.e., the jump in the helicity modulus does not depend explicitly on the strength of the nematic coupling, but relates directly to the vorticity of the vortex excitations that drive the phase transition. We comment on previous work on related models, where Berzinskii-Kosterlitz-Thouless transition temperatures were based on scaling assumptions contradicted by our findings.
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 .
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.
The Berezinskii-Kosterlitz-Thouless (BKT) phase transition in two-dimensional planar rotator and XYXYXY models on a square lattice, diluted by randomly placed vacancies, is studied here using hybrid Monte Carlo simulations that combine single spin flip, cluster, and over-relaxation techniques. The transition temperature Tc{T}_{c}Tc​ is determined as a function of vacancy density ${\ensuremath{\rho}}_{\mathrm{vac}}$ by finite-size scaling of the helicity modulus and the in-plane magnetic susceptibility. The results for Tc{T}_{c}Tc​ are consistent with those from the much less precise fourth-order cumulant of Binder. Tc{T}_{c}Tc​ is found to decrease monotonically with increasing ${\ensuremath{\rho}}_{\mathrm{vac}}$, and falls to zero close to the square lattice percolation limit, ${\ensuremath{\rho}}_{\mathrm{vac}}\ensuremath{\approx}0.41$. The result is physically reasonable: the quasi-long-range orientational order of the low-temperature phase cannot be maintained in the absence of sufficient spin interactions across the lattice.
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.
The fully frustrated XY model with Villain interaction on a square lattice is studied by means of Monte Carlo simulations. On the basis of the universal jump condition it is argued that there are two distinct transitions in the model, corresponding to the loss of XY order and Z2{\mathrm{Z}}_{2}Z2​ order, respectively. The Kosterlitz-Thouless (KT) transition is analyzed by finite-size scaling of the helicity modulus at lattices of size L=32 through 128, giving TKT{\mathrm{T}}_{\mathrm{KT}}TKT​\ensuremath{\approx}0.8108(1). The vorticity-vorticity correlation function is used to determine two different characteristic lengths, the Z2{\mathrm{Z}}_{2}Z2​ correlation length \ensuremath{\xi}, and the screening length \ensuremath{\lambda}, associated with the KT transition and free vortices. The temperature dependence of \ensuremath{\xi} is examined in order to determine Tc{\mathrm{T}}_{\mathrm{c}}Tc​ and the correlation length exponent, \ensuremath{\nu}. The exponent is found to be consistent with the two-dimensional (2D) Ising value, \ensuremath{\nu}=1, and the obtained critical temperature is Tc{\mathrm{T}}_{\mathrm{c}}Tc​=0.8225(5). The determinations of both \ensuremath{\xi} and \ensuremath{\nu} are done carefully, first applying the techniques to the 2D Ising model, which serves as a convenient testing ground.
The Berezinskii-Kosterlitz-Thouless (BKT) transition in two-dimensional planar rotator and XY models on a square lattice, diluted by randomly placed vacancies, is studied here using hybrid Monte Carlo simulations that combine single spin flip, cluster and over-relaxation techniques. The transition temperature TcT_cTc​ is determined as a function of vacancy density ρvacρ_{vac}ρvac​ by calculations of the helicity modulus and the by finite-size scaling of the in-plane magnetic susceptibility. The results for TcT_cTc​ are consistent with those from the much less precise fourth-order cumulant of Binder. TcT_cTc​ is found to decrease monotonically with increasing ρvacρ_{vac}ρvac​, and falls to zero close to the square lattice percolation limit, ρvac≈0.41ρ_{vac}\approx 0.41ρvac​≈0.41 . The result is physically reasonable: the long-range orientational order of the low-temperature phase cannot be maintained in the absence of sufficient spin interactions across the lattice.